The ‘Extended’ Atkinson family: The class of multiplicatively decomposable inequality measures,and some new graphical procedures for analysts

This paper introduces and characterises a class of inequality measures which extends the Atkinson family. This class contains canonical forms of all aggregative inequality measures, each bounded above by one, provides a new dominance criterion for ordering distributions in terms of inequality and offers some new graphical procedures for analysts. The crucial axiom for the characterisation is an alternative to the standard additive decomposition property that we call ‘multiplicative decomposability,’ where the within-group component is a generalised weighted mean with weights summing exactly to one. An erratum to this article can be found at     

Complaints and inequality

Temkin (1986, 1993) set out a philosophical basis for the analysis of income inequality that provides an important alternative to the mainstream welfarist approach. We show that the Temkin principles can be characterised by a parsimonious axiomatic structure and we use this structure to derive a new class of inequality indices and an inequality ordering. This class of indices has a family relationship to well-known measures of inequality, deprivation and poverty. The ordering is shown to have properties analogous to second-order dominance results.We would like to acknowledge financial support by the Training and Mobility of Researchers Programme of the European Communities, grant #ERBFMRXC T980248 (Living Standards, Inequality and Taxation), and to thank STICERD for hosting Ebert in order to facilitate our collaboration. Ralph Bayer helped greatly in preparation of the figures. We are grateful for comments from Yoram Amiel, Kurt Devooght and Alain Trannoy and from participants at workshops in Warwick, Bocconi, the LSE and Venice. We also wish to acknowledge the helpful comments of the referees of this journal.     

Inequality orderings and unit consistency

The paper examines the implications of the newly proposed unit consistency axiom for partial inequality orderings. We first show that some intermediate Lorenz dominance conditions violate the axiom. We then characterize a class of intermediate Lorenz orderings and demonstrate that the only unit-consistent member is the one related to Krtscha (Models and measurement of welfare and inequality. Springer, Heidelberg, 1994)’s intermediate notion of inequality which has recently been investigated by Zoli (A surplus sharing approach to the measurement of inequality. Discussion paper no. 98/25, University of York, 1998; Logic, game, theory and social choice. Tilburg University Press, Tilburg, 1999) and Yoshida (Soc Choice Welf 24:557–574, 2005). Finally, we provide a general characterization for unit-consistent Lorenz orderings and the Krtscha-type dominance again turns out to be the only one that is intermediate and unit-consistent.     

A dissimilarity index of multidimensional inequality of opportunity

A recent literature on inequality of opportunity offers quantitative tools for comparisons and measurement based on stochastic dominance criteria and traditional inequality indices. In this paper I suggest an additional way of assessing inequality of opportunity with two indices of dissimilarity across distributions. The indices are based on a traditional homogeneity test of multinomial distributions and are similar to the square coefficient of variation (Reardon and Firebaugh, Sociol Method 32:33?C67, 2002). Their properties are studied, as well as their usefulness and limitations in applications when both circumstances and advantages/outcomes are multidimensional. An empirical application measures changes in inequality of opportunity from an old to a young cohort in Peru. The importance of assessing the sensitivity of the results to group definitions and group proportions is highlighted.     

The effect on inequality of changing one or two incomes

We examine the effect on inequality of increasing one income, and show that for two wide classes of indices a benchmark income level or position exists, dividing upper from lower incomes, such that if a lower income is raised, inequality falls, and if an upper income is raised, inequality rises. We provide a condition on the inequality orderings implicit in two inequality indices under which the one has a lower benchmark than the other for all unequal income distributions. We go on to examine the effect on the same indices of simultaneously increasing one income and decreasing another higher up the distribution, deriving results which quantify the extent of the ‘bucket leak’ which can be tolerated without negating the beneficial inequality effect of the transfer. Our results have implications for the inequality and poverty impacts of different income growth patterns, and of redistributive programmes, leaky or not, which are briefly discussed.     

