1. [Monopoly] Consider the following monopolists that face nonlinear demand functions.

(a) Suppose a monopolist faces a demand curve of Q = e^200-P, and has costs C(Q) = 3Q - ln(Q). Solve for equilibrium quantity and price.

(b) Suppose a monopolist faces an inverse demand curve of P = A- Q^v, and a cost function C(Q) = Q^v+1. Solve for equilibrium quantity and price, as a function of A and v. What happens to the equilibrium price as v approaches ∞? What does this tell you about the behavior of Q*as v → ∞?

2. [Monopoly and Price Discrimination] There are three submarkets, A, B, and C. Demand in submarket A is given by: Q_A = 200 - P/10. In B, it is QB = 160 - P/5, and in C it is Qc = 40 - P/5.

(a) There is a monopolist who has a marginal cost of production of MC(Q) = 20/33Q and no fixed costs. Suppose that she cannot price discriminate between these three markets.

i. What is her equilibrium price P* and equilibrium quantity Q*?

ii. What is her equilibrium producer surplus? What consumer surplus does each group receive? What is total surplus?

(b) Suppose now that this same monopolist can implement 3rd degree price discrimination (group pricing).

i. What is her equilibrium price in each market? (P*_A, P*_B, P*_C)? What quantity does she sell in each market (Q*_A, Q*_B, Q_C*) and how does that relate to Q* from before?

ii. What is her new equilibrium producer surplus? What consumer surplus does each group receive? What is total surplus?

iii. Suppose instead that the monopolist had a constant marginal cost μ. What value of μ, would result in the same equilibrium outcome (in terms of quantities sold to each group), as your answer to 2.b.i? How would this change total surplus?

3. [Duopoly] Suppose there are two firms in a market, Firm 1 and Firm 2. Firm 1 has a constant marginal cost of production of 12, while Firm 2 has a constant marginal cost of production of 8. They face a demand of Q = 50 - P/4.

(a) Suppose that these two firms engage in quantity competition (Cournot competition). What is the equilibrium quantity produced by each firm, and what is the prevailing market price?

(b) Suppose instead that these firms competed via prices (as in Bertrand competition). They have no capacity constraints. What is the equilibrium outcome of this game? [Note: If you wish, for this part of the problem, you may assume that firms set prices to two decimal places.]

(c) Suppose now that each firm has a capacity constraint. What is the equilibrium in this market when these constraints are q¯1 = q¯2 = 10?

(d) What happens when those constraints are increased to q¯1 = 10 q¯2 = 14?

(e) What happens when those constraints are increased to q¯1 = q¯2 = 300?

4. [Differentiated Demand - Price Competition] Two firms compete via price competition in a market with differentiated products. The demand for Firm l's good is Q₁ = 600 - 4P₁ + 3P₂, while the demand for Firm 2's good is Q₂ = 200 - 2P₂ + P₁. Firm 1 has costs C₁(Q₁) = 10Q, while Firm 2 has costs C₂(Q₂) = 2(Q₂)².

(a) Solve for the best response functions for each firm.

(b) Solve for the Nash equilibrium of this game.

5. [Spatial Bertrand] There are two firms in a linear town. Firm 1 is located at x = 0, and Firm 2 is located at x = 1. Firm 1 has a constant marginal cost of 1; Firm 2 has a constant marginal cost of 2. There is a unit mass of consumers in this town (N=1) uniformly distributed along the line. Each consumer gets a benefit V from buying the good (which we will assume is high enough that they always buy the good), but experiences transit costs equal to twice the distance from their location to the store from which they buy the good (in the notation we used in class, t = 2). That is, a consumer that is located distance d away from store i receives a surplus of V -p_i - 2d if they buy the good from store i.

(a) Solve for the marginal consumer as a function of p₁ and p₂.

(b) Solve for each firm's best response function.

(c) Solve for the equilibrium prices in this town.

(d) Suppose now that the prices of Firm 1 and Firm 2 are fixed, and cannot be changed. However, a new firm (Firm 3) is entering the town. This firm will be located at x = 0.5, and has a constant marginal cost of c.

i. Solve for the lowest consumer that purchases the good from Firm 3, as a function of Firm 3's price. (Remember, pi and /32 are fixed.) Solve for the highest consumer that purchases the good from Firm 3.

ii. Solve for Firm 3's optimal price, as a function of it's marginal cost c.

- Problem sets are due in class on Tuesday, October 18. Please bring your problem set to the section in which you are enrolled.

- You must submit a physical copy of your problem set.

- Be sure to put your name and student number on your solutions - problem sets submitted without a name will be discarded.

- Either type your solutions, or write them extremely neatly. The graders will be instructed to disregard any answers that are difficult to read.

- Show all of your work, and present it in a logical manner.

- You may work in groups of up to four. You do not have to work in a group, but you may do so if you choose.

- Each member of the group must write up their own solutions and turn them in. If you choose to work in groups, on the first page of your solutions, list the group members with whom you collaborated.