Chapter 1 introduces in statistics the notion of the barycenter of the distribution of a non-negative random variable Y with a positive finite mean µY and the quantile function Q(x). The barycenter is denoted by µX and defined as the expected value of the random variable X having the probability density function fX(x) = Q(x)/µY. For continuous populations, the Gini index is 2µX - 1, i.e., the normalization of the barycenter, which is in the range [0, 1/2], the concentration area is µX - 1/2, and the Gini’s mean difference is 4µY (µX - 1/2). The same barycenter-based formulae hold for discrete populations. The introduction of the barycenter allows for new economic, geometrical, physical, and statistical interpretations of these measures. For income distributions, the barycenter represents the percent of the expected recipient of one unit of income, as if the stochastic process that leads to the distribution of the total income among the population was observable as it unfolds. The barycenter splits the population into two groups, which can be considered as “the winners” and “the losers” in the income distribution, or “the rich” and “the poor”. We provide examples of application to thirty theoretical distributions and an empirical application with the estimation of personal income inequality in Luxembourg Income Study Database’s countries. We conclude that the barycenter is a new measure of the location or central tendency of distributions, which may have wide applications in both economics and statistics and provides a new statistical interpretation of the Gini index. Chapter 2 contributes to the literature on the decomposition of inequality indices into contributions of population subgroups in five ways. First, we propose a new axiomatization of the decomposition by subgroups of the population inequality that is the first in the literature and is applicable to any inequality index. Second, we propose a new two-term decomposition by subgroups of the Gini, Bonferroni, and De Vergottini indices that is exact, i.e., the sum of the within and between components is equal to the overall inequality, and independent of the order in which the subgroups are sorted, i.e., the decomposition gives the same results whatever the ordering of the subgroups. Third, using the elements of the proposed decomposition method, we provide a new graphical representation of population and subgroups income distributions. Fourth, through a comparison with the most important among the other methods proposed in the literature, we show that the proposed decomposition procedure is the only one satisfying all the proposed axioms. We provide an empirical application of the proposed methodology by studying income inequality in the Euro area between 2007 and 2019, i.e., between the financial crisis and the epidemic crisis. Chapter 3 contributes to the literature on the Kakwani concentration index in three ways. First, we show that the Kakwani concentration index belongs to the family of the center of mass-based concentration index that include the Gini, Bonferroni, and De Vergottini inequality, and we extend the concentration index proposed by Kakwani, based on the Gini index, to the Bonferroni and De Vergottini inequality indices. Second, we extend to the Kakwani index the Balance of Inequality graphical representation of center of mass-based inequality indices, which allows us to overcome the limitations of the Lorenz curve. Third, we extend to the Kakwani index the exact within-between decomposition by population subgroup proposed with reference to the Gini, Bonferroni, and De Vergottini inequality indices. We provide an empirical application of the proposed methodology by studying the global Kakwani indices of across countries distribution of life expectancy and carbon dioxide emissions with respect to energy consumption in 2019 and their decomposition by continents.

