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Urban Scale and Inequality: A Mean Field Game Approach to Zipf's Law, Gibrat's Law, and Endogenous City Growth

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  • Heng-fu Zou

Abstract

This paper develops a dynamic mean field game model to explain the emergence of heavy-tailed city size distributions and persistent urban inequality. Each city optimizes intertemporal consumption and investment in infrastructure to maximize utility from both consumption and population scale. City size follows a stochastic process with investment-driven drift, while productivity evolves as a mean-reverting diffusion. We derive the equilibrium using the Lasry–Lions Master Equation and simulate the resulting stationary distribution. The model generates high Gini coefficients and Pareto-like upper tails, consistent with Zipf’s law and Gibrat’s law. These patterns arise endogenously through capital accumulation, productivity shocks, and scale feedbacks, offering a unified framework to understand urban size inequality.

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  • Heng-fu Zou, 2025. "Urban Scale and Inequality: A Mean Field Game Approach to Zipf's Law, Gibrat's Law, and Endogenous City Growth," CEMA Working Papers 757, China Economics and Management Academy, Central University of Finance and Economics.
    • Handle: RePEc:cuf:wpaper:757
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    8. Xavier Gabaix, 1999. "Zipf's Law for Cities: An Explanation," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 114(3), pages 739-767.
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