Abstract
Individuals care not only about their own survival, but also about the survival of other persons. However, little attention has been paid so far to measuring the contribution of longer coexistence time to living standards. For that purpose, we develop a measure of coexistence time—the joint life expectancy—which quantifies the average duration of existence for a group of persons. Then, using a lifecycle model with risky lifetime, we construct an equivalent consumption measure incorporating gains in single and joint life expectancies. An empirical application to France (1820–2010) shows that, assuming independent individual mortality risks, the rise in joint life expectancies contributed to improve standards of living significantly. We examine the robustness of that result to the introduction of dependent mortality risks using copulas, and we show that equivalent consumption patterns are robust to introducing risk dependence.
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Notes
Note that the existence of coexistence concerns in real life can be represented by means of various microeconomic models, depending on what motivates coexistence concerns (either self-oriented concerns or altruism) and on how coexistence time is perceived/quantified by individuals. See Sect. 3 on this.
As usual, we assume \(S_{i0}=1\). Note that expression (3) presupposes that a person dying during a year dies at the beginning of that year. Alternatively, if one supposes that a person dying during a year dies, on average, in the middle of that year, Eq. (3) becomes:
$$\begin{aligned} E\left( L_{i}\right) = {\displaystyle \sum \limits _{x=0}^{T-1}} S_{ix+1}+0.5 \end{aligned}$$Note that we restrict ourselves here to a double lottery, in which one person cares about his survival and about the survival of another person. But in the real world individuals may care about the survival of more than one other person, implying a more complex lottery.
Similar calculations could be carried out for other groups of different size, including persons of unequal ages.
While that formula concerns the coexistence of two persons, joint life expectancies can also be defined for a larger number of individuals.
Alternatively, one may compute cohort joint life expectancies, on the basis of the survival conditions faced within a given cohort.
Section 5 is dedicated to the measurement and valuation of coexistence gains when mortality risks are dependent.
The form of the relation between the probabilities of joint survival (and, hence, the joint life expectancy) and individual survival probabilities may vary, depending on how dependent individual survival processes are. See Sect. 5 on this.
The utility function is assumed to exhibit self-oriented coexistence concerns: according to that function, individuals care about the survival of other persons only for themselves, without caring about the well-being of those other persons. It should be stressed here that this utility function does not constitute the only way to represent coexistence concerns. As an alternative to self-oriented coexistence concerns, we could have represented coexistence concerns by means of a utility function exhibiting altruism (individuals would be interested in the well-being of others, and, hence, in the survival of others).
Sources: The Human Mortality Database (2013), University of California, Berkeley (US), Max Planck Institute for Demographic Research (Germany). Available online at: http://www.mortality.org. Note that Sect. 4.1 presents demographic data over 1816–2010, whereas our income data only cover the period 1820–2010. Hence, demographic data for years 1816–1819 will not be used when computing consumption equivalents (Sect. 4.2).
Note that, if one carries out the same computation exercise for a pair composed of a man and a woman of age 50 years, one finds a similar pattern: stagnation of the joint life expectancy during the nineteenth century, and strong growth during the second part of the twentieth century. While a man and a woman of age 50 could expect to coexist about 15 years in the nineteenth century, their expected duration of coexistence is now equal to 28 years.
Fig. 4 
(Period) joint life expectancy for a woman and a man of age 25, France (1816–2010)
The income figures are GDP per capita expressed in International Geary-Khamis dollars (1990). Sources: The Maddison Project: http://www.ggdc.net/maddison/maddison-project/home.htm. Life expectancy figures come from the Human Mortality Database (2013).
Fig. 9 
Construction of the equivalent income for year 1972 in France
The indifference map on Fig. 9 is drawn in an arbitrary way. The next subsection explores the construction of a more realistic indifference map on the basis of the empirical literature on money/risk and risk/risk trade-offs.
The reference survival conditions—here the ones prevailing in 1950—were chosen arbitrarily. A similar construction could be carried out under alternative reference survival conditions.
The reason why we do not consider lower ages is that our consumption data do not cover ages inferior to 25 years (see the Appendix).
In the case of society M3, it is assumed that the representative individual cares about the survival of a spouse, of a child and of a parent. In the case of society M6, it is assumed that the representative agent cares about the survival of one spouse, of two parents and of two children. Finally, in the case of society M9, it is assumed that the representative individual cares about the survival of one spouse, two parents and two children, as well as of one person of his generation, one person of the previous generation and one person of the next generation.
The calibration of preference parameters under dependent mortality risks is examined in Sect. 5.
Survival probabilities are taken from the Human Mortality Database. For simplicity, we take survival probabilities for the total population (men and women).
See the Appendix for data on consumption profiles.
Given that Miller’s study covers the period 1974–1999, we use, for the calibration, the average levels of the variables (consumptions, survival probabilities) over that period.
We assume, here again, the independence of individual mortality risks.
Note that, as stressed by Costa and Kahn (2004), the VSL has grown during the twentieth century. In the light of this, one may compute an equivalent consumption on the basis of several—instead of one—sets of preference parameters. Such a measure of standards of living would incorporate both the improvement in survival conditions and the change in preferences. In this paper, we rely on a fixed set of preference parameters, to capture only the gain in survival conditions.
Table 1 Calibration of preference parameters Unadjusted consumption is here defined as the constant equivalent consumption level on the life cycle, that is, the hypothetical constant consumption level such that, if enjoyed at all periods of life, this would make the representative agent indifferent with his actual lifetime consumption profile.
