Zipfian and Lotkaian continuous concentration theory

Abstract

This paper studies concentration (i.e. inequality) aspects of the functions of Zipf and of Lotka. Since both functions are power laws (i.e. they are – mathematically the same) it suffices to develop one concentration theory for power laws and apply it twice for the different interpretations of the laws of Zipf and Lotka. After a brief repetition of the functional relationships between Zipf’s law and Lotka’s law, we prove that Price’s law of concentration is equivalent with Zipf’s law. The major part of the paper is devoted to the development of continuous concentration theory, based on Lorenz curves. We calculate the Lorenz curve for power functions and, based on this, calculate some important concentration measures such as the ones of Gini, Theil and the variation coefficient. We also show, using Lorenz curves, that the concentration of a power law increases with its exponent and we interpret this result in terms of the functions of Zipf and Lotka.