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working paper
department
of economics
HETEROGENEITY, STRATIFICATION, AND GROWTH
Roland Benabou
MIT, NBER and CEPR
No.
93-4
Dec.
1992
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
HETEROGENEITY, STRATIFICATION, AND GROWTH
Roland Benabou
MIT, NBER and CEPR
No.
93-4
Dec. 1992
'
:
;
'j
Heterogeneity, Stratification, and
Growth
Roland Benabou
MIT,
NBER
and CEPR.
June 1992
This version
I
am
Tirole,
-
December 1992
grateful to Olivier Blanchard, Ricardo Caballero, Peter
Yoram Weiss and
especially Julio
gratefully acknowledged. All errors are
my
Rotemberg
own.
Diamond, Steven Durlauf, John Heaton, Jean
for helpful discussions.
Financial support from the
NSF
is
Abstract
We
examine how economic
stratification affects inequality
where heterogeneous agents interact through
borhood
effects)
stratified into
We
study economies
goods or externalities (school funding, neigh-
and economy- wide linkages (complementary
growth and welfare when families are
integrated.
local public
and growth over time.
skills,
homogeneous
knowledge
local
spillovers).
We
compare
communities and when they remain
Segregation tends to minimize the losses from a given amount of heterogeneity, but integra-
tion reduces heterogeneity faster. Society
may
thus face an intertemporal tradeoff: mixing leads to slower
growth in the short run, but to higher output or even productivity growth
occurs in particular
cases of segregation
which structure
is
when comparing
and integration.
more
local
in the long run. This tradeoff
and national funding of education, which correspond to
More
efficient over short
generally,
we
identify the key parameters
special
which determine
and long horizons. Particularly important are the degrees of
complementarity in local and in global interactions.
Introduction
1
In the United States a family's income, assets, education level, ethnic background and lifestyle can be
predicted quite accurately from
its
zip code -and this in spite of the great diversity of the
American
population. 1 This strong degree of social and economic segregation, epitomized of late by the spread of
gated communities,
as
is
reflected in wide disparities in the funding
primary and secondary education or law enforcement.
It
and quality of
local public services, such
also manifests itself in the increasingly different
types of behavior and values to which the young are exposed during their formative years.
The production
of goods and services thus brings together on the factory floor and at the office workers
on the one hand, managers and professionals on the other, whose upbringing and
are
becoming increasingly disparate. Could
human
levels of
capital
and outcomes
this polarization of educational opportunities
be a contributing factor not only to the widening inequality in income and wealth observed over the
decade, but also to the slowdown of productivity growth?
One
reads for instance in the
MIT
last
Commission's
Report on Industrial Productivity:
"American and foreign students
not only in their average scores on standardized tests
differ
but also in the dispersion of those scores around the mean. The Japanese aim at bringing
students to a high
.
.
common
level of
competence, and they are largely successful; as a result
.new entrants to the Japanese work
In the U.S.
learn.
work
force,
force are generally literate, numerate,
employers have discovered high rates of
with basic mathematics and reading in workers with high school diplomas
of
(
The
young Americans are technologically
Made
in
in
literate
illiteracy
.
.
.
and
difficulty
Only a tiny fraction
and have some knowledge of foreign
is
of course
much more heterogeneous than
societies.
the Japanese. But this greater heterogeneity
good part endogenous, precisely because any exogenous differences
historical factors, immigration, or just plain luck) are magnified
1
and prepared to
America, (1989))
U.S. workforce
may be
all
Weiss (1989) provides a comprehensive and
in
human
by economic or social
lively description of the "clusters"
capital (due to
stratification.
In
used in commercial and political marketing.
fact,
a more diverse population
may
increase the value of integration, as
it
accelerates the homogenization
of the labor force. Motivated by these observations, this paper investigates the relationship between the
extent to which an
and
its
The
economy
is
stratified into
homogeneous communities,
its
degree of income inequality,
growth performance over time.
central question
efficient:
is
the following: given a heterogeneous population, which social structure
segregation by income and education, or integration?
is
more
includes as a special case the issue of
It
We
whether public education should be funded locally or nationally.
formalize these issues in a simple
but quite general growth model with both economy-wide linkages (such as complementarity
in
production
or knowledge spillovers) and local, community-level externalities or public goods (such as locally funded
schools or neighborhood effects).
We
show that the answer to the question raised above depends on two
basic effects, which can give rise to an interesting intertemporal tradeoff.
The
first effect
measures how
gating disparate levels of
human
efficient
each social structure
is
at processing heterogeneity,
capital into the production of goods and, ultimately,
show that when family background and community quality are complements
in
i.e.
at aggre-
new knowledge.
We
a child's education, a segre-
gated economy tends to have smaller instantaneous losses, hence faster growth, for any given amount of heterogeneity.
it
The second
effect is
dynamic: because an integrated society
is
better at reducing heterogeneity,
converges faster to a homogeneous outcome-or in the presence of shocks, converges to a
distribution of skills
and income. Integration thus
delivers
much
generations. Is this effect important enough for mixing to be
of
more
its
less
unequal
payoff over the course of several
efficient in
the long run, even
when
it
leads to losses in the short run?
We show
that the answer tends to be affirmative, so that integration at
eventually raises everyone's income.
It is
first
hurts the better off but
then Pareto improving, without need for redistribution, provided
agents have a low enough discount rate. Conversely, increased segregation leads to:
distribution of income;
(ii)
(i)
a widening in the
a short-lived burst of growth, benefiting mostly the better off households;
a decline in the economy's long-run level of output, or even in the long-run growth rate.
these results remain even as
all
(iii)
Remarkably,
global complementarities linking together rich and poor families
become
»
vanishingly small.
Of course,
stratification
is
much
need not be the case that mixing
it
remains more
efficient if the degree
is
preferable in the long run. For instance,
that
of complementarity between agents' levels of human capital
stronger in local interactions than in global interactions. Intuitively, this
knowledge at the community
we show
means that
level (e.g. in schooling) entail sufficiently greater losses
than
disparities in
at the aggregate
level (e.g. in production).
What
paper
this
stratification
offers
But
as
we move
Thinking about extreme
the spillover
is
closer to the
show that one cannot even
(the geometric
helps to clarify
It also
some
issues of aggregation
combined with heterogeneity. In models with a representative agent,
spillovers are specified.
cases, this
may seem obvious;
minimum, to some
identify -as
is
which
it is
arise
how
becomes
naturally the equilibrium will be different
average, or to the
common
when
irrelevant
to models with heterogeneity, the choice of aggregator
and arithmetic averages) without
rate, as well as the
practice- the
significantly
maximum
mean
of
all
agent's actions.
if
But we
of the logs and the log of the
mean
changing the economy's long run growth
normative conclusions about the efficiency of stratification and integration. One must
beyond rough intuitions and pin down the key parameters which determine how heterogeneity
therefore go
affects
thus a framework in which the costs and benefits of different degrees of
can be spelled out.
externalities are
crucial.
is
The aim
growth.
is
not only to clarify the properties implicitly embodied in the specifications of
previous models, but also to provide a guide for future empirical work on spillovers and neighborhood
or peer effects.
2
In addition to the overall degree of returns to scale
community and economy-wide inputs
role played
by the three
interactions,
in the
production of
elasticities of substitution
and between
all
human
and the
capital,
which operate within
relative weights of family,
we bring
local interactions, within global
three inputs.
This paper draws on previous work by Benabou (1991), Tamura (1991a,b) and
(1992).
Benabou (1991) demonstrated that
In a sense our effort
is
similar, in
mapping between various forms
to light the crucial
in
Glomm
and Ravikumar
a general equilibrium context, agents' incentives to segregate
a macroeconomic and dynamic context, to Arnott and Rowse's (1987) study of the
of peer effects
and optimal school
structure.
themselves due to the presence of local externalities or public goods can have important effects on aggregate
productivity and welfare, even with perfect capital markets. But the model was static, hence not suited
to study
growth or the idea that the merits of integration and segregation may look very
Being a model with ex-ante identical agents,
short and in the long run.
whether due to
issue of inequality,
could also not address the
it
conditions or ongoing shocks; nor could
initial
different in the
it
capture the notion of
homogenization over time.
Tamura
human
(1991a) studies endogenous growth
capital spillover,
and shows two main
to decreasing returns, heterogeneity slows
long run the
economy converges
to a
when heterogeneous agents
results.
issue of
how
in a general class of
by an economy- wide
because individual accumulation
First,
down growth. Second,
this effect
is
is
subject
only temporary, as in the
homogeneous outcome. Tamura uses simulations to demonstrate the
vanishing impact of heterogeneity. In this paper
dynamic path,
are linked
we provide
models with both
stratification affects growth,
local
and to answer
it
analytical solutions for the economy's entire
and global
spillovers.
by showing how the
This allows us to raise the
losses per unit of dispersion,
the convergence speed and their interaction differ under segregation and integration.
Glomm and Ravikumar
(1992) compare growth under private and public education. In a private system
parents buy education for their
own
children
and there are no
spillovers. In
Glomm and Ravikumar show
a nationally funded public good, generating again a global spillover.
a private system offers students better incentives to invest in
long-run growth rate.
economy
to
grow
by heterogeneity
adults'
growth.
human
If
also provide
in
is
human
capital,
is
that
and thus leads to a higher
an example where heterogeneity can cause a public education
faster for a while; but they
do not analyze why
this is so.
We make
clear the role played
each system's performance. Most importantly, we show that when children's ability or
capital
there
They
a public system education
is
subject to
random
shocks, public education
may
in fact lead to faster
long-term
even a very small amount of complementarity in the production sector, a move to public
education can be Pareto improving, provided families' intergenerational discount rate
is
low enough. In
any case, both private and nationally funded education systems dominate locally funded public education,
which has neither the incentive properties of the former nor the homogenization properties of the
latter.
Finally, this
paper
is
also closely related to Durlauf (1992)
and
S.
Cooper (1992), through a shared
how community
concern about the effects of stratification in a dynamic, stochastic economy. Durlauf shows
formation and local funding of education can generate path-dependence in lineage income, trapping some
families in pockets of poverty while others enjoy growth. S.
his model,
Cooper (1992) incorporates redistribution
into
by allowing communities linked through production externalities to vote on cross-subsidies. Our
main concern here
how
is
stratification affects the
growth performance and
efficiency of the
whole economy,
with particular emphasis on the potential tradeoff between the short and the long run.
