Heavy tails and upper-tail inequality: The case of Russia

Motivated, in part, by the recent surge of interest in robust inequality measurement, cross-country inequality comparisons, applications of heavy-tailed distributions and the study of global and upper-tail inequality, this paper focuses on robust analysis of heavy-tailedness properties and inequality in the upper tails of income distribution in Russia, as measured, mainly, by its tail indices. The study is based on recently developed approaches to robust inference on the degree of heavy-tailedness and their implications for the analysis of upper-tail inequality discussed in the paper. Among other results, the paper provides robust estimates of heavy-tailedness parameters and tail indices for Russian income distribution and their comparisons with the benchmark values in developed economies reported in the previous literature. The estimates point out to important similarity between heavy-tailedness properties of income distribution and their implications for the analysis of upper-tail income inequality in Russia and those in developed markets.     

Income distribution and redistribution through taxation: An international comparison

Income tax progressivity is studied with Generalized Entropy measures of inequality. Luxembourg Income Study data sets for nine countries are used for international comparison and analysis. progressivity indices are generated by using the Generalized Entropy family as well as Atkinson measures. We further our understanding by examining pre-tax and post-tax measures of inequality based respectively on gross and disposable household incomes. The decomposition property is shown to be desirable for enhancing our knowledge of income inequality and the redistributive effect of income taxes. Thus decomposition based on family size and number of earners is conducted. We learn that countries vary in their emphasis regarding redistributive effects of income taxes.     

Is the Pareto–Lévy law a good representation of income distributions?

Mandelbrot (Int Econ Rev 1:79–106, 1960) proposed using the so-called Pareto–Lévy class of distributions as a framework for representing income distributions. We argue in this article that the Pareto–Lévy distribution is an interesting candidate for representing income distributions because its parameters are easy to interpret and it satisfies a specific invariance-under-aggregation property. We also demonstrate that the Gini coefficient can be expressed as a simple formula of the parameters of the Pareto–Lévy distribution. We subsequently use income data for Norway and seven other OECD countries to fit the Pareto–Lévy distribution as well as the Generalized Beta type II (GB2) distribution. The results show that the Pareto–Lévy distribution fits the data better than the GB2 distribution for most countries, despite the fact that GB2 distribution has four parameters whereas the Pareto–Lévy distribution has only three.     

The Intensity and Shape of Inequality: The ABG Method of Distributional Analysis

Inequality is anisotropic: its intensity varies by income level. We here develop a new tool, the isograph, to focus on local inequality and illustrate these variations. This method yields three coefficients which summarize the shape of inequality: a main coefficient, α, which measures inequality at the median; and two correction coefficients, β and γ, which pick up any differential curvature at the top and bottom of the distribution. The analysis of a set of 232 microdata samples from 41 different countries in the LIS datacenter archive allows us to provide a systematic overview of the properties of the ABG (α β γ) coefficients, which are compared to a set of standard indices including Atkinson indices, generalized entropy, Wolfson polarization, and the GB2 distribution. This method also provides a smoothing tool that reveals the differences in the shape of distributions (the strobiloid) and how these have changed over time.     

A TREATMENT OF ABSOLUTE INDICES OF POLARIZATION*

Absolute polarization indices remain unchanged under equal absolute augmentation in all incomes. This paper identifies the class of absolute polarization indices whose orderings of alternative income distributions agree with the rankings generated by nonintersecting absolute polarization curves. We explore the possibility of using the Kolm (1976) – Blackorby‐Donaldson (1980) ethical absolute inequality index in polarization measurement. We establish that although inequality and polarization are dissimilar concepts, different absolute inequality indices can be employed to design alternative absolute polarization indices. A numerical illustration is provided using Indian data and it is shown that inequality and polarization are different issues in income distribution analysis.     

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.     

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.     

HOW DO EUROPEANS EVALUATE INCOME DISTRIBUTIONS? AN ASSESSMENT BASED ON HAPPINESS SURVEYS

The purpose of this paper is to measure the evaluation of income inequality by European citizens. Starting from the concept of a social welfare function defined on income distributions the paper estimates the degree and nature of inequality aversion of Europeans. It uses subjective well-being (SWB) as an empirical measure of welfare and estimates how SWB is related to average income and measures of income inequality (from an appropriate class). The estimated relationship is used to determine those inequality measures which qualify as proper representations of people's inequality aversion.     

