Extreme Values, Means, and Inequality Measurement

• Published date: 01 September 2021
• Author: Walter Bossert,Conchita D’Ambrosio,Kohei Kamaga
• Date: 01 September 2021
• DOI: http://doi.org/10.1111/roiw.12490

© 2020 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of

International Association for Research in Income and Wealth

564

EXTREME VALUES, MEANS, AND INEQUALITY MEASUREMENT

by Walter bossert

CIREQ,University of Montreal

ConChita D’ambrosio

INSIDE,University of Luxembourg

AND

Kohei Kamaga*

Faculty of Economics,Sophia University

We examine some ordinal measures of inequality that are familiar from the literature. These measures

have a quite simple structure in that their values are determined by combinations of specific summary

statistics such as the extreme values and the arithmetic mean of a distribution. In spite of their com-

mon appearance, there seem to be no axiomatizations available so far, and this paper is intended to fill

that gap. In particular, we consider the absolute and relative variants of the range, the max-mean and

the mean-min orderings, and quantile-based measures. In addition, we provide some empirical observa-

tions that are intended to illustrate that, although these orderings are straightforward to define, some of

them display a surprisingly high correlation with alternative (more complex) measures.

JEL Codes: H24, I31

Keywords: economic index numbers, Luxembourg Income Study, mean values

1. introDuCtion

The measurement of income inequality has been an active field of investiga-

tion for over a century, and early classical contributions include those of Lorenz

(1905), Gini (1912), Pigou (1912), and Dalton (1920). While much of the litera-

ture focuses on a relative notion of inequality (i.e. on scale-invariant measures),

absolute indices (which are translation-invariant) are examined as well. Centrist

or intermediate measures that represent compromises between the relative and the

absolute approach are discussed in Kolm (1976a,b), Pfingsten (1986), and Bossert

and Pfingsten (1990). The normative approach connects inequality to welfare and

can be traced back to Kolm (1969), Atkinson (1970), and Sen (1973) in the case

Note: We are grateful to Giorgia Menta for excellent research assistance. We thank Takuya Hasebe,

seminar participants at Fukuoka University and Sophia University, three referees, and the edi-

tor-in-charge, Prasada Rao, for comments and suggestions. Financial support from the Fonds de

Recherche sur la Société et la Culture of Québec, the Fonds National de la Recherche Luxembourg

(Grant C18/SC/12677653) and KAKENHI (20K01565) is gratefully acknowledged.

*Correspondence to: Kohei Kamaga, Faculty of Economics, Sophia University, Tokyo, Japan

(kohei.kamaga@sophia.ac.jp).

Review of Income and Wealth

Series 67, Number 3, September 2021

DOI: 10.1111/j.1475-4991.2020.12490.x

This is an open access article under the terms of the Creative Commons Attribution License, which

permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Review of Income and Wealth, Series 67, Number 3, September 2021

565

© 2020 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of

International Association for Research in Income and Wealth

of relative measures, and to Kolm (1969) and Blackorby and Donaldson (1980) if

an absolute notion of inequality is adopted. Ethical measures of inequality in an

ordinal setting are analyzed by Blackorby and Donaldson (1984), Ebert (1987),

and Dutta and Esteban (1992).

In this paper, we follow an ordinal approach to inequality measurement and,

therefore, focus on inequality orderings. Our main results provide characteriza-

tions of some simple measures of inequality that are familiar from the literature.

The first of these are range-based measures that perform inequality comparisons

by means of the difference between maximal and minimal income in the abso-

lute case, and the ratio of the maximum and the minimum in a relative setting.

The max-mean orderings use the difference and the ratio of the maximum and the

arithmetic mean, and the mean-min measures employ the arithmetic mean and the

minimal income. In addition, we examine inequality orderings that focus on the

income gaps (in the absolute case) or the income shares (for relative measures) of

the top or bottom quantile of an income distribution. All of these inequality order-

ings satisfy three standard axioms, namely, S-convexity, continuity, and replication

invariance. However, as far as we are aware, they have not been axiomatized yet.

The primary motivation of our analysis is rooted in the observation that many

of the measures discussed here are well known and well established in the literature.

Despite this fact, there are no characterizations available so far, and it seems to us

that this gap ought to be filled. With this objective in mind, clearly the properties

we employ in our axiomatizations cannot but reflect the nature of these indices.

As a consequence, whatever perceived shortcomings there are in the comparisons

according to these measures are inevitably mirrored in the corresponding recom-

mendations of (some of) the axioms.

Clearly, the measures discussed here are rather coarse because of their lim-

ited use of income distribution statistics and, therefore, we do not mean to advo-

cate their use over all competing suggestions. Nevertheless, as discussed by Leigh

(2009, p.162) in the context of justifying the use of the top income shares, when

some data are absent or reliable estimates of the entire income distribution are not

available, they can serve as a useful proxy for measuring inequality. In particular,

in light of the interdependence between different parts of the income distribution

resulting from economic activities, they could be a useful and easy-to-use tool

for drawing inferences about overall inequality from limited data; see Atkinson

(2007, pp.19–25) and Atkinson etal. (2011, pp.7–12) for discussions regarding top

income shares. Alvaredo (2011) examines connections between the Gini coefficient

and top income shares from a theoretical perspective. Therefore, we think that it is

worthwhile to provide axiomatic characterizations of those inequality orderings.

Our results also clarify under which circumstances we may safely rely on the prox-

ies provided by our orderings. While some of the axioms we employ may appear to

have somewhat controversial recommendations, they mirror the coarse nature of

the underlying inequality orderings. The analysis carried out in this paper suggests

that, in the presence of data limitations, the relatively coarse measures character-

ized here are capable of providing quite close approximations.

Among the orderings we consider, the range-based inequality orderings that

compare the distance between (or the ratio of) the maximal and the minimal income

do not utilize the average income. In this sense, these inequality orderings are