Does the Lorenz Curve Really Measure Inequality?Another Look at Inequality Measurement]

This version, January 2005.

Alain Chateauneuf† and Patrick Moyes‡

Abstract

We propose in the paper an alternative approach to inequality and welfare measurement whichexploits the flexibility of the dual model of choice under risk of M. Yaari (Econometrica 55 (1987),99–115). Typical welfare and inequality measurement is required to be Lorenz consistent whichguarantees that inequality decreases and welfare increases as a result of a progressive transfer. Wechallenge this conventional practice by substituting the weaker differentials, deprivation and satis-faction [absolute] quasi-orderings for the Lorenz criterion. To shed light on the normative contentof these new inequality quasi-orderings we identify for each criterion the elementary transforma-tions – analogous to progressive transfers in the traditional approach – that allow one to derive ina finite number of steps the dominating distribution from the dominated one. Restricting next at-tention to distributions of equal means, we show that the utilitarian model – the so-called expectedutility model in the theory of risk – does not permit one to make a distinction between the viewsembedded in the differentials, deprivation, satisfaction and Lorenz quasi-orderings. In contrast itis possible within the Yaari model to derive the restrictions to be placed on the weighting functionwhich guarantee that the corresponding welfare orderings are consistent with these quasi-orderings.Finally we drop the equal mean condition and indicate the implications of our approach for theabsolute ethical inequality indices.Journal of Economic Literature Classification Number: D31, D63. Keywords: Income Differentials,Deprivation, Satisfaction, Lorenz Dominance, Progressive Transfers, Expected Utility, Dual Modelof Choice Under Risk, Generalized Gini Social Welfare Functions.

1. Introduction

1.1. Motivation and Relationship to the Literature

Following Atkinson (1970) and Kolm (1969) there is a wide agreement in the literature toappeal to the Lorenz curve for measuring inequality. A distribution of income is typicallyconsidered as being no more unequal than another distribution if its Lorenz curve lies nowherebelow that of the latter distribution. Besides its simple graphical representation, much of thepopularity of the so-called Lorenz criterion originates in its relationship with the notion ofprogressive transfers. It is traditionally assumed that inequality is reduced by a progressivetransfer i.e., when income is transferred from a richer to a poorer individual whithout affectingtheir relative positions on the ordinal income scale. The principle of tranfers, which capturesthis judgement, is closely associated with the Lorenz quasi-ordering of distributions of equal

] The present document constitutes the basis of a communication entitled “Gini or Lorenz: Does it Make aDifference?” to be presented by the second author at the International Conference to Commemorate C. Giniand M.O. Lorenz Centenary Scientific Research, University of Siena, Italy, 23–26 May 2005.

† CERMSEM, Universite Paris 1 Pantheon-Sorbonne, 106-112 Boulevard de l’Hopital, F-75647 Paris Cedex13, France, Tel. (+33) [0]1.55.43.42.97, Fax. (+33) [0]1.55.43.43.01, Email. chateaun@univ-paris1.fr.

‡ CNRS, IDEP and GRAPE, Universite Montesquieu Bordeaux IV, Avenue Leon Duguit, F-33608 Pessac,France, Tel. (+33) [0]5.56.84.29.05, Fax. (+33) [0]5.56.84.29.64, Email. moyes@u-bordeaux4.fr, HomePage:http://moyes.u-bordeaux4.fr.

means. Indeed half a century ago, Hardy, Littlewood and Polya (1952) have demonstratedthat, if a distribution Lorenz dominates another distribution, then the former can be obtainedfrom the latter by means of a finite sequence of progressive transfers, and conversely1. Thisrelationship between progressive transfers and the Lorenz quasi-ordering constitutes the cor-nerstone of the modern theory of welfare and inequality measurement. As a consequence, theliterature has concentrated on Lorenz consistent inequality measures i.e., indices such that aprogressive transfer is always recorded as reducing inequality or increasing welfare.

Notwithstanding its wide application in theoretical and empirical work it must be recog-nized that the approach based on the Lorenz curve is neither the only possibility nor it isimmune to criticism. On the one hand, there is evidence that the social status of an individ-ual – approximated by her position in the social hierarchy – plays an important role in herfeeling of well-being [see e.g. Weiss and Fershtman (1998)]. Attitudes such as envy, depriva-tion and resentment have been argued to be important components of individual judgementsand they might be taken into account as far as distributive justice is concerned. The notionof individual deprivation originating in the work of Runciman (1966) accommodates suchviews by making the individual’s assessment of a given social state depend on her situationcompared with the situations of all the individuals who are treated more favourably than her.The deprivation profile, which indicates the level of deprivation felt by each individual, mighttherefore constitute the basis of social judgement. Drawing upon previous work by Yitzhaki(1979, 1982), Hey and Lambert (1980), Kakwani (1984), and Chakravarty (1997), one canpropose different deprivation quasi-orderings depending on the way individual deprivation isdefined. Individual deprivation in a given state formally ressembles the aggregate povertygap where the poverty line is set equal to other individuals’ incomes2. So stated, one mayconceive of absolute individual deprivation, which is simply the sum of the gaps between theindividual’s income and the incomes of all individuals richer than her, and relative depriva-tion, where the income gaps are deflated by the individual’s income [see Chakravarty andMoyes (2003)]. Then the deprivation quasi-ordering is based on the comparisons of the indi-vidual deprivation profiles – equivalently deprivation curves – and social deprivation decreasesas the curves move downwards.

Rather than comparing herself with individuals who are richer than her – equivalentlywho occupy a higher position on the social status scale – an individual can consider as thereference group those who are poorer. Then she may find some comfort by realizing thatthere are individuals who receive less income than she does and, the larger the aggregate gapbetween her income and the incomes of the poorer individuals, the higher her satisfaction

1Although parts of this general result appeared in different places in Hardy, Littlewood and Polya (1952),one had to wait until Berge (1963) who collected these scattered statements and provided a self-containedproof of what is known now as the Hardy-Littlewood-Polya theorem. Related results in the field of inequalitymeasurement have been provided by Kolm (1969), Atkinson (1970), and Fields and Fei (1978) among others[see also Dasgupta, Sen and Starrett (1973), Sen (1973), and Foster (1985)].2Most scholars take for granted that individual deprivation is simply the sum – possibly normalized in asuitable way – of the income gaps between the individual’s income and the incomes of all individuals richerthan her. An axiomatic characterization of the absolute deprivation profile is provided by Ebert and Moyes(2000).

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will be. Absolute individual satisfaction will be assimilated with the sum of the gaps betweenthe individual’s income and the incomes of all individuals poorer than her, while relativeindividual satisfaction obtains whene the income gaps are deflated by the individual’s ownincome [see Chakravarty (1997), Chakravarty and Moyes (2003)]. The notion of satisfactionmay be considered the dual of the notion of deprivation: for satisfaction the reference groupevery individual compares herself consists of the poorer individuals, while it comprises allricher individuals in the case of deprivation. Then the satisfaction quasi-ordering is based onthe comparisons of the individual satisfaction curves and social satisfaction decreases as theindividual satisfaction curve moves downwards. Such feelings as deprivation and satisfactionmay exacerbate the tensions in the society and ultimately lead to conflicts. A natural objectiveof the society will be to make individual satisfaction and deprivation as small as possible, theminimum being attained when all incomes are equal.

On the other hand whereas most of the literature on inequality and welfare measurementimposes the principle of transfers, one may however question the ability of such a conditionto capture the very idea of inequality in general. Indeed, though a progressive transferdefinitively reduces inequality between the individuals involved in the transfer, it generallywidens at the same time the income gaps between each of these two individuals and restof the society3. Anyone who considers that all [pairwise] inequalities have to be countedwould find difficult to admit that inequality on the whole has declined. It is to some extentsurprising that the profession has been assimilating overall inequality reduction with localpairwise inequality reduction for such a long time. The fact that progressive transfers arenot universally approved has been confirmed by recent experimental studies which find thata majority of respondents reject the principle of transfers [see e.g. Amiel and Cowell (1992),Ballano and Ruiz-Castillo (1993), Harrison and Seidl (1994), Gaertner and Namezie (2003)among others].

However the experimental studies fail to provide information about the subjects’ prefer-ences towards equality with the exception that these preferences are at variance with theviews captured by the principle of tranfers used in the theory of inequality measurement.Different ideas come to mind in order to reconcile the theory with the conclusions of theseexperimental studies. The first idea that comes to mind is to declare that inequality un-ambiguously decreases if and only if the income differentials between any two individuals inthe population are reduced, assuming that all individuals occupy the same positions on theincome scale in the situations under comparison. This is a kind of unanimity point of view:overall inequality decreases if and only if the inequalities between any two individuals in thesociety decrease. This rules out the limitation of the principle of transfers we pointed outabove since now, not only should the gap between the donor and the recipient of a transfer bereduced, but also the gaps between these two individuals and the individuals not taking partin the transfer. This still leaves open the question to know which kind of income differentialsare thought of relevance when making inequality judgements. The relative and absolute dif-

3The only situation where this cannot happen is when the progressive transfer takes place between the richestand the poorest individuals or when the society comprises only two individuals, which are very special cases.

