Lorenz Dominance Research Articles

Inequality in green space distribution and its association with preventable deaths across urban neighbourhoods in the UK, stratified by Index of Multiple Deprivation

BackgroundThis study investigated inequalities in the distribution of green space (GS) and the association between inequalities in amounts of GS and preventable deaths across urban neighbourhoods with different Index of...

Redistribution with needs

AbstractWe take an axiomatic approach to study redistribution problems when agents report income and needs. We formalize axioms reflecting ethical and operational principles such as additivity, impartiality and individual rationality. Different combinations of those axioms characterize three focal rules (laissez faire, full redistribution, and need‐adjusted full redistribution) as well as compromises among them. We also uncover the structure of those compromises exploring the Lorenz dominance criterion as well as majority voting. Our analysis provides an axiomatic justification for a linear income tax system. We conclude our analysis resorting to Eurostat's Household Budget Survey from where we illustrate the different redistribution patterns accounting for needs across European countries.

Test of dominance relations based on kernel smoothing method

New test statistics for weakly Lorenz dominance and weakly generalised Lorenz dominance are proposed, and some asymptotic properties of test statistics are obtained. The simulation results show that our test statistics can improve test power in comparison with the non-smoothed empirical methods. Finally, we apply our inference framework to an actual example.

Likelihood Ratio Tests for Lorenz Dominance

In testing hypotheses pertaining to Lorenz dominance (LD), researchers have examined second- and third-order stochastic dominance using empirical Lorenz processes and integrated stochastic processes with the aid of bootstrap analysis. Among these topics, analysis of third-order stochastic dominance (TSD) based on the notion of risk aversion has been examined using crossing (generalized) Lorenz curves. These facts motivated the present study to characterize distribution pairs displaying the TSD without second-order (generalized Lorenz) dominance. It further motivated the development of likelihood ratio (LR) goodness-of-fit tests for examining the respective hypotheses of the LD, crossing (generalized) Lorenz curves, and TSD through approximate Chi-squared distributions. The proposed LR tests were assessed using simulated distributions, and applied to examine the COVID-19 regional death counts of bivariate samples collected by the World Health Organization between March 2020 and February 2021.

The probability of multidimensional poverty: A new approach and an empirical application toEU‐SILCdata

AbstractThis paper proposes a novel method to analyze multidimensional poverty by using a large set of feasible weights to summarize the information about the poor, which enables remaining agnostic about the relative importance given to different poverty dimensions. This method allows for the calculation of the individual probability of being poor in a multidimensional perspective. The distribution of individual probabilities can then be combined with Generalized Lorenz dominance techniques to derive unanimous consent for a wide class of social welfare functions with a minimum load of value judgments. The innovations proposed here allow to move from a dual definition of poverty, where poor and non‐poor individuals are classified in a mutually exclusive context, to a continuous measure of deprivation capturing both the extensive and intensive margin of multidimensional poverty. The empirical application of the method consists of measuring multidimensional poverty in ten selected countries using four waves of EU‐SILC data (2008–2014).

Inequality in Education: A Comparison of Australian Indigenous and Nonindigenous Populations

Abstract Data from the Household Income and Labour Dynamics in Australia Survey is used to estimate distributions for the level of educational attainment for Australian indigenous and nonindigenous populations for the years 2001, 2006, 2014 and 2017. Bayesian inference is used to analyse how these ordinal categorical distributions have changed over time and to compare indigenous and nonindigenous distributions. Both the level of educational attainment and inequality in educational attainment are considered. To compare changes in levels over time, as well as inequality between the two populations, first order stochastic dominance and an index of educational poverty are used. To examine changes in inequality over time, two inequality indices and generalised Lorenz dominance are considered. Results are presented in terms of posterior densities for the indices and posterior probabilities for dominance for the dominance comparisons. We find some evidence of improvement over time, especially in the lower parts of the indigenous distribution and that inequality has significantly increased from 2001 to 2017.

