The Cost of Uncertain Life Span

Abstract

A considerable amount of uncertainty surrounds the length of human life. The standard deviation in adult life span is about 15 years in the U.S., and theory and evidence suggest it is costly. I calibrate a utility-theoretic model of preferences over length of life and show that one fewer year in standard deviation is worth about half a mean life year. Differences in the standard deviation exacerbate cross-sectional differences in life expectancy between the U.S. and other industrialized countries, between rich and poor countries, and among poor countries. Accounting for the cost of life-span variance also appears to amplify recently discovered patterns of convergence in world average human well-being. This is partly for methodological reasons and partly because unconditional variance in human length of life, primarily the component due to infant mortality, has exhibited even more convergence than life expectancy.

1 Introduction

Over the last two centuries, the demographic transition has produced vast reductions in human mortality rates and roughly a doubling in the average length of life among advanced countries (Lee, 2003). Since the 1950s, period life expectancy at birth, e₀, has exhibited much convergence among developed nations, and the average across those countries has increased at a roughly linear annual rate (White, 2002). The maximum female life expectancy at birth has exhibited similar trends over a much longer period (Oeppen and Vaupel, 2002). In the developing world, progress against mortality was delayed until the twentieth century, and in some regions it has been at least partially reversed by the HIV/AIDS epidemic or the fall of communism (Wilson, 2001; Deaton, 2004; Cutler, Deaton and Lleras-Muney, 2006). But there is also much evidence of convergence in demographic conditions, particularly when the focus is period life expectancy at birth (Wilson, 2001; Becker, Philipson and Soares, 2005).

The mean of the distribution of length of life is arguably the most informative, but the shape of the distribution is also interesting and, as I argue in this paper, economically important. The three panels in Figure 1 depict three representations of the same underlying period mortality data for the U.S. in 1900 and 2000, between which life expectancy at birth grew from 47.7 to 76.7 years. Probability distributions of ages at death appear in panel A, survivorship probabilities are in panel B, and age-specific log mortality rates are shown in panel C.¹ All three panels reveal a large amount of historical shape-shifting, but the dynamics in the distribution of length of life, shown in panel A, are the primary subject of this paper. The distribution moved rightward, as one would expect given the increase of 29 years in e₀ over this period. But a very large part of this improvement was due to the massive reduction in infant mortality that is visible in the graph. For individuals surviving infancy and childhood, past say the age of 10, the mean length of adult life, which I denote M₁₀, rose by a more modest 17 years, from 60.6 to 77.4, while the modal adult age at death increased from 72 to 85.² As the old-age mode moved rightward, the amount of variance around that mode fell notably. The standard deviation in length of life above age 10, or S₁₀, fell 5.3 years, from 20.0 to 14.7. These standard deviations in adult life span, a measure introduced by Edwards and Tuljapurkar (2005) that will become useful later, are superimposed in Figure 1A as horizontal bars extending on either side of the adult mean.³

Figure 1. Distributions of life span, survivorship, and log mortality in the U.S. in 1900 and 2000.

Data sources are period life tables from Bell and Miller (2005) and author’s calculations, and they are unweighted averages of sex-specific life-table entries. In panel A, the density of deaths at age 0 was 0.1328 in 1900 and is left off the chart for expositional clarity. The horizontal bars in panel A depict one standard deviation above age 10, S₁₀, on either side of the mean above age 10, M₁₀.

Once we examine the shape of the distribution rather than just its mean, the simple story of broad cross-country convergence in mortality becomes more complicated. Edwards and Tuljapurkar (2005) reveal large differences among industrialized countries since 1960 in levels and trends in the variance of adult age at death as measured by S₁₀. While e₀ may have converged among these countries (White, 2002), the variance accounts for an increasing amount of mortality divergence between them. Separately, Edwards (2011) examines length of life distributions in a broad cross section of rich and poor countries in 1970 and 2000, reporting evidence of declining within-country inequality, or the average level of S₁₀, but also indications of rising between-country inequality, especially in length of adult life. Infant mortality has exhibited much convergence across countries and has substantially narrowed differentials in e₀, the unconditional average length of human life (Wilson, 2001; Moser, Shkolnikov and Leon, 2005). But that narrative omits several interesting dynamics: relatively broad-based reductions in S₁₀ among many developing countries owing to the demographic transition, a new divergence in S₁₀ among industrialized countries, and widening differences in length of adult life across development groups.

What, if any, are the implications of these less obvious trends in mortality for human well-being? Are there welfare effects associated with S₁₀ and infant mortality above and beyond the benefits reflected in e₀? In this paper, I show that the answer to the second question is yes, and that valuing longevity or life extension based on the mean alone, without accounting for the shape of the longevity distribution, typically leads to downward bias in estimates of the benefits of mortality decline. This is because the basic model of intertemporal choice over a finite lifetime, developed by Yaari (1965), implies that individuals are averse to risk or inequality over length of life under standard parameter assumptions. The basic intuition behind this result proceeds as follows. Consider a consumer whose subjective discount rate equals the real interest rate. If perfect annuities are available, the individual will plan constant consumption and thus constant flow utility in each period of life. Expected lifetime utility then equals a constant times the survivorship function shown in Figure 1B, a downward sloping concave curve. By Jensen’s Inequality, a mean-preserving spread in length of life reduces expected lifetime utility in such a scenario.

Risk and inequality are conceptually related in choice theory (Atkinson, 1970), but they are not strictly the same in this context. The total variance in length of life in the population, as shown in Figure 1A, consists of between-group and within-group inequality. The former includes well-known socioeconomic gradients in life expectancy (Kitagawa and Hauser, 1973), while the latter would include true uncertainty faced by individuals who know their socioeconomic status and other characteristics that are important for longevity. But while differences between groups are certainly important, within-group inequality in length of life tends to be large (Edwards and Tuljapurkar, 2005). In this paper I will parsimoniously treat variance in length of life as though it were equivalent to uncertainty, but in reality different groups face different levels of uncertainty.⁴

For typical parameter values, the standard intertemporal model of Yaari (1965) suggests that variance in adult life span is costly, and that infant mortality is extremely costly, considerably more so than its effect on e₀ alone implies. In industrialized countries, each year in adult standard deviation is currently worth about half a year in the average; that is, an individual would agree to give up half a year in mean life span to obtain a standard deviation that is one year lower. This result lends new salience to the persistent differences we see across advanced countries today in their levels of S₁₀, which can be as large as 2 years. My results suggest that reductions in S₁₀ in the U.S. were worth perhaps 9 percent of the total value of gains against mortality prior to 1950 but only 5 percent since then, because progress against S₁₀ has slowed significantly. I also find that explicitly accounting for infant mortality and S₁₀ amplifies the global convergence results of Becker, Philipson and Soares (2005), who priced the value of gains in income and e₀ between 1960 and 2000. Further, it might partially reverse some of the findings of Jones and Klenow (2010), who build on the work by Becker, Philipson and Soares by including consumption inequality and leisure in their measure of consumption-equivalent welfare. Jones and Klenow also account only for e₀ and report reduced welfare in Africa and Latin America between 1980 and 2000.⁵

More broadly, my results suggest that because variance in length of life is costly, medical decision making should probably take it into account in addition to average length of life. Increased interest in end-of-life care, which must balance patients’ wishes, quality of life, and the potential reward of risky medical procedures, is a manifestation of a similar perspective (Kwok et al., 2011; Kelley, 2011). For populations, reducing aggregate health inequality takes on additional salience given the relatively high costs of variance in life span compared to the benefits of longevity increases.

The structure of the paper is as follows. In section 2, I contextualize my work by discussing previous research on choice under uncertain survivorship, the value of life, and bequests. In addition to Lucas (1987), Becker, Philipson and Soares (2005), and Jones and Klenow (2010), my work is also related to recent research on preferences over periods of life by Bommier (2006, 2008), and Bommier and Villeneuve (2011). In section 3, I examine a simple utility-theoretic model of the cost of uncertain length of adult life based on the framework of Yaari (1965), and I recover a convenient analytic solution for the cost of S₁₀ that compares favorably to solutions from numerical simulations. I also discuss the less tractable cost of infant mortality, which proves to be very important in assessing the value of longevity gains during the demographic transition. In section 4, I explore the implications of these insights for decomposing the value of mortality declines and assessing convergence in average human well-being. The final section offers some concluding remarks.

