Effects of Demographic Change on Economic Growth in an OLG Economy with Childrearing Costs and Exogenous Fertility

Effects of Demographic Change on Economic Growth in an OLG Economy with Childrearing Costs and Exogenous Fertility

Hiroyuki HASHIMOTO ^#

September, 2020

ABSTRACT

This paper explore the relationship between population aging and economic growth by developing an endogenous growth model of overlapping generations, where human capital accumulation is the engine of income growth and childrearing costs for all working periods are explicitly taken into account. We demonstrate that a fall in fertility rate deters economic growth when the childrearing cost by parents is equal to or greater than that by grandparents. In contrast, it is possible that a fall in fertility rate fosters economic growth if the childrearing cost by parents is strictly less than that by grandparents.

Keywords: Childrearing Cost; Human Capital; Population Aging; Economic Growth JEL classification: H22, H23, J13, J24, J31, O41

# University of Hyogo8-2-1 Gakuennishi-machi Nishi-ku Kobe Hyogo, 6512197 JAPAN

E-mail hiro@mba.u-hyogo.ac.jp

1. Introduction

This paper develops an overlapping generations model in which fertility is

given exogenously, but the cost of raising children is explicitly considered, and

examines the impact of lower fertility on economic growth in the long run. Since its

seminar work, Solow (1956), many types of economic growth models have been

developed to uncover the underlying mechanisms of economic growth. On the other

hand, many empirical studies have tested theoretical hypotheses about the sources of

economic growth and have found empirical evidence to share in the field of economic

growth. Human capital accumulation has been recognized as one of the important

factors for sustaining economic growth, especially as emphasized by Lucas (1988). The

economic growth literature has highlighted the decrease in fertility rates as a

characteristic feature of the demographic issue that needs to be explained in the

framework of economic growth theory. A number of previous studies have postulated

that there is a negative link between population growth and economic growth, while one

study has indicated there to be a positive connection between them. In the exogenous

economic growth literature, the Solow model has been used to explain this negative

relationship, which is referred to as the “capital dilution effect.” Diamond(1965)

developed an overlapping generations model which posseces the "capital dilution

effect". In endogenous economic growth literature, although there are numerous studies

that have investigated the various channels through which population growth and

economic growth are linked, such studies comprise two strands of literature. One

demonstrates the demographic phenomena in the setting of Malthusian-to-Neoclassical

growth, finding that an increase in fertility fosters economic growth (e.g., Galor and

Weil(2000); Galor and Moav(2002)). ¹ The other develops models to display the negative link between population growth and economic growth in a neoclassical growth framework, which corresponds to “modern economic development” (Barro and Becker(1989); Becker et al.(1990); Moav(2005)). Although there has been an increase in the studies that incorporate more realistic features into certain stylized economic growth models, the modeling framework itself substantially decides whether or not population growth enhances economic growth. Numerous such types of economic growth models, in which human capital is accumulated, also have been developed to depict various aspects of economic growth. For example, Becker, Murphy, and Tamura (1990) indicate the existence of a poverty trap. Many theoretical models that study the interaction between population aging and the social security system have adopted the

"overlapping generations" model. For example, Kaganovich and Zilcha (1999) and Grozen et al (2003) deal with the issue of government financing of public pensions.

In order to explore effects of popuiation aging in an endogenously growing overlapping generations economy, we develop an overlapping-generations model with two types of capital, physical and human capital, where the engine of growth is human capital accumulation and different two types of workers engage in production. Our model is basically a modified version of Hashimoto et al (1997) that is very similar to Lucas (1988). Moreover as employed in Cipriani (2014), we take the childrearing cost into account in order to study how the cost affects economic growth, although the number of children is exogenously given. In such a model, we demonstrate that a fall in fertility rate deters economic growth when the childrearing cost by parents is equal to or greater than that by grandparents. In contrast, it is possible that a fall in fertility rate

1 See Doepke(2008) for an example of a work that summarizes this issue well.

fosters economic growth if the childrearing cost by parents is strictly less than that by grandparents.

The rest of this paper is organized as follows. Section 2 set the model and Section3 considers the relationship between population growth and economic growth.

Section 4 gives summary and concluding remarks.

