Economic and mathematical methods in economics. The role of applied economics and mathematics research

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION

FEDERAL AGENCY FOR EDUCATION

State educational institution of higher professional education

RUSSIAN STATE TRADE AND ECONOMICS UNIVERSITY

TULA BRANCH

(TF GOU VPO RGTEU)

Abstract in mathematics on the topic:

"Economic and mathematical models"

Completed:

2nd year students

"Finance and Credit"

day department

Maksimova Kristina

Vitka Natalya

Checked:

Doctor of Technical Sciences,

Professor S.V. Yudin _____________

Introduction

1.Economic and mathematical modeling

1.1 Basic concepts and types of models. Their classification

1.2 Economic and mathematical methods

Development and application of economic and mathematical models

2.1 Stages of economic and mathematical modeling

2.2 Application of stochastic models in economics

Conclusion

Bibliography

Introduction

Relevance.Modeling in scientific research began to be used in ancient times and gradually captured new areas of scientific knowledge: technical design, construction and architecture, astronomy, physics, chemistry, biology and, finally, social sciences. The modeling method of the 20th century brought great success and recognition in almost all branches of modern science. However, modeling methodology has been developed independently by individual sciences for a long time. There was no unified system of concepts, no unified terminology. Only gradually the role of modeling as a universal method of scientific knowledge began to be realized.

The term “model” is widely used in various fields of human activity and has many semantic meanings. Let us consider only such “models” that are tools for obtaining knowledge.

A model is a material or mentally imagined object that, in the process of research, replaces the original object so that its direct study provides new knowledge about the original object.

Modeling refers to the process of constructing, studying and applying models. It is closely related to such categories as abstraction, analogy, hypothesis, etc. The modeling process necessarily includes the construction of abstractions, inferences by analogy, and the construction of scientific hypotheses.

Economic and mathematical modeling is an integral part of any research in the field of economics. The rapid development of mathematical analysis, operations research, probability theory and mathematical statistics contributed to the formation of various types of economic models.

The purpose of mathematical modeling of economic systems is to use mathematical methods to most effectively solve problems arising in the field of economics, using, as a rule, modern computer technology.

Why can we talk about the effectiveness of using modeling methods in this area? Firstly, economic objects at various levels (starting from the level of a simple enterprise and ending with the macro level - the national economy or even the world economy) can be considered from the perspective of a systems approach. Secondly, such characteristics of the behavior of economic systems as:

-variability (dynamism);

-inconsistent behavior;

-tendency to deteriorate performance;

-environmental exposure

predetermine the choice of method for their research.

The penetration of mathematics into economics involves overcoming significant difficulties. Mathematics, which developed over several centuries mainly in connection with the needs of physics and technology, was partly to blame for this. But the main reasons still lie in the nature of economic processes, in the specifics of economic science.

The complexity of the economy was sometimes seen as a justification for the impossibility of modeling it and studying it using mathematics. But this point of view is fundamentally wrong. You can model an object of any nature and any complexity. And it is precisely complex objects that are of greatest interest for modeling; This is where modeling can provide results that cannot be obtained by other research methods.

The purpose of this work- reveal the concept of economic and mathematical models and study their classification and the methods on which they are based, as well as consider their application in economics.

Objectives of this work:systematization, accumulation and consolidation of knowledge about economic and mathematical models.

1.Economic and mathematical modeling

1.1 Basic concepts and types of models. Their classification

In the process of researching an object, it is often impractical or even impossible to deal directly with this object. It may be more convenient to replace it with another object similar to this one in those aspects that are important in this study. In general modelcan be defined as a conventional image of a real object (processes), which is created for a deeper study of reality. A research method based on the development and use of models is called modeling. The need for modeling is due to the complexity and sometimes impossibility of directly studying a real object (processes). It is much more accessible to create and study prototypes of real objects (processes), i.e. models. We can say that theoretical knowledge about something, as a rule, is a combination of different models. These models reflect the essential properties of a real object (processes), although in reality reality is much more meaningful and richer.

Model- this is a mentally represented or materially realized system that, displaying or reproducing an object of study, is capable of replacing it so that its study provides new information about this object.

To date, there is no generally accepted unified classification of models. However, from a variety of models, verbal, graphic, physical, economic-mathematical and some other types of models can be distinguished.

Economic and mathematical models- these are models of economic objects or processes, the description of which uses mathematical means. The purposes of their creation are varied: they are built to analyze certain prerequisites and provisions of economic theory, logical justification of economic patterns, processing and bringing empirical data into the system. In practical terms, economic and mathematical models are used as a tool for forecasting, planning, managing and improving various aspects of the economic activity of society.

Economic and mathematical models reflect the most essential properties of a real object or process using a system of equations. There is no unified classification of economic and mathematical models, although their most significant groups can be identified depending on the classification attribute.

By purposemodels are divided into:

· Theoretical-analytical (used in the study of general properties and patterns of economic processes);

· Applied (used in solving specific economic problems, such as problems of economic analysis, forecasting, management).

Taking into account the time factormodels are divided into:

· Dynamic (describe an economic system in development);

· Statistical (an economic system is described in statistics in relation to one specific point in time; it is like a snapshot, slice, fragment of a dynamic system at some point in time).

According to the duration of the time period under considerationmodels are distinguished:

· Short-term forecasting or planning (up to a year);

· Medium-term forecasting or planning (up to 5 years);

· Long-term forecasting or planning (more than 5 years).

According to the purpose of creation and usemodels are distinguished:

· Balance sheet;

· Econometric;

· Optimization;

·Network;

· Queuing systems;

· Imitation (expert).

IN balance sheetmodels reflect the requirement of matching the availability of resources and their use.

Options econometricmodels are assessed using mathematical statistics methods. The most common models are systems of regression equations. These equations reflect the dependence of endogenous (dependent) variables on exogenous (independent) variables. This dependence is mainly expressed through the trend (long-term trend) of the main indicators of the modeled economic system. Econometric models are used to analyze and forecast specific economic processes using real statistical information.

Optimizationmodels allow you to find the best option for production, distribution or consumption from a variety of possible (alternative) options. Limited resources will be used in the best possible way to achieve the goal.

Networkmodels are most widely used in project management. The network model displays a set of works (operations) and events, and their relationship over time. Typically, the network model is designed to perform work in such a sequence that the project completion time is minimal. In this case, the task is to find the critical path. However, there are also network models that are focused not on the time criterion, but, for example, on minimizing the cost of work.

Models queuing systemsare created to minimize the time spent waiting in queues and downtime of service channels.

ImitationThe model, along with machine decisions, contains blocks where decisions are made by a human (expert). Instead of direct human participation in decision making, a knowledge base can act. In this case, a personal computer, specialized software, a database and a knowledge base form an expert system. Expertthe system is designed to solve one or a number of problems by simulating the actions of a person, an expert in a given field.

Taking into account the uncertainty factormodels are divided into:

· Deterministic (with uniquely defined results);

· Stochastic (probabilistic; with different, probabilistic results).

By type of mathematical apparatusmodels are distinguished:

· Linear programming (the optimal plan is achieved at the extreme point of the range of changes in the variables of the system of restrictions);

· Nonlinear programming (there may be several optimal values of the objective function);

· Correlation-regression;

·Matrix;

·Network;

·Game theories;

· Queuing theories, etc.

With the development of economic and mathematical research, the problem of classifying the models used becomes more complicated. Along with the emergence of new types of models and new features of their classification, the process of integrating models of different types into more complex model structures is underway.

modeling mathematical stochastic

1.2 Economic and mathematical methods

Like any modeling, economic-mathematical modeling is based on the principle of analogy, i.e. the possibility of studying an object through the construction and consideration of another, similar to it, but simpler and more accessible object, its model.

The practical tasks of economic and mathematical modeling are, firstly, the analysis of economic objects, secondly, economic forecasting, foreseeing the development of economic processes and the behavior of individual indicators, and thirdly, the development of management decisions at all levels of management.

The essence of economic-mathematical modeling is to describe socio-economic systems and processes in the form of economic-mathematical models, which should be understood as a product of the economic-mathematical modeling process, and economic-mathematical methods as a tool.

Let us consider the issues of classification of economic and mathematical methods. These methods represent a complex of economic and mathematical disciplines, which are an alloy of economics, mathematics and cybernetics. Therefore, the classification of economic and mathematical methods comes down to the classification of the scientific disciplines that make up them.

With a certain degree of convention, the classification of these methods can be presented as follows.

· Economic cybernetics: system analysis of economics, theory of economic information and theory of control systems.

· Mathematical statistics: economic applications of this discipline - sampling method, analysis of variance, correlation analysis, regression analysis, multivariate statistical analysis, index theory, etc.

· Mathematical economics and econometrics, which studies the same issues from the quantitative side: theory of economic growth, theory of production functions, input balances, national accounts, analysis of demand and consumption, regional and spatial analysis, global modeling.

· Methods for making optimal decisions, including operations research in economics. This is the most voluminous section, including the following disciplines and methods: optimal (mathematical) programming, network methods of planning and management, theory and methods of inventory management, queuing theory, game theory, theory and methods of decision making.

Optimal programming, in turn, includes linear and nonlinear programming, dynamic programming, discrete (integer) programming, stochastic programming, etc.

· Methods and disciplines specific separately for both a centrally planned economy and a market (competitive) economy. The first includes the theory of optimal pricing of the functioning of the economy, optimal planning, the theory of optimal pricing, models of material and technical supply, etc. The second includes methods that allow us to develop models of free competition, models of the capitalist cycle, models of monopoly, models of the theory of the firm, etc. . Many of the methods developed for a centrally planned economy can also be useful in economic and mathematical modeling in a market economy.

· Methods of experimental study of economic phenomena. These usually include mathematical methods of analysis and planning of economic experiments, methods of machine imitation (simulation modeling), and business games. This also includes methods of expert assessments developed to assess phenomena that cannot be directly measured.

Economic-mathematical methods use various branches of mathematics, mathematical statistics, and mathematical logic. Computational mathematics, theory of algorithms and other disciplines play a major role in solving economic and mathematical problems. The use of mathematical apparatus has brought tangible results in solving problems of analyzing expanded production processes, determining the optimal growth rate of capital investments, optimal placement, specialization and concentration of production, problems of choosing optimal production methods, determining the optimal sequence of launching into production, problems of preparing production using network planning methods and many others .

