Economic Dynamics Phase Diagrams and their Economic Application Second Edition AMBRIDGE mor? informal mn ■ wwwcambridgexrgi"97ßÜ521816&47 This page intentionally left blank Economic Dynamics Phase Diagrams and Their Economic Application Second Edition This is the substantially revised and restructured second edition of Ron Shone's successful undergraduate and graduate textbook Economic Dynamics. The book provides detailed coverage of dynamics and phase diagrams including: quantitative and qualitative dynamic systems, continuous and discrete dynamics, linear and nonlinear systems and single equation and systems of equations. It illustrates dynamic systems using Mathematica, Maple and spreadsheets. It provides a thorough introduction to phase diagrams and their economic application and explains the nature of saddle path solutions. The second edition contains a new chapter on oligopoly and an extended treatment of stability of discrete dynamic systems and the solving of first-order difference equations. Detailed routines on the use of Mathematica and Maple are now contained in the body of the text, which now also includes advice on the use of Excel and additional examples and exercises throughout. The supporting website contains a solutions manual and learning tools. ronald shone is Senior Lecturer in Economics at the University of Stirling. He is the author of eight books on economics covering the areas of microeconomics, macroeconomics and international economics at both undergraduate and postgraduate level. He has written a number of articles published in Oxford Economic Papers, the Economic Journal, Journal of Economic Surveys and Journal of Economic Studies. Economic Dynamics Phase Diagrams and Their Economic Application Second Edition RONALD SHONE University of Stirling CAMBRIDGE UNIVERSITY PRESS cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521816847 © Ronald Shone 2002 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2002 iSBN-13 978-0-511-07833-0 eBook (NetLibrary) isbn-io 0-511-07833-1 eBook (NetLibrary) iSBN-13 978-0-521-81684-7 hardback isbn-io 0-521-81684-x hardback iSBN-13 978-0-521-01703-9 paperback isbn-io 0-521-01703-3 paperback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface to the second edition page xi Preface to the first edition xiii PART I Dynamic modelling 1 Introduction 3 1.1 What this book is about 3 1.2 The rise in economic dynamics 5 1.3 Stocks, flows and dimensionality 8 1.4 Nonlinearities, multiple equilibria and local stability 12 1.5 Nonlinearity and chaos 15 1.6 Computer software and economic dynamics 17 1.7 Mathematica and Maple 20 1.8 Structure and features 24 Additional reading 25 2 Continuous dynamic systems 26 2.1 Some definitions 26 2.2 Solutions to first-order linear differential equations 37 2.3 Compound interest 39 2.4 First-order equations and isoclines 41 2.5 Separable functions 45 2.6 Diffusion models 53 2.7 Phase portrait of a single variable 54 2.8 Second-order linear homogeneous equations 59 2.9 Second-order linear nonhomogeneous equations 64 2.10 Linear approximations to nonlinear differential equations 66 2.11 Solving differential equations with Mathematica 70 2.12 Solving differential equations with Maple 73 Appendix 2.1 Plotting direction fields for a single equation with Mathematica Appendix 2.2 Plotting direction fields for a single equation with Maple 79 Exercises 80 Additional reading 84 vi Contents 3 Discrete dynamic systems 85 3.1 Classifying discrete dynamic systems 85 3.2 The initial value problem 86 3.3 The cobweb model: an introduction 87 3.4 Equilibrium and stability of discrete dynamic systems 88 3.5 Solving first-order difference equations 99 3.6 Compound interest 105 3.7 Discounting, present value and internal rates of return 108 3.8 Solving second-order difference equations 110 3.9 The logistic equation: discrete version 118 3.10 The multiplier-accelerator model 123 3.11 Linear approximation to discrete nonlinear difference equations 127 3.12 Solow growth model in discrete time 130 3.13 Solving recursive equations with Mathematica and Maple 131 Appendix 3.1 Two-cycle logistic equation using Mathematica 135 Appendix 3.2 Two-cycle logistic equation using Maple 137 Exercises 138 Additional reading 141 4 Systems of first-order differential equations 142 4.1 Definitions and autonomous systems 142 4.2 The phase plane, fixed points and stability 145 4.3 Vectors of forces in the phase plane 149 4.4 Matrix specification of autonomous systems 156 4.5 Solutions to the homogeneous differential equation system: real distinct roots 160 4.6 Solutions with repeating roots 162 4.7 Solutions with complex roots 164 4.8 Nodes, spirals and saddles 166 4.9 Stability/instability and its matrix specification 178 4.10 Limit cycles 179 4.11 Euler's approximation and differential equations on a spreadsheet 183 4.12 Solving systems of differential equations with Mathematica and Maple 186 Appendix 4.1 Parametric plots in the phase plane: continuous variables 194 Exercises 196 Additional reading 200 5 Discrete systems of equations 201 5.1 Introduction 201 5.2 Basic matrices with Mathematica and Maple 204 5.3 Eigenvalues and eigenvectors 208 Contents vii 5.4 Mathematica and Maple for solving discrete systems 214 5.5 Graphing trajectories of discrete systems 220 5.6 The stability of discrete systems 223 5.7 The phase plane analysis of discrete systems 235 5.8 Internal and external balance 239 5.9 Nonlinear discrete systems 245 Exercises 247 Additional reading 250 6 Optimal control theory 251 6.1 The optimal control problem 251 6.2 The Pontryagin maximum principle: continuous model 252 6.3 The Pontryagin maximum principle: discrete model 259 6.4 Optimal control with discounting 265 6.5 The phase diagram approach to continuous time control models 270 Exercises 283 Additional reading 285 7 Chaos theory 286 7.1 Introduction 286 7.2 Bifurcations: single-variable case 287 7.3 The logistic equation, periodic-doubling bifurcations and chaos 293 7.4 Feigenbaum's universal constant 301 7.5 Sarkovskii theorem 302 7.6 Van der Pol equation and Hopf bifurcations 304 7.7 Strange attractors 307 7.8 Rational choice and erratic behaviour 312 7.9 Inventory dynamics under rational expectations 315 Exercises 319 Additional reading 321 PART II Applied economic dynamics 8 Demand and supply models 325 8.1 Introduction 325 8.2 A simple demand and supply model in continuous time 326 8.3 The cobweb model 332 8.4 Cobwebs with Mathematica and Maple 338 8.5 Cobwebs in the phase plane 339 8.6 Cobwebs in two interrelated markets 346 8.7 Demand and supply with stocks 349 8.8 Stability of the competitive equilibrium 353 8.9 The housing market and demographic changes 358 8.10 Chaotic demand and supply 363 viii Contents Appendix 8.1 Obtaining cobwebs using Mathematica and Maple 367 Exercises 371 Additional reading 374 9 Dynamic theory of oligopoly 375 9.1 Static model of duopoly 375 9.2 Discrete oligopoly models with output adjusting completely and instantaneously 377 9.3 Discrete oligopoly models with output not adjusting completely and instantaneously 389 9.4 Continuous modelling of oligopoly 398 9.5 A nonlinear model of duopolistic competition (R&D) 405 9.6 Schumpeterian dynamics 414 Exercises 419 Additional reading 423 10 Closed economy dynamics 424 10.1 Goods market dynamics 425 10.2 Goods and money market dynamics 429 10.3 IS-LM continuous model: version 1 431 10.4 Trajectories with Mathematica, Maple and Excel 437 10.5 Some important propositions 442 10.6 IS-LM continuous model: version 2 447 10.7 Nonlinear IS-LM model 453 10.8 Tobin-Blanchard model 455 10.9 Conclusion 465 Exercises 467 Additional reading 469 11 The dynamics of inflation and unemployment 470 11.1 The Phillips curve 470 11.2 Two simple models of inflation 472 11.3 Deflationary'death spirals' 484 11.4 A Lucas model with rational expectations 490 11.5 Policy rules 493 11.6 Money, growth and inflation 494 11.7 Cagan model of hyperinflation 500 11.8 Unemployment and job turnover 506 11.9 Wage determination models and the profit function 509 11.10 Labour market dynamic s 513 Exercises 516 Additional reading 518 12 Open economy dynamics: sticky price models 519 12.1 The dynamics of a simple expenditure model 519 12.2 The balance of payments and the money supply 524 Contents ix 12.3 Fiscal and monetary expansion under fixed exchange rates 532 12.4 Fiscal and monetary expansion under flexible exchange rates 539 12.5 Open economy dynamics under fixed prices and floating 545 Exercises 551 Additional reading 552 13 Open economy dynamics: flexible price models 553 13.1 A simplified Dornbusch model 554 13.2 The Dornbusch model 559 13.3 The Dornbusch model: capital immobility 564 13.4 The Dornbusch model under perfect foresight 567 13.5 Announcement effects 573 13.6 Resource discovery and the exchange rate 581 13.7 The monetarist model 586 Exercises 589 Additional reading 592 14 Population models 593 14.1 Malthusian population growth 593 14.2 The logistic curve 596 14.3 An alternative interpretation 601 14.4 Multispecies population models: geometric analysis 603 14.5 Multispecies population models: mathematical analysis 619 14.6 Age classes and projection matrices 626 Appendix 14.1 Computing a and b for the logistic equation using Mathematica 630 Appendix 14.2 Using Maple to compute a and b for the logistic equation 631 Appendix 14.3 Multispecies modelling with Mathematica and Maple 632 Exercises 634 Additional reading 637 15 The dynamics of fisheries 638 15.1 Biological growth curve of a fishery 638 15.2 Harvesting function 644 15.3 Industry profits and free access 647 15.4 The dynamics of open access fishery 650 15.5 The dynamics of open access fishery: a numerical example 654 15.6 The fisheries control problem 658 15.7 Schooling fishery 661 15.8 Harvesting and age classes 669 x Contents Exercises 673 Additional reading 676 Answers to selected exercises 677 Bibliography 688 Author index 697 Subject index 700 Preface to the second edition I was very encouraged with the reception of the first edition, from both staff and students. Correspondence eliminated a number of errors and helped me to improve clarity. Some of the new sections are in response to communications I received. The book has retained its basic structure, but there have been extensive revisions to the text. Part I, containing the mathematical background, has been considerably enhanced in all chapters. All chapters contain new material. This new material is largely in terms of the mathematical content, but there are some new economic examples to illustrate the mathematics. Chapter 1 contains a new section on dimensionality in economics, a much-neglected topic in my view. Chapter 3 on discrete systems has been extensively revised, with a more thorough discussion of the stability of discrete dynamical systems and an extended discussion of solving second-order difference equations. Chapter 5 also contains a more extensive discussion of discrete systems of equations, including a more thorough discussion of solving such systems. Direct solution methods using Mathematica and Maple are now provided in the main body of the text. Indirect solution methods using the Jordan form are new to this edition. There is also a more thorough treatment of the stability of discrete systems. The two topics covered in chapter 6 of the first edition have now been given a chapter each. This has allowed topics to be covered in more depth. Chapter 6 on control theory now includes the use of Excel's Solver for solving discrete control problems. Chapter 7 on chaos theory has also been extended, with a discussion of Sarkovskii's theorem. It also contains a much more extended discussion of bifurcations and strange attractors. Changes to part II, although less extensive, are quite significant. The mathematical treatment of cobwebs in chapter 8 has been extended and there is now a new section on stock models and another on chaotic demand and supply. Chapter 9 on dynamic oligopoly is totally new to this edition. It deals with both discrete and continuous dynamic oligopoly and goes beyond the typical duopoly model. There is also a discussion of an R&D dynamic model of duopoly and a brief introduction to Schumpeterian dynamics. Chapter 11 now includes a discussion of deflationary 'death spirals' which have been prominent in discussions of Japan's downturn. Cagan's model of hyperinflations is also a new introduction to this chapter. The open economy was covered quite extensively in the first edition, so these chapters contain only minor changes. Population models now include a consideration of age classes and Leslie projection matrices. This material is employed xii Preface to the second edition in chapter 15 to discuss culling policy. The chapter on overlapping generations modelling has been dropped in this edition to make way for the new material. Part of the reason for this is that, as presented, it contained little by the way of dynamics. It had much more to say about nonlinearity. Two additional changes have been made throughout. Mathematica and Maple routines are now generally introduced into the main body of the text rather than as appendices. The purpose of doing this is to show that these programmes are 'natural' tools for the economist. Finally, there has been an increase in the number of questions attached to almost all chapters. As in the first edition, the full solution to all these questions is provided on the Cambridge University website, which is attached to this book: one set of solutions provided in Mathematica notebooks and an alternative set of solutions provided in Maple worksheets. Writing a book of this nature, involving as it does a number of software packages, has become problematic with constant upgrades. This is especially true with Mathematica and Maple. Some of the routines provided in the first edition no longer work in the upgrade versions. Even in the final stages of preparing this edition, new upgrades were occurring. I had to make a decision, therefore, at which upgrade I would conclude. All routines and all solutions on the web site are carried out with Mathematica 4 and Maple 6. I would like to thank all those individuals who wrote or emailed me on material in the first edition. I would especially like to thank Mary E. Edwards, Yee-Tien Fu, Christian Groth, Cars Hommes, Alkis Karabalis, Julio Lopez-Gallardo, Johannes Ludsteck and Yanghoon Song. I would also like to thank Simon Whitby for information and clarification on new material in chapter 9. I would like to thank Ashwin Rattan for his continued support of this project and Barbara Docherty for an excellent job of copy-editing, which not only eliminated a number of errors but improved the final project considerably. The author and publishers wish to thank the following for permission to use copyright material: Springer- Verlag for the programme listing on p. 192 of A First Course in Discrete Dynamic Systems and the use of the Visual D Solve software package from Visual D Solve; Cambridge University Press for table 3 from British Economic Growth 1688-1959, p. 8. The publisher has used its best endeavours to ensure that the URLs for external websites referred to in this book are correct and active at the time of going to press. However, the publisher has no responsibility for the websites and can make no guarantee that a site will remain live or that the content is or will remain appropriate. March 2002 Preface to the first edition The conception of this book began in the autumn semester of 1990 when I undertook a course in Advanced Economic Theory for undergraduates at the University of Stirling. In this course we attempted to introduce students to dynamics and some of the more recent advances in economic theory. In looking at this material it was quite clear that phase diagrams, and what mathematicians would call qualitative differential equations, were becoming widespread in the economics literature. There is little doubt that in large part this was a result of the rational expectations revolution going on in economics. With a more explicit introduction of expectations into economic modelling, adjustment processes became the mainstay of many economic models. As such, there was a movement away from models just depicting comparative statics. The result was a more explicit statement of a model's dynamics, along with its comparative statics. A model's dynamics were explicitly spelled out, and in particular, vectors of forces indicating movements when the system was not in equilibrium. This led the way to solving dynamic systems by employing the theory of differential equations. Saddle paths soon entered many papers in economic theory. However, students found this material hard to follow, and it did not often use the type of mathematics they were taught in their quantitative courses. Furthermore, the material that was available was very scattered indeed. But there was another change taking place in Universities which has a bearing on the way the present book took shape. As the academic audit was about to be imposed on Universities, there was a strong incentive to make course work assessment quite different from examination assessment. Stirling has always had a long tradition of course work assessment. In the earlier period there was a tendency to make course work assessment the same as examination assessment: the only real difference being that examinations could set questions which required greater links between material since the course was by then complete. In undertaking this new course, I decided from the very outset that the course work assessment would be quite different from the examination assessment. In particular, I conceived the course work to be very 'problem oriented'. It was my belief that students come to a better understanding of the economics, and its relation to mathematics, if they carry out problems which require them to explicitly solve models, and to go on to discuss the implications of their analysis. This provided me with a challenge. There was no material available of this type. Furthermore, many economics textbooks of an advanced nature, and certainly the xiv Preface to the first edition published articles, involved setting up models in general form and carrying out very tedious algebraic manipulations. This is quite understandable. But such algebraic manipulation does not give students the same insight it may provide the research academic. A compromise is to set out models with specific numerical coefficients. This has at least four advantages. It allowed the models to be solved explicitly. This means that students can get to grips with the models themselves fairly quickly and easily. Generalisation can always be achieved by replacing the numerical coefficients by unspecified parameters. Or alternatively, the models can be solved for different values, and students can be alerted to the fact that a model's solution is quite dependent on the value (sign) of a particular parameter. The dynamic nature of the models can more readily be illustrated. Accordingly concentration can be centred on the economics and not on the mathematics. Explicit solutions to saddle paths can be obtained and so students can explicitly graph these solutions. Since it was the nature of saddle paths which gave students the greatest conceptual difficulty, this approach soon provided students with the insight into their nature that was lacking from a much more formal approach. Furthermore, they acquired this insight by explicitly dealing with an economic model. I was much encouraged by the students' attitude to this 'problem oriented' approach. The course work assignments that I set were far too long and required far more preparation than could possibly be available under examination conditions. However, the students approached them with vigour during their course work period. Furthermore, it led to greater exchanges between students and a positive externality resulted. This book is an attempt to bring this material together, to extend it, and make it more widely available. It is suitable for core courses in economic theory, and reading for students undertaking postgraduate courses and to researchers who require to acquaint themselves with the phase diagram technique. In addition, it can also be part of courses in quantitative economics. Outside of economics, it is also applicable to courses in mathematical modelling. Finally, I would like to thank Cambridge University Press and the department of economics at Stirling for supplying the two mathematical software programmes; the copy editor, Anne Rix, for an excellent job on a complex manuscript; and my wife, Anne Thomson, for her tolerance in bringing this book about. January 1997 PARTI Dynamic modelling CHAPTER 1 Introduction 1.1 What this book is about This is not a book on mathematics, nor is it a book on economics. It is true that the over-riding emphasis is on the economics, but the economics under review is specified very much in mathematical form. Our main concern is with dynamics and, most especially with phase diagrams, which have entered the economics literature in a major way since 1990. By their very nature, phase diagrams are a feature of dynamic systems. But why have phase diagrams so dominated modern economics? Quite clearly it is because more emphasis is now placed on dynamics than in the past. Comparative statics dominated economics for a long time, and much of the teaching is still concerned with comparative statics. But the breakdown of many economies, especially under the pressure of high inflation, and the major influence of inflationary expectations, has directed attention to dynamics. By its very nature, dynamics involves time derivatives, dx/dt, where x is a continuous function of time, or difference equations, xt — xt-\ where time is considered in discrete units. This does not imply that these have not been considered or developed in the past. What has been the case is that they have been given only cursory treatment. The most distinguishing feature today is that dynamics is now taking a more central position. In order to reveal this emphasis and to bring the material within the bounds of undergraduate (and postgraduate) courses, it has been necessary to consider dynamic modelling, in both its continuous and discrete forms. But in doing this the over-riding concern has been with the economic applications. It is easy to write a text on the formal mathematics, but what has always been demonstrated in teaching economics is the difficulty students have in relating the mathematics to the economics. This is as true at the postgraduate level as it is at the undergraduate level. This linking of the two disciplines is an art rather than a science. In addition, many books on dynamics are mathematical texts that often choose simple and brief examples from economics. Most often than not, these reduce down to a single differential equation or a single difference equation. Emphasis is on the mathematics. We do this too in part I. Even so, the concentration is on the mathematical concepts that have the widest use in the study of dynamic economics. In part II this emphasis is reversed. The mathematics is chosen in order to enhance the economics. The mathematics is applied to the economic problem rather than the 4 Economic Dynamics (simple) economic problem being applied to the mathematics. We take a number of major economic areas and consider various aspects of their dynamics. Because this book is intended to be self-contained, then it has been necessary to provide the mathematical background. By 'background' we, of course, mean that this must be mastered before the economic problem is reviewed. Accordingly, part I supplies this mathematical background. However, in order not to make part I totally mathematical we have discussed a number of economic applications. These are set out in part I for the first time, but the emphasis here is in illustrating the type of mathematics they involve so that we know what mathematical techniques are required in order to investigate them. Thus, the Malthusian population growth model is shown to be just a particular differential equation, if population growth is assumed to vary continuously over time. But equally, population growth can be considered in terms of a discrete time-period model. Hence, part I covers not only differential equations but also difference equations. Mathematical specification can indicate that topics such as A, B and C should be covered. However, A, B and C are not always relevant to the economic problem under review. Our choice of material to include in part I, and the emphasis of this material, has been dictated by what mathematics is required to understand certain features of dynamic economic systems. It is quite clear when considering mathematical models of differential equations that the emphasis has been, and still is, with models from the physical sciences. This is not surprising given the development of science. In this text, however, we shall concentrate on economics as the raison d'etre of the mathematics. In a nutshell, we have taken a number of economic dynamic models and asked: 'What mathematics is necessary to understand these?' This is the emphasis of part I. The content of part I has been dictated by the models developed in part II. Of course, if more economic models are considered then the mathematical background will inevitably expand. What we are attempting in this text is dynamic modelling that should be within the compass of an undergraduate with appropriate training in both economics and quantitative economics. Not all dynamic questions are dealt with in this book. The over-riding concern has been to explain phase diagrams. Such phase diagrams have entered many academic research papers over the past decade, and the number is likely to increase. Azariades (1993) has gone as far as saying that Dynamical systems have spread so widely into macroeconomics that vector fields and phase diagrams are on the verge of displacing the familiar supply-demand schedules and Hicksian crosses of static macroeconomics, (p. xii) The emphasis is therefore justified. Courses in quantitative economics generally provide inadequate training to master this material. They provide the basics in differentiation, integration and optimisation. But dynamic considerations get less emphasis - most usually because of a resource constraint. But this is a most unfortunate deficiency in undergraduate teaching that simply does not equip students to understand the articles dealing with dynamic systems. The present book is one attempt to bridge this gap. I have assumed some basic knowledge of differentiation and integration, along with some basic knowledge of difference equations. However, I have made great Introduction 5 pains to spell out the modelling specifications and procedures. This should enable a student to follow how the mathematics and economics interrelate. Such knowledge can be imparted only by demonstration. I have always been disheartened by the idea that you can teach the mathematics and statistics in quantitative courses, and you can teach the economics in economics courses, and by some unspecified osmosis the two areas are supposed to fuse together in the minds of the student. For some, this is true. But I suspect that for the bulk of students this is simply not true. Students require knowledge and experience in how to relate the mathematics and the economics. As I said earlier, this is more of an art than a science. But more importantly, it shows how a problem excites the economist, how to then specify the problem in a formal (usually mathematical) way, and how to solve it. At each stage ingenuity is required. Economics at the moment is very much in the mould of problem solving. It appears that the procedure the investigator goes through1 is: (1) Specify the problem (2) Mathematise the problem (3) See if the problem's solution conforms to standard mathematical solutions (4) Investigate the properties of the solution. It is not always possible to mathematise a problem and so steps (2)-(4) cannot be undertaken. However, in many such cases a verbal discussion is carried out in which a 'story' is told about the situation. This is no more than a heuristic model, but a model just the same. In such models the dynamics are part of the 'story' -about how adjustment takes place over time. It has long been argued by some economists that only those problems that can be mathematised get investigated. There are advantages to formal modelling, of going beyond heuristics. In this book we concentrate only on the formal modelling process. 1.2 The rise in economic dynamics Economic dynamics has recently become more prominent in mainstream economics. This influence has been quite pervasive and has influenced both microeconomics and macroeconomics. Its influence in macroeconomics, however, has been much greater. In this section we outline some of the main areas where economic dynamics has become more prominent and the possible reasons for this rise in the subject. 1.2.1 Macroeconomic dynamics Economists have always known that the world is a dynamic one, and yet a scan of the books and articles over the past twenty years or so would make one wonder if they really believed it. With a few exceptions, dynamics has been notably absent from published works. This began to change in the 1970s. The 1970s became a watershed in both economic analysis and economic policy. It was a turbulent time. 1 For an extended discussion of the modelling process, see Mooney and Swift (1999, chapter 0). 6 Economic Dynamics Economic relationships broke down, stagflation became typical of many Western economies, and Conservative policies became prominent. Theories, especially macroeconomic theories, were breaking down, or at best becoming poor predictors of economic changes. The most conspicuous change was the rapid (and accelerating) rise in inflation that occurred with rising unemployment. This became a feature of most Western economies. Individuals began to expect price rises and to build this into their decision-making. If such behaviour was to be modelled, and it was essential to do so, then it inevitably involved a dynamic model of the macro-economy. More and more, therefore, articles postulated dynamic models that often involved inflationary expectations. Inflation, however, was not the only issue. As inflation increased, as OPEC changed its oil price and as countries discovered major resource deposits, so there were major changes to countries' balance of payments situations. Macro-economists had for a long time considered their models in the context of a closed economy. But with such changes, the fixed exchange rate system that operated from 1945 until 1973 had to give way to floating. Generalised floating began in 1973. This would not have been a problem if economies had been substantially closed. But trade in goods and services was growing for most countries. Even more significant was the increase in capital flows between countries. Earlier trade theories concentrated on the current account. But with the growth of capital flows, such models became quite unrealistic. The combination of major structural changes and the increased flows of capital meant that exchange rates had substantial impacts on many economies. It was no longer possible to model the macroeconomy as a closed economy. But with the advent of generalised floating changes in the exchange rate needed to be modelled. Also, like inflation, market participants began to formulate expectations about exchange rate movements and act accordingly. It became essential, then, to model exchange rate expectations. This modelling was inevitably dynamic. More and more articles considered dynamic models, and are still doing so. One feature of significance that grew out of both the closed economy modelling and the open economy modelling was the stock-flow aspects of the models. Key-nesian economics had emphasised a flow theory. This was because Keynes himself was very much interested in the short run - as he aptly put it: 'In the long run we are all dead.' Even growth theories allowed investment to take place (a flow) but assumed the stock of capital constant, even though such investment added to the capital stock! If considering only one or two periods, this may be a reasonable approximation. However, economists were being asked to predict over a period of five or more years. More importantly, the change in the bond issue (a flow) altered the National Debt (a stock), and also the interest payment on this debt. It is one thing to consider a change in government spending and the impact this has on the budget balance; but the budget, or more significantly the National Debt, gives a stock dimension to the long-run forces. Governments are not unconcerned with the size of the National Debt. The same was true of the open economy. The balance of payments is a flow. The early models, especially those ignoring the capital account, were concerned only with the impact of the difference between the exports and imports of goods and Introduction 7 services. In other words, the inflow and outflow of goods and services to and from an economy. This was the emphasis of modelling under fixed exchange rates. But a deficit leads to a reduction in the level of a country's stock of reserves. A surplus does the opposite. Repeated deficits lead to a repeated decline in a country's level of reserves and to the money stock. Printing more money could, of course, offset the latter (sterilisation), but this simply complicates the adjustment process. At best it delays the adjustment that is necessary. Even so, the adjustment requires both a change in the flows and a change in stocks. What has all this to do with dynamics? Flows usually (although not always) take place in the same time period, say over a year. Stocks are at points in time. To change stock levels, however, to some desired amount would often take a number of periods to achieve. There would be stock-adjustment flows. These are inherently dynamic. Such stock-adjustment flows became highly significant in the 1970s and needed to be included in the modelling process. Models had to become more dynamic if they were to become more realistic or better predictors. These general remarks about why economists need to consider dynamics, however, hide an important distinction in the way dynamics enters economics. It enters in two quite different and fundamental ways (Farmer 1999). The first, which has its counterpart in the natural sciences, is from the fact that the present depends upon the past. Such models typically are of the form where we consider just a one-period lag. The second way dynamics enters macroeconomics, which has no counterpart in the natural sciences, arises from the fact that economic agents in the present have expectations (or beliefs) about the future. Again taking a one-period analysis, and denoting the present expectation about the variable y one period from now by Eyt+\, then Let us refer to the first lag as a past lag and the second a future lag. There is certainly no reason to suppose modelling past lags is the same as modelling future lags. Furthermore, a given model can incorporate both past lags and future lags. The natural sciences provide the mathematics for handling past lags but has nothing to say about how to handle future lags. It is the future lag that gained most attention in the 1970s, most especially with the rise in rational expectations. Once a future lag enters a model it becomes absolutely essential to model expectations, and at the moment there is no generally accepted way of doing this. This does not mean that we should not model expectations, rather it means that at the present time there are a variety of ways of modelling expectations, each with its strengths and weaknesses. This is an area for future research. 1.2.2 Environmental issues Another change was taking place in the 1970s. Environmental issues were becoming, and are becoming, more prominent. Environmental economics as a subject began to have a clear delineation from other areas of economics. It is true that yt =f{yt-\) (1.1) yt = g(Eyt+i) (1.2) 8 Economic Dynamics environmental economics already had a body of literature. What happened in the 1970s and 1980s was that it became a recognised sub-discipline. Economists who had considered questions in the area had largely confined themselves to the static questions, most especially the questions of welfare and cost-benefit analysis. But environmental issues are about resources. Resources have a stock and there is a rate of depletion and replenishment. In other words, there is the inevitable stock-flow dimension to the issue. Environmentalists have always known this, but economists have only recently considered such issues. Why? Because the issues are dynamic. Biological species, such as fish, grow and decline, and decline most especially when harvested by humans. Forests decline and take a long time to replace. Fossil fuels simply get used up. These aspects have led to a number of dynamic models - some discrete and some continuous. Such modelling has been influenced most particularly by control theory. We shall briefly cover some of this material in chapters 6 and 15. 1.2.3 The implication for economics All the changes highlighted have meant a significant move towards economic dynamics. But the quantitative courses have in large part not kept abreast of these developments. The bulk of the mathematical analysis is still concerned with equilibrium and comparative statics. Little consideration is given to dynamics - with the exception of the cobweb in microeconomics and the multiplier-accelerator model in macroeconomics. Now that more attention has been paid to economic dynamics, more and more articles are highlighting the problems that arise from nonlinearity which typify many of the dynamic models we shall be considering in this book. It is the presence of nonlinearity that often leads to more than one equilibrium; and given more than one equilibrium then only local stability properties can be considered. We discuss these issues briefly in section 1.4. 1.3 Stocks, flows and dimensionality Nearly all variables and parameters - whether they occur in physics, biology, sociology or economics - have units in which they are defined and measured. Typical units in physics are weight and length. Weight can be measured in pounds or kilograms, while length can be measured in inches or centimetres. We can add together length and we can add together weight, but what we cannot do is add length to weight. This makes no sense. Put simply, we can add only things that have the same dimension. DEFINITION Any set of additive quantities is a dimension. A primary dimension is not expressible in terms of any other dimension; a secondary dimension is defined in terms of primary dimensions.2 2 An elementary discussion of dimensionality in economics can be found in Neal and Shone (1976, chapter 3). The definitive source remains De Jong (1967). Introduction 9 To clarify these ideas, and other to follow, we list the following set of primary dimensions used in economics: (1) Money [M] (2) Resources or quantity [Q] (3) Time [T] (4) Utility or satisfaction [S] Apples has, say, dimension [Ql] and bananas [Q2]. We cannot add an apple to a banana (we can of course add the number of objects, but that is not the same thing). The value of an apple has dimension [M] and the value of a banana has dimension [M], so we can add the value of an apple to the value of a banana. They have the same dimension. Our reference to [Q1 ] and [Q2] immediately highlights a problem, especially for macroeconomics. Since we cannot add apples and bananas, it is sometimes assumed in macroeconomics that there is a single aggregate good, which then involves dimension [Q]. For any set of primary dimensions, and we shall use money [M] and time [T] to illustrate, we have the following three propositions: (1) If a e [M] and b e [M] then a ± b e [M] (2) If a e [M] and b e [T] then ab e [MT] and a/b e [Mr-1] (3) If y =f(x) and y e [M] then/(jc) e [Af]. Proposition (1) says that we can add or subtract only things that have the same dimension. Proposition (2) illustrates what is meant by secondary dimensions, e.g., [Mr-1] is a secondary or derived dimension. Proposition (3) refers to equations and states that an equation must be dimensionally consistent. Not only must the two sides of an equation have the same value, but it must also have the same dimension, i.e., the equation must be dimensionally homogeneous. The use of time as a primary dimension helps us to clarify most particularly the difference between stocks and flows. A stock is something that occurs at a point in time. Thus, the money supply, Ms, has a certain value on 31 December 2001. Ms is a stock with dimension [M], i.e., Ms e [M]. A stock variable is independent of the dimension [T]. A flow, on the other hand, is something that occurs over a period of time. A flow variable must involve the dimension [r-1]. In demand and supply analysis we usually consider demand and supply per period of time. Thus, qd and qs are the quantities demanded and supplied per period of time. More specifically, qd e [QT~l] and qs e [QT~X]. In fact, all flow variables involve dimension [r-1]. The nominal rate of interest, i, for example, is a per cent per period, so i e [r-1] and is a flow variable. Inflation, it, is the percentage change in prices per period, say a year. Thus, it e [r-1]. The real rate of interest, defined as r = i — it, is dimensionally consistent since r e being the difference of two variables each with dimension [r-1]. Continuous variables, such as x(t), can be a stock or a flow but are still defined for a point in time. In dealing with discrete variables we need to be a little more careful. Let xt denote a stock variable. We define this as the value at the end of period t? Figure 1.1 uses three time periods to clarify our discussion: t — 1, t and 3 We use this convention throughout this book. 10 Economic Dynamics Figure 1.1. t-1 t+l Ax~x,-x,_i—> xl period t+l. Thus xt-i is the stock at the end of period t — 1 and xt is the stock at the end of period t. Now let zt be a flow variable over period t, and involving dimension [T-1]. Of course, there is also zt-\ and zt+\. Now return to variable x. It is possible to consider the change in x over period t, which we write as Axt = xt Xt—i This immediately shows up a problem. Let xt have dimension [Q], then by proposition (1) so would Axt. But this cannot be correct! Axt is the change over period t and must involve dimension [T-1]. So how can this be? The correct formulation is, in fact, (1.3) 4^= *fT*f~i\ ^[Q7"1] At t - (t - 1) Implicit is that At = 1 and so Axt = xt — xt-\. But this 'hides' the dimension [r-1]. This is because At e [T], even though it has a value of unity, Axt/At e tor-1]. Keeping with the convention Axt = xt — xt-\, then Axt e [ßr_1] is referred to as a stock-flow variable. Axt must be kept quite distinct from Zt- The variable zt is a flow variable and has no stock dimension. Axt, on the other hand, is a difference of two stocks defined over period t. Example 1.1 Consider the quantity equation MV = Py. M is the stock of money, with dimension [M]. The variable y is the level of real output. To make dimensional sense of this equation, we need to assume a single-good economy. It is usual to consider y as real GDP over a period of time, say one year. So, with a single-good economy with goods having dimension [Q], then y e [QT~1 ]. If we have a single-good economy, then Pis the money per unit of the good and has dimension [MQ~1 ]. V is the income velocity of circulation of money, and indicates the average number of times a unit of money circulates over a period of time. Hence V e [T-1]. Having considered the dimensions of the variables separately, do we have dimensional consistency? MV e [MW1] = [Mr-1] Py e [MQ-^iQT-1] = [M^1] and so we do have dimensional consistency. Notice in saying this that we have utilised the feature that dimensions 'act like algebra' and so dimensions cancel, as with [QQ~1]. Thus Py e [MQ-^iQT-1] = [MQ~lQT~l] = [MT~l] Introduction 11 Example 1.2 Consider again the nominal rate of interest, denoted i. This can more accurately be defined as the amount of money received over some interval of time divided by the capital outlay. Hence, [Mr"1] , i e--- = [J"1] [M] Example 1.3 Consider the linear static model of demand and supply, given by the following equations. qd = a — bp a, b > 0 qs = c + dp d>0 (1.4) qd = qs = q with equilibrium price and quantity a — c „ ad + be b + d' * b + d and with dimensions qd,qs e[QT~\ pe[MQ~1} The model is a flow model since qd and qs are defined as quantities per period of time.4 It is still, however, a static model because all variables refer to time period t. Because of this we conventionally do not include a time subscript. Now turn to the parameters of the model. If the demand and supply equations are to be dimensionally consistent, then a,c e [QT~l] and b,de [<22r_1M_1] Then a-ce [QT-1] b + de [Q2T-lM-1] ror_1i Also ad e [QT-l][Q2T-lM-1] = [Q3T~2M~l] be e [Q2T-lM-l][QT-1] = \Q^T~2M~X\ q* e —:-r = [QT'1] Where a problem sometimes occurs in writing formulas is when parameters have values of unity. Consider just the demand equation and suppose it takes the 4 We could have considered a stock demand and supply model, in which case qd and qs would have dimension [Q]. Such a model would apply to a particular point in time. 12 Economic Dynamics form q = a — p. On the face of it this is dimensionally inconsistent, a e [QT ] andp e [M<2_1] and so cannot be subtracted! The point is that the coefficient of p is unity with dimension [<22r_1M_1], and this dimension gets 'hidden'. Example 1.4 A typically dynamic version of example 1.3 is the cobweb model qf = a — bpt a, b > 0 (1.5) q\ = c + dpt-i d > 0 qdt=q\ = qt Here we do subscript the variables since now two time periods are involved. Although qf and q\ are quantities per period to time with dimension [QT-1], they both refer to period t. However, p e [MQ~X] is for period t in demand but period t — 1 for supply. A model that is specified over more than one time period is a dynamic model. We have laboured dimensionality because it is still a much-neglected topic in economics. Yet much confusion can be avoided with a proper understanding of this topic. Furthermore, it lies at the foundations of economic dynamics. 1.4 Nonlinearities, multiple equilibria and local stability Nonlinearities, multiple equilibria and local stability/instability are all interlinked. Consider the following simple nonlinear difference equation (1.6) xt=f(xt-1) An equilibrium (a fixed point) exists, as we shall investigate fully later in the book, if x* =f(x*). Suppose the situation is that indicated in figure 1.2(a), then an equilibrium point is where f(xt-i) cuts the 45°-line. But in this example three such fixed points satisfy this condition: x\, x\ and x\. A linear system, by contrast, can cross the 45°-line at only one point (we exclude here the function coinciding with the 45°-line), as illustrated in figures 1.2(b) and 1.2(c). It is the presence of the nonlinearity that leads to multiple equilibria. If we consider a sequence of points {xt} beginning at xq, and if for a small neighbourhood of a fixed point x* the sequence {xt} converges on x*, then x* is said to be locally asymptotically stable. We shall explain this in more detail later in the book. Now consider the sequence in the neighbourhood of each fixed point in figure 1.2(a). We do this for each point in terms of figure 1.3. In the case of x\, for any initial point (°r Xo) in the neighbourhood of x\, the sequence {xt} will converge on x^. This is also true for the fixed point x^. However, it is not true for the fixed point x\, represented by point b. The fixed point x\ is locally asymptotically unstable. On the other hand both x\ and x\ are locally asymptotically stable. Suppose we approximate the nonlinear system in the neighbourhood of each of the fixed points. This can be done by means of a Taylor expansion about the appropriate fixed point. These are shown by each of the dotted lines in figure 1.3. Introduction 13 (a) Nonlinear Figure 1.2. r-l (b) Linear (slope < 1) x,_, (c) Linear (slope > 1) X, X,_j Observation of these lines indicates that for equilibrium points x* and x^ the linear approximation has a slope less than unity. On the other hand, the linear approximation about x\ has a slope greater than unity. It is this feature that allows us to deal with the dynamics of a nonlinear system - so long as we keep within a small neighbourhood of a fixed point. 14 Economic Dynamics Figure 1.3. Xr=x._ f\ linear approximation X, (-1 ,« lpiear I ^approximation X0 X 2 X() X,- X, x, x,_. - ' ' 'x^linear r—XrAxt-\) approximation c Although a great deal of attention has been given to linear difference and differential equations, far less attention has been given to nonlinear relationships. This is now changing. Some of the most recent researches in economics are considering nonlinearities. Since, however, there is likely to be no general solutions for nonlinear relationships, both mathematicians and economists have, with minor Introduction 15 exceptions, been content to investigate the local stability of the fixed points to a nonlinear system. The fact that a linear approximation can be taken in the neighbourhood of a fixed point in no way removes the fact that there can be more than one fixed point, more than one equilibrium point. Even where we confine ourselves only to stable equilibria, there is likely to be more than one. This leads to some new and interesting policy implications. In simple terms, and using figure 1.2(a) for illustrative purposes, the welfare attached to point x\ will be different from that attached to x\. If this is so, then it is possible for governments to choose between the two equilibrium points. Or, it may be that after investigation one of the stable equilibria is found to be always superior. With linear systems in which only one equilibrium exists, such questions are meaningless. Multiple equilibria of this nature create a problem for models involving perfect foresight. If, as such models predict, agents act knowing the system will converge on equilibrium, will agents assume the system converges on the same equilibrium? Or, even with perfect foresight, can agents switch from one (stable) equilibrium to another (stable) equilibrium? As we shall investigate in this book, many of the rational expectations solutions involve saddle paths. In other words, the path to equilibrium will arise only if the system 'jumps' to the saddle path and then traverses this path to equilibrium. There is something unsatisfactory about this modelling process and its justification largely rests on the view that the world is inherently stable. Since points off the saddle path tend the system ever further away from equilibrium, then the only possible (rational) solution is that on the saddle path. Even if we accept this argument, it does not help in analysing systems with multiple equilibrium in which more than one stable saddle path exits. Given some initial point off the saddle path, to which saddle path will the system 'jump'? Economists are only just beginning to investigate these difficult questions. 1.5 Nonlinearity and chaos Aperiodic behaviour had usually been considered to be the result of either exogenous shocks or complex systems. However, nonlinear systems that are simple and deterministic can give rise to aperiodic, or chaotic, behaviour. The crucial element leading to this behaviour is the fact that the system is nonlinear. For a linear system a small change in a parameter value does not affect the qualitative nature of the system. For nonlinear systems this is far from true. For some small change (even very small) both the quantitative and qualitative behaviour of the system can dramatically change. Strangely, nonlinearity is the norm. But in both the physical sciences and economics linearity has been the dominant mode of study for over 300 years. Nonlinearity is the most commonly found characteristic of systems and it is therefore necessary for the scientist, including the social scientist, to take note of this. The fact that nonlinear systems can lead to aperiodic or chaotic behaviour has meant a new branch of study has arisen - chaos theory. It may be useful to point out that in studying any deterministic system three characteristics of the system must be known (Hilborn 1994, p. 7): 16 Economic Dynamics (1) the time-evolution values, (2) the parameter values, and (3) the initial conditions. A system for which all three are known is said to be deterministic. If such a deterministic system exhibits chaos, then it is very sensitive to initial conditions. Given very small differences in initial conditions, then the system will after time behave very differently. But this essentially means that the system is unpredictable since there is always some imprecision in specifying initial conditions,5 and therefore the future path of the system cannot be known in advance. In this instance the future path of the system is said to be indeterminable even though the system itself is deterministic. The presence of chaos raises the question of whether economic fluctuations are generated by the 'endogenous propagation mechanism' (Brock and Malliaris 1989, p. 305) or from exogenous shocks to the system. The authors go on, Theories that support the existence of endogenous propagation mechanisms typically suggest strong government stabilization policies. Theories that argue that business cycles are, in the main, caused by exogenous shocks suggest that government stabilization policies are, at best, an exercise in futility and, at worst, harmful, (pp. 306-7) This is important. New classical economics assumes that the macroeconomy is asymptotically stable so long as there are no exogenous shocks. If chaos is present then this is not true. On the other hand, new Keynesian economics assumes that the economic system is inherently unstable. What is not clear, however, is whether this instability arises from random shocks or from the presence of chaos. As Day and Shafer (1992) illustrate, in the presence of nonlinearity a simple Keynesian model can exhibit chaos. In the presence of chaos, prediction is either hazardous or possibly useless - and this is more true the longer the prediction period. Nonlinearity and chaos is quite pervasive in economics. Azariadis (1993) has argued that much of macroeconomics is (presently) concerned with three relationships: the Solow growth model, optimal growth, and overlapping generations models. The three models can be captured in the following discrete versions: (l-8)kt + sf(kt) (0 kt+i = -;—;- 1 + n (1.7) (ii) kt+i = f(kt) + (l-8)kt- ct u'(ct) = pu'(ct+1)[f(kt+1) + (1 - 8)] (Hi) (1 + n)kt+1 = z[f(kt+1) + (1 - 8), w(kt)] The explanation of these equations will occur later in the book. Suffice it to say here that Azariadis considers that the business of mainstream macroeconomics amounts to 'complicating' one of [these] dynamical systems ... and exploring what happens as new features are added, (p. 5) 5 As we shall see in chapter 7, even a change in only the third or fourth decimal place can lead to very different time paths. Given the poor quality of economic data, not to mention knowledge of the system, this will always be present. The literature refers to this as the butterfly effect. Introduction 17 All these major concerns involve dynamical systems that require investigation. Some have found to involve chaotic behaviour while others involve multiple equilibria. All three involve nonlinear equations. How do we represent these systems? How do we solve these systems? Why do multiple equilibria arise? How can we handle the analysis in the presence of nonlinearity? These and many more questions have been addressed in the literature and will be discussed in this book. They all involve an understanding of dynamical systems, both in continuous time and in discrete time. The present book considers these issues, but also considers dynamic issues relevant to microeconomics. The present book also tries to make the point that even in the area of macroeconomics, these three systems do not constitute the whole of the subject matter. As one moves into the realms of policy questions, open economy issues begin to dominate. For this reason, the present book covers much more of the open economy when discussing macroeconomic issues. Of importance here is the differential speeds of adjustment in the various sectors of the economy. Such asymmetry, however, is also relevant to closed economy models, as we shall see. 1.6 Computer software and economic dynamics Economic dynamics has not been investigated for a long time because of the mathematical and computational requirements. But with the development of computers, especially ready-made software packages, economists can now fairly easily handle complex dynamic systems. Each software package has its comparative advantage. This is not surprising. But for this reason I would not use one package to do everything. Spreadsheets -whether Excel, QuattroPro, Lotus 1-2-3, etc. - are all good at manipulating data and are particularly good at displaying sequential data. For this reason they are especially useful at computing and displaying difference equations. This should not be surprising. Difference equations involve recursive formulae, but recursion is the basis of the copy command in spreadsheets, where entries in the cells being copied have relative (and possibly absolute) cell addresses. If we have a difference equation of the form xt =f{xt-\), then so long as we have a starting value xq, it is possible to compute the next cell down as/(^o)- If we copy down n— 1 times, then xn is no more than/(x„_i). Equally important is the fact that/(.*;,_ i) need not be linear. There is inherently no more difficulty in copying f{xt-i) = a + bxt-i than in copying= a + bxt-i + cx]_x or/(*,_i) = a + b sin(x,_i). The results may be dramatically different, but the principle is the same. Nonlinear equations are becoming more important in economics, as we indicated in the previous section, and nonlinear difference equations have been at the heart of chaos. The most famous is the logistic recursive equation xt =f(xt-i, A) = Xxt-i(\ -xt-\) (1.8) It is very easy to place the value of A in a cell that can then be referred to using an absolute address reference. In the data column all one does is specify and then x\ is computed from/(^o, which refers to the relative address of xq and 18 Economic Dynamics Figure 1.4. $ indicates absolute address formula being investigated absence of $ indicates relative address spreadsheet version of formula initial value values derived from copying cell B6 down > interactive graph B5 is the initial value B6 is where the formula is first entered This is then copied to the clipboard The clipboard is then copied down eight times in a single operation (as shown by the blocking action) the absolute address of k. This is then copied down as many times as one likes, as illustrated in figure 1.4.6 This procedure allows two things to be investigated: (1) different values for A (2) different initial values (different values for xq). Equally important, xt can be plotted against t and the implications of changing X and/or can immediately be observed. This is one of the real benefits of the Windows spreadsheets. There is no substitute for interactive learning. In writing this 6 In this edition, all spreadsheets are created in Microsoft Excel. Introduction 19 book there were a number of occasions when I set up spreadsheets and investigated the property of some system and was quite surprised by the plot of the data. Sometimes this led me to reinvestigate the theory to establish why I saw what I did. The whole process, sometimes frustrating, was a most satisfying learning experience. The scope of using spreadsheets for investigating recursive equations cannot be emphasised enough. But they can also be used to investigate recursive systems. Often this is no more difficult than a single equation, it just means copying down more than one column. For example, suppose we have the system xt = axt-\ + byt-\ yt = cxt-i + dyt-i Then on a spreadsheet all that needs to be specified is the values for a, b, c and d and the initial values for x and y, i.e., xq and yo. Then x\ and y\ can be computed with relative addresses to and yo and absolute addresses to a, b, c and d. Given these solutions then all that needs to be done is to copy the cells down. Using this procedure it is possible to investigate some sophisticated systems. It is also possible to plot trajectories. The above system is autonomous (it does not involve t explicitly) and so {x(t), y(t)} can be plotted using the spreadsheet's x-y plot. Doing this allows the display of some intriguing trajectories - and all without any intricate mathematical knowledge.7 Having said this, I would not use a spreadsheet to do econometrics, nor would I use Mathematica or Maple to do so - not even regression. Economists have many econometrics packages that specialise in regression and related techniques. They are largely (although not wholly) for parameter estimation and diagnostic testing. Mathematica and Maple (see the next section) can be used for statistical work, and each comes with a statistical package that accompanies the main programme, but they are inefficient and unsuitable for the economist. But the choice is not always obvious. Consider, for example, the logistic equation xt =/(x,_i) = 3.5x,_i(l (1.10) It is possible to compute a sequence {xt} beginning at xo = 0.1 and to print the 10th through to the 20th iteration using the following commands in Mathematical clear[f] f[x_] :=3.5x(l-x) ; StartingValue: . 1; FirstIteration=l0: Lastlteration=2 0; i=0; y=N[StartingValue]; While[i<=LastIteration, If[i>=FirstIteration, Print[i, " N[y,8] ] ]; Y = f[y]; i =i+l] 7 See Shone (2001) for an introductory treatment of economic dynamics using spreadsheets. 8 Taken from Holmgren (1994, appendix Al). 20 Economic Dynamics which would undoubtedly appeal to a mathematician or computer programmer. The same result, however, can be achieved much simpler by means of a spreadsheet by inputting 0.1 in the first cell and then obtaining 3.5xq{\ — *o) in the second cell and copying down the next 18 cells. Nothing more is required than knowing how to enter a formula and copying down.9 There are advantages, however, to each approach. The spreadsheet approach is simple and requires no knowledge of Mathematica or programming. However, there is not the same control over precision (it is just as acceptable to write N [ y, 9 9] for precision to 99 significant digits in the above instructions). Also what about the iteration from the 1000th through to 1020th? Use of the spreadsheet means accepting its precision; while establishing the iterations from 1000 onwards still requires copying down the first 998 entries! For the economist who just wants to see the dynamic path of a sequence {xt}, then a spreadsheet may be all that is required. Not only can the sequence be derived, but also it can readily be graphed. Furthermore, if the formula is entered as/(.*;) = rx{\ — x), then the value of r can be given by an absolute address and then changed.10 Similarly, it is a simple matter of changing xq to some value other than 0.1. Doing such manipulations immediately shows the implications on a plot of {xt}, most especially its convergence or divergence. Such interactive learning is quick, simple and very rewarding. The message is a simple one. Know your tools and use the most suitable. A hammer can put a nail in a plank of wood. It is possible to use a pair of pliers and hit the nail, but no tradesman would do this. Use the tool designed for the task. I will not be dealing with econometrics in this book, but the message is general across software: use the software for which it is 'best' suited. This does beg the question of what a particular software package is best suited to handle. In this book we intend to answer this by illustration. Sometimes we employ one software package rather than another. But even here there are classes of packages. It is this that we concentrate on. Which package in any particular class is often less important: they are close substitutes. Thus, we have four basic classes of software: (1) Spreadsheets Excel, QuattroPro, Lotus 1-2-3, etc. (2) Mathematics Mathematica, Maple, MatLab, MathCad, DERIVE, etc. (3) Statistical SPSS, Systat, Statgraphics, etc. (4) Econometrics Shazam, TSP, Microfit, etc. 1.7 Mathematica and Maple An important feature of the present book is the ready use of both Mathematica and Maple.11 These packages for mathematics are much more than glorified calculators because each of them can also be applied to symbolic manipulation: they can expand the expression (x + y)2 into x2 + y2 + 2xy, they can carry out differentiation and integration and they can solve standard differential equations - and much 9 Occam's razor would suggest the use of the spreadsheet in this instance. 10 We use r rather than X to avoid Greek symbols in the spreadsheet. 11 There are other similar software packages on the market, such as DERIVE and MathCad, but these are either more specialised or not as extensive as Mathematica or Maple. Introduction 21 output r Kernel —^ input \ Figure 1.5. Front End requested as required Libraries more. Of course, computer algebra requires some getting used to. But so did the calculator (and the slide rule even more so!). But the gains are extensive. Once the basic syntax is mastered and a core set of commands, much can be accomplished. Furthermore, it is not necessary to learn everything in these software packages. They are meant to be tools for a variety of disciplines. The present book illustrates the type of tools they provide which are useful for the economist. By allowing computer software to carry out the tedious manipulations - whether algebraic or numeric - allows concentration to be directed towards the problem in hand. Both Mathematica and Maple have the same basic structure. They are composed of three parts: (1) a kernel, which does all the computational work, (2) a front end, which displays the input/output and interacts with the user, and (3) a set of libraries of specialist routines. This basic structure is illustrated in figure 1.5. What each programme can do depends very much on which version of the programme that is being used. Both programmes have gone through many upgrades. In this second edition we use Mathematica for Windows version 4 and Maple 6 (upgrade 6.01).12 Each programme is provided for a different platform. The three basic platforms are DOS, Windows and UNIX. In the case of each programme, the kernel, which is the heart of the programme, is identical for the different platforms. It is the front end that differs across the three platforms. In this book it is the Windows platform that is being referred to in the case of both programmes. The front end of Maple is more user friendly to that of Mathematica, but Mathematical kernel is far more comprehensive than that of Maple}1' Both have extensive specialist library packages. For the economist, it is probably ease of use 12 Mathematica for Windows has been frequently upgraded, with a major change occurring with Mathematica 3. Maple was Maple V up to release 5, and then become Maple 6. Both packages now provide student editions. 13 Mathematical palettes are far more extensive than those of Maple (see Shone 2001). 22 Economic Dynamics that matters most, and Maple's front end is far more user friendly and far more intuitive than that of Mathematica. Having said this, each has its strengths and in this book we shall highlight these in the light of applicability to economics. The choice is not always obvious. For instance, although the front end of Maple is more user friendly, I found Mathematical way of handling differential equations easier and more intuitive, and with greater control over the graphical output. Certainly both are comprehensive and will handle all the types of mathematics encountered in economics. Accordingly, the choice between the two packages will reduce to cost and ease of use. Having mentioned the front end, what do these look like for the two packages? Figure 1.6 illustrates the front end for a very simple function, namely y = x3, where each programme is simply required to plot the function over the interval —3 < x < 3 and differentiate it. Both programmes now contain the graphical output in the same window.14 In Mathematica (figure 1.6a) a postscript rendering of the graph is displayed in the body of the page. This can be resized and copied to the clipboard. It can also be saved as an Encapsulated Postscript (EPS), Bitmap (BMP), Enhanced Metafile (EMF) and a Windows Metafile. However, many more graphical formats are available using the Export command of Mathematica. To use this the graphic needs to have a name. For instance, the plot shown in figure 1.6 could be called plotl6, i.e., the input line would now be plotl6=Plot [ (xA3, {x,-3,3}] Suppose we wish to export this with a file name Fig01_06. Furthermore, we wish to export it as an Encapsulated Postscript File (EPS), then the next instruction would be Export ["Fig01_06.eps" ,plotl6, "EPS" ] In the case of Maple (figure 1.6b) the plot can be copied to the clipboard and pasted or can be exported as an Encapsulated Postscript (EPS), Graphics Interchange Format (GIF), JPEG Interchange Format (JPG), Windows Bitmap (BMP) and Windows Metafile (WMF). For instance, to export the Maple plot in figure 1.6, simply right click the plot, choose 'Export As', then choose 'Encapsulated Postscript (EPS)...' and then simply give it a name, e.g., Fig01_06. The 'eps' file extension is automatically added. Moving plots into other programmes can be problematic. This would be necessary, for example, if a certain degree of annotation is required to the diagram. This is certainly the case in many of the phase diagrams constructed in this book. In many instances, diagrams were transported into CorelDraw for annotation.15 When importing postscript files it is necessary to use CorelDraw's '.eps,*.ps (interpreted)' import filter. In this book we often provide detailed instructions on deriving solutions, especially graphical solutions, to a number of problems. Sometimes these are provided in the appendices. Since the reader is likely to be using either Mathematica or Maple, then instructions for each of these programmes are given in full in the body 14 This was not always the case with Maple. In earlier versions, the graphical output was placed in separate windows. 15 CorelDraw has also gone through a number of incarnations. This book uses CorelDraw 9.0. Introduction 23 (a) Mathematica Figure 1.6. (b) Maple I Muplv 6 - IFigDl mws] ■-la1*l g] Eta EM Wsw Onions Window tfalp Mal^l IfTlxltH ^wi--Hie.iH^iHni*i l»J*ISw|;| ___ > plot(f,x=-3.,3); -3 -2 / ■ '0 3x* / / 1 Tims- lOt ByT8s_J.lt M AueiloWl 1 -7G of the text for the most important features useful to the economist. This allows the reader to choose whichever programme they wish without having to follow instructions on the use of the alternative one, with which they are probably not familiar. Although this does involve some repeat of the text, it seems the most sensible approach to take. The routines contained here may not always be the most efficient - at 24 Economic Dynamics least in the eyes of a computer programmer - but they are straightforward and can readily be reproduced without any knowledge of computer programming. Furthermore, they have been written in such a way that they can easily be adapted for any similar investigation by the reader. 1.8 Structure and features This book takes a problem solving, learning by doing approach to economic dynamics. Chapters 2-5 set out the basic mathematics for continuous and discrete dynamical systems with some references to economics. Chapter 2 covers continuous single-equation dynamics, while chapter 3 deals with discrete single-equation dynamics. Chapter 4 covers continuous dynamical systems of equations and chapter 5 deals with discrete dynamical systems of equations. Chapters 6 and 7 cover two quite distinct dynamical topics that do not fit into the continuous/discrete categorisation so neatly. Chapter 6 deals with control theory and chapter 7 with chaos theory. Both these topics are more advanced, but can be taken up at any stage. Each deals with both continuous and discrete modelling. Chapters 1-7 constitute part I and set out the mathematical foundation for the economic topics covered in part II. Part II contains chapters 8-15, and deals with problems and problem solving. Each subject intermingles continuous and discrete modelling according to the problem being discussed and the approach taken to solving it. We begin with demand and supply in chapter 8. Chapter 9 also deals with a topic in microeconomics, namely the dynamics of oligopoly. This chapter is new to this edition. We then introduce the basic modelling of macroeconomics in terms of closed economy dynamics, emphasising the underlying dynamics of the IS-LM model and extending this to the Tobin-Blanchard model. Next we consider the important topics of inflation and unemployment. Here we are more restrictive, considering just certain dynamic aspects of these interrelated topics. Chapters 12 and 13 deal with open economy dynamics, a much-neglected topic in macroeconomics until recently. Chapter 12 deals with the open economy under the assumption of a fixed price level, while Chapter 13 deals with open economy dynamics under the assumption of flexible prices. It will be seen that the modelling approach between these two differs quite considerably. In chapter 14 we consider population models, which can be considered a microeconomic topic. Not only does it deal with single populations, but it also considers the interaction between two populations. Finally, chapter 15 on fisheries economics also deals with a microeconomic topic that is a central model in the theory of environmental economics. All the topics covered in part II are contained in core courses in economic theory. The main difference here is the concentration on the dynamics of these topics and the techniques necessary to investigate them. All chapters, with the exception of this one, contain exercises. These not only enhance the understanding of the material in the chapter, but also extend the analysis. Many of these questions, especially in part II, are problem solving type exercises. They require the use of computer software to carry them out. Sometimes this is no more than using a spreadsheet. However, for some problems the power of a mathematical programme is required. It is in carrying out the exercises that one learns Introduction 25 by doing. In a number of the exercises the answers are provided in the question. When this is not the case, answers to a number of the questions are supplied at the end of the book. The present book has a number of features. The coverage is both up-to-date and deals with discrete as well as continuous models. The book is fairly self-contained, with part I supplying all the mathematical background for discussing dynamic economic models, which is the content of part II. Many recent books on dynamic economics deal largely with macroeconomics only. In this book we have attempted a more balanced coverage between microeconomics and macroeconomics. Part I in large part treats continuous models and discrete models separately. In part II, however, the economics dictates to a large extent whether a particular model is discrete or continuous - or even both. A feature of both part I and part II is a discussion of the phase diagram for analysing dynamic models. A major emphasis is problem solving, and to this end we supply copious solved problems in the text. These range from simple undergraduate economic models to more sophisticated ones. In accomplishing this task ready use has been made of three software packages: Mathematica, Maple and Excel. The text has detailed instructions on using both Mathematica and Maple, allowing the reader to duplicate the models in the text and then to go beyond these. In order to reinforce the learning process, the book contains copious exercises. Detailed solutions using both Mathematica and Maple are provided on the Cambridge University website. Additional reading Additional material on the economic content of this chapter can be found in Azariades (1993), Brock and Malliaris (1989), Bullard and Butler (1993), Day and Shafer (1992), De Jong (1967), Farmer (1999), Mizrach (1992), Mooney and Swift (1999), Mullineux and Peng (1993), Neal and Shone (1976) and Scheinkman (1990). Additional material on Mathematica can be found in, Abell and Braselton (1992, 1997a, 1997b), Blachman (1992), Brown, Porta and Uhl (1991), Burbulla and Dodson (1992), Coombes et al. (1998), Crandall (1991), Don (2001), Gray and Glynn (1991), Huang and Crooke (1997), Ruskeepaa (1999), Schwalbe and Wagon (1996), Shaw and Tigg (1994), Shone (2001), Skeel and Keiper (1993), Varian etal. (1993), Wagon (1991) and Wolfram (1999). Additional material on Maple can be found in Abell and Braselton (1994a, 1994b, 1999), Devitt (1993), Ellis et al. (1992), Gander and Hrebicek (1991), Heck (1993), Kofler (1997), Kreyszig and Norminton (1994) and Nicolaides and Walkington (1996). CHAPTER 2 Continuous dynamic systems 2.1 Some definitions A differential equation is an equation relating: (a) the derivatives of an unknown function, (b) the function itself, (c) the variables in terms of which the function is defined, and (d) constants. More briefly, a differential equation is an equation that relates an unknown function and any of its derivatives. Thus -^+3xy = ex ax is a differential equation. In general dy -T=f(x,y) dx is a general form of a differential equation. In this chapter we shall consider continuous dynamic systems of a single variable. In other words, we assume a variable x is a continuous function of time, t. A differential equation for a dynamic equation is a relationship between a function of time and its derivatives. One typical general form of a differential equation is dx (2.1) —=f(t,x) dt Examples of differential equations are: dx , (i) — + 3x = 4 + e~' dt d d^c o (ii) — +4t— -3(l-tz)x = 0 dtz dt dx (iii) — = kx V ' dt du dv (iv) — + — + 4u = 0 dt dt Continuous dynamic systems 27 In each of the first three examples there is only one variable other than time, namely x. They are therefore referred to as ordinary differential equations. When functions of several variables are involved, such as u and v in example (iv), such equations are referred to as partial differential equations. In this book we shall be concerned only with ordinary differential equations. Ordinary differential equations are classified according to their order. The order of a differential equation is the order of the highest derivative to appear in the equation. In the examples above (i) and (iii) are first-order differential equations, while (ii) is a second-order differential equation. Of particular interest is the linear differential equation, whose general form is dnx dn~lx a°^~df> + ai^~df^ + • • • + an{t)x = g^ (2-2) If ao(t), a\(i), ... , an(t) are absolute constants, and so independent of t, then equation (2.2) is a constant-coefficient «th-order differential equation. Any differential equation not conforming to equation (2.2) is referred to as a nonlinear differential equation. The rath-order differential equation (2.2) is said to be homogeneous if g(t) = 0 and nonhomogeneous if g(t) is not identically equal to zero. Employing these categories, the examples given above are as follows: (i) a linear constant-coefficient differential equation with nonhomogeneous term g(t) = 4 + e~' (ii) a second-order linear homogeneous differential equation (iii) a linear constant-coefficient homogeneous differential equation. In the present book particular attention will be directed to first-order linear differential equations which can be expressed in the general form dx hit)— + k(t)x = g(t) dt by dividing throughout by h(t) we have the simpler form ^ + a(t)x = b(t) (2.3) dt The problem is to find all functions x(t) which satisfy equation (2.3). However, in general equation (2.3) is hard to solve. In only a few cases can equation (2.1) or (2.3) be solved explicitly. One category that is sometimes capable of solution is autonomous or time-invariant differential equations, especially if they are linear. Equation (2.1) would be autonomous if df/dt = 0 and nonautonomous if df/dt ^ 0. In the examples of ordinary differential equations given above only (iii) is an autonomous differential equation. A solution to a rath-order differential equation is an ra-times differential function x = 0(0 which when substituted into the equation satisfies it exactly in some interval a < t < b. 28 Economic Dynamics Example 2.1 Consider (iii) above. This is an autonomous first-order homogeneous differential equation. Rearranging the equation we have dx 1 dt x Integrating both sides with respect to t yields ^kdt f--dt= Ik J dt x J In x(t) = kt + Co where Co is the constant of integration. Taking exponentials of both sides yields x(t) = cekt where c = ec°. It is readily verified that this is indeed a solution by differentiating it and substituting. Thus kca — Jcjc — kca which holds identically for any a < t < b. Example 2.2 To check whether x(t) = 1 + t + ce' is a solution of dx/dt = x — t, we can differentiate x with respect to t and check whether the differential equation holds exactly. Thus dx , — = l + cet dt ;. 1 + ce1 = 1 + t + ce1 - t Hence x(t) = 1 + t + ce' is indeed a solution. Example 2.3 Check whether ap0 P(t) = bpo + (a - bp0)e at is a solution to the differential equation -j- =p(a- bp) dt Differentiating the solution function with respect to t we obtain dp -1-0 — = -ap0[bpo + (a- bp0)e at] z(-a(a - bp0)e at) dt a2po(a — bpo)e~at [bpo + (a- bpo)e at] atl2 Continuous dynamic systems 29 while substituting for p we obtain ) 2 ap — bp2 h apo bpo + (a- bp0)e ' a2p0(a - bp0)e~ bpo + (a- bpo)e —at [bpo + (a- bpo)e-at]2 which is identically true for all values of t. Equation x(t) = cekt is an explicit solution to example (iii) because we can solve directly x(t) as a function of t. On occasions it is not possible to solve x(t) directly in terms of t, and solutions arise in the implicit form Solutions of this type are referred to as implicit solutions. A graphical solution to a first-order differential equation is a curve whose slope at any point is the value of the derivative at that point as given by the differential equation. The graph may be known precisely, in which case it is a quantitative graphical representation. On the other hand, the graph may be imprecise, as far as the numerical values are concerned; yet we have some knowledge of the solution curve's general shape and features. This is a graph giving a qualitative solution. The graph of a solution, whether quantitative or qualitative, can supply considerable information about the nature of the solution. For example, maxima and minima or other turning points, when the solution is zero, when the solution is increasing and when decreasing, etc. Consider, for example, dx/dt = t2 whose solution is where c is the constant of integration. There are a whole series of solution curves depending on the value of c. Four such curves are illustrated in figure 2.1, with solutions F(x, 0 = 0 (2.4) x(t) = — + c x(t) = — + 8, x(t) = -+2, x(t) =--3 x(0=03/3)+c Figure 2.1. 30 Economic Dynamics A general solution to a differential equation is a solution, whether expressed explicitly or implicitly, which contains all possible solutions over an open interval. In the present example, all solutions are involved for all possible values of c. A particular solution involves no arbitrary constants. Thus, if c = 2 then x(t) = (t3/3) + 2 represents a particular solution. It is apparent that a second-order differential equation would involve integrating twice and so would involve two arbitrary constants of integration. In general the solution to an rath-order differential equation will involve n arbitrary constants. It follows from this discussion that general solutions are graphically represented by families of solution curves, while a particular solution is just one solution curve. Consider further the general solution in the above example. If we require that x = 0 when t = 0, then this is the same as specifying c = 0. Similarly if x = 2 when t = 0, then this is the same as specifying that c = 2. It is clear, then, that a particular solution curve to a first-order differential equation is equivalent to specifying a point (xo, to) through which the solution curve must pass (where to need not be zero). In other words, we wish to find a solution x = x(t) satisfying x(to) = xq-The condition x(to) = xq is called the initial condition of a first-order differential equation. A first-order differential equation, together with an initial condition, is called a first-order initial value problem. In many applications we find that we need to impose an initial condition on the solution. Consider the following first-order initial value problem dx (2.5) — = kx x(t0) = x0 dt Rearranging and integrating over the interval to to t\ we obtain This is a particular solution that satisfies the initial condition. We shall conclude this section with some applications taken from economics and some noneconomic examples. At this stage our aim is simply to set out the problem so as to highlight the type of ordinary differential equations that are involved, the general or specific nature of the solution and whether the solution satisfies some initial value. Example 2.4 A simple continuous price-adjustment demand and supply model takes the form: qd = a + bp b < 0 qs = c + dp d > 0 x(t) = xoe' ,k(t-t0) dp a(q - qs) a > 0 dt Continuous dynamic systems 31 c-a, r ,c-a, -a(rf-*y Figure 2.2. where quantities, gd and and price, /?, are assumed to be continuous functions of time. Substituting the demand and supply equations into the price adjustment equation we derive the following dp --a(b — d) = a(a — c) dt which is a first-order linear nonhomogeneous differential equation. Using a typical software programme for solving differential equations, the solution path is readily found to be c — a Po - c — a b-d -a(d—b)t which satisfies the initial condition. For d - b > 0 the solution path for different initial prices is illustrated in figure 2.2 Example 2.5 Suppose we have the same basic demand and supply model as in example 2.4 but now assume that demand responds not only to the price of the good but also to the change in the price of the good. In other words, we assume that if the price of the good is changing, then this shifts the demand curve. We shall leave open the question at this stage of whether the demand curve shifts to the right or the left as a result of the price change. The model now takes the form qd = a + bp+f— b<0,f^0 dt qs = c + dp d>0 (2.7) ^ = a(qd -qs) a > 0 dt 32 Economic Dynamics This is effectively a stock-adjustment model. Stocks (inventories) change according to the difference between supply and demand, and price adjusts according to the accumulation-decumulation of stocks. Thus, if i(t) denotes the inventory holding of stocks at time t, then di dt = qs-qd i = k+ / ( 0 which is the third equation in the model. Substituting the demand and supply equations into the price-adjustment equation results in the following first-order linear nonhomogeneous differential equation dp dt with solution Pit)-- a(b — d) 1 - af P = a(a — c) c — a b-d + Po c — a b-d which satisfies the initial condition p(0) = po. For this model there are far more varieties of solution paths, depending on the values of the various parameters. Some typical solution paths are illustrated in figure 2.3. We shall discuss the stability of such systems later. Figure 2.3. C-a c-a {-a(d-bM\-af)} Continuous dynamic systems 33 Example 2.6 Assume population, p, grows at a constant rate k, where we assume that p is a continuous function of time, t. This means that the percentage change in the population is a constant k. Hence dp 1 -r- = k (2.8) dt p which immediately gives the first-order linear homogeneous differential equation dp with solution pit) = p0ekt which satisfies the initial condition p(0) = p0. Typical solution paths for this Malthusian population growth are illustrated in figure 2.4. Example 2.7 In many scientific problems use is made of radioactive decay. Certain radioactive elements are unstable and within a certain period the atoms degenerate to form another element. However, in a specified time period the decay is quite specific. In the early twentieth century the famous physicist Ernest Rutherford showed that the radioactivity of a substance is directly proportional to the number of atoms 34 Economic Dynamics present at time t. If dn/dt denotes the number of atoms that degenerate per unit of time, then according to Rutherford dn (2.9) — = -kn X > 0 V ' dt where A is the decay constant of the substance concerned and n is a continuous function of time. This is a first-order linear homogeneous differential equation and is identical in form to the exponential population growth specified in example 2.6 above. We shall return to this example later when we consider its solution and how the solution is used for calculating the half-life of a radioactive substance and how this is used to authenticate paintings and such items as the Turin shroud. Example 2.8 In this example we consider a continuous form of the Harrod-Domar growth model. In this model savings, S, is assumed to be proportional to income, Y; investment, /, i.e., the change in the capital stock, is proportional to the change in income over time; and in equilibrium investment is equal to savings. If s denotes the average (here equal to the marginal) propensity to save, and v the coefficient for the investment relationship, then the model can be captured by the following set of equations S = sY (2.10) I = K = vf I = S where a dot above a variable denotes the first-time derivate, i.e., dx/dt. Substituting, we immediately derive the following homogeneous differential equation vt = sY 1>-(^ = ° with initial condition Io = S0 = sY0 It also follows from the homogeneous equation that the rate of growth of income is equal to s/v, which Harrod called the 'warranted rate of growth'. The solution path satisfying the initial condition is readily established to be Y(t) = Y0e(s/V)t Example 2.91 It is well known that the Solow growth model reduces down to a simple autonomous differential equation. We begin with a continuous production function 1 We develop this model in detail here because it has once again become of interest and is the basis of new classical growth models and real business cycle models. A discrete version of the model is developed in chapter 3. Continuous dynamic systems 35 Y = F(K, L), which is twice differentiable and homogeneous of degree one (i.e. constant returns to scale), hetk = K/L denote the capital/labour ratio and y = Y/L the output/labour ratio. Then Y F(K,L) (K \ = F[-,l)=F(k,l)=f(k) L L \L i.e. y=f(k) with/(0) = 0,f(k) > 0,f"(k) <0,k>0 We make two further assumptions: 1. The labour force grows at a constant rate n, and is independent of any economic variables in the system. Hence L = nL L(0) = L0 2. Savings is undertaken as a constant fraction of output (S = sY) and savings equal investment, which is simply the change in the capital stock plus replacement investment, hence I =K + SK S = sY K + SK = sY K(0) = K0 Now differentiate the variable k with respect to time, i.e., derive dk/dt, dK dh Ai L--K— 0 dx b With no more information we cannot solve this equation. Suppose, then, that y=f(x) and f(x) = ax-bf(x) /(0) = \ b Since, by assumption,/^) is differentiable, then so is/'(.*;). Thus fix) = a- bfix) = a — b[ax — bf(x)] = a — [abx — b2f(x)] Since each derivative can be reduced to functions of x and fix), then so long as f(x) is differentiable, all order differentials exist. But why consider the existence of such differentials? The reason is that they give information about f(x), the domain of x. Now consider y =f(x) for the range x > 0. Since/(0) = a/b ba f'(0) = a.O--= -a < 0 b Then we know that/(.*;) crosses the v-axis at a/b and for x near zero the function is decreasing. This decrease will continue until a turning point occurs. A turning point requires that/'(.*;) = 0. Let x* denote the value of x at the turning point, then f(x*) = ax* - bfix*) = 0 ax or — =/(**) b i.e. where fix) cuts the line y = ax/b. To establish whether the turning point at x = x* is a minimum or a maximum we turn to fix*) = a- [abx* - b2fix*)] b2ax*' a abx* b a — [abx* — abx*] a > 0 42 Economic Dynamics Figure 2.6. Rough sketch of J{x), the solution to dy/dx = ax - by, y(0)=a/b and a,b>0 Hence, f(x) reaches a minimum at x = x* where f(x) cuts the line y = ax/b. It must follow, then, that for x > x*,f(x) is positively sloped. This can be verified immediately f'(x) = ax- bf(x) x > x* implying ••• /'(*)>0 ax > f(x) or ax > bf{x) All the analysis so far allows us to graph the properties, as shown in figure 2.6. The curve f(x) cuts the v-axis at a/b, declines and reaches a minimum where f(x) cuts the line y = ax/b, and then turns up. Although we cannot identify/(jc) or the solution value of x*, we do know that x* is nonzero. But can we obtain additional information about the shape of/(.*;)? Yes - if we consider isoclines. Given Isoclines and direction fields dy — = ax — by dx then for every (^:,v)-combination this equation specifies the slope at that point. A plot of all such slopes gives the direction field for the differential equation, and gives the 'flow of solutions'. (The slopes at given points can be considered as small lines, like iron filings, and if many of these are drawn the direction field is revealed - just like iron filings reveal magnetic forces.) However, it is Continuous dynamic systems 43 y Figure 2.7. y=2x-2 not possible to consider all points in the (^:,v)-plane. One procedure is to consider the points in the (.*;,}>)-plane associated with a fixed slope. If m denotes a fixed slope, then f(x, y) = m denotes all combinations of x and y for which the slope is equal to m. f(x, y) = m is referred to as an isocline. The purpose of constructing these isoclines is so that a more accurate sketch of f(x) can be obtained. For dy/dx = ax — by = m the isoclines are the curves (lines) These are shown in figure 2.7. Of course, the slope of/(.*;, y) at each point along an isocline is simply the value of m. Thus, along y = ax/b the slope is zero or inclination arctanO = 0°. Along y = (ax/b) — (I/b) the slope is unity or inclination arc tanl = 45°; while along y = (ax/b) — (2/b) the slope is 2 or inclination arc tan2 = 63°. Hence, for values of m rising the slope rises towards infinity (but never reaching it). We have already established that along y = ax/b the slope is zero and so there are turning points all along this isocline. For m negative and increasing, the slope becomes greater in absolute terms. Consider finally m = a/b. Then the isocline is with intercept —a/b2. Then along this isocline the slope of the directional field is identical to the slope of the isocline. Hence, the direction fields look quite different either side of this isocline. Above it the solution approaches this isocline asymptotically from above. Hence, the function/^) takes the shape of the heavy curve in figure 2.7. In general we do not know the intercept or the turning point. In this instance we consider the approximate integral curves, which are the continuous lines drawn in figure 2.7. Such integral curves can take a variety of shapes. ax — by = m ax m y b b 44 Economic Dynamics We can summarise the method of isoclines as follows: (1) From the differential equation dy — = 0(*, y) dx determine the family of isoclines 0 dt then we can re-write this equation dn — = —Xdt n Integrating both sides, and letting cq denote the coefficient of integration, then j— = — jkdt + Co In n = —Xt + cq n = e~Xt+c° = ce~Xt c = ec° At t = to, n = no. From this initial condition we can establish the value of c n0 = ce~Xt° c = noeXt° n = n0e~XteXt° = noe~xit~to) The half-life of a radioactive substance is the time necessary for the number of nuclei to reduce to half the original level. Since no denotes the original level then half this number is no/2. The point in time when this occurs we denote t\/2. Hence ^ = noe-x^-to) }_ _ e—Ut\p.—to) 2 — In 2 = -Uti/2 - to) In 2 0.693 •"• h/2 = to H—— = to + A A Usually, to = 0 and so 0.693 h/2 / These results are illustrated in figure 2.11. Continuous dynamic systems 49 Figure 2.11. Example 2.14 Testing for art forgeries All paintings contain small amounts of the radioactive element lead-210 and a smaller amount of radium-226. These elements are contained in white lead which is a pigment used by artists. Because of the smelting process from which the pigment comes, lead-210 gets transferred to the pigment. On the other hand, over 90 per cent of the radium is removed. The result of the smelting process is that lead-210 loses its radioactivity very rapidly, having a half-life of about 22 years; radium-226 on the other hand has a half-life of 1,600 years (see example 2.7). For most practical purposes we can treat radium-226 emissions as constant. Let 1(f) denote the amount of lead-210 per gram of white lead at time t, and Iq the amount present at the time of manufacture, which we take to be to. The disintegration of radium-226 we assume constant at r. If A is the decay constant of lead-210, then dl = -Xl + r l(t0) = l0 dt with solution r '- Ht-t0)\ , i„-Ut-t0) l{t) = -(1 -e-x(t-t0>) +Iq Although 1(f) and r can readily be measured, this is not true of Iq, and therefore we cannot determine t — to. We can, however, approach the problem from a different perspective. Assume that the painting of interest, if authentic, is 300 years old and if new is at the present time t. Then t — to = 300. If we substitute this into the previous result and simplify we obtain Xlo = Xl(t)e300X - r(e300x - 1) 5 This is based on the analysis presented in Braun (1983, pp. 11-17). 50 Economic Dynamics It is possible to estimate kl$ for an authentic painting. It is also possible to estimate klo for the lead in the painting under investigation. If the latter is absurdly large relative to the former, then we can conclude that it is a forgery. A very conservative estimate would indicate that any value for XIq in excess of 30,000 disintegrations per minute per gram of white lead is absurd for an authentic painting aged 300 years. Using 22 years for the half-life of lead-210, then the value of X is (ln2/22) and g300A. _ g(300/22)ln2 _ 2(150/11) To estimate the present disintegration rate of lead-210 the disintegration rate of polonium-210 is used instead because it has the same disintegration rate as lead-210 and because it is easier to measure. In order, then, to authenticate the 'Disciples at Emmaus', purported to be a Vermeer, it is established that the disintegration rate of polonium-210 per minute per gram of white lead in this particular painting is 8.5 and that of radium-226 is 0.8. Using all this information then we can estimate the value of XIq for the'Disciples at Emmaus' as follows: kk = (8.5)215°/n - 0.8(215°/n - 1) = 98,050 which is considerably in excess of 30,000. We, therefore, conclude that the 'Disciples at Emmaus' is not an authentic Vermeer. Example 2.15 The logistic curve (2.20) In this example we shall consider the logistic equation in some detail. Not only does this illustrate a separable differential equation, but also it is an equation that occurs in a number of areas of economics. It occurs in population growth models, which we shall consider in part II, and in product diffusion models. It is the characteristic equation to represent learning, and hence occurs in a number of learning models. We shall justify the specification of the equation in part II; here we are concerned only with solving the following growth equation for the variable x dx — = kx(a dt x) The differential equation is first separated dx = kdt (a — x)x Integrating both sides, and including the constant of integration, denoted cq f, ^ \ = [kdt + co J (a- x)x J However 1 (a — x)x a 1 1 ' - +- x a — x Continuous dynamic systems 51 Hence 1 a 1 a /dx f dx 1 f --h /- = / kdt + co x J a-x J -[Inx — In \ a — x\] = kt + cq 1 -In a In a — x x = kt + Cq = akt + aco a — x Taking anti-logs, we have _ akt+acQ _ cicq akt _ rfia^ a — x where c = eac°. Substituting for the initial condition, i.e., t = to then x = xo, we can solve for the constant c, as follows ,akto a — xo c 11 X° \e-akt0 ^a — xo, Substituting, then x I xo a — x \a — xq g—akto gCikt Solving for x a x = a — xq a — xo X0 \eak(t-t0) 7ak(t—to) 1 + *0 yak(t—to) a — Xq, Which can be further expressed6 axo x = (a - xo)e-ak{t-^ + x0 (2.21) From the logistic equation (2.21) we can readily establish the following results, assuming that is less than a: 1. For t = to then x = xq 2. As t oo then x a 6 The logistic growth equation is a particular example of the Bernoulli function and can be solved in a totally different way using a simple transformation. See n. 2 and exercise 6. 52 Economic Dynamics Figure 2.12. x 3. An inflexion occurs at the point 1 / a — xq t = t0-\--In ak \ Xq a x = — 2 The logistic curve is shown in figure 2.12. Example 2.16 Constant elasticity of demand Let a commodity x be related to price p with a constant elasticity of demand e, then dx p --= — e e > 0 dp x We can rearrange this as dx dp x p /dx f ( - = -eJ- Integrating both sides and adding a constant of integration, then "dp P \nx = —elnp + co = — e Inp + In c where Co = In c = In cp~£ Therefore x = cp~E which is the general expression for a demand curve with constant elasticity of demand. Continuous dynamic systems 53 2.6 Diffusion models In recent years we have seen the widespread use of desktop computers, and more recently the increased use of the mobile phone. The process by which such innovations are communicated through society and the rate at which they are taken up is called diffusion. Innovations need not be products. They can just as easily be an idea or some contagious disease. Although a variety of models have been discussed in the literature (e.g. Davies 1979; Mahajan and Peterson 1985), the time path of the diffusion process most typically takes the form of the S-shaped (sigmoid) curve. Considering the mobile phone, we would expect only a few adoptions in the early stages, possibly business people. The adoption begins to accelerate, diffusing to the public at large and even to youngsters. But then it begins to tail off as saturation of the market becomes closer. At the upper limit the market is saturated. Although this is a verbal description of the diffusion process, and suggests an S-shaped mathematical formulation of the process, it supplies no exact information about the functional form. In particular, the slope, which indicates the speed of the diffusion; or the asymptote, which indicates the level of saturation. Furthermore, such diffusion processes may differ between products. The typical diffusion model can be expressed dN(t) —^=g(t)(m-N(t)) (2.22) dt where N(t) is the cumulative number of adopters at time t, m is the maximum number of potential adopters and g(t) is the coefficient of diffusion. dN(t)/dt then represents the rate of diffusion at time t. Although we refer to the number of adopters, the model is assumed to hold for continuous time t. It is possible to think of g(t) as the probability of adoption at time t, and so g(t)(m — N(t)) is the expected number of adopters at time t. Although a number of specifications of g(t) have been suggested, most are a special case of g(t) = a + bN(t) So the diffusion equation generally used is dN(t) -— = (a/m + bN(t))(m - N(t)) (2.23) dt If we divide (2.23) throughout by m and define F(t) = N(t)/m, with F(t) =N(t)/m, then dF(t) —^=(a + bF(t))(l-F(t)) (2.24) dt This is still a logistic equation that is separable, and we can re-arrange and integrate by parts (see example 2.15) to solve for F(t) I — e~(a+b)t F® ~ 1 + (b/a)e-(a+b» (2'25) This specification, however, is not the only possibility. The Gompertz function also exhibits the typical S-shaped curve (see exercise 2), and using this we can 54 Economic Dynamics express the diffusion process as dN(t) (2.26) —— = bN(t)(\n m - lniV(O) dt or dF{t) —^=bF(t)(- In F(t)) dt Suppressing the time variable for convenience, then the two models are F = (a + bF)(l — F) and F = bF{-\nF) Pursuing the logistic equation, we can graph F against F. When F = 0 then F = a and when F = 0 then {a + bF){\ — F) = 0 with solutions F\ = —b/a and F2 = 1 Since F denotes the rate of diffusion, then the diffusion rate is at a maximum (penetration is at its maximum rate) whenF = 0, i.e., when d2F/dt2 = 0. Differentiating and solving for F, which we denote Fp (for maximum penetration rate), we obtain la (1 a \ m am Fp=2-2b implylng Np = m.Fp = m^---j = -- — In order to find the time tp when F{tp) is at a maximum penetration rate, we must first solve for F{t). This we indicated above. Since we need to find the value of t satisfying F{t) = Fp, then we need to solve 1 _ e~(a+b)t l a 1 + (b/a)e-(a+b» 2 2b for t, which we can do using a software package. This gives the time for the maximum penetration of In - \ a (2.27) tp = a + b Since d2F/dt2 = 0 at Fp, then this must denote the inflexion point of F(t). The stylised information is shown in figure 2.13. Notice that the time for the maximum penetration is the same for both F(t) and N(t). Also note that F(t) involves only the two parameters a and b; while N(t) involves the three parameters a, b and m. 2.7 Phase portrait of a single variable This book is particularly concerned with phase diagrams. These diagrams help to convey the dynamic properties of differential and difference equations - either single equations or simultaneous equations. To introduce this topic and to lay down some terminology, we shall consider here just a single variable. Let x denote Continuous dynamic systems 55 dF/dt Figure 2.13. -b/a Fp=(b-a)l2b F(t ) 1h tp=ln(b/a)/(a+b) N(t ) m sot-■ tp=\n(b/a)/(a+b) a variable which is a continuous function of time, t. Let denote an autonomous differential equation, so that x'{t) is just a function of x and independent of t. Assume that we can solve for x'{t) for any point in time t. Then at any point in time we have a value for x'{t). The path of solutions as t varies is called a trajectory, path or orbit. The ^-axis containing the trajectory is called the phase line. If x'{t) = 0 then the system is at rest. This must occur at some particular point in time, say to. The solution value would then be x(to) = x*. The point x* is referred to variedly as a rest point, fixed point, critical point, equilibrium point or steady-state solution. For the Malthusian population equation p'{t) = kp, there is 56 Economic Dynamics Figure 2.14. dx/dt > 0 x dx/dt < 0 dx/dt < 0 x dx/dt > 0 "4 4 attractor repellor dx/dt > 0 x" dx/dt > 0 dx/dt < 0 4 4 dx/dt < 0 (right) shunt (left) shunt only one fixed point, namely p* = 0. In the case of the logistic growth equation x'(t) = kx(a — x) there are two fixed points, one at x\ = 0 and the other at x\ = a. In example 2.4 on demand and supply the fixed point, the equilibrium point, is given by which is also the fixed point for example 2.5. For the Harrod-Domar growth model (example 2.8) there is only one stationary point, only one equilibrium point, and that is Y* = 0. For the Solow growth model, in which the production function conforms to a Cobb-Douglas (example 2.9), there are two stationary values, one at k* = 0 and the other at Whether a system is moving towards a fixed point or away from a fixed point is of major importance. A trajectory is said to approach a fixed point if x{t) x* as t oo, in this case the fixed point is said to be an attractor. On the other hand, if x{t) moves away from x* as t increases, then x* is said to be a repellor. Fixed points, attractors and repellors are illustrated in figure 2.14. Also illustrated in figure 2.14 is the intermediate case where the trajectory moves first towards the fixed point and then away from the fixed point. Since this can occur from two different directions, they are illustrated separately, but both appear as a shunting motion, and the fixed point is accordingly referred to as a shunt. Consider once again the logistic growth equation x'{t) = kx(a — x), as illustrated in figure 2.15. Figure 2.15(a) illustrates the differential equation, figure 2.15(b) illustrates the phase line7 and figure 2.15(c) denotes the path of x{t) against time. The stationary points on the phase line are enclosed in small circles to identify them. The arrows marked on the phase line, as in figure 2.15(b), indicate the direction of change in x{t) as t increases. In general, x* = 0 is uninteresting, and for any initial value of x not equal to zero, the system moves towards x* = a, as illustrated in figure 2.15(c). Even if x initially begins above the level x* = a, the system moves over time towards x* = a. In other words, x* = a is an attractor. 7 Some textbooks in economics confusingly refer to figure 2.15(b) as a phase diagram. Continuous dynamic systems 57 If any trajectory starting 'close to' a fixed point8 stays close to it for all future time, then the fixed point is said to be stable. A fixed point is asymptotically stable if it is stable as just defined, and also if any trajectory that starts close to the fixed point approaches the fixed point as t —► oo. Considering the logistic equation as shown in figure 2.15, it is clear that x* = a is an asymptotically stable rest point. Figure 2.15 also illustrates another feature of the characteristics of a fixed point. The origin, x* = 0, is a repellor while x* = a is an attractor. In the neighbourhood of the origin, the differential equation has a positive slope. In the neighbourhood 8 We shall be more explicit about the meaning of 'close to' in section 4.2. 58 Economic Dynamics of the attractor, the differential equation has a negative slope. In fact, this is a typical feature of instability/stability. A fixed point is unstable if the slope of the differential equation in the neighbourhood of this point is positive; it is stable if the slope of the differential equation in the neighbourhood of this point is negative. If there is only one fixed point in a dynamic system, then such a fixed point is either globally stable or globally unstable. In the case of a globally stable system, for any initial value not equal to the fixed point, then the system will converge on the fixed point. For a globally unstable system, for any initial value not equal to the fixed point, then the system will move away from it. Consider example 2.4, a simple continuous price-adjustment demand and supply model with the differential equation dp — = a(a — c) + a(b — d)p a > 0 dt For a solution (a fixed point, an equilibrium point) to exist in the positive quadrant then a > c and so the intercept is positive. With conventional shaped demand and supply curves, then b < 0 and d > 0, respectively, so that the slope of the differential equation is negative. The situation is illustrated in figure 2.16(a). Continuous dynamic systems 59 Given linear demand and supply then there is only one fixed point. The system is either globally stable or globally unstable. It is apparent from figure 2.16 that the fixed point is an attractor, as illustrated in figure 2.16(b). Furthermore, the differential equation is negatively sloped for all values of p. In other words, whenever the price is different from the equilibrium price (whether above or below), it will converge on the fixed point (the equilibrium price) over time. The same qualitative characteristics hold for example 2.5, although other possibilities are possible depending on the value/sign of the parameter/. Example 2.6 on population growth, and example 2.7 on radioactive decay, also exhibit linear differential equations and are globally stable/unstable only for p = 0 and n = 0, respectively. Whether they are globally stable or globally unstable depends on the sign of critical parameters. For example, in the case of Malthusian population, if the population is growing, k > 0, then for any initial positive population will mean continuously increased population over time. If k < 0, then for any initial positive population will mean continuously declining population over time. In the case of radioactive decay, k is positive, and so there will be a continuous decrease in the radioactivity of a substance over time. The Harrod-Domar growth model, example 2.8, is qualitatively similar to the Malthusian population growth model, with the 'knife-edge' simply indicating the unstable nature of the fixed point. The Solow growth model, example 2.9, on the other hand, exhibits multiple equilibria. There cannot be global stability or instability because such statements have meaning only with reference to a single fixed point system. In the case of multiple fixed points, statements about stability or instability must be made in relation to a particular fixed point. Hence, with systems containing multiple equilibria we refer to local stability or local instability, i.e., reference is made only to the characteristics of the system in the neighbourhood of a fixed point. For instance, for the Solow growth model with a Cobb-Douglas production function homogeneous of degree one there are two fixed points The first is locally unstable while the second is locally stable, as we observed in figure 2.10. The first fixed point is a repellor while the second fixed point is an attractor. The slope of the differential equation in the neighbourhood of the origin has a positive slope, which is characteristic of a repellor; while the slope of the differential equation in the neighbourhood of the second fixed point is negative, which is characteristic of an attractor. These characteristics of the slope of the differential equation in the neighbourhood of a system's fixed points and the features of the phase line are illustrated in figure 2.17. 