Screening: Consider the market for used cars:

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:| Value to Seller Value to Buyer Lemon 1 3 Crea m Puff 7 13 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 1/2 probability of being a lemon and a 1/2 probability of being a cream puff the buyer would be willing to pay at most 1/2\"3 + 1/2"'l3 = 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=l1, P_lemon=2. The inspector's word is always taken as true. Ifthe 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 pu's)? Problem 2. Screening: In class, we considered price discrimination via quality. The same model also works for price discrimination via lump sum discount (buy more at a lower price per unit). Consider a small yogurt shop that sells fresh-made yogurt to customers. The shop owner is facing two types of customers and is currently designing a price menu and want to max. expected revenue by giving a larger discount to the customers who buy more. 0 Buyer's utility function is given by: 11(1), q, t) = v - q p - q o In which: 0 Her willingness to pay for quality: 12 E {1,2}. 0 q: quantity of yogurt she is going to buy 0 p: the price she has to pay for each unit she buys 0 Seller has cost of production for each customer: 60;) = qr2 o The prot is given by: p - q (:03) 0 Seller does not know buyer's willingness to pay but knows the distribution: Prob(v = 1) = g; Prob(u = 2) =% . Solve for the second degree price discrimination pricing strategy. int: Seller wants the low type to participate and does not want high type to pretend to be low type]