From Zipf to Price and beyond

Consider a society comprising n members that are ranked in a decreasing order of their personal wealths. For this society: Zipf’s Law manifests the case in which the members’ ranks and wealths display an inverse power-law relation; and Price’s Law manifests the case in which the n richest members possess, collectively, half of all the wealth. This paper goes from Zipf’s Law to Price’s Law and beyond, and it does so by introducing and exploring a novel Generalized Price’s Law (GPL): a general allometric scaling of the quantiles of rank distributions. Akin to multifractals, the GPL is governed by spectrums of powers. The GPL spectra are investigated, and are shown to be fractal objects: from a socioeconomic perspective, the spectra are power-law Lorenz curves that characterize poor fractality and rich fractality; from a probabilistic perspective, the spectra are characterized by Pareto statistics and by Lindy Laws. The intersection of Zipf’s Law and of the GPL is also investigated, and it is shown to be: (i) the phase transition between the two markedly different fractal regimes of the GPL spectra; (ii) the phase transition between two markedly different macroscopic regimes of Zipf’s Law; (iii) Price’s Law. Metaphorically, this paper establishes navigation directions in the space of rank distributions: how to get from the ‘Zipf street’ to the ‘Price junction’, from there to the new ‘GPL avenue’, and back.