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Cones with Semi-interior Points and Equilibrium




We study exchange economies in ordered normed spaces (X, ||.||) where agents have possibly different consumption sets. We define the notion of semi-interior point of the positive cone X+ of X, a notion weaker than the one of interior point. We prove that if X+ has semi-interior points, then the second welfare theorem holds true and a quasi equilibrium allocation exists. In both cases the supporting price is continuous with respect to a new norm |||.||| on X which is strongly related with the initial norm and the ordering, and in some sense can be considered as an extension of the norm adopted in classical equilibrium models. Many examples of cones in normed and Banach spaces with semi-interior points but with empty interior are provided, showing that this class of cones is a rich one. We also consider spaces ordered by strongly reflexive cones where we prove the existence of a quasi equilibrium without the closedness condition (i.e. without the condition that the utility space is closed). The results in the case of semi-interior points derive from those concerning the case of ordering cones with nonempty interior.

Suggested Citation

  • Achille Basile & Maria Gabriella Graziano & Maria Papadaki & Ioannis A. Polyrakis, 2016. "Cones with Semi-interior Points and Equilibrium," CSEF Working Papers 443, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy.
  • Handle: RePEc:sef:csefwp:443

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    References listed on IDEAS

    1. Aliprantis, Charalambos D. & Monteiro, Paulo K. & Tourky, Rabee, 2004. "Non-marketed options, non-existence of equilibria, and non-linear prices," Journal of Economic Theory, Elsevier, vol. 114(2), pages 345-357, February.
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    5. Mas-Colell, Andreu, 1986. "The Price Equilibrium Existence Problem in Topological Vector Lattice s," Econometrica, Econometric Society, vol. 54(5), pages 1039-1053, September.
    6. Araujo, Aloisio, 1985. "Lack of Pareto Optimal Allocations in Economies with Infinitely Many Commodities: The Need for Impatience," Econometrica, Econometric Society, vol. 53(2), pages 455-461, March.
    7. Florenzano, Monique, 1983. "On the existence of equilibria in economies with an infinite dimensional commodity space," Journal of Mathematical Economics, Elsevier, vol. 12(3), pages 207-219, December.
    8. Mas-Colell, Andreu & Zame, William R., 1991. "Equilibrium theory in infinite dimensional spaces," Handbook of Mathematical Economics, in: W. Hildenbrand & H. Sonnenschein (ed.),Handbook of Mathematical Economics, edition 1, volume 4, chapter 34, pages 1835-1898, Elsevier.
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    cones; equilibrium; ordered spaces; second welfare theorem; strongly reflexive cones;

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies

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