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 Q = K1/3L1/3M1/3. The prices 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 cost minimization problem given that the firm wants to produce Q units of output?
b) What is the solution to the firm’s short-run cost minimization problem when the firm wants to produce Q units of output and capital is fixed at K?
c) When Q = 4, the long-run cost-minimizing quantity of capital is 4. If capital is fixed at K = 4 in the short run, show that the short-run and long-run cost-minimizing quantities of labor and materials are the same.