2
The Formal Demography of Population Aging, Transfers, and the Economic Life Cycle

Ronald D. Lee

INTRODUCTION

The human life-cycle has two stages of dependency—childhood and old age—separated by a long stage of surplus production. Dependent age groups are sustained by flows of resources upwards and downwards by age from the more productive stage in the middle. These resource flows occur through three institutional channels: the family, the public sector, and financial markets. Through each channel, resource reallocation takes one of three forms: capital formation, credit transactions, and interage transfers. As fertility and mortality decline, the population age distribution shifts toward older ages, which changes the terms on which these resource flows take place. The public sector, through which some of the transfer flows are channeled, is particularly sensitive to the consequences of these age distribution changes. The governments of many industrial nations are concerned that transfers to the elderly, which are already costly, will be raised rapidly by population aging in the early twentieth century. Many Third World nations, preoccupied in the past with the costs of their young populations'

The author is grateful to Robert Willis for many helpful discussions and for the extensive use made of his earlier work, and to Michael Anderson for excellent research assistance. James Poterba and Samuel Preston provided helpful comments on an earlier draft. Research support from the Institute for International Studies of the University of California at Berkeley is gratefully acknowledged.



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Demography of Aging 2 The Formal Demography of Population Aging, Transfers, and the Economic Life Cycle Ronald D. Lee INTRODUCTION The human life-cycle has two stages of dependency—childhood and old age—separated by a long stage of surplus production. Dependent age groups are sustained by flows of resources upwards and downwards by age from the more productive stage in the middle. These resource flows occur through three institutional channels: the family, the public sector, and financial markets. Through each channel, resource reallocation takes one of three forms: capital formation, credit transactions, and interage transfers. As fertility and mortality decline, the population age distribution shifts toward older ages, which changes the terms on which these resource flows take place. The public sector, through which some of the transfer flows are channeled, is particularly sensitive to the consequences of these age distribution changes. The governments of many industrial nations are concerned that transfers to the elderly, which are already costly, will be raised rapidly by population aging in the early twentieth century. Many Third World nations, preoccupied in the past with the costs of their young populations' The author is grateful to Robert Willis for many helpful discussions and for the extensive use made of his earlier work, and to Michael Anderson for excellent research assistance. James Poterba and Samuel Preston provided helpful comments on an earlier draft. Research support from the Institute for International Studies of the University of California at Berkeley is gratefully acknowledged.

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Demography of Aging high child dependency burden, are now beginning to worry instead about the costs of impending population aging. This chapter develops an accounting framework for evaluating systems of interage transfers, and examines how such systems are affected by changing population age distributions. To understand the role of transfers in achieving a desirable allocation of consumption over the life-cycle, it is necessary to consider them in relation to the other forms of reallocation: credit and capital. There is a rich and controversial literature on the relation between transfer systems and capital accumulation: Does life-cycle saving account for the capital stock of industrial nations (Modigliani, 1988; Tobin, 1967)? Or is the desire to leave bequests responsible (Kotlikoff and Summers, 1981, 1988)? Do public sector pension systems undermine private saving (Feldstein, 1974)? Or do elderly parents simply increase their familial transfers to their children to offset the pensions (Barro, 1974)? Transfers may be used to achieve efficient allocations over the life-cycle that are unattainable via competitive market mechanisms (Samuelson, 1958), and if transfers upwards or downwards by age are needed to achieve efficient allocations, that fact tells us that the population growth rate is less than or more than the optimal rate (Samuelson, 1975, 1976; Willis, 1988; Lee, in press, b). These are issues that a coherent accounting framework may help to clarify. The theoretical basis for a comprehensive framework for studying the reallocation of resources across age in general, and transfers in particular, has been laid by economic and demographic research over the past 35 years. Macroeconomic models with "overlapping generations" sprang from the seminal work of Samuelson (1958) and, later, Diamond (1965). The literature has developed to the point that there is now a textbook that teaches macroeconomic theory entirely from the point of view of a simple model of economies with overlapping generations (McCandless, 1991). The models have been used to explore such diverse topics as the existence of money, the rate of interest, aggregate savings rates, the Ricardian equivalence theorem, optimal population growth rates, economic fluctuations, and so on. These important developments in economic theory pave the way for a deeper integration of demography and macroeconomics than has yet proven possible. However, perhaps because of the wish to examine nonsteady-state situations, the demographic models used by most mainline economists are very simplistic: the life-cycle typically consists of two broad age groups, workers and retirees, or young and old, with perfect survival until the end of the second. Childhood is often ignored, and life really begins at labor market entry. This life-cycle incorporates only one period of dependency rather than two. In such a demographic world (used all the way through the McCandless textbook), some of the most basic questions cannot be properly posed or will receive misleading answers. This is true of most questions

