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 = 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 ) ∼ 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 ≤ 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 x 1 ≈ 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.