Adult Longevity and Growth Takeoﬀ
Daishin Yasui
Graduate School of Economics, Kobe University
August 22, 2012

Abstract
This paper develops an overlapping generations model in which agents make educational and fertility decisions under life-cycle considerations, and retirement from work
is distinguished from death. This model sheds light on a novel mechanism that links
life expectancy, retirement, education, fertility, and growth. Gains in adult longevity
induce agents to save more for retirement, reduce fertility, invest in education, andachieve sustained growth. Even if the length of working life is shortened by early retirement, this mechanism works as long as adult longevity increases suﬃciently. Ourmodel replicates the stylized facts of the transition from stagnation to growth in termsof longevity, time in retirement, fertility, education, and income, as well as reconciles
the theory that gains in life expectancy trigger a growth takeoﬀ by increasing education with the observation that the length of working life is not substantially prolonged
because of retirement. This study provides a framework for considering the joint
determination of education, fertility, and retirement.
JEL classiﬁcation: J13; O11
Keywords: Fertility; Growth; Human capital; Life expectancy; Retirement

Email address: yasui@econ.kobe-u.ac.jp
Tel: +81-78-803-7245
Fax: +81-78-803-7293

1

1

Introduction

It is sometimes argued that increased life expectancy triggers rises in investment in humancapital, a demographic transition, and a takeoﬀ from stagnation to growth.1 This argument is based on the life-cycle model à la Ben-Porath (1967): increased life expectancy
prolongs the working life over which investments in human capital are paid oﬀ, thereby
positively aﬀecting human capital investment. This intuitively appealing mechanism has
attracted many growth theorists, and a number of growth theories have been constructedbased on it. Examples include Ehrlich and Lui (1991), Meltzer (1992), de la Croix and
Licandro (1999), Kalemli-Ozcan et al. (2000), Blackburn and Cipriani (2002), Boucekkine
et al. (2002, 2003), Kalemli-Ozcan (2002), Chakraborty (2004, 2005), Cervellati and Sunde(2005, 2007), Soares (2005), Boucekkine et al. (2007), Soares and Falcão (2008), and de la
Croix and Licandro (2011). However, this conventional theory is challenged by the factthat even if mortality declines, the period of working life, over which the investments inhuman capital are paid oﬀ, might not be prolonged, because of early retirement. The estimation of Lee (2001) shows that over the last 150 years, the expected length of retirement
life at age 20 for U.S. male workers increased considerably, while their expected length ofworking life exhibited little change.2 Hazan (2009) ﬁnds a similar trend.3 Gendell andSiegel (1992) estimate that age at ﬁnal retirement has fallen by 4 to 5 years for both men
and women since 1950.4 Whether increased life expectancy promotes education through1

Life expectancy generally reﬂects both child mortality and adult longevity. The focus of this paper ison adult longevity. We use the term “life expectancy” to mean adult longevity unless otherwise noted.
2
According to Lee (2001), the expected length of the retirement period at age 20 rose from 2.65 years
for the cohort born in 1830 to 5.50 years for the cohort born in 1880 and 13.13 years for the cohort born in
1930, while the expected lengths of working life for these cohorts were 41.05, 41.79, and 41.89, respectively.
3
According to Hazan (2009), in the United States, the expected length of retirement life increased morethan that of working life in the 19th and 20th centuries: 20-year-old men who belong to the cohort bornin 1840 were expected to live for another 43.2 years and work for 37.23 years, while their counterpartsborn in 1930 were expected to live for another 53.01 and work for 41.73 years. Note that Hazan (2009)
explores not only the expected number of years in the labor market but also yearly hours worked andshows that the expected total working hours over a lifetime decreased because of declining hours worked
per year. In our model, as will be apparent, decreases in child-rearing time result in increases in working
time, stimulating educational investments. However, taking into account that female participation in paid
work increased and female participation in housework decreased over the last century (see Greenwood et al.
2005, Ramey and Francis 2009, and Kimura and Yasui 2010) and that women have taken the central rolein child rearing, on which this paper focuses, our mechanism would not contradict the historical evidence
on households’ time allocation.
4
Although the studies referred to above are based on U.S. data, there is little doubt that most developed
countries have experienced both rises in life expectancy and declines in labor force participation by the2the Ben-Porath channel mentioned above is controversial, but there is little doubt that
increased life expectancy has resulted in a considerable increase in the length of retirement
life.5 This paper aims to construct a growth model focusing on the role of the length ofretirement life, rather than that of working life, in individual life-cycle behavior.
The key mechanism in our model is as follows. Gains in life expectancy induce agents
to save more for their retirement life and to reduce fertility to increase their labor supply
and earn more income.6 With strong motives to save, agents allocate resources to theirown education, thereby escaping stagnation and reaching a state of sustained growth.
Even if the length of working life is shortened by early retirement, this mechanism works
as long as adult longevity increases suﬃciently.7 That is, our model reconciles the theory
that gains in life expectancy trigger a growth takeoﬀ by increasing educational investment
with the observation that because of retirement, the length of working life has not been
substantially prolonged.8 Incorporating fertility choice is crucial for deriving the resultthat increased adult longevity promotes education without the prolongation of working life:
declines in fertility induced by gains in adult longevity imply increases in labor supply, andthus the return on education can rise despite falls in retirement age. Our results indicatethat economies (dynasties) with low longevity are likely to stagnate and those with highelderly. See the World Bank’s World Development Indicators for life expectancy and Kalemli-Ozcan andWeil (2010) and the references therein for labor force participation of the elderly.
5
Regarding the controversy on the Ben-Porath channel, see Hazan (2009), Sheshinski (2009), Cervellati
and Sunde (2010), and Cai and Lau (2011) and the references therein. Sheshinski (2009), Cervellati andSunde (2010), and Cai and Lau (2011) aim to overcome the challenge raised by Hazan (2009) by generalizing
survival law. On the other hand, we focus on fertility choice, that is, household time allocation, while
keeping the rectangularity on survival function as in Hazan (2009). Our model will complement rather
than contradict the attempts of these studies.
6
Empirical analyses in Bloom et al. (2003) and Zhang and Zhang (2005) indicate that life expectancy
has positive eﬀects on savings rate.
7
Bar and Leukhina (2010) present a growth model in which reductions in adult mortality trigger a
growth takeoﬀ by improving knowledge transmission and encouraging innovation activity. Their mechanism
also works without rises in the length of working life.
8
Ferreira and Pessôa (2007) have a similar motivation. They attempt to reconcile the increases in
education and rises of retirement by incorporating labor-leisure choice, not fertility choice, into a life-cycle
model. In their model, to reconcile the two facts, agents must work more intensively at the expense ofleisure when they are young. According to the estimation of Ramey and Francis (2009), however, leisuretime did not decrease for all age groups in the twentieth-century United States. Furthermore, it should be
noted that there is more to incorporating fertility choice, not labor-leisure choice, than consistency withthe data. Under the conventional assumption that the utility from leisure is discounted at the same rate
as the utility from consumption, as in the model of Ferreira and Pessôa (2007), increased longevity does
not alter the relative marginal utility of consumption compared to leisure, and thus the eﬀect proposed inthis paper does not exist (see also the discussion below equation (1) in Section 2).

