Discussion Paper No.319
Public education, endogenous fertility
and economic growth

Ko Shakuno

May 14, 2014

TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND
MANAGEMENT TOHOKU UNIVERSITY
27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANPublic education, endogenous fertility
and economic growth

Ko Shakuno∗†

May 16, 2014

Abstract
Using a closed-economy overlapping generations model with endogenous fertility, child quality choice and human capital accumulation, this paper examine the eects of public investment in education
on fertility rate and per capita output growth rate under a pay-as-yougo (PAYG) social security system. Parents face a trade-o between thequantity and quality of children. Dierently from previous studies, thispaper shows that there is an inverted U-shaped relation between public investment in education and fertility. Small sized public education
policy stimulates fertility and impedes growth.

Keywords Public education・Private education・Fertility・Human
capital

JEL-Classication J13・I22・H52

I am grateful to Ryo Horii for insightful comments and frequent suggestions. I would
like to thank Akiomi Kitagawa, Masao Nakagawa, Yunfang Hu, Hiroshi Yoshida, Keishun
Suzuki and the seminar participants held at Tohoku University on April, 2014. All remaining errors are my own.
†
Graduate School of Economic and Management, Tohoku University, 27-1 Kawauchi,
Aoba-ku, Sendai-shi 980-8576, Japan. E-mail address: b4ed1002@s.tohoku.ac.jp∗

1

1 Introduction
Recently many developed countries face the declining birth rate and increasing longevity. Decreasing young population is a matter of public concern
in these countries because it decreases labor force in the future and reduces
the contributions to the social security system. Policy makers aim at increasing the population growth rate and the ratio of economically active workers
to total population by various public investments in children e.g., child allowances, social security system and public education. Some literature dealt
with public nanced child care policy in an overlapping generations model.
For example, van Groezen et al (2003) analyzes the relation between childcare subsidy, pay-as-you-go (PAYG) social security system and fertility. But
these literature ignore a child quality issue. Becker and Lewis (1973) showsa quality-quantity trade-o of fertility choice. Parents face a trade-o between having many children and spending large resources on their quality
e.g., health and education, of each children. If parents decide fertility andeducation jointly, government investment in children not only has an eect
on their fertility choice also aects human capital accumulation.
Few studies examine the eects of public education policy on private
investment in education and fertility and growth under PAYG social security system. de la Croix and Doepke (2004) extended Kaganovich and Zilcha(1992) to examine the choice of an education regime in an endogenous growthmodel with endogenous fertility choice. They show that at public education
regime economic growth rate is higher and income inequality is lower than at
private education regime. But their analysis is limited to two polar schooling systems i.e., the government or household only supplies the educational
investment in their articles. Under public education regime, parents are individually not allowed to choose the level of education for their children. In
reality, in many developed countries private education coexists with public
education. Table 1 shows the percentage of private investment in education
to total educational expenditure in OECD countries.
2

prime.& second.
Japan
10.2
Korea
19.2
USA
8.3
Italy
2.2
Germany
12.9
France
7.4
Sweden
0.1

ter.
65.6
76.7
64
32.4
11.8
15.6
8.7

Table 1: The percentage of private investment in education to total educational expenditure in OECD countries by level of education (primary, secondary and post-secondary non-tertiary education and tertiary education).
The data are from OECD (2010a).
The issue addressed in this paper are as follows: how a change in the
allocation between public investment in education and PAYG social security
benets aects fertility and growth ? This paper extends de la Croix andDoepke (2003) to investigate the eect of public education on fertility byintroducing the parental fertility decision and the child quality issue into athree-period-lived overlapping generations model with PAYG social security
system. Dierently from previous works, I assume that private education
coexists with public education. Government nances two public program;
PAYG social security program and public education program.This paper investigates the eects of an increase in public funding for children's educationon the parent's fertility and educational decision1 .
The main results are as follows. There is an inverted U-shaped relation between public investment in education and fertility. And an V-shaped relation
between public education and growth appear. Small sized public education
policy generates a substitution eect from quality of children to quantity ofchildren in contradiction to Zhang (1997). On the contrary, in the case inwhich the size is suciently high, increasing public investment in education
decreases fertility and stimulates per capita output growth for a given PAYG
1

