Explicit expressions of the Pietra index for the generalized function for the size distribution of income

The importance of the Pietra index in socioeconomic systems and econophysics has been highlighted by Eliazar and Sokolov (2010). In this paper, we obtain closed expressions for the Pietra index for the generalized function for the size of income proposed by McDonald (1984). This family is composed of three classes of distributions: the generalized gamma distribution (GG), the generalized beta of the first kind (GB1) and the generalized beta of the second kind (GB2). For the different distributions, we obtain closed and simple expressions of the Pietra index, which can be easily computed. We also obtain the Pietra index for other relevant income models including finite mixtures of distributions and the κ-generalized distribution (Clementi et al., 2008). Finally, two empirical applications with real income data are given.