3

longevity are likely to grow, and within an intermediate range of longevity, multiple longrun equilibria arise; whether an economy (a dynasty) is stuck in a poverty trap or grows
persistently depends on the initial amount of ancestral human capital.
Particularly important results in the growth literature are (I) signiﬁcant increases inlife expectancy are needed to promote education and generate a growth takeoﬀ (i.e. a small
increase might have no eﬀect on schooling and growth); (II) under a given life expectancy,
some economies (dynasties) succeed in growth while others remain poor, depending on theinitial amount of ancestral human capital; and (III) rises in education, declines in fertility,and falls in retirement age can occur simultaneously. The ﬁrst two results can account for
two historical facts often viewed as evidence against the growth models that focus on life
expectancy. Result (I) is consistent with the fact that in the Western world, life expectancy
began to rise more than a century prior to the signiﬁcant increases in schooling that took
place in the second half of the nineteenth century.9 Result (II) can account for the fact that
although the spread of health and medical knowledge to less-developed countries has led todramatic cross-country convergence in life expectancy, convergence in per-capita income
has not been observed.10 Result (II) also has important policy implications: policies
promoting education as well as those improving health are needed to put an economy
stuck in a poverty trap onto a growth path. Result (III) captures a salient feature of theprocess of economic growth broadly observed across industrialized economies in the lastcentury. To the best of our knowledge, this is the ﬁrst study to generate rises in education,
declines in fertility, and falls in retirement age simultaneously in a single framework.
We share with Soares (2005) the motivation to explain the changes in fertility andeducational attainment in terms of exogenous changes in life expectancy.11 Soares (2005)

9

We are aware that in Western Europe, there were gains in literacy prior to the establishment of formal
schooling in the 19th century (see, for instance, Boucekkine et al. 2003). On the other hand, secondary
schooling did not rise in those days. Since this paper addresses the agents’ decision on their own education,
it is reasonable to focus on the data on secondary schooling. Galor (2005a) provides a comprehensive survey
on demographic changes and related facts.
10
See Deaton (2004), Becker et al. (2005), and Acemoglu and Johnson (2007) for details.
11
Although our theory, as well as that of Soares (2005), concentrates on exogenous reductions in mortality
as the main driving force behind the demographic transition and growth takeoﬀ, we do not deny thepossibility that individuals can invest in their own health to reduce mortality. However, several studies
support the view that a large fraction of mortality declines are unrelated to economic conditions andexogenous to individuals and countries (see, for instance, Preston 1975, 1980; Becker et al. 2005).

4

provides the cross-country data for 1960 and 1995, suggesting that for constant levels of
income, life expectancy and educational attainment rose and fertility declined, and changes
in fertility and educational attainment closely followed the changes in life expectancy. We
are also motivated by this evidence. Our model departs from the model of Soares (2005)
by explicitly considering individual life-cycle behavior. In particular, since our model
distinguishes between retirement from the labor market and death, we can separately
explore the eﬀects of the prolongation of working life and those of retirement life. Thisdistinction is important because the prolongation of working life and that of retirement
life might provide very diﬀerent incentives for individuals, for example, our model suggeststhat the prolongation of working life increases fertility while that of retirement life reduces
fertility.
Many data, including those presented in Soares (2005), suggest a strong correlation
between life expectancy and economic development and growth.12 Nevertheless, the causal
interpretation of this relationship is controversial. In recent studies, for example, Acemoglu
and Johnson (2007) ﬁnd that increases in life expectancy had either an insigniﬁcant or asmall negative eﬀect on income per capita while Lorentzen et al. (2008) ﬁnd that life expectancy is a signiﬁcantly positive predictor of income per capita.13 Cervellati and Sunde(2011) reconcile the seemingly contradictory empirical ﬁndings by allowing for a nonmonotonic eﬀect of life expectancy on income growth. Their estimation suggests that theeﬀect of life expectancy on income per capita may be ambiguous before the demographictransition, but should be unambiguously positive after its onset. Our theory predicts thatsigniﬁcant increases in life expectancy and/or parents’ human capital are required to generate a growth takeoﬀ, and that increased life expectancy promotes education after the
takeoﬀ but does not promote education before the takeoﬀ, a prediction consistent with
the non-monotonicity result of Cervellati and Sunde (2011).
12
See, for instance, Shastry and Weil (2003) for the cross-country relationship between adult longevity
and income per capita. Jayachandran and Lleras-Muney (2009) ﬁnd that life expectancy positively aﬀects
educational attainments at the individual level using data for Sri Lanka.
13
The diﬀerence between their results might partly come from the diﬀerence between the indicators
used in their estimation. The focus of Acemoglu and Johnson (2007) is largely on child mortality whilethat of Lorentzen et al. (2008) is on adult mortality. Lorentzen et al. (2008) ﬁnd that adult mortality
strongly aﬀects growth by aﬀecting investment and fertility even when controlling for infant mortality.
The prediction of our model is consistent with their empirical results.

5This paper is not the ﬁrst study to focus on the motives for savings as a channel
through which increased life expectancy reduces fertility. This channel was already observed by Zhang and Zhang (2005) and Chen (2010). What diﬀerentiates our work fromtheirs is the detailed investigation into life-cycle behavior. We employ a continuous-timeoverlapping-generations model, and the periods of working life and retirement each have apositive length. In contrast, Zhang and Zhang (2005) and Chen (2010) use a discrete-timeoverlapping-generations model in which an individual’s life consists of childhood, adulthood, and old age, and increased life expectancy is modeled as the increased probabilityof surviving to old age. Since increased life expectancy is tantamount to prolonged retirement life in their model, they could not separately explore the eﬀects of the prolongationof working life and that of the retirement period. We can conduct such an explorationand consider the endogeneity of the length of working life (Subsection 3.1) and the eﬀects
of increased longevity in the earlier stages of development where there is little retired life
(Subsection 3.3).14
This paper follows a large body of literature that addresses the interaction of educationand fertility in economic development and growth (e.g. Becker et al. (1990), Tamura(1996), Galor and Weil (2000), Galor and Moav (2002), Greenwood and Seshadri (2002),
Hazan and Berdugo (2002), Lucas (2002), de la Croix and Doepke (2003, 2004), KalemliOzcan (2003), Lagerlöf (2003), Doepke (2004, 2005), Doepke and Zilibotti (2005), Soares(2005), Moav (2005), Soares and Falcão (2008), and de la Croix and Licandro (2011)).
With respect to the transmission of poverty across generations and its relationship withfertility, this paper is similar to Hazan and Berdugo (2002) and Moav (2005). Their models
and ours predict that dynasties within a country can converge to one of two long-run
equilibria: poor dynasties with human capital below a threshold level converge to a loweducation and high-fertility equilibrium, whereas wealthy dynasties with human capital
above the threshold level converge to a high-education and low-fertility equilibrium.15 This14

