The Society for Economic Studies
The University of Kitakyushu
Working Paper Series No.2012-6(accepted in December 5, 2012)1

How Is the Child Allowance to be Financed? By Income Tax orConsumption Tax?∗
Masaya Yasuoka† Naohisa Goto‡
December 5, 2012

Abstract
In economically developed countries, family support policies are provided by the government toraise fertility. Because of an increase in the dependency ratio in many aging societies, governments
must pay ever-increasing pension beneﬁts. Therefore, the government must somehow increase thenumber of younger people to make the pension system sustainable. This paper presents an examination of whether child allowances can raise fertility or not. This issue has been analyzed in numerous
earlier studies. However, this paper presents examination of the means used to ﬁnance such a childallowance: income taxes and consumption taxes. The diﬀerent means of taxation exert substantiallydiﬀerent eﬀects on fertility.
The results presented in this paper are as follows. First, a child allowance ﬁnanced by an income
tax can not always raise fertility. However, such an allowance ﬁnanced by a consumption tax canalways raise fertility. Second, this paper presents an examination of an optimal tax policy to maximizesocial welfare. An optimal child allowance and an optimal income transfer from younger people toolder people diﬀer according to whether they are ﬁnanced by an income tax or by a consumption tax.

JEL Classification:D10, H55, J13, J14, J18, J26
Keywords: Child allowances, Endogenous fertility, Pay-as-you-go pension, Optimal taxation∗ Research for this paper was supported ﬁnancially by a Grant-in-Aid for Scientiﬁc Research (No.23730283) and the
University of Kitakyushu. The authors are thankful to Mototsugu Fukushige, Hiroyuki Hashimoto, Kei Hosoya, Kenichiro
Ikeshita, Wataru Kobayashi, Shigeki Kunieda, Motohiro Sato, Toshiyuki Uemura and seminar participants of the public
ﬁnance seminar held at Hitotsubashi University, Konan University and 2012 Autumn Meeting of Japan Association forApplied Economics. Nevertheless, any remaining errors are the authors’ responsibility.
† Corresponding to: Faculty of Economics and Business Administration, The University of Kitakyushu, 4-2-1 Kitagata,
Kokura-minami, Kitakyushu, Fukuoka 802-8577 Japan, Tel.:+81-93-964-4318, E-mail: yasuoka@kitakyu-u.ac.jp‡ Faculty of Economics and Business Administration, The University of Kitakyushu

21

Introduction

Some OECD countries are experiencing an aging society with low birth rates, as described by Sleebos(2003). The total fertility rate in Japan in 2010 was 1.39. Italy and Germany also suﬀered a decrease infertility. In contrast, France and Sweden provide extensive ﬁscal support for families with policies thathave apparently halted the decrease in fertility. Figure 1 shows the correlation between the total fertility
rate and ﬁscal support for families in some OECD countries.
[Insert Fig. 1 around here.]
As Fig. 1 shows, increased ﬁscal support for families brings about higher fertility than that in countries
that do not support families actively.1 Therefore, it is necessary that the government give a child allowance
actively to parents, as is done in France and Sweden, and thereby bear child-care costs publicly andvisibly: Japan must increase expenditures for family policies actively to raise the fertility rate in Japan.
Macdonald (2006) surveys some earlier papers that describe examinations of family policy. Some earlier
papers describe that ﬁnancial incentives can be eﬀective for increasing fertility (Lutz (1999), Milligan(2002), Laroque and Salanie (2005), Lyssiotou (2012)). In earlier studies, Zhang (1997), Oshio (2001),
van Groezen et al. (2003), Yasuoka (2006), van Groezen and Meijdam (2008), Yasuoka and Goto (2011)
show that child allowances can raise fertility because of a decrease in the net cost of caring for children.
However, Fanti and Gori (2009) show that the child allowance reduces fertility because it prevents capital
accumulation by decreasing saving by younger people and by temporarily increasing population growth.2
However, these earlier studies do not suﬃciently examine how a child allowance is ﬁnanced. Manyof these studies consider an income tax or lump-sum tax as the appropriate means to ﬁnance childallowances. As another tool to ﬁnance child allowances, we consider a consumption tax. A few studies
have been undertaken to examine the eﬀects of a child allowance. Yasuoka and Goto (2011) examine
whether a child allowance can raise fertility or not if a consumption tax is used as a tool for ﬁnance in asmall open economy. Yasuoka (2006) examines whether a child allowance ﬁnanced by a consumption tax
1 The child allowance level in France is higher than that in Japan.

In Japan, the child allowance is given for childrenof junior high school age. In France, the child allowance is given for children under 20 years old, except for the ﬁrst-born
child. The amount of the child allowance is 10,000 yen per month in Japan (10,000 yen per month for third-born child of
elementary school.). The amount in France is 19,000 yen for the second child (25,000 yen for the third child). In addition
to this allowance, children who are older than 11 years old are given a larger allowance (Data: Cabinet Oﬃce, Governmentof Japan (2007):“White Paper on Birthrate-Declining Society”).
2 Originally, Fanti and Gori (2009) report that taxes for children (negative child allowance) can raise fertility. This result
is the same as that obtained when the positive child allowance reduces fertility.

