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EDUCATION, INCOME DISTRIBUTION, AND GROWTH:
THE LOCAL CONNECTION
Roland Benabou
MIT
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
EDUCATION, INCOME DISTRIBUTION, AND GROWTH:
THE LOCAL CONNECTION
Roland Benabou
MIT
94-16
May 1994
M.I.T.
LIBRARIES
JUL 1
2 1994
RECEIVED
Education, Income Distribution, and Growth:
The Local Connection
Roland Benabou
MIT
September 1993
Revised
am
May
1994
Diamond, Dick Eckaus, Jonathan Gruber,
Michael Kremer, Glenn Loury, Julio Rotemberg, Richard Startz and Jean Tirole for useful comments.
Financial support from the NSF is gratefully acknowledged.
I
grateful to Olivier Blanchard, Patrick Bolton, Peter
Abstract
This paper develops a simple model of human capital accumulation and community formation by heterogeneous families, which provides an integrated framework for analyzing the local determinants of inequality
and growth. Five main conclusions emerge.
First,
minor differences
or wealth can lead to a high degree of stratification.
will
compound
Imperfect capital markets are not necessary, but
these other sources. Second, stratification
persistent across generations.
Whether
or not the
in education technologies, preferences,
same
makes inequality
is
in
education and income more
true of inequality in total wealth depends on
the ability of the rich to appropriate the rents created by their secession. Third, the polarization of urban
areas resulting from individual residential decisions can be quite inefficient, both from the point of view of
aggregate growth and in the Pareto sense, especially in the long run. Fourth,
of school expenditures
poor communities
is
much
and income growth both
and
efficiency. Fifth,
insufficient to
less
than
fall.
it
Yet
reduce stratification,
lowers
it
may
it
in richer
still
it
when state-wide
may improve
educational achievement in
communities; thus average academic performance
be possible for education policy to improve both equity
because of the cumulative nature of the stratification process,
harder to reverse once
it
has run
its
equalization
course than to arrest at an early stage.
it is
likely to
be much
Education, Income Distribution, and Growth:
The Local Connection
by Roland Benabou
Abstract
This paper develops a simple model of
human
capital accumulation
and community formation by
heterogeneous families, which provides an integrated framework for analyzing the local determinants of
inequality and growth. Five
main conclusions emerge.
First,
minor differences
in education technologies,
preferences, or wealth can lead to a high degree of stratification. Imperfect capital
essary,
but
will
compound
these other sources. Second, stratification
income more persistent across generations. Whether the same
on the
ability of the rich to appropriate the rents created
by
is
makes
markets are not nec-
inequality in education and
true of inequality in total wealth depends
their secession. Third, the polarization of urban
areas resulting from individual residential decisions can be quite inefficient, both from the point of view of
aggregate growth and in the Pareto sense, especially in the long run. Fourth,
of school expenditures
poor communities
insufficient to reduce stratification,
is
much
less
and income growth both
and
efficiency. Fifth,
than
fall.
it
Yet
lowers
it
may
it
in richer
still
it
when
may improve
state-wide equalization
educational achievement in
communities; thus average academic performance
be possible for education policy to improve both equity
because of the cumulative nature of the stratification process,
harder to reverse once
it
has run
its
course than to arrest at an early stage.
it is
likely to
be much
Introduction
The accumulation
growth.
of
human
As demonstrated most
vividly
by the physical blight and
certain fundamental inputs in this process are of a local nature.
level of individual families
decentralized, but also with
Not only
many forms
They
are determined neither at the
is
this the case with school resources
fiscal
determined in large part by the set of adults to which she
and incomes
is
directly
when funding
of "social capital": peer effects, role models, job contacts,
norms of behavior, crime, and so on. Through these
distribution of skills
social pathology of inner-city schools,
nor that of the whole economy, but at the intermediate level of communities,
neighborhoods, firms or social networks.
is
income inequality and productivity
capital underlies the evolution of both
and sociological
is
spillovers,
a
child's education
is
"connected". Therefore, the next generation's
shaped by the manner
in
which today's population organizes
itself into differentiated clusters.
The aim
of this paper
is
twofold.
First, to
provide a unified analysis of the forces that
socioeconomic segregation, income distribution, and productivity growth.
We
tie
together
develop a simple model
where heterogeneous families form communities, choose local public expenditures, and accumulate human
capital. It allows us to elucidate the roles played
by endowments, preferences, the technology of education,
capital markets, school funding, neighborhood effects,
us to incorporate
The second
some of the main
objective of this paper
insights
is
and discrimination.
from the previous
on the transmission of inequality across generations;
of inner-city poverty,
and on
we examine some
The other
new
two important questions. One
effect of stratification
policy:
synthetic quality also enables
literature, while obtaining several
to further our understanding of
social mobility in general.
Its
it
ones.
is
the
bears directly on the problem
central issue concerns education finance
of the distributional and aggregate implications of the growing trend towards
court-mandated, state-wide equalization of school expenditures.
The
literature to
which the paper belongs has
its
sources in the classic works of Tiebout (1956) on
local public
goods and Schelling (1978) on segregation and externalities. More recent prominent influences
include the
work of Loury (1977), (1987) on
racial inequality
and of sociologists such as Coleman (1988),
Wilson (1987) and Jencks and Mayer (1990) on group interactions. Another important motivation
current debate over local disparities in school funding, perhaps best publicized by Kozol (1991).
developed here builds on
De Bartolome
(1990),
Benabou
(1993), (1992)
is
the
The model
and Durlauf (1992). But the paper
is
also closely related to Borjas (1992a), (1992b), Durlauf (1993),
(1993), through the link between
Tamura
community composition and the
(1993) and Lundberg and Startz
persistence of inequality;
and to
Glomm
and Ravikumar (1992), Fernandez and Rogerson (1992), (1993b) and Cooper (1992), through the issue of
education finance reform.
We now
outline the structure of the paper
In Section 2,
we
identify the forces which
show how even very small
and
its
main
The model
results.
is
presented in Section
promote or hinder segregation and explain
The
differences in education technologies, preferences or wealth lead to a high degree
rest of the
on productivity, we examine
compound
the
paper studies the potentially large effects of these small causes. Focusing first
in Section 3
how
the total surplus generated by a metropolitan area reflects
organization into local communities. In particular,
its
We
their interactions.
of stratification. Capital market imperfections are not necessary, but even minor ones will
other factors.
1.
we
explain
why
the typical pattern of city-suburb
polarization can be very inefficient.
We
We
then take up the issue of whether socio-economic segregation makes inequality more persistent.
demonstrate
the long run this
in Section 4
may
how income convergence
reduce income growth for
is
slowed down, how ghettos
all families.
We
arise,
and how
in
present in particular a very simple
prototype of the "local poverty trap" and "self-defeating secession" phenomena, which makes apparent the
general principles at work in more complex models
(e.g.,
Benabou (1993), Durlauf (1992),
because our framework allows for financial bequests as well as
human
capital,
we
(1993)).
But
are also able to point
out an important qualification which the previous literature has generally overlooked. While stratification
exacerbates inequality in education and income, the same need not be true for inequality in total wealth.
The determinant factor
is
the cost paid by the rich to separate themselves from the poor, or conversely the
extent to which they are able to appropriate the rents generated in the process of segregation.
various collective practices and institutions by which this
may come
We discuss
about, showing in particular
how
it
jure racial segregation turns into de facto (economic) segregation once legal barriers to mobility are lifted.
The other main
issue studied in the
tralization of school funding
it
paper
is
education finance.
We show
in Section 5 that the decen-
and control constitutes an additional segregating
force,
and that
in general
need not improve efficiency In Section 6 we then examine the implications of policies, adopted by a
growing number of states, aimed at reducing disparities in education expenditures between rich and poor
communities. Benabou (1992) and Fernandez and Rogerson (1993b) predict very significant gains to moving from a system where school funding
is
based on local resources to a national or state-wide scheme. In
addition to reducing inequality, this would raise the economy's long-term output, or even growth rate; given
a low enough intergenerational discount rate,
it
would be Pareto-improving. In
this
paper we temper this
optimistic scenario by downplaying capital market imperfections, and emphasize instead the interaction
between purchased inputs and
cast doubt
on the effectiveness of redistributive funding
in reducing earnings inequality
Indeed, several pieces of evidence seem to
social spillovers in education.
in raising the
and augmenting aggregate surplus.
We
model can help explain some of these puzzles.
performance of poor schools, hence
We
show how a simple version
determine when equalization works and when
counterproductive, and, in the latter case, identify alternative policies which
and
efficiency.
of our
Because these policies work through changes
may still
it is
improve both equity
community composition, however, they may
in
be constrained by a form of irreversibility which we show to be inherent in the stratification process.
We
conclude this preamble with some suggestive evidence on metropolitan stratification and growth.
While the
neighborhood and school composition on individual outcomes have been extensively
effects of
documented, there
is
as yet
no empirical study of
provocative data which bears on the issue; since
for the
their aggregate implications.
it is
very incomplete,
paper rather than supporting evidence for specific
areas, selected
by Rusk; the data
is
also plotted in Figure
results.
1. It
we
Table
1
Rusk (1993) presents some
include
are respectively .74
as
Rusk
and
.91,
income (1969-1989) and
shows a strong positive relationship between
total
may
mean incomes, and
employment (1973-1988). The
or .60 and .45, depending on whether
does, or in 1970, which
as general motivation
examines fourteen metropolitan
the lack of economic segregation, measured by the ratio of central city to suburban
area's growth in both per capita
it
we measure income
be more appropriate. 1 This striking stylized fact
the
correlations
disparities in 1989,
is
no substitute
a systematic econometric analysis, and does not allow any inference about causality. But
it
for
does suggest
that city-suburb stratification and metropolitan economic performance are interdependent processes, and
that the underlying
mechanisms deserve
careful theoretical
and empirical investigation.
'I am grateful to Ed Glaeser for kindly providing the 1970 figures, from the Census Bureau's County and City
Data Books. The above result seems opposite to that of Glaeser, Scheinkman and Shleifer (1993), who find that
racial segregation in a city affects population growth positively. Note, however, that the focus is here on city-suburb
economic dichotomy, rather than on racial segregation and population growth within city limits.
