The Society for Economic Studies
The University of Kitakyushu
Working Paper Series No.2012-5(accepted in December 5, 2012)1

Pension System Sustainability and Public Debt∗
Minoru Hayashida†Masaya Yasuoka‡
December 5, 2012

Abstract
This paper presents consideration of a pension system not as managed using a balanced budget
in each period but rather with a budget system in which issuance of public bonds is allowed. In an
endogenous fertility model, these analyses reveal how a child allowance aﬀects the stock of public debt
per capita and whether the government can maintain constant beneﬁt and constant contribution rates
or not while maintaining ﬁscal sustainability. Results obtained for a small open economy reveal thatchild allowances can always bring about a steady state with constant public debt per capita. However,child allowances can not always bring about a steady state in a closed economy. Moreover, this paper
presents examination of whether child allowances can reduce public debt per capita or not. In botha small open economy and a closed economy, child allowances are not always suﬃcient to reduce thepublic debt per capita. According to numerical examples given by the realistic parameter condition,a steady state equilibrium with public debt or public asset can be obtained; child allowances can
reduce the public debt per capita (or raise the public asset). However, that steady state equilibrium
is not locally stable. Other parameter conditions have a stable steady state in which child allowances
can raise public assets per capita.

Keywords: Pension, Public debt, Endogenous fertility, Child allowances
JEL Classifications:J13, H55, G23∗ This paper was presented at the 66th Annual Congress of the International Institute of Public Finance (IIPF 2010),
the 66th Annual Congress of Japanese Institute of Public Finance and 2012 Autumn Meeting of Japan Association forApplied Economics. We are grateful to Nobuo Akai, Jørgen Andersen, Katsufumi Fukuda, Mototsugu Fukushige, Hiroyuki
Hashimoto, Tomoya Ida, Hideki Nakamura, Tamotsu Nakamura, Kazumasa Oguro, Ryoji Ohdoi, Takashi Oshio, Pierre
Pestieau, Masaya Sakuragawa, and seminar participants for their useful comments. Any remaining errors are ours. Research
for this paper was ﬁnancially supported by Grants-in-Aid for Scientiﬁc Research (No.21730159, No. 23730283).
† Faculty of Economics and Business Administration, The University of Kitakyushu‡ Corresponding address: Faculty of Economics and Business Administration, The University of Kitakyushu, 4-2-1 Kitagata, Kokuraminami, Kitakyushu, Japan 802-8577, Tel.: +81-93-964-4318, E-mail: yasuoka@kitakyu-u.ac.jp2

1

Introduction

In Japan, the share of the population older than 65 years old among the total population, which reached20.1% in 2005 1 , has continued increasing. Nevertheless, total fertility in Japan has remained low.

2

Because of this rapidly aging society with fewer children, which decreases pension contributions concomitantly with the decrease in younger people, and because of the rapid increase in older people (withincreasing pension beneﬁts), the pension system in Japan was reformed in 2004. Those reforms increased
the contribution rate and national tax burden, with the intention of eventually ﬁxing the contribution
rate at some future date.

3

The government also applies a macroeconomic sliding scale by which it

examines economic and social factors (e.g., aging, fewer children) and controls beneﬁts. We regard the
public pension system in Japan, reformed as described above, as a Deﬁned Contribution (DC) system.
Recently, social security payments became the greatest ever recorded in recent national ﬁscal budgets
in Japan.4 Payments for social security include the national tax burden for pension beneﬁts, which
continue to increase. 5 The annual expenditure can not be ﬁnanced using tax revenues alone. Therefore,
the government issues public debt. The stock of public debt continues to increase; it has reached 600
trillion Japanese yen.6 The government can not collect tax revenue and contribute it to provision ofpension beneﬁts. Therefore, the government must issue public debt. However, even if the government
were allowed to continue its issuance of public bonds, the government would be compelled eventually toincrease tax revenues to avoid bankruptcy through an increased contribution rate, a decrease in pension
beneﬁts, or child support policies to halt the decrease in birth rates.
This paper describes a model incorporating children as consumption goods and examines a pension
system allowing the issuance of public bonds. Many analyses described in earlier papers have subsumed
a balanced budget pension. Because of a balanced budget, the government must change the beneﬁt

1 Data: ‘A National Census in Japan’
2 Although the total fertility rate in Japan decreased to 1.26 in 2005, the fertility rate increased slightly to 1.37 in 2008.
The total fertility rate in Japan remains less than that in either France or Sweden (Data: Ministry of Health, Labour andWelfare ‘Vital Statistics’).
3 Before reforming the pension system, the contribution rate of employees’ pensions was 13.58%. Payments to the national
pension were 13.3 thousand Japanese yen. Following the 2004 pension reform, payments will increase step-by-step. Finallyin 2017, the contribution rate will be 18.3% and the payment of the national pension will be 16.9 thousand Japanese yen.
4 Social security payments’ share in the original national ﬁscal budget in 2009 was 28%.
5 Social security expenditures increase every year (Data: Fiscal in Japan). In the National General Account for ﬁscal
year 2008 in Japan, social insurance accounts for 80.4% of social security expenditures.
6 According to the Ministry of Finance, Japan, the stock of public debt reached 581 trillion Japanese yen in ﬁscal year
2009.

