Elements of Economic Analysis II Lecture II: Production Function and Profit Maximization Kai Hao Yang 09/26/2017 1 Production Function Just as consumer theory uses utility function a function that assign a utility level to a vector of consumption bundles to represent preference, firm theory uses production function a function that gives a certain amount of outputs for a given vector of inputs to describe technology. We will introduce the notion of production function and some basic properties in this section. Formally, let F : R 2 + R + be a twice differentiable function. The function F can be interpreted as a production of two factors (mostly labor and capital) since for any L 0, K 0, y = F (L, K) can be thought of as the number of outputs produced by using the technology F with L units of labors and K units of capitals. 1.1 Examples of Production Function Cobb-Douglas Production Function: F (L, K) = AL α K β, A > 0, α > 0, β > 0. Department of Economics, University of Chicago; e-mail: khyang@uchicago.edu 1
2 Perfect Substitutes: F (L, K) = al + bk, a > 0, b > 0. Fixed Proportion: F (L, K) = min{al, bk}, a > 0, b > 0. Constant Elasticity of Substitution Production Function: F (L, K) = A(αL σ + (1 α)k σ ) 1/σ, A > 0, α (0, 1). Exercise 1. In consumer theory, we didn t see the term A in Cobb-Douglas utility function, can we drop the term A for production functions too without loss of generality? Why? (Hard) Take a CES production function, let r = 1/(1 σ), if σ 1, what is r and what does the function become? If σ 0, what is r and what does the function become? If σ, what is r and what does the function become? Based on the answers above, can you guess what is the economic interpretation of r? 1.2 Isoquant, Production Possibility Set and the Technical Rate of Substitution Just as utility function can be described by indifference curves, it is sometimes useful to describe the production functions by isoquants. Given any output level y, we say the the y-isoquant of a production function F is the curve: I y := {(L, K) R 2 + F (L, K) = y}. That is, the isoquant at y is the combination of production factors that gives output level y. Exercise 2. Draw the isoquants for Cobb-Douglas, Perfect Substitute and Fixed Proportion technologies. Analogous to the marginal rate of substitution as utility function, we use the term technical rate of substitution to describe the slopes of isoquants. That is: T RS(L, K) := F L(L, K) F K (L, K). Just as marginal rate of substitution, technical rate of substitution can be interpreted as the amount of capital that one needs to gain in order to maintain the same level of production
3 when labor decreases by one unit. We often assume that the technical rate of substitution is decreasing. Given a production function, we can easily describe the production possibility set the set of inputs and outputs that can possibly arise given the technology. That is, we can define: Y := {(y, L, K) R 3 y F (L, K)}. We often assume that the production function F must be such that the set Y is convex. This is in fact not a demanding assumption, but a natural consequence of concavity of production function. 1.3 Marginal Productivity Similar to marginal utility, sometimes we are also interested in the marginal productivity of a production function F. That is, we use: F L (L, K) = F (L, K), L F K(L, K) = F (L, K) K to denote the marginal productivity of labor and capital, respectively. In words, marginal productivity measures how much more one can produce by adding a infinitesimal amount of input while fixing another input. Practically, a technology often exhibits diminishing marginal productivity. On the other hand, whether or not the marginal productivity decreases as the other factor increases (i.e. the sign of F LK ) is uncertain. (Think about producing a cup of coffee in a cafe and agricultural production). Exercise 3. Are decreasing marginal productivity and diminishing technical rate of substitution the same? 1.4 Return to Scale Suppose that a firm can produce y units of goods by using L units of labor and K units of capital. A natural question to ask is: When the firm s size is doubles, how will the output change? Also doubled? More than doubled? Or less than doubled? We say that a production function exhibits increasing return to scale if for any L > 0, K > 0, any r > 1, F (rl, rk) > rf (L, K).
4 That is, a production function is with increasing return to scale if when the inputs grows r times larger, the output increases more that r times more. Similarly, we say that a production function exhibits decreasing return to scale if for any L > 0, K > 0, r > 1, F (rl, rk) < rf (L, K), and F is with constant return to scale if for any L > 0, K > 0, r > 0. F (rl, rk) = rf (L, K) Two significant properties of constant return to scale production functions, which will be used later, are as following: F L (rl, rk) = F L (L, K) and F K (rl, rk) = F K (L, K). (Euler s formula) F (L, K) = LF L (L, K) + KF K (L, K). 2 Profit Maximization With the basic knowledge of the production function, we can now begin studying the simplest form of a firm s decision. 2.1 Short-Run Profit Maximization In the short run, the firm cannot adjust the amount of capital but can only adjust the amount of labor used in production. Let w denote the wage of workers and p denote the price of the good that the firm is producing. The firm s short run problem is then given by: max pf (L, K) wl, L 0 for a fixed K > 0. Using first-order condition, a necessary condition for optimal level of employment L is that: pf L (L, K) = w, or equivalently, F L (L, K) = w p.
5 Furthermore, if we assume that F has diminishing marginal productivity on labor so that F LL < 0, then the condition F L (L, K) = w p characterizes the solution to the firm s short run problem. Notice that the left hand side is the marginal productivity of labor and the right hand side is the real wage. As such, at optimum, it must be that marginal productivity equals to real wage. In fact, the first-order condition above conveys more information about comparative statics. In particular, suppose that we are interested in how the short-run labor demand for a firm changes as wage changes. By the analysis above, for any w, let L (w) denote the amount of labor that solves F L (L (w), K) = w p. Differentiate both sides with respect to w, we then have: F LL (L (w), K)L (w) = 1 p L (w) = 1 F LL (L (w), K)p. As such, we can conclude that L (w) < 0 as we assumed diminishing marginal return of labor. Therefore, the short run labor demand curve is downward sloping. 2.2 Long-Run Profit Maximization In the long run, however, the firm can change its capital level while making production decisions. The firm s decision problem then becomes: max pf (L, K) wl rk, L,K 0 where r is the rental rate of capital. Using first-order condition, a necessary condition for optimal level of labor and capital is: pf L (L, K) = w, and pf K (L, K) = r, or equivalently, F L (L, K) = w p, and F K(L, K) = r p. We can now see that the principle that marginal productivity of a factor equals to the real cost of that factor at optimum still holds even in the long run. In the long run, now only
6 marginal productivity of labor equals to real wage, marginal productivity of capital equals to real rental too. If a firm s technology exhibits constant return to scale, recall that by Euler s formula, F (L, K) = LF L (L, K) + KF K (L, K). Suppose that L and K is the optimal amount of labor and capital for the firm under prices p, w, r. From the first-order condition, F L (L, K ) = w p, and F K(L, K ) = r p. Combining this with the Euler s formula when evaluated at L, K, we have pf (L, K ) = wl + rk. Notice that the left hand side is exactly the revenue of the firm at optimum and the right hand side is the cost of the firm at optimum. The above equation says that for a firm with constant return to scale technology, at optimum, profit must be zero! Exercise 4. Give an intuitive explanation for the result above.