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ferentials quasi-orderings introduced by Marshall, Olkin and Proschan (1967) constitute twopossible candidates4.

1.2. The Theoretical Approach Developed in the Paper

The paper aims at examining the consequences for the measurement of inequality and welfareof substituting the differentials, deprivation and satisfaction quasi-orderings for the Lorenzone. The preceding discussion suggests that these alternative criteria are more likely to besupported by the public than the Lorenz criterion. However the precise value judgements theyintroduce as well as the way they shape preferences towards equality remain vague. A firstobjective of the paper is to identify the elementary transformations of income distributionsthat imply inequality reduction according the differentials, deprivation and satisfaction quasi-orderings, respectively. We further want these transformations to have the property that,if one distribution is ranked higher than another one by a criterion, then a sequence ofsuch transformations can be found that guarantees that the dominating distribution can beobtained from the dominated one. This makes things difficult especially because we would likethese transformations to be as simple as possible. Although these transformations compriseprogressive transfers as particular cases, we will show in the paper that they involve in generalmore than just one progressive transfer. Contrary to progressive transfers that impose noconstraints on the choice of the individuals taking part into the transfer – with the exceptionthat the beneficiary is poorer than the donor – these new transformations require somesolidarity among the participants involved.

While most experimental studies indicate that a majority of respondents reject the transferprinciple they do not tell anything about the way their preferences look like. They do notgive indications about the principles that explain the subjects’ attitudes regarding inequalitythat we observe in the experimental studies. Assuming that we subscribe to these alternativenotions of inequality, a second objective of the paper is to identify the preferences that areconsistent with the differentials, deprivation and satisfaction quasi-orderings, respectively.The literature on economic inequality has mainly focused up to now on two general classes ofpreferences: those that can be represented by the utilitarian social welfare function and thosethat can be rationalized by the [generalized] Gini social welfare function. These approachesare related respectively to the expected utility model and the dual model of choice due toYaari (1987) in the risk literature, which constitute particular cases of the rank-dependentexpected utility model introduced by Quiggin (1993)5. We will follow this practice andexplore the implications for the utilitarian and the generalized Gini social welfare functions of

4There are other possible views – e.g. along the lines suggested by Bossert and Pfingsten (1990) – that mightconstitute alternative grounds for constructing a theory of inequality measurement less at variance with thepublic’s attitudes.5The utilitarian model has a long tradition in the inequality literature and goes back at least to Dalton (1920),who suggested to assimilate the degree of inequality with the welfare loss caused by unevenly distributedincomes. However one had to wait until Atkinson (1970) who refined this suggestion and definitively imposedit in the field of inequality measurement. The dual model was introduced in the fields of choice under riskand subsequently applied to the measurement of inequality in Yaari (1988). Independently Ebert (1988) andWeymark (1981) developed related approaches in the inequality literature.

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requiring the underlying preferences to conform with the views expressed by the differentials,deprivation and satisfaction quasi-orderings. It will be argued that the standard expectedutility model does not permit to distinguish between the different concepts of inequalitydiscussed above. In other words, the utilitarian social welfare function is not sufficientlyflexible to accommodate such distinct attitudes as those encompassed by the differentials,deprivation and satisfaction quasi-orderings. On the contrary the dual model of choice permitsto characterize the preferences that are consistent with these quasi-orderings. More precisely,the paper identifies the restrictions to be imposed on the weighting function that guaranteethat welfare does not decrease and inequality will not increase when incomes are more equallydistributed according to the three former quasi-orderings when one considers equal meandistributions6. The concepts of inequality and welfare are no longer unambiguously relatedwhen the distributions under comparison have different means since now an increase in welfaredoes not necessarily imply that inequality decreases and conversely. The ethical approach toinequality measurement, which derives the inequality index from the social welfare function,guarantees that the results above carry over in the case of inequality measurement7.

1.3. Organization of the Paper

Section 2 presents our conceptual framework consisting of distributions for finite populationsof possibly different sizes where every individual is associated with a given income. In addi-tion to the Lorenz criterion, we distinguish different inequality views which we identify withquasi-orderings defined on the set of income distributions. Section 3 introduces the Lorenzcriterion and the differentials, deprivation and satisfaction quasi-orderings, which are consid-ered alternative criteria for making inequality comparisons. After having formally definedthese criteria we explore their relationships and show that they are all compatible with theLorenz quasi-ordering when distributions have equal means. We examine in Section 4 dif-ferent ways of weakening the notion of a progressive transfer or equivalently of strengheningthe principle of transfers. For each of our criteria we identify the elementary transformationswhich imply dominance and which make one able to derive the dominating distribution fromthe dominated one in a finite number of steps. Section 5 investigates the implications for thesocial welfare functions of the inequality views captured by the differentials, deprivation andsatisfaction quasi-orderings in the particular case where distributions have equal means. It isshown that the utilitarian model does not allow to distinguish between these views and thetraditional one captured by the Lorenz quasi-ordering. On the contrary the Yaari model –equivalently the generalized Gini social welfare function – allows the ethical planner to makea distinction between these competing views and we identify the classes of welfare functionsthat are consistent with each of these inequality views. Section 6 indicates how the analysis

6Up to now the standard principle of transfers has always been imposed in the dual model and its flexibilityhas yet not been fully exploited in the inequality and welfare literature. The flexibility properties that havebeen considered up to now are related to: (i) the impact of favorable composite transfers on inequality [Zoli(2002), Chateauneuf and Wilthien (1999)], and (ii) the evaluation of welfare in the case of heterogeneouspopulations [Zoli (1999)].7See Blackorby, Bossert and Donaldson (1999) for a recent survey of the literature on the ethical approachto inequality measurement.

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can be extended in order to cover the general case where the distributions under comparisondo not necessarily have the same mean. Finally we summarize our results in Section 7 whichalso hints at some directions for future work.

2. Preliminary Notation and Definitions

We assume throughout that incomes are drawn from an interval D which is a compact subsetof R. An income distribution or situation for a population consisting of n identical indi-viduals (n ≥ 2) is a list x : = (x1, x2, . . . , xn) where xi ∈ D is the income of individual i.For later reference we indicate by 1n : = (1, . . . , 1) the unit vector in Rn. Letting Yn(D)represent the set of income distributions for a population of size n, the set of all incomedistributions of finite size will be denoted as Y(D) :=

⋃∞n=2 Yn(D). The dimension of dis-

tribution x ∈ Y(D) is indicated by n(x) and its arithmetic mean by µ(x) : =∑n(x)

i=1 xi/n(x).Given x : =

(x1, . . . , xn(x)

) ∈ Y(D), we use x( ) : =(x(1), x(2), . . . , x(n(x))

)to indicate its non-

decreasing rearrangement defined by x( ) = Px for some n(x) × n(x) permutation matrixP such that x(1) ≤ x(2) ≤ · · · ≤ x(n(x))

8. We denote as F ( · ;x) the cumulative distributionfunction of x ∈ Y(D) defined by F (z;x) : = q(z;x)/n(x), for all z ∈ (−∞, +∞), whereq(z;x) : = #

{i ∈ {1, 2, . . . , n(x)} ∣∣ x(i) ≤ z

}. We let F−1( · ;x) represent the inverse cumu-

lative distribution function – equivalently the quantile function – of x obtained by lettingF−1(0;x) := x(1) and

(2.1) F−1(p;x) : = Inf{z ∈ (−∞, +∞)

∣∣ F (z;x) ≥ p}, ∀ p ∈ (0, 1]

[see Gastwirth (1971)].

We are interested in the comparisons of income distributions from the point of view ofsocial welfare and inequality. To this purpose we assume the existence of an ethical observerendowed with an [ethical] preference ordering – a complete, reflexive and transitive binaryrelation on the set of income distributions – that captures her views or attitudes concerningthe desirability of alternative income distributions. Her [ethical] preferences are representedby a social welfare function W : Y(D) −→ R that associates to every distribution a realnumber W (x) that represents the social welfare attained in situation x ∈ Y(D). WhenW (x) ≥ W (y) we will say that situation x is at least as good as y from the point of viewof W . Under certain conditions it is possible to associate to the social welfare function aninequality index I : Y(D) −→ R that indicates for every distribution the degree of inequalityattained with the convention that I(x) ≤ I(y) means that situation x is no more unequalthan situation y9. Here social welfare functions and inequality indices are just particularcardinal representations of the ethical observer’s preferences, and no cardinal significance

8A permutation matrix P : = [ pij ] is an n × n (some n ≥ 2) matrix such that (i) for all i, j, either pij = 0,or pij = 1, (ii) for all i, pij = 1 implies pik = 0, for all k 6= j, and (iii) for all j, pij = 1 implies phj = 0, for

all h 6= i.9This is the so-called ethical approach to inequality measurement where inequality is assimilated with theincome loss due to the fact that incomes are unevenly distributed in the society [see Blackorby, Bossert andDonaldson (1999) for a survey].