3

can raise fertility or not with numerical examples obtained for a closed economy. Our paper analyticallyand theoretically examines the eﬀect of a child allowance ﬁnanced in a closed economy.3
Figure 2 shows the national burden rate in some OECD countries.
[Insert Fig. 2 around here.]
The national burden in European countries is about 50% − −60%. In contrast, the national burden in
Japan is less than 40%. Therefore, Japan can not provide suﬃcient ﬁscal support for families, as Fig. 1
shows. In Japan, the share of older people who are more than 65 years old to the total population is at
23.0% in 2010.4 and it is expected that the aging society will be progressing steadily. Therefore, it will
be increasingly diﬃcult to provide a child allowance ﬁnanced by income tax because the share of younger
people to total population will decrease.
Why must the government restore declining fertility? It must do so because the government is obligated to manage the social security system. Social security systems such as those of health insurance
and pension systems must be supported by younger generations. Unless the population of the young
generation is suﬃciently large, the social security system can not be maintained. Concretely, the government must collect a contribution from younger people to run the pay-as-you-go pension system. If
the population of the younger generation is small, then the pension that older people receive under aconstant contribution rate would be small, too. The burden of the younger people would become quite
large if the government were to ﬁx the pension paid to older people. Therefore, unless the population
is suﬃciently large, the government can not aﬀord to pay pensions to older people suﬃciently and then,can not maintain the system.
[Insert Fig. 3 around here.]
Figure 3 shows government social spending and social spending for people of advanced age. In European
countries, public social expenditures are larger than for people in Japan. However, regarding old age
social expenditure, that in Japan is nearly same as that in European countries: the share of old age
social expenditures to government social spending in Japan is greater than that in European countries.
3 In a closed economy, we consider capital accumulation, which aﬀects household income, the wage rate, and the interest
rate. These changes also aﬀect fertility. The eﬀects of an income tax and consumption tax on capital accumulation diﬀer.
In a small open economy, there is substantially no diﬀerence between the two tax systems because we do not consider capital
accumulation.
4 Data: A 2012 Declining Birthrate White Paper (2012)4

Actually, we must consider the following question. How is optimal social spending for both younger
people and older people determined? For a small open economy, van Groezen et al. (2003) examine
the optimal child allowance. Yasuoka and Goto (2011) describe how fertility is determined under asmall open economy that has adopted a pay-as-you-go pension system. Moreover, they describe how thechild allowance aﬀects fertility and the welfare level, how the child allowance is ﬁnanced, and how bothincreased fertility by the child allowance and the direct income transfer to older people aﬀect the welfareof older people. Endogenous fertility with a pay-as-you-go pension engenders market failure. Oshio(2001) and van Groezen and Meijdam (2008) set a closed economy model and examine the optimal childallowance. Diﬀerent from our manuscript is the assumption of income taxation alone. Our paper presents
consideration not only of income tax but also of a consumption tax. Yasuoka (2006) examines how thechild allowance is ﬁnanced: with a labor income tax, a capital income tax, and a consumption tax, andderives the result that the child allowance should not be ﬁnanced by a labor income tax because of alarge decrease in social welfare. However, Yasuoka (2006) does not calculate the optimal child allowance,
the income transfer from younger people to older people.5
The aims of our paper are the following. First, we present an examination of the eﬀect of a childallowance ﬁnanced by income tax or consumption tax on fertility. Second, our analyses derive an optimalchild allowance and an optimal income transfer from younger people to older people. We set an endogenous
fertility model with a child allowance and pay-as-you-go pension system in a closed economy. The
endogenous fertility models presented by Nishimura and Zhang (1992), Zhang and Zhang (1998) andOshio and Yasuoka (2009) incorporate children as investment goods.6 These papers show derivations
demonstrating that public pensions decrease fertility because the parent does not need children to carefor their life when they are old. However, if children are regarded as consumption goods, then public
pensions might increase fertility, as shown in the model by van Groezen et al. (2003) and others.
The results presented in theoretical papers collectively imply a positive correlation between fertility
and income (van Groezen et al. (2003) and Fanti and Gori (2009) and others). Galor and Weil (1996)
5 Some earlier studies examine the pension system to improve social welfare.

Hirazawa and Yakita (2009) report theirresults, showing that the tax cut improves social welfare in endogenous fertility model in pay-as-you-go pension. Lin andTian (2003) examine whether consumption taxes raise social welfare to increase pension beneﬁts or not in the model withendogenous fertility model considering children as investment goods. However, pension beneﬁts ﬁnanced by consumption
taxes are ambiguous.
6 Becker and Barro (1988) and Barro and Becker (1989) set the dynamic general equilibrium model with endogenous
fertility. Based on Becker and Barro (1988) and Barro and Becker (1989), many papers present examinations of factors
determining fertility.

