Paper: arXiv 2604.22976

Authors: Robert T. Nachtrieb

Abstract

The distribution of income and wealth in developed economies exhibits a robust two-class structure: an exponential (Boltzmann–Gibbs) bulk covering $\sim!97%$ of the population, and a power-law (Pareto) tail in the upper $\sim!3%$. We derive this structure from first principles via an explicit mechanistic chain: Gibrat’s law for firm growth implies a Zipf firm-size distribution; maximum entropy applied to within-firm wages, combined with mixture aggregation across firms, yields a Boltzmann–Gibbs income distribution with temperature $T_y$ for employees; additive-noise wealth dynamics with a reflecting wall at zero produce a Boltzmann–Gibbs employee wealth distribution with temperature $T_w$. For firm owners, multiplicative capital returns produce a Pareto wealth tail with exponent $α_w = 1/θ$, where $θ$ encodes how total returns scale with firm size. The empirical value $α_w \approx 1.30$ \cite{Yakovenko2009} is reproduced with no tuned parameters from the observed firm value scaling $V = V_0(s/s_0)^{0.77}$~\cite{Axtell2001}, and simultaneously yields the first quantitative estimate of the returns-per-employee size exponent: $ζ= θ- 1 \approx -0.23$. For empirical values $ν\approx 0.3$, $c \approx 0.81$, $k \approx 0.15$ (BEA long-run savings rate $\approx 5%$), the model gives $T_w/T_y \approx 1.7,\text{yr}$, i.e.\ lower-class households hold roughly 1–2 years of income as wealth, with the precise ratio depending on savings and tax rates and testable cross-country. As a parameter-free empirical test, firms near zero profit have a cash martingale whose first-passage time gives establishment exit rate $\sim t^{-1/2}$; convolving with the Zipf firm-size distribution yields firm-level exit rate $\sim t^{-1/2}!\log t$, with apparent exponent $b = 0.295 \pm 0.03$, confirmed against BDS firm-age data with no free parameters.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper leverages advanced statistical mechanics, Fokker-Planck equations, and entropy-based derivations, while simultaneously grounding these theoretical models in specific empirical data from the BEA, Census Bureau, and Compustat to validate parameters without tuning.

Research Flowchart

  flowchart TD
    Goal[Research Goal: Explain 2-class income/wealth distribution]
    Methodology[Methodology: MaxEnt & Mixture Aggregation]
    Inputs[Inputs: Gibrat's Law, Firm Size, Firm-Specific Data]
    Dynamics[Mechanistic Chain: Firm growth -> Individual income/wealth]
    Validation[Empirical Tests: BDS data & Firm exit rates]
    Findings[Outcomes: Boltzmann-Gibbs bulk + Pareto tail + $ζ \approx -0.23$]

    Goal --> Methodology
    Inputs --> Dynamics
    Methodology --> Dynamics
    Dynamics --> Validation
    Validation --> Findings

Cited References (1)

  1. 0 (1)

0 of 1 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.