The following problem of planning economic growth with multiplicity of goals is the starting point of our considerations. In interval of the social reproduction process, considered in finite planning period (O, T), n different measurable economic goais are realized. They may be expressed as a vector, called the "generał economical activity goal" . The dynamie Pareto optimum is defined as the situation in which it is not possible to increase the degree of attainment of one goal without decreasing the degree of attainment of at least one of the other goals in the finał moment of planning period. If the stocks of n goods in moment T are the econoinical activity goals, the choice problem of optimal strategy of the economic growth can be formulated in foliowin gway : what conditions must be fulfilled to maximize the production of one good during the planning period (O, T) without decreasing stocks of the other goods below the given level in moment T. Our purpose may be formulated as the maximization problem of a Lagrange functional, where the boundary conditions are determined. The strategy which maximizes the functional is the optima! strategy in the sense of dynamie Pareto optimum. The necessary conditions for the maximum of this functional may be obtained from modified Pontriagin conditions. Beside the other conditions, they consist of differentia! equations describing the dynamics of the conjugate variables which may be treated as specific dual prices. Then the maximization of mentioned Lagrange functional with given boundary conditions may be treated as the maximization of weighted sum in interval (O, T;, where the weights are the dual prices in moment T, when the other conditions are fulfilled. lt may be proved that this problem is equivalent to the maximization of the hamiltonian which may be treated as the product ~income) · flow in every moment of planning interval; it is the weighted sum of current goals (particular good ftows) . We prove that the strategy maximizing the hamiltonian in every moment of planning period (O , T) is the strategy t1/hich corresponds to the dynamie Pareto optimum if only the weights in the hamiltonian arc the conjugate variables obtained from the Pontriagin conditions. The maximum of hamiltonian in given moment may be treated as a •s•tatic Pareto optimum in this moment.
Operational Program Digital Poland, 2014-2020, Measure 2.3: Digital accessibility and usefulness of public sector information; funds from the European Regional Development Fund and national co-financing from the state budget.
Oct 15, 2021
Jul 19, 2021