Metro per capita income
growth (%)
Metro employment
growth (%)
1969-1989
1973-1988
23
8
53
26
11
83
81
34
40
Milwaukee, Wis.
84
62
25
22
Harrisburg, Pa.
87
72
33
29
Louisville, Ky.
91
79
30
21
Richmond, Va.
91
83
45
46
Nashville, Tenn.
92
98
49
55
Syracuse, N.Y.
98
77
29
27
Madison, Wis.
104
95
34
51
Houston, Tex.
107
89
36
67
Indianapolis, Ind.
108
90
32
34
Raleigh, N.C.
119
103
62
81
Albuquerque, N. Mex.
148
118
41
75
Metro Area
City/Suburb per capita
income ratio (%)
1989
1970
Cleveland, Ohio
69
53
Detroit, Mich.
80
Columbus, Ohio
Table
1:
City-Suburb Inequality and Metropolitan Area Performance.
Source: see
Rusk
(1993),
and footnote
1
flfura la
Flgnra lb
a
a
a
79
100
U5
TBATI089
Flfun
lc
Flfur* Id
TEATI070
TRATI070
2.5
3.0
s
a
>•
City-Suburb Inequality and Metropolitan Area Performance
YRATIO= relative city /suburb mean income; YGROWTH=annual growth rate of metropolitan income
per capita; NGROWTH= annual growth rate of metropolitan employment. All values are percentages
Figure
1:
The model
1
1.1
Agents and communities
There
is
a continuum of families, with unit measure. Parents are of two types,
White and Black,
n and
city
1
—
n.
etc.),
human
with
The average
level of
capital
human
A
and
B
capital
is
=
j
denoted h
=
n h&
+ (1 — n) /»b-
1,2, each holding the
These agents
from
All land belongs to absentee landowners; departures
as the richer one: x 1
> In —
A
x2
.
In the
with two types and two communities
associate in the provision of formal
of a city with
many
populations; almost
assumption that
1.2
A
maximize the resulting
denoted as i J and community
1 is
,
would be the
is
the minimal framework in which to study
and informal education.
— hB)/h
is
denned
we
It
reverse.
how
agents
can also be viewed as a representative
"slice"
focus on any pair of contiguous towns and their
small.
parent with type h €
utility
c
first
period, she
=
amount borrowed
+ p>+V{h)
= u (h) +
c',
C1
,
j
=
1,2, so as
y(h)
h'
=
F{h,ti,E>)
c
and pays rent
debt
d.
The
(P
h')
,
subject to
d
=
level of
or saved.
max<i U(c,
d + P{h,d)
consumes
augmented by her chosen
chooses a community
{/lyt./ifi} initially
U^(h):
U' (h)
In the
is
Preferences and technologies
There are two periods.
to
(h A
not essential.
extend to such a model. Thus we shall occasionally make the convenient
results
Ah/h =
is
it
types and communities, where
all
a
American context the two locations would be referred to as
"the suburbs" and "the urban center"; in Europe
A model
live in
this neutral allocation are considered later on.
agents in each community
=
i1
are
same number 1/2 of single-family
homes. The inelasticity of land supply and population density simplifies the analysis but
of rich, or type
and poor,
endowments h* > ha- The proportions of the two types
composed of two towns or communities,
The proportion
(rich
plus taxes t'(h) out of her initial resources w(A),
interest rate r(h,d)
In the second period, the resources
may depend on
d available
for
her type as well as the
consumption or bequest
(both interpretations are possible) equal second period income y(h), minus debt repayments P(h,d)
+ r(h,d)).
d(l
Finally, the child's
education), by the quality
to
its
schools,
The
V
local spillover
by the
affected
capital h'
is
determined by that of the parent (through at-home
of social interactions in the chosen community, and by the resources devoted
measured by the per student budget E* ?
Neighborhood
1.3
V
human
=
social
effects
captures the non-fiscal channels through which a child's acquisition of
mix of neighboring families. These sources
Coleman
of "social capital" (Loury (1977),
(1988)) include: peer effects between students from different backgrounds, both in
skills is
and out of the classroom
(Banerjee and Besley (1991)); the fact that neighboring adults enforce norms of behavior and provide role
models, as well as networking contacts, for the young (Wilson (1987), Streufert (1991), Montgomery (1991));
and crime or other
V
that
=
adults in
L(x J h A ,h,B)
;
C
J
(in the sense
and
activities
is
homogeneous of degree one
share the same level h, then
V
—
We
h.
D
of
L
also require
human
Another key feature of the
how
beneficial
is
spillover
L
=
capital,
>
human
0,
the
CES
it is
complements, L
is
=
if all
with h A
capital; its value for a
L(n; h A hs),
How
heterogeneity within a school or neighborhood? Put in another way, do stronger
(x h A '
convex
rises
response to a mean-preserving spread in the distribution.
is its
A
important to allow both cases. 4
index L(x)
and that
and L
individuals tend to "pull up" the average to their level, or do weaker ones tend to "drag
Clearly,
we assume
any improvement
to increase with
as an average of local residents' levels of
representative sample of population will be denoted
costly, or
in the distribution of
loss of generality,
of first-order stochastic dominance) in the local distribution: L'(x)
One can thus think
/ifi.
which disrupt education (Sah (1991)). 3 Without
+ (1 — i) h B
in i,
'
convenient specification which
)'-»
and heterogeneity
.
When
is
1/e
>
a source of
0,
we
it
shall
individual levels of
loss:
L <
h.
When
down"
to theirs?
sometimes use
is
human
capital are
<
individual
1/e
0,
2
More generally, F(h,L,E) should be understood as the child's earnings potential, net of her optimally chosen
studying effort (measured in the numeraire good). Incorporating private purchases of education is also straightforward: c and E are replaced in (1) by c + e and E + e, and parents maximize over both e and d.
3
Jencks and Mayer (1990) provide an extensive survey of the empirical evidence on neighborhood effects, and
Manski (1993) a discussion of methodology. Recent studies include Datcher (1982), Dynarski et al. (1989), Corcoran
(1990), Crane (1991), Borjas (1992a), (1992b), Case and Katz (1991), and Brooks-Gunn et al. (1993).
Dynarski, Schwab and Zampelli (1989) find that average student performance in a school district rises, ceteris
paribus, with dispersion among family incomes. Hamilton and Macauley (1991), on the other hand, find that
income dispersion raises the unit cost of education. Brooks-Gunn et al. (1993) find that it is primarily the upper
et
al.
4
tail
of a neighborhood's
income distribution which
affects the
development of children and adolescents raised
there.
i
levels are substitutes,
L
is
concave in x, and heterogeneity
(minimum) and "best-shot" (maximum)
generally, the concavity of L(x) will be
1.4
Local school funding
The
other local input into education
and poor agents
of rich
is
a source of gains: L
>
cases are obtained in the limit as 1/e goes to
h.
+00
The
Leontieff
More
or —00.
an important determinant of the efficiency of equilibrium. 5
decentralized school expenditures. 6 Given the distribution (z J
community -hence, given
in the
is
social capital
D
and rents p>-
,
1—
J
)
their preferences
over school budgets and taxation are aggregated by some local political mechanism into funding decisions
& = E(x
3
)
and tax schedules V{h,p)
x')t{hB,f>i ,xi)
political
=
E(xi).
mechanism
1
+ (l —
The
be majority voting, the outcome of lobbying, or delegation to an
moment we
For the
mechanisms which promote
t(h,p,x'), subject to the budget constraint i J t(hA,p' a^)
thus allow taxes to depend on individual income and property values. 7
in question could
efficient local planner.
of
We
=
shall leave
it
unspecified, so as to identify the general features
or hinder segregation.
Equilibrium
1.5
Let us rewrite the utility of an adult of type h, living in a community with percentage x of rich households
and paying p
in land rent, as:
V(h,p,x)
=
max{U(u,(h)
+ d-p-t(h,p,x),y(h)-P(h,d),F(h,L(x),E(x))}
(2)
a
Equilibrium in the land market
to bid
more than the poor
will result in stratification if the rich (in
for land in a richer
community. Formally,
human
capital) are willing or able
this simple sorting condition in the
space of community quality and price means that a rich family's iso-utility curve, or bid-rent,
5
For a more detailed discussion, see Benabou (1992). Similar functional forms are used by
studying the provision of public goods with non-additive individual contributions.
As
for social composition, the link
Comes
is
steeper
(1993) in
between expenditures and students' achievement or future earnings
is
the
much
empirical debate. See Hanushek (1986) and Card and Krueger (1992) for opposing views.
Note that the 3 and t J (h, p) chosen by 3 residents may depend on the equilibrium rent p3 (as opposed to an
subject of
E
C
individual's bid p), through its effects on their wealth
To keep
the notation simple, this dependence on j
is
and on potential revenues from alternative forms of taxation.
in the dependence on x 3
subsumed
.
than a poor one's:
V*(h,p,z)
J?,(A,p,x)sg
—VP (h,p,x)
v(h,p',z')=v(h,p,x)
increases with h
1
In that case, the slightest divergence from the symmetric allocation x
equilibrium) in favor of
for land in
C
community
1
will set in
motion a cumulative process:
thereby further raising x 1 ,/? 1 and lowering x 2 ,p2
1
,
poor by the rich in
C
1
=
and more concentration of poverty
in
C2
,
.
x2
=
(3)
n (which
is
always an
rich agents outbid
poor ones
This leads to more displacements of the
until at least
homogeneous. The three possible configurations are indicated on Figure
1,
one community
is
completely
together with the equilibrium
s
conditions on p 1 and p 2
.
Rh x {h,p,x) >
Proposition 1
1.
If
the rich all live
m
communtty
1,
— In—
x2
2.
it is
If
1).
for
all
umque
p,x, the
2n, x 2
—
0); if
The symmetric equilibrium x 1
=
x2
Rhz(h,p,x)
<
for
1
all
(i 1
=
=
>
n
n
stable equilibrium, is stratified.