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Demography of Aging concerning aging, such as the effect of mortality change on saving behavior or capital accumulation, or the effect of slower growth due to lower fertility, which will alter the share of life-cycle resources devoted to children and alter saving behavior in complex ways (see Mason, 1987). Many interesting empirical questions are also overlooked, including those that involve the interaction of age distribution with institutional context. Largely ignored by mainline economists, a few economic demographers have begun to explore the interface of richer demographic models and the overlapping generation models of economists. Arthur and McNicoll (1978) published a brief but seminal comment on a paper by Samuelson (1975). Willis (1988), building on Gale (1973), published an important paper incorporating familial and public transfer systems into an accounting framework that related flows to stocks, and on which the work proposed here leans heavily. Lee (1980, in press, a) and Lee and Lapkoff (1988) also contributed analytic and empirical studies and developed early versions of the approach proposed here. On the more demographic side, Keyfitz (1985, 1988) and Bourgeois-Pichat (1978) developed formal analyses of the demography of funded and unfunded pension systems, and Preston (1982) developed and applied analytic tools relating the distribution of characteristics over the individual life-cycle to the distribution of characteristics in a stable population, and showed how these distributions are affected by changes in fertility and mortality. This chapter develops a conceptual and accounting framework that builds on these two approaches and seeks to bridge the gap between them. Parts are heavily influenced by Willis (1988). The past literature in both demography and economics has paid insufficient attention to mortality change, and this chapter also offers some advances in that direction. At the same time, the analysis here is limited to comparative steady states and mainly to the special case of steady states with optimal saving and investment. These are serious limitations in the United States since recent decades have been marked by major changes in mortality, rates of inflation, real interest rates, regulations governing Social Security and private pensions, rates of real wage growth, rates of appreciation of housing, and so on. SOME ANALYTICS OF AGING IN STABLE POPULATIONS In a closed population, population aging can occur due either to decline in fertility or to decline in mortality, and these have quite different effects. Nonetheless, the distinction between the effects of changing fertility and changing mortality is not the most helpful one. Let p(x) be the probability of survival from birth to age x, let B(t) be the number of births at t (actually, the number between t and t + dt), and let n be the growth rate of the stable population. Then the stable population age

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Demography of Aging distribution at time t is N(x,t) = B(t)e-nxp(x). We can divide by population size to find the proportional age distribution, be-nxp(x), where b is the crude birth rate; b is simply a scaling factor here, the same for every age. The natural decomposition, therefore, is into a "rate of growth" effect, due to n, and a "life-cycle" effect or "individual aging" effect, due to the survival schedule, p(x). When fertility changes, this affects the rate of growth, n, but not p(x). Therefore fertility has only a rate of growth effect. Higher fertility raises n, which increases the size of more recently born cohorts relative to older ones and therefore makes the population younger. A mortality decline is of course reflected in p(x), which leads to individual aging, tending to make the population older. However, lower mortality also raises the population growth rate, n, since more women survive to childbearing age. In this way, lower mortality tends to make the population younger. The net effect of mortality decline is therefore ambiguous, and can in fact make the population either younger or older, depending on the pattern of mortality change, which in turn depends on the initial level of mortality. Figure 2-1 depicts this decomposition. The Rate of Growth Effect Fertility Change Let us consider more formally the way that fertility and mortality affect the population age distribution through the rate of growth and individual Figure 2-1 Fertility, mortality, and stable age distributions.

