Proceedings of the Steklov Institute of Mathematics

, Volume 291, Issue 1, pp 127–145 | Cite as

Proportional economic growth under conditions of limited natural resources

  • A. V. Kryazhimskiy
  • A. M. Tarasyev
  • A. A. Usova
  • Wei Wang


The paper is devoted to economic growth models in which the dynamics of production factors satisfy proportionality conditions. One of the main production factors in the problem of optimizing the productivity of natural resources is the current level of resource consumption, which is characterized by a sharp increase in the prices of resources compared with the price of capital. Investments in production factors play the role of control parameters in the model and are used to maintain proportional economic development. To solve the problem, we propose a two-level optimization structure. At the lower level, proportions are adapted to the changing economic environment according to the optimization mechanism of the production level under fixed cost constraints. At the upper level, the problem of optimal control of investments for an aggregate economic growth model is solved by means of the Pontryagin maximum principle. The application of optimal proportional constructions leads to a system of nonlinear differential equations, whose steady states can be considered as equilibrium states of the economy. We prove that the steady state is not stable, and the system tends to collapse (the production level declines to zero) if the initial point does not coincide with the steady state. We study qualitative properties of the trajectories generated by the proportional development dynamics and indicate the regions of production growth and decay. The parameters of the model are identified by econometric methods on the basis of China’s economic data.


Hamiltonian System Optimal Control Problem STEKLOV Institute Capital Stock Production Factor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  • A. V. Kryazhimskiy
    • 1
    • 2
  • A. M. Tarasyev
    • 3
    • 4
  • A. A. Usova
    • 3
  • Wei Wang
    • 5
  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.International Institute for Applied Systems AnalysisLaxenburgAustria
  3. 3.Krasovskii Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesYekaterinburgRussia
  4. 4.Institute of EconomicsUral Branch of the Russian Academy of SciencesYekaterinburgRussia
  5. 5.Center for Industrial Ecology, Department of Chemical EngineeringTsinghua UniversityBeijingChina

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