2.1 Model to evaluate gains in longevity

The economic model to assess longevity improvements is based on the hypothesis that people assign a high monetary value to a small increase in the probability of survival. As a result, they are willing to sacrifice some vital consumption in exchange for a longer lifetime. A formula to valuate health status was developed by Murphy and Topel (²⁰⁰⁶) and is the same formula applied to any asset in economic models.

In equation (1), H(t) measures health status, c(t) is consumption of goods different from health services, .(t) is leisure, and S (t,a) is the probability that an individual in cohort . survives to date t. Parameter r reflects inter-temporal preferences and . is the interest rate at which the person can save or borrow. The time horizon is infinite, but the chances of survival are very low at ages just below 100. Income and expenditure must comply with a budget constraint, expressed in equation (2), in which variable . measures the value of assets.

Subject to:

The solution to this problem produces the VLY, v(t), as the sum of the financial flow and the monetary valuation of utility in a period (equation 3). As an intuition on why the first term on the right side is interpreted as a monetary value, we can note that the numerator is an unobserved quantity, while the denominator is the reciprocal of the money required to buy additional utility; thus, the term is a price times a quantity.

The model uses two assumptions concerning the period-specific utility function to facilitate the calculations. First, the relative demands for leisure and consumption are independent of a person’s level of wealth (zero income effects). Second, the utility function includes parameters relative to “survival consumption” (z₀) and intertemporal substitution (σ). These assumptions are imbued in equation (4) and imply the surplus function Φ(z) in equation (5).

The variable Z is a composite commodity that adds consumption to leisure using prices as weights. This is possible because there are no income effects: Z = Z_cC + Z_II.The parameter σ of intertemporal elasticity of substitution plays a main role: if it has a value close to zero, lifetime consumption profiles do not change following alterations in the relative cost of consumption between periods; for example, unexpected, exogenous news of an increase in longevity does not alter planed consumption. An alternate way to express the implications of low σ is that consumption does not respond to the interest rate (the inter-temporal price of consumption). On the other hand, very large values of σ mean that consumers are not willing to sacrifice consumption to increase longevity.

The calculations of v(t) involve: (i) estimation of consumption, income, and leisure profiles using a cross-section econometric model; and (ii) selection of parameters of intertemporal substitution and minimum consumption through price theory and a review of the extant literature. The values of v(t), together with data on probabilities of survival (S) and disability (of which H is a function), and an assumption of the intertemporal discount parameter (ρ) are plugged into equation (1) to obtain the VSL.

Willingness to pay (WTP) for an increase in longevity is defined as the difference in VSL under two different patterns of survival. The calculations are based on the discrete time version shown in equation (6). Superscripts 1 and 2 denote alternative survival scenarios. Timewise, the product of a change in survival probabilities at age α ( ΔS_{a t}, ), the value of a life year and the discount factor R^t-a -generates changes in the WTP(α). This is the value of the increase in longevity ΔS_{a t}, .

A change in the lifetime distribution of survival probabilities can take an arbitrary form. Two scenarios are defined to measure reasonable levels of improvement. One is a homogenous improvement for all countries and age groups, the other assumes convergence towards the levels of the longest-lived countries. The first scenario is a shift proportional to all future ages, and can be described through a parameter λ.

A second scenario follows a pattern of convergence in which all countries approach the longest-lived, in proportion to life expectancy; the country with lowest life expectancy improves proportionally more, the longest lived shows no improvement, and the rest show improvement that is somewhere in between. To construct this, counterfactual survival probabilities for each country and age, S²(a,i), are obtained by applying the following calculation, where T refers to the country with the highest life expectancy:

The VSL includes the variable H(t). H-health is approximated through data on Years lost to Disability (YLD), which is in turn based on disability weights obtained from the Global Burden of Disease database (^{Salomon, et al., 2015}). This assumes that disability weights measure the loss in welfare due to a condition by decreasing the VLY.

A review of the causality framework exposes the potential bias of the results. As stated, it says that the survival patterns are independent of consumption-work decisions, rates of interest and discount, and disability patterns. If data were for individuals, these assumptions would be less acceptable because an individual who learns that she has an improved likelihood of survival or lower expected disability is likely to change her work-consumption decisions. The results below assume implicitly that, for populations, changes in survival probabilities arise in such a way that populations cannot respond quickly to them. Thus, estimates can be thought of as being short-term.

Aldy and Viscusi (²⁰⁰⁷) discuss the calculation of VLY and VSL by age. These studies usually define a hedonic wage equation and use a fatality risk variable determined by the remaining years of life. The VSL is also built as the discounted sum of VLY. However, a main difference with the calculation in this paper is that we do not estimate a wage equation. Instead, we assume a structure of preferences and impose parameter values that vary with per capita income. Thus, we have the equivalent of a hedonic wage equation in the relation between income and survival probabilities. It may be noted that we do not aim to estimate only one value of VSL, but rather a set of values, one for each country.