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should be attributed to the values taken by these indices. We denote as W(D) and I(D) thesets of social welfare functions and inequality indices respectively.

Although our primary concern is to make comparisons of arbitrary distributions whosedimensions may differ, it is worth emphasizing that there is no loss of generality restrictingattention to distributions with the same dimension. This is a consequence of the principleof population according to which a replication does not affect inequality and welfare [seeDalton (1920)]10. Similarly because all individuals are identical in all respects other thantheir incomes, one usually imposes the condition of symmetry which requires that exchangingincomes between two individuals would not affect the levels of welfare and inequality11.

In a number of cases it is impossible to reach a unanimous agreement regarding theappropriate ordering of situations and the largest consensus that one might reasonably expectmay only consist of a partial ranking of the situations under comparison. Such limitedagreements are represented by quasi-orderings i.e., reflexive, transitive and incomplete binaryrelations defined on the set of distributions12. Given the quasi-ordering ≥J over Y(D), wedenote as >J and ∼J its asymmetrical and symmetrical components defined in the usualway. We assume that a quasi-ordering expresses primitive views concerning inequality thatthe ethical observer may adhere to or not. Precisely, given a quasi-ordering ≥J over Y(D)and a social welfare function W ∈ W(D) [resp. an inequality index I ∈ I(D)], we will saythat W [resp. I] is consistent with ≥J , if

(2.2) ∀ x,y ∈ Y(D) : x ≥J y =⇒ W (x) ≥ W (y) [resp. I(x) ≤ I(y)].

All the quasi-orderings ≥J we consider in the paper have the property that (i) xr ∼J x, forall x ∈ Y(D) and all r ∈ N (r ≥ 2), and (ii) Px ∼J x, for all x ∈ Y(D) and all n(x)× n(x)permutation matrices P . Therefore there is no loss of generality in restricting attention todistributions in Yn(D) with n ≥ 2 that are non-decreasingly arranged.

3. From Lorenz to Alternative Inequality Views

3.1. An Introductory Example and Some Definitions

It is typically assumed in normative economics that inequality is reduced and welfare increasedby a transfer of income from a richer individual to a poorer individual. More precisely, wehave:

Definition 3.1. Given two income distributions x,y ∈ Yn(D), we will say that x is obtainedfrom y by means of a progressive transfer , if there exists ∆ > 0 and two individuals i, j such

10Given two distributions x,y ∈ Y(D) we will say that x is a replication of y if there exists r ∈ N (r ≥ 2)

such that x = yr : = (y; . . . ;y) ∈[Yn(y)(D)

]r. Then the index M ∈ W(D) ∪ I(D) satisfies the principle of

population if, for all x ∈ Y(D) and all r ∈ N (r ≥ 2): M (xr) = M(x).11Precisely the index M ∈ W(D) ∪ I(D) satisfies the condition of symmetry if, M (Px) = M(x), for allx ∈ Y(D) and all n(x)× n(x) permutation matrices P .12We adopt throughout the paper the terminology proposed by Sen (1970, Chap. 1*), but we recognize thatthere are other possibilities.

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that

xk = yk, ∀ k 6= i, j;(3.1.a)

xi = yi + ∆; xj = yj −∆; and(3.1.b)

∆ ≤ (yj − yi)/ 2.(3.1.c)

By definition, a progressive transfer does not reverse the relative positions of the individualsinvolved. However, although the donor cannot be made poorer than the recipient, it may bethe case that their positions relative to the positions of the other individuals are modified.It is convenient to assume that the progressive transfer is rank-preserving in the sense thatthe relative positions of all the individuals are unaffected, which amounts to impose theadditional condition:

(3.2) (xk − xh) (yk − yh) ≥ 0, ∀ h 6= k.

Principle of Transfers. For all x,y ∈ Yn(D), we have W (x) ≥ W (y) and I(x) ≤ I(y),whenever x is obtained from y by a [rank-preserving] progressive transfer.

The notion of a progressive transfer is closely associated with that of the Lorenz quasi-ordering. The Lorenz curve of distribution x ∈ Yn(D), which we denote as L(p;x), is definedby

(3.3) L(p;x) : =∫ p

0

F−1(s;x) ds, ∀ p ∈ [0, 1].

By definition L(p;x) represents the total income possessed by the p fraction of poorest indi-viduals deflated by the population size in situation x13.

Definition 3.2. Given two income distributions x,y ∈ Yn(D), we will say that x Lorenzdominates y, which we write x ≥L y, if and only if

(3.4) L(p;x) ≥ L(p;y), ∀ p ∈ (0, 1) and L(1;x) = L(1;y).

The higher its associated Lorenz curve, the less unequal a distribution is according to theLorenz criterion. Condition (3.4) can be equivalently rewritten as

(3.5)1n

k∑

j=1

xj ≥ 1n

k∑

j=1

yj , ∀ k = 1, 2, . . . , n− 1, and µ(x) = µ(y).

As we already insisted, much of the popularity of the Lorenz criterion originates in the factthat it is closely associated with progressive transfers. Hardy, Littlewood and Polya (1952)were the first to show that a distribution Lorenz dominates another one if and only if it canbe obtained from the latter by means of successive applications of progressive transfers [seealso Berge (1963), Kolm (1969), Fields and Fei (1978) and Marshall and Olkin (1979) amongothers]. Precisely, they proved the following:

13Our definition of the Lorenz curve is different from the standard one which requires that the total incomepossessed by the p fraction of poorest individuals is deflated by the total income. The difference is immaterialas long as we are interested in the comparison of distributions with equal means.

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Proposition 3.1. Let x,y ∈ Yn(D) such that µ(x) = µ(y). Then, the following two

statements are equivalent:

(a) x is obtained from y by means of a finite sequence of progressive transfers.

(b) x ≥L y.

It is important to note that there is no particular restriction imposed on the way the progres-sive transfers are combined: any sequence of progressive transfers results in an improvementin terms of Lorenz dominance. A direct implication of Proposition 3.1 is that any measurethat verifies the principle of transfers is Lorenz-consistent [see e.g. Foster (1985)].

However despite its wide use in theoretical and empirical work on inequality the Lorenzquasi-ordering is not the only criterion that can be used in order to rank distributions inan unambiguous way. Other criteria have appeared in the literature that pay attention todifferent features of the distributions under comparisons and that may be considered poten-tial candidates for measuring inequality. Among these most noteworthy are the deprivationand satisfaction indices – due to Kakwani (1984) and Chakravarty (1997) respectively – thatare concerned with the feelings of the individuals with respect to their personal situationscompared to the situations of the other individuals. On the other hand in some circum-stances the application of the Lorenz quasi-ordering has implications that go far beyond thejudgements embedded in this criterion. A typical example is given in public finance, where ithas been shown that a non-decreasing average tax rate is a necessary and sufficient conditionfor the after tax distribution to Lorenz dominate the before tax distribution whatever thecircumstances [see Jakobsson (1976)]. Actually the application of the Lorenz criterion impliesthat all pairwise relative income differences be not larger in the after tax distribution than inthe before tax one [see Moyes (1994)]. Such a test involving the comparisons of all pairwisedifferences – leaving aside for the moment the way we measure these differerences – appearsto be an uncontroversial criterion for passing inequality judgements.

Therefore the Lorenz quasi-ordering does not exhaust all the possibilities for measuringinequality and one may equally think of alternative criteria. However substituting another cri-terion such as one of those mentioned above for the Lorenz quasi-ordering calls into questionthe very principle of transfers that is at the heart of the theory of inequality measurement.Indeed someone who accepts the views embedded in either of the differentials, deprivationand satisfaction quasi-orderings may not agree on the fact that a progressive transfer de-creases inequality in all circumstances. This is a direct implication of Proposition 3.1 whichestablishes the equivalence between Lorenz dominance and the existence of a sequence ofprogressive transfers. This has been best exemplified in a number of experimental studies bymeans of questionnaires where it has been demonstrated that the principle of transfers is re-jected by a majority of respondents [see Amiel and Cowell (1992), Ballano and Ruiz-Castillo(1993), Harrison and Seidl (1994), Gaertner and Namezie (2003) among others]. The follow-ing example, which captures the main features of the situations presented to the interviewedin these experiments, might help to illustrate this point.

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Example 3.1. Let n = 4 and consider the distributions x1 = (1, 3, 5, 7), x2 = (1, 3, 6, 6), x3 =(1, 4, 4, 7), x4 = (2, 2, 5, 7), and x5 = (2, 3, 5, 6). It is immediate that each of the distributionsx2, x3, x4 and x5 obtains from x1 by means of a single [rank-preserving] progressive transferof one income unit. Proposition 3.1 ensures that xg ≥L x1, for all g = 2, 3, 4, 5, so thateveryone who subscribes to the principle of transfers – equivalently to the Lorenz criterion –will consider that distributions x2, x3, x4 and x5 are less unequal than distribution x1.

