A note on the relationship between additive separability and decomposability in measuring income inequality

Abstract

This note introduces original technical results in the theoretical measurement of inequality by specifying the relationships between additive separability and homotheticity (of measures of welfare closely related to measures of inequality), and decomposability and homogeneity in measures of inequality. More specifically, an interrogation is made of the resonances and dissonances between the classic contributions of Atkinson and Shorrocks, which are key representatives, respectively, of the 'social welfare function' and the 'axiomatic' approaches to measuring inequality. In brief, in the presence of otherwise common assumptions, it is shown that additive separability and homotheticity of welfare are stronger combined conditions than decomposability and homogeneity (of degree zero) of income inequality. The gap between the two, however, can be closed by adding an extra term around total income to the measure of welfare, allowing for wider considerations of the relationship between social welfare, total income, and the distribution of individual incomes.

Keywords:

• Inequality measurement

Acknowledgments

Thanks to Tony Shorrocks for correspondence and to the editor and the two anonymous referees, who have provided valuable comments. The usual disclaimers apply.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The measures Shorrocks (1980) derives have come to be known as the ‘generalised entropy’ family, although in his original article he referred to them as the class of additively-decomposable measures.

2 For a range of inequality-aversion parameters, Atkinson's inequality index is ordinally, but not cardinally, equivalent to the class of indices Shorrocks derives. For details, see Cowell (2000).

3 We thank one of the anonymous referees for indicating this. Briefly, these results were obtained by interpreting the three propositions below – which are in Hardy, Littlewood and Polya's theorem – in the context of inequality and welfare measurement (for details, see Le Breton, 1999). If a and b are ordered n-length vectors, so that $a_1\le a_2\dots \le a_n$, then:

1. there exists a bi-stochastic matrix that is not a permutation matrix Q such that $\mathbf{a}=\mathbf{Q}\mathbf{b}$ (i.e. a can be obtained from b through a series of progressive transfers);

2. $\sum _{i=1}^k{a}_i\ge \sum _{i=1}^k{b}_i\mathrm{\forall }1\le k\le n$, with strict inequality for at least one k, and $\sum _{i=1}^n{a}_i=\sum _{i=1}^n{b}_i$ (i.e. a Lorenz-dominates b);

3. for any strictly concave increasing function v, $\sum _{i=1}^{n}v\left(a_i\right)>\sum _{i=1}^{n}v\left(b_i\right)$ (i.e. the social welfare of a, evaluated by the summation of individual utilities through v, is greater than that of b).

4 It has been shown in the literature that one can construct social welfare functions that underpin inequality indices derived in the ‘axiomatic approach’ (Ebert, 1988; Shorrocks, 1988). What this contribution does is to work with the specific class of social welfare functions that are implicit in Shorrocks' measures, and the explicit social welfare function Atkinson proposes, to show under which conditions they are equivalent or lead to equivalent measures of inequality.

5 For Atkinson without symmetry, see Fine (1985). It merely results in the replacement of each f by ${\mathrm{b}}_{i}f$, for constants ${\mathrm{b}}_i$, i.e. the functional form does not change showing what a powerful condition is additive separability with or without symmetry.

6 Throughout, where possible, we deal in two individuals/incomes alone to ease presentation and understanding.

7 Homotheticity is usually defined in terms of a monotonic increasing function of a homogeneous function.

8 Leaving aside A and B, which serve as scaling factors, see below.

9 Notice that Atkinson's measure is defined as the loss of mean income affordable to maintain total social welfare constant, if income were equally distributed: $I=\left((\mu -y_{ede})/\mu \right)$, where $y_{ede}$ is the implicitly defined ‘equally distributed equivalent level of income’ for a given c, so that $\sum W\left(x_i\right)=nW\left(y_{ede}\right)$. Note also that in Atkinson (1970) the inequality-aversion parameter $ϵ=1-\mathrm{c}$ in our notation.

10 Note that Atkinson's inequality measure (see note 9) is $I=1-{\left[\frac{1}{n}\sum \left[{\left(\frac{x_i}{\mu }\right)}^{\mathrm{c}}\right]\right]}^{1/c}.$ Atkinson's definition is a monotonic transformation of ours, given the differences explained in note 9, which does not otherwise affect our results.