2.2 Data

This section explains the construction of variables used to calculate the VLY (equation 4): c, l and Y. The information comes from two classes of sources: labor, consumption, and health data are taken from international databases, while parameter values are obtained from extant literature and price theory criteria.

Health data come from IHME (²⁰¹⁷), a rich database that contains information on mortality, illness, and disabilities for almost all countries. Consumption data are from the National Transfer Accounts Project; 23 countries are included in the analysis, using the most recent data (^{NTA, 2019}; ^{Lee and Mason, 2011}). NTA data are mostly cross-sectional and not updated periodically for most countries. Labor data are from the ILO (²⁰²⁰), which has information on labor force participation for all countries in the NTA database.

The selection of countries is dictated mainly by data on individuals’ consumption by age. These are not produced directly in administrative or market process, and the way resources are distributed among household members is not homogeneous in relation to household size and structure. For example, increased longevity does not always imply longer working lives and the reasons are complex due to strong wealth effects on the demand for leisure and to the institutional environment for retirement (including social security and the internal arrangement of households in relation to multi-generational families). Improved data would provide consumption levels by individuals for several years.

For health, the GBD database (^{IHME, 2017}) provides information on death rates and on years lost due to disability (^{YLD; WHO, 2018}) by age.³The values of survival probabilities are obtained directly for 2017. The variable H is built as 1 minus the percentage rate of YLD by country and age.⁴

Leisure is measured using data on rates of participation in the labor market by age, which is available for most countries for several years. Data on hours of work by age is less common because it is produced from high-quality national labor surveys and relatively few have them; ILO data do not provide systematic information on hours by age. We calculate hours of work by age as the product of hours and labor force participation by age. To construct the variable on leisure hours we take the number of hours in a year minus the hours of work.

For earnings, NTA data includes variables by age group. Comprehensive global data sets to study earnings over the lifecycle are not available. To explore patterns across countries, researchers have selected countries with richer data. In a recent study, Lagakos, et al. (²⁰¹⁸) focus on 18 countries with better data and, within that set, on 8 core countries with cross-sectional data over 15 years or more. However, even for that selected group, they focus on income from salaried work. The ILO and the World Bank data have information on earnings from different sources by sex, occupation, and sector, but have not produced harmonized data by age.

In sum, there are no systematic data on leisure, earnings, and consumption variables comparable in quality to the IHME health database. The NTA provides the larger sample of countries with harmonized data on earnings and consumption by age and determines the set of countries in the study. We use the more recent data from the NTA for each country; few countries have more than one year of data available, making it impossible to develop models with time effects.

2.3 Specification of consumption, earnings, and leisure profiles

A regression model is used to obtain values on expected consumption and income over the lifecycle and by age-group. Available data panels are unbal­anced, less so for health data, but more so for labor data, and even more so for consumption data. This explains the strategy of calculating regression models to build life-cycle profiles of consumption, leisure, and labor earnings for each country.

The assumption is that consumption, income, and leisure profiles can be represented by a common model across countries and over time, and the regional parameters are useful to measure differences in behavior across countries. The heuristic is that the shapes of consumption, income and leisure profiles change slowly, and are stationary over time except for a shift in levels determined by national income levels and regional effects. The main argument against this assumption is that longevity increases productive age and time effects on the average consumption and income profiles. However, the lack of panel data for most countries and the long interval of time required to identify those effects precludes their identification.

The regression model relates consumption, earnings, and leisure (j=c,e,l), to powers of age (α) and regional dummies Index i refers to countries and refers to regions of the world. The error term uij is assumed to be independent of age and country, iid across countries). Time effects are not included, a decision dictated by data availability on consumption by age group.

The model provides consumption profiles over the life cycle in intervals of five years (α = 1,…,20) because that is the framework of NTA data. Thus, the profiles are assumed to have age coefficients common to all countries (β_h), and regional effects (γ_hi). The choice of the order of the regression polynomial (h) is made on empirical considerations.

Equation (9) describes the models for per capita consumption, leisure, and earnings at each age α, and region i. The models include dummies for five regions, defined in part by the availability of consumption data: Latin America and Caribbean (Argentina, Colombia, Costa Rica, El Salvador, and Jamaica), Europe (Austria, Finland, France, Hungary, Moldova, Slovenia, Turkey, and United Kingdom), Asia (Cambodia, Singapore, South Korea, and Taiwan), North America (Canada and United States), and Africa (Ethiopia, Ghana, Nigeria, and Senegal). Estimates are weighted by population by age group for each country. For earnings, NTA data have variables for total labor income, and subcategories for employed and self-employment labor income. Here, we use the variable on total income. Calculations use values at ages 15 to 65 relative to the 30-49 age group.

