Economic-mathematical methods and models of analysis. Economic and mathematical methods (EMM)

Heuristic methods

Operations Research Methods

Mathematical theory of optimal processes

Methods of economic cybernetics

Classical methods of mathematical analysis

Methods of mathematical statistics

Econometric Methods

Methods of mathematical programming

Economic and mathematical methods of economic activity analysis.

Methods of elementary mathematics are used in conventional traditional economic calculations when substantiating resource needs, accounting for production costs, developing plans, projects, in balance calculations, etc. Allocation methods of classical higher mathematics on the diagram is due to the fact that they are used not only within the framework of other methods, for example, methods of mathematical statistics and mathematical programming, but also separately. Thus, factor analysis of changes in many economic indicators can be carried out using differentiation and integration.

Methods of mathematical statistics widely used in economic analysis. They are used in cases where the change in the analyzed indicators can be represented as a random process. Statistical methods, being the main means of studying mass, recurring events, play an important role in predicting the behavior of economic indicators. When the relationship between the analyzed characteristics is not deterministic, but stochastic, then statistical and probabilistic methods are practically the only research tool. The most widespread of the mathematical and statistical methods in economic analysis are methods of multiple and pair correlation analysis.

For studying one-dimensional statistical populations used: variation series, distribution laws, sampling method. For studying multivariate statistical populations apply correlations, regressions, dispersion, covariance, spectral, component, factorial types of analysis studied in the courses of the theory of statistics.

The next group of economic and mathematical methods - econometric methods.Econometrics- a scientific discipline that studies the quantitative aspects of economic phenomena and processes by means of mathematical and statistical analysis based on the modeling of economic processes. Accordingly, econometric methods are based on the synthesis of three areas of knowledge: economics, mathematics and statistics. The foundation of econometrics is economic model, which is understood as a schematic representation of an economic phenomenon or process using scientific abstraction, reflecting them characteristic features. Of the econometric methods, the method of "cost-output" analysis has received the greatest distribution in modern economics. For its development, the outstanding economist V. Leontiev received the Nobel Prize in 1973. Input-output analysis method- this is an econometric method of analysis, which consists in building matrix (balance) models, according to a chess scheme and allowing in the most compact form to present the relationship between costs and production results. The convenience of calculations and the clarity of economic interpretation are the main advantages of using matrix models. This is important when creating mechanized data processing systems, when planning the production of products using a computer.

Methods of mathematical programming in economics- these are numerous methods for solving problems of optimizing the production and economic and, above all, the planned activities of an economic entity. In essence, these methods are means of planned calculations. Their value for the economic analysis of the implementation of business plans lies in the fact that they allow us to assess the tension of planned targets, determine the limiting groups of equipment, types of raw materials and materials, obtain estimates of the scarcity of production resources, etc.

Under Operations Research is understood as a method of purposeful actions (operations), a quantitative assessment of the solutions obtained and the choice of the best of them. The subject of operations research is economic systems, including the production and economic activities of enterprises. The goal is such a combination of structural interrelated elements of systems, which is most suitable for the task of obtaining the best economic indicator from a number of possible ones.

As a division of operations research game theory- this is a theory of building mathematical models for making optimal decisions in the face of uncertainty or conflict of several parties with different interests.

Queuing Theory - it is a theory that develops mathematical methods for quantifying queuing processes based on probability theory. So, any of the structural divisions of an industrial enterprise can be represented as an object of a service system.

A common feature of all problems associated with queuing is the random nature of the phenomena under study. The number of service requests and the time intervals between their arrival are random in nature, they cannot be predicted with unambiguous certainty. However, in their totality, many of these requirements are subject to certain statistical patterns, the quantitative study of which is the subject of queuing theory.

Methods of economic cybernetics are being developed economic cybernetics - a scientific discipline that analyzes economic phenomena and processes as very complex systems, from the point of view of the laws and mechanisms of control and the movement of information in them. Of the methods of economic cybernetics, the most widely used in economic analysis are

31methods modeling and system analysis.

IN last years in economics, interest has increased in methods of empirical search for optimal conditions for the course of the process, using human experience and intuition. This is reflected in the application heuristic methods (solutions), which are non-formalized methods for solving economic problems related to the current economic situation, based on intuition, past experience, expert assessments of specialists, etc.

For the analysis of production, economic, commercial activities, many methods from the above exemplary scheme have not found practical application and are only being developed in the theory of economic analysis. At the same time, this scheme did not reflect some of the economic and mathematical methods considered in the special literature on economic analysis: fuzzy set theory, catastrophe theory etc. In this study guide attention is focused on the basic economic and mathematical methods that have already received wide application in the practice of economic analysis.

The application of one or another mathematical method in economic analysis is based on methodology of economic and mathematical modeling business processes and scientifically based classification of methods and problems of analysis.

According to the classification criterion of optimality, all economic and mathematical methods (tasks) are divided into two groups: optimization and non-optimization. Optimization Methods- a group of economic and mathematical methods of analysis that allow you to search for a solution to a problem according to a given optimality criterion. Non-optimization methods- a group of economic and mathematical methods of analysis used to solve problems without an optimality criterion.

On the basis of obtaining an exact solution, all economic and mathematical methods are divided into exact and approximate ones. TO precise methods refer to a group of economic and mathematical methods, the algorithm of which allows you to get only one solution according to a given optimality criterion or without it. TO approximate methods include a group of economic and mathematical methods used in the case when stochastic information is used in the search for a solution and the solution of the problem can be obtained with any degree of accuracy, as well as those in the application of which it is not guaranteed to obtain a unique solution according to a given optimality criterion or without it.

