TY - JOUR

T1 - Analysis of a Growth Model with a CES Production Function

AU - Usova, A.

AU - Tarasyev, A.

N1 - This work was supported by the Russian Science Foundation, project no. 19-11-00105, https://rscf.ru/en/project/19-11-00105/ .

PY - 2023

Y1 - 2023

N2 - Abstract: The paper investigates a growth model with a production function of constant elasticity of substitution, which generalizes the Cobb–Douglas or Leontief functions. The investment indicators of the model are considered as control parameters chosen to maximize the utility functional. An infinite-horizon optimal control problem is formulated. A Hamiltonian function and a Hamiltonian system are constructed by applying the Pontryagin maximum principle. A qualitative analysis of the Hamiltonian system is provided. Next, the existence and uniqueness of a steady state in the system are proved, and an algorithm for its search based on solving a nonlinear equation in one special variable is given. The Hamiltonian system near the steady state is stabilized by applying a regulator that can be constructed due to the saddle-point character of the steady state. A numerical example is given that illustrates the obtained analytical results. © Pleiades Publishing, Ltd. 2023. ISSN 1064-5624, Doklady Mathematics, 2023, Vol. 108, Suppl. 1, pp. S157–S165. Pleiades Publishing, Ltd., 2023. Russian Text The Author(s), 2022, published in Matematicheskaya Teoriya Igr i Ee Prilozheniya, 2022, Vol. 14, No. 4, pp. 96–114.

AB - Abstract: The paper investigates a growth model with a production function of constant elasticity of substitution, which generalizes the Cobb–Douglas or Leontief functions. The investment indicators of the model are considered as control parameters chosen to maximize the utility functional. An infinite-horizon optimal control problem is formulated. A Hamiltonian function and a Hamiltonian system are constructed by applying the Pontryagin maximum principle. A qualitative analysis of the Hamiltonian system is provided. Next, the existence and uniqueness of a steady state in the system are proved, and an algorithm for its search based on solving a nonlinear equation in one special variable is given. The Hamiltonian system near the steady state is stabilized by applying a regulator that can be constructed due to the saddle-point character of the steady state. A numerical example is given that illustrates the obtained analytical results. © Pleiades Publishing, Ltd. 2023. ISSN 1064-5624, Doklady Mathematics, 2023, Vol. 108, Suppl. 1, pp. S157–S165. Pleiades Publishing, Ltd., 2023. Russian Text The Author(s), 2022, published in Matematicheskaya Teoriya Igr i Ee Prilozheniya, 2022, Vol. 14, No. 4, pp. 96–114.

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U2 - 10.1134/S1064562423600859

DO - 10.1134/S1064562423600859

M3 - Article

VL - 108

SP - S157-S165

JO - Doklady Mathematics

JF - Doklady Mathematics

SN - 1064-5624

IS - S1

ER -