The Spillover Effect of Compulsory Insurance

The Geneva Papers on Risk and Insurance Theory, 19:23-34 (1994) 91994 The Geneva Association The Spillover Effect of Compulsory Insurance CHRISTIAN GOLLIER GREMAQ and IDEI, University of Toulouse, and Department of Finance, Groupe HEC (Paris), F-78350 Jouy-en-Josas, France PATRICK SCARMURE Catholic University of Mons Abstract The assumption usually made in the insurance literature that risks are always insurable at the desired level does not hold in the real world: some risks are n o t - - o r are only partially--insurable, while others, such as civil liability or health and workers' injuries, must be fully insured or at least covered for a specific amount. We examine in this paper conditions under which a reduction in the constrained level of insurance for one risk increases the demand of insurance for another independent risk. We show that it is necessary to sign the fourth derivative of the utility function to obtain an unambiguous spillover effect. Three different sufficient conditions are derived if the expected value of the exogenous risk is zero. The first condition is that risk aversion be standard--that is, that absolute risk aversion and absolute prudence be decreasing. The second condition is that absolute risk aversion be decreasing and convex. The third condition is that both the third and the fourth derivatives of the utility function be negative. If the expected value of the exogenous risk is positive, a wealth effect is added to the picture, which goes in the opposite direction if absolute risk aversion is decreasing. Key words: Insurance demand, multiple sources of risk, compulsory insurance, standard risk aversion, prudence. 1. Introduction The interdependence of different sources of risk affecting final wealth has been recently examined within the framework of expected utility. This interdependence has long been pointed out in finance, but assumptions were made, such as the normality of the distribution of returns, to simplify research. Typically, if risks are normally distributed and independent, the demand for each specific risk depends only on the characteristics of this risk. However, the separability of decisions relative to independent risks does not hold in general. Under more general distributions, such a separability property holds only if the utility function is quadratic or exponential. For example, without any other assumption than risk aversion, two independent risks that are undesirable when considered in isolation can be jointly desirable. Samuelson [1963] pointed out that a large proportion of the population could view this possibility as a consequence of the law of large numbers, but this is clearly a misinterpretation of it. Pratt and Zeckhauser [1987] described the necessary and sufficient condition on the utility function--called properness--that guarantees that two independent undesirable risks can never be 24 CHRISTIAN GOLLIER AND PATRICK SCARMURE jointly desirable. A sufficient condition for properness is that all odd derivatives of the utility function u be positive and all even derivatives negative. A weakness of the concept of properness is that it appeared to be useless to analyze the effect of a change in one risk on the demand for another risk. Kimball [1993] examined a stronger restriction on utility functions that yields unambiguous comparative statics properties for this problem. Namely, risk aversion is "standard" if any loss-aggravating risk I always lowers the demand for any other independent risk. Kimball shows that the necessary and sufficient condition for standardness is that both absolute risk a v e r s i o n - u"(w)/u'(w)--and absolute prudence - u"(w)/u"(w)--be decreasing in wealth. 2 In this paper, we consider a specific class of changes in risk. Namely, we examine the case of a change in the scale of the risk that is, a change from risk k:~ to risk k'$. Changes in scale arise in many real-world problems. Elmendorf and Kimball [1992] examine the problem of the change in the labor income tax rate (1 - k) on the demand for risky assets. Since gross labor incomes ~ are risky, this is a typical application of the problem addressed above. Alternatively, consider a mutual fund that can invest in different financial instruments under some regulatory constraints as the existence of upper limits of investment in some specific classes of assets. The question is, what is the effect of a change in the upper limit for one asset on the demand for other assets? We will illustrate this analysis by considering the problem of a change in the compulsory rate of insurance in one risk on the demand of insurance for other independent risks? The introduction or change in the compulsory rate of insurance raises several interesting issues about welfare effects of insurance programs. Indeed, compulsory insurance is usually required to ensure adequate judgements to third parties or to reduce negative externalities that would arise in the absence of social security programs. If compulsory insurance produces compensating changes on the freely chosen insurance coverages, this in turn affects welfare. For instance, if compulsory insurance requirements reduce the demand for insurance for other risks, it creates the externality one was trying to remove by introducing compulsory insurance, and it can undermine policy objectives. The question of complementarity or substitution between compulsory insurance and other insurance lines is thus a potentially important one. The paper is organized as follows. In Section 2, we present the general model. We then consider in Section 3 the effect of imposing full insurance on one risk either in the small and in the large. Section 4 is allocated to the more general problem of analyzing the spillover effect of a marginal change in the compulsory rate of insurance, and some concluding remarks are provided in the last section. 2. The model We consider the problem of a risk-averse individual who faces two sources of risk. In the absence of insurance, the agent's final wealth is Woo - ~ - 22, where 21 25 SPILLOVER EFFECT OF COMPULSORY INSURANCE and ~2 are two independent random variables representing potential losses on specific assets. An insurance policy on risk i is characterized by a couple (ai, P~(a~)) where a; is the rate o f retention and P~ is the premium paid prior to the realization of the loss. We assuume that the insurance pricing is linear with )ti _> 0 denoting the loading factor. Namely, the insurance tariff on risk i takes the following form: Pi(a~) = (1 + ~,i)(1 - a~)tt,, (1) where #i is the expected loss for risk i. The net wealth when retaining a share of al of risk 1 and a share a2 of risk 2 is a random variable that is defined as 2 ~ ( a , , a~) = Woo - 2 Y~ [a.~, + e,(a,)] = ,'o + Y~ aye, i--1 i=l (2) where w0 = Woo - ~ _ 1 (1 + Zi)/~iis the net wealth in case of full insurance, and yi = (1 + Zi)gi - 2i is the net gain of not buying insurance. 4 Because Zi is a (...truncated)