THEIL, INEQUALITY INDICES AND

DECOMPOSITION

Frank A. Cowell

ABSTRACT

Theil’s approach to the measurement of inequality is set in the context of

subsequent developments over recent decades. It is shown that Theil’s

initial insight leads naturally to a very general class of decomposable

inequality measures. It is thus closely related to a number of other com-

monly used families of inequality measures.

1. INTRODUCTION

Henri Theil’s book on information theory (Theil, 1967) provided a land-mark in the development of the analysis of inequality measurement. Thesignificance of the landmark was, perhaps, not fully realised for some time,although his influence is now recognised in standard references on theanalysis of income distribution. Theil’s insight provided both a method forthinking about the meaning of inequality and an introduction to an im-portant set of functional forms for modelling and analysing inequality.Theil’s structure laid the basis for much of the work that is done on de-composition by population subgroups. The purpose of this paper is to setTheil’s approach in the context of the literature that has since developed and

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PAGN: SCAN:14/4/06 12:54

Dynamics of Inequality and Poverty

Research on Economic Inequality, Volume 13, 345–360

Copyright r 2006 by Elsevier Ltd.

All rights of reproduction in any form reserved

ISSN: 1049-2585/doi:10.1016/S1049-2585(06)13012-4

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to demonstrate that its contribution is more far-reaching than is commonlysupposed.

We first introduce a framework for analysis (Section 2) and considerTheil’s approach to inequality in Section 3. Section 4 introduces a generalclass of inequality indices foreshadowed by Theil’s work and Section 5 itsproperties. Section 6 concludes.

2. ANALYTICAL FRAMEWORK

2.1. Notation and Terminology

Begin with some tools for the description of income distribution. The realnumber x denotes an individual’s income: assume that issues concerning thedefinition of the income concept and the specification of the income receiverhave been settled. Then we may speak unambiguously of an income dis-tribution. Represent the space of all valid univariate distribution functionsby F; income is distributed according to F 2 F where F has support X; aninterval on the real line R: for any x 2 X; the number F(x) represents theproportion of the population with incomes less than or equal to x.

Standard tools used in distributional analysis can be represented as func-tionals defined on F: The mean m is a functional F 7!R given by mðF Þ :¼R

xdF ðxÞ: An inequality measure is a functional I : F 7!R which is givenmeaning by axioms that incorporating criteria derived from ethics, intuition,or mathematical convenience.

2.2. Properties of Inequality Measures

Now consider a brief list of some of the standard characteristics of ine-quality measures.

Definition 1. Principle of transfers. I(G)4I(F) if distribution G can be

obtained from F by a mean-preserving spread.

In order to characterise a number of alternative structural properties ofthe functional I consider a strictly monotonic continuous function t : R 7!R

and let XðtÞ :¼ ftðxÞ : x 2 Xg \ X: A structural property of inequality meas-ures then follows by determining a class of admissible transformations T:1

Every t 2 T will have an inverse t�1 and so, for any F 2 F; we may definethe t-transformed distribution F ðtÞ 2 F such that

FRANK A. COWELL346

8x 2 XðtÞ : F ðtÞðxÞ ¼ F ðt�1ðxÞÞ (1)

F(t) is the associated distribution function for the transformed variable t(x).A general statement of the structural property is

Definition 2. T-Independence. For all t 2 T : IðF ðtÞÞ ¼ IðF Þ:

Clearly, not all classes of transformations make economic sense. How-ever, two important special cases are those of scale independence, where T

consists of just proportional transformations of income by a strictly positiveconstant, and translation independence, where T consists of just transfor-mations of income by adding a constant of any sign.

The following restrictive assumption makes discussion of many issues ininequality analysis much simpler and can be justified by appeal to a numberof criteria associated with decomposability of inequality comparisons(Shorrocks, 1984; Yoshida, 1977).

Definition 3. Additive separability. There exist functions f : X 7!R and

c : R2 7!R such that

IðF Þ ¼ c mðF Þ;Z

fðxÞdF ðxÞ

� �

(2)

Given additive separability, most other standard properties of inequalitymeasures can be characterised in terms of the income-evaluation function fand the cardinalisation function c.

