36.6 (1) Each year, 1,000 citizens of New Crankshaft, Pennsylvania, sell their used cars and buy new cars. The original owners of the old cars have no place to keep second cars and must sell them. These used cars vary a great deal in quality. Their original owners know exactly what is good and what is bad about their cars, but potential buyers can't tell them apart by looking at them. Lamentably, though they are in other respects model citizens, the used-car owners in New Crankshaft have no scruples about lying about their old jalopies Each car has a value, V, which a buyer who knew all about its qualities would be willing to pay. There is a very large number of potential buyers, any one of which would be willing to pay $V for a car of value $V. The distribution of values of used cars on the market is quite simply described. In any year, for any V between 0 and $2,000, the number of used cars available for sale that are worthless than $V is V/ 2. Potential used-car buyers are all risk-neutral. That is if they don't know the value of a car for certain, they value it at its expected value, given the information they have. NAME 437 Rod's Garage in New Crankshaft will test out any used car and nd its true value V. Rod's Garage is known to be perfectly accurate and perfectly honest in its appraisals. The only problem is that getting an accurate appraisal costs $200. People with terrible cars are not going to want to pay $200 to have Rod tell the world how bad their cars are. But people with very good cars will be willing to pay Rod the $200 to get their cars appraised, so they can sell them for their true values. Let's try to gure our exactly how the equilibrium works, which cars get appraised, and what the unappraised cars sell for. (a) If nobody had their car appraised, what would the market price for used cars in North Crankshaft be and what would be the total revenue received by used-car owners for their cars? They'd all sell for $1,000 for total revenue of $1,000,000. (b) If all the cars that are worth more than 95X are appraised and all the cars that are worth less than $X are sold without appraisal, what will the market price of unappraised used cars be? (Hint: 1What is the expected value of a random draw from the set of cars worth less than $X?) $X/2. (c) If all the cars that are worth more than $X are appraised and all the cars that are worth less than $X are sold without appraisal, then if your car is worth $X, how much money would you have left if you had it appraised and then sold it for its true value? 3X 200 . How much money would you get if you sold it without having it appraised? $X/2 . (d) In equilibrium, there will be a car of marginal quality such that all cars better than this car will be appraised and all cars worse than this car will be sold without being appraised. The owner of this car will be just indi'erent between selling his car unappraised and having it appraised. \"That will be the value of this marginal car? Solve X / 2 = X 200 to get X = $400. {6) In equilibrium, how many cars will be sold unappraised and what will they sell for? The worst 200 cars will be unappraised and will sell for $200. 438 INFORMATION (Ch. 36] (f) In equilibrium, what will be the total net revenue of all owners of used cars, after Rod's Garage has been paid for its appraisals? $1, 000,000 800 x 200 = 840,000. 36.7 (2) In Pot Hole, Georgia, 1,000 people want to sell their used cars. These cars vary in quality. Original owners know exactly what their cars are worth. All used cars look the same to potential buyers until they have bought them; then they nd out the truth. For any number X between 0 and 2,000, the number of cars of quality lower than X is X f 2. If a car is of quality X, its original owner will be willing to sell it for any price greater than X. If a buyer knew that a car was of quality X, she would be willing to pay X + 500 for it. \"Then buyers are not sure of the quality of a car, they are willing to pay its expected value, given their knowledge of the distribution of qualities on the market. {(1) Suppose that everybody knows that all the used cars in Pot Hole are for sale. 1What would used cars sell for? $1 , 500 . Would every used car owner be willing to sell at this price? NO . Which used cars would appear on the market? Those worth less than $ 1 , 500 . {5) Let X * be some number between 0 and 2,000 and suppose that all cars of quality lower than X ' are sold, but original owners keep all cars of quality higher than X *. 1What would buyers be willing to pay for a used car? Kill/2 + 500. At this price, which used cars would be for sale? Cars worth less than X * / 2 + 500. {(2) Write an equation for the equilibrium value of X ', at which the price that buyers are willing to pay is exactly enough to induce all cars of quality less than X * into the market. X*/2 + 500 = X* . Solve this equation for the equilibrium value of X '. )C': = $1, 000