Intersecting Lorenz Curves, the Degree of Downside Inequality Aversion, and Tax Reforms

This paper clarifies the conceptual distinction of downside inequality aversion (or transfer sensitivity) as a normative criterion for judging income distributions from the Pigou-Dalton principle of transfers. We show that when the Lorenz curves of two income distributions intersect, how the change from one distribution to the other is judged by an inequality index exhibiting downside inequality aversion often depends on the relative strengths of its downside inequality aversion and inequality aversion. For additive inequality indices or their monotonic transformations, a measure characterizing the strength of an index’s downside inequality aversion against its inequality aversion is shown to determine the ranking by the index of two distributions whose Lorenz curves cross once. The precise condition under which the same result generalizes to the case of multiple-crossing Lorenz curves is also identified. The results are particularly useful in understanding the distributional impact of tax reforms. I received exceptionally helpful comments from Mike Hoy, Peter Lambert, the Editor, Buhong Zheng, and an anonymous referee. The remaining errors and shortcomings are my own – W.H. Chiu     

A class of distribution and association sensitive multidimensional welfare indices

This paper axiomatically characterizes a class of two-parameter generalized mean social welfare indices having two or more dimensions of well-being. These indices, under appropriate parametric restrictions, are sensitive to two distinct forms of inter-personal inequality. The first form of inequality is concerned with the dispersion of each dimensional achievement across the population. The second is concerned with the association or correlation across dimensions, reflecting the observation that the correlation of individual components of well-being across dimensions is relevant for social welfare evaluation. It is shown that many existing multidimensional welfare indices are closely related to this new class. Statistical tests are developed to verify the significance of the evaluations generated by these indices. Finally, the class of indices is applied to Indonesian data to analyze the impact on financial crisis on the country’s social welfare using two waves of Indonesian Family Life Surveys (IFLS).     

Multilevel interterritorial convergence and additive multidimensional inequality decomposition

In this paper we justify the use of the between-groups S-convex relative inequality indices to measure interterritorial convergence. In addition, we provide two appealing extensions to the use of the additive decomposable General Entropy inequality indices. First, multilevel decomposition along several subpartitions is satisfied and a natural application to convergence decomposition along different territorial level arises. Second, further general inequality decomposition along several dimensions is also satisfied. Thus, this framework provides a better explanation of the determinants of convergence (and inequality) when it is decomposed into different dimensions in comparison with other indices available in the literature. Received: 21 May 1999/Accepted: 21 September 2000     

Poverty measures incorporating variable rate of alleviation due to population growth

If the absolute number of poor people goes up, but the fraction of people in poverty comes down, has poverty gone up or gone down? Issues like lack of scale homogeneity in resource allocation for poverty eradication lead practitioners and economists to have two different views of poverty when population size changes. This paper attempts to put these two conceptions of poverty into a common framework. We present an axiomatic development of a family of poverty measures, which is a generalization of the well known FGT class of measures, without a population replication axiom. This family has an intuitive link to standard measures, but it also allows intermediate positions between one or other of “the absolute numbers” or the “fraction in poverty” conception. We hope that this family will prove useful in empirical and policy work.     

Multidimensional inequality and multidimensional generalized entropy measures: An axiomatic derivation

This paper generalizes the axiomatic approach to the design of income inequality measures to the multiattribute context. While the extension of most axioms considered desirable for inequality indices is straightforward, it is not entirely clear when a situation is more unequal than another when each person is characterised by a vector of attributes of well-being. We explore two majorization criteria which are partial orders ranking distributions of attributes by their degree of inequality. The two criteria are motivated by the Pigou-Dalton Transfer Principle in the unidimensional context and its equivalent formulation. These criteria gauge inequality loosely speaking with respect to the dispersion of the multidimensional distribution of the attributes. They, however, fail to address a different dimension of multivariate inequality pertaining to an increase in the correlation of the attributes. In this connection, this paper introduces a correlation-increasing majorization criterion proposed by Boland and Proschan (1988). Finally, in conjunction with other axioms commonly invoked in the literature on inequality, the majorization criteria lead inexorably to the class of multidimensional generalized entropy measures. Received: 15 June 1995 / Accepted: 30 September 1997     