Income tax incidence in an ageing population: An examination of the measurement of income redistribution

The distribution of income in a population consists of the different distributions in the various age groups weighted by the age structure. In using inequality statistics to measure the redistributive effect of taxation, it is desirable to remove the arbitrary effect of the age-weighting. Formulae for ‘conventional’ and for ‘correct’ indices of incidence are given, and are evaluated in a stochastic-process model that captures many features of actual income distributions. For the lognormal distribution, the direction of bias of conventional incidence indices is determined simply by writing them as functions of the lognormal dispersion parameter.     

Estimating inequality in the distribution of welfare using demand models

In this paper two demand models with general household equivalence scales (GES) are estimated. These GES are identifiable, since they have the independence of base utility (IB) or equivalence scales exactness (ESE) property. Estimates of household characteristics adjusted income can then be calculated relative to a specific household type. This “individual equivalent income (IEI) is then used to calculate measures of inequality in the distribution of welfare. As more than one model is estimated, the sensitivity of these estimates to model specification changes can be considered. Comparisons are also made to estimates of inequality based on household income. It is found that absolute inequality is sensitive to model specification, but relative inequality is not. This is true using indices of inequality of the Atkinson (1970), Kolm (1976a,b) and Sen (1973) type, or more general measures of inequality based on estimates of Lorenz curve decile ordinates, constructed using methods introduced by Beach and Davidson (1983). Thus, if absolute measures of inequality are required, these results suggest some further research is required to determine a class of models which is less sensitive to model specification error. However, if only relative measures of inequality are needed, then estimates of inequality appear to be very robust to the choice of model specification.     

Economic policy and income distribution in China

"Using the Kakwani interpolation method to reconstruct rural and urban size distributions, the authors present Lorenz curve estimates for Chinese income distribution by period, 1952-1986. The level of inequality is one of the lowest in the world. The change in inequality over time conforms to the experience of other less-developed countries. Although rural inequality increased after 1978, national inequality fell, due to a reduction in the rural-urban income gap."     

A Different Perspective on the Evolution of UK Income Inequality

This paper scrutinizes the conventional wisdom about trends in UK income inequality and also places contemporary inequality in a much longer historical perspective. We combine household survey and income tax data to provide better coverage of all income ranges from the bottom to the very top (and make our estimates available to other researchers). We make a case for studying distributions of income between tax units (i.e. not assuming the full income sharing that goes with the use of the household as the unit of analysis) for reasons of principle as well as data harmonization. We present evidence that income inequality in the UK is as least as high today as it was just before the start of World War 2.     

Estimation of Concentration Measures and Their Standard Errors for Income Distributions in Poland

Measures of concentration (inequality) are often used in the analysis of income and wage size distributions. Among, them the Gini and Zenga coefficients are of greatest importance. It is well known that income inequality in Poland increased significantly in the period of transformation from a centrally planned economy to a market economy. High income inequality can be a source of serious problems, such as increasing poverty, social stratification, and polarization. Therefore, it seems especially important to present reliable estimates of income inequality measures for a population of households in Poland in different divisions. In this paper, some estimation methods for Gini and Zenga concentration measures are presented together with their application to the analysis of income distributions in Poland by socio-economic groups. The basis for the calculations was individual data coming from the Polish Household Budget Survey conducted by the Central Statistical Office. The standard errors of Gini and Zenga coefficients were estimated by means of the bootstrap and the parametric approach based on the Dagum model.     

The Pro‐Poorness,Growth and Inequality Nexus: Some Findings From a Simulation Study

An income growth pattern is pro‐poor if it reduces a (chosen) measure of poverty by more than if all incomes were growing equiproportionately. Inequality reduction is not sufficient for pro‐poorness. In this paper, we explore the nexus between pro‐poorness, growth, and inequality in some detail using simulations involving the displaced lognormal, Singh–Maddala, and Dagum distributions. For empirically relevant parameter estimates, distributional change preserving the functional form of each of these three‐parameter distributions is often either pro‐poor and inequality reducing, or pro‐rich and inequality exacerbating, but it is also possible for pro‐rich growth to be inequality reducing. There is some capacity for each of these distributions to show trickle effects (weak pro‐richness) along with inequality‐reducing growth, but virtually no possibility of pro‐poorness for growth which increases overall inequality. Implications are considered.     

DEPRIVATION,INEQUALITY AND WELFARE*

Welfare ranking of income distributions involves a trade‐off between equity and efficiency. A person's feeling of deprivation about higher incomes may be of a relative or absolute type. We consider an intermediate notion of deprivation, a convex mix of relative and absolute deprivations. We then look at the problem of welfare ranking of income distributions when welfare increases under a globally equitable redistribution and under an income increase that keeps intermediate deprivation fixed. All deprivation indices can be regarded as inequality indices but the converse is not true. We also provide a numerical illustration of our results.     

