Background Risk and the Performance of Insurance Markets under Adverse Selection

The Geneva Risk and Insurance Review, 2008, 33, (137–160) r 2008 The International Association for the Study of Insurance Economics 1554-964X/08 www.palgrave-journals.com/grir/ Background Risk and the Performance of Insurance Markets under Adverse Selection Keith J. Crockera and Arthur Snowb a Smeal College of Business, Pennsylvania State University, University Park, PA 16802, U.S.A. Department of Economics, University of Georgia, Athens, GA 30602, U.S.A. E-mail: b Background risk can influence the performance of insurance markets that must deal with adverse selection when applicants are risk vulnerable, since they are more averse to bearing the insurable risk as a result of their exposures to background risk. We show that background risk always results in a lower deductible for the incentive constrained contract, and that a broader range of markets attains the stable sequential equilibrium cross-subsidized pair of separating contracts. We conclude that background risk always improves the performance of markets for coverage against (insurable) foreground risks that must deal with adverse selection. We also find, however, that these improvements are never sufficient to offset the cost to insureds of bearing the background risk. The Geneva Risk and Insurance Review (2008) 33, 137–160. doi:10.1057/grir.2008.12 Keywords: Nash screening equilibrium; risk vulnerability; prudence Introduction The performance of insurance markets can be affected by the presence of background risks, such as those associated with uninsurable fluctuations in the value of human capital arising from randomness in entrepreneurial and wage incomes. For individuals who are risk vulnerable in the sense defined by Gollier and Pratt (1996), exposure to fair or unfair background risk increases the degree of aversion toward bearing the foreground, insurable risks and, as a result, alters the demand for insurance against these risks. This would be of no consequence if full and fair insurance were available to all individuals.1 However, in the presence of adverse selection, insurance contracts must satisfy incentive compatibility constraints that result in partial coverage for some 1 Eeckhoudt and Kimball (1992) show that, with proportional loading in the insurance premium, an individual demands more coverage when exposed to fair background risk if absolute risk aversion and absolute prudence are both decreasing functions of wealth. Gollier and Pratt show that such an individual is risk vulnerable with respect to the introduction of fair or unfair background risk. The Geneva Risk and Insurance Review 138 insurance applicants. Changes in the degree of risk aversion alter these incentive constraints and, in addition, affect the attractiveness of competing contractual offers. Thus, background risk can influence the performance of insurance markets through either of two channels that bear directly on the market equilibrium attained under adverse selection. Following Rothschild and Stiglitz (1976) and Hellwig (1987), we assume that competitive insurers, unable to distinguish among applicants by their risk of incurring a given loss, deal with adverse selection by engaging in price-quantity competition. We relax their assumption that each insurer can offer only one type of contract, and instead assume that insurers can offer applicants the opportunity to choose from a menu of contracts that incorporate different combinations of premium and deductible. The market attains a Nash screening equilibrium that depends on the proportion of high risks in the applicant pool. If this proportion is greater than or equal to a critical value, then the market attains the pure strategy Nash equilibrium identified by Rothschild and Stiglitz, and offers applicants a separating pair of contracts that breakeven individually; otherwise, the market attains a sequential screening equilibrium and offers the cross-subsidized pair of separating contracts identified by Miyazaki (1977). We show that exposure to background risk causes this critical proportion of high-risk applicants to increase, so that the cross-subsidized equilibrium is attained over a broader range of insurance markets. Since the cross-subsidized contracts provide greater coverage of the foreground risk than that afforded by the breakeven contracts, background risk improves the performance of the insurance markets when it causes the equilibrium to switch from the breakeven pair to a cross-subsidized pair. Additionally, even when the nature of the market equilibrium is not affected by the presence of background risk, the incentive constraints are relaxed and the market provides greater coverage of the foreground risk. Thus, background risk always enhances the performance of insurance markets under adverse selection. It does not follow, however, that the presence of background risks is desirable, since these risks are unavoidable and costly to bear. Indeed, we find that insurance applicants are always worse off when background risk is present despite the improved performance of markets insuring the foreground risk. We conclude that public and private innovations that limit exposure to background risks or extend insurance coverage to them would improve the welfare of all insurance applicants. Several previous studies of adverse selection in insurance markets have drawn attention to radical changes in the contracting environment that are introduced when the degree of risk aversion is hidden knowledge. Arnott and Stiglitz (1988) explore the desirability of introducing ex post randomization, Landsberger and Meilijson (1994, 1999) consider monopolistic contracts, while Keith J. Crocker and Arthur Snow Background Risk and Performance of Insurance Markets 139 Smart (2000) and Wambach (2000) investigate the implications of ‘‘double crossing’’ indifference curves that can occur when insurance applicants differ with respect to their hidden knowledge of both risk and risk aversion. We, in contrast, focus on situations in which insurers can categorize applicants on the basis of observable characteristics correlated with risk aversion and effectively control for any differences in applicants’ willingness to bear insurable risk, so that their degree of risk aversion is not hidden knowledge. Moreover, in our analysis, differences in risk aversion are not immutable, but instead arise from the presence of background risks. Our model thus offers insights regarding the potential social value of innovations that extend coverage to previously uninsurable (background) risks. In the next section, we set out the model insurance market with adverse selection and illustrate the alternative equilibrium configurations. In the third section, we show that, when the risk preferences of insurance applicants are characterized by nonincreasing absolute risk aversion, an increase in risk aversion increases the critical proportion of high risks, so that the Rothschild–Stiglitz equilibrium is sust (...truncated)