Scenario 7.3:
Use the production function: Q = 4L 1/2 K 1/2 .
-Refer to Scenario 7.3. Suppose that the price of labor is $5 and the price of capital is $20. Your firm desires to produce 200 units of output. How much labor will be hired to minimize the costs of producing 200 units of output?
A) 25
B) 50
C) 100
D) 200
E) none of the above

Question

Quizplus · Accepted Answer

To minimize cost, the firm needs to hire the quantity of labor such that the marginal product of labor (MPL) divided by the wage rate is equal to the marginal product of capital (MPK) divided by the rental rate. In other words:
MPL/5 = MPK/20

Taking the derivative of the production function with respect to labor gives us the MPL:
MPL = 4S1^(1/2)U1P11/2*P0^(-1)*K*S1^(1/2)*U1*Q^(-1)

Plugging in Q=200, K=1, P0=1, and P1=5:
MPL = 4(S1^(1/2))(U1)(5.6)

Therefore, MPL/5 = (S1^(1/2))(U1)(2.24).

Similarly, taking the derivative of the production function with respect to capital gives us the MPK:
MPK = 2S1^(1/2)S1^-1*U1*P11/2*P0^(1/2)*K^(-1/2)*U1*Q^(-1/2)

Plugging in Q=200, K=1, P0=1, and P1=20:
MPK = 2(S1^(1/2))(U1)(8)

Therefore, MPK/20 = (S1^(1/2))(U1)(0.4).

Setting MPL/5 equal to MPK/20 and solving for S1^(1/2)U1 gives:
(S1^(1/2))(U1) = 11.2

Therefore, the firm needs to hire:
L = (S1^(1/2))(U1)(Q/K) = (11.2)(200) = 2,240 / 5 = 448 units of labor.

However, since the firm desires to produce only 200 units of output, it can reduce the amount of labor hired and increase the amount of capital rented until it reaches the desired level of output. Given that the firm is trying to minimize costs, it will rent the amount of capital such that the MPK divided by the rental rate is equal to the MPL divided by the wage rate. In other words:
MPK/20 = MPL/5

Plugging in the above expressions for MPK and MPL:
2S1^(1/2)S1^-1*U1*P11/2*P0^(1/2)*K^(-1/2)*U1*Q^(-1/2) = 4S1^(1/2)U1P11/2*P0^(-1)*K*S1^(1/2)*U1*Q^(-1)

Canceling some of the factors:
2K = 20Q^(-1/2)

Plugging in Q=200:
K = 10

Therefore, the firm will rent one unit of capital and hire: 
L = (S1^(1/2))(U1)(Q/K) = (11.2)(200) / 10 = 224 units of labor.

Since 224 is the quantity of labor that the firm should hire given its production function and input prices, the correct answer is choice C.

Scenario 7.3: Use the Production Function: Q = 4L1/2K1/2

Scenario 7.3: Use the Production Function: Q = 4L^1/2K^1/2