Solving Internal Habit Formation Models Through Dynamic Programming in Infinite Dimension

Emmanuelle Augeraud-Véron, Mauro Bambi () and Fausto Gozzi

Journal of Optimization Theory and Applications, 2017, vol. 173, issue 2, No 11, 584-611

Abstract: 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.

Keywords: Optimal control problems with delay; Dynamic programming; Habit formation; 49L20; 49K25; 34K35 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (6)

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DOI: 10.1007/s10957-017-1073-8

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