This is my question that is very important

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 7"; 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 y units of output, K (g) and L(y). 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? (0) 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 y units of output? What is the short run cost function of the rm, CSR (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 (c)) are tangent at y = y*. Explain why. Problem 3 (15 pts) Explain your answers! Bicycle frames are produced by a number of identical rms under perfect competition. Each rm's long-run cost function is C(y) = y3 20342 + 1003; -I 8,000 Where y is the number of frames produced. The market demand for frames is q = 2, 500 3p Where p is the market price per frame. (a) What is the long run equilibrium price? (b) What is the long run equilibrium level of industry supply? (c) What is the long run equilibrium number of rms in the industry? How many frames does each of them produce