Ronald Lee
Demography and Economics
University of California
2232 Piedmont Ave
Berkeley, CA 94720
e-mail: rlee@demog.berkeley.edu
Shripad Tuljapurkar
Morrison Institute for Population and Resource Studies
Biological Sciences
Stanford University
Stanford, CA 94305
We are grateful to Timothy Miller and Michael Anderson for assistance. John Wilmoth and Kenneth Wachter made helpful comments on an earlier draft. Lee's research for this paper was funded by a grant from NIA, AG11761. Tuljapurkar's research for this paper was funded by a grant from NICHD, HD32124. The authors would also like to acknowledge support by Berkeley's Center for the Economics and Demography of Aging.
We analyze in three steps the influence of the projected mortality decline on the long run finances of the Social Security System. First, on a theoretical level, mortality decline adds person years of life which are distributed across the life cycle. The interaction of this distribution with the age distribution of taxes minus benefits determines the steady state financial consequences of mortality decline. Second, examination of past mortality trends in the US and of international trends in low mortality populations, suggests that mortality will decline much faster than foreseen by the Social Security Administration's forecasts. Third, based on recent work on stochastic demographic forecasting, stochastic forecasts of the system's actuarial balance are derived, indicating a broader range of demographic uncertainty than in the latest Social Security Administration forecasts, and a relatively greater contribution to uncertainty from fertility than mortality.
The long run outlook for the federal budget is deep red, largely due to projections of steeply rising per capita health care expenditures and of increases in the relative size of the elderly population. These two interact, since Medicare and Medicaid largely serve an elderly clientele.
Demographers used to emphasize the dominant role of fertility levels and change in the population aging process. Now they recognize the increasingly important role of declining mortality, at least for populations which have already achieved life expectancies above the mid-60s. For example, in the most recent report by the Board of Trustees of the Social Security Administration (Board of Trustees, 1995), sensitivity analysis for the Trust Fund balance 75 years in the future indicates that uncertainty about mortality plays a more important role than uncertainty about fertility or immigration, as shown in Table 1. By this measure, the effect of uncertainty due to mortality is 70 percent greater than the effect of uncertainty due to fertility, and almost six times that of uncertainty due to immigration.
Despite the growing recognition of the importance of mortality decline as a factor in population aging, our analytic understanding of the role it plays in the economic life cycle and in transfer systems like Social Security is not fully developed, perhaps as a result of the discipline's prior emphasis on fertility decline. By contrast, the influence of different levels of fertility across stable populations and steady state economies is quite well understood: fertility affects the population growth rate, and the population growth rate enters into many economic calculations in a way completely analogous to an interest rate. For example, the implicit rate of return earned through any pay as you go transfer system like Social Security equals the population growth rate in steady state (plus the rate of productivity growth). A change in steady state fertility leads to a change in the implicit rate of return. But for mortality, the situation is much more complicated.
This paper is about the influence of mortality decline on the long run finances of the Social Security System, excluding Medicare. After discussing the effect of mortality decline on the economic life cycle in a general way, it develops these ideas in the specific context of Social Security, and evaluates the effect of life expectancy gains on the finances of the system. The paper then considers the mortality forecasts of the Office of the Actuary of the Social Security Administration in relation to past mortality trends in the US, and in relation to international experience. An alternate set of stochastic mortality forecasts by Lee and Carter (1992) is discussed. The paper ends by using stochastic demographic forecasts by Lee and Tuljapurkar (1994) to develop stochastic forecasts of the Social Security Reserve Fund, which are compared to the latest Social Security Administration forecasts, and used to assess the relative roles of uncertainty in mortality and fertility.
In a stable population, the age distribution is proportional to e-nxp(x) where x is age, n is the population growth rate, and p(x) is the probability of surviving from birth to age x. From this observation, it follows that the age distribution of the stable population depends both on the population growth rate and on the shape of the survival schedule, p(x). For a new born, p(x) describes the expected number of person years lived at different ages in the life cycle, and is given by the nLx column in the relevant life table.[1] A difference in fertility affects the age distribution only by altering the growth rate, n. A difference in mortality, however, has two effects: First, it affects the proportion of births which survives to reproductive ages, and thereby affects the population growth rate, n. Second, it affects the shape of the individual survivorship schedule, p(x). Lower mortality raises the growth rate, tending to make populations younger, but it also raises p(x) relatively more at older ages, tending to make populations older. In populations which start with high mortality, the first effect dominates when mortality declines, and populations become younger. In countries with low mortality like the US, however, survival to childbearing is so high that further mortality decline has little effect on it, and the second effect dominates, making populations older (see Lee, 1994a, for a more detailed discussion).
Individuals, of course, simply live until they die. But the average person in a cohort lives fractional years at each age and dies very gradually, as described by p(x). The influence of p(x) on the demographic life cycle can be summarized by the number of person years lived in each of the three stages of the economic life cycle, or by the proportions of the total life expectancy that are spent in each of these stages. These proportions change as life expectancy changes.
In broadest terms, the economic life cycle of the average person begins with a period of dependency, in which a substantial amount is consumed by children but little is produced. This first stage is followed by a second, in which labor earnings far exceed own consumption, and then by a third stage in which consumption again exceeds labor earnings. Let us mark the boundaries of the economic life cycle stages conventionally at ages 15 and 65. Then when life expectancy is 20, 36 percent of the average life is spent in childhood dependency, 61 percent is spent in the ages of surplus production, and only three percent is spent in dependent old age.[2] By the time life expectancy rises to 75, roughly where the US is now, 20 percent of life is spent in dependent childhood, 63 percent is spent in the surplus producing ages, and 17 percent is spent in dependent old age (see Lee, 1994a for further details).
In order to understand the economic consequences of mortality decline, we must consider the interaction of the economic life cycle with changes in the demographic life cycle. Figure 1 plots the life cycle curves of consumption and labor earnings based on cross-sectional US data for 1987.[3]
Consumption possibilities over the life cycle are governed by a life cycle budget constraint. When mortality declines, there are incremental gains to person years of life lived at different ages. Some of these incremental gains in life occur during years when people earn more than they consume; such gains loosen the life cycle budget constraint, and permit more consumption age for age over the life cycle. Other incremental gains occur at ages when more is consumed than is earned. Such gains tighten the life cycle budget constraint, and require that consumption be reduced at some ages. To assess the effect of declining mortality on the life cycle budget constraint, we must examine the interaction of these incremental gains in life with the average economic life cycle, as plotted in Figure 1.
