1,000 people want to sell their used cars. These cars vary in quality. Original owners know exactly what their cars are worth. All used cars look the same to potential buyers until they have bought them; then they find out the truth. For any number X between 0 and 2,000, the number of cars of quality lower than X is X/2. If a car is of quality X, its owner will be willing to sell it for any price greater than X. If a buyer knew that a car was of quality X, she would be willing to pay X + 500 for it.When buyers are not sure of the quality of the car they are wiling to pay its expected value, given their knowledge of the distribution of qualities on the market.

1. Suppose that everybody believes that all the used cars are for sale. What would used cars sell for? Would every used car owner be willing to sell at this price? Which used cars would appear on the market?

2. Let X* be some number between 0 and 2,000. Suppose that all used cars of quality lower than X* are sold, but owners keep all cars of quality higher than X*. What would buyers be willing to pay for a used car? At this price which used cars will be for sale?

3. show an equation for the equilibrium value of X*, at which the price that buyers are willing to pay is exactly enough to induce all cars of quality less than X* into the market.

4. Find the equilibrium value of X*