For the following production function:

y = K^x L^x

a. Show whether the function exhibits constant, increasing or decreasing returns to scale for different parameter values of x.

b. Consider the following three parameter values: x = 1 x = 1/2 and x = 1/4 For each of these parameter values, draw two isoquants in a graph, one for y = 1 and one for y = 2: Use the graph to show the returns to scale of the production function.

c. Define the long-run cost minimization problem for the firm.

d. Obtain the long-run input demands, total cost and average cost all as a function of output y, input prices w1, w2 and technology parameter x. Also, obtain the short-run average costs for K = 1.

e. Graph the (long-run) average cost functions for all three cases in part (b). They should have a very essential difference. Why? (your answer should point towards returns to scale).

f. Consider the case of x = (7/16). Also, assume that the market operates under perfect competition. Obtain long-run quantity produced and profits for the firm (as a function of parameters w1; w2; p). Also obtain the (long-run) unconditional input demands (as a function of parameters w1, w2, p).

g. For the case of constant returns to scale in part (b) obtain the long-run output price p (as a function of input prices w1; w2) that implies zero profits for the firm. Argue that a higher price is not consistent with perfect competition and a lower one would induce the firm to shut down.

h*. Consider the case of w1 = w2 = 1 and x = 1. Also assume that the firm has a monopoly. The inverse demand function is: p = (20/3)y^-(7/10). Obtain the long-run profit maximizing quantity and price.