Because life expectancy is just the sum over all ages of the expected person years lived, a one year gain in life expectancy is just the sum over all ages of the gains in person years lived. It is a simple matter, therefore, to allocate a one year increase in life expectancy across age. Figure 2 does this, by five year age group, starting at the US life table for 1995 (based on the Lee-Carter, 1992, model). Very little of the one year is gained during childhood, only .02 years. Much more is gained during the working stage, .28 years. Most, however, is gained later, after age 65: .70 years. Thus seventy percent of the gain occurs at a life cycle stage characterized by leisure combined with a substantial level of consumption.
Figure 2 also plots the difference between consumption and labor earnings at each age, derived from the data plotted in Figure 1. It is clear that most of the person years of life are gained (across steady states) at the ages when people consume more than they produce. Consumption during these additional years of life is funded partially by the increased years spent alive during the surplus producing ages, but this offset is only partial. We can find the net cost of living this longer life by multiplying the distribution of person years gained and the difference between consumption and labor earnings at each age, and then summing these products.[4] This sum may conveniently be expressed as a fraction of initial life time consumption. The result for the data plotted in Figure 2 is .8 percent. That is, in order to maintain the same life cycle profiles of labor earning and consumption when life expectancy rises from 75.5 years to 76.5 years would require an additional .8 percent of the value of life time consumption.
This .8 percent must come from somewhere, and it can in principle be paid in many different ways. One way would be for people to work .8 percent longer each week--say by .008x40 hours = 19 minutes per week, or about 4 minutes per day--for each year of life expectancy gained. Another way would be to postpone retirement, so as to work .8 percent longer. If the average person works now from 20 to 65, then working another .36 of a year after 65, or about four extra months, would do the trick. Alternatively, people could consume .8 percent less at every age, for example. Or they could pay a higher payroll tax rate into a pay as you go pension program, like Social Security.
Here is a simple way to think about it. If everyone initially lived to the average life expectancy of 75.5, and people consumed the same amount every year they lived, then we would expect consumption needs to increase by 1/75.5 = .013, or by 1.3 percent. But only .7 years of the gain occurs after age 65, and about .28 of it occurs during the working years, and therefore helps to offset the costs of longer life. The net result is that the 1.3 percent is reduced to .8 percent. A bargain!
We should not think, however, that this .8 percent result is generally applicable. In fact, the plot of person years gained against age as shown in Figure 2 looks entirely different for a high mortality population, where most of the gains are concentrated in the working years rather than in the retirement years. For a population with very high mortality, the result of this calculation shows a life cycle gain in age-by-age consumption possibilities as a result of mortality decline, rather than a net loss (see Lee, 1994a).[5]
One might wonder about discounting. Do these results assume a zero rate of interest? This is an important question to which we now turn.
First note that these calculations were based on cross-sectional age profiles, whereas the true longitudinal profiles would shift multiplicatively upwards at the rate of labor productivity growth. Also, there may be a positive or negative population growth rate in the real world. Positive productivity growth and positive population growth both reduce the numerical importance of the number of older people compared to younger people, and therefore act just like discounting at a positive rate. For this reason, our calculations have implicitly assumed discounting at a rate equal to the sum of the long term productivity growth rate and long term population growth rate. For past decades, this sum has been two or three percent per year, but for the future it is expected to be somewhat lower. With a higher rate of discount, the costs of longer life would be less than .8 percent of life time consumption.
Earlier we explained that mortality decline tends to make a population younger by raising the growth rate, but also tends to make it older through greater individual aging. A mortality change is said to be "neutral" when death rates at every age change by the same additive amount. For a neutral mortality decline, these two effects are exactly offsetting, and there is no change in the age distribution of the stable population.[6] In the general case, however, one or the other effect dominates, and both have useful economic interpretations. The individual aging effect, or the distribution of person years lived over the life cycle, has a powerful influence on the funds that will be needed to provide for consumption in retirement, as calculated above for the US. The rate of growth effect has a powerful influence on the terms on which unconsumed output during the working years can be transformed into income for retirement, because the implicit real rate of return to a pay as you go transfer system, whether familial or public sector, equals the population growth rate plus the rate of productivity growth. More rapid population growth means a higher rate of return on foregone consumption, and therefore easier provision for old age consumption.
With a neutral mortality decline, the consumption needs of longer life can be paid for entirely out of the higher rate of return on the preexisting level of foregone consumption, with no needed change in a familial or public sector transfer system. More realistically, the rate of growth effect dominates in a high mortality setting, while in a low mortality setting, the individual aging effect dominates. Of course, the rate of growth has other consequences as well: more rapid growth means a higher burden of child dependency, and it may mean reduced capital per head or other resources per head, as well (see Lee, 1994b).
Having considered in a general way how longer life affects the economic life cycle, we will now turn to the specific question of how it affects the US Social Security system.
The same analytic machinery can be used to assess the effect of mortality decline on Social Security finances. While the effect on the demographic life cycle is unaltered, we need a new picture of the economic life cycle, which is here confined to interactions with the Social Security system. Figure 3 plots the payroll taxes and the benefits by age for the OASDI (Old Age, Survivors, and Disability Insurance, not including "H" which is the hospital insurance portion of Medicare).
Payroll taxes are paid in non-negligible amounts from about age fifteen to age 80, peaking in the mid-40s. The age distribution of benefits is, of course, much less symmetric--that is the whole point of the system. The vast bulk of the benefits comes after age 62, rising throughout the 60s and 70s, peaking around age 80, and then declining. This pattern reflects several influences. First, more recent retirees have had higher average earnings throughout their lives, and therefore qualify for higher benefits, tending to make benefit levels decline with age. Second, some older people continue to work after reaching age 65, and this leads to deferral of some benefits, tending to make the benefit levels rise with age. Third, the benefit to a couple is much less than twice the benefit to an individual, other things equal, so as more and more couples are transformed by death into individual recipients of benefits, the average level of individual benefits tends to rise. This tends to make the benefit level rise with age, due to increasing widow(er)hood. Fourth, some generations were affected by botched legislation in the early 1970s, which mistakenly double indexed benefits to inflation, a mistake which was rectified after a few years, but which has permanently elevated benefit levels for certain generations, and which led to the so-called "notch" generation. It is also worth noting the benefits to surviving dependent children, showing up as a rise with age in the teen years, and the rise in benefits in the 40s and 50s which reflects both increased payments to surviving spouses, and increased disability benefits.
