Population aging and endogenous economic growth
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DOI: 10.1007/s0014801204419
 Cite this article as:
 Prettner, K. J Popul Econ (2013) 26: 811. doi:10.1007/s0014801204419
Abstract
We investigate the consequences of population aging for longrun economic growth perspectives. Our framework incorporates endogenous growth models and semiendogenous growth models as special cases. We show that (1) increases in longevity have a positive impact on per capita output growth, (2) decreases in fertility have a negative impact on per capita output growth, (3) the positive longevity effect dominates the negative fertility effect in case of the endogenous growth framework, and (4) population aging fosters longrun growth in the endogenous growth framework, while its effect depends on the relative change between fertility and mortality in the semiendogenous growth framework.
Keywords
Demographic change Technological progress Longrun economic growth1 Introduction
Population aging in industrialized countries has been identified as a central topic regarding future economic development. It has gained attention in academic research as well as in the public debate (see, for example, Bloom et al. 2008, 2010a, 2011; The Economist 2009, 2011, for an overview). While declining fertility—even below the replacement level—triggers increases in the mean age of a certain population and slows down population growth, decreasing old age mortality allows individuals to enjoy the benefits of retirement for longer time periods. Just to get an impression of the severity of the demographic changes, we are facing the following: on the global scale, the total fertility rate has dropped from five children per woman in 1950 to 2.5 children per woman today, while life expectancy has increased from 48 years in 1950 to 68 years today (cf. United Nations 2011; Bloom et al. 2011). The economic consequences of these developments are expected to be huge. To mention only the most well known examples, support ratios will decline such that fewer and fewer workers will have to carry the burden of financing more and more retirees (see for example Gertler 1999; Gruescu 2007); overall productivity levels will change because workers have agespecific productivity profiles and the age decompositions of societies will shift (see Skirbekk 2008, for an overview); and the savings behavior of individuals will change because they expect to live longer (see for example Heijdra and Ligthart 2006; Heijdra and Romp 2008).
However, as regards the implications of population aging for per capita output growth in a setting with diminishing marginal products of capital, there are only transient effects of changing support ratios, changing savings behavior of households, and changing aggregate productivity profiles. The reason is that a shift from high to low fertility cannot lead to a permanently changing age decomposition of a certain population (cf. Preston et al. 2001), and the induced change in the savings behavior of households has only level effects on per capita output (cf. Solow 1956; Cass 1965; Koopmans 1965; Diamond 1965). By contrast, we are interested in the implications of population aging for per capita output growth over a long time horizon. Since technological progress has been identified as the main determinant of longrun economic prosperity (see for example Romer 1990; Aghion and Howitt 1992; Jones 1995a; Segerström 1998), we are particularly concerned with the effects of changing age decompositions on research and development (R&D) intensities. The natural model classes to investigate these effects are endogenous and semiendogenous growth frameworks, where the R&D effort of a society is determined by general equilibrium forces due to the interaction between utilitymaximizing households and profitmaximizing firms.
Endogenous growth models with purposeful R&D investments (see for example Romer 1990; Grossman and Helpman 1991; Aghion and Howitt 1992) state that, aside from other influences, the population size of a certain country is crucial for its longrun economic development. The argument is that larger countries are able to grow faster because they have more scientists to employ and feature larger markets with more profit opportunities for innovative firms. This is called the scale effect, which was, however, questioned by Jones (1995b) because it had not been supported by empirical evidence. In another contribution, Jones (1995a) paved the way for semiendogenous growth models (see also Kortum 1997; Segerström 1998), where longrun economic performance is affected by population growth rather than population size. The basic idea of semiendogenous growth models is that developing a constant share of new technologies becomes more and more difficult with an expanding technological frontier. Consequently, ever more scientists have to be devoted to R&D activities in order to sustain a certain pace of technological progress. In the long run, this can only be achieved by having positive population growth.
Despite that the described models examine the economic growth effects of changes in demographic patterns as represented by population size and population growth, they remain silent when it comes to the consequences of population aging. The reason is their common underlying assumption that economies are populated by representative identical individuals who live forever. We introduce agespecific heterogeneity of individuals by generalizing these frameworks to account for finite individual planning horizons and overlapping generations in the spirit of Blanchard (1985) in case of the endogenous growth paradigm and in the spirit of Buiter (1988) in case of the semiendogenous growth paradigm. In so doing, we assume that individuals do not live forever but that they have to face a certain probability of death at each instant. Furthermore, we allow for endogenous fertility choices inspired by Barro and Becker (1989), Sato and Yamamoto (2005), and Miyazawa (2006), where parents want to have children, while they also face the associated costs in terms of foregone consumption. The standard endogenous and semiendogenous growth models are then special cases of our framework with the probability of death being equal to zero and fertility being exogenously given.
