7.29. Suppose that the firm uses three inputs to produce its output: capital K, labor L, and materials M. The firm’s production function is given by For this production function, the marginal products of capital, labor, and materials are and MPM ! . The prices 1 3K 1 3L 1

3M$2 MP 3 L ! 1 3K 1 3L$2 3M1 3, MPK ! 1 3K$2 3L 1

3M1 3, Q ! K1 3L 1

3M 1 3 .

of capital, labor, and materials are r ! 1, w ! 1, and m ! 1, respectively.

a) What is the solution to the firm’s long-run costminimization problem given that the firm wants to produce Q units of output?

b) What is the solution to the firm’s short-run costminimization problem when the firm wants to produce Q units of output and capital is fixed at

c) When Q ! 4, the long-run cost-minimizing quantity of capital is 4. If capital is fixed at in the short run, show that the short-run and long-run cost-minimizing quantities of labor and materials are the same.