Measuring Inequalities without Linearity in Envy: Choquet Integrals for Symmetric Capacities

The (generalized) Gini indices rely on the social welfare function of a decision maker who behaves in accordance with Yaari's model, with a function f that transforms frequencies. This SWF can also be represented as the weighted sum of the welfare of all the possible coalitions in the society, where the welfare of a coalition is defined as the income of the worst-off member of that coalition. We provide a set of axioms (Ak) and prove that the three following statements are equivalent: (i) the decision maker respects (Ak), (ii) f is a polynomial of degree k, (iii) the weight of all coalitions with more than k members is equal to zero. Journal of Economic Literature Classification Number: D63.     

A new class of deprivation-based generalized Gini indices

Summary. This paper attempts to merge two strands of the literature on relative deprivation and generalized Gini indices by designing a new class of generalized Gini indices based on the concept of relative deprivation. In this connection, rank-dependent marginal deprivation functions are introduced. A new class of generalized Gini indices is derived axiomatically. It turns out to be the sum of modified versions of the S-Gini and the I-Gini introduced by Donaldson and Weymark (1980). Received: December 23, 1998; revised version: November 9, 1999     

Salience and social choice

The axioms of expected utility and discounted utility theory have been tested extensively. In contrast, the axioms of social welfare functions have only been tested in a few questionnaire studies involving choices between hypothetical income distributions. In a controlled experiment with 100 subjects placed in the role of social planners, we test five fundamental properties of social welfare functions to determine the efficacy of traditional social choice models in predicting social planner allocations when presented with choice sets designed to test the axioms of the theory. We find that three properties of the standard social welfare functions tested are systematically violated, producing an Allais paradox, a common ratio effect, and a framing effect in social choice. We find support for scale invariance and a preference for tail-increasing transfers. Our experiment also enables us to test a model of salience-based social choice which predicts the systematic deviations and highlights the close relationship between these anomalies and the classical paradoxes for risk and time.

     

The measurement of income mobility: A partial ordering approach

Summary. Given a set of longitudinal data pertaining to two populations, a question of interest is the following: Which population has experienced a greater extent of income mobility? The aim of the present paper is to develop a systematic way of answering this question. We first put forth four axioms for income movement-mobility indices, and show that a familiar class of measures is characterized by these axioms. An unambiguous (partial) ordering is then defined as the intersection of the (complete) orderings induced by the mobility measures which belong to the characterized class; a transformation of income distributions is “more mobile” than another if, and only if, the former is ranked higher than the latter for all mobility measures which satisfy our axioms. Unfortunately, our mobility ordering depends on a parameter, and therefore, it is not readily apparent how one can apply it to panel data directly. In the second part of the paper, therefore, we derive several sets of parameter-free necessary and sufficient conditions which allow one to use the proposed mobility ordering in making unambiguous income mobility comparisons in practice. Received: July 12, 1995; revised version: May 13, 1997     

Prioritarian Poverty Comparisons with Cardinal and Ordinal Attributes

The ethical view of prioritarianism holds that if an extra bundle of attributes is to be allocated to either of two individuals, then priority should be given to the worse off among the two. We consider multidimensional poverty comparisons with cardinal and ordinal attributes and propose three axioms that operationalize the prioritarian view. Each priority axiom, in combination with a handful of standard properties, characterizes a class of poverty measures.     

Averaging Income Distributions

Various inequality and social welfare measures often depend heavily on the choice of a distribution of income. Picking a distribution that best fits the data involves throwing away information and does not allow for the fact that a wrong choice can be made. Instead, Bayesian model averaging utilizes a weighted average of the results from a number of income distributions, with each weight given by the probability that a distribution is ‘correct’. In this study, prior densities are placed on mean income, the mode of income and the Gini coefficient for Australian income units with one parent (1997–8). Then, using grouped sample data on incomes, posterior densities for the mean and mode of income and the Gini coefficient are derived for a variety of income distributions. The model‐averaged results from these income distributions are obtained.     

A Note on α‐Gini Measures

This note provides a characterization of α‐Gini inequality measures. These measures generalize the standard Gini index by including one sensitivity parameter α, which captures different value judgments. The α‐Gini measures are shown to be weakly decomposable and unit consistent. Weak decomposition provides within‐group and between‐group inequalities. Unit consistency keeps unchanged the ranking of two income distributions when the income units vary. It is shown that the α‐Gini measures are relevant with either “leftist” or“rightist” views.     