The focus of Zhang and Zhang (2005) is rather empirical, while ours is theoretical. Despite thediﬀerences between their model and ours mentioned above, their empirical results that life expectancy has
a signiﬁcant positive eﬀect on savings rate, secondary school enrollment, and growth, but a signiﬁcantnegative eﬀect on fertility, support our theory.
15
This prediction implies that if the initial average level of human capital is above the threshold, thena larger fraction of dynasites converge to a high-education and low-fertility equilibrium in a more equal6prediction implies that poverty can persist even in rich countries, while wealthy households
can carry through even in poor countries.
We focus on adult longevity, whereas several studies (e.g. Kalemli-Ozcan (2003),
Lagerlöf (2003), Doepke (2005), Soares (2005), Hazan and Zoabi (2006), and Soares andFalcão (2008)) explore the eﬀects of declining child mortality on the quantity-quality
tradeoﬀ ﬁrst introduced by Becker (1960).16 There is controversy, however, as to whether
declining child mortality induces parents to substitute quality for quantity and reduce
fertility.17 Nevertheless, this controversy does not detract from the contribution of ourpaper because we address the individual’s own education and the eﬀects of rising adult
longevity.18 Our paper is related to Boucekkine et al. (2002) and de la Croix and Licandro(2011) in that the role of adult longevity is considered in a continuous-time overlappinggenerations framework. Our paper diﬀers from these studies as our model jointly considers
the endogeneity of fertility and the presence of the retirement period.19 The joint consideration of fertility and retirement enables us to derive some interesting results: for example,
under the assumption of exogenous retirement, the prolongation of working life and retirement life might provide very diﬀerent incentives for individuals; under the assumptionof endogenous retirement, fertility reduction induced by gains in adult longevity decreases
the retirement age.
This paper also provides a new insight concerning the individual choice of retirement
date. In previous decades, explaining the decline in labor force participation of the elderly

economy. In this respect, our paper, like Hazan and Berdugo (2002) and Moav (2005), is related to theliterature on the eﬀect of income inequality on economic development and growth (e.g. Banerjee andNewman (1993) and Galor and Zeira (1993)).
16
Soares (2005) and Soares and Falcão (2008) explore the eﬀects of both adult longevity and childmortality.
17
Hazan and Zoabi (2006) argue that declining child mortality does not aﬀect the quantity-quality
tradeoﬀ because it aﬀects not only the returns to quality but also the returns to quantity. Kalemli-Ozcan(2003) shows that when individuals decide on fertility in the face of uncertainty concerning child survival
and a precautionary demand for children exists, declining child mortality reduces fertility. Doepke (2005)
argues that precautionary demand should not have a strong eﬀect because deceased children can be replaced
through higher subsequent fertility. Galor (2005b) points out the implausibility of precautionary demand
from an evolutionary perspective.
18
Dahan and Tsiddon (1998), Blackburn and Cipriani (2002), Soares (2005), Zhang and Zhang (2005),
Kimura and Yasui (2007), Soares and Falcão (2008), Chen (2010), and de la Croix and Licandro (2011)
also consider agents’ investments in their own education in a framework with endogenous fertility.
19
Boucekkine et al. (2002) distinguish retirement from death, but assume exogenous fertility; de la Croix
and Licandro (2011) endogenize fertility, but do not distinguish retirement from death.

7has been a major concern for economists. For example, a well-developed public pension
system, income eﬀects of rising wages, and technological progress that makes obsolete theskills of the elderly are well known as possible causes of the decline in retirement age.
Our work is related to Boucekkine et al. (2002), Ferreira and Pessôa (2007), and KalemliOzcan and Weil (2010), which explore the eﬀects of increased longevity on the retirement
decision.20 We show that declining fertility reduces the retirement age, an eﬀect not
identiﬁed in the literature as a possible mechanism that induces early retirement. Declines
in fertility mean increases in working time and increases in earnings for agents when they
are young adults, thereby enabling them to save more for the future and retire early via theincome eﬀect. We refer to this as the ‘fertility eﬀect’ on the individual retirement decision.
This eﬀect allows our model to simultaneously generate rises in education, declines in
fertility, and falls in retirement age.
The remainder of this paper is organized as follows. Section 2 presents the basic modeland illustrates the main results of our paper. In Section 3, we consider the individual retirement decision and explore the joint determination of retirement and fertility, introduce
a quantity-quality tradeoﬀ as an extension, and discuss the possibility that our model isapplicable to an explanation of the long-run fertility transition. Section 4 concludes.

2

Basic Model

2.1

Economic Environment

Consider a small-open economy in which time is continuous and the population consistsof a discrete number of overlapping generations.
An individual’s life consists of three periods, childhood, adulthood, and old age, eachof which has a time interval with positive measure. In childhood, agents do not make any
decisions and only consume a ﬁxed quantity of time from their parents. In adulthood,they invest in their own education, raise children, supply labor to the market, and con20
See Kalemli-Ozcan and Weil (2010) and the references therein for details on such works. Kalemli-Ozcan
and Weil (2010) present a novel explanation for the rise in retirement, which they call the ‘uncertainty
eﬀect’: as mortality falls, the risk of dying before consuming retiremet savings falls, and it becomes optimal
to plan and save for retirement.

8

sume goods. In old age, they only consume goods. We denote the lengths of childhood,
adulthood, and old age by D, W , and R, respectively. That is, the length of the period
over which agents consume goods is T ≡ W + R. We treat T as a proxy variable ofadult longevity: ‘increases in life expectancy’ and ‘the prolongation of lifetime’ are deﬁned
as rises in T in the model.21 To keep our primary mechanism as simple as possible, we
assume that there is no uncertainty about the length of each period. All decisions are
made at the beginning of adulthood (adult age 0): agents decide the number of children,
the amount of investment in their own education, and the consumption plan over their
lifetime.22 For simplicity, the time required for education and childbirth is assumed to
be zero: both events are instantaneously completed at adult age 0. The educational investment entails only pecuniary costs, and the expenditure on education must be repaid
during the remainder of life.23 The world rate of interest is equal to r > 0, which isconstant over time. Individuals can lend and borrow any amount at this rate. Having
children entails only time costs for child rearing. Each adult is endowed with a unit oftime that can be devoted to market work and child rearing at each time. Rearing a childrequires a fraction z ∈ (0, 1) of the time endowment at every time from adult age 0 to D,
during which period children are dependent family members. Agents retire from work at
adult age W and die at adult age T ≡ W + R. We assume D < W , which seems realistic.
Figure 1 illustrates the timing of the model.
Agents receive utility from the number of children they have and from the consumption

21

In this section, we assume that D, W , and R are exogenously ﬁxed. In Section 3, as an extension, we
endogenize the length of working life, W , under the assumption that T is exogenously ﬁxed.
22
The basic model excludes the educational investments made for children to concentrate on our main
contribution. In Section 3, we incorporate the investments in a child’s education, showing that the so-called
quantity-quality interaction does not change our main results.
23
As suggested by a large body of literature on human capital accumulation, for example, Bils and Klenow(2000), the opportunity cost constitutes a major part of education costs in the real world. The assumptionthat education is goods-consuming is for analytical convenience; we can introduce time-consuming education in place of goods–consuming education without aﬀecting the qualitative results. A possible way is todivide childhood into two sub-periods, early childhood and later childhood, and assume that agents canchoose how much time to spend in schooling and work in later childhood; human capital in later childhood
is exogenously given and schooling in later childhood increases human capital used in adulthood. Under
such assumptions, the results remain qualitatively the same as in the case of goods-consuming education.