To see how mortality decline affects the finances of the Social Security system, we can again interact the changes in person years lived with the age profile of net taxes (taxes minus benefits). Social Security is a complex system. While this calculation will capture the most important effects of mortality decline, there are also secondary effects which will be missed. These include a reduction in the costs of survivor's benefits, and a reduction in the average retiree's benefit level due to a reduced incidence of widow(er)hood.
Figure 4 plots the age distribution of net Social Security taxes together with the increase in person years lived when life expectancy rises by one year (note that the two have different scales). It is hardly surprising that most person years of life are gained at the ages of receiving benefits, rather than at the ages of paying taxes. Multiplying and summing over the life cycle, we find that the total is $4462, or 3.6 percent of the annual flow of benefits or taxes per capita.[7] With life expectancy higher by one year, it would therefore be necessary either to raise payroll taxes by 3.6 percent, or to reduce benefits by 3.6 percent. A combination of these can be achieved by raising the age of qualification for full benefits, along the lines currently planned for the Social Security system.
Having just considered the way in which a one year gain in life expectancy would affect the finances of the Social Security system, it is now natural to consider by how much can we realistically expect life expectancy to rise in the coming decades. A good starting point is to examine the mortality forecasts of the Office of the Actuary of the Social Security Administration (henceforth, SSA).
The most recent SSA forecasts (Board of Trustees, 1995:12) are for a sexes-combined life expectancy of 80.7 in 2070. Within the range of forecasts made in recent years for the US, this is definitely at the low end in terms of gains. Even relative to the upper limit of 85 years claimed by Fries (1980), and supported by Olshansky et al (1990), this figure is very low. Some national populations have already attained e0 very close to this: Japan's is 80, and Sweden's is 79, for example. Some analysts believe that a life expectancy of 100 is attainable by the middle of the next century, if not sooner (see Ahlburg and Vaupel, 1990, and Manton, Stallard and Tolley, 1991). Lee-Carter (1992) forecast life expectancy for 2050 at 86.1, plus or minus about 4.5 years. In this section, we will consider the SSA forecasts in historical and international context.
The SSA forecasts are based on trends in rates for ten causes of death. An historical average rate of decline for each cause is calculated over a twenty year base period. An ultimate rate of decline is assumed to hold for each cause, for each of four broad age groups, based on an assessment of various factors believed to influence the rate of decline for each cause in the long run. The projected rate of decline is initially set equal to the historical average rate, then is assumed to trend towards the ultimate rate. About twenty five years into the projection, the ultimate rates are fully in effect, and continue throughout the projection period (see Social Security Administration, 1992). Wilmoth (1995) has shown that a trend extrapolation forecast of aggregate death rates is bound to forecast more rapid declines than is a trend extrapolation of cause-disaggregated death rates, for the simple reason that the most slowly declining (or most rapidly rising) cause specific death rate will receive a rising weight in the total. This observation does not tell us, however, which approach will be more accurate. In any event, the SSA cause specific death rate forecast is not based on trend extrapolation, so this argument does not apply to it.
The forecasts that result from the SSA approach imply a sharp slowing of the rates of decline of mortality at most ages, relative both to the previous two decades, and to longer run historical trends, from most decadal start points back to 1900. Table 2 shows the difference between long run historical trends, and the rates of decline assumed in the SSA forecast. The first column shows rates of decline of death rates between 1900 and 1988, for broad age groups and by sex. The second column shows the average rate of decline forecast by SSA (1992) from 1990 to 2070. We see, for example, that whereas the death rate for males aged 0-14 declined at 3.25 percent per year, SSA forecasts that for the next 80 years it will decline at only 1.21 percent per year, and similarly for females 0-14. The third column shows the difference between these rates, and the last column shows the ratio of SSA rate forecasts to the trend extrapolated rates over a 78 year horizon to 2066. We see that for the 0-14 age group of males and females, the SSA forecasted rates are about five times as great as the trend extrapolations. These are the most extreme cases, and the differences diminish with age. For males, the difference for 65+ is negligible. For females 65+, however, the SSA rates are 37 percent higher in 2066 than the trend extrapolated rates. For the age adjusted total death rate, the ratio of SSA projection to trend extrapolation is 1.3 for males, and 1.9 for females. The very low rates of decline forecast by SSA for death rates at younger ages have a substantial impact on the proportion forecast to survive to age 65, and therefore on life expectancy at birth. Analysis of international mortality trends provides reason to expect some reduction in the differences in rate of decline between for mortality at younger and older ages (Horiuchi and Wilmoth, 1995), but the change in age patterns of decline assumed by SSA and reflected in Table 2 appear extreme, and no justification is given.
Mortality decline at the older ages is particularly important for the future of life expectancy, since there is so little room for death rates to decline at younger ages. We have made more detailed calculations for rates of decline of mortality for ages above 60, as shown in Figures 5 and 6. These figures plot the average rate of decline of the single year of age death probabilities (q) for ages 60, 70, 80 and 90, calculated over progressively shorter periods: first for 1900 to 1988; then from 1910 to 1988; and so on, up to 1980 to 1988. The purpose of this presentation is to show the effect of basing an extrapolation on varying degrees of historical depth, all the way back to the beginning of the century. Markers at the far right of the figures indicate the average rate of decline forecast by SSA for 1990 to 2070.[8]
Figure 5 plots the data for males. We see that the rates of decline for q60 and q70 have been accelerating throughout the century, while those for q80 and q90 have held steady; in both cases, there has been a slight slowing of gains in the past decade. Comparing the SSA forecasts with the historical record, we see that the forecasts for male mortality at these ages are very much in line with the average rates of decline for 1900 to 1988 (the points at the far left of the graph), but that as the period of the historical average becomes shorter, the SSA forecasts diverge by greater and greater amounts for q60 and q70.
Figure 6 plots the data for females, revealing a different pattern. While there is a slight acceleration in the first half of the century, curves are generally much flatter, and the rates of decline for q60 and q70 fall off distinctly after 1940, while all the rates of decline are sharply reduced in the 1980s. For females, the SSA forecasts are in line with the experience of the 1980s, but are way out of line with the rates of decline averaged over any longer historical period.
Table 3 draws out the implications of the discrepancy between the SSA forecasts and the long run historical averages. We have extrapolated in two different ways. The first is based on the rate of decline from 1900 to 1988. However, mortality data before 1933 are based on only a subset of the states, and the quality of data for old ages is highly suspect (see Alter, 1990). For this reason, it is also of interest to extrapolate from 1930 to 1988. The ratios for males are all quite close to unity, indicating that the SSA forecasts for older males are close to the extrapolated rates. For females, however, the SSA forecasts are systematically higher, and for the extrapolations since 1930, they are as much as twice as great.
