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

We study the distribution P(ω) of the random variable ω=x1/(x1+x2), where x1 and x2 are the wealths of two individuals selected at random from the same tempered Paretian ensemble characterized by the distribution Ψ(x)∼ϕ(x)/x1+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≤x≤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 x1≈x2. For 0<α<1 we observe a more complicated behavior which depends on the value of δ=L/H. In particular, for δ<δc–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 x1 and x2 are disproportionately different.