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(Horizontal Product differentiation) Consider a horizontally differentiated product market in which two firms are located at any points 1, and l, on the real line, respectively, with the notation 1, 51,. 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. A consumer located at x 8 [0,1] obtains indirect utility v= max{v,,v,} with v;=r-t(x-4)? - p, if she buys one unit from firm 1 and v2 =r-7(12 x)? p, if she buys from firm 2. a. Suppose that prices are regulated at P; = 2c. In the game in which firms simultaneously decide where to locate their product, characterize the Nash equilibrium. Determine the demand function for each firm for each admissible price pair (P1, p2) given locations l, and 12. b. Suppose that the two firms simultaneously set prices. Determine the market equilibrium for all possible combinations of l, and 12. c. Suppose that the social planner chooses the first-best optimal prices. Explain any price pairs P1 = p2 would be socially optimal given the pair of locations 1 = 0 and 12 2 1 d. Compare the results obtained in b and c for locations 1 = 0 and 12 Is the equilibrium socially efficient? Depending on your answer, elaborate on the sources of the inefficiency or give the reason for efficiency. 1 (Horizontal Product differentiation) Consider a horizontally differentiated product market in which two firms are located at any points 1, and l, on the real line, respectively, with the notation 1, 51,. 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. A consumer located at x 8 [0,1] obtains indirect utility v= max{v,,v,} with v;=r-t(x-4)? - p, if she buys one unit from firm 1 and v2 =r-7(12 x)? p, if she buys from firm 2. a. Suppose that prices are regulated at P; = 2c. In the game in which firms simultaneously decide where to locate their product, characterize the Nash equilibrium. Determine the demand function for each firm for each admissible price pair (P1, p2) given locations l, and 12. b. Suppose that the two firms simultaneously set prices. Determine the market equilibrium for all possible combinations of l, and 12. c. Suppose that the social planner chooses the first-best optimal prices. Explain any price pairs P1 = p2 would be socially optimal given the pair of locations 1 = 0 and 12 2 1 d. Compare the results obtained in b and c for locations 1 = 0 and 12 Is the equilibrium socially efficient? Depending on your answer, elaborate on the sources of the inefficiency or give the reason for efficiency. 1

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