Risk aversion and the value of risk to life.
ABSTRACT
The standard literature on the value of life relies on Yaari's (1965) model, which includes an implicit assumption of risk neutrality with respect to life duration. To overpass this limitation, we extend the theory to a simple variety of preferences that are not necessarily additively separable sep·a·ra·ble adj. Possible to separate: separable sheets of paper. sep . The enlargement enlargement, n an increase in size. enlargement, Dilantin, n.pr See hyperplasia, gingival, Dilantin. enlargement, idiopathic, n we propose is relevant for the evaluation of lifesaving programs: current practice, we estimate, puts too little weight on mortality risk reduction of the young. Our correction exceeds in magnitude that introduced by the switch from the notion of number of lives saved to the notion of years of life saved. INTRODUCTION Billions of dollars are spent every year on mortality reduction programs. Issues like the allocation of funds to medical research or prevention, the design of safety rules, or the wording of environmental bills raise intense debate on the relevance of the choices made by governments and their agencies. For economists, the baseline is that alternative projects should be evaluated with objective criteria to avoid pure waste or dramatic underinvestment in less popular issues. To back public decisions, some inquiry into individual valuation of life is indispensable. In practice, if we leave apart contingent valuation Contingent valuation is a surveybased economic technique for the valuation of nonmarket resources, such as environmental preservation or the impact of contamination. While these resources do give people utility, certain aspects of them do not have a market price as they are not , the analysis of the wagerisk tradeoff is the major source of estimates of people's behavior with respect to risk to life. These surveys are primarily informative about industrial workers. Because public programs affect wider populations whose characteristics may vary considerably and given that the mortality changes considered are often beyond the range experienced by the reference sample, a theoretical support for the interpretation of the data is indispensable. The choice of the structural lifecycle model that minimizes bias at estimation and extrapolation (mathematics, algorithm) extrapolation  A mathematical procedure which estimates values of a function for certain desired inputs given values for known inputs. If the desired input is outside the range of the known values this is called extrapolation, if it is inside then stages is capital. The standard approach uses additively separable lifecycle models. The intertemporal additivity assumption, which involves an implicit assumption of risk neutrality with respect to length of life, is extremely constraining (Bommier, 2006). Although this model has been severely criticized in other branches of the literature, (1) it remains an almost universal assumption for applied theory papers on the value of life. (2) In this article, we develop an alternative model, based on recursive See recursion. recursive  recursion von NeumannMorgenstern utility functions, which relaxes the additivity assumption and thereby introduces what we shall call mortality risk aversion risk aversion The tendency of investors to avoid risky investments. Thus, if two investments offer the same expected yield but have different risk characteristics, investors will choose the one with the lowest variability in returns. (MRA MRA Medical Record Administrator. MRA Magnetic resonance angiography, see MR angiography ). (3) Although this extension complicates intermediate calculations, practical difficulties are kept at a reasonable level: formulas for the value of statistical lives are almost as simple as those obtained with the standard additive model. There are therefore no technical difficulties for applying this novel approach to concrete issues. Above all, relaxing additivity warrants a significant gain in accuracy. As a proof of concept, we use empirical results on the wagerisk tradeoff to calibrate To adjust or bring into balance. Scanners, CRTs and similar peripherals may require periodic adjustment. Unlike digital devices, the electronic components within these analog devices may change from their original specification. See color calibration and tweak. both the additive and nonadditive models. While the additive model proves unable to fit the data, the generalization gen·er·al·i·za·tion n. 1. The act or an instance of generalizing. 2. A principle, a statement, or an idea having general application. proposed provides an excellent fit with reasonable estimated parameters. To emphasize the importance of accounting for MRA, we compare the benefits of (fictitious Based upon a fabrication or pretense. A fictitious name is an assumed name that differs from an individual's actual name. A fictitious action is a lawsuit brought not for the adjudication of an actual controversy between the parties but merely for the purpose of ) lifesaving policies using different methods. The magnitude of the bias caused by the additive separability sep·a·ra·ble adj. Possible to separate: separable sheets of paper. sep assumption appears to be uncomfortably big. The type of costbenefit analysis that is currently recommended for lifesaving programs is likely to be strongly biased in favor of the elderly if the decline of the VSL VSL Vessel (shipping) VSL Value of Statistical Life VSL Virtual Software Library VSL Variable Speed of Light (theoretical cosmology/physics) VSL Vector Statistical Library VSL Straight Line Velocity with age is underestimated. The correction we suggest could exceed in magnitude that introduced by the switch from the notion of number of lives saved to the notion of years of life saved. The empirical wagerisk tradeoff is used as a test of alternative theories of the lifecycle preferences. Potentially, a better understanding of lifecycle behaviors would be instructive for many applications not directly related to the value of life literature. For example, this may to help to design contributions and benefits in life insurance in order to respond more adequately to individuals' needs and thereby increase market performance. RELATED LITERATURE Most of the economic literature on the VSL is based on a particular model whose standard version (e.g., Arthur, 1981; Shepard and Zeckhauser, 1984; Rosen, 1988) relies on elements developed in Yaari (1965). Several extensions have recently been suggested. In Murphy and Topel (2006), health multiplies the instantaneous utility derived from the flow of consumption. Because health is assumed to be exogenous Exogenous Describes facts outside the control of the firm. Converse of endogenous. in the part of their paper assessing the gain from mortality risk reduction, their approach is equivalent to assuming that agents have additively separable utility functions whose (exogenous) discount function is not necessarily exponential 1. (mathematics) exponential  A function which raises some given constant (the "base") to the power of its argument. I.e. f x = b^x If no base is specified, e, the base of natural logarthims, is assumed. 2. . Hall and Jones (2007) also extend Yaari's model by introducing a health component in the utility function. Still, health being unobserved, they end up assuming in applications that it equals the inverse of the mortality rate. Though sensible, this amounts to assuming that instantaneous utility depends on mortality through a particular functional form. Ehrlich and Yin (2005) model a technology through which protection expenditures increase longevity; the authors also introduce a bequest motive A bequest motive seeks to provide an economic justification for the phenomenon of gratuitous, intergenerational transfers of wealth. In other words, to explain why people leave money behind when they die. . The above contributions extended Yaari's model in several directions but have in common that they all maintain the assumption of additive separability of preferences. It is precisely that later assumption that we shall relax. Our contribution is thus of a different nature: instead of incorporating additional variables to Yaari's model (such as health or bequest bequest: see legacy. ), we explore the potential of a less straightly structured specification. As we shall see, this provides different insights, especially on the speed at which VSL may or may not decline with age at old ages. The effect of age on the VSL is controversial. (4) Simple simulations of the original models exhibit either a decline with age or an inverse Ushape. When careful calibration is achieved to match empirical consumption profiles, the inverse Ushape is generally found, with a rather slow decline at old ages. The aforementioned theoretical extensions of Murphy and Topel (2006) and Ehrlich and Yin (2005) tend to confirm this prediction. Empirical works, however, do not converge to a consensus on the relation between age and VSL. The hedonic he·don·ic adj. 1. Of, relating to, or marked by pleasure. 