IDEAS home Printed from https://ideas.repec.org/a/eee/mateco/v71y2017icp36-48.html
   My bibliography  Save this article

Cones with semi-interior points and equilibrium

Author

Listed:
  • Basile, Achille
  • Graziano, Maria Gabriella
  • Papadaki, Maria
  • Polyrakis, Ioannis A.

Abstract

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 and we study the existence of equilibrium in the case where X+ has semi-interior points. In Section 4, we study the case where X+ has interior points and we prove a second welfare theorem and the existence of equilibrium. Subsequently we apply these results in the case where X+ has semi-interior points. In the case of semi-interior points the supporting price vectors are 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. In the last section we apply our results to strongly reflexive cones.

Suggested Citation

  • Basile, Achille & Graziano, Maria Gabriella & Papadaki, Maria & Polyrakis, Ioannis A., 2017. "Cones with semi-interior points and equilibrium," Journal of Mathematical Economics, Elsevier, vol. 71(C), pages 36-48.
    • Handle: RePEc:eee:mateco:v:71:y:2017:i:c:p:36-48
    DOI: 10.1016/j.jmateco.2017.03.002
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304406817300642
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmateco.2017.03.002?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Mas-Colell, Andreu & Richard, Scott F., 1991. "A new approach to the existence of equilibria in vector lattices," Journal of Economic Theory, Elsevier, vol. 53(1), pages 1-11, February.
    2. Mas-Colell, Andreu, 1986. "The Price Equilibrium Existence Problem in Topological Vector Lattice s," Econometrica, Econometric Society, vol. 54(5), pages 1039-1053, September.
    3. 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.
    4. Bewley, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," Journal of Economic Theory, Elsevier, vol. 4(3), pages 514-540, June.
    5. BEWLEY, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," LIDAM Reprints CORE 122, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. 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.
    7. Khan, M. Ali & Tourky, Rabee & Vohra, Rajiv, 1999. "The supremum argument in the new approach to the existence of equilibrium in vector lattices," Economics Letters, Elsevier, vol. 63(1), pages 61-65, April.
    8. Foivos Xanthos, 2014. "Non-existence of weakly Pareto optimal allocations," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(2), pages 137-146, October.
    9. Podczeck, Konrad & Yannelis, Nicholas C., 2008. "Equilibrium theory with asymmetric information and with infinitely many commodities," Journal of Economic Theory, Elsevier, vol. 141(1), pages 152-183, July.
    10. Charalambos Aliprantis & Kim Border & Owen Burkinshaw, 1996. "Market economies with many commodities," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 19(1), pages 113-185, March.
    11. 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.
    12. 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.
    13. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, November.
    14. Xanthos, Foivos, 2014. "A note on the equilibrium theory of economies with asymmetric information," Journal of Mathematical Economics, Elsevier, vol. 55(C), pages 1-3.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Huaping Huang, 2019. "Topological Properties of E -Metric Spaces with Applications to Fixed Point Theory," Mathematics, MDPI, vol. 7(12), pages 1-14, December.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Charalambos Aliprantis & Kim Border & Owen Burkinshaw, 1996. "Market economies with many commodities," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 19(1), pages 113-185, March.
    2. Charalambos Aliprantis & Rabee Tourky, 2009. "Equilibria in incomplete assets economies with infinite dimensional spot markets," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 38(2), pages 221-262, February.
    3. Aliprantis, Charalambos D. & Border, Kim C. & Burkinshaw, Owen, 1997. "Economies with Many Commodities," Journal of Economic Theory, Elsevier, vol. 74(1), pages 62-105, May.
    4. Aliprantis, C. D. & Tourky, R. & Yannelis, N. C., 2000. "The Riesz-Kantorovich formula and general equilibrium theory," Journal of Mathematical Economics, Elsevier, vol. 34(1), pages 55-76, August.
    5. Marakulin, Valeri M., 1998. "Production equilibria in vector lattices with unordered preferences : an approach using finite-dimensional approximations," CEPREMAP Working Papers (Couverture Orange) 9821, CEPREMAP.
    6. Monique Florenzano & Valeri Marakulin, 2000. "Production Equilibria in Vector Lattices," Econometric Society World Congress 2000 Contributed Papers 1396, Econometric Society.
    7. Messaoud Deghdak & Monique Florenzano, 1999. "Decentralizing Edgeworth equilibria in economies with many commodities," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 14(2), pages 297-310.
    8. Podczeck, Konrad, 1996. "Equilibria in vector lattices without ordered preferences or uniform properness," Journal of Mathematical Economics, Elsevier, vol. 25(4), pages 465-485.
    9. Manh-Hung Nguyen & San Nguyen Van, 2005. "The Lagrange multipliers and existence of competitive equilibrium in an intertemporal model with endogenous leisure," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00194723, HAL.
    10. Foivos Xanthos, 2014. "Non-existence of weakly Pareto optimal allocations," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(2), pages 137-146, October.
    11. Abramovich, Y A & Aliprantis, C D & Zame, W R, 1995. "A Representation Theorem for Riesz Spaces and Its Applications to Economics," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 5(3), pages 527-535, May.
    12. Carlos Hervés-Beloso & V. Martins-da-Rocha & Paulo Monteiro, 2009. "Equilibrium theory with asymmetric information and infinitely many states," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 38(2), pages 295-320, February.
    13. He, Wei & Yannelis, Nicholas C., 2015. "Equilibrium theory under ambiguity," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 86-95.
    14. Aloisio Araujo & Jean-Marc Bonnisseau & Alain Chateauneuf & Rodrigo Novinski, 2014. "Optimal Risk Sharing with Optimistic and Pessimistic Decision Makers," Working Papers 2014-579, Department of Research, Ipag Business School.
    15. Burke, Jonathan L., 2000. "General Equilibrium When Economic Growth Exceeds Discounting," Journal of Economic Theory, Elsevier, vol. 94(2), pages 141-162, October.
    16. Aloisio Araujo & Jean-Marc Bonnisseau & Alain Chateauneuf & Rodrigo Novinski, 2017. "Optimal sharing with an infinite number of commodities in the presence of optimistic and pessimistic agents," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 63(1), pages 131-157, January.
    17. Aliprantis, Charalambos D. & Tourky, Rabee & Yannelis, Nicholas C., 2001. "A Theory of Value with Non-linear Prices: Equilibrium Analysis beyond Vector Lattices," Journal of Economic Theory, Elsevier, vol. 100(1), pages 22-72, September.
    18. Aliprantis, Charalambos D. & Florenzano, Monique & Tourky, Rabee, 2006. "Production equilibria," Journal of Mathematical Economics, Elsevier, vol. 42(4-5), pages 406-421, August.
    19. Aase, Knut K., 2010. "Existence and Uniqueness of Equilibrium in a Reinsurance Syndicate," ASTIN Bulletin, Cambridge University Press, vol. 40(2), pages 491-517, November.
    20. Fuentes, Matías N., 2011. "Existence of equilibria in economies with externalities and non-convexities in an infinite-dimensional commodity space," Journal of Mathematical Economics, Elsevier, vol. 47(6), pages 768-776.

    More about this item

    Keywords

    Cones; Equilibrium; Ordered spaces; Second welfare theorem; Strongly reflexive cones;
    All these keywords.

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:mateco:v:71:y:2017:i:c:p:36-48. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jmateco .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.