We examine how child-allowance policies with pay-as-you-go systems affect fertility and growth rates. This study demonstrates that when a government initiates a child-allowance policy using some part of the pension budget, the fertility rate declines in aging economies.Uncertain lifetime, Child allowances, Fertility, Social security, Human capital

Megumi Mochida

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 4 
  
Child-allowances, fertility, and uncertain lifetime 
  
 
 
Megumi Mochida  
Kagawa University 
Abstract 
We examine how child-allowance policies with pay-as-you-go systems affect fertility and growth rates. This study 
demonstrates that when a government initiates a child-allowance policy using some part of the pension budget, the 
fertility rate declines in aging economies.
I would like to thank Koichi Futagami, Ryo Horii, Kazuo Mino, Tetsuo Ono, Takashi Oshio, Akira Yakita, and seminar participants at Osaka 
University for their helpful comments. I acknowledge the financial support from a JSPS Grant-in-Aid for Young Scientists (B) (No. 20730182) 
and from the research fund for Kagawa University Young Scientists 2008-2009. All remaining errors are, of course, my own. 
Citation: Megumi Mochida, (2009) ''Child-allowances, fertility, and uncertain lifetime'', Economics Bulletin, Vol. 29 no.4 pp. 2712-2721. 
Submitted: Oct 25 2009.   Published: October 28, 2009. 
 
