Properties of Hamiltonian systems in the Pontryagin maximum principle for economic growth problems

Krasovskiĭ, A. A.; Tärasyev, A.M.

doi:10.1134/s0081543808030103

Cited by 18 publications

(9 citation statements)

References 6 publications

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“…Problem analysis is conducted within the Pontryagin maximum for the problems over the infinite time interval [3,4]. Similar problems with smooth production function y = f(k) are investigated in the papers [10,11,12,16,17,18], where the existence of the unique optimal solution having the property of growth saturation is established. In order to apply the ideas proposed in these papers, we need to replace the non-smooth production function y = f(k) with its estimate…”

Section: Growth Model and Control Problemmentioning

confidence: 99%

“…Using the calibrated models and the approximated production function, we consider control problems on optimal distribution of investments in the capital stock of country's economy. The quality of the control process is estimated by the integral consumption index discounted on the infinite time interval [3,12,16]. The problem analysis, provided in the paper, is based on the Pontryagin maximum principle [13] for the problems with the infinite time horizon [1,3,10].…”

Section: Introductionmentioning

confidence: 99%

“…The paper includes numerical results for forecast scenarios of development of the Japan's economy that are constructed using the algorithms proposed in the papers [12,17,18] for solving the Hamiltonian systems. In the numerical experiment, the original model considers the non-smooth production function that is glued from two functions of the Cobb-Douglas type.…”

Section: Introductionmentioning

confidence: 99%

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An optimization model of the economic growth under presence of structural changes

Tärasyev

Usova

Tarasyev

2020

International Conference of Numerical Analysis and Applied Mathematics Icnaam 2019

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The paper investigates economic growth models with non-smooth production functions. The non-smoothness of a production function can be caused, for example, by structural changes in the economy of a region (or a country). Another possible reason is the self-adjustment property of the model, which implies that estimated parameters of the production function may change after the econometric analysis conducted for statistical data augmented with data of new periods. Application of the Pontryagin maximum principle leads to analysis of the Hamiltonian system that includes derivatives of the production function, therefore, the smoothness property of the production function is important for this application. The paper proposes the technique that makes the production function differentiable and, under some additional assumptions, does not affect the qualitative behavior of optimal solutions. Theoretical results are supported by the numerical example.

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Section: Growth Model and Control Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An optimization model of the economic growth under presence of structural changes

Tärasyev

Usova

Tarasyev

2020

International Conference of Numerical Analysis and Applied Mathematics Icnaam 2019

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“…Note that the conditions of the existence theorem (see [8,11]) are fulfilled for control problem (7.1)- (7.4). Moreover, we can formulate the necessary optimality conditions for the control problems with finite horizon [7] and with infinite horizon [8,14] in the form of the Pontryagin maximum principle. Theorem 1.…”

Section: The Pontryagin Maximum Principlementioning

confidence: 99%

“…The model implements the results of investigating the optimization of investments in production factors [12][13][14][15][16][17][18][19][20][21] and essentially supplements and renews them by applying multilevel optimization constructions.…”

Section: Introductionmentioning

confidence: 99%

Optimal control for proportional economic growth

Kryazhimskii

Taras’ev

2016

Proc. Steklov Inst. Math.

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The research is focused on the question of proportional development in economic growth modeling. A multilevel dynamic optimization model is developed for the construction of balanced proportions for production factors and investments in a situation of changing prices. At the first level, models with production functions of different types are examined within the classical static optimization approach. It is shown that all these models possess the property of proportionality: in the solution of product maximization and cost minimization problems, production factor levels are directly proportional to each other with coefficients of proportionality depending on prices and elasticities of production functions. At the second level, proportional solutions of the first level are transferred to an economic growth model to solve the problem of dynamic optimization for the investments in production factors. Due to proportionality conditions and the homogeneity condition of degree 1 for the macroeconomic production functions, the original nonlinear dynamics is converted to a linear system of differential equations that describe the dynamics of production factors. In the conversion, all peculiarities of the nonlinear model are hidden in a time-dependent scale factor (total factor productivity) of the linear model, which is determined by proportions between prices and elasticities of the production functions. For a control problem with linear dynamics, analytic formulas are obtained for optimal development trajectories within the Pontryagin maximum principle for statements with finite and infinite horizons. It is shown that solutions of these two problems differ crucially from each other: in finite horizon problems the optimal investment strategy inevitably has the zero regime at the final stage, whereas the infinite horizon problem always has a strictly positive solution. A remarkable result of the proposed model consists in constructive analytical solutions for optimal investments in production factors, which depend on the price dynamics and other economic parameters such as elasticities of production functions, total factor productivity, and depreciation factors. This feature serves as a background for the productive fusion of optimization models for investments in production factors in the framework of a multilevel structure and provides a solid basis for constructing optimal trajectories of economic development.

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