Fuzzy measurement of income inequality: a class of fuzzy inequality measures

The typical practice of income inequality evaluation is by using inequality indices. Yet there is no best measure of inequality as we know it, and hence, any conclusion based on an inequality index must be subject to some doubts. Some authors even argued that the concept of inequality, by its very nature, is vague, and thus, cannot be measured like an exact concept. Motivated by these considerations, this paper studies axiomatic fuzzifications of inequality measures. Consequently, a systematic method of viewing the conclusions of inequality comparisons in terms of truth value statements is developed. Furthermore, it is shown that this method (or in fact, any other fuzzy inequality measure) can be used to construct confidence intervals for the crisp conclusions of inequality indices.I wish to thank Kaushik Basu, Larry Blume, Gary Fields, Tapan Mitra, Maurice Salles, Anthony Shorrocks, Sinan Unur, the participants of the 1993 Midwest Mathematical Economics Conference held in University of Wisconsin at Madison and those of the 2nd International Meeting of the Society for Social Choice and Welfare held at University of Rochester, and two anonymous referees for insightful comments and suggestions. Janos Aczel and Karol Baron kindly provided some help about a mathematical problem I faced, I am grateful to them. Needless to mention, however, the usual disclaimer applies.     

Illustrating the Implications of How Inequality is Measured: Decomposing Earnings Inequality by Race and Gender

This paper makes three distinct contributions: it presents a novel modification to an established methodology for assessing inequality using the CPS ASEC data, it illustrates how valuable a multi-metric inequality analysis is by reconciling some open questions regarding the trend in inequality and the role of the composition of income along the distribution, and it provides a baseline assessment of the trend in earnings inequality for four distinct groups of income earners. The evolution of earnings inequality from 1995 to 2010 is compared to increasing inequality in total income as documented by Thomas Piketty and Emmanuel Saez to show that earnings inequality has followed a qualitatively similar, though less extreme trend. In the process, the disconnect between the trend in the Gini coefficient and inequality assessed via the share of income going to the top 1 % of income earners is reconciled through the use of several alternative inequality indices. Finally, the evolution of the earnings distribution for black women, black men, white women, and white men are considered separately, which shows that there are important differences in the experience of inequality. The main findings are that only white men have experienced changes in within-group earnings inequality that parallel the changes in inequality seen in the overall distribution. By contrast, black income earners have seen no notable increase in within-group inequality by any measure, suggesting that they may rightly perceive growing inequality as primarily a between-group phenomena.     

Constructing indices of multivariate polarization

Multivariate indices of polarization are constructed to measure effects of non-income attributes like wealth and education. Polarization is considered as the presence of groups which are internally homogeneous, externally heterogeneous, and of similar size. We propose a class of polarization indices which is built from measures of relative groups size and from decomposable indices of socio-economic inequality. For the latter, we employ the special inequality indices of Maasoumi (Econometrica 54:991–997, 1986), Tsui (J Econ Theory 67:251–265, 1995; Soc Choice Welf 16:145–157, 1999) and Koshevoy and Mosler (J Multivar Anal 60:252–276, 1997). Then, postulates for multidimensional polarization measurement are stated and discussed. The approach is illustrated by an empirical application to the population of the East and West Germany with polarization defined on income and education.     

Transitive and acyclic rationality indicators of fuzzy choice functions on base domain

In the present paper we introduce the indicators of the fuzzy transitive congruence axiom, fuzzy direct-revelation axiom and fuzzy acyclic congruence axiom. These indicators measure the degree to which a fuzzy choice function satisfies these axioms. We use the indicators of fuzzy transitive congruence axiom and fuzzy acyclic congruence axiom to calculate the minimum degree to which the direct fuzzy revealed preference relation is the transitive and acyclic respectively. We established that (i) the degree to which the fuzzy choice function is full rational is the degree to which it satisfies fuzzy transitive congruence axiom and (ii) the degree to which the fuzzy choice function is acyclic rational is the minimum degree to which it satisfies fuzzy direct-revelation axiom and its fuzzy revealed preference is acyclic. We show that a similarity relation on the set of fuzzy choice functions preserves the indicators of fuzzy transitive congruence axiom, fuzzy direct-revelation axiom, fuzzy acyclic congruence axiom and (transitive and acyclic) rationality indicators.     