Solving standard problems is characterized by clarity of purpose, the ability to develop procedures and rules for conducting calculations in advance.

There are the following prerequisites for using methods of economic and mathematical modeling, the most important of which are a high level of knowledge of economic theory, economic processes and phenomena, the methodology of their qualitative analysis, as well as a high level of mathematical training and mastery of economic and mathematical methods.

Before starting to develop models, it is necessary to carefully analyze the situation, identify goals and relationships, problems to be solved, and the initial data for solving them, maintain a notation system, and only then describe the situation in the form of mathematical relationships.

2. Development and application of economic and mathematical models

2.1 Stages of economic and mathematical modeling

The process of economic and mathematical modeling is a description of economic and social systems and processes in the form of economic and mathematical models. This type of modeling has a number of significant features associated both with the modeling object and with the apparatus and modeling tools used. Therefore, it is advisable to analyze in more detail the sequence and content of the stages of economic and mathematical modeling, highlighting the following six stages:

.Statement of the economic problem and its qualitative analysis;

2.Construction of a mathematical model;

.Mathematical analysis of the model;

.Preparation of background information;

.Numerical solution;

Let's look at each of the stages in more detail.

1.Statement of the economic problem and its qualitative analysis. The main thing here is to clearly formulate the essence of the problem, the assumptions made and the questions to which answers are required. This stage includes identifying the most important features and properties of the modeled object and abstracting from minor ones; studying the structure of an object and the basic dependencies connecting its elements; formulating hypotheses (at least preliminary) explaining the behavior and development of the object.

2.Building a mathematical model. This is the stage of formalizing an economic problem, expressing it in the form of specific mathematical dependencies and relationships (functions, equations, inequalities, etc.). Usually, the main design (type) of a mathematical model is first determined, and then the details of this design are specified (a specific list of variables and parameters, the form of connections). Thus, the construction of the model is in turn divided into several stages.

Excessive complexity and cumbersomeness of the model complicate the research process. It is necessary to take into account not only the real capabilities of information and mathematical support, but also to compare the costs of modeling with the resulting effect.

One of the important features of mathematical models is the potential for their use to solve problems of different qualities. Therefore, even when faced with a new economic problem, there is no need to strive to “invent” the model; first you need to try to apply already known models to solve this problem.

.Mathematical analysis of the model.The purpose of this stage is to clarify the general properties of the model. Purely mathematical research methods are used here. The most important point is the proof of the existence of solutions in the formulated model. If it is possible to prove that the mathematical problem has no solution, then the need for subsequent work on the original version of the model disappears and either the formulation of the economic problem or the methods of its mathematical formalization should be adjusted. During the analytical study of the model, questions are clarified, such as, for example, whether the solution is unique, what variables (unknown) can be included in the solution, what will be the relationships between them, within what limits and depending on the initial conditions they change, what are the trends in their change, etc. d. An analytical study of a model, compared to an empirical (numerical) one, has the advantage that the conclusions obtained remain valid for various specific values of the external and internal parameters of the model.

4.Preparation of initial information.Modeling places stringent demands on the information system. At the same time, the real possibilities of obtaining information limit the choice of models intended for practical use. In this case, not only the fundamental possibility of preparing information (within a certain time frame) is taken into account, but also the costs of preparing the corresponding information arrays.

These costs should not exceed the effect of using additional information.

In the process of preparing information, methods of probability theory, theoretical and mathematical statistics are widely used. In system economic and mathematical modeling, the initial information used in some models is the result of the functioning of other models.

5.Numerical solution.This stage includes the development of algorithms for the numerical solution of the problem, compilation of computer programs and direct calculations. The difficulties of this stage are due, first of all, to the large dimension of economic problems and the need to process significant amounts of information.

Research carried out by numerical methods can significantly complement the results of analytical research, and for many models it is the only feasible one. The class of economic problems that can be solved by numerical methods is much wider than the class of problems accessible to analytical research.

6.Analysis of numerical results and their application.At this final stage of the cycle, the question arises about the correctness and completeness of the modeling results, about the degree of practical applicability of the latter.

Mathematical verification methods can identify incorrect model constructions and thereby narrow the class of potentially correct models. Informal analysis of theoretical conclusions and numerical results obtained through the model, comparing them with existing knowledge and facts of reality also makes it possible to detect shortcomings in the formulation of the economic problem, the constructed mathematical model, and its information and mathematical support.

2.2 Application of stochastic models in economics

The basis for the effectiveness of banking management is systematic control over the optimality, balance and sustainability of functioning in the context of all elements that form the resource potential and determine the prospects for the dynamic development of a credit institution. Its methods and tools require modernization to take into account changing economic conditions. At the same time, the need to improve the mechanism for implementing new banking technologies determines the feasibility of scientific research.

The integral coefficients of financial stability (IFS) of commercial banks used in existing methods often characterize the balance of their condition, but do not allow them to give a complete description of the development trend. It should be taken into account that the result (CFU) depends on many random reasons (endogenous and exogenous), which cannot be fully taken into account in advance.

In this regard, it is justified to consider the possible results of a study of the stable state of banks as random variables having the same probability distribution, since the studies are carried out using the same methodology using the same approach. In addition, they are mutually independent, i.e. the result of each individual coefficient does not depend on the values of the others.

Taking into account that in one trial the random variable takes one and only one possible value, we conclude that the events x1 , x2 , …, xnform a complete group, therefore, the sum of their probabilities will be equal to 1: p1 +p2 +…+pn=1 .

Discrete random variable X- coefficient of financial stability of bank “A”, Y- bank “B”, Z- bank “C” for a given period. In order to obtain a result that gives grounds to draw a conclusion about the sustainability of banks' development, the assessment was carried out on the basis of a 12-year retrospective period (Table 1).

Table 1

Serial number of the year Bank “A” Bank “B” Bank “C”11,3141,2011,09820,8150,9050,81131,0430,9940,83941,2111,0051,01351,1101,0901,00961,0981,1541,01771,1121,1151,02981,3111,328 1.06591, 2451,1911,145101,5701,2041,296111,3001,1261,084121,1431,1511,028Min0,8150,9050,811Max1,5701,3281,296Step0,07550,04230,0485

For each sample for a specific bank, the values are divided into Nintervals, the minimum and maximum values are defined. The procedure for determining the optimal number of groups is based on the application of the Sturgess formula:

N=1+3.322 * log N;

N=1+3.322 * ln12=9.525?10,

Where n- number of groups;

N- the number of the population.

h=(KFUmax- KFUmin) / 10.

table 2

Boundaries of intervals of values of discrete random variables X, Y, Z (financial stability coefficients) and the frequency of occurrence of these values within the designated boundaries

Interval number Interval boundaries Frequency of occurrence (n )XYZXYZ10,815-0,8910,905-0,9470,811-0,86011220,891-0,9660,947-0,9900,860-0,90800030,966-1,0420,990-1,0320,908-0,95702041,042-1,1171,032-1,0740,957-1,00540051,117-1,1931,074-1,1171,005-1,05412561,193-1,2681,117-1,1591,054-1,10223371,268-1,3441,159-1,2011,102-1,15131181,344-1,4191,201-1,2431,151-1,19902091,419-1,4951,243-1,2861,199-1,248000101,495-1,5701,286-1,3281,248-1,296111

Based on the found interval step, the boundaries of the intervals were calculated by adding the found step to the minimum value. The resulting value is the boundary of the first interval (the left boundary is LG). To find the second value (the right boundary of PG), the step is again added to the found first boundary, etc. The last interval boundary coincides with the maximum value:

LG1 =KFUmin;

PG1 =KFUmin+h;

LG2 =PG1;

PG2 =LG2 +h;

PG10 =KFUmax.

Data on the frequency of occurrence of financial stability coefficients (discrete random variables X, Y, Z) are grouped into intervals, and the probability of their values falling within the specified boundaries is determined. In this case, the left value of the boundary is included in the interval, but the right one is not (Table 3).

Table 3

Distribution of discrete random variables X, Y, Z

IndicatorIndicator valuesBank “A”X0,8530,9291,0041,0791,1551,2311,3061,3821,4571,532P(X)0,083000,3330,0830,1670,250000,083Bank "B"Y0,9260,9691,0111,0531,0961,1381,1801,2221,2651,307P(Y)0,08300,16700,1670,2500,0830,16700,083Bank "C"Z0,8350,8840,9330,9811,0301,0781,1271,1751,2241,272P(Z)0,1670000,4170,2500,083000,083

By frequency of occurrence of values ntheir probabilities were found (the frequency of occurrence is divided by 12, based on the number of units in the population), and the midpoints of the intervals were used as values of discrete random variables. Laws of their distribution:

Pi= ni /12;

Xi= (LGi+PGi)/2.

Based on the distribution, one can judge the probability of unsustainable development of each bank:

P(X<1) = P(X=0,853) = 0,083

P(Y<1) = P(Y=0,926) = 0,083

P(Z<1) = P(Z=0,835) = 0,167.

So, with a probability of 0.083, bank “A” can achieve a financial stability coefficient value of 0.853. In other words, there is an 8.3% chance that its expenses will exceed its income. For Bank “B”, the probability of the ratio falling below one was also 0.083, however, taking into account the dynamic development of the organization, this decrease will still be insignificant - to 0.926. Finally, there is a high probability (16.7%) that the activities of Bank C, other things being equal, are characterized by a financial stability value of 0.835.

At the same time, from the distribution tables one can see the probability of sustainable development of banks, i.e. the sum of probabilities, where the coefficient options have a value greater than 1:

P(X>1) = 1 - P(X<1) = 1 - 0,083 = 0,917

P(Y>1) = 1 - P(Y<1) = 1 - 0,083 = 0,917

P(Z>1) = 1 - P(Z<1) = 1 - 0,167 = 0,833.

It can be observed that the least sustainable development is expected in bank “C”.

