The Society for Economic Studies
The University of Kitakyushu
Working Paper Series No.2012-1(accepted in July 6,2012)1

Can Child-Care Support Policies Halt Decreasing Fertility?∗
Masaya Yasuoka†
July 6, 2012

Abstract
Some earlier papers examine whether child allowances can raise fertility or not in an endogenous
fertility model with a defined contribution pension system. They derive that a child allowance canraise fertility. This paper, which sets an endogenous fertility model with a defined benefit pension
system, is aimed at deriving the level of child allowances or education subsidies to make the pension
system sustainable. A child allowance can raise fertility instantaneously. However, in the long run,fertility might continue decreasing and the pension system might not be sustainable if less childallowance is provided. In a defined benefit system, tax burdens for pension benefits are heavy inan aging society with fewer children. A heavy tax burden reduces the household income and thendecreases fertility. Therefore, child allowances must be provided to halt decreasing fertility in thelong run. Nevertheless, given parametric conditions, education subsidy of more than a certain level
can not halt the decrease of fertility in the long run.

Keywords: Child allowances, Education subsidy, Deﬁned beneﬁt, Endogenous fertility
JEL Classifications: G23 H55 J13∗ I would like to thank Makoto Hirazawa, Koji Kitaura, Tomoya Sakagami, Yurika Shiozu and seminar participants atthe 2012 Spring Meeting of Japan Association for Applied Economics for their helpful comments and suggestions. However,any remaining errors are mine. Research for this paper was ﬁnancially supported by a Grant-in-Aid for Scientiﬁc Research(No.23730283).
† 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

This paper presents examination of whether child care support policies can raise fertility and education
investment for children or not in an endogenous fertility model with a pay-as-you-go pension (unfundedpension), which is ﬁxed beneﬁt (deﬁned beneﬁt (DB)). A pay-as-you-go pension reduces fertility because
older people do not need a gift from their children, as reported by Zhang (1995), Zhang and Zhang (1998),
Wigger (1999), Oshio and Yasuoka (2009), and others. Child-care support policies to halt the decreasein fertility must be provided to sustain the pension system in an aging society with fewer children. Oshio(2001), van Groezen, Leers and Meijdam (2003), van Groezen and Meijdam (2008), and Mochida (2009)derived that child allowances can raise fertility in a deﬁned contribution pension (deﬁned contribution(DC)). However, fertility can not always be raised by child allowances in a closed economy because of thedecrease in income per capita, as presented by Fanti and Gori (2009).1
Child allowances are a subsidy that is given based on the number of children. We can consider an
education subsidy as another child care support policy which is a subsidy for the quality of children.
Zhang (1997) and Zhang and Casagrande (1998) examined the eﬀects of child care policies on fertility
and human capital accumulation and derived that child allowances can increase fertility, but decreased
human capital of children and education subsidies can increase children human capital but decrease
fertility through substitution between the quality and quantity of children.2 However, an education
subsidy does not always reduce fertility. Yasuoka and Miyake (2008) derived that an education subsidy
raises both human capital accumulation and fertility in a model with a pay-as-you-go pension. Fanti andGori (2010, 2011) considered a model with public education and derived a result in which an increase inpublic education raises fertility.3
Oshio (2001), van Groezen, Leers and Meijdam (2003), Yasuoka (2006), van Groezen and Meijdam(2008), Mochida (2009), and Yasuoka and Goto (2011) examine the eﬀects of child allowances on fertility
in a model with a DC pension. Some earlier papers present examination of the eﬀect of child allowanceson fertility. Oshio and Yasuoka (2009) set an endogenous fertility with a DB pension and derived the1 Some earlier papers have described that an increase in income per capita decreases fertility. Galor and Weil (1996) set
a model in which the child care time to have children is necessary and derived that an increase in opportunity cost to stop
working caused by an increase in income per capita reduces fertility.
2 de la Croix and Doepke (2003) set a model with quality and quantity of children. An increase in human capital
accumulation reduces fertility because of an increase in opportunity costs of having children.
3 Public education is examined by Glomm and Ravikumar (1992) and others.

