Discussion Papers In Economics
And Business

Rising Longevity, Fertility Dynamics, and R&D-based
GrowthKoichi Futagami and Kunihiko KonishiDiscussion Paper 17-26

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANRising Longevity, Fertility Dynamics, and R&D-based
GrowthKoichi Futagami and Kunihiko KonishiDiscussion Paper 17-26

August 2017

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANRising Longevity, Fertility Dynamics, and R&D-based
Growth∗Koichi Futagami†

Kunihiko Konishi‡§August 18, 2017

Abstract
This study constructs an overlapping-generations model with endogenous fertility, mortality, and R&D activities. We demonstrate that the model explains theobserved fertility dynamics of developed countries. When the level of per capita
wage income is either low or high, an increase in such income raises the fertility
rate. When the level of per capita wage income is in the middle, an increase in suchincome decreases the fertility rate. The model also predicts the observed relationship between population growth and innovative activity. At ﬁrst, both the rates of
population growth and technological progress increase, that is, there is a positive
relationship. Thereafter, the rate of population growth decreases but the rate oftechnological progress increases, showing a negative relationship.

JEL classiﬁcation: D91, J13, O10
Keywords: Fertility, Mortality, R&D∗

We would like to thank Real Arai, Masaru Inaba, Koji Kitaura, Kazutoshi Miyazawa, Akihisa
Shibata, Kouki Sugawara, Kizuku Takao, Takashi Unayama, Akira Yakita, and the seminar participants
at Kansai University, Kyoto University, and Nagoya Gakuin University for their useful comments. We
gratefully acknowledge the ﬁnancial support from the Research Fellowship for Young Scientists of theJapan Society for the Promotion of Science (JSPS) (No.16J09472). Any errors are our responsibility.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
JAPAN; e-mail: futagami@econ.osaka-u.ac.jp‡
Institute of Economic Research, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501,
JAPAN; e-mail: k.konishi.econ@gmail.com§
Research Fellow of the Japan Society for the Promotion of Science (JSPS).

1

1

Introduction

Over the past two centuries, many developed countries have experienced changes in fertility patterns. Galor (2005) identiﬁes, what he calls, two distinct regimes the postMalthusian regime and the sustained growth regime. In the post-Malthusian regime,
technological progress showed a marked acceleration and the rising per capita income
has a positive impact on fertility. That is, this period saw the positive Malthusian eﬀectof per capita income on fertility.1 In the sustained growth regime, although technologicalprogress continues, the increase in per capita income has a negative impact on fertility.
Empirical studies in recent years have found that some developed countries are experiencing a reversal of the fertility pattern. For example, Myrskylä et al. (2009) analyze
the cross-sectional relationship between total fertility rates (TFR) and the Human Development Index (HDI) in 1975 and 2005.2 They look at 2005 and ﬁnd that a higher
HDI implies a higher fertility rate in suﬃciently high HDI countries. Additionally, LuciGreulich and Thévenon (2014) analyze the relationship between TFR and per capita
GDP by using OECD countries from 1960 to 2007. They ﬁnd results that are similar to
those found by Myrskylä et al. (2009): a higher per capita GDP implies a higher fertility
rate in several highly developed countries.3
In addition to fertility dynamics, Strulik et al. (2013) explore historical and empiricalevidence to show that ‘the erstwhile positive correlation between population growth andinnovative activity turns negative during economic development.’ They construct a uniﬁed growth model that incorporates R&D-based innovation to explain this “populationproductivity reversal.” The model proposes the following: At ﬁrst, a high fertility rate
promotes innovative activity; that is, the relation between population growth and innovative activity is positive. Subsequently, the emergence of mass education reduces thefertility rate. However, because the accumulation of human capital outweighs the eﬀect of
declining fertility, there is both R&D-based innovation and economic growth. Therefore,
1

Galor (2005) refers the period before the post-Malthusian regime as the Malthusian epoch. In thisepoch, although there is a positive relationship between per capita income and fertility, the average
growth rate of per capita income is very low. However, our study does not focus on the Malthusianepoch and the source of take-oﬀ from the Malthusian epoch.
2
HDI measures human development and is composed of per capita gross domestic product (GDP),life expectancy, and school enrolment.
3
Hirazawa and Yakita (2017) also hint at a similar relationship between TFR and per capita GDP.

2

the relation between population growth and innovative activity becomes negative in thecourse of economic development.
The purpose of this study is to propose a simple model which explains this fertility dynamics of developed countries and provides an alternative mechanism to explain
the population-productivity reversal. To examine the fertility dynamics, we constructa three-period-lived overlapping-generations (OLG) model with endogenous fertility andmortality. Furthermore, we integrate our model with the R&D-based growth model ofJones (1995). The two deﬁning characteristics of our model are as follows: First, we consider two types of rearing costs: goods and time. When parents raise children, they mustbuy some ﬁnal goods and sacriﬁce their time. Second, following Hirazawa and Yakita(2017), we assume that a young individual probably survives to old age and an increasein per capita wage income aﬀects this probability positively. As discussed in Cutler et al.
(2006), in developed countries, improved nutrition, vaccinations, and medical treatments
are important factors in declining mortality during the course of economic development.4
In our model, the only income earned during one’s youth is wage income. Thus, a higher
wage income can be taken as implying better nutrition. Furthermore, in our model, an
increase in per capita wage income is induced by technological progress; that is, we canconsider a low-income (high-income) economy as an economy with low (high) knowledgeof health and technology.
Using this framework, we obtain the following results: When per capita wage income islow, an increase in such income leads to an increase in the number of children people have(this is the “income-eﬀect”). When per capita income is at a middle level, an increase inincome leads to a fall in fertility rates because the eﬀect of an increasing probability ofsurvival into old age comes into play. When per capita wage income increases, parents
increase their savings to provide for consumption in their old age and decrease the numberof children they have. When per capita wage income is high, the survival probability in
old age reaches a suﬃciently high level and increasing the survival probability in old age
becomes smooth. Thus, when per capita wage income increases, the income-eﬀect once

4

Cutler et al. (2006) also emphasize the importance of public health. For example, Chakraborty(2004) incorporates this aspect by assuming that the probability of survival into old age depends on
public health expenditure. However, for simplicity, we do not consider this aspect.

3

again comes into play; parents increase the number of children they have. In this study,
an increase in per capita wage income is induced by technological progress. Hence, theeconomy experiences this fertility dynamics in the course of economic development.
With regard to the population-productivity reversal, our model makes the following
predictions. At ﬁrst, an increase in the fertility rate increases the size of the workingage population, which accelerates technological progress because the aggregate savings
increase; that is, there is a positive relationship between population growth and innovative
activity. Thereafter, an increase in the probability of survival into old age reduces thefertility rate and raises per capita savings. Although declining fertility rate has a negativeeﬀect on aggregate savings, the increase in per capita savings outweighs this negativeeﬀect. As a result, aggregate savings increase, which promotes technological progressand economic growth. Thus, our model predicts the population-productivity reversal as
discussed in Strulik et al. (2013). In addition, our theoretical predictions are consistent
with empirical ﬁndings (e.g., Bloom et al., 2003; Gehringer and Prettner, 2014).5
The remainder of this paper is organized as follows. Section 2 describes the relevant
literature. Section 3 establishes the model. Section 4 investigates the relationship between
fertility and per capita wage income. Section 5 examines the equilibrium dynamics of theeconomy, the sustainability of economic growth, and the balanced growth path (BGP).
Section 6 investigates the model numerically. Section 7 concludes the paper.