The impact of income inequality on values and attitudes

Scholars in several social science disciplines scholars have argued from their respective disciplinary perspectives that income inequality has a considerable impact on economic and social performance of a nation. This essay investigates the possible impact of income inequality on 290 values and attitudes in forty industrial nations from an economic perspective. The results show that inequality has a significant impact on values and attitudes especially concerning religion and the family.     

Source decomposition of changes in income inequality: the integral-based approach and its approximation by the chained Shapley-value approach

This paper proposes source decomposition of the change in income inequality between two time points using the integral-based approach (IBA) and the (chained) Shapley-value approach as its approximation. In comparison to static Shapley-value decompositions and traditional decompositions for the square of the coefficient of variation and the Gini index, the new dynamic Shapley-value approach is intuitively appealing as a decomposition procedure for changes in inequality. It is expected to yield relatively small differences among various inequality measures, essentially maintaining consistency with income source classification. Path dependency, a possible drawback of the new decompositions, is not expected to be a particular problem in the usual cases. The properties are illustrated for application to the increase in US family income inequality during 1979–1996. In this empirical study, the new decompositions showed a tendency that was clearly different from those of the existing decompositions, indicating that the proposed procedures shed new light on analysis of the causes of inequality changes. An extension to incorporate other factors such as family structure is also successful without loss of the desirable properties.     

Inequality decomposition values: the trade-off between marginality and efficiency

This paper presents a general procedure for decomposing income inequality measures by income sources. The methods of decomposition proposed are based on the Shapley value and extensions of the Shapley value of transferable utility cooperative games. In particular, we find that Owen’s value can find an interesting application in this context.We show that the axiomatization by the potential of Hart and Mas-Colell remains valid in the presence of the domain restriction of inequality indices. We also examine the properties of these decomposition rules and perform a comparison with Shorrocks’ decomposition rule properties.     

Income inequality games

The paper explores different applications of the Shapley value for either inequality or poverty measures. We first investigate the problem of source decomposition of inequality measures, the so-called additive income sources inequality games, based on the Shapley value, introduced by Chantreuil and Trannoy (1999) and Shorrocks (1999). We show that multiplicative inequality games provide dual results compared with Chantreuil and Trannoy’s ones. We also investigate the case of multiplicative poverty games for which indices are non additively decomposable in order to capture contributions of sub-indices, which are multiplicatively connected with, as in the Sen-Shorrocks-Thon poverty index. We finally show, in the case of additive poverty indices, that the Shapley value may be equivalent to traditional methods of decomposition such as subgroup consistency and additive decomposition.     

Population growth and poverty measurement

If the absolute number of poor people goes up, but the fraction of people in poverty comes down, has poverty gone up or gone down? The economist’s instinct, framed by population replication axioms that undergird standard measures of poverty, is to say that in this case poverty has gone down. But this goes against the instinct of those who work directly with the poor, for whom the absolute numbers notion makes more sense as they cope with more poor on the streets or in the soup kitchens. This paper attempts to put these two conceptions of poverty into a common framework. Specifically, it presents an axiomatic development of a family of poverty measures without a population replication axiom. This family has an intuitive link to standard measures, but it also allows one or other of “the absolute numbers” or the “fraction in poverty” conception to be given greater weight by the choice of relevant parameters. We hope that this family will prove useful in empirical and policy work, where it is important to give both views of poverty—the economist’s and the practitioner’s—their due.     

Accounting for income distribution trends: A density function decomposition approach

This paper develops methods for decomposing changes in the income distribution using subgroup decompositions of the income density function. Overall changes are related to changes in subgroup shares and changes in subgroup densities, where the latter are broken down further using elementary transformations of individual incomes. These density decompositions are analogous to the widely-used decompositions of inequality indices by population subgroup, except that they summarize multiple features of the income distribution (using graphs), rather than focusing on a specific feature such as dispersion, and are not dependent on the choice of a specific summary index. Nonetheless, since inequality and poverty indices can be expressed as PDF functionals, our density-based methods can also be used to provide numerical decompositions of these. An application of the methods reveals the multi-faceted nature of UK income distribution trends during the 1980s.