A Theory of Persistent Income Inequality
Author(s): Steven N. Durlauf
Source: Journal of Economic Growth, Vol. 1, No. 1 (Mar., 1996), pp. 75-93
Published by: Springer
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ofEconomicGrowth,
1: 75-93 (March,1996)
Journal
ffi
1996Kluwer
Academic
Boston.
Publishers,
A TheoryofPersistent
IncomeInequality
STEVEN N. DURLAUF
of Wisconsin,Madison, Wl 53706 and
DepartmentofEconomics, University
National Bureau of Economic Research
and
This paperexploresthedynamicsof incomeinequalityby studyingtheevolutionof humancapitalinvestment
of their
neighborhoodchoice fora populationof families. Parentsaffecttheconditionalprobabilitydistribution
children'sincomethroughthechoice of a neighborhoodin whichto live. Neighborhoodlocationaffectschildren
boththroughlocal public financeof educationas well as throughsociological effects.These forcescombine to
createincentivesforwealthierfamiliesto segregatethemselvesintoeconomicallyhomogeneousneighborhoods.
combines withstrongneighborhoodwidefeedbackeffectsto transmiteconomic status
Economic stratification
across generations,leadingto persistentincome inequality.
Keywords:inequality,neighborhoods,spillovers
J24,J62,015, 040
JEL Classification:
1. Introduction
haveexamined
withBeckerandTomes(1979)andLoury(1981),manyresearchers
Starting
incomedistribution
a nondegenerate
cross-section
modelsexplaining
(see GalorandZeira,
Inmuchofthis
recent
contributions).
1993,andB6nabou,1993,1994,forsomeimportant
inchildren
inhumancapitalinvestment
differences
literature,
playa majorrole
byparents
aretakento be
humancapitalmarkets
cross-section
in generating
inequality.
Generally,
cannot
be
financed
human
sense
that
formation
in
the
byissuingclaims
capital
incomplete
of
such
contracts.
As a
of
due
to
the
lack
future
a
child's
enforceability
earnings
against
in
human
to
invest
families
are
better
than
families
able
result,
capital,
poor
high-income
human
arepassedon acrossgenerations.
andincomedisparities
imperfect
Consequently,
of
in
the
time-series
income
substantial
correlation
induce
serial
can
markets
profile
capital
incomerankings
as relative
distribution
changeslowlyovertime.
income
inrelative
ofthesemodelstoexplainsomestickiness
rankings,
Despitetheability
of
fortheaveragebehavior
on modelswitha striking
thisworkhascentered
implication
models
these
Zeira
and
of
Galor
With
the
overtime.
families
(1993),
generally
exception
whencomputed
oversufficiently
thataverageincomesareequalforall families,
predict
forone
is noasymptotic
themodelsimplythatthere
Further,
tendency
longtimehorizons.
in
that
this
Tomes
and
in
income.
Becker
another
above
rank
to
fact,
(1986),
argue
family
andchildincome
as a number
ofstudiescomparing
is empirically
feature
accurate,
parent
that
.3 or below,suggesting
correlation
havefoundsmallintergenerational
coefficients,
incomesconverge
rapidly.1
family
incomedistribution
exhibits
there
existsevidencethattheoverallcross-section
Although
in thedistribution's
evidenceofpersistence
therealso existssubstantial
meanreversion,
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DURLAUF
76
tails. For example,Brittian(1977) has foundsubstantialcorrelationin relativeeconomic
statusbetweenfathersand sons in theUnitedStates. Amongfatherswhose relativestatus
rankingwas in thetop 10 percentofthesample,theaverageson's percentilerankingwas 13
percent,whereasforfatherswhosepercentilerankingwas 90 percentor below,theranking
was 71.8 percent. Similarly,Cooper, Durlauf,and Johnson(1994) have foundevidence
correlationcoefficientvaries widelyacross familiesdepending
thatthe intergenerational
of thecountytheyresidedin, rangingfrom.02 to over .4, withhigher
on characteristics
or poor countries.In a relatedliterature,
values associated withrelativelyaffluent
many
scholarshave arguedin favoroftheexistenceof a class ofchronicallypoorpeople who are
trappedin ghettos.Wilson(1987) has documentedthegrowthand persistenceofthechronicallypoor in a numberof studies.Wilson's workhas emphasizedtheidea thatas middlethe
and upper-classblacks have moved outsideof historicallysegregatedneighborhoods,
a
breakdown
of
social
and
ecohave
found
themselves
confronted
residents
by
remaining
This
thathas renderedpovertyin theseneighborhoods
nomicinstitutions
self-perpetuating.
to economicfactorssuchas thelack of an adequatetax base
breakdownhas been attributed
to supportschools, as well as to sociological factorssuch as the lack of successfulrole
modeledin Streufert,
modelsto motivatechildrento tryto leave theghetto(formally
1991)
on thepartof some
or peer groupeffectsthatimplythatthelack of educationalattainment
studentswill hurtthe abilityof othersto learn(formallymodeled in deBartolome,1991,
and Benabou, 1993). These ideas are consistentwiththeempiricalfindingsof Datcher
(1982) and Corcoran,Gordon,Laren,and Solon (1989) thatneighborhoodcharacteristics
of individualincomelevels. One important
are an important
determinant
implicationof the
whether
for
statistic
a
a
sufficient
work
is
that
income
is
not
determining
family's
empirical
across
generations.
povertypersists
a dynamic
incomeinequalitybyconstructing
to understand
This paperattempts
persistent
model of incomedistribution.Our model containstwo key features.First,we explicitly
Educationis lowide influenceson individualoccupationalattainment.
model community
of
cally financed;intercommunity
borrowingis ruled out.2In addition,the distribution
shocks among offspring
is allowed to dependon neighborhoodcomposition
productivity
in orderto capturevarioussociological influences.These factorscreatea feedbackfrom
thecommunityincomedistribution
to therealized incomeof offspring.Second, families
choose whichneighborhoodsin whichto live,subjectto minimumand maximumincome
whichproxyforzoningrestrictions.Homogeneousneighborhoodsbenefit
requirements,
thewealthydue to thepositivespillovereffectsinducedbyhighpercapitaincomeswhereas
providetheadvantageof loweraveragecosts to edularger,heterogeneousneighborhoods
ofintergenerational
thesefeaturesinducea complexpattern
cation. Together,
neighborhood
and incomedynamics.We describeconditionsunderwhichuniformly
formation
poor and
Furfamilies.
of
can
a
prosperouscommunities emergeamong population initiallynonpoor
cross-sectional
a
link
between
creates
the
stratification
economic
of
ther,
neighborhoods
and intertemporal
inequality.The basic modelis also able todescribetheprocessby which
intoghettosas wealthierfamilies
heterogeneousurbancommunitiescan be transformed
factorsstronglyinflumove to suburbs. Together,theseresultsindicatehow community
ence whethera familyis trappedin poverty.
as functionsof neighborhood
By modelingindividualeducationlevels and productivity
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A THEORYOF PERSISTENTINCOMEINEQUALITY
77
a mechanism
we introduce
setis affected
behavior,
bywhicheachfamily's
opportunity
by
of
thechoicesofothers.Thisidea hasbeenthebasisformuchrecentworkon theories
and
and
coordination
failure
One
John,
1988).