2

Literature review

In this section, we review the relevant literature from the following two points of view:

2.1

Fertility dynamics

de la Croix and Licandro (2013) construct an OLG model with endogenous fertility,
longevity, and education. The model explains the observed fertility dynamics from theMalthusian and post-Malthusian regimes to the sustained growth regime. Our study is
diﬀerent from theirs in the following respects. First, they do not propose the rebound of

5

Bloom et al. (2003) ﬁnd that an increase in life expectancy leads to higher savings rates. Gehringer
and Prettner (2014) ﬁnd a positive eﬀect of rising longevity on total factor productivity.

4the fertility rate. Second, they assume that longevity is extended by parents’ investment
in a child’s physical development. Third, in their model, a demographic transition is
induced by a Beckerian child quantity-quality trade-oﬀ. Finally, they do not consider
R&D activities, or in other words, economic growth is not driven by R&D activities.
Hirazawa and Yakita (2017) attempt to explain the observed facts that an increasein per capita income raises both life expectancy and the fertility rate in several highly
developed countries. They construct an OLG model with endogenous elderly labor supply
and fertility. Their model proposes that rising longevity increases elderly labor supply,
which increases per capita wage income in old age. Hence, young individuals decreasesavings and raise their consumption and the number of children they have. Our study
is diﬀerent in the following respects. First, our model does not allow for elderly labor
supply and our result is based on the notion of an increase in per capita wage income
when people are young. Second, their model does not propose the initial increase in thefertility rate; that is, they do not consider the post-Malthusian regime. Third, in theirmodel, economic growth is not driven by R&D activities.
Maruyama and Yamamoto (2010) integrate endogenous fertility into the Grossman
and Helpman (1991) type of the variety expansion growth model and explain a newmechanism for fertility decline. In their model, variety expansion derived from innovation induces parents to raise consumption expenditure on diﬀerentiated goods and reduce
the number of children they have. Hence, their result of fertility decline is derived diﬀerently from our mechanism and the fertility rate decreases monotonically in the course of
economic development.

2.2

Population-productivity reversal and the eﬀects of demographic change on economic growth

As mentioned in the Introduction, Strulik et al. (2013) propose a model that explains the
population-productivity reversal. Our study is diﬀerent in the following respects: First, in
their model, a demographic transition is induced by the Beckerian child quantity-quality
trade-oﬀ and the fertility rate decreases monotonically. Second, they do not consider
endogenous mortality. Third, their mechanism of population-productivity reversal isdiﬀerent from ours.
5This study is also related to studies showing that the growth rate can increase evenunder declining fertility in R&D-based growth models (e.g, Prettner and Trimborn, 2016;
Hashimoto and Tabata, 2016). Prettner and Trimborn (2016) integrate the OLG modelof Blanchard (1985) with the R&D-based growth model of Jones (1995) and examine
the growth eﬀects of demographic change during the transitional dynamics. In theirmodel, demographic change is represented by an exogenous decrease in mortality rate andan exogenous reduction in fertility. They show that falling mortality increases savings,
which outweighs the eﬀect of declining fertility and promotes technological progress andeconomic growth in the short and medium run. Although their model describes the samemechanism as ours to show how these eﬀects play out, they do not investigate the growtheﬀect of demographic change under endogenous fertility and mortality.6 Hashimoto andTabata (2016) construct an OLG model with R&D-based growth and endogenous fertility
and education. In their model, young individuals choose the number of children they have,the amount of education their children receive, and the amount of investment in their own
education. They show that an increase in the probability of survival into old age reduces
the fertility rate but raises young individuals’ investment in their own education. Whenthe probability of survival into old age is suﬃciently low (high), the eﬀect of humancapital accumulation outweighs (falls short of) the eﬀect of declining fertility; that is,
an increase in the probability of survival into old age enhances (worsens) technologicalprogress and economic growth. Their result is based on a Ben-Porath mechanism which
our model does not consider.7 In addition, they do not show the population-productivity
reversal during the transitional dynamics.

3

Model

We consider a three-period overlapping-generations model following Diamond (1965). An
individual lives for three periods–childhood, youth, an old age. In childhood, they do notmake any economic decision. In youth, every individual supplies labor to the market and6

Prettner (2013) integrates the OLG model of Blanchard (1985) with the R&D-based growth modelsof Romer (1990) and Jones (1995) and investigates the eﬀects of demographic change on long-run growth.
However, he does not take into account the transitional dynamics.
7
The Ben-Porath mechanism works as follows: rising life expectancy prolongs active working lives,
which has a positive impact on investment in human capital.

6earns labor income. Young individuals also raise children. From youth to old age, anindividual dies with a probability 1 − λt ∈ [0, 1). That is, the probability of an individual
surviving to the next period is λt ∈ (0, 1]. In old age, individuals retire, consume theirsavings, and have no bequest motive.

3.1

Individuals

Each young individual at period t maximizes the following utility:
Ut = log cyt + βλt log cot+1 + γ log nt ,(1)

where cyt is their consumption when young, cot+1 is their consumption when they are old,
nt is the number of children each young individual has, β is the subjective discount rate,and γ is the weight on utility derived from the number of children an individual has.8
We assume that each young individual has one unit of time endowment. Raising children
involves two kinds of costs: time and goods. When each young individual raises nt unitsof children, he or she has to incur δnt units of ﬁnal goods and ρnt units of time. Let wt
be the wage rate. The disposable working income becomes (1 − ρnt )wt . Additionally, we
normalize the price of ﬁnal goods to one. Thus, the budget constraint for each young
individual can be expressed as,cyt + st + δnt = (1 − ρnt )wt ,δ > 0, ρ ∈ (0, 1),(2)

where st represents the savings in youth. The disposable working income (1 − ρnt )wt isdivided among consumption, savings, and child-rearing expenses.
As in Yaari (1965) and Blanchard (1985), we assume that there are actuarially fairinsurance companies and the annuities market is perfectly competitive. At the end of the
youth period, each individual deposits his or her savings with an insurance company. The
company invests them and pays the return on that investment to the surviving insured old
individuals. Owing to perfect competition, the rate of return on the annuities becomes(1 + rt+1 )/λt , where rt+1 is the interest rate in period t + 1. Hence, the consumption of8

Following Galor (2005) and Strulik et al. (2013), we assume that nt is the number of surviving
children because we do not want to consider child mortality, for the sake of analytical simplicity.