It is clear from this discussion that the SSA forecasts for both males and females, for age groups up to 25-64, assume a substantial slowing relative to the longer term rates of decline. For age 60 and above, the SSA forecasts rates of decline for males that are in line with long run historical trends but are much slower than recent experience. For females 60 and above, SSA forecasts rates of decline that are in line with experience since 1980 or so, but which are much slower than longer run experience. There is nothing intrinsically wrong with forecasting a slowing of mortality gains, and SSA evidently believes it right to do so, based on their cause specific analysis.
The kind of deceleration in mortality decline that the SSA projections foresee could occur for a number of reasons, some of which might be related to the specifics of the US situation. Many other possible reasons for predicting a slowing of declines are not specific to the US, however, but are rather based on more general consideration of the likelihood of medical advances against the various causes of death, or on some notion of lower limits to death rates which, as approached, make further gains more difficult.
To assess the plausibility of reasons of this more general kind, it is revealing to look at rates of decline of mortality in populations of other industrial nations with good mortality data at old ages, in some of which mortality is already substantially lower than our own. The United Kingdom, France, Sweden, the Netherlands, and Japan are such countries.[9] They range in life expectancy from the UK, which is about the same as that in the US, to Japan and Sweden, which have life expectancy three or four years higher. According to SSA projections, not until 2045, or fifty five years from now, will the U.S. attain the life expectancy of 79.6 years which is currently observed for Japan (Manton and Vaupel, 1995). Therefore the rates of decline of mortality in these countries are highly relevant for assessing the prospects for mortality decline now and over the next five or six decades for the US.[10]
Figure 7 compares the SSA projected rate of decline for male mortality rates at older ages to recent rates of decline for these five populations (based on Horiuchi and Wilmoth, 1995, Table 1). We see that for males, the rates of decline projected by SSA are very substantially lower than any other country has experienced in the period 1975-79 to 1985-89, except for 75-79 year olds in the Netherlands. In fact, the projected rates of decline are only half as great as the average for these five populations at 60-64, and only a third as great as the average at 75-79. For females, shown in Figure 8, the contrast is even greater, with projected rates of decline that are less than a third of the average at 60-64, and less than a fifth of the average at 75-79. The very low rates of decline for US females shown in this table reflect the sharp slowdown in declines that occurred in the 1980s, marking a break with the longer term trends (see Figure 6). The rates of decline calculated from earlier dates are far more rapid; for example, from 1960-64 to 1985-89, the rates of decline are .0115 for age 60-64 and .0154 for age 75-79, roughly twice the rates of decline shown in the figure.
We should note that the average rates of decline for the same countries at earlier dates, 1955-59 to 1965-69, are similar for females 60-64, but only 60 percent as fast for 75-79, leaving the qualitative conclusion unchanged; the SSA projected rate of decline is still only about a third of the average international rate. For males, however, the picture is different. The average rate of decline of mortality for males 60-64 from 1955-59 to 1965-69 was only .15 percent per year, far lower than the SSA projection, and at 75-79 it was only .41 percent per year, slightly lower than the SSA projection.
Kannisto et al (1994) present evidence on rates of decline from a larger collection of countries with reliable data on old age mortality, for ages 80 to 100. They find that a) "In most developed countries outside of Eastern Europe, average death rates at ages above 80 have declined at a rate of 1 to 2 percent per year for females and 0.5 to 1.5 percent per year for males since the 1960s." (Kannisto et al, 1994:794). b) Rates of decline at these older ages have been accelerating throughout the 20th century, not slowing. c) There is at most a very modest tendency for countries with higher death rates to experience more rapid declines since the 1960s. According to Manton and Vaupel (1995), the US death rates at these ages are lower than those in other developed countries. However, the Kannisto et al (1994) study gives no reason to expect that this should mean that they will decline more slowly.
Figures 5 and 6 also plot the average rate of change of mortality at extreme old age for 19 populations for the decade of the 1980s for ages 80 to 89, and 90 to 99 (Kannisto et al, 1994:801). It can be seen from the figures that the long run rate of decline forecast by SSA is considerably less than these averages for males, and a great deal less for females.
SSA projects an average rate of decline up to 2070 for females age 80 of .6 percent per year, or one third the average of 13 countries with highly reliable data, and also for age 90, where it is half the average rate of decline. For males, SSA forecasts a decline of .5 percent per year at age 80 and again at age 90, which are about half the average rates of decline in this international sample. In the context of the past experience of these 13 populations, the SSA forecasts appear to be for very slow gains.
Generalization is hazardous, but these international comparisons provide strong evidence that US mortality decline is not yet pushing up against biological limits, or against limits imposed by already existing medical technology. In our view, the SSA forecasts of mortality decline are far too low, and even the SSA upper bracket for rates of decline of mortality is too low.
Recently, one of us, with Lawrence Carter, developed a new approach to forecasting mortality, and used it to forecast for the US (Lee and Carter, 1992). It is based on the extrapolation of historical trends using time series analysis, while using a simple model to maintain a coherent age pattern in the rates. Although the forecast was based on data from 1900 to 1989, the rates of mortality decline in the long term forecasts are actually much closer to the rates of decline from 1933 to 1989.[11] LC forecast sexes combined life expectancy of 86.1 in 2065, versus 80.5 by Board of Trustees (1995), which reflects more than double the gain forecast by SSA.
The LC average rates of decline by sex for 1990 to 2065 (taken from Carter and Lee, 1992) are plotted in Figures 5 and 6, along with the international data and the SSA forecasts. For males, the forecasts are for substantially slower declines than occurred in these countries--at about half of the international average rate, although still rather faster than in the SSA forecasts. For females, the forecast rate of decline equals the international average for 60-64, and is substantially below it for 75-79, although three times as rapid as in the SSA forecasts.
One important feature of the LC method is that it provides probability bounds for the forecasts. These are based on several assumptions: that the model is correctly specified, that the data are accurate, and that future structural breaks are in some sense similar to those that have occurred in the past. For this reason, the probability bounds should be viewed as expressing a lower limit on the uncertainty of the forecasts.
The LC sexes combined forecast of life expectancy for 2065 is 86.1, with a 95 percent probability interval of 80.9 to 90.2. Thus the intermediate forecast of SSA falls just outside this range. The bracket for the SSA forecast in 2070 is 77.9 to 84.4, so the LC point forecast falls above this range. However, it is important to understand that the LC probability bounds refer to mortality in a single year, whereas the SSA type scenario refers to mortality that always follows the high or the low trajectory. These concepts of uncertainty are quite different, although this matters much more for fertility variation than mortality variation due to the strong trend in mortality (see Lee and Tuljapurkar, 1994, henceforth LT, for a discussion of these issues).