2 Background

2.1 The value of a statistical life and mortality reduction

This paper examines the welfare cost of variance in length of life, which depends critically on how consumers value life years. A robust literature explores the valuation of life and the willingness to pay for mortality reduction using theoretical and empirical perspectives (Rosen, 1988; Viscusi, 1993; Tolley, Kenkel and Fabian, 1994; Johansson, 2002; Viscusi and Aldy, 2003; Aldy and Viscusi, 2008). These papers typically assume that agents maximize expected utility, and studies recover empirical estimates of the value of a statistical life (VSL) by observing wage differentials associated with mortality risks that vary across occupations. Resulting estimates of the VSL life vary widely, as reported by Viscusi and Aldy (2003) in their meta-analysis, with a range of $4 to $9 million in the U.S. They also appear to vary systematically across time and space, probably because income and consumption are changing, but the literature does not agree on precisely how the VSL varies. Viscusi and Aldy (2003) report an income elasticity of the VSL of about 0.5 based on international cross sections, while Costa and Kahn (2004) recover an elasticity of about 1.6 in longitudinal U.S. data.

In theory, the value of a statistical life should be a function of the value of remaining life years. Researchers have combined this principle with assumptions about the shape of intertemporal preferences, typically the framework of additive separability introduced by Yaari (1965) and discussed below, in order to recover measures of the value of a statistical life year (VSLY) from estimates of the VSL derived from hedonic wage regressions. In its most simple and common form, this approach posits that the age slope of the VSLY is either flat, if the discount rate is zero, or downward sloping if it is positive. But it can also be hump-shaped when calibrated with life-cycle patterns of consumption and earnings (Murphy and Topel, 2006; Aldy and Viscusi, 2008). Using such an approach and a posited income elasticity of the VSL, several studies measure the consumption-equivalent value of long-term improvements in longevity (Cutler and Richardson, 1997; Nordhaus, 2003; Murphy and Topel, 2006), typically finding that the value of mortality decline rivals the value of economic growth. Others have combined the value of longevity extension with growth in income or consumption in order to assess convergence in total human well-being (Becker, Philipson and Soares, 2005; Jones and Klenow, 2010). Most of these efforts utilize realistic survivorship probabilities like those shown in panel B of Figure 1, and thus are accounting for changes in the shape of the life-span distribution such as depicted in panel A. But studies of international well-being typically cannot due to data limitations (Becker, Philipson and Soares, 2005; Jones and Klenow, 2010). As a result, these studies measure life span using e₀ alone and none of the higher moments.

2.2 Preferences over length of life

To be sure, higher moments do not matter if there is risk neutrality over length of life, and the degree of that risk aversion is not trivially clear and could be zero or even negative. This topic is particularly salient for the medical profession, where decisions regarding life and death and the costs, benefits, and riskiness of procedures must be weighed by physicians and patients alike. Assessing net benefits requires assumptions about preferences over health states in different future periods, and the medical literature recognizes that the degree of risk aversion over remaining years of life will affect this calculation (Ried, 1998; Bleichrodt and Quiggin, 1999). Measuring the concavity of preferences over life span is neither a common nor straightforward activity, but the consensus view based on empirical research in the last two decades seems to be that individuals are risk averse. Bleichrodt and Johannesson (1997) report that four out of five empirical studies directly examining this question reject risk neutrality over life years in favor of risk aversion (McNeil, Weichselbaum and Pauker, 1978; Stiggelbout et al., 1994; Verhoef, Haan and van Daal, 1994; Maas and Wakker, 1994). These investigations typically ask respondents, sampled either from medical treatment programs or from the community, to assess the desirability of various probabilistic scenarios regarding survival in perfect health versus death. For many but not all respondents, certainty equivalents are concave in life years, implying risk aversion.⁶

Do standard models of preferences based on the work of Yaari (1965) imply risk aversion over length of life? In the next section, I show that the answer is yes, given certain reasonable conditions. But my approach is not the first nor the last word on the topic, which is an interesting new area of research. In several recent contributions, Bommier (2006, 2008) and Bommier and Villeneuve (2011) develop and use a generalized version of the additively separable preferences introduced by Yaari (1965). By specifying a recursive utility function, they introduce an extra preference parameter that governs the curvature of expected utility over length of life in addition to the time discount rate, which already provides curvature under reasonable assumptions, as I will show in the next section. In contrast, Bommier (2006) argues that Yaari’s framework even with discounting implies risk neutrality over length of life, but his definition of risk neutrality is zero utility curvature along a special life-cycle consumption path that may or may not be representative.⁷ Because preferences over length of life depend on consumption in a mechanical way, Bommier defines risk attitudes around a “constant flow of satisfaction consumption profile,” in which discounted flow utility, which is also the marginal utility of an additional year of life, is constant over time. Under such circumstances, standard measures of risk aversion in length of life, which depend on the second derivative of utility, must be zero because the marginal utility of additional years of life is assumed to be constant. Viewed this way, Bommier’s risk neutrality result seems tautological and far from general.

But it is clear that this new literature fills an obvious gap and raises some important questions. The contributions of Bommier (2006, 2008) and Bommier and Villeneuve (2011) are similar in methodology and motivation to those of Kreps and Porteus (1978) and Epstein and Zin (1989 and Epstein and Zin (1991), who use recursive models to separate preferences over financial risk from intertemporal substitution, and their success in fitting empirical patterns may be similar. Bommier and Villeneuve (2011) find that additional risk aversion over length of life than is implied by time discounting alone provides a better fit to the empirical age trajectory in the value of a statistical life reported by Aldy and Viscusi (2003), which has an inverted-U shape. But Murphy and Topel (2006), Hall and Jones (2007) and others reproduce similar trajectories by modeling health in the utility function, or by allowing for quality adjustment of life years by a means other than injecting more risk aversion. As Bommier and Villeneuve remark, the debate over the degree of risk aversion in length of life, like the ongoing debate over risk aversion in financial decisions, is far from settled but is worth engaging. If risk aversion is indeed greater than implied by time discounting alone, then my estimates will understate the true cost of life-span uncertainty. But Nordhaus (2003), Becker, Philipson and Soares (2005), Murphy and Topel (2006), Hall and Jones (2007); Jones and Klenow (2010), and others in the sub-field on valuing longevity extension all assume additive separability as posited by Yaari (1965). My contribution follows in their footsteps and highlights the independent and interesting cost of the variance in the widely used baseline model; it makes no progress in answering similarly interesting questions about nonseparable preferences in theory and practice.

The inverted-U-shaped age trajectory of the VSL and the VSLY in recent empirical studies is both controversial and independently interesting for this study. As Aldy and Viscusi (2008) discuss, the result contrasts with the basic intuition that newborns should value their lives most highly because they have the most years left to live. The simple model I propose in the next section incorporates that dynamic as a reasonable baseline. But if the true VSLY follows an inverted-U shape, then the cost of variance in adult length of life may be overstated in the simple model. This is because a mean-preserving spread increases old life years at the expense of young life years, which in the simple model are more dear because of discounting. But the reason why the VSLY rises before declining with age is arguably important in a welfare study. If the accumulation and depreciation of a broad version of human capital including knowledge, skills, and health is the driving force, then measuring a heightened loss of welfare with death at middle age seems appropriate. If instead the VSLY follows the hump-shaped earnings age profile because of binding liquidity constraints, as Aldy and Viscusi (2008) seem to suspect, then it is less clear whether a welfare study ought to price a market failure that probably reduces well-being though ideally it should not. My approach values young human life the most, a reasonable but not uncontroversial baseline.

2.3 Annuities and bequests

Even if individuals are inherently risk averse over life span, they may be able to diversify life-span risk through contingent claims. Indeed, observed heterogeneity in stated preferences may reflect differential access to or use of markets for contingent claims. Annuities and life insurance are two examples of market instruments that diversify risks associated with uncertain life spans, while bequests are a nonmarket instrument that could also conceivably hedge against life-span uncertainty. In this paper, I show that while full annuitization removes all consumption risk associated with uncertain life span, and therefore improves welfare, annuities cannot remove the utility risk. Even under full annuitization, life-span uncertainty is costly. Whether life insurance helps offset S₁₀ is more difficult to say, since it affects individual utility only through the bequest motive. That is, actuarially fair life insurance is like a precommitted bequest, and to assess its benefits we must understand bequests.