The joint life expectancies taken into account when considering the M3, M6 and M9 cases are those between persons of age 25 and 0, ages 25 and 25, as well as between persons of ages 25 and 50 years.
Fig. 11 
Unadjusted and adjusted consumption for France (1820–2010), societies M0, M3, M6 and M9 (VSL lower bound)
For a detailed proof of this, see Georges et al. (2001).
When \(\theta \) tends to \(-\infty \), Frank’s copula reflects the maximal positive dependence between the survival of the two individuals.
Assuming a constant structure of dependence simplifies the picture. As shown by Luciano et al. (2010), the degree of dependence of mortality risks within couples tends to diminish across generations. Relying on a constant dependence structure is thus a simplification.
That value differs from the one appearing in Table 1 (equal to 4.031), for which calculations were based on representative agents without gender differentiation (unlike here, where the two members of the couple are modeled as a woman and a man).
The relative gap is defined as the ratio: [equivalent consumption under dependent risks minus equivalent consumption under independent risks]/equivalent consumption under dependent risks. The higher relative gaps in war times are due to the lowest levels of joint life expectancy for those years.
Fig. 14 
Relative gap between adjusted consumptions under dependent and independent mortality risks
On this, see Gustafsson (2001).
In particular, the reduction in coexistence time induced by birth postponement may be non negligible since the 1970s.
Consumption profiles are estimated by regressing the logarithm of consumption on a constant, on age, (age)\(^{2}\), (age)\(^{3}\), (age)\(^{4}\) and (age)\(^{5}\), as well as on dummy variables for the different years of birth. The latter control variables are used to neutralize the impact of cohort effects.
On the link between life expectancy and the propensity to save, see Chakraborty (2004).
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The author would like to thank Koen Decancq, Marc Fleurbaey, André Masson, Julien Tomas, Alain Villemeur, Hélène Xuan, Stéphane Zuber, two anonymous referees, as well as participants of seminars at University Paris Dauphine and at the Caisse des Dépôts et Consignations for helpful comments on this paper.
G. Ponthiere acknowledges the financial support of the ANR Equirisk (Equity in Risky Intertemporal Economic Environments) (ANR-12-INEG-0006-01).
Appendix
Appendix
Demographic data On the demographic side, we use data from the lifetables of the Human Mortality Database. In order to compute the equivalent income incorporating the value of changes in global survival conditions, we use individual probabilities of survival at all ages of life, for all years under study (from 1820 to 2010). We also use the probabilities of joint survival, for different pairs of individuals (varying with the age and the sex of the members of the pair), also for the period 1820–2010.
Consumption profiles As far as consumption statistics are concerned, we use consumption profiles by age (between age 25 and age 80) estimated for France by Boissinot (2007). Those lifetime consumption profiles were estimated on the basis of 5 successive waves of the INSEE Survey on Household Budgets.Footnote 36 Each wave includes about 10,000 households, which implies a total sample size of between 25,000 and 30,000 individuals. On the basis of that pseudo-panel, consumption profiles were estimated on an age interval between 25 and 80 years. Figure 15 below shows the consumption profile obtained by Boissinot on the basis of 15 cohorts, born between 1960 and 1975. That consumption profile has been normalized, a unitary value being assigned to consumption at the beginning of the lifecycle (i.e. at age 25).
Normalized consumption profile (age 25–80)
The consumption profile estimated by Boissinot exhibits the standard inverted-U shape.Footnote 37 The maximum consumption is achieved around age 45. After age 50, consumption starts falling continuously, to reach, at age 80, a level equal to about 70 % of the consumption at age 25. The consumption profile on Fig. 15 stops at age 80. This is problematic for the monetization of longevity gains, since a significant part of those gains concern ages higher than 80. This is the reason why we extended the consumption profile estimated by Boissinot until the age of 110. For that purpose, we calculated the growth rate of consumption between ages 75 and 80, and we extrapolated the rest of the consumption profile by applying that growth rate to the age interval 80–110 (see Fig. 16).
Extended consumption profile (age 25–110)
The lifetime consumption profile on Fig. 16 was then used to derive, by extrapolation, lifetime consumption profiles for all years considered in this study (1820–2010). For that purpose, we started from the lifetime consumption profile estimated by Boissinot (2007) for year 2000, and we used the levels of income per head for all years between 1820 and 2010 (sources: the Maddison Project 2015), in such a way as to derive the corresponding lifetime consumption profiles associated to those years. For each year, the consumption profile is assumed to keep the same general shape, which is characterized by the proportionality coefficients appearing on Fig. 16. Figure 17 shows some extrapolated lifetime consumption profiles, for years between 1820 and 2010.
Lifetime consumption profiles, 1820–2010 (in 2000 euros)
This extrapolation amounts to assuming that intertemporal trade-offs did not change significantly over time, and that differentials in income per head explain all differences observed, year after year, in consumption levels at a given age. Obviously, that assumption can only be regarded as a first proxy. On a period as large as two centuries, the strong improvement of survival conditions may have affected savings behaviors, and, thus, the shape of the lifetime consumption profile.Footnote 38 However, this extrapolation is required given the lack of historical data on lifetime consumption profiles over that period.
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Ponthiere, G. The contribution of improved joint survival conditions to living standards: an equivalent consumption approach. Soc Choice Welf 46, 407–449 (2016). https://doi.org/10.1007/s00355-015-0919-y
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DOI: https://doi.org/10.1007/s00355-015-0919-y