Inequality and welfare: Some axiomatic characterizations

• Author: Liu Qingbin,Satya R. Chakravarty,Nachiketa Chattopadhyay
• DOI: http://doi.org/10.1111/ijet.12169
• Published date: 01 June 2019
• Date: 01 June 2019

Inequality and welfare: Some axiomatic characterizations

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Satya R. Chakravarty



, Nachiketa Chattopadhyay

y

and Liu Qingbin

z

We characterize social welfare functions, which, under ceteris paribus assumptions, rank income

distributions in exactly the same way as the negative of the Atkinson, Kolm–Pollak, and Theil

mean logarithmic deviation indices of inequality of income distributions. While the first two

characterizations rely on a general social welfare function, the third employs the symmetric

utilitarian structure of the welfare function. Implications of several subsets of the set of axioms

considered in the paper are also investigated.

Key words inequality, Atkinson index, Kolm–Pollak index, Theil index, social welfare,

characterization

JEL classification C43, D31, D63, O15

Accepted 20 April 2017

1 Introduction

Simple comparison of the income inequality levels of two distributions (using some index of

inequality) does not involve the differences between the mean incomes of the distributions under

consideration and therefore ignores an important factor which affects the welfare of a population. It

is now common practice to interpret alternative indices of inequality in terms of social welfare. The

influence of mean income on the welfare of a population may be so significant that it can reverse the

inequality ranking between two distributions. A social welfare function is increasingly related to the

mean income (efficiency) of a distribution and some decreasing transformation of its inequality

(equity). This ensures that when efficiency considerations are absent (mean income is fixed), an

increase in equity (i.e. a reduction in inequality) increases social welfare. Likewise, given the level of

inequality, an increase in mean income increases social welfare. A welfare function of this type is

popularly known as a reduced or abbreviated form of welfare function in the literature (see, for

example, Graaff 1977; Ebert 1987; Lambert 2001; Amiel and Cowell 2003; Chakravarty 2009a,b).

Now, a particular index of inequality can be interpreted in terms of welfare using more than one

welfare function. For instance, several authors, including Newbery (1970), Sheshinski (1972), Kats

(1972), Sen (1973, 1974), Chipman (1974), Kondor (1975), Blackorby and Donaldson (1978, 1980),

Donaldson and Weymark (1980, 1983), Lambert (1985), Weymark (1981), Kakwani (1985) and

Chakravarty (1988, 2009a,b), made attempts to relate the Gini index to different social welfare

functions. It may often be desirable to understand the implicit value judgments involved in the

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Economic Research Unit, Indian Statistical Institute, Kolkata, India. Email: satya@isical.ac.in

y

Sampling and Official Statistics Unit, Indian Statistical Institute, Kolkata, India.

z

University of International Business and Economics, Beijing, China.



The authors thank an anonymous referee for many helpful comments and suggestions. Chakravarty thanks the University

of International Business and Economics, Beijing, China, for support.

doi: 10.1111/ijet.12169

International Journal of Economic Theory xxx (2018) 1–16 IAET 1

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choice of a particular social welfare function. An axiomatic characterization can be helpful for this

purpose. An axiomatic characterization of a social welfare function provides a set of necessary and

sufficient conditions that uniquely isolate the welfare function. A characterization enables us to

understand the welfare function in terms of the axioms employed in the characterization exercise.

Consequently, axiomatic characterizations of different welfare functions can be helpful to separate

two functions on the basis of the axiom sets.

One objective of this paper is to characterize the Theil (1972) mean logarithmic deviation index

of inequality from a welfare-theoretic perspective. This is because no welfare-theoretic

characterization of the index exists in the literature. Our paper fills this gap. This index of

inequality is a scale-invariant or relative index; it remains invariant under equi-proportionate

changes in all incomes. It has many advantages. It is easy to compute from grouped data. An income-

by-income replication of the population does not change the value of the index. This property makes

the inequality index convenient for cross-population comparison of inequality. A (progressive)

transfer of income from anyone to someone with a lower income reduces the value of the inequality

index by a larger amount the lower the income of the recipient is. Given any partitioning of the

population into two or more subgroups using some homogeneous attribute, say age, sex, etc., among

all relative indices, it is the only index that can be expressed as the sum of the between-group and

within-group components of inequality, where the within-group factor is defined as the sum of the

population share weighted average of subgroup inequality levels and the between-group term is the

level of inequality that arises because of variation of mean incomes across subgroups (Bourguignon

1979). This particular decomposition of the mean logarithmic deviation index enables us to use the

two decomposed components to construct a reduced-form index of polarization, where between-

group inequality represents alienation across subgroups and within-group inequality is inversely

related to identification of individuals in respective subgroups (see Zhang and Kanbur 2001;

Rodriguez and Salas 2003; for a recent review, see also Chakravarty 2015). These considerations also

have led us to focus on the Theil (1972) index.

The second objective of the paper is to characterize the welfare function underlying the Atkinson

(1970) relative index of inequality using a set of axioms different from that employed by Atkinson

(1970). While Atkinson (1970) used the symmetric utilitarian criterion, the sum of identical

individual utility functions, as the welfare function, we do not impose any such restriction at the

outset. Rather additivity is derived from more primitive axioms in our characterization. It may be

noted that the Atkinson index is the only relative index which satisfies a notion of subgroup

decomposability introduced by Blackorby et al. (1981), when the subgroups are constituted by

partitioning the population in two or more subgroups. They used the well-known Atkinson–Kolm–

Sen equally distributed equivalent welfare incomes associated with different subgroups of the

income distribution to define within- and between-group components of inequality.

Kolm (1976) provided welfare-theoretic characterizations of the Atkinson (1970) index and the

Kolm–Pollak (Kolm 1976; Pollak 1971) absolute or translation-invariant index, which remains

unaltered under equal absolute translation of all incomes. While for the former it has been assumed

that the social welfare function is the product of equity and efficiency, for the latter it has been taken

as the difference between these two statistics. Chakravarty (2009b) provided much simpler proofs of

Atkinson’s (1970) and Kolm’s (1976) characterizations using the symmetric utilitarian framework.

Generality is lost to a large extent because of the specific forms of welfare functions used in Kolm’s

(1976) and Chakravarty’s (2009b) characterizations. For absolute indices, the Kolm–Pollak index

turns out to be the only index satisfying the notion of decomposition and subsequent aggregation

introduced by Blackorby et al. (1981). We therefore develop an alternative social welfare-based

characterization of the Kolm–Pollak index also using a more general structure and a more primitive

Inequality and welfare Satya R. Chakravarty et al.

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