If individuals are altruistic, it is conceivable that bequests could hedge lifespan risk relatively well if the bequest motive is strong. With utility deriving solely from consumption rather than from other aspects of living, then the disutility of early death could in theory be balanced by increased utility among survivors under altruistic bequests. Similarly, the additional utility deriving from late death could be offset by the impact of diminished bequests. But it is difficult to see why such fully altruistic individuals with utility only from consumption would care about any moment of life span, including the average. Probably in part for this reason, the value-of-life literature typically ignores bequest motives altogether (Chang, 1991; Johansson, 2002).

In any event, the literature on bequests is mixed with regard to the strength of the motive, with some research indicating they are generally not intended (Hurd, 1987, 1989) and other research suggesting otherwise (Kopczuk and Lupton, 2007). A prevailing view in economics is that bequests are simply unused precautionary savings (Dynan, Skinner and Zeldes, 2004). Findings in the medical literature of risk aversion over length of life certainly suggest that bequest motives are either not universal or not strong enough to hedge against the risk of death. Another perspective on bequests is that they can be strategic, a quid pro quo promised in exchange for elderly care (Bernheim, Shleifer and Summers, 1985). Leaving aside the problem that living too long risks depleting bequeathable wealth as well as requiring informal care, we might interpret strategic bequests as merely another form of annuitization, if they are set aside up front.

In the next section, I explore the theoretical cost of uncertain life span in the standard intertemporal model of Yaari (1965), with special attention paid to the role of annuitization. I find that individuals who discount their future well-being in the standard way should be risk averse over life span, even when they can fully annuitize. Consistent with the value-of-life literature, I do not model a bequest motive explicitly. I express the cost of variance in life span in terms of the value of the mean, which probably lessens the sensitivity of my results to the assumption of no intended bequests. A bequest motive would work to reduce both the value of the mean length of life and the cost of variance simultaneously.

3 Modeling the cost of life-span uncertainty

In this section, I show how the basic intertemporal model of Yaari (1965) with time discounting and additive separability implies there is a large welfare cost associated with adult life-span uncertainty for reasonable parameter values. There is also a very high cost attached to infant mortality.

3.1 Setup of the model

Consider an expected utility maximizer at time t = 0 with an implicit rate of time discounting equal to δ and no bequest motive.⁸ Following Yaari (1965), lifetime expected utility is the discounted sum of period utilities drawn from consumption, u(c(t)), weighted by the force of time discounting, e^−δt, and the probability that the individual is alive, ℓ(t):

$$EU={\int }_0^{\infty }u(c(t))e^{-\delta t}\ell (t)dt.$$ (1)

The survivorship function, ℓ(t), shown in panel B of Figure 1, is one minus the cumulative density function of life span, which is shown in panel A.

The individual has a financial endowment W that can be consumed or saved at a fixed market rate of interest, r. For simplicity, there is no labor, education, capital, or financial risk in this model.⁹ In a market without annuities, the budget constraint requires the individual to finance the present value of all future consumption out of wealth:

$$W={\int }_0^{\infty }c(t)e^{-rt}dt.$$ (2)

Under this budget constraint, the model will produce unintended bequests whose size varies inversely with length of life. If instead actuarially fair annuities are available, the budget constraint takes a different form:

$$W={\int }_0^{\infty }c(t)e^{-rt}\ell (t)dt.$$ (3)

An annuity pays off in future periods only if the individual is alive. This allows the buyer to finance future consumption more cheaply than through saving, but at the expense of unintended bequests.¹⁰

The individual maximizes equation (1) subject either to (2) or (3) depending on whether annuities are available. The Euler condition that describes intertemporal choice is

$$u^{\prime }(c(t+1))=u^{\prime }(c(t))e^{\delta -r+D·q(t)},$$ (4)

where D is an indicator of the lack of annuities, and it multiplies q(t), the mortality rate.¹¹ Under full annuitization, D = 0 and mortality cancels out of equation (4) because survivorship appears in both the objective and the constraint, and consumption will be flat through age if δ = r. But without annuities, D = 1, and the mortality rate, q(t), which typically increases exponentially through age, will pull marginal utility higher and consumption lower over age through a type of precautionary saving (Hubbard and Judd, 1987), producing a consumption trajectory that looks like the survivorship curve.

My analytical strategy is to assume full annuitization, normally distributed life spans, and power utility in order to find a convenient closed-form solution for the cost of life-span variance. Later, I employ numerical simulations to assess its fit. Because flow utility is a function of consumption, which typically depends on age and thus the life-span distribution, it is convenient to assume a specific functional form. Power utility is a standard assumption and a reasonable baseline.

3.2 Full annuitization and normally distributed life spans

Let u(·) take the familiar form of power utility with constant relative risk aversion over consumption plus a constant utility shifter K.¹²

$$u(c(t))=\frac{c{(t)}^{1-\gamma }}{1-\gamma }+K.$$ (5)

Under full annuitization, survivorship weights appear both in lifetime utility and in the budget constraint. Thus mortality cancels out of the Euler equation (4), which simplifies to

$$c(t)=c(0)e^{[(r-\delta )/\gamma ]t},$$ (6)

where c(0) is a function of wealth, the parameters, the annuitization indicator, D, and the moments of life span. I will proceed by assuming that c(0) remains constant over small changes in the moments of life span, an assumption that I later relax in numerical simulations.¹³

With the simplified Euler equation, I can completely solve the model by reformulating the budget constraint through a change in the order of integration:

$$W=E\left[{\int }_0^{T}c(t)e^{-rt}dt\right],$$ (7)

where T is a random variable, the realization of life span, and E is the expectations operator. The survivorship weights are now implicit in the expectation. For a given distribution of T, I could use equation (6) to solve the integral and then find c(0) by taking the expectation.¹⁴ But I am primarily interested in the relative price of variance in life-span in this model, which is governed by its relative marginal utility. To proceed, I change the order of integration in lifetime expected utility, equation (1):

$$EU=E\left[{\int }_0^{T}u(c(t))e^{-\delta t}dt\right],$$ (8)

where as before, T is a random variable. With the power utility formulation in equation (5) and the consumption function in equation (6), expected lifetime utility under full annuitization is

$$EU=E\left[\frac{c{(0)}^{1-\gamma }}{(1-\gamma )\widehat{\delta }}\left(1-e^{-\widehat{\delta }T}\right)+\frac{K}{\delta }(1-e^{-\delta T})\right],$$ (9)

where

$$\widehat{\delta }=\delta -\frac{1-\gamma }{\gamma }(r-\delta ).$$ (10)

When r is close to δ, I have δ̂ ≈ δ; and δ̂ = δ when either r = δ or γ → 1.

3.2.1 Risk aversion over life span

Examination of equation (9) reveals that individuals are risk averse over life span in this model if the rate of time discounting, δ, is positive and not too different from the real interest rate, r.¹⁵ The Arrow-Pratt coefficient of absolute risk aversion over T, −EU_TT /EU_T, is approximately equal to the rate of time discounting, δ, and exactly equal when r = δ. That is, absolute risk aversion in life span is roughly constant under reasonable assumptions.

One would expect individuals who are risk averse over life span to be hurt by uncertainty in life span, and this is more easily seen if I assume that length of life is normally distributed:¹⁶ T ~ N(M, S²). In that case, expected lifetime utility in this model is

$$EU=\frac{c{(0)}^{1-\gamma }}{(1-\gamma )\widehat{\delta }}\left[1-e^{-\widehat{\delta }M+{\widehat{\delta }}^2{S}^2/2}\right]+\frac{K}{\delta }\left[1-e^{-\delta M+{\delta }^2{S}^2/2}\right],$$ (11)

by virtue of the properties of lognormality. Expected lifetime utility is a decreasing function of S, provided that δ > 0, r and δ are not too dissimilar, and period utility is positive.¹⁷

3.2.2 Pricing the variance in life span

It is convenient to recover the price p_S of a standard deviation in life span, S, in terms of the mean, M, by constructing the ratio of their marginal lifetime utilities:

$$p_S=\frac{\partial EU/\partial S}{\partial EU/\partial M}.$$ (12)

When the utility shifter K is nonzero, this ratio is mathematically complicated.¹⁸ A first-order Taylor expansion around K = 0 reveals

$$p_S\approx -\widehat{\delta }S+{\left(\frac{c{(0)}^{1-\gamma }}{(1-\gamma )}\right)}^{-1}\left(\widehat{\delta }-\delta \right)e^{(\widehat{\delta }-\delta )M+({\delta }^2-{\widehat{\delta }}^2)S^2/2}SK.$$ (13)

