Growing inequality in systems showing Zipf’s law
Giordano De Marzo, Federico Attili and Luciano Pietronero

Published 20 March 2023
Journal of Physics: Complexity, Volume 4, Number 1

Focus on Fundamental Theory and Application of Complexity Economics

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 the Gini index, one of the most used inequality measures, in the presence of Zipf’s law. This situation occurs in many complex financial and economic systems. First, we show that applying asymptotic formulas to finite-size systems always leads to overestimating inequality. We thus compute finite size corrections and show that depending on Zipf’s exponent, two distinct regimes can be observed: low inequality, where the Gini index is less than one, and maximal inequality, where the Gini index asymptotically tends to its maximal value of one. In both cases, the inequality of an expanding system slowly increases just as an 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 an increase in inequality entirely explained by Zipf’s law and the systems expanding. This shows that in growing complex systems, finite size effects must be considered to properly assess if inequality is increasing due to natural growth processes or if a change in the economic structure of the systems produces it. Finally, we discuss how such effects must be carefully considered when analyzing survey data.

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