Fractals, Wealth Distributions, and Non-Euclidean Geometry: Quadrant Lorenz Curves

Lorenz curves are a socioeconomic method of coding the statistics of positive-valued random variables. Standing at the confluence of fractals, wealth distributions, and non-Euclidean geometry, this paper presents quadrant Lorenz curves (QLCs): Lorenz curves whose shapes are that of the symmetric quarters of the p-norm unit circle. On the one hand, QLCs have a single parameter, and a clear geometric meaning. On the other hand, a comprehensive exploration establishes that QLCs display an assortment of socioeconomic and statistical behaviors. (1) They span a continuum of socioeconomic states ranging from perfect equality to perfect inequality, with three phase transitions along the continuum. (2) They display local rich fractality, local poor fractality, and double-Pareto tails. (3) They have inherent socioeconomic and statistical symmetries. The exploration includes a detailed socioeconomic-inequality analysis that illuminates QLCs from multiple inequality-indices 'angles'. In a nutshell: QLCs manifest a novel class of 'fractal' statistical distributions, and the QLC features are unmatched by those of any other such class.