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

Aoki, Takaaki (2013): Some Mathematical Properties of the Dynamically Inconsistent Bellman Equation: A Note on the Two-sided Altruism Dynamics.

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).

• Item Type: MPRA Paper
• Language: English
• Keywords: Bellman equation, Two-sided altruism, Dynamic inconsistency, Self-consistent policy functions, Indeterminacy, Overlapping generations model.
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis; D - Microeconomics > D9 - Intertemporal Choice > D91 - Intertemporal Household Choice ; Life Cycle Models and Saving; O - Economic Development, Innovation, Technological Change, and Growth > O4 - Economic Growth and Aggregate Productivity > O41 - One, Two, and Multisector Growth Models

References

Abel A. B. (1987) "Operative Gift and Bequest Motives", American Economic Review, 77/5, 1037-1047.

Aoki T. (2011) "On the Implications of Two-Sided Altruism in Human Capital Based OLG Model", Asian Journal of Law and Economics, 2/2.

Araujo A. (1991) "The once but not twice differentiability of the policy function", Econometrica, 59/ 5, 1383-1393.

Bellman R. (1957) Dynamic Programming, Princeton, Princeton University Press.

Benveniste, LM. and Scheinkman J. A. (1979) "On the Differentiability of the Value Function in Dynamic Models of Economics", Econometrica, 47/3, 727-732.

Blackwell D. (1965) "Discounted Dynamic Programming", Annals of Mathematical Statistics, 36, 226-235.

Boldrin M. and Montrucchio L. (1986) "On the Indeterminacy of Capital Accumulation Paths", Journal of Economic Theory, 40/1, 26-39.

Geanakoplos J. and Brown D. J. (1985) "Comaparative Statics and Local Indeterminacy in OLG Economics: An Application of the Maultiplicative Ergodic Theorem", Cowles Foundation Discussion Paper, 773.

Hori H. (1997) "Dynamic Allocation in an Altruistic Overlapping Generations Economy", Journal of Economic Theory, 73/2, 292-315.

Hori H. and Kanaya S. (1989) "Utility Functionals with Nonpaternalistic Intergenerational Altruism", Journal of Economic Theory, 49, 241-265.

Kimball M. S. (1987) "Making Sense of Two-Sided Altruism", Journal of Monetary Economics, 20/2, 301-326.

Mitra T. (2000) "Introduction to Dynamic Optimization Theory", Optimization and Chaos, Majumdar, Mukal, Tapan Mitra and Kazuo Nishumura eds, Studies in Economic Theory, Springer.

Montrucchio L. (1987) "Lipschiz continuous policy functions for strongly concave optimization problems", Journal of Mathematical Economics, 16, 259-273.

Pontryagin, L. S. and Associates (1962) The Mathematical Theory of Optimal Processes, New York: Interscience.

Santos M. S. (1991) "Smoothness of the Policy Function in Discrete Time Economic Models", Econometrica, 59/5, 1365-1382.

Stokey N. and Lucas R. E. (1989) Recursive Methods in Economic Dynamics, Harvard University Press.