Economics and Finance

Axiomatic Derivation of Pareto Wealth Distribution via Arbitrage Dynamics and the Fokker—Planck Equatio

Authors: Sayo Asanagi
Comments: 28 Pages.

The upper tail of the individual wealth distribution in human societies universallyfollows a power law P (w) ∝ w−κ with Pareto exponent κ ∈ (1.5, 2.5), yet a fullyendogenous derivation of κ from economic first principles has remained an open prob-lem. We introduce the Arbitrage-Driven Wealth Distribution (ADWD) framework,founded on four base axioms (rationality, wealth-proportional information access,time-scale separation, and finite arbitrage lifetime) and three developmental axioms(determination of trading time, arbitrage arrival rate, and lifetime—trading-time con-sistency). From these axioms alone we derive a Fokker—Planck equation governingthe evolution of the wealth density P (w, t) with fully endogenous drift and diffusioncoefficients. The stationary solution yields a Pareto distribution whose exponent isgiven byκ = 1 + 2μlossαδ0,where every parameter carries a precise economic meaning. The dissipation rate μlossis decomposed without additional axioms into three components: consumption dis-sipation μC = ρ (from the rationality axiom via Ramsey optimisation), depreciationdissipation μD = 1/γ (from the finite lifetime axiom), and equilibration dissipationμE = αW 2∞/(N δ0) (from the information-access and time-scale axioms). The ag-gregate wealth ceiling W∞ emerges endogenously from a logistic structure implicitin the axioms, satisfying a self-consistency equation. A growth-correction term (theβ-term) is derived in three stages: a dynamic extension of the information-accessaxiom, a Nash equilibrium argument that identifies β = (1 − κ)β1/2, and a pertur-bative verification showing that the β-term vanishes at stationarity while governingthe transient approach to the power law. We establish two bridging theorems. The-orem P shows that Piketty’s qualitative criterion r > g is a special case of ourframework, recovered in the limit μD, μE → 0, and is sharpened to the quantita-tive condition αηW∞ > μloss. A correspondence theorem identifies the Bouchaud—Mézard exchange model as the limiting case ρ → 0, γ → ∞ of ADWD, providingthe economic micro-foundation that the physical model lacks. Finally, we proposea three-equation empirical protocol for estimating the sole non-directly-observableparameter α from wealth-distribution data, transaction frequencies, and aggregategrowth rates, enabling in-principle falsification of the theory
Category: Economics and Finance