On the Measurement of Polarization
Author(s): Joan-Maria Esteban and Debraj Ray
Source: Econometrica, Vol. 62, No. 4 (Jul., 1994), pp. 819-851
Published by: The Econometric Society
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Econometrica, Vol. 62, No. 4 (July, 1994), 819-851
ON THE MEASUREMENT OF POLARIZATION
BY JOAN-MARIA ESTEBAN AND DEBRAJ RAY'
Suppose that a population of individuals may be grouped according to some vector of
characteristics into "clusters," such that each cluster is very "similar" in terms of the
attributes of its members, but different clusters have members with very "dissimilar"
attributes. In that case we say that the society is polarized. Our purpose is to study
polarization,and to provide a theory of its measurement.Our contention is that
polarization,as conceptualizedhere, is closelyrelatedto the generationof socialtensions,
to the possibilitiesof revolutionand revolt, and to the existence of social unrest in
general. We take special care to distinguishour theory from the theory of inequality
measurement.We derive measuresof polarizationthat are easily applicableto distributions of characteristicssuch as income and wealth.
KEYWORDS:Clustering,convergence,inequality,income distribution,polarization,so-
cial conflict.
1. INTRODUCTION
THIS PAPER IS CONCERNED WITH THE CONCEPTUALIZATION and
measurement of
polarization.We shall argue that the notion is fundamentallydifferent from
inequality,which has received considerableattention in the literatureon economic developmentand elsewhere.2
Let Y be a set of characteristicsor attributes.An individualwill be described
by an element in this set. The prevailingstate of affairsin a society can then be
captured by a distributionover Y, which describes the proportions of the
populationpossessingattributesin any subset of Y.
Consider a particular distribution on Y. Suppose that the population is
groupedinto significantly-sized"clusters,"such that each cluster is very "similar" in terms of the attributes of its members, but different clusters have
memberswith very "dissimilar"attributes.In that case we would say that the
society is "polarized."Our purpose is to study these intuitivecriteriacarefully,
and to providea theory of measurement.
When introducinga new concept, one should start by showing that it is
differentfrom other standardsummarystatisticsand by persuadingthe reader
1
Research on this project was started when Esteban was visiting the Indian Statistical Institute in
1987. The results reported here were obtained when Ray (then at the I.S.I.) visited the Instituto de
Analisis Economico under the Sabbatical Program of the Ministry of Education, Government of
Spain. We are grateful for funding from the Ministerio de Educacion y Ciencia (Grant No.
PB90-0172) and EC FEDER Contract No. 92 11 07 005, and for supportive facilities at the I.S.I. We
have benefited from long discussions with David Schmeidler, and have also received useful
comments from Anthony Atkinson, Partha Dasgupta, Bhaskar Dutta, Michael Manove, Isaac
Meilijson, Nicholas Stern, and participants of numerous seminars and conferences. The current
version owes much to the comments of Andreu Mas-Colell and two anonymous referees.
2 For theoretical studies, see Kolm (1969), Atkinson (1970), Sen (1973), Dasgupta, Sen, and
Starrett (1973), Fields and Fei (1978), and the numerous references in the excellent survey by Foster
(1985). For empirical studies, the reader may consult Kuznets (1955), Kravis (1960), Adelman and
Morris (1973), Weisskoff (1970), Ahluwalia (1976), and Fields (1980), to mention only a few.
819
820
J.-M. ESTEBAN AND D. RAY
that the phenomenoncapturedby the new concept is indeed relevant.Our task
is not easy for at least three reasons. First, one cannot rely on pre-existing
intuitionor axiomatization.We shall have to introducethe concept and develop
its intuition at the same time. Second, the theory of economic inequality-the
obvious contender to the theory developed here-is based on a long and
venerable tradition in welfare economics. This tradition is by now so deeply
rooted in our way of thinkingabout distributions,that sometimeswe take the
propositionsproven two decades ago as unquestionablyintuitive axioms.3Finally, while we see polarizationas a particularlyrelevantcorrelateof potential
or open social conflict, "mainstream"economics has thus far paid little attention to this last issue.4
We begin with the obviousquestion:why are we interestedin polarization?It
is our contention that the phenomenonof polarizationis closely linked to the
generation of tensions, to the possibilitiesof articulatedrebellion and revolt,
and to the existence of social unrest in general. This is especially true if the
underlyingset of attributesis a variablesuch as income or wealth.A societythat
is dividedinto groups,with substantialintra-grouphomogeneityand inter-group
heterogeneityin, say, incomes,is likelyto exhibitthe featuresmentionedabove.
At the same time, measured inequality in such a society may be low. The reader
unconvinced of this last point may turn at once to Section 2, where the
distinctionbetween polarizationand inequalityis made via a series of examples.
The idea above is not novel at all. Marxwas possiblythe first social scientist
to give a coherent interpretationof society as characterizedby the buildingup
of conflictand its resolution.We are not interestedhere in his specifictheoryof
the developmentof the capitalistsociety, as much as in his view of intergroup
conflict dynamics,and the use of a concept such as polarizationto describe
these dynamics.5Deutsch (1971)'saccountof the Marxiantheoryis particularly
relevantfor our purpose:
"As the struggleproceeds,'the whole societybreaksup moreand moreinto
two hostile camps,two great, directlyantagonisticclasses:bourgeoisieand
proletariat.'The classes polarize, so that they become internallymore
homogeneousand more and more sharplydistinguishedfrom one another
in wealth and power"(Deutsch(1971,p. 44)).
This way of understandingsocial conflicthas been later taken up by contemporarysociologists(e.g., Simmel(1955) and Coser(1956))and politicalscientists
3 We find particularlyrevealingthe questionnaireresultsobtainedby Amiel and Cowell(1992).
The majorityof students with an education in Economics ranked distributionsin a manner
consistentwith the Lorenzordering,while studentswith other backgroundsproducedrankingsthat
violatedthis ordering.
4 There is a strandof recent literaturethat sharesour view that social conflict,and in particular
distributionalconflict,plays a crucialrole in the explanationof macroeconomicphenomena.For
instance,Alesina et al. (1992), Alesina and Rodrik (1991), Benhabiband Rustichini(1991), and
Perssonand Tabellini(1991)focus on the relationshipbetweendistributionalconflictand economic
growth.
5 See also Lenin (1899),Chayanov(1923),and Shanin(1966).
POLARIZATION
821
(e.g. Gurr(1970, 1980) and Tilly (1978)).6In fact, the broadview that a society
split up into two well-definedand separated camps exhibits a high degree of
potential (if not open) social conflict is neither new nor specificallyMarxist.It
has also been shared by such differentpolitical analystsas BenjaminDisraeli,
the nineteenth century Tory prime minister, or the US Kerner Commission,
twentyfive years ago.7
This literaturediffersfrom studies on inequalityin one simpleyet fundamental way:in the formationof categoriesor groups,or "subsocieties,"thepopulation frequency in each category also carries weight.8
The following simple example is illustrative. Consider an initially equal
society of peasant farmers. Suppose, now, that through a process that redistributes a certain fixed amount of agriculturalsurplus, a fraction a of these
peasantsbecome rich kulaks at the expense of the remainder.Given the value
of the surplus, for which values of a would we say that a "high" degree of
differentiationis takingplace? Of course, there is no precise numericalanswer
to this question,but certainlyvalues of a very close to zero or unitywould not
be construedas creatinga sharpdivisionamongthe population.However,note
that under every known measure of inequality, lower values of a would
unambiguouslygive rise to higher inequality.9By effectively neglecting the
population frequencyin each category, inequalitymeasurementdeparts from
the study of differentiation.
Anotherway to see the differenceis to recognizethat the axiomsof inequality
measurement(or equivalently,second-orderstochastic dominance for meannormalizeddistributions)fail to adequatelydistinguishbetween "convergence"
to the global mean and "clustering"around "local means."'0 Consider the
following example, related to the question of economic "convergence"in the
world economy,a topic that has recentlyreceived considerableattention." To
be precise, consider the frequency distributionof national growth rates in
6Marx'scriticR. Dahrendorf(1959)himselfdoes not attackthis view of conflict,but ratherthat
Marxoverlookedthe fact that individualcharacteristicsare multidimensional,so that differences
within and similaritiesbetween social classes restrainedthe polarizationprocess (see Deutsch
(1971)).Indeed, polarizationin some attributesuch as incomemay be counterbalancedby dissimilaritiesin other attributes,an objectioncommonto the theoryof inequality.
Disraelicoined the term"twonations"to describethe Englishsocial structureof his time in his
novel Sybil,publishedin 1845.The KernerCommission(1968)expressedconcernaboutthe ongoing
divisionin the United States into two societies.
8By focussing on this one difference,we do not mean to deny the existence of others. In
particular,it is difficultto remain unimpressedby the richnessof the political and sociological
studies alluded to above. Nevertheless,this very richness opens the door to a possible lack of
sharpnessat the conceptuallevel. The theoryof inequalitycan certainlynot be criticizedon these
grounds.Our objective,then, is to retainthe sharpnessof a formalconstruction,capturingthe new
featuresin the cleanestpossibleway.
inequalitymeasure.
