A.A. Usova

Construction of a regulator for the Hamiltonian system in a two-sector economic growth model

We consider an optimal control problem of investment in the capital stock of a country and in the labor efficiency. We start from a model constructed within the classical approaches of economic growth theory and based on three production factors: capital stock, human capital, and useful work. It is assumed that the levels of investment in the capital stock and human capital are endogenous control parameters of the model, while the useful work is an exogenous parameter subject to logistic-type dynamics. The gross domestic product (GDP) of a country is described by a Cobb-Douglas production function. As a utility function, we take the integral consumption index discounted on an infinite time interval. To solve the resulting optimal control problem, we apply dynamic programming methods. We study optimal control regimes and examine the existence of an equilibrium state in each regime. On the boundaries between domains of different control regimes, we check the smoothness and strict concavity of the maximized Hamiltonian. Special focus is placed on a regime of variable control actions. The novelty of the solution proposed consists in constructing a nonlinear stabilizer based on the feedback principle. The properties of the stabilizer allow one to find an approximate solution to the original problem in the neighborhood of an equilibrium state. Solving numerically the stabilized Hamiltonian system, we find the trajectories of the capital of a country and labor efficiency. The solutions obtained allow one to assess the growth rates of the GDP of the country and the level of consumption in the neighborhood of an equilibrium position.

Stabilizing the Hamiltonian system for constructing optimal trajectories

An infinite-horizon optimal control problem based on an economic growth model is studied. The goal in the problem is to optimize the mechanisms of investment in basic production assets in order to increase the growth rate of the consumption level. The main output variable-the gross domestic product (GDP)-depends on three production factors: capital stock, human capital, and useful work. The first two factors are endogenous variables of the model, and the useful work is an exogenous factor. The dependence of the GDP on the production factors is described by the Cobb-Douglas power-type production function. The economic system under consideration is assumed to be closed, so the GDP is distributed between consumption and investment in the capital stock and human capital. The optimal control problem consists in determining optimal investment strategies that maximize the integral discounted relative consumption index on an infinite time interval. A solution to the problem is constructed on the basis of the Pontryagin maximum principle adapted to infinite-horizon problems. We examine the questions of existence and uniqueness of a solution, verify necessary and sufficient optimality conditions, and perform a qualitative analysis of Hamiltonian systems on the basis of which we propose an algorithm for constructing optimal trajectories. This algorithm uses information on solutions obtained by means of a nonlinear regulator. Finally, we estimate the accuracy of the algorithm with respect to the integral cost functional of the control process.

Optimal Two Sector Growth Models with Three Factors

The paper is devoted to construction of optimal trajectories in the model, which balances growth trends of investments in capital and labor efficiency. The model is constructed within the framework of classical approaches of the growth theory. It is based on three production factors: capital, educated labor and useful work. GDP level is described by a production function of the Cobb–Douglas type. The utility function of the growth process is given by an integral consumption index discounted on the infinite horizon. The optimal control problem is posed to balance investments in capital and labor efficiency. The problem is solved on the basis of dynamic programming principles. A novelty of the solution consists in constructing nonlinear stabilizers constructed on the feedback principle, which leads the system from any current position to a steady state. Growth and decline trends of the simulated trajectories are studied for all components included in the model.

Capital vs Education: Assessment of Economic Growth from Two Perspectives

Abstract The paper is devoted to construction of optimal trajectories in the model which balances growth trends of investments in capital and labor efficiency. The model is constructed within the framework of classical approaches of the growth theory. It is based on three production factors: capital, educated labor and useful work. It is assumed that capital and educated labor are invested endogenously, and useful work is an exogenous flow. The level of GDP is described by an exponential production function of the Cobb-Douglas type. The utility function of the growth process is given by an integral consumption index discounted on the infinite horizon. The optimal control problem is posed to balance investments in capital and labor efficiency. The problem is solved on the basis of dynamic programming principles. Series of Hamiltonian systems are examined including analysis of steady states, properties of trajectories and their growth rates. A novelty of the solution consists in constructing nonlinear stabilizers based on the feedback principle which lead the system from any current position to an equilibrium steady state. Growth and decline trends of the model trajectories are studied for all components of the system and their proportions including: dynamics of GDP, consumption, capital, labor efficiency, investments in capital and labor efficiency.

Proportional economic growth under conditions of limited natural resources

The paper is devoted to economic growth models in which the dynamics of production factors satisfy proportionality conditions. One of the main production factors in the problem of optimizing the productivity of natural resources is the current level of resource consumption, which is characterized by a sharp increase in the prices of resources compared with the price of capital. Investments in production factors play the role of control parameters in the model and are used to maintain proportional economic development. To solve the problem, we propose a two-level optimization structure. At the lower level, proportions are adapted to the changing economic environment according to the optimization mechanism of the production level under fixed cost constraints. At the upper level, the problem of optimal control of investments for an aggregate economic growth model is solved by means of the Pontryagin maximum principle. The application of optimal proportional constructions leads to a system of nonlinear differential equations, whose steady states can be considered as equilibrium states of the economy. We prove that the steady state is not stable, and the system tends to collapse (the production level declines to zero) if the initial point does not coincide with the steady state. We study qualitative properties of the trajectories generated by the proportional development dynamics and indicate the regions of production growth and decay. The parameters of the model are identified by econometric methods on the basis of China’s economic data.

Global Stabilization of the Hamiltonian System in the Two-Sector Economic Growth Model

Abstract In optimal control problems with infinite time horizon, arising in economic growth models, the analytical solution can be derived in specific cases only. This fact is explained, first of all, by nonlinear character of the Hamiltonian system arising in the Pontryagin maximum principle. Another difficulty is connected with the so-called transversality condition which describes the asymptotic behavior of adjoint variables at the infinite time. In the paper, a synthesis of optimal trajectories is carried out. Obtained results allow to conclude that the nonlinear stabilizer ensures global stabilization of the Hamiltonian dynamics. The structure of nonlinear stabilizer is based on the qualitative theory of differential equations. Under assumption on existence of a saddle steady state an “eigen–plane” is constructed by two eigenvectors corresponding to negative eigenvalues. Relations describing “eigen-plane” allow to exclude adjoint variables from (a) the Hamiltonian dynamics and (b) optimal control representations at the steady state neighborhood. So we obtain (a) the stabilized dynamics independent from adjoint variables, and (b) the structure of the nonlinear stabilizer. The provided analysis of the Hamiltonian system argues that the suggested nonlinear stabilizer ensures global stabilization of the Hamiltonian dynamics.

The Value Function as a Solution of Hamiltonian Systems in Linear Optimal Control Problems with Infinite Horizon

Abstract The paper deals with analytical construction of the value function for a linear control problem with infinite horizon arising in problems of economic growth. The proposed algorithm is based on analysis of asymptotic properties of the Hamiltonian system in the Pontryagin maximum principle. The method of indeterminate coefficients is applied for identification of parameters of the value function. Sensitivity analysis of parametric solutions is implemented with respect to qualitative properties of steady states of the Hamiltonian system. The structure of optimal feedbacks is outlined and asymptotic behavior of optimal trajectories is analyzed. Applications to economic growth modeling are discussed.