26 August 2016

Serguei Kaniovski visited ASA, IIASA

Serguei Kaniovski (Austrian Institute of Economic Research (WIFO), Vienna) visited the Advanced System Analysis program on 26 August, 2016 and gave a talk on "**The Optimal Use of Exhaustible Resources under Non-constant Returns to Scale" (joint work with Sergey Aseev (ASA) and Konstantin Besov (**Steklov Mathematical Institute, Moscow, Russia)).

**Abstract**

The 1972 Club of Rome's report on the `Limits to Growth' painted a gloomy future and led to an ongoing controversial debate. Essential ingredients of the rise and decline scenario were the rising resource scarcity and pollution. The central question was whether the scarcity of natural resources such as fossil fuels will limit growth and cause a substantial decline in standards of living. The report has been subject to utmost scrutiny by academics and journalists. Recent re-examinations lend credibility to the conclusions reached in the report.

Following publication of the report, a group of distinguished economists countered the scarcity argument in a series of papers now commonly referred to as the Dasgupta-Heal-Solow-Stiglitz (DHSS) model. Their effort resulted in a substantial theoretical contribution to the theory of economic growth and a standard model with resource constraints. One argument against the pessimism was that man-made capital can substitute the resource in making of consumption goods. This led to the concept of a weakly sustainable path (Hartwick), on which the expansion of man-made capital offsets the depletion of resources.

Despite extensive theoretical investigation, a complete analysis of the model has been presented only recently by Benchekroun and Withhagen for constant returns to scale. The centerpiece of the paper is a complete and rigorous study of the welfare-maximizing investment and depletion policies in the DHSS model under decreasing or increasing returns to scale. The non-constant returns to scale and constraints related to a finite stock of exhaustible resource preclude the application of Arrow’s sufficient optimality conditions - the standard approach to solving such models in economics.

This study follows a more general approach based on an existence theorem and necessary optimality conditions. We establish a general existence result and show that an optimal admissible policy may not exist if the output elasticity of the resource equals to one. Using an appropriate version of the maximum principle for infinite horizon optimal control problems, we are able to characterize the optimal policies for all possible model parameters and any initial stocks. We finish the paper with an economic interpretation and a discussion of the welfare-maximizing policies.

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PUBLICATIONS

Aseev, S.M. & Veliov, V.M. (2015). Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions. Proceedings of the Steklov Institute of Mathematics 291 (1), 22-39. 10.1134/S0081543815090023.

Aseev, S.M. (2015). On the boundedness of optimal controls in the infinite-horizon process. Proceedings of the Steklov Institute of Mathematics 291, 38-48. 10.1134/S0081543815080040.

Aseev, S.M. (2014). On some properties of the adjoint variable in the relations of the Pontryagin maximum principle for optimal economic growth problems. Proceedings of the Steklov Institute of Mathematics 278 (S1), 11-21. 10.1134/S0081543814090028.

Aseev, S.M. & Veliov, V.M. (2014). *Maximum Principle for Infinite-horizon Optimal Control Problems under Weak Regularity Assumptions.* Research Report 2014-06, Research Unit ORCOS, Institute of Mathematical Methods in Economics, Vienna University of Technology, Austria (June 2014)

Aseev, S.M. & Veliov, V.M. (2014). Needle variations in infinite-horizon optimal control. In: Variational and Optimal Control Problems on Unbounded Domains. Eds. Wolansky, G & Zaslavski, AJ, RI: American Mathematical Society (Providence. ISBN 978-1-4704-1077-310.1090/conm/619.

Aseev, S.M. (2013). On some properties of the adjoint variable in the relations of the Pontryagin maximum principle for optimal economic growth problems. Proceedings of the Institute of Mathematics and Mechanics UrB RAS 19 (4), 15-24.

Aseev, S.M., Besov, K.O., & Kaniovski, S. (2013). The problem of optimal endogenous growth with exhaustible resources revisited. In: Green Growth and Sustainable Development. Eds. Cuaresma, J Crespo, Palokangas, T & Tarasyev, A, pp. 3-30 Berlin/Heidelberg: Springer. 10.1007/978-3-642-34354-4_1.

Aseev, S.M. & Veliov, V.M. (2012). *Needle Variations in Infinite-Horizon Optimal Control.* Research Report 2012-04, Operations Research and Control Systems, Institute of Mathematical Methods in Economics, Vienna University of Technology, Austria (September 2012)

Aseev, S.M., Besov, K.O., Ollus, S.-E., & Palokangas, T. (2012). Optimal growth in a two-sector economy facing an expected random shock. Proceedings of the Steklov Institute of Mathematics, 4-34. 10.1134/S0081543812020022.

