I present an overlapping generations model, with formal education as the engine of growth, close to Glomm and RaviKumar (1992). Contrary to Glomm and Ravikumar, I Show that public schooling, when compared to a private system, may stimulate economic growth.

Luís Francisco Aguiar-Conraria

English

Research Papers in Economics

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“ “P Pu ub bl li ic c   v vs s   P Pr ri iv va at te e   S Sc ch ho oo ol li in ng g      
i in n   a a   E En nd do og ge en no ou us s   G Gr ro ow wt th h   M Mo od de el l” ”   
   
   
   
L Lu uí ís s   A Ag gu ui ia ar r- -C Co on nr ra ar ri ia a   
   
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* NIPE – Núcleo de Investigação em Políticas Económicas – is supported by the Portuguese Foundation for 
Science and Technology through the Programa Operacional Ciência, Teconologia e Inovação (POCTI) of the 
Quadro Comunitário de Apoio III, which is financed by FEDER and Portuguese funds. Public vs Private Schooling in an Endogenous
Growth Model
Luís Aguiar-Conraria
Department of Economics, Cornell University
NIPE, Universidade do Minho
E-mail: lgc9@cornell.edu
July 13, 2004
Abstract
I present an overlapping generations model, with formal education as
the engine of growth, close to Glomm and Ravikumar (1992). Contrary to
Glomm and Ravikumar, I show that public schooling, when compared to a
private system, may stimulate economic growth.1. Introduction
The question addressed in this paper is whether public schooling is growth en-
h a n c i n go rg r o w t hi n h i b i t i n gw h e nc o m p a r e dt oap r i v a t es y s t e m . I nm ym o d e l
t h ee n g i n eo fg r o w t hi sf o r m a ls c h o o l i n g . I ti sk n o w nf r o mt h el i t e r a t u r e ,s e e
Glomm and Ravikumar (1992) and Zhang (1996) as the leading references – also
see Bräuninger and Vidal (2000), that one of the eﬀects of the public system is
the reduction in inequality. This happens because all agents face the same quality
of education, while in a private system richer families have better schooling. So,
on distributional grounds, there is a consensus that a public system is superior to
a private system, at least if one considers equality as a goal.
A more intriguing result is that they conclude that a public education sys-
tem reduces economic growth. Even when in the presence of homogenous agents
this result is true due to a negative ﬁscal externality. According to Glomm and
Ravikumar (1992), students study more in a private system and hence accumu-
late more human capital. This happens because agents include the money they
invest on the education of their children in their utility function. This eﬀect is
internalized only in a private system.
In this paper, I argue that if altruistic behavior of the parents is not allowed,
2the results are reversed, with public schooling becoming growth enhancing relative
to private schooling.
2. The Model
I consider a basic Overlapping Generations (OG) model in which agents maximize
utility over a life time of two periods. Each generation consists of a continuum
of agents. In each period we have a generation of old agents and a generation of
newborn agents.
Contrary to Glomm and Ravikumar (1992) and Zhang (1996),´I consider the
young generation to be responsible for ﬁnancing their education. I do not in-
troduce gifts in the utility function of the parents. Hence, the decision is fully
internal. Young agents have access to international capital markets and can use
their human capital as a collateral to ﬁnance their education spending.
To simplify, I assume, as in Galor and Zeira (1993), a small open economy
with perfect access to the international capital markets. Agents can borrow at an
exogenous interest rate r. I assume r =0 .W ec a ni n t e r p r e tt h i sa s s u m p t i o na s
capturing the student loans at subsidized interest rates, common in some countries
(although assuming r>0 does not change the results).
I assume that agents only consume in their second period of life. The utility
3function of the agents born in period t is given by:
u(ct,t,c t,t+1)=ct,t+1 (2.1)
In the ﬁrst period of their lives, agents have to decide whether to go to the
university or not and how much money and time they spend on education. The
human capital of an agent born in t depends on these choices and also on the
human capital of the parent and on the average human capital of the old agents
in the economy:
ht = θt (et)
α (Ht−1)
β (ht−1)
