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.

中文翻译：

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洛伦兹曲线可以做的任何事情，顶级股票都可以做

美国和世界各地最高收入轨迹的最新精确证据依赖于份额和比率，促使对其作为不平等衡量指标的财产进行新的调查。目前的证据表明，最高份额与基尼系数之间存在数学联系，经验联系也延伸到阿特金森度量。本文报告的工作加强了这一证据，并做出了以下贡献：首先，它将份额和比率形式化，表明作为彼此的单调变换，它们是单一不平等测度（此处称为 TopBot）的不同表现形式。其次，提出了满足归一化原则的两种标准形式的TopBot。第三，它提出了最高份额和基尼系数之间的新联系，显示与洛伦兹曲线相关的属性和结果也适用于顶级股票。第四，它在数学上指定的概率分布中研究了 TopBot，表明 TopBot 与经典度量（例如 Gini、Atkinson 和 Theil 度量）以及变异系数单调相关。因此，TopBot 似乎是一个真正的不平等措施。此外，TopBot 的另一个特点是易于计算和易于解释，使其成为衡量不平等的一种有吸引力的人民衡量标准。这项工作还提供了新的见解，例如，鉴于不平等措施之间（单调）关系的非线性，Spearman 相关性比 Pearson 相关性更合适，并且相关性减弱表示分布形式的差异和变化，