IDEAS home Printed from https://ideas.repec.org/p/hal/psewpa/halshs-03920386.html
   My bibliography  Save this paper

On the uniqueness of the optimal path in a discrete-time model à la Lucas (1988)

Author

Listed:
  • Stefano Bosi

    (EPEE - Centre d'Etudes des Politiques Economiques - UEVE - Université d'Évry-Val-d'Essonne - Université Paris-Saclay, Université Paris-Saclay, UEVE - Université d'Évry-Val-d'Essonne)

  • Carmen Camacho

    (PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, PJSE - Paris Jourdan Sciences Economiques - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Thai Ha-Huy

    (EPEE - Centre d'Etudes des Politiques Economiques - UEVE - Université d'Évry-Val-d'Essonne - Université Paris-Saclay, Université Paris-Saclay, UEVE - Université d'Évry-Val-d'Essonne)

Abstract

In a simple discrete-time version of the Lucas (1988) model, we prove, first, that the Balanced Growth Path (BGP) is optimal and, second, that the optimal solution is unique. After briefly discussing the results obtained in some of the continuous-time versions of the Lucas model, we address the issues of existence and uniqueness of the optimal solution in discrete time. Making use of the supermodularity of the value function, we prove that the optimal solution must be monotonically increasing. This feature does indeed suit the BGP, although nothing can ensure yet its optimality. After providing the exact expression of the BGP, we do prove both its optimality and its uniqueness.

Suggested Citation

  • Stefano Bosi & Carmen Camacho & Thai Ha-Huy, 2023. "On the uniqueness of the optimal path in a discrete-time model à la Lucas (1988)," PSE Working Papers halshs-03920386, HAL.
    • Handle: RePEc:hal:psewpa:halshs-03920386
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-03920386
    as