Section 2 presents a model of education and production which motivates and sets
It
basic problem.
then shows how similar issues arise in several other models, and provides a general framework in which
to study them. Section 3
in
up the
examines how heterogeneity
affects short
an integrated economy. Section 4 shows how randomness
and long run growth
in innate ability magnifies
in
a segregated and
the long-run effects
of stratification. Section 5 considers the model's implications for the efficiency of nationally funded public
education, locally funded public education, and private education.
forms used in the text are generalized
2
2.1
in
Appendix A; proofs
Local and Global Interactions in
A
First
Section 6 concludes.
are gathered in
Human
the accumulation of
in production.
functional
Appendix B.
Capital
Model: Education and Production
We first consider the most natural channels through which group-specific
ities arise in
The
There
is
human
capital: local
and economy-wide complementar-
funding of education, and imperfect substitutability
a continuum of overlapping generation families
i
€ 0, of unit measure. During each
period adults work, consume, and spend time rearing their single child. At time zero, the adult
of dynasty
i
faces the following problem:
member
maximize
=
Uq
Eq
2~Jp
<u ( c <)
subject to:
i
<=o
(1)
c\
=
(2)
y>
= v\w\
ft«
(3)
and h
l
given.
There are no
She spends a fraction
v\
{l-ri)y\
= K-ci-ai-riWiEiy-*
+1
human
financial assets, only
endowment
of her unit time
The term
the rest to helping her child learn.
h't in (3)
unpredictable component of the child's innate ability
in the
production of
human
capital
is
is
At time
capital.
t,
adult
i
at work, earning the hourly
has
human
capital h\.
wage w\, and devotes
could also be due in part to inherited ability; the
represented by the
a local public good, which
i.i.d.
shock Q. The other input
financed by taxing the labor income of
is
local residents. Per capita expenditures are therefore:
E\=ri.YJ =
(4)
Ti.
r ydm\(y)
Jo
where m\
at time
t:
is
the distribution of income and
city,
suburban town,
state, etc.
secondary schooling. But law enforcement,
To
simplify the model,
we
shall take
Y
'
t
its
average, in the
The most obvious
community
and voting choices
"ceteris paribus"
both the fraction of time
3
This in shown in Section
and tax
rates in
will lead, in equilibrium, to
Tamura
5.2;
on some new
is
belongs
primary and
u\
spent working and the tax rate
If
effects
r\ to
agents have logarithmic preferences,
such invariant rules. 3 But these are really
in the political
economy
.
the values of u and T are given in Proposition
(1991a) and
good
i
libraries, etc., are also relevant.
assumptions by which we abstract from the issues explored
literature, in order to focus
to which family
skill-enhancing public
be constant over time and independent of community composition.
their decisions
fl|
Glomm and Ravikumar
6.
Log-utility also leads to constant investment
(1992) respectively. Voting models of education with variable
tax rates are analyzed by Perotti (1990), Saint-Paul and Verdier (1991) and Fernandez and Rogerson (1992),
among
others.
,
We now
turn to the production sector. All workers take part in the production of a numeraire good,
performing complementary tasks or specializing in imperfectly substitutable intermediate inputs.
total
output
is:
Y =
(5)
t
where
fi t
v(i h^dn {h)\'~
X
=v-Hu
t
denotes the distribution of
human
and managers
will
mean
positively
(6)
lagging wages for basic workers.
and the same
on the economy-wide
but
level of
t
finite.
human
is
will
advances in knowledge
Such interdependence
Given
(5),
any worker's
capital:
,
true for any community's level of per capita income:
is
Y =J\dm\(y) = v-(H )i-(j °°h^1
i
(7)
large,
= vw\ = v(H )$(h\) st1
y\
This complementarity
etc. Conversely, lagging
seems quite plausible, especially as we allow a to be arbitrarily
wage and labor income depend
4
production or clerical workers
insufficiently skilled
drag down the productivity of engineers, managers, doctors,
scientists, engineers
(7>1
capital in the whole labor force Q.
meant to capture the idea that poorly educated,
by
Thus
t
i
dfi\(h?)
=
v •(//,)'•
(L\)^
o
where
fi\
is
the distribution of
human
capital in
community
to increase with the level of skills. Incorporating (4)
and
£l\
Note that a
.
(7) into (3), the
>
1 is
required for income
accumulation of
human
capital
takes the form:
(8)
where a
=
Equation
(8) involves
4
We
6,
=
(1
-
6)(a
both a
-
l)/<r,
h\
+1
7
=
= e-c -(h\nL\f(H y
t
t
(1
-
8)/a
,
with
local linkage L\, because public
a+
/?
+7 =
1
and
=
k
(1
Tamura (1991b)
offers
6
u) (vt)
1
-6
.
goods are funded by community income, and a
develop in appendix a variant of Ethier's (1982) model of specialization which leads to (5) and
proof of Proposition 6.
-
(6)
below; see the
a model leading to an aggregate production function closely related to
Kremer (1992) and R. Cooper (1992) study equilibrium and optimal task assignment
in firms or production teams.
(5).
global linkage
H
= Y /is,
t
t
because
workers are complementary in production. Both are
all
CES
aggregates,
with the same elasticity of substitution. As workers become better substitutes, communities become
interdependent:
/?
rises
and 7
falls.
This model allows us to ask the following questions.
Is it
more
maximizing aggregate output and growth, to have the population
=
(L\
h\) or
mix into
identical, integrated
takes a short-run or long-run perspective?
transfers to richer families?
These
Suppose that households are
of public education funding
reflects the
income of
communities (L\
Can
then more
stratify into
homogeneous communities
Does the answer depend on whether one
for the specific
more
directly policy-relevant
where each community's school budget
where expenditures uniformly
reflect national
model developed above, we show that similar
a wide class of models from the growth and
human
capital literatures.
a Puzzle
aggregate income clearly matters
rise to
a model with local and economy-wide interactions
either production or education uses
if
some nationally
provided public good, such as defense or infrastructure. Second, technological spillovers a la
Lucas (1988)
y\
—
•
R&D,
it
(h\)
a
may
(H
affect workers' productivity, leading to individual
b
t
)
will again
knowledge directly
.
As long
as the accumulation of
be affected by
H
t
.
if
new human
spillovers to
Romer
in schooling or
(1991a) assumes that the aggregate level of
capital: h\ +1
=
(h't )
a
(H
b
t
)
.
be confined to a smaller sphere than the whole econ-
only because geographical distance limits frequency of interaction. Indeed, sociologists have long
described, and economists recently modelled, a variety of channels through which a community's
capital
(1986)-
production functions of the type
knowledge uses produced resources,
Tamura
Alternatively,
affects the generation of
One would expect some knowledge
omy,
manner.
by education and income. Which system
local funding,
efficient:
There are many potential channels which can give
First,
t
from the point of view of
integration be Pareto improving even without compensating
or national funding,
its residents,
More Models and
like (8).
H )t
efficient,
issues can also be rephrased in a
income? Before answering these questions
2.2
=
in fact geographically segregated
is
issues arise very naturally in
less
makeup
affects the educational
outcome of
its
8
young people.
These sources of
human
"social capital"
(Loury (1977), Coleman (1990)) include: peer
effects
between students of
and Besley (1991)); the
or within the school (Banerjee
good
or bad, as well as networking contacts for the
mery
(1990));
and crime or other
activities
different ability in the classroom
fact that neighboring adults provide role models,
young (Wilson (1987), Streufert (1991) and Montgom-
which interfere with education. There
empirical evidence of such peer or neighborhood effects; see
Benabou (1991)
is
also a fair
amount
of
for a brief review. In contrast
to fiscal spillovers, pure neighborhood effects can generally not be remedied by simply improving access to
capital markets or
by redistributing income across communities. 5
Of course any combination
But
likely.
aggregate
their
H
efficient -in
t
common
of the channels discussed here and in the preceding section
feature
is
possible, even
is
the presence of a local aggregate L\ and perhaps an economy-wide
in the production function for
new human
capital.
The
question then arises again:
is it
more
the sense of increased output and in the Pareto sense- for society to stratify into homogeneous
clusters, or for
each community to
reflect the
nation-wide distribution of
human
capital?
Naturally, one expects the answer to depend on the form which interactions take.
prising extent to which this
is
true, introduce the idea of local elasticity of substitution,
empirical flesh to the discussion of non-fiscal spillovers,
(1992a) investigates whether
To show
human
let
the sur-
and give some
us consider the following example.
Borjas
capital externalities operate within ethnic groups. Using longitudinal
data, he estimates the model:
\og(h\ +1 )
(9)
where
h\ +l
capital;
is
a son's
and L\
is
level of
=
human
"ethnic capital"
,
5
a and
/?
capital,
+ (3\og(L\) + control
measured as
variables
his hourly wage; h\
defined as the geometric average of
in the father's ethnic group: log(L{)
Borjas estimates both
alog(/ij)
= / °° \og(h)dfj,\(h).
Finally,
r/J
is
+
is
human
r)\.
his father's level of
human
among
adults
capital levels
an unpredictable individual shock.
to be between .25 and 0.30 and statistically significant. 6
Benabou (1991) shows that they can lead
to inefficient self-stratification even in a representative agent
model with perfect
capital markets.
6
Borjas also uses years of education instead of log-wages.
The remarks made below concerning
the consequences of
These results are very interesting
in
and of themselves, adding to the body of evidence that group
interactions influence the acquisition of skills.
But they
(narrow) point of view of maximizing aggregate income,
also raise the following question.
is it
more
together, study together, etc., or that they remain separate?
to answer this question. Unfortunately,
we
answer a priori: for any values of a and
is
exposed to his own group's average
shall see that
7
efficient
that ethnic groups mix,
One might hope
human
capital than
the long-run the two paths converge to the same level
if
if all
a
is
always higher
accumulation
+ (3 <
will
now be
generally
segregation, especially in the long run. In this instance, the
the
mean
of the logs and the log of the
mean can be
This point
each child
1.
more
common
log(/
hdfi't (h)).
under mixing than under
quite misleading: with heterogeneous agents, Jensen's
in fact
is
efficient
=
capital
practice of not distinguishing between
inequality implies that both individual and the economy's growth rates
which the externality operates.
if
are exposed to the population average. In
operates through the geometric average, one had used the arithmetic average: log(L{)
later on, capital
live
(9) constrains the
But suppose that instead of assuming that ethnic
This conclusion seems rather distressing.
