31 A firm produces output (y), using capital (K ) and labor (L). The per-unit price of capital is r, and the per-unit price of labor is w. The firm's production function is given by, y = Af(L, K), where A > 0 is a parameter reflecting the firm's efficiency. (40 points) (a) Let p denote the price of output. In the short run, the level of capital is fixed at K. Assume that the marginal product of labor is diminishing. Using comparative statics analysis, show that the short-run profit-maximizing choice of labor is increasing in the parameter, A. (10 points) For the remainder of the question, assume that A = 1, f(L, K) = L3Ki, r = 3 and W = 4. (b) Solve for the firm's long-run conditional factor demands, L*(y) and K*(y), and solve for the firm's long-run cost function. (15 points) (c) Let p denote the price of output. Solve for the firm's factor demands, L*(p) and K* (p), and the supply function, y*(p). (15 points)