Thirdhand car markets. Consider now two sequential used-car markets. The first is a market for secondhand 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:

where M is the amount of non-car goods consumed, x_j 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:

where again M is the amount of non-car goods consumed, xj 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

Let P₁ 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.

a. What will be the average quality of cars offered on the market by Original Owners under these conditions, as a function of P₁ and a?

b. For what values of b will Original Buyers be willing to buy the cars, as a function a?

c. 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 Secondhand Owners and they are in a thirdhand car market with new buyers, which we call Secondhand Buyers.

The Secondhand Buyers know all about the first market for secondhand cars.

The Secondhand Owners, after driving around their new cars, have learned the quality of the cars, but the Secondhand Buyers only know the average quality of the cars on the market. The Secondhand Buyers own no cars and have the following utility function:

where c is a parameter in the utility function. Let P₂ be the price of used cars put up for sale by Secondhand Owners in equilibrium. For what values of c will Secondhand Buyers be willing to buy the cars, as a function of b?

d. Suppose b=3. For what price P2 will the cars available on the thirdhand car market be uniformly distributed between 0 and 50?

e. Suppose also that c =

5. Will the Secondhand Buyers buy any cars in equilibrium if car quality is distributed uniformly between and 100?

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