3level of child allowances to sustain the DB pension. Child allowances can raise fertility in the short run.
However, if fertility does not increase to any great degree, then child allowances can not stop decreasing
fertility because of a tax burden to provide pension beneﬁts. This is not derived in the model with a DC
pension.4
This paper presents consideration of an endogenous fertility model with DB pension in a small open
economy and aims to derive the level of child allowances examined by Oshio and Yasuoka (2009) and thelevel of an education subsidy to sustain a DB pension. The conclusions presented in this paper are shownas follows. Under some parametric conditions, two steady state equilibria exist: one for low fertility
and the other for high fertility. Depending on an initial condition, fertility converges to zero or a highfertility level. However, without a steady state equilibrium, fertility continues decreasing to zero. Child
allowances and education subsidies are necessary to prevent continued decreasing. As shown in the resultspresented in this paper, more than a certain level of child allowances can halt the decrease in fertility andcan sustain a DB pension. However, a certain level of education subsidy can not always bring about asteady state with constant fertility to sustain a DB pension. Given some parametric conditions, fertility
converges to zero in the long run because of the great amount of the education subsidy.
In Japan, the ratio of older people to the total population is 23.1%. The total fertility rate was
1.39 in 2010.5 The contribution rate for pensions is increasing, creating a heavy burden for households.
Pension reform in 2004 in Japan was aimed at regulating pension beneﬁts by the government to avoid
imposing a heavy burden on younger people (Macroeconomic Slide System). However, reforms havestill not been conducted. Current circumstances are not sustainable because pension beneﬁts are notadequately ﬁnanced by payments from younger people if this burden per household is heavy. Therefore,
it is necessary that an increase in the future working population and income per capita be made to
decrease the tax burden of households and to create a pension system that is sustainable. Child-care
support policies must be provided.6 This paper presents examination of the level of child allowances and
4 Lin and Tian (2003) describe an endogenous fertility model with a DB pension and examined whether tax reformﬁnanced by consumption tax can raise social welfare or not. Borgmann (2005) considered an exogenous fertility model withuncertainty and whether social welfare in DB pension is greater than that in a DC pension.
5 Data: Demographic Statistics (Ministry of Health, Labour and Welfare, Japan), Population Estimates (Ministry ofInternal Aﬀairs and Communications, Japan), White Paper on Birthrate-Declining Society (Cabinet Oﬃce, Japan)
6 In Japan, child-care support policies are insuﬃciently provided. Fiscal support for families, which is measured bygovernment expenditure for child care to GDP, is 0.79% at 2007 (Data: Cabinet Oﬃce, Government of Japan (2011)). Thislevel is less than that in France (3.0%), Sweden (3.35%), and other European countries. The ratio of public education
expenditure to GDP is 3.3% in 2007, which is less than that of the OECD average (4.8%) (Data: OECD (2010) Education
at a Glance 2010).

4

education subsidies to make the pension system sustainable. This analysis is important.
This paper consists of the following. Section 2 sets an endogenous fertility model with the DB pension
model. Section 3 derives the equilibrium and examines the eﬀects of child care support policies on fertility
and human capital accumulation. Section 4 concludes with a presentation of the results obtained in thispaper.

2

Model

This model economy consists of a two-period (young and old) overlapping generations model and assumesa small open economy. An interest rate 1 + r and wage rate w is given exogenously. In the following
subsection, we explain each agent.

2.1

Households

Each household lives in three periods–childhood, young, and old periods–and supplies labor to earn an
income during the young period. Young people supply labor inelastically for consumption during theyoung period and use savings to pay for consumption during the old period in addition to caring forchildren and providing education for their children. A government provides not only a pension systemthat gives older people a ﬁxed beneﬁt but also a child allowance and education subsidy for younger people.
Consequently, a household’s lifetime budget constraint is given as7

c1t +

c2t+1
p
+ (z − q)nt + (1 − i)et nt = (1 − τt − θt )wht +
.
1+r
1+r

(1)

Therein, nt and et respectively represent the number of children and the quality of children (education
investment). Necessary goods to bring up a child are represented as z. Both q and i denote child allowances (subsidy for the quantity of children) and education subsidy (subsidy for the quality of children).
In addition, c1t and c2t+1 respectively denote consumption during the young period and that during theold period. ht denote the human capital stock. Younger people face labor income taxation (tax rate orcontribution rate τt for pension beneﬁt and a tax rate θt for child allowances and education subsidies).
Older people receive pension beneﬁt p. Furthermore, t signiﬁes the period. A household’s utility function
7 Becker and Barro (1988) and Barro and Becker (1989) assume a utility function reﬂecting parents’ concern about theirchildren’s welfare. However, Eckstein and Wolpin (1985) assume a utility function that reﬂects parents’ concerns when
having children.

