There are many potential buyers for used cars

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There are many potential buyers for used cars. All of them are willing to pay $1,000 for a lemon and $2,000 for a good used car. There are 1,000 owners of lemons and 1,000 owners of good used cars. The reservation price of owners of lemon is $750. Among the 1,000 owners of good used cars, 600 of them have reservation price at $1,250 and 400 of them have reservation price at $1,750.

Suppose we are back in a time when there is no CARFAX or other car reports; the buyers of used cars cannot identify which car is a lemon and which car is of good quality (and the sellers know the exact quality of their cars.)

a. Although the buyers do not know the exact quality of a car, they know that there are 1,000 lemons and 1,000 good cars. They assess that with probability \frac{1}{2} a car is a lemon and with probability 1212 a car is of good quality. Suppose the buyers are risk-neutral. How much they are willing to pay for a car?

b. At the amount the buyers are willing to pay that you find in (a), how many lemon owners (l), and how many good-car owners (g) are willing to sell their cars?

c. Given what you find in (b), what will be the new amount the buyers are willing to pay for a car? (Assume that the buyers, by referring to the l and g you find in (b), update their probability assessment of a lemon and a good car being on the market.)

d. At the new amount that the buyers are willing to pay for a new car, how many lemon owners and how many good-car owners are willing to sell their cars?

e. Given (c) and (d), is there an equilibrium in this used-car market? In other words, is the new amount that buyers are willing to pay for a car an equilibrium price? Explain by referring to the notion of mutual best responses as equilibrium.