REDUCING INEQUALITIES AMONG UNEQUALS

• Author: Mathieu Faure,Nicolas Gravel
• Date: 01 February 2021
• DOI: http://doi.org/10.1111/iere.12490
• Published date: 01 February 2021

INTERNATIONALECONOMIC REVIEW

Vol. 62, No. 1, February 2021 DOI: 10.1111/iere.12490

REDUCING INEQUALITIES AMONG UNEQUALS∗

By Mathieu Faure and Nicolas Gravel1

Centre de Sciences Humaines (Delhi) & Aix-Marseille University, CNRS, AMSE

This article establishes an equivalence between four incomplete rankings of distributions of income among

agents who are vertically differentiated with respect to some nonincome characteristic (health, household size,

etc.). The f‌irst ranking is the possibility of going from one distribution to the other by a f‌inite sequence of in-

come transfers from richer and more highly ranked agents to poorer and less highly ranked ones. The second

ranking is the unanimity among utilitarian planners who assume that agents’ marginal utility of income is de-

creasing with respect to both income and the source of vertical differentiation. The third ranking is the Bour-

guignon (Journal of Econometrics, 42 (1989), 67–80) Ordered Poverty Gap dominance criterion. The fourth

ranking is a new dominance criterion based on cumulative lowest incomes.

1. introduction

When can a distribution of income among a group of homogeneous agents be considered

more equal than another? An important achievement of the modern theory of inequality

measurement is the demonstration made by Hardy et al. (1952)—and popularized among

economists by Dasgupta et al. (1973), Kolm (1969), Sen (1973), and Fields and Fei (1978)—

that the following four answers to this question are equivalent.

(1) Ais more equal than Bif it can be obtained from Bby means of a f‌inite sequence of

bilateral Pigou–Dalton transfers.

(2) Ais more equal than Bif all utilitarian ethical observers who assume that individuals

convert income into well-being by the same concave utility function so agree.

(3) Ais more equal than Bif poverty, as measured by the poverty gap, is lower in Athan in

Bfor every def‌inition of the poverty line.

(4) Ais more equal than Bif the kpoorest agents have a larger cumulated income in A

than in Bwhatever kis (i.e., if the distribution of income in ALorenz dominates that

in B).

Any of these answers provides an incomplete, but yet very robust, answer to the basic

question. The f‌irst answer identif‌ies an elementary transformation of the distribution that in-

tuitively captures in a crisp fashion the very notion of inequality reduction—Pigou-Dalton

here—that is sought. The second answer links inequality measurement to a set of explicit

normative principles and seeks consensus among them. Although answer (2) is formulated

in terms of the ethically contentious doctrine of utilitarianism, it can actually be shown (see,

∗Manuscript received August 2019; revised September 2020.

1With the usual disclaiming qualif‌ication, we are indebted to Sebastian Bervoets, Yann Bramoullé, Alain

Chateauneuf, and, especially, Patrick Moyes for many insightful comments and discussions. We have also received

extremely valuable comments and suggestions from Masaki Aoyagi and three referees from this journal. Finally, we

gratefully acknowledge the f‌inancial support of the French Agence Nationale de la Recherche (ANR) through four

contracts: Measurement of Ordinal and Multidimensional Inequalities (ANR-16-CE41-0005), Challenging Inequali-

ties: A Indo-European perspective (ANR-18-EQUI-0003), Communication and Information in Games on Networks

(ANR-15-CE38-0007), and The European University of Research AMSE (ANR-17-EURE-020). Please address cor-

respondence to: Nicolas Gravel, Centre de Sciences Humaines, 2, Dr APJ Abdul Kalam Road, 110011 Delhi, India;

and Aix-Marseille University, CNRS,AMSE. Phone: +911130410090. E-mail: nicolas.gravel@univ-amu.fr.

