ASSIGNMENT 5 (Econ 357)

Consider a second-hand car market. There are two types of cars, “lemons” and “peaches”. A seller values a lemon at ✩100 and a peach at ✩200. A buyer values a lemon at ✩120 and a“peach”at ✩300.

1. What are the gains-to-trade per car?  If the probability of a car being a lemon is  , what are the expected gains-to-trade?

Solution: The gains-to-trade are 20 per lemon and 100 per peach. If the probability of a lemon is  , the expected gains-to-trade (per sold car) are 20 +  100 = 40.

2. Suppose that the information is perfect and the the gains-to-trade are split equally between the buyers and the sellers for each type of car. What are the prices and is the allocation pareto-efficient? Suppose that all gains-of- trade are assigned to the buyers. What are the corresponding prices?

Solution: The price for a lemon is 110 and the price for a peach is 250. The allocation is pareto efficient as any other price results in redistribution of the gains from one side of the market to the other. If the buyers have all gains-to-trade, the price is 100 for a lemon and 200 for a peach.

3. Suppose that the buyers cannot observe the quality of the car.

(a) What is the expected value of a car for a buyer if  of the cars are lemons? What is the maximum price a buyer is willing to pay?        Solution: The expected value

120 + 300 = 255

corresponds to the maximum price a buyer is willing to pay.

(b) If  of the cars are lemons, does there exist a pooling equilibrium in which both types of cars are sold?  What is the minimal fraction of lemons such that an equilibrium does not involve any peaches sold? Solution: There exists such a pooling equilibrium in case  of cars are lemons.  Denote by q the fraction of lemons in the market.  The corresponding condition is

120q + 300(1 − q) ≤ 200

or q ≥  . So the minimal fraction of lemons is  .

(c) Suppose that the fraction of lemons in the market is higher than the threshold obtained in (b).  What cars are traded in equilibrium?  Is the outcome pareto-efficient?

Solution:  There is a separating equilibrium in which only lemons are traded and the sellers of the peaches leave the market since they cannot expect a price higher than their valuation of the car.  The outcome is not pareto-efficient as gains-to-trade are lost.