9To be precise,this would happenin the case of everyLorenz-consistent
10Landsbergerand Meilijson(1987)also note (in the contextof risk)that second-orderstochastic
dominanceis not necessarilyassociatedwith transfersof probabilitymass from the center of the
supI?ortof a distributionto its tails.
See, for example,Barro(1991),Barroand Sala-i-Martin(1992),and Mankiw,Romer,andWeil
(1992).
822
J.-M. ESTEBAN AND D. RAY
per capita GDP. Suppose that over time, we were to observethat "low"growth
rates were convergingto their commonmean, while the same is true for "high"
growth rates. In terms of growth rates, then, two "growthpoles" would be
forming-a clear emergenceof North and South. Polarizationin our sense may
well be rising. But every inequality measure defined on growth rates, and
consistentwith the Dalton TransfersPrinciplewould register an unambiguous
decline!12
Indeed, there are a numberof social and economicphenomenafor which the
knowledgeof the degree of clusteringor polarizationcan be more telling than a
measure of inequality.Quite apart from the distributionof income, wealth, or
growth rates, relevant economic examples include labor market segmentation,
conceptsof the dual economyin developingcountries,or distributionof firmsby
size in a given industry.In the broader arena of the social sciences, there are
issues of social class or significantproblemsconcerningracial, religious,tribal,
and nationalisticconflicts,which clearlyhave more to do with the clusteringof
attributesthan with the inequalityof their distributionover the population.
All this suggestsa need to develop a theory of the measurementof polarization. Indeed, the theory of inequalitymeasurementis possibly on far weaker
ground in its claim to independent existence. Given the underpinningsof
inequalitytheory as simplywelfare economicsapplied to a restricteddomain,13
one mightquestionthe value added of this specificattention.A good part of the
literaturehas found its justificationin the assertionthat one should decompose
distributionalquestions from questions of the aggregate product (or GDP).
Even if the social welfarefunctionwere to enjoythis curiousproperty,'4there is
still no need to treat the theory of inequality as separate from welfare economicsin general,distinguishedby propertiesintrinsicto the theoryitself. More
closely related to the subject of this paper is the motivation for studying
inequalityas a phenomenonclosely relatedwith social unrest.The openingpage
of Sen's celebrated book, On Economic Inequality,asserts that "the relation
between inequalityand rebellionis indeed a close one, and it runs both ways."
We do not disagree with this position, but it remains to be seen empirically
whether this connectionis specificallywith inequality,or with features that are
12
See Section2 for a numericalexample.Our preliminarystudiesusingthe WorldBankdata set
reveal that the distributionby frequenciesover the annual growthindices by countriesis multimodal. We comparedthe behaviorof the polarizationmeasures developed in this paper with
inequalityindices applied to the same data. Specifically,we find that while the Gini coefficient
continuallydiminished over the period 1979-86, polarizationfalls too in 1981 but increases
thereafter.We might add that we plan to applythe ideas developedin this studyto the empirical
questionof "internationalconvergence."But that is the subjectof anotherpaper. For interesting
recent researchthat bearson the issue of multimodalityin this context,see Quah(1992).
13 For measuresof relativeinequality,this is the domainover whichthe total sum of incomesis
constant.
14 For studiesin this area, see Sheshinski(1972),Blackorbyand Donaldson(1978),Ebert(1987),
and Dutta and Esteban(1992).
POLARIZATION
823
commonto both polarizationand inequality,15and perhapsmore closely tied to
the formerin cases of disagreement.16
We end our introductorydiscussion with a remark on the basic set of
attributesthat we use for the developmentof our measurementtheory.Ideally,
this set must include all individualattributesthat are relevant (at the social
level) for creatingdifferencesor similaritiesbetween persons, possiblyrelative
to a given announcedsocioeconomicpolicy.'7 But we will simplifythe analysis
enormouslyby assumingthat the attribute can be captured by the values of
some scalar variable. Indeed, we work with the natural logarithmof income
(thoughthis particularinterpretationis formallyquite unnecessary).The idea is
that percentage differences in income are a good proxy for socioeconomic
differencesor similarities.More generally,even if the attributespace is multidimensional,our analysisis undisturbedif the differencesbetween attributescan
be representedby a metric.But such mathematicalsleight of hand only sweeps
a serious dimensionalityissue under the rug. So we do not defend our restriction except on groundsof tractability.We note, moreover,that exactlythe same
criticismsmay be levelled againstthe theory of inequalitymeasurement.18
The rest of the paper is organized as follows. Section 2, which takes the
reader througha series of examples,is designed to make two points. First, we
arguethat there is an analyticaldistinctionbetween polarizationand inequality.
Second, we submit that there is no simple (and acceptablyrich) partial order
(like the Lorenz criterion in inequality theory) that will capture increasing
polarization.Specifically,any measure of polarizationmust be "global" and
"nonlinear,"in a sense that is made precise in Section 2. In Section 3, we
formally develop and study polarization measures. Section 3.1 deduces an
allowable class of measures, obtained by combininga behavioralmodel with
certain intuitive assumptionsor axioms.Theorem 1 reveals that the allowable
class of measures is quite sharp and amenable to empiricaluse. Section 3.2
(Theorem2) studies the maximalelements of polarizationmeasures,and reveals
15After all, there are a numberof situations(see Section 2) where polarizationand inequality
measures agree in their rankings.In this intersectionfalls, for instance, the phenomenonof a
disappearingmiddleclass, a matterof concernto sociologistsand some economistsin the contextof
U.S. society.See, for example,Kostersand Ross (1988),Horriganand Haugen(1988),and Duncan,
Smeeding,and Rodgers(1991).
16 For instance,the empiricalfindingsof Nagel (1974)show that social conflictis low both under
complete equalityand under extremeinequality.Indeed, politicalscientistsworkingon inequality
and conflict, such as Midlarski(1988) and Muller, Seligson, and Fu (1989) have alreadyserved
notice that standardnotionsof inequalitymaybe inadequatefor the studyof conflict.Comingacross
their work after a first draft of this paper was written,we see that their alternativenotions of
patterned inequality and bifurcated inequality seem to be motivated by the same arguments we put
forwardin this paper. A thoroughempiricalexplorationof our contentionsis the subject of a
forthcomingpaper.
17 Indeed,the idea of usingspacesof policiesand measuringpolarizationas a state of affairsthat
yield high resistanceto policieschosen at randomformsthe basis of an alternativedevelopmentof
the theory.But this is beyondthe scope of the presentpaper.
18 While multidimensional
variantsof inequalitytheory do exist, to go beyond the one-dimensional theorythey must imposeparticularrestrictionson how dimensionsare to be compared.
824
J.-M. ESTEBAN AND D. RAY
them to be degenerate,symmetricbimodal distributions.In Section 3.3 (Theorem 3), we sharpenthe class of allowablemeasureseven furtherby the use of an
additionalplausible assumption.In Section 3.4, we illustrate how these measures work by applyingthem to a set of examples, including the motivating
examplesin Section 2. In Section 4, we extend our class of measuresbeyondthe
simple model of Section 3, in a form that might be even more suitable for
empiricalwork. Section 5 describespossible extensionsof the basic theory,and
concludesthe paper. This section also containsremarkson the issue of population and income normalizationfor the measureswe obtain.
2. POLARIZATION:
OVERVIEW AND SOME EXAMPLES
The objectiveof this section is to study a numberof examplesin an informal
way. Some of these examples will be elevated in the sequel to the status of
axioms.At present, however,our statementsare not meant to be precise. They
are meant to align your intuition to ours, to form a preliminaryjudgement
regardingthe formalconcept we shall later introduce.
Loosely speaking, every society can be thought of as an amalgamationof
groups,where two individualsdrawn from the same group are "similar,"and
from differentgroups,are "different"relativeto some given set of attributesor
characteristics.The polarizationof a distributionof individualattributesmust
exhibitthe followingbasic features.
FEATURE
1: There must be a high degree of homogeneitywithineach group.
FEATURE
2: There must be a high degree of heterogeneityacross groups.
FEATURE 3: There must be a small number of significantlysized groups. In
particular,groups of insignificantsize (e.g., isolated individuals)carry little
weight.
We start by comparingtwo alternativedistributionsof a given attribute(say
income)over a population.Example1 below is designedto illustrateFeature 1.
1
2
3
4
5
6
7
8
9
10
FIGURE 1A
FIGURE 1.-Diagrams
1
2
3
4
5
6
FIGURE 1B
to illustrate Example 1.
7
8
9
10
POLARIZATION
825
EXAMPLE
1: In Figure la, the population is uniformlydistributedover ten
values of income,spaced apartequallyat the points 1,2,..., 10. In Figure lb, we
collapse this distributioninto a two-spikeconfigurationconcentratedequallyon
the points 3 and 8. This diagramformalizesthe discussionof per capita GDP in
the Introductionas a specificexample,or per capita growthrates if the attribute
is reinterpretedappropriately.
Which distributionexhibitsgreater"polarization"?We would argue that it is
the latter distribution.Two groupsare now perfectlywell formed in the second
diagram,while in the formerdiagramthe sense of groupidentityis more fuzzy.
