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Elementary results on solutions to the bellman equation of dynamic programming: existence, uniqueness, and convergence

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  • Takashi Kamihigashi

Abstract

We establish some elementary results on solutions to the Bellman equation without introducing any topological assumption. Under a small number of conditions, we show that the Bellman equation has a unique solution in a certain set, that this solution is the value function, and that the value function can be computed by value iteration with an appropriate initial condition. In addition, we show that the value function can be computed by the same procedure under alternative conditions. We apply our results to two optimal growth models: one with a discontinuous production function and the other with “roughly increasing” returns. Copyright Springer-Verlag Berlin Heidelberg 2014

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  • 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.
    • Handle: RePEc:spr:joecth:v:56:y:2014:i:2:p:251-273
    DOI: 10.1007/s00199-013-0789-4
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    More about this item

    Keywords

    Dynamic programming; Bellman equation; Value function; Fixed point; C61;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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