Economic-mathematical methods and modeling. Economic and mathematical methods and models

Economic and mathematical methods are based on the use of correlation and regression analysis, which makes it possible to establish the closeness of the relationship and the type of dependence of the average value of any value on some other or several values. In our case, this is the establishment of the dependence of the development of demand on the influence of the most important factors. in the practice of forecasting the commodity-group structure of demand, trend and regression models are most often used:

Trend models of demand forecasting are equations that formalize sustainable processes of its development. They are used to predict the most stable patterns for large commodity sub-sectors (for example, the ratio of demand for food and non-food products). The main parameter of trend models is time, that is, in essence, we are also talking about extrapolation of trends and patterns of the base period to the forecast period.

Regression (factorial) models reflect the quantitative relationship of one indicator with another or with a group of others (multiple regression). The variables are the factors that determine the dynamics of demand. The mathematical basis for building models is the most important provisions of the theory of probability, mathematical statistics and higher mathematics. The process of building such models consists of several successive stages.

The first and most important stage in modeling the development of the commodity-group structure of the population's demand is the selection of factors. They should reflect the objective processes of the phenomenon under study, be quantitatively measurable and independent of each other.

At the second stage, the strength of influence or the closeness of the relationship between factors and demand in the base period is calculated. It is determined using correlation coefficients and goodness of fit tests.

At the third stage, the mathematical form of the relationship or the type of dependence of demand on factors is revealed, functions are selected, and the process of demand development is most accurately described.

Fourth stage: calculation of the parameters of the equation. The parameters of the equations express the degree and direction of the impact of each factor on demand and are calculated using the least squares method.

Fifth stage: assessment of the predictive value of the model based on retrospective calculations.

Economic and mathematical methods are effectively used in short-term forecasting. Since the objective reality of our economy is that it is rather difficult to identify and quantify more or less stable factors that affect the predicted process. Therefore, the preparation of medium-term and, especially, long-term forecasts seems to be quite difficult in modern conditions. And as a rule, forecasting for short-term periods prevails. Economic and mathematical modeling is the basis of economic forecasting. It allows, on a strictly quantitative basis, to reveal the nature of the links between the individual elements of the market and those factors that influence its development. What is especially important is that mathematical models make it possible to observe how events will develop under certain initial assumptions.

In the economic and mathematical modeling of demand, a group of methods can also be used - exponential smoothing and forecasting, based on the use of already made forecasts of demand development trends and the latest data on the sale of goods.

Mathematical methods help to reveal quantitative phenomena and relationships. But they are only a continuation of economic analysis, the final result primarily depends on the choice of the base period, the selection of factors, on whether the degree of stability of the phenomenon is correctly determined.

Graphical methods are connected by a geometric representation of functional dependence using lines on a plane. With the help of a coordinate grid, dependency graphs are constructed, for example, the level of costs on the volume of manufactured and sold products, as well as graphs that can depict correlations between indicators (comparison diagrams, distribution curves, time series diagrams, statistical cartograms).

Example: building a network diagram for the construction and installation of enterprises. A table of works and resources is compiled, where their characteristics, volume, performer, shift, need for materials are indicated in the technological sequence. The duration of the task and other information. Based on these indicators, prepare a network schedule. Graph optimization is carried out by shortening the critical path, i.e. minimization of terms of performance of works at the given levels of resources, minimization of the level of consumption of resources at fixed terms of performance of works.

The method of correlation-regression analysis is used to determine the closeness of the relationship between indicators that are not in functional dependence. The tightness of the connection is measured by the correlation ratio (for a curvilinear dependence). For a straight-line dependence, the correlation coefficient is calculated. The method is used in solving problems on the "launch-release".

Example: to determine the dependence of the output of products on average from their launch, making up the appropriate regression control.

The linear programming method is reduced to finding the extreme values (maximum and minimum) of some functions of variables. Based on the solution of a system of linear equations, when the dependence between phenomena is strictly functional.

Example: problems of rational use of the operating time of production equipment.

Dynamic programming methods are used in solving optimization problems in which the objective function and constraints are characterized by nonlinear dependencies.

Example: fill a vehicle with carrying capacity X with a load consisting of certain items so that the cost of the entire load is maximum.

Mathematical game theory explores optimal strategies in game situations. The decision requires certainty in the formulation of the conditions: establishing the number of players, possible payoffs, determining the strategy.

Example: to maximize the average value of income from the sale of manufactured products, taking into account the vagaries of the weather.

Mathematical theory of queuing.

Example: providing workers with the necessary tools.

The matrix method is based on linear and vector-matrix algebra, and is used to study complex and high-dimensional structures at the industry level, at the level of enterprises.

Example: to identify the distribution between the shops of products for domestic consumption, and the total volume of output, if the parameters of direct costs and the final product are given.

Consider the features of the methodology of economic analysis in relation to the study of demand for goods.

Demand forecasting can be carried out by various methods, in particular, three main groups can be distinguished: methods of economic and mathematical modeling (extropolation methods), normative methods, methods of expert assessments.

Methods of simple (formal) extrapolation consist in the transfer to the future period of past and present trends in the development of the commodity-group structure of demand based on the analysis of the dynamic series.

For extrapolation, information about market dynamics is presented in one form or another - graphical, statistical, mathematical, logical. In any case, it is believed that economic processes are characterized by "inertia" or the obligatory continuation of the direction of their flow in the near future. Extrapolations require extreme discretion from the market researcher. It is not enough to study past market trends - it is necessary to take into account new conditions and factors that were not typical for the past, but may appear in the future. At the same time, it is necessary to get rid of taking into account factors and circumstances that have lost their relevance and no longer have an impact on the course of events in this market.

This method is quite simple and accessible, but it is advisable to use it only for a period in which a change in trends is unlikely, that is, for a short term, and for enlarged product groups.

Simple extrapolation methods also include calculations of the elasticity of demand depending on changes in any factor.

1. Economic and mathematical methods used in the analysis of economic activity

List of sources used

1. Economic and mathematical methods used in the analysis of economic activity

One of the ways to improve the analysis of economic activity is the introduction of economic and mathematical methods and modern computers. Their application increases the efficiency of economic analysis by expanding the studied factors, substantiating managerial decisions, choosing the best option for using economic resources, identifying and mobilizing reserves to increase production efficiency.