7the surviving old individuals becomes
cot+1 =

1 + rt+1
st .
λt

(3)

By using (1), (2), and (3), the intertemporal utility maximization problem can bestated as,)([]
1 + rt+1
st + γ log nt .
max log (1 − ρnt )wt − δnt − st + βλt log
st ,ntλtSolving this maximization problem, we obtain
βλt wt,1 + βλt + γ
γwt
nt =
.
(1 + βλt + γ) (ρwt + δ)
st =

3.2

(4a)
(4b)

Final goods sector

Following Romer (1990) and Jones (1995), we consider three production sectors: a ﬁnal
goods sector, an intermediate goods sector, and an R&D sector. We assume that theﬁnal goods market is perfectly competitive. The production technology of the ﬁnal goodis given by∫ At
Yt = L1−αY,txαj,t dj,

0 < α < 1,

(5)0

where Yt , LY,t , At , and xj,t represent the output level, labor input, the variety of intermediate goods, and the input of the jth intermediate good, respectively. Because markets
are perfectly competitive, factor prices equal their marginal products:
∫ At
Yt
−α
,
wt = (1 − α)LY,t
xαj,t dj = (1 − α)
LY,t
0
α−1
qj,t = αL1−αY,t xj,t ,

(6a)
(6b)

where qj,t is the price of the jth intermediate good. From (6b), the demand function forintermediate good j is given as,(
xj,t =

α
qj,t

8

1
) 1−α

LY,t .

(7)

3.3

Intermediate goods sector

We make the following assumptions about the intermediate goods sector: Each diﬀerentiated intermediate good is produced by a single ﬁrm because an intermediate goodis inﬁnitely protected by a patent. That is, the intermediate goods market is monopolistically competitive. Firm entering the intermediate goods market must acquire new
blueprints. They issue shares to raise funds.
We further assume that one unit of labor input produces one unit of a diﬀerentiatedintermediate good. Therefore, a ﬁrm manufacturing an intermediate good j (ﬁrm j)
maximizes its own proﬁt, πj,t = qj,t xj,t − wt xj,t , subject to the demand function (7).
Solving this maximization problem, we obtain the following price charged by ﬁrm j:
qj,t = qt =

1
wt .
α

(8)

Thus, all intermediate goods are priced equally. From (7) and (8), the output andmonopoly proﬁts of ﬁrm j are given by,() 1
α2 1−α
LY,t ,
xj,t = xt =wt
1−απj,t = πt =wt xt .
α

(9a)
(9b)

Because the market value of new blueprints is equal to the sum of the discounted present
value of the proﬁt ﬂow after period t, we can express it as,
vt =

∞
∑
τ =t+1πτ,
u=t+1 (1 + ru )∏τwhere vt is the market values of new blueprints. Using this, we obtain the following
no-arbitrage condition:
rt+1 vt = πt+1 + vt+1 − vt .
The return on stock purchase is equal to the sum of dividend πt+1 and capital gains orlosses vt+1 − vt .

3.4

R&D sector

Next, we consider the technology involved in developing a new intermediate good. R&D
activities require labor inputs and the R&D sector is also perfectly competitive. New
9blueprints developed by R&D activities are sold to intermediate good ﬁrms. We assume
the following R&D technologies:
At+1 − At = θt LA,t ,(10)where θt and LA,t represent the productivity of R&D and the amount of labor devoted to
R&D. At+1 − At measures new intermediate goods. Following Romer (1990), Grossman
and Helpman (1991), and Jones (1995), we assume the productivity of R&D depends on
existing knowledge produced through previous R&D activities. We assume the following
productivity for R&D:
θt = Aϕt .

(11)Thus, At also represents the stock of technological knowledge. As discussed in Jones(1995), we consider a parameter range where 0 < ϕ < 1. The proﬁt of R&D ﬁrms isgiven by πtA = vt (At+1 − At ) − wt LA,t = (vt θt − wt ) LA,t . R&D activities are perfectlycompetitive. Hence, when R&D is undertaken, the following equality holds:
vt =

wt
Aϕt

.

(12)

On the other hand, when vt θt < wt , R&D is not conducted because πtA < 0.

3.5

Market clearing condition and dynamic system

Labor is used for production of ﬁnal and intermediate goods and for R&D activities. As
mentioned above, the labor supply of each individual is 1 − ρnt . Let Nt be the population
size of young individuals at period t. The labor market clearing condition becomes
LY,t + At xt + LA,t = (1 − ρnt )Nt .

(13)

The aggregate savings of young individuals in period t must be used for investment in
R&D (At+1 − At )vt or for the purchase of existing stocks vt At . Hence, the asset market
clearing condition is given by,(At+1 − At )vt + At vt = At+1 vt = st Nt .

10

(14)

Finally, we consider the equilibrium condition for the ﬁnal goods market. The ﬁnal
goods are used for consumption and for raising children. The ﬁnal goods-market clearing
condition is as follows:
Yt = cyt Nt + cot λt−1 Nt−1 + δnt Nt .
Note, because the probability of survival of old individuals in period t is λt−1 , the population size of old individuals in period t becomes λt−1 Nt−1 .
Next, we characterize the dynamic system of this economy. Substituting (9a) into (5)yields
(
Yt =

α2
wt

α
) 1−α

At LY,t .

(15)

Using (6a) and (15), we obtain,
wt = α̂A1−α
,
t

(16)

where α̂ ≡ α2α (1 − α)1−α . Let us deﬁne the growth rate of At as gA,t ≡ (At+1 − At )/At .
(4a), (12), and (14) yield
1 + gA,t =st Nt ϕ−1βλt
At =Nt Atϕ−1 .
wt
1 + βλt + γ(17)Following Hirazawa and Yakita (2017), we assume that the probability of survival rate
λt depends on the wage rate wt , that is, λt = λ(wt ). (16) shows there is a one-toone relationship between the wage rate wt and the stock of technological knowledge At .
Therefore, we can say that the probability of the survival rate λt depends on the stockof technological knowledge.
Because At+1 > At (At+1 = At ) corresponds to gA,t > 0 (gA,t = 0) and the dynamics
of the young individual population size becomes Nt+1 = nt Nt , equations (4b), (16), and(17) characterize the dynamic system with respect to At and Nt .9 Note that both At and
Nt in period t are predetermined variables.

9

In this study, At+1 < At does not hold because there is no product obsolescence.

