You would like to buy a used car. Suppose there are three types of quality (a): low (a = 0), medium (a = 1) and high (a = 2), each equally likely (i.e. each with probability 1/3). Only the seller knows the exact quality of his/her car. Also suppose (as in class) that for any quality of the car, your utility is 1.5 times the seller's utility, that is, for a car of quality a, the seller's utility from it is a, but your utility from it is 1.5a.

(i) The market price for a used car is p. For what values of p, all three types of sellers want to sell? For what values of p, exactly two types want to sell? For what values of p, only one type wants to sell?

(ii) Is there any price at which trade is mutually agreeable? Namely, given your answers in (i), is there any price at which you want to buy a used car?

(iii) Suppose you have help from a mechanic, so now you can tell whether a car is low quality, but cannot tell between medium and high quality cars. Is there any price at which trade is mutually agreeable? If yes, give an ex of such a price; if no, describe why. Justify your answers.