Characterizations of Lorenz curves and income distributions

The purpose of this paper is to propose and justify the use of a few measures of inequality for summarizing the basic information provided by the Lorenz curve. By exploiting the fact that the Lorenz curve can be considered analogous to a cumulative distribution function it is demonstrated that the moments of the Lorenz curve generate a convenient family of inequality measures, called the Lorenz family of inequality measures. In particular, the first few moments, which often capture the essential features of a distribution function, are proposed as the primary quantities for summarizing the information content of the Lorenz curve. Employed together these measures, which include the Gini coefficient, also provide essential information on the shape of the income distribution. Relying on the principle of diminishing transfers it is shown that the Lorenz measures, as opposed to the Atkinson measures, have transfer-sensitivity properties that depend on the shape of the income distribution.