# Spatial Pattern and City Size Distribution

## Summary and Keywords

Many large cities are found at locations with certain geographic and historical advantages, or the first nature advantages. Yet those exogenous locational features may not be the most potent forces governing the spatial pattern and the size variation of cities. In particular, population size, spacing, and industrial composition of cities exhibit simple, persistent, and monotonic relationships that are often approximated by power laws. The extant theories of economic agglomeration explain some aspects of this regularity as a consequence of interactions between endogenous agglomeration and dispersion forces, or the second nature advantages.

To obtain results about explicit spatial patterns of cities, a model needs to depart from the most popular two-region and systems-of-cities frameworks in urban and regional economics in which the variation in interregional distance is assumed away in order to secure analytical tractability of the models. This is one of the major reasons that only few formal models have been proposed in this literature. To draw implications about the spatial patterns and sizes of cities from the extant theories, the behavior of the *many-region extension* of the existing two-region models is discussed in depth.

The mechanisms that link the spatial pattern of cities and the diversity in size as well as the diversity in industrial composition among cities are also discussed in detail, thought the relevant theories are much less available. For each aspect of the interdependence among spatial patterns, size distribution and industrial composition of cities, the concrete facts are drawn from Japanese data to guide the discussion.

Keywords: city size distribution, spatial patterns, agglomeration, racetrack geography, interregional distance, power laws, central place theory

Introduction

In the years since the formal analysis of city formation started, around the time of Alonso (1964),^{1} the spatial pattern of cities has remained a relatively minor subject in urban economics,^{2} despite the fact that economic geographers in the past (e.g., Christaller, 1933; Lösch, 1940; von Thünen, 1826) have commonly suggested that there is an inseparable correspondence between (population) size and spatial distribution of cities (see, e.g., Fujita, 2010).^{3}

The mainstream theories in urban economics have abstracted from the heterogeneity in intercity/regional space by adopting the *systems-of-cities model*, following on the pioneering work by J. Vernon Henderson (Henderson, 1974),^{4} or by simply assuming the presence of only two regions in an economy (see a collection of two-region models presented in Baldwin, Forslid, Martin, Ottaviano, & Robert-Nicoud, 2003). The mechanism that determines the size of a city/region has always been a major subject in most of these theories, and for this purpose, the abstraction from interregional space in these approaches substantially simplifies the analysis.

As a consequence of this particular evolution of the field, there exists a rather limited theoretical as well as empirical literature that relates to the spatial pattern and size of cities. To my knowledge, there are two major strands of formal models that explicitly deal with the spatial pattern of cities: *new economic geography* (*NEG*) and *social-interactions models*. The former explains city formation by the externalities that arise from monopolistically competitive markets (see, e.g., Baldwin et al., 2003; Fujita, Krugman, & Venables, 1999a, for surveys), whereas the latter does so by the externalities that arise from direct interactions among agents outside the markets (see, e.g., Beckmann, 1976; Fujita & Ogawa, 1982; Fujita & Smith, 1990; Mossay & Picard, 2011). This article focuses on the basic structure and implications of these theoretical models in connection with the observed sizes and spatial patterns of cities.

I start by making observations on the relationships between size, spatial pattern, and industrial structure of cities in reality by using data from Japan. Then, generic properties of the canonical models (to be made in the section "Theories") of the extant theories are discussed. In particular, while most models are investigated in the context of the two-region set-up in their original studies, in this article, by extensively drawing from the work of Akamatsu, Mori, Osawa, and Takayama (2019), I summarize their behavior in a many-region set-up in which the spatial pattern of cities can be more properly studied. Finally, the article concludes with some remarks.

Facts about Size, Location, and Industrial Composition of Cities

In order to summarize and classify the extant theoretical models for the size and spatial patterns of cities, it is useful to have a concrete idea about the basic relationships observed between them in reality. Given that the intercity space has been largely abstracted in the literature, however, systematic research on this subject is scarce, and the results published so far provide little decisive evidence (e.g., Dobkins & Ioannides, 2001; Ioannides & Overman, 2004; Overman & Ioannides, 2001). To demonstrate the strong correspondence between theories and facts, rather than trying to put together subtle pieces of evidence from the existing empirical literature, I attempt to establish a set of clear-cut facts using data from Japan.

There are two major reasons to focus on Japan as a real-world example. One is the availability of the micro data for industrial locations. The other is the fact that the highway and high-speed railway networks in Japan were developed simultaneously, almost from scratch, to become full-fledged nation-wide networks between 1970 and 2015. The changes in size and spatial patterns of Japanese cities in this period provide a useful test case to verify the implications from the theoretical models of endogenous city formation. By utilizing these data, the key facts on all the aspects regarding size and spatial patterns as well as industrial structure of cities can be obtained for the same set of cities.

Throughout this section, a *city* is defined to be a contiguous set of (approximately) 1 square kilometer cells with at least 1,000 people per km^{2} and a total population of at least 10,000.^{5} The advantage of this simple definition of a city is that the basic regional units (1 km^{2} cells) are consistent in the cross sections of a given country, and across different points in time, unlike more commonly used definitions of metropolitan areas based on administrative regions. Under this definition of a city, the set of all cities in a country account for the population (area) share in the country of 43.6% (2.4%), 44.6% (1.6%), 77.1% (12.4%), 48.7% (2.9%), and 47.0% (3.8%) for the United States, Europe, Japan, China, and India, respectively.^{6}

It is to be noted that the evidence for Japanese cities presented here is not specific to Japan. For size and spatial patterns of cities, as discussed in the sections “Size and Spacing of Cities” and “Size Distribution of Cities,” Mori, Smith, and Hsu (2019) have shown that qualitatively the same (and more extensive) results hold for Japan and five other countries—China, France, Germany, India, and the United States—under the same definition of a city. For the size and industrial structure of cities, as discussed in the section “Size and Industrial Structure of Cities,” the qualitatively similar results have also been presented for the case of the United States (see Hsu, 2012; Schiff, 2014) under the standard metro areas and industrial classifications.

## Size and Spacing of Cities

Many large cities are found at locations with certain first nature advantages.^{7} Yet those exogenous locational features may not be the most potent forces governing the spatial pattern of cities. In particular, population size, spacing, and industrial composition of cities exhibit a simple, persistent, and monotonic relationship that has long been recognized by economic geographers, including von Thünen (1826), Christaller (1933), and Lösch (1940). They (especially Christaller) proposed *a central place pattern* in the relation between the size and location of cities such that *larger cities tend to serve as centers around which smaller cities are grouped*. Moreover, this relation is recursive so that some of the small cities serve as centers around which even smaller cities are grouped. This central place pattern of cities naturally implies that *larger cities are spaced further apart*.

To see this in the actual data, let $\mathcal{U}$ be the set of all 450 cities identified in Japan in 2015, *s _{i}* be the share of city $i\in \mathcal{U}$ in the national population, and $||i,j||$ for $i,\phantom{\rule{0.2em}{0ex}}j\in \mathcal{U}$ be the road distance between cities

*i*and

*j*.

