Ecn2201 intmicro DES ch18

ECON 2201 Intermediate Microeconomics David Simon Technologies A technology is a process by which inputs are converted to an output. E.g. labor, a computer, a projector, electricity, and software are being combined to produce this lecture. Technologies Usually several technologies will produce the same product -- a blackboard and chalk can be used instead of a computer and a projector. Which technology is “best”? How do we compare technologies? Input Bundles xi denotes the amount used of input i; i.e. the level of input i. An input bundle is a vector of the input levels; (x1, x2, … , xn). E.g. (x1, x2, x3) = (6, 0, 9×3). Production Functions y denotes the output level. The 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 A production plan is an input bundle and an output level; (x1, … , xn, y). A production plan is feasible if The 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 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 What does a technology look like when there is more than one input? The two input case: Input levels are x1 and x2. Output level is y. Suppose the production function is Technologies with Multiple Inputs Technologies with Multiple Inputs Output, y x1 x2 (8,1) (8,8) Technologies with Multiple Inputs The 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 Isoquants 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 More 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 The complete collection of isoquants is the isoquant map. The isoquant map is equivalent to the production function -- each is the other. E.g. 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 Fixed-Proportions Technologies t l min{l,2t} = 14 4 8 14 2 4 7 min{l,2t} = 8 min{l,2t} = 4 l = 2t Perfect-Substitutes Technologies A perfect-substitutes production function is of the form Perfect-Substitution Technologies 9 3 18 6 24 8 l l + 3t = 18 l + 3t = 36 l + 3t = 48 All are linear and parallel Technology Vocabulary Technology: Process by which inputs are changed into an output. Input Bundles: A set of inputs, denoted: (x1,x2,x3,…) Production Function: A mathematical function that describes the maximum amount of output that can be produced from an input bundle Technical Efficiency / Inefficiency: A production plan is technically efficient if the combination of inputs used leads to max output. Technology Vocabulary Isoquant: all combinations of inputs that lead to the same output Marginal Physical Product: the change in output from a small change in an input Diminishing Marginal Product: A marginal product is diminishing in some input if it decreases as the amount of the input increases. Simple Example Cobb Douglas Given this production function, finding the marginal physical product is straight forward. Technical Rate-of-Substitution At 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 How is a technical rate-of-substitution computed? 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. Rules of Production Each production line “makes” tennis balls by moving them to the other container Work is done in 30 second shifts All workers in the line must handle each ball on its way to the other container Dropped balls cannot be sold (they are worthless) Workers can only touch one ball at a time. Line manager chooses team of workers Total # of workers is set by me (initially) Auditors count & report output per shift Auditors help make sure the buckets stay in the same place between shifts. * Use online stop watch http://www.online-stopwatch.com/full-screen-stopwatch/ Production Data Track firms’ output as workers change * Analysis: Productivity Dynamics * Key Financial Numbers Each unit produced brings in a fixed amount of revenue (i.e. “transfer price”) $10.00 per tennis ball Additional Financial information: Workers paid on a per-shift basis $15 per worker per 30-second shift Auditor is required, fee paid by the line $40 per shift regardless of production level We will apply this when we do profit max and costs. * Marginal Product Marginal Revenue Thoughts What is the Technology for “producing” tennisballs. What is the main input used? How could the inputs be used inefficiently? We could describe our production through a production function after collecting data. Returns-to-Scale Marginal products describe the change in output level as a single input level changes. Returns-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), For k>1 F(kx1,kx 2) = k F(x1,x 2) 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), For k>1 F(kx1,kx 2) < k F(x1,x 2) 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), For k>1 F(kx1,kx 2) > k F(x1,x 2) 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 A 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 has constant returns to scale Expand all input levels proportionately by k. The output level becomes k F(x1,x 2) Show general form. Returns to scale and perfect substitutes example: Examples of Returns-to-Scale The Cobb-Douglas production function has returns to scale based on the values Of a, b. Examples of Returns-to-Scale The Cobb-Douglas production function is The Cobb-Douglas technology’s returns- to-scale is constant if a1+ a2= 1 F(x1,x 2) 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. Example Fisheries Producing fish requires labor (l) and boats (b). Fishing is more productive in Spring than in Summer and is least productive in winter. Spring: f(l,b) = l*b Summer: f(l,b) = l^(1/2)*b^(1/2) Examples of Returns-to-Scale A quick note that perfect complements Also exhibits CRS Returns-to-Scale So a technology can exhibit increasing returns-to-scale even if all of its marginal products are diminishing. Why? Returns-to-Scale A marginal product is the rate-of-change of output as one input level increases, holding all other input levels fixed. Marginal 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 When 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. The Long-Run and the Short-Runs The long-run is the circumstance in which a firm is unrestricted in its choice of all input levels. There are many possible short-runs. A 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 Examples of restrictions that place a firm into a short-run: temporarily being unable to install, or remove, machinery having to meet domestic content regulations. The Long-Run and the Short-Runs A useful way to think of the long-run is that the firm can choose as it pleases in which short-run circumstance to be. Short run choose labor given capital. Long run, choose capital and then make short run choice. The Long-Run and the Short-Runs What do short-run restrictions imply for a firm’s technology? Suppose the short-run restriction is fixing the level of capital (can’t build more buildings / ovens right away). Capital is thus a fixed input in the short-run. Labor remains variable. 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 Returns to scale: your turn Does the following Cobb Douglas production function have constant, increasing, or decreasing returns to scale. Formally show your answer: Winter: f(l,b) = l^(1/3)*b^(1/2) Extra Slides Returns to scale Perfect Complements Ex 2. F(x1,x2) = min{3x1,2x2}