Statistical Mechanics of Household Income and Wealth: Derivation from Firm Dynamics via Maximum Entropy and Mixture Aggregation
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| Format: | Preprint |
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2026
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| _version_ | 1866915956157054976 |
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| author | Nachtrieb, Robert T. |
| author_facet | Nachtrieb, Robert T. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_22976 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Statistical Mechanics of Household Income and Wealth: Derivation from Firm Dynamics via Maximum Entropy and Mixture Aggregation Nachtrieb, Robert T. Physics and Society Statistical Mechanics Statistical Finance 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. |
| title | Statistical Mechanics of Household Income and Wealth: Derivation from Firm Dynamics via Maximum Entropy and Mixture Aggregation |
| topic | Physics and Society Statistical Mechanics Statistical Finance |
| url | https://arxiv.org/abs/2604.22976 |