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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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    1. Jan Eeckhout, 2004. "Gibrat's Law for (All) Cities," American Economic Review, American Economic Association, vol. 94(5), pages 1429-1451, December.
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    3. Gilles Duranton & Diego Puga, 2000. "Diversity and Specialisation in Cities: Why, Where and When Does it Matter?," Urban Studies, Urban Studies Journal Limited, vol. 37(3), pages 533-555, March.
    4. Rosen, Kenneth T. & Resnick, Mitchel, 1980. "The size distribution of cities: An examination of the Pareto law and primacy," Journal of Urban Economics, Elsevier, vol. 8(2), pages 165-186, September.
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    6. Edward L. Glaeser & Joshua D. Gottlieb, 2009. "The Wealth of Cities: Agglomeration Economies and Spatial Equilibrium in the United States," Journal of Economic Literature, American Economic Association, vol. 47(4), pages 983-1028, December.
    7. Duncan Black & Vernon Henderson, 2003. "Urban evolution in the USA," Journal of Economic Geography, Oxford University Press, vol. 3(4), pages 343-372, October.
    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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