11

4

Relationship between fertility and per capita wage
incomeIn this section, we examine the relationship between nt and wt . First, following Hirazawa
and Yakita (2017), we specify λ(wt ) as follows:λ(wt ) =ν
,
1 + χe−ψwt(18)where ν ∈ (0, 1] and χ, ψ > 0. Under this functional form, we haveνψχe−ψwt> 0,(1 + χe−ψwt )2)
νψ 2 χe−ψwt ( −ψwt′′
λ (wt ) =χe−1.
(1 + χe−ψwt )3ν
lim λ(wt ) ≡ λ =
,
wt →0
1+χ
lim λ(wt ) ≡ λ = ν.
λ′ (wt ) =wt →+∞

(19a)
(19b)
(19c)
(19d)

From (19b), we obtain the following properties of λ(wt ). If 0 < χ ≤ 1, λ′′ (wt ) < 0 holds.
That is, λ(wt ) is concave for any wt . If χ > 1, λ′′ (wt ) R 0 holds for any wt Q wλ , wherewλ ≡ ψ1 log χ. When wt is either low or high, an increase in wt raises λt but this increaseis not large. On the other hand, when wt is at a middle level, an increase in wt increases
λt substantially.
Next, using (4b), we diﬀerentiate nt with respect to wt :∂ntγδγβλ′ (wt )wt
=
−
.
∂wt
[1 + βλ(wt ) + γ] (ρwt + δ)2 [1 + βλ(wt ) + γ]2 (ρwt + δ)

(20)

The ﬁrst term shows that an increase in per capita wage income increases the fertility
rate. Because we normalize the price of ﬁnal goods to one, an increase in the wage rate
decreases the relative price of ﬁnal goods, which reduces the child-rearing costs by ﬁnal
goods and raises the number of children. We call this the “income-eﬀect.” The second
term shows that a rise in per capita wage income raises the probability of surviving into
old age, which decreases the fertility rate and increases the savings meant for consumption
in old age. As shown in Appendix A, we clarify which one is larger, and thus state thefollowing proposition:

12

Proposition 1.
{
1. If χ > max

ψδ−2ρ
, χ̃
ψδ+2ρ

}
, an increase in wt raises nt for any 0 < wt < w̄1 or wt > w̄2 ,while an increase in wt reduces nt for any w̄1 < wt < w̄2 , where χ̃, w̄1 , and w̄2 aredeﬁned in Appendix A.
2. If 0 < χ < ψδ−2ρ
or 0 < χ < χ̃, an increase in wt raises nt for all wt .
ψδ+2ρFigure 1: Relationship between nt and wt .

In addition, by using (4b) and (19d), nt converges to the following value:
lim nt ≡ n∗ =

wt →∞

γ
.
(1 + βν + γ)ρ

Figure 1 illustrates the relationship between nt and wt , corresponding to the ﬁrst case of
Proposition 1. As discussed above, an increase in per capita wage income has opposite
eﬀects on fertility. When per capita wage income is low, the income-eﬀect is relatively
large. That is, rising per capita wage income has a positive impact on the number ofchildren. When per capita wage income is at a middle level, the eﬀect of an increasing
probability of survival into old age comes into play. Thus, when per capita wage income
increases, parents increase their savings for consumption in old age and decrease consumption in their youth and the number of children they have. When per capita wage
income is high, the probability of survival into old age becomes smooth. Thus, when per13

capita wage income increases, the income-eﬀect is higher, or in other words, parents once
again increase the number of children they have.
At the end of this section, we mention the probability of survival rate λ(wt ). Intuitively, it seems that the ﬁrst case of Proposition 1 does not hold if λ(wt ) is a concave
function. However, in Section 6, we provide a numerical example and show that this caseholds even under a concave function λ(wt ). Moreover, our result of the fertility reboundis derived from the following: the survival probability in old age attains a suﬃciently
high level and increasing the survival probability in old age becomes smooth. To show
the empirical plausibility, Figure 2 presents the ratio of survivors at diﬀerent age points
in high-income countries from 1950 to 2100. In addition, Table 1 shows the average rateof increase for the data presented in Figure 2. In the ﬁrst decade of 2000, the ratio ofsurvivors at ages 50, 60, and 70 is suﬃciently high, and the average rate of increase begins
to decline from 2015 to 2100.

Figure 2: Ratio of survivors at each age in high-income countries.
Source: World Population Prospects: The 2017 Revision (United Nations).
Note: World Population Prospects presents the number of survivors at diﬀerent age
points assuming that there are 100,000 people at birth. We calculate the ratio ofsurvivors, using this data.

14

age 50

age 60age 701950-1985

0.01640.02270.03701985-2015

0.00570.01060.02402015-2100

0.00190.00400.0086

Table 1: Average rise rate of the data presented in Figure 2

5

Equilibrium path

5.1

Phase diagram

We consider a phase diagram in the (At , Nt ) plane. For our purpose, we impose χ > ψδ−2ρψδ+2ρ
and χ > χ̃, which corresponds to the ﬁrst case of Proposition 1. To investigate whether
At+1 > At or At+1 = At at each point of the (At , Nt ) plane, we set gA,t = 0 in (17) as
follows:Nt =1 + βλ(wt ) + γ 1−ϕAt ≡ Γ(At ).
βλ(wt )

(21)

We have At+1 > At above this locus and At+1 = At below this locus. Hereafter, we referto the region where At+1 > At as R&D region and the region where At+1 = At as no
R&D region. Using (16), we diﬀerentiate Γ(At ) with respect to At as follows:(1 + γ)λ′ (wt ) 1−ϕ ∂wt 1 + βλ(wt ) + γ+(1 − ϕ)A−ϕ
t ,
2 At
∂A
βλ(w
)
β [λ(wt )]tt{}λ′ (wt )wt
A−ϕt=−(1 − α)(1 + γ)
+ (1 − ϕ) [1 + βλ(wt ) + γ] .
βλ(wt )λ(wt )Γ′ (At ) = −(22)

The ﬁrst term of (22) implies that an increase in At increases the probability of survivingλ(wt ), from (16) and (19a). This increases the per capita savings, and thus, the no
R&D region shrinks. The second-term of (22) implies that an increase in At reduces themarginal productivity of the R&D sector derived from the knowledge spillover, which
enlarges the no R&D region. If the ﬁrst-term’s eﬀect is larger, Γ′ (At ) < 0 holds. On theother hand, if the second-term’s eﬀect is larger, Γ′ (At ) > 0 holds. As shown in Appendix
B, we clarify which one is larger. If 0 < χ < χ̂, we obtain
Γ′ (At ) > 0

f or all At ,15

where χ̂ is as deﬁned in Appendix B. We depict this case in the left-side panel of Figure
3. The shaded area represents the no R&D region. In contrast, if χ > χ̂, we obtain
Γ′ (At ) > 0 when 0 < At < AΓ,1 , At > AΓ,2 ,Γ′ (At ) < 0 when AΓ,1 < At < AΓ,2 ,where AΓ,1 and AΓ,2 are deﬁned as Γ′ (AΓ,1 ) = 0 and Γ′ (AΓ,2 ) = 0. We depict this case in
the right-side panel of Figure 3. As discussed below, the properties of the phase diagram
remain the same in both cases.

Figure 3: At+1 > At and At+1 = At regions on the (At , Nt ) plane.
Next, we examine whether Nt+1 > Nt or Nt+1 < Nt at each point of the (At , Nt )plane. As discussed below, if n∗ is less than 1, the population size of young individuals
continues to decrease in the long run, and as a result, the economy deﬁnitely falls into the
no R&D region in the long run. To ensure sustainable growth, we impose the following
assumption:
Assumption 1. n∗ =γ
> 1.
(1 + βν + γ)ρ

We show in Appendix C the following lemma:

16{Lemma 1. Suppose that χ > max

}

ψδ−2ρ
, χ̃
ψδ+2ρ

and Assumption 1 holds.