As we show
i.e.
the
to use Borjas' estimates
by using a geometric average,
the path of total labor income
/?,
From
depend on the aggregator through
independent of the stratification
issue:
if
there
is
heterogeneity within each ethnic group, an equation like (9) will be misspecified unless the particular
aggregator which
it
imposes happens to be the correct one.
These remarks demonstrate the importance of the
elasticity of substitution
among
individual inputs into
is
developed below.
this parameter, rather
than constraining
the production of a peer effect or neighborhood externality; the underlying intuition
It will
it
therefore be quite important in empirical
to either one or infinity as
is
usually done.
work to estimate
8
within-group inequality and inter-group mixing are quite general, and apply to that specification as well.
This question
is
not purely hypothetical; one suspects, and Borjas' (1992b) later work indeed tends to indicate, that
"ethnic capital" really arises from neighborhood effects combined with ethnic segregation.
One
(9) the
could specify L| as a
group's variance (A|)
2
CES
index and estimate
its elasticity t
from a non-linear regression, or more simply include
of log-human capital. Its coefficient will provide
10
an estimate of — 1/e; see Section
3.1
in
2.3
A
General Framework
common
Recognizing in the various examples discussed above a
underlying structure, we shall consider
from here on the general model of knowledge accumulation:
h\+1
(10)
where
Q
is
a random shock, h\
economy-wide index of human
parental
is
= F(Clh\,L\,H
t)
human
capital,
and L't
capital. In general, equation (10)
is
,
H
t
and an
are respectively a local
not a purely technological assumption,
but a reduced form which already embodies a variety of market and non-market interactions: equilibrium
wages, financing of local public goods, technological spillovers, peer effects in schooling, etc.
examples incorporated at most one
local
externalities can be reduced to (10),
The two
and one global
where L\ and
levels of interaction in (10)
H
t
open up the
link at a time, but
we
discuss below
how
heterogeneity. It
The
possibility of
human
value, positive or negative, of a
CES
capital inputs in L\
h'^
t
(J°°
While
H
t
individual
is
i
more homogeneous work-
t.
and
H
t
respectively.
We
and global
level,
therefore specify
<WoV~
computed over the whole population, L\ only
belongs at time
equalized,
h^dMh)Y"
1
H =
(12).
is
het-
averages, with potentially different elasticities of substitution:
L\=(j°°
(11)
When
an intertemporal tradeoff.
force in the next generation. Intuitively, heterogeneity causes greater losses at the local
the external effects as (symmetric)
multiple
net loss, positive or negative, represents the efficiency cost of local
must be weighted against the
the less substitutable are individual
earlier
are appropriate composite indices.
erogeneous families share the same school or community, or when some input into education
the rich lose and the poor gain.
Our
Depending on the context
a country.
11
reflects the
this
composition of the group
fi{
to which
can be a school, a community, a region, even
We
will
show that the
Figure
indicated on
costs of heterogeneity in L\
we allow them
1,
substitutes rather than complements,
H
—00,
H =
t
and
max{h\,i £
The
fi}.
t
and inequality
is
As
are indeed measured by 1/c and l/<r.
to take any values -even negative ones.
a source of gains.
spans the whole range from a Leontieff technology,
t
H
9
As
In that case agents are
1/cr decreases
from +00 to
H = min{h\,i G fl}, to a "frontier" technology,
latter case corresponds to the
t
model of Murphy,
Shleifer
and Vishny (1991),
where the best innovation becomes embodied into the next generation of technology or know-how. Similarly
we
at the local level,
allow
cases between peer effects of the type "one
all
models where the best individual
may
(
sets the standard.
More
bad apple
bunch" to
spoils the
generally, the accumulation of
human
role
capital
involve several interactions at each level, say:
h \+\
10 ')
H\
For instance,
>
-
costly (l/o !
t
i
0)
= F{Q\IA
l
t
,...L Kt \Hi tt ,...HNtt )
could arise from complementarity in the production of goods, which makes heterogeneity
and
priced through wages, while i72 ,t could be associated to the generation of non-rival,
is
non-excludable new ideas, where inequality
is efficient
(I/02
<
But
0).
all
local
and global
spillovers will
matter only through two weighted averages: we show in appendix that (10') reduces to (10), where L\
and
H
are appropriately defined. Finally, another important specification
t
function for
new human
(with either
j3
equal one.
To
or
The
capital.
is
that of F(-), the production
assumes the multiplicative form
literature almost universally
zero), constraining the elasticity of substitution between h\, L\
7 equal to
simplify the exposition,
appendix, we generalize F(-) to a
CES
we
retain a
Cobb-Douglas specification
aggregator, and
show how the
for
and
H
(8)
to
t
most of the paper. In
effects of heterogeneity
and
social
structure also depend importantly on the extent to which parental, local and national inputs in a person's
education are substitutes or complements.
For a
measures
The
last
CES
index with constant returns such as (12), or just H(x,y)
(in the
neighborhood of x
property -which
as in (8)) whereas the
det(H")
= #n
y) the
two do not. Indeed, H12
first
2
)
to
7
2
(1
-
(.jr «
+
«
j y
complementarity between inputs, the concavity of
relevant here- remains true with
is
H22 - (H12
=
=
is
any return to
then proportional to 7
i)/cr, but log((if(2±i, ^-)/H(x,y))
12
—
is
1
+
1/<t,
,
1/ct
simultaneously
H and the cost of heterogeneity.
(when the index
scale
)»-»
is
raised to
H\\ and H22
to
some power
7 —
1
—
1/cr
7,
and
simply proportional to 7/17; see Section 3.1.
L = min{hM
1+
£
local
complements,
local
H =
maxfh 1
}
and
global complements
global substitutes
->
«-
o
local
and
local substitutes,
global complements
global substitutes
L = maxfh 1
}
local and global
Figure 1: The costs of heterogeneity:
degrees of complementarity 1/e and 1/a.
12A
H
= minfh 1 }
Community Composition
2.4
Given the general model described by (10)-(12), we
mulation and welfare under two regimes of interest. The
composition
is
the
same
endogenous community formation. This
environment leads high and
The same
local public
to
perfect integration, where each
is
is
we do not seek
commu-
basic force
is
work
here:
with d 2 F/dhdL
goods or externalities L\ are strategic complements
make more educated parents
paper to
offer
a theory of
differential sensitivity to the quality of
low-skill workers to segregate as
at
in this
for three reasons.
Benabou (1991) already provides such a model, where a
First,
permit.
perfect stratification, where each type of
as that of the population at large. Intermediate cases of partial segregation
could easily be considered using the same methods; but
their
first is
The second
agent lives in a separate, homogeneous community.
nity's
compare the dynamics of human capital accu-
shall
much
>
0,
in the
as technology and institutions
parental
human
capital h\
production of h\ +l
and
This tends
.
willing to outbid less educated ones for land or housing in a "better"
community. One could thus obtain once again segregation as the only stable equilibrium, sustained by land
rent differentials. Alternatively, one could follow Durlauf (1992) and allow each community's residents to
vote on zoning or
minimum income
requirements. In the absence of significant fixed costs, the rich have no
desire to let in the poor, so this
would again lead to
would require tying oneself to a
specific choice of preferences
elasticity of
stratification.
Implementing either approach, however,
and of the "technology" of segregation:
housing supply in each location, school district boundaries,
costs for a school or a
The second reason
community, mobility
is
feasibility of zoning, size of
many
setup
costs, etc.
that the mixing and sorting regimes correspond to alternative policies: local or
national funding of schools, tracking or busing, mixed income housing, etc.
are
price-
The
last
reason
is
that there
sources of stratification which are unrelated to parents' concern for their children's education:
differences in income, tastes, racial segregation, etc.
We
therefore choose to be agnostic about the causes
of stratification and focus on the growth performances of two "pure" cases which deliver the
complete segregation and complete integration.
13
main
insights:
Stratification
3
and Growth: the Short and the Long Run
Let us start with the simplest possible case. There are two types of agents,
each.
They
differ
only by their
initial
endowments of human
the initial variance of log-human capital.
10
We
In a stratified economy, the local environment
h\ +1
(13)
In an integrated economy,
t
Denoting
.
all
all
= e-(h\) a +P(Ht y,
variables in the integrated
h\ +1
=
local level thus has
human
+ /? to
capital
/?;
from a
the effect of
accumulation
H
will alter
a way which we
make
a,
a
Q=
>
0.
Thus
A2
is
1.
two
effects.
results,
with
= A,B.
level of local externality or public
we
hat,
First,
i
it
good,
Lf = Lf =
have:
=
A, B.
decreases the return to scale on parental
raises the return to scale
on the
local aggregate
human
from
capital
the impact of any given amount of heterogeneity on the economy's growth rate, in
precise below. In other words, one of the
effect of
distribution with small
i
remains unchanged. These changes in the effective technology of
t
mixing
is
A =
y/a(l
—
a)
two
social structures will
be more
efficient
knowledge than the other.
to accelerate convergence to a
'As L\ depends on group composition but not group
same
differences:
(L t y(Ht r,
and correspondingly
at aggregating heterogeneous levels of
The second
same
economy with a
(h\)
Mixing agents at the
the
| log(fto /hf?)
uncertainty:
all
compounds family
agents share in the
(14)
to
abstract from
and B, with measure 1/2
Dynamics and Losses from Heterogeneity
3.1
L
A=
capital:
A
->
size,
homogeneous
any proportions (a,l
—
a) of
society.
A and B
Denoting A*
=
families will lead to
log(h^/h^). More generally, the Taylor approximations used below apply to any
enough dispersion.
14
1
\
in the segregated
\og(h*/hf)
economy and
A«
(15)
two
Intuition suggests that these
less efficient for
effects
=
may
heterogeneity affects growth.
We
t is
is
+ /?)
log(hf/hf)
(13)
we
capital,
but
still
more
investigated below; but
efficient in the
first
A =
t
(hf
A
-f
t
levels of
and agents are complements
but
it
it
human
distribution of
capital.
hf)/2, which brings out the role played by
\
0--1
level of human capital, the right
= 6 + (a + P+y-l) log(A
knowledge are unequal, however, the two bracketed terms
face decreasing returns
case,
The
and any aggregate index of human
economy where everyone had the average
The
is
long run because
be
...
to Jensen's inequality.
heterogeneity
for instance
get:
would be 0- Af +P -AJ and the growth rate \og(A t+ i/A t )
When
may
- ..(.