2.8 Second-order linear homogeneous equations A general second-order linear homogeneous differential equation with constant coefficients is 0 and (2.28) 60 Economic Dynamics repellor attractor dk/dt > 0 dk/dt < 0 m- •m- kt=0 fixed points Or ay"{t) + by'(t) + cy(t) = 0 If we can find two linearly independent solutions9 y\ and _>>2 then the general solution is of the form y = cm + where c\ and C2 are arbitrary constants. Suppose y = ext. Substituting we obtain ax2ext + bxext + cext = 0 ext(ax 2 + bx + c) = 0 Hence, y = ext is a solution if and only if ax + bx + c = 0 9 See exercises 9 and 10 for a discussion of linear dependence and independence. Continuous dynamic systems 61 which is referred to as the auxiliary equation of the homogeneous equation. The quadratic has two solutions -b + -Jb2 - Aac -b - -Jb2 - Aac r = -, s = - 2a 2a If b2 > Aac the roots r and s are real and distinct; if b2 = Aac the roots are real and equal; while if b2 < Aac the roots are complex conjugate. There are, therefore, three types of solutions. Here we shall summarise them. 2.8.1 Real and distinct (b2 > 4ac) If the auxiliary equation has distinct real roots r and s, then en and est are linearly independent solutions to the second-order linear homogeneous equation. The general solution is y(t) = ciert + c2est where c\ and c2 are arbitrary constants. If y(0) and /(O) are the initial conditions when t = 0, then we can solve for c\ and C2 y(0) = Cler(0) + c2es(0) = cx + c2 y'(t) = rc\en + sc2est /(0) = rcxem + sc2es(0) = rcx + sc2 Hence /(0) - sy(0) y'(0) - ry(0) c\ = -, c2 = - r — s s — r and the particular solution is \ r — s J \ s — r J which satisfies the initial conditions y(0) and y'(0). Example 2.17 Suppose d2y dy -f+4-f -5v = 0 dt2 dt Then the auxiliary equation is x2 + Ax - 5 = 0 (jc + 5)(jc - 1) = 0 Hence, r = —5 and 5=1, with the general solution y(t) = cxe~5t + c2e' 62 Economic Dynamics lfy{0) = 0 and /(O) = 1, then 1 1 c\ = -5-1 6 1 1 1-C-5) 6 So the particular solution is 2.5.2 Real and equal roots (b2 — 4ac) If r is a repeated real root to the differential equation ay" {t) + by'{t) + c = 0 then a general solution is y(t) = cxen + c2tert where c\ and c2 are arbitrary constants (see exercise 9). If y(0) and y'(0) are the two initial conditions, then y(0) = Cl+ c2(0) = Cl y'(t) = rc\ert + rc2tert + c2ert y(0) = rcx + c2 Hence ci = y(0), c2 = /(0) - ry(0) So the particular solution is y(t) = y(0)ert + [/(O) - ry(0)]tert Example 2.18 y"{t) + 4/(0 + 4y(t) = 0 Then the auxiliary equation is x2 + Ax + 4 = 0 (jc + 2)2 = 0 Hence, r = — 2 and the general solution is y(t) = cie~2t + c2te~2t lfy(0) = 3 and /(0) = 7, then ci = y(0) = 3 c2=y(0)-ry(0) = 7-(-2)(3) = 13 Continuous dynamic systems 63 so the particular solution is y(t) = 3e~2t + Ute-= (3 + I3t)e-2t 2.8.3 Complex conjugate (b2 < 4ac) If the auxiliary equation has complex conjugate roots r and s where r = a + ifi and s = a — i/3 then e°" cos(0f) and e°" sin(0O are linearly independent solutions to the second-order homogeneous equation (see exercise 10). The general solution is y(t) = Cleat cos(/30 + c2eat sin(/30 where c\ and c2 are arbitrary constants. If y(0) and /(0) are the initial conditions when ? = 0, then we can solve for c\ and C2 y(0) = c\ cos(0) + c2 sin(0) = c\ y'{t) = {aci + pc2)eat cos(pt) + (ac2 - fici)eat sin(0O y(0) = (aci + pc2)e° cos(0) + (ac2 - pc{)e° sin(0) = aci + f5c2 i.e. y(0)-«y(0) ci = y(0) and c2 =--- P Hence, the particular solution is y(t) = y(0)eat cos(^) + |V(0) sin(^) Example 2.19 fit) + 2y'{t) + 2y{t) = 0, y(0) = 2 and y'(0) = 1 The auxiliary equation is x2 + 2x + 2 = 0 with complex conjugate roots -2 + y4-4(2) r = - = — 1 + i 2_ -2 - V4 - 4(2) . s = -= — 1 — i 2 The general solution is y(t) = c\e~l cos(r) + c2e~l sin(f) 64 Economic Dynamics The coefficients are ci = y(0) = 2 /(0) - ay(0) = 3 ß Hence the particular solution is y(t) = 2e~' cos(f) + 3e~' sin(0 2.9 Second-order linear nonhomogeneous equations A second-order linear nonhomogeneous equation with constant coefficients takes the form Let L(y) = ay"(t) + by'(t) + cy(t) then equation (2.29) can be expressed as L(y) = g(t). The solution to equation (2.29) can be thought of in two parts. First, there is the homogeneous component, L(y) = 0. As we demonstrated in the previous section, if the roots are real and distinct then yc = cxert + c2est The reason for denoting this solution as yc will become clear in a moment. Second, it is possible to come up with a particular solution, denoted yp, which satisfies L(yp) = g(t). yc is referred to as the complementary solution satisfying L(y) = 0, while yp is the particular solution satisfying L(yp) = g(t). If both yc and yp are solutions, then so is their sum, y = yc + yp, which is referred to as the general solution to a linear nonhomogeneous differential equation. Hence, the general solution to equation (2.29) if the roots are real and distinct takes the form y(t) = yc + yP = cxen + c2est + yp The general solution y(t) = yc + yp holds even when the roots are not real or distinct. The point is that the complementary solution arises from the solution to L(y) = 0. As in the previous section there are three possible cases: (1) Real and distinct roots (2.29) _+,_ + ,v = gW or ay" {t) + by'{t) + cy(t) = g(t) yc = cxe + c2eL ,st (2) Real and equal roots yc = c\e + c2te' ft Continuous dynamic systems 65 (3) Complex conjugate roots yc = Cleatcos(fit) + c2eat sin(/80 In finding a solution to a linear nonhomogeneous equation, four steps need to be followed: Step 1 Find the complementary solution yc. Step 2 Find the general solution y^ by solving the higher-order equation Lh(yh) = 0 where y^ is determined from L(y) and g(t). Step 3 Obtain yq = yh - yc. Step 4 Determine the unknown constant, the undetermined coefficients, in the solution yq by requiring Uyq) = g(t) and substituting these into yq^ giving the particular solution yp. Example 2.20 Suppose y"(t)+y'(t) = t Step 1 This has the complementary solution yc, which is the solution to the auxiliary equation x2+x = 0 x(x + 1) = 0 with solutions r = 0 and s = — 1 and yc = cieQt + c2e~' = ci+ c2e~l Step 2 The differential equation needs to be differentiated twice to obtain Lhiyh) = 0. Thus, differentiating twice y(4)(0 + y(3)(0 = o with auxiliary equation x4 + x3 = 0 with roots 0, -1, 0, 0. Hence10 yh = ae0t + c2e~l + c3te0t + c4t2e0t = c\ + c2e~' + c^t + c^t2 10 We have here used the property that en, tert and tlen are linearly independent and need to be combined with a root repeating itself three times (see exercise 9(h)). 66 Economic Dynamics Step 3 Obtain yq = yh — yc. Thus yq = (ci + c2e~l + c3t + c4t2) - (ci + c2e~v) = c3t + c^t2 Step 4 To find c3 and c\, the undetermined coefficients, we need L(yq) = t. Hence y'^t)+y'q{t) = t But from step 3 we can derive y'q = c3 + 2c4t fq = 2^4 Hence 2C4 + C3 + 2c\t = t Since the solution must satisfy the differential equation identically for all t, then the result just derived must be an identity for all t and so the coefficients of like terms must be equal. Hence, we have the two simultaneous equations 2c4 + C3 = 0 2c4 = 1 with solutions C4 = V2 and C3 = — 1. Thus yp = -t+ \t2 and the solution is y(t) = ci + c2e~' -t+\t2 It is also possible to solve for c\ and c2 if we know y(0) and y'(0). Although we have presented the method of solution, many software packages have routines built into them, and will readily supply solutions if they exist. The economist can use such programmes to solve the mathematics and so concentrate on model formulation and model features. This we shall do in part II. 2.10 Linear approximations to nonlinear differential equations Consider the differential equation x = f(x) here/ is nonlinear and continuously differentiable. In general we cannot solve such equations explicitly. We may be able to establish the fixed points of the system by solving the equation f(x) = 0, since a fixed point is characterised byi = 0. Depending on the nonlinearity there may be more than one fixed point. Continuous dynamic systems 67 Iff is continuously differentiable in an open interval containing x = x*, then we approximate / using the Taylor expansion /(*)=/(**)+/(**)(*-**) f"(x*)(x - x*) fn(x*)(x - x*) + -LJ^1-- + --+Rn(x,x*) 21 n\ where Rn(x, x*) is the remainder. In particular, a first-order approximation takes the form f(x) =/(**) + f(x*)(x - x*) + R2(x, x*) If the initial point xq is sufficiently close to x*, then R2(x, x*) — 0. Furthermore, if we choose x* as being a fixed point, then f(x*) = 0. Hence we can approximate f(x) about a fixed point x* with f(x)=f(x*)(x-x*) (2.30) Example 2.21 Although we could solve the Solow growth model explicitly if the production function was a Cobb-Douglas by using a transformation suggested by Bernoulli, it provides a good example of a typical nonlinear differential equation problem. Our equation is k=f(k) = saka -(n + S)k This function has two fixed points obtained from solving k[saka~l -(n + S)] = 0 namely , sa \ ~(~) k* = 0 and k% = - \n + 8. Taking a first-order Taylor expansion about point k*, we have f(k)=f(k*)+f(k*)(k-k*) where f(k*) = asaik*)01'1 -(n + S) and f(k*) = 0 Consider first k* = k\ = 0, then f(k*) = limf(k) = limiasak01-1 - (n + 8)] = oo Next consider k = k\ > 0, then/(^) = 0 and r . N _c_!_^-i«-i / sa. \ V"-'/ (n + S) f(k*2) = asa(k*2f~l - (n + 8) = asa n + 8 = a(n + 8) — (n + 8) = -(n + 8)(l - a) 68 Economic Dynamics Figure 2.18. fik)=sak"-(n+5)k Hence f(k) = -(n + 8)(l - a)(k - k*) Since 0 < a < 1 and n and 8 are both positive, then this has a negative slope about fci; and hence k\ is a locally stable equilibrium. The situation is shown in figure 2.18. The first-order linear approximation about the non-zero equilibrium is then k =f(k) = -in + 8)(l - a)(k - k*) with the linear approximate solution k(t) = k* + (k(0) - k*)e-(n+S)(1-a)t As t oo then k(f) -± k\. What we are invoking here is the following theorem attributed to Liapunov THEOREM 2.1 If x = f(x) is a nonlinear equation with a linear approximation f(x)=f(x*)+f(x*)(x-x*) about the equilibrium point x*, and ifx* is (globally) stable for the linear approximation, then x* is asymptotically stable for the original nonlinear equation. Care must be exercised in using this theorem. The converse of the theorem is generally not true. In other words, it is possible for x* to be stable for the nonlinear system but asymptotically unstable for its linear approximation. Continuous dynamic systems 69 x> 0 x< 0 ■m- Example 2.22 Consider x = fix) = a(x — x*)3 — co < x < co, a > 0 There is a unique equilibrium at x = x* = 0 which is globally stable. This is readily seen in terms of figure 2.19, which also displays the phase line. Now consider its linear approximation at x = x* f(x) = 3aix - x*)2 fix*) = 0 and so x =fix) =fix*) +fix*)ix -x*) = 0 which does not exhibit global stability. This is because for any xq ^ x* then x = xq for all t since x = 0. Consequently, xq does not approach x* in the limit, and so x* = 0 cannot be asymptotically stable. 70 Economic Dynamics We shall return to linear approximations in chapter 3 when considering difference equations, and then again in chapters 4 and 5 when we deal with nonlinear systems of differential and difference equations. These investigations will allow us to use linear approximation methods when we consider economic models in part II. 2.11 Solving differential equations with Mathematica 2.11.1 First-order equations Mathematica has two built in commands for dealing with differential equations, which are the DSolve command and the NDSolve command. The first is used to find a symbolic solution to a differential equation; the second finds a numerical approximation. Consider the following first-order differential equation dy i =-«->'•0 In particular, we are assuming that y is a function of t, y(t). Then we employ the DSolve command by using DSolve[y' [t]== f [y[t] ,t] ,y [t] ,t] Note a number of aspects of this instruction: (1) The equation utilises the single apostrophe, so y'{t) denotes dy/dt (2) The function f(y(t), t) may or may not be independent of t (3) y(t) is written in the equation rather than simply y (4) The second term, y(t), is indicating what is being solved for, and t denotes the independent variable. It is possible to first define the differential equation and use the designation in the DSolve command. Thus Eq = y'[t]==f[y[t],t] DSolve[Eq,y[t] , t] If Mathematica can solve the differential equation then this is provided in the output. Sometimes warnings are provided, especially if inverse functions are being used. If Mathematica can find no solution, then the programme simply repeats the input. The user does not need to know what algorithm is being used to solve the differential equation. What matters is whether a solution can be found. What is important to understand, however, is that a first-order differential equation (as we are discussing here) involves one unknown constant of integration. The output will, therefore, involve an unknown constant, which is denoted C[l]. Consider the examples of first-order differential equations used in various places throughout this chapter shown in table 2.1. Mathematica has no difficulty solving all these problems, but it does provide a warning with the last stating: 'The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way.' What is also illustrated by these solutions is that the output may not, and usually is not, provided in Continuous dynamic systems 71 Table 2.1 First-order differential equations with Mathematica Problem Input instructions DSolve[x'[t]==kx[t],x[t],t] DSolve[x'[t]==l+cExp[t],x[t],t] DSolve[p'[t]-a (b-d)p[t]==a(a-c),p[t],t] DSolve[x'[t]==kx[t](a-x[t]),x[t],t] DSolve[k' [t]==sak [t]a-(n+S) k[t],k[t],t] Table 2.2 Mathematica input instructions for initial value problems Problem Input instructions DSolve[{p' [t]==p[t] (a-bp[t]),p[0]==pO}, p[t] ,t] DSolve [ {n' [t] ==-A.n [t] , n [0] ==n0 } , n [t] , t] DSolve[{y'[x]==x2-2x+l,y[0]==1},y[x],x] a way useful for economic interpretation. So some manipulation of the output is often necessary. It will be noted that none of the above examples involve initial conditions, which is why all outputs involve the unknown constant C[l]. Initial value problems are treated in a similar manner. If we have the initial value problem, ^=f(y,t) y(0) = yO dt then the input instruction is DSolve[{y'[t]==f[y[t],t],y[0]==y0},y[t],t] For example, look at table 2.2. 2.11.2 Second-order equations Second-order differential equations are treated in fundamentally the same way. If we have the homogeneous second-order differential equation (i) dx — — kx dt (ii) dx — - 1 + ce1 dt (iii) dp --a(b — d)p — a(a — c) dt (iv) dx — = kx(a — x) dt (v) k = saka-(n + S)k (i) dp -j- =P(a~ bp), p(0) = pO dt (ii) dn dt = -kn, n(0) = nO (iii) dy 0 — =x2 - 2x+l,y(Q) = 1 dx 72 Economic Dynamics Table 2.3 Mathematica input instructions for second-order differential equations Problem Input Instructions (i) d2y dy TT+4-r ~5y = 0 DSolve[y" [t]+4y' [t]-5y[t]==0,y[t],t] dtz dt (ii) fit) + 4/(f) + 4/0 = 0 DSolve [y' ' [t] +4y' [t] +4y [t] ==0, y [t] , t] (iii) f(t) + 2/(f) + 2y(t) = 0 DSolve [y' ' [t] +2y' [t] +2y [t]==0,y [t] , t] (iv) f(t) + y,(t) = t DSolve [y" [t]+y' [t]==t,y[t] ,t] Table 2.4 Mathematica input instructions for initial value problems Problem Input instructions (i) d2y dy -j + 4-j- - 5y = 0, DSolve[ {y" [t]+4y' [t]-5y[t]==0,y[0]==0, 3a,{a,-2,2,.5}] Note the following: (a) the solution to the differential equation is evaluated by letting C[l], the constant of integration, take the value of a. This is accomplished by adding the term 7. C[l]->a' (b) a is then given values between —2 and 2 in increments of 0.5. Step 6 Plot the trajectories using the Plot and Evaluate commands plottraj=Plot [ Evaluate[trajectories], {x,-2,2} ] Note that it is important to give the domain for x the same as in the direction field plot. Step 7 Combine the direction field plot and the trajectories plot using the Show command (not available prior to version 2.0) Show[arrows,plottraj] This final result is shown in figure 2.8. Figure 2.9(a) (p. 46) This follows similar steps as for figure 2.8, and so here we shall simply list the input lines, followed by a few notes. (1) Input <{0,0}, AxesLabel->{"t","p"} ] Input (2) and (3) are simply to check the initial population size and the final population size. Input (6) has {1, kp} (with k = 0.01) as the first element in the Continuous dynamic systems 79 (1) Input (2) Input (3) Input (4) Input (5) Input (6) Input PlotVectorField. Input (7) indicates some options that can be used with the Show command. These too could be employed in (a) above. Figure 2.9(b) (p. 46) Before we can plot the logistic function we need to solve it. In this example we shall employ the figures for a and b we derive in chapter 14 for the UK population over the period 1781-1931. a = 0.02 and b = 0.000436 and with pO = 13. <{0,0}, AxesLabel->{"t","p"} ] Note again that the PlotVectorField has the first element in the form {I, (a — bp)p} (with a = 0.02 and b = 0.000436). Appendix 2.2 Plotting direction fields for a single equation with Maple Figure 2.8 (p. 45) Given the differential equation dy . — = 2x - y dx the direction field and isoclines can be obtained using Maple as follows: Step 1 Load the DEtools subroutine with the instruction with (DEtools) : Note the colon after the instruction. Step 2 Define the differential equation and a set of points for the isoclines. Eq: = diff(y(x),x)=2*x-y Points: = { [-2,2], [-1,1], [-1,0.5], [-0.5,-2], [0,-2], [0.5,-1.5], [0.5,-1], [1,-1], [1.5,-0.5] }; Step 3 Obtain the direction field and the integral curves with the instruction DEplot(Eq,y(x),x=-2..2,Points,y=-2..2, arrows=slim, linecolour=blue); 80 Economic Dynamics Note that the direction field has six elements: (i) the differential equation (ii) y(x) indicates that x is the independent variable and y the dependent variable (iii) the range for the x-axis (iv) the initial points (v) the range for the v-axis (vi) a set of options; here we have two options: (a) arrows are to be drawn slim (the default is thin) (b) the colour of the lines is to be blue (the default is yellow). Figure 2.9(a) (p. 46) This follows similar steps as figure 2.8 and so we shall be brief. We assume a new session. Input the following: (1) with (DEtools ) : (2) equ:=pO*exp (k*t); (3) newequ:=subs(p0=13,k=0.01,equ); (4) inisol:=evalf(subs(t=0,newequ)); (5) finsol:=evalf(subs(t=150,newequ)); (6) DEplot(diff(p(t),t)=0.01*p,p(t),t=0..150, {[0,13]}, p=0..60, arrows=slim,linecolour=blue); Instructions (2), (3), (4) and (5) input the equation and evaluate it for the initial point (time t = 0) and at t = 150. The remaining instruction plots the direction field and one integral curve through the point (0, 13). Figure 2.9(b) (p. 46) The logistic equation uses the values a = 0.02 and b = 0.000436 and pO = 13. The input instructions are the following, where again we assume a new session: (1) with (DEtools ) : (2) DEplot(diff(p(t),t)=(0.02-0.0 00436*p)*p,p(t), t=0..150,{[0,13]},p=0..50,arrows=slim, linecolour=blue); Exercises 1. Show the following are solutions to their respective differential equations (i) ^ = ky y = ce^ ax Continuous dynamic systems 81 2. Analyse the qualitative and quantitative properties of the Gompertz equation for population growth p = p'(t) = kp(a — In p) 3. Solve the following separable differential equations (i) d^=x{\-y2) -\ 0} is called the (positive) orbit of yo. 3.3 The cobweb model: an introduction To highlight the features so far outlined, and others to follow, consider the following typical cobweb model in which demand at time t, qd, depends on the price now ruling on the market, pt, while the supply at time t, qst, depends on planting, which in turn was governed by what the price the farmer received in the last period, pt-\. The market is cleared in any period, and so qf = q\. Assuming linear demand and supply curves for simplicity, the model is, then, qf = a — bpt a, b > 0 qst = c + dpt-i d > 0 (3.5) q? = q< Substituting, we obtain a — bpt = c + dpt-i or which is a first-order nonhomogeneous dynamic system. It is also an autonomous dynamic system since it does not depend explicitly on t. This model is illustrated in figure 3.1. The demand and supply curves are indicated by D and S, respectively. Because we have a first-order system, we need one initial starting price. Suppose this is po. This gives a quantity supplied in the next period of q\, read off the supply curve, and indicated by point a. But since demand equals supply in any one period, this gives a demand of also q\, while this demand implies a price of p\ in period 1. This in turn means that supply in period 2 is q2. And so the sequence continues. We shall refer to this model frequently in this chapter. 88 Economic Dynamics 3.4 Equilibrium and stability of discrete dynamic systems If yt+l = f(yt) is a discrete dynamic system, then y* is a fixed point or equilibrium point of the system if (3.7) f(yt)=y* for all; A useful implication of this definition is that y* is an equilibrium value of the system yt+\ = f(yt) if and only if /=/(/■) For example, in the cobweb model (3.6) we have, '-^-(§)*• Hence a — c p = - where p > 0 if a > c 1 b + d 1 With linear demand and supply curves, therefore, there is only one fixed point, one equilibrium point. However, such a fixed point makes economic sense (i.e. for price to be nonnegative) only if the additional condition a > c is also satisfied. As with fixed points in continuous dynamic systems, a particularly important consideration is the stability/instability of a fixed point. Let y* denote a fixed point for the discrete dynamic system yt+\ =f(yt). Then (Elaydi 1996, p. 11) Discrete dynamic systems 89 (i) The equilibrium point y* is stable if given s > 0 there exists 8 > 0 such that \y0 - y*\ < 8 implies \fn(yo) -y*\ < s for all n > 0. If y* is not stable then it is unstable. (ii) The equilibrium point y* is a repelling fixed point if there exists e > 0 such that 0 < \yo - y*\ < £ implies \f(yo) -y*\ > \yo-y*\ (iii) The point y* is an asymptotically stable (attracting) equilibrium point2 if it is stable and there exists r\ > 0 such that \yo — y*\ < V implies lim yt = y* If r] = oo then y* is globally asymptotically stable. All these are illustrated in figure 3.2(a)-(e). In utilising these concepts we employ the following theorem (Elaydi 1996, section 1.4). THEOREM 3.1 Let y* be an equilibrium point of the dynamical system yt+\ =f(yt) where f is continuously differentiable aty*. Then (i) if \f'(y*)\ < 1 then y* is an asymptotically stable (attracting) fixed point (ii) if \f'(y*)\ > 1 then y* is unstable and is a repelling fixed point (iii) if\f'(y*)\ = 1 and (a) iff"(y*) 7^ 0, theny* is unstable (b) iff'iy*) = 0 andf"'(y*) > 0, theny* is unstable (c) if f"{y*) = 0 and f"'{y*) < 0, then y* is asymptotically stable (iv) iff(y*) = -1 and (a) if —2f"'(y*) - 3[f"(y*)]2 < 0, then y* is asymptotically stable (b) if-2f"(y*) - 3[f"(y*)]2 > 0, theny* is unstable. The attraction and repulsion of a fixed point can readily be illustrated for a first-order system. Suppose f(yt) is linear for the first-order system yt+\ = f(yt). This is represented by the lines denoted L in figures 3.3(a) and (b), where yt+\ is marked on the vertical axis and yt on the horizontal axis. The equilibrium condition requires yt+\ = yt for all t, hence this denotes a 45°-line, denoted by E in figures 3.3(a) and (b). The fixed point in each case, therefore, is y*. 2 Sometimes an asymptotically stable (attracting) equilibrium point is called a sink. 90 Economic Dynamics (e) Globally asymptotically stable t Consider first figure 3.3(a). We require an initial value for y to start the sequence, which is denoted yo. Given yo in period 0, then we have y\ in period 1, as read off from the line L. In terms of the horizontal axis, this gives a value of yi as read off the 45°-line (i.e. the horizontal movement across). But this means that in period 2 the value of y is y2, once again read off from the line L. In terms of the horizontal axis this also gives a value y2, read horizontally across. Regardless of the initial value yo, the sequence converges on y*, and this is true whether yo is below y*, as in the figure, or is above y*. Using the same analysis, it is clear that in figure 3.3(b), starting from an initial value of y of y0, the sequence diverges from y*. If y0 is below y* then the system creates smaller values of y and moves away from y* in the negative direction. On the other hand, if yo is above y*, then the sequence diverges from y* with the sequence diverging in the positive direction. Only if Discrete dynamic systems 91 3.3(b) (d>\) repellor y0 = y* will the system remain at rest. Hence, y* in figure 3.3(a) is an attractor while y* in figure 3.3(b) is a repellor. It is apparent from figure 3.2 that the essential difference between the two situations is that the line in figure 3.3(a) has a (positive) slope less than 45°, while in figure 3.3(b) the slope is greater than 45°. Another feature can be illustrated in a similar diagram. Consider the following simple linear dynamic system yt+i = -y, + k 92 Economic Dynamics Given this system, the first few terms in the sequence are readily found to be: yt+i = -y, + k yt+2 = -yt+i + k = -(-yt + k) + k = yt yt+3 = -yt+2 + k = -yt + k yt+4 = -yt+3 + k = -(-yt + k) + k = yt It is apparent that this is a repeating pattern. If yo denotes the initial value, then we have yo = yi = y4 = ■ ■ ■ and yi =y3=y5 = ... We have here an example of a two-cycle system that oscillates between — yo + k and yo. There is still a fixed point to the system, namely y* = -y* + k but it is neither an attractor nor a repellor. The situation is illustrated in figure 3.4, where again the line L denotes the difference equation and the line E gives the equilibrium condition. The two-cycle situation is readily revealed by the fact that the system cycles around a rectangle. Return to the linear cobweb model given above, equation (3.5). Suppose the slope of the (linear) demand curve is the same as the slope of the (linear) supply Discrete dynamic systems 93 curve but with opposite sign. Then b = d and Pt = ( a — c a — c d d or a — c Pt+\ = —pt + k where k = d which is identical to the situation shown in figure 3.4, and must produce a two-cycle In general, a solution yn is periodic if yn+m = yn for some fixed integer m and all n. The smallest integer for m is called the period of the solution. For example, given the linear cobweb system it is readily established that the price cycles betweenpo and 3-po, while the quantity cycles between 4 + 2po and 10 — 2po (see exercise 12). In other words Po = Pi = Pa = ■ ■ ■ and px = p3 = p5 = ... so that yn+2 = yn for all n and hence we have a two-cycle solution. More formally: DEFINITION If a sequence (ytj has (say) two repeating values y\ andy2, then y\ and >>2 are called period points, and the set (y\,y2J is called a periodic orbit. Geometrically, a ^-periodic point for the discrete system yt+\ =f(yt) is the v-coordinate of the point where the graph offk(y) meets the diagonal line yt+\ = yt. Thus, a three-period cycle is wheref3(y) meets the line yt+\ = yt. In establishing the stability/instability of period points we utilise the following theorem. THEOREM 3.2 Let b be a k-period point off. Then b is result. 10 - 2pt 4 + 2pt_1 ( Hi) (i) (ii) 94 Economic Dynamics In deriving the stability of a periodic point we require, then, to compute [fk(y)Y, and to do this we utilise the chain rule Uk(y)]' =f(ydf(y*2)---f(y*n) where y*, y^, ■ ■ ■, yl are the fc-periodic points. For example, if y\ and y^ are two periodic points of/2( y), then \U2(y)]'\ = \fW(y*2)\ and is asymptotically stable if \f'W(y*2)\ < i All other stability theorems hold in a similar fashion. Although it is fairly easy to determine the stability/instability of linear dynamic systems, this is not true for nonlinear systems. In particular, such systems can create complex cycle phenomena. To illustrate, and no more than illustrate, the more complex nature of systems that arise from nonlinearity, consider the following quadratic equation yt+i = ay, - by] First we need to establish any fixed points. It is readily established that two fixed points arise since y* = ay* - by*2 = ay* - ^ which gives two fixed points a — \ y* = 0 and y* = - b The situation is illustrated in figure 3.5, where the quadratic is denoted by the graph G, and the line E as before denotes the equilibrium condition. The two equilibrium points, the two fixed points of the system, are where the graph G intersects the line E. Depending on the values for a and b, it is of course possible for the graph G to be totally below the line E, in which case only one equilibrium point exists, namely y* = 0. Whether one or more equilibria exist, the question of interest is whether such a fixed point is stable or unstable. Suppose we attempt to establish which by means of a numerical example yt+i = 2yt - y] The situation is illustrated in figure 3.6, where G denotes the graph of the difference equation, and the line E the equilibrium condition. The two equilibrium values are readily found to be y* = 0 and y* = 1. As in the linear system, we need to consider a starting value, which we denote yo, then y\ = 2vo — y^. But this is no more than the value as read off the graph G. In terms of the horizontal axis, this value is read off by moving horizontally across to the E-line, as shown more clearly in figure 3.7. Given y\ then y2 = 1y\ — y\ as read off the graph G, which gives y2 on the horizontal axis when read horizontally Discrete dynamic systems 95 off the E-line. And so on. It would appear, therefore, that y* = 1 is an attractor. Even if yo is above y* = 1, the system appears to converge on y* = 1. Similarly, _y* = 0 appears to be a repellor. It is useful to use a spreadsheet not only to establish the sequence {yn}, but also to graph the situation. A spreadsheet is ideal for recursive equations because the relation gives the next element in the sequence, and for given initial values, the sequence is simply copied to all future cells. A typical spreadsheet for the present example is illustrated in figure 3.8, where we have identified the formulas in the initial cells. 96 Economic Dynamics Figure 3.7. G y*=\ Discrete dynamic systems 97 Given such a spreadsheet, it is possible to change the initial value yo and see the result in the sequence and on the various graphs that can be constructed.3 For instance, considering yt+1 = 3.2yt - O.Sy2 readily establishes that the equilibrium value is y* = 2.75, but that this is not reached for any initial value not equal to it. For any initial value not equal to the equilibrium value, then the system will tend towards a two-cycle with values 2.05 and 3.20, as can readily be established by means of a spreadsheet. It is also easy to establish that for any value slightly above or slightly below 2.75, i.e., in the neighbourhood of the equilibrium point, then the system diverges further from the equilibrium. In other words, the equilibrium is locally unstable. What is not apparent, however, is why the system will tend towards a two-cycle result. We shall explain why in section 3.7. Nor should it be assumed that only a two-cycle result can arise from the logistic equation. For instance, the logistic equation yt+1 =3.84>v(l -yt) has a three-cycle (see exercise 13). We can approach stability/instability from a slightly different perspective. Consider the first-order difference equation yt+\ = f(yt) with fixed points satisfying a = f(a). Let y denote yt+\ and x denote yt, then the difference equation is of the form y = f(x). Expanding this equation around an equilibrium point (a, a) we have y — a = f(a)(x — a) or y = a[l-f(a)]+f(a)x which is simply a linear equation with slope f'{a). The situation is illustrated in figure 3.9. This procedure reduces the problem of stability down to that of our linear model. There we noted that if the absolute slope of f(x) was less than the 45°-line, as in figure 3.9, then the situation was stable, otherwise it was unstable. To summarise, If \f'(a) | < 1 then a is an attractor or stable If \f'(a) | > 1 then a is a repellor or unstable If = 1 then the situation is inconclusive.4 We can use such a condition for each fixed point. Many spreadsheets now allow graphics to be displayed within the spreadsheet, as shown here - especially those using the Windows environment. Hence, any change in initial values or parameter values results in an immediate change in the displayed graph. This is a very interactive experimentation. However, it is possible to utilise higher derivatives to obtain more information about the fixed point a, as pointed out in theorem 3.1 (p. 89). 98 Economic Dynamics Figure 3.9. 0 y^a X Example 3.1 yt+i = 2yt - y] The fixed points can be found from a = 2a — a2 a2 — a = 0 a(a — 1) = 0 a = 0 and a = 1 To establish stability, let y = fix) = 2x-x2 then f(x) = 2-2x f(0) = 2 and f(l) = 0 Since |/'(0) | > 1 then a = 0 is unstable Since l/'Cl)! < 1 then a = 1 is stable Example 3.2 yt+1 = 3.2yt - O.Sy2 Discrete dynamic systems 99 The fixed points can be found from a = 3.2a — 0.8a2 0.8a2 - 2.2a = 0 a(0.8a - 2.2) = 0 a = 0 and a = 2.75 To establish stability let y = f(x) = 3.Ix-O.Sx2 then f(x) = 3.2- 1.6* /'(0) = 3.2 and /'(2.75) = -1.2 Since |/'(0) | > 1 then a = 0 is unstable Since |/'(2.75)| > 1 then a = 2.75 is unstable. Although a = 2.75 is unstable, knowledge about fix) does not give sufficient information to determine what is happening to the sequence {yn} around the point a = 2.75. 3.5 Solving first-order difference equations For some relatively simple difference equations it is possible to find analytical solutions. The simplest difference equation is a first-order linear homogeneous equation of the form yt+i = ay, (3.8) If we consider the recursive nature of this system, beginning with the initial value yo, we have yi = ay0 yi = ay i = a(ay0) = a2y0 y3 = ay2 = a(a2y0) = a3y0 yn = a"y0 The analytical solution is, therefore, yn = any0 (3.9) satisfying the initial value yo- The properties of this system depend only on the value and sign of the parameter a. There is only one fixed point to such a system, y* = 0. For positive yo, if a exceeds unity, then the series gets larger and larger over time, tending to infinity in the limit. If 0 < a < 1, then the series gets smaller 100 Economic Dynamics and smaller over time, tending to zero in the limit. If a is negative, then the series will alternate between positive and negative numbers. However, if —1 < a < 0 the values of the alternating series becomes smaller and smaller, tending to zero in the limit. While if a < —1, then the series alternates but tends to explode over time. The various solution paths are plotted in figure 3.10. Example 3.3 A number of systems satisfy this general form. Consider the Malthusian population discussed in chapter 2, but now specified in discrete form. Between time t and t + 1 the change in the population is proportional to the population size. If p, denotes the population size in period t, then Apt+\ = pt+\ — p, is proportional to pt. If k denotes the proportionality factor, then Apt+i = kpt Or pt+i = (1 + k)pt which has the analytical solution pt = (1 + lifpo Discrete dynamic systems 101 wherepo is the initial population size. If population is growing at all, k > 0, then this population will grow over time becoming ever larger. We shall discuss population more fully in chapter 14. Example 3.4 As a second example, consider the Harrod-Domar growth model in discrete time St = sYt It = v(Yt - Yt_x) St = It This gives a first-order homogeneous difference equation of the form with solution If v > 0 and v > s then v/(v — s) > 1 and the solution is explosive and nonoscil-latory. On the other hand, even if v > 0 if s > v then the solution oscillates, being damped if s < 2v, explosive if s > 2v or constant if s = 2v. The analytical solution to the first-order linear homogeneous equation is useful because it also helps to solve first-order linear nonhomogeneous equations. Consider the following general first-order linear nonhomogeneous equation yt+i =ayt + c A simple way to solve such equations, and one particularly useful for the economist, is to transform the system into deviations from its fixed point, deviations from equilibrium. Let y* denote the fixed point of the system, then y* = ay* + c Subtracting the equilibrium equation from the recursive equation gives yt+i -y* = a(yt - y*) Letting xt+i = yt+i - y* and xt = yt mogeneous difference equation in x Xt+l = axt with solution — y* then this is no more than a simple ho- Xf — o^xq Hence, yt-y* = a'(yo - y*) 102 Economic Dynamics or yt = -r~— +at[y0- —-— ) I — a \ 1 — a / which clearly satisfies the initial condition. Example 3.5 Consider, for example, the cobweb model we developed earlier in the chapter, equation (3.5), with the resulting recursive equation a — c ( d Pt = —--1^1 Pt-i and with equilibrium a — c * P b + d Taking deviations from the equilibrium, we have d Pt-P = -j(Pt-\ ~P ) b which is a first-order linear homogeneous difference equation, with solution Pt-P* = (Po-P*) or 'a — c\ ( d (3-12) » = l.^) + (-5 P0 b + d With the usual shaped demand and supply curves, i.e., b > 0 and d > 0, then d/b > 0, hence {—d/bf will alternate in sign, being positive for even numbers of t and negative for odd numbers of t. Furthermore, if 0 < | —d/b\ < 1 then the series will become damped, and in the limit tend towards the equilibrium price. On the other hand, if | — d/b\ > 1 then the system will diverge from the equilibrium price. These results are verified by means of a simple numerical example and solved by means of a spreadsheet, as shown in figure 3.11. The examples we have just discussed can be considered as special cases of the following recursive equation: (3.13) yn+i = anyn y0atn = 0 The solution to this more general case can be derived as follows: y\ = am y2 = aiyi = aiaQyQ y3 = a2yi = a2a1a0y0 yn = an-\an-2 ... a\aoyo Discrete dynamic systems 103 026_-j » " -in: it Figure 3.11. 20 c = 4 d = 1.8 - d/b = -1.5 t price Divergent 0 1 1 3 2 0 Path of price 3 4.5 400 i 4-2725 300 | 5 7.875 200 ■ 6 -7.3125 .„ 100 ■ A 1 7 15.46875 P(t> Q ' v\/\i 8 -18 7031 -100 1 2 4 6 8 10 » W 9 32.55469 -200 • V 10 -44'332 -300 11 70,99805 t 12 -101997 13 1574956 14 -231.743 1=j 1-1^-1 •tr NUM or yn = 'n-l Y\ak k=0 yo (3.14) Hence, if a^ = a for all k, then 'n-l n k=0 ak a and yn = a y0 Consider an even more general case: that of the nonhomogeneous first-order equation given by Then yn+i = anyn + gn a0, go, y0atn = 0 yi = a0yo + go y2 = ciiyi +g\= ax(aoyo + go) + gi = fliaoyo + «1^0 + gi y3 = a2y2 + gi = a2(a1a0yo + axg0 + gi) + g2 = a2a1a0yo + a2a\go + a2gi + g2 (3.15) with solution for yn of 'n-l yn I [ < k=0 n-l i=0 " n-l Yl ak k=i+l (3.16) 104 Economic Dynamics We can consider two special cases: Case A : ak = a for all k Case B : ak = a and gk = b for all k Case A a.k = a for all k In this case we have yn+i = ayn + gn go, yo at n = 0 Using the general result above, then n—l n—l Y\ak = an and ]~~[ ak = an~l~l k=0 k=i+l Hence, n-l (3.17) yn = any0 + J2a'l~i~18i i=0 Case B ak = a and gk = bfor all k In this case we have yn+i = ayn + b y0atn = 0 We already know that if ak = a for all k then n—l n—l Y[ak = an and ]~~[ ak = an 1 1 k=0 k=i+l and so n-l yn = anyo + b^n~l~l i=0 This case itself, however, can be divided into two sub-categories: (i) where a = 1 and (ii) where a ^ 1. Case (i) a = 1 If a = 1 then n-l n — i— 1 n i=0 and so yn = yo + bn Case (ii) a ^ 1 Let n-l S = J2 a"-'-1 Discrete dynamic systems 105 then aS n-l i=0 S-aS=(l-a)S=l-an 1 -an I — a and yn = any0 + b I — a Combining these two we can summarise case B as follows yo + bn a = 1 (\-an anyo + b-- \ 1 — a a ^ 1 (3.18) These particular formulas are useful in dealing with recursive equations in the area of finance. We take these up in the exercises. These special cases can be derived immediately using either Mathematica or Maple with the following input instructions5: Mathematica RSolve[{y[n+l]==y[n]+b, y[0]==y0},y[n],n] RSolve[{y[n+1]==a y[n]+b, y[0]==y0},y[n],n] Maple rsolve({y(n+1)=y(n)+b, y (0)=y0},y (n) ); rsolve({y(n+1)=a*y(n)+b, y (0)=y0},y (n)); 3.6 Compound interest If an amount A is compounded annually at a market interest rate of r for a given number of years, t, then the payment received at time t, Pt, is given by Pt=A(l+r)' On the other hand, if it is compounded m times each year, then the payment received is Pt=A(l + -) \ m/ If compounding is done more frequently over the year, then the amount received is larger. The actual interest rate being paid, once allowance is made for the compounding, is called the effective interest rate, which we denote re. The relationship between re and (r, m) is developed as follows / r \m A(l + re)=A[l + -) / r\m i.e. re = I 1 H--) — 1 V m' It follows that re > r. 5 See section 3.13 on solving recursive equations with Mathematica and Maple. 106 Economic Dynamics Figure 3.12. re 0.082 0.08 0.078 0.076 0 . 074 0. 072 r=8% r=7% Example 3.6 A bank is offering a savings account paying 7% interest per annum, compounded quarterly. What is the effective interest rate? 0.07\4 re=\l +- ) - 1 = 0.072 or 7.2%. If we assume that m is a continuous variable, then given an interest rate of say, 7%, we can graph the relationship between re and m. A higher market interest rate leads to a curve wholly above that of the lower interest rate, as shown in figure 3.12. Returning to the compounding result, if an amount is compounded at an annual interest rate r, then at time t we have the relationship Yt = (1 + r)Yt_\. If we generalise this further and assume an additional deposit (or withdrawal) in each period, a,, then the resulting recursive equation is Yt = (1 + r)7,_i + at_x Or more generally, we have the recursive equation Yt = at_1+ bYt_x Many problems reduce to this kind of relationship. For example, population of a species at time t may be proportional to its size in the previous period, but predation may take place each period. Or, human populations may grow proportionally but immigration and emigration occurs in each period. Solving the recursive equation can be achieved by iteration. Let the initial values be Yq and oq, respectively, then Yx = a0 + bY0 Y2 = a\ + bY\ = a\ + b(ao + bYo) = a\ + bao + b Yq y3 = a2 + bY2 = a2 + b(a\ + bao + b Yq) = a2 + ba\ + b ao + h Yq 2 3 y4 = «3 + bYi = «3 + b(a2 + ba\ + b üq + fr Yq) 2 3 4 = 03+ ba2 + b a\ + fr üq + b Yq Discrete dynamic systems 107 and so on. The general result emerging is Yt = at_x + bat_2 + b2at_3 + ••• + b'^ao + b% or t-i k=0 Having derived the general result two cases are of interest. The first is where a*: = a for all k; the second is where a^ = a and b = 1 for all k. Case (i) a;t = a for all k In this case we have Yt = a + bYt-x with the general result t-i Yt = aJ2bt~1~k + b% k=0 or 1 -bv Y' = a^T^b)+blYo It is useful for the economist to see this result from a different perspective. In equilibrium Yt = Y for all t. So Y = a + bf — a l-b Re-arranging the result for case (i), we have a ab' Yt =---+ b'Yo l-b l-b . , a \ a Yt = b'[Y0---- + l-b) l-b It is clear from this result that if | b \ < 1 then the series converges on the equilibrium. If 0 < b < 1, there is steady convergence; while if —1 < b < 0, the convergence oscillates. If \b\ > 1, the system is unstable. Case (ii) ak = a and b = 1 for all k In this case Yt = a + Yt-i with result t-i Yt = aJ2^)t~1~k + Yo k=0 i.e. Yt = at + Y0 108 Economic Dynamics Example 3.7 An investor makes an initial deposit of £10,000 and an additional £250 each year. The market interest rate is 5% per annum. What are his accumulated savings after five years? For this problem, Yq = £10,000, a^ = £250 for all k and b = (1 + r) = 1.05. Hence /1-(1.05)5\ , Y5 = 250---- + (1.05)5(10000) = £14,144.20 3.7 Discounting, present value and internal rates of return Since the future payment when interest is compounded is Pt = Po(l + r)1, then it follows that the present value, PV, of an amount P, received in the future is Pt PV =--— (1 + r)' and r is now referred to as the discount rate. Consider an annuity. An annuity consists of a series of payments of an amount A made at constant intervals of time for n periods. Each payment receives interest from the date it is made until the end of the rath-period. The last payment receives no interest. The future value, FV, is then FV = A(l + r)n~l + A(l + r)n~L + • • • + (1 + r)A + A Utilising a software package, the solution is readily found to be "(l + rf- 1 FV = A ---- r On the other hand, the present value of an annuity requires each future payment to be discounted by the appropriate discount factor. Thus the payment A received at the end of the first period is worth A/(I + r) today, while a payment A at the end of the second period is worth A/(l + r)2 today. So the present value of the annuity is A A A A n-2 PV + (1 + r) (1 + r)2 + ••• + (1 + r) n-l + (1 + r)n with solution PV = A 1 -(1 +r)- Example 3.8 £1,000 is deposited at the end of each year in a savings account that earns 6.5% interest compounded annually. (a) At the end of ten years, how much is the account worth? (b) What is the present value of the payments stream? Discrete dynamic systems 109 (a) (b) FV = A PV = A (l + r)n - 1 1 - (1 + r)~ 1000 1000 (1 +0.065)10 - 1 0.065 - (1 + 0.065)"10 0.065 £13494.40 £7188.83 Discounting is readily used in investment appraisal and cost-benefit analysis. Suppose Bt and Ct denote the benefits and costs, respectively, at time t. Then the present value of such flows are Bt/(l + rj and Ct/(l + r)1, respectively. It follows, then, that the net present value, NPV, of a project with financial flows over ra-periods is NPV E i=0 Bt E E Bt-Q (1 + rj to i1 + r)' to i1 + r> Notice that for t = 0 the benefits Bq and the costs Co involve no discounting. In many projects no benefits accrue in early years only costs. If NPV > 0 then a project (or investment) should be undertaken. Example 3.9 Bramwell pic is considering buying a new welding machine to increase its output. The machine would cost £40,000 but would lead to increased revenue of £7,500 each year for the next ten years. Half way through the machine's lifespan, in year 5, there is a one-off maintenance expense of £5,000. Bramwell pic consider that the appropriate discount rate is 8%. Should they buy the machine? 10 NPV = -40000 + 7500 5000 t=\ (1 + rj (1 + rf The second term is simply the present value of an annuity of £7,500 received for ten years and discounted at 8%. The present value of this is -io- i — i i -t- w.wn i = 7500 PV = A 1 - (1 + r)~ 0.08 Hence NPV = -40000 + 7500 1 - (1.08)-0.08 -10' 5000 (1.08)5 £6922.69 Since NPV > 0, then Bramwell pic should go ahead with the investment. Net present value is just one method for determining projects. One difficulty, as the above example illustrates, is that it is necessary to make an assumption about the appropriate discount rate. Since there is often uncertainty about this, computations are often carried out for different discount rates. An alternative is to use the internal rate of return (IRK). The internal rate of return is the discount rate that leads to a zero net present value. Thus, the internal rate of return is the value of r satisfying E t=0 Bt-Q (1 + r)' 0 110 Economic Dynamics Although software programmes can readily solve for the internal rate of return, there is a problem in the choice of r. Bt — Ct is a polynomial with the highest power of n, and so theoretically there are n possible roots to this equation. Of course, we can rule out negative values and complex values. For example, the choice problem for Bramwell pic involves r10 as the highest term and so there are ten possible solutions to the equation -40000 + 7500 1 - (1 + r) -10 ■ 5000 (1 + r)5 Eight solutions, however, are complex and another is negative. This leaves only one positive real-valued solution, namely r = 0.1172 or r = 11.72%. Since such a return is well above the typical market interest rate, then the investment should be undertaken. The point is, however, that multiple positive real-valued solutions are possible. 3.8 Solving second-order difference equations 3.8.1 Homogeneous Consider the following general second-order linear homogeneous equation (3.19) yn+2 = ayn+i + byn Similar to the solution for a first-order linear homogeneous equation, we can suppose the solution takes the form yn = cxrn + c2sn for some constants r and s and where c\ and c2 depend on the initial conditions yo and yi. If this indeed is correct, then Clr"+2 + c2sn+2 = a{Clrn+l + c2sn+l) + b(cxrn + c2sn) Re-arranging and factorising, we obtain cxrn(r2 -ar-b) + c2sn(s2 - as - b) = 0 So long as r and s are chosen to be the solution values to the general quadratic equation x2 — ax — b = 0 i.e..*; = rand.*; = s, where r ^ s, theny„ = c\rn + c2sn is a solution to the dynamic system. This quadratic equation is referred to as the characteristic equation of the dynamical system. If r > s, then we call y\ = c\rn the dominant solution and r the dominant characteristic root. Furthermore, given we have obtained the solution values r and s, and given the initial conditions, yo and yi, then we can solve for the two unknown coefficients, Discrete dynamic systems 111 c\ and c2. Since yo = cir° + c2s° = c\ + c2 yi = c1r + c2s then yi - Wo , y\- ryo c\ = - and c2 = - r — s s — r The solution values r and s to the characteristic equation of the dynamic system are the solutions to a quadratic. As in all quadratics, three possibilities can occur: (i) distinct real roots (ii) identical real roots (iii) complex conjugate roots Since the solution values to the quadratic equation are -a ± V«2 + 4b then we have distinct real roots if a2 > —4b, identical roots if a2 = —4b, and complex conjugate roots if a2 < —4b. Example 3.10 (real distinct roots) Suppose yn+2 = yn+\ + ^yn The characteristic equation is given by x2 - x - 2 = 0 i.e. (jc - 2)(x + 1) = 0 Hence, we have two real distinct roots, x = 2 and x = —I, and the general solution is yn = Cl(2)n + C2(-\)n If we know _yo = 5 and y\ = 4, then yi-sy0 _ 4-(-l)(5) _3 r-s 2-(-l) yi-ryp _4- (2)(5) _ s-r (-l)-2 Hence, the particular solution satisfying these initial conditions is given by y„ = 3(2)B + 2(-l)B As figure 3.13 makes clear, this is an explosive system that tends to infinity over time. The limiting behaviour of the general solution yn = c\rn + c2sn is determined by the behaviour of the dominant solution. If, for example, r is the dominant c\ = c2 = 112 Economic Dynamics Figure 3.13. j,=3(2)"+2(-l)" 120000h 100000 80000 y„ 60000-40000 20000 0 —•—f—•—»—' f —i-r- 2 4 6 8 10 12 14 n characteristic root and \r\ > \s\, then yn = rn [Cl+C2(£)"] Since \s/r\ < 1, then (s/r)n 0 as n oo. Therefore, lim y„ = lim c\rn n^oo n^oo There are six different situations that can arise depending on the value of r. (1) r > 1, then the sequence {c\rn} diverges to infinity and the system is unstable (2) r = 1, then the sequence {c\rn} is a constant sequence (3) 0 < r < 1, then the sequence {c\rn} is monotonically decreasing to zero and the system is stable (4) — 1 < r < 0, then the sequence {c\rn} is oscillating around zero and converging on zero, so the system is stable (5) r = — 1, then the sequence {c\rn} is oscillating between two values (6) r < — 1, then the sequence {c\rn} is oscillating but increasing in magnitude. Identical real roots If the roots are real and equal, i.e., r = s, then the solution becomes yn = (ci + c2)rn = c3rn But if c3rn is a solution, then so is cajif1 (see Chiang 1992, p. 580 or Goldberg 1961, p. 136 and exercise 14), hence the general solution when the roots are equal is given by yn = c3rn + c4nrn We can now solve for c3 and c\ given the two initial conditions yo and y\ yo = c3r° + c4(0)r° = c3 yi = c3r + c4(l)r = (c3 + c4)r Discrete dynamic systems 113 Hence C3 = yo yi c4 =--c3 r y\ - ryo Therefore, the general solution satisfying the two initial conditions, is „ , i y\ - ryo , „ yn =yor + I - I nr Example 3.11 (equal real roots) Let yn+2 = 4yn+1 - Ayn This has the characteristic equation x2 - 4x + 4 = (jc - 2)2 = 0 Hence, r = 2. yn = c3(2)n + c4n(2)n Suppose yo = 6 and y\ = 4, then c3 =y0 = 6 c _ yi ~ ry0 _ 4 - (2)(6) _ C4 r 2 Hence, the particular solution is yn = 6(2)" - 4n(2)n which tends to minus infinity as n increases, as shown in figure 3.14. In the case of the general solution yn = (c3 + c\n)rn (1) If \r\ > 1, then yn diverges monotonically (2) If r < — 1, then the solution oscillates (3) If \r\ < 1, then the solution converges to zero y« 200000 0 -200000 -400000 -600000 H -800000 -1000000 -1200000 -1400000 -1600000 -1800000 H -2000000 y~6(T)-4n(T) Figure 3.14. 8 10 114 Economic Dynamics Complex conjugate roots6 If the roots are complex conjugate then r = a + fii and s = a — fii and Rn cos(fJt) and Rn sin(fJt) are solutions and the general solution is yn = ciRn cos(9n) + c2Rn sin(9n) where R = ja2 + ß2, a ß cosö = — and sinö = — R R sin 9 ß or tanö = - = — cosö a Example 3.12 (complex conjugate) Consider yn+2 - 4yn+1 + I6yn = 0 The characteristic equation is x2- Ax +16 = 0 with roots 4± V16- 64 /V48\ r, s = - = 2 ± - i i.e. r = 2+lVÄSi öl = 2 s = 2-±V48z ß = ±V48 and polar coordinates R = J22 + (±V48)2 = V4+ 12 = 4 a 2 1 f3 ±V48 cos 9 = — = - = - and sin9 = — = - = - R 4 2 i? 4 2 Implying 0 = jt/3. Hence /njT\ „ /njT\ yn = Cl4ncos (-) + c24n sin (-) 6 In this section the complex roots are expressed in polar coordinate form (see Allen 1965 or Chiang 1984). Discrete dynamic systems 115 Given yo and yi, it is possible to solve for c\ and c2. Specifically c\ = yo c2 = yi -y04cos (^j 4 sin in(!) If r and s are complex conjugate, then yn oscillates because the cosine function oscillates. There are, however, three different types of oscillation: (1) R > 1. In this instance the characteristic roots r and s lie outside the unit circle, shown in figure 3.15(a). Hence yn is oscillating, but increasing in magnitude. The system is unstable. (2) R = 1. In this instance the characteristic roots r and s lie on the unit circle, and the system oscillates with a constant magnitude, figure 3.15(b). (3) R < 1. In this instance the characteristic roots r and s lie inside the unit circle and the system oscillates but converges to zero as n oo, figure 3.15(c). The system is stable. unit circle R = 1 Figure 3.15. s=a-$i Im unit — ----. 7 A I \ 116 Economic Dynamics 3.8.2 Nonhomogeneous A constant coefficient nonhomogeneous second-order difference equation takes the general form (3.20) yn+2 + ayn+i + byn = g(n) If g(n) = c, a constant, then yn+2 + ayn+i + byn = c which is the form we shall consider here. As with second-order differential equations considered in chapter 2, we can break the solution down into a complementary component, yc, and a particular component, yp, i.e., the general solution yn, can be expressed yn = yc + yP The complementary component is the solution to the homogeneous part of the recursive equation, i.e., yc is the solution to yn+2 + ayn+i + byn = 0 which we have already outlined in the previous section. Since yn = y* is a fixed point for all n, then this will satisfy the particular solution. Thus y* + ay* + by* = c * c y =- l + a + b so long as 1 + a + b ^ 0. Example 3.13 yn+i - 4yn+i + I6yn = 26 Then y* - Ay* + I6y* = 26 y* = 2 Hence, yp = 2. The general solution is, then y. = Cl4-c«(^)+<*4-(^)+2 Example 3.14 yn+2 - 5yn+1 +4yn = 4 Discrete dynamic systems 117 In this example, 1 + a + b = 0 and so it is not possible to use y* as a solution. In this instance we try a moving fixed point, ny*. Thus (n + 2)y* - 5(n + l)y* + Any* = A -3y* = 4 -An For the complementary component we need to solve the homogeneous equation yn+2 - 5yn+1 + Ayn = 0 whose characteristic equation is x2 - 5x + A = 0 with solutions 5 ± V25 - 16 5 ±3 r, s = - = - 2 2 i.e. r = 4 and 5=1. Hence yn = ClAn + c2ln - (An/3) = c\An + c2 - (An/3) Example 3.15 yn+2 + yn+i - 2yn = 12 yo = A and yi = 5 The particular solution cannot be solved for y* (since 1 + a + b = 0) and so we employ ny* (n + 2)y* + (n+ l)y* - 2ny* = 12 (n + 2 + n + 1 - 2n)y* = 12 12 y* = — = A y 3 Hence, ny* = An = yp. The complementary component is derived by solving the characteristic equation x2 + x - 2 = 0 (jc + 2)(x - 1) = 0 giving r = 1 and s = —2. Giving the complementary component of yc = c\rn + c2sn = Cl(iy + c2(-2)n = Cl + c2(-2f Hence, the general solution is yn = yc + yP = ?! + c2(-2f + An 118 Economic Dynamics Given Vo = 4 and y\ = 5, then y0 = ci + c2 = 4 yi = c\ - 2c2 + 4 = 5 with solutions ci = 3 and c2 = 1 Hence, the general solution satisfying the given conditions is yn = 3 + (-2)" + An For the nonhomogeneous second-order linear difference equation yn+2 + ayn+i + byn = c yn v*, where y* is the fixed point, if and only if the complementary solution, yc, tends to zero as n tends to infinity; while yn will oscillate about y* if and only if the complementary solution oscillates about zero. Since the complementary solution is the solution to the homogeneous part, we have already indicated the stability of these in section 3.8.1. In the case of the second-order linear difference equations, both homogeneous and nonhomogeneous, it is possible to have explicit criteria on the parameters a and b for stability. These are contained in the following theorem (Elaydi 1996, pp. 87-8). THEOREM 3.3 The conditions 1 + a + £ > 0, 1 - a + > 0, \-b> 0 are necessary and sufficient for the equilibrium point of both homogeneous and nonhomogeneous second-order difference equations to be asymptotically stable. 3.9 The logistic equation: discrete version Suppose (3.21) Ayt+1 = ayt - by2 where b is the competition coefficient.7 Then yt+i = (1 +a)yt - by2 This is a nonlinear recursive equation and cannot be solved analytically as it stands. However, with a slight change we can solve the model.8 Let y] - ytyt+i 1 We shall discuss this coefficient more fully in chapter 14. 8 This approximate solution is taken from Griffiths and Oldknow (1993, p. 16). Discrete dynamic systems 119 then yt+\ = (1 + a)y, - bytyt+i Solving we obtain {l+a)yt 1 +byt This can be transformed by dividing both sides by y,+iy, 1_ 1 l+a yt yt+\ 1 + byt i.e. 1 1 + by, 1 1 b yt+\ (l+a)y, (I + a) yt l+a hetxt = l/yt, then / 1 \ b xt+i = 1—,— xt + l+a/ l + a In equilibrium xt+\ = x, = ... = x*, hence x*= I —^ )x* + I+ a) l + a Solving for x* we obtain the fixed point x* = -a Subtracting the equilibrium equation from the recursive equation we obtain l + a which has the general solution f —|— ) (x0 - x*) \ 1 + a or b _. ( b xt = - + (1 + a) [x0-- a \ a Substituting back x, = 1 /y, for all t lb , / 1 b - = _+(i+fl)-'--- yt a \y0 a Hence, 1 yt = b\ (lb - +(l+a)"M--- a) \y0 a or ay0 yt =-- (3-22) byo + (1 + a) '(a - byo) 120 Economic Dynamics It is readily established that a hm yt = - t^OO 0 Three typical plots are shown in figure 3.16, for jo < a/b,yo = a/bandyo > a/b. Return to the original formulation yt+i -yt = ay, - by] i.e. yt+\ = (1 + a)yt - by2 It is not possible to solve this nonlinear equation, although our approximation is quite good (see exercise 6). But the equation has been much investigated by mathematicians because of its possible chaotic behaviour.9 In carrying out this investigation it is normal to respecify the equation in its generic form (3.23) xt+1 = kxt(l - xt) It is this simple recursive formulation that is often employed for investigation because it involves only a single parameter, A. The reader is encouraged to set up this equation on a spreadsheet, which is very straightforward. If A = 3.2 it is readily established that the series will, after a sufficient time period, oscillate between two values: a\ = 0.799455 and a2 = 0.513045. This two-cycle is typical of the logistic equation for a certain range of A. To establish the range of A is straightforward but algebraically tedious. Here we shall give the gist of the solution, and leave appendices 3.1 and 3.2 to illustrate how Mathematica and Maple, respectively, can be employed to solve the tedious algebra. Let f(x) = kx(\ — x) then a two-cycle result will occur if a=f(f(a)) We shall investigate chaos in chapter 7. Discrete dynamic systems 121 where a is a fixed point. Hence a = f[Xa(l — a)] = X[Xa(l — a)][l — Xa(l — a)] = X2a(l — a)[l — Xa(l — a)] It is at this point where Mathematica or Maple is used to solve this equation. The range for a stable two-cycle is established by solving10 -1 < f{al)f{a2) < 1 where a\ and a2 are the two relevant solutions. Since f(x) = X(l-x)-Xx then we can compute f(a\)f(a2), which is a surprisingly simple equation of the form 4 + 2X - X2 Hence, we have a stable two-cycle if -1 < 4 + 2X- X2 < 1 Discarding negative values for A, we establish the range to be 3 < X < 3.449. Given we have already a\ and a2 solved for any particular value of X, then we can find these two stable solutions for any X in the range just established. Thus, for X = 3.2 it is readily established using Mathematica or Maple, that a\ = 0.799455 and a2 = 0.513045, which are the same results as those established using a spreadsheet. For X < 3 we have a single fixed point which is stable, which again can readily be established by means of the same spreadsheet. Finally, if A = 3.84 the system converges on a three-cycle result with a\ = 0.149407, a2 = 0.488044 and a? = 0.959447 (see exercise 13). Example 3.16 As an application of the logistic equation, different from its normal application in population models (see chapter 14), we turn to the issue of productivity growth discussed by Baumol and Wolff (1991). Let qt denote the rate of growth of productivity outside of the research development industries; y, the activity level of the information producing industry (the R&D industries); andp, the price of information. The authors now assume three relationships: (1) Information contributes to productivity growth according to: (i) yt+i = a + byt (2) The price of information grows in proportion to productivity in the sector outside of the R&D industries, so: .... Pt+i ~Pt (n) - = vqt+1 Pt See theorem 3.2, p. 93 and Sandefur (1990, chapter 4). 122 Economic Dynamics (3) Information demand has a constant elasticity, so: (in) yt+\ - yt Pt+i -pt yt \ pt Substituting (i) into (ii) and the result into (iii), we obtain yt+i - yt yt -sv(a + byt) Assume ev = k > 0 then yt+i - yt yt k(a + byt) i.e. yt+i = (1 - ak)yt - kby] which is no more than a logistic equation. In equilibrium yt = y* for all t, hence y* = (1 - ak)y* - kby*2 y*(ak + kby*) = 0 and y* = Oory* = --y b It is possible to consider the stability in the locality of the equilibrium. Since yt+i = (1 - ak)yt - kby] let yt+\ = y and yt = x, then y = (1 — ak)x — kbx2 = f(x) f(x) = (l-ak)- 2kbx and dy dx dyt+i y*=—a/b y*=-a/b dyt = (1 - ak) - 2kb(-a/b) = 1 + ak Hence the stability is very dependent on the sign/value of ak. Letting yt+i = Ay, - By2, A = (1 - ak), B = kb then using our earlier approximation (equation (3.22)) we have Ay0 i.e. yt yt By0 + (l+A)-'(A-By0) (1 - ak)y0 kbyo + (2 — ak) '(1 — ak — kbyo) Various paths for this solution are possible depending on the values of v and e. For instance, if v = 1 and e < 2, then k = ev < 2, and if a < 1 then ak < 2, which Discrete dynamic systems 123 is highly probable. Even with ak < 1, two possibilities arise: with various paths for yt. This should not be surprising because we have already established that the discrete logistic equation has a variety of paths and possible cycles. 3.10 The multiplier-accelerator model A good example that illustrates the use of recursive equations, and the variety of solution paths for income in an economy, is that of the multiplier-accelerator model first outlined by Samuelson (1939). Consumption is related to lagged income while investment at time t is related to the difference between income at time t — 1 and income at time t — 2.11 In our formulation we shall treat government spending as constant, and equal to G in all periods. The model is then Ct = a + bYt_i it = v(y(_i - Yt_i) Gt = G for all t Et = Ct + It + Gt Yt = Et which on straight substitution gives rise to the second-order nonhomogeneous recursive equation Yt - (b + v)Yt_1 + vYt_2 = a + G The particular solution is found by letting Yt = Y* for all t. Hence Y* - (b + v)Y* + vY* = a + G In other words, in equilibrium, income equals the simple multiplier result. The complementary result, Yc, is obtained by solving the homogeneous component Yt - (b + v)Yt_1 + vYt_2 = 0 which has the characteristic equation x2 - (b + v)x + v = 0 with solutions (i) (ii) if a < 0 then ak < 2 if 0 < a < 1 then 0 < ak < 2 i.e. 7* = a + G l-b r, s = (b + v)± y/(b + v)2 - 4v 2 Samuelson originally related investment to lagged consumption rather than lagged income. 124 Economic Dynamics Example 3.17 Determine the path of income for the equations C, = 50 + 0.75Ff_! /, = 4(7,_i - Yt_i) G = 100 The equilibrium is readily found to be Y* = 600, which is the particular solution. The complementary solution is found by solving the quadratic x2 - (19/4)jc + 4 = 0 i.e. r = 3.6559 and 5= 1.0941 Since r and s are real and distinct, then the solution is Yt = ci(3.6559)' + c2(1.0941)' + 600 and c\ and c2 can be obtained if we know Yq and Y\. Of more interest is the fact that the model can give rise to a whole variety of paths for Yt depending on the various parameter values for b and v. It is to this issue that we now turn. From the roots of the characteristic equation given above we have three possible outcomes: (i) real distinct roots (b + v)2 > 4v (ii) real equal roots (b + v)2 = 4v (iii) complex roots (b + v)2 < 4v In determining the implications of these possible outcomes we use the two properties of roots r + s = b + v rs = v It also follows using these two results that (1 - r)(l - s) = 1 - (r + s) + rs = l-(b + v) + v = l-b and since 0 < b < 1, then 0 < (1 — r)(l — s) < 1. With both roots real and distinct, the general solution is Yt = c1rt + c2s' + Y* where r is the larger of the two roots. The path of Y, is determined by the largest root, r > s. Since b > 0 and v > 0, then rs = v > 0 and so the roots must have the same sign. Furthermore, since r + s = b + v>0, then both r and s must be positive. The path of income cannot oscillate. However, it will be damped if the largest root lies between zero and unity. Thus, a damped path occurs if0<5 s > 1, which implies 0 < b < 1 and rs = v > 1. Discrete dynamic systems 125 With only one real root, r, the same conditions hold. Hence, in the case of real roots with 0 < b < 1, the path of income is damped for 0 < v < 1 and explosive for v > 1. If the solution is complex conjugate then r = a + /3i and s = a — /3i and the general solution Yt = ciR* cos(>6>) + C2R1 sin(6>0 + Y* exhibits oscillations, whose damped or explosive nature depends on the amplitude, R. From our earlier analysis we know R = ^Ja2 + fi2. But b + v J n +V4v -(b + v)2 a = - and p = - 2 H 2 Hence R _ (b + v) + 4v_ (b + v)2 For damped oscillations, R < l,i.e.,v < 1; while for explosive oscillations, R > 1, i.e., v > 1. All cases are drawn in figure 3.17. The dividing line between real and complex roots is the curve (b + v)2 = 4v, which was drawn using Mathematical Implicit-Plot command and annotated in CorelDraw. A similar result can be derived using Maple. The instructions for each software are: Mathematica < 1, respectively. The accelerator model just outlined was utilised by Hicks (1950) in his discussion of the trade cycle. The major change was introducing an autonomous component to investment, Iq, which grows exogenously at a rate g. So at time t, Figure 3.17. 126 Economic Dynamics the autonomous component of investment is 7o(l + gf • Hicks' model can then be expressed Ct = bYt_x (3.24) It = 70(1 + g)< + v(rf_! - 7,_2) Yt = Ct + It Substituting, we get Yt = bYt-! + 70(1 + g)' + v(Yt.x - Yt-2) = (b + v)Yt-l-vYt-2+h(\+g)t Since the model involves a moving equilibrium, then assume equilibrium income at time f is 7(1 + gf and at time t — 1 it is 7(1 + gf~l, etc. Then in equilibrium (3.25) 7(1 + g)1 -(b + v)7(l + g)1-1 + v7(l + g)^2 = 70(1 + g)1 Dividing throughout by (1 + g)'~2, then 7(1 + g)2 -(b + v)7(l + g) + vf = 70(1 + g)2 (3.26) (3.27) i.e. /o(l+^)2 (l+S)2-(£ + v)(l+£) + v Note that in the static case where g = 0, that this reduces down to the simple result Y = h/{\-b). The particular solution to equation (3.25) is then y =F(l + rf = _k{l + S)'+1_ ' (l+g)2-(fc + v)(l+g)+v Since the homogeneous component is 7(1 + gy -(b + v)7(l + g)1-1 + v7(l + g)1-2 = 0 then the complementary function, Yc is 7C = cir1 + c2s' where -(b + v) ± y/(b + v)2 - 4v r,s=-2- The complete solution to equation (3.25) is then Yt = cir1 + c2sl + (l+g)2-(b + v)(l+g) + v -{b + v) + y/{b + v)2 - 4v r = - 2 -(b + v) - y/(b + v)2 - 4v s = 2 Once again the stability of (3.27) depends on the sign of (b + v)2 — 4v, and the various possibilities we have already investigated. Discrete dynamic systems 127 3.11 Linear approximation to discrete nonlinear difference equations In chapter 2, section 2.7, we considered linear approximations to nonlinear differential equations. In this section we do the same for nonlinear difference equations. A typical nonlinear difference equation for a one-period lag is xt -xt-\ = g(xt-i) Axt = g(xt-i) However, it is useful to consider the problem in the recursive form xt = g(xt-i)+xt-i i.e. xt =/(*,_ i) because this allows a graphical representation. In this section we shall consider only autonomous nonlinear difference equations and sof(xt-\) does not depend explicitly on time. We have already established that a fixed point, an equilibrium point, exists if jc* =/(**) for all t and that we can represent this on a diagram with xt-\ on the horizontal axis and xt on the vertical axis. A fixed point occurs where f(xt-i) cuts the 45°-line, as shown in figure 3.18, where we have three such fixed points. Since/(:ic) = x3 then y =f(y) and satisfies y = y3 or y(y2 — 1) = 0. This results in three values for y, y = 0, — 1 and 1. It is to be noted that we have drawn xt = f{xt-\) as a continuous function, which we also assume to be differentiable. We have also established that x* is an attractor, a stable point, if there exists a number e such that when |*o — x* \ < e then xt approaches x* in the limit, otherwise it is unstable. In the present illustration we can consider only local stability or 128 Economic Dynamics instability, and so we take s to be some 'small' distance either side of x* or x\ or x\. In order to establish the stability properties of each of the equilibrium points, we take a Taylor expansion of / about x*. Thus for a first-order linear approximation we have =/(**) +f(x*)(xt-1 - x*) + R2(xt-1x*) Ignoring the remainder term, then our linear approximation is xt=f(x*)+Ax*)(xt-1-x*) Furthermore, we have established that: if |/'(:*:*) | < 1 then x* is an attractor or stable if |/'(:*:*) | > 1 then x* is a repellor or unstable if |/'CO | = 1 then the stability of x* is inconclusive. Example 3.18 Consider xt = y/4xt-i - 3 This has two equilibria found by solving x2 — Ax + 3 = 0, i.e., x* = 1 and x\ = 3, and shown by the points a and b in figure 3.19. The linear approximation is Figure 3.19. Discrete dynamic systems 129 Take first x* = 1, then /(*?) = 1 f{x\) = 2{Ax\ - 3)"1/2 = 2 Hence xt = 1 + 2(x,_i - 1) = -1 + 2xt-x which is unstable since f'(x\) = 2 > 1. Next consider x\ = 3 M) = 3 = 2(4** - 3)"1/2 = ^ Hence x, = 3 + 0^ (x,_i - 3) 2 = 1 + 3^-1 which is stable since /'(x^) = 2/3 < 1. Example 3.19 y,+i =f(yt) = 3.2yt-0.8y2 Letting yt = y* for all t we can readily establish two equilibria: y\ = 0 and y2 = 2.75. Considering the nonzero equilibrium, then Kyi) = 2-75 /(^) = 3.2-1.6^ = -1.2 Hence, the linear approximation is yt+1 = 2.75 - l.2(yt - 2.75) = 6.05 - \.2yt The situation is shown in figure 3.20. The solution to this model is yt+1 =2.75 + (-1.2)'(y0- 2.75) which is oscillatory and explosive. Although the linear approximation leads to an explosive oscillatory equilibrium, the system in its nonlinear form exhibits a two-cycle with values 2.0522 and 3.1978.12 What the linear approximation reveals is the movement away from y* = 2.75. What it cannot show is that it will converge on a two-cycle. This example, therefore, illustrates the care required in interpreting the stability of nonlinear difference equations using their linear approximations. This can be established quite readily with a spreadsheet or as explained in appendices 3.1 and 3.2. 130 Economic Dynamics Figure 3.20. y -2 ^ 7,+1=6.05-1.2y, 2 . 3 ^2 ^,+,=3.2^,-0.8// 3.12 Solow growth model in discrete time We have already established in chapter 2, example 2.9, that a homogeneous of degree one production function can be written y = f(k), where y is the output/labour ratio and k is the capital/labour ratio. In discrete time we have13 yt=f(kt-i) whereyt = Yt/Lt-\ andfc,_i = Kt-\/Lt-\. Given the same assumptions as example 2.9, savings is given by St = sYt and investment as It = Kt — Kt-\ + 8Kt-\, where 8 is the rate of depreciation. Assuming saving is equal to investment in period t, then sYt =Kt- Kt_x + 8Kt-X =Kt-(l- 8)Kt_x Dividing both sides by L,_i, then Kt (1 - S)Kt-i ill Lt-A Lt-i Kt ( Lt U- -(1-5)- K Lt \Lt-iJ Lt-\ But if population is growing at a constant rate n, as is assumed in this model, then Lt — Lt-\ Lt- l.e. -4-1 1 + n 13 A little care is required in discrete models in terms of stocks and flows (see section 1.3). Capital and labour are stocks and are defined at the end of the period. Hence, Kt and Lt are capital and labour at the end of period t. Flows, such as income, investment and savings are flows over a period of time. Thus, Yt, It and St are flows over period t. Discrete dynamic systems 131 Hence or syt = kt(l + n) - (1 - 8)kt_i (l + n)kt-(l-8)kt-l=sf(kt-i) which can be expressed (l-8)kt-l+sf(kt-l) kt = - 1+n i.e. k, = h(kt-\) With constant returns to scale and assuming a Cobb-Douglas production function, then yt = f(kt-\) = ak"_x a > 0, 0 < a < 1 Example 3.20 This can be investigated by means of a spreadsheet, where we assume a = 5, a = 0.25, 5 = 0.1, n = 0.02, 8 = 0 and let ko = 20. Alternatively, using a Taylor expansion about k* > 0, then (1 - 8)(kt-i - k*) + asaik^f-^kt-i - k*) kt = h(k*) + = h(k*) + 1 + n '(1 — 8) + asa(k*)a~l k* + 1 + n (1 - 8) + asa(k*)a~1 (kt-i - k*) 1 + n The situation is illustrated in figure 3.21. (kt-i - k*) 3.13 Solving recursive equations with Mathematica and Maple Both Mathematica and Maple come with a solver for solving recursive equations. RSolve in Mathematica and rsolve in Maple. They both operate in fundamentally the same way, and both can solve only linear recursive equations. While rsolve is built into the main kernel of Maple, the RSolve command of Mathematica is contained in the DiscreteMath package, and so must first be loaded with the following command. Needs["DiscreteMath"RSolve" (Note the back single-quote on RSolve.) These solvers are particularly useful for solving many difference equations. There are, however, some differences in the two 132 Economic Dynamics solvers. One difference is shown immediately by attempting to solve the recursive equation xt = axt-\. The input and output from each programme is as follows. Mathematica RSolve[x[t]==ax[t-1] , x[t] , t] {{x[t]->a1+tC[l] Maple rsolve(x(t)=a*x(t-l),x(t)); x (0) afc While Maple's output looks quite familiar, Mathematical looks decidedly odd. The reason for this is that Mathematica is solving for a 'future' variable. If the input had been RSolve[x[t+1]== ax[t],x[t],t] Then the solution would be {{x[t]->atC[l] }} which is what we would expect. Note also that while Mathematica leaves unsolved the unknown constant, which it labels C[l], Maple assumes the initial condition is *(0) for t = 0. If attempting to solve yt = ayt_\ + byy-2 for example, then when using Mathematica, this should be thought of as yt+2 = ayt+\ + byt when solving for yt. With this caveat in mind, we can explore the RSolve and rsolve commands in more detail. Discrete dynamic systems 133 When an initial condition is supplied the caveat just alluded to is of no consequence. Thus, if we wish to solve xt = axt-i x0 = 2 then the instructions are: Mathematica RSolve[{x[t]==ax[t-1],x[0]==2}, x[t] , t] with result {{x[t]->2a<}} Maple rsolve({x(t)=a*x(t-1),x(0)=2},x (t)); with result 2afc So no difference arises when initial conditions are supplied. Using either the RSolve command of Mathematica or the rsolve command of Maple, we can readily check the following equations used in this chapter: yt+\ = ay, pt+i = (1 + k)pt yt+\ = ayt + c (a — c\ (d\ Pt=\—)-\b)Pt-1 The following observations, however, should be borne in mind. (1) When using both Mathematica and Maple to solve the Harrod-Domar model, problem (iii), the recursive equation should be thought of as Yt+\ = (v/(v + s))Y, and solved accordingly. (2) On some occasions it is necessary to use additional commands, especially the Simplify command (Mathematica) or the simplify command (Maple). (3) Mathematica sometimes supplies 'If conditions in the solutions, usually to do with t > — 1 for example. This partly arises from the caveat mentioned above. Many of these can be avoided by writing the equations in terms of future lags, as in the case of the Harrod-Domar model. (4) A number of solutions involve complex output that is not always meaningful. This is especially true of general algebraic problems, such as solving yt+2 = ayt+i + byt. (5) Even when results have been simplified, it is not always possible to interpret the results in an economically meaningful way. For instance, in problem (v), it is impossible for a computer software package to 'know' that (a — c)l(b + d) is the equilibrium price and that it is more economically meaningful to take the difference (po — (a — c)/(b + d)). Economic insight is still a vital element. (i) (ii) (iii) (iv) (v) 134 Economic Dynamics Problems (i)-(v) are all recursive equations of the first-order. The same basic form is used to solve higher-order recursive equations. Given the recursive equation yt+2 = ayt+i + byt then this can be solved with the instructions: Mathematica RSolve[y[t+2]==ay[t+1]+by[t],y[t],t] Maple rsolve(y (t + 2)=a*y(t + 1)+b*y(t),y(t)); But because this is a general recursive equation the output in each case is quite involved. Mathematical output even more so, since it involves Binomial equations! What is revealed by the output is the need to know two initial conditions to solve such second-order recursive equations: Solving yt+2 = yt+\ + 2yt with initial conditions y(0) = 5andv(l) = 4,we have Mathematica RSolve[{y[t + 2]==y[t + l]+2y[t],y[0]==5,y[l]==4},y[t] ,t] with output {{y[t]->2 (-l)fc + 3 2t}} Maple rsolve({r (t + 2)=y(t + 1)+2*y (t),y(0)=5,y(1)=4},y(t) ); with output 2 (-1)t + 3 2t Furthermore, there is no difficulty with repeated roots, which occur in solving yt+2 = 4yt+\ — 4yt. For initial conditions y(0) = 6andv(l) = 4, we have solutions Mathematica: { {y [t] ->-21+t (-3 + 2t)}} Maple: (-4t-4)2fc + 10 2t Here we see that output in the two packages need not look the same, and often does not, yet both are identical; and identical to 6(2)' - 4t(2)' which we derived in the text. Complex roots, on the other hand, are solved by giving solutions in their complex form rather than in trigonometric form. The RSolve and rsolve commands, therefore, allow a check of the following equations in this chapter. (i) yt+2 = ayt+i + byt (Ü) yt+2 = yt+l + 2yt y(0) = 5,y(l) = 4 (in) yt+2 = 4yt+1 - 4yt (iv) yt+2 = 4yt+l-4yt y(0) = 6, y(l) = 4 (v) yt+2 = 4^+1 - 16}V (vi) yt+2 = ayt+1 - byt + c (vii) yt+2 = 4yt+l - \6yt + 26 (viii) yt+2 = 5yt+1 -4yt + 4 (ix) yt+2 = -yt+1+2yt+l2 y(0) = 4,y(l) (x) Yt = (b + v)Yt.1 - vYt-2 + (a + G) (xi) Y,= 4.75F,_1 + 4Yt_2 + 150 Discrete dynamic systems 135 Neither Mathematica nor Maple, however, can solve directly the logistic equation (l+a)yt yt+1 = 1 , u— 1 +byt This is readily accomplished using the substitution provided in section 3.9. It is worth pointing out that in the case of numerical examples, if all that is required is a plot of the sequence of points, then there is no need to solve the recursive (or difference) equation. We conclude this section, therefore, with simple instructions for doing this.14 The equation we use as an example is pt = 5.6 - 0Apt-i po = 1 Mathematica Clear [p]; p[0]=l; p[t_] :=p[t]=5.6-0.4p[t-l] ; data=Table[{t,p[t] }, {t,0,20} ]; ListPlot[data,PlotJoined->True,PlotRange->All]; Maple t:='t' : p : —'p' : p:=proc (t)option remember; 5.6-0.4*p(t-1)end: p(0):=1: data:=seq( [t,p(t)], t=0 ..20)]; plot(data,colour=black,thickness=2); Notice that the instructions in Maple require a 'small' procedural function. It is important in using this to include the option remember, which allows the programme to remember values already computed. Higher-order recursive equations and nonlinear recursive equations are dealt with in exactly the same way. With discrete dynamic models, however, it is often easier and quicker to set the model up on a spreadsheet (see Shone 2001). Appendix 3.1 Two-cycle logistic equation using Mathematica THEOREM The number a satisfies the equation a=f{f{a)) if a is either a fixed point or is part of a two-cycle for the dynamical system Xn+l =f(xn) 14 I am grateful to Johannes Ludsteck, Centre for European Economic Research (ZEW), for the method of computing tables from recursive equations in Mathematica, which is more efficient than the one I provided in the first edition. 136 Economic Dynamics Example (the generic logistic equation) ln[l]:= f[x_]=rx(l-x) Out[l]= r (1-x) x In [2]: = eql=f[f[x]] 0ut[2]= r2 (1-x) x (1-r (l-x)x) In [3]:= soll=Solve[eql==x,x] 0ut[3]= {{x^O}, {x^^},{x^r+r2-rl23--^}, L 2r r _^ r + r2 + r v-3-2r+r" i i tX 2r2 U In [4]:= al=sollE3, 1, 2J Outf4J= r + r2-rV-23-2r+-^ 2r2 In [5]:= a2=soll[[4, 1, 2J „ . rt-7 r + r2+r ^-3-2r + r2 Out[5] = -9- 2r2 In[61:= g[x_]=dxf [x] Out[6]= r (1-x)-rx In [7] .-= eq2=Simplify[g[al]g[a2] ] 0ut[7]= 4+2r-r2 In [8]:= sol2=Nsolve[eq2==0,r] 0ut[8]= { {r ^-1.23607 }, {r^ 3. 23607}; In [9]: = rstar=sol2112, 1, 2J Out[9]= 3.23607 In [10] ;= al/.r -> rstar Out [10]= 0.5 In [11] := a2/.r -> rstar Out [11]= 0.809017 In [12] := al/.r -»3.2 Out [12]= 0.513045 In [13] := a2/.r -»3.2 Out [13]= 0.799455 In [14]:= Nsolve[-l==4 + 2r-r2, r] Out[14]= { {r ^-1. 44949}, {r^ 3.44949; In [15]:= Nsolve[4+2r-r2 = = l, r] Out [151= {{r->-l. }, {r^3. }} Considering only positive roots, we have: V-V?r + r2 r = 3 and r = 3.44949 Discrete dynamic systems 137 Appendix 3.2 Two-cycle logistic equation using Maple > f:=x->r* x* (1-x); /:= x —► rx{\ — x) > > eql:=f(f(x) ) ; egi := r2x(\ — x)(l — rx(l — x)) > soil:=solve(eql=x,x); 1 1 1 i ii I-r — 1 + r 9 99 So/i := 0,-, A-*-*-, r 1 1 1 I-z —r H----v — 3 — 2r + r2 2 2 2 _ r > al:=soll[3]; 1 1 1 r H---h -V-3 - 2r + r2 fli:= 2-2_^ > a2:=soll[4]; 1 1 1 /-r —r H----v — 3 — 2r + r2 a2:= 2-2__2l- r > g:=diff(f(x),x); g:= r(l — x) — rx > eq2:=expand(subs(x=al,g)*subs(x=a2,g)); eq2:= 4 + 2r — r2 > sol2:=solve(eq2=0,r); sol2:= 1 - V5, 1 + V5 > rstar:=sol2[2]; rstar:= 1 + V5 > evalf(subs(r=rstar,al)); .8090169946 > evalf(subs(r=rstar,a2)); .4999999997 > evalf(subs(r=3.2,al)); .7994554906 > evalf(subs(r=3.2,a2)) ; .5130445094 138 Economic Dynamics > solve(eq2 = -l,r) ; 1 - V6, 1 + V6 > evalf(%); -1.449489743, 3.449489743 > solve(eq2=l,r); -1,3 Exercises 1. Classify the following difference equations: (i) yt+2 = yt+i - 0.5yt + 1 (ii) yt+2 = 2yt + 3 r... yt+i -yt . (m) -= 4 yt (iv) yt+2 - 2yt+i + 3yt = t 2. Suppose you borrow an amount Pq, the principal, but you repay a fixed amount, R, each period. Formulate the general amount, Pt+\, owing in period t + 1, with interest payment r%. Solve for Pn. 3. In question 2, suppose the repayment is also variable, with amount repaid in period t of Rt. Derive the solution Pn. 4. Establish whether the following are stable or unstable and which are cyclical. (i) yt+1 = -0.5yt + 3 (ii) 2yt+1 = -3yt + 4 (iii) yt+i = -yt + 6 (iv) yt+i = Q.5yt + 3 (v) 4yt+2 + 4yt+1 -2 = 0 5. Consider yt+i =y] -y2 + l (i) Show that a = 1 is a fixed point of this system. (ii) Illustrate that a = 1 is a shunt by considering points either side of unity for yo. 6. Use a spreadsheet to compare yt+i = (1 + a)yt - by] and (1 + a)yt using (i) a = 1.5 b = 0.1 yo = l (ii) a = 1.5 b = 0.1 yo = 22 (iii) a = 2.2 b = 0.1 yo = 1 (iv) a = 2.2 b = 0.1 yo = 25 (v) a = 1.8 b = 0.15 yo = 11.5 Discrete dynamic systems 139 7. Derive the cobweb system for the price in each of the following demand and supply systems, and establish whether the equilibrium price is (i) stable, (ii) unstable, or (iii) oscillatory. (i) qf =10-3Pt (ii) qf = 25 - 4pt (iii) qf = 45 - 2.5p, qst = 2+Pt_1 q° = 3 + 4pt_1 q\ = 5 + 7.5/Vi qf = q* qf = q* qf = q* 8. Suppose we have the macroeconomic model Ct = a + bYt-i Et = Ct + It + Gt Yt = Et where C and Y are endogenous and I and G are exogenous. Derive the general solution for Yn. Under what conditions is the equilibrium of income, Y*, stable? 9. Verify your results of question 8 by using a spreadsheet and letting 7=10, G = 20, a = 50, Yq = 20, and b = 0.8 and 1.2, respectively. For what period does the system converge on Y* — Yq within 1 % deviation from equilibrium? For the same initial value Yo, is the period longer or shorter in approaching equilibrium the higher the value of bl 10. Given qf = a- bpt q\ = c + dpet Pet =Pt-\- e(pt-i -Pt-i) (i) Show that if in each period demand equals supply, then the model exhibits a second-order nonhomogeneous difference equation forpt. (ii) Use a spreadsheet to investigate the path of price and quantity for the parameter values a =10 c = 2 e = 0.5 b=3 d=l 11. In the linear cobweb model of demand and supply, demonstrate that the steeper the demand curve relative to the supply curve, the more damped the oscillations and the more rapidly equilibrium is reached. 12. Using a spreadsheet, verify for the linear cobweb model of demand and supply that whenever the absolute slope of the demand curve is equal to the absolute slope of the supply curve, both price and quantity have a two-period cycle. 13. Given the following logistic model yt+l = 3.84v,(l -yt) set this up on a spreadsheet. Set _yo = 0.1 and calculate yn for the first 100 elements in the series. Use the 100th element as the starting value and then re-compute the next 100 elements in the series. Do the same again, and verify that this system tends to a three-cycle with fli = 0.149407 a2 = 0.488044 a3 = 0.959447 140 Economic Dynamics 14. f\{x) and^C*) are linearly dependent if and only if there exist constants b\ and b2, not all zero, such that bifiix) + b2f2(x) = 0 for every x. If the set of functions is not linearly dependent, then/i (x) and f2(x) are linearly independent. Show that yx = Y* and y2 = tY* are linearly independent. 15. Given the following version of the Solow model with labour augmenting technical progress Yt = F(Kt,AtLt) Kt+l =Kt + SKt St = sYt h = St Lt+i — Lt - = n U At = y'Ao (i) show that f (1 - 8)kt + sf{kt) kt+\ = - t+i y(l + n) where k is the capital/labour ratio measured in efficiency units, i.e., k = (K/AL). (ii) Approximate this result around k* > 0. 16. Given the model qf = a — bpt b > 0 q\ = c + dpet d > 0 Pt = Pet-i ~ HPt-i -Pet-i> 0 < A < 1 (i) Show that price conforms to a first-order nonhomogeneous difference equation. (ii) Obtain the equilibrium price and quantity. (iii) Show that for a stable cobweb then 2b 0 < X < b + d 17. A student takes out a loan of £8,000 to buy a second hand car, at a fixed interest loan of 7.5% per annum. She intends to pay off the loan in three years just before she graduates. What is her monthly payment? 18. A bacterial cell divides every minute. A concentration of this bacterium in excess of 5 million cells becomes contagious. Assuming no cells die, how long does it take for the bacteria to become contagious? Discrete dynamic systems 141 19. Show that neither Mathematica nor Maple can solve the following difference equation xt xt+i = —— 1 +xt Use either programme to generate the first elements of the series up to t = 10, and hence show that this indicates a solution. 20. A Fibonacci series takes the form xn = xn-\ + xn-2 xq = I and x\ = 1 (i) Use a spreadsheet to generate this series, and hence show that the series is composed of integers. (ii) Solve the recursive equation with a software package and show that all of its factors are irrational numbers but that it takes on integer values for all n, which are identical to those in the spreadsheet. Additional reading For additional material on the contents of this chapter the reader can consult Allen (1965), Baumol (1959), Baumol and Wolff (1991), Chiang (1984), Domar (1944), Elaydi (1996), Farmer (1999), Gapinski (1982), Goldberg (1961), Griffiths and Oldknow (1993), Hicks (1950), Holmgren (1994), Jeffrey (1990), Kelley and Peterson (2001), Samuelson (1939), Sandefur (1990), Shone (2001), Solow (1956) and Tu (1994). CHAPTER 4 Systems of first-order differential equations 4.1 Definitions and autonomous systems In many economic problems the models reduce down to two or more systems of differential equations that require to be solved simultaneously. Since most economic models reduce down to two such equations, and since only two variables can easily be drawn, we shall concentrate very much on a system of two equations. In general, a system of two ordinary first-order differential equations takes the form dx ~T=x = /(*>0 dt dy — = y = g(x, y, t) dt Consider the following examples in which x and y are the dependent variables and t is an independent variable: (0 X = y = ax — by — ce' rx + sy — qe' (ii) X = y = ax — by rx + sy (iii) X = y = ax — bxy rx — sxy Examples (i) and (ii) are linear systems of first-order differential equations because they involve the dependent variables x and y in a linear fashion. Example (iii), on the other hand, is a nonlinear system of first-order differential equations because of the termxy occurring on the right-hand side of both equations in the system. Examples (ii) and (iii) are autonomous systems since the variable t does not appear explicitly in the system of equations; otherwise a system is said to be nonautonomous, as in the case of example (i). Furthermore, examples (ii) and (iii) are homogeneous because there is no additional constant. Example (i) is nonhomogeneous with a variable term, namely ce1'. A solution to system (4.1) is a pair of parametric equations x = x(t) and y = y(t) which satisfy the system over some open interval. Since, by definition, they satisfy the differential equation system, then it follows that the solution functions are differentiable functions of t. As with single differential equations, it is often necessary to impose initial conditions on system (4.1), which take the form xo = x(t0) and y0 = y(t0) Systems of first-order differential equations 143 Our initial value problem is, then x = fix, y, t) y = g(x, y, t) (4.2) x(to) = x0,y(t0) = y0 Economic models invariably involve both linear and nonlinear systems of equations that are autonomous. It is, therefore, worth exploring the meaning of autonomous systems in more detail because it is this characteristic that allows much of the graphical analysis we observe in economic theory. In order to elaborate on the ideas we need to develop, consider an extremely simple set of differential equations. Example 4.1 x = 2x y = y (4.3) x(t0) = 2, y(t0) = 3 We can capture the movement of the system in the following way. Construct a plane in terms of x and y. Then the initial point is (xo, yo) = (2,3). The movement of the system away from this initial point is indicated by the systems of motion, or transition functions, xf(t) and y'(t). If we can solve the system for x(t) and y(t), then we can plot the path of the system in the (x, v)-plane. In the present example this is easy to do. Here we are not so concerned with solving systems of autonomous equations, but rather in seeing how such solutions appear in the (x, v)-plane. The solutions are x(t) = 2e2t and y(t) = 3el The path of the system in the (x, v)-plane is readily established by eliminating the variable t and expressing y as a function of x. Thus which is defined for x > 2, y > 3. Over time, x(t) increases beyond the initial value of ^(0) = 2 and y(t) increases beyond the initial value of y(0) = 3. Hence, the system moves along the trajectory shown in figure 4.1(a) and in the direction indicated by the arrows. More significantly, for any initial point (xq, yo) there is only one trajectory through this point. Put another way, no matter when the system begins to move, it will always move along the same trajectory since there is only one trajectory through point (xo, yo)- In terms of figure 4.1(a), there is only one trajectory through the point (2,3). Example 4.2 But now consider a similar system of equations x = 2x + t y = y (4.4) x(0) = 2, y(0) = 3 144 Economic Dynamics Figure 4.1. (a) 2 4 6 8x 10 12 14 with solutions 9e2t - 1 t x(t) =--- w 4 2 y(t) = 3ef which clearly satisfy the initial conditions. Eliminating e', we can express the relationship between x and y as y = *Jl + 4x + 2t x>2,y>3 which clearly depends on t. This means that where a system is at any point in time in the (x,v)-plane depends on precisely the moment the system arrives at that point. For instance, in figure 4.1(b) we draw the system for three points of time, t = 0, 5, 10. There is no longer a single trajectory in the (x,v)-plane, but a whole series of trajectories, one for each point in time. The system is clearly time-dependent. Autonomous systems are time-independent. One must be careful of the meaning here of being 'time-independent'. All that is meant is that the time derivatives are not changing over time (the transition functions are independent of time); however, the solution values for the dependent variables x and y will be functions of time, as we clearly saw from the above example. With autonomous systems there is only a single trajectory in the (x,v)-plane which satisfies the Systems of first-order differential equations 145 initial conditions. Of course, with different initial conditions, there will be different trajectories in the (^:,v)-plane, but these too will be unique for a given initial condition. 4.2 The phase plane, fixed points and stability In chapter 2 we introduced the phase line. This was the plot of x(t) on the ^-line. It is apparent that figure 4.1(a) is a generalisation of this to two variables. In figure 4.1(a) we have plotted the path of the two variables x and y. At any point in time we have a point such as (x(t), y(t)), and since the solution path is uniquely defined for some initial condition (xo, yo), then there is only one path, one function y = (p(x), which satisfies the condition yo = (p(xo). A solution curve for two variables is illustrated in figure 4.2(a), whose coordinates are (x(t), y(t)) as t varies over the solution interval. This curve is called a trajectory, path or orbit of the system; and the (*,v)-plane containing the trajectory is called the phase plane of the system. The set of all possible trajectories is called the phase portrait. It should be noted that in the case of autonomous systems as t varies the system moves along a trajectory (x, y) through the phase plane which depends only on the coordinates (x, y) and not on the time of its arrival at that point. As time increases, the arrows show the direction of movement along the trajectory. The same is true for an autonomous system of three variables, x, y and z. Figure 4.2(b) illustrates a typical trajectory in a three-dimensional phase plane which passes through the point (xo, yo, zo). Again the arrows indicate the movement of this system over time. Since for autonomous systems the solution curve is uniquely defined for some initial value, then we can think of y as a function of x, y = (p(x), whose slope is given by dy dy/dt dx dx/dt and which is uniquely determined so long as dx/dt is not zero. For autonomous systems, this allows us to eliminate the variable t. To see this, return to example 4.1 dy dy/dt y dx dx/dt 2x This is a separable equation and so can be solved by the method developed in chapter 2. Solving, and solving for the constant of integration by letting the initial conditions be = 2 and yo = 3, we find again that y = «J9x/2 (see exercise 1). If (x*, y*) is a point in the phase plane for which f(x, y) = 0 and g(x, y) = 0 simultaneously, then it follows that dx/dt = 0 and dy/dt = 0. This means that neither x nor y is changing over time: the system has a fixed point, or has an equilibrium point. For example 4.1 it is quite clear that the only fixed point is (jc*, y*) = (0, 0). In the case of example 4.2, although y* = 0, x* = t/2 and so x* depends on the point in time the system arrives at x*. Consider another example. 146 Economic Dynamics Figure 4.2. (a) y (Wo) y=¥& s lx(i),y(t)) (b) (4.6) Example 4.3 We can establish the fixed point of the following simultaneous equation system x = x — 3y y = —2x + y xo = 4, y0 = 5 by setting x = 0 and y = 0, which has solution x* = 0 and y* = 0. Systems of first-order differential equations 147 It should be quite clear from these examples that independent homogeneous linear equation systems have a fixed point at the origin. Also, there is only the one fixed point. Having established that such a system has a fixed point, an equilibrium point, the next step is to establish whether such a point is stable or unstable. A trajectory that seems to approach a fixed point would indicate that the system was stable while one which moved away from a fixed point would indicate that the system was unstable. However, we need to be more precise about what we mean when we say 'a fixed point (x*, y*) is stable or unstable'. A fixed point (x*, y*) which satisfies the condition f(x, y) = 0 and g(x, y) = 0 is stable or attracting if, given some starting value (xo, yo) 'close to' (x*, y*), i.e., within some distance 8, the trajectory stays close to the fixed point, i.e., within some distance e > 8. It is clear that this definition requires some measure of 'distance'. There are many ways to define distance, but the most common is that of Liapunov. In simple terms we define a ball around the fixed point (x*, y*) with radius 8 and e, respectively. Thus, define B$(x*, y*) to be a ball (circle) centred on (x*, y*) and with radius 8. Define a second ball, BE(x*, y*) to be a ball (circle) centred on (x*, y*) and with radius e > 8. The situation is illustrated in figure 4.3(a) and (b). We have a starting value (xo, yo) 'close to' the fixed point (x*, y*), in the sense that (xo, yo) lies in the ball B$(x*, y*). The solution value (path) starting from (xo, yo) stays 'close to' the fixed point, in the sense that it stays within the ball BE(x*, y*). The solution paths in figure 4.3(a) and (b) both satisfy this condition, and so are both stable. But a careful consideration of the statement of stability will indicate that there is nothing within the definition that insists that the trajectory has to approach the fixed point. All that is required is that it stay within the ball BE(x*, y*). A look at figure 4.3(b) will indicate that this satisfies the definition of stability just outlined. However, the solution path is periodic, it begins close to the fixed point (i.e. the starting point lies within the ball B$(x*, y*)) but cycles around the fixed point while staying 'close to' the fixed point (i.e. stays within the ball BE(x*, y*)). Such a limit cycle is stable but not asymptotically stable. We shall find examples of this when we consider competing population models in chapter 14. A fixed point that is not stable is said to be unstable or repelling. A fixed point is asymptotically stable if it is stable in the sense just discussed, but eventually approaches the fixed point as t oo. Thus to be asymptotically stable, it must start close to (x*, y*) (i.e. within 8), it must remain close to the fixed point (i.e. within e), and must eventually approach (x*, y*) ast—* oo. Hence, the situation shown in figure 4.3(a) is asymptotically stable. Also notice that the trajectory can move away from the fixed point so long as it stays within the ball BE(x*, y*) and approaches the fixed point in the limit.1 Asymptotic stability is a stronger property than stability. This is clear because to be asymptotically stable then it must be stable. The limit condition on its own is not sufficient. A system may start 'close to' (x*,y*) (i.e. within B&(x*,y*)) and approach the fixed point in the limit, but diverge considerably away from (go beyond the ball BE(x*, y*)) in the intermediate period. 1 A fixed point that is stable but not asymptotically stable is sometimes referred to as neutrally stable. Figure 4.3(b) illustrates a periodic trajectory around the fixed point, which is accordingly neutrally stable. This type of system is typical of competing populations, as we shall illustrate in chapter 14. 148 Economic Dynamics Figure 4.3. (a) y x If a system has a fixed point (x*, y*) which is asymptotically stable, and if every trajectory approaches the fixed point (i.e. both points close to the fixed point and far away from the fixed point), then the fixed point is said to be globally asymptotically stable. Another way to consider this is to establish the initial set of conditions for which the given fixed point is asymptotically stable, i.e., the largest ball from which any entering trajectory converges asymptotically to the fixed point. This set of initial conditions is called the basin of attraction. A fixed point is locally asymptotically stable if there exists a basin of attraction, Be(^:*, y*), within which all trajectories entering this ball eventually approach the fixed point (x*, y*). If the basin of attraction is the whole of the (^:,v)-plane, then the system is globally asymptotically stable about the fixed point (x*, y*). Systems of first-order differential equations 149 Mathematicians have demonstrated a number of properties for the trajectories of autonomous systems. Here we shall simply list them. (1) There is no more than one trajectory through any point in the phase plane (2) A trajectory that starts at a point that is not a fixed point will only reach a fixed point in an infinite time period (3) No trajectory can cross itself unless it is a closed curve. If it is a closed curve then the solution is a periodic one. 4.3 Vectors of forces in the phase plane We established in chapter 2, when considering single autonomous differential equations, that we could establish the direction of x when t varies from the sign of x. In the case of a system of two differential equations we can establish the direction of x from the sign of x and the direction of y from the sign of y. Such movements in x and y give us insight into the dynamics of the system around the equilibrium. In this section we shall pursue three very simple examples in some detail in order to investigate the dynamic properties of each system. Although each can be solved explicitly, this will not always be the case, and the qualitative dynamics we shall be developing will be particularly useful in such circumstances. Example 4.4 This continues example 4.3, equation system (4.6), but we shall repeat it here. Consider the following first-order autonomous system with initial conditions x = x — 3y y=-2x + y (4.7) xo = 4, y0 = 5 The solution to this system, which satisfies the initial values, is the following 8eU-V6> + 5^(1-76); + Se(l+V6)t _ 5^ed+V6)t x(t) = y(t) = 4 l5e(i-V6); + ^/96ed-V6> + i5ed+V6)t _ J96e(i+V6)t 6 For the moment we are not concerned about how to derive these solutions, which we shall consider later, all we are concerned about here is to show that these are indeed the solution values that satisfy the initial conditions. We verify this by differentiating both x and y with respect to t and substituting into the system of equations. A rather tedious exercise, which can be accomplished with software packages like Mathematica or Maple. Doing so shows that each equation in the system is identically true. Hence the equations are indeed the solutions to the system. It will be noted that the solution values are not straightforward. We could plot the solution values for x and y against time, as shown in figure 4.4. Alternatively, and much more informatively, we can plot the path of points {x(t), y(t)} in the phase plane as the independent variable t varies over the solution interval. Generally, this will involve a phase portrait, which again can be done using Mathematica (4.8) 150 Economic Dynamics Figure 4.4. Figure 4.5. or Maple.2 Such a solution path to the system in the (x,y)-plane is illustrated in figure 4.5. This is a specific solution since it satisfies the initial conditions. But we want to know what is happening to the path in relation to the fixed point, the equilibrium point, of the system. We already know that a fixed point is established by setting x = 0 and y = 0. The fixed point is readily established by solving the simultaneous equations 0 = x - 3y 0 = -2x + y which has solution x* = Oandy* = 0. Another way to view the fixed point is to note that 0 = x — 3y is the equilibrium condition for the variable x; while 0 = — 2x + y is the equilibrium condition for the variable y. The fixed point is simply where the two equilibrium lines intersect. The equilibrium lines are y = % (* = 0) 3 y = 2x (y = 0) See section 4.12. Systems of first-order differential equations 151 Figure 4.6. Consider the first equilibrium line. Along this line we have combinations of x and y for which x = 0. But this means that for any value of x on this line, x cannot be changing. This information is shown by the vertical dotted lines in figure 4.6 for any particular value of x. Similarly, for the line denoted y = 0 (y = 2x) the value of y on this line cannot be changing. This information is shown by the horizontal dotted lines in figure 4.6 for any particular value of y. Next consider points either side of the equilibrium lines in the phase plane. To the right of the x-line we have x y < — implying x > 0 Hence, for any point at which x lies below the x-line, then x is rising. Two are shown in figure 4.6 by the horizontal arrows that are pointing to the right. By the same reasoning, to the left of the x-line we have x y > — implying x < 0 Hence, for any point at which x lies above the x-line, then x is falling. Two are shown in figure 4.6 by the horizontal arrows pointing to the left. By the same reasoning we can establish to the right of the v-line y < 2x implying y < 0, hence y is falling while to its left y > 2x implying y > 0, hence y is rising Again these are shown by the vertical arrows pointing down and up respectively in figure 4.6. It is clear from figure 4.6 that we have four quadrants, which we have labelled I, II, III and IV, and that the general direction of force in each quadrant is shown by the arrow between the vertical and the horizontal. It can be seen from figure 4.6 that in quadrants I and III forces are directing the system towards the origin, towards the fixed point. In quadrants II and IV, however, 152 Economic Dynamics the forces are directing the system away from the fixed point. We can immediately conclude, therefore, that the fixed point cannot be a stable point. Can we conclude that for any initial value of x and y, positioning the system in quadrants I or III, that the trajectory will tend over time to the fixed point? No, we cannot make any such deduction! For instance, if the system began in quadrant I, and began to move towards the fixed point, it could over time pass into quadrant IV, and once in quadrant IV would move away from the fixed point. In fact, this is precisely the trajectory shown in figure 4.5. Although the trajectory shown in figure 4.5 moves from quadrant I into quadrant IV, this need not be true of all initial points beginning in quadrant I. Depending on the initial value for x and y, it is quite possible for the system to move from quadrant I into quadrant II, first moving towards the fixed point and then away from it once quadrant II has been entered. This would be the situation, for example, if the initial point was (xq, yo) = (4, 2) (see exercise 2). This complex nature of the solution paths can be observed by considering the direction field for the differential equation system. The direction field, along with the equilibrium lines are shown in figure 4.7. Why the dynamic forces seem to operate in this way we shall investigate later in this chapter. Example 4.5 The following system of linear differential equations x = —3x + y y = x-3y with initial condition xq = 4 and yo = 5 has solution equations 9e2t - 1 1 + 9e2t X(t) = y(t) = which gives rise to a trajectory which approaches the fixed point (x*, y*) = (0, 0), as shown in figure 4.8. The path of x(t) and y(t), represented by the phase line, as t increases is shown by the direction of the arrows. Figure 4.7. y y=0 Systems of first-order differential equations 153 Figure 4.8. No matter what the initial point, it will be found that each trajectory approaches the fixed point (x*, y*) = (0, 0). In other words, the fixed point (the equilibrium point) is globally stable. Considering the vector of forces for this system captures this feature. The equilibrium solution lines are y = 3x for x = 0 x y = — for y = 0 y 3 y with a fixed point at the origin. To the right of the ^-line we have y < 3x or — 3x + y < 0 implying x < 0 so x is falling While to the left of the ^-line we have y > 3x or — 3x + y > 0 implying x > 0 so x is rising Similarly, to the right of the v-line we have x y < — or 0 < — 3y + x implying y > 0 so y is rising While to the left of the v-line we have x y > -or0> — 3y + x implying y < 0 so y is falling All this information, including the vectors of force implied by the above results, is illustrated in figure 4.9. It is clear that no matter in which of the four quadrants the system begins, all the forces push the system towards the fixed point. This means that even if the trajectory crosses from one quadrant into another, it is still being directed towards the fixed point. The fixed point must be globally stable. If the initial point is (xo, yo) = (4, 5), then the trajectory remains in quadrant I and tends to the fixed point (x*, y*) = (0, 0) over time. However, figure 4.9 reveals much more. If the system should pass from one quadrant into an adjacent quadrant, then the trajectory is still being directed towards the fixed point, but the movement 154 Economic Dynamics Figure 4.9. Figure 4.10. of the system is clockwise? This clockwise motion is shown most explicitly by including the direction field on the equilibrium lines, as shown in figure 4.10. Example 4.6 The two examples discussed so far both have fixed points at the origin. However, this need not always be the case. Consider the following system of linear 3 The nature of the trajectory can be established by noting that 2 2 _ (9e2t - l)2 (1 + 9e2')2 x +y = + _ 81 4det(A) (2) real and equal if tr(A)2 = 4det(A) (3) complex conjugate if tr(A)2 < 4det(A). 