(see Cooper
important
equilibria
multiple
ourmodelandprevious
workis thatwedo notrelyonuniform
between
distinction
positive
in long-run
all agentsto generate
behavior.Instead,our
between
feedbacks
multiplicity
- thatis, thetendency
of theeconomy
stratification"
for
analysisrelieson "endogenous
with
one
another.
characteristics
to
interact
stratiwith
similar
only
Endogenous
agents
whichin ourmodelmeansthattherichandpoorlivein separatecommunities,
fication,
inlong-run
behavior
difagentstoexperience
byallowingdifferent
multiplicity
produces
environments.3
interaction
ferent
a wayofunderstanding
howagentsevolvetoward
different
Ouranalysisalso provides
The
literature
coordination
has generfailure/multiple
equilibrium
long-run
equilibria.4
theexistence
ofmultiple
ondemonstrating
steadystatesinan economy,
allyconcentrated
come
about.
how
different
thisliterature
without
Further,
equilibriaactually
explaining
Our
all
end
at
the
results
illustrate
that
same
how
assumes
equilibrium.
agents up
generally
can emergeamonggroupsofagentsas a consequence
ofthe
distinct
equilibria
long-run
oftheeconomy.
samplepathrealization
particular
income.Section3 characterizes
a baseline
modeloffamily
Section2 ofthepaperdescribes
is private.
Section4
whenallhuman
incomedistribution
theequilibrium
capitalinvestment
is determined
at an
whenhumancapitalinvestment
equilibrium
analyzestheaggregate
the
with
of
behavior
5
characterizes
the
level.
Section
economy endogenous
economy-wide
areprovided
conditions
Somesufficient
andlocalpublicfinance.
formation
neighborhood
an
in
Section6
to
and
forpoverty,
emerge
economy.
inequality
prosperity, persistent
ofinner-city
Section7
ofurbancenters
andtheemergence
thebreakup
considers
poverty.
arefoundin a technical
andconclusions.Proofsforall theorems
containsa summary
theauthor
thatis availablefrom
uponrequest.
appendix
2, A ModelofEvolvingFamilies
Structure
2.1. Population
indexed
set/ offamilies,
ofa finite
consists
Thepopulation
byi. Familyi\t is composed
offamily
incomes
Thevector
oftheagenti , t - 1,whois bornatt - 1 andhisoffspring.
as Ytandis themainobjectofourstudy.
. . . Yij], is denoted
atr,{Y\Jy
whenyoungandhasonechildand
education
livetwoperiods.Eachagentreceives
Agents
indexed
liveinneighborhoods
whenold.Families
works
occupying
byd. Thesetoffamilies
intheneighborhood.
offamilies
number
denotes
the
t
d
at
is
Nd,t',#(NdJ)
neighborhood
offamilies.
is atleastas largeas thenumber
ofneighborhoods
Thenumber
2.2. Preferences
whenold,C/.,,andthe
is determined
byconsumption
£/,,/-i
Agenti, t - l's totalutility
arebasedon 3,, the
as
an
his
of
income
Expectations
offspring adult,YiJ+\.5
expected
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All use subject to JSTOR Terms and Conditions
78
DURLAUF
oftheeconomy
uptot:
history
Uu-i = E(u(Cu) + v(Yu+i) | 3,).
(1)
andconcaveandobey
Thefunctions
«(•) andu(-) arecontinuous,
increasing,
u(oo) = v(oo) = oo;
w'(oo)= i/(oo) = 0.
(2)
2.3. Production
Technology
laborendowment
Eacholdagenthasa fixed
L, whichisappliedtooneofa setofoccupations
the
indexed
k
denotes
of
Ytisdetermined
occupation agenti, t- 1. Aggregate
output
by ; Oitt
the
amount
of
to
each
a
function
of
labor
devoted
linear
L*t/,
by
occupation,
r,= f>*i*.,.
(3)
All workers
arepaidtheirmarginal
so thatif Oitt= fc,thenYitt= wkL. The
product,
thestatespaceforfamily
/,f's income.
sequenceY = w\L,Y = wiL, . . . thusdefines
as experiencing
incomeis lessthanorequalto Yp.
A family
is defined
ifa parent's
poverty
2.4. EducationConstraints
on Occupation
is determined
Agenti, t - Vs occupation
bytwofactors:theamountofhumancapital
shockff/.Formally,
inhimduring
anda humancapitalproductivity
invested
youth,
///,f-i,
< er+\thenOlt = O(r, frj).
3 numbers
e\. . . e^ suchthatifer< ///,r_i
(4)
minimum
threshold
fordifferent
levelsofhumancapital
The er's represent
requirements
aretreated
Thesethresholds
sincehumancapitalinvestment
investment.
as discrete
helps
choiceamonga discrete
itispossibletonordetermine
anindividual's
classofoccupations;
relatethethresholds
tooccupational
malizeOit so that0(r, 0) = r,whichwoulddirectly
training.
in both
on O(-, •). First,thefunction
is nondecreasing
We place severalrestrictions
for
fixed
Second,
arguments.
£,,
er+\- er
exhibits
noninis nonincreasing
in r, so thetransformation
ofhumancapitalintooutput
ofanequilibrium
returns
toscale;thisrestriction
theexistence
is usedinproving
creasing
a neighlevelas itensuresthatpreferences
overtaxrateswithin
education
neighborhood
has
bounded
aresingle-peaked.
borhood
Third,0(r, £,,,)- E(O(r, f,-t/)
| %t-\) uniformly
ofr),which
from
allowssufficient
education
toshielda person
support
(independent
poverty.
willnot
shocks
forfixedr, (5) is nondecreasing
in ft,,,so favorable
Fourth,
productivity
reducetherateofreturn
toeducation,
whichhelpsensurethatrichneighborhoods
spend
moreoneducation
thanpoorones.
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A THEORYOF PERSISTENTINCOMEINEQUALITY
79
Shocks
2.5. Properties
ofProductivity
ofa productivity
theempirical
shockdependson />,<*,„
Theconditional
probability
probincomes
over
for
families
in
d
at
t.
distribution
Thisdependence
neighborhood
ability
thatcreatea feedback
froma
is designedto capturepeergroupandrolemodeleffects
to
the
of
its
economic
outcomes
children:
composition
neighborhood's
Pwbttu+i I 3i) = Pn>btiu+\I hdj) ifi e NdJ.1
(6)
shocksis increased
shifts
ina neighoffavorable
Theprobability
byrightward
productivity
incomedistribution.
of
borhood'sempirical
LettingF\ and F2 denotetworealizations
IfFi(a)<F2(a)Vat
thenf^ProbUu+i \FY)d( < f^ Pn*(ft.,+i\F2)d^Wb.
K}
cancompensate
onegroupoffamilies
another
existswhereby
No mechanism
groupforits
a first
form
ofmarket
incomona neighborhood's
effect
shocks,introducing
productivity
pleteness.
shocksconsistof neighborhoodto productivity
Finally,we assumethatinnovations
and
components:
specific family-specific
&.,+,- £(£.,+i I Fr.dj)= Kf.,+i+ ifc.,+i.
