Question

Blue-eyed people are more likely to lose their expensive watches than are brown-eyed people. Specifically, there is an 80 percent probability that a blue-eyed individual will lose a \(\$ 1,000\) watch during a year, but only a 20 percent probability that a brown-eyed person will. Blue-eyed and brown-eyed people are equally represented in the population. a. If an insurance company assumes blue-eyed and brown-eyed people are equally likely to buy watch-loss insurance, what will the actuarially fair insurance premium be? b. If blue-eyed and brown-eyed people have logarithmic utility-of-wealth functions and current wealths of \(\$ 10,000\) each, will these individuals buy watch insurance at the premium calculated in part (a)? c. Given your results from part (b), will the insurance premiums be correctly computed? What should the premium be? What will the utility for each type of person be? d. Suppose that an insurance company charged different premiums for blue-eyed and brown-eyed people. How would these individuals' maximum utilities compare to those computed in parts (b) and (c)? (This problem is an example of adverse selection in insurance.

Short answer

Answer: a) The actuarially fair insurance premium can be calculated as: Expected Loss = (0.5 * 0.8 * 1000) + (0.5 * 0.2 * 1000) = 400 + 100 = $500 b) Individuals with logarithmic utility-of-wealth functions will buy the insurance if the utility with insurance is higher than without insurance, i.e., \(Utility_{Insurance} > Utility_{NoInsurance}\). c) The insurance company needs to determine if the calculated premium is correct, i.e., whether it should charge a different premium for each type of person, to maximize the utility. d) The utility will be maximized if the insurance company charges different premiums for blue-eyed and brown-eyed people: \(Premium_{BlueEyed} = 0.8 * 1000 = $800\) \(Premium_{BrownEyed} = 0.2 * 1000 = $200\) Comparing the utilities with different premiums will determine the optimal premium structure and maximum utility for each type of person.

Step-by-step solution

a. Actuarially Fair Insurance Premium Calculation

To calculate the actuarially fair insurance premium, we need to find the expected loss for the insurance company. The expected loss is the average loss that the insurance company expects to pay to its clients. As blue-eyed and brown-eyed people are equally represented in the population, the probability of a person being blue-eyed or brown-eyed is 50%. So, the expected loss can be calculated as: Expected Loss = (probability of blue-eyed person * loss probability for blue-eyed * watch value) + (probability of brown-eyed person * loss probability for brown-eyed * watch value) Let's calculate the expected loss: \(\textrm{Expected Loss} = (0.5 * 0.8 * 1000) + (0.5 * 0.2 * 1000)\)

b. Logarithmic Utility of Wealth and Insurance Purchase

To determine whether individuals with logarithmic utility-of-wealth functions will buy insurance at the actuarially fair premium, we need to compare their utilities with and without insurance. The logarithmic utility function is given by: \(U(W) = \log(W)\) Assuming the premium is equal to the expected loss calculated earlier (denoted as P), let's find the utility with insurance: \(Utility_{Insurance} = \log(W - P + L - 1000)\) Where: - \(W\) is the current wealth, which is \(\$10,000\) - \(P\) is the insurance premium, equal to the actuarially fair insurance premium - \(L\) is the expected loss And let's compute the utility without insurance: \(Utility_{NoInsurance} = 0.5 * \log(W - L) + 0.5 * \log(W)\) As the individuals will purchase the insurance if their utility with insurance is higher than without insurance, we will compare the utilities: \(Utility_{Insurance} > Utility_{NoInsurance}\)

c. Correct Premium, Utility and Related Comparison

Based on the comparison in part (b), we need to determine if the calculated premium is correct and what the utility for each type of person would be. If the premium is too high, the individuals will not purchase the insurance and the insurance company will need to recalculate the premium based on the actual population of its customers. Let's call the new premium \(P_{New}\). The insurance company can set a new premium by only considering the population that purchases the insurance: \(\textrm{Expected Loss}_{new} = (0.5 * 0.8 * 1000) + (0.5 * 0.2 * 1000)\) The utility for each type of person can be calculated using the new premium: \(Utility_{New} = \log(W - P_{New} + L - 1000)\)

