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Existence and uniqueness of equilibrium for a spatial model of social interactions

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  • Blanchet, Adrien
  • Mossay, Pascal
  • Santambrogio, Filippo

Abstract

We extend Beckmann’s spatial model of social interactions to the case of a two-dimensional spatial economy involving 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 minimiser of the functional, we obtain that of a spatial equilibrium. Under mild conditions on the primitives of the economy, the functional is shown to satisfy displacement convexity, a concept used in the theory of optimal transportation. This provides a variational characterisation of spatial equilibria. Moreover, the strict displacement convexity of the functional ensures the uniqueness of spatial equilibrium. Also, the spatial symmetry of equilibrium is derived from that of the spatial primitives of the economy. Several examples illustrate the scope of our results. In particular, the emergence of multiple of equilibria in the circular economy is interpreted as a lack of convexity of the problem.

Suggested Citation

  • Blanchet, Adrien & Mossay, Pascal & Santambrogio, Filippo, 2014. "Existence and uniqueness of equilibrium for a spatial model of social interactions," TSE Working Papers 14-489, Toulouse School of Economics (TSE).
  • Handle: RePEc:tse:wpaper:28212
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Takayama, Yuki & Kuwahara, Masao, 2017. "Bottleneck congestion and residential location of heterogeneous commuters," Journal of Urban Economics, Elsevier, vol. 100(C), pages 65-79.
    2. Ioannides, Yannis M., 2015. "Neighborhoods to nations via social interactions," Economic Modelling, Elsevier, vol. 48(C), pages 5-15.
    3. Pierre Degond & Jian-Guo Liu & Christian Ringhofer, 2014. "Evolution of wealth in a nonconservative economy driven by local Nash equilibria," Working Papers hal-00967662, HAL.
    4. Pierre Degond & Jian-Guo Liu & Christian Ringhofer, 2013. "Evolution of the distribution of wealth in an economic environment driven by local Nash equilibria," Papers 1307.1685, arXiv.org.
    5. Pierre Degond & Jian-Guo Liu & Christian Ringhofer, 2014. "Evolution of wealth in a nonconservative economy driven by local Nash equilibria," Papers 1403.7800, arXiv.org.
    6. Akamatsu, Takashi & Mori, Tomoya & Osawa, Minoru & Takayama, Yuki, 2017. "Spatial scale of agglomeration and dispersion: Theoretical foundations and empirical implications," MPRA Paper 80689, University Library of Munich, Germany.
    7. Akamatsu, Takashi & Fujishima, Shota & Takayama, Yuki, 2017. "Discrete-space agglomeration model with social interactions: Multiplicity, stability, and continuous limit of equilibria," Journal of Mathematical Economics, Elsevier, vol. 69(C), pages 22-37.
    8. Vincent Boitier, 2014. "Unemployment Dispersion and City Configurations: Beyond the Bid Rent Theory," Working Papers hal-00999559, HAL.

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    Keywords

    social interaction; spatial equilibria; multiple cities; optimal transportation; displacement convexity;

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