GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 269

Composition of Public Education
Expenditures and Human Capital
Accumulation

Katsuyuki Naito and Keigo Nishida

June 2012GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Composition of Public Education
Expenditures and Human Capital
Accumulation
Katsuyuki Naito ∗Keigo Nishida †Abstract
This paper provides a simple theory to study how the allocation of
public funds between primary and higher education aﬀects human capital accumulation. The allocation is endogenously determined throughmajority voting. Public funding for higher education is not supported
when a majority is poor. In some cases, higher education starts to berealized as a majority of individuals accumulate enough human capitalthrough primary education. Although the emergence of higher education can accelerate aggregate human capital accumulation, it widens
income inequality because the very poor are excluded from higher education and the declined budget share for primary education decreases
its quality.
JEL Classifications: D72, O11, O15, O40
Keywords: Public Education, Economic Development, Income Inequality, Majority Voting∗

Institute of Economic Research, Kyoto University, Yoshida Honmach, Sakyo-ku, Kyoto, 606-8501, Japan, Email: k.naito.71@gmail.com†
Graduate School of Economics, Kyoto University, Yoshida Honmach, Sakyo-ku, Kyoto,606-8501, Japan, Email: nishida.keigo.85x@st.kyoto-u.ac.jp1

1

Introduction

Human capital accumulation can be an engine of economic growth, and thegovernment has signiﬁcant roles in providing formal education. In many existing studies, human capital is produced in a single education sector, where
an amount of government expenditure is a direct input. In reality, however,an education system is divided into multiple stages, such as primary, secondary and tertiary education, and the government supports each of them.
Primary and secondary (K-12) education has a particularly big diﬀerence
from tertiary education in the sense that it is mandatory but serves everyone
for free although college education is optional and requires private spending.
It is relatively recently, however, that a number of studies have begun toanalyze how the allocation of government expenditures for diﬀerent education sectors aﬀects economic growth and income distribution (Restuccia and
Urrutia, 2004; Su, 2004; Blankenau, 2005; Blankenau et al., 2007; Arcalean
and Schiopu, 2010). In particular, with heterogeneity in income or innate
ability, a policy that changes budget allocations across diﬀerent education
sectors may increase the welfare of some individuals at the expense of others
as shown by Su (2004) and Blankenau et al. (2007), which provides fertile
ground for politico-economic analysis. For example, enhancing college education may most beneﬁt the rich but may not beneﬁt the poor who cannot
aﬀord to attend there or those whose private return from college is low dueto their low innate ability.
This paper proposes an overlapping generations model in which the allocation of a given level of public education expenditures for multiple education
sectors is determined via majority voting and analyzes its eﬀects on humancapital accumulation. Individuals live for two periods, childhood and adulthood, and they have children when they are adult. Individuals in adulthoodmake all the economic and political decisions but they care about humancapital of their children. The government operates two education sectors,
which we call primary and higher education sectors. Primary education iscompulsory but serves all childhood individuals free of charge. On the other
hand, higher education is optional and requires private expenditures. In eacheducation, private return on education depends on resource allocations fromthe government and parental human capital. Parents with low human capitalare unable to or unwilling to have their children receive higher education because its private return is low or they are too poor to cover its private costs.
Parents whose human capital is above some threshold are willing and able2

to have their children obtain higher education. We show that the individualwith median human capital is the decisive voter although preferences for the
allocation policy are not necessarily single-peaked.
Our model generates various dynamics of human capital accumulationand steady state income inequalities. As long as human capital of the medianvoter is below some threshold, all the resources are provided for primary education. In some cases, the median voter accumulates suﬃcient human capitalthrough primary education and a positive share of government expenditureis allocated to higher education. Although the emergence of higher education can facilitate aggregate human capital accumulation, it widens steadystate income inequality because the very poor are excluded from higher education and declined share for primary education diminishes its quality. In
other cases, public support for higher education is never realized if the initialhuman capital level of the median voter is low.
The relationship between the political economy of education policies andeconomic growth has been studied at least since Glomm and Ravikumar(1992) and Saint-Paul and Verdier (1993). A more recent example includes
Galor et al. (2009), who examine an incentive of agents who are better
endowed with production factors to prevent the implementation of public
education. They argue that landowners wish to block policy changes that
promote human capital formation and unequal distribution of landownershiphas an adverse eﬀect on growth.
The rest of this paper is organized as follows. Section 2 sets up a model,
and Section 3 analyzes individuals’ decisions of investment in higher education. Section 4 studies how the share of government budget for primary andhigher education is determined under majority voting. Section 5 examinesinteractions between the education policy and human capital accumulation.
Section 6 concludes.