2. The Model

We consider a small open economy which exists over infinite number of periods. ² The economy is composed of individuals who live for three periods without uncertainty on lifetime and many competitive firms that produce a single good. The production requires physical capital and two types of workers with human capital. We label the generation that was born at period t as “generation t”. Individuals in the same generation are assumed to be identical and each has n children. The number of children n is assumed to be exogenously determined. We assume that N _{t + 1} = + ( 1 n N ) _t holds, where N t is the size of generation t. In both the first and second period of life, each individual is endowed with one unit of time. In the first period of life, each individual is young worker and allocates the productive time between work and accumulation in human capital which is the engine of growth in this economy. In the second period of life, each is old worker and supplies the labor force inelastically to labor market. In the third period of life, each is retired. The members of the different two generations engage in production in any period t.

2 The assumption needs for the determinacy of equilibrium, even though it does not ensure the

determinacy all by itself. In other words, the closed-economy version of the following model

exhibits the determinacy of equilibrium.

2.1 Human Capital Accumulation

Only the young govern investment in human capital. Suppose that each young individual of generation t is endowed with human capital, h _t ^y . We assume that it is accumulated according to

o

h t _{+ 1} ⁼ H ( ) u t h t ^y , (1)

where u _t ∈[ , ] 0 1 is the part of time devoted to human capital accumulation, h _t ^o ₊₁ is the level of human capital of old worker of generation t at period t+1, and H(⋅) is assumed to be increasing, concave and continuously twice differentiable function such that

H(0)=1 and = +∞

→ ' ( )

lim

0 t

u H u

t

hold. Since one remarkable aspect as for human capital accumulation is a social activity as stated in his seminal work of Lucas (1988), we simply assume that each young individual at period t inherits a portion of average level of human capital of old individuals at period t:

y

h t = ( 1 − δ ) h _t ^o : 0≤δ≤1, (2) where h _t ^o is the average level of human capital of old individuals at period t and initial level h ₀ ^o is given. ³ Individuals between neighboring generations complete “the inheritance” expressed in (2) without intention, therefore without costly activities. Thus the inheritance mechanism is intergenerational externality. The parameter δ stands for Lucasian intergenerational externality, which could be interpreted as the extent of inheritance of human capital. ⁴ For simplifying the following analysis, we set δ to be zero. Note that the consistency condition h _t ^o = h _t ^o is required in equilibrium, since individuals are identical within generation. Thus ^h t

y = h _t ^o must hold in equilibrium.

3 Lucas (1988, p19) says: …the initial level each new member begins with is proportional to (not equal to!) the level already attained by older members of the family….

4 If δ is unity, it means the absence of intergenerational externality.

2.2 Individuals

We label the utility of each individual of generation t as U _t , which depends on consumption at the second and the third period of life, c _t+1 ² and c _t+2 ³ . For the analytical simplicity, consumption is assumed not to occur in the first period of life. We give each individual a log-linear utility function

3 2 1 2

1 ln

ln ₊ + ⁻ ₊

= _{t t}

t c c

U ρ ^{, ρ>} _1. ⁽³⁾ where ρ represents the subjective discount factor on the third-period utility, ln c _t ³ _{+ 2} . Each young individual with ^h t

y allocates u _t units of time for accumulating own human capital, while supplying1 − u _t units of time to firm through competitive labor market in order to earn income w _t ^y ( 1− u h _t ) _t ^y . Since individuals are assumed not to need consumption in the first period of life, they save all of wage income in that period and earn the going rate of return R _{t + 1} . Each old individual with h _t ^o ₊₁ supplies one unit of time to firm inelastically to earn income w h _t ^o _{+ 1 t} ^o _{+ 1} , and receives the saving with accrued interest, R _{t + 1} w _t ^y ( 1 − u _t ) h _t ^y . Although the number of children n is exogenous in this model, we assume that raising children is costly activity, and the parents and the grandparents bear the child-rearing cost as employed in Cipriani (2014). The amount of resources for parents to raise each child is assumed to be a fixed proportion to income earned in the first period of life, 𝑞𝑞 ^𝑦𝑦 𝑤𝑤 _𝑡𝑡 ^𝑦𝑦 (1 − 𝑢𝑢 𝑡𝑡 )ℎ _𝑡𝑡 ^𝑦𝑦 . ⁵ Since the parents have n children, the child-rearing cost is written as 𝑞𝑞 ^𝑦𝑦 𝑤𝑤 _𝑡𝑡 ^𝑦𝑦 (1 − 𝑢𝑢 _𝑡𝑡 )ℎ _𝑡𝑡 ^𝑦𝑦 𝑛𝑛 . Each old individual consumes a part of wealth, bears the child-rearing cost 𝑞𝑞 ^𝑜𝑜 𝑤𝑤 _𝑡𝑡+1 ^𝑜𝑜 ℎ _𝑡𝑡+1 ^𝑜𝑜 𝑛𝑛 and saves the rest s _{t + 1} to earn the going rate of return R _{t + 2} . Each retired individual spends all of the saving with

