Anything Lorenz Curves Can Do, Top Shares Can Do: Assessing the TopBot Family of Inequality Measures

2018 ◽  
Vol 49 (4) ◽  
pp. 947-981 ◽  
Author(s):  
Guillermina Jasso

Newly precise evidence of the trajectory of top incomes in the United States and around the world relies on shares and ratios, prompting new inquiry into their properties as inequality measures. Current evidence suggests a mathematical link between top shares and the Gini coefficient and empirical links extending as well to the Atkinson measure. The work reported in this article strengthens that evidence, making several contributions: First, it formalizes the shares and ratios, showing that as monotonic transformations of each other, they are different manifestations of a single inequality measure, here called TopBot. Second, it presents two standard forms of TopBot, which satisfy the principle of normalization. Third, it presents a new link between top shares and the Gini coefficient, showing that properties and results associated with the Lorenz curve pertain as well to top shares. Fourth, it investigates TopBot in mathematically specified probability distributions, showing that TopBot is monotonically related to classical measures such as the Gini, Atkinson, and Theil measures and the coefficient of variation. Thus, TopBot appears to be a genuine inequality measure. Moreover, TopBot is further distinguished by its ease of calculation and ease of interpretation, making it an appealing People’s measure of inequality. This work also provides new insights, for example, that, given nonlinearities in the (monotonic) relations among inequality measures, Spearman correlations are more appropriate than Pearson correlations and that weakening of correlations signals differences and shifts in distributional form, themselves signals of income dynamics.

Author(s):  
W. Henry Chiu

Abstract This paper defines and characterizes the concept of an increase in inverse downside inequality and show that, when the Lorenz curves of two income distributions intersect, how the change from one distribution to the other is judged by an inequality index exhibiting inverse downside inequality aversion often depends on the relative strengths of its aversion to inverse downside inequality and inequality aversion. For the class of linear inequality indices, of which the Gini coefficient is a member, a measure characterizing the strength of an index’s aversion to inverse downside inequality against its own inequality aversion is shown to determine the ranking by the index of two distributions whose Lorenz curves cross once. The precise condition under which the same result generalizes to the case of multiple-crossing Lorenz curves is also identified.


2021 ◽  
Author(s):  
◽  
Margaret Nell Galt

<p>This thesis examines the level and distribution of wealth and income in New Zealand between about 1870 and 1939. To do so it draws upon the available aggregate statistics on wealth and income, and it uses a sample of wealth holders especially constructed to alleviate the data deficiencies which have arisen through New Zealand not having a wealth census. The evidence available suggests that New Zealand was correctly portrayed as having a high level of wealth with an egalitarian distribution. In 1893, the first year in which average wealth could be estimated, New Zealand was definitely wealthier than Victoria. This wealth was not evenly distributed but the gini coefficient of about 0.75 suggested that New Zealand was an egalitarian economy compared to the United States, Britain, or even Australia. Over the period to 1939 the average level of wealth increased by about 100 percent. Most of this increase took place between 1900 and 1922; the late 1920's and 1930's were periods of slow growth. But this increase was not sufficient to maintain New Zealand's high position relative to Australia, and probably to other countries. The growth of real wealth was accompanied by a redistribution of wealth and by the 1930's, the gini coefficient was only about 0.73. Most of this decline was due to the declining assets held by the very rich. In 1890 to 1895 the top one percent of wealth holders owned 55 - 60 percent of all assets, but by 1935 to 1939 this had fallen to 25 - 30 percent. The very rich had, in fact, never been rich by international standards. The case studies in the thesis did not include one millionaire. As a rule they were first generation wealthy men who came from a well-to-do background, who had superior education, but who had to achieve being wealthy through their own efforts. There were few women among the top wealth holders, and those who did appear inherited their wealth from their father or or husband. The wealthy did not show signs of being a closed elite. There was a considerable amount of upward mobility in the group, and the Scots especially tended to come from poor backgrounds. The practise of equal inheritance among all the children meant that few families remained very wealthy for more than one generation. The same social and occupational mobility was clear among our sample of estate holders. Only 50 percent of sons had the same social status as their fathers. The remaining sons were fairly evenly divided between those who rose and those who fell in status. The sample, which was constructed from probate valuations and death certificate records, suggests some of the factors which assisted and hindered upward mobility. Being born female at a time when women did not pursue careers, or own family property obviously influenced the wealth holdings of a considerable proportion of the population. For men, the place of their birth proved to be significant. The Scottish showed a marked tendency to be upwardly mobile, while being Irish or New Zealand born was a definite handicap. Those who were born overseas did better if they arrived as young adults between 1860 and 1880. Assisted migrants produced proportionately less probatable estates, but those who did had about the same estates as those not assisted. Wealth was concentrated among those involved in farming, trading and the professions throughout most of our period, but over time agricultural wealth showed signs of being replaced by industrial fortunes. The professions had the advantage of a comparatively high income which enabled people to accumulate fortunes. Lifetime income undoubtedly had the major influence on wealth at death. The level of average income increased probably three-fold in the period. Again most of this rise came between 1900 and 1920. It is probable that the distribution also became more equal, through the reduced incomes to the top earners. There was a strong trend for margins for skill to decline over time, even though they were already small relative to those found in the United States. The exception to this was teachers' salaries, which showed a marked rise as the occupation became more professional. The rise of teachers' wages, shop work and clerical jobs all changed the employment structure for women, which was reflected in a changed attitude towards higher education. The 1930's saw a reduction in incomes largely through unemployment and short-time. However, the reduction was heaviest among those in the top 10 percent. The depression had mixed effects on production levels, prices and wages, but only one of our three sample industries, butter and cheese making, showed strong evidence of wage overhang. In 1939 New Zealand was still a wealthy nation, though probably she would not have ranked as highly on an international scale as in 1890. The distribution of both wealth and income had changed over our period to being substantially more egalitarian.</p>


