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

  • Adrien BLANCHET
  • Pascal MOSSAY
  • Filippo SANTAMBROGIO

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 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 multiplicity of equilibria in the circular economy is interpreted as a lack of convexity of the problem.

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Paper provided by Research Institute of Economy, Trade and Industry (RIETI) in its series Discussion papers with number 13055.

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Length: 31 pages
Date of creation: Jun 2013
Date of revision:
Handle: RePEc:eti:dpaper:13055
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  1. Marcus Berliant & Shin-Kun Peng & Ping Wang, 2000. "Production Externalities and Urban Configuration," Vanderbilt University Department of Economics Working Papers 0011, Vanderbilt University Department of Economics.
  2. Robert E. Lucas, Jr., 2001. "Externalities and Cities," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 4(2), pages 245-274, April.
  3. Behrens, Kristian & Murata, Yasusada, 2007. "General equilibrium models of monopolistic competition: A new approach," Journal of Economic Theory, Elsevier, vol. 136(1), pages 776-787, September.
  4. Paul Krugman, 1990. "Increasing Returns and Economic Geography," NBER Working Papers 3275, National Bureau of Economic Research, Inc.
  5. Kokovin, Sergey & Thisse, Jacques-Fran├žois & Zhelobodko, Evgeny, 2010. "Monopolistic Competition: Beyond the CES," CEPR Discussion Papers 7947, C.E.P.R. Discussion Papers.
  6. Robert E. Lucas & Esteban Rossi-Hansberg, 2002. "On the Internal Structure of Cities," Econometrica, Econometric Society, vol. 70(4), pages 1445-1476, July.
  7. Fujita, Masahisa & Ogawa, Hideaki, 1982. "Multiple equilibria and structural transition of non-monocentric urban configurations," Regional Science and Urban Economics, Elsevier, vol. 12(2), pages 161-196, May.
  8. Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
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