A rm has a production technology involving two inputs, capital (K } and labor (L), f (K ; L) = K 1331.31; 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. (1)) 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(y) 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? (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 y units of output? What is the short run cost function of the rm, c33(y)? 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 = 3.". Explain Why