However, this is just a list of properties that may or may not be satisfiedby some arbitrarily specified index. In order to make progress let us brieflyconsider the alternative ways in which the concept of inequality has beenmotivated in the economics literature.

3. THE BASIS FOR INEQUALITY MEASUREMENT

3.1. Standard Approaches

It is useful to distinguish between the method by which a concept of in-equality is derived and the intellectual basis on which the approach isfounded. The principal intellectual bases used for founding an approach toinequality can be roughly summarised as follows. First, there are ‘‘funda-mentalist’’ approaches including persuasive ad hoc criteria such as the Ginicoefficient and those based on some philosophical principle of inequality

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Theil, Inequality Indices and Decomposition 347

such as Temkin’s ‘‘complaints’’ (Temkin, 1993). Typically such approachesfocus on a concept that quantifies the distance between individual incomepairs or between each income and some reference income. Second, there areapproaches derived from an extension of welfare criteria (Atkinson, 1970;Sen, 1973). These build on standard techniques such as distributional dom-inance criteria and usually involve interpreting inequality as ‘‘welfare-waste.’’ A third approach is based on an analogy with the analysis of choiceunder uncertainty (Harsanyi, 1953, 1955; Rothschild & Stiglitz, 1973). Thisleads to methods that produce inequality indices on F that are very similarto measures of risk defined on the space of probability distributions.

3.2. The Theil Approach

Theil added a further intellectual basis of his own. He focused on inequalityas a by-product of the information content of the structure of the incomedistribution. The information-theoretic idea incorporates the followingmain components (Kullback, 1959):

1. A set of possible events each with a given probability of its occurrence.2. An information function h for evaluating events according to their as-

sociated probabilities, similar in spirit to the income-evaluation function(‘‘social utility’’?) in welfarist approaches to inequality.

3. The entropy concept is the expected information in the distribution.

The specification of h uses three axioms:

Axiom 1. Zero-valuation of certainty: h(1) ¼ 0.

Axiom 2. Diminishing-valuation of probability: p4p0 ) hðpÞohðp0Þ:

Axiom 3. Additivity of independent events: hðpp0Þ ¼ hðpÞ þ hðp0Þ:

The first two of these appear to be reasonable: if an event were consideredto be a certainty (p ¼ 1) the information that it had occurred would bevalueless; the greater the assumed probability of the event the lower thevalue of the information that it had occurred. It is then easy to establish:

Lemma 1. Given Axioms 1–3 the information function is h(p) ¼ �log(p).

In contrast to the risk-analogy approach mentioned above Theil’s appli-cation of this to income distribution replaced the concept of event-prob-abilities by income shares, introduced an income-evaluation function thatplayed the counterpart of the information function h and specified a com-

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FRANK A. COWELL348

parison distribution, usually taken to be perfect equality. The focus onincome shares imposes a requirement of homotheticity – a special case of T-independence – on the inequality measure and the use of the expected valueinduces additive separability.

Given an appropriate normalisation using the standard population prin-ciple (Dalton, 1920) this approach then found expression in the followinginequality index:

ITheilðF Þ :¼

Zx

mðF Þlog

x

mðF Þ

� �

dF ðxÞ (3)

and also the following (which has since become more widely known as themean logarithmic deviation):

IMLDðF Þ :¼ �

Zlog

x

mðF Þ

� �

dF ðxÞ (4)

The second Theil index or MLD is an example of Theil’s application of theconcept of conditional entropy; conditional entropy in effect introduces al-ternative versions of the comparison distribution and has been applied tothe measurement of distributional change (Cowell, 1980a).

3.3. Decomposition

The measures founded on the different intellectual bases discussed in Sec-tions 3.1 and 3.2 contrast sharply in their implications for inequality.