^{8}Define the

*spacing*of city

*i*by the distance to the closest city of the same or a larger size class:

^{9}

Figure 1(a) shows the relationship between *d _{i}* and

*s*in log scale for each city $i\in \mathcal{U}$. The correlation between them is as high as 0.67.

_{i}^{10}This confirms the spacing-out property of cities mentioned previously.

If the number, *n _{i}*, of cities within the distance ${d}_{i}$ from city $i\in \mathcal{U}$ is counted by

as shown in Figure 1(b), it also has a strong correlation, 0.86, with the city size, *s _{i}*, in log scale. Thus it is clear that larger cities are surrounded by smaller ones.

Mori et al. (2019) conduct a formal test for this central place pattern of cities, that is, if the largest cities are spaced out relative to the whole set of cities in a country. Specifically, they first fix the number *L* of the largest cities in a given country, and form a Voronoi partition with respect to a set of a given number *K* (**≥** 2) of randomly selected cities. The test statistic is the count of partition cells containing at least one of these *L* largest cities. If there is substantial spacing between the largest cities in reality, then this count is expected to be larger for Voronoi partitions than for fully random partitions (i.e., without any regard to spatial relations among cities) of the same cell sizes. For a range of (*L*, *K*) values, they found strong evidence for the spacing-out of large cities in all the six countries considered (China, France, Germany, India, Japan, and the United States).^{11}

## Size Distribution of Cities

It is well known that city size distribution in a well-urbanized country exhibits an approximate power law (e.g., Batty, 2006; Bettencourt, 2013; Gabaix & Ioannides, 2004).^{12} Formally, if a given set of *n* cities is postulated to satisfy a power law, and if these city sizes are ranked as ${s}_{1}\ge {s}_{2}\ge \xb7\xb7\xb7\ge {s}_{n}$, so that the rank, ${r}_{i}$, of city $i$ is given by ${r}_{i}=\phantom{\rule{0.2em}{0ex}}i$, then for

for $b=\phantom{\rule{0.2em}{0ex}}\text{ln}\left(cn\right)/\alpha $.

In Figure 2, Panel (a) shows the rank-size distributions of cities every five years from 1970 to 2015, where *s _{i}* indicates the share of city

*i*in the national population; Panel (b) shows the change in

*α*, that is, the Zipf’s coefficient, 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 total city population, where Tokyo alone accounts for 26% in 2015.

There is a strand of literature that informally argues that *Zipf’s law* (after Zipf, 1949) holds, that is, the power law with *α* = 1 holds for city size distribution in a country (see, e.g., Gabaix & Ioannides, 2004; Ioannides, 2012, §8.2). But there is abundant evidence against it (e.g., Black & Henderson, 2003; Mori et al., 2019; Nitsch, 2005; Soo, 2005), as is also clear in the Japanese case shown in Figure 2.

While the power laws for city size distributions are best known at the country level (see, e.g., Gabaix & Ioannides, 2004), Mori et al. (2019) have shown that similar power laws appear recursively in spatial hierarchies of regions within a country that reflect the central place patterns, as discussed in the section “Size and Spacing of Cities.” Specifically, they construct a spatial hierarchy in a country by first constructing a Voronoi partition of the set of all cities in the country using a given number of their largest cities as cell centers, and then continuing this partitioning procedure within each cell recursively. They found that city size distributions in different parts of these spatial hierarchies exhibit power laws that are significantly similar than would be expected by chance alone. Thus, their result suggests that city systems have a spatial fractal structure within countries.

## Size and Industrial Structure of Cities

Many studies (e.g. Baum-Snow & Paven, 2013; Bettencourt, Lobo, Helbing, Kühnert, & West, 2007; Combes, Duranton, & Gobillon, 2008; Combes, Duranton, Gobillon, Puga, & Roux, 2012; Davis & Dingel, 2019; Glaeser & Maré, 2001; Glaeser & Resseger, 2010) have indicated strong correlations between socioeconomic quantities and sizes of cities (e.g., wages, education level, gross domestic product, industrial diversity, number of patent applications, amount of crime, level of traffic congestion). This section presents one of the clearest representations of such correlations by focusing on industrial location.

Let $\mathcal{I}$ be the set of all industries that operate in at least one of the cities, and for a given industry $i\in \mathcal{I}$, call a city a *choice city* of this industry if industry *i* is in operation in the city. These choice cities exhibit a systematic variation in their average population size across industries. To see this, denote by ${\mathcal{U}}_{i}(\subseteq \mathcal{U})$ the set of all choice cities of industry $i\in \mathcal{I}$, then the average size of choice cities for industry *i* is given by

where $\#{\mathcal{U}}_{i}$ means the cardinality of set ${\mathcal{U}}_{i}$.

Now, consider three-digit secondary and tertiary industries of the Japanese Standard Industrial Classification (JSIC) that are present both in 2000 and 2015. Of all the 237 such industries, there are 162 and 175 industries that have at least one establishment in cities in 2000 and 2015, respectively.^{13} Figure 3 shows the relationship between ${\overline{s}}_{i}$ and *N _{i}* for $i\in \mathcal{I}$ in log scale, where ${N}_{i}=\#{\mathcal{U}}_{i}$. The dashed curves indicate the upper and lower bound for the average size of choice cities in 2015, where for each $i\in \mathcal{I}$, the former (latter) is the average size of the largest (smallest)

*N*cities.

_{i}There are two key features in these plots. First, the number *N _{i}* and average size ${\overline{s}}_{i}$ of choice cities exhibit a strong

*power law*, which is persistent between 2000 and 2015. Second, the average sizes of choice cities are almost hitting their upper bound, meaning that the choice cities of an industry $i\in \mathcal{I}$ are roughly the largest

*N*cities, which in turn implies that there is roughly a

_{i}*hierarchical relationship*in the industrial composition between larger and smaller cities.

^{14}

To see this, let ${\mathcal{I}}_{i}$ represent the set of industries that are present in city $i\in \mathcal{U}$, and for cities *i* and $j\in \mathcal{U}$ such that *s _{i} > s_{j}*, define the

*hierarchy share*for city

*j*with

*i*by

where a larger value of ${H}_{ij}$ indicates a higher consistency with the hierarchical relationship, and ${H}_{ij}=1$ means the perfect hierarchical relationship, that is, ${\mathcal{U}}_{j}\subseteq {\mathcal{U}}_{i}$. The average values of the hierarchy shares for all the relevant city pairs,

where $\overline{H}\equiv \#\{(i,j):i,j\in \mathcal{U},{s}_{i}>{s}_{j}\}$, can be taken as an aggregate measure of spatial coordination among industries. A larger value of *H* indicates a higher degree of spatial coordination, and the coordination is perfect if *H* = 1. The actual values of *H* are 0.76 and 0.80 in 2000 and 2015, respectively, which are quite high.^{15}

Together with the central place pattern discussed previously (see Figure 1), the fact that the spatial coordination of diverse economic activities leads to the diversity in city size has already been suggested informally by Christaller (1933) and Lösch (1940).

