Modelling income distributions using Tsallis statistics

The World Bank provides data sets for income distributions of over 140 countries. We demonstrate that the large majority of these can be described by a distribution derived from Tsallis statistics, which is a generalisation of Boltzmann statistics, applicable to non-equilibrium systems. (For nine countries the log-normal distribution is statistically preferred.) The result of our least square fits of the income distributions suggests a roughly linear variation of the two Tsallis fit parameters, $\lambda$ (an inverse temperature), and the index of non-extensivity, $q$, for $q\lesssim 1.5$ (with $q=1$ corresponding to Boltzmann statistics). Values of the Gini index (a measure of inequality) for the different countries, obtained from our least square fits, are in good agreement with published World Bank data. Finally, we present an expression for the cumulative distribution for income data which is normalised with respect to the average income, to allow for an estimation of the power law exponent describing its tail.