Abstract

A parametric family of inequality measures yielding Theil’s entropy index and variance of the logarithms as special cases is derived. The problem of selecting one particular measure within this family is solved as an estimation problem using the Maximum Likelihood Method of Box and Cox for choosing a power transformation.
One important feature of this approach is that the inequality measure so derived becomes intrinsically related to a symmetric and unimodal (approximately Normal) distribution, which becomes the standard of comparison, rather than the usual egalitarian distribution. Besides, essential descriptive measures of location and dispersion of a data distribution arise naturally as a result of applying the proposed methodology. An interesting fact is that the family of inequalities here derived combines dispersion with asymmetry of the data.
Some principles satisfied by the parametric family are the following: (1) Pigou-Dalton´s Transfer Principle (inequality increases or at least does not decrease with an income transfer from a poor to a rich individual, while it decreases when a rich individual makes a transfer to a poor one); (2) Population Principle (relativity to the size of the population); (3) Relative Income Principle (invariance to proportional changes of income of the whole population), so that it can be considered a member of the class of relative indices of inequality. Furthermore, the family of inequalities also satisfies the (4) Anonymity Principle (so that income is the only characteristic of the individuals under study to be considered), and finally it has the property of (5) Decomposition of Total inequality of several groups into Between group inequality and Within group inequality.
An empirical application of the methodology associated to the new family of measures of inequality is carried out with Mexican data of the income distribution for year 2018. There, it is shown some closeness between the Gini coefficient and the measure here derived, except when decomposing total inequality into between groups and within groups, where the result provided by the new family is clearer than Gini´s.