The Footloose Entrepreneur model with a finite number of equidistant regions

• DOI: http://doi.org/10.1111/ijet.12215
• Date: 01 December 2020
• Author: José M. Gaspar,Sofia B.S.D. Castro,João Correia‐da‐Silva
• Published date: 01 December 2020

doi: 10.1111/ijet.12215

The Footloose Entrepreneur model with a finite number of

equidistant regions

Jos´

eM.Gaspar,∗Sofia B.S.D. Castro†and Jo˜

ao Correia-da-Silva‡

Westudy the Footloose Entrepreneur model with a finite number of equidistant regions, focusing

on the analysis of stability of agglomeration, total dispersion, and boundary dispersion. As the

number of regions increases, there is more tendency for agglomeration and less tendency for

dispersion. As it tends to infinity, agglomeration always becomes stable while dispersion always

becomes unstable. These results are robust to any composition of the global workforce and its

dependence on the number of regions. Numerical evidence suggests that boundary dispersion is

never stable. Weintroduce exogenous regional heterogeneity and obtain a general condition for

stability of agglomeration.

Key wor ds core-periphery, Footloose Entrepreneur,multiple regions, agglomeration

JEL classification R10, R12, R23

Accepted 22 December 2018

1 Introduction

The secular tendency for spatial agglomeration of economic activity is well known and has always

been a matter of profound debate. Recent developments have allowed a morer igoroustreatment of

such phenomena, with recourse to microeconomic foundations.1The benchmark in this literature is

the Core-Periphery (CP) model, introduced by Krugman (1991). However, several issues have been

raised in the recent literature about possible shortcomings due to some simplifying assumptions.

One of these shortcomings is the prevalent focus on the 2-region framework.

∗Lisbon Schoolof Economics and Management, University of Lisbon, Lisbon, Católica Porto Business School, Universidade

Católica Portuguesa and CEF.UP, University of Porto, Porto,Portugal. Email: jgaspar@porto.ucp.pt

†CMUP and Faculty of Economics, University of Porto,Porto, Portugal.

‡CEF.UPand Faculty of Economics, University of Porto , Porto, Portugal. Email: jgaspar@porto.ucp.pt

We thank Kristian Behrens, PasqualeCommendatore, Laura Gardini, Paulo Guimar˜

aes, Kiyohiro Ikeda, Ingrid Kubin,

VascoLeite, Pascal Mossay,Minoru Osawa, Cesaltina Pires, Jos ´

e PedroPontes, Irina Sushko, two anonymous referees and

the Associate Editor for very useful comments and suggestions. Wealso thank participants in the 2nd Ph.D. Workshop

on Industrial and Public Economics at Reus, the 3rd Industrial Organization and Spatial Economics conference at

Saint Petersburg, and the International Conference ESCOs 2018 at Naples. This work also benefited from discussions

while Jos´

e Gaspar visited Tohoku University. This research was financed by the European Regional Development Fund

through COMPETE 2020 – Programa Operacional Competitividade e Internacionalização (POCI) and by Portuguese

Public Funds through Fundação para a Ciência e Tecnologia in the fr amework of projects POCI-01-0145-FEDER-

006890, PEst-OE/EGE/UI4105/2014, PEst-C/MAT/UI0144/2013, PTDC/EGE-ECO/30080/2017, and Ph.D. scholarship

SFRH/BD/90953/2012. Part of this work was developed while Jo˜

ao Correia-da-Silva was a Marie CurieFellow at Toulouse

School of Economics, financed by the EuropeanCommission (H2020-MSCA-IF-2014-657283).

1See Fujita, Krugman, and Venables(1999), Ottaviano, Tabuchi, and Thisse (2002), Baldwin et al. (2004), Robert-Nicoud

(2005) and the references therein.

International Journal of Economic Theory (2019) 1–27 © IAET 1

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International Journal of Economic Theory

Multi-region FE model Jos´

e M. Gaspar et al.

Theoretical insights on a model with three or more regions are interesting for different reasons.