5 Cipriani (2014) studies the case where only parents bear the childrearing cost .

accrued interest R _{t + 2} s _{t + 1} on the third-period consumption. Thus the budget constraints of each individual of generation t are written as

+𝑤𝑤 _𝑡𝑡+1 ^𝑜𝑜 ℎ _𝑡𝑡+1 ^𝑜𝑜 − 𝑞𝑞 ^𝑜𝑜 𝑤𝑤 _𝑡𝑡+1 ^𝑜𝑜 ℎ _𝑡𝑡+1 ^𝑜𝑜 𝑛𝑛 − 𝑠𝑠 _𝑡𝑡+1 ^, (4) and

c _t+2 ³ = R _t+2 s _t+1 . (5)

An individual born at period t is to maximize (3) subject to (4), (5) and (1). It is straightforward to obtain the first-order conditions for the choice of s t+ 1 and u t to be optimal. The conditions are

c _t+2 ³ c _t+1 ² = R _t+2

ρ ^, (6)

𝑅𝑅 𝑡𝑡+1 (1 − 𝑞𝑞 ^𝑦𝑦 𝑛𝑛) (1 − 𝑞𝑞 ^𝑜𝑜 𝑛𝑛)

𝑤𝑤 _𝑡𝑡 ^𝑦𝑦

𝑤𝑤 _𝑡𝑡+1 ^𝑜𝑜 = 𝐻𝐻′(𝑢𝑢 _𝑡𝑡 ) (7)

together with the budget constraints. (6) means that the marginal loss of the second-period utility by saving one unit of income must equal the marginal gain of the third-period utility by receiving R _{t+ 2} units of income. The LHS of (7) represents the marginal opportunity cost of allocating additional time for accumulation in human capital and the RHS indicates the marginal benefit of such time allocation. Thus equation (7) displays that each individual allocates one unit of endowed time so as to maximize his lifetime income equating the marginal benefit with the marginal cost. The optimal decision of each individual is characterized by (7) and

𝑠𝑠 𝑡𝑡+1 = �𝑅𝑅 𝑡𝑡+1 (1 − 𝑢𝑢 𝑡𝑡 )(1 − 𝑞𝑞 ^𝑦𝑦 𝑛𝑛)𝑤𝑤 _𝑡𝑡 ^𝑦𝑦 + (1 − 𝑞𝑞 ^𝑜𝑜 𝑛𝑛)𝑤𝑤 _𝑡𝑡+1 ^𝑜𝑜 𝐻𝐻(𝑢𝑢 _𝑡𝑡 )�ℎ _𝑡𝑡 ^𝑦𝑦 1 + 𝜌𝜌

(8) (8) is derived by using (6) and the budget constraints. Note that u t in (8) satisfies (7), in other words, the numerator in the RHS of (8) is the maximized lifetime income. Thus an individual’s decision making is of step-wise. The first decision

𝑐𝑐 _𝑡𝑡+1 ² = 𝑅𝑅 _𝑡𝑡+1 �𝑤𝑤 _𝑡𝑡 ^𝑦𝑦 (1 − 𝑢𝑢 _𝑡𝑡 )ℎ _𝑡𝑡 ^𝑦𝑦 − 𝑞𝑞 ^𝑦𝑦 𝑤𝑤 _𝑡𝑡 ^𝑦𝑦 (1 − 𝑢𝑢 _𝑡𝑡 )ℎ _𝑡𝑡 ^𝑦𝑦 𝑛𝑛�

is to maximize the lifetime income by choosing the optimal time allocation. The second one is to choose the optimal saving with the maximized lifetime income.

2.3 Firms

In each period, firm is assumed to be profit-maximizer in competitive markets.

Assuming constant-returns-to-scale technology in each firm, the production function at the aggregate level at period t is represented as the following technology:

Y _t = AK _t ^α [ (1 − u _t )H _t ^y ] ^β [ ] ^{H t o γ , α + β + γ =1,} (9)

where A is technological parameter, K _t denotes aggregate physical capital stock, H _t ^y is aggregate human capital stock of young workers given by N h _{t t} ^y , and H _t ^o is aggregate human capital stock of old workers given by N _{t −1} h _t ^o , at period t, respectively.

