Optimal control two-sector economy

The analytical solution of a dynamic problem of optimum control by the two-sector economy on a finite interval of time is received. Control consists in distribution of the produced product into accumulation (investment) and non-productive consumption. The problem consists in selecting such a control, at which, for a planned finite interval of time, the maximum of non-productive consumption is provided. Both sectors of economy are characterized by the values: k is the fixed capital, U is the labor resources, and Y _t are the Cobb-Douglas production functions (i = 1,2). The Y, is the gross product produced by the i-th sector at a time, i.e. Y _iAt is gross product produced during At. As a result, we obtain a system of equations describing the behavior of the two-sector economy к = - к ₁ + W ₁, к ₁(0) = к ₁₀ > 0, 2 = Ml 2 + 2 2 = 20 > С = 5С + W ₃, С(0) = 0. Here, C(t) is non-productive consumption, ц (>0) is damping coefficients, 5 (>0) is discount rate, W ₁, W ₂, W ₃ are shares of gross product, aimed at investing sectors of the economy and an increase in non-productive consumption. Control by the two-sector economy consists in choosing values W ₁, W ₂, W ₃. In this paper, we consider such distributions of gross product when W ₁, W ₂, W ₃ are shares of the total gross domestic product produced by both sectors, i.e. Wi=Ui(Y1+Y ₂) (=1,2,3), where u ₁+u ₂+u ₃=1, 03=1-u ₁-u ₂. In this case, the non-productive consumption in the interval [0, T] is equal to т т J = J e -t) W ₃ (t) dt = J e -t )(1 - u ₁ - u ₂) Y (t)dt. 0 0 In the interval [0, T], the main problem is to find such controls u ₁ and u ₂, subjected to the restrictions, under which this functional is maximized. The problem solution is carried out by using the Pontryagin maximum principle. Since the Hamiltonian is linear with respect to the controls u ₁ и u ₂, then, as shown, there are the special controls u _1oc and u _2oc when the capital of both sectors remains constant. The interval [t ₁, t ₂], in which the special control takes place, corresponds to a site of the balanced equilibrium state of the economy, which is called as the highway. As a result, in all cases the optimal control is piecewise and constant. The interval [0, T] is broken into three parts: [0, t ₁] is an exit to the highway, [t ₁, t ₂] is the highway, [t ₂, Т] is a descent from the highway. The problems of the optimal exit to the highway and optimal descent from the highway are solved. The exit to the highway is carried out by the relay control with one switching point. At a descent from the highway, the produced product is invested only in one sector of economy depending on task parameters. Also, obtained results allow to formulate necessary conditions of optimum control existence for our problem, i.e. restrictions on the length of a interval [0, T], the entry conditions, and the model parameters. These conditions are connected with need of fulfillment of the inequalities 012

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