Under the second distributionpopulation is either "rich" or "poor,"with no
"middle class" bridgingthe gap between the two, and one may be inclined to
perceive this situationas more conflictualthan the initial one.'9
If you are still hesitant,thinkof the attributeas an opinion index over a given
political issue, running from right to left. Many would agree that political
conflict is more likely under a two-spike distribution-with perhaps not completely extremepolitical opinions,but sharplydefined and involvingpopulation
groupsof significantsize-rather than under the uniformdistribution.
But the point is this. If you admit the possibilityof greater polarizationin
Figurelb, you are forced to departfromthe domainof inequalitymeasurement.
For under any inequalitymeasure that is consistent with the widely accepted
Lorenz ordering,inequalityhas come down in Figure lb relativeto la!
EXAMPLE
2: This example illustrates Feature 2 above. It also shows that there
are cases where increasingpolarizationalso conformsto Lorenz-worseningof
the underlyingdistribution.ConsiderFigures2a and 2b. Think of the attribute
as income. Figure2a is just the old Figure lb. In Figure2b, we also have a two
point income distribution,but this time concentratedequallyon the incomes 1
and 10. Which has more polarization?Well, as far as intra-grouphomogeneity
goes, there is nothingto choose between the two diagrams.But there is greater
inter-groupheterogeneityin Figure 2b. So it should be fairly easy to conclude
that there is more polarizationin Figure 2b. But note that this time income
inequality,too, has worsened in Figure 2b (relativeto 2a). In particular,we do
not claim that the notion of polarizationalways conflictswith that of inequality.
EXAMPLE
3: With the help of Figure 3, we illustrate Feature 3. In Figure 3a,
the population is distributedaccordingto a three-point distribution,with the
attributesspaced equally. Think of the central and the right-handmasses as
being of approximatelythe same size. Considera small shift of populationmass
from the left extreme to the right extreme.The problemis immediatelyseen to
be more complexthan those of the previousexamples.While it is true that the
shift creates a greatertendencytowardsformingtwo sharplydefinedgroups,the
19
In this context,see Proposition9 in Coser(1956)(whichdrawson Simmel(1955)),wherebythe
internalcohesionof groupsis associatedwith increasedconflictbetweengroups.
826
1
J.-M. ESTEBAN AND D. RAY
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
FIGURE 2B
FIGURE 2A
FIGURE 2.-Diagrams
FIGURE 3A
to explain Example 2.
FIGURE 3B
FIGURE 3.-Diagrams
to explain Example 3.
left mass maywell be instrumentalin creatingpart of the social tensionsthat do
exist, and the net effect is far from clear.
Consider,however,the case illustratedin Figure 3b, where the left mass is
very tiny indeed comparedto the other two. In this case, the initial contribution
of the left mass is vanishinglylow and it mightmake sense to arguethat the very
same move now serves to increasepolarization.This is the substanceof Feature
3. While we do not adopt the example of Figure 3b as a basic assumptionfor
our main result, we do explore its implicationsfor the choice of measure (see
Section 3.3 and Theorem3). To us, it is reasonableas a basic axiom,thoughit is
of interest to see how far we can go without it.
The examples above are meant to show that the concept of polarization
(whateverit is) is logically separate from that of economic inequality,and to
bringout certaindefiningfeatures of the concept.The two remainingexamples
in this section are meant to illustrate an entirely different set of issues.
Specifically,we wish to argue that any reasonablemeasureof polarizationmust
be global in nature,in a way that inequalitymeasuresare not. To see this point,
recall the Dalton Principleof Transfersthat underliesthe Lorenzordering.The
principle states that startingfrom any distributionof income, any transferof
income from an individualto one richerthan him must increaseinequality.The
principle is a local one. To apply it, it is unnecessaryto take account of the
827
POLARIZATION
1
2
3
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5
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7
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9
10
FIGURE 4A
FIGURE 4.-Diagrams
1
2
3
4
5
6
7
8
9
10
FIGURE 4B
to illustrate Example 4.
original distribution. In our thinking about the subject, we have found it
impossibleto come up with a similarlocal prescriptionfor increasingpolarization. The examplesthat follow try to indicatewhy this is so.
EXAMPLE 4: Consider the two income distributionsin Figures 4a and 4b.
Each distributionhas point masses at the income levels 1, 5, and 10. In Figure
4a, half the population is equally divided between incomes 1 and 5, and the
other half is at income 10. Consider,now, the fusion of the firsttwo groupsinto
a single mass at the income level 3 (see arrowsin 4a). Has polarizationgone up?
Note that the collectionof individualsat incomelevels 1 to 5 no longerhave any
interpersonalanimosity;they are all united at income level 3. They now see as
their "commonenemy"the sizeable groupat income level 10. Moreover,so far
as the group at income 10 is concerned, they now see facing them a united
group of individualsof similarsize. On average,this group is as differentfrom
the 10-groupas it was before.
It would, indeed, be difficult to defend the view that polarization has
decreasedin this situation!
But now turn to Figure4b. Here, the only differenceis that almostthe entire
populationis dividedbetween the two income levels of 1 and 5, and there is a
tiny fractionof individualsat income level 10. Performthe same experimentas
before. You will notice that it is now practicallyimpossibleto make the same
argumentas we made for the precedingcase. Because of the small size of the
10-group,"most"of the "polarization"in Figure 4b comes from the fact that
there are two groupssituated at incomes 1 and 5. Put them together and you
have wiped out most of the social tension. It is true that there is a group at 10,
but it is a very small group, and any notion of continuity20would requirethat
the intra-grouptension also be small in the new situation,so in Figure 4b the
opposite argument-polarization has come down-is more convincing.
20 Simply send E to 0 in Figure 4b. At E = 0, polarization must come down by the argument of
Example 2. So this must also happen for ? > 0 but small!
828
J.-M.
1
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7
ESTEBAN
8
9
10
AND
1
D. RAY
2
3
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FIGURE SB
FIGURE SA
1
5
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3
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FIGURE Sc
FIGURE
5.-Diagrams
to illustrate Example 5.
EXAMPLE 5: Reconsider Example 2. Imagine moving from Figure 2a to
Figure2b by a series of smallchanges.In each change,equal fractionsof people
at incomes 3 and 8 are removedand replacedat 1 and 10 respectively.The end
result is, of course, Figure 2b. Figures 5a and 5c reproduce Figure 2, with
Figure 5b displayingan intermediatescenariowhere the populationis equally
divided among the income levels 1, 3, 8, and 10. Note that these changes
generate a sequence of regressiveLorenz dominateddistributions.
We have alreadyarguedthat 5c displaysmore polarizationthan 5a. But does
5b displaymore polarizationthan 5a? We claim that there is no unambiguous
answerhere. While it is true that there are very differentgroupsin 5b (relative
to 5a), there is also little group homogeneity.It all depends on whether one
gives more weight to intergroupdifferences at the expense of within group
homogeneity. We are not making any particularclaim as to a direction of
change, but are only tryingto convince you that there is a genuine ambiguity
here.
These examples (4 and 5) are meant to illustrate the possibility that the
searchfor a (reasonablyrich)partialorderfor increasesin polarizationcan be a
difficultone. For one thing, the effect (on polarization)of a given change may
POLARIZATION
829
depend on factorsthat are not directlyassociatedwith the change(e.g., the size
of the 10-groupsin Figures 4a and 4b). This is what we meant above by the
global nature of the concept. Moreover,the same directionsof change can be
associatedwith different effects on polarization,dependingon the initial configurations.This point is illustrated by both Examples 4 and 5. Both these
features are absent in the theory of inequalitymeasurement,embodied in the
Lorenzordering.
In the next section, we present a formal analysisthat attempts to capture
some of the issues raised here.
3. POLARIZATION
MEASURES
3.1. Axiomatic Derivation of a Class of Measures
3.1.1. Distributions and Polarization Measures
The previous section should have already made clear that the concept of
polarizationis somewhatcomplicated,and no single simple axiom(such as the
Dalton Axiom in the contextof inequalitymeasurement)will serve to captureits
essence. Accordingly,our aim is limited. We introduce a model of individual
attitudes in a society that leads naturally to a broad class of polarization
measures. On this class, we place certain axioms and narrowdown the set of
allowablemeasuresconsiderably.Finally,we show that the measurewe obtain
performs"well" in the context of the examplesearlier described.
Our basic perceptualvariableis the naturallogarithmof income, denoted by
y, and we shall presumethat its values lie in DR
(zero incomesare not permitted).
By an individual9 we shall mean an individualwho has income (with log equal
to) 9. The choice of this variable is based on the presumption that only
percentage differencesmatter. But any other scalar can be used as the basic
perceptualvariablewith little overallconceptualdifference.21
In this paper,we consideronly those distributionswith supporton some finite
set of incomes. This simplificationis made for expositoryconvenience.There
are nontrivialadditionalissues involvedin dealingwith the case of distributions
with infinitesupport.We commenton these brieflybelow, and refer the reader
to Estebanand Ray (1991)for a detailed developmentof the theoryin this case.
We begin with some backgrounddefinitionsand notation. For any positive
integer n, (rr, y) (n1r ***, yn;y1,* , yn) is a distribution if y E WR,yi $ yj for
all i, j, and nr> 0. The total population associated with (irr,y) is given by EL 11Ti.