But I ignore income inequality to maintain tractability of the model.
3contribution rate by reducing PAYG social security benets..
This paper is similar to some previous papers that study the relation between private and public investment in education. Kaganovich and Zilcha(1999) studies the eects of public education policy on private educational
choice, welfare and growth. Golmm and Kaganovich (2003) explores therelationship between parental educational and public investment in humancapital formation. But they ignore the response of fertility to public education policy. Omori (2010) built a overlapping generations model with twointergenerational transfer program and endogenous fertility choice. But with
their modeling public education policy have a monotonic eect on fertility
rate and it does not examine the eect of public education policy on percapita output growth.
The rest of paper is organized as follows: Section 2 presents the model usedin this article. Section 3 examines fertility rate and per capita output growthrate in the balanced growth path. Section 4 devotes to show the eects ofpublic education on fertility and growth analytically. The last section oers
some concluding remarks.

2 ModelTime is discrete and goes from 0 to innite. Consider a closed-economy thatis populated by overlapping generations of people who live for three periods.
Each individual lives for three periods; childhood, adulthood and old-age. He
receives education in the rst period, works, rears and educates children in
the second period and retires in the third period. The length of each period
is normalized to one. All decisions are made in the adulthood period of life.
Individuals care about adulthood consumption ct , old-age consumption dt+1 ,their number of children nt , and the human capital of children et . Total
population Pt evolves over time according to Pt+1 = nt Pt .

4

2.1

HouseholdPreference of any individual in generation t are given by:log ct + β log dt+1 + γ log nt ht+1 ,(1)where ct , dt+1 are consumption proles of generation t. The parameter 0 <β < 1 is a psychological discount factor and 0 < γ < 1 is a altruism factor.
Raising one child takes a constant fraction 0 < ϕ < 1 of an adult's time.
This fraction of time is exogenous and constant. An adult has to choose
consumption proles ct and dt+1 , saving for old-age st , number of children
nt , and private educational investment per child et . The budget constraint
for the adulthood period is2 :ct = (1 − τ ) {(1 − ϕnt )wt ht − et nt wt ht } − st ,The budget constraint for the old-age period is:
dt+1 = Rt+1 st + Tt+1 ,Using these budget constraints, I have a lifetime budget constraint as follows:(1 − τ ) {(1 − ϕnt )wt ht − et nt wt ht } + Tt+1 /Rt+1 = ct + dt+1 /Rt+1 .

(2)

The human capital of generation t + 1 depends on the human capital of theirparents ht , the public education investment Et and the private education
investment et . The human capital of generation t + 1 is assumed to be
produced as follows:
ht+1 = µ(Et + et )η ht .

(3)

I assume that private educational spending for children is tax-deductible in the same
way as de la Croix and Doepke (2003). This assumption is not essential to my main
results.
2

5

Here h ≡ H/P represents per capita human capital. µ is a eciency parameter and 0 < η < 1. The public educational investment in children is obtainedfor free and perfectly substitutes to the educational investment provided bytheir parents3 . The private investment in children's education e has to be
paid for by their parents.
Given the wage rate wt , the price of physical capital Rt+1 , the humancapital level ht , the child rearing time per child ϕ, the policies of government, individuals during adulthood period in generation t choose st , nt , et tomaximize their lifetime utility Eq. (1). To nd the solution to the household
maximization problem, I solve the following Kuhn-Tucker problem:
max log ct + β log dt+1 + γ log nt ht+1

c,d,e,n+λ1 {(1 − τ )(1 − ϕnt wt ht − et nt wt ht ) + Tt+1 /Rt+1 − ct − dt /Rt+1 }+λ2 et ,where λ1 and λ2 are shadow prices. The Kuhn-Tucker conditions are:1
= λ1 ,
ct
β
λ1
=
,
dt+1
Rt+1
γ
= λ1 (1 − τ )(ϕ + et )wt ht ,ntγη= λ1 (1 − τ )nt wt ht + λ2 ,Et + etdt+1
ct += (1 − τ )(1 − ϕnt wt ht − et nt wt ht ),Rt+1
λ2 et = 0.