The meaning of decomposability can be explained as follows. Supposethat individuals are characterised by a pair (x, a) of income and attributes;the attributes a may be nothing more than a simple indicator of identity. Letthe attribute space be A and let P be a partition of A:

P :¼ A1;A2; . . . ; AJ :[J

j¼1

Aj ¼ A;Aj \ Ai ¼+ if iaj

( )

(5)

Let the distribution of x within subgroup j (i.e. where a 2 Aj) be denoted byF(j) and let the proportion of the population and the mean in each subgroupbe defined by

pj ¼

Z

a2Aj

dF ðx; aÞ

and

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Theil, Inequality Indices and Decomposition 349

mj ¼1

pj

Z

a2Aj

xdF ðx; aÞ

Then the minimum requirement for population decomposability is that ofsubgroup consistency, i.e. the property that if inequality increases in a pop-ulation subgroup then, other things being equal, inequality increases overall:

Definition 4. The inequality index satisfies subgroup consistency if there is

a function F such that

IðF Þ ¼ FðI1; I2; . . . ; IJ ; p;mÞ (6)

where

I j :¼ IðF ðjÞÞ

p :¼ ðp1; p2; . . . ; pJ Þ

l :¼ ðm1; m2; . . . ; mJ Þ

and where F is strictly increasing in each of its first J arguments.Note that we only need the slightly cumbersome bivariate notation in

order to explain the meaning of decomposability. Where there is no am-biguity we shall continue to write F with a single argument, income x.

Now consider each of the types of inequality measure in Section 3.1 interms of decomposability. The first group of these measures (the funda-mentalist approaches) typically results in measures that are not strictly de-composable by population subgroups: for example, it is possible to findcases where the Gini coefficient in a subgroup increases and the Gini co-efficient overall falls, violating subgroup consistency. The second group canbe made to be decomposable by a suitable choice of welfare axioms. Thethird group appears to be naturally decomposable because they are based ona standard approach to choice under uncertainty that employs the inde-pendence assumption. But clearly this conclusion rests on rather specialassumptions: it would not apply if one used a rank-dependent utility cri-terion for making choices under risk.

However, the Theil indices based on the entropy concept are naturallydecomposable by population subgroup. This property does not depend onthe additivity of independent events in the information function (Axiom 3)but because of the aggregation of entropy from the individual informationcomponents which induces additive separability. This ease of decompositionof his indices was exploited by Theil in a number of empirical applications(Theil, 1979a, b, 1989).

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FRANK A. COWELL350

3.4. A Generalisation

In their original derivation, the Theil measures in Section 3.2 use an axiom(No. 3 in the abbreviated list above) which does not make much sense in thecontext of distributional shares. It has become common practice to define

IaGEðF Þ :¼1

a2 � a

Zx

mðF Þ

� �a� 1

� �

dF ðxÞ (7)

where a 2 ð�1;þ1Þ is a parameter that captures the distributional sen-sitivity: if a is large and positive the index is sensitive to changes in thedistribution that affect the upper tail; if a is negative the index is sensitive tochanges in the distribution that affect the lower tail. The indices (3) and (4)are special cases of (7) corresponding to the values a ¼ 1, 0, respectively.Measures ordinally equivalent to those in the class with typical member (7)include a number of pragmatic indices such as the variance and measures ofindustrial concentration (Gehrig, 1988; Hart, 1971; Herfindahl, 1950).

The principal attractions of the class (7) lie not only in the generalisationof Theil’s insights but also in the fact that the class embodies some of the keydistributional assumptions discussed in Section 2.2.

Theorem 1. A continuous inequality measure I : F 7!R satisfies the prin-

ciple of transfers, scale invariance, and decomposability if and only if it is

ordinally equivalent to (7) for some a.2

However it is useful to consider the class (7), and with it the Theil indices,as members of a more general and flexible class. To do this we move awayfrom Theil’s original focus on income shares, but retain the use of T-in-dependence and additive separability.

4. A CLASS OF INEQUALITY MEASURES

4.1. Intermediate Measures

Consider now the ‘‘centrist’’ concept of inequality introduced by Kolm(1969, 1976a, b). This concept has re-emerged under the label ‘‘intermediateinequality’’ (Bossert, 1988, 1990) QA :1. As the names suggest, centrist conceptshave been shown to be related in limiting cases to measures described as‘‘leftist’’ or ‘‘rightist’’ in Kolm’s terminology; intermediate inequality meas-ures in their limiting forms are related to ‘‘relative’’ and ‘‘absolute’’ meas-ures. However, a general treatment of these types of measures runs into a

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Theil, Inequality Indices and Decomposition 351

number of difficulties. In some cases, the inequality measures are well-de-fined only with domain restrictions. The nature of these restrictions is fa-miliar from the well-known relative inequality measures which are definedonly for positive incomes. In the literature, results on the limiting cases areavailable for only a subset of the potentially interesting ordinal inequalityindices.