Posterior Probabilities for Lorenz and Stochastic Dominance of Australian Income Distributions*

Using Household, Income and Labour Dynamics in Australia (HILDA) data for 2001, 2006, 2010, 2014 and 2017, we compute posterior probabilities for dominance for all pairwise comparisons of income distributions in these years. The dominance criteria considered are Lorenz dominance and first‐ and second‐order stochastic dominance. The income distributions are estimated using an infinite mixture of gamma density functions, with posterior probabilities computed as the proportion of Markov chain Monte Carlo draws that satisfy the inequalities that define the dominance criteria. We find welfare improvements from 2001 to 2006 and qualified improvements from 2006 to the later three years. Evidence of an ordering between 2010, 2014 and 2017 cannot be established.

Testing for multisample generalized Lorenz curves dominance

The generalized Lorenz curve is simply the ordinary Lorenz curve scaled up by the mean and its scaling reveals a dominance relationship that is not apparent from an examination of means and Lorenz curve on their own. Second order stochastic dominance is equivalent to generalized Lorenz dominance as well and discussed in past. [1] proposed test for generalized Lorenz dominance over a specified interval of interest. However, no test has been discussed for testing multisample generalized Lorenz dominance. We propose a test for testing multisample generalized Lorenz dominance using two different approaches viz. specified interval approach and confidence interval approach. The simulation study is also carried out to find the power of the proposed test.

Measuring Multidimensional Inequality of Opportunity

This paper develops a normative approach to the measurement of ex-ante inequality of opportunity in a multidimensional setting, i.e., when the individual outcome is represented by a multidimensional variable. We characterize three classes of social welfare functions, all endorsing ex ante compensation but each of them reflecting a specific reward principle: (1) utilitarian, (2) agnostic and (3) averse. The first class is implemented via generalized Lorenz Dominance applied to each attribute separately. The agnostic and inequality averse classes are implemented by a welfarist Lorenz ordering, namely, of type-aggregate utilities. In the case of inequality-averse class, utility functions are submodular, hence capturing the dependence between attributes. We also develop normative inequality indices (Atkinson, 1970; Kolm 1969; Sen, 1973) for the classes of welfare functions and study their properties. Finally, we propose an empirical applications of the methods developed in the paper: by using the National Longitudinal Study of Adolescent to Adult Health (Add Health) we evaluate inequality of opportunity in U.S. for the case of three dimensions of individual outcomes, namely, education, health and income.

Improved Nonparametric Bootstrap Tests of Lorenz Dominance

One income or wealth distribution is said to Lorenz dominate another when the Lorenz curve for the former is nowhere below that of the latter, indicating a (weakly) more equitable allocation of resources. Existing tests of the null of Lorenz dominance based on pairs of samples of income or wealth achieve the nominal rejection rate asymptotically when the two Lorenz curves are equal, but are conservative at other null configurations. We propose new nonparametric bootstrap tests of Lorenz dominance based on preliminary estimation of a contact set. Our tests achieve the nominal rejection rate asymptotically on the boundary of the null; that is, when Lorenz dominance is satisfied, and the Lorenz curves coincide on some interval. Numerical simulations indicate that our tests enjoy substantially improved power compared to existing procedures at relevant sample sizes. Supplementary materials for this article are available online.

The measurement of welfare change

We propose a class of measures of welfare change that are based on the generalized Gini social welfare functions. We analyze these measures in the context of a second-order dominance property that is akin to generalized Lorenz dominance as introduced by Shorrocks (Economica 50:3–17, 1983) and Kakwani (Advances in econometrics, vol 3. JAI Press, Greenwich, pp 191–213, 1984). Because we consider welfare differences rather than welfare levels, the requisite equivalence result involves affine welfare functions only, as opposed to the entire class of strictly increasing and strictly S-concave welfare indicators. Thus, our measures are associated with those members of the generalized-Gini class that are strictly increasing and strictly S-concave. Moving from second-order dominance to first-order dominance does not change this result significantly: for most intents and purposes, the generalized Ginis remain the only strictly increasing and strictly S-concave measures that are equivalent to this first-order dominance condition phrased in terms of welfare change. Our final result provides a characterization of our measures of welfare change in the spirit of Weymark’s (Math Soc Sci 1:409–430, 1981) original axiomatization of the generalized Gini welfare functions. Journal of Economic Literature Classification No.: D31.

