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Measuring how risk tradeoffs adjust with income

Journal of Risk and Uncertainty volume 40, pages 33–55 (2010)Cite this article

Abstract

Efforts to reconcile inconsistencies between theory and estimates of the income elasticity of the value of a statistical life (IEVSL) overlook important restrictions implied by a more complete description of the individual choice problem. We develop a more general model of the IEVSL that reconciles some of the observed discrepancies. Our framework describes how exogenous income shocks, such as unexpected medical expenditures, may affect labor supply decisions which in turn influence both the coefficient of relative risk aversion and the IEVSL. The presence of a consumption commitment, such as a home mortgage, also alters this labor supply adjustment. We use data from the Health and Retirement Study to explore the responsiveness of labor force exit decisions to spousal health shocks and the role of a home mortgage as a constraint on this response.

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Notes

  1. For example, see Lisa Heinzerling’s indictment of all of benefit cost analysis based on her dissatisfaction with the important role VSL estimates play for policy (Ackerman and Heizerling 2004). Heinzerling is the current Associate Administrator of EPA’s Office of Policy, Economics and Innovation.

  2. We are aware of four methodologies used to estimate the IEVSL: 1) meta-analyses of hedonic wage studies (see Mrozek and Taylor 2002; Viscusi and Aldy 2003; Bowland and Beghin 2001); 2) stated preference studies (see Hammitt and Graham 1999; Hammitt and Zhou 2000; Mitchell and Carson 1986); 3) comparisons of VSL estimates at different points in time for a single country (see Hammitt et al. 2003; Costa and Kahn 2004); 4) cross-country comparisons of VSL estimates (see Hammitt et al. 2003).

  3. In contrast, with respect to contemporaneous differences in income and wealth, EPA’s Science Advisory Board (U.S. EPA 2007) notes that “…the SAB Panel recognizes our current empirical abilities may not permit these adjustments. One is to adjust for differences in real income and wealth between study populations. Since the value of reducing mortality risk increases with income and wealth, differences in these factors are expected to yield differences in estimated valuation. However, the appropriate magnitude of adjustment is not clear, because of uncertainty about the value(s) of the income elasticity and very little empirical evidence concerning the relationship between wealth and mortality valuation” (p. D7-8). For additional discussion of considerations in using VSL for benefit analyses of rules see U.S. EPA (2000).

  4. Murphy and Topel do not report their implied income elasticity. We computed the implied IEVSL based on their assumptions about the preference specification and model parameters. They assume an elasticity of intertemporal substitution equal to 0.8 and a consumer surplus per unit of the composite of consumption and leisure at age 50 equal to about 2.11. These assumptions imply an income elasticity for the VSL of 1.57. See pp. 882–886 of Murphy and Topel (2006).

  5. Since prices are normalized to unity and x is a perfect substitute for c, the Kaplow formulation is equivalent to assuming an indirect utility function in describing preferences. The distinction between wealth and income for static models is largely one of terminology. The models we discuss do not have an inter-temporal dimension so there is no saving and asset accumulation.

  6. In fact, in a working paper version of his 2005 paper, Kaplow (2003) reports the measure given here as (1.6a).

  7. In an independent analysis, Smith et al. (2003) use labor supply elasticity measures for specific preference functions to measure VSL.

  8. Our assumption of p′(l) > 0 implies that an individual faces a relatively lower probability of death in non-work related activities so that substituting an hour of leisure with an hour of work increases the risk of death. Ruhm (2000) suggests that mortality rates are pro-cyclical. Specifically, he finds that an increase in a state’s unemployment rate is associated with a decrease in that state’s mortality rate. While this result is based on aggregate data and therefore is not specific to a particular working environment, the direction of the estimated effect is consistent with our assumption.

  9. A comparable result holds in Kaplow’s model if we allow x to affect utility and the marginal utility of consumption (c) to vary with x.

  10. Note our definitions of γ I,l fixed and R I,l fixed are constructed to be consistent with Kaplow (2003) in that these measures are defined with respect to expected utility, V. Conventional practice would define R in terms of the curvature properties of the utility function. With no labor supply adjustment, comparable measures defined with respect to utility, u, are identical. Thus, the distinction is irrelevant for Model I with l fixed.

  11. One potential explanation for this complementarity could be health related. Indeed, following research by Hall and Jones (2007), we recently argued (Evans and Smith 2008) that improved health increases complementarity between consumption and leisure (implying substitution between consumption and labor). Thus, for a given degree of risk aversion, those individuals in good health are likely to have higher income elasticities for their risk tradeoffs then those in poor health.

