Consider a strategic setting in which two geographically distinct firms (players 1 and 2) compete by setting prices. Suppose that consumers are uniformly distributed across the interval [0,1], and each will buy either one unit or nothing. Firm 1 is located at 0 and firm 2 is located at 1. Assume that the firms cannot change their locations; they can only select prices. Simultaneously and independently, firm 1 chooses a price p1 and firm 2 chooses a price p2 . Suppose that the firms produce at zero cost and that due to a government regulation, they must set prices between 0 and 6.
As in the standard location game, consumers are sensitive to the distance they have to travel in order to purchase. But they are also sensitive to price. Consumers get a benefit of 6 from the good that is purchased, but they also pay a personal cost of c times the distance they have to travel to make the purchase. Assume that c is a positive constant. If the consumer at location x ∈ [0, 1] purchases from firm 1, then this consumer’s utility is 6 − cx − p1 . If this consumer instead purchases from firm 2, then her utility is 6 − c(1 − x) − p2 . If this consumer does not purchase the good, her utility is 0.
(a) Suppose that for given prices p1 and p2, every consumer purchases the good. That is, ignore the case in which prices are so high that some consumers prefer not to purchase. Find an expression for the “marginal consumer” who is indifferent between purchasing from firm 1 or firm 2.
Denote this consumer’s location as x*( p1, p2 ).
(b) Continue to assume that all consumers purchase at the prices you are analyzing. Note that firm 1’s payoff (profit) is p1 x*( p1, p2 ) and firm 2’s payoff is p2[1 − x*( p1, p2 )]. Calculate each firm’s best response as a function of the other player’s strategy. Also graph the best-response functions for the case of c = 2.
(c) Find and report the Nash equilibrium of this game for the case in which c = 2.
(d) As c converges to 0, what happens to the firms’ equilibrium profits?
(e) What are the rationalizable strategies of this game for the case in which c = 2?
(f) Find the Nash equilibrium of this game for the case in which c = 8.