In what follows we consider a general structure that allows one to addressthese difficulties, that will be found to subsume many of the standard fam-ilies of decomposable inequality measures, and that shows the inter-rela-tionships between these families and Theil’s fundamental contribution.

4.2. Definitions

We consider first a convenient cardinalisation of the principal type of de-composable inequality index:

Definition 5. For any a 2 ð�1; 1Þ and any finite k 2 Rþ an intermediatedecomposable inequality measure is a function Ia;kint : F7!R such that

Ia;kint ðF Þ ¼1

a2 � a

Zxþ k

mðF Þ þ k

� �adF ðxÞ � 1

� �

(8)

Intermediate measures have usually appeared in other cardinalisations,for example,

½1þ k�½1� ½1þ ½a2 � a�Ia;kint ðF Þ�1=a�

(Bossert & Pfingsten, 1990; Eichhorn 1988).3 From (8) we may charac-terise a class of measures that are of particular interest.

Definition 6. The intermediate decomposable class is the set of functions

I :¼ fIa;kint : a 2 ð�1; 1Þ; 0oko1g (9)

where Ia;kint is given by definition 5.

The set I can be generalised in a number of ways. Obviously one couldrelax the domain restrictions upon the sensitivity parameter a and the lo-cation parameter k. But more useful insights can be obtained if the pos-sibility of a functional dependence of a upon k is introduced. Let T0 � T bethe subset of affine transformations and consider a 2 T0 such that

aðkÞ :¼ gþ bk (10)

where g 2 R;b 2 Rþ: Distributional sensitivity depends upon the location

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FRANK A. COWELL352

parameter k. Then the class T in (9) is equivalent to a subset of the followingrelated class of functions:

Definition 7. The extended intermediate decomposable class is the set of

functions.

I :¼ fIa;kextðF Þ : a 2 T0; k 2 Rg (11)

where

Ia;kextðF Þ :¼ yðkÞZ

xþ k

mðF Þ þ k

� �aðkÞ� 1

" #

dF ðxÞ (12)

and y(k) is a normalisation term given by

yðkÞ :¼1þ k2

aðkÞ2 � aðkÞ(13)

Note that (12) adopts two special forms for the cases a(k) ¼ 0, 1. Ifa(k)-0 applying L’Hospital’s rule shows that the limiting form (12) is

½1þ k2�

Zlog

mðF Þ þ k

xþ k

� �� �

dF ðxÞ

Likewise, if a(k) ¼ 1 (12) becomes

½1þ k2�

Zxþ k

mðF Þ þ k

� �

logxþ k

mðF Þ þ k

� �� �

dF ðxÞ

The class of I will be the primary focus of the rest of the paper.

5. PROPERTIES OF THE CLASS

The class of extended intermediate decomposable measures possesses severalinteresting properties and contains a number of important special cases.

First, it has the property that it is T0-independent.Second, the class is decomposable. Decomposition by population sub-

groups of any member of I can be expressed in a simple way. Again take apartition P consisting of a set of mutually exclusive subgroups of the pop-ulation indexed by j ¼ 1, 2,y, J as in (5), so that inequality in subgroup j is

Ia;kextðFðjÞÞ :¼ y ðkÞ

Zxþ k

mðF ðjÞÞ þ k

� �aðkÞ� 1

" #

dF ðjÞðxÞ (14)

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Theil, Inequality Indices and Decomposition 353

The inequality in the whole population can be broken down as follows:

Ia;kextðF Þ ¼XJ

j¼1

wjIa;kextðF

ðjÞÞ þ Ia;kextðFPÞ (15)

where wj is the weight to be put on inequality in subgroup j:

wj :¼mðF ðjÞÞ þ k

mðF Þ þ k

� �aðkÞ(16)

and FP represents the distribution derived concentrating all the populationin subgroup j at the subgroup mean m(F(j)) so that between-group inequalityis given by

Ia;kextðFPÞ ¼ yðkÞXJ

j¼1

mðF ðjÞÞ þ k

mðF Þ þ k

� �aðkÞ� 1

" #

(17)

The decomposition relation (15) is clearly easily implementable empiri-cally for any given value of the parameter pair (a, k).