1/2, the poor all live in
n
<
1/2,
community 2 (x 1
=
unstable.
is
p,x, the unique equilibrium
is
completely integrated
(i.e.
symmetric), and
stable.
The proof
is
given in appendix.
Examining the marginal rate of substitution between
consumption and child education thus brings to
light the
economic forces which lead to
integration. Using the Euler equation for the optimal level of borrowing,
_
Equation
goods.
8
If
(
U3 {c,c>,h>)
FL
-L'(x)
+ FE
-E'(x)
will
stratification or
we have:
m \\
n .i( h
(4) incorporates the contributions of technology, preferences, capital
The corresponding components
first-period
n r \\-i
U)
markets and local public
be discussed in turn.
As in much of the literature on community composition (e.g., De Bartolome (1990), Wheaton (1993)) we only
determine the rent differential which ensures that no agent wants to switch communities. The base level of these
rents could be pinned down by relaxing one of the simplifying assumptions of inelastic land supply and fixed number
of agents of each type, but this would needlessly complicate the model.
CD
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o
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n
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m
ro
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<
3"
>
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CD
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ro
ro
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ii
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3
GD
The determinants
2
Local complementarities
2.1
simplest version of the model
The
F=
quality,
d(l
of stratification
+ r);
so that
is
that where: (a) the only local input into education
is
community
F(h, L), so that no collective decisions are involved; 9 (b) capital markets are perfect, P(h, d)
(c) utility is linear,
U3/U2
bid-rent slope
is
U(c,d,h')
= c + (d + /»')/ (1 + r)
or
more generally d and
when d
constant. This last assumption seems especially appropriate
now
h' enter additively,
is
a bequest. The
simplifies to:
Rxi h,p,x)
Stratification occurs
those with lower
,
=
if
families with higher
human
wealth. In the words of
"The individual's remedy
Those who leave the
who maintain
human
intensifies the
= ^^-L'(x)
1 + r
capital are
Baumol
sensitive to
neighborhood quality than
(1967):
community's problems and each feeds upon the other.
city are usually the very persons
the houses,
more
(5)
who do not commit
who
care and can afford to care-the ones
crimes, and
who
are most capable of providing
the taxes needed to arrest the process of urban decay. Their exodus therefore leads to further
deterioration in urban conditions and so induces yet another
wave of emigration, and so on."
Proposition 2 A small amount of complementarity between family human
(FhL
>
0)
is
sufficient to cause stratification.
capital
The distribution of financial wealth
and community
is
quality
irrelevant.
This simplest specification of preferences and technologies will be referred to as the "basic model"
contrast to most of the literature on inequality,
it
shows that there might be very
policies to affect either the distribution of educational
9
little
.
In
role for redistributive
attainment or the efficiency of equilibrium.
is that agents care about community
impact on their child's education; crime is a good example, with richer agents
caring most about it. The second is that F = F(h, E) and communities finance schools by taxing income at a fixed
rate (such is then case when preferences are logarithmic; Glomm and Ravikumar (1992)). Then E(x) = r* L(x),
where L(x) 5 tw(^) + (1 — x) u»(fes) is simply average income in period 1.
This model also admits two interesting alternative interpretations. One
quality
L
for reasons other
than
its
8
Capital market imperfections
2.2
In reality,
it
may not
be easy for a poor family
We now show that differences in
well-educated community.
rich,
who values education
"ability" to
pay
to complement, or even replace, differences in "willingness" to
We
Proposition 3 Small imperfections
poor families (Phd
The
in generating
in capital markets, resulting in a higher opportunity cost of funds for
monitoring technologies, or a tax subsidy to
when only
A
families can afford a
The
=
with increasing marginal cost of borrowing (or decreasing marginal return on savings)
P" >
0,
>
0;
is
rich ones:
0.
10
earning
in their first period of life,
In order to live in the
same community,
d decreases with h for any given
A
agents are always lenders and
p,
less
wealthy
hence the
result.
B
agents always borrowers
<
Alternatively, the imperfection could take the form of a borrowing constraint, d
resources are such that
it
binds for the B's but not for the
former but an equality for the
latter,
implying again
j4's,
the Euler condition
is
Rx (h A ,p,x) > Rx (hB,p,x).
repeating that, absent offsetting forces, segregation will occur no matter
credit
same
that of a wedge between borrowing and lending rates, as in Galor and Zeira (1993);
endowments which ensure that
are easily found.
=
during retirement, y(h)
must then borrow more than
conditions on
for the
relative to
downpayment.
to endogenize this differential. Let everyone face the
is
simplest case
When
home ownership
P(d)
way
with u)'(h)
families
F(L), and
F'(L) L'(x) / Pd(h,d).
due to standard asymmetric information problems. Let agents work only
wj(/i),
— \,F =
interest schedule
better
+ r(d)),
U3/U2
0) are sufficient to cause stratification.
reflect different
renting (interest deductions),
A
<
d'ihjPjj
economic segregation. 10
most simply when each family faces a constant opportunity cost of funds, with r^ < tb
result arises
This could
in (5).
+
=
move into a
pay for community quality tend
abstract from inter-family differences in tastes or returns to education,
allow instead frictions in credit markets: Rx(h,d)
d(l
highly to borrow enough to
how small wealth
d.
an inequality
But
it
bears
differences
and
market imperfections.
The
empirical literature yields contradictory estimates of the sign of Fhl\ see Jencks and
most recent study (Brooks-Gunn
et al. (1993)) finds
Fhl >
9
0.
Mayer
(1990).
The
Wealth
2.3
We now
effects
focus on the term (Ua/U2)(c,c',h') in (4), which captures preferences for education relative to
To
old age consumption or financial bequest.
isolate
wealth
perfect. Finally, let parental utility be additively separable: U(c, d, h')
p h
^
(
„ ,->
(/l,p
x)
where u(z)
= max, {u(z -
Proposition 4 Small
s)
'
=
F(L) and
u(c)
capital markets be
+ v{d) + w(h').
Then:
- W '( F L )) F(L)L
V - W 'W L » F
F'(L\
V
{L) L
- -vgy F>(L\
u'(z(h)- P)
(
+ v(s(l + r))}
differences
F=
effects, let
m
and z(h)
= u(h) + y(h)/(l + r)
lifetime resources (w'(/i)
+y
/
(/»)/(l
is
lifetime wealth.
+ r) >
0) are sufficient to cause
stratification.
When F =
cation occurs
less
if
F{h,L), the marginal rate of substitution
This
is
U3/U2 = w'(F(h,L)) /u'(z(h) -
in the
marginal
the sense in which
utility of
consumption which
results
different levels or rates of taxation across
communities
(e.g.,
rich agents vote for high taxes to finance public goods,
This leads, through a Tiebout (1956)
good. Similar wealth effects
infinitely elastic-
but through
Femandez-Rogerson (1992)). Ceteris paribus,
such as education, whereas poor ones prefer lower
mechanism, to as much segregation by wealth as the
is
the case in Fershtman and Weiss's (1992) analysis of socio/ status, where agents derive
from being
in
a profession with a high proportion of well educated members. Wealthier agents are
willing to accept
a lower remuneration of human
capital, in
more of them become highly educated, and the labor market
of our
financial wealth
Such
differentials.
more
-like
is
model
We
yield simple versions of
have until now assumed that
exchange
stratifies.
for higher status.
Therefore
We show in appendix how variations
Femandez-Rogerson (1992) and Fershtman and Weiss (1992).
all
land rents accrue to absentee landowners. This
is
a "neutrality"
assumption with respect to the allocation of the capital gains and losses created by stratification
and
is
of communities will allow. Instead of rents or taxes, wages can also play the role of compensating
number
utility
from higher
community composition L must be a normal
underlie models where sorting occurs not through land rents -supply
levels.
p). Stratifi-
the reduction in the marginal utility of education due to better parental background h
than the reduction
z(h).
is
C2
.
It
also simplifies the problem, by
examine how relaxing
this
assumption
making agents'
initial
affects the results. Let
10
wealth exogenous: u(h)
each
A
agent
initially
=
u7(/i).
own vA
in
C
We now
units of land
C
in
T7B
and
1
=
T)
A units
in
C2
;
the corresponding
{l/2-ni) A )/(l-n). Total wealth
endowments
unchanged. The converse, perhaps more
initial allocation
where vA >
— n vA )/(l — n)
= A,B. UuA =
i
,
(1/2
t]
A
=
and
1/2,
=
l/2n,
tj
=
a
0.
More
generally, consider
any
A hence vb < VB- As communities become more concentrated, the wealth
,
Propositions 3 and 4 show that this in itself creates a
when
others are absent.
all
Local public goods, political mechanisms, and discrimination
2.4
mechanism through which communities choose taxes and
defer until Section 5 specifying a particular
But inspection of the corresponding terms
school expenditures.
dency toward
stratification, with a
Proposition 5
FhE >
Political
To
when Fhe < 0) tend
is
to
<
public choice
An
(e.g.,
community (thx{h,p,z) < 0) or with land
0).
which are closer to their preferred choices, stratification
numerous
Tiebout (1956).
produce stratification. So do mechanisms such thai
the extent that a higher proportion of rich agents
all
la
such that expenditures E(x) increase with x when
fall with the proportion of rich agents in the
values (th p (h,p,x)
in (4) already indicates a "natural" ten-
presumption of efficiency gains, a
mechanisms whose outcome
(respectively, decrease
income tax rates
not
1
2
p +T)i p
i/i
=
1/3
another neutral case where the results remain
vA
realistic case, is
segregating force, which can sustain stratification even
We
i
losses; this is
while that of the B's declines.
j4' s rises
of the
r)
agent are
=u(h ) +
levels are w(/».)
both classes share equally in capital gains and
B
for a
mechanisms are such that the
is
will
rich
reflected in taxation
occur as agents "vote with their feet"
become more powerful
majority voting), this requirement seems both fairly weak and
alternative interpretation of th x (h,p, x)
relative to the rich, of joining a rich
<
or thp(h,p,x)
community. These
and expenditure decisions
costs,
<
is
as they
.
