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 \(\$ 1000\) 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

Based on the given information, we followed these steps to find the solution: 1. Calculate the expected utility without insurance. 2. Calculate the expected utility with insurance. 3. Compare expected utilities to determine if insurance is beneficial. 4. Find the maximum amount Ms. Fogg would be willing to pay for insurance. By comparing the expected utilities, we determined if purchasing insurance was beneficial for Ms. Fogg. After calculating the maximum amount she would be willing to pay, we found the point at which she would be indifferent between purchasing insurance or not.

Step-by-step solution

Expected Utility without Insurance

First, calculate the expected utility without insurance, which considers two situations: 1. Ms. Fogg loses \(1000: With a 25% probability, she will spend \)9,000. The utility of this situation is \(U(9000) = \ln{9000}\). 2. Ms. Fogg does not lose any money (75% probability), and she will spend \(10,000. The utility of this situation is \)U(10000)=\ln{10000}$. To find the expected utility, we gather the probabilities and the utility of each situation: Expected Utility (without insurance) = \(0.25 \cdot \ln{9000} + 0.75 \cdot \ln{10000}\).

Expected Utility with Insurance

Now, we need to calculate the utility if she purchases an insurance with an actuarially fair premium of $250: The utility for this situation would be: \(U(9750)=\ln{9750}\), because she will spend \(10,000 - 250 = 9,750\) regardless of whether she loses $1000 or not (thanks to the insurance). The expected utility of this situation is just the utility with insurance, since there is no uncertainty anymore.

Comparing Expected Utilities

Now we need to see if her expected utility is higher with the insurance than without: Expected Utility (without insurance) = \(0.25 \cdot \ln{9000} + 0.75 \cdot \ln{10000}\) Expected Utility (with insurance) = \(\ln{9750}\) Let's compare them and see if purchasing insurance is beneficial for Ms. Fogg: \(0.25 \cdot \ln{9000} + 0.75 \cdot \ln{10000} < \ln{9750}\) Comparing with a calculator, we can determine if this inequality holds to see if purchasing insurance is beneficial.

Maximum Amount for Insurance

To find the maximum amount Ms. Fogg would be willing to pay for the insurance, we need to find the price that would make her indifferent between purchasing or not purchasing it. Let "\(x\)" be the maximum amount Ms. Fogg is willing to pay for insurance. The expected utility without insurance should be equal to the expected utility with insurance at this cost: \(0.25 \cdot \ln{9000} + 0.75 \cdot \ln{10000} = \ln{(10000-x)}\) Now, we need to solve for \(x\): \(x = 10000 - \exp{\left(0.25 \cdot \ln{9000} + 0.75 \cdot \ln{10000}\right)}\) Calculating this expression, we find the maximum amount Ms. Fogg would be willing to pay for the insurance.

Key concepts

Utility Function

Understanding the utility function is like unlocking the secret to a consumer's satisfaction. It's a mathematical tool that economists use to represent a consumer's preference for certain goods or experiences. Think of it as a personal happiness meter, where higher utility means more satisfaction.

For example, in our exercise with Ms. Fogg, her enjoyment of the trip is quantified using the utility function \(U(Y)=\text{ln}Y\). This particular form, a natural logarithm function, is commonly used because it captures the concept of diminishing marginal utility; the idea that each additional dollar spent brings a smaller increase in satisfaction than the last. Ms. Fogg's dilemma with the potential loss of money and decision on buying insurance is assessed through the lens of expected utility, showcasing the interplay between financial decisions and their impact on contentment.

Risk and Insurance in Microeconomics

Risk is an inevitable part of life, and microeconomics doesn't ignore this. It dives into how consumers deal with risk, like the potential of losing money, and how they can use tools such as insurance to manage it.

In our Ms. Fogg scenario, the risk is losing $1000 on her trip. Economists calculate the 'expected utility' to gauge how unpleasant this scenario could be for her. When Ms. Fogg considers purchasing insurance at an actuarially fair premium (a fancy way of saying the insurance cost equals the expected loss), she's acting like a textbook example of risk aversion – preferring to avoid uncertainty even at a small cost. By buying the insurance, Ms. Fogg locks in a known utility, eliminating the gamble and finding peace of mind.

Consumer Choice and Preference

At the crossroads of desires and decisions lie consumer choice and preference. It's all about what people prefer and how they choose among various options considering their limited resources. Consumers face multiple options and must decide which gives them the most bang for their buck, or in economic jargon, 'maximizes their utility.'

Relating back to our globetrotting Ms. Fogg, her preferences are laid out by her utility function. She can opt for buying insurance or risk losing part of her travel fund. The consumer's decision-making process, as reflected in the exercise, demonstrates the delicate balance between risk, cost, and the pursuit of happiness. By determining the maximum amount she's willing to pay for insurance, Ms. Fogg's choices depict a real-world application of microeconomic principles.