When K = 0 or when δ̂ = δ, this reduces exactly to a parsimonious relationship:

$$p_S=-\widehat{\delta }S.$$ (14)

The price of a standard deviation in life span, p_S, is negative when δ̂ > 0 because S is a bad. An individual who faces higher variance in life span must be compensated by a longer mean life span. In addition, p_S increases linearly with the level of S, with the magnitude of the slope equal to δ̂, approximately the rate of time discounting. That is, the costliness of a standard deviation in life span in terms of the mean rises with the level of uncertainty. Mathematically speaking, this follows directly from the lognormality of lifetime utility. Intuitively, the willingness to bear additional risk falls with the level of risk because its marginal disutility rises.¹⁹

It is also convenient to write out the marginal utilities in the special case of K = 0:

$$p_S=\frac{\partial EU/\partial S}{\partial EU/\partial M}=\frac{-{\widehat{\delta }}^{2}S}{\widehat{\delta }}\frac{\frac{c{(0)}^{1-\gamma }}{(1-\gamma )\widehat{\delta }}{e}^{-\widehat{\delta }M+{\widehat{\delta }}^2{S}^2/2}}{\frac{c{(0)}^{1-\gamma }}{(1-\gamma )\widehat{\delta }}{e}^{-\widehat{\delta }M+{\widehat{\delta }}^2{S}^2/2}}=\frac{-{\widehat{\delta }}^{2}S}{\widehat{\delta }}=-\widehat{\delta }S.$$ (15)

The end result is the same as in the Taylor expansion, but this formulation reveals a subtle point that will become important in section 4. The marginal utility of mean life years, the denominator, is decreasing in the mean but increasing in the variance, so that improvements in life expectancy of a given size are more valuable when variance is high. This somewhat counterintuitive result again derives from discounting. Gaining an additional year at the mean under full certainty is not worth as much as gaining an additional year in expectation when there are higher survivorship weights on earlier, less heavily discounted years, and lower weights on later years.

Is p_S = −δ̂S high or low? It clearly depends on the level of the discount rate. If we choose r = δ = 0.03, their standard values in calibration exercises (Hubbard, Skinner and Zeldes, 1994; Becker, Philipson and Soares, 2005), then at the current U.S. level of S = 15, which I measure using the adult variance S₁₀, we find that p_S = −0.45 year. That is, the average citizen would be willing to give up almost half a year in mean life span in order to obtain a standard deviation in life span that was one year lower. For now, I simply remark that this cost seems large, and later I provide some context for assessing the cost relative to levels of population health across time and space.

3.3 Numerical solutions of the full model

In order to examine the sensitivity of the analytical result in equation (14) to alternative assumptions, I set parameters to match those used by Becker, Philipson and Soares (2005): r = δ = 0.03, γ = 0.8, and K = −16.2. Initial wealth equaling $800,000 is consistent with the parameter values, U.S. life spans, and per capita consumption of $26,650 per year. I also fully endogenize consumption. I begin by modeling life span as normally distributed and omitting infant mortality, and then I reintroduce realistic adult and infant mortality.²⁰ For convenience, I set adult mean and variance equal to U.S. levels in 1994 and search for the decrease in M that compensates for the observed decrease in S.²¹

3.3.1 Normally distributed life spans

The two panels in Figure 2 plot p_S as given by equation (14), shown by the thick solid line, on the same axes with two other loci that I obtain from numerical simulation of the model with normally distributed life spans. The thin solid lines depict the numerical model with annuities, while the dashed line shows results without annuities. In the top panel, I fix r = 0.03 and examine how varying δ changes p_S, while in the bottom panel I fix δ = 0.03 and vary r.

Figure 2. The price of S, a standard deviation in life span, in terms of mean life span when life span is normally distributed.

In both panels, the thick black lines show p_S, the price of a standard deviation in life span in terms of mean life span, either as a function of the rate of time preference, δ (panel A), or of the interest rate, r (panel B). The standard deviation of life span is set to S₁₀ = 15.66, the level prevailing in the U.S. in 1994. In panel A, r is fixed at 0.03, and in panel B, δ is fixed at 0.03. The thin solid lines shows the locus in the numerical model with normally distributed length of life and complete annuitization of wealth. The dashed lines depicts the locus in a model with no annuities.

In both panels, the two solid lines are fairly similar, revealing limited differences between the analytical and numerical versions of the model with annuities, except at the extremes. Both panels reveal that when δ is small relative to r = 0.03, the price of variance p_S draws close to zero. This is especially clear at the upper left in panel A, where δ is less than 0.01 while r = 0.03. This is essentially the risk neutrality result found by Bommier (2006) when consumption growth is rapid enough to keep the marginal utility of length of life constant, all because r exceeds δ by a lot. When time discounting is very low relative to the rate of interest and intertemporal substitutability is high, it is optimal to consume more in the future. A mean-preserving spread in life span, which trades away earlier years for later years, could actually improve expected well-being for somebody with heavily back-loaded consumption.

The dashed lines in Figure 2 show that the cost of uncertain life span is higher without annuities. It is striking that annuitization removes only a little over one third of the total cost of life-span uncertainty under baseline parameter values. That is, in Yaari’s (1965) framework, the direct utility cost of S is relatively more important than the cost of consumption uncertainty. The period utility curvature parameter, γ, can play a role here but is relatively uninteresting.²²

3.3.2 Realistic adult life spans without infant mortality

As shown in Figure 1A, modern distributions of human life span are skew-left and leptokurtic, not normal. Skewness implies that a mean-preserving spread in life span lowers survivorship probabilities asymmetrically; leftward skewness distributes an increase in variance more broadly below the mode or mean. Because of discounting, skewness should amplify the cost of uncertain life span. Figure 3A is the analogue of Figure 2A with realistic adult mortality.²³ It depicts the same three loci of p_S against δ for r = 0.03 and uses the same vertical scale for easier comparison. The thick solid line, which shows the analytical model’s p_S, is the same in each, but the other two schedules are lower here, especially for larger δ. That is, the cost of life-span uncertainty is indeed higher under realistic adult mortality with leftward skewness.

Figure 3. The price of S, a standard deviation in life span, in terms of mean life span with realistic mortality.

In both panels, the thick black lines show p_S, the price of a standard deviation in life span in terms of mean life span, as a function of the rate of time preference, δ, when r is fixed at 0.03. The standard deviation of life span is set to S₁₀ = 15.66, the level prevailing in the U.S. in 1994. The thin solid lines shows the loci in the numerical model with annuities, while the dashed lines show them without. In panel A, the numerical model features realistic adult mortality and zero infant mortality, while in panel B mortality is fully realistic. Survivorship weights are from life tables provided by the Human Mortality Database (2009) and modified in order to reveal the results using a translation method described in the text. In panel A where infant mortality is omitted, life-table deaths at ages under 10 are set to equal deaths at age 10, and the entire distribution is rescaled to sum to unity. In panel B, the entire distribution is used, holding infant mortality constant as described in the text.

3.3.3 Fully realistic life spans with infant mortality

Does the presence of infant mortality affect p_S? Given a fixed pattern of adult survivorship, higher infant mortality reduces the average length of life, raising the marginal utility of M. But if the marginal disutility of adult variance rises with total variance, infant mortality could amplify the marginal disutility of S. The effect on p_S is ambiguous. To proceed, I numerically simulate the cost of adult variance under fully realistic survivorship with infant mortality. I treat infant mortality as a completely separate dynamic, fixing infant deaths at their relative probability in 1999 and reestimate the compensating change in mean adult life span.²⁴ Panel B in Figure 3 depicts p_S as a function of δ with fully realistic mortality. Including infant mortality does not change the trajectories much, but it does slightly attenuate p_S compared to panel A. The presence of infant mortality must increase the marginal utility of mean life more than it raises the marginal disutility of a standard deviation in life span.²⁵ But the bottom line is that the analytical solution remains a conservative estimate of the true cost.

3.4 The cost of infant mortality

A separate but related issue that arises with cross-country comparisons is the cost of infant mortality. The previous section shows it does not greatly affect p_S, and differences in infant mortality between advanced countries today are minor. But infant mortality is much higher and more volatile across time and space, and like S₁₀, it is costly because of the curvature of expected lifetime utility. To see why, assume for simplicity that infant mortality is completely described by q(0), the probability of death between ages zero and one. A change in q(0) equal to i = −0.01 raises survivorship, life expectancy at birth e₀, and lifetime expected utility each by 1 percent.²⁶ By comparison, proportional increases in e₀ brought about solely by reductions in adult mortality typically produce less than proportional increases in lifetime utility.²⁷ This is because discounting reduces the value of gains that are centered around the average length of life relative to the value of gains derived from infant mortality, which are broadly distributed across all ages. The implication is that declines in infant mortality are considerably more valuable than is implied by their effect on life expectancy at birth alone.