Mathematical methods are based on the methodology of economic and mathematical modeling and scientifically substantiated classification of problems in the analysis of economic activity. Depending on the goals of economic analysis, the following economic and mathematical models are distinguished: in deterministic models - logarithm, equity participation, differentiation; in stochastic models - correlation-regression method, linear programming, queuing theory, graph theory, etc.

Stochastic analysis is a method for solving a wide class of statistical estimation problems. It involves the study of mass empirical data by building models of changes in indicators due to factors that are not in direct relationships, in direct interdependence and interdependence. A stochastic relationship exists between random variables and manifests itself in the fact that when one of them changes, the law of distribution of the other changes.

In economic analysis, the following most typical tasks of stochastic analysis are distinguished:

The study of the presence and tightness of the relationship between the function and factors, as well as between factors;

Ranking and classification of factors of economic phenomena;

Revealing the analytical form of connection between the studied phenomena;

Smoothing the dynamics of changes in the level of indicators;

Identification of the parameters of regular periodic fluctuations in the level of indicators;

The study of the dimension (complexity, versatility) of economic phenomena;

Quantitative change of informative indicators;

Quantitative change in the influence of factors on the change in the analyzed indicators (economic interpretation of the equations obtained).

Stochastic modeling and analysis of relationships between the studied indicators begin with a correlation analysis. Correlation consists in the fact that the average value of one of the features varies depending on the value of the other. An attribute on which another attribute depends is called a factor attribute. The dependent sign is called effective. In each specific case, in order to establish the factorial and effective characteristics in unequal sets, it is necessary to analyze the nature of the relationship. So, when analyzing various features in one set, the wages of workers in connection with their work experience acts as a productive feature, and in connection with indicators of living standards or cultural needs - as a factor. Often, dependencies are considered not from one factor sign, but from several. For this, a set of methods and techniques is used to identify and quantify the relationships and interdependencies between features.

In the study of mass socio-economic phenomena, a correlation is manifested between factor signs, in which the value of the effective sign is influenced, in addition to the factor, by many other signs acting in different directions simultaneously or sequentially. Often, a correlation is called incomplete statistical or partial, in contrast to functional, which is expressed in the fact that for a certain value of a variable (independent variable - argument), another (dependent variable - function) takes on a strict value.

Correlation can be identified only in the form of a general trend in the mass comparison of facts. Each value of the factor attribute will correspond not to one value of the effective attribute, but to their combination. In this case, to open the connection, it is necessary to find the average value of the effective attribute for each factor value.

If the relationship is linear:

The values of the coefficients a and b are found from the system of equations obtained by the least squares method according to the formula:

N is the number of observations.

In the case of a rectilinear form of relationship between the studied indicators, the correlation coefficient is calculated by the formula:

If the correlation coefficient is squared, then we get the coefficient of determination.

Discounting is the process of converting the future value of capital, cash flows or net income into the present value. The rate at which discounting is done is called the discount rate (discount rate). The basic premise underlying the concept of discounted real money flow is that money has a time value, that is, an amount of money available today is more valuable than the same amount in the future. This difference can be expressed as an interest rate that characterizes relative changes over a certain period (usually equal to a year).

Many tasks that an economist has to face in everyday practice when analyzing the economic activities of enterprises are multivariate. Since not all options are equally good, among the many possible options, you have to find the best one. A significant part of such problems for a long time was solved on the basis of common sense and experience. At the same time, there was no certainty that the variant found was the best one.

In modern conditions, even minor mistakes can lead to huge losses. In this regard, it became necessary to involve optimization economic and mathematical methods and computers in the analysis and synthesis of economic systems, which creates the basis for making scientifically based decisions. Such methods are combined into one group under the general name "optimization methods of decision making in economics". To solve an economic problem by mathematical methods, first of all, it is necessary to build a mathematical model adequate to it, that is, to formalize the goal and conditions of the problem in the form of mathematical functions, equations and (or) inequalities.

In the general case, the mathematical model of the optimization problem has the form:

max (min): Z = Z(x),

under restrictions

f i (x) Rb i , i = ,

where R are relations of equality, less than or greater than.

If the objective function and the functions included in the constraint system are linear with respect to the unknowns included in the problem, such a problem is called a linear programming problem. If the objective function or system of constraints is not linear, such a problem is called a non-linear programming problem.

Basically, in practice, non-linear programming problems are reduced by linearization to a linear programming problem. Of particular practical interest among non-linear programming problems are dynamic programming problems, which, due to their multi-stage nature, cannot be linearized. Therefore, we will consider only these two types of optimization models, for which there is good mathematical and software today.

The dynamic programming method is a special mathematical technique for optimizing nonlinear problems of mathematical programming, which is specially adapted to multi-step processes. A multi-step process is usually considered a process that develops over time and breaks down into a number of "steps" or "stages". At the same time, the dynamic programming method is also used to solve problems in which time does not appear. Some processes fall into steps in a natural way (for example, the process of planning the economic activity of an enterprise for a period of time consisting of several years). Many processes can be divided into stages artificially.

The essence of the dynamic programming method is that instead of finding the optimal solution for the entire complex problem at once, they prefer to find optimal solutions for several simpler problems of similar content, into which the original problem is divided.

The dynamic programming method is also characterized by the fact that the choice of the optimal solution at each step must be made taking into account the consequences in the future. This means that while optimizing the process at each individual step, in no case should you forget about all subsequent steps. Thus, dynamic programming is far-sighted planning with a perspective.

The principle of decision choice in dynamic programming is defining and is called the Bellman principle of optimality. We formulate it as follows: the optimal strategy has the property that, whatever the initial state and the decision made at the initial moment, subsequent decisions should lead to an improvement in the situation relative to the state resulting from the initial decision.

Thus, when solving an optimization problem using the dynamic programming method, it is necessary at each step to take into account the consequences that the current decision will lead to in the future. The exception is the last step, which ends the process. Here you can make such a decision to ensure the maximum effect. Having optimally planned the last step, one can “attach” the penultimate step to it so that the result of these two steps is optimal, and so on. It is in this way - from the end to the beginning - that the decision-making procedure can be deployed. The optimal solution found under the condition that the previous step ended in a certain way is called a conditionally optimal solution.