For analytical tractability, the mortality rate has to be ageindependent in our model. Consequently, the interactions between child mortality and fertility decisions cannot be studied satisfactorily within its realm. Nevertheless, we acknowledge that this is a very important topic: Doepke (2005) analyzes the implications of decreasing child mortality for the predictions of the Barro and Becker (1989) model and concludes that a decrease in child mortality alone cannot explain the slowdown of population growth within this framework. By contrast, KalemliOzcan (2002, 2003) assume that parents are risk averse and use this framework to investigate the interactions between child mortality and human capital investments. The finding is that a decline in the exogenous child mortality rate reduces precautionary demand for children, which has the potential to reduce population growth. The slowdown in population growth, in turn, allows parents to raise investments in children’s human capital, which enables countries in a preindustrial stage of development to escape from the Malthusian trap. Furthermore, Cigno (1998) focuses on the implications that endogenous child mortality has on the fertility decision of parents and finds that mortality and fertility are positively related if parents realize that they can increase the survival probabilities of their children by spending more for child nutrition and health. If child mortality is high (less developed countries), governmental investments aimed at reducing child mortality raise parent’s investments in the number of children as well as in the nutrition and health of these children, which altogether promotes population growth. By contrast, if child mortality is low (developed countries), further governmental investments in reducing child mortality mainly lead to lower investments of parents both in the number of their kids as well as in their nutrition and health. Thus, at this stage of development, governmental policies that reduce child mortality primarily represent a subsidy on parent’s consumption. We conclude that very interesting and relevant interactions occur once that (endogenous) child mortality is taken into account. It should therefore be considered in a followup simulation study, where analytical tractability is less important and therefore mortality can be allowed to vary with age.
Altogether, our results indicate that a more realistic demographic structure in traditional endogenous and semiendogenous growth models is desirable because it allows to disentangle the growth effects of a changing population size from those of a changing population age structure. We can also show that the growth effects of population aging differ between the endogenous growth paradigm and the semiendogenous growth paradigm. In particular, we find that decreasing mortality has a positive effect on longrun growth, while the converse holds true for decreasing fertility. The positive effects of decreases in mortality outweigh the negative effects of decreases in fertility in case of the Romer (1990) model, while the positive and negative effects exactly offset each other in the Jones (1995a) framework. Furthermore, population aging positively impacts longrun economic growth in the Romer (1990) case, whereas its particular effect in the Jones (1995a) case depends on the relative change between fertility and mortality.
Two other branches of the literature closely relate to our efforts. The first one (Reinhart 1999; Futagami and Nakajima 2001; Petrucci 2002) basically follows the Romer (1986) assumption that there are knowledge spillovers in the production process, and, hence, there are no diminishing returns of capital in the aggregate production function. This assumption allows them to draw conclusions on the effects of demographically induced changes in individual savings behavior even on longrun economic growth performance. A very interesting recent contribution (Schneider and Winkler 2010) uses this framework to endogenize the rate of mortality and to analyze the welfare implications of individual health investments. However, the knowledge spillover model of Romer (1986) has been criticized because empirical evidence rather points toward a diminishing marginal product of capital (cf. Mankiw et al. 1992). Furthermore, one cannot analyze the effects of aging on purposeful R&D investments within such a framework and, as we will see later on, the transmission mechanism of the impact of aging on economic growth within these types of models differs to our approach because we also allow for an endogenous interest rate.
The second related branch to our work (KalemliOzcan et al. 2000; Cervellati and Sunde 2005; Hazan and Zoabi 2006) focuses on the implications of population aging on human capital accumulation and basically states that an increase in the life expectancy of individuals renders investments in human capital more profitable. Consequently, human capital accumulation increases, which fosters economic growth via the particular link that these models establish between human capital accumulation and economic development.^{1} However, also these models do not consider the effects of aging on purposeful R&D investments and therefore the transmission mechanism of the effects of aging on economic growth is, by its very nature, different to ours.