A large value of *H*, as in the case of Japan, means not only that industries have different numbers of agglomerations (i.e., choice cities), but also that their locations tend to coincide, that is, a more localized industry chooses to locate in cities in which more ubiquitous industries are present. The case of perfect coordination (i.e., $H=\phantom{\rule{0.2em}{0ex}}1$) corresponds to the *hierarchy principle* in Christaller (1933).

To close this subsection, it is worth pointing out that while there is a strong tendency toward a hierarchical relationship in the industrial composition between larger and smaller cities, it is by no means the rule. Figure 4 shows the distribution of ${H}_{ij}$ of all the relevant city pairs in 2015. While the mean value is *H* = 0.80, the standard deviation is 0.13, and the range is from 0.18 to 1. Low hierarchy shares are realized for *specialized cities* in which only a small, specific set of industries is concentrated. As will be discussed in the section “Diversity in City Size,” the standard systems-of-cities models (e.g., Henderson, 1974; Rossi-Hansberg & Wright, 2007) associate the size of a city with that in scale economies specific to the industries in which the city is specialized, thereby explaining the diversity in city size in terms of the variation in industry-specific scale economies.

## Growth of City Sizes

Finally, we look at the characteristics of the growth of individual city sizes in Japan between 1970 and 2015. It is of particular interest to quantify the evolution of city sizes in this period, since it coincides with the period in which the highway and high-speed railway networks were developed almost from scratch to the extent that they cover almost the entire nation, where the total highway (high-speed railway) length increased from 879 km (515 km) by more than 16 (10) times to 14,146 km (5,350 km).

The level of interregional transport access has been one of the key parameters to determine the size and spatial patterns of cities in the literature. The evolution of the sizes of individual cities is expected to reflect the response to the improved interregional transport access, although the benefit for each city may vary depending on their relative location. Thus, the changes in size and spatial patterns experienced by Japanese cities in this period provides an ideal test case for the theoretical models of endogenous agglomeration.

There was substantial movement of population among cities in these 45 years. In particular, there is a clear tendency of *global agglomeration* toward a smaller number of cities, as the number of cities has decreased from 503 to 450.^{16}

Figure 5 reveals key facts about the change in individual city sizes for the 302 cities that have remained throughout the entire period. Panel (a) provides additional evidence for global agglomeration: the size of the remaining cities in terms of population share (in the country) has grown by 21% on average.^{17} Note that it is more meaningful to look at the population share of a city rather than the population size itself to understand the tendency of global agglomeration, because the population share removes the effects of general population growth and/or urbanization from the population sizes.^{18}

Despite the tendency of global agglomeration, there is also a clear tendency of *local dispersion* as the areal size of an individual city has almost doubled (Panel b), while the population density has decreased by 22% on average (Panel c).^{19}

This simultaneous occurrence of global agglomeration and local dispersion, given an improvement in interregional access, may seem paradoxical. But it can be explained by integrating the extant theories of endogenous agglomeration discussed in the next section, “Theories.”

Theories

A model capable of explaining the spatial patterns of cities necessarily involves many regions with large variations in interregional distance, such that some cities are close to while others are far from one another. But the majority of the extant models adopt either two-region or systems-of-cities set-ups in which there is no variation in interregional distance.^{20} Thus, no explicit spatial patterns reflecting the relation among the number, size, and spacing of cities can be expressed by these models.

A recent work by Akamatsu et al. (2019) represented a breakthrough by showing that a wide variety of the extant models of endogenous agglomeration can be reformulated in a many-region set-up and formally analyzed in a unified framework. Specifically, they focus on a canonical model, that is, a static model with (a) a continuum of homogeneous mobile agents, each of whom chooses a single location; (b) there is a single type of interregional transport cost; and (c) transport costs are subject to the iceberg technology. The reformulated models are shown to boil down to one of the three reduced forms in terms of the spatial pattern of agglomeration and dispersion.

The canonical model covers a wide range of standard models in urban and regional economics. It includes the class of NEG models based on the Dixit–Stiglitz type constant-elasticity-of-substitution (CES) sub-utility function for love of variety (e.g., Forslid & Ottaviano, 2003; Helpman, 1998; Krugman, 1991; Murata & Thisse, 2005; Pflüger, 2004; Pflüger & Südekum, 2008; Puga, 1999; Redding & Stürm, 2008; Tabuchi, 1998); the social-interactions model of city-center formation based on technological externalities (e.g., Beckmann, 1976; Blanchet, Mossay, & Santambrogio, 2016; Mossay & Picard, 2011); and the economic geography models in the “universal gravity” framework by Allen, Arkolakis, and Takahashi (Forthcoming), including the Armington (1969) model with labor mobility by Allen and Arkolakis (2014), a standard formulation in the recent quantitative spatial economics (see, e.g., Redding & Rossi-Hansberg, 2017, for a survey).

Important classes of models that are outside their scope include city-center formation models in which firms and households have different location incentives (violation of (a)) (e.g., Fujita & Ogawa, 1982; Lucas & Rossi-Hansberg, 2002; Ahlfeldt, Redding, Sturm, & Wolf, 2015; Monte, Redding, & Rossi-Hansberg, 2018; Osawa and Akamatsu, 2019); those with multiple industries with industry-specific transport costs (violation of (b)) (e.g., Fujita & Krugman, 1995; Fujita, Krugman, & Mori, 1999b; Tabuchi & Thisse, 2011); and NEG models with additive transport costs (violation of (c)) by Ottaviano, Tabuchi, and Thisse (2002).

Drawing largely from Akamatsu et al. (2019), the section “Spatial Pattern of Cities” reviews the mechanism underlying the relation between population/areal size and spacing of cities in reality, as discussed in the sections “Size and Spacing of Cities” and “Growth of City Sizes.” To explain the observed diversity in the size and industrial structure of cities discussed in the sections “Size Distribution of Cities” and “Size and Industrial Structure of Cities,” respectively, and their relation to the spatial pattern of cities, a model needs to go beyond the canonical model considered by Akamatsu et al. (2019), and incorporate variations in the degree of increasing returns (and/or those in transport costs). At present, there are only a handful of models that have succeeded in such extensions. The section “Diversity in City Size” reviews the theoretical developments in this direction.

## Spatial Pattern of Cities

By formalizing and generalizing the idea proposed by Krugman (1996, ch. 8) based on Turing (1952), Akamatsu, Takayama, and Ikeda (2012) proposed an analytical framework for many-region models of endogenous agglomeration under the symmetric racetrack geography with the help of discrete Fourier transformation. While Akamatsu et al. (2012) has focused on a many-region extension of the model by Pflüger (2004), Akamatsu et al. (2019) have generalized their framework, and have shown that a wide variety of the extant models can be classified by the three distinct reduced forms, despite the difference in their specific mechanisms underlying agglomeration and dispersion. I start by describing the basic set-up of this approach.