  12. Our derivation of the CRR with variable labor supply parallels a related measure developed by Chetty (2006). However, our expression for the CRR is different from Chetty’s because his model does not consider risk as a function of labor supply. Thus in his model, the CRR defined in terms of utility is identical to the measure we define with respect to expected utility (as in Kaplow).

  13. See the Appendix for a proof of this result. It is important to recognize that this conclusion depends on defining the CRR in terms of expected utility as Kaplow has proposed. Arrow’s (1971) overview of the theory underlying the definition of the coefficient of risk aversion describes it as a feature of the utility function not the expected utility function. We have adopted the Kaplow convention in order to facilitate direct comparisons between our results and those of Kaplow and others. Chetty and Szeidl (2007) also define a CRR in terms of expected utility (p. 844).

  14. See Chetty and Szeidl (2007) pp. 845–846 for the derivation in a closely related model and further discussion.

  15. This is the point of arguments for using preference calibration in other types of benefits transfer.

  16. The HRS is a national panel study intended to be representative of individuals who fell in the age cohort of 51 to 61 years old in 1992 (wave 1) and their spouses. The HRS (Health and Retirement Study) is sponsored by the National Institute of Aging (grant number NIA U01AG09740) and conducted by the University of Michigan. We rely on the RAND Corporation’s cleaned version of the HRS.

  17. See Chetty (2006) for citations of additional studies that use indirect methods to estimate risk coefficients.

  18. Subjects faced two sequences of hypothetical survey questions, beginning with the Barsky et. al. (1997) job choice questions. After answering the Barsky sequence, subjects faced a second set of hypothetical gamble questions, identical to the Barsky sequence except for the context. The second set of questions related to tradeoffs involving a hypothetical inheritance, rather than a job. The authors changed the format of the original Barsky questions by asking multiple variations to the same respondent. They varied the odds of the job and inheritance outcomes and asked each version of the question to each respondent. In addition, since subjects were students they were told to use the income of their parents in evaluating the job prospects. Both of these changes are important departures from the format used in asking these questions of HRS respondents as well as from the format used in the Smith and Mansfield (2009) work. See Anderson and Mellor (2009) for a more detailed discussion of the results with respect to the inheritance questions.

  19. The text of the survey question is: “Now I want to ask how your health affects paid work activities. Do you have any impairment or health problem that limits the kind or amount of paid work you can do?”

  20. Chronic health conditions include diabetes or high blood sugar, lung disease such as chronic bronchitis or emphysema, arthritis or rheumatism, and high blood pressure or hypertension. We define an acute health condition as a stroke or transient ischemic attack, cancer or a malignant tumor of any kind except skin cancer, heart attack, coronary heart disease, angina, congestive heart failure, or other heart problem. These definitions are broadly consistent with Coile (2004).

  21. All reported dollar figures are nominal.

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Authors and Affiliations

  1. The Robert Day School of Economics and Finance, Claremont McKenna College, Claremont, CA, USA

    Mary F. Evans

  2. Department of Economics, Arizona State University, Tempe, AZ, USA

    V. Kerry Smith

  3. Resources for the Future, Washington, DC, USA

    V. Kerry Smith

  4. National Bureau of Economic Research, Cambridge, MA, USA

    V. Kerry Smith

Authors
  1. Mary F. Evans
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  2. V. Kerry Smith
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Corresponding author

Correspondence to Mary F. Evans.

Additional information

Thanks are due to Jon Valentine and Christina Stoddard for their help in the preparing and editing this manuscript. Josh Abbott, Kelly Maguire, J.R. DeShazo, W. Kip Viscusi, and participants at the 2008 Southern Economic Association Meetings and the Vanderbilt Law School Heterogeneity of the Value of Statistical Life Conference, and seminar attendees at Claremont McKenna College provided helpful comments on earlier drafts of this paper. The usual disclaimer applies.

The U.S. Environmental Protection Agency (EPA) provided funding for this research under STAR grant RD-83159502-0. The research has not been subjected to EPA review and therefore does not necessarily reflect the views of the Agency, and no official endorsement should be inferred.

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Evans, M.F., Smith, V.K. Measuring how risk tradeoffs adjust with income. J Risk Uncertain 40, 33–55 (2010). https://doi.org/10.1007/s11166-009-9085-x

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Keywords

  • Value of a statistical life
  • Risk aversion
  • Consumption commitment
  • Labor supply

JEL Classifications

  • Q51
  • D01
  • J26