IDEAS home Printed from https://ideas.repec.org/p/ctl/louvir/2014011.html
   My bibliography  Save this paper

On the Mitra-Wan Forest Management Problem in Continuous Time

Author

Listed:
  • Giorgio FABBRI

    (EPEE,Université d'Evry-Val-d'Essonne, Département d'Economie)

  • Silvia FAGGIAN

    (Dipartmento di Economia, Università di Venezia “Ca'Foscari”)

  • Giuseppe FRENI

    (Department of Business and Economics, University of Naples “Parthenope”)

Abstract

The paper provides a continuous-time version of the discrete-time Mitra-Wan model of optimal forest management, where trees are harvested to maximize the utility of timber flow over an infinite time horizon. The available trees and the other parameters of the problem vary continuously with respect to both time and age of the trees, so that the system is ruled by a partial differential equation. The behavior of optimal or maximal couple is classified in the cases of linear, concave or strictly concave utility, and positive or null discount rate. All sets of data share the common feature that optimal controls need to be more general than functions, i.e. positive measures. Formulas are provided for golden-rule configurations (uniform density functions with cutting at the ages that solve a Faustmann problem) and for Faustmann policies, and their optimality/maximality is discussed. The results do not always confirm the corresponding ones in discrete time.

Suggested Citation

  • Giorgio FABBRI & Silvia FAGGIAN & Giuseppe FRENI, 2014. "On the Mitra-Wan Forest Management Problem in Continuous Time," LIDAM Discussion Papers IRES 2014011, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
    • Handle: RePEc:ctl:louvir:2014011
    as

