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Solving the income fluctuation problem with unbounded rewards

Author

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  • Li, Huiyu
  • Stachurski, John
Abstract
This paper studies the income fluctuation problem without imposing bounds on utility, assets, income or consumption. We prove that the Coleman operator is a contraction mapping over the natural class of candidate consumption policies when endowed with a metric that evaluates consumption differences in terms of marginal utility. We show that this metric is complete, and that the fixed point of the operator coincides with the unique optimal policy. As a consequence, even in this unbounded setting, policy function iteration always converges to the optimal policy at a geometric rate.

Suggested Citation

  • Li, Huiyu & Stachurski, John, 2014. "Solving the income fluctuation problem with unbounded rewards," Journal of Economic Dynamics and Control, Elsevier, vol. 45(C), pages 353-365.
    • Handle: RePEc:eee:dyncon:v:45:y:2014:i:c:p:353-365
    DOI: 10.1016/j.jedc.2014.06.003
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    Citations

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    Cited by:

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    2. Pierri, Damian Rene & Reffett, Kevin, 2021. "Memory, multiple equilibria and emerging market crises," UC3M Working papers. Economics 32871, Universidad Carlos III de Madrid. Departamento de Economía.
    3. David Staines, 2023. "Stochastic Equilibrium the Lucas Critique and Keynesian Economics," Papers 2312.16214, arXiv.org, revised Jun 2024.
    4. Tjeerd de Vries & Alexis Akira Toda, 2022. "Capital and Labor Income Pareto Exponents Across Time and Space," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 68(4), pages 1058-1078, December.
    5. Ma, Qingyin & Toda, Alexis Akira, 2021. "A theory of the saving rate of the rich," Journal of Economic Theory, Elsevier, vol. 192(C).
    6. Stachurski, John & Toda, Alexis Akira, 2019. "An impossibility theorem for wealth in heterogeneous-agent models with limited heterogeneity," Journal of Economic Theory, Elsevier, vol. 182(C), pages 1-24.
    7. Lehrer, Ehud & Light, Bar, 2018. "The effect of interest rates on consumption in an income fluctuation problem," Journal of Economic Dynamics and Control, Elsevier, vol. 94(C), pages 63-71.
    8. Ma, Qingyin & Toda, Alexis Akira, 2022. "Asymptotic linearity of consumption functions and computational efficiency," Journal of Mathematical Economics, Elsevier, vol. 98(C).
    9. Ma, Qingyin & Stachurski, John & Toda, Alexis Akira, 2022. "Unbounded dynamic programming via the Q-transform," Journal of Mathematical Economics, Elsevier, vol. 100(C).
    10. Ma, Qingyin & Stachurski, John & Toda, Alexis Akira, 2020. "The income fluctuation problem and the evolution of wealth," Journal of Economic Theory, Elsevier, vol. 187(C).
    11. Acikgoz, Omer, 2015. "On the Existence of Equilibrium in Bewley Economies with Production," MPRA Paper 69061, University Library of Munich, Germany.
    12. Acikgoz, Omer, 2015. "On the Existence and Uniqueness of Stationary Equilibrium in Bewley Economies with Production," MPRA Paper 71066, University Library of Munich, Germany, revised 30 Apr 2016.
    13. Açıkgöz, Ömer T., 2018. "On the existence and uniqueness of stationary equilibrium in Bewley economies with production," Journal of Economic Theory, Elsevier, vol. 173(C), pages 18-55.
    14. Thomas J. Sargent & John Stachurski, 2024. "Dynamic Programming: Finite States," Papers 2401.10473, arXiv.org.
    15. Qingyin Ma & John Stachurski, 2019. "Dynamic Optimal Choice When Rewards are Unbounded Below," Papers 1911.13025, arXiv.org.
    16. Qingyin Ma & John Stachurski & Alexis Akira Toda, 2018. "The Income Fluctuation Problem with Capital Income Risk: Optimality and Stability," Papers 1812.01320, arXiv.org.
    17. Bar Light, 2018. "Precautionary Saving in a Markovian Earnings Environment," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 29, pages 138-147, July.
    18. Alexis Akira Toda, 2024. "Unbounded Markov dynamic programming with weighted supremum norm Perov contractions," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 12(2), pages 141-156, December.
    19. Bar Light, 2020. "Uniqueness of equilibrium in a Bewley–Aiyagari model," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 69(2), pages 435-450, March.

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    More about this item

    Keywords

    Coleman operator; Policy iteration; Time iteration; Global convergence;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • E21 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Consumption; Saving; Wealth

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