In general, the distribution law specifies a random variable, but more often it is more appropriate to use numbers that describe the random variable in total. They are called the numerical characteristics of a random variable, and they include the mathematical expectation. The mathematical expectation is approximately equal to the average value of the random variable, and the more tests are carried out, the more it approaches the average value.

The mathematical expectation of a discrete random variable is the sum of the products of all possible values and its probability:

M(X) = x1 p1 +x2 p2 +…+xnpn

The results of calculating the values of mathematical expectations of random variables are presented in Table 4.

Table 4

Numerical characteristics of discrete random variables X, Y, Z

BankExpectationDispersionMean square deviation“A”M(X) = 1.187D(X) =0.027 ?(x) = 0.164"V"M(Y) = 1.124D(Y) = 0.010 ?(y) = 0.101 "С" M(Z) = 1.037D(Z) = 0.012? (z) = 0.112

The obtained mathematical expectations allow us to estimate the average values of the expected probable values of the financial stability coefficient in the future.

So, according to calculations, we can judge that the mathematical expectation of sustainable development of bank “A” is 1.187. The mathematical expectation of banks “B” and “C” is 1.124 and 1.037, respectively, which reflects the expected profitability of their work.

However, knowing only the mathematical expectation, which shows the “center” of the expected possible values of the random variable - CFU, it is still impossible to judge either its possible levels or the degree of their dispersion around the obtained mathematical expectation.

In other words, the mathematical expectation, due to its nature, does not fully characterize the sustainability of the bank’s development. For this reason, it becomes necessary to calculate other numerical characteristics: dispersion and standard deviation. Which allow us to assess the degree of dispersion of possible values of the financial stability coefficient. Mathematical expectations and standard deviations allow us to estimate the interval in which the possible values of the financial stability coefficients of credit institutions will lie.

With a relatively high characteristic value of the mathematical expectation of stability for bank “A”, the standard deviation was 0.164, which indicates that the bank’s stability can either increase by this amount or decrease. In case of a negative change in stability (which is still unlikely, given the obtained probability of unprofitable activity equal to 0.083), the bank’s financial stability coefficient will remain positive - 1.023 (see Table 3)

The activity of Bank “B” with a mathematical expectation of 1.124 is characterized by a smaller range of coefficient values. Thus, even under unfavorable circumstances, the bank will remain stable, since the standard deviation from the predicted value was 0.101, which will allow it to remain in the positive profitability zone. Therefore, we can conclude that the development of this bank is sustainable.

Bank “C”, on the contrary, with a low mathematical expectation of its reliability (1.037), ceteris paribus, will encounter an unacceptable deviation equal to 0.112. In an unfavorable situation, and also taking into account the high percentage of probability of unprofitable activities (16.7%), this credit institution will most likely reduce its financial stability to 0.925.

It is important to note that, having made conclusions about the sustainability of development of banks, it is impossible to confidently predict in advance which of the possible values the financial stability coefficient will take as a result of the test; it depends on many reasons, which cannot be taken into account. From this position, we have very modest information about each random variable. In this connection, it is hardly possible to establish patterns of behavior and the sum of a sufficiently large number of random variables.

However, it turns out that under some relatively broad conditions the overall behavior of a sufficiently large number of random variables almost loses its random character and becomes natural.

When assessing the sustainability of banks' development, it remains to estimate the probability that the deviation of a random variable from its mathematical expectation does not exceed a positive number in absolute value ?.The inequality of P.L. allows us to give the estimate we are interested in. Chebysheva. The probability that the deviation of a random variable X from its mathematical expectation in absolute value is less than a positive number ? not less than :

or in case of reverse probability:

Taking into account the risk associated with loss of stability, we will evaluate the probability of a discrete random variable deviating from the mathematical expectation downward and, considering deviations from the central value both downward and upward to be equally probable, we will rewrite the inequality again:

Next, based on the task, it is necessary to estimate the probability that the future value of the financial stability coefficient will not be lower than 1 from the proposed mathematical expectation (for bank “A” the value ?let’s take it equal to 0.187, for bank “B” - 0.124, for “C” - 0.037) and calculate this probability:

jar":

Bank "C":

According to the inequality of P.L. Chebyshev, the most stable in its development is Bank “B”, since the probability of deviation of the expected values of a random variable from its mathematical expectation is low (0.325), while it is comparatively less than for other banks. Bank A is in second place in terms of comparative sustainability of development, where the coefficient of this deviation is slightly higher than in the first case (0.386). In the third bank, the probability that the value of the financial stability coefficient deviates to the left of the mathematical expectation by more than 0.037 is an almost certain event. Moreover, if we take into account that the probability cannot be more than 1, exceeding the values according to the proof of L.P. Chebyshev must be taken as 1. In other words, the fact that the bank’s development may move into an unstable zone, characterized by a financial stability coefficient of less than 1, is a reliable event.

Thus, characterizing the financial development of commercial banks, we can draw the following conclusions: the mathematical expectation of a discrete random variable (the average expected value of the financial stability coefficient) of bank “A” is equal to 1.187. The standard deviation of this discrete value is 0.164, which objectively characterizes the small spread of coefficient values from the average number. However, the degree of instability of this series is confirmed by the fairly high probability of a negative deviation of the financial stability coefficient from 1, equal to 0.386.

Analysis of the activities of the second bank showed that the mathematical expectation of the CFU is equal to 1.124 with a standard deviation of 0.101. Thus, the activities of a credit institution are characterized by a small spread in the values of the financial stability coefficient, i.e. is more concentrated and stable, which is confirmed by the relatively low probability (0.325) of the bank moving into the unprofitable zone.

The stability of bank “C” is characterized by a low value of the mathematical expectation (1.037) and also a small spread of values (standard deviation is 0.112). L.P. inequality Chebyshev proves the fact that the probability of obtaining a negative value of the financial stability coefficient is equal to 1, i.e. the expectation of positive dynamics of its development, all other things being equal, will look very unreasonable. Thus, the proposed model, based on determining the existing distribution of discrete random variables (values of financial stability coefficients of commercial banks) and confirmed by assessing their equally probable positive or negative deviation from the obtained mathematical expectation, allows us to determine its current and future level.

Conclusion

The use of mathematics in economic science gave impetus to the development of both economic science itself and applied mathematics, in terms of methods of economic and mathematical models. The proverb says: “Measure twice - Cut once.” Using models requires time, effort, and material resources. In addition, calculations based on models are opposed to volitional decisions, since they allow us to assess in advance the consequences of each decision, discard unacceptable options and recommend the most successful ones. Economic and mathematical modeling is based on the principle of analogy, i.e. the possibility of studying an object through the construction and consideration of another, similar to it, but simpler and more accessible object, its model.

The practical tasks of economic and mathematical modeling are, firstly, the analysis of economic objects; secondly, economic forecasting, forecasting the development of economic processes and the behavior of individual indicators; thirdly, the development of management decisions at all levels of management.

The work revealed that economic and mathematical models can be divided according to the following criteria:

· intended purpose;

· taking into account the time factor;

· the duration of the period under consideration;

· purposes of creation and use;

· taking into account the uncertainty factor;

· type of mathematical apparatus;

The description of economic processes and phenomena in the form of economic and mathematical models is based on the use of one of the economic and mathematical methods that are used at all levels of management.

Economic and mathematical methods are becoming especially important as information technologies are introduced in all areas of practice. The main stages of the modeling process were also considered, namely:

· formulation of an economic problem and its qualitative analysis;

· building a mathematical model;

· mathematical analysis of the model;

· preparation of background information;

· numerical solution;

· analysis of numerical results and their application.

The work presented an article by Candidate of Economic Sciences, Associate Professor of the Department of Finance and Credit S.V. Boyko, which notes that domestic credit institutions exposed to the influence of the external environment are faced with the task of finding management tools that involve the implementation of rational anti-crisis measures aimed at stabilizing the growth rate of the basic indicators of their activities. In this regard, the importance of adequately determining financial stability using various methods and models increases, one of the varieties of which is stochastic (probabilistic) models, which allow not only to identify the expected factors of growth or decline in stability, but also to formulate a set of preventive measures to preserve it.

The potential possibility of mathematical modeling of any economic objects and processes does not mean, of course, its successful feasibility with a given level of economic and mathematical knowledge, available specific information and computer technology. And although it is impossible to indicate the absolute limits of the mathematical formalizability of economic problems, there will always be still unformalized problems, as well as situations where mathematical modeling is not effective enough.

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Math modeling

One of the types of formalized sign modeling is mathematical modeling, carried out using the language of mathematics and logic. To study any class of phenomena in the external world, a mathematical model of it is built, i.e. an approximate description of this class of phenomena, expressed using mathematical symbolism.

The process of mathematical modeling itself can be divided into four main stages:

Istage: Formulating laws connecting the main objects of the model, i.e. recording in the form of mathematical terms formulated qualitative ideas about the connections between model objects.

IIstage: Study of mathematical problems to which mathematical models lead. The main question is the solution of the direct problem, i.e. obtaining, as a result of the analysis of the model, output data (theoretical consequences) for their further comparison with the results of observations of the phenomena being studied.

IIIstage: Adjustment of the accepted hypothetical model according to the criterion of practice, i.e. clarification of the question of whether the results of observations are consistent with the theoretical consequences of the model within the limits of observational accuracy. If the model was completely defined - all its parameters were given - then determining the deviations of theoretical consequences from observations provides solutions to the direct problem with subsequent assessment of the deviations. If deviations go beyond the accuracy of observations, then the model cannot be accepted. Often, when building a model, some of its characteristics remain undefined. Application of the practice criterion to the evaluation of a mathematical model allows one to draw a conclusion about the correctness of the provisions underlying the (hypothetical) model to be studied.

IVstage: Subsequent analysis of the model in connection with the accumulation of data on the studied phenomena and modernization of the model. With the advent of computers, the method of mathematical modeling took a leading place among other research methods. This method plays a particularly important role in modern economic science. Studying and forecasting any economic phenomenon using the method of mathematical modeling allows us to design new technical means, predict the impact of certain factors on this phenomenon, and plan these phenomena even in the presence of an unstable economic situation.

