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Spatial patterns and city size distribution

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Many large cities are found at locations with certain first nature advantages. Yet, those exogenous locational features are not the most potent forces governing the spatial pattern of cities. In particular, the (population) size, spacing, and industrial composition of cities exhibit a simple, persistent, monotonic relationship. Theories of economic agglomeration suggest that this regularity is a consequence of interactions between endogenous agglomeration and dispersion forces. Size and spacing of cities Cities typically exhibit a central place pattern in the country, in which larger cities are surrounded by smaller ones, and are more spaced apart. Figure 1 shows the relationship between the size si of each city i and the distance di to its closest city of comparable size or

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Many large cities are found at locations with certain first nature advantages. Yet, those exogenous locational features are not the most potent forces governing the spatial pattern of cities. In particular, the (population) size, spacing, and industrial composition of cities exhibit a simple, persistent, monotonic relationship. Theories of economic agglomeration suggest that this regularity is a consequence of interactions between endogenous agglomeration and dispersion forces.

Size and spacing of cities

Cities typically exhibit a central place pattern in the country, in which larger cities are surrounded by smaller ones, and are more spaced apart. Figure 1 shows the relationship between the size si of each city i and the distance di to its closest city of comparable size or larger for the 450 Japanese cities in 2015.1 The correlation between them is as high as 0.67, unconditional on the topographical heterogeneity of city location. 

Figure 1 Size and spacing of Japanese cities in 2015

Spatial patterns and city size distribution

Size distribution of cities

It is well-known that city size distribution in a relatively self-sufficient economic region – typically a country – exhibits an approximate power law (e.g. Gabaix and Ioannides 2004). More specifically, if a given set of n cities satisfy a power law, the city size and their rankings have an approximately log-linear relation. Japan is an obviously mono-polar economy organised around Tokyo, and is a typical case in which the approximate power law for city size distribution holds at the country level.2

In Figure 2, Panel (a) shows the rank-size distributions of cities every five years from 1970 to 2015, where si indicates the share of city i in the national population. Panel (b) shows the change in the Zipf coefficient – the elasticity of rank with respect to city size – over these 45 years. One can see that the city size distribution exhibits an approximate power law in each year, although agglomeration towards larger cities has been accelerated. The variation in city size is remarkably large, as exhibited by the largest three cities – Tokyo, Osaka and Nagoya – accounting for 45% of the national population.

Figure 2 Size distribution in Japanese cities for 1970-2015

Spatial patterns and city size distribution

Size and industrial composition of cities

It is well-known that the size and socioeconomic quantities (e.g. wage, education level, GDP, industrial diversity, numbers of patent applications) of cities exhibit strong correlations.3 The clearest such instance is industrial location. For the three-digit secondary and tertiary industries of the Japanese Standard Industrial Classification that are present in both 2000 and 2015, Figure 3 shows the relation between the number and average size of the cities in which industry i is located. The dashed curves indicate the upper and lower bounds for the average size, where the former is the average size of the largest Ni cities, and the latter is the average size of the smallest Ni cities.

Figure 3 Number and average size of cities in which an industry is located

Spatial patterns and city size distribution

The figure indicates two key relationships between industrial location and city size. First, the number and average sizes of the cities exhibit a strong and persistent power law, designated as the number-average size rule by Mori et al. (2008). Second, the average sizes are almost hitting their upper bound, meaning that the set of the Ni cities comprises roughly the largest Ni cities. Hence, a hierarchical principle à la Christaller (1933) roughly holds, i.e. an industry found in a smaller city is also found in all the larger cities.

Mori et al. (2008) have formally shown that the number-average size rule is indeed a consequence of the simultaneous presence of the power law for city size distribution and the hierarchy principle.

Theory for central place pattern

Akamatsu et al. (2018) have shown that a wide variety of the extant models of agglomeration, once extended to a many-region setup, can account for the basic mechanism underlying the central place pattern.4 Most existing theoretical models for city formation are static and involve a single type of mobile agent subject to the same degree of increasing returns (e.g. Krugman 1991, Beckmann 1976). In this context, they proposed a formal framework utilising a symmetric discrete racetrack geography to identify the spontaneous spatial pattern of cities that arises purely from agglomeration and dispersion forces. 