Formally, I can derive a formula for p_i, the cost of infant mortality in terms of the mean. Assuming complete annuities, normally distributed life spans, and K = 0 for simplicity,

$$p_i=\frac{\partial EU/\partial i}{\partial EU/\partial M}=-\frac{(1-e^{-\widehat{\delta }M+{\widehat{\delta }}^2{S}^2/2})/\widehat{\delta }}{e^{-\widehat{\delta }M+{\widehat{\delta }}^2{S}^2/2}}=-\frac{e^{\widehat{\delta }M-{\widehat{\delta }}^2{S}^2/2}-1}{\widehat{\delta }},$$ (16)

where flow utility cancels. Equation (16) states that a unit reduction in i is preferable to a unit increase in M, which occurs when p_i < −1, if the compounded risk-adjusted return to investing at the discount rate for M years, the numerator, is higher than the discount rate, the one-period return. An interesting quantity is p_i/100, the price of one percentage point in the infant survival probability in terms of the mean length of life. By equation (16), when δ = 0.03, p_i/100 equals one third of the risk-adjusted total return at δ over M years. The latter reaches and begins to exceed 1 in absolute value around M = 40 when there is realistic adult variance. Thus when life expectancy exceeds 40, as it typically does in human populations, a decline in infant mortality of a percentage point is more valuable than an additional year in the adult mean. A quick glance at Tables 1 and 2, which I discuss next, reveals that gains against infant mortality, loosely approximated by 1 − ℓ₁₀ or the probability of death before age 10, have been on the order of tens of percentage points last century.

Table 1.

Changes in U.S. life expectancy, adult life-span variance, and survivor-ship since 1900

		1900	1950	2000
	Life expectancy, e0	47.7	68.4	76.7
	Std. dev. in adult life span, S10	20.0	15.6	14.7
	Survivorship to age 10, ℓ10	0.782	0.963	0.991
		1900–2000	1900–1950	1950–2000
[1]	Average S10	17.4	17.8	15.2
[2]	Average pS	0.52	0.53	0.45
[3]	Change in S10, ΔS10	5.3	4.4	0.9
[4]	Life year benefit of ΔS10, [2] × [3]	2.8	2.3	0.4
[5]	Average ℓ10	0.886	0.872	0.977
[6]	Life year benefit wtd. by avg. ℓ10, [4] × [5]	2.4	2.0	0.4
[7]	Change in life expectancy, Δe0	29.0	20.7	8.3
[8]	Total improvement in life years, [6] + [7]	31.4	22.7	8.7
[9]	Share due to ΔS10, [6] ÷ [8]	0.078	0.090	0.046

Demographic data are simple averages of sex-specific period life tables presented by Bell and Miller (2005) and are based on age-specific mortality rates measured in the given year. Life expectancy at birth, e₀, is the familiar average number of years lived starting from birth or age 0. The standard deviation of length of life conditional on survival to age 10, S₁₀, is calculated using rescaled probabilities of death above age 10 and is measured around the mean length of life conditional on survival to age 10, M₁₀, which equals e₁₀ +10. The survivorship probability at age 10 is ℓ₁₀. The price of S₁₀ in terms of mean life years is given by p_S = −δS₁₀, as described in the text, where it is assumed that δ = 0.03 in all periods.

Table 2.

Value of survivorship gains by region of the world and groups of countries, 1970–2000

	1970				2000
	Life exp. e0	Life std. dev. S10	Survivorship ℓ10	GDP per capita	Life exp. e0	Life std. dev. S10	Survivorship ℓ10	GDP per capita
East Asia & Pacific	58.3	16.4	0.871	695	69.7	15.1	0.959	3,908
Europe & Central Asia	67.3	15.9	0.947	5,498	68.2	16.2	0.970	6,838
High income	70.6	15.0	0.971	12,951	77.7	14.4	0.992	25,954
Latin America & Caribbean	60.4	16.8	0.881	4,839	71.5	16.7	0.965	7,085
Middle East & North Africa	53.7	17.9	0.822	3,202	67.7	15.2	0.947	4,569
South Asia	47.8	17.7	0.757	1,183	60.9	17.1	0.898	2,510
Sub-Saharan Africa	45.8	19.8	0.741	1,565	50.7	19.4	0.827	1,667
Poorest 50% countries in 1970	53.6	17.4	0.820	1,098	64.1	16.8	0.919	3,326
Richest 50% countries in 1970	67.9	15.6	0.948	10,105	73.4	16.0	0.978	17,024
World	58.8	17.0	0.867	4,360	66.9	16.8	0.937	7,505

	Value of survivorship gains in annual income calculated with:			Yearly growth rate of full income (%) calculated with:
	only e0	e0 and S10	entire ℓ(x)	only e0	e0 and S10	entire ℓ(x)
East Asia & Pacific	477	720	987	6.3	6.5	6.7
Europe & Central Asia	60	54	281	0.8	0.8	0.9
High income	2,035	2,766	3,343	2.6	2.7	2.8
Latin America & Caribbean	926	1,299	2,009	1.7	1.8	2.1
Middle East & North Africa	838	1,474	1,903	1.8	2.1	2.4
South Asia	438	699	877	3.1	3.4	3.6
Sub-Saharan Africa	105	197	235	0.4	0.6	0.7
Poorest 50% countries in 1970	425	651	865	4.2	4.4	4.6
Richest 50% countries in 1970	1,067	1,290	2,020	2.0	2.0	2.1
World	800	1,159	1,684	2.2	2.3	2.5

Notes: The sample, fully described in Edwards (2011), comprises 180 countries including Taiwan. Regions are otherwise as defined by the World Bank, and regional averages are weighted by population. GDP per capita is measured in 2000 international prices, adjusted for terms of trade. Measures are collected from the Penn World Table, Maddison (2003), and the IMF. Life expectancy at birth, e₀, the standard deviation of length of life above age 10, S₁₀, survivorship to age 10, ℓ₁₀, and the entire ℓ(x) distributions are derived from official data and estimates collected by Edwards (2011). The value of life expectancy gains and full income are calculated three ways using the methodology and parameter values of Becker, Philipson and Soares (2005) and 1970 as the base year. When only e₀ is used, survivorship is assumed to be 1 until dropping to zero at age e₀. When e₀ and S₁₀ are used, deaths are assumed to be distributed normally around a mean of e₀. When the entire ℓ(x) curve is used, the survivorship schedule is fully realistic and reflects everything: the mean, the spike in deaths due to infant and child mortality, the old-age hump, and all its skewness and kurtosis.

4 Discussion and Extensions

4.1 The role of the discount rate

The key element in the model is the discount rate, δ, which is approximately the coefficient of absolute risk aversion over gambles in life span under reasonable assumptions. The discount rate is a latent preference parameter, but it is standard in the literature to set it equal to 3 percent, roughly the real rate of return on government bonds, and here I have followed suit. Viscusi and Aldy (2003) review estimates of the discount rate in the U.S. and report a very wide range of 1–17 percent, making it difficult to reject the hypothesis that market rates of interest and the discount rate are the same (Picone, Sloan and Taylor Jr., 2004). Via an evolutionary argument, Rogers (1994) suggests that the discount rate in human populations should equal roughly 2 percent in the long run, which is in the same ballpark.

Discount rates appear to vary over individuals within and across countries (Barsky et al., 1997; Becker and Mulligan, 1997; Bishai, 2004), as does length of life (Edwards and Tuljapurkar, 2005; Edwards, 2011). Complications may arise if discount rates and life spans are related to one another, but the implications are difficult to assess because the literature does not speak with one voice on the subject. Fuchs (1982) views the discount rate as determinining health investments, while Becker and Mulligan (1997) see wealth, uncertainty, and health or the length of life as jointly determining the discount rate. Bishai (2004) presents empirical evidence of reductions in the discount rate, i.e., increases in patience, with age and schooling. In related work, Satchell and Thorp (2011) argue that mortality and fertility patterns imply that family trusts ought to first increase and then decrease patience as it ages. At the population level, these concerns seem unlikely to be problematic for assessing the average cost of uncertain life span. But microeconomic implications of life-span variance may be very different if lines of causality running between the discount rate and mortality are important.