An inequality measure for stochastic allocations

Few papers in the literature on inequality measurement deal with uncertainty, particularly when the ranking of cohorts may not be fixed. We present a set of axioms implying such a class of inequality measures under uncertainty that is a one-parameter extension of the generalized Gini mean over the distribution of average allocations. The extension consists of a quadratic term accounting for inter-personal correlations. In particular, our measure can simultaneously accommodate a preference for “shared destiny”, a preference for probabilistic mixtures over unfair allocations, and a preference for fairness “for sure” over fairness in expectation.     

Ethically flexible gini indices for income distributions in the continuum

A single-parameter generalization of the Gini coefficient (S-Gini) is presented for income distributions defined in the continuum. Special cases are Dorfman's formula for the Gini, and the authors' S-Ginis for finite populations. We exploit the duality between indices of relative inequality and homothetic social-evaluation functions to construct these new indices.     

INEQUALITY OF OPPORTUNITIES VS. INEQUALITY OF OUTCOMES: ARE WESTERN SOCIETIES ALL ALIKE?

This paper analyzes the relationship between income inequality and inequality of opportunities for income acquisition in nine developed countries during the 1990s. Equality of opportunity is defined as the situation where income distributions conditional on social origin cannot be ranked according to stochastic dominance criteria. We measure social origin by parental education and occupation and use the database built by Roemer et al. (2003 ). Stochastic dominance is assessed using nonparametric statistical tests. Our results indicate strong disparities in the degree of equality of opportunity across countries and a strong correlation between inequality of outcomes and inequality of opportunity. The U.S. and Italy show up as the most unequal countries in terms of both outcome and opportunity. At the opposite extreme, income distributions conditional on social origin are almost the same in Scandinavian countries even before any redistributive policy. We complement the ordinal comparison by resorting to an original scalar “Gini” index of opportunities, which can be decomposed into a risk and a return component. In our sample, inequality of opportunity is mostly driven by differences in mean income conditional on social origin, and differences in risk compensate the return element in most countries.     

A consistent multidimensional Pigou-Dalton transfer principle

The unidimensional Pigou-Dalton transfer principle demands that a regressive transfer in income—a transfer from worse-off (poor) to better-off (rich)—decreases social welfare. In a multidimensional setting the direct link between income (or any other attribute) and individual well-being is absent. We interpret the social welfare level of a distribution in which each individual has the same bundle as the individual well-being level. We define regressivity on the basis of this individual well-being ranking. In a setting with both transferable and non-transferable attributes, the imposition of the ensuing “consistent” Pigou-Dalton principle forces individual well-being to have a quasi-linear structure in the transferable attributes. Since we allow for transferable and non-transferable attributes, our result provides a normative underpinning for criteria in the distinct literatures of multidimensional inequality measurement (only transferable attributes) and of needs (one transferable and one non-transferable attribute).     

Estimation of Concentration Measures and Their Standard Errors for Income Distributions in Poland

Measures of concentration (inequality) are often used in the analysis of income and wage size distributions. Among, them the Gini and Zenga coefficients are of greatest importance. It is well known that income inequality in Poland increased significantly in the period of transformation from a centrally planned economy to a market economy. High income inequality can be a source of serious problems, such as increasing poverty, social stratification, and polarization. Therefore, it seems especially important to present reliable estimates of income inequality measures for a population of households in Poland in different divisions. In this paper, some estimation methods for Gini and Zenga concentration measures are presented together with their application to the analysis of income distributions in Poland by socio-economic groups. The basis for the calculations was individual data coming from the Polish Household Budget Survey conducted by the Central Statistical Office. The standard errors of Gini and Zenga coefficients were estimated by means of the bootstrap and the parametric approach based on the Dagum model.     

The Choquet Bargaining Solutions

We axiomatically investigate the problem of rationalizing bargaining solutions by social welfare functions that are linear in every rank-ordered subset of Rⁿ₊ . Such functions, the so-called Choquet integrals, have been widely used in the theories of collective and individual choice. We refer to bargaining solutions that can be rationalized by Choquet integrals as Choquet bargaining solutions. Our main result is a complete characterization of Choquet bargaining solutions. As a corollary of our main result, we also obtain a characterization of the generalized Gini bargaining solutions introduced by Blackorby et al. (1994, Econometrica62, 1161–1178). Journal of Economic Literature Classification Numbers: D71, C78.     

