Question

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

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

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: \(Y_2 = 10,000\).

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: \(U(Y_2) = \ln(10,000)\)

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

Calculate Spending with Insurance Premium

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

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

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. \(U(Y - x) = E[U]\)

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.

Key concepts

Risk Aversion

Risk aversion is a concept that describes a preference for certainty over uncertainty when it comes to potential losses or gains. Individuals who are risk averse prefer to avoid taking risks if possible, especially when it involves the possibility of losing resources like money.
In the context of Ms. Fogg’s travel plans, her potential to lose $1,000 poses a risk. A risk-averse individual like Ms. Fogg would naturally want to mitigate this potential loss. By considering the purchase of traveler's checks or insurance, she demonstrates a behavior consistent with risk aversion. This way, she can ensure that her travel budget remains mostly untouched, despite the potential loss.

Understanding risk aversion is crucial as it influences decision-making. Individuals will assess scenarios based on their comfort with potential losses, often opting for strategies that secure their well-being, even if it means incurring a small loss (such as the insurance premium).

Utility Function

A utility function is a mathematical representation that allows individuals to rank their preferences based on the satisfaction or utility they derive from consuming goods and services. In Ms. Fogg’s case, her utility function is given by the natural logarithm of her spending, denoted as \( U(Y) = \ln Y \).
Utility functions help capture the level of satisfaction one gets from spending their money, in this case, on an around-the-world trip. The logarithmic function reflects diminishing marginal utility, meaning each additional dollar spent has less impact on satisfaction than the previous one. This relationship is typical in real-world scenarios, where initially, spending more provides significant happiness, but over time, the added satisfaction decreases.

For Ms. Fogg, the utility function is essential in determining how much utility she derives under various scenarios, such as losing money or not. It helps calculate her expected utility and how different choices, like buying insurance, affect her overall satisfaction from the trip.

Actuarially Fair Insurance

Actuarially fair insurance refers to a premium that exactly equals the expected value of a potential loss. It is a theoretically ideal insurance scenario where the cost of the premium reflects the probability of the risk occurring without any additional charges for profit by the insurer.
In Ms. Fogg’s situation, purchasing insurance with an actuarially fair premium means the cost of \(\\(250\) aligns with the expected loss from the \(25\%\) probability event where she could lose \(\\)1,000\). Therefore, \({\text{Premium}} = 0.25 \times 1,000 = \\(250\). This type of insurance is crucial for a risk-averse individual as it eliminates the uncertainty of a potential loss while being fairly priced.

Buying this insurance raises Ms. Fogg's expected utility because it transitions her spending from a probabilistic outcome to a certain one. Despite the expense of the insurance premium, the peace of mind and the insurance benefit increase her overall utility as it removes the financial risk of losing \(\\)1,000\) during her trip.