Question

There are many sellers of used cars. Each seller has exactly one used car to sell and is characterised by the quality of the used car he wishes to sell. The quality of a used car is indexed by θ, which is uniformly distributed between 0 and 1. If a seller sells his car of quality θ for price p, his utility is p − θ². If he does not sell his car, 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.

Find the highest equilibrium market price p of used cars.

Answer 1

This is a case of adverse selection.

Only those sellers with a car with quality below p are willing to sell. Sellers with a better car prefer to keep it. (sellers are aware of quality). However in doing so, there utility is nil.

No buyer will be willing to buy at any p > 0 (based on their utility as they are not aware of car quality; expected quality is always less than the price paid)

At minimum equilibrium: p = 0, and no cars are sold.

Average car quality index is θ = (0+1) / 2 = 0.5 (uniformly distributed between 0 and 1). Buyer is not aware of the car quality and is a price taker. The seller based on his utility function sells when P = θ² or P = (0.5)² = 0.25 (maximum equilibrium price)