Existence and Uniqueness of Equilibrium for a Spatial Model of Social Interactions

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]  Masahisa Fujita,et al.  Multiple equilibria and structural transition of non-monocentric urban configurations , 1982 .

[2]  Stability of a spatial model of social interactions , 2016 .

[3]  Kristian Behrens,et al.  General equilibrium models of monopolistic competition: A new approach , 2007, J. Econ. Theory.

[4]  Mtw,et al.  Mass Transportation Problems: Vol. I: Theory@@@Mass Transportation Problems: Vol. II: Applications , 1999 .

[5]  Jacques-François Thisse,et al.  Monopolistic Competition: Beyond the Constant Elasticity of Substitution , 2012 .

[6]  Jacques-François Thisse,et al.  Monopolistic Competition: Beyond the CES , 2010 .

[7]  Robert E. Lucas,et al.  Externalities and Cities , 2001 .

[8]  A. Venables,et al.  The Spatial Economy: Cities, Regions, and International Trade , 1999 .

[9]  Ping Wang,et al.  Production Externalities and Urban Configuration , 2002, J. Econ. Theory.

[10]  M. K. Jensen.  Aggregative games and best-reply potentials , 2010 .

[11]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

[12]  L. Shapley,et al.  Potential Games , 1994 .

[13]  Ivan P. Gavrilyuk,et al.  Variational analysis in Sobolev and BV spaces , 2007, Math. Comput..

[14]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[15]  E. Glaeser,et al.  Non-Market Interactions , 2000 .

[16]  Y. Brenier.  Polar Factorization and Monotone Rearrangement of Vector-Valued Functions , 1991 .

[17]  F. Santambrogio.  Gradient flows in Wasserstein spaces and applications to crowd movement , 2010 .

[18]  Esteban Rossi-Hansberg,et al.  ON THE INTERNAL STRUCTURE OF CITIES , 2002 .

[19]  L. Shapley,et al.  REGULAR ARTICLEPotential Games , 1996 .

[20]  Pradeep Dubey,et al.  Strategic complements and substitutes, and potential games , 2006, Games Econ. Behav..

[21]  Jacques-François Thisse,et al.  Economics of Agglomeration: Cities, Industrial Location, and Regional Growth , 2002 .

[22]  Pierre Picard,et al.  On spatial equilibria in a social interaction model , 2011, J. Econ. Theory.

[23]  Jacques-François Thisse,et al.  Agglomeration and Trade Revisited , 2002, World Scientific Studies in International Economics.

[24]  S. Rachev,et al.  Mass transportation problems , 1998 .

[25]  Antje Baer,et al.  Direct Methods In The Calculus Of Variations , 2016 .

[26]  Martin J. Beckmann,et al.  SPATIAL EQUILIBRIUM IN THE DISPERSED CITY , 1976 .

[27]  E. Glaeser,et al.  Advances in Economics and Econometrics: Nonmarket Interactions , 2003 .

[28]  M. Breton,et al.  Games of social interactions with local and global externalities , 2011 .