Lorenz Dominance and Non-welfaristic Redistribution

Our concern is for income inequalities that may result from non-welfaristic redistribution schemes. We show that for large classes of income functions Lorenz dominance results can be found in the comparison of two egalitarian equivalent mechanisms. Comparisons of different conditionally egalitarian mechanisms only yield poverty dominance results. In general, no egalitarian equivalent mechanism can be Lorenz dominated by a conditionally egalitarian mechanism. Our analysis stresses the need for accurate empirical estimates of the pre-tax income function and of the distributions of responsibility and compensation characteristics. We thank the Editor, Marc Fleurbaey and two anonymous referees, Geert Dhaene and seminar/conference participants at UAP-workshop (Namur, 2003), ‘Welfarist and non-welfarist approaches to public economics’ (Ghent, 2004), SED (Palma, 2004), SSC&W (Osaka, 2004) and IIPF (Milan, 2004) for helpful comments and suggestions. Financial support from the Federal Public Planning Service Science Policy, Interuniversity Attraction Poles Program – Belgian Science Policy [Contract No. P5/21] is gratefully acknowledged.     

Measuring and implementing equality of opportunity for income

Departing from the welfarist tradition, recent theories of justice focus on individual opportunities as the appropriate standard for distributive judgments. To explore how this philosophical conception can be translated into concrete public policy, we select the income as relevant outcome and the income tax as the relevant redistributive policy, and we address the following questions: (i) what is the degree of opportunity inequality in an income distribution? (ii) how to design an opportunity egalitarian income tax policy? Several criteria for ranking income distributions on the basis of equality of opportunities are derived. Moreover, we characterize an opportunity egalitarian income tax and we formulate criteria for choosing among alternative tax schedules. I would like to thank Peter Lambert for his encouragement and guidance, and Walter Bossert, Maurice Salles, and partecipants at the 1998 Social Choice and Welfare Conference, Vancouver, and at the Conference on Non–Welfaristic Issues in Normative Economics in Caen for their helpful comments and suggestions. Financial support of the Ministero dellIstruzione, Università e Ricerca (Italy) is gratefully acknowledged. Responsibility for any remaining deficiency is mine.     

Lorenz undominated allocations for TU-games: The weighted Coalitional Lorenz Solutions

In this paper we introduce and study the w-Coalitional Lorenz Solutions to identify the similarities and differences between the prenucleolus and the Shapley value. The similarity is that they both use egalitarian criteria over coalitions. The two main differences are: the prenucleolus and the Shapley value use different egalitarian criteria, and they weight the coalitions differently when applying the criteria. Received: 27 October 2000/Accepted: 2 October 2001     

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.     

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     

An Ex-Post View of Inequality of Opportunity in France and its Regions

This paper proposes an ex-post measure of inequality of opportunity in France and its regions by assessing the inequality between individuals exerting the same effort. To this end, we define a fair income that fulfils ex-post equality of opportunity requirements. Unfairness is measured by an unfair Gini based on the distance between the actual income and the fair income. Our findings reveal that the measures of ex-post inequality of opportunity largely vary across regions, and that this is due to differences in reward schemes and in the impact of the non responsibility factors of income. We find that most regions have actual incomes closer to fair incomes than to average income, excepted Ile de France where the actual income looks poorly related to effort variables. Finally, we find that income inequality and inequality of opportunity are positively correlated among regions.     

Richness orderings

An index of richness in a society is a measure of the extent of its affluence. This paper presents an analytical discussion on several indices of richness and their properties. It also develops criteria for ordering alternative distributions of income in terms of their richness. Given a line of richness, an income level above which a person is regarded as rich, and depending on the redistributive principle, it is shown that the ranking relation can be implemented by seeking dominance with respect to the generalized Lorenz curve of the rich or the affluence profile of the society. When the line of richness is assumed to be variable, we need to employ the stochastic dominance conditions for ordering the income distributions.     

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.      

Beyond Oaxaca–Blinder: Accounting for differences in household income distributions

This paper develops a method to decompose differences across distributions of household income, based on counterfactual distributions that ‘lie between’ the actually observed distributions. Our approach decomposes differences between any two income distributions (or functionals such as inequality or poverty measures) into shares due to price effects; occupational structure effects; and endowment effects. Comparing the household income distributions of the USA and Brazil in 1999, we find that most of Brazil’s excess inequality (of 13 Gini points) is accounted for by underlying inequalities in the distributions of education and of non-labor income, notably pensions (between four and six Gini points each). Steeper returns to education in Brazil also make an important contribution (of two to five points). Differences in occupational structure and in racial and demographic composition are much less important. An erratum to this article can be found at     

Gini’s nuclear family

The purpose of this paper is to justify the use of the Gini coefficient and two close relatives for summarizing the basic information of inequality in distributions of income. To this end we employ a specific transformation of the Lorenz curve, the scaled conditional mean curve, rather than the Lorenz curve as the basic formal representation of inequality in distributions of income. The scaled conditional mean curve is shown to possess several attractive properties as an alternative interpretation of the information content of the Lorenz curve and furthermore proves to yield essential information on polarization in the population. The paper also provides asymptotic distribution results for the empirical scaled conditional mean curve and the related family of empirical measures of inequality.      

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.     

