Let X be a continuous or discrete random variable with values in [0,M] and consider all functions (here called transformations) \(q:[0,M]\to [0,\infty )\) that are increasing and have given bounded rates \(B \le \frac {q(v)-q(u)}{v-u} \le A\) for u < v. We prove that among such transformations, there is a transformation q that minimizes the Gini index of q(X), and such a q can be chosen as piecewise linear with only two rates, namely A and B. In the motivation for the study, X represents the incomes of a population. Our results imply that among all such tax policies with fixed allowable minimum and maximum tax rates, there is a tax policy that minimizes the Gini index of the disposable incomes of the population and such a tax policy has only two brackets with the given minimum and maximum rates.

设X是一个连续或离散的随机变量，其值在 [0, M ] 并考虑所有函数（这里称为转换）\(q:[0,M]\to [0,\infty )\)增加并具有给定有界率\(B \le \frac {q(v)-q(u)}{vu} \le A\)对于u < v。我们证明在这样的变换中，有一个变换q使q ( X )的基尼指数最小化，并且这样的q可以选择为只有两个比率的分段线性，即A和B。在研究动机中，X代表人口的收入。我们的结果表明，在所有这些具有固定可允许的最低和最高税率的税收政策中，有一项税收政策使人口可支配收入的基尼指数最小化，而这种税收政策只有两个等级，给定的最低和最高税率率。