The paper proceeds as follows: Section 2 describes a model that nests the Romer (1990) and the Jones (1995a) frameworks as special cases and features a richer demographic structure. Section 3 examines the effects of demographic change for longrun economic growth perspectives in the nested specifications. Section 4 draws conclusions and highlights scope for further research.
2 The model
This section characterizes the basic model of R&Dbased economic growth with overlapping generations and endogenous fertility. It nests the Romer (1990) framework with strong spillovers in the research sector and a constant population size and the Jones (1995a) framework with weaker spillovers in the research sector and a growing population size as special cases (cf. Strulik 2009).
2.1 Basic assumptions and their implications
The basic structure of our model economy is that there are three sectors: final goods production, intermediate goods production, and R&D. The economy has two productive factors at its disposal: capital and labor. Labor and machines are used to produce final goods for a perfectly competitive market, capital and blueprints are used in the Dixit and Stiglitz (1977) monopolistically competitive intermediate goods sector to produce machines, and labor is used to produce the machinespecific blueprints in the R&D sector. Note that the R&D sector competes for labor with the final goods sector on the perfectly competitive labor market.
Technological progress is considered to be horizontal, that is, it is represented by the introduction of new product varieties in the intermediate goods sector. Using a model with vertical innovations by assuming that technological progress takes the form of quality improvements of existing product varieties would not change our qualitative results with respect to the longrun growth rate. The reason is that the drivers of longrun growth in the corresponding endogenous growth literature (Grossman and Helpman 1991; Aghion and Howitt 1992) and semiendogenous growth literature (Segerström 1998) are qualitatively very similar to the generic models of horizontal innovation. However, while the longrun growth rate in models of horizontal innovation is always too low from a social perspective, this is not true for models of vertical innovation. In these frameworks, a quality improvement of an existing product by a contesting firm drives the incumbent out of business. This so called “business stealing effect” reduces the profits of established firms and therefore has a negative impact on overall welfare. Consequently, the possibility arises that—from a social perspective—innovation could be too fast and longrun economic growth too high. This means that the welfare effects of demographic change might be different if models of vertical innovations were considered as baseline frameworks in our study.^{2}
In contrast to the representative agent assumption on which the Romer (1990) and Jones (1995a) frameworks rely, we introduce overlapping generations in the spirit of Blanchard (1985) to the Romer (1990) case and in the spirit of Buiter (1988) to the Jones (1995a) case.^{3} We assume that the population of an economy consists of different cohorts that are distinguishable by their date of birth denoted as t _{0}. Each cohort consists of a measure N(t _{0}, t) of individuals at a certain instant t > t _{0}. In addition, individuals have to face a constant ageindependent instantaneous risk of death which we denote by μ. Due to the law of large numbers, this expression also refers to the fraction of individuals dying at each instant. Furthermore, we endogenize fertility decisions of individuals in the sense that their utility increases in the number of children they have, while raising children requires parents to forego consumption (cf. Barro and Becker 1989; Sato and Yamamoto 2005; Miyazawa 2006).^{4} The fertility decisions of individuals allow for population growth, population stagnation and population decline, depending on the level of mortality. We will examine the growth effects of all these possible outcomes for both specifications of the underlying R&Dbased economic growth model. Since the longrun growth rates tended to be constant in industrialized countries over the last decades (cf. Kaldor 1957; Jones 1995a; Acemoglu 2009), we will, in the next step, focus our attention on balanced growth paths (BGPs). By its definition, a BGP implies constant longrun growth, which requires the population to stay constant in the Romer (1990) framework and to grow in the Jones (1995a) framework. In our framework, both cases are possible outcomes for different parameter specifications with respect to the fertility decisions of individuals. Along the BGP of the overlapping generations version of the Romer (1990) model, the parameters have to be such that the birth rate equals μ, whereas in the overlapping generations version of the Jones (1995a) model, the parameters have to be such that the population grows at rate n = β − μ, where β > μ denotes the birth rate.^{5}
In the Romer (1990) framework, demographic change can then be analyzed by contemporaneous proportional shifts in both fertility and mortality. In this context, proportional means “in the same order of magnitude,” such that a constant population level is preserved. Decreasing fertility and mortality therefore leads to population aging and leaves the population size constant. In the Jones (1995a) framework, demographic change can be analyzed by changing the birth rate and the mortality rate independently of each other. Decreases in the birth rate ceteris paribus lead to population aging and a slowdown in the population growth rate, while decreases in mortality ceteris paribus increase the population growth rate but have no effect on the age structure.
