4. Consider a horizontally differentiated product market in which two rms are located at any points 11 and 12 on the real line, respectively, with the notation ll 5 [2. 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 00. A consumer located at x e [0; 1] obtains indirect utility v = max (121; 122) with v1 = r 1(x ll)2 p1 if she buys one unit from rm 1 and 122 = r 7(12 x)2 1);; if she buys from rm 2. Firms have marginal costs equal to c. a. Suppose that prices are regulated at p,- = 2c. In the game in which rms simultaneously decide where to locate their product, characterize the Nash equilibrium. b. Determine the demand function for each rm for each admissible price pair (p1; p2) given locations 11 and 12. c. Suppose that the two rms simultaneously set prices. Determine the market equilibrium for all possible combinations of (II; 12). d. Suppose that the social planner chooses rst-best optimal prices. Which price pairs would be socially optimal for the pair of loca- tions 11 = 0 and 12 = 1/2