Insurance pricing with complete information, state-dependent utility, and production costs

Abstract

We consider a group of identical risk-neutral insurers selling single-period indemnity insurance policies. The insurance market consists of individuals with common state-dependent utility function who are identical except for their known accident probability q. Insurers incur production costs (commonly called expenses or transaction costs by actuaries) that are proportional to the amount of insurance purchased and to the premium charged. By introducing the concept of insurance desirability, we prove that the existence of insurer expenses generates a pair of constants qmin and qmax that naturally partitions the applicant pool into three mutually exclusive and exhaustive groups of individuals: those individuals with accident probability q∈[0,qmin) are insurable but do not desire insurance, those individuals with accident probability q∈[qmin,qmax] are insurable and desire insurance, and those individuals with accident probability q∈(qmax,1] desire insurance but are uninsurable. We also prove that, depending on the level of q and the marginal rate of substitution between states, it may be optimal for individuals to buy complete (full) insurance, partial insurance, or no insurance at all. Finally, we prove that when q is known in monopolistic markets (i.e., markets with a single insurer), applicants may be induced to “over insure” whenever partial insurance is bought.