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Demography of Aging aging. Let F be the total fertility rate, let sr be the proportion of births that is female, and let Af be the mean age of childbearing in the stationary population; p(Af) is then the proportion of female births surviving to the mean age of childbearing. To a linear approximation, the population growth rate is given by1 The effect on n of a change in F is found by differentiating this expression for n Note that while the effect of fertility change depends on its initial level, it is independent of the level of mortality, to a first approximation. Mortality Change Let i be some index of the level of mortality, such that the survival probabilities vary with i. For any level of i there will be some corresponding level of life expectancy at birth, e0. The scaling of index i is arbitrary. It will be convenient to choose a scale such that in the neighborhood of any given level of life expectancy, a one-unit change in i corresponds to a change of 1 year in life expectancy; that is, d(e0)/di = 1. The effect on n of a change in mortality, indexed by i, is given by The effect of mortality decline on the population growth rate is independent of the level of fertility, to a first approximation. Note that p(Af) is bounded above by unity, so that once mortality is already very low, further declines in mortality can have very little effect on the population growth rate. Figure 2-2 plots ∂n/∂i as calculated from Coale-Demeny model life tables, where i is scaled so that a unit change corresponds to a gain in e0 by one year. Figure 2-2 shows how ∂n/∂i changes, depending on the initial level of life expectancy, for life expectancy from 20 to 80 years. When mortality is initially very high, a 1-year gain from a life expectancy of 20 years to a life expectancy of 21 years leads to an increase in the population growth rate by 0.16 percent per year. When mortality is very low, a 1-year gain in life expectancy from 79 years to 80 years would raise the population growth rate by less than 0.01 percent per year, a tiny amount. 1   In the following expression, In [p(Af)]/Af equals minus the mean age-specific growth rate between ages 0 and Af.

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Demography of Aging Figure 2-2 The effect of a 1-year gain in life expectancy on the population growth rate, from differing initial levels of life expectancy (∂n/∂i). SOURCE: Calculated from Coale-Demeny (1983) model life tables (west females). The Life-Cycle Effect, or Individual Aging The shape of the individual life-cycle is described, from a demographic point of view, by the distribution of expected person years lived by age. Ex post, every individual simply lives until death, which is a discrete event. But in prospect, the individual faces a series of probabilities of survival, p(x), which change with age. We can also think of p(x) as describing the density of expected person-years lived at age x for an individual at birth. Expectation of life at birth, e0, is simply the integral over all ages of p(x); put differently, it is the sum over all ages of the expected person-years lived. The shape of the demographic life-cycle depends on the severity of the mortality regime: under high mortality, the proportion of the life-cycle lived in the third stage, old age, is relatively small, and under very low mortality it is relatively high. Figure 2-3 plots the number of person-years lived in each of the three stylized life-cycle stages for different mortality regimes indexed by e0, life expectancy at birth.2 When life expectancy is 2   I have used the Coale-Demeny (1983) model life-table system, west female.

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Demography of Aging Figure 2-3 Person-years lived in each life-cycle stage, by life expectancy. SOURCE: Calculated from Coale-Demeny (1983) model life tables (west females). 20, only 0.6 of a person-year is lived at age 65 and above; when life expectancy is 80, 16 years are. This is a 27-fold increase. Person-years lived in childhood increase from 7.2 to 14.9, just more than doubling. Person-years lived in the working ages rise from 12.2 to 49.0, quadrupling. This is a different pattern than most of us would expect, since it is well known that, historically, mortality has declined most rapidly in childhood and least rapidly at the older ages.3 Figure 2-4 shows how the proportion of the life-cycle spent in each stage changes as we pass from the high-mortality regime to a low-mortality regime. The proportion of the life-cycle spent in the working years changes little; the proportion spent in childhood declines markedly; and the proportion spent in old age increases dramatically. Recall that life expectancy is the sum over all ages of p(x). If life 3   It is easy to see, however, that proportional change in person-years lived at any age is nondecreasing with age. If mortality falls at all ages, then this proportion will increase monotonically with age, even if declines are greatest at younger ages.

