Control theory in infinite dimension for the optimal location of economic activity: The role of social welfare function

Raouf Boucekkine (), Giorgio Fabbri, Salvatore Federico and Fausto Gozzi
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Salvatore Federico: UNISI - Università degli Studi di Siena = University of Siena

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Abstract: In this paper, we consider an abstract optimal control problem with state constraint. The methodology relies on the employment of the classical dynamic programming tool considered in the infinite dimensional context. We are able to identify a closed-form solution to the induced Hamilton-Jacobi-Bellman (HJB) equation in infinite dimension and to prove a verification theorem, also providing the optimal control in closed loop form. The abstract problem can be seen an abstract formulation of a PDE optimal control problem and is motivated by an economic application in the context of continuous spatiotemporal growth models with capital di usion, where a social planner chooses the optimal location of economic activity across space by maximization of an utilitarian social welfare function. From the economic point of view, we generalize previous works by considering a continuum of social welfare functions ranging from Benthamite to Millian functions. We prove that the Benthamite case is the unique case for which the optimal stationary detrended consumption spatial distribution is uniform. Interestingly enough, we also find that as the social welfare function gets closer to the Millian case, the optimal spatiotemporal dynamics amplify the typical neoclassical dilution population size effect, even in the long-run.

Keywords: INFINITE DIMENSION; HAMILTON-JACOBI-BELLMAN EQUATION; SPATIOTEMPORAL GROWTH MODEL; BENTHAMITE SOCIAL WELFARE FUNCTION; MILLIAN SOCIAL WELFARE FUNCTION; IMPERFECT ALTRUISM; PARTIAL DIFFERENTIAL EQUATION; PDE (search for similar items in EconPapers)
Date: 2021
New Economics Papers: this item is included in nep-dge, nep-geo and nep-gro
Note: View the original document on HAL open archive server: https://hal.science/hal-02548170v2
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Citations: View citations in EconPapers (1)

Published in Pure and Applied Functional Analysis, 2021, 6 (5), pp.871-888

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Working Paper: Control theory in infinite dimension for the optimal location of economic activity: The role of social welfare function (2020)
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