IDEAS home Printed from https://ideas.repec.org/p/vnm/wpaper/174.html
   My bibliography  Save this paper

Optimal investment models with vintage capital: Dynamic Programming approach

Author

Listed:
  • Silvia Faggian

    (Department of Applied Mathematics, University of Venice)

  • Fausto Gozzi

    (LUISS Guido Carli)

Abstract

The Dynamic Programming approach for a family of optimal investment models with vintage capital is here developed. The problem falls into the class of infinite horizon optimal control problems of PDE's with age structure that have been studied in various papers (see e.g. [11, 12], [30, 32]) either in cases when explicit solutions can be found or using Maximum Principle techniques. The problem is rephrased into an infinite dimensional setting, it is proven that the value function is the unique regular solution of the associated stationary Hamilton-Jacobi-Bellman equation, and existence and uniqueness of optimal feedback controls is derived. It is then shown that the optimal path is the solution to the closed loop equation. Similar results were proven in the case of finite horizon in [26][27]. The case of infinite horizon is more challenging as a mathematical problem, and indeed more interesting from the point of view of optimal investment models with vintage capital, where what mainly matters is the behavior of optimal trajectories and controls in the long run. The study of infinite horizon is performed through a nontrivial limiting procedure from the corresponding finite horizon problems

Suggested Citation

  • Silvia Faggian & Fausto Gozzi, 2008. "Optimal investment models with vintage capital: Dynamic Programming approach," Working Papers 174, Department of Applied Mathematics, Università Ca' Foscari Venezia.
    • Handle: RePEc:vnm:wpaper:174
    as

    Download full text from publisher

    File URL: http://virgo.unive.it/wpideas/storage/2008wp174.pdf
    File Function: First version, 2008
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Feichtinger, G. & Hartl, R.F. & Kort, P.M. & Veliov, V., 2001. "Dynamic Investment Behavior Taking into Account Ageing of the Capital Good," Other publications TiSEM 1e12e7c6-11c2-4632-a8e2-1, Tilburg University, School of Economics and Management.
    2. Feichtinger, Gustav & Hartl, Richard F. & Kort, Peter M. & Veliov, Vladimir M., 2006. "Capital accumulation under technological progress and learning: A vintage capital approach," European Journal of Operational Research, Elsevier, vol. 172(1), pages 293-310, July.
    3. 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.
    4. Boucekkine, Raouf & Licandro, Omar & Puch, Luis A. & del Rio, Fernando, 2005. "Vintage capital and the dynamics of the AK model," Journal of Economic Theory, Elsevier, vol. 120(1), pages 39-72, January.
    5. Almeder, Christian & Caulkins, Jonathan P. & Feichtinger, Gustav & Tragler, Gernot, 2004. "An age-structured single-state drug initiation model--cycles of drug epidemics and optimal prevention programs," Socio-Economic Planning Sciences, Elsevier, vol. 38(1), pages 91-109, March.
    6. Benhabib, Jess & Rustichini, Aldo, 1991. "Vintage capital, investment, and growth," Journal of Economic Theory, Elsevier, vol. 55(2), pages 323-339, December.
    7. Silvia Faggian* & Fausto Gozzi, 2004. "On The Dynamic Programming Approach For Optimal Control Problems Of Pde'S With Age Structure," Mathematical Population Studies, Taylor & Francis Journals, vol. 11(3-4), pages 233-270.
    8. Feichtinger, Gustav & Hartl, Richard F. & Kort, Peter M. & Veliov, Vladimir M., 2006. "Anticipation effects of technological progress on capital accumulation: a vintage capital approach," Journal of Economic Theory, Elsevier, vol. 126(1), pages 143-164, January.
    9. 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.
    10. Chari, V V & Hopenhayn, Hugo, 1991. "Vintage Human Capital, Growth, and the Diffusion of New Technology," Journal of Political Economy, University of Chicago Press, vol. 99(6), pages 1142-1165, December.
    11. Silvia Faggian, 2008. "Equilibrium Points for Optimal Investment with Vintage Capital," Working Papers 182, Department of Applied Mathematics, Università Ca' Foscari Venezia.
    12. Barucci, Emilio & Gozzi, Fausto, 1998. "Investment in a vintage capital model," Research in Economics, Elsevier, vol. 52(2), pages 159-188, June.
    13. Silvia Faggian, 2008. "Maximum Principle for Boundary Control Problems Arising in Optimal Investment with Vintage Capital," Working Papers 181, Department of Applied Mathematics, Università Ca' Foscari Venezia.
    Full references (including those not matched with items on IDEAS)