The estimated coefficients are used to obtain profiles of the variables for each region. For consumption and earnings these are multiplied by the per capita income of each country to obtain consumption and earnings profiles by country (equation 10):

3.1 Parameter selection

In addition to the consumption, earnings and leisure profiles, the calculations involve six other parameters: the time discount rate (ρ), the interest rate (r), the elasticities of substitution between consumption and leisure, and between consumption at different points in time (σ), the parameter for minimum consumption (z_o/z), and the number of hours over which the calculation is made.

Murphy and Topel (²⁰⁰⁶) use 1.1 for σ and 0.05 for z_o/z. In a lower income country, a smaller gap between consumption and subsistence consumption is justified (i.e., a higher z_o/z). This follows the meta-analysis of Havranek et al. (²⁰¹³), which indicates that lower income populations are less likely to substitute consumption over time as they are closer to the minimum acceptable levels of consumption. It will be useful to investigate the role of the assumption on z_o/z. On ., Havranek, et al. (²⁰¹³) find average values of 0.5 in an evaluation of estimates for 104 countries. They find that richer households and households in countries with more developed financial markets show higher values of intertemporal elasticity. On the other hand, in a recent review of the literature, Low and Meghir (²⁰¹⁷) refer to the “practice” of using σ at a slightly higher value than 1.

To support the selection of parameters, Table 1 shows estimated average values of one year of life relative to GDP per worker for a country for different combinations of σ and z_o/z(this covariation has the same sign for all countries). When σ approaches zero, the value of one year of life reaches very high values; this is not meaningful because, if true, individuals place a lot of value on the possibility to live a little longer, the VSL is thousands of times higher than market income, but at the same time individuals are not willing to reduce consumption profiles by much. On the other hand, if the elasticity of substitution is large (say, 10), individuals’ may not care whether they consume in one period or another, the VLY approximately equals income, and individuals do not care about longevity. Note also that variations of the VLY near σ = 1.1 are not small because poverty (defined by high values of z_o/z) reduces the value of longevity. Our analysis uses values of z_o/z between 0.2 to 0.05; country-specific values are assigned linearly between those extremes according to GDP per worker. Thus, the wealthiest country has 0.05, and the poorest has 0.2; for example, the values for Mozambique, Chile, Spain, and United States are 0.20, 0.16, 0.11 and 0.05, respectively. For the elasticity of substitution, we use σ = 1.1 for consistency with other studies and because values far from unity lead to absurd counterfactuals (^{McCloskey, 1987}). However, we continue to consider the arguments in favor of values closer to 0.5 (^{Hall, 1988}; Hall and Jones, 2005).

For the interest rate we assume a value of 3% as reference, although the long-term cycles of interest rates and the cost of liquidity for families may vary substantially over time and within and across populations. Rates on government bonds have varied between the nineties and the first two decades of the millennium. In wealthier countries, real rates went from levels above 3% to near-zero or negative, while in emerging economies they went from near 10% to values not far from zero. Additionally, families rarely have access to the interest rates paid by government debt and may face steep charges for borrowing money. In wealthier countries financial markets offer lower costs of access to financing (including in the form of health insurance), while in other countries families usually find themselves in a range between no access to formal financing to expensive and constrained financing. For families, access to health insurance defines the interest rates that apply to the consumption of expensive health services, and lack of insurance often means that access to such procedures is closed.

The value of 8,766 hours available per year corresponds to 24 hours a day times 365.25 days; Murphy and Topel (²⁰⁰⁶) use a parameter of 4,000. A related issue is that the marginal value of leisure time is approximated with the salary. For workers, the argument relates to the analysis of the marginal decision to work longer hours, not the value of the time available to the person at other margins. How to value leisure time remains a pending issue. Aspects related to the home economy must be included in order to model the alternative margins, which are not only work and leisure, but also domestic work, sleep, and possibly others (^{Rosen, 1974}).

3.2 Estimation of c, l, e, VLY and VSL

This section shows the estimates for equation 9 and explains the selection of the benchmark specifications, providing the data used to calculate VSLs. An Appendix details the results and is available from the author upon request.

In general, consumption follows a flag pattern (increases from childhood to early adulthood). Figure 1 shows consumption by age groups, standardized by the age group 30-49. Wealthier countries tend to have increases after prime ages, while other countries have flat or even declining profiles, as is the case of African countries. The causes may be benefits of the welfare state that promote consumption in old age and higher private savings in wealthier countries, and shorter life expectancies and lower qualities of life in old age that promote downward patterns in poorer countries. For example, the top left panel shows that compared with Canada and the United States, consumption in El Salvador is lower at older and younger ages, and is flatter over the lifecycle.