5

show a negative correlation between fertility and wage income. In contrast, Ahn and Mira (2002) show
a positive correlation between fertility and the female labor participation rate in OECD countries. Apps
and Rees (2004), Martı́nez and Iza (2004) and Yasuoka and Miyake (2011) consider the child-care sector
and derived a theoretical model that is consistent with the positive correlation. This positive correlationengenders a positive correlation between fertility and income. Galor and Weil (1996) consider a highopportunity cost for child care. Opportunity costs for child-care costs determine the correlation between
fertility and income. In fact, van Groezen et al. (2003) and others imply no large opportunity cost andtherefore positive correlation between fertility and income. The analyses described in this paper assume
an endogenous fertility model, implying a positive correlation.
The results derived from our analyses and presented in this manuscript are as follows. First, a childallowance ﬁnanced by a consumption tax can always raise fertility, although a child allowance ﬁnanced byincome tax will not always raise fertility. Second, based on how the child allowance is ﬁnanced, an optimal
income transfer from younger people to older people and an optimal child allowance diﬀer between a childallowance ﬁnanced by an income tax and one ﬁnanced by a consumption tax.
The remainder of this paper is presented as follows. Section 2 introduces the model. Section 3 presents
a description of the equilibrium. Section 4 presents derivation of the ﬁrst best solution, which maximizessocial welfare and optimal tax rate and optimal child allowance to achieve the ﬁrst-best solution. The
ﬁnal section presents conclusions of this study.

2

The Model

The model economy is based on a two-period (young and old) overlapping generations model. Thiseconomy has agents of three types: households, ﬁrms, and a government.

2.1

Households

Households experience two periods: young and old. During the young period, each household supplies
labor to earn labor income. Households have one unit of time, which is assumed to be supplied for labor
inelasticity. Younger people consume and raise children. These analyses assume that it is necessary forhouseholds (parents) to invest in child care to have children. During the older period, each household
only consumes. The government imposes a payroll tax to provide income transfer to older people. In

6addition, the government imposes a payroll tax and consumption tax to provide a child allowance. Eachhousehold distributes its labor income among child-care goods and other consumption. Consequently, we
obtain the following budget constraint.
(1 + τct )c1t +(1 + τct+1 )c2t+1
pt+1+ (z − q)nt = (1 − θ − τt )wt +.
1 + rt+1
1 + rt+1(1)

Therein, τct denotes the consumption tax rate in t period (0 < τct < 1). For these analyses, we assume
that the government does not impose a consumption tax for child-care goods, and therefore does not
aﬀect fertility negatively.7 Younger people face a payroll tax by tax rate in t period τt (0 < τt < 1) toprovide a child allowance q and θ (0 < θ < 1) to provide income transfer to older people. The pension
received by older people is pt+1 . In addition, wt and rt+1 respectively represent the wage rate in t period
and interest rate in t + 1 period. Furthermore, z denotes the child-care cost per child. Parents are givena child allowance q for a child, which is assumed as z > q. In addition, c1t , c2t+1 and nt respectively
represent consumption by younger people, older people, and the number of children.
A household’s utility function ut is given as follows:8
ut = α ln c1t + β ln c2t+1 + (1 − α − β) ln nt , 0 < α, β < 1, α + β < 1.

(2)

An individual chooses consumption during young and old life c1t , c2t+1 and chooses the number of children
nt to maximize lifetime utility (2) subject to the lifetime budget constraint (1). The optimal allocations
are determined as
(
)
α
pt+1
(1 − τt − θ)wt +,1 + τct
1 + rt+1()β(1 + rt+1 )
pt+1
c2t+1 =
(1 − τt − θ)wt +,1 + τct+1
1 + rt+1
(
)
pt+1
1−α−β
(1 − τt − θ)wt +
.
nt =
z−q
1 + rt+1
c1t =

2.2

(3)
(4)
(5)

Firms

The production function of ﬁnal goods is given as a neoclassical constant-returns-to-scale function,
Yt = AKtϵ Nt1−ϵ , 0 < ϵ < 1, 0 < A,

(6)

7 In economically developed countries with a consumption tax rate for necessities, child-care service is zero or discounted

by the government. Therefore, our analyses assume a zero tax rate for child-care cost per child z.
8 This assumption is conventional in the modeling of endogenous fertility: Eckstein and Wolpin (1985), Galor and Weil
(1996) and van Groezen et al. (2003), and others.

7where Yt and Kt respectively represent the ﬁnal goods and capital stock. Moreover, Nt denotes labor
input and younger population size. Each ﬁrm determines the demand for capital stock and labor to
maximize proﬁt. Assuming perfect competition, the wage rate wt and the interest rate rt are
wt = A(1 − ϵ)ktϵ ,1 + rt = Aϵktϵ−1 ,

(7)
(8)

t
where kt ≡ K
Nt . The capital stock is assumed to be fully depreciated in a single period.

