Discussion Papers In Economics
And Business
Dynamic analysis of demographic change and humancapital accumulation in an R&D-based growth model

Kohei Okada

Discussion Paper 19-18

December 2019

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANDynamic analysis of demographic change and human capital
accumulation in an R&D-based growth model∗
Kohei Okada†

Abstract
Employing an overlapping-generations model of R&D-based growth with endogenous fertility, mortality, and education choice, we examine how demographic changes and human capital
accumulation influence R&D activity. We show that multiple steady states can exist in thiseconomy. One steady state has a high level of human capital and the other has a low level.
In the steady state with high (low) level of human capital, there is a high (low) level of R&D
activity, a low (high) fertility rate, and a high (low) old-age survival rate. In addition, we showthat the government can steer an economy away from a poverty trap trajectory by investing inpublic health. We also show that an improvement in the government’s public health policy has
an inverted U-shaped eﬀect on the growth rate at the steady state.
Keywords: Demographic change, Human capital accumulation, R&D
JEL Classification: I25, J10, O10, O30∗
I especially thank Koichi Futagami for his helpful comments and encouragement. I also thank Ken-ichi Hashimoto,
Ken Tabata, Takaaki Morimoto, Tasturo Iwaisako, and Yoshiyasu Ono for their helpful comments and suggestions.
In addition, I thank participants of the 2018 Japanese Economic Association Autumn Meeting held at Gakushuin
University and the PET 2019 Annual Meeting of the Association for Public Economic Theory held at the Universityof Strasbourg. I acknowledge financial support from a Grant-in-Aid for JSPS Fellows of the Japan Society for the
Promotion of Science (No. JP18J15399).
†
Graduate School of Economics, Osaka University. Address: 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail: koheiokada257@gmail.com

1

R&D expenditure (% of GDP)

4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0

10

20

30

40

50

60

70

80

90

100

Enrollment ratio, tertiary(%)

Figure 1: R&D expenditure and enrollment rate of tertiary education. Source: World Bank. Crosscountry data for 2015. Simple ordinary least squares estimation shows that there is statisticallysignificant positive correlation between the enrollment ratio and the level of R&D expenditure perunit of GDP.

1

Introduction

In most developed countries, people receive higher levels of education than do people in developing
countries. Moreover, most developed countries face low fertility rates and population aging. On thecontrary, developing countries face high fertility rates and high mortality rates. In addition, mostdeveloped countries more actively undertake R&D and realize more rapid technological change thandeveloping countries. Figure 1 shows the relationship between the enrollment ratio and the level of
R&D expenditure per unit of GDP. According to Figure 1, we find positive correlation between theenrollment rate and R&D activity. Figure 2 shows the relationship between the fertility rate and thelevel of R&D expenditure per unit of GDP. We find negative correlation between the fertility rateand R&D activity. Meanwhile, Figure 3 shows the relationship between the old-age dependency ratio
and the level of R&D expenditure per unit of GDP. As Figure 3 shows, there is positive correlation
between the old-age survival rate and R&D activity. This remarkable empirical evidence suggeststhat there are mutual relationships among demographic changes, education, and R&D activity.
Therefore, this study examines how demographic changes and human capital accumulation influence
R&D activity. We show that education and the human capital accumulation play an important rolein economic development in an R&D-based growth model. We clarify the historical conditions that2

R&D expenditure (% of GDP)

4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0

1

2

3

4

5

Fertility rate

Figure 2: R&D expenditure and fertility rate. Source: World Bank. Cross-country data for 2015.
Simple ordinary least squares estimation indicates that there is statistically significant negative
correlation between the fertility rate and the level of R&D expenditure per unit of GDP.

R&D expenditure (% of GDP)

4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0

5

10

15

20

25

30

35

40

45

Old age dependency ratio (%)

Figure 3: R&D expenditure and old-age dependency ratio. Source: World Bank. Cross-country
data for 2015. Simple ordinary least squares estimation shows that there is statistically significant
positive correlation between the old-age dependency ratio and the level of R&D expenditure perunit of GDP.

3

determine whether a country becomes a developed country or a developing country.
We construct a simple overlapping-generations model of R&D-based growth with endogenous
fertility, mortality, and education choice. Each individual lives for three periods. In the first
period of their lives, individuals do not make any decisions. In the second period, individuals raisechildren and invest in their education, supply eﬃcient units of labor, pay income tax, consume
diﬀerentiated goods, and save any remaining income. In the final period, individuals retire andconsume diﬀerentiated goods. Old individuals live through old age with an endogenous survival rate,
which is determined by the government’s public health policy. In this model, there is a production
sector and an R&D sector. In the production sector, a single firm produces a diﬀerentiated goodby using eﬀective labor. Similarly, in the R&D sector, R&D firms use the eﬀective labor to invent
new diﬀerentiated goods. The equilibrium dynamics of this economy are characterized by the levelof human capital. We show that multiple steady states can exist in this economy. One of themhas a high level of human capital and the other has a low level. In the steady state with the high(low) level of human capital, there is a high (low) level of R&D activity, a low (high) fertility rate,
and a high (low) old-age survival rate. In addition, we examine how government policy aﬀects thiseconomy. We show that the government can steer an economy away from a poverty trap trajectoryby investing in public health. We also show that an improvement in the government’s public healthpolicy has an inverted U-shaped eﬀect on the growth rate at the steady state. We examine how the
improved public health policy aﬀects welfare and show that the government may face a trade-oﬀ
between improving the welfare levels of the current generation and those of future generations.
This study is related to research on demographic change, human capital accumulation, andR&D-based growth. In particular, this study is related to research in the unified growth literature,such as Galor and Weil (2000), Galor and Moav (2002), and Galor (2011). They explain the transition from the Malthusian regime to the sustained growth regime by considering the interrelationshipof demographic change, human capital accumulation, and technological change. These studies argue
that demographic change, human capital accumulation, and technological change play an importantrole in shifting from the Malthusian regime to the sustained growth regime. In addition, they argue
that the diﬀerence between developed and developing countries is their timing in taking oﬀ from
the Malthusian regime. Specifically, developing countries do not stay in the Malthusian regime
and automatically move to the sustained growth regime. However, many developing countries face4