2

The Model

We study an overlapping generations model. Each generation consists of aunit mass of individuals who live for two periods, childhood and adulthood.
Every individual, who belongs to a lineage indexed by i, has one child inthe second period. The only source of heterogeneity across individuals is thehuman capital of their parents. Individual i born in period t derives utility

3

from consumption in her adulthood and human capital of her child:
U (cit+1 , hit+1 ) = cit+1 + hit+1 ,(1)where cit+1 is the consumption in her adulthood and hit+1 is the humancapital of her child.
In the childhood, individuals make no decisions but receive compulsory
primary education supplied by the government. After the completion of the
primary education, their parents may decide to have them receive further
education, or higher education, which is also supplied by the government.
Higher education requires a private cost, whose size is ﬁxed and normalized
to one. Parents cannot borrow against future human capital of their children
and therefore must self-ﬁnance the cost. In the adulthood, individuals makeall the economic and political decisions. They supply their human capital
to a ﬁrm that transforms one unit of human capital to one unit of a ﬁnal
good. The ﬁnal good market is perfectly competitive and therefore the wage
per unit of human capital is one. Individuals also vote for an allocation of
government expenditures on primary and higher education, which determines
the quality of each education. After that, they decide, as parents, whether
they have their children receive higher education and then consume all theremaining wealth.
If individual i born in period t + 1 receives only primary education, she
accumulates human capital according to(hit+1 = B

P

GPt+1
ht

)β
β ∈ (0, 1), γ ∈ (0, 1),hγit ,(2)

where hit+1 is the human capital output, B P is the productivity parameterof primary education, GPt+1 is the government expenditure on primary education, hit is her parental human capital, and ht is the average (and aggregate)
human capital of the parental generation. The term GPt+1 /ht should be interpreted as the number of teachers hired at primary schools. Since the wage
per unit of human capital paid by the ﬁnal good sector is one, the government
must compensate the same wage rate to hire an old individual as a teacher.
The cost of hiring teachers are thus ht and GPt+1 /ht represents the number ofteachers.
If individual i receives higher education, she accumulates human capital4

according to(hit+1 = B

P

GPt+1
ht

)β

(
hγit + B A

GA
t+1
ht

)β
hγit ,(3)

A
where GAt+1 is the government expenditure on higher education and B is its
Aproductivity parameter. The term Gt+1 /ht represents the number of teachers
employed at higher education institutions. We assume thatBA ≥ BP .

(A.1)

Income is taxed at the rate of τ ∈ (0, 1), and the tax revenue is τ ht . The
government allocates the revenue to primary and higher education, whichleads toGPt+1 = xt+1 τ ht ,GA
t+1

= (1 − xt+1 )τ ht ,

(4)
(5)

where xt+1 ∈ [0, 1] is the share of government expenditure on primary education. In period t + 1, individuals born in period t vote for the value of xt+1 ,which aﬀects the human capital of their children. Substituting out GPt+1 andGAt+1 from (2) and (3) gives the human capital level of individual i born inperiod t + 1:
hit+1 = B P xβt+1 τ β hγit(6)

if she receives only primary education, andhit+1 = B P xβt+1 τ β hγit + B A (1 − xt+1 )β τ β hγit(7)

if she receives higher education.

3

Investment in Education

Given the share of government expenditure on primary and higher education,
each individual decides whether she invests in higher education for her child.
When individual i with hit has her child obtain higher education, (1) and (7)give her utility, V E (xt+1 , hit ), asV E (xt+1 , hit ) = [(1 − τ )hit − 1] + [B P xβt+1 + B A (1 − xt+1 )β ]τ β hγit .
5(8)The ﬁrst term is the consumption in her adulthood and the second term ishuman capital of her child. When individual i decides not to have her childreceive higher education, (1) and (6) give
V N (xt+1 , hit ) = (1 − τ )hit + B P xβt+1 τ β hγit ,(9)where V N (xt+1 , hit ) is her utility.
Individual i with hit is able to invest in higher education for her child if(1 − τ )hit ≥ 1, which is equivalent to
1
≡ H̄.
1−τ

hit ≥

(10)

Individual i is willing to have her child receive higher education if and only
if V E (xt+1 , hit ) ≥ V N (xt+1 , hit ). Simple calculations show
[

1
hit ≥
A
B (1 − xt+1 )β τ β] γ1≡ H(xt+1 ).