4.5 Solutions to the homogeneous differential equation system: real distinct roots Suppose we have an w-dimensional dynamic system (4.21) x = Ax where *2 a\\ a\2 X\ *2 and suppose u1, u2, ... , un are n linearly independent solutions, then a linear combination of these solutions is also a solution. We can therefore express the general solution as the linear combination X = C\\Xl + c2u2 + ... + cnun where c\, c2, ... , cn are arbitrary constants. In the case of just two variables, we are after the general solution X = ClU1 + C2U2 In chapter 2, where we considered a single variable, we had a solution x = cen This would suggest that we try the solution5 x = eXt\ where A is an unknown constant and v is an unknown vector of constants. If we do this and substitute into the differential equation system we have XeXt\ = AeXt\ eliminating the term eXt we have X\ = Av i.e. (A - kl)\ = 0 For a nontrivial solution we require that det(A - XI) = 0 5 Here u = ert\. Systems of first-order differential equations 161 We investigated this problem in the last section. What we wish to find is the eigenvalues of A and the associated eigenvectors. Return to the situation with only two variables, and let the two roots (the two eigenvalues) be real and distinct, which we shall again label as r and s. Let vr be the eigenvector associated with the root r and \s be the eigenvector associated with the root s. Then so long as r ^ s u1 = erW and u2 = esW are independent solutions, while x = ciertyr + c2estys (4.22) is a general solution. Example 4.8 Find the general solution to the dynamic system x = x + y y = -2x + 4y We can write this in matrix form X ' l r X y _ -2 4_ y _ The matrix A of this system has already been investigated in terms of example 4.7. Note, however, that det(A) > 0. In example 4.7 we found that the two eigenvalues were r = 3 and s = 2 and the associated eigenvectors were v = "1" vs — "1" _2_ i v — _1_ Then the general solution is x = c\e' or, in terms of x and y + c2e it x(f) = cie3t + c2e2t y(t) = 2Cle3t + c2e2t Given an initial condition, it is possible to solve for c\ and c2. For example, if ^(0) = 1 andv(0) = 3, then 1 = C\ + c2 3 = 2c\ + c2 which gives c\ = 2 and c2 = — 1. Leading to our final result x{t) = 2eil - e y(t) = 4e3t ?2t n2t 162 Economic Dynamics 4.6 Solutions with repeating roots In chapter 2 we used ceXt and cteXt for a repeated root. If X = r which is a repeated root, then either there are two independent eigenvectors v1 and v2 which will lead to the general solution x = ciertyl + c2erty2 or else there is only one associated eigenvector, say v. In this latter case we use the result (4.23) x = ciertyl + c2(ertt\ + erty2) In this latter case the second solution satisfies ertt\ + ert\2 and is combined with the solution ertvl to obtain the general solution (see Boyce and DiPrima 1997, pp. 390-6). We shall consider two examples, the first with a repeating root, but with two linearly independent eigenvectors, and a second with a repeating root but only one associated eigenvector. Example 4.9 Consider x = x y = y Then X "l 0" X y_ 0 l y_ where A = '1 0' 0 1 det(A) = 1, A-Xl = 1 — X 0 0 I — X Hence, det(A — XT) = (1 — X)2 = 0, with root X = r = 1 (twice). Using this value of X then (A-I) 0 0 0 0 Since (A — rl)v = 0 (r = 1) is satisfied for any vector v, then we can choose any arbitrary set of linearly independent vectors for eigenvectors. Let these be v and then the general solution is x = c\e 1 0 + c2er v2 = "0" 1 Systems of first-order differential equations 163 or x(t) = c\ert y(t) = c2en Let Then with Example 4.10 x = x — y y = x + 3y X "i -l" X ý _ 1 3 _ y _ 1 1 1 3 Hence, det(A — AI) = Using k = r = 2 det(A) = 4, A - XI = 1 l ^ 3 _^ X2 - 4k + 4 = (k - 2)2, with root X = r = 2 (twice). and A-rl (A - rl) ■1 ■1 X "-i -l" X y _ _ i i y _ which implies — x — y = 0. Given we normalise x to x = 1, then y = — 1. The first solution is then „2i 1 ■1 To obtain the second solution we might think of proceeding as in the single variable case, but this is not valid (see Boyce and DiPrima 1997, pp. 390-6). What we need to use is e2tt\ + e2ty2 where we need to find the elements of v2. Since we know v, then the second solution u2 = (x2, y2) is , +e2t -1 v2 *2 yi = e2tt i.e. Hence x2 = e2t{t + vi) y2 = e2t(-t + v2) x = 2e2\t + vi) + e2t = e2\2t + 2vi + 1) y = 2e2\-t + v2) + e2t = e2\-2t + 2v2 + 1) 164 Economic Dynamics Substituting all these results into the differential equation system we have e2\2t + 2vi + 1) = e2\t + vi) - e2\-t + v2) e2t(-2t + 2v2 - 1) = e2t(t + vi) + 3e2'(-r + v2) Eliminating e2' and simplifying, we obtain Vi + V2 = — 1 Vi + V2 = -1 which is a dependent system. Since we require only one solution, set v2 = 0, giving vi = —1. This means solution x2 is " 1 x2 = e2tt ■1 + e it -1 0 Hence, the general solution is c\e it 1 + c2 I e2tt + e it -1 0 or x = c\e + c2(t — l)e y = it it it -c\e — c2te (4.24) 4.7 Solutions with complex roots For the system x = Ax with characteristic equation det(A—XI) = 0, if tr(A)2 < 4det(A), then we have complex conjugate roots. Return to our situation of just two roots, X = r and X = s. Then r = a + f3i s = a — ^i But this implies that the eigenvectors vr and \s associated with r and s, respectively, are also complex conjugate. Example 4.11 Consider Then with x = —3x + 4y y = —2x + y X "-3 4" X y _ -2 1 y_ A = -3 4 -2 1 det(A) = 5, A-Xl = —3 — X -2 4 1 — X Systems of first-order differential equations 165 and det(A — k\) = X2 + 2X + 5, which leads to the roots -2 + V4 - 20 -2 - V4 - 20 r = - = — 1 + 2i and s = - The associated eigenvectors are (A - kl)\r = i.e. -(2 + 20V1 + Avr2 = 0 -2v\ + (2 - 2i)vr2 = 0 Let v\ = 2, then v\ = 2(2 + 2/)/4 = 1 + i. Thus = -1 - 2/ 2-2/ 4 i/ — "0" -2 2-2/_ V — 0 u ,(-l+2i)i 2 1 + / Turning to the second root. With A = s ■■ -2 + 2/ 4 (A - A.I)vs = -2 2 + 2/ 1 - vr y2. i.e. (-2 + 2i)v\ + 4v^ = 0 -2v\ + (2 + 2i)v| = 0 Choose v\ = 2, then v2 = —(—2 + 2/)(2)/4 = 1 — /. Hence the second solution is 2 1 -/ i.e. vs is the complex conjugate of vr. Hence the general solution is u2 = e-(l+2i)t x = Cle(-1+2i)i 2 1 + / + c2 \s\. Suppose both roots are negative, then r < s < 0. Further, suppose the associated eigenvectors vr and Vs are as shown in figure 4.16 by the heavy arrows. Thus it is quite clear that as t oo, en 0 and est 0, and so x —► 0 regardless of the value of c\ and c2. Of particular significance is that if the initial point lies on vr, then c2 = 0 and the system moves down the line through vr and approaches the origin over time. Similarly, if the initial point lies on \s, then c\ = 0 and the system moves down the line Vs, approaching the origin in the limit. The critical point is called a node.7 In the present case we have a stable node. If r and s are both positive, then the system will move away from the fixed point over time. This is because both x and y grow exponentially. In this case we have an unstable node. Example 4.12 Let with x = —2x + y y = x — 2y -2 1 1 -2 det(A) = 3, A - XI —2 — X 1 1 —2 — X 1 Sometimes called an improper node. 168 Economic Dynamics Figure 4.16. Hence det(A - XI) = X2 + 4X + 3 = (X + 3)(X + 1) = 0, which leads to roots X = r = — 3 and X = s = —I. Using these values for the eigenvalues, the eigenvectors are 1 ■1 and " 1 "1" -1 + c2e 1 1 which gives the general solution x = c\e~3t or x(t) = c\e~3t + c2e~' y(t) = -cie~3t + c2e~' The solution is illustrated in figure 4.17,8 where the solution paths are revealed by the direction field, indicating quite clearly that the origin is a stable node. Case 2 (Real distinct wots of opposite sign) Consider again x = c\ertyr + c2est\s where r and s are both real but of opposite sign. Let r > 0 and s < 0. Suppose the eigenvectors are those as shown in figure 4.18. If a solution starts on the line 8 Notice that the solution paths tend towards the eigenvector Vs. Systems of first-order differential equations 169 through vr then c2 = 0. The solution will therefore remain on vr. Since r is positive, then over time the solution moves away from the origin, away from the fixed point. On the other hand, if the system starts on the line through Vs, then c\ = 0, and since s < 0, then as t oo the system tends towards the fixed point. For initial points off the lines through the eigenvectors, then the positive root will dominate the system. Hence for points above vr and \s, the solution path will veer towards the line through vr. The same is true for any initial point below vr and above Vs. On the other hand, an initial point below the line through \s will be dominated by the larger root and the system will veer towards minus infinity. In this case the node is called a saddle point. The line through vr is called the unstable arm, while the line through \s is called the stable arm. Saddle path equilibria are common in economics and one should look out for them in terms of real distinct roots of opposite sign and the fact that det(A) is negative. It will also be important to establish the stable and unstable arms of 170 Economic Dynamics the saddle point, which are derived from the eigenvectors associated with the characteristic roots. Because of the importance of saddle points in economics, we shall consider two examples here. Example 4.13 Let then with x = x + y y = 4x + y X "i r X ý _ 4 1_ y _ A = 1 1 4 1 det(A) = -3, A-k\ = 1 -k 1 4 1 -k giving det(A - XI) = k2 - 2k - 3 = (k - 3)(k + 1) = 0. Hence, k = r = 3 and k = s = —I. For k = r = 3 then (A - k\)yr 1 0 i.e. Let v\ For k -2v\ + vr2 = 0 4v[ -2vr2 = 0 -- 1, then vr2 = 2. Hence, one solution is u and s = — 1, then (A - kl)ys = \s = 0 i.e. Let vs, = 2vj + vs2 = 0 4v\ + 2v\ = 0 1, then v2 = —2. Hence, a second solution is u2 = est 1 -2 and 1 -2 The situation is illustrated in figure 4.19. The solution paths are revealed by the direction field. The figure quite clearly shows that the unstable arm of the saddle is the line through the eigenvector vr, while the stable arm of the saddle is the line through the eigenvector \s. Systems offirst-order differential equations 171 Example 4.14 Let x = 3x — 2y y = 2x- 2y then X "3 -2" X y 2 -2 y _ with A = 3 -2 2 -2 det(A) = -2, A - XI 3 — X -2 2 —2 — X giving det(A — XI) = X2 — X — 2 = (X — 2){X + 1) = 0. Hence, X = r = 2 and A = 5 = — 1. For A = r = 2 then (A - Xl)yr 1 -2 2 -4 v' = 0 i.e. v\ -2vr2 = 0 2v\ -4vr2=0 Let v'i = 2, then v'2 = 1. Hence, one solution is u1 = en For A = 5 = — 1, then (A - Xl)\s = and 4 -2 2 -1 vs = 0 172 Economic Dynamics Figure 4.20. i.e. 4v° 2vs2 2v\ -vs2 = 0 Let v\ = 1, then v u 0 1 2. Hence, a second solution is and = The situation is illustrated in figure 4.20. The solution paths are revealed by the direction field. The unstable arm of the saddle is the line through the eigenvector vr, while the stable arm of the saddle is the line through the eigenvector \s. Case 3 (Real equal roots) In this case k = r = s. Throughout assume the repeated root is negative. (If it is positive then the argument is identical but the movement of the system is reversed.) There are two sub-cases to consider in line with our earlier analysis: (a) independent eigenvectors, and (b) one independent eigenvector. The two situations were found to be: (a) x = c\ertyl + c2ert\2 (b) x = ciertv + c2[ertt\ + ert\2] Consider each case in turn. In example 4.9 we found for two independent eigenvectors x(t) = c\ert y(t) = c2ert Hence, x/y = c\/c2 is independent of t and depends only on the components of vr and Vs and the arbitrary constants c\ and c2. This is a general result and so all solutions lie on straight lines through the origin, as shown in figure 4.21. In this case the origin is a proper node that is stable. Had the repeated root been positive, then we would have an unstable proper node. It is this situation we gave an example Systems of first-order differential equations 173 Figure 4.21. x Figure 4.22. \ 3» \ \ 3 T * f f t / / y N \ ■s » f f f. / y V N \ \ t f f / /■ y y y \ \ -X ! '/ 5* y y yr X \ t !* / .- is X \ % ! f / ,^ \ v n > ,xr ___v \ V .---^ .,--P T y -r '* .......V. .---* ■«}---- -tf------ *......N __» ""1 V- - .....- ......- 1 . -S------->»—' -<<*■■■ or"' "j *>' ^-HT-*-f* ^ —* \ \ '"'-A "'—i» —* if' ^, f ^1 .>-- if"' ■/ i \ \ X is"' * f ! i, X \ \ "'ii "v. ft' £/ . / * 5? \ \ "'-vi J • j i \ \ x ■ \ 'X V / l "i *■ 7 I t \ h \ \ \ ^ / / /' -a -3 ^ \ \ of at the beginning of section 4.6. The direction field, along with the independent vectors is shown in figure 4.22 for this example. For the second sub-case, where again r < 0, for large t the dominant term must be C2ertt\, and hence as t —► oo every trajectory must approach the origin and in such a manner that it is tangent to the line through the eigenvector v. Certainly, if C2 = 0 then the solution must lie on the line through the eigenvector v, and approaches the origin along this line, as shown in figure 4.23. (Had r > 0, then every trajectory would have moved away from the origin.) The approach of the trajectories to the origin depends on the eigenvectors v and v2. One possibility is illustrated in figure 4.23. To see what is happening, express 174 Economic Dynamics Figure 4.23. y / the general solution as x = [cierty + c2eny2 + c2ertt\] = [(civ + c2y2) + c2t\]ert = uert Then u = (civ + c2\2) + c2ty which is a vector equation of a straight line which passes through the point c\\ + c2\2 and is parallel to v. Two such points are illustrated in figure 4.23, one at point a (c2 > 0) and one at point b (c2 < 0). We shall not go further into the mathematics of such a node here. What we can do, however, is highlight the variety of solution paths by means of two numerical examples. The first, in figure 4.24, has the orientation of the trajectories as illustrated in figure 4.23, while figure 4.25 has the reverse orientation. Whatever the orientation, the critical point is again an improper node that is stable. Had r > 0, then the critical point would be an improper node that is unstable. Case 4 (Complex roots, a ^ 0 and f5 > 0) In this case we assume the roots X = r and X = s are complex conjugate and with r = a + /3i and s = a — /3i, and a ^ 0 and fi > 0. Systems having such complex roots can be expressed x = ax + fy y = — fix + ay Systems of first-order differential equations 175 x- -Ax-y y=x-2y Figure 4.24. \ 1; „2 -1 ° -2 K 1 \ \ \ 2 j)= -x-5y Figure 4.25. X ' a ß~ X y _ -ß a_ y _ or Now express the system in terms of polar coordinates with R and 9, where R2 = x2 + y2 and tan 9 = - x and R = aR which results in R = ceat where c is a constant Similarly e = -e giving 9 = -fit + 90 where 9(0) = 90 What we have here are parametric equations R = ce°" 9 = -Bt + 90 176 Economic Dynamics Figure 4.26. (a) x= -x+4y y= -4x-y (b) x=x+4y y= ~4x+y , y 40 20 -20 ■ß X -40 in polar coordinates of the original system. Since ^ > 0 then 9 decreases over time, and so the motion is clockwise. Furthermore, as t oo then either R 0 if a < 0 or R oo if a > 0. Consequently, the trajectories spiral either towards the origin or away from the origin depending on the value of a. The two possibilities are illustrated in figure 4.26. The critical point in such situations is called a spiral point. Case 5 (Complex roots, a = 0 and p> > 0) In this case we assume the roots X = r and X = s are complex conjugate with r = pi and s = — fii (i.e. a = 0). In line with the analysis in case 4, this means X ' 0 ß' X y _ -ß o_ y_ resulting in R = 0 and 9 = giving R = c and 9 = — fit + 9o, where c and 9q are constants. This means that the trajectories are closed curves (circles or ellipses) with centre at the origin. If ^ > 0 the movement is clockwise while if ^ < 0 the movement is anticlockwise. A complete circuit around the origin denotes the phase Systems of first-order differential equations 111 Figure 4.27. -4 of the cycle, which is 2jt//0. The critical point is called the centre. These situations are illustrated in figure 4.27. Summary From the five cases discussed we arrive at a number of observations. 1. After a sufficient time interval, the trajectory of the system tends towards three types of behaviour: (i) the trajectory approaches infinity (ii) the trajectory approaches the critical point (iii) the trajectory traverses a closed curve surrounding the critical point. 2. Through each point (xq, yo) in the phase plane there is only one trajectory. 3. Considering the set of all trajectories, then three possibilities arise: (i) All trajectories approach the critical point. This occurs when (a) tr(A)2 > 4det(A), r < s < 0 (b) tr(A)2 < 4det(A), r = a + /3i, s = a — /3i and a < 0. 178 Economic Dynamics (ii) All trajectories remain bounded but do not approach the critical point as t oo. This occurs when tr(A)2 < 4det(A) and r = /3i and s = -pi(a = 0). (iii) At least one of the trajectories tends to infinity as t oo. This occurs when (a) tr(A)2 > 4det(A), r > 0 and s > 0 or r < 0 and s > 0 (b) tr(A)2 < 4det(A), r = a + fii, s = a — fii and a > 0. 4.9 Stability/instability and its matrix specification Having outlined the methods of solution for linear systems of homogeneous autonomous equations, it is quite clear that the characteristic roots play an important part in these. Here we shall continue to pursue just the two-variable cases. For the system x = ax + by y = cx + dy where b d and A — XI a — X b c d — X we have already shown that a unique critical point exists if A is nonsingular, i.e., det(A) ^ 0 and that (4.26) r, s = Furthermore, if: tr(A) ± Vtr(A)2 - 4det(A) (i) (ii) (iii) tr(A)2 > 4det(A) the roots are real and distinct tr(A)2 = 4det(A) the roots are real and equal tr(A)2 < 4det(A) the roots are complex conjugate. This leads to our first distinction. To illustrate the variety of solutions we plot the tr(A) on the horizontal axis and the det(A) on the vertical, which is valid because these are scalars. The plane is then divided by plotting the curve tr(A)2 = 4det(A) (i.e. x2 = Ay), which is a parabola with minimum at the origin, as shown in figure 4.28. Below the curve tr(A)2 > 4det(A) and so the roots are real and distinct; above the curve the roots are complex conjugate; while along the curve the roots are real and equal. We can further sub-divide the situations according to the sign/value of the two roots. Take first real distinct roots that lie strictly below the curve. If both roots are negative then the tr(A) must be negative, and since det(A) is positive, then we are in the region below the curve and above the ^-axis, labelled region I in figure 4.28. In this region the critical point is asymptotically stable. In region II, which is also below the curve and above the ^-axis, both roots are positive and the system is unstable. Systems of first-order differential equations 179 det(A) Figure 4.28. tr(A)=4det(A) Asymptotically stable Unstable 0 Uns1 able, saddk point III tr(A) If both roots are opposite in sign, we have found that the det(A) is negative and the critical point is a saddle. Hence, below the *-axis, marked region III, the critical point is an unstable saddle point. Notice that this applies whether the trace is positive or negative. The complex region is sub-divided into three categories. In region IV the sign of a in the complex conjugate roots a ± pi is strictly negative and the spiral trajectory tends towards the critical point in the limit. In region V a is strictly positive and the critical point is an unstable one with the trajectory spiralling away from it. Finally in region VI, which is the y-axis above zero, a = 0 and the critical point has a centre with a closed curve as a trajectory. It is apparent that the variety of possibilities can be described according to the tr(A) and det(A) along with the characteristic roots of A. The list with various nomenclature is given in table 4.1. 4.10 Limit cycles9 A limit cycle is an isolated closed integral curve, which is also called an orbit. A limit cycle is asymptotically stable if all the nearby cycles tend to the closed orbit from both sides. It is unstable if the nearby cycles move away from the closed orbit on either side. It is semi-stable if the nearby cycles move towards the closed orbit on one side and away from it on the other. Since the limiting trajectory is a periodic orbit rather than a fixed point, then the stability or instability is called an orbital stability or instability. There is yet another case, common 9 This section utilises the VisualDSolve package provided by Schwalbe and Wagon (1996). It can be loaded into Mathematica with the Needs command. This package provides considerable visual control over the display of phase portraits. 180 Economic Dynamics Table 4.1 Stability properties of linear systems Matrix and eigenvalues T^pe of point T^pe of stability tr(A) < 0, det(A) > 0, tr(A)2 > 4det(A) Improper node Asymptotically stable r < s < 0 tr(A) > 0, det(A) > 0, tr(A)2 > 4det(A) Improper node Unstable r > s > 0 det(A) < 0 Saddle point Unstable saddle r>0, 5*<0orr<0, 5*>0 tr(A) < 0, det(A) > 0, tr(A)2 = 4det(A) Star node or proper node Stable r — s < 0 tr(A) > 0, det(A) > 0, tr(A)2 = 4det(A) Star node or proper node Unstable r — s > 0 tr(A) < 0, det(A) > 0, tr(A)2 < 4det(A) Spiral node Asymptotically stable r = a + ßi, s — a — ßi, a < 0 tr(A) > 0, det(A) > 0, tr(A)2 < 4det(A) Spiral node Unstable r = a + ßi, s — a — ßi, a > 0 tr(A) = 0, det(A) > 0 Centre Stable r = ßi, s = —ßi in predatory-prey population models. If a system has closed orbits that other trajectories neither approach nor diverge from, then the closed orbits are said to be stable. Geometrically, we have a series of concentric orbits, each one denoting a closed trajectory. In answering the question: 'When do limit cycles occur?' we draw on the Poincare-Bendixson theorem. This theorem is concerned with a bounded region, which we shall call R, in which the long-term motion of a two-dimensional system is limited to it. If for region R, any trajectory starting within R stays within R for all time, then two possibilities arise: (1) the trajectory approaches a fixed point of the system as t oo; or (2) the trajectory approaches a limit cycle as t oo. When trajectories that start in R remain in R for all time, then the region R is said to be the invariant set for the system. Trajectories cannot escape such a set. The following points about limit cycles are worth noting. (1) Limit cycles are periodic motions and so the system must involve complex roots. (2) For a stable limit cycle, the interior nearby paths must diverge from the singular point (the fixed point). This occurs if the trace of the Jacobian of the system is positive. (3) For a stable limit cycle, the outer nearby paths must converge on the closed orbit, which requires a negative trace. (4) Points (2) and (3) mean that for a stable limit cycle the trace must change sign in the region where the limit cycle occurs. (5) The Poincare-Bendixson theorem holds only for two-dimensional spaces. (6) If the Poincare-Bendixson theorem is satisfied, then it can be shown that if there is more than one limit cycle they alternate between being stable and unstable. Furthermore, the outermost one and the innermost one must Systems of first-order differential equations 181 be stable. This means that if there is only one limit cycle satisfying the theorem, it must be stable. Example 4.15 The following well-known example has a limit cycle composed of the unit circle (see Boyce and DiPrima 1997, pp. 523-7): x' = y + x — x(x2 + y2) y' = -x + y - y(x2 + y2) Utilising the VisualDSolve package within Mathematica, we can show the limit cycle and two trajectories: one starting at point (0.5,0.5) and the other at point (1.5,1.5). The input instructions are: In [2] := PhasePlot [{x7 [t] ==y[t] + x[t] - x[t] (x[t]~2 + y[t]~2), y' [t] == -x[t] + y[t] - y[t] (x[t]~2 + y[t]^2)}, {x[t], y[t]}, {t, 0, 10}, {x, -2, 2}, {y, -2, 2}, InitialValues -> {{0.5, 0.5}, {1.5, 1.5}}, ShowInitialValues -> True, FlowField -> False, FieldLength -> 1.5, FieldMeshSize -> 25, WindowShade -> White, FieldColor -> Black, Nullclines -> True, PlotStyle -> AbsoluteThickness [1.2], InitialPointStyle -> AbsolutePointSize [3], ShowEguilibria -> True, DirectionArrow -> True, AspectRatio -> 1, AxesLabel -> {x, y}, PlotLabel -> "Unit Limit Cycle"]; which produces figure 4.29 showing a unit limit cycle. Example 4.16 (Van der Pol equation) The Van der Pol equation is a good example illustrating an asymptotically stable limit cycle. It also illustrates that a second-order differential equation can be reduced to a system of first-order differential equations that are more convenient for Unit limit cycle Figure 4.29. y 2 1 0 -1 x -10 12 182 Economic Dynamics Figure 4.30. Van der Pol equation solving. The Van der Pol equation takes the form: (4.27) x - /x(l - x2)x + x = 0 Let y = x, then y = x, so we have the two equations, (4.28) (4.29) x = y y = /x(l — x2)y — x To illustrate the limit cycle, let \x = 1. The phase portrait that results is shown in figure 4.30. Here we take two initial points: (a) point (0.5,0.5), which starts inside the limit cycle; and (b) point (1.5,4), which begins outside the limit cycle. Example 4.17 Walrasian price and quantity adjustment and limit cycles The presence of limit cycles is illustrated by a Walrasian model which includes both price and quantity adjustments (see Flaschel et al. 1997 and Mas-Colell 1986). Let 7 denote output of a one good economy and L labour input. Y = f(L) is a production function which is twice differentiable and invertible with L = f~l(Y) = (p(Y) and (p'(Y) > 0. In equilibrium the price, p, is equal to marginal wage cost, where marginal wage cost is also given by (p'(Y). Thus, p* = (p'(Y). For simplicity we assume that the marginal wage cost is a linear function of Y, with (p'(Y) = c\ + c{Y. Aggregate demand takes the form D(p, L) and in equilibrium is equal to supply, i.e., D[p*, (p(Y*)] = Y*. Finally, we have both a price and a quantity adjustment: p = a[D(p, 0(F)) - Y] a > 0 Y = 0[p-'(¥)] P>0 These establish two differential equations in p and Y. Consider the following numerical example. Let 0'(7) = 0.87 + 0.57 D{p) = -0.02p3 + 0.8p2 - 9p + 50 then (p*, Y*) = (13, 24.26) with isoclines: p = 0 Y = -0.02p3 + O.Sp2 -9p + 50 7 = 0 p = 0.87 + 0.57 or 7=1.74 + 2;? Figure 4.31 reproduces the figures derived in Flaschel et al. 1997 using Mathematical for a = 1 and different values of the parameter /3. Not only do the figures Systems of first-order differential equations 183 beta=2 beta=2.5 Figure 4.31. 5 10 15 20 2 beta=3 5 10 15 20 25P beta=3.2 10 15 20 25* 5 10 15 20 25* illustrate a stable limit cycle, but they also illustrate that the limit cycle shrinks as f> increases. 4.11 Euler's approximation and differential equations on a spreadsheet10 Although differential equations are for continuous time, if our main interest is the trajectory of a system over time, sometimes it is convenient to use a spreadsheet to do this. To accomplish this task we employ Euler's approximation. For a single variable the situation is shown in figure 4.32. We have the differential equation dx -r =f(x, t) dt x(t0) = Xq Let x = (to)-We also know dx/dt at to, which is simply f(xo, to). If we knew x = (p(t), then the value at time t\ would be 0(t\). But if we do not have an explicit form for x = (p(t), we can still plot (p(t) by noting that at time to the slope at point P isf(xo, to), which is given by the differential equation. The value of x\ at time t\ (point R) is given by x\ = x0 +f(xo, t0)At At = h- t0 This process can be repeated for as many steps as one wishes. Iff is autonomous, so dx/dt = f(x), then xn = Xn-i +/(*„_! )Af It is clear from figure 4.32 that point R will deviate from its 'true' value at point Q, the larger the step size, given by At. If the step size is reduced, then the approximation is better. (4.30) (4.31) 10 See Shone (2001) for a treatment of differential equations with spreadsheets. 184 Economic Dynamics Figure 4.32. *='»\ Hi™™ ■ j/ L.i) / - 76.96298 ,4 A Si-.nl/ 6,445106 13.14782 2.916051 ill .. .. . i >\r (4.34) Example 4.19 (The Lorenz curve) The Lorenz equations are given by: dx —- = a(y- x) dt dry dt rx — y — xz dz — = xy - bz dt with parameter values a = 10, r = 28, b = 8/3 and we take a step size of At = 0.01. In this example we take 2,000 steps, however figure 4.34 only shows the first few steps. In figure 4.35 we have three generated plots, (i) (x,y), (ii) (x, z) and (iii) (y,z). These diagrams illustrate what is referred to as strange attractors, a topic we shall return to when we discuss chaos theory. 4.12 Solving systems of differential equations with Mathematica and Maple Chapter 2 sections 2.11 and 2.12 outlined how to utilise Mathematica and Maple to solve single differential equations. The method for solving systems of such equations is fundamentally the same. Consider the system, dx -r =f(x,y, t) (4.35) at dy -r = g(x, y, t) Systems of first-order differential equations 187 Figure 4.35. ¥(t) Lorenz Curvi 60 -. -25 -20 -15 -10 -5 0 5 10 15 20 35 Lorenz cutva 60 i -30 -20 -10 0 10 20 30 40 ylü Then the solution method in each case is: Mathematica DSolve[{x' [t]== f [x[t] ,y[t] ,t] ,y' [t]==g[x[t] , y[t] ,t] }, {x[t] ,y[t] },t] Maple dsolve ({diff(x(t),t)=f(x(t),y(t),t),diff(y(t),t)=g(x(t), y(t) ,t) }, {x(t) ,y(t) }) ; If initial conditions ^(0) = xO and y(0) = yO are provided, then the input instructions are: 188 Economic Dynamics Mathematica DSolve[{x' [t]== f [x[t] ,y[t] ,t] ,y' [t]==g[x[t] ,y[t] ,t] , x(0)==xO,y(0)==yO},{x[t],y[t]},t] Maple dsolve(({diff(x(t),t)=f(x(t),y(t),t),diff(y(t),t) =g(x(t) ,y(t) ,t) , x(0)=xO,y (0)=y0}, {x(t) ,y (t) }) ; It is often easier to define the equations, variables and initial conditions first. Not only is it easier to see, but much easier in correcting any mistakes. For instance in Mathematica define eq: = {x' [t]==f[x[t],y[t],t],y' [t]==g[x [t],y[t], t], x[0]==xO,y[0]==y0} var: = {x[t],y [t] } and then solve using DSolve[eq,var,t] In Maple define: eq:=diff (x(t) ,t)=f (x(t) ,y (t) ,t) ,diff (y (t) ,t) = g(x(t) ,y (t) ,t) ; init:=x(0)=x0,y(0)=y0; var:={x (t),y (t) }; and then solve using dsolve ({eq,init}, var) ; Example 4.4 in the text can then be solved in each package as follows Mathematica eq: = {x' [t]==x[t]-3y[t], y' [t] x[0]== 4,y [0]== 5} var:={x[t],y[t]} DSolve[eq,var,t] Maple eq:=diff(x(t),t)=x(t)-3*y(t), -2*x(t)+y (t) ; init:=x(0)=4,y(0)=5; var:={x (t),y (t) }; dsolve({eq,init},var); Although the output looks different in the two cases, they are equivalent and identical to that provided in the text. ==-2x[t]+y[t], diff (y(t) ,t) = Systems of first-order differential equations 189 So long as solutions exist, then the packages will solve the system of equations. Thus, the system of three equations with initial values: x'(t) = x(t) y'(t) = x(t) + 3y(t) - z(t) z'(t) = 2y(t) + 3x(t) x(0)= l,y(0) = l,z(0) = 2 can be solved in a similar manner with no difficulty. In the case of nonlinear systems of differential equations, or where no explicit solution can be found, then it is possible to use the NDSolve command in Math-ematica and the dsolve(..., numeric) command in Maple to obtain numerical approximations to the solutions. These can then be plotted. But often more information can be obtained from direction field diagrams and phase portraits. A direction field shows a series of small arrows that are tangent vectors to solutions of the system of differential equations. These highlight possible fixed points and most especially the flow of the system over the plane. A phase portrait, on the other hand, is a sample of trajectories (solution curves) for a given system. Figure 4.36(a) shows a direction field and figure 4.36(b) a phase portrait. In many instances direction fields and phase portraits are combined on the one diagram - as we have done in many diagrams in this chapter. The phase portrait can be derived by solving a system of differential equations, if a solution exists. Where no known solution exists, trajectories can be obtained by using numerical (4.36) \ \ \ MM M M * M M M \ \ \ \ MM k k V k * M M M \ \ 1 s \ \ \ t k k k k k k k k k M M \ \ » k k L k k k k k k k k k k * M \ t * k k k k k k k k k k k k k k k k i> t k k k 1 i k k k k k k k k k k k k k » * k k k k k k k k k k k k k k k k t k k , k k k k k k k k k k ' k k k k k k 1 k k k k k k k k k k k k k k i i ' ' " k k k t k k 1 i k j i i i i i k i k » ' i:: \\\\ 4 4 4 4 4 r * * 4 4 4 » ' '' \ 4 4 4 4 * * r r * r * * 4 4 4 t * ***444tt **44444* -2 -1 0 1 2 y Figure 4.36. 190 Economic Dynamics solutions. These are invariably employed for systems of nonlinear differential equation systems. 4.12.1 Direction fields and phase portraits with Mathematica Direction fields in Mathematica are obtained using the PlotVectorField command. In order to use this command it is first necessary to load the PlotField package. There is some skill required in getting the best display of direction fields using the PlotVectorField command, and the reader should consult the references supplied on using Mathematica in chapter 1. Given the system of differential equations (4.35), then a direction field can be obtained with the instructions Needs [ "" Graphics "PlotField" " ] dfield=PlotVectorField[{f(x,y,t),g(x,y,t)}, {x,xmin,xmax}, {y,ymin,ymax}, DisplayFunction->Identity] Show[dfield, DisplayFunction->$DisplayFunction] To obtain a 'good' display it is often necessary to adjust scaling, change the arrow lengths and change the aspect ratio. All these, and other refinements, are accomplished by optional instructions. Thus, figure 4.36(a) can be obtained from the following input Needs ["" Graphics"PlotField" " ] dfield=PlotVectorField[{1-y,x2+y2}, {x,-2,2}, {y,-1,3}, Frame->True, PlotPoints->20, DisplayFunction->Identity] Show[dfield, DisplayFunction->$DisplayFunction] The phase portrait is not straightforward in Mathematica and requires solving the differential equations, either with DSolve command, if an explicit solution can be found, or the NDSolve command for a numerical approximation. If an explicit solution can be found with the DSolve command, then phase portraits can be obtained with the ParametricPlot command on supplying different values for the constants of integration. On the other hand, if a numerical approximation is required, as is often the case with nonlinear systems, then it is necessary to obtain a series of solution curves for different initial conditions. In doing this quite a few other commands of Mathematica are needed. Consider the Van der Pol model, equation (4.28), a simple set of instructions to produce a diagram similar to that of figure 4.30 is eql:= {x' [t]==y[t],y' [t]== (1-x[t]A2)y[t]-x[t], x[0]==0.5,y[0]==0.5} eq2:= {x' [t]==y[t],y' [t]== (1-x[t]A2)y[t]-x[t], x[0]==0.5,y [0]==4} var:={x,y} trange:={t,0,20} Systems of first-order differential equations 191 soll=NDSolve[eql,var,trange] sol2=NDSolve[eq2,var, trange] graphl=ParametricPlot[Evaluate[{x[t],y[t]} /.soli], {t,0,2 0},PlotPoints->50 0, DisplayFunction-Mdentity] ; graph2=ParametricPlot[Evaluate[{x[t],y[t]} /.sol2], {t,0,2 0},PlotPoints->50 0, DisplayFunction-Mdentity] ; Show[{graph1,graph2},AxesLabel->{" x" ," y " }, DisplayFunction->$DisplayFunction] ; The more trajectories that are required the more cumbersome these instructions become. It is then that available packages, such as the one provided by Schwalbe and Wagon (1996), become useful. For instance, figure 4.36(b) can be produced using the programme provided by Schwalbe and Wagon with the following set of instructions: PhasePlot[{x'[t]==l-y[t],y'[t]==x[t]A2+y[t],A2}, {x[t],y[t]},{t,0,3},{x,-2,2},{y,-l,3}, InitialValues->{{-2,-1},{-1.75,-1},{-1.5,-1}, {-1,0},{-1,-1},{-0.5,-1},{0,-1},{-1.25,0}, {0.5,-1},{1,-1}}, PlotPoints->500, ShowInitialValues->False, DirectionArrows->False, AspectRatio->l, AxesLabel->{x,y}] When considering just one trajectory in the phase plane, the simple instructions given above can suffice. For instance, consider the Lorenz curve, given in equation (4.34), with parametric values a = 10, r = 28, and b = S/3. We can construct a three-dimensional trajectory from the initial point (*0, yO, zO) = (5,0, 0) using the following input instructions: eqs: = {x' [t]==10(y [t]-x[t] ),y' [t]==28x[t]-y [t]-x[t] z [t], z' [t]==x[t]y[t]-(8/3)z[t], x[0]==5,y[0]==0,z [0]==0} var:={x,y,z} lorenzsol=NDSolve[eqs,var,{t,0,3 0},MaxSteps->3000] lorenzgraph=ParametricPlot3D[ Evaluate[x[t],y[t],z[t]} /.lorenzsol], {t,0,30},PlotPoints->2 0 0 0,PlotRange->All]; The resulting phase line is shown in figure 4.37. This goes beyond the possibilities of a spreadsheet, and figure 4.37 should be compared with the three two-dimensional plots given in figure 4.35. 192 Economic Dynamics Figure 4.37. 4.12.2 Direction fields and phase portraits with Maple Direction fields and phase portraits are more straightforward in Maple and use the same basic input commands. Given the system of differential equations (4.35), then a direction field can be obtained with the instructions with(DEtools): with(plots): Dfield:=dfieldplot( [diff (f (x(t) ,t)=f (x(t) ,y (t) ,t) , diff (y (t) ,t)=g(x(t) ,y(t) ,t) ] , [x(t),y(t)], t=tmin..tmax, x=xmin..xmax, y=ymin..ymax); display(Dfield); Notice that the instruction 'with(plots):' is required for use of the display command. To obtain a 'good' display it is often necessary to add options with respect to arrows. For example, a Maple version of figure 4.36(a) can be achieved with the following input with(DEtools): with(plots): Dfield:=dfieldplot( [diff(x(t),t)=l-y(t),diff(y(t),t)=x(t)A2 +Y(t)A2], [x(t),y(t)], t=0..1, x=-2..2, y=-1..3, arrows=SLIM): display(Dfield); The phase portrait, not surprisingly, uses the phaseportrait command of Maple. This particular command plots solution curves by means of numerical methods. Systems of first-order differential equations 193 In a two-equation system, the programme will produce a direction field plot by default if the system is a set of autonomous equations. Since we require only the solution curves, then we include an option that indicates no arrows. To illustrate the points just made, consider the Van der Pol model, equation (4.28), a simple set of instructions to produce a Maple plot similar to figure 4.30 is phaseportrait( [D(x) (t)=y(t), D(y) (t) = (l-x(t) A2) *y (t) -x(t) ] , [x(t),y(t)], t=0..10, [ [x(0)=0.5,y(0)=0.5],[x(0)=0.5,y(0)=4] ], stepsize=.05 linecolour=blue, arrows=none, thicknes s = l); Producing more solution curves in Maple is just a simple case of specifying more initial conditions. For instance, a Maple version of figure 4.36(b) can be produced with the following instructions: with (DEtools) : phaseportrait( [D(x) (t)=l-y(t) ,D(y) (t)=x(t) A2+y(t) A2] , [x(t),y(t)], t=0..3, [ [x(0)=-2,y (0)=-l] , [x(0)=-1.75,y (0)=-l] , [x(0)=1.5,y (0)=-l] , [x(0)=-l,y (0)=0] , [x(0)=-l,y (0)=-l] , [x(0)=-0.5,y (0)=-l] , [x(0)=0,y (0)=-l] , [x(0)=-1.25,y (0)=0] , [x(0)=0.5,y (0)=-l, [x(0)=l,y (0)=-l] ] , x=-2..2, y=-l. . 3, stepsize=.05, linecolour=blue, arrows=none, thicknes s = l); Trajectories for three-dimensional plots are also possible with Maple. Consider once again the Lorenz curve, given in equation (4.34), with parameter values a = 10, r = 28and£> = 8/3. We can construct a three-dimensional trajectory from the initial point (^0, yO, z0) = (5,0, 0) using the following input instructions: with (DEtools) : DEplot3d( [diff(x(t),t)=10*(y(t)-x(t) ) , diff (y (t) ,t)=28*x(t)-y (t)-x(t) *z (t) , diff (z (t) ,t)=x(t) *y(t)-(8/3) *z (t) ] , [x(t) ,y(t) , z (t) ] , t=0..30, [ [x(0)=5,y (0)=0, z (0)=0] ] , stepsize=.01, linecolour=BLACK, thicknes s = l); 194 Economic Dynamics Figure 4.38. The resulting phase line is shown in figure 4.38. This goes beyond the possibilities of a spreadsheet, and figure 4.38 should be compared with the three two-dimensional plots given in figure 4.35. It is worth noting that figure 4.38 is the default plot and the orientation can readily be changed by clicking on the figure and revolving. Appendix 4.1 Parametric plots in the phase plane: continuous variables A trajectory or orbit is the path of points {x(t), y(t)} in 2-dimensional space and {x(t), y(t), z(t)} in 3-dimensional space as t varies. Such plots are simply parametric plots as far as computer programmes are concerned. There are two methods for deriving the points (x(t), y(t)) or (x(t), y(t), z(t))\ (1) Solve for these values (2) Derive numerical values by numerical means: (a) by solving numerically, or (b) deriving by recursion. Method 2(a) is used particularly in the case of differential equations, while method 2(b) is used for difference (or recursive) equations. In each of these cases initial conditions must be supplied. 4A.1 Two-variable case Consider the solution values for x and y in example 4.1, which are x(t) = 2e2t and y(t) = 3e' Both x and y are expressed in terms of a common parameter, t, so that when t varies we can establish how x and y vary. More specifically, if t denotes time, then (x(t), y(t)) denotes a point at time nnthe(*,y)-plane, i.e., a Cartesian representation of the parametric point at time t. If the differential equation system which generated x(t) and y(t) is autonomous, then there is only one solution curve, and we can express this in the form y = (p(x), where yo = (*o) and (xq, yo) is some initial point, i.e., *(0) = xq and y(0) = yo at t = 0. In the present example this is readily Systems of first-order differential equations 195 found since x 2e2t 2e2t 2 y2 (3er)2 9e2t 9 Hence y = Whether or not it is possible to readily find a Cartesian representation of the parametric curve, it is a simple matter to plot the parametric curve itself using software packages. Example 4.1 with Mathematica The two commands used in this set of instructions, DSolve and ParametricPlot are now both contained in the main package:11 Clear[x,y] sol=DSolve [ {x'[t]==2x[t],y'[t]==y[t],x[0]==2, y[0]==3}, {x[t] ,y[t] },t] solx=sol[[1,1,2]] soly=sol[ [1,2,2] ] x [t_] :=solx y [t_] :=soly traj=ParametricPlot[{x[t],y[t]},{t,0,l}] If the equations for x(t) and y(t) are already known, then only the last instruction need be given. For example, if it is known that x(t) = 2e2t and y(t) = 3e' then all that is required is traj=ParametricPlot [ {2e2t, 3efc}, {t,0,1} ] Example 4.1 with Maple To use Maple's routine for plotting parametric equations that are solutions to differential equations it is necessary to load the plots package first. The following input instructions will produce the trajectory for example 4.1: restart; with(plots): sys:={diff(x(t),t)=2*x(t),diff(y(t),t)=y(t), x(0)=2,y (0)=3} vars:={x(t),y(t)}: sol:=dsolve(sys,vars,numeric); odeplot (sol, [x (t) ,y(t) ] ,0. .l,labels=[x,y] ) ; 11 In earlier versions, DSolve and ParametricPlot needed to be loaded first since these were contained in the additional packages. This is no longer necessary, since both are contained in the basic built in functions. 196 Economic Dynamics Notice that we have placed a semi-colon after the 'sol' instruction so that you can observe that Maple produces a procedural output, which is then used in the odeplot. If the equations for x(t) and y(t) are already known, then the plot command can be used. For example, if it is known that x(t) = 2e2t and y(t) = 3e* then all that is required is plot ( [2*exp (2*t) , 3*exp (t) , t=0 . . 1] , labels= [x, y] ) ; 4A.2 Three-variable case Plotting trajectories in 3-dimensional phase space is fundamentally the same, with just a few changes to the commands used. Equation (4.36) with Mathematica The input instructions are Clear[x,y,z] sol=DSolve[{x'[t]==x[t],y'[t]==x[t]+3y[t]-z[t], z [t]==2y [t]+3x[t] ,x[0]== l,y [0]==1, z [0]— 2}, {x[t],y[t],z[t]},t] solx=sol[[1,1,2]] soly=sol[ [1,2,2] ] solz = sol[ [1,3,2] ] x [t_] :=solx y [t_] :=soly z [t_] := solz traj=ParametricPlot3D[{x[t],y[t],z[t]},{t,0,5}] If the equations for x(t), y(t) and z(t) are already known, then only the last instruction need be given. For example, if it is known that x(t) = e',y(t) = 2e' — e2t + 2te' and z(t) = Ate1 — e2t + 3e* then all that is required is traj=ParametricPlot3D [ {efc, 2et-e2t + 2tet, 4tet-e2t + 3et}, {t,0,5}] Equation (4.36) with Maple The input instructions are restart; with(plots) : sys: = {diff(x(t),t)=x(t),diff(y(t),t)=x(t)+3*y(t)-z(t), diff (z (t) ,t)=2*y(t)+3*x(t) ,x(0)=l,y (0)=1, z (0)=2}; vars:={x(t),y(t),z(t)}: sol:=dsolve(sys,vars,numeric); odeplot(sol, [x (t),y(t),z(t)],0..5,labels=[x,y,z] ); Systems of first-order differential equations 197 If the equations for x(t), y(t) and z(t) are already known, then we use the spacecurve command, as illustrated in the following instructions: traj=spacecurve([exp(t),2*exp(t)-exp(2*t)+2*t*exp(t), 4*t*exp (t)-exp (2*t)+3*exp (t)], t = 0 ..5,labels=[x,y,z]); Exercises 1. (i) Show that y« = i is a separable function, and solve assuming ^(0) = 2 and y(0) = 3. (ii) Verify your result using either Mathematica or Maple. 2. For the system x = x — 3y y = —2x + y use a software package to derive the trajectories of the system for the following initial values: (a) (xq, yo) = (4, 2) (b) (xo,y0) = (4,5) (c) (x0, y0) = (-4, -2) (d) (xo,yo) = (-4,5) 3. For the system x = —3x + y y = x-3y (i) Show that points (^o, yo) = (4, 8) and (^o, yo) = (4, 2) remain in quadrant I, as in figure 4.9. (ii) Show that points (^0, yo) = (—4, —8) and (^0, yo) = (—4, —2) remain in quadrant III, as in figure 4.9. (iii) Show that points (*o, yo) = (2, 10) and (xo,yo) = (—2, —10) pass from one quadrant into another before converging on equilibrium. (iv) Does the initial point (^o, yo) = (2, —5) have a trajectory which converges on the fixed point without passing into another quadrant? 4. For the system x = -2x -y + 9 y = —y + x + 3 establish the trajectories for each of the following initial points (i) (x0,yo) = (1, 3), (ii) (x0,yo) = (2, 8), and (iii) (x0,yo) = (3, 1), showing that all trajectories follow a counter-clockwise spiral towards the fixed point. 5. Given the dynamic system x = 2x + 3y y = 3x + 2y 198 Economic Dynamics (i) Show that the characteristic roots of the system are r = 5 and s = -1. (ii) Derive the eigenvectors associated with the eigenvalues obtained in (i). (iii) Show that the solution values are: x(t) = c\e5t + c2e~' y(t) = c\e5t — c2e~t and verify that Cle5t are linearly independent. (iv) Given x(0) = 1 andy(O) = 0, show that "1" —/ ' 1 1 c2e -1 x(t) y(t) = \e5t - \e-f For the dynamic system x = x + 3y y = 5x + 3y Show: (i) that the two eigenvalues are r = 6 and s = —2 (ii) that the two eigenvectors are 1 5/3 and 1 -1 (iii) and that the general solution satisfying x(0) = 1 and y(0) = 3 is ¥• - i< ¥' + v Let V = [v1 v2 ] denote a matrix formed from the eigenvectors. Thus, if x(t) = le6t - \e~2t y(t) = le6t+ \e~2t then V = 1 -2 and 1 1 -2 2 The determinant of this matrix is called the Wronksian, i.e., W(v1, v2) = det(V). Then v1 and v2 are linearly independent if and only if W(v1, v2) is nonzero. Show that for the system x = x + y y = -2x + 4y the Wronksian is nonzero. Systems of first-order differential equations 199 8. Given x = x y = 2x + 3y + z z = 2y + 4z (i) Find the eigenvalues and eigenvectors. (ii) Provide the general solution. (iii) Show that the Wronksian is nonzero. 9. For each of the following systems (a) find the eigenvalues and eigenvectors; (b) solve the system by finding the general solution; (c) obtain the trajectories for the specified initial points; and (d) classify the fixed points. x = —3x + y W y = x-3y initial points = (1, 1), (-1, 1), (-1, -1), (1, -1), (2, 0), (3,1), (1,3) x = 2x-4y (H) . o y = x — 3y initial points = (1, 1), (-1, 1), (4, 1), (-4, -1), (0, 1), (0,-1), (3, 2), (-3, -2) x = y (m) . „ y = —4x initial points = (0, 1), (0, 2), (0, 3) x = -x + y (iv) . y=-x-y initial points = (1, 0), (2, 0), (3, 0), (-1, 0), (-2, 0), (-3, 0). 10. For the following Holling-Tanner predatory-prey model 1 x\ 6xy 6/ ~ (8 + 8jc) 0.4y y = 0.2y[l--- (i) Find the fixed points. (ii) Do any of the fixed points exhibit a stable limit cycle? 11. Consider the Rossler attractor x = —y — z y = x + 0.2y z = 0.2 + z(x - 2.5) (i) Show that this system has a period-one limit cycle. (ii) Plot x(t) against t = 200 to 300, and hence show that the system settles down with x having two distinct amplitudes. 200 Economic Dynamics 12. Consider the following Walrasian price and quantity adjustment model <(>'(J) = 0.5 + 0.257 D(p) = -0.025;?3 + 0J5p2 - 6p + 40 p = 0.75[D(p, 0(F)) - Y] Y = 2[p- 0'(7)] (i) What is the economically meaningful fixed point of this system? (ii) Does this system have a stable limit cycle? 13. Reconsider the system in question 12, but let the quantity adjustment equation be given by Y = 0\p-'(¥)] Let ft = 2, 2.5, 3 and 3.2. What do you conclude about the long-run behaviour of this system? 14. Consider the following system <(>'(J) = 0.5 + 0.257 D(p) = -0.025;?3 + 0J5p2 - 6p + 40 p = a[D(p, $(¥))-¥] Y = 2[p- $'(¥)] Let a = 0.5, 0.75 and 1. What do you conclude about the long-run behaviour of this system? 15. Set up the Rossler attractor x = —y — z y = x + ay z = b + zix — c) on a spreadsheet with step size At = 0.01 and a = 0.4, b = 2 and c = 4. Plot the system for initial point (x,y,z) = (0.1, 0.1, 0.1) in (i) (*,;y)-plane (ii) (*,z)-plane (iii) (v,z)-plane Additional reading Additional material on the contents of this chapter can be obtained from Arrow-smith and Place (1992), Beavis and Dobbs (1990), Borrelli etal. (1992), Boyce and DiPrima (1997), Braun (1983), Chiang (1984), Flaschel et al. (1997), Giordano and Weir (1991), Jeffrey (1990), Lynch (2001), Mas-Colell (1986), Percival and Richards (1982), Schwalbe and Wagon (1996), Shone (2001) and Tu (1994). CHAPTER 5 Discrete systems of equations 5.1 Introduction In chapter 3 we considered linear difference equations for a single variable, such as xt = 2x, t-i, xt = Axt-i + Axt-2, xt = axt-i + b Each of these equations is linear and autonomous. But suppose we are interested in such systems as the following: (i) xt = axt-\ + by t-i yt = cxt-i + dyt-\ (ii) xt = Axt-i + 2 yt = -2y,-i - 3xt-i + 3 (iii) xt = 2xt-i + 3yt-i + Azt-i yt = Q.5xt-\ zt = 0Jyt-\ All these are examples of systems of linear autonomous equations of the first order. As in previous chapters, we shall here consider only autonomous equations (i.e. independent of the variable t), but we shall also largely restrict ourselves to linear systems. If all the equations in the system are linear and homogeneous, then we have a linear homogeneous system. If the system is a set of linear equations and at least one equation is nonhomogeneous, then we have a linear nonhomogeneous system. If the equations are homogeneous but at least one equation in the system is nonlinear, then we have a nonlinear homogeneous system. If at least one equation is nonlinear and at least one equation in the system is nonhomogeneous, then we have a nonlinear nonhomogeneous system. In this chapter we shall concentrate on linear homogeneous equation systems. In terms of the classification just given, systems (i) and (iii) are linear homogeneous systems, while (ii) is a linear nonhomogeneous system. A more convenient way to express linear systems is in matrix form. Hence the three systems can equally be written in the form: (i) xt a b xt-\ yt. c d (or ut = Au,_i) 202 Economic Dynamics (ü) (iü) yt Zt "4 0 " Xt-l + "0" — _-3 -2_ _3_ Jt-i _ 2 3 4 0.5 0 0 0 0.7 0 Xt-l yt-i Zt-l (or ut = Au,_i + b) (oru, = Au,_i) In general, therefore, we can write first-order linear homogeneous systems as (5.1) ut = Au,_i and a first-order linear nonhomogeneous system as (5.2) ut = Aut-x + b where u is a n x 1 vector, Aanxn square matrix and basx 1 vector. Consider the system xt a b~ Xt-l yt. c d_ .yt-i. Then in equilibrium xt = xt-\ = x* for all t and yt = yt-\ = y* for all t. Hence * X a b~ * X c d_ or u* = Au* An equilibrium solution exists, therefore, if u* - Au* = 0 i.e. (I - A)u* = 0 or u* = (I - A)"1*) = 0 An equilibrium for a first-order linear homogeneous system is, therefore, u* = 0. This is a general result. For a first-order linear nonhomogeneous system ut = Au,_i + b an equilibrium requires ut = ut-\ = u* for all t, so that u* = Au* + b (I - A)u* = b u* = (I - A)_1b and so an equilibrium exists so long as (I — A)-1 exists. The solution u* = (I — A)_1b is the general equilibrium solution for a first-order linear nonhomogeneous system. Discrete systems of equations 203 Example 5.1 xt = 2xt-\ + 3yt-i yt = -2xt-i +yt-i or xt " 2 3" Xt-l .yt. -2 1_ i.e. ut = Au t-i where I-A = -1 -3 2 0 "-1 -3" ~x*~ "0" _ 2 0 0_ and (I - A)u* = the only values for x and y satisfying this system are x* = 0 and y* = 0. i.e. Then Example 5.2 xt = Axt-i + 2 yt = -2yt-\ - 3xt-\ + 3 Xt "4 0 " xt-\ + "2" — _-3 -2_ _3_ yt. yt-i. = (I -A)_1b "-3 0" -l "2" "-2/3" . 3 3. _3_ . 5/3 . i.e.** = -2/3 andv* = 5/3. Having established that an equilibrium exists, however, our main interest is establishing the stability of such systems of equations. In establishing this we need to solve the system. This is fairly straightforward. For the first-order linear homogeneous equation system we have ut = Au,_i = A(Au,_2) = AV-2 = A2(Auf_3) = AV_3 204 Economic Dynamics with solution (5.3) ut = A'u0 where uo is the initial values of the vector u. Given uo and the matrix A, then we could compute uioo = A100uo, or any such time period. Similarly, with the first-order nonhomogeneous linear equation system we have ut = Au,_i + b = A(Au,_2 + b) + b = A2ut-2 + Ab + b = A2(Au,_3 + b) + Ab + b = A3u,-3 + A2b + Ab + b with solution (5.4) ut = A'uo + (I + A + A2 + ... + A'"1)b Although solution (5.3) and (5.4) are possible to solve with powerful computers, it is not a useful way to proceed. We require to approach the solution from a different perspective. It will be recalled from our analysis of differential equation systems in chapter 4 that a linear nonhomogeneous system can be reduced to a linear homogeneous system by considering deviations from equilibrium. Thus for ut = Au,_i + b, with equilibrium vector u* we have u* = Au* + b. Subtracting we obtain ut - u* = A(u,_i - u*) or zt = Azt-\ which is a linear first-order homogeneous system. In what follows, therefore, we shall concentrate more on linear homogeneous systems with no major loss. 5.2 Basic matrices with Mathematica and Maple Both Mathematica and Maple have extensive facilities for dealing with matrices and matrix algebra. The intention in this section is to supply just the briefest introduction so that the reader can use the packages for the matrix manipulations required in this book. It is assumed that the reader is familiar with matrix algebra. Both programmes treat matrices as a list of lists - a vector is just a single list. While most of the basic matrix manipulations are built into Mathematica, it is necessary to load one or even two packages in Maple. The two packages are (1) linalg and (2) Linear Algebra, and are loaded with the instructions: with (linalg) : with(LinearAlgebra): The lists in Mathematica use curly braces, while those in Maple use straight (table 5.1). Both programmes have palettes that speed up the entry of vectors and matrices, although Mathematical is far more extensive than that of Maple. Discrete systems of equations 205 Table 5.1 Representations of matrices in Mathematica and Maple Mathematica Maple Conventional representation Vector {a, b, c] [a, b, c] a, b, c] or a b c Matrix {{a, b], {c, d}} [ [a, b], [c, d] ] a b c d 5.2.1 Matrices in Mathematica To illustrate Mathematical package, let a T3 2 4 " |_1 -2 -3 ' mB=[2 3 0. , mC = " 2 -1 2 r 0 3 then in Mathematica use: mA={{3,2,4},{1,-2,-3}} mB={{0,-1,1}, {2,3,0} } mC={{2,l},{-1,0},{2,3}} mA+mB (to add) mA-mB (to subtract) mA.mC (to multiply) Notice that mA cannot be multiplied by mB. Any such attempt leads to an error message indicating that the matrices have incompatible shapes. Square matrices have special properties. For illustrative purposes, let 2 1 -1 3 0 2 -1 2 1 Typical properties are shown in table 5.2. A special square matrix is the identity matrix. To specify a 3 x 3 identity matrix in Mathematica one uses Identity Matrix[3]. To construct the characteristic polynomial for the above square matrix, then we use1 mA-AldentityMatrix[3] and the characteristic equation is obtained using Det[mA-AldentityMatrix[3]]==0 which in turn can be solved using Solve[Det[mA-AldentityMatrix[3]]==0] 1 The characteristic polynomial can be obtained directly with the command CharacteristicPolyno-mial[mA]. 206 Economic Dynamics Table 5.2 Properties of matrices and Mathematica input Property Mathematica input Trace Transpose Inverse Determinant Eigenvalues Eigenvectors Characteristic polynomial Matrix Power (power n) Tr[mA] Transpose[mA] Inverse[mA] Det[mA] Eigenvalues[mA] or Eigenvalues[N[mA]] Eigenvectors[mA] or Eigenvectors[N[mA] CharacteristicPolynomial[mA,X] MatrixPower[mA,n] or Solve[N[Det[mA-AldentityMatrix[3]]]==0] As one gets familiar with the package, long strings of instructions can be entered as a single instruction, as in the final solve. To verify the results of example 4.12 in chapter 4, input the following, where we have added the instruction 7/ MatrixForm' to display the matrix in more familiar form mA={{-2,1}, {1,-2} } Det[mA] mA-AldentityMatrix[2] //MatrixForm Eigenvalues[mA] Eigenvectors[mA] All results are indeed verified. 5.2.2 Matrices in Maple To illustrate Maple's package, let mA = 3 2 4 1 -2 -3 mB = 0 2 -1 1 3 0 then in Maple use: mA—matrix ( [ [ 3, 2 , 4 ] , [ 1, -2 , -3 ] ] ) ; mB —matrix ([[0,-1,1], [2,3,0]]); mC:=matrix([[2,1], [-1,0], [2,3]]); evalm(mA+mB) (to add) evalm(mA-mB) (to subtract) evalm(mA&*mC) (to multiply) mC 1 r -1 o 2 3 Notice that mA cannot be multiplied by mB. Any such attempt leads to an error message indicating that the matrices have non-matching dimensions. Discrete systems of equations 207 Table 5.3 Properties of matrices and Maple input Property Maple input Trace trace (mA) ; Transpose transpose (mA) ; Inverse inverse (mA) ; Determinant det (mA) ; Eigenvalues eigenvals (mA) ; or evalf (eigenvals (mA) ) ; Eigenvectors eigenvects (mA) ; or evalf (eigenvects (mA) ) Characteristic polynomial charpoly (mA, ' lambda' ) ; Matrix Power (power n) evalm (mAAn) Square matrices have special properties. For illustrative purposes, let mA 2 1 -1 3 0 2 -1 2 1 Typical properties are shown in table 5.3. The characteristic polynomial in Maple simply requires the input charpoly(mA,'lambda'); which in turn can be solved using solve(charpoly(mA,'lambda')=0); or fsolve(charpoly(mA,'lambda')=0,lambda,complex); As one gets familiar with the package, long strings of instructions can be entered as a single instruction, as in the final fsolve. Notice too that the final fsolve required the option 'complex' to list all solutions. Using fsolve(charpoly(mA,'lambda')=0); gives only the real solution. To verify the results of example 4.12 in chapter 4, input the following: with (linalg) : with(LinearAlgebra): mA—matrix ([[-2,1], [1, -2] ] ) ; det(mA); evalm(mA-lambda*IdentityMatrix(2)); eigenvals(mA); eigenvects(mA); All results are indeed verified, when it is realised that vr = [1 — 1 ] is fundamentally the same as vr = [ — 1 1 ]. 208 Economic Dynamics 5.3 Eigenvalues and eigenvectors Let us concentrate on the first-order linear homogeneous equation system a b xt-\ yt. c d yt-i. or ut = Au,_i with solution ut = A1uq. We invoke the following theorem. THEOREM 5.1 If the eigenvalues of the matrix A are r and s obtained from | A — kl\ () such that r ^ s, then there exists a matrix V = [ vr Vs ] composed of the eigenvectors associated with r and s, respectively, such that D = r 0 0 s = V_1AV We shall illustrate this theorem by means of an example. Example 5.3 Let A = 2 1 1 2 The characteristic equation is | A — kl\ = 0, i.e. (2 - X)2 - 1 = X2 - 4k + 3 = 0 2-k 1 1 2-k Hence, k = r = 1 and k = s = 3. For k = r = 1 we have the equation (A - rl)vr = 0 or 2 1 1 2 1 0 0 1 "0" o_ i.e. "1 r '< "0" 1 i_ o_ Hence v\-\-vr2 = 0. Let v\ = 1 then vr2 = —v\ = — 1. Thus, Discrete systems of equations 209 For k = s = 3 we have 2 1 1 2 3 0 0 3 "0" A. o_ i.e. "-1 1 '< "0" 1 -1_ A. o_ Hence, —v\ + v2 = 0. Let v\ = 1, then vs2 = v\ = 1. Thus, the second eigenvector is 1 1 -1 1 Our matrix, V, is therefore V=[vr vs] = From the theorem we have D = V_1AV, i.e V_1AV = i r -i "2 r i r "i 0" -i i 1 2 -i i 0 3_ which is indeed the matrix D formed from the characteristic roots of A. Since D = V_1AV then VDV"1 = V(V"1AV)V-1 = A Furthermore A2 = (VDV_1)(VDV_1) = VD2V1 A3 = (VDV"1)(VD2V"1) = VD^"1 Hence or A' = (VDV^XVD'^V-1) = VD'V"1 ut = Au0 = VD'V-1!*) (5.5) u, 0 0 We can summarise the procedure as follows: (1) Given a first-order linear homogeneous equation system ut = Aut-i, where u is a 2 x 1 vector and A is a 2 x 2 matrix, with solution ut = A'uo, obtain the eigenvectors r and s (assumed to be distinct). (2) Derive the eigenvector vr associated with the eigenvalue r and the eigenvector Vs associated with the eigenvalue s, and form the matrix V = [V, vs]. (3) From (2) we have the general solution ut = ar'Y + bs'\s 210 Economic Dynamics and we can find a and b for t = 0 from uo = a\r + b\s where uo is known. This can either be done by direct substitution, or using the fact that a\r + b\s = [ \r \s] i.e. V u0 u0 or (4) Write the solution ut = ar'Y + bs'\s But we can do this whole process in one step. First we note ut = artyr + bs'\s = [ \r \s] = VD' V 0" a _0 s'_ b_ But Hence = V^uo ut = VD'V"1!*) which is the result we proved above. The gain, if there is one, in doing the four steps is the need to solve for a and b. Since this can often be done by direct substitution, then the four steps involve no inverse matrix computation. Example 5.4 Let xt+i = -S-xt+yt yt+1 = 4 - 0.3xt + 0.9yt setting xt+i = xt = x* and yt+\ = yt = y* for all t, the fixed point is readily shown to be = (6.4, 20.8). Now consider the system in terms of deviations from equilibrium, then xt+i - x* = -(xt - x*) + (yt - y*) yt+1 -y* = -0.3(xt - x*) + 0.9(yt - y*) or ut = Auř_i Discrete systems of equations 211 where A = -1 1 -0.3 0.9 -1 - k 1 -0.3 0.9 — k Solving for the eigenvalues from A-kl = we have |A - kl\ = -(1 + k)(0.9 — k) + 0.3 = k2 + OAk - 0.6 = 0 giving r = 0.7262 and s = —0.8262. Given r = 0.7262 then (A - 0.7262I)vr = 0 so 1.7262 1 "0" -0.3 0.1738 _ o_ i.e. Let Vo -1.7262vJ + vr2 = 0 -0.3^+0.1738^ = 0 1 then v\ = 0.5793. For 5 = -0.8262 0.1738 1 "0" -0.3 1.7262 _ A. 0_ i.e. -0.1738v5 + v| = 0 -0.3v5 + 1.7262v* = 0 Let v\ = 1 then v\ = 5.7537. Hence V=[vr vs] 0.5793 5.7537 1 1 and u, (0.7262)' 0 0 (-0.8262)' Suppose xq = 2 and yo = 8, i.e. u0 = -4.4 -12.8 Then Xt+l ~ x yt+i - y* 0.5793 5.7537 1 1 (0.7262)' 0 0 (-0.8262)' 0.5793 5.7537 1 1 -4.4 -12.8 212 Economic Dynamics Figure 5.1. Example 5.4: Original system yt 25 - 20 - 15 - 5 - ,-.-9- I I 1 —r ...............— i.e. xt+l - x* = -7.7526(0.7262)' + 3.3526(-0.8262)' yt+1 -y* = -13.3827(0.7262)' + 5.827(-0.8262)' This procedure does give insight into the dynamics and it is possible to plot the solutions. However, if interest is purely in the dynamics of the trajectory, this can be obtained immediately using a spreadsheet. Once the equations for xt+i and yt+\ have been entered in the first cells they are then copied down for as many periods as is necessary and the {xt,yt} coordinates plotted on the x-y line plot, as shown in figure 5.1. This simple procedure also allows plots of x(t) and y(t) against time.2 The solution generalises to more than two equations. If A is a 3 x 3 matrix with distinct roots q, r and s, then the solution is V 0 0" ut = V 0 0 0 0 s' here V = [ vq vr \s ]. Example 5.53 In this example we shall also illustrate how Mathematica ox Maple can be employed as an aid. Let i 2 1 xt-i yt -l 1 0 yt-i Zt 3 -6 -1 Zt-l See Shone (2001) and section 5.5 below. 3 Adapted from Sandefur (1990, chapter 6). Discrete systems of equations 213 Then A-k\ = 1 - k 2 1 -1 l-k 0 3 -6 -l-k Within Mathematica carry out the following instructions, where we have replaced A by a m = {{l-a,2,l}, {-l,l-a,0}, {3,-6,-1-3}} sols = Solve[ Det[m]==0, 3] or in Maple m:=matrix( [ [1-3,2,1], [-1,1-3,0], [3,-6,-1-3] ] ); sols:=solve(det(m)=0,3); which gives the three eigenvalues4 q = 0, r = — 1 and s = 2. The next task is to obtain the associated eigenvectors. For q = 0, then 1 2 1 i v\ 0 (A - 0I)vr = -1 1 0 i V2 0 3 -6 -1 1 V3 0 which leads to the equations v? + 2v\ + v\ = 0 -v\ + vq2 = 0 3v\ -6vq2-vq3 = 0 We can solve this system within Mathematica with the instruction Solve[{x+2y+z==0, -x+y==0, 3x-6y-z==0}, {x,y,z}] or in Maple with the instruction solve( {x+2*y+z=0, -x+y=0, 3*x-6*y-z=0}, {x,y,z}); which provides solutions x = 1, y = 1 and z = — 3, where x is set arbitrarily at unity. Carrying out exactly the procedure for r = — 1 and s = 2 we obtain the results r = — 1 implies x = 2, y = 1 and z = —6 s = 2 implies x = 1, y = — 1 and z = 3 Hence our three eigenvectors and the matrix V are: 1 ~ 2 ~ 1 1 2 1 v^ = 1 , yr = 1 , v* = -1 , v = 1 1 -1 -3 -6 3 -3 -6 3 4 This could be obtained directly using the command Eigenvalues [m] in Mathematica or eigenvals(m) in Maple. 214 Economic Dynamics Hence our solution is (oy 0 0 ut = V o (-iy o 0 0 (2J V^uo Suppose Then i.e. or u0 yt = 2 1 1 -1 -6 3 0' 0 0' 0 (-1)' 0 0 0 2' 1 2 1 1 1 -1 -3 -6 3 -1 " 3 " -4 _ 3 _ jc, = 6(-1)' + 2(20 y, = 3(-l)'-2(20 z, = -18(-l)' + 6(20 xt " 2 " 1 yt = 3(-iy 1 + 2'+1 -1 -zt _ _-6_ _ 3 _ 5.4 Mathematica and Maple for solving discrete systems Mathematica and Maple can be used in a variety of ways in helping to solve systems of discrete equations. Here we consider two: (i) Solving directly using the RSolve/rsolve command. (ii) Solving using the Jordan form. 5.4.1 Solving directly Mathematical RSolve command and Maple's rsolve command can each handle systems of linear difference equations besides a single difference equation. In each case the procedure is similar to that outlined in chapter 3, section 3.13. Suppose we wish to solve example 5.4 with initial condition (xq, yo) = (2, 8), i.e., the system xt+i = -8 -xt + yt yt+1 = 4 - 0.3xt + 0.9yt x0 = 2, y0 = S Discrete systems of equations 215 then the instructions in each case are: Mathematica equ={x[t+1]==-8-x[t]+y[t], y [t + 1] —4-0 . 3x [t] +0 . 9y [t] , x[0]==2, y[0]==8} var={x[t],y[t]} RSolve[equ,var,t] Maple equ:=x (t + 1)=-8-x (t)+y (t), y(t + 1)=4-0.3*x (t)+0.9*y (t); init:=x(0)=2, y(0)=8; var:={x(t),y(t)}; rsolve({equ,init},var); The output from each programme looks, on the face of it, quite different - even after using the evalf command in Maple to convert the answer to floating point arithmetic. Maple gives a single solution to both x(f) and y(t). Mathematica, however, gives a whole series of possible solutions depending on the value of t being greater than or equal to 1, 2 and 3, respectively, and further additional conditional statements. In economics, with t representing time, the value of t must be the same for all variables. This means we can ignore the additional conditional statements. What it does mean, however, is that only for t > 3 will the solution for x(t) provided by Mathematica and Maple coincide; while y(t) will coincide for t > 2. This should act as a warning to be careful in interpreting the output provided by these packages. Turning to the three-equation system (example 5.5) with initial condition (x0,yo,zo) = (3, -4, 3) xt = xt-i + 2y,-i + Zt-\ yt = -xt-\ +yt-i zt = 3xt-i - 6yt-\ - Zt-\ xq = 3, y0 = -A, zo = 3 then we would enter the following commands in each programme: Mathematica equ={x[t]==x[t-1]+2y[t-l]+z[t-l], y[t]==-x[t-l]+y[t-1], z[t]==3x[t-1]-6y[t-1]-z[t-1] x[0]==3, y[0]==-4,z[0]==3} var={x[t] ,y[t] ,z [t] } RSolve[equ,var,t] Maple equ :=x (t)=x(t-l)+2*y(t-l)+z(t-l) , y (t)=-x(t-l) +y (t-1) , z(t)=3*x(t-1)-6*y(t-1)-z (t-1); init:=x(0)=3, y(0)=-4, z(0)=3; var-)x(t) ,y (t) , z (t) }; rsolve({equ,init},var); 216 Economic Dynamics In this instance the output in both programmes is almost identical. Mathematica, however, qualifies the solution for z [ t ] by adding 15if[t==0,l,0].If?is time, then this will not occur, and so this conditional statement can be ignored, in which case the two programmes give the same solution - which is also the one provided on p. 214. (5.6) (5.7) (5.8) 5.4.2 Solving using the Jordan form In section 5.3 we found the eigenvalues of the matrix A and used these to find the matrix V formed from the set of linearly independent eigenvectors of A. The diagonal matrix J = diagOi, ...,kn) is the Jordan form of A and V is the transition matrix, such that V_1AV = J From this result we have At = yj'V"1 and since the solution to the system ut = Au,_i is ut = A'uq, then u, = VJ'V_1Uo where J' = 0 0 k'2 0 0 0 0 So our only problem is to find the matrices J and V. Both Mathematica and Maple have commands to supply these matrices directly. In Mathematica one uses the command JordanDecomposition[mA]; while in Maple it is necessary to first load the linalg package, and then to use the command jordan(mA, ' V), where mA denotes the matrix under investigation and V is the transition matrix. To illustrate how to use these commands consider example 5.3, where mA = 2 1 1 2 Mathematica mA={{2,l},{1,2}} {V,J}=JordanDecomposition[mA] MatrixForm /@ {V,J} MatrixForm[[Inverse[V].mA.V]] Maple with (linalg) : mA:=matrix( [ [2,1], [1,2] ]); J:=jordan (mA, 'V ) ; print(V); evalm(VA(-1)&*mA&*V) ; Discrete systems of equations 217 In each of these instructions the last line is a check that undertaking the matrix multiplication does indeed lead to the Jordan form of the matrix. In each package we get the Jordan form 1 0 0 3 However, the transition matrix in each package on the face of it looks different. More specifically, Mathematica V Maple V = -1 1 1 1_ " 1 i- 2 2 1 1 _ 2 2 _ But these are fundamentally the same. We noted this when deriving the eigenvectors in the previous section. We arbitrarily chose values for v\ or v\ (along with the values associated with the eigenvalue s). In Maple, consider the first column, which is the first eigenvector. Setting v\ = 1, means multiplying the first term by —2, which gives a value for v\ = — 1. Similarly, setting v\ = 1 in Maple, converts vs2 also to the value of unity. Hence, the two matrices are identical. In each case the last instruction verifies that V-1 AV = J. Using Maple verifies all the results in section 5.3. However, Mathematica seems to give inconsistent results for a number of the problems. In particular, it appears the transition matrices provided by Mathematica for examples 5.4, 5.6 and 5.7 are not correct. This shows up with the last instruction, since for these examples MatrixForm[Inverse[V].ma.V] does not give the matrix J! It should be noted that all the examples in section 5.3 involve real and distinct roots. Even in the case of complex roots, these are distinct. A more general theorem than Theorem 5.1 is the following: THEOREM 5.2 If A is a n x n square matrix with distinct eigenvalues X\, ..., Xn, then the matrix A is diagonalisable, such that V_1AV = J and J = diag(Ai, ..., Xn). Since k\, ... ,kn are distinct eigenvalues of the matrix A, then it is possible to find n linearly independent eigenvectors v1, ..., v™ to form the transition matrix V. Systems that have repeated roots involve linear dependence. Such systems involve properties of Jordan blocks, which is beyond the scope of this book. However, a complete study of the stability of discrete systems would require an understanding of Jordan blocks, see Elaydi (1996) and Simon and Blume (1994). When the matrix A has repeated roots, then it is not diagonalisable. It is, however, possible to find an 'almost diagonalisable' matrix which helps in solving systems with repeated roots. As indicated in the previous paragraph, for a general system 218 Economic Dynamics of n equations, this requires knowledge of Jordan blocks. Here we shall simply state a result for a 2 x 2 system. THEOREM 5.3 If A is a 2 x 2 matrix, then there is a transition matrix V such that (a) V^AV = Ji = (b) V^AV = J2 = (c) V^AV = J3 = r 0 0 s X 1 0 X for real distinct roots r and s for repeated root X a + ßi 0 0 a-ßi for complex conjugate roots X = a ± ßi In each case, J, is the Jordan form of the particular matrix A. We shall use theorem 5.3 when discussing the stability of discrete systems in section 5.6. Section 5.2 dealt with case (a) in detail. Here we shall consider just one example of cases (b) and (c), using both Mathematica and Maple. Example 5.6 Consider the matrix in example 4.10, which is A2 1 -1 1 3 then the instructions in each programme are: Mathematica A2={{1,-1},{1,3}} Eigenvalues[A2] {V2,J2}=JordanDecompos ition[A2~ MatrixForm /@ {V2,J2} MatrixForm[Inverse[V2].A2.V2] Maple with (linalg) : A2:=matrix([ [1,-1], [1,3]] ); eigenvals(A2); J2:=jordan(A2,'V2'); print(V2); evalm(V2A(-1)&*A2&*V2); With each programme we get the Jordan form as J2= 2 1 J 0 2 Discrete systems of equations 219 Example 5.7 Next consider the matrix in example 4.11, which is A3 : -3 4 -2 1 then the instructions in each programme are: Mathematica A3={{-3,4},{-2,1}} Eigenvalues[A3] {V3,J3}—JordanDecomposition[A3] MatrixForm /@ {V3,J3} MatrixForm[Inverse[V3].A3.V3] Maple with (linalg) : A3:=matrix([ [-3,4], [-2,1]]); eigenvals(A3); J3:=jordan(A3,'V3'); print(V3); evalm(V3A(-1)&*A3&*V3); With each programme we get the Jordan form as J3 = ■1 + 2/ 0 0 -1 -2/ Verifying the results in theorem 5.3. When considering the stability of the system u, Au t-i (5.9) we can approach this from a slightly different perspective, which can provide some valuable insight into the phase portrait of discrete systems. What we intend to do is to transform the system using the matrix V. Thus, define zt = v-y This implies ut = Yzt. We can therefore write system (5.9) in the form Yzt = AVz,_i premultiplying by the matrix V-1, we have z, = V_1AVz,. (5.10) Jz t-i where J r 0 0 s (5.11) System zt = Jzt-\ is referred to as the canonical form of the system ut = Au,_i. The important point is that the stability properties of (5.11) are the same as those of (5.9). The solution to the canonical form is simply zt = J'z0 r 0 0 zo where zq = V 1 u0 220 Economic Dynamics When considering the phase space of this canonical form it is useful to consider the following: Zlt V 0" Zw -Z2i_ 0 s'_ _Z20_ Now take the ratio of Zi/zi, then Z\t rlzw ^r) \ZwJ and so the path of the system is dominated by the value/sign of s/r. 5.5 Graphing trajectories of discrete systems The mathematics of solving simultaneous equation systems is not very straightforward and it is necessary to obtain the eigenvalues and the eigenvectors. However, it is possible to combine the qualitative nature of the phase plane discussed in the previous section and obtain trajectories using a spreadsheet or the recursive features of Mathematica and Maple. 5.5.1 Trajectories with Excel Example 5.8 Consider the following system of equations xt = -5 + 0.25x,-! + OAyt-x vř = 10 - xt-i + y,_i x0 = 10, yo = 5 In cells B8 and C8 we place the initial values for x and y, namely = 10 and yo = 5. In cells B9 and C9 we place the formulas. These are B9 = -5 + 0.25* B8 + 0.4* C8 C8 = 10 - B8 + C8 These cell entries contain only relative addresses. Cells B8 and C8 are then copied to the clipboard and pasted down in cells B10:C28. Once the computations for (xt, yt) have been obtained, then it is a simple matter of using the x-y plot to plot the trajectory. Given the discrete nature of the system the trajectories are not the regular shapes indicated by the phase plane diagram. They constitute discrete points that are joined up. Even so, the nature of the system can readily be investigated. Figure 5.2 shows the initial values of = 10 and y0 = 5. Always a good check that the equations have been entered correctly is to place the equilibrium values as the initial values. The equilibrium point is (x*, y*) = (10, 31.25). Placing these values in cells B8 and C8 leads to them being repeated in all periods. One of the advantages of this approach, besides its simplicity, is the ready investigation of the system for various initial conditions. The graphics plot can sometimes change quite dramatically! This procedure allows quite complex discrete dynamic systems of two equations to be investigated with the minimum mathematical knowledge. Of course, to fully Discrete systems of equations 221 ß Eta 6fa yaw >isar< Format loots Quia arrJow tfäfc QUB^aa? i^etm mi N25 J_ ^■■■HKlül Figure 5.2. -HI*! :: - b / m m m « ? A B C O 1 Figure 5.2, Example 5.8 2' 3 4 5 K "7" = -5 + 0.25*M+0.4.yH = 10-^+7,., 7 t x(t) y(t> S 9 0 1 10 -0 5 5 5 10 2 -3.125 15.5 11 3 041875 28 gas" 12 4 6.554688 38.20625 13 5 11 92117 41 66156 14 6 14.64092 39.73039 15 7 14.55239 35 08947 16 8 12 67389 30 53709 1/ 9 10.38331 27 8632 18 10 8 741107 27 4799 19 11 8.177235 28.73879 20 12 8.539824 30 56155 21 13 9 359577 32 02173 22 14 10.14859 32 66215 23 15 10.60201 32 51357 24 16 in 65593 911fifi •4 > Example 5.8 45 - 40 - 35 - 30 - 25/ ym - X15 " \o - ,-8- Ctaw»-Read» NUM. appreciate what is happening requires an understanding of the material in many of the chapters of this book. Consider the following nonlinear system, which is used to produce the Henon map and which we shall investigate more fully in chapter 7. Example 5.9 The system is xt = 1 - ax]_x + yt-i yt = bxt-\ Our purpose here is not to investigate the properties of this system, but rather to see how we can display trajectories belonging to it. We begin with the spreadsheet, as shown in figure 5.3. We place the values of a and b in cells E3 and E4, where a = 1.4 and b = 0.3. In cells B8 and C8 we place the initial values for x and y, which are xq = 0.01 and yo = 0. The formulas for the two equations are placed in cells B9 and C9, respectively. These take the form B9 1-$E$3*B8A2+C8 C9 $E$4*B8 The cells with dollar signs indicate absolute addresses, while those without dollar signs indicate relative addresses. Cells B9 and C9 are then copied to the clipboard and pasted down. After blocking cells B8:C28 the graph wizard is then invoked and the resulting trajectory is shown in the inserted graph. The most conspicuous feature of this trajectory is that it does not have a 'pattern'. In fact, given the parameter values there are two equilibrium points: (x*, y*) = (—1.1314, —0.3394) and (xj, y2) = (0.6314, 0.1894), neither of which is approached within the first twenty periods. Why this is so we shall investigate in chapter 7. 222 Economic Dynamics Figure 5.3. HS e*t:öS*-jtttt Faraat lo*-teö iffndäw fclsfc 025 j_ - - .......------ -lalxi / W W € .1 - * • * A B ! C ■ Figure 5.3: Example 5.9 X(t): O.Ol; 0 9938S; -0 39561 1.079741; y(t) o 0.003 0 299958 -0.11898 -0 75118 ! 0.323922 0.533987 0.375454 0. -0.22535 0.160196 0.112636 -0.18528 0.288853 1.240804 -0.05558 10 -1.21101 11 -0 68092 12 -001241 13 0.795509 14 0.110308 0.238653 15 1.221617 0.033093 1R .rl.Cmi. n,3B6485. 0.372241 -0.3633 -0.20428 -0.00372 14 0.3 Readr i -ir 5.5.2 Trajectories with Mathematica and Maple The spreadsheet is ideal for displaying recursive systems and the resulting trajectories. But occasionally it is useful to display these trajectories within Mathematica or Maple. In doing this care must be exercised in writing the simultaneous equations for computation so that the programmes remember earlier results and do not recompute all previous values on each round. This leads to more cumbersome input instructions - which is why the spreadsheet is so much easier for many problems. We shall consider once again examples 5.8 and 5.9. Example 5.8 (cont.) The input instructions for each programme are Mathematica Clear[x,y,t] x[0]:=10; y[0]:=5; x[t_] :=x[t]=-5 + 0.25x[t-l]+0.4y[t-1] y[t_]:=y[t]=10-x[t-l]+y[t-l] data—Table [{x[t],y[t]},{t,0,20}]; ListPlot[data,PlotJoined->True,PlotRange->All] Maple t:='t': x:='x': y:='y': x:=proc(t) option remember;-5 + 0.25*x(t-1)+0.4*y(t-1)end: y:=proc(t) option remember; 10-x(t-1)+y (t-1) end: x(0) —10: y (0) :=5: data— [seq( [x(t) ,y(t) ] , t=0. .20) ] ; plot (data); The Maple instructions join the points by default. If just a plot of points is required Discrete systems of equations 223 with Maple, then the last line becomes plot (data, plotstyle=point); The resulting trajectories are similar to that shown in the chart in figure 5.2 (p. 221). As one might expect, both Mathematica and Maple allow more control over the display of the trajectories than is available within Excel. Furthermore, both these programmes allow more than one trajectory to be displayed on the same diagram. This is not possible within spreadsheets. Spreadsheets can display only one (x, y)-trajectory at a time. Example 5.9 (cont.) The input instructions for each programme for producing discrete plot trajectories are Mathematica Clear[x,y,t,a,b] x [0] :=0.01; y[0] :=0; a:=1.4; b:=0.3; x [t_] :=x[t]=l-a x[t-1]A2+y[t-1] y [t_] :=y[t]=b x[t-l] data=Table[{x[t],y[t]},{t,0,20}]; ListPlot[data,PlotJoined->True,PlotRange->All] Maple t:='t': x:='x': y:='y': a:='a': b:='b': x:=proc(t) option remember; l-a*x(t-1)A2+y(t-1) end: y:=proc(t) option remember; b*x(t-l) end: x (0) :=0.01: y(0):=0: a:=1.4: b:=0.3: data:=[seq([x(t),y(t)], t=0 ..20) ] ; plot (data); The resulting trajectories are similar to that shown in the chart in figure 5.3. Spreadsheets do not allow three-dimensional plots, but it is very easy to adapt the instructions just presented for Mathematica and Maple to do this. The only essential difference is the final line in each programme. Assuming 'data' records the list of points {x(t), y(t), z(t)}, then a three-dimensional plot requires the instruction Mathematica ListPlot3D[data,PlotJoined->True] Maple plot3d(data); 5.6 The stability of discrete systems 5.6.1 Real distinct roots For systems with real distinct roots, r and s, which therefore have linearly independent eigenvectors, we can establish the stability properties of such systems by 224 Economic Dynamics considering the general solution ^_^* ut = arlyr + bsl\s where ut = ' Ut-y _ If \r\ < 1 and \s\ < 1 then ar'\r 0 and bs'\s 0 as t oo and so ut -> 0 and consequently the system tends to the fixed point, the equilibrium point. Return to example 5.4 where r = 0.7262 and s = —0.8262. The absolute value of both roots is less than unity, and so the system is stable. We showed this in terms of figure 5.1, where the system converges on the equilibrium, the fixed point. We pointed out above that the system can be represented in its canonical form, and the same stability properties should be apparent. To show this our first task is to compute the vector zq. Since zo = V_1Uo, then -13.3827" 0.5827 and lit = (0.7262)'(-13.3827) Z2t = (-0.8262)'(0.5827) Setting this up on a spreadsheet, we derive figure 5.4. The canonical form has transformed the system into the {z\, Z2)-plane, but once again it converges on the fixed point, which is now the origin. Now consider the situation where \ r\ > 1 and \s\ > 1 thenu, —► ±oo depending on the sign of the characteristic roots. But this result occurs so long as at least one root is greater than unity in absolute value. This must be so. Let \r\ > 1 and let |si < 1. Then over time bs'\s 0 as t oo, while ar'\r ±oo as t oo, which means ut -> ±oo as t oo. The system is unstable. Return to example 5.3 where r = 1 and s = 3. Both roots are distinct, positive and at least one is greater than unity. It follows from our earlier argument that this system must be unstable. We have already demonstrated that the diagonal matrix Zw 0.5793 5.7537 -4.4 Z20 1 1 -12.8 Discrete systems of equations 225 (the Jordan form) and the transition matrix are: "1 0" i r v 0 3_ _-i i_ Suppose for this system (u\o, U20) = (5, 2), then Hit i r 'V 0" i r -i "5" _U2t_ _-i i_ 0 3'_ _-i i_ _2_ i.e. 3 1 ,t - + -3' 2 2 3 7 , U2t =---h -3 2 2 Therefore, as r increases u\t -> +oo and «2* ~> +oo. Turning to the canonical form, zq = V_1Uq, hence "3/2" _Z20 _ _7/2_ and zu = 1' 22/ = 3' 3 2 Furthermore, z2t /3y7/2= /7 1/3/2 \3 Zl and so for each point in the canonical phase space, the angle from the origin is increasing, and so the direction of the system is vertically upwards, as illustrated in figure 5.5(b). Once again, the same instability is shown in the original space, figure 5.5(a), and in its canonical form, figure 5.5(b). The fact that the trajectory in figure 5.5(b) is vertical arises from the fact that root r = 1. Suppose one root is greater than unity in absolute value and the other less than unity in absolute value. Let these be \r\ > 1 and \s\ < 1. The system remains unstable because it will be governed by the root \r\ > 1, and the system will be dominated by the term ar'\r. This means that for most initial points uo, the system diverges from the equilibrium, from the steady state (x*, y*). But suppose a = 0, then ut = bs'\s and since \s\ < 1, then the system will converge on (x*, y*). There exist, therefore some stable trajectories in the phase plane. Example 5.10 xt+1 = -0.85078*, - yt yt+l =x, + 2.35078V, 226 Economic Dynamics Figure 5.5. (a) Example 5.3: Original system 250000 -I 200000 - 150000-yt 100000- XX 50000 - n i U 1 ( r 1 1 ' 1 1 1 ) 50000 100000 150000 200000 250000 (b) Example 5.3: Canonical form 250000 -| 2.00 The fixed point is at the origin, i.e., x* = 0 and y* = 0. This system takes the matrix form Xt+1 "-0.85078 -1 _yt+i _ 1 2.35078 The eigenvalues of the matrix A of this system are readily found to be r = 2 and s = —0.5. We therefore satisfy the condition \r\ > 1 and \s\ < 1. Furthermore, we can obtain the eigenvectors of this system as follows (A - 2I)vr = 0 i.e. 2.85078 -1 "0" 1 0.35078 _ o_ Discrete systems of equations 221 or -2.85078vi V2 = 0 Let v\ v\ + 0.35078^ = 0 1 then vr2 = -2.8508. The second eigenvector is found from (A - 0.5I)v* = 0 i.e. 0.35078 -1 "0" 1 2.85078 _ A. 0_ or -0.35078V 0 + 2.85078v* = 0 Let Vn = 1 then v -2.8508. Hence v = 1 vs — "-2.8508" _-2.8508 _ v — 1 One eigenvector represents the stable arm while the other represents the unstable arm. But which, then, represents the stable arm? To establish this, convert the system to its canonical form, with zt+\ = V_1u,+i. Now take a point on the first eigenvector, i.e., point (1, —2.8508), then z0 = V = 1 -2.8508" -i 1 "1" 2.8508 1 _-2.8508 _ 0 Hence, 2'(1) (-0.5/(0) -+ +oo as t 0 Z\t = Therefore zu arm. Now take a point on the eigenvector \s, i.e., the point (—2.8508, 1), then oo. So the eigenvector vr must represent the unstable z0 = V 1 -2.8508 -2.8508 1 -,-1 "-2.8508" "0" 1 _1_ Hence, zu = 2'(0) = 0 z2t = (-0.5)'(1) Therefore zit -> 0 as t oo. So the eigenvector Vs must represent the stable arm of the saddle point. Figure 5.6 illustrates the trajectory of the original system starting from point (—2.8508, 1), and shows that it indeed converges on the point (x*,y) = (o, o).5 Before leaving this example, notice that when we considered point (1, —2.8508) on the eigenvector associated with r = 2, the point became (1, 0) in terms of its 5 The system is, however, sensitive. A plot beyond t = 10 has the system moving away from the fixed point. 228 Economic Dynamics canonical representation. Similarly, point (—2.8508, 1) on the eigenvector associated with s = —0.5 became point (0, 1) in its canonical representation. In other words, the arms of the saddle point equilibrium became transformed into the two rectangular axes in z-space. This is a standard result for systems involving saddle point solutions. So long as we have distinct characteristic roots these results hold. Here, however, we shall confine ourselves to the two-variable case. To summarise, if r and s are the characteristic roots of the matrix A for the system ut = Au,_i and derived from solving | A — Xl\ =0, then (i) if \r\ < 1 and \s\ < 1 the system is dynamically stable (ii) if \r\ > 1 and \s\ > 1 the system is dynamically unstable (iii) if, say, \r\ > 1 and \s\ < 1 the system is dynamically unstable. In the case of (iii) the system will generally be dominated by the largest root and will tend to plus or minus infinity depending on its sign. But given the fixed point is a saddle path solution, there are some initial points that will converge on the fixed point, and these are values that lie on the stable arm of the saddle point. As we shall see in part II, such possible solution paths are important in rational expectations theory. Under such assumed expectations behaviour, the system 'jumps' from its initial point to the stable arm and then traverses a path down the stable arm to equilibrium. Of course, if this initial 'jump' did not occur, then the trajectory would tend to plus or minus infinity and be driven away from equilibrium. 5.6.2 Repeating roots When there is a repeating root, A, the system's dynamics is dominated by the sign/value of this root. If |A| < 1, then the system will converge on the equilibrium value: it is asymptotically stable. If |A| > 1 then the system is asymptotically unstable. We can verify this by considering the canonical form. We have already Discrete systems of equations 229 showed that the canonical form of ut = Au,_i is z* = J'zo In the case of a repeated root this is ' X1 tXl~l 0 X' zo Hence Zlt = X'zw + tX' 1Z20 Zit = AřZ20 Therefore if \X\ < 1, then | A'| —► 0 as t —► oo, consequently ziř —► 0 and Z2i —► 0 as ř —► oo. The system is asymptotically stable. If, on the other hand, \X\ > 1, then IX' J —► oo as ř —► oo, and zii —► ±oo and Z2i —► ±oo as t —► oo. The system is asymptotically unstable. We can conclude for repeated roots, therefore, that (a) if \X\ < 1 the system is asymptotically stable (b) if \X\ > 1 the system is asymptotically unstable. Example 5.11 Consider the following system xt+i =4 + xt-yt yt+i = -20 + xt + 3yt Then** = 12 and v* = 4. Representing the system as deviations from equilibrium, we have xt+i - x* = (xt - x*) - (y, - y*) yt+1 -y* = (xt - x*) + 3(yt - y*) This system can be represented in the form ut = Au,_i and the matrix of the system is 1 -1 1 3 We have already considered this in example 5.6 with results J = 2< a'-1 0 2' "2 r V i r 0 2 1 0_ The solution is then VJ'V-1!*) where J' u, Since the stability properties of u, = Au,_i are the same as those of its canonical form, let us therefore consider 230 Economic Dynamics Figure 5.7. as •15)Bte Bat !öaw Etsert Format loafc'Qsta SKfcdew Bp<%3 M24 4 • / mmm * ; A 6 C D 1 Figure 5.7: Example 5.11 3 . *»i=4 + * -y, .4...: 5 8 7 t x(t) y(t) 8 ! 0 0.1 0.1 9 1 4 -19.6 10 2 273 -74 8 11 3 1064 -2168 12 ; 4 327 2 -564 13 5 895.2 -1384,8 14 6 2284 -3279.2 15 ' 7 5567.2 -7573,6 16 8 13144 8 -17173.6 17 9 30322.4 -38396 18 10 68722.4 -84885.6 19 20 21 22 23 J* I * Readf Ih»t1 / Sfct«? X OhtM?,/ 20000 0 -20000 y< -40000 -60000 -80000 -100000 Example 5.11 l^"s»2G000 40000 60000 800 NUM i.e. zi, = 2'zio + ?2'_1Z20 22/ = 2'Z20 Clearly, z\t -> oo and zit ~> 00 as t -> oo regardless of the initial point. This is illustrated for the original system using a spreadsheet, as shown in figure 5.7 for the initial point (0.1, 0.1). Example 5.12 Consider xt+i = 8 + 1.5x, -yt yt+i = -l5 + xt- 0.5yt Then x* = 108 and y* = 62. Taking deviations from equilibrium xt+i -x* = l.5(xt - x*) - (y, - y*) yt+i-y* = (xt-x*)-0.5(yt-y*) and so the matrix of this system is Using either Mathematica or Maple we can establish that the eigenvalues are k = 0.5 (repeated twice) and the Jordan form and transition matrices are J = "0.5 1 "1 1" V = _ 0 0.5 _ 1 0_ Discrete systems of equations 231 ©E*> Bdlt Vbw Insul Format look Cab tp\p ■■■■■■I^HMHKII^ Figure 5.8. 10 . B / ill « ? A j B C Figure 5.8: Example 5.12 = 8+1.5*,-= -15+x, - ■0.5j-, x(t) 5 10.5 36.25 60 625 78.5625 y(t) 5 -12.5 1.75 20 375 35 4375 90 40625 45.84375 52.48438 56.52344 58 90234 60.27148 61 0459 107.4556 61.47803 107.7053 61,71655 107.8414 61.84705 107.9151 61.91791 15 107.9547 61.95615 6 97 76563 7 102 1641 8 104 7227 S 106 1816 10 107.001 11 12 13 14 j k l i£ Example 5.12 40 20 Readr 1-1 NUM Hence, 0.5' íO.5'"1 0 0.5' and the canonical form of the system is 0.5' iO.5'"1 0 0.5' zo Hence 20 zu = 0.5'zio + KO.^z. Z2t = 0.5'Z20 Therefore, zu —► 0 and zit 0 as t —► oo regardless of the initial point. This is illustrated for the original system using a spreadsheet, as shown in figure 5.8 for the initial point (5,5). 5.6.3 Complex conjugate roots Although complex conjugate roots involve distinct roots and independent eigenvectors, it is still necessary to establish the conditions for stability where r = a + /3i and s = a — /3i. For the system ut = Au,_i where A is a 2 x 2 matrix with conjugate roots a ± pi, then the Jordan form is (a + fiif 0 0 (a- pi)' J = a + ßi 0 0 a-ßi and J = 232 Economic Dynamics Considering the canonical form zt = \zt_\, then ~ (a + pi)1 0 z, = J'z0 0 (a- pi) To investigate the stability properties of systems with complex conjugate roots, we employ two results (see Simon and Blume 1994, appendix A3): (i) a±ip = R(cos 9 ± / sin 0) (ii) (a ± ifi)n = Rn[cos(n9) ± i sin(n0)] (De Moivre's formula) where R = y/a2 + ß2 and tan 9 a From the canonical form, and using these two results, we have zu = (a + pi)' zio = R'[cos(t9) + isin(tf)] Z2t = (a - Pi)1 z2o = R'icositO) - isin(^)] It follows that such a system must oscillate because as t increases, sin(t9) and cos(t9) range between +1 and —1. Furthermore, the limit of z\t and Z2t as t -> oo isgovernedby the term = |i?|'.If|i?| < 1, then the system is an asymptotically stable focus; if \R\ > 1, then the system is an unstable focus; while if \R\ = l,then we have a centre.6 We now illustrate each of these cases. Example 5.13 |R| < 1 Consider the system xt = 0.5xt-i + 0.3y,_i yt = -xt-i + yt-i Then the matrix of the system is A = 0.5 0.3 -1 1 with characteristic roots r = 0.75 + 0.48734/ and s = 0.75 - 0.48734/. Hence R = 7(0.75)2 + (0.48734)2 = 0.8944 Such a system should therefore have an asymptotically stable focus. We illustrate this in figure 5.9, where the original system is set up on a spreadsheet and the initial point is given by (xo, yo) = (10, 5). As can be seen from the inserted graph, the system tends in the limit to the origin. 6 See section 3.8 for the properties of R in the complex plane, especially figure 3.15. Discrete systems of equations 233 HICto tci W jscn fsnw loos aaa flWcw Ijrfp Figure 5.9. * 10 - » £ abc 1 Figure 5.9: Example 5.13 2 ' XJ jr, = 0.5xM + 0.3jM 4 5 A = S 7 "t x(t) y(t) 8 0 10 5 9 1 6.5 -5 ■to 2 1.75 -115 11 3 -2 575 -13 25 ■13-: 4 -5 2625 -10 875 18 5 -5 83375 -5 4125 14 S -4 54063 042125 15 : 7 -2.14394 4.961875 1« 8 0.416594 7.105813 17 9 2.340041 6.689219 1S 10 3.176785 4.349178 19 11 2.893146 1.172392 20 12 1798291 -1.72075 21 13 0.382919 -3.51905 22.; 14 -0.86425 -3.90196 23 15 -1.60272 -3.03771 2* , .=143489.. Craw- tj Ready ilL. NUM Example 5.14 |R| > 1 Consider the system = xt-\ + 2yt-\ yt = -xt-\ + yt-\ Then the matrix of the system is with characteristic roots r = 1 + is/l and 5=1 — i\pl. Hence R = 7(1)2 + (V2)2 = V3 = 1.73205 Such a system should therefore have an unstable focus. We illustrate this in figure 5.10, where the original system is set up on a spreadsheet and the initial point is given by (xq, yo) = (0.5, 0.5). As can be seen from the inserted graph, the system is an unstable focus, spiralling away from the origin. Example 5.15 |R| = 1 Consider the system xt = 0.5xt-i + 0.5y,_i yt = -xt-i + yt-i Then the matrix of the system is '0.5 0.5' A = -1 1 234 Economic Dynamics Figure 5.10. iufwiwyia gjElla EP« *i* tMI r9W Li* Q>» An*"" tj* 04 ' X - io - i»y 111 * .« u c c-t Figure 5.10: Example 5.14 2 $ " *j 5 x, = jcm + 2^ j'r = -*m+j'.- xt 0.5 1 1 C -2 -4 -4 0 "8 16 15 17 9: 18 10! 19 20 21 22 ?3 !« 4;IM*S./:Sk*«» / '.. " . v.:..:,;.,: .......................... .. new- fei- Asenan*»- \ xao||jiai a.^.^ Ready NUM Figure 5.11. *' L* t/h v«* Lnwr» '3/mnt Inrs Qa» yfcrrti» ly*> • l.; - B / Ii 1 « jj.». A » ■ C O Figure 5.11: Example 5.15 P 2 3 x, = 0.5i,_ 1 + 0.5/m 4 I l S » = "-Vi 6 7 t xt yt 8 0 5 5 9 1 5 0 10 i 2 2.5 -5 11 3 -1.25 -7.5 12 4 -4.375 -5.25 13 5 -5.3125 -1.875 14 S -3 59375 34375 1S 7 -0.07813 7.03125 18 8 3.476563 7.109375 17' 9 5.292989 3 832813 18 10 4.482891 -1.66018 19 11 1.401387 -6.12305 20 12 -2 36084 -7.52441 21 13 -4.94263! -5.16357 22 14 -50531 -0.22095 23 15 -2.63702 4.832153 'S* . 1fi 7/KW177 Ready [JS^oShapssr \ "» D014S *-,£•&• = •ir NUM with characteristic roots r = 0.75 + 0.661438/ and s = 0.75 — 0.661438/. Hence R = y(0.75)2 + (0.661438)2 = 1 Such a system should oscillate around a centre, where the centre is the origin. We illustrate this in figure 5.11, where the original system is set up on a spreadsheet and the initial point is given by (xq, yo) = (5, 5). As can be seen from the inserted graph the system does indeed oscillate around the origin. Discrete systems of equations 235 5.7 The phase plane analysis of discrete systems Consider the following system outlining discrete changes in x and y. Axt+1 = a0 + aixt + a2yt Ayt+\ = b0 + b\xt + b2yt where Axt+\ = xt+\ — ^andAvf+i = yt+\ — vf. In equilibrium, assuming it exists, at x = x* and y = y*, we have 0 = ao + a\x* + a2y* 0 = b0 + blX* + b2y* Hence, the system in terms of deviations from equilibrium can be expressed Axt+i = ax(xt - x*) + a2(yt - y*) Ayt+\ = b\{xt - x*) + b2{yt - y*) We can approach the problem in terms of the phase plane. As with the continuous model we first obtain the equilibrium lines Axt+\ = 0 and Ayt+\ = 0. Consider the following system with assumed signs on some of the parameters Axt+\ = ao + a\xt + a2yt a\, a2 > 0 Ayt+\ = b0 + b\xt + b2yt b\ > 0, b2 < 0 Then -a0 \ I a\ \ V If Axt+i = 0 then yt = - - — )x, «2 / \a2J If Aym = 0 then yt = ( -^j - (^jxt Consider now points either side of Axt+\ = 0. If Axt+\ > 0 then yt > ( —- ) — ( — ]xt a2 > 0 If Axt+\ < 0 then y, < ( —- ) — ( — )** a2 > 0 \ a2 J \a2J Similarly for points either side of Ayt+\ = 0 If Ayt+1 > 0 then yt < (f~^j ~ {j^jXt bl < 0 If Ayt+1 < 0 then yt > (f~^j ~ (j~)Xt bl < 0 The vector forces are illustrated in figures 5.12(a) and (b). Combining the information we have the phase plane diagram for a discrete system, illustrated in figure 5.13. The combined vector forces suggest that the equilibrium is a saddle point. 236 Economic Dynamics Figure 5.12. Example 5.16 Given Ax,+i = -12 + 0.3xt + 3yt Ayt+1 = 4 + 0.25*, - 1.5yt Then Axt+\ = 0 implies yt = 4 — 0. lxt Ayt+\ = 0 implies yt = 2.6667 + 0.1667.Y, and we obtain the equilibrium values x* = 5 y* = 3.5 Discrete systems of equations 237 In terms of deviations from the equilibrium we have for x xt+i = -12 + 1.3*, + 3yt x* = -12+ 1.3** +3/ (*m - x*) = l.3(xt - x*) + 3(yt - y*) and for y we have yt+l = 4 + 0.25*, - Q.5yt y* = 4 + 0.25** - 0.5/ (yt+1 - y*) = 0.25(xt - **) - 0.5(yt - y*) Therefore the system expressed as deviations from equilibrium is xt+l -x* = l.3(xt - **) + 3(yt - y*) yt+1 -y* = 0.25(xt - **) - 0.5(yt - y*) The dynamics of the system in the neighbourhood of (**, y*) is therefore determined by the properties of 1.3 0.25 3 -0.5 First we require to obtain the eigenvalues of the matrix A: |A - All = 1.3 — A 3 0.25 -(0.5 + X) = X2 — 0.8A - 1.4 = 0 with distinct eigenvalues r = 1.649 and s = —0.849. 238 Economic Dynamics Next we need to obtain the associated eigenvectors, vr and \s, respectively. Consider r = 1.649, then A-rI = "-0.349 0.25 3 -2.149 _ Then(A - rl)vr = Oand "-0.349 3 "0" 0.25 -2.149_ _o_ then -0.349v^ + 3vr2 = 0 0.25v^ -2.149v^ = 0 Let vr2 = 1, then v\ = 8.596, which arises from either equation. Similarly, for s = -0.849 then A-sl- 2.149 3 0.25 0.349 and (A - sl)\s = 2.149 3 0.25 0.349 giving 2.149v5 +3v| = 0 0.25v\ + 0.349v^ = 0 Let vs2 = 1, then v\ = —1.396. Hence the two eigenvectors are "8.596" "-1.396" 1 v = 1 8.596 1 with associated matrix V : Hence, our solution is ut = V ■1.396 1 (1.649)' 0 0 (-0.849)' v-i u0 Given some initial values (xq, yo) we could solve explicitly for ut. But we can gain insight into the dynamics of this system by looking closely at the phase plane. We have already established that for Axt+\ = 0 then yt = 4 — 0. lxt Ayt+1 = 0 then yt = 2.6667 + 0.1667^ These are drawn in figure 5.14. Discrete systems of equations 239 We can also readily establish that if Axt+\ > 0 then yt > 4 — 0.lxt and x is rising if Axt+i < 0 then y, < 4 — 0.lxt and x is falling and if Ayt+\ > 0 then _y, > 2.6667 + 0.1667.*;, and y is rising if A}v+i < 0 then y, < 2.6667 + 0.1667.*;, and y is falling These vector forces are also illustrated in figure 5.14. The figure also illustrates that a saddle point equilibrium is present with a stable arm S\S\ and an unstable arm S2S2. In general, the system will move away from the equilibrium point, except for initial values lying on the stable arm S\S[. Although the vector force diagram of discrete systems can provide much information on the dynamics of the system, the stability properties usually can be obtained only from its mathematical properties. 5.8 Internal and external balance We can illustrate the use of the phase plane by considering an important policy issue that has been discussed in the literature, namely internal and external balance.7 Having set up a macroeconomic model, a fixed target policy is then imposed on it. Two fixed targets are chosen: the level of real income and the balance on the balance of payments. Real income is considered set at the full employment level, 7 See Shone (1989, chapter 11) and the seminal article by Mundell (1962). 240 Economic Dynamics which denotes the condition of internal balance. External balance represents a zero balance on the combined current and capital account of the balance of payments. Following Tinbergen's analysis (1956), there are two policy instruments necessary for achieving the two policy objectives. These are government spending, which is used to achieve internal balance, and the interest rate, which is used to achieve external balance (by influencing explicitly net capital flows). Suppose we set up an adjustment on the part of the two instruments that assumes that the change in the policy variable is proportional to the discrepancy between its present level and the level to achieve its target. More formally we have A&+1 = gt+i -8t = h(gt ~ g*) h < 0 Art+1 = rt+i -rt = k2(rt - rf) k2 < 0 where g* is the target level of government spending in period t, and r* is the target interest rate in period t. Example 5.17 Following Shone (1989) we have the following two equations relating g, and r, derived from a macroeconomic model, IB rt = -3.925 + 0.5& XB rt = 7.958 + 0.186^ where IB denotes internal balance and XB denotes external balance. In the case of internal balance we require g* which is equal to g* = 7.85 + 2rt, while for external balance we require r* which is equal to r* = 7.958 + 0.186g,. Then Agt+1 = ki(gt - 7.85 - 2rt) kx < 0 Art+1 = k2{rt - 7.958 - 0.186&) k2 < 0 The isoclines are where Ag,+i = 0 and Art+\ = 0 and the equations of which no more than represent the internal balance and external balance lines, respectively. The stationary values of g and r are where the two isoclines intersect, giving g* = 37.84 and r* = 15. The situation is illustrated in figure 5.15. The vectors of force are readily established (noting k\ and k2 are negative): if Agt+\ > 0 then rt > —3.925 + 0.5gt and gt is rising if Agf+i < 0 then rt < —3.925 + 0.5gt and g, is falling Also if Art+\ > 0 then r, < 7.958 + 0.186g, and r, is rising if Art+1 < 0 then rt > 7.958 + 0.186g, and rt is falling which are also illustrated in figure 5.15. The use of the spreadsheet is convenient in considering this problem. To do this we need to express and rt+\ in terms of gt and rt. We assume that k\ = —0.5 8 Internal balance can also be considered as a suitable income-inflation combination. See Shone (1979). Discrete systems of equations 241 IB(Ag =0) Figure 5.15. XB(Ar;+1=0) Figure 5.16. and &2 = —0.75. We then have gt+1 = 3.925 + 0.5gt + rt rt+i = 5.9685 + 0.1395& + 0.25r, The quadrant that typified the UK economy for periods in the 1960s is quadrant IV (figure 5.15), which has the economy with a balance of payments deficit (below the XB curve) and unemployment (above the IB curve). Suppose, then, that the economy begins at point (go, fo) = (20, 9). The trajectory the economy follows is shown in figure 5.16. It is very easy to use this spreadsheet to investigate the path of the economy in any of the four quadrants, and we leave this as an exercise. What can readily be established is that, regardless of the initial point, the economy moves towards the equilibrium point where the two isoclines intersect. 242 Economic Dynamics Example 5.18 But we have presupposed that government spending is used to achieve internal balance and the rate of interest is used to achieve external balance. Suppose we assume the opposite assignment: set interest rates to achieve internal balance and government spending to achieve external balance. Then r*t = -3.925 + 0.5& and g*t = -42.785 + 5.376r, Then Art+l = k3(rt - rf) = k3(rt + 3.925 - 0.5gt) k3 < 0 Agt+1 = k4(gt - gf) = k3(gt + 42.785 - 5.376r,) k4 < 0 Setting Art+\ = 0 and Agt+\ = 0 gives rise to the internal balance isocline and the external balance isocline, respectively, leading to the same equilibrium point (g*, r*) = (37.85, 15), as shown in figure 5.17. However, the vector forces are now different: if Art+\ > 0 then r, < —3.925 + 0.5gt and r, is rising if Art+\ < 0 then rt > —3.925 + 0.5gt and rt is falling Similarly if Agt+\ > 0 then rt > 7.958 + 0.186^ and gt is rising if Agt+1 < 0 then rt < 7.958 + 0.186& and gt is falling From figure 5.17 it is apparent that the equilibrium point E is a saddle point. Under this assignment economies finding themselves in sectors I and III will converge on the equilibrium only so long as they remain in these sectors, although this is Figure 5.17. XB (Ag,.,= 0) 8 g, Discrete systems of equations 243 not guaranteed (see exercise 9). Any point in sectors II and IV, other than the equilibrium point, however, will move away from equilibrium (figure 5.17). Given this alternative assignment, and assuming £3 = —0.75 and £4 = —0.5 we obtain the two equations gt+1 = -21.3925 + 0.5gt + 2.688r, rt+1 = -2.94375 + 0.375& + 0.25r, Taking a point once again in quadrant IV, but now close to the equilibrium, point (37, 14), the spreadsheet calculations quite readily show the economy diverging from the equilibrium, as illustrated in figure 5.18, where we plot only up to period 9.9 Again it is very easy to use this spreadsheet to investigate the path of the economy for any initial point in any of the four quadrants. This we leave as an exercise. Comparing figures 5.15 and 5.17 leads to an important policy conclusion. It is not the slopes of the internal and external balance lines perse which governs the dynamics, but rather the policy assignment. This was Mundell's conclusion (1962). He put it differently and claimed that stability requires pairing the instrument with the target over which it has the greatest relative impact: the principle of effective market classification. Government spending has the greatest relative impact on income and hence on achieving internal balance. This immediately implies that the interest rate has the greatest relative impact on the balance of payments, and hence on achieving external balance. Consequently, the assignment represented in figure 5.15 is stable while that in figure 5.17 is unstable. This approach to dynamics is readily generalised. Consider again internal and external balance, but now using the two instruments government spending, g, and the exchange rate, S. The situation is shown in figure 5.19 (Shone 1989, chapter 11). We can capture this situation with the two linear equations IB St = a0 - aigt ai > 0 XB St = b0 + bigt bi>Q 9 Note how sensitive discrete systems can be to initial conditions. Placing the 'equilibrium' values (37.847,14.9985) as the initial conditions, still has the system diverging! 244 Economic Dynamics where ao > bp. Equilibrium is readily found to be + _ ap - bp ^ _ apb\ + a\bp a\ + b\ a\ + b\ However, in order to consider the dynamics of this point we need to specify some assignment. Suppose we assign government spending to internal balance and the exchange rate to external balance, satisfying Agt+i = gt+i -gt = Mgt -g*t) h < 0 ASt+1 = St+1 -St = k2(St - Sf) k2 < 0 The equations for g* and S* are g* = (ao/fli) - (1M)5, Sf = bp + b\gt Hence IB Agt+1 = ki[g, - (ao/ai) + (l/ax)St] XB ASt+1 = k2[St -bp- blgt] It is readily established (noting k\ and k2 are negative) that: if Agt+\ > 0 then St < ap + a\gt and gt is rising if Agt+\ < 0 then St > ap + a\gt and gt is falling if ASt+\ > 0 then St < bp + b\gt and St is rising if A5,+i < 0 then St > bp + b\gt and St is falling Discrete systems of equations 245 with resulting forces illustrated in figure 5.19. This quite clearly illustrates an anticlockwise motion. What we cannot establish is whether the system converges on the equilibrium or moves away from it. Example 5.19 To illustrate this suppose we have for internal and external balance the equations IB 5, = 20 - 2gt XB 5, = -4 + 4g, with equilibrium g* = 2.5 and S* = 10. Further, suppose adjustment is of the form Ag,+1 =-0.15(gt - g*) A5,+1 = -0.5(5, - 5f) where gf = -2.5 + 0.55, % = 20 - Agt then gt+1 = -1.875 + 0.25g, + 0.3755, 5,+1 = 10 - 2gt + 0.55, Given an initial point (go, So) = (4, 12), figure 5.20 shows the typical anticlockwise spiral trajectory that is tending towards the equilibrium. 5.9 Nonlinear discrete systems Just as we can encounter nonlinear equations of the form xt = f(xt-\), so we can have nonlinear systems of equations of the form xt =f(xt-\,yt-\) yt = g(pct-\,yt-\) 246 Economic Dynamics where we assume just a one-period lag. A steady state (x*, y*) exists for this system if it satisfies x*=f(x*,y*) *-g(x*,y*) y It is possible to investigate the stability properties of this nonlinear system in the neighbourhood of the steady state so long as / and g are continuous and differ-entiable. Under such conditions we can expand the system in a Taylor expansion about (jc*, y*), i.e. xt-x* = —-(xt-i -x)-\----(yt-i - y ) Let yt-y an an dxt-i dg(x*,y*) dxt-i df(x*,y*) dxt-i dg(x*,y*) tyt-i , , dg(x*,y*), (xt-i -x) + —-(yt-i - y ) oyt-i df(x*,y*) dX; an an t-i tyt-i dg(x*,y*) dyt-i then Xt-x* = an(xt-i - x*) + an(yt-i ~ y*) y,-y* = a2i(xt-i - x*) + a22(yt-i - y*) or xt an a\2 'xt-i -x*' Jt -y*. a2\ a22_ yt-\ — y* _ i.e. ut = Au,_i which is no more than a first-order linear system with solution ut = VD'V^uo and where D is the diagonal matrix with distinct eigenvalues r and s on the main diagonal, V = [ vr \s] is the matrix of eigenvectors associated with r and s, and Uo is a vector of initial values. It should be noted that the matrix A is simply the Jacobian matrix, J, of the nonlinear system evaluated at a fixed point (x*, y*). Under certain restrictions on A (or J), the linear system 'behaves like' the nonlinear system. The situation can be illustrated by means of figure 5.21. The nonlinear system, N, can be mapped into an equivalent linear system, L, by the mapping F, such that the qualitative properties of the linear system in the neighbourhood of 0 are the same as that of the nonlinear system in the neighbourhood of 0.10 In other words, the two systems are topologically equivalent. 10 F is then said to be a diffeomorphism. Discrete systems of equations 247 Nonlinear (N) Linear (L) Figure 5.21. What are these restrictions? We require tr(J) ^ 0 and det(J) ^ 0.11 Since the two systems are topologically equivalent, then the dynamics of N in the neighbourhood of 0 can be investigated by means of the linear system in the neighbourhood of 0. We do this by establishing the eigenvalues and eigenvectors of J. More specifically, we require conditions to be imposed on the eigenvalues. If r and s are the two eigenvalues then the systems are topologically equivalent if: (1) r and s are distinct real roots and \r\ < 1 and \s\ < 1. (2) r and s are distinct but complex and lie strictly inside the unit circle.12 If r and s in the neighbourhood of (x*, y*) satisfy this second condition, then (x*, y*) is said to be a hyperbolic fixed point. Exercises 1. Convert example 5.4 into a system of difference equations and establish the qualitative properties of the system in terms of the isoclines and vector forces. Does this confirm the stability established in section 5.2? 2. Convert example 5.6 into a system of difference equations and establish the qualitative properties of the system in terms of the isoclines and vector forces. Does this illustrate that the equilibrium point is a saddle path solution? 3. For example 5.4 use a spreadsheet to verify that the system converges on the fixed point (x*, y*) = (6.4, 20.8) for each of the following initial points: (3,10), (3,30), (10,10), (10,30). 4. Set up example 5.5 on a spreadsheet and investigate its characteristics. In particular, consider: (a) different initial values (b) plot x(t), y(t), and z{t) against t (c) plot y(t) against x(t) (d) plot z(t) against x(t) (e) plot x(t) against z(t). 11 An alternative way to state the condition is that J must be invertible. 12 See sub-section 3.8.1, p. 110. 248 Economic Dynamics 5. Use either Mathematica or Maple and do a 3D plot of the solution to example 5.5 for the initial point u° = (3, —4.3). 6. Use either Mathematica or Maple and derive a 3D directional field for example 5.5. Does this give you any more insight into the dynamics of the system over what you gained from question 4? 7. For each of the following systems: (a) find the eigenvalues of the system (b) find the eigenvectors of the system (c) establish the diagonal matrix of the system. (i) xt+1 = yt yt+i = -xt (ii) xt+\ = -2xt + yt yt+i =xt + 2yt (iii) xt+i = 3xt - 4yt yt+i = xt- 2yt 8. For the model IB: rt = -3.925 + 0.5ft XB: rt = 7.958 +0.186^ where Agt+l = -0.5(gt-g*) Art+1 = -0.75(r, - rf) use a spreadsheet to plot the trajectories for the following initial values: (a) (so.ro) = (20,12) (b) (g0,r0) = (20,20) (c) (g0,r0) = (50,10) (d) (g0,r0) = (50,20) 9. For the assignment of interest rates to achieve internal balance and government spending to achieve external balance, we have the set of equations gt+1 = -21.3925 + 0.5ft + 2.688r, rt+1 = -2.94375 + 0.375ft + 0.25r, (a) Take four initial points, one in each of the sectors represented in figure 5.17, and use a spreadsheet to investigate the economy's trajectory. (b) Take a variety of points in sectors I and III and establish whether the trajectories remain in these sectors. (c) Take a point on the stable arm and, using a spreadsheet, establish that the trajectory tends to the equilibrium point. 10. For the system IB: St = 5 + 2ft XB: St = 20 -4ft Discrete systems of equations 249 where 11. 12. 13. 14. 15. Ag,+i AS-. = and -0.15(gt-g*) -0.5(5, - 5?) 8t st -2.5 + 0.55, 20 - 4gt for the target equations, using a spreadsheet establish the trajectories for the following initial conditions: (a) (g0,S0) = (2.5,12) (b) (soA) = (3,10) (c) (soA) = (1,5) (d) (£0,5o) = (l,12) Let ITlA = 2 3 1 -2 mB = 4 -2 1 -1 mC = 3 2 1 -10 3 Using any software package, perform the following operations (i) Trace and determinant of mA x mB (ii) Transpose of mA x mC (iii) Inverse of mB (iv) Eigenvalues and eigenvectors of mA and mB (v) Characteristic polynomial of mA. Solve the following system using a software package xt+i = -5 + xt- 2yt yt+i = 4 + xt-yt xo = 1, yo = 2 What is the Jordan form, J, and the transition matrix, V, of the following matrix? mA 1 -2 1 -1 Hence show that V" .mA.V = J. For the system in question 12, set this up on a spreadsheet. (i) What is the fixed point of the system? (ii) Plot the trajectory from the initial point. Does this trajectory converge on the fixed point? For the following system, establish the Jordan form and the transition matrix. Represent the original system and its canonical form on a spreadsheet, and hence show whether the system is asymptotically stable. x, = 5.6 — 0Axt-i y, = 3.5 + OAx,-! - 0.5v,_i x0 = 2, y0 = 1 250 Economic Dynamics Additional reading Discrete systems of equations are discussed less frequently than continuous systems of equations, but additional material on the mathematical contents of this chapter can be found in Azariades (1993), Chiang (1984), Elaydi (1996), Goldberg (1961), Griffiths and Oldknow (1993), Holmgren (1994), Kelley and Peterson (2001), Lynch (2001), Sandefur (1990), Shone (2001), Simon and Blume (1994) and Tu (1994). On internal and external balance, references will be found in the main body of the chapter in section 5.8. CHAPTER 6 Optimal control theory 6.1 The optimal control problem Consider a fish stock which has some natural rate of growth and which is harvested. Too much harvesting could endanger the survival of the fish, too little and profits are forgone. Of course, harvesting takes place over time. The obvious question is: 'what is the best harvesting rate, i.e., what is the optimal harvesting?' The answer to this question requires an optimal path or trajectory to be identified. 'Best' itself requires us to specify a criterion by which to choose between alternative paths. Some policy implies there is a means to influence (control) the situation. If we take it that x(t) represents the state of the situation at time t and u{t) represents the control at time t, then the optimal control problem is to find a trajectory {x(t)} by choosing a set {u(t)} of controls so as to maximise or minimise some objective that has been set. There are a number of ways to solve such a control problem, of which the literature considers three: (1) Calculus of variations (2) Dynamic programming (3) Maximum principle. In this chapter we shall deal only with the third, which now is the dominant approach, especially in economics. This approach is based on the work of Pontryagin et al. (1962), and is therefore sometimes called the Pontryagin maximum principle. Since minimising some objective function is the same as maximising its negative value, then we shall refer in this chapter only to maximising some objective function. Second, our control problem can either be in continuous time or in discrete time. To see the difference and to present a formal statement of the optimal control problem from the maximum principle point of view, consider table 6.1. In each case, the objective is to maximise / and so find a trajectory {x(t)} by choosing a suitable value {«(?)}• What table 6.1 presents is the most general situation possible for both the continuous and discrete formulations of the optimal control problem under the maximisation principle. There are some special cases, the most important being the distinction between finite and infinite horizon models. In the latter case the terminal time period is at infinity. All the problems we shall discuss in this chapter involve autonomous systems, and so t does not enter explicitly into V,f or F. An important aspect of control problems is that of time preference. The 252 Economic Dynamics Table 6.1 The control problem Continuous Discrete f'1 max J= / V(x, u, t)dt + F(x\ t) x = /(x, u, t) x(f0) = x° x(fi) = X1 {u(0} e U f0 (or t = 0) is initial time t\ (or T) is terminal time x(f) — {x\(t), ..., x„(t)} or x, — {x\t, ..., xni} n-state variables x(?o) = x° or x, = x° for t = 0 is the initial state x(t\) = x1 or x, = xr for t = T is the final state (or terminal state) u(f) = {u\{i), ..., um(t)} or u, — {u\t, ..., umt] m-control variables {u(f)} is a continuous control trajectory to < t < t\ {u,} is a discrete control trajectory 0 < t < T U is the set of all admissible control trajectories x(f) =/(x, u, i) or xt+\ — xt =f(xt, ut, i) denote the equations of motion J is the objective function V(x(f), u(f), t) or V(xt, ut, t) is the intermediate function Fix1, i) or F(xT, t) is the final function r-i max J - V(xt, ut, i) + F(xT, i) {ui] xt+1 -xt =f(xt,ut,t) xt = x° when t = 0 x, = xr when t = T U, € U simplest models involve no discounting. It is sometimes easier to consider a model with no discounting, and then to consider the more realistic case of the same model with discounting. In many models the terminal value F(xT) is zero, but this need not always be so. A typical continuous optimal control problem incorporating the assumptions of (1) a finite time horizon, T, (2) only autonomous equations, (3) a zero function in the terminal state and, (4) only one state variable and one control variable is fT max / = / V(x, u)dt {«00} Jo (6.1) x=f(x,u) x(0) = x° x(T) = xT where the state variable, x and the control variable, u, are both functions of time t. The situation is illustrated in figure 6.1. The paths u* and u** both constitute solutions to the differential equation x = f(x, u). The problem, however, is to choose one path that maximises the relation / and that satisfies the terminal condition x(f) = xT and .*;(?**) = xT. 6.2 The Pontryagin maximum principle: continuous model As just pointed out, the objective is to find a control trajectory {u{t)} that maximises / and takes the system from its present state x° to its terminal state xT. What is required, therefore, is a 'set of weights' that allows a comparison of the different trajectories of alternative controls. Also note that the emphasis of this formulation Optimal control theory 253 Figure 6.1. of the control problem is to find the optimal control trajectory {u(t)}. Once this is known the optimal state trajectory {x(t)} can be computed. The 'weights' are achieved by defining a Hamiltonian for the control problem (6.1). As with Lagrangian multipliers, let k(t) denote the Lagrangian multiplier for the constraint x =f(x, u). This is referred to as the costate variable or adjoint variable. Then V(x, u)dt + I k[f(x, u) — x]dt Jo [V(x, u) + kf(x, u) — kx]dt -f Jo -f Jo The Hamiltonian function is defined as H(x, u) = V(x, u) + kf{x, u) Hence fT L = I [H(x, u) — kx]dt Jo Equation (6.3) can be further transformed by noting that (see exercise 2) - / Xdt= f xXdt- [X(T)x(T) - A(OMO)] Jo Jo which allows us to express L as L = f [H(x, u) + Xx\dt - [X(T)x(T) - X(0)x(0)] Jo Consider what happens to the state variable when the control variable changes, i.e., let {u(t)} change to {u{t) + Au(t)} with the result on the state trajectory from {x(t)} to {x(t) + Ax(t)}. Then the change in the Lagrangian, AL, is AL (6.2) (6.3) (6.4) (6.5) Jo Jo BH dH —dx H--du + kdx dx du dH , (dH . . , —du + I--\- X ]dx du \ dx dt - k(T)dxT dt - k(T)dxT 254 Economic Dynamics For a maximum AL = 0. This implies the necessary conditions: (i) dH du = 0 0 < t < T (Ü) k = dH dx 0 0 then H is a maximum at u = 3 the boundary, hence u*(t) = 3, as shown in Figure 6.2(a). From (ii) we have k = —k — 5 k*(t) = ke'1 - 5 k*(l) = ke~l -5 = 0 k = 5e1 .