METHODS OF COMPARATIVE STATICS AND DYNAMICS IN THE THEORY OF ECONOMIC CYCLES - Anatolii Voronin; Olga Gunko (Сучасний стан наукових досліджень та технологій в промисловості)
The subject of this work is the problem of describing the dynamic behavior of the price in the market of one product. The typical balance of interaction of supply and demand functions depending on the price is considered. The dynamic model of price evolution is based on the assumption that the demand function at a given time depends on the supply function at all previous points in time, i.e. there is a process with an aftereffect. The core of the integral transformation is a characteristic of the second order, which can initiate periodic regimes in price variables.  The aim of the work is to synthesize a mathematical model of price changes in the market of one product and study the stability of its equilibrium states with the manifestation of the structure of marginal cycles. The task of the study is to demonstrate the degree of connection between the problem of stability of equilibrium and the problem of obtaining fruitful results in comparative statics. This duality is the principle of conformity of P. Samuelson. The basic mathematical model of the studied process of price dynamics is a system of two nonlinear differential equations of the first order. The research methods are the nonlinear theory of analysis of dynamical systems, the mathematical theory of stability of systems of differential equations, the conceptual apparatus of analysis of typical bifurcations of birth (death) of the boundary cycle, known as the Andronov-Hopf bifurcation. As a result of a detailed analysis of the properties and parameters of self-oscillating modes, a double cycle is revealed, i.e. there is a fact of coexistence around the equilibrium state of stable and unstable limit cycles. Subsequent mathematical transformations prove that the line of demarcation of these two cycles is completely determined by the static parameters of the studied system, which illustrates the mechanism of action of the principle of conformity of P. Samuelson. Conclusions: on the example of the functioning of the labor market, a comprehensive analysis of the stability of two positive equilibrium states, which characterize the effects of substitution and income. Computer simulations are used to perform computational experiments that demonstrate self-oscillating modes of price changes. As a result of the analysis of the obtained numerical results, it is possible to draw a conclusion about the stability of the limit cycle in the vicinity of the equilibrium state, which corresponds to the substitution effect.