Question

1. Joe is currently unemployed and without health insurance coverage. He derives utility (U) from his interest income on his savings (Y) according to the following function: U = 5(Y1/2) Joe presently makes about $40,000 of interest income per year. He realizes that there is about a 5 percent probability that he may suffer a heart attack. The cost of treatment will be about $20,000 if a heart attack occurs.

A. Calculate Joe’s expected utility level without any health insurance coverage.

B. Calculate Joe’s expected income without any insurance coverage.

C. Suppose Joe must pay a premium of $1,500 for health insurance coverage which covers for the entire cost of treatment when he gets ill. Would he buy the health insurance? Why or why not?

D. Suppose now that the government passes a law that allows all people—not just the self-employed or employed—to have their entire insurance premium exempted from taxes. Joe is in the 33 percent tax bracket. Would he buy the health insurance at a premium cost of $1,500? Why or why not? What implications can be drawn from the analysis?

E. Suppose Joe purchases the health insurance coverage (at a premium cost of $1500) and represents the average subscriber, and his expectations are correct. Calculate the loading fee the insurance company will receive.

Answer 1

Joe presently makes about $40,000 of interest income per year. There is about a 5 percent probability that he may suffer a heart attack. The cost of treatment will be about $20,000 if a heart attack occurs.

A) We have the following probabilities P1 = 0.05 and P2 = 0.95

Joe's expected income is = (40,000*0.95) + (20,000*0.05) = 39,000

Joe's expected utility is E(U) = 5(Y)^(1/2) = 5*(39000)^(0.5) = 987.42

B) Joe’s expected income without any insurance coverage :

I = (0.95)(40,000) + (0.05)(20,000) = 38,000 + 1,000 = 39,000

C) Joe's expected income without insurance is $39,000 and with insurance is $40,000. Thus, he has an expected loss of $1,000. When he pays a premium, p1 will be zero and p2 becomes 1. So, in this case, expected utility is: 1*5*(40,000−1,500) ^1/2 = 981.07, which is less than 987.42. Hence, Joe should not purchase insurance.

D) If Joe is taxed at a 33% rate, he will pay $12,870 in taxes annually (33% of $39,000), taking home $26,130, which has a utility of 808.24. If Joe chooses to pay the premium of $1,500, his income increases to $40,000 and his taxable income drops to $38,500. He would pay $12,705 in taxes on this amount, taking home $27,295 and the utility of this income is 826.06. Joe’s income utility is greater when he chooses to buy insurance tax-free, therefore he will by insurance if the premium is tax-exempt.