Beyond the power law – a new approach to analyze city size distributions

This work proposes a new approach to analyze the city size distribution (CSD). We present a general equation for the rank size logarithmic plot, with a new positive exponent α. When α = 1, the Pareto distribution is yielded; when α ≠ 1, the log of the curves exhibits a concave distribution. We studied the CSDs of 41 cases in 35 countries (in several countries we examined cities and metropolitan areas or agglomerations) in order to apply our new equation. We determined accurately the exponent α for 31 cases. In 18 cases we received α = 1, in one case α < 1, and in 12 cases α > 1. However, for the other cases, either the distributions were not homogeneous, or the data exhibited significant fluctuations which precluded a good determination of the exponent α. Based on this analysis, we developed a series of models (based on the models of town growth of Gabaix and of Blank and Solomon) in order to describe the different CSDs. The results of these models include power laws as well as cases that are represented by concave distributions on a logarithmic plot of the rank size.