γ , 0 <λ,α,β,γ<1 and α + β + γ =1 (2.2)
where ht−1 is the human capital of the young agent’s parent, Ht−1 is the average
human capital at time t of the generation born in t−1, et ≥ 0 is the money invested
in education and θt ∈ [0,1] is the time spent in School. One can interpret the
inﬂuence of et and Ht−1 as measuring the quality of the school.
In a private system, if an agent wants to study she chooses et. In a public sys-
tem, the agent pays nothing when studying, and the total spending on education
in each period is ﬁnanced by taxes raised on all agents (young and old) working
4in the next period. The money spent per student in education, and hence the
necessary tax rate to support that expenditure, is decided by majority voting.
We can interpret this model as one in which basic education, say at a high
school level, is guaranteed (for example, through compulsory schooling), but above
that it is a private decision.
On the production, side I assume a linear technology (production of each
worker is equal to her human capital), and hence wages coincide with the human
capital.
To capture the heterogeneity of the agents, I assume that old agents in the
same generation are diﬀerentiated by their stock of human capital according to
some distribution function Gt.
Ia s s u m en oﬁxed costs in education and I am considering education to be
a pure rival good. Relaxing these assumptions would make the case for public
education stronger.
2.1. Equilibrium in the private system
The agent’s optimization problem is to choose θt,e t,c t,t+1 to maximize
u(ct,t,c t,t+1)=ct,t+1
5subject to
ct,t+1 = θt (et)
α (Ht−1)
β (ht−1)
γ +( 1− θt)θt (et)
α (Ht−1)
β (ht−1)
γ − etθt
The ﬁr s tt e r mi st h ev a l u eo ft h eh u m a nc a p i t a lo ft h ea g e n ta tt i m et+1,t h e
second term is the value of the human capital, times the time spent working in
period t, so it represents the wages earned in the ﬁr s tp e r i o d .T h el a s tt e r mi st h e
money the agent borrowed to ﬁnance her education. For a strictly positive level
of et,t h eo p t i m a lt i m es p e n to ns t u d y i n gi sg i v e nb y :
θt =1−
e
1−α
t
2H
β
t−1h
γ
t−1
(2.3)
For et =0the agent is indiﬀerent between any choice of θt. I assume that in
case of indiﬀerence the agent choose θt =1−
e1−α
t
2H
β
t h
γ
t
.
With this information we can ﬁnd the optimal level of et:
et =
µ
2α
2 − α
¶ 1
1−α
H
β
1−α
t−1 h
γ
1−α
t−1 (2.4)
6Plugging this result in equation 2.3:
θt =2
1 − α
2 − α
So the human capital of this individual is:
ht =
2 − 2α
2 − α
µ
2α
2 − α
¶ α
1−α
H
β
1−α
t−1 h
γ
1−α
t−1 (2.5)
2.2. Equilibrium in the public system
Under the public education regime there is a voting process to decide, by majority
rule, how much is spent on education, and, simultaneously, the tax rate necessary
to support that expenditure.
I solve the individual’s optimization problem in two steps. First, I take, as
given, the tax rate τt+1 (I assume a steady state tax rate, τt+1 = τt = τ)a n dt h e
education expenses et and determine the optimal time of schooling:
max
θt
ct,t+1
subject to
ct,t+1 = θt (et)
α (Ht−1)
β (ht−1)
γ (1 − τ)+( 1− θt)θt (et)
α (Ht−1)
β (ht−1)
γ (1 − τ)
7Solving we get θt =1 . The optimal tax rate, from this agent’s perspective,
can be determined by solving the problem:
max
ct,t+1,et,τt+1
ct,t+1
subject to
ct,t+1 = θt (et)
α (Ht−1)
β (ht−1)
γ (1 − τ)
et = τ
R
(et)
α (Ht−1)
β (ht−1)
γ dGt−1
where the second restriction is just a balanced government budget restriction,
which can be solved with respect to τ:
τ =
(et)
1−α
(Ht−1)
β R
(ht−1)
γ dGt−1
Given this result, the optimal et is:
et =
µ
α(Ht−1)
β
Z
(ht−1)
γ dGt−1
¶ 1
1−α
(2.6)
which implies:
τ = α (2.7)
The preferred tax rate and the education expenditure are independent of the
8human capital of the agent’s parent, and hence we have unanimous voting among
the young. The old at time t will be indiﬀerent about the taxes at t+1, and hence
they will not veto this tax rate. In a steady state equilibrium, the tax rate is α
and the education expenditure is given by 2.6.
T h el a wo fm o t i o no ft h ei n d i v i d u a l ’ sh u m a nc a p i t a li s :
ht = α
α
1−α (Ht−1)
β
1−α (ht−1)
γ
µZ
(ht−1)
γ dGt−1
¶ α
1−α
(2.8)
3. Comparisons between the two systems
3.1. Homogeneous agents
Consider the case of homogeneous agents. This hypothesis helps us to understand
the growth implications of each of the systems by abstracting from inequality
issues.
The distribution function Gt−1 is degenerate and Et (ht−1)
γ =( ht−1)
γ.U s i n g