Finally, we want to make some assumptions explicit that are implied by the overlapping generations and R&Dbased growth literature on which we base our analyses. These include that the labor supplies of different individuals, irrespective of their age, are perfect substitutes, individuals have perfect foresight, and they take the behavior of other agents as given, that is, they do not engage in strategic interactions. Furthermore, we abstract from bequests which would, in general, weaken the effects of generational turnover. However, unless full bequests were assumed (and therefore, the representative agent assumption would be reintroduced through the back door), our results would not change qualitatively.^{6}
2.2 Consumption side
2.2.1 Aggregation in case of a constant population
2.2.2 Aggregation in case of a growing population
2.3 Production side
2.4 Market clearing
3 Effects of demographic change on economic growth
From now on, we have to distinguish between the Romer (1990) case, where technological spillovers are strong and the population size is constant, and the Jones (1995a) case, where technological spillovers are weaker and the population grows at rate n. We derive the per capita growth rates along the BGPs in these cases and analyze their dependence upon demographic change. In the next two subsections, we focus on interior solutions, where growth rates are positive. When we summarize our results in Section 3.3, we will, however, also consider the boundary solutions.
3.1 The BGP growth rate in the Romer (1990) case
Proposition 1
In case of the endogenous growth in the spirit of Romer ( 1990 ), increasing longevity has a positive effect on the BGP growth rate of an economy.
Proof
The intuition for this finding is that a decrease in mortality slows down the turnover of generations, and so a lower market interest rate is required to sustain a given growth rate of aggregate consumption expenditures. Due to the fact that future profits of R&D investments are discounted with this market interest rate, the profitability of R&D investments rises. Consequently, R&D efforts increase which fosters longrun growth because intertemporal knowledge spillovers in the Romer (1990) case are large enough for the effect to be sustainable.
3.2 The BGP growth rate in the Jones (1995a) case
Proposition 2
In the case of semiendogenous growth in the spirit of Jones ( 1995a ), decreasing mortality raises the BGP growth rate of an economy, while decreasing fertility lowers it.
Proof
The interpretation for this finding is that a decrease in mortality, holding fertility constant, and an increase in fertility, holding mortality constant, both increase the population growth rate. This leads to a permanent increase in the flow of scientists into the R&D sector. Consequently, a faster growth rate of the number of patents can be sustained.
3.3 Comparison between the different frameworks
We start this subsection with a short summary regarding the possible effects of different demographic scenarios on longrun economic prosperity within the two different R&Dbased growth frameworks.
Remark 1
Since the empirically relevant cases for modern knowledgebased economies are those featuring positive nonaccelerating economic growth (cf. Kaldor 1957; Jones 1995a; Acemoglu 2009), we focus on the two relevant scenarios (2a, 3b) when comparing the effects of population aging on economic prosperity.
Considering scenario 2a, that is, the Romer (1990) model with demography, a decrease in mortality is accompanied by a proportional decrease in fertility. Both effects offset each other with regards to population growth such that the population size stays constant. Proposition 1 then allows us to conclude that the benefits of decreasing mortality for economic growth overcompensate the drawbacks of decreasing fertility. The intuitive explanation is that decreasing mortality also decreases the market interest rate by which future profits of R&D investments are discounted. This leads to a shift of resources to R&D and consequently fosters per capita output growth. The reason for the growth effect to be longlasting is that intertemporal knowledge spillovers are strong. By contrast, a contemporaneous proportional decrease in fertility and mortality does not change the growth rate along a BGP in scenario 3b, that is, in the Jones (1995a) model with demography, as evident from Eq. 39. The reason is that intertemporal knowledge spillovers are too weak for a onetime resource shift to have longlasting effects. We summarize this in the following remark:
Remark 2
In the case of endogenous growth in the spirit of Romer (1990), the benefits of decreasing mortality overcompensate for the drawbacks of proportional decreases in fertility with regards to longrun economic growth perspectives, while in case of semiendogenous growth in the spirit of Jones (1995a), the benefits and drawbacks exactly offset each other.
