Bivariate income distributions for assessing inequality and poverty under dependent samples

As indicators of social welfare, the incidence of inequality and poverty is of ongoing concern to policy makers and researchers alike. Of particular interest are the changes in inequality and poverty over time, which are typically assessed through the estimation of income distributions. From this, income inequality and poverty measures, along with their differences and standard errors, can be derived and compared. With panel data becoming more frequently used to make such comparisons, traditional methods which treat income distributions from different years independently and estimate them on a univariate basis, fail to capture the dependence inherent in a sample taken from a panel study. Consequently, parameter estimates are likely to be less efficient, and the standard errors for between-year differences in various inequality and poverty measures will be incorrect. This paper addresses the issue of sample dependence by suggesting a number of bivariate distributions, with Singh–Maddala or Dagum marginals, for a partially dependent sample of household income for two years. Specifically, the distributions considered are the bivariate Singh–Maddala distribution, proposed by Takahasi (1965), and bivariate distributions belonging to the copula class of multivariate distributions, which are an increasingly popular approach to modelling joint distributions. Each bivariate income distribution is estimated via full information maximum likelihood using data from the Household, Income and Labour Dynamics in Australia (HILDA) Survey for 2001 and 2005. Parameter estimates for each bivariate income distribution are used to obtain values for mean income and modal income, the Gini inequality coefficient and the headcount ratio poverty measure, along with their differences, enabling the assessment of changes in such measures over time. In addition, the standard errors of each summary measure and their differences, which are of particular interest in this analysis, are calculated using the delta method.     

Note

Functional classes of Lorenz curves are derived from a generalization of a relative poverty notion. All these Lorenz curves compare individual income to the average of all larger or all smaller incomes. The parameters of the Lorenz curves are effectively computed from empirical income data by least square regressions. Best fits are analyzed and resulting functional Gini indices are compared to empirical Gini indices.First version received: September 2002/Final version received: April 2003     

我国农村低收入群体贫困变动及其影响因素——基于福建省农村居民收入分组数据的分析

旨在考察一个较长的时间序列内农村低收入人口贫困状况的动态变化,探明经济增长和收入分配等宏观层面影响因素与贫困指数之间的定量关系与内在联系。基于GQ模型和（模型,选取具备可分解性的FGT指数,采用1990—2006年福建省农村居民收入分组面板数据,分别运用4条低收入标准线对低收入群体的H指数、PG指数、SPG指数和贫困弹性进行模拟。实证结果表明,福建省的经济增长有效减少了绝对低收入人口数量,而收入分配差距的扩大却降低了低收入群体的潜在福利,从而影响农村相对贫困的减少效果。     

RELATIVE OR ABSOLUTE POVERTY LINES: A NEW APPROACH

When measuring poverty over time analysts must choose the value of the income elasticity of the poverty line, which essentially determines whether an absolute or relative poverty line is being used. The choice of this parameter is ultimately a value judgement, but this paper suggests an approach which has some empirical basis. Borrowing from the life-style and deprivation approach to poverty, various dimensions of poverty and deprivation are identified and the income elasticity of these items is used as the income elasticity of the poverty line. Data from the 1987 and 1994 Irish Household Budget Surveys suggest an upper bound of 0.7 for this parameter. Poverty measures using a number of values of the income elasticity of the poverty line are presented and test statistics are presented to determine whether observed differences in poverty measures are statistically significant.     

经济增长、收入分配与贫困——对新疆农村居民贫困变动的实证分析

通过运用FGT贫困指数和在Lorenz曲线基础上的贫困指示增长曲线对新疆农村最近几年的贫困变动进行研究。研究结果表明,从2003年到2007年,经济增长减缓了新疆农村的贫困状况,但收入分配状况的恶化在一定程度上抵消了经济增长减缓贫困的积极作用。除2003～2004年以外,其余时期的经济增长都不是亲贫式的增长,新疆农村的贫困人口在经济增长中获得的利益少于非贫困人口。因此,为了有效地减少贫困,新疆不仅要注重通过经济增长减少贫困规模,同时也要重视缩小收入分配差距,并且把减少收入分配不均等的重点要放在提高最贫困人口的收入水平上。     