(8)
whereasinnovations
of one another,
are independent
acrossneighborhoods
Innovations
theeffects
ofa range
whichcancapture
within
a neighborhood
correlate,
maypositively
is madeforsimplicity
as it
Thisassumption
influences.
ofcommonwithin-neighborhood
ofeachneighborhood
in
aheadtransition
allowsus tostudytheone-generation
dynamics
isolation.
2.6. BudgetConstraint
Cir andtaxes7)-,,:
consumption
Agent1,f - 1 divideshisincomebetween
Yu = Cij + 7},,
(9)
2.7. HumanCapitalFormation
Thecostfuncreceivethesamehuman
ina neighborhood
Allchildren
capitalinvestment.7
toachieve
therequired
tionNHdj measures
capita
expenditures
per
within-neighborhood
a givenlevelofpercapitahuman
capitalind att:
NHdJ= NH(Hu,#(NdJ)tFYMJ).
(10)
on the
ofneighborhood
theeffect
in N //(•,-,•)reflects
The thirdargument
composition
areassumedtoreducethetotalandmarginal
shifts
ofFY,d,t
education
process;rightward
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80
DURLAUF
WhileNH is increasing
costsofeducation.8
andconvexwithrespectto //,,,,
thefunction
in neighborhood
forlargeneighborhoods.
is decreasing
an incentive
These
size,creating
whichwillturn
outtoensure
thatconaveragecostsareassumedtobe bounded,
decreasing
willemergegivensomerealizations
ditions
existunderwhichmorethanoneneighborhood
incomedistribution.
oftheempirical
Forall Hit and Fy,dj,thereexistsa levelM < oo
suchthat
0 < NH(Hiti,1, FYjj)-NH(HUf#(n,
Fy.d.t)< M.
(11)
2.8. Structure
ofTaxes
choosesa proportional
taxrate
Givenrealizedfamilyincomesat f,each neighborhood
are
Communities
cannot
whose
to
finance
education.
borrow
in
€
used
Xd,t (0, 1)
proceeds
whichis a secondformofmarket
ordertofinance
education,
incompleteness:
wherei e Nd,t.
Tit= TdJYitt
(12)
of Nd,t
A givenXd,t
is an equilibrium
rateforNd,tifat leastone halfofthemembers
therateto anyfixedalternative.
The highest
suchrateis alwayschosen.(We also
prefer
infavoroftheonewith
assumethatfamilies
resolveindifference
betweenneighborhoods
eachcommunity
human
lossofgenerality,
thehighest
Without
expected
capitalformation.)
is required
tochoosea highenoughtaxrateso thatHit > e\.
is deThe modelwillbe completeoncethedistribution
of familiesby neighborhood
on
termined.
We nowconsidertheimplications
of different
assumptions neighborhood
structure
ofincomeinequality.
forthedynamics
3. EquilibriumwithFamily-Specific
FeedbackEffects
Themodel,essentially
a modelofprivate
humancapitalformation.
Thissectiondescribes
arenocross-family
toLoury(1981),explores
thedynamics
ofincomewhenthere
equivalent
effects.
is a member
1. Humancapitalformation
is a privategood: Eachfamily
of
Assumption
a separateneighborhood.
thatthereexistequilibrium
human
itis straightforward
Giventhisassumption,
toverify
and
for
each
family.
sequences
capital consumption
Theorem1. Existenceand probability
structureof equilibriumin economywith
feedbackeffects:Forall familiesi, thereexistsa sequenceof income,
family-specific
andhumancapitallevelssuchthateacholdagentmaximizes
expected
utility
consumption
ineachperiod Eachfamily's
incomeobeys
| 3,) = Prob(YiJ+l\Yu).
Prob(Yitt+l
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(13)
A THEORY OF PERSISTENT INCOME INEQUALITY
81
incomeinthismodelwilldependontheproperties
ofrelative
oftheimplied
Thebehavior
income.In particular,
theevolution
offamily
Theorem
2
Markovchainthatcharacterizes
can
underwhichlong-run
in
this
theconditions
illustrates
inequality emerge
economy.
incomeinequality:Fortheprocess
forpresenceofpermanent
Theorem2. Conditions
withYnforanyn greater
individual
communicates
income,
family
ifYm
characterizing
i.e.
incomeinequality
is notpermanent,
thatanym,theneither
> 0) = 0
Prob(Yu+s- Yjj+s > OWs> 0 | Yu - YJtt
(14)
orforanyfinitef,
Theset[Y i , . . . Yf} is transient.9
( 15)
incomethan
of thetheorem
one familyto alwayshavegreater
The conditions
permit
withprobability
1. Permanent
incomesbecomeinfinite
butonlyifbothfamily
another,
arelowincome-absorbing
states
cantherefore
occuronlyifthere
duetopoverty
inequality
statesare
income.The low income-absorbing
fortheMarkovprocessdescribing
family
humancapitalin theiroffspring
to allow
cannotinvestsufficient
thosein whichfamilies
toeverattain
theirdescendants
high-income
occupations.
FeedbackEffects
4. EquilibriumwithEconomywide
whenall families
oftheeconomy
theevolution
Wenextconsider
occupya common
neigheducation
studied
of
the
an
extreme
version
model
The
borhood.
economy
public
represents
ofthissectionis thatallchildren
(1992).Thekeyfeature
expebyGlommandRavikumar
feedback
effects.
riencethesameintertemporal
is an economywide
2. Humancapitalformation
good: Allfamiliesare
Assumption
N at all timest.
members
neighborhood
ofa common
ofa sequence
totheexistence
isequivalent
inthiseconomy
ofanequilibrium
Theexistence
taxratemaynotexistdueto
an equilibrium
taxrates.As is wellknown,
ofequilibrium
with
aresingle-peaked
thatpreferences
Itis easytoestablish
behavior.
nontransitive
voting
To see this,
whichrulesoutnonexistence
problems.
respecttotaxratesinthiseconomy,
tothe
thechoiceofthetaxratethatagenti, t - 1 wouldmakeifhecoulddictate
consider
functions
and
restrictions
on
the
Our
entirecommunity.
jointlyimply
utility production
overr valuesbelowanindividual's
mustbemonotonically
thatindividual
increasing
utility
there
must
overr valuesabovethatlevel.Therefore,
taxrateanddecreasing
mostpreferred
wouldbe
existat leastonetaxrateineachperiodsuchthatat leasthalfofthepopulation
opposedtoanychangeintherate,whichmeansthatthereexistsat leastoneequilibrium
ofanequilibrium
taxrate.Theexistence
impliestheexistence
sequenceoftaxratesinturn
andhuman
stochastic
ofanequilibrium
income,
capital.
consumption,
processforfamily
of equilibriumwitheconomywide
structure
Theorem3. Existenceand probability
and
feedbackeffects:Thereexistsa jointstochastic
consumption,
processoverincome,
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82
DURLAUF
humancapitallevelsforallfamiliesat all dateswithtaxesdetermined
bymajority
voting
incomeobeys
eachperiod Eachfamily's
#('))-10
PrvbWj+x| 3,) = Prob(Yu+x| FY%N,u
(16)
cannotgenerate
incomeinequality,
unliketheeconomy
Thiseconomy
anyintertemporal
followsimmediately
fromtheelimination
ofanydifferences
in Section3. Thedifference
distributions
shockprobability
acrossthe
in thehumancapitalformation
andproductivity
howforcesin
young.ThemodelsinSections3 and4 thusrepresent
polarcasesillustrating
canaffect
ofneighborhoods
thataffect
thedegreeofincomestratification
intheeconomy
allow
different
income
classes
income
Homogeneous
neighborhoods
tertemporal
equality.
from
oneanother.
toevolveseparately
neighborhoods
promote
intertempoHeterogeneous
ofhuman
ralequality
thecontemporaneous
capitalas wellas through
through
equalization
Thewayin whichneighborcommonpeergroupandrolemodeleffects
acrosschildren.
thedynamics
therefore
in understanding
of
hoodsaredetermined
becomesa keyfeature
andthepersistence
andinequality.
theincomedistribution
ofpoverty
and
5. EquilibriumwithEndogenousNeighborhood
Formation
Local FeedbackEffects
intoneighborhoods.