1. If γ w̄2 > [1 + βλ(w̄2 ) + γ] (ρw̄2 + δ), the following holds. Nt+1 > Nt holds when
wt > ŵ1 , while Nt+1 < Nt holds when 0 < wt < ŵ1 , where ŵ1 is deﬁned as shownin Appendix C.
2. If γ w̄1 > [1 + βλ(w̄1 ) + γ] (ρw̄1 + δ) and γ w̄2 < [1 + βλ(w̄2 ) + γ] (ρw̄2 + δ), thefollowing holds. Nt+1 > Nt holds when ŵ1 < wt < ŵ2 or wt > ŵ3 , while Nt+1 < Ntholds when 0 < wt < ŵ1 or ŵ2 < wt < ŵ3 , where ŵ2 and ŵ3 are as deﬁned inAppendix C.
Figure 4 uses the results obtained so far to illustrate the relationship between nt
and wt and phase diagrams on the (At , Nt ) plane. The ﬁrst (second) case of Lemma 1
corresponds to the left-side (right-side) panels in Figure 4. Here, we deﬁne Āj and Âj asw̄j = α̂Ā1−αand ŵj = α̂Â1−α(j = 1, 2, 3). As shown in Appendix C, the deﬁnition of ŵjjj(j = 1, 2, 3) implies that Â1 < Ā1 and Â2 < Ā2 < Â3 . Since At and Nt are predeterminedvariables at time t, the initial state of the economy is given by point (A0 , N0 ) on the(At , Nt ) plane. If the economy is in the no R&D region (the lower side of the Nt = Γ(At )locus), At is constant, and as a result, the fertility rate nt is also constant from (4b), (16),and (19a). In this case, when Nt+1 > Nt , the economy can reach the R&D region in aﬁnite time period. On the other hand, when Nt+1 < Nt , the economy falls into a trap.
For our research purposes, we assume that the initial stock of technological knowledge
is given by A0 ∈ (Â1 , Ā1 ) in the discussions that follow. We describe the transition
dynamics in the R&D region (the upper side of Nt = Γ(At ) locus).
First, we consider the ﬁrst case of Lemma 1. Under A0 ∈ (Â1 , Ā1 ), both At and Nt
continue to increase, that is, the economy is on a sustainable growth path. With regard
to the fertility dynamics, the fertility rate decreases with time under Ā1 < At < Ā2 , andincreases with time under Â1 < At < Ā1 or At > Ā2 .
Next, we consider the second case of Lemma 1. If N0 is suﬃciently small, as represented by Q1 in the lower-right panel of Figure 4, R&D is undertaken and both the
population size of young individuals and the fertility rate increase, at ﬁrst. After exceeding the vertical line Ā1 , the fertility rate begins to fall. After exceeding the vertical line
17Figure 4: Relationship between nt and wt and phase diagram on the (At , Nt ) plane.

18

Â2 , the fertility rate drops to below 1; the population size of young individuals decreases
with time, and as a result, the economy falls into the no R&D region. If N0 is a speciﬁcvalue as represented by Q2 , the economy follows a path that is similar to that of the initialpoint Q1 . In this case, the economy converges to the point (Â3 , N̂3 ), where N̂3 ≡ Γ(Â3 ).
At this point, the fertility rate is 1 and since there is no R&D, the economy remains at
this point. If N0 is suﬃciently large as represented by Q3 , the economy initially follows apath similar to the one detailed above. However, after exceeding the vertical line Ā2 , thefertility rate increases with time. Hence, the economy goes beyond the vertical line Â3
and once again, the population size of young individuals increases. That is, the economy
attains a sustainable growth path.10 In summary, we can state that the path converging
to the point (Â3 , N̂3 ) is a threshold curve that sustains economic growth. If (A0 , N0 ) falls
into the lower side of this arm, the economy falls into a trap in the long run. These
results are summarized in the following lemma:{
Lemma 2. Suppose that χ > max}, A0 ∈ (Â1 , Ā1 ), and Assumption 1 holds.

ψδ−2ρ
, χ̃
ψδ+2ρ

1. If γ w̄2 > [1 + βλ(w̄2 ) + γ] (ρw̄2 + δ), the economy can be on a sustainable growthpath.
2. If γ w̄1 > [1 + βλ(w̄1 ) + γ] (ρw̄1 + δ) and γ w̄2 < [1 + βλ(w̄2 ) + γ] (ρw̄2 + δ), there
is a threshold curve that sustains economic growth. This curve is an inverted-U on
the (At , Nt ) plane.
When the economy is in the region where At ∈ (Â1 , Ā1 ) and Nt > Γ(At ), technologicalprogress is triggered by R&D activities. As mentioned earlier, both per capita wage
income and the fertility rate increase with time. Therefore, this region corresponds tothe post-Malthusian regime. After exceeding the vertical line Ā1 , the economy enters thesustained growth regime; that is, the economy experiences declining fertility although
per capita wage income continues to rise. As in Lemma 2, a reduction in the fertility
rate may lead the economy into a trap. However, if the initial population size of young
individuals is suﬃciently large, sustainable growth can be attained. Along a sustainable10

If we do not impose Assumption 1, Nt+1 < Nt holds for all At > Â2 . In this case, the economy
deﬁnitely falls into the no R&D region. Thereafter, At and nt become constant and nt falls to below 1.
Since Nt+1 < Nt , the economy cannot get out of the trap.

19

growth path, if the economy exceeds the vertical line Ā2 , the fertility rate rebounds, andthe economy converges to the BGP . In summary, we can state the following proposition:{
Proposition 2. Suppose that χ > max}, A0 ∈ (Â1 , Ā1 ), and Assumption 1

ψδ−2ρ
, χ̃
ψδ+2ρ

hold. If N0 is suﬃciently large, the economy can be on a sustainable growth path. Alongthe sustainable growth path, the fertility rate increases at ﬁrst and then starts to fall.
Subsequently, the fertility rate rebounds and converges to n∗ .

5.2

Growth rates

So far we have discussed the fertility dynamics and the sustainability of economic growth;
however, some important questions remain in our model. In this subsection, we discuss
the growth rate of At during the transitional dynamics, the value of GDP, and the growth
rates for each economic value along the BGP.
5.2.1

Growth rate of At

We ﬁrst consider the growth rate of At . (17) leads to the following remark:Remark 1. During the transitional dynamics, a lower At or a higher Nt or λt implies ahigher gA,t .
This can be intuitively explained as follows. A rising At reduces the eﬀect of knowledge
spillover and the marginal productivity of the R&D sector (see (11)). By using (4a), we
obtain
st
βλt
=
.
wt
1 + βλt + γ(23)