3

or contribution rate to balance the budget. However, such changes might be not desired because of
unexpected beneﬁts or contributions or intergenerational conﬂict. Pension beneﬁts or contribution rates
need not be changed if the government can continue ﬁnancing its obligations using public debt.
Some earlier papers have presented examinations of pension systems. Pension analyses include twoissues: One for the diﬀerence between Deﬁned Contribution (DC) and Deﬁned Beneﬁt (DB) and the otherfor a relation between the pension system and endogenous fertility. Borgmann (2005) compared DC with
DB and ascertained which pension system is better. Diﬀerent from a Deﬁned Beneﬁt (DB) system, thepension beneﬁt in DC depends on the dependency ratio and economic growth, which includes uncertainty
because older people can not know the beneﬁt in advance. Yasuoka and Oshio (2008) examined DC andderived an optimal contribution rate given an uncertain beneﬁt. These models incorporate the assumptionof exogenous population growth.
Some papers have presented examination of the pension system in an endogenous fertility model.
Zhang and Zhang (2007) examined the optimal contribution rate for a pension in an endogenous fertility
and DC pension model. Oshio and Yasuoka (2009) set an endogenous fertility model that included a
DB pension. With an increase in the number of older people with fewer children, the contribution rateincreases. Therefore, fertility decreases because the contribution rate increases. Finally, fertility continues
decreasing over time. For that reason, such a pension system is not sustainable. Lin and Tian (2003) also
examined diﬀerent means to ﬁnance the pension beneﬁt in DB.7 These papers all present examinationsof balanced budget systems. Considering public debt, the government can ﬁx not only pension beneﬁts
but also pension contributions. The pension system with ﬁxed pension beneﬁts and contributions in a
public debt model are not examined. Therefore, our paper presents an examination of them.
Some earlier works have presented examinations of a pension system and a child allowance with an
endogenous fertility model that incorporates children as consumption goods. Oshio (2001), van Groezen,
Leers, and Meijdam (2003), and van Groezen and Meijdam (2008) showed that the fertility level thatis chosen by households is lower than the socially optimal fertility in a pay-as-you-go pension system.

7 However, it is noteworthy that Lin and Tian (2003) and Oshio and Yasuoka (2009) regarded children as investment
goods. We should examine the diﬀerence of motivation to have children in examining pension and fertility. As explainedby Nishimura and Zhang (1992) and Zhang and Zhang (1998), by virtue of pensions, parents limit their number of children
because a pension guarantees an income during their old period on behalf of their children. However, if children are regarded
as consumption goods, then an increase in income by pensions raises fertility (e.g., Oshio (2001), van Groezen, Leers, andMeijdam (2003)).

4

Moreover, they asserted, based on those results, it is important to provide child allowances to raise fertility.
8

With a balanced pension budget, an increase in fertility increases future pension beneﬁts. However,households do not consider this positive externality eﬀect. Therefore, a child allowance is necessary. In
fact, a child allowance has been adopted to stop the trend of fewer children in developed countries. Haltingthat trend toward fewer children pulls up future tax revenues through increased working population size.
Examining whether a child allowance can raise fertility and how a child allowance aﬀects the dynamics ofpublic debt is important. Our paper presents consideration of a child allowance ﬁnanced by public debt
without an increase in the tax burden and examines whether a child allowance can raise fertility or not
and whether public debt per capita is reduced or not because of an increase in population size.
Diamond (1965) considered public debt in an overlapping-generations model. Samuelson (1958) andAzariadis (1993) examined whether a ﬁscal policy that brings about ﬁscal deﬁcit is sustainable or not.
Sustainability depends on the primary ﬁscal deﬁcit and the gap separating interest rate and population
growth rate. Government expenditure in these models is regarded as public consumption. Similarly,
Chalk (2000) and Bräuninger (2005) considered whether public bond ﬁnancing is sustainable or not in
the model incorporating government consumption.9 Ono (2003) examined a pension system with public
debt. Ono (2003) ﬁxed the contribution rate and beneﬁt and issued public debt to ﬁnance the wedge
between contributions and beneﬁts in a closed economy. The dynamics of public debt and capital can beshown to depend on parametric conditions and the initial public debt. Meijdam, van de Ven, and Verbon(1996) examined the dynamics of public debt in a small open economy and derived the mechanism bywhich taxation aﬀects the dynamics of public debt. This paper presents consideration of public debt
introduced by Meijdam, van de Ven, and Verbon (1996) and Ono (2003). However, in addition to public
debt, we set an endogenous fertility model.
Our paper presents examination not only of a small open economy but also of a closed economy. Ina small open economy, even if the public debt continues to increase because of a primary deﬁcit, thestock of public debt converges to a constant level by virtue of the child allowance. The child allowanceraises fertility. Therefore, the size of the younger population is large: tax revenues increase and the
8 However, Fanti and Gori (2009), which does not incorporate pensions, showed that taxation for children raises fertility
because of the income eﬀect in a closed economy. Simultaneously, this result signiﬁes that a child allowance lowers fertility
because of decreased capital per capita, i.e. income.
9 Yakita (2008) investigated public capital formation ﬁnanced by public debt and examined the sustainability of public
debt.

