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Solving Internal Habit Formation Models Through Dynamic Programming in Infinite Dimension

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Listed:
  • Emmanuelle Augeraud-Veron

    (University of La Rochelle)

  • Mauro Bambi

    (University of York)

  • Fausto Gozzi

    (LUISS University)

Abstract

In this paper, we study an economic model, where internal habits play a role. Their formation is described by a more general functional form than is usually assumed in the literature, because a finite memory effect is allowed. Indeed, the problem becomes the optimal control of a standard ordinary differential equation, with the past of the control entering both the objective function and an inequality constraint. Therefore, the problem is intrinsically infinite dimensional. To solve this model, we apply the dynamic programming approach and we find an explicit solution for the associated Hamilton–Jacobi–Bellman equation, which lets us write the optimal strategies in feedback form. Therefore, we contribute to the existing literature in two ways. Firstly, we fully develop the dynamic programming approach to a type of problem not studied in previous contributions. Secondly, we use this result to unveil the global dynamics of an economy characterized by generic internal habits.

Suggested Citation

  • Emmanuelle Augeraud-Veron & Mauro Bambi & Fausto Gozzi, 2017. "Solving Internal Habit Formation Models Through Dynamic Programming in Infinite Dimension," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 584-611, May.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:2:d:10.1007_s10957-017-1073-8
    DOI: 10.1007/s10957-017-1073-8
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    1. Boucekkine, R. & Fabbri, G. & Gozzi, F., 2010. "Maintenance and investment: Complements or substitutes? A reappraisal," Journal of Economic Dynamics and Control, Elsevier, vol. 34(12), pages 2420-2439, December.
    2. Faggian, Silvia & Gozzi, Fausto, 2010. "Optimal investment models with vintage capital: Dynamic programming approach," Journal of Mathematical Economics, Elsevier, vol. 46(4), pages 416-437, July.
    3. M. Bambi & G. Fabbri & F. Gozzi, 2012. "Optimal policy and consumption smoothing effects in the time-to-build AK model," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 50(3), pages 635-669, August.
    4. Augeraud-Veron, Emmanuelle & Bambi, Mauro, 2015. "Endogenous growth with addictive habits," Journal of Mathematical Economics, Elsevier, vol. 56(C), pages 15-25.
    5. Constantinides, George M, 1990. "Habit Formation: A Resolution of the Equity Premium Puzzle," Journal of Political Economy, University of Chicago Press, vol. 98(3), pages 519-543, June.
    6. Boucekkine, R. & Fabbri, G. & Gozzi, F., 2014. "Egalitarianism under population change: Age structure does matter," Journal of Mathematical Economics, Elsevier, vol. 55(C), pages 86-100.
    7. Fabbri, Giorgio & Gozzi, Fausto, 2008. "Solving optimal growth models with vintage capital: The dynamic programming approach," Journal of Economic Theory, Elsevier, vol. 143(1), pages 331-373, November.
    8. Giuseppe Freni & Fausto Gozzi & Neri Salvadori, 2006. "Existence of optimal strategies in linear multisector models," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 29(1), pages 25-48, September.
    9. Harl E. Ryder & Geoffrey M. Heal, 1973. "Optimal Growth with Intertemporally Dependent Preferences," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 40(1), pages 1-31.
    10. Bambi, Mauro & Gori, Franco, 2014. "Unifying Time-To-Build Theory," Macroeconomic Dynamics, Cambridge University Press, vol. 18(8), pages 1713-1725, December.
    11. Mauro Bambi & Cristina Di Girolami & Salvatore Federico & Fausto Gozzi, 2014. "On the Consequences of Generically Distributed Investments on Flexible Projects in an Endogenous Growth Model," Discussion Papers 14/15, Department of Economics, University of York.
    12. Becker, Gary S & Murphy, Kevin M, 1988. "A Theory of Rational Addiction," Journal of Political Economy, University of Chicago Press, vol. 96(4), pages 675-700, August.
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    Cited by:

    1. Morhaim, Lisa & Ulus, Ayşegül Yıldız, 2023. "On history-dependent optimization models: A unified framework to analyze models with habits, satiation and optimal growth," Journal of Mathematical Economics, Elsevier, vol. 105(C).
    2. Bambi, Mauro & Gozzi, Fausto, 2020. "Internal habits formation and optimality," Journal of Mathematical Economics, Elsevier, vol. 91(C), pages 165-172.
    3. Augeraud-Veron, Emmanuelle & Boucekkine, Raouf & Gozzi, Fausto & Venditti, Alain & Zou, Benteng, 2024. "Fifty years of mathematical growth theory: Classical topics and new trends," Journal of Mathematical Economics, Elsevier, vol. 111(C).
    4. Snezhana Kirusheva & Thomas S. Salisbury, 2023. "A greedy algorithm for habit formation under multiplicative utility," Papers 2305.04748, arXiv.org.
    5. Xuepin Wu & Jiru Han, 2021. "Psychological Needs, Physiological Needs and Regional Comparison Effects," Sustainability, MDPI, vol. 13(16), pages 1-21, August.

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