# (Optional, requires appendix) Return to the example of the 2011 Citrus in Section 4.B. The two types

(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.