AN EGALITARIAN SYSTEM BREEDS GENEROSITY: THE IMPACT OF REDISTRIBUTION PROCEDURES ON PRO‐SOCIAL BEHAVIOR

In a meritocratic system, people are compensated on the basis of their individual ability, whereas in an egalitarian system people are equally compensated. Essentially, in the latter system high performers are taxed and subsidize underperformers. Would differences in income redistribution procedures affect people's pro‐social behavior? In experiments, we found that people are more generous toward strangers in an egalitarian treatment than in a meritocratic treatment. Interestingly, being taxed does not reduce the generosity of high performers, whereas being subsidized significantly increases the generosity of low performers. (JEL C91, D63, D64)     

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.     

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     

Immigrant Wage Disadvantage in Sweden and the United Kingdom: Wage Structure and Barriers to Opportunity1

This article examines immigrant/native-born wage inequalities among workers in two Western European countries: Sweden, social democratic and with comparatively low wage dispersion, and the United Kingdom, economically liberal and with comparatively high wage dispersion. The analysis includes immigrants from 26 countries of origin. Findings demonstrate that inequalities in terms of real wages are smaller in more egalitarian Sweden. However, in terms of relative positions within the labor market, inequalities are if anything smaller in the UK. These findings highlight the role of wage dispersion in magnifying immigrant disadvantage, but also the limits of wage compression for ameliorating barriers to immigrant opportunity.     

Averaging Lorenz curves

A large number of functional forms has been suggested in the literature for estimating Lorenz curves that describe the relationship between income and population shares. The traditional way of overcoming functional-form uncertainty when estimating a Lorenz curve is to choose the function that best fits the data in some sense. In this paper we describe an alternative approach for accommodating functional-form uncertainty, namely, how to use Bayesian model averaging to average the alternative functional forms. In this averaging process, the different Lorenz curves are weighted by their posterior probabilities of being correct. Unlike a strategy of picking the best-fitting function, Bayesian model averaging gives posterior standard deviations that reflect the functional-form uncertainty. Building on our earlier work (Chotikapanich and Griffiths, 2002), we construct likelihood functions using the Dirichlet distribution and estimate a number of Lorenz functions for Australian income units. Prior information is formulated in terms of the Gini coefficient and the income shares of the poorest 10% and poorest 90% of the population. Posterior density functions for these quantities are derived for each Lorenz function and are averaged over all the Lorenz functions.     

Inequality aversion,impartiality and utilitarianism

Egalitarianism presents the problem of baseline-dependence: egalitarianism from where? No collective choice can be egalitarian for all choices of a baseline. So the question arises as to whether and how the egalitarian program can be weakened in order to be compatible with baseline-independence. I characterize the set of choices that satisfy a minimal degree of inequality-aversion, embodied in the generalized Lorenz criterion, independently of the choice of a baseline. It turns out that this set coincides with the Lorenz-Utilitarian set, the set of Utilitarian choices which are not Lorenz dominated for any choice of baseline. Either this set is empty or it coincides with the Utilitarian set (when this is a singleton). Moreover, I also characterize the full Utilitarian set by using a standard requirement of impartiality (the Suppes-Sen criterion).     

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.     

Perceived Freedom and its Sociological Effects: An Inquiry into the Relationship Between Liberalism and Inequality

Abstract:  Individual freedom has been regarded as one of the most important criteria for a desirable society and life. Recently, the theories of liberalism and distributive justice have received increasing attention as a theoretical basis for tackling problems pertaining to socioeconomic inequality and poverty. However, few quantitative analyses focus on the meaning of freedom in the context of socioeconomic inequality and political order in an existing liberal-democratic society. In this paper, I construct a self-rated measure of freedom—the overall perceived freedom scale—and examine (1) whether socioeconomic inequality is related to inequality of freedom, (2) whether freedom is related to subjective well-being, and (3) whether freedom affects people's political attitude toward inequalities. Analyses using data from a 2005 Japanese national survey ( N  = 1320) reveal the following results: (1) people with higher personal income (not household income), higher education, and better health tend to perceive greater freedom, (2) people with higher perceived freedom are more likely to be satisfied with their life, and (3) individuals with higher perceived freedom are more likely to oppose the government's egalitarian policy, and their opposition is based on the liberal principle of freedom and self-responsibility. On the basis of these findings, the double-edged possibility of individual freedom and liberalism as an "enhancer" and "moderator" of socioeconomic inequalities and political conflicts is suggested.     

Arbeitsteilung,Berufsorientierung und Partnerschaftsstabilität – Ehen und nichteheliche Lebensgemeinschaften im Vergleich*

The central finding of this paper is that the link between the division of labour and the risk of separation differs depending on the type of relationship. Marriages tend to be destabilised by the wife’s participation in the labour force and stabilised by the traditional division of labour. Empirical evidence of this is manifest in a positive effect of the wife’s income and a negative effect of the husband’s relative income on the likelihood of separation. In contrast to this, unmarried long-term relationships are stabilised by egalitarian division of labour. This is evident from the absence of negative effects of an employment of the female cohabiter on stability of relationship. Furthermore, in unmarried cohabitations the relative income of the man increases the risk of a separation. In addition to the income ratio, homogamous attitudes have explanatory value. Cohabitations are stable when both partners are career-oriented. This finding points to an egalitarian gender-role-orientation among unmarried men. It also implies that an egalitarian orientation has an impact on the living arrangement which needs to be accepted by both partners – for instance when family planning and timing of marriage are concerned.