In a representative agent
side
»
we must determine exactly how
g—
1(+1
have:
A>a'A = A,.
by the degree of inequality
From
we
in the integrated one,
lead to an intertemporal tradeoff: mixing
focus on the per capita stock of knowledge
12
f
•
Consider for instance the segregated economy.
fully described
heterogeneity most clearly.
|
t
any given distribution of human
reduces heterogeneity faster. 11 This issue
capital at time
(a
A =
differences represent the losses caused
a source of gain
if
a
+ /? >
1
or 1/<t
in the
<
0.
differ
t ),
=
log(0).
from A" + ^ and A] due
by heterogeneity when communities
production of the aggregate
We
with 6
hand
H
t
Conversely,
.
shall often focus the exposition
on the
first
should be kept in mind throughout the paper that the model allows for any configuration of
parameters: we do not impose that inequality be bad for growth.
In the general
model
(10)
we
define greater efficiency as increased total
production technology and preferences
It will
since
it is
Ht > Ht
be clear how to go from
unaffected by dispersion,
for
any a >
0,
(e.g.
Section 2.1)
we
and vice-versa
capital. In specific models,
shall also consider aggregate
this arithmetic average to
and
human
a <
any other aggregate index, such as Ht. At
0.
15
given a
output and individual welfare.
therefore not biased toward segregation or integration: as
for
i.e.
is
a logical choice
At > At, At
=
At implies
We
can simplify the expression for the growth rate under heterogeneity by using Taylor approxiFor any A and x,y such that
mations.
log
A
£ i
((
f
(16)
/(
)
\
£-js
^)
f^±\
og
)
A =
can be approximated as
\ \og(x/y)
tf A (A)
l0g
We
(i
see that the drag
Thus \og{Ht /A t )
w -A 2 /2<r and:
to:
*0 + (« + + 7-l)log(i*)-(«(l-a) + ^ +
/?
J
on each economy's growth
is
^^-a
the product of two factors.
be discussed below. The second
is
The
2t
first is
the economy's
the current variance
A2
or
A2
capital distribution.
The Short Run
3.2
We
human
m A(l-A)A 2 /2.
^(A) =
t
efficiency loss per unit of dispersion, to
of the
not too large, the loss function
a e+(a + p + 7 -l)log(A )-{{a + m-^-l3) + l)^--(a + 0) 2t
For the integrated economy, similar derivations lead
(17)
is
first
ask which economy grows faster, for any given amount of heterogeneity. In other words, suppose
that at time
t
—
higher or lower?
previously segregated populations
From
£=
intuition
is
clear:
(19)
is
(a
is
C A 2 /2,
human
C A 2 /2,
capital at
t
=
1
be
with:
+ /?)(l-a-/?) + 7/<7
losses reflect the concavity of the function h a+l3
agents' inputs in the aggregate
reduction in growth
integrated: will
(16), the efficiency loss in the segregated case
(18)
The
become
H, which has weight
7.
l/<7 of
In an integrated economy, the corresponding
where:
£ = a(l-a) + p/e + y/<r,
16
and the complementarity
with a similar interpretation involving returns to scale at the family rather than the community
both local and global
and only
if
C >
4>
<
1,
the education production function
capita
(e
=
is
enough
endowment At Such
.
is
clearly the case
Borjas (1992a), the mixed economy
when 2a
+
>
1,
The
Mixing
is
if
concave in the previous generation's
will
then accelerate growth even
poor do not drag
more vulnerable
hand
if
Lt
is
is
standard, static models of matching.
+7 =
a+
1, it
the short
too far below the per
definition,
mixing
is
in the short run.
13
means that parental human
best understood by showing
By
t
in
human
(e
=
1) as in
to heterogeneity than the segregated one. Finally
education than the economy-wide aggregate
intuition for Proposition 1
L
a geometric average
mixing tends to reduce human capital accumulation
child's
for any given amount of
the local spillover operates through the arithmetic average
(1992). On the other
quite plausible since under constant returns,
important to a
is less
local substitutability so that the
Glomm and Ravikumar
oo), as in
i.e.
= 0(l-2a-0-l/e)>O.
capital under integration than under segregation.
run, provided there
in the short run,
C, or:
(20)
When 2a +
and
elasticities of substitution.
Proposition 1 The mixed economy has higher growth
heterogeneity, if
level,
H
t
:
how
a >
<j>
This case
capital
is
is
more
7.
embodies the
effects at
inefficient if the losses of the rich
work
in
exceed the
gains of the poor, meaning that:
(21)
(A 1
-A
1
[itir -
13
The same
is
)/(H2/2)
(fc?)"]
=
(h*)°
[ctf
r
true for log(Hi/Hi)
[{hW -
(Lof] - (h*)°
- a?] + (A?r
w
fi
(A 2 /2)
((1
•
[ctf
- la -
f - (A?)"]
f + (*? y -24] + Krtr + (^)l
/3)(1
17
[(L
-
1/cr)
+
\jb
- 1/t),
unless 1/<t
-
1/e
is
Ap
-
Tp
sufficiently large.
positive.
is
The
first
It is positive since
term
arises
children from better backgrounds lose
to the per capita average Aq. For small dispersion this
The second term comes from the
It is
more from a given
term
human
(/?/e)
•
term
is
close to
— (3(1 —
it
A a+!3 A 2 /2. The
than mixing; the second one goes
stratification proportional to 2a/?
•
—
in the
/?(1
Because mixing equalizes knowledge
/?)
Aq + ^
•
—
<
up, so Lq
first
and
really matters
A To
-
C =
t
A 2 /2.
<
+ /?/e = —
:
A 2 /2.
if
1/e
The
>
community than
in
0).
a rich
term incorporates the
final
poorly educated agents drag
0,
Aq. For small dispersion this last term
tend to
Summing
opposite direction.
/?)
•
last effects
make
sorting
more
is
efficient
three yields a net impact of
all
<f>-
faster,
the drag on growth due to dispersion eventually becomes
shown by the
smaller in the integrated than in the segregated economy, as
t
•
The Long Run
3.3
what
h*
capital has a larger impact in a poor
well educated agents pull
approximately equal to
decline in La, such as from
approximately equal to 2a/? Aq
is
pure losses from heterogeneity in generating the local spillover Lq
down Lq more than
0).
local inputs (F12
decreasing impact of marginal improvements in local conditions (F22
negative since an extra unit of
one. For small dispersion this
>
from the complementarity of parental capital and
6
is
whether this
effect
is
sufficient for
A
t
to
make up
answer this question, we solve the difference equations (16) and
(1
-
i?')/(l
log (
^
-
R),
\
we
terms
its initial
(17).
in (16)-(17).
A 2 Vfl'-i-<W/?V*-r
„
Denoting
R = a + /? + 7
Rt -(
A2 R
A2
-(a
a + P) 2t
l
c
- c
t
-c:%p<-^ = c -c.%
t
Therefore:
18
-
R- -(a + /?) 2
B*- a 2t
R _ a2
But
handicap and overtake
have:
2
^Gi^)
last
and
Proposition 2 For any 7 >
the gap between the integrated
0,
and segregated economies
is,
for
t
large
enough:
^j„$.^( a +
io g
(22)
/j
+ 7 )«
where:
$=
(23)
(a
+ /?)(l-a-/?) + T /<r
+ /? + 7 - (a + /?) 2
In the long run, the gap shrinks to zero
to a finite limit
R=
if
The two numerators
denominators
1,
a(l
and explodes
total returns to scale
if
R>
if
- a) + 0/e + 7/tr
-«2
/? + 7 - <*
a+
a
1.
R=
a+
+7
are less than one, tends
Equation (23) embodies the main insights of the paper.
represent each economy's instantaneous losses per unit of heterogeneity.
reflect the different speeds of
convergence to a homogeneous society.
The
The two
tradeoff between
incurring the costs of local heterogeneity and reducing the losses from aggregate heterogeneity at a faster
rate
apparent
is
obvious factor
is
in the fact that
$
decreasing in /3/e and increasing in f/o-.
is
But an
additional, less
involved: the difference between the concavity of the technologies faced by a
and by a family, adjusted by the appropriate convergence speeds. This value of
$
for 1/e
=
community
1/cr
=
is
generally positive, as will be seen below.
We
are
now ready
when
to answer the question:
is
the long-run
human
capital stock larger under
mixing than under segregation? Let us start with a useful benchmark case, assuming:
returns; (b)
e
=
a:
heterogeneity
a "neutrality" assumption;
and
H
<
than between the three inputs
t
1:
there
h\,
L
t
,
is
H
more
quality
and
(c) also
seems plausible
education than workers' different
14
When
for
society's general level of
F(h,L,H)
is
a
CES
1,
constant
lA
t
.
substitutability within the composite inputs
Note that the model of Section
three restrictions; in particular, (c) was required for a worker's income to rise with her
Assumption
R=
equally costly or beneficial at the local and economy-wide levels,
is
(c) 1/(7
(a)
2.1
imposed
human
L
t
all
capital.
most alternative interpretations. Parental background, peer group
knowledge or income are
skill levels in
likely to
be poorer substitutes
in
a child's
the production of output or know-how. In the benchmark
function with elasticity A, the relevant comparison
19
is l/<7
<
l/A; see the appendix.
case equation (23) becomes:
>
t-
(24)
at
.xL«
so that integration raises the long-run level of
Figure 2
to,
given
stationary.
< $ and R =
Initially,
The common trend
1.
A
the richer
worsens, but the overall impact
is
to the fact that society remains
and
=
a
—
00. Let
of the rich at
5.6
%. The
A=
1,
or
first raises
initial
course,
it
•
boom
6
t
is
factored out from
all
The
stratified at
variables,
h* /h^ —
7.4; this
common
level
which
is
distribution of income
lower than what
orders of magnitude,
feel for
let
a
=
it
slows
it
.5, /?
=
corresponds to a coefficient of variation of 0.76.
is
rich
.3,
preferable in the long run.
is
when
The
converse
15
is
we show
true for
in
R>
=
.2
and
secession
We now
examine more
16
We shall
come back
to this feature
appendix that decreasing
1.