A higher λt implies a higher per capita savings relative to wage income. From (17) and(23), rising Nt or λt increases aggregate savings, which raises the supply of funds in theasset market (see (14)). These two eﬀects have a positive impact on investment in R&D.
Moreover, from individuals’ optimal decisions as discussed in Section 3, young individuals
allocate their wage income to consumption when young, and to savings for consumption
in old age and for raising children. Thus, we obtain the following relationship between
the fertility rate and the growth rate of the stock of technological knowledge.
20Remark 2. When the economy is in transitional dynamics, one sees the following staticand dynamic eﬀects. Let us consider any period t:
1. Increasing (decreasing) nt has a negative (positive) impact on st , which can reduce(raise) aggregate savings at period t and gA,t .
2. Increasing (decreasing) nt has a positive (negative) impact on Nt+1 , which can raise(reduce) aggregate savings at period t + 1 and gA,t+1 .
Because analytical investigation of the evolution of gA,t is too complicated, we explore
this through a numerical example (see Section 6).
5.2.2

GDP and BGP

In our model, the value of GDP is not equivalent to that of Yt . The correct value of GDP
is as given below:
GDP = Yt + vt (At+1 − At ).
As mentioned above, when R&D is undertaken, πtA = vt (At+1 − At ) − wt LA,t = 0 holds.
By using this and the ﬁnal goods-market clearing condition, we obtain11
GDP = Yt + wt LA,t ≡ Zt .
The calculation of Yt is shown in Appendix D. In this study, we consider three generations(children, young individuals, and old individuals). To calculate the per capita GDP, wedeﬁne the population size of the economy at time t as
Mt ≡ nt Nt + Nt + λt−1 Nt−1 .
11Gross domestic income (GDI) is calculated by
GDI = (1 − ρnt )wt Nt + πt At .

From (9b), (13), and (D.2), we obtain
GDI = wt (LY,t + At xt + LA,t ) +1−α1wt At xt =wt LY,t + wt LA,t .
α1−αBecause (6a) implies wt LY,t = (1 − α)Yt , GDI is as given below:
GDI = Yt + wt LA,t .
Thus, we conﬁrm that the value of GDI is equivalent to that of GDP.

21

Zt
.
Hence, we deﬁne per capita GDP as zt ≡ M
tIf the economy is on a sustainable growth path, At and Nt grow forever. We deﬁne
BGP as that state of the economy in which the growth rates and the employment shareof each sector is constant. Let us deﬁne the growth rate of a variable Xt as gX,t ≡(Xt+1 − Xt )/Xt . As mentioned above, the fertility rate converges to n∗ in the long run.
As shown in Appendix E, we can derive the growth rates along the BGP as follows:∗= n∗ ,
1 + gM1

1 + gA∗ = (n∗ ) 1−ϕ ,1−α

1 + gY∗ = 1 + gZ∗ = (n∗ ) 1−ϕ +1 ,1−α

1 + gz∗ = (n∗ ) 1−ϕ .
As in Jones (1995), the growth rates along the BGP are determined by the fertility rate.

6

Numerical example

In this section, we analyze the model by using numerical examples. We set the maximum
value of λt to 1 (that is, λ = ν = 1) and the minimum value of λt to 0.2 (that is,λ = ν/(1 + χ) = 0.2 leads to χ = 4). The time cost of raising children is assumed to beρ = 0.08, which lies between the value of Hirazawa and Yakita (2017) and Strulik et al.
(2013). To ensure sustained economic growth, we assume n∗ = 1.2. We set β = 0.5 andadjust γ to satisfy n∗ = 1.2. According to Alvarez-Pelaez and Groth (2005), a plausible
range of monopoly markup in US industry is the interval [1.05, 1.40]. To achieve this,we set α = 0.72. We let ϕ = 0.9 and obtain gz∗ = 0.67. This implies that the annual
growth rate of per capita GDP is about 1.5%, assuming one period to be 35 years. We
choose the remaining parameter values as δ = 0.05, ψ = 0.1, χ = 4 and set the initialvalues of At and Nt as A−1 = 500 and N−1 = 50. Figure 5 presents the evolution of thesurvival probability, fertility, the growth rate of the stock of technological knowledge, andthe growth rate of per capita GDP from period t = 0 to t = 10.
The upper left- and upper right-side panels in Figure 5 show the evolution of thesurvival probability, λt , and the evolution of fertility, nt , respectively. The numericalresult is consistent with our theoretical result in Sections 4 and 5. That is, the fertility
22rate ﬁrst rises and then begins to fall. Subsequently, the fertility rate rebounds andconverges to the BGP.
The lower left-side panel in Figure 5 shows that gA,t ﬁrst rises and then falls, converging
to the BGP. Moreover, we notice that the evolution of gz,t is similar to that of gA,t , fromthe lower right-side panel in Figure 5. As mentioned in the Introduction, our model
predicts the population-productivity reversal. Initially, the rising fertility rate increases
the population size of young individuals. This enlarges aggregate savings and the growthrate of the stock technological knowledge. Later, the increasing probability of survival
into old age decreases the fertility rate and raises per capita savings. As mentioned
in Remarks 1 and 2, the growth rate of the stock of technological knowledge has thefollowing eﬀects: (i) A decrease in nt has a negative eﬀect on gA,t+1 , (ii) an increasein λt has a positive eﬀect on gA,t , and (iii) an increase in At has a negative eﬀect on
gA,t . Our results show that the positive eﬀect (ii) outweighs the negative eﬀects (i) and(iii); that is, the growth rate of the stock technological knowledge increases with time.

Figure 5: The evolution of λt , nt , gA,t , and gz,t from period t = 0 to t = 10.

23

Figure 6: The evolution of λt , nt , gA,t , and gz,t at each value of χ from period t = 0 to
t = 10.
This suggests the population-productivity reversal. Subsequently, the negative eﬀect (iii)
becomes suﬃciently large, and the growth rate of the stock of technological knowledge
begins to decline and converge to the BGP.
Next, we show that our main results hold even if λ(wt ) is a concave function. Theψδ−2ρ
calculation of χ̃ and ψδ+2ρ
using the same parameter values as in Figure 5 yields thevalue of χ̃ ≈ 0.405 and ψδ−2ρ
≈ −0.939. As discussed in Section 4, the condition ofψδ+2ρ
concavity of λ(wt ) is 0 < χ ≤ 1. Thus, if 0.405 < χ ≤ 1, the ﬁrst case of Proposition 1
holds. To conﬁrm this result, we present Figure 6, which shows the evolution of λt , nt ,gA,t , and gz,t for each value of χ from period t = 0 to t = 10. The green line represents
the case where χ = 0.4, and the red and blue lines show the cases where χ = 0.7 andχ = 1. As we can see from Figure 6, the cases where χ = 0.7 and χ = 1 correspond tothe ﬁrst case of Proposition 1, whereas the case where χ = 0.4 corresponds to the secondcase of Proposition 1. In addition, the cases where χ = 0.7 and χ = 1 show that the
population-productivity reversal holds under the concave function λ(wt ).

