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 $M P _ { K } = 2 K L$. Further, the wage rate is $w = \$ 10$ and the rental rate of capital is $r = \$ 20$. Suppose that the firm spends exactly $\$ 1200$ in the most efficient way possible. How much output can the firm produce?
A) 50,000
B) $56,406.25$
C) 60,750
D) 64,000

Question

Quizplus · Accepted Answer

To maximize output, the firm should equate the ratio of marginal products to the ratio of input prices: $\frac{MP_L}{MP_K} = \frac{w}{r}$. Solving $\frac{K^2}{2KL} = \frac{10}{20}$ gives $K = 2L$. Substituting $K = 2L$ into the budget constraint $20K + 10L = 1200$, we find $L = 20$ and $K = 40$. Substituting these into the production function $Q = K^2L$, we get $Q = 40^2 \times 20 = 64,000$.

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

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