The essence of economic analysis

Analysis (decomposition, dissection, analysis) is a logical technique, a method of research, the essence of which is that the subject being studied is mentally dissected into its component elements, each of which is then studied separately as part of a dissected whole, in order to identify the elements isolated during the analysis connect using another logical technique - synthesis - into a whole enriched with new knowledge.

Under economic analysis understand an applied scientific discipline, which is a system of special knowledge that allows one to assess the effectiveness of the activities of a particular subject of a market economy.

Theory of economic analysis allows you to rationally justify, predict the development of the management object in the near future and evaluate the feasibility of making a management decision.

Main directions of economic analysis:

Formulating a system of indicators characterizing the performance of the analyzed object;

Qualitative analysis of the phenomenon being studied (result);

Quantitative analysis of this phenomenon (result):

For the development and adoption of management decisions, it is important that it is a means of solving the main problem of identifying reserves for increasing the efficiency of economic activity in improving the use of production resources, reducing costs, increasing profitability and increasing profits, i.e. aimed at the ultimate goal of implementing a management decision.

The developers of the theory of economic analysis emphasize it characteristic peculiarities:

1. A dialectical approach to the study of economic processes, which are characterized by: the transition of quantity into quality, the emergence of a new quality, the negation of the negation, the struggle of opposites, the withering away of the old and the emergence of the new.

2. The dependence of economic phenomena on causal relationships and interdependence.

3. Identification and measurement of relationships and interdependencies of indicators are based on knowledge of objective patterns of development of production and circulation of goods.

Economic analysis, first of all, is factorial, that is, it determines the influence of a set of economic factors on the performance indicator of an enterprise.

The influence of various factors on the economic indicator of the functioning of an enterprise or firm is carried out using stochastic analysis.

In turn, deterministic and stochastic analyzes provide:

Establishing cause-and-effect or probabilistic relationships between factors and performance indicators;

Identification of economic patterns of influence of factors on the functioning of an enterprise and their expression using mathematical dependencies;

The possibility of constructing models (primarily mathematical) of the influence of factor systems on performance indicators and using them to study the influence on the final result of the management decision made .

In practice, various types of economic analysis are used. Analyzes are especially important for management decisions: operational, current, long-term (by time periods); partial and complex (by volume); to identify reserves, improve quality, etc. (as intended); predictive analysis. Forecasts allow you to economically justify strategic, operational (functional) or tactical management decisions .

Historically, two groups of methods and techniques have developed: traditional and mathematical. Let's take a closer look at the application of mathematical methods in economic analysis.

Mathematical methods in economic analysis

The use of mathematical methods in the field of management is the most important direction for improving management systems. Mathematical methods speed up economic analysis, contribute to a more complete accounting of the influence of factors on business results, and increase the accuracy of calculations. The application of mathematical methods requires:

* a systematic approach to the study of a given object, taking into account interconnections and relationships with other objects (enterprises, firms);

* development of mathematical models that reflect quantitative indicators of the systemic activities of the organization's employees, processes occurring in complex systems such as enterprises;

* improving the information support system for enterprise management using electronic computer technology.

Solving problems of economic analysis using mathematical methods is possible if they are formulated mathematically, i.e. real economic relationships and dependencies are expressed using mathematical analysis. This necessitates the development of mathematical models.

In management practice, various methods are used to solve economic problems. Figure 1 shows the main mathematical methods used in economic analysis.

The selected classification criteria are quite arbitrary. For example, in network planning and management various mathematical methods are used, and many authors put different content into the meaning of the term “operations research”.

Methods of elementary mathematics used in traditional economic calculations when justifying resource needs, developing plans, projects, etc.

Classical methods of mathematical analysis are used independently (differentiation and integration) and within the framework of other methods (mathematical statistics, mathematical programming).

Statistical methods - the main means of studying mass repeating phenomena. They are used when it is possible to represent changes in the analyzed indicators as a random process. If the relationship between the analyzed characteristics is not deterministic, but stochastic, then statistical and probabilistic methods become practically the only research tool. In economic analysis, the best known methods are multiple and pair correlation analysis.

To study simultaneous statistical populations, the distribution law, variation series, and sampling method are used. For multidimensional statistical populations, correlations, regressions, dispersion, covariance, spectral, component, and factor types of analysis are used.

Economic methods are based on the synthesis of three areas of knowledge: economics, mathematics and statistics. The basis of econometrics is an economic model, i.e. schematic representation of an economic phenomenon or processes, reflection of their characteristic features using scientific abstraction. The most common method of economic analysis is “input-output”. The method represents matrix (balance sheet) models built according to a checkerboard pattern and clearly illustrating the relationship between costs and production results.

Mathematical programming methods - the main means of solving problems of optimizing production and economic activities. In essence, the methods are means of planning calculations, and they make it possible to assess the intensity of planned tasks, the scarcity of results, and determine limiting types of raw materials and groups of equipment.

Under Operations Research understands the development of methods of targeted actions (operations), quantitative assessment of solutions and selection of the best one. The goal of operations research is the combination of structural interconnected elements of the system that most provides the best economic indicator.

Game theory as a branch of operations research, it is a theory of mathematical models for making optimal decisions under conditions of uncertainty or conflict of several parties with different interests.

Methods of mathematical statistics

Rice. 1. Classification of the main mathematical methods used in economic analysis.

Queuing theory based on probability theory explores mathematical methods for quantifying queuing processes. A feature of all problems associated with queuing is the random nature of the phenomena being studied. The number of requests for service and the time intervals between their arrivals are random in nature, but in the aggregate they are subject to statistical laws, the quantitative study of which is the subject of queuing theory.

Economic cybernetics analyzes economic phenomena and processes as complex systems from the point of view of control laws and the movement of information in them. Modeling and system analysis methods are most developed in this area.

The application of mathematical methods in economic analysis is based on the methodology of economic-mathematical modeling of economic processes and a scientifically based classification of methods and problems of analysis. All economic and mathematical methods (problems) are divided into two groups: optimization decisions based on a given criterion and non-optimizing(solutions without optimality criterion).

On the basis of obtaining an exact solution, all mathematical methods are divided into accurate(with or without a criterion, a unique solution is obtained) and close(based on stochastic information).

Optimal exact methods include methods of the theory of optimal processes, some methods of mathematical programming and methods of operations research; optimization approximate methods include some of the methods of mathematical programming, operations research, economic cybernetics, and heuristics.

Non-optimization exact methods include methods of elementary mathematics and classical methods of mathematical analysis, economic methods; non-optimization approximate methods include the method of statistical tests and other methods of mathematical statistics.

Mathematical models of queuing and inventory management are especially often used. For example, the theory of queuing is based on the theory developed by scientists A.N. Kolmogorov and A.L. Khanchin's theory of queuing.

Queuing theory

This theory allows us to study systems designed to serve a massive flow of requirements of a random nature. Both the moments at which requirements arise and the time spent servicing them can be random. The purpose of the theoretical methods is to find a reasonable organization of service that ensures its specified quality, to determine optimal (from the point of view of the accepted criterion) standards of duty service, the need for which arises unplanned and irregularly.

Using the method of mathematical modeling, it is possible to determine, for example, the optimal number of automatically operating machines that can be serviced by one worker or a team of workers, etc.

A typical example of objects of queuing theory is automatic telephone exchanges - PBXs. The PBX randomly receives “requests” - calls from subscribers, and “service” consists of connecting subscribers with other subscribers, maintaining communication during a conversation, etc. Problems of the theory, formulated mathematically, usually come down to the study of a special type of random processes.

Based on the given probabilistic characteristics of the incoming call flow and the duration of service and taking into account the design of the service system, the theory determines the corresponding characteristics of the quality of service (probability of failure, average waiting time for the start of service, etc.).

Mathematical models of numerous problems of technical and economic content are also linear programming problems. Linear programming is a discipline devoted to the theory and methods of solving problems about extrema of linear functions on sets defined by systems of linear equalities and inequalities.

Enterprise planning problem

To produce homogeneous products, it is necessary to spend various production factors - raw materials, labor, machine tools, fuel, transport, etc. Usually there are several proven technological production methods, and in these methods the costs of production factors per unit of time for the production of products are different.

The amount of production factors consumed and the number of products manufactured depend on how long the enterprise will work using one or another technological method.

The task is posed of rational distribution of the enterprise’s operating time using various technological methods, i.e. such that the maximum number of products will be produced at the given limited costs of each production factor.

Based on the method of mathematical modeling in operational research, many important problems that require specific solution methods are also solved. These include:

· The problem of product reliability.

· The task of replacing equipment.

· Scheduling theory (the so-called scheduling theory).

· Resource allocation problem.

· Pricing problem.

· Theory of network planning.

Product reliability problem

The reliability of products is determined by a set of indicators. For each type of product, there are recommendations for choosing reliability indicators.

To evaluate products that may be in two possible states - operational and failed, the following indicators are used: average operating time before failure occurs (time to first failure), time between failures, failure rate, failure flow parameter, average time to restore an operational state, probability of failure-free operation during time t, availability factor.

Resource Allocation Problem

The issue of resource allocation is one of the main ones in the production management process. To solve this issue, operational research uses the construction of a linear statistical model.

Pricing problem

For an enterprise, the issue of pricing products plays an important role. The profit of an enterprise depends on how pricing is carried out. In addition, in the current conditions of a market economy, price has become a significant factor in competition.

Network planning theory

Network planning and management is a planning system for managing the development of large economic complexes, design and technological preparation for the production of new types of goods, construction and reconstruction, major repairs of fixed assets through the use of network diagrams.

The essence of network planning and management is to compile a mathematical model of the managed object in the form of a network diagram or a model located in the computer memory, which reflects the relationship and duration of a certain set of works. The network diagram, after its optimization by means of applied mathematics and computer technology, is used for operational management of work.

Solving economic problems using the method of mathematical modeling allows for effective management of both individual production processes at the level of forecasting and planning of economic situations and making management decisions based on this, and the entire economy as a whole. Consequently, mathematical modeling as a method is closely related to the theory of decision making in management.

Stages of economic and mathematical modeling

The main stages of the modeling process have already been discussed above. In various branches of knowledge, including economics, they acquire their own specific features. Let us analyze the sequence and content of the stages of one cycle of economic and mathematical modeling.