First, note
the degree of economy-wide linkage 7 tends
and poor are completely independent from one another and when
on. Second,
7
slowdown but higher steady-state output.
of stratification on the economy's long-run performance
arbitrarily small extent.
reduced,
erased two generations later. Integrating a previously segregated society leads
that the steady-state gap (24) remains positive even
when
is
growth by 4.5%, but eventually lowers the steady-state path of the economy by
need not be the case that mixing
The impact
down due
would have been
generally the relative performance of the two social structures, by varying key parameters.
to zero.
time
making them
more heterogeneous. Eventually, even the A's accumulation
to the converse scenario, with an initial growth
Of
$ A 2 /2. 15
favorable and growth accelerates. Over time, however,
had society remained integrated. To get a
e
by
'
agents benefit, while the poorer B's lose.
dynasties' capital stocks converge to a
all
capital
r) >1
what happens when a previously integrated economy becomes
illustrates
<
<j>
human
t
(
is
thus surprisingly different
their fates are linked to
when evaluating each dynasty's
total returns
R<
1
an
welfare later
raise the benefits of integration; the
Third, and most importantly, segregation remains preferable in the long run
Note that (24) allows for inequality to be
beneficial,
i.e.
for
\/a
=
l/t
<
0.
When
all externalities
if
operate through
geometric averages (Tamura (1991a), Borjas (1992a)), (24) shows that mixing and segregation lead to the same steady-state.
On
the other
hand asa
asymptote approaches
+
/3
= l— 7
approaches one, the speed at which the segregated economy tends toward
zero; see the expression for \og,(At/AoR') in appendix.
20
its
lower
hi
hA
A 00
•00
hB
0
t
integrated
stratified
Figure
2:
The
economy
economy
short and long run
effects of stratification
20A
disparities in
knowledge entail
sufficiently greater losses at the
the aggregate level, e.g. in production. Rearranging (23), with
*>0
<
(25)
The two
e
Figure
R=
level, e.g. in schooling,
than at
1:
I_I<(l-lV_L^)
>
o) \l+a
as
regions, illustrated on
community
a-
3, are quite intuitive.
area between the boundary and the two axes; since 1/a
+ /3j
V
<
<
Perhaps most noteworthy
is
the triangular
1/e, heterogeneity creates negative spillovers
at the local level but positive ones at the aggregate level. Nonetheless,
mixed communities lead to a superior
long-run outcome due to the differential combination of returns to scale in accumulation and convergence
speeds discussed
than 1/e. For instance,
is
much
order to overcome this effect and
earlier. In
easier for
it
will
be
efficient for the
good managers to make up
than for students from favorable backgrounds to
Similarly,
qualities
0,
1/<t
must be
managerial and working classes to
sufficiently larger
live separately if it
poorly qualified workers in the production of output,
offset the effect of
weaker schoolmates in peer interactions.
be optimal to sort successive generations of Ph.D. students into departments of
when complementarities
differential
are stronger during graduate studies than they are during research careers.
Welfare and Pareto Optimality
3.4
It is
will
it
for
make $ <
easy to go from the aggregates
segregation:
log(ft^)
=
log(A t )
+
A
t
or
A
log(2/(l
t
+
to each group's path of
e
_2A<
«
))
log(A t )
+
human
capital. For instance,
A« — A^/2. Given a
under
specification of
preferences and of the relationship between skills h\ and income y\, for instance as in Section 2.1,
we
could compute present values of each family's utility or of any planner's social welfare function. But the
main
(0
<
insights are clear even without doing so. First,
<j>
<
$),
and of any
levels.
it
will lead to
more
integration
is
more
a higher value in each period, not only of total
social welfare function
In the
if
which
interesting case
is
where
a
<f>
human
CES
aggregate of individual
<
<
planner has a low enough discount rate. Moreover,
efficient
even
capital,
human
in
the short run
but also of output
capital or
consumption
$, social welfare will be higher under mixing
if
21
individual agents' discount rate p
is
if
the
high enough,
.
Cost of
local
heterogeneity
O
:
<
-•***
,
,:
:
:
:
:
O
>
Cost of global
«*"
heterogeneity
:
1
<D
>
Stratification versus integration
Figure 3:
in the long run
21A
.
integration will be Pareto improving even without any redistribution, because in the long-run
are identical
and share the same
Random
4
it
and to
R >
1
capital Aoo or Aoo
and Long Run Growth
main parameters which determine how heterogeneity
First, the long-run distribution of
implies explosive growth.
which the economy was
agents
us to draw the essential distinction between the short and the long run effects of
identify the
had two drawbacks.
but then
human
Ability, Stratification
The previous model allowed
stratification,
level of
all
stratified
had no
This
effect
growth. But
+ ft >
income was always degenerate -unless a
clearly contrary to the evidence.
is
affects
on the long run growth
rate,
1,
Second, the way in
except again in the case of
increasing returns; see (22).
In this section
we
solve both problems by incorporating
returns to education, as in Loury (1981).
random shocks
Such random draws of luck constitute a permanent source of
inequality, but also of social mobility: the relative rankings of
forever, but will
will
On
have a
to children's ability or uncertain
any two dynasties
will
no longer be preserved
change with positive probability. Because a mixed society "undoes" inequality
less dispersed
the other hand,
it
asymptotic distribution of
may
still
be
human
capital
it
and income than a segregated one.
processing any given
less efficient at
faster,
amount of
heterogeneity.
Under
constant returns, the balance of these two effects will determine which of the integrated or segregated
economies has the higher long-run growth
4.1
rate.
Dynamics
Let the accumulation of
and the shocks
Q
are
human
i.i.d.
capital be given
with log(£J)
~
by
Af(0,s 2 ).
(8),
initial distribution
of
human
of this specification, which builds on
is
t
are defined as in (11)-(12)
log(/i'
)
~
M"(m,
A2
).
We
also
The advantage
Ravikumar's (1992) deterministic model,
22
little loss
already captured by the h\ term.
capital to be log-normal:
Glomm and
H
Assuming independent shocks involves
of generality, since intergenerational correlation of ability
take the
where L\ and
is
that h\
remains log-normally distributed. This allows
if
A
log(/i;)~^K,A 2 ),
t
=
(<r
—
H =
t
consider
{£ h^ <*(*))
human
capital at time
log(A»+i)
(27)
Human
log
°°
aggregates and loss functions to be computed exactly:
hdp
((/
X
t
(h))
//~
AA
=
d/i,(fc))
A(l
-
A)
•
A?/2. Setting
l)/«r yields:
(26)
Now
**(At ) =
then:
CES
capital at time
t
+
1 is
'"
= «P (m, +
t+1. Taking
(^) f ) = A,
logarithms in (8) with L\
e Xp
•
=
= 9 + log(C!) + (a + /?) log(AJ) + 7 (m + 2—1
t
[-§)
h\ (segregation):
.
M)
therefore also log-normally distributed: log(/iJ +1 )
~ A/"(m
t
+i,
A 2+1
),
with
m t+1 =0 + R-m +y(Zf±)£
t
(28)
A?+1
Integration yields similar expressions, with
The
steady-state variance of
expected.
make
We
human
capital
could solve (28) for the
mean
+ /?) 2 A 2 + s 2
=
(a
a
+
replaced by
then
A 2^ =
is
capital
A =f
t
(29)
log
(30)
log
hdfi t (h). Using (26) to (28),
(^f)
(^f)
C=
(a
,
it is
which
m
t
at
(e
is
—
l)/e
added to y (<r
lower than
any point
A 2^ =
in time.
1
But
—
1)/<t.
_/^,g\i
,
as
in order to
better to track once again the behavior of total
we obtain the growth
rates of the
two economies:
e+J + (R-tyog(A )-^-((a + 0)(l-<*-P)+l)
=
9
t
+ j + (R-l)\og(A )-^-(a(l-a) + ^ + l
+ /?)(1 — » — /?) + j/a
respective variances.
<>a
/?
=
t
These expressions are identical to (16) and
factors
*
1
of log-human capital
the losses from heterogeneity appear most clearly,
human
a and
and £
(17),
=
The comparison between
a(l
up to a constant.
— a) +
0/e
+ y/cr
In particular,
for
we recognize the
loss
each economy, multiplied by their
the two social structures' efficiency at aggregating levels of
23
knowledge,
between their capital stocks one period
i.e.
therefore unchanged: log(ii/>li)
To examine
and (A t
,
A
t
);
=
/?(1
- 2a -
-
after starting
1/e)
A 2 /2 =
•
<f>
from the same
A 2 /2.
the impact of stratification on long-term output and growth,
see the appendix.
We
first
initial conditions, is
consider the case where initial
we
solve (28)-(30) for
endowments
(A
t
,At)
are the only source
of inequality.
Proposition 3
The
:
effect
2
of initial inequality. If s
=
then for any j
>
and
t
large enough:
<»»Mr«n^ )£*-£ *
This
is
exactly the
same expression
as in the two-group case, so
all
the results derived previously extend
to this model.
Proposition 4
log(
$
effect
2
of ongoing inequality. If s
>
0,
then for large enough
t:
i
returns, the integrated economy's long run growth rate exceeds that of the segregated
economy
2
•
s /2.
As seen
by a
The
&~l('-±-'M¥kHT=fa-iiv)-H £s)
Under constant
by
:
$ >
unless the cost of local heterogeneity exceeds that of aggregate heterogeneity
margin.
Proposition 4 shows, quite intuitively, that recurrent inequality impacts the two
earlier,
sufficient
economies one
level higher
A2
than
the
same way
A2
vanishes asymptotically.
as
initial dispersion does.
affects long-run levels.
When R <
24
When R =
1,
s
2
1,
s
2
affects long-run
growth rates
impacts long-run levels whereas the
in
effect of
4.2
Discounting and Welfare
Let us
now examine
»
the welfare of individual families, asking in particular whether integration can bring
about a Pareto improvement without redistribution. To derive simple, closed form expressions, we assume
that agents have logarithmic utility and compute the expected present value of log-human capital for
Note that
each family.
if
labor income depends not only on
own human
capital but also on aggregate
productivity, as in (6), this will understate the relative benefit of integration with respect to segregation.
