Parametric Lorenz Curves and the Modality of the Income Density Function

Similar looking Lorenz curves can imply very different income density functions and potentially lead to wrong policy implications regarding inequality. This paper derives a relation between a Lorenz curve and the modality of its underlying income density: given a parametric Lorenz curve, it is the sign of its third derivative which indicates whether the density is unimodal or zeromodal (i.e., downward‐sloping). The density modality of several important Lorenz curves such as the Pareto, Weibull, Singh–Maddala parametrizations and hierarchical families of Lorenz curves are discussed. A Lorenz curve performance comparison with Monte Carlo simulations and data from the UNU–WIDER World Income Inequality Database underlines the relevance of the theoretical result: curve‐fitting based on criteria such as mean squared error or the Gini difference might lead to a Lorenz curve implying an incorrectly‐shaped density function. It is therefore important to take into account the modality when selecting a parametric Lorenz curve.     

Inequalities in Income Distribution in Slovenia

This article presents an empirical analysis of income distribution based on income tax data for Slovenia in 1991-2000. It presents evidence of rising inequalities in income distribution (gross income, gross wages and pensions). These results are supported by coefficients of variation, Gini coefficients and by the Lorenz curves. Inequalities increased rapidly in the 1991 to 1993 period. After a significant decrease in 1994 and a steady increase from 1995 to 2000, the peak value from 1993 was not surpassed. Atkinson's requirements for dominance comparisons are not violated. Inequalities are also present in the distribution of the benefits of economic growth among income groups.     

Fitting parametric lorenz curves to grouped income distributions — A critical note

Empirical Economics - The paper surveys various parametric Lorenz curves to be fitted to grouped income data in order to obtain an estimate for the Gini measure of inequality. The curves are fitted...     

A NEW FUNCTIONAL FORM FOR ESTIMATING LORENZ CURVES

There are several functional forms for estimating Lorenz curves from grouped data. Based on studies of the Spanish distribution of income, we propose a new functional form which provides very good fits. Our specification contains the Pareto Lorenz curve as a particular case, and allows one to compute easily, with the provided formulae, the Gini, Kakwani, and Chakravarty Inequality Indexes.     

现行社会保障制度对不同阶层收入影响的实证分析

文章利用洛伦兹曲线和基尼系数,首先测算了城镇不同收入阶层缴纳社会保险费后收入分布的变化及基尼系数的变化情况,在此基础上测算了不同收入阶层从政府得到不同社会保障收入后的收入分布的变化及基尼系数的变化情况,从而得出社会保障制度实施的综合效应。分析结果显示,不同类型的社会保障对不同收入阶层的影响是不同的,即使某些政策有利于基尼系数的缩小,但不利于收入最低的阶层。研究认为,有针对性地调整部分社会保障政策,对调整收入分配差距的目标有帮助。     

Approximating unanimity orderings: An application to Lorenz dominance

Stochastic dominance and Lorenz dominance are examples of orderings which require unanimous agreement among an infinite set of indices. This paper considers various subsets of inequality measures that respect Lorenz dominance, and assesses the extent to which a small number of indices can reproduce the Lorenz ordering. Using income data for 80 countries, our results suggest that Lorenz dominance can be predicted with 99% accuracy using just 3 or 4 inequality measures, as long as two of them focus on the extreme upper and lower tails of the distribution. In contrast, confining attention to the index families and parameter ranges normally considered may fail to detect the majority of occasions when Lorenz curves intersect. These results lead us to question the faith placed in procedures based on a finite set of inequality indices, and to suggest that similar lessons will apply to other types of unanimity orderings.     

Inverse stochastic dominance,inequality measurement and Gini indices

We investigate the relationship between the third degree inverse stochastic dominance criterion introduced in Muliere and Scarsini (1989) and inequality dominance when Lorenz curves intersect. We propose a new definition of transfer sensitivity aimed at strengthening the Pigou-Dalton Principle of Transfers. Our definition is dual to that suggested by Shorrocks and Foster (1987). It involves a regressive transfer and a progressive transfer both from the same donor, leaving the Gini index unchanged. We prove that finite sequences of these transfers and/or progressive transfers characterize the third degree inverse stochastic dominance criterion. This criterion allows us to make unanimous inequality judgements even when Lorenz curves intersect. The Gini coefficient becomes relevant in these cases in order to conclusively rank the distributions.     

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     

经济增长、收入分配对农村贫困变动的影响

文章利用了基于Lorenz曲线进行贫困测度及其分解的方法,实证研究了1985～2003年中国农村的贫困变动,以及经济增长和收入分配对贫困变动的影响.结果表明,经济增长使农村居民收入增加,大幅度减少了贫困,但农村居民的收入差距逐渐拉大,收入不平等加剧部分地抵消了经济增长的减贫成效.     

AN ESTIMATION OF U.S. FAMILY WEALTH AND ITS DISTRIBUTION FROM MICRODATA, 1973

This paper presents a method of estimating U.S. family net wealth across the entire population, utilizing capitalization of several income items available from income tax microdata. Other forms of wealth, and debt, are indirectly estimated using relationships gleaned from estate tax data. Concentration in the distribution of wealth, and assets such as corporate stock, are measured with Gini coefficients and Lorenz curve analysis and compared to similar estimates of concentration in the distribution of income. Comparisons of the results with previous estimates for the United States are made in the latter section of the paper.     

