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Chapter 9: Problem 2

In Problem \(8.5,\) Ms. Fogg was quite willing to buy insurance against a 25 percent chance of losing \(\$ 1,000\) of her cash on her around-the-world trip. Suppose that people who buy such insurance tend to become more careless with their cash and that their probability of losing \(\$ 1,000\) rises to 30 percent. What is the actuarially fair insurance premium in this situation? Will Ms. Fogg buy insurance now? (Note: This problem and Problem 9.3 illustrate moral hazard.)

Short Answer

Expert verified
Answer: The actuarially fair insurance premium for a 30% probability of losing $1,000 is $300. Ms. Fogg's decision to purchase insurance depends on her risk tolerance, as her expected loss with or without insurance is the same (\$300). If she is risk-averse, she will likely buy insurance; if she is risk-neutral or risk-seeking, she may choose not to buy insurance.

Step by step solution

01

Calculate the actuarially fair insurance premium

To determine the actuarially fair insurance premium, we can use the formula: Actuarially fair insurance premium = Probability of loss × Amount of loss In our case, the probability of loss has increased to 30%, and the amount of loss remains $1,000. Actuarially fair insurance premium = 0.3 × \(1,000 = \)300
02

Evaluate Ms. Fogg's decision in purchasing insurance

To determine if Ms. Fogg would still buy insurance, we need to compare her expected loss without insurance and her expected loss with insurance. 1. Without insurance: Expected loss = Probability of loss × Amount of loss Expected loss = 0.3 × \(1,000 = \)300 2. With insurance: Expected loss = Insurance premium + (Probability of loss × Excess amount) Since the insurance premium is actuarially fair, and the excess amount is zero, the expected loss with insurance is equal to the insurance premium. Expected loss with insurance = $300 The expected loss with or without insurance remains the same (\(300). The decision to purchase insurance depends on Ms. Fogg's risk tolerance. If she is risk-averse and prefers paying the premium to avoid the potential of a higher loss, she will buy insurance. If she is risk-neutral or risk-seeking and can bear the risk of potentially losing all \)1,000, she may choose not to buy insurance.

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Most popular questions from this chapter

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 \(\left(C_{1}\right)\) but that tomorrow's consumption \(\left(C_{2}\right)\) 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(C_{1}, C_{2}\right)=\sqrt{C_{1} C_{2}}.\\] 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. b. More generally, suppose that the individual's expected utility depends on the timing of the coin flip. Specifically, assume that \\[ \text { expected utility }=E_{1}\left[\left(E_{2}\left\\{U\left(C_{1}, C_{2}\right)\right\\}\right)^{\alpha}\right] \\] where \(E_{1}\) represents expectations taken at the start of day \(1, E_{2}\) represents expectations at the start of day \(2,\) and \(\alpha\) represents a parameter that indicates timing preferences. Show that if \(\alpha=1,\) the individual is indifferent about when the coin is flipped. c. Show that if \(\alpha=2\), the individual will prefer early resolution of the uncertainty-that is, flipping the coin at the start of day 1 d. Show that if \(\alpha=.5,\) the individual will prefer later resolution of the uncertainty (flipping at the start of day 2 ). e. 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 \(1978]: 185-200\).)

Suppose an individual knows that the prices of a particular color TV have a uniform distribution between \(\$ 300\) and \(\$ 400\). The individual sets out to obtain price quotes by phone. a. Calculate the expected minimum price paid if this individual calls \(n\) stores for price quotes. b. Show that the expected price paid declines with \(n\), but at a diminishing rate. c. Suppose phone calls cost \(\$ 2\) in terms of time and effort. How many calls should this individual make in order to maximize his or her gain from search?

Suppose there are two types of workers, high-ability workers and low-ability workers. Workers' wages are determined by their ability- high ability workers earn \(\$ 50,000\) per year, lowability workers earn \(\$ 30,000 .\) Firms cannot measure workers' abilities but they can observe whether a worker has a high school diploma. Workers' utility depends on the difference between their wages and the costs they incur in obtaining a diploma. a. If the cost of obtaining a high school diploma is the same for high-ability and low-ability workers, can there be a separating equilibrium in this situation in which high-ability workers get high-wage jobs and low-ability workers get low wages? b. What is the maximum amount that a high-ability worker would pay to obtain a high school diploma? Why must a diploma cost more than this for a low-ability person if having a diploma is to permit employers to identify high-ability workers?

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.

Problem 8.4 examined a cost-sharing health insurance policy and showed that risk-averse individuals would prefer full coverage. Suppose, however, that people who buy cost-sharing policies take better care of their own health so that the loss suffered when they are ill is reduced from \(\$ 10,000\) to \(\$ 7,000 .\) Now what would be the actuarial fair price of a cost-sharing policy? Is it possible that some individuals might prefer the cost-sharing policy to complete coverage? What would determine whether an individual had such preferences? (A graphical approach to this problem should suffice.)
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