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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

Investment in risky assets can be examined in the state-preference framework by assuming that \(W^{*}\) dollars invested in an asset with a certain return, \(r,\) will yield \(\mathrm{W}^{*}(\mathrm{l}+r)\) in both states of the world, whereas investment in a risky asset will yield \(\mathrm{W}^{*}\left(1+r_{g}\right)\) in good times and \(\left.\mathrm{W}^{*}\left(\mathrm{l}+r_{b}\right) \text { in bad times (where } r_{g}>r>r_{b}\right)\) a. Graph the outcomes from the two investments. b. Show how a "mxied portfolio" containing both risk-free and risky assets could be illustrated in your graph. How would you show the fraction of wealth invested in the risky asset? c. Show how individuals' attitudes toward risk will determine the mix of risk- free and risky assets they will hold. In what case would a person hold no risky assets? d. If an individual's utility takes the constant relative risk aversion form (Equation 8.62 ), ex plain why this person will not change the fraction of risky asset held as his or her wealth increases.

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.

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.)

Show that if an individual's utility-of-wealth function is convex (rather than concave, as shown in Figure 8.1 ), he or she will prefer fair gambles to income certainty and may even be willing to accept somewhat unfair gambles. Do you believe this sort of risk-taking behavior is common? What factors might tend to limit its occurrence?

In deciding to park in an illegal place, any individual knows that the probability of getting a ticket is \(p\) and that the fine for receiving the ticket is / Suppose that all individuals are risk averse (that is, \(U^{\prime \prime}(W)<0\), where Wis the individual's wealth) Will a proportional increase in the probability of being caught or a proportional increase in the fine be a more effective deterrent to illegal parking? \([\text {Hint}\) : Use the Taylor \(\text { series approximation }\left.U(W-f)=U(W)-f U^{\prime}(W)+-U^{\prime \prime}(W) .\right]\)
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