A price taking firm chooses its inputs to maximize short-run profits. Its Cobb-D

Question

A price taking firm chooses its inputs to maximize short-run profits. Its Cobb-Douglass production function has the following form: q(L, K) = L ^(1/2)K^(1/3) . The output price is 1,000 per unit and the cost of each unit of input is 10. In the short-run, capital is fixed at 27 units.

(a) Set up the profit function in terms of labor only.

(b) Find the optimal choice of labor, L* .

(c) Given your answer to part (b), do you think that there is excess capital compared to the optimal level of quantity?

Explanation / Answer

Q = L1/2K1/3

P = 1000

w = r = 10

Short-run K = 27

(a)

Q = L1/2K1/3 = L1/2(27)1/3

= 3L1/2

Total cost, TC = wL + rK = 10L + (10 x 27)

TC = 10L + 270

Total revenue = P x Q = 1000 x Q = 3000L1/2

So, Profit = Revenue - total cost = 3000L1/2 - 10L - 270

(b) Optimal combination is when

P x MPL = w

Where MPL = dQ / dL = 3 x (1/2) x L-1/2

= 1.5L-1/2

So, 1,000 x 1.5L-1/2 = 10

L1/2 = 150

L = 22,500

(c)

TC = 10 x 22,500 + 10 x 270 = 225,270

TR = 3000L1/2 = 450,000

So profit > 0.

In the short tun, firm makes supernormal profit, so capital should be increased to make economic profit zero.

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