     1 Introduction
Declining fertility rates threaten the feasibility of current social security sys-
tems, thereby posing a serious problem to many advanced countries. On
the other hand, as the active life after retirement becomes longer through a
higher life-expectancy, the importance of public pensions which stabilize a
retired lifestyle is increasing.
This paper describes how child-allowance policies with PAYG systems af-
fect fertility and growth rates by incorporating an uncertain lifetime. This
model has an endogenous growth mechanism where the engine of growth is
human capital accumulation. Groezen et al. (2003) show that a child-support
policy stimulates fertility rates but disregard human capital accumulation.
We show that a child-allowance policy increases the fertility rate, but de-
creases the growth rate.1 In addition, the social security budget for elderly
people is commonly much larger than that for young families in low-fertility
countries, such as Japan, Italy, and Spain.2 Therefore, we also analyze an
eﬀect when the government initiates a child-allowance policy by employing
some portion of revenue which has been used for pensions. This policy change
leads to a lower fertility rate in high life-expectancy economies with a larger
opportunity cost of having children.
The remainder of this paper is organized as follows. Section 2 presents
the model. Section 3 explains eﬀects of a child-allowance policy with PAYG-
systems. Section 4 shows eﬀects of the child-allowance policy by diminishing
the size of pension systems. Concluding remarks appear in Section 5.
2 The economy
We consider an overlapping-generations model of endogenous growth, in-
corporating an uncertain lifetime. The life of a representative individual is
divided into three periods: a childhood and a young-working period, both
with ﬁxed durations, and a retirement period. Each young individual has a
probability p∈(0,1] to survive to the retirement period. Competitive insur-
ance companies promise individuals a payment of
(1+rt+1)
p at in exchange for
having an estate at accruing to the companies.3
The government levies a tax τ ∈ [0,1) on labor wage and redistributes
1Zhang (1997) examines how a child-allowance policy aﬀects fertility and growth rates
in a simple model without savings, pensions, and an uncertain lifetime.
2The Japanese social security budget in 2006 forʠ children and families” is 3.4% in
contrast to 50.1% forʠ old-age people”.
3This is a simpliﬁed version of Blanchard’s (1985) model.
1µ part of the revenue to those young child-rearing individuals as child al-
lowances, and (1−µ) part of it to retired individuals as pensions. The value
of the allowance per child and the payment per retired at time t is described,
respectively, as Qt and At.
2.1 Households
In childhood, individuals only accumulate human capital. Individuals are
endowed with one divisible unit of time in their young periods, reproduce
asexually, and allocate their time toward labor and raising children. They
receive labor income, which is taxed away, and allowances in the end of their
young periods. They consume part of their income and invest the rest of
it in annuities. Subsequently, living individuals obtain the principal and
interest from their annuities and consume them with their pension beneﬁts
after retirement.
Each individual who is born at time t, and called generation t + 1, accu-
mulates human capital ht+1 according to
ht+1 = θe
γ
tht, θ ≥ 1, γ ∈ (0,1),
where et is parental-teaching time and ht is parents’ human capital.
The time constraint of generation t is
1 = lt + nt+1(q + et),
where lt, nt+1, and q respectively denote the labor time, the number of chil-
dren, and the rearing time per child.
The budget constraints of a member of generation t when young and
retired are given, respectively, by (1 − τ)wthtlt + Qtnt+1 = c
y
t + at and
(1+rt+1)
p at + At+1 = co
t+1, where c
y
t and co
t+1 is the consumption when young
and retirement.
The utility function of generation t(≥ 0) is4
ut = (1 − σ)ϕlnc
y
t + p(1 − σ)(1 − ϕ)lnc
o
t+1 + σlnnt+1ht+1. (1)
The parameter σ ∈ (0,1) measures the taste for children’s total human cap-
ital. The parameter ϕ ∈ (0,1) describes the subjective discount factor. By
solving individuals’ optimization problems, the optimal values are given by:
4With an uncertain lifetime, p∈(0,1], this utility form is employed by Pecchenino and
Utendorf (1999) and Yakita (2001), among others.
2nt+1 =
κσ(1 − γ)
{q(1 − τ)wtht − Qt}
It,
lt = 1 − κσ
{q(1 − τ)wtht − γQt}
{q(1 − τ)wtht − Qt}
1
(1 − τ)wtht
It,
et =
γq
1 − γ
−
γQt
(1 − γ)(1 − τ)wtht
,
c
y
t = κ(1 − σ)ϕIt,
c
o
t+1 = κp(1 − σ)(1 − ϕ)
(1 + rt+1)
p
It,
at = κp(1 − σ)(1 − ϕ)It −
pAt+1
(1 + rt+1)
,
where κ≡ 1
{1−(1−p)(1−σ)(1−ϕ)} and It≡ (1 − τ)wtht +
pAt+1
(1+rt+1).
2.2 Production
Competitive ﬁrms produce a single ﬁnal good. The aggregate production
function at time t is Yt = F(Kt,htltNt) = Kα
t (htltNt)1−α, where Yt, Kt, Nt,
and α∈(0,1) respectively denote the aggregate output, the physical capital
which fully depreciates in the production process, the working-age popula-
tion, and the share of physical capital.
The factor markets are presumed to be perfectly competitive. The factors
are paid by their marginal products:
wt = (1 − α)˜ kt
α
,
(1 + rt) = α˜ kt
α−1
,
where ˜ kt≡ Kt
htltNt is the physical capital per eﬀective-labor.
2.3 Equilibrium
The government faces the budget constraints of the policies:
child allowances; µτwthtltNt = Qtnt+1Nt,
public pensions; (1 − µ)τwthtltNt = AtpNt−1.
3Using capital market-clearing conditions, Kt+1 = atNt, we can get the values
at equilibrium as follows.5
nt+1 =
[µτΨ + κσ{1 + (1 − µ)τ
(1−α)
α }{(1 − γ)(1 − τ) − µγτ}]
q{1 − (1 − µ)τ}Ψ
≡ng,
lt =
(1 − τ)
{1 − (1 − µ)τ}
[1 −
κσ{1 + (1 − µ)τ
(1−α)
α }
Ψ
]≡lg,
et =
γqκσ{1 − (1 − µ)τ}{1 + (1 − µ)τ
(1−α)
α }
[µτΨ + κσ{1 + (1 − µ)τ
(1−α)
α }{(1 − γ)(1 − τ) − µγτ}]
≡eg,
where Ψ≡ [1 + κ{1 − (1 − σ)(1 − ϕ)}(1 − µ)τ
(1−α)
α ].
The capital per eﬀective-labor becomes ˜ kt+1 = ˜ kt = ˜ k∗ in the steady
state. Consequently, the per-capita growth rate in the balanced-growth path
depends only on the parental-teaching time. It is given by
(1 + g
∗) = θe
γ
g.
3 Policy eﬀects
The following proposition summarizes eﬀects of an introduction of the child-
allowance policy and the PAYG-pension system.
Proposition 1. Introduction of a child-allowance policy and a PAYG-
pension system increases the number of children and decreases the labor time.
When the government introduces the child-allowance policy, the education
time per child and the per-capita growth rate decrease. When the government
introduces the PAYG-pension system only, the introduction has no eﬀects on
the education time per child and the per-capita growth rate.
Proof. See Appendix A.
These results are explained as follows. The prices, in labor terms, of
raising and educating a child are, respectively, q −
Qt
(1−τ)wtht and nt+1. The
number of children increases because of income eﬀects and price eﬀects by
the intervention. The per-child education time decreases because negative
price eﬀects dominate positive income eﬀects. When only the pension system
exists, the negative price eﬀects are proportional to positive income eﬀects.
Therefore, there is no change in education time.
5Note that an increase in life expectancy lowers the fertility rate and education time.
44 Fertility and life expectancy
This section presents analysis of fertility eﬀects when the government intro-
duces a child-allowance policy using µ part of the revenue which has been
spent on the PAYG-pension systems.6 That is, the government partly di-
minishes the transfer from young to old and redistributes resources among
young individuals to increase contributors in the future.7
The eﬀect of such a policy change on the fertility rate can be recognized
by the sign of the following formula.
∂ng
∂µ
|µ=0 = s
2p
2 − [σ(1 − γ)(1 − τ)
(1 − α)
α
− {σ + 2(1 − σ)ϕ}{1 + τ
(1 − α)
α
}]sp
+(1 − σ)ϕ(1 − s){1 + τ
(1 − α)
α
}
2 ≡ f(p ;τ),
where s ≡ (1 − σ)(1 − ϕ) > 0.
This is a quadratic function of life expectancy, p. Herein, we shall see a
case in which the policy change engenders a lower fertility rate.8 The sign of
f(p ;τ) is either positive or negative depending on p, if both the labor-capital
distribution ratio and the labor-tax rate are
(1 − α)
α
> α1 and 0 < τ < τ1,
where α1≡ 1
σ(1−ϕ)(1−γ) and τ1≡
−[σ(1−γ)s+(1−σ){σ(1−ϕ)+2ϕ}]+
√
Υ
2(1−σ)ϕ(1−s)
(1−α)
α
.9 The critical
value of the old-age degree in the economy, p1, which is an intersection of
graph f(p ;τ) with the x-axis, is the smaller solution of f(p ;τ) = 0:10
p1≡
[σ(1 − γ)(1 − τ)
(1−α)
α − {σ + 2(1 − σ)ϕ}{1 + τ
(1−α)
α }] −
√
Φ
2s
.
6When the government introduces the child-allowance policy by levying a new labor
tax for the sole purpose of supporting the policy, the fertility rate always increases.
7Because individuals in each generation are assumed to be homogenous, this policy
change exactly eases the tax burden of the young generation.
8See Appendix B for another case.
9Υ≡σ2(1 − ϕ)2 + 4σ(1 − γ)ϕ(1 − ϕ)(1 − s)
(1−α)
α
+2(1 − σ)σ(1 − γ)s{σ(1 − ϕ) + 2ϕ} + σ2(1 − γ)2s2 > 0
10Φ≡{σ2 + 4(1 − σ)ϕ(1 − s)2(1 + s)}{1 + τ
(1−α)
α }2 + σ2(1 − γ)2(1 − τ)2(1−α
α )2
−2σ{σ + 2(1 − σ)ϕ}(1 − γ)(1 − τ){1 + τ
(1 − α)
α
}
(1 − α)
α
> 0
5Therefore, we obtain the following proposition.
Proposition 2. When the labor-capital distribution ratio is
(1−α)
α > α1 and
the labor-tax rate is 0 < τ < τ1, an introduction of the child-allowance policy
using some part of pension revenue
(2a) increases the fertility rate if p∈(0,p1),
(2b) decreases the fertility rate if p∈(p1,1],
(2c) has no eﬀects on the fertility rate if p = p1.
In the event of a larger labor share, which implies higher labor income,
or a lower labor-tax rate, the opportunity cost of having children becomes
large. In aging economies, where working-age individuals face a high proba-
bility that they will be alive in their retirement periods, pension and annuity
amount per capita is relatively small by the larger old-population. Therefore,
if the pension beneﬁt is decreased by the policy change, individuals have an
incentive to increase the labor supply for their retired-age consumption. For
that reason, even if individuals receive child allowances for having children,
they decrease the number of children those they have.
5 Concluding remarks
We have presented the eﬀects of child-allowance policies. Some empirical
studies discuss whether child-allowance policies actually increase the fertility
rate or not. It might be important that individuals are guaranteed suﬃcient
pension beneﬁts to stimulate the fertility rate in aging economies.
6Appendix A.
The eﬀect on the number of children is positive, as
∂ng
∂τ
|τ=0 =
1
q
[µ(1 − κσ) + (1 − µ)κ
2pσ(1 − σ)(1 − ϕ)(1 − γ)
(1 − α)
α
] > 0.
This sign is satisﬁed with any µ∈[0,1].
The eﬀect on the labor time is negative, as
∂lg
∂τ
|τ=0 = −[µ(1 − κσ) + (1 − µ)κ
2pσ(1 − σ)(1 − ϕ)
(1 − α)
α
] < 0.
This sign is also satisﬁed with any µ∈[0,1].
The eﬀect on the education time per child is given by
∂eg
∂τ
|τ=0 = −µ
(1 − κσ)γq
κσ(1 − γ)2 ≤ 0.
This sign is negative when µ∈(0,1], and zero when µ = 0.
Therefore, the eﬀect on the per-capita growth rate in the balanced-growth
path is given by
∂(1 + g∗
g)
∂τ
|τ=0 =
∂θeγ
g
∂τ
|τ=0 ≤ 0.
Appendix B.
The following proposition summarizes positive fertility eﬀects of an introduc-
tion of the child-allowance policy.
Proposition. An introduction of the child-allowance policy using some por-
tion of pension revenue increases the fertility rate, when
(a) the labor-capital distribution ratio is
(1−α)
α ≤α1,
(b) the labor-capital distribution ratio is
(1−α)
α > α1 and the labor-tax rate is
τ1 < τ < 1.
The introduction has no eﬀects on the fertility rate when the labor-capital dis-
tribution ratio is
(1−α)
α > α1 and the labor-tax rate is τ = τ1. The following
f(p ;τ) expresses the eﬀect on the fertility rate.11
f(p ;τ) =
∂nt+1
∂µ
|µ=0 = s
2[p − Γ]
2 + Λ.
11 Γ≡
[σ(1−γ)(1−τ)
(1−α)
α −{σ+2(1−σ)ϕ}{1+τ
(1−α)
α }]
2s ,
Λ≡ −
[σ(1−γ)(1−τ)
(1−α)
α −{σ+2(1−σ)ϕ}{1+τ
(1−α)
α }]
2
4 + (1 − σ)ϕ(1 − s){1 + τ
(1−α)
α }2.
7The graph of f(p ;τ) is convex downward and the value of f(0 ;τ) is posi-
tive.12 The condition by whichʠthe value of f(1 ;τ) is negative” allows the
graph of f(p ;τ) to intersect the x-axis once in p∈(0,1].13 .
The value of f(1 ;τ) is expressed by a function of τ:
f(1 ,τ) = (1 − s)(1 − σ)ϕ{
(1 − α)
α
}
2τ
2
+[(1 − σ){σ(1 − ϕ) + 2ϕ} + sσ(1 − γ)]
(1 − α)
α
τ
+[(1 − σ) − sσ(1 − γ)
(1 − α)
α
].
The sign of f(1 ,1) is positive.14 Because the ﬁrst and second terms of
f(1 ,τ) are positive, the sign of this function depends on the third term.
When the labor-capital distribution ratio is
(1 − α)
α
≤
1
σ(1 − ϕ)(1 − γ)
≡α1,
the third term is non-negative. Therefore, the sign of f(1 ,0) is positive;
thereby, that of f(1 ;τ) is positive for any τ(> 0). Consequently, the value
of f(p ;τ) is always positive in p∈(0,1].
When the third term is negative, as
(1−α)
α > α1, the graph of f(1 ,τ) has
an intercept in τ∈(0,1). The intersection is a larger solution of f(1 ,τ) = 0:15
τ1≡
−[σ(1 − γ)s + (1 − σ){σ(1 − ϕ) + 2ϕ}] +
√
Υ
2(1 − σ)ϕ(1 − s)
(1−α)
α
.
The value of f(1 ;τ) is negative when τ∈(0,τ1) and positive when τ∈(τ1,1).
Therefore, when
(1−α)
α > α1 and τ1 < τ < 1, the value of f(p ;τ) is always
positive in p∈(0,1]. When
(1−α)
α > α1 and 0 < τ < τ1, the value of f(p ;τ) is
positive or negative depending on p.
12f(0 ;τ) = (1 − σ)ϕ(1 − s){1 + τ
(1−α)
α }
2
> 0.
13To evade intricacy, we eliminate the case in which the graph intersects the x-axis twice.
14f(1 ,1) = (1 − σ)[1 + {σ(1 − ϕ) + 2ϕ}
(1−α)
α + (1 − s)ϕ(1−α
α )2] > 0.
15Υ≡σ2(1 − ϕ)2 + 4σ(1 − γ)ϕ(1 − ϕ)(1 − s)
(1−α)
α
+2(1 − σ)σ(1 − γ)s{σ(1 − ϕ) + 2ϕ} + σ2(1 − γ)2s2 > 0
8References
[1] Blanchard, O.J. (1985)ʠ Debt, deﬁcits, and ﬁnite horizons” Journal of
political economy 93, 223-247.
[2] Groezen, B., Leers, T., and L.Meijdam (2003)ʠ Social security and en-
dogenous fertility: pensions and child allowances as Siamese twins” Jour-
nal of Public Economics 87, 233-251.
[3] Pecchenino, R.A., and K.R. Utendorf (1999)ʠSocial security, social wel-
fare and the aging population” Journal of Population Economics 12, 607-
623.
[4] Yakita, A. (2001)ʠUncertain lifetime, fertility and social security” Jour-
nal of Population Economics 14, 635-640.
[5] Zhang, J. (1997)ʠ Fertility, growth and public investments in children”
Canadian Journal of Economics 30, 835-843.
9

Child-allowances, fertility, and uncertain lifetime

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Child-allowances, fertility, and uncertain lifetime

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