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 cur rent 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: If the insurance company charged different premiums for blue-eyed and brown-eyed people, blue-eyed individuals would pay a premium of $800, and brown-eyed individuals would pay a premium of $200. In this scenario, both blue-eyed and brown-eyed individuals would buy insurance at their respective premiums, as their expected utility is higher with insurance.

Step-by-step solution

Calculate Expected Losses

To calculate the actuarially fair insurance premium, we need to find the expected losses for both types of people. For blue-eyed people: E(LossB) = 0.80 * \$1,000 = \$800 For brown-eyed people: E(LossBR) = 0.20 * \$1,000 = \$200 Since the population has equal representation of both types, the combined expected loss is the average: E(Loss) = (\$800 + \$200) / 2 = \$500 Thus, the actuarially fair insurance premium should be $500. #b. Buying insurance at the premium calculated in part (a)#

Calculate Utility Without Insurance

Both blue-eyed and brown-eyed individuals have logarithmic utility functions and a current wealth of \$10,000. We first calculate their expected utility without insurance: Blue-eyed individuals: • With loss, 80% chance: U(W - Loss) = ln(\$9,000) • Without loss, 20% chance: U(W) = ln(\$10,000) Brown-eyed individuals: • With loss, 20% chance: U(W - Loss) = ln(\$9,000) • Without loss, 80% chance: U(W) = ln(\$10,000)

Calculate Utility With Insurance

Calculate the expected utility for both types with actuarially fair insurance premium calculated in part(a): Blue-eyed individuals: • With loss, 80% chance: U(W - Premium) = ln(\$9,500) • Without loss, 20% chance: U(W - Premium) = ln(\$9,500) Brown-eyed individuals: • With loss, 20% chance: U(W - Premium) = ln(\$9,500) • Without loss, 80% chance: U(W - Premium) = ln(\$9,500) Considering the calculated utilities, both blue-eyed and brown-eyed individuals would prefer buying insurance at the actuarially fair premium of \$500, as their expected utility is higher with insurance. #c. Correct premium and utilities#

Evaluate the Correctness of Premiums

As we saw, both types of individuals would buy insurance at the premium calculated based on their expected losses. But since blue-eyed people have a higher risk of losing the watch, and they are more likely to buy insurance, the calculated premium will be underestimated.

Calculate New Premium and Utilities

To find the correct premium, we can find the weighted average loss considering the number of insured individuals from each group. For simplicity, let's assume all blue-eyed individuals buy insurance, and none of the brown-eyed individuals do. The new premium would then be: Premium = E(LossB) = \$800 Now recalculate utilities for blue-eyed individuals with this premium: • With loss, 80% chance: U(W - Premium) = ln(\$9,200) • Without loss, 20% chance: U(W - Premium) = ln(\$9,200) For blue-eyed individuals, the new utility is higher without insurance. Thus, they would not buy insurance at the correct premium of \$800. #d. Charging different premiums and utilities#

Calculate Utilities with Different Premiums

If the insurance company charged different premiums for blue-eyed and brown-eyed people, the premiums would be equal to their expected losses. Blue-eyed individuals would have a premium of \$800, and brown-eyed individuals would have a premium of \$200. In this case, the expected utility with insurance for both types would be the same: Blue-eyed individuals: • With loss, 80% chance: U(W - Premium) = ln(\$9,200) • Without loss, 20% chance: U(W - Premium) = ln(\$9,200) Brown-eyed individuals: • With loss, 20% chance: U(W - Premium) = ln(\$9,800) • Without loss, 80% chance: U(W - Premium) = ln(\$9,800) Comparing these utilities to those without insurance, both blue-eyed and brown-eyed individuals would buy insurance at their respective premiums in this scenario.

Key concepts

Actuarially Fair Premium

The concept of an actuarially fair premium is crucial when it comes to insurance. It represents the average expected loss that an insured person may face over time. In simple terms, it is the premium amount that would perfectly balance the insurance company's expenses and its payouts to policyholders.
This idea of fairness ensures that the insurance company does not make a profit or a loss on average. It is calculated based on the probability of a loss occurring and the amount of the loss.
For example, in the exercise provided, blue-eyed individuals have an 80% chance of losing a $1,000 watch, leading to an expected loss of $800. Brown-eyed individuals have a 20% chance, which results in an expected loss of $200. With half of the population in each category, the combined actuarially fair premium would be $500. This approach integrates the risk for both groups, equating to the average expected loss across the entire insured population.

Expected Utility

Expected utility is a concept used in economics to measure the desirability of uncertain outcomes. It is the sum of utilities across all possible outcomes, weighted by their probability.
In this context, it helps individuals decide whether or not they should purchase insurance. Utility reflects satisfaction or happiness an individual gets from a particular outcome.
People are generally risk-averse, meaning they prefer a certain outcome over taking a gamble. Thus, they weigh their decision by considering potential losses and gains.
In our example, the expected utility without insurance for a blue-eyed individual considers both the chance of losing the watch and not losing it, calculated using a logarithmic utility function. By purchasing insurance, individuals can achieve a smoother and more predictable utility, potentially leading to better financial security and peace of mind.

Logarithmic Utility

Logarithmic utility is a specific type of utility function often used in economic models. It's formulated as the natural logarithm of wealth, expressed as \( U(W) = \ln(W) \).
This function captures risk aversion — individuals value incremental increases in wealth less as they become wealthier, a property known as diminishing marginal utility.
In our watch insurance scenario, both blue-eyed and brown-eyed individuals have logarithmic utility functions.
Their decision to purchase insurance at a given premium can be understood by comparing the expected utility of different choices. For instance, the utility with insurance is calculated by considering the reduction in wealth by the premium but eliminates the large uncertain loss.
Logarithmic utility functions model real-world attitudes toward risk by making clear why individuals may buy insurance to protect against significant financial losses.

Insurance Premium Computation

Insurance premium computation involves calculating the premium cost so that it reflects the risk each insured individual poses.
Initially determined using expected losses, it ensures that premiums are fair based on expected value calculations. In our example, it would initially appear that both groups pay $500. However, this does not account for adverse selection, where higher-risk individuals are more likely to purchase coverage.
Ultimately, correctly computed premiums might differ between individuals of differing risk levels to avoid adverse selection. In this scenario, blue-eyed people should face higher premiums of $800, while brown-eyed individuals would pay $200, proportional to their expected losses.
Separating premiums improves fairness and ensures that each individual pays in accordance with their actual risk, maintaining the balance in the insurance market. It is a delicate balance that requires understanding risk and economic behavior to be done effectively.