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

Wen Tai Hsu; Thomas J. Holmes; Frank Morgan

doi:10.1016/j.jet.2014.09.018

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

Wen Tai Hsu, Thomas J. Holmes, Frank Morgan

Economics (Twin Cities)

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

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.

Original language	English (US)
Pages (from-to)	245-273
Number of pages	29
Journal	Journal of Economic Theory
Volume	154
DOIs	https://doi.org/10.1016/j.jet.2014.09.018
State	Published - 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.

Keywords

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

Publisher link

10.1016/j.jet.2014.09.018

Find It

Find It at U of M Libraries

Cite this

@article{2efa9bdd56924f118661fb66af72bec2,

title = "Optimal city hierarchy: A dynamic programming approach to central place theory",

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.",

keywords = "Central place theory, City hierarchy, Dynamic programming, Fixed point, Principle of optimality",

author = "Hsu, {Wen Tai} and Holmes, {Thomas J.} and Frank Morgan",

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: {\textcopyright} 2014 Elsevier Inc.",

year = "2014",

month = nov,

day = "1",

doi = "10.1016/j.jet.2014.09.018",

language = "English (US)",

volume = "154",

pages = "245--273",

journal = "Journal of Economic Theory",

issn = "0022-0531",

publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - Optimal city hierarchy

T2 - A dynamic programming approach to central place theory

AU - Hsu, Wen Tai

AU - Holmes, Thomas J.

AU - Morgan, Frank

N1 - 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.

PY - 2014/11/1

Y1 - 2014/11/1

N2 - 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.

AB - 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.

KW - Central place theory

KW - City hierarchy

KW - Dynamic programming

KW - Fixed point

KW - Principle of optimality

UR - http://www.scopus.com/inward/record.url?scp=84922976344&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84922976344&partnerID=8YFLogxK

U2 - 10.1016/j.jet.2014.09.018

DO - 10.1016/j.jet.2014.09.018

M3 - Article

AN - SCOPUS:84922976344

SN - 0022-0531

VL - 154

SP - 245

EP - 273

JO - Journal of Economic Theory

JF - Journal of Economic Theory

ER -

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

Abstract

Bibliographical note

Keywords

Publisher link

Find It

Other files and links

Fingerprint

Cite this