### Basic Set-Up

Consider the location space consisting of a set of $K$ discrete regions, $\mathcal{K}\equiv \{0,\text{1},\mathrm{...},K-\text{1}\}$. There is a continuum of homogeneous mobile agents whose regional distribution is denoted by $\mathit{h}={\left({h}_{i}\right)}_{i\in \mathcal{K}}$, where ${h}_{i}$ is the mass of mobile agents located in region *i*. Their total mass is a given constant, $H\equiv {\displaystyle {\sum}_{i\in \mathcal{K}}{h}_{i}}$. All regions in $\mathcal{K}$ are featureless and are placed at an equal interval on a circle. In this racetrack economy, transportation is possible only along the circumference.^{21}

Let region index $0,1,\dots ,K\u20131$ represent the location on the racetrack in clockwise direction. Transport costs take iceberg form, that is, if a unit of product is shipped from region *i* to *j*, then only the fraction ${d}_{ij}={d}_{ji}\in \left[0,\text{1}\right)$ reaches *j*. The *spatial discounting matrix*,$\mathit{D}=\left[{d}_{ij}\right]$, expresses the underlying distance structure of the economy. Typically, iceberg costs are expressed as ${d}_{ij}=\text{exp}[-\tau {\mathcal{l}}_{ij}]$, where ${\mathcal{l}}_{ij}$ is the distance between regions *i* and *j* and $\tau \in (0,\infty )$ is the transport technology parameter.

The relocation of agents is assumed to be much slower than market reactions, so that the short-run equilibrium conditions (such as market clearing and trade balance) determine the pay-off (utility or profit) in each region as a function of a given regional distribution of agents, $\mathit{h}$. Specifically, given $\mathit{h}$, their *short-run* pay-off of choosing each region is determined, where the short-run pay-off function is denoted by $v\left(\mathit{h}\right)\equiv {\left({v}_{i}\left(\mathit{h}\right)\right)}_{i\in \mathcal{K}}$, with *v _{i}*(

**) representing the pay-off for an agent located in region $i\in \mathcal{K}$.**

*h*In the *long run*, agents are mobile and are free to choose their locations to improve their own pay-offs. In (*long-run*) *equilibrium*, it must hold that ${v}^{*}=\phantom{\rule{0.2em}{0ex}}{v}_{i}\left(\mathit{h}\right)$ for all regions $i$ with ${h}_{i}>0$, and ${v}^{*}\ge {v}_{i}\left(\mathit{h}\right)$ for any region $i$ with ${h}_{i}=0$, where ${v}^{*}$ is the equilibrium pay-off level.

A change in endogenous agglomeration pattern is treated as an instance of bifurcation of an equilibrium. To address the stability of equilibria, a standard approach in the literature is to introduce equilibrium refinement based on *local stability* under myopic evolutionary dynamics, where the rate of change in the number of residents ${h}_{i}$ in region $i$ is modeled on the basis of the regional distribution of agents, $\mathit{h}$, and that of pay-off, $v\left(\mathit{h}\right)$. Let a deterministic dynamic be denoted by $\dot{\mathit{h}}=\mathit{F}\left(\mathit{h},\mathit{v}\left(\mathit{h}\right)\right)$, where $\dot{\mathit{h}}$ represents the time derivative of $\mathit{h}$, and assume that (a) $F$ satisfies differentiability with respect to both arguments, (b) agents relocate in the direction of an increased aggregate pay-off under $F$, (c) the total mass of agents is preserved under $F$, and (d) any spatial equilibrium is a rest point of the dynamic, that is, if ${\mathit{h}}^{*}$ is an equilibrium, it must hold that $\dot{\mathit{h}}=\mathit{F}\left({\mathit{h}}^{*},\phantom{\rule{0.2em}{0ex}}\mathit{v}\left({\mathit{h}}^{*}\right)\right)=\mathbf{0}$. The stability of an equilibrium then is defined in terms of asymptotic stability under $\mathit{F}$.

### Formation of a City

With a racetrack geography, the uniform distribution of mobile agents is always an equilibrium when the pay-off function is symmetric across regions. Call an equilibrium with uniform distribution a *flat-earth equilibrium*, and denote it by $\overline{\mathit{h}}=\left(h,\phantom{\rule{0.2em}{0ex}}h,\mathrm{...},\phantom{\rule{0.2em}{0ex}}h\right)$ with $h\equiv H/K$.

If the adjustment dynamic is formulated so that the agents migrate in order to maximize their pay-off, it follows (Akamatsu et al., 2019, Appendix B) that each eigenvalue of Jacobian matrix $\mathit{J}$ of $F$ and that of the Jacobian matrix $\nabla v$ of ** v** are real, and have a perfect positive correlation at the flat-earth equilibrium. Thus, one can focus on $\nabla v$ instead of $\mathit{J}$ to investigate the stability of the flat-earth equilibrium. What remains is to identify the direction of the bifurcation at the flat-earth equilibrium, which is equivalent to finding the eigenvector of $\nabla v(\overline{\mathit{h}})$ whose eigenvalue changes its sign from negative to positive first among all the eigenvectors of $\nabla v(\overline{\mathit{h}})$.

The sign of the *k*-th eigenvalue of $\nabla v(\overline{h})$ has been shown to coincide with the sign of the model-specific function of the form:

where ${c}_{0},{c}_{1}$_{,} and ${c}_{2}$ are the constants specific to a given model, and ${f}_{k}$ is the *k*-th eigenvalue of the spatial discounting matrix ** D** which is known to be real, and common to all models. The eigenvector associated with ${f}_{k}$ is given by ${\mathit{\eta}}_{k}=\left({\eta}_{k,i}\right)=(\mathrm{cos}[\theta ki])$ for $i\in \mathcal{K}$ with $\theta \equiv \text{2}\pi /K$, and the bifurcation from the flat-earth equilibrium takes place in the direction given by $\mathit{h}=\overline{\mathit{h}}+\u03f5{\eta}_{k}$ with $\u03f5>0$.

The value *k* coincides with the number of equidistant regions toward which mobile agents migrate the most. For example, at $k=K/2$, the value ${\eta}_{K}{}_{/2,i}$ of each element $i\in \mathcal{K}$ in eigenvector, ${\mathit{\eta}}_{K/2}$, is given as depicted for the case of $K=\phantom{\rule{0.2em}{0ex}}16$ in Figure 6(a), so that agglomerations start to form at alternate regions, $0,2,4,\mathrm{...},K-2\left(=\phantom{\rule{0.2em}{0ex}}14\right)$.^{22} At $k=\phantom{\rule{0.2em}{0ex}}1$, as depicted in Figure 6(b), a unimodal agglomeration will form around region 0.^{23}

There are two key properties of ${{f}^{\prime}}_{k}s$ that are useful to investigate the stability of flat-earth equilibrium:

1. ${f}_{k}$ is monotonically increasing in transport cost, $\tau $.

2. ${f}_{1}=\phantom{\rule{0.2em}{0ex}}ma{x}_{k}{}_{=1,2,\mathrm{...},K}{f}_{k}$ and ${f}_{K}{}_{/2}={\text{min}}_{k}{}_{=1,2,\mathrm{...},K}{f}_{k}>0$.

