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.     

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.     

Distorted Lorenz curves: models and comparisons

The economic literature contains many parametric models for the Lorenz curve. A number of these models can be obtained by distorting an original Lorenz curve $L$ by a function $h$ , giving rise to a distorted Lorenz curve ${\widetilde{L}}=h\circ L$ . In this paper, we study, in a unified framework, this family of curves. First, we explore the role of these curves in the context of the axiomatic structure of Aaberge (2001) for orderings on the set of Lorenz curves. Then, we describe some particular models and investigate how changes in the parameters in the baseline Lorenz curve $L$ affect the transformed curve ${\widetilde{L}}$ . Our results are stated in terms of preservation of some stochastic orders between two Lorenz curves when both are distorted by a common function.     

Discontinuous transformations,Lorenz curves and transfer policies

This article reconsiders the effect of variable transformations on the redistribution of income. If the theorems hold for all income distributions, previously given conditions are both necessary and sufficient. Different versions of the conditions are compared and their equivalence are stated. It is proved that the continuity of the transformations can be implicitly included in the necessary and sufficient conditions. In general, continuity is a realistic restriction when the theory is applied to transfer problems. The effect of dropping the explicit continuity restriction on the transformations is analysed, since situations in which discontinuity can be assumed are also considered. The analyses have yielded the fundamental result that continuity is a necessary condition if income inequality persists or is reduced.     

Intersecting generalized Lorenz curves and the Gini index

As is well known, the use of the Gini coefficient in comparisons is inconsistent with an utilitarian approach. This paper analyzes the Gini coefficient's normative significance in welfare comparisons evaluating income distributions according to Yaari dual social welfare function. When generalized Lorenz curves cross once, the Gini coefficient is decisive in determining welfare rankings if we strengthen the Principle of Transfers applying a Positional version of the Principle of Transfer Sensitivity. This result can also be extended to the case of multiple crossings. Received: 28 August 1996 / Accepted: 22 October 1997     

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     

Preference reversals and the analysis of income distributions

It is known from the literature on uncertainty that in cases where individuals express a preference for a high win-probability bet over a bet with high winnings they nevertheless will bid more to obtain the bet with high winnings. We investigate whether a similar phenomenon applies in the parallel social-choice situation. Here decisions are to be made between a distribution with a large group of high-income people and a distribution with a small group of very high-income people. Results from a number of experimental designs are analysed. We are grateful for helpful comments from three referees of this journal, Peter Dolton, Graham Loomes, Dirk Van der gaer and seminar participants at the LSE and the University of Gent. We also thank Guillermo Cruces for computational assistance.     

Inequality and welfare evaluation of heterogeneous income distributions

This paper establishes the principles that should govern the welfare and inequality analysis of heterogeneous income distributions. Two basic criteria – the equity preference condition and the compensation principle – are shown to be fundamentally incompatible. The paper favours the latter, thereby vindicating the traditional method of dealing with heterogeneous samples. However, inequality and welfare comparisons will usually be well defined only if equivalent incomes are obtained using constant scale factors; and researchers will need to distinguish clearly between inequality of nominal incomes and inequality of living standards. Furthermore, household observations must always be weighted according to family size.     

Bivariate income distributions with lognormal conditionals

In this paper, the most general bivariate distribution with lognormal conditionals is fully characterized, using the methodology proposed by [3]. The properties of the new family are studied in detail, including marginal and conditional distributions, regression functions, dependence measures, moments and inequality measures. The new distribution is very broad, and contains as a particular case the classical bivariate lognormal distribution. Several subfamilies are studied and a generalization of the basic model is discussed. Finally, we present an empirical application. We estimate and compare the basic model proposed in the paper with a classical model, using data from the European Community Household Panel in different periods of time.     

Lorenz curves for various processes: A pluralistic approach to equity

Theoretical Lorenz curves for a bureaucracy withn seniority levels and similar curves for a simple stochastic economy with capitalistic opportunity to move up or down are derived. In both cases it is argued that equality of distribution is not an appropriate ideal. But the main thrust is that income alone is not a sufficient criterion of equity. A more general Aristotelian viewpoint is argued for. The marked increase in the equity of education between 1940 and 1984 in the United States is taken as a good example. But even within education, it is argued, detailed policies will realistically satisfy no simple set of equity principles.     

Simple adjustments of observed distributions for missing income and missing people

Correcting household survey distribution data for missing income or for undersampling may give an idea of the extent of possible biases in measuring inequality, especially when there are reasons to expect the missing income and people to belong to the top of the distribution. There are simple ways to do so when only an aggregate estimate of how much is missing is available. Atkinson had provided a formula to correct the Gini coefficient for the missing income, which was later generalized by Alvaredo (Econ. Lett. 110(3), 274–277 2011). This paper concentrates on the whole distribution and explores various alternative adjustment methods based on three key parameters: how much income, how many people are missing and on what range of income the correction should bear.     