5

is assumed as 8

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

(2)

Human capital in the subsequent period ht+1 is accumulated by education investment et and the parents’human capital ht based on the following equation.9
ht+1 = Aeϵt h1−ϵ
, 0 < A, 0 < ϵ < 1.
t

(3)

Under budget constraints (1) and (3), allocations of c1t , c2t+1 , nt , and et to maximize utility (3) shownas()
p
c1t = α (1 − τt − θt )wt ht +
,
1+r
(
)
p
c2t+1 = (1 + r)β (1 − τt − θt )wt ht +
,
1+r
(
)
p
(1 − ϵ)(1 − α − β) (1 − τt − θt )wt ht + 1+r
nt =
,
z−q
ϵ z−q
et =
.
1−ϵ 1−i

(4)
(5)

(6)
(7)

Child allowance q raises fertility nt and education subsidy i pulls up education investment et .

2.2

Government

The government executes two policies: one for the pension and one for child care support policies (childallowances and education subsidy). A payroll tax rate θt is levied on young people. This tax revenue
is used as child allowances and education subsidy. With a balanced budget, we obtain the following
equation.
qnt + iet nt = θt wht(8)

Moreover, the government collects payroll tax revenue at tax rate τt from younger people to give a pension
beneﬁt for older people. We consider a deﬁned beneﬁt (DB) by which pension beneﬁt p is ﬁxed and tax
rate τt is adjusted to hold the balanced budget. Then, the government budget constraint is shown as
τt =

p
.
nt−1 wht

8 This utility function contains n and htt+1 , as assumed by de la Croix and Doepke (2003) and others.
9 Zhang (1997) and Zhang and Casagrande (1998) assume the same human capital accumulation equation.

6(9)

The tax burden per capita τt becomes low if fertility nt−1 or human capital ht is large. Therefore, fertility
or human capital raised by child care support policies contributes to a decrease in the tax burden.

3

Equilibrium

In this section, we consider equilibria of two types: one for the equilibrium with child allowances and one
for that with an education subsidy. First, we consider the equilibrium with child allowances.

3.1

Child Allowances

Considering i = 0, (8) reduces to qnt = θt wht . Substituting this equation and (9) into (6), we obtain thefollowing equation.
(1 − ϵ)(1 − α − β)nt =z − (1 − (1 − ϵ)(1 − α − β))q

(

p
p
wht +
−
1 + r nt−1)
(10)

By substituting (7) into (3), human capital in t + 1 period is shown as(
ht+1 = Aϵ(z − q)
1−ϵ

)ϵ
h1−ϵ
.
t

(11)

Deﬁning ∆nt ≡ nt+1 − nt and ∆ht ≡ ht+1 − ht , we obtain the following equations.
(1 − ϵ)(1 − α − β)z − (1 − (1 − ϵ)(1 − α − β))q()ϵ
ϵ(z − q)∆ht = A
h1−ϵ
− ht
t
1−ϵ

∆nt =

(

(
wA

ϵ(z − q)
1−ϵ

)ϵ
h1−ϵ
+
t

p
p
−
1 + r nt)
− nt ,(12)(13)

The loci of ∆nt = 0 and ∆ht = 0 are depicted in Fig. 1.10
[Insert Fig. 1 around here.]
Based on parametric conditions, the dynamics of two types of nt and ht are shown. Figure 1-1 shows
the case with two steady state equilibria: E1 and E2 . The former, E1 , is a saddle-point stable steadystate equilibrium; E2 is a sink. Depending on initial conditions h0 and n0 , the dynamics of nt and ht
converge to E2 which brings about higher fertility than that at E1 or E3 , which brings no fertility. Figure
1-2 shows the case of no steady state equilibrium. Then, ht and nt converge to E3 for any h0 and n0 .