Although
become more
realistic.
that the poor incur higher costs,
which are particularly relevant when the
A's and B's correspond to different ethnic groups, can be pecuniary, due to discrimination by owners or
intermediaries in the housing market, or non-pecuniary, such as harassment. 11
when
11
come back
to
them
discussing inequality.
Of
agents
We
course, this
component
who have pure
of
t
does not enter into budget constraints.
"tastes" for the ethnic
mix of
their
community, as
11
Yet another interpretation
in Schelling (1978) or
Miyao
is
that of
(1978).
Stratification
3
and
efficiency: a simple case
We now examine whether it is efficient for different classes of agents to segregate in education. The criterion
we
use
that of aggregate productivity or surplus: are the output gains of rich communities greater or
is
12
smaller than the losses of poor ones? Distributional issues are taken up in Section 4.
exposition,
we
abstract from school funding and taxes; they are incorporated in Section
generated by a community with a fraction x of rich households
= xF{h A
S{x)
,
L(x))
+
is
,
solution
at a corner,
is
+
S(2n
—
and coincides with the
<
stratified
market equilibrium.
< min{2n,
concave, there are decreasing social returns to the concentration of
and symmetric: S'(x)
interior
comparing the gains 5(i 1 )
in the
—
=
S'(2n
—
x), so
=
x
n.
13
The
surplus
(6)
x) over
x
5.
L(x))
A planner choosing population allocations so as to maximize the productivity of the
area would therefore maximize S(x)
simplify the
thus S(x)/2, where:
- x)F(hB
(1
To
If
1}.
human
5
On
is
convex, the optimal
5
is
and the optimum
is
the other hand
capital,
The underlying
whole metropolitan
intuition
is
if
apparent when
S(n) from stratification in the rich community with the losses S(n)
—
5(x 2 )
poor one. To assess the efficiency of equilibrium, we therefore evaluate S":
S'
S"
=
= F(h A ,L)-F(h B ,L) + (xFL (h A ,L) + (l-x)FL {h B ,L))L';
2(FL (h A ,L)-FL (h B ,L))L'
+ (xFLL (h A ,L) + (l-x)FLL (h B ,L))(L') 2
(7)
+(xFL (h A ,L) +
where L
=
an increase
from
in
quality
when
is
The
first
term
community quality?
stratification.
resident,
12
L(x), etc.
in (7) represents the
If
Fm
>
0, it is
from a low or from a high
- x)FL (h B ,L)) L"
answer to the question: who
benefits
most from
the better educated family, hence an efficiency gain
The second term measures whether an
starting
(1
level; if
increase in
Fll <
0,
L
is
more valuable
to the average
the marginal productivity of
decreasing, implying an efficiency loss from stratification.
The
last
term
community
in (7) represents the
how economy-wide complementarities can translate aggregate results into Pareto comand how welfare might differ between the short and long run.
13
Naturally, if 5 changes curvature, the optimum may involve partial segregation.
There, we also explain
parisons,
12
answer to the question: where does a marginal well-educated agent, or family, contribute most to raising
the quality of
it
community?
its
L" <
If
joins than in the poor one which
it
0,
such a family
much
is
less valuable in the rich
leaves behind, leading to another loss
from
In the basic model, agents allocate themselves solely on the basis of the
how small
the private gains which
it
represents. 14
More
community which
stratification.
first
term
generally, wealth constraints
in (7),
no matter
and capital market
imperfections also shape the residential equilibrium.
Proposition 6 Agents
segregate or integrate depending on
factors unrelated to the productivity of education.
Fn
tration effects"
FhL<0, no matter how
The equilibrium can
and on other
small,
be very inefficient, if the "concen-
or L" are large in absolute value and have the opposite sign of Fhh, or more generally
of the incentive to segregate Rhx-
We
shall generally
tration of
be more interested in the case where the city
human wealth on one
side
and poverty on the other
is
stratifies,
and the resulting concen-
potentially inefficient -as Table
1
would
tend to suggest. But Proposition 6 also makes clear when inefficient mixing will occur.
4
Stratification
Does
families'
and inequality
tendency to segregate into homogeneous communities make inequality more persistent?
The simple answer
and community
is
positive, as stratification
levels.
The
result
is
compounds
greater income inequality than would have occurred
remained integrated. Assuming perfect segregation (n
h'A
in the
£'(1)
/h'B
0,
which
is
=
1/2),
we
if
When
the empirically relevant case. In Section 4.2
show how segregation slows down convergence among
they do, the inequality
we make
different groups,
V
14
had
(8)
)
role.
families
have:
= F(h A ,h A )/F(h B ,h B > F(h A ,L)/F{h B ,L),
simple case where purchased inputs play no
>
disparities in educational inputs at the family
the
model
is
truly
reinforced
if
dynamic and
and can even generate poverty
Half of the gains 2 {Fi,(h A L) — Fi(/ib, L))
represents the private gains from trade accruing to a pair of A
B agents who exchange places. The other half is external: a marginal rise in L is more valuable, ceteris paribus,
in the community with more //-sensitive agents; see the expression for S' Interestingly, the results of Brooks-Gunn
,
and
.
et al. (1993) suggest that
FhL >
but Fll, L"
<
0.
13
But
traps.
we point out
first
More
overlooked.
that this story
specifically,
we
measure of inequality and the cost
A
4.1
discuss
subject to an important qualification, which tends to be
is
what
are usually implicit assumptions about the appropriate
which the rich are able to exclude the poor.
at
and
caveat: inequality in what,
what price?
at
usual approach, in both the theoretical and empirical literatures,
The
is
we
education levels or professional status of successive generations, as
that this
cost,
is
also the costs of educational investments leads one to
bequest), or
more generally the
Proposition 7 Consider
the general
n < 1/2, then
U
1
(h A )
— U 2 (Iib) >
=
1/2, the ranking
The same
2.
is
is
family inequality
What
is
is
where the
F(/ifl),
is
m
question
proved in the appendix.
It raises
1
capital plus
V/, r
>
0,
Vh p
>
0,
ensuring
left
and
right
(3).
15
utility levels.
hand sides correspond
1/2, the inequality
is
reversed,
and
if
d+
h!
and 2
they are distributed evenly:
respectively.
all
housing
initially
use this stronger condition for simplicity.
types and communities (as in
Wheaton
when
utility is linear.
some new and more
Who owns the
Proposition 7
,
the old question of whether child inequality or
it
identifying the recipients of the capital gains
land in communities
We
n >
the relevant concept (Stiglitz (1973)), but also
the cost to the rich of seceding from the poor?
means
if
children's total wealth
which allow stratification to occur and earn the rents which
15
(human
ambiguous.
true of inequality
This simple result
—
at a
lead to a very different answer.
and assume
1,
and integrated equilibrium. But
respectively to the stratified
n
children's total wealth
may
of parents. This
TJ{h A )
not obvious
it is
Taking into account not only the payoff but
0).
compare
model of Section
But
Equilibrium stratification need not increase inequality between rich and poor families'
1.
If
utility levels
>
bill (if t x
did in (8).
community quality and better schools come
the appropriate measure of inequality. Better
namely a higher rent p 1 > p 2 and/or tax
to track across families the earnings,
is
may
and
assets,
16
practical ones.
property rights or legal rights
create? In our model, answering this
losses
which stratification generates on
derived under the "neutral" assumption that
belongs to absentee landowners, or alternatively to the
The same
result can
be shown in a model with a continuum of
(1993)) using only (3).
16
Interestingly, a similar indeterminacy can be demonstrated in a model where agents self-select through a tradebetween school quality and tax rates, rather than land rents (e.g., Fernandez- Rogerson (1992)). Even though
there is no leakage" of community resources to outsiders, stratification need not increase inequality in utilities.
off
14
city's residents
1/2,
i
=
vB
may
but with no correlation between an agent's type and the location of her property
= A,B). A more
=
»jfl
or
0.
may
assumption might be that
realistic
Yet even then,
agents
own
need not increase inequality
stratification
not offset capital losses in
A
C 2 What
.
is
required
is
vA
>
>
va, VB
and
institutions
l/2n,
in total wealth: capital gains in
C
1
a larger share of the corresponding
which allow the
the generation and persistence of inequality by various collective
among
these
is
by raising -even
rich to capture the benefits of their secession,
temporarily- the relative cost to the poor of joining or remaining in a
Section 2.4). Prime
=
Va
=
Va-
What this points to is the role played in
practices
=
77,-
that the rich capture a larger share of the
rents which their exodus creates, and/or that the poor be left holding
losses:
the land: va
all
=
(i/,
community which
is
appreciating (see
whether enforced by law or custom. As long as
racial discrimination,
Blacks are simply not allowed to bid for land in the suburbs to which White families move, the latter can
regroup without dissipating too
are allowed in, those
much
of the resulting rents on higher land values.
who do come must pay
When,
later on, Blacks
the full value of the community's social capital. Although
the education gap between their children and those of their white neighbors will close, the gap in total
wealth will not.
In fact, the inequality in wealth which occurs the
and property values adjust can
in itself
the specification of Section 2.3, with n
non-land wealth, and each
own
=
+
p
1
-
p
2
)
-
u(z)
de jure segregation
is lifted
be sufficient to sustain de facto economic segregation. Consider
1/2.
their houses in
Suppose White and Blacks have the same amount
C
1
and
groups will remain separated even once free mobility
that u(z
moment
< w(F(h A )) -
u>(F(/»b))
C2
is
<
2 of total
respectively. Proposition 4 implies that the
— p2
such
In recent years, the
main
allowed, though a rent differential p 1
u(z)
-
u(z
+ p2 -
p
1
).
two
device used to "keep out the poor" has been the right given to a community's residents to impose zoning
regulations, which essentially operate as
minimum income
requirements (Durlauf (1992),Wheaton (1993),
Fernandez and Rogerson (1993a)). 17
The empirical
implication of these observations
is
that studies which attempt to measure the role of
peer effects, neighborhood spillovers or school quality in the intergenerational persistence of inequality,
should try to take into account not only labor earnings but also financial wealth:
'
how much
of the
human
EquivaJently, Durlauf (1993) assumes that a neighborhood's land price adjusts to the (equilibrium) value which
keeps out the marginally poorer family, only once all richer families have moved in and bought the land at cost. In
all these models, it still remains that some of the higher human capital passed on to children in rich communities
reflects the higher taxes paid by their parents.
