A model of longevity, fertility and growth

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Abstract

Economic and demographic outcomes are determined jointly in a dynamic general equilibrium model of longevity, fertility and growth. Reproductive agents in overlapping generations mature safely through two periods of life and face an endogenous probability of surviving for a third period. Given this probability, each agent maximises her expected lifetime utility by choosing consumption and the number of children. Child-bearing is costly in the sense that time must be spent on child-rearing activities rather than on production or education. The model produces multiple development regimes which yield different predictions about life expectancy, fertility, timing of births and educational investment depending on initial conditions. These predictions accord strongly with the empirical evidence on demography and development.

Introduction

Greater life expectancy, lower fertility, later timing of births and higher levels of education are some of the most striking trends to occur during the demographic transition of economies. They are trends that account for major differences between developed and developing countries, and which may be seen as both causes and effects of changes in economic activity. As yet, there is no single theoretical model which is able to offer a unified explanation of these phenomena. In this paper we develop such a model, the hallmark of which is the joint, endogenous determination of longevity, child-bearing and human capital accumulation in dynamic general equilibrium.

Increases in life expectancy as an economy develops can be staggering, to say the least. Fogel (1994), for example, reports that, between 1850 and 1950, life expectancy at birth in the US rose by almost 75% from 40 yr of age to 68 yr of age. By 1995, the figure had moved closer towards 100% as average lifetime extended even further to 76 yr of age (Statistical Abstract of the United States, 1995). Similar dramatic changes have taken place in other countries and are well-documented in numerous studies, including Easterlin (1996), Fogel (1997), Livi-Bacci (1997), the United Nations (1991) and the World Bank (1993). In addition, there is considerable evidence, surveyed and contributed to by Mirowsky and Ross (1998), to suggest that such changes are symptomatic of higher levels of education and human capital accumulation which not only make more resources available for spending on life-preserving activities (by raising standards of living and fostering economic growth), but which also encourage the adoption of healthy lifestyles on the part of individuals for various socio-economic reasons. Indeed, it is possible to argue that personal education improves personal health primarily because it improves personal effective agency: that is, education allows people to develop knowledge, skills and abilities that make them better equipped to create a way of living that is conducive to their welfare and that is not mediated by economic status.1

At the same time as lifetimes have lengthened, there have been noticeable shifts in fertility patterns which reveal not only a fall in birth rates but also a growing tendency for births to occur later, rather than sooner, in life. The historical decline in fertility is probably the most well-known stylised fact of demographic transition (see, e.g., Barro and Sala-i-Martin, 1995; Coale and Watkins, 1986; Dyson and Murphy, 1985; Wrigley, 1969; World Bank, 1984). That the timing of fertility decisions has altered as well is also corroborated by a large body of evidence, as surveyed by Hotz et al. (1997). Some straightforward calculations, for example, show that, during the past 30 yr (both in the US and elsewhere), the probability of having a first child at age 20 has steadily fallen, while the probability of having a first child at age 35 has steadily risen. Like the decrease in mortality, both this intertemporal substitution of child-bearing and the general decline in fertility have often been studied in conjunction with observations about human capital accumulation. Of particular note has been the identification of a significant positive (negative) relationship between the timing of first births (number of births) and the length of time spent in education (see, e.g., Kravdal, 1994; Martinelle, 1990; Matthews et al., 1982; Rindfuss et al., 1996).

At the theoretical level, there exists a growing class of dynamic general equilibrium models which attend separately to one or more, though not all, of the above observations. Barro and Becker (1989) and Becker and Barro (1988) present the seminal analysis of fertility choice and growth, treating both the timing of births and lifetimes as given, and abstracting from human capital accumulation.2 Blackburn and Cipriani (1998) and Becker et al. (1990) extend this framework to allow for endogenous mortality and human capital accumulation, respectively. Galor and Stark (1993) introduce uncertain, but exogenous, lifetimes, along with human capital accumulation, while departing from endogenous fertility decisions. Ehrlich and Lui (1991) develop a similar model with fertility choice, but not the timing of fertility choice, re-included. And Iyigun (1996) considers the case of the joint determination of child-bearing, the timing of child-bearing and human capital accumulation, though still not the determination of lifetimes. Individually, these, and other, contributions are able to explain some, but not all, of the above evidence on demographic transition. The objective of the present paper is to develop a simple analytical model that is able to do so.3

