Consider a horizontally differentiated product market in which two firms are located at any points l1 and l2 on the real line, respectively, with the notation l1 l2. Firms produce at marginal costs c. There is a continuum of consumers of mass 1 who are uniformly distributed on the unit interval. They have unit demand and have an outside utility of . A consumer located at x [0; 1] obtains indirect utility v = max (v1; v2) with v1 = r (x l1) 2 p1 if she buys one unit from firm 1 and v2 = r (l2 x) 2 p2 if she buys from firm 2. Firms have marginal costs equal to c.

a. Suppose that prices are regulated at pi = 2c. In the game in which firms simultaneously decide where to locate their product, characterize the Nash equilibrium.

b. Determine the demand function for each firm for each admissible price pair (p1; p2) given locations l1 and l2.

c. Suppose that the two firms simultaneously set prices. Determine the market equilibrium for all possible combinations of (l1; l2).

d. Suppose that the social planner chooses first-best optimal prices. Which price pairs would be socially optimal for the pair of locations l1 = 0 and l2 = 1/2?

e. Compare your results obtained in (c) and (d) for locations l1 = 0 and l2 = 1/2. Is the equilibrium socially efficient? Depending on your answer elaborate on the sources of the inefficiency or give the reason for efficiency.