Public education policy act as the device that enhance the eciency of human capitalformation in Omori (2010) and Fanti and Gori (2011). In contrast, public investment ineducation is substituting for private input in Zhang (1997) and Azarnert (2010).
3

6

From these Kuhn-Tucker conditions, the saving level of generation t is given
by:
(
st =

1
1+β+γ

)
{β(1 − τ )wt ht − (1 + γ)Tt+1 /Rt+1 }.

(4)

If the return on private investment in education is higher than the returnof quantity of children, parents invest a part of their disposal income ineducation et > 0. However, if the return on private investment in education
is below the return of quantity of children, a corner solution et = 0 exists.
An optimal non-corner solution for the number of children and the parental
investment in children's education at time t is given by:(1 − τ )wt ht + Tt+1 /Rt+1,
nt = γ̂(1 − η)
(1 − τ ) (ϕ − Et ) wt ht
(
)
1
et =
{η(1 − τ )ϕwt ht − Et }.
1−η

(5)
(6)

The relation between private and public investment in education is illustrated
in Figure 1. Public education system generates a substitution eect from
quality of children to quantity of children and there is a negative relation
between private and public investment in education. The optimal corner
solution for the number of children and the parental investment in education
at time t is:(1 − τ )wt ht + Tt+1 /Rt+1,(1 − τ )ϕwt ht= 0,nt = γ̂

(7)

et

(8)

γ
where γ̂ ≡ 1+β+γ. Notice that fertility at equilibrium is positively related tolifetime income i.e., the wage earned during the adulthood period of life and
PAYG social security benets during the old-age period of life, and negatively
related to child-rearing costs i.e., the private investment in education and thexed child rearing-cost.

7

et
φη
1-η

̲̲̲̲̲

φη

Et

Figure 1: The relation between public and private investment in education
2.2

Firm

A single representative rm operates the technology:
Yt = Kt α Lt 1−α ,where Kt is the aggregate physical capital and Lt is the aggregate labor
supply at time t. Parameter 0 < α < 1 is expressed as the share of physicalcapital in production. Physical capital depletes completely after one period
use in production. The rm chooses inputs by maximizing prots Yt −wt Lt −Rt Kt at time t. From the rst order conditions for prot maximization, factor
prices are derived as follows:
(

Kt
Lt

)α

wt = (1 − α)( )α−1Kt,
Rt = α
Lt

,

(9)
(10)

where wt and Rt means the wage rate and the interest factor at time t
respectively.
8

2.3

Government

I assume that the government behaves under a balanced budget restriction.
Two policies are funded by income tax τ collected at constant rate. I denea parameter χ which represents the fraction of government revenue devoted
to public investment in education. Public investments in education policy
and PAYG social security policy in period t is given by:
Tt Pt−1 = τ (1 − χ)Lt wt ,Et wt ht Pt = τ χLt wt ,

(11)
(12)

where Tt is the per capita PAYG social security benet and Et is the per
child public nanced education expenditure in period t. In the following
discussions, the government predetermines the sequences of education tax
rate τ and fraction χ.
2.4

Markets

The equilibrium conditions in the physical capital market and the labormarket are written as:
Kt+1 = st Pt ,Lt = Pt (1 − ϕnt − et nt ) ht .

(13)
(14)

Now I can dene the competitive equilibrium in our model as follows:Denition 1 Competitive Equilibrium:

Given the initial aggregate physical capital K0 , the initial stock of aggregate human capital H0 , the initial population size P0 , an equilibrium consistsof sequences of household's decision rules{ct , dt+1 , st , nt , et }∞t=0 , sequences of∞∞
prices {wt , Rt }t=0 , government expenditure {Et , Tt+1 }t=0 and aggregate quantities {Kt+1 , Lt }∞t=0 such that9

1. the households' decision rules {ct , dt+1 , st , nt , et }∞t=0 maximizes lifetime
utility function (1) subject to the lifetime budget constraint (2) and the human capital formation function (3);
2. the factor prices {wt , Rt }∞t=0 are such that markets clear, i.e., (9) and (10)hold;
3. the government expenditures {Et , Tt+1 }∞t=0 hold (11) and (12);∞
4. the aggregate valuables {Kt+1 , Lt }t=0 hold (13) and (14).