'
15
wealth afforded to a child by a better educational environment
is
reflected in a reduction of the bequest
she receives? Answering this question might also help determine whether the peer effects found by Borjas
among members
(1992a)
of the
same ethnic group
truly represent "ethnic capital", or
whether they really
operate through local spillovers in ethnically segregated neighborhoods -as Borjas (1992b) would seem to
While the former are inherited
indicate.
a price in the form of rent or tax
through birth, the latter are subject to choice and carry
"for free"
differentials.
Bearing in mind this section's caveat about the links between stratification and different measures of
inequality,
we now extend
the analysis to a
to ghettos or poverty traps,
dynamic
and how those can
setting.
demonstrate how stratification gives
rise
turn adversely affect overall growth.
in
Overlapping generations
4.2
Let the two periods considered until
the distribution of
now
represent any two overlapping generations.
human wealth must be
more
capital markets are perfect (see Section 2.1
bequests: uj(h), y(h) represent labor income in the
The second condition
that the
first
things simple,
The
first
condition holds
generally), or alternatively
two periods of
life,
and
all
of
if
if utility is
linear
and
there are no financial
c' is
consumed
in old age.
requires that either inequality in children's earnings be the criterion of interest, or
generation of A's
who
secede capture the whole (present) value of rents created in community
leaving the J3's holding the less valuable land in
human
To keep
the only relevant state variable, determining both residential
equilibrium and the appropriate measure of inequality.
1,
We
community
2.
Thus
all
housing
is
owner-occupied, and
capital inequality translates directly into inequality of total wealth or utility.
For simplicity we abstract from purchased local public goods, so that
h'
=
F(h, L). Finally,
let
n
=
1/2;
thus whether integrated or segregated, the two groups always remain homogeneous. Also, in either case,
if tiA,t
>
/»b,i
the
same
is
true for the next generation.
The
generations of rich and poor segregate.
when F(h,L)
= Oh a L p
,
with
Xi+i
=
\og(h At+ i/hB,t+i)
\,+ i
=
log(hA,t+i/f^B,t+i)
a < a +
= axi
—
(o
+
/?
<
Therefore
resulting impact
1
<
9.
Xu
initial
is
satisfied for all (h,p,x), all
on the persistence of inequality
is
most
clear
In the integrated equilibrium, the level of inequality
converges to zero at the rate
/?)
if (3)
1
—
a.
In the stratified equilibrium,
conditions vanish at a slower rate, or even not at
16
all if
a
+ /? =
l.
18
Inequality can be even
more
returns in (h, L) jointly over
are
persistent
if
F(h,L) has decreasing returns
some range. Such
what underlies ghettos or
h alone, but increasing
in
local increasing returns, realized only through segregation,
(Benabou (1993), Durlauf (1992), (1993), Lundberg and
local poverty traps
Startz (1993)). Suppose for instance that local spillovers involve a threshold effect:
F(h, L)
where /
>
magnitudes
and L{x)
=
(h A ) x (hB) l
When
is
/ii,
(+ i
=
F(hi <t ,L t )\ given
geometric average. Hence the following result, which
f < hB,o < h*
and
h.A,t
=
//(l
—
0}
grow asymptotically
<
Lo-
at the rate
Under
—
"
(9)
is
fixed, basic level
we have
(9), this
1
>
hi,t+i
implies
illustrated in
capital accumulation be given by hi it +i
-1 /( 1- °))
1
Here
defined as the geometric average.
the two classes are segregated,
they remain mixed, we have
Proposition 8 Let human
/,
measured as deviations from some
h, L, F, etc., are
to be taken literally.
~x
= 6h° max{L -
=
—
h
>
0;
L t+ \ = F(L ,L
Figure
since
F{hi tt ,L3t ), as defined
integration, inequality converges to zero,
0.
Under segregation,
the
FhL >
=
is
not
= A,B. When
i
t )
capital
L
t
the
is
2.
human
in (9),
and
and both
let
hg,t
capital poor families
converges to h in finite time, while that of rich families grows asymptotically at the rate
In equilibrium, stratification occurs (due to
thus h'
F(hi tt ,hi t ),
t
human
all
6—1.
or the other factors discussed earlier) confining
poorer families to a steady state of low productivity and income, which cohabitation with better educated
dynasties would have allowed
them
to escape. 19
Two
recent papers provide supporting evidence for the
kind of non-convexity which underlies the ghetto phenomenon. Crane (1991) finds threshold effects in the
way a neighborhood's
professional
mix
influences local rates of high-school dropout
and teenage pregnancy.
Using a methodology specifically suited to detect non-linearities in income dynamics, Cooper, Durlauf and
Johnson (1993)
find that very
"Estimating such a
(1992b) finds that
low levels of mean county income generate poverty traps.
log-linear relationship for the descendants of
human
capital spillovers slow
immigrants to the United States, Borjas (1992a)
both equal about .25 and
down convergence markedly: a and
are statistically significant.
Tamura
(1991b) and Galor and Tsiddon (1992) present models which combine increasing returns at the indieconomy-wide technological externality, through which the poor eventually
catch up with the rich. The implicit assumption is that the rich do not segregate (i.e., limit interactions to their
vidual (country or family) level with an
own
group), as they do
when given the opportunity
in our model.
17
Figure 3a
Figure 3b
Stratification:
F(h[,h{)
+1 =
h{
Integration: hj
+1
= F(h(,L t ), L t =
(h^ (h{)^k
may
Stratification
even hurt the productivity and income of richer dynasties, when, in addition to
economy-wide
local interactions in education, the different classes interact at the global, or
simplest
way
in
B
educated
Hm =
nil*
is
as
more educated
A
agents be managers, and the
incomes, y A
=
A
agents workers.
managerial
is
The
which city and suburban residents can be bound together
the production of goods or knowledge. Let for instance the
less
level.
h A'
(Y
human
n)» ,yg
'
/
=
representative firm's output
capital
(Y
h B°
Hw =
and
/ (1
— n)hs
(1
— n))»
is
complementary factors
2
V =
(//"m
worker
"
-
+ Hw !~r~)
human
capital.
are clearly interdependent.
'
,
The
in
where
resulting
Task assignments are
exogenous, but a model with occupational choice and competitive labor markets would have very similar
implications (Benabou (1993)). 20 Instead of production complementarities, agents could be linked through
economy-wide knowledge
spillovers (Lucas (1988),
public goods. These ties can
all
Romer
h,
H
is
some economy-wide index of human
which represents a basic
to heterogeneity
is
shall
1
the
Under segregation, human
among
degree of complementarity
like L, is
(10)
measured as deviations from
local index L, the sensitivity of
H=
(/iyi)
as in Proposition 8,
and
and
all
When
agents' skill levels; see
a geometric average:
same conditions
as:
~x
capital. All variables are still
integration, inequality converges to zero,
In this
H
and earnings. Like that of the
level of skills
simply assume that H,
Proposition 9 Assume
Under
(1991b)) or nationally funded
a key determinant of the costs and benefits of stratification.
this sensitivity reflects the
we
Tamura
be captured by writing individual productivity
y(h)=h x
where
(1986),
n
(/iB)
H
is
Benabou
a
CES
H
index,
(1992). Here
1-n
.
the production technology (10).
earnings grow asymptotically at the rate
8—1.
capital inequality keeps rising, but all earnings converge to h in finite time.
economy, productivity growth slows down and eventually peters out because the secession
of "managerial" dynasties prevents "worker" dynasties from acquiring the skills necessary to keep up. 21
Alternatively,
Tamura
(1991a) and Benabou (1992) present models where increasing returns cause agents to
specialize in imperfectly substitutable intermediate inputs.
essentially the
'The
same
The induced mapping from human
results of Glaeser,
Scheinkman and
Shleifer (1993) confirm the
force, in contrast to a high
concentration of
human
income
is
importance of a well educated general labor
capital on a small elite. In explaining a city's growth, they find
the proportion of residents with 12 to 15 years of education (high school graduates and
more important and
capital to
as the one below.
statistically
more
some
significant than the proportion with higher education.
18
college) to be both
Making those
Douglas)
skills essential to
When
only a convenient shortcut.
is
general model h'
=
H
production (by defining
F(h,L,E), reductions
as the geometric average, or
education uses real resources (schools,
in productivity feed
and even
one,
in the
absence of threshold
and income to which
capital
self-defeating
depends
effects, stratification
rich families converge.
respectively by the degrees of complementarity \jz
9,
Section
There
3).
is
But over the long
society
may
more than
stratification benefits the rich
H
as in the
human
is
capital
greater than
can lower the steady-state path of
or not the secession of the rich
Cobb-
is
human
ultimately
on the relative costs of local and aggregate heterogeneity, measured
in particular
As shown by Proposition
Whether
/»b in
as a
R&D),
back into each family's
accumulation. Thus even when the elasticity of substitution a between h A and
Y
it
and l/a
in local
and global interactions.
be faced with an intertemporal tradeoff. In the short run,
hurts the poor, so that overall growth increases (S"
>
in
run, excessive heterogeneity acts as a drag on growth, affecting all lineages.
then an incentive for the
rich,
subsidize the education of the poor.
provided their intergenerational discount rate
They could
for instance agree to
of schools, neighborhoods, networks, and so on.
We
may be
low enough, to
inter-community transfers (Cooper
(1992)) or support switching to a system of national funding (Benabou (1992)).
that purchased inputs play no role in Proposition 9, these
is
But as shown by the
fact
poor substitutes for the actual integration
return to this issue
when
discussing policy later on.
School financing
5
We now
explicitly incorporate local school expenditures
and taxes into the
analysis.
We
show how they
represent an additional segregating force, then examine the consequences for efficiency and inequality.
These
results will also
5.1
The
We
most simple
schools financed by
case,
conducted
in the
next section.
where agents are risk-neutral, capital markets
lump-sum or property taxes (with
that each family's tax
is
for the policy evaluation
choice of education expenditures
start with the
ment
form the basis
bill
inelastic lot size, these are equivalent).
and
local
This implies
equals per-student expenditures E, and that a community's educational invest-
not constrained by
how a simple adaptation
frictionless,
its
of the
tax base. Section 5.4 considers income tax financing, and Section 5.5 shows
model can incorporate wealth
19
constraints.
