Optimal Insurance with Adverse Selection

Abstract

We solve the principal-agent problem of a monopolist insurer selling to an agent whose riskiness (chance of a loss) is private information, a problem introduced in Stiglitz (1977)'s seminal paper. We prove several properties of optimal menus: the highest type gets full coverage (efficiency at the top), all other types are underinsured (downward distortions elsewhere), the contract is non-negative and there are always gains to trade. The main novelty here is that we prove these basic properties for an arbitrary type distribution using elementary arguments. We also prove a new comparative static result for wealth effects, showing that the principal always prefers an agent facing a larger loss, and a poorer one if the agent's risk aversion decreases with wealth. We then specialize to the case of a continuum of types distributed according to an smooth density. We give sufficient conditions for complete sorting, exclusion, and quantity discounts. Our most surprising result is that, under two mild assumptions-log-concavity of the density and decreasing absolute risk aversion for the agent-the optimal premium is backwards-S shaped in the amount of coverage, first concave, then convex (a shape consistent with global quantity discounts). We contrast our results with the standard monopoly model with private values and quasilinear preferences and with competitive insurance models. We calculate a closed form solution for the CARA case and use it to illustrate these differences. Although we focus on the monopoly insurance problem, our proofs can be adapted to other screening problems with wealth e ffects and common values.