MODELING BY SOME SYSTEMS OF DIFFERENTIAL EQUATIONS

Abstract

This article examines macro models that evolve over time, along with the mathematical models that describe them. These are dynamic models, wherein all variables generally depend on time, which acts as an independent variable.
Dynamic models can be categorized into continuous and discrete time models, reflecting the nature of the system’s dynamics. In discrete dynamic models, systems of difference equations or individual difference equations are employed. Conversely, continuous dynamic models utilize differential equations, particularly systems of differential equations. In certain scenarios, systems with mixed dynamics exist, for which differential equations are also used.
Difference equations and systems thereof are effectively applied in modeling dynamic processes in fields such as economics and banking.
In this study, we focus on systems of differential equations and investigate their reduction to a single second-order differential equation. Specifically, we demonstrate their application to solving financial forecasting problems. Furthermore, we analyze a discrete second-order model, namely the business cycle model proposed by A. Hansen and P. Samuelson and later refined by J. Hicks.
This model subsequently became known as the Samuelson-Hicks model, and it is inherently discrete. Another notable model of the economic cycle is the Kaldor model, which has been the subject of extensive research in economic dynamics. The Kaldor model posits that investment in a single-sector economy is contingent upon national income, with the propensity to invest decreasing when income deviates from its equilibrium value. Additionally, it assumes that for a given level of income, investment diminishes with the growth of capital. This model is also represented by a system of first-order differential equations. Notably, we delve into the economic cycle model of A. Phillips, devoting considerable attention to its analysis. The Phillips model serves as a continuous analog of the Samuelson-Hicks model.