    Download full text from publisher

    File URL: https://shs.hal.science/halshs-03920386/document
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Romer, Paul M, 1986. "Increasing Returns and Long-run Growth," Journal of Political Economy, University of Chicago Press, vol. 94(5), pages 1002-1037, October.
    2. Benhabib Jess & Perli Roberto, 1994. "Uniqueness and Indeterminacy: On the Dynamics of Endogenous Growth," Journal of Economic Theory, Elsevier, vol. 63(1), pages 113-142, June.
    3. Casey B. Mulligan & Xavier Sala-i-Martin, 1993. "Transitional Dynamics in Two-Sector Models of Endogenous Growth," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 108(3), pages 739-773.
    4. Dirk Bethmann, 2013. "Solving Macroeconomic Models with Homogeneous Technology and Logarithmic Preferences," Australian Economic Papers, Wiley Blackwell, vol. 52(1), pages 1-18, March.
    5. A. Bucci & C. Colapinto & M. Forster & D. La Torre, 2011. "Stochastic technology shocks in an extended Uzawa–Lucas model: closed-form solution and long-run dynamics," Journal of Economics, Springer, vol. 103(1), pages 83-99, May.
    6. Arantza Gorostiaga & Jana Hromcová & Miguel-Ángel López-García, 2013. "Optimal taxation in the Uzawa–Lucas model with externality in human capital," Journal of Economics, Springer, vol. 108(2), pages 111-129, March.
    7. La Torre, Davide & Marsiglio, Simone, 2010. "Endogenous technological progress in a multi-sector growth model," Economic Modelling, Elsevier, vol. 27(5), pages 1017-1028, September.
    8. Ruiz-Tamarit, José Ramón, 2008. "The closed-form solution for a family of four-dimension nonlinear MHDS," Journal of Economic Dynamics and Control, Elsevier, vol. 32(3), pages 1000-1014, March.
    9. Ha-Huy, Thai & Tran, Nhat Thien, 2020. "A simple characterisation for sustained growth," Journal of Mathematical Economics, Elsevier, vol. 91(C), pages 141-147.
    10. Lucas, Robert Jr., 1988. "On the mechanics of economic development," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 3-42, July.
    11. Hiraguchi, Ryoji, 2009. "A note on the closed-form solution to the Lucas-Uzawa model with externality," Journal of Economic Dynamics and Control, Elsevier, vol. 33(10), pages 1757-1760, October.
    12. Dirk Bethmann, 2007. "A Closed-form Solution of the Uzawa-Lucas Model of Endogenous Growth," Journal of Economics, Springer, vol. 90(1), pages 87-107, January.
    13. Amir, Rabah, 1996. "Sensitivity analysis of multisector optimal economic dynamics," Journal of Mathematical Economics, Elsevier, vol. 25(1), pages 123-141.
    14. Ladron-de-Guevara, Antonio & Ortigueira, Salvador & Santos, Manuel S., 1997. "Equilibrium dynamics in two-sector models of endogenous growth," Journal of Economic Dynamics and Control, Elsevier, vol. 21(1), pages 115-143, January.
    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. Canova, Fabio & Sæterhagen Paulsen, Kenneth, 2023. "Symbolic stationarization of dynamic equilibrium models," Journal of Economic Dynamics and Control, Elsevier, vol. 154(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. Bosi, Stefano & Camacho, Carmen & Ha-Huy, Thai, 2023. "Balanced growth and degrowth with human capital," Economics Letters, Elsevier, vol. 232(C).
    2. De, Supriyo, 2014. "Intangible capital and growth in the ‘new economy’: Implications of a multi-sector endogenous growth model," Structural Change and Economic Dynamics, Elsevier, vol. 28(C), pages 25-42.
    3. Arantza Gorostiaga & Jana Hromcová & Miguel-Ángel López-García, 2013. "Optimal taxation in the Uzawa–Lucas model with externality in human capital," Journal of Economics, Springer, vol. 108(2), pages 111-129, March.
    4. Turnovsky, S., 2000. "Growth in an Open Economy: some Recent Developments," Papers 5, Warwick - Development Economics Research Centre.
    5. Kejak, Michal, 2003. "Stages of growth in economic development," Journal of Economic Dynamics and Control, Elsevier, vol. 27(5), pages 771-800, March.
    6. Lutz Arnold, 2007. "A generalized multi-country endogenous growth model," International Economics and Economic Policy, Springer, vol. 4(1), pages 61-100, April.
    7. Shiro Kuwahara, 2017. "Multiple steady states and indeterminacy in the Uzawa–Lucas model with educational externalities," Journal of Economics, Springer, vol. 122(2), pages 173-190, October.
    8. Brito, Paulo & Venditti, Alain, 2010. "Local and global indeterminacy in two-sector models of endogenous growth," Journal of Mathematical Economics, Elsevier, vol. 46(5), pages 893-911, September.
    9. Altar, Moisa & Necula, Ciprian & Bobeica, Gabriel, 2008. "Modeling The Economic Growth In Romania. The Role Of Human Capital," Journal for Economic Forecasting, Institute for Economic Forecasting, vol. 5(3), pages 115-128, September.
    10. C. Chilarescu & I. Viasu, 2019. "Uniqueness and Multiple Trajectories for the Case of Lucas Model," Computational Economics, Springer;Society for Computational Economics, vol. 54(3), pages 1157-1177, October.
    11. Thorsten Wichmann, 1995. "Food Consumption and Growth in a Two Sector Economy - A Theoretical Model and Numerical Simulations," Berlecon Research Papers 0001, Berlecon Research.
    12. Stephen Turnovsky, 1999. "Knife-Edge Conditions and the Macroeconomics of Small Open Economies," Discussion Papers in Economics at the University of Washington 0031, Department of Economics at the University of Washington.
    13. Neustroev, Dmitry, 2013. "The Uzawa-Lucas Growth Model with Natural Resources," MPRA Paper 52937, University Library of Munich, Germany.
    14. Constantin Chilarescu & Ioana Viasu, 2016. "A Closed-form Solution of a Two-sector Endogenous Growth Model with Habit Formation," Australian Economic Papers, Wiley Blackwell, vol. 55(2), pages 112-127, June.
    15. Pedro Garcia-Castrillo & Marcos Sanso, 2000. "Human Capital and Optimal Policy in a Lucas-type Model," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 3(4), pages 757-770, October.
    16. Benhabib, Jess & Gali, Jordi, 1995. "On growth and indeterminacy: some theory and evidence," Carnegie-Rochester Conference Series on Public Policy, Elsevier, vol. 43(1), pages 163-211, December.
    17. Alfred Greiner & Willi Semmler, 1996. "Multiple steady states, indeterminacy, and cycles in a basic model of endogenous growth," Journal of Economics, Springer, vol. 63(1), pages 79-99, February.
    18. Jones, Larry E. & Manuelli, Rodolfo E., 1997. "Endogenous growth theory: An introduction," Journal of Economic Dynamics and Control, Elsevier, vol. 21(1), pages 1-22, January.
    19. Alberto Bucci & Simone Marsiglio, 2019. "Financial development and economic growth: long‐run equilibrium and transitional dynamics," Scottish Journal of Political Economy, Scottish Economic Society, vol. 66(3), pages 331-359, July.
    20. Constantin Chilarescu, 2018. "The effect of externality on the transitional dynamics: the case of Lucas model," Economics Bulletin, AccessEcon, vol. 32767(3).

    More about this item

    Keywords

    Human capital; Balanced growth path;

    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:hal:psewpa:halshs-03920386. 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: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    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.