Abstract
Evaluation of projects that affect mortality risk usually assumes that risk changes are small and similar across individuals. In reality, risks differ among individuals and information about risk heterogeneity determines the extent to which affected lives are “statistical” or “identified” and influences the outcome of benefit-cost analysis (BCA). The effects of information about risk heterogeneity on BCA depend on, inter alia, whether information concerns heterogeneity of baseline or change in risk and whether valuation uses compensating or equivalent variation. BCA does not systematically favor identified over statistical lives. We suggest some political factors that may explain the apparent public bias.
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Notes
In her paper “Murdering Statistical Lives ...?” Linnerooth (1982: 231) attributes the provocative term to Howard Raiffa. It was subsequently used by Graham (1995) and Viscusi (2000b). Tal (1998: 170) cites an alternative use by Merrell and Van Strum (1990: 21) to describe the permitting of any activity (e.g., pesticide use) that imposes mortality risk on someone.
Howard (1984) proposed valuing risk increments in terms of “micromorts,” i.e., risk changes of one in a million. This alternative term emphasizes the focus on small risk changes. Equivalent terms include value of statistical life (Viscusi 1992), value per statistical life (Hammitt 2000), value per life saved (Jones-Lee 1976), and value of prevented fatality (Jones-Lee 2004).
In particular, Pratt and Zeckhauser (1996) examine the effects of risk and wealth on optimal public safety expenditures defined behind a Rawlsian veil of ignorance (before individuals have information about their own wealth and risk). Linnerooth (1982) and Hammitt (2002) compare VSL with effectiveness measures such as lives and life years saved.
Another critical source of heterogeneity is wealth. The effects of wealth heterogeneity are beyond the scope of this paper. We concentrate on the issue announced in the title, which motivates the focus on heterogeneity of risk.
Broome’s (1978) provocative paper raises a number of issues for BCA. Many of these issues are addressed in a set of critical papers (Buchanan and Faith 1979; Jones-Lee 1979; Williams 1979). On the issue of interest to this paper, Ulph (1982) suggests that Broome’s paradox arises due to an inappropriate social welfare function, i.e., a utilitarian framework. Our discussion differs from Ulph’s since we work within a utilitarian framework.
Broome (1978) presents another version of the paradox based on the timing of decisions: Suppose information about who will die is not available today but will be available tomorrow. If the project is up for consideration today, it may be acceptable, but tomorrow it will be deemed unacceptable by an infinite margin. Hence re-evaluating the project one day later would lead to rejecting it as “infinitely wrong.” Similarly, Heinzerling (2000) argues that the evaluation of projects that affect mortality risks should not depend on whether the lives at risk are identifiable.
Alternatively, we could consider terminal wealth with the project, w + s, and obtain net compensating variation, \( C{\left( x \right)} - s \).
Note that if one wishes to maximize social welfare it is optimal to transfer all the benefits of the project to the person with higher survival probability. That is, the maximum of \( {\left( {p - 2e} \right)}{\left( {w + s + y} \right)} + p{\left( {w + s - y} \right)} \) is obtained for y = −s. Since there is assumed to be no bequest motive, any wealth given to a person who dies contributes no well-being to the society. This shows that social welfare can be made higher under information about individual risks than under no information. In contrast, BCA can lead to the recommendation to adopt the welfare-enhancing project only in the no-information case. Information may have a negative welfare effect for a society that uses BCA as its decision rule. Suppose the condition in Eq. 9 holds. Then if society evaluates the project using BCA, information would lead it to reject the project. Without information, the project would be accepted and social welfare would increase.
See, e.g., Blackorby and Donaldson (1990), Johansson (1998), and in the context of mortality risk, Armantier and Treich (2004). Linnerooth (1982) suggests that the higher VSL of individuals facing larger mortality risk is not sufficient justification for disproportionately allocating public resources to protect them. Although the private opportunity cost of spending is low for such individuals (by the dead-anyway effect described in Section 2), the social opportunity cost is not. Pratt and Zeckhauser (1996) propose adjusting empirical estimates of VSL to account for differences between private and social opportunity costs of spending associated with heterogeneity of mortality risk.
When individual costs are weighted by the individual marginal utilities of wealth, per-capita cost is independent of information about risk heterogeneity. Without information, the marginal utility of wealth is (p − e) for both individuals; with information, the marginal utility of wealth is (p − 2e) for H and P for L. With and without information, per-capita weighted cost equals ew. If the marginal utility of the bequest is equal to the marginal utility of wealth given survival, then the expected marginal utility is independent of risk and the monetary compensation is linear in the incremental risk (e.g., let the utility if dead equal w − β instead of zero). In this case, maximization of social welfare and maximization of the sum of unweighted monetary compensation lead to the same outcome.
Published in French, Drèze’s paper is not well known among Anglophone economists although it was discussed by Jones-Lee (1974, 1976). Drèze modeled individual WTP to eliminate a one-time mortality risk and optimal spending on risk reduction using the now-standard state-dependent utility function (12).