Denote by -9 the space of all distributions.
A polarization measure (PM) is a mapping P: 9 -- R
21
It may be argued, for example, that large, percentage differencesin income make little
differenceto an individualaftersome threshold.In that case, the basicperceptualvariablemightbe
some strictlyconcavetransformof y.
830
J.-M. ESTEBAN AND D. RAY
Throughout,we will suppose that the ranking induced by a polarization
measure over two distributionsis invariantwith respect to the size of the
population.This homotheticitypropertyis standardin the theory of inequality
measurement(see, e.g., Foster (1985)). We refer to such a propertyas Condition H.
CONDITION
H: If P(rr, y) > P(rr', y') for two distributions (iir, y) and (r', y'),
then for all A > 0, P(Arr,y) > P(Ar', y').
We now turn to the study of a behavioralmodel that will yield a class of
polarizationmeasures.
3.1.2. A Model
Our first concept deals with the fact that intra-grouphomogeneityaccentuates polarization.We proposethat an individualy feels a sense of identification
with other individualswho have the same incomes as him. Thus the identification felt by an individualis an increasingfunctionof the numberof individuals
in the same income class of that individual.Formally,we introducea continuous
identification function I: R+-D>R+. Assume I(p) > 0 whenever p > 0. Formally,
no furtherassumptionneed be made, though it is no surprisethat we will later
deduce that I(p) is increasing(and more).
Two remarksare in order.First,we have postulateda particularlysharpform
of identificationwhereby individuals feel identified with people who earn
exactlythe same income,but with no one else. This is defensibleonly if we think
of incomes as point estimates of income classes or intervals.It is surely more
reasonableto also allow for (a perhapssmaller)identificationwith neighboring
individuals.Indeed, as we point out in detail in Section 4, this postulateleads to
a drawbackwith our derived class of measures.We have neverthelesschosen
this route in the interest of a cleaner model.
Second, the sense of identificationmay depend not only on the numberof
similarindividualsbut also on the commoncharacteristics(in this case, income)
that these individualspossess. We commenton this extensionin Section 5.3.
Next, we posit that an individualfeels alienated from others that are "far
away"from him. This concept will deal with the fact that inter-groupheterogeneity accentuates polarization. Let a: R;+
R+, with a(O)= 0, be a nonde-
creasingcontinuousfunction, to be called the alienationfunction. We assume
that individual y feels an alienation a(8(y, y')) with an individual y', where
8(y, y') stands simplyfor absolute distance Iy - y'f.
We pause here to explicitlytake note of an importantfeature subsumedin
the definition. We have assumed that the concept of alienation (as well as
identification)is perfectly symmetric.An individualwith low income feels the
POLARIZATION
831
same alienation towards an individualwith high income as the latter feels
towardsthe former. This is perhaps inappropriate,as it can be argued that a
polarizedsociety arises from the alienationthat the poor feel towardsthe rich,
and not the other way around.We discussan "asymmetric"model that captures
these features in Section 5.2.
Now we are in a position to put togetherthese two concepts.Whatwe wish to
capture is the effectiveantagonismthat y feels towards y'. This is, roughly
speakingjust the same as y's alienationvis-a-visy', but we wish to allow for the
possibility that an individual's feelings of identification may influence the
"effectivevoicing"of his alienation.22In the sequel, this influencewill turn out
to be the criticalfeature that distinguishespolarizationfrom inequality.
Accordingly,we allow for this influence (though we do not assume it). We
postulate that the effective antagonism felt by y towards y' is given by a
continuous function T(I, a), where a = a(8(y, y')) and I =I(p), with p the
measureof people in the income class of individualy. This function T is taken
to be strictly increasing in a whenever (I, a)>> 0. We assume further that
T(I, 0) = 0.23
Finally,the total polarizationin the society is postulatedto be the sum of all
the effective antagonisms:
(1)
P(7,
Y)
=
n
n
E
E
iri7jT(I(7i),a(6(yi,yj))).
i=1 j=1
The postulate (1) embodies two assumptions.First, we record that polarization depends on the "vector"of effectiveantagonismsin the society and on this
postulate of
alone; in this respect it is analogousto a Bergson-Samuelson-type
nonpaternalism.Second it goes further and proposes an analogue of additive
utilitarianism.Some justificationfor the additivepostulate may be found along
lines suggestedby Harsanyi(1953) and others: that an impartialobserverwho
might find himself in any position in the given distributionmight use the
expected value of his effective antagonismto judge overallpolarization.
On the other hand, we have left the structureof the functions I(-), a(-), and
T(&)very general indeed. The choice of any one functionalform for each will
yield a particularmeasureof polarization.
22
Thus, we are primarilyinterestedin notions of organized opinion,protest,etc. The notion of
an individualwho runs amok,perhapsprovokedby his sense of isolation,is an importantone that
we do not considerhere.
23 Note that these assumptionsformallypermit identificationto not have any influence at all
(T(I, a) = a is permitted,for instance), or perhaps even have negative influence, though these
featureswill disappearin the sequel. Note, moreover,that it is also possibleto weightone's feelings
of alienationby invokingthe visibilityof the group from which the individualfeels alienated.But
there is nothing so far in our discussionand exampleswhich appear to necessitate this. Indeed,
whethersuch a weightingshould enter positivelyor negativelyis far from intuitivelyclear. So we
ignorethis in the interestsof simplicity.See Section5.3 for more discussion.
832
J.-M. ESTEBAN AND D. RAY
3.1.3. Axiomatic Derivation
We are interested in narrowingdown the class of allowable measures by
imposingsome "reasonable"axioms.
AXIOM 1:
p
q-
q
x
0
q
y
Data: p, q>> 0, p > q, 0 <x <y.
Statement: Fix p > 0 and x > 0. There exists E > 0 and ,ut> 0 (possibly depending on p and x) such that if 8(x, y) < E and q < ,up, then the joining of the two q
masses at their mid-point, (x + y)/2, increases polarization.
The intuition behind this axiom should be very clear. The two right hand
masses are individuallysmaller than p (see Data). Moreover,they are "very
close" to each other (see Statement).In such a case, by pooling the two small
masses we "identify"them while not changingthe average distance from the
third mass. This should thereforeraise polarization.24
AXIoM
2:
p
r
q
0
24
-
x
I
y
The axiom is actually stated in very weak terms. The axiom is invoked only when the distance
between x and y is small, and also only when the q-masses are small relative to p.
833
POLARIZATION
Data:(p,q,r)>>O, p>r, x>Iy-xl.
Statement: There is E > 0 such that if the population mass q is moved to the
right (towards r) by an amount not exceeding e, polarization goes up.
This is also an intuitiveaxiom.The intermediatepoint mass q is at least as
close to the r-massas it is to the p-mass.Additionally,the p-mass is largerthan
the r-mass.So if only small locationalchangesin the q-massare permitted,the
direction that brings it closer to the nearer and smaller mass should raise
polarization.
We now turn to our final axiom.
AxIOM3:
r
q
p
0
x
-d
d
y
Data: (p,q)>>0, x=y-x-d.
Statement: Any new distributionformed by shifting population mass from the
central mass q equally to the two lateral masses p, each d units of distance away,
must increase polarization.
This axiom is so intuitiveit hardlyrequirescomment.The axiom states that
the disappearanceof a "middle class" into the "rich" and "poor" categories
must increasepolarization.
We may now state our main result.To do so, it will be necessaryto introduce
the function f: R22- R, definedby
(2)
f(z,a)
(1 +a)[z
-
Z +]2
It is possible to show that there exists a number a* > 0 such that
maxz> 0 f(z, a) < 0 (resp. < 0) if and only if a < a* (resp. < a*). Moreover,it
can be shownthat f(z, a*) < 0 for all but one point z*, where equalityholds. It
is also possible to show analyticallythat a* E (1,2). Numerical computation
reveals that a* - 1.6. Details of the proofs of these assertionsand the numerical computationare availableon request.
834
J.-M. ESTEBAN AND D. RAY
We now have the followingtheorem.
1: A polarization measure P* of the family defined in (1) satisfies
THEOREM
Axioms 1, 2, and 3, and Condition H, if and only if it is of the form
(3)
P*(wr,y)
n
n
i=l
j=l
=KE
r1+arj
Y-
j
for some constants K > 0 and ae (0, a* ] where a* - 1.6 (see above, equation
(2)).
This theorem dramaticallynarrowsthe class of allowablepolarizationmeasures. The only two degrees of freedom are in the constants K and a. The
former is simply a multiplicativeconstant which has no bearing on the order,
but which we use for population normalization(see Section 5.1 below). The
constant a reflectsour deductionthat the identificationfunctionmust be of the
form pa, where a > 0. Note that no a priori assumptionof how identification
affects effective antagonismhad been postulated in the model. The deduction
that this effect is indeed positive is not surprising,given the earlier discussion
and examples.This is preciselywhat will distinguishpolarizationmeasuresfrom
inequalitymeasures.