Inspection of the above distributions reveals that x2 is obtained from x1 by transferring oneunit of income from the richest individual to the second richest, which actually amounts toequalize the incomes of the two richest individuals. Inequality between individuals 3 and 4has therefore been eliminated but at the same time the income gap – or income differentials– between individuals 1 and 3 on the one hand, and individuals 2 and 3 on the other hand,has been widened. Although the Lorenz criterion declares that x2 is unambiguously moreequal than x1, there might be – and there are actually – people who disagree invoking forinstance the fact that not all pairwise income differentials are made smaller as a result of theprogressive transfer.

Suppose we agree with the above view according to which inequality unambiguously de-creases if and only if the absolute difference between any two incomes is reduced. Precisely,given the income distribution x ∈ Yn(D), we define:

(3.6) AD(p, s;x) := F−1(s;x)− F−1(p;x), ∀ 0 ≤ p < s ≤ 1.

Thus AD(p, s;x) measures the absolute income gap between the richer individual occupyingrank s and the poorer individual ranked p in situation x. We are confident that nobodycan seriously object that inequality unambiguously decreases when the absolute income gapsbetween any richer and any poorer individuals are made smaller. The following definitioncaptures precisely this idea.

Definition 3.3. Given two income distributions x,y ∈ Yn(D), we will say that x dominatesy in absolute differentials, which we write x ≥AD y, if and only if

(3.7) AD(p, s;x) ≤ AD(p, s;x), ∀ 0 ≤ p < s ≤ 1.

In our discrete framework condition (3.7) can be equivalently rewritten as:

(3.8)xk − xh

n≤ yk − yh

n, ∀ h = 1, 2, . . . , k − 1, ∀ k = 2, 3, . . . , n.

For later reference it is worthnoting that there is no loss of generality if we consider only thepairwise differences between adjacent individuals so that condition (3.8) is actually equivalentto:

(3.9)xk+1 − xk

n≤ yk+1 − yk

n, ∀ k = 1, 2, . . . , n− 1.

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This quasi-ordering, first introduced by Marshall, Olkin and Proschan (1967) in the fields ofmajorization [see also Bickel and Lehmann (1976)], has been considered a suitable inequalitycriterion by a number of scholars [see e.g. Thon (1987), Preston (1990), and Moyes (1994,1999)].

Although it appears difficult to object against its consensual nature, the absolute dif-ferentials quasi-ordering may be considered too strong a criterion. When comparing twodistributions by means of the absolute differentials quasi-ordering, every individual comparesher situation with that of all the individuals richer than her. It might be that what is im-portant is not really by how much every individual falls below every richer individual, butrather by how much on average she is away from the richer individuals. This is reminiscent ofthe notion of deprivation introduced by Runciman (1966) according to which the individual’sassessment of a given social state depends on her situation compared with the situations ofthe individuals who are treated more favourably than her. The absolute deprivation curve ofdistribution x ∈ Yn(D) – denoted as ADP (p;x) – is defined by

(3.10) ADP (p;x) :=∫ 1

p

[F−1(s;x)− F−1(p;x)

]ds, ∀ p ∈ [0, 1]

[see Kakwani (1984)]. We can interpret ADP (p;x) as a measure of the absolute deprivationfelt by individual with rank p in situation x. By definition, the best-off individual is neverdeprived and thus ADP (1;x) = 0, for all x ∈ Y(D). Following Chakravarty (1997), weintroduce:

Definition 3.4. Given two income distributions x,y ∈ Yn(D), we will say that there is nomore absolute deprivation in x than in y, which we write x ≥ADP y, if and only if

(3.11) ADP (p;x) ≤ ADP (p;y), ∀ p ∈ [0, 1).

Actually condition (3.11) simply states that overall deprivation decreases if the individualdeprivation felt by any member of the society decreases. Condition (3.11) can be equivalentlyrewritten as:

(3.12)n∑

j=k+1

xj − xk

n≤

n∑

j=k+1

yj − yk

n, ∀ k = 1, 2, . . . , n− 1.

In the notion of deprivation the reference group every individual compares herself withconsists of those individuals who are richer – or more generally who have a higher social status– than her. If we consider the set of poorer rather than richer individuals as the referencegroup, then we obtain the dual notion of satisfaction [see Chakravarty (1997)]. The startingpoint is the presumption that an individual derives satisfaction from the observation thatthey are other people not as well-off as she is, and that the larger the gap between her incomeand the average income of these poorer individuals the more satisfied she is. This idea is

11

related to the absolute satisfaction curve of distribution x ∈ Yn(D) – denoted as ASF (p;x)– defined by:

(3.13) ASF (p;x) : =∫ p

0

[F−1(p;x)− F−1(s;x)

]ds, ∀ p ∈ [0, 1].

We can interpret ASF (p;x) as a measure of the absolute satisfaction felt by individual rankedp in situation x. By definition, the worst-off individual is never satisfied since there is nobodyshe can compare herself with and ASF (0;x) = 0, for all x ∈ Y(D). A natural way toimprove well-being in the society is to reduce the overall satisfaction of its members. FollowingChakravarty (1997), we introduce:

Definition 3.5. Given two income distributions x,y ∈ Yn(D), we will say that there is nomore absolute satisfaction in x than in y, which we write x ≥ASF y, if and only if

(3.14) ASF (p;x) ≤ ASF (p;y), ∀ p ∈ (0, 1].

Actually condition (3.14) simply states that overall satisfaction decreases if the individualsatisfaction felt by any member of the society decreases. Condition (3.14) can be equivalentlyrewritten as:

(3.15)k−1∑

j=1

xk − xj

n≤

k−1∑

j=1

yk − yj

n, ∀ k = 2, 3, . . . , n.

The following remark points out properties of the deprivation and satisfaction curves thatwill prove useful later on.

Remark 3.1. Let n > 2 and x ∈ Yn(V ). Then we have:

(a) ADP

(k

n;x

)= ADP

(k + 1

n;x

)+

n− k

n[xk+1 − xk], for all k = 1, 2, . . . , n− 1.

(b) ASF

(k

n;x

)= ASF

(k − 1

n;x

)+

k − 1n

[xk − xk−1], for all k = 2, 3, . . . , n.

Since by definition the income distributions are non-decreasingly arranged, a direct implica-tion of Remark 3.1 is that the absolute deprivation and the absolute satisfaction curves arerespectively non-increasing and non-decreasing.

Applying the preceding quasi-orderings to the comparisons of the distributions introducedin Example 3.1 gives the rankings indicated in Table 3.1. The symbol “1” at the intersectionof row i and column j means that “xi >J xj”, while the occurence of the symbol “#”indicates that the distributions xi and xj are not comparable. Table 3.1 makes clear that,depending on the quasi-ordering we use, the change in inequality caused by a progressivetransfer may be ambiguous. In particular anyone who subscribes to the views captured byeither the differentials, or the deprivation or the satisfaction quasi-ordering may feel unable

12

to accept the common view that inequality decreases as a result of a progressive transfer.The inequality reducing impact of a progressive transfer as measured by the Lorenz criterionsurvives in three cases out of seven for the the deprivation and satisfaction quasi-orderingsand in only one case for the differentials quasi-ordering.

Table 3.1. Inequality Rankings of Distributions of Example 3.1.

≥AD

x1 x2 x3 x4

x2 #x3 # #x4 # # #x5 1 # # #

≥ADP

x1 x2 x3 x4

x2 1x3 # #x4 # # #x5 1 # # 1

≥ASF

x1 x2 x3 x4

x2 #x3 # #x4 1 # #x5 1 1 # #

≥L

x1 x2 x3 x4

x2 1x3 1 #x4 1 # #x5 1 1 1 1

3.2. Properties and Relationships Between the Inequality Quasi-Orderings

Table 3.1 indicates clearly that the differentials, the deprivation and the satisfaction quasi-orderings give different rankings of distributions and that these rankings are in turn distinctfrom the one that results from the application of the Lorenz criterion. But Table 3.1 alsosuggests that these inequality quasi-orderings might not but unrelated as the rankings oneobtains are more or less finer depending on the chosen quasi-ordering. Leaving aside the casen = 2, where the all preceeding quasi-orderings provide the same ranking of distributions,this is confirmed by the following result:

Remark 3.2. Let n > 2 and suppose all the distributions under comparison have equal

means. Then, we have: (i) ≥AD ⊂ ≥ADP ; (ii) ≥AD ⊂ ≥ASF ; (iii) ≥ADP ⊂ ≥L; (iv) ≥ASF

⊂ ≥L; and (v) ≥ADP 6= ≥ASF .

The preceding remark confirms that the three quasi-orderings we have introduced are relatedto – but distinct from – the Lorenz criterion and thus capture dimensions of inequality that arenot embedded in the latter criterion. Although these criteria have some appeal in themselvesit would be illuminating if one could uncover the normative judgements they build on. A firstdirection is to identify the elementary transformations – analogous to progressive transfers inthe case of the Lorenz criterion – that imply inequality reduction according to each of these

13

quasi-orderings. A second issue is to determine the structure of the individuals’ preferencesfor equality that are consistent with the views expressed by the differentials, satisfaction anddeprivation quasi-orderings. Both issues are of particular importance for understanding thenormative content of these three quasi-orderings and they will be the subject of the twofollowing sections.

