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.     

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.     

DOMINANCE TESTING OF TRANSFERS IN ROMANIA

In this paper we compare the progressivity of different government transfers made to households in Romania. We use distribution free standard errors to examine the difference between concentration curves that may be correlated, and thereafter employ statistical tests that take into account the covariance matrix for the ordinate estimates. In addition, we estimate extended Gini coefficients for the same transfers to check their consistency with the tests of inequality dominance. The results show that almost all transfer payments in Romania are progressive, and that they have an important effect on the distribution of income. Rankings among different transfer payments are, however, not robust. In particular, sensitivity analysis using different household equivalence scales indicates that many results are not consistent across scales, and that lower size elasticities contribute to changes in ranking of Ginis and loss of statistical significance in dominance tests.     

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.     

A stochastic dominance approach to the measurement of discrimination

This note suggests a bridge between stochastic dominance (Rothschild and Stiglitz, 1970 [17], 1973 [18]), inequality measurement (Atkinson, 1970 [1]) and discrimination measurement (Gastwirth, 1975 [10]). Discrimination orderings are defined and illustrated through discrimination curves, in the same spirit as stochastic dominance analysis. The main result, which links the second order discrimination curve and the Gastwirth discrimination index, also generalizes the equivalence between Generalized Lorenz dominance and second order stochastic dominance.     

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 concept of Lorenz domination

We introduce in this paper a new criterion for evaluating the inequality of incomes that is consistent and is implied by the unanimous partial order generated by the class of absolute inequality indexes introduced by Kolm (1976). The corresponding ranking is shown to be easily implementable through pairwise comparisons of absolute Lorenz curves. Absolute Lorenz curves are related to generalized Lorenz curves introduced by Shorrocks (1983) and are furthermore amenable to an intuitively appealing interpretation.     

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     

INEQUALITY OF OPPORTUNITIES VS. INEQUALITY OF OUTCOMES: ARE WESTERN SOCIETIES ALL ALIKE?

This paper analyzes the relationship between income inequality and inequality of opportunities for income acquisition in nine developed countries during the 1990s. Equality of opportunity is defined as the situation where income distributions conditional on social origin cannot be ranked according to stochastic dominance criteria. We measure social origin by parental education and occupation and use the database built by Roemer et al. (2003 ). Stochastic dominance is assessed using nonparametric statistical tests. Our results indicate strong disparities in the degree of equality of opportunity across countries and a strong correlation between inequality of outcomes and inequality of opportunity. The U.S. and Italy show up as the most unequal countries in terms of both outcome and opportunity. At the opposite extreme, income distributions conditional on social origin are almost the same in Scandinavian countries even before any redistributive policy. We complement the ordinal comparison by resorting to an original scalar “Gini” index of opportunities, which can be decomposed into a risk and a return component. In our sample, inequality of opportunity is mostly driven by differences in mean income conditional on social origin, and differences in risk compensate the return element in most countries.     

Second‐Order Discrimination and Generalized Lorenz Dominance

We propose a definition of second‐order discrimination that does not require the reference distribution to first‐order dominate the comparison one, and allows rankings of discrimination patterns when both the reference and the comparison distributions differ. It involves comparing the probabilities that randomly selected individuals in the reference and comparison distributions belong to subgroups having the same cumulative mean income, yields orderings of distributions equivalent to those from generalized Lorenz dominance, and allows orderings of discrimination patterns, partial or complete, across pairs of distributions. We compare discrimination against U.S. seniors (inter‐distributional inequality between seniors and non‐seniors) by ethnicity.     

Dominance testing for ‘pro-poor’ growth with an application to European growth

This article introduces statistical testing procedures to evaluate ??pro-poor?? growth. Our measure of ??pro-poorness?? follows Kakwani (J Quant Econ 16(1):67?C80, 2000), Kakwani and Pernia (Asian Dev Rev 18(1):1?C16, 2000), and Son (Econ Lett 82:307?C314, 2004), who decompose the generalized Lorenz ordinates into a growth effect and an inequality effect. We derive an asymptotic distribution-free covariance matrix for the decomposed generalized Lorenz curves. Using this decomposition (and our standard errors), we test for pro-poor dominance in the growth process. We illustrate our test for the pro-poor dominance by evaluating the degree of pro-poor growth in five European countries.     