We nowconsideran economywherefamiliesorganizethemselves
of thedistribution
The dynamics
of thiseconomydependcritically
on theevolution
of
of several
familiesbyneighborhood.
Thisevolution
in turndependson theinteraction
factors.Decreasing
income
ofhumancapitalpromote
percapitacostsin theproduction
within
taxassumption
On theotherhand,theproportional
neighborhoods.
heterogeneity
meansthatheterogeneity
alsoleadstoredistribution
from
richtopoor,creating
incentives
for
thewealthy
toformisolatedcommunities.
Therichmayalsochoosetoisolatethemselves
for
fromthepoorinordertoinducea morefavorable
distribution
conditional
probability
their
children's
wish
to
avoid
a
shocks.
may
high
productivity
Alternatively,
poorminority
taxratespreferred
inresponse
toforcesthat
families.
bymoreaffluent
emerges
Inequality
acttopromote
thestratification
ofcommunities
byincome.
5.1. Characterization
ofEquilibrium
areallowedto
Wefirst
define
a setofrulesfortheformation
Communities
ofneighborhoods.
actas a typeofzoning
erectincomerequirements
forentry.
incomerequirements
Minimum
maximum
income
thatpermits
restriction
tokeepoutpoorfamilies;
wealthy
neighborhoods
the
allow
one
would
want
to
live
in
if
choose.
isolation
Ideally,
requirements
poor
they
a modelinwhichhousepricedifferences
offamilies
couldacttoproducea configuration
intodifferent
wishestomove
defined
neighborhoods
byincomeclassessuchthatnofamily
toa different
existanequilibrium
with
theredoesnotnecessarily
However,
neighborhood.
thesecharacteristics.
Nonexistence
theremaynotexista
mayoccurfortworeasons.First,
setofhousepricesthatsupport
desiredsegregation
In thiscase,therichwill
bytherich.11
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A THEORYOF PERSISTENTINCOMEINEQUALITY
83
newneighborhoods
inresponseto theentryofthepoor,precludinganystable
keepforming
at
a
do not,in
neighborhoodconfiguration pointin time. Second, house pricedifferences
can
a
mechanism
which
avoid
with
families
rich
families.
isolation,provide
by
living
poor
no stableneighborhood
will
exist
if
thepoor
In theabsence of sucha barrier,
configuration
wish to be isolated. This can occur,forexample,whenthefixed-costcomponentof some
educationalthreshold
er is highso thattherichwantto live withthepoorin ordertoexpand
thetax base and reduceper capitacosts,whereasthepoor wantlow taxesdue to thehigh
marginalutilityof consumption.12
of neighborhoods,
familiesare
In orderto ensuretheexistenceof a stableconfiguration
assumed to sequentiallychoose neighborhoodsduringa timeperiod. Once a
therefore
thefamilymay notmove again.13This sequentialstructure
familyentersa neighborhood,
is only one of manypossible ways to ensureexistence;all subsequentanalysisholds for
formation
rulethatallows wealthierfamiliesto isolatethemselvesfrom
anyneighborhood
therestof thepopulation.
Assumption3. Rules forneighborhoodformation: (A) At the beginningof timet all
shocksf,t/are realized. All agentsknowthe realizationsof theseshocksas
productivity
ana*#(Nd,t-\)
well as Fy,d,t-\
for o//neighborhoodsd. (B) Families movesequentially
accordingto someknownorder,each eitherenteringa neighborhoodin whichitis eligible
to move or enteringan emptyneighborhood.No familymoves more thanonce. (C) A
majorityoffamilieswhohave enteredany neighborhoodduringtimet can voteto erecta
incomerequirement
and/ormaximum
minimum
forjoiningtheneighborhoodTheminimum
above
the
lowest
income
be
set
cannot
barrier
amongfamiliesthathave alreadyentered
barriercannotbe setbelow thehighestincomeamong
theneighborhoodand themaximum
in theneighborhood14
familiescurrently
of movesby familiesresolvesthechasingproblem:
Observethatthesequentialstructure
thepoor and richcan each isolate themselveswhen so desired. Different
orderingswill
all
of
families
different
however,
subsequent
by neighborhoods;
configurations
produce
theoremshold foranyorderingof families.Assumption3 leads to Theorem4.
Theorem 4. Existence and probabilitystructureof equilibrium in economy with
endogenous neighborhoodformationand local feedback effects: Thereexistsa joint
humancapital,and neighborhoodmemberstochasticprocess overincome,consumption,
old agentmaximizesexpectedutility.Each
that
at
all
dates
such
each
all
shipfor families
income
obeys
family's
#(Afc.,)).
Prob(Yu+{ | 3,) = Prob(Yu+l \FYtdJ,
(17)
This economycan exhibitcomplex dynamicsover time. Neighborhoodsemergeand
versushomogeforheterogeneous
neighborhoods
disappearas familiestradeoffincentives
Alternative
thelevel ofeconomicstratification.
to determine
neous neighborhoods
specifidifferent
ofourmodelcan produceextremely
formsandparameters
cationsofthefunctional
behavioracross familyincomes. We now describesome
cross-sectionand intertemporal
and persistent
under
which
conditions
sufficient
inequalitycan emerge
prosperity,
poverty,
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84
DURLAUF
whenall familiesare initially
of theeconomy,
alongdifferent
samplepathrealizations
to themodel,theseconditions
effects
areso fundamental
nonpoor.Sinceneighborhood
outcomes
to neighas conditional
areexpressed
statements
relating
offspring
probability
on the
Theserestrictions
can alwaysbe replacedwithrestrictions
borhoodcharacteristics.
of
the
of
and
Durlauf
model;
(1994) provideexamples specific
Cooper(1992)
primitives
fulfill
thesesufficient
conditions.15
ofeconomies
whoseequilibria
parameterizations
5.2. TheEmergence
ofPersistent
Poverty
howpersistent
can emergeinthismodel,itis necessary
to
In ordertounderstand
poverty
choicesofchildren
bornin
theoccupation
boundtheconditional
probability
characterizing
as
Thisboundmaybe written
uniformly
poorneighborhoods.
= 1 - €.