It should be noted, however,that a cannot take on arbitrarypositivevalues,
but must be bounded in the way indicatedby the theorem.Withoutthis bound,
all the axioms cannot be satisfied and furthermore,the bound is needed to
verifyother intuitivepropertiesof the measure(see below).
Indeed, each one of the restrictionsimplied by the theorem is importantin
signingthe directionof change in intuitiveexamples.The readermay turn right
awayto Section 3.2, where a numberof examplesare workedout and the use of
each of these conditionsis illustrated.
It should be noted that the measure P* bears a strikingresemblanceto the
Gini coefficient. Indeed barring the fact that we are using the logarithmof
incomes,our measure would be the Gini if a were equal to zero. It is precisely
the fact that populationweights are raised to a power exceeding unity in the
formula above that gives rise to distinctlydifferentbehaviorof a polarization
measure.
Thus a may be treated as the degree of "polarizationsensitivity"of the
derived measure. The larger is its value, the greater is the departure from
inequality measurement.For more on this, see the discussion following the
statementof Theorem3, and Section 3.4.2.
With these remarksmade, we now turn to a proof.
PROOF: (Necessity.) Define 0: lR2- R
by 0Gr, S) T(IOrr),a(a)). First, we
show that 0Qr, *) is linearfor all wr> 0. Considerthe case depicted in Axiom 1,
with S(x, y) < e and q < Ap. Total polarizationis given by
(4)
P1 _pq[O(p, x) + 0(p, y)]
+ pq[O(q, x) + O(q, y)] + 2q20(q, IY-xl).
835
POLARIZATION
When the two q's are put together at (x + y)/2, polarizationis
(5)
p2
pq [0( P
Y )]+ 2pq [0(2q, xy)]
So by Axiom 1 we have, combining(4) and (5),
pL
'P 2J)
2q,
2
]>P[(P,
x) + O( y)]
+p[O(q, x) + O(q, y)]
+ 2q0(q, Iy -xl).
Passing to the limit as q -O 0, we get for each p > 0,
(6)
20 p,
2
> (p,x)+
(P,Y)
The above holds, it should be noted, for each (p, x) > 0 when y is "sufficiently"close to x. Nevertheless,it is possible to show that because 0(p, *) is
continuousfor each p, 0(p, *) mustbe concave.The detailsof this argumentare
but tedious, and are thereforeomitted.
straightforward
Next, pick any (p, x, x') >> 0 with x > x'. We claim that there exists D > 0 such
that for all A E (0, D], 0(p, x + A) > 0(p, x' - A).
To establish this claim, consider Axiom 2. For given (p, x, x') >>0, choose
(q, r) >>O such that p > r and y such that y - x = x'. Then (p, q, r, x, y)
satisfies all the conditionsof that axiom.Now, polarizationis initiallygiven by
(7)
Pa
pr[O(p, y) + 0(r, y)] +pq[O(p, x +A) + O(q, x + A)]
+ qr[O(q, x'-A)
+ 0(r, x'-A)]
evaluatedat A = 0. By the axiom,it mustbe the case that Pa > POfor all values
of A E (O,x').25 Using this informationin conjunctionwith (7), we see that
pq[{0(p, x +,A) - 0(p, x)} + {O(q,x +,A) - O(q,x)}] > qr[{O(q,x') - O(q, x' -,A)}
+ {0(r, x') - 0(r, x' - A)}] for all A E (0, x'/2). Passing to the limit as
(q, r) -->(p, p), we deduce that
O(p, x +,A) - O(p, x) > O(p, x') - O(p, x' -,A),
for all 0 <,A < D x'/2. This establishesthe claim. Combiningthis claim with
the earlierdeductionthat 0(p, *) is concave(and continuous)for all p, we may
conclude at once26 that there exists a continuousfunction 4( ) with +(p) > O
for all p > 0 such that for all (p, 8) > 0,
O(P, a) = O( p)6
25
If this were not true, then keeping(p, q, r, y) fixed, there would exist some value x" E (x, y)
such that Axiom2 wouldbe violatedwith x" in place of x.
26We omit the detailsof this deduction.One easyway is to note that by the concavityof 0(p,),
o admitsright and left hand derivativesin its second argument.By interpretingthis claim as an
additionalrestrictionon these derivatives,we maydeduce the linearityof 0 in its second argument.
836
J.-M. ESTEBAN AND D. RAY
We show, now, that k(&)must be an increasingfunction. To do this, firstpick
q > 0 and then recall the scenario of Axiom 1, with (p, x, y) chosen such that
5(x, y) < E and q < ,up. Total polarizationis given by
Pi -pq[O(p)x +4(p)y]
+pq[O(q)x + O(q)y] + 2q20(q)ly
When the two q's are put together at (x + y)/2, polarizationis
x +y
x+y
+ 2pqb(2q)
=2pq4(p)
p2-
-xj.
2
2
By combining these two observations and using Axiom 1, it follows that
4(2q)> f(q). Since q was an arbitrarypositive number,the continuousfunction +(q) must be increasing.
We show, next, that 4(p) must be of the form Kpa, for constants (K, a) >> 0.
To prove this we proceed as follows. Considerany two-pointdistributionwhere
positivepopulationmasses p and q are situated at a distanceof one unit from
each other. Total polarizationis given by pq[b(p) + +(q)]. Now note that for
each (p, q, p') >>0, there exists q' > 0 such that
+ +(q)] =p'q'[O(p') + 4(q')].
By ConditionH, it follows that for all A > 0,
(8)
pq[b(p)
(9)
A2pq[O(Ap) + 4(Aq)] = A2p'q'[ (Ap') + O(Aq')].
Combining(8) and (9), we see that
(10)
o(p') + o(q')
O(Ap') + O(Aq')
+(P) + +(q)
O(Ap) + O(Aq)
Takinglimits in (10) as q -O 0 and noting that q'
0 as a result,we have for all
(p, p', A) >> O,
(
(
1)
)
--
()
O(Ap)
O(Ap')
Put A = i/p and r = p'/p in (11). Note that p and r can be varied freely over
Makingthe requiredsubstitutionsin (11), we arriveat the fundamental
DR++.
Cauchyequation
(12)
k(p)4(r) = 4(pr)4(l)
for all (p, r) >>0, where 4( ) is continuous and strictly increasing(this last
observationby our previous argument).The class of solutions to (12) (that
satisfythe additionalqualifications)is completelydescribedby +(p) = Kpa for
constants(K, a) >>0 (see, e.g., Aczel (1966, p. 41, Theorem3)).
To summarize,we have proved so far that O(p,8) = Kp for (p, 8) > 0. It
remainsto deduce the bounds on a. To do so, consider the case of Axiom 3.
Polarizationis given by
(13)
P()
2d[(p-)
+ 4d(p
1p+a
_
(q + 2)
A)2+a,
+ (1 + 2,)
1+a
-
A)
837
POLARIZATION
evaluatedat A = 0. The axiomimplies that the derivativeof P with respect to
A must be nonpositivefor all values of (p, q) >>0. Performingthe necessary
calculation and defining z as the ratio p/q, we see that this requirementis
equivalentto the conditionthat f(z, a) < 0 for all z, where f(z, a) is definedin
(2). But this implies that a < a*.
(Sufficiency.)ClearlyP* as describedin the statementof the theoremsatisfies
ConditionH. It remainsonly to verifyAxioms 1, 2, and 3.
(VerifyingAxiom 1): Considerp, q, x, y as in the Axiom. Initially,polarization
is
p = (pl+aq + q+ap)(x + y) + q2+aly-xl.
Putting the two q's together at (x + y)/2,
Pt = (2p
=
(pl+aq
+aq
+ 21+aql+ap)(x
+ ql +ap)(x
+y)/2
+ y) + (2 al)
(x + y) ql +ap,
so that P' >P whenever (2a - 1)(x +y)p > 2qIy -xl. It is easy, therefore, to
find (e, ,)? >>0 so that Axiom 1 is satisfied.
(VerifyingAxiom 2): Take p, q, r, x, y satisfyingthe conditions of Axiom 2.
Polarizationis given by
+ ql+ap) + (y -x)(q1+ar
+ rl+ap).
P=x(pl+aq
(14)
+rl+aq)
+y(pl+ar
Considera small increase in x by A. Then, using (14), the change in polariza-
tion is
A{(p1+aq
+ ql+ap)
-(ql+ar
+ rl+aq)}.
It is easy to verifythat this expressionis positive if p > r.
(VerifyingAxiom 3): Considerthe case depicted in Axiom 3. Recall (14). One
can verify that it will be sufficient to prove that for every (p, q)>>0, the
derivative of P(A) evaluated at A = 0 is nonpositive, and that it is strictly
negative for all but at most one ratio of p to q. For each ratio z =p/q, this
derivative is given by f(z, a). Now refer to the assertions following (2) to
complete the verification.
Q.E.D.
This completesthe proof of the theorem.
3.2. Maximal Elements of the Polarization Measure
This subsection is devoted to examiningthe partial orderingsgenerated by
the class of measures given by Theorem 1. Moreover,we show that for given
income classes Y1,Y2,..., Yn and total population,the distributionthat assigns
half the populationto the lowest income class and the remainderto the highest
income class is more polarized than any other distribution,under any of the
measuresgivenby Theorem1. In what follows,we shall take as fixedthe income
classes and identifydistributionsby the vector of populationmasses.
