Question

1. There are many sellers of used cars. Each seller has exactly one used car to...

1.

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 − θ 2 . 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.

2.

Consider a second price auction for a single item with two bidders. Suppose the bidders have independent private values, uniformly drawn in the interval [0, 1]. Suppose the seller sets a reserve price p = 0.5; that is, only bids above p = 0.5 can win. If a bidder bids above p and the other bids below p, then the first bidder wins and pays a price p. If both bid above p, then the highest bidder wins and pays the second highest price.

In the Bayesian equilibrium in undominated strategies, what is the probability that the item will not be sold?

3.

Consider a second price auction for a single item with two bidders. Suppose the bidders have independent private values, uniformly drawn in the interval [0, 1]. Suppose the seller sets a reserve price p = 2/5; that is, only bids above p = 2/5 can win. If a bidder bids above p and the other bids below p, then the first bidder wins and pays a price p. If both bid above p, then the highest bidder wins and pays the second highest price.

In the Bayesian equilibrium in undominated strategies, what is the seller’s expected revenue?

4.

Consider a second price auction for a single item with two bidders. Suppose bidder 1 has value uniformly drawn in the interval [0, 1], while bidder 2 has value 0.5.

In the Bayesian equilibrium in undominated strategies, what is the seller’s expected revenue?

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