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Recursive Utility and Thompson Aggregators, II: Uniqueness of the Recursive Utility Representation

Author

Listed:
  • Robert Becker

    (Indiana University)

  • Juan Pablo Rincon-Zapatero

    (Universidad Carlos III de Madrid)

Abstract

We reconsider the theory of Thompson aggregators proposed by Mari-nacci and Montrucchio [30]. We demonstrate the Koopmans equation has a unique utility function solution given a Thompson aggregator. Uniqueness holds only on the interior of the commodity spaces positive cone. Our proof veries the Koopmans operator is a u0 concave operator. We verify this using general sufficient conditions due to Liang, et al [28]. Previous published results apply variants of the contraction mapping theorem to the space of possibly utility functions endowed with the Thompson metric. Concave operator methods work on the possible utility function space with its norm topology. Our approach combines order and metric structures to demonstrate uniqueness differently than in the existing literature.

Suggested Citation

  • Robert Becker & Juan Pablo Rincon-Zapatero, 2018. "Recursive Utility and Thompson Aggregators, II: Uniqueness of the Recursive Utility Representation," CAEPR Working Papers 2018-008, Center for Applied Economics and Policy Research, Department of Economics, Indiana University Bloomington.
    • Handle: RePEc:inu:caeprp:2018008
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    File URL: https://caepr.indiana.edu/RePEc/inu/caeprp/caepr2018-008.pdf
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    References listed on IDEAS

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    1. Le Van, Cuong & Vailakis, Yiannis, 2005. "Recursive utility and optimal growth with bounded or unbounded returns," Journal of Economic Theory, Elsevier, vol. 123(2), pages 187-209, August.
    2. 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.
    3. Lucas, Robert Jr. & Stokey, Nancy L., 1984. "Optimal growth with many consumers," Journal of Economic Theory, Elsevier, vol. 32(1), pages 139-171, February.
    4. Bloise, Gaetano & Vailakis, Yiannis, 2018. "Convex dynamic programming with (bounded) recursive utility," Journal of Economic Theory, Elsevier, vol. 173(C), pages 118-141.
    5. 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.
    6. Massimo Marinacci & Luigi Montrucchio, 2017. "Unique Tarski Fixed Points," Working Papers 604, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    7. Cuong Le Van & Lisa Morhaim & Yiannis Vailakis, 2008. "Monotone Concave Operators: An application to the existence and uniqueness of solutions to the Bellman equation," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00294828, HAL.
    8. Robert Becker & Juan Pablo Rincon-Zapatero, 2018. "Recursive Utility and Thompson Aggregators, I: Constructive Existence Theory for the Koopmans Equation," CAEPR Working Papers 2018-006, Center for Applied Economics and Policy Research, Department of Economics, Indiana University Bloomington.
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    11. V. Filipe Martins-da-Rocha & Yiannis Vailakis, 2010. "Existence and Uniqueness of a Fixed Point for Local Contractions," Econometrica, Econometric Society, vol. 78(3), pages 1127-1141, May.
    12. Cuong Le Van & Lisa Morhaim & Yiannis Vailakis, 2008. "Monotone Concave Operators: An application to the existence and uniqueness of solutions to the Bellman equation," Working Papers hal-00294828, HAL.
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    14. Boud, John III, 1990. "Recursive utility and the Ramsey problem," Journal of Economic Theory, Elsevier, vol. 50(2), pages 326-345, April.
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    1. Stachurski, John & Wilms, Ole & Zhang, Junnan, 2024. "Asset pricing with time preference shocks: Existence and uniqueness," Journal of Economic Theory, Elsevier, vol. 216(C).

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    More about this item

    Keywords

    Recursive Utility; Thompson Aggregators; Koopmans Equation; u0 – Concave Operator Theory;
    All these keywords.

    JEL classification:

    • D10 - Microeconomics - - Household Behavior - - - General
    • D15 - Microeconomics - - Household Behavior - - - Intertemporal Household Choice; Life Cycle Models and Saving
    • D50 - Microeconomics - - General Equilibrium and Disequilibrium - - - General
    • E21 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Consumption; Saving; Wealth

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