IDEAS home Printed from https://ideas.repec.org/p/igi/igierp/561.html
   My bibliography  Save this paper

On the equality of Clarke-Rockafellar and Greenberg-Pierskalla differentials for monotone and quasiconcave functionals

Author

Listed:
  • Simone Cerreia-Vioglio
  • Fabio Maccheroni
  • Massimo Marinacci

Abstract

We study monotone, continuous, and quasiconcave functionals defifined over an M-space. We show that if g is also Clarke-Rockafellar differentiable at x and 0 62 @CRg (x), then the closure of Greenberg- Pierskalla differentials at x coincides with the closed cone generated by the Clarke-Rockafellar differentials at x. Under the same assumptions, we show that the set of normalized Greenberg-Pierskalla differentials at x coincides with the closure of the set of normalized Clarke-Rockafellar differentials at x. As a corollary, we obtain a differential characterization of quasiconcavity a la Arrow and Enthoven (1961) for Clarke-Rockafellar differentiable functions.

Suggested Citation

  • Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci, 2015. "On the equality of Clarke-Rockafellar and Greenberg-Pierskalla differentials for monotone and quasiconcave functionals," Working Papers 561, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    • Handle: RePEc:igi:igierp:561
    as

    Download full text from publisher

    File URL: https://repec.unibocconi.it/igier/igi/wp/2015/561.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Paolo Ghirardato & Marciano Siniscalchi, 2012. "Ambiguity in the Small and in the Large," Econometrica, Econometric Society, vol. 80(6), pages 2827-2847, November.
    2. Cerreia-Vioglio, S. & Maccheroni, F. & Marinacci, M. & Rustichini, A., 2015. "The structure of variational preferences," Journal of Mathematical Economics, Elsevier, vol. 57(C), pages 12-19.
    3. ,, 2016. "Objective rationality and uncertainty averse preferences," Theoretical Economics, Econometric Society, vol. 11(2), May.
    4. J. P. Crouzeix, 1980. "Conditions for Convexity of Quasiconvex Functions," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 120-125, February.
    5. Cerreia-Vioglio, S. & Maccheroni, F. & Marinacci, M. & Montrucchio, L., 2011. "Uncertainty averse preferences," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1275-1330, July.
    6. Hanany Eran & Klibanoff Peter, 2009. "Updating Ambiguity Averse Preferences," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 9(1), pages 1-53, November.
    7. Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci & Luigi Montrucchio, 2008. "Complete Monotone Quasiconcave Duality," Carlo Alberto Notebooks 80, Collegio Carlo Alberto.
    8. Simone Cerreia-Vioglio & Paolo Ghirardato & Fabio Maccheroni & Massimo Marinacci & Marciano Siniscalchi, 2011. "Rational preferences under ambiguity," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 48(2), pages 341-375, October.
    9. Jean-Paul Penot & Michel Volle, 1990. "On Quasi-Convex Duality," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 597-625, November.
    10. Martinez-Legaz, J. -E., 1991. "Duality between direct and indirect utility functions under minimal hypotheses," Journal of Mathematical Economics, Elsevier, vol. 20(2), pages 199-209.
    11. Crouzeix, J. -P., 1983. "Duality between direct and indirect utility functions : Differentiability properties," Journal of Mathematical Economics, Elsevier, vol. 12(2), pages 149-165, October.
    12. Luca Rigotti & Chris Shannon & Tomasz Strzalecki, 2008. "Subjective Beliefs and ex ante Trade," Econometrica, Econometric Society, vol. 76(5), pages 1167-1190, September.
    13. Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci & Luigi Montrucchio, 2011. "Complete Monotone Quasiconcave Duality," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 321-339, May.
    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. Centrone, Francesca & Rosazza Gianin, Emanuela, 2018. "Capital allocation à la Aumann–Shapley for non-differentiable risk measures," European Journal of Operational Research, Elsevier, vol. 267(2), pages 667-675.