Unobserved components in an error-correction model of consumption for Southern European countries

In this paper we show how the potential misspecification of the consumption function can be ameliorated by approximating any unmodelled long run variation with an unobserved component in the form of a time-varying trend. This methodology is applied to Greek, Portuguese and Spanish consumption functions during the post-second World war period. The empirical evidence suggests that there are many determinants of long-run consumption in these countries, in addition to income and inflation, and these unobserved long-run effects are captured by a nonstationary stochastic component. The long-run elasticity of consumption with regards to the unobserved component is greater than unity in all countries. First version received: January 1999/Final version received: June 2000     

Increasing returns to savings and wealth inequality

In this paper I present an explanation to the fact that in the data wealth is substantially more concentrated than income. Starting from the observation that the composition of households' portfolios changes towards a larger share of high-yield assets as the level of net worth increases, I first use data on historical asset returns and portfolio composition by wealth level to construct an empirical return function. I then augment an Overlapping Generation version of the standard neoclassical growth model with idiosyncratic labor income risk and missing insurance markets to allow for returns on savings to be increasing in the level of accumulated assets. The quantitative properties of the model are examined and show that an empirically plausible difference between the return faced by poor and wealthy agents is able to generate a substantial increase in wealth inequality compared to the basic model, enough to match the Gini index and all but the top 1 percentile of the US distribution of wealth.     

A tale of two Ginis in the US, 1921–2012

Following a methodology by Jantzen and Volpert (2012), we use IRS Adjusted Gross Income data for the US (1921–2012) to estimate two Gini indices representing inequality at the bottom and the top of the income distribution, and to calculate the overall Gini as a function of the parameters underlying the two indices. A steady increase in the overall Gini since the Second World War actually hides two different periods of distributional changes. First, the increase in inequality from the mid 1940s to the late 1970s is driven by rising inequality at the bottom of the income distribution that more than offsets a decrease in inequality at the top. The implication is that middle-income earners gained relative to high-incomes, and especially relative to low-income earners. Second, the rise in the Gini after 1981 is driven by rising inequality at the top. Third, top-driven inequality follows a U-shaped trajectory consistent with Piketty and Saez (2003, 2006). Fourth, the welfare effects of the different distributional changes behind increasing inequality can be evaluated in light of the Lorenz-dominance criterion by Atkinson (1970): we argue that the rise in inequality since 1981 is much more likely to be associated with a social welfare loss net of compensating growth.     

Robust estimation of models of Engel curves

Both standard and robust methods are used here to estimate models of Engel curves for three household commodities, namely, food, transport, and tobacco and alcohol in Canada. The income elasticities of demand computed from the various methods differ significantly for the transport and tobacco-alcohol consumption where there are obvious outliers and zero expenditures problem. Robust estimators point to lower income elasticities and have better performance than the standard LS and Tobit estimator. These results are analyzed in the light of the information on finite-sample performance obtained in a previous Monte Carlo study. First version received: July 2000/Final version received: July 2001 RID="*" ID="*"  I wish to thank Victoria Zinde-Walsh, John Galbraith, Clint Coakley, two anonymous referees and an associate editor for helpful comments. I would also like to thank Anastassia Khouri for kindly providing the 1992 Family Expenditure Survey of Canada data.     

Empirical likelihood confidence intervals for the Gini measure of income inequality

Gini coefficient is among the most popular and widely used measures of income inequality in economic studies, with various extensions and applications in finance and other related areas. This paper studies confidence intervals on the Gini coefficient for simple random samples, using normal approximation, bootstrap percentile, bootstrap-t and the empirical likelihood method. Through both theory and simulation studies it is shown that the intervals based on normal or bootstrap approximation are less satisfactory for samples of small or moderate size than the bootstrap-calibrated empirical likelihood ratio confidence intervals which perform well for all sample sizes. Results for stratified random sampling are also presented.     

DUAL‐INDEX MEASUREMENT OF INCOME INEQUALITY

This study introduces the concept of unequally distributed income and proposes a dual‐index measurement of income inequality that evaluates the magnitude and dispersion of unequally distributed income. We use the Rawlsian index as a magnitude index and employ current income inequality indices as a dispersion index. We describe the properties of the Rawlsian and dispersion indices, and apply these indices to real income data. Further, we show that the Gini dispersion index is a weighted average of Rawlsian indices for the sub‐distributions of the unequally distributed income distribution.     

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.     

PROGRESSIVITY AND REDISTRIBUTION IN NON‐REVENUE NEUTRAL TAX REFORMS: THE LEVEL AND DISTANCE EFFECTS

Kakwani and Reynolds–Smolensky indices are used in the literature to measure the progressivity and redistributive capacity of taxes. These indices may, however, show some limits when used to make normative assessments about non‐revenue neutral tax reforms. Two approaches have traditionally been taken to overcome this problem. The first of these consists of comparing after‐tax income distributions through generalized Lorenz (concentration) curves. The second approach is based on the decomposition of changes in the Reynolds–Smolensky index into changes in the average tax rate and variations in progressivity. Nonetheless, this decomposition between the average tax rate and progressivity may be further exploited to obtain some information that can be relevant to assess tax reforms. The main aim of this study is to draw up some indicators that can be useful to quantify the effects of non‐revenue neutral tax reforms. These indicators are used to investigate the last personal income tax reforms that have taken place in Spain.     

基尼系数理论最佳值及其简易计算公式研究

本文针对《中国统计年鉴》有关城镇和农村居民收入数据单列而难以直接测算城乡合一基尼系数的情况 ,证明了 :在收入五分法下基尼系数的计算方法 ,可以五分法中最高收入组与最低收入组各自所占的收入比重之差来简易计算。文章以严格的数学推导证明出基尼系数的理论最佳值为三分之一 ,并用推导出的简易公式对世界银行利用收入五分法所公布的 3 2个国家的统计数据进行了重新计算 ,结果发现 ,以笔者的简易公式所计算出的基尼系数与世界银行公布的基尼系数极其接近。本文关于基尼系数的简易计算公式及其理论最佳值的分析 ,为中国城乡合一的基尼系数的测算及分析提供了一种新的思路。