Suppose you work for an insurance company. You know that there are equal numbers of individuals who will get in an accident with probability 0.2 and 0.3 and that the loss from getting in an accident is $4,000. Suppose you wish to screen individuals by offering full-coverage insurance as well as 30 percent coinsurance (i.e., the insurance pays 30 percent of the loss amount, or $1,200, in the event of an accident). You wish to include a 5 percent premium over the actuarially fair values for these policies to help cover overhead and provide a profit margin for your company.
a. What price will the insurance company charge for each type of policy?
b. Will this pair of policies screen individuals into risk classes if we assume that all insured individuals have utility function U(X) = 2000 ∙ (X/4000)° 5 where X is income? For each of the three options (no insurance, full-coverage insurance^ and coinsurance) for both types of individuals and compare expected utility levels to see what each will choose
Background on Expected Utility: In this conceptualization of utility, we can compare uncertain outcomes with certain outcomes. For example, if an individual does not have insurance and does not get into an accident, he or she has a utility level of 2,000 (because X = $4,000) or zero if he or she gets into an accident (because then X = $0). The expected utility (EU) is based on the likelihood of each event occurring. A person facing a 10 percent chance of having an accident would have an expected utility of 1,800 using this utility function because EU = 0.9 ∙ 2,000 + 0.1-0. This person would be indifferent between facing this "lottery" and having $3,240 with certainty (because U(3240) = 1,800). This certainty equivalent value was obtained by solving for X in the equation 1800 = 2000 ∙ (X/4000)05. The lottery would be preferred to amounts less than $3,240 offered with certainty.