Income Share Elasticity and Stochastic Dominance

Income share elasticity is a function π which can describe the size distribution of income (Esteban in Intern Econ Rev 27:439–444, 1986). On the other hand, the conventional density representation of the latter gives parameters of first or second order stochastic dominance (SD), widely used to describe shifts in income distribution, to which inequality measures are attached. The paper draws a link between the two, by providing conditions such that a given shift to π is equivalent to a first or second order SD shift of the distribution of income. Some applications to Lorenz rankings are also provided.     

Robust inequality comparisons

This paper is concerned with the problem of ranking Lorenz curves in situations where the Lorenz curves intersect and no unambiguous ranking can be attained without introducing weaker ranking criteria than first-degree Lorenz dominance. To deal with such situations, Aaberge (Soc Choice Welf 33:235–259, 2009) introduced two alternative sequences of nested dominance criteria for Lorenz curves, which proved to characterize two separate systems of nested subfamilies of inequality measures. This paper uses the obtained characterization results to arrange the members of two different generalized Gini families of inequality measures into subfamilies according to their relationship to Lorenz dominance of various degrees. Since the various criteria of higher degree Lorenz dominance provide convenient computational methods, these results can be used to identify the largest subfamily of the generalized Gini families, and thus the least restrictive social preferences, required to reach unambiguous ranking of a set of Lorenz curves. We further show that the weight-functions of the members of the generalized Gini families offer intuitive interpretations of higher degree Lorenz dominance, which generally has been viewed as difficult to interpret because they involve assumptions about third and higher derivatives. To demonstrate the usefulness of these methods for empirical applications, we examine the time trend in income and earnings inequality of Norwegian males during the period 1967–2005.     

Ranking intersecting Lorenz curves

This paper is concerned with the problem of ranking Lorenz curves in situations where the Lorenz curves intersect and no unambiguous ranking can be attained without introducing weaker ranking criteria than first-degree Lorenz dominance. To deal with such situations two alternative sequences of nested dominance criteria between Lorenz curves are introduced. At the limit the systems of dominance criteria appear to depend solely on the income share of either the worst-off or the best-off income recipient. This result suggests two alternative strategies for increasing the number of Lorenz curves that can be strictly ordered; one that places more emphasis on changes that occur in the lower part of the income distribution and the other that places more emphasis on changes that occur in the upper part of the income distribution. Both strategies turn out to depart from the Gini coefficient; one requires higher degree of downside and the other higher degree of upside inequality aversion than what is exhibited by the Gini coefficient. Furthermore, it is demonstrated that the sequences of dominance criteria characterize two separate systems of nested subfamilies of inequality measures and thus provide a method for identifying the least restrictive social preferences required to reach an unambiguous ranking of a given set of Lorenz curves. Moreover, it is demonstrated that the introduction of successively more general transfer principles than the Pigou–Dalton principle of transfers forms a helpful basis for judging the normative significance of higher degrees of Lorenz dominance. The dominance results for Lorenz curves do also apply to generalized Lorenz curves and thus provide convenient characterizations of the corresponding social welfare orderings.     

Detecting a change in wealth concentration without the knowledge of the wealth distribution

Consumption is commonly used as a proxy for permanent income. We go a step further by establishing the link between the distribution of consumption and that of permanent income in terms of dominance orderings. We introduce two new dominance orderings, the Generalized Top Lorenz test and the related affluence ordering. If consumption is a concave function of permanent income, we get an indirect and robust method to detect a change in permanent income concentration when the full stream of income receipts along the life cycle is unknown and only consumption data are available. Our application on US data for the period 1980?C2002 points out the difficult start in life of people belonging to the ??Baby loser generation?? (people born in the sixties) with respect to the previous and following cohort.     

Utility-gap Dominances and Inequality Orderings

The justification for using Lorenz dominance as an inequality ranking condition has been based on the aggregate social welfare comparison and the Pigou–Dalton principle of transfers. Since both the aggregating aspect of the social welfare function and certain implications of the principle of transfers are debatable, ordering conditions stronger than Lorenz dominance are worth exploring. A particularly interesting direction to pursue is to follow the frequently invoked notion that inequality is the “gap” between the rich and the poor. This paper follows this notion to formally propose a unified utility-gap concept and characterizes several utility-gap based conditions as general stronger-than-Lorenz-dominance ranking criteria. Specifically, we propose utility-gap dominance which requires all pair-wise utility-gaps in one distribution to be uniformly smaller than those of the other distribution. We then explore a conceptually weaker dominance concept – quasi dominance – which imposes conditions only on the gap between each person’s utility and some reference utility point of the distribution. I am grateful to two anonymous referees and Peter Lambert for their very constructive comments and suggestions on an earlier version of the paper. The usual caveat applies.     

