Abstract
This chapter provides a review of the link between central place theory and the power laws for cities. A theory of city size distribution is proposed via a central place hierarchy a la Christaller (1933) either as an equilibrium results or an optimal allocation. Under a central place hierarchy, it is shown that a power law for cities emerges if the underlying heterogeneity in economies of scale across good is regularly varying. Furthermore, we show that an optimal allocation of cities conforms with a central place hierarchy if the underlying heterogeneity in economies of scale across good is a power function.
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Notes
- 1.
This is often called “hierarchy principle” in the literature.
- 2.
A fractal structure is a structure in which smaller parts of it resemble the entire structure.
- 3.
We do not discuss how to decide the optimal distance \(\ell _{1}\) here, since our focus is the city hierarchy between any two largest cities. Nevertheless, this is very crucial question for presenting a complete and meaningful model, please see Sect. 3.5 in Hsu et al. (2014) for the solution.
- 4.
For example, let \(t=z_{1}=\ell _{1}=1\). Consider a discontinuous setup cost requirement function: for an arbitrarily small \(e\in \left( 0,1\right) ,\phi \left( y\right) =1/13\) for \(y\in \left[ 0,e\right] \) and \(\phi \left( y\right) =1\) for \(y\in (e,1]\). It is readily verified that, in between two \(z_{1}\)-cities, the per capita cost is minimized by evenly placing two immediate sub-cities with \(z^{\prime }=e\).
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Hsu, WT., Zou, X. (2019). Central Place Theory and the Power Law for Cities. In: D'Acci, L. (eds) The Mathematics of Urban Morphology. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-12381-9_3
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