Equations of nonlinear dynamics of development of industrial enterprises, taking into account the amount of its maximum profit

Cite item


In the published article, new modifications of economic and mathematical models of the dynamic development of enterprises are proposed, the production of which is being restored due to the introduction of their own investments. The developed models are presented in the form of systems of differential equations for an arbitrary number of production factors. Stationary solutions of these systems of equations correspond to the equilibrium states of the operation of enterprises and represent the limiting values of the factors of production. Two versions of systems of differential balance equations for enterprises, describing the growth of factors of production and output, have been established. In the first case, the growth of resources and output is limited by the limiting values of the factors of production. In the second case, the growth of resources and output is limited by the pre-calculated values of the factors of production that correspond to the value of the maximum profit of the enterprise. It is shown that the growth of production factors of the enterprise should not exceed the values corresponding to the value of the maximum profit. Otherwise, the company starts to operate at a loss. In the presented models, proportional, progressive and digressive depreciation deductions are considered. The constructed models make it possible to describe various modes of operation of enterprises. Such regimes include a stable output of products by enterprises, a temporary suspension of the work of enterprises during its technical re-equipment, and a temporary partial curtailment of production.

About the authors

Alexander L. Saraev

Samara National Research University

Author for correspondence.
Email: alex.saraev@gmail.com
ORCID iD: 0000-0002-9223-6330

Candidate of Economic Sciences, associate professor of the Department of Mathematics and Business Informatics

Russian Federation, 34, Moskovskoye shosse, Samara

Leonid A. Saraev

Samara National Research University

Email: saraev_leo@mail.ru
ORCID iD: 0000-0003-3625-5921

Doctor of Physical and Mathematical Sciences, professor, head of the Department of Mathematics and Business Informatics