3 Fertility and Growth
In this section, I derive the fertility rate and the per capita output growthrate in the balanced growth path.
3.1

Fertility

From Eq. (4) and (13), the equilibrium condition in the capital market attime t can be rewritten as:(

Kt+1 =1
1+β+γ){β(1 − τ )wt ht − (1 + γ)Tt+1 /Rt+1 } Pt .

(15)

Using Eq. (9) and Eq. (11), the present value of per capita social security
benet at time t is given by:(
Tt+1 /Rt+1 = τ (1 − χ)

1−α
α

)
Kt+1 /Pt .

(16)

Substituting Eq. (16) into (15) for the aggregated physical capital dynamics,
I have:Kt+1 = S(τ, χ)(1 − τ )wt ht Pt ,

10

(17)

where S(τ, χ) represents the saving rate at equilibrium:
)
}−1
{
(
1−α
τ (1 − χ)S(τ, χ) ≡ β (1 + β + γ) + (1 + γ).
αFrom Eq. (5), (6), (7), (8) and (14), the labor supply at time t is given by:(18)

Lt = L(τ, χ)ht Pt .

where L(τ, χ) represents the working time of parents and is given by:{(L(τ, χ) ≡ 1 − γ̂ 1 + τ (1 − χ)

1−α
α

)

}
S(τ, χ) .

Notice that the labor supply does not depend on the private educational
decision of parents. Using Eq. (12) and (18), per child public education
expenditure is given by:
Et (τ, χ) =Ẽ(τ, χ),nt

where Ẽ = τ χL(τ, χ). Next I derive the fertility rate at the competitive
equilibrium. First, suppose that parents in period t invest a part of theirdisposal income in education i.e., et > 0. I term this case as the mixed
education case. From Eq. (5), the fertility rate at the equilibrium is givenby:{}nt = γ̂(1 − η)(1 + T (τ, χ)) + Ẽ(τ, χ) ϕ−1 ≡ n∗mix ,

where

(
T (τ, χ) ≡ τ (1 − χ)

1−α
α

(19)

)
S(τ, χ).

Second, suppose that parents in period t invest nothing in their children's
human capital accumulation i.e., et = 0. In this situation, parents nd it
11

optimal to leave their children to the public education institution completely.
They use all adulthood period resources to consumption, saving and childrearing. I term this situation as the complete public education case. From
Eq. (7), optimal number of children at the equilibrium is as follows:
nt = γ̂(1 + T (τ, χ))ϕ−1 ≡ n∗pub .

(20)

The number of children is constant over time in this model.

Lemma 1 For all contribution rate τ ∈ (0, 1) and fraction χ ∈ [0, 1], a risein the PAYG social security contribution rate τ (1 − χ) increases fertility n∗ .

3.2

Investment in educationτχcomplete public
educationΦ(τ,χ)

Φ(τ,1)

mixed education

τ(1-χ)

Figure 2: The function Φ(τ, χ)In order to investigate under what conditions parents privately invest
in their children's education, I derive a inner solution condition for private
education investment. From the Kuhn-Tucker conditions, the condition is12

written as follows:ηϕ −Ẽ(τ, θ)
> 0.
n∗mix

Arranging above condition, the following lemma provides a necessary andsucient condition for parental private investment in the balanced growthpath.

Lemma 2 Parents invest in their children's quality if and only if the size of
public investment in education τ χ satises:τχ <ηγ̂(1 + T (τ, χ))≡ Φ(τ, χ).
1 − γ̂(1 + T (τ, χ))(21)

Φ(τ, χ) is a positive function of the PAYG social security contribution rateτ (1 − χ). Figure 2 illustrates the threshold level Φ(τ, χ). Φ(τ, 1) is thethreshold value before applying the PAYG social security system. If the sizeof public education τ χ is larger than the threshold level Φ(τ, χ), parents do
not invest their children's quality privately. However, if the size of public
education is lowers than it, parents invest a part of their disposal income intheir children's education. From Eq. (19) and (20) I obtain the fertility ratein the balanced growth path as follows:
{
∗

n =

{γ̂(1 − η)(1 + T (τ, χ)) + E(τ, χ)} ϕ−1 ≡ n∗mix (χ < χ̃),γ̂(1 + T (τ, χ))ϕ−1 ≡ n∗pub(χ̃ ≤ χ).