Finally, we know that population aging is described by contemporaneous proportional decreases in fertility and mortality in scenario 2a (the Romer 1990 model with demography), whereas in scenario 3b (the Jones 1995a model with demography), it can be triggered by decreases in fertility only. Therefore, population aging has a positive impact on longrun economic growth if endogenous growth models are the accurate description of underlying growth processes, while it has a negative impact in semiendogenous growth models as long as the fall in birth rates is not (over)compensated by (more than) proportional exogenous decreases in mortality.^{17} We summarize this finding in the following proposition:
Proposition 3
In the case of endogenous growth in the spirit of Romer ( 1990 ), population aging has a positive impact on the longrun economic growth rate.

If mortality is constant or decreases less than proportional to fertility, population aging negatively impacts longrun economic growth.

If mortality declines proportional to fertility, population aging has no effect on longrun economic growth.

If mortality declines more than proportional to fertility, population aging positively impacts longrun economic growth.
Proof
This follows from Propositions 1 and 2 and the fact that population aging is represented by a decrease in μ in the Romer (1990) model and by a decrease in β in the Jones (1995a) model.^{18} However, a decrease in fertility whose effect on the population growth rate is fully compensated by proportional decreases in mortality has no growth effect in the Jones (1995a) case as evident from Eq. 39. In case that the fertility decline is associated with a more than proportional decline in mortality, population growth and longrun economic growth even accelerate. □
Altogether, we have been able to describe some important effects of demographic change on economic development within the realms of R&Dbased economic growth models. In general, decreases in fertility negatively affect longrun growth, whereas decreasing mortality fosters longrun growth. In case of the Romer (1990) framework, the positive effects of decreasing mortality overcompensate for the negative effects of decreasing fertility, while in the Jones (1995a) framework, the positive and negative effects of contemporaneous proportional declines in both fertility and mortality offset each other. Furthermore, we have been able to show that the effects of population aging crucially depend on the underlying model used to describe the growth process. While population aging is in general beneficial in the Romer (1990) environment, the effect in the Jones (1995a) environment also depends on the relative change between fertility and mortality. If declining fertility is not fully compensated by declining mortality, population aging has a negative effect on economic growth, while the converse holds true if the mortality rate falls even stronger than fertility. In case that both mortality and fertility decrease proportionally, then the longrun economic growth rate is not affected at all.
4 Conclusions
We set up a model of endogenous technological change that nests the Romer (1990) and the Jones (1995a) frameworks. We generalized this model by introducing finite individual planning horizons and thereby allowing for overlapping generations and agespecific heterogeneity of individuals. Furthermore, we introduced endogenous fertility decisions of individuals such that they care for the number of kids they have, while taking into account the associated costs. Altogether, we showed that the underlying demographic processes play a crucial role in describing the R&D intensity and thereby longrun economic growth perspectives of industrialized countries.
Our results regarding the effects of demographic change on longrun economic growth perspectives have been the following: (a) decreasing mortality positively affects longrun growth, (b) decreasing fertility negatively affects longrun growth, (c) the negative effects of decreases in fertility are overcompensated by the positive effects of decreases in mortality in the case of the Romer (1990) model, while they exactly offset each other in the Jones (1995a) framework, (d) population aging is beneficial for longrun economic growth in the Romer (1990) case, whereas it depends on the relative change between fertility and mortality whether it is associated with increasing or decreasing longrun economic growth in the Jones (1995a) case.
Our main conclusion is that currently ongoing demographic changes do not necessarily hamper technological progress and therefore economic prosperity. Simultaneously decreasing birth and death rates can even lead to an increase in the economic growth rate. These results, while holding in a stylized theoretical modeling framework, are also in line with empirical studies claiming that the negative effects of population aging on economic prosperity might not be as severe as often argued (cf. Bloom et al. 2008, 2010a, b).
In our framework, we relied on the notion of horizontal innovations. However, the results would carry over to endogenous and semiendogenous growth models with vertical innovations (Grossman and Helpman 1991; Aghion and Howitt 1992; Segerström 1998) because the mechanisms causing economic growth in these models are very similar to those of the underlying models we used. It is, however, important to keep in mind that the welfare properties of models with horizontal innovation and of models with vertical innovation differ considerably. While faster economic growth is always beneficial in models with horizontal innovation, this is not true in models of vertical innovation. The reason is that—in the latter model class—introducing new products drives technologically inferior firms out of business. The associated negative welfare effect could potentially be strong enough to overcompensate for the positive welfare effects of technological progress through economic development. Frameworks that integrate horizontal and vertical innovation (cf. Young 1998; Peretto 1998; Dinopoulos and Thompson 1998) feature a balanced economic growth rate that positively depends on population growth and on the fraction of labor devoted to R&D. Therefore, elements of both cases that we considered would be present when introducing demography to such a framework. However, Jones (1999) showed that these models require very strong parameter restrictions for a balanced growth path to exist which limits their generality.
