RISK POOLING IN THE PRESENCE OF MORAL HAZARD

Transcription

1 # Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA Bulletin of Economic Research 56:1, 2004, RISK POOLING IN THE PRESENCE OF MORAL HAZARD Joseph G. Eisenhauer Department of Economics, Canisius College ABSTRACT The size of an insurance pool that minimizes average risk per policy is derived for cases in which moral hazard offsets the benefits of pooling. Keywords: Insurance, moral hazard JEL classification numbers: D81, G22 I. INTRODUCTION Insurance is widely regarded as a means not only of transferring risk, but of reducing risk by pooling numerous exposure units facing independent, identically distributed losses. It is also well known, however, that insurance can induce moral hazard on the part of the insured, thereby increasing the level of risk. This can occur in several ways. First, there is the possibility of outright fraud, as when an insured exaggerates a loss when filing a claim; indeed, this was the original meaning of moral hazard. Second, utility maximizers will substitute insurance for precautionary activities (Pauly, 1968; Ehrlich and Becker, 1972); Lee and Ligon (2001) have shown that this effect becomes more severe as the size of a mutual insurance pool increases. Additionally, moral hazard can create negative externalities, as when each of several insured drivers, exhibiting moral hazard, increases the probability of loss for him- or herself and for Correspondence: Joseph G. Eisenhauer, Department of Economics, Canisius College, 2001 Main Street, Buffalo, NY , USA. eisenhauer@canisius.edu. I am grateful to two anonymous referees for helpful comments; any errors are my own. 107

2 108 BULLETIN OF ECONOMIC RESEARCH all others (Eisenhauer, 1996). Moral hazard can thereby offset the riskmitigating effect of pooling; this can potentially occur to such an extent that the optimal size of the insurance pool is finite. While others have examined this trade-off from a social welfare perspective (see, for example, Manning and Marquis, 1996), the number of insureds needed to achieve the minimum level of risk per policy has not been determined in previous research. The present note extends this literature by examining the conditions under which moral hazard is sufficient to fully negate the risk-reducing property of pooling, and by characterizing the risk-minimizing pool size under such circumstances. The paper considers two forms of moral hazard: one affecting probabilities and one affecting the value of claims. II. ANALYSIS Consider a group of n homogeneous individuals, each of whom faces a probability p of experiencing a loss for which a claim of L will be filed, and a complementary probability (1 p) of no loss. The assumption that individuals display homogeneity (as opposed to heterogeneity) is adopted for convenience, as it greatly simplifies the formal analysis. As it is generally the case in practice, assume that a loss is less likely than a favourable state of nature, so that 0 < p < 0.5. Each individual s claim has an expected value of pl, with a variance given by 2 ¼ L 2 p(1 p) and a corresponding standard deviation of ¼ Lp 0.5 (1 p) 0.5. If both L and p are fixed, the expected value of total claims in the group of insureds taken collectively is npl, with a variance of n 2 and a standard deviation of n 0.5. Thus, the total risk, or what Cummins (1991) describes as the insurer s absolute risk, naturally increases as more policy-holders are added to the risk pool. But in the absence of moral hazard, pooling reduces the standard deviation of the average claim to /n 0.5. This increases the level of accuracy in predicting claims, and thus represents a reduction of the insurer s average risk per policy. This decrease in the average risk is a central feature of all pooling arrangements those of stock corporations as well as mutual insurers and it is the focus of the present analysis. Moral hazard, however, alters both the expected value and the standard deviation, by increasing the probability of loss and/or the magnitude of the potential claim. Where monitoring is impossible, insurers may use deductibles and coinsurance clauses to discourage moral hazard; then an insured who reduces precautionary effort and increases the probability of a claim also raises expected out-of-pocket costs. By exaggerating the size of the loss when filing the claim, however, the insured may attempt to recoup the out-of-pocket expense. It is thus worthwhile to consider both probability and claim size as variables

3 RISK POOLING IN THE PRESENCE OF MORAL HAZARD 109 subject to moral hazard. Following Lee and Ligon (2001), moral hazard is treated as an increasing function of n. Thus, rises as n increases, > 0 > 0. Consequently, the pool size that minimizes the average risk per policy is found by solving: Min n =n 0:5 ¼ Lp 0:5 (1 p) 0:5 n 0:5 ð1þ where both p and L are non-negative functions of n. The first-order condition is given by: np(1 p)(@l=@n) þ 0:5nL(1 2p)(@p=@n) 0:5p(1 p)l ¼ 0 and the second-order condition to guarantee a minimum is given by: 0:5p(1 p)@l=@n þ 1:5n(1 2p)(@p=@n)(@L=@n) þ np(1 p)@ 2 L=@n 2 ð3þ þ 0:5nL(1 2p)@ 2 p=@n 2 nl(@p=@n) 2 > 0: Note that no simple conditions or restrictions on the p and L functions can ensure that (3) will always hold, though some illustrative cases in which the second-order condition is satisfied are presented below. From (2), the risk-minimizing pool size is: n ¼ ð2þ 0:5p(1 p)l p(1 p)(@l=@n) þ 0:5(1 2p)L(@p=@n) ; ð4þ beyond this size, if the second-order condition holds, pooling fails to reduce the amount of risk per policy. (In the extreme, of course, a solution of n ¼ 1 would imply that there are no benefits at all to pooling risk.) Equations (1) (4) provide a general solution for the two-state model of pooling in the presence of moral hazard. However, to make the analysis more tractable, it is convenient to consider two separate cases. First, suppose p is constant and moral hazard operates exclusively by raising L. Then the solution becomes: n ¼ ; ð5þ minimization of the average risk occurs where the elasticity of potential claims with respect to pool size is ½, provided that the second-order condition 2 2 > (0:5=n)(@L=@n): ð6þ In this case, there is an interior solution if the potential claim increases at an increasing, constant, or even slightly decreasing rate. As a simple

