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

Abstract

We study the distribution P(ω) of the random variable ω= 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 Ψ(x) \sim ϕ(x)/x^{1 + α}, where α> 0 is the Pareto index and $ϕ(x)$ is the cut-off function. We consider two forms of ϕ(x): a bounded function ϕ(x) = 1 for L \leq x \leq H, and zero otherwise, and a smooth exponential function ϕ(x) = \exp(-L/x - x/H). In both cases Ψ(x) has moments of arbitrary order. We show that, for α> 1, P(ω) always has a unimodal form and is peaked at ω= 1/2, so that most probably x_1 \approx x_2. For 0 < α< 1 we observe a more complicated behavior which depends on the value of δ= L/H. In particular, for δ< δ_c - a certain threshold value - P(ω) has a three-modal (for a bounded ϕ(x)) and a bimodal M-shape (for an exponential ϕ(x)) form which signifies that in such ensembles the wealths x_1 and x_2 are disproportionately different.

9 pages, 8 figures, to appear in Physica A