Public versus Private Investment in Human Capital:
Endogenous Growth and Income Inequality
Gerhard Glomm and B. Ravikumar
JPE 1992
Presented by Prerna Dewan and Rajat Seth
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Objectives
To model human capital investment through formal schooling as the
engine of growth
To compare public education and private education regimes with
respect to per capita income, growth and income inequality
To endogenize the choice of educational system in an economy
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Models such as Romer (1986) and Lucas (1988) study human capital
as a factor of growth, however fail to account for the large
involvement of the public sector in human capital investment
Most models of long run growth are representative agent models, and
do not address income distribution issues
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ESSENCE OF THE MODEL
Two period overlapping generations economy wherein parents have a
bequest motive; they value the quality of education of their offspring
Quality of education in the two regimes determined by
1
2
tax on the income of the old: public education
private choices of individuals: private education
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THE BASIC FRAMEWORK
OLG model with constant population
Each generation consists of a continuum of agents who are
differentiated by the stock of human capital of their parents
Initial generation of old agents at t = 0; j th member endowed with
knowledge hj0
Knowledge of initial generation distributed according to pdf Go (. )
which is lognormal with parameters µ0 and σ02 .
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THE BASIC FRAMEWORK
Individuals have identical preferences given by a warm glow utility
function:
Ut = ln nt + ln ct+1 + ln et+1
(1)
Individuals are endowed with one divisible unit of time in their youth
which they allocate between leisure and human capital accumulation
Human capital accumulation takes the form:
ht+1 = θ(1 − nt )β etγ htδ
(2)
θ>0
β, γ, δ ∈ (0, 1)
At time t + 1, an individual’s income is the same as his human
capital ht+1
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PUBLIC EDUCATION REGIME
Each individual’s earning at time t + 1 is taxed at the rate τt+1
Total tax revenues determine the quality of public schools at time t +
1:
Et+1 = τt+1 Ht+1
(3)
Z
where Ht+1 ≡
ht+1 dGt+1 (ht+1 )
Tax rate is determined endogenously by the old agents in each period
through majority voting
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PRIVATE EDUCATION REGIME
Quality of education is chosen individually
Each individual allocates her income ht+1 between own consumption,
ct+1 , and the quality of education for the offspring, et+1
ct+1 + et+1 = ht+1
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(4)
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EQUILIBRIUM UNDER PUBLIC EDUCATION
The equilibrium for the public education economy is a set of sequences
∞
∞
∞
∞
. ∞
{nt }∞
t=0 ,{ht+1 }t=0 , {ct }t=0 , {Gt+1 ( )}t=0 , {Et }t=0 , and {τt }t=0 such
that
1
nt and ct+1 are the optimal choices of an agent born at time t whose
parent’s human capital is ht ;
2
the human capital of each agent is determined by
ht+1 = θ(1 − nt )β etγ htδ ;
3
4
5
given the distribution Gt (. ) at time t, the distribution of income
Gt+1 (. ) at time t + 1 is determined by the transformation of variables
ht+1 = θ(1 − nt )β Etγ htδ ;
the tax rate τt is preferred by a majority of old agents at time t; and
R
the quality of schools at time t is Et = τt ht dGt (ht )
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EQUILIBRIUM UNDER PUBLIC EDUCATION
We solve individual j’s optimization problem in two steps.
1
Solve for optimal effort, consumption and human capital investment
in the first period
max ln nt + ln ct+1 + ln Et+1
nt ,ct+1
subject to:
1
2
ct+1 = (1 − τt+1 )ht+1 ,
ht+1 = θ(1 − nt )β Etγ htδ ,
given Et , ht , Et+1 , and τt+1 .
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EQUILIBRIUM UNDER PUBLIC EDUCATION
2
Solve for the agent’s preferred tax rate in the second period
max ln[(1 − τt+1 )ht+1 ] + ln τt+1 Ht+1
τt+1
where Ht+1 is the mean income at time t + 1
Note that the old agent’s choice of tax rate does not alter his income
but affects the fraction of income he can consume.
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EQUILIBRIUM UNDER PUBLIC EDUCATION
The public education equilibrium is characterized by:
1 − nt =
ht+1 = θEtγ (
β
1+β
(5)
β β δ
) ht
1+β
(6)
τt+1 =
1
2
(7)
Note that time allocated to human capital investment by an individual
born at time t is independent of the tax rate and the individual type
because of log preferences and Cobb Douglas learning technology. Also,
each individual’s preferred tax rate is independent of individual income and
constant over time because of log preferences over consumption and
bequests.