-. k*(t) = 5el~f - 5 Since u*(t) = 3 x* = x* + 3 x*(t) = -3 + ke1 x(0) = -3 + ke° = 2 256 Economic Dynamics Figure 6.2. (a) H 0.2 0.4 0.6 0.8 1 / Hence x*(t) = -3 + 5e' Although the control variable remains constant throughout, the state variable increases from ^(0) = 2, as shown in figure 6.2(b). Example 6.2 The control problem is max / u2 dt M Jo x = —u x(0) = 1 ^(1) = 0 The Hamiltonian for this problem is H(x, u) = V(x, u) + kf(x, u) = u2 + k(—u) Optimal control theory 257 with first-order conditions: dH (i) — = 2u - X = 0 du dH (ii) X =--=0 dx (iii) x = — u (iv) x(0) = 1 (v) x(l) = 0 From (i) 2u = X Thus u = ^X ■ _ k X~~2 X = 0 Solving these with a software package we obtain Xt x(t) = ci - — Ht) = c2 But ^(0) = 1 so 0 1 = c\-- or c\ = 1 2 Similarly ^(1) = 0 X x(l) = 1 - - = 0 V ' 2 :. X = 2 or c2 = 2 x* = 1--= 1 - t 2 u* = \X = 1 These optimal paths are illustrated in figure 6.3. Example 6.3 The control problem is max— / \(xl + ul)dt {»} Jo X = X + u x(0) = 2, x(l) = 0 258 Economic Dynamics Figure 6.3. Optimal path of control variable Optimal path of state variable The Hamiltonian for this problem is H(x, u) = V(x, u) + kf(x, u) -(x2 + u2) + k(x + u) With first-order conditions m u (1) — = --+k = 0 implying du 2 (Ü) (in) u = 2k dH -x * , k =--= ---\-k) = ±x- k dx \ 2 I 2 X = X + u implying x = x + 2k Substituting (i) into (iii) and eliminating u, we arrive at two differential equations in terms of x and k x = x + 2k k k Although a simple set of differential equations, the solution values are rather involved, especially when solving for the constants of integration. The general solution is2 x(t) = cie^1 + c2e~^2t k(t) = Cj-(V2 - l)e^2t - j(V2 + X)e-^2t However we can solve for c\ and c2 by using the conditions ^(0) = 2 and ^(1) = 0 as follows *(0) = c\ + c2 = 2 x(l) = de^21 + c^-^21 = 0 2 The software packages give, on the face of it, quite different solutions. They are, however, identical. The results provided here are a re-arrangement of those provided by Maple. Optimal control theory 259 Solving we get c\ = —0.1256 and c2 = 2.1256. All this can be done with the help of computer software programs, with the resulting trajectories for x* and u* shown in figure 6.4(a) and (b). What these examples show is a pattern emerging for solving the control problem. The steps are: (1) Specify the Hamiltonian and obtain the maximisation conditions (2) Use the equation dH/du to solve for u in terms of the costate variable X (3) Obtain two differential equations: one for the state variable, x, and one for the costate variable, A (4) Solve the differential equations deriving general solutions (5) Use the conditions on ^(0) and x(T) to obtain values for the coefficients of integration (6) Substitute the optimal path for A* into the equation for u to obtain the optimal path u* for the control variable. 6.3 The Pontryagin maximum principle: discrete model The discrete time control model based on the maximum principle of Pontryagin takes a similar approach to the continuous time formulation so we can be brief, 260 Economic Dynamics (6.8) although some care must be exercised in the use of time periods. Again we let x denote the only state variable, u the only control variable and k the costate variable. Our problem amounts to: r-i max / = ^ V(xt, Ut) {u,] t=0 %t+l ~ xt =f(Xt, Ut) xq = a The Lagrangian is then r-i (6.9) L = ^ {V&t, ut) + kt+1 [f(xt, ut) - (xt+i - xt)]} t=o Define the discrete form Hamiltonian function (6.10) H(xt, ut) = V(xt, ut) + kt+1f(xt, ut) then r-i L=^2 [H(xt, ut) - kt+1(xt+i - xt)] t=0 which can be maximised by satisfying the conditions dL dH — = — = 0 t = o,..., r - i dut dut dL dH — = — +kt+1-kt = 0 t=l,...,T-l dxt oxt dL dH - (xt+i -xt) t = 0, ..., T - 1 dkt+\ dkt+\ dL -kT = 0 3^7- More succinctly: dH (i) — = 0 t = 0, ..., T - 1 dut dH (n) kt+i-kt = -— t=\,...,T -\ dxt (6-11) dH (m) xt-i -xt = —- =f(xt, ut) t = 0, 3^+i (iv) kT = 0 (v) x0 = a It is useful to verify these conditions for, say, T = 3, most especially noting the range for t for condition (ii). But how do we go about solving such a model? Unlike the continuous time model it is not simply solving two differential equations. It is true that in each time period we have two difference equations for the state and costate variables that require solving simultaneously. One solution method is to program the problem, as Optimal control theory 261 in Conrad and Clark (1987). A simpler method in the case of numerical examples is to use a spreadsheet. To illustrate the solution method by means of a spreadsheet, consider the following example. Example 6.43 Iron ore sells on the market at a constant price p per period but costs ct = byt /xt, where xt denotes the remaining reserves at the beginning of period t and yt is the production in period t. The mine is to be shut down in period 10. What is the optimal production schedule {y*} for t = 0, ... ,9 given p = 3, b = 2 and the initial reserves xq = R = 600 tons? (Assume no discounting over the period.) Let us first set up the model in general terms, replacing ut by yt. The objective function V(xt, yt) is no more than the (undiscounted) profit, namely \t( \ bylt ( byt\ V(xt, yt)=pyt--= p--\yt xt \ xt J Next we note that if xt denotes the remaining reserves at the beginning of period t, then xt+\ = xt — yt or xt+\ — xt = —yt. Thus, our Hamiltonian function is ( byt\ H(xt, yt) = I p - — \yt - h+\yt Our optimality conditions are therefore: 2byt —- - Xt+l =0 t = 0, ..., 9 '•" f"'! ,= 1....,9 (i) dH dyt = p- (Ü) -h (in) xt-\ -xt (iv) Xq = R (v) Xj = -- 0 dxt i -2 -yt t = 0,...,9 To solve this problem for a particular numerical example, let/? = 3, b = 2 and R = 600. The computations are set out in detail in figure 6.5. In doing these computations we begin in period 10 and work backwards (see exercise 1 on backward solving). Since Aio = 0 then from (i) we know 3-41 - I = 0 yXg which allows us to compute yg/xg. Having solved for yg/xg we can then use condition (ii) to solve for kg. We do this repeatedly back to period 0. This gives us columns 2 and 3 of the spreadsheet. Since xq = R = 600, we have the first entry in the x(t) column. Then yo is equal to xo(yo/xo) and finally x\ = xq — yo. This allows us to complete the final two columns. The optimal production path {y*} is therefore given by the final column in figure 6.5 and its path, along with that of the reserves, is shown in figure 6.6(a). 3 This is adapted from Conrad and Clark (1987, p. 20). 262 Economic Dynamics Figure 6.5. set at 600 copied up. =D6-E6. w 4E 5 6 _. 9 W 11" 12 13 14 15 16 17 Figure 6.5 formula copied ~7 s ~ 2.4084 2.3567 2.2945 2.2181 2.1216 1.9955 1.8221 15645 ', 1.1250' i.OOOO 0.1955 0.2196 0.2511 0.2945 0.3589 04688 T75ÖÖ: 132 353.3703 284.2924 221.8636 166.1480 117.2196 75.1510 39.9240 9.9810 75.6730 69.0778 62.4288 55.7156 48.9284 42.0686 35.2270 29.9430 0.0000 Example 6.4 ron ore sells on the market at a price of p per period but costs c(t}=2*[y(t)/x(t)]A2. where x(t) denotes the remaining reserves at trie beginning of period t and y(t) the production in period t. The mine is to be shut down in period 10. What is the optimal production schedule {y*(t)} for t=0_____9 given p=3 and the initial reserves are x(0)=R=600? (Assume no discounting over the period.) set ai zero =B16+$C$4*(C15A2) copied down Given the computations the trajectory for (k*, x*) can also be plotted, which is shown in figure 6.6(b), which are direct plottings from a spreadsheet. In this example we solved the discrete optimisation problem by taking account of the first-order conditions and the constraints. We employed the spreadsheet merely as a means of carrying out some of the computations. However, spreadsheets come with nonlinear programming algorithms built in. To see this in operation, let us re-do the present example using ExceVs nonlinear programming algorithm, which is contained in the Solver add-on package.4 The initial layout of the spreadsheet is illustrated in figure 6.7. It is important to note that when setting out this initial spreadsheet we place in cells B7 to B16 some 'reasonable' numbers for extraction. Here we simply assume a constant rate of extraction of 60 throughout the 10 periods t = 0 to t = 9. Doing this allows us to compute columns D and E. Column D sets k\o = 0 and then copies backwards the formula kl = kl+1 + (^) for cells D16 to D8 (no value is placed in cell D7). The values in column E are the values for the objective function V(xt, yt). The value for L, which is the sum of the values in column E for periods t = 0 to t = 9, is placed in cell El9. At the moment this stands at the value 1448.524. 4 On using Excel's Solver see Whigham (1998), Conrad (1999) and Judge (2000). Optimal control theory 263 a RgllSJI? A B c D ' E F 0 3 1 Figure 6.7 X 3 P = 3 R - 600 4 b = 2 6 t y[t) x(t) a-tt) V(x,y) 7 0 60.0000 600 168 0000 8 1 60.0000 540 0000 3.0735 166.6667 9 2 60.0000 4800000 3.0548 165.0000 10 3 60.0000 420 0000 3.0236 162.8571 11 60 0000 360 0000 2.9828 160.0000 12 5 60.0000 3000000 2 9272 156.0000 13 G 60.0000 240.0000 2 8472 150.0000 14 7 60.0000 180.0000 2.7222 140,0000 15 8 60.0000 120.0000 2.5000 120 0000 16 9 60.0000 60 0000 2.0000 60.0000 17 10 0 00000 0 18 19 L = Sum V = 1448 524 20 i -ir A 1 KH\SI... ■«2 / intMj / hi Figure 6.7. 264 Economic Dynamics Figure 6.8. 1 2 3 4 5 6_ 7 8 9 10 11 12 13 14 15 Figure 6.7 sej Target Cell: : . ]sE$:9 e^lfo-: :■ <* Max :"rMG'; ;:TöIl»:ö^; |ö~ r Bi? Changing ösllss1 i j$B$7:$B$16 Syfcjact to the Constraints: $B$7:$8$16 >= 0 -»3 Apö Change Qelate 16 17 tf 19 20 91 i Ii—Drrjonrr— L = Sum V 1448.524 Options ß.eset Ail btelp Of course it would be most unlikely if L were at a maximum with such arbitrary numbers for extraction. The maximum control problem is to maximise the value in cell E19, i.e., maximise L, subject to any constraints and production flows. The constraints are already set in the spreadsheet, although we do require others on the sign of variables. First move the cursor to cell El9 and then invoke the solver. By default this is set to maximise a cell value, namely cell El9. We next need to inform the programme which is the control variable and hence which values can be changed, i.e., what cells it can change in searching for a maximum. These are cells B7 to B16. In specifying the above problem we implicitly assumed xt and y, were both positive. In particular, we assumed the level of production, the control variable, was positive. We need to include this additional constraint in the Solver so that any negative values are excluded from the search process. The Solver window is shown in figure 6.8. Once all this information has been included the Solver can do its work. The result is shown in the spreadsheet in figure 6.9. As can be observed this gives more or less the same results as figure 6.5, as it should. The value of the objective function has also increased from 1448.52 to 1471.31. It should be noted in figure 6.9 that in period 10 we have A(10) = 0 and at this value x(!0) = 9.9811. We have to assume that the reserves in period 10 are therefore 9.9811 and that these are simply left in the ground. In other words, x(T) is free. The shadow price of a free product is zero, hence A(10) = 0, and this implies it is not optimal to mine the remaining reserves. Hence F(xT) = 0 or xT is free. We have spent some time on this problem because it illustrates the use of spreadsheets without having to handle algebraically the first-order conditions. It also has the advantage that it can handle comer solutions.5 Most important of all, it provides a way of solving real-life problems. 5 Corner solutions would require setting out the Kuhn-Tucker conditions for optimisation. See Chiang (1984), Simon and Blume (1994) and Huang and Crooke (1997). Optimal control theory 265 JOS (13 A B C E F ~ 1 Figur« 6.9 1 7 3 P = 3 R = 600 4 2 5 6 t y(t) x(t] * Iron ore sells on the market at a price of p 7 0 182 2140 600 435.9639 per pe-iod but costs ;{tj - 2,y(ti/>2V't). 8 1 127 8322 417.78S0 1 7654 277,5179 wtiere x(t) denotes the remaining reserves 9 2 89.6113 230.9537 1.7767 176.4004 at Die beginning of pettod t and y(t) the 10 3 63.0999 200 3424 1.7634 112.3604 production in period t. The mine is to be 11 4 44 3863 137.24 26 1.7413 71 3397 shut down in period 10. What is trie 12 5 31.3973 92 8563 1 7062 4i> 302(] optimal production schedule. y*[t). for 13 6 22 2845 61.4539 3 6482 236150 t=Q 9 given p = 3, the initial reserves are 14 7 15.9201 39. T 745 1.5501 17 3686 R=600 and the discount rate is 10%? 15 8 11.4942 23.2543 1.3746 10.7356 16 9 8.8238 11.7601 1 0236 5 6109 17 10 0 2.9362 0 13 ■ 19 L = Sum V - 1131.769 20 ■j 21 22 i 1 * H\=>i~« hi ) Figure 6.10. (iv) kT = 0 (v) x0=R = 600 It should be noted that the only difference between this and the undiscounted conditions is in terms of condition (i), where kt+\ is multiplied by the discount factor. Once again we use Excel's Solver to handle the computations of this problem, with the results shown in figure 6.10, which should be compared with figure 6.9. Notice once again that in period 10 we have A(10) = 0 and that at this value *(10) = 2.9362. This level of reserves in period 10 is simply left in the ground, x(T) is free. The shadow price of a free good is zero, hence A(10) = 0, and this implies it is not optimal to mine the remaining reserves. Put another way, it is cheaper to leave the remaining reserves unmined than incur the costs of mining them. What these computations show is a similar trajectory for optimal production but starting from a much higher level of production. This is understandable. The future in a discounting model is weighted less significantly than the present. The comparison is shown in figure 6.11. Consider now discounting under a continuous time model. Consider the control problem max/= / e~StV(x, u)dt Wit)} Jo x=f(x,u) (6.18) jc(0) = 0 x(T) = xT The Lagrangian is L= f {e~StV(x, u) + k[f(x, u) - x]}dt Jo 268 Economic Dynamics Figure 6.11. Optimal production 200 No discounting - - ■ - ■ 10% discounting and the Hamiltonian is H{x, u) = e~StV(x, u) + kf{x, u) Define the current value Hamiltonian function, Hc, by Hc{x, u) = V(x, u) + fxf(x, u) then Hc = He' St or H = Hce -St ix = Xe&t or X = \xe &t Now reconsider our five optimality conditions. Since eu is a constant for a change in the control variable, then condition (i) is simply dHc/du = 0. The second condition is less straightforward. We have dH dH dx dx ce-St From A = \xe St X = (ie~&t - 8[ie~u Equating these we have dH ce~St = ße~St - 8[ie-St dx dHc or fi =---h 8{i dx Condition (iii) is , _ dH _ dHc _st _ dHc dk dk dfx while condition (iv) becomes k(T) = ix{T)e~&t = 0 and condition (v) remains unchanged. : fix, u) Optimal control theory 269 To summarise, define the current value Hamiltonian and current value Lagrangian multiplier, i.e. Hc(x, u) = H(x, u)eSt = V(x, u) + iif(x, u) where X = fie~St. Then the optimality conditions are: dHc (i) —- = 0 0 < t < T du dHc (ii) fi =--- + SfM 0 0 implies x > —3 X < 0 implies X > —5 so we know that the optimal trajectory starting from x(0) = 2 will lead to a rise in the state variable x and a fall in the costate variable A. This is verified in figure 6.12. The system begins from point (x(0), A(0)) = (2, 8.5914), satisfying the initial condition on the state variable x; and has a terminal point (x(l), A(l)) = (10.5914, 0), which satisfies the terminal condition on the costate variable, X. Of all possible trajectories in the phase plane, this is the optimal trajectory. Example 6.2 (cont.) In example 6.2 we derived the following two differential equations x = -\X X = 0 There is only one isocline for this problem. When x = 0 then X = 0 and so the x-isocline coincides with the x-axis. Our initial point is (x(0), A(0)) = (1,2) and Optimal control theory 271 Figure 6.12. \ \ \ \ \\\\\\S\ \ \\\\\\\\\\\ \\\\\\\\\\ \\\\\\\\\\ , ,\\\\\\\\\\ \ \ \\\\w WW \w \\\\ V A, 3« 2.5* Z 1.5: t 0.5« 0.2 0.4 0.6 Figure 6.13. 0.8 1 X for X > 0 we have x < 0 and so the trajectory is moving to the left. Earlier we demonstrated that A remains at the value of 2 throughout the trajectory. When t = 1 then *(1) = 0, which satisfies the condition on the terminal point, which in the phase plane is the point (*(1), A(l)) = (0, 2). As can be seen in terms of figure 6.13, the optimal trajectory in the phase plane is the horizontal line pointing to the left. Example 6.3 (cont.) The two differential equations we derived for example 6.3 were x = x + Tk k '=- ^x — k 272 Economic Dynamics When x = 0 then k = — ^x and when k = 0 then k = ^x. We have therefore two distinct isoclines in this example. Furthermore, if x > 0 then x + 2k > 0 implying k > — ^x Hence, above the ^-isocline, x is rising while below it is falling. Similarly, if k > 0 then ^x — k > 0 implying k < ^x Hence, below the A-isocline, k is rising while above it is falling. This suggests that we have a saddle-point solution. This is also readily verified by considering the eigenvalues of the system. The matrix of the system is with eigenvalues r = s/2 and s = —s/2. Since these are real and of opposite sign, then we have a saddle point solution. When t = 0 we already have ^(0) = 2 but we need to solve for k(0). But A(0) = |(V/2-l)-|(V/2+l) and we know that c\ = —0.1256 and c2 = 2.1256. Substituting these values we get k(0) = —2.6. The initial point (^(0), k(0)) = (2, —2.6) therefore begins below the ^-isocline, and so the vector forces are directing the system up and to the left. The optimal trajectory is shown in figure 6.14. It is apparent from example 6.3 and 6.6 that the maximisation approach of Pontryagin gives us first-order conditions in terms of the Hamiltonian which, in the present simple models, leads to two differential equations in terms of the state variable x and the costate variable k (or /x). Control problems, however, pose two difficulties: (1) the differential equations are often nonlinear (2) in economics functional forms are often unspecified.6 Even the most simple control problem can lead to nonlinear differential equations, and although we have developed techniques elsewhere for dealing with these,7 until the advent of the computer they were largely left to the mathematician. When the functional forms are not even specified then there are no explicit differential equations to solve. However, the qualitative properties of the fixed points can still be investigated by considering the system's qualitative properties in the phase plane. First consider a simple example for which we have an explicit solution. 6 What we often know are certain properties. Thus we may have a production function y = f(k) where f(k) is unspecified other than being continuous, differentiable and where f'(k) > 0 and f"(k) < 0. 7 See sections 2.7 and 3.9. Optimal control theory 273 Figure 6.14. \\\\\\\ \ \ WW \ \ \ W.N.^^.--. \, \ \ \ \ X=Q (k= (l/2)x) x=0 (X=~( 1/2» Example 6.78 Our problem is poo max/= / (201nx - 0Au2)dt {"} Jo i = u — O.lx jc(0) = 80 The Hamiltonian for this problem is H = 20 lnx - 0.1a2 + X(u - O.Ijc) with first-order conditions — = -0.2a + X = 0 3a 3/f /20 A =--=---OAX 3x \ x i = a — O.lx which can be reduced to two differential equations in terms of x and X x = —O.lx + 5X -20 X = -+ 0.1X x The fixed point of this system is readily found by setting x = 0 and X = 0, giving x* = 100 and X* = 2. Furthermore, the two isoclines are readily found to be X = 0.02x (x = 0) 200 X = - (X = 0) x and illustrated in figure 6.15. 8 Adapted from Conrad and Clark (1987, pp. 46-8). 274 Economic Dynamics Figure 6.15 also shows the vector of forces in the four quadrants, which readily indicate a saddle point solution. This can be verified by considering a linearisation about the fixed point (x*, X*) = (100, 2). This gives the linear equations x = -0.1(x - x*) + 5(X - X*) X = 0.002(jc - jc*) + 0A(X - X*) The resulting matrix of the linear system is _r -o.i 5" ~ [ 0.002 0.1 _ with eigenvalues r = 0.14142 and s = —0.14142, confirming a saddle point solution. To establish the equations of the arms of the saddle point solution, take first the eigenvalue r = 0.14142. Then (A - rl)vr = 0 i.e. /r —o.i 5 \L 0.002 0.1 or "-0.24142 5 0.002 -0.04142 Using the first equation, -0.24142vi + 5V2 = 0 -0.14142 1 0 \ vi 0 0 1 ) v5 0 vi 0 0 Figure 6.15. Optimal control theory 275 Let v\ = 1, then 5vr2 = 0.24142 vr2 = 0.048284 Therefore, (k - k*) = 0.048284(jc - jc*) (k — 2) = 0.048284(jc - 100) i.e. k = -2.8284 + 0.48284* and since this is positively sloped it represents the equation of the unstable arm. Now consider the second eigenvalue, s = —0.14142 '-0.1 5 0.002 0.1 (A - rl)\s + 0.14142 1 0 0 0 1 ) 0 i.e. 0.04142 5 0.002 0.24142 Using again the first equation, then 0. 04142V} + 5v^ = 0 1, then "0" A. o_ Let Vi 5v* -0.04142 vs2 = -0.008284 Therefore, (k - k*) = -0.008284(* - jc*) (k — 2) = -0.008284(* - 100) i.e. k = 2.8284 - 0.00828284* and since this is negatively sloped it represents the equation of the stable arm. If *(0) = 80 then the value of k on the stable arm is k(0) = 2.16568. The trajectory, along with isoclines and the stable arm, are shown in figure 6.16. Although the point begins on the stable arm, it gets pulled away before it reaches the equilibrium! What this diagram reveals is that this system is very sensitive to initial conditions. But the direction field does show a clear saddle point equilibrium. Example 6.8 (Ramsey growth model) In this example we shall consider the Ramsey growth model,9 which is the basis of much of the optimal growth theory literature. We shall consider the model in terms 9 Ramsey (1928). See also Burmeister and Dobell (1970), Barro and Sala-i-Martin (1995) and Romer (2001). 276 Economic Dynamics (6.20) of continuous time. We begin with simple definitions of income and investment, namely Y(t) = C(t) + I(t) I(t) = K(t) + SK(t) Hence Yjt) = Cjt) Kjt) SKjt) Lit) Lit) Lit) Lit) Kit) i.e. yit) = dt) ++ Skit) ^_d/K\ LK-KL K (K\ L But dt\L) L2 L \L J L _K L ~ L~ L We assume population grows at a constant rate n, so that L/L = n hence K — = k + kn L and yit) = c(t) + kit) + in + 8)k(t) If we have a homogeneous of degree one production function then we can express output, y, as a function of k. Thus, y = f(k). Dropping the time variable for convenience, we therefore have the condition (6.21) k=fik)-in + S)k-c In order to consider the optimal growth path we require to specify an objective. Suppose Uic) denotes utility as a function of consumption per head. The aim is Optimal control theory 277 to maximise the discounted value of utility subject to the equation we have just derived, i.e. /•OO max/= / e~ßtU(c)dt M Jo k=f(k)-(n + 8)k-c HO) = k0 0 0 then f(k*) > (n + 8 + fi) which implies k < k* as seen in terms of the upper diagram of figure 6.17. Hence, to the left of the c = 0 isocline, c is rising; to the right of c = 0, then c is falling. Similarly, if k > 0 then f(k*) — (n + 8)k* > c. Thus below the k = 0 isocline k is rising, while above the k = 0 isocline is falling. The vector forces clearly indicate that (k*, c*) is a saddle point solution. The only optimal trajectory is that on the stable arm. For any ko the only viable level of consumption is that represented by the associated point on the stable arm. Given the initial point on the stable arm, the system is directed towards the equilibrium. Notice that in equilibrium k is constant and so capital is growing at the same rate as the labour force. Furthermore, since k is constant in equilibrium then so is y, and hence Y is also growing at the same rate as the labour force. We have, therefore, a balanced-growth equilibrium. Example 6.9 (Ramsey growth model: a numerical example) Consider the optimal growth problem max / = k=f(k)-(n + 8)k-c k(0) = k0 0 0 and expectations take the form of adaptive expectations, i.e. 7te = fj(7t - Tt") f3>0 The government has just won the election at t = 0 and the next election is in 5 years' time. It assumes that voters have poor memories, and weight more heavily the economic situation the closer it is to the election. It accordingly assumes a weighting factor of e°-05t. 8. Solve the following control problem x = —x + u x(0) = 5, jc(1) = 10 Plot the trajectory in (x, A)-space. 9. Solve the following control problem max / (3x2 — u2)dt {"} Jo x = 2x + u x(0) = 10, jc(1) = 15 Plot the trajectory in (x, A)-space 10. Solve the equilibrium for the following Ramsey model. Linearise the system about the equilibrium and establish its stability properties /•OO max/= / e~omU(c)dt M Jo Optimal control theory 285 k = k03 - (n + 8)k-c k(0) = 1 where U(c) = 4c1/4, n = 0.02, 8 = 0.03 Additional reading Beavis and Dobbs (1990), Blackburn (1987), Bryson, Jr. and Ho (1975), Burmeister and Dobell (1970), Chiang (1992), Conrad (1999), Conrad and Clark (1987), Fryer and Greenman (1987), Intriligator (1971), Kirk (1970), Leonard and Long (1992), Pontryagin et al. (1962), Ramsey (1928), Romer (2001) and Takayama (1994). CHAPTER 7 Chaos theory 7.1 Introduction The interest and emphasis in deterministic systems was a product of nineteenth-century classical determinism, most particularly expressed in the laws of Isaac Newton and the work of Laplace. As we pointed out in chapter 1, if a set of equations with specified initial conditions prescribes the evolution of a system uniquely with no external disturbances, then its behaviour is deterministic and it can describe a system for the indefinite future. In other words, it is fully predictable. This view has dominated economic thinking, with its full embodiment in neoclassical economics. Furthermore, such systems were believed to be ahistoretic. In other words, such systems were quite reversible and would return to their initial state if the variables were returned to their initial values. In such systems, history is irrelevant. More importantly from the point of view of economics, it means that the equilibrium of an economic system is not time-dependent. Although the physical sciences could in large part undertake controlled experiments and so eliminate any random disturbances, this was far from true in economics. This led to the view that economic systems were subject to random shocks, which led to indeterminism. Economic systems were much less predictable. The random nature of time-series data led to the subject of econometrics. The subject matter of econometrics still adheres to the view that economic systems can be captured by deterministic components, which are then augmented by either additive or multiplicative error components. These error components pick up the stochastic nature of the data series, most especially time-series data. For instance, the classical linear model takes the form (7.1) y = X/3 + £ £~iV(0,a2I) Not only is X/3 assumed linear but, more significantly from the point of view of our present discussion, it is assumed to be deterministic. All randomness is attributed to the error term. Even where the error term has distributions that are not normally distributed, the econometric approach effectively partitions the problem into a deterministic component and a random component. Randomness cannot and does not arise from the deterministic component in this approach. Such a view of the world is a 'shotgun wedding of deterministic theory with "random shocks'" (Mirowski 1986, p. 298). Mandelbrot (1987), in particular, was an ardent critic of the way econometrics simply borrowed classical determinism and added a random Chaos theory 287 component. The emphasis of econometrics on the Central Limit Theorem was, in Mandelbrot's view, flawed. It is not our intention here, however, to expand on these views. Suffice it to say that the dichotomy between deterministic economic elements of a system and additive random components is still the mainstay of the econometric approach. Three important considerations came out of these early discussions that are relevant for this chapter. 1. Linearity, far from being the norm, is the exception. Linear economic models lead to unique equilibrium points, which are either globally stable or globally unstable. On the other hand, nonlinear systems can lead to multiple equilibria and hence, local stability and instability. However, nonlinear systems also tend to lead to complexity. Until the development of chaos theory, there was little formal way of handling complex systems. On a more practical note, the study of complex systems could not have occurred without the development of computers and computer software. 2. Many economic time series are generated by discrete processes. The second consideration, once again highlighted by Mandelbrot, is that many economic time series are generated by discrete processes, and therefore should not be modelled as continuous processes. The emphasis of continuous processes comes, once again, from the physical sciences. We already noted in earlier chapters that a discrete equivalent of a continuous system could exhibit instability while its continuous version is stable! Although this is not always the case, it does highlight the importance of modelling systems with discrete models if discrete processes generate the time series. 3. The occurrence of bifurcations. A third strand was consideration of a system's equilibrium to changes in the value of important parameters. It became clear that for some systems, their behaviour could suddenly change dramatically at certain parameter values. This led to the study of bifurcation. This chapter is concerned with how deterministic systems can exhibit chaos, and to all intents and purposes have the characteristics of randomness: a randomness, however, which does not occur from random shocks to the system. We find that such behaviour occurs when parameters of the system take certain values. The parameter value at which the system's behaviour changes is called a bifurcation value. We therefore begin our discussion with bifurcation theory. 7.2 Bifurcations: single-variable case In this and the next section we shall confine ourselves to studying some of the properties of first-order systems that depend on just one parameter. We shall represent this with xt+1 =f(xt, X) (7.2) 288 Economic Dynamics in which/ is nonlinear. Two examples are : (i) xt+i = l.5xt(l -xt)-k (ii) xt+i = kxt(l -xt) and we shall make great use of these two equations to illustrate bifurcation theory and chaos. But what do we mean by the terms 'bifurcation' and 'chaos' ? Bifurcation theory is the study of points in a system at which the qualitative behaviour of the system changes. In terms of our general representation,/^*, k) denotes an equilibrium point, a stationary point, whose value depends on the precise value of the parameter k. Furthermore, the stability properties of the equilibrium point will also depend on the value of A. At certain values of k the characteristics of the system change, sometimes quite dramatically. In other words, the qualitative behaviour of the system either side of such values is quite different. These points are called bifurcation points. The types of bifurcations encountered in dynamic systems are often named according to the type of graph they exhibit, e.g., cusp bifurcation and pitchfork bifurcation, to name just two. But such classifications will become clearer once we have described how to construct a bifurcation diagram. As we shall note in a moment when we consider the two examples in detail, for certain values of the parameter k a system may settle down to a periodic cycle: cycles of 2, 4, 8, etc. or cycles of odd-numbered periods, like 3. However, there comes a point, a value of k beyond which there is no regular cycle of any period. When this happens the system becomes irregular or chaotic. As we shall see, the bifurcation diagram is most useful in showing the occurrence of chaotic behaviour of dynamic systems. Example 7.1 We shall now consider the first example in detail to highlight some points about bifurcation theory, and consider the logistic equation in detail in the next section. First, we need to establish the fixed points of the system. These are found by solving x* = l.5x*(l - x*) - k or solving I5x*2 - 5x* + 10k = 0 i.e. * 1 ± VI - 24k x* = — 6 If 1 — 24k < 0, i.e., k > 1/24, then no equilibrium exists. If 1 — 24k > 0, i.e., k < 1 /24, then two equilibria exist „ 1 - VI - 24k „ 1 + VI - 24k X-, = - and Xo = - 1 The first example is adapted from Sandefur (1990), while the second example has been widely investigated by mathematicians. A good starting point, however, is May (1976). Chaos theory 289 In order to investigate the stability of these equilibria, we need to consider/'^*). This isf'(x*) = 1.5 — 3**. Substituting the lower value we have f(x\) = 1 + 0.5V1 - 24k > 1 for all A. < 1 /24 Hence, x\ is unstable or repelling. Next consider the stability of x\ f(x*2) = 1 - 0.5V1 - 24k < 1 for all A. < 1 /24 The system is stable or attracting if — 1 < f'(x*) < 1, i.e., if —0.625 < k < 1/24 or -0.625 < k < 0.041667. The third and final situation is where k = 1 /24. In this case the two fixed points are the same with value 1 /6. Furthermore,/^ 1/6) = 1, and so the stability of this fixed point is inconclusive or semistable. The value k = 1 /24 is a bifurcation value. We can combine all this information about the equilibrium points and their attraction or repelling on a diagram which has the parameter k on the horizontal axis, and the equilibrium point x* on the vertical axis. Such a diagram is called a bifurcation diagram, and such a diagram is shown in figure 7.1 for the present problem. It is to be noted that the heavy (continuous and dotted) line denoting the equilibrium values for various values of the parameter A is a parabola, which satisfies the equation (6** - if = 1 - 24k -1 290 Economic Dynamics with vertex (X, x*) = (1/24, 1 /6) occurring at the bifurcation value X = 1/24 with corresponding equilibrium value x* = 1/6. The vertical arrows show the stability properties of the equilibria. Notice that inside the parabola the arrows point up while outside of the parabola the arrows point down. In particular, for X > 1 /24 the system will tend to minus infinity; if X. = 1 /24 then there is only one equilibrium value (x* = 1 /6) which is semistable from above; and if —0.625 < X. < 0.041667 then there are two equilibrium values, the greater one of which is stable and the lower one unstable. We can be more precise. Let iV\ denote the number of equilibrium values of the system when the parameter is equal to X., then for any interval (Xo — e, Xo + e) if Nx is not constant, Xo is called a bifurcation value, and the system is said to undergo a bifurcation as X. passes through Xq. For the example we have been discussing (7.3) Nx 2, for X. < 1/24 1, for A = 1/24 0, for X. > 1/24 and so X. = 1 /24 is a bifurcation. Furthermore, this is the only value of X. for which Nx is not a constant, and so the system has just this one bifurcation value. Figure 7.1 also illustrates what is called a saddle node bifurcation. It is called this because at the value Xq the fixed points of the system form a U-shaped curve that is opened (in this instance opened to the left). Example 7.2 (saddle-node bifurcation) In this example we shall take a similar case, but consider a continuous model. Let (7.4) x\t) = X — x(t)2 For equilibrium we have 0 = X — x*2 x* = VI If X < 0 then no equilibrium exists. For X > 0 there are two fixed points one for +VX and another for —sfx. In order to consider the stability conditions for continuous systems we need to consider/7^*) in the neighbourhood of the fixed point. If/'(.*;*) < 0, then x* is locally stable; and iff'(x*) > 0, then x* is locally unstable. Since f(x*) = -2x*, then f'(x\) = f(+«/x) = -2jX < 0 for X > 0, and so x\ = +Vx is stable. On the other hand, f'ixi,) =f'(-*jx) = +2^/x > 0 for X > 0, and so x\ = —s/x is unstable. At X = 0 the two fixed points coincide and the fixed point is stable from above. The situation is shown in figure 7.2. Summarising in the neighbourhood of the point X = 0, (7.5) Nx 2, for X > 0 1, for X = 0 0, for X < 0 and once again we have a saddle node bifurcation occurring this time at X = 0. Chaos theory 291 Figure 7.2. Example 7.3 (transcritical bifurcation) Consider another example of a continuous nonlinear dynamical system x'(t) = kx — x2 = x(k — x) with fixed points x\ = 0 and x\ = k Obviously, the two fixed points become identical if k = 0. Summarising in the neighbourhood of k = 0, we have (7.6) N* = 2, for k < 0 1, for A = 0 2, for k > 0 (7.7) and so k = 0 is a bifurcation value. Turning to the stability properties, we have f(x*) = k — 2x* and f(0) = k > 0 for k > 0 hence unstable < 0 for k < 0 hence stable For the second fixed point we have f{k) = -k < 0 for k > 0 hence stable > 0 for k < 0 hence unstable Another way to view this is to consider x\ = 0 being represented by the horizontal axis in figure 7.3, and x\ = k being represented by the 45°-line. The two branches intersect at the origin and there takes place an exchange of stability. This is called a transcritical bifurcation. The characteristic feature of this bifurcation point is that the fixed points of the system lie on two intersecting curves, neither of which bends back on themselves (unlike the saddle-node bifurcation). 292 Economic Dynamics Example 7.4 (pitchfork bifurcation) Consider the following continuous nonlinear dynamical system (7.8) x\t) = kx(t) - x(t)3 = x(t)(k - x(t)2) This system has three critical points: X* = 0, x\ = + Vk *3 = -Vk where the second and third fixed points are defined only for positive k. Summarising in the neighbourhood of A = 0 we have (7.9) Nx 1 for k < 0 3 for k > 0 and so k = 0 is a bifurcation value. Since f'{x*) = k — 3x*2, then at each fixed point we have f(0) = k < 0 for k < 0 hence stable > 0 for k > 0 hence unstable /'(+Vk) = -2k < 0 for k > 0 hence stable f(—\fk) = — 2k < 0 for k > 0 hence stable The characteristic feature of this bifurcation is that at the origin we have a U-shaped curve, which here is open to the right, and another along the horizontal axis that crosses the vertex of the U. It forms the shape of a pitchfork, as shown in figure 7.4. It is therefore called a pitchfork bifurcation.2 It is important to recall in all this discussion of bifurcation points that the properties, especially the stability/instability properties, are defined only for neighbourhoods of the bifurcation point. There may be other bifurcation points belonging to the system with different properties. Example 7.4 illustrates what is sometimes referred to as a 'supercritical pitchfork'. If we have the same continuous nonlinear system but with —x3 replaced with +x3, then we have a 'subcritical pitchfork'. Chaos theory 293 7.3 The logistic equation, periodic-doubling bifurcations and chaos Chaos theory is a new and growing branch of mathematics that has some important implications for economic systems. As an introduction to this subject return to the generic form of the logistic difference equation xt+l =f(x„ k) = kxt(l -xt) 0 3 then/2(^:) begins to intersect the 45°-line in three places. The instability of the single solution and the stability of the two-cycle is more clearly shown in figure 7.7(d), where X = 3.4. A number of features should be noted about this figure. First, f(x) and/2(^:) intersect at the common point Eo = 0.70588. This is unstable. We can establish this by computing/2^), which has a value of 1.96, and since/2(^q) > 1, = 0.70588 is unstable. Second, there are two stable equilibrium points Ei and E2 with the following characteristics Ei : x\ = 0.451963 f(x*, X) = -0.76 E2 : x\ = 0.842154 f(x\ X) = -0.76 since f2'(x\) =f2' (x2) = —0.76, then both Ei and E2 are stable. However, at X = 3.449 the two-cycle becomes unstable. But what occurs at and beyond the point where X = 3.449? What occurs is two period-doubling bifurcation points, i.e., each of the two solutions themselves become unstable but divide into two stable solutions, leading to a total of four solutions. In other words, we have a four-cycle solution. The range for a fourcycle result can be found in a similar manner to the range for a two-cycle result, i.e., by finding the values of a which satisfy the equation «=/(/(/(/(«)))) or a=f\a) (7.12) and whose stability is established by solving4 -1 3.57. With such a diversity of equilibrium solutions depending on the value of k, it would be interesting to know what the bifurcation diagram for the logistic equation would look like over the range 0 < k < 4. We need to plot the relationship between x* and k between the values of 0 and 4. Before the advent of computers, this would be virtually impossible, but now it is relatively easy. The result is shown in figure 7.12,5 while a closer look at the range 3 < k < 4 is shown in figure 7.13. Notice in figure 7.13 the 'windows' occurring. Three are marked on the diagram. The first window marked is for the occurrence of a six-period cycle; the second is the occurrence of a five-period cycle, while the third is for a three-period cycle. 5 The bifurcation diagram plots only stable equilibrium points, so in the range 3 < k < 3.449 only two curves are plotted. The bifurcation point k — 3 is not a saddle node bifurcation but rather a pitchfork bifurcation, as shown in figure 7.6. It is therefore more useful to think of this point as a period-doubling bifurcation. 300 Economic Dynamics Figure 7.11. (c) xM=3.59xf(l-*() (d) xM=Ax£\-xü Figure 7.12. 12 3 4 Such windows represent stable periodic orbits that are surrounded by chaotic behaviour (the dark regions). These two diagrams show an amazing diversity of equilibria for such a simple deterministic equation. Such results direct attention to three observations: (1) the presence of nonlinearity can give rise to deterministic chaos, (2) in the presence of chaos there exists the sensitive dependence to initial conditions, and (3) in the presence of chaos prediction, even for a simple deterministic system, is virtually impossible. Bifurcation diagrams require a considerable amount of routine computations and there are now a growing supply of such routines written by mathematicians Chaos theory 301 Figure 7.13. i - 3.2 3.4 3.6 3.6 A period-5 A. window period-3 window and computer programmers, both in Mathematica and Maple. This is also true of strange attractors, such as the Henon map and the Lorenz strange attractor, which we discuss in section 7.7. The interested reader should consult the various sources at the end of this chapter. In general for the present text, the bifurcation diagrams in Mathematica utilise the routine provided by Gray and Glynn (1991), while a number of the chaos diagrams in Maple utilise the routines provided by Lynch (2001).6 7.4 Feigenbaum's universal constant In discussing the logistic equation we noted that the first bifurcation occurred at value 3, the second at value 1 + = 3.44949, while a third occurs at value 3.54409. We can think of these values as representing the point at which a 2^-cycle first appears. Thus for k = 0 a 2°-cycle occurs at ko = 3 for k = 1 a 21-cycle occurs at kx = 1 + = 3.44949 for k = 2 a 2 -cycle occurs at k2 = 3.54409 and so on. If k^ denotes such occurrences of a 2^-cycle, then three results have been shown to hold for large k: (i) kk+1 « 1 + V3TXI k = 2, 3,... (ii) lim kk = 3.57 kh — At_i (hi) lfdk = ^-k-± k = 2,3,4,... kk+i — kk then lim dk = 8 = 4.669202 6 The printing of such diagrams can be problematic and will certainly depend on the internal RAM of the printer. 302 Economic Dynamics Table 7.2 Approximations for the occurrence of the 2fc-cycle k 0 1 2 3 4 5 6 A.* 3 3.44949 3.54409 3.56441 3.56876 3.56969 3.56989 dk 4.75148 4.655512 4.671264 4.677419 4.65 S is called the Feigenbaum constant after its discoverer, and is important because it is a universal constant. The first result is only approximate, and there are a number of ways of approximating the value of A^+i (see exercise 2). Although these approximations are supposed to hold only for large k, they are reasonable even for small k. Using a procedure provided by Gray and Glynn (1991, p. 125) we derive a more accurate estimate of the two-cycle bifurcation points. In particular, table 7.2 provides the first seven points and computes kk and dk. As can be seen, Xk is rapidly approaching the limit of 3.57, while dk approaches the limit of 4.6692, although not so rapidly and not uniformly. 7.5 Sarkovskii theorem In this section we have a very limited objective. Our intention is to present the background concepts necessary to understand the significance of the Sarkovskii theorem, which is central to periodic orbits of nonlinear systems. The ideas and concepts are explained by means of the logistic equation, most of the properties of which we have already outlined. In table 7.3 we list the first 30 positive integers, where we have expressed some of the numbers in a way useful for interpreting a Sarkovskii ordering. Using the numbers in table 7.3 as a guide, we can identify the following series, where a >- b means a precedes b in the order and 'odd number' means the odd numbers except unity (table 7.4). Every possibly positive integer is accounted for only once by all series taken together. A Sarkovskii ordering is then So >- S\ >~ S2 Sk 24 >- 23 >- 22 >- 2 >- 1 We are now in a position to state the theorem. THEOREM 7.1 (Sarkovskii) Let f be a continuous function defined over a closed interval [a,b] which has a periodic point with prime period n. If n >~ m in a Sarkovskii ordering, then f also has a periodic point with prime period m. Another theorem occurring just over a decade later is the following.7 7 Sarkovskii's paper of 1964 was not known to Western mathematicians until the publication of Li and Yorke's paper in 1975. Chaos theory 303 Table 7.3 Expressing the first 30 integers for a Sarkovskii ordering 1 11 21 2 12 22.3 22 2.11 3 13 23 4 22 14 2.7 24 23.3 5 15 25 6 2.3 16 24 26 2.13 7 17 27 8 23 18 2.9 28 22.7 9 19 29 10 2.5 20 22.5 30 2.15 Table 7.4 The series of a Sarkovskii ordering Series Numbers in the series Description of the series 50 3 >- 5 >- 7 >- ... odd numbers 51 2.3 >- 2.5 >- 2.7 >- ... 2.(odd numbers) 52 22.3 >- 22.5 >- 22.7 > ... 22.(odd numbers) Sk 2k3 > 2k.5 > 2k.l > ... 2<\(odd numbers) >- 24 >- 23 >- 22 >- 2 >- 1 Powers of 2 in descending order* Note: *Recall 21 = 2 and 2U = 1. THEOREM 7.2 (Li-Yorke) If a one-dimensional system can generate a three-cycle then it can generate cycles of every length along with chaotic behaviour. The Li-Yorke theorem is a corollary of the Sarkovskii theorem. If m = 3 in the Sarkovskii theorem, then n = 5, say (n >- m) also has a periodic point. Therefore for all n >- m f will have a periodic point. Hence, if a one-dimensional system can generate a three-cycle, it must be capable of generating a cycle of any length. The windows in figure 7.13 represent period-6, period-5 and period-3 cycles, respectively. The period-5 lies to the left of period-3, with period-3 being the highest ordering. But why is period-6 to the left of period-5? Period-6 is equivalent to period-2.3 in the Sarkovskii ordering and so belongs to the series Si. All periods in Si are to the left of all periods in So. Hence, period-6 is to the left of period-5, which in turn is to the left of period-3. Suppose a continuous function / over the closed interval [a,b] has a period-5 cycle, then according to the Sarkovskii theorem it has cycles of all periods with the possible exception of period-3. Notice that the possibility of a period-3 is not ruled out. Similarly, iff has no point of period-2, then there do not exist higher-order periodicities, including chaos. The Sarkovskii theorem, and to some extent the Li-Yorke theorem, demonstrates that even systems that exhibit chaotic behaviour still have a structure. The word 304 Economic Dynamics 'chaos' conjures up purely unsystematic patterns and unpredictability. Although the movement of an individual series may be aperiodic, chaotic systems themselves have structural characteristics that can be identified. 7.6 Van der Pol equation and Hopf bifurcations (7.16) (7.17) (7.18) We met the Van der Pol equation in chapter 4 when we considered limit points. In this section our interest is in the Van der Pol equation and its bifurcation features. The nonlinear equation is a second-order differential equation of the form 'x = /x(l — x )x — X Second-order differential equations can be expressed in the form of a system of first-order differential equations with suitable transformations. Let x = y then y = x, hence we have the system of first-order equations: x = y y = /x(l — x2)y — x and the only unknown parameter is \x. The fixed points of the system are established by setting x = 0 and y = 0, which is the singular point P = (0, 0). Furthermore, the linearisation of the system can be expressed as 1 0 -(1 + 2\xxy) /x(l — x2) Expanding the system around the fixed-point, P = (0, 0), we have X y _ X y _ X " 1 0" X y _ — 1 \x y _ Hence the matrix of the linearised system is 1 0 -1 \x whose eigenvalues are y/P2 and A2 2 2 Using these eigenvalues we can identify five cases, as shown in table 7.5. Figure 7.14 illustrates each of these cases. If we concentrate on the equilibrium values for x and y, say x* and y*, then for \x < 0, x* = 0 and y* = 0 and the system moves along the /x-axis. At \x = 0 the system changes dramatically taking on the shape of a circle at this value. Then, as fx continues in the positive direction the system takes on a limit cycle in the x-y plane for any particular positive value of /x, the shape of which is no longer a circle. All of these are schematically illustrated in figure 7.15, which also shows the movement of the system by means of arrows. Clearly the system exhibits a bifurcation at the value \x = 0. This is an example of a Hopf bifurcation. Chaos theory 305 Table 7.5 Properties of the Van der Pol equation Cases Parameter values Properties I (i < —2 both eigenvalues are real and negative, P is a stable node II —2 < fi < 0 eigenvalues are complex with negative real parts, P is a stable focus III fi — 0 eigenvalues are purely imaginary, P is a centre IV 0 < fi < 2 eigenvalues are complex, with positive real parts, P is an unstable focus V fi > 2 both eigenvalues are real and positive, P is an unstable node (e) Stable node (u=2.5) 306 Economic Dynamics Figure 7.15. More generally, a Hopf bifurcation occurs when there is a change in the stability of a fixed point into a limit cycle. Clearly a limit cycle can occur only if the system is defined for two variables and at least one parameter. All these conditions are satisfied by the Van der Pol equation. As \x passes through \x = 0, the system changes from a stable equilibrium at the origin for \x < 0 into a limit cycle for \x > 0. Hence \x = 0 gives rise to a Hopf bifurcation that occurs at the origin in the (x,y)-plane. There are in fact two types of Hopf bifurcations, one in which the limit cycles are created about a stable point (a subcritical Hopf bifurcation) and one in which the limit cycles are created around an unstable critical point (a supercritical Hopf bifurcation). Limit cycles of finite amplitude may also suddenly appear as the parameter of the system is varied. In the physical sciences such large-amplitude limit cycles are more common than supercritical Hopf bifurcations (Lynch 2001). More importantly for economics, these systems exhibit multiple stable equilibria for which the system may jump from one stable equilibrium to another as the parameter of the system is varied. Equally significant for economics is that the existence of multistable solutions allows for the possibility of hysteresis. Example 7.5 (Large-amplitude limit cycle bifurcation)9. Consider the system x = x(k + x2 — x4) (7.19) y = -i For system (7.19) the only critical point is x* = 0. Let/(.*;) = x(k + x2 — x4) then f'(x) = k + 3x2 — 5x4 and/'(.*;* = 0) = A. Therefore the critical point x* = 0 is 8 Lynch (2001). Chaos theory 307 Figure 7.16. stable for k < 0 and unstable for X > 0. The system undergoes a subcritical Hopf bifurcation at A = 0. However, for a certain range of A, say A > Aj, the system also has a stable large-amplitude limit cycle. The bifurcation diagram for this system is illustrated in figure 7.16.9 As can be seen in figure 7.16, over the range X.s < X. < 0, there exist two steady-state solutions. A number of important implications follow from this feature. First, which steady state is approached depends on the initial conditions. If the equations represented some economic system, then in all probability the welfare attached to one of the stable equilibrium would be quite different from that of the other. Policy-makers may, therefore, attempt to push the system in the direction of one particular stable equilibrium by changing the initial conditions. Second, the system exhibits hysteresis. Suppose the value of A started at X-o < Xs, figure 7.16. As X is increased then x remains at x* = 0 until A = 0. At A = 0, however, there is a sudden jump to the large-amplitude limit cycle. This is so because A = 0 is a subcritical Hopf bifurcation. As X continues to increase, the value of x* follows the upper path, along RS. The path traversed is then PQRS. Now suppose the value of X is decreased to its former level Xq. The system moves down along the upper path SRT. Once X = Xs, the system jumps down to x* = 0 (point U) and then remains there as X is decreased further. The 'return trip' is therefore SRTUP, which is quite different from its outward journey. The system is hysteretic. 7.7 Strange attractors We noted in the last section how with a two-dimensional system a Hopf bifurcation could arise. In the Van der Pol equation, once \x > 0 then the system gets attracted to a limit cycle. But other two- or higher-dimensional systems can have 'strange' attractors. We shall discuss the concept of strange attractors by way of examples. The examples we consider are the Henon map and the Lorenz attractor.10 Part of the bifurcation diagram can be constructed using the implicit plot routines in either Mathe-matica or Maple. Just do the implicit plot of X + x2 — x4 = 0 for — 1 < X < 2 and 0 < x < 2, and the result is as portrayed in figure 7.16. See also the Rossler attractor in exercises 9 and 10. 308 Economic Dynamics 7.7.1 The Henon map (7.20) (7.21) (7.22) The Henon map arises from a set of two equations involving two parameters, which are real numbers Xt+i = 1 - ax2 + yt a > 0 \b\ < 1 yt+i = bxt We can think of this as the function "1 -2 Ha,b(xt, yt) = axz + yt bxt or simply H. To establish some of the properties of the Henon map, let f{x, y) = 1 - ax2 + y g(x, y) = bx Then the Jacobian, J, is J = whose determinant is — b and eigenvalues fx fy —2ax 1 _gx gy _ b 0_ X = —ax ± 7a2x2 + b which is readily obtained using either Mathematica or Maple - in fact all the mathematical properties we are about to discuss are obtained using either of these software programmes. Hence, the eigenvalues are real only if 7a2x2 + b > 0. Furthermore, the fixed points of the Henon map are found to be 2a V 2a b 2a y h / — \b 2a V 1 + 7(1 - b)2 + 4a) 1 + 7(1- b)2 + 4a) 1 - 7(1 - b)2 + 4a) 1 - 7(1 - b)2 + 4a) which exist ifa>—1(1 — b)2. Turning now to the stability properties of the fixed points, we recall that the fixed point is attracting if the eigenvalue is less than unity in absolute value. It can be established (Gulick 1992, pp. 171-2) that (1) (2) If a < — t(1 — b)2, then H has no fixed points if-ki-b)2 < a < r(l — b) and a^O, then H has two fixed points, P\ and P2, of which P\ is attracting and P2 is a saddle point If the parameter b is set fixed over the interval [0,1] and a is allowed to vary, then there will be two bifurcation values for a: one at —(1 — b)2/A and another at 3(1 — b)2/4, with the system changing from an attracting fixed point to one of a saddle point. Figure 7.17 shows the Henon map with parameters a = 1.4 and b = 0.3 and with initial point (jc0, yo) = (0.1, 0). The two fixed points are Px = (0.6314, 0.1894) Chaos theory 309 and P2 = (—1.1314, —0.3394), and the figure illustrates the existence of a strange attractor. Why then is it called a strange attractor? For b = 0.3 the Feigenbaum constant of a for the Henon map is 1.0580459 (Gulick 1992, p. 173). One would assume that for a > 1.06 with chaos present that the iterates would virtually fill the whole map. But this is not the case. For example, given the parameter values a = 1.4 and b = 0.3, then no matter what the starting values for x and y, the sequence of points {x(t),y(t)} is attracted to the orbit shown in figure 7.17, and such an orbit seems rather a 'strange' shape. At the same time, however, the trajectory {x(t), y(t)} is very sensitive to initial conditions. This is illustrated in figure 7.18 in the case of variable x, for the same parameter values and (xq, yo) = (0.001, 0), where we plot the first 100 observations.11 So although the two sequences converge on the attractor illustrated in figure 7.17, they approach it quite differently. Because This plot is derived using Excel rather than Mathematica or Maple, since spreadsheets are good for plotting time-series or discrete trajectories quickly and easily. See section 5.5. 310 Economic Dynamics this attractor is sensitive to initial conditions, then it is called a chaotic attractor. The word 'strange' refers to the geometrical shape of the attractor, while the word 'chaotic' indicates sensitivity to initial conditions and therefore refers to the dynamics of the attractor (Hommes 1991). We have already noted that when a series is chaotic then it is very sensitive to initial conditions. We have also noted in terms of the Henon map, that regardless of the initial value, the orbit will settle down to that indicated in figure 7.17 if a = 1.4 and b = 0.3. Now if two series are chaotic then they will diverge from one another, and such a divergence will increase exponentially, as illustrated in figure 7.18. If it is possible to measure the divergence between two series then we can obtain some measure of chaos. Furthermore, looking at the Henon map, if we divide the rectangular box into very tiny rectangles then, by means of a computer, it is possible to establish how many times points in the attractor are visited by trajectories of various points. The Lyapunov dimension or Lyapunov number does just this. For instance, the Lyapunov number for the Henon map is 1.26. The following theorem has been demonstrated (Gulick 1992). THEOREM 7.3 (1) If an attractor has a non-integer Lyapunov number then it is a strange attractor. (2) If an attractor has sensitive dependence on initial conditions then it has a Lyapunov number greater than unity. The Henon map satisfies (1), so it is a strange attractor; but it also satisfies (2), so it is also called a chaotic attractor. It is possible to have a strange attractor that is not chaotic and a chaotic attractor that is not strange. However, most strange attractors are also chaotic. The Lyapunov dimension is important in the area of empirical chaos, and is used in the economics literature to detect chaos.12 7.7.2 The Lorenz attractor The Lorenz attractor was probably the first strange attractor to be discussed in the literature, and was certainly the origin of the term 'butterfly effect'.13 The Lorenz system is composed of three differential equations x = — 1, where Po = P\ = Pi f°r r = 1. There is therefore a bifurcation at r = 1. As r passes through this value the critical point Po bifurcates and the critical points P\ and P2 come into existence. A typical plot of the Lorenz system is shown in figure 7.19 for a = 3, r = 26.5 and b = 1. Starting at the value (xq, yo, zo) = (0, 1, 0) the system first gets drawn to point P2 then after moving around and away from this point it gets drawn to point P\. This process keeps repeating itself. In order to consider the stability properties of the Lorenz system we need to consider the linear approximation. The coefficient matrix of the linearised system is A = ' -a a 0 ' r — z —1 — x y x —b This leads to a set of cubic characteristic equations, one for each of the three critical points. There has been much investigation into the properties of these for various values of the parameters, which is beyond the scope of this book. Asymptotic stability is assured, however, if the real parts of the eigenvalues are negative. It can be shown that if 1 < r < rH where a (a + b + 3) o — b — 1 then the real parts of the eigenvalues are negative and so the three critical points are asymptotically stable. Furthermore, if r = rH then two of the roots are imaginary and there occurs a Hopf bifurcation. When r > r# there is one negative real eigenvalue and two complex eigenvalues with positive real parts and the three critical points are unstable. Figure 7.20 shows the paths of two Lorenz systems. Both are drawn for a = 10 and b = 8/3. With these values then r# = 24.7368. In figure 7.20(a) we have r = 22.4 with initial point (7,7,20), which is 'close to' the strange attractor; and in figure 7.20(b) we have r = 28. The first shows what an asymptotically stable 312 Economic Dynamics Figure 7.20. (a) a = 10, b = 8/3, r = 22.4 (b)a= 10,6 = 8/3, r = 28 system looks like, with the system being attracted to one of the fixed points. What is not so clear is what an unstable Lorenz system would look like. Figure 7.20(b) shows one such possibility, with the system starting from point (5,5,5) constantly being first attracted to one fixed point and then the other repeatedly. The Lyapunov dimension for the Lorenz system is 2.07. Then by theorem 7.3 it is a strange attractor (the Lyapunov number is noninteger) and it is also chaotic (the Lyapunov number exceeds unity). 7.8 Rational choice and erratic behaviour In this first economic application of chaos theory we consider the situation where preferences depend on experience, and is based on the paper by Benhabib and Chaos theory 313 Day (1981). In such a situation choices can show cyclical patterns or even erratic (chaotic) patterns. The type of situations envisaged in which preferences depend on experience is where an individual reads a novel, then sees the movie rather than read the book a second time, but as a consequence of seeing the movie is stimulated to reread the novel. This consumption pattern is intertemporal. Furthermore, although habit is a strong pattern of human behaviour, so is novelty. So from time to time we do something quite different. For example, the person who holidays each year in Majorca, but then suddenly decides a holiday in the Alps is what is required. Or the individual who alternates between a beach holiday and one in the mountains or the country. The feature here is a shift in consumption pattern that then shifts back. Such choice behaviour is ruled out in neoclassical consumer theory. Neoclassical consumer theory cannot handle novelty in choice behaviour. The analysis begins with the typical Cobb-Douglas utility function that is maximised subject to the budget constraint. Here x and y denote the consumption of the two goods, p and q their prices, respectively, and m is the individual's level of income max U = xay1~a s.t. px + qy = m Setting up the Lagrangian (7.25) L = xayl a - k(m - px - qy) leads to the first-order conditions dL/dx = aU/x — kp = 0 dL/dy = (1 - a)U/y - kq = 0 dL/dk = m — px — qy = 0 From these conditions we readily establish the demand curves m m x = —a, y = —(1 - a) (7.26) p q Now assume that the parameter a in the utility function, which represents a property of preferences, depends endogenously on past choices. More specifically, assume at+1 = bxtyt (7.27) The parameter b in equation (7.27) denotes an 'experience-dependent' parameter. The greater the value of b the greater the value of the parameter a in the next period and so the more preferences swing in favour of good x. Substituting (7.27) into the demand equations (7.26), then m m xt+i = —bxtyt, yt+\ = —(1 - bxtyt) (7.28) p q Concentrating on xt+\, then from the budget equation we have m — pxt yt =- 314 Economic Dynamics so mbxt(l -pxt) *t+i = - q Taking the short run and normalising prices at unity, so that p = q = 1, then (7.29) xt+1 = bmxt{m - xt) The fixed point of equation (7.29) is found by solving x* = bmx*(m — x*) which is bm2 — 1 x* = - bm So such consumption will only be positive if bm2 > 1. Furthermore, since xt+\ = f(xt) = bmxt(m - xt) represents consumption of good x, then this is at a maximum when/'(.*;) = 0, i.e. f(x) = bm2 - 2bmx = 0 m X ~ 2 and maximum consumption is f(m/2) = bm3/4. Since maximum consumption cannot exceed total income (recall p = 1), then f(m/2) < m, or bm3 2 —— < m implying bm < 4 To summarise, we have established two sets of constraints bm2 > 1 and bm2 < 4. Putting these together, then (7.30) 1 < bm2 < 4 Benhabib and Day (1981) in considering equation (7.29) and the constraints (7.30), establish the following results: (1) A three-period cycle exists and so by the Li-Yorke theorem, period cycles of every order exist, including chaos. (2) Chaos begins at a critical value c = 2.57 over the interval x e [0, m] where c < bm2 < 4. Benhabib and Day also make the following observation. The smaller the experience parameter b, the greater the income endowment m must be in order to generate chaotic behaviour. What this suggests is that for individuals with low income, long-run patterns of consumption tend to be stable. However, as income grows, so does the possibility of instability, and erratic (chaotic) behaviour becomes more likely at very high levels of income.14 14 Does this explain the erratic consumption patterns of individuals like the DJ Chris Evans, Sir Elton John or that of Victoria and David Beckham? Chaos theory 315 7.9 Inventory dynamics under rational expectations This section discusses a disequilibrium inventory model by Hommes (1991), where a full discussion can be found (1991, chapter 28). In this model the short side of the market determines market values. We discuss this concept in chapter 8, but the situation is illustrated in figure 7.21 for both the labour market and the product market. Labour supply, Ut, is assumed constant at the value of c. Labour demand, Lf, is less straightforward and we shall approach a discussion of this by considering first aggregate supply. For the moment, we assume it is a downward sloping curve, as shown in figure 7.21.15 Actual labour employed, Lt, is then given by the short side of the market, so Lt = min{Lf, L\}, and is shown by the heavy line in figure 7.21(a). Figure 7.21(b) shows aggregate demand, yf, and aggregate supply, yst. It denotes the level of inventories, and is positive when there is excess demand, otherwise it is zero, i.e., It = max{0, yst — yf}. We denote expected aggregate demand by E{yf) and the desired level of inventories by if. It is assumed that desired inventories are proportional to expected aggregate demand, if = /3E(yf). Production is assumed proportional to labour employed, 8Lt, and so 8 denotes labour productivity. Aggregate supply, therefore, is inventories over from the last period plus current production, i.e., yst = It-\ + 8Lt. On the other hand, aggregate supply is based on expected aggregate demand plus desired inventories. Thus y\ = 7,_i + 8Lt yf = E (yf) +lf = E (yf) + /3E (yf) = (i + P)E(y?) Setting these equal to each other gives {\ + m(yf)-h-i L'~ 8 Figure 7.21 presents the terms in a more familiar setting, but wages and prices do not enter this modelling framework. 316 Economic Dynamics (7.32) (7.33) (7.34) (7.35) Hence labour demand is given by (l + ß)E(yf)-It.1 max 0, Aggregate demand in the economy is assumed to be a linear function of labour employed, yf = a + bLt, where b can be thought of as the marginal propensity to consume, and we assume labour productivity is greater than the marginal propensity to consume, 8 > b. The final element of the model is our rational expectations assumption. We assume E{yf) = yf, i.e., perfect foresight. The model can therefore be summarised by six equations (1) Ldt = max 0, (l + ß)E(yf) (2) (3) (4) (5) (6) H = c where c is a constant yf yst a + bLt It-i + SLt It = max{0,yf - yf] E (yf) = yf Consider now the labour market in terms of figure 7.21(a). At a wage rate, wq, say, actual labour employed, Lt must be positive and equal to labour demand. Therefore _ (1 + 0)E (yf) - It-i _ (1 + + bLt) - It-i 8 8 Solving (7.34) for Lt gives = (1 + 0)a - It-i 8-Wl+p) This value must lie between 0 and c. Assume 8 — b(l + /3) > 0, then (l + /3)fl-/f_i &-b(\ + p) which implies (1 + /3)fl - c[8 - b(l + /?)] < It-i Let yi = (1 + fi)a - c[8 - b(l + ft)] then It_x > yx. Similarly, if Lt > 0 then It-\ < a(l + /3). Let 72 = a(l + P), then putting these together, we have that if Lt lies between 0 and c, then y\ < It-\ < Y2- We need, however, to consider three situations: (i) It-i < yi,(ii)yi < It-\ < yi, and (iii) It-\ > y2. (i) It-i < Yi If It-\ < y\ then Lt = c since this is the short side of the market in these circumstances. Then it = yt- yf = It-\ + 8c — a — be Chaos theory 317 i.e. (ii) Then = It-i +(S -b)c-a Yi < 7,_i < y2 it = yst- yf = It-i + 8L, - a - bL, ^(l + ß)a-I,-1 (7.36) = It-i + 8 - b 8-b(l + ß) — a On expanding and simplifying I, = -bßh t-i + a8ß 8-b(\+p) 8-b(l+/3) (iii) 7,_i > y2 Under these circumstances L, given by equation (7.35) is negative and so L, = Lf = 0. Hence h = yst-ydt = it-i - a Combining all three cases, then = f{It-\) is a piecewise function of the form (7.37) (7.38) h = It-i + (8 — b)c — a -bßlt t-i + a8ß h-\ < 71 7l < < 72 8-H\ + ß) 8-H\ + ß) h-i - a I,-\ > 72 (7.39) and describes the inventory dynamics of the present model. Equilibrium investment is denned only for the range y\ < It-\ < Yi- In this instance -bßl* a8ß i.e. / 8-b(l + ß) + 8- b{\ + ß) a8ß 8-b and since we have assumed 8 > b, then this is positive. We shall now pursue this model in terms of a numerical example. Example 7.6 Let a = 0.2, b = 0.75, c = 1, 8 = 1 and for the moment we shall leave /3 unspecified. If /3 = 0.2, then y\ = 0.14 and 72 = 0.24. Equilibrium investment is given by /* = 0.16, which lies between the two parameter values. The piecewise difference equation is then given 318 Economic Dynamics Figure 7.23. 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 by I, = It-! + 0.05 -1.5/f_i+0.4 It-i ~ 0-2 /,_! < 0.14 0.14 < It-! < 0.24 It-i > 0.24 The situation is shown in figure 7.22. The equation It = f(It-i) is clearly nonlinear, even though it is made up of linear segments. Although the equilibrium level of investment, I* = 0.16, exists and is identified in terms of figure 7.22 where the diagonal cuts the function f(It-i), it is not at all obvious that this fixed point will be approached in the present case. In fact, as figure 7.23 illustrates, the system is chaotic, as shown by the cobweb not settling down at the fixed point from a starting value of Iq = 0.14.16 In constructing figure 7.23 we assumed that ft = 0.2. If, as we just indicated, this results in chaotic behaviour, then it would be useful to see the bifurcation diagram for I* against ft. In doing this we allow ft to range over the interval 0 to 1 /3, this The procedure for constructing cobwebs for piecewise functions is covered in chapter 8. However, Mathematica can be used to produce this function with the Which command as follows f[x_] :=Which[x<=0.14,x+0.05,x>0.14 && x<0.24,0.4-1.5x, x>=0.24,x-0.2] Maple's, definition for the piecewise function is f:=x->piecewise(x<=0.14,x+0.05,x>0.14 and x<0.24,0.4-1.5*x, x>=0.24,x-0.2); Chaos theory 319 0.3i- -— Figure 7.24. 0.05 latter value comes from setting 8 — b(l + /3) = 0 for the values given above. In constructing the bifurcation diagram it is important to realise that the intervals for the piecewise function vary with the change in /3. Given the values above, we have the piecewise difference equation I,= /f-i+0.05 7,_i < 0.2(l+/3)-[l-0.75(l+/3)] -O.750/,_i 0.20 l-0.75(l+č) + l-0.75(l+č) 0.2(1+/J)-[1-0.75(1+/J)] 0.2(1+0) which we use to construct the bifurcation diagram shown in figure 7.24. It is apparent from this figure that although the system exhibits chaos over certain ranges for 0, there is regular alternating behaviour. For /3 = 0.25 the system is chaotic. The model offers much more possibilities depending on the values of the parameters. All we have done here is illustrate the presence of chaos in such a dynamic inventory model. Exercises 1. The logistic equation xt+\ = Xx,(l — xt) has Xq = 3 and X\ = 3.4495. Given the Feigenbaum constant of 8 = 4.669 use this to predict the value of X at which k = 2 doubling takes place. Compare it with the value derived by the computer programme provided in Gray and Glynn (1991, p. 125). 2. Given Xo, X\ and the universal Feigenbaum constant 8 = 4.669, establish that k\ — Xq Xi = -+ Ai 8 X2-Xx / 1 1 \ A3 =---+A2 = (A!-Ao)( ^ + -1 +Aj and find the value for k^. Show that lim X.k =--h X\ k^oo 8—1 and hence show that if Xq = 3 and X\ = 3.4495 then the approximate value at which chaos begins for the logistic equation is 3.572. 320 Economic Dynamics (i) Show that xn-\-\ = kxn(\ xn) has the same properties as yn+i =y2n + c if k(2 — k) k c =--- and yn = - - kxn (ii) Verify by constructing bifurcation diagrams for each function. Obtain the bifurcation diagram for the following tent function. T(x) = 2x 0 < x < \ 2(1-jc) \0 d>0 a > 0 Demand and supply models 327 where quantities demanded and supplied, qd and qs, and price, p, are assumed to be continuous functions of time. The fixed point, the equilibrium point, of this system is readily found by setting dp/dt = 0, which gives * a-c P = - 1 b + d and equilibrium quantity ad + be H b + d For a solution (a fixed point, an equilibrium point) to exist in the positive quadrant, then a > c and ad + be > 0. We can solve for the price path by substituting the demand and supply equations into the price adjustment equation giving the following first-order linear nonho-mogeneous differential equation dp — + a(b + d)p dt a(a — c) with solution a — c m = ITTd + PO ~ a — c b + d -a(b+d)t which satisfies the initial condition p(0) = po. It may be thought that the dynamics of this model are quite explicit, but this is not in fact the case. To see this consider the movement of the quantity over time. The first thing we must note is that there are two different quantities qd{t) and qs{t). So long as the price is not the equilibrium price, then these quantities will differ, and it is this difference that forces the price to alter. Thus 328 Economic Dynamics Figure 8.2. P P Po t q„ q q\ t or, more succinctly q(t) = mm(qd(t), qs(t)) The logic behind assumption 1 is that the model contains no stocks, and when the price is below the equilibrium price, the current production is all that can be supplied onto the market. Some demand will go unsatisfied. If, on the other hand, the price is above the equilibrium price, and current production is in excess of demand, suppliers cannot force people to purchase the goods, and so will sell only what is currently demanded. Although when price is below the equilibrium price and the quantity traded is what is currently produced, the excess demand gives a signal to suppliers to increase future production and to raise the price. The model is less satisfactory when interpreting what is happening when the price is above the equilibrium price. In this instance there will be unsold goods from current production. The model has nothing to say about what happens to these goods. All we can say is that in the future suppliers will decrease their current production and lower the price. If stocks were included in the model and the stocks, which accumulated when the price was above the equilibrium price, could be sold off when there is excess Demand and supply models 329 demand, then we move away from assumption 1 to a different assumption. By way of example suppose we make the following assumption: Stocks are sufficiently plentiful to allow all demands to be met at any price, and price adjusts in proportion to the change in stock levels. Thus, if i(t) denotes the inventory holding of stocks at time t, then On the face of it this appears to be the same model, resulting as it does in the same first-order linear nonhomogeneous differential equation. It is true that the solution for the price gives exactly the same solution path. The difference in the model arises in terms of the quantity traded. The time path of the quantity demanded, and the time path of the quantity supplied, are as before. For a price above the equilibrium price it is still the case that the quantity traded is equal to the quantity demanded (the short side of the market), and the unsold production goes into stock holdings. It is this rise in stocks that gives the impetus for suppliers to drop the price. On the other hand, when the price is below the equilibrium price, the quantity traded is equal to the quantity demanded (the long side of the market!). The excess demand over current production is met out of stocks. The fall in stock holdings is the signal to producers to raise the price and raise the level of production. Under this second assumption, therefore, we have q(t) = qd(t) for all p What is invariably missing from elementary discussions of demand and supply is the dynamics of the quantity traded. At any point in time there can be only one quantity traded, and whether this is equal to the quantity demanded, the quantity supplied or some other quantity depends on what is assumed about the market process when in a disequilibrium state. To illustrate the significance of the arguments just presented let us consider the labour market. Here we are not concerned with the derivation of the demand for labour and the supply of labour, we shall simply assume that labour demand, LD, is negatively related to the real wage rate, w, and labour supply, LS, is positively related to the real wage rate. Since we are concerned here only with the dynamic process of market adjustment, we shall assume labour demand and labour supply are linear functions of the real wage rate, and we set up the model in continuous time. Thus ASSUMPTION 2 A Jo and price adjusts according to — = -a— = -a(qs - qd) = a(qd - qs) a > 0 dt dt LD(t) = a - bw(t) LS(t) = c + dw(t) LD(t) = LS(t) d>0 b>0 (8.8) 330 Economic Dynamics Figure 8.3. LS LD2 LD, 0 E, E2 LD,LS,E The flexible wage theory asserts that wages are highly flexible and will always alter to achieve equilibrium in the labour market, i.e., the wage rate will adjust to establish LD{t) = LS{t). Let us be a little more specific. A typical textbook version (Parkin and King 1995, chapter 29) is that if the real wage is below the equilibrium wage (w < w*), then the demand for labour is above the supply of labour. Employers will raise the wage rate in order to attract labour, and that this will continue until a wage rate of w* is established. On the other hand, if the wage rate is above the equilibrium wage rate, then households will not be able to find jobs and employers will have many applicants for their vacancies. There will be an incentive for employers to lower the wage rate, and for labour to accept the lower wage in order to get employed. This will continue until the wage rate of w* is established. The dynamic adjustment just mentioned is assumed to take place very quickly, almost instantaneously. For this reason the model is often referred to as a market clearing model.1 At any point in time the wage rate is equal to the equilibrium wage rate, and the quantity of employment is equal to labour demand that is equal to labour supply. Let employment at time t be denoted E(t), then in the flexible wage theory, the wage will adjust until LD{t) = LS{t) = E{t). A shift in either the demand curve for labour or the supply curve of labour will alter the equilibrium wage rate and the level of employment. Thus a rise in the capital stock will increase the marginal product of labour and shift the labour demand curve to the right. This will lead to a rise in the real wage rate and a rise in the level of employment, as illustrated in figure 8.3. A different wage theory, however, is also prevalent in the literature, called the sticky wage theory. This asserts that money wage rates are fixed by wage contracts and adjust only slowly. If this assumption is correct, then it does not follow that the real wage will adjust to maintain equality between labour demand and labour 1 This is the basic assumption underlying the approach by Barro and Grilli (1994) in their elementary textbook, European Macroeconomics. Demand and supply models 331 LS Figure 8.4. LD 0 E, E2 E' LD,LS,E supply. Suppose at the ruling money wage and price level the real wage is above the market clearing wage, as shown in figure 8.4 at the real wage rate w\. The labour market is in disequilibrium and there is no presumption in this theory that the real wage will fall, at least in the short and (possibly) medium term. At this real wage rate there is an excess supply of labour. But employment will be determined by the demand for labour, and the excess supply will simply increase the level of unemployment. Employment will be at the level E\ and unemployment will be increased by U\. It is just like our earlier model of stock accumulation. However, in this model it is not so easy for employers to reduce the wage on offer. At a real wage rate of w2, which is below the market clearing wage rate of w*, there is an excess demand for labour. Employment, however, is no longer determined by the demand curve of labour. If individuals are not willing to put themselves on to the labour market at that real wage, then employers will simply be faced with vacancies. Employment will be at the level E2 and vacancies will rise by V2. Of course, if such a situation prevails there will be pressure on the part of firms to increase the nominal wage, and hence increase the real wage, in order to attract individuals into the labour market. There will be dynamic forces present which will push up the nominal, and hence the real, wage rate. What we observe, then, is that the short side of the market determines the level of employment. But the labour market illustrates another implicit dynamic assumption. When the real wage is below the market clearing wage then there will be pressure on firms to raise the nominal wage in order to fill their vacancies. When the real wage is above the market clearing wage employers may wish to reduce their nominal, and hence their real, wage but may be prevented from doing so because of contractual arrangements. In other words, there is an asymmetric market adjustment: real wages may rise quicker when there is excess demand for labour than they will fall when there is an equivalent excess supply of labour. Such asymmetric market adjustment takes us yet further into realms of additional assumptions specific to the labour 332 Economic Dynamics market. What appeared to be a straightforward demand and supply model takes on a rather complex pattern depending on the assumptions of dynamic adjustment. 8.3 The cobweb model In the previous section we referred to the possibility that price would be altered in the next period in the light of what happened in this period. Whenever decisions in one period are based on variables in another period we inherently have a dynamic model. The simplest of such models is the cobweb model of demand and supply. This model is also set out most usually in a discrete form. This is understandable. The model was originally outlined for agriculture (Ezekiel 1938), and concentrated on the decision-making of the farmer. First we shall consider a simple linear version of the model. Demand at time t is related to the price ruling at time t. Letting qf denote the quantity demanded at time t and pt the actual price at time t, then we have qdt = a — bpt b > 0 However, the farmer when making a decision of what to grow or what to produce will need to make a decision much earlier, and he will make a decision of what quantity to supply, q\, based on what price he expects to receive at time t, i.e., pet. Accordingly, the supply equation takes the form q\ = c + dpet d > 0 At any point in time, the market is considered to be in equilibrium and so the quantity traded, qt, is equal to the quantity demanded, which is equal to the quantity supplied. The model is, then qf = a- bpt (8.9) qst = c + dpet qf = q\ = qt As the model stands it cannot be solved because of the unobservable variable pet. We therefore have to make a further assumption about how the supplier forms his or her expectation or makes a decision about the expected price. The simplest assumption of all is that he or she expects the price at time t to be what it was in the previous period. This assumption, of course, amounts to assumingpet = pt-\-The model now becomes qf = a- bpt (8.10) q*t = c + dpt-i qf = q\ = qt Substituting the demand and supply equations into the equilibrium condition we obtain a — bpt = c + dpt-i (8.11) Demand and supply models 333 which is a first-order nonhomogeneous dynamic system. Notice that it is also an autonomous dynamic system because it does not depend explicitly on the variable t. The system is in equilibrium when the price remains constant for all time periods, i.e.,/?, = Pt-i = ■ ■ ■ = p*. Thus a — c p* = - where p *>0 if a>c (8.12) b + d With linear demand and supply curves, therefore, there is only one fixed point, one equilibrium point. However, such a fixed point makes economic sense (i.e. for price to be nonnegative) only if the additional condition a > c is also satisfied. To solve this model, as we indicated in part I, we can reduce the nonhomogeneous difference equation to a homogeneous difference equation by taking deviations from the equilibrium. Thus a — c\ ( d Pt=\—)-\b]Pt-1 * ,a-c\ (d\ # p =\—r- i-yi-ip Pt-P* = -[fj (Pt-l ~p*) with solution Pt-P* = ( -^j (Po-P*) which satisfies the initial condition pt = po when t = 0. More fully ' a — c\ ( d\' a — c P° [b + d With the usual shaped demand and supply curves, i.e., b > 0 and d > 0, then d/b > 0, hence {—d/bf will alternate in sign, being positive for even numbers of t and negative for odd numbers of t. Furthermore, if 0 < | —d/b\ < 1 then the series will become damped, and in the limit will tend towards the equilibrium price. On the other hand, if \—d/b\ > 1 then the system will diverge from the equilibrium price. Finally, if \b\ = \d\ (or \—d/b\ = 1), then the system will neither converge nor diverge and will exhibit a two-period cycle. These results were verified by means of a simple numerical example and solved by means of a spreadsheet in chapter 3, figure 3.11. (8.13) Example 8.1 By way of variation, here we shall solve a numerical version of the system using Mathematica and Maple (see Eckalbar 1993). The input instructions for each 334 Economic Dynamics package are Mathematica {a,b,c,d}={20,4,2,2.5} A=-b/d ptl [pt_] :=A*pt price [t_,pO_] :=ListPlot[NestList[ptl,pO,t], PlotJoined->True, PlotRange->All, AxesOrigin->{0,0}] price[20,1] Maple A:=20: b:=4: c:=2: d:=2.5: A:=-d/b; sol:=rsolve(p(t)=A*p(t-1),p): equ:=subs(p(0)=l,sol): f:=t->equ: points: =[seq([t,f(t)],t=0..20): plot (points); In Mathematica, we first set values for the parameters a, b, c and d, namely {a, b, c, d} = {20,4,2,2.5}. We then define the ratio (-d/b) = A, resulting in A = -0.625, and define the recursive function ptl[pt_] := A*pt. It now remains to generate the series of values for the price using the NestList command, and to plot the resulting series using the ListPlot command. The result is figure 8.5. Notice that this version of the cobweb allows different periods to be specified and a different initial price. Thus, price[50,2] would indicate a plot of 50 periods with an initial price of 2. Figure 8.5 can also be generated using the Maple commands given above. Again, we input the values for the four parameters and define A. In Maple we approach the next part slightly differently. We solve the difference equation, using the rsolve command. Then define the equation for the initial price being unity, and use this to define the mapping /. Next we create a series of points using the sequence command and plot these to create figure 8.5. It is readily observed from figure 8.5 that this system is dynamically stable, with the price converging on the equilibrium price - since deviations from equilibrium converge on zero. Using a software package such as Mathematica or Maple different values for the parameters can readily be investigated. For instance, parameter Figure 8.5. p(t) i 0 . 75 0.5 0.25 -0.25 -0.5 Demand and supply models 335 values {20, 4, 2, 6} and price [20,1] will readily be shown to exhibit an unstable system with price diverging from its equilibrium. This stability and instability is purely dependent on the value of A. In figure 8.5 the system is stable and A = —0.625 while the alternative parameter values gives A = —1.5 leading to an unstable system. All this we investigated in chapter 3. Nothing in the analysis so far would indicate why it is referred to as the 'cobweb' model. This is because we have concentrated solely on the time path of price in the system. There are two ways to exhibit the cobweb. One is to show the sequence of points on a demand and supply diagram (figure 8.6(a))2 and the other is to plot the differential equation for price in relation to the 45°-line (figure 8.6(b)). In many ways figure 8.6(a) is more revealing because it shows the behaviour of both price and quantity. Here we have a convergent cobweb. The same convergence pattern is shown in figure 8.6(b), but here concentration is on the price sequence. The second approach, however, is more useful for mathematical investigation (especially of nonlinear systems). So far we have concentrated on the very simple linear model. A number of avenues can be explored. Within the confines of the linear model, it is possible to specify a different behaviour for the expected price. It may be argued that expecting the price in the current period to be what it was in the previous period is very naive and takes no account of the trend in prices. It is possible, therefore, to specify an 2 Notice that quantity is on the vertical axis and price on the horizontal axis. 336 Economic Dynamics (8.14) (8.15) adaptive expectation in which the expected price is an adjustment of the forecast error in the previous guess. More specifically, we can write Pt =Pet-x-Wt-\ -Pt-i) If k = 1 then this amounts to our previous model. In other words, the previous model can be considered as a special case of the present model in which k = 1. The model in full is qf = a — bp, b > 0 q\ = c + dp,-\ d > 0 pe, = pU - ^{pU ~ Pt-i) 0 0 then price is expected to move in the same direction as in the past; while if r\ < 0, then price is expected to reverse itself. The extent of Demand and supply models 337 i.e. p, = ( ) - ( - )(1 + ri)pt_x + ( — Ipt-2 (8.20) these price movements is very dependent on the magnitude of 77. The full model is qf = a — bpt b > 0 q\ = c + dpt-i d > 0 Pet =Pt-\ + v(Pt-i -Pt-i) The model can be solved algebraically, but it can also be easily investigated by means of a spreadsheet. To do this we first do some algebraic manipulation. Substituting the expectations equation into the supply equation, and then equating this with demand, we find a- bpt = c + d[pt-\ + v(Pt-i ~Pt-i)] 'a-c\ /d\„ , N , fdt]\ (8-21) ~b~) ~ U which is a second-order difference equation. Example 8.2 Suppose we set up the initial spreadsheet with the following parameter values, and the necessary two initial prices a =100 c=-20 77 = -1.2 b = 2 d=1.25 po = 20 pi = 24 The model is illustrated in figure 8.7. Having set up the model, it is quite easy to change the value of the parameters, but most especially 77, and see the result on the price series. In most spreadsheets the price plot will change interactively as the parameter values are changed. It is of course possible to solve the model algebraically and investigate the restrictions on the parameter values (see Gandolfo 1971, pp. 91-6), but any student of economics can investigate the properties of this model by means of a spreadsheet. 338 Economic Dynamics 8.4 Cobwebs with Mathematica and Maple The intention of this section is to provide routines for creating cobwebs using Mathematica and Maple. There are many ways to do this and here we shall provide only one for each package. Mathematical routine is based on Gray and Glynn (1991, chapter 7), while Maple's routine is an adaptation of this. The Maple routine is written as a procedure, a mini-programme. The reason for this is so many cobwebs can be then created with the minimum of input instructions. A different Maple routine can be found in Lynch (2001).3 Both the routines below can handle linear and nonlinear equations and can also be adapted to deal with stepwise functions. Since the Mathematica routine is the basis for the two instructions, a detailed explanation of this is provided in appendix 8.1. This appendix also provides the Maple routine just as a set of input instructions rather than as a procedure. We consider only the recursive form of the cobweb, as illustrated in figure 8.6(b) and begin with the linear recursive equation (8.22) xt = f(xt-i) = a + bxt-i The instructions for each programme are as follows, where we use the recursive equation in figure 8.6(b) as an illustration: Mathematical f[x_]:=a+b*x {a=4.5,b=-0.625} xO = l points=Rest[Partition[Flatten[Transpose[ {NestList[f,x0,20],NestList [f,x0,2 0] }] ],2,1]]; web=ListPlot[points,PlotJoined->True] lines=Plot[{f[x],x},{x,0,5}] cobweb=Show[web, lines] Maple restart; with(linalg) : with(plots) : cobweb:=proc(f,xO,n,xmin, xmax) local fk,listl,lis12,lis13,list4,web,lines; fk:=(x,k)->simplify ( (f@@k) (x)); listl:=transpose(array( [ [seq(fk(x0,k),k=0..n)], [seq(fk(x0,k),k=0..n)]])): list2:=convert(convert(listl,vector),list): list3:=convert(transpose(array( [list2 [1. .nops (list2)-1], list2 [2..nops (list2) ]])),listlist) : list4:=[list3[2..nops (list3)]] : 3 See Lynch (2001, pp. 249-50 ). Here Lynch provides a routine for the tent function, which is a stepwise function, but this is readily adapted for any linear or nonlinear function. 4 Intermediate displays in Mathematica can be suppressed by including the instruction: DisplayFunction-> Identity, then in the final display using the Show command, include DisplayFunction->$DisplayFunction. See figure 8.8(a) and appendix 8.1. Demand and supply models 339 web:=plot(list4): lines:=plot ({f(x), x},x=xmin. .xmax,colour=blue) : display({web,lines}); end: f:=x->4.5-0.625*x; cobweb(f,1,20,0,5); The cobweb procedure for Maple requires you to first define the function / and then to supply the values for xO, n, xmin and xmax. The penultimate line therefore defines the function and then the input instruction cobweb(/, 1,20,0,5) indicates to use the procedure for the denned function, give xO a value of unity, n a value of 20, and xmin and xmax values of 0 and 5, respectively. These instructions produce a similar plot to the Mathematica instructions given above. Nonlinear equations too are readily handled. If we are considering, for example, the nonlinear logistic equation xt =f(xt-i) = rxt-i(l - xt-i) x0 = 0.1 with r = 3.85, then in Mathematica we replace the first three lines with f[x_]:=r*x*(1-x) {r=3.85} x0=0.1 and we also need to change the range for x in the 'lines' input to {x,0,1}. In Maple, on the other hand, we simply replace the last two lines with f:=x->3.85*x* (1-x); cobweb (f,0.1,20,0,1); Figure 8.8 shows screen shots of the final output for each programme. As can be seen from the figure, the results are virtually the same. It is, of course, possible to include instructions for labelling the axes and provide headings, but we have not done that here. It is also fairly straightforward to deal with piecewise functions, and we shall illustrate exactly how in the next section. 8.5 Cobwebs in the phase plane Return to our linear cobweb model in which supply is based on last period's price. The resulting difference equation is ( a — c\ ( d\ Pt=\—)-\b)Pt-X or pt = A - Bpt-i a — c d where A = - and B = — b b Hence, the functionpt = f(p,-\) = A — Bp,-\ is linear in the phase space in which pt-i is on the horizontal axis and pt is on the vertical axis. Figure 8.9 illustrates 340 Economic Dynamics Figure 8.8. (a) Mathematiea m FigOB.OBa nb ■ In13] - points = Rest [Partition [Flatten [TiailJpOie \ (HeitLiit\f.. xO, 20], He«tL±it[f, xO, 20]>]], 2, 1)1 r" ln[4J= wsb ■ ListPlotrpoitltB, Plot Joined -t True, f>i ■playFunation ■+ Idanti] \n\5f= UnM .Plot [{flt|, i}, {a, 0, 1}, DIiplayFunotlon-* Identity] ; ln[6| = cobweb = Show [w^b , linoi, DisplayFuaction -4 $Di BplayKTunction.] ; D'.l D.2 J 3o0k »jL_I (6) Maple Eilfl Edfl ^tw |nntfi Fäpiwn Qpiwnfl. window r!,: :; üigi^iBjgi oas Ian tuns eh ^ 'ffi ^^j.m ^ I web:-plot(list4)j 1 iries ;-p lot ( [f{n)fX)j jt=xraxn . . XEiau , colour-blue: J ; display( {web,linfls}),■ > J .B5*x* {1-jt) i -t 3.S5 j (I - j) > cobw*b[f,0.1,20,0,1>; 11 0.1 06 0.8 ■ J»l-T Tim: (Si S^titVU AwvlobüB-1 Jffi two possibilities in which demand and supply have conventional slopes. The linear mapping is shown by the line denoted L. The 45°-line, denoted E, satisfies the condition pt = pt-i for all t and p* denotes a fixed point, an equilibrium point. Demand and supply models 341 (b) repellor Whether such a fixed point is stable, unstable or periodic can be established from the slope of L. If 0 < \—B\ < 1, as in figure 8.9(a), thenp* is an attractor, and the price sequence {pt}, starting at po, converges on p*. Starting at po, then, pi = A — Bpo, which is read off the line L. This is the same as p\ on the E-line. At pi, then p2 = A — Bp\, which again is read off the line L. The sequence will continue until p* is reached. On the other hand, the sequence {pt} starting at po in figure 8.9(b), diverges from p*. This is because \—B\ > 1, and so p* is a repellor. What happens when | —B\ = 1, where the slope of the demand curve in absolute value is equal to the slope of the supply curve? We then have pt =A-pt-\ The situation is illustrated in figure 8.10. In chapter 3, section 3.4, we defined a solution yn as periodic if yn+m = vra, and the smallest integer for m is the period 342 Economic Dynamics Figure 8.10. P, Po p' Pi A-i of the solution. In the present example, beginning at po, we have Pi = A -Po Pi = A -Pi = A -(A -Po) = Po Pi = A -Pi = A -Po P4 = A -P3 = A -(A -Po) = P0 If follows, therefore, that Po=Pi=P4 = ■■■ and pi = p3 = p5 = ... We have a two-period cycle solution. In fact, with linear demand and supply, with equal slopes in absolute value, there can only be a two-period cycle and no higher one is possible. Discrete systems have come under increasing investigation in recent years because of the possibility of chaotic behaviour to which they can give rise (see chapter 7). By way of introduction to this analysis we shall continue with our simple linear example and consider how Mathematica can be used to investigate cobweb models. It will be found that the phase plane plays an important part in this approach. Although a little repetitive, we shall provide both Mathematica and Maple instructions for deriving the cobwebs. However, since the Maple procedure never changes, we shall simply write 'cobweb procedure'. Example 8.3 We shall illustrate the technique by means of the following simple cobweb model qf = 24- 5pt qst = -4 + 2pt_1 <£ = <£ = % Demand and supply models 343 which has equilibrium valuesp* = 4 and q* = 4. The resulting difference equation is Pt = 5.6 - 0.4p,_i The objective is to plot a sequence of points in the phase plane. Joining up these points forms the web of the cobweb. Superimposed on this web is the line/(p) = 5.6 — OAp and the 45°-line. A full explanation of the instructions for Mathematica and Maple are given in appendix 8.1. Mathematica f [p_] :=5.6-0.4p pO = l points=Rest[Partition[Flatten[Transpose[ web=ListPlot[points,PlotJoined->True] lines=Plot[ {f[p],p}, {p,0,10} ] cobweb=Show[web,lines, AxesLabel->{"pt-1","pt"} ] Maple "cobweb procedure' f:p->5.6-0.4*p cobweb(f,1,20,0,10) Notice in the Maple instructions that although the procedure defines the function in terms of the variable x, we define / in terms of p. We can do this, and we do it repeatedly throughout this chapter, because the variables within the procedure are defined locally. The result is shown in figure 8.11. To illustrate the generality of this approach, and to highlight a nonlinear cobweb, consider the following model. Example 8.4 {NestList[f,p0,20],NestList[f ,p0, 20] } ] ] , 2, 1 ] ] ; = 4 - 3pt = $ = % pt Figure 8.11. 10 6 2 8 2 6 10 pt- 1 344 Economic Dynamics The equilibrium, assuming a positive price, is given by p* = 1 and q* = 1, and the difference equation is given by Pt = (4/3) - (1/3)/^ The question now arises as to whether the solution p* = 1 is stable or not. Using Mathematica we can input the following instructions (where we have included solving for equilibrium price): Mathematica f [p_] : = (4/3)-(l/3)pA2 EquPrice=Solve[p==f[p] , p] p0=1.5 points=Rest[Partition[Flatten[Transpose[ {NestList[f,p0,20],NestList[f,p0,20]} ] ] , 2, 1 ] ] ; web=ListPlot[points,PlotJoined->True] lines=Plot[ {f[p],p}, {p,0,2} ] cobweb=Show[web,lines, AxesLabel->{"pt-1" , "pt"} ] Maple's instructions are Maple "cobweb procedure' f:=p->(4/3)-(1/3)*pA2 EquPrice:=solve(p=f(p),p); cobweb (f,1.5,20,0,2); The resulting cobweb is illustrated in figure 8.12. The model, therefore, has a locally stable solution for a positive price equilibrium. It should be noted5 that at the equilibrium point we have /'(/>*) = 2(-l/3)p* =-2/3 i.e. \f(p*)\ < 1 5 See chapter 3, section 3.4. Demand and supply models 345 which verifies the local stability of p* = 1. Although this particular nonlinearity leads to a stable solution, some authors (e.g. Waugh 1964) have argued that other types will lead to a two-period cycle and may well be the norm. One especially important nonlinearity can arise where there is a price ceiling set (Waugh, 1964). Consider the following linear demand and supply model. Example 8.5 qf = 42- 4pt qst = 2 + 6pt-x qdt=q\ = qt The difference equation from this model is p,= lO- l.5pt-i with equilibrium values p* = 4 and q* = 26. Since the slope of the function/(p) = 10—1.5p is greater than unity in absolute terms, then the cobweb is unstable. This is readily verified starting with a price of po = 3.5. Suppose a price ceiling pc = 6 is set, then the price in any period cannot exceed this ceiling. What we have, then, is a function which is kinked at p = 8/3, the value where pc = f(p). This can be expressed as f(p) 6 p < 8/3 10-1.5p P>S/3 Within Mathematica there is only a slight difference in denning the function. We define it using the If command, i.e. f[p_]:=If[ p<8/3,6,10-1.5p] i.e. if p < 8/3 then/(p) takes the value 6, otherwise it takes the value 10 — l.5p. The remaining instructions for creating the cobweb remain unaffected, although we have added the price ceiling to the lines, i.e. lines=Plot[ {f[p], p, 6}, {p,0,6}] In Maple we use a similar instruction using the piecewise function, i.e. f:p->piecewise(p<8/3,6,10-1.5*p); which says for p less than 8/3 take the value 6, else take the value 10-1.5p for P > 8/3. All other instructions remain the same. The full instructions for both packages, including a plot of f(p), are: Mathematica f [p_] :=If[p<8/3,6,10-1.5p] Plot[f[p], {p,0,6} ] p0=3.5 points=Rest[Partition[Flatten[Transpose[ {NestList[f,p0,10],NestList[f,p0,10]} ] ] , 2, 1 ] ] ; web=ListPlot[points,PlotJoined->True] 346 Economic Dynamics Figure 8.13. pt lines=Plot[ {f[p],p,6}, {p,0,6} ] cobweb=Show[web,lines, AxesLabel->{"pt-1", "pt"} ] Maple "cobweb procedure' f:p->piecewise(p<8/3,6,10-1.5*p); cobweb(f,3.5,10,0,6); which results in the cobweb shown in figure 8.13. What figure 8.13 shows is that initially, starting from a price of po = 3.5, the market is unstable, moving away from the equilibrium price p* = 4. However, once the price ceiling is reached, the system settles down to a two-period cycle, oscillating between p = 1 andp = 6. Although the equilibrium is not attained, the ceiling does limit the price variation. 8.6 Cobwebs in two interrelated markets6 Interrelated markets with time lags illustrate the application of discrete dynamic systems. An early example was the corn-hog cycle mentioned by Ezekiel (1938) and Waugh (1964). The model amounts to two markets: Corn market dct = a\ — b\pct b\ > 0 (8.23) sct = c\ + d\pct_x d\ > 0 Hog market dht =a2- b2pht b2>0 Ji _ „ , j Ja (8.24) snt = c2 + d2pht_x + epct_x di > 0, e < 0 d? = si where dc = demand for corn dh = demand for hogs sc = supply of corn sh = supply of hogs pc = price of corn ph = price of hogs 6 This section requires knowledge of chapter 5. Demand and supply models 347 The corn market is our typical cobweb with a one-period lag on supply, i.e., farmers base the supply in period t on the expected price of corn, which they assume is the same as it was last period. The corn market is independent of the hog market. However, the hog market besides having a similar time lag on the supply also depends on what farmers expect the corn price to be - since this is food for the hogs. It is assumed that farmers expect the price of corn to be what it was last period. This model can apply to any animal-feed interaction. The model can be handled by deriving a system of difference equations, which is accomplished by substituting the respective demand and supply equations into the equilibrium conditions and suitably re-arranging the results. The results are P,= a\ — c\ bi a2 — c2 d\ -\vh b2)Pl-1 (8.25) Pt-i Let p* and p*h denote the equilibrium prices in the corn and hog markets, respectively. Then taking deviations from equilibria, we have Pi * ■Pc »> = - ^i) W-i-A) ~ (£) ("«->-p') Let Pi ■d\/b\ B ■pU -e/b2 yt=Pt-pt A2 = -d2/b2 implies xt-\ implies yt-i = pht_x - p*h Then the system can be written more succinctly as xt = Axxt-i yt = A2yt-i + Bxt-i or in matrix notation Xt 0 " xt-\ yt. B Ai_ yt-i. Equilibria in the two markets require xt = xt-\ = x* for all t, and yt = yt-\ for all t. Using these conditions, the equilibria are readily found to be * Pc = * Ph a\ — c\ b\ + d\ a2 — c2 b2 + d2J b2 + d2 \b\ + d\ a\ — c\ (8.26) We investigated the properties of such systems in chapter 5. Let A denote the matrix of the system, where A = B 0 ' A2 348 Economic Dynamics with characteristic equation |A — k\\ = (A\ — k)(A2 — X) = 0, then the characteristic roots are r = A\ and s = A2. Stability in the corn and hog market is assured if \r\ < 1 and \s\ < 1, i.e., \—d\/b\\ < 1 and | —<^?21 < 1- It should be noted that \—d\/b\ | < 1 is the condition for stability in the corn market, while |— d2lb2\ < 1 is the condition for stability in the hog market, assuming constant corn prices. Example 8.6 To pursue the analysis further we shall consider the following numerical example. For corn we have dct=2A-5pct sct = -4 + 2pU dct = < while for the hog market we have dht = 20 - 5$ sht = 2.5 + 2.5/^ - 2pU dht = sht Substituting and solving for pct and pht, respectively, we obtain pct = 5.6 - 0.4^_! p) = 3.5 + OAp^ - 0.5/^! The equilibrium price in each market is readily found to be p* = 4 andp*h = 3.4. The question now arises as to whether this combined equilibrium is stable. Taking deviations from equilibrium, the system reduces down to "-0.4 0 Xt-1 Jt. . °-4 -0.5 _ yt-i with characteristic equation |A - AI| = (-0.4 - A)(-0.5 - X) = 0 and characteristic roots r = —0.4 and s = —0.5. For r = —0.4 we have (A - rl)vr = 0 so that 0.4 + 0.4 0 "0" 0.4 -0.5 + 0.4 _ o_ or " 0 0 "0" 0.4 -0.1 _ o_ then 0.4v^ -O.lv^ = 0 Demand and supply models 349 Let Vj = 1 then vr2 = 0.4vj/0.1 = 4. Hence the eigenvector associated with r = -0.4 is v I or s = —0.5 we have (A - sl)\s = 0 giving 0.4 + 0.5 0 "0" 0.4 -0.5 +0.5 _ A. 0_ or "0.1 0" "0" 0.4 0 A. 0_ Hence, 0.1 v| = 0 giving v\ = 0. v\ can be anything, so let it be unity. Then Hence, the matrix composed of both eigenvectors, denoted V, is given by V = [vr \s] = 1 0 4 1 Hence or uř = VDV^uo "1 0" "(-0.4)' 0 "1 0" -l xo yt. 4 1 0 (-0.5)'_ 4 1 yo As t oo then each of the terms (—0.4)' and (—0.5)' tends to zero, and so xt -> 0 and y, —► 0, which in turn means pct p* andpht p*h. The interrelated market is, therefore, dynamically stable. 8.7 Demand and supply with stocks In section 8.1 we indicated that suppliers often change prices in response to their level of stocks. This, of course, applies only to non-perishable goods. It is surprising, therefore, that there are so few demand and supply models with stock behaviour built in. Here we shall consider just a few simple stock models. Stocks can be built up only when there is excess supply. Furthermore, stocks are specified for a point in time. Let Qt denote the level of stocks at the end of period t. Then the change in stocks over period tis Q, — Qt-\ and this arises from the excess supply over period t, i.e. AQt = Qt- Qt-i =qst-qdt 350 Economic Dynamics We continue to assume a linear demand and supply model. What happens in such a model depends on how suppliers alter prices in response to stock changes. We consider two alternative assumptions ASSUMPTION 1 Suppliers raise the price if stocks in the previous period fall, and the rise in price is set proportional to the fall in stocks Pt -Pt-\ = -yiAQt-x yi > 0 ASSUMPTION 2 Suppliers raise the price if stocks in the previous period fall below a given level, Q, and the rise in price is set proportional to the deviation of stocks from the specified level Pt-pt-i = -YiiQt-\ ~ Q) 72 > 0 Under assumption 1 we have the model qf = a- bpt (8.27) qst=c + dpt Pt-Pt-i = -YiAQt-i where AQ, = q\ — qf for any period t. Therefore, Pt-pt-i = -YiiqUi ~ 4-\> pt = pt-i - yi(c + dpt-\ -a + bpt-\) = Pt-i + Y\{a - c) - yi{b + d)pt-\ i.e. (8.28) pt = y\{a — c) + [1 — y\{b + d)]pt-i The fixed point of this system, the equilibrium point, is found by setting pt = p* for all t. With result P = - 1 b + d the same equilibrium we found in earlier models. Although the behaviour of suppliers in response to stock changes does not affect the equilibrium price (and hence also the equilibrium quantity), it does have an impact on the path to equilibrium. The solution to the difference equation (8.28) is (8.29) pt = + [1 - Yi(b + d)Y (po b+d \ b+d where po is the price at time t = 0. There are three possible time paths for price: (i) If 0 < 1 — y\{b + d) < 1 then the system converges steadily on the equilibrium value. This occurs if 0 < Y\ < 1 /(b + d). Demand and supply models 351 (ii) If —1 < 1 — y\(b + d) < 0 then the system converges on the equilibrium in terms of damped oscillations. This occurs if l/(b + d) < y\ < 2 Kb + d). (iii) If 1 — y\(b + d) < — 1 then the system has explosive oscillations. This occurs if Y\ > 2/(b + d). Example 8.7 Consider the following model qf = 20 - 4pt q° = 2 + 2.5p, pt-pt-i = -0.2 AQt then pt = 3.6 - 0.3/?,_i with fixed point p* = 2.769 and solution Pt = 2.769 + (-0.3)'(p0 - 2.769) and the system converges on the equilibrium in terms of damped oscillations. Turn now to assumption 2. Our model is qf = a- bpt q\ = c + dpt (8.30) Pt-pt-i = -Yi(Qt-\ ~ Q) Substituting, we have pt = pt-\ - YiiQt-i ~ Q) However, we need to establish Qt-\ — Q. Lag this equation by one period, then Pt-\ = pt-2 - YliQt-2 ~ Q) Hence, Pt -Pt-\ = Pt-\ - Pt-2 ~ YiiQt-i ~ Qt-i) pt = 2pt-x - pt-2 ~ Y2{qst-i ~ 0, i=l,2 Equilibrium requires p1 = 0 and p2 = 0 (which automatically implies p0 = 0) and this will lead to equilibrium prices p\ and p2, i.e., E\(l,p*,p2) = 0 and Eii^,P\,P2) = 0. Define the vector p* = (p*,p2). Assume such an equilibrium exists. Here we are not concerned with existence but rather with the stability of the competitive equilibrium. Stability can be considered in terms of the phase plane (pi,p2), obtaining the two equilibrium lines px = 0 and p2 = 0, which divide the phase plane into regions, and then considering the vector forces in the various regions. This we now do. Demand and supply models 355 Figure 8.15. Consider px = 0 first. Differentiate E\ with respect to p\ andp2, i.e. Endpi + E12dp2 = 0 Then -E n dpi dpi pi=0 r 0 (b) £22 < 0 and £21 > 0 It follows from condition (a) that the slope of the px = 0 isocline is positive, and it follows from condition (b) that the slope of the p2 = 0 isocline is positive. We further assume that commodities 0 and 1 and 0 and 2 are gross substitutes. This means £01 > 0 and £02 > 0. But how does this help us? Since Et are homogeneous of degree zero, then from Euler's theorem8 we have (a) £10 + p\En + p2El2 = 0 (b) £20 + P1E21 + P1E21 = 0 7 Gross substitutability refers to the uncompensated demand curve while net substitutability refers to the compensated demand curve. See Shone (1975, section 4.4). 8 See Chiang (1984, p. 418). 356 Economic Dynamics From condition (a), and taking account of £10 > 0, we have p\En +P2E12 < 0 . . . Pi -En implying — < —— Pi E12 Similarly, from condition (b), and taking account of £"20 > 0, we have P1E21 + P2E22 < 0 P2 —E21 implying — > - (since £"22 < 0) Pi E22 From these results it follows —£21 P2 —En - < — < - £22 Pl E\2 i.e. dpi dpi < p2=o dp2 dpi Pi=0 as illustrated in figure 8.16. Now consider the situation each side of the isocline px = 0. In figure 8.16(a) we move from point a to point b. The only price changing is p\, hence we have £11 < 0 and £21 > 0. Differentiatingpx = &i£iwith respect top\, we obtain dpi dpi k\E\\ < 0 or signCdpi) = signihEndpx) Hence, to the right of p1 = 0, p1 < 0 and p\ is falling. To the left of p1 = 0, where dpi < 0, then sign(dp1) > 0 and sopx > 0. Carrying out the same logic for figure 8.16(b) we have p2 = ^£2 dp2 7 7-1 n dpi sign(dp2) = ^gn{k2E2idpi) Hence, to the right of p2 = 0 we have p2 > 0 andP2 is rising; to the left of p2 = 0, P2 is falling. Combining all these vector forces, we have the situation shown in figure 8.17. A system which begins in region I will either directly converge on p*, or will move into either quadrants II and IV and then converge on p*. Similarly, any initial point lying in quadrant III will either converge directly on p* or will move into either quadrant II and IV and then converge on p*. Points of the system beginning in region II will remain in that region with a trajectory converging on p*. Similarly, an initial point in region IV will have a trajectory remaining in this region and converging on p*. Example 8.10 The market can be illustrated by means of the following numerical example, where we postulate only the excess demand curves for commodities 1 and 2. Suppose, Demand and supply models 357 P,=0 Figure 8.16. Figure 8.17. 358 Economic Dynamics Figure 8.18. then E\ = 3 - 6pi + 3p2 E2=16 + Api - 8p2 with dynamic adjustments Pi = 2E\ Pi = 3£2 We first must establish whether an equilibrium exists (and with meaningful prices). In equilibrium p1 = 0 and p2 = 0 implying £"1=0 and £2 = 0. Solving the two linear equations readily reveals p* = (p>i,p2) = (2, 3). Furthermore, the two isoclines are readily shown to be p1=0 implying p2 = -1 + 2pi p2 = 0 implying p2 = 2 + 0.5pi The trajectories can be derived by solving the two dynamic equations px = 2(3 - 6p\ + 3p2) = 6 - 12px + 6p2 p2 = 3(16 + 4Pl - 8p2) = 48 + 12Pl - 24p2 as outlined in part I. In figure 8.18, derived using Maple and annotated, we have four trajectories with starting values (0.5, 1), (3, 2), (3, 4), (1,4) along with the direction field. This figure shows quite clearly the stability of the competitive equilibrium. 