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Chapter 7: 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 \(\$ 1,000\) of her cash on the trip, what is the trip's expected utility? b. Suppose that Ms. Fogg can buy insurance against losing the \(\$ 1,000\) (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 \(\$ 1,000\) without insurance. c. What is the maximum amount that Ms. Fogg would be willing to pay to insure her \(\$ 1,000 ?\)

Short Answer

Expert verified
Answer: The expected utility of Ms. Fogg's trip without insurance is approximately 9.02. With an actuarially fair insurance premium of $250, her expected utility increases to approximately 9.08. Ms. Fogg would be willing to pay a maximum amount of approximately $317 to insure her $1,000.

Step by step solution

01

Define Spending Scenarios

There are two possible scenarios for Ms. Fogg's spending on this trip: 1. She loses $1,000 with a probability of 25%; 2. She does not lose any money with a probability of 75%. Thus, her total spending is: - Case 1: \(Y_1 = 10,000 - 1,000 = 9,000\) - Case 2: \(Y_2 = 10,000\).
02

Calculate the Utility for Each Scenario

Utilizing the given utility function, \(U(Y) = ln(Y)\), we find: - Utility of Scenario 1: \(U(Y_1) = \ln(9,000)\) - Utility of Scenario 2: \(U(Y_2) = \ln(10,000)\)
03

Calculate Expected Utility

Expected Utility = \(0.25 \times U(Y_1) + 0.75 \times U(Y_2)\) \(E[U] = 0.25 \times \ln(9,000) + 0.75 \times \ln(10,000)\) \(E[U] ≈ 9.02\) a. The trip's expected utility without insurance is approximately 9.02. b. Expected Utility with Insurance
04

Calculate Spending with Insurance Premium

Ms. Fogg spends \( \$250\) on insurance, which leaves her with a guaranteed spending of \(Y = 10,000 - 250 = 9,750\).
05

Calculate the Utility with Insurance

Using the utility function, \(U(Y) = ln(Y)\), we find: - Utility of Scenario with Insurance: \(U(Y_{insured}) = \ln(9,750)\) Since insurance eliminates the possibility of losing $1,000, and an actuarially fair premium is paid, by definition, the expected utility with insurance should be higher. \(U(Y_{insured}) ≈ 9.08\) b. Ms. Fogg's expected utility with insurance is approximately 9.08, which is higher than the expected utility without insurance. c. Maximum Amount for Insurance
06

Define the Utility Equality

Ms. Fogg would be willing to pay an amount 'x' for insurance if the utility with insurance is equal to the expected utility without insurance. \(U(Y - x) = E[U]\)
07

Solve for x

\(\ln(10,000 - x) = E[U] ≈ 9.02\) To solve for x, we take the exponent of both sides: \(10,000 - x = e^{9.02}\) Now, we solve for x: \(x = 10,000 - e^{9.02}\) \(x ≈ \$317\) c. Ms. Fogg would be willing to pay a maximum amount of approximately \(317 to insure her \)1,000.

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Most popular questions from this chapter

In Example 7.3 we showed that a person with a CARA utility function who faces a Normally distributed risk will have expected utility of the form \(E[U(W)]=\mu_{W}-(A / 2) \sigma_{W}^{2},\) where \(\mu_{W}\) is the expected value of wealth and \(\sigma_{W}^{2}\) is its variance. Use this fact to solve for the optimal portfolio allocation for a person with a CARA utility function who must invest \(k\) of his or her wealth in a Normally distributed risky asset whose expected return is \(\mu_{r}\) and variance in return is \(\sigma_{r}^{2}\) (your answer should depend on \(A\) ). Explain your results intuitively.

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 \(W^{+}\left(1+r_{g}\right)\) in good times and \(W^{*}\left(1+r_{b}\right)\) in bad times (where \(r_{g}>r>r_{b}\) ). a. Graph the outcomes from the two investments. b. Show how a "mixed 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 7.42 ), explain why this person will not change the fraction of risky assets held as his or her wealth increases. \(^{25}\)

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 the eggs carried on any one trip will be broken during the trip. The individual considers two strategies: (1) take all 12 eggs in one trip; or (2) take two trips with 6 eggs in each trip. a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that, on 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 then 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?

In deciding to park in an illegal place, any individual knows that the probability of getting a ticket is \(p\) and that the fine for receiving the ticket is \(f\). Suppose that all individuals are risk averse (i.e., \(U^{\prime \prime}(W)<0\), where \(W\) is the individual's wealth). Will a proportional increase in the probability of being caught or a proportional increase in the fine be a more effective deterrent to illegal parking? Hint: Use the Taylor series approximation \(U(W-f)=U(W)-f U^{\prime}(W)+\left(f^{2} / 2\right) U^{\prime \prime}(W)\)
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