Assuming that physical capital fully depreciates at one period and using consistency condition h _t ^y = _{h t} ^o , the profit maximization gives the following first-order conditions:

R _t = α A k _t h _t ^o



  

 

α −1

(1 − u _t ) ^β n ^{α +β −1} , (10)

w _t ^y = β A k _t h _t ^o



  

 

α

(1 − u _t ) ^{β −1} n ^{α +β −1} , and (11)

w _t ^o = (1 − α − β )A k _t h _t ^o



  

 

α

(1 − u _t ) ^β n ^{α +β} , （12）

where _{k t} is defined as

t t

N

K , R _t stands for the rental price of capital, and w _t ^y and w _t ^o are wage rates to be spent for young and old workers, at period t, respectively.

These conditions require that firms must employ productive factors in order to equate

the marginal product with the market price they face in the competitive markets.

2.4 Equilibrium

The equilibrium condition in the capital market is given as

𝐾𝐾 _𝑡𝑡+1 = 𝑁𝑁 _𝑡𝑡−1 𝑠𝑠 _𝑡𝑡 + 𝑁𝑁 _𝑡𝑡 (1 − 𝑢𝑢 _𝑡𝑡 )(1 − 𝑞𝑞 ^𝑦𝑦 𝑛𝑛)𝑤𝑤 _𝑡𝑡 ^𝑦𝑦 ℎ _𝑡𝑡 ^𝑦𝑦 (13) where _{N t − 1} and _{N t} stand for the number of the old and the young workers in period t, respectively. The LHS of (1) is the capital stock at period t+1, while the RHS displays aggregate saving in period t.

Since h _t ^y = _{h t} ^o holds in equilibrium, it is useful to denote h _t instead of h _t ^y or _{h t} ^o in equilibrium. Using (1) and (2), we obtain the following equation.

A competitive equilibrium must satisfy the equation of motion for human capital (14) and the capital-market-clearing condition (13), together with households’

and firms’ optimality conditions, (7), (8), (10), (11), (12), consistency condition,

o t y

t h

h = ≡ h _t , and _{r = r t} , at every period. Substituting all of these conditions into (13) and (14) generates the following dynamical system about the motion of k _t ^, h _t ^and u _t

and

ℎ 𝑡𝑡+1 = 𝐻𝐻 𝑡𝑡 (𝑢𝑢 _𝑡𝑡 )ℎ _𝑡𝑡 . (14)

𝑅𝑅 (1 − 𝑞𝑞 ^𝑦𝑦 𝑛𝑛) (1 − 𝑞𝑞 ^𝑜𝑜 𝑛𝑛)

𝛽𝛽 1 − 𝛼𝛼 − 𝛽𝛽

1 𝑛𝑛 �

𝑘𝑘 _𝑡𝑡 ℎ 𝑡𝑡 �

𝛼𝛼

(1 − 𝑢𝑢 𝑡𝑡 ) ^𝛽𝛽−1 = 𝐻𝐻 ^′ (𝑢𝑢 _𝑡𝑡 ) � 𝑘𝑘 _𝑡𝑡+1 ℎ 𝑡𝑡+1 �

𝛼𝛼

(1 − 𝑢𝑢 𝑡𝑡+1 ) ^𝛽𝛽 ,

𝑛𝑛 ² (1 + 𝜌𝜌) 𝑘𝑘 𝑡𝑡+1

ℎ 𝑡𝑡+1 = � 𝛽𝛽 𝛼𝛼 𝑅𝑅 ²

(1 − 𝑞𝑞 ^𝑦𝑦 𝑛𝑛) (1 − 𝑞𝑞 ^𝑜𝑜 𝑛𝑛)

𝑘𝑘 𝑡𝑡−1

ℎ 𝑡𝑡−1 + 𝐻𝐻(𝑢𝑢 _𝑡𝑡−1 ) 1 − 𝛼𝛼 − 𝛽𝛽 𝛼𝛼 𝑅𝑅𝑛𝑛 𝑘𝑘 𝑡𝑡

ℎ 𝑡𝑡 � 1

𝐻𝐻(𝑢𝑢 _𝑡𝑡 )𝐻𝐻(𝑢𝑢 _𝑡𝑡−1 ) +𝑛𝑛(1 + 𝜌𝜌) 𝛽𝛽

𝛼𝛼 𝑅𝑅

(1 − 𝑞𝑞 ^𝑦𝑦 𝑛𝑛) (1 − 𝑞𝑞 ^𝑜𝑜 𝑛𝑛)