4. Inequality, Solidarity and Equalizing Transformations

4.1. A Preliminary Result

Since the differentials, satisfaction and deprivation quasi-orderings are all subrelations of theLorenz quasi-ordering, it follows from Proposition 3.1 that, if a distribution is ranked aboveanother one according to either of the former quasi-orderings, then a sequence of progres-sive transfers will be needed in order to construct the dominating distribution starting fromthe dominated one. However Example 3.1 demonstrated that not all progressive transfers –and thus not all sequences of such transfers – reduce inequality as measured by the differen-tials, satisfaction or deprivation quasi-orderings. Therefore the precise way these progressivetransfers have to be combined for such domination to hold has to be determined.

It might be helpful to begin with a benchmark result that constitutes a first step to-wards a more general solution. By construction distributions x2, x3, x4 and x5 in Example3.1 obtain from x1 by means of a single rank-preserving progressive transfer of one incomeunit. Close inspection reveals that the only case where the resulting distribution dominatesthe original distribution according to the differentials quasi-ordering is when the progressivetransfer involves the richest and the poorest individual. The three cases where the progres-sive transfers generate an improvement according to the deprivation quasi-ordering is whenincome is taken from the richest individual and given to someone poorer. Finally a transferfrom any richer individual to the poorest one results in a reduction of inequality as measuredby the satisfaction quasi-ordering. The positions on the income scale of the individuals takingpart in the transfer seem to play a crucial role in the redistributive impact of the progressivetransfer. The result below – which we state without proof – indicates the restrictions one hasto introduce for inequality to decrease as a result of a single progressive transfer in a veryparticular case.

Remark 4.1. Let n > 2 and x,y ∈ Yn(D) such that, such that y1 < y2 < . . . < yn−1 < yn.

Suppose we are only permitted to use a single rank-preserving progressive transfer in order

to obtain x from y. Then, we have:

(a) x ≥AD y if and only if 1 = i < j = n.

(b) x ≥ADP y if and only if i < j = n.

(c) x ≥ASF y if and only if 1 = i < j.

(d) x ≥L y if and only if i < j.

This result uncovers the rationale behind the construction of Example 3.1 and also hints atpotential explanations why the public might reject the principle of transfers in some given

14

situations. But above all Remark 4.1 confirms that there is little room for reducing inequalityas measured by either of the differentials, deprivation and satisfaction quasi-orderings if oneis only allowed to make use of single progressive transfers.

4.2. General Results

Given Remark 4.1 we know that, for inequality as measured by any of the differentials, depri-vation and satisfaction quasi-orderings to decrease, a single progressive transfer will generallynot be sufficient and progressive transfers will have to be combined in one way or another.Therefore our first task is to identify possible transformations of distributions which combineprogressive transfers in such a way that domination in terms of the differentials, deprivationand satisfaction quasi-orderings obtains. A related task is to derive the appropriate sequenceof such transformations that permits to obtain the dominating distribution from the domi-nated one. We impose to ourselves two requirements: (i) the transformations must admit as aparticular case the progressive transfers exhibited in Remark 4.1, and (ii) the transformationsmust be elementary in the sense that they are as simple as possible. The latter requirementhas the effect that in general a single transformation would not suffice to convert the domi-nated distribution into the dominating one: successive applications of such transformationswill be needed.

Considering first those transformations, which successive applications of result in a distri-butional improvement according to the differentials quasi-ordering, we propose:

Definition 4.1. Given two income distributions x,y ∈ Yn(D), we will say that x is obtainedfrom y by means of a T1-transformation, if there exists δ, ε > 0 and two individuals h, k

(1 ≤ h < k ≤ n) such that condition (3.2) holds and:

xg = yg, ∀ g ∈ {h + 1, . . . , k − 1};(4.1.a)

xi = yi + δ, ∀ i ∈ {1, . . . , h}; xj = yj − ε, ∀ j ∈ {k, . . . , n};(4.1.b)

hδ = (n− k + 1)ε.(4.1.c)

Therefore T1-transformations comprise as a particular case the progressive transfers identifiedin statement (a) of Remark 4.1. Such elementary transformation impose a lot of solidarityin the society. There is solidarity among the rich: if some income is taken from a richindividual, then the same amount has to be taken from every richer individual. Symmetrically,there is solidarity among the poor: if some income is given to a poor individual, then thesame amount has to be given to every poorer individual. This solidarity among the donorsand the beneficiaries is typically broken down in the progressive transfer. The followingresult indicates that the T1-transformations and the absolute differentials quasi-ordering areintrinsically related.

Proposition 4.1. Let n > 2 and x,y ∈ Yn(D) such that µ(x) = µ(y). Then, the following

two statements are equivalent:

15

(a) x is obtained from y by means of a finite sequence of T1-transformations.

(b) x ≥AD y.

Turning now attention to those transformations, which successive applications of wouldresult in a distributional improvement according to the deprivation quasi-ordering, we pro-pose:

Definition 4.2. Given two income distributions x,y ∈ Yn(D), we will say that x is obtainedfrom y by means of a T2-transformation, if there exists δ, ε > 0 and two individuals h, k

(1 ≤ h < k ≤ n) such that condition (3.2) holds and:

xg = yg, ∀ g ∈ {1, . . . , h− 1} ∪ {h + 1, . . . , k − 1};(4.16.a)

xh = yh + δ; xj = yj − ε, ∀ j ∈ {k, . . . , n};(4.16.b)

δ = (n− k + 1)ε.(4.16.c)

Again a progressive transfer constitutes a particular case of a T2-transformation, whichin turn is a particular T1-transformation. Although solidarity is still present in a T2-transformation, it is now limited to the donors. If some income is taken from a rich individual,then the same amount is to be taken from every richer individual. However it is no longernecessary that individuals poorer than the transfer recipient do benefit also from some [equal]additional income. The result below establishes the connection between dominance in termsof the absolute deprivation quasi-ordering and T2-transformations.

Proposition 4.2. Let n > 2 and x,y ∈ Yn(D) such that µ(x) = µ(y). Then, the following

two statements are equivalent:

(a) x is obtained from y by means of a finite sequence of T2-transformations.

(b) x ≥ADP y.

Finally we consider the transformations, which successive applications of would result ina distributional improvement according to the satisfaction quasi-ordering. We propose:

Definition 4.3. Given two income distributions x,y ∈ Yn(D), we will say that x is obtainedfrom y by means of a T3-transformation, if there exists δ, ε > 0 and two individuals h, k

(1 ≤ h < k ≤ n) such that condition (3.2) holds and:

xg = yg, ∀ g ∈ {h + 1, . . . , k − 1} ∪ {k + 1, . . . , n};(4.39.a)

xi = yi + δ, ∀ i ∈ {1, . . . , h}; xk = yk − ε; and(4.39.b)

hδ = ε.(4.39.c)

The progressive transfers identified in statement (c) of Remark 4.1 are particular instances ofa T3-transformation. A T3-transformation is to some extent the dual of a T2-transformation:solidarity concerns the beneficiaries of the transfer. If some additional income is given to a

16

poor individual, then the same amount has to be given to every poorer individual. But thereis no need that individuals richer than the donor give away some [equal] amount of income.Dominance in terms of the absolute satisfaction quasi-ordering and T3-transformations arerelated as it shown below.

Proposition 4.3. Let n > 2 and x,y ∈ Yn(D) such that µ(x) = µ(y). Then, the following

two statements are equivalent:

(a) x is obtained from y by means of a finite and non-empty sequence of T3-transformations.

(b) x ≥ASF y.

Propositions 4.1 to 4.3 show that it is possible to associate to the differentials, deprivationand satisfaction quasi-orderings relatively simple transformations, which by successive appli-cations allow one to derive the dominating distribution from the dominated one. Althoughthese transformations comprise as particular cases the usual progressive transfers, they arein general more complicated. However invoking Remark 3.2 it is clear that any such trans-formation can be decomposed into a finite sequence of progressive transfers.

5. Welfare Comparisons for Equal Mean Distributions

5.1. Two Classes of Social Welfare Functions

Basically, two general families of social welfare functions have been studied in the inequalityliterature up to now. The first approach – the expected utility model or the utilitarian socialwelfare function – assumes linearity in the weights while the second approach – the Yaarimodel – assumes linearity in incomes. It is convenient to define

Pi : =i

n,(5.1.a)

Qi : = 1− Pi−1 =n− i + 1

n,(5.1.b)

for all i = 1, 2, . . . , n, with P0 = 0 and Qn+1 : = 0. According to the utilitarian model socialwelfare in situation x ∈ Yn(D) is given by

(5.2) WU (x) =n∑

i=1

[Pi − Pi−1] U (xi) ≡ 1n

n∑

i=1

U (xi) ,

where the utility function U is increasing and defined up to an increasing and affine trans-formation. Letting x0 : = 0, the Yaari social welfare function is defined by

(5.3) Wf (x) =n∑

i=1

[f (Qi)− f (Qi+1)] xi ≡n∑

i=1

f (Qi) [xi − xi−1] ,

where f ∈ F : = {f : [0, 1] −→ [0, 1] | f continuous, non-decreasing, f(0) = 0 and f(1) = 1}is the weighting function. Inequality aversion is fully captured by the utility function U inthe utilitarian framework and by the weighting function f in the Yaari model. The two formermodels are actually particular cases of the rank-dependent expected utility model [see e.g.Quiggin (1993)].

