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 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 allocate its budget such that the ratio of marginal products to input prices is equal: $ \frac{MP_L}{w} = \frac{MP_K}{r} $. Solving $ \frac{K^2}{10} = \frac{2KL}{20} $ gives $ K = L $. With a budget of $1200, $ 10L + 20K = 1200 $, substituting $ K = L $ gives $ 30L = 1200 $, so $ L = 40 $ and $ K = 40 $. Substituting into the production function $ Q = K^2L = 40^2 \times 40 = 64,000 $.

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

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