Aseev, S.M., Besov, K.O., & Kryazhimskiy, A.V. (2012). Infinite-horizon optimal control problems in economics. Russian Mathematical Surveys 67 (2), 195-253. 10.1070/RM2012v067n02ABEH004785.

Aseev, S.M. & Veliov, V.M. (2012). Maximum principle for infinite-horizon optimal control problems with dominating discount. Dynamics of Continuous, Discrete and Impulsive Systems, Series B (DCDIS-B) 19 (1), 43-63.

Aseev, S.M. & Veliov, V.M. (2012). Necessary optimality conditions for improper infinite-horizon control problems. In: Operations Research Proceedings 2011. Eds. Klatte, D, Luthi, HJ & Schmedders, K, Berlin-Heidelberg: Springer. 10.1007/978-3-642-29210-1_4.

Aseev, S.M., Besov, K.O., & Kaniovski, S. (2010). Optimal Endogenous Growth with Exhaustible Resources. IIASA Interim Report. IIASA, Laxenburg, Austria: IR-10-011

Aseev, S.M., Besov, K.O., Ollus, S.-E., & Palokangas, T. (2010). Optimal economic growth with a random environmental shock. In: Dynamic Systems, Economic Growth, and the Environment. Eds. Cuaresma, J. Crespo, Palokangas, T. & Tarasyev, A., Heidelberg: Springer-Verlag. ISBN 978-3-642-02131-210.1007/978-3-642-02132-9_6.

Aseev, S.M. (2009). *Infinite-Horizon Optimal Control with Applications in Growth Theory (Lecture Notes).* MSU CMC Publications Department, MAKS Press, Moscow, Russia

Aseev, S.M. & Kryazhimskiy, A.V. (2008). Shadow prices in infinite-horizon optimal control problems with dominating discounts. Applied Mathematics and Computation 204 (2), 519-531. 10.1016/j.amc.2008.05.031.

Aseev, S.M. & Kryazhimskiy, A.V. (2008). On a class of optimal control problems arising in mathematical economics. Proceedings of the Steklov Institute of Mathematics 262 (1), 10-25. 10.1134/S0081543808030036.

Aseev, S.M. & Kryazhimskiy, A.V. (2007). The Pontryagin maximum principle and optimal economic growth problems. Proceedings of the Steklov Institute of Mathematics 257 (1), 1-255. 10.1134/S0081543807020010.

Aseev, S.M. & Kryazhimskiy, A.V. (2007). On optimal labor allocation policy for technological followers. In: Structured Models in Population and Economic Dynamics, Viennese Vintage Workshop, 26-27 November 2007.

Aseev, S.M. & Kryazhimskiy, A.V. (2007). Optimal labor allocation policy for technological followers. In: IIASA-Tokyotech Workshop on Hybrid Management of Technology in the 21st Century.

Aseev, S.M. & Kryazhimskiy, A.V. (2007). The Pontryagin maximum principle and optimal economic growth problems. In: Proceedings of Ninth Workshop on Optimal Control, Dynamic Games and Nonlinear Dynamics, 7-9 May 2007.

Aseev, S.M. & Kryazhimskiy, A.V. (2007). A dynamic model of optimal capital accumulation for an enterprise. In: IIASA-Tokyotech Workshop on Hybrid Management of Technology in the 21st Century, 8-9 September 2007.

Aseev, S.M., Hutschenreiter, G., & Kryazhimskiy, A.V. (2005). A dynamical model of optimal investment in R&D. Journal of Mathematical Sciences 126 (6), 1495-1535. 10.1007/s10958-005-0040-3.

Aseev, S.M. (2005). *Optimal Control and Dynamic Models in Biology.* IIASA DYN-NEA Biologizing Control Theory Workshop, 19-20 December 2005, Laxenburg, Austria

Aseev, S.M., Hutschenreiter, G., Kryazhimskiy, A.V., & Lysenko, A. (2005). A dynamic model of optimal investment in research and development with international knowledge spillovers. Mathematical and Computer Modeling of Dynamical Systems 11 (2), 125-133. 10.1080/1387395050500067361.

