Power-law distributions in economics: a nonextensive statistical approach

Abstract

The cornerstone of Boltzmann-Gibbs ($BG$) statistical mechanics is the Boltzmann-Gibbs-Jaynes-Shannon entropy $S_{BG} \equiv -k\int dx f(x)\ln f(x)$, where $k$ is a positive constant and $f(x)$ a probability density function. This theory has exibited, along more than one century, great success in the treatment of systems where short spatio/temporal correlations dominate. There are, however, anomalous natural and artificial systems that violate the basic requirements for its applicability. Different physical entropies, other than the standard one, appear to be necessary in order to satisfactorily deal with such anomalies. One of such entropies is $S_q \equiv k (1-\int dx [f(x)]^q)/(1-q)$ (with $S_1=S_{BG}$), where the entropic index $q$ is a real parameter. It has been proposed as the basis for a generalization, referred to as {\it nonextensive statistical mechanics}, of the $BG$ theory. $S_q$ shares with $S_{BG}$ four remarkable properties, namely {\it concavity} ($\forall q>0$), {\it Lesche-stability} ($\forall q>0$), {\it finiteness of the entropy production per unit time} ($q \in \Re$), and {\it additivity} (for at least a compact support of $q$ including $q=1$). The simultaneous validity of these properties suggests that $S_q$ is appropriate for bridging, at a macroscopic level, with classical thermodynamics itself. In the same natural way that exponential probability functions arise in the standard context,power-law tailed distributions, even with exponents {\it out} of the L\'evy range, arise in the nonextensive framework. In this review, we intend to show that many processes of interest in economy, for which fat-tailed probability functions are empirically observed, can be described in terms of the statistical mechanisms that underly the nonextensive theory.