4310 : Macroeconomics
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4310 : Macroeconomics
Espen Henriksen
September 17, 2008
4310 : Macroeconomics
Introduction Administrative 2/25
Administrative
1 Questions???
4310 : Macroeconomics
Introduction Refresher in microeconomics and prob. theory 3/25
The envelope theorem
Consider an arbitrary maximization problem where the objective
function f depends on some parameter a:
M(a) = max
x
f(x, a),
Let x(a) be the value of x that solves the maximization problem in
terms of the parameter a. Then M(a) = f(x(a), a). The envelope
theorem tells us how M(a) changes as the parameter a changes,
namely:
dM(a)
da
=
f(x, a)
a
x=x(a)
.
I.e. the derivative of M with respect to a is given by the partial
derivative of f(x, a) with respect to a, holding x fixed, and then
evaluating at the optimal choice (x).
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Introduction Refresher in microeconomics and prob. theory 4/25
The Fundamental Theorems of Welfare Economics
First Fundamental Welfare Theorem: every competitive
equilibrium is Pareto-optimal.
Second Fundamental Welfare Theorem: every Pareto-optimal
allocation can be achieved as a competitive equilibrium after a
suitable redistribution of initial endowments.
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Introduction Refresher in microeconomics and prob. theory 5/25
Solving the Social Planner's problem
When the first welfare theorem holds the solution to the social
planner's problem is the competitive equilibrium allocation.
That is, there exists a set of prices such that the optimum solution
can be decentralized as a competitive equilibrium with a price
system that has an inner product representation.
The social planner's problem is much easier to solve since we get
rid of the prices and the individuals' budget constraint.
Remember: 1WT does in general not hold in an overlapping
generations economy.
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Introduction Refresher in microeconomics and prob. theory 6/25
Conditional expectation
In probability theory, a conditional expectation (also known as
conditional expected value or conditional mean) is the expected
value of a real random variable with respect to a conditional
probability distribution.
Usual notation either E (Xt+1 = x | Xt = xt), the conditional
expectation of Xt+1 given that Xt is equal to xt, or simply
Et (Xt+1).
t + 1
R O S
R 0.50 0.25 0.25
t O 0.25 0.50 0.25
S 0.50 0.50 0.00
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Introduction Refresher in microeconomics and prob. theory 7/25
Fixed-point Theorems and The Contraction Mapping
Theorem
Varian: Brouwer Fixed-Point Theorem
Other fixed-point theorems like Kakutani.
The Contraction Mapping Theorem.
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Introduction Overview 8/25
Building the toolbox
A B
C D
No uncertainty Uncertainty
No decisions
Decisions
A : Solow growth model
B : Value iteration on a Markov process
C : Ramsey growth model
D : Stochastic neoclassical growth model
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Introduction Overview 9/25
The Euler Equation
Rt+1
u(ct+1)
u(ct)
= 1
where for the neoclassical growth model
Rt+1 = 1 + f
(kt+1) - .
u
(ct)
marginal disutility of saving one more unit
= 
discount factor between two periods rate of ret
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Introduction Overview 10/25
The Euler Equation in a Stochastic Model Economy
Et Rt+1
u(ct+1)
u(ct)
= 1
where for the neoclassical growth model
Rt+1 = 1 + zt+1f
(kt+1) - .
ct = Et Rt+1u
(ct+1)
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Introduction Dynamic programming 11/25
Recursive formulation
Mathematical strategy:
Seek the optimal savings/investment function directly and
then to use this function to compute the optimal sequence of
investments from any initial capital stock.
This way of looking at the problem ­ decide on the immediate
action to take as a function of the current situation ­ is called
a recursive formulation.
It exploits the observation that a decision problem of the same
structure recurs every period.
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Introduction Dynamic programming 12/25
Why bother?
Ramsey completely solved this problem using variational
methods ­ so why bother?
Consider the stochastic variation of the problem:
In that case it makes no sense to choose a deterministic plan
of investments for all future dates.
The best future choices will depend on how much output is
available at the time ­ which again will depend on the current
capital stock, the state of the technology, and the properties of
the technology process.
So in order solve stochastic dynamic decision problems we
need recursive methods.
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Introduction Uncovering the value function 13/25
Contraction mapping
In general, v() is unknown, but the Bellman equation can be used
to find it.
In most of the cases we will deal with, the Bellman equation
satisfies a contraction mapping theorem, which implies that
1 There is a unique function v() which satisfies the Bellman
equation.
2 If we begin with any initial function v0(k) and define
vi+1(k) = max
c,k
[u(c) + vi(k
)]
subject to
c + k
= f(k) + (1 - d)k
for i = 0, 1, 2, . . . then limi vi+1(k) = v(k).
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Introduction Uncovering the value function 14/25
How to find the true value function
The above two implications give us two alternative means of
uncovering the value function.
1 Implication 1 above, if we are fortunate enough to correctly
guess the value function v(); then we can simply plug v(kt+1)
into the right side and then verify that v(kt) solves the
Bellman equation. This procedure only works in a few cases.
2 Implication 2 above is useful for doing numerical work. One
approach is to find an approximation to the value function
and iterate.
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Approach 15/25
Approach
1 Pose a question
2 Use well-tested theory
3 Construct a model economy
4 Estimate the structural parameters of the model economy
5 Run the computational experiment and compare the
performance of the model with what we have observed
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Business cycles What are they? 16/25
What are business cycles?
Burns and Mitchell (1913, 1927, 1946): expansions occurring at
the same time in many economic sectors, followed by similarly
general recessions, contractions and revivals.
fluctuations occurs in aggregate activity, not in particular
sectors.
cycles are recurrent, but not periodic.
cycles have at least two different stages: expansions and
contractions.
once the economy enters into one of the stages, it stays there
for some time.
there are regular and predictable co-movements between
variables over the cycles.
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Business cycles What are they? 17/25
Lucas (1981): deviations of real aggregate output from a
trend, where there is co-movement of deviations from trend of
various economic aggregates with those of output.
NBER: a recession is a persistent period of decline in total
output, income, employment, and trade, usually lasting from
six months to one year, and marked by wide spread contracts
in many sectors of the economy.
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Business cycles What are they? 18/25
Business cycle facts
For each variable, study the following properties
Procyclical, countercyclical or acyclical
Leading or lagging
More or less variable than the output (volatility)
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Business cycles What are they? 19/25
Some definitions
Procyclical: a variable that usually increases in booms,
decreases in recession. For example, productivity is procyclical.
Countercyclical: a variable that usually decreases in booms,
increases in recession. For example, unemployment.
Acyclical: a variable that shows no systematic relationship to
the business cycle.
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Business cycles Filtering out low-frequency components 20/25
Hodrick-Prescott filter
The Hodrick-Prescott filter is a standard curve-fitting technique
used to obtain a smoothed non-linear representation of a time
series in order to filter out low-frequency components of the series.
min
T
t=1
(yt - t)2
+ 
T-1
t=2
[(t+1 - t) - (t - t-1)]2
.
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Business cycles U.S. Business Cycles 21/25
Facts of U.S. Business Cycles
The magnitude of fluctuations in output and aggregate hours
of work are nearly equal.
Employment fluctuates almost as much as output.
Productivity is slightly procyclical but varies less than output.
Consumption of non-durables and services is much smoother
than output.
Investment in both producers' and consumers' durables
fluctuations more than output
Wages vary less than productivity and therefore output.
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Business cycles U.S. Business Cycles 22/25
Co-movement over the Business Cycle
Consumption, investment, inventories and imports are
strongly procyclical but exports largely acyclical.
Government expenditures are essentially uncorrelated with
output.
Average hourly compensation is uncorrelated with output.
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Business cycles Approach 23/25
Approach
1 Pose a question
2 Use well-tested theory
3 Construct a model economy
4 Estimate the structural parameters of the model economy
5 Run the computational experiment and compare the
performance of the model with what we have observed
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Business cycles Approach 24/25
Question
To what extent can business cycles fluctuations be accounted for
by a model economy populated by rational (not perfect foresight)
agents responding optimally (not perfectly) to real shocks (random
changes in technology, government purchase, taxes, taste,
government regulation, terms of trade, energy prices, etc.) given
their information set and then re-optimize when new information
becomes available?
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Business cycles A few concepts ­ from 1st lecture 25/25
"Rational expectations"
"I should like to suggest that expectations, since they are
informed predictions of future events, are essentially the
same as the predictions of the relevant economic theory.
At the risk of confusing this purely descriptive hypothesis
with a pronouncement as to what firms ought to do, we
call such expectations rational."
John F. Muth. (1961). "Rational Expectations and the Theory of Price Movements"
Not normative
Consistency. Ensure that the behavior of the model is
consistent with individual decision makers' beliefs about the
behavior of the model economy.