sustained challenges of low economic growth, high fertility rate, and high mortality rate. Therefore,
developing countries may be caught in a poverty trap, thereby hindering the transition from theMalthusian regime to the sustained growth regime. Hence, many studies allow the existence of multiple steady states with the poverty trap, including Blackburn and Cipriani (2002), Moav (2005),
Kimura and Yasui (2007), Chen (2010), and Chakraborty and Chakraborty (2018), taking endogenous fertility and education choice into account. They argue that individuals’ education decisions
play an important role in economic development. However, they do not consider R&D activities.
In contrast to this research, we consider the relationship between demographic change, education,
and R&D activity simultaneously. This study is also related to Chakraborty (2004), Hashimoto andTabata (2005), and Agénor (2015). whose works endogenize the old-age survival rate by focusingon public health policy and exploring the dynamic relationship between mortality and economic
growth. However, they do not consider fertility, education, and R&D simultaneously. In contrast
to this research, we consider these factors and can explain the facts shown in Figures 2 and 3.
Furthermore, this research is related to Chu et al. (2013), Strulik et al. (2013), and Hashimoto andTabata (2016), who investigate the relationship between demographic change, education, and R&D
activity simultaneously. However, Chu et al. (2013) and Strulik et al. (2013) do not consider mortality. Hashimoto and Tabata (2016) consider mortality but regard it as exogenous; moreover, they
do not refer to a multiplicity of steady states. In contrast to this research, we consider endogenous
mortality and show the existence of multiple steady states. Lastly, this study is related to Futagami
and Konishi (2017), who explore the dynamic relationship between endogenous fertility, mortality,
and R&D. In addition to their research, we study how education and human capital accumulationaﬀect the economy.
This rest of this paper is organized as follows. Section 2 shows the basic structure of the model,
Section 3 analyzes equilibrium and dynamics. Section 4 analyses government policy. Finally, theconclusion is described in Section 5.

5

2

The model

2.1

Individuals

Time is discrete and denoted by t = 0, 1, 2, · · · . Each individual lives for three periods (childhood,
adulthood, and old age). In the first period (childhood), individuals do not make any decisions, andthey are raised by their parents. In the second period (adulthood), individuals raise their children
and invest in their education, supply eﬃcient units of labor, pay income tax, consume diﬀerentiatedgoods, and save any remaining income. In the final period (old age), individuals retire and consume
diﬀerentiated goods. An individual dies at the beginning of his or her old age with probability
1 − λt ∈ [0, 1] and lives through old age with probability λt ∈ [0, 1]. Members of the cohort born inperiod t − 1 become active workers in period t. Thus, we call this cohort generation t and use Nt
to represent the number of adults who are alive in period t. Let nt denote the number of children
for each adult. Therefore, the relationship between the sizes of the adult populations during any
two consecutive periods can be expressed as Nt+1 = nt Nt . Individuals derive their utility from thenumber of children nt , their children’s level of human capital ht+1 , their own consumption during
adulthood C1,t , and their own consumption during old age C2,t+1 . The lifetime utility of individuals
in generation t is expressed as

ut = η log nt + log C1,t + βλt (γ log ht+1 + log C2,t+1 ),(1)

where the positive parameters η and γ denote the weights of the number of their children andthe children’s level of human capital, respectively. β ∈ (0, 1] denotes the discount factor. Following
Mariani et al. (2010), we assume that intergenerational altruism is eventually magnified by a higher
λt , because the success of children has a greater eﬀect on those parents who will live long enoughto witness it. We specify the subutility function Ck,t for k ∈ {1, 2} as

Ck,t ≡[∫ A t] α1ck,t (i) di ,
α

(2)

0

where ck,t (i) represents the consumption of diﬀerentiated good i ∈ [0, At ]. At denotes the variety ofdiﬀerentiated goods or the level of technological knowledge in this economy, which grows through6

R&D. In individuals’ second period of life (adulthood), they are endowed with one unit of time,
which is devoted to working lt in the labor market, raising nt identical children, and educating eachchild et . We assume that a fixed amount of time z is required to bear and raise a child. Individuals
divide their disposable income (1 − τ )wt ht lt between consumption and saving st for their old age1 .
Following Yaari (1965), we assume that insurance companies are risk neutral and that the private
annuities market is competitive. Insurance companies promise individuals a payment (Rt /λt )st , inexchange for which the estate st accrues to the companies, where λt is the average probability ofsurviving and Rt+1 represents the gross interest rate. In the absence of a bequest motive, individuals
are willing to invest their assets in such insurance. Here, wt and τ ∈ [0, 1) are the wage rate foreﬃcient units of labor and the income tax rate, respectively. Thus, the budget and time constraints
for individuals in generation t are expressed as follows:
E1,t = (1 − τ )wt ht lt − st ,Rt+1
E2,t+1 =
st ,
λt
∫ At
E1,t =
pt (i)c1,t (i)di,0∫ At+1E2,t+1 =pt+1 (i)c2,t+1 (i)di,

(3)
(4)
(5)
(6)

0

lt + znt + et nt = 1,

(7)where E1,t , E2,t+1 , and p(i) denote the expenditure of an individual in adulthood, the expenditureof an individual in old age, and the price of good i, respectively2 .
We assume that the human capital production function is given by the following expression:
ht+1 = ϕ(et ht )σe h̄σt h ,ϕ > 0,σe , σh ∈ (0.1].

(8)ϕ, σe , and σh are parameters. ht reflects externalities from the human capital stock of parents andh̄t reflects externalities from the human capital stock of the society. In equilibrium, h̄t = ht holds,
1

In this model, we do not take account of the goods cost of child rearing. If we were to consider this, the dynamics
of per capita human capital ht and the variety of diﬀerentiated goods At would depend on each other and the model
would be quite complicated. Therefore, we do not take account of the goods cost of child rearing to simplify the
analysis. However, this simplification is quite common in the literature, for example, Galor and Moav (2002).
2
In our specifications, we ignore the utility cost of labor eﬀort. However, our results still hold even if we takeaccount of the utility cost of labor eﬀort and the substitution among leisure and time spent doing other things.