357

© (2020) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of So-

cial and Economic Research Association

358 faure and gravel

e.g., Gravel and Moyes, 2013) to hold for the much larger class of aggregations of concave

utilities by increasing and utility-inequality-averse ethical observers. Moreover, even the wel-

farist or utilitarian interpretation underlying answer (2) is not required. The utility function

can also be interpreted as ref‌lecting the “social value” attached by the ethical observer to

the income received by the agent. This social value may not be related to the agent’s per-

sonal welfare or happiness. Finally, the third and fourth answers provide empirically imple-

mentable tests—poverty gap or Lorenz dominance—to determine whether or not one dis-

tribution is more equal than another. Comparing income distributions by means of Lorenz

dominance has become a routine practice followed by thousands of researchers all over the

world. Many sophisticated inference techniques—see, for example, Anderson (1996), Beach

and Davidson (1983), Bishop et al. (1989a), Bishop and Formby (1999), and Davidson and

Duclos (2000)—have also been proposed to assess the robustness of Lorenz dominance or

stochastic dominance comparisons when applied to samples instead of the whole population

of interest. Moreover, despite their possible incompleteness, these criteria have been shown

in empirical applications—see for example Bishop et al. (1989b), Duclos et al. (2006), Gravel

et al. (2009), or Gravel and Mukhopadhyay (2010)—to rank conclusively a signif‌icant fraction

of all the possible pairs of distributions. When such conclusive rankings cannot be obtained,

and the Lorenz curves associated to the two distributions cross, it is common to compare dis-

tributions using (much) more ethically demanding inequality indices. In such a case, the re-

quirement that the ranking provided by the inequality indices be compatible with any of those

four answers is considered to be very important (see, e.g., Foster, 1985).

Remarkable and foundational to inequality measurement as it is, this equivalence only con-

cerns distributions of income, or any other cardinally measurable attribute, between otherwise

perfectly homogeneous agents. Yet, income is not the only ethically relevant source of dif-

ferentiation between economic agents. If these agents are collectivities such as households or

jurisdictions, they differ not only by their total income but also by the number of members

among whom the income must be shared. If the agents are individuals, they may also differ by

nonincome characteristics such as age, health, education, or effort. What does “reducing in-

equalities” in one characteristic mean when applied to agents who are differentiated with re-

spect to another characteristic? In short, how can one def‌ine reducing inequality among un-

equals? This is the basic question addressed in this article.

Specif‌ically, we establish an equivalence between four notions of inequality reduction

among unequals, each of which being analogous in nature to one of the above four notions

of inequality reduction among equals. The elementary transformation that we propose to cap-

ture inequality reduction among unequals is like a Pigou–Dalton transfer, but with the stip-

ulation that the donor must be both richer and more highly ranked than the receiver. More-

over, contrary to what is usually required in a Pigou-Dalton transfer—see however Atkinson

(1987), Chakravarty and Muliere (2003), or Zheng (2007) for alternative formulations—we do

not restrict the transfer to being lower than half the income difference between the giver and

the receiver. The quantity transferred can be as large as the full income difference. The nor-

mative principles that we examine are those generated by comparisons of distributions by a

utilitarian ethical observer who assumes that agents convert income into utility by the same

function exhibiting a marginal utility of income that is decreasing with respect to both income

and the source of vertical differentiation. The empirically implementable criterion that we

consider is the Bourguignon (1989) Ordered Poverty Gap (OPG) dominance criterion. This

criterion requires that poverty, measured by the income poverty gap, be smaller in the domi-

nating distribution than in the dominated distribution for any collection of poverty lines that

are decreasing with respect to the agent’s vertical standing. We f‌inally introduce a “cumulated

lowest incomes” dominance criterion—which generalizes Lorenz dominance to our setting—

and prove its equivalence with OPG dominance.

This article can clearly be seen as a contribution to the multidimensional—in fact two-

dimensional—inequality measurement literature, which has emerged in the last 40 years or so.

