Optimal Insurance with Adverse Selection

We solve the principal-agent problem of a monopolist insurer selling to an agent whose riskiness (probability of an accident) is private information, a problem introduced in Stiglitz (1977)'s seminal paper. We first derive several important properties of optimal menus for an arbitrary type distribution. Besides some standard ones such as no overinsurance and no pooling at the highest type, we show that the optimal premium and indemnity are nonnegative for all types and that the principal always makes positive expected profit. More importantly, we provide a novel comparative static analysis of wealth eects, showing that the principal always prefers an agent facing a larger potential loss, as well as a poorer one if the agent's risk aversion decreases with wealth. We then specialize to the case with a continuum of types distributed according to an smooth density function. We give simple sucient conditions for complete sorting, optimal exclusion, and quantity discounts. The main result here is that, under two mild assumptions-the monotone likelihood ratio property for the density and decreasing absolute risk aversion for the agent-the optimal premium is 'S- shaped' in the amount of coverage, first concave, then convex. We contrast these results with those of the standard monopoly pricing model with quasilinear preferences, and with the competitive insurance model.