5

increased population decreases the public debt per capita directly. Meijdam, van de Ven, and Verbon(1996) analyzed the dynamics of public debt in a small open economy. Meijdam, van de Ven, and Verbon(1996) showed that public debt per capita continues increasing because of a decrease in the tax burden.
However, analyses described herein show that an increase in government expenditure as a child allowancecan stop increasing and public debt per capita converges to a certain level.
The public debt decreases physical capital supplied to the product sector, which raises the interest
rate in a closed economy. This is a crowding-out eﬀect. An increase in public debt for a child allowanceraises the interest rate and enlarges the primary deﬁcit. Therefore, even if a child allowance raises tax
revenue and decreases public debt per capita directly, a child allowance can not always stop increasing
public debt because of an increase in the interest rate.
The numerical examples given by the realistic parameter conditions show a stable steady state equilibrium in which child allowances can reduce the public debt per capita (or raise the public asset). However,this steady state equilibrium is not stable. If other parameter conditions are given, then we obtain thestable steady state equilibrium with positive public assets, showing that child allowances can raise public
assets. Our paper describes the possibility that child allowances ﬁnanced by a public asset, that is pension
funds, can increase public assets. Ono (2003) examined how aging aﬀects the dynamics of public debt
per capita in exogenous fertility and a closed economy and showed that the eﬀects of aging on public
debt are ambiguous.10 Our paper is intended to examine whether fertility that is increased by a childallowance as a child-care policy halts the increase of public debt or not.
The remainder of this paper is the following. Section 2 of this paper establishes the model. Section 3
derives equilibrium in a small open economy and closed economy. Section 4 examines policy eﬀects suchas an increase in the child allowance in a closed economy with numerical examples. The ﬁnal section
presents results.

2

The ModelThis model economy consists of a two-period (young and old) overlapping generations model with three
agents: households, ﬁrms, and a government. In the following subsection, we explain each agent.

10 The eﬀect of an increase in the population growth rate is ambiguous, too.

6

2.1

Households

Each household lives in three periods—childhood, young, and old—and supplies labor to earn an income
during the young period. Young people supply labor inelastically for consumption during the young
period and use savings for consumption during the old period in addition to caring for children. A
government provides not only a pension system that gives older people a ﬁxed beneﬁt but also a childallowance for younger people. The budget constraint is given as

c1t +

pt+1
c2t+1
+ (zt − qt )nt = (1 − τ )wt +.
1 + rt+1
1 + rt+1(1)

Therein, qt denotes the child allowance. Furthermore, nt represents the number of children. Necessary
goods to bring up a child are represented as zt . In addition, c1t and c2t+1 respectively denote consumption
during young and old periods. Here, wt shows the wage rate; interest rate 1 + rt+1 is given for savings.
Younger people face income taxation (tax rate or contribution rate τ ). Older people receive pension
beneﬁt pt+1 . Furthermore, t signiﬁes the period. We assume that the child-care cost zt depends on wage
income such as zt = ẑwt (ẑ > 0).11 Moreover, the government provides a child allowance as qt = q̂wt(ẑ > q̂ > 0) and pension beneﬁt as pt+1 = x̂wt (x̂ > 0).12 A household’s utility function is assumed as

ut = α ln nt + β ln c1t + (1 − α − β) ln c2t+1 , 0 < α, β < 1, α + β < 1.

(2)

Under the budget constraint (1), the allocation of c1t , c2t+1 and nt to maximize their utility is shownas(
c1t = βwt (1 − τ ) +x̂
1 + rt+1

)

,
(
)
x̂
c2t+1 = (1 + rt+1 )(1 − α − β)wt (1 − τ ) +,1 + rt+1()
α (1 − τ ) + 1+rx̂t+1.
nt =ẑ − q̂

(3)
(4)

(5)

11 van Groezen, Leers, and Meijdam (2003), Fanti and Gori (2009) and Oshio (2001) also assumed the same child-carecost. van Groezen and Meijdam (2008) examined an economy with child-care cost z as a wage increasing function.
12 Zhang and Zhang (2007) explained that the assumed pension beneﬁt is practiced by many developed countries such as
France and Germany. x̂ denotes replacement rate.

7

2.2

Firms

A representative ﬁrm produces ﬁnal good Yt with constant returns to scale or a neoclassical productfunction, shown as

Yt = Ktθ Nt1−θ , 0 < θ < 1.

