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Optimal Insurance with Adverse Selection

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  • Hector Chade
  • Edward Schlee

Abstract

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 \emph{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 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 \emph{backwards-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.
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Suggested Citation

  • Hector Chade & Edward Schlee, 2008. "Optimal Insurance with Adverse Selection," Levine's Working Paper Archive 122247000000002175, David K. Levine.
  • Handle: RePEc:cla:levarc:122247000000002175
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Daniel McFadden & Carlos Noton & Pau Olivella, "undated". "Remedies for Sick Insurance," Working Papers 620, Barcelona Graduate School of Economics.
    2. Nathaniel Hendren, 2013. "Private Information and Insurance Rejections," Econometrica, Econometric Society, vol. 81(5), pages 1713-1762, September.
    3. Johan N. M. Lagerlöf & Christoph Schottmüller, 2013. "Facilitating Consumer Learning in Insurance Markets - What Are the Welfare Effects?," Discussion Papers 13-12, University of Copenhagen. Department of Economics.
    4. Attar, Andrea & Mariotti, Thomas & Salanié, François, 2014. "Multiple Contracting in Insurance Markets," TSE Working Papers 14-532, Toulouse School of Economics (TSE), revised Sep 2016.
    5. Georges Dionne & Casey Rothschild, 2014. "Economic Effects of Risk Classification Bans," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 39(2), pages 184-221, September.
    6. Renaud Bourlès & Dominique Henriet, 2012. "Risk-sharing Contracts with Asymmetric Information," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 37(1), pages 27-56, March.
    7. Mingli Zheng & Chong Wang & Chaozheng Li, 2016. "Insurance Contracts with Adverse Selection When the Insurer Has Ambiguity about the Composition of the Consumers," Annals of Economics and Finance, Society for AEF, vol. 17(1), pages 179-206, May.

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    JEL classification:

    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

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