For the SSA forecast, one can obtain high and low bounds for an item such as the population 85+ simply by calculating the appropriate survivors according to either the low mortality scenario or the high mortality scenario. This is the classic scenario based method commonly used in demographic forecasting. But the forecasts generated in this way are intrinsically incapable of providing probabilistically consistent indications of uncertainty. As an example, consider the forecasts of the total dependency ratio in the 1992 population projections by SSA. The forecast for 2070 gives a range for the population 0-19 is 34 percent; for the population 20-64, the range is 20 percent; and for the population 65+ it is 9 percent. Yet for the total dependency ratio, which is the sum of the first and the third divided by the second, the range is only 5 percent. That is because SSA bundles high fertility and high mortality together in its scenarios, and low fertility and low mortality, to generate a wide range for the old age dependency ratio; in doing so, they generate an impossibly narrow range for the total dependency ratio. The Bureau of the Census bundles low fertility with high mortality to generate a wide range for population growth rates and size, but consequently has implausibly narrow ranges for some other variables (see Lee, 1972, and Lee and Tuljapurkar, 1994). It is impossible to avoid problems of this sort with scenario based forecasting, which is the reason for developing the much more complicated approach of stochastic forecasting.
With the LT population forecasts, based on the LC mortality forecast model and a similar stochastic model for fertility, one must instead take into account the probabilities of all the possible mortality trajectories, involving high mortality in some years, and low in others, with consequent cancellation or reinforcement of variations, and the same for fertility. It is simplest to do this by repeated stochastic simulation of the population, although LT also presents analytic results.
By restricting ourselves to forecasts of numbers of people already born by 1990, we can see the isolated influence of mortality. Table 4 shows the difference between population projections based on the LC forecasts and those of SSA for 65+ and 85+. Differences in the 85+ age group always reflect only mortality (migration is handled deterministically, and numbers in the table are adjusted for differences in the definition of the base population). Differences in the 65+ population reflect only mortality through 2055. For the 65+ population, differences in the forecasts are fairly small until around 2040, by which time they are important. The biggest differences, however, are for the population 85+, which is almost 30 percent larger by 2020, and over 60 percent larger by 2060. The table does not show the range of uncertainty in the forecasts, because the ranges turn out to be quite similar in the two sets of forecasts despite the differences in methods and concepts (see LT, Figure 7). The really striking differences in the brackets emerge for forecasts of the total dependency ratio, for which the SSA brackets are impossibly narrow, a kind of problem which is inherent to scenario based population forecasts (see LT, Figure 10).
LT describes a stochastic approach to demographic forecasting which leads to probability distributions for demographic quantities of interest, reflecting probabilistic forecasts of fertility and mortality, but taking net migration rates to be fixed at the levels assumed in the Middle assumption used in the US Bureau of the Census forecasts. These stochastic population forecasts can be used to generate forecasts of probability distributions of the Social Security Trust Fund balance and actuarial balance reflecting demographic uncertainty. To do this, calculate age profiles of payroll tax payments made, and benefits received, based on data from the 1991 Consumer Expenditure Survey, suitably adjusted to match national totals for payroll taxes and benefits, when weighted by the US population age distribution (see Lee and Miller, 1995). These age profiles are then shifted over time so as to reflect legislated changes in the age of retirement with full benefits, and to reflect assumed rates of productivity growth. The rate of productivity growth is taken to equal the SSA medium assumption, as is the real interest rate used to generate income on the Trust Fund balance. Thus the only uncertainty reflected in our probabilistic forecasts is uncertainty about the future course of fertility and mortality. In future work, other sources of uncertainty will be incorporated.
Here we will consider the results of three sets of stochastic forecasts of the actuarial balance: first, when both fertility and mortality are stochastic; second, when only mortality is stochastic and fertility follows its trajectory of point forecasts; and third, when only fertility is stochastic, and mortality follows its trajectory of point forecasts. Our comparisons will be limited to the period 1995 to 2069 and will take the "actuarial balance" as the criterion. The actuarial balance is the amount by which present value of expected revenues (taxes plus interest) exceeds or falls short of the present value of expected benefit payments, allowing for a reserve fund at the end of the period equal to one year's benefit payments, with this difference expressed as a percentage of taxable payroll. The middle forecast by SSA for the actuarial balance is -2.17 percent. The point forecast by LT is -2.21 percent, very nearly identical. The similarity in the result, despite the more rapid mortality decline in the LT forecast, is presumably due to the slightly higher level of fertility in the LT forecast.[12] Because the SSA forecast is so much more detailed than that of LT, there are doubtless other important differences as well.
Table 5 gives the difference between the upper and lower bounds of a 95 percent probability interval for the actuarial balance up to 2069, for the three sets of forecasts. There are two important points to observe. First, in the LT forecasts, mortality contributes substantially less to the uncertainty of the actuarial balance up to 2069 than does fertility--about 35 percent less, in fact (2.13/3.29 = .65). This is in sharp contrast to the SSA sensitivity analysis reported in Table 1, which implied that mortality contributed 70 percent more uncertainty (1.48/.87 = 1.70). Thus the relative importance of uncertainty about mortality in the SSA forecasts is about two and a half times as great as in ours (1.7/.65 = 2.6). The results of the SSA sensitivity analyses depend on two factors: the sensitivity of the outcome to variation in fertility or mortality, and the size of the variation in fertility or mortality that is reflected in the values used--generally, in the High-Low range of the forecast scenarios. The discrepancy in the relative importance of uncertainty about mortality and fertility is probably due to the different treatments of fertility. SSA considers only a rather narrow range of TFRs (1.6 to 2.2) and rules out possible fluctuations in fertility by considering only flat trajectories for fertility. The stochastic forecasts of LT reflect an explicit empirical assessment of the uncertainty in forecasts of mortality and fertility, based on a model fit to the historical record. Their fertility model implies a much broader range for fertility, and takes account of the possibility of long swings like the US baby boom and bust.