8.9 The housing market and demographic changes In this section we shall consider a model of Mankiw and Weil (1989) which deals with the impact of the 'baby boom' of earlier years and its impact on the housing Demand and supply models 359 market and later the 'baby bust' and its impact on the housing market. The model is captured by the following set of equations: H = stock of housing Hd = demand for housing R = real rental price N = adult population h = H/N = housing per adult P = real price of a standardised housing unit rP = operating cost of owning a home (r assumed constant) d = rate of depreciation H = net investment in housing H + dH = gross investment in housing n = N/N = growth in population The first equation denotes the demand for housing in which the variable N acts as a shift parameter, and here attempts to capture demographic changes in the population. From this equation it follows that the demand for housing per adult is given by Hj/H =f(R). Consequently the rental associated with such demand is given by the inverse off, i.e., Rj = f~l{h) = R(h). The market clearing condition is that R = Rd, hence R = R(h), which is the second equation in the model. In interpreting the third equation we note that rP is defined as the operating cost of owning a home (where r is assumed constant), i.e., the user cost, and this needs to be adjusted for the change in the price of the house - invariably the capital gain. In a perfectly functioning housing market, this should be equal to the real rental price. If this was not the case then there would be a movement out of home ownership into rented accommodation if R < rP — p and into home ownership if the reverse inequality was true. Gross investment (net investment, H, plus depreciation, dH) is assumed to be an increasing function of the price of housing, g(P) and g' > 0, and proportional to the adult population. Rather than deal with the model in terms of H it is easier to manage by considering h = H/N. Differentiating h with respect to time, then (i) (ii) (iii) (iv) Hd=f(R)N R = R(h) R' > 0 R(h) = rP-p H + dH = g(P)N g>>0 (8.41) where H =--nh N g(P)N - dH — nh N i.e. h = g(P) -(d + n)h 360 Economic Dynamics The model can therefore be expressed in terms of two differential equations (8.42) P = rP- R(h) h = g(P) -(d + n)h which allows a solution for P and h. Suppose there exists a fixed point (h*, P*) such that P = 0 and h = 0. Such a fixed point denotes an equilibrium combination of P and h. In order to see what is happening in this model we need to investigate it in terms of the phase plane. Consider the price equation first. In a steady state we have P = 0, which implies rP = R{h) n R(h) dP dh r R'{h) < 0 since R\h) < 0 Hence the price stability condition gives rise to a downward sloping line in the phase plane where P is on the vertical axis and h on the horizontal axis, as shown in figure 8.19. To the right (and above) this line we have the condition that P > R(h)/r, which implies P > 0, and so house prices are rising. Similarly, to the left (and below) the line shown in figure 8.19 we have P < R(h)/r, which implies P < 0 and house prices are falling. Now consider the second equation. In equilibrium there is no change in the number of houses per adult of the population, i.e., h = 0. Hence g(P) = (d + n)h , dP g'{P)—=d + n dh i.e. dP dh d + n > 0 since g\P) > 0 Figure 8.19. 0 P=0 h Demand and supply models 361 0 h=0 h<0 h Figure 8.20. h=0 h Figure 8.21. This stability condition gives rise to an upward sloping line in the phase plane, as shown in figure 8.20. Above and to the left of the stability condition h > 0 and so h is rising, while below and to the right of h = 0, h is falling. We are now in a position to put the information together, as illustrated in figure 8.21. The phase diagram illustrates the vector of forces in the various quadrants. It also illustrates the presence of a saddle point, point E, with SS' denoting the stable arm of the saddle point. A market which starts out of equilibrium, such as point A in figure 8.21, will initially exhibit a rapid rise in house prices, pushing the system from point A to point B on the saddle path SS', and then over time the system will move down the stable arm of the saddle path to the equilibrium point E. Consider the market in equilibrium and a 'baby boom' occurs, resulting in a rise in the shift parameter n. For the moment let us concentrate solely on the equilibrium lines. The price equation is independent of n and so this has no effect on this line. However, the change in n will raise the absolute value of the slope 362 Economic Dynamics Figure 8.22. (d + n). Consequently there will be a new housing line to the left of the original. In figure 8.22 we have labelled this h\. There is a saddle path SiSj associated with the saddle point A, and there is another saddle path associated with the saddle point D, each denoting stable arms of their respective saddle points. Suppose it is announced that there is a 'baby boom' and that the population is now growing at 1 per cent higher than previously, but it is also indicated that this will last for only a short period, say ten years, and then the growth will return to its previous level. What will be the movement of house prices, P, and housing per adult, hi The market, having perfect foresight, will know that there will be a rise in house prices, and so the system will move vertically up from point A to point B, this movement being purely the announcement effect. At this stage there is no increase in housing demand (since the babies have not yet become adults), and so the system is still governed by the dynamic forces associated with point A. Accordingly, the system will move in the north easterly direction, along a path like BC. Once the 'baby boom' takes place and the increase in population takes place and becomes part of the housing market, then the market will be governed by the new saddle path S2S2. Accordingly the system will move along this stable arm of the saddle point towards point D. However, if there is then a reduction in population growth, which returns to its former level, then the system will move along the path indicated by the arrows and towards point A. (Notice that point D, although governing the movement after point C is reached, will not necessarily be achieved.) This analysis is based on perfect foresight and movements are dominated by the stable arms of the saddle points. In comparison the authors consider a naive model in which house prices are assumed to remain constant at any moment of time - a very naive version indeed! In this instance there are no capital gains (since P = 0) and so P = R(h)/r. Since prices cannot be changing at any level of h, then the market is always positioned along the line P = 0. In terms of figure 8.22, there would be no movements of Demand and supply models 363 Pit) Figure 8.23. ( \ time forward looking model nave model the system when the announcement was made, it would simply remain at point A. Once the baby boom became of adult age and entered the housing market, there would be a shift left in the h\ = 0 line, moving the system towards point D. The system would move up the P = 0 line towards point D, reaching point D at the height of the growth, and then returning down the P = 0 line towards point A. The difference in price movement of this naive version in comparison with the perfect foresight version is illustrated in figure 8.23. Two observations can be made in the light of figure 8.23: (1) A model of perfect foresight would always indicate a rise in house it would be in the naive model. This is not so readily testable since it depends on a counterfactual question. 8.10 Chaotic demand and supply9 We have pointed out that chaotic behaviour can arise only in the presence of nonlinearity. Although it is possible to have both nonlinear demand and supply, the most significant nonlinearity is in supply, most especially when supply involves a time lag. Take a typical case that at low prices supply increases slowly, say because of start-up costs and fixed costs of production. Furthermore, suppose at high prices then again supply increases only slowly, say because of capacity constraints. This suggests a S-shaped supply curve. The logistic equation exhibits such a S-shape. An alternative specification, which is frequently employed in modelling, is the arctan function. One such specification of supply in terms of expected prices is q\ = arctan {ixpf) (8.43) 9 This section draws heavily on Hommes (1991, section 1.5). (2) prices prior to the 'baby boom' becoming of adult age. This is a testable hypothesis. The range of price movement in the perfect foresight model is less than 364 Economic Dynamics Figure 8.24. (a)n=l (b)|i = 3 ■1.5 (8.44) (8.45) This specification sets the origin at the inflection point, which is unique, and so prices and quantities can be negative as well as positive. The parameter \x determines the steepness of the S-shape. For instance, figure 8.24(a) has \x = 1 while figure 8.24(b) has \x = 3. The higher the value of \x the steeper the S-shape. Note in figure 8.24 that price is on the horizontal axis and quantity is on the vertical axis - the reverse of conventional economics texts. For simplicity we assume demand is linear and a function of actual prices, i.e. 4 a bpt b>0 A typical nonlinear cobweb is then qf = a- bpt q\ = arctan (npf) Pet =pet_l+X(pt.l -pet_i) qf The expression for expected prices is a typical adaptive expectations assumption, which can be written (8.46) pet = Xpt-i + (1 - X)pet_l Demand and supply models 365 From the first, second and fourth equation of (8.45) we have Pt = a — qt a — qt a — arctan (/xpf) b b From (8.46) we have pet+l = kpt + (1 pet+l-d-m k)pet which on re-arrangement gives Pt / Hence i.e. pet+1 — (1 — k)pet a — arctan (ixpf) Pet+i k (1 - k)pet + ka k arctan (/xpf) b b orpet+l = f(pet). Notice that (8.47) is a recursive equation for expected prices. For given values of the parameters a, b, k and \x we can establish the fixed point(s) of equation (8.47). But is this value unique? More specifically, what happens to the fixed point of expected prices, pe*, when the demand curve shifts? A shift in the demand curve is captured by a variation in the parameter a. If we allow the parameter a to vary between -1.25 and +1.25 we can establish the equilibrium points for pe by plotting the bifurcation diagram of pe against a. (8.47) Let then Example 8.11 k = 0.3, b = 0.25, ix = 3 0.3a 0.3 arctan (3pet) 0.25 0.25 The resulting bifurcation diagram is shown in figure 8.25. H = 3 Figure 8.25. 0.5 P -0.5- 366 Economic Dynamics For low values of a, there is a unique equilibrium for expected price. Around the value a = —0.9 a period-doubling bifurcation occurs. The stable two orbit remains until a reaches about 0.9, and then a period-halving occurs. Thereafter the system settles down again to a unique stable equilibrium. An important observation to make about this diagram is that it is symmetrical about the origin.10 This is a characteristic feature of arctan. The question now arises: What happens when /x takes on different values? A most complex set of results can occur. For instance, figure 8.26 sets out three possibilities: \x = 3.5, \x = 4 and \x = 4.5. In figure 8.26(a)-(c) we have a ranging over the interval -1.25 to +1.25. In figure 8.26(d) we have \x = 4.5 and a ranging over a smaller interval, namely 0.4 to 1.25. Figure 8.26(a) (/x = 3.5) shows a doubling bifurcation turning into a period-four orbit, which then turns into a period-two orbit and finally to a stable equilibrium. In Figure 8.26(b) (/x = 4) there is this same basic pattern, but within the period-four orbit there occurs periods of chaos. Such periods of chaos within the period-four orbit become larger as /x rises. A closer look at the chaotic region on the right of figure 8.26(c) (/x = 4.5) shows some unusual patterns within the chaotic region, as shown in figure 8.26(d). What is clearly illustrated by this example is that in the presence of nonlinearity markets can exhibit a variety of price behaviour, including a chaotic one. It is important, therefore, to have some clear estimation of a system's parameter values (including their margin of error!) and to establish the likely equilibria of the system for such values. If chaos is present, then the system is sensitive to initial conditions. This applies also to the diagrams in figure 8.26. Demand and supply models 367 In such circumstances regression results, which are often employed in empirical work, may not be very meaningful. Appendix 8.1 Obtaining cobwebs using Mathematica and Maple 8A.1 Mathematica11 To employ Mathematica to derive linear and nonlinear cobwebs, the main problem is deriving the sequence of points that make up the web. Here we derive the method for the linear case, since it is easier to represent and follow through. But once it has been obtained, it can be used for both linear and nonlinear cases. The example is from section 8.4. First we define the price function: f [p_] := 5.6-0.4p It is, of course, possible to be general and input this in the form f[p_]:=a -bp and then define a = 5.6 and b = 0.4. However, we shall continue with the numerical example. The next step is to generate a series of prices starting from the initial price, pO (in this appendix we shall not subscript the t indicators since this will not be done within Mathematica). This can be accomplished quite easily with the NestList command and using the instruction NestList[f,p0,3] This will generate a sequence {pt} beginning at pO and supplying a further three observations, i.e., it will generate the series {p0,pl,p2,p3}. For any price p we know that the value on the line is f(p). For instance at pO the point on the line is pi =/(p0). Thus, point A in figure 8A.1 is (p0,/(p0)) or (p0,pl). Point B is the associated point on the 45°-line, and is therefore represented by point (p 1 ,p 1). Point C, once again, is read off the line and is (p 1 ,/(p 1)) = (p 1 ,p2). Point D is (p2,p2) since it is on the 45°-line; while point E is (p2,p3). The points emerging, therefore, are: (p0,pl), (pl.pl), (pl,P2), (P2,p2), (P2,p3), ... Since these points consist solely of prices from the series {pt}, then it must be possible to generate these sequences of points with a suitable transformation of {pt}. This indeed can be done, but it requires a little manipulation. Consider the sequence derived using NestList[f,p0,3]. The sequence would be {p0,pl,p2,p3}. Now form the series {NestList [f,p0,3], NestList[f,p0,3] } The result would be: {{p0,pl,p2,p3}, {p0,pl,p2,p3}} 11 A neater more efficient version is provided in Gray and Glynn (1991, chapter 7). The less efficient version provided here is more intuitive. 368 Economic Dynamics Figure 8A.1 This can be considered as a matrix of dimension 2x4. Now transpose this matrix using Transpose[ {NestList[f,pO,3], NestList[f,pO,3]} ] which will give the list {{P0,p0}, {pl.pl}, {P2,p2}, {P3,p3}} which is a 4 x 2 matrix. Now we need to collapse this list by getting rid of the inner brackets. We do this using the Flatten command. Thus Flatten[Transpose[{NestList[f,pO,3], NestList [f,pO,3] } ]] gives the series {pO, pO, pi, pi, p2, p2, p3, p3}. The next step is to take groups of 2 and move along the sequence one element at a time, i.e., we wish to form the sequence {{P0,p0}, {P0,pl}, {pl,pl}, {pl,P2}, {P2,p2}, {P2,p3}, {P3,p3}} We accomplish this using the Partition command. Thus Partition[list,2,1] will take a 'list' and convert it into another list each element composed of two elements, and moving along 'list' one element at a time. Consequently, the sequence just given would be generated from the instruction Partition[Flatten[Transpose[ {NestList[f,p0,3], NestList [f,p0,3; , 2,1 where we have written this over two lines for ease of viewing. The final step is to remove the first element {pO,pO}, since this is not part of the cobweb. This is accomplished using the Rest command. Thus, Rest[5] deletes the first element of the series denoted V. Our list of points, therefore, is achieved with the rather Demand and supply models 369 involved instruction points=Rest[Partition[Flatten[Transpose[ {NestList[f,pO,3], NestList[f,p0,3]} ] ] , 2,1 ] ] ; where the semicolon at the end of the instruction suppresses the points being listed. Having obtained the list of points, we need to join them up. Let us refer to this as 'web'. Then the web is the plotting of the list of points just derived and joined together. We do this with the ListPlot command, and qualifying it with the instruction PlotJoined->True, which instructs the programme to join up the points. Thus web=ListPlot[points, PlotJoined->True ] To complete the picture we need to draw the line f(p) = 5.6 — 4.Op and the 45°-line. These can be done together with the instruction lines=Plot[ {f[p],p}, {p,0,10} ] Our cobweb is then achieved with the final instruction Show[web,lines, AxesLabel->{"pt-1","pt"} ] which also includes an instruction to label the respective axes. To summarise, where we have specified pO to have a value of unity and increased the run to twenty: f [p_] :=5.6-0.4p pO = l points=Rest[Partition[Flatten[Transpose[ {NestList[f,p0,20],NestList[f,p0,20]} ] ] , 2, 1 ] ] ; web=ListPlot[points,PlotJoined->True, DisplayFunction->Identity] lines=Plot[ {f[p],p}, {p,0,10}, DisplayFunction-Mdentity ] cobweb=Show[web,lines, AxesLabel->{"pt-1", "pt"}, DisplayFunction->$DisplayFunction ] In these final set of instructions we have suppressed the list of points by using the semicolon and we have suppressed intermediate displays by using the instruction: DisplayFunction->Identity. The display is turned on again in the final line by adding, Display Function->$Display Function. The result is shown in figure 8A.2. This set of instructions can be used for any cobweb, whether linear or nonlinear. In fact, with only a slight modification, it can be used to investigate any first-order difference equation model. For instance, we used a variant of it in chapter 7 when we investigated the properties of the logistic equation. 370 Economic Dynamics Figure 8A.2 pt 10 1 2 4 6 8 TO 8 6 4 2 Pt - 1 8A.2 Maple The logic and development here is similar to the previous outline but there are some differences. Since we shall be using linear algebra and plotting routines then these two libraries need to be called. The first few lines of the programme are then with (linalg) : with(plots): f:=p->5.6-0.4*p; fn:=(p,n)->simplify( (f@@n)(p) ); where the last line allows us to specify the initial values of p and the number n. In what follows we take these to be 1 and 20, respectively. Our next task is to repeat the sequence twice and to transpose it into a 21 x 2 matrix. In the following instructions this is called listl. Next we convert to an array by converting listl into a vector and converting this in turn into a list. This we call Ust2. The next stage is to make up two lists from Ust2, one in which the first element is dropped and the second in which the last element is dropped. These two lists are combined, made into an array, then transposed, and finally converted to a list of lists. This we label Ust3. The final list, Ust4, is derived by dropping the first element. The four lists are listl:^transpose(array([[seq(fn(1,n),n=0..20)], [seq(fn(l,n), n=0..20) ] ] ) ); list2:^convert(convert(listl,vector),list); list3:^convert(transpose(array( [ list2 [1. .nops (list2)-1], list2 [2 . .nops (list2) ] ] list4:=[list3[2..nops(list3)] ]; In these instructions the term 'nops' denotes the number of elements in the specified list. Thus, nops(list2) denotes the number of elements in Ust2. The web is simply the plot of Ust4, since by default the points are joined. We add finally the line/(p) = 5.6 — 4.Op and the 45°-lines with the instruction lines:=plot( {f(p),p}, p=0..10): ) ), listlist); Demand and supply models 371 (Notice that we suppress the output by using the colon - which we also do with the web). The full instructions are: with (linalg) : with (plots) : f:=p->5.6-0.4*p fn:=(p,n)->simplify( (f@@n)(p) ); listl:^transpose(array ( [ [seq(fn(1,n) ,n=0. .20) ], [seq(fn(l,n), n=0. .20) ] ] ) ) : list2:^convert(convert(listl,vector),list): list3:^convert(transpose(array( [ list2 [1. .nops(list2)-1] , list2 [2. .nops (list2) ] ] ) ), listlist) : Iist4: = [list3[2. .nops (list3)] ]: web:=plot (list4) : lines:=plot( {f(p),p}, p=0..10): display( {web,lines} ); In these instructions we have suppressed all lists by placing a colon at the end of the instructions that generate them. In section 8.4 we present these instructions as a procedural mini-programme. An alternative derivation of a cobweb with Maple can be found in Lynch (2001, chapter 14). As withMathematica this routine can be used to derive a cobweb for any function f(p), whether it is linear or nonlinear (including the possibility that f(p) is kinked -as in the case of a price ceiling). Exercises 1. (i) Show that if b < Oinqf = a - bp, andd > 0in<7* = c + dPt-\ then the cobweb is still convergent if 0 < d/{—b) < 1 and divergent if d/{-b) > 1. (ii) Is the behaviour oscillatory? 2. Establish the convergence, divergence, or oscillation of the following systems. (i) qf = 100 - 2pt (ii) qf = 5 + 2pt qst = -20 + 3pt-i qst = 35 +pt-i Po = 10 Po = 10 (iii) qf = 100 - 2pt (iv) qf = 18 - 3Pt t = -4 + 2pet pet = pu ~ A- {Pt-i ~ Pt-i) 0 3 the system is unstable. (2) Increasing marginal costs are a stabilising influence. 9.1 Static model of duopoly The model we shall consider in this chapter has a very simple linear demand curve and, at this stage, constant marginal costs. The model is as follows. 376 Economic Dynamics (9.1) (9.2) p = 9-Q Q = qi+q2 TC\ = 3q\ TC2 = 3q2 Since our interest is with stability and the impact of increasing the number of firms in the industry, or changing the specification of marginal cost, we assume for simplicity that all firms are identical for any size n, where n represents the number of firms in the industry. Since this model of duopoly is dealt with in most intermediate microeconomic textbooks, we shall be brief. Total revenue and profits for each firm are: Firm 1 TRi = pqi = (9 - q\ - q2)q\ 7i\ = (9 - q\ - q2)q\ - 3q\ Firm 2 TR2 = pq2 = (9 - qx - q2)q2 7t2 = (9 - qx - q2)q2 - 3q2 Since the conjectural variation is that firm 1 will maximise its profits under the assumption that firm 2 holds its output constant, then we can differentiate the profit function of firm 1 with respect to q\, holding q2 constant. The same conjectural variation holds for firm 2, so it will maximise its profits under the assumption that firm 1 will hold its output level constant, so here we differentiate the profit function of firm 2 with respect to q2, holding q\ constant. Doing this we obtain dqi dJT2 = 6 - 2qi - q2 = 0 = 6 - q\ — 2q2 = 0 dq2 Solving we obtain the two reaction functions Firm 1 R\ q\ = 3 — \q2 Firm 2 R2 q2 = 3 - \q\ The Cournot solution, then, is where the two reaction curves intersect, i.e., where (<7i> q2) = (2, 2). The situation is shown in figure 9.1. Notice that the isoprofit curves for firm 1 are at a maximum, for any given level of output for firm 2, at the point on the reaction curve for firm 1. Furthermore, the preference direction is in the direction of the arrow on the reaction curve. The highest level of profits for firm 1 is at point A, where it is a monopolist. Similarly, the isoprofit curves for firm 2 are at a maximum, for any given level of output for firm 1, at the point on the reaction curve for firm 2. Firm 2's preference direction is in the direction of the arrow on its reaction curve, and the highest level of profits it can reach is indicated by point B, where firm 2 is a monopolist. The reaction functions play an important role in our dynamic analysis. In this static duopoly model, they have also been interpreted as Nash solutions for each firm, respectively, and so the Nash solution to the game is where they are consistent: where they geometrically intersect. Hence, the solution is often called a Cournot-Nash solution. What figure 9.1 also shows is that the Cournot-Nash solution is not optimal. This is illustrated by the fact that the isoprofit curves through the Cournot-Nash solution reveal that higher profits can be achieved, as shown by the Dynamic theory of oligopoly 377 shaded area, but that this would require some form of cooperation (collusion!) on the part of the two firms. But how do we know whether from some arbitrary starting position the Cournot-Nash solution will be achieved? In other words, is the Cournot-Nash solution dynamically stable? In order to answer this question we must set up the model in dynamic terms. Only then can we answer this question. Whatever the answer happens to be, the same question applies when we increase the number of firms in the industry. As we do so, we must move away from the diagrammatic formulation of the model and concentrate on its mathematical specification. In the next section we consider a discrete model with output adjusting completely and instantaneously. Our main concern is with the dynamic stability of oligopoly as the number of firms in the industry increases. 9.2 Discrete oligopoly models with output adjusting completely and instantaneously 9.2.1 Constant marginal costs Two-firm case (n = 2) In the static model the assumption was that firm 1 would maximise its profits under the assumption that firm 2 would hold its output level constant. A similar condition applies also to firm 2. Here we assume that in time period t its rivals will choose the same output level they chose in time period t — 1, and chooses its own output at time t so as to maximise its profits at time t. More specifically, q\t is chosen so as to maximise firm l's profits in time period t, under the assumption that firm 2 has output in time period t the same level it was in time period t — 1, so that #2,f = a2,t-h While for firm 2, q2j is chosen so as to maximise firm 2's profits 378 Economic Dynamics (9.3) in time period t, under the assumption that firm 1 has output in time period t the same level it was in time period t — 1, so that q\t = q\j-\. These dynamic specifications for each firm change the form of the total revenue function, and hence the profit functions. Total costs are unaffected. The profit function for each firm is Firm 1 tti,, = (9 - qu - q2,t-i)qi,t ~ 3qu Firm 2 7t2,t = (9 - q\,t-\ - qi,t)q2,t ~ 3qi,t Again, in the spirit of Cournot, each firm is maximising its profits under the conjectural variation that the other firm is holding its output level constant. Therefore, dqij dqi,t = 6 - 2qu - q2,t-i = 0 6 - qi,t-i - 2g2, 0 which results in the following dynamic adjustments -2 1 qi,t = j — 22 + 2~1~t (ql0-q2 0+ (-1)t (-4+ql0+q2 0) ) , ->2 + 2 -l-t (-ql0+q2 0+ (-1)t (-4+ql0+q2 0) ) It is useful to do this generalisation first because the result can then readily be set up on a spreadsheet, which often allows plots of the trajectories much easier than either Mathematica or Maple. The following instructions using Maple rsolve({ql(t)=3-(1/2)*q2(t-1),q2(t)=3-(1/2)*ql(t-1), ql(0)=ql0,q2(0)=q20},{ql(t),q2(t)}); gives the result q2(t) 1 + 2 + 2 + -IV 1 q20 qlO q\{t) ■1 + 2 ^20 + 1 ^10 l/-iV i/-r' + 2 - «20+2 - ^10 Dynamic theory of oligopoly 379 This second result allows us to see more clearly that we can express it in the form #u = 2-2 (/-pj + \ {^pj ^10 + ^o) + \ Q) ^10 _ ^2°) I2,t = 2-2 + \ (^P) + ^o) + \ Q) ^_^10 + ^o) Since these solutions involve the terms (1 /2)'and (—1 /2)', then as t tends to infinity these terms tend to zero, and so the system converges on the equilibrium point (q*, q*) = (2, 2) regardless of the initial values. Figure 9.2 shows this convergence for four different initial values: (tfio> 3. What we now need to investigate is whether this stability holds for n > 3. Three-firm case (n = 3) We continue with our example, which assumes linear demand and constant marginal costs. Our model is now Recall, a quick check on the correct entry of the formulas is to place the equilibrium value as the initial value, and then all entries for output of a given firm should be the same. Dynamic theory of oligopoly 381 p = 9-Q Q = qi + qi + q-i Td = 3qx (9.5) TC2 = 3q2 TC3 = 3q3 Profits are readily found to be 7Ti = (9 - q\ -qi - q3)q\ - 3«i = (9 - q\ -qi - q3)qi - 3g2 *3 = (9 - qi -qi - True, AxesLabel={* ~ ql" , " q2 " , " q3 " } , PlotStyle->Thickness[0.01]]; Maple with (plots) : ql:=t->(3/2)-(3/2)*(-1)-t+(1/3)*( (-1)At)*6+ (1/3)* ( (1/2)-t)* (-3); q2:=t->(3/2)- (3/2)* (-1)At + (1/3)* ( (-1)At)*6+ (1/3)* ( (1/2)"t)* (0); q3:=t->(3/2)-(3/2)*(-1)At+(1/3)*((-1)At)*6+(1/3)*((1/2)At)*3; points:=[seq([ql(t),q2(t),q3(t)],t=0..10)]; pointplot3d(points, axes=BOXED, connect=true,thickness=2, labels= P * ql" , " q2" , " q3" ] , colour=black, orientation=[-17,79]); Figure 9.5 was in fact generated by the Maple instructions.2 2 Orientation in either programme requires some trial and error to get a perspective that is the most revealing of the 3-dimensional plot. Figure 9.5(b) has orientation =[—56,50]. It is usually necessary to join up the points because this gives a better view of the discrete trajectory. 384 Economic Dynamics (9-9) Four-firm case (n = 4) Our model is now P = 9 - Q Q = qi + qi + 1 (\ (9.10) + - I - ) (- 3, the system is unstable. Firm 1 30 - 20 • 10 n n qi U \ ' / \ ' -10 • I 2 4 6 V \10 -20 -30 • t Firm 2 30 -I 20 • * 10 ■ ^ /K A q2 0 < W \ 1 -10 - I 2 4 1 v v -20 ■ \ -30 ■ t ▼ Figure 9.6. 386 Economic Dynamics Figure 9.7. (9.11) Concentrating on firm 1 for the moment, we can plot the time path of output for n = 2, 3 and 4 starting at the initial same monopoly value, i.e., q\ = 3, while all other output levels are zero no matter how many firms there are in the industry. This is illustrated in figure 9.7, where we have the situation for n = 2 and 3 in relation to the left axis and n = 4 refers to the right axis. It quite readily reveals that the stability exhibited for n = 2 is the exception rather than the rule! 9.2.2 Increasing marginal costs Two-firm case (n = 2) We retain the basic model, with the exception that each firm has rising marginal cost. In particular, we assume p = 9-Q Q = qi+q2 rci = 1q\ TC2 = 3q22 Profits are then Firm 1 7tu = (9 - qu - q2,t-i)qi,t ~ 3q\t Firm 2 7t2j = (9 - qU-i - q2j)q2j - 3qj t Differentiating dJt2t = 9- Sqi,t - qi,t-\ = 0 = 9 - qi,t-i - Sqi,t = 0 dq2j which results in the following dynamic adjustments Dynamic theory of oligopoly 387 9 1 qu = g - g .72,1-1 9 1 qij = - - -<7u-i Setting q\t = q\ and ,t = g - g 0 qi,t - qi,t-i = k2(x2,t - qi,t-i) k2 > 0 where x\,t and x2j are the desired output levels for each firm. What these adjustment equations indicate is that each firm adjusts its previous period's output by a proportion of the discrepancy between its desired output level at time t and its output level in the previous period. Note also, however, that the optimal value at time t is adjusted according to the information at time t — 1. Output at time t can therefore be considered a two-step procedure. The adjustment is illustrated in figure 9.9. The system at time t — 1 is at point A. Given the adjustment equations (9.16) the system moves in the next period to point B, and so on. What is not obvious is whether it will converge on the Cournot solution, point C. Nor is it obvious what shape the trajectory will take. These two issues we need to investigate. The desired output levels on the part of each firm are given by their reaction function, so that Dynamic theory of oligopoly 391 -2 1 *l,t = J — 7a2,t-l \ (9-17) X2,t = 3 — 2~qi,t-i Substituting and simplifying we have qi,t = 3h + (1 - h)qU-x - \k\q2,t-\ (9.18) qi,t = 3k2 - jhqu-i + (1 - k2)q2,t-i Although these difference equations can be solved, the output is long and unwieldy. We can however, obtain some additional insight if we assume that k\ = k2 = k and solve for initial values q\o = q\o and q2o = q2$. The solutions are -3k V 1 (-3k x' qu = 2- 2( — + - y— + 1 ) (qw + q2o) 1 (-k V + 2 I ~Y + 1 ) (?io -?2o) -3k V 1 (-3k x' (9.19) ^ = 2-2 ^— + ^— + 1 j (qw + q20) i (-k v + 2 I -y + 1) (-^10 + ^20) Since k > 0, then both roots are less than +1. Accordingly, stability will be assured if the roots are greater than -1. Furthermore, (-p + 1) is less than (^ + 1) and so the root (z|^ + 1) dominates the system. Stability requires that (^ + 1) > — 1 or 392 Economic Dynamics k=1.S qi k < 4/3. If k = 4/3 then the root is —1, and the system oscillates, after a certain time period. If k > 4/3 then the system is explosive. Although the duopoly model with complete and instantaneous adjustment is stable, the same cannot be said of partial adjustment. In this instance, duopoly can exhibit stable, oscillatory and explosive adjustment paths depending on the size of the adjustment coefficient k. One example of each is illustrated in figure 9.10. When k = 1 the system converges on the fixed point. For k = 4/3 the system soon converges on the two values which the system oscillates between, namely 1.5 and 2.5. However, when k = 1.5 the system is explosive, oscillating with greater amplitude either side of the fixed point. Dynamic theory of oligopoly 393 Three-firm case (n = 3) Retaining our assumption of identical adjustment coefficients, then (k > 0) qi,t - 0, then each root is less than +1. In addition, {—2k + 1) is less than (^r + 1) and so the root {—2k + 1) dominates the system. Stability requires that —2k + 1 > — 1 or < 1. If k = 1 then the system oscillates; and if k > 1 the equilibrium values is never attained. One example of each is illustrated in figure 9.11. These three-dimensional plots were produced as discussed earlier. In figure 9.11(a) k = 0.8 and the system which has firm 1 as a monopolist converges on the equilibrium. In figure 9.11(b), with k = 1, the system soon begins to oscillate between two values. When k = 1.2, however, the system oscillates with (9.22) 394 Economic Dynamics Figure 9.11. (a) k = 0.8 ever-increasing amplitude. Although output cannot be negative, once again what we are attempting to illustrate is the inherent instability when k > 1. Four-firm case (n = 4) Following exactly the same analysis as for n = 3, we obtain solutions 6 6 /-5k V 1 /-5k x' #U = - - - ( + 1 J + - J + 1 ) (#10 + #20 + #30 + #40) l/-k W + - I — + 1 ) (3#10 - #20 - #30 - #40) Dynamic theory of oligopoly 395 6 6 /-5k V 1 /-5fc x ' qi,t = 5 ~ 5 l^^- + 1J + 4 ~y~ + 1 I («io + #20 + #30 + #4o) 1 /-k V + - I — + 1 I (-#10 + 3^20 - #30 - #40) 6 6 /-5k V 1 /-5fc x' q3,t = 5 ~ 5 l^^- + 1J + 4 l^^- + 1 I («io + #20 + #30 + #4o) 1 (-k V (9.23) + - I — + 1 I (-#10 - #20 + 3 0, stability requires (^ + 1) > -1 or k < 4/5. If k = 4/5 then the system oscillates; while if k > 4/5 it is explosive. Figure 9.12 illustrates each of these, where we show only the results for output q\. 9.3.2 Increasing marginal costs Two-firm case (n = 2) Returning to the model P=9-Q Q = " +f (9.24) TCi =3q\ 2 TC2 = 3qz2 396 Economic Dynamics (9.25) (9.26) with reaction functions 9 1 Xu = 8 _ 8^2''-1 9 1 X2,t = g - g«l,»-l and once again assume for k > 0 that 0, then stability requires ^ + 1 > — 1, or k < 16/9. The system exhibits oscillations and explosive behaviour if k = 16/9 and k > 16/9, respectively. Three-firm case (n = 3) Following the same procedure as in sub-section 9.3.1 we derive the results 9 9 (—5k Y 1 (—5k Y qu = to - to + V + 3 It + V 1 /-7fc Y + 3 I + 1 ) (2 0, then the smallest root is (-|^ + 1), which must be larger than -1 for stability, i.e., k < 8/5. Four-firm case (n = 4) Following the same procedure as we did to derive equations (9.23) we obtain the results 9 9 (-Ilk V 1 /-H*; V qu = jY ~ Tj v "IT" + 1 / + 4 \ "IT" + 1 / ^10 + q2°+ q3°+ qAo) 1 (-Ik x ' 9 9 (-Ilk Y 1 / — HA: V qu = jY ~ yy I "IT" +1 / + 4 v "IT" +1 / ^10 + 920 + q3°+ qAo) 1 (-Ik x ' 9 9 /-Hit V 1 (-Hk V ?3,f = jY ~ YTv "IT" +1 / + 4 v +1 / ^10 + q20 + q3°+ qAo) 1 (-Ik x ' + - --h 1J (~ 0 and the smallest root is (^jp + 1), then for stability we require k < 16/11. Table 9.1 sets out the value of k for stability for n = 2,3 and 4 and under constant and increasing marginal costs. What this table reveals is the following: (1) As the number of firms in the industry increases, the size of k falls, so the likelihood of instability is increased with the number of firms in the industry. (2) The presence of increasing marginal costs acts as a stabilising influence, raising the critical value of k, and so increasing the range over which the systems exhibit stability. Table 9.1 Stability values for k Constant MC Increasing MC n — 2 k < 4/3 k < 16/9 n = 3 k < 1 k < 16/10 n - 4 k < 4/5 k < 16/11 (9.29) 398 Economic Dynamics (3) The presence of increasing marginal costs does not rule out oscillations or instability. Oscillations and instability can occur for any number of firms in the industry if k is sufficiently large enough. 9.4 Continuous modelling of oligopoly So far the discrete form of the oligopoly model has been investigated. We now turn to its continuous representation. It cannot be guaranteed that the results which hold for n = 2, 3 and 4 for the discrete model hold for its continuous counterpart. As with the discrete model, consideration will be given to the case of both constant and increasing marginal costs. (9.30) 9.4.1 Constant marginal costs Two-firm case (n = 2) The same example is used; paying particular attention to what happens when there is an increase in the number of firms and what happens when the assumption of constant marginal costs is changed to one of increasing marginal costs. For the two-firm case our model is p{t) = 9 - Q(t) Q(t) = qi(t) + q2(t) rCi(0 = 3?i(0 TC2(t) = 3q2(t) This leads to total revenue and profits for each firm of Firm 1 77?!(0 = (9 - qi(t) - q2(t))qi(t) 7Tl(0 = (9 - qi(t) - q2(t))q\(t) - 3qx(t) Firm 2 TR2(t) = (9 - qi (t) - q2(t))q2(t) *2(t) = (9 - qi(t) - q2(t))q\(t) - 3q2(t) We now specify the dynamics. We assume that for firm 1 output is adjusted continuously in proportion to the discrepancy between the desired level and the actual level. The same applies to firm 2. Hence dq\(t) Firm 1 - = k\{x\{i) — q\{t)) k\ > 0 dt Firm 2 ^(0 _ _ k2 > 0 dt The desired level of output for each firm is the output level that maximises profits under the assumption that the other firm does not alter its output level. Differentiating each profit function under the assumed conjectural variation, and setting Dynamic theory of oligopoly 399 equal to zero 37Ti(0 3c?i(0 d7T2(t) dq2(t) Then = 6 - (0 - q2(t) = 0 = 6-^1(0-2^2(0 = 0 *i(0 = 3 - \q2(t) x2(t) = 3-\qi(t) Substituting these into the dynamic adjustment equations, we obtain (9.31) 3r 3c?2(0 3k\ — k\q\(t) k2 3k2 - —qi(t) k2q2(t) dt 2 First consider the fixed point of the system. This is where output levels do not change. These represent two isoclines, which are given by qi(t) = 3 - \q2{t) q2(t) = 3 - ±4i(0 which are no more than the same reaction functions we have above and in the previous models, and which intersect at (q*, q*) = (2, 2). The equilibrium is therefore unaffected. Consider now the model in matrix form. We have q\(t) qi(f) 3fci 3k2 + -h -| -I "*2. ?i (0 qi(f) and the matrix of the system is A = ■k2 k 2 ^k > 0, which hold with tr(A) = -(&!+ k2) < 0 and det(A) = kxk2 - ^ regardless of the values that k\ and &2 take, so long as they are both positive. Furthermore, tr(A)2 - 4det(A) = (kx + k2)2 - 3kxk2 = (h - k2)2 + hk2 which is positive since k\ > 0 and k2 > 0. These results indicate that the equilibrium is stable, and that any cyclical behaviour is ruled out. Although both Mathematica and Maple can solve the linear differential equation system, the solution is once again unwieldy. Letting k\ = k2 = k, however, we obtain the following more insightful solutions f 3k\t 1 qi(t) = 2- 2ev 2 ~>Xqw + qio) + ~eK ~2' (qw - q2o) q2(t) = 2- 2e("T> + ^e(-f)'(q 10 + (- (9.32) ■qio + qio) 400 Economic Dynamics which tend to the equilibrium point (q*, q2) = (2, 2) as t tends to infinity. We illustrate this for k = 0.5 in figure 9.13, which shows the phase diagram for this two-firm model under continuous adjustment. As can be seen from figure 9.13, all paths converge on the equilibrium regardless of the initial values. It should be noted that this is in marked contrast to the discrete model, in which stable, unstable and oscillatory behaviour is feasible depending on the size of the adjustment coefficient. In the present formulation, the size of the adjustment coefficient has no bearing on the stability/instability of the system, all it does is change the speed of convergence. Three-firm case (n = 3) Performing the same analysis as in the previous case, the model reduces down to the following (9.33) ~qi(t)~ ~3kx~ = 3k2 + _3k3_ -ki h 2 h 2 2 -k2 h 2 2 h 2 ~qi(t)~ qi(t) _qs(t)_ and the matrix of the system is A3 = h. 2 h 2 2 2 2 h 2 with tr(A3) = — (k\ + k2 + k^) < 0 and det(A3) = — jk\k2k3 < 0, which holds regardless of the values of the adjustment coefficients. Since det(A3) < 0 a saddle-point solution results. Dynamic theory of oligopoly 401 If we assume that k\ = k2 = k3 = k, then the solutions are qi(t) = 3 2 ~ 3 —e 2 -2kt + 1 — e 3 (410 + 420 + 0, which holds regardless of the values of the adjustment coefficients. We now see the saddle point solution does not hold for n = 4. If we assume k\ = k2 = h = h = k then we can express the solutions in the form (9.35) q\(0 \ - \e-^ + l-e-(5k/2)\qw + q20 + 430 + q4o) -(k/2)t (3^10 — 420 — 430 — 44o) 42(0 1 + r -5 - \e-^ + \e-^\qi0 + 420 + 430 + 44o) 1 -(k/2)t (-410 + 3420 - 430 - 440) 43(0 = \- ~5e-(5k^ + l-e-(5k/2)\qW + 420 + 430 + 440) (9.36) 1 + 7 0 3 -(ki + k2 + h) < 0 -(l/2)kik2k3 < 0 4 -(ki + k2 + h + k4) < 0 (5/16)kik2k3k4 > 0 5 -(ki + k2 + k3 + k4 + k5) < 0 -(3/l6)kik2k3k4k5 < 0 6 -(A:! + k2 + k3 + k4 + k5 + k6) < 0 (7/64)kik2kik4k5k6 > 0 Since tr(A4) is now —4k < 0, det(A4) is now j^k4 > 0 and tr(A4)2 - 4det(A4) = 16k2 - \kA = \k2{64 - 5k2) then the system has an improper node if k < ^/f and a spiral node if k > y^f, where the node is at point {q\, q*2, q\, #4) = (f, f, f, f )• The pattern emerging for the continuous model with constant marginal costs appears complex. The pattern emerging can be seen in terms of table 9.2. Although the trace remains negative, the determinant alternates in sign - positive for even numbers and negative for odd numbers. Nor is cyclical behaviour ruled out, as we noted in the case of four firms. What appears to emerge is saddle point solutions whenever n is odd, and either an improper node or a spiral node whenever n is even. What can be concluded with some confidence from this fairly exhaustive example is that the asymptotic stability exhibited for duopoly is a rather special case. 9.4.2 Increasing marginal costs Two-firm case (n = 2) Returning to the model p{t) = 9 - Q(t) Q(t) = qi(t) + q2(t) rCi(0 = 3ql(t) TC2(t) = 3q2(t) the desired output levels are 9 1 *i(0 = g - g<72(0 9 1 xi(t) = - - -qi(t) Dynamic theory of oligopoly 403 while the dynamics are the same as in the situation of constant marginal costs. The adjustment equations become dqi(t) dt dq2(t) 9ki ki —--hqiit) - —q2(t) 9k2_k2 dt 8 8 Equilibrium is at = (1, 1). q\(t) ~ hqiif) The system can be expressed, q\(f) qi(t) 9h 8 9ki 8 + -h k 8 _k 8 -k2 qi(t) qi(t) and the matrix of the system is B2 -t -k2 with tr(B2) = -{k\ + k2) < 0 and det(B2) = ^hk2 > 0. Furthermore, tr(B2)2 - 4det(B2) = (kx + k2)z -k\k2 63k\k2 16 = (*i - k2y + 16 which is positive. Once again for duopoly in the presence of increasing marginal costs, the equilibrium is stable and any cyclical behaviour is ruled out. Under the simplifying assumption that k\ = k2 = k, we obtain the solutions (9k\ 1 / 9k \ 1 / Ik \ v, - - - T)t + -e~^T)t(qw + q20) + -e~^~>\qio - q2o) (9k\ 1 / 9k \ 1 / Ik \ w ~ - - T)t + ^e (l™ + 12o) + -e~^>\-qio + q2o) which tends to the equilibrium point {q\,q2) = (1, 1) as t tends to infinity. The stability of duopoly under increasing marginal costs is illustrated in figure 9.14. Once again, all paths converge on the equilibrium regardless of the initial values. (9.38) •I 4, 4, 4 4 / / / / •/ / f *s f S 4> 4 J 4 » 'J f i* f ^ *f 'S ■' / / / / / / / / y s ' s / •i 41 ■/ J \* ■/ v- f * y "f" jS 4- ^ 4 * \i f f * 8? ^ ' Figure 9.14. 404 Economic Dynamics Three-firm case (n = 3) Following a similar procedure, the system for three firms can be expressed ~qi(t)~ r9k> i 8 (9.39) qi(t) = 9k2 8 + 9k3 L 8 -1 " —ATI k 8 8 8 ^2 8 8 k 8 "?i(0" 92(0 and the matrix of the system is B3 _k _k~i ^1 8 8 8 8 ""2 8 k 8 £3 _*3 _k 8 8 ^3. with tr(B3) = -{ki + k2 + k3) < 0 and det(B3) = -W-kfah < 0. Since the de- terminant is negative we have a saddle point solution, with node at (q*, q2,q\) v m' in' ^(^'■ 10' 10' 10' Assuming k\ = k2 = k3 = k then we have the solutions qi(t) (9-40) q2{t) 93(0 9 9 — e -(f> To ~ 10 9 9 —e -(?> To ~ 10 9 9 — e -(f> To ~ 10 (4 > _ e (4 )\qm + q20 + ^30) + Ig (?>(2^io - ^20 - 0. The saddle point solution disappears once again, and we have either an improper node or a spiral node, with the node at point (q*, q2, q\, q\) = (yy, yy, jy, jy). Dynamic theory of oligopoly 405 Table 9.3 Roots for firm sizes 2, 3 and 4 n Constant marginal costs -3k -k ~' ~2 -k 3 -2k, — 2 -5k -k Increasing marginal costs -9k -Ik -5k -Ik -\\k -Ik 8 ' ~ Assuming k\ = k2 = k3 = k\ = k then we have the solution 9 9 _(uk\t 1 / ii*\f 4i(0 = — -—e v s )' + -e ^ )\q10 + q20 + q30 + ' + -e ^ >Xqm + q20 + q30 + (-^io + 3 SB). 9.5.1 No learning Each firm is assumed to adjust the resource allocation to R&D according to their competitive advantage. Parker et al. consider two possible policy alternatives: Reinforcing policy (a > 0, b > 0) Research funds are increased when performance is good and reduced when performance is bad. The change is set proportional to the differential in performance, i.e. Rf+l - Rf = a(Sf - Sf) a>0 Rf+1 - Rf = b(Sf - Sf) b>0 Counteracting policy (a < 0, b < 0) Research funds are increased when performance is bad and decreased when performance is good. The change is set proportional to the differential in performance, i.e. Rf+1 - Rf = a(Sf - Sf) a<0 Rf+l - Rf = b(Sf - Sf) b<0 Of course, one firm could be reinforcing while the other is pursuing a counteracting policy, which results in four basic interactions. The critical relationship is that between product quality standard and R&D allocations. A typical relationship is S-shaped indicating low improvements in standard for low levels of R&D, much greater improvements for higher levels, but a tailing off in improvements in standard once diminishing returns set in, which occurs at high levels of R&D. Such an S-shaped curve can be captured by arctan.4 Thus nA for firm A (9.43) nB for firm B are the functions employed to represent quality standard by each firm. The higher the value of m the steeper the S-function, while a positive n shifts it down. The inflexion point occurs on the v-axis at the value of n. 4 We employed a similar relationship in the previous chapter, section 8.10. Sf = (^-J arctan (Rf) Sf = (^)aictan(tff) 408 Economic Dynamics (9.44) (9.45) (9.46) (9.47) Substituting the quality standard into the resource function we have '2mA\ Rf+1 = max 0,Rf + a I - I arctan (Rf) TT I — a 2ml TV RB+l = max 0,R? + arctan (Rf) - a(nA - nB) 2mB TV arctan (Rf) /2mA\ -bi-J arctan (Rf) - b{nB - nA) for firms A and B, respectively. Given the parameters m^, mB, nA and nB, then we have the recursive equations Rf+1=fA(Rf,Rf) Rf+l=fB(Rf,Rf) which allows us to plot the trajectories {Rf,Rf} for alternative policy options given {RA,RB}. Given the values for Rf and Rf, we can compute the product quality standard for firm A and B, respectively. Given some initial market share {k^, kB] it is possible to compute the market shares for the next period. To do this, however, we need to define the average product quality standard for the industry, S. This is defined as the weighted sum of the product quality standard for each firm, where the weight is the respective market share. We therefore have iS* / — k^ S j ~\~ S j k^ ~\~ kj — 1 and kf+1=kf + y{St-St) t+1=kf + y(Sf-St) kB Although there are no known methods for solving equations (9.44), we can readily employ a spreadsheet to carry out simulations. For simplicity the authors treat the functions (9.43) as identical for each firm and give values to the parameters m and n of 100 and 40, respectively. If both firms engage in a reinforcing policy, then whichever firm starts with the greater resources devoted to R&D will have the greater product quality standard and hence the competitive advantage. The market share for the firm with the competitive advantage will rise while that for the other firm will fall until it goes out of business. Given the symmetrical nature of the firms, if the same resources are devoted to R&D, then the market share will remain unaffected regardless of the values of a and b. Example 9.1 (Reinforcing policy by both firms) To see the model in operation, consider the following values rrfi =mB = 100 RA = 10 RB = 8 nA = nB = 40 k% = 0.5 kB = 0.5 y = 0.01 Dynamic theory of oligopoly 409 Figure 9.16. 1 2 3 ' a = 0 1 4 b = . 0,2 5". 6 . t RA(t) 7 0 10 8" 1 10.15716 9 2 10.15917 10 3 10.16164 11 4 10.16474 12 : 5 10.16876 1-3": 6 10.17429 14 7 10.18277 15 8 10.20039 16 : 9; 20.19415 17 10: 30.191 18 : 11: 40.18889 19: 12 50.18731 20 . 13 60,18604 21 14 70.18498 22 15 80.18407 mA = mB = RB(t) 8 7 685684 7 287813 6.800841 6.190946 5.400529 4 317409 2.66685 0: 0: 0 0 0 0 0 0 100 100 SA(t) 53.6549 53.75244' 53 75367: 53.75518 53.75707 53.75952 53.76289 53.76805 53.77875 5685008; 57.89213: 58,41626 58 73168 58 94234 59.093 59.20609 nA = nB = SB(t) 52.08332 51.76308 51.31881 50.7057 49 80498 48.34392 45.5101 37.16139 0 0 0 0 0 0 0 0 40: 40 kA(t) 0.5 0.507858 0.517648 0.529393 0'543744 0.561776 0.585508 0 619715 0.682868 0.853418 0.93675 0.973367 0.988925 0.995429 0.998123 0.999232 l-l...................... kB(t) 0.5: 0492142 0482352 0.470607 0456256 0.438224 0.414492 0.380285 0.317132 0.146582 0.06325 0.026633 0.011075 0.004571 0.001877 0.000768 0.01 Sbar 52.86911 52.77339 52.57921 52.32007 51.9539 51.38627 50.34217 47.45279 36.72378 48.51685 54.23044 56.86044 58.08122 58.67295 58.98211 59.16064 The model is illustrated in the spreadsheet in figure 9.16. The parameter values are set out at the top of the spreadsheet. The initial conditions Rq and RB allow for columns Rf and Rf to be computed recursively. Once these values have been obtained, then the columns for Sf and Sf are computed. We have added one additional constraint, however, in computing columns D and E of the spreadsheet. Since quality standard cannot be negative, we have for columns D and E the formulas max = max 2mA\ jarctan (Rf) -nA,0 71 J 2mB TV ) Jarctan (iff) - nB, 0 for firms A and B, respectively. Given the values for the product quality standard, we compute the market shares and the average product quality standard for the industry using the formulas in equations (9.46) and (9.47). Once again, however, in constructing columns F and G of the spreadsheet, we add the constraint that the market share cannot be negative and must sum to unity, i.e., columns F and G have the formulas = max [kf + y (Sf — St), 0] for column F = 1 — kf for column G for kf and kf, respectively. It is now easy to plot resource allocations, product quality standard and market shares. For example, given firm A has more resources devoted initially to improving product quality standard, firm A will soon dominate the market. Figure 9.17(b) shows that by period 10 firm A has a virtual monopoly. 410 Economic Dynamics If one firm engages in a reinforcing policy while the other engages in a counteracting policy, then the system will oscillate - oscillations occurring in particular with the amount of resources devoted to R&D and in terms of market share. Such oscillations depend not only on the value and sign of a and b but also on the initial resources devoted to R&D. Various possibilities are readily investigated once the model is set up on a spreadsheet or in mathematical software packages. The authors plot the path of Rf and Rf against t after the system settles down, i.e., they plot the paths for t = 200... 250. Since the equations are symmetrical, then oscillations of one period will show up in the phase plane (once the system has settled down) as simply two points; two-period oscillations as four points, and so on. Choosing the same four paired combinations {a, b) and initial resource allocations Rq = 10 and Rq = 15 as Parker et al. with m = 100 and n = 40 for both firms, period 1, 2 and 4 along with chaotic behaviour can be observed in figure 9.18, where the trajectories are plotted for t from 200 to 250. Dynamic theory of oligopoly 411 (a) Period 1 (b) Period 2 Figure 9.18. (a,b)=<0.1,-3.8) and (R*0,RB0)=(10,15) 9.75 9.8 9.85 9.9 9.95 10 10.05 (a,b)=(0.1,-4.0) and (R*o,RB0)»{10,15) 20 15 RB10 5 9.7 9.8 9.9 10 10.1 10.2 (c) Period 4 (d) Chaos (a,b)=(0.1,-4.18) and (RA0,RB0)=(10,15) 25 20-15- 3 10 5 - 9,7 9.8 9.9 10 10.1 10.2 Figure 9.18 was produced with Excel. If Mathematica or Maple is used for constructing this and other figures, then the following instructions can be used. In these instructions we provide a plot of Rf for t = 200 to 250 and a plot of the trajectory {Rf, Rf] for the same time period: Mathematica Clear [RA, RB, t, a, b, mA, mB, nA, nB] RA[0]:=10; RB[0]:=15; a:=0.1; b:=-3.8; mA:=100; mB:=100; nA:=40; nB:=40; RA[t_]:=RA[t]= Max [0,RA[t-l] + (2 a mA/it) (Arc Tan [RA [t —1 ] ] ) — (2 a mB/jr) (ArcTan [RB [t-1] ] )-a (nA-nB) ] RB[t_]:=RB[t]= Max [0, RB [t-1] + (2 b mB/jr) (ArcTan [RB [t —1 ] ] ) — (2 b mA/jr) (ArcTan [RA[t-l] ] )-b (nB-nA) ] dataRA=Table[{t,RA[t]},{t,200,250}]; dataRARB=Table[{RA[t],RB[t]},{t,200,250}]; ListPlot[dataRA, PlotJoined->True] ListPlot[dataRARB,PlotStyle->PointSize[0.02]]; Maple RA:='RA' : RB:='RB' : t:='t': a:='a': b:='b': mA:='mA': mB:='mB': nA:='nA': nB:='nB': a:=0.1: b:=-3.8: mA:=100: mB:=100: nA:=40: nB:=40: 412 Economic Dynamics RA:=proc(t) option remember; max(0,evalf(RA(t-l)+(2*a*mA/Pi)*(arctan (RA(t-l)))-(2*a*mB/Pi)*(arctan(RB(t-1)))-a*(nA-nB))) end; RB:=proc(t) option remember; max(0,evalf(RB(t-1)+(2*b*mB/Pi)*(arctan (RB(t-1)))-(2*b*mA/Pi)*(arctan(RA(t-l)))-b*(nB-nA))) end; RA(0) —10 : RB (0) :=15: dataRA:=[seq([t,RA(t)],t=200..250)]: dataRARB:=[seq([RA(t),RB(t)],t=2 00..250)]: plot(dataRA); plot(dataRARB,style=point); The only item in these instructions that needs to be changed in producing figure 9.18 is the value of the parameter b, which takes on the four values b = -3.8, -4.0, -4.18 and -4.5. 9.5.2 Learning The model is further extended by the authors to take account of learning. The possibility of adaptation is taken into account by allowing the policy parameters a and b to change. In establishing when to change the policy a simple rule is chosen. Let / denote the frequency of choosing when to change policy (assumed constant), e.g., every year or every quarter. Then define M, the sum of the difference in product quality standard between the last decision point and the present one, assumed to be at time t, i.e. If M > 0 then the current policy is assumed satisfactory and no change in policy is made. If, however, M < 0, then a change is considered necessary. Since a positive value of M for firm A implies a negative value of M for firm B and vice versa, then at any decision point one firm will always be altering its policy. The authors consider two policy adaptations. Proportional policy adaptation A change amounting to a proportion of the existing policy, i.e. A change amounting to a fixed amount is applied to the existing policy; this amount can be either positive or negative, i.e. t-f+i as = aas_\ 0 < a < 1 bs = ßbs-! 0<ß 0 Ac' j = —ß(c\ — c™) ß > 0 k\ — market share of firm at time t percentage change in market share 2+1" *i Tit — average industry profit at time t a — speed of selection parameter j3 — speed of imitation parameter Absolute adaptation mechanism Firm A RA 10.4 20 40 60 t 80 100 120 Figure 9.20. Absolute adaptation mechanism Firm B 20 40 60 t 80 100 120 the selection process. The second element is imitation, i.e., firms with inferior technology can imitate firms with superior technology. Put another way, a firm can reduce the efficiency gap through imitation. We shall refer to this as the imitation process. The model is captured in terms of the relationships set out in table 9.5. 416 Economic Dynamics Consider first the selection process. We assume that the percentage change in market share of the zth firm is dependent on its relative fitness compared with the industry average. We measure fitness in terms of profit margins. Hence (9.48) k~l+1 = a (jtJ - Wt) i=l,...,n a>0 We assume the speed of selection parameter, a, is the same for all firms. As in the previous section, we measure the average profit as a weighted sum of the profits of all firms in the industry, where the weights are the respective market shares, i.e. n n (9.49) Wt = K^'t kt = 1 for a11 f i=i i=i where it] = p\ — c\ is the profit margin for the zth firm at time t. Notice that if a = 0 market shares remain unaffected and so what equation (9.48) is attempting to capture is the extent to which firms with above average profits increase their market share while those with below average profits lose some of their market share. Next we allow for imitation. We assume this is captured by a reduction in a firm's unit costs. A firm's unit costs can be reduced if it imitates the 'best' firm in the industry. We assume that the 'best' firm is that with the lowest unit cost at time t. We further assume that firms have full information on the technology of their competitors and that imitation is costless. Imitation, then, is captured by the relationship (9.50) Ac\+l = -jl(c\-c™) 0>O where c™ is the minimum unit cost at time t and /3 measures the speed of imitation. Of course, if /3 = 0 no imitation takes place. Example 9.4 (Constant price, selection, no imitation) The basic workings of the model are illustrated in the spreadsheet in figure 9.21. Price for all three firms is set at 18; unit costs are initially set at 8, 10 and 12 for firms 1, 2 and 3, respectively, which in this example remain constant for all time periods. This leads to profit margins of 10 for firm 1, 8 for firm 2 and 6 for firm 3. These too remain constant for all time periods. Firm 1 is the firm with the lowest unit costs. Market shares are initially set at 1 /3 for each firm. Subsequent market shares are calculated using (9.51) k\+l = *j(i + *j) where klt+l is given by formula (9.48). The final two columns compute average profits and average unit costs for the industry. Figure 9.22(a) shows the profile of market shares. Given no imitation, the process is governed purely by selection. What we readily observe is 'survival of the fittest'. By about period 8, firms 2 and 3 have virtually zero market share and firm 1 is a virtual monopolist. As can be seen in terms of figure 9.22(b), in this example Dynamic theory of oligopoly 417 S3 Ells Edit ile* tissrt Far mat looks Cau 023 j.' " " / in « 3j » A » ?' Figure 9.21. 5 O 1 2 J Pi = 18 alpha - 0.25 4 p2 = 18 beta = 0 ■5 6 P3 = 18 7: t d(t) c2(t) c3(t) cm(t) k1(t) k2(t) *3(t) *1 0. Consider the following values c1 - 8 k1 -KQ — 1/3 P = 18 c2 - 10 k2 - 1/3 a = 0.25 c3 - 12 k3 - 1/3 P = 0.5 Now firms 2 and 3 can prevent themselves being pushed out of the market by imitating firm 1, the most efficient firm. Over time, the industry emerges with a more-or-less identical product produced by each firm. All firms' unit costs converge on the unit costs of the most efficient firm, which implies that the average does also, as shown in figure 9.23(b). Since price is constant, then it also follows that profits of each firm converge to the same level, namely that of firm 1. This must also be the case for average industry profits. As figure 9.23(a) illustrates, in the early period firm 1, the most efficient firm, gains in market share at the expense of firms 2 and 3. But as firms 2 and 3 begin to imitate the most efficient firm, and so achieve a lowering of unit costs, market shares stabilise. After about period 8, firm 1 has 66% of the market, firm 2 has 27% while firm 3 has 7%. The initial cost advantage of firm 1 leads to an increase in market share and a permanent market advantage. However, the process of imitation prevents firms 2 and 3 from being 418 Economic Dynamics driven out of the market, and eventual market shares reflect the initial differences in cost advantage. The model allows for other considerations, such as product differentiation. In this instance brand loyalty may allow a firm to charge a higher price. The spreadsheet shown in figure 9.21 readily allows for such a consideration. Models incorporating any combinations of the following are possible. Selection a > 0 Imitation /3 > 0 Product differentiation Product prices different However, none of these models exhibits oscillatory behaviour, the type of behaviour we encountered in the previous section involving R&D. This follows from the fact that both speed parameters a and /3 are positive. The selection parameter reinforces the advantage of the firm with higher than average profits and diminishes consistently the market share of firms with below average profits. Where imitation Dynamic theory of oligopoly 419 takes place, it is always leading to a lowering of unit costs as the firm imitates the most efficient firm. Although to some extent offsetting the selection process, it cannot lead to oscillatory behaviour. Exercises 1. For the general linear demand model with constant marginal costs p = A — BQ Q = qi+q2 TC\ = a\q\ TC2 = a2q2 Show that if a\ = a2 then q* = q\ for the Cournot solution. 420 Economic Dynamics 2. For the n-firm oligopoly model with constant and equal marginal costs p = A — BQ Q = tqt 1=1 TCi = aqi i = 1... n Show (i) q* is the same for all i. (ii) the reaction curves can be expressed (A-a)-BJ2 qj * =-2B- 3. The text states that for linear demand and constant marginal costs the duopoly model is dynamically stable. Set up a spreadsheet which allows for the following parameter values: A, B,a\, and a2 for the model p = A-BQ B > 0 Q = qi + q2 TC\ = a\q\ TC2 = a2q2 Hence show (i) equilibrium q* and q\ are A - 2a i + a2 q^ =-3^- A + a\ — 2a2 q2 =-3^- (ii) that no matter what the initial value for (410, 420X the system always converges on the equilibrium. 4. For the duopoly model p = 9-Q Q = qi+q2 TC\ = a\q\ TC2 = a2q2 (i) Establish the equilibrium for q\ and q2 in terms of ai and a2. Show that if ai < a2 then q\ > q\. (ii) Let a\ = 3 and a2 = 5 and consider initial points (a) firm 1 the monopolist (b) firm 2 the monopolist From which initial point does the system reach equilibrium sooner? 5. Consider the model set out in equation (9.5). Let the costs, however, be TQ = 5qi, i= 1, 2, 3. (i) Is the equilibrium point (q*, q2, q\) closer to the origin? (ii) Establish the reaction curves for this model. (iii) Does this system also oscillate with constant amplitude? Dynamic theory of oligopoly 421 6. Consider the model p = 9-Q Q = qi+q2 + q3 TC\ = 3q\ TC2 = 2q2 TC3 = 1 and so income will diverge from the new equilibrium. This is shown in figure 10.1. The initial level of income is the original equilibrium level of £1,640. In period 1 autonomous investment is raised by £20, which is maintained for all periods thereafter, so income in period 1 is £1,660. Income in period 3 and beyond is then specified according to the recursive equation (10.14). However, as figure 10.1 reveals, income never reaches the new equilibrium of £1,720. Of course, this is not the only possibility and the resulting path of income depends very much on whether y/(b + v)2 — 4v is real or complex. Closed economy dynamics 429 10.2 Goods and money market dynamics2 The previous section considered only the goods market, and even then only in simple terms. The essence of the IS-LM model is the interaction between the goods market and the money market. This interaction is even more significant when there are lags in the system. Again we illustrate this by introducing a lag into the goods market of the form Yt = Et-\. On the other hand, we assume the money market adjusts in the same time period t, so that the demand for real money balances in time t is equal to the supply of real money balances in time t. This is a reasonable assumption. Algebraically, our model is Goods market ct = a + bydt ydt =yt- tax, taxt = t0 + hyt it = j'o - hrt 8t = 8 et = c, + it + gt 0 < b < 1 o < h < l h > 0 Money market mf = mo + kyt — urt k > 0, u > 0 where c = real consumption y = real income yd = real disposable income tax = real taxes i = real investment r = the nominal rate of interest g = real government spending e = real total expenditure m d m rrt m m. the demand for real money balances ms = the supply of real money balances On substitution, we arrive at the difference equation yt = (a- bt0 + io + g) Or more simply yt = A + By,-! mo — m u + b(l - h) kh u yt-i where A = a - bt0 + i'o + g - h , kh B = b{\-H)-\ — mo m setting yt yt-\ = y* the equilibrium level of income is found to be /mo — m (a -bt0 + i0 + g)-h f = l-b(l-h) + kh u (10.15) (10.16) (10.17) The model in this section is based on Teigen (1978, introduction to chapter 1). 430 Economic Dynamics Define xt = yt — y*, then xt = Bxt-i with solution xt = Blx0 or yt = y*+B\y0-y*) The stability of the equilibrium now depends on whether B < 1 or B > 1. If B < 1 then this amounts to l-b(l-h) h k < -u or But why express the condition in this way? The equation for the IS curve is the solution for goods market equilibrium. This takes the form yt = a - bt0 + i0 + g + b(l - h)yt - hrt _ a-bt0 + i0 + g _ [1 - b(l - h)]yt r' ~ h h The LM curve is the solution for the money market. This takes the form mo — m ( k \ n =-+ [-)yt Hence, the stability condition [l-fr(l-fi)] k B < 1 or--< - h u amounts to the slope of the IS curve being less steep than the slope of the LM curve. This is definitely satisfied for the usual case where the IS curve is negatively sloped and the LM curve is positively sloped. With 0 < b < 1 and 0 < t\ < 1, then 0 < 1 - b{\ - h) < 1. With h > 0 then -[1 - b{\ - h)]/h < 0 while k/u > 0. A shift, say, in the IS curve to the right will lead to a rise in income over time, converging on the new equilibrium level. Example 10.3 This is illustrated in table 10.2, which is based on the following numerical model. ct = 110 + 0.75vf ydt =yt- tax, tax, = -80 + 0.2v, it = 320 - Art gt = 330 for all t et = ct + it + gt yt = et-i 20 + 0.25yt - \0rt Closed economy dynamics 431 Table 10.2 Dynamic impact of a rise in government spending by 20 million t yt taxt yf ct rt h mt bd, St k, 0 2000.00 320.00 1680.00 1370.00 5.00 300.00 470.00 10.00 310.00 0.00 1 2020.00 324.00 1696.00 1382.00 5.50 298.00 470.00 26.00 314.00 1.00 2 2030.00 326.00 1704.00 1388.00 5.75 297.00 470.00 24.00 316.00 1.50 3 2035.00 327.00 1708.00 1391.00 5.88 296.50 470.00 23.00 317.00 1.75 4 2037.50 327.50 1710.00 1392.50 5.94 296.25 470.00 22.50 317.50 1.88 5 2039.75 327.75 1711.00 1393.25 5.97 296.13 470.00 22.25 317.75 1.94 6 2039.38 327.88 1711.50 1393.63 5.98 296.06 470.00 22.13 317.88 1.97 7 2039.69 327.94 1711.75 1393.81 5.99 296.03 470.00 22.06 317.94 1.98 8 2039.84 327.97 1711.88 1393.91 6.00 296.02 470.00 22.03 317.97 1.99 9 2039.92 327.98 1711.94 1393.95 6.00 296.01 470.00 22.02 317.98 2.00 10 2039.96 327.99 1711.97 1393.98 6.00 296.00 470.00 22.01 317.99 2.00 11 2039.98 328.00 1711.98 1393.99 6.00 296.00 470.00 22.00 318.00 2.00 12 2039.99 328.00 1711.99 1393.99 6.00 296.00 470.00 22.00 318.00 2.00 13 2040.00 328.00 1712.00 1394.00 6.00 296.00 470.00 22.00 318.00 2.00 14 2040.00 328.00 1712.00 1394.00 6.00 296.00 470.00 22.00 318.00 2.00 15 2040.00 328.00 1712.00 1394.00 6.00 296.00 470.00 22.00 318.00 2.00 m\ = 470 mdt=mst Table 10.2 shows income gradually rising from the initial equilibrium level £2,000 million to the new equilibrium level of £2,040 million arising from a sustained increase in government spending of £20 million, occurring in period 1. As income rises the demand for real money balances increases, leading to a rise in the rate of interest. The rate of interest gradually rises from 5 % to 6%. The rise in the rate of interest leads to a gradual fall in investment. Table 10.2 also shows the path of other endogenous variables - such as taxes, disposable income, consumption, etc. It also illustrates the path of the budget deficit, denoted bdt = gt — taxt, along with the dynamic multiplier in the final column. 10.3 IS-LM continuous model: version 1 We shall begin with the simplest formulation of the model. Real expenditure is the sum of consumer expenditure, investment expenditure and government expenditure (where we assume the economy is closed). Consumers' expenditure is related to real disposable income, investment expenditure is negatively related to the rate of interest, and government expenditure is assumed to be exogenous. We therefore postulate a very simple linear expenditure function e(t) = a + b(l - h)y(t) - hr(t) a > 0, 0 < & < 1, 0 < fi < 1, h > 0 where e = real expenditure a = autonomous expenditure b = marginal propensity to consume t\ = marginal rate of tax (10.18) 432 Economic Dynamics y = real income h = coefficient of investment in response to r r = nominal interest rate The demand for real money balances is assumed to be positively related to real income and negatively related to the nominal interest rate (10.19) md(t) = ky(t) - ur{t) k,u>0 The nominal money supply is assumed exogenous at Ms = Mo and the price level is assumed constant. Hence, real money balances are exogenous at mo = Mo/P. It is now necessary to be more precise on the adjustment assumptions in each of the markets. We assume that in the goods market, income adjusts according to the excess demand in that market and that interest rates adjust according to the excess demand in the money market, i.e. (10.20) (10.21) y = y'(t) = a{e{t) - y{t)) a > 0 r = r'{t) = p(md(t) -mo) p > 0 These differential equations can be expressed explicitly in terms of y and r, where we now assume these variables are continuous functions of time, and that we drop the time variable for convenience y = a[b(l — t\) — l]y — ahr + aa r = fiky — flur — ftmo The equilibrium lines in the (v,r)-phase plane are established simply by setting y = 0 and f = 0 respectively. For y = 0 we derive the equilibrium line —a[l — b(l — t\)] y — ahr + aa = 0 a - [I - b(l - h)]y i.e. r = - h which is no more than the IS curve. This equilibrium line has a positive intercept (a/h) and a negative slope (—(1 — b(l — t\))/h). Similarly, for r = Owe derive the equilibrium line which is no more than the LM curve. This equilibrium line has a negative intercept —mo/a and a positive slope k/u. The model has just one fixed point for which y = 0 and r = 0. This is the point a + (h/u)mo —(mo/u)(\ — b(\ — t{j) + (k/u)a (10.22) (y*,r*) l-b(l- h) + (khju) 1 - b(l - h) + (khju) and is shown by point Eo in figure 10.1. More importantly, we need to consider the dynamic forces in operation when each of the markets are not in equilibrium. First consider the goods market. For points to the right of the IS curve, as drawn in figure 10.2, we have a-n-bd-t^y r > - h 0 > a + b(l — t\)y — hr — y implying y < 0. Hence, to the right of the IS curve income is falling. By the same reasoning it is readily established that for points to the left of the IS curve income Closed economy dynamics 433 Figure 10.2. is rising. Considering next the money market, for points to the right of the LM curve ky — mo r > - u 0 > ky — ur — mo implying f > 0, and so interest rates are rising. Similarly, to the left of the LM curve it is readily established that interest rates are falling. The implied vectors of force in the four quadrants are illustrated in figure 10.2, which clearly indicate a counter-clockwise movement. Suppose the economy is in all-round equilibrium, shown by point Eo in figure 10.3. Now consider the result of a fall in the nominal money supply. This will shift the money market equilibrium line to the left. The new equilibrium will be at point Ei. But what trajectory will the economy take in getting from Eo to Ei ? Four possible paths are drawn, labelled Ti, T2, T3 and T4, respectively. Trajectory Ti makes a very extreme assumption on the part of adjustment in the money market and the goods market. It assumes that the money market adjusts instantaneously, with interest rates adjusting immediately to preserve equilibrium in the money market. With such immediate adjustment, then in the first instance the economy must move from Eo vertically up to point A. This is because income has not yet had a chance to change, and is still at the level yo. With the sharp rise in interest rates, investment will fall, and through the multiplier impact on income, income will fall. As income falls, the demand for money declines, and so too does the rate of interest. The interest rate will fall always in such a manner that equilibrium is preserved in the money market. This means that the adjustment must take place along the new LM curve, as shown by trajectory Ti. Under this assumption of instantaneous adjustment in the money market, the interest rate 434 Economic Dynamics Figure 10.3. LM,(m,) 0 y, y„ y overshoots its new equilibrium value and then settles down at the new equilibrium rate. Real income, on the other hand, falls continually until the new equilibrium level is reached. Trajectory T2, on the other hand, indicates that both markets adjust imperfectly in such a manner that the economy gradually moves from Eo to Ei, with interest rates rising gradually until they reach the new level of r\, and income falling gradually until it reaches its new level of yi. If the economy conforms to this trajectory, then no overshooting occurs. But our analysis in part I indicates that there is no reason to assume that this is the only possible trajectory - given the vector of forces present. For instance, trajectory T3 shows a sharper rise in interest rates than in trajectory T2, and overshooting of interest rates and income, with a resulting counter-clockwise spiral towards the new equilibrium Ei. If we assume that the money market, although not adjusting instantaneously, is very quick to adjust, and that the goods market is also adjusting quickly, then trajectory T3 is more likely. This is an important observation. A spiralling trajectory to the new equilibrium (trajectory T3) is more likely if both markets have quick adjustment speeds, and consequently the more likely overshooting will be observed in both endogenous variables y and r. Even so, a counter-clockwise spiral is not the most likely outcome; it is more likely to be trajectory T4. This is because, in general, the money market is relatively much quicker to adjust than the goods market and the adjustment path will be contained within the triangle Eo AEi, being drawn towards trajectory Ti. A similar analysis holds for a monetary expansion, shown in figure 10.4, where the economy is initially at equilibrium point Eo. Under instantaneous adjustment in the money market, the trajectory is Ti. Interest rates fall to point A on the new LM curve. The sharp fall in interest rates stimulates investment, which, through the multiplier, stimulates the level of income. As income rises the demand for money rises and so too do interest rates, but in such a manner that the money market clears Closed economy dynamics 435 IMJjhq) Figure 10.4. 0 y0 y, y continually. Hence the economy moves along the new LM curve until equilibrium Ei is reached. Once again, interest rates overshoot their new equilibrium level, but the level of income adjusts gradually until its new equilibrium level is achieved. If both markets show a fair degree of adjustment, then path T2 will be followed. However, this would require the goods market to adjust quite quickly. In this instance, interest rates fall gradually until the new level of r\ is reached, and income rises gradually until the new level of y\ is reached. There is no overshooting either of the interest rate or of income. If both the money market and the goods market are quick to adjust, then the economy is more likely to follow the trajectory illustrated by T3 in figure 10.4. In other words, a spiral path to the new equilibrium, moving in a counter-clockwise direction, and such that both the rate of interest and the level of income overshoot their equilibrium values. However, with the dominance of adjustment in the money market, a counter-clockwise movement will be observed but it is not likely to be a spiral path. The most likely trajectory is T4. It is apparent from this discussion that the speed of adjustment is very much to do with the values of the reaction coefficients a and /3 in the dynamic system. The higher the value of the coefficient, the quicker the market responds to a disequilibrium. To some extent, it is the relative values of these coefficients that will determine which trajectory the economy will take. To clarify this point, let us consider a numerical example. Example 10.4 Since throughout the price level is constant, we shall assume that this has a value of unity. The assumed parameter values and the initial level of the money stock are a = 50 k = 0.25 b = 0.75 m0 = 8 436 Economic Dynamics Figure 10.5. (a) 56 58 60 ,, 62 0 20 30 40 50 r t h = 0.25 u = 0.5 h = 1.525 The economy's equilibrium is (yo, fo) = (62, 15), shown by point Eo in figure 10.5(a). A fall in the real money stock to mi = 5 leads to the new equilibrium point3 (yi, r\) = (54, 17) and shown by point Ei. The resulting differential 3 More exactly Cy1; n) = (54.375, 17.1875). Closed economy dynamics 437 equation system, with unspecified values for a and p>, is y = -0.4375ay - 1.525ar + 50a r = 0.25/3y - 0.5/3r - 5/3 The trajectory the economy takes to the new equilibrium will depend very much on the values of a and /3. Consider three possible combinations, leading to three possible trajectories Ti: a = 0.05 T2: a = 0.1 T3: a = 0.5 /3 = 0.8 /3 = 0.8 /3 = 0.8 If the money market is quicker to adjust than the goods market, as is the most likely situation, then typical trajectories are Ti and T2 in figure 10.5(a). In these cases the economy will exhibit overshooting of the interest rate, first rising above the equilibrium level and then falling, with the new equilibrium interest rate higher than initially, as shown in figure 10.5(a). On the other hand, there will be a gradual decrease in the level of income to the new lower equilibrium level. A counterclockwise spiral pattern, as shown by trajectory T3, will occur only if both the money market and the goods market are quick to adjust, as illustrated in figure 10.5(c). Although a counter-clockwise spiral is possible, therefore, it is not the most likely outcome of this dynamic system because the goods market is not likely to be quick to adjust. 10.4 Trajectories with Mathematica, Maple and Excel Figure 10.5(a) set out three trajectories employed in example 10.4. In this and later chapters we shall be producing a number of trajectories for both continuous and discrete systems of equations. We shall therefore take a digression and outline exactly how to do this with three different software packages: Mathematica, Maple and (for discrete systems) Excel? Figure 10.5 will be used throughout as an example. 10.4.1 Mathematica To produce trajectories and other plots with Mathematica, two commands of importance are used, namely the NDSolve command and the ParametricPlot command. The first command is used to obtain a numerical solution to the differential equation system, which it does by producing an InterpolatingFunction. The second command is then used to plot the values of the InterpolatingFunction. The input instructions are as follows: soll=NDSolve[ y'[t]==2.5-0.07625r[t]-0.021875y[t], r' [t]==-4.0-0.4r[t]+0.2y [t] , y[0]==62,r[0]==15}, {y,r},{t,0,50}] trl=ParametricPlot[ {y[t],r[t]} /. soil, {t,0,50}, PlotPoints->200]; 4 See Shone (2001) for a demonstration of how to produce trajectories on a spreadsheet for continuous systems of two equations employing Euler's approximation. 438 Economic Dynamics sol2=NDSolve[ y'[t]==5-0.1525r[t]-0.04375y[t], r' [t]==-4.0-0.4r[t]+0.2y [t] , y[0]==62,r[0]==15}, {y,r},{t,0,50}] tr2=ParametricPlot[ {y[t],r[t]} /. sol2, {t,0,50}, PlotPoints->200]; sol3=NDSolve[ y'[t]==25-0.7625r[t]-0.21875y[t], r'[t]==-4.0-0.4r[t]+0.2y[t], y[0]==62,r[0]==15}, {y,r},{t,0,50}] tr3=ParametricPlot[ {y[t],r[t]} /. sol3, {t,0,50}, PlotPoints->200]; trajectories=Show[tri,tr2,tr3]; pathyl=Plot[ y[t] /.soll, {t,0,50}, PlotPoints->200; pathy2=Plot[ y[t] /.sol2, {t,0,50}, PlotPoints->200; pathy3=Plot[ y[t] /.sol3, {t,0,50}, PlotPoints->200; pathy=Show[pathyl,pathy2,pathy3]; pathrl=Plot[ r[t] /.soll, {t,0,50}, PlotPoints->200; pathr2=Plot[ r[t] /.sol2, {t,0,50}, PlotPoints->200; pathy3=Plot[ r[t] /.sol3, {t,0,50}, PlotPoints->200; pathr=Show[pathr1,pathr2,pathr3]; Note: 1. We use the NDSolve rather than DSolve because we are deriving a numerical solution. 2. The simultaneous equations include the two initial values for y and r, which in the present example denotes the initial equilibrium before a disturbance. 3. The parameter values include the fall in the money supply to mo = 5 and we are deriving trajectory Ti, so a = 0.05 and p5 = 0.8. 4. ParametricPlot is a built in command in Mathematica v2.0 and higher, and so can be employed without recourse to other subroutines. 5. There is no comma after {y[t] ,r[t]} because these coordinates are specified for the solution values derived earlier. Thus, the qualifier 7. soil' instructs the programme to plot the coordinates using each value derived from the output of soil. 6. Interim displays can be suppressed by including the option Display Function-> Identity in each and then in the Show command include the option DisplayFunction-> $DisplayFunction. For instance trajectory 1 can be written trl=ParametricPlot[ {y[t],r[t]} /. soil, {t,0,50}, PlotPoints->2 0 0, DisplayFunction->Identity] and trajectories can be written trajectories=Show[trl,tr2,tr3, DisplayFunction->$Di splayFunction]; Closed economy dynamics 439 10.4.2 Maple In some respects it is easier to produce trajectories in Maple, but more involved to produce the values for plotting y(t) and r(t). The reason for this is because we can use Maple's phaseportrait command to produce the trajectories. This implicitly uses the numerical solution for the differential equations. Thus, the three trajectories and their combined display for figure 10.5 is as follows: with (DEtools) : with (plots) : trl:^phaseportrait( [D (y) (t)=2.5-0.07625*r(t)-0.021875*y(t), D(r) (t)=-4-0.4*r (t)+0.2*y(t) ] , [y(t) ,r (t) ] , t = 0. .50, [ [y(0)=62,r (0)=15] ] , stepsize=.05, linecolour=black, arrows=none, thickness=2): tr2:^phaseportrait( [D(y) (t)=5-0.1525*r(t)-0.04375*y (t), D(r) (t)=-4-0.4*r (t)+0.2*y(t) ] , [y(t) ,r (t) ] , t = 0. .50, [ [y(0)=62,r (0)=15] ] , stepsize=.05, linecolour=red, arrows=none, thickness=2): tr3:=phaseportrait( [D(y) (t)=25-0.7625*r(t)-0.21875*y (t), D(r) (t)=-4-0.4*r(t)+0.2*y(t)], [y(t) ,r (t) ] , t = 0. .50, [ [y(0)=62,r (0)=15] ] , stepsize=.05, linecolour=blue, arrows=none, thickness=2): display(trl,tr2,tr3); Notes: 1. It is necessary to load the DEtools and plots subroutines first. 2. Using phaseportrait implicitly uses a numerical solution to the differential equations. 3. A small stepsize, here 0.05, produces a smoother plot. 4. Having arrows set at none means the direction field is not included. 440 Economic Dynamics Figures 10.5(b) and 10.5(c) can be produced with a similar set of instructions, except now we use DEplot with the option 'scene'. The instructions are: pathyl=DEplot( [D(y) (t)=2.5-0.07625*r(t)-0.021875*y(t) , D(r) (t)=-4-0.4*r (t)+0.2*y(t) ] , [y(t),r (t) ], t = 0. .50, [ [y(0)=62,r(0)=15] ] , stepsize=.05, linecolour=black, arrows=none, thickness=2, scene=[t,y]): pathy2=DEplot( [D (y) (t)=5-0.1525*r (t)-0.04375*y(t), D(r) (t)=-4-0.4*r(t)+0.2*y(t)], [y(t),r (t) ], t = 0. .50, [ [y(0)=62,r(0)=15] ] , stepsize=.05, linecolour=black, arrows=none, thickness=2, scene=[y,t]): pathy3=DEplot( [D(y) (t)=25-0.7625*r (t)-0.21875*y (t), D(r) (t)=-4-0.4*r(t)+0.2*y(t)], [y(t),r (t) ], t = 0. .50, [ [y(0)=62,r(0)=15] ] , stepsize=.05, linecolour=blue, arrows=none, thickness=2, scene=[t,y]): display (pathyl,pathy2,pathy3); pathrl=DEplot( [D(y) (t)=2.5-0.0 7 625*r(t)-0.021875*y (t), D(r) (t)=-4-0.4*r (t)+0.2*y(t) ] , [y(t),r (t) ], t = 0. .50, [ [y(0)=62,r(0)=15] ] , stepsize=.05, linecolour=black, arrows=none, thickness=2, scene=[t,r]): pathr2=DEplot( [D (y) (t)=5-0.1525*r (t)-0.04375*y(t), D(r) (t)=-4-0.4*r(t)+0.2*y(t)], [y(t),r (t) ], t = 0. .50, Closed economy dynamics 441 [ [y(0)=62,r (0)=15] ] , stepsize=.05, linecolour=black, arrows=none, thickness=2, scene=[r,t]): pathr3=DEplot( [D(y) (t)=25-0.7625*r(t)-0.21875*y (t), D(r) (t)=-4-0.4*r(t)+0.2*y(t)], [y(t) ,r (t) ] , t = 0. .50, [ [y(0)=62,r (0)=15] ] , stepsize=.05, linecolour=blue, arrows=none, thickness=2, scene=[t,r]): display(pathrl,pathr2,pathr3); 10.4.3 Excel Discrete trajectories can also be derived using Excel, although there are some limitations. Consider a discrete variant of example 10.4. yt+i — yt = —0.4375aVf — 1.525ar, + 50a rt+1 -rt = 0.25fyt - 0.5^rt - 5/3 or the recursive form yt+1 = (1 - 0.4375a)v, - 1.525ar, + 50a rm = 0.25/3^+ (l-0.5/3)r(-5/3 This numerical example is set out in the spreadsheet shown in figure 10.6. The spreadsheet shows the data computations which can be used to produce a given trajectory or a multiple time plot of y(t) or r(t). The initial values are the equilibrium values y* = 62 and r* = 15. Cells B13 and CI3 write out the formulas using both absolute addresses for the parameters a and 6 and relative addresses for y(0) and r(0). These cells are then copied to the clipboard and pasted down up to t = 50. A similar procedure is done for columns F and G along with columns J and K. Unfortunately spreadsheets cannot plot more than one trajectory on the same graph. Selecting cells B12 : C62 and invoking the chart wizard and selecting the x-y plot option produces a plot of trajectory Ti. Similarly, selecting cells F12: G62 produces trajectory T2 and selecting J12 : K62 produces trajectory T3. To produce the discrete equivalent of figure 10.5(b) first select cells A12 : B62 and while holding down the Ctrl-key, select cells F12 : F62 and, while continuing to hold down the Ctrl-key, select cells J12: J62. Invoking the chart wizard and selecting the x-y plot option produces figure 10.5(b). In the same manner figure 10.5(c) can be produced for a multiple plot of r(t) against t. 442 Economic Dynamics lillllll iiisiitlii Stiiifei! lilllili q h 1(11111! j .-... 1 Figure 10.6 - discrete version of figure 10.5 2 J yM = (l-0.4373ar).)>(-1.525arr(+JOa! 4 g r!+i = Q.lSßy, + (l-O.Jj0fc 8 7 8 T1 alpha = 0.05 T2 alpha =; Oil t3 alpha =: 0.5 9 beta = 0,8 beta = I oW" beta = 0.8. 10 11 t y(t) r(t) t y(t) r(t) t y(t) r(t) 12 0 62.00 15.00 0 62.00 15.00! 0 62.00^ 15.00 13 1 62 00 17 40 1 62.00 17.40] 1 62.00: 17.40 14 2 6182 13 if- 2 61 63 2! 60.17 '■' 18.84 15 3 6153 19.67 3 61.06 19.63! 3 57.64: 19.34 16 4 61.18 20 11 4 6040: 19.99! 4! 55.29 19 13 17 5 60.81 20 30" 5 59.71 20.07! 5! 53.61! 18.54 ie 6 60.43 20.34 6 59.03! 19.99! 6! 52.75, 17.84 19 7 60.06 20.29; 7 58.40: 19.80: .... _., 52.60 17 25 20 8 59.70 20.19: 8 57.83^ 19.56! ...j7 ......'.52':.94'. .. 16 87 ...... The instructions provided in this section allow the reproduction of all two-dimensional trajectories provided in this book. They can be used to produce trajectories for any similar set of differential or difference equations. 10.5 Some important propositions Similar results can be derived for an increase in the money supply (see exercise 7). The most likely trajectory to the new equilibrium point is for the economy to exhibit an overshoot with regard to its interest rate response (falling sharply and then rising somewhat), while income will gradually rise to its new higher equilibrium level. Does the economy exhibit the same type of dynamic behaviour for a shock to the goods market, i.e., a shift in the IS curve? The situation is shown in figures 10.7 and 10.8. Consider first a fiscal expansion {a rising from 50 to 55) which shifts the IS curve from ISo to ISi, as illustrated in figure 10.7. The economy moves from equilibrium point Eo to equilibrium point Ei. But what dynamic path does it take to the new equilibrium? If we again assume that the money market adjusts instantaneously, then there will be a gradual rise in income as the multiplier impact of the expansion moves through the economy. The increase in income will raise the demand for money and hence raise the rate of interest. This rise in interest rate will be such as to maintain equilibrium in the money market. Hence, the economy will move along the LM curve until the new equilibrium is reached. There is no overshooting either of income or of interest rates. With less than instantaneous adjustment in the money market (/3 = 0.8), and a sluggish adjustment in the goods market (a = 0.1), then the economy will follow trajectory T2, with interest rates rising gradually until the new level r\ is reached, and income adjusting gradually until the new level of y\ is reached. Again the economy exhibits no overshooting. Only in the unlikely event that the goods market adjusts very rapidly (e.g. a = 0.5) along with the money market will the economy exhibit a spiral path following a counter-clockwise movement to the new equilibrium, trajectory T3, and with the economy exhibiting overshooting behaviour (see exercise 9). In the case of a fiscal contraction (a falling from 50 to 45), illustrated in figure 10.8, the economy will follow trajectory Ti with instantaneous adjustment Closed economy dynamics 443 T2(a=0.1,(1=0.8) Figure 10.8. T,(a=0.5, (1=0.8) 58 60 62 64 in the money market, with interest rates and income declining steadily until the new equilibrium is reached. Similarly, if the money market is quick to adjust (but not instantaneous, e.g., ft = 0.8) and the goods market is sluggish in its adjustment (a = 0.1), then path T2 will be followed. Only in the unlikely event that the goods market is very quick to adjust (e.g. a = 0.5) as well as the money market (e.g. ft = 0.8) will a spiral path like T3 be followed (see exercise 10). We can make a number of important propositions about the dynamic behaviour of (closed) economies concerning money market shocks and goods market shocks. PROPOSITION 1 If the money market is quick to adjust and the goods market is sluggish in its adjustment, then a monetary shock will most likely lead to a counterclockwise movement with the interest rate overshooting its equilibrium value and income gradually changing to its new equilibrium level. COROLLARY 1 A counter-clockwise spiral to a new equilibrium arising from a monetary shock is only likely to occur in the event that both the money market and goods market are quick to adjust to disequilibrium states. 444 Economic Dynamics PROPOSITION 2 If the money market is quick to adjust and the goods market is sluggish in its adjustment, then a goods market shock will most likely lead to a gradual movement of the economy to its new equilibrium, with the economy exhibiting no overshooting of either interest rates or income. COROLLARY 2 A counter-clockwise spiral to a new equilibrium arising from a fiscal shock is only likely to occur in the event that both the money market and goods market are quick to adjust to disequilibrium states. Can we make any observations about the dynamic behaviour of this economy when there is a combined fiscal and monetary shock? In carrying out this particular analysis we shall simply assume that the money market is quick to adjust, but not instantaneous, and that the goods market is sluggish in its adjustment. In figure 10.9 we illustrate a fiscal and monetary expansion, a rising from 50 to 55 and m rising from 8 to 12. In figure 10.10 we illustrate a fiscal and monetary contraction, a falling from 50 to 45 and m falling from 8 to 5. Under the assumption made about relative adjustment, it is very likely that the trajectory of the economy in each case is a counter-clockwise movement to the new equilibrium, with major overshooting of interest rates and a gradual change in income to the new equilibrium level. Overshooting of income will, once again, occur only if the goods market adjusts Figure 10.9. r LM0 20 18 LM 16 14 ~~ IS 12 10 IS0 65 70 75 80 y Figure 10.10. r 22 . 5 Closed economy dynamics 445 quickly to a disequilibrium along with the money market, trajectory T3. Similarly, a combined fiscal and monetary contraction, which is illustrated in figure 10.10, leads to a sharp rise in interest rates in the short period, and as income begins to fall, interest rates too are brought down. There is unlikely to be any overshooting of income. Only in the unlikely event that the goods market adjusts quickly to a disequilibrium along with the money market will this occur, trajectory T3. These results should not be surprising. The initial impact on interest rates comes about because of the shift in the LM curve. Only when income begins to adjust will this effect be reversed. In the case of fiscal and monetary shocks opposing each other, and under the same assumption about relative adjustment behaviour, the dynamic path to the new equilibrium can have various possibilities depending on which shock is the greater. Figure 10.11 illustrates a fiscal expansion and a monetary contraction, with equilibrium points Eq and Ei, respectively. If the fiscal expansion is the more Figure 10.11. Monetary contraction and fiscal expansion 0 y 446 Economic Dynamics dominant of the two shocks (figure 10.11(a)), then the economy will traverse a smooth path from Eo to Ei with a rise in interest rates and a rise in income. On the other hand, if the monetary contraction dominates (figure 10.11(b)), then the economy will move counter-clockwise, with interest rates overshooting their new equilibrium level and income gradually falling. Similarly, in figure 10.12 we show a fiscal contraction and a monetary expansion. If the fiscal contraction dominates (figure 10.12(a)), then the economy will decline gradually from equilibrium point Eo to Ei. On the other hand, if the monetary expansion dominates, the decline in the interest rate may very well overshoot its equilibrium level, although income will gradually rise. We arrive, then, at two further propositions: PROPOSITION 3 If the money market is quick to adjust and the goods market is sluggish in its adjustment, then a fiscal expansion (contraction) combined with Closed economy dynamics 447 a monetary expansion (contraction) will more likely lead to a counterclockwise movement, with interest rates rising (falling) initially and then falling (rising) into the medium and long term; while income will gradually rise (fall) until its new equilibrium position is reached. PROPOSITION 4 If the money market is quick to adjust and the goods market is sluggish in its adjustment, then a fiscal expansion (contraction) combined with a monetary contraction (expansion) will give rise to a gradual change in interest rates and income if the fiscal shock dominates, but will exhibit interest rate overshooting if the monetary shock dominates. There is one important observation we can draw from this analysis about the dynamic behaviour of the economy. Given the assumption about relative speeds of adjustment, then interest rate volatility is far more likely to be observed than income volatility. 10.6 IS-LM continuous model: version 2 In this section we shall extend the investment function to include real income. In other words, business will alter the level of investment according to the level of income; the higher the level of income the more business undertakes new investment. We shall continue with a simple linear model, but this simple extension will lead to the possibility that the IS curve is positively sloped. In considering the implications of this we shall consider some explicit numerical examples in order to see the variety of solution trajectories. In one case we shall derive an explicit saddle path solution. Since the formal derivation is similar to the previous section we can be brief. The model is5 e = a + b(\ — t)y — hr + jy md = ky — ur (10.23) y = a(e- y) r = p>(md — mo) where a > 0, 0 < b < 1, 0 < t < 1, h > 0, j > 0, k > 0, u > 0, a > 0, f3 > 0 which gives the two differential equations y = a[b(l — t) +j — l]y — ahr + aa (10.24) f = ftky — fiur — p>m§ with the IS curve obtained from setting y = 0 as a-[l-b(l-t)-j]y r = - h 5 Although we use t for the marginal rate of tax, there should be no confusion with the same letter standing for time. 448 Economic Dynamics and an LM curve obtained from setting r = 0 as ky — mo r = - u The major difference between this version and the one in the previous section is that now the IS curve can have either a negative slope (if b(l — t) + j < 1) or a positive slope (if b(l — t) +j > 1), and that the positive slope is more likely the larger the value of the coefficient j. Since we dealt with a negatively sloped IS curve in the previous section, let us consider here the implications of a positively sloped IS curve, i.e., we assume b(l — t) +j > 1. For a positively sloped IS curve, points to the left of this line represent a-[l-b(l-t)-j]y r > - h 0 > a + [b(l -t)+ j]y -hr-y implying y < 0. Hence, to the left of the IS curve income is falling. Similarly, by the same reasoning, for points to the right of the IS curve income is rising. There is no change for the LM curve, and we have already established that for points to the right of the LM curve interest rates are rising while to the left of the LM curve interest rates are falling. With a positively sloped IS curve, there are two possibilities: (i) the IS curve is less steep than the LM curve (ii) the IS curve is steeper than the LM curve. The two possibilities, along with the vector of forces outlined above, are illustrated in figure 10.13(a) and (b). Figure 10.13(a) reveals a counter-clockwise trajectory while figure 10.13(b) reveals an unstable situation, although it does indicate that a trajectory might approach the equilibrium point. Neither situation is straightforward. Although figure 10.13(a) indicates a counter-clockwise trajectory, is the trajectory tending towards the equilibrium or away from it? There is nothing within the model as laid down so far to indicate which is the case. In order to see what the difficulty is, consider the following two numerical examples. Example 10.5 Example 10.6 a = 2 k = 0.25 a = 0.05 a = 2 k = 0.25 a b = 0.75 u = 0.5 ß = 0.8 £ = 0.8 u = 0.25 ß t = 0.25 mo = 8 7 = 0.2 mo = 8 h= 1.525 h= 1.525 j = 0.8 j = 0.95 Solution Solution y* = 66 r* = 17 y* = 55.3 r* = 22.3 Closed economy dynamics 449 LM Figure 10.13. Intercepts and slopes IS intercept =1.3 IS slope = 0.238 LM intercept = —16 LM slope = 0.5 Intercepts and slopes IS intercept =1.3 IS slope = 0.387 LM intercept = -32 LM slope = 1 Both examples typify the situation in figure 10.13(a), with a positive IS curve, and the IS curve less steep than the LM curve. However, the dynamics of both these examples is different. Both lead to a counter-clockwise path. However, example 10.5 leads to a stable path which appears to traverse a straight line path after a certain time period, while example 10.6 leads to an unstable spiral, as illustrated in figure 10.14(a) and 10.14(b) (see exercise 11). But, then, what is it that is different between these two examples? To answer this question we need to consider the differential equation system in terms of deviations from equilibrium, and then to consider the trace and determinant of the dynamic system.6 Return to the general specification of the differential equations y = a[b(l — t) +j — l]y — ahr + aa r = fiky — flur — fimo 6 See chapter 4. 450 Economic Dynamics Figure 10.14. (a) r (10.25) (10.26) (10.27) and consider the equilibrium values of the variables, i.e. 0 = a[b(l -t)+j- l]y* - ahr* + aa 0 = ßky* - ßur* - ßm0 Subtracting the second set from the first we have y = a[b(l -t)+j- l](y - y*) - ah(r - r*) r = ßk(y - y*) - ßu{r - r*) and the matrix of this system is A = 'a[b(l - t) +j - 1] -ah ßk -ßu whose trace and determinant are tr(A) = a[b(l -t)+j-l]-ßu det(A) = -aßu[b{\ - t) + j - 1] + aßkh Using these results we can summarise the properties of the systems in examples 10.5 and 10.6 in terms of the values for their trace and determinant. These are Closed economy dynamics 451 Example 10.5 tr(A) = -0.382 det(A) = 0.008 where tr(A)2 > 4det(A) Example 10.6 tr(A) = 0.043 det(A) = 0.014 where tr(A)2 < 4det(A) In terms of table 4.1 in part I (p. 180), it is clear that example 10.5 satisfies the conditions for an asymptotically stable node while example 10.6 satisfies the condition of an unstable spiral, verifying what is shown in figure 10.13(a). Notice that in both examples the determinant of the system is positive. Although this is necessary for a spiral path, it is not sufficient to determine whether the path is stable or unstable. This requires information on the sign of the trace. A stable spiral requires the trace to be negative; while an unstable spiral arises if the trace is positive -and in both cases the condition that tr(A)2 < 4 det(A) needs to be satisfied. Unfortunately, there is no geometric representation of the trace requirement. It can be ascertained only from the system itself. Even so, a comparison of the two examples indicates quite clearly that for an unstable spiral to be more likely, it is necessary for the coefficient of induced spending (b(l — t) + j) to be high and for there to be quick adjustment in both markets (large values for the reaction coefficients a and If the IS curve is positively sloped, and is less steep than the LM curve, the most likely result is a counter-clockwise stable spiral. Let us now consider a third example for which the IS curve is positively sloped but is steeper than the LM curve. Example 10.7 Parameter values Solution Intercepts and slopes a = -25 k = 0.22 a = 0.05 y* = 65.4 IS intercept = — 25 b = 0.75 u = 0.75 ß = 0.8 r* = 10.5 IS slope = 0.5125 h = 0.25 m0 = 8 h = 1 LM intercept = -10.7 j ■ = 0.95 LM slope = 0.293 tr(A) = -0.574375 det(A) = -0.006575 where tr(A)2 > 4det(A) This example typifies the situation in figure 10.13(b). But in order to see what is happening, we need to derive the characteristic equations of the system. In terms of deviations from the equilibrium we have the general results indicated already in terms of the equation system given above. Substituting the numerical values given in example 10.7, we obtain the following differential equation system y = 0.025625(v - y*) - 0.05(r - r*) f = 0.176(v-v*)-0.6(r-r*) 452 Economic Dynamics whose characteristic roots can be obtained from |A - All = 0 i.e. 0.025625 - X -0.05 0.176 —0.6 — X = 0 Which leads to the quadratic equation X2 + 0.574375A - 0.006575 with solutions 0.0112277 Using the first solution, we have -0.585603 0.025625 0.176 -0.05 -0.6 y-y r — r* = 0.0112277 y-y r — r* which leads to the relationship r - r* = 0.287945(v - y*) On the other hand, using the second characteristic root, and following through the same procedure, we find r-r* = 12.224555(v - v*) These two results indicate two saddle paths; one of which is stable and the other is unstable. To verify this, we use Mathematica to plot ten trajectories of example 10.7, which are illustrated in figure 10.15. Given the vectors of force already established for figure 10.13(b), which typifies example 10.7, it is clear that the first characteristic root leads to an unstable saddle path, while the second characteristic root leads to a stable saddle path. The dynamics of this system, then, is schematically illustrated in figure 10.16, showing the saddle paths (denoted SiS'i and S2S'2 associated with r and s, respectively) in relation to the IS and LM curves. The equilibrium of this system, then, is unstable except for the unlikely event that the initial point lies on the stable saddle path denoted S2S'2 in figure 10.16. Also notice Closed economy dynamics 453 Figure 10.16. 40 60 80 100 from figure 10.16 that one of the saddle paths is almost identical to the LM curve. This is a result of the assumption of rapid adjustment in the money market relative to the goods market. With perfect adjustment in the money market, then one saddle path would be identical with the LM curve, and this would be the unstable saddle path in the present context. 10.7 Nonlinear IS-LM model In this section we shall consider a nonlinear version of the IS-LM model. We can be brief because much of the analysis has already been carried out. In this version consumption spending in real terms is related to real income (where we assume disposable income has been eliminated); investment is inversely related to the nominal rate of interest (we assume expected inflation is zero) and positively to the level of real income; government spending is assumed exogenous. Our expenditure function, in real terms, is then e = c(y) + i(r, y) + g 0 < cy < 1, ir < 0, iy > 0 (10.28) The demand for real money balances, md, is assumed to be positively related to real income (the transactions demand for money) and inversely related to the rate of interest (the speculative demand for money). Thus md = l(y,r) ly > 0, lr < 0 (10.29) The dynamics are in terms of excess demand in the goods market and excess demand for real money balances, i.e. y = a(e — y) a > 0 (10.30) f=B(l(y,r)-m0) 3>0 where mo is the supply of real money balances, and mo is assumed exogenous. Equilibrium in the goods market requires y = 0 or e = y, while equilibrium in the money market requires f = 0 or l(y, r) = mo. Suppose such a fixed point exists and is denoted iy*, r*). The question arises is whether such an equilibrium is dynamically stable. Given the nonlinear nature of the system, and the fact that no explicit functional forms are specified, then it is not possible to establish this in any absolute sense. We can, however, use the linearisation technique discussed in 454 Economic Dynamics (10.31) (10.32) (10.33) (10.34) part I to establish the stability in the neighbourhood of the equilibrium point.1 As we mentioned in part I, when employing such a linearisation, only local stability can be established. But even this is better than having nothing to say on the matter. Expanding the above system around the fixed point iy*, r*) gives y = a ß 'd(g-y), , d(e-y) ' —ä—(y - y ) + —ä—(r ~r) dy dr 3(7-m0) 3(7-m0) ' -^-(y - y ) +-7i-(r - r ) dy dr But 3(g -y) dy 3(7 - m0) dy - Cy ~\~ iy 1 , = lyt 3(g -y) dr 3(7 - m0) dr = h = lr Hence ~y~ ř y = a(cy + iy-l)(y- y*) + air(r - r*) f = Ply(y-y*) + Plr(.r-r*) which can be written as a matrix dynamic system in the form a(cy + iy — 1) air y — y* and where the matrix of the system is a(cy + iy — 1) air fily filr The dynamics of the system can now be determined from the properties of A. These are tr(A) = a(cy + iy- 1) + filr det(A) = aB(cy + iy — l)lr — ap>irly = —aB[lr(l - cy - iy) + irly] Can we interpret any economic meaning to the tr(A) and the det(A)? To see if we can, let us consider the slopes of the IS and LM curves. For the IS curve we have y = e, hence y = c(y) + i(y, r) + g Totally differentiating this expression with respect to y and r we obtain dy = Cydy + iydy + irdr and so the slope of the IS curve, denoted dr/dy, is given by (1 or dr dy iy)dy = irdr ry ~ ly ir 1 See section 2.7. Closed economy dynamics 455 The slope of the LM curve is established in the same manner (and noting that mo is exogenous) 0 = lydy + lrdr dr —ly dy lr If the IS curve is less steep than the LM curve, then 1 Cy ly ly i.e. lr(l — Cy — iy) + iyly < 0 — aP[lr(l — Cy — iy) + irly] > 0 Hence, det(A) > 0. This is certainly satisfied in the usual case of a negatively sloped IS curve and a positively sloped LM curve. But we have already established in the previous section that a stable solution will occur if both the IS and the LM curves are positively sloped but that the IS curve is less steep than the LM curve and that the trace of the system is negative in sign. 10.8 Tobin-Blanchard model 10.8.1 The model in outline9. There has been some interest by economists as to whether stock market behaviour can influence income and interest rates - at least in the short run. The IS-LM model so far outlined does not allow for any such link. It is plausible to think that investment will, in some way, be influenced by stock market behaviour. Such a link was considered by Blanchard (1981) following on the approach to investment suggested by Tobin (1969), and what is referred to as the q-theory of investment. The variable q represents the market value of equities as a ratio of the replacement cost. It can be understood as follows.9 If all future returns are equal, and denoted R, and are discounted at the interest rate r, then the present value of equities, V say, is equal to R/r. On the other hand, firms will invest until the replacement cost of any outstanding capital stock, RC, is equal to the return on investment, R/p, where p is the marginal efficiency of capital. Then V _ R/r _ p q~RC~ R/p ~ 7 Consequently, net investment is a positive function of q, which still means that it is inversely related to r. In the long run r = p and so q = 1, and there is no net investment. The upshot of this approach is that investment, rather than being inversely related to r is positively related to q. This in turn means aggregate expenditure (and hence aggregate demand) is positively related to q. (10.35) 8 A different treatment than the one presented here, also utilising phase diagrams, is provided in Romer (2001, chapter 8). See also Obstfeld and Rogoff (1999, section 2.5.2). 9 See Stevenson, Muscatelli and Gregory (1988, pp. 156-9) for a fuller discussion. 456 Economic Dynamics We can accordingly express aggregate expenditure, e, as a(t) = a\y(t) + a2q(t) + g 0 < a\ < 1, a2 > 0 where g is real government spending. The goods market is assumed to adjust with a lag, with reaction coefficient a > 0, thus y(t) = a(e(t) - y(t)) a > 0 The money market, on the other hand, is assumed to adjust instantaneously, and so the demand for real money balances is equal to the supply of real money balances, i.e. ky(t) — ur(t) = mo k > 0, u > 0 The next equation relates the rate of interest (on bonds) to the yield on equities, which are equal because it is assumed that bonds and equities are perfect substitutes, i.e. biy{t) + qe{t) r(t) = - qit) where b\y constitutes the firms' profits, which are assumed proportional to output, and qe constitutes expected capital gains. Finally, we assume rational expectations, which in the present model is equivalent to perfect foresight, and so qe = q. Suppressing the time variable, then the model can be stated in terms of five equations e = axy + a2q + g niQ = ky — ur y = a(e-y) b\y + qe r = - q qe = q which can be reduced to two nonlinear nonhomogeneous differential equations, namely y = o(a\ — l)y + oa2q + og (10.37) fkq \ qm0 q =--bi \y-- \u J u First we need to establish the existence of a fixed point, an equilibrium point. We do this by setting y = 0 and q = 0, and solving for y and q. This is no more than where the two isoclines intersect. So let us first look at these separately. First consider the y = 0 isocline, which we shall refer to as the IS curve since it implies goods market equilibrium. We have y = a{a\y + a2q + g - y) = 0 -(1 -ax)y + a2q + g = 0 (\-a{)y-g i.e. q = - a2 (10.36) Closed economy dynamics 457 which is linear with intercept on the g-axis of —g/a2 and slope of (1 — a\)la2. Since we have assumed that a\ lies between zero and unity, then the slope of this line is positive. Next consider the q = 0 isocline, which we shall refer to as the LM curve since it implies money market equilibrium. We have which is nonlinear, and has an asymptote at y = mo/k, which means that q is positive only if y > mo/k. This we shall assume to be the case. Also, as y oo, then q ub\/k. Although it is possible to solve for y and q, the solution involves a quadratic and does not reveal anything new. What we have here, however, is a nonlinear nonhomogeneous differential equation system. To establish the nature of the equilibrium we need to consider the vectors of forces in the four quadrants. We have already established that the y = 0 isocline is positively sloped. Furthermore, if y > 0 then Hence, above the y = 0 isocline, y is rising while below it y is falling, as illustrated in figure 10.17. In establishing the nature of the forces either side of the q = 0 isocline we first need to establish its slope. We find this with a little manipulation as follows ub\y i.e. q = (ky - mo) (1 - a{)y-g q > ub\y q = (ky - m0) Figure 10.17. y 458 Economic Dynamics dq dy (ky — mo)ub\ — ub\yk (ky - m0)2 (ky - m0)/u (ky - m0)2 r The slope of the LM curve in (g,y)-space is therefore ambiguous. The slope is positive if b\ > qk/u and negative if b\ < qk/u. To interpret these two situations, consider a rise in income, shown by the movement from point A to point B in figures 10.18(a) and (b). From the money market equation this will raise the rate of interest, r; from the yield on equities equation, this will raise profits and hence the equity yield. If the rise in income raises the yield on equities by less than it raises r then q must fall in order to re-establish equilibrium between r and the yield on equities ({b\y + q)/q), as shown in figure 10.18(a) by the movement from point B to point C. This Blanchard called the 'bad news' case because the rise in Figure 10.18. (a) 'bad news' V y (b) 'good news' y Closed economy dynamics 459 income led to a fall in stock market prices. On the other hand, if the increase in income increases r by less than the yield on equities {{b\y + q)/q), then q must rise, as shown in figure 10.18(b) by the movement from point B to point C. This Blanchard called the 'good news' case, since the rise in income leads to a rise in stock market prices. Whether the q = 0 isocline is negatively sloped ('bad news') or positively sloped ('good news'), if q > 0 then ub\y q > - (ky - m0) and so above the q = 0 isocline q is rising while below it q is falling, as shown by the arrows in figure 10.18. The combined vector forces in both the 'bad news' case and the 'good news' case are illustrated in figures 10.18(a) and (b). In each case, the vector forces indicate y 460 Economic Dynamics Figure 10.20. 34 35 36 37 y a saddle path solution. Given the assumption of rational expectations, and given that for any value of y there is a unique point, (a unique value of q) on the saddle path, then the economy will be at this value of q and will, over time, converge on the equilibrium.10 Example 10.8 Let us illustrate the model with a numerical example. In this example we consider only the 'bad news' case. The model is e = O.Sy + 0.2q + 1 8 = 0.25y - 0.2r y = 2(e- y) O.ly + q r = - q leading to the two nonlinear nonhomogeneous differential equations y = 14 - 0.4y + 0.4^ q = 1.25qy - O.ly - 40q with equilibrium values y* = 35.76 and q* = 0.76 (and r* = 4.7). The solution with vector forces is shown in figure 10.20. Let us take this numerical example further and consider the linear approximation. Taking a Taylor expansion around the equilibrium, we have y = -0.4(y - y*) + 0.4(4 - q*) q = \.25q*(y - y*) - 0.1(y - y*) - 40(q - q*) + \.25y*{q - q*) i.e. y = -0.4(y - y*) + 0A(q - q*) q = 0.85(y - y*) + 4.1(q - q*) 10 There is a problem if the LM curve is everywhere steeper than the IS curve (see Scarth 1996). Closed economy dynamics 461 Figure 10.21. The matrix of the system is T-0.4 0.4" ~|_0.85 4.7 _ with characteristic equation X2 — 4.43A — 2.22 = 0 and characteristic roots r = 4.7658 and s = —0.4658. The fact that the characteristic roots have opposite signs verifies the saddle point equilibrium (as does the fact that det(A) is negative, i.e., det(A) = —2.22). The general solution is y(t) = y* + CleAn65%t + C2e-°A65Ht q(t) = q* + C3e4J658t + c^"0-4658' The saddle paths are readily found by solving (A - rl)vr = 0 and (A - sl)\s = 0 giving the two respective eigenvectors 1 Vs — 1 12.9145 , v -0.1645 _ where \s is the stable arm of the saddle point. These results, using the above linearisation, are shown in figure 10.21, which includes the direction field for the linearisation.11 In this example, the stable arm is almost identical with the linear approximation to q — 0 at (y*, q*), see exercise 13. 