We examine how an introduction of education subsidies affects growth rates, incorporating an uncertain lifetime. We demonstrate that the introduction of subsidies engenders higher growth rates in aging economies, except when the education-tax rate is sufficiently low.Education subsidies; Social security; Uncertain lifetime

Megumi Mochida

English

Research Papers in Economics

 
 
 
Discussion Papers In Economics 
And Business 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Graduate School of Economics and 
Osaka School of International Public Policy (OSIPP) 
Osaka University, Toyonaka, Osaka 560-0043, JAPAN
 
The Effect of Education Subsidies in an Aging 
Economy 
 
Megumi MOCHIDA
 
 
Discussion Paper 05-30  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
October 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 and 
Osaka School of International Public Policy (OSIPP) 
Osaka University, Toyonaka, Osaka 560-0043, JAPAN 
 
The Effect of Education Subsidies in an Aging 
Economy 
 
Megumi MOCHIDA
 
 
Discussion Paper 05-30 The Eﬀect of Education Subsidies in an Aging
Economy¤
Megumi MOCHIDAy
Abstract
We examine how an introduction of education subsidies aﬀects growth
rates, incorporating an uncertain lifetime. We demonstrate that the intro-
duction of subsidies engenders higher growth rates in aging economies, except
when the education-tax rate is suﬃciently low.
Keywords: Education subsidies; Social security; Uncertain lifetime
JEL: H31; H52; H55
℡℡℡℡℡
¤I would like to thank Koichi Futagami, Ryo Horii, Kazuo Mino, Akira Momota, Tetsuo Ono,
and Akira Yakita for their helpful comments. All remaining errors are, of course, my own.
yGraduate 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).
11 Introduction
Industrial countries are aging rapidly. When we intend to maintain pay-as-you-go
ﬁnanced pension systems with moderate beneﬁts, the tax burdens of the young-
working generation become much heavier. Therefore, young-working households
who save some after-tax income for a longer retired period cannot aﬀord to invest
much in their children. In this paper, we examine eﬀects of education subsidies to
complement parental educational investment, considering aging eﬀects.
The eﬀectiveness of education subsidies has been widely discussed by Zhang
and Casagrande (1998), Kaganovich and Zilcha (1999), and Wigger (2004), among
others. Kaganovich and Zilcha (1999) show that the introduction of education
subsidies can engender higher growth rates only if (i) parents have overly low
preference levels for their children to invest privately in the children’s education,
and (ii) parents happen to have a medium preference level for children, and the
government has large education revenue (and relatively small pension revenue).
In other cases, the usual negative eﬀects of subsidies, which crowd out parental
private investment, are so large that the introduction of education subsidies has a
negative eﬀect, or none at all, on the growth rate.
Our motivation is to determine the eﬀectiveness of education subsidies in aging
countries where parents cannot aﬀord to invest much in their children’s education.
2Incorporating an uncertain lifetime to a model that is based on Kaganovich and
Zilcha (1999), we demonstrate that, in the case of aging economies, the introduction
of education subsidies engenders a higher growth rate, except when the education-
tax rate is suﬃciently low.
2 The model
We consider 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 (each with ﬁxed duration), and
a retirement period (of uncertain length). The individual is alive at the beginning
of the third period with probability p 2 (0;1]. Let Nt denote the number of