2: The bimodal distribution (H/2, 0,. . ., 0, H/2) is more polarized
THEOREM
than any other distribution
(rr,...
measure P* in the class given by (3).
,
n) with total population
H, under any
838
J.-M. ESTEBAN AND D. RAY
PROOF: Because of the continuityof any given measureP* in the class given
by Theorem 1, and given the compactness of the space of distributions,it
suffices to show that any distribution not equal to (H/2, 0, .. ., 0, H/2) can be
dominated by another in terms of polarization as measured by P*. In the
argumentsto follow, we will considermeasureswith K normalizedto unity:no
loss of generalityis involved.The followinglemmaswill be useful.
LEMMA1: Supposethat thereexistsk, 1 < k < n, with thepropertythat Tk > 0
and {EW`?lw1- E>k? 1w) and {Ek_l'l '+a - E1 =k+ 1T1I+a}are nonzeroand of the
samesign. Then,if the sign is positive(resp.negative),the newdistributioncreated
by movingall massfrom k to k + 1 (resp. to k - 1) raisespolarizationunderP*.
In what follows,we will denote Iyi - yjIby a(ij).
Suppose that the condition of the lemma is met with sign positive (the
negative case uses a symmetricargument).Denote by P and P' the values of
polarizationbefore and after the change. By singling out the indices k and
k + 1, we can write P as
PROOF OF LEMMA 1:
(15)
P-
n
n
n
E
Erja(ij)
+ 7Tk+' E 7ja(ki)
j=1
i=1
j=l
iok,k+1
jok,k+1
n
+
kE wl+aa(jk)
j=1
n
+ .1+al E Trja(k +
1,j)
j=l
jok
n
a(k+1,j).
+Wrk+ l E, wJil+a
j=1
jik
Now observe that for j > k, a(kj) = a(k + 1, j) + a(k + 1, k), while for j < k,
a(kj)
=
a(k + 1, j) - a(k + 1, k). Consequently, we may rewrite (15) as
n
(16)
P
n
n
1I rja(i;)
E
i=1
i$k,k+1
j=1
j?k,k+1
k+
k+
E 7rja(k + 1 j)
j=1
jok
n
+ (Wk+Wk+1)
E w+aa(k
+ 1,j)
j=1
j#k
n
+ 7rl+aa(k + 1, k)
E
k-1
v-Ev
j=k+l
+ lTka(k + 1,
k)
T1+a
(
j=k+l
j=1
k-1
n
-
E
+a )
j=1
Recall now that P' is the measure achieved by shifting all mass from k to k + 1.
839
POLARIZATION
Therefore
n
(17)
P
-r
n
n
+
rr1+arja(ij)
E
E
i=1
i#k,k+l
+ 7k+l)
(7k
E
j=1
j#k,k+l
j=1
j#k
rja(k +1, j)
n
+(k
+ 1, j).
l+a a(k
E
+Wk+1)
j=1
jik
Subtracting (16) from (17), we have
n
(18)
P
+
P1[(Wk
1k+l)a
(k
Ewvja(k+1,j)
k1+a)]
+
j=l
(k-1
+7kl?aa(k+l,k)
n
7rj-
vi
E
j=k+l
k-1
n
r+_
+ wrka(k + 1,j=+
k)
j
j=l
j
;1
wJ+
,
E
j=k+l
Combiningthis
(7)+a
+ w))
Note that because a > 0, (7k + Wk+k)l+a>
observationwith (18) and the conditions of the lemma, we see that P' >P.
Q.E.D.
LEMMA2: Suppose that an income distribution has at least four nonzero mass
points. Then there exists k, 1 < k < n, such that the conditions of Lemma 1 are
met.
PROOFOF LEMMA2: Let k1 be the second nontrivial mass point counting
from the left, and k2 be the second nontrivialmass point counting from the
right. Then, because there are at least four nonzero mass points, k1 < k2. We
claim that the conditionsof Lemma1 hold for at least one of these two indices.
Clearly,for either k = k1 or k = k2, i = 1 - Ei=k+l1i # 0.
Case 1: Suppose that k = k1. Consider the subcase where
n
ki-1
(19)
E
E wii=1
wi>O.
i=kl+l
Because wi > 0 for at most one index i for 1 < i < k1, we have
Eik 1-1+a.
Consequently, by (19),
k - 1
S7r 1+a=
i=l
1 +a
kl-l
S
\i=1l
i
>l
\i=
1 +a
n
E
i=kl+l
i
(E
1Yl
li)l+a
=
n
>
E
r1i+a.
i=kl+l
This means that the conditionsof Lemma 1 are met for k = k1.
Next, considerthe subcasewhere (19) fails. Then it certainlymust be the case
that (19) holds when k1 is replaced by k2. Now carry out exactly the same
argumentfor k2 as we did for k1.
840
J.-M. ESTEBAN AND D. RAY
Case2: Suppose that k = k2. Then use exactlythe same argumentas in Case
Q.E.D.
1, with the roles of k1 and k2 reversed.
We may now complete the proof of the theorem. If the number of mass
points is at least 4, use Lemmas1 and 2. If the numberof mass points is 3, use
Axiom 3. Finally,if the numberof mass points is 2, then use the computationin
Q.E.D.
Section 3.4.1 below.
Lemmas1 and 2 are of independentinterestfor polarizationanalysis.Lemma
1 establishes a criterion of polarizationdominancewhich generates a partial
ordering.Roughly speaking,this criterionsays that polarizationis raised if we
shift a populationmass to a neighboringclass wheneverthe distributionof the
populationon the side to which this neighborbelongs satisfiesthe followingtwo
conditions:total population is less, and more "dispersed"than on the other
side. Lemma 2 demonstrates that for every distributionthere always is an
income class such that the previous conditions are satisfied. For additional
discussionof this point, see Section 3.4.3, item (3).
3.3. An Additional Restriction
The conclusionof Theorem 1 maybe sharpenedfurtherby the inclusionof an
additionalassumption,whichwe find plausible.27This assumptionconcernsthe
case studied in Example3 of Section 2, which, it will be recalled,was designed
to illustratethe insignificanceof small groupsin the concept of polarization.
AXIOM
4:
r
q
Pi
0
x
y
Data: (p, q, r, ) >>O and q > r.
Statement: There is ,u > 0 such that if p < ,ur and q - r < ,u, then a transfer of
population mass from p to r will not decrease polarization.
27We decided against using this assumption directly in Theorem 1. Theorem 1 does achieve a
dramatic simplification, and it may be useful to explicitly see which assumptions are driving the
result obtained there.
841
POLARIZATION
Note that Axiom 4 only applies if p is small relativeto the masses q and r,
and if q - r. We note, too, that mass is transferredfrom p to only the smaller
of the two remainingmasses,so the Axiomis quite weak. We refer the readerto
Section 2, Example3, for a discussionof this point.
THEOREM 3: Under Condition (H) and Axioms 1-4, every polarization measure must be of the form described in Theorem 1, with the additional restriction
that a> 1.
We note that Theorem 1 is deficient in one respect. Because of the weak
natureof the axiomsemployed,Theorem1 fails to rule out the possibilitythat a
maybe arbitrarilyclose to zero. As noted above,smallervalues of a indicate an
increasinglack of "polarizationsensitivity,"and a greater degree of concordance with inequality measurement.The limiting case of a = 0, as we have
alreadyobserved,is one of inequalitymeasurement,with our measureconverging to the Gini coefficient (defined on log incomes). En route, to the limit,
Axioms 1 to 3 are satisfied, but in progressivelyweaker terms (with failure
occurring,of course, only at the limit). The additionalrestrictionachieved in
Theorem 3 rules out these cases altogether,boundingthe permissiblevalues of
a below by unity, and impartinga minimaldegree of "polarizationsensitivity"
to the entire class of measures.See Section 3.4.2 for additionalillustration.
PROOF OF THEOREM
3: For given (p, q, r, A) as in Axiom 4, polarizationis
given by
(20)
P
A{[p
1+aq
+ q1 +ap]
r r+l+aq]
+
+ [q'
+ rl+ap]}.
+2[p1+ar
Differentiating(20) with respect to p under the understandingthat dq = 0 and
dr = -dp, we see that
(21)
--=
+(1+ )paq +q1+a
+ 2(1 +a)par-2pl
(1 +a)raq
-ql+a
+a-2(1
+ a)rap
+ 2r'
+a.
Using Axiom 4, it follows that evaluated at p = 0 and q = r, (1/A)(dP/dp)
Passing to the limit as p -> 0 and r -p q in (21), we see that
< 0.
1 dP
a dp (=
O,r=q)
= -(1 +a)ql
+
2q'+
So the requiredinequalityholds if and only if a > 1. This completes the proof
of the proposition.
Q.E.D.
842
J.-M. ESTEBAN AND D. RAY
3.4. Propertiesof the Polarization Measure in Particular Cases
In this subsection,we apply the polarizationmeasurethat we have obtained
to some simple and (intuitively) not-so-simple cases, to examine how the
measure behaves. These cases include an analysis of the various "desirable"
features of a polarizationmeasure, discussedin Section 2. In all the examples
that follow, we normalizepopulationto a unit mass and set K = 1 in (3).