Of course we acknowledge that our modeling approach was only a first step toward a more thorough understanding of the impact of demographic change on longrun economic development as caused by purposeful R&D investments. In order to guarantee analytical solutions and to focus on the main channels by which demographic change can impact upon longrun economic growth, we abstracted from imperfect annuity markets, agedependent mortality, educational decisions (and therefore human capital investments), and agespecific and occupationspecific heterogeneities of workers. We believe that these assumptions represent a sensible choice regarding the tradeoff between analytical tractability and a concise exposition on the one hand and a detailed description of reality on the other hand. However, it makes clear that there is scope for further research.
There are various channels by which human capital accumulation can foster economic growth (see for example Lucas 1988; Galor and Weil 2000).
For an interesting study of physical capital taxation and labor income taxation in an overlapping generation growth model, see Chin and Lai (2009). However, longrun growth in their framework is not driven by R&D investments.
I would like to thank Timo Trimborn for his helpful suggestions regarding the implementation of fertility decisions within these types of models.
Note that the period fertility rate is equivalent to the birth rate in such a demographic setting (cf. Preston et al. 2001, p. 93).
For an interesting study on the effects of accidental bequests on inequality and growth, see Miyazawa (2006).
An alternative specification would be to allow for time costs of children such that each child requires the parent to spend a fraction of time at home and therefore not supplying labor on the labor market (cf. Galor and Weil 2000). This would, however, lead to perpetually increasing fertility in our case. The reason is that the expression for the birth rate involves the ratio of individual consumption to wages, which grows in an overlapping generations framework, even along the BGP.
Note that we abstract from indivisibilities of children similar to the standard Barro and Becker (1989) formulation.
This implicitly assumes that each worker is endowed with one unit of human capital. For alternative specifications, see, for example, Eicher (1996) and Arnold (2002).
If positive overall profits were present, new firms would enter the market until these profits had vanished.
Note that we do not allow for the possibility of duplication in the research process for the sake of comparability between the Romer (1990) model and the Jones (1995a) model. However, allowing different researchers to develop the same blueprint would not change our results qualitatively.
We solved the system using Mathematica. The corresponding file is available upon request. Note that there are two solution triples for g, r, and ξ. One of them features a negative ξ, and therefore it can be ruled out by economic arguments because neither the aggregate capital stock nor aggregate consumption expenditures can become negative. We therefore restrict our attention to the economically meaningful solution triple.
Strictly interpreted, in this case and for t → ∞, the corresponding societies would become extinct.
Recall that proportional means “in the same order of magnitude”, such that a constant population level is preserved.
Note that changes in the demographic structures are either triggered by changes in γ or by changes in ψ.
Acknowledgements
I would like to thank Anton Belyakov, David Canning, Alessandro Cigno, Dalkhat Ediev, Günther Fink, Inga Freund, Theresa GrafenederWeissteiner, Ingrid Kubin, Michael Kuhn, Alexia Prskawetz, Michael Rauscher, Maik Schneider, Holger Strulik, Timo Trimborn, Vladimir Veliov, Sebastian Vollmer, Katharina Werner, Ralph Winkler, Stefan Wrzaczek, two anonymous referees, and the participants of the Annual Meeting of the Austrian Economic Association 2010, the Conference of the European Society for Population Economics 2010, the Conference of the European Economic Association 2010, the Annual Meeting of the German Economic Association 2010, the “Work in Progress Seminar” at the Harvard Center for Population and Development Studies, and the “Macro Lunch Seminar” at the Department of Economics, Brown University, for useful comments and suggestions. Financial support by the Vienna Science and Technology Fund (WWTF) in its “Mathematics and...” call 2007, by the Max Kade foundation for financing the postdoctoral fellowship 30393 “Demography and Longrun Economic Growth Perspectives”, and by the National Institute on Aging is gratefully acknowledged.
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