1. Statement of the economic problem and its qualitative analysis. The main thing here is to clearly formulate the essence of the problem, the assumptions made and the questions to which answers are required. This stage includes identifying the most important features and properties of the modeled object and abstracting from minor ones; studying the structure of an object and the basic dependencies connecting its elements; formulating hypotheses that explain the behavior and development of the object.

2. Construction of a mathematical model. This is the stage of formalizing an economic problem, expressing it in the form of specific mathematical dependencies and relationships (functions, equations, inequalities, etc.). Usually, the main design (type) of a mathematical model is first determined, and then the details of this design are specified (a specific list of variables and parameters, the form of connections). Thus, the construction of the model is in turn divided into several stages.

It is wrong to believe that the more facts a model takes into account, the better it “works” and gives better results. The same can be said about such characteristics of the complexity of the model as the forms of mathematical dependencies used (linear and nonlinear), taking into account factors of randomness and uncertainty, etc. Excessive complexity and cumbersomeness of the model complicate the research process. It is necessary to take into account not only the real capabilities of information and mathematical support, but also to compare the costs of modeling with the resulting effect (as the complexity of the model increases, the increase in costs may exceed the increase in effect).

One of the important features of mathematical models is the potential for their use to solve problems of different qualities. Therefore, even when faced with a new economic problem, there is no need to strive to “invent” the model; First, you need to try to apply already known models to solve this problem.

In the process of building a model, a comparison of two systems of scientific knowledge is carried out - economic and mathematical. It is natural to strive to obtain a model that belongs to a well-studied class of mathematical problems. Often this can be done by somewhat simplifying the initial assumptions of the model, without distorting the essential features of the modeled object. However, a situation is also possible when the formalization of an economic problem leads to a previously unknown mathematical structure. The needs of economic science and practice in the mid-twentieth century. contributed to the development of mathematical programming, game theory, functional analysis, and computational mathematics. It is likely that in the future the development of economic science will become an important stimulus for the creation of new branches of mathematics.

3. Mathematical analysis of the model. The purpose of this stage is to clarify the general properties of the model. Purely mathematical research methods are used here. The most important point is the proof of the existence of solutions in the formulated model (existence theorem). If it can be proven that the mathematical problem has no solution, then there is no need for subsequent work on the original version of the model; either the formulation of the economic problem or the methods of its mathematical formalization should be adjusted. During the analytical study of the model, questions are clarified, such as, for example, is there a unique solution, what variables (unknowns) can be included in the solution, what will be the relationships between them, to what extent and depending on what initial conditions they change, what are the trends in their change and etc. An analytical study of a model, compared to an empirical (numerical) one, has the advantage that the conclusions obtained remain valid for various specific values of the external and internal parameters of the model.

Knowing the general properties of a model is so important, often in order to prove such properties, researchers deliberately idealize the original model. And yet, models of complex economic objects are very difficult to study analytically. In cases where analytical methods fail to determine the general properties of the model, and simplifications of the model lead to unacceptable results, they move on to numerical research methods.

4. Preparation of background information. Modeling places stringent demands on the information system. At the same time, the real possibilities of obtaining information limit the choice of models intended for practical use. In this case, not only the fundamental possibility of preparing information (within a certain time frame) is taken into account, but also the costs of preparing the corresponding information arrays. These costs should not exceed the effect of using additional information.

5. Numerical solution. This stage includes the development of algorithms for numerical solution of the problem, compilation of computer programs and direct calculations. The difficulties of this stage are due, first of all, to the large dimension of economic problems and the need to process significant amounts of information.

Typically, calculations using an economic-mathematical model are multivariate in nature. Thanks to the high speed of modern computers, it is possible to conduct numerous “model” experiments, studying the “behavior” of the model under various changes in certain conditions. Research carried out by numerical methods can significantly complement the results of analytical research, and for many models it is the only feasible one. The class of economic problems that can be solved by numerical methods is much wider than the class of problems accessible to analytical research.

6. Analysis of numerical results and their application. At this final stage of the cycle, the question arises about the correctness and completeness of the modeling results, about the degree of practical applicability of the latter.

References

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Theory

1.

Model- this is a simplified representation of a real device and the processes and phenomena occurring in it . Modeling is the process of creating and researching models. Modeling facilitates the study of an object with the aim of its creation, further transformation and development. It is used to study an existing system when it is impractical to conduct a real experiment due to significant financial and labor costs, as well as when it is necessary to analyze the designed system, i.e. which does not yet physically exist in this organization.

The modeling process includes three elements: 1) subject (researcher), 2) object of research, 3) model that mediates the relationship between the cognizing subject and the cognizable object.

The model has the following functions:

1) a means of understanding reality 2) a means of communication and learning 3) a means of planning and forecasting 3) a means of improvement (optimization) 4) a means of choice (decision making)

During modeling, knowledge about the object under study is expanded and refined, and the original model is gradually improved. Any deficiencies found after the first simulation cycle are corrected and the simulation is run again. Thus, the modeling methodology contains great opportunities for self-development.

2.

Modeling in economics is an explanation of socio-economic systems using symbolic mathematical means. The practical tasks of economic and mathematical modeling are: analysis of economic objects and processes, economic forecasting, prediction of the development of economic processes, preparation of management decisions at all levels of economic activity.

Features of the economy as an object of modeling are:

1) the economy, as a complex system, is a subsystem of society, but, in turn, it consists of production and non-production spheres that interact with each other;

2) emergence, meaning that economic objects, processes and phenomena have properties that none of the elements that form them has;

3) probabilistic, uncertain, random nature of the occurrence of economic processes and phenomena;

4) the inertial nature of economic development, according to which the laws, patterns, trends, connections, dependencies that took place in the past period continue to operate for some time in the future.

All of the above and other properties of the economy complicate its study, the identification of patterns, dynamic trends, connections and dependencies. Mathematical modeling is a tool whose skillful use allows one to successfully solve problems in the study of complex systems, including such complex ones as economic objects, processes, and phenomena.

3.

Economic system it is a complex dynamic system, including the processes of production, exchange, distribution, redistribution and consumption of goods (a system of subjects of economic relations interacting in the market).

Microeconomic systems - (corporations and associations; enterprises; organizations; institutions; individual subjects of economic relations).

Macroeconomic systems - (region; national economy; world economy; system of interacting markets;)

Methodology: a branch of knowledge that studies the conditions, principles, structure, logical organization, methods and methods of activity.

Mechanism: a system of practical methods aimed at ensuring the practical use of methods and models for solving problems of managing economic systems.

Method: a set of tools aimed at solving a specific problem.

Mathematical method: a method of research aimed at analyzing, synthesizing, optimizing or forecasting the state, structure, functions or behavior of an economic system, the consequences and prospects of its functioning, management or development, using formal methods and apparatus of mathematical research.

Mathematical model: a mathematical description of an object (process or system), used in research instead of the original object, for the purpose of analysis, determination of quantitative or logical connections between its parts.

Complex of mathematical models: a collection of collaborative mathematical models that use or exchange common data and are aimed at achieving a common goal or solving a common problem.

4.

There are two basic approach to economic modeling: microeconomic and macroeconomic. Microeconomic approach reflects the functioning and structure of individual elements of the system being studied (for example, when studying the banking sector, such an element is a commercial bank) or the state and development of individual socio-economic processes occurring in it, and is implemented, first of all, through the development of applied methods for analyzing performance results. So, for example, in relation to a bank, this is an analysis of the bank’s liquidity, assessment of banking risks, etc. Tasks within the framework of the microeconomic approach are also implemented through the development of special economic and mathematical models. Macroeconomic approach involves analyzing the specifics of the functioning of the system under study in connection with the main macroeconomic indicators of the development of the national economy. In relation to the analysis of the activities of the banking sector, this approach consists of considering it in interaction with various segments of the financial market and, accordingly, in the relationship between the indicators of the banking sector and the macroeconomic indicators of the economy as a whole. In this case, the macroeconomic approach can practically be implemented by constructing factor analysis models, such as the factor model of the market for government short-term obligations, the model of the loan capital market, as well as in constructing and assessing forecast values for the dynamics of individual indicators of the banking sector.

A number of areas in modeling are based on microeconomics, while others are based on macroeconomics. There are no clear boundaries, for example, we can say that the economics of an industrial enterprise, labor economics, public utilities economics belong to microeconomics, monetary economics, investment, consumption are macroeconomics, and the financial market, international trade, economic development is an area of overlap.

5.

In its most general form, equilibrium in the economy is the balance and proportionality of its main parameters, in other words, a situation where participants in economic activities have no incentive to change the existing situation.

Market equilibrium is a situation in the market when the demand for a product is equal to its supply. Typically, equilibrium is achieved by either limiting needs (in the market they always appear in the form of effective demand) or increasing and optimizing the use of resources.

A. Marshall considered equilibrium at the level of an individual economy or industry. This is a micro level that characterizes the features and conditions of partial equilibrium. But general equilibrium is the coordinated development (correspondence) of all markets, all sectors and spheres, the optimal state of the economy as a whole.

Moreover, the balance of the national system. economy is not only market equilibrium. Because disruptions in production inevitably lead to disequilibrium in markets. And in reality, the economy is influenced by other, non-market factors (wars, social unrest, weather, demographic shifts).

The problem of market equilibrium was analyzed by J. Robinson, E. Chamberlin, J. Clark. However, the pioneer in the study of this issue was L. Walras.

As for the state of equilibrium, according to Walras, it presupposes the presence of three conditions:

1) supply and demand for factors of production are equal; a constant and stable price is set for them;

2) supply and demand for goods (and services) are also equal and are sold on the basis of constant, stable prices;

3) prices of goods correspond to production costs.

There are three types of market equilibrium: instantaneous, short-term and long-term, through which supply sequentially passes in the process of increasing its elasticity in response to increasing demand.

6.

CLOSED ECONOMY- a model of a closed economic system focused on the exclusive use of its own resources and the rejection of foreign economic relations. This model was implemented, as a rule, in conditions of preparation for war or war. In particular, the economy of Nazi Germany and the pre-war economy of the USSR were approaching it.

