Suppose a firm's production function is given by Q = F(L, K) = 5LK, where L is the amount of labor and K is the amount of capital. The wage rate is $100 per unit of labor and the rental rate of capital is $50 per unit of capital. a. What is the least-cost combination of capital and labor if the firm produces 1000 units of output? b. What is the firm's long run cost function? c. If the firm currently uses 10 units of capital, what is its short-run cost function?

Question

What will be an ideal response?

Answer 1

a. The tangency condition requires that the MRTSLK equal the ratio of input prices. For a Cobb-Douglass production function, MRTSLK = (?/?)(K/L), so the tangency condition is (?/?)(K/L) = W/R, where W is the wage rate and R is the rental rate of capital. Substituting gives K/L = $100/$50 = 2, so K = 2L. Substituting this into the production function yields Q = 5L(2L) = 10L2, which implies that L = (Q/10)0.5. To produce 1,000 units of output, the firm would require 10 workers. Since K = 2L, the firm would employ 20 units of capital.
b. From the answer to part a, we know that L = (Q/10)0.5. We also know that the firm uses K = 2L = 2(Q/10)0.5 units of capital. The cost function is therefore:
C(Q) = 50 × 2 × (Q/10)0.5 + 100 × (Q/10)0.5 = 200(Q/10)0.5 = 20(10Q)0.5
c. The short-run production function is written as Q = 5L(10) = 50L, so Q/50 = L. The short-run cost function is CSR(Q) = 50(10) + 100(Q/50) = 500 + 2Q.