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Solving Internal Habit Formation Models Through Dynamic Programming in Infinite Dimension

Author

Listed:
  • Emmanuelle Augeraud-Veron

    (GREThA - Groupe de Recherche en Economie Théorique et Appliquée - UB - Université de Bordeaux - CNRS - Centre National de la Recherche Scientifique)

  • Mauro Bambi
  • Fausto Gozzi

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," Post-Print hal-02871232, HAL.
    • Handle: RePEc:hal:journl:hal-02871232
    DOI: 10.1007/s10957-017-1073-8
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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. Lisa Morhaim & Ayşegül Yıldız Ulus, 2023. "On history-dependent optimization models: A unified framework to analyze models with habits, satiation and optimal growth," Post-Print hal-04103627, HAL.
    4. Giorgio Fabbri & Silvia Faggian & Salvatore Federico & Fausto Gozzi, 2025. "Optimal Control in Infinite Dimensional Spaces and Economic Modeling: State of the Art and Perspectives," Working Papers 2025-06, Grenoble Applied Economics Laboratory (GAEL).
    5. 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).
    6. Giorgio Fabbri & Emmanuelle Augeraud-Véron, 2025. "Economic insecurity, memory effects and allocations choices," Working Papers hal-05398465, HAL.
    7. Snezhana Kirusheva & Thomas S. Salisbury, 2023. "A greedy algorithm for habit formation under multiplicative utility," Papers 2305.04748, arXiv.org.
    8. 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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