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The spatial AK model and the Pontryagin maximum principle

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  • Ballestra, Luca Vincenzo
Abstract
We are concerned with the endogenous growth model, namely the spatial AK model, that has recently been proposed and analyzed by Boucekkine et al. (2013a,b). From the mathematical standpoint, this model consists of an infinite-horizon parabolic optimal control problem, which is excellently solved in Boucekkine et al. (2013b) by means of dynamic programming. Nevertheless, one of the main aims of Boucekkine et al. (2013a,b) is also to show that the spatial AK model cannot be dealt with using the maximum principle of Pontryagin. More precisely, according to the analysis carried out by Boucekkine, Camacho and Fabbri, the Pontryagin conditions, albeit necessary, would not allow one to determine the unique solution of the optimal control problem. In the present paper, we show that such a conclusion needs to be reconsidered. In particular, if a Michel-type transversality condition is imposed and the fact that the adjoint variable must be non-negative is taken into account, the maximum principle is capable of yielding the unique solution of the spatial AK model.

Suggested Citation

  • Ballestra, Luca Vincenzo, 2016. "The spatial AK model and the Pontryagin maximum principle," Journal of Mathematical Economics, Elsevier, vol. 67(C), pages 87-94.
    • Handle: RePEc:eee:mateco:v:67:y:2016:i:c:p:87-94
    DOI: 10.1016/j.jmateco.2016.09.012
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    2. Emmanuelle Augeraud-Véron & Raouf Boucekkine & Vladimir Veliov, 2019. "Distributed Optimal Control Models in Environmental Economics: A Review," AMSE Working Papers 1902, Aix-Marseille School of Economics, France.
    3. Paulo B. Brito, 2022. "The dynamics of growth and distribution in a spatially heterogeneous world," Portuguese Economic Journal, Springer;Instituto Superior de Economia e Gestao, vol. 21(3), pages 311-350, September.
    4. Raouf Boucekkine & Giorgio Fabbri & Salvatore Federico & Fausto Gozzi, 2019. "A spatiotemporal framework for the analytical study of optimal growth under transboundary pollution," LIDAM Discussion Papers IRES 2019016, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
    5. Xepapadeas, Anastasios & Yannacopoulos, Athanasios N., 2023. "Spatial growth theory: Optimality and spatial heterogeneity," Journal of Economic Dynamics and Control, Elsevier, vol. 146(C).
    6. Spyridon Tsangaris & Anastasios Xepapadeas & Athanasios Yannacopoulos, 2022. "Spatial externalities, R&D spillovers, and endogenous technological change," DEOS Working Papers 2225, Athens University of Economics and Business.

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