We assume
that in a
community
with composition (x,
1
—
x) and spillover quality L, education expenditures are determined as:
Er{z,L)
We
tinuous
as the
= aigmax{zF(h A ,L,E) + (l-z)F(h B ,L,E)-(l + r)E}.
choose this decision rule for two reasons. First, unlike majority voting,
manner both the proportions and the preferences of
outcome of unrestricted vote-trading. 22 Second,
munity composition;
(11)
E
it
local residents. It
it
a simple, con-
reflects in
can
in fact
be interpreted
yields the efficient outcome, conditional
this allows us to highlight the issue of inter- rather
than inira-community
on com-
efficiency.
Equation (11) implies:
EE
_
dE-(x.L)
with
F},
all
results,
E
,
i
= A,B
stands for FhE(hi,L,E'(L,x)), etc.
simplifies notation
varies with
g (a) =
Ah
EE
.
~
Fju
-Fee
The approximations hold when Ah/h
Although
this condition
is
+ fegfc + a - *^ L 'w
u - a-**&-(!-«)*&
are complements, rich communities
may
*i>i*L.
- fbe
Ah+ jj^. L
E*(x, L(x))
>
(
-Fee
i2)
thus spend more on their students than poor
same environment, students from poor backgrounds would have
a higher marginal product of education expenditures than those from richer families (FhE
<
0)-
Stratification
Having determined E(x) andt(/i,p,i)
or equivalents in
SEG =
(1
+
=
E(x), we
now
replace
families
who
them in
the bid-rent differential
r){Rt {h A ,p,x) - Rx(h B ,p,x)) = (F£ - Fg)
expression defines the net private gains, in terms of children's
B
=
as:
ones, even though, were they placed in the
5.2
small,
is
not required for any of the
and makes intuitions more transparent. The school budget E(x)
community composition
When E and L
*/*
xF$,.+(l-*)Fg
derivatives evaluated at (h, L(n), E(n)).
it
I*
-xF*
EE~^"-~*->*EE
E -(l-z)F§E
dL
where
nR
F$-F§
9E'{t,L)
human
U + (Fg
capital, to a
Rh x (h, p,x),
- F§)E'.
marginal pair of
A
This
and
switch places during the stratification process:
a weighted average of each group's preferred level, E*(x,L) = 0(x)E A (L) +
have implications very similar to those of (11). Note that we assume Fee <
throughout. Also, the case of most interest is when the equilibrium involves a high degree of stratification; as
stressed by Tiebout (1956), there is then near-unanimity in each community.
22
(1
A political process resulting in
— 0(x)) E (L), 0'(x) > 0, would
20
SFG
The term
-
(F A - F B + (F A - F§) ££bl±Q=£2£m
*
((^L + ^ff^Ji' +
in brackets represents the
education and community quality.
")
l<
<*fi- ff?'
+
(13)
^ffA^A/i.
comp/emen<an<y, both direct and indirect through E, between parental
The
last
term, always positive, captures the greater incentive of each
type of agent to join a community where individuals with similar preferences, being more numerous, have
greater weight in the setting of policy.
the absence of local externalities
stratification
(FL
The
-
implication of (13)
first
The second one
0).
that stratification always occurs in
that local funding of education
is
< SEG). 23
(Fm <
which would otherwise not have occurred
is
This
is
may
induce
probably an important
piece of the explanation for the difference in the extent of stratification between the United States
and
most other industrialized countries.
Efficiency
5.3
Recall that E(x)
is
chosen optimally, given
x.
thus again determined by the concavity of net
S(x)
From
(11)
=
efficient
S'
(surplus-maximizing) residential allocation
community output, now defined
= x F(hA ,L(x), E(x)) + (l-x) F(h B
and the envelope theorem,
S"
The
= F A - FB +
x
L(x), E(x))
,
F£ +
(1
-
i)
Ff
as:
- (1 + r) E(x)
.
is
(14)
Then, using (12)
yields:
2(F£-F?)L' + (xF£L + (l-x)FBL )(L'y + (xF£ + (l-x)F?)L"
(15)
-{xF£B + (l-x)F§E
The
first
)
(£') 2
three terms are due to local spillovers, which affect education both directly and through their
interaction with E. Their interpretation
is
the optimal adjustment of expenditures to
the
same
as in (7).
The
last
term, always positive, arises from
community composition i and quality
L.
Absent local
spillovers
(and imperfections, say, in capital markets), the tendency to segregate in response to differing preferences
over education promotes efficiency (Tiebout (1956)). In general, however,
23
in
De Bartolome
(1990) considers the case where Fhl
<
it
may have
= Fel < Fhe and where these two opposing effects result
an asymmetric but incompletely stratified equilibrium. Although his political mechanism
,
is different, this
can be
< min{2n, 1} to the equation f^n _ x SEG(x)dx = F(Iia,L(x ),E(x
- [F(h B Lix ), Eix )) - F(h B L(x*),E(x 2 ))] = 0.
understood as interior solution n
F{h A L(x 2 ), E(x 2 ))
the opposite effect:
<
1
,
x
1
i
1
,
21
))
—
Proposition 10 The
Indeed,
< SEG and S" <
FhL <
suffices that
decentralization of school funding can trigger stratification which
we can
is inefficient:
it
0.
rewrite:
= SEG + (F£-FB)L> + (xF£L + (l-z)F*L )(L')* + (xF£ + (l-x)F?)L"
S"
(16)
+ (zF£E + (l-x)F£E )L'E'
The term
SEG
represents the net private
were discussed
in (13).
gams from
stratification; the
All others terms constitute wedges
between
elements which determine
social
and private
three of those involve only the local spillover, and were discussed in Section 3.
interaction of
E
The
values.
last
>
paribus, for
L
sign
The
first
one involves the
and L. As agents congregate into more homogeneous groups, they ignore they own
on community quality L, including the resulting impact on the productivity of E. Assuming,
that E'(x)
its
the effect
0,
is
favorable
to be high in the
E
L and
if
are complements:
community where
E
=
We
is
high.
It is
realistically,
a planner would also
detrimental
if
effect
like, ceteris
they are substitutes. 24
Income taxes
5.4
We now
consider income taxes: t(h,x)
r(x) h.
continue to assume that in each community, school
expenditures E(x) are chosen efficiently with respect to local surplus, that
income tax rate
A(x)
— x hA +
is
(1
is,
according to (11).
then simply determined by the balanced budget constraint: r(x)
—
x)
hs
is
=
E(x)
/
The
A(x), where
average community income. This leads to the following result, proved in the
appendix.
Proposition 11 Let
CES
Ah/h
the local spillover
L
be
education production function F(h, L, E)
is
homogeneous of degree no greater than one, or more generally
if the
any
index,
and assume that
return to marginal expenditures decreases with the scale of educational inputs
Fs(h,L,E) for A >
1),
the school budget rises less than one for one with average
is
small.
If the
(Fs(^h,\L,\E) <
community income.
Therefore the income tax rate r(x) decreases with x, promoting stratification.
24
Equation (16) readily yields De Bartolome's (1990) result that an incompletely stratified equilibrium is inefgiven his assumptions Fhl < 0, Fll <
and L" = Fel = 0. As seen in the previous footnote, such an
ficient,
equilibrium
is
defined by
f**_ xl
SEG(x) dx =
0; this
implies S'fi
optimality.
22
1
)
- S'(2n - x 1 ) = f^_ xl
S"(x) dx
<
0,
violating
and borrowing constraints
5.5
Wealth
When
agents have concave utility or face imperfect capital markets, their preferences over education spend-
effects
ing reflect not only
The
of funds.
its
contribution to their child's
effect of equilibrium land rents
be dealt with, as in
De Bartolome
capital, but also their different opportunity costs
on wealth makes
There
(1990).
human
is,
tax base. Following Becker (1975),
funds, r(h A )
<
let rich
ones, other things equal.
It
SEG(x);
<
r'(h)
also tends to
F(h,L,E) by F(h,L,E) =F{h,L,E)
its essential
(1
0.
promoting
stratification, as
still
im-
due to their lower
costs of
+ r)/(l + r(h))
then implies that poorer communities invest
make Fhl >
efficiency analysis the actual products F(h, L,
shown
less
in
than rich
earlier. In the
E) should be used (hence another wedge between SEG{x)
and S"(x)), unless the planner cannot provide loans more
sum
ability to invest,
can
it
and poor agents have different but constant opportunity
r(h B ). 25 Then, simply replace
the derivations of E'{x) and
more complicated, but
however, a simpler way to incorporate
namely that communities with poorer residents have a lower
plication,
the
this case
efficiently
than the market, or seeks to maximize
of agents' utilities rather than net output. In such cases she should internalize agents' different
costs of funds
and the resulting gains from
and use F(h, L, E) to compute
stratification,
social surplus.
Policy implications
6
6.1
Background
In the
United States, an increasing number of states are being forced by court decisions to close the wide
gaps which exist between the school budgets of different communities. The underlying view
disparities,
stemming from the lower property tax base of poorer towns, do not give equal
to their young.
is
that these
starting chances
Equalizing transfers, or state control of education funding, are then needed to reduce
26
inequality as well as possible inefficiencies.
There
is
also a contrarian view, of
representative.
Under a 1986 court order
existing school system
25
More
which a recent
.
.
.and replaced
specifically, let r(h)
it
to
remedy
article
on Kansas City
in
The Economist
(1993)
is
racial segregation in its schools, the city "scrapped its
with the best
be a decreasing function of
.
.
.
money could
buy," at the cost of an extra $1.3
initial resources w(fc),
so that wealth redistributions do
affect educational choices.
26
The most recent example is Michigan, where voters recently approved a proposal to replace local property taxes
by an increase in the state sales tax, as the primary source of funding for schools.