3.4.1. Two-pointDistributions
Our first example studies aspects of Example 2 (Section 2). Consider the
two-point income distributionwith weights 7r and 1 - r concentrated at incomes x and y. Polarizationis given by
+ (1w-g) ?awJ
P[ w1+a(1-r)
(y-x).
It is immediatethat pulling apart the two groupswill increase polarization.
Furthermore,by using the expression above, it can be verified that for given
(x, y), polarizationis maximizedat rr= 0.5. This last feature uses the condition
(provedin Theorem 1) that a < a.28
3.4.2. Three-pointDistributions
We now take a look at different three-point distributions.Without loss of
generality, denote by (0, x, y) the incomes (0 < x < y) and by (p, q, r) the
correspondingpopulationweights.The general form of the measurenow reads
P-[pl+q+ql+apx
+ [q1+a
[pl+pr+rl+p]y
r + r1+aq](y-x).
One special case is where p = r. This yields a symmetricdistribution.As
populationweight on this distributionis shifted from the center to the sides, we
expect polarizationto rise. This is exactlythe sentimentof Axiom 3, which our
polarizationmeasureis knownto satisfy.
Another specialcase is that illustratedby Example4 in Section 2. Here q = r.
The idea is to examinewhat happenswhen the two q-massesare fused at the
midpointof their incomes.We arguedin Section 2 that polarizationshould rise
if p is "large"relativeto q, but fall if the relativesizes were reversed.Denoting
the polarizationvalues before and after the changeby P and P' respectively,we
can reproduceexactlythe same argumentin the proof of Theorem 1 to see that
P > P' if and only if
p
q
2(y-x)
(2 - hl)(x + y)
28 Indeed, this feature holds as long as xx< 2.
843
POLARIZATION
p
q
q
d
qd-
FIGURE 6.-Weight-shifting fromcenterto sides.
The behaviorof the measurethereforereinforcesour intuition.In particular,if
p is large relativeto q, polarizationrises, while exactlythe opposite is true if p
is small.29This is preciselythe kind of global sensitivitythat we referredto in
Section 2.
Apropos the discussionfollowingTheorems 1 and 3, note how clearly a acts
as a measureof "polarizationsensitivity."In particular,the smallerthe value of
a the highermust be the ratio of p to q before polarizationcan be said to have
increasedunder the measure.
3.4.3. Distributions with Four or More Mass Points
(1) We start by reproducingExample5 of Section 2. This example,it will be
recalled,was designed(along with Example4), to show the globalnature of the
measure.ConsiderFigure 6.
Imagine that weight is being transferredequally from the two mass points
nearer the center towards the sides: thus, we have initially the configuration
(0, ', ', 0) and finally, the more polarized distribution (2, 0,0, '). Our purpose is
to investigateintermediatebehavior.
Use Figure6 to examinethe formulato be givenbelow. Initially,polarization
is given by
p
= 2[p
l+aq
+ 2(pl+aq
+ ql+ap]
d + 2p2+a
+ ql+ap)(1
d
+ r7)d + 2q2+a(2
+ 71)d
= 2p2+a7d
+ 2[ p1+aq + ql +ap + q2+a] (2 + q)d
= 2p2+a7d
+ 2[ pl+aqqlq 1+a](2 + j) d,
using the fact that p + q = 1/2. With dp/dq
=
-1,
the derivative of this
29Of course,the precisethresholdvaluesdependon the valuesof x and y. But that is exactlyas
it shouldbe.
844
J.-M. ESTEBAN AND D. RAY
expressionwith respect to q is
-2(2 + a)pl+a 7d+ 2[ pl+a-(1
+a)
paq
+
2(1
+ a)qa]
(2 + 7)d.
We wish to examinethe sign of this as q rangesover [0, ]. First note that at
q = 2 (and p = 0), the derivativeis unambiguouslypositive. This reflects the
intuitivelyappealingfeaturethat when the populationis alreadylargelybunched
at the two extreme points, furtherbunchingwill serve to accentuatepolarization.
At q = 0 (and p = 1), the derivative has the value d[2 - (1 + a)-q]2-a. This
could be of either sign, dependingon the values of -1 and a. For instance,if -q
is "small,"then the initial polarizationis relativelysmall and all moves to the
extremes raise polarization.However, if qj> 2/(1 + a), initial polarizationis
relativelylarge. In this case, the above analysis shows that as population is
moved away from the central masses to the extremes, polarizationfirst goes
down and then goes up,30ultimatelyattaininga higherlevel than the initialone.
Under the conditionsyieldingTheorem3, this requirementis veryweak, saying
only that -q> 1. The reader may wish to compare this with our intuitive
discussionof Example5.
(2) We turn next to the question of Dalton transfersthat are equalizing,to
checkwhetherpolarizationcan rise in this case. Numericalexamplesare easy to
find. Consider,for instance,the case of Example1 in Section 2. To be concrete,
think of incomes in this example as absolute values (and not logarithms),to
retain an explicit notion of Lorenz domination. It is possible to verify by
numerical methods that polarizationwill rise as long as a exceeds approximately 0.3. This conditionis more than amplysatisfiedunder Theorem3.
Indeed, it is easy to provide a general formula that checks whether the
"bunching"of incomes initiallyuniformlydistributedover a finite number of
income classes, into a smallernumberof income points, increasespolarization.
Apart from being of possible interest for its own sake, this exercise shows that
our polarizationmeasureis readilyamenableto analyticalcalculation.
Consider n (logarithmic)income points spaced uniformlywith pairwisedistance d, and suppose that the population is uniformlydistributedon these
points. Defining P(n, d) to be the value of polarizationthus obtained,we see
that
P(
d)
(2+
a)
E
()
(
+
(n-k)(n-
k+
2
k=1\2
1))
where in deriving the above expression, we recall that the sum 1 + 2 +
*..
+m
m(m + 1)/2 for any positive integer m. Using, in addition, the formula
1 + 22 + * +m2 = m(m + 1)(2m + 1)/6 for any positive integer m, the above
simplifiesto yield
=
(22)
30
P(n, d) -dn-(2+a)n(n2
-
1)/3.
It can be checkedthat a higherdegree of nonlinearityis not possiblein this case.
845
POLARIZATION
Consider positive integers M and K. Suppose, now, that we start with MK
income points spaced at uniformdistance, normalizedto unity. We "collapse"
this distributionby taking adjacent sets of K points each and concentrating
these at the averageincome levels of these sets. We now have a new uniform
distributionover M points spaced K units apart.Using (22), we may conclude
that polarizationhas increasedif
(KM)
(2+a)(K2m2
-
1) KM/3 >JKMQ-(2+a)(M2
-
1) M/3,
which on simplificationyields the condition
(23)
K 2+a(M2
-
1) >K 2M2 - 1.
It is immediate that for (23) to be satisfied for any given K, a > 0. This is
intuitive.Recall that for a = 0, the polarizationmeasureessentiallyreducesto a
Gini measure of inequality(in the perceptualvariablelog-income),so that all
bunchingof incomes will bring the measure down by the Dalton principleof
transfers.Note, too, that if bunchingresultsin a singlegroup(M = 1), polarization cannot go up, and this too is reflectedby the condition(23).
Next, observethat for any K and M # 1, condition(23) is most stringentwhen
M = 2. It follows, therefore,that for givenK, if polarizationis to rise under all
such circumstanceswhich involve the bunchingof K groups, a necessaryand
sufficientconditionfor this is
(24)
3K2+a > 4K2 _ 1.
Note that if the polarizationmeasuresatisfiesAxiom4, then a > 1 and (24) is
automaticallysatisfiedfor all values of K > 2.
(3) We end this section of the paper by observing that our polarization
measure combines a preference for numerical equality at different income
points with a preferencefor creatingbunchedpopulations.Thus, providedthat
groups can somehowbe defined, "intra-group"equalityenhances polarization.
One way of examiningthis is to generalizethe conditionsof Axiom 2. Consider
a particularincome class k and examinethe distributionson either side of this
class. Supposethat the followingtwo conditionshold: (i) total populationto the
left of this class exceeds total populationto the right of this class, and (ii) the
income distributionon the space of incomes to the left of this class is Lorenzdominated by the income distributionto the right of this class. Informally,
higherincomes are more scarcelypopulated and more bunchedtogether.Then
a shift of population weight from income class k to class k + 1 must raise
polarization as defined by (3). The proof of this observation is implied by
Lemma 1 and the fact that a > 0, and is omitted.
4. CROSS-IDENTIFICATION:AN EXTENDED CLASS OF MEASURES
It is appropriateat this stage to reconsidera serious assumptionthat drives
the derivationof the polarizationmeasure(3) in the previoussection. It will be
recalled that our model posited a particularlysharp form of the identification
846
J.-M.
