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Chapter 8: Problem 4

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

Expert verified
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

01

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)#
02

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)
03

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#
04

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.
05

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#
06

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.

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Most popular questions from this chapter

Suppose there is a \(50-50\) chance that a risk-averse individual with a current wealth of \(\$ 20,000\) will contact a debilitating disease and suffer a loss of \(\$ 10,000\) a. Calculate the cost of actuarially fair insurance in this situation and use a utility-of-wealth graph (such as shown in Figure 8.1 ) to show that the individual will prefer fair insurance against this loss to accepting the gamble uninsured. b. Suppose two types of insurance policies were available: (1) A fair policy covering the complete loss. (2) A fair policy covering only half of any loss incurred. Calculate the cost of the second type of policy and show that the individual will generally regard it as inferior to the first.

In some cases individuals may care about the date at which the uncertainty they face is resolved. Suppose, for example, that an individual knows that his or her consumption will be 10 units today \((Q)\) but that tomorrow's consumption \(\left(\mathrm{C}_{2}\right)\) will be either 10 or \(2.5,\) depending on whether a coin comes up heads or tails. Suppose also that the individual's utility function has the simple Cobb-Douglas form \\[U\left(Q, C_{2}\right)=V^{\wedge} Q.\\] a. If an individual cares only about the expected value of utility, will it matter whether the coin is flipped just before day 1 or just before day \(2 ?\) Explain. More generally, suppose that the individual's expected utility depends on the timing of the coin flip. Specifically, assume that \\[\text { expected utility }=\mathbf{f}^{\wedge}\left[\left(\mathrm{fi}\left\\{17\left(\mathrm{C}, \mathrm{C}_{2}\right)\right\\}\right) \text { ) } \text { ? } |\right.\\] where \(E_{x}\) represents expectations taken at the start of day \(1, E_{2}\) represents expectations at the start of day \(2,\) and \(a\) represents a parameter that indicates timing preferences. Show that if \(a=1,\) the individual is indifferent about when the coin is flipped. Show that if \(a=2,\) the individual will prefer early resolution of the uncertainty - that is, flipping the coin at the start of day 1 Show that if \(a=.5,\) the individual will prefer later resolution of the uncertainty (flipping at the start of day 2 ). Explain your results intuitively and indicate their relevance for information theory. (Note: This problem is an illustration of "resolution seeking" and "resolution-averse" behavior. See D. M. Kreps and E. L. Porteus, "Temporal Resolution of Uncertainty and Dynamic Choice Theory," Econometrica [January \(1978]: 185-200\).

For the constant relative risk aversion utility function (Equation 8.62 ) we showed that the degree of risk aversion is measured by \((1-R)\). In Chapter 3 we showed that the elasticity of substitution for the same function is given by \(1 /(1-R) .\) Hence, the measures are reciprocals of each other. Using this result, discuss the following questions: a. Why is risk aversion related to an individual's willingness to substitute wealth between states of the world? What phenomenon is being captured by both concepts? b. How would you interpret the polar cases \(R=1\) and \(R--^{\circ}\) in both the risk-aversion and substitution frameworks? c. A rise in the price of contingent claims in "bad" times \(\left(P_{b}\right)\) will induce substitution and income effects into the demands for \(W_{g}\) and \(W_{h}\). If the individual has a fixed budget to devote to these two goods, how will choices among them be affected? Why might \(W_{g}\) rise or fall depending on the degree of risk aversion exhibited by the individual? d. Suppose that empirical data suggest an individual requires an average return of 0.5 per cent if he or she is to be tempted to invest in an investment that has a \(50-50\) chance of gaining or losing 5 percent. That is, this person gets the same utility from \(W_{o}\) as from an even bet on \(1.055 W_{o}\) and \(0.955 W_{o}\) i. What value of \(R\) is consistent with this behavior? ii. How much average return would this person require to accept a \(50-50\) chance of gaining or losing 10 percent? Note: This part requires solving nonlinear equations, so approximate solutions will suffice. The comparison of the risk/reward trade-off illustrates what is called the "equity premium puzzle," in that risky investments seem to actually earn much more than is consistent with the degree of risk-aversion suggested by other data. See N. R. Kocherlakota, "The Equity Premium: It's Still a Puzzle" Journal of Economic Literature (March 1996 ): \(42-71\)

A farmer believes there is a \(50-50\) chance that the next growing season will be abnormally rainy. His expected utility function has the form \\[ \begin{array}{c} \mathbf{1} \\ \text { expected utility }=-\boldsymbol{I} \boldsymbol{n} \boldsymbol{Y}_{N R}+-\boldsymbol{I} \boldsymbol{n} \boldsymbol{Y}_{R} \end{array} \\] where \(Y_{N R}\) and \(Y_{R}\) represent the farmer's income in the states of "normal rain" and "rainy," respectively. a. Suppose the farmer must choose between two crops that promise the following income prospects: $$\begin{array}{lcr} \text { Crop } & Y_{H} & Y_{R} \\ \hline \text { Wheat } & \$ 28,000 & \$ 10,000 \\ \text { Corn } & 19,000 & 15,000 \end{array}$$ Which of the crops will he plant? b. Suppose the farmer can plant half his field with each crop. Would he choose to do so? Explain your result. c. What mix of wheat and corn would provide maximum expected utility to this farmer? d. Would wheat crop insurance, available to farmers who grow only wheat, which costs \(\$ 4000\) and pays off \(\$ 8000\) in the event of a rainy growing season, cause this farmer to change what he plants?

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 \(\$ 1000 ?\)
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