Suppose that a firm uses three inputs to produce its output: capital K, labor L, and materials M. The firm’s production function is given by q = K0.5L0.25M0.25. For this production function, the marginal products for labor and materials, respectively, are MPL = .25K0.5L-0.75M0.25 and MPM = (.25)K0.5L0.25M-0.75. The prices of capital, labor, and materials are r = $15, w = $2, and m = $1, respectively. Suppose that capital is fixed at K = 36 in the short run.
(a) Find the quantities of labor and materials that the firm will use to minimize the cost of producing an output level q in the short run. Do not assign the output level q a specific numerical value. (Hint: This is a short-run cost problem with two variable inputs. In class (but in the context of the long run, so no fixed third factor), we went through the technique you need to answer part (a). We also, in the long-run context, did the special case of q = 40.)
(b) Making use of your results in part (a), derive the short-run total cost function, the short-run variable cost function, and the short-run fixed cost function for this firm.
(c) Making use of your results in part (b), derive the short-run average cost function and the short-run average variable cost function.