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?

Another price taking firm chooses its inputs to maximize long-run profits. Labor and capital are substitutes in production and both exhibit decreasing returns to scale, q(L, K) = L^(1/2) + K^(1/2) . The output price is 100 and the price of each of the inputs is 10.

(a) Set up the firm’s profit function and provide the two conditions for profit maximization.

(b) Find the profit maximizing levels of capital and labor.

(c) How does a change in labor change the optimal choice of capital?

Explanation / Answer

(Question - 1)

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.

(Question - 2)

Q = L1/2 + K1/2

P = 100, w = r = 10

TR = P x Q = 100 x (L1/2 + K1/2)

TC = wL + rK = 10(L + K)

(a)

Profit = TR - TC = 100 x (L1/2 + K1/2) - 10(L + K)

Profir maximization conditions are:

(i) d(TR) = d(TC)

MR = MC, and

(ii) MRTS = MPL / MPK = w / r

NOTE: There are 2 multi-part questions. The 1st question is answered with all sub-parts & the 1st sub-part of 2nd question has been answered.

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