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Demography of Aging Figure 2-4 Proportion of life cycle lived in life-cycle stage, by life expectancy. SOURCE: Calculated from Coale-Demeny (1983) model life tables (west females). expectancy increases by 1 year, then the sum of p(x) must also increase by 1 year. Put differently, the plot of the changes in p(x) when life expectancy changes by 1 year gives us an additive decomposition of that 1 year into gains in person-years lived at the various ages. Figure 2-5 plots dp(x)/di for various initial levels of life expectancy, showing how these gains in person-years are distributed across the three life-cycle stages, and how that distribution varies from initially low to initially high levels of life expectancy.4 For example, by starting at a life expectancy of 20, if life expectancy were 4   We could, for example, think of i as equaling 0.4 of one ''level" in the Coale-Demeny model life-table system, since one level corresponds to an increment of 2.5 years of e0. More generally, however, we could simply take two survival schedules, call them p(x) and p^(x), from two actual life tables with life expectancies at birth of e0 and e0 ^. Then dp(x,i)/di is estimated by [p(x)-p^(x)]/(e0-e0^). We could also readily derive dp(x)/di under simple assumptions about mortality change, such as the "neutral" mortality assumption of constant additive changes in the force of mortality at all ages, or alternatively on the assumption that the force of mortality changes proportionately at all ages. Both of these assumptions are quite poor as approximations to the age pattern of actual mortality change, however.

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Demography of Aging Figure 2-5 Person-years gained in each life-cycle stage when life expectancy increases by 1 year, by initial level of life expectancy. SOURCE: Calculated from Coale-Demeny (1983) model life tables (west females). to increase to 21 then, according to the Coale-Demeny life-table system, this would consist of 0.22 year for children, 0.70 year for the working ages, and 0.08 year for the elderly. When initial life expectancy is 75, however, as it is now in the United States, then a gain of 1 year in life expectancy would be distributed as only 0.04 year to children, only 0.34 year to the working ages, and 0.62 year to the elderly. We have now considered the effects of fertility and mortality on the population age distribution through the rate of growth effect and the life-cycle effect. While I have described the sign and magnitude of each effect, I have not combined them all in an appropriately weighted combination to find their net effect on population aging. I am not doing this here, in part because it is not useful for the analysis later in this chapter, and, in part, because it has been done elsewhere (see, for example, Coale, 1972). Suffice it to say here that fertility decline unambiguously causes population aging. Mortality decline starting from very high mortality actually makes populations younger, as the rate of growth effect overwhelms the individual aging effect. However, when starting from lower levels of mortality, the