The second point to note is the relative amounts of uncertainty in the forecasts. SSA does not attach any probabilistic interpretation to their forecasts, and it is generally not possible to do so with scenario based forecasting except by ex post analysis.[13] We note that the 95 percent probability band of the LT forecast, reflecting uncertainty only from fertility and mortality, has a width of 3.78 percent. This is narrower than the 6.42 percent of SSA, which reflects many sources of uncertainty in addition to fertility and mortality.[14] If we sum the SSA brackets from Table 1 for fertility and mortality only, we find an interval of 2.35 percent, which is 62 percent of the LT fertility and mortality interval of 3.78. For these sources of variation, for this measure of actuarial balance, and for a 75 year horizon, it appears that the probability coverage of the SSA interval for fertility and mortality combined is around 80 percent (roughly plus or minus 1.25 standard deviations). Recalling that the uncertainty incorporated in our mortality forecast is at best a lower bound on the true uncertainty, we might conclude that the probability coverage of the range of demographic assumptions in the SSA forecast is actually somewhat lower than 80 percent.
Intergenerational inequities have been much discussed in the recent budget debates. It is expected that future generations will have to pay far higher taxes than we pay now, and that the growth in spending for entitlements is a big part of the reason. Auerbach and Kotlikoff (1994) project that for future generations, net taxes will take up 82 percent of the present value of their life time labor earnings, for example. Some of the expected increase is due to rising interest charges on the national debt, if deficits continue. Much of it is due to the rising costs of publicly funded health care services. An important part, however, is due to population aging, reflecting both declining mortality and low fertility.
The tax costs of population aging can be viewed as a demographic "accident" against which generations should be insured. In this case, it might be thought unfair that future generations should have to pay higher taxes simply because they happen to live in a time when the age distribution is old. Here we take a different view: that it is appropriate that future generations pay more in taxes to help finance consumption during the longer years of retirement that they will enjoy due to lower mortality. Alternatively, they could postpone retirement and leave taxes for pensions unchanged, or they could reduce benefit levels. But longer life is costly, because incremental years lived come largely at ages that are traditionally spent in leisure. Future generations must pay for the gift of longer life in one way or another. Indeed longer life is not really a "gift", and future generations should also be prepared to pay for the higher medical costs that help produce the longer lives.
Let us focus on just the consumption costs of longer life. Lee and Carter forecast a ten year gain in life expectancy. The rough calculations we reported earlier suggest that this would require an increase in the payroll tax rate by 36 percent of its current level in steady state. But steady state is not the appropriate assumption when the gains are forecast to occur gradually over the next 75 years; the population age distribution in 2070 will not fully reflect the period life expectancy of that year. From the LT forecasts, it appears that each additional year of life expectancy gained will result in a 2 percent or so increase in the population 65+ in 2065, which will be offset to some degree by increases in the working age population with lower mortality.[15]
A similar argument can be made about that portion of increased taxes that results from lower fertility. Recent generations have chosen to have fewer children, and future generations are expected to as well. This is a choice that involves substantial savings in private childrearing costs as well as reductions in public costs of education and health care. Childbearing and rearing involve substantial positive externalities for the public sector (see Lee, 1990), and it is inevitable that when generations choose to reduce their fertility, they thereby choose either to impose on their children higher per capita costs of supporting the elderly (if benefits are fixed) or lower benefits for themselves (if tax rates are fixed). From the point of view of the children, the higher tax rates might be viewed as an unfortunate accident, although they benefit in many other ways as members of smaller sibships and smaller generations. The tradeoff between fertility and old age support is explicitly internalized within parental decision making in traditional societies, where the number of children that a couple bears is directly related to that couple's support in old age. However, with the public sector in industrial nations taking over much of the role of supporting the elderly, this tradeoff is no longer internal to the couple's decision making process, but it remains nonetheless.[16]
Another set of issues concerns the size of future gains in life expectancy. We have argued that the SSA forecasts, on which planning for the Social Security system is based, foresee implausibly small gains to life expectancy over the next 75 years. This assessment is based both on comparisons to other countries which have achieved much more rapid gains while traversing the same range, and to US history over the 20th century, which also shows more rapid gains.
In this paper we have restricted ourselves to a contrast between the SSA forecasts, which foresee small gains in mortality compared to most other forecasts, and the LC forecasts, which we view as middle of the road. A move from the SSA forecasts to the LC forecasts would have considerable implications for projections of the financial stability of the system; for example, Table 4 shows the projected population at age 65 and above is 21 percent larger under the LC point forecast than under the SSA middle forecast. How much more jarring, then, would the implications be if we switched from the SSA forecasts of small gains to forecasts by others who see the possibility of truly big gains? While SSA gives a point forecast for life expectancy at 80.7 in 75 years, and LC at 86.1, others like Ahlburg and Vaupel (1990), Manton et al (1991), and Schneider and Guralnik (1990), view a life expectancy of 90 to 100 or more as a real possibility (for an overview of approaches to forecasting mortality, see Lee and Skinner, in press). The dramatic implications of such long life for retirement systems and health care scarcely need to be emphasized.
(1) The probability of survival from birth into an age group (x,x+n) is given by nLx/(n l0). The average number of person years lived in this age interval per birth is given by nLx/l0.
(2) Using Coale-Demeny (1983) MWF life tables.
(3) These profiles could be transformed to reflect better the longitudinal pattern by shifting both upwards by a factor of erx where x is age and r is the rate of labor productivity growth. Consumption in this figure is defined broadly to include not only the usual consumer expenditures, but also the costs of public education and of public sector health care; and it reflects consumption of the services of household capital items such as houses, cars, and other durables, rather than the age at which they were purchased. Labor income is pretax, and includes the self employment income and the contributions of employers to Social Security as well as fringe benefits.
(4) This is not to be confused with the cost of extending life through medical interventions, and so on. In the present calculation, such costs are ignored except as they are implicit in the consumption schedule, and the focus is instead on the cost of net consumption in the additional years.
(5) That is, the effect of mortality decline operating through the life cycle budget constraint in this way is beneficial. However, will be discussed below, there is also an effect of mortality decline on the rate of population growth, and in the case of a high mortality country this effect is very powerful and typically works strongly in the opposite direction.
(6) Neutral mortality change is a change which alters the force of mortality schedule by the same additive amount at every age; it is an analytically convenient but empirically implausible special case of mortality change.
(7) For purposes of this calculation, the profiles were altered to correspond to a PAYGO system given a stationary population age distribution, by reducing taxes proportionately at all ages so that aggregate taxes equal aggregate benefits.
(8) We are using these somewhat older SSA forecasts because the report which contains them provides considerable detail on both the forecasts and on the base period data, making the calculations for the figures possible. The forecasts appear to be almost identical to those in Board of Trustees, 1995.
(9) The data for these comparisons are drawn from Horiuchi and Wilmoth (1995), Table 1. We present rates of change for all the five countries that they analyze, and they chose these countries because "These countries have relatively high reputations about the accuracy of reported ages of old persons and old decedents" (p.4).
