Discussion Papers In Economics
And Business

The Effect of Education Subsidies in an Aging
EconomyMegumi MOCHIDA
Discussion Paper 05-30

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe Effect of Education Subsidies in an Aging
EconomyMegumi 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 andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe Effect of Education Subsidies in an Aging
Economy∗Megumi MOCHIDA†

Abstract
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.
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.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 5600043, Japan. E-mail address：cg088mm@srv.econ.osaka-u.ac.jp (M. Mochida).

1

1Introduction

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 youngworking 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 tocomplement 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 thegovernment 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 anegative 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.

2

Incorporating an uncertain lifetime to a model that is based on Kaganovich andZilcha (1999), we demonstrate that, in the case of aging economies, the introductionof education subsidies engenders a higher growth rate, except when the educationtax 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), anda retirement period (of uncertain length). The individual is alive at the beginningof the third period with probability p ∈ (0, 1]. Let Nt denote the number ofworking-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 + ρt+1 ) = (1+rpt+1 ) , 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

1

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

3both education subsidies and public schools. Secondly, we show the equilibriumof 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 education expenditures for their children, net , their current consumption, cyt , and investing in annuities, at , for their post-retirement consumption, cot+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 tand called generation t + 1. Human capital is accumulated according to:

ht+1 = (et + νt )γ (egt )1−γ .

In that equation, γ ∈ (0, 1) denotes the efficiency of education input provided
privately, such as textbooks and tutors. The education subsidy given by the government is νt , and egt is the public-school quality provided by the government.2
The subsidies are used directly for human capital accumulation as in the methods

We call egt ”public-school quality”, as do Glomm and Ravikumar (1992), to distinguish strictly
between νt , which is subsidized to private parental investment, and egt . We can also regard egt asother public educational investment like libraries, museums, and so on.
24

of Kaganovich and Zilcha (1999). Individuals determine the amount of private
investment for their children, taking the value of νt and egt as given.
The budget constraints of a member of generation t when young and retired are
given, respectively, as (1 − τt − ω)wt ht = cyt + net + at and (1 + ρt+1 )at + Tt+1 = cot+1 ,where τt is the social-security-tax rate, and ω is the education-tax rate. The lifetime
utility function of generation t is represented as3

ut = lncyt + plncot+1 + δnlnht+1 .

By solving individuals’ optimization problems, the optimal values are given as
(1+p)
γδ
p
1
et = (1+p+γδn)
It − (1+p+γδn)νt , cyt = (1+p+γδn)
It , cot+1 = (1+p+γδn)(1 + ρt+1 )It , and
Tt+1
Tt+1
p
at = (1+p+γδn)
It − (1+ρ, where It ≡ (1 − τt − ω)wt ht + (1+ρ
.
t+1 )
t+1 )

The aggregate production function at time t is given as Yt = AKtα (ht Nt )1−α ,where Yt , A, Kt , and α∈(0, 1) respectively denote the aggregate output, the productivity parameter, the physical capital that fully depreciates in the production process, and the share of physical capital. Because the factor markets are presumed toαt) ,be perfectly competitive, the firms take factor prices as given: wt = A(1−α)( hK
t Nt

α−1

t
(1 + rt ) = Aα( hK
)
t Nt

.

The government allocates the µ ∈ [0, 1) portion of the education-tax revenue to3With an uncertain lifetime, p∈(0, 1], this utility form is employed by Pecchenino and Pollard(2002) and Yakita (2001), among others.

5education subsidies and the (1−µ) portion of it to public schools.4 Here, µ is treatedas a predetermined parameter and is constant over time. Budget constraints of thegovernment at time t are〔Education subsidies〕;

νt =

µω
w t ht ,n〔Public schools〕;

egt =(1 − µ)ω
w t ht .
nSocial security payments are specified as a replacement rate, φ ∈ (0, 1), on
current workers’ wage income, as

Tt = φwt ht .

(1)

Total pension benefits must be balanced by pension revenue:

Tt pNt−1 = τt wt ht Nt .

From (1) and (2), the social-security-tax rate is determined as

τt =

4

pφ
≡ τ.
n

When µ = 0, it means PSP is operated.

6(2)

Note that this contribution rate is increasing in the degree of aging, p.
By employing the capital market-clearing condition, Kt+1 = at Nt , the humancapital level of generation t + 1 is represented ass∗ {1 + (1−α)τ }γδ γ (1 − µ)ω 1−γαht+1 = ( ν) () wt ht ≡h∗ν wt ht ,
p
n

where
s∗ν ≡

p{1 − τ − (1 − µ)ω}{(1 + p + γδn) + (1 + γδn) (1−α)τ}α.