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Demography of Aging individual aging effect outweighs the rate of growth effect, and the population becomes older. THE ECONOMIC LIFE-CYCLE AND AGGREGATE CONSTRAINTS We now turn from the formal demography of aging to the economics of population age distributions. We can carry out our analysis at the level of either the individual or the household. Individuals live in households, and consumption in households is subject to resource pooling, scale economies, and public goods. Furthermore, children are not responsible for financing or choosing their own level of consumption; instead, this is done by their parents. For these reasons, the household accounting framework is appealing. However, the household framework also presents serious analytic difficulties: the dissolution and reconstitution of households over the life-cycle, the presence of multiple adults of different ages in the household, and covariation of household headship propensities with earnings or wealth of individuals. Because of these difficulties with the household framework, I employ the individual accounting framework predominantly here, despite its occasional artificiality in dealing with children. For a household-based analysis, see Lee (in press, b). Some technical details on the household accounting framework are provided in the appendix to this chapter. The Economic life-cycle The human life-cycle begins and ends with stages of dependency, in the sense that consumption exceeds labor earnings. This generalization applies on average to age groups, but not necessarily to individuals so far as old age is concerned. The average shape appears to be universal, although ages and extent of dependency may vary widely from population to population. It arises from the combined influence of physiology, culture, institutions, and economic choice, in ways that we take as given.5 Figure 2-6 shows profiles of labor earning and consumption for U.S. households by age of respondent for 1987. Earnings are before taxes and include employers' contributions to Social Security, as well as fringe benefits. Quinn and Burkhauser, in this volume, discuss some of the economic and social factors shaping this age profile and the way it has changed over time. If labor markets are competitive, then this age profile reflects the 5   There is an extensive literature on the economics of retirement (Hurd, 1990) and on the physiology of aging. A delayed exit from childhood dependency is not simply a function of the educational system; long delayed transitions to economic adulthood are also observed in some traditional preindustrial societies (Sahlins, 1972; Bledsoe and Cohen, 1993).

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Demography of Aging Figure 2-6 Labor income and consumption by age of household respondent. SOURCE: Calculated from 1987 U.S. Consumer Expenditure Survey. efficiency of labor at each age, as well as the hours of labor supplied, for surviving individuals at each age, averaged across sex. Consequently this age profile can be used to calculate the aggregate quantity of labor in efficiency units as a weighted sum of the population age distribution. When the population age distribution changes, for example as a result of population aging, this age profile permits assessment of the consequences for aggregate production. Consumption includes in-kind public sector transfers (health, education, food stamps, housing benefits) and the imputed value of services from owner-occupied housing, automobiles, and consumer durables (for details, see Lee, in press, b). Reallocation Across Age and Time The household age profiles in Figure 2-6 indicate that old households are able, on average, to consume far more than they produce through their own labor, whereas young households consume slightly more than they produce. Evidently there has been a reallocation of output from more productive to less productive age groups. Such reallocations take one of three forms: credit, capital accumulation, or transfers. The defining feature of transfers is that unlike borrowing or lending, they involve no quid pro quo.6 6   As I am using the term, one could not ask whether there was an exchange motive for a transfer. If there is an exchange motive for a familial transaction then it involves some form of familial credit. To the extent that implicit interest rates for the transaction differ from the market interest rate, a transfer takes place.

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Demography of Aging Figure 2-10 Person-years of life gained for 1-year increase in life expectancy versus labor earnings minus consumption, U.S. data, 1985. SOURCE: The distribution of person-years gained is based on recent Swedish life tables. The data on earnings minus consumption are taken from Lee and Lapkoff (1988), based on the 1985 Consumer Expenditure Survey. It is instructive to plot ∂p(x)/∂i against y1(x) -c(x) for actual data. Figure 2-10 does this for U.S. data. It is striking that the greater part of the gains in person-years lived occurs at ages for which consumption vastly exceeds earning. Indeed, 66 percent of the gains occur at age 65 or over.15 Computing the population-weighted integral as described by the right side of the equation, and dividing by the present value of life-cycle consumption so that both sides are expressed as proportions, yields a value of -0.009, or nearly -1 percent. The interpretation is that a 1-year increase in life expectancy requires adjustments to life-cycle consumption or labor earnings equal to 0.9 percent of their present value at birth. These adjustments could take the form of a 0.9 percent reduction in consumption at every age; an increase in labor supply by 0.9 percent at every age, say from 40 hours per week to 40.36 hours per week; or a postponement of retirement by five months from 15   For the Swedish mortality on which this figure is based.

