A stochastic dominance approach to the measurement of discrimination

This note suggests a bridge between stochastic dominance (Rothschild and Stiglitz, 1970 [17], 1973 [18]), inequality measurement (Atkinson, 1970 [1]) and discrimination measurement (Gastwirth, 1975 [10]). Discrimination orderings are defined and illustrated through discrimination curves, in the same spirit as stochastic dominance analysis. The main result, which links the second order discrimination curve and the Gastwirth discrimination index, also generalizes the equivalence between Generalized Lorenz dominance and second order stochastic dominance.     

Fuzzy measurement of income inequality: Some possibility results on the fuzzification of the lorenz ordering

Summary This paper starts from the premise that the concept of income inequality is ill-defined, and hence, it studies the measurement of income inequality from a fuzzy set theoretical point of view. It is argued that the standard (fuzzy) transitivity concepts are not compatible with fuzzy inequality orderings which respect Lorenz ordering. For instance, we show that there does not exist a max-min transitive fuzzy relation on a given income distribution space which ranks distributions unambiguously according to the Lorenz criterion whenever they can actually be ranked by it. Weakening the imposed transitivity concept, it is possible to escape from the noted impossibility theorems. We introduce some alternative transitivity concepts for fuzzy relations, and subsequently, construct a class of fuzzy orderings which preserve Lorenz ordering and satisfy these alternative transitivities. It is also shown that fuzzy measurement can be used to construct confidence intervals for the crisp conclusions of inequality indices.I wish to thank Ashish Banerjee, Kaushik Basu, Larry Blume, Gary Fields, Semih Koray, Tapan Mitra, Antony Shorrocks, Sinan Unur and two anonymous referees of this journal for insighthul comments and suggestions. I am also grateful to the participants of the 1993 Midwest Mathematical Economics Conference held in University of Wisconsin at Madison and the 2nd International Meeting of the Society for Social Choice and Welfare held in University of Rochester.     

Second‐Order Discrimination and Generalized Lorenz Dominance

We propose a definition of second‐order discrimination that does not require the reference distribution to first‐order dominate the comparison one, and allows rankings of discrimination patterns when both the reference and the comparison distributions differ. It involves comparing the probabilities that randomly selected individuals in the reference and comparison distributions belong to subgroups having the same cumulative mean income, yields orderings of distributions equivalent to those from generalized Lorenz dominance, and allows orderings of discrimination patterns, partial or complete, across pairs of distributions. We compare discrimination against U.S. seniors (inter‐distributional inequality between seniors and non‐seniors) by ethnicity.     

Stochastic dominance and decomposable measures of inequality and poverty

In this paper, we characterize some new links between stochastic dominance and the measurement of inequality and poverty. We show that: for second-degree normalized stochastic dominance (NSD), the weighted area between the NSD curve of a distribution and that of the equalized distribution is a decomposable inequality measure; for first-degree and second-degree censored stochastic dominance (CSD), the weighted area between the CSD curve of a distribution and that of the zero-poverty distribution is a decomposable poverty measure. These characterizations provide graphical representations for decomposable inequality and poverty measures in the same manner as Lorenz curve does for the Gini index. The extensions of the links to higher degrees of stochastic dominance are also investigated.     

Estimating inequality in the distribution of welfare using demand models

In this paper two demand models with general household equivalence scales (GES) are estimated. These GES are identifiable, since they have the independence of base utility (IB) or equivalence scales exactness (ESE) property. Estimates of household characteristics adjusted income can then be calculated relative to a specific household type. This “individual equivalent income (IEI) is then used to calculate measures of inequality in the distribution of welfare. As more than one model is estimated, the sensitivity of these estimates to model specification changes can be considered. Comparisons are also made to estimates of inequality based on household income. It is found that absolute inequality is sensitive to model specification, but relative inequality is not. This is true using indices of inequality of the Atkinson (1970), Kolm (1976a,b) and Sen (1973) type, or more general measures of inequality based on estimates of Lorenz curve decile ordinates, constructed using methods introduced by Beach and Davidson (1983). Thus, if absolute measures of inequality are required, these results suggest some further research is required to determine a class of models which is less sensitive to model specification error. However, if only relative measures of inequality are needed, then estimates of inequality appear to be very robust to the choice of model specification.     

A TREATMENT OF ABSOLUTE INDICES OF POLARIZATION*

Absolute polarization indices remain unchanged under equal absolute augmentation in all incomes. This paper identifies the class of absolute polarization indices whose orderings of alternative income distributions agree with the rankings generated by nonintersecting absolute polarization curves. We explore the possibility of using the Kolm (1976) – Blackorby‐Donaldson (1980) ethical absolute inequality index in polarization measurement. We establish that although inequality and polarization are dissimilar concepts, different absolute inequality indices can be employed to design alternative absolute polarization indices. A numerical illustration is provided using Indian data and it is shown that inequality and polarization are different issues in income distribution analysis.     

