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Optimal city hierarchy: A dynamic programming approach to central place theory

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  • Hsu, Wen-Tai
  • Holmes, Thomas J.
  • Morgan, Frank

Abstract

Central place theory is a key building block of economic geography and an empirically plausible description of city systems. This paper provides a rationale for central place theory via a dynamic programming formulation of the social planner's problem of city hierarchy. We show that there must be one and only one immediate smaller city between two neighboring larger-sized cities in any optimal solution. If the fixed cost of setting up a city is a power function, then the immediate smaller city will be located in the middle, confirming the locational pattern suggested by Christaller [4]. We also show that the solution can be approximated by iterating the mapping defined by the dynamic programming problem. The main characterization results apply to a general hierarchical problem with recursive divisions.

Suggested Citation

  • Hsu, Wen-Tai & Holmes, Thomas J. & Morgan, Frank, 2014. "Optimal city hierarchy: A dynamic programming approach to central place theory," Journal of Economic Theory, Elsevier, vol. 154(C), pages 245-273.
    • Handle: RePEc:eee:jetheo:v:154:y:2014:i:c:p:245-273
    DOI: 10.1016/j.jet.2014.09.018
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    1. Lederer, Phillip J & Hurter, Arthur P, Jr, 1986. "Competition of Firms: Discriminatory Pricing and Location," Econometrica, Econometric Society, vol. 54(3), pages 623-640, May.
    2. Yingyi Qian, 1994. "Incentives and Loss of Control in an Optimal Hierarchy," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 61(3), pages 527-544.
    3. Esteban Rossi-Hansberg & Mark L. J. Wright, 2007. "Urban Structure and Growth," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 74(2), pages 597-624.
    4. Jan Eeckhout, 2004. "Gibrat's Law for (All) Cities," American Economic Review, American Economic Association, vol. 94(5), pages 1429-1451, December.
    5. Gilles Duranton, 2007. "Urban Evolutions: The Fast, the Slow, and the Still," American Economic Review, American Economic Association, vol. 97(1), pages 197-221, March.
    6. Hernán D. Rozenfeld & Diego Rybski & Xavier Gabaix & Hernán A. Makse, 2011. "The Area and Population of Cities: New Insights from a Different Perspective on Cities," American Economic Review, American Economic Association, vol. 101(5), pages 2205-2225, August.
    7. Tabuchi, Takatoshi & Thisse, Jacques-François, 2011. "A new economic geography model of central places," Journal of Urban Economics, Elsevier, vol. 69(2), pages 240-252, March.
    8. Takatoshi Tabuchi & Jacques-François Thisse, 2006. "Regional Specialization, Urban Hierarchy, And Commuting Costs," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 47(4), pages 1295-1317, November.
    9. Wen‐Tai Hsu, 2012. "Central Place Theory and City Size Distribution," Economic Journal, Royal Economic Society, vol. 122(563), pages 903-932, September.
    10. Steven C. Salop, 1979. "Monopolistic Competition with Outside Goods," Bell Journal of Economics, The RAND Corporation, vol. 10(1), pages 141-156, Spring.
    11. Fujita, Masahisa & Krugman, Paul & Mori, Tomoya, 1999. "On the evolution of hierarchical urban systems1," European Economic Review, Elsevier, vol. 43(2), pages 209-251, February.
    12. Eaton, B Curtis & Lipsey, Richard G, 1982. "An Economic Theory of Central Places," Economic Journal, Royal Economic Society, vol. 92(365), pages 56-72, March.
    13. 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.
    14. Tomoya Mori & Koji Nishikimi & Tony E. Smith, 2008. "The Number‐Average Size Rule: A New Empirical Relationship Between Industrial Location And City Size," Journal of Regional Science, Wiley Blackwell, vol. 48(1), pages 165-211, February.
    15. Tomoya Mori & Tony E. Smith, 2011. "An Industrial Agglomeration Approach To Central Place And City Size Regularities," Journal of Regional Science, Wiley Blackwell, vol. 51(4), pages 694-731, October.
    16. Duranton, Gilles, 2006. "Some foundations for Zipf's law: Product proliferation and local spillovers," Regional Science and Urban Economics, Elsevier, vol. 36(4), pages 542-563, July.
    17. Quinzii, Martine & Thisse, Jacques-Francois, 1990. "On the Optimality of Central Places," Econometrica, Econometric Society, vol. 58(5), pages 1101-1119, September.
    18. Robert E. Lucas & Esteban Rossi-Hansberg, 2002. "On the Internal Structure of Cities," Econometrica, Econometric Society, vol. 70(4), pages 1445-1476, July.
    19. Luis Garicano, 2000. "Hierarchies and the Organization of Knowledge in Production," Journal of Political Economy, University of Chicago Press, vol. 108(5), pages 874-904, October.
    20. Luis Garicano & Esteban Rossi-Hansberg, 2006. "Organization and Inequality in a Knowledge Economy," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 121(4), pages 1383-1435.
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    4. Kikuchi, Tomoo & Nishimura, Kazuo & Stachurski, John, 2018. "Span of control, transaction costs and the structure of production chains," Theoretical Economics, Econometric Society, vol. 13(2), May.
    5. de Palma, André & Papageorgiou, Yorgos Y. & Thisse, Jacques-François & Ushchev, Philip, 2019. "About the origin of cities," Journal of Urban Economics, Elsevier, vol. 111(C), pages 1-13.
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    9. Kikuchi, Tomoo & Nishimura, Kazuo & Stachurski, John & Zhang, Junnan, 2021. "Coase meets Bellman: Dynamic programming for production networks," Journal of Economic Theory, Elsevier, vol. 196(C).

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    More about this item

    Keywords

    Central place theory; City hierarchy; Dynamic programming; Principle of optimality; Fixed point;
    All these keywords.

    JEL classification:

    • R12 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General Regional Economics - - - Size and Spatial Distributions of Regional Economic Activity; Interregional Trade (economic geography)
    • R13 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General Regional Economics - - - General Equilibrium and Welfare Economic Analysis of Regional Economies
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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