Question

We have seen that the assumption of preferences indicating risk aversion lead to consumers being willing to pay more than the actuarially fair price for health insurance. Darrell has a utility function for daily income given by u(I) = 10* square root of I . Thus, an income of $36 a day would give Darrell a utility of u($36) = 10* square root of 36 = 10*6 = 60.

Darrell has a job that pays him $100 a day. Darrell goes snowboarding on weekends, so with probability 10% (p = 0.10) Darrell has an accident that results in him breaking his leg and unable to work.

- What is the actuarially fair price of an insurance policy that will pay Darrell $100 a day if he injures himself snowboarding and is unable to work (just think of this policy paying Darrell for a single day of missed work to simplify calculations)? What is Darrell’s expected utility for a given day in the case that he doesn’t have insurance (think of what he expects his utility to be on Monday when he leaves work on Friday)?

- Would Darrell be willing to buy the insurance policy in part (a) at the actuarially fair price you calculated? Would Darrell be willing to pay more than this actuarially fair price? If so, what is the most that Darrell would pay for such a policy? What does this tell you about Darrell’s risk premium given his preferences? Is he risk averse?

- Now consider Tommy who also likes to snowboard but is much more reckless and makes more money than Darrell. Tommy has the same utility function as Darrell, but has a daily income of $225. Tommy will break his leg with probability 60% (so p = 0.6) and be fine for work on Monday only 40% of the time. What is an actuarially fair price of insurance for Tommy that will pay him $225 per day in the event of an accident? What is his expected income without insurance? Calculate the risk premium that Tommy would be willing to pay for insurance given this scenario. How does his risk premium compare to Darrell’s?

Answer 1