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Demography of Aging age 65 (if the productivity at age 65 is assumed to equal the life-cycle average, and the rate of population growth is zero; if productivity is below average, or the growth rate is positive, then retirement would have to be postponed longer).16 I call this the life-cycle effect, or individual aging effect, of declining mortality because it reflects the simple need to provide for more years of life, in this case life lived in retirement. One should not think, however, that mortality decline is necessarily costly in this way. A similar calculation can be carried out for a hypothetical high-mortality population with initial life expectancy of 20 years, and earning and consumption profiles as reported in Mueller (1976), intended to be representative of Third World agricultural populations. In this case, the life-cycle effect is actually positive! Person-years of life gained are predominantly in the working ages, as can be confirmed by reference to Figure 2-5, which shows that at a life expectancy of 20, 69 percent of the gain accrues to the ages 15-64. Mortality Decline and Transfers There is a corresponding equation constraining adjustments to the transfer system, which holds not only for the golden rule case, but for the general case as well: for fixed n. Figure 2-11 shows the information needed to make this calculation for Social Security. The integral equals about $3,200 over an individual life-cycle. To put this on an annual per capita basis, we can simply multiply by b, which yields $43 per year. The interpretation is similar to previous ones. When person-years are gained at ages that, on net, receive transfers, this is costly. The integral of gains and losses on the right must be balanced by adjustments to age-specific transfer flows on the left. The effect of mortality change on health costs cannot be treated this simply, however. An appropriate analysis must treat separately the health costs of those who survive and the health costs of those who die. Those who die impose the heavy costs of a terminal illness. When mortality 16   If the population growth rate is positive, then earnings at older ages are more heavily discounted over the life-cycle (or older people make up a smaller share of the population in a cross-sectional interpretation), and therefore retirement ages would have to rise by more. Lower survival to old age has a similar effect.

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Demography of Aging Figure 2-11 Annual net Social Security benefits and person-years of life gained (for a gain of 1 year in life expectancy) by age of individual. SOURCE: Tax and benefit data calculated from 1987 U.S. Consumer Expenditure Survey. Mortality data calculated from recent Swedish life tables. changes, the distribution of deaths by age changes, and hence the age profiles of health costs themselves change. This can all be handled in a straightforward way, but is not done here. Mortality Decline and the Rate of Growth Effect The derivatives above hold the population growth rate fixed. In practice, as discussed earlier, mortality decline leads to more rapid population growth, because more female births survive to the reproductive ages. The full derivative is as follows:

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Demography of Aging What is added here is the factor ∂n/∂i which is multiplied by the rate of growth effect discussed earlier and has been shown to equal transfer wealth, T/b. We have seen earlier that ∂n/∂i is close to zero in low-mortality populations, but that it is substantial in high-mortality settings. In low-mortality settings such as the United States, therefore, the effect of mortality decline is essentially equal to the life-cycle, or individual aging, effect: people live longer, and their additional years are years of leisure. They must either consume less at each age or work longer in order to pay for the gift of longer life. We have seen that each additional year of life expectancy requires a reduction in consumption, or an increase in labor supply, of about 1 percent. In high-mortality settings, ∂n/∂i is substantial, so that mortality decline makes populations younger. Because transfers are downwards in such populations, a younger population is costly. However, because there is little capital in such societies, capital dilution is presumably relatively unimportant. (Increased pressure on fixed resources probably is important, but is not included in this analysis.) Finally, the life-cycle effect is relatively small and positive, since mortality decline adds years of life mainly during the working years. The net effect is that mortality decline in both high-mortality and low-mortality settings has similar consequences, but for very different reasons: a 1-year gain in life expectancy entails a 1- percent reduction in the present value of consumption or a corresponding increase in earnings. We see, then, that when mortality declines in high-mortality settings, the rate of growth effect dominates. Higher rates of return are earned on life-cycle wealth, which is held mainly in the form of transfer wealth. But since net allocational flows are downwards by age, such populations hold net transfer debt, and a higher rate of return is therefore costly and leads to reduced life-cycle consumption. When mortality declines in low-mortality settings, however, there is very little effect on the rate of growth, so the life-cycle effect dominates. The rate of return to life-cycle wealth is unchanged, but individuals must provide for more years of retirement, which is costly in terms of forgone consumption or leisure earlier in the life-cycle. In the special and historically inaccurate case of a neutral mortality decline, in which the absolute decline in the force of mortality at every age is equal, the rate of growth effect on the age distribution exactly offsets the life-cycle effect. Longer years of retirement could be paid for exactly by the increased rate of return on life-cycle wealth, requiring no readjustment of life-cycle consumption or earning profiles—if we ignore the effect of capital dilution. The results of this section can be compared to those of Kotlikoff (1989:359),