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EQUILBRIUM UNDER PUBLIC EDUCATION
Since ht is lognormally distributed with mean µt and variance σt2 , ht+1 is
2 , where
also lognormally distributed with mean µt+1 and variance σt+1
µt+1 = ln[θEtγ (
β β
) ] + δµt
1+β
(8)
2
σt+1
= δ 2 σt2
(9)
Substituting the values of Et from equation (3) and τt+1 from equation
(7) in equation (6), we get:
β β δ
1
) ht ≡ AHtγ htδ
ht+1 = θ( )γ Htγ (
2
1+β
For lognormal distribution, Ht = e µt +(
σt2
)
2
. So:
µt+1 = ln A + γ ln(Ht ) + δµt = ln(A) + (γ + δ)µt +
Gerhard Glomm and B. Ravikumar
(10)
γσt2
2
(11)
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EQUILIBRIUM UNDER PRIVATE EDUCATION
The equilibrium for the private education economy is a set of sequences
∞
∞
∞
. ∞
{nt }∞
t=0 , {ct }t=0 , {Gt+1 ( )}t=0 , {et }t=0 , and {ht+1 }t=0 such that
1
nt , ct+1 and et+1 are the optimal choices of an agent born at time t
whose parent’s human capital is ht ;
2
the human capital of each agent is determined by
ht+1 = θ(1 − nt )β etγ htδ ;
3
given the distribution Gt (. ) at time t, the distribution of income
Gt+1 (. ) at time t + 1 is determined by the transformation of variables
ht+1 = θ(1 − nt )β etγ htδ ;
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EQUILIBRIUM UNDER PRIVATE EDUCATION
The young individual’s optimization problem at time t is as follows:
max
nt ,ct+1 ,et+1
ln nt + ln ct+1 + ln et+1
subject to:
1
ct+1 = ht+1 − et+1
2
ht+1 = θ(1 − nt )β etγ htδ ,
given et and ht .
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EQUILIBRIUM UNDER PRIVATE EDUCATION
The private education equilibrium is characterized by:
1
ct+1 = et+1 = ht+1
2
[Note that the quality of education is agent specific.]
1 − nt =
1
2
β
+β
(12)
(13)
[Note that the time devoted to human capital accumulation is different for
the two economies. (Why? )]
1
β β γ+δ
ht+1 = θ( )γ [ 1
] ht = Bhtγ+δ
2 2 +β
(14)
ht+1 is lognormally distributed with mean µt+1 = ln B + (γ + δ)µt and
2
variance σt+1
= (γ + δ)2 σt2
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HOMOGENEOUS AGENTS
Comparison of the two equilibrium paths with respect to levels and
growth rates of income when initial generation is homogeneous
Evolution of income in the two regimes is given by
u
ht+1
= A(htu )γ+δ
(15)
(since per capita income at time t coincides with the representative
agent’s income in the case of homogeneous agents economy)
r
= B(htr )γ+δ
ht+1
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HOMOGENEOUS AGENTS
Conditions for the existence and uniqueness of the (nontrivial) steady-state
income/human capital:
PROPOSITION 1
(a) If γ + δ 6= 1, then there exists a unique steady state given by
1
2
3
u
= hsu whenever htu = hsu ,
hsu > 0 such that ht+1
r
r
hs > 0 such that ht+1 = hsr whenever htr = hsr , and
for γ + δ < 1, hsr > hsu , and for γ + δ > 1, hsr < hsu .
(b) If γ + δ = 1 and A 6= 1, then there does not exist a steady state
for the public education economy.
(c) If γ + δ = 1 and B 6= 1, then there does not exist a steady state
for the private education economy.
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HOMOGENEOUS AGENTS
Figure: Human Capital accumulation in a homogeneous household economy :
Decreasing Returns
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HOMOGENEOUS AGENTS
Figure: Human Capital accumulation in a homogeneous household economy :
Constant Returns
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HOMOGENEOUS AGENTS
Figure: Human Capital accumulation in a homogeneous household economy :
Increasing Returns
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HOMOGENEOUS AGENTS
Assume that both private and public education economies start off
with the same (positive) level of initial income h0 .
PROPOSITION 2
htr > htu for all t > 0
X Along the equilibrium path, a private education economy yields higher
income levels in all periods than a public education economy.