9

Figure 1: Timing of events

stream over their lifetime. The utility function of the agents of generation t is
γ ln nt + (1 − γ) T0e−ρτ ln ct (τ ) dτ,(1)where nt and ct (τ ), respectively, represent the number of children and consumption at
adult age τ , and γ ∈ (0, 1) and ρ > 0 denote the relative weight given to children andthe subjective discount rate, respectively. The ﬁrst term of (1) denotes the utility thatparents derive from children and the second term denotes the utility that they derivefrom consumption. The main result of this paper comes directly from this speciﬁcationof preferences: the prolongation of lifetime implies that more weight is given to self consumption. Alternatively, if parents simply regard their children as consumer durables fromwhich they receive utility at each time, the ﬁrst term also must be integrated from 0 to
T and discounted at rate ρ. Given that there are motives to demand children other thanthe consumption motive (e.g. the procreation motive), however, it seems reasonable toassume the utility function given by (1) to concentrate on our main mechanism.24 For
24
A speciﬁcation such as (1) is only one extreme, as is the speciﬁcation that the term on children is
integrated from 0 to T and discounted at rate ρ; the reality might be somewhere between these two
speciﬁcations. The eﬀect proposed in this paper does not vanish as long as the latter extreme is notassumed.

10example, Soares (2005), Zhang and Zhang (2005), Doepke et al. (2007), and Chen (2010)
also assume a utility function in which the utility from children is not discounted in thesame way as the utility from consumption. Note that our main result is robust as long as
the prolongation of lifetime raises the relative weight given to self consumption (e.g. thelog utility is just for convenience).
Denote the wage rate by w > 0. Agents with human capital ht and nt children earn
(1 − znt ) wht at each time from adult age 0 to D and earn wht at each time from D to
W . The production function of human capital is given by


,
h
,
ht = max ηxθt h1−θ
t−1

(2)

where xt and ht−1 are educational investment and parents’ human capital, respectively,and η > 0, θ ∈ (0, 1) and h > 0 are the parameters. This formulation of human capital
production technology implies that the minimum level of human capital, h, is guaranteed
irrespective of educational investment and parents’ human capital. Alternatively, we caninterpret that there exist two sectors, a modern market sector and a traditional agriculture
sector (or home production), and agents can freely decide which sector to choose. Human
capital matters, and earnings depend on the amount of human capital in the former sector
but not in the latter.
The accumulation of assets is described byȧt (τ ) = rat (τ ) + (1 − znt ) wht − ct (τ ) for τ ∈ [0, D] ,(3)ȧt (τ ) = rat (τ ) + wht − ct (τ ) for τ ∈ [D, W ] ,(4)ȧt (τ ) = rat (τ ) − ct (τ ) for τ ∈ [W, T ] ,

(5)

and

where at (τ ) is the quantity of assets at adult age τ and ȧt (τ ) denotes its time derivative.
The diﬀerences among (3), (4), and (5) indicate that agents can supply labor in adulthood
but not in old age, and that they must spend time znt on child rearing until adult age D.

11

The initial and terminal conditions for assets are given by, respectively,
at (0) = −xt and at (T ) = 0.

Agents start their working lives with education expenditure debts, and they cannot leave
debts or bequests for their oﬀspring.
The intertemporal budget constraint for the agents of generation t is
 T0e−rτ ct (τ ) dτ + xt = D0e−rτ (1 − znt ) wht dτ + WDe−rτ wht dτ.

The present discounted value of expenditure on consumption and education must equalthat of lifetime earnings.

2.2

Agent’s Problem

Consider the agent’s problem. The optimal path of consumption is given by the familiar
Euler equation:ċt (τ )= r − ρ.
ct (τ )

(6)

It follows that consumption at adult age τ can be written as
ct (τ ) = Ct e(r−ρ)τ for τ ∈ [0, T ] ,where Ct > 0 is the initial consumption.
Therefore, the problem faced by the agent can be written as
max γ ln nt + (1 − γ) Tnt ,Ct ,xt

 T
s.t.
0

e−ρτ Ct dτ + xt =

 D
0

0

e−ρτ ln Ct e(r−ρ)τ dτ,e−rτ (1 − znt ) wht dτ +



,
h
.
and ht = max ηxθt h1−θ
t−1

 W
D

(7)

e−rτ wht dτ,(8)

We focus on the interior solutions for fertility choice throughout this paper. Depending12

on the parameters, the time constraint at each time might be binding for τ ≤ D, that is,znt = 1. We concentrate on the parameter conﬁgurations such that the time constraint
does not bind.25
The solution to the above problem can either be interior or at a corner for educational
choice, that is, xt > 0 or xt = 0.26 First, consider the case of an interior solution. The
ﬁrst-order conditions imply
nE
t =

CtE =

θ
1−θ



γ 1 − e−rW,
1−e−ρTz (1 − e−rD ) γ + 1−γ1−θρ(1 − θ)ρθ
1 − e−ρT

and


xE
t =



1 − e−rWr1−γ 1−e−ρTwη
1−θρ1−γ 1−e−ρT
γ + 1−θ ρ−ρT

1−γ 1−e
1 − e−rW 1−θ ρ wηθ−ρTr
γ + 1−γ 1−e
1−θ

(9)

1
1−θ

ht−1 ,

(10)

1
1−θ

ht−1 ,(11)ρ

where the superscript E represents ‘educated.’ Consumption and education increase withparents’ human capital, ht−1 , whereas fertility is constant with respect to the changes in
ht−1 because the positive income and negative substitution eﬀects on fertility cancel eachother out.
The important comparative-static results are that fertility, nEt , is decreasing in thelength of lifetime, T , increasing in the length of working life, W , and decreasing in thelength of child-rearing period, D; and the amount of educational investment, xE
t , is increasing in T and W . Let us elaborate the mechanism behind this. The prolongationof lifetime stimulates the demand for consumption relative to the demand for children
because agents must consume goods over a longer period. In response to such a change,
agents shift their time from child rearing to work.27 The resultant increases in working
25
As will be apparent, the number of children is exclusively determined by the parameters (see nEt givenEUby (9) and nUt given by (12)). Since nt < nt (see Proposition 2), we can exclude the corner case byconcentrating on the parameter conﬁgurations under which znUt < 1.
26
In contrast to fertility choice, we cannot exclude the corner solutions for educational choice by imposing
restrictions on the parameters, except for limiting cases (e.g. the case where θ → 0). Whether the agent
chooses an interior solution or a corner one depends on the amount of parental human capital, ht−1 , whichis the state variable of this economy.
27
Since nE
t is constant with respect to ht−1 and decreasing in T , the model predicts that the amountof resources allocated to children decreases as economy develops and adult mortality declines. One might13

time raise the return on education, and thus the amount of educational investment increases. Conversely, the prolongation of working life raises fertility. Since agents havealready been liberated from child care at the retirement date (adult age W ), the prolongation of working life has only a positive income eﬀect on fertility. An increase in W has two
opposing eﬀects on education: the prolongation of working life directly increases working
time while the increase in fertility induced by the prolongation of working life implies adecline in working time. The former direct eﬀect dominates the latter eﬀect, and thus theamount of educational investment increases.
Next, consider the case of a corner solution. The ﬁrst-order conditions imply
nU
t =



