Consider a firm with the production function f(K, L) = (K1/4 )(L 2/4 )

(a) Showing your work, derive the long run labor demand curve.

(b) Verify that labor demand slopes downward.

(c) Determine whether capital is a gross complement to labor, a gross substitute to labor, or neither.

(d)In light of your answer in part (c), use an isoquant graph to illustrate how a decrease in the price of capital will impact the firm's labor demand. (Your graph does not have to be perfectly to scale. Be sure to label all pieces of your graph, and be clear about what is happening as the price of capital changes.)

(e) Suppose that p = 8, r = 8, w = 2. How many units of labor will the firm hire in the long run?