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Optimal investment models with vintage capital: Dynamic programming approach

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  • Faggian, Silvia
  • Gozzi, Fausto

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 ([Barucci and Gozzi, 1998], [Barucci and Gozzi, 2001], [Feichtinger et al., 2003] and [Feichtinger et al., 2006]) 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 by (Faggian, 2005b) and (Faggian, 2008a). 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. Finally it is explained how the results can be applied to improve the analysis of the optimal paths previously performed by Barucci and Gozzi and by Feichtinger et al.

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  • 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.
  • Handle: RePEc:eee:mateco:v:46:y:2010:i:4:p:416-437
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    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. 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.
    3. 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.
    4. 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.
    5. Benhabib, Jess & Rustichini, Aldo, 1991. "Vintage capital, investment, and growth," Journal of Economic Theory, Elsevier, vol. 55(2), pages 323-339, December.
    6. 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.
    7. 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.
    8. Gustav Feichtinger & Richard F. Hartl & Suresh P. Sethi, 1994. "Dynamic Optimal Control Models in Advertising: Recent Developments," Management Science, INFORMS, vol. 40(2), pages 195-226, February.
    9. 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.
    10. Silvia Faggian, 2008. "Equilibrium Points for Optimal Investment with Vintage Capital," Working Papers 182, Department of Applied Mathematics, Università Ca' Foscari Venezia.
    11. Barucci, Emilio & Gozzi, Fausto, 1998. "Investment in a vintage capital model," Research in Economics, Elsevier, vol. 52(2), pages 159-188, June.
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    Cited by:

    1. 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.
    2. Enrico Biffis & Beniamin Goldys & Cecilia Prosdocimi & Margherita Zanella, 2015. "A pricing formula for delayed claims: Appreciating the past to value the future," Papers 1505.04914, arXiv.org, revised Jan 2019.
    3. Faggian, Silvia & Gozzi, Fausto & Kort, Peter M., 2021. "Optimal investment with vintage capital: Equilibrium distributions," Journal of Mathematical Economics, Elsevier, vol. 96(C).
    4. Boucekkine, R. & Fabbri, G. & Gozzi, F., 2010. "Maintenance and investment: Complements or substitutes? A reappraisal," Journal of Economic Dynamics and Control, Elsevier, vol. 34(12), pages 2420-2439, December.
    5. Raouf Boucekkine & David de la Croix & Omar Licandro, 2011. "Vintage capital growth theory: Three breakthroughs," UFAE and IAE Working Papers 875.11, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
    6. Silvia Faggian & Luca Grosset, 2009. "Optimal investment in age-structured goodwill," Working Papers 194, Department of Applied Mathematics, Università Ca' Foscari Venezia.
    7. 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).
    8. Ballestra, Luca Vincenzo, 2016. "The spatial AK model and the Pontryagin maximum principle," Journal of Mathematical Economics, Elsevier, vol. 67(C), pages 87-94.
    9. Raouf Boucekkine & David de la Croix & Omar Licandro, 2011. "Vintage capital growth theory: Three breakthroughs," UFAE and IAE Working Papers 875.11, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
    10. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
    11. 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.
    12. repec:spr:compst:v:78:y:2013:i:2:p:259-284 is not listed on IDEAS
    13. 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.

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    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;

    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

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