(22)

where χ̃ means the fraction which satises the equation τ χ̃ = Φ(τ, χ̃). Next
I examine the dynamics in this model.

13

3.3

Dynamics

I dene a capital/labor ratio as k ≡ K/L. Using Eq. (17), (18) and (22),I derive:{kt+1 =(1 − τ )S(τ, χ)L(τ, χ)n∗ (ht+1 /ht )}(1 − α)ktα .

(23)

From the assumption 0 < α < 1, this dynamics always converges to thecapital/labor ratio k monotonously.
{k=(1 − τ )S(τ, χ)(1 − α)L(τ, χ)n∗ (ht+1 /ht )

1
} 1−α

.

(24)

Using the Production technology and Eq. (18), per capita output growthrate at time t is written as follows:
Yt+1 /Nt+1,
Yt /Nt()α
ht+1 kt+1
=
.
ht
kt

1 + gt =In the balanced growth path, the capital/labor ratio is constant kt = kt+1 =k . Per capita output growth rate in the balanced growth path can be written
as follows:
{

∗

1+g =

∗
µη̂(ϕ − Emix (τ, χ))η ≡ 1 + gmix(χ < χ̃),η∗µEpub (τ, χ) ≡ 1 + gpub(χ̃ ≤ χ),where
Emix (τ, θ) ≡ Ẽ(τ, χ)/n∗mix ,Epub (τ, θ) ≡ Ẽ(τ, χ)/n∗pub .
η
η̂ ≡
1−η

14

(25)

4 Policy experiments
In this section, I examine the eect of reallocating public funds fromPAYG social security benets to public investment in education on fertility
in the balanced growth path. I impose following parameter restrictions of
the model:

Assumption 1γ̂ < α , τ < 0.5, η < 0.5.

I can show the eect of public education on fertility in the balanced growth
path as following proposition:
Panel (A)Panel (B)n*(1+g*)(1+g0)

n０
〜

χ

〜

χ

χ

χ

Figure 3: The eect of reallocating public funds from PAYG social security
benets to public investment in education on fertility rate and per capita
output growth rate.

Proposition 1 Assuming that the tax rate is constant, if the size of public
investment in education is suciently small i.e., χ < χ̃, a reallocation publicfunds from PAYG social security benets to pubic education increases fertility

15

rate n∗mix . On the contrary, if the size of public investment in education issuciently large i.e., χ̃ ≤ χ, a reallocation public funds from PAYG social
security benets to pubic education decreases fertility rate n∗pub .

Proof 1 See Appendix.
Figure 3(A) summarizes the impact of public investment in education
on fertility, where n0 denotes the fertility rate if public education scheme
does not exist. The intuition behind Proposition 1 is as follows. If χ < χ̃,reallocating public nancial resources from PAYG social security policy topublic education policy has three eects on fertility: (1) a positive directeect that increases fertility by reducing per child education cost, (2) a negative direct eect that decreases fertility by reducing parent's lifetime budget,(3) a positive indirect eect that increases fertility by increasing per childpublic education expenditure. Two positive eects dominate a negative effect under Assumption 1. This results is similar to Azarnert (2010) which
studies the eect of free public education on fertility, private educational investments, and human capital accumulation at dierent stages of economic
development. In this paper, the availability of free public schooling crowds
out public education investment, stimulates fertility and impedes growth atadvanced stages of development.If χ̃ ≤ χ, a change in the allocation fromPAYG social security policy to public education policy has a direct negativeeect on fertility by reducing PAYG social security benets.
Next, I examine how reallocating public nancial resources from PAYG
social security benets to public investment in education aects per capita
output growth rate in the balanced growth path. From Eq. (22), I derive
the following proposition.

Proposition 2 Assuming that the wage tax rate is constant, if the size of

public investment in education is small i.e., χ < χ̃, a reallocation public funds
from PAYG social security benets to public investment in education slows∗). On the contrary, if the size of public
per capita output growth (1 + gmix
investment is suciently large i.e., χ̃ ≤ χ, a reallocation public funds from

16

PAYG social security benets to pubic investment in education stimulates per∗
capita output growth (1 + gpub).

