Optimal Insurance with Adverse Selection

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. 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. Terms of use: Documents in EconStor mayFor 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.

Risk-sharing Contracts with Asymmetric Information

“…We show that the mutual structure copes better with asymmetric information since the asymmetry of information does not necessarily here lead to a loss of efficiency. Moreover, when the asymmetry of information leads to a loss of efficiency, the loss is entirely borne by low-risk-type agents, as in the case of insurance companies (Rothschild and Stiglitz, 1976;Chade and Schlee, 2008).In addition to these positive results, our work offers a framework for the design of risk-sharing contracts between financial institutions or insurance companies. In particular, it can be used to precisely design the direction of conditional financial cash flows.…”

“…Compared to the case of complete information, it also induces changes in the direction of transfer (from low-risk to high risk) in some states of nature for most utility functions of the Harmonic Absolute Risk Aversion (HARA) class. Finally, when the asymmetry of information leads to a loss of efficiency, this loss is entirely borne by low-risk agents, as in the case of insurance companies (see Rothschild and Stiglitz, 1976;Chade and Schlee, 2008).…”

“…When expected risk heterogeneity is high, the asymmetry of information induces a loss of efficiency by reducing insurance when both agents are low risk. In this sense, asymmetric information here has the same impact on risk-sharing, as it has on insurance companies (see Rothschild and Stiglitz, 1976;Chade and Schlee, 2008): a reduction in the coverage offered to low-risk agents.For the agreement to remain attractive to low-risk agents, however, it has to compensate for previous distortions. This is done partly by specifying transfers from the high-risk to the low-risk individual when neither of the two agents suffers damage.…”

“…Following Rothschild and Stiglitz (1976), many authors have analysed how insurance companies deal with asymmetric information on the distribution of wealth. As we focus on a principal-agents model, our paper is more closely linked with works on monopoly insurance (see, for example, Stiglitz, 1977;Landsberger and Meilijson, 1994;Chade and Schlee, 2008). The main result of these papers is to show that the optimal contract amounts to providing the "high type" (high-risk-type or the most risk-averse agents) with full insurance and the other type(s) with only partial insurance.…”

“…This is because the optimal risk-sharing agreement depends on the risk types and realizations of both agents, whereas under private insurance the optimal allocations (e.g. Rothschild and Stiglitz, 1976;Bisin and Gottardi, 2006;Chade and Schlee, 2008) only depend on the risk type of the contracting agent. …”