More on the measurement of inequality

The article, written in 1973, examines what comparisons of income distributions can be made when Lorenz curves cross, employing the concept of third-order stochastic dominance. “More on the measurement of inequality” as reviewed by Prof. P. Lambert will appear in the Rediscovered Classics section, along with the editorial.     

Ranking Income Distributions According to Equality of Opportunity

According to the opportunity egalitarian ethics, economic inequalities due to factors beyond the individual responsibility are inequitable and to be compensated by society, whereas inequalities due to personal responsibility are equitable and not to be compensated. In this paper we derive welfare and inequality criteria that respect this principle of justice. We characterize axiomatically classes of opportunity egalitarian social evaluation functions. Then, by requiring unanimous agreement among such classes, we provide simple distributional conditions to rank income distributions. These criteria extend the Lorenz and generalized Lorenz partial orderings, commonly used in the unidimensional case of income distributions, to the current context of equality of opportunity.     

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.     

The measurement of opportunity inequality: a cardinality-based approach

We consider the problem of ranking distributions of opportunity sets on the basis of equality. First, conditional on a given ranking of individual opportunity sets, we define the notion of an equalizing transformation. Then, assuming that the opportunity sets are ranked according to the cardinality ordering, we formulate the analogues of the notions of the Lorenz partial ordering, equalizing (Dalton) transfers, and inequality averse social welfare functions – concepts which play a central role in the literature on income inequality. Our main result is a cardinality-based analogue of the fundamental theorem of inequality measurement: one distribution Lorenz dominates another if and only if the former can be obtained from the latter by a finite sequence of rank preserving equalizations, and if and only if the former is ranked higher than the latter by all inequality averse social welfare functions. In addition, we characterize the smallest monotonic and transitive extension of our cardinality-based Lorenz inequality ordering. Received: 2 May 1995 / Accepted: 11 October 1996     

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     

Characterizations of egalitarian binary relations as transitive closures with a special reference to Lorenz dominance and to single-crossing conditions

This article considers, in the context of the fixed-population constant-sum comparison of income distributions, a number of intransitive binary relations smaller than Lorenz dominance. We determine their transitive closure, and we study how they relate to each other and to other relations that have appeared in the literature. Among other results, we provide alternative characterizations of Lorenz dominance.     

Using equivalent income of equivalent adults to rank income distributions

The paper proposes the consistent and rigorous use of equivalence scales for households of different size. Equivalence scales are employed for deriving the income of an equivalent adult representing the household and for weighting the different household types. The proceeding takes into account the possibilities and the needs of different household types at the same time and seems to be adequate if income distributions are to be compared in terms of social welfare or inequality. The approach allows to define and to investigate the usual concepts (Lorenz curve, Lorenz dominance, social welfare function, progressive transfers etc.) for heterogeneous populations. They can be interpreted in an obvious way, and the results, well known for homogeneous populations, can be generalized. Furthermore, some unpleasant and unsatisfactory paradoxa or impossibility results, which can be found in the literature, vanish in the framework considered. Received: 7 March 1997/Accepted: 4 November 1997     

Continuity and egalitarianism in the evaluation of infinite utility streams

There exists a utilitarian tradition à la Sidgwick of treating equal generations equally. Diamond showed that there exists no social evaluation ordering over infinite utility streams in the presence of the Pareto principle, the Sidgwick principle, and continuity. Instead of requiring the Sidgwick principle of procedural fairness, we focus on two principles of distributional egalitarianism along the line of the Pigou–Dalton transfer principle and the Lorenz domination principle, and show that there exists no social evaluation relation satisfying one of these egalitarian principles and the weakened continuity and rationality axioms even in the absence of the Pareto principle. This paper is a substantially revised version of our foregoing paper, “On the Possibility of Continuous, Paretian and Egalitarian Evaluation of Infinite Utility Streams,” which was based on our joint research conducted as a part of the Project on Intergenerational Equity under the auspices of the Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan. Thanks are due to Geir Asheim, Kaushik Basu, Walter Bossert, Hajime Hori, Tomoki Inoue, Mitsunori Noguchi and Koichi Tadenuma with whom we had several discussions on this and related issues. We are also grateful to the anonymous referees and the Associate Editor in charge of our earlier paper for their helpful comments and suggestions. Needless to say, no one other than ourselves should be held responsible for any remaining defect of this paper.     

An extended Gini approach to inequality measurement

It is well-known that, when the Lorenz curves do not cross, the ranking of distributions provided by the Gini index is identical to the one implied by the Lorenz criterion. This does not preclude inequality as measured by the Gini index to increase while the Lorenz curves cross. A suitable modification of the Gini coefficient allows the Lorenz quasi-ordering to coincide with the ranking generated by the application of unanimity over the class of extended Gini indices. Recently the Lorenz quasi-ordering and the underlying principle of transfers have come under attack, while new criteria – the differentials, deprivation and satisfaction quasi-orderings – have been proposed for providing unambiguous rankings of distributions. We suggest to weaken the principle of transfers by imposing additional restrictions on the progressive transfers, which take into account the positions on the income scale of the donors and beneficiaries. We identify the subclasses of extended Gini indices that satisfy these weaker versions of the principle of transfers and we show that the application of unanimity among these classes generate rankings of distributions that coincide with those implied by the differentials, deprivation and satisfaction quasi-orderings.      

