(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.