10 See for Appendix for detail proof about the locus of ∆nt = 0.

7

Fertility n and the human capital stock h in the steady state are shown as follows if this economy has
two steady state equilibria E1 and E2 , as
(
(
)
)
√
1
(1 − ϵ)(1 − α − β)pwh +− Dq f or E1 , and2 z − (1 − (1 − ϵ)(1 − α − β))q
1+r
(
(
)
)
√
1
(1 − ϵ)(1 − α − β)
p
n=
wh +
+ Dq f or E2 ,2 z − (1 − (1 − ϵ)(1 − α − β))q
1+r

n=

(14)
(15)

1

h=
(
where Dq ≡A ϵ ϵ(z − q),1−ϵp(1−ϵ)(1−α−β)(wh+ 1+r)z−(1−(1−ϵ)(1−α−β))q(16))24(1−ϵ)(1−α−β)p− z−(1−(1−ϵ)(1−α−β))q. With Dq > 0, we obtain the two steadystate equilibria, depicted in Fig. 1-1. However, if Dq < 0, then no steady state exists, as depicted in Fig.
1-2. Now, we assume the economy with Dq < 0 at q = 0.11 That is, fertility converges to zero in thelong run. However, by virtue of child allowances, Dq changes from a negative value to a positive one andthen two steady state equilibrium might be brought about. Child allowances can bring about the steadystate equilibrium which does not continue decreasing fertility if Dq > 0. The child allowances which hold
Dq > 0 are depicted in Fig. 2.12
[Insert Fig. 2 around here.]If a greater than q ∗ child allowance is provided, then the two steady state equilibria are brought about even
if no steady state equilibrium existed before provision of child allowances. Child allowances can not bringz.
about the steady state equilibrium because nt given by (10) must not be negative if q ∗ > 1−(1−ϵ)(1−α−β)Then, we establish the following proposition.

Proposition 1

We assume no steady state economy without child allowances. Child allowances canzbring about the steady state and stop decreasing fertility if q > q ∗ , where q ∗ < 1−(1−ϵ)(1−α−β).13Proposition 1 shows that if child allowances that amount to more than a certain level are provided,
then the steady state equilibrium is brought about, as shown in Fig. 1-1. Eq. (10) shows that childallowances can raise fertility nt for any nt−1 . This result is the same as that reported by van Groezen,
)2
( 1
11 D

q < 0 at q = 0 is (1 − ϵ)(1 − α − β)pwA ϵ ϵz+ 1+r1−ϵ< 4pz.

12 D

q < 0 at q = 0. Therefore, q exists to hold Dq = 0 as shown in Fig. 2.
z
13 If the preference for children is small, that is 1 − α − β is small, thenis small. Therefore, q ∗ might be1−(1−ϵ)(1−α−β)zgreater than 1−(1−ϵ)(1−α−β) when 1 − α − β is small: it is diﬃcult to bring about the steady state equilibrium with childallowances.

8

Leers and Meijdam (2003). In the model with deﬁned contribution examined by van Groezen, Leers
and Meijdam (2003), child allowances can raise fertility both in the short term and in the long run witha DC pension. However, in the model with a DB pension examined in this paper, fertility might notincrease in the long run. Without a steady state equilibrium, even if child allowances can raise fertility
in the short run, no steady state equilibrium makes fertility continue decreasing. If the pension system
is managed as a DC pension, this result is not derived. Even if child allowances can raise fertility in the
short run, the tax burden is heavy when fertility is low. Then a household’s income decreases. A decreasein the household’s income reduces fertility because the household decreases the payment for child care.
Finally, fertility continues decreasing and converges to zero in the long run because of the heavy tax
burden. Therefore, the child allowances must be provided at more than a certain level because of the
cessation of a decrease in the long run. Fanti and Gori (2009, 2011) reported that child allowances can
not raise fertility with a DC pension in the long run. However, the reason diﬀers from that described inthis paper. Fanti and Gori (2009) explained that child allowances reduce the per-capita capital stock andthat household incomes decrease. Fanti and Gori (2010) explained that an increase in fertility created bychild allowances reduces the quality of public education. Then human capital accumulation is prevented.
Finally household incomes decrease. Substantially, proposition 1 in this paper is derived similarly by
Oshio and Yasuoka (2009).