(11)H ′ (x) > 0 since smaller share of government expenditure on higher education
lowers its productivity, which makes more individuals unwilling to invest in
education for their children. H(x) satisﬁes(
H(0) =

1
A
B τβ

) γ1
and

lim H(x) = ∞.

(12)

x→1

We assume thatH̄ > H(0),(A.2)

which means that higher education is not aﬀordable for some individuals. In
the following analysis, it is useful to deﬁne x(hit ) by1
H [x(hit )] = hit ⇔ x(hit ) = 1 −
τ

(

1
A
B hγit

) β1
.

(13)

Figure 1 sketches H(x) as well as H̄.
Let us consider decisions of parents born in period t and their resulting
utility under given xt+1 . If 0 ≤ hit < H̄, individual i is unable or unwillingto have her child receive education and her utility is therefore V N (xt+1 , hit ).
If hit ≥ H̄, then there are two cases that must be considered; (i) 0 ≤ xt+1 ≤6

H(xt+1 )hit
H̄0x(hit ) 1

xt+1

Figure 1: Feature of H(x)x(hit ) and (ii) x(hit ) < xt+1 ≤ 1. In the case where 0 ≤ xt+1 ≤ x(hit ),individual i is willing and able to have her child get higher education, andher utility is V E (xt+1 , hit ). On the other hand, if x(hit ) < xt+1 ≤ 1, she isable but unwilling to have her child receive education because small shareon higher education, 1 − x, makes its productivity low. Her utility is hence
V N (xt+1 , hit ).

4

Political Equilibrium

Based on the analysis so far, this section identiﬁes the most preferred share foreach individual and characterizes political equilibrium. First of all, higher
education is not aﬀordable for any xt+1 ∈ [0, 1] if 0 ≤ hit < H̄. Since
V N (xt+1 , hit ) is increasing in xt+1 , an individual with hit ∈ [0, H̄) prefers
xt+1 = 1. In order to identify the most preferred share of an individual withhit ≥ H̄, let us deﬁne x∗ byx∗ = arg max V E (xt+1 , hit ).

(14)

xt+1

By the ﬁrst order condition,
1

∗

x =

(B P /B A ) 1−β1

1 + (B P /B A ) 1−β∈ (0, 1).

(15)

The remaining task in order to ﬁnd out the most preferred share is tocompare V E (x∗ , hit ) with V N (1, hit ). Although individuals prefer xt+1 = x∗under the presumption that they invest in higher education for their children,the individuals may prefer to set xt+1 = 1 and to choose not to invest in
7

Vit
VN
VE

xt+1
1
Figure 2: Welfare of individuals with hit ∈ [H̄, H(x∗ ))0 x(hit ) x∗

higher education. If x∗ > x(hit ), or equivalently hit < H(x∗ ), then individual
i with hit is unwilling to have her child receive higher education. Hence, sheobviously prefers xt+1 = 1 to xt+1 = x∗ . Figure 2 draws the welfare of suchindividuals as a function of xt+1 . If x∗ ≤ x(hit ), or hit ≥ H(x∗ ), then an
individual with hit is willing to have her child obtain higher education at
xt+1 = x∗ . However, V N (1, hit ) may exceed V E (x∗ , hit ). Simple calculations
show that V E (x∗ , hit ) ≥ V N (1, hit ) is equivalent to
{ A
[
]}
(16)
B (1 − x∗ )β − B P 1 − (x∗ )β τ β hγit ≥ 1.
Since we assume B A ≥ B P , the inequality can be rewritten as
[

1
hit ≥
A
∗
β
{B (1 − x ) − B P [1 − (x∗ )β ]} τ β] γ1≡ H̃ > H(x∗ ).