Chapter 1 introduces in statistics the notion of the barycenter of the distribution of a non-negative random variable Y with a positive finite mean µY and the quantile function Q(x). The barycenter is denoted by µX and defined as the expected value of the random variable X having the probability density function fX(x) = Q(x)/µY. For continuous populations, the Gini index is 2µX - 1, i.e., the normalization of the barycenter, which is in the range [0, 1/2], the concentration area is µX - 1/2, and the Gini’s mean difference is 4µY (µX - 1/2). The same barycenter-based formulae hold for discrete populations. The introduction of the barycenter allows for new economic, geometrical, physical, and statistical interpretations of these measures. For income distributions, the barycenter represents the percent of the expected recipient of one unit of income, as if the stochastic process that leads to the distribution of the total income among the population was observable as it unfolds. The barycenter splits the population into two groups, which can be considered as “the winners” and “the losers” in the income distribution, or “the rich” and “the poor”. We provide examples of application to thirty theoretical distributions and an empirical application with the estimation of personal income inequality in Luxembourg Income Study Database’s countries. We conclude that the barycenter is a new measure of the location or central tendency of distributions, which may have wide applications in both economics and statistics and provides a new statistical interpretation of the Gini index. Chapter 2 contributes to the literature on the decomposition of inequality indices into contributions of population subgroups in five ways. First, we propose a new axiomatization of the decomposition by subgroups of the population inequality that is the first in the literature and is applicable to any inequality index. Second, we propose a new two-term decomposition by subgroups of the Gini, Bonferroni, and De Vergottini indices that is exact, i.e., the sum of the within and between components is equal to the overall inequality, and independent of the order in which the subgroups are sorted, i.e., the decomposition gives the same results whatever the ordering of the subgroups. Third, using the elements of the proposed decomposition method, we provide a new graphical representation of population and subgroups income distributions. Fourth, through a comparison with the most important among the other methods proposed in the literature, we show that the proposed decomposition procedure is the only one satisfying all the proposed axioms. We provide an empirical application of the proposed methodology by studying income inequality in the Euro area between 2007 and 2019, i.e., between the financial crisis and the epidemic crisis. Chapter 3 contributes to the literature on the Kakwani concentration index in three ways. First, we show that the Kakwani concentration index belongs to the family of the center of mass-based concentration index that include the Gini, Bonferroni, and De Vergottini inequality, and we extend the concentration index proposed by Kakwani, based on the Gini index, to the Bonferroni and De Vergottini inequality indices. Second, we extend to the Kakwani index the Balance of Inequality graphical representation of center of mass-based inequality indices, which allows us to overcome the limitations of the Lorenz curve. Third, we extend to the Kakwani index the exact within-between decomposition by population subgroup proposed with reference to the Gini, Bonferroni, and De Vergottini inequality indices. We provide an empirical application of the proposed methodology by studying the global Kakwani indices of across countries distribution of life expectancy and carbon dioxide emissions with respect to energy consumption in 2019 and their decomposition by continents.

Three Essays on The Measurement of Economic Inequality / Giorgio Di Maio , 2024 May 13. 36. ciclo, Anno Accademico 2021/2022.