17

5.2. An Almost Impossibility Result

The egalitarian nature of the utilitarian rule has long been recognized when all individualshave the same concave utility function. Precisely a progressive transfer improves social welfareas measured by the utilitarian rule if and only if the common utility function is concave. Incombination with Proposition 3.1 this implies that Lorenz domination holds if and onlyif, whatever the concave utility function one chooses, the sum of utilities generated by thedominating distribution is always greater than the sum of utilities generated by the dominateddistribution. However the equalizing implications of the utilitarian social welfare function arebased on a particular notion of inequality reduction and one may wonder what happens whenalternative inequality views – such as the ones considered in this paper – are substitutedfor the ones captured by the Lorenz criterion. Because the quasi-orderings we are interestedin are weaker than the Lorenz quasi-ordering – they imply but are not implied by it – oneexpects that the class of utility functions that guarantee that the utilitarian rule is equalizingis larger than the class of concave utility functions. Actually the utilitarian model is notflexible enough to distinguish between the views captured by the differentials, deprivationand satisfaction quasi-orderings as the following result demonstrates.

Proposition 5.1. Let n > 2 and J ∈ {AD, ADP, ASF, L}. Then, the following two state-

ments are equivalent:

(a) For all x,y ∈ Yn(D) such that µ(x) = µ(y): x ≥J y =⇒ WU (x) ≥ WU (y).

(b) U is concave.

The consistency of the utilitarian social welfare function with the different inequality viewscaptured by our three quasi-orderings leads to the same restriction as the one implied byLorenz-consistency: the utility function has to be concave. Thus the utilitarian model doesnot permit to distinguish between the views embedded in the Lorenz quasi-ordering and itsthree competitors: the differentials, the deprivation and the satisfaction quasi-orderings. Onthe contrary, as we will demonstrate in a while, an ethical observer endowed with the Yaarisocial welfare function will be able to make a difference between these alternative views.

5.3. Consistency of the Yaari Model with Different Inequality Views

Before we state our main results, we need first introduce some definitions concerning theweighting function. Given a function g : R −→ R and an interval V ⊆ R, we will say that g

is convex over V if

(5.8) ∀ u, v ∈ V, ∀ λ ∈ [0, 1] : g((1− λ)u + λv) ≤ (1− λ)g(u) + λg(v).

Given V : = (v, v) ⊆ R and ξ ∈ V , we will say that g is star-shaped from above at ξ if

(5.9) ∀ u, v ∈ (v, ξ) ∪ (ξ, v) : u < v =⇒ g(u)− g(ξ)u− ξ

≤ g(v)− g(ξ)v − ξ

18

[see e.g. Landsberger and Meilijson (1990)]. Each of the four following classes of weightingfunctions will play a crucial role in subsequent developments:

FAD : = {f ∈ F | f(Q) ≤ Q, ∀Q ∈ (0, 1)};(5.10.a)

FADP : = {f ∈ F | f is star-shaped from above at 0};(5.10.b)

FASF : = {f ∈ F | f is star-shaped from above at 1};(5.10.c)

FL : = {f ∈ F | f is convex over [0, 1]}.(5.10.d)

It is a straightforward exercise to check that the classes of weighting functions defined aboveare nested in the way indicated below.

Remark 5.1. (i) FL ⊂ FADP ⊂ FAD; (ii) FL ⊂ FASF ⊂ FAD.

Typical members of the classes FAD, FADP , FASF and FL are represented in Figures 5.1.a,5.1.b, 5.1.c and 5.1.d, respectively. One can easily check that the weighting function repre-sented in Figure 5.1.d is convex and thus star-shaped from above at 0 and 1, and that italso verifies f(Q) ≤ Q, for all Q ∈ (0, 1). Figure 5.1.c depicts a weighting function that isstar-shaped from above at 1, verifies f(Q) ≤ Q, for all Q ∈ (0, 1), but is neither star-shapedfrom above at 0 nor convex. On the other hand the weighting function represented in Figure5.1.b is star-shaped from above at 0 and thus verifies f(Q) ≤ Q, for all Q ∈ (0, 1), but isneither star-shaped from above at 1 nor convex. Finally the weighting function depicted inFigure 5.1.a is not star-shaped but it lies everywhere below the main diagonal.

We are interested in identifying the restrictions to be placed on the weighting functionf ∈ F for the Yaari social welfare function to be consistent with our absolute quasi-orderings.Actually we are able to provide more general results that establish the links between thedifferent equalizing transformations we introduced in Section 4, subclasses of the Yaari socialwelfare functions and the inequality quasi-orderings. Considering first the inequality viewcaptured by the absolute differentials quasi-ordering, we obtain:

Proposition 5.2. Let n > 2 and consider two distributions x,y ∈ Yn(D) such that µ(x) =µ(y). Then, the following three statements are equivalent:

(a) x is obtained from y by means of a finite sequence of T1-transformations.

(b) Wf (x) ≥ Wf (y), for all f ∈ FAD.

(c) x ≥AD y.

The conditions that ensure inequality reduction in terms of the absolute income differentialsare rather weak: the weighting function must lie below the main diagonal in the [0, 1]× [0, 1]space. Therefore the class of Yaari social welfare functions that are consistent with the viewsexpressed by the differentials quasi-ordering is quite large.

We turn now to the absolute deprivation quasi-ordering and search for the restrictions tobe placed on the weighting function f that guarantee that the Yaari social welfare functionis consistent with this criterion.

19

Proposition 5.3. Let n > 2 and consider two distributions x,y ∈ Yn(D) such that µ(x) =µ(y). Then, the following three statements are equivalent:

(a) x is obtained from y by means of a finite sequence of T2-transformations.

(b) Wf (x) ≥ Wf (y), for all f ∈ FADP .

(c) x ≥ADP y.

The conditions that ensure inequality reduction in terms of the absolute deprivation quasi-ordering are weaker than convexity: the weighting function must be star-shaped at the originon the interval (0, 1).

The next result identifies the restrictions to be placed on the weighting function f thatguarantee that the Yaari social welfare function is consistent with the absolute satisfactionquasi-ordering.

Proposition 5.4. Let n > 2 and consider two distributions x,y ∈ Yn(D) such that µ(x) =µ(y). Then, the following three statements are equivalent:

(a) x is obtained from y by means of a finite sequence of T3-transformations.

(b) Wf (x) ≥ Wf (y), for all f ∈ FASF .

(c) x ≥ASF y.

The conditions that ensure inequality reduction in terms of the absolute satisfaction quasi-ordering are weaker than convexity: the weighting function must be star-shaped at 1 over(0, 1).

Finally for the sake of completeness we recall the conditions to be met by the weightingfunction for the Yaari social welfare function to be consistent with the Lorenz quasi-ordering.

Proposition 5.5. Let n > 2 and consider two distributions x,y ∈ Yn(D) such that µ(x) =µ(y). Then, the following three statements are equivalent:

(a) x is obtained from y by means of a finite sequence of single progressive transfers.

(b) Wf (x) ≥ Wf (y), for all f ∈ FL.

(c) x ≥L y.

The convexity of the weighting function is therefore necessary and sufficient for welfare asmeasured by the Yaari social welfare function to increase as the Lorenz curve moves upwards.

Contrary to the utilitarian model, which does not allow to make a distinction between theinequality views considered in this paper, the Yaari social welfare function permits to sepa-rate these different attitudes towards inequality. This is achieved by means of the weightingfunction which captures the planner’s concern for inequality. Under the equal mean condi-tion, Propositions 5.2 to 5.5 identify the restrictions to be placed on the weighting functionthat guarantee that welfare does not decrease when inequality as measured by our four quasi-orderings goes down. The propositions also identify the appropriate sequences of transfor-mations that are needed in order to convert the dominated distribution into the dominating

20

one for the differentials, deprivation and satisfaction quasi-orderings. These transformationsare more complicated than – and are generally distinct from – the traditional progressivetransfers.

6. Extensions When Distributions Have Different Means

So far we have restricted our attention to the comparison of distributions with equal means, inwhich case the notions of inequality and welfare are related in a one-to-one manner: decreasesin inequality and welfare improvements are different sides of the same coin. This is nolonger true when the distributions under comparison do not have the same mean. Then therelationship between inequality and welfare is no longer unambiguous and we will focus hereon inequality.

Nothing in the definitions of the differentials, deprivation and satisfaction quasi-orderingsprevents comparisons from being made between income distributions of unequal means. Onthe contrary the Lorenz quasi-ordering does not allow one to pass judgements when thesituations under comparisons have different means. We indicate by x : = (x1, . . . , xn) themean-reduced distribution of x ∈ Yn(D) obtained by letting xi : = xi − µ(x), for all i =1, 2, . . . , n. The absolute Lorenz curve of distribution x ∈ Yn(D) – denoted as AL(p;x) – isdefined by

(6.1) AL(p;x) : = L (p; x) =∫ p

0

[F−1(s;x)− µ(x)

]ds, ∀ p ∈ [0, 1].

