Consider a production function of three inputs, labor, capital, and materials, given by Q = LKM. The marginal products associated with this production function are as follows: MPL = KM, MPK = LM, and MPM = LK. Let w = 5, r = 1, and m = 2, where m is the price per unit of materials.
a) Suppose that the firm is required to produce Q units of output. Show how the cost-minimizing quantity of labor depends on the quantity Q. Show how the cost minimizing quantity of capital depends on the quantity Q. Show how the cost-minimizing quantity of materials depends on the quantity Q.
b) Find the equation of the firm's long-run total cost curve.
c) Find the equation of the firm's long-run average cost curve.
d) Suppose that the firm is required to produce Q units of output, but that its capital is fixed at a quantity of 50 units (i.e., K = 50). Show how the cost-minimizing quantity of labor depends on the quantity Q. Show how the cost-minimizing quantity of materials depends on the quantity Q.
e) Find the equation of the short-run total cost curve when capital is fixed at a quantity of 50 units (i.e., K = 50) and graph it along with the long-run total cost curve.
f) Find the equation of the associated short-run average cost curve.