A firm's production function is given by $Q = K ^ { 2 } L$ . The marginal products of labor and capital are, respectively, $M P _ { L } = K ^ { 2 }$ and $M P _ { X } = 2 K L$ . Further, the wage rate is $w = \$ 10$ and the rental rate of capital is $r = \$ 20$ . Suppose that the firm wants to produce 27,000 units of output in the most efficient way possible. How much does the firm spend?
A) $600
B) $900
C) $1,500
D) $2,100

Question

Quizplus · Accepted Answer

To produce 27,000 units of output in the most efficient way possible, the firm should use the input combination that minimizes the cost of production while producing 27,000 units. 
The isoquant for 27,000 units is given by $Q = K ^ { 2 } L = 27,000$. 
Taking the natural logarithm on both sides and differentiating with respect to both K and L yields: 
$$\frac{1}{L}=\frac{2}{K}$$ 
This implies that the firm should use capital and labor in a ratio of 2:1 to minimize the cost of production. 
Let the firm use K units of capital and L units of labor to produce 27,000 units, hence the production function becomes $27,000 = K ^ { 2 } L$. Substituting the ratio L=2K into the production function gives $27,000=4K^3$, which implies that K=15.09. Hence, L=30.18. 
The total cost of production is then given by $TC = wL + rK = (\$10)(30.18) + (\$20)(15.09) = \$603$, which is closest to option B ($900).

A Firm's Production Function Is Given By Q=K2LQ = K ^ { 2 } LQ=K2L

A Firm's Production Function Is Given By $Q = K ^ { 2 } L$