    Download full text from publisher

    File URL: http://sites.uclouvain.be/econ/DP/IRES/2014011.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. 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.
    2. Tahvonen, Olli & Salo, Seppo, 1999. "Optimal Forest Rotation within SituPreferences," Journal of Environmental Economics and Management, Elsevier, vol. 37(1), pages 106-128, January.
    3. Boucekkine, Raouf & Germain, Marc & Licandro, Omar & Magnus, Alphonse, 1998. "Creative Destruction, Investment Volatility, and the Average Age of Capital," Journal of Economic Growth, Springer, vol. 3(4), pages 361-384, December.
    4. Tapan Mitra & Henry Y. Wan, 1985. "Some Theoretical Results on the Economics of Forestry," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 52(2), pages 263-282.
    5. W. A. Brock & A. Haurie, 1976. "On Existence of Overtaking Optimal Trajectories Over an Infinite Time Horizon," Mathematics of Operations Research, INFORMS, vol. 1(4), pages 337-346, November.
    6. Raouf Boucekkine & David de la Croix & Omar Licandro, 2006. "Vintage Capital," Economics Working Papers ECO2006/8, European University Institute.
    7. 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.
    8. Boucekkine, Raouf & Germain, Marc & Licandro, Omar, 1997. "Replacement Echoes in the Vintage Capital Growth Model," Journal of Economic Theory, Elsevier, vol. 74(2), pages 333-348, June.
    9. David Gale, 1967. "On Optimal Development in a Multi-Sector Economy," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 34(1), pages 1-18.
    10. Salo, Seppo & Tahvonen, Olli, 2002. "On Equilibrium Cycles and Normal Forests in Optimal Harvesting of Tree Vintages," Journal of Environmental Economics and Management, Elsevier, vol. 44(1), pages 1-22, July.
    11. Lionel W. McKenzie, 2008. "Equilibrium, Trade, and Growth: Selected Papers, Lionel W. McKenzie," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262135019, April.
    12. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    13. 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.
    14. Ali Khan, M. & Piazza, Adriana, 2012. "On the Mitra–Wan forestry model: A unified analysis," Journal of Economic Theory, Elsevier, vol. 147(1), pages 230-260.
    15. 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.
    16. Emilio Barucci & Fausto Gozzi, 2001. "Technology adoption and accumulation in a vintage-capital model," Journal of Economics, Springer, vol. 74(1), pages 1-38, February.
    17. 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).
    18. Salant, Stephen W., 2013. "The equilibrium price path of timber in the absence of replanting: does Hotelling rule the forests too?," Resource and Energy Economics, Elsevier, vol. 35(4), pages 572-581.
    19. Mitra, Tapan & Wan, Henry Jr., 1986. "On the faustmann solution to the forest management problem," Journal of Economic Theory, Elsevier, vol. 40(2), pages 229-249, December.
    20. R. M. Solow & J. Tobin & C. C. Weizsäcker & M. Yaari, 1971. "Neoclassical Growth with Fixed Factor Proportions," Palgrave Macmillan Books, in: F. H. Hahn (ed.), Readings in the Theory of Growth, chapter 9, pages 68-102, Palgrave Macmillan.
    21. Natali Hritonenko & Yuri Yatsenko, 2008. "From Linear to Nonlinear Utility in Vintage Capital Models," Mathematical Population Studies, Taylor & Francis Journals, vol. 15(4), pages 230-248.
    22. Mitra, Tapan & Ray, Debraj & Roy, Rahul, 1991. "The economics of orchards: An exercise in point-input, flow-output capital theory," Journal of Economic Theory, Elsevier, vol. 53(1), pages 12-50, February.
    23. Foley, Duncan K, 1975. "On Two Specifications of Asset Equilibrium in Macroeconomic Models," Journal of Political Economy, University of Chicago Press, vol. 83(2), pages 303-324, April.
    24. Heijdra, Ben J. & Romp, Ward E., 2009. "Retirement, pensions, and ageing," Journal of Public Economics, Elsevier, vol. 93(3-4), pages 586-604, April.
    25. W. A. Brock, 1970. "On Existence of Weakly Maximal Programmes in a Multi-Sector Economy," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 37(2), pages 275-280.
    26. Terry Heaps, 2014. "Convergence of Optimal Harvesting Policies to a Normal Forest," Discussion Papers dp14-01, Department of Economics, Simon Fraser University.
    27. McKenzie, Lionel W., 2005. "Optimal economic growth, turnpike theorems and comparative dynamics," Handbook of Mathematical Economics, in: K. J. Arrow & M.D. Intriligator (ed.), Handbook of Mathematical Economics, edition 2, volume 3, chapter 26, pages 1281-1355, Elsevier.
    28. E. Barucci & F. Gozzi, 1999. "Optimal advertising with a continuum of goods," Annals of Operations Research, Springer, vol. 88(0), pages 15-29, January.
    29. Olli Tahvonen, 2004. "Optimal Harvesting Of Forest Age Classes: A Survey Of Some Recent Results," Mathematical Population Studies, Taylor & Francis Journals, vol. 11(3-4), pages 205-232.