2.3

Government

The government imposes labor income taxation at a tax rate τt and consumption at a tax rate τct toprovide a child allowance q. In addition, the government imposes labor income taxation at a tax rate θto provide beneﬁts for older people pt+1 . The budget of the child allowance and pension are assumed to
be separate. The analyses described in this paper assume a balanced budget in each period.9 Then, thegovernment budget constraint for pension is presented as Nt+1 θwt = Nt pt+1 , i.e.,
pt+1 = θwt nt ,(9)

= nt . Therein, Nt+1 denotes the population size of younger people in t + 1 period.
because of NNt+1
t
Therefore, in the t+1 period, the population size of older people is Nt . The government budget constraint
for a child allowance is presented as Nt τt wt + τct (Nt c1t + Nt−1 c2t ) = Nt qnt , that is,
)
(
c2t
.
qnt = τt wt + τct c1t +
nt−1

3

(10)

Equilibrium and Policy Eﬀect

In this section, we consider two equilibria: one for the equilibrium with a child allowance ﬁnanced byincome tax and the other for the equilibrium with a child allowance ﬁnanced by the consumption tax.

3.1

Equilibrium with Income Tax

First, we derive the equilibrium in which the child allowance is ﬁnanced by income taxation. Consideringτct = 0, Eq. (10) can be transformed to the following equation
qnt = τt wt .
9 Many earlier papers assume a balanced budget that is similar to that described in this paper (e.g.

(2003)). Ono (2003) considers public debt and alleviated the assumption of balanced budget.

8(11)
van Groezen et al.

In that equation, τt is determined to hold. Considering Eqs. (5), (9), and (11), fertility nt is shown asnt =(1 − α − β)(1 − θ)wt
t+1
z − (α + β)q − (1−α−β)θw
1+rt+1

.

(12)

t+1
Therein, z − (α + β)q − (1−α−β)θw
must be positive because nt > 0. If this model economy is given as
1+rt+1a small open economy with wage rate wt , wt+1 and an interest rate rt+1 are ﬁxed, then a child allowance
can always raise fertility nt , as shown by Eq. (12), which is obtained by van Groezen et al. (2003)
and Yasuoka and Goto (2011). Our paper is intended to examine the eﬀect of a child allowance in aclosed economy for which capital accumulation is considered. The dynamics of capital per capita kt isrepresented by the capital market equilibrium condition Kt+1 = Nt st , where st denotes an individual’ssaving. Dividing this equation by Nt , thenkt+1 =

st
.
nt

(13)

Substituting an individual saving st = (1 − τt − θ)wt − (z − q)nt − c1t , Eqs.(3), (5), (9), and (11) into
(13), we obtain the dynamics equation of kt .
kt+1 =(1 − β)zαq(1 − θ)wt
−
+
nt
1−α−β
1−α−β

(14)

We consider the steady state equilibrium which holds kt+1 = kt . We deﬁne n and k respectively as
fertility and the capital per capita in the steady state. Considering Eqs. (7), (8), (12), and (14), we
obtain n and k, which hold the following two equations:(1 − α − β)(1 − θ)A(1 − ϵ)k ϵ,z − (α + β)q − θ(1−ϵ)(1−α−β)kϵ(1 − θ)(1 − ϵ)Ak ϵ(1 − β)z
αq
k=
−
+
.
n
1−α−β
1−α−β

n=

(15)
(16)

Now, we examine whether a child allowance ﬁnanced by an income tax can raise fertility or not. Diﬀerentiating n and k about q at the approximation of q = 0, we obtain the following equations as
dn
dk
+ a12
= (α + β)n,
dq
dq
dn
dk
α
a21
+ a22
=
dq
dq
1−α−β
a11

where
a11 = z −θ(1 − ϵ)(1 − α − β)
k,
ϵ

9

(17)
(18)

(
)
θ
ϵ−1
a12 = −(1 − α − β)(1 − ϵ) (1 − θ)Aϵk
+ n ,
ϵ
kϵ
a21 = (1 − θ)A(1 − ϵ) 2 ,nk ϵ−1
a22 = 1 − ϵ(1 − θ)A(1 − ϵ).
ndndq is shown as follows,αa22 (α + β)n − a12 1−α−βdn=dq
a a − a12 a21( 11 22)n2ϵ (α + β)ϵ + α(1 − ϵ)θ − βϵ − (1 − β)ϵ(ϵ + θ(1 − ϵ)).
=a11 a22 − a12 a21(19)

Considering child allowance q = 0, Eqs. (12) and (14), then the dynamics equation of kt+1 are given as
kt+1 =

ϵβz.
(1 − α − β)(ϵ + θ(1 − ϵ))(20)

The sign of the denominator of (19) is positive.10 Therefore, if the sign of the numerator of (19) is
positive, then the sign of dn
dq is positive, so a child allowance can raise fertility. This condition is shownas the followingβ>(ϵ + θ(1 − ϵ))(ϵ − α).
ϵ(1 + θ(1 − ϵ))(21)Then, as shown by Eq. (21), the following proposition is established.

Proposition 1

A child allowance ﬁnanced by an income tax can raise fertility in the steady state if

Eq. (21) is held.

