Problem 2 (30 pts) Explain your answers! A rm has a production technology involving two inputs, capital (K) and labor (L), f (K ; L) = K1/3L1/3. The price of capital is r; the price of labor is w, and the output price is p. (a) Does the production function of the rm exhibit constant / increasing / decreasing returns? Justify your answer. (b) Let r = 1 and w = 4 and assume both inputs are variable. Derive the rm's conditional input demands for producing 3; units of output, K(y) and My). What is the rm's cost function C(y)? What are the long-run average and marginal cost functions? For any output price p, derive the rm's long run supply function y = S (p) If p = 4 how much output will the rm produce? (c) Now assume that in the short run the rm's quantity of capital is xed at K = 1. What is the conditional input demand for labor for producing 3; units of output? What is the short run cost function of the rm, (:33 (3;)? What are the rm's short run average and marginal cost functions? Are there any xed costs? For any output price p derive the rm's short run supply function y = SSR(p). (d) For what output quantity, y* is K = 1 the optimal long run level of capital? Show that the long run average cost curve (from (b)) and the short run average cost curve (from (6)) are tangent at y = y*. Explain why