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

The equilibrium price is the only price where the plans of consumers and the plans of producers agree—that is, where the amount consumers want to buy of the product, quantity demanded, is equal to the amount producers want to sell, quantity supplied. This common quantity is called the equilibrium quantity. At any other price, the quantity demanded does not equal the quantity supplied, so the market is not in equilibrium at that price.

The word equilibrium means balance. If a market is at its equilibrium price and quantity, then it has no reason to move away from that point. However, if a market is not at equilibrium, then economic pressures arise to move the market toward the equilibrium price and the equilibrium quantity.

Here in this case, 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. Also, sellers know the quality of the car they are selling, but buyers do not know its quality. We have to determine the highest equillibrium market price of used cars.

Suppose for car A, which is used car and the seller is willing to sell it; θ is 0.5, as it is mentioned above that θ is uniformly distributed between 0 and 1. Now, the seller's utility would be p - (0.5)². And the buyer will receive utility 0.5 - p. Price will be p.

Keep in mind that competitive equilibrium is achieved when profit-maximizing producers and utility-maximizing consumers settle on a price that suits all parties. If the utility of both seller and buyer matches at the set price p, this will be called as the highest price.