^{24}

Canonical models typically have a positive value of ${c}_{1}$. Since ${f}_{1}>\phantom{\rule{0.2em}{0ex}}0$, it means that the second term on the right-hand side (RHS) in (7) represents the agglomeration force, as it works to destabilize the flat-earth equilibrium. In these models, if a stable flat-earth equilibrium exists, then one must have either *c*_{0} < 0 or *c*_{2} < 0, or both, so that all the eigenvalues of $\nabla v(\overline{\mathit{h}})$ can be negative at the flat-earth equilibrium. In particular, since ${f}_{k}$ is positive and increasing in $\tau $ for each $k=\phantom{\rule{0.2em}{0ex}}1,2,\mathrm{...},K\u20131$, the flat-earth equilibrium is stable for sufficiently small transport costs if ${c}_{0}<\phantom{\rule{0.2em}{0ex}}0$, and for sufficiently large transport costs if ${c}_{2}<\phantom{\rule{0.2em}{0ex}}0$.

The bifurcation from the flat-earth equilibrium leading to the city formation under ${c}_{0}<\phantom{\rule{0.2em}{0ex}}0$ and that under ${c}_{2}<\phantom{\rule{0.2em}{0ex}}0$ are, however, qualitatively different in two aspects. The first aspect is the timing at which the bifurcation takes place. The bifurcation under ${f}_{k}$ takes place in the increasing phase of transport costs, whereas that under ${c}_{2}<\phantom{\rule{0.2em}{0ex}}0$ occurs in their decreasing phase.

The second aspect is the spatial scale of agglomeration and dispersion. Provided that *c*_{2} < 0, the bifurcation takes place in the direction of ${\eta}_{K}{}_{/2}$, that is, every other region along the racetrack attracts in-migration of mobile agents, when $G({f}_{K}{}_{/2})$ becomes positive (refer to Figure 6(a)). The regional distribution of mobile agents that arises in this bifurcation is $\overline{\mathit{h}}+\u03f5{\mathit{\eta}}_{K/2}$ (for$\u03f5>0$) as illustrated in Figure 7(a). In other words, small cities (i.e., agglomerations) form locally, while they are dispersed globally all over the location space.

Provided that ${c}_{0}<\phantom{\rule{0.2em}{0ex}}0$, the bifurcation takes place in the direction of ${\mathit{\eta}}_{1}$ when $G\left({f}_{1}\right)$ turns to positive (refer to Figure 6(b)). The regional distribution of mobile agents that arises in this case is given by $\overline{\mathit{h}}+\u03f5{\mathit{\eta}}_{1}$ as illustrated in Figure 7(b). In other words, the agglomeration takes place globally, and forms a single gigantic city, while the dispersion takes place locally around the city center (region 0) so that the city stretches over the entire location space.

A crucial difference between the two cases is the dependence of dispersion force on the distance structure of the model. The third term ${c}_{\text{2}}{f}_{k}^{2}$ on the RHS of (7) *depends* on the distance structure of the economy (through ${f}_{k}$). As discussed, this force leads to *global dispersion* (with local agglomeration) as in Figure 7(a). On the contrary, the first term on the RHS of (7) is the dispersion force when ${c}_{0}<\phantom{\rule{0.2em}{0ex}}0$ which is *independent* of the distance structure of the economy. As discussed above, this force leads to *local dispersion* (with global agglomeration) as in Figure 7(b).

In Akamatsu et al. (2019), the models with only global dispersion force, that is, ${c}_{0}\ge 0$ and ${c}_{2}<\phantom{\rule{0.2em}{0ex}}0$, are called Class (i). The models of this class are shown to exhibit *period doubling bifurcations* as transport costs decrease, leading to *a smaller number of larger cities with a larger spacing between neighboring cities, until all mobile agents concentrate in one region* (Figure 8a).^{25} The models with only local dispersion force, that is, ${c}_{0}<\phantom{\rule{0.2em}{0ex}}0$ and ${c}_{2}\ge 0$, are called Class (ii). The Class (ii) models involve at most one bifurcation when the flat-earth equilibrium loses stability. In the models of this class, *keeping unimodal regional distribution, the concentration of mobile agents proceeds as transport costs increases, until all mobile agents concentrate in one region* (Figure 8b). The models that incorporate both types of dispersion force, that is, ${c}_{0}<\phantom{\rule{0.2em}{0ex}}0$ and ${c}_{2}<\phantom{\rule{0.2em}{0ex}}0$, may be the most realistic, and account for the formation of multiple cities with a positive internal space (Figure 8c). These are called Class (iii).

Two implications are worth mentioning. First, the heterogeneity among interregional distances is an essential feature of a model to investigate the spatial pattern of cities. In the context of a two-region model or a systems-of-cities model in which there is no variation in interregional distance, the dispersion of mobile agents in Class (i) and Class (ii) models look exactly the same. But, as indicated by the middle panels of Figure 8(a)(b), these are qualitatively different in spatial scale. The dispersion takes place at the global scale in Class (i) models, in the form of an increase in the number of cities, and at the local scale in Class (ii) models, in the form of a larger spatial extent of a city.

Second, the responses of agglomeration/dispersion to the level of transport costs are *opposite* between global and local spatial scales. More specifically, given the lower interregional transport costs, the agglomeration proceeds at global scale, that is, the number of cities decreases, the sizes and the spacing of the remaining cities increase, while the dispersion proceeds at local scale, that is, the average population density within a city decreases and the spatial extent of a city increases.^{26}

Notice that the behavior of Class (i) models essentially accounts for the larger cities being spaced further apart as discussed in the section “Size and Spacing of Cities,” and the behavior of Class (iii) models, that is, the combination of Classes (i) and (ii), can account for the evolution of city growth of Japan as discussed in the section “Growth of City Sizes.”

I will now review a variety of extant models that fall into one of these three classes, as well as those that do not.

### New Economic Geography

NEG (e.g., Fujita et al., 1999a) commonly utilizes the monopolistic competition together with scale economies in production to explain the endogenous agglomeration. On the one hand, the love of product variety by consumers and the presence of transport costs give an incentive for consumers to locate closer to firms. On the other hand, each indivisible firm subject to scale economies at the plant level has an incentive to locate and supply near the concentration of consumers.^{27}

In this context, the global dispersion force associated with ${c}_{2}<\phantom{\rule{0.2em}{0ex}}0$ in (7) is introduced typically by assuming immobile consumers in each region who generate dispersed demand for the differentiated products (e.g., Forslid & Ottaviano, 2003; Krugman, 1991, 1993; Pflüger, 2004). The assumption of immobility of consumers is nothing but simplification to assure the dispersed demand. It can be obtained endogenously, for example, by introducing land-intensive sectors that also require labor inputs (e.g., Fujita & Krugman, 1995; Puga, 1999), which in turn generates dispersed demand from workers. With transport costs, the proximity to demand matters, and hence, the spatial dispersion of consumers results in the formation of multiple cities, where the firms in each city mainly serve their nearby local market.