462 Economic Dynamics 10.8.2 Unanticipated fiscal and monetary expansion We are now in a position to consider the effects of fiscal and monetary policy. In this sub-section we shall concentrate on unanticipated changes in policy, leaving anticipated changes to sub-section 10.8.3. Fiscal expansion Consider first a fiscal expansion, a rise in g. This has no impact on the q = 0 isocline but decreases the intercept of the y = 0 isocline, i.e., it shifts this isocline right (down). The situation for both the 'bad news' case and the 'good news' case is illustrated in figure 10.22 (where we assume that the q = 0 isocline is less steep than the y = 0 isocline). In each case the initial equilibrium is at point Ei where y1 = 0 intersects q = 0. The associated stable arm of the saddle point is SiSj. A rise in g shifts the IS curve down to y2 = 0. Initially income does not alter, and the system 'jumps' to the new saddle path at point A, and then over time Figure 10.22. (a) Bad news y (b) Good news Closed economy dynamics 463 moves along S2S2 to the new equilibrium point E2. Although income rises in both situations, in the 'bad news' case asset prices decline, while in the 'good news' case they rise. Monetary expansion Consider next monetary expansion, a rise in mo. This has no impact on the IS curve, but shifts the q = 0 isocline up, since The system 'jumps' from Ei to point A on the new saddle path S2S2 and then moves along this until the new equilibrium point E2 is reached, as shown in figure 10.23. In each case income rises and asset prices rise from one equilibrium point (a) Bad news Figure 10.23. y (b) Good news y=0 y 464 Economic Dynamics to the next but the path of asset prices is different between the 'bad news' case and the 'good news' case.12 Even though the fiscal and monetary changes were unanticipated, it is assumed that the moment they are implemented the economy 'jumps' from the initial equilibrium to a point on the saddle path, and then adjusts over time along the stable arm of the saddle point. But what happens if the changes are announced in advance? 10.8.3 Anticipated fiscal and monetary policy Suppose some policy change is announced at time to and to be implemented in some future time t\. In this instance the policy change is anticipated, and some response can occur now in anticipation of what is known to occur once the policy is actually implemented. However, what occurs now is governed by the dynamics of the original equilibrium, since the new equilibrium has yet to come about. Fiscal expansion Consider first a fiscal expansion. We have already established that this will not shift the q = 0 isocline but will shift the IS curve down. In anticipation of what will happen to stock market prices, the system will move from point Ei to point A' (where A' falls short of point A on the saddle path), as shown in figure 10.24. In the 'bad news' case, stock market prices fall while in the 'good news' case they rise. This movement, of course, is simply anticipating the final implication of the policy change. But from the time the policy is announced until the time the policy is implemented, the economy is driven by the dynamic forces associated with the initial equilibrium point Ei. Hence, the system moves from point A' to point B' (on the saddle path S2S2). The policy is now carried out, and the system moves along S2S2 from point B' to point E2. The impact on income in the two cases is now different. In the 'bad news' case income falls and then rises, while in the 'good news' case it continually rises over time. On the other hand, asset prices gradually fall (if rather irregularly) in the 'bad news' case, and gradually rise (if rather irregularly) in the 'good news' case. Monetary expansion Finally consider the case of monetary expansion, which is announced in advance, and shown in figure 10.25. As in the previous situation, in the first instance the asset price will move part way towards its new equilibrium value, shown by point A'. It will then be governed by forces associated with the initial equilibrium point Ei, and so will move along the trajectory with points A'B'. Point B' is associated with the time the policy is implemented. Thereafter, the system will move along the stable arm of the saddle point, i.e., along S2S2, until point E2 is reached. For a fuller discussion of what is taking place over the adjustment path, see Blanchard (1981). Closed economy dynamics 465 (a) Bad news Figure 10.24. y y Comparing figure 10.24 with 10.22 and figure 10.25 with 10.23 shows quite a different behaviour and that it makes quite a difference to the dynamic path of the economy whether policies are announced (anticipated) or not. This is important. There has been a growing tendency on the part of policy-makers to announce in advance their policy intentions - and, at least in the UK, this applies to both monetary and fiscal policy. 10.9 Conclusion Although the IS-LM model is considered in some detail in intermediate macroeconomics, little attention has been paid to its dynamic characteristics. In this chapter we have concentrated on discussing the dynamics of the IS-LM model - both in discrete terms and by means of continuous time variables. Such a treatment has allowed us to consider possible trajectories for income and the rate of interest. Although we have not dealt with other endogenous variables, it is quite clear 466 Economic Dynamics that we can obtain their paths from a knowledge of y(t) and r(t). For instance, given y(t) we can compute tax{t) = to + hy{t), which in turn allows us to compute disposable income, yd{t) = y(t) — tax(t). This in turn allows us to compute consumption, c(t) = a + byd(t), and so on. However, this is possible only when we have explicit functional forms for all relationships in the model. What we observe from this chapter is the importance of different adjustment speeds in the goods market relative to the money market, where the latter adjusts more quickly than the former. Although a number of trajectories exhibit a counter-clockwise movement towards equilibrium, a counter-clockwise spiral, although possible, is not so likely given a quick adjustment in the money market. Overshooting, however, especially of the rate of interest, is likely to be a common occurrence, as is interest rate volatility. With investment related to both the rate of interest and the level of income, it is possible to have a positively sloped IS curve. If the IS curve is steeper than the LM curve then the most likely outcome is an unstable saddle path. This result not only depends on investment being significantly and positively related to income, but also on the (realistic) assumption that the money market is quicker to adjust Closed economy dynamics 467 than the goods market. Although we have considered this possibility in the confines of a simple (linear) model, it does beg the question of whether it will occur in a more complex linear model or even in a nonlinear model. We do not, however, investigate these questions in this text. Finally, we extended the IS-LM analysis to allow for stock market behaviour employing the Tobin-Blanchard model. Once again the differential speeds of adjustment in the goods market relative to the asset market was shown to be important for dynamic trajectories. This model also highlighted the importance of unanticipated against anticipated policy changes. Exercises 1. Consider the model C, = 110 + 0.757, /, = 300 Et = C, + I, 7, = Et-X Plot the solution path for income and consumption for three different adjustment lags, j=1,2,3 for a permanent increase in investment of £ 10 million beginning in period 1. 2. Consider the numerical model in example 10.2, but assume that it = 320 - 4r,_i Show that this leads to a second-order difference equation for income. Either solve this second-order equation for yo = 2000, y\ = 2010 and >>2 = 2010. Hence plot the path of y(t) and r(t); or else set the problem up on a spreadsheet and plot y(t) and r(t). 3. Reconsider the model in exercise 1 and establish the dynamic multiplier for each of the three time lags in response to a rise in investment of £20 million. What can you conclude from these results? 4. Consider the model Ct = 110 + 0.757, /, = 4(7, - Yt-,) Et = Ct + It Yt = Et-x (i) Show that this results in a second-order difference equation for income. Solve this equation. (ii) Suppose /, = 4(C,-C,_!) Does this lead to a different time path for income? 5. For the numerical model in example 10.2 set up a spreadsheet and derive the solution path for all endogenous variables resulting from a rise in 468 Economic Dynamics real money balances of £20 million. Compare your results with those provided in table 10.2. 6. Set up a spreadsheet to derive trajectories for the discrete model yt+i -yt = a[b(l - i) - l]yt - ahrt + aa rt+i -r, = flkyt - fiurt - f3m0 Derive the equilibrium income and interest rate, by setting yt+\ = yt = ... and rt+\ = rt = ... and place cells on the spreadsheet to compute such equilibria. Use the parameter values in the text to derive the three trajectories Ti (a = 0.05, f3 = 0.8), T2 (a = 0.1, f3 = 0.8) and T3 (a = 0.5, p = 0.8). How would you use this specification to: (i) show a goods market shock? (ii) show a money market shock? 7. Show that for the same system outlined for figure 10.4 that a monetary expansion from mo = 8 to m\ = 12 leads to a new equilibrium point (y, r) = (72.2, 12.1). Using either Mathematica ox Maple, establish three trajectories for the same combinations of a and ft as in exercise 6. Or, using the discrete form of the model outlined in exercise 6, set up the model on a spreadsheet and obtain the three trajectories. 8. Use the spreadsheet model of exercise 6 to investigate the implications for the three trajectories Ti, T2 and T3 of (i) a higher marginal propensity to consume (ii) a lower marginal rate of tax (iii) investment being more interest-sensitive (higher h) (iv) a higher income velocity of circulation of money (lower k) (v) a more interest-sensitive demand for money (higher u). 9. Use your model in exercise 6 and verify that if the parameter a rises from 50 to 55 the adjustment path exhibits overshooting of both y and r if a = 0.5 and fj = 0.8. 10. Use your model in exercise 6 and verify that if the parameter a falls from 50 to 45 the adjustment path exhibits overshooting of both y and r if a = 0.5 and fj = 0.8. 11. Use Mathematica or Maple to derive the trajectory {y(t),r(t)} for examples 10.5 and 10.6 in section 10.6. Verify the statements in the text, namely (i) Example 10.5 leads to a stable path which appears to traverse a straight line path after a certain time period. (ii) Example 10.6 leads to an unstable spiral. 12. Reconsider example 10.6 in section 10.6. Does the same saddle pathresult if a = 0.1 and fj = 0.8? 13. (i) Show that the linear approximation to the q = 0 isocline in the Tobin-Blanchard model is * q = q - b\mou (ky* - mo)2 (y-y*) Closed economy dynamics 469 (ii) In example 10.8 show that the equation for the linear approximation to q = 0 at (y*, q*) is q = 7.23352 - 0.181008v while the equation for the saddle path is q = 6.64988 -0.164683v 14. Consider the following IS-LM model e = a + b(l — t)y — hr + jy a = 5 k = 0.5 md = ky — ur b = 0.75 u = 0.3 y = a(e- y) r = fi{md -mo) t = 0.25 a = 0.25 h = 03 p = 0.4 j = 0.4 m0 = 10 (i) Findy* and r*. (ii) What are the equations for the IS curve and the LM curve? (iii) Obtain the trace and determinant of the system, and hence establish whether a stable or unstable spiral is present. 15. Given the Tobin-Blanchard model e = O.Sy + 0.2q + 1 16 = 0.5v - 0.25r y = 2(e- y) 0A5y + q r = - q (i) Find y* and q*. (ii) Show that the fixed point (y*, q*) is a saddle point equilibrium. (iii) Derive the equation of the stable arm of the saddle point. Additional reading Additional material on the contents of this chapter can be obtained from Blanchard(1981),McCafferty (1990), Obstfeldand Rogoff (1999), Romer (2001), Scarth (1996), Shone (1989, 2001), Stevenson, Muscatelli and Gregory (1988), Teigen (1978) and Tobin (1969). CHAPTER 11 The dynamics of inflation and unemployment 11.1 The Phillips curve At the heart of most discussions of inflation is the Phillips curve which, in its modern formulation, stipulates a relationship between price inflation, it, and unemployment, u, augmented for inflationary expectations, ite. Thus (11.1) it=f(u) + ^ite 0<£<1 This relationship is the expectations-augmented Phillips curve. The only slight difference from standard textbook treatments is the presence of £ lying between zero and unity. The reason for introducing this will become clear. Next we make a simple assumption about expected inflation, namely (11.2) ite = p(it - ite) > 0 This is no more than a continuous version of adaptive expectations. When the actual rate of inflation exceeds the expected rate, expectations are revised upwards and when the actual rate is below the expected rate, then expectations are revised downwards. Suppose the government attempts to maintain unemployment at some constant level, a*.1 We further suppose that they are successful and so f(u*) is a constant and known. To establish the implications of such a policy, differentiate the Phillips curve relationship (11.1) with respect to time under the assumption that u = u* and substitute in equation (11.2). Then fc =^fCe = $p(7t ~ 7t, = p($7t - $7Ze) But %ite = 7t — /(«*), and so It = /3[$7t - It +f(u*)] i.e. (11.3) 7t = 0(1 -£)7t which is linear with intercept ftf(u*) and slope —0(1 — §). Relationship (11.3) is illustrated in figure 11.1. 1 u* is often assumed to be the level of unemployment associated with full employment. This was the type of policy pursued in the UK between 1945 and 1979. The dynamics of inflation and unemployment 471 71 Figure 11.1. First we need to establish whether a fixed point exists. For such a point ft = 0, i.e. or it f(u*) Only if 0 < § < 1, however, will 7T* exist. In particular, if § = 1 then 7T* is undefined. Second, if 0 < § < 1, then 7T* is asymptotically globally stable since the relationship between ft and 7T is negatively sloped. Third, if § = 1 inflation is always correctly anticipated and it = ite and fce = 0. In this instance the rate of unemployment is constant regardless of the rate of inflation. This unemployment rate, following the work of Friedman and Phelps, is referred to as the natural rate of unemployment (or the non-accelerating inflation rate of unemployment, NAIRU), and denoted un. This rate satisfies f(un) = 0. The situation is illustrated in the more conventional diagram, figure 11.2. However, we can go further. Since ft = pf{u) and f'{u) < 0, if u = u* < un then it follows that ft > 0; while if u = u* > un, then ft < 0. This implies that if the government maintains the level of unemployment below the natural level (here we ignore u* > un) then permanent inflation will be the result. This is because expected inflation always lags behind actual inflation and the economy is forever trying to catch up with what it observes. It is common in a number of studies to assume a relationship between inflation, it, and real income, y, and expected inflation, ite. In particular it is common to express this relationship in the form jt = a(y — y„) + Jte a > 0 where y and yn are in natural logarithms and yn is the natural level of income (the income level associated with un), and where the relationship is referred to as 'the (11.4) (11.5) 472 Economic Dynamics Figure 11.2. ÄK)=0 0 u u Phillips curve'. This is not the relationship between inflation and unemployment and in fact embodies two reaction functions.2 It is worth spelling these out in detail because of the common occurrence of this equation. Following the original formulation of the Phillips curve, we postulate a relationship of the form it = —y\(u — un) + ite yi > 0 This is the first reaction function indicating the response of price inflation to the unemployment gap. It implies a specific functional form for f{u). The second reaction function is a formulation of Ohm's law3 and is given by u-un = -y2(y - yn) yi > 0 Substituting this into the previous equation we obtain * = ym(y-yn) + *e Although both it, in terms of the unemployment gap, and it, in terms of the output gap, are both referred to as 'the Phillips curve', the second is more suspect because it involves an additional behavioural relationship, namely Okun's law - which is far from being a law. We shall, however, conform to common usage and refer to both as the Phillips curve. 11.2 Two simple models of inflation Macroeconomic modelling has generally incorporated the Phillips curve within an IS-LM framework. In this section we shall consider the simplest of these models to 2 See Shone (1989, chapter 3) for a more detailed discussion on this. 3 See Shone (1989, appendix 3.2). or (11.6) it = a(y — yn) + ite a > 0 The dynamics of inflation and unemployment 473 highlight the dynamics. Basically, the goods market and money market combine to give the aggregate demand curve (see Shone (1989, chapter 2). To see this, consider the following simple linear model, where variables (other than inflation and rates of interest) are in logarithms. Goods market c = a + b(l — t)y i = i0 - h(r - 7te) (11.7) y = c + i + g Money market nid = ky — ur ms = m — p (11-8) md = ms where c = real consumption y = real GDP i = real investment r = nominal rate of interest 7Te = expected inflation g = real government spending ma = real money demand ms = real money supply m = nominal money stock p = price level Solving for y and r we obtain (a + j'o + g) + (h/u)(m — p) + hite f = r l-b(l-t) + (hk/u) ky* — (m — p) u The main focus of attention is on y*, the equilibrium level of real income. It should be noted that this is a linear equation in terms ofm—p and ite, i.e. y = ao + a\{m — p) + a27te a\ > 0, a2 > 0 (11.10) and this represents the aggregate demand curve, the AD curve. Why? Because it denotes equilibrium in both the goods market and the money market. In other words, all points along the AD curve denote equilibrium in both the goods market and the money market. We can express the aggregate demand curve in the usual way as a relationship between p and equilibrium y, with p on the vertical axis and y on the horizontal axis. Then V fli / \aij \aij i.e. p = cq - c\y + c2JTe 474 Economic Dynamics where ao + a\m 1 a2 Co = -, ci = —, c2 = — a\ a\ a\ which clearly indicates an inverse relationship between the price level, p, and the level of real income, y. It is at this point we introduce inflation, it. We assume that the rate of inflation is proportional to the output gap and adjusted for expected inflation, as outlined in the previous section, i.e. it = a(y — yn) + ite a > 0 yn is the output level for which it = ite = 0. It represents the long-run situation where prices are completely flexible. Under this condition the equilibrium price level is p* and y = yn regardless of p and so the long-run aggregate supply curve is vertical at yn. The situation is illustrated in figure 11.3. Although figure 11.3 expresses p as a function of y, the more interesting and revealing relationship is that between y and real money balances, m — p, i.e., y = ao + a\(m — p) + a2ite. Only when there is a change in real money balances (a change 'mm— p) will there be a shift in AD. This is important. In elementary courses in economics it is quite usual to say something like 'a decrease in the money supply shifts LM left, raising r and reducing v'. But money supply has hardly ever decreased! What has decreased is the growth in the money supply. This leads to a fall in p. So long as m falls more than p, then real money balances will fall, i.e., m — p < 0. It is this which shifts the LM curve to the left.4 In other words, only when m-p^O will the aggregate demand curve shift. 4 See chapter 10 on the IS-LM model. The dynamics of inflation and unemployment 475 Example 11.1 To illustrate this model, let ite = 0 and let y = 9 + 0.4(m — p) m = 5 it = 0.2(y - yn) yn = 6 then p = 27.5 - 2.5y In equilibrium y = yn hence it = 0, i.e., y* = 6 and /?* = 12.5. The situation is illustrated in figure 11.4. At a price level below (or above) p* = 12.5, forces will come into play to move the economy towards equilibrium. To illustrate these dynamic forces, consider the following discrete version of the model. We have (noting we have the natural logarithm of prices) vf_i = 9 + 0.4(mf_! -Pt_i) *t = Pt-Pt-i = 0.2(y,_i - yn) i.e. 7tt = 0.2[9 + 0.4(mf_! -A_!)-6] = 0.6 + 0.08(m,_i 476 Economic Dynamics But mt-\ = mt = 5 for all t and so jtt = 1 - 0.08p,_i Since prices are in natural logarithms, then 7tt = pt — pt-\, hence Pt -Pt-\ = 1 - O.OSpt-i i.e. pt = 1 + 0.92pt-i We can either use the original formulation of the model in a spreadsheet, or this linear relationship5 Pt = f{Pt-\) = 1 + 0.92pt-\, as shown in figure 11.5. Either way, we can first solve for the fixed point pt = p* for all t so that p* = 1 + 0.92p* p* = 12.5 The spreadsheet representation is illustrated in figure 11.6, which shows the system converging on equilibrium for an initial price of po = 5. Convergence to equilibrium in the neighbourhood of p* = 12.5 is assured because \f'(p*)\ < 1 (see n. 5). The model just discussed has a major weakness and that is that in the long run the only acceptable level of inflation is zero, since only this is consistent with the (assumed) zero expectations value of inflation. But can a situation arise in which it = 7te at some positive value and the economy is in long-run equilibrium with income at the natural level? To answer this question, first return to our aggregate demand relation y = ao + a\{m — p) + a27te If we take the time derivative of this relationship6 we obtain the demand pressure curve, with formula y = a\{m — 7t) + a2ite 5 Given Pt =f(pt_i) = 1 + 0.92pt-i then f'ip) = 0.92 < 1, which is the requirement for stability as indicated in part I. 6 We assume that the variables are in logarithms and so dp/dt = dlnP/dt = jz . The dynamics of inflation and unemployment All E3 Microsoft Excel Elle Edit iew Insert Formal Took Qata öndtjw yelp Figure 11.6. 4 z 51 at ttl L19 -| • HRE3! A B C H J K 3 Figure 11.6 y =a0 +ai(m-p)+aif^ n=a(y-y„} + Ti' t P(t) y(t) pi(t) 0 5.0 9 0 0.30 1 6.5 84 0.24 2 8.1 7.8 0.18 3 9.5 7 2 0.12 4 10.6 6 7 0^07 5 11.4 6.4 004 6 11.9 6 2 0.02 7 12.2 6 1 0.01 8 12 3 6 1 0.01 9 12.4 6 0 0 00 10 12.5 6 0 0.00 Ready k < f M\sh**M/8 2 J, Sfw«t3 / a1 = a2 = 9 m; 0.4 yn --0 alpha = E[pi) = 5 6 0 1 0 12.5 Price level and rate of inflation -■ 0.35 -■ 0-30 -- 0 25 -- 0.20 o - 0 15 c -- 0 10 -- 0D5 T 0 00 -Price level — ■ -Inflation AtfDSHapes. \ V □ o 141 * £' s a S U . 1 ■:r MM where monetary growth, m, is exogenously given. We can now combine this with the Phillips curve and a dynamic adjustment for inflationary expectations, giving the model y = a\(m — tt) + a2tte a\ > 0, a2 > 0 tt = a(y - yn) + 7Te a > 0 (11-11) jte = p(iz -7te) p > 0 The model is captured in terms of figure 11.7 in its more traditional form. The demand pressure curve intersects the short-run Phillips curve on the long-run Phillips curve. Since in this situation tt = ite, then it follows y = yn and fce = 0, which implies m = tt. 478 Economic Dynamics To consider the dynamics of the model, it can be reduced to two differential equations.7 From the Phillips curve and the dynamic adjustment equations we immediately obtain ite = ap(y-yn) For the demand pressure curve we substitute the short-run Phillips curve for it and the result just obtained for rte i.e. y = a\(m — it) + a2rte = aim - ai[a{y - yn) + 7te] + a2afi(y - yn) = a\m — a(a\ — a2f5)(y — yn) — a\Xe Thus, we have the two differential equations ite = ap(y - yn) y = a\m — a(a\ — a2f5)(y — yn) — a\7te which can be solved for y* and 7te*. Notice that the model solves for the time path of expected inflation, but the time path of actual inflation is readily obtained from the short-run Phillips relationship, i.e. 7t(t) = a(y(t) - yn) + 7te(t) To solve for equilibrium, a steady state, we set fte = 0 and y = 0. From the first condition it immediately follows that y = yn. Combining this result with fte = 0 andj = 0 immediately gives the result rte* = m. (In what follows we shall suppress the asterisk.) First consider the fte = 0 isocline. In this instance it readily follows that y = yn and so the isocline is vertical at the natural level of income. If y > yn then fte > 0 and hence ite is rising, and so to the right of the vertical isocline we have vector forces pushing up expected inflation. Similarly, when y < v»jthen7re < 0 and there are forces pushing down the rate of expected inflation. These forces are illustrated in figure 11.8(a). Consider next the y = 0 isocline. In this case a\m — a(a\ — a2f5)(y — yn) = a\7te which is negatively sloped if 1 — (a2 fi/ a\) > 0, which we assume to be the case. If y > Othen 7 This is a simpler version of a similar model discussed in McCafferty (1990, chapter 7). The dynamics of inflation and unemployment 479 %=0 Figure 11.8. y=y. y and so to the left (below) the y = 0 isocline there are forces present increasing y. Similarly, to the right (above) this isocline there are forces decreasing y. These forces are illustrated in figure 11.8(b). Combining the two isoclines leads to four quadrants with vector forces as shown in figure 11.9. What this shows is a counter-clockwise movement of the system. Hence, starting at any point such as point A, the system will move in an anticlockwise direction either converging directly on the equilibrium point, as shown by trajectory Ti, or converging on the equilibrium point with a counter-clockwise spiral, as shown by trajectory T2. Which of these two trajectories materialises depends on the values of the exogenous variables and parameters of the dynamic system. Of course, there is nothing in the qualitative dynamics preventing the counter-clockwise spiral diverging from the equilibrium. All we know from figure 11.9 is that the equilibrium is a spiral node. We can illustrate the model with a numerical example. We shall present this model first in continuous time and then in discrete time. The discrete time version has the merit that the system's dynamics can readily be investigated on a spreadsheet. 480 Economic Dynamics Figure 11.9. h"=0 n —m y=0 0 y, y Example 11.2 Consider the numerical model y= 10(15 - it) + 0.5ire it = 0.2(y- l5) + 7te ft" = \.5(lt ~7te) Equilibrium income and expected inflation is readily found to be y* = 15 and 7te* = 15, which is equal to the actual rate of inflation and to the growth of the money supply. We have already established that the fte = 0 isocline is vertical at the natural level of income, namely y* = yn = 15. On the other hand, the demand pressure curve y = 0 is given by i.e. 7te = 17.775 -0.185y and it is readily verified that 7te* = 15 when y* = 15. Furthermore, the two differential equations take the form y= 111.15- 1.85y- IOtt" jte = -4.5 + 0.3y which in terms of deviations from equilibrium are y = -1.85(y - y*) - 10(jre - 7te*) 7te = 0.3(y-/) Hence, the matrix of this system is A = -1.85 -10 0.3 0 The dynamics of inflation and unemployment 481 s y with tr(A) = —1.85 and det(A) = 3. From chapter 4, table 4.1 (p. 180), since tr(A) < 0, det(A) > 0 and tr(A)2 < 4det(A) then we have a spiral node. Furthermore, the characteristic roots of A are r, s = —0.925 ± 1.4644/ and since a in the characteristic roots r, s = a ± Bi is negative, then the system is asymptotically stable. We verify this by using a software package to derive the direction field of this system along with a trajectory beginning at point (vn, 7Tq) = (12, 12), as shown in figure 11.10. Consider the system in equilibrium at7T* = 7Te = 15 and y = yn = 15. Now let monetary growth decline from mo = 15 to m\ = 12. The result is shown in figure 11.11. In line with our previous analysis, we have an anticlockwise spiral that converges on the new equilibrium point Ei. We noted above that although the model solves for 7Te(t) we can derive jr(t) from the short-run Phillips curve. What is the difference between the path of 7Te(t) and the path of 7t{t)l These paths for a reduction in monetary growth just analysed 482 Economic Dynamics are shown in figure 11.12, which also shows the path of y(t) over part of the adjustment period. What the lower diagram illustrates is not only the cycle nature of actual and expected inflation, but that actual inflation is initially below expected inflation. This is because actual income initially falls short of the natural level and so dampens inflation. When, however, income is above the natural level then actual inflation is above expected inflation and so pushes up actual inflation. Example 11.3 Next consider a discrete version of the model with the same parameter values. The model is yt - yt-i = 10(m,_i - 7tt-\) + 0.5(71/ - itet_^) 7tt = 0.2(v, - yn) + 7tf < - <_i = 1.5(7T,_i - <_!) which leads to the two difference equations yt = 177.75 - 0.85v,_i - lOjrf.j xf = -4.5 + 0.3v,_i + itet_x The dynamics of inflation and unemployment 483 Ote gdlt ttfew Spatrt FgriMt look Bab Whto* tMP CSia Figure 11.13. K2 ' r[ -' - 10 - B / 1 UJ 3 4'! 5": 6 ■ 7 8 9 10 iiTjl 12| i£j 16 IV 18 19 20. 21..; h 1 »t A B Example 11.3 al = a2 = A0 = A1 = A2 = 10 alpha = 0.5 beta = 151.85 BO = 0876567 B1 = -10 B2 = y(t) 12.0000 42 3700 69.5944 87.9871 93.1925 83; 1584 6 58.7234 7 23 6704 8 -15.8041 9 -52.1441 0 -77.8415 1 -86 9406 2 -76.3492 Draw v [4 Ready 13 -46.6760 pie(t) 12.0000 11.9400 12 •'•871 13.5793 15.0390; 16.6029: 17.9660: 18.8405; 19.0139 18.3978 17.0550 15.1981 13.1593 11.3323 pHt) 11.4000 17.4140 23.4063 28.1767 30.6775 30.2346 26.7107 20.5746 12.8531 4.9690 -1.5133 -5.1900 -5.1105 -1.0029 0.2 yn = 1 5 mdot = -0.3 0.02 1 j k \2 1 15 15 Figure 11.13 AUtoShapee. \ V. P O H 4 Sf ■ • S ■ M MJM These are readily set out on a spreadsheet as shown in figure 11.13, which includes the dynamic path of the system from a starting value of (vo, 7Tq) = (12, 12). The system, however, now diverges from the fixed point in a counter-clockwise direction! Why is this? The matrix of the system is A = -0.85 -10 0.3 1 with characteristic equation X2 — 0.15A + 2.15 = 0 and complex roots, r, s a ± 8i, i.e. r = 0.075 + 1.4644/ s = 0.075 - 1.4644/ For discrete systems, stability requires8 < 1 y/a2 + B2 However, in this example ^Ja2 + fi2 = 1.4663 and so the system, illustrated in figure 11.13, is explosive. This example should act as a warning. It is not possible to attribute the same properties to discrete systems as occur in continuous systems. The more complex the system the more likely the discrete system will exhibit different properties from its continuous counterpart. 8 See section 3.8 and Azariadis (1993, pp. 36-8). 484 Economic Dynamics 11.3 Deflationary 'death spirals'9 At the time of writing (mid-2001), Japan was in a recession and the USA began to experience a serious downturn - enough for some economists to wonder whether a major deflation worldwide was likely. In explaining such a possibility, interest has returned to the concept of the liquidity trap. Not in the sense of the early literature that considered a low positive nominal interest rate so that the demand for real money balances became infinitely elastic at this value, but because the nominal interest rate cannot be negative. These two types of liquidity trap are conceptually different. The floor of zero on the nominal interest rate leads to what Groth (1993) has called a dynamic liquidity trap. Here we shall present a simplified version of the model outlined in Groth (1993) and similar to the one utilised by Krugman (1999). The model is in natural logarithms, except for all inflation rates and the nominal interest rate. (1) c ■ a + b(l - t)y C = : consumption (2) I = io — h(r — ite) y = : income (3) y = ■ c + i + g i investment (4) md = ky — ur r = nominal interest rate (5) ms = m — p 7te = expected inflation (6) md = ms md = demand for real money balances (7) TV = = a(y - yn) + 7te ms = supply of real money balances (8) fce = ß(lt - 7te) m = = nominal money supply p = price level yn = natural level of income it = inflation fte = ditejdt g, yn and m are assumed constant, as are all autonomous expenditures (a and io) and all parameters (b, t, h, k, u, a and /3). The first six equations are the familiar IS-LM model, equation (7) is the expectations augmented Phillips curve and equation (8) specifies adaptive expectations. The dynamics of the model is analysed in terms of (ms, 7Te)-phase space, i.e., we need to derive two equations of the form ms =f(ms, 7te) fte = g(ms, 7te) Although the algebra is a little tedious, it does allow us to investigate various numerical versions of the model. From equation (5), and noting m is constant, we have ms = —it and substituting equation (7) into this we have (11.13) ms =-[a(y - yn) + 7te] I am grateful to Christian Groth, University of Copenhagen, for drawing my attention to the literature on which this section is based. The dynamics of inflation and unemployment 485 From equation (7) we immediately have it — ite = a(y — yn), which on substitution into equation (8), gives ite = af3(y - yn) In order to eliminate income, y, in each dynamic equation, we require to solve the IS-LM component of the model embedded in equations (l)-(6). Combining (1), (2) and (3) we derive the IS-curve in exactly the same way we did in chapter 10. This is (11.14) a + i0 + h [1 - b(l - t)]y h (11.15) From equations (4), (5) and (6) we obtain the LM-curve -ms ky r ■ — u u Substituting equation (11.16) into equation (11.15) we derive an expression for equilibrium income (11.16) „ (a + i0 + g) + hite + (h/u)ms y = l-b(l-t) + (kh/u) Substituting equation (11.17) into equation (11.13) we obtain (11.17) -a(a + i0 + g) 1 - b(l - t) + (kh/u) ah l-b(l-t) + (kh/u) + ayn +1 a(h/u)ms l-b(l-t) + (kh/u) 7T which is a linear function of ms and ite. Substituting equation (11.17) into equation (11.14) we obtain it ■ af}(a + iQ + g) \-b(\-t) + (kh/u) af>hite 1 - b(\ - t) + (kh/u) + af>(Ji/u)ms 1 - b(l - t) + (kh/u) which is also linear in ms and ite. We shall simplify these linear equations by writing them in the form ms = A + Bms + Citt (11.18) (11.19) ite = D + Ems + Fit' Using these equations we can define the (ms, 7Te)-phase plane with isoclines ms = 0 and ite = 0. We shall now pursue this model by means of a numerical example. 486 Economic Dynamics Example 11.4 Consider the model c = 60 + 0.75(1 -0.2)v i = 430 - 4(r - 7te) y=c+i+g md = 0.25v - lOr ms = 450 — p k = 0.25 u m = 450 p yn = 2000 a = 0.1 b = 0.75 t i0 = 430 h £ = 330 a = 60 0.2 4 10 0 md = ms TV =0.1(v- 2000) + 7re Tte = 0.08(7T - 7te) ß = 0.08 Then ms = 36- 0.08m, - 1.87Te 7te = -2.88 + 0.0064m, + 0.064jre Setting ms = 0 and fce = 0 we derive the two isoclines = 0 7Te = 20 - 0.0444m, rce = 0 7Te = 45-0.1m, with fixed point (m*, 7Te*) = (450, 0). The two isoclines identify four quadrants, as shown in figure 11.14. To derive the vector forces in each quadrant, we note that ms > 0 implies ite < 20 - 0.0444m, therefore below m, = 0 and m, is rising while above it it is falling. Similarly, rce > 0 implies 7Te > 45 — 0.1m, therefore above fce = 0 and ite is rising while below it it is falling. The vector forces, therefore, indicate a counter-clockwise movement around the fixed point. To consider (local) stability, consider the linear system in terms of deviations from equilibrium, then m, = -0.08(m, - m*) - l.S(jre - O 7te = 0.0064(m, - m*) + 0.064(jre - O The dynamics of inflation and unemployment 487 Figure 11.15. v y /=y=2000 N NIS(Tle=-5) whose matrix is _ T—0.08 -1.8" ~ |_0.0064 0.064 _ with tr(A) = -0.016 det(A) = 0.0064 which indicates local stability.10 In fact, as Groth (1993) indicates, stability is guaranteed if /3u/m* < 1, and in the present example /3u/m* = 0.00178 and so stability is assured. Before continuing with this example, it is useful to display the results in the more familiar IS-LM model. The equations for the IS-curve and the LM-curve are IS: r = 205 -0.lv LM : r = -45 + 0.025v which intersect at the point (y*, r*) = (2000, 5) with it = ite = 0, as shown in figure 11.15. So far, however, we have not taken account of the nominal interest rate floor of zero. If the equilibrium interest rate is r = 0, then m^ = kyn = 500, which is equal to the money supply, ms. But if r = 0, then ite must be equal to minus the real rate of interest, where rreal = r — ite. Therefore in our numerical example it follows that ite = —5. This is illustrated by the dotted line in figure 11.15, which passes through point y = yn = 2000 for r = 0. The eigenvectors are —0.008 ± 0.0796; and since the real part is negative, the system is asymptotically stable. See chapter 4. 488 Economic Dynamics (11.21) (11.22) The resulting kink in the money demand curve at r = 0 results in a kink in both isoclines ms = 0 and fce = 0. To establish exactly where these kinks occur we note that the equilibrium interest rate for the general model is r = (k/u)(a + i0 + g) l-b(l-t) + (kh/u) (kh/u)ite + (h/u)(k/u) 1 - b(l - t) + (kh/u) mv + 1 - b(l - t) + (kh/u) i.e. r* = G + Hms + Jjte For our numerical example, this expression is r* = 41 -0.08m, + 0.2jre and so the relationship between ite and ms when r* = 0 is given by 7te = -205 + 0.4m, Equating equation (11.22) with each equation in (11.20) gives the kinks at the following values ms = 0 (ms, 7Te) = (506.3, -2.48) 7te = 0 (ms, 7te) = (500, -5) At these values the isoclines become horizontal, as illustrated in figure 11.16. Note in particular, that ire is equal to the real rate of interest that we established above, namely —5. Now return to equation (11.14) where jce = a^(y — yn).lt immediately follows that 7t 0 implies y = yn fce > 0 implies y > yn fce < 0 implies y < yn The dynamics of inflation and unemployment 489 But we established earlier that for jfe < 0 the economy is below the fce = 0 isocline. So the recessionary region is shown by the area below this isocline, as illustrated in figure 11.17, and identified by the shaded area. Consider a situation where the economy is in recession, and at point A in figure 11.17, where y < yn and there is excess capacity. Suppose the line marked Ti shows the trajectory of the economy. But at point B, the economy hits the nominal interest rate floor, and thereafter moves in the southeast direction and always away from the fixed point. It cannot get out of the dynamic liquidity trap. The output gap feeds expectations of deflation, and since the nominal interest rate cannot fall any further below zero, this implies a rise in the real interest rate. This in turn worsens the output gap. The economy falls into a deflationary spiral that it cannot escape. More significantly, raising the money supply to expand the economy will not alleviate the situation. Now consider an independent Central Bank's solution to the economy's problem at point A. Given the economy is in recession, it could expand the money supply. At point A the nominal rate of interest is positive. If it expands the money supply immediately, we may suppose the economy moves along trajectory T2. It passes into the corridor (what Krugman calls the 'window of opportunity') and can manoeuvre the economy to equilibrium. On the other hand, if it misses the corridor and follows trajectory Ti, then deflation passes the point of no return. What Krugman argues is that if the Central Bank increases monetary growth rapidly then trajectory Ti is more likely. It is interesting in this regard to comment on the behaviour of the European Central Bank (ECB) in April-May 2001. The USA was concerned about a recession and the Fed (Federal Reserve) lowered interest rates in a set of steps. The independent Bank of England also lowered UK interest rates. However, the ECB kept interest rates constant, i.e., refused to expand the money supply. Europe at the time was in a situation of below full capacity (y < yn), with high levels of unemployment, especially in Germany. If, then, the economy (Europe in this instance) 490 Economic Dynamics follows path Ti by the time the ECB decides to act, it may be too late. As Krugman (1999) says, conservative monetary policy may seem prudent and responsible to the European Central Bank today, just as it did to the Bank of Japan not long ago, but in retrospect that supposed prudence may look like disastrous folly. 11.4 A Lucas model with rational expectations In line with earlier sections, our aim in this one is to introduce a simple macro-economic model to illustrate how rational expectations are employed in macro-economic modelling. From the outset we need to be absolutely clear about variables for which expectations are formed. In particular, we need to specify the date the expectation was formed, and second the future time period about which the expectation is being formed. To be more precise, suppose we have a variable X about which expectations are being formed. If the expectation is made at time t, then we write Et to denote an expectation being formed at time t. But it is possible to formulate an expectation about X one period ahead, i.e., EtXt+i, or two periods ahead, EtXt+2, etc. In fact, we can formulate an expectation for any future time period. By the same reasoning, an expectation about Xt+\ may have been made two periods ago, i.e., Et_\Xt+\ is an expectation made at time t — 1 about variable X at time t + 1. So far we are simply specifying a notation to express expectations. No statement has been made about how such expectations are formed. Thus, if it denotes inflation, then Etitt+\ denotes expected inflation11 next period having been made in period t. Since prices pt are usually expressed in natural logarithms, as we shall be doing in this section, then Et7tt+i = Etpt+i — pt The model we shall investigate is yf = a0 + a\(mt -pt) + st a0 > 0, a\ > 0 yst = yn + biipt - Et-ipt) + vt b\ > 0 This model has a variety of new features that are worth commenting on. First, aggregate demand is the same as our earlier section but has a random component added to it. Second, the Phillips curve is in the Lucas form, i.e., the natural level of income is adjusted by deviations of prices from expected prices. Third, the aggregate supply also involves random shocks. Fourth, the random components are normally distributed with zero mean and constant variance. For those readers less familiar with stochastic equations, the random terms simply act like shocks to the AD and AS curves. On average, since their means are zero, the likely expected curve is the respective deterministic component. 11 Although it is common to write 7zet+l, this does not make it explicit when the expectation was formed. It is implicitly assumed to be at time t. (11.23) 37 = yst = yt £~iV(0,a£2), The dynamics of inflation and unemployment 491 First we solve the model under the assumption that expectations are given. Thus, we can express the equations in matrix form as follows 1 a\ ~yt~ ao + a\m + s, _1 -bx_ yn - b\Et_xpt + vt Using Cramer's rule we can solve for yt and pt a\b\Et-\pt a0b\ + a2yn a\b\mt yt =-—:— + «1 + b\ ao - yn a\ + b\ a\mt a\ + b\ + b\8t + a\vt a\ + b\ b\Et-ipt st-v, Pt = -— + a\ + b\ a\ + b\ a\ + b\ a\ + b\ These are the reduced form equations under the assumption that expected prices are exogenous. The next step in the rational expectations procedure is to take the expected value at time t—1 for the variable pt. In other words, the expectation of the variable p is derived in the same manner that determines the variable p itself. Thus Et-iPt = But Et-\Et a0-yn aiEt-\mt bxEt_xpt Et_xet - Et_xvt a\ + b\ a\ + b\ Et_\vt = 0, hence «i + b\ a\ + b\ a0-yn axEt_xmt b\Et_xpt Et-ipt = -—r- H--—--h Et-iPt = a\ + b\ a0 a\ +b\ a\ + b\ yn a\ + Et-xm, which is the rational expectations solution for Et_xPt-Now having solved for Et_xPt we can substitute this into the reduced form equations. Doing so, and simplifying, we obtain the solutions for y, and p, as follows axbx(mt - Et_xmt) bxst + axvt yt = yn-\--—:--1- ax + bx ax + bx ao ~ yn axm, - bxE,_xmt et - v, Pt =--1--—--h ax ax + bx ax + bx To see that this model is consistent with our earlier results, consider the following two cases: (11.24) (11.25) (i) constant money supply and correct expectations (ii) constant monetary growth and correct expectations. To analyse these two cases we first need to obtain the rate of inflation itt = pt — pt_ x ao - yn axmt - bxEt_xmt st - vt Pt =--1--—:--h Pt-i ax ax + bx ' a\ + bx ao-yn , axmt-x - b\Et-2mt-\ , St-i ~ vt-x + ax ax + bx + ax + bx 492 Economic Dynamics (11.27) But 7tt =Pt-Pt-\, hence fli(m,-m,_i) b\(Et-imt - Et_2mt-i) 7t t = -:--h + a\ + b\ a\ + b\ (et - et-i) - (v, - v,_i) a\ + b\ Under the condition that mt = m,-\ and Et-\mt = E,-2mt-\, then (et - st-\) - (v, - v,_i) (11.26) 7t-, a\ + b\ with no random shocks (et = st-i =0 and vt = vt_\ = 0) then 7tt = 0, which was the first model we analysed in section 11.2. Under the condition of constant monetary growth, X, which is expected, then mt - mt_i = X Et-\mt - Et-2mt-i = X so that a\X b\X (st - st-i) - (v, - v,-i) 7tt =--1---1-- a\ + b\ a\ + b\ a\ + b\ . (et- st-i) - (vt - vt-i) i.e. 7tt = X H-- «i + b\ with no random shocks inflation is equal to monetary growth, X, the result next analysed in section 11.2. It is worth summarising a number of features of this model. (1) Since axb\(mt - Et_imt) b\St + axvt yt = y* +-7-7-+ ——7— a\ +e>i a\ + b\ then the correct forecast on the part of market participants means that income can still deviate from the natural level in the short run, but only due to random factors either on the demand side or on the supply side. (2) The deviation of yt from yn depends not only on the level of the random elements, but also on: (a) The parameters of the economic system (both AD and AS). (b) The correctness of forecasting government monetary policy. Assuming no shocks (et = vt = 0) then income can still be above/below the natural level if forecasters underestimate/overestimate the money supply. (3) A positive monetary surprise (i.e. mt > E,-\mt) means a rise in yt, p, and 7tt. (4) Although pt includes forecast errors these are random in nature and so there are no systematic forecast errors. To see this, note ai(mt - Et_imt) st - vt pt - Et-ipt = -—--h a\ + b\ a\ + b\ If the money stock is constant (i.e. mt = E,-\mt), or if the money stock is forecasted correctly {Et_\mt = mt), or if the money stock itself is subject The dynamics of inflation and unemployment 493 to random shocks (which then means mt — Et-\mt is a random variable), then pt — Et-\pt is purely a random variable. Thus, the (mathematical conditional) expectation is E{pt - Et-ipt) = £fe)~fVf) = 0 a\ + b\ (5) It can be shown (see exercise 5) that any systematic component of a money supply rule has no bearing on the solution value of output, i.e., systematic elements of policy which are known have no impact on real output. (6) A systematic component of a money supply rule can have a bearing on the solution of pt, and hence on itt (see exercise 6). (7) The procedure here adopted for deriving the rational expectations is possible only if the reduced form equations can be derived. Where this cannot be done, then other procedures are necessary. (See Holden, Peel and Thompson 1985; Leslie 1993). (8) Most attention has been on the result derived in (1) indicating policy impotence with regard to influencing the level of real income. So long as the money supply is correctly forecasted (i.e. there are no monetary surprises), then income can deviate from the natural level only as a result of random shocks to either aggregate demand or aggregate supply. 11.5 Policy rules In the previous section we pointed out the possibility of policy impotence. Let us make this more precise. Consider some policy rule for the money supply. A variety has been considered - some active and some passive. An active policy rule is one in which the policy in period t depends on the performance of the economy in the previous periods. A passive policy rule is completely independent of recent economic performance. For our present analysis we shall consider policy rules based only on variables in the previous period. This in no way invalidates the conclusions. Let x denote the policy instrument used for monetary control12 and let q denote a vector of economy variables. Then an active policy rule takes the form mt=f(xt-i,qt-i) (11.28) where f(x, q) is nonstochastic and can be linear or nonlinear. A passive policy rule, on the other hand, can take the form mt = g{xt-i) (11.29) where g(x) is nonstochastic which can be linear or nonlinear. Return now to the result in the previous section for income, given in equation (11.25) aibi{mt - Et-\mt) b\St + axvt yt = y>* +-—r-+ ——7— a\ + b\ a\ + b\ 12 A typical choice for x is either the money base or the rate of interest. 494 Economic Dynamics (11.30) This result suggests that income will deviate from its natural level for two basic reasons: (1) deviation of mt from Et_\mt (2) random occurrences to either aggregate demand or aggregate supply. Here our concentration is on the first. Given either of the two rules above, so long as they are nonstochastic then Et-imt = E,-if(xt-i, q) =f(xt-i, q) Et-imt = E,-ig(xt-i) = g(xt-i) which implies that deviations mt — Et-\mt = 0 for both the active and the passive policy rule. It does not matter therefore whether the rule is active or passive nor whether it is simple or complex, the result is still the same. Deviations will be nonnegative only when forecasters get the government's policy rule wrong. This would especially be true when the government 'changed' the rule without announcing it. Under rational expectations theory, however, market participants would soon come to know the rule as they attempted to minimise their errors. If the policy rule involved a random component, wu which we again assume is normally distributed with zero mean and constant variance, then the two rules can take the form mt =f(xt-i, q) + wt mt = g(xt-i) + wt wherew, ~ iV(0, er2). Taking expectations at time t — 1, i.e., Et_\, we immediately arrive at the nonstochastic component since Et-\wt = 0. Hence mt - Et_imt = wt in both cases. The result on income is the same, namely aib\Wt + b\et + a\vt yt = yn + a\ + b\ The only reason why income deviates from its natural level is because of random shocks, including random elements to the policy rule. Before leaving this topic a warning is in order. The impotence of policy may appear to be solely because of the assumption of rational expectations. But this is not true. It also depends on the model chosen to illustrate the problem. In particular the result crucially depends on the assumption of completely flexible prices and a vertical long-run Phillips curve. (See Attfield, Demery and Duck 1985, chapter 4). 11.6 Money, growth and inflation In this section we turn to yet another model where inflation takes place in a growing economy. In such models it is common to establish that along the equilibrium growth path, the expected rate of inflation equals the rate of monetary expansion The dynamics of inflation and unemployment 495 minus the warranted rate of growth.13 Furthermore, in models involving rational expectations (which amount to perfect foresight models) then expected inflation equals actual inflation. The model we use is that given by Burmeister and Dobell (1970, chapter 6) and taken up by George and Oxley (1991). In this model agents have perfect foresight and all markets are assumed to clear continuously.14 The model assumes a fixed money growth rule of the type advocated frequently by Milton Friedman. The goods market is captured by the following set of equations Y = F(K, L) y = f(k) (11.31) C = cY where Y = cY + K + SK Y = output K = capital stock L = labour y = Y/L k = K/L C = consumption c = marginal propensity to consume K = dK/dt = net investment 8 = depreciation The final equation in (11.31) is no more than income equals consumption plus investment, and is the condition for equilibrium in the goods market. This condition is assumed to hold continuously. In line with our discussion of the Solow growth model in section 2.7, we can derive the following differential equation k = sf(k) - (n + S)k (11.32) where s = 1 — c n = L/L (the rate of growth of the labour force) Turning now to the money market we assume a constant monetary growth rule M — = X or M = XM (11.33) M The real demand for money per capita, m = M/L, is given by M m= — =PG(y,r) (11.34) L 13 For an analysis of the production function and the resulting differential equation, see section 2.7. 14 As we have pointed out elsewhere, these are two quite separate assumptions. 496 Economic Dynamics where m = per capita nominal money balances P = price level r = nominal interest rate Gy = dG/dy > 0 Gr = dG/dr < 0 The model is more easily analysed in terms of per capita real money balances, namely x = m/P, where m Equation (11.35) is assumed to hold continuously because the money market is assumed to be always in equilibrium. From equation (11.35) we assume we can derive the implicit function In this model there are only two assets: money and physical capital. The real rate of interest is the nominal rate, r, minus the rate of inflation, it (where ite = it). In equilibrium this will be equated with the marginal product of capital, f'(k), adjusted for the rate of depreciation, 8, i.e. r-it = f(k)-8 or r = f(k) -8 + it We now require to obtain a differential equation for x. Since x = m/P then x is (11.35) x = — = G(y, r) r = H(y, x) or x m --itx P i.e. x=p+ (f'(k) r 8)x But n)— = (X — n)m = (X — n)Px and so x = = (f(k) + X-8-n-r)x i.e. (11.37) x = (f(k) + X-8-n- H(f(k), x))x The dynamics of inflation and unemployment 497 We can establish the first isocline quite easily. In equilibrium k = k* and x = x* with the result that k = 0 and x = 0. Consider k = 0, then sf(k*) = (n + 8)k* and for positive k this is unique, as illustrated in figure 11.18. It is also independent of x. Hence, in (x, fc)-space this gives rise to a vertical isocline at k*, as shown in figure 11.19. For k < k* then k is rising while for k > k*,kis falling, leading to the vector forces shown in Figure 11.19. The isocline x = 0 is less straightforward, and in general is nonlinear. We shall pursue this isocline by means of a numerical example, using the figures in George and Oxley (1991, p. 218). y (n+S)k Figure 11.18. 498 Economic Dynamics Let Then Example 11.5 y = 2k0-25 In x = In y — 0.25 In r s = 0.2, S = 0.03, X = 0.05, n = 0.02 k = OAk025 - 0.05/c If k = 0 then /c(0.4/T0-75 - 0.05) = 0 i.e. k* = Oor/c* = 16. Before considering i = 0, we note that —0 25 * = yr .-. r = A"4 = (2£0-25)4;T4 = 16fcT4 and f(k) = 0.5/T0-75 Hence x = (0.5k-°15 - I6kx~4)x If x = 0 then (O^-1-'3 - 160^ = 0 So x = 0 if jc = 0 or (0.5/T1-75 - 16*-4) = 0 The second term leads to the isocline x = 2.3784£a4375 which is nonlinear, and is shown in figure 11.20. Figure 11.20. x 10 4 The dynamics of inflation and unemployment 499 Figure 11.21. k=0 Figure 11.22. ^ ^ ^ ^ ^ ^ ^ ^ j^t ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ A f f f f f f f l rfffffffffff ttttfffffffi ^% ^ ^ ^ ^ ^ & ^ ^ t t t t r r r r 'r t t r •fi ji 4 *a ^ ■* ~ "till Mr i ^ 1 I I J i j i I l 3 I } I I 3 3 I 3 J i \ ^ 1 2 \ \ M \ \ Et _ / Since i > 0 when x > 2.3784 &0-4375 then above the isocline vector forces are pushing x up; while below the isocline they are pushing x down. These forces are also illustrated in figure 11.20. The system's dynamics are shown in figure 11.21. The two isoclines intersect at the equilibrium point E, where (k*, x*) = (16, 8). What figure 11.21 shows is not only a nonlinear isocline, but that the equilibrium is a saddle-point solution whose stable arm is given by the equation satisfying x = 0. The full dynamics of the system are shown in figure 11.22, which uses Maple's phaseportrait command to 500 Economic Dynamics plot the vector forces and the x = 0 isocline, and which has then been annotated to complete the figure. 11.7 Cagan model of hyperinflation 11.7.1 Original model Cagan argued that during periods of hyperinflation the main determinant of the demand to hold money balances was the expected rate of inflation - the higher the rate of expected inflation the lower the demand to hold real money balances. Income and interest rates could be thought of as constant during periods of hyperinflation relative to the impact of expected inflation. The Cagan (1956) model consists of two equations, a demand for money equation (where nominal demand is equated with nominal supply) and an equation for adaptive expectations m(t) — pit) = —a7te(t) a > 0 (11.38) fce(t) = y[7t(t)-7te(t)] y>0 where m = In M = logarithm of nominal money stock p = In P = logarithm of prices 7Te = expected inflation it = inflation Also note that since p(t) is a logarithm then p(t) = 7t{t). In order to appreciate the dynamics of this model, differentiate the first equation in (11.38) with respect to time, holding the money stock constant at some level, then -p(t) = -OlTCe(t) = -ay [7r(0 - 7Te(t)] But p(t) = 77/(0 SO —7r(0 = -ay [77/(0 - 7Te(t)] -ayjte{t) y[m(t)-p(t)] i.e. 77/(0 = -= - 1 — ay 1 — ay or y [m(0 - pit)] (11.39) p(t)=ny pyn 1 — ay which is a first-order differential equation. The fixed point of equation (11.39) satisfies p(t) = 0, which means p(t) = m(t). Differentiating this result with respect to time t gives the typical monetarist result p(t) = 77/(0 = m(t), i.e., inflation is equal to the growth of the money supply. The The dynamics of inflation and unemployment 501 fixed point is stable only if the coefficient of p(t) is negative, which requires 1 — ay > 0 or ay < 1 The situation is shown in figure 11.23(a). Cagan's stability condition emphasises that with a highly sensitive demand for money function (a high), then stability requires inflationary expectations to adapt slowly to past inflation rates (y = l/a small). If this is not the case, then the system is unstable, as shown in figure 11.23(b), and the economy will exhibit either accelerating inflation or accelerating deflation depending on the initial price level. Now consider the Cagan model with rational expectations as represented by perfect foresight. Then m(t) — p{t) = —a7te(t) a > 0 jTe(t) = 7t(t) Hence i.e. m(t) — p(t) = —a7t(t) = —ap(t) (11.40) p(t) = --[m(t)-p(t)] (11.41) a Since the coefficient of p(t) is l/a > 0, then this system is globally unstable. The dynamics is captured in terms of figure 11.24. Assume the system is in equilibrium with po = p* with money supply mo. Now suppose there is a rise in the money supply from mo to m\. To restore equilibrium in the money market the demand for real money balances must also increase. In the present model this can occur only if expected inflation (equal to actual inflation) falls. But as the inflation rate falls the price level starts to fall (it < 0). With the money stock now fixed at mi, real money balances rise. To re-establish equilibrium in the money market means that 502 Economic Dynamics Figure 11.24. P (m0-p) 0 P {™-p) a et the demand for real money balances must fall, which means ixe = ix also falls. The result is a continuing fall in the price level. 11.7.2 Cagan model with sluggish wages15 This version of the model consists of the following equations, where we have suppressed the time variable m — p = ky — ajte a > 0 7te = TV = p where all variables are in natural logarithms except for inflation rates and where m = stock of money p = price level y = income 7Te = expected inflation n = labour w = money wages n = natural level it = p = inflation The first equation adds an income component to the Cagan demand for money equation. The second equation arises from a Cobb-Douglas production function, while the third equation is that real wages is equal to the marginal physical product of labour. The fourth equation is the Phillips curve while the fifth equation is the assumption of perfect foresight. (11.42) y = c + (1 - 0)n w — p = a — On w = ß(n — n) 0 < 0 < 1 ß > 0 This draws on Turnovsky (1995, section 6.2). The dynamics of inflation and unemployment 503 From the first and last equations we have m — p = ky — ap ky (m — p) p=--- a a From the second and third equations we have On = a — (w — p) a w — p y = C + (l-6) 1 = C + a(\-6) (\ (w-p) Substituting this result into the previous one, we have a k P = -a c + c + a(\-6) a(\-6) k (\ a m --+ a (w-p) m — p k (\ a 1 + -a a k (\ P-- a Turning to the second differential equation, w = f5(n — n) a w — p n =--- w w = p a w — p _ ----n R(a P w The model therefore comprises the following two differential equations a c + a(\-6) m + a k (\ a w ft (a 13 Or in matrix form P w k a c + a(l-9) m a <-B) + k (\ -a where the matrix of the system is rk (\ a 1 + -a k a + 1 a k (\ a 1 k (\- +--- a a w (11.43) P w (11.44) and where det(A) = —fi/aO < 0. Hence this system is represented by a saddle point solution. To illustrate the model consider the following numerical example. 504 Economic Dynamics Example 11.6 The equations are m-p = 0.25y - 2ire y = 3 + 0.75« w — p = 2 — 0.25« w = 0.3(n - 20) 7te = It = p If m = 20 then we have p = -8.875 + 0.875;? - 0.375w w = —3.6 + l.2p — l.2w The fixed point of the system is found by setting p = 0 and w = 0 with result (w*,p*) = (12.5, 15.5). The resulting isoclines are p = 0 p = 10.1429 + 0.4286w w = 0 p = 3 + w In matrix form the system is p "-8.875" + "0.875 -0.375" P w -3.6 1.2 -1.2 w and so the matrix of the system is [0.875 -0.375" L 1.2 -1.2 _ with tr(A) = —0.325 and det(A) = —0.6. Since det(A) < 0 then we have a saddle point equilibrium. The saddle point equilibrium is also verified by computing the eigenvalues. The eigenvalues are r = —0.954 and s = 0.629, and since these are opposite in sign we have a saddle point solution. The eigenvectors associated with the eigenvalues are "0.2009" 1 — "0.8361" "1.5241 0.9796 _ 4.8761 v = _ 0.5486 _ 1 To derive the saddle paths we need to consider (A — rl)vr = 0 and (A — sl)\s = 0. Consider r = —0.954 first. Then (A - rl)vr = 1.829 -0.375 1.2 -0.246 P -p "0" w — w 0_ Using the first equation, then 1.829(j? - 15.5) - 0.375(w - 12.5) = 0 p = 12.9371 +0.205w The dynamics of inflation and unemployment 505 Next consider s = 0.629, then (A - s\)ys = 0.246 -0.375" p-p "0" 1.2 -1.829 w — w 0 Again using the first equation, then 0.2460? - 15.5) - 0.375(w - 12.5) = 0 p = -3.5549+ 1.5244w i.e. S? : p= 12.9371 + 0.205w stable arm S° : p = -3.5549 + 1.5244w unstable arm The vector forces of this extended Cagan model are illustrated in figure 11.25. When p > 0 then p > 10.1429 + 0.4286w, so above the p = 0 isocline p is rising while below p is falling. Similarly, when w > 0 then p > 3 + w, and so above the w = 0 isocline w is rising while below w is falling. Consider now a one-off rise in the money supply, so m = 25. This has an impact only on the p = 0 isocline, which shifts up. The equation of this new isocline is p = 13 + 0.4286w resultingin a new equilibrium of (w*,p*) = (17.5, 20.5). The situation is illustrated in figure 11.26. Notice that dp* = dw* = dm = 5. This readily follows from the equation of w = 0. The new stable saddle path is S\S\ which passes through the fixed point Ei. But what trajectory does the economy follow? In this model we assume that prices are flexible and can 'jump' to the new stable arm immediately. This is an implication of the assumption of rational expectations with perfect foresight. Wages are assumed to alter continuously but sluggishly owing to wage contracts. The path the economy follows, therefore, is En->A->Ei as shown by the heavy line in figure 11.26. 506 Economic Dynamics 11.8 Unemployment and j ob turnover In order to introduce the dynamics of unemployment (and employment) we consider in this section a very extreme model in which we assume that at the ruling wage there is full employment in the sense that the number of jobs is matched by the number of households seeking employment. The working population, N, is assumed fixed and the number of jobs available is constant. At any instant of time a fraction s of individuals become unemployed and search over firms to find a suitable job. Let/ denote the probability of finding a job, i.e., the fraction finding a job. At any moment of time, if u is the fraction of the participating labour force unemployed, then s(l — u)N = individuals entering the unemployment pool fuN = individuals exiting the unemployment pool. The change in the unemployment pool, uN, is therefore given by the differential equation d(uN) (11.45) ——- = s(l - u)N-fuN 0 0, -=-- < 0 ds (s+f)2 df (s+f)2 In other words, the equilibrium unemployment rate - the natural rate in this model -rises the more individuals enter the unemployment pool to actively search for a The dynamics of inflation and unemployment 507 job; and falls the greater the job-finding rate. But this simple model says more than this about the equilibrium (natural) level of unemployment. It says that the level of u* occurs because individuals need to seek alternative employment and that the search for a new job takes time. The time path of unemployment is readily found by solving the differential equation (11.46). If u(0) = uq, then u(t) = u* + (u0 - u*)e-(s+f)t u* = s+f Since both s and/ are positive then this solution implies that unemployment tends to its equilibrium value over time. The situation is illustrated in figure 11.27. In this model concentration is on the level of unemployment. Of course, if N is fixed then employment, E, is simply E = (l—u)N or e = E/N = (l—u), where e is the employment rate. In order to lay the foundation for other dynamic theories it is worth noting that at any moment of time there will be an unemployment rate of u = U/N, and a vacancy rate of v = V/N. Since N is constant throughout we can concentrate on the rates u, v and e. 508 Economic Dynamics (11.48) (11.49) (11.50) At any moment of time there will be an unemployment rate u and a vacancy rate v, where those individuals who are unemployed are attempting to match themselves with the available vacancies. Since we have assumed that the number of jobs is matched by the number seeking employment, then u = v. The problem is one of matching the unemployed to the vacancies. Accordingly the literature refers to the matching rate or 'exchange technology' (Mortensen 1990). In other words, the unemployed and the jobs that employers are seeking to fill are 'inputs' into the meeting process. Let this be denoted m(u, v). Given m(u, v), then for such a meeting to take place there must either be some unemployment or some vacancies. More formally m(0,v) = m(u,0) = 0. Furthermore, the marginal contribution of each 'input' is positive, i.e., dmldu >0 and dmldv > 0. Following Diamond (1982) it is further assumed that the average return to each 'input' is diminishing, i.e., m/u and m/v diminishes with u and v, respectively. Finally, and purely for mathematical convenience, we assume that m(u, v) is homogeneous of degree k, so that m(u, v) = ukm(l, v/u) Using this analysis we can write the change in employment as the total match Nm(u, v) minus those losing a job s(l—u)N, i.e. dE d(eN) — = —-1 = Nm(u, v) - s(l - u)N dt dt de or — = e = m(u, v) — se dt Although the time path of employment, e(t), must mirror the time path of the unemployment rate, u(t), since e = 1 — u, the present formulation directs attention to the matching rate m(u, v). In general (Mortensen 1990), the equilibrium hiring frequency m(u, v)/u will be a function of the present value of employment per worker to the firm, q, and the employment rate, e. This can be established by noting that m(u,v) ukm(l,v/u) k_x - = - = u m(l, v/u) u u = (1 - e)k~lm(\, v/u) = h(q, e) The hiring function h(q, e) is a function of q since the value of v/u in (11.49) is that determined in equilibrium. In equilibrium, the return on filling a vacancy (mq/v) is equal to the cost of filling a vacancy, c, i.e. m(u, v) which gives (1 - e) k-\ cv/u m(l, v/u) which means that the hiring frequency is related to both q and e. Furthermore, we can establish from this last result that hq > 0 and he < 0 if k > 1 and he > 0 if The dynamics of inflation and unemployment 509 k < 1. Hence m(u, v) u = Kq, e) hq > 0, he < 0 if k > 1 he > 0 if k < 1 m(w, v) = a/i(g, e) = (1 — e)h(q, e) which in turn leads to the following equilibrium adjustment equation e = (1 — e)h{q, e) — se The profit to the firm of hiring an additional worker is related to q and the employment rate, e. i.e., jr(q, e), and will be different for different models of the labour market. This profit arises from the difference in the marginal revenue product per worker, MRPl, less the wage paid, w. If we denote the MRPl by x(e), then jr(q, e) = x{e) — w.16 However, the future profit stream per worker to the firm is rq = x(e) — w — s(q — kv) + q where rq represents the opportunity interest in having a filled vacancy and kv is the capital value of a vacant job, i.e., the present value of employment to the firm is the profit from hiring the worker less the loss from someone becoming unemployed plus any capital gain. Since in equilibrium no vacancies exist, then kv = 0 and so rq = 7t{q, e) — sq + q or q = (r + s)q — jr(q, e) To summarise, we have two differential equations in e and q, i.e. e = (1 — e)h{q, e) — se q = (r + s)q — 7t(q, e) Whether a unique equilibrium exists rests very much on the degree of homogeneity of the match function, i.e., the value of k in m(u, v) = ukm(l, v/u), and the productivity per worker x(e). 11.9 Wage determination models and the profit function In order to establish the properties of this differential equation system we need to have information on the partial derivatives of the profit function, i.e., itq and ite. But this in turn requires a statement about wage determination, and there are a variety of wage determination models. Here we shall consider just two: a market clearing model and a shirking model.17 16 In the case of the shirking model of wage determination MRPl = x(e) — ak, where X denotes the average number of times that the effort of each worker is checked and a the fixed cost required to do the checking. 17 This analysis draws heavily on Mortensen (1990) who also considers an insider-outsider model of wage determination. (11.51) (11.52) (11.53) 510 Economic Dynamics For instance, in the simplest model, Diamond (1971), all workers are identical and all have the same reservation wage and so the wage must equal the value of leisure forgone when employed, denoted b. Thus, w = b and the profit function is jr(q, e) = x{e) — b, with itq = 0 and ite < 0 if x'{e) < 0, i.e., if we have diminishing returns to labour employed. In the case of the shirking model, an individual can receive a wage w and if successful at shirking can receive a value b in leisure. If, however, the employer monitors the worker with a frequency X and fires them if they are found shirking, then the equilibrium wage must exceed b to ensure that the expected worker cost of shirking per period is no less than the benefit b. If ye denotes the expected present value of a worker's income when employed and yu the expected present value of a worker's future income when unemployed, then in equilibrium Hye -yu) = b Furthermore The first equation states that the opportunity interest from holding a job must equal the wage received plus the income she receives when unemployed, which she faces with probability s, plus any capital gain. The second equation states that the opportunity interest on being unemployed must equal the return from not working (including any unemployment benefit) plus the income she receives when employed, which she faces with an average match of m(u, v)/u, plus any capital gain. In equilibrium ye = 0 and w + s(yu - ye) + ye (ye - yu) + ý, u sb Hence In other words the wage rate is (11.54) The dynamics of inflation and unemployment 511 Using this result we can obtain the optimal values for X, w and 7t(q, e) (see exercise 6), i.e., 1 l X, = (b/a) 2 [r + s + h(q, e)] 2 w = b + (ab)2[r + s + h(q,e)]2 (11.55) l l 7t(q, e) = x{e) -b- 2(ab) 2 [r + s + h(q, e)] 2 Thus the optimal wage paid exceeds the market clearing wage and is an increasing function of h(q, e), so long as a > 0. We therefore have two alternative dynamic systems: Model 1 Market clearing (?): 2 I h(q, e) = (l- e)-3 (-) 3 hq > 0, he > 0 For e = 0 then c(se)3 (11.56) e = (1 — e)h{q, e) — se q = (r + s)q — x(e) + b Model 2 Shirking model e = (l-e)h(q,e)-se j j q = (r + s)q - x(e) + b + 2(ab)2[r + s + h(q, e)]2 Both systems are nonlinear and the dynamics depend on the value of k, and hence on the properties of h(q, e), and on the properties of x(e). An equilibrium steady state requires e = 0 and q = 0. So in both models equilibrium satisfies (1 - e*)h(q*, e*) = se* in other words, the hire flow must equal the turnover flow. The isocline e = 0 is called by Mortensen (1990) the employment singular curve and for x'{e) < 0 and k < 1 this curve is upward sloping. For instance, if m(u, v) = (uv)* so that m(u, v) = m{v/u)* with k = l/2 we derive the following results (see exercise 7). v = ( - )3 u3 q = l-e To pursue this analysis further, consider the following example in which we derive explicitly the isocline q = 0, called the value singular curve by Mortensen. 512 Economic Dynamics Let Example 11.7 l m(u, v) = (mv)4 -0.2 then x(e) = 3e~ r = 0.05, a = 0.1, c = 1, 5 = 0.2, = 3.2 _2 1 = (1 -e) 3^r3 and for e = 0 0.008e3 1 — e Considering q = 0 for each model we have Model 1 q = 0.25q - 3e~02 + 3.2 = 0 i.e. q = I2e~0-2 - 12.8 Hence the equilibrium level of employment is found from solving 0.008e3 ___02 1 - e = I2e~uz - 12.8 which can be done by means of a software package.18 The solution is found to be e* = 0.7212. The stylised situation is shown in figure 11.28. Recall that if you do not have a software package like Mathematica or Maple, you can use the Solver of Excel's spreadsheet. The dynamics of inflation and unemployment 513 Model 2 In the case of model 2, the shirking model, the q = 0 isocline is given by q = 0.25q - 3e~02 + 3.2 + 2[(0.1)(3.2)] 2 0.25 + 0.008e: l-e i.e. I2e -0.2 12.8 - (2.56)2 0.25 + 0.008e3 1 Qualitatively this leads to the same situation as in figure 11.28 except that the q = 0 isocline is below that of model 1, so leading to a smaller level of equilibrium employment. In fact, given the parameter values this is found to be e* = 0.3207. These equilibrium values are consistent with the equilibrium wages in the two models, which are: Model 1 w = b = 3.2 l l Model 2 w = b + (ab)2 [r + s + h(q, e)] 2 = 3.4836 Since the two models are qualitatively identical, we shall pursue here only the simple market clearing model illustrated in figure 11.28. 11.10 Labour market dynamics The situation we have developed so far for the simple market clearing model is a set of differential equations given by e = (1 — e)h(q, e) — se q = (r + s)q — x(e) + b which reproduces equations (11.56). The isocline e = 0 is upward sloping and q = 0 is downward sloping. Given the parameter values in example 11.7 we have the equilibrium point (e*, q*) = (0.7212, 0.0108) which is unique. The isoclines are given by e = 0 q = 0 furthermore implying implying when e > 0 then q > 0.008e3 q = 1- 1 — e q = I2e-02 - 12.8 0.008e3 1 so employment is rising above the e = 0 isocline and falling below this isocline. Similarly when q > 0 then q > \2e -0.2 12.8 so the present value of the future profit stream of the marginal worker is rising above the q = 0 isocline and falling below this isocline. These vector forces are 514 Economic Dynamics illustrated in figure 11.29 and indicate that the equilibrium point E is a saddle point solution. This property is also true for model 2 in which wages are determined within a shirking model.19 Given the saddle point nature of the equilibrium in all models the only solution trajectories are those which lie on the saddle path SS'. Suppose the present level of employment is eo, as shown in figure 11.30, then the only rational expectations trajectory must be the starting point (eo, qo) at point A and the path along SS' to point E. Any point below SS', such as point B, tends the system to zero present value profit stream from the marginal worker; or, such as point C, to an ever expanding profit, i.e., an unstable speculative bubble. The solution value so far is unique because we have assumed k < 1 andx'(e) < 0. A number of labour economists, however, have been investigating the situation of increasing returns in the production exchange process, which allows various possibilities for x(e). Consider the situation shown in figure 11.31 in which the e = 0 isocline is upward sloping while the isocline q = 0 takes a variety of shapes. There are now three solutions: a low (L), medium (M) and high (H) (e, q)-pair. The medium employment level is unstable. But for any level of employment such as eo in figure 11.31, there are two values of q consistent with the rational expectations behaviour of the system: point A on SiSj and point B on S2S2. In the case of point A, the system will tend to solution point L; while for point B, the system will tend to solution point H. It is also possible that in the neighbourhood of point M a stable limit cycle can occur. In fact, the insider-outsider model of wage determination also leads to the same qualitative model with a corresponding saddle-point solution (Mortensen 1990). The dynamics of inflation and unemployment 515 This second version of the model, exhibiting as it does multiple equilibria arising from increasing returns, illustrates a point we made in chapter 1. Rational expectations alone is not sufficient to determine outcomes. At eo points A and B are equally likely and yet the solution points L and H, respectively, involve quite different welfare implications. This suggests quite strongly that some policy coordination is necessary to fix the system on one or other of the solution paths. 516 Economic Dynamics Exercises (i) Solve the nonhomogeneous differential equation ^ = 8f(u) - 8(1 - $)7T at for 7r(0) = ttq. (ii) Show that for 8 > 0 and 0 < § < 1 the equilibrium 7T* is asymptotically stable. Given the model yt-i = 9 + 0.2(m,_! 7T, = a(y,_i -yn) a > 0 if = 5 and y„ = 6, use a spreadsheet to investigate the dynamics of the system for different values of a. For the system y = -1.85(y - y*) - 10(jre - 7te*) jte = 0.3(y - y*) establish that the characteristic roots are complex conjugate and that r, s = a ± Bi has a < 0. Show that if EtPt+1 - Et_xPt = (1 - k)(Pt - Et_xPt) 0 < k < 1 then (b \ 00 Consider the model yf = a0 + ai(mt-pt) yst =yn + bi(pt-Et-ipt) ydt=yst=yt where expectations are formed rationally. (i) Show that if money supply follows a systematic component such that mt = no which is correctly anticipated by market participants, theny = yn. (ii) Show that if mt = /x0 + zt where zt ~ N(0, a2), then aibiZt yt = yn + a\ + b\ Interpret this result. In the shirking model of wage determination the firm chooses the optimal value of k. Given rq = max{^:(e) — ak — w — sq + q] The dynamics of inflation and unemployment 517 (i) Show that A = I - I [r + s + h(q, e)] 2 (ii) Hence show that 1 1 w = b + (ab)2[r + s + h(q, e)]2 1 n(q, e) = x(e) — b — 2(ab)2[r + s + h(q, e)] 1 7. Given m(u, v) = (wv)4 and [m(u, v)/v]q = c (i) Show that -(f)1-* (ii) Hence show that 2 I e) = (l-e)-3 (^)3 (iii) Verify hq > 0 and he > 0. (iv) Show that the e = 0 isocline is given by: c(se)3 1 — e 8. Given m(u, v) = s/uv and [m(u, v)/v]q = c (i) Show that v = (1 - e)(c/q)2 (ii) Hence show that h(q, e) is independent of e. 9. For the numerical model (11.4) establish the new steady-state equilibrium values for k and x for each of the following, and illustrate diagrammati-cally the trajectory the economy follows (i) A rise in s from 0.2 to 0.3 (ii) A rise in n from 0.05 to 0.06 (iii) A rise in technology such that y = 5 k0'25 ■ 10. For the numerical model (11.4) establish the new steady-state equilibrium values for k and x for a rise in monetary growth from A = 5% to A = 6%. What trajectory does the economy traverse? 11. Consider the model t = 0.25 (l)c = ■■ a + b(l - t)y a = 100 b = 0.8 (2) i = z'o— h(r — 7te) io = 600 h = 2.5 (3)v = ■■ c + i + g g = 525 (4) md = ky — ur k: = 0.25 u = 5 (5) ms = m — p m = 700 P = 0 (6) md = ms a = 0.2 (l)jt = = a(y - yn) + ite yn = 3000 = f,(it - 7te) P = 0.05 518 Economic Dynamics (i) What is the fixed point of the system? (ii) Derive equations for the two isoclines (iii) Derive an equation for r* = 0 and hence establish the presence of a corridor. 12. In the Cagan model with perfect foresight we have the model m — p = —ajre 7te = It Given seigniorage is defined as AM and money grows at a constant rate X (i) Express In S in terms of X. (ii) Establish that the value of X which maximises seigniorage is _ 1 a Additional reading Further material on the contents of this chapter can be found in Attfield, Demery and Duck (1985), Azariades (1993), Burmeister and Dobell (1970), Cagan (1956), Carter and Maddock (1984), Diamond (1971, 1982), Frisch (1983), George and Oxley (1991), Groth (1993), Holden, Peel and Thompson (1989), Krugman (1999), McCafferty (1990), Mortensen (1990), Pissarides (1976, 1985), Scarth (1996), Sheffrin (1983), Shone (1989) and Turnovsky (1995). CHAPTER 12 Open economy dynamics: sticky price models In this chapter and chapter 13 we shall consider a number of open economy models that exhibit dynamic behaviour. We shall start with the very simplest - the income-expenditure model considered at the beginning of all courses on macroeconomics. This model assumes a fixed exchange rate. Simple as it is, it will allow us to set the scene and illustrate, in the simplest possible terms, how instability may occur, but is less likely to occur in an open economy in comparison to a closed one. We then do the same in the context of the IS-LM model we discussed in chapter 10, extending it to the open economy, but considering the situation under both a fixed and a flexible exchange rate. This forms the basis of the Mundell-Fleming model. This model was originally concerned with the relative impact of monetary and fiscal policy under fixed and floating exchange rate regimes, but with perfect capital mobility. It has become the standard model of open economy macroeconomics, and so we shall look into its dynamic properties in some detail - for models with some (but not perfect) capital mobility and for situations of perfect capital mobility. We shall find that the assumption about the degree of capital mobility is quite important to the dynamic results. As in earlier chapters, we shall be particularly interested in what happens out of equilibrium, and hence in the dynamic forces in operation in an open economy. 12.1 The dynamics of a simple expenditure model The simplest macroeconomic model for an open economy is the one where prices are assumed constant, and so we need not distinguish between real and nominal variables. Expenditure, E, is the sum of consumption expenditure, C, investment expenditure, /, government expenditure G, and expenditure on net exports, NX -where net exports are simply the difference between exports, X, and imports, M. We make four behavioural assumptions with respect to consumption expenditure, net taxes, investment expenditure and imports. Consumption expenditure is assumed to be a linear function of disposable income, where disposal income, Yd, is defined as the difference between income, Y, and net taxes, T, and we make a further behavioural assumption that net taxes is linearly related to income. Investment expenditure is assumed to be positively related to the level of income (we shall consider investment and interest rates more fully in the IS-LM dynamic model). Finally, we assume that imports are linearly related to the level of income. We treat 520 Economic Dynamics government spending and exports as exogenous. The definitions and behavioural equations of our model are, then E = C + I + G + NX C = a + bYd a > 0,0 < b < I Yd = Y — T (12.1) T = T0-tY 0 < t < 1 I = Io+jY j>0 M = Mq + mY 0 < m < 1 NX = X — M We now make a dynamic assumption about how the model adjusts over time. We assume that national income will adjust continuously over time in response to the excess demand in the goods market. More explicitly, we assume dY (12.2) — = X(E-Y) X > 0 dt In other words, when expenditure exceeds income, then there will be pressure in the economy for income to rise. This is because firms can sell all they wish, and with stocks running down then they will expand their production, take on more labour and so raise the overall level of economic activity. On the other hand, if expenditure falls short of income, then there will be a build up of stocks. Firms will cut back on production, possibly lay off workers, and generally lead to a reduction in economic activity. Equilibrium in this model is therefore defined to be a situation where income is not changing, or where E = Y. Substituting the equations in (12.1) into equation (12.2), we obtain the following differential equation (123) ^ =X(a-bT0+Io + G + X-M0)- X[l - b(l - t) -j + m]Y = XA — X[l — b(l - t) -j + m]Y where A denotes all autonomous expenditures. First consider equilibrium in this model. This requires dY/dt = 0, i.e. XA — X[l - b(\ - t) -j + m]Y = 0 (12-4) r A 1 — b(l — t) — j + m But our main concern is whether this equilibrium is stable or unstable. Since there is only one equilibrium, one fixed point, in this model then the situation will either be globally stable or globally unstable. Two situations are illustrated in figures 12.1 and 12.2. In figure 12.1 the growth line is negatively sloped. Hence, for income less than the equilibrium level, income will rise; while for income above the equilibrium level, income will fall. Hence, the fixed point is a stable one. In figure 12.2, on the other hand, the growth line is positively sloped. In this case, if income is below the equilibrium level then it will decline, and decline continually. If, on the other hand, income is above the equilibrium level, then income will rise continually. In other words, the equilibrium is globally unstable. Open economy dynamics: sticky price models 521 ___.AE Figure 12.1. Phase Y line It is clear from the differential equation in (12.3) that the slope of the growth line will be negative if b(l — t) + j; — m < 1. This also ensures that the simple expenditure multiplier, k, is positive, i.e. k = 1 1 — b(l — t) — j + m > 0 But there is no reason for b(l — t) + j — m < 1. Suppose investment responds quite readily to income, with j = 0.3. The marginal propensity to consume can during some periods be quite high. Suppose then, that b = 0.95. Further assume that the marginal rate of tax is t = 0.25. Finally, assume the marginal propensity to import is 0.2. Then b(l — t) +j — m = 0.8125, and since this is less than unity the economy exhibits stability. However, in the absence of trade (with m = 0), then we have b(l — t) +j = 1.0125, which is greater than unity and the economy would exhibit an unstable situation. Why is there this difference? Take the closed economy first, and assume that income begins initially below the equilibrium level. As illustrated in figure 12.2 at Y = Yq, the change in 522 Economic Dynamics Figure 12.2. Phase Y line income is negative and income would decline. The reason is that at this initial level of income, income exceeds aggregate expenditure (Yq > AEq). There is a build up of stocks and so firms lay off workers. Because h is high, they lay off quite a number of workers. But the loss in income of the workers means that they in turn have less disposable income. With a high marginal propensity to spend, this means a major cut in consumer spending. But this will itself lead to a further excess supply of goods, and so firms will respond with further cuts. Hence, the economy goes into continuous decline. If income had begun above the equilibrium level, at Y = Y\, with stocks running down, then firms would expand their production, disposable income would rise and consumption expenditure would rise. The economy would expand. Of course, once it reached full employment, then this would eventually manifest itself in rising prices (which we have assumed constant so far). Open economy dynamics: sticky price models 523 Why is the open economy different? Again begin with income below the equilibrium level, as typified by the situation in figure 12.1 for 7 = Yq. At this level of income, if the economy were not importing then there would be a running down of stocks as expenditure is in excess of income. But with the economy open, part of this demand is directed abroad and so the run-down in stocks at home is not as great. Hence openness tends to dampen the multiplier. Furthermore, the greater the marginal propensity to import the more likely stability because the greater the stabilising influence.1 We shall state this in the form of a proposition: PROPOSITION 1 The higher the marginal propensity to import, the more likely the economy will exhibit a stable equilibrium. We can consider this proposition in more detail by considering a simple numerical discrete model that we can investigate by means of a spreadsheet. The model is an extension of that provided in table 10.2 (but here we ignore the money market). In line with our discussion in chapter 10, we introduce dynamics into this discrete model by assuming that income in period t adjusts according to total expenditure in the previous period. Our model is Et = Ct + It + G0+NX Ct = 100 + 0J5Y? Y? = Yt- Taxt Taxt = -80 + 0.27, /, = 320 + 0.17, M, = 10 + 0.27, NXt =Xo-Mt Yt = Et-X where we assume government spending remains constant at Go = £330 million for all periods and exports remain constant at X§ = £440 million for all periods -unless either is shocked. Equilibrium income is readily found to be equal to 7* = £2500 million, which can be found from the resulting difference equation 7, = 1250 + 0.57,_! A rise in government spending from £330 million to £400 million results in a new equilibrium level of income of £2640 million. The movement of the economy over time in terms of the main variables is illustrated in table 12.1, which also includes the dynamic multiplier. What the table shows is that all variables gradually tend to their new levels as the multiplier impact comes closer to its final value. A marginal propensity to import of m = 0.3 (and with autonomous exports at £690 million) also leads to an initial equilibrium level of income of £2500 million. For the same rise in government spending from £330 million to 1 Exactly the same argument holds for the tax rate. The higher the marginal rate of tax the greater the stabilising influence on the economy, and the more likely the equilibrium is stable. 524 Economic Dynamics Table 12.1 Impact of a rise in government spending of £70 million t E, Y, Tax, Yd, Q It M, NX, 0 2500. .00 2500. .00 1 2570. .00 2500. .00 420. .00 2080. .00 1670 .00 570. .00 512. .00 -70. .00 0 .00 2 2605. .00 2570. .00 434. .00 2136. .00 1712. .00 577. .00 524. .00 -84. .00 1. .00 3 2622. .50 2605. .00 441. .00 2164. .00 1733. .00 580. .50 531. .00 -91. .00 1. .50 4 2631. .25 2622. .50 444. .50 2178. .00 1743. .50 582. .25 534. .50 -94. .50 1. .75 5 2635. .63 2631. .25 446. .25 2185. .00 1748. .75 583. .13 536. .25 -96. .25 1. .88 6 2637. .81 2635. .63 447. .13 2188. .50 1751. .38 583. .56 537. .13 -97. .13 1. .94 7 2638. .91 2637. .81 447. .56 2190. .25 1752. .69 583. .78 537. .56 -97. .56 1. .97 8 2639. .45 2638. .91 447. .78 2191. .13 1753. .34 583. .89 537. .78 -97. .78 1. .98 9 2639. .73 2639. .45 447. .89 2191. .56 1753. .67 583. .95 537. .89 -97. .89 1. .99 10 2639. .86 2639. .73 447. .95 2191. .78 1753. .84 583. .97 537. .95 -97. .95 2. .00 11 2639. .93 2639. .86 447. .97 2191. .89 1753. .92 583. .99 537. .97 -97. .97 2. .00 12 2639. .97 2639. .93 447. .99 2191. .95 1753. .96 583. .99 537. .99 -97. .99 2. .00 13 2639. .98 2639. .97 447. .99 2191. .97 1753. .98 584. .00 537. .99 -97. .99 2. .00 14 2639. .99 2639. .98 448. .00 2191. .99 1753. .99 584. .00 538. .00 -98. .00 2. .00 15 2640. .00 2639. .99 448. .00 2191. .99 1753. .99 584. .00 538. .00 -98. .00 2. .00 16 2640. .00 2640. .00 448. .00 2192. .00 1754. .00 584. .00 538. .00 -98. .00 2. .00 17 2640. .00 2640. .00 448. .00 2192. .00 1754. .00 584. .00 538. .00 -98. .00 2. .00 18 2640. .00 2640. .00 448. .00 2192. .00 1754. .00 584. .00 538. .00 -98. .00 2. .00 19 2640. .00 2640. .00 448. .00 2192. .00 1754. .00 584. .00 538. .00 -98. .00 2. .00 20 2640. .00 2640. .00 448. .00 2192. .00 1754. .00 584. .00 538. .00 -98. .00 2. .00 £400 million, the economy also gradually approaches its new equilibrium level of income (namely £2617 million) marginally sooner than with a lower marginal propensity to import (see exercise 4). This can also be seen in terms of figure 12.3, which captures the movement of the economy in the first few periods. It is clear that the new level of income is lower. The economy is inherently more stable the higher the marginal propensity to import. Of course, the corollary of this is that government spending has less influence on the domestic economy. Or, put another way, the more open an economy the greater the change in government spending necessary to achieve a given change in national income. 12.2 The balance of payments and the money supply We precede our discussion of open economy models under fixed and flexible exchange rates with a consideration of two interrelated variables: the balance of payments and the money supply. Both of these play a prominent role in the modelling to follow, and it is important that they are fully understood. This is quite important because we shall be setting up the models in real terms, even though we shall be assuming that prices at home and abroad are constant. This assumption of constant prices will be relaxed in chapter 13. 12.2.1 The balance of payments We define the balance of payments in real terms, bp, as the sum of real net exports, nx, and real net capital flows, cf i.e. (12.5) bp = nx + cf Consider first net exports. Open economy dynamics: sticky price models 525 Figure 12.3. Y=E ÄE[(m=0.2) t . ~— ^£,(^=0.2) AE2(m=0.3) Net exports in nominal terms, NX, is the value of exports minus the value of imports. Since we do not distinguish between goods in our modelling, then exports and imports in domestic currency are priced in terms of the domestic price level P. Making a clear distinction between real and nominal variables we have NX = Px- Pz where x is real exports and z real imports. Defining S as the spot exchange rate expressed as domestic currency per unit of foreign currency,2 and letting P* denote the price level abroad, then NX = Px- SP*z Dividing throughout by P to bring everything into real terms NX SP* or nx = x — Rz where R = SP* and where R define the real exchange rate, a variable which features prominently in later models.3 (12.6) This means that a rise is S denotes a devaluation of the domestic currency while a fall in S indicates a revaluation of the domestic currency. This variable denotes the competitiveness of home goods relative to those abroad. (12.7) (12.8) (12.9) (12.10) Economic Dynamics Real exports depend on income abroad and the real exchange rate (competitiveness). We assume a simple linear function x = x0+fR f>0 The constant can be considered as relating to income abroad, but we shall be holding this constant throughout. The second term captures competitiveness. Suppose the home currency depreciates, so S rises and hence so does R. Then domestic prices fall relative to those abroad and hence exports are stimulated. There is then a positive relationship between real exports and the real exchange rate.4 In the case of real imports we assume Rz = Zo + my — gR 00 where m is the marginal propensity to import, and real imports decline with a devaluation of the domestic currency (a rise in S). Combining the results we can now express real net exports as nx = (xq +fR) - (z0 + my - gR) = (xo - Zo) + (f + g)R - my = nx0 + (/ + g)R - my where nxo = xq — zo- Turning now to real net capital flows, cf, we assume that cf = cfo + v(r — r*) v > 0 where cfo is real net capital flows independent of interest rates, and r and r* are the nominal interest rates at home and abroad (which are here equal to the real rates since we shall be holding prices at home and abroad constant). At this point we do not need to consider expected changes in the exchange rate.5 The modelling is for a fixed exchange rate with no expected devaluation or revaluation, i.e., dSe jdt = 0. In chapter 13 we shall relax this assumption. Under this assumption, capital flows 4 We are assuming here that the Marshall-Lerner condition is satisfied. Differentiating net exports with respect to S we have where Ex and Ez are the export and import price elasticities, respectively. Assuming initially x = z, dNX dx ^ dz = P--SP*--P*z dS dS dS = P*xEx - P*zEz - P*z then dNX = P*x(Ex -Ez-l) dS dNX or dS > 0 if Ex + Ez > 1 i.e. \EX\ + \EZ\ > 1 5 Had we done so then the net capital flow equation would become cf = cfo + v(r - r* - S") Open economy dynamics: sticky price models 527 in real terms according to the uncovered interest differential, r — r*, with an inflow if r > r* and vice versa. Combining net exports and the capital flow equation, we arrive at an expression for the balance of payments where bpo = nxo + cfo. Balance of payments equilibrium occurs when bp = 0, a deficit when bp < 0 and a surplus when bp > 0. Recall that in chapter 10 we discussed the IS-LM model. We can now introduce a third relationship into the framework. Under the assumption of fixed exchange rates, the BP curve denotes combinations of income and interest rates for which the balance of payments is in equilibrium. Setting bp = 0 and expressing the result as r a function of y, we have hence the BP curve is, in general, positively sloped. But also note that if bp < 0 then In other words, below the BP curve the balance of payments is in deficit, while above the BP curve the balance of payments is in surplus. This information is displayed in figure 12.4. One way to account for the situations off the BP curve is to take a point on the BP curve, such as point A in figure 12.4, and move horizontally across to point B, moving to a point below the BP curve. Since the rate of interest remains constant, then so too do net capital flows. On the other hand, the rise in the level of income raises the level of imports, and hence worsens the current account. Hence, if initially the balance of payments was zero, then it must now be negative as a result of the worsening current account balance. Note also that this helps to explain why the BP curve is positively sloped. Given the deficit at point B, then this can be eliminated by raising the rate of interest. This will increase the net capital inflow, so bringing the balance of payments back into equilibrium (at point C). A similar argument applies to points above the BP curve. Care must be exercised in interpreting the BP curve, and points off it. The BP curve denotes combinations of income and interest rates for which bp = 0. In other words, we interpret external equilibrium as a situation where the current account is matched by the capital account but with opposite sign (i.e. nx = — cf or bp = 0). This is not the only definition of external equilibrium, but it is the one we shall use throughout this chapter. But what about the vectors of force either side of the BP curve? It is here we must be especially careful. If the exchange rate is fixed, then although there is some force acting on the market exchange rate, there is no change in the parity rate, and it is the parity rate that determines the bp = nx + cf = nx0 + (f + g)R -my + cf0 + v(r - r*) = bp0 + (f + g)R -my + v(r - r*) (12.11) (12.12) bpo + (f + g)R -my + v(r - r*) < 0 528 Economic Dynamics Figure 12.4. bp> 0 (surplus) reserves building up BP(i5p=0) bp<0 (deficit) reserves running down y intercept of the BP curve.6 If the economy is below the BP curve, then the balance of payments is in deficit. What is occurring in this situation is a running down of the country's reserves. Such a situation can persist in the short term, but not necessarily in the medium and long term. A similar situation arises in the case of a surplus, which occurs above the BP curve. Here the economy is adding to its reserves. The implication of the change in the reserve position of the economy depends on a number of factors. These include: (i) The change in the money supply resulting from a change in the reserves. (We shall take this up in the next sub-section.) (ii) The extent to which the authorities sterilise the impact on the money supply. (We shall also take this up in the next sub-section.) (iii) Whether a change in the parity rate is considered a possibility. What we observe here are asset market forces that act on the economy in the medium and long term. We shall return to these where appropriate. It is also worth noting some special cases: (i) If v = 0 then the BP curve is vertical at income level bpo + (f + g)R y =- m 6 The market exchange rate is determined by the demand and supply of foreign exchange, but the parity rate is set by the authorities. Under the Bretton Woods system, where exchange rates were fixed vis-á-vis the dollar, the market exchange rate could fluctuate either side of the parity rate by ± 1 per cent. Open economy dynamics: sticky price models 529 (ii) If v = oo there is perfect capital mobility and the BP curve is horizontal at r = r*. (iii) With some, but not perfect, capital mobility then the BP curve is positively sloped. However, there are two further categories which can be distinguished here, depending on the relative slopes of the BP and LM curves, which are both positively sloped: (a) the BP curve is steeper than the LM curve (b) the BP curve is less steep than the LM curve. Situations (i) and (ii) are illustrated in figure 12.5. Before we consider the IS-LM-BP model we need to take note of the fact that the expenditure function has now altered, since it includes net exports, and hence (a) Perfect capital immobility Figure 12.5. 0 BP (v=0) y (b) Perfect capital mobility r—r BP (v= oo) 0 y 530 Economic Dynamics (12.13) (12.14) so too has the IS-curve that we developed in chapter 10. To be specific e = a + [b(l — i) + j]y — hr + nx0 + (f + g)R — my = [a + nx0 + (f + g)R] + [b(l - t) +j - m]y - hr which leads to an IS curve of _ a + nx0 + (f + g)R _ [l-b(l-t)-j + m]y r~h h which indicates a change in the position of the IS curve and in its slope relative to that in the closed economy. 12.2.2 The money supply in an open economy In the model developed in chapter 10 the money supply was exogenous and fixed. In an open economy with a fixed exchange rate this is no longer the case. To see why this is so, we need to be clear on the definition of money for an open economy. Here we shall consider just the narrow definition of money, the money base, and denoted MO, and a broader definition of money supply, namely Ml. Specifically MO = Cp + CBR Ms = Cp + D where MO = money base Cp = cash held by the public CBR = commercial bank reserves at the Central Bank Ms = money supply (here Ml) D = sight deposits We shall further assume a simple money multiplier relationship between Ms and MO,7 i.e. (12.15) Ms = qM0 Return to the money base MO = Cp + CBR. This is the money base from the point of view of Central Bank liabilities. It is possible to consider a consolidated banking system from the point of view of the asset side.8 The money base from the asset side denotes Central Bank Credit, CBC, and international reserves, IR.9 Thus (12.16) MO = Cp + CBR = CBC + IR Hence (12.17) Ms = qMO = q(CBC + IR) 7 If Cp = cD and CBR = rD then Ms = cD + D = (1+ c)D and MO = cD + rD = (c + r)D. Hence, Ms/MO = (1 + c)/(c + r) or Ms = qMO. See Shone (1989, pp.147-51). 8 See Copeland (2000, pp. 120-8). 9 International reserves, IR, should not be confused with commercial bank reserves at the Central Bank, CBR. Open economy dynamics: sticky price models 531 Looking at the money base from the point of view of assets means that any change in the money base can occur from two sources: (i) Open market operations (including sterilisation) which operates through changes in CBC. (ii) Changes in the foreign exchange reserves that, under a fixed exchange rate, is equal to the balance of payments. Open market operations, ACBC, can usefully be thought of in terms of two components, (a) Open market operations which have nothing to do with the balance of payments, denoted fx, and which we shall refer to as autonomous open market operations, (b) A component that is responding to the change in the reserves. Let, then ACBC = ix- XAIR 0 < k < 1 (12.18) where k denotes the sterilisation coefficient. If k = 0 then regardless of the change in reserves, no sterilisation occurs; if k = 1, then we have perfect sterilisation. Thus, for a surplus on the balance of payments and a rise in the money base of AIR, the Central Bank reduces the money base by an equal amount. If the country has a deficit, leading to a reduction in the money base, then the Central Bank increases the money base by an equal amount. Where some, but not perfect, sterilisation occurs, then 0 < k < I. We are now in a position to consider the money supply in more detail. Ms = q(CBC + IR) AMs = q(ACBC + AIR) = q(/x - kAIR + AIR) = q[/x + (1 - k)AIR] Hence AMs /xq q(\- k)AIR = — + —--- (12.19) P P P Consider the two extreme cases: (i) ix = 0 and k = 0, no autonomous open market operations and no sterilisation AMs qAIR AIR -= -= q.bp where bp = - P P H P i.e. real money balances change by a multiple of the balance of payments (in real terms). A deficit leads to a fall in the money supply, while a surplus leads to a rise in the money supply. (ii) fx = 0 and k = 1 no autonomous open market operations and perfect sterilisation AMs -= 0 P i.e. under no autonomous open market operations and perfect sterilisation there is no change in the money supply regardless of the balance of payments situation. 532 Economic Dynamics It should be noted that retaining the assumption of an exogenous and constant money supply for an open economy is equivalent to assuming no autonomous open market operations and perfect sterilisation (i.e. case (ii)). In general, this is not true, and so for an open economy the money supply should be treated as endogenous. With no sterilisation, a deficit on the balance of payments under a fixed exchange rate leads to a shift left in the LM curve, while a surplus on the balance of payments leads to a shift right in the LM curve, as illustrated in figure 12.6. We are now in a position to consider the dynamics of monetary and fiscal policy under a fixed exchange rate. 12.3 Fiscal and monetary expansion under fixed exchange rates 12.3.1 Fiscal expansion In chapter 10 we have already established the vectors of force either side of the IS curve and the LM curve. Even with the IS curve re-specified for an open economy, as outlined in the previous section, the forces either side remain the same. In subsection 12.2.1 we established the deficit/surplus situation either side of the BP curve. In a dynamic context, the BP curve is the condition for which bp = 0. There is no equivalent to the adjustment functions in the goods market or the money market. Why is this? The exchange rate, S, is assumed to be fixed. Prices at home, P, and abroad, P*, are assumed constant. Hence the real exchange rate, R = SP*/P, is constant. Once income and interest rates are determined by the dynamics of the Open economy dynamics: sticky price models 533 Table 12.2 Parameter values and equilibrium points for figure 12.8 Equations: e = a + (f + g)R + b(l - i)y - hr+jy - my mj — Md/P — ky — ur ms = Ms/P = q(CBC0 + IR0) + q[fi + (1 - ; R = SP*/P nx = (x0 - z0) + (/ + g)R ~ my cf = cfo + v(r - r*) bp — nx + cf dy/dt — y — a(e — y) a > 0 dr/dt — r — /J(mj — ms) /J > 0 Intercepts and slopes: IS intercept = 27.924 IS slope = -0.3375 LM intercept — —6 LM slope = 0.5 BP intercept = 6.152 BP slope = 0.2 isi intercept = 32.924 ISi slope = -0.3375 LMi intercept = -8.791 LMj slope = 0.5 Parameter values: a = 43.5 m = 0.2 S = 1.764 / = 5 P = 1 P* = 1 g = 2 k = 0.25 x0 = 0 R = SP*/P = S u = 0.5 zo = 24 b = 0.75 CBC = 0 cfo = 20.5 t = 0.3 IRo = 3 v = 1 h = 2 q=l r* = 15 j = o X = 0, fi = 0 a = 0.05 B = 0.8 Solutions for point Eq: y = 40.506 r = 14.253 nx = -19.753 cf= 19.753 bp = 0 Solutions for point Ej: y = 46.476 Ms — 3 r = 17.238 bp = 1.791 Solutions for point E2: 3; = 49.809 Ms = 4.395 r = 16.114 bp = 0 goods market and the money market, the balance of payments is automatically determined from bp = bpo + (f + g)R -my + v(r - r*) But this is a short-run result. Why? Because a deficit leads to a fall in the reserves and hence to a reduction in the money supply, while a surplus leads to a rise in the reserves and hence to an expansion in the money supply, as explained in subsection 12.2.2. In the long run, with no sterilisation, interest rates and income will change until the deficit/surplus is eliminated. Geometrically, the LM curve will shift until it intersects the IS curve on the BP curve. To see this adjustment consider the following numerical model outlined in table 12.2, where CBCq and IRq denote the initial level for Central Bank credit and international reserves, respectively. In this model all three curves intersect at the same point, namely (y, r) = (40.506, 14.253). The situation is shown in figure 12.7, in which it should be noted that the BP curve is less steep than the LM curve. Consider a rise in autonomous spending by 10, e.g., because of a rise in government spending. The situation is shown in figure 12.8. In the short run the economy moves from equilibrium point Eo to Ei. Since the money market always clears, or is very quick to clear, then the economy moves along either the LM curve or close to it. At Ei the economy is in surplus. This follows from the new IS curve (whose 534 Economic Dynamics Figure 12.8. \ 46.476 y 40.506 49.809 Open economy dynamics: sticky price models 535 intercept and slope are indicated in the table 12.2), intersects the LM curve above the BP curve. This must be a short-run result. Assuming no sterilisation, then the money supply must rise as the balance of payments surplus leads to a rise in international reserves. Of course, equilibrium Ei will persist into the medium term if perfect sterilisation occurs and the money supply remains constant.10 In the case of no sterilisation, then the money supply will rise, the LM curve will shift right, and this will continue until the balance of payments becomes zero once again. This requires the final LM curve to cut the BP curve and the IS curve on the BP curve. This is shown by LMi in figure 12.8, where all three curves (ISi, LMi and BPo) all intersect at point E2. Given the fact that the money supply under these circumstances is endogenous, it is possible to establish that it must increase from Ms = 3 to Ms = 4.395. We have so far concentrated on the comparative statics. But what type of trajectory will such an economy follow? From our analysis so far we know that the money market is quick to adjust and the initial movement will be close to the initial LM curve. But as the economy goes into surplus the money supply will rise so shifting the LM curve right, income will adjust and the interest rate will be brought down because of the monetary expansion. The expected trajectory, therefore, is shown by the path indicated in figure 12.8 on which the arrow heads are marked. To the extent that any sterilisation takes place, then the actual path the economy follows will deviate from the trajectory shown. For instance, with perfect sterilisation and instantaneous clearing in the money market, then the path will be along LMo, between Eo and Ei. Under perfect capital mobility the BP curve is horizontal at r = r* (v = 00). The qualitative results are similar. The fiscal expansion leads to a rise in interest rates, which in turn leads to an immediate capital inflow. This will continue until the interest rate is brought into line with the interest rate abroad. During this process the balance of payments is in surplus because of the favourable capital account. The resulting surplus on the balance of payments leads to a rise in the money supply. However, since adjustment is quite quick the trajectory is either along the BP curve or close to it, as shown in figure 12.9. This will occur, however, so long as no sterilisation takes place. 12.3.2 Monetary expansion Consider next a monetary expansion for the model outlined in table 12.2. Suppose Central Bank credit is raised from zero to CBC = 2, raising the money supply from Ms = 3 to Ms = 5. This results in a new LM curve given by LMi r=-10 + 0.5v This cuts ISo with solution values y = 45.282 r = 12.641 bp = -2.567 10 The situation is more serious where the BP curve is steeper than the LM curve, then the rise in autonomous spending leads to a deficit. Perfect sterilisation will eventually lead to a running out of gold and overseas currency. 536 Economic Dynamics Figure 12.9. where the deficit on the balance of payments results because LMi cuts ISn below the BP curve, as shown in figure 12.10. In the long run, however, the deficit leads to a fall in international reserves and a fall in the money supply, shifting the LM curve back to LMn. The final equilibrium, E2, is the same as En. What about the dynamic path of this result? This is quite different from a situation of a fiscal expansion. To see this, consider a situation of instantaneous adjustment in the money market. The initial impact is a fall in the rate of interest to r = 10.253 (point A). This not only overshoots the short run equilibrium point Ei, but leads to a greater deficit because of the larger capital outflow. Two forces now come into operation. With a fall in the rate of interest investment rises which, through the multiplier, raises the level of income. Simultaneously, however, the deficit leads to a fall in international reserves and a fall in the money supply. The economy moves along a shifting LM curve, with a trajectory shown by the arrows pointing from position A to E2. How 'bowed out' the trajectory is depends on the extent to which the money supply is slow to fall as a result of the deficit (i.e. as a consequence of the fall in the level of reserves). Also, the trajectory will be more bowed out the more the Central Bank engages in any sterilisation in an attempt to move (or keep) the economy at point Ei. Perfect capital mobility does not change the qualitative nature of the results just discussed. The major difference is that the fall in the rate of interest below the world level r* will lead to a rapid outflow of capital and a more immediate reduction in the money supply. The economy is more likely to return to En more quickly, with a less 'bowed out' return trajectory. Open economy dynamics: sticky price models 537 Figure 12.10. 