A closed economy is an economy fenced off from the world economic community by a high level of customs duties and non-tariff barriers. An increasing number of developing countries are moving from closed to open economies. The economies of some countries in the poor South, primarily the countries of sub-Saharan Africa, remain closed. The economies of these countries are not affected by the increase in international economic exchanges and capital movements. The closed nature of the economy reinforces deep underdevelopment, which, in turn, does not allow them to adapt to structural changes in world markets.

OPEN ECONOMY- the country's economy is closely connected with the world market and the international division of labor. It is the opposite of closed systems. The degree of openness is characterized by such indicators as: the ratio of exports and imports to GDP; movement of capital abroad and from abroad; currency convertibility; participation in international economic organizations. In modern conditions, it is becoming a factor in the development of the national economy, a guide to the best world standards.

Many directions of economic thought in the West (representatives of open economy countries) developed their own model of an open economy. This topic remains relevant to this day because... open economy models open up a range of issues such as interaction between national economies, the combination of macroeconomic and foreign economic policies, and in the case of its non-equilibrium level, the issue of developing one’s own stabilization policy.

Closed and open economy models:

Fundamental imbalance of the economy (uneven development)

Government intervention (protectionism and anti-dumping policy) and globalization (competition for resources)

Import and export are signs of an open economy

Mutual dependence of countries (international division of labor)

Transnational corporations (capital flows)

7.

The development of technological models is one of the most consistent methods in macroeconomic modeling.

These models directly link production outputs and costs with its technology, allow the use of material and financial balance ratios, and carry out forecasting, optimization and development analysis.

Technological models can be static And dynamic .

-Static models operate with constant values A and B, describe the existing balance of inputs and outputs and are intended for short-term forecasts or optimization (for example, Leontief MOB model)

- Dynamic models include price dynamics (and possibly autonomous technological progress), make it possible to study economic growth and economic sustainability ( model of von Neumann, Morishima and etc.)

At the same time, the technological approach has a number of disadvantages: in technological models usually not considered: -Geographical location of the object; -Real technical progress; -Price dynamics; -Limited labor resources, etc.

The von Neumann model is expanding economy model , in which all outputs and costs increase in the same proportion. The model is closed, that is, all outputs of one period become expenses of the next period. It also does not use primary factors and considers consumption as a cost in the technological process, so all costs are reproducible and there is no need to consider primary resources.

Model assumptions: The real level of wages corresponds to the subsistence level and all excess income is reinvested; The real level of wages is given and incomes are of a residual nature; There is no distinction between primary factors of production and production volumes; There are no “initial” factors of production, such as labor in traditional theory.

The model describes an economy characterized by linear technology of production processes.

modeling V economy. 2.1. The concept of “model” and “ modeling" With the concept “ modeling economic systems” (and also mathematical etc.) are connected...

Economic-mathematical modeling as a way to study and evaluate economic activities

Abstract >> Economics

Ed. L. N. Chechevitsyna - M.: Phoenix, 2003 Mathematical modeling V economy: Textbook / ed. E.S. Kundysheva... ed. L. T. Gilyarovskaya - M.: Prospekt, 2007 Mathematical modeling V economy: Textbook / ed. IN AND. Mazhukina...

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Test >> Economic and mathematical modeling

... : "Economic-mathematical methods and modeling" 2006 Contents Introduction Mathematical modeling V economy 1.1 Development of methods modeling 1.2 Modeling as a method of scientific knowledge 1.3 Economic-mathematical ...

1. Modeling as a method of scientific knowledge.

Modeling in scientific research began to be used in ancient times and gradually captured new areas of scientific knowledge: technical design, construction and architecture, astronomy, physics, chemistry, biology and, finally, social sciences. The modeling method of the 20th century brought great success and recognition in almost all branches of modern science. However, modeling methodology has long been developed independently by individual sciences. There was no unified system of concepts, no unified terminology. Only gradually the role of modeling as a universal method of scientific knowledge began to be realized.

The term “model” is widely used in various fields of human activity and has many semantic meanings. Let us consider only such “models” that are tools for obtaining knowledge.

A model is a material or mentally imagined object that, in the process of research, replaces the original object so that its direct study provides new knowledge about the original object

The main feature of modeling is that it is a method of indirect cognition using proxy objects. The model acts as a kind of cognition tool that the researcher puts between himself and the object and with the help of which he studies the object of interest to him. It is this feature of the modeling method that determines the specific forms of using abstractions, analogies, hypotheses, and other categories and methods of cognition.

The need to use the modeling method is determined by the fact that many objects (or problems related to these objects) are either impossible to directly study, or this research requires a lot of time and money.

The modeling process includes three elements: 1) the subject (researcher), 2) the object of research, 3) a model that mediates the relationship between the cognizing subject and the cognizable object.

Let there be or need to create some object A. We construct (materially or mentally) or find in the real world another object B - a model of object A. The stage of constructing a model presupposes the presence of some knowledge about the original object. The cognitive capabilities of the model are determined by the fact that the model reflects any essential features of the original object. The question of the necessity and sufficient degree of similarity between the original and the model requires specific analysis. Obviously, the model loses its meaning both in the case of identity with the original (then it ceases to be an original), and in the case of excessive difference from the original in all significant respects.

Thus, the study of some sides of the modeled object is carried out at the cost of refusing to reflect other sides. Therefore, any model replaces the original only in a strictly limited sense. It follows from this that for one object several “specialized” models can be built, concentrating attention on certain aspects of the object under study or characterizing the object with varying degrees of detail.

At the second stage of the modeling process, the model acts as an independent object of study. One of the forms of such research is conducting “model” experiments, in which the operating conditions of the model are deliberately changed and data on its “behavior” are systematized. The end result of this step is a wealth of knowledge about the R model.

At the third stage, knowledge is transferred from the model to the original - the formation of a set of knowledge S about the object. This process of knowledge transfer is carried out according to certain rules. Knowledge about the model must be adjusted taking into account those properties of the original object that were not reflected or were changed during the construction of the model. We can with sufficient reason transfer any result from a model to the original if this result is necessarily associated with signs of similarity between the original and the model. If a certain result of a model study is associated with the difference between the model and the original, then it is unlawful to transfer this result.

The fourth stage is the practical verification of the knowledge obtained with the help of models and their use to build a general theory of the object, its transformation or control.

To understand the essence of modeling, it is important not to lose sight of the fact that modeling is not the only source of knowledge about an object. The modeling process is “immersed” in a more general process of cognition. This circumstance is taken into account not only at the stage of constructing the model, but also at the final stage, when the combination and generalization of research results obtained on the basis of diverse means of cognition occurs.

Modeling is a cyclical process. This means that the first four-step cycle may be followed by a second, third, etc. At the same time, knowledge about the object under study is expanded and refined, and the initial model is gradually improved. Deficiencies discovered after the first modeling cycle, due to poor knowledge of the object and errors in model construction, can be corrected in subsequent cycles. Thus, the modeling methodology contains great opportunities for self-development.

2. Features of the application of the method of mathematical modeling in economics.

Most objects studied by economic science can be characterized by the cybernetic concept of a complex system.

The most common understanding of a system is as a set of elements that interact and form a certain integrity, unity. An important quality of any system is emergence - the presence of properties that are not inherent in any of the elements included in the system. Therefore, when studying systems, it is not enough to use the method of dividing them into elements and then studying these elements separately. One of the difficulties of economic research is that there are almost no economic objects that could be considered as separate (non-systemic) elements.

The complexity of a system is determined by the number of elements included in it, the connections between these elements, as well as the relationship between the system and the environment. The country's economy has all the hallmarks of a very complex system. It combines a huge number of elements and is distinguished by a variety of internal connections and connections with other systems (natural environment, economies of other countries, etc.). In the national economy, natural, technological, social processes, objective and subjective factors interact.

3. Features of economic observations and measurements.

For a long time, the main obstacle to the practical application of mathematical modeling in economics has been filling the developed models with specific and high-quality information. The accuracy and completeness of primary information, the real possibilities of its collection and processing largely determine the choice of types of applied models. On the other hand, economic modeling studies put forward new requirements for the information system.

Depending on the objects being modeled and the purpose of the models, the initial information used in them has a significantly different nature and origin. It can be divided into two categories: about the past development and current state of objects (economic observations and their processing) and about the future development of objects, including data on expected changes in their internal parameters and external conditions (forecasts). The second category of information is the result of independent research, which can also be performed through simulation.

Methods for economic observations and the use of the results of these observations are developed by economic statistics. Therefore, it is worth noting only the specific problems of economic observations associated with the modeling of economic processes.

In economics, many processes are massive; they are characterized by patterns that are not apparent from just one or a few observations. Therefore, modeling in economics must rely on mass observations.

Another problem is generated by the dynamism of economic processes, the variability of their parameters and structural relationships. As a result, economic processes must be constantly monitored, and it is necessary to have a steady flow of new data. Since observations of economic processes and processing of empirical data usually take quite a lot of time, when constructing mathematical models of the economy it is necessary to adjust the initial information taking into account its delay.

Knowledge of quantitative relationships of economic processes and phenomena is based on economic measurements. The accuracy of measurements largely determines the accuracy of the final results of quantitative analysis through simulation. Therefore, a necessary condition for the effective use of mathematical modeling is the improvement of economic measures. The use of mathematical modeling has sharpened the problem of measurements and quantitative comparisons of various aspects and phenomena of socio-economic development, the reliability and completeness of the data obtained, and their protection from intentional and technical distortions.

During the modeling process, interaction between “primary” and “secondary” economic indicators arises. Any model of the national economy is based on a certain system of economic measures (products, resources, elements, etc.). At the same time, one of the important results of national economic modeling is the obtaining of new (secondary) economic indicators - economically justified prices for products in various industries, assessments of the efficiency of different-quality natural resources, and indicators of the social utility of products. However, these measures may be influenced by insufficiently substantiated primary measures, which forces the development of a special methodology for adjusting the primary measures for business models.

