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Solving Internal Habit Formation Models Through Dynamic Programming in Infinite Dimension

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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.

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Notes

  1. Other contributions to the literature have studied external habits, i.e., habits formed over the whole economy average of past consumption (e.g., Augeraud-Veron and Bambi [1]). In this article, we study only the case of internal habits and we will often refer to them simply as habits.

  2. The equity premium puzzle refers to the inability of models without habit formation, to explain the differential between the risky rate of return of the stock market and the riskless rate of interest, within reasonable parameter choices.

  3. See also [10] for a discrete versus continuous time comparison. Also [11,12,13] for the application of the dynamic programming technique to models with age structure.

  4. The case \(\gamma =1\) can be treated exactly as the other ones. Section 3.6 explains the main features of this case.

  5. The space \(L^1_{loc}([0,+\infty [;{\mathbb {R}}_+)\) is the set of all functions from \([0,+\infty [\) to \({\mathbb {R}}_+\) that are Lebesgue measurable and integrable on all bounded intervals.

  6. In the following we will indicate with \(W^{1,2}\) the Sobolev space defined as

    $$\begin{aligned} W^{1,2}:=\{f\in L^{2}\left( \left[ -\tau ,0\right] ;{\mathbb R}\right) ,Df\in L^{2}\left( \left[ -\tau ,0\right] ;\mathbb {R}\right) \}. \end{aligned}$$

  7. We refer to Bambi et al. [9, 23] for the definitions of \(\varPi \) and of the solutions in \(\varPi .\)

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Authors and Affiliations

  1. University of La Rochelle, La Rochelle, France

    Emmanuelle Augeraud-Veron

  2. University of York, York, UK

    Mauro Bambi

  3. LUISS University, Rome, Italy

    Fausto Gozzi

Authors
  1. Emmanuelle Augeraud-Veron
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  2. Mauro Bambi
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  3. Fausto Gozzi
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Correspondence to Fausto Gozzi.

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Communicated by Vladimir Veliov.

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Augeraud-Veron, E., Bambi, M. & Gozzi, F. Solving Internal Habit Formation Models Through Dynamic Programming in Infinite Dimension. J Optim Theory Appl 173, 584–611 (2017). https://doi.org/10.1007/s10957-017-1073-8

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  • DOI: https://doi.org/10.1007/s10957-017-1073-8

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