Statistical game theory is an integral part of general game theory, which is a section of modern applied mathematics that studies methods for substantiating optimal solutions in conflict situations. In the theory of statistical games, such concepts as the original strategic game and the actual statistical game are distinguished. In this theory, the first player is called "nature", which is understood as a set of circumstances under which the second player has to make decisions - "statistics". In a strategic game, both players act actively, assuming that the opponent is a "reasonable" player. A strategy game is characterized by complete uncertainty in the choice of strategy by each player, that is, the players do not know anything about each other's strategies. In a strategic game, both players act on the basis of deterministic information defined by the loss matrix.

In the actual statistical game, nature is not an active player in the sense that it is "not intelligent" and does not try to counteract the maximum payoff of the second player. The statistician (the second player) in the statistical game seeks to win the game against an imaginary opponent - nature. If in a strategic game the players operate under conditions of complete uncertainty, then a statistical game is characterized by partial uncertainty. The fact is that nature develops and "acts" in accordance with its objectively existing laws. The statistician has the opportunity to gradually study these laws, for example, on the basis of a statistical experiment.

Queuing theory is an applied area of the theory of random processes. The subject of her research is probabilistic models of real service systems, where at random (or not at random) times there are service requests and there are devices (channels) for fulfilling requests. Queuing theory explores mathematical methods for quantifying queuing processes, the quality of systems functioning, where both the moments of appearance of requirements (applications) and the time spent on their execution can be random.

The queuing system finds application in solving the following problems: for example, when requests (requirements) for service are received en masse with their subsequent satisfaction. In practice, this can be the receipt of raw materials, materials, semi-finished products, products to the warehouse and their issuance from the warehouse; processing of a wide range of parts on the same technological equipment; organization of adjustment and repair of equipment; transport operations; planning of reserve and insurance reserves of resources; determination of the optimal number of departments and services of the enterprise; processing of planning and reporting documentation, etc.

The balance model is a system of equations that characterize the availability of resources (products) in physical or monetary terms and the direction of their use. At the same time, the availability of resources (products) and the need for them quantitatively coincide. The solution of such models is based on the methods of linear vector-matrix algebra. Therefore, balance methods and models are called matrix methods of analysis. The clarity of images of various economic processes in matrix models and elementary methods for resolving systems of equations allow them to be used in various production and economic situations.

The mathematical theory of fuzzy sets, developed in the 60s of the XX century, is now increasingly used in the financial analysis of the enterprise, including the analysis and forecast of the financial position of the enterprise, the analysis of changes in the working capital, free cash flows, economic risk, and assessment of the impact of costs on profit , calculating the cost of capital. This theory is based on the concepts of "fuzzy set" and "membership functions".

In the general case, solving problems of this type is rather cumbersome, since there is a large amount of information. The practical use of the theory of fuzzy sets makes it possible to develop traditional methods of financial and economic activity, adapt them to the new needs of accounting for the uncertainty in the future of the main indicators of enterprises.

Task 1

Based on the given data on the number of personnel of an industrial enterprise, calculate the turnover rate for the admission and departure of workers and the turnover rate. Draw conclusions.

Decision:

Let's define:

1) acceptance rate (K pr):

Last year: Kpr \u003d 610 / (2490 + 3500) \u003d 0.102

Reporting year: Cpr. = 650 / (2539 + 4200) = 0.096

In the reporting year, the coefficient of external turnover on acceptance decreased by 0.006 (0.096 - 0.102).

2) the coefficient for the dismissal (retirement) of employees (K uv):

Last year: Kvyb. = 690 / (2490 + 3500) = 0.115

Reporting year: Kvyb. = 725 / (2539 + 4200) = 0.108

In the reporting year, the coefficient of external turnover on retirement also decreased by 0.007 (0.108 - 0.115).

3) staff turnover rate(K tech):

Last year: Ktek. = (110 + 30) / (2490 + 3500) = 0.023

Reporting year: Ktek. = (192 + 25) / (2539 + 4200) = 0.032

In the reporting year, the staff turnover rate also increased by 0.009 (0.032 - 0.023), which is a negative trend in the movement of staff.

4) the coefficient of the total turnover of labor(K about):

Last year: Cob = (610 + 690) / (2490 + 3500) = 0.217

Reporting year: Kob. = (650 + 725) / (2539 + 4200) = 0.204

The coefficient of the total turnover of the labor force decreased by 0.013 (0.204 - 0.217).

Task 2

Create an initial production volume model. Determine the type of factorial model. Calculate the influence of factors on the change in the volume of production by all known methods.

Decision:

The effective indicator is capital productivity.

Initial mathematical model:

FO \u003d VP / OF.

Model type - multiple. The total number of performance indicators used for the calculation is 3, since the influence of 2 factors is calculated (2 + 1 = 3). The number of conditional performance indicators is 1, since it is equal to the number of factors minus 1.

For this model, the following techniques are applicable: chain substitution, index and integral.

1. Calculate the level of influence of the factors of change in the effective indicator by the method of chain substitution.

Solution algorithm:

FO pl \u003d VP pl / OF pl \u003d 20433 / 2593 \u003d 7.88 rubles.

FO conditional 1 \u003d VP f / OF pl \u003d 20193 / 2593 \u003d 7.786 rubles.

FO f \u003d VP f / OF f \u003d 20193 / 2577 \u003d 7.836 rubles.

The calculation of the factors that influenced the change in capital productivity, we will issue in the table.

number of factors	Name of factors	Calculation of the level of influence of factors	The level of influence of factors of change in the total amount of profit
	Change the return on assets by changing the volume of production	7,786-7,88 =-0,094
	Change capital productivity by changing fixed assets	7,836-7,786 = 0,05
	TOTAL (balance linkage)

2. Calculate the level of influence of the factors of change in the effective indicator in an integral way.

VP \u003d VP f - VP pl \u003d 20193 - 20433 \u003d -240;

OF \u003d OF f - OF pl \u003d 2577 - 2593 \u003d -16.

FO pl \u003d 20433 / 2593 \u003d 7.88 rubles.

FO f \u003d 20193 / 2577 \u003d 7.836 rubles.

FD vp = = 15 ln|0.99| = -0.09284

FD of \u003d?