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Demography of Aging who reaches an apparently very different conclusion: "... increasing the length of life, including productive life, appears to permit a higher level of consumption in every year that an individual is alive." The principal difference, I believe, lies in Kotlikoff's assumption that labor supply increases at some ages over the life-cycle; I treat this as a costly reduction in leisure. CONCLUSION: RESEARCH DIRECTIONS This chapter has developed an economic-demographic age accounting framework with strong links to formal demography, as well as links to various models and themes in economics. These economic links include overlapping generations, optimal population growth, life-cycle saving, the bequest motive, generational accounting, and private responses to public transfer programs. I believe that further development of this interface of formal demography and macroeconomics holds promise for theory, measurement, empirical work, and policy-oriented research. Research needs can be divided into empirical applications of the basic framework, on the one hand, and further development of the accounting and analytic framework, on the other. Empirical Applications The framework described here can be used to organize, summarize, and interpret data on transfer systems and the life-cycle. As long as we use synthetic cohort methods, which assume that cohort profiles can be constructed from cross sections, the data necessary to implement the framework appear to be widely available in both developed and Third World populations. Because the analysis requires only aggregate age profiles rather than individual data, information can be drawn from differing sources and pooled, with a basic household expenditure survey providing much of what is needed. Data on bequests are an important exception, and for Third World populations, it may be necessary to develop measures of within-household transfers. More experience with application of the framework will indicate whether currently available data are adequate. If we abandon the synthetic cohort assumption, data requirements become very severe, because it is then necessary to reconstruct the life histories of each cohort. Some of the work by Kotlikoff and his collaborators makes only partial synthetic cohort, steady-state assumptions (Kotlikoff and Summers, 1981; Auerbach et al., 1991).17 17   For example, Kotlikoff and Summers (1981) assume that the shapes of the age profiles of labor earnings and consumption are fixed, while allowing the levels of the profiles to vary by historical period based on estimated national aggregates for labor income and consumption. This is a partial synthetic cohort assumption.

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Demography of Aging One particularly promising use of the framework would be to shape and inform the development of age-based national accounts, continuing the work begun by generational accounting (Auerbach et al., 1991). Comparative cross-cultural and historical estimation also appears possible and should provide an interesting view of the transition from familial to public transfer systems as the welfare state develops, and perhaps also afford some insights into how the growth of public sector transfers affects life-cycle consumption profiles and capital accumulation. Some aspects of the resource flows from one ethnic group to another can be investigated within this framework; for example, do transfers flow from younger nonwhite populations to older white populations in the United States? It would also be very interesting to incorporate immigration, because the transfer patterns of subpopulations arriving as young adults would be quite different. There is a literature on the effect of migration on population age distribution (e.g. Schmertmann, 1992). Similarly, it may be possible to address gender issues. To do so properly would require introducing time use into the accounts, so as to treat explicitly the productive use of home time. Most of these extensions would require further methodological research. Methodological Research Although the basic framework is quite general, its implementation here is confined to a doubly special case: first, to steady states, and second, to golden rule economies with r = n. For calculations of the various forms of wealth, it is a simple matter to relax the golden rule assumption. However, to analyze the comparative statics of aging without the golden rule assumption would require making additional behavioral assumptions in order to determine ∂r/∂n, for example, by specifying a life-cycle utility function as in Tobin (1967). Serious policy-oriented work requires relaxation of the steady-state assumption, so that transitional phases can be considered. There are two distinct aspects of the steady-state assumption. The first is the assumption that age profiles are changing only at a constant exponential rate, so that synthetic cohort estimation of the profiles is possible from a single cross section. Relaxing this assumption would impose very heavy demands for detailed longitudinal data over many decades. The second is that the population and economy are in steady state; this assumption is difficult to relax analytically, but it can in principle be handled by appropriate macrosimulation, elaborating on the methods used by Auerbach and Kotlikoff (1987). There are a number of other issues that need to be resolved, some straightforward and some more difficult. Both the individual life-cycle framework and the household life-cycle framework require attention to conceptual as well as measurement issues. Education should be incorporated as