Third, notice that the measure (12) can be written in the form (2) thus

yðkÞZ

fðxÞfðmðF ÞÞ

dF ðxÞ � 1

� �

(18)

where the income-evaluation function f is given by

fðxÞ ¼1

aðkÞ½xþ k�aðkÞ (19)

and a(k), y(k) are as defined in (10) and (13): the income-evaluation functioninterpretation is useful in examining the behaviour of the class of inequalitymeasures in limiting cases of the location parameter k. The important spe-cial cases of Ia;kext (F) correspond to commonly used families of inequalitymeasures: the generalised entropy indices are given by Ia;0ext

� �(Cowell, 1977);

the Theil indices (Theil, 1967) are a subset of these given by the casesa(0) ¼ 1 and a(0) ¼ 0 (see Eqs. (3) and (4) respectively); the Atkinson indices(Atkinson, 1970) are ordinally equivalent to a subset of Ia;0ext

� �:

1� 1þ1

yð0ÞIa;0ext

� �1=að0Þ; að0Þo1

There are other measures that can be shown to belong to this class forcertain values of the location parameter k. However, here we encounter aproblem of domain for the income-evaluation function f. This problem

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FRANK A. COWELL354

routinely arises except for the special case where a(k) is an even positiveinteger4; otherwise one has to be sure that the argument of the power func-tion used in (19) is never negative. Because of this it is convenient to discusstwo important subcases.

5.1. Restricted Domain: x Bounded Below

We first consider the case that corresponds to many standard treatments ofthe problem of inequality measurement: �k � infðXÞ: This restriction en-ables us to consider what happens as the location parameter goes to (pos-itive) infinity.

Theorem 2. As k-N the extended intermediate inequality class (11) be-

comes the class of Kolm indices

IbKðF Þ : b 2 Rþ

n o

where

IbKðF Þ :¼

1

b

Zeb½x�mðF Þ�dF ðxÞ � 1

� �

(20)

Proof. To examine the limiting form of (12) note that the parameterrestriction ensures that, for finite x 2 X and k sufficiently large, we havexk2 ð�1; 1Þ: So, consider the function

wðx; y; a; kÞ :¼ logfðxÞfðyÞ

� �

¼ aðkÞ log 1þx

k

� �� log 1þ

y

k

� �h i(21)

Using the standard expansion

logð1þ tÞ ¼ t�t2

2þ

t3

3� ::. (22)

and (10) we find

wðx; y; a; kÞ ¼ bþgk

h ix� y�

x2

2kþ

y2

2kþ

x3

2k2�

y3

2k2� . . .

� �

(23)

For finite g, b, x, y we have:

limk!1

wðx; y; a; kÞ ¼ b½x� y� (24)

and

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Theil, Inequality Indices and Decomposition 355

limk!1

yðkÞ ¼ limk!1

1þ 1k2

bþ gk

2� 1

kbþ g

k

¼1

b2(25)

So we obtain

limk!1

Ia;kextðF Þ ¼1

b2

Z½expðb½x� mðF Þ�Þ � 1� dF ðxÞ (26)

’

This family of Kolm indices form the translation-invariant counterpartsof the family (7) (Eichhorn & Gehrig, 1982; Toyoda, 1980).5

Theorem 3. As k-N and b-0 (12) converges to the variance.

Proof. An expansion of (26) gives

Z1

2!½x� mðF Þ�2 þ

1

3!b½x� mðF Þ�3 þ

1

4!b2½x� mðF Þ�4 þ :::

� �

dF ðxÞ

As b-0 this becomes the variance. ’

5.2. Restricted Domain: x Bounded Above

A number of papers in the mainstream literature make the assumption thatthere is a finite maximum income.6 If we adopt this assumption then itmakes sense to consider parameter values such that �k � supðXÞ: However,it is immediate that the new parameter restriction again ensures that, forfinite x 2 X and (�k) sufficiently large, we have x

k2 ð�1; 1Þ: Therefore, the

same argument can be applied as in Eqs. (21)–(26) above: again the eval-uation function converges to that of the Kolm class of leftist inequalitymeasures.