(6)

The ﬁrm inputs capital stock Kt and labor (population size of younger people) Nt . With a perfectlycompetitive market, the wage rate wt and the interest rate rt are
wt = (1 − θ)ktθ ,1 + rt = θktθ−1 ,

(7)
(8)

t
where kt ≡ K
Nt and the capital stock depreciates fully in a single period.

2.3

Government

The government executes two policies: one for the pension and the other for a child allowances. A payroll
tax rate τ , which we can regard as the contribution rate, is levied on younger people. Older people receive
pension beneﬁt pt . Assuming a balanced budget in each period, the government must change the tax
rate to balance the budget. However, allowing issuance of public debt, the government need not change
the tax rate in each period. The governmental budget is shown as13
bt+1 =

3

1 + rtx̂wt−1
τ wt
bt ++ q̂wt −.
nt
nt nt−1nt(9)Equilibrium

Before deriving the equilibrium in closed economy, we derive the equilibrium in a small open economy.

3.1

Small Open Economy

In a small open economy, the wage rate wt and interest rate 1 + rt are ﬁxed as ŵ and 1 + r̂. Then, thedynamics of the public debt stock per capita (9) can be shown asbt+1 =

1 + r̂
x̂ŵ (
τ)
bt + 2 + q̂ −
ŵ,
n̂
n̂
n̂

13 See Appendix for a detailed proof.

8

(10)

where n̂ =x̂()α(1−τ + 1+r̂)
. Even if 1+r is larger than n and the primary balance is in deﬁcit, x̂n̂ŵ2 + q̂ − n̂τ >ẑ−q̂

0, the public debt stock bt converges to b̂ because of child allowances as a result of ∂∂n̂q̂ , as shown in Fig.
1.
[Insert Fig. 1 around here.]If the public debt per capita converges to b̂, then the child allowance q̂ can decrease b̂ if the following
condition holds:
db̂
=
dq̂

(
)
(q̂− n̂x̂2 )ŵ−b̂n̂ ŵ +ẑ−q̂n̂ − 1 + r̂

< 0.

(11)

Meijdam, van de Ven, and Verbon (1996) showed that the public debt stock per bt continues increasing
because of a decrease in tax revenue. An increase in the child allowance increases the government
expenditure under constant tax revenue, thereby it is considered that bt does not converge to a certain
level. However, child allowances raise fertility n̂; then bt converges to a certain level even if government
expenditures are expanded.14
However, this result is derived in a small open economy, which is not considered private capital
accumulation. If the public debt stock increases in a closed economy, then the private capital stock might
decrease. This eﬀect decreases the wage rate and increases the interest rate. Therefore, child allowances
cannot halt the continual increase in the public debt per capita because tax revenue decreases and theinterest rate increases. In the next subsection, we derive the equilibrium and examine whether childallowances can stop continue increasing public debt per capita and whether child allowances can decrease
public debt per capita.

3.2

Closed Economy

The equilibrium in this model is speciﬁed by two dynamics equations: one for public debt per capita
bt and the other for capital stock per capita kt . The dynamics of bt can be given as (9). Considering
nt given by (5), we obtain the dynamics of bt as a function of kt . Deﬁning st as household saving, theequilibrium of capital market is given as bt+1 + kt+1 = nstt . Then, the dynamics of kt can be represented
14 Theoretically, the discussion presented in this paper shows that fertility increases as far as the child allowances increase.
However, we think that the fertility can not be raised more than the biological limit level. Therefore, if an interest rate isextremely high, then an increase in the fertility provided by child allowances does not contribute to convergence of public
debt per capita.

9

as

[
bt+1 + kt+1 =]1−τ(α + β)(ẑ − q̂)
−
wt .
nt
α

(12)

Therefore, the equilibrium in this economy is speciﬁed by (5), (7), (8), (9), and (10). This equation shows
the capital market equilibrium condition. Deﬁning k and b as k = kt+1 = kt and b = bt+1 = bt , k and b
in the steady state are determined according to the following equations:((α + β)(ẑ − q̂)
1−τ
−
k+b =
n
α
( x̂
)
w n + q̂n − τ.
b=n − (1 + r)

)
w,

(13)
(14)

Therein,w = (1 − θ)k θ ,1 + r = θk θ−1 ,[]x̂α (1 − τ ) + 1+rn=.
ẑ − q̂

(15)
(16)
(17)

Capital per capita k in the steady state is given such that the following condition holds:x̂+ q̂n − τk 1−θ1−τ(α + β)(ẑ − q̂)
=
−
− n
.
1−θ
n
α
n − (1 + r)

(18)

This paper presents an examination of the proposition that child allowances ﬁnanced by public debt candecrease the public debt stock in the long run because of the eﬀect of increase in tax revenue brought
about by younger people in the future period.
We examine how the respective dynamics of physical capital stock kt and public debt bt are determined.
db
Calculating dkdq̂ and dq̂ yields

dk
a22 b1 − b2=,dq̂
a11 a22 − a21db
a11 b2 − a21 b1=,
dq̂
a11 a22 − a21
where
k + b ∂w (1 − τ )w ∂n+,
a11 = 1 −n2∂k ) (( w ∂k)∂n ∂rn − (1 + r) ∂w
x̂w∂n
a21 = b
−
−
+
−
q̂
,
∂k
∂k
w
∂k
n2
∂k
a22 = n − (1 + r),