One reason is the understanding of interdependenciesamong many regions to guide empirical studies

(Bosker, Brakman, Garretsen, and Schramm 2010). A two-region set-up overlooksthe var iability of

market access across regions (Fujita and Thisse 2009), thus more complex spatial patterns mayarise

in a multi-regional set-up compared to a two-region one (Fujita and Mori 2005; Fujita and Thisse

2009; Behrens and Robert-Nicoud 2011; Akamatsu, Takayama, and Ikeda 2012; Gaspar 2018). As

pointed out byFujita, Krugman, and Venables (1999) and Berliant and Mori (2017), the consideration

of only two regions stems from the advantage of dealing with more tractable problems, although it

seems implausible that the geographical dimension of economic activity can be reduced to a 2-region

framework. It is important, therefore, to understand to what extent the main conclusions obtained

using 2-region models extend to models with more regions.

This has motivated a number of different studies. Castro, Correia-da-Silva, and Mossay (2012)

studied a 3-region version of the CP model by Krugman (1991), and also a version with an even

number of regions equally spaced around a circle. Comparing the 3-region model with the 2-region

model, they concluded that the additional region favors the agglomeration of economic activity and

hinders the dispersion of economic activity.

Akamatsu, Takayama, and Ikeda (2012) and Ikeda, Akamatsu, and Kono (2012) studied agglom-

eration processes in a CP model with 2nequally spaced regions around a circle. They applied the

discrete Fourier transformation to identify eigenvalues of the Jacobian matrix of the dynamical sys-

tem as a unimodal function of a spatial discounting matrix which accounts for the spatial patternsdue

to agglomeration and dispersion forces that operate differently in every region. This allowed them

to determine which equilibrium configurations arise after a decrease in transport costs that deems

the symmetric equilibrium unstable. They uncovered a spatial period doubling bifurcation whereby

the number of regions in which firms locate is reduced by half and the spacing between each pair

of adjacent “core” regions doubles after each bifurcation. If consumers are homogeneous regarding

preferences for residential location, the economy may eventually converge to full agglomeration. If

they are heterogeneous, a bifurcation may occur at some late stage that reverts the spatial economy

to the fully symmetric outcome, thus lending support to the idea of a bell-shaped curve of spatial

development in other models with heterogeneous preferences such as Tabuchi and Thisse (2002) and

Murata (2003).

Oyama (2009) incorporated self-fulfilling expectations in migration decisions in a multi-regional

variant of the CP model and studied global stability of agglomeration. He found that a fully symmet-

ric setup deems the choice of forward looking expectations and myopic migration for the dynamics

irrelevant. In this case, initial advantages and history determine the resulting spatial distribution.

With exogenous asymmetries, perfect foresight of migrants originates a unique globally stable equi-

librium corresponding to agglomeration in a region that is either relatively protected (lower trade

barriers) or has the potentially largest market size, irrespective of the initial state.

Tabuchi and Thisse (2011) studied the rise of a hierarchical system of centr al places in a multi-

location space. Barbero and Zof´

ıo (2016) considered different network topologies to study how

the interplay between centripetal and centrifugal forces changes according to the heterogeneity of

location space. They showed that more heterogeneous configurations increase the likelihood of

agglomeration in regions with higher centrality; that is, those that are relatively better located.

The role of heterogeneousdistances between regions has also been addressed in other frameworks,

such as the racetrack economy,with either a discrete or a continuous uniform distribution of locations

(Krugman 1993; Fujita, Krugman, and Venables1999; Picard and Tabuchi 2010; Mossay 2013; Ago,

Hamoudi, and Lefouili 2017), three equally spaced regions locatedon a line (Ago, Isono, and Tabuchi

2006), or hexagonal distributions (Ikeda, Murota, and Takayama 2017a). Ikeda, Murota, Akamatsu,

2International Journal of Economic Theory (2019) 1–27 © IAET

International Journal of Economic Theory 16 (2020) 420–446 © IAET 421