On the estimation of the Lorenz curve under complex sampling designs

This paper focuses on the estimation of the concentration curve of a finite population, when data are collected according to a complex sampling design with different inclusion probabilities. A (design-based) Hajek type estimator for the Lorenz curve is proposed, and its asymptotic properties are studied. Then, a resampling scheme able to approximate the asymptotic law of the Lorenz curve estimator is constructed. Applications are given to the construction of (i) a confidence band for the Lorenz curve, (ii) confidence intervals for the Gini concentration ratio, and (iii) a test for Lorenz dominance. The merits of the proposed resampling procedure are evaluated through a simulation study.

Lorenz dominance based algorithms to solve a practical multiobjective problem

The set of Pareto nondominated solutions obtained in some practical cases of multiobjective optimization problems can be huge, rendering decision making difficult. Applying Lorenz dominance instead of Pareto dominance during the optimization process can help to alleviate this difficulty. Lorenz dominance is a refinement of Pareto dominance that integrates fairness in multiobjective optimization when objectives are considered equal and can help select only the well located solutions. By introducing a partial order among a set of Pareto-nondominated solutions, Lorenz dominance reduces the size of the nondominated front by keeping only fair solutions. In this work, we investigate the use of the infinite order Lorenz dominance within three new methods to solve a practical case of the multiobjective knapsack problem, which involves elaborating efficient action plans in social and medico-social structures. We assess the proposed methods on large problem instances with up to 8 objectives and 500 candidate actions and show their effectiveness in comparison with four leading reference algorithms.

Statistical inference for the difference of two Lorenz curves

ABSTRACTTesting the Lorenz dominance is of importance in economic and social sciences. In this article, we propose new tools to do inferences for the difference of two Lorenz curves. The asymptotic normality of the proposed smoothed nonparametric estimator is proved. We also propose a smoothed jackknife empirical likelihood (JEL) method which avoids to estimate the complicate asymptotic variance. It is proved that the proposed JEL ratio statistics converge to the standard chi-square distribution. Simulation studies and real data analysis are also conducted, and show encouraging finite-sample performance.

A reply to Ponte et al (2016) Supply chain collaboration: Some comments on the nucleolus of the beer game

Purpose: The aim of the paper is to pick up the result of a previously published paper in order to deepen the discussion. We analyze the solution against the background of some well-known concepts and we introduce a newer one. In doing so we would like to inspire the further discussion of supply chain collaborationDesign/methodology/approach: Based on game theoretical knowledge we present and compare seven properties of fair profit sharing.Findings: We show that the nucleolus is a core-solution, which does not fulfil aggregate monotonicity. In contrast the Shapley value is an aggregate monotonic solution but does not belong to the core of every cooperative game. Moreover, we present the Lorenz dominance as an additional fairness criteria.Originality/value: We discuss the very involved procedure of establishing lexicographic orders of excess vectors for games with many players.

Almost Lorenz dominance

This paper extends Leshno and Levy’s (Manag Sci 48:1074–1085, 2002) approach of “almost stochastic dominance” to inequality orderings. We define and characterize the notion of “almost Lorenz dominance” (ALD) and illustrate it with the US income data. An income distribution is said to “almost” Lorenz dominate another distribution when its Lorenz curve lies almost everywhere but not entirely above the other Lorenz curve. We show that this condition is equivalent to requiring that “almost” all Gini-type inequality measures rank the former distribution to have less inequality than the latter distribution; the condition on the Gini-type inequality measures has a clear interpretation and is easy to apply. We further define an almost composite transfer (ACT) and show that ALD amounts to a sequential application of such transfers. The empirical illustration with the US income data (1967–1986) demonstrates the utility of this generalized notion of inequality ordering.