Finally,
in
we use the benchmark
specification 1/e
which direction deviations from
=
1/cr
<
1
and
R=
t,
differences
and ultimately
its
U'
-U'
= E
£>*
P
(
p/3{m
lQg(fcj/AJ)
I
*'o
(1
a~
\l-p'
The
first
term
the
mean
lose
reflects the
distribution of
its initial
l
PS
2
*
+ (W)A
2(1
impact of the
initial
-
2
-p)
endowment;
-
pa)(l
p(a
(A±2
+ /?))
T
l-fa + 0)
\l-pa*
ceteris paribus, dynasties
2
s
)
of the unconditional average of (log)
the expectation of the asymptotic distribution facing each dynasty,
reflects their
\A
))
which start above
made
better off by integration.
sufficiently patient.
1/cr
>
it
makes
human
all
is
always
(through
level
A2
)
capital. Since this
better off to an extent
degree of patience or intergenerational altruism. As p tends to one, so does the fraction
point in time has finite support, integration
'Assuming
2
from integration, while those which start below the mean, gain. The second term
and possibly the growth rate (through
of dynasties
capital at
log(/i' ))
-
do not remain paribus: integration raises the
which
human
endowment. Taking
positive, reflecting the fact that ceteris
is
3.3.
stratified cases yields:
t=o
,
z's
expected intertemporal welfare, conditional on
between the mixed and
(31)
should be clear from Section
this case will pull the results.
Straightforward but tedious derivations allow us to derive family
any time
It
1.
17
0;
Note that
the bias
is
this has
reversed
Since in practice the distribution of
is
for all practical
0.
25
wealth at any
purposes Pareto improving
nothing to do with any insurance
when \/a <
human
effect:
in (31),
if
agents are
dynasty
i's
own
shocks log(Q) are set to their expected value of zero under both integration and segregation.
H^
Let us next examine the role played in this result by the global externality
accumulation paths of rich and poor (due
Suppose that
2.1).
this link
for instance to
Recall from (24) that the asymptotic difference in growth rates
economy converges to
opposing
effects into
in the limit;
down
to zero.
-\og(hj)
+
1-pa
(<r-l\
\
<r
tributes positively to each dynasty's net welfare.
=
2
,
this
is
from
initial
+ f3 tends
$ A 2 /2
•
at
to
1.
between the
which the
stratified
Equation (31) incorporates these two
it
becomes:
\ / />s 2
+
mean
(l-/>)A 2 \j
2(l-p)
of (log)
)\
human
capital
still
con-
endowments are the only source of inequality,
When
factor. Integration is not Pareto
s
initial conditions,
2
>
0,
improving unless richer families receive com-
however, mixing has a growth rate
provided the discount factor
is
effect.
makes almost
all
families better
off,
This dominates any
Thus we
high enough.
the fraction of net gainers becomes arbitrarily close to one as p tends to one.
integration
1, at
,
pensating transfers.
effect
When
together the
only a level effect so only a bounded fraction of the population has a positive net gain Uq — Uq,
any value of the discount
for
+ /?)'
J \l-pofl) V
In the limit, integration's long run effect on the unconditional
s
2
1-a
f
R=
s /2 or in levels
discounted present values; when 7 goes to zero,
u'-U^PJl^l
(32 )
$
on the other hand, the speed (a
trajectory slows
its inferior
ties
complementarity in production, as in Section
becomes very weak: 7 tends to zero and, maintaining
two economies remains positive
which
If
level
see that once again
agents are patient enough,
even with an arbitrarily small degree of complementarity
between rich and poor. 18
18
<
k'
if
The formal statement corresponding
p
>
ho are better off under integration.
p*(ho) then
all
to (31)
The
is:
V/i
,
V"K
>
0,
3p(ho,f) such that
if
p
stronger statement corresponding to (32)
dynasties starting with h'
<
ho are better off under integration.
26
>
p(ho,~f) all dynasties starting with
is:
V^o, 3p*(ho) such that:
V7 >
0,
Local versus National Funding of Public Education
5.1
What do
the results of the general model imply for the relative efficiency of locally and nationally funded
Recall from the model of Section 2.1 that
public schooling?
education and school expenditures in a child's
income segregation
with:
in the general
= <r>l, a = 6,
e
Proposition 5
funding
/?
=
(1
-
model
6)(a
-
is
$
l)/<x,
y
=
(1
-
1
=
R=l.
6)/a and
—
6 are the weights of parental
funding (E\
r
-Y )
t
=
r
Y ')
t
is
equivalent to
equivalent to integration,
Therefore, by (20) and (24):
human
initial
endowments
capital
The higher
= -6(l-S)(l-js) <0
capital accumulation than local
A
t
is
+ cr) >
more
l.
-
19
efficient in the long run, since:
=( 2T4TTi)(ttI)(^1 )
2>0
are the only source of inequality, national funding raises the long run levels of
and output Yt by $• A 2 /2. When
raises the long run growth rate of
for
capital, local
and national funding (E\
(8),
true for output growth if 6(1
(2) National funding, however, is
human
and
6
in the short run, since:
The same
When
human
if
(1) National funding of education leads to slower
*
it
Model
Applications of the General
5
human
capital
children's ability or returns to education are uncertain,
and output by
$
2
•
s /2.
agents' intergenerational discount factor, the larger the proportion of
a national rather than a local system; for a high enough
p,
them who would
vote
there would be unanimity. Being derived
from a very simple model, Proposition 5 should of course be taken with caution. In particular, one should
keep in mind two important maintained assumptions.
'Using (26) yields: log^i/Vi)
=
(1
-
6(1
+ <r))
(1
-
S)
(<r
27
-
l)
2
•
(A 2 /2<r 2 ).
tax rates are kept constant across the two regimes. In practice, richer families
First,
redistribution inherent in national funding by voting for lower taxes;
likely to internalize
economy-wide
spillovers
may respond
to the
on the other hand, agents are more
when voting on a national education budget
rather than that
of their school district. Richer families could also withdraw their children from the public school system,
Variations in tax rates or a switch to private education will
thus preserving or restoring stratification.
essentially
be reflected in different values of the constant factor 0. This
Second, the underlying source of inefficiency in local funding
is
is
examined
in the
next section.
the absence of a capital market where
poor communities can borrow from richer ones to finance schools. National equalization of expenditures
amounts to a
partial
and gradual redistribution of human
capital, with
some payback to the
form of a higher Ht. Given decreasing returns to dynastic accumulation (a
intuitive that
it
should increase aggregate
efficiency.
true in the long run: early on, output and
families lose
more than poor ones
when dynasties
almost perfect substitutes, /im (T _t0O ($)
=
|
(1
—
5.2
Public versus Private Education
Let us
now
is
is
we simply
may
—
is
(1
—
6)/<r), it is
that this
only
actually be reduced, as rich
face returns arbitrarily close to one: as workers
6)/(l
is
the result that the steady-state gain from a
+ 6) >
become
0.
consider privately purchased education, along the lines of
the model of Section 2.1,
1
perhaps most surprising
capital accumulation
Also unexpected
gain.
national scheme remains finite even
human
So what
+P —
rich in the
Glomm and Ravikumar
(1992). In
replace equation (4) by:
e\=ri.y\
(4')
where
t\
now
represents the fraction of her income which adult
corresponding input
e|
i
spends on her child's education, and the
takes the place of the per capita school budget E\ in the accumulation equation (3).
We
again keep the preferences side of the model as simple as possible by assuming logarithmic
We
also
assume log-normally distributed
initial
conditions and shocks, as in Section
28
4,
utility.
or a small enough
.
With
dispersion that the Taylor approximations of Section 3 are legitimate.
these conditions,
we prove
in
appendix:
Proposition 6 Under private
education, the fractions
1
—v
and t of their time and income which
adults
devote to their children are given by:
v
=
1
— p8
Under national public education,
by voters
levels in
is
output Yt
=
v
unit of
m
unchanged. The tax rate unanimously preferred
-
(*-l\
1
(") <
6)
Pi 11-*)
-
-6)(a-l)/a
l-p6 + p(l-6)(a-l)/(T
is
a
\
T
J
o-
that the private marginal value of
This
capital enables the adult to not only
is
< Kl-*) _
\-p6
human
higher under private than under public education. This
human
offspring.
is
1-pS
is
consume more, but
very similar to the effect identified by
purchased. In our model, time
is
savings rates.
in
human
in
an extra
first case,
buy more education
also to
Glomm and Ravikumar
shows up not
(1992),
in different values of
this implies that private
Whether the accumulation
who allow
when education
for her
the young
is
privately
factor
6
The
difference
but in different preferred
i>,
under private education
=
f or f
—
is
higher or lower than
human
capital allows an
(as in the case leading to f), or even in that of all her
29
its
r*
In principle, voters should realize that a marginal increase in any agent's
own consumption
offspring.
and
education need not lead to higher investment
a publicly funded system depends on whether f
increase not only in her
.
allocated not between studying and leisure but between production
capital therefore
More importantly,
capital.
counterpart
human
r
because in the
at-home education, both of which allow adults to pass on more instruction to their
marginal values of
capital
hence also the return on
capital,
a choice between leisure and effort at studying, and show that they work harder
in
human
Ht-'
The reason why f < r
is
adults' time allocation
- P(l-g)
=
r* or f, depending on whether or not they internalize the complementarity of
f
savings,
T
,
dynasty
(as under private education, leading to r), but in that of all dynasties. If they do, they will base their vote
on the
social rather
>
than the private return to educational investment, leading to r*
On
r.
the other
hand, private underinvestment in education could also be addressed by uniformly taxing consumption and
This would be equivalent to raising r without going to publicly,
subsidizing education.
funded education. For this reason, but also to highlight the
still
be considered the most
private education
is
likely one, as
we turn
and
Proposition 7 Let
might
to comparing growth rates. Proposition 6 shows that
<f>
<
and
$>
economy
(L\
given by Proposition
still
=
h\),
5:
denote the trend factors reflecting the differential incentive effects of private
and national public education. Public education
leads to faster growth in
>
effects of heterogeneity, the case
equivalent to locally funded public education in a segregated
except for the value of 0. Therefore we have, with
uniformly
i.e.
human
capital
is less efficient
and output
in the long
in the short
run
if
$
run
2
•
(s /2)
if
>
—
—
>
<j>
A 2 /2,
but
0.
This result formalizes some of the main arguments in the debate about public versus private education
or related voucher schemes, at least where efficiency
of incentive and stratification effects,
and perhaps
is
concerned.