2. Of or relating to hedonism or hedonists. [Greek h regressions on wages in Aldy and Viscusi (2003), Kniesner, Viscusi, and Ziliak (2006), and Viscusi and Aldy (2007) also show an inverse Ushape relation between age and VSL, with a rather rapid decline of VSL at old ages. Other recent works (Alberini et al., 2004; Smith et al., 2004; Aldy and Viscusi, 2008), based either on contingent valuation or wagerisk tradeoffs, tend to minimize the significant decline that was apparent in previous estimates. The debate seems far from being closed. The present article contributes to it by showing that when the assumption of additive separability of preferences is relaxed in order to account for MRA, then a rapid decline of VSL at old ages becomes theoretically plausible. LIFETIME PREFERENCES Basic Concepts and Notation Consider individuals of age a. We define a life as an infinite consumption profile c and a (finite) age at death T. In life (c, T), c is a continuous function mapping the age interval [[alpha], +[infinity]] into a(n unspecified) closed interval of R. Consumption at age t is denoted by [c.sub.t]. Note that consumption is not a priori a priori In epistemology, knowledge that is independent of all particular experiences, as opposed to a posteriori (or empirical) knowledge, which derives from experience. constrained to equal zero for t > T; we just assume that individuals do not care for consumption after death. Agents are assumed to be expected utility maximizers, and we denote de·note tr.v. de·not·ed, de·not·ing, de·notes 1. To mark; indicate: a frown that denoted increasing impatience. 2. [U.sub.a] (c, T) the utility associated to the life (c, T) as assessed at age a. Assuming that individuals do not care for consumption after death amounts to posing [U.sub.a] (c, T) = [U.sub.a] (c', T) for any two c, c' that are equal on [a, T]. This enables us to normalize normalize to convert a set of data by, for example, converting them to logarithms or reciprocals so that their previous nonnormal distribution is converted to a normal one. [U.sub.a] so as to have [U.sub.a](c,a) = 0, [for all]c. We work with the recursive model throughout the article, where [U.sub.a](c, T)= [[integral].sup.T.sub.a] u([c.sub.t]) exp exp abbr. 1. exponent 2. exponential ([[integral].sup.t.sub.a] [upsilon up·si·lon or yp·si·lon n. Symbol The 20th letter of the Greek alphabet. ]([c.sub.t])d[tau]) dt. (1) This timeconsistent specification first appeared in the economic literature in Uzawa (1969) in the case of immortal agents (with T replaced by infinity). In the case of agents whose life duration is finite with probability one, and with preferences defined over consumption and life duration, this recursive specification was derived from axioms This is a list of axioms as that term is understood in mathematics, by Wikipedia page. In epistemology, the word axiom is understood differently; see axiom and selfevidence. Individual axioms are almost always part of a larger axiomatic system. covering a standard notion of stationarity in Bommier (2005). Two special cases of the recursive model (1) must be highlighted. They are equally simple and the empirical part of this article will show a clear difference (in favor of the second) in their abilities to fit data. The first one is simply the additive one. Let us take [upsilon](x) = [lambda], a constant, and find [U.sup.add.sub.c] (c, T) = [[integral].sup.T.sub.a] u([c.sub.t])[e.sup.[lambda](ta)]dt, (2) where u is a wellbehaved instantaneous utility function; [lambda] is the subjective discount factor. The additive specification is by far the most popular in the economic literature. It contains an assumption of risk neutrality with respect to life duration (Bommier, 2006), which may be too restrictive when one studies endogenous endogenous /en·dog·e·nous/ (endoj´enus) produced within or caused by factors within the organism. en·dog·e·nous adj. 1. Originating or produced within an organism, tissue, or cell. choices of mortality risk and hence the value of life. The second one is the multiplicative model in which [upsilon](c) = k u(c), [for all]c, for some constant k; Equation (1) can be integrated to give [U.sup.multi.sub.a](c, T) = 1  exp (k [[integral].sup.T.sub.a] u([c.sub.t])dt) / k. (3) The term multiplicative refers to the fact that the exponentials of the instantaneous utilities multiply each other. Being a concave Concave Property that a curve is below a straight line connecting two end points. If the curve falls above the straight line, it is called convex. transformation of an additive utility function, this latter specification maintains the assumption of weak separability of preferences. Increasing k amounts to increasing risk aversion in the sense of Kihlstrom and Mirman (1974). This specification is therefore particularly appropriate to illustrate the impact of risk aversion on the value of risk to life. Uncertain Lifetime Let us consider now the case where lifetime is uncertain in order to model the tradeoff between mortality and consumption. A given consumption profile c associated with a distribution of life duration m(x) provides the expected utility [E.sub.m][U.sub.a](c) = [[integral].sup.+[infinity].sub.a] [U.sub.a](c, T)m(T)dT. (4) This expected utility will be simply denoted by E [U.sub.a] (c) in the rest of the article, when this cannot be a source of confusion. We shall assume that all the distribution functions m(x) that we will consider along the article are smooth over R+. To a distribution function m(T) corresponds the survival function [s.sup.T.sub.a] = 1  [[integral].sup.T.sub.a] m(t) dt, (5) where [s.sup.T.sub.a] is probability of being alive at age T, conditional on being alive at age a and the hazard rate of death [[mu].sub.t] = m(t) / [s.sup.T.sub.a]. Hazard rate of death and survival function are then related by [s.sup.T.sub.a] = exp ([[integral].sup.T.sub.a] [[mu].sub.t] dt). (6) Through a simple integration by parts In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. The rule arises from the product rule of differentiation. , the expected utility [E.sub.m] [U.sub.a] (c) provided in (4) can be reformulated as [E.sub.m][U.sub.a] (c) = [[integral].sup.+[infinity].sub.a] [s.sup.T.sub.a] [partial derivative] / [partial derivative]T [U.sub.a] (c, T) dT. (7) With recursive utilities as in (1) that yields E[U.sub.a] = [[integral].sup.+[infinity].sub.a] [s.sup.t.sub.a] u([c.sub.t]) exp ([[integral].sup.T.sub.a] [upsilon]([c.sub.[tau]]) d[tau]) dt, (8) a formula that we will take as starting point Noun 1. starting point  earliest limiting point terminus a quo commencement, getgo, offset, outset, showtime, starting time, beginning, start, kickoff, first  the time at which something is supposed to begin; "they got an early start"; "she knew from the for most of our computations. In order to guarantee that the above integral converges, we make two purely technical assumptions. Assumption 1: [[mu].sub.t] tends to infinity as t tends to infinity. Assumption 2: c is bounded in the long run; that is, there is an interval [[c.sub.min], [c.sub.max]] with [c.sub.min] > 0 and [c.sub.max] < + [infinity] on which c is supported after some arbitrary date. This article will not discuss the consequences, for given mortality, of recursive preferences on the intertemporal allocation of wealth. Such aspects are indeed discussed in Bommier (2005). We focus instead on issues related to endogenous mortality choices, a typical example of which being the wagerisk tradeoff. THE VALUE OF STATISTICAL LIVES A natural concept to deal with choices involving mortality changes is the marginal rate of substitution In economics, the marginal rate of substitution (MRS) is the leastfavorable rate at which an agent is willing to exchange units of one good or service for units of another. between mortality and consumption, or to get positive values its opposite: Definition 1 (VSL): The value of a statistical life at age t > a is defined by (5) VSL(c, t) [equivalent to]  ([partial derivative] E [U.sub.a] / [partial derivative] [[mu].sub.t]) / ([partial derivative] E [U.sub.a] / [partial