2021 ◽  
Vol 37 (4) ◽  
pp. 1047-1058
Author(s):  
Marion van den Brakel ◽  
Reinder Lok

Abstract Indisputable figures on income and wealth inequality are indispensable for politics, society and science. Although the Gini coefficient is the most common measure of inequality, the straightforward concept of the Robin Hood index (namely, the income share that has to be transferred from the rich to the poor to make everyone equally well off) makes it a more attractive measure for the general public. In a distribution with many negative values – particularly wealth distributions – the Robin Hood index can take on values larger than 1, indicating an intuitively impossible income transfer of more than 100%. This article proposes a method to normalise the Robin Hood index. In contrast to the original index, the normalised Robin Hood index always takes on values between 0 and 1 and ends up as the original index in a distribution without negatives. As inequality measures are commonly applied to equivalised income, we also introduce a method for adequately transferring equivalised incomes from the rich to the poor within the framework of the (normalised) Robin Hood index. An empirical application shows the effect of normalisation for the Robin Hood index, and compares it to the normalisation of the Gini coefficient from previous research.


2019 ◽  
Vol 5 ◽  
pp. 237802311988128 ◽  
Author(s):  
Ernesto F. L. Amaral ◽  
Shih-Keng Yen ◽  
Sharron Xuanren Wang-Goodman

We provide an overview of associations between income inequality and intergenerational mobility in the United States, Canada, and eight European countries. We analyze whether this correlation is observed across and within countries over time. We investigate Great Gatsby curves and perform metaregression analyses based on several papers on this topic. Results suggest that countries with high levels of inequality tend to have lower levels of mobility. Intergenerational income elasticities have stronger associations with the Gini coefficient compared to associations with the top 1 percent income share. Once models are controlled for methodological variables, country indicators, and paper indicators, correlations of mobility with the Gini coefficient lose significance but not with the top 1 percent income share. This result is an indication that recent increases in inequality at the top of the distribution might be negatively affecting mobility on a greater magnitude compared to variations across the income distribution.


Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 488
Author(s):  
Hang K. Ryu ◽  
Daniel J. Slottje ◽  
Hyeok Y. Kwon

The Gini coefficient is generally used to measure and summarize inequality over the entire income distribution function (IDF). Unfortunately, it is widely held that the Gini does not detect changes in the tails of the IDF particularly well. This paper introduces a new inequality measure that summarizes inequality well over the middle of the IDF and the tails simultaneously. We adopt an unconventional approach to measure inequality, as will be explained below, that better captures the level of inequality across the entire empirical distribution function, including in the extreme values at the tails.


1984 ◽  
Vol 23 (2-3) ◽  
pp. 365-379 ◽  
Author(s):  
Zafar Mahmood

To study the consequences of an economic change on income distribution we rank distributions of income at different points in time and quantify the degree of income inequalities. Changes in income distribution can be ascertained either through drawing the Lorenz curves or through estimating different inequality indices, such as Gini Coefficient, coefficient of variation, standard deviation of logs of in• comes, Theil's Index and Atkinson's Index. Ranking the distributions of income through Lorenz curves is, of course, possible only as long as they do not intersect. Moreover, when Lorenz curves do not intersect each other, all inequality measures rank income distributions uniformly. However, if the Lorenz curves do intersect each other. different inequality measures may rank income distributions differently and thus the direction of change cannot be determined unambiguously. For this reason , the use of a single measure would be misleading. Accordingly , the use of a 'package' of inequality measures becomes essential.


2021 ◽  
Vol 114 ◽  
pp. 01019
Author(s):  
Oleg I. Pavlov ◽  
Olga Yu. Pavlova

We study how the presence of the middle class in the sense of Gevorgyan-Malykhin affects the value of income inequality measures including the Gini coefficient J and the Hoover index H. It is proved that in the presence of the middle class (1) $J \leqslant \frac{1}{2}\frac{{L'\left( 0 \right)}}{2}$ (where L is the Lorenz function), (2) $H \leqslant \frac{1}{2}$, (3) the longest vertical distance between the diagonal and the Lorenz curve (which is equal to H) is attained at ${z_0} < \frac{3}{4}$ A tight upper bound for P90/P10 ratio is found assuming L′(0)>0. Tight upper and lower bounds for the differential deviation in terms of the Gini coefficient are found as well.


2020 ◽  
Vol COVID-19 ◽  
pp. e2020157
Author(s):  
James B. Davies

The cross-country relationship of COVID-19 case and death rates with previously measured income inequality and poverty in the pandemic’s first wave is studied, controlling for other underlying factors, in a worldwide sample of countries. If the estimated associations are interpreted as causal, the Gini coefficient for income has a significant positive effect on both cases and deaths per capita in regressions using the full sample, and for cases although not for deaths when OECD and non-OECD subsamples are treated separately. The Gini coefficient for wealth has a significant positive effect on cases, but not on deaths, in both subsamples and the full sample. Poverty generally has weak positive effects in the full and non-OECD samples, but a relative poverty measure has a strong positive effect on cases in the OECD sample. Analysis of the gap between COVID-19 first-wave cases and deaths per capita in Canada and the higher rates in the United States indicates that 37% of the cases gap and 28% of the deaths gap could be attributed to the higher income Gini in the U.S. according to the full sample regressions.


2020 ◽  
Vol 8 ◽  
Author(s):  
Suchismita Banerjee ◽  
Bikas K. Chakrabarti ◽  
Manipushpak Mitra ◽  
Suresh Mutuswami

We provide a survey of the Kolkata index of social inequality, focusing in particular on income inequality. Based on the observation that inequality functions (such as the Lorenz function), giving the measures of income or wealth against that of the population, to be generally nonlinear, we show that the fixed point (like Kolkata index k) of such a nonlinear function (or related, like the complementary Lorenz function) offer better measure of inequality than the average quantities (like Gini index). Indeed the Kolkata index can be viewed as a generalized Hirsch index for a normalized inequality function and gives the fraction k of the total wealth possessed by the rich 1−k fraction of the population. We analyze the structures of the inequality indices for both continuous and discrete income distributions. We also compare the Kolkata index to some other measures like the Gini coefficient and the Pietra index. Lastly, we provide some empirical studies which illustrate the differences between the Kolkata index and the Gini coefficient.


Sign in / Sign up

Export Citation Format

Share Document