Discussion Papers In Economics
And BusinessChild Allowances, Fertility, and Uncertain
Lifetime
Megumi Mochida
Discussion Paper 05-11

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANChild Allowances, Fertility, and Uncertain
Lifetime
Megumi Mochida
Discussion Paper 05-11

May 2005

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANChild Allowances, Fertility, and Uncertain
Lifetime∗
Megumi Mochida†
January 2005
Abstract
We examine how child-allowance policies with pay-as-you-go systems affect fertility and growth rates. A current method to subsidize
child-rearing households, which determines benefits according to family’s number of children, increases the fertility rate but decreases thegrowth rate. This study also demonstrates that when a government
initiates a child-allowance policy using some part of the pension budget, the fertility rate declines in aging economies.
Keywords: Fertility, Social security, Uncertain lifetime.
JEL: D91, H55, J13.

∗

I would like to thank Koichi Futagami, Ryo Horii, Kazuo Mino, Takashi Oshio, Akira
Yakita, and seminar participants at Osaka University for their helpful comments. Allremaining errors are, of course, my own.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail address：cg088mm@srv.econ.osaka-u.ac.jp (M. Mochida).

1

1Introduction

Declining fertility rates threaten the feasibility of current social security systems, thereby posing a serious problem to many advanced countries. On
the other hand, as the active life after retirement becomes longer through ahigher life-expectancy, the importance of public pensions which stabilize aretired lifestyle is increasing. Pay-as-you-go financed (PAYG-) public pensions, which are the dominant scheme in countries with aging populations,
are supported by working-age people. For that reason, reproduction of theimmediately subsequent generation poses a social problem as long as people
anticipate receiving PAYG-pension benefits after their own retirement. However, for raising children, who will be contributors to future PAYG-pension
systems, people virtually depend on the expenses of individuals with theirown children.1
This paper describes how child-allowance policies with PAYG systems
affect fertility and growth rates by incorporating an uncertain lifetime. Thismodel has an endogenous growth mechanism where the engine of growth is
human capital accumulation by parental-teaching education. Children aretreated as consumer durables, as in Becker and Barro (1988) and Barro andBecker (1989).2 Children’s human capital is also assumed to be a source
1

Folbre (1994) describes this as Individuals who devote relatively little time or energy
to child-rearing are free-riding on parental labor”.
2
When we consider motives for having children, we abstract from the old-age security
hypothesis, because most low-fertility countries have already constructed social security
systems and the degree of such motives in developed countries is much lower relative to2

of utility to the parent, as in Eckstein and Zilcha (1994). Groezen et al.
(2003) show that a child-support policy stimulates fertility rates in a small
open economy but disregard human capital accumulation. We show that achild-allowance policy increases the fertility rate, but decreases the growthrate by retarding human capital accumulation.3
In addition, the social security budget for elderly people is commonlymuch larger than that for young families with children in low-fertility countries, such as Japan, Italy, and Spain.4 Therefore, we also analyze an effectwhen the government initiates a child-allowance policy by employing some
portion of revenue which has been used for pensions. That is, the government diminishes the size of pension systems and redistributes it among theworking generation, who are rearing future contributors. This policy change
leads to a lower fertility rate in high life-expectancy economies with a larger
opportunity cost of having children. This is true because a smaller pension
size in aging economies compels individuals to increase the labor supply toprovide for their post-retirement consumption. Therefore, even if individuals
receive child allowances, they decrease the number of children they choose to
have.
The remainder of this paper is organized as follows. Section 2 presents

that in developing economies.
3
Zhang (1997) examines how a child-allowance policy affects fertility and growth rates
in a simple model without savings, pensions, and an uncertain lifetime.
4
The Japanese social security budget in 2001 for children and families” is 3.7% in
contrast to 68.7% for old-age people”.

3the model. Section 3 explains effects of a child-allowance policy with PAYGsystems. 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 use an overlapping-generations model of endogenous growth, incorporating 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 of uncertain length. For simplicity, we
assume that the individual is either alive or dead at the beginning of thethird period. The probability that the individual is alive at the beginning of
the retirement period is denoted as p∈(0, 1], which is known and common toindividuals in the same generation.
Assuming that insurance companies are risk-neutral and that private annuities markets are competitive, the insurance companies promise individuals
a payment of (1+rpt+1 ) at in exchange for having an estate at accruing to thecompanies. In the absence of bequest motives, individuals are willing toinvest their assets in such insurance.5
The government levies a tax τ ∈ [0, 1) on wages of young individuals andredistributes µ part of the revenue to those young child-rearing individuals as

5This is a simplified version of Blanchard’s (1985) model.

4child allowances, and (1 − µ) part of it to retired individuals as PAYG-public
pensions. The value of the allowance per child at time t is described as Qt
and the payment per retired person at time t is described as At .