This proposition always holds if ϵ < α. With small α + β, then 1 − α − β becomes large, indicating ahigh preference for having children. An increase in the child allowance can greatly raise fertility, as shownby (5). We infer that the larger 1 − α − β is, the greater the eﬀect of increasing fertility is. However, an
increase in fertility and tax burden prevent capital accumulation per capita. This reduces a wage rate,i.e., a child allowance has the eﬀect of decreasing fertility via a decrease in household’s income. As longas Eq. (21) holds, which means that α (to be more than ϵ) or β is large (to hold Eq. (21) even if ϵ > α),i.e., the preference for having children is not large, the latter eﬀect is small, then a child allowance canraise fertility. Based on parametric conditions, Fanti and Gori (2009) derived that the tax for children
10 See for Appendix for a detailed proof.

10(negative child allowance) can raise fertility. Substantially, the result in this paper is the same as that in
Fanti and Gori (2009). We can ﬁnd the result in the model given also by Oshio (2001), Yasuoka (2006)
and van Groezen and Meijdam (2008).11

3.2

Equilibrium with Consumption Tax

Second, we derive the equilibrium in which a child allowance is ﬁnanced by consumption taxation. Considering τt = 0, Eq. (10) changes to the following equation,()
c2tqnt = τct c1t +
.
nt−1

(22)

Inserting Eqs. (3), (4), and (5) into (22), we obtain
qnt =

τct
z−q
(αnt + 1 + rt ) .
1 + τct 1 − α − β(23)Then, fertility nt is given asnt =A(1 − α − β)(1 − ϵ)(1 − θ)ktϵz − q − (1−α−β)(1−ϵ)θ
kt+1
ϵ

.

(24)

The dynamics of capital per capita kt is given by Eq. (13). Substituting an individual saving st ≡(1 − θ)wt − (z − q)nt − (1 + τc )c1t , Eqs. (3), (5), into (13), we obtain the dynamics equation of kt .
kt+1 =(1 − θ)wt(1 − β)(z − q)
−
nt
1−α−β

(25)

The capital stock per capita k and fertility n in the steady state are given as the following equations:n=(1 − α − β)(1 − θ)A(1 − ϵ)k ϵ,z − q − (1−α−β)(1−ϵ)θ
kϵ(1 − θ)A(1 − ϵ)k ϵ(1 − β)(z − q)
k=
−
.
n
1−α−β

(26)
(27)

Now, we examine whether a child allowance ﬁnanced by an consumption tax can raise fertility or not.
Diﬀerentiating n and k for q at the approximation of q = 0, we obtain the following equations:
dn
dk
+ a12
= n,
dq
dq
dn
dk
1−β
a21
+ a22
=
.
dq
dq
1−α−β

a11

(28)
(29)

11 As shown by (32), aggregate consumption and child care in the steady state is shown by c + c2 +zn. The maximization1nϵof this aggregate consumption reduces to 1 + r = n, that is, β = 1−ϵ+ θ. Then, substituting this equation into (21), we((1−θ)(1−ϵ)−1)ϵobtain α >, which is always held. That is, given 1 + r > n, an increase in child allowance ﬁnanced by1−ϵincome tax can raise the fertility.

11

12
Here, dn
dq is shown as follows,
1−βna22 − 1−α−β
a12
dn
=
dq
a11 a22 − a12 a21()
n (1 − ϵ)(1 − (1 − β)θ) + (1−β)(1−ϵ)θ
ϵ=> 0.
a11 a22 − a12 a21(30)

As long as the denominator a11 a22 − a12 a21 > 0, which is derived by a locally stable condition, the signof dn
dq is always positive. A child allowance can always raise fertility. Then, the following proposition isestablished as demonstrated by the proof above.13

Proposition 2 A child allowance ﬁnanced by a consumption tax can always raise fertility.

Apart from the case of an income tax, if a child allowance is ﬁnanced by a consumption tax, thenfertility always increases for the following reason. First, the consumption tax does not aﬀect fertility.
Therefore, under constant capital stock per capita, child allowance raises fertility. Second, consumption
tax aﬀects not only fertility but also the capital stock per capita k. An income tax reduces the household’s
disposable income directly and indirectly via prevention of capital accumulation because of decreasedsavings. This eﬀect decreases fertility. However, this consumption tax eﬀect is small even if an increasein population size prevents capital accumulation per capita. Therefore, if the government seeks to raise
fertility always, then the child allowance is ﬁnanced not by an income tax but by a consumption tax. This
proposition is also obtained by Yasuoka and Goto (2011). However, Yasuoka and Goto (2011) derived this
proposition in a small open economy, which capital accumulation does not consider. Consumption taxes
aﬀect not only fertility but also the capital stock per capita, which changes household income and whichaﬀects fertility indirectly. Even if this eﬀect is included, the child allowance ﬁnanced by a consumption
tax can always raise fertility, as derived in this paper.
However, a child allowance should be provided to maximize not fertility but social welfare. In the next
section, we examine the optimal child allowance ﬁnanced by the income tax and an optimal consumption
tax rate to bring about maximization of social welfare.
12 See Appendix for a detailed proof.
13 Even if the government levies consumption tax not only consumption but also child care z, proposition 2 is established.

The budget constraint (1) changes to (1 + τct )c1t +

(

1−β

)

pt+1
(1+τct+1 )c2t+1+ (z − q)nt = (1 − θ − τt )wt + 1+r. Then, we obtain
1+rt+1t+1

na22 − 1−α−β a12z(αn+1+r)dn> 0.
= 1−α−βz(αn+1+r)+zn
dq
1−α−β

12

4

Optimal Taxation and Child Allowance

This section explains policy eﬀects. The model brings about an externality through a pay-as-you-go
pension system, as described by van Groezen et al. (2003), van Groezen and Meijdam (2008) andYasuoka and Goto (2011). Because of the externality, we show that the allocations in a decentralized
economy diﬀer from those of a command optimum. The social welfare function Wt is deﬁned as14
Wt =

∞
∑

ρs−t (α ln c1s−1 + β ln c2s + (1 − α − β) ln ns−1 ) ,

(31)

s=t

where 0 < ρ < 1 signiﬁes the social discount factor. In a closed economy, the resource constraint percapita is shown as
Aksϵ = c1s + zns +c2s+ ns ks+1 .
ns−1

(32)

We derive the optimal allocation to maximize the social welfare subject to a resource constraint. Deﬁningc∗1s , c∗2s , and n∗s respectively as the optimal consumption of younger people, that of older people, and the
optimal fertility, we derive optimal allocations subject to (32) and given c1t−1 , c2t−1 , nt−1 and kt asfollows:15
c∗1t+1ρ(1 + rt+1 )
=
,
∗
c1t
n∗t
c∗2t+1β(1 + rt+1 )
=
,
∗
c1t
α
α(z + kt+1 )
c∗1t
=
.
n∗t
1−α

(33)
(34)
(35)

β(1+rt+1 )Considering allocations (3) and (4) in a decentralized economy, we derive c2t+1: The saving
c1t =α

allocation to maximize social welfare is consistent with that in a decentralized economy. In light of(32)–(35), we derive these allocations in the steady state as reduced forms asαρA ∗ϵk ,ρ+ββρ(1 + r∗ )A ∗ϵ
c∗2 =
k ,
ρ+β

c∗1 =

n∗ = ρ(1 + r∗ ).

(36)
(37)
(38)

14 As shown in van Groezen et al. (2003), if the social welfare function is deﬁned not by the Millian criterion but by the∑∞ s−t ∏s
Benthamite welfare criterion given as Wt =ρnu, then the command optimum does not coincide
s=t
i=t s−2 s−1
with Pareto-eﬃcient allocations. Therefore, we use the Millian criterion as van Groezen et al. (2003) use it.
15 See Appendix for a detailed proof.

13

Therein, k ∗ denotes the capital stock per capita to maximize social welfare in the steady state and1 + r∗ = Aϵk ∗ϵ−1 . Inserting these equations, Eqs. (36)–(38) into (32) in the steady state, we obtain k ∗ ask∗ =zϵ(ρ + β).
1 − α − ϵ(ρ + β)(39)

The sign of 1 − α − ϵ(ρ + β) must be positive to hold positive k ∗ . In the following subsection, we show an
optimal child allowance ﬁnanced by income tax or consumption tax and income transfer from younger
people to older people.

4.1

Income Taxation

First, we derive a child allowance q i∗ ﬁnanced by income tax τ (τc = 0) and the transfer for older people θi∗α(z−q)to accommodate Eqs. (36)–(38) in the steady state. Eqs. (3) and (5) provide c1 = 1−α−βn. Substituting

Eqs. (36), (38), and (39) into this equation, we obtain an optimal child allowance ﬁnanced by income
tax q i∗ asq i∗ =(β − ϵ(ρ + β))z.
1 − α − ϵ(ρ + β)(40)Considering Eqs. (3), (7), (9), (11), (36), and (39), we obtain an optimal income transfer θi∗ as follows
θi∗ =(1 − (1 − ρ)ϵ)(ρ + β) − ρ(1 + β).
(1 − ρ)(1 − ϵ)(ρ + β)(41)Then, the following proposition is established.

Proposition 3

Child allowance q i∗ and income transfer θi∗ should be given as (40) and (41), respectively

to achieve social welfare maximization in the steady state if the government provides a child allowanceﬁnanced by an income tax.

An optimal child allowance is not always positive, as shown by van Groezen and Meijdam (2008).
Our paper presents examination of an optimal child allowance in closed economy, but this result is theβsame with Yasuoka and Goto (2011). The child allowance should be provided if the inequality ϵ < ρ+βis held. However, when this inequality is not held, the child allowance should not be provided but the
government imposes a child tax. An income transfer θi∗ from younger people to older people should beβ. Otherwise, the income transfer from older people to younger people
provided, if the inequality ϵ < ρ+βshould be adopted. This proposition was obtained already by van Groezen and Meijdam (2008), who

14examine a closed economy. However, our manuscript presents examination of a child allowance ﬁnanced
not only by an income tax but also by a consumption tax. In the following subsection, we examine thecase of a consumption tax.

4.2

Consumption Taxation

We deﬁne q c∗ as a child allowance ﬁnanced by consumption tax τc (τ = 0) and θc∗ as the transfer for
older people in this case to holds Eqs. (36)–(38) in the steady state. By Eqs. (3) and (5), we obtainn. Substituting Eqs. (36) and (38) into this equation, we obtain
c1 = (1+τα(z−q)
c )(1−α−β)k∗ϵ(z − q c∗ )=.
ρ+β
(1 + τc )(1 − α − β)(42)Substituting Eqs. (36)–(38) and 1 + r∗ = Aϵk ∗ϵ−1 into (23) in the steady state, we obtain an optimal
consumption tax rate asτc =q c∗ ρϵ(ρ + β) 1.
αρ + β k ∗(43)

An optimal child allowance ﬁnanced by consumption tax q c∗ is given by Eqs. (39), (42) and (43) as shownbyq c∗ = ((β − ϵ(ρ + β))z).
(1 − α − ϵ(ρ + β))(44)

1 + ρ(1−α−β)αρ+βIn the steady state, inserting Eqs. (3) into (7), (22), and (36)-(38), we obtain an optimal income transfer
θc∗ .
θc∗ =

(
)
(1 − ϵ)(ρ + β) − ρ 1 + ρ(β−ϵ(ρ+β))(1−β)ρ+β(1 − ϵ)(1 − ρ)(ρ + β)

.

(45)

Then, the following proposition is established.

Proposition 4Child allowance and income transfer should be given by (44) and (45), respectively, tomaximize social welfare in the steady state if the government provides a child allowance ﬁnanced using a
consumption tax.

This proposition shows that child allowance ﬁnanced by consumption tax can also bring about social
welfare maximizing allocations. Similarly to the case of income taxation, based on parametric condition,β
the government should provide child allowance or impose a child tax. As is true also with q c∗ , if ϵ < ρ+β,a child allowance is expected to be provided unless the government imposes a tax for children. Moreover,15

βan income transfer θi∗ is also provided from younger people to older people if ϵ < ρ+β. Then, the proof

shown above establishes the following proposition.

Proposition 5βWhichever the kind of taxation, if ϵ < ρ+β, then the government should provide a childallowance and income transfer from younger people to older people to maximize social welfare in thesteady state.

The child allowance and income transfer directly aﬀect not only fertility and consumption but also
the capital stock per capita. The change of capital stock alters fertility and consumption indirectly. Withβϵ < ρ+β, a direct eﬀect dominates the indirect eﬀect; then the direct eﬀects of child allowance and income

transfer can bring about socially optimal allocations. Alternatively, an indirect eﬀect created by a childtax and income transfer from older people to younger people changes the capital stock. It can bringabout socially optimal allocations.
Finally, we compare (40) with (44), which reveals that the absolute value of (40) is less than that of(44): |q i∗ | > |q c∗ |. Then, we obtain the following proposition.

Proposition 6

Child allowance ﬁnanced by consumption tax is smaller than that by income tax if thegovernment is expected to provide a child allowance to maximize social welfare in the steady state.

Why is this proposition established? Optimal fertility is given as shown in (38). The child allowance
increases fertility and decreases capital stock per capita because of an increase in the population size.
However, the consumption tax reduces capital stock to less than the case of an income tax. An income
tax reduces household’s saving and then capital stock decreases. Income tax reduces the fertility andthe fertility diverges largely from optimal fertility. Therefore, so a large amount of child allowance must
be provided if the child allowance is ﬁnanced by income tax. This proposition is not obtained by van
Groezen et al. (2003), van Groezen and Meijdam (2008) and Yasuoka and Goto (2011). This proposition
is derived originally herein.

16

5Conclusions and Remarks

This paper presented an examination, using an endogenous fertility model, of how a child allowancefor younger people is to be collected. Concretely, this paper presented a description of how the childallowance aﬀects fertility. In addition, this paper examined the optimal tax policy to maximize social
welfare. The analyses described herein yielded the following results.
First, the child allowance ﬁnanced by a child allowance can not always raise fertility as shown by
earlier studies. However, a child allowance ﬁnanced by a consumption tax can always raise fertility inspite of a closed economy. Therefore, we can ﬁnd that the eﬀect of child allowance for fertility changes
based on how it is ﬁnanced.
Second, this paper has presented derivation of the allocations to maximize social welfare presented
an examination of the optimal taxation to achieve optimal allocations. The optimal child allowance andan optimal income transfer from younger people to older people diﬀers between the case of an income
tax and the case of a consumption tax. An optimal policy to achieve social welfare maximization diﬀers
according to the mode of taxation.

