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Bernoulli substitution in the Ramsey model: Optimal trajectories under control constraints

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Abstract

We consider a neoclassical (economic) growth model. A nonlinear Ramsey equation, modeling capital dynamics, in the case of Cobb-Douglas production function is reduced to the linear differential equation via a Bernoulli substitution. This considerably facilitates the search for a solution to the optimal growth problem with logarithmic preferences. The study deals with solving the corresponding infinite horizon optimal control problem. We consider a vector field of the Hamiltonian system in the Pontryagin maximum principle, taking into account control constraints. We prove the existence of two alternative steady states, depending on the constraints. A proposed algorithm for constructing growth trajectories combines methods of open-loop control and closed-loop regulatory control. For some levels of constraints and initial conditions, a closed-form solution is obtained. We also demonstrate the impact of technological change on the economic equilibrium dynamics. Results are supported by computer calculations.

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References

  1. F. P. Ramsey, “A mathematical theory of saving,” Econ. J. 38 (152), 543–559 (1928).

    Article  Google Scholar 

  2. D. Acemoglu, Introduction to Modern Economic Growth (Princeton Univ. Press, Princeton, 2008).

    MATH  Google Scholar 

  3. D. Romer, Advanced Macroeconomics (McGraw-Hill, New York, 2011; Vysshaya Shkola Ekonomiki, Moscow, 2014).

    Google Scholar 

  4. N. N. Moiseev, “Theory of optimal control in an infinite time interval,” USSR Comput. Math. Math. Phys. 14 (4), 33–43 (1974).

    Article  MATH  Google Scholar 

  5. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, 2nd ed. (Nauka, Moscow, 1969; Gordon and Breach, New York, 1986).

    Google Scholar 

  6. S. M. Aseev, K. O. Besov, and A. V. Kryazhimskii, “Infinite-horizon optimal control problems in economics,” Russ. Math. Surv. 67 (2), 195–253 (2012).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. I. G. Malkin, Theory of Motion Stability, 2nd ed. (Nauka, Moscow, 1966) [in Russian].

    MATH  Google Scholar 

  9. V. I. Maksimov and Yu. S. Osipov, “Infinite-horizon boundary control of distributed systems,” Comput. Math. Math. Phys. 56 (1), 14–25 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler, and D. A. Behrens, Optimal Control of Nonlinear Processes (Springer, Berlin, 2008).

    Book  MATH  Google Scholar 

  11. R. M. Solow, “Technical change and the aggregate production function,” Rev. Econ. Stat. 39 (3), 312–320 (1957).

    Article  MathSciNet  Google Scholar 

  12. W. T. Smith, “A closed form solution to the Ramsey model,” BE J. Macroecon. 6 (1), 1–27 (2006).

    Google Scholar 

  13. K. Shell, “Applications of Pontryagin’s maximum principle to economics,” Mathematical Systems: Theory and Economics, Ed. by H. W. Kuhn and G. P. Szego (Springer, Berlin, 1969).

    Google Scholar 

  14. A. A. Krasovskii and A. M. Tarasyev, “Properties of Hamiltonian systems in the Pontryagin maximum principle for economic growth problems,” Proc. Steklov Inst. Math. 262, 121–138 (2008).

    Article  MathSciNet  Google Scholar 

  15. R. L. Keeney and H. Raiffa, Decisions with Multiple Objectives: Preferences and Value Tradeoffs (Wiley, New York, 1976; Radio i Svyaz’, Moscow, 1981).

    MATH  Google Scholar 

  16. A. A. Krasovskii and A. M. Tarasyev, “Dynamic optimization of investments in the economic growth models,” Autom. Remote Control 68 (10), 1765–1777 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  17. A. A. Krasovskii and A. M. Tarasyev, “Construction of nonlinear regulators in economic growth models,” Proc. Steklov Inst. Math. 268, Suppl. 1, 143–154 (2010).

    Article  MathSciNet  Google Scholar 

  18. P. Hartman, Ordinary Differential Equations (Wiley, New York, 1964; Mir, Moscow, 1970).

    MATH  Google Scholar 

  19. A. A. Krasovskii, A. M. Tarasyev, and C. Watanabe, “Optimization of functionality development,” Appl. Math. Comput. 217 (3), 1125–1134 (2010).

    MathSciNet  MATH  Google Scholar 

  20. P. D. Lebedev and A. A. Uspenskii, “Procedures for calculating the nonconvexity measures of a plane set,” Comput. Math. Math. Phys. 49 (3), 418–427 (2009).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. International Institute for Applied Systems Analysis (IIASA), Schlossplatz 1, A-2361, Laxenburg, Austria

    A. A. Krasovskii

  2. Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, 620990, Russia

    P. D. Lebedev & A. M. Tarasyev

  3. Ural Federal University, Yekaterinburg, 620002, Russia

    A. M. Tarasyev

Authors
  1. A. A. Krasovskii
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  2. P. D. Lebedev
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  3. A. M. Tarasyev
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Correspondence to A. A. Krasovskii.

Additional information

Published in Russian in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 5, pp. 768–782.

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Krasovskii, A.A., Lebedev, P.D. & Tarasyev, A.M. Bernoulli substitution in the Ramsey model: Optimal trajectories under control constraints. Comput. Math. and Math. Phys. 57, 770–783 (2017). https://doi.org/10.1134/S0965542517050050

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  • DOI: https://doi.org/10.1134/S0965542517050050

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