3. Calculate the level of influence of the factors of change in the effective indicator by the index method.

I FO \u003d I VP I OF.

I FO \u003d (VP f / OF f): (VP pl / OF pl) \u003d 7.836 / 7.88 \u003d 0.99

I VP \u003d (VP f / OF pl): (VP pl / OF pl) \u003d 7.786 / 7.88 \u003d 0.988

I OF \u003d (VP f / OF f): (VP f / OF pl) \u003d 7.836 / 7.786 \u003d 1.006

I FD \u003d I VP I OF \u003d 0.988 1.006 \u003d 0.99.

If we subtract the denominator from the numerator of the above formulas, then we will obtain absolute increases in capital productivity as a whole and due to each factor separately, i.e., the same results as in the chain substitution method.

Task 3

Determine what will be the average level of yield if the amount of fertilizer applied is 20 centners. Determine the closeness of the relationship between the indicator "y" and the factor "x".

Given: Regression Equation

where y is the average change in yield, c / ha

x - the amount of fertilizers applied, c.

The coefficient of determination is 0.92.

Decision:

The average level of productivity is 62 q/ha.

Regression analysis aims at the conclusion, definition (identification) of the regression equation, including the statistical evaluation of its parameters. The regression equation allows you to find the value of the dependent variable if the value of the independent or independent variables is known.

The correlation coefficient is calculated by the formula:

It is proved that the correlation coefficient is in the range from minus one to plus one (-1< R x, y <1). Коэффициент корреляции в квадрате () называется коэффициентом детерминации. Коэффициент корреляции R for this sample is 0.9592 (). The closer it is to unity, the closer the relationship between the features. In this case, the relationship is very close, almost absolute correlation. Determination coefficient R 2 equals 0.92. This means that the regression equation is determined by 92% by the variance of the resulting attribute, and 8% are accounted for by third-party factors.

The coefficient of determination shows the proportion of the scatter taken into account by the regression in the total scatter of the resulting attribute. This indicator, equal to the ratio of the factorial variation to the total variation of the trait, makes it possible to judge how "successfully" the type of function is chosen. The more R 2 , the more the change in the factor attribute explains the change in the resultant attribute and, therefore, the better the regression equation, the better the choice of function.

List of sources used

Analysis of the economic activity of the enterprise: Proc. allowance / Under the total. ed. L. L. Ermolovich. - Minsk: Interpressservice; Ecoperspective, 2001. - 576 p.

Savitskaya G. V. Analysis of the economic activity of the enterprise, 7th ed., Rev. - Minsk: New knowledge, 2002. - 704 p.

Savitskaya GV Theory of economic activity analysis. - M.: Infra-M, 2007.

Savitskaya GV Economic analysis: Proc. - 10th ed., corrected. - M.: New knowledge, 2004. - 640 p.

Skamai LG, Trubochkina MI Economic analysis of the enterprise. - M.: Infra-M, 2007.

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION

FEDERAL AGENCY FOR EDUCATION

State educational institution of higher professional education

RUSSIAN STATE TRADE AND ECONOMIC UNIVERSITY

TULA BRANCH

(TF GOU VPO RGTEU)

Essay on mathematics on the topic:

"Economic and mathematical models"

Completed:

2nd year students

"Finance and Credit"

day department

Maksimova Kristina

Vitka Natalia

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 all new areas of scientific knowledge: technical design, construction and architecture, astronomy, physics, chemistry, biology and, finally, social sciences. Great success and recognition in almost all branches of modern science brought the modeling method of the twentieth century. However, modeling methodology has been developed independently by individual sciences for a long time. There was no unified system of concepts, a 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 meanings. Let us consider only such "models" that are tools for obtaining knowledge.

A model is such a material or mentally represented 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 building, 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, and conclusions 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 kinds of economic models.

The purpose of mathematical modeling of economic systems is the use of mathematical methods for the most effective solution of problems that arise in the field of economics, using, as a rule, modern computer technology.

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

-variability (dynamics);

-inconsistency of behavior;

-tendency to degrade performance;

-environmental exposure

predetermine the choice of the method of their research.

The penetration of mathematics into economics is associated with overcoming significant difficulties. This was partly "guilty" of mathematics, which has been developing over several centuries, mainly in connection with the needs of physics and technology. 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 considered as a justification for the impossibility of its modeling, study by means of mathematics. But this point of view is fundamentally wrong. You can model an object of any nature and any complexity. And just complex objects are of the greatest interest for modeling; this is where modeling can provide results that cannot be obtained by other methods of research.

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

Tasks 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 studying an object, it is often impractical or even impossible to deal directly with this object. It is more convenient to replace it with another object similar to the given one in those aspects that are important in this study. In general modelcan be defined as a conditional 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 the impossibility of direct study of 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 various models. These models reflect the essential properties of a real object (processes), although in reality reality is much more meaningful and richer.

Modelis a mentally represented or materially realized system, which, displaying or reproducing the object of study, is able to replace it in such a way that its study provides new information about this object.

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

Economic and mathematical models- these are models of economic objects or processes, in the description of which mathematical means are used. The goals of their creation are varied: they are built to analyze certain prerequisites and provisions of economic theory, to provide a rationale for economic patterns, to process and bring empirical data into a 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 it is possible to single out their most significant groups depending on the attribute of the classification.

For the intended purposemodels are divided into:

· Theoretical and 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).

By taking into account the time factormodels are divided into:

· Dynamic (describe the economic system in development);

· Statistical (the 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 considered period of timedistinguish models:

· 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 applicationdistinguish models:

· Balance;

· econometric;

· Optimization;

· Network;

· Queuing systems;

· Imitation (expert).

AT balance sheetModels reflect the requirement of matching the availability of resources and their use.

OptimizationModels make it possible to find the best variant of production, distribution or consumption from the set 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 in time. Typically, the network model is designed to perform work in such a sequence that the project timeline is minimal. In this case, the problem is to find the critical path. However, there are also network models that are focused not on the criterion of time, but, for example, on minimizing the cost of work.