    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. Raouf Boucekkine & David de La Croix & Omar Licandro, 2011. "Vintage capital theory: Three breakthroughs," Working Papers halshs-00599074, HAL.
    2. Faggian, Silvia & Gozzi, Fausto & Kort, Peter M., 2021. "Optimal investment with vintage capital: Equilibrium distributions," Journal of Mathematical Economics, Elsevier, vol. 96(C).
    3. Mauro Bambi & Cristina Girolami & Salvatore Federico & Fausto Gozzi, 2017. "Generically distributed investments on flexible projects and endogenous growth," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 63(2), pages 521-558, February.
    4. Silvia Faggian & Luca Grosset, 2009. "Optimal investment in age-structured goodwill," Working Papers 194, Department of Applied Mathematics, Università Ca' Foscari Venezia.
    5. Silvia Faggian, 2008. "Equilibrium Points for Optimal Investment with Vintage Capital," Working Papers 182, Department of Applied Mathematics, Università Ca' Foscari Venezia.
    6. Kredler, Matthias, 2014. "Vintage human capital and learning curves," Journal of Economic Dynamics and Control, Elsevier, vol. 40(C), pages 154-178.
    7. Cagri Saglam & Vladimir M. Veliov, 2008. "Role of Endogenous Vintage Specific Depreciation in the Optimal Behavior of Firms," International Journal of Economic Theory, The International Society for Economic Theory, vol. 4(3), pages 381-410, September.
    8. Fabbri, Giorgio & Gozzi, Fausto & Zanco, Giovanni, 2021. "Verification results for age-structured models of economic–epidemics dynamics," Journal of Mathematical Economics, Elsevier, vol. 93(C).
    9. Feichtinger, Gustav & Hartl, Richard F. & Kort, Peter M. & Veliov, Vladimir M., 2008. "Financially constrained capital investments: The effects of disembodied and embodied technological progress," Journal of Mathematical Economics, Elsevier, vol. 44(5-6), pages 459-483, April.
    10. d’Albis, Hippolyte & Augeraud-Veron, Emmanuelle & Venditti, Alain, 2012. "Business cycle fluctuations and learning-by-doing externalities in a one-sector model," Journal of Mathematical Economics, Elsevier, vol. 48(5), pages 295-308.
    11. Raouf Boucekkine & Giorgio Fabbri & Patrick-Antoine Pintus, 2011. "On the optimal control of a linear neutral differential equation arising in economics," Working Papers halshs-00576770, HAL.
    12. Fabbri, Giorgio & Faggian, Silvia & Freni, Giuseppe, 2015. "On the Mitra–Wan forest management problem in continuous time," Journal of Economic Theory, Elsevier, vol. 157(C), pages 1001-1040.
    13. Raouf Boucekkine & David Croix & Omar Licandro, 2004. "MODELLING VINTAGE STRUCTURES WITH DDEs: PRINCIPLES AND APPLICATIONS," Mathematical Population Studies, Taylor & Francis Journals, vol. 11(3-4), pages 151-179.
    14. Silvia Faggian & Luca Grosset, 2013. "Optimal advertising strategies with age-structured goodwill," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(2), pages 259-284, October.
    15. Emmanuelle Augeraud-Veron & Mauro Bambi, 2012. "Does habit formation always increase the agents' desire to smooth consumption?," Discussion Papers 12/12, Department of Economics, University of York.
    16. BOUCEKKINE, Raouf & FABBRI, Giorgio & PINTUS, Patrick, 2012. "On the optimal control of a linear neutral differential equation arising in economics," LIDAM Reprints CORE 2449, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    17. Goetz, Renan-Ulrich & Hritonenko, Natali & Yatsenko, Yuri, 2008. "The optimal economic lifetime of vintage capital in the presence of operating costs, technological progress, and learning," Journal of Economic Dynamics and Control, Elsevier, vol. 32(9), pages 3032-3053, September.
    18. Mauro Bambi & Cristina Di Girolami & Salvatore Federico & Fausto Gozzi, 2014. "On the Consequences of Generically Distributed Investments on Flexible Projects in an Endogenous Growth Model," Discussion Papers 14/15, Department of Economics, University of York.
    19. Feichtinger, Gustav & Hartl, Richard F. & Kort, Peter M. & Veliov, Vladimir M., 2006. "Capital accumulation under technological progress and learning: A vintage capital approach," European Journal of Operational Research, Elsevier, vol. 172(1), pages 293-310, July.
    20. Gamboa, Franklin & Maldonado, Wilfredo Leiva, 2014. "Feasibility and optimality of the initial capital stock in the Ramsey vintage capital model," Journal of Mathematical Economics, Elsevier, vol. 52(C), pages 40-45.

    More about this item

    Keywords

    Optimal investment; vintage capital; age-structured systems; optimal control; dynamic programming; Hamilton-Jacobi-Bellman equations; linear convex control; boundary control;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • E22 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Investment; Capital; Intangible Capital; Capacity

    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:vnm:wpaper:174. 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: Daria Arkhipova (email available below). General contact details of provider: https://edirc.repec.org/data/dmvenit.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.