In particular, they have shown that the existing models can be classified into three reduced forms in terms of their spatial patterns of cities, despite the difference in their specific mechanisms underlying agglomeration and dispersion forces. Figure 4 depicts the two polar cases out of the three cases. The class of models incorporating only global dispersion force are able to account for the formation of multiple distinct cities (Panel a), whereas only one city can form in the other class of models incorporating local dispersion force without global dispersion force (Panel b). 

Formally, the dispersion force is global if it is dependent on, and local if it is independent of, interregional distance. An example of the global dispersion force is the demand for urban goods from dispersed consumers. Due to transport costs, urban producers are encouraged to locate near the dispersed consumers. As a consequence, multiple cities form so that firms in each city mainly serve their nearby markets. On the other hand, the local dispersion force accrues from, for example, locally scarce land for housing and congestion externalities. Since these dispersion forces are confined within a given city, and are independent of interregional distance, the local dispersion force pushes out mobile agents from inside the city. As a result, the dispersion in this case takes the form of spatial sprawl of a given city, rather than the formation of another city. The last class of models with both global and local dispersion forces can account for the formation of multiple cities, each with positive spatial extent. 

Figure 4 Multipolar versus monopolar agglomeration

Spatial patterns and city size distribution

To account for the central place pattern, the global dispersion force is essential. In this class of models, the mobile agents completely disperse over all locations for sufficiently high transport costs, and as transport costs decrease, the agglomeration proceeds in the form of spatial-period doubling bifurcation, until all mobile agents concentrate in one city. Hence, the central place pattern is a typical consequence in this class of model.5

Theory to account for hierarchy principle and city size diversity

To account for the large variation in size and industrial composition of cities, in addition to the central place pattern, models need to incorporate diversity in increasing returns. The straightforward extensions of the models with global dispersion force discussed above include Fujita et al. (1999) and Tabuchi and Thisse (2011) in the new economic geography framework, whereas Hsu (2012) proposed an entirely new spatial competition model of firm entry. In these models, the different degrees of increasing returns among industries result in the different spatial frequencies of agglomeration in them.

In these models, the mechanism underlying city size diversity is the spatial coordination of agglomerations among industries through inter-industry demand externalities that arise from shared consumers. Industries subject to larger increasing returns agglomerate in a smaller number of cities that are farther apart. Larger cities are formed at the location in which a larger number of industries co-locate to share consumers. This spatial coordination of industries leads to the positive correlation between the size, spacing, and industrial diversity of a city.

Figure 5 depicts an example of such a coordination of four industries, “red”, “blue”, “green” and “black,” in an economy with 16 locations. Each circle with a given color indicates the location of the industry corresponding to the color. Agglomerations of individual industries are equidistant, where the “red” industry is the most agglomerated, while the “black” one is ubiquitous. In the presence of inter-industry positive externalities, agglomerations of different industries coordinate spatially, and the hierarchy principle holds among cities6.

Figure 5 Spatial coordination of industries and hierarchy principle

Spatial patterns and city size distribution

Conclusion

The location, size, and industrial composition of cities in reality can be characterised in terms of the central place pattern, power law, and hierarchy principle, respectively, as illustrated for the case of Japan. Despite the multiplicity of equilibria, the regularity appears to be a natural consequence of the self-organisation of an economy with endogenous agglomeration and dispersion forces, whereas no evidence thus far is available for concrete exogenous (or historical) factors accounting for the regularity. It follows that a place-based policy targeting an individual region is necessarily constrained by this endogenous regularity, so that the relative growth of an individual region tends to imply a relative decline of another region in the maintenance of the regularity. This column thus urges regional economists and policymakers to see the theories of agglomeration that have been developed in the past quarter of a century in a fresh light. See, for example, the recent development of the framework for systematic numerical analysis based on the numerical and group-theoretic bifurcation theory (e.g. Ikeda et al. 2012, 2017). Their approach enables us to explore the models of endogenous agglomeration with asymmetric geography as well as two-dimensional location space in a many-region setup.

Authors note: This column is based RIETI Discussion Paper 18-E-053.

References

Akamatsu, T, Y Takayama, and K Ikeda (2012), “Spatial discounting, Fourier, and racetrack economy: A recipe for the analysis of spatial agglomeration models,” Journal of Economic Dynamics and Control, 99 (11), 32-52.

Akamatsu, T, T Mori, M Osawa, and Y Takayama (2018), “Spatial scale of agglomerations: Theoretical foundations and empirical implications,” Discussion paper, no. 84145, Munich Personal RePEc Archives.

