We thank Stephan Lauermann, Wei Li, Larry Samuelson, Ilya Segal, Lones Smith, and audience members at the University of Montreal, CEA-Universidad de Chile, Miami, Southern California, Georgia, Santa Barbara, Ohio State, Arizona, Arizona State, Duke, Riverside, Johns Hopkins, Iowa, the 2007 Midwest Economic Theory Meetings at Michigan, the 2008 Winter Meetings of the Econometric Society, the 2008 NSF/NBER Decentralization Conference at Tulane, the 2008 SWET Conference at UCSB, the 2008 SED Conference at Michigan, 2009 Cowles Summer Theory Conference, and 2011 Barcelona Jocs for comments. We are also grateful to Jeff Ely (Associate Editor) and three anonymous referees for their detailed suggestions.
Optimal insurance with adverse selection
Article first published online: 3 OCT 2012
Copyright © 2012 Hector Chade and Edward Schlee
Volume 7, Issue 3, pages 571–607, September 2012
How to Cite
Chade, H. and Schlee, E. (2012), Optimal insurance with adverse selection. Theoretical Economics, 7: 571–607. doi: 10.3982/TE671
- Issue published online: 3 OCT 2012
- Article first published online: 3 OCT 2012
- Submitted 2009-11-2. Final version accepted 2011-8-15. Available online 2011-8-16.
- Principal–agent model;
- monopoly insurance;
- common values;
- wealth effects;
- quantity discounts;
- empirical tests for adverse selection
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.