We consider an overlapping generations economy in which the life expectancy of agents extends probabilistically to three periods. Agents are bearers of children, investors in education and producers and consumers of output. Child-bearing is costly in the sense that time must be spent on non-productive child-rearing activities, while educational investment is the means of accumulating human capital which raises the future productivity of labour. An exogenous increase in life expectancy (i.e., the probability of surviving for three periods) raises the opportunity costs of current work and reproduction by raising the future returns to human capital accumulation. Under such circumstances, agents devote more of their time to education and have fewer numbers of children when young, implying a higher growth rate of output and a lower growth rate of population. The innovation of our analysis is to endogenise life expectancy by allowing the probability of survival to depend on the level of development of the economy itself. That this may have important implications was evident to Hammermesh (1985) who also made the observation that individuals do, indeed, tend to extrapolate past improvements in longevity when determining their expected lifetimes (and, on average, tend to predict their horizons rather well).4 Extending our model in this way not only allows us to account for all of the above facts about demographic transition, but also has the effect of creating multiple development regimes such that the limiting outcomes of the economy depend critically on initial conditions. As development now takes place, there is an increase in life expectancy, an increase in education, a decrease in fertility and a decrease in child-bearing early on in life. This process of transition is not smooth, however, there being a threshold level of capital, below which the economy is on a low development path and above which the economy is on a high development path. Correspondingly, there is a low-steady-state equilibrium in which life expectancy is low, education is low, fertility is high and early child-bearing is high, and a high-steady-state equilibrium (or even an equilibrium with positive long-run growth) in which life expectancy is high, education is high, fertility is low and early child-bearing is low. These results complement those obtained in certain other models of fertility and growth (see, e.g., Becker et al., 1990; Galor and Weil, 1996; Iyigun, 1995; Nelson, 1956; Pavilos, 1995; Raut and Srinivasan, 1994), as well as those in the broader literature on poverty traps and threshold externalities. They are also related to the growing body of work on the development of economies over the very long run and the transition from pre-industrial to post-industrial societies (see, e.g., Galor and Weil, 1998; Kremer, 1993; Jones, 1999; Tamura, 1996, Tamura, 1999).5

The model is set out in Section 2. In Section 3 we solve for the optimal decisions of individuals. Section 4 contains our analysis of growth and demographic transition. Concluding remarks appear in Section 5.

Section snippets

The model

Time is discrete and indexed by t=0,…,. There is an endogenous population of reproductive agents belonging to overlapping generations with finite but uncertain lifetimes. Each agent matures safely through two periods of life and has a probability of surviving for a third period. After being raised by her parent in the first period of life, an agent becomes active as an investor in education, a bearer of children and a producer and consumer of output. Education is undertaken during the second

Individual fertility, education and production

An agent is faced with the problem of maximising (1) subject to (2)–(4), together with ytt+i=ctt+i (i=1,2). Solving this problem for any given πtt+1 yields the following optimal decision rules for labour supply and child demand:ltt+1=11+γ+πtt+1θ,ltt+2=11+γ+ψ,ntt+1=γq(1+γ+πtt+1θ),ntt+2=γq(1+γ+ψ).The decision rule for education follows as1−ltt+1−qntt+1=πtt+1θ1+γ+πtt+1θ.

These decision rules depend on the parameters q (the cost of child-rearing), γ (the utility weight on offspring), ψ (the utility

Demographics and development

The expression in (8) describes the equilibrium path of development of the economy. Changes in life expectancy cause changes in this path, affecting both the transitional dynamics and steady state of the economy. This result is notable as it stands, but it is all the more significant when one allows for the endogenous determination of life expectancy itself, as we do shortly.

If the probability of survival was constant, πtt+1=π for all t, then there would be a unique equilibrium in which the

Conclusions

It is natural to presume that life expectancy is an important factor in determining the life-cycle behaviour of individuals. At the same time, it is unnatural to presume that life expectancy is wholly exogenous and independent of economic conditions. Until now, models of fertility choice and growth have been based on both presumptions, meaning that they have given part, but not the whole, of the picture. The model developed in this paper is a first attempt at filling in some of the gaps by

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    The authors are grateful for the comments and suggestions of two anonymous referees on an earlier version of the paper, and for the financial assistance provided by the Leverhulme Trust (Grant No. F/120/BE) and the European Commission (TMR Grant No. ERB-FMB1-CT95-0058). The usual disclaimer applies.

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