(Optional, requires appendix) Return to the example of the 2011 Citrus in Section 4.B. The two types of Citrus—the reliable orange and the hapless lemon—are outwardly indistinguishable to a buyer. In the example, if the fraction f of oranges in the Citrus population is less than 0.65, the seller of an orange will not be willing to part with the car for the maximum price buyers are willing to pay, so the market for oranges collapses. But what if a seller has a costly way to signal her car’s type? Although oranges and lemons are in nearly every respect identical, the defining difference between the two is that lemons break down much more frequently. Knowing this, owners of oranges make the following proposal. On a buyer’s request, the seller will in one day take a 500-mile round-trip drive in the car. (Assume this trip will be verifiable via odometer readings and a time-stamped receipt from a gas station 250 miles away.) For the sellers of both types of Citrus, the cost of the trip in fuel and time is $0.50 per mile (that is, $250 for the 500-mile trip). However, with probability q a lemon attempting the journey will break down. If a car breaks down, the cost is $2 per mile of the total length of the attempted road trip (that is, $1,000). Additionally, breaking down will be a sure sign that the car is a lemon, so a Citrus that does so will sell for only $6,000. Assume that the fraction of oranges in the Citrus population, f, is 0.6. Also, assume that the probability of a lemon breaking down, q, is 0.5 and that owners of lemons are risk neutral.
(a) Use Bayes’ rule to determine fupdated, the fraction of Citruses that have successfully completed a 500-mile road trip that are oranges. Assume that all Citrus owners attempt the trip. Is fupdated greater than or less than f? Explain why.
(b) Use fupdated to determine the price, pupdated, that buyers are willing to pay for a Citrus that has successfully completed the 500-mile road trip.
(c) Will an owner of an orange be willing to make the road trip and sell her car for pupdated? Why or why not?
(d) What is the expected payoff of attempting the road trip to the seller of a lemon?
(e) Would you describe the outcome of this market as pooling, separating, or semiseparating? Explain.