The local dispersion force associated with ${c}_{0}<\phantom{\rule{0.2em}{0ex}}0$ in (7) is introduced by assuming consumption of locally scarce land (e.g., Helpman, 1998; Redding & Rossi-Hansberg, 2017; Redding & Sturm, 2008), sometimes together with commuting costs (e.g., Murata & Thisse, 2005).^{28} All these costs of concentration are confined within a given region, and thus are independent of interregional distance. The dispersion in this case takes the form of overflow of mobile agents from a given city to the nearby regions, rather than the formation of new distinct cities at distant regions.

There are models that incorporate both global and local dispersion forces (Pflüger & Südekum, 2008; Tabuchi, 1998), that is, of Class (iii) with ${c}_{0}<0$ and ${c}_{2}<0$ in (7). While these themselves treat only the two-region case, their many-region extensions can generate a more realistic spatial pattern of cities that involves both global and local dispersion, as shown in Figure 8(c) (see Akamatsu et al., 2019, §5).

### Social-Interactions Model

In the 1970s and 1980s, several attempts were made to explain the endogenous formation of the central business district (CBD) *within a city*. The development of the models of this type was initiated by Solow and Vickrey (1971) and Beckmann (1976), then followed by several others (e.g., Borukhov & Hochman, 1977; Fujita, 1980, 1988, 1990; Ogawa & Fujita, 1980; Fujita & Ogawa, 1982; Imai, 1982; Kanemoto, 1990; Tauchen & Witte, 1983; O’Hara, 1977; Tabuchi, 1986).

In these models, the formation of CBD is explained by introducing positive technological externalities generated from the interaction between each pair of individual agents. While the previously mentioned models vary in the specification of positive externalities, Fujita and Smith (1990) have shown that their formulations are essentially equivalent, and reformulated commonly by the so-called *additive interaction function*, ${S}_{i}\left(\mathit{h}\right)\equiv {\displaystyle {\sum}_{j\in K}{d}_{ij}{h}_{j}}$.

In the simplest specifications (as in, e.g., Beckmann, 1976), this additive interaction function enters the utility function of consumers directly. This model assumes land consumption by mobile agents, while the production sector is abstracted, that is, they incorporate only local dispersion force, and hence belong to Class (ii). The model by Takayama and Akamatsu (2011), includes global dispersion force by introducing mobile firms and immobile consumers in each region. This model thus contains both local and global dispersion force, that is, of Class (iii).^{29}

### Other Relevant Models

In the NEG literature, a particularly important deviation from the canonical models is to consider different transport cost structures by industry. For example, Fujita and Krugman (1995) included transport costs for (urban) differentiated products as well as land-intensive (rural) homogeneous products. In the presence of rural goods that are costly to transport, the delivered price for such goods is lower in regions farther away from cities, which generates a dispersion force. This is similar to the local dispersion force in that even a small deviation from an urban agglomeration will decrease the price of rural goods and increase the pay-off of the deviant. However, the advantage of dispersion persists outside the agglomeration, that is, it depends on the distance structure of the model. This type of dispersion force has been shown to result in the formation of an *industrial belt*, a continuum of agglomeration associated with multiple atoms of agglomeration as demonstrated by the simulations in Mori (1997) and Ikeda, Murota, Akamatsu, and Takayama (2017). The formal characterization of industrial belts, however, remains to be carried out.

Among the extant social-interactions models, some distinguish location incentives between firms and consumers/workers, unlike the canonical models discussed previously (e.g., Fujita & Ogawa, 1982; Lucas & Rossi- Hansberg, 2002; Ogawa & Fujita, 1980; Ota & Fujita, 1993). This distinction is especially crucial for explaining the location patterns within a city, while it may be less relevant for the purpose of explaining the spatial pattern of cities. At present, few formal results have been obtained regarding the spatial pattern of cities that arise in these models. One remarkable exception is the recent study by Osawa and Akamatsu (2019) which has fully characterized the stable equilibria in a simplified version of the model by Fujita & Ogawa (1982). Unlike the social interactions models with a single type of mobile agents discussed previously, this model allows multiple agglomerations to form when either or both of commuting costs and interaction costs (${d}_{ij}$) is high. As their model is highly stylized, its generality and relation to the results under the single type of mobile agents considered by Akamatsu et al., (2019) is yet to be studied.

Other relevant models that were not covered so far include the *spatial oligopoly* models designed to explain the agglomeration of retail stores (e.g., Dudey, 1990; Konishi, 2005; Wolinsky, 1983). In these models, consumers have imperfect information on the types and prices of goods sold by stores before they visit them. The greater the agglomeration of stores, the more likely it is that consumers will find their favorite commodities. The concentration of stores is explained by the market-size effect due to taste uncertainty and/or lower price expectation. The dispersion force is a global one given by the exogenous and spatially dispersed demand. Thus, these models are expected to behave similarly to Class (i) models, although no extensive analyses have been conducted in this direction (see Konishi (2005, §5) for a discussion on the spacing of retail clusters).^{30}

## Diversity in City Size

The most popular theoretical explanation of power law for city size distribution at this point may be the *random growth theory* (e.g., Duranton, 2006, 2007; Gabaix, 1999; Ioannides, 2012, §8.2; Rossi-Hansberg and Wright, 2007), which postulates that the growth rates of individual cities follow Gibrat’s law (Gibrat, 1931), that is, they are independently and identically distributed random variables.

This theory is highly compatible with systems-of-cities models. For example, in the model by Rossi-Hansberg and Wright (2007), individual industries are subject to city-level positive externality from agglomeration, but do not benefit from collocation with other industries, so that the externality is industry-specific. Then, each city would specialize in a single industry in the presence of urban costs due to scarcity of land and the need for commuting in a city. If the industry- (or city-) specific productivity growth rates satisfy the basic assumptions of random growth theory (including Gibrat’s law), the model generates the power law for city size distributions. It is a plausible explanation, since we have seen in the section “Size and Industrial Structure of Cities” that specialized cities are rather ubiquitous (refer to Figure 4 and the corresponding discussion) despite the strong evidence for the hierarchy principle à la Christaller (1933).

A key implication of the random growth theory is that similar power laws hold for all (sufficiently large) random subsets of cities in a country, that is, *without any regard to spatial relation among cities*. Thus, this theory essentially denies the mutual dependence of size and spatial patterns of cities. But, as discussed in the section “Size Distribution of Cities,” Mori et al. (2019) have shown that the similarity in power laws for city size distributions is much stronger among the cells in the spatial hierarchical partitions of cities that are consistent with the central place patterns than among random subsets of cities, that is, if city sizes were generated by a random growth process.^{31}

To account for the large diversity in city size observed in reality by the many-region models described in the section “Spatial Pattern of Cities,” one needs to incorporate diversity in increasing returns (and/or that in transport costs). While Class (i) models with a global dispersion force can account for the formation of multiple cities, there is little variation in the sizes of cities to be realized in equilibrium, since each model has only one type of increasing returns.

