EXISTENCE AND UNIQUENESS OF EQUILIBRIUM FOR A SPATIAL MODEL OF SOCIAL INTERACTIONS*

• Published date: 01 February 2016
• Author: Adrien Blanchet,Filippo Santambrogio,Pascal Mossay
• Date: 01 February 2016
• DOI: http://doi.org/10.1111/iere.12147

INTERNATIONAL ECONOMIC REVIEW

Vol. 57, No. 1, February 2016

EXISTENCE AND UNIQUENESS OF EQUILIBRIUM FOR A SPATIAL MODEL

OF SOCIAL INTERACTIONS∗

BYADRIEN BLANCHET,PASCAL MOSSAY,AND FILIPPO SANTAMBROGIO1

Universit´

e de Toulouse (TSE, GREMAQ), France; Newcastle University, U.K. and CORE,

Belgium; Universit´

e Paris Sud, France

We extend Beckmann’s spatial model of social interactions to the case of a two-dimensional spatial economy

with a large class of utility functions, accessing costs, and space-dependent amenities. We show that spatial equilibria

derive from a potential functional. By proving the existence of a minimizer of the functional, we obtain that of spatial

equilibrium. Under mild conditions on the primitives of the economy, the functional is shown to satisfy displacement

convexity. Moreover, the strict displacement convexity of the functional ensures the uniqueness of equilibrium. Also,

the spatial symmetry of equilibrium is derived from that of the primitives of the economy.

1. INTRODUCTION

Since Marshall (1920), it has been known that both market and nonmarket forces play an

important role in shaping the distribution of economic activities across space. The new economic

geography literature has reemphasized the role of localized pecuniary externalities mediated by

the market in a general equilibrium framework; see Krugman (1991). Social interactions through

face-to-face contacts also contribute to the gathering of individuals in villages, agglomerations,

or cities; see Glaeser and Scheinkman (2003). In Beckmann (1976), the urban structure results

from the interplay between a spatial communication externality and the land market.

When studying the role of agglomeration forces on the urban structure, the existing literature

traditionally relies on specific functional forms regarding utility functions or transportation

costs. New economic geography models make a wide use of Constant Elasticity of Substitution

(CES) or quadratic preferences over manufacturing varieties and of iceberg transport costs;

see Fujita et al. (1999) and Ottaviano et al. (2002). In Beckmann’s spatial model of social

interactions, the preference for land is logarithmic, and the cost of accessing agents is linear;

see Fujita and Thisse (2002).

More recently, some efforts have been made to build models allowing for more general pref-

erences over goods, with internal or external increasing returns to scale. For instance, some

works have extended the CES preferences traditionally used in general equilibrium models

of monopolistic competition to the case of preferences with variable elasticity of substitution;

see Behrens and Murata (2007) and, more generally, Zhelobodko et al. (2012). Also, in a

multidistrict model with external increasing returns in the spirit of Fujita and Ogawa (1982),

Lucas and Rossi-Hansberg (2002) have proved the existence of a symmetric spatial equilibrium

from standard neoclassical assumptions on preferences and technology. Despite these various

∗Manuscript received December 2012; revised May 2014.

1The authors acknowledge the support of the Agence Nationale de la Recherche through the project EVaMEF

ANR-09-JCJC-0096-01, the Ram´

on y Cajal program at the Universidad de Alicante, and the COST Action IS1104.

The authors wish to thank Marcus Berliant, Masahisa Fujita, Andr´

e Grimaud, Michel Le Breton, Yasasuda Murata,

Daisuke Oyama, J´

erˆ

ome Renault, Franc¸ois Salani ´

e, Yasusada Sato, as well as participants at the IAST LERNA—

Eco/Biology Seminar, the Tokyo workshop on Spatial Economics organized by RIETI and the University of Tokyo,

the Public Economic Theory (PET) conference in Seattle, and the Urban Economics Association (UEA) conference

in Washington for helpful comments. Please address correspondence to: Pascal Mossay, Newcastle University Business

School, 5 Barrack Road, Newcastle, NE1 4SE, UK. E-mail: pascal.mossay@ncl.ac.uk.

31

C

(2016) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

32 BLANCHET,MOSSAY,AND SANTAMBROGIO

efforts in extending models addressing agglomeration forces mediated by the market mech-

anism, little progress has been made to extend further spatial models where agglomeration

externalities are driven by nonmarket forces. The aim of this article is to fill this gap by address-

ing the existence and uniqueness of equilibrium for general spatial economies involving social

interactions.

Our main results are the following: We generalize Beckmann’s spatial model of social interac-

tions to the case of a two-dimensional spatial economy with a large class of preferences for land,

accessing costs, and space-dependent amenities. We prove the existence and the uniqueness

of spatial equilibrium. So as to get our results, we start our analysis by providing conditions

under which spatial equilibria derive from a potential. Stated differently, we build a functional

of which the critical points correspond to the spatial equilibria of the economy. In this context,

the conditions ensuring the existence of a minimizer of the functional also ensure the existence

of a spatial equilibrium of the economy. As the functional is not convex in the usual sense, we

introduce another notion of convexity, referred to as displacement convexity, a concept widely

used in the theory of optimal transportation. Under mild conditions on the primitives of the

economy, the functional is shown to be displacement convex, and we obtain an equivalence

between the minimizers of the functional and the spatial equilibria of the economy. This pro-

vides a variational characterization of spatial equilibrium. Moreover, if the functional displays

strict displacement convexity, we get the uniqueness of minimizer, and hence that of spatial

equilibrium. Also, the spatial symmetry of equilibrium is derived from that of the primitives

of the economy. We present several examples with the purpose of illustrating the scope of our

existence and uniqueness results. In particular, one- and two-dimensional geographical spaces,

linear and quadratic accessing costs, and linear and power residence costs are examined. Fi-

nally, the circular spatial economy is revisited so as to illustrate the role of nonconvexities in

explaining the emergence of multiple equilibria. A direct method allows us to derive all the

spatial equilibria arising along the circle. The analysis completes the work initiated by Mossay

and Picard (2011).

The remainder of the article is organized as follows: Section 2 presents the economic environ-

ment and generalizes Beckmann’s spatial model of social interactions. In Section 3, we prove the

existence of a spatial equilibrium. Section 4 is devoted to the variational characterization and

the uniqueness of equilibrium as well as its spatial symmetry properties. In Section 5, we present

several examples of spatial economies so as to illustrate the scope of our results. Section 6 is

devoted to the analysis of the circular economy. Section 7 summarizes the main results of the

article and concludes.

2. SPATIAL MODEL

In this section, we present the economic environment. We consider a closed spatial economy

Eextending along a one- or two-dimensional geographical space K⊂Rd,d=1,2, hosting a

unit-mass of agents distributed according to the spatial density λ∈M(K)={λ∈L1(K):λ≥

0,Kλ=1}, the set of nonnegative, Lebesgue measurable and integrable functions with a unit

norm. Agents meet each other so as to benefit from social contacts. The social utility S(x) that

an agent in location x∈Kderives from interacting with other agents is given by

S(x)=B−W∗λ(x),(1)

where the constant Bdenotes the total benefit from interacting with other agents, W:Rd→

R∪{+∞}the cost function of accessing them, and W∗λ(x) the convolution of Wwith λ, KW

(x−y)λ(y)dy, representing the accessing cost from location x. To ensure that social interactions

are global, Bis assumed to be large enough, B>maxxW∗λ(x).