3.2

Education Subsidy

Considering q = 0, (8) reduces to iet nt = θt wht . Substituting this equation and (9) into (6), we obtainthe following equation:

)
(
p
p
+ 1+r
(1 − ϵ)(1 − α − β) wht − nt−1
(
)
nt =
.
1 + ϵi(1−α−β)
z
1−i

(17)

By substituting (7) into (3), human capital in t + 1 period is shown as(
ht+1 = Aϵz(1 − ϵ)(1 − i)

)ϵ
h1−ϵ
.
t

Then, ∆nt and ∆ht are shown by the following equations.
)ϵ
)
(
(
p
p
ϵz
h1−ϵ
−
+
(1 − ϵ)(1 − α − β) wA (1−ϵ)(1−i)
t
nt−1
1+r
)
(
− nt ,∆nt =ϵi(1−α−β)
z
1+
1−i
(
)ϵ
ϵz
∆ht = Ah1−ϵ− ht .
t(1 − ϵ)(1 − i)

9

(18)

(19)

(20)

As shown by the case of child allowances, the loci of ∆nt = 0 and ∆ht = 0 are depicted in Fig. 1.
The fertility n and the human capital stock h in the steady state are shown as follows if this economy
has two steady state equilibria E1 and E2 .
()√(1 − ϵ)(1 − α − β)− Di()
f or E1 ,n=2z 1 + iϵ(1−α−β)
(1−i)
(
)
1
√
p
A ϵ ϵzw(1 − ϵ)(1 − α − β) (1−ϵ)(1−i)
+ 1+r
+ Di
(
)
n=
f or E2 ,2z 1 + iϵ(1−α−β)
(1−i)
1

p
A ϵ ϵzw(1−ϵ)(1−i) + 1+r

(21)

(22)

1

A ϵ ϵzh=,(1 − ϵ)(1 − i)[(where Di ≡ (1 − ϵ)(1 − α − β)

(23)
)]2

1

p
A ϵ ϵzw(1−ϵ)(1−i) + 1+r()− 4(1 − ϵ)(1 − α − β)pz 1 + iϵ(1−α−β). With(1−i)

Di > 0, we obtain the two steady state equilibria depicted in Fig. 1-1. However, if Di < 0, then no
steady state exists, as depicted in Fig. 1-2. Now, we assume the economy with Di < 0 at i = 0.14 Thatis, we consider the economy without education subsidy where fertility converges to zero in the long run.
The condition to have two steady state equilibria is shown as])2
4pz(1 − ϵ(1 − α − β)) 2
p
i
−
1+r
(1 − ϵ)(1 − α − β)
[(
]
)2
1
p
pA ϵ ϵzw
4pz(2 − ϵ(1 − α − β))
−2
+
−
i
1+r
(1 − ϵ)(1 + r)
(1 − ϵ)(1 − α − β)
( 1
)2
(
)2
1
p
2pA ϵ ϵzw
A ϵ ϵzw
4pz
+
+
+
−
> 0.
1+r
(1 − ϵ)(1 + r)1−ϵ(1 − ϵ)(1 − α − β)[(We deﬁne Y ≡

(

p
1+r

)2

(24)

− 4pz(1−ϵ(1−α−β))(1−ϵ)(1−α−β) . Then, if Y > 0, then the education subsidy i to be Di > 0

exists, as shown in Fig. 3-1.
[Insert Fig. 3 around here.]
Two steady state equilibria which stop decreasing fertility if an education subsidy is provided by morethan i∗ < i. If 1 < i∗ , then the education subsidy brings about no steady state equilibrium because imust be less than one. We deﬁne the following equation.
[(
Z≡4

p
1+r

)2

1

pA ϵ ϵzw
2pz(2 − ϵ(1 − α − β))+−(1 − ϵ)(1 + r)
(1 − ϵ)(1 − α − β)14 This condition is the same with D

q < 0 at q = 0.

10

]2


− 4Y 

(

p
1+r

)2

1
ϵ

+

2pA ϵzw+(1 − ϵ)(1 + r)(

We consider the case of Y < 0. If Z > 0 and

p
1+r

(

)2

1
ϵ

A ϵzw
1−ϵ



)2
−

4pz
.
(1 − ϵ)(1 − α − β)

(25)

1
ϵ

pA ϵzw+ (1−ϵ)(1+r)− 2pz(2−ϵ(1−α−β))< 0 hold, then an(1−ϵ)(1−α−β)education subsidy set as ilow < i < ihigh brings about two steady state equilibria, as shown in Fig. 3-2.15
[(
]
)2
1
2pz(2−ϵ(1−α−β))ppA ϵ ϵzw
However, if Z < 0 or Z > 0 and −2 1+r + (1−ϵ)(1+r) − (1−ϵ)(1−α−β)< 0, then the education
subsidy can not bring about the steady state. Consequently, the following proposition is established.