7

because all individuals are homogeneous in this economy. Individuals maximize their utility giventhese externalities3 .
We next consider the individual’s utility maximization. By maximizing the subutility function(2) subject to the budget constraint (5), we obtain the demand for diﬀerentiated good i as follows:

ck,t (i) =pt (i)−ϵEk,t ,
Pt1−ϵ1where ϵ ≡ 1−αand Pt is the price index defined by Pt ≡[∫At
1−ϵ dj0 pt (j)

(9)
] 1

1−ϵ

. This demand function

implies that indirect utility becomes a linear function of expenditure as follows:

Ck,t =

Ek,t
.
Pt

(10)

Let us denote the total demand for good i as xt (i). xt (i) is given by

xt (i) = c1,t (i)Nt + c2,t (i)λt−1 Nt−1 ,=pt (i)−ϵ(E1,t Nt + E2,t λt−1 Nt−1 ).
Pt1−ϵ

(11)

By maximizing (1) subject to (3), (4), (7), (8), and (10), we obtain the following solution:η − βγσe λt,(1 + η + βλt )zβγσe zλtet =,η − βγσe λtβ(1 − τ )wt ht λt
st =,1 + η + βλt
1 + βλtlt =.
1 + η + βλt
nt =

(12)
(13)
(14)
(15)

t
According to (12) and (13), the fertility rate decreases with the old-age survival rate λt (i.e., ∂n∂λt ),∂etwhereas investment in education for each child increases with λt (i.e., ∂λ). An increase in thetold-age survival rate stimulates demand for consumption relative to demand for both quantity andquality of children, because this increase induces individuals to anticipate the need to consume
3

Yakita (2010) and Hashimoto and Tabata (2016) discuss these assumptions of human capital production function
in detail.

8goods over a longer period of time. In response to this change, individuals shift their time fromchild rearing to work. Therefore, the fertility rate decreases and working time and savings increase
with the old-age survival rate. On the other hand, an increase in the old-age survival rate has threeeﬀects on investment in education for each child. First, it motivates parents to shift their timefrom education to work (i.e., the first positive eﬀect). Second, if the old-age survival rate increases,
parents invest more in education for each child because of the inter-generational externality ofeducation (i.e., the second positive eﬀect). Third, the decrease in the fertility rate that resultsfrom the rise in the old-age survival rate decreases the opportunity costs that parents incur fromproviding their children with an education, which motivates parents to invest more in education foreach child (i.e., the negative eﬀect). In equilibrium, the first positive eﬀect and the negative eﬀect
oﬀset one another. Consequently, investment in education for each child increases with the old-age
survival rate because of the second positive eﬀect.
We make the following assumption, which is a suﬃcient condition for ensuring nt > 0.
Assumption 1

η>γThis assumption implies that having a family must be more important than investing in the education of children4 .

2.2

Production

There are diﬀerentiated goods indicated by i ∈ [0, At ]. A single firm produces each good. Eachfirm supplies a diﬀerentiated good monopolistically and sets its price. The monopoly is protected
through perfect patent protection. Each monopolistic firm produces one unit of good by using one
unit of eﬀective labor. The producer of good i maximizes the following profit:
πt (i) = pt (i)xt (i) − wt xt (i),4This assumption ensures the existence of a consistent solution; see Strulik (2004).

9

(16)

subject to the total demand function for good i (11). From the profit maximization condition, theprice of good i is

pt (i) =1
wt ≡ p t .
α(17)Hence, all goods have the same price. Thus, the firm-specific index i in the diﬀerentiated goods
sector can be dropped. By substituting (17) into the demand function (11), we obtain the output
level of the diﬀerentiated good:E1,t Nt + E2,t λt−1 Nt−1p−ϵ
xt = ∫ At t1−ϵ (E1,t Nt + E2,t λt−1 Nt−1 ) =
.
At pt
p
dj
t
0

(18)

The total expenditure is treated as a numeraire (E1,t Nt + E2,t λt−1 Nt−1 = 1). Therefore, we canrewrite (18) by using (17) as follows:

xt =

1
α
=
.
At p twt At(19)

The profit of each diﬀerentiated good firm is given by
(
πt =

2.3

)
1
− 1 wt x t .
α

(20)

R&D

R&D firms use the eﬀective labor to invent new diﬀerentiated goods. After invention, the firms sella blueprint of a new good to an entrepreneur. Development of At+1 − At new blueprints requires
lR,t units of eﬀective labor input. Let us define ∆At as ∆At ≡ At+1 − At . Thus, given research
productivity of ψt , output is expressed as follows:∆At = ψt At lR,t ,(21)

At implies the spillover from general knowledge accumulated by past innovation. To remove thestrong scale eﬀect, we follow Laincz and Peretto (2006) and Chu et al. (2013). We assume that ψt
ψlν−1

R,t
is decreasing in the scale of the economy ht Nt . We specify productivity as ψt ≡ (ht Nν where ψ > 0t)10and the parameter ν ∈ (0, 1) inversely measures the negative duplication externality discussed
in Jones (1995) and Jones and Williams (2000). In accordance with Jones (1995), the researchtproductivity is given for each firm. gt ≡ ∆A
At is the growth rate of product variety. Given lR,t , thegrowth rate of product variety features decreasing returns to scale in lR,t . The R&D firms’ profits
πtR are given by
πtR = vt ∆At − wt lR,t ,()
wt= vt −
∆At ,
ψt At

(22)

where vt is the price of a blueprint of a newly invented good. Because an entrepreneur pays theprice of the blueprint of the new good to the R&D firm, this price corresponds to the price of equity
that is sold to the household. Free entry into R&D races leads to the following zero-profit condition:
vt ≤

wt
ψt At

with equality if∆At > 0.

(23)

As shown in Appendix A, R&D is always undertaken and (23) holds with equality. The value of vt
equals the present value of future profits as follows:

vt =

∞
∑

πT
.
T
Π
R
ν
ν=t+1
T =t+1After some manipulations, we obtain the following no-arbitrage condition.

Rt+1 =

vt+1 + πt+1
.
vt

(24)

Each individual saves at the gross rate of interest Rt+1 determined by (24).