Redistribution in the age of austerity: evidence from Europe 2006–2013

We examine the relationship between changes in a country’s public sector fiscal position on inequality at the top and bottom of the income distribution during the age of austerity from 2006 to 2013. We use a parametric Lorenz curve model and Gini-like indices of inequality as our measures to assess distributional changes. Based on Statistics of Income and Living Conditions (EU-SILC) and IMF data for 12 European countries, we find that more severe adjustments to the cyclically adjusted primary balance (i.e., more austerity) are associated with a more unequal distribution of income driven by rising inequality at the top. The data also weakly suggests a decrease in inequality at the bottom. The distributional impact of austerity measures reflects the reliance on regressive policies and likely produces increased incentives for rent-seeking while reducing incentives for workers to increase productivity.     

Inverse stochastic dominance,inequality measurement and Gini indices

We investigate the relationship between the third degree inverse stochastic dominance criterion introduced in Muliere and Scarsini (1989) and inequality dominance when Lorenz curves intersect. We propose a new definition of transfer sensitivity aimed at strengthening the Pigou-Dalton Principle of Transfers. Our definition is dual to that suggested by Shorrocks and Foster (1987). It involves a regressive transfer and a progressive transfer both from the same donor, leaving the Gini index unchanged. We prove that finite sequences of these transfers and/or progressive transfers characterize the third degree inverse stochastic dominance criterion. This criterion allows us to make unanimous inequality judgements even when Lorenz curves intersect. The Gini coefficient becomes relevant in these cases in order to conclusively rank the distributions.     

The concept and measurement of horizontal inequity

This paper discusses the concept of horizontal equity and presents three formal principles as a basis for making comparisons of the extent of horizontal inequity produced by different redistributions. A condition analogous to the Lorenz criterion used for inequality comparisons is established under which the horizontal inequities of redistributions can be ranked without recourse to a cardinal measure of inequity. Ordinal comparisons are not possible in many cases. The three principles suggest three properties that ‘reasonable’ measures of horizontal inequity should possess. A class of measures which satisfies the properties is proposed. Most other existing measures are found not to satisfy them.     

DIVERGENT INEQUALITIES: THEORY AND EMPIRICAL RESULTS

Widely used summary measures of inequality or the idea of the "disappearing middle class" are potentially misleading. Divergences between evidence cited and conclusions drawn include failing to distinguish between the concepts of inequality and polarization, and using scalar "inequality" measures which are not consistent with rankings based on Lorenz curves. In addition, inappropriate claims about trends in inequality can arise from focusing on only a sub-population such as full-time male workers, and failing to account for sampling variability. These divergences are illustrated using Canadian data on labour incomes over the 1967 to 1994 period.     

Inequality foundations of concentration measures: An application to the Hannah-Kay indices

The aim of this paper is to provide a connection between concentration and inequality. In particular, we show that the general entropy inequality indices are, among inequality measures, those consistent with the whole set of Hannah-Kay concentration indices. We isolate the inequality component underlying concentration measures, obtaining an explicit additive decomposition of the change in concentration into the change in its two components, i.e., inequality and the number of firms. This relationship proves to be valid for the whole set of Hannah-Kay concentration indices, and embodies as particular cases some others previously found in the literature. We are particularly grateful to Emilio Cerdá for helpful comments on the mathematical formulation of the paper. The second author also acknowledges support through the European Commission Project #ERBCHRXCT980248.     

Carbon dioxide emission and income: A temporal analysis of cross-country distributional patterns

This paper explores the relationship between the inter-country income inequality and CO₂ emission and temporal shifts in such a relationship. It also examines how the mean per capita CO₂ emission and its distributional inequality are related to the corresponding mean and the distributional inequality of income. The analysis is based on a cross-country panel data set at the level of country-group. Here environmental damage is treated as a private good and the technique of Lorenz and specific concentration curve analysis have been used as the basic analytical framework to argue that distributional inequality of income should be an explanatory variable in the Environmental Kuznets Curve relationship, along with the mean income level. In the empirical exercise, Johansen's cointegration analysis technique is used to explore existence of statistically significant cointegrating vector(s) relating mean emission and Specific Concentration Ratio of emission to mean income level and Lorenz Ratio of income, using a set of country-group specific time series data set which covers four country-groups (viz., Africa, America, Asia and Europe) and the World as a whole. The empirical results confirm that the inter-country income inequality has significant effect on the mean emission level and inter-country inequality of emission level for most of the country-groups considered.     

THE LORENZ ORDER AND THE EFFECTS OF TAXATION POLICIES

To any finite population of n individuals with associated incomes x ₁…, x _n we can associate a Lorenz curve. By associating this population with a random variable X representing the income of a randomly chosen individual of the population, the concept of a Lorenz curve and the associated partial order (the Lorenz Order) based on nested Lorenz curves is readily extended to be defined in the class of all non-negative integrable random variables. In this context well known results on inequality attenuating and inequality rank preserving taxation policies are found to admit simple more general proofs. Some results on the effects of random taxation are also reviewed. The effects of applying different taxation policies within subpopulations lead one to consider questions regarding inequality attenuation results in mixture settings. It is observed that, more generally, inequality comparisons can be unambiguously made between any non-negative variables even if measured in dissimilar units.     

