Modelling Production

Transcription

1 Modelling Production David N. DeJong University of Pittsburgh Econ. 1540, Spring 2010 DND () Production Econ. 1540, Spring / 23

2 Introduction The production function is the foundation upon which models of economic growth are based. The models we will study are based on the Cobb-Douglas production function, speci ed under two scenarios: No technological progress Technological progress DND () Production Econ. 1540, Spring / 23

3 Scenario 1: Cobb-Douglas Production, No Technological Progress Y = K α L 1 α, 0 < α < 1. Y : real GDP K: aggregate stock of physical capital L: aggregate labor hours α: capital s share of output DND () Production Econ. 1540, Spring / 23

4 Critical Features Y = K α L 1 α K, L both necessary for production; i.e., Y = 0 if either K = 0 or L = 0 For xed K, as L increases, Y increases, but at a decreasing rate. I.e., the marginal product of labor is decreasing in L: As K increases, MP L increases. MP L = Y L = (1 α) K α L α K α = (1 α) L DND () Production Econ. 1540, Spring / 23

5 Critical Features, cont. Y = K α L 1 α For xed L, as K increases, Y increases, but at a decreasing rate. I.e., the marginal product of capital is decreasing in K: As L increases, MP K increases. MP K = Y K = αk α 1 L 1 α L 1 α = α K DND () Production Econ. 1540, Spring / 23

6 Critical Features, cont. Y = K α L 1 α The production function features constant returns to scale: increasing (K, L) by a factor z leads to an increase in Y by the same factor z. Mathematically, f (zk, zl) = (zk ) α (zl) 1 α = z α z 1 α K α L 1 α = zy. Intuition: the production process is replicable. DND () Production Econ. 1540, Spring / 23

7 Returns to Scale To determine the returns to scale implied by an arbitrary function Y = f (K, L), compare zy versus f (zk, zl). If zy = f (zk, zl), constant returns to scale If zy > f (zk, zl), decreasing returns to scale (e.g., doubling inputs leads to LESS than a doubling of output) If zy < f (zk, zl), increasing returns to scale (e.g., doubling inputs MORE than doubles output) DND () Production Econ. 1540, Spring / 23

8 Returns to Scale, cont. Exercise: consider the single-input function Y = a + bl, b > 0. Calculate returns to scale for a = 0 a > 0 a < 0 DND () Production Econ. 1540, Spring / 23

9 Graphical Representation Returning to the Cobb-Douglas function Y = K α L 1 α, graph Y as a function of L. Next, indicate the impact on the graph of a change in K. Repeat for the relationship between Y and K, indicating the impact on the relationship of a change in L. DND () Production Econ. 1540, Spring / 23

10 Production from the Perspective of a Firm Consider the problem of a hypothetical rm seeking to choose (K, L), and thus Y, in order to maximize pro ts: Π = PY {z} Revenue (WL + RK ), {z } Cost where Π : rm s nominal pro ts (i.e., the dollar value of rm s pro ts) P : aggregate price level W : aggregate nominal wage rate R : aggregate nominal rental price of capital DND () Production Econ. 1540, Spring / 23

11 Firm s Perspective, cont. The rm s problem in real terms: where π = Y (wl + rk ), π = Π P w = W P r = R P DND () Production Econ. 1540, Spring / 23

12 Pro t Maximization from a Cost-Bene t Analysis Bene t of hiring an additional worker: (additional output generated by additional worker) x (price per unit of output), i.e., marginal revenue: MR L = MP L P Cost of hiring an additional worker: marginal cost: MC L = W So long as MR L > MC L, L should be increased (thus causing MR L to decrease). Likewise, if MR L < MC L, L should be deceased (thus causing MR L to increase). DND () Production Econ. 1540, Spring / 23

13 Cost-Bene t Analysis, cont. Implication: the pro t-maximizing (optimal) choice of L satis es MR L = MC L, or or or or P MP L = W, MP L = w, K α (1 α) = w, L L = (1 α) 1 α 1 1 α K. w DND () Production Econ. 1540, Spring / 23

14 Cost-Bene t Analysis, cont. Likewise, the optimal choice for K satis es MR K = MC K, or or L 1 α α = r, K K = α 1 1 α α L. r DND () Production Econ. 1540, Spring / 23

15 Cost-Bene t Analysis, cont. Implication for pro ts: π = Y (wl + rk ) = Y (MP L L + MP K K ) K α = Y (1 α) L L = Y (1 α) K α L 1 α α K α L 1 α = 0.! L 1 α α K K Thus the revenue generated by production activities is returned entirely to workers and the owners of capital. DND () Production Econ. 1540, Spring / 23

16 Cost-Bene t Analysis, cont. Payment made to labor: MP L L = K α (1 α) L L = (1 α) Y Payment made to capital:! L 1 α MP K K = α K K = αy Interpretation of α : capital s share of labor. DND () Production Econ. 1540, Spring / 23

17 Graphical Representation of Pro t Maximization Isoquant Curve: (K, L) combinations such that Y remains xed at Y 0 To derive, for K depicted on the vertical axis, re-write Y = K α L 1 α as K = f (L; Y = Y 0 ). Exercise: derive, graph (calculate slope explicity, examine its properties). DND () Production Econ. 1540, Spring / 23

18 Graphical Representation of Pro t Maximization, cont. Isocost Curve: (K, L) combinations such that total cost C remains xed at C 0 To derive, for K depicted on the vertical axis, re-write as C = wl + rk K = f (L; C = C 0 ). Exercise: derive, graph (calculate slope explicity, examine its properties). DND () Production Econ. 1540, Spring / 23

19 Graphical Representation of Pro t Maximization, cont. Pro t maximization (cost minimization): Using isoquant, isocost curves, determine how to choose (K, L) optimally to produce a given level of output. DND () Production Econ. 1540, Spring / 23

20 A Note on the Behavior of Output per Worker Let Then y = Y /L. y = K α L 1 α L = K α L 1 α L 1 = K α L 1 α L 1 = K α L α K α = L = k α DND () Production Econ. 1540, Spring / 23

21 On the Behavior of Output per Worker, cont. Exercise: calculate intercept, slope of y = k α, graph y as a function of k DND () Production Econ. 1540, Spring / 23

22 Cobb-Douglas Production with Technological Progress A: total factor productivity Y = A K α L 1 α, 0 < α < 1. Note that a change in A has a proportional impact on the productivity of (K, L) DND () Production Econ. 1540, Spring / 23

23 Technological Progress, cont. An alternative speci cation: Y = K α (H L) 1 α, 0 < α < 1. H: labor-speci c technological productivity Note: the two speci cations are observationally equivalent. Exercise: examine how movements in H a ect (Y, K ), (Y, L) relationships DND () Production Econ. 1540, Spring / 23