Chapter 9 3. Suppose that a profit-maximizing firm has the production function Q

Question

Chapter 9 3. Suppose that a profit-maximizing firm has the production function Q(K, L)-VKVL. (a) If the firm's current level of capital is fixed in the short run at K4, write the optimality condition that the firm's choice of labor must satisfy in order for the firm to maximize profit in the short run b) Find the firm's profit-maximizing level of labor LsR as a function of w and P (c) Now suppose that P 16, r = 2 and w = 2. Find the firm's short-run profit-inaximizing level of labor, and the profit maximizing quantity of output. d) Find the firm's short-run profit (e) Write the two optimality conditions that the firm's choice of capital and labor must (f) Using the prices above, find the firm's profit maximizing quantity of labor and capital, (g) Find the firm's long-run profit. Explain why the firm's profit increased in the long run satisfy in order for the firm to maximize profit in the long run as well as the firm's profit maximizing quantity of output as compared to the short run

Explanation / Answer

Q = K0.5 + L0.5

Total cost: C = wL + rK

When K = 4,

Q = (4)0.5 + L0.5 = 2 + L0.5

C = wL + 4r

(a) The short-run optimality condition is

Output price x marginal product of labor (MPL) = wage rate

P x MPL = w [Required condition]

(b) MPL = dQ/dL = 0.5 / L0.5

P x (0.5 / L0.5) = w

L0.5 = 0.5P / w

Squaring both sides,

LSR* = 0.25P2 / w2

(c) P = 16, r = 2, w = 2

LSR* = 0.25 x (16 x 16) / (2 x 2) = 16

Q = 2 + L0.5 = 2 + (16)0.5 = 2 + 4 = 6

(d) Short run profit = (P x Q) - C

= (16 x 6) - [(2 x 16) + (2 x 4)]

= 96 - (32 + 8) = 96 - 40

= 56

NOTE: First 4 parts are answered.

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