A firm's production function is given by $Q = K L$. The marginal products of labor and capital are, respectively, $M P _ { L } = K$ and $M P _ { K } = L$. The wage rate of labor is $w = \$ 10$ and the rental rate of capital is $r = \$ 20$. The firm spends exactly $\$ 1000$ in the most efficient way possible. How much output can the firm produce?
A)450
B)800
C)1000
D)1250

Question

Quizplus · Accepted Answer

To minimize cost, the firm should equate the ratio of marginal products to the ratio of input prices: $ \frac{M P _ { L }}{w} = \frac{M P _ { K }}{r} $. This gives $ \frac{K}{10} = \frac{L}{20} $, or $ K = 2L $. Substituting into the budget constraint $ 10L + 20K = 1000 $, we get $ 10L + 20(2L) = 1000 $, which simplifies to $ 50L = 1000 $, so $ L = 20 $ and $ K = 40 $. The output is $ Q = K \times L = 40 \times 20 = 800 $.

A Firm's Production Function Is Given By Q=KLQ = K LQ=KL

A Firm's Production Function Is Given By $Q = K L$