4.2 Uncertainty in actual life spans

Different groups and individuals face substantially different amount of life-span uncertainty, not all of which can be due to behavioral differences. Edwards and Tuljapurkar (2005) show that S₁₀ is systematically lower by about 1 year among females compared with males, and that it is 2–3 years higher among African Americans relative to whites. Individuals in the lowest quintile of household income had 2.4 more years in standard deviation than those in the upper 80 percent, while those without a high school degree had 2.1 years more than high school graduates. A subgroup difference of 3 years in standard deviation implies an increase in the costliness of life-span uncertainty of 20 percent.²⁸

With this much subgroup variation, one should be a little cautious about interpreting trends in aggregate uncertainty. An increase in aggregate S₁₀ could reflect increasing between-group inequality, increasing within-group inequality, or both, and there are different implications of each. If an increase in life expectancy were enjoyed only by a more advantaged subgroup, it would technically increase aggregate S₁₀ by raising between-group inequality, but it would also be Pareto-improving if the less advantaged did not lose. Although aggregate S₁₀ in the U.S. has displayed little trend (Edwards and Tuljapurkar, 2005), recent evidence on widening educational differentials in life expectancy suggests between-group inequality is indeed increasing (Meara, Richards and Cutler, 2008). By implication, within-group inequality should have been decreasing, and if that were true, disadvantaged groups could be gaining from reductions in their S₁₀ even while the rich are getting richer. These patterns pose interesting questions for future research.

What about members of different birth cohorts? Up to now, I have proxied the actual, or cohort life spans of individuals with those based on period mortality rates. Although period S₁₀ is the appropriate variance analogue of period life expectancy, the most commonly cited population health statistic, we would also like to know the levels of variance faced by actual cohorts. Differences between cohort and period S₁₀ do not appear to be large, as shown by Figure 4. I plot both measures for the U.S. since 1900 using decennial period life tables from Bell and Miller (2005) prior to 1930 and annual life tables from the Human Mortality Database (2009) starting in 1933, and decennial cohort life tables from Bell and Miller. The two series track each other relatively well, with both showing the enormous impact of the epidemiological transition early last century. Cohorts alive today, who likely face S₁₀ = 15, face drastically less uncertainty in their life spans than those born around 1900, for whom S₁₀ = 21.²⁹

Figure 4. The standard deviation in life span above age 10 in the U.S. by year and by birth cohort.

The underlying data are period and cohort life table death distributions, both historical and forecast, taken from Bell and Miller (2005), labeled SSA in the graph, and Human Mortality Database (2009), labeled HMD. The statistics in the figure are standard deviations in length of life above age 10, S₁₀, which are calculated as described by Edwards and Tuljapurkar (2005).

4.3 Population health over time and space

By providing a new method of converting units of variance in length of life into units of mean, equation (14) facilitates a more complete assessment of population health in a variety of settings. In particular, we can reexamine either the differences between advanced countries such as the U.S. and Sweden at a point in time, the differences between observations of the same advanced country at different points in time, or the differences between rich and poor countries across time. Applied intertemporally, this new perspective also provides insights into the nature and timing of demographic and epidemiological transitions that have occurred and are proceeding today in various parts of the world.

4.3.1 Contemporary differences between advanced countries

In 1999, individuals in the U.S. experienced a standard deviation in life spans conditional on survival to age 10 equal to about S₁₀ = 15 years. At that level, each year of standard deviation is worth about p_S = −0.45 year of mean life span in this model, assuming r = δ = 0.03, the standard value in calibration exercises using U.S. data. In Sweden that same year, S₁₀ was about 13. According to this model, individuals in the U.S. would be willing to give up almost 0.9 year in mean life span to have the lower S₁₀ of their Swedish counterparts.³⁰ The mean life span conditional on survival to age 10, M₁₀, was 77.7 years in the U.S. and 80.0 in Sweden in 1999. If we account for differences in S₁₀, the total difference in population health between the U.S. and Sweden, as measured by “effective” life expectancy, is more like 3.2 life years per person rather than 2.3, an increase of more than a third.

A similar result emerges when we examine a group of advanced economies. Among the 27 members of the OECD that are also designated as high-income countries by the World Bank, life expectancy at birth averaged 78.4 with a standard deviation of about 2 years in 2000, while S₁₀ averaged 13.5 with a standard deviation of 0.5 year. Translating the differences in S₁₀ from the average into penalties in life expectancy using p_S = −0.45 widens the standard deviation in effective life expectancy from 2 to about 2.2 years, or by roughly 10 percent. Measured inequality among advanced countries increases when we account for different variances, but the effect is moderate in size.

4.3.2 Temporal trends within a country

Because the underlying trends in mean and adult variance are so different, as shown by Edwards and Tuljapurkar (2005), the findings in this paper suggest it is worthwhile to decompose the economic value of historical gains against mortality into portions attributable to reductions in variance versus improvements in mean life span. Nordhaus (2003) and Murphy and Topel (2006) measure the total value of mortality improvement in the U.S. over historical periods using the entire survivorship curve. They find that the value of health improvements is very large, rivaling the value of GDP. But how much is due to increases in the mean, which show no signs of stopping (White, 2002; Oeppen and Vaupel, 2002), and how much is due to declines in adult variance, which have largely stalled since 1960?

The top two rows of Table 1 list e₀ and S₁₀ in 1900, 1950, and 2000. Survivorship to age 10, ℓ₁₀, which acts as an importance weight for characteristics of the life table above age 10, is shown in the third row. Changes over time in these three life-table parameters appear in the bottom panel. Between 1900 and 1950, life expectancy rose by 20.7 years from 47.7 to 68.4, an increase of over 40 percent, while S₁₀ fell by 4.4 years, from 20 to 15.6, a reduction of about 20 percent. Life expectancy continued to increase steadily after 1950, rising an additional 8.3 years by 2000, but further reductions in S₁₀ were slight, totaling only 0.9 year over the second half of the century.

The bottom panel in Table 1 translates the observed declines in S₁₀ into equivalent gains in mean life years using the formula for p_S derived earlier, assuming δ = 0.03 in all periods. The relevant price of a discrete change in S₁₀ is δ times the average S₁₀ during the interval, and the resulting value of the decline in S₁₀ must be discounted by the average level of survivorship to age 10 in order to capture the benefit to the average individual. This latter step is especially important when ℓ₁₀ is relatively low, as it was in the U.S. early last century. Combining the value of decreased S₁₀ with the change in e₀ produces a rough measure of the total gains against mortality in terms of mean life years. By this accounting, reductions in life-span variance prior to 1950 accounted for 9 percent of the gains in effective life expectancy, but after 1950 that figure fell to 4.6 percent.

4.3.3 Convergence in “full income” across countries

In a widely cited paper, Becker, Philipson and Soares (2005) reveal relatively more global convergence in human well-being when it is measured by “full income,” a statistic they devise that combines GDP per capita with the monetized value of life, than when it is measured by GDP per capita alone. Owing to data constraints in their panel of 96 countries measured since 1960, they can only account for changes in life expectancy at birth rather than changes in higher moments or in entire survivorship schedules. Given the costliness of variance, has its omission biased their results, and if so which way?

Complete life tables are not widely available for a broad cross section of countries because the underlying mortality data are typically of poor quality. For many developing countries where data is scarce, surveillance agencies have estimated levels of life expectancy at birth, e₀, using model life tables, but historical databases do not include complete model life tables. Recently, Lopez et al. (2002) and the World Bank have produced estimates of current life tables for virtually all countries, but historical coverage has remained lacking. New work by Edwards (2011) collects and reconstructs historical estimates of life tables in developing countries and allows a reassessment of convergence in the full income measure after accounting for changes in variance and the shape of the survivorship function.³¹ As discussed by Edwards, these new data are of varying quality, but they are no worse than much of the original data on life expectancy commonly found in databases and used by Becker, Philipson and Soares (2005) and many others.

