1. A firm has production function y = f(x1, x2) = x1/31 x2/32, where y is the amount of output, x1, x2 are the amount of input 1 and 2 respectively.
(a) Suppose the firm chooses to produce with inputs x01, x02. Calculate the marginal product with respect to input 1 and input 2. (Express them in terms of x01, x02.)
(b) What's the firm's technical rate of substitution given input level x01, x02?
(c) Suppose the prices for input 1 and input 2 are respectively w1 = 8, w2 = 2. The market price for the output is p = 50. In order to produce a fixed level of output y0 = 8, what's the optimal amount of each input that the firm chooses to use for production?
2. A firm has production function y = f(x1, x2) = min{2x1, x2}, where y is the amount of output, x1, x2 are the amounts of input 1 and 2 respectively. Suppose the prices for input 1 and input 2 are respectively w1 = 2, w2 = 40. The market price for the output is p = 50. In order to produce a fixed level of output y0 = 30, what's the optimal amount of each input that the firm chooses to use for production?
3. Consider a monopoly faces a market demand function q(p) = 200 - p, where p is the price that the monopoly charges, q is the amount demanded by the consumer. The monopoly has marginal cost function MC(q) = q.
(a) At the price level of 100, what's the consumer's demand elasticity?
(b) Suppose the monopoly sets the price at the level p = 150, calculate the consumer surplus in this case.
(c) Suppose the monopoly sets the price at the level p = 150, calculate the monopoly's net profit.
(d) In order to maximize its net profit, what's the optimal price level that the monopoly should charge?
(e) Consider a social regulator who aims to maximize the social welfare (which equals the summation of consumer surplus and the monopolist's profit). What's the optimal price level that the social regulator should set in order to maximize the social welfare?