Multidimensional Inequality Across Three Developed Countries

This paper produces comparable estimates of multidimensional inequality for the U.S., Germany, and Australia. Two alternative approaches with differing interpretations are employed. The first method projects all facets of welfare onto a single variable which is then analyzed using standard univariate techniques. The second approach establishes equivalent‐income distributions that would lead to an equalization of welfare, such that the difference between this counterfactual and the true income distribution can be measured. This difference is then interpreted as the degree of income redistribution required to offset welfare inequality. Using data on permanent incomes, health scores, years of education, and leisure times, we observe much higher levels of inequality in the U.S. than in Germany or Australia. Our results are highly statistically significant and hold over a large variety of weighting specifications.     

Effects of Changes in Family Composition and Employment Patterns on the Distribution of Income in Australia: 1981–1982 to 1997–1998

We examine income distributions over the last two decades, presenting both non-parametric kernel density estimates and summary measures. Standard errors of summary measures are also reported to facilitate statistical inference. We find a significant increase in private income inequality, but only a modest increase in disposable income inequality, implying an increase in the inequality-reducing effects of income taxes and transfers. Using a semi-parametric procedure developed by DiNardo et al. (1996), we then examine the effects of changes in family characteristics on the distribution of private income, finding that half the increase in inequality is explained by changes in the distribution of employment.     

西部大开发有助于缩小西部地区的收入不平等吗——基于双倍差分法的经验研究

本文采用双倍差分法考察了西部大开发对中国西部地区收入不平等的影响效应及其作用机制。结果表明:西部大开发显著地缩小了中国西部地区的收入不平等,在选用衡量收入不平等的不同指标和控制其它影响因素后,这一结论依然稳健;通过进一步研究发现,西部大开发主要通过吸收外资、提高城市化水平和财政支农支出规模的途径来缩小西部地区的收入不平等。     

Measuring inequality change in an economy with income growth

This paper analyzes how to measure changes in inequality in an economy with income growth. The discussion distinguishes three stylized kinds of economic growth:

• 1.(1) high income sector enrichment,
• 2.(2) low income sector enrichment,
• 3.(3) high income sector enlargement, in which the high income sector expands and absorbs persons from the low income sector.

Th e two enrichment types pose no problem for assessing inequality change in the course of economic growth: for high income sector enrichment growth, inequality might reasonably be said to increase, whereas for low income sector enrichment, inequality might be said to decrease. These adjustments are non-controversial and non-problematical. Where problems arise is in the case of high income sector enlargement growth. In that case, the two alternative approaches have been shown in this paper to yield markedly results:

• 1.(1) The traditional inequality indices generate an inverted-U pattern of inequality. That is, inequality rises in the early stages of high income sector enlargement growth and falls thereafter.
• 2.(2) The new approach suggested here, based on axioms of gap inequality and numerical inequality, generates a U pattern of inequality. That is, inequality falls in the early stages of high income sector enlargement growth and rises thereafter.

The discrepancy between the familiar indices and the alternative approach based on axioms of gap inequality and numerical inequality bears further scrutiny. Two courses of action are possible. One might try to axiomatize inequality in ways that generate an inverted-U pattern in high income sector enlargement growth, thereby rationalizing the continued use of the usual inequality indices with the inverted-U property. Alternatively, one might retain the axioms proposed here, embed them into a more formal structure, and construct a family of inequality indices consistent with them. Others might wish to pursue the first course; I am at work on the second.     

The Population Sub-Group Income Effects on Inequality: Analytical Framework and an Empirical Illustration

A method is developed to examine the population sub-group income effects on inequality using the generalized Theil indices. The proposed method is illustrated by computing the effects of marginal changes in the occupation-specific incomes on per capita income inequality in Australia based on data for 7197 sample households relating to the 1988-89 Household Expenditure Survey conducted by the Australian Bureau of Statistics (ABS). The study shows that the growth of (distribution neutral) per capita income in four occupations, namely, unemployed and pensioners, trade-persons, machine operators and laborers, is inequality reducing whereas the growth of per capita income in all other occupations is inequality augmenting. The occupation-specific income effects on between-group inequality are, however, stronger than those on within-group inequality. These findings are invariant to the choice of alternative distributional weights used in the generalized Theil indices.