24

7Conclusion

In this study, we construct an overlapping-generations model with endogenous fertility,
mortality, and R&D activities. Our model demonstrates the observed fertility dynamics
from the post-Malthusian regime through the sustained growth regime to the reboundof fertility rate. Furthermore, our model depicts another mechanism for the populationproductivity reversal.
There are several interesting directions for future research. First, we do not consider
human capital investment, even though human capital investment is an important factor
in fertility and economic growth. Most existing studies examine various important rolesof the Beckerian child quantity-quality trade-oﬀ and the Ben-Porath mechanism. Moreover, in recent years, many developed countries’ TFR has fallen to below the so-called
replacement level (2.1 children per women) even after the fertility rebound. Our model
predicts that the economy will deﬁnitely fall into a trap if the long-run fertility rate islower than the replacement level. However, as pointed out by Strulik et al. (2013), human capital investment can ensure economic growth if human capital growth exceeds thedecline in population. Future research could examine the implication of human capital in
our model. Second, we do not calibrate the model. As shown in Figure 5, our numericalresult does not match the actual data. In future research, incorporating human or physical capital could help match the results to actual data as in Strulik et al. (2013). Third,
Luci-Greulich and Thévenon (2013) state that family policies (paid leave, childcare services, and ﬁnancial transfers) of developed countries have a positive impact on fertility;that is, family policies can contribute to the recent fertility rebound. Thus, investigating
the eﬀects of child-rearing policies oﬀers another area for further research. Fourth, asmentioned in footnote 8, we do not consider child mortality. According to Cutler et al.
(2006), the child mortality rate declined signiﬁcantly before 1960. Incorporating childmortality could provide interesting insights into fertility dynamics. Fifth, in life-cycle
models, an expansion of public pension is crucial to individual decisions on fertility andsavings. If there is a pay-as-you-go pension system, individual savings may decrease,which works against the population-productivity reversal. Future research should examine the eﬀects of public pension in our model.
25

Appendix
A

Proof of Proposition 1∂nt, we rearrange (20) as follows:
To investigate the sign of ∂wt∂nt
Φ(wt ),=γ∂wt[1 + βλ(wt ) + γ]2 (ρwt + δ)2∂nt
where Φ(wt ) ≡ δ[1 + βλ(wt ) + γ] − βλ′ (wt )wt (ρwt + δ). The sign of ∂w
is determined byt∂ntthat of Φ(wt ), that is, Φ(wt ) R 0 implies ∂w
R 0. Diﬀerentiating Φ(wt ) with respect tot

wt and using (19a) and (19b) yield[]Φ′ (wt ) = −βwt 2ρλ′ (wt ) + λ′′ (wt )(ρwt + δ) ,{}[] −ψwtβνψχwt e−ψwt
2ρ − ψ(ρwt + δ) + χ 2ρ + ψ(ρwt + δ) e,=−
(1 + χe−ψwt )3βνψχwt e−ψwt≡−Ω(wt ).
(1 + χe−ψwt )3Here, Ω(wt ) satisﬁesΩ(0) = 2ρ(1 + χ) + ψδ(χ − 1),[]Ω′ (wt ) = −ψρ − ψχ ρ + ψ(ρwt + δ) e−ψwt < 0.
If Ω(0) < 0, we have Φ′ (wt ) > 0 for all wt . On the other hand, if Ω(0) > 0, we haveΦ′ (wt ) Q 0 for any wt Q w̃, where w̃ is deﬁned as Φ′ (w̃) = 0. Here, the condition ofΩ(0) > 0 is as follows:χ>ψδ − 2ρ.
ψδ + 2ρ

These results imply that the relationship between Φ(wt ) and wt has a U-shape and w̃
minimizes the value of Φ(wt ).
We then substitute (18) and (19a) into Φ(wt ) as follows:δβνβνψwt (ρwt + δ)χe−ψwt.
Φ(wt ) = δ(1 + γ) +−1 + χe−ψwt(1 + χe−ψwt )226

Using this, we obtain
δβν
> 0,
wt →0
1+χ
lim Φ(wt ) = δ(1 + γ) + δβν > 0.
lim Φ(wt ) = δ(1 + γ) +wt →∞In addition, if Φ(w̃) < 0, we obtain the following relationship:
Φ(wt ) > 0 when 0 < wt < w̄1 or wt > w̄2 ,Φ(wt ) < 0 when w̄1 < wt < w̄2 ,where w̄1 and w̄2 are deﬁned as Φ(w̄1 ) = 0 and Φ(w̄2 ) = 0 hold. In contrast, if Φ(w̃) > 0,
we have Φ(wt ) > 0 for all wt .
Next, we examine the condition of Φ(w̃) < 0. Because w̃ satisﬁes Φ′ (w̃) = 0, we
obtain[]2ρ − ψ(ρw̃ + δ) + χ 2ρ + ψ(ρw̃ + δ) e−ψw̃ = 0,2ψ(ρw̃ + δ).
⇔ 1 + χe−ψw̃ =2ρ + ψ(ρw̃ + δ)

(A.1)

An investigation of (A.1) gives us the following properties:dw̃
[2ρ + ψ(ρw̃ + δ)]e−ψw̃
=
> 0,
dχ
ψρ + ψχ[ρ + ψ(ρw̃ + δ)]e−ψw̃
w̃ → ∞ when χ → ∞.

(A.2)
(A.3)

Using (A.1), we rearrange the condition of Φ(w̃) < 0 as given below:
Φ(w̃) < 0,δβνβνψ w̃(ρw̃ + δ)χe−ψw̃−< 0,
1 + χe−ψw̃(1 + χe−ψw̃ )22ρ + ψ(ρw̃ + δ)
⇔ δ(1 + γ) + δβν2ψ(ρw̃ + δ)[
]2
ψ(ρw̃ + δ) − 2ρ 2ρ + ψ(ρw̃ + δ)
− βνψ w̃(ρw̃ + δ)
< 0,2ρ + ψ(ρw̃ + δ)
2ψ(ρw̃ + δ){}
⇔ 4δ(1 + γ)ψ(ρw̃ + δ) + 2δβν[2ρ + ψ(ρw̃ + δ)] < βν w̃ [ψ(ρw̃ + δ)]2 − 4ρ2 ,[] []2(1 + γ)2(1 + γ)ψδ1
ψ 2 (ρw̃ + δ)2 − 4ρ2
⇔ ψρ
+1 +
+ (2ρ + ψδ)
<
.
(A.4)
βν
βν
w̃
2δ
⇔ δ(1 + γ) +Let us deﬁne the left- and right-hand sides of (A.4) as ηL (w̃) and ηR (w̃). ηL (w̃) is[]2(1+γ)
decreasing in w̃ and limw̃→0 ηL (w̃) = ∞ and limw̃→∞ ηL (w̃) = ψρ βν + 1 hold. On the27other hand, ηR (w̃) is increasing in w̃ and limw̃→0 ηR (w̃) = ψ δ 2δ−4ρ and limw̃→∞ ηR (w̃) = ∞2 2

2hold. From (A.2) and (A.3), we ﬁnd that there is a unique χ̃ that satisﬁes ηL (w̃) = ηR (w̃).
Hence, χ > χ̃ implies ηL (w̃) < ηR (w̃). In contrast, if 0 < χ < χ̃, ηL (w̃) > ηR (w̃) holds(that is, Φ(w̃) > 0 holds).

