A firm has production function y = f(x1, x2) = x 1^1/3 x 2 ^2/3 , where y is the amount of output, x1, x2 are the amount of input 1 and 2 respectively. (a) Suppose the firms chooses to produce with inputs x1^0 , x2^0 . Calculate the marginal product with respect to input 1 and input 2. (Express them in terms of x1^0 , x2^0 .)
(b) What’s the firm’s technical rate of substitution given input level x1^0 , x2^0 ?
(c) Suppose the prices for input 1 and input 2 are are respectively w1 = 8, w2 = 2. The market price for the output is p = 50. In order to produce a fixed level of output y 0 = 8, what’s the optimal amount of each input that the firm chooses to use for production?