The methodological component in Becker, Philipson and Soares (2005) that captures the effects of mortality is the value of an annuity based on the survivorship function, ℓ(t), given by

$$A(\ell )={\int }_0^{\infty }{e}^{-rt}\ell (t)dt.$$ (17)

With full annuitization and the time discount rate δ equal to the interest rate r, indirect utility is the product of A(ℓ) and the period utility function. When only life expectancy e₀ is known, survivorship must implicitly be rectangular, or equal to 1 at all ages until falling to 0 at e₀. Under those conditions, equation (17) simplifies to

$$A^{\ast }(\ell )\approx \frac{1-e^{-r·e_0}}{r}.$$ (18)

As seen and discussed earlier, for example in equation (11), the presence of variance in adult length of life will reduce the value of the annuity:

$$A^{†}(\ell )\approx \frac{1-e^{-r·e_0+r^2{S_{10}}^2/2}}{r},$$ (19)

where S₁₀ is the standard deviation in adult length of life. If there is any adult variance, A(ℓ) and A^†(ℓ) are both less than A^*(ℓ), so equation (19) should be a closer approximation to (17) than (18).

As discussed in section 3, high infant mortality also tends to reduce A(ℓ), holding other things including e₀ equal. Indeed, infant mortality is additional variance in human life span that is highly non-Gaussian and non-central. In the simple model based on Yaari (1965), living with complete certainty from birth until death at age e₀ is more valuable than facing a nonzero probability d of dying in infancy followed by living with certainty until e₀/(1 − d) > e₀, even though life expectancy at birth is exactly the same in either case. This is because the cost of infant mortality outweighs the benefit of living past e₀ due to discounting, the same reason why S₁₀ is costly. Both equations (18) and (19) will typically overestimate A(ℓ) because each formula omits infant mortality, which has been and remains high in the developing world. If complete life tables are available, of course, it is possible to avoid this problem by measuring A(ℓ) exactly. In order to gauge the importance of life-span uncertainty for convergence, I estimate A(ℓ) and full income using all three methods and then compare results.

Tables 2 and 3 reproduce their counterparts in Becker, Philipson and Soares (2005) using a wider sample of 180 countries covering virtually all of the world’s population. Results using only the original 96 countries examined by Becker, Philipson and Soares are similar and indicate slightly more convergence than is shown here.³² The top panel in Table 2 shows population weighted averages of aggregates from life tables and national income accounts for a set of world regions as defined by the World Bank, for the poorest and richest countries in 1970, and for the world as a whole, using mortality data from Edwards (2011) and income data from the Penn World Table, Maddison (2003), and the IMF. All regions gained income per capita during the interval, at an average annual rate of about 1.8 percent worldwide, and all but one gained in terms of life expectancy, which rose 0.27 year per annum on average. The world standard deviation in adult life span remained fairly steady, declining only 0.2 year from 17.0 to 16.8. Survivorship to age 10 rose 7 percentage points, from 86.7 to 93.7 percent. Although almost all regions experienced gains against mortality, there was also clear heterogeneity. Europe and Central Asia, a World Bank category that does not include the high income European countries, experienced only a very small 0.9 year total gain in life expectancy and actually suffered a small increase in S₁₀, from 15.9 to 16.2, although ℓ₁₀ increased somewhat. Sub-Saharan Africa gained almost 5 years in life expectancy, due to a large increase in ℓ₁₀, which rose from 74.1 to 82.7 percent. But variance in adult life dropped only 0.4 to 19.4.

Table 3.

Evolution of cross-country inequality in full income, 1970–2000

	Income per capita		Full income in 2000 calculated with:
	1970	2000	e0 only	e0 and S10	entire ℓ(x)
Relative mean deviation	0.4816	0.4227	0.4117	0.4038	0.3970
Coefficient of variation	1.2041	1.1801	1.1524	1.1342	1.1076
Std. dev. of logs	1.2216	0.9910	0.9874	0.9791	0.9819
Gini coefficient	0.5439	0.4999	0.4904	0.4836	0.4781
Regression to the mean since 1970		−0.3507	−0.3681	−0.3805	−0.3872

Notes: See notes to Table 2. Inequality measures are weighted by country population. Regression to the mean is the coefficient from a weighted OLS regression of the change in the natural log of income over the period on its initial logged level, with 1970 populations as weights. All four regression coefficients are statistically significant at the 1% level.

The bottom panel of Table 2 translates gains in income and survivorship into gains in the “full income” measure proposed by Becker, Philipson and Soares (2005) using the three different methods of valuing life span discussed above. When only e₀ is used, the annual value of gains is an additional $800 (in 2000 international dollars) for the world as a whole, and the growth in full income is 2.2 percent, or 0.4 faster than the growth rate of income per capita during the period. Accounting for both e₀ and S₁₀ increases the contributions of mortality gains, to $1,159 and 0.5 percentage point faster annual growth respectively. Using the entire survivorship schedule including infant mortality more than doubles the gains, to $1,684 and an extra 0.7 percentage point in annual growth.

The hefty importance these comparisons seem to attach to S₁₀ is odd in light of the relatively small decrease in world S₁₀ since 1970, and it is in fact misleading. The presence of any S₁₀ at all raises the value of gains in e₀ by increasing their marginal utility. Equation (15) shows this clearly: the marginal utility of the mean length of life, which appears in the denominator, is a decreasing function of the mean but an increasing function of the variance. When I account for S₁₀ in addition to e₀ in Table 2, I am basically turning variance on after being off, which has a large effect on measured trends in well-being even when the trend in variance is negligible.

A more appropriate way to decompose growth in full income into portions attributable to changes in individual moments is by measuring gains using counterfactual but fully realistic survivorship curves that hold one or more moments constant over time. Without any change in world S₁₀, the value of total annual survivorship gains would be about 5 percent lower than the $1,685 we observe in Table 2. This share is similar to what I found in Table 1 for the U.S. after 1950. Indeed, average annual declines in S₁₀ for the U.S. since 1950 and for the world since 1970 are both only around 0.01 to 0.02, a sobering remark on recent progress against high adult variance in the developing world. Declines in infant and child mortality, on the other hand, have been large and more prevalent among poor countries (Moser, Shkolnikov and Leon, 2005). To gauge their importance, I generated a fictitious world survivorship curve for 2000 assuming there was change since 1970 only in ℓ₁₀, and not in the shape of the life-span distribution above age 10. Based on this counterfactual, the value of improvements spurred by increases in ℓ₁₀ accounted for more than 75 percent of the total value of gains against mortality.

Heterogeneity across regions in the progress against mortality is certainly evident in the lower panel in Table 2, where results broadly amplify the patterns reported by Becker, Philipson and Soares (2005). In dollar terms, the benefits of mortality reduction were felt considerably more strongly by the richest 50 percent of countries, but growth rates of full income were faster for the poorest 50 percent because their initial money incomes were so much lower. Sub-Saharan Africa remains a basket case even when improvements in the entire survivorship curve are priced, raising growth in full income to only 0.7 percent. But a similar picture emerges for Europe and Central Asia, where the negative influence of a slight increase in S₁₀ was more than offset by improvements in ℓ₁₀, but not by enough to raise growth past 0.9 percent. Elsewhere, economic growth that was already fairly robust is further enhanced by accounting for improvements in survivorship. In East Asia and the Pacific, growth in full income is a stout 6.7 percent after accounting for much improvement in all three mortality statistics, up 0.4 from the e₀-only reading. The Middle East, North Africa, and South Asia benefit the most, about 0.6 percentage point in annual growth, from accounting for all changes in ℓ(x) as opposed to e₀ alone.

Table 3 descends past regions to examine inequality in average human well-being using countries weighted by population as the unit of observation. By any one of several measures, inequality across individuals in full income in 2000 is less than inequality in income per capita in 2000 or in 1970. The Gini coefficient, a widely cited statistic, registers 0.4781 when measured using full income in 2000 derived with countries’ entire ℓ(x) schedules. By comparison, the Gini on per capita income in 2000 was 0.4999. Accounting for successively higher moments of the life-span distribution reinforces this basic convergence result, as it did among regions in Table 2. Another upshot of the math in equation (15) is that the value of a given increase in mean life span will be larger when S₁₀ is higher, such as in a poor country. Part of what we see here is thus surely the mechanical effect of S₁₀ and infant mortality raising the measured value of gains in e₀ disproportionately more among poor countries with high variance. But surely also at work are continued global convergence in infant mortality even as convergence in e₀ turned to divergence after 1980 (Moser, Shkolnikov and Leon, 2005), as well as some convergence in country S₁₀ revealed by Edwards (2011) as narrowing within-country inequality.

