Scale Invariance in Economics and in Finance

Abstract

“Scale invariance” is not a common expression in economics, and expressions like “self similarity” or “self affinity” are scarcely used. Even “stable non gaussian laws” (or “Lévy laws”, or “Pareto-Lévy” laws) and “fractals”, introduced in economics by B. Mandelbrot [1–2], remain relativily unheard outside financial markets and research centers devoted to finance. Actually, history of economics thought is full of scale invariant distributions, but under another name: Pareto. This famous sociologist and economist of the end of the XIXth century found numerous examples of power laws in income size-distributions [3]. Hence the common name “Pareto” for such a type of economic distributions. Nevertheless, income distributions are not the unique family of scale invariant economic distributions. Firm sizes and assets sizes [4–6], industrial insurance claim sizes [7–9], economic damages due to natural catastrophes, such as hurricanes or earthquakes [10–11], stock market prices fluctuations in the short run and in the long run [2, 6, 12–14] etc. display either one tail with a power law (for obvious reasons most economic size-distributions can only take a positive value) or two tails (all stock market prices, since they have positive and negative variations). In that case they have either a power law on both tails, a typical behavior of all stable non gaussian distributions fitting the whole distribution, or a gaussian law fitting the whole distribution. As regards stock market, scale invariance is not a new scientific discovery. Bachelier [12] in his now famous thesis on Paris “Bourse” settled in 1900 the founding stone of brownian motion, the prototype of gaussian scale invariance. In order to sort out the origins and implications of scale invariance in economics, we will restrict our analysis to one-tail economic phenomena, deferring the analysis of two tails scale invariant economic phenomena to another lecture.