40.506 45.282 V This section verifies the Mundell-Fleming results: PROPOSITION 2 Under fixed exchange rates, fiscal policy is effective at changing the level of income but monetary policy is totally ineffective. Also, our analysis indicates three dynamic forces in operation: (1) pressure on income to change whenever expenditure differs from income (2) pressure on interest rates to change whenever the demand and supply of money are not equal (3) pressure on the money supply to adjust whenever there is a balance of payments disequilibrium, and where the extent of this change depends on the degree of sterilisation being undertaken by the Central Bank. The change in income is likely to be slow since the goods market takes time to adjust to any disequilibrium. On the other hand, interest rates are likely to adjust quite quickly. This supposition, however, assumes that the Central Bank is not attempting to control the rate of interest. The speed of the change in the money supply arising from changes in the balance of payments (the level of reserves) is likely to lie between that of the change in the rate of interest arising from capital 538 Economic Dynamics flows and that of the level of income. What is being hinted at here is the need for another adjustment function: namely, the rate at which the authorities change the money supply in response to a change in the level of reserves. This is a much more sophisticated analysis than we propose to investigate in this book. 12.3.3 Rise in the foreign interest rate A less frequently discussed shock, but an important one, arises from a change in the foreign interest rate. Consider a rise in the foreign interest rate, a rise in r*. Such a rise shifts only the BP curve in the first instance. From equation (12.12) we note that a rise in r* raises the intercept of the BP curve - in fact by exactly the rise in r*. The situation is shown in figure 12.11, where we start from the same initial position. The rise in the foreign interest rate from r* = 15 to r* = 18 shifts the BP curve up to BPi. Given this situation, the economy is still at Eo and so experiences a deficit on the balance of payments, bp = —3. Under a fixed exchange rate and no sterilisation, the deficit leads to a capital outflow and a fall in the money supply. LM shifts left to LMi and the economy settles down at Ei. But what trajectory does the economy follow on its path to Ei ? The immediate impact of the deficit is a fall in the money supply. If the money market adjusts instantaneously to this fall in the money supply, then the economy moves vertically up to point B on LMi. Thereafter, as income falls in response to the rise in the rate of interest, money demand falls putting pressure on the interest rate to fall until point Ei is reached. In this scenario the trajectory of the economy is Eo->B->Ei, and labelled trajectory Ti. What we observe is an overshoot of the domestic interest rate. If the shift in the LM curve is not complete or not so immediate, and depending on income adjustment, another trajectory is possible, shown by T2. Also note one other feature. The change in interest rate abroad is shown by the vertical distance between BPo and BPi while the rise in the domestic interest rate is less. Why is this? The fact that income is falling means a fall in imports and so net exports are rising. Hence the size of the capital outflow does not have to be as great. Even with perfect capital mobility, the same basic logic holds. The only difference is that eventually the domestic interest rate will rise in line with the foreign interest rate. With instantaneous adjustment of money supply to the resulting deficit and instantaneous adjustment in the money market, the interest rate will once again overshoot the final rise. In this section we have concentrated on the impact of changes in fiscal and monetary policy under a fixed exchange rate system - typical of the situation under Bretton Woods. However, since 1973 the exchange rate in most countries has been floating.11 In the next section we consider the IS-LM-BP model under the assumption that the exchange rate is allowed to float. However, we retain the assumption that prices at home and abroad are constant. This reminder is important. A change in the exchange rate is most likely to lead to a change in the price level in the medium and longer term. We shall take up this question of the link between 11 Britain floated its exchange rate in June 1972. Open economy dynamics: sticky price models 539 LM,(A&=0.663) Figure 12.11. 16.137 14.253 / / / i LM0(My=3) B i i i / „ BP,(r=18) >* T> /' i J- ^,BPc(r*=15) „ ■- i - 7- f~~~>-W <- bp=-3 i i , i / i / i / \is ' 40.506 y 34.925 the exchange rate and changes in the price level in chapter 13. Even so, what it does mean is that the real exchange rate, R = SP*/P is changing because of the change in S. 12.4 Fiscal and monetary expansion under flexible exchange rates 12.4.1 Fiscal expansion In this section we shall consider monetary and fiscal policy under floating exchange rates. In doing this we need to be clear on the implications of floating. With the spot exchange rate floating, S variable, and with fixed prices at home and abroad (P and P* constant), then the real exchange rate, R, will vary directly with S. Whatever is happening in the economy, the exchange rate will vary so that the balance of payments is always in equilibrium, bp = 0. If we assume instantaneous adjustment in the foreign exchange market and the money market, then the full impact of any change in the economy will initially fall on interest rates and the exchange rate. Only over time will the economy adjust to the situation as income changes. In terms of the diagrammatic treatment we have been using, the BP curve will shift continuously so that it always passes through the intersection between the IS and LM curves. Consider the initial situation depicted in table 12.2. Once again let autonomous spending rise by 10. This shifts the IS curve to ISi, as shown in figure 12.12, which 540 Economic Dynamics Table 12.3 Exchange rate values and equilibrium points for figure 12.12 5 = 1.764 5=1.764 150 r = 27.924 - 0.3375); Point E0 Point Ei LM0 r = -6 + Q.5y y = 40.506 y = 46.476 BP0 r = 6.152 + 0.2y r = 14.253 r = 17.238 bp = 1.791 151 r = 32.924 - 0.3375); 5 = 1.547 5=1.547 152 r = 32.165 - 0.3375); Point E2 LMo r = -6 + 0.5); y = 45.57 BP2 r = 7.671 + Q.2y r = 16.785 is the same ISi curve indicated in table 12.2. The resulting surplus on the balance of payments leads to an immediate appreciation of the domestic currency. The BP curve shifts up, and the resulting appreciation results in the IS curve shifting left to IS2. The final results are set out in table 12.3 and illustrated in figure 12.12. Our discussion, however, concentrates on the comparative statics. Let us for a moment turn to the dynamics of adjustment. In doing this we shall, as already indicated, assume instantaneous adjustment in all asset markets (money and foreign exchange), but slow adjustment in the goods market. The initial impact of the fiscal expansion is to move the economy to point Ei, with a trajectory moving along LMo from Eo to Ei, as shown in figure 12.12. Because of the resulting surplus, the domestic currency appreciates shifting the BP curve up to BPi. It should be noted Open economy dynamics: sticky price models 541 that BPi passes through point Ei, which it must do to eliminate any surplus on the balance of payments. The appreciation, however, leads to an appreciation of the real exchange rate, a fall in R, which leads over time to a reduction in net exports. As net exports decline, so too does income through the multiplier impact. As income falls, so too does the demand for money, and this leads to a fall in the rate of interest. What we observe, since the money market is continuously in equilibrium, is a movement along LMn from Ei to E2. As the interest rate falls, however, the amount of net capital inflows declines and so the exchange rate must depreciate. This shifts the BP curve down from BPi to BP2 which occurs as the IS curve shifts from ISi to IS2. In other words, in the (y, r)-plane, the economy gradually moves down LMn from Ei to E2, establishing itself at the final equilibrium point E2. The most likely trajectory, therefore, is a movement along LMo from Eo to E2 as all these forces take effect. There is some difference in the results for the situation of perfect capital mobility. This is illustrated in figure 12.13. We can be brief because the formal analysis is similar. The fiscal expansion shifts IS to ISi and the economy from Eo to Ei. The domestic currency appreciates as a result of the balance of payments surplus, shifting the BP line up to BPi. The resulting appreciation leads to a fall in net exports shifting IS left. As this takes place, income falls, interest rates fall, and the exchange rate depreciates, with the economy moving down the LM curve from Ei to E2 = Eo, and with the situation returning to its initial position, with no impact on the level of income. Two observations are worth noting about all these results, which we shall put in the form of two propositions. The first is the typical Mundell-Fleming result concerning a fiscal expansion under the assumption of a floating exchange rate; the second proposition is in the spirit of Dornbusch and overshooting. 542 Economic Dynamics PROPOSITION 3 Under flexible exchange rates, fiscal policy is effective in changing the level of income where there is some degree of capital mobility, but totally ineffective where there is perfect capital mobility. PROPOSITION 4 Under flexible exchange rates, a fiscal expansion leads to an overshooting of interest rates and an overshooting of the exchange rate, and this result holds with some degree of capital mobility or with perfect capital mobility. The important results are those with regard to some degree of capital mobility, since this is likely to capture the real world. A fiscal expansion would lead to a rise in income, a rise in interest rates and an appreciation of the domestic currency. This would be followed by interest rates falling, income falling and the exchange rate depreciating - but all such that the initial impact outweighs the secondary impacts. 12.4.2 Monetary expansion A monetary expansion under imperfect capital mobility and under perfect capital mobility is illustrated in figures 12.14 and 12.15, respectively. The adjustment is similar in both cases. However, figure 12.14 illustrates a numerical example. Figure 12.14. 40.506 45.282 46.582 y Open economy dynamics: sticky price models 543 Table 12.4 Equilibrium points for figure 12.14 IS, LM and BP curves Solution values 150 r = 27.924 - 0.3375); E0 y = 40.506 Ei y= 45.282 LM0 r = -6 + 0.5y r = 14.253 r = 12.641 BP0 r = 6.152 + 0.2); 5= 1.764 bp =-2.561 LMi r=-10 + 0.5); E2 y = 46.582 151 r = 29.013 -0.3375); r= 13.291 BP2 r = 3.975 +0.2); 5 = 2.075 Which illustrates the result of increasing the money supply by 2. The relevant information is given in table 12.4 A rise in the money supply shifts the LM curve right to LMi and moves the economy from point Eo to point Ei. At Ei the balance of payments is in deficit. Since the exchange rate is flexible and adjusts instantaneously, it will depreciate, shifting the BP curve down from BPo to BPi, where it intersects both the IS curve and LM curve at point Ei. The depreciation leads to a rise in the real exchange rate and hence to a stimulus to net exports. This leads to a shift right in the IS curve. In terms of figure 12.14, this will be to IS i, and the resulting rise in the rate of interest leads to an appreciation of the exchange rate, but not enough to swamp the original depreciation. The economy accordingly moves to point E2 (the intersection point between ISi, LMi and BP2). The situation in figure 12.15 is somewhat similar. The depreciation leads to a shift right in the IS curve to ISi, but this will cut the LMi curve on the original BP curve because interest rates will have to be brought back into line with world interest rates, which is accomplished by an expected appreciation of the currency, which returns BPi to BPo. Again we have concentrated on the comparative statics. But what will the trajectory of the economy look like in the short run and in the long run? The analysis is similar for both figure 12.14 and figure 12.15. The immediate impact of the monetary expansion is a sharp drop in the rate of interest, to point A on LMi. This 544 Economic Dynamics is because the money market adjusts immediately, while the goods market is yet to alter. But there is another immediate result. The sharp fall in the rate of interest leads to a major depreciation of the exchange rate. There will be another BP curve (not shown) that passes through point A. As the goods market adjusts to the lower interest rate, stimulating investment, and through the multiplier stimulating the level of income, the economy will move to point Ei, the movement taking place along LMi and the BP curve continuously adjusting upwards until BPi is reached. In the longer run, however, the depreciation which originally occurred will begin to shift the IS curve because of the stimulus to net exports. This will lead to a further movement along LMi and a further shift up in the BP curve until the economy moves to point E2. Again we arrive at two propositions: PROPOSITION 5 Under flexible exchange rates, monetary policy is effective in changing the level of income, and the effect is greater the greater the degree of capital mobility. PROPOSITION 6 Under flexible exchange rates, a monetary expansion leads to overshooting of interest rates and overshooting of exchange rates, and the less the degree of capital mobility the greater the overshooting of the exchange rate. The important results are those with regard to some degree of capital mobility, since this is likely to capture the real world. A monetary expansion would lead to a rise in income, a fall in interest rates and a depreciation of the domestic currency. This would be followed by interest rates rising, income rising and the exchange rate appreciating - but all such that the initial impact outweighs the secondary impacts. 12.4.3 A rise in the foreign interest rate Finally, consider the situation where the foreign interest rate is increased under floating. As earlier, the initial impact is to raise the BP curve by the amount of the increase. The initial situation, point Eo in figure 12.16, now represents a deficit on the balance of payments. This leads to a depreciation of the domestic currency, which improves competitiveness. The improvement in competitiveness stimulates net exports, so shifting IS to the right (to ISi) and BP down to BP2 in figure 12.16. The final equilibrium is at Ei. But what is the trajectory of the economy over the adjustment period? As the depreciation stimulates net exports shifting IS right and BP down, the economy will move along LMo, since the money market clears in every period. The economy will have a trajectory along LMo between Eo and Ei, shown by the arrows. We arrive, then, at the important conclusion that under a floating exchange rate the economy's adjustment exhibits no overshooting. The same basic conclusion holds with perfect capital mobility, except that the final equilibrium must have the domestic interest rate equal to the new (higher) foreign interest rate, the trajectory remains along LM with no overshooting. Open economy dynamics: sticky price models 545 r LM0 .-BP, ' BP2 Figure 12.16. IS„ 0 y 12.5 Open economy dynamics under fixed prices and floating So far we have concentrated on the comparative statics with some reference to the dynamics. Keeping within the simple linear model, let us consider the dynamics in more detail. We begin by stating the adjustment functions for the three markets explicitly, namely Goods market y = a(e — y) a > 0 Money market f = 3(md - ms) ft > 0 (12.20) Foreign market S = y{bp) y > 0 To simplify we set P = P* = 1 so that R = S and hence e = (a + nx0) + [b(l -t)+j- m]y - hr + (f + g)S giving y = a(a + nxo) + a[b(l — t) +j — m — l]y — ahr + a(f + g)S Equilibrium in the goods market is where y = 0 which no more than traces out the IS curve in (y, r)-space. If y > 0 then e > y and r is below the value on the line y = 0. Hence, below and to the left of y = 0 then y is rising while above and to the right income is falling, as shown in figure 12.17(a). or y = A0 + Axy + A2r + A3S (12.21) where An = a(a + nxo) Ax = a[b(l - t) + j - m - 1] A2 = —ah A3 = a(f + g) 546 Economic Dynamics Figure 12.17. Ql-_ y The money market is unchanged. Money market equilibrium is where r = 0 which traces out the LM curve in (y, r)-space. If r > 0 then Mj/P > Ms/P and r is below and to the right of the value on the f = 0 line. Hence, below and to the right of f = 0 then r is rising, while above and to the left r is falling, as shown in figure 12.17(b). The greater the value of 6 the faster interest rates rise or fall to clear the money market. Finally, the foreign exchange market is in equilibrium when bp = 0 or S = 0, which simply traces out the BP curve in (y, r)-space. Here we need to be careful. Open economy dynamics: sticky price models 547 Below 5 = 0 then bp < 0 and 5 is rising representing a depreciating of the domestic currency, resulting in the 5 = 0 line shifting down, as shown in figure 12.17(c). The fact that the 5 = 0 line shifts should be clear. The BP curve is drawn in (y, r)-space for a fixed exchange rate. When the exchange rate varies, which it will do for either a deficit or a surplus, then the result is a shift in the BP curve. However, there is also a shift in the IS curve. A depreciation of the domestic currency shifts IS right while an appreciation shifts it left - assuming the Marshall-Lerner conditions are satisfied.12 The higher the value of y, the more the BP curve shifts for any given deficit or surplus. Turning to the money market we have f = P(md -ms) ft > 0 = f5(ky — ur — mo) i.e. r = —fimo + f3ky — flur or where f = B0+B1y + B2r (12.22) B0 = -fimo Bi = pk B2 = -pu Finally, for the foreign exchange market 5 = y(bp) = y[bp0 + (f + g)S - my + v(r - r*)] = y(bpo — vr*) — ymy + yvr + y(f + g)S or where 5 = Co + Ci y + C2r + C35 (12.23) C0 = y(bpo - vr*) C\ = —ym C2 = yv C3 = y(f + g) Our model, then, amounts to three differential equations y = A0 + A\y + A2r + A3S r = B0+Biy + B2r (12.24) 5 = C0 + C\y + C2r + C35 The fixed point is where y = 0, r = 0 and 5 = 0 and can be solved for the equilibrium values y*, r* (not the interest rate abroad)13 and 5*. See n. 4, p. 526. There should be no confusion between the foreign interest rate and the equilibrium interest rate both being referred to as r*. 548 Economic Dynamics It is clear from the specification of the initial adjustment functions that the three parameters a, /3 and y have no bearing on the existence of a fixed point. What they do have a bearing on is the dynamic path or trajectory of the system from some initial value. Such a trajectory is less straightforward than any we have encountered so far because in (y, r)-space the trajectory is also governed by the movement of the exchange rate. The fiscal expansion we outlined in the previous section illustrates the problem. Suppose we start from an equilibrium. A fiscal expansion will shift the y = 0 line to the right. Interest rates will be pushed up and there will be a capital inflow resulting in a balance of payments surplus. The extent of the interest rate rise depends on the value of /3. The resulting surplus on the balance of payments leads to an appreciation of the domestic currency. The extent of the appreciation depends on the value of y, which in turn will influence the trajectory arising from changes in the rate of interest and changes in the level of income. Finally, the changes in income will be governed by the parameter a. The difficulty, of course, is that we are attempting to reduce a three-variable problem into a two-dimensional plane. To appreciate some of the difficulties suppose we have the parameter values given in table 12.2 along with a = 0.05, p = 0.8 and y = 0.0001. The fixed point is (y*, r*, 5*) = (40.506, 14.253, 1.764) Now let autonomous spending rise by 10 as before. We have already established (see table 12.3) that the new fixed point is (y*, r*, 5*) = (45.570, 16.785, 1.547) But is this new fixed point attained? The first thing we note is that Eo becomes our initial position and the dynamics of the system are governed by point E2. The differential equation system associated with point E2 is y = 2.675 - 0.03375v - O.lr + 0.355 r = -2.4 + 0.2y - 0.4r 5 = -0.00185 - 0.00002v + O.OOOlr + 0.00075 whose trajectory we can establish using a software package. One possible trajectory is shown in figure 12.18. The trajectory must begin in the shaded triangle and move anticlockwise. But there is nothing in the qualitative analysis precluding the system overshooting and spiralling towards E2 (or even away from E2!). Such a possibility is very dependent on the reaction coefficients a, /3 and y. For instance, with the same fiscal expansion but now a = 0.1 (higher than before) then the trajectory lies outside that for a = 0.05, as shown in figure 12.19. This should not be surprising because a higher a is indicating a greater response on income in the goods market for any given level of excess demand. On the other hand, a higher value for /3 (say yS = 1.5 rather than 0.8) leads to a trajectory inside that for p = 0.8, as shown in figure 12.20. Again this should not be surprising Open economy dynamics: sticky price models 549 since a higher value for B will push the trajectory towards the LM curve. Notice, however, that the trajectory in figure 12.19 overshoots the equilibrium E2 while it is difficult to see whether this is the case in figure 12.20. We have not considered the reaction coefficient y. The greater y the more the BP curve and IS curve shift for any given level of balance of payments disequilibrium. The greater the value of y the more likely the system over-reacts and equilibrium E2 not attained. By way of example, compare the following two situations for a fiscal change. The first situation is as before, while situation II has a higher 14 Recall that with instantaneous adjustment in the money market the system would move along the LM curve. 550 Economic Dynamics level of y, with the exchange rate reacting significantly to balance of payments disequilibria. Situation I a= 0.05, B = 0.8, y = 0.0001 y = 2.675 - 0.03375y - O.lr + 0.355 r = -2.4 + 0.2y - 0.4r 5 = -0.00185 - 0.00002y + O.OOOlr + 0.00075 Situation II a= 0.05, B = 0.8, y = 0.05 y = 2.675 - 0.03375y - O.lr + 0.355 r = -2.4 + 0.2y - 0.4r 5 = -0.925 - O.Oly + 0.05r + 0.355 As can be seen from figure 12.21, the trajectories for these two situations are quite different. More significantly, although point E2 exists, it is not attained in situation II. Why is this? We noted that the fiscal expansion led to a surplus and to an appreciation. The resulting rise in income led to a subsequent depreciation. But this is swamped in the present situation and the appreciation begins to dominate the dynamics pushing the system along the dotted line trajectory in figure 12.21. Of course, these parameter values are purely illustrative. But they act as a warning in the present example that the comparative statics is not sufficient to establish Open economy dynamics: sticky price models 551 the likely trajectory of the system. It also suggests that the larger y the greater the possibility that the system is dynamically unstable. Exercises 1. Set up the same numerical model as in section 12.1, but have the adjustment lag Yt = Et_2- Compare your results for the time path of Yt with that in table 12.1. 2. Consider the simple dynamics of section 12.1 with exactly the same parameter values but with the following alternative lags for the import function (take each one separately): (i) Mt = 10 + 0.27,_i (ii) Mt = 10 + O.lEt-x 3. Show that ANXt = -mktAG and that lim mkt = mk where Xt = X0 and Mt = M0 + mYt. 4. Consider the numerical model in section 12.1 for three alternative marginal propensities to import: m = 0.2, m = 0.3 and m = 0.4, all other parameters constant. (i) Show that the three total expenditure lines emanate from the same point on the vertical axis but that their slopes differ by deriving the equation for each total expenditure curve. (ii) Obtain the equilibrium income in each case. (iii) Assuming income begins at Yq = £2000 million, plot on the same graph the path of income over 10 periods for each marginal propensity. (iv) What conclusions do you draw from your analysis? 5. Using the data in table 12.2, consider the new equilibrium income and interest rate for the following changes, assuming no sterilisation, constant prices and a fixed exchange rate. Draw each situation and the likely path to the new equilibrium. (i) Rise in a from 43.5 to 50.0. (ii) Fall in IR (hence fall in the money supply) from 3 to 2. (iii) Devaluation of the exchange rate from 1.764 to 2. 6. Using the data in table 12.3, consider the new equilibrium income and interest rate for the following changes, assuming no sterilisation and that P and P* are fixed. Draw each situation and the likely path to the new equilibrium. (i) autonomous spending rises by 20 and under floating S = 1.33. (ii) Ms rises from 3 to 4, i.e., LM becomes r = —8 + 0.5y and under floating 5= 1.92. 552 Economic Dynamics 7. Consider the following discrete version of the numerical model of section 12.5, where again the figures refer to point E2. yt+l -yt = 2.675 - 0.03375v, - O.lr, + 0.355, r,+1 - r, = -2.4 + 0.2v, - 0Art S,+1 - S, = -0.00185 - 0.00002v, + 0.0001r, + 0.00075, (i) Set the system up on a spreadsheet and show that (v*, r*, 5*) = (45.570, 16.785, 1.547) is an equilibrium. (ii) Show that this equilibrium is not attained from the initial position (yo, r0, So) = (40.506, 14.253, 1.764). (iii) Is the equilibrium attained for initial values very close to the equilibrium? 8. For situation II in section 12.5, the equivalent discrete model is yt+1 -y, = 2.675 - 0.03375v, - O.lr, + 0.355, r,+1 - r, = -2.4 + 0.2v, - 0.4r, 5,+1 - 5, = -0.925 - O.Olv, + 0.05r, + 0.355, Given (yo, tq, So) = (40.506,14.253,1.764) show that point E2 represented by(v*, r*, 5*) = (45.570,16.785,1.547) is not attained and that the system is explosive. Additional reading Additional material on the contents of this chapter can be obtained from Copeland (2000), Dernburg (1989), Gapinski (1982), Gärtner (1993), Karakitsos (1992), McCafferty (1990), Pilbeam (1998) and Shone (1989). CHAPTER 13 Open economy dynamics: flexible price models Since the advent of generalised floating in 1973 there have been a number of exchange rate models, most of which are dynamic. In this chapter we shall extend our discussion of the open economy to such models. Besides having the characteristic of a flexible exchange rate they also have the essential feature that the price level is also flexible, at least in the long run. This is in marked contrast to chapter 12 in which the price level was fixed. The models are often referred to, therefore, as fix-price models and flex-price models, respectively. The majority of the flex-price models begin with the model presented by Dornbusch (1976). Although the model emphasised overshooting, what it did do was provide an alternative modelling procedure from the Mundell-Fleming model that had dominated international macroeconomic discourse for many years. It must be stressed, however, that the model and its variants are very monetarist in nature. Although the Mundell-Fleming model assumed prices fixed, which some saw as totally inappropriate, the models in the present chapter assume full employment, and hence a constant level of real income. This too may seem quite inappropriate. Looked at from a modelling perspective, it allows us to concentrate on the relationship between the price level and the exchange rate. Of particular importance, therefore, in such models is purchasing power parity. It does, of course, keep the analysis to just two main variables. Purchasing power parity (PPP) indicates that prices in one country are equal to those in another after translating through the exchange market. There is a vast literature on this topic that we shall not go into here. All we shall do is stipulate that this is supposed to hold at the aggregate level. Hence, if P is the price level at home, P* the price level abroad, and S the exchange rate (quoted in terms of domestic currency), then SP* = P Taking natural logarithms, and setting the foreign price to unity (i.e. P* = 1), which throughout is held constant, then InS-lnP* = InP i.e. s = p since In P* = In 1 = 0 where lower case letters denote natural logarithms. A rise in S (or s) is an 554 Economic Dynamics appreciation of the foreign currency, i.e., a depreciation of the domestic currency. For purchasing power parity to hold, therefore, we require s = p. So long as s differs from p, then purchasing power parity does not hold.2 It is the purchasing power parity condition that drives the long-run result in the models to be discussed in this chapter. In other words, in the short run it is possible for the economy to deviate from purchasing power parity but in the long run purchasing power parity must hold.3 One of the essential differences between the present models and those of chapter 12 is that they are presented in terms of natural logarithms. Accordingly we shall denote all variables in natural logarithms with lower case letters. The exception is interest rates. These are percentages and the home interest rate will be denoted r and the foreign rate r*, as in chapter 12. In section 13.1 we consider a simplified Dornbusch model in which the goods market is independent of the rate of interest. This captures most of the characteristics of the original Dornbusch model but is easier to follow. In section 13.2 we consider Dornbusch's (1976) model. Both these models assume perfect capital mobility. In section 13.3 we consider what happens when capital is immobile (but not perfectly immobile). Next we consider the Dornbusch model under the assumption of perfect foresight, which gives a rational expectations solution (section 13.4). One of the main features of rational expectations modelling is the possibility of considering the impacts of government announcements. This topic we consider in section 13.5. The discovery of gas and then oil in the North Sea led to major impacts on the exchange rate, which in turn influenced adversely the non-oil sector. Section 13.6 presents a popular model for considering any resource discovery and its impact on the exchange rate. The final section 13.7 considers the dynamics of a simple monetarist model. Throughout we concentrate on the economic dynamics, illustrating this with many numerical examples. 13.1 A simplified Dornbusch model4 We begin with a simplified Dornbusch model that captures nearly all the features of the original but is more manageable. We can then go on to further complications once this is fully understood. All the Dornbusch models begin with three markets. There is the goods market, the money market and the foreign exchange market (or the balance of payments). The goods market reduces down to two simple relationships, a total expenditure equation and a price adjustment equation, where income is assumed constant at the full employed level. The money market is a 1 The reader needs to be vigilant concerning which currency is appreciating and which depreciating. Since S (or s) is the price of overseas currency in terms of domestic currency (the European convention of quoting exchange rates, other than the UK), then a rise in the price is an appreciation of the foreign currency. However, most discussion takes place in terms of the price of domestic currency. 2 In terms of the analysis of chapter 12 purchasing power parity requires the real exchange rate, R, to equal unity. 3 There is something wholly unsatisfactory in this modelling. Although in the short run deviation from PPP is possible, but not in the long run, income cannot deviate from its full employment level either in the short run or in the long run, which is quite unrealistic. 4 Based on a model presented in Gärtner (1993). Open economy dynamics: flexible price models 555 Table 13.1 Model 13.1 Goods market e — cy + g + h(s — p) 0 < c < I, h > p — a(e — y) a > 0 Money market nid = p + ky — ur k > 0, u > 0 ms — nid — m International asset market r=r* + se se — v(s — s) v > 0 e = total expenditure y — real income (exogenous) g — government spending s — spot exchange rate p — domestic price level p — inflation rate (since p — In P) nid — demand for money r = domestic interest rate ms — supply of money m — exogenous money balances r* — interest rate abroad se — change in expected spot rate (expected depreciation/appreciation) "s — purchasing power parity rate (equilibrium rate) straightforward demand for money and a constant level of real money balances. The international asset market varies considerably from one model to another. Here we assume perfect capital mobility and therefore the domestic interest rate is equal to the foreign interest rate adjusted for any expected change in the exchange rate. The expected change in the exchange rate, in turn, depends on the extent of the deviation of the exchange rate from its purchasing power parity level. The model, then, is captured by the set of equations in table 13.1. The model can be captured diagrammatically by deriving two equilibrium lines in (5,p)-space. A goods market equilibrium line, which denotes combinations of p and s for which the price level is not changing, i.e., p = 0, which we shall denote GM; and an asset market line which denotes combinations of p and s which maintains equilibrium in the money market and satisfies the condition on the expected change in the exchange rate, which we shall denote AM. Substituting the expenditure function into the price adjustment relation p = a(e — y), and setting p equal to zero, gives the following relationship between the price level and the exchange rate (1 ~c)y g p = s---— 1 h h Equation (13.1) is a positive relationship between p and s with a slope of unity. If we impose the condition of purchasing power parity, which we shall do, then in the long run p = s, and so the intercept of the GM line must be zero. Hence, in figure 13.1 we have drawn the GM line through the origin with a slope of unity. Furthermore, p > 0 if e > y, i.e. (1 ~c)y g PQ,h>Q p — a(e — y) a > 0 Money market nid — p + ky — ur k > 0, u > 0 ms — nid — m International asset market r=r* +se se — v(s — s) v > 0 e = total expenditure y — real income (exogenous) g — government spending s — spot exchange rate p — domestic price level p — inflation rate (since p — In P) nid — demand for money r = domestic interest rate ms — supply of money m — exogenous money balances r* — interest rate abroad se — change in expected spot rate (expected depreciation/appreciation) s — purchasing power parity rate (equilibrium rate) (13.6) (13.7) (13.8) different, this does add a significant complication. The change is to the expenditure function, which now assumes that investment (a component of expenditure) is inversely related to the rate of interest; hence a component — dr (d > 0) is added to the expenditure function. This has the immediate implication that the goods market and the asset market are interdependent, and this interdependence arises through the rate of interest. The asset market line remains unaffected and therefore can be expressed as before, i.e., the AM line is m = p + ky — u[r* + v(s — s)] i.e. p = (m — ky + ur* + uvs) — uvs However, the goods market line, GM, now takes the form P (I - c) + (dk/u)' h + (d/u) y + g + (dm/u)' h + (d/u) + hs h + (d/u) Notice in particular that the slope of the GM line (where p is on the vertical axis and s on the horizontal axis) is now h 1 slope GM < 1 h + (d/u) 1+(d/uh) since d, v and h are all positive. It is also still the case that below the GM line the goods market has expenditure in excess of income, and so there is pressure on prices to rise. Above the GM line, expenditure is less than income, and there is pressure on prices to fall. The situation is illustrated in figure 13.4. In this figure we have both markets in equilibrium at point E, which in this instance is both a short-run equilibrium and a long-run equilibrium. It is a short-run equilibrium because the solution lies at the intersection of GM and AM, but it is also a long-run equilibrium because this also satisfies the purchasing power parity condition, which is given by the 45°-line, and denoted PPP. It is no longer the case, therefore, that the GM line coincides with the 45°-line. Now consider a monetary expansion once again. This shifts the AM line to the right, as before. Since we have retained the assumption of an instantaneously adjusting asset market and a sluggish goods market adjustment, the economy Open economy dynamics: flexible price models 561 0 PPP(s=/>) ey(p>0) AM yv5° Figure 13.4. PPPO=p) Figure 13.5. moves initially to point C on AMi in figure 13.5. This movement is because of the immediate capital outflow. But now two responses come into play. At point C expenditure is in excess of income and so there is pressure on prices to rise, so moving the economy up AMi. However, the rise in the nominal money supply has initially led to a fall in the rate of interest. This fall in the rate of interest shifts the GM curve to the left (it raises the intercept). One can think of the shift in the GM line as follows. If the money supply rises, this puts pressure on domestic interest rates to fall. This fall stimulates investment which increases expenditure. At the existing exchange rate prices are now higher, and so the GM line has shifted up (left) as a result of the impact on r from the rise in m. This result can be seen in 562 Economic Dynamics terms of equation (13.7). A rise in m leads to a rise in the intercept term, i.e., a shift up in the GM line. Expectations will change so long as the exchange rate differs from its purchasing power parity level. Hence, the system will come to a long-run equilibrium once the goods market line has shifted from GMn to GMi, establishing a new equilibrium Ei once again on the PPP line. Prices rise and the domestic currency depreciates. Because of our assumption about perfect capital mobility the rate of interest must return to its former level, which is equal to the foreign interest rate. Consider the following numerical example. Example 13.2, Model 13.2 e = O.Sy - O.lr + 5 + 0.01(5 -p) p = 0.1(e-y) md = P + 0.5y — 0.5r ms = md = 105 r = r* + se se = 0.2(5 - s) y = 20, r* = 10 This gives the GM and AM lines as GM p = 95.2 + 0.04765 AM p= 110-0.15 with equilibrium point En given by (so,p0) = (100, 100), which satisfies the purchasing power parity condition, i.e., 5 = p. Furthermore, r = r* = 10, as illustrated in figure 13.6. Now consider an increase in the money supply from 105 to 110. As we indicated above, this shifts both the asset market line and the goods market line. The new lines are GM p = 100 + 0.04765 AM p= 115.5 - 0.15 Figure 13.6. P AM,(/n=110) PPP 98 100 102 104 106 108 110 s Open economy dynamics: flexible price models 563 with the new equilibrium point, Ei, given by (s\,p{) = (105, 105), which also satisfies the purchasing power parity condition and the condition that r = r* = 10. As in the previous model, comparing equilibrium point Eo with Ei, we notice that ds = dp = dm = 5 As we would expect with an unchanged AM line, point C has an exchange rate of s = 155 at the price levelp = 100. Again there is overshooting of the exchange rate, first the domestic currency is depreciating and then appreciating, with an overall depreciation in the long run. The fact that the interest rate affects the GM line must mean that although the system moves along AM, as in the previous model, it must do so at a different speed. We can establish this in the present example as follows. First we note (see exercise 3) that we can express the change in prices as a first-order autonomous homogeneous differential equation, i.e. This is consistent with our previous result. If d = 0, then this reduces to the same differential equation we considered in example 13.1. The adjustment coefficient Price level Figure 13.7. P 104 103 102 101 100 0 10 20 30 40 t 50 120 130 140 150 S 564 Economic Dynamics is therefore / h d\ k = alh+ — + -= 0.031 \ uv u ) which is greater than 0.011 as we would expect. In terms of the dynamics, the only essential difference is the adjustment coefficient k. The price level and the exchange rate still adjust the same, since e~Xt applies to both, as in example 13.1, and so adjustment still takes place along a trajectory determined by the AM line. What makes the difference from example 13.1 is the change in the interest rate during the adjustment period. Although this moves the goods market line to the left, the change in the rate of interest speeds up the adjustment process. Eventually, however, the rate of interest returns to its former level of r = r* = 10. This difference in the adjustment path for the exchange rate and the price level is illustrated in figure 13.7. 13.3 The Dornbusch model: capital immobility The two versions of the Dornbusch model so far outlined assume that capital is perfectly mobile. This has the effect of leaving interest rates equal to the world level in long-run equilibrium. Furthermore, in the previous two models the exchange rate will always overshoot its long-run equilibrium when the money supply is changed. Allowing exchange rate immobility leads to the possibility of undershooting rather than overshooting. To see this we need to change the relationship between the domestic interest rate and the foreign interest rate. In doing this we need to define the balance of payments. This is given by (13.9) bp = h(s-p) + b(r-r* - se) h>0,b>0 Equation (13.9)saysno more than the balance of payments is the sum of the current account and the net capital flow. The current account element is the same as that in the expenditure function,5 while the second element denotes net capital flows which is responding to the difference between the two interest rates, adjusted for any expected change in the exchange rate. Perfect capital mobility implies b = oo, while a value of b close to zero implies more extreme capital immobility. We need to make one further observation concerning this equation. Given a perfectly floating exchange rate, then the balance of payments is always in balance and so bp = 0. We retain the assumption about the expected change in the exchange rate, namely that it adjusts to the difference between the purchasing power parity level and the actual level. Since nothing else in the model is different, then there is no change in the goods market line. Only the specification of the asset market line is changed. Consider example 13.1 again, which excludes any interest rate impact on the goods market, and so the GM line is the same as the purchasing power parity line, and is a 45°-line through the origin. The model is set out in detail below in table 13.3. 5 Gärtner (1993) has a different coefficient on (s — p) in the expenditure function and the balance of payments equation. There is no real need for this. Both arise from net exports, which occurs identically in both equations. Open economy dynamics: flexible price models 565 Table 13.3 Model 13.3 Goods market e = cy + g + h(s ■ p = a(e- y) ■p) 0 0 a > 0 Money market nid = p + ky — ur k > 0, u > 0 ms — nid — m International asset market bp = h(s-p) + b(r-r*-se) h > 0, b > 0 se — v(s — s) v > 0 e = total expenditure y — real income (exogenous) g — government spending s — spot exchange rate p — domestic price level p — inflation rate (since p — In P) nid — demand for money r = domestic interest rate ms — supply of money m — exogenous money balances bp — balance of payments r* — interest rate abroad se — change in expected spot rate (expected depreciation/appreciation) s — purchasing power parity rate (equilibrium rate) As we have just indicated, the essential change is to the asset market line. This now takes the form m — ky + ur* uvs [uv + (uh/b)]s p=---+-----——— (13.10) 1 1 - (uh/b) 1 - {uh/b) 1 - (uh/b) Notice that this is consistent with model 13.1. If —► oo then this equation reduces to the asset market equation of section 13.1. Of particular importance in this model is the slope of the asset market line, which is uv + (uh/b) slope of AM =--K—±-L (13.11) 1 —(uh/b) A very high value of b, a high degree of capital mobility, will mean the typical negatively sloped asset market line, with analysis identical to that in section 13.1. However, with a very low degree of capital mobility, a value of b close to zero, can mean a positively sloped asset market line. The situation is illustrated in figure 13.8. 566 Economic Dynamics (13.12) A rise in the money supply will shift the asset market line to the right, from AMo to AMi, and the equilibrium will move from Eo to Ei on the goods market line, which coincides with the purchasing power parity condition. The movement of the economy now, however, is quite different. Although the trajectory is still along the new asset market line, with sticky prices initially the economy moves to point C on AMi. Since again there is excess expenditure over income, prices will rise. The rise in the price level, although reducing real money balances and raising the rate of interest at home, will have only a small effect on capital inflows. In order, therefore, to maintain balance of payments equilibrium the domestic currency must also depreciate (s must rise). Hence, the economy moves along AMi from point C to point Ei. In this version of the model, therefore, the exchange rate undershoots its long-run equilibrium level. There is initially a rapid depreciation of the domestic currency (a movement from point Eo to point C), followed by a further gradual depreciation in response to the price rise (a movement from point C to point Ei). In this version of the model the rate of interest both before and after the change in the money supply will equal the interest rate abroad. Since purchasing power parity implies s = p, and since in long-run equilibrium se = 0, then it follows that r = r* in long-run equilibrium. Once again price movements and exchange rate movements can be expressed by the equations pit) = p + (p0-p)e-xt s(t) = s + (s0- s)e~Xt but now / 1 - (uh/b) (13.13) k = ahl- + 1 \uv + (uh/b) Notice that for b oo this reduces to the adjustment coefficient of model 13.1 (see exercise 5 for a numerical example illustrating this model). The analysis is very similar in the original Dornbusch model, but with capital immobility. This model, model 13.4, is presented in table 13.4. Table 13.4 Model 13.4 Goods market e = total expenditure e = cy — dr + g + h(s — p) 00, h>0 y = real income (exogenous) p = a(e — y) a > 0 g = government spending s = spot exchange rate p = domestic price level p = inflation rate (since p = In P) Money market = demand for money wid = p + ky — ur k > 0, u > 0 r = domestic interest rate ms = rtid = tn ms = supply of money m = exogenous money balances International asset market bp = balance of payments bp = h(s — p) + b(r — r* — se) h > 0, b > 0 r* = interest rate abroad se = v(s — s) v > 0 se = change in expected spot rate (expected depreciation/appreciation) s ^purchasing power parity rate (equilibrium rate) Open economy dynamics: flexible price models 567 The situation is illustrated in figure 13.9. A rise in the money supply shifts both the asset market line and the goods market line. But because capital is very immobile, the exchange rate initially undershoots its long-run equilibrium value, s2 <~s\. In this model we have (see exercise 6) -a (h + dv)(l-(uh/b)) + h (P-P) uv + (uh/b) so prices and the exchange rate have the same adjustment coefficient '(h + dv)(l - (uh/b)) uv + (uh/b) + h This is consistent with all our previous results. If d —► 0 the model reduces to model 13.3; if b oo the model reduces to model 13.2; and if d 0 and b oo the model reduces to model 13.1. (13.14) (13.15) 13.4 The Dornbusch model under perfect foresight One of the advantages of the Dornbusch model is that it readily lends itself to different specifications of exchange rate expectations. One such specification is perfect foresight. This model has a number of formal advantages. It can be shown that rational expectations is formally the same as expectations under perfect foresight, and since it is easier to handle models under the assumption of perfect foresight, then all the features of modelling rational expectations can be captured by this version. Second, the assumption that se = v(s — s) with v > 0, is the same as the assumption of perfect foresight - so long as v is correctly chosen (see exercise 7). Again we shall begin with the simplified Dornbusch model in which expenditure is independent of the rate of interest. The model is captured in model 13.5, and set out in table 13.5, where we have replaced the assumption about exchange 568 Economic Dynamics Table 13.5 Model 13.5 Goods market e — cy + g + h(s — p) 0 < c < l, h > 0 p — a(e — y) a > 0 Money market nid — p + ky — ur k > 0, u > 0 ms — nid — m International asset market r=r* +se se — s e = total expenditure y — real income (exogenous) g — government spending s — spot exchange rate p — domestic price level p — inflation rate (since p — In P) nid — demand for money r = domestic interest rate ms — supply of money m — exogenous money balances r* — interest rate abroad se — change in expected spot rate (expected depreciation/appreciation) s — change in spot exchange rate rate expectations of model 13.1 with that of perfect foresight. This is the only difference from model 13.1, but it will be seen that it has significant implications for the dynamic behaviour of prices and exchange rates. Since the formal algebraic manipulations are the same in deriving the goods market line and the asset market line, we shall be brief. There has been no change to the goods market, this remains the same as model 13.1, and under the assumption of purchasing power parity is a 45°-line through the origin. The dynamics of the goods market is still specified by the relationship (13.16) p = a[h(s - p) - (1 -c)y + g] a > 0, h > 0, 00 if p>m — ky + ur* s <0 if p Oifp < s + (g - (I - c)y)/h p < Oifp > s + (g - (1 - c)y)/h In other words, below the horizontal line the exchange rate is falling (the domestic currency is appreciating), while above the horizontal line the exchange rate is rising (the domestic currency is depreciating). On the other hand, below the goods market line prices are rising while above it prices are falling, consistent with our earlier analysis. The vector forces in figure 13.10 illustrate all this information. What is quite clear from figure 13.10 is that we have a saddle point equilibrium. This can be established as follows. Consider the system in terms of deviations from equilibrium, which is particularly useful. p = a[h(s -p)-(l-c)y + g] 0 = a[h(s-p)-(l-c)y + g] and p = —ah(p — p) + ah(s — s) and s —{p + ky — m) — r* u 0 — {p + ky — m) — r* u 1 s = -(p -p) u 570 Economic Dynamics Hence, the system can be written in matrix form as follows. (13.20) p —ah ah P-P s _l/u 0 s — s Letting A denote the matrix of the system, then we immediately have A = —ah l/u ah 0 and det(A) = -ah < 0 Since det(A) < 0 then the critical point, E in figure 13.10, is a saddle point. What appears conspicuously absent from this analysis is any discussion of the asset market line, which was so prominent in model 13.1. But this is not the case. The saddle point solution along with the vector forces illustrated in figure 13.10 suggests there is one line through point E and passing through quadrants I and III which is a stable arm of the saddle. This is indeed the case. But more significantly, this stable arm is no more than the asset market line. We shall not prove this algebraically but rather show it is the case by means of a numerical example. Example 13.3 is a slight variant on example 13.1, where perfect foresight replaces the exchange rate expectation formation. Example 13.3, Model 13.5 The model is e = 0.8y + 4 + 0.01(5 - p) p = 0.l(e-y) md = P + 0.5y — 0.5r ms = m^ = 105 r = r* + se se = s y = 20, r* = 10 We can express this in the form of deviations from equilibrium. p = -0.00l(p-p) + 0.001(5 - s) s = 2(p- p) The goods market line is the 45°-line through the origin as before. But this line simply denotes the conditionp = 0 while the horizontal line in figure 13.10 denotes the condition s = 0. It is readily established that the equilibrium point is given by (s, p) = (100, 100). In other words, the trajectories when passing over these lines do so with infinite slope and zero slope, respectively, in the phase plane. To establish the arms of the saddle point we need to consider the matrix of the system and its associated eigenvalues and eigenvectors. The system can be written -0.001 2 0.001 0 p-p s — s Open economy dynamics: flexible price models 571 with associated matrices A = -0.001 0.001 2 0 A-X\ = '-(0.001 + X) 0.001 2 —X 0.0452242 0.001 P-P "0" 2 0.0442242 _ s — s 0 Hence, det(A - XI) = X2 + 0.00U - 0.002 = 0, with roots r = 0.0442242 and s = —0.0452242. The fact that the roots are of opposite sign verifies that the equilibrium point is a saddle point solution. For r = 0.0442242 we have (A - rl)vr = This leads to the equation -0.04522420? -p) + 0.001(5 -s) = 0 i.e. p = 97.7888 + 0.02215 This is the line that would pass through the equivalent of quadrants II and IV in figure 13.10, and denotes the unstable arm of the saddle point. Or equivalently the eigenvector 1 45.2242 which emanates from point En. This solution is shown by the saddle path denoted SP1 in figure 13.11. Similarly, using s = —0.0452242 we obtain 0.0442242 0.001 ~P ~P~ "0" 2 0.0452242 5 — 5 0 (A - sl)\s = This leads to the equation 0. 04422420? -p) + 0.001(5 - s) = 0 1. e. p = 102.2612 - 0.02265 572 Economic Dynamics Figure 13.12. Po 0 sn=0 This is the line which would pass through the equivalent of quadrants I and III in figure 13.10, and denotes the stable arm of the saddle point, and is denoted SP2 in figure 13.11. It leads to the eigenvector 1 -44.2242 with the resulting general solution Pit) sit) 1 45.2242 e°-04422t + c2 1 -44.2242 -0.04522. The dynamics of the situation are revealed by considering an increase in the money supply. The initial equilibrium is at point Eo, with associated saddle paths SPq and SP2. An increase in the money supply shifts the 5 = 0 line up as shown in figure 13.12, from so = 0 to s\ = 0. There is a new equilibrium point Ei with its associated saddle paths,6 namely SP} and SP2. But what is the trajectory of the economy in this situation? With perfect foresight the market knows that there is the saddle path through Ei, and so moves immediately to point C on this saddle path. At this stage prices have not moved and the domestic currency has depreciated to 52. With excess demand in the goods market, the economy moves along the stable arm of the saddle path reaching point Ei as prices begin to rise and the domestic currency appreciates, with the economy moving along trajectory Ti. Unfortunately in this model, any lack of perfection will send the system away from point Ei. For example, if the market under-estimates the depreciation and the system moves to point D, then it will over time diverge and move along trajectory SPj has the equation p m = 110. 107.3743 - 0.02265* for an increase in the money supply from m — 105 to Open economy dynamics: flexible price models 573 T2, with the economy heading into a major slump. Similarly, if the market over-adjusts and moves to point F, then the system will become explosive, with prices rising and the domestic currency depreciating, as shown by trajectory T3. Given all our previous analysis we can outline the original Dornbusch model under perfect foresight quite readily. All we need to recall is that the goods market line is positively sloped and with a slope less than unity, and that the 45°-line now indicates only purchasing power parity. The situation is captured in figure 13.13. Again, an increase in the money supply moves the economy from point Eo to point C as prices remain sticky. All adjustment initially falls on the exchange rate (and the rate of interest). There is a large depreciation of the domestic currency. As prices rise in response to excess demand in the goods market, the economy adjusts along the stable arm SP2, eventually re-establishing purchasing power parity at the long-run equilibrium point Ei. Like the simpler version, any mis-judgement by the market will send the system away from point Ei towards the unstable arm SP}. 13.5 Announcement effects This section has three objectives: (i) To present a discrete formulation of the Dornbusch model under perfect foresight. (ii) To deal with policy announcements which: (a) are actually carried out (b) are not carried out as promised. (iii) To provide some implications for price and exchange rate variability. 574 Economic Dynamics Table 13.6 Model 13.6 e, = cyt+g + h(s, - pt) Pt+i -Pt = a(et - y) mdt - pt + ky - ur, mf = mst = m rt = r* + (set+1 - set) s = 0.999p(t)+0.001s{t! m - !C5 m = 110 A = s(t)+2p(t;-2m+1Ö''" p(0)= " 100° p(ö)= ........100 5 s(Ö) - ......120';" s(Ö) = 326 121 6 (1) (2) (3) (4) (5): (5): (7) (8): (9) t. P(t) s(t) P(t) s(t). SP[p(t)]: diff p(t): s(t) 8 0 100.00- 120.00 100.00: 326.12; 104.66; 4.66 100.00! 120.00! •1 1 100.02 120.00 100.23i' 316.12; 104.66! 4.64 100.02! 120.00: i: 2 100.04: 120.04 100.44^ 306.57! 104.661 4.62 100.04^ 120.04: Ii ' 3:" .....iÖÖ':Ö6. 120.12 100.65: 297.46! 104:66! ' 4.60 100.06: 120:12 i; 4 100.08; 120.24 ........-c;84 288.75: 104.66; 4.58 100.08 120.24 13 I 5 iÖÖVi'Ö'r"'" ..t2p-40 101.03; ' 28044! 104.65; 4.55 'ioo.iov 120.40' 14 "e 100.12: 120.60 101.21! 272.51 104:65! 4.53 100.12" meo 15. 7 "'" ibö"i4T" 12Ö".84 10138!" 264 93 104.64 '4.5O 100.14 120.84 16 8 100.16: 121.12 101.55; 257.70: 104.64 4.47 100,15 121.12 1' 9 100.18 121.45 101.70 250.80! 104.63: 4.45 100.18 121.45 > 10 " IOO'.2Ö'" 121.81 101.85 244.20: 104.62: 4.42 "1OÖ.2O 121.81 11 100.23 122.22 101.99: 237:91: "04.61 439. 20 12 100.25 122.87 102 13: J.:\ _ioa-*=<~— 21 13 100.27 _ 22 50 51 "'52" 53 "54 55' 58 "57" 58 59 60 61 62 63 64 65 66 67 68 69 101.92! 102.01" IO2.IO: 102.20; X2 3C 102.40: 102.51 102:62 102.74 102.86 102.99 103.12 103.26 103.40 "lÖ3"56" 103.71 103.88 104.05 104.23; 104.42 185.30; 188.98; 192.83: 196.85: 20t06! 2Ö5'45r 210.04! 214.84; 219.85; ' 225.09; .230.56: "'236.29-242.25 248.49: 255.00 "2g^81" 288.92 ■ 276.35: 284.10; 292.20: 300.67: -—ra^ü^ 104.48: 104:51;' ''104.S3V 104.55' 104:57; 104.61: 104.63 104.64 104.66" 104.67 104.69 104.70 104.72 104.73 104:74 104.75 104.76: 104.77: 104.79! 104.79! -T28798 127.90 123 86 "125:87" 124.93 124.03 ''123.17 122.35 121.56 120.81 '120:1Ö' 119.42 118.75 118.14 117.55 116.98 116.44 115.92 115.43 114.96 114.51 114.08 M ~1Ü372B~ 103.18. 103.10 '1Ö3.OI 102.92 102.83 102.62 102.52 "l'Ö2:40' '102.28 102.16 102.03 101.90 101.76 101.81 "101:45 101.29 101.13' "iÖO.95 10077 100.58 TW 1.34 1.18 1.00 0.82 0.63 0 43 0.23 0.01 101.84 101.92; 102.011 102.10! 102.20! ''ToilöT' 102.40: 102 51: 185.30 188.98 192:83 196.85 20: C6 20545 210.04 214,84 -0.22 -O.45" -0.70: -0.95' -1.22 -1.50 -1,80: -2.10: -2.42: -2.75; -3.1Ö': -346: -3.84 102.51: 102.62 10273 102 83 102.93 103.02 103.11 " 103.20 103.28 103.36 "1Ö343 103.50 103.57 215.29 210:30 205.54 200.99 196.65 192.50 188 55 184.77 181.16 177.72 174.43 171.29 168.29 J Figure 13.18. Open economy dynamics: flexible price models 579 and column (2), to give column (7). When this difference is zero we have establish the point at which the economy begins to move along the stable arm of the saddle point associated with Ei. In other words, all the observations in columns (8) and (9) at or above where the difference column is positive are a copy of the observations in columns (2) and (3). After that point, shown by a bar across columns (8) and (9), we plot observations along the asset market line. We do this by first taking the price immediately above the bar and then using this to compute the corresponding point on SP2. The remaining values in columns (8) and (9) then conform to the model with m = 110. Columns (8) and (9) constitute the observations for figure 13.18 for period 0 to 120. This use of the spreadsheet allows us to consider different time periods of announcement. Columns (4) and (5) indicate the path of the economy where no announcement at all is made, and there is an unexpected increase in the money supply. This is the version first considered by Dornbusch. In terms of figure 13.19 --———-1 Figure 13.19. Announcement effects - s(t) 350 -i ioo [....................r—i-1-1-1———i 0 20 40 60 80 100 120 t No announcement----Short announcement..........Long announcement 580 Economic Dynamics Table 13.7 Response periods Period (s =150) Period (p =102) No announcement 34 11 Short announcement 76 51 Long announcement 113 88 this gives the 'No announcement' path for the exchange rate and the price level, and is a plot of columns (5) and (4), respectively, of figure 13.17. A period that is relatively short in announcing government intentions will lead to a point like F, which is 'close to' the asset market line through point Ei. This is the trajectory computed in figure 13.17 and illustrated in figure 13.18. This leads to the path for the exchange rate and the price level marked 'Short announcement' in figure 13.19. A period of 'long announcement' will lead to a point F that is closer to point Eo. The computations of this are not given here, but follow exactly the same reasoning. This leads to the path for the exchange rate and the price level denoted 'Long announcement' in figure 13.19. But we can go further in our analysis with this numerical example. Suppose we take a point of reference for the exchange rate, say s = 150. We now ask the question for each of these announcements, how long does it take the system to reach s = 150, when moving along the asset market line?8 Similarly, how long does it take the exchange rate to reach s = 150 once the policy has actually been implemented? Similarly, how long does it take the price level to reach p = 102? The results are tabulated as shown in table 13.7. What we observe from figure 13.19 and from table 13.7 is three important observations. First, the exchange rate varies less the longer the time period of the announcement. Second, the greater the time period for the announcement, the longer it takes for the exchange rate and the price level to reach the new equilibrium. Thus, increasing the announcement period increases the adjustment period. Policy-makers therefore need to weigh these two possibilities. Third, the price level gradually approaches its new level, with just a minor kink in the case of a short and long announcement. In other words, price changes do not show the same dramatic changes that can occur with the exchange rate. But there is a further problem to consider. Policy-makers are often known to renege on their announcement. They may announce they will increase the money supply, but when the time comes, they decide not to do so! Does this in any way change the results? The situation is illustrated in figure 13.20. On the announcement of an increase in the money supply the economy immediately moves to point F in anticipation of the changes that are expected. As before, the economy then moves along the trajectory between F and G, which is dominated by the saddle path SPq which passes through equilibrium point Eo. At point G, the moment when the change should take place, the government announces that it does not intend to change the money supply after all! Given perfect foresight, and given instantaneous adjustment in the asset market, the economy will move immediately 8 We cannot ask the time for it to reach point Ej, since this is at infinity. However, any common point of reference will do for this comparison. Open economy dynamics: flexible price models 581 (PPP) Figure 13.20. from point G to point H on SPq (since SPq is the asset market line through the original equilibrium point Eo). Since this is a stable arm of the saddle point Eo, the economy will accordingly move down this line approaching Eo in the limit. Hence, the economy will have a trajectory Eo-F-G-H-Eo. Prices no longer show a gradual movement to the new equilibrium, but on the contrary rise and fall. But even more dramatic is the movement in the exchange rate. We can illustrate the movement in prices and the exchange rate using the same technique we developed in relation to figure 13.19. We consider a short announcement period which positions F at (s, p) = (100, 120), as before. Also as before, the economy moves along the same trajectory until point G is reached. Now, however, the situation changes, as shown in figure 13.21.9 Realising the government has reneged on their decision, market participants move money back into the economy, leading to a sharp appreciation of the domestic currency. The economy is now above the PPP line, and income is in excess of expenditure. This leads to a gradual fall in prices, which in turn leads to a depreciation of the currency. The conclusion we come to, therefore, is that although policy announcements lead to less variation in prices and exchange rates, reneging on such policy announcements leads to more variation in prices and exchange rates than would have occurred without any such announcement. 13.6 Resource discovery and the exchange rate The analysis so far presented, with some modification, allows us to consider some of the implications of a major resource discovery like North Sea Oil. We assume that 9 Although in this numerical example the exchange rate becomes negative, which is not possible, the general path of the exchange rate, however, is as displayed in the figure. 582 Economic Dynamics the discovery leads to an increase in wealth and hence to an increase in permanent income. We capture this effect by adding a tQxmfxp{f > 0) to the expenditure equation, where xp denotes the permanent income stream from the new wealth and / is a positive coefficient. Thus, our expenditure equation now takes the form (13.25) e = cy + g + h(s - p) +fxp On the other hand, x is the current income from oil that adds additional demand for money balances, which is captured by a term jx(j > 0) in the demand for money equation. But we need to make an additional change to the demand for money equation. To understand this, return to prices and exchange rates in unlogged form. We assume that the domestic price level (the RPI) is a weighted average of domestically produced goods, P, and imported goods, Pf = SP*, i.e. (13.26) Q = Pa(SP*)1~a taking natural logarithms and denoting these by lower case letters, then (13.27) q = ap + (l -a)(s+p*) But if we set P* = 1, then p* = 0, hence (13.28) q = ap + (1 - a)s Open economy dynamics: flexible price models 583 Table 13.8 Model 13.7 Goods market e = cy + g + h(s - p) +fxp 0 < c < l,h> 0, / > 0 p = a(e- y) Money market = q + ky — ur + jx k > 0, u > 0, j > 0 q — ap + (1 — a)s 0 < a < 1 International asset market r=r* + se se — s e = total expenditure y = real income (exogenous) g — government spending s — spot exchange rate p — price of domestic goods p — inflation rate (since p — In P) nid — demand for money ms — supply of money r = domestic interest rate m — exogenous money balances q — domestic price level a =weight of domestic goods in q r* — interest rate abroad se — change in expected spot rate (expected depreciation/appreciation) s — change in spot exchange rate Finally, we deflate money balances by the domestic price level Q. Thus the demand for money equation becomes md = q + ky — ur + jx where q = ap + (1 — a)s. The complete model (model 13.7), under the assumption of perfect foresight and no interest rate effect on expenditure, is given in table 13.8. Carrying out the same manipulations as for model 13.5 we obtain p = a[g — (1 — c)y +fxp] — ahp + ahs (13.29) p + 1 — a s + ky +jx — m - u / \ u In equilibrium p = 0 and s = 0, hence 0 = a[g — (1 — c)y +fxp] — ahp + ahs ky +jx — m 0 p + 1 — a s + — r So taking deviations from the equilibrium we have p = —ah{p —p) + ah{s — s) 1 — a + (s-S) Or in matrix notation —ah a ah 1 — a u u Hence, the matrix of this system is P-P s — š A = -ah a u ah 1 — a (13.30) (13.31) (13.32) 584 Economic Dynamics (13.33) (13.34) with det(A) = —ah/u < 0. Since det(A) is negative the equilibrium point is a saddle point. From the conditions p = 0 and s = 0 given above, we can solve for p and s using Cramer's rule. These are m m + •(l-c)(l-«) h a(l — c) h + k . (1 - <*)g . f(l ~ a)xp . * y H--;--1--;--h ur h h ■jx <*g , <*fxp , * . y- — + —— +ur -jx h h It is apparent, therefore, that the discovery of a major resource leading to terms xp and x will influence the equilibrium price and exchange rate.10 To see this more clearly we need to consider the model in more detail. To do this we need to consider the equilibrium lines associated with p = 0 and 5 = 0. With some algebraic manipulation these are p = s + g ~ (1 ~ c)y +fxp h P = m — Icy — jx + ur* a for p = 0 1 — a a for s = 0 Consequently, the goods market line, the line associated withp = 0, is a 45°-line. The second equilibrium line, that associated with s = 0, is negatively sloped. Initially we assume that purchasing power parity is satisfied. This is best considered as the situation before any resource is discovered. Hence, the goods market line These results are consistent with those of model 13.5. If a = 1 and x = xp = 0, then we have the same equilibrium results as model 13.5. Open economy dynamics: flexible price models 585 passes through the origin, as shown in figure 13.22. This figure also shows the vector forces and the saddle paths associated with the equilibrium point E, and denoted SP1 and SP2. Note in particular that SP2 denotes the asset market equilibrium. In this model, as in previous models, we assume asset markets are always clearing while the goods market takes time. Now consider the discovery of a natural resource, such as oil, as shown in figure 13.23. The change in xp (from zero to some positive amount) shifts thep = 0 line up (i.e. leads to a rise in the intercept) to px = 0. But this will generate an income stream and so raise x (from zero to some positive amount). This in turn will shift the s = 0 line left (i.e. will reduce the intercept) to s\ = 0. The economy will move from equilibrium point En to equilibrium point Ei. But what trajectory will such an economy take? Initially prices do not change, and so the economy moves horizontally from point Eo to point C, point C being on the new saddle path SP2 through Ei. The domestic currency has accordingly appreciated taking the full impact of the adjustment in the short run. Point C is on the new asset market line. The resulting increase in permanent income raises consumers' expenditure. Since we have full employment this results in excess demand and hence to a rise in the price level. Accordingly the economy moves up the new asset market line from point C to point Ei. In this example there is no overshooting. This occurs, however, only where the goods market impact is greater than the money market impact. To see this consider figure 13.24, which shows the situation where the money market impact exceeds that of the goods market. Again the economy moves horizontally from point Eo to point C on SP2, but then moves down the new asset market line until point Ei is reached. This is because the significant effect of the current income stream on the demand for money leads to a significant rise in the rate of interest. In order to maintain the condition r = r* + se, the domestic 586 Economic Dynamics Figure 13.24. p currency must depreciate, moving the economy along the path C to Ei on SP2. Although prices gradually fall, the home currency first appreciates, overshooting its long-run equilibrium value, and then depreciates but leading to an eventual appreciation of the exchange rate. The difference in the two results is determined by the rate of resource depletion. In figure 13.23 the rate of depletion is slow and so the permanent income stream outweighs the current income from the oil extraction. Figure 13.24, however, indicates that resource depletion is quick and there is a relatively large current income from resource sales. In either case, the discovery of a resource, although having ambiguous results on the price level, does lead to an appreciation of the home currency, with the possibility of overshooting the quicker the resource depletion. Given the discovery of North Sea Oil, we may hypothesise that market participants with perfect foresight would know that the domestic currency would appreciate in the long run and would act accordingly. The situation is similar to the analysis in the previous section, and the result is shown in figure 13.25. Because of the anticipated appreciation, the economy moves to point F. We can think of this as the situation the moment the discovery is made. The economy then moves along the trajectory F to G, which is governed by the unstable arm SP}, and where point G is determined by the point in time that the oil comes on-stream. The economy then moves along the asset market line, SP2, from point G to point Ei. 13.7 The monetarist model An early flex-price model of exchange rate determination was the simple monetarist model. The model is set out in table 13.9. All variables are in natural logarithms except for interest rates.11 11 Since s is In S, then s is the percentage change in the exchange rate. Open economy dynamics: flexible price models 587 m — p = ky — ur k > 0, u > 0 r=r* + se p = s+p* se — s m = nominal money supply y — real income r = nominal interest rate at home r* — nominal interest rate abroad p — domestic price level p* — foreign price level s — exchange rate The first equation is no more than real money balances is equal to real demand for money balances, where we assume a simple demand for money equation. The second equation is the interest parity condition under perfect capital mobility, while the third equation is purchasing power parity. The final equation is rational expectations under perfect foresight. Real income is assumed constant at the natural level. Also m, r* and p* are assumed constant. Substituting, we have m — s — p* = ky — u(r* + se) or s= (-)y-r*--(m-p*)+ (- )s (13.35) \u J u \u J which is a first-order differential equation. The dynamics of the model are illustrated in figure 13.26. Since m, y, r* andp* are constant, then so is fk\ * !, \-]y-r--(m-p ) \u J u which is the intercept on the vertical axis. The slope is 1 /u. We have labelled the line A to denote asset market. The fixed point is readily established by setting 588 Economic Dynamics Figure 13.27. s = 0, hence 0=(-)y-r*- -(m-p*) + (-] s \u/ u \u/ or (13.36) s = (m-p*) + ur* - ky Since the slope of the asset market line is positive, the system is dynamically unstable. Also we have a linear differential equation, so the system is globally unstable. To solve differential equation (13.35) we normally require an initial condition, say s(0) = sq. But any so < s leads the system to a continual appreciation of the home currency (fall in s); while for sq > s the home currency continually Open economy dynamics: flexible price models 589 depreciates (s rises). Since in this rational expectations model market participants have perfect foresight, then s jumps immediately to s. The effect of a rise in nominal money balances is shown in figure 13.27. The asset market line shifts down (by {\/u)Am) and the new equilibrium exchange rate increases to s2. From equation (13.36) it immediately follows that As = Am, i.e., the domestic currency depreciates by exactly the same percentage as the rise in nominal money balances. Under perfect foresight, expected depreciation and actual depreciation are identical and the system immediately jumps from si to s2. Exercises 1. For the model outlined in table 13.1 we have the result pit) = p + (p0-p)e-Xt (i) Show that s(t) = s + (s0 - s)e~Xt where so is the initial exchange rate after the shock, but associated with the price level po. (ii) In terms of example 13.1, show that point C is represented by (s,p) = (155,100) and that s(t) = 105 + 50e-°011' (iii) Plot on the same graph p(t) and s(t) for A = 0.011 and A = 0.02. 2. Given the model in section 13.1 (example 13.1), establish the comparative static and dynamics of a rise in the foreign interest rate from r* = 10 to r* = 12. 3. For the Dornbusch model given in table 13.2 (model 13.2), show that 1 (i) s-s =--(p-p) uv (ii) hence show ( h d\ p = -a\h-\---\- - )(p-p) \ uv u J and ( h d k = a I h H---1— \ uv u 4. Consider the following discrete version of the Dornbusch model of table 13.2 et = O.Sy - O.lr, + 5 + 0M(st -pt) pt+i -pt = 0.2(et-y) mf =pt + 0.5y - 0.25rt mdt =m\ = 105 r, = r* + f+1-f 590 Economic Dynamics set+1-set =0.25(s-st) y 20 10 (i) Derive an expression for the GM line and the AM line of the form pt = (p(st) and establish the fixed point of the model. (ii) Set up the model on a spreadsheet and establish its fixed point, starting from the initial value p, = 100. (iii) Letm^ rise from 105 to 110 establish the new equilibrium and demonstrate that dmt = ds, = dp,. Given the numerical model based on table 13.3 e = 0.Sy + 4 + 0.01(5 - p) p = 0.2(e-y) md = P + 0.5v — 0.5r nid = ms = 105 bp = 0.01(5 -p) + b(r - r* - se) se = 0.2(5 - 5) y = 20 r* = 10 (i) If b = 0.0045 establish that the initial equilibrium is (sp) = (100,100). (ii) For a rise in the money supply to ms = 110, establish that point C is represented by (s,p) = (104.54128, 100). (iii) Confirm that the new equilibrium satisfies dp = ds = dm. For the model in table 13.4 (i) Show that '(h + dv)(l - (uh/b)) P and k uv + (uh/b) (h + dv)(l - (uh/b)) + h (P-P) + h uv + (uh/b) (ii) If d 0 thenp and k reduce to the values of model 13.3 (table 13.3). (iii) If b oo then p and k reduce to the values of model 13.2 (table 13.2). (iv) If d 0 and b oo then p and k reduce to the values of model 13.1 (table 13.1). 7. A numerical version of model 13.4 (table 13.4) is e = O.Sy - O.lr + 5 + 0.01(5 -p) p = 0.l(e-y) md = P + 0.5v — 0.5r nid = ms = 105 bp = 0.01(5 -p) + 0.004(r - r* - se) Open economy dynamics: flexible price models 591 se = 0.2(5 - s) y = 20 r* = 10 (i) Establish the following at the initial equilibrium. (a) Equilibrium is (s,p)= (100,100). (b) GM0: p = 95.238095 + 0.0476190s (c) AM0: p = -440 + 5As (ii) Let ms rise to 110. Find point C on AMi and establish that for the new equilibrium dm = ds = dp. 8. Using the model in example 13.3, and assuming r* = 18, all other parameters the same, then establish that (i) the initial equilibrium is (s,p) = (104,104) (ii) the characteristic roots are r = 0.04422415 and s = -0.0452242 (iii) the saddle path equations are: (a) unstable arm: p = 101.70034 + 0.022112085 (b) stable arm: p = 106.35166 - 0.0226121s (iv) for a rise in the money supply to 110 the intercepts of the saddle paths only alter to 106.58978 for the unstable arm and to 111.46472 for the stable arm, respectively. 9. In the model outlined in table 13.8 suppose we have the following numerical version of the model e = O.Sy + 4 + 0.01(5 - p) + 2xp p = 0.l(e-y) m = q + 0.5y — 0.5r + x q = ap + (1 — a)s nid = ms = 105 r = r* + se se = s y = 20 r* = 10 (i) If initially xp = x = 0 and a = 0.8, show that the initial equilibrium, E0, is given by (s,p) = (100, 100). (ii) Show that the stable and unstable arms of the saddle point Eo are: stable arm p = 125.31 — 0.2535 unstable arm p = 99.7531 + 0.002469s (iii) Now assume a resource discovery which leads to xp = 0.5 and x = 0.3. With a = 0.8, (a) show that equilibrium (s, p) = (19.7', 119.7) (b) the unstable arm is given by p = 119.65135 + 0.00246943s (c) the stable arm is given by p = 124.68608 - 0.2531005s 592 Economic Dynamics 10. Use the model surrounding figure 13.25 to analyse the UK's position in 1979 when the Conservative government under Mrs Thatcher took office. The basic information at the time was as follows. (a) Oil had been discovered in the North Sea, was being drilled around 1975 and was known to come on-stream in 1979. (b) The Conservatives won the General Election in 1979 with Mrs Thatcher indicating: - removal of all UK exchange controls; and - a reduction in monetary growth to combat inflation. Take as your starting date 1975 when oil was being drilled. 11. For the monetarist model in table 13.8 let y = 20, m=106, p* = l, £ = 0.5, u = 0.5, r* = 10 (i) Derive the differential equation for this model. (ii) Solve for equilibrium s. 12. Suppose nominal money supply grows at a constant rate X and inflation abroad is constant at it*, i.e. m = X, and p* = it* Derive an expression for equilibrium s under the assumption that a stationary equilibrium is one in which real money balances are constant. Additional reading Additional material on the contents of this chapter can be obtained from Buiter and Miller (1981), Copeland (2000), Dornbusch (1976), Dernburg (1989), Ford (1990), Frenkel and Rodriguez (1982), Gärtner (1993), MacDonald (1988), Niehans (1984),ObstfeldandRogoff(1999),Pilbeam(1998),R0dseth(2OOO),Shone(1989, 2001). CHAPTER 14 Population models 14.1 Malthusian population growth Population growth is frequently considered by means of differential equations, where the growth can be of persons, animal species, or bacteria. Although the increase in population is discontinuous, if the population is very large, then the additions to its size will be very small and so it can be considered as changing continuously. Hence, we assume population size, p, changes continuously over time and that p{t) is differentiable. The simplest population growth model is to assume that population grows/declines at a constant rate. Thus — - = k dt p this means that the change in the population is proportional to the size of the population dp — = kp dt 1 where k is positive for a growth in the population and negative for a decline. The initial condition is that if at time to the population is po then p(to) = Po Although (14.1) is a simple equation to solve, let us investigate its qualitative properties by means of phase-space. For positive k the growth curve is linear, positively sloped, and passes through the origin, as shown in figure 14.1. It is clear, then, that the only equilibrium for this population is a population of zero, since this is the only value of p for which dp/dt = 0. Furthermore, for any population of size greater than zero, e.g., po, then dp/dt is positive and so population will be increasing over time. In other words, the arrows along the phase line indicate a continuously growing population. If, on the other hand, k is negative then equilibrium population size is still zero, but now for any population greater than zero means dp/dt is negative and so population will decrease over time until it is extinguished. Although not wholly realistic, let us solve for the population size explicitly. To do this integrate both sides of the differential equation 594 Economic Dynamics Figure 14.1. dp/dt dpi4t > 0 Po dp/dt < 0 p(k<0) (14.2) dp - = kd dt 1 [dp=[ J dt J \np P = c0e kdt kt + c kt po for where c is the constant of integration. Applying the initial condition p t = to we have p = poekt° implying c0 = poe~kt° Which leads to the result p=p0e-ktoekt =p0ek(t-to) and which clearly satisfies the initial condition. In this model, population grows/declines exponentially, and is referred to as the Malthusian model of population growth. Of interest in rapidly growing populations is the time necessary for the population to double in size.1 It is readily shown that for the Malthusian model this 1 Biologists refer to this as the mean generation time, i.e., the time necessary for a population to reproduce itself. Population models 595 period depends only on the rate of growth, k. To show this let the population be po initially at time to. Let the time period when the population has doubled be denoted t\. Then the length of time for the population to double is t\ — to. Furthermore, Pi = 2po, hence For example, if a population is growing at 2% per annum, then it will double approximately every 0.6931 /0.02 = 35 years regardless of the initial population size. Example 14.1 Table 14.1 gives the population of the UK from 1781 to 1931. Our first problem is to estimate the parameter k. Suppose we set po = 13 million for the initial year 1781. Further, take the population in year 1791 to be as in the table, namely 14.5 million. This allows us to estimate the value of k. Letting to = 0 to represent 1781, then h = 10 for 1791, i.e., h - t0 = 10 p(0) = po = 13 p(lO) = Poe10k = I3e10k = 14.5 In 14.5 - In 13 2.6741 - 2.5649 k= - = - 10 10 k = 0.01092 Using this estimate of k we compute the Malthusian estimate of population growth, as shown in column (3) of table 14.1. 0.6931 k Table 14.1 UK Population, 1781-1931 (million) Year Actual Malthusian Logistic 1781 1791 1801 1811 1821 1831 1841 1851 1861 1871 1881 1891 1901 1911 1921 1931 13.000 14.500 15.902 18.103 21.007 24.135 26.751 27.393 28.977 31.556 34.934 37.802 41.538 45.299 47.168 49.007 13.000 14.500 16.173 18.039 20.121 22.442 25.032 27.920 31.142 34.735 38.743 43.213 48.200 53.761 59.964 66.883 13.000 14.996 17.143 19.410 21.756 24.135 26.498 28.799 30.993 33.046 34.934 36.641 38.162 39.500 40.662 41.662 Source: Deane and Cole (1962, table 3, p. 8). 596 Economic Dynamics (14.3) (14.5) A discrete version of the model may appear more appropriate. This takes the form Apt+1 = kpt i.e. pt+i =Pt + kpt = (l + k)p, Using the analysis of chapter 3, we have the general solution (14.4) p, = (l + kypo Again, using po = 13 and p\o = (1 + k)10(l3) = 14.5, we obtain /14.5\^ k=l — \ - 1 = 0.0109798 Using this estimate of k, and the discrete solution, we compute an alternative series based on the Malthusian assumption. However, it is readily established that this gives exactly the same figures (to three places of decimal) as the continuous model. The model is reasonably accurate up to 1851 but thereafter the error becomes not only quite large but increasing. This should not be surprising. In the first instance, k was estimated from the first two observations. Second, the population increases at an ever-increasing rate, which is unrealistic. Third, for distant population there is no account taken of competition of the population for the limited resources available. It may be thought that the model is inappropriate because it does not take account of births and deaths. But this is not strictly true. If births are assumed to follow the Malthusian law as well as deaths, i.e., both grow at constant rates b and d, respectively, then dPb bt A dPd dt — = p0em and — = p0eai dt dt d-l = dJ^-d-^=p^=p^ dt dt dt Hence, the k we estimated using data from 1781 and 1791 would account for both births and deaths. This means that the problem lies elsewhere. Although we have considered births and deaths we have taken no account of immigration or emigration. Migration (immigration minus emigration), however, is usually fairly small relative to the total size of the population, or occurs only at specific times (most especially in human populations). This would suggest, therefore, that the exponential growth curve might not be the most appropriate specification of the growth process. 14.2 The logistic curve An alternative approach is to assume that not only does population grow with population size, but that as it grows its members come into competition with each other for the food or limited resources. In order to capture this 'competition' it is assumed that there are p(p — l)/2 interactions for a given population of size p. Assuming such interactions lead to additional deaths, for example because of disease or war, then we can assume that the growth in the population will also Population models 597 diminish in proportion to this element of interaction. In other words, population now changes by hpip ~ 1) dp dt kp kp + hp _ hp2 2 2 hp2 Therefore dp dt ap — bp = p(a — bp) a > 0, b > 0 (14.6) which is referred to as the logistic growth equation. In general the parameter b is small relative to the parameter a, so that the second term is often negligible. However, as the population size grows and competition becomes greater, the second term —bp2 becomes more significant. This is especially true as time moves further away from the initial level. As the second term becomes more significant, this dampens the growth in the population. This second formulation is referred to as the logistic law of growth. Before solving for population explicitly, let us investigate the qualitative properties of the population by considering the phase-space. The logistic growth equation p = p(a — bp) a > 0, b > 0 is an autonomous first-order differential equation. The qualitative properties of this equation are shown in the phase diagram in figure 14.2. dpldt Figure 14.2. \ P'o Po P dp/dt>0 dp/dt<0 A alb P'o Phase line 598 Economic Dynamics (14.7) (14.8) The equilibrium population is where p = 0, i.e., zero population growth, which occurs at a Pi=0 and * Pi = Since we are interested only in positive populations we can ignore p\ = 0 and so just refer to equilibrium rate p*. For po < a/b, where po is the initial population, then dp/dt > 0, and so p rises over time. For po > a/b then dp/dt < 0, and p falls over time. The arrows in figure 14.2 show these properties. It is clear thatp* = a/b is a (locally) stable equilibrium.2 Although population approaches the limit a/b, this is never in fact achieved (see exercise 5). We can solve for p explicitly as follows dp 2 — = ap — bp = p(a — bp) dt Jpo Piß - bp) JPo But 1 _ 1 p(a — bp) a "p dp a ./„„ p a -b a — bp p -bdp 1 f" dp 1 fp -bdp _ f a Jpo P a Jpo (a - bP) Jt0 dt Solving we have 1 1 a lap--(—b) ln(a — bp) a — bp to a \poJ In In p(a - bpo) a — bpo a(t - t0) Po to ^po(a - bp)/ :. po(a — bp)ea{f~to) = p(a — bpo) We can now solve for p(t) p0aea(t~to) = pbp0ea(t~to) + p(a - bp0) = p[bp0ea(t-to) + (a- bpo)] i.e. Pit) apo bpo + (a- bpo)e~a(t~to) This represents the logistic function, which is sketched in figure 14.3, and shows the logistic curve. This curve depends on the three parameters a, b and po. It has an upper limit of a hm pit) = - t^oo o 2 Expanding/) =f(p) in a Taylor series around/7* = a/b we obtain the following linear approximation p = —a ip — |), which has a positive intercept (a2/b) and a negative slope (—a). Population models 599 The zero population growth is, however, never reached. (This result is also established in exercise 5 using a linear approximation around the equilibrium.) A second property of the logistic function is that it has an inflexion point at 2b This is readily established from the logistic growth equation, since the inflexion point occurs where the logistic growth equation is at a maximum. Thus, if f(p) = ap- bp2 f(p) = a-2bp = 0 a P=2~b The shape of the logistic curve depends on whether the initial population is below or above the inflexion value of p, or even above the limit value a/b. Figure 14.3 illustrates three different paths. We can use the logistic function and the data provided in table 14.1 to compute the values of a and b for the logistic growth equation. Using figures for 1781, 1831 and 1881, respectively, for t(0), t(50) and r(100), we have the following two equations 13a 24.135 = 34.934 13b + (a- 13b)e~50a 13a 13b + (a- 13b)e-100a 600 Economic Dynamics Figure 14.4. UK Population Growth 1781-1931 0 i i-1-1-1-1-1— 1781 1801 1821 1841 1861 1881 1901 1921 -Actual---Malthusian----Logistic which provide two nonlinear equations in two unknowns. Using a mathematical software package for solving equations3 (and using the Malthusian value of k for a first approximation for a), it can be established that a = 0.02038302 b = 0.0004605 As indicated above, the value of b is very small and the population has to be large before this second term becomes significant. Even so, it implies an upper limit for the population of the UK of a/b = 46.745186 million. Using these values for a ando, we have the logistic results shown in column (4) of table 14.1. It is clear that these give significantly different results than those of the Malthusian growth law and that towards the end of the period they under-estimate the growth in the population of the UK. The different growth processors relative to the actual observations are illustrated in figure 14.4. This shows quite clearly that the Malthusian law grossly over-estimates the UK population in 1931, while the logistic growth equation under-estimates it. Of course, a possible reason for the under-estimate of the logistic growth equation is the choice of years to estimate the parameters a and b. We quite arbitrarily chose t\ to be fifty years on from to and t2 to be 100 years on. A different choice of years would give different computed values of a and b, and hence different values in column (4) of table 14.1. It is even possible to estimate a and b using nonlinear statistical estimation, which would use all the available data in table 14.1. However, the point being emphasised is that the logistic calculations are sensitive to the computed/estimated values of a and b, and most especially the limit in the growth of the population. We might, however, approach the logistic equation in terms of its discrete approximation we developed in chapter 3, section 3.7. It is assumed that the change in the population, Apt+\ conforms to the rule (14.10) Apt+1 = apt - bp] 3 After defining the equations, Mathematica can solve these equations using the FindRoot command and using initial guesses for a and b. Maple can do the same using the fsolve command and giving ranges for a and b. The two programmes give the same results (see appendices 14.1 and 14.2). The same results can be established using TK Solver. With all programmes, care must be exercised in providing initial guesses. Population models 601 which has the approximate solution Pt = -~-~z- (14-11) bpo + (l+a) '(a - bpo) This too has the limit a/b. Again using the figures for 1781, 1831 and 1881 we obtain two equations 13a 24.135 = 34.934 13fc + (l + 10)"50(a- 13fc) 13a 13fc + (l + 10)-100(a - Ub) which gives two slightly different estimates for a and b, namely a = 0.0205922 b = 0.00044052 However, once again using these estimates for a and b along with the discrete form for the population, we obtain exactly the same estimates as column (4) of table 14.1. Although the discrete approximation is good for forecasting population, care must be exercised in its use. The original model is nonlinear. As we showed in chapter 3, for certain values of the parameters a and b the model leads to cyclical behaviour. This is not true of the discrete approximation. Regardless of the values of a and b the discrete approximation leads to an equilibrium value of a/b in the limit for some arbitrary population size which is nonzero. For instance if we consider the two formulations4: pt+i = apt - bp] = 3.2pt - 2.2p] Pt+i 't (l+a)pt 4.2pt l+bpt 1 + 2.2pt i.e. a = 3.2 and b = 2.2, then system (i) goes to a 2-cycle with values oscillating between 0.74625 and 1.16284. On the other hand, system (ii) converges very quickly on the limiting value of 1.45455. These quite different stability characteristics of the two systems are a warning about the use of approximations when dealing with nonlinear systems. 14.3 An alternative interpretation In modelling population change it is useful to consider the process from a different perspective. Population at a point in time is a stock. This stock level will change depending on the difference between the inflow and the outflow. Depending on the population under investigation there will be different factors contributing to each of these flows. For example, a typical inflow will consist of births and immigration; while a typical outflow will consist of deaths and emigration. In the case of fish populations, however, there is also the extent of the harvesting over the period. We 4 See chapter 3, section 3.9 for a derivation of the second equation above. 602 Economic Dynamics shall consider fisheries in chapter 15, and here we shall concentrate on 'natural' changes to population. We have then: Net change in population = inflow — outflow = (births + immigration) — (deaths + emigration) But births and deaths can be considered as 'internal' to the population, while immigration and emigration can be considered as coming from outside the system, as 'external' influences on the population. We can, therefore, redefine the net change in the population as composed of internal change plus external change as follows: Net change in population = internal change + external change = (births — deaths) + migration where, of course, migration is immigration less emigration. Notice that this interpretation is particularly useful for open systems, for it is only in such systems that migration can take place. For example, when considering the population of the UK we can consider the internal change in terms of births and deaths of UK citizens, and we can consider the external change in terms of the migration of the population in and out of the UK. On the other hand, if we are considering world population, then this is a closed system (at least until planetary movements of population take place!). There can be only births and deaths in a closed system. Abstracting from the many characteristics that make up a population, like age, sex, density, fertility, etc., we can think of a representative unit that contributes a net amount to the internal change in the population, which we shall label n. The population size at a point in time is p(t), and denotes the number of individuals at time t. Hence, the internal change in the population is np{t). Letting m(t) denote the migration (immigration less emigration) over the same interval of time as we are measuring the internal change, and measured at time t, then m(t) denotes the external change. Accordingly, the change in the population, dp{t)/dt is given by (14.12) — =np(t) + m(t) dt Example 14.2 (Malthusianpopulation growth) In the case of the Malthusian population growth we considered earlier, there is no migration (m(t) = 0 for all t) and population is assumed to grow at a constant rate r. In other words, the net contribution of each member is assumed to be equal to r (i.e. n = r). Hence for n = r and m(t) = 0 for all t dp(t) —— = rp(t) dt with population at time t given by p(t) = p0e' Population models 603 Example 14.3 (Logistic growth curve) Again there is assumed to be no migration and m(t) = 0 for all t. Assume, as in the Malthusian case, that a population which is not influenced by other factors grows at a constant rate r. But now further assume that there is a restraint on the growth process that is proportional to the size of the population. In other words, the growth process r is reduced by a factor r\p(t). The net internal contribution is therefore given by n(t) = r — r\p(t) Notice in particular that the internal net contribution is a function of time since it is related to the stock size of the population. Under these two assumptions about migration and net internal change, we have for the growth of the population dp — = (r- np(t))p(t) dt ( P(t)\ r = r I 1--I p(t) where k = — \ k J n which is the logistic growth equation we discussed earlier. Notice first that r is the Malthusian growth of population and k denotes the carrying capacity of the population. This version of the logistic equation will be found particularly useful when we discuss fisheries in chapter 15. For this population its size at time t is given by Pit) = —7~r~—\— (14-14) ! + (--! \Po As we shall see in the next section, this alternative view of population change will be found very useful when considering multispecies populations that interact with each other in complex ways. (14.13) 14.4 Multispecies population models: geometric analysis Consider some closed system, a habitat, in which there are just two species. These two species can interact with each other in a variety of ways. They may be: (1) independent of each other, (2) in competition with each other, (3) one a predator and the other a prey, (4) both mutually supportive of each other. If both are independent of each other then the populations will grow according to the type of laws we have already considered. In this section we are more concerned with interacting species. But before we consider each of the possible interactions in turn, we need to model the problem. 604 Economic Dynamics (14.15) (14.16) Let the two species be denoted x(t) and y(t), respectively. Then we can posit that the growth of the two species, with no migration for each species, as x = Rx(t) y = Qy(t) where R denotes the net contribution of each individual in the ^-population and Q the net contribution of each individual in the y-population. The extent to which a typical member of the ^-population contributes to the stock depends not only on births and deaths, but also on its interaction with the y-population. The same holds for the v-population. Consider a very general interaction specification, namely R = a + Bx(t) + yy(t) Q = 8 + ey(t) + $x(t) For each population, a and 8 denote the natural growth coefficient of the species. The second term denotes the over-crowding (or self-limiting) coefficient of the species. As with the logistic growth equation, if /3 and e are negative, then overcrowding will occur and the species come into competition with themselves. On the other hand, if B and e are positive, then growth expands as the population size increases, i.e., there is an increase in fertility as population expands. This we refer to as mutualism. If y and £ are both zero then the two species are independent of each other. If y and £ are both negative, then each is in competition for the limited resources of the habitat. The growth of one species is at the expense of the other. On the other hand, if y and £ are both positive, then we have a mutually supportive closed system: the growth of each species is mutually beneficial. Finally we have a predatory-prey relationship. If y is positive and £ is negative then x is the predator and y is the prey; if y is negative and £ is positive, then x is the prey and y is the predator. The predatory-prey model has been discussed in some detail in the literature, and much of it is the model of Lotka and Volterra or its extension. Given the general specifications here, then it is possible, for example, to consider models that combine over-crowding and have predatory-prey features or only predatory-prey characteristics. We now turn to each of the various models to consider them in some detail. In doing this we shall employ Mathematica to illustrate, in particular, the numerical examples in the phase plane. Some of the basic instructions for doing this are provided in appendix 14.3, which also includes instructions for using Maple. Here we concentrate on the geometric features of the modelling, leaving the mathematical analysis of such models to the next section. 14.4.1 Competition with no over-crowding Consider the following model x = [a — by]x x(0) = xq a > 0, b > 0 y = [c — dx]y y(0) = yo c > 0, d > 0 The terms —by and —dx (where we suppress the time variable) show that each species is in competition for the limited resources of the habitat. We assume the habitat represents a closed system so there is no migration. Does such a system Population models 605 have an equilibrium? Stationary values occur when x = 0 and y = 0, i.e. x = [a — by(t)]x(t) = 0 implying y = a/b ox x = 0 y = [c — dx(t)]y(t) = 0 implying x = c/d or y = 0 Hence, there are two stationary points (x*, y*) = (0, 0) and (x%, y2) = (c/<^ a/b), as shown by points Eo and Ei, respectively, in figure 14.5. Figure 14.5 also illustrates the qualitative nature of the trajectories. In this problem only nonnegative values of x and y are meaningful. Consider first the trajectories in the neighbourhood of the origin. Since y < a/b, then 0 < a — by, and so x > 0 and hence x is increasing. Similarly, x < c/d means 0 < c — dx, and so y > 0 and hence y is increasing. In fact, this specifies the nature of trajectories in quadrant I in figure 14.5. Using the same reasoning, we can summarise the properties of the four quadrants as shown in table 14.2. The trajectories are looking complex. For some trajectories in quadrant I the system seems to tend towards the equilibrium point Ei. However, if it passes into quadrant II then it moves away from the equilibrium point Ei. This is because x dominates the habitat and fertility of y is now so low that it begins to decline. A similar problem occurs if the Figure 14.5. Table 14.2 Vector properties for competition with no over-crowding Quadrant I For x < c/d then c—dx < 0, hence y > 0 For y < a/b then 0 < a—by, hence x > 0 Quadrant III For x > c/d then 0 < c—dx, hence ý < 0 For y > a/b then 0 > a—by, hence x < 0 Quadrant II For x > c/d then 0 > c—dx, hence ý < 0 For y < a/b then 0 < a—by, hence x > 0 Quadrant IV For x < c/d then 0 < c—dx, hence ý > 0 For y > a/b then 0 > a—by, hence x < 0 606 Economic Dynamics trajectory moves from quadrant I into quadrant IV. In this instance, however, species y dominates the habitat and x declines to extinction. A similar logic holds if the system begins in quadrant III. An initial situation in either quadrant II or IV simply moves the system away from the equilibrium point Ei. Example 14.4 We can try to see what is happening to this system by considering a numerical example. Consider the following competitive model x = [4 - 3y]x y=[3- x]y Equilibrium points can readily be found by setting x = 0 and y = 0, which gives two equilibrium points To highlight the stability/instability properties of equilibrium Ei (here we ignore Eo), we can consider the direction field, which is illustrated in figure 14.6. This diagram illustrates a number of features. First, equilibrium Ei appears to be a saddle path solution. Second, the possible trajectories of the system conform to those highlighted by the qualitative discussion of figure 14.5, in particular the movement of the system in the various quadrants, and the likely paths as trajectories move from one quadrant into another. Third, the movement of the system is from quadrant I into quadrants II and IV; and from quadrant III into quadrants II and IV. Fourth, it is not obvious whether any path will lead to the equilibrium point Ei. E0:(4,^) = (0, 0) Ei: (*?,>>?) = (3, 4/3) Point Ei, in particular, is the solution to the two equations 4 y=3 < x = 3 y = 0 Figure 14.6. y=0 y Population models 607 Although it was not possible to solve the nonlinear system given in the general specification of the system, we can obtain more detailed information on the properties of the trajectories in the phase-plane by noting5 dy dy/dt (c — dx)y dx dx/dt (a — by)x which uses the chain rule. We can re-arrange this expression as follows --b)dy = (--d)dx y J \x J Integrating both sides we have a In y — by = c In x — dx + k\ a In y — c In x = by — dx + k\ yax~c = keby~dx k = eh where k\ is the constant of integration. Hence yax~c k gby—dx where k is a constant. For a given value of k this solution gives the solution trajectory in the phase-plane. Example 14. 4 (cont.) Returning to our numerical example, we can use Mathematica or Maple, to plot the trajectories for various values of k. We do this using Mathematical ContourPlot command or Maple's contourplot command (see appendix 14.3). Figure 14.7 shows a number of trajectories for different values of k. The trajectories in figure 14.7 verify the general features outlined in figures 14.5 and 14.6, most especially the saddle path nature of equilibrium Ei. 14.4.2 Predatory-prey model with no over-crowding (Lotka-Volterra model) Consider the following model x = (a — by)x = ax — bxy a > 0, b > 0 y = (—c + dx)y = —cy + dxy c > 0, d > 0 In this model y is the predator and x is the prey. Notice that if the stock of x is zero, then the predator has no food and is assumed to die out, as indicated by — c. The greater the food stock, the greater the ^-population, and hence the greater the growth in the predator. On the other hand, the natural growth of the ^-stock does not depend on the predator for food and so a is positive, but it is subject to prey, 5 This is possible only for autonomous systems, see chapter 4. (14.18) 608 Economic Dynamics Figure 14.7. 1 2 3 4 5 6 x and so the greater the y-population, the more the ^-population will be subject to prey, as indicated by —b. Our first task is to establish the equilibrium of the system, to find the stationary points. We do this by setting x = 0 and y = 0 and solving for x and y. Thus x = (a — by)x = 0 implying y = a/b or x = 0 y = (—c + dx)y = 0 implying x = c/d or y = 0 Hence, there are two stationary points: (x\, y*) = (0, 0) and (x2, y2) = (c/d, a/b), represented by points Eo and Ei, respectively, in figure 14.8. Figure 14.8 also illustrates the qualitative nature of the trajectories. We can summarise the properties of the four quadrants as shown in table 14.3. It would appear, then, that the trajectories follow some sort of anticlockwise spiral. Example 14.5 To see whether this is so, consider a numerical example at this stage, namely i = (2 " Too)1 The equilibrium (other than the origin) is readily found to be (x*, y*) = (100, 200). But the much more interesting question is what is happening to the species out of equilibrium. To obtain some initial insight into this obtain the direction field for this system. This is illustrated in figure 14.9. What is apparent from figure 14.9 is that the system has a cyclical pattern around the equilibrium point Ei, and that the movement of the system is anticlockwise. Population models 609 Table 14.3 Vector properties for predatory-prey model Quadrant I For x < c/d then — c + dx < 0, hence y < 0 For y < a/b then 0 < a — by, hence x > 0 Quadrant III 2 For x > c/d then — c + dx > 0, hence ý > 0 For y > a/b then 0 > a—by, hence x < 0 Quadrant II For x > c/d then —c + > 0, hence ý > 0 For y < fl/fc then 0 < a — by, hence x > 0 Quadrant IV For x < c/d then —c + dx < 0, hence ý < 0 For y > a/b then 0 > a—by, hence x < 0 a:=0 Figure 14.8. predator =0 prey y 400 300 200 100 Figure 14.9. 200 x 610 Economic Dynamics However, we can go further into the trajectories by noting that the predator must be a function of the prey, i.e., y = f(x). By the chain rule we have dy dy/dt dx dx/dt Substituting the specific general equations, we have dy (—c + dx)y dx (a — by)x or ^--b^jdy = ^--h d^jdx Integrating both sides, we have a In y — by = — c In x + dx + k\ a In v + c In x = by + dx + k\ yaxc = keby+dx k = ek> where k\ is the constant of integration. Hence, gby+dx where k is a constant. For a given value k this solution gives the solution trajectory in the phase-plane. Once again, using Mathematical ContourPlot command or Maple's contour-plot command, we obtain typical trajectories shown in figure 14.10, which clearly illustrates the cyclical pattern of the solution. Using the information in figure 14.9 we further note that the system moves in an anticlockwise direction. Suppose, however, we concentrate on just one trajectory with the initial situation shown by point Po in figure 14.11, where Po denotes the initial point C^o, yo) = (50, 300). Point Po is in the northwest quadrant. In this situation the predator is in excess of its equilibrium level while the prey is below its equilibrium level. But because the number of predators is contracting, the number of prey will soon begin to rise as the system moves into the southwest quadrant. Once into the southwest quadrant, the number of prey begins to rise since the number of predators is too small to be a major threat. Eventually, this moves the system into the southeast quadrant, allowing sufficient prey for the predator once again to expand towards its equilibrium. However, too great an expansion in the predatory population diminishes the prey as the system moves into the northeast quadrant. From figures 14.10 and 14.11 it is clear that the trajectories form closed curves. This means that neither the predator nor the prey becomes extinct. Each species cycles between its minimum and maximum level, as illustrated in figure 14.12. This figure plots the time path of the predator, y, and the prey, x. The starting point is represented by point Pq (i.e. x = 50, y = 300), the point shown in figure 14.11. Population models 611 Figure 14.10. 0 50 100 150 X 200 prey 0 50 100 150 200 x prey What is also clear from figure 14.12 is that the predator lags behind the prey in a cyclic pattern, and because of this the stationary state is never attained. 