Question

A building contractor approaching retirement is contemplating purchasing some coastal land along the Atlantic seaboard to build a retirement cottage for himself. However, there is a possibility that with rising seas, by the time he retires and is ready to build, the land may be designated within the "set-back line," meaning it would be too close to the water to build. If the contractor can build the cottage, his wealth will be $1,000,000. If no building is possible, the land still has some value as beach access, and the contractor’s wealth would be $640,000. If the contractor does not purchase the property, his wealth would be $810,000.

Assume that the contractor has a utility function: U(x) = 10w0.5, where w = wealth
A. Suppose the probability that the state Coastal Management Program will declare the land is in the set-back line is 10%. What is the contractor’s expected wealth if he purchases the land?
B. What is the contractor’s expected utility from purchasing the land?
C. What is the contractor’s utility if he does not purchase the land? Will the contractor purchase the land? Explain, including a discussion of his risk preference.
D. On a separate scrap sheet, show graphically the contractor’s utility function, including his expected utility if he purchases the land, and his utility if he does not. LABEL CAREFULLY.
E. What would the probability that the land is declared within the "set-back line" have to be in order for the contractor to be indifferent about purchasing the land (in percent)?

Answer 1

A. The contractor's expected wealth if he purchases the land is $1,000,000 * 0.1 + $640,000 * 0.9 = $676,000.

B. To calculate the expected utility, we multiply the utility of each outcome by its respective probability and sum them up:

Expected utility = U($1,000,000) * 0.1 + U($640,000) * 0.9

Expected utility = 10(1,000,000)^0.5 * 0.1 + 10(640,000)^0.5 * 0.9

C. The utility if he does not purchase the land is U($810,000) = 10(810,000)^0.5.

To determine whether the contractor will purchase the land, we compare the expected utility of purchasing ($676,000) with the utility of not purchasing ($810,000). If the expected utility is greater than the utility of not purchasing, the contractor will choose to purchase the land.

D. Please refer to the graph below.

E. To find the probability that would make the contractor indifferent about purchasing the land, we set the expected utility of purchasing equal to the utility of not purchasing and solve for the probability.

Expected utility of purchasing = Utility of not purchasing

10(1,000,000)^0.5 * p + 10(640,000)^0.5 * (1-p) = 10(810,000)^0.5

Solving for p, the probability that would make the contractor indifferent, will yield the answer in percent.