derivative] [c.sub.t]). (9) An agent of age t is ready to give up VSL(c,t) x d[mu] x dt in consumption to save d[mu] x dt statistical lives. This is how we construe construe v. to determine the meaning of the words of a written document, statute or legal decision, based upon rules of legal interpretation as well as normal meanings. the term "value of statistical life." As discussed in Johansson (2002), various definitions of VSL have been suggested. Another popular approach is to define VSL as being the opposite of the MRS MRS  Modifiable Representation System. An integration of logic programming into Lisp. ["A Modifiable Representation System", M. Genesereth et al, HPP 8022, CS Dept Stanford U 1980]. between mortality rate and wealth. Then VSL not only depends on individuals' preferences but also on intertemporal constraints. This latter approach coincides with ours whenever intertemporal constraints are as those detailed in Appendix B. From (8), one obtains VSL(c, t) = E[U.sub.t] / u'([c.sub.t])  [upsilon]'([c.sub.t])E[U.sub.t]. (10) The following expression relates VSL to survival probabilities and discount rates. Proposition 1: For any consumption profile, the VSL is a discounted sum of life years [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ]. (11) where the discount factor [rho](c, [tau]') = RD(c, [tau]')  MRA(c, [tau]') + 1 / [[sigma].sub.[tau]'] [[??].sub.[tau]'] / [c.sub.[tau]]. The terms RD(c, t), MRA(c, t) and [[sigma].sub.t], are, respectively, the (mortality adjusted) rate of time discounting, the MRA and the intertemporal elasticity of substitution Elasticity of substitution is the elasticity of the ratio of two inputs to a production (or utility) function with respect to the ratio of their marginal products (or utilities). Mathematical definition Let the utility over consumption be given by . They are formally defined in Appendix A. Proof: See Appendix A. In the case of the recursive utility functions we consider, one gets: RD(c, t) = [upsilon]([c.sub.t])u'([c.sub.t])  [upsilon]'([c.sub.t])(u([c.sub.t])  [[mu].sub.t] E [U.sub.t]) / u'([c.sub.t])  [upsilon]'([c.sub.t])E [U.sub.t], (12) MRA(c, t) = [upsilon]'([c.sub.t])u([c.sub.t]) u'([c.sub.t]). (13) In the additive case, RD(c, t) = [lambda] and MRA(c, t) = 0 so that with [c.sub.t] = c (a constant), the expression of VSL simplifies to: VSL(c, t) = u(c) / u'(c) [[integral].sup.+[infinity].sub.t] [s.sup.[tau].sub.t] [e.sup.[lambda]([tau]t)] d[tau]. (14) This formula has been known for years and its simplicity explains its success. It is considered very convenient because if we abstract from consumption variations, VSL is proportional to a discounted sum of life years. The relation between age and VSL is then computable from a standard life table and a discount rate. This way of accounting for age was initially introduced by Moore and Viscusi (1988) and has been used by agencies like the U.S. Environmental Protection Agency (EPA EPA eicosapentaenoic acid. EPA abbr. eicosapentaenoic acid EPA, n.pr See acid, eicosapentaenoic. EPA, n. ) and the Office of Management and Budget The Office of Management and Budget (OMB), formerly the Bureau of the Budget, is an agency of the federal government that evaluates, formulates, and coordinates management procedures and program objectives within and among departments and agencies of the Executive Branch. (OMB OMB abbr. Office of Management and Budget Noun 1. OMB  the executive agency that advises the President on the federal budget Office of Management and Budget ) for costbenefit analyses even though there remains an ongoing debate about the interest of such an adjustment (EPA, 2000; Dockins et al., 2004; OMB, 1996, 2003). Proposition 1 shows that allowing for recursive preferences instead of focusing on additive preferences is associated with a minor increase in complexity. Although the generalization makes intermediate calculations more fastidious fas·tid·i·ous adj. 1. Possessing or displaying careful, meticulous attention to detail. 2. Difficult to please; exacting. 3. Having complex nutritional requirements. Used of microorganisms. , we eventually find that the benefit of saving one statistical life among individuals of a given age is also proportional to the discounted sum of years at risk. Casually, we find that accounting for consumption variations is relatively simple, whether preferences are additive or not. Nonetheless, there are two notable differences between the additive and the recursive models. First, in the recursive model the mortality adjusted rate of discount RD is not constant. Instead of using a discount function [e.sup.[lambda]([tau]t)], as in the additive case, we have to use exp([[integral].sup.[tau].sub.t] RD(c, [tau]') d[tau]'). Actually, when we calibrate the model (see the "Data Fitting" section), we find that the variations of RD remain limited until advanced ages, so this first difference can be considered as minor. The second difference is much more significant: years of life have to be discounted with the mortality adjusted rate of discount (RD) minus MRA. Consequently, the greater MRA, the faster VSL declines as a function of age. This is fairly intuitive: a risk averse Risk Averse Describes an investor who, when faced with two investments with a similar expected return (but different risks), will prefer the one with the lower risk. Notes: A risk averse person dislikes risk. agent is willing to pay more to avoid the chance of a major loss. In terms of mortality, a major loss would be an early death. The additive model, which disregards MRA, may underestimate the speed at which VSL declines with age. The bias is estimated and confirmed in the "Data Fitting" section. WageRisk TradeOff The revealed preferences argument can be invoked to show how occupational choices provide information about utility functions. Assume that at all ages an individual has to choose between jobs that differ with respect to wage and instantaneous fatality fa·tal·i·ty n. 1. A death resulting from an accident or disaster. 2. One that is killed as a result of such an occurrence. risk. Labor income can be used for consumption or savings. Let [[mu].sup.0.sub.t] be the exogenous baseline mortality rate at age t. For an extra instantaneous mortality [[mu].sub.t] (total mortality being [[mu].sup.0.sub.t] + [[mu].sub.t]), the wage is denoted by w(t, [[mu].sub.t]). The marginal risk premium [partial derivative] w / [partial derivative] [mu] is denoted [w.sub.[mu]]. Proposition 2: Under fairly general conditions, detailed in Appendix B, the marginal risk premium equals the VSL: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15) Proof: See Appendix B. The observation of the wagerisk tradeoff reveals VSL and makes the calibration of the utility function possible. Compared to similar results, the strength of the proposition is that it is established under quite general conditions that does not require the existence of complete markets. In particular, the results holds even if individuals face borrowing constraints, which is often viewed as one of the main reasons for which individuals have low consumption at young ages. DATA FITTING Method A hedonic regression In economics, hedonic regression, or more generally hedonic demand theory, is a method of estimating demand or prices. It decomposes the item being researched into its constituent characteristics, and obtains estimates of the value of each characteristic. fits the envelope of the choices made by the workers in the sample (Viscusi and Aldy, 2003). Because the envelope is tangent tangent, in mathematics. 1 In geometry, the tangent to a circle or sphere is a straight line that intersects the circle or sphere in one and only one point. to individual indifference curves, the prediction based on the hedonic regression for a vector of individual characteristics can be interpreted as the VSL for the corresponding worker. We base the calculations on this fundamental observation. Several recent contributions estimated the relation between age and VSL from hedonic regressions and provided contrasting results (see the discussion in the "Related Literature" section). As an illustration, we use the result of one of them (Aldy and Viscusi, 2003, henceforth From this time forward. The term henceforth, when used in a legal document, statute, or other legal instrument, indicates that something will commence from the present time to the future, to the exclusion of the past. A&V) to calibrate our model. By doing so, we do not claim to provide undisputable estimates of the true preference parameters because they are conditional on the particular empirical ageVSL relationship we employ. Nevertheless, we comply with the objective of the article: showing that relaxing additivity parsimoniously can significantly improve the ability of the structural model to fit the data. (6) The consequences for policy recommendations are far from trivial. [FIGURE 1 OMITTED] We use the parameters given by A&V in their Table 4: [w.sup.AV.sub.