Elements of Economic Analysis II Lecture III: Cost Minimization, Factor Demand and Cost Function

Transcription

1 Elements of Economic Analysis II Lecture III: Cost Minimization, Factor Demand and Cost Function Kai Hao Yang 10/05/ Cost Minimization In the last lecture, we saw a firm s profit maximization problem. In fact, the profit maximization problem can be decomposed into two parts Fix a quantity to be produced and minimize total production cost, and then maximize profit by choosing an optimal quantity. Such decomposition is sometimes more informative about the firm s behavior. This is the topic for this lecture. 1.1 Long Run Cost Minimization Formally, let F be the firm s production function. Fix any y 0, the firm s cost minimization problem is given by: min wl + rk s.t. F (L, K) = y. That is, the cost minimization problem aims to find the optimal combination that minimizes the firm s total cost in order to produce y units of outputs. For any w, r, y, we let L (w, r, y) and K (w, r, y) denote the solution of the cost minimization problem and let c(w, r, y) := wl (w, r, y) + rk (w, r, y) Department of Economics, University of Chicago; khyang@uchicago.edu 1

2 2 be the optimal value of the cost minimization problem. We say the L (w, r, y) and K (w, r, y) are the conditional factor demand of the firm and that c(w, r, y) is the cost function of the firm. Indeed, fix any w, r, for a firm that will minimize total cost when producing y units, our theory predicts that this firm will need L (w, r, y) units of labor and K (w, r, y) units of capital. Furthermore, we can conclude from the cost minimization problem that the (smallest possible) total cost to produce y units of output is c(w, r, y). We can learn more about the conditional factor demands and the cost function by examining the first-order (and second-order) condition of the cost minimization problem. Recall that we can use the Lagrangian to solve a constrained optimization problem. That is, let L = wl + rk + µ(y F (L, K)). Assume that L (w, r, y) > 0 and K (w, r, y) > 0, first order condition then gives: w = µ(w, r, k)f L (L (w, r, y), K (w, r, y)), r = µ(w, r, y)f K (L (w, r, y), K (w, r, y)), which implies: F L (L (w, r, y), K (w, r, y)) F K (L (w, r, y), K (w, r, y)) = w r. That is, the technical rate of substitution must be equal to relative input price at optimum. Indeed, suppose that the technical rate of substitution is not the same as relative price, say, T RS < w. If the firm gives up δ amount of labor, it will reduce its cost by wδ. In order r to maintain the same production level y, the firm needs to rent T RS δ units more capital, which costs it r T RS δ. As T RS < w r, this cost is less then r w r firm can further reduce its total cost while maintaining y units of production. δ = wδ. As such, the Exercise 1. What does this remind you of? Can you think of any analogy of L and K? Example 1. Suppose the F (L, K) = L α K β becomes: for some α, β (0, 1). The Lagrangian then L = wl + rk µ(y x α y β ).

3 3 First-order condition then gives: [L] : w = µαl α 1 K β (1) [K] : r = µβl α K β 1 (2) [µ] : y = L α K β (3) After multiplying L, K on both sides of (1) and (2), respectively, we have: wl = µαl α K β = µαy, rk = µβl α K β = µβy. and therefore Substituting back into (3), we have: L = µαy w, K = µβy r. ( αµy w ) ( ) α β βµy = y. r Rearranging, µ = [α α β β w α r β y 1 α β ] 1. Thus, using (1) and (2), L (w, r, y) = ( ) β α ( w ) β y 1, K (w, r, y) = β r ( ) α α ( w ) α y 1 β r and c(w, r, y) = κw α r β y 1, where κ := ( ) β α β + ( α β ) α. There are few noticeable features of the solution L, K and the value c under this case. For instance, we can see that the conditional factor demand of labor and capital are decreasing in wage and rental rate, respectively. Also, the cost function is increasing as y increases and has constant return to scale with respect to the factor prices. Furthermore, if α + β = 1 (i.e. the production function is constant return to scale), the cost function is also constant return to scale in y. We will examine more about these features later.

