Consider a town where all 1,000 inhabitants are willing to sell their car for any price. The quality of the cars varies; the owners know the quality but potential buyers do not observe it. There are many potential buyers who are willing to pay V for a car with value V. While the buyers cannot detect the value of an individual car, they know the distribution of the value: it varies between 0 and 1,000 and it is uniformly distributed such that the number of cars, whose value is less than V, is V. The buyers are risk neutral.

There is also a car inspection station in the town, which is capable of detecting the value of the car truthfully and it will also do so if asked. Car owners can decide whether they want to get their car inspected for a price of 200 . There are no other costs of selling the car.

1. a) If no car is inspected, what is the price for a used car, the number of cars sold and the overall value of the used car market?

2. b) Assume next that all cars with value greater than X will be inspected. What are now the prices and profits of uninspected and inspected cars, respectively?

3. c) In the equilibrium, one can determine a value z: cars with value greater than this threshold value are inspected and the rest are not. Determine z. How many cars are left uninspected and what is their price? What is now the total value of the used car market?