[mu]] (t) = 1.92 x [10.sup.7] + 1.88 x [10.sup.6]t  4.54 x [10.sup.4] [t.sup.2] + 335.24 [t.sup.3], (16) where t [member of] [18, 62], expresses the individual's age in years, and [w.sub.[mu]] the yearly wage in 1996 dollars. The calibration strategy we pursue involves searching the parameters of the recursive model that best fit Equation (16). In order to calibrate the model, we also need the agespecific consumption profile [c.sup.*], which is not available in the data set used by A&V. The optimal consumption profile cannot be deduced from the theoretical model without specification of the intertemporal budget constraints, on which we have limited knowledge. Rather than posing specific constraints, we approximated [c.sup.*] with a smoothed version of the age specific individual consumption profile reported in Lee and Tuljapurkar (1997) (see Figure 1 for the original estimates and the smoothed profile that we use). (7) As we use consumption data from a different source, we search the best fit for the [20, 60] age interval instead of [18, 62]. Goodness of Fit Goodness of fit means how well a statistical model fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e. The first question that we may address is whether we can reproduce (16) with the standard additive model (namely, [upsilon] = [lambda] = Constant and u(c) = [c.sup.1[gamma]] / 1[gamma] [u.sub.0] for some constants [u.sub.0] and [gamma]). The answer is positive, but with very implausible im·plau·si·ble adj. Difficult to believe; not plausible. im·plausi·bil parameters. Indeed the distance minimizing discount rate is 8.1 percent, which explains 94 percent of the agerelated variance in equation (16). Had we constrained the RD to be greater than or equal to 3 percent (to approach values that are considered as reasonable), we would have at best explained 58 percent of the agerelated variance. At this point it is legitimate to wonder whether this poor fit is due to the fact that we only considered isoelastic instantaneous utility functions, or more fundamentally to the additive separability. We relax each of these assumptions in turn. If we simply require u to be increasing and concave rather than isoelastic, we can obviously improve the fit. By considering rates of discount greater than or equal to 3 percent, we can now explain 79 percent of the agerelated variance. The gain in explanatory power might seem significant, but in fact, it is quite disappointing when we recall that we added an infinity of degrees of freedom to the model (u is now nonparametric). This control stage adds weight to our view that structure (additive/nonadditive) matters much more that specification (isoelastic/nonparametric), which we now illustrate. In fact, keeping u isoelastic but opting for the recursive form appears to be a much more efficient way to improve the predictive power The predictive power of a scientific theory refers to its ability to generate testable predictions. Theories with strong predictive power are highly valued, because the predictions can often encourage the falsification of the theory. of the model. We explored the case where u(c) [c.sup.1[gamma]] / 1[gamma] [u.sub.0] [u.sub.0] and [upsilon] = [lambda] + [beta]u; compared to the standard additive model ([beta] = 0), this structure requires only one additional degree of freedom. Moreover it encompasses the multiplicative model (obtained when [lambda] = 0) described in the "Basic Concepts and Notation" section, which has the same number of degrees of freedom as the standard additive model. In Figure 2, we report the minimum distance (the sum of squares) between the theoretical predictions and the empirical estimates, the survival weighted average RD being constrained to take particular values given on the horizontal axis. The results obtained with the additive and the multiplicative models are also reported. The distance on the vertical axis has been normalized so that the distance between the empirical VSL and its mean equals 1. Opting for the recursive model dramatically increases the capacity of the theory to reproduce empirical VSL. Even if we constrain con·strain tr.v. con·strained, con·strain·ing, con·strains 1. To compel by physical, moral, or circumstantial force; oblige: felt constrained to object. See Synonyms at force. 2. the mortality adjusted RD to take reasonable positive values we still obtain an excellent fit. We can constrain the survivalweighted average RD to take any value between 1 and 5 percent, and still explain more than 96 percent of agerelated variability of the wagerisk tradeoff. This is much better than the additive model, which only explains from 49 to 66 percent thereof. Table 1 reports the model's performance (variance explained and parameters) for a range of discount factors. Figure 3 illustrates the fits obtained when the average mortalityadjusted RD is constrained to equal 3 percent in both models. Interestingly enough, one can see from Table 1 or Figure 2 that when RD is constrained to plausible positive values, the multiplicative model does a much better job than the additive one, with the same number of degrees of freedom. Therefore even if one is reluctant to increase the complexity of the model, a significant gain is obtained. [FIGURE 2 OMITTED] Evaluated Parameters For the recursive model, as apparent in Figure 2, the curve representing the distance between predicted and actual values exhibits a flat shape around the minimum; in practice this means that the combination of parameters that optimally fit the data is difficult to state. The observation of the relation between age and VSL may not suffice to calibrate all the parameters of the model with precision. This is not surprising given the theoretical results provided in the "The Value of Statistical Lives" section. From Equation (11) we know that what matters for determining the variations of [w.sub.[mu]], along the life cycle is mainly the combination of two elements: the mortalityadjusted RD minus MRA. If consumption were constant along the life cycle, we would expect empirical observation of VSL to be informative about the difference between RD and MRA, and not about each of them separately. Though in our case consumption is not constant, which in principle should solve the identification problem, our estimates suffer from the same kind of indeterminacy in·de·ter·mi·na·cy n. The state or quality of being indeterminate. Noun 1. indeterminacy  the quality of being vague and poorly defined indefiniteness, indefinity, indeterminateness, indetermination . For each value of RD we find the best value of MRA, but it is hard to tell what is the best pair of RD and MRA. [FIGURE 3 OMITTED] Ultimately, to discriminate more sharply between the several likely possibilities, we should investigate data beyond the wagerisk tradeoff. One possibility would be to look at consumptionsmoothing behavior in order to estimate RD from another source. Yet, our conclusions regarding the values of RD would be contingent on Adj. 1. contingent on  determined by conditions or circumstances that follow; "arms sales contingent on the approval of congress" contingent upon, dependant on, dependant upon, dependent on, dependent upon, depending on, contingent strong assumptions regarding the credit market and its imperfections, whereas these are not necessary for our analysis. Moreover, a single database that would be sufficiently rich to inform on both the wagerisk tradeoff and consumption smoothing seems out of reach. We preferred therefore to consider plausible range of values of RD rather than trying to evaluate a single value. Results thereafter are systematically reported for RD taking values 1, 3, and 5 percent. The last row of Table 1 provides the estimated values for the rate of discounting for life years (RDLY), which is formally defined in Appendix A. This RD provides information on how people would be willing to trade off survival probabilities at different ages. This is a crucial element when estimating the welfare benefits of mortality risk reductions occurring at different ages, as will be shown in the "Welfare Evaluation" section. Practical Consequences From the last two rows of Table 1, it is possible to get a first idea about the bias