The key
issue
is
also the relevant time horizon.
the relative importance
Our
results also suggest
that in the long run, both private and national systems dominate locally funded public education, which
has neither the incentive properties of the former nor the homogenization properties of the
latter.
these conclusions are based on a simple, stylized model and should be taken with caution.
is
indicative of the
major forces
our results to those of
Glomm and Ravikumar
have been building on. Their model leads to the reduced form
h\ +1
with 7
=
=
•
(h\)
a
(At) 13 under public education.
(no global interaction),
Ravikumar observe that
if
there
But the model
at play in each case.
It is also interesting to relate
and
Of course
is
e
—
is
=
•
inequality, the
+ /? <
1;
(h't )
whose work we
a+ P under private education,
another special case of our general model,
oo (perfect substitutability) and
enough
under public education, provided 2a
This
h't+1
(1992),
s
2
=
(no shocks).
economy can temporarily experience
Glomm
faster
and
growth
but they do not explain why. The analysis of the short run
effects of segregation in (22) provides the answer:
when
30
2 a/3
<
/?(1
—
/?),
the complementarity between
parental background and school quality
is
dominated by the decreasing returns to
accumulation equation under public schooling (integration)
less
quality.
concave in parents'
human
This makes'the
capital than its
counterpart under private schooling (segregation). Turning to the long run, the asymptotic growth rate
in
Glomm
and Ravikumar's model
a
over time (for
the situation
is
+ (5 <
1); it
reflects incentives only, as the effects of initial inequality
A2
vanish
therefore always higher under private education. Proposition 7 shows that
quite different
when one
random
of
some degree
of
allows for ongoing sources of heterogeneity such as
ability or parental altruism, unpredictable obsolescence of specialized skills etc.,
and
for
complementarity in production, no matter how small.
5.3
For
Immigration
many
countries, the immigration of workers
and
their families with lower levels of education than the
The
resident population constitutes another periodic or ongoing source of heterogeneity.
East and West
Germany
the following.
provides another example.
The economy's
What
unification of
the results of this paper tend to
long run performance
show
be superior, benefiting
in this
family
context
is
lines, if
immigrants and their descendants are integrated rapidly, meaning that they share neighborhoods,
is
likely to
schools and other public goods with the richer local population, than
neous "ghettos". 20 However, integration
established residents.
may
if
they remain isolated in homoge-
well have a negative impact
Their individual incentives
will
all
on the
first
few generations of
always push society toward segregation. Whether
they will collectively (through the political process) recognize and seize the long-run benefits of integration
will
20
depend on how they discount the welfare of future generations.
See Burda and Wyplosz (1991) for a model of migration flows with
richer country to accept immigration or unification
a larger population; see
Tamura
may be
human
capital spillovers.
One
possible motive for the
the standard gain from increasing specialization achievable with
(1991b).
31
6
Conclusion
The model developed
in this
paper
community and economy-wide
is
extremely simple. The acquisition of
effects.
The accumulation equation
is
human
capital reflects family,
general enough to encompass the
reduced forms of most previous models. The degrees of complementarity or substitutability
economy- wide interactions capture the direct costs or benefits of heterogeneity
In spite of
results.
We
its simplicity, this
may
slow
each
and
level.
framework allowed us to study several important issues and derive many
examined how economic
that integration
at
in local
stratification affects
down growth
in the short
growth and welfare, and showed
run but promote
it
in the long run.
We
in particular
also
compared
same
the performance of locally and nationally funded public education systems, which exhibited the
intertemporal tradeoff. Introducing private education lead to an additional tradeoff between incentive and
stratification effects.
The model could be extended
in
a number of directions. For instance,
it
would be interesting to
refine
the production and labor market side of the model, by distinguishing occupations which play asymmetric
roles in the
human
production process, such as managers and workers. This would introduce an optimal degree of
capital inequality in the labor force, which could then be related to those generated
integration
and alternative education systems. 21 The interplay between
and global interactions also seems
like
by
stratification,
local complementarities, clustering
a promising direction for future research. In Benabou (1991) this
idea allowed us to study the social structure and productivity of cities. In this paper
we extended
a dynamic framework with heterogeneous agents, and to the study of education funding.
It
it
to
should be
applicable to a variety of other problems.
Recall that under symmetry, the optimal degre of inequality
heterogeneity, the lower
is
Ht), or infinity,
when they
is
either zero,
are substitutes (the
32
more
when agents
are
complements (the more
heterogeneity, the higher
is J/t).
Appendix A: More General Education Technologies
Complementarity Within and Between Inputs
A.l
The Cobb Douglas
Moreover,
it
specification (8)
allows stratification to have long-run effects only
what should matter
L\ and
H
t
,
convenient but has no special theoretical or empirical justification.
is
when
1/<t or 1/c differ
from one.
Intuitively,
are the relative values of the elasticities of substitution operating within the aggregates
and between the three inputs entering
into h\ +1
=Q
L\,H
F(h\,
t
).
We
show here that such
is
the case. Let:
(A.l)
h\ +1
where the weights
When
sign.
a',
ft',
and
=
-C*
j'
sum
•
[a'(h\)^
A tends to one we obtain (8) in the
3,
x -'
t
R>
to one,
approximations similar to those of Section
+ P{L\)^ +Y(H )^\
is
any degree of returns to scale and A can take any
with a
limit,
=
Ra',
/?
one can show that the
=
Rf3'
and 7
=
Rj'. Using Taylor
loss factors for the segregated
and
integrated economies are:
^
+ n\-f-^
+W + nM-R) + ^)
(A.2)
C
S R
(A
c
s * (-3^22 +«.(!_*,+£+£)
.
3)
For instance in C, heterogeneity
parent, due to a'
H
<
1
is
costly because of: (a) concavity of a child's
more concave;
within
H
t
R >
1
has the opposite
complementarity within L t
.
<j>
=
C-t = Rp'
l
(
2 "'
if
>
beneficial; (b) decreasing total returns,
is
effect; (c)
In the short run, mixing raises the growth rate
(A.4)
capital in that of her
and to the complementarity of the three inputs into her education, 1/A
are substitutes, on the other hand, heterogeneity
F(-)
human
<j>
>
0,
where:.
t + (2a' + /?')(! - R) j
33
\
0; if h,
L and
which make
(d) complementarity
The
interpretation
is
that decreasing total returns
make mixing more
approximation, the law of motion (28) for
Proposition 8 The gap between
as in Propositions
^
We
see that
3 and
4,
the integrated
with
when A
C and C now
=e=a
and
economy (within H).
°
$=
(A.6)
As we saw
tends to
R-a
2
a
in the case A
make
is
=
gg + a
or
a
R=
1
.
This
effect. Intuitively,
7(1
given by the
Thus:
same expressions
(A^)
2
(1^) + f + X
+ y- a 2
+
manner
the only case where the
is
in
aggregation then causes similar losses
also tend to
1
as:
(« + ?)(! ~R)
+
R-(a + 0) 2
/
,
/
(,
1\
l\
V
\)'
\)
\
(l-fl)(2a
1
t
)
than within
and 1/e
make
=
\/<r
=
0,
We now
L
t
as
or
H
t
l
see that greater complementarity
(1/A
>
1/a) has a similar
shown on Figure
integration beneficial, except
3. Finally,
when 1/A
is
+ /?)
i?-(o + /3) 2
R-Xa
greater complementarity at the aggregate than at the local level
(h\,L\,H
=
$
-«) +
aj
integration superior in the long run.
R<
(
\/<r
(i.e.
>
1/e)
a higher
effect. In particular,
non-increasing returns
sufficiently greater
than one.
Multiple Spillovers
Denote by
and by
3 or
1,
when A
A. 2
1
can also rewrite
\\
e
why $ >
to scale,
We
L_}l + (}--}l\
cost of disparity) between
this
.
occurs at the level of a child's education (within F(h, L, H)), of a community (within L) or of
it
the whole
is
A2
given by (A. 2) and (A. 3), and:
=
either A
which agents are partitioned has no steady-state
whether
Equation (A.4) also shows
turn to the long run. To a second-order
and segregated economies
+ (Q + fl, (^1) ( 1-H) + X
Q + /? + 7-(a + /3) 2
$=
1.
remains unchanged, and similarly for
<« + /.Xl—g)
_
~
;
2,
A2
We now
efficient.
=
when A
similar to that of (20), to which (A.4) reduces
ejt
and an respectively the
/?fc,7„ their
CES
elasticities of substitution of the L'k
weights (partial elasticities)
in
F. Then, whether F(-)
as above, equations (18)-(19) or (A.2)-(A.3)
34
show that
(10')
is
's
t
is
and
Hn
,t
's
entering (10'),
Cobb-Douglas as
in Section
equivalent to (10) with
/?
=
.
K
(A.7)
/?/£
= £>*/<*,
j/<T
N
= Y,lJ"n.
n=l
fc=l
Appendix B: Proofs
Proof of Propositions
Solving (28) to (29) leads
<-)
*(&)
m
*(&)
hence the results as
<
Technology.
A 2 = A 2 + (a + /?) 2< (A 2 - A*,) A 2 = A^ + a 2t (A 2 - A 2*,)
—
any 7
oo, for
We first
for
>
and:
0.
(6).
fill
in a
few details with respect to the production sector of Section
any choice of hours worked
v\.
We
2,
so as to
drop time subscripts for simplicity. Output
produced by competitive firms with constant returns, according to the technology:
Y =
(B.3)
t
where x s
p,
,
*,
K-<4)G=S)-£ (*-*)(*=?)
compute labor income
is
to:
(4).
K-f)(^)-f<^>«±f)
-
Proof of Proposition
1.
and
(3)
=
is
the quantity of intermediate input
(x,/Y)~
'
,
yielding revenue y,
in a single input;
with
human
This
is
or, in
=
p,
x,
(x^dsY
(J
s.
=
The output
(x,)~z~
sector's inverse
(Y)*
We
.
demand curve
assume that workers must
each then chooses a different one, and we can replace the index
capital h' working v'
<
1
hours produces x'
=
v
x
for s is
h* units,
and earns
s
by
j/'
=
i
£
(i/'
fi.
then
specialize
ah
h')~z~
agent
(Y)*
her net income, as production entails only the opportunity cost of time (e.g. the value of leisure,
our case, parenting). Labor income also takes the form
be called the hourly wage; hence
(2).