Aseev, S.M. & Katsumoto, M. (2005). *Dynamic Optimization of Innovator's Behavior on Technological Products Market.* 6th IIASA-TITech Technical Meeting, May 2005, Laxenburg, Austria

Aseev, S.M. & Katsumoto, M. (2005). *Leader-leader Stochastic Innovation Race: Preliminary Results and Numerical Simulations.* 7th IIASA-TITech Technical Meeting, September 2005, Laxenburg, Austria

Aseev, S.M. & Kryazhimskiy, A.V. (2005). The Pontryagin Maximum Principle and Transversality Conditions for a Class of Optimal Control Problems with Infinite Time Horizons. IIASA Research Report (Reprint). IIASA, Laxenburg, Austria: RP-05-003. Reprinted from SIAM Journal on Control and Optimization, 43(3):1094-1119 [2004].

Katsumoto, M. & Aseev, S.M. (2005). *A Modeling Analysis of Business Strategies in the Innovative Industry.* 20th Conference of Japan Society for Science Policy and Research Management, National Graduate Institute for Policy Studies, 22-23 October 2005, Tokyo, Japan

Katsumoto, M. & Aseev, S.M. (2005). *A Modeling Approach to Innovation Race: Industrial Dynamics and Optimization Theory.* Spring IIASA Methodology Workshop, 3 May 2005, Laxenburg, Austria

Aseev, S.M. & Katsumoto, M. (2004). A Dynamic Model of Stochastic Innovation Race: Leader-Follower Case. IIASA Interim Report. IIASA, Laxenburg, Austria: IR-04-035

Aseev, S.M. (2004). *Problems of Dynamic Optimization under Risky Factors.* Working Paper No. 42, Free University of Bolzano, Italy [2004]

Aseev, S.M. & Kryazhimskiy, A.V. (2004). The Pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons. SIAM Journal on Control and Optimization 43 (3), 1094-1119. 10.1137/S0363012903427518.

Aseev, S.M. & Kryazhimskiy, A.V. (2004). The Pontryagin maximum principle for an optimal control problem with a functional specified by an improper integral. Doklady Mathematics 69 (1), 89-91.

Aseev, S.M. & Smirnov, A.I. (2004). The Pontryagin maximum principle for the problem of optimally crossing a given domain. Doklady Mathematics 69 (2), 243-245.

Aseev, S.M. & Kryazhimskiy, A.V. (2003). The Pontryagin Maximum Principle for Infinite-Horizon Optimal Controls. IIASA Interim Report. IIASA, Laxenburg, Austria: IR-03-013

Aseev, S.M., Hutschenreiter, G., Kryazhimskiy, A.V., & Lysenko, A. (2003). Optimization of economic growth via optimal investment in R&D. In: Proceedings 4th MATHMOD Vienna - Fourth International Symposium on Mathematical Modelling, 5-7 February 2003.

Aseev, S.M., Hutschenreiter, G., & Kryazhimskiy, A.V. (2002). A Dynamical Model of Optimal Allocation of Resources to R&D. IIASA Interim Report. IIASA, Laxenburg, Austria: IR-02-016

Aseev, S.M., Hutschenreiter, G., & Kryazhimskiy, A.V. (2002). *Optimal Investment in R&D with International Knowledge Spillovers.* WIFO Working Papers, No. 175 (March 2002)

Aseev, S.M., Kryazhimskiy, A.V., & Tarasyev, A.M. (2001). First Order Necessary Optimality Conditions for a Class of Infinite Horizon Optimal Control Problems. IIASA Interim Report. IIASA, Laxenburg, Austria: IR-01-007

Aseev, S.M. (2001). Extremal problems for differential inclusions with state constraints. Proceedings of the Steklov Institute of Mathematics, 1-63.

Aseev, S.M., Kryazhimskiy, A.V., & Tarasyev, A.M. (2001). The Pontryagin maximum principle and transversality conditions for a class of optimal economic growth problems. In: Preprints of the 5th IFAC Symposium on Nonlinear Control Systems, 4-6 July 2001.

Aseev, S.M., Kryazhimskiy, A.V., & Tarasyev, A.M. (2001). The Pontryagin maximum principle and transversality conditions for an optimal control problem with infinite time interval. Proceedings of the Steklov Institute of Mathematics, 64-80.

Aseev, S.M. (1999). Methods of regularization in nonsmooth problems of dynamic optimization. Journal of Mathematical Sciences 94 (3), 1366-1393.

International Institute for Applied Systems Analysis (IIASA)

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