Question 3 (50%): The Market for Lemons. Consider two sequential used car markets. The first is a market for second-hand cars (one previous owner). The second is a market for thirdhand cars (two previous owners). In the first market, there are two types of traders: Original Owners and Original Buyers. Original Owners own all the cars and have the following utility function: U - Tax + M (1) where M is the amount of non-car goods consumed, I, is the quality level of the jth car, and a is a parameter in the utility function. Original Buyers own no cars and have the following utility function: U - Tex, + M (2) where again M is the amount of non-car goods consumed, I, is the quality level of the jth car, and b is a parameter in the utility function. There is a uniform distribution over the quality of all cars, with: X -- Uniform 0, 100] (3) Let P, be the price of used cars put up for sale by Original Owners in equilibrium. Original Owners know the quality of the cars they are selling but Original Buyers only know the average quality of the cars on the market. Original Buyers know the utility function of the Original Owners. 1. (5%) What will be the average quality of cars offered on the market by Original owners under these conditions, as a function of Pi and a? 2. (5%) For what values of b will Original Buyers be willing to buy the cars, as a function 3. (10%) Suppose b satisfies the condition you found in the previous question, and some cars are sold from the Original Owners to the Original Buyers. Now the second market takes place. The Original Buyers become Second-hand Owners and they are in a third- hand car market with new buyers, which we call Second-hand Buyers. The Second-hand Buyers know all about the first market for Second-hand cars. The Second-hand Owners, after driving around their new cars, have learned the quality of the cars, but the Second- hand Buyers only know the average quality of the cars on the market. The Second hand Buyers own no cars and have the following utility function: U - Lex, + M (4)