Russian Federation, 34, Moskovskoye shosse, Samara


  1. Harrod R.F. The trade cycle. Oxford: Clarendon Press, 1936.
  2. Domar E.D. Capital expansion, rate of growth, and employment. Econometrica, April 1946, vol. 14, issue 2, pp. 137–147. Available at: https://laprimaradice.myblog.it/media/00/00/2491562877.pdf.
  3. Solow R.M. A Contribution to the Theory of Economic Growth. Quarterly Journal of Economics, February 1956, vol. 70, no. 1, pp. 65–94. Available at: http://piketty.pse.ens.fr/files/Solow1956.pdf.
  4. Swan T.W. Economic Growth and Capital Accumulation. Economic Record, November 1956, vol. 32, no. 2, pp. 334–361. Available at: https://www.csus.edu/indiv/o/onure/econ200A/Readings/Swan.pdf.
  5. Kuznets S. Long Swings in the Growth of Population and in Related Economic Variables. Proceedings of the American Philosophical Society, 1958, vol. 102, рр. 25–52. Available at: https://www.jstor.org/stable/985303.
  6. Kuznets S. Quantitative Aspects of the Economic Growth of Nations. Paper VIII: Distribution of Income by Size. Economic Development and Cultural Change, 1963, vol. 11, no 2, рр. 1–80. Available at: http://piketty.pse.ens.fr/files/Kuznets1963.pdf.
  7. Uzawa H. Optimum Technical Change in an Aggregative Model of Economic Growth. International Economic Review, 1965, vol. 6, pp. 18–31. DOI: http://doi.org/10.1017/CBO9780511664496.009.
  8. Arrow K.J. The Economic Implications of Learning by Doing. Review of Economic Studies, 1962, vol. 29, no. 1, pp. 155–173. DOI: http://doi.org/10.2307/2295952.
  9. Denison E.F. The Contribution of Capital to Economic Growth. The American Economic Review, vol. 70, no. 2; Papers and Proceedings of the Ninety-Second Annual Meeting of the American Economic Association, 1980, pp. 220–224.
  10. Romer P.M. Increasing Returns and Long-run Growth. Journal of Political Economy, October 1986, vol. 94, pp. 1002–1037. Available at: https://www.parisschoolofeconomics.eu/docs/darcillon-thibault/paul-romer-increasing-returns-and-long-run-growth.pdf.
  11. Lucas R.E. On the Mechanics of Economic Development. Journal of Monetary Economics, July 1988, vol. 22, no. 1, pp. 3–42. Available at: https://www.parisschoolofeconomics.eu/docs/darcillon-thibault/lucasmechanic
  12. seconomicgrowth.pdf.
  13. Romer P.M. Endogenous Technological Change. Journal of Political Economy, October 1990, vol. 98, no. 5, pp. 71–102. Available at: http://www.dklevine.com/archive/refs42135.pdf.
  14. Grossman G.M., Helpman E. Innovation and Growth in the Global Economy. Cambridge, MA: MIT Press. 1991.
  15. Mankiw N., Romer D., Weil D. A Contribution to the Empirics of Economic Growth. Quarterly Journal of Economics, 1992, vol. 107, no. 2, pp. 407–437. Available at: https://eml.berkeley.edu/
  16. ~dromer/papers/MRW_QJE1992.pdf.
  17. Grossman G.M., Helpman E. Endogenous Innovation in the Theory of Growth. Journal of Economic Perspectives, 1994, vol. 8 (1), pp. 23–44. DOI: http://dx.doi.org/10.1257/jep.8.1.23.
  18. Barro R.J., Sala-i-Martin X. Economic Growth. Cambridge MA: MIT Press, 1995, 672 p. Available at: http://piketty.pse.ens.fr/files/BarroSalaIMartin2004Chap1-2.pdf.
  19. Bruno M., Easterly W. Inflation Crises and Long-Run Growth: NBER Working Papers 5209. National Bureau of Economic Research, Inc, 1995. Available at: http://www.nber.org>papers/w5209. (accessed 06.03.2020).
  20. Gong G., Greiner A., Semmler W. The Uzawa – Lucas model without scale effects: theory and empirical evidence. Structural change and economic dynamics, 2004, vol. 15, no. 4, pp. 401–420. DOI: http://doi.org/10.1016/J.STRUECO.2003.10.002.
  21. Nizhegorodtsev R.M. Models of logistics dynamics as a tool for economic analysis and forecasting. In: Modeling of economic dynamics: risk, optimization, forecasting. Moscow, 1997, pp. 34–51. Available at: https://studylib.ru/doc/2206631/modeli-logisticheskoj-dinamiki-kak-instrument-e-konomicheskogo. (In Russ.)
  22. Badash Kh.Z. The economic-mathematical model of the economic growth of enterprises. Bulletin of Udmurt University. Series Economics and Law, 2009, no. 1, pp. 5–9. Available at: https://cyberleninka.ru/
  23. article/n/ekonomiko-matematicheskaya-model-ekonomicheskogo-rosta-predpriyatiya/viewer; https://elibrary.ru/item.asp?id=11700881. (In Russ.)
  24. Korolev A.V., Matveenko V.D. Structure of equilibrium time-varying trajectories in the Lucas endogenous growth model. Automation and Remote Control, 2006, vol. 67, pp. 624–633. DOI: http://doi.org/10.1134/S0005117906040102 (English; Russian original)
  25. Kuznetsov Yu.A., Michasova O.V. Comparative analysis of the application of simulation packages and computer mathematics systems for the analysis of models of the theory of economic growth. Economic Analysis: Theory and Practice, 2007, no. 5 (86), pp. 23–30. Available at: https://elibrary.ru/item.asp?id=9337066. (In Russ.)
  26. Kuznetsov Yu.A., Michasova O.V. The generalized model of economic growth with human capital accumulation. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control processes, 2012, no. 4, pp. 46–57. Available at: http://mi.mathnet.ru/vspui93. (In Russ.)
  27. Prasolov A.V. Mathematical methods of economic dynamics. Saint Petersburg: Lan', 2015, 352 p. Available at: https://klex.ru/uzv. (In Russ.)
  28. Saraev A.L. Equations of nonlinear dynamics of crisis phenomena for multifactor economic systems. Vestnik of Samara State University, 2015, no. 2 (124), pp. 262–272. Available at: https://journals.ssau.ru/eco/article/view/5635. (In Russ.)
  29. Saraev A.L., Saraev L.A. Indicators of nonlinear dynamics and the limiting condition of a manufacturing enterprise. Journal of Economy and Entrepreneurship, 2018, no. 11 (100), pp. 1237–1241. Available at: https://elibrary.ru/item.asp?id=36512728. (In Russ.)
  30. Saraev A.L. Equations of dynamics of unstable multifactor economic systems taking into account retardation effects of internal investment. Kazan economic vestnik, 2015, no. 3 (17), pp. 66–71. Available at: https://kpfu.ru/staff_files/F1593146947/KEV__3_17____statya.pdf. (In Russ.)
  31. Ilyina E.A., Saraev A.L., Saraev L.A. To the theory of modernization of manufacturing enterprises, taking into account the lag of internal investment. Journal of Economy and entrepreneurship, 2017, no. 9–4 (86), pp. 1130–1134. Available at: https://elibrary.ru/item.asp?id=30782945. (In Russ.)
  32. Saraev A.L., Saraev L.A. Economic and mathematical model of the development of industrial enterprises, taking into account the effect of lagging internal investment. Journal of Economy and entrepreneurship, 2019, no. 5 (106), pp. 1316–1320. Available at: https://elibrary.ru/item.asp?id=39238012. (In Russ.)
  33. Saraev A.L., Saraev L.A. Multivariate mathematical model of the development of a manufacturing enterprise through internal and external investments. Vestnik of Samara University. Economics and Management, 2020, vol. 11, no. 2, pp. 157–165. DOI: http://doi.org/10.18287/2542-0461-2020-11-2-157-165. (In Russ)
  34. Saraev A.L., Saraev L.A., Stochastic calculation of curves dynamics of enterprise. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, 2020, vol. 24, no. 2, pp. 343–364. DOI: http://doi.org/10.14498/vsgtu1700. (In Russ.)
  35. Ilyina E.A., Saraev L.A. Predicting the dynamics of the maximum and optimal profits of innovative enterprises. Journal of Physics: Conference Series, 2021, vol. 1784, p. 012002. DOI: http://doi.org/10.1088/1742-6596/1784/1/012002.
  36. Saraev A.L., Saraev L.A. Mathematical models of the development of industrial enterprises, with the effect of lagging internal and external investments. Journal of Physics: Conference Series, 2021, vol. 1784, p. 012010. DOI: http://doi.org/10.1088/1742-6596/1784/1/012010.

Copyright (c) 2021 Vestnik of Samara University. Economics and Management

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies