INTMIC9.jpg Chapter 19 Technology Technologies uA technology is a process by which inputs are converted to an output. uE.g. labor, a computer, a projector, electricity, and software are being combined to produce this lecture. Technologies uUsually several technologies will produce the same product -- a blackboard and chalk can be used instead of a computer and a projector. uWhich technology is “best”? uHow do we compare technologies? Input Bundles uxi denotes the amount used of input i; i.e. the level of input i. uAn input bundle is a vector of the input levels; (x1, x2, … , xn). uE.g. (x1, x2, x3) = (6, 0, 9×3). Production Functions uy denotes the output level. uThe technology’s production function states the maximum amount of output possible from an input bundle. Production Functions y = f(x) is the production function. x’ x Input Level Output Level y’ y’ = f(x’) is the maximal output level obtainable from x’ input units. One input, one output Technology Sets uA production plan is an input bundle and an output level; (x1, … , xn, y). uA production plan is feasible if uThe collection of all feasible production plans is the technology set. Technology Sets y = f(x) is the production function. x’ x Input Level Output Level y’ y” y’ = f(x’) is the maximal output level obtainable from x’ input units. One input, one output y” = f(x’) is an output level that is feasible from x’ input units. Technology Sets The technology set is Technology Sets x’ x Input Level Output Level y’ One input, one output y” The technology set Technology Sets x’ x Input Level Output Level y’ One input, one output y” The technology set Technically inefficient plans Technically efficient plans Technologies with Multiple Inputs uWhat does a technology look like when there is more than one input? uThe two input case: Input levels are x1 and x2. Output level is y. uSuppose the production function is Technologies with Multiple Inputs uE.g. the maximal output level possible from the input bundle (x1, x2) = (1, 8) is uAnd the maximal output level possible from (x1,x2) = (8,8) is Technologies with Multiple Inputs Output, y x1 x2 (8,1) (8,8) Technologies with Multiple Inputs uThe y output unit isoquant is the set of all input bundles that yield at most the same output level y. Isoquants with Two Variable Inputs y º 8 y º 4 x1 x2 Isoquants with Two Variable Inputs uIsoquants can be graphed by adding an output level axis and displaying each isoquant at the height of the isoquant’s output level. Isoquants with Two Variable Inputs Output, y x1 x2 y º 8 y º 4 Isoquants with Two Variable Inputs uMore isoquants tell us more about the technology. Isoquants with Two Variable Inputs y º 8 y º 4 x1 x2 y º 6 y º 2 Isoquants with Two Variable Inputs Output, y x1 x2 y º 8 y º 4 y º 6 y º 2 Technologies with Multiple Inputs uThe complete collection of isoquants is the isoquant map. uThe isoquant map is equivalent to the production function -- each is the other. uE.g. Technologies with Multiple Inputs x1 x2 y Technologies with Multiple Inputs x1 x2 y Technologies with Multiple Inputs x1 x2 y Technologies with Multiple Inputs x1 x2 y Technologies with Multiple Inputs x1 x2 y Technologies with Multiple Inputs x1 x2 y Technologies with Multiple Inputs x1 y Technologies with Multiple Inputs x1 y Technologies with Multiple Inputs x1 y Technologies with Multiple Inputs x1 y Technologies with Multiple Inputs x1 y Technologies with Multiple Inputs x1 y Technologies with Multiple Inputs x1 y Technologies with Multiple Inputs x1 y Technologies with Multiple Inputs x1 y Technologies with Multiple Inputs x1 y Cobb-Douglas Technologies uA Cobb-Douglas production function is of the form uE.g. with x2 x1 All isoquants are hyperbolic, asymptoting to, but never touching any axis. Cobb-Douglas Technologies x2 x1 All isoquants are hyperbolic, asymptoting to, but never touching any axis. Cobb-Douglas Technologies x2 x1 All isoquants are hyperbolic, asymptoting to, but never touching any axis. Cobb-Douglas Technologies x2 x1 All isoquants are hyperbolic, asymptoting to, but never touching any axis. Cobb-Douglas Technologies > Fixed-Proportions Technologies uA fixed-proportions production function is of the form uE.g. with Fixed-Proportions Technologies x2 x1 min{x1,2x2} = 14 4 8 14 2 4 7 min{x1,2x2} = 8 min{x1,2x2} = 4 x1 = 2x2 Perfect-Substitutes Technologies uA perfect-substitutes production function is of the form uE.g. with Perfect-Substitution Technologies 9 3 18 6 24 8 x1 x2 x1 + 3x2 = 18 x1 + 3x2 = 36 x1 + 3x2 = 48 All are linear and parallel Marginal (Physical) Products uThe marginal product of input i is the rate-of-change of the output level as the level of input i changes, holding all other input levels fixed. uThat is, Marginal (Physical) Products E.g. if then the marginal product of input 1 is Marginal (Physical) Products E.g. if then the marginal product of input 1 is Marginal (Physical) Products E.g. if then the marginal product of input 1 is and the marginal product of input 2 is Marginal (Physical) Products E.g. if then the marginal product of input 1 is and the marginal product of input 2 is Marginal (Physical) Products Typically the marginal product of one input depends upon the amount used of other inputs. E.g. if then, and if x2 = 27 then if x2 = 8, Marginal (Physical) Products uThe marginal product of input i is diminishing if it becomes smaller as the level of input i increases. That is, if Marginal (Physical) Products and E.g. if then Marginal (Physical) Products and so E.g. if then Marginal (Physical) Products and so and E.g. if then Marginal (Physical) Products and so and Both marginal products are diminishing. E.g. if then Returns-to-Scale uMarginal products describe the change in output level as a single input level changes. uReturns-to-scale describes how the output level changes as all input levels change in direct proportion (e.g. all input levels doubled, or halved). Returns-to-Scale If, for any input bundle (x1,…,xn), then the technology described by the production function f exhibits constant returns-to-scale. E.g. (k = 2) doubling all input levels doubles the output level. Returns-to-Scale y = f(x) x’ x Input Level Output Level y’ One input, one output 2x’ 2y’ Constant returns-to-scale Returns-to-Scale If, for any input bundle (x1,…,xn), then the technology exhibits diminishing returns-to-scale. E.g. (k = 2) doubling all input levels less than doubles the output level. Returns-to-Scale y = f(x) x’ x Input Level Output Level f(x’) One input, one output 2x’ f(2x’) 2f(x’) Decreasing returns-to-scale Returns-to-Scale If, for any input bundle (x1,…,xn), then the technology exhibits increasing returns-to-scale. E.g. (k = 2) doubling all input levels more than doubles the output level. Returns-to-Scale y = f(x) x’ x Input Level Output Level f(x’) One input, one output 2x’ f(2x’) 2f(x’) Increasing returns-to-scale Returns-to-Scale uA single technology can ‘locally’ exhibit different returns-to-scale. Returns-to-Scale y = f(x) x Input Level Output Level One input, one output Decreasing returns-to-scale Increasing returns-to-scale Examples of Returns-to-Scale The perfect-substitutes production function is Expand all input levels proportionately by k. The output level becomes Examples of Returns-to-Scale The perfect-substitutes production function is Expand all input levels proportionately by k. The output level becomes Examples of Returns-to-Scale The perfect-substitutes production function is Expand all input levels proportionately by k. The output level becomes The perfect-substitutes production function exhibits constant returns-to-scale. Examples of Returns-to-Scale The perfect-complements production function is Expand all input levels proportionately by k. The output level becomes Examples of Returns-to-Scale The perfect-complements production function is Expand all input levels proportionately by k. The output level becomes Examples of Returns-to-Scale The perfect-complements production function is Expand all input levels proportionately by k. The output level becomes The perfect-complements production function exhibits constant returns-to-scale. Examples of Returns-to-Scale The Cobb-Douglas production function is Expand all input levels proportionately by k. The output level becomes Examples of Returns-to-Scale The Cobb-Douglas production function is Expand all input levels proportionately by k. The output level becomes Examples of Returns-to-Scale The Cobb-Douglas production function is Expand all input levels proportionately by k. The output level becomes Examples of Returns-to-Scale The Cobb-Douglas production function is Expand all input levels proportionately by k. The output level becomes Examples of Returns-to-Scale The Cobb-Douglas production function is The Cobb-Douglas technology’s returns- to-scale is constant if a1+ … + an = 1 Examples of Returns-to-Scale The Cobb-Douglas production function is The Cobb-Douglas technology’s returns- to-scale is constant if a1+ … + an = 1 increasing if a1+ … + an > 1 Examples of Returns-to-Scale The Cobb-Douglas production function is The Cobb-Douglas technology’s returns- to-scale is constant if a1+ … + an = 1 increasing if a1+ … + an > 1 decreasing if a1+ … + an < 1. Returns-to-Scale uQ: Can a technology exhibit increasing returns-to-scale even though all of its marginal products are diminishing? Returns-to-Scale uQ: Can a technology exhibit increasing returns-to-scale even if all of its marginal products are diminishing? uA: Yes. uE.g. Returns-to-Scale so this technology exhibits increasing returns-to-scale. Returns-to-Scale so this technology exhibits increasing returns-to-scale. But diminishes as x1 increases Returns-to-Scale so this technology exhibits increasing returns-to-scale. But diminishes as x1 increases and diminishes as x1 increases. Returns-to-Scale uSo a technology can exhibit increasing returns-to-scale even if all of its marginal products are diminishing. Why? Returns-to-Scale uA marginal product is the rate-of-change of output as one input level increases, holding all other input levels fixed. uMarginal product diminishes because the other input levels are fixed, so the increasing input’s units have each less and less of other inputs with which to work. Returns-to-Scale uWhen all input levels are increased proportionately, there need be no diminution of marginal products since each input will always have the same amount of other inputs with which to work. Input productivities need not fall and so returns-to-scale can be constant or increasing. Technical Rate-of-Substitution uAt what rate can a firm substitute one input for another without changing its output level? Technical Rate-of-Substitution x2 x1 yº100 Technical Rate-of-Substitution x2 x1 yº100 The slope is the rate at which input 2 must be given up as input 1’s level is increased so as not to change the output level. The slope of an isoquant is its technical rate-of-substitution. Technical Rate-of-Substitution uHow is a technical rate-of-substitution computed? Technical Rate-of-Substitution uHow is a technical rate-of-substitution computed? uThe production function is uA small change (dx1, dx2) in the input bundle causes a change to the output level of Technical Rate-of-Substitution But dy = 0 since there is to be no change to the output level, so the changes dx1 and dx2 to the input levels must satisfy Technical Rate-of-Substitution rearranges to so Technical Rate-of-Substitution is the rate at which input 2 must be given up as input 1 increases so as to keep the output level constant. It is the slope of the isoquant. Technical Rate-of-Substitution; A Cobb-Douglas Example so and The technical rate-of-substitution is x2 x1 Technical Rate-of-Substitution; A Cobb-Douglas Example x2 x1 Technical Rate-of-Substitution; A Cobb-Douglas Example 8 4 x2 x1 Technical Rate-of-Substitution; A Cobb-Douglas Example 6 12 Well-Behaved Technologies uA well-behaved technology is –monotonic, and –convex. Well-Behaved Technologies - Monotonicity uMonotonicity: More of any input generates more output. y x y x monotonic not monotonic Well-Behaved Technologies - Convexity uConvexity: If the input bundles x’ and x” both provide y units of output then the mixture tx’ + (1-t)x” provides at least y units of output, for any 0 < t < 1. Well-Behaved Technologies - Convexity x2 x1 yº100 Well-Behaved Technologies - Convexity x2 x1 yº100 Well-Behaved Technologies - Convexity x2 x1 yº100 yº120 Well-Behaved Technologies - Convexity x2 x1 Convexity implies that the TRS increases (becomes less negative) as x1 increases. Well-Behaved Technologies x2 x1 yº100 yº50 yº200 higher output The Long-Run and the Short-Runs uThe long-run is the circumstance in which a firm is unrestricted in its choice of all input levels. uThere are many possible short-runs. uA short-run is a circumstance in which a firm is restricted in some way in its choice of at least one input level. The Long-Run and the Short-Runs uExamples of restrictions that place a firm into a short-run: –temporarily being unable to install, or remove, machinery –being required by law to meet affirmative action quotas –having to meet domestic content regulations. The Long-Run and the Short-Runs uA useful way to think of the long-run is that the firm can choose as it pleases in which short-run circumstance to be. The Long-Run and the Short-Runs uWhat do short-run restrictions imply for a firm’s technology? uSuppose the short-run restriction is fixing the level of input 2. uInput 2 is thus a fixed input in the short-run. Input 1 remains variable. The Long-Run and the Short-Runs x2 x1 y The Long-Run and the Short-Runs x2 x1 y The Long-Run and the Short-Runs x2 x1 y The Long-Run and the Short-Runs x2 x1 y The Long-Run and the Short-Runs x2 x1 y The Long-Run and the Short-Runs x2 x1 y The Long-Run and the Short-Runs x2 x1 y The Long-Run and the Short-Runs x2 x1 y The Long-Run and the Short-Runs x2 x1 y The Long-Run and the Short-Runs x2 x1 y The Long-Run and the Short-Runs x1 y The Long-Run and the Short-Runs x1 y The Long-Run and the Short-Runs x1 y Four short-run production functions. The Long-Run and the Short-Runs is the long-run production function (both x1 and x2 are variable). The short-run production function when x2 º 1 is The short-run production function when x2 º 10 is The Long-Run and the Short-Runs x1 y Four short-run production functions.