2.1

Households

In childhood, individuals only accumulate human capital by receiving parentalteaching. Individuals are endowed with one divisible unit of time in their
young periods, reproduce asexually, and allocate their time toward labor
and raising children. To raise a child, they are compelled to spend their timein child-rearing and parental-teaching, which helps their children to accumulate human capital.6 They receive wage income, which is taxed away, andallowances according to the number of children in the end of their young
periods. They consume part of their income and invest the rest of their income in annuities. Subsequently, living individuals obtain the principal andinterest 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, accumulates human capital through education by parents at time t. Given parents’human capital, ht , when a parent allocates time et toward parental-teaching

6

In this model, as in Eckstein and Zilcha (1994), the decision of education time is basedon parental altruism. Chamley (1993) and Göcke (2004), among others, present a modelin which adult agents divide their own time between schooling and working.

5

for one child, the child accumulates human capital ht+1 according toht+1 = θeγt ht ,where θ ≥ 1 and γ ∈ (0, 1).7
The endowment time in the young period is normalized to unity. The
time constraint of generation t is then given by

1 = lt + nt+1 (q + et ),where lt , nt+1 , and q respectively denote the labor time, the number of children, and the rearing time per child.8
The budget constraints of a member of generation t when young andretired are given, respectively, by (1 − τ )wt ht lt + Qt nt+1 = cyt + at and(1+rt+1 )at + At+1p= cot+1 , where cyt is the consumption when young, at is

the annuity, At+1 is the pension benefit, and cot+1 represents post-retirement
consumption.
All individuals have identical preferences. The utility function of genera-

7

Glomm and Ravikumar (1992) employ a similar structure.
Another type of endogenous labor-supply model where agents choose their labor length
considering social security benefits is analyzed in Hu (1979), Momota (2003), and Cremer
et al. (2004).
8

6

tion t(≥ 0) is represented by9

ut = (1 − σ)ϕlncyt + p(1 − σ)(1 − ϕ)lncot+1 + σlnnt+1 ht+1 .

(1)

The parameter σ ∈ (0, 1) can be interpreted as the degree of preferenceovertotal human capital” of their children. The parameter ϕ ∈ (0, 1)

describes the degree of time preference for consumption while young. By
solving individuals’ optimization problems, the optimal values are given by:

nt+1 =

lt = 1 − κσet =κσ(1 − γ)It ,{q(1 − τ )wt ht − Qt }{q(1 − τ )wt ht − γQt }1
It ,{q(1 − τ )wt ht − Qt } (1 − τ )wt ht
γq
γQt
−
,
1 − γ (1 − γ)(1 − τ )wt ht
cyt = κ(1 − σ)ϕIt ,cot+1 = κp(1 − σ)(1 − ϕ)

(1 + rt+1 )It ,p

at = κp(1 − σ)(1 − ϕ)It −pAt+1,(1 + rt+1 )pAt+11
where κ≡ {1−(1−p)(1−σ)(1−ϕ)}and It ≡ (1 − τ )wt ht + (1+r
.
t+1 )

9

With an uncertain lifetime, p∈(0, 1], this utility form is employed by Pecchenino andPollard (1997), Pecchenino and Utendorf (1999), and Yakita (2001), among others.

7

2.2

Production

Competitive firms produce a single final good by employing both physicalcapital and effective labor input. The aggregate production function at time
t is given by Yt = F (Kt , ht lt Nt ) = Ktα (ht lt Nt )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 population, andthe share of physical capital. The production function, in intensive form, canbe expressed asy˜t = k˜tα ,where y˜t = htYlttNt is the output per effective-labor and k̃t = htKlt tNt is the physical capital per effective-labor.
The factor markets are presumed to be perfectly competitive. Therefore,
the firms take factor prices as given. Firms hire labor inputs and physicalcapital up to the point where their marginal products equal their factor
prices:αwt = (1 − α)k̃t ,(1 + rt ) = αk̃t

2.3

α−1

.

Equilibrium

The government faces the budget constraints of the policies:8child allowances;µτ wt ht lt Nt = Qt nt+1 Nt ,public pensions;(1 − µ)τ wt ht lt Nt = At pNt−1 .