There have been attempts at formalizing the central place theory of Christaller (1933) by introducing multiple industries subject to different degrees of increasing returns. The initial formal attempt was made by Beckmann (1958), but his model lacked a microeconomic foundation. Later models with more explicit mechanisms were developed by Fujita et al. (1999b), Tabuchi and Thisse (2011) in the context of the NEG, and by Hsu (2012) in the context of spatial competition model. In these models, the different degrees of increasing returns among industries result in the different spatial frequencies of agglomeration among industries.

The key to account for the diversity in city size in these models is the *spatial coordination* of agglomerations among industries through inter-industry demand externalities that arise from common consumers among industries. An industry subject to larger increasing returns agglomerates in a smaller number of cities that are farther apart. What is crucial is that these cities are chosen from the ones in which more ubiquitous industries subject to smaller increasing returns are located. Consequently, larger cities are formed at the location in which the coordination of a larger number of industries takes place. This spatial coordination of industries accounts for the positive correlation between the size, spacing, and industrial diversity of a city as observed in reality (see the sections “Size and Spacing of Cities” and “Size and Industrial Structure of Cities”).

In particular, Hsu (2012) proposed a unique spatial competition model with product differentiation and scale economies in production, and provided at this point the most far-reaching formal explanation for the mutual dependence between spatial pattern and size diversity of cities. When the distribution of scale economies in production of each firm (which is expressed in terms of the industry-specific fixed cost for production in his model) is *regularly varying*, then his model replicates the power law for city size distribution (see the section “Size Distribution of Cities”) together with the positive correlation between size and spacing of cities (see the section “Size and Spacing of Cities”), the power law for the number and the average size of choice cities of industries (see the section “Size and Industrial Structure of Cities”), as well as the hierarchy principle observed in Japan (see the section “Size and Industrial Structure of Cities”).

Davis and Dingel (2019) offer an alternative mechanism of spatial coordination among industries, which in turn results in the hierarchy principle and the diversity in city sizes in the context of a systems-of-cities model. Specifically, the hierarchy principle in this model arises from vertical heterogeneity in skill level among workers and skill requirement by industries together with inter-industry positive externality that is confined within the same city. The mechanism underlying the spatial coordination among industries in this model is different from the central place models discussed previously. On the one hand, a small city attracts only low-skill industries and workers as it offers only small city-level agglomeration externality. On the other hand, a large city attracts both high- and low-skill industries and workers. High-skilled workers have an incentive to live there, since the city offers a large city-level agglomeration externality and they can afford to live there. Although residential locations near the city center are occupied by high-skilled workers, low-skilled workers still can afford to live in locations with low land rent (due to longer commuting) near the city fringe, while enjoying the large city-level externality.

Alternatively, Desmet and Rossi-Hansberg (2009, 2014, 2015) and Desmet, Nagy, and Rossi-Hansberg (2018) incorporated dynamic externalities through endogenous innovation and spillover effects. These models are fundamentally different from all the models discussed so far in that the exogenous heterogeneity among regions is essential for city formation, that is, agglomerations do not form spontaneously. The uneven distribution of mobile agents resulting from the exogenous regional heterogeneity is magnified by the spillover effects over time. One exception in this strand of literature is Nagy (2017), who incorporated the same dynamic externalities into the NEG framework, so that his model is capable of explaining the spontaneous formation of multiple cities together with the diversity in city sizes. While this model has been applied to replicate the evolution of U.S. cities in the 19th century, the properties of agglomeration and dispersion in this model have not been formally analyzed.

Concluding Remarks

This article reviewed the models that explain the mutual dependence of spatial pattern and sizes of cities. A many-region geography with variations in interregional distance is an essential feature of a model in which the spatial pattern of cities is the subject of the study. Naturally, there have been very few formal attempts that explicitly deal with this high-dimensional problem until recently, with a notable exception by Hsu (2012).

A breakthrough was made about by Akamatsu et al. (2012), who proposed to focus on the racetrack economy, which involves many regions with heterogeneous interregional distances, while preserving symmetry among the regions. By utilizing the discrete Fourier transformation, they demonstrated that the spatial patterns of agglomeration that arise in the NEG models in a many-region set-up can be formally analyzed to a large extent. The same group of researchers have also developed a framework for systematic numerical analysis on a many-region geography based on the *numerical bifurcation theory* and *group-theoretic bifurcation theory* (e.g., Ikeda, Akamatsu, & Kono, 2012; Ikeda et al., 2017). Their numerical approach makes it possible to explore asymmetric geography (e.g., the presence of edges and heterogeneity in regional advantages) as well as two-dimensional location space in a many-region set-up.

In this article, drawing largely from Akamatsu et al. (2019), which applies the analytical tool developed by Akamatsu et al. (2012) to a wide variety of extant agglomeration models, I have reviewed the spatial pattern of cities and its relation to city sizes implied by these models. But Hsu (2012) continues to be the only tractable model that can account for the large diversity in city size in association with the observed spatial pattern of cities. Thus, much is to be expected in future developments in this respect.

Finally, no models so far have been successful in integrating intra- and intercity space. In the models aiming to explain intra-city spatial patterns, the location behavior of firms and that of workers are typically distinguished, and land consumption and/or land inputs by firms together with commuting are considered (e.g., Fujita & Ogawa, 1982; Lucas & Rossi-Hansberg, 2002; Ota & Fujita, 1993; Picard & Tabuchi, 2013). The models aiming to explain intercity spatial patterns, on the contrary, typically ignore different location incentives between firms and workers (most models discussed in this article belong to this group). But it is not trivial to integrate these two spatial scales in one model.

Some extant NEG models consider commuting and land consumption (e.g., Anas, 2004; Murata & Thisse, 2005). But such urban structure is by assumption confined within a given region, and does not extend beyond a single region. As discussed in the section “Spatial Pattern of Cities,” in a many-region geography with variations in interregional distance, these models belong to Class (ii), which means that at most unimodal agglomeration forms. Although each region in these models has monocentric urban structure *by assumption*, and hence, it is tempted to be interpreted as a “city”, they can generate essentially at most one “true” city.

To fully account for the spatial pattern of cities, the distinction between inside and outside each city should also be endogenized.

Acknowledgement

The author thanks the two anonymous referees for their constructive and careful comments. This research was conducted as part of the project “An empirical framework for studying spatial patterns and causal relationships of economic agglomeration,” undertaken at the Research Institute of Economy, Trade and Industry. This research has been partially supported by the Grant in Aid for Research (Nos. 17H00987,16K13360, 16H03613,15H03344) of the MEXT, Japan.

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## Notes:

(1.) There is a large body of literature on location theory that preceded urban economics and that has important implications for city and agglomeration formation (see, e.g., Thisse, Button, and Nijkamp, 1996 for a survey), although it was not designed to explain city formation per se.

(2.) A notable exception is Isard (1949, 1956). While no formal models have been proposed by Isard, he foresaw the necessity of increasing returns and imperfect competition in order to explain the formation of cities and their spatial pattern. In particular, he envisaged the emergence of new economic geography, which played a central role in this literature, as will be discussed in the section “Theories” (see Fujita, 2010, for further discussions).