    30. Dasgupta, Swapan & Mitra, Tapan, 2010. "On Optimal Forest Management: A Bifurcation Analysis," Working Papers 10-04, Cornell University, Center for Analytic Economics.
    31. Khan, M. Ali & Piazza, Adriana, 2010. "On the non-existence of optimal programs in the Robinson-Solow-Srinivasan (RSS) model," Economics Letters, Elsevier, vol. 109(2), pages 94-98, November.
    32. Halkin, Hubert, 1974. "Necessary Conditions for Optimal Control Problems with Infinite Horizons," Econometrica, Econometric Society, vol. 42(2), pages 267-272, March.
    33. Heaps, Terry & Neher, Philip A., 1979. "The economics of forestry when the rate of harvest is constrained," Journal of Environmental Economics and Management, Elsevier, vol. 6(4), pages 297-319, December.
    34. Dastidar, Krishnendu Ghosh & Mukhopadhyay, Hiranya & Sinha, Uday Bhanu (ed.), 2011. "Dimensions of Economic Theory and Policy: Essays for Anjan Mukherji," OUP Catalogue, Oxford University Press, number 9780198073970.
    35. M. Ali Khan & Adriana Piazza, 2010. "On uniform convergence of undiscounted optimal programs in the Mitra–Wan forestry model: The strictly concave case," International Journal of Economic Theory, The International Society for Economic Theory, vol. 6(1), pages 57-76, March.
    36. Samuelson, Paul A, 1976. "Economics of Forestry in an Evolving Society," Economic Inquiry, Western Economic Association International, vol. 14(4), pages 466-492, December.
    37. Heaps, Terry, 1984. "The forestry maximum principle," Journal of Economic Dynamics and Control, Elsevier, vol. 7(2), pages 131-151, May.
    38. repec:cor:louvrp:-2148 is not listed on IDEAS
    39. HALKIN, Hubert, 1974. "Necessary conditions for optimal control problems with infinite horizons," LIDAM Reprints CORE 193, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    40. Salo, Seppo & Tahvonen, Olli, 2003. "On the economics of forest vintages," Journal of Economic Dynamics and Control, Elsevier, vol. 27(8), pages 1411-1435, June.
    41. Cass, David, 1973. "On the Wicksellian Point-Input, Point-Output Model of Capital Accumulation: A Modern View (or, Neoclassicism Slightly Vindicated)," Journal of Political Economy, University of Chicago Press, vol. 81(1), pages 71-97, Jan.-Feb..
    42. Peleg, Bezalel, 1973. "A Weakly Maximal Golden-Rule Program for a Multi-Sector Economy," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 14(3), pages 574-579, October.
    43. Magill, Michael J. P., 1977. "Some new results on the local stability of the process of capital accumulation," Journal of Economic Theory, Elsevier, vol. 15(1), pages 174-210, June.
    44. Tahvonen, Olli, 2009. "Economics of harvesting age-structured fish populations," Journal of Environmental Economics and Management, Elsevier, vol. 58(3), pages 281-299, November.
    45. Barucci, Emilio & Gozzi, Fausto, 1998. "Investment in a vintage capital model," Research in Economics, Elsevier, vol. 52(2), pages 159-188, June.
    46. Kemp, Murray C. & Moore, Elvin J., 1979. "Biological capital theory: a question and a conjecture," Economics Letters, Elsevier, vol. 4(2), pages 141-144.
    47. Cass, David & Shell, Karl, 1976. "Introduction to Hamiltonian dynamics in economics," Journal of Economic Theory, Elsevier, vol. 12(1), pages 1-10, February.
    48. 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).
    49. Wan, Henry, Jr, 1994. "Revisiting the Mitra-Wan Tree Farm," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 35(1), pages 193-198, February.
    50. Tahvonen, Olli & Salo, Seppo & Kuuluvainen, Jari, 2001. "Optimal forest rotation and land values under a borrowing constraint," Journal of Economic Dynamics and Control, Elsevier, vol. 25(10), pages 1595-1627, October.
    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. Giorgio Fabbri & Silvia Faggian & Giuseppe Freni, 2016. "Non-existence of Optimal Programs in Continuous Time," Working Papers 2016:21, Department of Economics, University of Venice "Ca' Foscari".
    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. Galo Nuno & Benjamin Moll, 2018. "Social Optima in Economies with Heterogeneous Agents," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 28, pages 150-180, April.
    4. David Desmarchelier & Alexandre Mayol, 2022. "To seed, or not to seed? An endogenous labor supply approach in a simple overlapping generation economy," Bulletin of Economic Research, Wiley Blackwell, vol. 74(1), pages 25-38, January.
    5. Fabbri, Giorgio & Faggian, Silvia & Freni, Giuseppe, 2017. "Non-existence of optimal programs for undiscounted growth models in continuous time," Economics Letters, Elsevier, vol. 152(C), pages 57-61.
    6. Khan, M. Ali, 2016. "On a forest as a commodity and on commodification in the discipline of forestry," Forest Policy and Economics, Elsevier, vol. 72(C), pages 7-17.
    7. Silvia Faggian & Giuseppe Freni, 2015. "A Ricardian Model of Forestry," Working Papers 2015:12, Department of Economics, University of Venice "Ca' Foscari", revised 2015.

