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Uniqueness of equilibrium and redistributive policies: a geometric approach to efficiency

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  • Andrea Loi
  • Stefano Matta
  • Daria Uccheddu

Abstract

This paper examines the relationship between resource reallocation, uniqueness of equilibrium and efficiency in economics. We explore the implications of reallocation policies for stability, conflict, and decision-making by analysing the existence of geodesic coordinate functions in the equilibrium manifold. Our main result shows that in an economy with M = 2 consumers and L goods, if L coordinate functions, representing policies, are geodesics on the equilibrium manifold (a property that we call the finite geodesic property), then the equilibrium is globally unique. The presence of geodesic variables indicates optimization and efficiency in the economy, while non-geodesic variables add complexity. Finally, we establish a link between the existing results on curvature, minimal entropy, geodesics and uniqueness in smooth exchange economies. This study contributes to the understanding of the geometric and economic properties of equilibria and offers potential applications in policy considerations. Keywords: Uniqueness of equilibrium, redistributive policies, geodesics, equilibrium manifold, equilibrium selection, curvature, geodesics.

Suggested Citation

  • Andrea Loi & Stefano Matta & Daria Uccheddu, 2023. "Uniqueness of equilibrium and redistributive policies: a geometric approach to efficiency," Papers 2308.03706, arXiv.org.
    • Handle: RePEc:arx:papers:2308.03706
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    References listed on IDEAS

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    1. 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).
    2. Loi, Andrea & Matta, Stefano, 2008. "Geodesics on the equilibrium manifold," Journal of Mathematical Economics, Elsevier, vol. 44(12), pages 1379-1384, December.
    3. Andrea Loi & Stefano Matta, 2021. "Minimal entropy and uniqueness of price equilibria in a pure exchange economy," Papers 2102.09827, arXiv.org.
    4. Loi, Andrea & Matta, Stefano, 2018. "Curvature and uniqueness of equilibrium," Journal of Mathematical Economics, Elsevier, vol. 74(C), pages 62-67.
    5. Debreu, Gerard, 1970. "Economies with a Finite Set of Equilibria," Econometrica, Econometric Society, vol. 38(3), pages 387-392, May.
    6. Marriott,Paul & Salmon,Mark (ed.), 2000. "Applications of Differential Geometry to Econometrics," Cambridge Books, Cambridge University Press, number 9780521651165.
    7. 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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