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HOMOGENEOUS AGENTS
PROPOSITION 3
Vtr > Vtu for all t ≥ 0, where Vtr is the equilibrium level of utility of an
agent born at time t in the private education economy and Vtu is the
corresponding utility in the public education economy.
X The representative agent in the private education economy is better
off than his counterpart in the public education economy.
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HOMOGENEOUS AGENTS
PROPOSITION 4
(a) If γ + δ < 1, then limt→∞ (
(b) If γ + δ = 1, then
r
ht+1
htr
r
ht+1
htr )
= limt→∞ (
=B>A=
u
ht+1
htu
u
ht+1
htu )
= 1.
for all t ≥ 0.
(c) For γ + δ > 1,
1
2
r
ht+1
htr
u
ht+1
htu
is greater than one and increasing over time if h0 > hsr and
is greater than one and increasing over time if h0 > hsu .
X Decreasing returns implies zero long run growth;
X Constant returns implies constant long run growth under certain
parameter restrictions;
X Increasing returns are neither necessary nor sufficient for long-run
growth in either economy.
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HOMOGENEOUS AGENTS
Private education economies with homogeneous population achieves
higher incomes and growth rates than the public education economy
whenever γ + δ ≥ 1.
If a policy of mandatory schooling is enforced, then the allocations in
the public education regime would be the same as in the private
education regime.
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HETEROGENEOUS AGENTS
The initial income distribution is lognormal with parameters µ0 and
σ02 ; income inequality at time t is described by the parameter σt .
PROPOSITION 5
(a) In the public education economy, income inequality declines over
time.
(b) In the private education economy, income inequality declines over
time if γ + δ < 1, increases over time if γ + δ > 1, and remains
constant over time if γ + δ = 1.
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HETEROGENEOUS AGENTS
Even if γ + δ < 1, income inequality in the private education economy
does not decline as fast as in the public education economy.
Conditions for long-run growth in per capita income with
heterogeneous agents are identical to those in the homogeneous agent
economy.
Along the balanced growth path (γ + δ = 1) income inequality
declines in the public education economy but stays the same in the
private education economy.
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HETEROGENOUS AGENTS
PROPOSITION 6
(a) Consider two distinct public education economies with the same
mean income at time t, that is, Ht = Ht0 . If σt0 > σt , then
0 .
Ht+1 > Ht+1
(b) Similarly, consider two distinct private education economies with
0
Ht = Ht0 and σt0 > σt . Then Ht+1 > Ht+1
if and only if γ + δ < 1.
X Although the proposition establishes an order relation between per
capita income levels only for the next period, the relation holds for all
future periods.
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HETEROGENOUS AGENTS: AN EXAMPLE
If the initial income inequality is sufficiently high, then the public
education economy may yield higher per capita income for some future
periods than the private education economy:
Two economies with the same income distribution at time t;
γ + 2δ < 1 ⇐⇒ (γ + δ)2 < γ + δ 2 .
u
) = ln A + (γ + δ)µt +
ln(Ht+1
(γ + δ 2 )σt2
2
r
ln(Ht+1
) = ln B + (γ + δ)µt +
(γ + δ)2 σt2
2
u
r .
If σt2 is sufficiently large, Ht+1
> Ht+1
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CHOICE OF EDUCATIONAL REGIME
Parents decide by majority vote whether the educational system should be
private or public and no one can opt out.
PROPOSITION 7
A majority of old agents at time t would prefer public over private
education if and only if σt2 > 0.
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CONCLUSION
Agents in the private education economy spend more time on human
capital accumulation than their counterparts in a public education
economy because they internalize the fact that more time spent on
human capital accumulation increases consumption as well as
bequests.
For a homogeneous agent economy, per capita income, indirect utility
and growth rate are higher in a private education regime than in a
public education regime.
Income inequality declines faster in public education regime than in
pvt education regime.
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CONCLUSION
If two economies have the same per capita income but differing levels
of inequality, the economy with higher inequality will have lower per
capita income in all subsequent periods.
If inequality is sufficiently high in an economy, then a public
education regime may yield higher per capita income for some future
periods than a pvt education economy.
If majority voting decides the choice of education regime in an
economy, public education is chosen over private if a majority of
agents have income below the average.
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Appendix:Proof of Proposition 3
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Appendix:Proof of Proposition 3
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Appendix:Proof of Proposition 6
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Appendix:Proof of Proposition 6
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The End
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