Geodesic
Analisys of a growth model with a production CES-function
Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 14 (2022) no. 4, pp. 96-114.

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The paper investigates a growth model with a production function of constant elasticity of substitution, which generalizes such functions as Cobb-Douglas or Leontief, when one of its parameter tends to zero or infinity respectively. The investment indicators of the model are considered as control parameters that are chosen in order to maximize the utility functional. An optimal control problem with an infinite planning horizon is formulated. Using the Pontryagin maximum principle, the paper studies a Hamiltonian function and a Hamiltonian system and provides its qualitative analysis. Next, the existence and uniqueness of a steady state are proven, and an algorithm for its search is given by solving a nonlinear equation of one special variable. The following section performs the stabilization of the Hamiltonian system in the vicinity of the steady state using a controller that can be constructed due to saddle character of the steady state. Finally, a numerical example is given that illustrates obtained analytical results.
Keywords: stabilization, nonlinear regulator, Hamiltonian system, production function, CES-function. 
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Anastasiia A. Usova; Alexander M. Tarasyev. Analisys of a growth model with a production CES-function. Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 14 (2022) no. 4, pp. 96-114. http://geodesic.mathdoc.fr/item/MGTA_2022_14_4_a5/

[1] Aseev S.M., Kryazhimskii A.V., “Printsip maksimuma Pontryagina i zadachi optimalnogo ekonomicheskogo rosta”, Trudy MIAN, 257, 2007

[2] Tarasev A.M., Usova A.A., “Stabilizatsiya gamiltonovoi sistemy dlya postroeniya optimalnykh strategii”, Trudy MIAN, 277, 2012, 257–274

[3] Usova A.A., Issledovanie svoistv gamiltonovykh sistem i funktsii tseny v dinamicheskikh modelyakh rosta, Dissertatsiya na soiskanie stepeni kandidata fiz.-mat. nauk, 2012

[4] Krasovskii A., Kryazhimskiy A., Tarasyev A., “Optimal control design in models of economic growth”, Evolutionary Methods for Design, Optimization and Control, CIMNE, Barcelona, 2008, 70–75

[5] Krasovskii A., Tarasyev A., “Conjugation of Hamiltonian Systems in Optimal Control Problems”, IFAC-PapersOnLine, 17:1 (2008), 7784–7789

[6] Klump R., McAdam P., Willman A., Factor substitution and factor augmenting technical progress in the US: a normalized supply-side system approach, ECB Working Paper No 367, 2004

[7] Pontryagin L., Boltyanskii V. and et al., The Mathematical Theory of Optimal Processes, Interscience, New York, 1962

[8] Russkikh O.V., Usova A.A., Tarasyev A.M., “Sensitivity of the qualitative behavior of the Hamiltonian trajectories with respect to growth model parameters”, CEUR Workshop Proceedings, 1662 (2016), 110–120

[9] Tarasyev A., Usova A., “Construction of a regulator for the Hamiltonian system in a two-sector economic growth model”, Proc. of the Steklov Inst. of Math., 271 (2010), 1–21

[10] Tarasyev A., Usova A., “Structure of the Jacobian in economic growth models”, IFAC-PapersOnLine, 48:25, Proc. of the 16th IFAC Workshop CAO (2015), 191–196

[11] Tarasyev A., Usova A., “Cyclic behaviour of optimal trajectories in growth models”, IFAC-PapersOnLine, 49:18, Proc. of the 10th IFAC Symposium NOLCOS (2016), 1048–1053

[12] Tarasyev A., Usova A., “Robust methods for stabilization of Hamiltonian systems in economic growth models”, IFAC-PapersOnLine, 51:32 (2018), 7–12

[13] Tarasyev A., Usova A., Wang W., “Hamiltonian Trajectories in a Heterogeneous Economic Growth Model for Optimization Resource Productivity”, IFAC-PapersOnLine, 48:25 (2015), 74–79

[14] Tarasyev A., Zhu B., “Optimal proportions in growth trends of resource productivity”, IFAC-PapersOnLine, 45:25, Proc. of the 15th IFAC Workshop CAO 12, 182–187

[15] Usova A., Tarasyev A., “Structure of a Stabilizer for the Hamiltonian Systems”, Stability, Control and Differential Games, Proceedings, Lecture Notes in Control and Information Sciences, eds. Tarasyev A., Maksimov V., Filippova T., Springer, Cham, 2020