Visual Dominance and Visual Egalitarianism: Individual and Group-Level Influences of Sex and Status in Group Interactions

This study investigated visual dominance and visual egalitarianism of men and women (N = 94; 17 teams) in team meetings at diverse workplaces. Two novel gaze-related measures were developed: (a) a group visual dominance ratio (group-VDR) assessing each member’s visual dominance vis-à-vis all other members, and (b) a gaze distribution index (GDI) assessing each member’s visual egalitarianism to all group members. Multilevel analyses were conducted to account for influences of the team members’ sex and status on the individual level and for influences of sex and status composition of the teams, and the team leaders’ sex on the group level. Results suggested that high-status individuals displayed more visual dominance than low-status individuals. The significant interaction of individuals’ sex and status indicated that the positive relationship of status and visual dominance applied particularly to women. The more women in a team, the more visual dominance was displayed. The team leader’s sex significantly influenced visual egalitarianism: Gaze distribution was less egalitarian when the team leader was male.     

Distributive interdependencies in liberal egalitarianism

Liberal egalitarian theories of justice share the egalitarian ideal that inequalities due to factors beyond individual control should be eliminated. This justifies that individuals sometimes receive more (or less) than their marginal productivity, which in turn implies that a change in one individual’s effort sometimes will affect the post-tax income of others. What restrictions should we place on such distributive interdependencies? The article provides characterizations of three main classes of redistribution mechanisms on the basis of how this question is answered.     

Association of Egalitarian Attitudes with Income,Poverty, and Entitlement Program Utilization

This article examines the relationship between egalitarian gender attitudes and its effect on income, poverty, and social service utilization. In a sample of married fathers in the Fragile Families data (N = 717), we conducted regression analyses on these questions, five years after initial interviews. Our results indicate that egalitarian attitudes were positively associated with income and social service utilization and negatively associated with poverty. Implications of these findings are discussed, including possible explanations for these findings, their importance for social work practice, and recommendations for social policy.     

Intra-household inequality in transitional Russia

The paper proposes an original strategy for analyzing household sharing of income and satisfaction. Using two different subjective questions from the Russian data RLMS (Russia Longitudinal Monitoring Survey), we assume a correspondence between, first, the perception of income that household members report and their true income sharing, and, second, between their answer to a satisfaction question and their utility. We show that answers given by different members of the household bring pertinent information on income sharing and utility in the household. In particular, we find a significant effect of the female–male wage ratio in reported income perception and satisfaction differentials between household members. Our data covers the transition period (1994–2003) characterized by massive economic and social changes in Russia. We investigate the evolution of intra-household welfare distribution by exploiting the episode of the financial crisis of 1998 to distinguish two periods: pre- and post-crisis. Our empirical strategy consists of applying probit-type models allowing the parameters to change in a discrete manner between two periods and accounting for discrete unobserved heterogeneity.     

Comparing degrees of inequality aversion

We propose a straightforward dominance procedure for comparing social welfare orderings (SWOs) with respect to the degree of inequality aversion they express. Three versions of the procedure are considered, each of which uses a different underlying criterion of inequality comparisons: (i) a concept based on the Lorenz quasi-ordering, which we argue to be the ideal version, (ii) a concept based on a minimalist criterion of inequality, and (iii) a concept based on the relative differentials quasi-ordering. It turns out that the traditional Arrow–Pratt approach is equivalent to the latter two concepts for important classes of SWOs, but that it is profoundly inconsistent with the Lorenz-based concept. With respect to the problem of combining extreme inequality aversion and monotonicity, concepts (ii) and (iii) identify as extremely inequality averse a class of SWOs that includes leximin as a special case, whereas the Lorenz-based concept (i) concludes that extreme inequality aversion and monotonicity are incompatible.     

The Lorenz zonotope and multivariate majorizations

 The distribution of d commodities among n individuals is described by an n×d row stochastic matrix. We present a geometric approach to order such matrices. For a row stochastic matrix the Lorenz zonotope is investigated, which is a higher dimensional generalization of the Lorenz curve. The Lorenz zonotope is a convex polytope. The inclusion of Lorenz zonotopes defines an ordering between row stochastic matrices, which is a multivariate majorization. For a cone in nonnegative d-space, a cone extension of the Lorenz zonotope and the respective inclusion ordering are introduced. We study this class of orderings and establish equivalence with known majorizations. It is provided a finite set of inequalities to which the ordering is equivalent. Received: 16 February 1994/Accepted: 22 May 1996