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

Imitationthe model, along with machine decisions, contains blocks where decisions are made by a person (expert). Instead of the direct participation of a person 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 tasks by simulating the actions of a person, an expert in this 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 apparatusdistinguish models:

· Linear programming (the optimal plan is achieved at the extreme point of the region of change of the variables of the constraint system);

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

· Correlation-regression;

· Matrix;

· Network;

· game theory;

· Theories of queuing, etc.

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

simulation mathematical stochastic

1.2 Economic and mathematical methods

Like any modeling, economic and mathematical modeling is based on the principle of analogy, i.e. the possibility of studying an object by constructing and considering 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 and mathematical modeling lies in the description of socio-economic systems and processes in the form of economic and mathematical models, which should be understood as a product of the process of economic and mathematical modeling, and economic and mathematical methods - as a tool.

Let's consider questions of classification of economic and mathematical methods. These methods are a complex of economic and mathematical disciplines, which are an alloy of economics, mathematics and cybernetics. Therefore, the classification of economic and mathematical methods is reduced to the classification of the scientific disciplines included in their composition.

With a certain degree of conventionality, the classification of these methods can be represented as follows.

· Economic Cybernetics: System Analysis of Economics, Economic Information Theory and Control Systems Theory.

· 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 quantitative econometrics: economic growth theory, production function theory, input-output balances, national accounts, demand and consumption analysis, regional and spatial analysis, global modeling.

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

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

· Methods and disciplines that are specific to both a centrally planned economy and a market (competitive) economy. The former include the theory of optimal pricing of the functioning of the economy, optimal planning, the theory of optimal pricing, models of logistics, etc. The latter include methods that allow developing 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 include, as a rule, mathematical methods of analysis and planning of economic experiments, methods of machine simulation (simulation), business games. This also includes methods of expert assessments developed to evaluate phenomena that cannot be directly measured.

Various branches of mathematics, mathematical statistics, and mathematical logic are used in economic and mathematical methods. An important role in solving economic and mathematical problems is played by computational mathematics, the theory of algorithms and other disciplines. The use of the mathematical apparatus has brought tangible results in solving the problems of analyzing the processes of expanded production, determining the optimal growth rates of capital investments, optimal location, specialization and concentration of production, the problems of choosing the best production methods, determining the optimal sequence of launching into production, the problem of preparing production using network planning methods, and many others. .

Solving standard problems is characterized by a clear goal, the ability to develop procedures and rules for conducting calculations in advance.

There are the following prerequisites for the use of 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, knowledge of economic and mathematical methods.

Before starting to develop models, it is necessary to carefully analyze the situation, identify goals and relationships, problems that need to be solved, and the initial data for their solution, maintain a system of notation, 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 with both the object of modeling and the apparatus and means of modeling 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.Building a mathematical model;

.Mathematical analysis of the model;

.Preparation of initial information;

.Numerical solution;

Let's consider each of the stages in more detail.

1.Statement of the economic problem and its qualitative analysis. The main thing here is to clearly articulate the essence of the problem, the assumptions made and the questions that need to be answered. This stage includes highlighting the most important features and properties of the object being modeled and abstracting from minor ones; studying the structure of the object and the main dependencies connecting its elements; formulation of hypotheses (at least preliminary) explaining the behavior and development of the object.

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

It is wrong to assume that the more facts the 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 non-linear), taking into account the factors of randomness and uncertainty, etc.

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

One of the important features of mathematical models is the potential possibility of their use for solving problems of different quality. Therefore, even when faced with a new economic challenge, one should not strive to "invent" a model; First, it is necessary to try to apply already known models to solve this problem.

.Mathematical analysis of the model.The purpose of this step is to clarify the general properties of the model. Here purely mathematical methods of research are applied. 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 there is no need for subsequent work on the initial version of the model, and either the formulation of the economic problem or the methods of its mathematical formalization should be corrected. During the analytical study of the model, such questions are clarified as, for example, is the solution 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 of their change, etc. d. The analytical study of the model compared to the 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 imposes strict requirements on the information system. At the same time, the real possibilities of obtaining information limit the choice of models intended for practical use. This takes into account not only the fundamental possibility of preparing information (for a certain period of time), but also the costs of preparing the relevant 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 systemic 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, the compilation of computer programs and direct calculations. The difficulties of this stage are due, first of all, to the large dimension of economic problems, the need to process significant amounts of information.

A study carried out by numerical methods can significantly supplement the results of an analytical study, 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 simulation results, about the degree of practical applicability of the latter.

Mathematical verification methods can reveal incorrect model constructions and thereby narrow the class of potentially correct models. An informal analysis of the theoretical conclusions and numerical results obtained by means of the model, their comparison with the available knowledge and facts of reality also make it possible to detect the shortcomings of the formulation of the economic problem, the constructed mathematical model, 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 stability 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 need to be modernized to meet changing economic conditions. At the same time, the need to improve the mechanism for the implementation of new banking technologies determines the feasibility of scientific research.

The integrated financial stability ratios (CFS) of commercial banks used in existing methods often characterize the balance of their condition, but do not allow a complete description of the development trend. It should be borne in mind that the result (KFU) depends on many random causes (endogenous and exogenous) that cannot be fully taken into account in advance.

In this regard, it is justified to consider the possible results of the study of the steady state of banks as random variables with the same probability distribution, since the studies are carried out according to the same methodology using the same approach. Moreover, 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 on 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- the coefficient of financial stability of the 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 the development of banks, the assessment was carried out on the basis of a 12-year retrospective period (Table 1).

Table 1

Ordinal number of the year Bank "A" Bank "B" Bank "C"11.3141.2011.09820.8150.9050.81131.0430.9940.83941.2111.0051.01351.1101.0901.00961.0981.1541.01771.1121.1151.02981.3111.3281.0 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 particular bank, the values are divided into Nintervals, the minimum and maximum values are determined. The procedure for determining the optimal number of groups is based on the application of the Sturgess formula:

N\u003d 1 + 3.322 * ln 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 the intervals of values of discrete random variables X, Y, Z (financial stability coefficients) and the frequency of occurrence of these values within the indicated boundaries

Interval numberInterval boundariesFrequency of occurrences (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 interval step found, 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 (left boundary - LG). To find the second value (the right border of PG), the i step is again added to the found first border, and so on. The boundary of the last interval coincides with the maximum value:

LG1 =KFUmin;

PG1 =KFUmin+h;

LG2 =PG1;

PG2 =LG2 +h;

PG10 =KFUmax.