<YpVi e NdJ\Yitt<YpVi e NdJ,#(NdJ))
Prob(Yitt+x
(18)
when€ = 0, thepoverty
is persistent;
When€ is small,uniform
neighborhood
poverty
In terms
oftheunderlying
is permanent.
stochastic
income,
family
processcharacterizing
withhighincomelevels.Thecase
thismeansthatlowincomelevelsdo notcommunicate
where€= 0 is thestochastic
ofthepoverty
byGalorandZeira
trapderived
generalization
(1993),Benabou(1993),andothers.
oftheprevious
work
themainfeatures
modelcancapture
Theendogenous
neighborhoods
income
distribution
to
the
feedback
of
the
onpoverty
neighborhood
trapsthrough
empirical
restrictions
what
must
to
used
theincomeofoffspring.
be
identify joint
Equation(18) may
ofproductivity
shocks
andthedistribution
existon thelevelofhumancapitalinvestment
is low- thatis,
inpoorchildren
in poorneighborhoods.
First,humancapitalinvestment
suchthat
incomedistribution)
thereexistsa b (whichmaydependontheneighborhood
d choosesHitt< eb.
If Yitt< Yp Vi € NdJ,thenneighborhood
(19)
thatall educational
constraints
inthemodel,whichrequire
Giventheborrowing
expenif
willhold,forexample,
condition
this
ditures
arefinanced
taxes,
bywithin-neighborhood
tothehumancapitalproduction
fromtheneighborhood
incomedistribution
thefeedback
function
function
(equation(4)) is suchthathuman
(equation(10)) and theoccupation
or ifthecostsof
investment
is
unproductive,
bypoorneighborhoodssufficiently
capital
levelsthat
aresufficiently
education
largeatlowconsumption
highandk'(-) is sufficiently
forpoorcommunities.
is notutility
Second,
maximizing
highhumancapitalinvestment
when
shocksmustbe suchthatifthehuman
thedistibution
ofproductivity
capitalreceived
(almostalways)leadstoadultpoverty:
youngis lessthanorequaltoe^ thisbarrier
Prob{fa+i such thatO(b, £,,+,) <pVie
NdJ \Yu < V Vi € NdJ) = 1 €.
(20)
onthedistribution
willholdfor€smallorzeroiftheeffect
ofuniform
Thisequation
poverty
ofproductivity
shocksas captured
inequations(6) and(7) is strong
enough.
mean
statesdoesnot,however,
oflow-income
Theexistence
orabsorbing
near-absorbing
forsubstantial
thatanyfamiliesare actuallytrappedin poverty
lengthsof time,when
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A THEORYOF PERSISTENTINCOMEINEQUALITY
85
of poverty
theeconomystartsoffwithall nonpoorfamilies.The emergence
requires
One
of sometypeofdownward
of
downward
thepossibility
is
mobility. type
mobility
incomeamongfamilies
inAssumption
within
embodied
4, whichstatesthatthemaximum
belowsomethreshold.
candecrease
overtime,
forincomes
Thisassumption
a neighborhood
ofthewealthiest
weakas itis equivalent
torequiring
is extremely
onlythattheoffspring
canhavea lowerincomethanhisparent.
Thealternative
ofrequiring
ina community
family
downward
ofneighborhood
canexperience
thatallfamilies
every
mobility
periodregardless
sincethepoorestfamily
in a
andincomeis,on theotherhand,a verystrong
assumption,
effects
inducedbyitsmoreaffluent
fromthepositivespillover
willbenefit
neighborhood
willexista widerangeofspecifications
ofpreferences
there
Foranyspecification
neighbors.
theassumption
of £,,whichfulfill
distributions
a
of O(-, •) andprobability
byinducing
of
for
the
obtainable
with
a
distribution
occupations
probability
given
largeenoughsupport
level.
education
fordownwardincomemovements:Thereexistsan income
4. Conditions
Assumption
r"
> Yp suchthat
level7
Nd,t,
foranyneighborhood
Prob(maxYt,+1 < max7,, | Thr"hl > maxY<, > F) > 0.
(21)
4 impliesThoerem
5.
Assumption
i, there
3 and4,foranyfamily
exists
Theorem5. Persistent
poverty:Under
Assumptions
andvalueofe inequation(18) such
a setAi ofrealizations
ofYtwithallfamiliesnonpoor
overan arbitrarily
thefamilybecomestrappedinpoverty
that,withpositive
probability,
€
>
tf
a
set
some
there
exists
For
interval.
time
t,
B\withProb(Yt' B\ I Yt€ A\) > 0
long
suchthat
foranyfixedS and£ > 0,
Prob(Yu,+s< YpVj € (1, . . . , S) | Yt>e Bx) > 1 - £.
(22)
thekeytoour
human
Theorem
5 allowsfortemporary
amongall families;
capitalgrowth
is
as wellas thoseon persistent
ofpersistent
resultson thepossibility
inequality
poverty
itselfalongsome
can slowdownandreverse
humancapitalaccumulation
thata family's
oftheeconomy.
samplepathrealizations
5.3. TheEmergence
ofProsperity
If€ = 0 inequation(18), itis
howsomefamilies
We nextconsider
alwaysavoidpoverty.
theonlylong-run
otherwise
canremain
thatsomefamilies
equilibrium
nonpoor,
important
is thatthereexists
in themodelis permanent
amongall families.Onepossibility
poverty
income
incomeexceedsthislevel,family
suchthatifneighborhood
an incomethreshold
either
that
would
move
down.
an
Such
canneversubsequently
negative
require
assumption
families
invest
inchildren
orthatnonpoor
shocksdisappear
amongthenonpoor
productivity
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86
DURLAUF
Theelimination
atsucha ratethatnegative
shocksnevercausedownward
ofany
mobility.
seemsunnatural
evenifpoverty
werea persistent
downward
state,sincearguments
mobility
thatimplythatpositive
shocksamongpoorchildren
areunlikely
do notimplythatwealthy
children
willalwaysdo as wellas theirparents.Further,
an assumption
thatsufficiently
will
invest
families
human
to
eliminate
downward
would
wealthy
enough
capital
mobility
restrict
strongly
preferences.
An alternative
toexplaining
howsomefamilies
approach
escapepoverty
maybe based
on theidea thatnonpoorfamilies
insuretheirchildren
shocks
againstnegative
partially
overtime,generating
sequenceof humancapitalinvestments
bychoosingan increasing
infamily
income.Growth
infamily
attenuates
theprobability
ofa
income,inturn,
growth
everbecoming
thesequenceofshocksnecessary
todrivea family
poorbyrendering
family
less andless likely.Assumption
thelink
intopoverty
5 providesa meansofformalizing
andlong-run
thatsufficiently
between
wealthy
neighborhoods
growth
prosperity
bystating
as thelowestincome
exhibita weakformofexpectedincomegrowth
amongoffspring,
is
to
income
exceed
the
lowest
amongoffspringexpected
amongparents.16
forevolution
5. Conditions
offamilyincomesoutsideofpoverty:(A) For
Assumption
thelowestincomeamongoffspring
exceedsthelowest
uniformly
neighborhoods,
nonpoor
incomeamongparentswithpositive
probability:
f, > Yp Vj e Nd,, #(Nd,)) > 0.