(10) At the same time, it should be noted that Manton and Vaupel (1995) find that death rates in the US above age 80 are lower in the US than in other countries such as Japan and Sweden.
(11) Lee and Carter (1992) and Carter and Lee (1992) actually use age specific data only for 1933 to 1987, for the period when all states were in the death registration area, and one hopes a period when the quality of the age specific data was somewhat better than in early years. George Alter (1990) uses variable r methods to reestimate mortality from 1900 to 1935, and finds that a) e(65) was actually .3 to .6 years lower than indicated in the official data; and b) both males and females ôshow declining expectation of life in the first decades of this century and increases after 1920ö (p.12). These results suggest that the Lee- Carter model, which basically assumes that age specific death rates move together, probably does violence to the true patterns of old age mortality in the first 20 years of the century. Whether this is a more serious problem than, say, the pattern of rising young adult male mortality in parts of the post W.W.II period, is not clear. In any event, by using the age specific data from 1933 to 1987, and using an indirect method (the Lee-Carter second stage estimation technique) to estimate the trend in k before 1933, based on total numbers of deaths and the population age distribution, the rates of decline of old age mortality may well have been slightly overstated. This difficulty arises not from the method itself, but from its use as an indirect estimation technique when reliable age specific data are lacking. When the method is applied to the SSA data set for mortality since 1900, the forecasts of life expectancy in 2065 change hardly at all.
(12) When fertility in the stochastic forecast is constrained to have the same ultimate level--a TFR of 1.9--as in the SSA forecasts, then the forecasted actuarial balance for 2069 is somewhat reduced from -2.1 percent to -2.4 percent.
(13) The report of the Board of Trustees (1995) says: "While it is reasonable to assume that actual trust fund experience will fall within the range defined by the three alternative sets of assumptions used in this report, no definite assurance can be given that this will occur because of the uncertainty inherent in projections of this type and length." (1995:12)
(14) Additional sources of uncertainty include net migration, productivity growth rate, interest rate, inflation, disability incidence and disability termination.
(15) Note, however, that according to Table 4, the 5.5 year difference between the LC and the SSA forecasts of life expectancy lead by 2070 to a 21 percent difference in the population 65+, for a gain of nearly 4 percent for each difference of a year lived. It is not clear what explains this, but presumably it is due to the difference in age pattern of mortality decline in the two forecasts.
(16) In the cases of both familial and public sector old age support, in a defined contribution plan higher fertility increases the level of support for the elderly, and in a defined benefit plan, higher fertility reduces the per capita cost for the working age population.
Ahlburg, Dennis and James Vaupel (1990) "Alternative Projections of the U.S. Population," Demography v.27, n.4 (November) pp.639-652.
Alter, George (1990) "Old Age Mortality and Age Misreporting in the United States, 1900-1940," paper presented at the Annual Meetings of the Population Association of America, Toronto, Ontario, May 3.
Auerbach, Alan J., and Laurence J. Kotlikoff (1994) "The United States' fiscal and saving crises and their implications for the baby boom generation," report to Merrill Lynch & Co. (February).
Board of Trustees, Federal Old-Age and Survivors Insurance and Disability Insurance Trust Funds (1995) 1995 Annual Report of the Board of Trustees of the Federal Old-Age and Survivors Insurance and Disability Insurance Trust Funds (U.S. Government Printing Office, Washington, D.C.)
Carter, Lawrence and Ronald D. Lee (1992) "Modeling and Forecasting U.S. Mortality: Differentials in Life Expectancy by Sex," in Dennis Ahlburg and Kenneth Land, eds, Population Forecasting, a Special Issue of the International Journal of Forecasting, v.8, n.3 (November) pp.393-412.
Coale, Ansley J. and Paul Demeny (1983) Regional Model Life Tables and Stable Populations second edition (Princeton University Press, Princeton N.J.)
Fries, J.F. (1980) "Aging, Natural Death , and the Compression of Morbidity" New England Journal of Medicine 303:130-136.
Horiuchi, Shiro and John Wilmoth (1995) "Aging of Mortality Decline", paper presented at the 1995 Annual Meetings of the Population Association of America, in San Francisco.
Kannisto, Vaino, Jens Lauritsen, A. Roger Thatcher and James W. Vaupel (1994) "Reductions in Mortality at Advanced Ages: Several Decades of Evidence from 27 Countries" Population and Development Review v.20 n.4 (December), pp.793-810.
Lazear, Edward P. and Robert T. Michael (1988), Allocation of Income Within the Household (University of Chicago Press, Chicago and London).
Lee, Ronald (1990) "Population Policy and Externalities to Childbearing," Annals of the American Academy of Political and Social Science, special issue edited by Samuel Preston World Population: Approaching the Year 2000 (July), pp. 17-32.
Lee, Ronald (1994a) "The Formal Demography of Population Aging, Transfers, and the Economic Life Cycle," in Linda Martin and Samuel Preston, eds.; The Demography of Aging (National Academy Press) pp.8-49.
Lee, Ronald (1994b) "Fertility, Mortality and Intergenerational Transfers: Comparisons Across Steady States," in John Ermisch and Naohiro Ogawa, eds, The Family, the Market and the State in Ageing Societies (Oxford University Press), pp.135-157.
Lee, Ronald D. and Lawrence Carter (1992) "Modeling and Forecasting the Time Series of U.S. Mortality," Journal of the American Statistical Association v.87 n.419 (September) pp.659-671.
Lee, Ronald D. and Timothy Miller (1995) "Interage Resource Flows in the US: A Descriptive Account" paper presented at the 1995 annual meetings of the Population Association of America, in San Francisco.
Lee, Ronald and Skinner, Jonathan (in press) "Assessing Forecasts of Mortality, Health Status, and Health Costs During Baby-Boomers' Retirement" in Collection of Papers Prepared for a Workshop of the National Academy of Sciences on Modeling Retirement Income Security for the Baby Boom Generation.
Lee, Ronald D. and Shripad Tuljapurkar (1994) "Stochastic Population Forecasts for the U.S.: Beyond High, Medium and Low," Journal of the American Statistical Association (December) v. 89, no. 428, pp.1175-1189.
Manton, Kenneth, Eric Stallard and H. Dennis Tolley (1991) "Limits to Human Life Expectancy: Evidence, Prospects, and Implications," Population and Development Review v.17 n.4 (December) pp.603-638.
Manton, Kenneth G. and James W. Vaupel (1995) "Survival After the Age of 80 in the United States, Sweden, France, England and Japan", The New England Journal of Medicine v.333 no.18 (Nov. 2), pp.1232-1235.
