Refer to the following:
The price of capital is $12 per unit.

-How many units of labor should the firm use in order to produce 200 units of output at the least cost?
A) 12 units of labor
B) 18 units of labor
C) 30 units of labor
D) 40 units of labor
E) 22 units of labor

Question

Refer to the following:
The price of capital is $12 per unit.
 
-How many units of labor should the firm use in order to produce 200 units of output at the least cost?
A) 12 units of labor 
B) 18 units of labor 
C) 30 units of labor 
D) 40 units of labor 
E) 22 units of labor

Quizplus · Accepted Answer

To minimize cost, the firm must choose the input combination that gives the same level of output at the lowest cost. This is achieved by finding the point of tangency between the isoquant for 200 units of output and the isocost line with a slope of -$12/$6 = -2, representing the cost of one unit of labor relative to the cost of one unit of capital.

The MRTS (Marginal Rate of Technical Substitution) between labor and capital is the ratio of their marginal products, which is MPL/MPK = 20/30 = 2/3.

The optimal input combination is found where MRTS = w/r = 2/3 = - (MPL/MPK) = - (w/r), where w is the wage rate and r is the rental rate of capital. Solving for w/r = -2/3 and r = $12, we get w = -$8. Therefore, the firm should use 22 units of labor (at a cost of $-176) and 30/2 = 15 units of capital (at a cost of $180) to produce 200 units of output at the least cost.

Choice E is the only one that matches the optimal input combination.

Refer to the Following