(17)

Figure 3 (a) and (b) depict the welfare of an individual with hit < H̃ andhit ≥ H̃, respectively. The following lemma summarizes the obtained results.
Lemma 1 Individual i with hit prefers xt+1 = x∗ if hit ≥ H̃, while prefers
xt+1 = 1 if hit < H̃.
It is easy to show that the individual
income }is the decisive{ N with medianE
voter although V (xt+1 , hit ) ≡ max V (xt+1 , hit ), V (xt+1 , hit ) is not necessarily single peaked. This result immediately follows from lemma 1. Let hmt
denote the human capital of the individual with median income. If hmt < H̃,
then she prefers xt+1 = 1 the most. Since individuals whose human capital isless than H̃, who comprises of 50 percent of the total population, also prefer
xt+1 = 1 the most, it is chosen in majority voting. Similarly, if hmt ≥ H̃,
8

Vit
VE

Vit

VE

V

VN

N

x
0 x∗ x(hit ) 1 xt+1x∗x(hit )
1 t+1
(b)
(a)
Figure 3: Welfare of individuals with hit ≥ H(x∗ )0then the median voter prefers xt+1 = x∗ . Since individuals whose humancapital is greater than H̃, who consist of 50 percent of the total population,
also prefer xt+1 = x∗ , it is chosen in majority voting.
Proposition 1 Under majority voting, xt+1 = 1 if hmt < H̃, and xt+1 = x∗
if hmt ≥ H̃.

5

Dynamic Analysis

This section analyzes equilibrium dynamics. It should be remembered thatthe realization of higher education only relies on whether or not hmt exceeds
H̃. Let h∗1 , h∗2 , and h∗3 denote
1

F P (h∗1 , 1) = h∗1 ⇔ h∗1 = (B P τ β ) 1−γ ,[] 1F P (h∗2 , x∗ ) = h∗2 ⇔ h∗2 = B P τ β (x∗ )β 1−γ ,[] 1F A (h∗3 , x∗ ) = h∗3 ⇔ h∗3 = B P τ β (x∗ )β + B A τ β (1 − x∗ )β 1−γ ,

(18)
(19)
(20)

respectively. There are three cases that need to be considered; (i) H̃ ≤ h∗1 ,(ii) h∗1 < H̃ ≤ h∗3 , and (iii) h∗3 < H̃.
First, let us consider the case where H̃ < h∗1 . Figure 4 (a) sketches
the dynamics of hmt . If hm0 < H̃, higher education is not realized at theﬁrst place and all individuals consequently accumulate their human capital
according to F P (hit , 1). However, human capital of lineage m eventually
exceeds H̃ at some time period, say t̂, following which higher education9

is always realized and hmt converges to h∗3 . Human capital dynamics ofindividuals in the other lineages can also be easily observed. As shown in
Figure 4 (b), individuals in lineage i with hit̂ ≥ H(x∗ ) always have theirchildren receive higher education since period t̂, and human capital in suchlineages converges to h∗3 .
For an individual in lineage i such that hit̂ < H(x∗ ), there are two possibilities. If h∗2 ≥ H(x∗ ) as in Figure 4 (c), then human capital in the lineage
converges to h∗3 although individuals in the lineage obtain only primary education for some periods after period t̂. In this case, human capital in any
lineage converges to h∗3 and there is hence no steady state income inequality.
If h∗2 < H(x∗ ), human capital in the lineage converges to h∗2 because it evolves
according to F P (hit , x∗ ) at period t̂ and the subsequent periods. This dynamics is shown in Figure 4 (d). As contrasted with the case of h∗2 ≥ H(x∗ ),there is considerable income inequality in the long-run. In the steady state,some individuals obtain both primary and higher education while the others
obtain only primary education, and moreover, the quality of primary education is diminished since a positive share of the government budget is used toﬁnance higher education. This creates high income inequality. Notice that if
hm0 ≥ H̃, then t̂ = 0 and the above analysis remains intact.
Second, we need to discuss the case where h∗1 < H̃ ≤ h∗3 . In this case,the realization of higher education is dependent on the value of hm0 . Figure
5 (a) depicts the dynamics of hmt . If hm0 < H̃, hmt converges to h∗1 < H̃.
Higher education is hence never realized, and human capital of all lineages
converges to h∗1 . There is no income inequality in the long-run. If hm0 ≥ H̃,on the other hand, higher education is realized at the beginning, and hmt
converges to h∗3 . Dynamic analysis on the other lineages is essentially thesame as that in the case where H̃ ≤ h∗1 and hm0 ≥ H̃ (t̂ = 0). For lineage isuch that hi0 ≥ H(x∗ ), all individuals obtain higher education all the timeand hit converges to h3 . For lineage i such that hi0 ≤ H(x∗ ), the dynamic
behavior of hit depends on the value of h∗2 and H(x∗ ). If h∗2 ≥ H(x∗ ), hit
follows the path illustrated in Figure 4 (c). Every individual in the economy
obtain h∗3 in the steady state and there is income inequality in the long-run.
If h∗2 < H(x∗ ), all individuals in a lineage with hi0 < H(x∗ ) receive only
primary education and hit converges to h∗2 as shown in Figure 4 (d). This
generates substantial long-run income inequality.
Finally, in the case where h∗3 < H̃, higher education is not realized in
the long-run. If hm0 ≥ H̃, then higher education obtains majority support
in the beginning, but hmt eventually falls below H̃, following which higher
10hmt+145◦
F A (hit , x∗ )

hit+1

45◦
F A (hmt , x∗ )F P (hit , 1)

