Open AccessOptimal insurance with adverse selection
Author(s) -
Chade Hector,
Schlee Edward
Publication year - 2012
Publication title -
theoretical economics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 4.404
H-Index - 32
eISSN - 1555-7561
pISSN - 1933-6837
DOI - 10.3982/te671
Subject(s) - adverse selection , risk aversion (psychology) , private information retrieval , principal (computer security) , regular polygon , economics , loss aversion , distribution (mathematics) , curvature , mathematical economics , econometrics , microeconomics , mathematics , expected utility hypothesis , computer science , statistics , mathematical analysis , geometry , operating system
We solve the principal–agent problem of a monopolist insurer selling to an agent whose riskiness (loss chance) is private information, a problem introduced in Stiglitz's (1977) seminal paper. For an arbitrary type distribution, we prove several properties of optimal menus, such as efficiency at the top and downward distortions elsewhere. We show that these results extend beyond the insurance problem we emphasize. We also prove that the principal always prefers an agent facing a larger loss and prefers a poorer one if the agent's risk aversion decreases with wealth. For the standard case of a continuum of types and a smooth density, we show that, under the mild assumptions of a log‐concave density and decreasing absolute risk aversion, the optimal premium is backward‐S‐shaped in the amount of coverage—first concave, then convex. This curvature result implies that quantity discounts are consistent with adverse selection in insurance, contrary to the conventional wisdom from competitive models.