Abstract
We examine the effects of population aging due to declining fertility and rising elderly life expectancy on consumption possibilities in the presence of intergenerational transfers. Our analysis is based on a highly tractable continuous-time overlapping generations model in which the population is divided into three groups (youth dependents, workers, and elderly dependents) and lifecourse transitions take place in a probabilistic fashion. We show that the consumption-maximizing response to greater longevity in highly developed countries is an increase in fertility. However, with larger transfer payments, the actual fertility response will likely be the opposite, leading to further population aging.
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Notes
The mechanism that we examine here shares the feature of the model by Easterlin (1987) that reductions in living standards trigger compensating reductions in fertility. However, we focus on the fiscal effect of transfers to the elderly on after-tax earnings rather than on the effect of cohort size on the pre-tax wage of young workers—in principle, the two mechanisms could easily co-exist. The model by Cerda (2005) of social security crises is similar in spirit to ours, but our model permits a broader understanding of the interrelatedness of consumption possibilities and age structure.
Gertler (1999) extends the model of Blanchard (1985), allowing for two age groups (working age and retired). Grafenhofer et al. (2007) present a model of probabilistic aging involving eight age groups, which they use to quantitatively assess the effects of population aging on savings, consumption, and the tax rate. Both models, however, take fertility as exogenous. Bommier and Lee (2003) develop a model of an economy with a continuous age distribution. They derive some interesting aggregate steady-state results, but make limited progress in analyzing aggregate dynamics.
In our model, we ignore childhood and early-adult mortality, assuming that death occurs only among the retired. The model could easily be adapted to incorporate these other forms of mortality by changing the basic equations to:
$$ {\begin{array}{*{20}l} {\dot{{A}}_{\rm Y} (t)=N(t)-(\lambda_Y +\mu_{\rm Y} )A_{\rm Y} (t),} \\[2pt] {\dot{{A}}_{\rm M} (t)=\lambda_{\rm Y} A_{\rm Y} (t)-(\lambda _{\rm M} +\mu_{\rm M} )A_{\rm M} (t),} \\[2pt] {\dot{{A}}_{\rm O} (t)=\lambda_{\rm M} A_{\rm M} (t)-\mu _{\rm O} A_{\rm O} (t).} \\ \end{array} } $$With this set-up, λ j is the transition probability to the next group, and μ j is the death rate in the group. Analysis of this model would parallel that in the main text, with qualitatively similar results.
An additional set of equations, consistent with those governing the economic age structure, could be specified to allow ϕ to vary over time. However, such a specification results in a model that is substantially more complicated to analyze while adding very little qualitatively to our analysis.
A companion paper (Hock and Weil 2007) considers the additional effects of changes in the retirement age.
The GRR is the number of children individuals can expect to have, assuming that they are subject to the currently prevailing rate of fertility for the entirety of their childbearing years. Given our assumptions regarding the stock of fecund persons, the GRR works out to G = T F n = ϕT M n.
A stable population is one in which the relative size of the age groups is constant. When fertility is at the replacement rate (G = 1), the equilibrium dependency ratios are \(\bar{{y}}^r=T_{Y} /T_{M} \) and \(\bar{{o}}^r=T_{O} /T_{M} \), paralleling basic results from stable population theory.
The calculations in Hock and Weil (2007) use lifecycle consumption data from Japan and the United States as a benchmark and are derived from a simulated numerical analogue of the model presented here. Based on a range of mortality and retirement profiles that span what is observed in OECD member nations, consumption-maximizing fertility rates are generally well above replacement while actual fertility rates are substantially lower.
In the United States, for example, life expectancy at age 65 has risen by approximately 3.5 years since 1950 (Board of Trustees, Federal Old-Age and Survivors Insurance and Disability Insurance Trust Funds 2005) and is projected to rise by an additional 4 years by 2050 (Lee and Carter 1992). Other highly developed countries have seen reductions in old-age mortality at least as large and can be expected to follow a similar trend in the future (Munnell et al. 2004).
This can be established analytically since the marginal rate of demographic substitution rises for any value of y as T O increases.
Kohler et al. (2002) provide a more comprehensive review.
Although somewhat stylized, this distinction mirrors the substantial difference in the pattern of transfers to the old and young observed in highly developed countries. For example, private transfers account for over 61% of the transfer-based consumption of persons under age 20 in the United States, but only 11% of total transfers to persons over age 65 (Mason et al. 2009). One consequence of this assumption is that we abstract from a self-interested old-age support motive for childbearing of the type highlighted by Cigno (1992).
Today, Japan is the best example of a country where falling population size interacts in a toxic fashion with a high level of per-capita government debt.
Given the stochastic nature of aging in our model, parental payments are made upfront into a trust fund that pays for χ units of childcare services per period for the life of the child. To ensure that children always receive the requisite care, an actuarially neutral trust fund will set ξ = χ T Y /(1 − g T Y ), where g is the expected rate of wage growth. For the path of trust fund payments to remain bounded, g must be less than 1/ T Y . Taking 25 years as an upper limit on the average length of youth dependency, g must be less than 4% over the long run, which is in line with observed growth rates.
A proof is available on request.
In the case where all expenditures on children are privately funded, as in van Groezen et al. (2003), subsidies would be pegged to the child’s expected lifetime contribution to the pension system. In our model, child subsidies would be determined by the expected reduction in the tax burden associated with childbearing.
Cerda (2005) describes a similar mechanism. However, in the absence of public transfers to the young in his model, low fertility may produce ever-rising payroll tax rates, eventually resulting in a social security melt-down.
For analytic simplicity, we continue to treat public support for the young, α, as fixed and exogenous, although the model could accommodate bargaining over both parameters.
In general, the \(\tilde{{Z}}_y (y)\) locus either stays below the horizontal axis or becomes undefined after the initial crossing; we ignore the trivial set of values of μ for which the locus would start to rise after meeting the horizontal axis.
If the decisiveness parameter is less than one, then the multiplier effect on longevity will be amplified in the short run but dampened in the long run.
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Responsible editor: Junsen Zhang
Heinrich Hock acknowledges financial support for early work on this paper from an NICHD training grant (T32-HD07338) to the Population Studies and Training Center at Brown University. This paper has benefited from comments by participants at the Brown University Department of Economics Macro Lunch and the 2006 Conference on Population Aging, Intergenerational Transfers, and the Macroeconomy. We are particularly grateful to Alexia Prskawetz and two anonymous referees for valuable feedback. All errors are our own.
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Hock, H., Weil, D.N. On the dynamics of the age structure, dependency, and consumption. J Popul Econ 25, 1019–1043 (2012). https://doi.org/10.1007/s00148-011-0372-x
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DOI: https://doi.org/10.1007/s00148-011-0372-x