𝑘𝑘 _𝑡𝑡 ℎ _𝑡𝑡

1 𝐻𝐻(𝑢𝑢 _𝑡𝑡 ),

ℎ _𝑡𝑡+1 = 𝐻𝐻 _𝑡𝑡 (𝑢𝑢 _𝑡𝑡 )ℎ _𝑡𝑡 ,

where

k 0 and

h 0 are given as initial value. Though, on the face of it, the equilibrium dynamics is of complicated nonlinear difference equations, introducing a new variable, X _t ≡ k h _{t t} , can reduce the system to a two-dimensional first-order difference equations.

and

where

0 0

0 k h

X ≡ ⁱ s given. 𝜙𝜙 in (15) is well-defined function to satisify 𝑅𝑅 ^{(1−𝑞𝑞} _{(1−𝑞𝑞} ^𝑦𝑦 _{𝑜𝑜 𝑛𝑛)} ^𝑛𝑛) _{1−𝛼𝛼−𝛽𝛽} ^𝛽𝛽 _𝑛𝑛 ¹ 𝑋𝑋 _𝑡𝑡 ^𝛼𝛼 (1 − 𝑢𝑢 _𝑡𝑡 ) ^𝛽𝛽−1 = 𝐻𝐻 ^′ (𝑢𝑢 _𝑡𝑡 )𝑋𝑋 _𝑡𝑡+1 ^𝛼𝛼 (1 − 𝜑𝜑(𝑢𝑢 _𝑡𝑡 )) ^𝛽𝛽 and (16), where 𝜑𝜑 in (16) satisfies _{𝐻𝐻′(𝑢𝑢} ^{𝐻𝐻(𝑢𝑢 𝑡𝑡+1 )}

𝑡𝑡+1 ) = ^{(1−𝛼𝛼−𝛽𝛽)} _{𝛼𝛼𝛽𝛽(1+𝜌𝜌)} ^{2 (1−𝑞𝑞} _{(1−𝑞𝑞} ^𝑦𝑦 _𝑜𝑜 ^𝑛𝑛) _𝑛𝑛) + ^{1−𝛼𝛼−𝛽𝛽} _𝛼𝛼 + ^{(1−𝛼𝛼−𝛽𝛽)} _{𝛼𝛼𝛽𝛽(1+𝜌𝜌)} ^{2 (1−𝑞𝑞} _{(1−𝑞𝑞} ^𝑦𝑦 _𝑜𝑜 ^𝑛𝑛) _𝑛𝑛) ^{𝐻𝐻′(𝑢𝑢} _{𝐻𝐻(𝑢𝑢} ^{𝑡𝑡 )}

𝑡𝑡 ) . It is hereafter

assumed that ( )

1

) 1 )(

1 ) (

( _t qn H u _t

u

H α β

ρ αβ

−

−

−

> +

′ to assure the determinacy of

equilibrium. We focus the equilibrium on the balanced growth path in the following analysis. The balanced growth path is such that both _{X t} and _{u t} take constant value.

Though the whole system is depicted by (15) and (16), it is useful to employ (16) and (10) instead of (15) to characterize the balanced growth path. Thus equilibrium balanced growth path is characterized by

The assumptions imposed on H （・） ensures the existence and the uniqueness of the balanced growth path. The production function (9) implies that the growth rate of per-worker income, _{y t + 1 / y t} , is H (u ) on the balanced growth path. This shows that the

) ,

1 ( t t

t X u

X ₊ = φ (15)

)

1 ( t

t u

u ₊ = ϕ , (16)

(1 − 𝑢𝑢)𝐻𝐻 ^′ (𝑢𝑢 _𝑡𝑡 ) = (1 − 𝑞𝑞 ^𝑦𝑦 𝑛𝑛) (1 − 𝑞𝑞 ^𝑜𝑜 𝑛𝑛)

𝛽𝛽𝑅𝑅 1 − 𝛼𝛼 − 𝛽𝛽

1

𝑛𝑛 (17)

γ β

α α ⁻ − + ⁻

= AX ¹ ( 1 u ) ( 1 n )

R (18)

driving force of this economy is the accumulation of human capital and an increse in u means enhancing growth on the balanced growth path.

2.5 The Futurity of Aging Economy

Using (17) , we obtain

Since the sign of denominator in (19) is negative and that of second term of the numerator is positive, the sign of (19) depends on that of 𝑞𝑞 ^𝑜𝑜 − 𝑞𝑞 ^𝑦𝑦 . Then we have two cases: 𝑞𝑞 ^𝑜𝑜 ≤ 𝑞𝑞 ^𝑦𝑦 and 𝑞𝑞 ^𝑜𝑜 > 𝑞𝑞 ^𝑦𝑦 .