Allen, T, and C Arkolakis (2014), “Trade and the topography of the spatial economy,” Quarterly Journal of Economics 129 (3): 1085-1140.

Alonso, W (1964), Location and Land Use: Toward a General Theory of Land Rent, Cambridge, MA: Harvard University Press.

Baldwin, R, R Forslid, P Martin, I P Ottaviano, and F Robert-Nicoud (2003), Economic Geography and Public Policy, Princeton, NJ: Princeton University Press.

Beckmann, M J (1976), Mathematical Land Use Theory, Lexington, MA: Lexington Books. 

Bettencourt, L M A, J Lobo, D Helbing, C Kühnert, and G B West (2007), “Growth, innovation, scaling, and the pace of life in cities,” Proceedings of the National Academy of Sciences of the United States of America, 104 (7), 7301-7306.

Christaller, W (1933), Die Zentralen Orte in Süddeutschland, Jena: Gustav Fischer.

Combes, P-P, G Duranton, D Puga, and S Roux (2012), “The productivity advantage of large cities: Distinguishing agglomeration from selection,” Econometrica, 80 (6), 2543-2594.

Davis, D, and J I Dingel (2019), “The comparative advantage of cities,” Unpublished manuscript.

Gabaix, X, and Y M Ioannides (2004), “The evolution of city size distributions,” in J V Henderson and J-F Thisse (eds.), Handbook of Regional and Urban Economics Vol. 4, Elsevier Ch. 53, 2341-2378.

Glaeser, E L, and D C Maré (2001), “Cities and skills,” Journal of Labor Economics, 19 (2), 316-342.

Henderson, J V (1974), “The sizes and types of cities,” American Economic Review, 64 (4), 640-656.

Ikeda, K, K Murota, and T Akamatsu (2012), “Self-organization of Lösch’s hexagons in economic agglomeration for core-periphery models,” International Journal of Bifurcation and Chaos, 22 (8).

Ikeda, K, K Murota, and Y Takayama (2017), “Stable economic agglomeration patterns in two dimensions: Beyond the scope of central place theory,” Journal of Regional Science, 57 (1), 132-172.

Krugman, P (1991), “Increasing returns and economic geography,” Journal of Political Economy ,99 (3), 483-499.

Krugman P (1996), The Self-Organizing Economy, Cambridge, MA: Blackwell.

Lösch, A (1940), Die räumliche Ordnung der Wirtschaft. Eine Untersuchung über Standort, Wirtschaftsgebiete und internationalem Handel, Jena: Gustav Fischer.

Mori, T (2018), “Spatial pattern and city size distribution,” Discussion paper 18-E-053, RIETI.

Mori, T, K Nishikimi, and T E Smith (2008), “The number-average size rule: A new empirical relationship between industrial location and city size,” Journal of Regional Science, 48 (1), 165-211.

Redding S J, and D Sturm (2008), “The cost of remoteness: Evidence from German division and reunification,” American Economic Review, 98 (5), 1766-1797.

Turing, A (1952), “The chemical basis of morphogenesis,” Philosophical Transactions of the Royal Society, 237 (641), 37-72.

von Thünen, J H (1826), Der Isolierte Staat in Beziehung auf Landwirtschaft und Nationalökonomie, Wirtschaft & Finan. 

Endnotes

[1] Throughout this column, a city is defined to be a contiguous set of 1km-by-1km cells with at least 1000 people per km2 and total population of at least 10,000.

[2] Not surprisingly, large countries often have multiple large cities, and one can see that similar power laws hold in a sub-region around each of these cities. Under the same definition of a city, for example, the US has two comparable subsystem of cities around New York and Los Angeles; China around Shanghai and Hong Kong; and India has three around New Delhi, Kolkata and Mumbai.

[3] Among others, see, for example, Glaeser and Maré (2001), Bettencourt et al. (2007), Combes et al. (2012), Davis and Dingel (2019).

[4] Their approach originates from Turing (1952) and its application to regional economics by Krugman (1996, Ch.8), and then its formalization by Akamatsu et al. (2012).

[5] Among the existing models, these include Krugman (1991), Puga (1999), Forslid and Ottaviano (2003) and Pflüger (2004), whereas the models with only local dispersion force include Helpman (1998), Redding and Sturm (2008), Allen and Arkolakis (2014).

[6] In fact, Hsu (2012) has shown the formal condition under which all of the central place pattern, hierarchy principle, power law for city size distribution, and hence, the number-average size rule for industrial location holds.

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