Proposition 2

We assume no steady state economy. For Y > 0, an education subsidy can bring about

the steady state and stop decreasing fertility if education subsidy i∗ < i is provided. in the case of Y < 0,
the education subsidy provided by ilow < i < ihigh can bring about two steady state equilibria as long as()21
ppA ϵ ϵzw
Z > 0 and 1+r+ (1−ϵ)(1+r)− 2pz(2−ϵ(1−α−β))(1−ϵ)(1−α−β) < 0 hold.

Being diﬀerent from the analysis of child allowances, the analysis of education subsidy presents someinteresting results. An education subsidy provided at more than i∗ can bring about the steady state
equilibrium if Y > 0. This case is the same as that of the result for child allowances. However, if Y < 0,then an education subsidy provided between ilow and ihigh can bring about the steady state equilibrium:a high amount of education subsidy can not bring about the steady state equilibrium because fertility
is decreased greatly by a high amount of education subsidy, as shown by (17). The sign of Y becomes
negative if the child care cost z is large: in the case of high child care cost, the education subsidy must not
be provided beyond the certain level and child allowances should be provided to bring about the steadystate equilibrium. Yasuoka and Miyake (2008) and Fanti and Gori (2010) derived that an education
subsidy raises fertility because of an increase in human capital (therefore, household income) in the longrun. However, an education subsidy higher than a certain level eliminates the steady state equilibrium
and fertility converges to zero in this model.

4Conclusions

This paper presents an examination of whether child care support policies can stop decreasing fertility
or not in the long run by considering an endogenous fertility model with a pay-as-you-go pension. Child
15 No education subsidy giving two steady state equilibria exists if 1 < ilow .

11allowances and education subsidy can raise fertility in the long run, as derived in earlier studies, if thepension system is a deﬁned contribution (DC) pension. However, if the pension system is a deﬁned
beneﬁt (DB) pension, although child allowances and education subsidy can raise fertility instantaneously,child allowances and education subsidies can not always halt decreasing fertility in the long run. In a DB
pension scheme, the tax burden for pension beneﬁts is too heavy in an aging society with fewer children.
Even if fertility is raised by child allowances in the short run, fertility continues decreasing in the longrun because child allowances and education subsidies can not halt a continued increase of the tax burden
per capita as long as pension beneﬁts are ﬁxed. Households reduce the number of children because of alack of disposable income. The result related to education subsidies is noteworthy. More than a certain
level of education subsidy can not halt decreasing fertility, because of the substitution between quality
and quantity of children, if child care cost is high.

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15

Appendix
Locus of ∆nt = 0
Considering ∆nt = 0 and diﬀerentiating of ∆nt = 0 at nt and ht , we obtain the following equation.
dht
=
dnt

(1−ϵ)(1−α−β)p1 − n12 z−(1−(1−ϵ)(1−α−β))q
t
(
)ϵ
2

(1−ϵ) (1−α−β)wA
z−(1−(1−ϵ)(1−α−β))q

ϵ(z−q)
1−ϵ

h−ϵ
t

.

(26)

We deﬁne the numerator of (26) as X(nt ). Therefore, we obtain
√
√
(
)(
)
1
(1 − ϵ)(1 − α − β)p(1 − ϵ)(1 − α − β)pX(nt ) = 2 nt +
nt −
.
nt
z − (1 − (1 − ϵ)(1 − α − β))qz − (1 − (1 − ϵ)(1 − α − β))q√
Therefore, if nt >√(1−ϵ)(1−α−β)pdht
z−(1−(1−ϵ)(1−α−β))q , then X(nt ) > 0, that is dnt(1−ϵ)(1−α−β)pz−(1−(1−ϵ)(1−α−β))q , then X(nt ) < 0.

16

(27)

dht
> 0. That is dn< 0 if nt <t∆nt = 0

ht

E3

h

E1

∆ht = 0E2

nt

0Fig. 1-1: Dynamics of nt .

hth

∆nt = 0E3∆ht = 0nt

0Fig. 1-2: Dynamics of nt (no steady state).

17

Dq

0

q

q∗

Fig. 2: Child allowance q to hold Dq > 0.

Di

i∗

0

i

Fig. 3-1: Education subsidy to hold Di > 0 (Y > 0).

18

Di

0

ilow

ihigh

i

Fig. 3-2: Education subsidy to hold Di > 0 (Y < 0).

19