10

(19)
(20)

α+β
1 − τ ∂n−,αn2 ∂ q̂)(x̂w∂nb2 = wn + q̂ − 2 − b
,
n
∂ q̂
b1 =

θ−2

x̂αθ(1−θ)k∂rn
θ−1 ∂nand where ∂k= θ(θ − 1)k θ−2 , ∂w
, ∂ q̂ = ẑ−q̂and ∂n∂k = θ(1 − θ)k∂k = (ẑ−q̂)(1+r)2 . An increase inchild allowance q̂ directly raises public debt per capita b as shown by the ﬁrst term wn in b2 . However,an increase in the child allowance aﬀects public debt per capita b through the change of capital stockdb
is negative, thenper capita k, as shown by a ﬁrst term of the right-hand-side of (20). If the sign of dq̂

child allowances can reduce the public debt per capita because an increase in population size brings aboutdb
additional tax revenue. However, we can not show the sign of dq̂
and whether the steady state equilibrium

is locally stable or not. Therefore, given some parametric condition, our paper presents an examinationdbof whether the parameter set of dq̂
exists or not in the following section.

4

Numerical Examples

This section presents an examination, with numerical examples, of whether child allowances can reduce
public debt stock per capita or not. However, without parametric conditions, many cases show thedb. Therefore, we set some parameter conditions as follows. As reported by de la Croix
positive sign of dq̂

and Doepke (2003), the discount rate of the utility from the consumption during the older period is set
as about 0.3 because the quarterly discount rate is 0.99 and one period in the overlapping generations
model is regarded as 30 years and calculated as 0.99120 . Therefore, we consider the weight of utility fromconsumption in old period is 1 − α − β = 0.3, i.e., (i) α + β = 0.7. Second, we set the capital income
share (ii) θ = 0.3 as observed in Japan. Third, the replacement rate of pension beneﬁt in Japan is set
as about ﬁfty percent for the young generation’s wage income. Therefore, we set (iii) x = 0.5. Fourth,
we set the income tax rate as (iv) τ = 0.2. The contribution rate for pension rises to 18.3% in Japan.
Thereby, we set τ = 0.2.15 Fifth, we set the child-care cost ẑ as (v) ẑ = 0.08, which is nearly same as theparameter that de la Croix and Doepke (2003) set.16
15 Thepension reform at 2004 in Japan decided that the replacement rate is kept at about a half of
the working generation’s wage income and the contribution rate of pension is raised step by step in every year up to 18.3% and is ﬁxed at this contribution rate (Data: Ministry of Health, Labour and Welfare(http://www.mhlw.go.jp/topics/bukyoku/nenkin/nenkin/pdf/kaisei-h16.pdf).
16 de la Croix and Doepke (2003) consider child-care costs not as goods cost but as opportunity costs to stop working
because of child care and set ẑ = 0.075. As described by de la Croix and Doepke (2003), Haveman and Wolfe (1995) and
Knowles (1999) report ẑ = 0.15. Jinno and Uemura (2008) consider not only the goods costs but also opportunity cost to
have children, and set ẑ = 0.2.

11

db
Given these parametric conditions (i)–(v), the parameter set that brings about dq̂
< 0 is shown as

the following table.
[Insert Table 1 around here.]db
Table 1 shows that the parameter set exists that brings about dq̂
< 0 and then the steady statethat public debt per capita bt and private capital stock per capita kt converge to certain level b and k.
db
Theoretically, the parameter set to hold dq̂
< 0 exists. However, the fertility rate n and interest rate

1 + r are unrealistic values.
Therefore, we set a parameter set that is more realistic. First, the fertility rate in Japan at 2010 is1.39. Then, we set additional parameter conditions to obtain n = 0.7. Furthermore, a recent interest
rate is about 1% per year in Japan. One period is regarded as thirty years in the overlapping generations
model and the interest rate 1 + r = (1.01)30 = 1.35. Given ẑ = 0.08, τ = 0.2, x̂ = 0.5, and q̂ = 0, we
obtain α = 0.048. Then, we ﬁnd no steady state equilibrium under these parametric conditions, as shownin Fig. 2.17
[Insert Fig. 2 around here.][Insert Fig. 3 around here.]
Therefore, without a steady state equilibrium, the public debt stock bt might continue increasing or
decreasing, depending on the initial condition. This result is consistent with the fact that public debt inJapan is increasing.
Fig. 3-1 portrays two steady state equilibria (A and B) with parameter conditions (α = 0.2, β =db0.5, q̂ = 0.01, θ = 0.3, τ = 0.2, ẑ = 0.08). This steady state equilibrium (point A) holds dq̂
< 0, i.e.,

