Agglomeration patterns in a long narrow economy of a new economic geography model: Analogy to a racetrack economy

• Author: Yuki Takayama,Kiyohiro Ikeda,Kazuo Murota,Takashi Akamatsu
• DOI: http://doi.org/10.1111/ijet.12120
• Published date: 01 March 2017
• Date: 01 March 2017

doi: 10.1111/ijet.12120

Agglomeration patterns in a long narrow economy of a new

economic geography model: Analogy to a racetrack economy

Kiyohiro Ikeda,∗Kazuo Murota,†Takashi Akamatsu‡and Yuki Takayama§

The mechanism of self-organization of agglomerations in a long narrow economy of a new

economic geography model is elucidated by a theoretical comparative study with a racetrack

economy. Computational bifurcation theory is used to systematically obtain the equilibria of

these economies. A chain of spatially repeated core–periphery patterns `alaChristaller and

L¨

osch emerges when agglomeration forces are large. Peripheral zones are enlarged recurrentlyto

engender an agglomeration shadow en route to an atomic mono-center.A megalopolis with two

core places connected by an industrial belt emerges when agglomeration forces are small.

Key wor ds bifurcation, break point, long narroweconomy, new economic geography,racetrack

economy,t ransport cost

JEL classification R12, R13, C65, F12

Accepted 27 April 2016

1 Introduction

Narrow industrial belts serve as cradles of development and prosperity and engender megalopolises

worldwide. A chain of cities, for example, is distributed from Boston to Washington, DC in a closed

narrow corridor between the Atlantic Ocean and the Appalachian Mountains. A megalopolis, such

as New York in this chain, grows as a core of an economic agglomeration.

A long narrow economy of a spatial economy model is highlighted in this paper as a spatial

platform of a chain of cities. As expounded in Section 2, there is a characteristic agglomeration

pattern that hitherto has been observed fragmentarily in the literature: a chain of spatially repeated

core–periphery patterns `alaChristaller and L ¨

osch (e.g., Fujita and Mori 1997). A study of this pattern

would be a benchmark to examine the validity of the underlying models for empirical studies and

would also give a hint at the economic implications of agglomeration shadows.1It is desirable to set

∗Department of Civil and Environmental Engineering, TohokuUniversity, Aoba,Sendai, Japan. Email: kiyohiro.ikeda.b4@

tohoku.ac.jp

†School of Business Administration, Tokyo Metropolitan University, Tokyo, Japan.

‡Graduate School of Information Sciences, TohokuUniversity, Aoba, Sendai, Japan.

§Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa, Japan.

This study was conducted as a part of the project “The formation of economic regions and its mechanism: Theory and

evidence,”undertaken at RIETI.

1Arthur (1990) stated: “Locations with large numbers of firms therefore cast an ‘agglomeration shadow’ in which little or

no settlement takes place. This causes separation of the industry” (see also Fujita et al. 1999; Ioannides and Overman

2004; Fujita and Mori 2005).

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Agglomeration patterns Kiyohiro Ikeda et al.

forth the mechanism underpinning the self-organization of the pattern, just as Christaller’s (1933)

distributions were founded on geometrical principles.

In this paper, this mechanism is elucidated by a racetrackeconomy analog y and bycomputat ional

bifurcation theory. In the racetrack economy analogy, it is shown that recurrent bifurcations of the

racetrack economy serve as fairly good predictorsfor the onset of agg lomerations producing the chain

of patterns in the long narrow economy. The break point2is analytically predicted in the racetrack

economy and is successfully employed to index the agglomerations in the long narrow economy.

The analogy is in agreement with the underlying intuitive belief that both economies would tend

to behave similarly as the number of places increases although the long narrow economy is slightly

more asymmetric due to the presence of the borders.

In the study of the agglomeration properties of these economies, equilibria of the economies are

obtained by a numerical approach based on computational bifurcation theory (Ikeda et al. 2012,

appendix E). This approach can be applied to a wider class of many-region models beyond the new

economic geography (NEG), other than the long narrow and racetrack economies of the Forslid–

Ottaviano model employed in this paper. This paper, in this sense, is superior to other existing

approaches on many-region/continuous location models that cannot exhaustivelyobtain equilibr ia;

furthermore, numerical experiments on parameter dependence in these models are quite limited and

pre-existing numerical results are quite fragmented (e.g., Krugman 1993; Brakman et al. 1996; Fujita

and Mori 1997; Bosker et al. 2010). Note also that the numerical approach enables us to investigate

the equilibria of more asymmetric economies exhaustively by systematically changing some of their

parameters and by implementing realistic heterogeneities.