2.4

Government

The government collects income tax from individuals and invests it in the public health service Gt .
We assume that the public health service is produced by eﬀective labor lG,t and, for simplicity, that11

the production function of the public health service is linear, as follows:

Gt = δlG,t(25)

where δ > 0 is a constant parameter5 . The old-age survival rate λt is improved by the per capita
Gtpublic health service bt ≡ N. The old-age survival rate λt is given byt

λt = f (bt ),(26)

which satisfies f ′ (bt ) ≥ 0, f ′′ (bt ) < 0, f (0) ≥ 0, and limbt →∞ f (bt ) = µ ≤ 1. Public healthexpenditure is financed by the government’s balanced budget. Therefore, the government’s budget
constraint is

wt lG,t = τ wt ht lt Nt .

3

Equilibrium

3.1

Market equilibrium(27)

We first describe the equilibrium condition of the asset market. Individuals’ savings must be directed
to purchase either newly issued stocks for R&D or existing stocks of operating firms owned by thepreceding generation. Therefore, the asset market equilibrium condition is

st Nt = wt lR,t + vt At .

(28)

The left-hand side of (28) is the saving volume of the adult individuals. The first term on theright-hand side is the total investment in newly issued stocks, which is equal to the total cost of
R&D activities. The second term is the purchase of existing stocks. Note that the second term
5

We consider that nurses and doctors provide the public health service. Eﬀective labor lG,t includes their humanlcapital and we can express their raw labor as G,t; see (29). Therefore, we consider the skills of nurses and doctors
ht
in this simple liner functional specification. In addition, the public health service is produced without goods. If wetake account of goods input in the production of public health service in addition to the labor input, then λt depends
on both ht and At and the dynamics become quite complicated. This simplification allows us to undertake a simple
analysis. Futagami and Konishi (2017) analyze the situation in which the old-age survival rate depends on the varietyof goods in R&D-based growth model without human capital.

12implicitly assumes that ex-dividend stocks are traded between adult and old generations in eachperiod. In other words, the old generation always receives all dividends before selling the stocks.
We next describe the labor market equilibrium condition. Eﬀective labor is used for productionof diﬀerentiated goods, R&D, and public health services. The labor market-clearing condition is

ht lt Nt = lR,t + At xt + lG,t ,(29)

where lR,t , At xt , and lG,t denote the quantities of eﬀective labor engaging in R&D, production
activities, and improving public health, respectively.
From (15), (25), and (27), we obtainbt =(1 + βλt )δτ ht≡ b(λt ; ht , τ ).
1 + η + βλt

(30)
2

δht
t ;ht ,τ )t ,τ )
Note that b(0; ht , τ ) = τ1+η≥ 0, limλt →∞ b(λt ; ht , τ ) = δτ ht ≤ ∞, ∂b(λ∂λ≥ 0, ∂ b(λ∂λt ;h
≤ 0.
2
t
t

Let us define f −1 (λt ) as the inverse function of (26). This inverse function satisfies the following
characteristics:∂f −1 (λt )
∂λt

2 −1

≥ 0, ∂ f∂λ2(λt ) > 0, and f −1 (0) ≤ 0. λt and bt are determined by thetfollowing equation:f −1 (λt ) = b(λt ; ht , τ ).

(31)

We define λ∗t and b∗t as the solution of (31) with respect to λt and b∗t ≡ b(λ∗t , ht , τ ), respectively.
In addition, we can regard λ∗t as a function of ht and τ , and denote λ∗t = λ(ht ; τ ). Figure 4 shows
the relationship between f −1 (λt ) and b(λt , ht , τ ). Figure 4 shows the determination of (λ∗t , b∗t ).
t ;ht ,τ )As shown in Figure 4, (λ∗t , b∗t ) is uniquely determined. From (30), we find that ∂b(λ∂h
≥ 0.
t

∂λ∗

Therefore, if ht increases, the line of b(λt ; ht , τ ) shifts upward in Figure 5. We find that ∂htt > 0
from Figure 5 and obtain the following Proposition 1.
Proposition 1 An increase of per capita capital ht increases the old-age survival rate λ∗t .
If per capita human capital ht increases, the income of individuals increases. This income increase
raises the government’s income, because the government imposes income tax. Therefore, the government can invest more in public health and the old-age survival rate rises. We obtain the following

13

𝑓 %&(𝜆" )

𝑏"

𝛿𝜏ℎ"
𝑏"∗

𝑏(𝜆" ; ℎ" , 𝜏)

0

𝜆∗"

𝜆"

𝜇

Figure 4: Determination of (λ∗t , G∗t )𝑓 %&(𝜆" )

𝑏"
𝑏"∗ ′

ℎ" ↑or 𝜏↑𝛿𝜏ℎ"𝑏(𝜆" ; ℎ" , 𝜏)

𝑏"∗

0

𝜆∗"

𝜆∗" ′ 𝜇𝜆"Figure 5: The eﬀect of Ht and τ on (λ∗t , G∗t )14

proposition.
Proposition 2 An increase of per capita human capital ht decreases the fertility rate nt .
∂λ∗tProof. From (12) and ∂htt > 0, we find ∂n∂ht < 0.

From Proposition 1, if per capita human capital ht increases, the old-age survival rate λ∗t rises.
If the old survival rate λ∗t rises, individuals put more weight on old-age consumption. Therefore,
individuals increase their eﬃcient units of labor to save more. With increased eﬃcient units oflabor, individuals have to decrease the number of children owing to time constraints. Therefore, if
per capita human capital ht increases, the fertility rate nt decreases.
From (14), (21), (23), and (28), the growth rate of product variety gt is determined by thefollowing equation:1βψ ν (1 − τ )λt − 1−ν
gt + 1 =g ν .
1 + η + βλt t(32)This defines the relationship between gt and λt . We obtain the following proposition.
Proposition 3 An increase of per capita human capital ht increases the growth rate of productvariety gt .
Proof. see Appendix B.
If per capita human capital ht increases, the old-age survival rate increases. This increases savings and thus, investment in R&D increases. Consequently, the growth rate of product variety gt
increases.
t ;ht ,τ )≥ 0. Therefore, if τ increases, the line of b(λt ; ht , τ ) shifts
From (30), we find that ∂b(λ∂τ∂λ∗upward in Figure 5. We find that ∂τt > 0 from Figure 5 and obtain the following Lemma 1.
Lemma 1 An increase of the tax rate τ increases the old-age survival rate λ∗t .
If the government raises the tax rate, the government can collect more tax revenue and invest morein public health. Therefore, the old-age survival rate λ∗t rises.

