Abstract
The government often provides relief against large risks, such as disasters. A simple, general rationale for this role of government is considered here that applies even when private contracting to share risks is not subject to market imperfections. Specifically, the optimal private sharing of large risks will not result in complete coverage against them. Hence, when such risks eventuate, the marginal utility to individuals of government relief may exceed the marginal value of public goods. Consequently, social welfare may be raised if the government reduces public goods expenditures and directs these freed resources toward individuals who have suffered losses.
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Notes
On the role of the government in disaster assistance, see generally the historical account of Moss (1999) and, for example, the website of the Federal Emergency Management Agency (FEMA), www.fema.gov; on government programs to foster flood and earthquake insurance, see Federal Emergency Management Agency (2002) and the website of the California Earthquake Authority, www.earthquakeauthority.com. The government also promotes and furnishes other types of insurance, such as disability and health insurance, of course, but the basis for these types of aid appears to be different from that discussed here.
The policy of government relief will, however, result in some crowding out of payments made to accident victims under private risk-sharing contracts.
As will be seen, a well-designed ex post subsidy of coverage payments made to accident victims will be socially beneficial, but an ex ante subsidy of risk-sharing contracts cannot improve social welfare because it does not make public goods expenditures depend on the occurrence of accidents.
Regarding the paucity (and high cost) of private insurance coverage against catastrophic events, see generally Froot (1999, 2001), and also, for example, Jaffee and Russell (1997) and Cummins (2006). As Froot emphasizes, the scarcity of coverage due to these correlated losses is significant, even though one might have expected otherwise, given the existence of the reinsurance industry and the size of global capital markets.
See, for example, the discussion of the coverage of terrorism risk in Cummins (2006), p. 375.
The reader can verify that if there is no demand for public goods, the first-best outcome described in Proposition 1 is achieved under privately optimal risk-sharing described in Proposition 2.
This argument is developed by Arrow and Lind (1970), who show under certain assumptions not only that the risk-bearing cost per individual tends to zero with the population size, but also that the aggregate risk-bearing cost tends to zero.
See, for example, Kunreuther et al. (1978).
See, for example, Dionne et al. (2000) for a survey on adverse selection in insurance markets.
See Kaplow (1991).
The importance of this assumption is that it implies that the socially optimal level of individual consumption and of the public good are each rising in total wealth.
I do not consider non-monetary harm, such as that due to pain and suffering from an injury. In the central case of non-monetary harm—in which the harm is a decline in utility but does not involve any change in the utility from wealth—there is no demand for insurance. See Cook and Graham (1977). Hence, in that case, the issue studied in the model would be moot; and if harm were both a monetary loss and a utility loss, the results to be obtained here would be unaltered.
In considering only the feasible allocations given by (1), I am making two harmless simplifications: that total wealth is exhausted by an allocation; and that allocations do not depend on complete accident states (rather than only on accident events m).
In particular, let the feasible allocation scheme that maximizes the sum of expected utilities be denoted S, and let e be the expected utility of each person under this scheme, so that ne is the maximum sum of expected utilities. If S is not the scheme that maximizes the expected utility of each person subject to the constraint that each has the same expected utility, there must exist another scheme S′ under which each person obtains expected utility e′ > e. But then ne′ > ne, contradicting the assumption that S maximized the sum of expected utilities.
This follows because dy*(m)/dm = (dy*(m)/dk) (dk/dm). But (dy*(m)/dk) >0 by (13) and dk/dm = −h <0.
This follows by the logic of the previous note.
In this case, for each i, p i (1, n − 1) = p and p i (1, j) = 0 for j < n − 1, and also p i (0, 0) = 1 − p and p(0, j) = 0 for j ≥1.
For each i, p i (1, m − 1) = (m/n)p and p i (1, j) = 0 for j ≠ m − 1, and also p i (0, 0) = 1 − p, p i (0, m) = (1 − m/n)p, and p(0, j) = 0 for j ≠ 0 or m. Similarly, the symmetry assumption could hold in a variation of this example, in which the possible subgroups of m individuals are neighbors or are in some other manner restricted.
For each i, p i (1, j) = p j+1(1 − p)n–j–1(n!/[(j +1)!(n − j − 1)!]) and p i (0, j) = p j(1 − p)n–j(n!/[j!(n − j)!]).
Nevertheless, the lessons from the model would carry over to a general context in which risks are not symmetric and differ across individuals. The reason is that the main argument to be developed depends on two factors that are unrelated to the symmetric risk assumption, namely, (a) the point that privately-optimal risk-sharing contracts do not lead to full coverage against large risks and (b) the assumption that the government provides public goods.
Other types of private risk-sharing contracts could be considered, notably contracts among only a subset of individuals, or contracts under which the payment received by an accident victim depends on the identity of other accident victims rather than just on their total number. However, examination of a different set of contracts would not alter the main qualitative conclusion to be reached—that the government can raise the expected utility of individuals by giving relief in certain circumstances. The reason, in essence, is that, whatever the nature of private contracting, it cannot control the expenditure of the government on public goods.
Note that v(nt* − mg(m)) does not enter into (28) because the term does not depend on the risk-sharing contract.
Expression (38) implicitly reflects the symmetry assumption, for ns is subtracted rather than a sum of different subsidy amounts for different individuals.
The federal flood insurance program involves both ex post subsidy, in that it can resort to borrowing from the Treasury, and ex ante subsidy of premiums. See Federal Emergency Management Agency (2002), pp. 22–28.
The insurance offered under the auspices of the California Earthquake Authority incorporates an implicit ex ante subsidy of premiums because the Authority does not pay taxes. However, the program of the Authority does not reflect an ex post subsidy in that it cannot draw on the state’s funds if its assets are not sufficient to pay the claims made in a large earthquake. See www.earthquakeauthority.com.
The income tax system in fact has a feature that gives relief from accident losses: casualty losses may be deducted from taxable income. However, the casualty loss deduction is not conditioned on whether the losses are correlated (and thus not on the availability or price of insurance coverage).
References
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Acknowledgments
I thank Peter Diamond, Georges Dionne, Kenneth Froot, Louis Kaplow, and A. Mitchell Polinsky for comments, Michael Belinsky and Jonathan Borowsky for research assistance, and the John M. Olin Center for Law, Economics, and Business at Harvard University for research support.
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Shavell, S. A general rationale for a governmental role in the relief of large risks. J Risk Uncertain 49, 213–234 (2014). https://doi.org/10.1007/s11166-014-9203-2
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DOI: https://doi.org/10.1007/s11166-014-9203-2