Prob{minY(,+, > minY{, | Yt
(23)
thelowestincome
exceedsthe
(B) Forsufficiently
wealthy
amongoffspring
neighborhoods,
Y
lowestincomeamongparentsin expected
value. Thereexistsa pairofnumbers
r"
> YPand v > 0, suchthat
and v,with7
£(min Ytl+1- minYtt\Yit> Yhresh2^i€ N
#(N ;)) > v.
(24)
canescapepoverty.
5 is sufficient
toprovethatsomefamilies
Assumption
Theorem6. Permanent
of
prosperity:LetA2denotethesetofall possiblerealizations
3 through
5,foranyfamilyi, if
Ytsuchthatallfamiliesarenonpoor.UnderAssumptions
inall periods,
outofpoverty
thefamilyremains
Yt€ A2,thenwithpositive
probability
>YpWs>0\YteA2)>0.
Prob(YLt+s
(25)
thatthefamily
Theconditional
outofpoverty
remains
forallfuture
probability
periodscan
closetoone.Forsomet' > ttthereexistsa set#2wif/i
becomearbitrarily
Prob(Yt'€ B2 \
Yt€ A2) > 0 suchthat
foranyfixedf > 0,
Prob(Yis+s>7pVs>0\Yf
e B2) > 1 - £.
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(26)
A THEORYOF PERSISTENTINCOMEINEQUALITY
87
andtheEmergence
5.4. EndogenousStratification
ofPersistent
Inequality
forverydifferent
5 and 6 providegeneralconditions
income
Theorems
typesoffamily
realizations
of
the
different
Neither
to
thebehavior developalong
economy.
samplepath
as
discussed
the
of
oremrequiresmultiple
below, probability
neighborhoods,
although
or
the
absence
is
affected
of
or
ecoby
presence
poverty prosperitystrongly
observing
arenecessary,
ofneighborhoods.
for
nomicstratification
however,
neighborhoods
Multiple
a neighborhood
Sincefamilieswithin
of inequality.
common
theemergence
experience
incomedifferences,
thateliminate
influences
persistent
expected
intertemporal
inequality
tocross-section
income
thatdistinct
neighborhoods
emergeovertimeinresponse
requires
inturn,
willdependon thewayin
ofmultiple
Theemergence
neighborhoods,
inequality.
offagainst
theimplicit
aretraded
costsofeducation
redistribution
whichdecreasing
average
In
willalways
in
thatoccurs heterogeneous
neighborhoods
neighborhoods.fact,multiple
rich
and
as stated
formwhenthereis a largeenoughgapbetween
relatively
poorfamilies,
7.
inTheorem
an income
exists
ForanyincomelevelY there
stratification:
Theorem7. Endogenous
lg
>
<
Y
or
incomes
either
have
all
that
such
levelY
Yit
Y(t Y °w,no
if families
'
withincomeYitt> Y
anda family
willformthatcontains
botha family
neighborhood
withincomeYjj < Y
suchthattheeconomy
incomes
offamily
there
existconfigurations
Thisresult
impliesthat
8.
as statedinTheorem
bothpersistently
exhibits
families,
poorandnonpoor
3 through
incomeinequality:UnderAssumptions
Theorem8. Persistent
5,foranypair
all
with
a
set
there
exists
i
and
realizations
Yt
families
nonpoor
of
A3
of
j,
offamilies
theeconomy
evolves
anda valueofe inequation
(18) suchthat,withpositive
probability,
whereastheother
so thatonefamilyexhibits
persistent
familyexhibits
persistent
poverty
a set#3with
Forsomet' > t,thereexists
overan arbitrarily
longtimeinterval.
prosperity
>
and
>
S
€ #3I Yt€ A3) 0 suchthat
£ 0,
foranyfixed
Prob(Ytf
Prob(Yij.+5>YP>
€ (1, . . . , 5) | Yj € B3) > 1 - ? .
Yjtt.+sVs
(27)
betweenan economy'scross-section
7 and 8 providea directrelationship
Theorems
economicstratification,
and
the
of
incomedistribution
By inducing
dynamics inequality.
andthereby
for
environments
can
alter
the
interaction
cross-section
offspring
inequality
a
As
a
different
income
on
families
result,
sufficiently
large
degree
very
trajectories.
place
Since
overmanygenerations.
topersist
ata pointintimecancauseinequality
ofinequality
with
thefunction
onecanalwaysparameterize
O(r,fa) so thatthesetA3canbe obtained
incomesabovethepoverty
withidentical
evenifall families
start
line,
probability
positive
of
canemergeas a consequence
meansthatlong-run
stratification
inequality
endogenous
initial
conditions.
of
a
wide
oftheeconomy
thesamplepathrealization
range
Finally,
given
families
occursthrough
forindividual
ofpersistent
thatsincetheemergence
observe
poverty
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88
DURLAUF
theemergenceof uniformly
themodel also illustrateshow ghettoscan
poor communities,
endogenouslydevelopfromnonpoorinitialconditions.
The privatehumancapitalformation
economyof Section 3 and thesingleneighborhood
model of Section4 are special cases of theendogenousneighborhoodsmodel. When the
effectsofincreasedneighborhood
size on requiredpercapitaexpenditure
fora fixedhuman
to
a community
are
small
there
no
incentive
for
share
level
is
any family
enough,
capital
witha poorerfamily,whichmeanseach familymustoccupya distinctneighborhood.The
generalendogenousneighborhoodsmodel, however,possesses farricherdynamicsthan
the privateeducationspecial case. For example,unlikethe privateeducationmodel, the
modelpredictsthatthepersistenceofpovertywilldiffer
generalendogenousneighborhoods
poorneighborhood.This difference
accordingto whethera familyis locatedin a uniformly
is an implicationof boththe empiricalwork on neighborhoodeffectsas well as of the
evidence of Card and Krueger(1992) thatonce school qualityis controlledfor,thereis
littlerelationbetweenparentaleducationand income and the rate of returnto years of
schooling.In addition,theemergenceof uniformly
poor neighborhoodsitselfdependson
fortheentireeconomy,inducinga complexsequence
thecross-sectionincomedistribution
of interactions
betweenfamiliesovertime.
whensufficiently
highfixedhumancapital productioncosts and mobility
Alternatively,
costs are present,one can similarlyshow thattheendogenousneighborhoodsmodel will
collapse to thesingle neighborhoodmodel. The abilityof therichto isolate themselves
fromthepoor is essentialnotonlyin allowinginequality,butin increasingtheprobability
of observingsome familiesexperiencing
poverty.To see this,suppose thatall productivity
so thatwhenthe numberof familiesis large,it is likelyat
innovationsare idiosyncratic,
each pointin timethata nonnegligibleset of familiesexistswho have experiencedlarge
negativeinnovations.The childrenofthesefamilieswillexperiencelowerincomefeedback
effectsonlyiftheyareisolatedfromotherchildren.This allows a situationtoarise,forsome
ofthemodel,wherethelaw oflargenumbersimpliesthatwhenall familieslive
specifications
in thesame neighborhood,
thefeedbackeconomywide neighborhoodincomedistribution
makespovertyamongoffspring
whereaswhentherichisolatethemselvesfromthe
unlikely,
from
the
makes povertylikelyfor
the
feedback
lower
tail
of
theincomedistribution
poor,
theoffspring
of theless affluent.Put differently,
by destroying
endogenousstratification,
theimplicitinsurancethatexistsbetweenfamilieslocated in heterogeneouscommunities,
can increasetheprobability
thatan individualexperiencespovertybothas a childand as an
adult.
can lead to a very
The analysisof thissectionhas shownhow endogenousstratification
can coexistforlong
strongformofinequalityinthesensethatpoorandwealthycommunities
periodsoftime.The modelis also capable ofproducinglongperiodsof inequalitybetween
nonpoorneighborhoods.To see this,observe thatrightwardshiftsin a neighborhood's
theeffectsembodiedin equations(5) and (10), will
empiricalincomedistribution,
through
increasethemarginalproductof each level of per capita educationalexpenditure.Unless
this increaseis offsetby enough concavityin the functionthatconvertseducationinto
occupations,wealthycommunitieswill, fora wide range of preferences,choose higher
averageincomegrowthratesthanless wealthycommunities,
causingrelativeinequalityto
growovertime.