generated by the additive assumption. While the additive model constrains MRA to be absent, the recursive model gives estimates of MRA that range from 8.7 to 9.6 percent. In other words Adv. 1. in other words  otherwise stated; "in other words, we are broke" put differently , when people discount consumption with rates of 1, 3, and 5 percent, life years in VSL should be discounted with rates of 7.7, 5.9, or 4.6 percent, respectively. The additive model, which imposes the same rate of discount for consumption as for life years, is likely to cause a huge bias. In order to tell whether this is likely to lead to a major shift in policy recommendations, one may look at RDLY, which, as is explained below, is the rate of discount to be used for estimating the welfare equivalent of a statistical life. While the additive model constrains RDLY to equal the rate of discount, the more general model shows values of RDLY that exceed those of rate of discount by several percentage points. This means that the additive model puts too much relative weight on mortality risk reduction at old ages. Let us now explore how large the bias can be in practice. WELFARE EVALUATION Objective In order to evaluate the social benefits of mortality risk reductions, a well defined social objective is required. The utilitarian approach axiomatized by Blackorby, Bossert, and Donaldson (1997) involves assuming that the social planner In welfare economics, a social planner is a decisionmaker who attempts to achieve the best result for all parties involved. In neoclassical welfare economics, this means the maximization of a social welfare function. maximizes a stationary weighted sum of individuals' utilities at birth. The social welfare function is then given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17) where the sum is taken over all individuals, [[lambda].sub.S] is the social discount rate, [b.sub.i] is the birth year of individual i, and [U.sup.i.sub.0] is his expected utility at birth. We use Arthur's (1981) terminology. The welfare equivalent of a statistical life for individual i is defined by WE(c,t) [equivalent to]  [partial derivative][U.sup.i.sub.0] / [partial derivative][[mu].sub.t], (18) where c and [mu] are individual i's consumption and mortality. WE has a fairly simple expression in the general recursive case: (8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (19) where RDLY is the rate of time disounting for life years whose formal definition is provided in Appendix A. Like the VSL, the welfare equivalent is a discounted sum of life years. With the additive model RDLY = RD, thus, it is correct to use the discount rate inferred from empirical studies Empirical studies in social sciences are when the research ends are based on evidence and not just theory. This is done to comply with the scientific method that asserts the objective discovery of knowledge based on verifiable facts of evidence. on consumption smoothing to estimate the welfare equivalent of a statistical life. With the recursive model, RDLY is typically greater than the rate of time preferences estimated in studies on consumption smoothing. Thus, omission of MRA generates a prooldage bias in the welfare evaluation of mortality risk reduction. Methods We describe now the five evaluation methods for a program that we will compare in the following subsection subsection Noun any of the smaller parts into which a section may be divided Noun 1. subsection  a section of a section; a part of a part; i.e. . Method 1. The number of lives saved. Though there is no economic support for this method, it has been frequently used. Actually, in their most recent guidelines, EPA and OMB still recommend to use an ageindependent VSL for costbenefit analyses. In the absence of other source of heterogeneity het·er·o·ge·ne·i·ty n. The quality or state of being heterogeneous. heterogeneity the state of being heterogeneous. , this involves quantifying the benefits of reducing mortality by counting the number of lives saved, just as with this method (Dockins et al., 2004; OMB, 2003). Method 2. Utilitarianism utilitarianism (y'tĭlĭtr`ēənĭzəm, y with the additive utility function. The benefit of a program is measured by the social welfare function (17). Individuals are assumed to have the same additive utility function, with a rate of time preference of 1, 3, and 5 percent, the other parameters being drawn from the "Data Fitting" section. The social rate of discount is taken equal to the individual rate of time preference. Method 2'. Aggregate WTP WTP Web Tools Platform (Eclipse) WTP Willingness To Pay WTP Water Treatment Plant WTP We the People WTP Waste Treatment Plant WTP Wireless Transaction Protocol WTP Winnie The Pooh WTP Washington Transportation Plan with additive utility function. Assumptions on individuals are the same as for method 2. The benefit of a program is now evaluated by the sum of the individuals' willingness to pay Willingness to pay (WTP) generally refers to the value of a good to a person as what they are willing to pay, sacrifice or exchange for it. See also
Method 3. Utilitarianism with the recursive utility function. This method is similar to method 2, with the recursive model as estimated in the "Data Fitting" section. The average survivalweighted RD and the social rate of discount are constrained to 1, 3, and 5 percent. Method 3'. Aggregate WTP with the recursive utility function. This method is similar to method 2, with the recursive model as estimated in the "Data Fitting" section. The average survival weighted RD and the social rate of discount are constrained to 1, 3, and 5 percent. We could also define two additional methods that parallel methods 2 and 2' but make use of the multiplicative model. However, as it happens that the recursive model estimated in the "Data Fitting" section is practically multiplicative, the results are very close to those obtained with methods 3 and 3. In principle, method 2' (respectively 3') amounts to method 2 (respectively 3) only if one presumes that the marginal social value of consumption is equal across people of different ages; in other words, if redistribution is perfect. In practice, because the distribution of wealth is far from ideal with respect to the social welfare function, it has been argued that aggregate willingness to pay cannot be considered as a relevant policy indicator. The issue is not specific to lifesaving programs but general to any costbenefit analysis (see, e.g., the discussion in Blackorby and Donaldson, 1990). In the case of mortality reduction, Pratt and Zeckhauser (1996) stressed that because of the strong heterogeneity in mortality rates, aggregating individual willingness to pay may actually be a particularly misleading indicator. More recently Baker et al. (2008) discussed the possible justifications for relying on method 2', which they describe as "fairly restrictive." Despite these shortcomings, method 2' remains the most commonly used when intending to account for the agerelated heterogenity of VSL. Application To show the magnitude of distortion in the evaluation of safety programs, we consider two fictitious programs that are assumed to have the same cost: one that decreases mortality rates proportionally and another that decreases mortality rates uniformly. For example, we could think of air quality alerts (9) on the one hand and of earthquake surveillance on the other. We denote these hypothetical interventions as A and B. Policy A is characterized by a proportional reduction of mortality rates [[mu].sub.t] [right arrow] (1  [[epsilon].sub.A]) [[mu].sub.t], (20) and policy B by a uniform reduction of mortality rates [[mu].sub.t] [right arrow] [[mu].sub.t]  [[epsilon].sub.B], (21) where [[epsilon].sub.A] and [[epsilon].sub.B] are positive constants. We take the age structure of the population and the baseline mortality rates observed in the United States United States, officially United States of America, republic (2005 est. pop. 295,734,000), 3,539,227 sq mi (9,166,598 sq km), North America. The United States is the world's third largest country in population and the fourth largest country in area. in 1999. We also assume that A saves twice as many (statistical) lives as B. Policy A is mostly effective for older people (and babies) while policy B saves lives uniformly. Figure 4 shows the age distribution of lives saved (it has been scaled so that A saves 2,000 statistical lives while B saves only 1,000). We assume that the consumption profile is [c.sup.