Consider a competitive firm with a Cobb-Douglas Production function: Y = f (K, L) = K ^1/2 L ^1/3 . Let P be the price of output, W the wage, and R the rental rate on capital.

(a) Prove that this production function exhibits diminishing marginal products.

(b) Prove that the production function exhibits decreasing returns to scale for any z > 1.

(c) Solve the profit maximization problem and find expressions for labor demand, capital demand, and the supply of output.

(d) Plot the labor demand curve when P = 1, R = 1. Plot the supply curve when W = R = 1.

(e) Solve the cost minimization problem and find expressions for conditional labor demand and conditional capital demand.

(f) Find the minimized cost function from (e).

(g) Suppose that W = R = 1 and suppose that fixed costs are equal to F C = 4. Find and plot Average fixed costs (AFC), Average Variable costs (AVC), Average Total Cost (ATC), and Marginal costs (MC).