4 110 BULLETIN OF ECONOMIC RESEARCH numerical illustration, suppose the potential claim is linear in n, so that L ¼ a þ bn, with a > 0 and b > 0. ¼ b 2 L/@n 2 ¼ 0, so condition (6) holds everywhere. From (5), the risk-minimizing pool size is found by setting n ¼ 0.5L/b ¼ 0.5(a þ bn)/b ¼ 0.5(a/b) þ 0.5n, with the solution n ¼ a/b. If, for example, L ¼ 5000 þ 25n, the risk-minimizing pool is 200 policy-holders. Beyond this size, the pool ceases to reap any benefit by adding exposure units; indeed, the average risk per policy will actually increase with the number of insureds. Importantly, this occurs even if the insurer knows the functional relationship between L and n and marks up the insurance premium to reflect the increase in expected claims; the problem is that the variance around the expected claim will rise with n, making forecasting more difficult. The second case occurs if L is fixed and moral hazard operates exclusively by raising p. Then the general solution (4) reduces to: n ¼ p(1 p) (1 2p)@p=@n ð7þ provided that the second-order condition 2 2 > 2 : ð8þ 1 Since the right-hand side of (8) is positive, this case occurs only if the probability of filing a claim rises at a sufficiently increasing rate as the size of the pool increases. As an illustration, suppose p ¼ 0.02 þ (n 2 / 1,000,000). Then it is easily shown that (8) holds for all n < 400, and (7) can be solved to obtain n ¼ 148 as the risk-minimizing pool size. This is, to be sure, a local minimum, but lower levels of /n 0.5 do not occur until a loss is quite likely indeed, not until p > 2/3 situations in which it is highly doubtful that insurance would be either feasible or affordable, and which can therefore be safely disregarded (as initially assumed). 1 III. CONCLUSION It should be noted that these results are concerned only with the minimization of an insurer s average risk, which does not necessarily 1 Note that when p 0.5, further increases in p reduce, so in this range moral hazard manifested through p could, theoretically, contribute to lowering average (and possibly even total) risk: if the insurer knew the moral hazard function, forecast accuracy could always be enhanced by adding policy-holders. In the example, /n 0.5 would fall below its local minimum when n 810 and p , and in the limit, of course, /n 0.5 ¼ 0 when p ¼ 1. But this is an unrealistic scenario, as insurance is likely to be precluded by the high loss probabilities in this region: relatively modest loading factors would make premiums exceed insurance benefits.

5 RISK POOLING IN THE PRESENCE OF MORAL HAZARD 111 imply the maximization of utility, profit, or other objective functions. Depending on their preferences, insurers whether stock or mutual organizations may be willing to tolerate greater than minimal risk per policy in order to obtain a different degree of skewness in the loss distribution, increase market share, or satisfy other goals in their pursuit of profit or utility maximization. Nonetheless, to the extent that the average level of risk matters, this type of analysis has important implications for both practitioners and policy-makers. If the efficiency gains from pooling persist as the size of the pool rises, insurance can exhibit a tendency toward natural monopoly. Alternatively, if the benefits from pooling diminish due to moral hazard, then it becomes more difficult to accurately predict claims as the number of policy-holders rises beyond some limit. In that event, insurers may seek to insure a finite pool, in which case market concentration will be lower than it would be in the absence of moral hazard. An obvious direction for future research would be undertaking empirical estimation of moral hazard, to determine whether and where these limits exist in practice. REFERENCES Cummins, J. D. (1991). Statistical and financial models of insurance pricing and the insurance firm, Journal of Risk and Insurance, vol. 58, pp Ehrlich, I. and Becker, G. S. (1972). Market insurance, self-insurance, and selfprotection, Journal of Political Economy, vol. 80, pp Eisenhauer, J. G. (1996). Insurance and externalities, Journal of Insurance Issues, vol. 19, pp Lee, W. and Ligon, J. A. (2001). Moral hazard in risk pooling arrangements, Journal of Risk and Insurance, vol. 68, pp Manning, W. G. and Marquis, S. M. (1996). Health insurance: The tradeoff between risk pooling and moral hazard, Journal of Health Economics, vol. 15, pp Pauly, M. V. (1968). The economics of moral hazard: Comment, American Economic Review, vol. 58, pp