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

## 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).

## Notes

### 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.

## 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)CrossRefGoogle Scholar - S.M. Aseev, On the boundedness of optimal controls in infinite-horizon problems. Proc. Steklov Inst. Math.
**291**, 38–48 (2015b)CrossRefGoogle 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)CrossRefGoogle 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)CrossRefGoogle Scholar - S.M. Aseev, A.V. Kryazhimskii, The Pontryagin maximum principle and optimal economic growth problems. Proc. Steklov Inst. Math.
**257**, 1–255 (2007)CrossRefGoogle 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., 2016Google 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. 1–17Google 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)CrossRefGoogle 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)CrossRefGoogle Scholar - G.B. Asheim, T. Mitra, Sustainability and discounted utilitarianism in models of economic growth. Math. Soc. Sci.
**59**(2), 148–169 (2010)CrossRefGoogle Scholar - E.J. Balder, An existence result for optimal economic growth problems. J. Math. Anal. Appl.
**95**, 195–213 (1983)CrossRefGoogle 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, 1987Google Scholar
- D.A. Carlson, A.B. Haurie, A. Leizarowitz,
*Infinite Horizon Optimal Control. Deterministic and Stochastic Systems*(Springer, Berlin, 1991)CrossRefGoogle 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)CrossRefGoogle 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)CrossRefGoogle 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)CrossRefGoogle Scholar - P. Michel, On the transversality conditions in infinite horizon optimal problems. Econometrica
**50**,975–985 (1982)CrossRefGoogle 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)CrossRefGoogle Scholar - R.M. Solow, A contribution to the theory of economic growth. Q. J. Econ.
**70**(1), 65–94 (1956)CrossRefGoogle Scholar - S. Valente, Sustainable development, renewable resources and technological progress. Environ. Resour. Econ.
**30**(1), 115–125 (2005)CrossRefGoogle Scholar