Growing inequality in systems showing Zipf's law

A central problem in economics and statistics is the assessment of income or wealth inequality starting from empirical data. Here we focus on the behavior of Gini index, one of the most used inequality measures, in presence of Zipf's law, a situation which occurs in many complex financial and economical systems. First, we show that the application of asymptotic formulas to finite size systems always leads to an overestimation of inequality. We thus compute finite size corrections and we show that depending on Zipf's exponent two distinct regimes can be observed: low inequality, where Gini index is less than one and maximal inequality, where Gini index asymptotically tends to its maximal value one. In both cases, the inequality of an expanding system slowly increases just as effect of growth, with a scaling never faster than the inverse of the size. We test our computations on two real systems, US cities and the cryptocurrency market, observing in both cases an increase of inequality that is completely explained by Zipf's law and the systems expanding. This shows that in growing complex systems finite size effects must be considered in order to properly assess if inequality is increasing due to natural growth processes or if it is produced by a change in the economical structure of the systems. Finally we discuss how such effects must be carefully considered when analyzing survey data.