F P (hmt , 1)

0
hit+1

h∗3

H̃ hmt̂ h∗1
(a)

hmt

0

45◦
F A (hit , x∗ )hit+1

H(x∗ )hit̂ H̃ h∗1h∗3 hit(b)45◦
F P (hit , 1)P

F (hit , 1)
F P (hit , x∗ )F P (hit , x∗ )0hit̂H(x∗ )h∗2

h∗1

h∗3

hit

0

(c)

hit̂

h∗2 H(x∗ ) h∗1

hit

(d)

Figure 4: Dynamics of hmt and hit in the case of H̃ ≤ h∗1

education is not realized as illustrated in Figure 5 (b). The level of humancapital in any lineage consequently converges to h∗1 and there exists no income
inequality in the long-run.

6

Conclusion

The government plays an important role of funding both primary and higher
education. This paper provides a theoretical examination on how the allocation of public expenditures across the each education inﬂuences the dynamics
of human capital accumulation if the allocation is determined through majority voting. Primary education is compulsory while higher education is
optional and requires private expenditures. When a majority of individuals
11

45◦

hmt+1

45◦

hmt+1

F A (hmt , x∗ )F A (hmt , x∗ )F P (hmt , 1)

F P (hmt , 1)

0

h∗1

H̃

h∗3

hmt

0

(a)

h∗1

h∗3 H̃

hmt(b)Figure 5: Dynamics of hmt in the case of h∗1 < H̃ ≤ h∗3 and h∗3 < H̃

have not accumulated enough human capital, public funding for higher education is not realized. In some cases, however, the individuals accumulate
human capital suﬃciently, and higher education begins to obtain majority
support. This enables a majority of individuals to accelerate their humancapital accumulation although it decreases the budget share for primary education and declines its quality. As an economy grows, income inequality
can expand between individuals who obtain higher education and those who
do not.
We have assumed that individuals must cover private costs to obtainhigher education supplied by the government. The logic of our model would
be applied to other public policies by considering a situation in which accessto publicly provided services requires private spending. For example, thegovernment may consider two policies: (i) public support to enhance productivity in high-tech industries that employ skilled workers, and (ii) lumpsum transfer of a tax revenue to all individuals. Rich individuals who cancover education costs to be a skilled worker would prefer public support for
high-tech industries while poor individuals would prefer lump-sum transfer.
The implication of such a situation for economic growth would be a topic offuture research.

12

Acknowledgement
The authors thank Akihisa Shibata, Carlos Bethencourt, and participants of
PET 12 (Taipei, 2012) for their helpful comments. This research is ﬁnancially
supported by the Global COE program of Osaka University.

References
Arcalean, C., Schiopu, I., 2010. Public versus private investment and growthin hierarchical education system. Journal of Economic Dynamics and Control 34, 604-622.
Blankenau, W., 2005. Public schooling, college subsidies and growth. Journalof Economic Dynamics and Control 29, 487-507.
Blankenau, W., Cassou, S.P., Ingram, B., 2007. Allocating government education expenditures across K-12 and college education. Economic Theory
31, 85-112.
Galor, O., Moav, O., Vollrath, D., 2009. Inequality in landownership, theemergence of human-capital promoting institutions, and the great divergence. review of Economic Studies 76, 143-179.
Glomm, G., Ravikumar, B., 1992. Public versus private investment in humancapital: endogenous growth and income inequality. Journal of Political
Economy 100, 818-834.
Restuccia, D., Urrutia, C., 2004. Intergenerational persistence of earnings:
the role of early and college education. American Economic Review 94,
1354-1378.
Saint-Paul, G., Verdier, T., 1993. Education, democracy and growth. Journalof Development Economics 42, 399-407.
Su, X., 2004. The allocation of public funds in a hierarchical educational
system. Journal of Economic Dynamics and Control 28, 2485-2510.

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