d. Maximized Utilities with Different Premiums

Suppose the insurance company charged different premiums for blue-eyed and brown-eyed people. We can calculate the actuarially fair insurance premium for each type: \(Premium_{BlueEyed} = 0.8 * 1000\) \(Premium_{BrownEyed} = 0.2 * 1000\) To compare the individuals' maximum utilities, we calculate their utilities with the different premiums: \(Utility_{BlueEyed} = \log(W - Premium_{BlueEyed} + L - 1000)\) \(Utility_{BrownEyed} = \log(W - Premium_{BrownEyed} + L - 1000)\) We then compare these utilities to the ones found in parts (b) and (c).

Key concepts

Actuarially Fair Premium

When we talk about an 'Actuarially Fair Premium,' we are referring to a scenario where the insurance premium exactly equals the expected loss. It is calculated by determining the average amount an insurance company expects to pay out to policyholders. By calculating this, an insurance company aims to cover expected losses without making a profit or loss.

To compute this premium, we begin by figuring the expected loss, which is a blend of various probabilities and potential loss amounts. Suppose we have two groups—blue-eyed and brown-eyed—with an equal chance (50% each) of losing a \(\$1,000\) watch.

• Blue-eyed: 80% chance of loss = \(0.5 \times 0.8 \times 1000\)
• Brown-eyed: 20% chance of loss = \(0.5 \times 0.2 \times 1000\)

By adding these results, we get the expected loss, which gives us the actuarially fair premium: \(\textrm{Expected Loss} = (0.5 \times 0.8 \times 1000) + (0.5 \times 0.2 \times 1000)\).

This expected loss or fair premium balances the scales between the insurance payouts and collected premiums, ensuring planned sustainability for the insurer.

Logarithmic Utility

When assessing the inclination of individuals towards buying insurance, understanding 'Logarithmic Utility' is crucial. In economics, utility represents the satisfaction or benefit an individual receives from a choice or asset. Logarithmic utility is one specific way to calculate this satisfaction, commonly represented as \(U(W) = \log(W)\), where \(W\) is the wealth.

Under this framework, individuals' decisions are based on potential changes in wealth and how those influence their overall utility or satisfaction. When we're talking about individuals with logarithmic utility functions both with and without insurance, we're comparing which scenario maximizes their satisfaction.

• With insurance: \(Utility_{Insurance} = \log(W - P + L - 1000)\)
• Without insurance: \(Utility_{NoInsurance} = 0.5 \times \log(W - L) + 0.5 \times \log(W)\)

These expressions help to illustrate whether purchasing insurance maximizes an individual's utility compared to not purchasing it.
The presence of logarithmic utility also underscores risk aversion, showing that individuals often prefer outcomes that minimize loss of utility, reflecting realistic decision-making tendencies.

Insurance Premium Calculation

To identify how insurance premiums should be computed effectively, one needs to mix both understanding of risk distribution and actual consumer behavior. If average expected loss is the baseline, deviations occur since not all risk groups may purchase insurance equally.

In the scenario where different premiums for each eye color are charged, it's calculative adjustments showcased. Charging distinct premiums based on risk—like \(Premium_{BlueEyed} = 0.8 \times 1000\) for blue-eyed people compared to \(Premium_{BrownEyed} = 0.2 \times 1000\) for brown-eyed—leads to more precise matching of premium with potential payout.

This differentiation guards against adverse selection, a common issue where higher-risk individuals are more likely to purchase insurance than their lower-risk counterparts. By calculating separate premiums, insurers align more closely with the actual risk, not only addressing fairness but potentially improving utility for insured parties:

• It ensures that premiums reflect protected loss per group.
• It maintains insurance funds balance through mortality-matching pricing.
• Circumspect premium setting promotes trust and participation across demographic groups.

Overall, accurate insurance premium calculations close potential gaps between expected and actual outcomes, positively dictating market equilibrium.