15

ℎ'()
45°
ℎ'() = Φ ℎ'

0

ℎ∗

ℎ∗∗∗

ℎ∗∗

ℎ'

Figure 6: Dynamics of ht

3.2

Dynamics

The dynamics of this economy are characterized by per capita human capital ht . From (8), (13),and λ∗t = λ(ht ; τ ), we obtain]
βγσe zλ(ht ; τ ) σe σe +σhht
ht+1 = ϕ.
η − βγσe λ(ht ; τ )
|
{z
}
[

(33)

≡Φ(ht )Let us define the right-hand side of (33) as Φ(ht ). Figure 6 shows the phase diagram of ht when theeconomy has multiple steady states6 . In this case, the economy has three steady states. The firstone is stable and its per capita human capital is h∗ . The second one is the unstable steady state.
We denote the per capita human capital of this steady state as h∗∗ . The third one is stable and its
per capita human capital is high. We denote the per capita human capital of this steady state as
7th∗∗∗ . If we specify f (bt ) = ρ+µb
θ+bt , we obtain the following proposition .

Proposition 4 Suppose σh ∈ (1 − 2σe , 1 − σe ), and ρ is suﬃciently low. There exists ϕ such that
6

A numerical example reveals that there is a parameter configuration such that Assumption 1 is satisfied andtmultiple steady states exist. For example, if we specify f (Gt ) = ρ+µb
and assume that β = (0.98)25 , δ = 1, γ = 0.14,
θ+btη = 0.21, θ = 5, µ = 1, ρ = 1, σe = 0.635, σh = 0.165, τ = 0.1, ϕ = 34.02, and z = 0.088, then multiple steady states
exist. The justifications for this parameter setting are described in Section 4. If ϕ is in [30.6, 35.6], multiple steadystates exist.
7t
The functional form f (bt ) = ρ+µb
satisfies f ′ (bt ) ≥ 0, f ′′ (bt ) < 0, f (0) ≥ 0, and limbt →∞ f (bt ) = µ ≤ 1.
θ+bt

16

the economy has multiple steady states. When the economy has three steady states, if the initial per
capita human capital is low (h0 < h∗∗ ), per capita human capital ht converges to h∗ . However, if
the initial per capita human capital is high (h0 > h∗∗ ), per capita human capital ht converges toh∗∗∗ .
Proof. see Appendix C
If the initial per capita human capital is suﬃciently high (i.e., h0 > h∗∗ ), the old-age survival rate ishigh because of Proposition 5. Then, investment in education for each child is also high. Therefore,
per capita human capital converges to steady state h∗∗∗ . However, if the initial per capita humancapital is suﬃciently low (i.e., h0 < h∗∗ ), the old-age survival rate is low. Then, the investment ineducation for each child is also low. Therefore, per capita human capital converges to steady stateh∗ . This complementarity results in the existence of multiple steady states.
From Propositions 1, 2, 3, and, 4, we obtain the following results. In the steady state with ahigh level of human capital h∗∗∗ , the level of R&D activity is high, the fertility rate is low, and theold-age survival rate is high. However, in the steady state with a low level of human capital h∗ , thelevel of R&D activity is low, the fertility rate is high, and the old-age survival rate is low. These
results correspond to the empirical results, as shown in Figures 1, 2, and 3.

4

Government policy

In this section, we consider how government policy aﬀects this economy. We concentrate on the casein which multiple steady states exist in this economy. The government can set the income tax ratein this model. Therefore, we focus on how changes of the tax rate aﬀect this economy. First, weconsider the eﬀects on economic growth. Let us denote g ∗∗∗ as the growth rate of product varietyof the steady state with a high level of human capital h∗∗∗ and λ∗∗∗ as the old-age survival rate of
the steady state with h∗∗∗ . We obtain the following proposition8 .
Proposition 5 If the tax rate τ increases, the level of human capital of the unstable steady stateh∗∗ decreases and that of the steady state with a low level of human capital h∗ and a high level ofhuman capital h∗∗∗ increases.
8

Propositions 5 and 6 do not need to specify the functional form of f (bt ).

17

ℎ'()
45°
𝜏↑
ℎ'() = Φ ℎ'

ℎ∗

0

ℎ∗,

ℎ∗∗,

ℎ∗∗

ℎ∗∗∗

ℎ∗∗∗,

ℎ'

Figure 7: Eﬀect of change of the tax rate on development
Proof. see Appendix D
Let us consider an economy whose initial per capita human capital h0 satisfies h∗ < h0 < h∗∗ .
If the government does not implement any policy, this economy can converge to the steady statewith low per capita human capital h∗ . However, if the government sets a suﬃciently high tax rateand implements a health-improving policy, investment in education for each child increases. Then,human capital accumulation is stimulated and this economy converges to the steady state with highper capita human capital h∗∗∗ . In addition, if the government sets a higher tax rate, the steady-statelevel of human capital h∗ and h∗∗∗ increases.
We next obtain the following proposition.
2 ∗∗∗λProposition 6 Suppose ∂ ∂τ2

< 0, in which an increase of tax rate τ has an inverted U-shaped

eﬀect on the steady-state growth rate g ∗∗∗ .
Proof. see Appendix E
As shown in Appendix E, if the tax rate τ increases, the steady state old-age survival rate λ∗∗∗increases. Therefore, individuals save more and R&D investment increases. Then, the growthrate increases. However, if the tax rate τ increases, the disposable income decreases. Therefore,
individuals’ saving volume decreases, R&D investment decreases, and the growth rate decreases. In
2 ∗∗∗λaddition, if ∂ ∂τ
< 0 holds, the first positive eﬀect on the growth rate decreases when τ increases.
218

