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Optimal Exploitation of Renewable Resources: Lessons in Sustainability from an Optimal Growth Model of Natural Resource Consumption

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Control Systems and Mathematical Methods in Economics

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 687))

Abstract

We study an optimal growth model for a single resource based economy. The resource is governed by the standard model of logistic growth, and is related to the output of the economy through a Cobb-Douglas type production function with exogenously driven knowledge stock. The model is formulated as an infinite-horizon optimal control problem with unbounded set of control constraints and non-concave Hamiltonian. We transform the original problem to an equivalent one with simplified dynamics and prove the existence of an optimal admissible control. Then we characterize the optimal paths for all possible parameter values and initial states by applying the appropriate version of the Pontryagin maximum principle. Our main finding is that only two qualitatively different types of behavior of sustainable optimal paths are possible depending on whether the resource growth rate is higher than the social discount rate or not. An analysis of these behaviors yields general criterions for sustainable and strongly sustainable optimal growth (w.r.t. the corresponding notions of sustainability defined herein).

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References

  • D. Acemoglu, Introduction to Modern Economic Growth (Princeton University Press, Princeton NJ, 2009)

    Google Scholar 

  • S.M. Aseev, Adjoint variables and intertemporal prices in infinite-horizon optimal control problems. Proc. Steklov Inst. Math. 290, 223–237 (2015a)

    Article  Google Scholar 

  • S.M. Aseev, On the boundedness of optimal controls in infinite-horizon problems. Proc. Steklov Inst. Math. 291, 38–48 (2015b)

    Article  Google Scholar 

  • S.M. Aseev, Existence of an optimal control in infinite-horizon problems with unbounded set of control constraints. Trudy Inst. Mat. i Mekh. UrO RAN 22(2), 18–27 (2016) (in Russian)

    Article  Google Scholar 

  • S.M. Aseev, A.V. Kryazhimskiy, The Pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons. SIAM J. Control Optim. 43, 1094–1119 (2004)

    Article  Google Scholar 

  • S.M. Aseev, A.V. Kryazhimskii, The Pontryagin maximum principle and optimal economic growth problems. Proc. Steklov Inst. Math. 257, 1–255 (2007)

    Article  Google Scholar 

  • S. Aseev, T. Manzoor, Optimal growth, renewable resources and sustainability, International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria, WP-16-017, 29 pp., 2016

    Google Scholar 

  • S.M. Aseev, V.M. Veliov, Maximum principle for infinite-horizon optimal control problems with dominating discount. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 19, 43–63 (2012)

    Google Scholar 

  • S.M. Aseev, V.M. Veliov, Needle variations in infinite-horizon optimal control, in Variational and Optimal Control Problems on Unbounded Domains, ed. by G. Wolansky, A.J. Zaslavski. Contemporary Mathematics, vol. 619 (American Mathematical Society, Providence, 2014), pp. 1–17

    Google Scholar 

  • S.M. Aseev, V.M. Veliov, Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions. Proc. Steklov Inst. Math. 291(supplement 1), 22–39 (2015)

    Article  Google Scholar 

  • S.M. Aseev, K.O. Besov, A.V. Kryazhimskii, Infinite-horizon optimal control problems in economics. Russ. Math. Surv. 67(2), 195–253 (2012)

    Article  Google Scholar 

  • G.B. Asheim, T. Mitra, Sustainability and discounted utilitarianism in models of economic growth. Math. Soc. Sci. 59(2), 148–169 (2010)

    Article  Google Scholar 

  • E.J. Balder, An existence result for optimal economic growth problems. J. Math. Anal. Appl. 95, 195–213 (1983)

    Article  Google Scholar 

  • R.J. Barro, X. Sala-i-Martin, Economic Growth (McGraw Hill, New York, 1995)

    Google Scholar 

  • Brundtland Commission, Our common future: report of the world commission on evironment and development, United Nations, 1987

    Google Scholar 

  • D.A. Carlson, A.B. Haurie, A. Leizarowitz, Infinite Horizon Optimal Control. Deterministic and Stochastic Systems (Springer, Berlin, 1991)

    Book  Google Scholar 

  • L. Cesari, Optimization – Theory and Applications. Problems with Ordinary Differential Equations (Springer, New York, 1983)

    Google Scholar 

  • A.F. Filippov, Differential Equations with Discontinuous Right-Hand Sides (Kluwer, Dordrecht, 1988)

    Book  Google Scholar 

  • P. Hartman, Ordinary Differential Equations (J. Wiley & Sons, New York, 1964)

    Google Scholar 

  • H. Hotelling, The economics of exhaustible resources. J. Polit. Econ. 39, 137–175 (1974)

    Article  Google Scholar 

  • T. Manzoor, S. Aseev, E. Rovenskaya, A. Muhammad, Optimal control for sustainable consumption of natural resources, in Proceedings, 19th IFAC World Congress, vol.19, part 1 (Capetown, South Africa, 24–29 August, 2014), ed. by E. Boje, X. Xia, pp. 10725–10730 (2014)

    Article  Google Scholar 

  • P. Michel, On the transversality conditions in infinite horizon optimal problems. Econometrica 50,975–985 (1982)

    Article  Google Scholar 

  • L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, The Mathematical Theory of Optimal Processes (Pergamon, Oxford, 1964)

    Google Scholar 

  • F.P. Ramsey, A mathematical theory of saving. Econ. J. 38, 543–559 (1928)

    Article  Google Scholar 

  • R.M. Solow, A contribution to the theory of economic growth. Q. J. Econ. 70 (1), 65–94 (1956)

    Article  Google Scholar 

  • S. Valente, Sustainable development, renewable resources and technological progress. Environ. Resour. Econ. 30(1), 115–125 (2005)

    Article  Google Scholar 

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Acknowledgements

This work was initiated when Talha Manzoor participated in the 2013 Young Scientists Summer Program (YSSP) at IIASA, Laxenburg, Austria. T. Manzoor is grateful to Pakistan National Member Organization for financial support during the YSSP. Sergey Aseev was supported by the Russian Science Foundation under grant 15-11-10018 in developing of methodology of application of the maximum principle to the problem.

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

    Sergey Aseev

  2. International Institute for Applied Systems Analysis, Laxenburg, Austria

    Sergey Aseev

  3. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, Yekaterinburg, Russia

    Sergey Aseev

  4. Department of Electrical Engineering, Center for Water Informatics & Technology, Lahore University of Management Sciences, Lahore, Pakistan

    Talha Manzoor

Authors
  1. Sergey Aseev
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  2. Talha Manzoor
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Correspondence to Sergey Aseev .

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Editors and Affiliations

  1. Operations Research and Control Systems, Vienna University of Technology , Vienna, Austria

    Gustav Feichtinger

  2. Operations Research and Control Systems, Vienna University of Technology, Vienna, Austria

    Raimund M. Kovacevic

  3. Operations Research and Control Systems, Vienna University of Technology, Vienna, Austria

    Gernot Tragler

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Aseev, S., Manzoor, T. (2018). Optimal Exploitation of Renewable Resources: Lessons in Sustainability from an Optimal Growth Model of Natural Resource Consumption. In: Feichtinger, G., Kovacevic, R., Tragler, G. (eds) Control Systems and Mathematical Methods in Economics. Lecture Notes in Economics and Mathematical Systems, vol 687. Springer, Cham. https://doi.org/10.1007/978-3-319-75169-6_11

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