an increase in child allowances can reduce the public debt stock per capita. However, the public debt
stock is negative, i.e., the government has assets. In the steady state equilibrium, even if the primarybalance is in deﬁcit nx̂ + q̂n − τ > 0, the government can continue having assets by virtue of the highinterest rate. Moreover, an increase in child allowances can reduce the public debt per capita (raise thepublic asset per capita). An increase in the child allowance increases government expenditures directly,
17 Given k = 0.15 and b = 0 as an initial condition, the private capital per capita k and public debt per capita b are
0
0
1
1
given by k1 = 1.5 and b1 = −1.788 in the next period. Child allowances q̂ = 0.01 render these variables as k1 = 1.533 andb1 = −1.7866, which means that child allowances prevent accumulation of public assets. Given k0 = 0.1 and b0 = 0, thepublic debt per capita increases greatly. We can not derive the positive value of private capital stock per capita.

12

which magniﬁes the primary balance deﬁcit. However, an increase in tax revenues occurs because of theincreased population size.
Given the parameter condition (α = 0.3, β = 0.4, q̂ = 0.01, θ = 0.3, τ = 0.2, ẑ = 0.08), we can depict
dbFig. 3-2, and an increase in child allowances can reduce the public debt stock per capita, dq̂
< 0 brings

about a steady state equilibrium (point A). Point A in Fig. 3-2 shows the steady state equilibriumwith 1 + r > n. Then, the primary balance is surplus nx̂ + q̂n − τ < 0. Although a high interest rateincreases public debt per capita, the public debt per capita stays at a constant level because of the
primary balance surplus. Moreover, the child allowance can reduce the public debt per capita even if thechild allowance decreases the primary balance surplus directly. However, an increase in population size
raises tax revenues; ﬁnally constant public debt per capita is maintained.
In developed countries, it might be not meaningful to examine the case of b < 0 because the public
debt stock is positive. However, considering b as a pension fund, the case of b < 0 is worth examining.
However, these results are obtained in a locally unstable steady state. Therefore, it is apparentdb< 0 exists, given by thethat no steady state equilibrium exists or no stable steady state that holds dq̂

realistic parameter condition. However, given a parameter condition that is small, diﬀerent from realisticdb
parameter, we obtain dq̂
< 0 and b < 0 at a locally stable steady state, as shown in Table 2.

[Insert Table 2 around here.](i) and (ii) in Table 2 are applied to the parameter condition α + β = 0.7 and ẑ = 0.08. However, thereplacement rate x and τ diﬀer from the parameter conditions in Table 1. With small α, the stable steadystate equilibrium is brought about by controlling the replacement rate x and contribution rate τ . These
stable steady states are given by point B in Fig. 3-1. Furthermore, (iii) and (iv) in Table 2 have thestable steady state equilibrium given by other parameter conditions that do not hold in the condition of
α + β = 0.7. If β is low, implying a high household savings rate, then the stable steady state equilibriumis shown as point B in Fig. 3-2.
Yasuoka and Miyake (2012) theoretically derive that child allowances can not decrease the public debt
per capita in the AK model that marginal productivity of capital is constant. In Yasuoka and Miyake(2012), child allowances always raise the public debt per capita (or reduce the public assets). Diﬀerent
results described in this paper and those reported by Yasuoka and Miyake (2012) are attributable to the13assumption of a production function.

5

Concluding Remarks

This paper describes an endogenous fertility model with a pay-as-you-go pension model including public
debt and examines how public debt is aﬀected by the income tax rate, the pension beneﬁt rate, and achild allowance. Given a balanced budget in each period, which does not allow public bond issuance,
the government must set the income tax rate or pension beneﬁt to balance the budget in response to
an increased number of older people. However, being able to issue public debt, even if younger people
become fewer or older people become more numerous, the government need not change the income tax
rate and pension beneﬁt to balance the budget as long as the pension is sustainable. Such a pension
system is desired because it prevents circumstances under which a certain young generation suﬀers froma heavy burden or that a certain older generation suﬀers from low pension beneﬁts because of a balanced
budget in each period. Therefore, it is worth examination to ascertain whether such a pension system issustainable or not.
This paper presented derivation of results for both a small open economy and a closed economy. In asmall open economy, child allowances can halt the increase of the public debt stock per capita. In addition,
given some parameter conditions, child allowances can reduce the public debt per capita in the steadystate. However, in a closed economy, child allowances can not always stop the increase of the public debt
stock per capita, as derived by numerical examples given by realistic parameter condition. Furthermore,this numerical example derives the steady state equilibrium by which child allowances can reduce thepublic debt per capita (or raise the public asset (pension fund)). However, given a realistic parameter,
no steady state equilibrium exists or no stable steady state equilibrium exists. If the parameter condition
that is not consistent with the realistic condition is given, then we have a stable steady state in which
the government has negative debt, i.e., public asset and child allowances can reduce public debt (or raise
public assets).

