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Curvature and uniqueness of equilibrium

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  • Loi, Andrea
  • Matta, Stefano

Abstract

Let E(r) be the equilibrium manifold associated to a smooth pure exchange economy with fixed total resources r. Balasko (1980) has shown that if the equilibrium price is unique for every economy, then the price is constant, hence the curvature of E(r) is zero. By endowing E(r) with the metric induced from its ambient space, we show that, in the case of two commodities and an arbitrary number of agents, if the curvature of E(r) is zero then there is a unique equilibrium for every economy. Hence the zero curvature condition is sufficient to guarantee the uniqueness of equilibrium.

Suggested Citation

  • Loi, Andrea & Matta, Stefano, 2018. "Curvature and uniqueness of equilibrium," Journal of Mathematical Economics, Elsevier, vol. 74(C), pages 62-67.
    • Handle: RePEc:eee:mateco:v:74:y:2018:i:c:p:62-67
    DOI: 10.1016/j.jmateco.2017.11.002
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    References listed on IDEAS

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    1. Shapley, Lloyd S & Shubik, Martin, 1977. "An Example of a Trading Economy with Three Competitive Equilibria," Journal of Political Economy, University of Chicago Press, vol. 85(4), pages 873-875, August.
    2. Balasko, Yves, 1980. "Number and definiteness of economic equilibria," Journal of Mathematical Economics, Elsevier, vol. 7(3), pages 215-225, December.
    3. Loi, Andrea & Matta, Stefano, 2011. "Catastrophes minimization on the equilibrium manifold," Journal of Mathematical Economics, Elsevier, vol. 47(4), pages 617-620.
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    Cited by:

    1. Loi, Andrea & Matta, Stefano & Uccheddu, Daria, 2023. "Equilibrium selection under changes in endowments: A geometric approach," Journal of Mathematical Economics, Elsevier, vol. 108(C).
    2. Andrea Loi & Stefano Matta & Daria Uccheddu, 2023. "Uniqueness of equilibrium and redistributive policies: a geometric approach to efficiency," Papers 2308.03706, arXiv.org.
    3. Andrea Loi & Stefano Matta, 2021. "Minimal entropy and uniqueness of price equilibria in a pure exchange economy," Papers 2102.09827, arXiv.org.
    4. Andrea Loi & Stefano Matta & Daria Uccheddu, 2022. "Equilibrium selection: a geometric approach," Papers 2208.10860, arXiv.org.
    5. Andrea Loi & Stefano Matta, 2021. "Risk aversion and uniqueness of equilibrium: a polynomial approach," Papers 2107.01947, arXiv.org, revised Oct 2021.
    6. Loi, Andrea & Matta, Stefano, 2019. "Minimality and uniqueness of equilibrium," MPRA Paper 98055, University Library of Munich, Germany.
    7. Giménez, Eduardo L., 2022. "Offer curves and uniqueness of competitive equilibrium," Journal of Mathematical Economics, Elsevier, vol. 98(C).
    8. Loi, Andrea & Matta, Stefano, 2021. "Minimal entropy and uniqueness of price equilibria in a pure exchange economy," Journal of Mathematical Economics, Elsevier, vol. 97(C).

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