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

Table 3

Distribution of discrete random variables X, Y, Z

IndicatorValues of the indicatorBank "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 are found (the frequency of occurrence is divided by 12, based on the number of population units), and the midpoints of the intervals were used as values of discrete random variables. The 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 the value of the financial stability ratio equal to 0.853. In other words, there is an 8.3% chance that his expenses will exceed his income. For Bank B, the probability of the coefficient falling below one also amounted to 0.083, however, taking into account the dynamic development of the organization, this decrease will still turn out to be insignificant - to 0.926. Finally, there is a high probability (16.7%) that the activity of Bank C, other things being equal, will be characterized by a financial stability value of 0.835.

At the same time, according to 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 expedient to use numbers that describe the random variable in total. They are called the numerical characteristics of a random variable, they include the mathematical expectation. The mathematical expectation is approximately equal to the average value of a random variable and it approaches the average value the more the more tests have been carried out.

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

M(X) = x1 p1 +x2 p2 +…+xnpn

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

Table 4

Numerical characteristics of discrete random variables X, Y, Z

BankExpectationDispersionStandard deviation"A" M (X) \u003d 1.187 D (X) \u003d 0.027 σ (x) \u003d 0.164 "B" M (Y) \u003d 1.124 D (Y) \u003d 0.010 σ (y) \u003d 0.101 "C" M (Z) \u003d 1.037 D (Z) \u003d 0.012 σ (z) = 0.112

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

So, according to the calculations, it can be judged that the mathematical expectation of the 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, showing the "center" of the alleged possible values of the random variable - KFU, 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 stability of the bank's development. For this reason, it becomes necessary to calculate other numerical characteristics: dispersion and standard deviation. Which allow to estimate the degree of dispersion of possible values of the coefficient of financial stability. Mathematical expectations and standard deviations make it possible to estimate the interval in which the possible values of the financial stability ratios 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. With 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 ratio 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) will face, other things being equal, with an unacceptable deviation for it, equal to 0.112. In an unfavorable situation, and also given the high probability of loss-making activity (16.7%), this credit institution is likely to reduce its financial stability to 0.925.

It is important to note that, having drawn conclusions about the sustainability of the development of banks, it is impossible to predict in advance which of the possible values the financial stability ratio 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 certain relatively broad conditions, the total behavior of a sufficiently large number of random variables almost loses its random character and becomes regular.

Assessing the stability of the development of banks, it remains to estimate the probability that the deviation of a random variable from its mathematical expectation does not exceed the absolute value of a positive number ε. The estimate we are interested in can be given by P.L. Chebyshev. The probability that the deviation of a random variable X from its mathematical expectation in absolute value is less than a positive number ε not less than :

or in the case of inverse probability:

Taking into account the risk associated with the loss of stability, we will estimate the probability of a discrete random variable deviating from the mathematical expectation to the smaller side and, considering the deviations from the central value both to the smaller and the larger side to be equiprobable, we rewrite the inequality once again:

Further, based on the task set, it is necessary to estimate the probability that the future value of the financial stability ratio will not be lower than 1 from the proposed mathematical expectation (for bank "A" the value ε let's take equal to 0.187, for bank "B" - 0.124, for "C" - 0.037) and calculate this probability:

jar":

Bank "C"

According to 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 relatively less than in other banks. Bank A is in second place in terms of comparative stability of development, where the coefficient of this deviation is somewhat higher than in the first case (0.386). In the third bank, the probability that the value of the financial stability ratio deviates to the left of the mathematical expectation by more than 0.037 is a practically certain event. Moreover, if we take into account that the probability cannot be greater than 1, exceeding the values, according to the proof of L.P. Chebyshev should be taken as 1. In other words, the fact that the development of a bank can 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 1.187. The standard deviation of this discrete value is 0.164, which objectively characterizes a small spread of coefficient values from the average number. However, the degree of instability of this series is confirmed by a rather 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 KFU is 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 ratio, i.e. is more concentrated and stable, which is confirmed by the relatively low probability (0.325) of the bank's transition to the loss zone.

The stability of bank "C" is characterized by a low value of mathematical expectation (1.037) and a small spread of values (standard deviation is 0.112). Inequality L.P. 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, other things being equal, will look very unreasonable. Thus, the proposed model, based on determining the existing distribution of discrete random variables (the values of the financial stability ratios of commercial banks) and confirmed by assessing their equiprobable positive or negative deviation from the obtained mathematical expectation, makes it possible to determine its current and future level.

Conclusion

The use of mathematics in economics gave impetus to the development of both economics itself and applied mathematics, in terms of methods of the economic and mathematical model. The proverb says: "Measure seven times - Cut once." The use of models is time, effort, material resources. In addition, calculations based on models are opposed to volitional decisions, since they allow us to evaluate the consequences of each decision in advance, 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 by constructing and considering 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; thirdly, the development of managerial decisions at all levels of management.

In the work, it was found that economic and mathematical models can be divided according to the following features:

· intended purpose;

· taking into account the time factor;

· the duration of the period under consideration;

· purpose of creation and application;

· 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.

· formulation of the economic problem and its qualitative analysis;

· building a mathematical model;

· mathematical analysis of the model;

· preparation of initial information;

· numerical solution;

· analysis of numerical results and their application.

The paper 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 subject 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 an adequate definition of financial stability using various methods and models, one of the varieties of which are stochastic (probabilistic) models, which allow not only to identify the expected factors of growth or decrease in stability, but also to form a set of preventive measures to preserve it, is increasing.

The potential possibility of mathematical modeling of any economic objects and processes does not, of course, mean its successful feasibility at a given level of economic and mathematical knowledge, available specific information and computer technology. And although it is impossible to indicate the absolute boundaries 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.

Bibliography

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)Gataulin A.M., Gavrilov G.V., Sorokina T.M. and other Mathematical modeling of economic processes. - M.: Agropromizdat, 1990.

)Ed. Fedoseeva V.V. Economic-Mathematical Methods and Applied Models: Textbook for High Schools. - M.: UNITI, 2001.

)Savitskaya G.V. Economic Analysis: Textbook. - 10th ed., corrected. - M.: New knowledge, 2004.