A general class of additively decomposable inequality measures

Summary. This paper presents and characterizes a two-parameter class of inequality measures that contains the generalized entropy measures, the variance of logarithms, the path independent measures of Foster and Shneyerov (1999) and several new classes of measures. The key axiom is a generalized form of additive decomposability which defines the within-group and between-group inequality terms using a generalized mean in place of the arithmetic mean. Our characterization result is proved without invoking any regularity assumption (such as continuity) on the functional form of the inequality measure; instead, it relies on a minimal form of the transfer principle – or consistency with the Lorenz criterion – over two-person distributions. Received: October 27, 1997; revised: March 25, 1998     

Intermediate Inequality and Welfare: The Case of Spain, 1980–81 to 1990–91

The 1990–91 household expenditures distribution in Spain dominates, in the relative ("rightist") Lorenz sense, the 1980–81 distribution, but the latter dominates the former in the absolute ("leftist") Lorenz sense. This situation constitutes a textbook case for intermediate or "centrist" notions of inequality and social welfare. This paper presents the first empirical application of this sort, using the intermediate inequality concept introduced in Del Río and Ruiz-Castillo (2000). The data reveal that there is a decrease in household expenditures inequality for a relatively small set of centrist attitudes.     

Multidimensional welfare distributions: empirical application to household panel data from Vietnam

This article explores the empirical application of theoretical multidimensional welfare distribution analysis techniques to real household welfare distributions. The article operationalizes recent conceptual developments in multidimensional distribution theory and assesses their usefulness for the measurement of multidimensional household inequality. The results strongly highlight the importance of bringing nonmonetary aspects of household welfare into the forefront of inequality analysis. Agreement over the various approaches to the measurement of multidimensional inequality entails, however, nontrivial decisions that may limit the practical usefulness of these measures. We suggest that the use of multidimensional inequality ranges and the application of restrictive dominance criteria to multidimensional welfare distributions may open significant scope for further developments in the empirical analysis of multidimensional inequality.     

A Model of Procedural Decision Making in the Presence of Risk*

We introduce a procedural model of risky choice in which an individual is endowed with a core preference relation that may be highly incomplete. She can, however, derive further rankings of alternatives from her core preferences by means of a procedure based on the independence axiom. We find that the preferences that are generated from an initial set of rankings according to this procedure can be represented by means of a set of von Neumann–Morgenstern utility functions, thereby allowing for incompleteness of preference relations. The proposed theory also yields new characterizations of the stochastic dominance orderings.     

Bayesian assessment of Lorenz and stochastic dominance

We introduce a Bayesian approach for assessing Lorenz and stochastic dominance. For two income distributions, say X and Y, estimated via Markov chain Monte Carlo, we describe how to compute posterior probabilities for: (i) X dominates Y, (ii) Y dominates X and (iii) neither Y nor X dominates. The proposed approach is applied to Indonesian income distributions using mixtures of gamma densities that ensure flexible modelling. Probability curves depicting the probability of dominance at each population proportion are used to explain changes in dominance probabilities over restricted ranges relevant for poverty orderings. They also explain some seemingly contradictory outcomes from the p-values of some sampling theory tests.     

The Intensity and Shape of Inequality: The ABG Method of Distributional Analysis

Inequality is anisotropic: its intensity varies by income level. We here develop a new tool, the isograph, to focus on local inequality and illustrate these variations. This method yields three coefficients which summarize the shape of inequality: a main coefficient, α, which measures inequality at the median; and two correction coefficients, β and γ, which pick up any differential curvature at the top and bottom of the distribution. The analysis of a set of 232 microdata samples from 41 different countries in the LIS datacenter archive allows us to provide a systematic overview of the properties of the ABG (α β γ) coefficients, which are compared to a set of standard indices including Atkinson indices, generalized entropy, Wolfson polarization, and the GB2 distribution. This method also provides a smoothing tool that reveals the differences in the shape of distributions (the strobiloid) and how these have changed over time.     

Parametric Lorenz Curves and the Modality of the Income Density Function

Similar looking Lorenz curves can imply very different income density functions and potentially lead to wrong policy implications regarding inequality. This paper derives a relation between a Lorenz curve and the modality of its underlying income density: given a parametric Lorenz curve, it is the sign of its third derivative which indicates whether the density is unimodal or zeromodal (i.e., downward‐sloping). The density modality of several important Lorenz curves such as the Pareto, Weibull, Singh–Maddala parametrizations and hierarchical families of Lorenz curves are discussed. A Lorenz curve performance comparison with Monte Carlo simulations and data from the UNU–WIDER World Income Inequality Database underlines the relevance of the theoretical result: curve‐fitting based on criteria such as mean squared error or the Gini difference might lead to a Lorenz curve implying an incorrectly‐shaped density function. It is therefore important to take into account the modality when selecting a parametric Lorenz curve.