Figure 8: The inverted U-shaped eﬀect of the tax rate on the steady-state growth rate. We specify
25tf (Gt ) = ρ+µb
θ+bt and assume that β = (0.98) , δ = 1, γ = 0.14, η = 0.21, θ = 5, µ = 1, ρ = 1,σe = 0.635, σh = 0.165, ϕ = 34.02, and z = 0.088.
Hence, an increase of tax rate τ has an inverted U-shaped eﬀect on the steady-state growth rateg ∗∗∗ . Figure 8 shows a numerical example of the inverted U-shaped eﬀect of the tax rate on the∗∗∗steady-state growth rate. Considering ∂λ∂τ> 0 and Proposition 5, there is an inverted U-shaped

eﬀect between λ∗∗∗ and the steady-state growth rate g ∗∗∗ . Our simple regression supports thisinverted U-shaped relationship9 . More formal empirical evidence in support of this pattern can befound in An and Jeon (2006). Their regression using panel data from OECD countries over theperiod 1960–2000 also reflects an inverted U-shaped relationship between the old-age dependency
ratio and economic growth.
Next, we calibrate the model to examine how the change of tax rate aﬀects welfare. There aretnine structural parameters {α, β, δ, γ, η, σe , σh , ϕ, ψ, ν, z}. We specify f (Gt ) = ρ+µb
θ+bt and normalize
9

The World Development Indicators (World Bank 2019) are used to calculate the old-age dependency ratio andthe per capita output growth rate. We use data on 157 countries for the period from 1991 to 2016 and estimate simple
regressions, in which the average per capita output growth rate (Growth) as the dependent variable is a functionof the average old-age dependency ratio (Old1) and the value of its square (Old2). The following equation provides
simple estimation results using ordinary least squares:
Growth = −0.896(−1.134) + 0.535(4.032)Old1 − 0.017(−3.780)Old2,
where the figures in parentheses are the values of the t-statistics. The equation above suggests that there is an inverted
U-shaped correlation between the old-age dependency ratio and the per capita GDP growth rate. Hashimoto andTabata (2016) show a similar result.

19Figure 9: Response to change of tax rate. The dotted lines denote the steady-state value and thesolid lines denote the tax change response. The t-th generation’s total consumption is the sum of
its consumption level for adult age and that for old age.

20δ, µ, and ρ to 1. Following Futagami and Konishi (2017), we set the minimum value of λt to 0.2(i.e., θ = 5). Considering the range of the values estimated by Norrbin (1993) and Basu (1996),
we set α at 1/1.05. We regard one period in this overlapping-generations economy as 25 years andset β = (0.98)25 . Following Strulik et al. (2013), we set γ = 0.14 and z = 0.088. We choose
η to make the fertility rate 2.1/2 in the steady state with h∗∗∗ (i.e., η = 0.21). Following de la
Croix and Doepke (2003), we set σe = 0.635. Following Cardak (2004), we set σe + σh = 0.8 (i.e.,σh = 0.165). The average old-age dependency ratio of OECD countries from 1991 to 2016 is 0.2076.
Therefore, we choose ϕ to make the old-age dependency ratio 0.2067 in the steady state with h∗∗∗(i.e., ϕ = 34.02)10 . Following Jones and Williams (2000), we set ν = 0.5. We choose the valuesof the R&D productivity parameter ψ to make the annual growth rate of output 2% in the steadystate with h∗∗∗ (i.e., ψ = 2.69). We set the initial population size N0 = 1 and the initial economy
size A0 = 1.
Suppose that the economy is in the steady state with h∗∗∗ at time 0 and the government raises
the tax rate τ = 0.1 to τ = 0.13 at time 1. Figure 9 shows the levels of variables at the initialsteady state and those at the transition pass after the tax change. From Figure 9, we find that if thegovernment raises the tax rate, the welfare level of individuals of the first generation (i.e., people who
become adults at time 1) increases from the steady-state level. However, after the second generation(i.e., people who become adults after time 2), the welfare levels decrease from the steady-state levels
of individuals. Therefore, the government faces a trade-oﬀ between improving the welfare levels of
the current versus future generations. Let us consider the reason for this conflict. From Lemma
1, the old-age survival rate increases if the government raises the tax rate. This increases the
expected utility of adults of the t-th generation. However, from (12), the fertility rate decreases if
the old-age survival rate increases. If the fertility rate decreases, the utility level of adults of thet-th generation decreases. Therefore, if this negative eﬀect is suﬃciently small, the welfare level ofadults of the t-th generation increases. On the contrary, if the tax rate τ increases, the disposable
income decrease. Therefore, individuals’ savings decreases and R&D investment decreases. If R&D
10

The old-age dependency ratio is the ratio of people older than 64 years to the working-age population of those
aged 15–64 years. Noting that one period in this overlapping generations economy as 25 years, we calculate the old-age
dependency ratio of the steady state with h∗∗∗ by following equation:
old age dependency ratio = 25−15
25∗∗∗where n

∗∗∗

and λ

75−64 ∗∗∗
λ
25
,
∗∗∗
(n )2 + n∗∗∗ + 64−50
λ∗∗∗25denote the fertility rate and old-age survival rate of the steady state with h∗∗∗ , respectively.

21

investment decreases, the growth rate also decreases. Hence, individuals’ total consumption levels
increase more slowly over time owing to the low growth rate. Therefore, the welfare level gradually
becomes lower from the steady-state level after the tax increases. Hence, if the government raises
the tax rate, the welfare level of the current generation increases and the welfare levels of future
generations decrease. The government tends to decide the tax rate by placing more weight on
the current generation (i.e., adult agents) than on other generations, because children do not havevoting rights and some old agents are dead at time t. However, such a decision tends to harm social
welfare, because the welfare levels of future generations are sacrificed for the welfare gain of thecurrent generation.

