Skip to main content
SpringerLink
Log in

Multidimensional Economic-Growth Models with an Integral Utility Function

  • Published:
Moscow University Computational Mathematics and Cybernetics Aims and scope Submit manuscript

Abstract

An \(n\)-dimensional economic model is considered that has a Cobb–Douglas production function on the infinite planning horizon such that the utility function is an integral-type functional with a discount and a logarithm-type integrant. It is assumed that all of the model’s amortization factors are equal to one another. The constructed optimum control contains \(n-1\) special segments that are described analytically. A special sweep procedure for consecutively solving two Cauchy problems on each segment is developed to find moments of switching between segments and the shape of the optimum trajectory. On the last segment, the optimum trajectory lies along a special ray; from the viewpoint of economy, this ray can be interpreted as the mode of equilibrium growth. A Pontrjagin maximum-principle problem with a special transversality condition is used to construct the optimum solution. Optimality is confirmed using the Kiselev theorem on sufficient conditions. Moving to problems of large dimensions considerably increases the number of technical difficulties. The description of the optimality verification procedure is therefore presented in detail for methodological reasons.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. Yu. N. Kiselev, M. V. Orlov, and S. M. Orlov, ‘‘Boundary value problem of Pontryagin’s maximum principle in a two-sector economy model with an integral utility function,’’ Comput. Math. Math. Phys. 55 (11), 1779–1793 (2015).

    Article  MathSciNet  Google Scholar 

  2. Yu. N. Kiselev, M. V. Orlov, and S. M. Orlov, ‘‘Optimal resource allocation program in a two-sector economic model with an integral type functional for various amortization factors,’’ Differ. Equations 51 (5), 683–700 (2015).

    Article  MathSciNet  Google Scholar 

  3. Yu. N. Kiselev and M. V. Orlov, ‘‘Optimal resource allocation program in a two-sector economic model with a Cobb-Douglas production function,’’ Differ. Equations 46 (12), 1750–1766 (2010).

    Article  MathSciNet  Google Scholar 

  4. Yu. N. Kiselev and M. V. Orlov, ‘‘Optimal resource distribution program in a two-sector economic model with a Cobb-Douglas production function with distinct amortization factors,’’ Differ. Equations 48 (12), 1607–1622 (2012).

    Article  MathSciNet  Google Scholar 

  5. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961; Wiley-Interscience, New York, 1962).

  6. Yu. N. Kiselev, ‘‘Sufficient conditions of optimality in terms of constructions of the Pontryagin maximum principle,’’ in Mathematical Models in Economics and Biology (MAKS Press, Moscow, 2003), pp. 57–67 [in Russian].

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. D. A. Zyukin and I. E. Gurova, ‘‘Research of the level of usage of production factors in a pharmaceutical company based on the Cobb-Douglas function,’’ Sovremennye Nauchnye Issledovaniya i Innovatsii (Modern Scientific Researches and Innovations) [Electronic journal], No. 5 (61), 1–4 (2016) [in Russian].

  9. V. A. Gnevesheva, ‘‘Employment forecast using the Cobb-Douglas function,’’ Znanie. Ponimanie. Umenie (Knowledge. Understanding. Skill), No. 1, 117–122 (2005) [in Russian].

  10. A. M. Tarasyev and A. A. Usova, ‘‘Construction of a regulator for the Hamiltonian system in a two-sector economic growth model,’’ Proc. Steklov Inst. Math. 271 (1), 265–285 (2010).

    Article  MathSciNet  Google Scholar 

  11. Y. Iwasa and J. Roughgarden, ‘‘Shoot/root balance of plants: optimal growth of a system with many vegetative organs,’’ Theor. Popul. Biol. 25 (1), 78–105 (1984).

    Article  Google Scholar 

Download references

Funding

This work was supported by the Russian Foundation for Basic Research, project no. 18-31-00454.

Author information

Authors and Affiliations

  1. Department of Computational Mathematics and Cybernetics, Moscow State University, 119991, Moscow, Russia

    Yu. N. Kiselev, M. V. Orlov & S. M. Orlov

Authors
  1. Yu. N. Kiselev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. V. Orlov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. M. Orlov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Yu. N. Kiselev, M. V. Orlov or S. M. Orlov.

Additional information

Translated by A. Muravnik

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kiselev, Y.N., Orlov, M.V. & Orlov, S.M. Multidimensional Economic-Growth Models with an Integral Utility Function. MoscowUniv.Comput.Math.Cybern. 44, 73–86 (2020). https://doi.org/10.3103/S0278641920020041

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0278641920020041

Keywords:

Search

Navigation