Using capital market-clearing conditions, Kt+1 = at Nt , we can get the values
at equilibrium as follows.10}{(1 − γ)(1 − τ ) − µγτ }][µτ Ψ + κσ{1 + (1 − µ)τ (1−α)α
nt+1 =≡ng ,q{1 − (1 − µ)τ }Ψ
κσ{1 + (1 − µ)τ
(1 − τ )lt =[1 −{1 − (1 − µ)τ }
Ψ
et =

(1−α)
}
α

]≡lg ,

γqκσ{1 − (1 − µ)τ }{1 + (1 − µ)τ (1−α)}α[µτ Ψ + κσ{1 + (1 − µ)τ (1−α)}{(1 − γ)(1 − τ ) − µγτ }]
α

≡eg ,

].
where Ψ≡ [1 + κ{1 − (1 − σ)(1 − ϕ)}(1 − µ)τ (1−α)α
The growth rate of per-capita output at time t is
Yt+1
Nt+1
(1 + gt ) ≡ Yt
Nt=(

k̃t+1 α ht+1
)
.
ht
k̃t

˜ = k̃t = k˜∗ in the steady state.11
The capital per effective labor becomes kt+1
10
11Note that an increase in life expectancy lowers the fertility rate.
The equilibrium sequence of capital per effective labor isk̃t+1 =

κp(1 − σ)(1 − ϕ)(1 − α) αk̃t ,Eg lg ng

where Eg ≡θeγg . This dynamics is always stable due to assumption of α ∈ (0, 1). In thesteady state, the physical capital per effective labor becomesk̃ ∗ = (1
κp(1 − σ)(1 − ϕ)(1 − α) 1−α).
Eg lg ng9

Consequently, the per-capita growth rate in the balanced-growth path depends only on the parental-teaching time. It is given by(1 + g ∗ ) = θeγg .

3Policy effects

The following proposition summarizes effects of an introduction of the childallowance 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. Initially, the effect on the number of children is positive, as∂ng
1(1 − α)|τ =0 = [µ(1 − κσ) + (1 − µ)κ2 pσ(1 − σ)(1 − ϕ)(1 − γ)
] > 0.
∂τqαThis sign is satisfied with any µ∈[0, 1]. Consequently, this form of governmental intervention always increases the fertility rate.

10Secondly, the effect on the labor time is negative, as(1 − α)∂lg|τ =0 = −[µ(1 − κσ) + (1 − µ)κ2 pσ(1 − σ)(1 − ϕ)
] < 0.
∂ταThis sign is also satisfied with any µ∈[0, 1].
Lastly, the effect on the education time per child is given by∂eg(1 − κσ)γq|τ =0 = −µ≤ 0.
∂τκσ(1 − γ)2This sign is negative when µ∈(0, 1], and zero when µ = 0. That is, theintervention with child allowances (µ > 0) decreases the education time per
child. However, the introduction of a public pension without child allowances(µ = 0) has no effects on the education time per child.
Therefore, the effect on the per-capita growth rate in the balanced-growth
path depends on the per-child education time; it is given by∂(1 + gg∗ )∂θeγg|τ =0 =
|τ =0 ≤ 0.
∂τ∂τ

These results are explained as follows. The prices, in labor terms, oftraising and educating a child are, respectively, q − (1−τQ)w
and nt+1 . Thet ht

number of children increases because of income effects and price effects by
11

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.

4

Fertility and life expectancy

This section presents analysis of fertility effects when the government introduces a child-allowance policy using µ part of the revenue which has beenspent on the PAYG-pension systems.12 That is, the government partly diminishes the transfer from young to old and redistributes resources among
young individuals to increase contributors in the future.13
The effect of such a policy change on the fertility rate can be recognizedby the sign of the following formula.
∂ng(1 − α)(1 − α)|µ=0 = s2 p2 − [σ(1 − γ)(1 − τ )− {σ + 2(1 − σ)ϕ}{1 + τ
}]sp
∂µ
α
α
(1 − α) 2+(1 − σ)ϕ(1 − s){1 + τ} ≡ f (p ; τ ),α
where s ≡ (1 − σ)(1 − ϕ) > 0.
This is a quadratic function of life expectancy, p. Herein, we shall see a

12

We can easily see that the per-child education time always decreases, as in section 3.
When 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.
13
Because individuals in each generation are assumed to be homogenous, this policychange exactly eases the tax burden of the young generation.

12

case in which the policy change engenders a lower fertility rate.14 The signof f (p ; τ ) is either positive or negative depending on p, if both the laborcapital distribution ratio and the labor-tax rate are(1 − α)
> α1
α

and

0 < τ < τ1 ,

√

Υ 15
1
where α1 ≡ σ(1−ϕ)(1−γ)
and τ1 ≡ −[σ(1−γ)s+(1−σ){σ(1−ϕ)+2ϕ}]+.
The critical(1−α)
2(1−σ)ϕ(1−s)α

value of the old-age degree in the economy, p1 , which is an intersection ofgraph f (p ; τ ) with the x-axis, is the smaller solution of f (p ; τ ) = 0:16
− {σ + 2(1 − σ)ϕ}{1 + τ (1−α)}] −[σ(1 − γ)(1 − τ ) (1−α)
α
α
p1 ≡
2s

√

Φ

.