*] (see the "Method" section), for ages above 20. For ages below 20, and especially for babies and children, the assumption that preferences are independent of age becomes problematic. The low levels of consumption that are typically observed in the very first years of life would then imply very high marginal utility marginal utility In economics, the additional satisfaction or benefit (utility) that a consumer derives from buying an additional unit of a commodity or service. The law of diminishing utility implies that utility or benefit is inversely related to the number of units of consumption, and therefore very low values of statistical lives. This is hard to buy. To circumvent cir·cum·vent tr.v. cir·cum·vent·ed, cir·cum·vent·ing, cir·cum·vents 1. To surround (an enemy, for example); enclose or entrap. 2. To go around; bypass: circumvented the city. this difficulty, we maintain the assumption that preferences are independent of age and assume that consumption is the same between birth and 20. Of course this option is arbitrary, one of its merits being that most of the difference between A and B is based on effects on the adults, for which estimates are more reliable. [FIGURE 4 OMITTED] Intuitively, it is not very clear whether A or B should be preferred. On the one hand A saves more lives. On the other hand B saves younger people, who still have many years of life before them. We use the above five types of benefit evaluation. The results are summarized in Table 2. By assumption, A is twice as efficient as B from the viewpoint of method 1. The additive model in methods 2 and 2' provides an ageadjusted value of a statistical life, so the conclusion is different. Methods 2 and 2 predict that the benefits of A and B are of about the same size. The fact that B saves fewer lives than A is approximately compensated by the fact that it saves younger people. The question now is whether this age adjustment and this conclusion are correct. Methods 3 and 3' suggest that they are not. With the recursive model, the benefits of B appear to be much greater than those of A. The correction related to the introduction of MRA is anything but negligible. Passing from the additive model to the nonadditive one is a bigger step than passing from the traditional method (number of lives saved) to the additive model. EPA guidelines advise performing sensitivity analysis by calculating the results of both methods 1 and 2. As the results of method 2 are known to depend on the RD, about which there is no general agreement, they advise reporting the results for different rates. We report results for RDs lying in the 15 percent interval, which is generally considered as providing a reasonable confidence interval confidence interval, n a statistical device used to determine the range within which an acceptable datum would fall. Confidence intervals are usually expressed in percentages, typically 95% or 99%. . Unfortunately, the additive model is so restrictive that the truth may be way outside this interval. The methods currently used by EPA and OMB (and indirectly by policymakers) are likely to be significantly distorted in favor of the old. CONCLUSION Most economists would agree that predicting saving behavior under the assumption of risk neutrality would make little sense. They would also vehemently criticize a fund manager who decides to "optimize" investment under the assumption that members are risk neutral. However, the economic literature on the value of a statistical life has endorsed a similar choice. It focused on a specification that paid little attention to the fact that mortality makes our life akin to an extraordinary lottery. Is it reasonable to assume that individuals are risk neutral with respect to length of life? And to evaluate life saving programs under this assumption? These questions have been addressed in this article. On the theoretical side, the story is clear. MRA makes individual willingness to pay for mortality risk reduction decline more rapidly with age. Although intermediate calculations are sometimes fastidious, we eventually found that accounting for MRA is fairly simple. Just like with the standard additive model, estimating VSL and welfare benefits associated to mortality risk reduction simply involves computing weighted sums of lifeyears saved. The rates of discount to be used must however account for both time preferences and MRA. The key issue is therefore to estimate MRA. The difficulty of the task should not be underestimated. Since Arrow's (1971) and Pratt's (1964) seminal contributions, about 40 years have passed and a number of empirical studies tried to measure risk aversion with respect to lotteries on wealth. No consensus has emerged. There is no reason to believe that preferences with respect to lotteries on the length of life will be easier to assess. It would be excessively optimistic op·ti·mist n. 1. One who usually expects a favorable outcome. 2. A believer in philosophical optimism. op to expect that a single study could provide a robust estimate of MRA. This should be rather seen as a longterm objective that will probably require the collection of specific data. However, in order to clarify the ideas at stake, we used results from a recent empirical study on the relation between VSL and age to estimate plausible values of MRA. The theoretical extension neatly improved the quality of fit. We found that this index of risk aversion is likely to be positive and greater than the rate of time discounting. In other words, accounting for MRA may even be more important than accounting for time preferences. The contrast between our findings and the dominant economic approach is striking. While the notion of time preferences has been pointed out as being a critical element to estimate the value of a statistical life, the standard method simply rules out MRA. It seems that "the paradigm of optimizing a simple functional form" (to take Rubinstein's, 2003, words) has led economists to ignore a key ingredient of individual preferences. The consequence is that costbenefit analysis produced for the allocation of public money across lifesaving programs is likely to be strongly distorted. APPENDIX A: PROOF OF PROPOSITION 1 Definitions and Properties Along the article we make use of recursive utility functions which implies that agents' expected utility is given by: E [U.sub.a] = [[integral].sup.+[infinity].sub.a] [s.sup.t.sub.a]u([c.sub.t]) exp ( [[integral].sup.t.sub.a] v([c.sub.[tau]])[d.sub.[tau]] dt. (A1) As we depart from the additive case recalled below, the meanings of u and v are not straightforward and it is no longer immediate to relate these functions to properties that could be inferred from empirical observation. It is, for example, incorrect to interpret the integral [[integral].sup.t.sub.a] v([c.sub.[tau]]) [d.sub.[tau]] as an "accumulated rate of time preference," as was done in Uzawa (1969). The rate of time discounting is a welldefined marginalist concept that can be defined independently of the structure of the utility function, as in Epstein (1987), and that needs to be computed with the general recursive specification. In presence of mortality it is however useful to adjust the definition as follows: Definition 2 (RD): The mortality adjusted rate of time discounting at age t is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A2) In the absence of mortality at age t (i.e., if [s.sup.t.sub.a] were constant around t), RD(c, t) would correspond to the standard definition of rate of time discounting in continuous time. The correction 1/[S.sup.t.sub.a] simply neutralizes the uncertainty effect that mortality risk has on consumption (consumption is contingent on survival). With the recursive model, calculations yield RD(c, t) = v([c.sub.t])u'([c.sub.t])  v'([c.sub.t])(u([c.sub.t])  [[mu].sub.t] E [U.sub.t])/u'([c.sub.t])  v'([c.sub.t])E [U.sub.t] (A3) where E [U.sub.t] is defined in (8). Note that although the definition of RD(c, t) is conditional on a, the current age of the individual, RD(c, t) only depends on consumption and mortality at ages greater than or equal to t, a natural consequence of the recursive structure of the utility functions. Remark also that with additive utilities, that is, when v(.) = [lambda], this equation simplifies to RD = [lambda], which is consistent with the fact that the parameter X is generally introduced as the "rate of time preference" in studies that used the additive specification. But in the more general recursive setting the rate of time discounting, which is a key element when looking at optimal consumption smoothing, is endogenous and has a complex expression. A similar complication occurs when looking at the intertemporal elasticity of