Question: Consider a firm producing an output Q using the two inputs “labor” (L) and “capital” (K) according to the production function Q = K^.5L^.5 (i.e., the amount of output produced Q is equal to the square root of K times the square root of L). Suppose that the firm is in a short-run situation in which K is fixed at one unit (K = 1) but labor is variable. Suppose that the user cost of capital, r, is $48 for each unit of K and that the wage rate, w, the cost of one unit of labor, is equal to $3.

a) Determine this firm’s TVC, TC, AVC, AC, and MC for the output levels Q = 0, 1, 2, 3, 4, 5, 6, 7, and 8 (some costs may not be defined for Q = 0). (8 points)

b) Sketch AVC, AC, and MC (you can “connect the dots” to obtain curves) for this case on a single graph. Explain how the usual relationships between MC and AVC and AC hold in this case. Explain (no calculations are required) what would happen to these three curves if the wage rate w increased. (3 points)

c) Determine the Average Product of Labor (i.e., AP_L = Q/L) for this case (you can obtain a general equation or, alternatively, simply compute AP_L for several values of L). What happens to AP_L is this case as L increases? Explain how this is consistent with the nature of AVC for this case. (3 points)

d) Suppose now that capital K increases from one to four units (i.e., capital is now fixed at the level K = 4). Given this increase in K, determine the new values of TVC and TC (call them TVC* and TC*) for Q = 0, 2, 4, 6, and 8. How does TVC* compare with TVC? How does TC* compare with TC? How is MC affected by this increase in K? Explain. (3 points)

e) Suppose that capital is now fixed at its original unit level (i.e., K = 1) but that the user cost of capital r increases from $48 to $192. Explain how (no calculations are required) this change in r affects the firm’s MC curve. Since the values of TFC (= rK) for this new situation and for part ‘d’ are the same, why is it that MC is affected in one case but not in the other? (3 points)