Actually −AL(p;x) represents the average income short-fall of the p fraction of poorestindividuals in situation x [see Moyes (1987)].

Definition 6.1. Given two income distributions x,y ∈ Yn(D), we will say that x absoluteLorenz dominates y, which we write x ≥AL y, if and only if

(6.2) AL(p;x) ≥ AL(p;y), ∀ p ∈ (0, 1),

which in our framework is equivalent to:

(6.3)k∑

j=1

[xj − µ(x)] ≥k∑

j=1

[yj − µ(y)] , ∀ k = 1, 2, . . . , n− 1.

The absolute differentials, deprivation and satisfaction quasi-orderings as well as the absoluteLorenz quasi-ordering are absolute [inequality] measures in the sense that equal additions toall incomes leave inequality unchanged. More precisely, they satisfy:

Translation Invariance. The binary relation ≥J on Zn(D) is translation-invariant if, forall x,y ∈ Yn(D) and all γ, ξ ∈ R:

(6.4) x ∼J y =⇒ (x + γ1n) ∼J (y + ξ1n) .

Using translation invariance and Remark 3.2 we obtain:

21

Remark 6.1. When the distributions under comparison do not necessarily have equal means

and n > 2, we have: (i) ≥AD ⊂ ≥ADP ; (ii) ≥AD ⊂ ≥ASF ; (iii) ≥ADP ⊂ ≥AL; (iv) ≥ASF ⊂≥AL; and (v) ≥ADP 6= ≥ASF .

We would like to know whether it is possible to find inequality indices that are consistentwith the views reflected by the absolute differentials, deprivation, satisfaction and Lorenzquasi-orderings.

Following the ethical approach to inequality measurement, we restrict attention to thoseinequality indices that can be constructed from particular social welfare functions [see Black-orby, Bossert and Donaldson (1999)]. Given the social welfare function W : Yn −→ R, we letΞ represent the social evaluation function implicitly defined by

(6.5) W(x1, . . . , xn︸ ︷︷ ︸

n

)= W

(Ξ(x), . . . , Ξ(x)︸ ︷︷ ︸

n

), ∀ x ∈ Yn(D),

where Ξ(x) is the equally distributed equivalent income corresponding to distribution x. Solv-ing (6.5) when W is the utilitarian social welfare function defined in (5.2), we get

(6.6) ΞU (x) = U−1

(1n

n∑

i=1

U (xi)

), ∀ x ∈ Yn(D),

while, in the case of the Yaari social welfare function given by (5.3), we obtain

(6.7) Ξf (x) =n∑

i=1

[f (Qi)− f (Qi+1)] xi, ∀ x ∈ Yn(D).

The absolute inequality index , which measures the average income loss due to inequality, isdefined by

(6.8) IU (x) := µ(x)− ΞU (x), ∀ x ∈ Yn(D),

in the case of the utilitarian social welfare function, and by

(6.9) If (x) := µ(x)− Ξf (x), ∀ x ∈ Yn(D),

in the case of the Yaari social welfare function, respectively. Inequality aversion is fullycaptured by the utility function U in the utilitarian framework and by the weighting functionf in the Yaari model.

Actually substituting the inequality index for the social welfare function in the utilitar-ian framework does not change anything. Before we present the formal results, we find itconvenient to introduce some technicalities.

22

Remark 6.2. Let n ≥ 2. Then, the following two statements are equivalent:

(a) For all x,y ∈ Yn(D): x ≥L y =⇒ IU (x) ≤ IU (y).

(b) U is concave.

Remark 6.2 essentially says that the inequality index IU is Lorenz consistent if and only ifthe utility function U is concave. The next result is the crucial step here and it goes back toKolm’s (1976) plea for the absolute inequality view.

Remark 6.3. Let n ≥ 2. Then, the following two statements are equivalent:

(a) For all x ∈ Yn(D) and all γ ∈ R: IU (x + γ1n) = IU (x) 14.

(b) U(y) = 1η exp(ηy) for some η ∈ R and all y ∈ D.

This result basically says that inequality is not affected by adding or substracting the sameamount to all incomes if and only if the utility function U is of the CARA type and itis proved in its most general version in Ebert (1988, Theorem 4). The family of utilityfunctions exhibited in statement (b) of Remark 6.3 is closely associated with what is knownas the Kolm-Pollak family of inequality indices [see Kolm (1976)] We are now in a positionto state the analogue to Proposition 5.1 for the measurement of inequality:

Proposition 6.1. Let n > 2 and J ∈ {AD,ADP, ASF, AL}. Then, the following two

statements are equivalent:

(a) For all x,y ∈ Yn(D): x ≥J y =⇒ IU (x) ≤ IU (y).

(b) Either U(y) = 1η exp(ηy) for some η < 0, or U(y) = y, for all y ∈ D.

Proposition 6.1 confirms in the case of situations with possibly differing mean incomes whatwe learnt from Proposition 4.1: the utilitarian model does not allow one to distinguish betweenthe views associated with the differentials, the deprivation, the satisfaction and the Lorenz[absolute] quasi-orderings.

In contrast the inequality indices derived from the Yaari social welfare function will enableus to separate these different views, and one obtains the same restrictions on the weightingfunction as those pointed out in Propositions 5.2 to 5.5. Precisely, letting FAL = FL, weobtain:

Proposition 6.2. Let n > 2 and J ∈ {AD,ADP, ASF, AL}. Then, the following two

statements are equivalent:

(a) For all x,y ∈ Yn(D): x ≥J y =⇒ If (x) ≤ If (y).

(b) f ∈ FJ .

14In other words this means that the inequality index IU is translatable of degre zero over Yn(D).

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7. Concluding Remarks

We have challenged the traditional approach to welfare and inequality measurement basedon the Lorenz curve and we have considered three alternative criteria that rank distributionson the basis of comparisons of income differences. The differentials, deprivation and satis-faction quasi-orderings imply Lorenz dominance and the value judgements they reflect aretherefore more likely to be supported by the public than the ones embedded in the Lorenzcriterion. For each of these criteria we have identified the elementary transformations appro-priate combinations of which imply dominance. Progressive transfers are particular instancesof such transformations but in general these transformations involve more than one progres-sive transfer. We have shown next that the utilitarian model, that frames most of the theoryof welfare and inequality measurement, is unable to represent the preferences towards equalitywhen these are required to be consistent with the differentials, deprivation and satisfactionquasi-orderings. This is due the lack of flexibility of the utilitarian social welfare functionthat does not permit to distinguish between the views captured by these alternative crite-ria. On the contrary the Yaari model offers sufficient room to draw a distinction betweenthe preferences that are consistent with either of the differentials, deprivation and satisfac-tion quasi-orderings. The corresponding classes of Yaari social welfare functions have beenidentified and shown to be subclasses of the general class of Lorenz consistent social welfarefunctions. It is expected that some of the views represented by these new criteria are more inline with the public’s perception of inequality. However the extent to which these criteria arecloser to the public’s views is a matter of experimental investigation which lies outside thescope of this paper. It is possible to consider other [absolute] inequality views – equivalentlyquasi-orderings – and investigate their implications for the properties of the social welfarefunction. We would rather point at four other directions which the present analysis could beextended in.

Given the predominance of the relative approach in the inequality literature it would beinteresting to see if it were possible to find analogous results when one considers the relativeversions of the criteria examined in the paper. The relative counterparts of the quasi-orderingsexamined in this paper can easily be derived. However the identification of the restrictionsto be imposed on the weighting function which guarantee that social welfare increases as theresult of more equally distributed incomes in this case is a more difficult task. There is somepresumption that the Yaari model is not well designed for capturing the views embeddedin the relative versions of the differentials, deprivation and satisfaction quasi-orderings [seehowever Chateauneuf (1996)]. The utilitarian and the Yaari models provide only a oneparameter – the utility function and the weighting function, respectively – representationof the preferences towards equality. The rank-dependent expected utility model introducedby Quiggin (1993), which comprises as particular cases the two approaches examined in thispaper, offers more flexibility as the chosen value judgements can be reflected by the utilityfunction and the weighting function. One possible direction of research would be to investigatethe implications of this more general model for the appraisal of inequality attitudes. This

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might benefit from previous results obtained in the literature on risk which show the possibletrade-off between the concavity of the utility function and star-shapeness of the weightingfunction [see Chateauneuf, Cohen and Meilijson (1997)]. Another route that is worth pursuingwould be to characterize by means of additional conditions particular elements of the differentclasses we have identified. Indeed one may raise doubts about the ability of the differentialsquasi-ordering – and to a less extent the deprivation and satisfaction quasi-ordering – togenerate conclusive verdicts in practice. Although it must be stressed that these criteriaprovide guidance in some cases such as taxation design [see e.g. Chakravarty and Moyes(2003)], it is equally true that their ability to rank arbitrary real-world distributions is limited.They must therefore be considered a first round approach, which should be supplemented bythe use of particular indices in the general classes we have identified. Finally it must berecognized that inequality and welfare are multidimensioned phenomena and that income isonly one – though important – dimension among many others. The analysis would certainlybe enriched and also made more operational if it incorporates variables such as family size orhealth status in addition to income. Zoli (1999) has indicated how the Yaari model can begeneralized by introducing family needs – or more generally a handicap variable – into theanalysis when one confines attention to Lorenz consistent social welfare functions. A nextstep could be to investigate the implications of imposing consistency with the differentials,deprivation and satisfaction quasi-orderings in this more general setting.