γ 1 − e−rW−ρTz (1 − e−rD ) γ + (1 − γ) 1−eρ

(12)

and
CtU =

(1 − γ) wh
1 − e−rW−ρT ,r
γ + (1 − γ) 1−e

(13)

ρ

where the superscript U represents ‘uneducated.’ As in the case of an interior solution,
fertility, nUt , is decreasing in T , increasing in W , and decreasing in D. These comparativestatic results are derived using a mechanism identical to that in the case of positive educational investment. It should be noted that in either case, interior or corner, increases
in T and W yield opposite eﬀects on fertility, suggesting the importance of distinguishing
between retirement from the labor market and death.
Comparing the case of an interior solution with that of a corner solution, we obtainthe following two propositions.
Proposition 1 There is a threshold level of parents’ human capital, h∗ , above which
agents choose an interior solution of investing in education.
Proof. Denote the indirect utility function of the educated by V E (ht−1 ) and that of theuneducated by V U . Substituting (9) and (10) into (7), and (12) and (13) into (7), we
think that this prediction is unreasonable, but such a criticism is not crucial. If we incorporate educational
investment for children, the model predicts that the amount of resources allocated to children is increasing
in ht−1 and not always decreasing in T (see equations (26), (27), and (28) in Subsection 3.2).

14

obtain, respectively,
V E (ht−1 ) = A − γ ln γ +1 − γ 1 − e−ρT1−θρθρθ 1−θ (1 − θ)
1 − e−ρTln+ (1 − γ)ρ
1 − e−ρT−ρT

1−γ 1−e
1 − e−rW 1−θ ρ wη−ρTr
γ + 1−γ 1−e
1−θ

1
1−θ

ht−1

ρ

and


1 − e−ρT
V U = A − γ ln γ + (1 − γ)
ρ
−ρT
−rW
1−e
(1 − γ) wh
1−eln+ (1 − γ)−ρT ,ρrγ + (1 − γ) 1−eρ

where A is a constant term common to V E (ht−1 ) and V U .
V E (ht−1 ) is increasing in ht−1 , limht−1 →0 V E (ht−1 ) = −∞, and limht−1 →∞ V E (ht−1 ) =∞, while V U does not vary with ht−1 . Thus, there exists a unique h∗ such that the agentswith parents whose human capital level is h∗ are indiﬀerent to the choice between investing in education and remaining uneducated; that is, V E (h∗ ) = V U , V E (ht−1 ) < V U if
ht−1 < h∗ , and V E (ht−1 ) > V U if ht−1 > h∗ .
The larger the amount of parents’ human capital, the higher the return on educational
investment. The agents whose parents’ human capital level is high are more likely to be
educated.
Proposition 2 (i) The fertility of the educated is lower than that of the uneducated. (ii)
The threshold, h∗ , is decreasing in T and W .
Proof. (i) It follows from (9) and (12) thatUnEt − nt = − γ θ1−e−ρTz 1−θ (1 − γ)ρ1−eγ + 1−γ
1−θ
ρ

−ρT



−ρT

γ + (1 − γ) 1−eρ< 0.

(ii) Totally diﬀerentiating V E (h∗ ) = V U , we obtainγ
ρ2 e−ρT h∗dh∗=dT
1 − γ (1 − e−ρT )2 
1 − γ 1 − e−ρT
1 − e−ρT− ln γ +ln γ + (1 − γ)
ρ
1−θ
ρ
15


<0

and
dh∗
θ re−rW h∗
< 0.
=−
dW
1 − θ 1 − e−rW

Result (i) simply comes from the fact that the opportunity costs for child rearing arelarger for the educated than for the uneducated. When choosing to be educated, agents
face a tradeoﬀ between increasing the eﬃciency units of labor at each time and savingon educational expenditure. The educated with larger eﬃciency units of labor have acomparative advantage in earning income, whereas the uneducated with smaller eﬃciency
units have a comparative advantage in raising children. Therefore, the utility-maximizing
behavior of the educated (resp. uneducated) is characterized by a lesser (resp. higher)
number of children.
Result (ii) states that as lifetime and working life increase in length, agents are more
likely to choose to be educated: as T rises, agents come to attach importance to consumption relative to children. Then, it is advantageous for agents to acquire education
because it gives them a comparative advantage in earning income for consumption. As
such, dh∗ /dT < 0. The result dh∗ /dW < 0 reﬂects the conventional Ben-Porath mechanism: prolongation of working life over which investments in human capital are paid oﬀ
promotes human capital investment.

2.3

Dynamics of Human Capital

It follows from the results obtained above that the dynamic equation of human capital isgiven byht = h if ht−1 ≤ h∗and

ht = η

−ρT1−γ 1−e
1 − e−rW 1−θ ρ wηθ
1−e−ρTr
γ + 1−γ
1−θ
ρ

θ
1−θ

ht−1 ≡ Ωht−1 if ht−1 > h∗ .

Using this dynamic equation, we obtain the following proposition.

16

(14)

(15)

Proposition 3 (i) If Ω ≤ 1, the economy is characterized by stagnation. If Ω > 1 andh∗ ≥ h, multiple long-run equilibria exist: whether the economy persistently grows depends
on the initial amount of ancestral human capital. If Ω > 1 and h∗ < h, the economy ischaracterized by sustained growth. (ii) Rises in T and W increase Ω and decrease h∗ ,thereby facilitating the growth takeoﬀ.
Proof. (i) It is evident that depending on the parameters, Ω can be larger or smaller than
1 and h∗ can be larger or smaller than h. Thus, it suﬃces to show that Ωh∗ > h. Supposethat Ωh∗ ≤ h. Then, the relationship V E (h/Ω) ≥ V E (h∗ ) = V U must hold; that is, thefollowing must hold:
γ ln nEt + (1 − γ)1−γ
1 − e−ρTln
1−γ1−e−ρT
ρ
γ+
1−θ

ρ

1−γ
1 − e−ρTln≥ γ ln nU
+
(1
−
γ)
t
−ρT .
ρ
γ + (1 − γ) 1−eρU∗This relationship cannot hold because nEt < nt . It follows that Ωh > h.

(ii) It follows from diﬀerentiating (15) that ∂Ω/∂T > 0 and ∂Ω/∂W > 0. The results∂h∗ /∂T < 0 and ∂h∗ /∂W < 0 have already been obtained in Proposition 2.
Figure 2 depicts the dynamic system of human capital. Depending on the parameter
conﬁguration, we can classify the dynamic system into three cases. Panel (a) describes
the case in which the only long-run equilibrium is characterized by stagnation; panel (b)describes the case in which multiple long-run equilibria occur, characterized by stagnationand sustained growth; and panel (c) describes the case in which the only long-run equilibrium is characterized by sustained growth. The result that rises in T and W increaseΩ and decrease h∗ suggests that as T and W increase, the characteristics of the dynamic
system evolve as depicted in panel (d).28 The dynasties with low longevity are likely tostagnate and those with high longevity are likely to grow, and within an intermediate rangeof longevity, multiple long-run equilibria occur: whether a dynasty is stuck in a poverty
trap or takes oﬀ depends on the initial amount of ancestral human capital. Multiplicity
28

Whether Ωh∗ increases with T and W is ambiguous.

