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Existence and uniqueness of a fixed-point for local contractions

Author

Listed:
  • Vailakis, Yiannis
  • Martins-da-Rocha, Victor Filipe

Abstract

This paper proves the existence and uniqueness of a fixed-point for local contractions without assuming the family of contraction coefficients to be uniformly bounded away from 1. More importantly it shows how this fixed-point result can apply to study existence and uniqueness of solutions to some recursive equations that arise in economic dynamics.

Suggested Citation

  • Vailakis, Yiannis & Martins-da-Rocha, Victor Filipe, 2008. "Existence and uniqueness of a fixed-point for local contractions," FGV/EPGE Economics Working Papers (Ensaios Economicos da EPGE) 677, FGV/EPGE - Escola Brasileira de Economia e Finanças, Getulio Vargas Foundation (Brazil).
  • Handle: RePEc:fgv:epgewp:677
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    File URL: http://bibliotecadigital.fgv.br/dspace/bitstream/10438/1620/10/ECMA-MS7920-Supplemental-Material-07-12-10.pdf
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    References listed on IDEAS

    as
    1. Janusz Matkowski & Andrzej Nowak, 2011. "On discounted dynamic programming with unbounded returns," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 46(3), pages 455-474, April.
    2. Juan Rincón-Zapatero & Carlos Rodríguez-Palmero, 2007. "Recursive utility with unbounded aggregators," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 33(2), pages 381-391, November.
    3. Juan Pablo RincÛn-Zapatero & Carlos RodrÌguez-Palmero, 2003. "Existence and Uniqueness of Solutions to the Bellman Equation in the Unbounded Case," Econometrica, Econometric Society, vol. 71(5), pages 1519-1555, September.
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    Cited by:

    1. Robert A. Becker & Juan Pablo Rincón-Zapatero, 2017. "Arbitration and Renegotiation in Trade Agreements," Caepr Working Papers 2017-007 Classification-D, Center for Applied Economics and Policy Research, Economics Department, Indiana University Bloomington.
    2. Janusz Matkowski & Andrzej Nowak, 2011. "On discounted dynamic programming with unbounded returns," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 46(3), pages 455-474, April.
    3. V. Filipe Martins-da-Rocha & Yiannis Vailakis, 2013. "Fixed point for local contractions: Applications to recursive utility," International Journal of Economic Theory, The International Society for Economic Theory, vol. 9(1), pages 23-33, March.
    4. Philippe Bich & Jean-Pierre Drugeon & Lisa Morhaim, 2017. "On Temporal Aggregators and Dynamic Programming," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01437496, HAL.
    5. Anna Jaśkiewicz & Andrzej Nowak, 2011. "Stochastic Games with Unbounded Payoffs: Applications to Robust Control in Economics," Dynamic Games and Applications, Springer, vol. 1(2), pages 253-279, June.
    6. Philippe Bich & Jean-Pierre Drugeon & Lisa Morhaim, 2015. "On Aggregators and Dynamic Programming," Documents de travail du Centre d'Economie de la Sorbonne 15053, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    7. repec:spr:etbull:v:5:y:2017:i:2:d:10.1007_s40505-016-0112-1 is not listed on IDEAS
    8. Takashi Kamihigashi, 2014. "Elementary results on solutions to the bellman equation of dynamic programming: existence, uniqueness, and convergence," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 56(2), pages 251-273, June.
    9. Takashi Kamihigashi, 2011. "Existence and Uniqueness of a Fixed Point for the Bellman Operator in Deterministic Dynamic Programming," Discussion Paper Series DP2011-23, Research Institute for Economics & Business Administration, Kobe University.
    10. Takashi Kamihigashi & Cuong Le Van, 2015. "Necessary and Sufficient Conditions for a Solution of the Bellman Equation to be the Value Function: A General Principle," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01159177, HAL.
    11. Takashi Kamihigashi, 2014. "An order-theoretic approach to dynamic programming: an exposition," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(1), pages 13-21, April.
    12. Kazuo Nishimura & Myrna Wooders & Makoto Yano, 2013. "Macroeconomic dynamics and its micro foundation: A special issue in honor of Cuong Le Van," International Journal of Economic Theory, The International Society for Economic Theory, vol. 9(1), pages 1-3, March.
    13. Takashi Kamihigashi & Masayuki Yao, 2015. "Infnite-Horizon Deterministic Dynamic Programming in Discrete Time: A Monotone Convergence Principle," Discussion Paper Series DP2015-32, Research Institute for Economics & Business Administration, Kobe University.
    14. Matthias Messner & Nicola Pavoni & Christopher Sleet, "undated". "Contractive Dual Methods for Incentive Problems," GSIA Working Papers 2012-E26, Carnegie Mellon University, Tepper School of Business.
    15. Philippe Bich & Jean-Pierre Drugeon & Lisa Morhaim, 2015. "On Aggregators and Dynamic Programming," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01169552, HAL.
    16. repec:eee:jetheo:v:173:y:2018:i:c:p:181-200 is not listed on IDEAS
    17. Łukasz Balbus & Kevin Reffett & Łukasz Woźny, 2015. "Time consistent Markov policies in dynamic economies with quasi-hyperbolic consumers," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(1), pages 83-112, February.
    18. Nicole Bauerle & Anna Ja'skiewicz, 2015. "Stochastic Optimal Growth Model with Risk Sensitive Preferences," Papers 1509.05638, arXiv.org.
    19. repec:hal:journl:halshs-01169552 is not listed on IDEAS
    20. Jenö Pál & John Stachurski, 2011. "Fitted Value Function Iteration With Probability One Contractions," ANU Working Papers in Economics and Econometrics 2011-560, Australian National University, College of Business and Economics, School of Economics.
    21. Takashi Kamihigashi & Masayuki Yao, 2016. "Infinite-Horizon Deterministic Dynamic Programming in Discrete Time: A Monotone Convergence Principle and a Penalty Method," Discussion Paper Series DP2016-05, Research Institute for Economics & Business Administration, Kobe University, revised May 2016.
    22. repec:eee:jetheo:v:173:y:2018:i:c:p:118-141 is not listed on IDEAS
    23. Takashi Kamihigashi & Masayuki Yao, 2015. "Deterministic Dynamic Programming in Discrete Time: A Monotone Convergence Principle," Discussion Paper Series DP2015-15, Research Institute for Economics & Business Administration, Kobe University.
    24. João Brogueira & Fabian Schütze, 2017. "Existence and uniqueness of equilibrium in Lucas’ asset pricing model when utility is unbounded," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 5(2), pages 179-190, October.
    25. repec:hal:journl:halshs-01159177 is not listed on IDEAS

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