5

Conclusion

We constructed an overlapping-generations model of R&D-based growth with endogenous fertility,
mortality, and education choice. We showed that multiple steady states can exist in this economy.
One of them has a high level of human capital and the other has a low level. In the steady state withthe high (low) level of human capital, there is a high (low) level of R&D activity, a low (high) fertility
rate, and a high (low) old-age survival rate. In addition, we showed that the government can steer
the economy away from a poverty trap trajectory by investing in public health. We also showedthat an improvement in the government’s public health policy has an inverted U-shaped eﬀect on
the growth rate at the steady state. We examined how the improved public health policy aﬀects
welfare and showed that the government faces a trade-oﬀ between improving the welfare levels of thecurrent generation and future generations. In this study, we did not take account of the goods costof child rearing and, for simplification, assumed that the public health service is produced without
goods. However, these factors are worth analyzing in this framework. The expansion of the cost ofchild rearing and the provision of public health service is a promising direction for future research.

22

𝑔$ + 1

ℎ$ ↑

)'*
' *

Λ(ℎ$ )𝑔$

1
0

𝑔$∗

𝑔$

𝑔$∗′

Figure 10: Determination of gt and the eﬀect of Ht on gt

Appendix
Appendix A: Proof of that R&D is always undertaken
We show that R&D is always undertaken in this model. Suppose that R&D is not undertaken and∆At = 0. Then, (23) holds with strict inequality as follows:

vt <

wt
.
ψ t At(A1)

From (22), we have
∆At
=ψ
gt =
At

(

lR,t
ht Nt

)ν
.

(A2)

From (A2), if ∆At = 0, lR,t = 0. By using (15), (19), (21), (27), and (29), we obtain the wage rate
wt as follows:αwt = [(1+βλt )(1−τ )−
1+η+βλt( )1 ]
gt
ψ

ν

,

(A3)

Ht

where Ht ≡ ht Nt is aggregate human capital. From (A3), when ∆At = 0 (i.e., gt = 0), wt =α(1+η+βλ∗t )Ht(1+β+βλ∗t )(1−τ ) .

In addition, when lR,t = 0, ψt approaches infinity. Therefore, the right-hand sideof (A1) converges to 0 when ∆At = 0. Meanwhile, when ∆At = 0 (i.e., lR,t = 0), we obtain the23

following equation from (28),

vt =

st Nt
β(1 − τ )wt λ∗t Ht
=
> 0.
At
(1 + η + βλ∗t )At(A4)These results contradict (A1). Therefore, ∆At > 0 always holds in this economy.

Appendix B: Proof of Proposition 3
We check the eﬀect of ht on gt . From (32) and λ∗t = λ(ht ; τ ), we obtain1

gt + 1 =βψ ν (1 − τ )λ(ht ; τ ) − 1−νg ν .
1 + η + βλ(ht ; τ ) t
|
{z
}

(A5)

≡Λ(ht )1
ν(1−τ )λ(ht ;τ )Let us define Λ(ht ) as Λ(ht ) ≡ βψ1+η+βλ(h. Figure 10 shows the relationship between the leftt ;τ )hand and right-hand sides of (A5). From Figure 10, we find that (A5) has a unique solution of gt
and we denote this solution as gt . We obtain
1∂λ(ht ; τ )β(1 + η)ψ ν (1 − τ ) ∂λ(ht ; τ )
=
> 0.
∂ht
(1 + η + βλ∗t )2
∂ht
− 1−ν
ν

Therefore, when ht increases, the line of Λ(ht ; τ )gt(A6)

shifts upward and gt increases, as shown in

Figure 10.

Appendix C: Proof of Proposition 4t
Suppose f (bt ) = ρ+µb
θ+bt and from this equation, we obtainf −1 (λt ) =θλt − ρ.
µ − λt(A7)

Substituting (A7) into (31) and rearranging them, we obtainht =(1 + η + βλ∗t )(θλ∗t − ρ)
(1 + βλ∗t )(µ − λ∗t )δτ

24

(A8)

ℎ"#$

45°
ℎ"#$ = Φ ℎ"

𝐿𝐻𝑆 𝜆∗"𝑅𝐻𝑆 𝜆∗"#$

0𝜂

Panel A𝜇 𝛽𝛾𝜎7𝜆∗" , 𝜆∗"#$

0

ℎ∗∗

ℎ"

Panel C

𝜆∗"

45°

ℎ∗∗∗

𝜆 ℎ"

0

𝜆∗∗∗

𝜆∗∗

𝜆∗" , 𝜆∗"#$

0

ℎ∗∗∗

ℎ∗∗

ℎ"

Panel D

Panel B

Figure 11: Derivation of the phase diagram of ht
If ht is given, λ∗t is determined to satisfy (A8). From λ∗t = λ(ht ; τ ), (33), and (A8), we obtain thefollowing inequalities
ht+1 ≥ ht ,→1[
] 1−σe −σhσeβγσe ϕ σe zλ∗t
(1 + η + βλ∗t )(θλ∗t − ρ)
≥
.
∗
∗
∗
η − βγσe λt(1 + βλt )(µ − λt )δτ
|
{z
} |
{z
}

≡LHS(λ∗t )(A9)≡RHS(λ∗t )Let us define the left-hand side of (A9) as LHS(λ∗t ) and the right-hand side of (A9) as RHS(λ∗t ).
From (A9), we find that LHS(0) = 0, RHS( ρθ ) = 0, limλ∗t →ηβγσe

LHS(λ∗t ) = ∞, and limλ∗t →µ RHS(λ∗t ) =∞. Therefore, limλ∗t →µ RHS(λ∗t ) > limλ∗t →µ LHS(λ∗t ), because µ ≤ 1 and η > γ. From (A9), we25

obtain the following:1βγησe ϕ σe z
∂LHS(λ∗t )
=
≥ 0,
(A10)
∂λ∗t
(η − βγσe λ∗t )2∗ ∗∂RHS(λ∗t )θ[(1 + βλ∗t )2 µ + η(µ + βλ∗2t )] − {1 + (1 − µ)η + [2(1 + η) + βλt ]λt }ρ
=∂λ∗t
σe (1 + η + βλ∗t )2[] 1−σh∗1−2σe −σh
σe
1
+
η
+
βλ
t
∗
× δ(1 − σe − σh )τ (θλt − ρ) σe. (A11)∗∗(1 + βλt )(µ − λt )δτFirst, let us consider the case in which ρ = 0 as the benchmark. When ρ = 0, we obtain thefollowing equation from (A11)[µ(1 + βλ∗t )2 + η(µ + βλ∗2
∂RHS(λ∗t )t )]δ(1 − σe − σh )τ
=∗∂λt
θσe (1 + η + βλ∗t )2
ρ=0[] 1−σh∗ )θ
1−2σe −σh
σe
(1
+
η
+
βλ
t
.
× (λ∗t ) σe
(1 + βλ∗t )(µ − λ∗t )δτ(A11’)

