Consider the following market for used cars. There are N potential sellers and M potential buyers, and M > N. Each seller has exactly one used car to sell and is characterized by the quality of the used car he has. Let θ∈[0; 1] index the quality of used car. If a seller of type θ sells his car for a price of P, his payo is u(P; θ), and is 0 if he does not sell his car. The payoff for the buyer is θ-P if he buys a car of quality θ at price P, and is 0 if he does not buy. Information is asymmetric: Sellers know the quality of used cars but buyers do not. However, buyers know the quality of used car is uniformly distributed on [0,1].
(a) Argue that in a competitive equilibrium under asymmetric information, we must have E[θ|P] = P.
(b) Find all equilibrium prices when u(P; θ) = P - θ/2.
(c) Find all equilibrium prices when us(P; θ) = P -θ^(1/2). Describe the equilibrium in words. In particular, which cars are traded in equilibrium?
这是我们高微的一道作业题，主要是第一小题没什么思路，所以希望高手能提供下思路。Thanks~~~