Proof 2 See Appendix.
Figure 3(B) summarizes the impact of public investment in education on
per capita output growth rate, where (1 + g 0 ) denotes the growth rate if public education scheme does not exist. The intuition behind Proposition 2 is asfollows. From Eq. (25), per capita output growth rate depends on per childpublic investment in education E in the balanced growth path. If the sizeof public investment in education is small, a reallocating from PAYG social
security benets to public investment in education has three eects on per
child public education expenditure: (1) a positive direct eect that increases
public education expenditure by increasing the educational contribution rate,(2) a negative indirect eect that decreases public education expenditure bydecreasing labor supply and the tax base, (3) a negative indirect eect that
decreases per child public education expenditure by stimulating fertility rate.
A rst positive direct eect dominate other negative eects.
On the contrary, if the size of public investment in education is suciently
large, reallocating public funds from PAYG social security benets to public investment in education has three dierent eects on per capita public
education expenditure: (1) a positive direct eect that increases public education expenditure by increasing the educational contribution rate, (2) anegative indirect eect that reduces household's labor supply and tax base,(3) a positive indirect eect that increases per capita out public education
expenditure by reducing the number of child. Two positive eects always
dominates a negative indirect eect.

5 Conclusion
This paper examines the eects of an increase in public funding for education on the parental fertility and educational decision and per capita output
17

growth rate in the balanced growth path. The model used in this study is
the three-period-lived overlapping generation model, incorporating both thetrade-o between the quality and quantity of children. Government nances
PAYG social security program and public education program. Similar to
Zhang (1997) and Azarnert (2010), this paper assumes public input in education is substituting for private input at producing children's human capital.
First, this paper showed that under what conditions parents invest in theirchildren's quality privately. Dierently from previous studies, there are twodierent cases. When the size of public education expenditure is suciently
low and per child public investment in education is not large, parents educate their children privately. On the other hand, when per child public
educational investment is suciently large, parents leave their children's education to public education institution completely.
Second, I examine the eect of reallocating public funds from PAYG social
security benets to public investment in education on fertility and per capita
output growth in the balanced growth rate. When parents invest their children's education privately, public investment in education increases fertility
and impede growth. However when parents do not invest their children's
education at all, a higher public investment in education decreases fertility
rate and stimulate growth.

Appendix18

Proof of Proposition 1

1. Assuming that χ < χ̃, the proof of the rst part of proposition 1 uses thefollowing derivative:[]∂n∗mix∂T (τ, χ)−1=ϕ{γ(1 − η) − τ χγ}+ τ {(1 + β) − γT (τ, χ)} ,
∂χ
∂χ
= ϕ−1
[
{
}]
∂S(τ, χ)× 1 − γ̂ + γ̂ α̂ {τ (1 − χ) + (1 − η) − τ χ} S(τ, χ) + {(1 − η) − τ χ} (1 − χ),∂χwhere α̂ ≡ 1−α. Under assumption 1 the rst part of proposition 1 is obvious.
α2. Now the proof of second part of Proposition 1 straightforwardly derivesby the following derivative:∂n∗pub∂T (τ, χ)= γ̂ϕ−1
.
∂χ
∂χ
(τ,χ)
Since ∂T∂χ
< 0, the second part of proposition 1 is obvious.

2

Proof of Proposition 2

1. I derive the following derivative from Eq. (25):∂Emix= γ̂(1 − η)ϕ−1∂χ

{}∂ Ẽ(τ, χ)∂T (τ, χ)(1 + T (τ, χ)) −Ẽ(τ, χ) n∗−2
min .
∂χ
∂χ

(τ,χ)
Since ∂T∂χ
< 0 and 0 < ∂E(τ,χ), the rst part of proposition 2 is obvious.
∂χ

2. Next I consider the latter part of proposition 2. I calculate the following

derivative from Eq. (25):∂Epub= γ̂ϕ−1∂χ{}∂ Ẽ(τ, χ)∂T (τ, χ)(1 + T (τ, χ)) −Ẽ(τ, χ) n∗−2
pub .
∂χ
∂χ

19

In the same way as the rst part of proposition 2, the second part is obvious.
2

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