Non-anonymous growth incidence curves, income mobility and social welfare dominance

The distributional incidence of growth is generally analyzed by comparing the quantiles of the pre- and post-growth income distribution—e.g. the so-called Growth Incidence Curves. Such an approach based on an implicit re-ranking of individual incomes ignores income mobility by assuming that only post-growth income matters in social welfare. By contrast, this paper takes the view that “status quo matters” and that social welfare should logically be defined on both inital and terminal income. This leads to consider ’non-anonymous’ Growth Incidence Curves that plot income growth rates against the various quantiles of the initial distribution. Dominance criteria that generalize those available for standard growth incidence curves are derived, which account for the inequality of individual income changes, conditional on initial income. An application to the cross-country distributional feature of global growth illustrates the analysis.     

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.     

A characterization of income distributions in terms of generalized Gini coefficients

Most commonly used parametric models for the size distribution of incomes possess only a few finite moments, and hence cannot be characterized by the sequence of their moments. However, all income distributions with a finite mean can be characterized by the sequence of first moments of the order statistics. This is an attractive feature since the generalized Gini coefficients of Kakwani (1980), Donaldson and Weymark (1980, 1983) and Yitzhaki (1983) are simple functions of expectations of sample minima. We present results which streamline these characterizations motivated by Aaberge (2000). Received: 8 March 2001/Accepted: 31 July 2001     

Social welfare rankings of income distributions A new parametric concept of intermediate inequality

This paper installs a new concept of intermediate inequality, which we refer to as η-inequality equivalence, in the notable form of equivalence of the Lorenz partial ordering and social welfare dominance. The η-inequality equivalence is a parameterized generalization of Krtscha’s (1994) non-linear compromise between the relative and absolute inequality views. For each η ∈ [0,1], we place a class of social evaluation functions satisfying the S-concavity as well as the property that an increase in incomes while leaving η-inequality intact raises welfare. We prove that one income distribution dominates another for all social evaluation functions in iff the former has a higher mean and a higher η-Lorenz curve. We prove also that the class is strictly increasing in the sense of inclusion as η decreases.I am grateful to Kiyoshi Kuga for his helpful comments and suggestions. I am also grateful to an anonymous referee and an associate editor for many valuable comments and suggestions that have much improved the paper. A previous version of this paper was presented at the annual meeting of the Japanese Economic Association, October 7, 2001, Tokyo, Japan. I wish to thank Takashi Toyoda for his helpful comments and suggestions at the meeting. This research was supported in part by the Ministry of Education, Culture, Sports, Science and Technology in Japan (Grant-in-aid for Scientific Research No.12630032).     

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     

Social capital and household income distributions in the United States: 1980, 1990

This paper asked if changes in social capital influence the level and disparity of household income in the United States. Social capital is defined in this paper as one's sympathy (antipathy) for others and one's idealized self. Changes in social capital are expected to produce the following economic consequences. First, increases in social capital are expected to alter the terms of trade and to increase the likelihood of trades between friends and family. Second, increases in social capital are expected to increase an economic agent's concerns for the external consequences of his or her choices, internalizing what otherwise would be considered externalities. Third, increases in social capital between firms are expected to increase the likelihood that they will act in their collective interest. Fourth, increases in social capital are expected to increase the opportunities for specialization and the likelihood of trade. Finally, increases in social capital are expected to raise the average level of income and reduce the disparity of income.This paper empirically tested the relationship between changes in social capital indicator variables and changes in the average and coefficient of variation (CVs) of household income. State CVs and averages of household income were calculated for all 50 states and for different races/ethnic groups using the U.S. Census data for 1980 and 1990. Social capital indicator variables selected to measure changes in social capital included measures of family integrity including the percentages of households headed by a single female with children; educational achievement variables including high school graduation rates; crime rate variables including litigation rates; and labor force participation rates. The social capital indicator variables appeared to be significantly correlated with each other. However, in 1980, the percentages of households headed by a single female with children was not significantly related to the birth rates of single teens. By 1990, however, a strong correlation was found between the percentages of households headed by a single female with children and the birth rate of single teens.Income inequality among U.S. households measured using CVs increased between 1980 and 1990 in all 50 states. The largest increase in CVs was among white households. The smallest increase in CVs was among Asian households. The states with the largest increase in the ratio of 1990 and 1980 CVs were Arizona, Wyoming, Maine, Vermont, and Texas. Half of the states reported decreases in real household income between 1980 and 1990. Those states with the largest percentage decrease in real income were Wyoming, Alaska, Montana, Louisiana, and West Virginia. The largest percentage increase in real income was reported by Connecticut, New Jersey, Rhode Island, and Massachusetts.State CVs and averages of household income were regressed on four factors or subsets of social capital indicator variables. The four factors used to predict CVs and averages of household income were generally statistically significant. The findings of this report support the conclusion that changes in social capital have a significant effect on the disparity and level of household income.     

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.     

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