Therefore, we obtain the following proposition.
Proposition 2.
When the labor-capital distribution ratio is (1−α)> α1 and the labor-tax rateα
is 0 < τ < τ1 , an introduction of the child-allowance policy using some partof pension revenue(3a) increases the fertility rate if p∈(0, p1 ),(3b) decreases the fertility rate if p∈(p1 , 1],

14

See Appendix for another case.
Υ≡σ 2 (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} + σ 2 (1 − γ)2 s2 > 016
222
2 1−α 2Φ≡{σ 2 + 4(1 − σ)ϕ(1 − s)2 (1 + s)}{1 + τ (1−α)α } + σ (1 − γ) (1 − τ ) ( α )(1 − α) (1 − α)}>0−2σ{σ + 2(1 − σ)ϕ}(1 − γ)(1 − τ ){1 + τ
α
α

15

13

(3c) 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 becomeslarge. In aging economies, where working-age individuals face a high probability 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. When child allowances are given to parents depending on the number of children, parents
have an incentive to increase theing thequantity” of children without maintain-

quality” of each child. 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 tostimulate the fertility rate in aging economies.

14

Appendix.
The following proposition summarizes positive effects of an introduction of
the child-allowance policy on the fertility rate.
Proposition.
An introduction of the child-allowance policy using some portion of pension
revenue increases the fertility rate, when(2a) the labor-capital distribution ratio is (1−α)≤α1 ,α(2b) the labor-capital distribution ratio is (1−α)> α1 and the labor-tax rateαis τ1 < τ < 1.
The introduction has no effects on the fertility rate when the labor-capital> α1 and the labor-tax rate is τ = τ1 .
distribution ratio is (1−α)α
Proof. The following f (p ; τ ), which is a quadratic function of p, expresses
the effect on the fertility rate.17

f (p ; τ ) =∂nt+1|µ=0 = s2 [p − Γ]2 + Λ.
∂µThe graph of f (p ; τ ) is convex downward and the value of f (0 ; τ ) is positive.18 The condition by which17Γ≡[σ(1−γ)(1−τ )Λ≡ −the value of f (1 ; τ ) is negative” allows the

(1−α)
(1−α)
−{σ+2(1−σ)ϕ}{1+τ α }]α2s[σ(1−γ)(1−τ )

,

(1−α)
(1−α)
−{σ+2(1−σ)ϕ}{1+τ α }]2
α

4

2
18
f (0 ; τ ) = (1 − σ)ϕ(1 − s){1 + τ (1−α)α } > 0.

15

2
+ (1 − σ)ϕ(1 − s){1 + τ (1−α)α } .

graph of f (p ; τ ) to intersect the x-axis once in p∈(0, 1].19 .
The value of f (1 ; τ ) is expressed by a function of τ :f (1 , τ ) = (1 − s)(1 − σ)ϕ{(1 − α) 2 2}τα+[(1 − σ){σ(1 − ϕ) + 2ϕ} + sσ(1 − γ)]+[(1 − σ) − sσ(1 − γ)

(1 − α)τ
α(1 − α)].
αThe sign of f (1 , 1) is positive.20 Because the first and second terms off (1 , τ ) are positive, the sign of this function depends on the third term.
When the labor-capital distribution ratio is(1 − α)1≤≡α1 ,ασ(1 − ϕ)(1 − γ)

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 valueof f (p ; τ ) is always positive in p∈(0, 1].
When the third term is negative, as (1−α)> α1 , the graph of f (1 , τ ) has
αan intercept in τ ∈(0, 1). The intersection is a larger solution of f (1 , τ ) = 0:21

τ1 ≡−[σ(1 − γ)s + (1 − σ){σ(1 − ϕ) + 2ϕ}] +2(1 − σ)ϕ(1 − s) (1−α)
α

19

√

Υ

.

To evade intricacy, we eliminate the case in which the graph intersects the x-axis twice.
2
f (1 , 1) = (1 − σ)[1 + {σ(1 − ϕ) + 2ϕ} (1−α)+ (1 − s)ϕ( 1−ααα ) ] > 0.
(1−α)
21
2
2
Υ≡σ (1 − ϕ) + 4σ(1 − γ)ϕ(1 − ϕ)(1 − s) α+2(1 − σ)σ(1 − γ)s{σ(1 − ϕ) + 2ϕ} + σ 2 (1 − γ)2 s2 > 0

20

16

The value of f (1 ; τ ) is negative when τ ∈(0, τ1 ) and positive when τ ∈(τ1 , 1).
Therefore, when (1−α)> α1 and τ1 < τ < 1, the value of f (p ; τ ) is always
αpositive in p∈(0, 1]. When (1−α)> α1 and 0 < τ < τ1 , the value of f (p ; τ ) isαpositive or negative depending on p.

17

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