17

Figure 2: Dynamics of human capitalof long-run equilibria is based on the eﬀect of parental human capital on the returns on
education. The larger the parents’ human capital, the larger is the oﬀspring’s educational
investment, and thus the diﬀerence in the amount of ancestral human capital is inheritedby the following generation. However, even if a dynasty is initially caught in a poverty
trap, a suﬃciently large increase in life expectancy can liberate the dynasty from poverty.
The dynamic system has an important implication for the links among life expectancy,
human capital, and growth takeoﬀ. Some have argued that increased life expectancy
cannot result in growth taking oﬀ based on the observation that life expectancy in today’s

18

developing countries is comparable to that during the onset of the takeoﬀ of growth intoday’s developed countries, but they remain poor and their growth seems not to havetaken oﬀ. Our model can provide a reasonable explanation for this observation without
denying the role of life expectancy as the driving force of growth. According to our model,
increased life expectancy alone is not suﬃcient for an economy to take oﬀ from stagnation
if the dynamic system is characterized by multiple long-run equilibria; human capitalis also necessary. Since the late nineteenth century, universal education systems havebeen established in many developed countries. Our theory suggests that such education
reforms, combined with increased adult longevity, may have put these countries onto a
growth path.29
We conclude this subsection by investigating the eﬀect of simultaneous changes in
T and W . To see whether such changes promote education and growth, we need only
examine the eﬀect on xE
t :
∂xE
∂xE
t
dW + t dT∂W∂T(1 − γ) wθΩht−1
=1−e−ρT(1 − θ)2 γ + 1−γ
1−θ
ρ

dxE
t =

1 − e−rW −ρT
1 − e−ρT −rWeedW + γdT
ρr

Early retirement implies dW < 0 and increased life expectancy implies dT > 0. Even if
early retirement has a negative eﬀect on education through the conventional Ben-Porathmechanism, the total eﬀect on education can be positive if life expectancy suﬃciently
increases. According to Lee (2001), over the last 150 years, the expected length of postretirement period at age 20 for U.S. male workers considerably increased, while their
expected length of working life exhibited little change. Our model predicts that agentsincrease educational investment in response to such changes.

29

Galor and Moav (2006) provide a concise survey on the history of education reforms in the Westernworld.

19

3

Extensions and Discussion

3.1

Endogenous Retirement

Although life expectancy, which relies on knowledge and technological development in
the medical sciences, is largely exogenous for individuals (see, for example, Soares 2005),
retirement is usually an individual choice. Here, we extend the basic model by endogenizing
the retirement date. We reformulate the utility function of agents so that they receive
disutility from work:

γ ln nt + (1 − γ)

 T
0

−ρτ

e

(r−ρ)τ

ln Ct e

dτ −

 Wt
0

e−ρτ f (τ ) dτ,(16)

where f (τ ) represents the disutility from work at adult age τ and Wt is the retirement
date, which is determined by individual choice. Note that the consumption path givenby (6) is already incorporated in this expression. We assume the following instantaneous
disutility function:f (τ ) =

⎧
⎪
⎨ 0

if

τ ≤ D,⎪⎩ eλτ σifτ > D,(17)

where λ > ρ and σ > 0. This function indicates that work becomes increasingly hard withage.30 The assumption that f (τ ) = 0 for τ ≤ D ensures that agents do not retire before
completing child care, which is employed for simpliﬁcation. All the other assumptions are
the same as in the basic model.
The agent maximizes (16) with respect to nt , Ct , xt , and Wt , subject to equations (2),(8), and (17). We concentrate on the interior solutions for retirement choice, that is, we
assume that λ and σ are suﬃciently large that agents retire before death.
As in the basic model, the solution can either be interior, xt > 0, or at a corner, xt = 0.
First, consider the case of an interior solution. The ﬁrst-order conditions with respect to30

One interpretation is that the disutility of work (and the relative utility of leisure) is higher whenhealth is poorer. This is a convenient and occasionally implemented mechanism to model the retirement
motive. See, for instance, Bloom et al. (2007) and Hazan (2009).

20

nt and Wt are given by, respectively,Eγ 1 − e−rWt


nE
t =
1−e−ρTz (1 − e−rD ) γ + 1−γ
1−θ
ρ
and

1−γ 1−e−ρT −rWtEe1−θρ

1−e−rDr(λ−ρ)WtE e−rD −e−rWtE = e
E
1 − znt +
r

(18)

σ.

(19)

Equation (19) gives that agents choose their retirement date so that the disutility fromwork at retirement (RHS) is equal to the marginal cost of retiring, measured in termsof the loss in utility from foregone consumption (LHS). Solving these two equations, we
obtainE

1 − γ 1 − e−ρT
γ+
1−θ
ρ

−(λ−ρ+r)WtEe

1 − e−rWt
=
σ,
r

(20)

which implicitly deﬁnes the retirement date, WtE . The other choice variables, CtE , xEt ,and nEt , are given by (10), (11), and (18). We see that other than the result that theretirement date is endogenously determined by (20), the individual decision is the sameas in the basic model.
Next, consider the case of a corner solution. The ﬁrst-order conditions with respect to
nt and Wt are given by, respectively,

nU
t =

and



U
γ 1 − e−rWt

(21)

−ρT

z (1 − e−rD ) γ + (1 − γ) 1−eρ−ρTU

(1 − γ) 1−eρ e−rWt(λ−ρ)WtU
=
e
σ.
U


−rWt
1−e−rDU + e−rD −e
1
−
zn
t
r
r

(22)

The interpretation of (22) is the same as that of (19). Solving (21) and (22), we obtain−(λ−ρ+r)WtUe

1 − e−ρT
γ + (1 − γ)
ρ



U

=

1 − e−rWt
σ,
r

(23)

which implicitly deﬁnes the retirement date, WtU . The other choice variables, CtU and nUt ,21

are given by (13) and (21). We see that other than the result that the retirement dateis endogenously determined by (23), the individual decision is the same as in the basicmodel.
Totally diﬀerentiating (20) and (23), we have dWtE /dT > 0 and dWtU /dT > 0; in either
case, interior or corner, the length of working life increases with the length of lifetime. On
the other hand, in contrast to the basic model, the eﬀect of increases in T on fertility is
ambiguous. Let us elaborate on the interaction between fertility and retirement decisions
by exploiting the ﬁrst-order conditions. We proceed by focusing on the case of an interior
solution because similar reasoning is applicable to the corner case.
E
Equation (18) indicates the positive relationship between nEt and Wt , reﬂecting that,as mentioned in Section 2, the prolongation of working life has only positive income eﬀects
on fertility because agents have already been liberated from child care at the retirement
E
date. Equation (19) indicates the positive association between nEt and Wt , reﬂecting thatrises in fertility imply declines in working time and decreases in earnings, thereby increasing
the loss of utility from foregone consumption because of the diminishing marginal utility
and inducing agents to work longer. As depicted in Figure 3, the individual choice on
fertility and retirement is characterized by the intersection of these two equations.31
Consider the eﬀect of a rise in T . Equation (18) indicates that the rise in T decreases
nt for any given Wt . The prolongation of lifetime stimulates the demand for consumption
relative to the demand for children, which is the main mechanism in this paper, inducing
agents to have fewer children. Equation (19) indicates that the rise in T increases Wt for
any given nt . The rise in T implies the prolongation of lifetime over which agents must
consume goods, and thus an increase in the loss in utility from foregone consumption,
thereby extending the length of working life. This eﬀect is what Kalemli-Ozcan and Weil(2010) call the ‘horizon eﬀect,’ through which gains in adult longevity delay retirement.
In the diagram, the total eﬀects on fertility and retirement are ambiguous. As mentioned
above, however, it follows from (20) that dWtE /dT > 0, whereas the eﬀect on nt is analytically ambiguous. Depending on the parameters, fertility might rise or fall as a result
31