working-aged individuals, who have n children, in period t. Savings of individuals
who have died at the onset of the third period are distributed among the retired
individuals as an actuarially fair annuity.1 The expected rate of return to savings is
(1+½t+1) =
(1+rt+1)
p , where (1+rt+1) is the return of direct holdings of capital. The
government supplies education to children and social security beneﬁts to retired
individuals by taxing the labor income of young-working individuals. First, we
examine an education-subsidy policy (ESP), by which the government provides
1This is a simpliﬁed version of Blanchard’s (1985) model.
3both education subsidies and public schools. Secondly, we show the equilibrium
of a public-school policy (PSP) in which the government supplies public schools
alone. Thereafter, we compare the two growth rates in the next section.
In childhood, individuals only accumulate human capital. Young individuals
receive a wage income, which is taxed away. They divide their income among ed-
ucation expenditures for their children, net, their current consumption, c
y
t, and in-
vesting in annuities, at, for their post-retirement consumption, co
t+1. Subsequently,
living individuals obtain principal and interest from their annuities and consume
them with their pension beneﬁts, Tt+1, after retirement.
Let ht+1 be the human capital level of each individual who is born at time t
and called generation t + 1. Human capital is accumulated according to:
ht+1 = (et + ºt)
°(e
g
t)
1¡°:
In that equation, ° 2 (0;1) denotes the eﬃciency of education input provided
privately, such as textbooks and tutors. The education subsidy given by the gov-
ernment is ºt, and e
g
t is the public-school quality provided by the government.2
The subsidies are used directly for human capital accumulation as in the methods
2We call e
g
t ”public-school quality”, as do Glomm and Ravikumar (1992), to distinguish strictly
between ºt, which is subsidized to private parental investment, and e
g
t. We can also regard e
g
t as
other public educational investment like libraries, museums, and so on.
4of Kaganovich and Zilcha (1999). Individuals determine the amount of private
investment for their children, taking the value of ºt and e
g
t as given.
The budget constraints of a member of generation t when young and retired are
given, respectively, as (1¡¿t¡!)wtht = c
y
t +net+at and (1+½t+1)at+Tt+1 = co
t+1,
where ¿t is the social-security-tax rate, and ! is the education-tax rate. The lifetime
utility function of generation t is represented as3
ut = lnc
y
t + plnc
o
t+1 + ±nlnht+1:
By solving individuals’ optimization problems, the optimal values are given as
et =
°±
(1+p+°±n)It ¡
(1+p)
(1+p+°±n)ºt, c
y
t = 1
(1+p+°±n)It, co
t+1 =
p
(1+p+°±n)(1 + ½t+1)It, and
at =
p
(1+p+°±n)It ¡
Tt+1
(1+½t+1), where It´ (1 ¡ ¿t ¡ !)wtht +
Tt+1
(1+½t+1).
The aggregate production function at time t is given as Yt = AK®
t (htNt)1¡®,
where Yt, A, Kt, and ®2(0;1) respectively denote the aggregate output, the produc-
tivity parameter, the physical capital that fully depreciates in the production pro-
cess, and the share of physical capital. Because the factor markets are presumed to
be perfectly competitive, the ﬁrms take factor prices as given: wt = A(1¡®)( Kt
htNt)
®,
(1 + rt) = A®( Kt
htNt)
®¡1:
The government allocates the ¹ 2 [0;1) portion of the education-tax revenue to
3With an uncertain lifetime, p2(0;1], this utility form is employed by Pecchenino and Pollard
(2002) and Yakita (2001), among others.
5education subsidies and the (1¡¹) portion of it to public schools.4 Here, ¹ is treated
as a predetermined parameter and is constant over time. Budget constraints of the
government at time t are
⅌Education subsidies⅍; ºt =
¹!
n
wtht;