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When constructing economic models, significant factors are identified and details that are not essential for solving the problem are discarded.

Economic models may include models:

economic growth
consumer choice
equilibrium in the financial and commodity markets and many others.

Model is a logical or mathematical description of the components and functions that reflect the essential properties of the modeled object or process.

The model is used as a conditional image designed to simplify the study of an object or process.

The nature of the models may be different. Models are divided into: real, sign, verbal and tabular description, etc.

Economic and mathematical model

In the management of business processes, the most important are, first of all, economic and mathematical models, often combined into model systems.

Economic and mathematical model(EMM) is a mathematical description of an economic object or process for the purpose of their study and management. This is a mathematical record of the economic problem being solved.

Main types of models

Extrapolation Models
Factorial econometric models
Optimization Models
Balance models, Inter-Industry Balance Model (ISB)
Expert assessments
Game theory
network models
Models of queuing systems

Economic and mathematical models and methods used in economic analysis

R a \u003d PE / VA + OA,

In a generalized form, the mixed model can be represented by the following formula:

So, first you need to build an economic-mathematical model that describes the influence of individual factors on the general economic indicators of the organization. Widespread in the analysis of economic activity received multifactorial multiplicative models, since they allow us to study the influence of a significant number of factors on generalizing indicators and thereby achieve greater depth and accuracy of analysis.

After that, you need to choose a way to solve this model. Traditional ways: the method of chain substitutions, the methods of absolute and relative differences, the balance method, the index method, as well as the methods of correlation-regression, cluster, dispersion analysis, etc. Along with these methods and methods, specific mathematical methods and methods are also used in economic analysis.

Integral method of economic analysis

One of these methods (methods) is integral. It finds application in determining the influence of individual factors using multiplicative, multiple, and mixed (multiple additive) models.

Under the conditions of applying the integral method, it is possible to obtain more reasonable results for calculating the influence of individual factors than when using the chain substitution method and its variants. The chain substitution method and its variants, as well as the index method, have significant drawbacks: 1) the results of calculating the influence of factors depend on the accepted sequence of replacing the basic values of individual factors with actual ones; 2) an additional increase in the generalizing indicator, caused by the interaction of factors, in the form of an indecomposable remainder, is added to the sum of the influence of the last factor. When using the integral method, this increase is divided equally between all factors.

The integral method establishes a general approach to solving models of various types, regardless of the number of elements that are included in this model, and also regardless of the form of connection between these elements.

The integral method of factorial economic analysis is based on the summation of the increments of a function defined as a partial derivative multiplied by the increment of the argument over infinitely small intervals.

In the process of applying the integral method, several conditions must be met. First, the condition of continuous differentiability of the function must be observed, where some economic indicator is taken as an argument. Secondly, the function between the start and end points of the elementary period must change in a straight line G e. Finally, thirdly, there must be a constancy of the ratio of the rates of change in the values of the factors

dy / dx = const

When using the integral method, the calculation of a definite integral over a given integrand and a given integration interval is carried out according to the available standard program using modern computer technology.

If we are solving a multiplicative model, then the following formulas can be used to calculate the influence of individual factors on a general economic indicator:

∆Z(x) = y 0 * Δ x + 1/2Δ x *Δ y

Z(y)=x 0 * Δ y +1/2 Δ x* Δ y

When solving a multiple model to calculate the influence of factors, we use the following formulas:

Z=x/y;

Δ Z(x)= Δ x/Δ y Lny1/y0

Δ Z(y)=Δ Z- Δ Z(x)

There are two main types of problems solved using the integral method: static and dynamic. In the first type, there is no information about changes in the analyzed factors during this period. Examples of such tasks are the analysis of the implementation of business plans or the analysis of changes in economic indicators compared to the previous period. The dynamic type of tasks takes place in the presence of information about the change in the analyzed factors during a given period. This type of tasks includes calculations related to the study of time series of economic indicators.

These are the most important features of the integral method of factorial economic analysis.

Log method

In addition to this method, the method (method) of logarithm is also used in the analysis. It is used in factor analysis when solving multiplicative models. The essence of the method under consideration lies in the fact that when it is used, there is a logarithmically proportional distribution of the value of the joint action of factors between the latter, that is, this value is distributed between the factors in proportion to the share of influence of each individual factor on the sum of the generalizing indicator. With the integral method, the mentioned value is distributed among the factors equally. Therefore, the logarithm method makes the calculation of the influence of factors more reasonable than the integral method.

In the process of taking logarithms, not absolute values of the growth of economic indicators are used, as is the case with the integral method, but relative ones, that is, indices of changes in these indicators. For example, a generalizing economic indicator is defined as the product of three factors - factors f = x y z.

Let us find the influence of each of these factors on the generalizing economic indicator. So, the influence of the first factor can be determined by the following formula:

Δf x \u003d Δf lg (x 1 / x 0) / log (f 1 / f 0)

What was the impact of the next factor? To find its influence, we use the following formula:

Δf y \u003d Δf lg (y 1 / y 0) / log (f 1 / f 0)

Finally, in order to calculate the influence of the third factor, we apply the formula:

Δf z \u003d Δf lg (z 1 / z 0) / log (f 1 / f 0)

Thus, the total amount of change in the generalizing indicator is divided between individual factors in accordance with the proportions of the ratios of the logarithms of individual factor indices to the logarithm of the generalizing indicator.

When applying the method under consideration, any types of logarithms can be used - both natural and decimal.

Method of differential calculus

When conducting factor analysis, the method of differential calculus is also used. The latter assumes that the overall change in the function, that is, the generalizing indicator, is divided into separate terms, the value of each of which is calculated as the product of a certain partial derivative and the increment of the variable by which this derivative is determined. Let's determine the influence of individual factors on the generalizing indicator, using as an example a function of two variables.

Function is set Z = f(x,y). If this function is differentiable, then its change can be expressed by the following formula:

Let us explain the individual elements of this formula:

ΔZ = (Z 1 - Z 0)- the magnitude of the function change;

Δx \u003d (x 1 - x 0)- the magnitude of the change in one factor;

Δ y = (y 1 - y 0)- the amount of change of another factor;

is an infinitesimal value of a higher order than

In this example, the influence of individual factors x and y to change the function Z(generalizing indicator) is calculated as follows:

ΔZx = δZ / δx Δx; ΔZy = δZ / δy Δy.