In this paper, we investigatethe agg lomeration patterns of a long narrow economy with equally

spaced discrete places on a line segment. This paper employs a many-region version of the model

by Forslid and Ottaviano (2003) in favor of its analytical solvability, which plays a key role in

deriving the formulas for break points. There are unskilled workers who are immobile and skilled

ones (footloose entrepreneurs) who migrate between places to maximize utility. Nowadays, the

immobile workers can be interpreted as a population attached to certain amenities or to traditional

housing.

Agglomeration patterns are found to be dependent on agglomeration forces. When these forces

are large, essential properties of the model prevail and, in turn, the chain of spatially repeated core–

periphery patterns tends to emerge. When agglomeration forces are small, spatial heterogeneity due

to border effects become dominant. A megalopolis-like diffuse agglomeration pattern is observed in

both the replicator dynamics and the logit dynamics.3When transportation costs become very small,

the latter has more diffuse agglomeration patterns: a megalopolis-like hump-shaped distribution and

redispersion thereafter.

This paper is organized as follows. Related studies are discussed in Section 2. The analytically

solvable model is presented in Section 3. Bifurcation of the racetrack economy is described in Sec-

tion 4. A numerical procedure based on computational bifurcation theory is presented in Section 5.

Spatial agglomeration in the long narrow economy is investigated based on the racetrack economy

analogy in Section 6. Agglomerations in the logit dynamics are observed in Section 7.

2The break point is the value of the transport cost at the onset of bifurcation in the two-place economy(Fujita et al. 1999).

3In this paper, the replicatordynamics is employed as the standard case, while the logit dynamics (Fudenberg and Levine

1998) is used for comparison.

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Kiyohiro Ikeda et al. Agglomeration patterns

2 Related studies

In the early development of the NEG,4diverse agglomeration patterns of the long narrow economy

were observed. A megalopolis which consists of large core cities that are connected by an industrial

belt, that is, a continuum of cities, associated with lower transport costs was found by Mori (1997).

A discretized highly regular central place system `alaChristaller and L¨

osch5was observed when

population size increases (Fujita and Mori 1997). Yet little development occurred thereafter, while

microeconomic structures of NEG models have been well established.

The racetrack economy has come to be employedincreasingly as a spatial platform of an idealized

uniform trading space along the circumference of a circle. The emergence of discrete agglomerations

out of the uniformity was demonstrated. A spatial alternation of a core place with a large population

and a peripheral place with zero population `alaChristaller and L¨

osch (cf. Section 4) was found by

considering the hierarchy of different industries in Tabuchi and Thisse (2011). The emergence of this

pattern was explained by recurrent bifurcations in Ikeda et al. (2012) and Akamatsu et al. (2012).

Anas (2004) demonstrated the presence of balanced agglomeration, concentrated agglomeration,

and de-agglomeration by removing agriculture and treating congestion and prices of land and labor

as the main dispersion forces.

A comparative study of the long narrow and the racetrack economies was conducted by Mossay

and Picard (2011) for Beckman’s(1976) CBD formation model in a continuous space to display the

difference in agglomeration patterns: a single city in the long narrow economy and multiple equilibria

with an odd number of cities in the racetrack economy. This study, however, was conducted without

the stability analysis of equilibria and, therefore, its validity requires further verification.

This paper, in contrast to these studies, makes a contribution by elucidating the mechanism of

agglomerations in the long narrow economy by a comparative study with racetrack economy that

is based on a firm theoretical background. The racetrack economy analogy has been successfully

introduced to describe the mechanism of the onset of agglomerations.

3 Model of spatial economy

This paper employs an analytically solvable NEG model by Forslid and Ottaviano (2003). This model

replaces the production function of Krugman (1991) with that of Flam and Helpman (1987). The

fundamental logic and equilibrium equation of a many-region version of the model are presented

based on work of Akamatsu et al. (2016), as well as that of Ikeda et al. (2014). Basic assumptions are

presented in Section 3.1. Market equilibrium is introduced in Section 3.2 and spatial equilibrium in

Section 3.3.

3.1 Basic assumptions

The economy of this model is composed of nplaces (labeled i=1,...,n), two factors of production

(skilled and unskilled labor) and two sectors (manufacturing, M, and agriculture, A). Both Nskilled

4The evolution of spatial agglomeration was studied by Krugman (1991), leading to the subsequent development of NEG

models various kinds (for a review, see Baldwin et al. 2003). Real economic activities allow models of various kinds that

entail diverse agglomerations (P߬

uger and S¨

udekum 2008; Berliant and Yu2014).

5For central place theory, see Christaller (1933) and L¨

osch (1940).

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