From the point of view of the “interests” of economic modeling, currently the most pressing problems of improving economic indicators are: assessing the results of intellectual activity (especially in the field of scientific and technical developments, the computer science industry), constructing general indicators of socio-economic development, measuring feedback effects (impact economic and social mechanisms on production efficiency).

4. Randomness and uncertainty in economic development.

For economic planning methodology, the concept of uncertainty of economic development is important. In studies of economic forecasting and planning, two types of uncertainty are distinguished: “true”, due to the properties of economic processes, and “information”, associated with the incompleteness and inaccuracy of available information about these processes. True uncertainty cannot be confused with the objective existence of various options for economic development and the possibility of consciously choosing effective options among them. We are talking about the fundamental impossibility of accurately choosing a single (optimal) option.

In economic development, uncertainty is caused by two main reasons. Firstly, the course of planned and controlled processes, as well as external influences on these processes, cannot be accurately predicted due to the action of random factors and the limitations of human cognition at each moment. This is especially typical for forecasting scientific and technological progress, the needs of society, and economic behavior. Secondly, general state planning and management are not only not comprehensive, but also not omnipotent, and the presence of many independent economic entities with special interests does not allow us to accurately predict the results of their interactions. Incomplete and inaccurate information about objective processes and economic behavior increases true uncertainty.

At the first stages of research on economic modeling, models of the deterministic type were mainly used. In these models, all parameters are assumed to be exactly known. However, deterministic models are misunderstood in a mechanical sense and identified with models that are devoid of all “degrees of choice” (opportunities for choice) and have a single feasible solution. A classic representative of strictly deterministic models is the optimization model of the national economy, which is used to determine the best option for economic development among many feasible options.

As a result of the accumulation of experience in the use of strictly deterministic models, real opportunities have been created for the successful use of more advanced methodology for modeling economic processes that take into account stochasticity and uncertainty. Two main areas of research can be distinguished here. Firstly, the methodology for using strictly deterministic models will be improved: conducting multivariate calculations and model experiments with variations in the model design and its initial data; studying the stability and reliability of the resulting solutions, identifying the zone of uncertainty; inclusion of reserves in the model, the use of techniques that increase the adaptability of economic decisions to probable and unforeseen situations. Secondly, models are becoming widespread that directly reflect the stochasticity and uncertainty of economic processes and use the appropriate mathematical apparatus: probability theory and mathematical statistics, the theory of games and statistical decisions, queuing theory, stochastic programming, and the theory of random processes.

5. Checking the adequacy of models.

The complexity of economic processes and phenomena and other features of economic systems noted above make it difficult not only to construct mathematical models, but also to verify their adequacy and the truth of the results obtained.

In the natural sciences, a sufficient condition for the truth of the results of modeling and any other forms of knowledge is the coincidence of the research results with the observed facts. The category “practice” coincides here with the category “reality”. In economics and other social sciences, the principle “practice is the criterion of truth” understood in this way is more applicable to simple descriptive models used for passive description and explanation of reality (analysis of past development, short-term forecasting of uncontrollable economic processes, etc.).

However, the main task of economic science is constructive: the development of scientific methods for planning and managing the economy. Therefore, a common type of mathematical models of the economy are models of controlled and regulated economic processes used to transform economic reality. Such models are called normative. If normative models are oriented only towards confirming reality, then they will not be able to serve as a tool for solving qualitatively new socio-economic problems.

The specificity of verification of normative economic models is that they, as a rule, “compete” with other planning and management methods that have already found practical application. At the same time, it is not always possible to carry out a pure experiment to verify the model, eliminating the influence of other control actions on the modeled object.

The situation becomes even more complicated when the question of verification of long-term forecasting and planning models (both descriptive and normative) is raised. After all, you can’t passively wait 10-15 years or more for events to happen in order to check the correctness of the model’s premises.

Despite the noted complicating circumstances, the model’s compliance with the facts and trends of real economic life remains the most important criterion that determines the directions for improving models. A comprehensive analysis of the identified discrepancies between reality and the model, comparison of the results from the model with the results obtained by other methods help to develop ways to correct the models.

A significant role in checking models belongs to logical analysis, including by means of mathematical modeling itself. Such formalized methods of model verification as proving the existence of a solution in the model, checking the truth of statistical hypotheses about the relationships between the parameters and variables of the model, comparing the dimensions of quantities, etc., make it possible to narrow the class of potentially “correct” models.

The internal consistency of the model's premises is also checked by comparing the consequences obtained with its help with each other, as well as with the consequences of “competing” models.

Assessing the current state of the problem of the adequacy of mathematical models to economics, it should be recognized that the creation of a constructive comprehensive methodology for model verification, taking into account both the objective features of the objects being modeled and the features of their cognition, is still one of the most pressing tasks of economic and mathematical research.

6. Classification of economic and mathematical models.

Mathematical models of economic processes and phenomena can be more briefly called economic-mathematical models. Different bases are used to classify these models.

According to their intended purpose, economic and mathematical models are divided into theoretical and analytical, used in studies of the general properties and patterns of economic processes, and applied, used in solving specific economic problems (models of economic analysis, forecasting, management).

Economic and mathematical models can be intended to study different aspects of the national economy (in particular, its production, technological, social, territorial structures) and its individual parts. When classifying models according to the economic processes and substantive issues under study, one can distinguish models of the national economy as a whole and its subsystems - industries, regions, etc., complexes of models of production, consumption, generation and distribution of income, labor resources, pricing, financial relations, etc. .d.

Let us dwell in more detail on the characteristics of such classes of economic and mathematical models, which are associated with the greatest features of the methodology and modeling techniques.

In accordance with the general classification of mathematical models, they are divided into functional and structural, and also include intermediate forms (structural-functional). In studies at the national economic level, structural models are more often used, since the interconnections of subsystems are of great importance for planning and management. Typical structural models are models of intersectoral links. Functional models are widely used in economic regulation, when the behavior of an object (“output”) is influenced by changing the “input”. An example is the model of consumer behavior in the conditions of commodity-money relations. The same object can be described simultaneously by both a structure and a functional model. For example, to plan a separate industry system, a structural model is used, and at the national economic level, each industry can be represented by a functional model.

The differences between descriptive and normative models have already been shown above. Descriptive models answer the question: how does this happen? or how this could most likely develop further?, i.e. they only explain observed facts or provide a plausible prediction. Normative models answer the question: how should this be?, i.e. involve purposeful activity. A typical example of normative models are optimal planning models, which formalize in one way or another the goals of economic development, opportunities and means of achieving them.

The use of a descriptive approach in economic modeling is explained by the need to empirically identify various dependencies in the economy, establish statistical patterns of economic behavior of social groups, and study the likely paths of development of any processes under constant conditions or occurring without external influences. Examples of descriptive models are production functions and consumer demand functions built on the basis of statistical data processing.

Whether an economic-mathematical model is descriptive or normative depends not only on its mathematical structure, but on the nature of the use of this model. For example, the input-output model is descriptive if it is used to analyze the proportions of the past period. But this same mathematical model becomes normative when it is used to calculate balanced options for the development of the national economy that satisfy the final needs of society at planned production cost standards.

Many economic and mathematical models combine features of descriptive and normative models. A typical situation is when a normative model of a complex structure combines individual blocks, which are private descriptive models. For example, a cross-industry model might include consumer demand functions that describe consumer behavior as income changes. Such examples characterize the tendency to effectively combine descriptive and normative approaches to modeling economic processes. The descriptive approach is widely used in simulation modeling.

Based on the nature of the reflection of cause-and-effect relationships, a distinction is made between strictly deterministic models and models that take into account randomness and uncertainty. It is necessary to distinguish between uncertainty described by probabilistic laws and uncertainty for which the laws of probability theory are not applicable. The second type of uncertainty is much more difficult to model.

According to the methods of reflecting the time factor, economic and mathematical models are divided into static and dynamic. In static models, all dependencies relate to one moment or period of time. Dynamic models characterize changes in economic processes over time. Based on the duration of the time period under consideration, models of short-term (up to a year), medium-term (up to 5 years), long-term (10-15 or more years) forecasting and planning differ. Time itself in economic and mathematical models can change either continuously or discretely.

Models of economic processes are extremely diverse in the form of mathematical dependencies. It is especially important to highlight the class of linear models that are most convenient for analysis and calculations and, as a result, have become widespread. The differences between linear and nonlinear models are significant not only from a mathematical point of view, but also from a theoretical and economic point of view, since many dependencies in the economy are fundamentally nonlinear in nature: efficiency of resource use with increased production, changes in demand and consumption of the population with increased production, changes in demand and consumption of the population with rising incomes, etc. The theory of "linear economics" differs significantly from the theory of "nonlinear economics". Conclusions about the possibility of combining centralized planning and economic independence of economic subsystems significantly depend on whether the sets of production possibilities of subsystems (industries, enterprises) are assumed to be convex or non-convex.

According to the ratio of exogenous and endogenous variables included in the model, they can be divided into open and closed. There are no completely open models; the model must contain at least one endogenous variable. Completely closed economic and mathematical models, i.e. not including exogenous variables, are extremely rare; their construction requires complete abstraction from the “environment”, i.e. serious coarsening of real economic systems that always have external connections. The vast majority of economic and mathematical models occupy an intermediate position and differ in the degree of openness (closedness).

For models at the national economic level, the division into aggregated and detailed is important.

Depending on whether national economic models include spatial factors and conditions or not, a distinction is made between spatial and point models.

Thus, the general classification of economic and mathematical models includes more than ten main features. With the development of economic and mathematical research, the problem of classifying the models used becomes more complicated. Along with the emergence of new types of models (especially mixed types) and new features of their classification, the process of integrating models of different types into more complex model structures is taking place.

7. Stages of economic and mathematical modeling.

1. Statement of the economic problem and its qualitative analysis. The main thing here is to clearly formulate the essence of the problem, the assumptions made and the questions to which answers are required. This stage includes identifying the most important features and properties of the modeled object and abstracting from minor ones; studying the structure of an object and the basic dependencies connecting its elements; formulating hypotheses (at least preliminary) explaining the behavior and development of the object.

