Question

Suppose output, Q, is produced by labor, L, and capital, K, according to the following function: Q = K ½ L½.. Suppose the firm sells each unit of output in a competitive market for a price P = $100. Suppose the firm hires each unit of labor in a competitive market for a wage W = $25. Suppose the firm has to make do for now with a stock of capital K = 49; moreover, suppose each unit of capital costs R = $75.

A. How much labor will be demanded by the firm? Demonstrate and explain.

B. At the “optimal” quantity of labor, what is the capital-to-labor ratio K/L? Demonstrate and explain.

C. Utilizing the “optimal” quantity of labor, how much profit will the firm earn?

Answer 1

Production function is given by Q = K ½ L½.. This gives us the information that MPL is 0.5(K/L)^0.5 and MPK is 0.5(L/K)^0.5. This implies the MRTS is MPL/MPK or K/L

Output price is P = $100, wage rate is W = $25 and rental price of capital is $75. Current stock of capital is K = 49.

A. How much labor will be demanded by the firm? Demonstrate and explain.

Labor demand decision is taken using the rule of VMPL or MRP = wage

Here K is fixed at 49 so production function is Q = 49^1/2 x L^1/2 or Q =7L^1/2. Now MPL is 7*1/2 x L^(-1/2).

VMPL is given by Price x MPL so labor demand rule becomes

100 x 7*1/2 x L^(-1/2) = 25

350/25 = L^0.5

This gives L* = 196. This is the quantity of labor demanded.

B. At the “optimal” quantity of labor, what is the capital-to-labor ratio K/L? Demonstrate and explain.

Capital labor ratio is (K/L) = (49/196) = 0.25.

C. Utilizing the “optimal” quantity of labor, how much profit will the firm earn?

Profit = revenue - cost. Find the output as Q = (49^0.5)*(196^0.5) = 98 units. Cost function is composed of wage bill and capital cost.

= 100 x 98 - (196 x 25 + 49 x 75)

= $1225.