We chose the quartic model to construct the profiles of lifetime consumption. The quartic terms are significant only for the African region, but they seem to capture part of the difference in behavior between wealthier long-lived countries and poor short-lived countries. Figure 2 shows the predicted and actual values using the quartic models for four countries in the sample. We are evidently not capturing the upward trend that persists in wealthier countries after 80. On the other hand, data for those age groups is scarce and the weight for the population is low, so we do not expect our calculation of VLY to be affected very much. Many countries will reach life expectancies of above 80 during the present century, and the way savings, consumption, and labor will change will be a main issue for research, but something that we do not deal with here.

Longer lifespans mechanically change the pattern of consumption, and the relation between wealth and consumption over the life cycle cannot be expected to be homogenous: expenditures on education, health entertainment, leisure, and other goods likely have a complex relation to wealth, and also involve marital behavior and other factors from which we must abstract. Human capital decisions affect health in complex ways as pointed out by McFadden (²⁰⁰⁸). Thus, the regression model is only an approximation to the pattern of consumption over the lifecycle within regions where countries face similar wealth and longevity constraints. The endogeneity of these patterns is certainly a major concern in this research.

In sum, we use the quartic model to predict the profile of consumption over the life cycle for all countries. We assign a regional profile to each country in the GBD sample following a simple regional criterion and multiplying the national consumption per capita by the profile to obtain consumption per capita by age.

To model leisure, we use data on labor force participation rates (Figure 3) and average hours of work in 2010 (Figure 4). Figure 5 shows the calculated average leisure hours. Leisure is a conventional term, but it is important to know that it does not mean unproductive time. Young individuals do not work mostly because they are investing in human capital, and older adults that retire can also invest time and a larger share of their income in healthier lifestyles. GDP per worker is the variable used to shift the level of the consumption and income profiles to monetary levels (/ in equation 10) and is shown in Figure 6. Figure 7 illustrates the profiles of labor income from the NTA database. Important surveys on the topic have been produced by Willis (¹⁹⁸⁶), Card (¹⁹⁹⁹), and Heckman, Lochner and Todd (²⁰⁰⁶). A key message is that a concave earnings pattern results mainly from school and on-the-job human capital decisions. We expect longevity to affect investment in human capital. Thus, longevity affects investments in human capital, generating potential biases in the estimation of VLY.

Figure 8 illustrates the results for some of the countries in the calculated model. The solid lines correspond to average observed labor earnings, and the dotted lines to predicted values. Projections are based on the cubic model (column c in Table 2).

The present values of income and consumption are assumed to be equal across the life cycle. For each country, predicted consumption and income over life are Equality means that a scalar α makes While no special meaning is assigned to α here, a concern is that measurement errors that determine the gap between average earnings and consumption correlate with longevity. In particular, the intergenerational transmission of wealth and capital income are likely higher in wealthier societies and can covary with health and longevity. Figure 8 shows the resulting profiles for a subsample of countries. The distance between the original consumption profile (these are predicted values from the regression model) and the profile that equates present values of consumption and income is positive in all cases, which means that true income is likely higher than the values reported by the variable we use. On the other hand, the distance is larger for wealthier countries, possibly due to higher capital income, which is not included in the variable we use to measure income. The consumption and income profiles are used to calculate full income and full consumption using data on amount of work. Figure 10 shows the results scaled to monetary units using GDP per worker.

The discounted and non-discounted VSL for the benchmark case is shown in Figure 9. The remaining VSL by age is shown in Figure 10, which shows that wealthier countries have a much higher VSL. This does not mean that the lives of their citizens are more valuable, only that they are willing to pay more for improvements in longevity. Interest rates also have a large impact on the calculations for the remaining VSL, especially for children, due to the longer discount horizon (Figure 11). Under real rates above 2.5% that prevailed in the eighties and nineties, values for children are one half or less than those obtained at rates below 1%, which more closely reflect the reality of financial markets since 2008.

VLY estimates are higher when the elasticity of substitution is smaller (time provided by extra longevity is a more highly-valued good), and the ratio of GDP per worker to VLY is lower in wealthier countries; an extra year of life is worth only 75% of a year of productivity in Australia, but around 103% in Argentina and Brazil, and 133% in Senegal (Table 2, with σ = 1.1). Table 1 made an additional point: given a value of σ, the impact of being closer to subsistence consumption is significant, as shown in Table 3.

This section has described the data and calculations needed to obtain {y,c,l} for each country at each age, and shows calculations of v and VLY.