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Some Mathematical Properties of the Dynamically Inconsistent Bellman Equation: A Note on the Two-sided Altruism Dynamics

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  • Aoki, Takaaki

Abstract

This article describes some dynamic aspects on dynastic utility incorporating two-sided altruism with an OLG setting. We analyzed the special case where the weights of two-sided altruism are dynamically inconsistent. The Bellman equation for two-sided altruism proves to be reduced to one-sided dynamic problem, but the effective discount factor is different only in the current generation. We show that a contraction mapping result of value function cannot be achieved in general, and that there can locally exist an infinite number of self-consistent policy functions with distinct steady states (indeterminacy of self-consistent policy functions).

Suggested Citation

  • Aoki, Takaaki, 2013. "Some Mathematical Properties of the Dynamically Inconsistent Bellman Equation: A Note on the Two-sided Altruism Dynamics," MPRA Paper 44994, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:44994
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    File URL: https://mpra.ub.uni-muenchen.de/44994/1/MPRA_paper_44994.pdf
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    References listed on IDEAS

    as
    1. Araujo, A, 1991. "The Once but Not Twice Differentiability of the Policy Function," Econometrica, Econometric Society, vol. 59(5), pages 1383-1393, September.
    2. Benveniste, L M & Scheinkman, J A, 1979. "On the Differentiability of the Value Function in Dynamic Models of Economics," Econometrica, Econometric Society, vol. 47(3), pages 727-732, May.
    3. Santos, Manuel S, 1991. "Smoothness of the Policy Function in Discrete Time Economic Models," Econometrica, Econometric Society, vol. 59(5), pages 1365-1382, September.
    4. Aoki Takaaki, 2011. "On the Implications of Two-Sided Altruism in Human Capital Based OLG Model," Asian Journal of Law and Economics, De Gruyter, vol. 2(2), pages 1-40, July.
    5. Kimball, Miles S., 1987. "Making sense of two-sided altruism," Journal of Monetary Economics, Elsevier, vol. 20(2), pages 301-326, September.
    6. Montrucchio, Luigi, 1987. "Lipschitz continuous policy functions for strongly concave optimization problems," Journal of Mathematical Economics, Elsevier, vol. 16(3), pages 259-273, June.
    7. Abel, Andrew B, 1987. "Operative Gift and Bequest Motives," American Economic Review, American Economic Association, vol. 77(5), pages 1037-1047, December.
    8. Hori, Hajime, 1997. "Dynamic Allocation in an Altruistic Overlapping Generations Economy," Journal of Economic Theory, Elsevier, vol. 73(2), pages 292-315, April.
    9. Boldrin, Michele & Montrucchio, Luigi, 1986. "On the indeterminacy of capital accumulation paths," Journal of Economic Theory, Elsevier, vol. 40(1), pages 26-39, October.
    10. Hori, Hajime & Kanaya, Sadao, 1989. "Utility functionals with nonpaternalistic intergenerational altruism," Journal of Economic Theory, Elsevier, vol. 49(2), pages 241-265, December.
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    More about this item

    Keywords

    Bellman equation; Two-sided altruism; Dynamic inconsistency; Self-consistent policy functions; Indeterminacy; Overlapping generations model.;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D91 - Microeconomics - - Micro-Based Behavioral Economics - - - Role and Effects of Psychological, Emotional, Social, and Cognitive Factors on Decision Making
    • O41 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - One, Two, and Multisector Growth Models

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