Olshansky, S. Jay, Bruce A. Carnes and C. Cassel (1990) "In Search of Methuselah: Estimating the Upper Limits to Human Longevity," Science v.250, pp.634-640.
Schneider, Edward L., and Jack M. Guralnik (1990) "The aging of America: Impact on health care costs," Journal of the American Medical Association 263(17) (May 2).
Social Security Administration, Office of the Actuary, U.S. Department of Health and Human Services (1992) "Life Tables for the United States Social Security Area, 1990-2080," Actuarial Study No. 107, SSA Pub. No. 11-11536 (August).
United States, Social Security Administration, Office of Research And Statistics (1994) Social Security Bulletin, Annual Statistical Supplement (Washington D.C., Government Printing Office).
Wilmoth, John (in press) "Are Mortality Projections Always More Pessimistic When Disaggregated by Cause of Death?", Mathematical Population Studies.
Table 1. How Demographic Uncertainty Affects the Projected Actuarial Balance for the period 1995 to 2069.
Range of Demographic (Higher) - (Lower)
Assumptions Projected Actuarial
Balance 1995-2069
Mortality Reduction in Death 1.48 %
Rates, 1994-2069:
16% to 54%
Fertility Ultimate TFR: .87 %
1.6 to 2.2
Immigration Annual .26 %
Net-Immigration
(000s):
750 to 1,150
Source: Calculated from sensitivity tests presented in Board of Trustees (1995):133-135.
Table 2. Average Annual Rate of Decline in Mortality for Base Period versus Forecast, by Age and Sex, Percent per Year
1900-1988 1988-2066 Forecast Ratio of forecast
Age Group (base (forecast rate to trend
period) period) - base rate extrapolation in
2066
Male 0-14 3.25 1.21 -2.04 4.9
15-24 1.54 .65 -.89 2.0
25-64 1.09 .71 -.38 1.3
65+ .52 .54 +.02 1.0
Total .95 .60 -.35 1.3
Female 0-14 3.39 1.24 -2.15 5.3
15-24 2.52 .61 -1.91 4.4
25-64 1.59 .61 -.98 2.1
65+ .95 .55 -.40 1.4
Total 1.38 .58 -.80 1.9
Source: The first two columns of data are taken directly from Table 4 in SSA, 1992, page 9. The third column is the second minus the first. The last column is calculated as exp(-78*entry from previous column). It represents the ratio of the SSA forecast of mortality levels in 2066 to the death rate in 2066 that would result from extrapolating the historical trend from the base period 1900-1988 to 2066 (or over any other 78 year period in which the rates of decline differed by the amounts shown).
Table 3. Ratio of Social Security Forecasts to Trend Extrapolation for Old Age Mortality by Age and Sex, in 2068, from 1900 and from 1930.
Age Ratio of SSA forecast Ratio of SSA forecast
Group to trend to trend
extrapolation extrapolation
1900-1988 in 2088 1930-1988 in 2088
Male 60 .94 1.14
70 .98 1.11
80 1.00 1.10
90 .96 .99
Female 60 1.67 2.11
70 1.70 2.16
80 1.37 1.76
90 1.06 1.20
Table 4. Ratio of Forecasts based on Lee-Carter to those of SSA for Population 65+ and 85+
Ratios 1990 2000 2020 2040 2060 2070 65+ 1.00 1.01 1.10 1.09 1.16 1.21 85+ 1.00 1.12 1.28 1.34 1.64 1.61
Note: Because the SSA forecasts cover a slightly different population than the Lee-Tuljapurkar (1994) forecasts from which these are drawn, the base period population sizes in 1990 are different, with SSA a few percent larger. All forecasts have been adjusted by a constant to bring the baseline numbers into agreement. The Lee-Carter based forecasts for 65+ starting in 2060 reflect fertility as well as mortality.
Table 5. How Demographic Uncertainty Affects the Projected Actuarial Balance for the period 1995-2069, in LT Stochastic Forecasts and SSA Projections
Source of (Higher) - (Lower) Demographic Projected Actuarial Uncertainty in Balance 1995-2069 Forecast Lee-Tuljapurkar Stochastic Forecasts Mortality and 3.78 % Fertility Mortality 2.13 % Fertility 3.29 % Social Security Admin Forecasts All Sources 6.21 % Fertility and 2.35 % Mortality
Note: The middle forecast of the actuarial balance of the system for the period 1995 to 2069 according to SSA is -2.17 % of taxable payroll, meaning that the payroll tax rate would have to be raised by this amount to achieve exact balance. The LT mean forecast of actuarial balance in 2069 is -2.21 %. In the SSA forecasts, the middle forecast does not vary in the sensitivity tests. In the LT forecasts, because of interactions and nonlinearities, the mean varies slightly according to which variables are treated as stochastic.
Titles for figures as are as printed on them.
Notes to figures:
Figure 1. Based on data from the 1987 Consumer Expenditure Survey. Individual consumption is estimated from household consumption and household composition using procedures based on Lazear and Michael (1988). Consumption includes in kind public transfers (principally health care and education), and services imputed to owned housing, automobiles, and consumer durables. Labor income is before taxes, and includes all employers contributions including payroll tax, as well as self employment income.
Figure 2. Net earnings are calculated as the difference between the age schedules plotted in Figure 1. The age distribution of person years of life gained is calculated from the Lee-Carter (1992) mortality forecasts.
Figure 3. The data on Social Security benefits (OASDI, including pensions, survivors benefits, and disability benefits, but excluding Medicare) by single years of age are taken from United States (1994). The data on taxes by single years of age are calculated from data on taxable labor earnings taken from the 1993 Current Population Survey of the US. Both age profiles were multiplicatively adjusted by a small amount so that when multiplied times the population age distribution and summed, they total to the aggregate amounts for benefits and taxes reported in official sources.
Figure 4. Net Social Security Taxes are calculated as the difference between the age schedules plotted in Figure 3, averaged for five year age groups. The age distribution of person years of life gained is the same as in Figure 2.
Figure 5. Calculated from data in Social Security Administration (1992:Table 5), and two data points taken from Kannisto et al (1994:801).
Figure 6. Calculated from data in Social Security Administration (1992:Table 5), and two data points taken from Kannisto et al (1994:801).
Figure 7. Calculated from data in Horiuchi and Wilmoth (1995), Social Security Administration (1992), and Lee and Carter (1992).
Figure 8. Calculated from data in Horiuchi and Wilmoth (1995), Social Security Administration (1992), and Lee and Carter (1992).