2.8 we have the law of motion for human capital under the public system:
h
public
t = α
α
1−α (Ht−1)
β
1−α
³
h
public
t−1
´ γ
1−α
9For the private system we have:
h
private
t =
2 − 2α
2 − α
µ
2α
2 − α
¶ α
1−α
H
β
1−α
t−1
¡
h
private
t−1
¢ γ
1−α
Proposition 1. if h
public
0 = h
private
0 and α<0.615 then for t ≥ 1 we always have
h
public
t >h
private
t
Proof. To prove this result I only need to show that
α
α
1−α > 2−2α
2−α
¡
2α
2−α
¢ α
1−α
⇔ 0 <α<0.615018
The last step was solved numerically using Maple V.
This is the key result of the paper. Contrary to Glomm and Ravikumar (1992)
and Zhang (1996), the public school system may be growth enhancing relatively
to a purely private system. To know which of the systems is actually better
for growth, one has to estimate the elasticity of private income with respect to
education spending. It is unlikely that the elasticity is larger than 60%, so, for
the rest of the paper, I assume that 0 <α<0.615018.
103.2. Heterogeneous agents
As in Glomm and Ravikumar (1992) and Zhang (1996), I assume that Gt is
lognormal with parameters (µt,σ2
t). The evolution of these parameters can be
calculated using the human capital motion equations and the properties of the
lognormal distribution.
For the public schooling, using 2.8:
ht = α
α
1−α (Ht−1)
β
1−α (ht−1)
γ exp
µ¡
γµt−1 + γ
2σ
2
t−1
¢ α
1 − α
¶
µt =
α
1 − α
lnα +
µ
β + γ
1 − α
¶
µt−1 +
αγ2 (1 − α)+β
2
(1 − α)
2
σ2
t−1
2
σ
2
t = γ
2σ
2
t−1
With private schooling, using 2.5:
µt =l n
Ã
2 − 2α
2 − α
µ
2α
2 − α
¶ α
1−α
!
+
µ
β + γ
1 − α
¶
µt−1 +
µ
β
1 − α
¶2 σ2
t−1
2
σ
2
t =
µ
γ
1 − α
¶2
σ
2
t−1
Proposition 2. Consider two economies, with the same education system and
the same average human capital in period t.I np e r i o dt+1the economy with the
lowest variance will have a higher average human capital.
11Proof. See proposition 6 of Glomm and Ravikumar (1992).
Proposition 3. Consider two identical economies with diﬀerent education regimes
at time t. If income inequality is not too large then the economy with public
schooling will have a higher average human capital at period t +1 .
Proof. Let ut and σ2
t characterize the economy with public schooling, and u0
t
and σ02
t characterize the economy with private schooling. Given our assumptions
(µt,σ2
t)=( µ0
t,σ02
t ).
Ht+1 =e x p
µ
α
1 − α
lnα +
µ
β + γ
1 − α
¶
µt +
(1 − α)γ2 + β
2
(1 − α)
2
σ2
t
2
¶
H
0
t+1 =e x p
Ã
ln
Ã
2 − 2α
2 − α
µ
2α
2 − α
¶ α
1−α
!
+
µ
β + γ
1 − α
¶
µt +
β
2 + γ2
(1 − α)
2
σ2
t
2
!
So Ht+1 >H 0
t+1 iﬀ α
1−α lnα +
(1−α)γ2
(1−α)2
σ2
t
2 > ln
³
2−2α
2−α
¡
2α
2−α
¢ α
1−α
´
+
γ2
(1−α)2
σ2
t
2
Given our assumption that 0 <α<0.615018 we have α
1−α lnα>ln
³
2−2α
2−α
¡
2α
2−α
¢ α
1−α
´
and hence, if σ2
t is not too large, we will have Ht+1 >H 0
t+1.
We are left, again, with an empirical question. What is the meaning of saying
that σ2
t is not too large? Consider for example: α = 1
3 and γ = 1
3.I n t h i s c a s e
the theorem holds if σt < 3.1675. Taking the United States as a benchmark, and
assuming that the income distribution follows a log normal distribution, then,
12table3 of Gottschalk and Smeeding (1997) imply a value of σ around 0.6.
4. Conclusion
In this paper, I challenged the conclusion of Glomm and Ravikumar (1992) and
Zhang (1996) that public education hinders economic growth when compared to a
private education system. The conclusion of Zhang (1996) that public education
may be growth enhancing only if income inequality is very big was also contested.
The key to revert their results was to fully internalize the investment decisions in
education.
References
[1] Bräuninger, M. and Vidal, J.-P., 2000. Private versus Public Financing of
Education and Endogenous Growth. Journal of Population Economics 13, 387—
401. Convergence”, Journal of Population Economics, vol. 13, 403—424.
[2] Galor, O. and Zeira, J., 1993. Income Distribution and Macroeconomics. Re-
view of Economic Studies 60, 35-52.
13[3] Glomm, G. and Ravikumar, B., 1992, Public versus Private Investment in Hu-
man Capital: Endogenous Growth and Income Inequality. Journal of Political
Economy 4, 818—834.
[4] Gottschalk, Peter, and Timothy M. Smeeding. 1997. Cross-National Compar-
isons of Earnings and Income Inequality. Journal of Economic Literature 35,
633—687.
[5] Zhang, J., 1996. Optimal Public Investments in Education and Endogenous
Growth, Scandinavian Journal of Economics 98, 387—404.
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Public vs Private Schooling in an Endogenous Growth Model

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