The behaviour of the evaluation function f as the location parameterchanges is illustrated in Fig. 1: the limiting form is the heavy line in themiddle of the figure. As k-+N, the evaluation functions of the I classapproach this from the bottom right; as k-�N, the evaluation functionsapproach it from the top left. Fig. 2 shows the relationship of overall in-equality to the parameter k when income is distributed uniformly on the unitinterval: note that the limiting case (where the inequality measure is ordin-ally equivalent to the ‘‘leftist’’ Kolm index) is given by the point 1/k ¼ 0.

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FRANK A. COWELL356

5.3. Interpretations

The reformulation (12) is equivalent to (8) in that, given any arbitrary valuesof the location parameter k and the exponent in the evaluation function (19),one can always find values of g, b such that a(k) ¼ g+bk. Clearly, there is aredundancy in parameters (for finite positive k one can always arbitrarily fix

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0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-1 -0.5 0 0.5 1

1/k

Ineq

ualit

y

Fig. 2. Inequality and k For a Rectangular Distribution on [0,1].

φ(x

)/φ(

µ(F

))

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.2 0.4 0.6 0.8 1x

k = -1k = -2k = -10limitk = 10k = 2k = 1

Fig. 1. Values of f(x)/f(m(F)) as k Varies: X½0; 1�; g ¼ 0.5, b ¼ 2, and m(F) ¼ 0.5.

Theil, Inequality Indices and Decomposition 357

either g or b), but that does not matter because the important special casesdrop out naturally as we let k go to 0 (Generalised Entropy) or to N

(Kolm). Of course the normalisation constant y(k) could be specified insome other way for convenience, but this does not matter either.

The general formulation allows one to set up a correspondence betweenthe Generalised Entropy class of measures, including the Theil indices andthe Kolm leftist class of measures (k ¼N). Consider, for example, thesubclass that is defined by the restriction b ¼ g

Ib;kext ðF Þ :¼ yðkÞ

Zxþ k

mðF Þ þ k

� �b½1þk�

� 1

" #

dF ðxÞ (27)

By Putting k ¼ 0 one immediately recovers the Generalised Entropy classwith parameter b. However, letting k-N Theorem 2 gives the Kolm indexwith parameter b.

6. CONCLUSION

Theil’s seminal contribution led to a way of measuring inequality that hasmuch in common with a number of families of indices that have becomestandard tools in the analysis of income distribution. Indeed, in examiningsome of the widely used families of inequality indices, it is clear that arelatively small number of key properties characterise each family and thesets of characteristic properties bear a notable resemblance to each other.However, Theil’s approach has a special advantage in that his basis formeasuring inequality naturally leads to a decomposable structure, whereasdecomposability has to be imposed as an extra explicit requirement in al-ternative approaches to inequality. This paper has further shown that manyof these standard families of inequality measures are in fact related to theoriginal Theil structure.

UNCITED REFERENCES

Chakravarty & Tyagarupananda (1998); Chakravarty & Tyagarupananda(2000); Cowell (1988).

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NOTES

1. See Ebert (1996) for a detailed discussion of this concept.2. See Bourguignon (1979), Cowell (1980b), and Shorrocks (1980, 1984).3. See for example Bossert and Pfingsten (1990 p. 129) where the definition (in the

present notation) is given as ½1þ k�½1�R½xþ k=mðF Þ þ k�adF ðxÞ�1=a: Kolm’s stand-

ard formulation (Kolm, 1976a, p. 435) is found by multiplying this by a factormðF Þ þ k=1þ k; Kolm has suggested a number of other cardinalisations (Kolm,1996, p. 17).4. This condition is very restrictive. Indices with values of a(k)Z4 are likely to be

impractical and may also be regarded as ethically unattractive, in that they are verysensitive to income transfers amongst the rich and the super-rich.5. See Foster and Shneyerov (1999).6. See for example Atkinson (1970).

ACKNOWLEDGMENT

I am grateful to Udo Ebert for helpful discussions on an earlier version.

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