APPLICATION OF THE SAMUELSON EQUATION TO THE EVANS MODEL

Abstract

Mathematical modeling of economic processes is an actual direction of research, because the well-being of citizens and the country as a whole depends on it. In the case of the market, the prices of most goods and services are not planned centrally, are not directly regulated by the state, but are freely set and changed by the market itself. The main factors that control the movement of prices in the market are the demand and supply of goods. In the economy, the most important are dynamic models, the parameters of which change over time. The Evans model (Walras-Evans-Samuelson model) is currently one of the basic concepts explaining the dynamic establishment of the equilibrium price in the market of one product under the influence of supply and demand. This is due to the fact that knowing the dynamics of the economic parameter we are interested in, we can try to build a forecast of its further evolution. The article examines the market of one product. For convenience, we will assume that the functions of the dependence of demand and supply on the price are given by linear relationships. The construction of the Evans model is that the change in price is directly proportional to the excess of demand over supply and the duration of this excess. The Samuelson differential equation with an initial condition, that is, the Cauchy problem, is obtained. Samuelson's equation has a stationary (equilibrium) point, which is a positive price at which supply and demand will be equal. The analysis of the obtained solution of the problem shows that over a long enough time (relatively speaking, at ) the price asymptotically approaches the equilibrium value. If we are not interested in the temporal dependence, but only in the equilibrium price, then it can be found from the differential equation immediately by setting the condition , this is the so-called limit stationary mode. The solution of the Cauchy problem is found by the method of variation of the constant. The parameterization of the model using the fractional differentiation operator in the sense of Gerasimov-Caputo is considered. The resulting solution was analyzed depending on the parameter . For this, the asymptotic representation of the function was used for large values of the argument. Conclusions are made regarding price dynamics in time relative to the equilibrium price when demand and supply are equal.