A GENERAL METHOD FOR CREATING LORENZ CURVES

A general method to construct parametric Lorenz models of the weighted‐product form is offered in this paper. Initially, a general result to describe the conditions for the weighted‐product model to be a Lorenz curve, created by using several component parametric Lorenz models, is given. We show that the key property for an ideal component model is that the ratio between its second derivative and its first derivative is increasing. Then, a set of Lorenz models, consisting of a basic group of models, along with their convex combinations, is proposed, and it is shown that any model in the set possesses this key property. We introduce the concept of balanced fit, which provides a means of assigning weights, according to the preferences of the practitioner, to two alternative objectives for developing Lorenz curves in practice. These objectives are generating an acceptable Lorenz curve and improving the accuracy of the density estimation. We apply the balanced fit approach to income survey data from China to illustrate the performance of our models. We first show that our models outperform other popular traditional Lorenz models in the literature. Second, we compare the results generated by the balanced fit approach applied to one of the Lorenz models that we develop with those generated by the kernel method to show that the approach proposed in the paper generates plausible density estimates.     

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.     

Stochastic dominance and decomposable measures of inequality and poverty

In this paper, we characterize some new links between stochastic dominance and the measurement of inequality and poverty. We show that: for second-degree normalized stochastic dominance (NSD), the weighted area between the NSD curve of a distribution and that of the equalized distribution is a decomposable inequality measure; for first-degree and second-degree censored stochastic dominance (CSD), the weighted area between the CSD curve of a distribution and that of the zero-poverty distribution is a decomposable poverty measure. These characterizations provide graphical representations for decomposable inequality and poverty measures in the same manner as Lorenz curve does for the Gini index. The extensions of the links to higher degrees of stochastic dominance are also investigated.     

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.     

Pareto–lognormal distributions: Inequality,poverty, and estimation from grouped income data

The (double) Pareto–lognormal is an emerging parametric distribution for income that has a sound underlying generating process, good theoretical properties, and some limited favorable evidence of its fit to data. We extend existing results for this distribution in 3 directions. (1) We derive closed form formula for its moment distribution functions, and for various inequality and poverty measures. (2) We show how it can be estimated from grouped data using GMM. (3) Using grouped data from ten countries, we compare its performance with that of another leading 4-parameter income distribution, the generalized beta-2 distribution. The results confirm that, when using grouped data, both distributions provide a good fit, with the double Pareto–lognormal distribution outperforming the beta distribution in 4 out of 10 cases.     

经济增长、收入分配的减贫效应探析——以新疆为例

运用FGT贫困指数,以新疆为例,基于Lorenz曲线对农村贫困变动进行测算。结果表明,经济增长具有明显的减贫作用,而收入分配不平等加剧则部分地抵消了经济增长的减贫作用。     

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.     

Welfare Measurement and Measurement Error

The approximate effects of measurement error on a variety of measures of inequality and poverty are derived. They are shown to depend on the measurement error variance and functionals of the error–contaminated income distribution, but not on the form of the measurement error distribution, and to be accurate within a rich class of error–free income distributions and measurement error distributions. The functionals of the error–contaminated income distribution that approximate the measurement error induced distortions can be estimated. So it is possible to investigate the sensitivity of welfare measures to alternative amounts of measurement error and, when an estimate of the measurement error variance is available, to calculate corrected welfare measures. The methods are illustrated in an application using Indonesian household expenditure data.     

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.     

