[go: up one dir, main page]
IDEAS home Printed from https://ideas.repec.org/p/gbl/wpaper/2020-02.html
   My bibliography  Save this paper

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

Author

Listed:
  • Boucekkine, R.
  • Fabbri, G.
  • Federico, S.
  • Gozzi, F.
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.

Suggested Citation

  • Boucekkine, R. & Fabbri, G. & Federico, S. & Gozzi, F., 2020. "Control theory in infinite dimension for the optimal location of economic activity: The role of social welfare function," Working Papers 2020-02, Grenoble Applied Economics Laboratory (GAEL).
    • Handle: RePEc:gbl:wpaper:2020-02
    as

    Download full text from publisher

    File URL: https://gael.univ-grenoble-alpes.fr/sites/default/files/Mediatheque/doc-recherche/WP/A2020/gael2020-02.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Boucekkine, R. & Camacho, C. & Fabbri, G., 2013. "Spatial dynamics and convergence: The spatial AK model," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2719-2736.
    2. Boucekkine, Raouf & Camacho, Carmen & Zou, Benteng, 2009. "Bridging The Gap Between Growth Theory And The New Economic Geography: The Spatial Ramsey Model," Macroeconomic Dynamics, Cambridge University Press, vol. 13(1), pages 20-45, February.
    3. Raouf Boucekkine & Giorgio Fabbri, 2013. "Assessing Parfit’s Repugnant Conclusion within a canonical endogenous growth set-up," Journal of Population Economics, Springer;European Society for Population Economics, vol. 26(2), pages 751-767, April.
    4. Raouf Boucekkine & Giorgio Fabbri & Salvatore Federico & Fausto Gozzi, 2019. "Growth and agglomeration in the heterogeneous space: a generalized AK approach," Journal of Economic Geography, Oxford University Press, vol. 19(6), pages 1287-1318.
    5. Krugman, Paul, 1991. "Increasing Returns and Economic Geography," Journal of Political Economy, University of Chicago Press, vol. 99(3), pages 483-499, June.
    6. Faggian, Silvia & Gozzi, Fausto, 2010. "Optimal investment models with vintage capital: Dynamic programming approach," Journal of Mathematical Economics, Elsevier, vol. 46(4), pages 416-437, July.
    7. Treb Allen & Costas Arkolakis, 2014. "Trade and the Topography of the Spatial Economy," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 129(3), pages 1085-1140.
    8. Palivos, Theodore & Yip, Chong K., 1993. "Optimal population size and endogenous growth," Economics Letters, Elsevier, vol. 41(1), pages 107-110.
    9. Fabbri, Giorgio, 2016. "Geographical structure and convergence: A note on geometry in spatial growth models," Journal of Economic Theory, Elsevier, vol. 162(C), pages 114-136.
    10. Robin Cubitt & Chris Starmer & Robert Sugden, 2001. "Discovered preferences and the experimental evidence of violations of expected utility theory," Journal of Economic Methodology, Taylor & Francis Journals, vol. 8(3), pages 385-414.
    11. Fabbri, Giorgio & Gozzi, Fausto, 2008. "Solving optimal growth models with vintage capital: The dynamic programming approach," Journal of Economic Theory, Elsevier, vol. 143(1), pages 331-373, November.
    12. 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.
    13. World Bank, 2016. "World Development Indicators 2016," World Bank Publications - Books, The World Bank Group, number 23969.
    14. Marc Nerlove & Assaf Razin & Efraim Sadka, 1985. "Population Size: Individual Choice and Social Optima," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 100(2), pages 321-334.
    15. Lopez, Humberto, 2008. "The social discount rate : estimates for nine Latin American countries," Policy Research Working Paper Series 4639, The World Bank.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Albeverio, Sergio & Mastrogiacomo, Elisa, 2022. "Large deviation principle for spatial economic growth model on networks," Journal of Mathematical Economics, Elsevier, vol. 103(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Boucekkine, R. & Fabbri, G. & Federico, S. & Gozzi, F., 2020. "Control theory in infinite dimension for the optimal location of economic activity: The role of social welfare function," Working Papers 2020-02, Grenoble Applied Economics Laboratory (GAEL).