The sum of the influence of both of these factors is the main, linear part of the increment of the differentiable function, that is, the generalizing indicator, relative to the increment of this factor.

Equity method

In the conditions of solving additive, as well as multiple-additive models, the method of equity participation is also used to calculate the influence of individual factors on the change in the general indicator. Its essence lies in the fact that the share of each factor in the total amount of their changes is first determined. Then this share is multiplied by the total change in the summary indicator.

Suppose we are determining the influence of three factors − a,b and with for a summary y. Then, for the factor a, determining its share and multiplying it by the total value of the change in the generalizing indicator can be carried out according to the following formula:

Δy a = Δa/Δa + Δb + Δc*Δy

For the factor in the considered formula will have the following form:

Δyb =Δb/Δa + Δb +Δc*Δy

Finally, for the factor c we have:

∆y c =∆c/∆a +∆b +∆c*∆y

This is the essence of the equity method used for the purposes of factor analysis.

Linear programming method

See below:

Queuing Theory

See below:

Game theory

Game theory also finds application. Just like queuing theory, game theory is one of the branches of applied mathematics. Game theory studies the optimal solutions that are possible in situations of a game nature. This includes such situations that are associated with the choice of optimal management decisions, with the choice of the most appropriate options for relationships with other organizations, etc.

To solve such problems in game theory, algebraic methods are used, which are based on a system of linear equations and inequalities, iterative methods, as well as methods for reducing this problem to a specific system of differential equations.

One of the economic and mathematical methods used in the analysis of the economic activity of organizations is the so-called sensitivity analysis. This method is often used in the process of analyzing investment projects, as well as in order to predict the amount of profit remaining at the disposal of this organization.

In order to optimally plan and forecast the activities of the organization, it is necessary to foresee those changes that may occur in the future with the analyzed economic indicators.

For example, it is necessary to predict in advance the change in the values of those factors that affect the amount of profit: the level of purchase prices for acquired material resources, the level of selling prices for the products of a given organization, changes in customer demand for these products.

Sensitivity analysis consists in determining the future value of a generalizing economic indicator, provided that the value of one or more factors influencing this indicator changes.

So, for example, they establish by what amount the profit will change in the future, subject to a change in the quantity of products sold per unit. Thus, we analyze the sensitivity of net profit to a change in one of the factors affecting it, that is, in this case, the sales volume factor. The rest of the factors affecting the profit margin remain unchanged. It is possible to determine the amount of profit also with a simultaneous change in the future of the influence of several factors. Thus, sensitivity analysis makes it possible to establish the strength of the response of a generalizing economic indicator to changes in individual factors that affect this indicator.

Matrix method

Along with the above economic and mathematical methods, they are also used in the analysis of economic activity. These methods are based on linear and vector-matrix algebra.

Network planning method

See below:

Extrapolation Analysis

In addition to the considered methods, extrapolation analysis is also used. It includes consideration of changes in the state of the analyzed system and extrapolation, that is, the extension of the existing characteristics of this system for future periods. In the process of implementing this type of analysis, the following main stages can be distinguished: primary processing and transformation of the initial series of available data; choice of the type of empirical functions; determination of the main parameters of these functions; extrapolation; establishing the degree of reliability of the analysis.

In economic analysis, the method of principal components is also used. They are used for the purpose of a comparative analysis of individual components, that is, the parameters of the analysis of the organization's activities. Principal components are the most important characteristics of linear combinations of constituent parts, that is, the parameters of the analysis performed, which have the most significant values of dispersion, namely, the largest absolute deviations from the average values.

All models that a person uses in various areas of his activity can be conditionally divided into two groups: material and abstract. The first are objective, they can really be touched by hands. The latter exist only in the human mind. Within the framework of this article, only mathematical methods and models in the economy will be considered. They are used to analyze the processes and phenomena occurring in this area. Their use allows setting new economic tasks. Thanks to them, management makes decisions regarding the management of the organization, firm, enterprise.

Mathematical operations in economics are the most effective tool for studying problems in this area. In modern scientific and technical activities, they are becoming an important form of modeling. And in the practice of planning and management, this method is the main one.

Economic-mathematical methods and models are the basis on which various programs are implemented, originally designed to solve the problems of planning, analysis and management. Together with technical means, with databases, they are part of the human-machine system. It allows you to use models and knowledge to solve various kinds of problems (both unstructured and weakly structured).

Depending on the criteria that underlie the division, economic and mathematical methods and models are classified as follows.

1. By purpose, they are:

Applied, that is, with their help, specific tasks are solved;

Theoretical and analytical (they are used when it is necessary to investigate the general patterns and signs of the development of processes occurring in the economy).

2. By what causal relationships they reflect:

deterministic;

Probabilistic (take into account the factor of emerging uncertainty).

3. According to the level of those processes in the economy that they study:

Production and technological;

Socio-economic.

4. According to the way the time factor is reflected:

Dynamic, they show the ongoing changes;

Static, all dependencies here reflect only one period of time or moment.

5. By level of detail:

Macromodels (aggregated);

Micromodels (detailed).

6. According to the form in which mathematical dependencies are expressed:

non-linear;

Linear - they are very convenient to use for calculation and analysis, which has led to their wider distribution.

Economic and mathematical methods and models have their own principles of construction. These include:

1. The principle of unambiguous data. According to him, the information that is used at the beginning of the simulation should not depend on those parameters of the future system that are not even known at this stage of the study.

2. The principle of completeness of initial information. It means that the initial information used must be very accurate, since the results obtained depend on it.

3. The principle of succession. He says that those features of the object that were reflected or established in the first models should be preserved in each subsequent one.

4. The principle of effective implementation. Each model must be used in practice. The latest computing tools should help in its implementation.

Economic and mathematical methods and models are always built in several stages:

1) Definition of the problem, its analysis.

2) Design This is its expression in the form of functions, schemes, equations.

3) Analysis of the resulting model using mathematical techniques.

4) Preparation of initial information.

5) This is the actual development of programs, the compilation of algorithms and the conduct of calculations.

6) Analysis of the obtained results, their practical application.

Each of these stages may have its own specifics depending on the area of knowledge under consideration.