4 4 Exercise 2. Derive the conditional factor demands and cost functions for the following production functions: F (L, K) = al + bk. F (L, K) = min{al, bk}. F (L, K) = [αl σ + (1 α)k σ ] 1 σ. 1.2 Short Run Cost Minimization The discussions above assume that the firm can choose labor and capital flexibly. This is often not the case for a firm in the short run. As introduced in the last lecture, in the short run, firms often are not able to adjust the amount of capital. As such, for a fixed amount of capital K, the firm s cost minimization problem in the short run is given by: min wl + r K s.t. F (L, K) = y. L 0 Since we are focusing on the case with two inputs, the short run cost minimization problem is in fact very straightforward. Indeed, if the marginal productivity of labor is strictly positive so that F L (L, K) > 0, L 0, then there exists a unique L SR (y K) such that F (L SR (w, r, y K), K) = y. As such, there is only one feasible amount of labor in the cost minimization problem and hence the solution is clearly L SR (w, r, y K). Even if F L (L, K) = 0 for some range of L, whenever it is still possible to reach output level y by some L, the set F 1 (y, K) := {L 0 F (L, K) = y} is still nonempty. As such, the solution is simply: L SR (y K) = min F 1 (y, K).

5 5 In either cases, we say that L SR (y K) is the short run conditional labor demand and that c SR (w, r, y K) := wl SR (y K) + r K is the short run cost function. Furthermore, since the firm cannot adjust K, we also say that the term r K is the fixed cost. 2 Properties of Conditional Factor Demand and Cost Function As discussed above, we say that the solution L (w, r, y), K (w, r, y) to the cost minimization problem min wl + rk s.t. F (L, K) = y is the conditional factor demand function and that the optimal value c(w, r, y) is the cost function. By examining the nature of this cost minimization problem, we can actually see some noticeable properties of these functions. First, suppose that the production function F is concave. Then it is always true that the cost function c(w, r, y) is also concave in w and r and convex in y. As a result, we must have c ww 0, c rr 0 and c yy 0. Second, recall that the Lagrangian of the cost minimization problem is L = wl + rl + µ(y F (L, K)). By the envelope theorem, we have the following: c w (w, r, y) = L w (L (w, r, y), K (w, r, y), µ(w, r, y)) = L (w, r, y) c r (w, r, y) = L r (L (w, r, y), K (w, r, y), µ(w, r, y)) = K (w, r, y) c y (w, r, y) = L y (L (w, r, y), K (w, r, y), µ(w, r, y)) = µ(w, r, y) Combining the two observations above, we can now see that L w (w, r, y) = c ww (w, r, y), K r (w, r, y) = c rr (w, r, y), which means that the conditional factor demands must be downward-sloping in their own prices. This observation is called Shephard s Lemma.

6 6 On the other hand, since we defined c(w, r, y) as the cost function, c y can be interpreted as the marginal cost the change of total cost in production when the output produced increases in an infinitesimal amount. The observations above imply that c y (w, r, y) = µ(w, r, y) > 0 and that c yy (w, r, y) = µ y (w, r, y) 0, meaning that the cost function is increasing and convex in y, or equivalently, the marginal cost function is nonnegative and increasing in y. Furthermore, since by definition, for any y 0, w, r > 0, c(w, r, y) = for any λ > 0, we then have: c(λw, λr, y) = min wl + rk s.t. F (L, K) = y, min (λw)l + (λr)k s.t. F (L, K) = y = min λ[wl + rk] s.t. F (L, K) = y Notice that since λ > 0, this problem yields exactly the same solution L (w, r, y), K (w, r, y). Therefore, c(λw, λr, y) = λc(w, r, y). That is, c is of constant return to scale with respect to w and r. Finally, if we further suppose that F is of constant return of scale. Again, from for any y 0, any λ > 0, c(w, r, y) = c(w, r, λy) = min wl + rk s.t. F (L, K) = y, min = min = min λ L λ 0, K λ 0 wl + rk s.t. F (L, K) = λy ( L wl + rk s.t. F λ, K ) = y λ ( L λ + K ) ( L s.t. F λ λ, K ) = y λ = min λ[wl + rk] s.t. F (L, K) = y = λc(w, r, y).

7 7 That is, the cost function c(w, r, y) is also of constant return to scale if F exhibits constant return to scale. In particular, c(w, r, y) = y c(w, r, 1), y. This then implies that c y (w, r, y) = c(w, r, 1) for all y. That is, the marginal cost is always a constant. Furthermore, this constant marginal cost is exactly the minimized value of the problem: min wl + rk s.t. F (L, K) = 1. Exercise 3. Give an economic explanation and interpretation of this observation. 3 Summary Technical rate of substitution equals to relative factor prices at optimum. (Shephard s Lemma) L w (w, r, y) = c ww (w, r, y), K r (w, r, y) = c rr (w, r, y). Conditional factor demands are decreasing in their own prices. Assuming that F is concave, cost function is: Concave in (w, r). Constant return to scale in (w, r). Strictly increasing and convex in y. Linear in y if F has constant return to scale and strictly convex if F is not constant return to scale.