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A THEORYOF PERSISTENTINCOMEINEQUALITY
89
6. Urbanand SuburbanInteractions
we examinea variant
oftheendogenous
modeltosee how
In thissection,
neighborhoods
canemerge
inthecontext
ofurbancommunities.
ofthebreakup
incomeinequality
Much
ontheurbanpoor(seeWilson,1987)hasemphasized
thatthisgroupis,to
oftheliterature
members
ofheterogeneous
urbanneighborhoods
a largeextent,
composedoftheresidual
members
havedeparted
a number
forsuburbs.Further,
of authors
wherethewealthier
citiesis welldescribed
ofmanyAmerican
as a
havearguedthatthehistorical
experience
withdrawal
ofwealthy
urbancenters
inresponse
tosmallchanges
whitesfrom
precipitous
a phenomenon
andracialcomposition
ofneighborhoods,
known
as tipping
intheeconomic
Thesefeatures
can be captured
andreferences).
(see Schelling,1971,fora discussion
in favorof
thedecisionoffamilies
to leavea heterogeneous
citycommunity
bystudying
neighborhoods.
homogeneous
underwhichwealthier
conditions
WealterthemodelinSection5 inordertocharacterize
willdisperseto suburbs.Assumption
urbanpopulation
6
of a heterogeneous
members
whichwe treatas an urban
common
forall families,
an original
introduces
neighborhood
modelas itis opento
theendogenous
center.The urbancenter
neighborhoods
augments
ofincome.
all families
regardless
aremembers
ofurbancenter:(A)Allfamilies
6. Properties
ofneighborhood
Assumption
totherulesofAssumption
3. However,
areformed
1 attime0. (B) Neighborhoods
according
minimum
1
or
maximum
income
nofamilymaybe excluded
fromneighborhoodthrough
requirements.
redistribution
fromrichto poor
whenproportional
taxescausetheimplicit
As before,
have
an
incentive
to leave and
in theurbancenterto be largeenough,wealthy
agents
If thetaxbase of theinitially
formtheirownneighborhoods.
heterogeneous
neighbortheurbancentercan becomea poverty
hooddeteriorates
trap. Theorem9
sufficiently,
cancausetheurbancenter
to
incomedistribution
howtherealizedcross-section
illustrates
breakup.
model: Supposethat
Theorem9. Propertiesof urbanand suburbancommunities
3 through
1. GivenAssumptions
all familiesi,t - 1 inhabit
6, thereexist
neighborhood
will
be
center
urban
then
the
such
that
and
e
setsA4
(A) ifYt A4,
As
preserved,
nonempty
4 through
8 all
and (B) ifYt € As, thentheurbancenterwillbreakup at t. Theorems
hold.
onthesamplepathrealization
willdepend
ofa residual
Theemergence
poorneighborhood
toremain
the
decision
of
nature
the
zero/one
oftheincomedistribution;
byeachfamily
given
canleadtolarge
incomedistribution
inthecross-section
smallchanges
intheurbancenter,
ofthefactthatsome
ofN\tt.
Thisis an immediate
implication
changesinthecomposition
sincetheunionofthesesetsis thespaceof
ofA4andA5mustbe"near"eachother
elements
9 doesnotexclusively
all possibleincomes.Unlikemosttipping
models,Theorem
relyon
outofanurbancenter,
oflessaffluent
families
todrivethewealthy
thepercentage
although
intheurbancenter.
toremain
oftheaffluent
thewillingness
affects
thispercentage
strongly
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DURLAUF
90
7. Summaryand Conclusions
ofincomeandthepossibleemergence
theevolution
ofthedistribution
Thispaperdescribes
andinwhichtheempirical
inwhicheducation
is locallyfinanced
inaneconomy
ofpoverty
ina community
statusofoffspring.
affects
theeventual
incomedistribution
occupational
from
the
rest
of theeconomy
an
themselves
have
incentive
to
isolate
families
Wealthy
at thelowestcost.
in orderto providethehighestlevelof educationfortheirchildren
on theotherhand,createincentives
Decreasing
averagecostsinhumancapitalformation,
in incomeacrossagents.Whentheforces
toemergewithheterogeneity
forcommunities
oftheeconomywill
arestrong
stratification
endogenous
enough,
leadingtohomogeneity
Thisisolation
can
tobeisolated
therestofthepopulation.
from
occur,causingpoorfamilies
as theyareunabletojointly
orpermanent
inducepersistent
poverty
amongsomefamilies
toescapefromlow-paying
in theirchildren
humancapitalinvestment
sufficient
generate
occupations.
extensions
existto thecurrent
Severalinteresting
paper. First,thecriticalroleof the
makesit important
in inducing
to
incomedistribution
cross-section
inequality
persistent
itwouldbe valuableto
howthisdistribution
is determined.
understand
better
Specifically,
fordifferent
cross-section
whenO (r,fl>r)
is likelytoproduce
substantial
inequality
identify
willalsopermit
onetocompute
initialconditions.
ofthisfunction
aspects
analysis
Explicit
for
ofthemodelsuchas themeanfirst
ofthetransition
passagetimeoutofpoverty
dynamics
howa societyshouldstructure
families.
individual
Second,themodelmaybe usedtostudy
criterion.17
One
incomeredistribution
policiesin orderto maximizesomesocialwelfare
thatact to completethosemissing
schemesas mechanisms
can thinkof redistribution
Thedynamic
structure
of
manifest
whoseeffects
themselves
markets
through
inequality.
embedded
in
tradeoffs
different
that
the
and
themodel,however,
suggests
equity efficiency
of equalization
ofeducational
policieswillbe verycomplex.Forexample,theefficacy
ontheinteraction
willdependcritically
ineliminating
andinequality
poverty
expenditures
onsomeofthesequestions
withvarious
Research
effects.
oftheseexpenditures
sociological
underway.
is currently
Acknowledgments
I thankKenneth
MordecaiKurz,
RussellCooper,SuzanneCooper,PaulJohnson,
Arrow,
MorganKelly,SusanNelson,UdayRajan,JeroenSwinkels,Joseph
BoyanJovanovic,
reviewer
atBostonUniversity,
as wellas seminar
Brown,
Zeira,ananonymous
participants
FederalReserveBoardof
FRB
Dartmouth,
Institute, Minneapolis,
EuropeanUniversity
HebrewUniversity
atJerusalem,
Southern
CalRochester,
Governors,
Iowa,MIT,Oregon,
formanyhelpful
UC Davis,UCLA, Western
ifornia,
Stanford,
Ontario,andWisconsin
comments
on an earlierversionofthispaper.TheNationalScienceFoundation
hasprovidedfinancial
support.