14

References[1] Azariadis C. (1993), Intertemporal Macroeconomics, Blackwell, Oxford.
[2] Borgmann C. (2005), “Social Security, Demographics, and Risk,” Springer, Berlin.
[3] Bräuninger M. (2005), “The Budget Deﬁcit, Public Debt, and Endogenous Growth,” Journal of Public
Economic Theory, Vol.7, pp.827-840.
[4] Chalk N.A. (2000), “The Sustainability of Bond-Financed Deﬁcits: An Overlapping Generations
Approach,” Journal of Monetary Economics, Vol.45, pp.293-328.
[5] de la Croix D. and Doeple M. (2003), “Inequality and Growth: Why Diﬀerential Fertility Matters,”
American Economic Review, Vol.93-4, pp.1091-1113.
[6] Diamond P. (1965), “National Debt in a Neoclassical Growth Model,” American Economic Review,
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[7] Fanti L. and Gori L. (2009), “Population and Neoclassical Economic Growth: a New Child Policy
Perspective,” Economics Letters, Vol.104, pp.27-30.
[8] Futagami K., Iwaisako T. and Ohdoi R. (2008), “Debt Policy Rule, Productive Government Spending,and Multiple Growth Paths,” Macroeconomic Dynamics, Vol.12, pp.445-462.
[9] Groezen B., Leers, van T. and Meijdam L. (2003),“Social Security and Endogenous Fertility: Pension
and Child Allowances as Siamese Twins,” Journal of Public Economics, Vol.87, pp.233-251.
[10] Groezen B., Leers, van T and Meijdam L. (2008), “Growing Old and Staying Young: Population
Policy in an Ageing Closed Economy,” Journal of Population Economics, Vol.21, pp.573-588.
[11] Haveman R. and Wolfe B. (1995), “The Determinants of Children’s Attainments: A Review of
Methods and Findings,” Journal of Economic Literature, Vol.33-4, pp.1829-1878.
[12] Ihori T. (1996) Public Economic Theory (in Japanese), Yuhikaku.
[13] Jinno M. and Uemura T. (2008), “The Management of Public Pension and Economic Eﬀects of Child
Allowances – The Comparison between Deﬁned Contribution and Deﬁned Beneﬁt in HeterogeneousHouseholds,” Fiscal Studies (in Japanese), Vol.4, pp.184-200.

15

[14] Knowles J. (1999), “Can Parental Decisions Explain U.S. Income Inequality?” mimeo, University ofPennsylvania.
[15] Lin S. and Tian X. (2003), “Population Growth and Social Security Financing,” Journal of Population Economics, Vol.16, pp.91-110.
[16] Meijdam L., van de Ven M. and Verbon H. (1996) “The Dynamics of Government Debt,” European
Journal of Political Economy, Vol.12-1, pp.67-90.
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Journal of Public Economics, Vol.48, pp.239-258.
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Economics, Vol.33, pp.549-577.
[19] Oshio T. (2001), “Child-Care Reform in Pension System and the Fertility Rate (in Japanese),”Quarterly of Social Security Research, Vol.36-4, pp.535-546.
[20] Oshio T. and Yasuoka M. (2009), “Maximum size of social security in a model of endogenous fertility,”
Economics Bulletin, Vol. 29-2, pp.656-666.
[21] Samuelson P.A. (1958), “An Exact Consumption-Loan Model of Interest with or without the Social
Contrivance of Money,” Journal of Political Economy, Vol.66, pp.467-482.
[22] Thøgersen O. (2003), “A Note on Intergenerational Risk Sharing and the Design of Pay-As-You-Go
Pension Programs,” Journal of Population Economics, Vol.11, pp.373-378.
[23] Wigger B.U. (2009), “A Note on Public Debt, Tax-Exempt Bonds, and Ponzi Games,” Journal of
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16

[26] Yasuoka M. and Oshio T. (2008), “The Optimal and Acceptable Sizes of Social Security under
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17

Appendix
Appendix A: Stable Condition in a Closed Economy
The capital per capita k and public debt per capita b in the steady state are given as (13) and (14).
Then, using (5), (7), (8), (9), and (12), we obtain the dynamics equation of bt as)(
1x̂(bt + kt )− τ wt + q̂wt .
bt+1 =(1 + rt )bt +nt(1 − α − β)(1 − τ ) − (α+β)x̂

(21)

1+rt

We deﬁne (12) as F (kt+1 , bt+1 , kt , bt ) = 0 and (21) as G(kt+1 , bt+1 , kt , bt ) = 0. Then, we derive thefollowing equations at the approximation of the steady state, as
(
) (
)(
)
kt+1 − kc11 c12kt − k=,bt+1 − bc21 c22bt − bwhere
c11 = −
c21 = −
C=