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Demography of Aging a form of capital formation. Mortality change should be modeled in such a way that morbidity and labor efficiency change at the same time. Bequests and terminal illnesses should be made to depend not on age-group membership, but rather on age at death. Behavioral Theory This chapter has primarily addressed questions of accounting and has paid scant attention to behavioral issues. But there are important related research literatures on why people make familial transfers, on the rationale for public sector transfer systems (Becker and Murphy, 1988), and on the relations between public sector and familial transfers, as discussed earlier. There is also an important literature on the relation of familial and public transfer systems to fertility behavior (Caldwell, 1982; Willis, 1980, 1987). Advances in the conceptualization and description of transfer systems may also inform and stimulate further work on these important issues. Although a great deal of work evidently remains to be done, the time is ripe for a synthesis of work carried out over the past three decades by formal demographers, economic demographers, and economists working on age-distributed macroeconomic models. APPENDIX: HOUSEHOLDS As shown earlier, in a stable population the proportion of people age x is be-nx p(x). Let h(x) be the household headship rate at age x. The age distribution of household heads will then be bh e-nx p(x)h(x), where bh is the birth rate for households, given by This could be viewed as 1 divided by the discounted expected years of household headship over the life-cycle. The undiscounted expected number of years of headship is simply which for the United States in 1987 was about 28.5 years. The average headship rate, h, is given by the integral over all ages of the stable age distribution for individuals in the population, weighted by h(x). Adult individuals often move through a succession of households as they age over the life-cycle. This does not cause problems for the accounting of budgetary flows. It does cause problems, however, for the account-

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Demography of Aging ing of stocks. The approach taken here is to use household headship rates to translate the household level flows into average pseudoindividual flows by multiplying by headship rates. Then these pseudoindividual flows are cumulated to get stocks, calculated per original member of the birth cohort. These stocks can then be reexpressed on a per household basis, if desired, or left at the individual level. For example, by letting the superscripts h and i denote household and individual flows, the equation for cumulation of life-cycle wealth would be To find W we would weight this by the size of individual birth cohorts as in the main text. Then if desired we could calculate the aggregate life-cycle wealth per household by dividing this individual measure by the aggregate household headship rate. Unfortunately, these analytic expressions gloss over two serious complications. First, not all adults in the household need be the same age. When adults of very different ages are grouped together in households, as is sometimes the case, the age profiles for households will be far less informative about the life-cycle profiles of individuals—and it is ultimately individuals in whom we are interested. In many data sets, women will rarely be designated head if a male is present, so female ages will be systematically misrepresented by these procedures. The kind of flipping back and forth between accounts for households and for individuals that was just done for wealth accounting may then be incorrect and give misleading results. This problem will be particularly acute in Third World settings. Second, household headship is typically not distributed randomly across individuals of a given age. Instead, it may be that household headship is associated with economic variables of interest, such as labor earnings, consumption, or receipt of transfers. In this case, observing the economic flows into and out of households headed by a fraction of the population at some age, and then implicitly allocating a share of those flows to all people at that age, will be misleading. One way to lessen the distortions arising in these ways is to take the ages of all adults in the household into account. A simple way to do this would be to randomly choose one of the adult household members to designated the head, or better, the household reference person. A more efficient way would be to allocate a share of the household resources to each adult, keeping track of their individual ages. Such procedures can eliminate much of the bias in working at the household level.

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