Pricing high-risk and low-risk insurance contracts with incomplete information and production costs

We consider the traditional model of an insurance market that consists of high-risk and low-risk individual customers who are identical except for their accident probabilities. Though insurers know the values of the high-risk and low-risk accident probabilities, each individual customer’s accident probability is unknown to the insurer. It is well known that if individual customers have state-independent utility functions, and insurers incur neither production costs nor interest costs, then in competitive markets with imperfect information on accident probabilities, if an equilibrium exists it entails separate contracts with the high-risk individuals obtaining complete insurance and low-risk individuals obtaining partial insurance. While in monopolistic markets with imperfect information on accident probabilities, the following four properties hold: (i) high-risk and low-risk individuals never purchase the same insurance policy (i.e., pooling is never optimal); (ii) the optimal contract for the high-risk individual is complete insurance; (iii) if the low-risk individual buys insurance, his/her utility is essentially the same as it would have been had he/she not purchased any insurance; and (iv) there exists a critical (finite) ratio of high- to low-risk individuals such that if the actual ratio exceeds the critical ratio, the low-risk individuals purchase no insurance. In this paper we will extend the traditional model by assuming that individual consumers have a common state-dependent utility function and assume insurers incur production costs that are proportional to the amount of insurance purchased and to the premium charged as well as interest costs. We derive results for both competitive markets and monopolistic markets with imperfect information on accident probabilities. We prove that even though pooling is never optimal in the traditional framework, it may be optimal in our model and high-risk individuals may optimally choose partial insurance. In addition, we develop extensions to the four properties listed above.