    2. Raouf Boucekkine & Giorgio Fabbri & Salvatore Federico & Fausto Gozzi, 2020. "Optimal location of economic activity and population density: The role of the social welfare function," AMSE Working Papers 2003, Aix-Marseille School of Economics, France.
    3. Raouf Boucekkine & Giorgio Fabbri & Salvatore Federico & Fausto Gozzi, 2019. "Growth and agglomeration in the heterogeneous space: a generalized AK approach," Journal of Economic Geography, Oxford University Press, vol. 19(6), pages 1287-1318.
    4. 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.
    5. Boucekkine, Raouf & Fabbri, Giorgio & Federico, Salvatore & Gozzi, Fausto, 2022. "A dynamic theory of spatial externalities," Games and Economic Behavior, Elsevier, vol. 132(C), pages 133-165.
    6. Raouf Boucekkine & Giorgio Fabbri & Salvatore Federico & Fausto Gozzi, 2020. "A dynamic theory of spatial externalities," Working Papers halshs-02613177, HAL.
    7. Xepapadeas, Anastasios & Yannacopoulos, Athanasios N., 2023. "Spatial growth theory: Optimality and spatial heterogeneity," Journal of Economic Dynamics and Control, Elsevier, vol. 146(C).
    8. Calvia, Alessandro & Gozzi, Fausto & Leocata, Marta & Papayiannis, Georgios I. & Xepapadeas, Anastasios & Yannacopoulos, Athanasios N., 2024. "An optimal control problem with state constraints in a spatio-temporal economic growth model on networks," Journal of Mathematical Economics, Elsevier, vol. 113(C).
    9. 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.
    10. Ballestra, Luca Vincenzo, 2016. "The spatial AK model and the Pontryagin maximum principle," Journal of Mathematical Economics, Elsevier, vol. 67(C), pages 87-94.
    11. Giorgio FABBRI, 2014. "Ecological Barriers and Convergence: a Note on Geometry in Spatial Growth Models," LIDAM Discussion Papers IRES 2014014, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
    12. Torre, Davide La & Liuzzi, Danilo & Marsiglio, Simone, 2021. "Transboundary pollution externalities: Think globally, act locally?," Journal of Mathematical Economics, Elsevier, vol. 96(C).
    13. Fabbri, Giorgio, 2016. "Geographical structure and convergence: A note on geometry in spatial growth models," Journal of Economic Theory, Elsevier, vol. 162(C), pages 114-136.
    14. 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.
    15. Faggian, Silvia & Gozzi, Fausto & Kort, Peter M., 2021. "Optimal investment with vintage capital: Equilibrium distributions," Journal of Mathematical Economics, Elsevier, vol. 96(C).
    16. Giorgio Fabbri, 2014. "Ecological Barriers and Convergence: A Note on Geometry in Spatial Growth Models," Documents de recherche 14-05, Centre d'Études des Politiques Économiques (EPEE), Université d'Evry Val d'Essonne.
    17. Carmen Camacho & Alexandre Cornet, 2021. "Diffusion of soil pollution in an agricultural economy. The emergence of regions, frontiers and spatial patterns," PSE Working Papers halshs-02652191, HAL.
    18. Carmen Camacho & Alexandre Cornet, 2021. "Diffusion of soil pollution in an agricultural economy. The emergence of regions, frontiers and spatial patterns," Working Papers halshs-02652191, HAL.
    19. Raouf Boucekkine & Giorgio Fabbri & Fausto Gozzi, 2010. "Life span and the problem of optimal population size," Working Papers halshs-00536073, HAL.
    20. Raouf Boucekkine & Giorgio Fabbri & Fausto Gozzi, 2011. "Revisiting the Optimal Population Size Problem under Endogenous Growth: Minimal Utility Level and Finite Life," Asia-Pacific Journal of Accounting & Economics, Taylor & Francis Journals, vol. 18(3), pages 287-305.

    More about this item

    Keywords

    SPATIOTEMPORAL GROWTH MODEL; BENTHAMITE SOCIAL WELFARE FUNCTION; MILLIAN SOCIAL WELFARE FUNCTION; IMPERFECT ALTRUISM; PARTIAL DIFFERENTIAL EQUATIONS; PDE; HAMILTON-JACOBI-BELLMAN EQUATIONS; INFINITE DIMENSION;
    All these keywords.

    JEL classification:

    • R10 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General Regional Economics - - - General
    • O40 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - General
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gbl:wpaper:2020-02. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Adrien Hervouet (email available below). General contact details of provider: https://edirc.repec.org/data/inragfr.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.