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Wages and Capital returns in a generalized Pólya urn

Thomas Gottfried, Stefan Grosskinsky

TL;DR

This work introduces a generalized Pólya urn model that splits wealth growth into a fixed wage-driven component and a capital-return component, enabling increasing-returns dynamics to generate the observed Pareto-like tail in wealth distributions. Using stochastic approximation and a Lyapunov framework, the authors show convergence to fixed points of the field $G(x)=(1-r)p(x)+r\gamma-x$, with a deterministic limit when $r$ is large and $\beta>1$, and they characterize stability via a Lyapunov function $L$. Calibrating parameters to Germany’s 2021 wealth data, they demonstrate that the model reproduces the empirical wealth distribution, including the bottom tail aligned with scaled wages and a heavier top tail due to capital returns, and they provide time-evolution predictions via an ODE approximation. The results highlight the role of increasing returns and the labor share in wealth aggregation, while also noting the model’s strong rank-ordering tendency and suggesting avenues for refinement through investment-skill heterogeneity and shocks to better match empirical correlations between wages and wealth.

Abstract

It is a widely observed phenomenon that wealth is distributed significantly more unequal than wages. In this paper we study this phenomenon using a new extension of Pólyas urn, modelling wealth growth through wages and capital returns. We focus in particular on the role of increasing return rates on capital, which have been identified as a main driver of inequality, and labor share, the second main parameter of our model. We fit the parameters from real-world data in Germany, so that simulation results reproduce the empirical wealth distribution and recent dynamics in Germany quite accurately, and are essentially independent from initial conditions. Our model is simple enough to allow for a detailed mathematical analysis and provides interesting predictions for future developments and on the importance of wages and capital returns for wealth aggregation. We also provide an extensive discussion of the robustness of our results and the plausibility of the main assumptions used in our model.

Wages and Capital returns in a generalized Pólya urn

TL;DR

This work introduces a generalized Pólya urn model that splits wealth growth into a fixed wage-driven component and a capital-return component, enabling increasing-returns dynamics to generate the observed Pareto-like tail in wealth distributions. Using stochastic approximation and a Lyapunov framework, the authors show convergence to fixed points of the field , with a deterministic limit when is large and , and they characterize stability via a Lyapunov function . Calibrating parameters to Germany’s 2021 wealth data, they demonstrate that the model reproduces the empirical wealth distribution, including the bottom tail aligned with scaled wages and a heavier top tail due to capital returns, and they provide time-evolution predictions via an ODE approximation. The results highlight the role of increasing returns and the labor share in wealth aggregation, while also noting the model’s strong rank-ordering tendency and suggesting avenues for refinement through investment-skill heterogeneity and shocks to better match empirical correlations between wages and wealth.

Abstract

It is a widely observed phenomenon that wealth is distributed significantly more unequal than wages. In this paper we study this phenomenon using a new extension of Pólyas urn, modelling wealth growth through wages and capital returns. We focus in particular on the role of increasing return rates on capital, which have been identified as a main driver of inequality, and labor share, the second main parameter of our model. We fit the parameters from real-world data in Germany, so that simulation results reproduce the empirical wealth distribution and recent dynamics in Germany quite accurately, and are essentially independent from initial conditions. Our model is simple enough to allow for a detailed mathematical analysis and provides interesting predictions for future developments and on the importance of wages and capital returns for wealth aggregation. We also provide an extensive discussion of the robustness of our results and the plausibility of the main assumptions used in our model.
Paper Structure (14 sections, 9 theorems, 44 equations, 19 figures)

This paper contains 14 sections, 9 theorems, 44 equations, 19 figures.

Table of Contents

  1. Introduction
  2. The model and some rigorous results
  3. The two and three agent case
  4. Simulations for homogeneous feedback
  5. The wage-vector $\gamma$
  6. The labor share $r$
  7. The reinforcement parameter $\beta$
  8. Time evolution and predictions for the future
  9. Unequal investment skills as an alternative explanation
  10. Summary and Discussion
  11. The case $r=0$
  12. Heuristics on the stability of fixed points
  13. A functional law of large numbers for the dynamic
  14. Supplementary figures

Key Result

Theorem 2.1

For all $i\in[A]$ let $F_i(k)=\alpha_i k^\beta$ for $\alpha_i>0,\,\beta\in\mathbb{R}$, $r\in[0, 1]$. Then $\chi(n)\to\chi(\infty)$ converges almost surely to a stable fixed point of $G$ for $n\to\infty$.

Figures (19)

  • Figure 1: $1-CDF$ (Cumulative Distribution Function) of net personal wealth (purchasing power parity, equal split adults) in Germany and the USA in 2011 and 2021 in Euro resp. US-Dollar according to wid. Least square fit (red line) estimates a Pareto exponent of 1.44 for Germany 2021.
  • Figure 2: Qualitative illustration of the number of stable fixed points of $G$ for homogeneous feedback and different $r$ and $\beta$. $\circ$ marks the classical Pólya urn, which exhibits either deterministic (weak) monopoly or a Dirichlet distributed limit.
  • Figure 3: The line $g$ (see \ref{['eq: lineg']}) and the field $G_0$ (see (\ref{['eq: fieldG0']})) against wealth $x$ of agent 1. $\bullet$ marks stable and $\circ$ unstable fixed points. The arrows indicate the direction of the field $G$.
  • Figure 4: The line $g$ (see \ref{['eq: lineg']}) and the field $G_0$ (see (\ref{['eq: fieldG0']})) against wealth $x$ of agent 1 for various parameters.
  • Figure 5: The field $G$ (see \ref{['eq: gg']}) with $A=3$, $\beta=2, \alpha_1=\alpha_2=\alpha_3=1$ and $\gamma=(0.4, 0.4, 0.2)$ for various labor shares $r$. Stable fixed points are marked with $\bullet$ and unstable fixed points with $\circ$. Their exact position has been computed with Mathematica.
  • ...and 14 more figures

Theorems & Definitions (15)

  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 5 more