11 As discussed by Bourguignon (1979), Atkinson's measure is aggregative, but not additively decomposable, even if starting from being additively separable in individual welfare. This means that overall inequality can be expressed as a function of inequality within groups and their aggregate characteristics (e.g. their population- and income-shares), without having to resort to the exact distribution of incomes within each group in order to compute overall inequality. As such, Atkinson's measure (see note 10) is aggregative, but not additively decomposable (see also Cowell, 2011).

12 Unfortunately, we cannot work with just two individuals as previously, as the concern is with between and within group inequalities for which we need numbers of individuals.

13 There are also some minor technical assumptions which we do not cover, for Atkinson as well as for Shorrocks, not least to exclude lexicographic orders and the like for lack of continuity and differentiability of the measures and functions.

14 The proofs are made by forging equalities across cleverly chosen distributions of incomes, partitions, changes in individual incomes, possibly at the margins (i.e. differentiating changes), through reliance upon decompositions and symmetry. Thus, informally for example, $I(x,x,y)$ would equal $wI(x,y)+I(x,(x+y)/2,(x+y)/2)$, using decompositions into $\left(x\right)$ and $(x,y)$ and given that $I\left(x\right)=0$.

15 Shorrocks shows, furthermore, that additive decomposability, homogeneity and symmetry are sufficient conditions for the resulting measures to satisfy the Pigou-Dalton principle of transfers (which specifies that transfers from richer to poorer individuals, which do not invert their relative position to each other, must reduce inequality). For Atkinson, this also emerges directly from the specification of a strictly convex welfare function and an inequality measure that refers to the loss of income affordable to maintain social welfare constant.

16 As a Shorrocks function, see above, Atkinson has $f\left(x\right)=x^{\mathrm{c}}$ and $\theta (\mu ,n)=n^{\mathrm{c}}{\mu }^{\mathrm{c}}$, and similarly for the case $\mathrm{c}=0$.

17 It is possible to skip the derivation and go to the resulting functional forms towards the end of the section, although the derivation sheds some light on where the two methods differ.

18 Bourguignon (1979) arrived at essentially comparable results, deriving the only forms that are additively decomposable and homogeneous of degree zero in income (although he also relaxed the latter constraint). This, as explained in note 11, stands in contrast to Atkinson's inequality measure, which is aggregative but not additively decomposable. Interestingly, a focus of Bourguignon's (1979) contribution is on the weights of the additive decompositions that obtain, indicating that only expressions (19(19) $\begin{array}{rl}f\left(x\right)& =-\mathrm{A}\log x+\mathrm{B}x+\mathrm{C},\mathrm{c}=0\end{array}$(19) ) (20(20) $\begin{array}{rl}f\left(x\right)& =\mathrm{A}x\log x+\mathrm{B}x+\mathrm{C},\mathrm{c}=1\end{array}$(20) ) have weights in the within-group term of overall inequality that sum to unity. Shorrocks, although devoting less space to the matter of weights, further indicated that only the inequality index based on (19(19) $\begin{array}{rl}f\left(x\right)& =-\mathrm{A}\log x+\mathrm{B}x+\mathrm{C},\mathrm{c}=0\end{array}$(19) ) produces an unambiguous interpretation of the resulting decomposition, which would make it preferable for such a purpose even if at the cost of fixing the degree of inequality aversion (to a value derived based on other considerations than inequality-aversion per se, and hence arbitrary).

Additional information

Notes on contributors

Ben Fine

Ben Fine is Emeritus Professor of Economics at the School of Oriental and African Studies, University of London, UK, and Visiting Professor, Wits School of Governance, University of Witwatersrand, South Africa. He has published around 300 articles and 30 books, and been in receipt of both the Deutscher and Myrdal book prizes. His most recent books include Material Cultures of Financialisation, co-edited with K. Bayliss and M. Robertson, Routledge, 2018; and Race, Class and the Post-Apartheid Democratic State, co-edited with John Reynolds and Robert van Niekerk, University of KwaZulu-Natal Press, 2019, with A Guide to the Systems of Provision Approach: Who Gets What, How and Why, with K. Bayliss, Palgrave, 2020, forthcoming. He is Chair of the International Initiative for Promoting Political Economy (iippe.org).

Pedro Mendes Loureiro

Pedro Mendes Loureiro is Lecturer in Latin American Studies at the Centre of Latin American Studies and Department of Politics and International Studies (CLAS-POLIS) at the University of Cambridge, and Fellow of Fitzwilliam College, University of Cambridge. An interdisciplinary scholar, his research interests encompass political economy, the dynamics of multidimensional inequalities, critical approaches to development, and Latin American studies.