Problem 1. Screening: Consider the market for used cars. There are lemons (low quality cars) and cream puffs (high quality cars). You are provided with the following valuations of buyers and sellers of each type:

The value to the buyer represents the maximum price she will pay for a car of the given type. Assume buyers are risk neutral. (e.g. if a car has a ½ probability of being a lemon and a ½ probability of being a cream puff the buyer would be willing to pay at most ½*3 + ½*13 = 8.) The seller’s valuation is the minimum price for which she will sell the car.

a. Suppose there is only one price for cars. Compute the maximum proportion of lemons that the market can sustain (i.e. so that cars of both types will be traded)?

b. Now car inspection is possible for a fee, and different prices are allowed for each type: P_cream puff=11, P_lemon=2. The inspector's word is always taken as true. If the inspector says a car is a lemon, the buyer believes her. But the sellers know that the inspector is only right 90% of the time when inspecting a cream puff and 80% of the time when inspecting a lemon. What fee, if any, can be set that will allow the two types of cars to separate themselves (i.e. buyers willing to buy lemons at the lemon price, and sellers willing to sell at that price; and the same for cream puffs)?