When
v\
—
v
for all
35
i
y\
G
=
fi,
l
v w', where w'
=
p' h'
we obtain equations
=
dY/di/' can
(5) (given
N=
1
agents in
1.
fl;
Y
otherwise,
To
Private education.
binding.
The Bellman equation
W
(B.4)
t
first
rates:
i/f
for agent
= max
(h\)
{l, tT)
=
i is
v and
t{
and
we
first
—f. We
assume that
agents except
all
show that these
later
i
choose
restrictions are not
then:
{log{il-T).(i,h\)'-^(Yt ^)
1
-6
)}
order conditions are:
(B.5)
i
v\
a
\
)
•£
(B.6)
l-r
1+1
*
'
economy
Since the state of the
value function as:
W (h) = a
\og(h)
t
l
(B.7)
1
+ p(l-
At+i and A^+1
for
fB81
V
;
a=
=
log^j)
•
=
v and t
—
S, /?
-
-
p6
-
p(l
=
(1
-
6)(a
—
c
•
Markov process (A t
A^
+ d.
t.
(ftj+i)
t+l)
A2
,
This leads to
T
1)
dWt+i
dh
«+i
'
=
l
),
j/J
we guess the form
=
v,
t\
=
of the
r, with:
p(l-6)a
+ p(l-8)a
Replacing in (B.4) and using the recursion equations (28)
identifies the constants. In particular:
(*-!)/*
1
'
+6
+ p(l-6)a
S)a + p6aa/(a -
Equilibrium then requires that v
and (29)
characterized by the
is
(
dh
t
where a
>
(6).
p-EWt^^.Q-dl-^hiY-ir^h^iY^)
+
The
simplify the exposition,
and savings
invariant participation
N 7^),
should be multiplied by
6)((r
-
-
a
'
l)/a
l)/a, 7
=
(1
-
+b=
b
1
l-p'
6)/a and C,
t
2a 2
'
pC + (l-p)/<r
l-p(a + /3) 2
'
are given by (18)-(19). Replacing a in (B.7)
yields the desired results.
In this equilibrium,
all
agents choose the same constant values v and r. This
at least in the following sense.
Consider any
finite
horizon (T
36
<
is
in fact the only solution,
00) version of the model, without the
&
restrictions v\
=
=
period: (a) v\
v,
,rf
vt and
=
=
t\
A
f.
backwards induction similar to the derivations above shows that
rt ensuring
,
Y =
t
vt -Ht]
W (h) = a
(b)
t
t
game thus has a unique Markov
perfect equilibrium,
and
it
,
6, c,
d) calculated above.
involves symmetric strategies.
As
T—
all
j
The
oo,
*
tends to the equilibrium derived earlier.
it
We
Public education.
2.
that the
Markov process (A t
start again with
A
2
,
)
some
=
useful simplifications. First, let iPx
fully describes the state of
We
economy.
v for
t
+i through
Q =
t
k
(1
—
D)
6
(Dft)
1-1
but does not
,
expected to be the decisive voter (or a dictator) not just at
to the
same preferred tax
decisive at
t
i's
Bellman equation and
V(h\,At ,Al)
(B.9)
= ma*
+
1
first
t'
>
agent
choose
t\ as if
she
i
chooses rt' as
if
she were
same outcome. With these
lead to
{bg((l-r).(i/A{) ^(y«)')
t
Qi
P6
1-**
E
h \+i
dV
-
-Qfr(
they do not, the corresponding rate
form of the value function: V(h,A,A 2 )
0, f\
i
influences
s
( „ >r)
,
)}
r
-
2
ht+i>At+i,A t+1
)
,
=
agent
t
future periods. Because this leads
(e.g.,
t) will
so
order conditions are therefore:
+p -&T' E K+\
Equation (B.ll), which determines t\* corresponds to voters
v\
game
t,
l-v\
p(l-S)
If
in all
let
p-Ev(K-ci-((i-^)h\y.(rY y- s i f+1) A?+1
(<T-\
(B.10)
(B.ll)
but
t
rate for all agents, any other voting
but expected the median voter to prevail at
assumptions, agent
A^. Second,
affect
^
relax this restriction later on.
Note from (17) and the analog of (28) under integration that the tax rate ft implemented at
A
sach
-\og(h)+bf\og(At) — c t -A 2 +dt where
the (a t ,b t ,ct,dt)'s satisfy linear difference equations whose fixed points are (a,
finite
in
=
f, t\*
=
f\ is
=
a
who
+
r*, where:
1
(B.12)
l+p6a<r/(o--
1)
37
6
•
log(i)
-
-Q^iht+ii
A t+i,
internalize the effect of taxes
obtained by dropping the
log(ft)
dV
•
c
last
term.
A 2 + d.
Then
t
+i)
on At+\.
Again, we guess the
(B.lO)-(B.ll) imply
.
mim
r
1
^
While f
a
+ b.
is
t
l+p(l-*)a
a=
<
in (B.12)
+ S)
T^-'
a
+ 6 -rT7-
and (B.13) leads to the desired
d,
r* reflects the full social marginal value
c
results.
A?+1 )
-2^'
leads to:
i- ptt 2
Note that a
<
a
<
a
+
b,
implying that
other equilibria, consider again the finite horizon game, without the restriction \?%
agent's value function and (Markov) strategy
capital.
_
+ p(l-«)(a + S)
l
r*
To exclude
Whether voters
each period: (a)
i.e.
g(i^j)(a
<j
Replacing in (B.9) and using the recursion equations for (A t +i,
Replacing
<
Pi\-W
based on the private marginal value of human capital
(B14)
f
-
"
there
is
v\
—
depend on
internalize the effect of t\ on
vt ensuring in particular that
,
fi t
t
where
(i t
is
the distribution of
(c)
V (h,
t
,c t ,d t ys satisfy linear difference equations
fi t )
=
whose
above. Letting the horizon tends to infinity concludes the proof.
38
human
+i or not, backwards induction easily shows that in
remains log-normal and that
p, t
unanimity over the sequence of tax rates;
where the (at,b
(h\,\ix),
=v. An
at
•
\og(h)
+ bt
Y = VfHu
t
log(At)
—
(b) t\
ct
=
ft
A? + d
t
,
,
fixed points are (a,b,c,d) calculated
References
Arnott, R. and Rowse, R. (1987) "Peer Group Effects and Educational Attainment", Journal of Public
Economics, 32,
p. 287-305.
Banerjee, A. and Besley, T. (1990) "Peer
Group
Externalities
Nerd Behavior", Princeton University Working Paper No.
and Learning Incentives:
A
Theory of
68.
Benabou, R. (1991) "Workings of a City: Location, Education and Production",
MIT Working
Paper
No. 582, May.
Borjas, G. (1992a) "Ethnic Capital and Intergenerational Mobility", Quarterly Journal of Economics,
108, p. 123-150.
Borjas, G. (1992b) "Ethnicity, Neighborhoods,
and
Human
Capital Externalities", U.C. San Diego
mimeo, September.
Burda, M. and Wyplosz, C. (1991)
rope",
INSEAD mimeo,
Coleman,
J.
"Human
Capital, Investment
and Migration
in
an Integrated Eu-
September.
(1990) "Foundations of Social Theory", Chapter 12. Harvard University Press, Cambridge.
Cooper, R. (1992) "Team Dynamics", Boston University mimeo, July.
Cooper,
S. (1992)
Dertouzos,
Report of the
L., Lester,
W.
Positive
Theory of Income Redistribution", Stanford University mimeo,
July.
R. and Solow, R. (1989), "Made in America: Regaining the Productive Edge",
MIT Commission
Durlauf, S. (1992)
Ethier,
"A
on Industrial Productivity",
"A Theory
of Persistent
MIT
Income Inequality",
Press, Cambridge.
NBER Working
(1982) "National and International Returns to Scale in the
Paper No. 4056, April.
Modern Theory
of International
Trade", American Economic Review, 72, 389-405.
Fernandez, R. and Rogerson, R. (1992) "Income Distribution, Communities and the Quality of Public
Education:
A
Policy Analysis", mimeo, Boston University, July.
Glomm, G. and Ravikumar,
B. (1992) "Public vs. Private Investment in
39
Human
Capital:
Endogenous
Growth and Income
Inequality", Journal of Political
Economy,
100, 818-834.
Kremer, M. (1992) "An O-Ring Theory of Economic Growth", Harvard University mimeo, January.
Loury, G. (1977) "A
Minorities and
Dynamic Theory
of Racial
Income Differences",
in
A. Le
Mond
(eds.)
"Women,
Employment Discrimination" Lexington Books, MA.
,
Loury, G. (1981)
"
Intergenerational Transfers and the Distribution of Earnings", Econometrica, 49, 4,
p. 843-867.
Lucas, R. Jr. (1988)
"On the Mechanics
of
Economic Development", Journal of Monetary Economics,
22, 3-42.
Montgommery,
J. (1990) "Social
Networks and Persistent Inequality
in the
Labor Market", Northwest-
ern University mimeo, December.
Murphy, K.,
Shleifer,
A. and Vishny, R. (1991) "The Allocation of Talent: Implications for Growth",
Quarterly Journal of Economics, 106, 503-530.
Perotti, R. (1990) "Political Equilibrium,
Income Distribution, and Growth"
MIT mimeo,
December.
Roiner, P. (1986) "Increasing Returns and Long-Term Growth", Journal of Political Economy, 94, p.
1002-10037.
Saint-Paul, G. and Verdier, T. (1991) "Education,
Streufert, P. (1991)
"The
Effect of Underclass Social Isolation
Wisconsin-Madison Working Paper
DP
DELTA
July.
on Schooling Choice", University of
in
an Endogenous Growth Model", Journal of Political
99, 522-540.
Tamura, R. (1991b)
"Efficient Equilibrium Convergence:
Heterogeneity and Growth", University of
Iowa Working Paper No. 91-16, August.
Weiss,
mimeo,
954-91, August.
Tamura, R. (1991a) "Income Convergence
Economy,
Democracy and Growth",
M. (1989) "The Clustering of America," Harper and Row, New York.
Wilson,
W.
(1987) "The Truly Disadvantaged," University of Chicago Press, Chicago.
40
593S 079
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FEB.2 7
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