17

References[1] Ahn, N. and P. Mira (1999). “A Note on the Changing Relationship between Fertility and Female
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18

[13] Lissiotou P. (2012), “Is Fertility Aﬀected by Family Policy? Evidence from the Introduction of Child
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20

Appendix
The sign of a11 a22 − a12 a21()(1 − α − β)(1 − θ)(1 − ϵ)Ak ϵϵ(1 − θ)(1 − ϵ)Ak ϵ−1
a11 a22 − a12 a21 =
1−
n
n
(
)
2
ϵ
(1 − α − β)(1 − θ)(1 − ϵ) Akθ
ϵ−1+
n + (1 − θ)ϵAkn2ϵ(1 − α − β)(1 − θ)(1 − ϵ)Ak ϵ ϵ + (1 − ϵ)θ
> 0.
=
n
ϵ

The sign of (19)(α + β)na22α(α + β)ϵ + α(1 − ϵ)θ
a12 =n − β(1 − ϵ)(1 − θ)ϵAk ϵ−1
1−α−β
ϵ
(
)
θ(1−α−β)(1−ϵ)
k(α + β)ϵ + α(1 − ϵ)θ βϵ z −
ϵ
=n
−
ϵ
k
1−α−β
(α + β)ϵ + α(1 − ϵ)θ
βϵ(1 − β)z 1− βϵ −
ϵ1−α−β k)1(=
(α + β)ϵ + α(1 − ϵ)θ − βϵ2 − (1 − β)ϵ(ϵ + θ(1 − ϵ)) .
ϵ=

The bracket of this equation is positive if Eq. (21) holds.

The sign of (30)na22 −

1−β(1 − β)(1 − ϵ)θ
a12 = n +n − β(1 − ϵ)(1 − θ)ϵAk ϵ−1
1−α−β
ϵ
(
)
(1−α−β)(1−ϵ)θ
k
ϵ + (1 − β)(1 − ϵ)θ βϵ z −
ϵ
=n
−
ϵ
k
1−α−β
(
(
))
ϵ + (1 − β)(1 − ϵ)θ βϵ(1 − β)z
=n
−
k+
ϵ
k
1−α−β
(
)
(1 − β)(1 − ϵ)θ
= n (1 − ϵ)(1 − (1 − β)θ) +> 0.
ϵCommand Optimum Allocations
We set the Lagrange equation as follows
L=∞∑ρs−t (α ln c1s−1 + β ln c2s + (1 − α − β) ln ns−1 )

s=t

(
)
c2s−1
ϵ
λs−1 c1s−1 + zns−1 +++ ns−1 ks − Aks−1 ,
ns−2
s=t
∞
∑

21

where λi−1 denotes a Lagrange multiplier. We obtain the ﬁrst order conditions as∂Lαρ=+ λt = 0,
∂c1t
c1t
∂L
βρ
λt+1
=
+
= 0,
∂c2t+1c2t+1nt(1 − α − β)ρ
λt+1 c2t+1∂L=+ λt (z + k) −
= 0,
∂nt
nt
n2t
∂L
= λt−1 nt−1 − λt ϵAktϵ−1 = 0,
∂kt
c2t
∂L
= c1t + znt ++ nt kt+1 − Aktϵ = 0.
∂λt
nt−1

(46)
(47)
(48)
(49)
(50)

Substituting Eqs. (46) and (47) into (49), we obtain Eq. (34). Substituting Eqs. (46) and (47) into (48),
we obtain Eq. (35). Considering∂L
αρ2=+ λt+1 = 0,∂c1t+1c1t+1(51)

Eqs. (46) and (51), we obtain (33). In the steady state, Eq. (33) changes to (38). Inserting c∗2 = β(1+r)
c∗1αand n∗ = ρ(1 + r) into (32) in the steady state, we obtain c∗1 , as shown by (36). Then, Eq. (36) intoc∗2 = β(1+r)
c∗1 , we obtain c∗2 .
α22Fiscal Support for Families to GDP (%)
4
3.5
3

%

2.5
2
1.5
1
0.5
0
Japan
1.39

U.S.A.
1.93

Italy
1.4

Germany
1.39

France
2.01

U.K.
2.00

Sweden1.98

Fig. 1: Fertility (below the country) and Fiscal Support for Family (share of Gross Domestic Product)(Data: OECD Social Expenditure Database (November 2008), A 2012 Declining Birthrate White Paper(2012), Demographic Yearbook (UN) and Vital Statistics in Japan (Ministry of Health, Labour andWelfare (in Japan).) Data of Fiscal Support for Families are those of 2007. Fiscal Support for Family
includes beneﬁts in kind (day-care/home help and other beneﬁts in kind) and cash beneﬁts (family
allowance, maternity and parental leave and other cash beneﬁt). Data of the total Fertility Rate arethose of 2010.)

23

National Burden Rate
70
60

Social Security Burden
Tax Burden

50
40
30
20
10
0
Japan
5%

U.S.A.
8.875%

Italy
21%

Germany
19%

France
19.6%

U.K.
20%

Sweden25%

Fig. 2: National Burden Rate (Data: National Accounts (Cabinet Oﬃce, Government of Japan),Consumption Tax Rate in foreign countries (Ministry of Finance Japan), National Accounts (OECD),
Revenue Statistics (OECD). Data are for 2009. The percentage below country name denotes the consumption tax rate. The consumption tax rate for the U.S.A. is that of New York City.)24Government Social Spending and Old Age Social Spending
30

Government Social Spending
Old Age Social Spending

25

%

20
15
10
5
0
Japan

U.S.A.

Italy

Germany France

U.K.

SwedenFig. 3: Government Social Spending and Old Age Social Spending (data show public social expenditure as a percentage of GDP. (Data: Social Expenditure: Aggregated data, OECD Social Expenditure
Statistics (database)))25