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Chapter 8: Problem 5

Ms. Fogg is planning an around-the-world trip on which she plans to spend \(\$ 10,000 .\) The utility from the trip is a function of how much she actually spends on it \((Y),\) given by \\[ U(Y)=\ln Y \\] a. If there is a 25 percent probability that Ms. Fogg will lose \(\$ 1000\) of her cash on the trip, what is the trip's expected utility? b. Suppose that Ms. Fogg can buy insurance against losing the \(\$ 1000\) (say, by purchasing traveler's checks) at an "actuarially fair" premium of \(\$ 250 .\) Show that her expected utility is higher if she purchases this insurance than if she faces the chance of losing the \(\$ 1000\) without insurance. c. What is the maximum amount that Ms. Fogg would be willing to pay to insure her \(\$ 1000 ?\)

Short Answer

Expert verified
Question: Calculate the expected utility of Ms. Fogg's trip without insurance given a 25% probability of losing $1000 on the trip. Answer: The expected utility without insurance is approximately 9.107547.

Step by step solution

01

a. Expected utility without insurance

To calculate the expected utility without insurance, we will use the formula for expected utility EF[U(Y)] = pU(Y1) + (1-p)U(Y2), where p is the probability of losing \(1000 (25%), Y1 = Y - \)1000, and Y2= Y. In this case, Y=\(10000 so Y1=\)9000 and Y2=$10000. EF[U(Y)] = (0.25) * U(9000) + (0.75) * U(10000) EF[U(Y)] = (0.25) * ln(9000) + (0.75) * ln(10000) Now we can evaluate this expression for expected utility: EF[U(Y)] ≈ 9.107547
02

b. Expected utility with insurance

To calculate the expected utility with insurance, we need to account for the premium cost, P=\(250, Ms. Fogg will pay to insure her cash. With insurance, there is no probability of losing \)1000, so Y1 will now be Y - P. EF[U(Y)_with_insurance] = U(Y - P) EF[U(Y)_with_insurance] = U(10000 - 250) EF[U(Y)_with_insurance] = ln(9750) Now we can evaluate this expression for expected utility with insurance: EF[U(Y)_with_insurance] ≈ 9.186124 Now we can compare the expected utility with and without insurance. Since 9.186124 > 9.107547, Ms. Fogg's expected utility is higher if she purchases the insurance at an actuarially fair premium of $250.
03

c. Maximum amount Ms. Fogg is willing to pay for insurance

To find the maximum amount Ms. Fogg would be willing to pay for insurance, we need to find the value of P such that expected utility with insurance equals expected utility without insurance. EF[U(Y)] = ln(Y - P) Substitute the value of EF[U(Y)] we calculated in part (a): 9.107547 = ln(10000 - P) Now we can solve for P: P = 10000 - exp(9.107547) P ≈ 332.60 Ms. Fogg is willing to pay a maximum amount of approximately $332.60 for the insurance.

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Most popular questions from this chapter

Investment in risky assets can be examined in the state-preference framework by assuming that \(W^{*}\) dollars invested in an asset with a certain return, \(r,\) will yield \(W^{*}(1+r)\) in both states of the world, whereas investment in a risky asset will yield \(W^{*}\left(1+r_{\mathrm{g}}\right)\) in good times and \(\left.W^{*}\left(1+r_{b}\right) \text { in bad times (where } r_{g}>r>r_{b}\right)\) a. Graph the outcomes from the two investments. b. Show how a "mxied portfolio" containing both risk-free and risky assets could be illustrated in your graph. How would you show the fraction of wealth invested in the risky asset? c. Show how individuals' attitudes toward risk will determine the mix of risk- free and risky assets they will hold. In what case would a person hold no risky assets? d. If an individual's utility takes the constant relative risk aversion form (Equation 8.62 ), explain why this person will not change the fraction of risky asset held as his or her wealth increases. \(^{20}\)

Suppose there is a \(50-50\) chance that a risk-averse individual with a current wealth of \(\$ 20,000\) will contact a debilitating disease and suffer a loss of \(\$ 10,000\) a. Calculate the cost of actuarially fair insurance in this situation and use a utility-of-wealth graph (such as shown in Figure 8.1 ) to show that the individual will prefer fair insurance against this loss to accepting the gamble uninsured. b. Suppose two types of insurance policies were available: (1) A fair policy covering the complete loss. (2) A fair policy covering only half of any loss incurred. Calculate the cost of the second type of policy and show that the individual will generally regard it as inferior to the first.

An individual purchases a dozen eggs and must take them home. Although making trips home is costless, there is a 50 percent chance that all of the eggs carried on any one trip will be broken during the trip. The individual considers two strategies: Strategy 1: Take all 12 eggs in one trip. Strategy 2: Take two trips with 6 in each trip. a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that on the average, 6 eggs will remain unbroken after the trip home under either strategy. b. Develop a graph to show the utility obtainable under each strategy. Which strategy will be preferable? c. Could utility be improved further by taking more than two trips? How would this possibility be affected if additional trips were costly?

Show that if an individual's utility-of-wealth function is convex (rather than concave, as shown in Figure 8.1 ), he or she will prefer fair gambles to income certainty and may even be willing to accept somewhat unfair gambles. Do you believe this sort of risk-taking behavior is common? What factors might tend to limit its occurrence?

A farmer believes there is a \(50-50\) chance that the next growing season will be abnormally rainy. His expected utility function has the form \\[ \text { expected utility }=\frac{1}{2} \ln Y_{N R}+\frac{1}{2} \ln Y_{R} \\] where \(Y_{N R}\) and \(Y_{R}\) represent the farmer's income in the states of "normal rain" and "rainy," respectively. a. Suppose the farmer must choose between two crops that promise the following income prospects: $$\begin{array}{lcc} \text { Crop } & \boldsymbol{Y}_{N R} & \boldsymbol{Y}_{R} \\ \hline \text { Wheat } & \$ 28,000 & \$ 10,000 \\ \text { Corn } & 19,000 & 15,000 \end{array}$$ Which of the crops will he plant? b. Suppose the farmer can plant half his field with each crop. Would he choose to do so? Explain your result. c. What mix of wheat and corn would provide maximum expected utility to this farmer? d. Would wheat crop insurance, available to farmers who grow only wheat, which costs \(\$ 4000\) and pays off \(\$ 8000\) in the event of a rainy growing season, cause this farmer to change what he plants?
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