Non-dominated Sorting Genetic Algorithms for a Multi-objective Resource Constraint Project Scheduling Problem

Abstract The resource constraint project scheduling problem (RCPSP) has attracted growing attention since the last decades. Precedence constraints are considered as well as resources with limited capacities. During the project, the same resource can be required by several in-process jobs and it is compulsory to ensure that the consumptions do not exceed the limited capacities. In this paper, several criteria are involved, namely makespan, total job tardiness, and workload balancing level. Our problem is firstly solved by the non-dominated sorting genetic algorithm-II (NSGAII) as well as the recently proposed NSGAIII. Giving emphasis to the selection procedure, we apply both the traditional Pareto dominance and the less documented Lorenz dominance into the niching mechanism of NSGAIII. Hence, we adopt and modify L-NSGAII to our problem and propose L-NSGAIII by integrating the notion of Lorenz dominance. Our methods are tested by 1350 randomly generated instances, considering problems with 30–150 jobs and different configurations of resources and due dates. Hypervolume and C-metric are considered to evaluate the results. The Lorenz dominance leads the population more toward the ideal point. As experiments show, it allows improving the original NSGA approach.

A data science based standardized Gini index as a Lorenz dominance preserving measure of the inequality of distributions.

The Gini index is a measure of the inequality of a distribution that can be derived from Lorenz curves. While commonly used in, e.g., economic research, it suffers from ambiguity via lack of Lorenz dominance preservation. Here, investigation of large sets of empirical distributions of incomes of the World’s countries over several years indicated firstly, that the Gini indices are centered on a value of 33.33% corresponding to the Gini index of the uniform distribution and secondly, that the Lorenz curves of these distributions are consistent with Lorenz curves of log-normal distributions. This can be employed to provide a Lorenz dominance preserving equivalent of the Gini index. Therefore, a modified measure based on log-normal approximation and standardization of Lorenz curves is proposed. The so-called UGini index provides a meaningful and intuitive standardization on the uniform distribution as this characterizes societies that provide equal chances. The novel UGini index preserves Lorenz dominance. Analysis of the probability density distributions of the UGini index of the World’s counties income data indicated multimodality in two independent data sets. Applying Bayesian statistics provided a data-based classification of the World’s countries’ income distributions. The UGini index can be re-transferred into the classical index to preserve comparability with previous research.

Capturing preferences for inequality aversion in decision support

We investigate the situation where there is interest in ranking distributions (of income, of wealth, of health, of service levels) across a population, in which individuals are considered preferentially indistinguishable and where there is some limited information about social preferences. We use a natural dominance relation, generalised Lorenz dominance, used in welfare comparisons in economic theory. In some settings there may be additional information about preferences (for example, if there is policy statement that one distribution is preferred to another) and any dominance relation should respect such preferences. However, characterising this sort of conditional dominance relation (specifically, dominance with respect to the set of all symmetric increasing quasiconcave functions in line with given preference information) turns out to be computationally challenging. This challenge comes about because, through the assumption of symmetry, any one preference statement (“I prefer giving $100 to Jane and $110 to John over giving $150 to Jane and $90 to John”) implies a large number of other preference statements (“I prefer giving $110 to Jane and $100 to John over giving $150 to Jane and $90 to John”; “I prefer giving $100 to Jane and $110 to John over giving $90 to Jane and $150 to John”). We present theoretical results that help deal with these challenges and present tractable linear programming formulations for testing whether dominance holds between any given pair of distributions. We also propose an interactive decision support procedure for ranking a given set of distributions and demonstrate its performance through computational testing.

Remittances, Lorenz Dominance in the Distribution of Income and Redistribution

We analyze how changes in the distribution of income, characterized by the comparison of Lorenz curves, affect public redistribution for an economy with international interfamily transfers (remittances). Our analysis suggests that a fall in the inequality of income might increase or reduce the government’s ability to collect tax revenue and its electoral costs from inefficient taxation which in turn affect public redistribution. The main contribution of this paper is to characterize conditions in which a shift towards a dominant Lorenz curve can lead to an increase or fall in public redistribution. We also find that the composition of a change in the distribution of income, promoted by a change in the distribution of labor income or remittances, leads to different effects on the size of public redistribution.