    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. Khan, M. Ali, 2016. "On a forest as a commodity and on commodification in the discipline of forestry," Forest Policy and Economics, Elsevier, vol. 72(C), pages 7-17.
    2. Ali Khan, M. & Piazza, Adriana, 2012. "On the Mitra–Wan forestry model: A unified analysis," Journal of Economic Theory, Elsevier, vol. 147(1), pages 230-260.
    3. Newman, D.H., 2002. "Forestry's golden rule and the development of the optimal forest rotation literature," Journal of Forest Economics, Elsevier, vol. 8(1), pages 5-27.
    4. Salo, Seppo & Tahvonen, Olli, 2003. "On the economics of forest vintages," Journal of Economic Dynamics and Control, Elsevier, vol. 27(8), pages 1411-1435, June.
    5. Silvia Faggian & Giuseppe Freni, 2015. "A Ricardian Model of Forestry," Working Papers 2015:12, Department of Economics, University of Venice "Ca' Foscari", revised 2015.
    6. Khan, M. Ali & Piazza, Adriana, 2011. "Classical turnpike theory and the economics of forestry," Journal of Economic Behavior & Organization, Elsevier, vol. 79(3), pages 194-210, August.
    7. Terry Heaps, 2014. "Convergence of Optimal Harvesting Policies to a Normal Forest," Discussion Papers dp14-01, Department of Economics, Simon Fraser University.
    8. Faggian, Silvia & Gozzi, Fausto & Kort, Peter M., 2021. "Optimal investment with vintage capital: Equilibrium distributions," Journal of Mathematical Economics, Elsevier, vol. 96(C).
    9. Ken-Ichi Akao, 2011. "Optimum forest program when the carbon sequestration service of a forest has value," Environmental Economics and Policy Studies, Springer;Society for Environmental Economics and Policy Studies - SEEPS, vol. 13(4), pages 323-343, December.
    10. Salo, Seppo & Tahvonen, Olli, 2002. "On Equilibrium Cycles and Normal Forests in Optimal Harvesting of Tree Vintages," Journal of Environmental Economics and Management, Elsevier, vol. 44(1), pages 1-22, July.
    11. Emilio Barucci & Fausto Gozzi, 2001. "Technology adoption and accumulation in a vintage-capital model," Journal of Economics, Springer, vol. 74(1), pages 1-38, February.
    12. Piazza, Adriana & Roy, Santanu, 2015. "Deforestation and optimal management," Journal of Economic Dynamics and Control, Elsevier, vol. 53(C), pages 15-27.
    13. Raouf Boucekkine & David De la Croix & Omar Licandro, 2011. "Vintage Capital Growth Theory: Three Breakthroughs," Working Papers 565, Barcelona School of Economics.
    14. Sahashi, Yoshinao, 2002. "The convergence of optimal forestry control," Journal of Mathematical Economics, Elsevier, vol. 37(3), pages 179-214, May.
    15. Adriana Piazza, 2009. "The optimal harvesting problem with a land market: a characterization of the asymptotic convergence," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 40(1), pages 113-138, July.
    16. Adriana Piazza & Santanu Roy, 2020. "Irreversibility and the economics of forest conservation," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 69(3), pages 667-711, April.
    17. Xabadia, Angels & Goetz, Renan U., 2010. "The optimal selective logging regime and the Faustmann formula," Journal of Forest Economics, Elsevier, vol. 16(1), pages 63-82, January.
    18. Heaps, Terry, 2015. "Convergence of optimal harvesting policies to a normal forest," Journal of Economic Dynamics and Control, Elsevier, vol. 54(C), pages 74-85.
    19. 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.
    20. M. Ali Khan & Adriana Piazza, 2010. "On uniform convergence of undiscounted optimal programs in the Mitra–Wan forestry model: The strictly concave case," International Journal of Economic Theory, The International Society for Economic Theory, vol. 6(1), pages 57-76, March.

    More about this item

    Keywords

    Optimal harvesting problems; Forest Management; Measure-valued 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
    • D90 - Microeconomics - - Micro-Based Behavioral Economics - - - General
    • Q23 - Agricultural and Natural Resource Economics; Environmental and Ecological Economics - - Renewable Resources and Conservation - - - Forestry

    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:ctl:louvir:2014011. 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: Virginie LEBLANC (email available below). General contact details of provider: https://edirc.repec.org/data/iruclbe.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.