8.22. 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., Show how the cost-minimizing quantity of labor depends on the quantity Q. Show how ?