Abstract
This article analyzes the effects of compulsory insurance on the demand for self-insurance. We show that although a risk lover invests neither in insurance nor in self-insurance when insurance is voluntary, she invests in self-insurance when insurance is compulsory. On the contrary, when insurance is mandatory, a risk averter would substitute self-insurance for insurance. Economic policy implications of these antagonistic effects on self-insurance are discussed.
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Notes
As in Ehrlich and Becker (1972), self-insurance relates to the expenses reducing the size of the loss in case of an accident (for example, seat belts or health screening tests), while self-protection is an investment intended to reduce the loss probability.
Except when insurance prices are subsidized.
Without this assumption, a full insurance contract would crowd out all self-insurance opportunities. As shown in the Lemma 1 of Sect. 5, when facing full compulsory insurance, a decision maker, whether risk lover or risk averter, would invest in self-insurance in such a way that \({\text{SI}}^{\prime}\left( {a^{*} } \right) = \frac{1}{q}.\)
\(\frac{{\partial ^{2} {\text{EU}}}}{{\partial a^{2} }} = \left( {1 - q} \right)U^{{\prime \prime }} \left( {W_{0} - p\bar{I} - a} \right) + {\text{SI}}^{{\prime \prime }} \left( a \right)qU^{\prime } \left( {W_{0} - p\bar{I} - a - x_{0} + {\text{SI}}\left( a \right) + \bar{I}} \right) + \left[ {1 - {\text{SI}}^{\prime } \left( a \right)} \right]^{2} qU^{{\prime \prime }} \left( {W_{0} - p\bar{I} - a - x + {\text{SI}}\left( a \right) + \bar{I}} \right) < 0\).
It is worth noting that complementary insurance, if available, would eliminate this shortage. The policyholder could combine voluntary complementary insurance (Ic) with \(\bar{I}\), so as to achieve the optimum \(\left( {I^{*} = \bar{I} + I_{c} } \right)\). However, we have kept this case since it is entirely plausible that complementary insurance would be impossible to organize (adverse selection, moral hazard, narrow market, regulation, etc.). Moreover, even if provided, it would not necessarily be available to everyone.
Once \(p \ge q\), a risk lover has no incentive to invest in voluntary insurance activity.
Figure 1 remains an example and many other scenarii could arise, such as full insurance or over-insurance.
We simulated and obtained this kind of equilibrium with the following parameters: CRRA (Constant Relative Risk Aversion) utility function with a risk aversion of − 1.8, W0 = 10, x0 = 5, q = 0.2, loading rate of 30% (p = 0.26) and with a self-insurance technology characterized by \({\text{SI}}\left( a \right) = \ln \left( {30a + 1} \right).\)
At point D, the level of self-insurance is zero (a = 0) and the risk lover faces the maximum possible loss, x(0) = x0.
For ease of exposition, we focus on the case of a single maximizer of expected utility. However, our comparative statics argument, based on the single crossing property, still holds in case of multiple solutions.
This last effect would be reversed with \(\bar{I} > x_{0} - {\text{SI}}\left( {a^{*} } \right)\), an amount of compulsory insurance leading to an over-insurance coverage.
The first-order condition (3) characterizes a maximum only in the case of risk averters.
By symmetry, if there is over-insurance, the marginal cost of self-insurance is increasing, and the total effect of an increase in the insurance coverage on the marginal perception of self-insurance is not clear.
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Acknowledgements
The authors would like to thank Claude Montmarquette (CIRANO and Université de Montréal), Georges Dionne (HEC Montréal), Rachel Huang (National Central University, Taïwan), Valentin Baudin (ENS Paris-Saclay), and participants at the 2017 ARIA (American Risk and Insurance Association) annual meeting held in Toronto for valuable discussions and comments on the paper. The comments of two anonymous reviewers have substantially improved the paper. This work is partially supported by a grant overseen by the French National Research Agency (ANR) in the context of the “Investissement d’Avenir” program through the “iCODE Institute project” funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02.
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Pannequin, F., Corcos, A. Are compulsory insurance and self-insurance substitutes or complements? A matter of risk attitudes. Geneva Risk Insur Rev 45, 24–35 (2020). https://doi.org/10.1057/s10713-019-00043-x
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DOI: https://doi.org/10.1057/s10713-019-00043-x