2 Problem Consider the following variation of the model of the market for used cars that we studied in class. Used cars come in two different qualities: 0 = 1/2 or 0 = 1. The owner of a used car of quality 0 gets utility 0 from keeping the car, and gets utility p from selling the car at price p. A potential buyer of a used car gets utility 30 - p from buying a used car of quality 0 at price p. Assume that sellers know the quality of their used car, but buyers cannot observe this quality. Buyers only know that the probability that 0 = 1 is 1/5 and the probability that 0 = 1/2 is 4/5. 1. Suppose first that the price for used cars is p 2 1. What will be the average quality of the cars for sale at this price? What is the expected utility of a buyer who buys a car at this price p z 1? 2. Suppose next that p E [1/2, 3/4]. What will be the average quality of the cars for sale at this price? What is the expected utility of a buyer who buys a car at this price p? 3. Consider a different market, which is exactly equal to the one above, except that now the fraction of used cars of quality 0 = 1 is equal to a E (0, 1), and the fraction of used cars of quality 0 = 1/2 is equal to 1 - a. Show that, if a 2 1/3, there exists an equilibrium in which all cars are traded. Hint: show that, if a 2 1/3, buyers get a weakly positive expected utility from buying a car at price p = 1