(3.) See an intriguing review by Fujita (2012) on von Thünen’s work and ideas about spatial organization of economy.

(4.) See, for example, Abdel-Rahman and Anas (2004) for a survey. See also Behrens and Robert-Nicoud (2015) for more recent applications of this framework. The standard systems-of-cities models assume zero intercity transport cost. There are a few variations assuming an equal distance between any pair of cities (see, e.g., Anas and Xiong, 2003; Anas, 2004). In either case, there is only a single intercity distance, as in the case of a two-region model.

(5.) This definition of a city is a variation of that proposed by Rozenfeld, Rybski, Gabaix, and Makse (2011). The results to be presented here are not sensitive to the density and total population thresholds, unless they are set extremely high or low so that only a few high-density cities or a few spatially gigantic cities are identified.

(6.)
The estimated population count data at the 1 km^{2} cell level are obtained from the Statistics Bureau, Ministry of Internal Affairs and Communication of Japan (2015) for Japan, and from the LandScan by Oak Ridge National Laboratory (2015) for the other countries.

(7.) For the role of the natural advantage in city formation, see, for example, Davis and Weinstein (2002) for the case of Japan, Bleakley and Lin (2012) and Cronon (1991) for the United States, and Michaels and Rauch (2018) for France and the United Kingdom.

(8.)
The road distance is based on the OpenStreetMap data as of July 2017. The distance between cities is computed as the distance between the centroids of the most densely populated 1 km^{2} cells in these cities. The computation was done using the Stata interface, osrmtime, of Open Source Routing Machine by Huber and Rust (2016).

(9.) The lower threshold share, 0.75, defining the “same size class” in (1) is arbitrary. But the choice of the threshold value does not alter the qualitative result as long as it is not too far from 1.0.

(10.) The dashed line in the figure is the fitted line by Ordinary Least Squares (OLS) regression.

(11.)
Dobkins and Ioannides (2001) found a negative correlation between the size and spacing of cities in the United States for the period 1900–1980. But the specific feature of the U.S. cities that needs to be taken into account is their historical development. The formation of cities started in the northeastern region of the United States in the 19th century, and then expanded gradually west and then south. But the *effective* distance kept changing in the meantime in response to advancements in transport technology. As a consequence, the spacing of the same size class of cities has increased over time. Such underlying heterogeneity across regions is to some extent taken into account in the construction of counterfactuals in the test by Mori et al. (2019).

(12.) Dittmar (2011) shows evidence that power laws for city size distributions in Europe emerged after 1500, that is, after the dependence of city production on land relaxed substantially.

(13.) Data for the locations of establishments were obtained from Statistics Bureau, Ministry of Internal Affairs and Communication of Japan (2001, 2014).

(14.) These features were first recognized by Mori, Nishikimi, and Smith (2008) and Mori and Smith (2011) for the case of Japan, and Hsu (2012, Appendix A1) and Schiff (2014) for the case of the United States. See also Davis and Dingel (2019) for evidence of the hierarchical industrial structure of U.S. cities based on an alternative approach.

(15.)
These values are much higher than the values of *H* that can be realized under random location of industries after controlling for the industrial diversity (i.e., $\#{\mathcal{I}}_{i}$ for $i\in \mathcal{U}$) of cities and locational diversity (i.e., $\#{\mathcal{U}}_{i}$ for $i\in \mathcal{I}$) of industries (see, e.g., Mori, 2017; Mori et al., 2008; Mori & Smith, 2011).

(16.) Cities may emerge, disappear, split, and merge over time. Cities identified in the consecutive two years are considered to represent the same city if they mutually account for the largest population among all the overlapping cities.

(17.) “S.D.” in the panels means the standard deviation.

(18.) Overman and Ioannides (2001) have shown evidence that there is a mild tendency toward the decrease in population size of relatively large cities (i.e., metropolitan areas with urban core of at least 50,000 population) of the United States for the period 1920–1980. Their result is not directly comparable to the case of Japan here, since their results may be biased for relatively large cities, and the factors driving city sizes during the studied period were not made clear.

(19.) The suburbanization in response to the decrease in interregional transport access is one realization of local dispersion, and its evidence for the case of the U.S. metro areas has been reported by Baum-Snow (2007, 2017). For the global agglomeration and dispersion, no clear consensus has been attained at this point in the extant literature (e.g., Baum-Snow, 2017; Duranton & Turner, 2012; Faber, 2014). This is rather evident from the discussion in the section “Theories” that the effect of interregional transport access on each individual city size is not monotonic. See Akamatsu et al. (2019, §5) for an extensive discussion on this respect. Ioannides and Overman (2004) examined the change in the distance from each city to its nearest neighbor, and found it was decreasing in the period from 1900 to 1990, which should essentially imply global dispersion. But there is no discussion on the potential causes of this change in their paper.

(20.) See, for example, Baldwin et al. (2003) for a survey of NEG models, and Anas and Xiong (2003), Anas (2004), and Tabuchi, Thisse, and Zeng (2005) for systems-of-cities.

(21.) The racetrack location space is obviously counterfactual, as it is edgeless. Although the presence of the edge tends to make the agglomeration on the edge larger, since there is no competing agglomeration beyond the edge (see, e.g., Fujita & Mori, 1997; Ikeda et al., 2017), this effect becomes negligible for a large economy, and the agglomeration patterns can be approximated by that in the edgeless economy.

(22.)
It is equally likely that agglomerations take place at regions, 1,3,…, *K*—1.

(23.) It is equally likely that the agglomeration takes place around any region in $\mathcal{K}$.

(24.)
*f*_{0} whose corresponding eigenvector is *η*_{0} = (1,1,…, 1) is irrelevant for the stability of equilibria as the total mobile population is preserved. For simplicity, it is assumed that *K* is an even integer, although it is not essential.

(25.) See Akamatsu et al. (2012) for the formal analyses on the period doubling bifurcations of Class (i) models.

(26.) Of course, the actual evolution of the spatial patterns under the changing level of transport costs is more complicated, as neighboring cities may eventually merge in the case of Class (iii) models. See Akamatsu et al. (2019, §5).

(27.) An alternative formulation assumes the product variety in intermediate goods. See, e.g., Fujita et al. (1999a, ch.14).

(28.) A similar effect can be obtained by assuming local congestion externality that is effective within a given region.

(29.) The social interactions model by Picard and Tabuchi (2013) with non-iceberg transport costs can be shown to belong to Class (iii) (see Akamatsu et al., 2019, Appendix D).

(30.) See Economides and Siow (1988) for a related model that explains the spacing of market areas in which markets are formed due to matching externalities that arise in the exchange of consumption goods.

(31.) See Rozenfeld et al. (2008) and Rybski and Ros (2013) for other evidence against Gibrat’s law for city sizes. There still are possibilities to extend random growth models by adding spatial relations among cities, thereby accounting for the spatial fractal structure of city systems in terms of power laws of city size distributions. See, for example, Ioannides (2012, §8.2.5) for a review of related attempts.