ESTEBAN
AND
D. RAY
3/4
1/2
1/4
1/4
FIGURE 7A
FIGURE 7.-Diagrams
1/4
FIGURE 7B
1/2
1/4
1/4
FIGURE
7c
to illustratethe discontinuityproblem.
function;namely,that an individualidentifieswith another in exactlythe same
income class and with no one else. This extreme postulate is echoed in the
measurein the form of an unsatisfactorydiscontinuity.3'
Figure 7 reconsidersExample 4 in more detail. Begin with two population
masses, say 1/4 and 3/4 at two income points, as shown in Figure 7a. Now
suppose that the first mass and a subset of the second mass (of measure 1/4)
begin to move towardseach other, leavingthe remaininghalf of the population
where it is. Figures 7b and 7c illustratethis process. Initially,we would expect
polarizationto fall and finally,when the two populationmasses are fused (just
as in Example 4) we would have polarizationrise. All this is satisfiedby our
derivedmeasure.The point is that when the masses are alreadyclose (say, as in
Figure 7c) but not exactlysuperimposed,we would also expect polarizationto
be increasingin the describedmovement(supportingthis by an argumentthat
the identificationbetween the two 1/4-masses is growing).But our measure
does not incorporatecross-identificationbetween groups, and so continuesto
fall, only to jump up "in the limit" as the two masses exactlyfuse.
The problem is easily resolved by permitting identificationacross income
groupsthat are "sufficiently"close.32 But this resolutionis not costless. Admitting cross-identificationopens up a set of new issues. In particular, it is
impossible to specify (or deduce) a priori the domain over which a sense of
identificationprevails.
Nevertheless,such an extended class of measures(indexed by the choice of
the identificationzone) can be defined for the purpose of empiricalwork, and
can be shown to satisfy the axioms in the previous section, as well as the
properties discussed in Section 3.4. The choice of this zone must be left to the
empirical researcher, for it depends on the degree of grouping or averaging in each
of the "income classes" of the available data.
31
We are grateful to a referee for making this point, and for suggesting a discussion of it in the
paper.
32 Indeed, an earlier version of our paper (Esteban and Ray (1991)) does just this.
847
POLARIZATION
To illustrate an extended class, let D > 0 be such that if an income y' is
within D of an income y, then there is some identificationbetween the two
incomes. Formally,let w(d) be a positiveweightingscheme on [0, D] such that
w( ) is a decreasingfunction with w(D) = 0. For a given distribution(r, y),
define the identificationfelt by any memberof income class i as
Ii--~~~~17
j( W yj) .
j: IyI-yjLD
The new measure induced by the weightingfunction w is then some scalar
multipleof
n
(25)
PW(7IY
n
E
rjIia
i=1 j=1
max{ yi - y,I- D, 0)
A complete characterizationof such measures is beyond the scope of this
paper. But note that these measures do resolve the discontinuitydescribed
above.
5. OTHER EXTENSIONS
AND CONCLUDING
REMARKS
5.1. Income and Population Normalization
As stated earlier,we use the logarithmof incomes as the perceptualvariable
in our model. While this may appear nonstandardin the theory of inequality
measurement,where absolute differences in income are noted (and subsequently normalizedby mean income), we find our approachsuitable for two
reasons. The first is conceptual. We find it natural that individualsreact to
percentagedifferencesin income ratherthan absolute differences,and we wish
to keep these bilateral differencessymmetric.33The second reason is one of
convenience.A numberof our axioms and examplesinvolvethe movementof
population from one income class to another, which then alter aggregate
income. With absolute income, the measurewould have to be renormalizedat
each step before the axiomsdictatehow we want the measureto behave.This is
unnecessaryin the present formulation.A corollaryis that the measurecomes
prenormalizedin incomes.
The same is not true of population.The naturalnormalizationto choose, of
course, is to replace the populationweights rri,i = 1,.. ., n, by the population
frequencies. This is tantamount to choosing K = [ELn ,i](2+a)
in (3). But it is
importantto be clear as to what such a normalizationimplies in the context of
our model.
In our model, polarizationamong small populationsis smaller than among
larger groups,ceteris paribus.But a careful readingof the model will make it
33Thus, for example, the percentage difference in incomes, normalized by the income of the
individual concerned, is not symmetric. While more complicated formulations are symmetric (such
as the absolute difference of income divided by the sum of the two incomes), we felt that the
logarithm would do just as well.
848
J.-M. ESTEBAN AND D. RAY
clear that this refers to the relative importanceof the small population in a
contextwhere it is embeddedwithin a largerpopulation.Thus polarizationin a
small regionof a large countrymaymean little relativeto polarizationoccurring
in that same countryat a national level. But if this small region is an entirely
separatecountry,then the comparisonof the two countriesrequiresa scalingof
the population. Population homogeneity of degree zero may then be legitimatelyinvokedwithoutviolatingthe basic tenets of the model.
5.2. Asymmetric Polarization Measures
Earlier,we noted that a perfectlysymmetricmodel of polarizationmay not be
entirely appropriate,especially in a context where the variable of interest is
incomeor wealth.While the poor feel alienationfromthe rich,it maybe argued
that the correspondingsense of distance felt by the rich from the poor should
not enter in a symmetricway into the polarizationmeasure.We brieflydiscuss
an extremeversion of this observation:the alienationfunctionfor individualy
registerspositivevalues only for income values y' that are greaterthan that of
individualy.
Fortunately,we can exploit the formal structureabove with no change of
notation if we don't mind changingthe concept a bit. Interpret 3(y, y') now as
max{y'-y,O} (instead of Iy'-yI). The general class of measures studied in
Section 3 may now be also put forwardfor this case. As before, a set of axioms
may be imposed to narrow down this class of measures. This set will be
somewhatdifferentfrom the axiomsof that section, insofaras the latter has an
implicitbias towards symmetry(for example,Axiom 2). This class of polarization measures will certainly not yield symmetricdistributionsas its maximal
elements.
This last observationcan be verifiedeven without imposingany axiom at all.
Considerthe two point distributionin Section 3.3.1, with the mass points p and
(1 -p) situated at a distance d. Any asymmetricmeasure in the general class
yields polarizationequal to
P =p(1 -p)6(p,
d).
Suppose we were to maximize P with respect to a choice of p E [0,1]. We
observethat if 0 is increasingand differentiablein p, with ao/ap> 0, then the
maximum of the polarization measure must be attained at values of p that lie
strictlybetween2 and 1.34
34Here is a quickproof. Note that any value of p that maximizesP mustbe strictlybetween 0
and 1. So the first-ordernecessarycondition:
0( p*, d)(1l-
2p* ) + p*l (-P
*)
dO(p*, d) =
d
?
op
must hold at any values of p* that maximizeP. But now observe that because dO/Op> 0, this
conditioncannot hold for any P* < 2
POLARIZATION
849
This shows that asymmetricpolarization models will not yield symmetric
distributionsas maxilhalelements.
5.3. WeightedIdentification and Alienation Functions
We commenton two possible extensions.First, the sense of identificationof
an individualmay depend not only on the number of other individualswith
similar attributes,but also on the attributesthemselves. In other words, the
attributesmay confer "power"on the group to effectivelymanifestits sense of
identification.In the case where the space of attributesis income,this consideration may be of particular relevance. An equal number of people in two
differentincome categoriesmay possess an unequal degree of "effectiveidentification."The resourcesof the richergroupmay contributeto the abilityof that
groupto form lobbies,organizeprotests,push particularpolicies, and in general
to manifest itself as a unified entity. This extension is easily incorporatedinto
the model that we have set up here. The identificationfunctionof an individual
with income yi must now be extended to include not only the number rriwith
that income but also yi itself: I = I(ri, yi). A typical member of the extended
familyof polarizationmeasureswould then be
n
P*(7r,y) =KE
n
E
I+a
jYlYi-Yjl,
i=1 j=1
which reducesto the currentfamilyin case /8 = 0. We have not axiomatizedthis
extended family in the paper, though we conjecturethat this would not pose
serious difficulties.
Second, the alienationfelt by an individualy towardsanotherperson y' may
also depend upon the numberof individualswith the latter income. In a certain
sense, the notion of identificationalreadyincorporatesthis feature:the effective
antagonismof the latter group towards y will have been weightedby its sense
of identification.But it is certainlypossible to create additionalweighting.The
point is that we have little to say, a priori, regardingthe effect of such a
weighting on effective antagonism.Does it increase or decrease the effective
voicingof protest?In our mind,the answeris unclear,thoughsuch an extension
merits inquiry.
5.4. Methodological Remark
Perhaps the most important drawbackof our work is that it combines a
behavioralstoryof individualattitudeswith axiomaticrestrictionsplaced not on
the story itself but separatelyon distributionsof characteristics.In this sense,
there is perhaps a "deeper"axiomatizationof the concept of polarization,one
that is parallelto the theoryof inequalitymeasurement.Our effortsso far have
not been successful in uncoveringthis more fundamentaltheory. While we
believe that some progress has been achieved by motivating, defining, and
850
J.-M. ESTEBAN AND D. RAY
characterizinga new concept and its measure,it should certainlybe possible to
do better.
Instituto de Ana'lisis Economico, Universidad Autonoma de Barcelona, 08193
Bellaterra, Barcelona, Spain
and
Department of Economics, Boston University, 270 Bay State Rd., Boston, MA
02215, U.S.A.
Manuscript received March, 1991; final revision received October, 1993.
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