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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).

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File URL: http://mpra.ub.uni-muenchen.de/44994/
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Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 44994.

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Date of creation: 10 Mar 2013
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Handle: RePEc:pra:mprapa:44994

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

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  1. 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 727-32, May.
  2. Andrew B. Abel, 1988. "Operative Gift and Bequest Motives," NBER Working Papers 2331, National Bureau of Economic Research, Inc.
  3. Montrucchio, Luigi, 1987. "Lipschitz continuous policy functions for strongly concave optimization problems," Journal of Mathematical Economics, Elsevier, vol. 16(3), pages 259-273, June.
  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. Hori, Hajime, 1997. "Dynamic Allocation in an Altruistic Overlapping Generations Economy," Journal of Economic Theory, Elsevier, vol. 73(2), pages 292-315, April.
  7. Hori, Hajime & Kanaya, Sadao, 1989. "Utility functionals with nonpaternalistic intergenerational altruism," Journal of Economic Theory, Elsevier, vol. 49(2), pages 241-265, December.
  8. Araujo, A, 1991. "The Once but Not Twice Differentiability of the Policy Function," Econometrica, Econometric Society, vol. 59(5), pages 1383-93, September.
  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. Santos, Manuel S, 1991. "Smoothness of the Policy Function in Discrete Time Economic Models," Econometrica, Econometric Society, vol. 59(5), pages 1365-82, September.
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