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The Economics of Optimal Insurance Design

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Handbook of Insurance

Abstract

This chapter provides a survey on optimal insurance when insurers and policyholders have symmetric information about the distribution of potential damages. Under general conditions on the policyholder risk aversion and on transaction costs, the optimal insurance contract contains full insurance of losses above a straight deductible. This is proven without assuming expected utility. The use of expected utility generates additional results, e.g., in the case of nonlinear transaction costs.

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Notes

  1. 1.

    Eeckhoudt and Gollier (1995) provide a complete analysis of the insurance problem under EU.

  2. 2.

    For simplicity, we assume a finite number of states. All results remain true under continuous or mixed distribution functions.

  3. 3.

    Gollier (1987a) and Breuer (2004) allow for negative indemnities by assuming that transaction costs depend upon the absolute value of the indemnity. Surprisingly enough, in most cases, removing the constraint on the nonnegativity of claims has no effect on the optimal contract.

  4. 4.

    Cohen (1995) provides an excellent analysis of the various definitions of risk aversion and their connexions to each other.

  5. 5.

    It has been proven by Eeckhoudt et al. (1991) for the specific case of a binomial distribution.

  6. 6.

    Zilcha and Chew (1990) and Karni (1992) used the restriction of Frechet differentiability, whereas Gollier and Schlesinger (1996) did not make any restriction on the model.

  7. 7.

    The equivalence between this characterization of an MPC and the definition using white noises is in Rothschild and Stiglitz (1970).

  8. 8.

    For the definition of the order of risk aversion, see Segal and Spivak (1990).

  9. 9.

    Machina (1995) extends this result to non-expected utility models with Frechet differentiability.

  10. 10.

    Eeckhoudt and Gollier (1999) extend this result to non-expected utility models with second-order risk aversion.

  11. 11.

    See Mookherjee and Png (1989) for a first result on this topic. The literature on optimal auditing is not covered in this survey. This is because our basic assumption is symmetric information, ex ante and ex post.

  12. 12.

    Risk vulnerability is linked to the third and the fourth derivative of the utility function. All familiar utility (exponential, power, logarithmic) functions satisfy this property.

  13. 13.

    The analogies are numerous. For example, the fact that λ = 0 implies that D = 0 is equivalent in finance to the fact that risk-averse investors will not invest in the risky asset if its expected return does not exceed the risk-free rate.

  14. 14.

    See Schlee (1995) for some insights about how to test EU and NEU models with insurance demand data.

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Acknowledgements

This chapter is an updated version of Gollier (2000). The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013) Grant Agreement no. 230589.

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Authors and Affiliations

  1. Toulouse School of Economics (LERNA and IDEI), Toulouse, France

    Christian Gollier

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Correspondence to Christian Gollier .

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  1. Finance, HEC Montreal, Montreal, Québec, Canada

    Georges Dionne

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Gollier, C. (2013). The Economics of Optimal Insurance Design. In: Dionne, G. (eds) Handbook of Insurance. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0155-1_4

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