2.5.1The Case A: 𝑞𝑞 ^𝑜𝑜 ≤ 𝑞𝑞 ^𝑦𝑦

The sign of (19) is strictly positive as the sign of the numerator is definitely negative.

We immediately obtain the following.

Result 1: Suppose that the childrearing cost by parents is equal to or greater than that by grandparents. Then a fall in fertility rate deters economic growth.

Contrary, it is possible that a fall in fertility rate fosters economic growth if the childrearing cost by parents is strictly less than that by grandparents, i.e., 𝑞𝑞 ^𝑜𝑜 > 𝑞𝑞 ^𝑦𝑦 . Though the sign of (19) is indefinite, but we could postulate the following result by focusing the second term of the numerator: the range of 𝑢𝑢 and the properties of 𝐻𝐻 ^′ (𝑢𝑢) . Result 2: Suppose that the childrearing cost by parents is less than that by grandparents.

Then a fall in fertility rate is likely to foster economic growth as time devoted to

working becomes fewer and vice versa.

𝑑𝑑𝑢𝑢 𝑑𝑑𝑛𝑛 =

𝑞𝑞 ^𝑜𝑜 − 𝑞𝑞 ^𝑦𝑦

(1 − 𝑞𝑞 ^𝑜𝑜 𝑛𝑛) ² 𝛽𝛽𝑅𝑅

1 − 𝛼𝛼 − 𝛽𝛽 − (1 − 𝑢𝑢)𝐻𝐻 ^′ (𝑢𝑢)

𝑛𝑛[(1 − 𝑢𝑢 _𝑡𝑡 )𝐻𝐻 ^′′ (𝑢𝑢) − 𝐻𝐻 ^′ (𝑢𝑢)] (19)

The intuition for these outcomes is as follows. A decrease in the number of children n leads to a rise in the ratio of the old worker to the young worker,

t t

N N _{− 1}

for any period t on the balanced growth path. A rise in the old-young ratio means a rise in relative scarcity of the young worker in the labor market. The scarcity leads to a rise in the young worker’s wage rate that also stands for the opportunity cost of human capital accumulation. Moreover, a fall in the number of children means directly a decrease in the child-rearing cost implying a rise in the young workers’ net wage rate, (1 − 𝑞𝑞 ^𝑦𝑦 𝑛𝑛)𝑤𝑤 _𝑡𝑡 ^𝑦𝑦 ℎ _𝑡𝑡 ^𝑦𝑦 . This also increases the opportunity cost of human capital accumulation. In contrast，such a decrease in the child-rearing cost means a rise in the old workers’ net wage rate, (1 − 𝑞𝑞 ^𝑜𝑜 𝑛𝑛)𝑤𝑤 _𝑡𝑡+1 ^𝑜𝑜 ℎ _𝑡𝑡+1 ^𝑜𝑜 , which decreases the opportunity cost of human capital accumulation. Summing up the above, whether the change in incentive to accumulate human capital fosters economic growth or not is related to ^{(1−𝑞𝑞 𝑦𝑦 𝑛𝑛)}

(1−𝑞𝑞 ^𝑜𝑜 𝑛𝑛) 𝑤𝑤 _𝑡𝑡 ^𝑦𝑦

𝑤𝑤 _𝑡𝑡+1 ^𝑜𝑜 as shown in (7):

the degree of the gap between two childrearing costs 𝑞𝑞 ^𝑜𝑜 − 𝑞𝑞 ^𝑦𝑦 matters to understand the mechanism behind the relationship between population growth and economic growth in this model.

4. CONCLUDING REMARKS AND COMMENTS

We have explored the effect of population aging on the rate of economic

growth in an endogenously growing small open economy with three-period-lived

non-altruistic individuals, where the accumulation of human capital is the engine of

economic growth. We have shown that a fall in fertility rate deters economic growth

when the childrearing cost by parents is equal to or greater than that by grandparents. In

contrast, it is possible that a fall in fertility rate fosters economic growth if the

childrearing cost by parents is strictly less than that by grandparents. We can consider

endogenous fertility choice that might be a straightforward extension of the model

presented in this paper. We also postulate that the formulation of child-rearing cost must

be careful in the sense that child-rearing cost itself could change opportunity cost of

human capital accumulation. It is interesting to examine other types of child-rearing

cost. Introduction of public pension is also interesting. All of these issues are worthy of

further investigation.

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