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Chapter 8: 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 in this situation is $300. Moral hazard affects Ms. Fogg's decision to purchase insurance because her increased risk-taking behavior while insured leads to a higher insurance premium. Her decision will depend on her willingness to bear this cost and her perception of the increased risk.

Step by step solution

01

Calculate the actuarially fair insurance premium without increased risk

The actuarially fair insurance premium is the expected loss, which is the product of the probability of loss and the amount of the loss. In this case, without insurance, the probability of loss is 25% (0.25). The potential loss is $1,000. The expected loss without insurance would be: Expected loss = Probability of loss x Amount of the loss Expected loss = 0.25 x \(1,000 = \)250
02

Calculate the actuarially fair insurance premium with increased risk

When purchasing insurance, the probability of loss increases to 30% (0.30). We need to re-calculate the expected loss in this situation: Expected loss = Probability of loss x Amount of the loss Expected loss = 0.30 x \(1,000 = \)300 The actuarially fair insurance premium in this situation would be $300.
03

Discuss the implications of moral hazard and whether Ms. Fogg will purchase insurance

Moral hazard refers to the increased risk-taking behavior when a person is protected by insurance. In this case, Ms. Fogg is more careless with her cash and faces a higher probability of losing $1,000 when purchasing insurance. Given that the actuarially fair insurance premium increased to \(300, which reflects the increased risk of loss due to moral hazard, Ms. Fogg's decision to purchase insurance will depend on her willingness to bear this cost. If Ms. Fogg is still willing to pay \)300 to avoid the 30% chance of losing $1,000, she will purchase insurance. However, if she perceives the increased insurance premium as too high or the increased risk as not significant enough, she may decide not to purchase the insurance. The problem does not provide enough information to definitively determine Ms. Fogg's decision, but we can conclude that the actuarially fair insurance premium in this situation is $300.

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Most popular questions from this chapter

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.)

Suppose Molly Jock wishes to purchase a high-definition television to watch the Olympic Greco-Roman wrestling competition. Her current income is \(\$ 20,000,\) and she knows where she can buy the television she wants for \(\$ 2,000\). She has heard the rumor that the same set can be bought at Crazy Eddie's (recently out of bankruptcy) for \(\$ 1,700,\) but is unsure if the rumor is true. Suppose this individual's utility is given by \\[\text { utility }=\ln (Y).\\] where Fis her income after buying the television. a. What is Molly's utility if she buys from the location she knows? b. What is Molly's utility if Crazy Eddie's really does offer the lower price? c. Suppose Molly believes there is a \(50-50\) chance that Crazy Eddie does offer the lowerpriced television, but it will cost her \(\$ 100\) to drive to the discount store to find out for sure (the store is far away and has had its phone disconnected). Is it worth it to her to invest the money in the trip?

An individual purchases a dozen eggs and must take them home. Although making trips home is costless, there is a 50 percent chance that all of the eggs carried on any one trip will be broken during the trip. The individual considers two strategies: Strategy 1: Take all 12 eggs in one trip. Strategy 2: Take two trips with 6 in each trip. a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that on the average, 6 eggs will remain unbroken after the trip home under either strategy. b. Develop a graph to show the utility obtainable under each strategy. Which strategy will be preferable? c. Could utility be improved further by taking more than two trips? How would this possi bility be affected if additional trips were costly?

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, low ability 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?

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.
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