Determine the firms profit maximizing quantity show your

Question

Provide complete work that leads to each answer. Do the problems in the order given.

Question 1. Consider the following technology:

C(y_i) = y_i³ - 8y_i² + 36y_i (1)

Many firms have access to this technology, in fact so many that there is not room for all to profitably operate in the industry. Market demand for the product is given by

P = 2420 - Y (2)

where Y is market quantity.

a. If these firms are price takers, and entry/exit is costless, what is the long run free entry equilibrium per-firm quantity produced by each firm in the industry? Explain.

b. Do you have enough information to determine the number of firms who operate in such a long run free entry equilibrium? If so what is the number and justify your solution. If not explain what other information you would need.

c. Is the price-taking behavioral assumption sensible for this industry?

Question 2. A monopolist has access to an industry with market demand

P = 10 - y    (3)

where y is the firm's quantity. Its cost function is

C(y) = 2y     (4)

a. Determine the firm's profit maximizing quantity. Show your outcome on a graph. What is the firm's profit? Compute the point-elasticity of demand at the profit-maximizing output.

b. Now suppose the firm's cost function is

C(y) = 4y     (5)

Again determine the profit-maximizing quantity, profit and the elas- ticity at the profit-maximizing quantity. (No graph is required in this case.)

c. Essentially, we have two types of monopolist. Which monopolist type operates at the higher level of elasticity? Why?

d. Prove that for any linear demand, p = a -by, and constant marginal cost, c, that a monopolist would never ever operate at a point elas- ticity less than 1.

Question 3. A monopolist faces a market evenly split between high valua- tion consumers, with individual demand function

P^h = 10 - y^h     (6)

and low valuation consumers with individual demand function

P^l = 8 - 2y^l     (7)

For convenience, suppose the firm has a marginal cost equal to zero. Ar- bitrage is impossible for this good.

a. Suppose the firm can observe group status. Determine the firm's profit-maximizing third degree pricing scheme for these groups. Show this outcome on a graph.

b. Now suppose the firm cannot observe group status and so must use second degree price discrimination. Determine the firm's best second- degree pricing scheme. Show this outcome on a graph, and explain why your answer is the best outcome for the firm.

c. Which of these two outcomes (second degree and third degree) is better from the economy's view?