substitution, another key determinant determinant, a polynomial expression that is inherent in the entries of a square matrix. The size n of the square matrix, as determined from the number of entries in any row or column, is called the order of the determinant. of the marginal tradeoffs involved in consumption smoothing. In continuous time the intertemporal elasticity of substitution can be defined as the limit of the direct elasticity of substitution (as defined in McFadden, 1963) between consumptions at two different dates whose time distance tends to zero. Definition 3 (IES): The intertemporal elasticity of substitution at age t, which we denote [[sigma].sub.t], is defined by: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A4) where [[delta].sub.t] is the Dirac delta function The Dirac delta or Dirac's delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x . (10) With the recursive model, [[sigma].sub.t] = 1/[c.sub.t] u'([c.sub.t])  v'([c.sub.t])E [U.sub.t]/u" ([c.sub.t])  v'([c.sub.t])E [U.sub.t] (A5) When preferences are additive or multiplicative, this formula simplifies to [[sigma].sub.t] = u'([c.sub.t])/[C.sub.t]u"([c.sub.t]). Another interesting concept of time discounting simply expresses how people are willing to trade off survival probabilities at different ages. Definition 4 (RDLY): The rate of time discounting for life years is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A6) With the recursive model, RDLY(c, t) = v([c.sub.t]). (A7) Finally, we introduce a new concept, which is at the center of our analysis, and requires more comments and clarifications. Definition 5 (MRA): Mortality risk aversion is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A8) This coefficient is unaffected by an affine transformation (mathematics) affine transformation  A linear transformation followed by a translation. Given a matrix M and a vector v, A(x) = Mx + v is a typical affine transformation. of [U.sub.a], meaning that it represents a fundamental characteristic of individual preferences, independent of the specific representation that was chosen. If the marginal utility of life extension is decreasing in past consumption (i.e., if [[partial derivative].sup.2][U.sub.a](c, T)/[partial derivative][c.sub.t] [partial derivative]T] < 0 for all T > t) then MRA(c, t) [greater than or equal to] 0. The terminology "MRA" emphasizes that MRA(c, t) corresponds to a coefficient of risk aversion with respect to length duration along particular (and generally not constant) consumption paths. Indeed, writing [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A9) one obtains [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A10) The first term on the righthand side (RHS RHS Royal Horticultural Society RHS Right Hand Side RHS Rural Housing Service RHS Rickards High School (Tallahassee, FL) RHS Red Hat Society RHS Ridgewood High School (New Jersey) ) is recognizable as a coefficient of risk aversion with respect to life duration. When consumption profiles such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A11) are considered, MRA(c, t) and the ArrowPratt coefficient are equal. Consumption profiles that comply with (A11) are characterized by the fact that the marginal rate of substitution between additional life years and consumption just before death is independent of the age at death. In particular, (All) amounts to having [u([c.sub.t])e.sup.[lambda]t] constant in the additive model, and [c.sub.t] is constant with the multiplicative model. In both cases, this can be interpreted as having a constant flow of felicity (Bommier, 2006). The decomposition decomposition /de·com·po·si·tion/ (dekom?pahzish´un) the separation of compound bodies into their constituent principles. de·com·po·si·tion n. 1. into two terms is important for understanding the origin of MRA(c, t), but quite remarkably, with the recursive model any consumption profile leads to the following simple expression MRA(c, t) = v'([c.sub.t])u([c.sub.t])/u'([c.sub.t]), (A12) which depends only on consumption at time t. Remark that MRA(c, t) > (<) 0 if v(x) is increasing (decreasing) and is null A character that is all 0 bits. Also written as "NUL," it is the first character in the ASCII and EBCDIC data codes. In hex, it displays and prints as 00; in decimal, it may appear as a single zero in a chart of codes, but displays and prints as a blank space. with the additive model. Proof of Proposition 1 In the proof, VSL stands for VSL(c, t) and RD for RD(c, t). We start from (10) and we use the fact that d E [U.sub.t]/dt = ([[mu].sub.t] + v([c.sub.t]))E [U.sub.t]  u([c.sub.t]), (A13) to compute [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A14) Using (A3) and (A5), we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A15) From (10), we obtain E [U.sub.t] = u'([c.sub.t])VSL/1 + v'([c.sub.t])VSL, (A16) thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A17) Combining (A17) with (A15) yields [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A18) that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A19) We show now in three steps that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A20) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A21) Step 1. It is easy to see that the RHS of (A20), if it converges, is a solution to the ODE ode, elaborate and stately lyric poem of some length. The ode dates back to the Greek choral songs that were sung and danced at public events and celebrations. (A19). Step 2. Remark that E [U.sub.t] > 0. Indeed, a natural assumption is that the marginal value Marginal value is a term widely used in economics, to refer to the change in economic value associated with a unit change in output, consumption or some other economic choice variable. of life years, which is proportional to u, is positive, and u > 0 implies E [U.sub.t] > 0. Given Assumptions 1 and 2, E [U.sub.t] tends to zero as t tends to infinity. This and (10) imply that VSL [right arrow] 0 as t [right arrow] + [infinity]. We can also conclude from this, (A3) and E [U.sub.t] > 0, that RD is bounded below in the long run. Consequently, p(c, t) [right arrow] + [infinity] as t [right arrow] + [infinity]. This implies that the RHS of (A20) [right arrow] 0 as t [right arrow] + [infinity]. VSL and the RHS of (A20) have therefore the same limit when t [right arrow] + [infinity]. Step 3. The ODE (A19) being linear, if we denote by y the difference between the VSL and the RHS of (A20), we have y' = [rho](c, t)y. (A22) Given that [rho](c, t) [right arrow] + [infinity] as t [right arrow] + [infinity], y goes to infinity when t [right arrow] + [infinity] if it is not null. This fact, combined with the result on limits (step 2), proves that (A20) is true. Q.E.D. APPENDIX B: PROOF OF PROPOSITION 2 We denote by k = [([k.sub.t]).sub.t] [greater than or equal to] the agespecific saving profile defined by [k.sub.t] [equivalent to] w(t, [[mu],sub.t])  [C.sub.t]. (B1) For our purpose, we do not need to fully specify the lifetime budget constraints that are related to the intertemporal markets and their possible imperfections. We will simply assume that these constraints (possibly infinitely many) only bear on the function k and that each of them is Volterra differentiable dif·fer·en·tia·ble adj. 1. That can be differentiated: differentiable species. 2. Mathematics Possessing a derivative. . We denote the set of constraints by K. We may think of different kinds of constraints. With nonstorable commodities and no intertemporal markets, [k.sub.t] = 0 for all t. Another possibility would be a single constraint of the form [[integral].sup.[infinity].sub.0] [k.sub.t][h.sub.t][e.sup.rt]dt = 0 with r being the rate of interest and h = [([h.sub.t]).sub.t [greater than or equal to] 0] an exogenous function. This includes the important case of intertemporal markets, in particular, life annuities. (11) We could also imagine that the constraints K have the form [[integral].sup.t.sub.0] [k.sub.t][e.sup.r[tau]] [d.sub.[tau]] [greater than or equal to] 0 for all t. That would be the case in a world where there is no annuity market, no borrowing, and a rate of return on savings equal to r. More complex market imperfections can be thought of. Undoubtedly, allowing any kind of constraints on k leaves us with a fairly high degree of generality gen·er·al·i·ty n. pl. gen·er·al·i·ties 1. The state or quality of being general. 2. An observation or principle having general application; a generalization. 