. Answer parts a, b and c of this question using the Cobb-Douglas production function of the following form:

Y=AK L^{(1- )} where A>0 represents the level of technology, and K and L are capital and labor respectively, Y is output. is a constant parameter taking a value between 0 and 1. Assume that P is the price of output, W is the nominal wage and R is the nominal rental price of capital.

a. Show that the marginal product of labor (MPL) is diminishing in labor and increasing in capital. (8 points)

b. Assume that a profit-maximizing competitive firm has the production function mentioned above. Using calculus, show that the first-order conditions for profit-maximization for the firm are MPL=W/P (the marginal product of labor=real wage rate) and MPK=R/P (the marginal product of capital = real rental price). That is, set up a firm's optimization problem and derive corresponding first order optimality conditions. (14 points)

c. Use the optimality conditions from your answer in part (b) to explain what happens to the real wage and the real rental price when (i) technological progress improves the production function (that is, A increases); (ii) all nominal prices double in the economy. (8 points)