Proportionate vs disproportionate distribution of wealth of two individuals in a tempered Paretian ensemble

We study the distribution P(omega) of the random variable omega = x(1)/(x(1) + x(2)), where x(1) and x(2) are the wealths of two individuals selected at random from the same tempered Paretian ensemble characterized by the distribution psi(x) similar to phi(x)/x(1+alpha), where alpha > 0 is the Pareto index and phi(x) is the cut-off function. We consider two forms of phi(x): a bounded function phi(x) = 1 for L < x <= H, and zero otherwise, and a smooth exponential function phi(x) = exp(-L/x - x/H). In both cases psi(x) has moments of arbitrary order. We show that, for alpha > 1, P(omega) always has a unimodal form and is peaked at omega = 1/2, so that most probably x(1) approximate to x(2). For 0 < alpha < 1 we observe a more complicated behavior which depends on the value of delta = L/H. In particular, for delta < delta(c) - a certain threshold value - P(omega) has a three-modal (for a bounded phi(x)) and a bimodal M-shape (for an exponential phi(x)) form which signifies that in such ensembles the wealths x(1) and x(2) are disproportionately different. (C) 2011 Elsevier B.V. All rights reserved.