1

From (A10), limλt →0that∂RHS(λ∗t )∂λ∗tσe∂LHS(λ∗t )= βγσeηϕ∂λ∗t> 0, limλ∗t →0< ∞. Suppose that σh ∈ (1 − 2σe , 1 − σe ). Then, we find

∂RHS(λ∗t )∂λ∗t= ∞ and limλ∗t →µ∂RHS(λ∗t )= ∞ from (A11’).
∂λ∗t
ρ=0

ρ=0
ρ=0
∂LHS(λ∗t )increases. Therefore, there exists ϕ such that LHS(λ∗t ) and RHS(λ∗t )|ρ=0
If ϕ increases,∂λ∗thave their intersections at λ∗t = 0, λ∗∗ , λ∗∗∗ , as drawn in Panel A of Figure 11. If LHS(λ∗t ) andRHS(λ∗t )|ρ=0 have three intersections, there exist three steady states with respect to ht , as shownin Panel C of Figure 11. Next, let us consider the case in which ρ > 0. From (A9), we find that
RHS(λ∗t ) shifts left, as shown in Panel A of Figure 12, if ρ increases. From (A11), we find that
∂RHS(λ∗t )∂RHS(λ∗t )∂RHS(λ∗t )> 0 if ρ is not so high. We also find that limλ∗t → ρ= ∞ and limλ∗t →µ
=
∗
∂λ∗t
∂λ
∂λ∗t
θ
t

∞ from (A11). From Figure 12, if ρ > 0 and ρ is not so high, LHS(λ∗t ) and RHS(λ∗t ) havethree intersections and those values are positive (λ∗∗∗ > λ∗∗ > λ∗ > 0). Then, if ρ is not sohigh, three steady states exist with respect to ht and the values of these steady states are positive(h∗∗∗ > h∗∗ > h∗ > 0), as shown in Panel C of Figure 12.
In addition, if ht+1 ≥ ht holds, LHS(λ∗t ) ≥ RHS(λ∗t ) holds from (A8). Therefore, we obtain
Panel C of Figure 12, which is equivalent to Figure 6.

26

ℎ"#$

45°
ℎ"#$ = Φ ℎ"

𝐿𝐻𝑆 𝜆∗"𝑅𝐻𝑆 𝜆∗"#$
𝜌↑
0

𝜂

𝜌
𝜃

Panel A𝜇 𝛽𝛾𝜎7𝜆∗" , 𝜆∗"#$
45°

0

ℎ∗

ℎ∗∗

ℎ∗∗∗

ℎ"

Panel C

𝜆∗"

𝜆 ℎ"

0

𝜆∗

𝜆∗" , 𝜆∗"#$

𝜆∗∗∗

𝜆∗∗

0

ℎ∗

ℎ∗∗∗

ℎ∗∗

ℎ"

Panel D

Panel B

Figure 12: Derivation of Figure 4

Appendix D: Proof of Proposition 5t ;τ )
From Lemma 1, λ∗t is a function of τ such that λ∗t = λ(ht ; τ ) and ∂λ(h> 0. From (33), we obtain∂τ∂Φ(ht )=ϕ∂τ{t ;τ )
βγσe z ∂λ(h∂τ[η − βγσe λ(ht ; τ )]2}σe
hσt e +σh > 0.

(A12)

From (A12), we find that the line of Φ(ht ) shifts upward in Figure 7 if τ increases. Then, h∗∗decreases and h∗ and h∗∗∗ increase when τ increases.

Appendix E: Proof of Proposition 6
t ;τ )
From Lemma 1, λ∗t is a function of τ such that λ∗t = λ(ht ; τ ) and ∂λ(h> 0. If τ increases, the line∂τt ;τ )> 0. Therefore,of Φ(ht ) shifts upward in Figure 7 and h∗∗∗ increases. From Proposition 1, ∂λ(h
∂ht

27

Γ(𝜏)

Υ(𝜏)

𝜏∗

0

1

𝜏

Figure 13: Determination of τ ∗∂λ∗∗∗∂τ > 0 holds. Totally diﬀerentiating (32) with respect to gt and τ , we obtain

[
]
1
∗∗∗
βψ ν νg ∗∗∗ (1 + η)(1 − τ ) ∂λ∂τ − (1 + η + βλ∗∗∗ )λ∗∗∗dg ∗∗∗
].
[
=
1
1
dτ
(1 + η + βλ∗∗∗ ) (1 + η + βλ∗∗∗ ]ν(g ∗∗∗ ) ν + βψ ν (1 − ν)(1 − τ )λ∗∗∗∗∗∗Let us define Γ(τ ) as Γ(τ ) ≡ (1 + η)(1 − τ ) ∂λ∂τ∗∗∗larger than Υ(τ ), dgdτ(A13)and Υ(τ ) as Υ(τ ) ≡ (1 + η + βλ∗∗∗ )λ∗∗∗ . If Γ(τ ) is

2 ∗∗∗

λ
> 0 holds. Suppose ∂ ∂τ
< 0, we can obtain
2

[ ∗∗∗
]
∂Γ(τ )∂λ∂ 2 λ∗∗∗= −(1 + η)
− (1 − τ )
< 0,
∂τ
∂τ
∂τ 2
∂Υ(τ )∂λ∗∗∗= (1 + η + 2βλ∗∗∗ )
> 0.
∂τ
∂τ

(A14)
(A15)

From (A14) and (A15), we find the relationship between Γ(τ ) and Υ(τ ) in Figure 13. From Figure
13, there is a unique intersection between Γ(τ ) and Υ(τ ) and we denote the value of this intersection
( ∗∗∗
)
∗∗∗
as τ ∗ . From (A13), if τ < τ ∗ (τ > τ ∗ ), dgdτ > 0 dgdτ < 0 .

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