Assume that the production function is Cobb-Douglas:

Y=AK L^{1 -}

where is the parameter that satisfies 0< <1

a) Formulate a profit maximization problem for a perfectly competitive firm that takes nominal wage rateW;rental price of capitalR;and the price of final goodP;as given.

b) Derive the first order conditions that characterize the firm's optimal decisions with respect to capital and labor inputs.

c) Show that the marginal product of capital(MPK)is an increasing function of laborL;and a decreasing function of capitalK.

d) Using the firm's optimality conditions, show that average labor productivity(Y/L)is proportionate to real wage (W/P)

e) Using the optimality conditions show that in equilibrium the firm earns zero economic profits.