E2
The uniqueness can be easily veriﬁed from (20). We also have d2 nE< 0 from (18) andt /dWtE
E2
d Wt /dnt < 0 from (19).
222Figure 3: Individual choice of nt and Wt . Notes: n̄E is deﬁned by limWtE →∞ nEt in (18).

of increased adult longevity.
There are many factors aﬀecting the individual retirement decision. For example, theestablishment of a public pension system, income eﬀects of rising wages, and technologicalprogress obsoleting the skills of the elderly are well known as possible causes of early
retirement. It is natural that our model that excludes these factors cannot explain the
declining trends in the labor force participation of the elderly. What is more important
here is that endogenizing fertility mitigates the ‘horizon eﬀect’ of increased adult longevity.
Declining fertility, that is, increasing labor supply, enables agents to earn more and savemore for the future when they are young adults and retire early through the income eﬀect.
This mechanism possibly induces early retirement, although this has not been identiﬁedin the literature. This ‘fertility eﬀect,’ as will be shown below, allows us to generate asalient feature of the process of economic growth: simultaneous rises in education, declines
in fertility, and declines in the labor force participation of the elderly.
As a thought experiment, consider a technological change prejudicial to the elderly.32
In our model, such a change can be expressed as a rise in λ, which represents the paceat which the disutility from work increases with age.33 The rise in λ shifts (19) so that32

Introducing a transfer promoting retirement also permits us to conduct a similar thought experiment.
Such an formulation of the eﬀect of technological change can be interpreted as meaning that technological progress promotes skill obsolescence and calls for increased eﬀorts to continue working. Alternatively,
33

23

Wt decreases for any given nt , but does not shift (18). Hence, by increasing T and λappropriately, we can decrease nt and Wt simultaneously. Note that fertility and retirementaﬀect the return on education only by aﬀecting the present discounted value of the labor
WD
supply, 0 e−rτ (1 − znt ) dτ + D t e−rτ dτ , and thus simultaneous changes in nt and Wt
that do not change this value do not change the expenditure on education.34 Suppose thatsuch a change is caused by simultaneous increases in T and λ. This change then leads todecreases in nt and Wt but does not change xt . Then, suppose an inﬁnitesimal increasein T . As a result, xt increases while nt and Wt are still lower than their levels before theinitial changes in T and λ. It follows that we obtain the following proposition.
Proposition 4 There are changes in T and λ such that rises in schooling, declines in
fertility, and declines in retirement age are simultaneously generated
Finally, we explore the implication of endogenizing the retirement date for the dynamics
of human capital. As in the basic model, there is a threshold level of parents’ humancapital, above which agents choose an interior solution of investing in education. The
dynamic system is still given by (14) and (15), except that Wt in (15) is endogenously
determined by (20). It is obvious that result (i) of Proposition 3 holds entirely even in thecase of endogenous retirement. Thus, the dynamic system is still classiﬁed into three cases:
stagnation, multiple long-run equilibria, and sustained growth. It follows from (11) thatExEt positively depends on T and Wt , and thus, rises in T have the eﬀects of increasingExE
t directly and by raising Wt . This implies that result (ii) of Proposition 3 (rises in T

facilitate the growth takeoﬀ) also holds.

3.2

Quantity-Quality Model

It is rather standard in the literature to include in the utility of parents not only the numberof children, but also some measure of the quality of children, the so-called quantity-quality
model à la Becker (1960). In this subsection, we extend the model so that parents care
we can assume that human capital depreciates with age and technological change increases the depreciation
rate; then we obtain a qualitatively similar result, but the model becomes analytically complicated.
34
There exists an upward isoquant curve of the present discounted value of labor supply in the (nt , Wt )space.

24

about the amount of education for their children, and we explore the robustness of ourmodel to such an extension. The utility function and the production function of humancapital are modiﬁed as, respectively,
γ ln nt + γφ ln bt + (1 − γ)

and

 T
0

e−ρτ ln ct (τ ) dτ



1−θ−ξ
,h ,
ht = max ηxθt bξt ht−1(24)where bt is the amount of education for each child and φ ∈ (0, 1) and ξ ∈ (0, 1) areparameters.3536
The problem faced by the agent can be rewritten as

max

nt ,Ct ,xt ,bt

 T
s.t.
0

γ ln nt + γφ ln bt + (1 − γ)e−ρτ Ct dτ + xt + bt nt =

 D
0

 T
0

e−ρτ ln Ct e(r−ρ)τ dτ,e−rτ (1 − znt ) wht dτ +



1−θ−ξ
,h .
and ht = max ηxθt bξt−1 ht−1

 W
De−rτ wht dτ,(25)

The product of the amount of a child’s education and the number of children, bt nt
in the LHS of (25), represents the quantity-quality tradeoﬀ. The larger the amount ofeducation, the higher the cost of children; the larger the number of children, the higher
the cost of education.
As in the basic model, the solution can either be interior, xt > 0, or at a corner, xt = 0.
First, consider the case of an interior solution. The ﬁrst-order conditions imply
nE
t =



γ 1 − e−rW
γφ
1−γ
1−e
z (1 − e−rD ) γ + (1−φ)(1−θ)+ (1−φ)(1−θ)
ρ

−ρT

35

,

(26)

We assume that parents value the wealth, in the form of education, they pass on to their descendants.
This motive is much more tractable than other bequest motives and is often used in the development andgrowth literature, for example, Banerjee and Newman (1993) and Galor and Zeira (1993).
36
The assumption φ ∈ (0, 1) is conventional in literature. This assumption guarantees that the secondorder condition for maximization is satisﬁed in the household’s maximization problem: if φ ≥ 1, households
obtain inﬁnite utility by reducing the number of children close to zero and thereby increasing their educationlevel close to inﬁnity.