To the best of our knowledge, no contribution to this literature has succeeded in establishing

reducing inequalities among unequals 359

a foundational equivalence between an empirically implementable criterion (such as Lorenz

or poverty gap dominance), a welfarist (or otherwise) unanimity over a class of functions that

transform the attributes into achievement and an elementary operation that captures in an in-

tuitive way the nature of the equalization sought.

For instance, Atkinson and Bourguignon (1982) (and before them Hadar and Russell

(1974)) show that f‌irst- and second-order multidimensional stochastic dominance imply util-

itarian dominance over a class of individual utility functions that is specif‌ic to the order of

dominance. They also suggest (without providing proof) that there could be an equivalence

between their multidimensional stochastic dominance criteria and utilitarian unanimity over

their class of individual utility functions. But they do not identify an elementary operation

that can be implied by their criteria or that can imply them. Atkinson and Bourguignon (1987)

propose a nice interpretation of one of the Atkinson and Bourguignon (1982) stochastic dom-

inance criteria in the specif‌ic case of two attributes, one of which interpreted as an ordinal in-

dex of needs (such as household size). Yet, they do not identify the elementary operation that,

when performed a f‌inite number of times, would coincide with the criterion. It is in the very

same setting that Bourguignon (1989) introduces his OPG criterion. Bourguignon (1989) also

identif‌ies the class of utility functions over which utilitarian unanimity is equivalent to his cri-

terion. However, he does not identify the elementary operation that is equivalent to it.

Elementary transformations believed to lie behind the criteria proposed by Atkinson and

Bourguignon (1982), Atkinson and Bourguignon (1987), and Bourguignon (1989) have been

discussed by various authors, including Atkinson and Bourguignon (1982) themselves, Ebert

(1997), Fleurbaey et al. (2003), and Moyes (2012). Yet, none of these papers demonstrates

that performing these elementary operations a f‌inite number of times is equivalent to the im-

plementable criteria. In a related vein, Muller and Scarsini (2012) establish an equivalence

between a class of elementary transformations—multidimensional transfers and correlation-

reducing permutations, to be discussed below—and utilitarian unanimity over the class of in-

creasing and submodular utility functions.2However, they do not provide an implementable

test—such as Lorenz or poverty gap dominance—that coincides with either their elementary

transformations or the utilitarian unanimity over their class of utility functions. Another at-

tempt to propose an elementary transformation that would capture a plausible notion of in-

equality reduction in a multidimensional context has been made by Kolm (1977). This au-

thor def‌ines equalization by the fact of transferring from one agent to another an identical

fraction of both income and the source of vertical differentiation. Making sense of this no-

tion of equalization obviously requires that this source of vertical differentiation be cardinally

measurable. Although this measurability may be conceivable in some context (e.g., when the

agents are workers vertically differentiated by their number of hours worked), it is less so in

others (e.g., households differentiated by the number of their members). Kolm (1977) proves

that obtaining one distribution from another by means of a f‌inite sequence of such transfor-

mations is equivalent to having the two distributions ranked by all utilitarian ethical observers

who evaluate the agents’ well-being by means of the same concave utility function. However,

Kolm (1977) does not identify an empirically implementable criterion that is equivalent to his

notion of equalization.

Another approach to multidimensional equalization is followed by Koshevoy (1995) who

suggests that distributions of several attributes be compared on the basis of the inclusion of

their Lorenz zonotope. This Lorenz zonotope inclusion criterion is a clear generalization, to

any number of dimensions, of the usual unidimensional Lorenz dominance. It is a (relatively)

easy-to-check criterion that is applicable to any two distributions of several attributes. Ko-

shevoy (1995) proves that performing a f‌inite number of times the elementary transformation

proposed by Kolm (1977) is a suff‌icient condition for obtaining Lorenz zonotope inclusion.

However, this does not tell us much about the implicit equalization process embedded in this

criterion because the converse statement does not hold.

2See, for example, Marinacci and Montrucchio (2005) for a def‌inition of these properties.