These results mirror and amplify those of Becker, Philipson and Soares (2005) on convergence in average human well-being, but some circumspection is in order. Considerably less progress against mortality in poor but more sparsely populated areas like Sub-Saharan Africa, parts of Europe, and Central Asia, was not sufficient to reduce overall convergence in average human full income due to considerable progress in larger countries. Still, population-weighted estimates suggest more convergence in average human well-being since 1970 than in average income alone. Accounting for gains against life-span variance, either in the form of S₁₀ or infant mortality, strengthens the findings of Becker, Philipson and Soares for two reasons. Mechanically speaking, correctly accounting for even a stable level of variance raises the value of increases in life expectancy, and that effect increases with the level of variance, which is higher in poor countries. In addition, total variance was also strongly declining over this period, primarily due to robust declines in infant mortality but also due to moderate declines in S₁₀. The gains against variance, broadly defined, contributed a large amount to the value of progress against mortality.

4.3.4 A new perspective on the demographic transition

The insights of this paper also suggest a new interpretation of the historical timing of age-specific gains against disease and mortality during the demographic and epidemiologic transitions. As summarized by Wilmoth (2003), the classic transition begins with a decline in infant mortality and early death, brought about by progress against infectious disease. That is, the first stage of progress drastically lessens the unconditional variance in life spans. The second stage of the transition is characterized by a shift in focus away from infectious disease and toward treating chronic degenerative diseases afflicting the elderly. This works to lengthen the average adult life span but probably does not reduce the variance much if at all.

The current framework suggests this sequence was probably optimal, if it were not practically required.³³ High levels of variance inflate its cost considerably. Combating infectious disease reduces unconditional variance directly by reducing infant and child mortality, and it also reduces uncertainty in adult life. Certainly in the context of the modern developing world, in which life expectancy is short, variance is high, both infectious and chronic diseases claim lives, and technologies and practices to combat either one are readily available via the developed world, revealed preference seems to indicate that public health priorities largely center on the unconditional variance first, i.e., infant and premature adult mortality. An outlier based on this perspective, HIV/AIDS is a hybrid disease afflicting old and young alike.

Historically, these two very different stages of the transition produced a seamless pattern of steady increases in life expectancy at birth over time (Oeppen and Vaupel, 2002), which fits well with steady growth in per capita incomes (Hall and Jones, 2007) and shows that average health outcomes were rising consistently. But the technologies, cost structures, and incidence of benefits during each phase were entirely different. How societies set priorities in achieving mortality decline is a major question. A key insight of this paper, that variance in life span is costlier relative to mean life span when variance is higher, suggests that declines in total variance should precede sustained progress in the adult mean or mode.³⁴

4.4 Uncertain life span and economic behavior

This paper, like others in the literature on the value of gains against mortality, explores only one economic perspective on the cost of uncertainty in life span: the willingness to pay. Although willingness to pay should in principle encompass or account for all expected behavioral responses to changes in uncertainty, my simple theoretical model only captures the response of consumption and saving and not other behavioral responses that we think might be very interesting. The latter are a burgeoning field of research. Kalemli-Ozcan and Weil (2010) examine the effects of uncertain life span in a model with endogenous labor supply. Their results suggest that an optimal policy under high uncertainty might be low saving and working until death, or what macroeconomists might consider extreme rule-of-thumb as opposed to life-cycle behavior. Patterns in retirement behavior early in the 20th century may support this view (Costa, 1998). But in modern data, Hurd, Smith and Zissimopoulos (2004) find small effects of low subjective survivorship probabilities on retirement behavior among older Americans in the Health and Retirement Study. Education, another form of saving, is also endogenous and may react to uncertainty in length of life. Building on earlier work by Kalemli-Ozcan, Ryder and Weil (2000) that focuses primarily on the average length of life, Li and Tuljapurkar (2004) show that uncertainty has an effect on educational attainment in a general equilibrium setting. These and other directions are promising avenues for further research.

5 Conclusion

In the standard model of time-separable utility introduced by Yaari (1965), uncertainty in life span is costly when the force of time discounting, δ, which is also approximately the coefficient of absolute risk aversion in life span, is positive and sufficiently near the real interest rate. Even when wealth is fully annuitized, individuals with these preferences are hurt by uncertainty in life span and would be willing to trade away p_S = −δS years of mean life span in return for one less year in standard deviation, S. Because average life expectancy is increasing linearly over time in advanced countries while the standard deviation in adult life span now roughly fixed (Edwards and Tuljapurkar, 2005), constant absolute risk aversion is consistent with stable risk premia and thus is an intuitive result.³⁵ Risk aversion over length of life also fits the relatively scant empirical evidence on stated preferences and behavior in medical settings. If δ = r = 0.03, which is a standard parameterization, the average American would be willing to give up 0.45 life year in return for one year less in standard deviation, which is currently about 15 when measured by S₁₀, the standard deviation in length of life above age 10. This is large, implying that differences in population health between the U.S. and Sweden are more like 3.2 life years, or 40 percent higher than the difference of 2.3 years we find in life expectancy alone.

Adult variance is interesting because of the large differences in levels and trends we see among industrialized countries today (Edwards and Tuljapurkar, 2005), but it is not the only source of variance. Infant mortality is low in advanced countries today, but it was very high during historical periods and is high in many modern developing countries. Discounting places a very high cost on infant mortality indeed, making it two to three times more costly than the reduction in mean length of life it represents. This is because all of the probability weight at play occurs so early in life, during years that are extremely valuable in the simple model of Yaari (1965). This result would be moderated if the empirical age trajectory of the value of a statistical life year were superimposed, because it is hump-shaped (Aldy and Viscusi, 2008). But without accounting for the unconditional variance in length of life represented by infant deaths, valuing gains in life expectancy at birth will significantly understate the benefit of declining infant mortality.

The framework developed in this paper allows me to decompose the overall value of progress against mortality into parts attributable to reductions in infant mortality, increases in the adult mean, and reductions in adult variance. This decomposition is interesting because gains against adult variance and infant mortality have largely stopped in developed countries (Edwards and Tuljapurkar, 2005), while increases in the mean continue apace. Worldwide, moderate declines in adult variance combined with broad-based reductions in infant mortality, the larger source of unconditional variance in human life span, have produced much global convergence in average human well-being, more than is implied by trends in e₀ alone. These findings amplify those in the pioneering work of Becker, Philipson and Soares (2005), who due to data limitations could only consider life expectancy. If the insights in this paper were applied to the broader definition of welfare adopted by Jones and Klenow (2010), who also measure only e₀, it is possible their evidence of divergence among countries since 1980 may be partially reversed.

Several limitations of my approach are worth highlighting. I do not account for education or for physical capital, both of which are considered in a general equilibrium setting by Li and Tuljapurkar (2004). I also do not consider variation in morbidity or the quality of life. Bequests are a potentially key omission, because they could significantly reduce the marginal disutility of life-span uncertainty. But even if bequests were intended, which is unclear, they would also reduce the marginal utility of mean life span, leaving an ambiguous effect on the price of life-span variance relative to the mean. I also make no allowance for the special psychology that we know surrounds death (Slemrod, 2003). One could argue that knowing the precise date of death is actually not a good, at least in terms of emotional benefit, and that some uncertainty is preferable. Still, the economic cost of uncertain length of life in terms of retirement and estate planning is real and plausibly much larger. Questions remain about time discounting and its relationship to risk preferences over periods of life. Bommier (2006, 2008) and Bommier and Villeneuve (2011) develop a more general model than Yaari’s and report a greater degree of curvature than implied by exponential time discounting alone. In their study of 30 women in perfect health asked to rank lotteries over life span, Verhoef, Haan and van Daal (1994) report evidence supporting prospect theory: risk-seeking behavior over small gambles and risk aversion over large. My estimate of the cost of uncertain life span is probably a conservative estimate, and by presenting it I hope to motivate further research into this topic.

The primary implication for policy is that uncertainty in length of life is costly and should be targeted when it is high, whether in entire countries or among specific subgroups. Because its cost rises with its level, and since the level of uncertainty also diminishes the value of gains in life expectancy, policies that reduce uncertainty should be favored over those that increase average length of adult life when uncertainty is high, such as when infant mortality is high. A more provocative interpretation of this bottom line is that a high-variance country like the U.S. may stand to gain by focusing more on spreading the existing benefits of health treatments and technology more broadly across its citizens, at the expense of investing in discovery of new treatments and technology that extend life. Given the key role attached to U.S. pharmaceutical innovation in driving old-age mortality decline worldwide (Lichtenberg, 2007), such a refocusing may not come without cost. A less provocative implication is that developing countries are better served by attacking the causes of high uncertainty first, infectious diseases and infant mortality, before extending assistance that extends life for adults. In large part, development assistance is already configured to do this.