3. , although certain cases are not covered not covered Health care adjective Referring to a procedure, test or other health service to which a policy holder or insurance beneficiary is not entitled under the terms of the policy or payment system–eg, Medicare. Cf Covered. (e.g., nonlinear A system in which the output is not a uniform relationship to the input. nonlinear  (Scientific computation) A property of a system whose output is not proportional to its input. consumption taxes). Using (6) and (8), we rewrite the lifetime utility function of an agent of age a as E [U.sub.a](c,u) = [[integral].sup.+[infinity].sub.a] u([c.sub.t])exp ( [[integral].sup.t.sub.a] ([[mu].sub.[tau]] + ([[mu].sup.0.sub.[tau]] + v([C.sub.[tau]])) d[tau]. (B2) A rational agent solves the maximization program [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B3) The derivative [w.sub.[mu]]((t, [mu]) = [partial derivative]w(t,[mu])/[partial derivative][mu]) is the "wagerisk tradeoff." Even without an explicit formulation of the constraints K, we can show that at the optimal choice the wage risk tradeoff and the VSL are equal. Indeed, differentiating (B1), for all t, [tau], we have ([partial derivative]/[partial derivative][[mu].sub.t] + [w.sub.[mu]]((t, [[mu].sub.t]) [partial derivative]/[partial derivative](c.sub.t]) [k.sub.[tau]] = 0. (B4) Let [c.sup.*] and [[mu],sup.*] denote the optimal consumption and mortality paths. As we assumed that all constraints can be written as functions of k, the firstorder conditions ensure that for all t, utility cannot be improved without violating the constraints. Thus, because of (B4), it must be the case that at the optimum ([partial derivative]/[partial derivative][[mu].sub.t] + [w.sub.[mu]]((t, [[mu].sub.t]) [partial derivative]/[partial derivative](c.sub.t]) E [U.sub.a]] = 0. (B5) Therefore: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B6) Q.E.D. REFERENCES Alberini, A., M. Cropper CROPPER, contracts. One who, having no interest in the land, works it in consideration of receiving a portion of the crop for his labor. 2 Rawle, R. 12. , A. Krupnick, and N. Simon, 2004, Does the Value of a Statistical Life Vary With Age and Health Status? Evidence From the U.S. and Canada, Journal of Environmental Economics and Management, 48(1): 769792. Aldy, J. E., and W. K. 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Aldy, 2003, The Value of a Statistical Life: A Critical Review of Market Estimates Throughout the World, Journal of Risk and Uncertainty, 27(1): 576. Viscusi, W. K., and J. E. Aldy, 2007, Labor Market labor market A place where labor is exchanged for wages; an LM is defined by geography, education and technical expertise, occupation, licensure or certification requirements, and job experience Estimates of the Senior Discount for the Value of Statistical Life, Journal of Environmental Economics and Management, 53(3): 377392. Yaari, M. E., 1965, Uncertain Lifetime, Life Insurance, and the Theory of the Consumer? Review of Economic Studies, 32(2): 137150. (1) Even when mortality is not an issue, theoretical arguments underlined unpleasant consequences of the additive separability assumption (e.g., Richard, 1975; Deaton, 1974, 1992; Epstein and Zin, 1991). Moreover, the additive model's inability to fit intertemporal choice Intertemporal choice is the study of the relative value people assign to two or more payoffs at different points in time. This relationship is usually simplified to today and some future date. has been repeatedly underlined by empirical studies (Hayashi, 1985; Muellbauer, 1988; Browning, 1991; Carrasco, Labeaga, and LopezSalido, 2005). (2) See, for example, the recent contributions of Murphy and Topel (2006) and Hall and Jones (2007). (3) It should be clear that the nonadditive model we use introduces a variety of risk aversion toward life length that is to be distinguished from financial risk aversion as in Eeckhoudt and Hammitt (2004) and Kaplow (2005). These article discuss the impact of the curvature curvature Measure of the rate of change of direction of a curved line or surface at any point. In general, it is the reciprocal of the radius of the circle or sphere of best fit to the curve or surface at that point. of the instantaneous utility function on the value of statistical life (VSL). This issue matters particularly for understanding the income elasticity of the VSL documented in Kaplow (2005). (4) See the discussion in Aldy and Viscusi (2008) and the references to press articles therein. (5) Because of our continuous time modeling, we use Volterra derivatives. They measure utility changes when consumption (or mortality) varies by an infinitesimal value during an infinitesimally in·fin·i·tes·i·mal adj. 1. Immeasurably or incalculably minute. 2. Mathematics Capable of having values approaching zero as a limit. n. 1. short lapse of time. For example [partial derivative] [U.sub.a] / [partial derivative] [[mu].sub.t] d[mu] dt gives the change in [U.sub.a] when mortality rates increase by d[mu] during dt around t. A first application of Volterra derivatives to economics is Ryder and Heal (1973). (6) Using one of the regressions in Aldy and Viscusi (2008) is an alternative. The qualitative results they show are similar (invertedUshaped relationship between age and VSL with similar rates of growth), but they suggest an overall higher level of the VSL. A consensus on the ideal database and estimates is premature, and different readers may have different views, as we experienced. (7) Lee and Tuljapurkar (1997) is one the few studies that provide individual (not household) agespecific consumption profiles. (8) From (6), it follows that [partial derivative][s.sup.[tau].sub.a] / [partial derivative][[mu].sub.t] = 0 if [tau] < t, and [partial derivative][s.sup.[tau].sub.a] / [partial derivative][[mu].sub.t] = [s.sup.[tau].sub.a] if [tau] [greater than or equal to] t. Differentiating (8) then gives (19). (9) Assuming a marginal impact of air pollution proportional to baseline mortality seems reasonable to epidemiologists (Pope et al., 1995). (10) The presence of the Dirac delta function is a purely technical point related to continuous time modeling. This function appears when second order derivatives are involved. See also footnote 5. (11) To be more specific, exogenously priced life annuities are considered. Endogenous prices would mean that prices change as the consumer changes his mortality, for example, via activity choice. This case is not included here; if h were equal to the (endogenous) survival function, as with perfect intertemporal markets, the VSL at age a would be reduced by the wealth held at age a. Quantitatively speaking, the correction is minor (average wealth is typically much lower than the VSL). DOI (Digital Object Identifier) A method of applying a persistent name to documents, publications and other resources on the Internet rather than using a URL, which can change over time. : 10.1111/j.15396975.2010.01390.x Antoine Bommier is at ETHZurich. Bertrand Villeneuve is at Universite ParisDauphine (LEDA) and CREST (Laboratoire de Finance Assurance). The authors can be contacted via email: abommier@ethz.ch and bertrand.villeneuve@dauphine dau·phine n. The wife of a dauphin. [French, feminine of dauphin; see dauphin.] .fr, respectively. Antoine Bommier acknowledges financial support from Swiss Re Swiss Re is the world’s largest reinsurer, now that it has acquired GE Insurance Solutions (Ligi 2006). Founded in 1863, Swiss Re now operates in more than 30 countries. General Electric owns 8.9% of the firm. . TABLE 1 Calibration and Performance Additive ([beta] = 0) RD Model 1% 3% 5% Var. explained 66% 58% 49% [??] 0.72 0.22 0.011 [??] 1% 3% 5% [u.sub.0]/([[bar.c].sup.1[gamma]]/ 1.23 7.51 13.7 1[gamma]) (b) Average MRA 0 0 0 Average RDLY 1% 3% 5% Recursive Average RD Model 1% 3% 5% Var. explained 97% 96% 96% [??] 5.25 4.15 3.25 [??] 0.12% 0.04% 0.07% [u.sub.0]/([[bar.c].sup.1[gamma]]/ 6.22 5.46 4.51 1[gamma]) (b) Average MRA 8.7% 8.9% 9.6% Average RDLY 7.9% 8.4% 9.3% Multiplicative (lambda] = 0] Average RD Model 1% 3% 5% Var. explained 90% 95% 96% [??] 3.01 3.70 3.77 [??] 0 0 0 [u.sub.0]/([[bar.c].sup.1[gamma]]/ 6.37 5.52 4.47 1[gamma]) (b) Average MRA 5.2% 8.3% 10.4% Average RDLY 5.1% 7.9% 9.7% (a) Elasticity of substitution constrained to be nonnegative. (b) [bar.c]: (survival weighted) average consumption. TABLE 2 Benefits of B/Benefits of A Discount Rate Method for Benefit Evaluation 1% 3% 5% 1. Number of lives saved 0.5 0.5 0.5 2. Utilitarianism with additive utility 1.34 1.11 0.97 3. Utilitarianism with recursive utility 3.88 3.23 2.64 2. Aggregate WTP with additive utility 1.18 1.06 0.97 3. Aggregate WTP with recursive utility 1.72 1.95 1.75 

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