B

Properties of Γ(At )Using (18), (19a), and (22), we obtain{}[] −ψwtA−ϕt′Γ (At ) =(1 − ϕ)(1 + γ + βν) + (1 − ϕ) − (1 − α)(1 + γ)ψwt χe
,
βν
≡

A−ϕ
t
Θ(wt ).
βν

Here, Θ(wt ) satisﬁes
Θ(0) = (1 − ϕ)(1 + γ + βν) + (1 − ϕ)χ > 0,[]Θ′ (wt ) = − (1 − α)(1 + γ) + 1 − ϕ − (1 − α)(1 + γ)ψwt ψχe−ψwt .
Hence, Θ′ (wt ) Q 0 when wt Q wΘ , where wΘ ≡ (1−α)(1+γ)+1−ϕ. Using these results, we(1−α)(1+γ)ψ
can deﬁne the following two cases: When Θ(wΘ ) > 0, we obtain
Θ(wt ) > 0

f or all wt .

On the other hand, when Θ(wΘ ) < 0, we obtain
Θ(wt ) > 0 when 0 < wt < wΓ,1 , wt > wΓ,2 ,Θ(wt ) < 0 when wΓ,1 < wt < wΓ,2 ,where wΓ,1 and wΓ,2 are deﬁned as Θ(wΓ,1 ) = 0 and Θ(wΓ,2 ) = 0. Because the sign ofΓ′ (At ) is the same as that of Θ(wt ) and wt is determined by At , we can state the propertiesof Γ(At ) as in Subsection 5.1.
We then investigate the condition Θ(wΘ ) < 0. Rearranging Θ(wΘ ) < 0 yields
Θ(wΘ ) < 0,
⇔ (1 − ϕ)(1 + γ + βν) − (1 − α)(1 + γ)χe−ψwΘ < 0.
28This implies that Θ(wΘ ) < 0 holds when χ > χ̂, where χ̂ is deﬁned as follows:χ̂ ≡(1 − ϕ)(1 + γ + βν).
(1 − α)(1 + γ)e−ψwΘ

In contrast, we obtain Θ(wΘ ) > 0 when 0 < χ < χ̂.

C

Proof of Lemma 1

From Proposition 1, nt has a local maximum (minimum) value at wt = w̄1 (wt = w̄2 ).
Because Nt+1 R Nt is equivalent to nt R 1, we examine the following two cases:12 First,
if nt > 1 at wt = w̄2 , we obtain
Nt+1 R Nt

when wt R ŵ1 ,where ŵ1 is deﬁned as nt |wt =ŵ1 = 1. This case corresponds to the panels on the left sideof Figure 4. We next consider the other case. From Assumption 1, if nt > 1 at wt = w̄1
and nt < 1 at wt = w̄2 , we obtain
Nt+1 > Nt when ŵ1 < wt < ŵ2 or wt > ŵ3 ,Nt+1 < Nt when 0 < wt < ŵ1 or ŵ2 < wt < ŵ3 ,where ŵ2 and ŵ3 are deﬁned as nt |wt =ŵj = 1 (j = 2, 3) and ŵ1 < ŵ2 < ŵ3 . This casecorresponds to the panels on the right side of Figure 4. Using (4b), we obtain the following
relationship:
nt R 1 ⇔ γwt R [1 + βλ(wt ) + γ] (ρwt + δ).
Hence, the case where nt > 1 at wt = w̄2 corresponds to γ w̄2 > [1 + βλ(w̄2 ) + γ] (ρw̄2 +δ).
On the other hand, the case where nt > 1 at wt = w̄1 and nt < 1 at wt = w̄2 corresponds
to γ w̄1 > [1 + βλ(w̄1 ) + γ] (ρw̄1 + δ) and γ w̄2 < [1 + βλ(w̄2 ) + γ] (ρw̄2 + δ), respectively.

12

We rule out the case where nt < 1 at wt = w̄1 because this runs contrary to the observed data.

29

D

Calculation of Yt

Substituting (16) into (15), we obtain( 2 )ααYt =A1−αLY,t ,
t
1−α

(D.1)

We consider the labor market clearing condition to derive LY,t . By using (9a) and (16),the labor input into production of intermediate goods becomes
α2
LY,t .
1−α

(D.2)

]
1−α [
(1
−
ρn
)
N
−
L
if LA,t > 0.
ttA,t1 − α + α2

(D.3)

At xt =
From (13) and (D.2), we obtain
LY,t =

Furthermore, (10), (11), and (17) yield
LA,t =βλ(wt )Nt
− A1−ϕ.
t
1 + βλ(wt ) + γ(D.4)

By using (4b), (16), (D.1), (D.3), and (D.4), we can calculate the output of ﬁnal goods
Yt .

E
E.1

Derivation of the growth rates along the BGP∗
Derivation of gM

Because Nt+1 = n∗ Nt and λt = λ̄ hold along the BGP, the growth rate of population
along the BGP is given as,:∗=1 + gME.2

n∗ Nt+1 + Nt+1 + λ̄Nt(n∗ )2 + n∗ + λ̄=
= n∗ .
n∗ Nt + Nt + λ̄Nt−1n∗ + 1 + nλ̄∗

Derivation of gA∗

By using (D.4) and (19d), we obtain the employment share of R&D along the BGP as
follows:
β λ̄
A1−ϕ
LA,t
=
− t .
Nt
Nt
1 + β λ̄ + γ
30Because the employment share of R&D is constant, the term A1−ϕ
/Nt also becomestconstant along the BGP. Thus, the growth rate of At along the BGP is as follows:((1 + gA∗ )1−ϕ =
E.3

At+1
At

)1−ϕ
=

Nt+1
= n∗ .
Nt

Derivation of gY∗

From (D.1), the growth rate of Yt is determined by those of At and LY,t . (D.3) implies
(
)
1−α
LA,t
LY,t
∗
=
1 − ρn −.
Nt
1 − α + α2
Nt
Because the employment share of ﬁnal goods production becomes a constant, the growthrate of LY,t along the BGP is equal to n∗ . Therefore, the growth rate of Yt along the
BGP is given by
1−α

1 + gY∗ = (1 + gA∗ )1−α n∗ = (n∗ ) 1−ϕ +1 .
E.4

Derivation of gZ∗ and gz∗

From the deﬁnition of Zt , the growth rate of Zt is as follows:
1 + gZ,t =1
Yt+1 + wt+1 LA,t+1
wt+1 1−α LY,t+1 + LA,t+1
Zt+1
=
=
,
1
Zt
Yt + wt LA,t
wt
L + LA,t1−α Y,tLL

1−α Nt

Nt

Y,t+1
A,t+1
1
wt+1 1−α Nt+1 + Nt+1 Nt+1=.
L
1 LY,t
wt
Nt
+ A,t

L

Along the BGP, NY,tt and

LA,t
Nt

become a constant. These results and (16) imply
1−α

1 + gZ∗ = (1 + gA∗ )1−α n∗ = (n∗ ) 1−ϕ +1 .
Zt
, the growth rate of zt along the BGP is given by
Because zt = M
t

1 + gz∗ =1−α
zt+1
Zt+1 Mt
1 + gZ∗=== (n∗ ) 1−ϕ .
∗zt
Zt Mt+1
1 + gM31

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