Formally, VSL is equal to the marginal compensating and equivalent variations \( C\prime {\left( 0 \right)},\;P\prime {\left( 0 \right)},\;C^{\prime }_{n} {\left( 0 \right)},\;{\text{and}}\;P^{\prime }_{n} {\left( 0 \right)} \) that are defined below in Eqs. 16, 18, 20 and 23, respectively. Jones-Lee (1976) also evaluated these four measures of value.
The term value per statistical life may be motivated by noting that if a population of N individuals would each pay on average dw for a small risk reduction dp, a total of Ndw would be paid to prevent Ndp expected fatalities.
Identifiability may impair rational decision making when the threat of death is present. This is recognized by Schelling (1968): “The avoidance of a particular death—the death of a named individual—cannot be treated straightforwardly as a consumer choice. It involves anxiety and sentiment, guilt and awe, responsibility and religion. If the individuals are identified, there are many of us who cannot even answer whether one should die that two may live.” We abstract from this issue but recognize it may play an important role for practical BCA, e.g. for elicitation of individual WTP in surveys. Observe however that opposite arguments may be used with statistical lives due to the small probabilities involved, like the insensitivity of WTP to probabilities (Hammitt and Graham 1999).
The dead-anyway effect was recognized by Jones-Lee (1974, 1976), Weinstein et al. (1980), Linnerooth (1982), and others but the moniker was bestowed by Pratt and Zeckhauser (1996). It depends on \( u\prime > v\prime \), a condition that may not hold with perfect contingent-claims markets (Jones-Lee 1976; Weinstein et al. 1980; Breyer and Felder 2005) or when desired bequests are constrained because human capital or other components of wealth do not survive the individual’s death (Breyer and Felder 2005).
Note that the results rely on only the convexity of the indifference curves and that the conditions exhibited in (13) are sufficient, but not necessary, for this convexity property to hold. This suggests broader applications of the results than to pure mortality risks.
In model (12), a natural interpretation of the utility function v(·) is as a bequest function. This may be viewed as a form of altruism, e.g., toward one’s children. But this is not the form of altruism that is important here. We study altruistic preferences over other people’s mortality risks.
See Jones-Lee (1992) for an analysis of the implications of both pure (i.e., benevolent, non-paternalistic) and paternalistic altruism (e.g., safety-focused altruism) for the valuation of life.
A QALY represents a year of life adjusted by an index of quality of life or health (Gold et al. 1996; see Zeckhauser and Shepard (1976), Broome (1993), and Hammitt (2002) for economic discussions). The US Office of Management and Budget (2003) requires federal agencies to evaluate major regulations using CEA as well as BCA and an Institute of Medicine panel recommended that QALYs be used as the measure of effectiveness in this context (Miller et al. 2006).
Fernandez and Rodrik (1991) show that individual-specific uncertainty may change the identity of the median voter and hence the projects that will be selected by majority vote. We do not address how this effect might be influenced by information about heterogeneity of risk.
Richardson and McKie (2003), quoting Fried (1969: 1418), indicate that the director of a mining company refusing to spend money to save a trapped man may be civilly and criminally liable even if she has invested large sums of money to prevent such a catastrophe. Similar legal responsibilities may exist in hospitals, e.g., to care for terminally ill patients.
This practice is critically discussed by Viscusi (2000b). Note that a simple model of public provision of safety would actually predict quite an opposite result. According to this model, it is optimal to devote more resources to prevent risks faced by the sub-population of, say, richer people. Such a practice is efficient since it rests on the assumption that the government is also able to implement lump-sum transfers to redistribute income and finance the safety provision across the population. Richer people will be taxed more heavily to compensate poorer people so that, at the optimum, marginal utilities are equal. Even without redistribution, spending more to protect rich people may be justified by considering what contract people in an original position (behind a veil of ignorance) might reach (Pratt and Zeckhauser 1996).
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Acknowledgments
The authors thank Matthew Adler, Joanne Linnerooth-Bayer, John Graham, W. Kip Viscusi, an anonymous referee, and seminar participants at the universities of Montréal, Paris (Panthéon-Sorbonne), Ohio State, Kyoto, Lille, Mannheim, and Harvard for helpful comments. James K. Hammitt thanks the Université de Toulouse (LERNA-INRA, IDEI), Institut pour une Culture de Sécurité Industrielle (ICSI), and the Région Midi-Pyrénées for hospitality and support.
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This work began while James K. Hammitt was appointed to a Pierre-de-Fermat Chaire d’Excellence at the Université de Toulouse.
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Hammitt, J.K., Treich, N. Statistical vs. identified lives in benefit-cost analysis. J Risk Uncertainty 35, 45–66 (2007). https://doi.org/10.1007/s11166-007-9015-8
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DOI: https://doi.org/10.1007/s11166-007-9015-8