23
billion, or
$36,111 per student over the normal school budget. Three quarters of the cost were born by the
state, resulting in severe
"So
far,
however,
cutbacks in other
all this
same today
as
have seen no improvement whatsoever
reading and
math
.
.
.
produced few of the desired results
lavish expenditure has
racial balance in the schools is the
six years
districts.
it
was when the order was
issued.
.
.
.
The
The
past
standardized tests of
in children's scores in
Pupils in elementary schools which have not been turned into magnet
schools regularly outperform pupils in generously funded
has risen every year, without
fail,
since the decree
magnet schools
.
.
.
The drop-out
rate
was handed down, and now stands at a
disgraceful 60%."
Not
surprisingly,
Kansas City school
officials
vigorously dispute this bleak assessment.
of evidence appear consistent with the view that lack of financial resources
is
faced by mediocre or failing schools. Expenditures per student are higher in
But other pieces
not the main impediment
some urban school
districts
than in neighboring suburban or rural ones, thanks to a larger industrial tax base or to state subsidies.
The
latter nonetheless achieve better test scores
and dropout
rates.
Reviewing the empirical literature
on education production functions, Hanushek (1986) observes that most studies
link
between school expenditures and student achievement.
centralization of education funding experienced by
many
fail
to demonstrate any
Perhaps most importantly, the increased
states (starting with the 1971 California
Supreme
Court "Serrano" decision) appears to have significantly reduced spending inequalities, but not achievement
inequalities, as
measured from
by a concomitant decline
in
SAT scores (Downes (1992)).
Moreover,
it
seems to have been accompanied
those states' average test scores (Pelzman (1992)).
Opponents of the "lack of resources" hypothesis often point instead to
culprits such as teachers' unions,
the education bureaucracy, or the disincentive effects of the welfare system.
different explanation of the evidence,
social inputs in education.
We
We
shall
advance a very
based in particular on complementarities between purchased and
identify in particular a case
where equalizing school expenditures
fails to
reduce segregation and leads to a decline in overall academic performance, hurting the rich more than
helps the poor. Yet a better designed policy
may
still
24
improve both equity and
efficiency.
it
Equalizing school budgets
6.2
We
model equilibrium
introduced in Section
in
community
level.
This
is
j.
prior to state intervention using the simple specification of education expenditures
5.1:
E>
=
E'{x j L(x')), where
,
x>
human
the fraction of high
is
capital families
thus deliberately abstract from financing constraints at the individual or community
We
not because we consider the credit market imperfections and wealth effects discussed earlier
as unimportant; they played a central role in our previous analysis of local
and centralized school funding
27
What we seek to do here
(Benabou (1992)), as well as that of Fernandez and Rogerson (1993b).
highlight the limits of redistributive policies,
and explore whether there remains a
role for the state
is
to
even
absent such constraints.
We first
level.
mention some ways
in
which attempts to equalize school budgets might be defeated at the local
Clearly, outright redistributions of wealth leave
Mandating that
E
say, athletic facilities or
as education-related
and administrator
let
us
now
either through
unaffected; only consumption adjusts.
rich
a library from the school budget to some other line item.
A
poor one might claim
increased
may
some expenditures which have mostly consumption value
salaries). Finally, taxing
contribution to education by the
But
E2
and
A
E2
block grant does reduce spending in
by Downes (1992)
1
induce creative accounting.
be reduced and
1
E
C
1
E
1
amount
of the transfer.
and so do
for California following Serrano),
take as given that the state
is
As a
C2
,
in
where the population
on education (E'(x)
>
is
community
2 as a
where residents simply reduce their
result, total expenditures fall (as
observed
test scores.
successful in enforcing equalization of school expenditures,
28
matching grants (Feldstein (1975)) or simply by centralizing funding.
stable equilibrium
shift,
(nicer buildings, teacher
and rebating the proceeds to schools
but changes nothing in
,
town might
stratified
(SEG >
in (12)). After state intervention,
E
1
in (13))
is
We
start
from a
and the richer town spends more
reduced, and
E2
increased, to a
common
value E. 29 There are two cases to consider.
27
Credit constraints were also at the center of earlier models where education was privately purchased, such as
Glomm
and Ravikumar (1992).
combined State and Federal share in elementary and secondary school spending
rose from 35% to 50%, with a corresponding decline in reliance on local and private resources (Hanusheck (1986)).
29
£ could be either the mid-point (E 1 + E 2 )/2, or the optimal value E(n) = E'(n, L(n)) which would be chosen
Loury (1977) or
"Between 1960 and
1983, the
by integrated communities, or the optimal value given the stratified configuration.
25
F(h A ,h A ,E)
F(h B ,h B ,E)
E
E*
Figure 4a
E'
E
stratification persists
k
h''
,
F(hA ,hA ,E)
F(h AX,E)
F(h B ,L,E)
Ahl{
^-*"
f^r^T^
/f^r
Ah 2
F(h B 4i B 3)
""
..,
i
Im
...---
E
1
Figure 4b
E
E
2
integration occurs
E
Case
If
1.
Segregation persists
Fhl >
0,
the composition of each community remains unchanged. This could also be due to the other
stratifying forces discussed earlier, particularly
on each i ;
What
is
,
interesting
F(h A ,L 1 ,'E))
=
is
the
way
1/2).
In
which
in
C
1
students'
,
l
1
x 2 ) (F(h B ,L 2 ,E)
(11).
InC 2
human
1
- F(h B ,L 2 E 2 )) <
,
Proposition 12 When Fhl >
0,
1
capital
,'S))
>(l
(1
+ r) (E - E 2 ),
+ r)
is
for the case of perfect
-AA 1 = x l (F(h A ,L l ,E y ) -
reduced by
is
the corresponding increase
,
by Figure 3a
this occurs, as illustrated
+ (l-x )(F(hB,L ,E )-F(hB,L
maximization problem
-
for racial separation. Conditional
expenditures were previously set optimally, so state intervention necessarily reduces surplus.
stratification (n
(1
some group's preference
{E 1 -E), due
A/i 2
=
to the concavity of the
x 2 (F(h A L 2
,
,
E)—F(h A L 2
,
,
E 2 ))+
by the same argument.
community composition unchanged
equalizing school budgets leaves
reduces total surplus. Student achievement and subsequent earnings decline more
m
rich
but
communities than
they improve in poor ones (per dollar of cutback or subsidy). If state-wide expenditures are kept constant,
~E
= (E + E 2 )/2,
1
average student performance declines.
So while income inequality
The
intuition
is
is
reduced,
it is less
that school expenditures are
by
lifting
more productive
up the poor than bringing down the
in a
human
complementarity either between h and E, or more interestingly, between
teachers,
computers and other educational resources
may
not do
much
capital-rich
E
in
and L; see
rich.
community, due to
(12).
30
Top-notch
a school plagued by discipline
problems, the lack of motivation from role models in the community, peer pressure not to study, gangs,
etc.
Redistributing expenditures without simultaneously "redistributing" social capital
Case
If
2.
FhL <
Segregation
is
is
ineffective.
undone
(and the other stratifying factors are not too strong), equalizing expenditures removes the
incentive to segregate.
The
stable equilibrium then shifts from the stratified to the integrated configuration.
Both purchased and non-purchased inputs
E
and L into education are equalized across students, achieving
a greater reduction in inequality than in the previous case. There remains the question of whether academic
30
Downes and Pogue (1993) provide evidence that communities with a higher fraction of students from disadvantaged backgrounds have higher per-student costs, for a given level of educational achievement. While one must be
careful about identifying average and marginal costs, this suggests that school expenditures in these communities
have a lower marginal product.
26
1
achievement, income and the surplus from education
The answer
rich ones.
integrated community,
where
S
is
is
E=
E*(n,L(n)). The net change in surplus
L
small and
l
is
E
poor students than they
fall for
set at the optimal level for
is
then S(n)
— (5(i 1 ) + 5(2n —
an
1
))/2,
given by equation (15); hence the following results, proved in the appendix.
E=
are chosen optimally,
(E +
less for
where
easily obtained in the natural case
Proposition 13 When F^l <
is
more or
rise
E 2 )/2.
is
a
CES
equalizing school budgets leads to integration. If state-wide expenditures
0,
E*(n,L(n)),
total surplus increases provided
S
is
concave.
If,
moreover,
when state-wide expenditures remain
index, surplus increases even
Ah/h
E =
fixed,
In that case average student achievement also rises.
The conditions under which 5
were analyzed
earlier.
This case
is
concave, so that integration improves both efficiency and equality,
Figure 3b: the equalization of L induced by the new
illustrated in
is
policy shifts up the returns to educational expenditure on type
the returns to expenditure on type
A
curves and the two dashed ones
measured
three terms in (15).
The
is
distortions
agents.
The
B
much more than
agents,
vertical difference
it
shifts
down
between the average of the two
solid
by the sum of the
first
(to a second-order approximation),
from constraining expenditures to be the same across communities
and individuals are captured by the remaining term.
6.3
Reversing segregation
Consider
now
the case where equalizing expenditures
is
not sufficient to prevent segregation. This could
be due to any of the stratifying forces identified in Section
complementarity F^l >
13
and
illustrated
0.
Note that integration
by Figure 3b;
all
that matters
concerned with either inequality or inefficiency
communities
directly,
may
sufficiently attractive to families
(if
is
still
that
S" <
with high
through tax incentives or housing subsidies
1;
for convenience
to focus on the
yield the surplus gains described in Proposition
S
be concave. In principle, a state government
about integration by making poor
0) could bring
human
(tft x
we continue
capital,
(h,i)
>
and
vice-versa. This could be
0) or indirectly,
done
through a contingent
allocation of educational resources E(x):
SEC = FhL (h, L(x),
E(x)) L'(x)
+ FhE (h, L(x),
27
E(x)) E'(x)
-
(1
+ r) t hx (h, *) <
,
(17)
,
for all x.