References

Amiel, Y., and F.A. Cowell, (): “Measurement of income inequality: Experimentaltest by questionnaire”, Journal of Public Economics, Vol. 47, pp. 3–26.

Atkinson, A.B., (): “On the measurement of inequality”, Journal of Economic Theory ,Vol. 2, pp. 244–263.

Ballano, C., and J. Ruiz-Castillo, (): “Searching by questionnaire for the meaningof income inequality”, Revista Espanola de Economia, Vol. 10, pp. 233–259.

Berge, C., (): Topological Spaces, Including a Treatment of Multi-Valued Functions,

Vector Spaces and Convexity , Oliver and Boyd, Edinburgh.

Bickel, P.J., and E.L. Lehmann, (): “Descriptive statistics for non-parametric modelIII. Dispersion”, Annals of Statistics, Vol. 4, pp. 1139–1158.

Blackorby, C., Bossert, W., and D. Donaldson, (): “Income inequality measure-ment: The normative approach”, Chapter 3 in J. Silber (Ed.), Handbook of Income Inequality

Measurement, Kluwer Academic Press, Dordrecht, pp. 133–157.

Bossert, W., and A. Pfingsten, (1990): “Intermediate Inequality: Concepts, Indices andWelfare Implications”, Mathematical Social Sciences, 19, 117–134.

Chakravarty, S.R., (): “Relative deprivation and satisfaction orderings”, Keio Eco-

nomic Studies, Vol. 34, pp. 17–32.

Chakravarty, S.R., and P. Moyes, (): “Individual welfare, social deprivation and

25

income taxation”, Economic Theory , Vol. 21, pp. 843–869.

Chateauneuf, A., (): “Decreasing inequality: An approach through non-additive mod-els”, Cermsem Discussion Paper, Universite Paris I.

Chateauneuf, A., Cohen, M., and I. Meilijson, (): “More pessimism than greed-iness: A characterization of monotone risk aversion in the rank dependent expected utilitymodel”, Cermsem Discussion Paper, Universite Paris I.

Chateauneuf, A., and P.-H. Wilthien, (): “Third inverse stochastic dominance,Lorenz curves and favourable double-transfers’, Cermsem Discussion Paper, Universite ParisI.

Dalton, H., (): “The measurement of the inequality of incomes”, Economic Journal,Vol. 30, pp. 348–361.

Dasgupta, P., Sen, A.K., and D. Starrett, (): “Notes on the measurement ofinequality”, Journal of Economic Theory , Vol. 6, pp. 180–187.

Ebert, U., (): “Measurement of inequality: An attempt at unification and generaliza-tion”, Social Choice and Welfare, Vol. 5, pp. 59–81.

Ebert, U., and P. Moyes, (): “An axiomatic characterization of the Yitzhaki’s indexof individual deprivation”, Economics Letters, Vol. 68, pp. 263–270.

Foster, J.E., (1985): “Inequality measurement”, in H.P. Young (ed.), Fair Allocation, Amer-ican Mathematical Society Proceedings of Applied Mathematics, Volume 33.

Fields, G.S., and J.C.H. Fei, (): “On inequality comparisons”, Econometrica, Vol. 46,pp. 305–316.

Gaertner, W., and C. Namezie, (): “Income inequality, risk, and the transfer princi-ple: A questionnaire–experimental investigation”, Mathematical Social Science, Vol. 45, pp.229–245.

Gastwirth, J.L., (1971): “A General Definition of the Lorenz Curve”, Econometrica, Vol.39, pp. 1037–1039.

Hardy, G.H., Littlewood, J.E., and G. Polya, (): Inequalities, Cambridge Uni-versity Press, Cambridge (2nd edition).

Harrison, E., and C. Seidl, (): “Perceptional inequality and preference judgements:An empirical examination of distributional axioms”, Public Choice, Vol. 79, pp. 61–81.

Hey, J.D., and P.J. Lambert, (): “Relative deprivation and the Gini coefficiengt:Comment”, Quarterly Journal of Economics, Vol. 95, pp. 567–573.

Jakobsson, U., (): “On the measurement of the degree of progression”, Journal of

Public Economics, Vol. 5, pp. 161–168.

Kakwani, N.C., (): “The relative deprivation curve and its applications”, Journal of

Business and Economic Statistbics, Vol. 2, pp. 384–405.

Kolm, S-.C., (): “The optimal production of social justice”, in J. Margolis and H.

26

Guitton (Eds.), Public Economics, Macmillan, London, pp. 145–200.

Kolm, S-.C., (): “Unequal inequalities I”, Journal of Economic Theory , Vol. 12, pp.416–442.

Landsberger, M., and I. Meilijson, (): “Lotteries, insurance, and star-shaped utilityfunctions”, Journal of Economic Theory , Vol. 52, pp. 1–17.

Marshall, A.W., and I. Olkin, (): Inequalities: Theory of Majorization and its

Applications. Academic Press, New-York.

Marshall, A.W., Olkin, I., and F. Proschan, (): “Monotonicity of ratios of meansand other applications of majorization”, in O. Shisha (Ed.), Inequalities, Academic Press,New-York.

Moyes, P., (): “A new concept of Lorenz domination”, Economics Letters, Vol. 23, pp.203–207.

Moyes, P. (): “Inequality reducing and inequality preserving transformations of in-comes: Symmetric and individualistic transformations”, Journal of Economic Theory , Vol.63, pp. 271–298.

Moyes, P., (): “Stochastic dominance and the Lorenz curve”, Chapter 6 in J. Silber(Ed.), Handbook of Income Inequality Measurement, Kluwer Academic Press, Dordrecht, pp.199–222.

Preston, I., (): “Ratios, differences and inequality indices”, Institute for Fiscal StudiesWorking Paper No. W90–9.

Quiggin, J., (): Generalized Expected Utility Theory. The Rank-Dependent Model,Kluwer Academic Publishers, Boston/Dorrecht/London.

Runciman, W.G., (): Relative Deprivation and Social Justice, Routledge and KeganPaul, London.

Sen, A.K., (): Collective Choice and Social Welfare, Holden-Day, San Francisco. Re-published North-Holland, Amsterdam, 1979.

Sen, A.K., (): On Economic Inequality , Clarendon Press, Oxford.

Thon, D., (): “Redistributive properties of progressive taxation”, Mathematical Social

Sciences, Vol. 14, pp. 185–191.

Weiss, Y., and C. Fershtman, (): “Social status and economic performance: A sur-vey”, European Economic Review , Vol. 42, pp. 801–820.

Weymark, J., (): “Generalized Gini inequality indices”, Mathematical Social Sciences,Vol. 1, pp. 409–430.

Yaari, M.E., (): “The dual theory of choice under risk”, Econometrica, Vol. 55, pp.99–115.

Yaari, M.E., (): “A controversial proposal concerning inequality measurement”, Journal

of Economic Theory , Vol. 44, pp. 381–397.

27

Yitzhaki, S., (): “Relative deprivation and the Gini coefficient”, Quarterly Journal of

Economics, Vol. 93, pp. 321–324.

Yitzhaki, S., (): “Relative deprivation and economic welfare”, European Economic

Review , Vol. 17, pp. 99–113.

Zoli, C., (): “Inverse sequential stochastic dominance, relative deprivation and povertymeasurement”, Discussion Paper, University of Nottingham.

Zoli, C., (): “Inverse stochastic dominance, inequality measurement and Gini indices”,in P. Moyes, C. Seidl and A.F. Shorrocks (Eds), Inequalities: Theory, Experiments and

Applications, Journal of Economics, Supplement #9, pp. 119–161.

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Figure 5.1.a. A Weighting Function Compatible with the Absolute Differentials Quasi-Ordering

0

0.2

0.4

0.6

0.8

1

f(Q)

0.2 0.4 0.6 0.8 1

Q

Figure 5.1.b. A Weighting Function Compatible with the Absolute

Deprivation Quasi-Ordering

0

0.2

0.4

0.6

0.8

1

f(Q)

0.2 0.4 0.6 0.8 1

Q

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Figure 5.1.d. A Weighting Function Compatible with the Lorenz Quasi-Ordering

0

0.2

0.4

0.6

0.8

1

f(Q)

0.2 0.4 0.6 0.8 1

Q

Figure 5.1.c. A Weighting Function Compatible with the Absolute

Satisfaction Quasi-Ordering

0

0.2

0.4

0.6

0.8

1

f(Q)

0.2 0.4 0.6 0.8 1

Q

30