⅌Public schools⅍; e
g
t =
(1 ¡ ¹)!
n
wtht:
Social security payments are speciﬁed as a replacement rate, Á 2 (0;1), on
current workers’ wage income, as
Tt = Áwtht: (1)
Total pension beneﬁts must be balanced by pension revenue:
TtpNt¡1 = ¿twthtNt: (2)
From (1) and (2), the social-security-tax rate is determined as
¿t =
pÁ
n
´ ¿:
4When ¹ = 0, it means PSP is operated.
6Note that this contribution rate is increasing in the degree of aging, p.
By employing the capital market-clearing condition, Kt+1 = atNt, the human
capital level of generation t + 1 is represented as
ht+1 = (
s¤
ºf1 +
(1¡®)
® ¿g°±
p
)
°(
(1 ¡ ¹)!
n
)
1¡°wtht´h
¤
ºwtht;
where
s
¤
º ´
pf1 ¡ ¿ ¡ (1 ¡ ¹)!g
f(1 + p + °±n) + (1 + °±n)
(1¡®)
® ¿g
:
Consequently, the per-capita growth rate at time t is constant over time:
(1 + gº;t) ´
Yt+1
Nt+1
Yt
Nt
=
A(1 ¡ ®)
n
s
¤
º
®(h
¤
ºn)
1¡® ´ (1 + gº): : ESP
When there are no education subsidies, ¹ = 0, the per-capita growth rate at time
t is given as
(1 + gt) ´
Yt+1
Nt+1
Yt
Nt
=
A(1 ¡ ®)
n
s
¤®(h
¤n)
1¡® ´ (1 + g); : PSP
where
s
¤ ´
p(1 ¡ ¿ ¡ !)
f(1 + p + °±n) + (1 + °±n)
(1¡®)
® ¿g
;
7and
h
¤ ´ (
s¤f1 +
(1¡®)
® ¿g°±
p
)
°(
!
n
)
1¡°:
3 Education subsidies vs. Public schools
We shall set the ratio of the growth rate of ESP, (1 + gº), to the growth rate of
PSP, (1 + g), as ℡
G(p) ´
(1 + gº)
(1 + g)
= (1 ¡ ¹)(
1 ¡ ¿ ¡ (1 ¡ ¹)!
(1 ¡ ¿ ¡ !)(1 ¡ ¹)
)
®+(1¡®)°:
Inferring that the social-security-tax rate, ¿, is inﬂuenced by the degree of aging,
p, we obtain the following proposition.
Proposition.
(1) When the education-tax rate is higher than !H, the ESP implies a higher growth
rate than the PSP.
(2) When the education-tax rate is medium, as !L · ! · !H, the ESP implies:
(2a) a higher growth rate if the degree of aging is higher than ˆ p
(2b) a lower growth rate if the degree of aging is lower than ˆ p
than the PSP.
(3) When the education-tax rate is lower than !L, the ESP implies a lower growth
8rate than the PSP.
These threshold values are:
!H =
1 ¡ (1 ¡ ¹)
1
®+°(1¡®)¡1
1 ¡ (1 ¡ ¹)
1
®+°(1¡®)
2 (0;1);
!L =
(1 ¡
Á
n)f1 ¡ (1 ¡ ¹)
1
®+°(1¡®)¡1g
1 ¡ (1 ¡ ¹)
1
®+°(1¡®)
2 (0;1);
ˆ p =
n
Á
[(1 ¡ !) ¡ f1 ¡ (1 ¡ ¹)!g(1 ¡ ¹)
1
®+°(1¡®)¡1]
f1 ¡ (1 ¡ ¹)
1
®+°(1¡®)¡1g
2 (0;1):
Proof. See Appendix.
0 1
1
p
!L !H !
ˆ p
(1)
(2a)
Figure 1: The growth rate of ESP is higher than that of PSP in shaded area.
If public-school investment by the government is overly large compared to pri-
vate investment, transferring some part of education revenue from public-school
9investment to education subsidies that complement private investment leads to
more eﬃcient allocation and yields higher growth rates. In our model, parental in-
vestment becomes smaller in such economies in which parents have smaller after-tax
income because of a higher education-tax rate (i.e. Proposition (1)), and those in
which parents intend to save a larger part of income for a longer retirement period
in spite of heavier social-security burdens (i.e. Proposition (2a)). In other words,
except when the education-tax rate is suﬃciently low (0 < ! < !L), parental pri-
vate investment is much smaller than public-school investment in aging economies.
In that event, the introduction of education subsidies leads to higher growth rates.
The implications of our results are similar to those of Kaganovich and Zilcha
(1999): when private investment is not made suﬃciently by parents, the com-
plements to private investment using education subsidies engender higher growth
rates under the same tax revenue.5 However, although the threshold levels of
parental preferences for their children hold important meaning for results revealed
in Kaganovich and Zilcha (1999), the levels of preferences vary among individuals