∂G ∂F
∂F ∂G
∂bt+1 ∂kt − ∂bt+1 ∂kt

C
∂F ∂G
∂F ∂G
∂kt+1 ∂kt − ∂kt ∂kt+1
C, c12 = −, c22 = −

∂G ∂F
∂F ∂G
∂bt+1 ∂bt − ∂bt+1 ∂bt

,
C
∂F ∂G
∂F ∂G
∂kt+1 ∂bt − ∂bt ∂kt+1
C

,

∂F ∂G
∂F ∂G
−
∂kt+1 ∂bt+1∂bt+1 ∂kt+1

or
c11 =

x̂
1+r
)
(
n + n2 1−τ − (α+β)(ẑ−q̂)
n

1−
(

α

x̂
∂r
αx̂((τ − n)w−(1+r)b) ∂k(n2 (ẑ−q̂)(1+r)2 1−

αx̂2 wb + n2 (ẑ−q̂)(1+r)
2

)

)
∂r

αx̂(1−τ )w
n2 (ẑ−q̂)(1+r)2 ∂k

1 ∂rn ∂k +(q̂ − nτ

c12 =

) ∂w

(

(

)
)(
)
(
(α+β)(ẑ−q̂)∂r
α 1−τ((τ − nx̂ )w−(1+r)b) ∂w
n −
α
∂k ∂k
) 1−
)2
(
α(1−τ )x̂w

αx̂(1−τ )w∂r
n2 (ẑ−q̂)(1+r)2 ∂k(α+β)(ẑ−q̂)
1−τ
n −
α

x̂ 1−

∂k + n21−

c21 = −

c22 =

,

(

αx̂((

x̂
τ− n

n2 (ẑ−q̂)(1+r)2()

n2 (ẑ−q̂)(1+r)2 1−)∂rw−(1+r)b ∂kαx̂(1−τ )w∂r
1− 2n (ẑ−q̂)(1+r)2 ∂kn2 (ẑ−q̂)(1+r)2

)

a11
,
αx̂(1−τ )w∂r
1 − n2 (ẑ−q̂)(1+r)2 ∂k(α+β)(ẑ−q̂)
1−τ
n −
α

)

∂w
∂k

)w
∂r
1 − n2αx̂(1−τ(ẑ−q̂)(1+r)2 ∂k+

c12 c21.
c11

We examine the following characteristic equation,
H(λ) = λ2 − (c11 + c22 )λ + (c11 c22 − c12 c21 ),(22)Here, λ denotes the eigenvalues. The two real eigenvalues are given by the condition (c11 + c22 )2 −4(c11 c22 − c12 c21 ). If the following condition holds, the steady state equilibrium is obtained.
H(0) = c11 c22 − c12 c21 < 0,

18

(23)

,

H(1) = 1 − (c11 + c22 ) + c11 c22 − c12 c21 > 0,(24)

H(−1) = 1 + (c11 + c22 ) + c11 c22 − c12 c21 > 0.

(25)

19

bt+1

bt+1 = btbtb̂
Fig. 1: Dynamics of bt in a Small Open Economy.

+,+,+,+,0
+,0
+,2
+,2

!
+,.
+,.
+,.
+,2
+,2
+,0
+,0

"
+,+'
+,++,+0+,+'+,++,+'+,+-#$%#"/+,++01.
/+,+4-.5/+,20'35/+,+4.3/+,'0+2/+,+45/+,'.011

db
Table 1: Parameter Set to Hold dq̂
<0

20

&
+,+'.2'+,+'044+,+'0.0+,++1'+,++442+,++.02+,++..''().,.3425
3,+-04.1
3,+510.3
1,400031
5,+'.43.
'',35+4.
'',204+1*-,.2-0+3-,520025
0,.-45.4
0,340503
2,-44-52,1'.1-.,3-2410

0.1
0.05
0.05
0.1

0
0.007
0.014
0.021
0.028
0.035
0.042
0.049
0.056
0.063
0.07
0.077
0.084
0.091
0.098
0.105
0.112
0.119
0.126
0.133
0.14
0.147

0

b 0.150.2
0.25

Equ.130.3

Equ.14

0.35
0.4

k

Fig. 2: No Steady State Equilibrium.

0.02
0.01

A

0.01

0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
0.024
0.026
0.028
0.03
0.032
0.034
0.036
0.038
0.04

0

b 0.02

B

0.03
0.04
0.05

Equ.13Equ.140.06

kFig. 3-1: Two Steady State Equilibrium (Negative Public Debt).

21

0.02
0.01

A
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
0.024
0.026
0.028
0.03
0.032
0.034
0.036
0.038
0.04

0
k

b
0.01

Equ.130.02

BEqu.14

0.03

Fig. 3-2: Two Steady State Equilibrium (Positive Public Debt).

.
..
...
.:

/0*
/0*
/0*
/0*

!
/01
/01
/0*
/0*

"
/0/2
/0/2
/0/2
/0/2

#
/0/3
/0/6
/0/4
/0/4

$
/04
/07
/07
/09

%
/03
/06
/04
/08

&'(&#
5*03667
5/036*7
5*04*41
5/07**2db
Table 2: Parameter Set to Hold dq̂
< 0 in a Stable Steady State.

22

)
*+,
/0/8*332 304/*4/9 *01*/942/0/69/89 309/6414 *0223*26/0/13472 30/29261 30667621/0/14389 30/49*3 30642/63