Consequently, the per-capita growth rate at time t is constant over time:
Yt+1
Nt+1
(1 + gν,t ) ≡ Yt
Nt=

A(1 − α) ∗ α ∗ 1−αsν (hν n)≡ (1 + gν ).
n

: ESP

When there are no education subsidies, µ = 0, the per-capita growth rate at time
t is given as
Yt+1
Nt+1
(1 + gt ) ≡ Yt
Nt=

A(1 − α) ∗ α ∗ 1−αs (h n)≡ (1 + g),n: P SPwheres∗ ≡p(1 − τ − ω){(1 + p + γδn) + (1 + γδn) (1−α)
τ}
α

7

,

and
s∗ {1 + (1−α)τ }γδ γ ω 1−γαh ≡() ( ) .
pn∗3

Education subsidies vs. Public schools

We shall set the ratio of the growth rate of ESP, (1 + gν ), to the growth rate ofPSP, (1 + g), as

G(p) ≡(1 + gν )1 − τ − (1 − µ)ω α+(1−α)γ= (1 − µ)().
(1 + g)(1 − τ − ω)(1 − µ)Inferring that the social-security-tax rate, τ , is influenced by the degree of aging,p, we obtain the following proposition.
Proposition.
(1) When the education-tax rate is higher than ωH , the ESP implies a higher growthrate than the PSP.
(2) When the education-tax rate is medium, as ωL ≤ ω ≤ ωH , the ESP implies:(2a) a higher growth rate if the degree of aging is higher than p̂(2b) a lower growth rate if the degree of aging is lower than p̂than the PSP.
(3) When the education-tax rate is lower than ωL , the ESP implies a lower growth8rate than the PSP.
These threshold values are:1ωH =1 − (1 − µ) α+γ(1−α) −1
1

1 − (1 − µ) α+γ(1−α)∈ (0, 1),1ωL =(1 − nφ ){1 − (1 − µ) α+γ(1−α) −1 }1

1 − (1 − µ) α+γ(1−α)∈ (0, 1),1n [(1 − ω) − {1 − (1 − µ)ω}(1 − µ) α+γ(1−α) −1 ]∈ (0, 1).
p̂ =1
φ
{1 − (1 − µ) α+γ(1−α) −1 }Proof. See Appendix.

p
1

(2a)

p̂

(1)

0

ωL

ωH

1 ω

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 private investment, transferring some part of education revenue from public-school
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investment to education subsidies that complement private investment leads tomore efficient allocation and yields higher growth rates. In our model, parental investment 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 inwhich parents intend to save a larger part of income for a longer retirement periodin spite of heavier social-security burdens (i.e. Proposition (2a)). In other words,
except when the education-tax rate is sufficiently low (0 < ω < ωL ), parental private 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 complements 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 revealedin 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

5

When parental investment is sufficiently large because of a lower education-tax rate and alower 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 introductionof education subsidies has no effect on the growth rate if parents invest much in their children
because of larger preference for their children.

10rates – 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 subsidies engenders higher growth rates in aging economies, except when the educationtax rate is sufficiently low.

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Appendix.
Proof. Using τ = pφ
, we recognize that the ratio, G(p), is increasing in aging
ndegree, p, as

sign(

∂G(p)
φ
{α + (1 − α)γ}µω) = sign() > 0.
∂pn {1 − τ − (1 − µ)ω}(1 − τ − ω)Initially, we shall see at p = 0. The range of the education-tax rate, which issatisfied withG(p = 0) ≥ 1,
is expressed as1ω≥

1 − (1 − µ) α+γ(1−α) −1
1

1 − (1 − µ) α+γ(1−α)≡ ωH .

Therefore, when the education-tax rate is higher than ωH , G(p) > 1 is always
satisfied in p ∈ (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 withG(p = 1) ≥ 1,
is given as1ω≥(1 − nφ ){1 − (1 − µ) α+γ(1−α) −1 }1

1 − (1 − µ) α+γ(1−α)
12≡ ωL .

That is, when the education-tax rate is ωL ≤ ω ≤ ωH , G(p) > 1 is satisfied at least
at p = 1. In contrast, when the education-tax rate is ω < ωL , G(p) remains lessthan 1 over p ∈ (0, 1]. Consequently, when the education-tax rate is lower than ωL ,the PSP always leads to a higher growth rate.
Finally, in the case where education-tax rate is medium, as ωL ≤ ω ≤ ωH , thevalue of G(p), which is an increasing function in p, is less than 1 at p = 0 andmore than 1 at p = 1. Here, a threshold value of p is satisfied with G(p) = 1. This
threshold value is expressed as1n [(1 − ω) − {1 − (1 − µ)ω}(1 − µ) α+γ(1−α) −1 ]p̂ =1
φ{1 − (1 − µ) α+γ(1−α) −1 }(ωL ≤ ω ≤ ωH ).

Accordingly, when the economy’s degree of aging is lower than p̂, the value of G(p)remains less than 1. When the degree of aging is higher than p̂, G(p) is greaterthan 1. Consequently, it is only in aging economies (p̂ < p) that the ESP leads toa higher growth rate when the education-tax rate is medium, as ωL ≤ ω ≤ ωH .

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References[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 human capital: endogenous growth and income inequality. Journal of Political
Economy 100, 818-834.
[3] Kaganovich, M., Zilcha, I., 1999. Education, social security, and growth. Journal 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 Macroeconomics 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 taxationfor education subsidies. Economics Letters 60, 209-216.

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