and are diﬃcult to measure. Our contribution is to have demonstrated that two
observable parameters – the degree of aging in the economy and the education-tax
5When parental investment is suﬃciently large because of a lower education-tax rate and a
lower aging degree (the lower left area in Fig. 1), the ESP engenders a lower growth rate than
the PSP. This result diﬀers from that of Kaganovich and Zilcha (1999), in which the introduction
of education subsidies has no eﬀect on the growth rate if parents invest much in their children
because of larger preference for their children.
10rates – become thresholds of determining the subsidies’ eﬀects by incorporating an
uncertain lifetime into the model of Kaganovich and Zilcha (1999).
4 Concluding remarks
We have examined how the introduction of education subsidies aﬀects growth rates,
incorporating an uncertain lifetime. We have demonstrated that introducing subsi-
dies engenders higher growth rates in aging economies, except when the education-
tax rate is suﬃciently low.
11Appendix.
Proof. Using ¿ =
pÁ
n , we recognize that the ratio, G(p), is increasing in aging
degree, p, as
sign(
@G(p)
@p
) = sign(
Á
n
f® + (1 ¡ ®)°g¹!
f1 ¡ ¿ ¡ (1 ¡ ¹)!g(1 ¡ ¿ ¡ !)
) > 0:
Initially, we shall see at p = 0. The range of the education-tax rate, which is
satisﬁed with
G(p = 0) ¸ 1;
is expressed as
! ¸
1 ¡ (1 ¡ ¹)
1
®+°(1¡®)¡1
1 ¡ (1 ¡ ¹)
1
®+°(1¡®)
´ !H:
Therefore, when the education-tax rate is higher than !H, G(p) > 1 is always
satisﬁed in p 2 (0;1]. The ESP always engenders a higher growth rate.
Secondly, in a similar fashion, we check at p = 1. The range of the education-tax
rate, which is satisﬁed with
G(p = 1) ¸ 1;
is given as
!¸
(1 ¡
Á
n)f1 ¡ (1 ¡ ¹)
1
®+°(1¡®)¡1g
1 ¡ (1 ¡ ¹)
1
®+°(1¡®)
´ !L:
12That is, when the education-tax rate is !L · ! · !H, G(p) > 1 is satisﬁed at least
at p = 1. In contrast, when the education-tax rate is ! < !L, G(p) remains less
than 1 over p 2 (0;1]. Consequently, when the education-tax rate is lower than !L,
the PSP always leads to a higher growth rate.
Finally, in the case where education-tax rate is medium, as !L · ! · !H, the
value of G(p), which is an increasing function in p, is less than 1 at p = 0 and
more than 1 at p = 1. Here, a threshold value of p is satisﬁed with G(p) = 1. This
threshold value is expressed as
ˆ p =
n
Á
[(1 ¡ !) ¡ f1 ¡ (1 ¡ ¹)!g(1 ¡ ¹)
1
®+°(1¡®)¡1]
f1 ¡ (1 ¡ ¹)
1
®+°(1¡®)¡1g
(!L · ! · !H):
Accordingly, when the economy’s degree of aging is lower than ˆ p, the value of G(p)
remains less than 1. When the degree of aging is higher than ˆ p, G(p) is greater
than 1. Consequently, it is only in aging economies (ˆ p < p) that the ESP leads to
a higher growth rate when the education-tax rate is medium, as !L · ! · !H.
13References
[1] Blanchard, O.J., 1985. Debt, deﬁcits, and ﬁnite horizons. Journal of Political
Economy 93, 223-247.
[2] Glomm, G., Ravikumar, B., 1992. Public versus private investment in hu-
man capital: endogenous growth and income inequality. Journal of Political
Economy 100, 818-834.
[3] Kaganovich, M., Zilcha, I., 1999. Education, social security, and growth. Jour-
nal of Public Economics 71, 289-309.
[4] Pecchenino, R.A., Pollard, P.S., 2002. Dependent children and aged parents:
funding education and social security in an aging economy. Journal of Macroe-
conomics 24, 145-169.
[5] Wigger, B.U., 2004. Are higher education subsidies second best? Scandinavian
Journal of Economics 106, 65-82.
[6] Yakita, A., 2001. Uncertain lifetime, fertility, and social security. Journal of
Population Economics 14, 635-640.
[7] Zhang, J., and Casagrande, J., 1998. Fertility, growth, and ﬂat-rate taxation
for education subsidies. Economics Letters 60, 209-216.
14

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