Consider the following strategic setting. Every fall, two neighboring elementary schools raise money for field trips and playground equipment by selling giant candy bars. Suppose that individuals in the surrounding communities love candy bars and care about helping the children from both schools but have a slight preference for purchase of candy from the closest school. (In other words, candy bars from the two schools are imperfect substitutes.) Demand for school i's candy, in hundreds of bars, is given by qi = 24 - 2pi + pj , where pi is the price charged by school i, and pj is the price charged by the other school j. Assume that the candy bars are donated to the school and there are no costs of selling the candy. The schools simultaneously set prices and sell the number of candy bars demanded, so school 1's payoff is the revenue p1q1 and school 2's payoff is the revenue p2q2 .
(a) Compute the schools' best-response functions and the Nash equilibrium prices. How much money (in hundreds) does each school raise?
(b) In an effort to raise more money, the schools decide to meet and work together to set a common price for the candy bars sold at both schools. What price should the schools charge to maximize their joint fundraising revenues? How much money (in hundreds) would each school raise if they charge this price?
(c) Suppose that there is no way to externally enforce the price-fixing agreement, so the schools must rely on repeated interaction and reputations to sustain cooperation. If the schools anticipate holding the same fundraiser each fall for 5 years (and no longer), will they be able to maintain the price obtained in part (b)? Explain how or why not.
(d) Now suppose that the schools anticipate holding the same fundraiser every year forever. Define d as the schools' discount factor for periods of a year. Derive a condition on d that guarantees the schools will be able to sustain a cooperative agreement to sell candy bars at the price obtained in part (b).