Income redistribution going awry: The reversal power of the concern for relative deprivation

We demonstrate that a rank-preserving transfer from a richer individual to a poorer individual can exacerbate income inequality (when inequality is measured by the Gini coefficient). This happens when individuals’ preferences depend negatively not only on work time (effort) but also on low relative income. It is rigorously shown that the set of preference profiles that gives rise to this perverse effect of a transfer on inequality is a non-empty open subset of all preference profiles. A robust example illustrates this result.     

Decomposition of s‐concentration curves

Abstract. For any given order of inverse stochastic dominance, standard concentration curves are decomposed into three components, called contribution curves. Those components correspond to within‐group inequalities, between‐group inequalities, and transvariational inequalities. We prove, for all orders, that contribution curve dominance implies systematically welfare‐improving tax reforms and conversely. Accordingly, as welfare expansions may be costly in terms of particular inequalities, we propose targeted fiscal reforms.     

When income differences hurt or excite: The nonlinear effect of regional inequality on subjective wellbeing

Combining information from the European Union Statistics on Income and Living Conditions and the European Social Survey, we investigate the relationship between subjective well-being and income inequality using regional inequality indicators and individual data. We assume that inequality aversion and perception of social mobility affect the impact of regional inequality on subjective well-being in opposite directions. We find evidence of an inverse U-shaped effect of inequality, where inequality starts to have a positive effect on subjective well-being that becomes negative with a switch point before the average of the Gini index for the entire sample. The rationale for our nonlinear finding is that Hirschman's tunnel effect (and the positive effect of perceived social mobility) prevails for low levels of inequality, while inequality aversion and negative relative income effects are relatively stronger when inequality is higher. Robustness checks on different sample splits are consistent with the hypothesis of the two drivers.     

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.     

Equality of opportunity and luck: Definitions and testable conditions,with an application to income in France

We offer a model of equality of opportunity that encompasses different conceptions expressed in the public and philosophical debates. In addition to circumstances whose effect on outcome should be compensated and effort which represents a legitimate source of inequality, we introduce a third factor, luck, that captures the random factors whose impact on outcome should be even-handed for equality of opportunity to be satisfied. Then, we analyze how the various definitions of equality of opportunity can be empirically identified, given data limitations and provide testable conditions. Definitions and conditions resort to standard stochastic dominance tools. Lastly, we develop an empirical analysis of equality of opportunity for income acquisition in France over the period 1979–2000 which reveals that the degree of inequality of opportunity tends to decrease and that the degree of risk of income distributions, conditional on social origin, appears very similar across all groups of social origins.     

Why has Income Inequality in Germany Increased From 2002 to 2011? A Behavioral Microsimulation Decomposition

This paper proposes a method to decompose changes in income inequality into the contributions of policy changes, wage rate changes, and population changes while considering labor supply reactions. Using data from the Socio‐Economic Panel (SOEP), this method is applied to decompose the increase in income inequality in Germany from 2002 to 2011, a period that saw tax reductions and a controversial overhaul of the transfer system. The simulations show that tax and transfer reforms have had an inequality‐reducing effect as measured by the mean log deviation and the Gini coefficient. For the Gini, these effects are offset by labor supply reactions. In contrast, policy changes explain part of the increase in the ratio between the 90th and the 50th income percentiles. Changes in wage rates have led to a decrease in income inequality. Thus, the increase in inequality was due to changes in the population.     

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.     

A Gini decomposition analysis of inequality in the Czech and Slovak Republics during the transition1

Disposable income inequality, as measured by the Gini coefficient and using the Family Budget Survey data, increased very little, and by a similar amount, from 1989 to 1993 in the Czech Republic and Slovakia. This surprising result is examined with an analysis of changes in the channels of redistribution and Gini decomposition. We find that the sizeable increase in overall inequality due to changes in the wage earnings component is mitigated by changes in the tax and transfer components in both republics. As for the relative effects of government policies, changes in the transfer component contributed more than changes in the tax component to lowering the growth of inequality in the Czech Republic, while the reverse was true for Slovakia.     

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.