### Three Essays on The Measurement of Economic Inequality

#### Abstract

Chapter 1 introduces in statistics the notion of the barycenter of the distribution of a non-negative random variable Y with a positive finite mean µY and the quantile function Q(x). The barycenter is denoted by µX and defined as the expected value of the random variable X having the probability density function fX(x) = Q(x)/µY. For continuous populations, the Gini index is 2µX - 1, i.e., the normalization of the barycenter, which is in the range [0, 1/2], the concentration area is µX - 1/2, and the Gini’s mean difference is 4µY (µX - 1/2). The same barycenter-based formulae hold for discrete populations. The introduction of the barycenter allows for new economic, geometrical, physical, and statistical interpretations of these measures. For income distributions, the barycenter represents the percent of the expected recipient of one unit of income, as if the stochastic process that leads to the distribution of the total income among the population was observable as it unfolds. The barycenter splits the population into two groups, which can be considered as “the winners” and “the losers” in the income distribution, or “the rich” and “the poor”. We provide examples of application to thirty theoretical distributions and an empirical application with the estimation of personal income inequality in Luxembourg Income Study Database’s countries. We conclude that the barycenter is a new measure of the location or central tendency of distributions, which may have wide applications in both economics and statistics and provides a new statistical interpretation of the Gini index. Chapter 2 contributes to the literature on the decomposition of inequality indices into contributions of population subgroups in five ways. First, we propose a new axiomatization of the decomposition by subgroups of the population inequality that is the first in the literature and is applicable to any inequality index. Second, we propose a new two-term decomposition by subgroups of the Gini, Bonferroni, and De Vergottini indices that is exact, i.e., the sum of the within and between components is equal to the overall inequality, and independent of the order in which the subgroups are sorted, i.e., the decomposition gives the same results whatever the ordering of the subgroups. Third, using the elements of the proposed decomposition method, we provide a new graphical representation of population and subgroups income distributions. Fourth, through a comparison with the most important among the other methods proposed in the literature, we show that the proposed decomposition procedure is the only one satisfying all the proposed axioms. We provide an empirical application of the proposed methodology by studying income inequality in the Euro area between 2007 and 2019, i.e., between the financial crisis and the epidemic crisis. Chapter 3 contributes to the literature on the Kakwani concentration index in three ways. First, we show that the Kakwani concentration index belongs to the family of the center of mass-based concentration index that include the Gini, Bonferroni, and De Vergottini inequality, and we extend the concentration index proposed by Kakwani, based on the Gini index, to the Bonferroni and De Vergottini inequality indices. Second, we extend to the Kakwani index the Balance of Inequality graphical representation of center of mass-based inequality indices, which allows us to overcome the limitations of the Lorenz curve. Third, we extend to the Kakwani index the exact within-between decomposition by population subgroup proposed with reference to the Gini, Bonferroni, and De Vergottini inequality indices. We provide an empirical application of the proposed methodology by studying the global Kakwani indices of across countries distribution of life expectancy and carbon dioxide emissions with respect to energy consumption in 2019 and their decomposition by continents.
##### Scheda breve Scheda completa Scheda completa (DC)
13-mag-2024
Chapter 1 introduces in statistics the notion of the barycenter of the distribution of a non-negative random variable Y with a positive finite mean µY and the quantile function Q(x). The barycenter is denoted by µX and defined as the expected value of the random variable X having the probability density function fX(x) = Q(x)/µY. For continuous populations, the Gini index is 2µX - 1, i.e., the normalization of the barycenter, which is in the range [0, 1/2], the concentration area is µX - 1/2, and the Gini’s mean difference is 4µY (µX - 1/2). The same barycenter-based formulae hold for discrete populations. The introduction of the barycenter allows for new economic, geometrical, physical, and statistical interpretations of these measures. For income distributions, the barycenter represents the percent of the expected recipient of one unit of income, as if the stochastic process that leads to the distribution of the total income among the population was observable as it unfolds. The barycenter splits the population into two groups, which can be considered as “the winners” and “the losers” in the income distribution, or “the rich” and “the poor”. We provide examples of application to thirty theoretical distributions and an empirical application with the estimation of personal income inequality in Luxembourg Income Study Database’s countries. We conclude that the barycenter is a new measure of the location or central tendency of distributions, which may have wide applications in both economics and statistics and provides a new statistical interpretation of the Gini index. Chapter 2 contributes to the literature on the decomposition of inequality indices into contributions of population subgroups in five ways. First, we propose a new axiomatization of the decomposition by subgroups of the population inequality that is the first in the literature and is applicable to any inequality index. Second, we propose a new two-term decomposition by subgroups of the Gini, Bonferroni, and De Vergottini indices that is exact, i.e., the sum of the within and between components is equal to the overall inequality, and independent of the order in which the subgroups are sorted, i.e., the decomposition gives the same results whatever the ordering of the subgroups. Third, using the elements of the proposed decomposition method, we provide a new graphical representation of population and subgroups income distributions. Fourth, through a comparison with the most important among the other methods proposed in the literature, we show that the proposed decomposition procedure is the only one satisfying all the proposed axioms. We provide an empirical application of the proposed methodology by studying income inequality in the Euro area between 2007 and 2019, i.e., between the financial crisis and the epidemic crisis. Chapter 3 contributes to the literature on the Kakwani concentration index in three ways. First, we show that the Kakwani concentration index belongs to the family of the center of mass-based concentration index that include the Gini, Bonferroni, and De Vergottini inequality, and we extend the concentration index proposed by Kakwani, based on the Gini index, to the Bonferroni and De Vergottini inequality indices. Second, we extend to the Kakwani index the Balance of Inequality graphical representation of center of mass-based inequality indices, which allows us to overcome the limitations of the Lorenz curve. Third, we extend to the Kakwani index the exact within-between decomposition by population subgroup proposed with reference to the Gini, Bonferroni, and De Vergottini inequality indices. We provide an empirical application of the proposed methodology by studying the global Kakwani indices of across countries distribution of life expectancy and carbon dioxide emissions with respect to energy consumption in 2019 and their decomposition by continents.
Inequality; Measurement
Inequality; Measurement
Three Essays on The Measurement of Economic Inequality / Giorgio Di Maio , 2024 May 13. 36. ciclo, Anno Accademico 2021/2022.
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Descrizione: Three Essays on The Measurement of Economic Inequality
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11383/2172771`