(Karloff) Consider the following market for used cars. There are many sellers of used cars. Each seller has exactly one used car to sell and is characterized by the quality of the used car he wishes to sell. Let θ ∈ [0, 1] index the quality of a used car and assume that θ is uniformly distributed on [0, 1]. If a seller of type θ sells his car (of quality θ) for a price of p, his utility is us (p,θ). If he does not sell his car, then his utility is 0. Buyers of used cars receive utility θ − p if they buy a car of quality θ at price p and receive utility 0 if they do not purchase a car. There is asymmetric information regarding the quality of used cars. Sellers know the quality of the car they are selling, but buyers do not know its quality. Assume that there are not enough cars to supply all potential buyers.

(a) Argue that in a competitive equilibrium under asymmetric information, we must have E(θ | p) = p

(b) Show that if us (p,θ) = p − θ/2, then every p ∈ (0, 1/2] is an equilibrium price.

(c) Find the equilibrium price when us (p,θ) = p − √θ. Describe the equilibrium in words. In particular, which cars are traded in equilibrium? (d) Find an equilibrium price when us (p,θ) = p − θ 3. How many equilibrium are there in this case? (e) Are any of the preceding outcomes Pareto efficient? Describe Pareto improvements whenever possible.