经济增长、收入分配与贫困规模变动相关性研究--来自北京市城市居民的经验数据

文章详细讨论了经济增长、不平等与贫困规模变动的相互关系,在确立三者关系基本框架的基础上,构造了应用Lorenz曲线及其相应弹性分析三者关系的理论模型,并提出了亲贫困增长指数的概念。在对北京城市居民的经验数据进行分析的基础之上得到的结论是:政府制定扶贫政策的选择应该使收入增长效应与收入不平等效应之和最大化,这为制定扶贫政策提供了决策依据。     

Carbon Lorenz curves

The purpose of this paper is twofold. First, it exhibits that standard tools in the measurement of income inequality, such as the Lorenz curve and the Gini-index, can successfully be applied to the issues of inequality measurement of carbon emissions and the equity of abatement policies across countries. These tools allow policy-makers and the general public to grasp at a single glance the impact of conventional distribution rules such as equal caps or grandfathering, or more sophisticated ones, on the distribution of greenhouse gas emissions. Second, using the Samuelson rule for the optimal provision of a public good, the Pareto-optimal distribution of carbon emissions is compared with the distribution that follows if countries follow Nash–Cournot abatement strategies. It is shown that the Pareto-optimal distribution under the Samuelson rule can be approximated by the equal cap division, represented by the diagonal in the Lorenz curve diagram.     

The Lorenz curve as an archetype: A historico-epistemological study

In 1905, Max O. Lorenz suggested a simple method of measuring the concentration of wealth, based on the visual representation of income distribution. The Lorenz curve is now very popular and can be considered as canonical. However, the path leading from Lorenz's original work to contemporary interpretations of his graph has been anything but simple. We thus propose to trace the origin, the evolution and the various subsequent interpretations of the Lorenz graph. We argue that the original Lorenz curve has been shifted in epistemological status as well as inverted in graphic appearance.     

Pro‐Poorness Orderings

An indicator of pro‐poorness of a growth profile associated with a distribution of income is a measure of the extent to which growth is biased towards the poor. This paper proposes a general approach to pro‐poorness, called the progressive sequential averaging principle (PSA), relaxing the requirement of rank preservation due to growth. An endogenous benchmark for evaluating the growth of poor comes out naturally from this principle. A dominance relation on the basis of the above approach for a class of growth profiles is introduced through a simple device, called the PSA curve and its properties are examined in relation to the standard dominances in terms of the generalized Lorenz curve and the inverse generalized Lorenz curve. The paper concludes with an application to evaluate growth profiles experienced by the United States between 2001–07 and 2007–13.     

“When Life Gives You Lemons”: Using Cross-Sectional Surveys to Identify Chronic Poverty in the Absence of Panel Data

At the core of poverty eradication is the need to eliminate that poverty that is persistent over time (chronic poverty). Unfortunately, traditional approaches to identifying chronic poverty require longitudinal data that is rarely available. In its absence, this paper proposes an alternative approach that only requires 1 year of cross-sectional data on monetary and non-monetary poverty. It puts forth two conjectures and contends that the combined profile of a household as both income poor and multidimensionally poor can be used as a proxy of that household being chronically income poor. To explore the viability of this approach, we use a probit model and longitudinal data for three Latin American countries to estimate households’ probabilities of remaining in income poverty based on their past income and multidimensional poverty statuses. We find empirical support for the approach that is significant, consistent across countries, and robust to various controls and periods of analysis.     

Living Decently

One of the reasons why poverty lines became popular at the turn of the century was their promise of a scientific technique that would dispense with moralising about poverty. We argue that a price paid in this quest has been an impoverishment of the richness of the notion of 'a decent life', the moral concept underlying poverty. In addition, poverty lines have in practice been more to do with inequality at the bottom end of the income distribution than with poverty. The purpose of this article is to rehabilitate the measurement of poverty, and to make it credible. We set out our preferred method of poverty measurement, and illustrate it using data from the Australian Standard of Living Study. A feature of our approach is to distinguish clearly between issues of inequality and issues of poverty. Questions such as who is on the bottom of the income distribution, whether this has changed over time, and how income levels of the worst off compare with the mean, are questions of inequality. As such, the answers tell us nothing at all about how the worst off are actually living. To answer that question, we require direct measures of consumption and of social participation. These measures are not as simple, but they provide us with knowledge about poverty that poverty lines have promised, but have not delivered in a credible fashion.