Unequal uncertainties and uncertain inequalities: an axiomatic approach

In this paper, we provide an axiomatic characterization of social welfare functions for uncertain incomes. Our most general result is that a small number of reasonable assumptions regarding welfare orderings under uncertainty rule out pure ex ante as well as pure ex post evaluations. Any social welfare function that satisfies these axioms should lie strictly between the ex ante and the ex post evaluations of income distributions. We also provide an axiomatic characterization of the weighted average of the minimum and the maximum of ex post and ex ante evaluations.     

Quality,quantity and duration of lives

The evaluation of development processes and of public policies often involves comparisons of social states that differ in income distributions, population sizes and life longevity. This may require social evaluation principles to be sensitive to the quality, the quantity and the duration of lives. This paper: 1) reviews some of the normative issues at stake, 2) proposes and discusses some specific methods to address them in a generalized utilitarian framework and 3) briefly illustrates the application of some of these methods to the global distribution of incomes, population sizes and longevity over the last century. Depending on the approach taken, it is found inter alia that global social welfare in 2010 can be deemed to be between 1.8 and 407 times that of 1910, the role given to the quantity of lives being particularly important in that assessment.     

离散分布收入数据基尼系数的矩阵向量形式及相关问题

本文旨在给出离散分布收入数据基尼系数的很一般的矩阵向量形式,并以此为基础,在一般的意义上讨论基尼系数的按组分解及按类分解问题。本文的结论是,借助矩阵向量形式,我们可以确切地说明基尼系数"完全分解"的含义,更直观地表述目前已有的各种并组或并类基尼系数按组或按类分解的意义,简洁地揭示它们内在的数理结构;通过矩阵向量的运算法则,我们可以直接发现,不可能一般性地实现并组或并类基尼系数按组或按类"完全分解",且使分解后各项均有明确的经济意义;除非进一步给出特殊假定,各种分解的运算量与直接计算并组或并类基尼系数的运算量是相当的。     

Capital Wealth Taxation: Theory and Application

Using an ls model analysis, the economic effects of implementing a modest taxation rate on capital wealth (3%) are found to be basically favorable: slightly higher output, lower inequality as measured by various Gini coefficients, and higher social welfare according to the three major social welfare functions—Bentham, Nash and Rawls. Implementing capital wealth taxation enables a compensating reduction in the labor-income taxation rate. The single most important consequence of this change is increased labor output among the wealthiest households, whose labor productivity is highest. Even though labor output is reduced among less-wealthy households, the overall effect on aggregate output is positive.     

The trend of welfare disparity among subgroups of population in Australia 1983-1984 - 1993-1994

The aim of this paper is to estimate the trend of social welfare for Australia using 1983–1984, 1988–1989 and 1993–1994 Household Expenditure Survey data. The functional form of the Social Welfare Function (SWF) was derived by Sen, Degum, Yitzhaki and Shesinski (all independently). Since the function contains the Gini coefficient as the inequality parameter, it could not be formally disaggregated by subgroups of population. This paper, using a method of subgroup decomposition of the Gini coefficient developed by Podder, attempts to disaggregate the SWF. With this method it is now possible to identify disadvantaged groups by their relative shares in total welfare. In addition the method is used to determine effect of economic growth on specific subgroups, and in turn, on total social welfare. This study is based on the Australian economy. Distribution of relative shares of total social welfare among various regional groups are identified, groups determined by occupational status and groups determined by country of birth. The effect on society's welfare for a percentage change in income of a group and the trend of relative welfare of a specific group are also computed. This information can be used in a variety of social decision making situations, including cost benefit analysis.     

中国区域人才集聚合理性及其评价——中美500强企业地区分布的对比视角

本文采用中美500强企业地区分布统计数据,根据企业分布曲线,计算了中美500强企业的地区分布系数,并将此系数作为中美人才集聚度的测量指标。研究发现,我国500强企业的地区分布系数为0．58,远远高于可接受的国民收入分布（基尼）系数,但低于美国500强企业的地区分布系数。本文分析了产生此种状况的原因,展望了我国人才聚集的发展趋势,并提出了相对应的建议。     

On multidimensional poverty rankings of binary attributes

We address the problem of ranking distributions of attributes in terms of poverty, when the attributes are represented by binary variables. To accomplish this task, we identify a suitable notion of “multidimensional poverty line” and characterize axiomatically the Head-Count and the Attribute-Gap poverty rankings, which are the natural counterparts of the most widely used income poverty indices. Finally, we apply our methodology and compare our empirical results with those obtained with some other well-known poverty measures.