A new inequality measure that is sensitive to extreme values and asymmetries

There is a vast literature on the selection of an appropriate index of income inequality and on what desirable properties such a measure (or index) should contain. The Gini index is the most popular. There is a concurrent literature on the use of hypothetical statistical distributions to approximate and describe an observed distribution of incomes. Pareto and others observed early on that incomes tend to be heavily right-tailed in their distribution. These asymmetries led to approximating the observed income distributions with extreme value hypothetical statistical distributions. But these income distribution functions (IDFs) continue to be described with a single index (such as the Gini) that poorly detect the extreme values present. This paper introduces a new inequality measure to supplement the Gini (not to replace it) that better measures the inherent asymmetries and extreme values that are present in observed income distributions. The new measure is based on a third order term of a Legendre polynomial from the logarithm of a share function (or Lorenz curve). We advocate using the two measures together to provide a better description of inequality inherent in empirical income distributions with extreme values. Using Current Population Survey data, we show we can better describe the overall IDF and better detect changes in the tails of the empirical IDF using the two measures concomitantly.