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19MATEMATHICAL MODELING OF SOCIO-ECONOMIC PROCESSES
Klara Amangeldiyevna Baymuratova
Karakalpak State University named after Berdakh
Keywords: mathematical modeling, economic processes, supply and demand, commodity production, inhomogeneous differential equation, equilibrium price, sustainability, efficiency, economic law
Modern scientists use methods of mathematical analysis, linear programming, matrix and vector calculus, etc. to study economic processes, which in turn are components of mathematical modeling. The basis of economic theory is formed by economic laws, expressed in the form of quantitative relationships between the quantities that characterize the economic system, or process. Such laws make it possible to study real economic systems based on mathematical models. The construction and study of these models is the subject of mathematical economics, which considers economics as a complex dynamic system. In the study, the modeling method is the most important universal method.
The concept of a model refers to the basic concepts of science, representing some reflection of the objects (processes) of research. A particular type of models are mathematical models that reflect an object (process) using mathematical symbols. Model - "is an object or phenomenon, similar, i.e. sufficiently repeating the properties of the modeled object or phenomenon, essential for the purposes of a particular modeling. " The resulting model allows the researcher to experiment with different parameters, variables, conditions and constraints and find out what possible results this leads to. [2]. To study mathematical models of economics, in addition to economic science, it is necessary to master mathematical methods, among which the apparatus of differential equations plays an important role. Economic patterns, as a rule, are complex nonlinear relationships between economic values, the explicit form of which is difficult to establish directly. In the presence of a stable pattern, small changes in values can be approximately replaced by differentials. Then the nonlinear relationships between the quantities, respectively, are replaced by simpler linear relationships between the quantities and their derivatives. These ratios are differential equations with the help of which a mathematical model of an economic system or process is built. The process of mathematization consists in the application of quantitative concepts and formal methods of mathematics to the qualitatively diverse content of special sciences. The use of mathematical methods in science and technology has recently expanded significantly, deepened, penetrated into previously considered inaccessible areas. The effectiveness of these methods depend both on the specifics of the subject of a given
science, the degree of its theoretical maturity, and on the improvement of the mathematical apparatus itself, which makes it possible to display more and more complex properties and patterns of qualitatively diverse phenomena. The use of the principle of consistency, without which effective management is impossible, includes, along with a meaningful analysis of the processes under study, the use of the method of mathematical modeling.
The applications of mathematics in the socio-economic sciences developed in parallel with the development of mathematics itself, and the first constructions of mathematical models in the social sciences are associated with the use of physical analogies in the study of soial processes in the 17th-18th centuries, which laid the foundation for "social physics." mathematical model of the economy was carried out in the book O..Cournot "Investigation of the mathematical principles of the theory of wealth" published in France in 1838.
Mathematical modeling as a method of analyzing macroeconomic processes was first applied by the physician of King Louis, Dr. François Quesnay, who in 1758 published the work "Economic Table". In this work, the first attempt was made to quantitatively describe the national economy.Successful applications of mathematics in economics stimulated the use of mathematical modeling in other social sciences. For example, F. Edgeworth published the book "Mathematical Psychology." modeling in the analysis of specific socio-economic processes is the complexity of the object of modeling, since the applied theoretical model may be too simplified in comparison with the original object. Mathematical modeling should be based on a deep understanding of the basic theoretical models used in the construction of these models, which determine the limits of their applicability. Along with subjective difficulties, there are also quite objective problems that limit the effectiveness of the mathematical modeling method in the analysis of socio-economic processes. include the exceptional diversity and heterogeneity of objects to be modeled. The analysis of these models became possible due to the improvement of computer technology, which is a stimulus for the development of mathematical modeling of socio-economic processes. An example of a model of economic processes based on differential equations is given in terms of the complexity of the equations used (from simple to complex). Consider a model of a process in which it becomes necessary to use the theory of differential equations with separable and separated variables. Problems of this type include, for example, the dependence of supply or demand on the price of a product.
Example. (Supply and demand). Supply and demand are economic categories of commodity production that arise and function on the market, in the field of commodity exchange.
dp
Consider a product. We denote by p the price of the product, and by — = p' - the so-
dt
called trend of price formation (price derivative over time). Consider the case when supply and demand depend on the rate of price change [3, p. 10]. Depending on different factors, supply and demand can be different functions of price and price trends. One of the economic laws of commodity production is the law of supply and demand, which consists in the interdependence of supply and demand and their objective striving for compliance. For the economy, of interest is the condition under which demand is equal to supply. In this case, the price p = p0 is called equilibrium. Both functions s and q are linear with respect to the variables. Consequently, solving problems on supply and demand leads to the need to use the theory of linear differential equations of the first order. A description of methods for solving such equations can be found in [4]. For
dp
example, the supply and demand functions have the form: s = 4 ■ —+ p +19;
dt
q = 3 ■ dp _ 2p + 28 - e"3t. dt
Equilibrium between supply and demand is maintained under the condition
4 ■ dp + p +19 = q = 3 ■ dp _ 2 p + 28 _ e"3t dt dt
or when equality holds
^ + 3 p = ^ + 9. dt (1)
As a result, a linear inhomogeneous differential equation of the first order with constant coefficients is obtained. To solve it, we use the method of I. Bernoulli: we seek the general solution in the form p = u ■ v where u = u (t), v = v (t) p' = u' v + uv'
Substituting p and p (1), we arrive at the equation
du f dv ^
■ v + u dt
— + 3v
v dt
= e~3t + 9.
(2)
1 • dv „ ^ dv „ 7 , i i _3t
Looking for v:--+ 3v = 0 ^ — = _3dt ^ ln v = _3t ^ v = e .
dt v 11
Substituting the found value of v into equation (2), we find and:
du du
--e"3t = e~3' + 9 ^ — = 1 + 9e3 ^ u = t + 3e3 + C. The general solution to equation (1)
dt dt
is written in the form p = e~3t (t + 3e3t + C ).
Let us find the dependence of the equilibrium price on time, if at the initial moment p = 23: 23 = 0 + 3 + C ^ C = 20.
Thus, the sought dependence has the form
p = e~3t (t + 3e3t + 20).
To find out whether a given equilibrium price is stable, we find limp
t^x
x
in this case, the resulting uncertainty in the course of calculations — we will disclose
x
t + 3e3t + 20 1 + 9e3t ,. 27e3t „ according to L'opital's rule. lim p = lim---= lim-— = lim—— = 3.
t^x t^x e t^x 3e 9e
Therefore, the equilibrium price is stable. REFERENCES
1. Pelikh A.S., Terekhov L.L., Terekhova L.A. Economic and mathematical methods and models in production management. Rostov n / D. Phoenix, 2005, 248p.
2. Gladysheva A.V. Research of economic processes by methods of mathematical economics // Socio-economic phenomena and processes. 2010. №6 (022). Pp. 56-62.
3. Some applications of ordinary differential equations in economics: guidelines for students of all forms of education [comp .: O.V. Avdeeva, O. I. Mikryukov]. Vologda: VoGU, 2015, 43 p.
4. Eletskikh IA Educational and methodological support of the discipline "Differential Equations": a teaching aid. Yelets: YSUim.I.A. Bunin, 2018, 63p.
