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ABSTRACTTo any finite population of n individuals with associated incomes x1…, xn we can associate a Lorenz curve. By associating this population with a random variable X representing the income of a randomly chosen individual of the population, the concept of a Lorenz curve and the associated partial order (the Lorenz Order) based on nested Lorenz curves is readily extended to be defined in the class of all non‐negative integrable random variables. In this context well known results on inequality attenuating and inequality rank preserving taxation policies are found to admit simple more general proofs. Some results on the effects of random taxation are also reviewed. The effects of applying different taxation policies within subpopulations lead one to consider questions regarding inequality attenuation results in mixture settings. It is observed that, more generally, inequality comparisons can be unambiguously made between any non‐negative variables even if measured in dissimilar units.

Abstract
This work introduces two new curves that are multivariate generalizations of the "classical" Lorenz curve. All data of d-variate distributions can be visualized by drawing these curves in the plane, whereas Koshevoy's and Mosler's generalization by a lift zonoid in ℝd+1 can only be drawn for d = 2. The generalizations of the Lorenz curve induce partial orderings of d-variate distributions. Furthermore, two inequality or heterogeneity measures that are consistent with the induced rankings are proposed. They can be considered as new generalizations of the univariate Gini coefficient. For deciding which of the two measures is more appropriate for measuring a sort of convergence concerning different countries of an union or of regions of a country, we establish systems of axioms. Although these systems are reflecting natural properties, several of the axioms are new. Moreover, by means of these axioms well- known inequality measures are tested, too.

In 2000, the United Nations adopted the Millennium Development Goals (MDGs), a set of eight global development goals to be achieved between 2000 and 2015. We estimated the Lorenz Curve and Gini Index for determining any changes in inequality at the global level with countries as a unit of analysis for eight development indicators (proportion of population undernourished, school enrollment rates, the percentage of women in parliament, infant mortality rates, maternal mortality rates, HIV (Human Immunodeficiency Virus) rates, access to improved water sources, and access to a cellular device), representing one MDG each. All of the selected indicators improved on average between 2000 and 2015. An average improvement in an indicator does not necessarily imply a decrease in inequality. For instance, the average infant mortality rate decreased from 39.17 deaths per 1000 births in 2000 to 23.40 in 2015, but the Gini Index remained almost stable over the same period, suggesting no reduction in inequality among countries. For other indicators, inequality among countries decreased at varying rates. A significant data gap existed across countries. For example, only 91 countries had data on primary school enrollment rates in 2000 and 2015. We emphasize developing a global data collection and analysis protocol for measuring the impacts of global development programs, especially in reducing inequality across social, economic, and environmental indicators. This study will feed into currently enacted Sustainable Development Goals (SDGs) for ensuring more inclusive and equitable growth worldwide.

ABSTRACT: This paper analyses the gross inland energy consumption (EC) in the European Union countries (EU-15) taking in account the period 2005-2014. The standard tools in the measurement of income inequality such as Lorenz curves, Gini index, Generalized Entropy indices and Atkinson ones are applied. The empirical results, obtained through the decomposition of the generalized entropy indices, confirm that there are a small inward shift in the corresponding Lorenz curves, that the inequality distribution of EC across the EU-15 countries has decreased (the Gini coefficient falls from 44,27% in 2005 to 42,16% in 2014) and there are differences among the countries' clusters: Mediterranean, Continental, Nordic and Anglo-Saxon. This paper makes a good contribution to knowledge: firstly, it is innovative since it puts together energy Consumption and inequality among the EU-15 countries, secondly, it uses a very up-to-dated database (Eurostat), and thirdly, it fills a gap in the literature.

This chapter is focused on the problem of income inequalities in contemporary China which is one of the biggest developmental challenges for this country. First this part of the analysis includes general overview on studied problem putting emphasis on instrument for measuring income inequalities, and general drivers of this phenomenon. Second part is concentered on the nature of this problem in China since the late 1970s, when country has faced a period of rapid economic development. Chapter also points out attempts taken by the Chinese government to reduce income inequalities.