31
The new
stable equilibrium will be symmetric, with equal expenditures
But what
can even be chosen optimally, to equal E*(n,L(n)).
stratification is likely to be
rich families to
come back
place. For instance,
first
to
undo once
to the cumulative nature of the process, the
Due
on.
much harder
when using only education
may
is
amount of
it
This kind of irreversibility
cities are
may
The dynamic
explain
to stop in
is
e)
that
tracks early
its
would have taken to make them stay
expenditures, with
» E(n -
it is
shows
transfers required to induce the first few
Fhe >
in the
0>
- E{n + e)
small. Large transfers between the school budgets of rich
well be politically infeasible.
some US
this differential condition
has occurred than
considerably larger than what
is
E(2n - x 1 ) - Eix 1 )
where e
it
E(n) everywhere; E(n)
and poor towns, even
if
only temporary,
aspects of such policies also raise problems of credibility.
why magnet
schools and similar programs recently implemented in
not very successful at fostering racial and economic desegregation of schools: the scope
of redistribution required for such policies to be effective (advocates speak of a "Marshall Plan" for the
inner cities) seems to be politically unacceptable. In metropolitan areas which are
new
or have retained a
balanced urban-suburban composition, on the other hand, modest policy interventions can be
Rusk (1993). But the point
perhaps most relevant
is
fairly egalitarian allocation of school resources,
for
European countries where despite a
economic and
with quasi-ghettos starting to emerge at the periphery of
effective; see
racial stratification
many
cities. If
action
before polarization has reached the point where large gains and losses have
is
centralized,
appears to be on the
to be taken,
become locked
it
rise,
should be
in.
Conclusion
7
The model developed
in this
paper allowed us to link together important issues of local public finance,
income distribution and productivity growth. Five main
results emerge. First,
minor differences
cation technologies, preferences, or wealth can lead to a high degree of stratification.
markets are not necessary, but
31
will
compound
these other sources. Second, stratification
in
edu-
Imperfect capital
makes
inequality
whether competition among communities allowed to charge different
prices t(h,p, x) to agents with different characteristics will also achieve efficiency; see Eden (1992) and Epple and
Romano (1993) for an analysis of school competition with student vouchers. The use of E to improve on an
inefficient residential equilibrium, which does not require family characteristics to be observable, was considered by
An
important
De Bartolome
issue, not
(1990).
It
taken up here,
is
can only be implemented in a centralized manner.
28
in education
in total
and income more persistent across generations. Whether or not the same
is
true of inequality
wealth depends on the ability of the rich to appropriate the rents created by their secession. Third,
the polarization or urban areas resulting from individual residential decisions can be quite inefficient, both
from the point of view of aggregate growth and
when state-wide equalization
in the
of school expenditures
educational achievement in poor communities
is
much
insufficient to
less
erage academic performance and income growth both
Pareto sense, especially in the long run. Fourth,
than
fall.
between purchased and non-purchased educational inputs
Fifth,
is
it
reduce stratification,
lowers
it
in richer
it
may improve
communities; thus av-
Yet policies better tailored to the interaction
may
still
improve both equity and
efficiency.
because of the cumulative nature of the stratification process (hence of any offsetting transfers),
likely to
be
much harder
to reverse once
Important issues not addressed
in the
it
has run
its
course than to arrest at an early stage.
paper include the
of voucher programs for education or housing.
It
it
role of private schools
and the effectiveness
would be interesting to try and extend the model
to
explore these matters as well. But perhaps the most intriguing direction for further research follows from
two of the points mentioned above, concerning dynamics. The
rents during the "real time" process of stratification.
political equilibrium constrain the
first is
The second
is
the creation and appropriation of
how
considerations of credibility and
time path of desegregation policy, or even
merit a more detailed analysis than the
first
steps taken here.
29
its feasibility.
These issues
Appendix
A-Proof of Proposition
For
1.
1
all
(p
V{h,p2 ,x 2 ). Standard reasoning shows that
implies
>
AV(h')
for all
ft'
>
AV(h A ) = AV(h B ) =
allocation
(b)
find
is
(respectively,
ft
ft'
(respectively,
<
)
such that
AV(h B
AV(h A ) =
|
This requires x 2
0.
2
1
p ,2n,p ,0)
> AV(h B ). This
AV(h A
always find p l p 2 such that
,
=
0.
,
,
,
V(h,p*,x l ) -
AV(h) >
0) everywhere, then
=
=
x2
=
n,
hence p 1
=
0,
x1
=
2n, hence 2n
When Rhx >
\
(1- i 1 ) /2 =
requires
p
l
,
1,
p
2n -
2
,
equilibrium (p 1 x 1 p 2 x 2 )
,
,
same number of
,
B
is
said to be stable
1)
=
We now
This concludes the derivation of equilibria.
the
<
=
x 1 p* x*)
l
with strict inequality
ft),
2
p
> x2
x1
if
then
It is
.
Conversely, this symmetric
.
if,
0,
is
<
B
agents in the reverse direction makes
0), the
rest of the
moving
|e|,
proof
e
A
other condition
is
cannot be) an equilibrium.
(respectively,
turn to stability, which
for all small
Clearly, one can always
1.
(l-x 2 )/2 = 1-n,
The
0.
<
(respectively,
automatically (respectively, never) satisfied, hence the allocation
(c)
p
|
a solution to one of the following three conditions.
is
This requires x 1
0.
AV(h
let
,
,
always an equilibrium.
AV(h A ) > AV(h B =
p\p 2
x 1 p2 x 2 )
Rhx >
if
straightforward to show that an equilibrium
(a)
,
hence 2n
One can
1.
identical to case (b).
is
is
>
defined very simply.
agents from
C2
to
C
1
agents the highest bidders for land in
An
and
C
1
,
and vice versa. Formally:
e(AV(h A
for
p\x +
x
|
-
e)
- AV(h B p\x l +e,p 2 ,x 2 -
when Rhx >
0, is
<
As a standard
if
V(h,T,E)
result, agents segregate as
stable
=
much
it
and only
if
satisfies
if
<
0,
Rhx <
0.
The
stratified
AV(h A — AV(h B ) >
)
0,
which
(which are the only feasible ones).
B- Alternative Wealth- Sorting Mechanisms.
For instance,
is
then stable since
preserved by continuity for small perturbations e
taxes or user fees.
e))
|
any feasible e^0. Clearly, the symmetric equilibrium
equilibrium, which exists only
is
2
2
e,p ,x
u(h
(1)
Taxes.
Consider
first
- T) + w{E), -Ve/Vt
as permitted by the
number
the case of
lump-sum
clearly increases with h.
of communities.
Sorting
through proportional taxes requires restrictions on preferences: the income effect must dominate the substitution effect.
is
For instance,
let
V(h,
x)
r,
average community income and E(x)
=
u((l
= ri(x)
-Vx /VT =r L'(x) w'(t L{x)) /( h u'((l -
r)
ft))
-
r)h)
+ w(t L(x)),
where L{x)
= xh A +
(1
- x)h B
school quality, as in Fernandez-Hogerson (1992).
increases with
-zu"{z)/u'{z)>\.
30
ft
if
and only
if
u has relative
Then
risk aversion
F
Let us re-label each community
(2) Status groups.
members:
wage pi
(i)
=
F
unit of
human
=
u(pi
+
h)
As a
The
)
more
>
l
Hi
capital input
w'(xi) / (h
a lower remuneration
in
u'(pi h))
exchange
for
so that at least one occupation has a homogeneous
somewhat from Fershtman and Weiss
applies
(1992), but the basic
A(x) {F£
11. First, t'(x)
-Fi + (xFiL +
(1
<
Therefore
V « Ah(hFhL + L FE l + E E e), where
A(x)
ss
h and for any
are evaluated at (h,L,E). Finally, (/(A)
Note
B
agents remain in
when
it is
C2
C
1
after stratification, so
which remains mixed
CES
13.
and only
if
E'{x)A(x)
index,
Lfah,
E=
L'(x)
«
Ah,
when F(h,
L,
<
(1
since
after agents sort
E(x)A'(x), or using
- x)F#£ ) Ah <
<LTgmaxE {F(h, L, E)-(l+r) E} and
E)
Only the case where
is
for all A
>
1
homogeneous of degree one,
E =
(E 1
(12):
to a second-order approximation.
= FE (Ah, XL, \E)- E (h, L, E) <
in particular that
E-Proof of Proposition
if
- x)FiL ) L'{x)) + (-xFgE -
Ah / h,
result.
some
immediate.
is
For small
hence the
a
(ii)
= V{h A ,p\xi) - V(h B ,p\x>) > V(hA ,p,x) - V(h B ,p,x)
(h B )
The opposite reasoning
p.
D-Proof of Proposition
=
stratifies,
Vx / V^ =
as above,
willing to accept
(1) In the first case,
7.
)
themselves. Part (2)
i>
human
its
result are similar.
U l (hA ) - U 2 (h B = U\h A - U
x and p 1
market
result, the labor
C-Proof of Proposition
>
with the sector's total
falls
Under the same condition
w(xi).
specification used here differs
mechanism and
x1
conferring on
,
from wages and from belonging to a higher status group:
utility
increases with h: better educated agents are
higher status.
which
capital,
due to diminishing returns. Agents derive
workforce.
a "professional occupation"
a social status which increases with the fraction x ; of highly educated (A) co-workers;
p(Hi) per
V(h,pi ,xi)
,
+ E 2 )/2
derivatives
all
<
implies ^'(1)
r'(i)
=
0,
0.
remains to be proved. With
expenditures fixed, the change in average student achievement equals the change in total surplus. This,
change in surplus achieved with the optimal E*(n,L(n)), minus the distortion from
in turn, equals the
choosing
(^(x 1 )
E
instead.
By
+ E(x 2 ))/2 -
when L
is
a
CES
— (5(x
E{(x l
+
x 2 )/2). But 6
with elasticity
differentiating (12),
gains 5(n)
the envelope theorem, this distortion
1
)
E"(x)
is
+ 5(x 2 )/2
e,
is
V a Ah
of order
(Ah) 2
.
itself of
and L"
The
is
of second order in 6
second order
ss
31
in
Ah,
as
is
E*{n, L(n))
Ah, as can be seen from
(Ah) 2 /2e; hence E'(x)
distortion 6 2
which are of second order
in
= E—
is
of order
therefore negligible
(12):
Ah, and,
compared
shown by the expression
=
to the
(15) giving
S"
.
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