Problem 2 (A Discrete Version of Lemon Market). Consider a market for used cars in which there is a continuum of sellers, of total mass equal to 1. There is also a continuum of buyers, of total mass also equal to 1. Each seller has exactly one car for sale, and each buyer buys at most one car. There are two types of cars for sale in the market: low quality (lemons) and high quality (peaches). A half of the sellers are endowed with a lemon, and the other half are endowed with a peach. Buyers value lemons at $50 and peaches at $120. Sellers value lemons at $40 and peaches at $96 (that is, 80% of the buyers' valuation). Suppose that the sellers know the qualities of their car, but buyers cannot tell.

(Note. In Problem 1, there is a single seller who has one object, and the quality of the object is randomly drawn from some distribution. In this problem, there is a continuum of sellers, each endowed with one object, and the qualities satisfy a certain distribution. These two models are mathematically equivalent, but with slightly different interpre- tations. We use the second interpretation in this question and study the "competitive equilibrium" of the lemon market.)

(a) Does there exist a competitive equilibrium where all cars are sold? (That is, can we find a single price that clears the market and all cars are sold?)

(b) Does there exist a competitive equilibrium where no cars are sold? (That is, can we find a single price that clears the market and no cars are sold?)

(c) Does there exist a competitive equilibrium where only lemons are sold? (Hint. Only one price can clear the market.)