25

⎡
⎣
xE
t =


1 − e−rW
r



⎤ 1

1−θ
γφ
1−γ
1−e−ρTwηθρ(1−φ)(1−θ) + (1−φ)(1−θ)ξ
1−θ−ξ ⎦
bt−1 ht−1,γφ1−γ1−e−ρTγ + (1−φ)(1−θ)+ (1−φ)(1−θ)
ρ

(27)

and
bE
t =

1 − e−rD φzwht
1 − e−rD φzw  E θ ξ
1−θ−ξ
.
=
η xt bt−1 ht−1
r
1−φ
r
1−φ

(28)

Next, consider the case of a corner solution. The ﬁrst-order conditions imply
nU
t =



γ (1 − φ) 1 − e−rW−ρTz (1 − e−rD ) γ + (1 − γ) 1−eρ

and
bU
t =

1 − e−rD φzwh
.
r
1−φ

(29)

Although the calculation is a bit more complex than in the basic model, the main
results concerning the household’s decision problem remain qualitatively the same. Thereis a threshold level of the return on own education, which is determined by ht−1 andbt−1 , and agents faced with returns higher than the threshold level choose an interior
solution of investing in own education. The comparative-static results are ∂nEt /∂T < 0,
E
E
U
U
E
U
∂nE
t /∂W > 0, ∂xt /∂T > 0, ∂xt /∂W > 0, ∂nt /∂T < 0, ∂nt /∂W > 0, and nt < nt ,all of which have the same sign of inequality as in the basic model. The quantity ofeducation for each child, bt , which is newly introduced here, is increasing in T in the caseof positive own education (through the increase in xEt ) and constant with respect to Totherwise. When adult longevity, T , increases, in spite of the decreased weight given tochildren, bt does not decrease. This is because there exists a quantity-quality interaction:
since increased adult longevity decreases the number of children, as in the basic model,
the price of education for children falls and parents are induced to educate children, whichoﬀsets the eﬀect of decreased weight given to children on bt .
Consider the dynamics of human capital. If initial values of ht and bt are given, thenin one period, the relationship between ht and bt becomes linear by the household’s choice,(28) and (29). Thereafter, the economy (the dynasty) evolves according to the dynamic
system similar to that of the basic model: substituting (27) and (28) into (24), we ﬁnd26

that ht linearly depends on ht−1 in the case of xt > 0, and ht = h in the case of xt = 0.

3.3

Hump-Shaped Fertility Transition

The model presented above predicts a negative, monotonic relationship between adultU
longevity and fertility: it follows from (9) and (12) that ∂nE
t /∂T < 0, ∂nt /∂T < 0 andUnEt < nt . While such a model may be a good description of the dynamics of fertility

in countries that have already experienced the onset of the demographic transition, forexample, in the United Kingdom or the United States over the last 150 years, it is notuniversal. It is widely known that many Western countries experienced upward fertility
trends before the onset of the demographic transition. In this subsection, we discuss thepossibility that our model provides a novel explanation of the reversal in the fertility
transition.
This paper is motivated by the view that increased life expectancy has led to theprolongation of retirement life, rather than the prolongation of working life. Lee (2001)
and Hazan (2009) support this view for the United States in the 19th and 20th centuries,
but this might not be the case for earlier stages of development. According to Lee (2001),
the expected length of the retirement period for the U.S. male workers born in 1830 was
2.65 years. Since there is a lower bound on the length of the retirement period, 0, it is likelythat the length of working life was closely linked with adult longevity in earlier times. As
the simplest case of such environments, now consider the case that W = T , that is, rises
in adult longevity lengthen working life by exactly the same length. Furthermore, suppose
for simplicity that the subjective discount rate is equal to the interest rate, ρ = r. Then,U
it follows from (9) and (12) that ∂nE
t /∂T > 0 and ∂nt /∂T > 0. As already mentioned in

the paragraph below equation (9), rises in T and W have opposing eﬀects on fertility. In
the case we are now considering, the positive income eﬀect of the prolongation of working
life dominates, and thus increased adult longevity raises fertility.
Our model suggests that the association between fertility and adult longevity is positive
when rises in T are accompanied by rises in W , and negative otherwise. The result ofLee (2001) indicates that in the 19th century in the United States, the expected length27

of working life began to cease being tied to adult longevity because of retirement. That
the U.S. fertility rate started to decline in the early 19th century may or may not be acoincidence.

4

Concluding Remarks

There is ample evidence that life expectancy is positively correlated with economic development and growth. Motivated by this evidence, a number of researchers have developed
growth models in which gains in life expectancy promote education and growth via the
Ben-Porath mechanism, which oﬀers a clear explanation for the positive association between life expectancy and education. However, since the mid-nineteenth century, thelength of working life has not increased substantially because of falls in retirement age,which casts doubt on the role of the Ben-Porath mechanism in the growth process. This
paper presents an overlapping generations model in which agents make educational andfertility decisions under a life-cycle consideration, and retirement from the labor market is
explicitly distinguished from death. We reconcile the theory that gains in life expectancy
trigger a growth takeoﬀ by increasing education with the observation that increased life
expectancy has predominantly resulted in the prolongation of retirement life rather thanthe prolongation of working life. In spite of its simple form, the model provides fairly richpredictions about the links among life expectancy, retirement, education, fertility, andgrowth, for example, the prolongation of working life and that of retirement life have opposing eﬀects on fertility choice; gains in adult longevity could shift the dynamic systemof economies (dynasties) from stagnation to growth; and rises in education, declines in
fertility, and falls in retirement age can be simultaneously generated.
The model does not consider several factors relevant to economic growth. A prominent
example is the eﬀect of technological progress. As initially pointed out by Galor and Weil(1999, 2000), technological progress must have played an important role in increasing thedemand for human capital and raising education during the transition from stagnation
to growth. If such a demand shift occurs, the amount of educational investment could
increase even without increases in working time, which in our model is caused by declines28

in child-rearing time. This argument might render moot our endeavor to establish themechanism through which higher longevity promotes education without the prolongationof working life. Historical evidence, however, suggests that the return on human capital
did not increase during the period when European countries transitioned into modern
growth regimes, indicating that the rise in education over this period cannot be explainedby the demand shift alone (see Clark 2005 and Galor 2005b for this discussion). This paper
provides an explanation for the supply shift in terms of changes in individual life-cycle
behavior without contradicting the observation that the length of working life has not been
substantially prolonged because of retirement. Our model is more likely to complement
rather than contradict the models focusing on demand shifts, such as that of Galor andWeil (2000). Models explicitly incorporating the life-cycle consideration, like ours, add
new insight into the literature.
This model is simple and can be extended in several ways to explain more observations.
We conclude by suggesting some potential directions for future research. In our model,
acquiring education entails only pecuniary costs, and the timing of birth is ﬁxed and does
not vary across generations. Incorporating time-consuming education and endogenizing
the timing of birth would allow us to study changes in individual life-cycle behavior andthe demographic structure of the population. Introducing endogenous physical capital
accumulation in a closed-economy framework would generate some interesting predictions.
For example, such an extension would allow the model to capture the diluting eﬀect of
fertility and longevity on per-capita capital, and possibly improve our understanding of
the results of the recent empirical literature on the relationship between life expectancy
and growth (e.g. Acemoglu and Johnson (2007), Lorentzen et al. (2008), and Cervellati
and Sunde (2011)). Furthermore, our model can be applied to the analysis of public policy,
for example, the eﬀects of improvement in a public pension system and public education.

Acknowledgements
The author is grateful to Masako Kimura, Ryosuke Okazawa, and Koji Yokota for valuablecomments and thanks seminar participants at Chukyo University, Tohoku University, and29

the 14th Labor Economics Conference.

30

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