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How Fixed-Amount Transactions and Liquidity Constraints Amplify Wealth Inequality: A Kinetic Model Deviating from the Maximum Entropy Benchmark

Jihyuan Liuh

TL;DR

This work analyzes a minimalist wealth-exchange model with fixed-amount transactions under a hard budget constraint and shows that the resulting dynamics generate a highly unequal stationary state with a substantial pauper class, in sharp contrast to the Boltzmann-Gibbs maximum-entropy benchmark. By formulating a self-consistent mean-field master equation and solving it numerically, complemented by agent-based simulations, the authors identify a poverty trap mechanism: non-ergodic dynamics where zero-wealth agents become passive receivers and escape requires unlikely consecutive positive shocks. The results demonstrate that substantial inequality can arise purely from exchange rules in a homogeneous population, providing a mechanistic explanation for poverty as an emergent property of basic economic constraints and informing policy perspectives on liquidity and credit constraints.

Abstract

This paper investigates the emergence of wealth inequality through a minimalist kinetic exchange model that incorporates two fundamental economic features: fixed-amount transactions and hard budget constraints. In contrast to the maximum entropy principle, which predicts an exponential Boltzmann-Gibbs distribution with moderate inequality for unconstrained wealth exchange, we demonstrate that these realistic trading rules drive the system toward a highly unequal steady state. We develop a self-consistent mean-field theory, deriving a master equation where agent income follows a Poisson process coupled to the poverty rate. Numerical solution reveals a stationary distribution characterized by a substantial pauper class, high Gini coefficient, and exponential tail--significantly deviating from the maximum entropy benchmark. Agent-based simulations confirm these findings. We identify the poverty trap as the key mechanism: the liquidity constraint creates asymmetric economic agency, where zero-wealth agents become passive recipients, unable to participate in wealth circulation. This work establishes that substantial inequality can emerge spontaneously from equal-opportunity exchanges under basic economic constraints, without requiring agent heterogeneity or multiplicative advantage, providing a mechanistic foundation for understanding poverty as an emergent property of exchange rules.

How Fixed-Amount Transactions and Liquidity Constraints Amplify Wealth Inequality: A Kinetic Model Deviating from the Maximum Entropy Benchmark

TL;DR

This work analyzes a minimalist wealth-exchange model with fixed-amount transactions under a hard budget constraint and shows that the resulting dynamics generate a highly unequal stationary state with a substantial pauper class, in sharp contrast to the Boltzmann-Gibbs maximum-entropy benchmark. By formulating a self-consistent mean-field master equation and solving it numerically, complemented by agent-based simulations, the authors identify a poverty trap mechanism: non-ergodic dynamics where zero-wealth agents become passive receivers and escape requires unlikely consecutive positive shocks. The results demonstrate that substantial inequality can arise purely from exchange rules in a homogeneous population, providing a mechanistic explanation for poverty as an emergent property of basic economic constraints and informing policy perspectives on liquidity and credit constraints.

Abstract

This paper investigates the emergence of wealth inequality through a minimalist kinetic exchange model that incorporates two fundamental economic features: fixed-amount transactions and hard budget constraints. In contrast to the maximum entropy principle, which predicts an exponential Boltzmann-Gibbs distribution with moderate inequality for unconstrained wealth exchange, we demonstrate that these realistic trading rules drive the system toward a highly unequal steady state. We develop a self-consistent mean-field theory, deriving a master equation where agent income follows a Poisson process coupled to the poverty rate. Numerical solution reveals a stationary distribution characterized by a substantial pauper class, high Gini coefficient, and exponential tail--significantly deviating from the maximum entropy benchmark. Agent-based simulations confirm these findings. We identify the poverty trap as the key mechanism: the liquidity constraint creates asymmetric economic agency, where zero-wealth agents become passive recipients, unable to participate in wealth circulation. This work establishes that substantial inequality can emerge spontaneously from equal-opportunity exchanges under basic economic constraints, without requiring agent heterogeneity or multiplicative advantage, providing a mechanistic foundation for understanding poverty as an emergent property of exchange rules.

Paper Structure

This paper contains 28 sections, 25 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Pervasiveness of Wealth Inequality
  3. Theoretical Background: Kinetic Models and the Maximum Entropy Benchmark
  4. Research Gap and Problem Formulation
  5. Our Work and Central Thesis
  6. Model and the Maximum Entropy Benchmark
  7. Model Definition
  8. Maximum Entropy Principle and the Ideal Benchmark Distribution
  9. Mean-Field Theory and Self-Consistent Master Equation
  10. Mean-Field Approximation and Poisson Income Hypothesis
  11. Derivation of the Master Equation
  12. Expenditure Stage
  13. Income Stage
  14. Combined Update
  15. Proof of Wealth Conservation
  16. ...and 13 more sections

Figures (2)

  • Figure 1: Comparison of stationary wealth distributions. The mean-field numerical solution and agent-based simulation results show a significant deviation from the maximum entropy benchmark, characterized by a large pauper class ($k=0$) and enhanced inequality.
  • Figure 2: Convergence history of the numerical iteration. The poverty rate $p_0$ and Gini coefficient $G$ converge smoothly to their stationary values, while the mean wealth remains constant at 1.0.