    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. Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci & Luigi Montrucchio, 2011. "Complete Monotone Quasiconcave Duality," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 321-339, May.
    2. Cerreia-Vioglio, S. & Maccheroni, F. & Marinacci, M. & Rustichini, A., 2015. "The structure of variational preferences," Journal of Mathematical Economics, Elsevier, vol. 57(C), pages 12-19.
    3. Cerreia-Vioglio, Simone & Maccheroni, Fabio & Marinacci, Massimo & Montrucchio, Luigi, 2012. "Probabilistic sophistication, second order stochastic dominance and uncertainty aversion," Journal of Mathematical Economics, Elsevier, vol. 48(5), pages 271-283.
    4. Samuel Drapeau & Michael Kupper, 2013. "Risk Preferences and Their Robust Representation," Mathematics of Operations Research, INFORMS, vol. 38(1), pages 28-62, February.
    5. Cerreia-Vioglio, Simone & Maccheroni, Fabio & Marinacci, Massimo & Montrucchio, Luigi, 2013. "Ambiguity and robust statistics," Journal of Economic Theory, Elsevier, vol. 148(3), pages 974-1049.
      • Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci & Luigi Montrucchio, 2011. "Ambiguity and Robust Statistics," Working Papers 382, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    6. ,, 2016. "Objective rationality and uncertainty averse preferences," Theoretical Economics, Econometric Society, vol. 11(2), May.
    7. Eran Hanany & Peter Klibanoff & Sujoy Mukerji, 2020. "Incomplete Information Games with Ambiguity Averse Players," American Economic Journal: Microeconomics, American Economic Association, vol. 12(2), pages 135-187, May.
    8. Cerreia-Vioglio, S. & Maccheroni, F. & Marinacci, M. & Montrucchio, L., 2011. "Uncertainty averse preferences," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1275-1330, July.
    9. Simone Cerreia Vioglio & Fabio Maccheroni & Massimo Marinacci, 2016. "Absolute and Relative Ambiguity Aversion: A Preferential Approach," Working Papers 578, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    10. Karni, Edi & Maccheroni, Fabio & Marinacci, Massimo, 2015. "Ambiguity and Nonexpected Utility," Handbook of Game Theory with Economic Applications,, Elsevier.
    11. Cerreia-Vioglio, Simone & Maccheroni, Fabio & Marinacci, Massimo, 2022. "Ambiguity aversion and wealth effects," Journal of Economic Theory, Elsevier, vol. 199(C).
    12. André, Eric, 2016. "Crisp monetary acts in multiple-priors models of decision under ambiguity," Journal of Mathematical Economics, Elsevier, vol. 67(C), pages 153-161.
    13. Madhav Chandrasekher & Mira Frick & Ryota Iijima & Yves Le Yaouanq, 2022. "Dual‐Self Representations of Ambiguity Preferences," Econometrica, Econometric Society, vol. 90(3), pages 1029-1061, May.
    14. Matthias Lang, 2017. "First-Order and Second-Order Ambiguity Aversion," Management Science, INFORMS, vol. 63(4), pages 1254-1269, April.
    15. Spyros Galanis, 2021. "Dynamic consistency, valuable information and subjective beliefs," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(4), pages 1467-1497, June.
    16. Éric André, 2014. "Crisp Fair Gambles," AMSE Working Papers 1410, Aix-Marseille School of Economics, France, revised 15 Mar 2014.
    17. Chiaki Hara, 2023. "Arrow-Pratt-Type Measure of Ambiguity Aversion," KIER Working Papers 1097, Kyoto University, Institute of Economic Research.
    18. Sigrid Källblad, 2017. "Risk- and ambiguity-averse portfolio optimization with quasiconcave utility functionals," Finance and Stochastics, Springer, vol. 21(2), pages 397-425, April.
    19. Gumen, Anna & Savochkin, Andrei, 2013. "Dynamically stable preferences," Journal of Economic Theory, Elsevier, vol. 148(4), pages 1487-1508.
    20. Daniele Pennesi, 2013. "Endogenous Status Quo," Carlo Alberto Notebooks 314, Collegio Carlo Alberto.

    More about this item

    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:igi:igierp:561. 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: the person in charge (email available below). General contact details of provider: http://www.igier.unibocconi.it/ .

    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.