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A THEORY OF PERSISTENT INCOME INEQUALITY
91
Notes
1. Solon (1992) and Zimmerman(1992) show thatthiscorrelationis actuallycloser to .45 whenone controlsfor
varioustypesof measurementerror.
economic statushas been shown in many studies. Card and Krueger
2. The role of education in determining
affectfutureearnings.Kozol ( 199 1)
( 1992) is a recentanalysisthatshowseducationalqualitycan significantly
documentshow inadequatelocal fundinghas affectededucationqualityin poor school districtsthroughout
the
UnitedStates. Per capita studentexpendituredifferenceson the orderof 75 percentto 100 percentbetween
the UnitedStates. Local financeaccounts
adjacent innercityand suburbanschools are commonthroughout
forapproximately50 percentof all publicexpenditureson elementaryand secondaryschools.
3. In orderforthemodel to exhibituniformpositivefeedbacks,itwould be necessarythata givenfamily'sutility
be increasingin theincome of otherfamilies.This will generallynothold,in two senses. First familiesmay
respondto higherincome by segregatingthemselvesfrompoorerfamilies,causing those familiesto lose any
positivespillovers. Second, an increasein income among some familieswithina neighborhoodwill change
theirpreferencesover taxes, whichcould change the equilibriumtax ratein such a way as to hurttheother
familiesin the neighborhood.
4. The model can exhibitmultiplelong-runequilibriaforfamiliesin the sense thata family'saverage income
dependson the sample pathrealizationfortheeconomy.
5. We abstractfrompreferencesover neighborhoodlocationand size. Incorporationof these factorswill make
heterogeneouscommunitiesmorelikelybutwill have no qualitativeeffecton ourresults.We also abstractfrom
income,whichis an implicationof
anyeffectof thelevel of parentincomeon themarginalutilityof offspring
is positive,thelikelihood
some ofthesociological workon the"cultureofpoverty."Whenthiscross-derivative
withwhichthe incomesof therichdivergefromthoseof thepoor is increased.
6. Throughout,Prob(x \y) denotestheconditionalprobabilityof x giveny.
ofthiseffectwould have no qualitative
investment.Introduction
rulesoutanyfamily-specific
7. This formulation
effecton our resultsconceringpersistentinequalitybetweenfamilieslocated in different
neighborhoodsbut
would providea way of explainingwithin-neighborhood
inequalitypersistence.
8. Crane ( 199 1) shows thata negativerelationshipexistsbetweenthehighschool dropoutrateand thenumberof
one way in whichcommunityfactorscan affectthe human
illustrating
professionalworkersin a community,
capital producedby a given level of expenditure.This effectis distinctfromthe impactof the community
which relatesto the way in which an individualtransforms
income distributionon individualproductivity,
trainingintoan occupation.
9. Income_stateYm communicateswithstate Yn if thereis positiveprobabilitythata familywithincome Ym
attains Yn in the future. The transientstates of a stochasticprocess are those thatthe process will, with
probabilityone, enteronly a finitenumberof times.
10. #(/) appearsbecause neighborhoodsize affectstheper capita cost of education.
of familypreferencescan yield
11. B6nabou (1993, 1994) and Durlauf(1994) show howcertainparameterizations
economiesin whichequilibriumneighborhoodssupportedby rentalor house pricesdifferenceswill emergein
economiesof thetypeunderstudy.For theUnitedStates,itis commonforwealthycommunitiesto implement
will actuallyexclude the poor, Hamilton (1975)
zoning restrictionsto ensure thathouse price differentials
to
are necessaryforhouse pricedifferences
and Wheaton(1991) provideconditionswherezoningrestrictions
produceeconomic segregation.
12. This problemalso applies when house prices are used to segregatefamilies. Existence resultsderived in
Westhoff
(1977), wheretherichand thepooreach wishto liveapartfromone another,requirestrongadditional
of utilityfunctionsand incomebeyondthosewe impose and are based on
on thejoint distribution
restrictions
a muchdifferent
specificationof theeconomy.
13. Metaphorically,enteringa neighborhoodrequiresone to investa highpercentageof income intoa house that
cannotbe resold exceptat a prohibitively
large loss.
means thatno familycan be votedout of a neighborhoodthatit has alreadyentered.
14. This restriction
15. The complex interactionstructureinducedby the presenceof neighborhoodeffectsand endogenous neighon theconditionalprobabilitiesto technologiesand
borhoodformationrendersthe mappingfromrestrictions
preferencestoo complicatedto characterizeexplicitly,so thecurrentapproachmaximizesgenerality.
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92
DURLAUF
ofgrowth
thattheproductivity
ofincreasesineducation
16.Thepossibility
requires
amongsufficiently
wealthy
suchas
families
obeysa condition
O(r + l,ft,)-O(r,6,,)
>^ y > 1.
to showthatthereexist
concave,itis straightforward
largeandu{) sufficiently
By choosingy sufficiently
in thelowest
lawsfortheproductivity
shocksamongoffspring
whichproduceexpectedgrowth
probability
inthecontext
relative
toparents.
incomeofoffspring
of
(JonesandManuelli,1990,deriveananalogousresult
toexperience
a growth
modelwithout
shocks.)Inorderforheterogeneous
productivity
neighborhoods
growth
itis necessary
restrict
so thatrichfamilies
inthelowestincomeacrossfamilies,
tofurther
do not
preferences
thatthelowestincomeamongoffspring
suchlowtaxes(inordertoavoidredistribution)
failstogrow.
prefer
inC, whichwouldmeanthat
Onewaytodo thisis torestrict
theutility
function
so thatCu'(C) is decreasing
infamily
income.Sincethesepreference
within
a neighborhood,
thepreferred
taxrateis increasing
restrictions
athighconsumption
incomelevels,theyarecompatible
evaluated
andoffspring
withthe
applyonlytoutility
forequation(18) tohold.
restrictions
necessary
5 maybe replacedwitha condition
thatstatesthataverageor medianincomegrowsacross
Assumption
withincomesabovesomethreshold.
willhold,forexample,
inneighborhoods
Sucha condition
generations
aremodified
so thatthemarginal
incomeis increasing
inparent
ifpreferences
ofoffspring
incometo
utility
families
wheretheiroffspring
willexperience
suchan extent
thatwealthy
onlyenterneighborhoods
positive
expectedincomegrowth.
fortherichto redistribute
17.Cooper(1992) showshowincentives
to thepoorexistiftheproductivity
ofeach
levelofthesociety.See alsoFernandez
andRogerson
worker
(1992)forananalysis
dependsontheeducation
oftheeffects
ofdifferent
schoolfinancing
rulesonthelevelofeducation
acrosscommunities.
B6nabou(1992)
ofeconomicstratification.
analysisofgrowth
givesan interesting
consequences
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