(20) (10 points) Consider an individual who faces a 10% chance of getting a seve
ID: 1116564 • Letter: #
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(20) (10 points) Consider an individual who faces a 10% chance of getting a severe disability and is not able to work. For both states of the world his utility is given by In(x) where x is income. If the individual works, he earns wage earnings = 10. If he does not work he has income x = 2 Suppose an insurance company offers insurance coverage c at the premium price of p- * c, where 0 1. A. Write this individual's expected utility function if he purchases insurance in terms of income, p, and c. (1 point) B. If the insurance company offers actuarially fair insurance, what will be? (1 point) C. Under actuarially fair insurance, what level of coverage (c) will the individual choose? (2 points) D. Suppose the insurer faces operational costs equal to 15 cents for every dollar of coverage. The insurer adds this cost to the premium so that p ( + 0.15) * c. If the insurer only offers full coverage, will the individual purchase insurance? (3 points) E. With this same premium p = ( + 0.15) * c, what level ofcoverage would the individual choose and explain the intuition behind this level? (3 points) (21) (15 points) Suppose there are two people, both of whom have income w, and have the utility function U = ln(w). One faces a high risk of getting sick (p 0.5) the other faces a low risk of getting sick (p = 0.1). Ifthe individual gets sick they must pay x for medical bills. As such the high risk individual's expected utility is 0.5 In(w) + 0.5ln (w-x). A. How do you know that these individuals are risk averse? (1 points) B. Given they are risk averse, what is the intuitive motivation for why they would like to buy insurance? (2 points) C. Suppose there is one insurance company, and this company cannot observe a person's type (i.e., high risk vs. low risk). This company offers two full coverage (c = x) insurance options, with the following premiums: p1 = 0.5c and p2-0.1 c Which option will the low risk individual choose? Which option will the high risk individual choose? Explain why. (4 points) D. Given the individuals' choices in (3), what is the insurance company's expected payout? What is the insurance company's revenue from the premiums? What is the insurance companies expected profit (premium revenue minus expected payout)? (4 points) E. Suppose a new technology is invented, so the insurance company can identify whether you are a high or low risk individual when you walk in the door to purchase insurance. What characteristic of insurance does this related to (consumption smoothing, adverse selection, or moral hazard)? Explain how. (4 points)Explanation / Answer
21.
A. These individuals are known to be risk averse because they construct their utility function by pre-asumming their medical expenditures as denoted by ln0.5(w-x). In other words, these individuals assume in advance that they would have to incur a specific medical expenditure (x) as they have high risk of falling sick.
B. Given that the individuals are risk-averse, the intuitive motivation of getting a risk coverage against the lumpsum medical expenditure in case if they fall sick would induce them to buy an insurance.
C. The high risk individual would choose P1= 0.5c whereas the low risk individual would choose P2= 0.1c. This is so because, the high risk individual would require more coverage as compared to a low risk individual. Thus for getting greater coverage, the high risk individual would have to pay more premium and vice versa.
D. The expected payout of the insurance company would directly relate to the probability of the individual of falling sick. The premium paid by the individuals would be the revenue of the insurance company. The insurance company would make positive profits because it would get revenue from all the individuals in form of premiums, but the company would have to payout only few individuals since there is a probability that all the individuals buying the insurance would not fall sick. Hence, the expected revenue earned would be more than the expected payout.
E. This relates to a characteristic, Moral Hazard. Moral Hazard is a situation when one party gets involved in a risky event knowing that it is protected against the risk and the other party will incur the cost.
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