14.4.3 Competitive model with over-crowding In section 14.4.1 we considered a competitive model in which two species were in competition for the limited resources. But suppose there is also competition within each species as well; in other words, there is the possibility of over-crowding. We 612 Economic Dynamics can capture this situation in the following model x = (a — by — ux)x y = (c — dx — vy)y where the terms — ux2 and — vy2 denote the over-crowding in the ^-species, and v-species, respectively; while —by and — dx denote the interactive competition between the two species. This system is nonlinear and much more complex than our earlier models. But we can still readily obtain the stationary points of the system by setting x and y equal to zero. This is certainly satisfied for x = 0 and y = 0, and so the origin denotes an equilibrium of the system, and the axes represent isoclines. Once again we can use the chain rule to specify the situation in the phase-plane, dy dy/dt (c — dx — vy)y dx dx/dt (a — by — ux)x We cannot solve this because the expression is not separable. We can, however, derive expressions for two further isoclines 0 when (c — dx — vy) = 0 0 when (a — by — ux) = 0 These represent two straight lines in the phase-plane, of which there are four configurations depending on the values of the six parameters, a, b, c, d, u, and v, as illustrated in figure 14.13. The markings along the isoclines indicate that x = 0 when a — by — ux = 0 implying dy/dx = oo and y = (alb) — (u/b)x y = 0 when c — bx — vy = 0 implying dy/dx = 0 and y = (c/v) — (d/v)x dy dx dy dx Population models 613 (a) (b) Figure 14.13. c/d* alu x (c) (d) alu x while above and below the isoclines we have the properties x > 0 when a — by — ux > 0 implying y < (a/b) — (u/b)x (below x = 0) x < 0 when a — by — ux < 0 implying y > (a/b) — (u/b)x (above x = 0) y > 0 when c — dx — vy > 0 implying y < (c/v) — (d/v)x (below y = 0) y < 0 when c — dx — vy < 0 implying y > (c/v) — (d/v)x (above y = 0) which are indicated by the vectors of force in figure 14.13. In the upper diagrams in figure 14.13 extinction will occur in one of the species. So long as the system does not begin at the origin, then the system will either move to equilibrium point Ei, in which the y-species dies out, or to equilibrium point E2, in which the ^-species dies out. In the lower diagrams it is also possible for the two species to coexist. Such a situation occurs where the two isoclines intersect, and is given by the solution (x*,y*) = av — be uv bd But an important question is whether such a coexisting equilibrium is a stable solution of the model. Figure 14.13(c) would suggest that E3 is not a stable equilibrium, 614 Economic Dynamics while in figure 14.13(d) E3 appears a stable equilibrium. In order to verify these results we shall continue our discussion with two numerical examples. This not only allows us to compare the two diagrams in the lower part of figure 14.13 but also to consider some trajectories in the phase-plane. In order to show such trajectories, however, we need to solve the nonlinear system using numerical solutions. We do this within Mathematica, using the NDSolve command and the ParametricPlot command. Similar plots can be derived with Maple.6 Example 14.6 x = (3 — y — x)x y = (4 - 2x - y)y The basic properties of this system are illustrated in figure 14.14, which displays the isoclines and the vectors of force in the various quadrants. These forces are based on the following observations X = 0 when 3 -y- - x = 0 implying y = 3 - - x and dy/dx = 00 y = 0 when 4 - 2x — y = 0 implying y = 4 — 2x and dy/dx = 0 X > 0 when 3 -y- - x > 0 implying y < 3 - - x (below x) X < 0 when 3 -y- - x < 0 implying y > 3 - - x(above x) y > 0 when 4 - 2x — y > 0 implying y < 4 — 2x (below y) y < 0 when 4 - 2x — y < 0 implying y > 4 — 2x(above y) 6 See Lynch (2001) for plotting multispecies models with Maple. Population models 615 Figure 14.15. This system in general leads to the extinction of one of the species, depending on the initial situation. This is clearly illustrated in figure 14.15. The trajectories in this figure required the use of a software package to solve numerically the nonlinear system of equations. What figure 14.15 clearly shows is that if the initial situation is not on the saddle path solution, then the system will tend either towards equilibrium Ei, where only the ^-species survives, or equilibrium E2, where only the v-species survives. Only in the unlikely event of the initial condition of the system being on the saddle path will the system converge to the coexistent equilibrium point E3. Example 14.7 x = (4 — y — x)x y = (6 - x - 2y)y The basic properties of this system are illustrated in figure 14.16, which displays the isoclines and the vectors of force in the various quadrants. These forces are based on the following observations X = 0 when 4 -y — x = 0 implying y = 4 - - x and dy/dx = 00 y = 0 when 6 — X — 2y = 0 implying y = 3 — ^x and dy/dx = 0 X > 0 when 4 -y — x > 0 implying y < 4 - - x (below x) X < 0 when 4 -y — x < 0 implying y > 4 - - x(above x) y > 0 when 6 — X — 2y > 0 implying y < 3 — \x (below y) y < 0 when 6 — X — 2y < 0 implying y > 3 — \x (above y) In this example, unlike the previous example, the system converges on the coexistent equilibrium point E3, so long as the system does not have an initial point equal to the other stationary values. This is illustrated quite clearly in figure 14.17, which shows a number of trajectories for this nonlinear system. It is also quite clear from figure 14.17 that this system does not have a saddle path, except for the axes, 616 Economic Dynamics Figure 14.17. corresponding to the equilibrium point (x*, y*) = (0, 0), which is an uninteresting case. We can give another interpretation to our results. First we note that a denotes the natural growth of the ^-species and c denotes the natural growth of the v-species. We can then make the following definitions: u/a the competitive effect of x on itself relative to the natural growth of x b/a the competitive effect of y on x relative to the natural growth of x v/c the competitive effect of y on itself relative to the natural growth of y d/c the competitive effect of x on y relative to the natural growth of y Population models 617 Consider, then, figure 14.13(a). Here we have c/v > a/b and c/d > a/u or b/a > v/c and u/a > d/c, i.e., the relative competitive effect of the y-species on x is greater than its relative competitive effect on itself; while the relative competitive effect of the ^-species on itself is greater than its relative effect on the y-species. Depending, therefore, on the starting position, either the ^-species will die out or the y-species will. Turning to figure 14.13(b) we have a/b > c/v and a/u > c/d or v/c > b/a and d/c > u/a, i.e., the relative competitive effect of the y-species on itself is greater than its effect on the ^-species; while the relative impact of the ^-species on the y-species is greater than the relative competitive effect on itself. Hence, depending on the starting position, one of the species will die out. Next consider figure 14.3(c) where we have c/v > a/b and a/u > c/d or b/a > v/c and d/c > u/a. In this instance the relative competitive effect of the y-species on x is greater than its relative competitive effect on itself; while the relative competitive effect of the ^-species on y is greater than on itself. This is unstable, and which species wins out depends on the initial conditions, but E3 cannot be attained unless the starting point lies on a saddle path.7 Finally, in figure 14.13(d) we have a/b > c/v and c/d > a/u or v/c > b/a and u/a > d/c, i.e., the relative competitive effect of the y-species on itself is greater than the relative impact of the y-species on x; while the relative competitive effect of the ^-species on itself is greater than the relative impact of the ^-species on y. Accordingly, the species will settle to some mutually coexistent level - namely at E3. 14.4.4 Predatory-prey model with over-crowding In sub-section 14.4.2 we considered the predatory-prey model (often referred to as the Lotka-Volterra model), which involved no competition from within the species. But suppose there are many predators and so they are in competition with themselves for the prey. Suppose, too, that the prey, besides being under attack from the predator is also in competition for the resources of the habitat from members of its own species. Consider then the most general situation of predatory-prey with over-crowding of both species in the model x = (a — by — ux)x y = (—c + dx — vy)y where the terms —ux2 and —vy2 denote the over-crowding in the prey (^-species), and the predator (y-species), respectively. The stationary points of the system are, once again, obtained by setting x = 0 and y = 0. This is certainly satisfied at the point (x*, y*) = (0, 0). Hence the origin denotes one equilibrium solution, but an uninteresting one. The other solution is found by setting the terms in brackets to zero, which provides two isoclines. x = 0 when a — by — ux = 0 implying y = (a/b) — (u/b)x y = 0 when — c + dx — vy = 0 implying y = —(c/v) + (d/v)x 1 We shall illustrate this in the next section. 618 Economic Dynamics which gives the nontrivial equilibrium Furthermore, we can use the chain rule to express the slope of the trajectory in the phase plane, i.e. dy dy/dt (—c + dx — vy)y dx dx/dt (a — by — ux)x which is nonlinear and cannot be solved. Using the isoclines, however, we can get some insight into the possible trajectories. However, this is a much more complex system than the straight predatory-prey model, and so we shall continue our discussion with a numerical example. Example 14.8 Let X = 2----A V 100 75/ V 50 200/ Then the nontrivial equilibrium point is (x*, y*) = (112.5, 50). The question arises, however, as to whether, like the Lotka-Volterra model, a closed cycle occurs around the equilibrium point. In fact, this is not the case in the present model. The fact that there is competition from within each of the species leads the system towards the equilibrium point in the limit. This is illustrated in figure 14.18, which portrays the direction field along with a number of typical trajectories in the phase plane. It is quite clear that, given the parameter values, this system will always converge on the equilibrium in the limit. Hence, point (x*, y*) = (112.5, 50) is asymptotically stable. Population models 619 Of course, it is not always the situation that the equilibrium is asymptotically stable. For different parameter values the fixed point can be asymptotically unstable (see exercise 12), but once again does not converge on a closed orbit around the equilibrium, as in the Lotka-Volterra model. This section has illustrated quite a variety of solution paths to systems involving the interaction between two species depending on whether the interaction is competitive, mutual or of the predatory-prey variety. With more than two species the variety of interactions becomes even more complex, but the nature of the solutions is basically similar. In the next section we shall consider the mathematical properties of these systems. 14.5 Multispecies population models: mathematical analysis8 In this section we shall set out the general approach, look at just two of the examples in the previous section in detail, and summarise all remaining examples. We shall then conclude with some general comments about such linear approximations to nonlinear systems. Suppose we have a general nonlinear system denoting the interaction between two species of the form x = fix, y) y = g(x, y) Suppose further that this system has at least one fixed point, denoted (x*, y*), at which x = 0 and y = 0. If we wish to consider the stability of the system in the neighbourhood of the fixed point then, following our treatment in chapter 4, we can expand the system in a Taylor expansion around the fixed point. Thus . * df(x*,y*)t y-y , , Mx*,y*), (x-x) +---iy dx dy dgix^f) dg(x*,y*), -a-(x ~x)+ -a-iy - dx dy Let fx, fy, gx and gy denote the partial derivatives, each evaluated at the fixed point (x*,y*). Then x - x* =fx(x - x*) +fyiy - y*) y -y* = gxix - x*) + gyiy - y") or, in matrix notation ~x - X* ' fx fy ~x - X* ' y-y*. _Sx gy _ y-y*. The matrix composed of elements/;,/,, gx and gy are evaluated at the fixed point and this is a square matrix9 and the system is a linear approximation of the original nonlinear system. 8 This section requires knowledge of chapter 4. 9 It is a Jacobian matrix. 620 Economic Dynamics Let Sx gy. We have already shown in chapter 4 that all the stability properties of this linear system can be established from the eigenvalues and eigenvectors of A, along with the trace and determinant of A, where tr(A) =fx + gy det(A) =fXgy -fygX These properties, however, are only local and apply only for the neighbourhood of the fixed point under investigation. For nonlinear systems with more than one fixed point, as in all the examples in the previous section, then the neighbourhood of each fixed point must be investigated individually. Example 14.4 (cont.) Example 14.4 has the nonlinear system x = f(x, y) = 4x — 3xy y = g(x, y) = 3y- xy Taking an arbitrary equilibrium point (x*, y*), then we can expand this system in a Taylor expansion around this value x-x*=Mx-x*)+fy(y-y*) y -y* = gx(x - x*) + gy(y - y) where fx = 4 — 3y evaluated at (x*, y*) fy = — 3x evaluated at (x*, y*) gx = —y evaluated at (x*, y*) gy = 3 — x evaluated at (x*, y*) We have already established two fixed points Eo = (0,0) and Ei = (3, 4/3) and we need to consider the system's behaviour in the neighbourhood of each. Take the point Eo = (0, 0). Thenfx = A,fy = 0, gx = 0 and gy = 3. Hence A-T4 °" A" 0 3 approximation Population models 621 in the neighbourhood of the origin. The eigenvalues and eigenvectors are found from A-X\ = 4-X 0 0 3 — X where det(A — XI) = (4 — X)(3 — X) = 0 with the two eigenvalues r = 4 and s = 3. Furthermore, tr(A) = 7 and det(A) = 12. (Note that tr(A)2 > 4 det(A)). For r = 4 then "0 0 " X "0" 0 -1_ y _ o_ Hence it does not matter what values x and y take in forming the eigenvector vr Let v = Next consider s = 3, then "1 0" X "0" 0 0 y _ o_ and again it does not matter what values x and y take in forming the eigenvector Vs. Since vr must be linearly independent of Vs, then let "0' 1 The general solution is, therefore, x(t) y(t) eAt + c2 and it is quite clear that this is asymptotically unstable. The situation is shown in figure 14.19, at the point Eo. For any value not the origin and in the positive quadrant will move the system away from the origin over time. (Also notice that the two independent eigenvectors form part of the axes.) Next consider the point Ei = (3, 4/3). Thenfx = 0, fy = —9, gx = —4/3 and gy = 0. Hence A = 0 -4/3 0 and our system has the linear approximation ~x - X*' 0 -9" ~x - x*' * y-y _ -4/3 0 _ * y-y _ where (x*, y*) = (3, 4/3). The eigenvalues and eigenvectors are found from A-XI -X -9 -4/3 —X wheredet(A — XT) = X2 — 12 = 0, with the two eigenvalues r = ->/T2 = 2^3 and s = — VT2 = — 2 V3. Furthermore, tr(A) = 0 and det(A)= —12. From our analysis 622 Economic Dynamics Figure 14.19. 4/3 stable arm ^ unstable arm in chapter 4 we already know that these results identify a saddle point, since the eigenvalues are real and of opposite sign.10 For r = 2v/3 then -2V3 -9 X "0" 2V3_ y. 0 giving -2V3x -9y = 0 Let x = \/3 then y = —2/3, hence V3 v = -2/3 _ For s = —2^3 then -9 " X "0" 2>/3_ y _ 0 giving 2V3x -9y = 0 Let ^ = V3 then y = 2/3, hence "V3 2/3 10 Another identifying feature of the saddle point is that det(A) < 0. Population models 623 The general solution is, therefore, x-x*~\ T V3 -2/3 y Suppose C2 ■y -- 0 then e^1 + c2 'V3 2/3 J,-Ws)t * x — X ' V3 - * y-y -2/3 _ * x — X ~v3~ y-y*. = c2 2/3 _ and so vr represents an unstable arm because the system if perturbed will move over time away from (x*, y*) along the vector vr. On the other hand, if c\ =0 then ,(-2V3> which converges on (x*, y*) over time. Hence, \s is a stable arm of the saddle point Ei = (3, 4/3). The behaviour of the system, therefore, in the neighbourhood of Ei is illustrated in figure 14.19. Unlike our analysis in the previous section, this present analysis indicates that if the system begins on the stable arm of the saddle point in the neighbourhood of the fixed point, then it will converge on the fixed point over time. However, for all other perturbations in the neighbourhood of the critical point, the system will diverge away from it. In which direction depends on how the system is disturbed, i.e., which of the four quadrants the system is moved into (but not along the arm through V). The next example has more equilibrium points to consider but the formal analysis is the same. Accordingly we shall be more succinct in our presentation. Example 14.6 (cont.) The system is x = f{x, y) = 3x — xy — x y = g(x, y) = 4y- 2xy - y2 which has four equilibrium points: E0 = (0, 0), Ei = (3, 0), E2 = (0,4), E3 = (l,2) In each case we shall consider a linear approximation of the system in that neighbourhood. The matrix of the linear system has elements fx = 3 - y - 2x gx = ~2y E0= (0,0) '3 0 0 4 fy = -X gy = A-2x-2y 3-X 0 0 4-X 624 Economic Dynamics Hence det(A — XT) = (3 — X)(4 — X) = 0 with eigenvalues r = 4 and s = 3. The eigenvectors are and and the general solution is = ci e4t + c2 and is asymptotically unstable. The behaviour of the system in the neighbourhood of En is shown in figure 14.20. In particular, since both x and y are positive then the system will move away from the origin. Ei= (3,0) A = -3 -3 0 -2 A-Xl = -(3 + X) -3 0 -(2 + X) Hence, det(A — XT) = (3 + X){2 + X) = 0, with eigenvalues r Using r = —3 the associated eigenvector is r rr V 0 -3 and s -2. while for s = —2 the associated eigenvector is Population models 625 and so the general solution in the neighbourhood of Ei is x — x y-y e~3t + c2 3 -1 -it Since both r and s are negative, then the system is asymptotically stable in the neighbourhood of Ei. E2= (0,4) -1 0 -8 -4 A- kl b2 b3~ ~*i(0~ x2(t+ 1) = S12 0 0 *2(0 x3(t + 1) 0 ■?23 0 *3(0 628 Economic Dynamics (14.21) (14.22) or more succinctly x(t + 1) = Ax(0 In general A = ~b\ 02 b3 S12 0 0 bn-l 0 bn 0 0 0 0 ... sn-hn 0 The matrix A is often called a Leslie matrix after P. Leslie who first introduced them. Of course (14.20) is just a recursive equation with solution x(0 = A'x(0) where x(0) denotes the vector of females in each zth-class in period 0. Before continuing, consider the following simple numerical example. (14.20) Example 14.9 Let 0.4, = 0.9, b2 0.8, = 0.8 b3 = 0.2 5i2 = U.y, 523 Suppose a population has 10 million females in each of the three age classes, giving a total female population of 30 million. Using the recursive relations specified in (14.19), by means of a spreadsheet we can derive the time profile of this population over, say, ten periods as shown in table 14.5(a). The ten periods cover 150 years, since the projection interval is 15 years. Figure 14.22 shows the time profile of this population in terms of the three classes. Alternatively, using result (14.22) we could derive a particular row of the spreadsheet. For example, for t = 4 we have "1.0048 .7680 .1568" .7056 .6912 .1584 .6336 .3456 .0576 x(4) A4x(0) "10" "19.2960" 10 = 15.5520 10 10.3680 which is exactly the same as the row for t = 4 in table 14.5(a). In table 14.5(b) we have computed the proportion of the total female population in each class. It should be noticed that these proportions are settling down to 42.7% in class 1, 33.7% in class 2 and 23.6% in class 3 by period 10. What we shall now illustrate is that the dominant eigenvalue of the matrix A establishes the growth rate of the population, while the eigenvector associated with the dominant eigenvalue allows a computation of the proportion to which each class stabilises. Such results are highly significant. Once we know the matrix A, it is relatively easy to establish with computer software the eigenvalues and eigenvectors. To show these properties we utilise the following theorem.11 11 This theorem itself utilises the Perron-Frobenious theorem. Population models 629 Table 14.5 Age class projections (a) Numbers t Years xi (0 X2(t) x3(f) Total 0 0 10 10 10 30 1 15 14 9 8 31 2 30 14.4 12.6 7.2 34.2 3 45 17.28 12.96 10.08 40.32 4 60 19.296 15.552 10.368 45.216 5 75 22.237 17.366 12.442 52.042 6 90 25.275 20.010 13.893 59.178 7 105 28.897 22.747 16.008 67.652 8 120 32.958 26.007 18.198 77.163 9 135 37.629 29.662 20.806 88.097 10 150 42.943 33.866 23.730 100.538 (b) Per cent t Years xi (0 X2(t) x3(f) 0 0 33.3 33.3 33.3 1 15 45.2 29.0 25.8 2 30 42.1 36.8 21.1 3 45 42.9 32.1 25.0 4 60 42.7 34.4 22.9 5 75 42.7 33.4 23.9 6 90 42.7 33.8 23.5 7 105 42.7 33.6 23.7 8 120 42.7 33.7 23.6 9 135 42.7 33.7 23.6 10 150 42.7 33.7 23.6 THEOREM 14.1 If A is a Leslie matrix of the form A = then (1) (2) (3) b\ b2 bi s12 0 0 0 0 0 bn-\ K 0 0 Sn—l,n 0 bt>0 i= I, ...n 0 < Si-ij < 1 i = 2, ...n there exists a unique dominant eigenvalue, Xd, which is positive, the eigenvector associated with the dominant eigenvalue has positive components, all other eigenvalues, Xj ^ Xj satisfy \Xj\ < Xj. Example 14.9 (cont.) The Leslie matrix for example 14.9 is A = "0.4 0.8 0.9 0 0 0.8 0.2' 0 0 630 Economic Dynamics with eigenvalues X\ = 1.14136 -0.4767 -0.26466 We see that A i is the dominant eigenvalue and | k2 \ < k\ and \k3\ < k\. The eigenvector, v1 associated with the eigenvalue k\ is v = '0.72033' 0.56800 0.39812 which clearly has all positive components. Given the three eigenvalues, the system has the general solution x(t + 1) = cik\yl + c2k'2\2 + c3k'3\3 which will be governed in the limit by the dominant root, k\. Since A.j > 1 then the system grows over time. This is clearly shown in figure 14.22. Furthermore, the growth of the system is given by k\ — 1 = 0.14136, which means a growth rate of 14.1%. Since the eigenvector v1 has elements x\(t) = 0.72033,^(0 = 0.56800 and x3{t) = 0.39812 then their sum is 1.68645 and so normalising the eigenvector by dividing each term by this sum, we arrive at the values '0.42713' 0.33680 0.23607 which in percentage terms are the values we obtained as the limiting values in table 14.5(b). Appendix 14.1 Computing a and b for the logistic equation using Mathematica When solving for a and b in the logistic growth equation do not use Solve or Nsolve command, rather use FindRoot. It is important to have 'good' initial estimates of Population models 631 a and b. First define the two equations: ln[l]:= eql := 24.135 == 13a ln[2]:= eq2 := 34.934 == 13b + (a-13b) e"50 a _13a_ 13b + (a-13b) e"100 a Then use the FindRoot command using initial estimates for a and b. In[3] —FindRoot[{eql, eq2}, {a, 0.02}, {b, 0.0004}] Out[3]= {a -» 0.020383, b -» 0.000436045} We do the same for the linear approximation 13a In [4]:= eq3 := 24.135 == - = 13b + a_13b In [5]:= eq4 := 34.934 == (l+a)50 13a 13b + a-13b (l+a)100 In[6]:= FindRoot[{eq3, eq4}, {a, 0.02}, {b, 0.0004}] Out[6]= {a -» 0.0205922, b -» 0.00044052} Appendix 14.2 Using Maple to compute a and b for the logistic equation When solving for a and b in the logistic growth equation do not use the solve command, rather use the fsolve command. Because solving can be problematic, include ranges for a and b, e.g. a = 0.02..0.03 and b = 0.0004..0.0005. First define the two equations: > eql:=24.135=13*a/(13*b+(a-13*b)*exp(-50*a)); eql := 24.135 = 13- ttb + (a - 13b)e(~50a) > eq2:=34.934=13*a/(13*b+(a-13*b)*exp(-100*a)); a eq2 := 34.934 = 13- 13b + (a- 13£)e(-100fl) Then use the fsolve command using ranges for both a and b > fsolve ({eql,eq2},{a,b}, {a=0.02..0.03,b=0.0004..0.0005}); {a = .02038301946, b = .0004360453198} We do the same for the linear approximation > eq3:=24.135=13*a/(13*b+((l+a)~(-50))*(a-13*b)); a eq3 := 24.135 = 13 a — 13b 13b+ (l+a) 50 632 Economic Dynamics > eq4:=34.934=13*a/(13*b+((1+a)~(-100))*(a-13*b)); a eq4 := 34.934 = 13 a — 13b 13b + (1 + a)100 > fsolve ({eq3, eq4}, {a,b}, {a=0.02..0.03, b=0.0004..0.0005}); {a = .02059217184, b = .0004405196282} Appendix 14.3 Multispecies modelling with Mathematica and Maple In this appendix we give detailed instructions for deriving direction fields and trajectories for example 14.4 employing both Mathematica and Maple. We also give some basic instructions for the linear approximation. All other problems in this chapter can be investigated in the same manner. Here we concentrate only on the input instructions. The equation system we are to investigate is x = (4 - 3y)x y = (3 - x)y This is a nonlinear system and cannot be solved directly by any known method. However, as we pointed out in sub-section 14.4.1, we can express the properties of x = (a — by)x a > 0, b > 0 y = (c — dx)y c > 0, d > 0 in the phase plane by plotting the solution trajectories yax~c k = ——— k a constant gby—dx Alternatively for the nonlinear system x = fix, y) y = g(x, y) we can investigate the linear approximation X fx fy ~x-x*~ y_ _gx gy _ y-y*. in the neighbourhood of a particular fixed point (x*, y*), and where/;,/,, gx and gy are evaluated at a fixed point. 14 A. 1 Mathematica To derive the contour plot in Mathematica input the following instructions: k[x_,y_]:=yA a xA (-c)/EA (b y - d x) {a=4, b=3, c=3, d=l} graphl=ContourPlot[ k[x,y], {x,0.5,6}, {y,0.5,4}, ContourShading->False, PlotPoints->50] Population models 633 The contour plot, however, does not indicate in which direction the vector forces go. For this purpose we need to invoke the PlotVectorField command and then combine this with the contour plot. Thus graph2=PlotVectorField[ {(4-3y)x, (3-x)y}, {x,0.5,6}, {y,0.5,4}] Show[graphl,graph2] There may be memory problems with showing the two graphs together. Turning to the linear approximation, the system can be investigated by means of the following input instructions roots=Solve[ {(4-3y)x==0, (3-x)y==0}, {x,y} ] eq3=(4-3y)x eq4=(3-x)y matrixA= { {D[eq3,x], D[eq3,y]}, {D[eq4,x], D[eq4,y]} }; MatrixForm[matrixA] matrixAl=matrixA /. roots[[1]] Eigenvalues[matrixAl] Eigenvectors[matrixAl] matrixA2=matrixA /. roots [[2]] Eigenvalues[matrixA2] Eigenvectors[matrixA2] Although Mathematica gives the eigenvectors for matrixA2 as '-3V3 1 r3V3' 2 1 and 2 1 these are, in fact, the same as those in the text. 14A.2 Maple The equivalent in Maple is not as satisfactory. The contour plots can be obtained using the following input instructions. with(plots): equ:=yAe*xA(-3)/exp(3*y-x); contourplot(equ, x=0.5..6, y=0.5..4, grid=[40,40]); This plot has only contour lines to the left and right of the fixed point (x*, y*) = (3, 4/3) and not above or below this value. A better rendition of the phase portrait is to utilise the following instructions. with(plots): with(DEtools): seql:=seq( [0,0.5,0.5+0.25*1], 1=0..10); seq2:=seq( [0, 6, 1+0.25*j], j=1..10); seq3:=seq( [ 0,1,0.1 + 0.1*k], k=1..10) 634 Economic Dynamics inits:={seql, seq2, seq3}; phaseportrait( equ, [x,y], 0..1, inits, x=0.5..6, y=0.5..4,arrows=THIN); Turning to the linear approximation, the system can be investigated by means of the following input instructions with (linalg) : soll:=solve( { (4-3*y)*x=0, (3-x)*y=0} ); equl:=(4-3*y)*x; equ2:=(3-x)*y; matrixA:^matrix( [ [diff(equl,x), diff(equl,y)], [diff(equ2,x), diff (equ2,y) ] ] ); matrixAl:^matrix( [ [ subs(soil[1],diff(equl,x) ), subs (soil [ 1], diff(equl,y) ) ], [subs ( soil [ 1], diff(equ2,x) ) , subs (soil [ 1], diff (equ2,y)) ] ] ); eigenvals(matrixAl); eigenvects(matrixAl); matrixA2:^matrix( [ [ subs (soil[2],diff(equl,x) ), subs (soil [2], diff(equl,y) ) ], [ subs ( soil [2], diff(equ2,x) ) , subs (soil [ 2], diff (equ2,y)) ] ] ); eigenvals(matrixA2); eigenvects(matrixA2); These instructions produce the same results as with Mathematica, with the same eigenvectors that, as indicated above, are the same as those in the text - which can readily be verified. Exercises 1. Given the following data for population in England and Wales over the period 1701-91, obtain the estimated population using the continuous Malthusian population model and compare your results with those provided. Why do you think the estimated population under-estimates the actual population in 1791? Year 1701 1711 1721 1731 1741 1751 1761 1771 1781 1791 Population (million) 5.8 6.0 6.0 6.1 6.2 6.5 6.7 7.2 7.5 8.3 Source: Tranter (1973, table 1). 2. Two countries, A and B, have populations of equal size, po, and are growing at the same net rate of 2% per annum. However, population A has a birth rate of 3% per annum and a death rate of 1% per annum Population models 635 3. 4. 5. 6. 7. while country B has a birth and death rate of 5% and 3%, respectively. Unfortunately, country A suffers a major spread of AIDS and its death rate rises to 2% per annum. Assuming both populations conform to the Malthusian model, how long will it take for the population of country B to be twice the size of country A? A population has births b, deaths d and migration m each growing exponentially. If b < (d + m), how long before the population is half its original size? (i) If a population conforms to the Malthusian population model and is growing at 3% per annum, how long will it take for the population to treble in size? (ii) Derive a general formula for the time interval necessary for an increase in population to grow by X. times its initial size, assuming it is growing at some general rate k% per annum? For the logistic equation p = p(a — bp) a > 0, b > 0 expand this as a Taylor series around the equilibrium a/b and hence show that the population in the neighbourhood of the equilibrium can be expressed Show that as t oo then p —► a/b. What does this imply about the achievement of equilibrium? Suppose p = p(a + cp) a > 0, c > 0 (i) Explain this equation. (ii) Draw the phase line for this population and show that the population tends to infinity. (iii) Derive an explicit solution for the population and use this to show that an infinite population is reached at a finite point in time. A population is thought to have the feature that if it falls below a minimum level, m, then it will die out and that there is a maximum carrying capacity of M for the same population. (i) Given an intrinsic growth rate of r, discuss the usefulness of p = r(M — p){p — m) to describe this population. (ii) Compare (a) p = rp(M — p) (b) p = r(M — p)(p — m) For the Gompertz equation p = rp(a — \xvp) a > 0 (i) Solve the equation subject to p(0) = po. (ii) Sketch this graph and its associated phase line. 636 Economic Dynamics 9. (iii) Obtain the fixed points and establish their stability/instability. (iv) What happens to p as t oo? Solve the following system for two competing species x and y x = -3y(t) y = -9x(t) 10. and derive explicitly the phase line. Trout, species T, and bass, species B, are assumed to conform to the following model T = a (1 - — ) T- bTB B = c\\--)B- dTB 11. Analyse this model in detail using a graphical analysis. Suppose N(t) denotes the biomass of halibut in the Pacific Ocean. It has been estimated that for the equation N0 + (K- N0)e-rt Note: N(t)/K = N0/(N0 + (K - N0)e~rt) = (N0/K)/((N0/K) + (1 -(No/ K))e~rt) r = 0.71 per year and K = 80.5 x 106 kg. If the initial biomass is one-quarter of the carrying capacity, (i) What is the biomass 2 years later? (ii) What is the time at which the biomass is (a) half the carrying capacity? (b) three-quarters of the carrying capacity? 12. In each of the following systems which describes the interaction between two species of population x and y, (i) Find the stationary values. (ii) Linearise each system in the neighbourhood of all critical points. (iii) Find the eigenvalues and eigenvectors for each linearisation and describe the nature of the critical point. (iv) Try to establish the nature of the system by plotting sufficient tra- jectories (a) x = x + 100 xy 13. (c) x = x — x2 — xy y = y — 2xy — 2y2 Consider the following discrete numerical predatory-prey model, where x is the prey and y is the predator xt+i -xt= 1.4(1 -yt)xt yt+i -yt = 0.6(1 - Ayt + xt)yt (i) Establish the critical points. (ii) Find the linearisation coefficient matrix, A, for each critical point. Population models 637 (iii) Establish the eigenvalues and eigenvectors of A. (iv) Set up the system on a spreadsheet and establish the limit value of x and y as t oo. 14. Investigate fully the discrete dynamical system xt+i = l.3xt - 0.3x2 - 0A5xtyt yt+i = l.3yt - 0.3;y2 - 0A5xtyt (i) Showing in particular that four critical points exit. (ii) Linearising the system about each critical point. (iii) Establishing the behaviour of the system by considering sufficient trajectories in relation to (i)-(ii). 15. The American bison population can be sub-divided into three categories (Cullen 1985): calves, yearlings and adults. These are denoted^i(r), *2(0 and x^it), respectively. In each year the number of newborns is 42% of the number of adults from the previous year. Each year 60% of the calves live to become yearlings, while 75% of yearlings become adults. Furthermore, 95% of adults survive to live to the following year. (i) Write out the system as a set of difference equations. (ii) Draw a state diagram for this population. (iii) Show that this system has one real eigenvalue and two conjugate complex eigenvalues. (iv) What is the eventual growth rate of the bison population? (v) What proportion of calves, yearlings and adults does this system settle down to? Additional reading Additional material covered in this chapter can be found in Boyce and DiPrima (1997), Braun (1983), Caswell (2000), Deane and Cole (1962), Haberman (1977), Hoppensteadt (1992), Lynch (2001), Meyer (1985), Mooney and Swift (1999), Renshaw (1991), Sandefur (1990), Tranter (1973) and Vandermeer (1981). CHAPTER 15 The dynamics of fisheries In this chapter we consider a renewable resource. Although we shall concentrate on fishing, the same basic analysis applies to any biological species that involves births and deaths. A fishery consists of a number of different characteristics and activities that are associated with fishing. The type of fish to be harvested and the type of vessels used are the first obvious characteristics and activities. Trawlers fishing for herring are somewhat different from pelagic whaling.1 In order to capture the nature of the problem we shall assume that there is just one type of fish in the region to be harvested and that the vessels used for harvesting are homogeneous and that harvesters have the same objective function. Because fish reproduce, grow and die then they are a renewable resource. But one of the main characteristics of biological species is that for any given habitat there is a limit to what it can support. Of course, harvesting means removing fish from the stock of fish in the available habitat. Whether the stock is increasing, constant or decreasing, therefore, depends not only on the births and deaths but also on the quantity being harvested. The stock of fish at a moment of time denotes the total number of fish, and is referred to as the biomass. Although it is true that the biomass denotes fish of different sizes, different ages and different states of health, we ignore these facts and concentrate purely on the stock level of fish. But like any renewable resource, over an interval of time the stock level will change according to births, deaths and harvesting. We shall deal with harvesting later. For the moment we shall concentrate purely on the biological characteristics of the fish stock. Our first aim is to represent the biological growth curve of a fishery. 15.1 Biological growth curve of a fishery We assume that the growth rate of the fish stock, denoted ds/dt, is related to the biomass (the stock level), denoted s. Although stock size and the growth in stock size are related to time, in what follows we shall suppress the time variable in the stock. Thus, the instantaneous growth process for fish can be represented by the equation ds (15.1) -r=/(s) See Shone (1981, application 11) for a review of pelagic whaling. The dynamics of fisheries 639 which is a representation of the births and deaths of the species in the absence of harvesting. In order to take the analysis further we need to assume something about the biological growth curve. A reasonable representation is given by the logistic equation, which we discussed in chapter 14. Thus The coefficient r represents the intrinsic instantaneous growth rate of the biomass, i.e., it is equal to the rate of growth of the stock s when s is close to zero. More importantly, the coefficient k represents the carrying capacity (or saturation level) of the biomass, i.e., it represents the maximum population that the habitat can support. This follows immediately from the fact that the stock size will be a maximum when ds/dt = 0, i.e., when s = k. In what follows we assume that both r and k are constant. These, and other features of the logistic equation representing fish growth, are illustrated in figure 15.1. In the upper section of the diagram we have the growth curve represented by the logistic equation, while in the lower section we draw the equation of the stock size against time, i.e., the solution equation (15.2) (15.3) Figure 15.1. msy carrying capacity 0 •f{s)=rs[\-{slk)} biotic potential environmental resistance time 640 Economic Dynamics In the lower section we also draw the curve denoting the biotic potential. This curve represents the growth of a species in which there is no negative feedback from overcrowding or environmental resistance. In other words, the biotic potential denotes the exponential growth curve (15.4) s = s0ert The shaded region in the lower section of figure 15.1 shows the environmental resistance, which increases sharply after the inflexion point. The environmental resistance occurs because of the carrying capacity of the habitat. From the upper diagram in figure 15.1 it is clear that the growth function has a maximum point. At the stock size denoted smsy the growth of the fish stock is at a maximum, and is referred to as the maximum sustainable yield. The maximum sustainable yield is readily found. Since the growth curve is at a maximum, then we can establish this maximum by differentiating the growth curve and setting it equal to zero to solve for smsy and then substituting this value 'mtof{s). Thus /(,) = ra(-i) + r (1-0=0 —rs + r(k — s) = 0 (15.5) _ k Smsy — ~Z i.e. the stock level of fish at the maximum sustainable yield of a particular species is exactly half of its carrying capacity. Furthermore, given the logistic equation, equation (15.2), the growth function is symmetrical about smsy. The importance of the maximum sustainable yield is in relation to harvesting. The situation is shown in figure 15.2. With no harvesting, the species will be in The dynamics of fisheries 641 biological equilibrium when there is no growth, i.e., when ds/dt = 0. There are two biological equilibria, s* = 0 and s* = k, as shown in figure 15.2. Any stock size above zero and below the carrying capacity will lead to positive growth and hence an increase in the stock. Any stock level above the carrying capacity will lead to excessive environmental resistance, and hence to a decline in the stock size. (This naturally follows since above s* = k, ds/dt < 0, and hence the stock size must be declining.) Now consider three constant levels of harvesting, h\,h2 and h3. Harvesting level h\ is above the growth curve, which means that the fish are being extracted faster than they can reproduce. The fish will accordingly be harvested to extinction. At a harvest level of h2, the harvest line just touches the growth curve at the maximum sustainable yield. What does this imply? Suppose the fish stock begins at the level of the carrying capacity. There will be no natural growth in the fish stock, but there will be a level of harvesting equal to h2, which will result in the fish stock declining. When the fish stock declines to the maximum sustainable yield, then the natural growth in the fish population is just matched by the level of harvesting, and so this level of fish stock can be sustained perpetually. However, if the fish stock should fall below the maximum sustainable yield, then the rate of harvesting will exceed the natural population growth, and the fish stock will decline, and extinction will eventually result. This suggests that a management policy to harvest at the level h2 is not necessarily a sensible one, especially with the uncertainty involved in estimating fish stocks. If harvesting were at a level of h3, there are two possible equilibria, s* and s2, given by the stock levels where the harvesting line cuts the growth curve. Both s* and s2 represent sustainable yields. This is because at each of these stock levels the growth rate equals the rate of harvesting, and so the fish stock will remain constant. There are a number of characteristics of the fishery in this instance: (1) IfO < s < then harvesting exceeds natural fish growth, and the species will decline to extinction. (2) If s* < s < s2 then harvesting is less than the natural fish growth, and so the stock size will increase, and increase until s2 is reached. (3) If s > s2 then again harvesting exceeds the natural fish growth and so the fish stock will decline until it reaches s2. (4) Any deviation of the stock size away from s j will lead to a further movement of the fish stock away from this level, either to extinction or to the level s2. Accordingly, s^ denotes a locally unstable equilibrium. (5) Any deviation of the stock size around the level s2 will lead to the stock size changing until it reaches s2. Hence, s2 denotes a stable equilibrium. Consider the following discrete form of the model2 Ast+1 = st+1 - st = rst (l -j) ~ ht (15-6) which can readily be investigated by means of a spreadsheet. Let r = 0.2 and k = 1000; further assume h = 20 for all time periods. The results are shown 2 It is well known that the discrete form of the model can produce far more complex behaviour than the continuous model depending on the value of r. See Sandefur (1990). 642 Economic Dynamics Figure 15.3. dsldt 50 20 0 s Xs)=0.2s[l-(jf/1000)] in figure 15.3. An initial stock size equal to the carrying capacity leads to an equilibrium stock size of s = 887 approached from below (i.e. s falls). On the other hand, an initial stock level equal to s = 500 (the msy stock level), leads to the same equilibrium but approached from above (i.e. s increases). In fact any initial stock in excess of s = 113 will lead to the stable equilibrium. An initial stock level below 5=113 leads to extinction.3 Any constant level of harvesting in excess of 50 will automatically lead to extinction. We can generalise the problem by letting h(t) denote the harvesting function, then the net growth in the fish population is given by Suppressing the time variable for convenience, this can be expressed more simply as Equilibrium is established by setting equation (15.7) equal to zero, which gives the steady-state equilibrium. The steady-state equilibrium is at the maximum sustainable yield only if the harvest is at the level h2. One of the simplest models, developed by Crutchfield and Zellner (1962), is to assume that the harvest level is partly determined in a demand and supply market. Demand for fish is determined by price while the supply of fish is determined by price and by fish stocks. The market is assumed to clear, which determines the 3 Solving the quadratic rs(\-s/k) = h gives s\ = 112.702 and s*2 = 887.298. ds(t) = f(s(t)) - h(t) dt (15.7) The dynamics of fisheries 643 harvest. Thus the model is ds / s\ — = rs 1--) — h dt V k) qd = <2q — a\p ao > 0, a\ > 0 (15.8) qs = b\p + b2s b\ > 0, b2 > 0 qd = qs = h From the market equations we can eliminate the price and solve for the harvest function in terms of stock levels. This gives aQbi + axb2s h = h(s) = —-—■- (15.9) b\ + CL\ which is linear with positive intercept and positive slope. As before, the model is captured by superimposing the harvest function on the biological growth curve, as shown in figure 15.4. Example 15.1 Using the following numerical discrete version of the model on a spreadsheet st+l = st + 0.2 (l--—) + \ 1000/ qf = 45-pt q\ = l.2pt + 0.05st 1? = 1t= h the two equilibrium values are readily found to be s\ = 172 and s^ = 715. At the stable equilibrium value the price is found to be approximately p* = 4.2, with 644 Economic Dynamics Figure 15.5. dsldt.h 41 0 equilibrium harvest of h* = 41.4 The stable equilibrium of the model is illustrated in figure 15.5. Although this model relates the harvest to the fish stock, through the supply equation, no account is taken of the number of vessels employed. A consequence of this is that no account is taken of the profitability of the fishing to the fishermen, and hence no account is taken of entry into and exit from the industry. In order to consider such possibilities we need to consider the harvesting function in more detail. 15.2 Harvesting function The harvesting function, or catch locus, is a form of production function of the fishermen. Considered from this point of view, the harvest function is the catch at 4 Using the continuous form of the model it is readily established, employing a software programme suchasMathematicaoiMaple,\hats* = 171.736,4 = 714.628,/?(.j*) = 4.213 and ^) = 40.787. j(s)=0.2s[l-(s/\000)] The dynamics of fisheries 645 h^aes-, Figure 15.6. h=aesi time t, denoted h(t), as a function of the inputs. We assume the inputs are of two kinds. First, the stock of fish available for catching, s(t); and, second, the 'effort' expended by the fishermen, e{t). Effort is here an index of all inputs commonly used for fishing - such as man-hours, trawlers, time spent at sea, nets, etc. Again we suppress the time variable and simply write the harvesting function as h = h(e,s) (15.10) A common, and very simple, harvesting function used in the literature is h = aes a > 0 (15.11) where a denotes the technical efficiency of the fishing fleet. This function is illustrated in figure 15.6, where we have drawn the harvesting (measured in terms of numbers of fish) against effort. For a given stock size, harvesting is a constant fraction of effort. It follows therefore that the marginal product to effort is constant and equal to the average product with respect to effort. Also shown in figure 15.6 is that for a higher stock size, the harvesting function is to the left, i.e., for given effort (eo say) the catch size is greater the greater the stock of fish in the habitat (h2 > h\ if 52 > S\). Although a common harvesting function, this is but a special case of the Cobb-Douglas type harvesting function that allows for a diminishing marginal product to effort and to stock size. Such a function would be h = aeasp a>0, 0 < a < 1, 0 h\ if e2 > e\). Second, as drawn in figure 15.7, we have a constant marginal product with respect to fish stock (in terms of the Cobb-Douglas function this means Now that we have outlined the harvesting function we can return to consider equilibria. Steady-state equilibria requires that ds/dt = 0, hence it requires the condition where we have suppressed the time variable. The situation is illustrated in figure 15.8. First consider effort at level e\ with a corresponding harvesting function h\. There are two equilibria, an unstable equilibrium at s = 0 and a stable equilibrium at s = s\. On the other hand, if effort is raised to the level e2, with the corresponding harvesting function h2, then the stable stock equilibrium falls to s2. But another feature is illustrated in figure 15.8. As drawn the harvesting function hi and h2 both yield the same level of harvest at the respective equilibrium stock sizes. Economic efficiency would imply that the same harvest level would always ß = l). (15.13) f(s) = h(s) The dynamics of fisheries 647 h, dsldt Figure 15.8. h2=ae2s 0 As) s hl—aels be undertaken at the lowest effort. This means that harvest function h\ would be chosen over harvest function h2. In fact, on the grounds of economic efficiency, no effort would be employed which led to a harvesting function resulting in an equilibrium fish stock size less than the maximum sustainable yield. However, we have so far assumed open access to the fishery by all companies. But are there any circumstances where an equilibrium in open access to the fishery would be at a stock level below the maximum sustainable yield? To answer this question it must be recalled that open access involves no restrictions on companies harvesting in the locality under study or of new firms entering the industry (or firms leaving the industry). What is clearly missing from the analysis so far is any consideration of profits to the industry. 15.3 Industry profits and free access We simplify our analysis by assuming that the unit cost of effort expended is constant. Let this be denoted w, and can be considered as the 'wage' for effort. Then the total cost is given by Turning to revenue, we assume that all fish are sold at the same price, denoted p. Hence, with the total fish caught being h, it follows that total revenue is TC = we (15.14) TR = ph = paes (15.15) It follows, then, that profits for the industry are it = TR-TC = paes — we = (pas — w)e (15.16) 648 Economic Dynamics What is the shape of the TR and TC functions? We wish to construct TR and TC against effort. TC is linear since TC = we and w is assumed constant. TR is less straightforward. As effort rises we have already established that the stock size in equilibrium falls. Return to figure 15.7. At zero effort the harvesting function lies along the horizontal axis and the stock size is at the level k, but total revenue is zero. As effort rises the harvesting line shifts left, h rises and, withp constant, total revenue rises. Once effort has risen to a level such that s = smsy, then total revenue must be at a maximum since the harvest is at the maximum level. Effort beyond this means a fall in harvesting and a fall in total revenue. Hence, TR takes a similar shape to f(s), adjusted by the factor p. More formally TR = ph. But in equilibrium rs (1--J — aes = 0 V k) / ae\ ors = k{1--) Hence we can express h = aes as a function of e / ae\ h = aek yl--J ak = —(r — ae)e r Hence pak (15.17) TR = ph = -—(r — ae)e r which is quadratic in e. It is readily established that for this TR function: (1) TR = 0 at e = 0 and e = r/a (2) TR is a maximum at e = r/2a. The situation is shown in figure 15.9. Figure 15.9 highlights two other features. A rise in the 'wage' to effort shifts the total cost function to the left. Second, a rise in the price of fish shifts the total revenue function up, but still passing through the points e = 0 and e = em. The results on the profits function are illustrated in figure 15.10. In figure 15.11(a) assume effort is at the level e2. At this level of effort TR exceeds total cost (TR2 > TC2). There are excess profits in the industry and there will be entry by firms to take advantage of the excess profits. The increase in firms is captured in this model by an increase in effort. Entry will continue in open access while total revenue exceeds total cost. As effort rises with entry, total revenue will rise initially beyond TR2 but will then fall. Furthermore, as effort rises we move along both TC and TR. Effort will rise (entry will continue) until effort level e\ is reached, where TR = TC. In figure 15.11(b) we note that at effort level e\ we have the harvest function h\ and the equilibrium stock is s\ which is less than Smsy. What we have established here is that although for the same harvest lower effort would be the most efficient, with open access and free entry, effort would be established at level e\ and the equilibrium stock size s\ < smsy. In other words, effort will always be adjusted until profits reduce to zero because only then will The dynamics of fisheries 649 TR,TC s s / S \ „ TC,=w2e / / / 1 --~ ^ ^ \ ___.TC,=w,e TR2=p2/ 0 dt (15.18) = v(pas — w)e Our fisheries model with open access can therefore be captured by means of two dynamic equations s = rs (1--) — aes v k/ (15.19) e = v(pas — w)e where we have used the dot notation to denote the derivative with respect to time. The two variables under consideration are the stock size, s, and the amount of effort, e. In equilibrium both variables must be jointly determined. When out of equilibrium, equations for s and e will determine the dynamic path taken. To this we now turn. The situation is shown in figure 15.12. We measure the stock size on the horizontal axis and effort on the vertical axis. Our first problem is to determine the equilibrium paths. In equilibrium we know that s = 0 for equilibrium stock size e = 0 for equilibrium effort First consider the effort equilibrium. Entry will occur until profits are zero, at which point effort is zero. There is only one stock size consistent with this result, 652 Economic Dynamics namely w (15.20) s* = — ap This is shown by the vertical line in figure 15.12. Now consider a stock size less than w/ap. In this case pas < w (or pas — w < 0), and so losses are being made. Firms leave the industry and so effort is reduced. In other words, to the left of the vertical line there is a force on effort to fall. Similarly, to the right of the vertical line pas — w > 0 and so profits are being made and firms enter the industry with a resulting increase in effort. Hence, to the right of the vertical line there is a force on effort to rise. These forces are shown by the vertical arrows in figure 15.12. Now consider the stock equilibrium. We derive this as follows s = rs (l — — aes = 0 (15.21) r{k-s) = kae r / r \ a \ka/ Result (15.21) indicates that the equilibrium situation for stock size is linear with a negative slope. The intercept on the effort axis is given by (r/a), the slope is given by —(r/ka) and the intercept on the stock axis is k. Consider next points either side of this equilibrium line. Above the stock equilibrium line we have the condition e > r / r \ a \ka/ aes > rs (1--) V k) which means the harvest exceeds the natural stock growth for a given stock size. This, in turn, means that the stock size will fall over time. Hence, above the stock equilibrium line the forces are shown by arrows pointing to the left. By similar reasoning, points below the stock equilibrium line lead to growth in excess of harvesting for any stock size, and so the stock size will increase over time. Hence, below the stock equilibrium line the forces are shown by arrows pointing to the right. All these vectors of forces are illustrated in figure 15.12. Finally, we can readily establish the equilibrium stock size and effort level by solving the two linear equations. The equilibrium stock size is given immediately as s* = w/pa and the equilibrium effort is readily found to be equal to r ( w (15.22) e* = -\\ kap/ One final observation to make concerning the dynamics is the slope of the path when it crosses either equilibrium line. In the case of the vertical line at any point on this line effort is unchanging and so the trajectory must have a zero slope when crossing the vertical line. Similarly, a trajectory crossing the stock equilibrium line must have the stock size unchanging; hence the trajectory must have no slope when crossing this line. The trajectory over time must depend on the starting position of the species. Suppose, for illustrative purposes, that the species is at the level of its carrying The dynamics of fisheries 653 capacity, i.e., s = k, as shown in figure 15.13. At this stock level profits are to be had and entry will occur. This entry will raise the effort of the industry and will simultaneously reduce the fish stock. Because profits are initially large there is a sizable entry into the industry that pushes effort beyond its eventual equilibrium. As a result, the path must cross the vertical line (which it does with a zero slope). In other words, the system moves from quadrant I into quadrant II. In this quadrant harvesting is still above the natural growth level and so the stock size is continuing to fall. On the other hand, profits are now negative and some firms will be leaving the industry, resulting in a reduction in effort. However, the reduction in effort results in the system moving from quadrant II into quadrant III (cutting the stock equilibrium line with a zero slope). In quadrant III losses are still being made and so effort is falling, but the reduction in effort leads to less harvesting and a rise in the stock of fish. This pushes the system into quadrant IV. Now profits are again positive and firms will enter the industry resulting in increased effort. Furthermore, the harvesting is less than the natural growth and so the stock size will be rising. It follows, then, that the trajectory of the system over time is shown by the heavy line, showing a counter-clockwise spiralling path. Although figure 15.13 illustrates a stable spiral we have implicitly assumed certain values on the parameters in the construction of the diagram. This can best be noted by considering the mathematical properties of the system (equations (15.19)). In order to do this, however, we need to make two adjustments to the dynamical system. First, we consider percentage changes rather than simply changes. Hence, we need to divide the first equation by the stock size, s, and the second equation by the effort, e. Using hats to denote percentage changes, then our dynamic system 654 Economic Dynamics takes the form (15.23) (15.24) s = r t--I — ae V k) e = v{pas — w) Next we note that in equilibrium the percentage changes are zero, hence 0 = r 1 ae s = — I - ) (s — s) — a(e — e) 0 = v{pas — w) Subtracting the equilibrium conditions from these equations gives e = vpa{s — s) The matrix of this system is 0 vpa where the trace and determinant of A are tr(A) = --k det(A) = a vp A stable spiral, as indicated in table 4.1 (p. 180), requires three conditions to be met (1) tr(A) < 0 i.e. tr(A) = -(r/k) < 0 (2) det(A) > 0 i.e. det(A) = a2vp > 0 (3) [tr(A)]2 < 4det(A) It is the third condition that we have implicitly assumed in graphing figure 15.15. This requires the condition (if «= 4fl2l'p to be met. In order to see this issue, we shall now consider a numerical example. 15.5 The dynamics of open access fishery: a numerical example Example 15.2 Consider the following numerical example of the open access fishery s = 0.5s (l - —) - 0.005es V 200/ p = 25, w = 4, v = 0.02 The dynamics of fisheries 655 The two dynamic equations are, then s = 0.5s (l - —) - 0.005es V 200/ e = 0.02(0.125es-4e) Which gives the two equilibrium lines s = 32 e = 100 - 0.55 and the equilibrium solutions s* = 32 and e* = 84. The diagrams consistent with these results are illustrated in figure 15.14. The stable spiral in this example is readily established. The matrix A, its trace and determinant are -0.0025 -0.005" 0.0375 0 -0.0025 = 0.0001875 from which it readily follows that not only are the first two conditions for a stable spiral met, but so is the third condition, since [tr(A)]2 < 4det(A). A change in p or w The model readily illustrates the result of either a change in the wage rate or a change in the price level. Neither of these changes does anything to the stock equilibrium line. Only the effort equilibrium is altered. Thus, a rise in the price of fish will result in the effort equilibrium line shifting to the left. Assuming the system was initially in equilibrium, the result is shown in figure 15.15, where effort rises and the fish stock falls. The assumed trajectory is shown by Ti. For example, in our numerical example if the price of fish rises from p = 25 to p = 32, then the system will settle down at s* = 25 and e* = 87.5. On the other hand, a rise in the wage paid to effort will shift the effort equilibrium line to the right. The situation is shown in figure 15.16, where effort falls and the fish stock rises. The assumed trajectory is shown by Ti. For example, in our numerical example, if the wage rate rises from w = 4 to w = 10, then the system will settle down at s* = 80 and e* = 60. Figure 15.15 and 15.16 highlight a potential misleading result if concentration is paid only to equilibrium values. The equilibrium stock size will lead to extinction only if the price rises infinitely. But this ignores the dynamic behaviour out of equilibrium. If the trajectory of the system is that shown by T2 in figure 15.15, then extinction occurs before some positive equilibrium stock size can occur. In the case of trajectory T2, the rise in the price of fish leads to a glut of firms entering the industry. The rise in effort that results leads to excess harvesting and fish harvested faster than they can reproduce, so leading to extinction. The system never reaches its eventual equilibrium! A = tr(A) = det(A) = 656 Economic Dynamics Figure 15.14. h A=o.42s 25 0 TR,TC 625 336 50 84 s (C) The same issue can arise with a rise in the wage rate. This is shown by trajectory T2 in figure 15.16. Effort is reduced to zero and the fishing industry effectively collapses before the new equilibrium can be reached. One of the parameters of significance in these last two results is v, which is a reaction coefficient of the industry to profits and losses. The larger v, then the more likely are the results shown by trajectory T2 in figures 15.15 and 15.16. This is because the larger v, then the more firms will enter the industry when profits are positive. This means that effort rises more for any given size of fish stock, hence, the more steep the trajectory resulting from a rise in the price of fish. Similarly, if wages rise, then the resulting losses lead to a more rapid exit from the The dynamics of fisheries 657 Figure 15.15. Figure 15.16. industry and a relatively greater reduction in effort. The resulting trajectory is fairly steep. The converse of these results is that the smaller the value of the reaction coefficient, v, the more likely the system will converge to its equilibrium without oscillations, shown by trajectory T3 in figures 15.15 and 15.16. 658 Economic Dynamics 15.6 The fisheries control problem In the next section we shall discuss school fisheries, i.e., fish populations that shoal in large numbers. However, before we can do this we need to consider the optimal control problem more closely. So far we have ignored the facts that profits are spread over time, they need discounting, and fish left in the sea is forgone revenue for the fisherman. In this section we shall consider only a continuous model formulation, leaving the discrete form as exercises. Since the aim of the fisherman or agency is to maximise discounted profits subject to the biological growth function, we have a typical control problem as outlined in chapter 6, sections 6.1-6.3. Knowledge of these sections is required for the present section and the next one. To recap before we consider fisheries, a typical maximisation principle problem is to5 -ti max nax / V(x, u, t)dt + Fix1, t\) M Jtn (15.25) s.t. x =f(x, u, t) x(t0) = x° x(t\) = x1 by a suitable choice of the control variable u. Here V(x, u, t) is the objective function, Fix1) is the value of the terminal state, x = f(x, u, t) denotes how the state variable changes over time, while x(to) = x° and x{t\) = x1 denote the initial and final values of the state variable. In solving this maximisation principle problem, we form the dynamic Lagrangian. V(x, u, t)dt + Fix1) + / k[fix, u, t) - x]dt -.) Jtn / Jtn h [Vix, u, t) + kfix, u, t) — kx]dt + Fix1) We further define the Hamiltonian function Hix, u, t) = Vix, u, t) + kf(x, u, t) which implies L = f [Hix, u, t) - kx]dt + Fix1) J t0 and using /t\ rt\ Xxdt = I xkdt — [k(ti)x(ti) — k(to)x(to)] ■ j J to (see exercise 2, chapter 6) then L = f [Hix, u, t) + Xx]dt + Fix1) - [k(h)x(h) - kit0)xit0)] Jtn 5 In chapter 6 we assumed Fix1, t\) = 0 and so X(t\) = 0. Here we assume a nonzero terminal state x(t\) = x1 with value -FXx1). The dynamics of fisheries 659 The necessary conditions for an (interior) solution are, then, dH (i) — = 0 t0 < t < h du dH (ii) * = —^- to 0 then fi > 0 ds i.e. there is need of a capital gain to compensate for the loss of interest forgone. The third equation is simply the constraint. A steady state requires e = 0, s = 0 and fi = 0 15.7 Schooling fishery It is well known that some fish move in shoals for purposes of migration, reproduction or to fend off predators. Although schooling activity is a defence against natural predators, it makes them especially vulnerable to human predation. Modern equipment means shoals are easy to locate and the fish easy to catch. This means that the stock size has little impact on the catch so dh/ds = 0. We can, therefore, define the catch function as h = h(e, s) = h(e) and the three necessary conditions 6 See Neher (1990, chapter 9). 662 Economic Dynamics as (i) (p - ix)h'(e) = w (15.30) (ii) /x = [8 -f(s)]fx (iii) s=f(s)-h(e) where we retain the assumption that the fish sell at a constant pricep and the 'wage' per unit of effort, w, is constant. The steady state requires three conditions to be met e = 0, s = 0 and fx = 0 The dynamics of the problem is solved in stages. First, two variables are chosen whose dynamics are 'solved' in terms of the phase-plane. Both the steady state (equilibrium) and out-of-equilibrium situations can be depicted. The third variable is then considered in the light of what is occurring with these two. The common approach is to consider the phase-plane in terms of fish stock, s, and its shadow price ix.1 In continuing our analysis we shall assume some specific functional forms. The catch locus we shall assume is (15.31) h = aeb a > 0, 0 < b < 1 while the biological growth f(s) we shall assume is logistic (15.32) f(s) = rs(i-ls) where k is the carrying capacity and r is the intrinsic growth. Using these specifications h'{e) = abeb~l Irs f(s) = r- — k and the three necessary conditions are (i) (p — ix)abeb~l = w (15.34) 2rs S-[r-- k (15.33) (u) £ = (iii) s = rs (l--^ — aeb If fi = 0 then 2rs", S-[r- — )=0 2r which is shown by the vertical line in figure 15.17. If fx > 0 then s > k(r — S)/2r, and fx is rising. Similarly, when s < k(r — S)/2r then fx is falling. We can conclude 7 This is the resource cost of the fish caught as distinct from the price at which the fish sells on the market. The dynamics of fisheries 663 ^=0 hjr-b) ' 2r Figure 15.17. that for s < s* then fx is falling8 while for s > s* then fx is rising. These results are shown by the vector forces in figure 15.17. In order to derive the second isocline s = 0, we first need to eliminate effort, e. From (15.33)(i) we have b{p — \x)ae h-l W .". e w i 6-1 _ab(p — ix) Substituting this into (15.33)(iii) gives s = rs (1--) — V k) w b 6-1 _ab(p — fi) If s = 0, then the above expression is equal to zero, which is a quadratic in s. There is little gain in solving this for \x in terms of s explicitly. We can, however, express it implicitly. Define (15.35) 0(5, ix) = -rs (l - 0 + a Then for a turning point dd> Irs — = -r+- =0 ds k w ab(p — fx) b 6-1 (15.36) i.e. s a2 k 2 920 2r —T = — > 0 ds2 k 8 For s* > 0 then S must be less than r, i.e., the discount rate must be less than the intrinsic growth rate. 664 Economic Dynamics hence it is a minimum. The relationship between /x and s for s = 0 is shown in figure 15.18. If s > 0 then (p(s, /x) is above the curve, i.e., above s = 0 and s is rising; while below 5 = 0, s is falling. These forces are represented by the arrows in figure 15.18. We can now combine all the results, as shown in figure 15.19. The fixed point, the equilibrium point, is given by (s*, /x*), which occurs at the intersection of the two steady-state conditions fi = 0 and s = 0. They lead to four quadrants with vector forces illustrated by the arrows. The typical trajectories arising from this problem are illustrated in figure 15.20, indicating the presence of a stable saddle path SiSj and an unstable saddle path S2S2. 666 Economic Dynamics s = v(s, ß), then dv(s*, ß*) dv(s*,ß*) (s - s ) H----(ß - ß ) ß ds dß dg(s*,fJL*)r dg(s\ß*) -7.-(s - s ) H----(ß - ß ) äs dß Or s = (0.2 - 0.004s*)(s - s*) + 0.97652(10 - ß*)(ß - ß*) ß = 0.004ß*(s - s*) + (-0.1 + 0.004s*)(ß - ß*) i.e. s = 0.1(5 - s*) + 2.34375(/x - /x*) (i = 0.0304(5 - s*) Writing the equations in matrix form, we have 0.1 2.34375] [ s-s* 0.0304 0 JL/x-/x* where the matrix of the system, A, is given by s ß A = 0.1 2.34375 0.0304 0 from which we can readily compute tr(A) = 0.1 det(A) = -(0.0304)(2.34375) = -0.07125 Since det(A) < 0, we know from chapter 4 that we must have a saddle point. This is also verified if the characteristic roots are of opposite sign. The roots of the characteristic equation are tri A I ± Vtr(A) -4det(A) 0.1 ± V(0.1)2 -4(-0.07125) ki = 0.32157 k2 = -0.22157 Using the first solution we have 0.1 2.34375 0.0304 0 which leads to the relationship ix = 5.2375 + 0.09455 Using the second solution we have ' s — s* = 0.32157 ' s — s* ß — ß* ß — ß* 0.1 2.34375 0.0304 0 s — s ß — ß* -0.22157 s — s ß — ß* The dynamics of fisheries 667 Figure 15.21. Figure 15.22. which leads to the relationship /x = 11.03 -0.13725 These saddle path solutions, along with the isoclines s = 0 and jx = 0 are shown in figure 15.21. Finally Mathematica was used to produce a number of trajectories, as shown in figure 15.22. In doing this we employed the linear approximation results, i.e., we used the NDSolve command on the simultaneous linear differential equations s = 0.1(5 - s*) + 2.34375(/x - /x*) fi = 0.0304(5 - s*) where (s*, /x*) = (25, 7.6). Notice that this linear approximation was reasonable. The stable and unstable saddle paths are given by ix = 11.03 -0.13725 ix = 5.2375 + 0.09455, respectively. Given s = 10, then the corresponding points on the saddle paths are /xq = 9.658 and /xo = 6.1825, respectively. Taking the trajectory through the point (10,9.658) did indeed lead to a trajectory straight towards the equilibrium (5*, ix*). Similarly, for the point (10,6.1825) the trajectory moved directly away 668 Economic Dynamics Figure 15.23. s0 s s\ k s from (s*, ix*). The same is true for /xo = 4.17 and /xo = 9.9625 on the stable and unstable saddle paths for s = 50. All other initial points were taken off the saddle paths. It is quite clear from figure 15.22 that the numerical results support our earlier qualitative results. What meaning can we give to these results? All the trajectories in figure 15.22 represent solution paths, in the sense that each satisfies the three conditions listed above. The decision-maker, whether it be the fishery manager or the manager of an agency, begins with a stock size sq at time t = 0. Suppose this is below s*, as shown in figure 15.23. The manager has control over effort, e(t), and by implication over the shadow price /x. In particular, the manager can control the initial shadow price /xo = /x(0). Given (so, /xo) then the particular trajectory starting at this point will move the system to reach sT. If the terminal time T is also a choice variable, then the manager has to choose both /xo and T. Consider first the choice of T. If T is finite, then sT would be a target. But this implies that net benefits beyond sT are of no concern to the manager. Since, under free choice, this is unlikely, then the only logical possibility is for T to be infinite. Now consider all infinite planning horizons and the choice of /xo. With an initial stock so, three possible choices are shown by /Xq, /Xq and /Xq. /Xq belongs to the stable arm of the saddle path S\S\. Hence, the trajectory is along this path tending to (s*, /x*) in the limit.10 A choice of /Xq (above S\S\) implies a shadow price of fish in the sea higher than /Xq. Fishing effort is not so great and the fish stock rises until it reaches its own biological carrying capacity of k. On the other hand, an initial shadow price of /Xq leads to increased effort and to eventual extinction. The trajectories emanating from /Xq and /Xq cannot, therefore, be maximising paths. Since the paths are infinite, the trajectory never actually reaches fi*. The dynamics of fisheries 669 Only /Xq is optimal. The same logic holds if sq is initially above s*, at so = s'Q, /Xq is the only optimal path. The conclusion we arrive at is that the optimal path is to choose an initial point on the stable arm of the saddle path. Although this result is appealing, there is here a warning. If the shadow price is incorrectly estimated, then a divergent path is the most likely outcome - either raising the stock size to its carrying capacity or diminishing the species to extinction. This is certainly the problem facing a fishing agency which is attempting to balance profit and conservation. 15.8 Harvesting and age classes 11 So far we have assumed the fish are homogeneous and in particular have not made any distinction between males and females or age composition. As we pointed out in chapter 14, section 14.6, we can model female populations in terms of age classes. In doing this here for fish populations, we shall simplify drastically and concentrate on sustainable harvesting. Let Xi(t) denote the population of fish in the zth-age class just before harvesting and assume harvesting takes place at discrete time intervals. Throughout we will consider just three age classes. As in section 14.6, let bj denote the birth rate for the zth-class and Sy the survival rate from class i to class j. This gives the Leslie matrix in the present example for i = 1, 2, 3 of b\ b2 b3 s12 0 0 0 s23 0 and without harvesting we have bi b2 S12 0 _ 0 S23 or ~Xx(t+ 1)~ x2(t + 1) = _x3(t + 1)_ b3 0 0 "*i(0" x2(t) x3{t) x(t + 1) = Lx(0 (15.37) Now suppose hf are harvested (killed) for the age class i (i number of females harvested is 1, 2, 3). Then the where h\xi{t) h2x2{t) h3x3{t) or Hx(0 H = hi 0 0 0 h2 0 0 0 h3 This section is based on the analysis in Lynch (2001, chapter 13). 670 Economic Dynamics (15.38) (15.39) This means that HLx(r) will be the total harvested and the remaining population will be x(t + 1) = Lx(0 - HLx(0 = (I - H)Lx(f) If harvesting is to be sustainable across the age classes, then we require x(t + 1) = x(t), i.e. x(0 = (I - H)Lx(f) Let M = (I — H)L, then M is also a Leslie matrix, and by theorem 14.1 (p. 629) there is a unique dominant eigenvalue kd which is positive. Also recall from section 14.6 that the population will then grow at a rate kd — 1. If we require the population to stabilise (no growth), then we require = 1 and there is a nonzero vector solution of (I - H)Lx(0 = kdx(t) = x(0 where x(t) is the eigenvector associated with kd = 1. This means that we require the dominant eigenvalue of the matrix M = (I — H)L to equal unity. In our present example of z = 1, 2, 3 we have M = (I - H)L = 1 -hi 0 0 0 l-h2 0 0 0 1 -h3 (\-h{)b2 (1 (1 - h2)s12 0 0 (1 - h3)s23 Since kd = 1 is to be the dominant root, then we require |M - kd\\ = |M - I| = 0 which will impose restrictions on the h-values. We have b\ b2 b3 512 0 0 . 0 523 0 - h{)b3 0 0 IM-II (l-Äi)fci-l (\-h{)b2 (l-h)b3 (l-h2)s12 -1 0 0 d-h3)s23 -1 i.e. [(l - höh - l] + (l - h)b2(i - h2)Sl2 + (1 - h{)b3[(\ ~ h2)(\ - h3)sl2s23] = 0 (15.40) or (1 - Äi)^i + (1 - Äi)(l - h2)b2sl2 + (1 -Äi)(l -h2){\ -h3)b3sl2s23 1 Only values of hf(i =1,2, 3) lying in the range 0 < hf < 1 will satisfy (15.40) in order to produce a sustainable policy. Once ht{i = 1, 2, 3) are found, the eigenvector of M associated with = 1 can be computed. Finally, from this the normalised vector can be obtained. The dynamics of fisheries 671 Example 15.4 A fish species is divided into three age classes with a Leslie matrix 0.4 0.8 0.9 0 0 0.8 0.2' 0 0 We have already solved for the eigenvalues of this in example 14.9, where we found ki = 1.14136, k2 = -0.4767, k3 = -0.26466 with dominant root k\ = 1.14136 and associated eigenvector 0.72033' 0.56800 0.39812 and associated normalised eigenvector 0.42713' 0.33680 0.23607 If no harvesting takes place, therefore, the fish population grows at kj — 1 = 0.14136 or about 14% every period. Furthermore, the female population will settle down at 42.7% in age group 1, 33.7% in age group 2 and 23.6% in age group 3. Now consider four harvesting policies: (i) Uniform harvesting h\ = h2 = h3 = h (ii) Harvesting only the youngest age class h\ ^ 0, h2 = 0, h3 = 0 (iii) Harvesting only the middle age class h\ = 0, h2 ^ 0, h3 = 0 (iv) Harvesting only the oldest age class h\ = 0, h2 = 0, h3 ^ 0 (i) Uniform harvesting h\ = h2 = h3 = h Under this policy, and given the Leslie matrix above, equation (15.40) becomes (1 - h)(0A) + (1 - /i)2(0.8)(0.9) + (1 - /i)3(0.2)(0.9)(0.8) = 1 with solution h = 0.123855. Given this value for h, the matrix M takes the form M„ = 0.350458 0.788531 0 0.700916 0.175229' 0 0.700916 0 0 and the associated eigenvector is v = 0.720329' 0.568001 0.398121 while the normalised eigenvector is 0.427127' 0.336803 0.236070 672 Economic Dynamics (ii) Harvesting the youngest age class h\ ^ 0, h2 = 0, h3 = 0 Under this policy, and given the Leslie matrix above, equation (15.40) becomes (1 - h)(0A) + (1 - /n)(0.8)(0.9) + (1 - /n)(0.2)(0.9)(0.8) = 1 with solution h\ = 0.208861. Given this value for h\, the matrix M takes the form Mi 0.316456 0.632911 0.158228' 0.9 0 0 0.8 0 0 and the associated eigenvector is v = 0.655347' 0.589812 0.471850 while the normalised eigenvector is x«i 0.381679' 0.343511 0.274809 (Hi) Harvesting the middle age class h\ = 0, h2 ^ 0, h3 = 0 Under this policy, and given the Leslie matrix above, equation (15.40) becomes (1)(0.4) + (1 - /*2)(0.8)(0.9) + (1 - /i3)(0.2)(0.9)(0.8) = 1 with solution h2 = 0.305556. Given this value for h2, the matrix M takes the form M, 0.4 0.8 0.2' 0.625 0 0 0 0.8 0 and the associated eigenvector is v = 0.78072' 0.48795 0.39036 while the normalised eigenvector is X«2 0.470588' 0.294118 0.235294 (iv) Harvesting the oldest age class h\ = 0, h2 = 0, h3 ^ 0 Under this policy, and given the Leslie matrix above, equation (15.40) becomes (1)(0.4) + (1)(0.8)(0.9) + (1 - fc3)(0.2)(0.9)(0.8) = 1 There is no solution for hj that lies in the range 0 < hj < 1. The normalised vectors xu, x^iand xn2 determine the long-term distribution of the fish in the different age categories. If, however, it was considered beneficial to leave as many of the young age fish in the sea as possible, then pursuing a policy of catching middle age fish should be undertaken. In this case, the female population of the youngest age class would settle down at about 47%. The dynamics of fisheries 673 Exercises 1. A discrete-time production function takes the form ht = st(l - Eae') where E is the exponential term. Show that this function exhibits diminishing returns to effort and constant returns with respect to stock size. 2. For the Gompertz growth function (k f(s) = rs In I - derive the sm level and the growth at this level. 3. Given the following two growth functions, f(s) and g(s), respectively, and the same harvest function h(e, s), f(s) = rs(l- S-) (k g(s) = rs In I -h(e, s) = aes (i) Show, after eliminating the stock size, that the yield to effort (h/e) is linear and log-linear, respectively. (ii) Demonstrate that ordinary least squares estimates of the parameters in (i) are not sufficient to determine the three parameters a, r and k. 4. In the following numerical model s = 0.2s (l--—) - h V 1000/ qd = 45 - p qs = 1.2;? +0.05s qd = qs = h establish whether it is possible for a regulatory authority to set the price to clear the market at a stable equilibrium which has a yield equal to the maximum sustainable yield. 5. A typical fisheries problem can be captured by the following three fundamental relationships, where the first denotes the biological growth process of a fish stock; the second a harvesting function (or catch function); while the third denotes the profits of a fishing agency. g = ks(su - s) h = aes tt = ph — we 674 Economic Dynamics where: 8 growth of fish stock k = stock-specific parameter s = fish stock maximum sustainable fish population Ii = harvest level a technology efficiency parameter i = per unit of 'effort' expended in fishing P = price per unit of fish w = "wage" rate (i) Interpret these three equations in detail using sketches where possible. For the remaining questions assume: g = 0.5^(25 - s) h = 2.5es (ii) Given the level of effort is e = 2. (a) Establish the maximum sustainable yield (msy) and the stock size at this level. (b) Establish the steady-state value of the stock size. (c) Explain why the steady-state value of s exceeds the msy value. (iii) Now assume that 'effort' is not known and must be determined along with the stock size. Assume p = 0.6 and w = 12. (a) Establish the equilibrium values of stock size and effort that maximise profits. What is this level of profits? What is the growth rate? (b) Under the same price and wage levels, calculate the profits under the condition e = 2 (i.e. use your answers in (ii)) and compare them with the present level of maximum profits. (c) Explain why the stock size in this maximisation problem is greater than under (ii)(b). (iv) Undertake the calculations in question (iii) (a) with the following values for the respective parameters: (a) p = 0.8, w = 12 and a = 2.5 (b) p = 0.6, w = 15 and a = 2.5 (c) p = 0.6, w =12 and a = 3 Discuss the implications of these results. (v) Suppose entry and exit occurs according to the following rule: e = V7T = v(ph — we) where the parameter v denotes the speed of entry and exit in response to profits. For v = 5 ,p = 0.6 and w =12, establish the stock level associated with equilibrium (i.e. no entry or exit). Establish the vector of forces each side of the 5 = 0 phase line drawn with e on the vertical axis and s on the horizontal axis. The dynamics of fisheries 675 (vi) For a steady-state solution s = 0, and using p = 0.6 and w = 12, show that e is a linear function of s, and hence establish the vector of forces either side of this phase line. (vii) Using the results in (v) and (vi), establish the equilibrium for stock size and effort. Show that for a starting stock size of so = 25, the dynamic path gives a stable oscillation to equilibrium. Verify this result by establishing the characteristic roots of the dynamic system and the trace and determinant. (viii) Discuss this fisheries model with entry and exit commenting, in particular, on the importance of the parameter v. (i) In the following model s = 0.2s (l--—) - h V 1000/ h = 0.125es what is the level of effort that should be set by the regulatory authority to have a stable equilibrium with the maximum sustainable yield? (ii) Show that the level of effort obtained in (i) is the level that maximises 125 -(0.2-0.125e)e 0.2 (iii) What is the stock size which maximises profits in equilibrium if TR = ph and TC = we, with p = 0.4 and w = 10? Show that in the case of open access fishery, a rise in either the price of fish or the productivity of the fishing industry leads to an increase in effort and a reduction in fish stocks. Show that if the aim is to ■ r max / Ttdt I TIC Jo :f(S)-h where it = ph — we, and p and w are constant and h = h(e, s). Then the first-order conditions for an interior maximum are: dh (l) (p - k)— = w de dh (ii) k = (k-p)—-kf(s) ds (iii) s =f(s)-h(e,s) where k is the Lagrangian multiplier (the shadow price of the resource). In a fishery the fish can be divided into three age groups, each one a year long. The Leslie matrix for this population is L = 0 3 40 0.1 0 0 0 0.5 0 (i) What is the growth rate of the fish population if no harvesting takes place? 676 Economic Dynamics (ii) What is the long-run behaviour of the system if 25% of each age class is harvested? 10. What is the optimal sustainable harvesting policy for the system in question 9, given the youngest age class is not harvested and groups two and three are harvested to the same extent? Additional reading Additional material on the contents of this chapter can be obtained from Conrad (1999), Conrad and Clark (1987), Crutchfield and Zellner (1962), Cunningham, Dunn and Whitmarsh (1985), Dasgupta and Heal (1979), Fisher (1981), Hamilton (1948), Hartwick and Olewiler (1986), Hilborn and Walters (1992), Lynch (2001), McVay (1966), Neher (1990), Peterson and Fisher (1987), Shone (1981) and Smith etal. (1977). Answers to selected exercises Chapter 2 4 5 7 8 11 12 13 14 15 16 17 18 p{i) = pfke-a{e'k-V p(Q) = po and p —► ea as r —► oo (i) v = (iii) V 1 cex + 1 1 — cx (ii) v = ^ — c (i) >,= __jt2_|_JC_|_l (Ü) y =\± y/A+x^+2^+2x (i) v = + c (ii) v = 1 —x + 1 + ce (iii) 1 + ce -2x (iii) £2,910 J 3 Table about 1220AD 1990BC (i) r (ii) P(t) = P0ert 14 years (i) x = 3 (repellor), ^ = — 5 (attractor) (ii) fc* = 0.908 (ii) 7(0 = Y0e(s/V)t D(t) = D0+ -Y0(ert - 1) r 1.11% (a) = 'Rich' Years to 'Poor' Years to countries double countries double France 26 China 29 Japan 14 India 35 West Germany 28 Uganda -347 UK 35 Zimbabwe 347 USA 50 (b) Years for the population to halve. 678 Answers to selected exercises 19 (a) 23.32% (b) 3 years (2.97) (c) 0.7506 x 1010 20 £292,507.52 5 7 8 10 16 17 18 Chapter 3 (i) second-order, linear, autonomous, nonhomogeneous (ii) second-order, linear, autonomous, nonhomogeneous (iii) first-order, linear, autonomous, homogeneous (iv) second-order, linear, non-autonomous, nonhomogeneous Pt = (l+r)Pt.1 -R n ( R\ R pn = (i + ry [Po--) + - \ r / r Pn = (l + r)nP0 - (1 + r)""1^! - (1 + r)n-2R2 -(1 +r)Rn_x -Rn (i) yn = yo 1 2|-^| +2, stable 3V 4 / 3V 4 (n) yn = yo[--J — - I —- 1 + -, unstable (iii) yn = yo(-W - 3(-l)B + 3, cyclical (iv) yn = y0 - 6 + 6, stable (v) yn = yo (-IT - \ (-If + i cyclical (i) a3 - a2 - a + 1 = (a + \)(a - l)2 8 1 (i) P'= 2~ 3P'-U Stable (ii) pt = 5.5 — Pt-\, oscillatory (iii) pt = 16 — 3pt-i, unstable Yn = a + I + G (i) Pt = (i) Pt l-b a — c + bn Yo- a + I + G l-b , stable if 0 < b < 1 d\ (de^ k(b + d) Pt-i + X(a — c) a — c (ii)^=^ + J' £248.85 22.25 minutes _ ad + be H b + d Answers to selected exercises 679 19 20 _ Xp 1 + nxo (ii) Mathematica (ii) Maple xn _ 2-l-n[-{\ - V5)1+n + (1 + V5)1+n] v5(-2t^)" , ^(-2TT7fJ 1 - V5 1 + v/5 Chapter 4 7 8 v = "1" v2 1 1 ■ v — -1 W = 4 (i) A.1 = 1, X2 = 2, A. 3 = 5 1 " 0 " "0" V1 = -3/2 , v2 = 1 , v3 = 1 1 -1 2 3 (ii) y{t) = --Cle' + c2e2' + c3e5t z(t) = cxe' - c2e2t + 2c3e5' (iii) W = 3 (i) (r, s) = (-2, -4), x(t) = c\e~2t — c2e~4t y(t) = cie~2t + c2e-4t "1" "-1" _1_ 1 improper node (ii) (r, s) = (1,-2), vr = "4" "1" _1_ Vs = _1_ '-iß' Vs — "72" 1 i v — 1 jc(f) = 4cie; + c2e 2t saddle point y(t) = ae' + c2e~2t (iii) (r, 5) = (2i, -20, V *(0 = c\ sin(2r) + c2 cos(2r) centre y(t) = 2c i cos(2r) - 2c2 sin(2r) (iv) (r,s) = (-l+/,-l-/), Y x(t) = c2e~' sm(t) — c\e~' cos(r) spiral y(t) = c\e~' sin(/) + c2e~' cos(t) —i z 1 Vs = 1 680 Answers to selected exercises 10 (i) Pj = (6,0) and P2 = (0.846,2.114). (ii) Fixed point P2, exhibits a stable limit cycle 12 (i) (p*, 7*) = (7.055, 26.221) (ii) Yes. 13 System has limit cycles that shrinks as /3 rises. 14 System has limit cycles that expand as a rises. 11 12 Chapter 5 (i) (a) (r, s) = (i, -i) (b) v' (c) D —i i _ 1 _ _1_ i 0 0 -i (ii) (a) (r, s) = (-3,-1) (c)D = (b) v' "-1" = "1" 1 1 -3 0 0 -1 (iii) (a) (r, s) = (2,-1) (b) Y = (c) D = "4" "1" _1_ vs = _1_ 2 0 0 -1 (i) trace =11, determinant =14 3 5 (ii) (iii) 4 2 11 -5 1/2 -1 1/2 -2 (iv) r = +Jl, s = for itiA 3 1 3 1 n-r = - + -VT7, 2 2 s =---v 17 for mB 2 2 v V 1 1 r 2 -V7 - -3 3 5 1 4 ~ 4 = V* = 1 r 2 —V7 - -3 3 1 - + -VT7 4 4 for itiA for mB (v) X2 - 7 Mathematica xt -9 + ( 5 + - ) (-0' + ( 5 l- + l- ) ((-5 + 5/) + (10 + 0(-0' - (1 + 10iV) Answers to selected exercises 681 Maple xt = -9 + 5(-i)' + 5i' + ^K-i)f ~ ^!' 5 9 ,11 , 9 , 11 , yt =---h -(-0 H--i(-i) + -i--H y 2 4 4 4 4 '-i 0" ~l-i 1 + f _ 0 i _ , v = 1 1 14 (i) (jc*, y*) = (-9, —), 4-period cycle results 15 J = "-0.5 0 V = " o r 0 -0.4 _ __4 4_ Chapter 6 1 (i) Minimum ABEHJ = 12, (ii) JHEBA, hence same. 8 x(t) = 5cos(1.118030 + 8.68811sin(1.118030 y(t) = 3.2697 cos(l.118030 + 0.688441 sin(l. 118030 5e-'(-3e + 2e~2 - 2e2t + 3e1+2t) e2 - 1 \0e~\-9e + 6e2 - 2e2t + 3e1+2t) 1 - e2 10 c = (OAk-01 - 0.1067)c jc = k03 - 0.05k - c k* = 6.6047, c* = 1.4316 9 x(t) = y(t) = Chapter 7 1 k2 = 3.5458 2 ,i = (ii^)(_L + _L + ..._1|+Al lim Xk =--h k\ Chapter 8 (i) pt = 60 — \.5pt-\, oscillatory and divergent (ii) pt = 15 + 0.5pt-\, convergent (iii) pt = 60-pt-u cyclical 28 4 (iv) pt = —— 2Pt~1, oscillatory and divergent (i) pt = 24 - 14(-0.25)' (ii) pt = 30 - 20(-0.9)' (i) pt = 4.5 — 0.2p,_i, cyclical and convergent (ii) pt = 100 + 0.2/?,_i, convergent 682 Answers to selected exercises 5 (i) P* = 4, Pt = 4- 0.5(p,_i - 4), stable P* = io, pt = 10 + 2.5(^_! - 10), unstable (ii) P* = 6, a = 6 + 0.5(a_i - 6), stable P* = S, pt = 8+ 1.5(^-1-8), unstable 6 (i) P* = 3.5, 14/3 Pt 1 + 2SX (Po ~ 1), stable Pt = + I 1 - y \Pt-i (i) p = r(P- P*) - R'(h*)(h - h*) h = g'(P*)(P - P*) -(d + n){h - h*) Chapter 9 9 — 2a\ + a.2 (i) q\ = q2 = 3 9 + a\ — 2^2 ~~3 (ii) Cournot solution (q*, q*) = (|, |) (i) (q*,q*,q*) = (1,1,1) (Ü) = 2 - |^2,f-l - 2"r) = (44.39, 16.19) (ii) (y,r) = (38.12, 15.06) (iii) (y, r) = (41.49, 14.75) (i) (y, r) = (52.447, 20.223) s fixed, s 1.7640145 (y, r) = (50.663, 19.316) n variable, s = 1.33 (ii) (y, r) = (42.894, 13.447) s fixed, s 1.7640145 (y, r) = (43.544, 13.772) n variable, s = 1.9195 Chapter 13 2 AM(r* = 12) p =111.1-0.15 4 (i) GM0 : pt = 95.122 + 0.02445, AM0 : pt = 103.5938 - 0.06255, (s*,p*) = (97.5, 97.5) (ii) GMj : pt = 100 + 0.02445, AMj : pt = 108.90625 - 0.06255, (s*,p*) = (102.5, 102.5) 7 (ii) C(s,p) = (104.07407, 100) 11 (i) 5 = -200 + 25 (ii) 5 = 100 12 5 = (m - p*) - ky + u(r* + X - 7t*) Chapter 14 2 69 years -In 2 Answers to selected exercises 685 4 5 9 10 11 12 13 (i) 37 years (ii) Ink years P^-a(p-l) P«) = l + (po-l)e-at a lim pit) = -, hence equilibrium never achieved in finite time period. (iii) p{t) = -a c — kae~ ■at k = constant of integration m(c/ak) pit) = oo at t = -which is finite —a -a(e-"-l) (i) pit) = p% e~ (iii) p = 0, unstable; p = ea, stable (iv) lim pit) = ea t^oo y = \/3x2 + c a ( a \ b= - - — rforr = o b \bkx) b = k2 - I — j t for b = 0 (i) 46.67 x 106 kg (ii) (a) 1.27 years (b) 3.095 years (i) (a) Ei = (0,0), E2 = (25, 100) (b) Ei = (0, 0), E2 = (0, 1), E3 = (0.375, 0.25) (c) Ei = (0,0), E2 = (1,0), E3 = (0,0.5) (ii) (a) oscillations around E2 = (25, 100) (b) limit cycle around E3 = (0.375, 0.25) (c) competing predators at E2 = (1,0) (i) Ei =(0,0) E2 = (3, 1) E3 = (0, |) ru ni r n -4 (ii) Ai "1.4 0 " " 0 -4.2" _ 0 0.6 A2 = 0.6 -2.4 1.05 0 0.15 -0.6 (iii) Ei =(0,0), (r, s) = (0.6, 1.4) "0" vs — "1" _1_ , v — 0_ E2 = (3, 1), (r, s) = (-1.2 + 1.039/, -1.2 - 1.039/) -1.732 + 2/ i -1.732- 2/ —/ E3=(0,i), (r, s) = (-0.6, 1.5) "0" 1 1 v 0.09 _ 686 Answers to selected exercises 14 15 (i) Ei = (0, 0) 3 E2 = (1,0) (ii) Aj — 0 10 0 3 To J — o E3 = (0, 1) E4 -3 -3 to 20 0 — (3' 3) 3 20 3 -3 ' A4_ -1 L 20 10 J L 10 (i) Xl(t+l) = 0.4*3(0 x2(t+ 1) = 0.6zi(0 x3(t + 1) = 0.95*2(0 + 0.75*3(0 (iv) 10.5% (v) 24%, 13% and 63% 3 20 J ■1 -1 10 -1 Tj Chapter 15 1 dA = St(aE-a^ > 0, ^ = -a2stE~ae- < 0 det def dht ne d2ht — = l-E-ae', -^-=0 dst dsf k rk 2 sm = —, f(sm) = — where E is the exponential E E h [a2k\ 3 (i) - = ak — I - e for f(s) e \ r J In I - I = ln(afc) — I - I e for g(s) (ii) Use h/e = a + fie and \n(h/e) = a + fie. 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(1999) Mathematica: A System for Doing Mathematics by Computer, 4th edn., Redwood City: Addison-Wesley Author index Abell, M.L., 25 Allen, R.G.D., 114n, 141,374 Arrowsmith, D.K., 84, 200 Attneld, C.L.F., 494, 518 Azaraides, C, 4, 16, 25, 250, 483n, 518 Baker, G.L., 321 Barro, R.J., 275n, 330n, 374 Baumol, W.J., 121, 141, 298, 320, 321, 374 Beavis, B., 200, 285 Benhabib, J., 298, 312-314, 320, 321 Berry, J., 84 Blachman, N., 25 Blackburn, K., 285 Blanchard, O.J., 455, 458-459, 464n, 469 Blume, L., 217, 233, 250, 264n Borelli, R.L., 84, 200 Boyce, W.E., 37, 84, 162-163, 181, 200, 637 Braselton, J.R, 25 Braun, M., 49n, 84, 200, 637 Brock, W.A., 16, 25, 310n, 321 Brown, D.R, 25 Bryson, A.E. Jr., 285 Buchanan, N.S., 374 Buiter,W.H.,592 Bullard, J., 25 Burbulla, D.C.M., 25 Burmeister, E., 84, 275n, 285, 495, 518 Butler, A., 25 Cagan, P., 500-502, 505, 518 Carter, M., 518 Caswell, H., 637 Chiang, A.C., 114n, 141, 157n, 200, 250, 264n, 285, 355n, 374 Clark, C.W.,260, 261n Cole, W.A., 595, 637 Conrad, J.M., 260, 261n, 262n, 285 Coombes, K.R., 25 Copeland, L.S., 530n, 552, 592 Crandall, R.E., 25 Crooke, P.S., 25, 264n Crutchfield, J.A., 642, 676 Cunningham, S., 676 Dasgupta, P.S., 676 Davies, S., 53, 84 Day, R.H., 16, 25.31-34, 321 Deane, P., 595, 637 Demery, D., 494, 518 Dernburg, T.F., 552, 592 Devitt, J.S., 25 Diamond, P.A., 508, 510, 518 DiPrima, R.C., 37, 84, 162-163, 181, 200, 637 Dobbs, I., 200, 285 Dobell, A.R., 84, 275n. 285, 495, 518 Dodson, C.T.J., 25 Domar, E.D., 83, 141 Don, E., 25 Dornbusch, R., 541, 553-554, 579, 592 Duck, N.W., 494, 518 Dunn, M.R., 676 Eckalbar, J.C., 333, 374 Elaydi, S.N., 88-89, 118, 141, 217, 250 Ellis, W.J., 25 Ezekiel, M., 332, 346, 374 Farmer, R.E.A., 7. 25, 141 Fisher, C.A., 676 Flaschel, P., 182, 200 Ford, J.L.,592 Frenkel, R., 592 Friedman, J., 375, 423 Frisch, H., 518 Fryer, M.J., 285 697 698 Author index Gander, W., 25 Gandolfo, G., 337, 374, 375, 423 Gapinski, J.H., 141, 552 Gärtner, M, 552, 554n, 564n, 592 Gehrig, W., 423 George, D.A.R., 495, 497, 518 Giordano, F.R., 36n, 37, 84, 165n, 200 Gleick, J., 321 Glynn, J., 25, 301, 302, 319, 321, 338, 367n Goldberg, S., 141,250 Gollub, J.R, 321 Goodwin, R.M., 336-337, 374 Gray, T.W., 25, 301, 302, 319, 321, 338, 367n Greenman, J.V., 285 Gregory, M., 455n, 469 Griffiths, H.B., 84, 118n, 141, 250 Grilli, V., 330n, 374 Groth, C, 484, 487, 518 Gulick, D., 308-310, 321 Haberman, R., 637 Hamilton, J.E., 676 Hartwick, J.M., 676 Heal, G.M., 676 Heck, A., 25 Henderson, J.M., 423 Hicks, J.R., 125-126, 141 Hilborn, R., 15, 321, 676 Ho, Y., 285 Holden, K., 493, 518 Holmgren, R.A., 19n, 141, 250, 341 Hommes, C.H., 293, 310, 315, 321, 363n, 374 Hoppensteadt, F.C., 637 Hrebicek, J., 25 Huang, C.J., 25, 264n Intriligator, M.D., 285 Jeffrey, A., 84, 141,200 Jones, C.I., 83 Jong, F.J. de, 8n, 25 Judge, G., 262n Karakitsos, E., 552 Keiper, J.B., 25 Kelley, W.G., 141,250 Kesley, D., 321 Keynes, J.M., 6 King, D.N., 330, 374 Kirk, D.E., 285 Kofier, M., 25 Kreyszig, E., 25 Krugman, P., 484, 489, 490, 518 Leonard, D., 285 Leslie, D., 493, 628 Li, T.Y, 302n, 303 Long, N.V., 285 Ludsteck, J., 135n Lynch, S., 84, 301, 306, 306n, 321, 614n, 637,669n,676 MacDonald, R., 592 Maddock, R.,518 Mahajan,V., 53, 84 Malliaris, A.G., 16, 25 Mandelbrot, B., 286-287, 321 Mankiw, N.G., 358, 374 Mas-Colell, A., 182, 200 May, R.M., 288n, 321 McCafferty, S., 469, 478n, 518, 552 McMannus, M., 423 McVay, S., 676 Medio, A., 321 Miller, M.H., 592 Mirowski, P., 286, 321 Mizrach, B., 25 Mooney, D., 5n, 25 Mortensen, D.T., 508, 509n, 511, 514n, 518 Mullineux, A., 25 Mundell, R.A., 239n, 243 Muscatelli, V., 455n, 469 Neal, F, 8n, 25 Neher, P.A., 661n, 676 Nerlov, H., 374 Nicolaides, R., 25 Niehaus, J., 592 Norwinton, E.J., 25 Obstfeld, M., 455n, 469, 592 Okuguchi, K., 423 Oldknow, A., 84, 118n, 141, 250 Olewiler, N.D., 676 Oxley, L., 495, 497, 518 Parker, D., 405, 410, 423 Parkin, M., 330, 374 Parta, H., 25 Peel, D.A., 493,518 Peng, W., 25 Percival, I., 84, 200 Peterson, A.C., 141, 250 Peterson, F.M., 676 Author index 699 Peterson, R.A., 53, 84 Pilbeam, K., 552, 592 Pissarides, C.A., 518 Place, CM., 84, 200 Pontryagin, L.S., 285 Pratt, J.W.,277n Quandt, R.E., 423 Ramsey, F.P., 275, 285 Renshaw, E., 637 Richards, D., 84, 200 Rodriguez, C.A., 592 R0dseth, A., 592 Rogoff, K., 455n, 469, 592 Romer, D., 275n, 285, 455n, 469 Ruskeepaa, H., 25 Sala-i-Martin, X., 275n Samuelson, P.A., 123, 141 Sandefur, J.T., 121n, 141, 212n, 250, 288n, 321,637, 641n Scarth, W.M., 460n, 469, 518 Scheinkman, J.A., 25 Schwalbe, D., 25, 179n, 191, 200 Shafer,W., 16, 25 Shaw, W.T., 25 Sheffrin, S.M., 518 Shone, R., 8n, 19n, 21n, 25, 135, 141, 183n, 200, 212n, 239n, 240, 243, 250, 277n, 355n, 374, 375, 437n, 469, 472n, 473,518, 552, 577n, 592, 638n, 649, 676 Simon, CP., 217, 233, 250, 264n Skeel, R.D., 25 Smith, V.L., 676 Solow, R.M., 141 Stevenson, A., 455n, 469 Swift, R., 5n, 25 Szidarovsky, F., 423 Takayama, A., 84, 285 Teigen, R.L., 429n, 469 Theocharis, R.D., 423 Thompson, J.L., 493,518 Tigg,J.,25 Tinbergen, J., 240 Tobias, A., 405, 423 Tobin, J., 455, 469 Tranter, N., 637 Tu, P.N.V., 84, 141, 200, 250, 321 Turnovsky, S.J., 518 Uhl, J., 25 Vandermeer, J., 637 Varian, J.R., 25 Wagon, S., 25, 179n, 191, 200 Walkington, N., 25 Walters, C.J., 676 Waugh, F.V., 345, 346, 374 Weil, D.N., 358, 374 Weir, M.D., 36n, 37, 84, 165n, 200 Whigham, D., 262n Whitby, S., 405, 423 Whitmarsh, D., 676 Wolff, E.N., 121, 141 Wolfram, S., 25 Yorke,J.A.,302n, 303 Zellner, A., 642, 676 Subject index absolute adaptation mechanism, 415 absolute policy adaptation, 412-414 active policy rule, 493-494 adaptive expectations, 364-365, 470, 484 see also hyperinflation adjoint variable, 253 aggregate demand (AD), 473-475, 476 shocks, 490, 493 aggregate supply curve, long-run, 474-475 shocks, 490, 493 annuities, 108 aperiodic series, 293-294 arctan function, 363-365, 366, 407 art forgery, testing, 49-50 asset market Dornbusch model, 555-557, 560-567, 568-570 monetarist model, 587-589 resource discovery, 583-586 asymptotic stability, 402, 618, 624, 625 see also fixed points asymptotically stable rest point, 57 attractors, 56-59, 60, 89-93, 95, 97, 127-128 chaotic, 310 Cobweb model, 341 see also strange attractor, Henon map, Lorenz [strange] attractor auxiliary equation, 61-64 baby boom, housing market, 358, 361-363 baby bust, 359 balance, 239-245 balance of payments, 524-530, 532-535 external balance, 239-243 see also IS-LM-BP model Bank of England, interest rates, 489 Bank of Japan, 490 basin of attraction, 148 Beckham, D. and V. erratic behaviour, 314n Bernoulli equation, 36, 44, 51n, 81 bifurcation chaotic behaviour, 298, 304-307 demand and supply, 365-366 diagram, 288-289, 299, 300-301, 318-319 Henon map, 308 period doubling, 296n, 297, 299n, 366 pitchfork, 292, 296, 299n: sub/supercritical 292n points, 288, 295, 299n saddle-node, 290, 299n transcritical, 291 value 287, 289-290, 295 see also Hopf bifurcation biological growth curve, see fisheries biomass, see fisheries biotic potential, 640 Bretton Woods, 528n, 538 Brock's residual test, 310n butterfly effect, 16n, 310 Cagan model, see hyperinflation calculus of variations, 251 canonical form, 219-220, 224-228 complex root, 232-234 repeated root, 229-231 capital flows, 6 capital mobility Dornbusch model, 558, 562, 564-567 IS-LM-BP model, 541-542, 544 carrying capacity, see fisheries Central Limit Theory, 287 centre point, 177 chaos, 288 rational choice, 314 theory, 293-301 700 Subject index 701 chaotic attractor see attractors chaotic behaviour of systems, 298-300 demand and supply, 363-367 chaotic series, 310 characteristic equation, 110, 205 characteristic polynomial Mathematica, 205 Maple, 207 characteristic roots, see roots Cobb-Douglas production function Cagan model, 502-503 Solow growth model, 36, 44, 56, 59, 67, 131 cobweb model, 12, 87-88, 92-93, 102, 332-337 interrelated markets, 346-349 using Mathematica and Maple, 338-339, 367-371 phase plane, 339-346 with stock behaviour, 352-353 competitive equilibrium, 353-358 complementary solution, 64, 65, 116, 117-118 complex conjugate roots, see roots compound interest, 39-40, 105-108 conjectural variation, 376 constant elasticity, 52 continuous control problem, 255-259 with discounting, 265-270 phase diagram, 270-283 Corel Draw, 22, 125 corn-hog cycle, 346-349 costate variable, 253 Cournot duopoly, 379 Cournot solution (Cournot-Nash solution), 375-377, 378, 381, 382, 384,387, 389, 390 Cramer's rule, 491, 584 critical point, 55, 167, 174, 177, 178-179 see also nodes Crutchfield and Zellner model, see fisheries De Moirre's formula, 232 deflationary spiral, 489, see also dynamic liquidity trap demand and supply chaotic behaviour, 363-367 continuous time, 326-332 labour market, 329-332 short market, 327-329 stock behaviour, 349-353 stock levels, 329 see also Cobweb model: competitive equilibrium demand equation, 11-12, 587 see also hyperinflation demand pressure curve, 476-478, 480 Derive, 20 deterministic systems, 15-16 difference equations, 4 autonomous, 86 first-order, 85, 99-105 first-order linear, 37-39 linear, 85 non-autonomous, 86 non-linear, 85 recursive, 85, 87 second-order homogeneous, 110-115 second-order non-homogeneous, 116 second-order solutions, 110-118 second-order linear difference solutions, 61-66, 110-115 simple, 99-105 differential equations, 4, 26-27 autonomous, 27-28 approximation, 66, 68-70 complex roots, 164-166 constant-coefficient rath-order, 27 first-order 142: initial condition, 30, 39; with chaos, 298, 300, 309-310 first-order linear, 37-39 general solutions, 27-37, 64, 65, 113, 114, 116-118, 124, 142-143 graphical solution, 29 implicit solution, 29, 41 homogeneous, 27 linear, 27, 142, 154-155 matrices, 156-161 non-homogeneous, 27 non-linear, 27, 67, 142 order classification, 27 ordinary, 27 partial, 27, 30, 64, 65, 116, 117 particular solution, 30 qualitative properties, 41^15 repeating roots, 162-164 second-order linear homogeneous, 59-64 second-order linear non-homogeneous, 64-66 separable, 45-48, 81: logistic curves 50-52; radioactive decay, 48-50 time-invariant, 27 diffusion, 53-54 702 Subject index dimensionality, 8-12 diminishing returns, 645-646 direction fields, 42-45, 152 Mathematica, 77-79, 189-191 Maple, 79-80, 189-190, 192-194 discount rate, 108-109 discounting, and optimal control, 265-270 discrete time control model, 259-264 disequilibrium inventory model, 315-319 dominant solution, 110 Dornbusch model, 559-564 capital immobility, 564-567 original, 559-564 perfect foresight, 567-573, 574-581 duopoly incomplete and non-instantaneous adjustment, 389-392, 395-396 output adjusting, 377-380, 386-387 static, 375-377 dynamic adjustment process, 326 constraint, 661 liquidity trap, 484-487, 489 modelling, 3-4 multiplier, 426-427, 431 programming, 251 dynamics, computer software, 17-24 dynamical systems autonomous, 86, 142-145, 149 matrix specification, 156-160 non-autonomous, 86, 142 random shocks, 286, 490, 492^193 effective interest rate, 105-106 eigenvalues, 158-161, 166, 168 fish harvests, 670-672 linear systems, 208-214, 226-228 Mathematica and Maple, 213-214, 626 phase plane, 237-239 population growth, 624-626, 628-630 eigenvectors, 158-162, 164-165, 166, 168, 170,172 fish harvests, 670-672 independent, 172-173 linear systems, 208-214 Mathematica and Maple, 213-214, 626 population growth, 624-626, 628-630 employment rate, 506-509 singular curve, 511 sticky wage theory, 331 endogenous propagation mechanism, 16 environmental economics, 7-8 environmental resistance, 640, 641 equilibrium lines, 432 see also trajectories equilibrium point, 55-56, 58 autonomous systems, 145-147, 153 bifurcation theory, 288-290 in discrete dynamic systems, 88-89, 94 stability, 454 Euler's approximation, 183-185, 437n Euler's theorem, 355 European Central Bank (ECB), 489-490 Evans, C. and erratic behaviour, 314n Excel, 17, 18,20, 25 discounting, 267 discrete optimisation problem, 262-264 employment level, 512n equilibrium value, 280n trajectories, 220-221, 309n, 441-442 exchange of stability, 291, 295, 299 exchange rate fixed, 532-539 floating, 6, 538, 539-544 internal and external balance, 243-245 exchange technology, 508 exogenous shocks, 16 expectations, 6-7 inflationary, 470-472, 476, 477-478, 482,501 expectations-augmented Phillips curve, 470,484 expenditure model, see income-expenditure model explicit solution, 29, 41 exponential growth curve, 596 external equilibrium, 527 Federal Reserve, 489 Feigenbaum's universal constant, 301-302, 309 Fibonacci series, 141 first-order initial value problem, 30 fiscal expansion fixed exchange rate, 532-535 Tobin-Blanchard model, 462-463, 464 fiscal shocks, 444-446 fisheries biological growth curve, 638-644, 646 biomass, 638-639, 645, 649, 651-652, 653 carrying capacity, 639, 640, 641, 668 Crutchfield and Zellner model, 642-644 entry and exit, 644, 648-649, 650-653 Subject index 703 harvesting, 601, 648, 640, 641, 642, 653, 669-672 harvesting function, 644-646, 649, 659 profits, 644, 647, 650, 658, 659: under open access, 647-657 optimal control problem, 658-661 schooling activity, 661-669 stable spiral, 653-654, 655 fixed points, 55-58, 60 asymptotically stable, 12, 57, 147 autonomous systems, 145-148, 150-153 in discrete dynamic system, 88, 98-99, 101, 127,224 fish stocks, 664 IS-LM model, 453 neutrally stable, 147n qualitative properties, 272-275 shunt, 56 stable, 56, 58, 8890, 93-94, 147 unstable, 147 see also attractor, repellor flexible wage theory, 330 flex-price model, 553 see also Dornbusch model, monetarist model floating exchange rate, see exchange rate flow of solutions, 42 flow variables, 10 foreign interest rate, 538-539 foreign market adjustment function, 545 see also balance of payments; Dornbusch model; IS-LM-BP model global stability, 58, 59, 68-69, 89-90, 148, 153 monetarist model, 588 Gompertz function, 53-54, 81, 635, 673 goods market, 473, 495 Dornbusch model, 554-557, 560-570 resource discovery, 583-586 see also IS-LM model; IS-LM-BP model goods market equilibrium condition, 424 Goodwin model, 337 government spending, internal balance, 240-245 half-life, 48-50 Hamiltonian function, 253-260, 266, 268, 272,273, 658, 660-661 current value, 266, 268-269, 277 Harrod-Domar growth model, 34, 56, 59, 83,101 using Mathematica and Maple, 133 Henon map, 221, 301, 307-310 hiring frequency, 508-509 Holling-Tanner predatory-prey model, 199 Hopf bifurcation, 304-307, 311 housing market, 358-363 demographic change, 358-363 hyperinflation Cagan model, 500-505 hysteresis, 306-307 imitation process, 415^19 income-expenditure model, 519-524 indeterminism, and random shocks, 286 inflation growth, 494-500 Lucas model, 490^93 Phillips curve see IS-LM model see also hyperinflation initial value problem, 30, 39, 81, 86-87 Mathematica, 71-72 Maple, 74-76 insider-outsider model, 509n, 514n integral curves, 43-44 integrating factor, 38 intercepts, see IS-LM model interest parity condition, 587 interest rate, and external balance, 240-243 internal rate of return, 109-110 interpolating function, 437 invariant set, 180 inventory model, and rational expectations, 315-319 investment function IS-LM model, 447 g-theory, 455 IS curve, see IS-LM model, Tobin-Blanchard model IS-LM model, 24, 424-425, 465-467, 484^85,487^88,519 BP curve, 529-530, 532-535 continuous model, 431^37, 447-453 lags, 425-426, 429 income, 424-428, 447-448 intercepts, 448-449 interest rate, 424, 447-448 investment function, 447 monetary expansion, 535-538 non-linear, 453-455 Phillips curve, 472-483 704 Subject index IS-LM model (cont.) shift in IS curve, 442^143 shocks, 443-447 IS-LM-BP model, 529-530 fiscal expansion, 532-535 fixed prices, 545-551 foreign interest rate rise, 538-539, 544-545 monetary expansion, 535-538, 542-544 isoclines, 42-44, 355-356, 358, 399, 456-457, 459, 462 Cagan model, 505 fish stocks, 663, 667 inflation, 478, 479, 480 liquidity trap, 485-486, 488-489 Mothematica, 11 Maple, 79-80 money growth, 497-498 population growth, 612-613, 615-616, 617-618 wage determination, 511, 513-514 isoprofit curves, 376-377 Jacobian matrix, 619n job-finding rate, 506-507 John, Sir E. and erratic behaviour, 314n Jordan blocks, 217-218 Jordan form xi, 216-217, 225, 230, 231 k-periodic point, 93-94 Kuhn-Tucker condition, 264n Lagrangian, 253-260, 266, 267, 658, 675 current value, 269 Leslie matrix xi, 628, 629, 669, 670-672, 675 Liapunov theorem, 68, 147 limit cycle, 147, 179-183, 514 Hopf bifurcation, 306 large-amplitude, 306-307 linear approximation, 127-130 dependence, 60-61 systems, 201-204 linearity, exception to norm, 287 Li-Yorke theorem, 303, 314 LM curve, see IS-LM model, Tobin-Blanchard model logistic equation, 118-123 chaos, 293-301 productivity growth, 121-123 Sarkovskii theorem, 302-304 logistic function, 598-599 logistic growth curves, 44, 50-52, 56-57, 299,597, 598 direction field, 46 fixed point, 56 logistic growth equation, 597-601, 603, 630-632, 638-639 see also fisheries long-run aggregate supply curve, 474-475 Lorenz [strange] attractor, 301, 307, 310-312 Lorenz curve, 186, 191, 193 Lotka-Volterra model, 604, 607-611, 617, 618,619 Lotus, 1,2,3, 17,20 Lucas model, 49-53 Lyapunov dimension, 310, 312 Malthusian population growth, 4, 33, 100-101, 593-596, 600, 602-603 direction fields, 44, 46 fixed points, 55, 59 Maple xi, xii, 19-23, 25 basic matrices, 204-205, 206-207 Cobweb, 370-371 complex roots, 134 DEtools, 79-80 differential equations, 73-77, 149-150, 186-190, 192-194 direction fields, 44 discrete systems, 214-216 dsolve, 73-77, 187-189 eigenvalues/vectors, 212-214, 230 employment level, 512n equilibrium values, 280 Jordan form, 216-217 linalg, 204 LinearAlgebra, 204 logistic equation, 118-119, 134, 137-138,631-632 optimal control problem, 25 8n oligopoly models, 378-379, 382, 383, 411 oscillations, 125 parametric plots, 195-196 phase plane in Cobweb, 342-343, 345-346 phaseportrait, 192-193 population growth, 600n, 604, 607, 610, 614: procedural function 76 multispecies model, 633-634 recursive equations, 105, 131-134: Cobweb 338-339, 340 Subject index 705 rsolve, 131-134, 214-216, 378-379, 382 trajectories, 222-223, 439-441, 499 marginal revenue product per worker, 509 market clearing model, 330, 509-513 market share, and R&D, 406, 409-410, 416,417-419 Marshall-Lerner condition, 526n, 547 MathCad, 20 Mathematica, xi, xii, 19-23, 25 basic matrices, 204-206 bifurcation diagrams, 301, 307 Cobweb model, 334, 342-343, 345-346, 367-369 complex roots, 134, 195-196 differential equations, 70-73, 149-150, 186-191 direction fields, 44 discrete systems, 214-216 DSolve, 70-71, 187-189, 195 eigenvalues/vectors, 212-214, 230 equilibrium values, 280 Henon map, 308 ImplicitPlot, 125 Jordan form, 216-217 NDSolve, 70, 72-73 oligopoly model, 378, 382-383, 411 oscillations, 125 parametric plots, 195-196 piecewise function, 318n PlotField, 77-79 PlotVectorField, 190 population growth, 600n, 604, 607, 610, 614: logistic equation, 630-631; multispecies model, 632-633 recursive equations, 105, 131-134 recursive Cobweb, 338-339, 340 RSolve, 131-134, 214-216, 378, 382 trajectories, 222-223, 437-438, 452, 667 VisualDSolve, 179n, 181 Mat Lab, 20 maximisation principle problem, 658 maximum principle, 661 see also Pontryagin maximum principle maximum sustainable yield, 640, 641 mean generation time, 594n Microfit, 20 migration, 596, 602-603, 604 monetarist model, 586-589 monetary expansion Dornbusch model, 560-562, 568 fixed exchange rates, 535-538 Tobin-Blanchard model, 463-465 monetary shocks, 443-447 money market, 473, 495, 501 adjustment function, 545 Dornbusch model, 554-555, 560, 565, 566 resource discovery, 58-64 see also IS-LM model; IS-LM-BP model money supply growth, 474, 493, 500 open economy, 530-532 policy rules, 493-494, 495 multiple equilibria, 12, 15 multiplier, 523 multiplier-accelerator model, 123-126, 427^128 Mundell-Fleming model, 519, 537, 541, 553 mutualism, 604 Nash solution, see Coumot solution natural growth coefficient, 604 natural unemployment rate, 506-507 net present value, 109 nodes, 167-168 improper, 174, 402, 404 proper, 172 spiral, 176, 402, 404, 479, 481 Van der Pol equation, 305 non-accelerating inflation rate of unemployment (NAIRU), 471-472 non-linear discrete systems, 245-247 nonlinearity, 8, 12-15 chaotic behaviour, 15-17, 300 North Sea gas and oil discovery, 554, 581-586 North Sea herring fisheries, 646 Occam's razor, 20n Okun's law, 472 oligopoly two-firm models, 375-380, 386-387, 389-392, 395-396 three-firm models, 380-383, 387-388, 393- 394, 396-397, 400-401, 404 four-firm models, 384-386, 388-389, 394- 395, 397-398 openness, 523-524 optimal control problem, 251-252 continuous model, 252-259 discounting, 265-270 maximum control, 259-264 see also fisheries optimal growth model, 16 706 Subject index orbit, 55, 145, 179 orbital stability, 179 ordinary differential equation, 27 oscillations, 125, 129 over-crowding, 605, 605 competition, 611-617 fish stocks, 640 predatory-prey model, 617-619 overlapping generation model, 16 overshooting, 442-444, 447, 466, 538, 541,544 flex-price models, 553, 557, 574 parametric plot, 194-196 parity rate, 527-528 partial differential equation, 27 particular solution, 30, 64, 65, 116, 117 passive policy rule, 493^194 paths, 55, 145 counter-clockwise, 449 counter-clockwise spiral, 437 counter-clockwise stable spiral, 451 spiral, 435, 442^143 stable, 449 stable spiral, 451 unstable spiral, 449, 451 see also trajectories pelagic whaling, 638 perfect foresight, 15, 495 Dornbusch model, 567-573, 574-581 monetary model, 587, 589 rational expectations, 502-503 periodic solution, 93 Perron-Frobenious theorem, 628n Peruvian anchoveta fisheries, 646 phase diagrams, 3, 4 control models, 270-283 single variable, 54-59 two-firm model, 400 phase line, 55, 56-57 phase plane, 45, 145, 166 discrete systems, 235-239 housing market, 360-363 internal and external balance, 239-245 optimal trajectory, 271 see also Cobweb model phase portrait, 145, 149, 189 direction fields, 190-194 discrete systems, 219 Phillips curve, 470-472, 477-478, 481, 494 and Cagan, 502-503 and Lucas, 490 Poincare-Bendixson theorem, 180 policy announcement, time periods, 579-581 Pontryagin maximum principle, 251-264 continuous model, 252-259, 272 discrete model, 259-264 population growth by age of women, 626-627 Malthusian, 593-596, 600, 603 multispecies analysis, 619-626, 632-634 natural changes, 601-602 see also fisheries, logistic growth equation, migration, predatory-prey relationship portfolio balance condition, 661 predatory-prey relationship, 604, 607-611, 617-619 present value, 108-109 price inflation, see Phillips curve price-ceiling Cobwel model, 345 principle of effective market classification, 243 product differentiation, 418 product innovation, 406 production function, 272n profit function, 509-510 proportional policy adaptation, 412-414 purchasing power parity (PPP), 553-555, 557, 560, 562-564, 568, 584-585, 587 g-theory of investment, 425, 455 QuattroPro, 17, 20 radioactive decay, half-life, 33-34, 48-50 fixed points, 59 Ramsey growth model, 275-283 rational expectations, 7, 15, 228, 494, 495 Cagan model, 501, 505 Dornbusch model, 567-573 Lucas model, 490-493 monetarist model, 587, 589 reaction coefficients, 435-437, 451, 656-657 real income level, internal balance, 239-240 real wages, 502-503 recursive equation, 85, 87 dominant, 405 Mathematical Maple, 134 multiplier-accelerator model, 123-126 solutions, 105-108 Subject index 701 regression, spreadsheets, 19 relative risk aversion, 277 repellor, 56-57, 59, 60, 89-93, 95, 97, 128, 147 Cobweb model, 341 resource depletion rate, 586 resources, 9 rest point, 55 roots characteristic, 461, 481, 666 complex conjugate, 63-64, 65, 111, 114-115, 125, 159-160, 164-166, 174-177, 178-179, 217-218, 231-234, 428 real and distinct, 61-62, 64, 111-112, 124-125, 159-161, 167-169, 178, 217-218,223-228,405 real and equal, 62-63, 64, 111-113, 159-160, 172-174, 178, 228-231 Rossler attractor, 199, 200 Rossler equations, 320-321 saddle path, 15, 169, 282, 447, 452-453, 460, 461, 464, 572, 575-576, 585, 606-607, 615, 664, 667-668 saddle point, 169-170, 179, 2325, 274, 278,281,461,622, 666 Cagan model, 503-505 housing market, 361-362 oligopoly model, 400-401, 402, 404 stable arm, 169-170, 172, 227-228, 237, 275,281-282,361-362, 461, 462-466, 499, 505, 570-572, 576-577, 623, 625, 668 unstable arm, 169-170, 172, 227, 237, 275,281 wage determination, 514 Sarkovskii's theorem xi, 302-304 Schumpeterian dynamics xi, 414-419 secondary dimensions, 9 selection process, 414-419 Shazam, 20 shirking model, 509-513, 514 slopes fish stocks, 652 IS-LM model, 430, 448-449, 451, 454-455, 527 Tobin-Blanchard model, 457-459 Solow growth model, 16, 34-37, 56, 67, 82, 495 direction fields, 44, 47 discrete time, 130-131: multiple equilibria, 59 speculative demand for money, 453 spiral point see nodes spreadsheets recursive systems, 19-20 SPSS, 20 stability problem, 97-99, 128-129 stability competitive equilibrium, 353-358 demand and supply models, 349-353 discrete systems, 223-234 expenditure model, 520-523 linear systems, 203-204, 219, 551, 619-620 local, 12-15, 59, 97, 127-128, 148, 454: Cobweb 344; fish stocks, 641; liquidity trap, 486^187 Lorenz system, 311 non-linear systems, 601, 620-626 oligopoly models, 385-386, 387-389, 392, 396, 397^100, 402, 403 see also asymptotic stability, global stability stable fixed point see attractor, repellor, fixed points state diagram, 627 Statgraphics, 20 sticky prices, 557, 566, 573 sticky wage theory, 330-332 stock behaviour demand and supply models, 349-353 stock market behaviour, see Tobin-Blanchard model stock variables, 9-10 stock-adjustment model, 32 stock-flow, 6-7, 10 strange attractor, 186, 307-312 survival of the fittest, see selection process Systat, 20 Taylor expansion, 67, 128, 460 non-linear discrete system, 246 Solow growth model, 131 tent function, 320 Thatcher, M., 592 time-independency, 144-145 time-series data discrete processes, 287 randomness, 286 Tobin-Blanchard model, 24, 424^125, 455^165, 467 trade-cycle model (Hicks), 125-126 trajectories, 55, 145, 149, 177-178 Cagan model, 505 708 Subject index trajectories {cont.) deflation, 489 discrete systems, 220-223 Dornbusch model, 559, 572-573, 577-579, 580-581 eigenvectors, 173-174, 390 using Excel, 441-442 IS-LM model, 433-435, 437, 442-445, 448 IS-LM-BP model, 54-50 using Maple, 439^141 using Mathematica, 437^138, 452 population growth, 605-607, 614-615: fish stocks, 652-653, 655-657, 664, 667-668; Lotka-Volterra model, 608, 610 rational expectations, 514, 535, 536, 541 see also paths transactions demand for money, 453 transient chaos, 293 TSP, 20 two-cycle result, 97 undershooting, 564, 566-567 undetermined coefficients, 65-66 unemployment level, 506-509 see also Phillips curve unit limit cycle, 181 utility, satisfaction, 9 vacancy rate, 507-508 value singular curve, 511 VanderPol equation, 181-182, 190, 193 bifurcation features, 304-307 vector forces, 151-156 Cagan model, 505 discrete systems, 235-236 flex-price models, 585 inflation, 478^480 IS-LM model, 433-435 liquidity trap, 486 market clearing model, 513-514 phase plane analysis, 239, 240, 356-357 population growth, 605-606, 615-616: fish stocks, 652, 66-74; predatory-prey model, 609 Tobin-Blanchard model, 459-460 Vermeer, 50 VisualDSolve, 179n wage determination, 509-513 Walrasian price and quantity adjustment, 182-183, 199, 354 warranted rate of growth (Harrod), 34 Wronksian, 198