2. Construction of a mathematical model. This is the stage of formalizing an economic problem, expressing it in the form of specific mathematical dependencies and relationships (functions, equations, inequalities, etc.). Usually, the main design (type) of a mathematical model is first determined, and then the details of this design are specified (a specific list of variables and parameters, the form of connections). Thus, the construction of the model is in turn divided into several stages.

One of the important features of mathematical models is the potential for their use to solve problems of different qualities. Therefore, even when faced with a new economic problem, there is no need to strive to “invent” the model; First, you need to try to apply already known models to solve this problem.

3. Mathematical analysis of the model. The purpose of this stage is to clarify the general properties of the model. Purely purely mathematical research methods are used here. The most important point is the proof of the existence of solutions in the formulated model (existence theorem). If it can be proven that the mathematical problem has no solution, then there is no need for subsequent work on the original version of the model; either the formulation of the economic problem or the methods of its mathematical formalization should be adjusted. During the analytical study of the model, questions are clarified, such as, for example, is there a unique solution, what variables (unknowns) can be included in the solution, what will be the relationships between them, to what extent and depending on what initial conditions they change, what are the trends in their change and etc. An analytical study of a model, compared to an empirical (numerical) one, has the advantage that the conclusions obtained remain valid for various specific values of the external and internal parameters of the model.

4. Preparation of background information. Modeling places stringent demands on the information system. At the same time, the real possibilities of obtaining information limit the choice of models intended for practical use. In this case, not only the fundamental possibility of preparing information (within a certain time frame) is taken into account, but also the costs of preparing the corresponding information arrays. These costs should not exceed the effect of using additional information.

5. Numerical solution. This stage includes the development of algorithms for numerical solution of the problem, compilation of computer programs and direct calculations. The difficulties of this stage are primarily due to the large size of economic problems and the need to process significant amounts of information.

6. Analysis of numerical results and their application. At this final stage of the cycle, the question arises about the correctness and completeness of the modeling results, about the degree of practical applicability of the latter.

Relationships between stages. Figure 1 shows the connections between the stages of one cycle of economic and mathematical modeling.

Let us pay attention to the reciprocal connections of the stages that arise due to the fact that during the research process shortcomings of the previous stages of modeling are discovered.

Already at the stage of building a model, it may become clear that the formulation of the problem is contradictory or leads to an overly complex mathematical model. In accordance with this, the original formulation of the problem is adjusted. Further, mathematical analysis of the model (stage 3) can show that a slight modification of the problem statement or its formalization gives an interesting analytical result.

Most often, the need to return to previous stages of modeling arises when preparing initial information (stage 4). You may find that the necessary information is missing or that the cost of preparing it is too high. Then we have to return to the formulation of the problem and its formalization, changing them so as to adapt to the available information.

Since economic and mathematical problems can be complex in structure and have a large dimension, it often happens that known algorithms and computer programs do not allow solving the problem in its original form. If it is impossible to develop new algorithms and programs in a short time, the original formulation of the problem and the model are simplified: the conditions are removed and combined, the number of factors is reduced, non-linear relationships are replaced with linear ones, the determinism of the model is strengthened, etc.

Deficiencies that cannot be corrected at intermediate stages of modeling are eliminated in subsequent cycles. But the results of each cycle also have a completely independent meaning. By starting your research by building a simple model, you can quickly obtain useful results, and then move on to creating a more advanced model, supplemented with new conditions, including refined mathematical dependencies.

As economic and mathematical modeling develops and becomes more complex, its individual stages are isolated into specialized areas of research, the differences between theoretical-analytical and applied models intensify, and models are differentiated according to levels of abstraction and idealization.

The theory of mathematical analysis of economic models has developed into a special branch of modern mathematics - mathematical economics. Models studied within the framework of mathematical economics lose direct connection with economic reality; they deal exclusively with idealized economic objects and situations. When constructing such models, the main principle is not so much to get closer to reality, but to obtain the largest possible number of analytical results through mathematical proofs. The value of these models for economic theory and practice is that they serve as a theoretical basis for applied models.

Quite independent areas of research are the preparation and processing of economic information and the development of mathematical support for economic problems (creation of databases and information banks, programs for automated construction of models and software services for user economists). At the stage of practical use of models, the leading role should be played by specialists in the relevant field of economic analysis, planning, and management. The main area of work for economists and mathematicians remains the formulation and formalization of economic problems and the synthesis of the process of economic and mathematical modeling.

8. The role of applied economic and mathematical research.

We can distinguish at least four aspects of the use of mathematical methods in solving practical problems.

1. Improving the economic information system. Mathematical methods make it possible to organize the system of economic information, identify shortcomings in existing information and develop requirements for the preparation of new information or its correction. The development and application of economic and mathematical models indicate ways to improve economic information aimed at solving a specific system of planning and management problems. Progress in information support for planning and management is based on rapidly developing technical and software tools of computer science.

2. Intensification and increase in the accuracy of economic calculations. The formalization of economic problems and the use of computers greatly speed up standard, mass calculations, increase accuracy and reduce labor intensity, and make it possible to carry out multivariate economic justifications for complex activities that are inaccessible under the dominance of “manual” technology.

3. Deepening the quantitative analysis of economic problems. Thanks to the application of the modeling method, the capabilities of specific quantitative analysis are significantly enhanced; study of many factors influencing economic processes, quantitative assessment of the consequences of changes in the conditions for the development of economic objects, etc.

4. Solving fundamentally new economic problems. Through mathematical modeling, it is possible to solve economic problems that are practically impossible to solve by other means, for example: finding the optimal version of the national economic plan, simulating national economic activities, automating control over the functioning of complex economic objects.

The scope of practical application of the modeling method is limited by the capabilities and effectiveness of formalizing economic problems and situations, as well as the state of information, mathematical, and technical support of the models used. The desire to apply a mathematical model at any cost may not give good results due to the lack of at least some necessary conditions.

In accordance with modern scientific ideas, systems for developing and making business decisions should combine formal and informal methods, mutually reinforcing and complementary to each other. Formal methods are primarily a means of scientifically based preparation of material for human actions in management processes. This makes it possible to productively use a person’s experience and intuition, his ability to solve poorly formalized problems.

Methods of economic theory

The study of human economic life has been in the field of interest of scientists since ancient times. The gradual complication of economic relations required the development of economic thought. Leaps in science have always been accompanied by challenges facing humanity at various stages of evolution. Initially, people obtained food, then began to exchange it. Over time, agriculture arose, which contributed to the division of labor and the emergence of the first craft professions. An important stage in the economic life of mankind was the industrial revolution, which gave impetus to the rapid growth of production and also influenced social changes in society.

Modern economic science was formed relatively recently, when scientists moved from solving problems facing the ruling class to studying the processes occurring in systems regardless of the interests of society.

The subject of economic theory is the optimization of the ratio of increasing demand in conditions when the volume of supply is limited due to limited resources.

It is worth noting that for a long time economic systems were considered in short-term periods, that is, in statics. Although new trends of the twentieth century demanded a new approach from economists, focused on the dynamic development of economic structures.

Economic systems are quite complex formations in which each subject simultaneously enters into many connections. They can be considered from the point of view of macroeconomic aggregate indicators, as well as as a result of the work of an individual economic agent. The science of economics uses various methods to facilitate the processes of research and analysis of economic phenomena. Most often used in practice:

abstraction method (separation of an object from its connections and operating factors);
synthesis method (combining elements into something common);
method of analysis (dividing the overall system into components);
deduction (study from the particular to the general) and induction (study of the subject from the general to the particular);
systematic approach (allows us to consider the object being studied as a structure);
mathematical modeling (building models of processes and phenomena in mathematical language).

Modeling in economics

The essence of modeling is to replace a real model of a process, phenomenon or system with another model that can simplify its research and analysis. It is important to maintain the proximity of the original model to its scientific analogue. Modeling is used for the purpose of simplification. Often in practice there are phenomena that cannot be studied without the use of visual scientific generalizations.

The following modeling goals can be distinguished:

Search and description of the reasons for the behavior of the original model.
Predicting the future behavior of the model.
Drawing up projects and plans for systems.
Process automation.
Search for ways to optimize the original model.
For training specialists, students and others.

At their core, models can also be of different types. A verbal model is based on a verbal description of a system or process. The graphical model is a visual representation of various dependencies on each other. It can also describe the behavior of the original model in dynamics. Natural modeling involves creating a model that can partially or completely reflect the behavior of the original. The most widely used is mathematical modeling. It makes it possible to use the entirety of mathematical tools and language. In mathematics, statistical models, dynamic and information models are used. Each of their types is used to achieve specific goals faced by specialists.

Note 1

The division of the economy into macro and micro levels has led to the fact that modeling also simulates systems at various levels of the organization. Econometrics, which uses statistics and probability theory, is most often used to study economic structures. It is worth noting that it is mathematical modeling that makes it possible to take into account the time factor, which is important in the dynamic development of systems.

Mathematical models in economics

Before starting economic and mathematical modeling, preparatory work is carried out, which may include the following stages:

Setting goals and objectives.
Carrying out formalization of the process or phenomenon being studied.
Finding the required solution.
Checking the resulting solution and model for adequacy.
If the results of the verification are satisfactory, these models can be applied in practice.

Mathematical models are distinguished by the use of the language of mathematics at the stage of their construction, as well as in further calculations. This language allows you to most accurately describe connections, dependencies and patterns. When the transition to solving models is made, various types of solutions can be used. For example, exact or analytical gives the final calculation indicator. The approximate value has a certain calculation error and is often used to build graphical models. The solution, expressed as a number, gives the final result, which is often derived using computer calculations. It is worth remembering that the accuracy of the solutions does not mean the accuracy of the calculated model.

An important stage in mathematical modeling is checking the obtained results and the simulation model for adequacy. Typically, verification work is based on comparing data from a real model with data from a constructed one. However, in mathematical and economic modeling it is quite difficult to perform this action. Typically, the adequacy of calculations is determined subsequently in practice.

Note 2

Mathematical modeling in economics allows one to simplify phenomena and processes in economic systems, make calculations and obtain relatively correct calculation results. It is important to remember that this approach is also not universal, as it has a number of the disadvantages listed above. The adequacy of modeling is often achieved through time-tested hypotheses and calculation formulas.