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

Wen Tai Hsu, Thomas J. Holmes, Frank Morgan

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)245-273
Number of pages29
JournalJournal of Economic Theory
StatePublished - Nov 1 2014

Bibliographical note

Funding Information:
We are grateful to an anonymous associate editor and three anonymous referees for their very thoughtful and helpful comments and suggestions that drastically improved this paper. For their helpful comments, we also thank Yi-Chun Chen, Tom Davidoff, Gilles Duranton, Yannis Ioannides, Samuel Kortum, Erzo G.J. Luttmer, Shin-Kun Peng, Will Strange, Shenghao Zhu, and the seminar participants at Academia Sinica, National Taiwan University, National University of Singapore, Singapore Management University, the 2007 Midwest Economic Theory Meetings, the 2008 Annual Meeting of the Urban Economics Association, the 2009 Econometric Society's North American Summer Meeting, the 2009 Annual Meeting of Society of Economic Dynamics, and the 2012 IRES Symposium. We also thank Paul Thompson for permission to use his central place theory graph. Frank Morgan acknowledges partial support from the National Science Foundation DMS 0803168 . All errors are ours.

Publisher Copyright:
© 2014 Elsevier Inc.


  • Central place theory
  • City hierarchy
  • Dynamic programming
  • Fixed point
  • Principle of optimality


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