Thus, based on the use of only two classification features, all economic and mathematical methods are divided into four groups:

1) optimization exact methods;

2) optimization approximate methods;

3) non-optimization exact methods;

4) non-optimization approximate methods.

Yes, to optimization exact methods include methods of the theory of optimal processes, some methods of mathematical programming and methods of operations research. TO optimization approximate methods include: individual methods of mathematical programming; methods of operations research, methods of economic cybernetics; methods of mathematical theory of planning extreme experiments; heuristic methods. TO non-optimization exact methods include: methods of elementary mathematics and classical methods of mathematical analysis, econometric methods. TO non-optimization approximate methods include: the method of statistical tests and other methods of mathematical statistics.

Of the enlarged groups of economic and mathematical methods presented by us, some methods from these groups are used to solve various problems - both optimization and non-optimization; both exact and approximate.

2 . Linear programming methods. All economic problems solved using linear programming methods are distinguished by alternative solutions and certain limiting conditions. Solving such a problem means choosing the best, optimal one from a significant number of all possible options. This is the importance and value of using linear programming methods in economics. Using other methods to solve such problems is almost impossible.

Linear programming is based on solving a system of linear equations (with transformation into equations and inequalities), when the dependence between the studied phenomena is strictly functional. It is characterized by: a mathematical expression variables, certain order, sequence of calculations (algorithm), logical analysis. It can be applied only in those cases when the studied variables and factors have mathematical certainty and quantitative limitations, when, as a result of a known sequence of calculations, the interchangeability of factors occurs, when the logic in the calculations, mathematical logic are combined with a logically sound understanding of the essence of the phenomenon under study.

With the help of linear programming methods in industrial production, for example, the optimal overall productivity of machines, units, production lines is calculated (for a given range of products and other given values), the problem of rational cutting of materials is solved (with an optimal yield of blanks). IN agriculture they are used to determine the minimum cost of feed rations for a given amount of feed (by type and nutrient content). The problem of mixtures can also find application in foundry production (the composition of a metallurgical charge). The same methods solve the transport problem, the problem of rational attachment of consumer enterprises to manufacturing enterprises.

3. Methods of dynamic programming. Dynamic programming methods are used to solve optimization problems in which the objective function and/or constraints are characterized by non-linear dependencies.

Signs of non-linearity are, in particular, the presence of variables /, in which the exponent differs from unity, as well as the presence of a variable in the exponent, under the root, under the sign of the logarithm.

In the economy in general and in the economics of the enterprise, in particular, there are a lot of examples of non-linear dependencies. Thus, the economic efficiency of production increases or decreases disproportionately to changes in the scale of production; the cost of producing a batch of parts increases with an increase in the size of the batch, but not proportionally to them. A non-linear relationship is characterized by a change in the amount of wear of production equipment depending on the time of its operation, the specific consumption of gasoline (per 1 km of track) - on the speed of vehicles and many other economic situations.

Economic and mathematical methods (EMM)- a generalizing name for a complex of economic and mathematical scientific disciplines, united to study the economy. Introduced by Academician V.S. Nemchinov in the early 60s. There are statements that this name is very conditional and does not correspond state of the art development economics, since "they (EMM. - ed.) do not have their own subject of study, different from the subject of study of specific economic disciplines" .

However, although the trend is correctly noted, it does not appear to be realized soon. EMM actually have common object research with other economic disciplines - economics (or more broadly: the socio-economic system), but a different subject of science: i.e. they study different aspects of this object, approach it from different positions. And most importantly, in this case, special research methods are used, developed so much that they themselves become separate scientific disciplines of a special methodological nature. Unlike disciplines in which ontological aspects predominate, and research methods act only to a greater or lesser extent as auxiliary means, in the "methodological" disciplines that make up a significant part of the EMM complex, the methods themselves turn out to be the object of research. Moreover, the real synthesis of economics and mathematics is yet to come, and it will take a long time before it is fully realized.

The generally accepted classification of economic and mathematical disciplines, which were an alloy of economics, mathematics and cybernetics, has not yet been developed. With a certain degree of conventionality, it can be represented in the form of the following scheme.

0. Principles of economic and mathematical methods:

theory economic and mathematical modeling, including economic and statistical modeling;

theory optimization of economic processes.

1. Mathematical statistics (its economic applications):

sampling method;

dispersion analysis;

correlation analysis;

regression analysis;

multivariate statistical analysis;

factor analysis;

index theory, etc.

2. Mathematical economy and econometrics:

theory of economic growth (models of macroeconomic dynamics);

theory of production functions;

intersectoral balances (static and dynamic);

national accounts, integrated material and financial balances;

analysis of demand and consumption;

regional and spatial analysis;

global modeling, etc.

3. Methods for making optimal decisions, including operations research:

optimal (mathematical) programming;

linear programming;

non-linear programming;

dynamic programming;

discrete (integer) programming;

block programming;

fractional linear programming;

parametric programming;

separable programming;

stochastic programming;

geometric programming;

branch and bound methods;

network methods of planning and management;

program-target methods of planning and management;

theory and methods of inventory management;

queuing theory;

game theory;

decision theory;

scheduling theory.

4. EMM and disciplines specific to a centrally planned economy:

theory of optimal functioning of the socialist economy (SOFE);

optimal planning:

economic;

perspective and current;

sectoral and regional;

theory of optimal pricing;

5. EMM specific to competitive economy:

models of the market and free competition;

business cycle models;

models of monopoly, duopoly, oligopoly;

indicative planning models;

models of international economic relations;

models of the theory of the firm.

6. Economic cybernetics:

system analysis economy;

economic Information Theory, including economic semiotics;

control systems theory, including theory automated systems management.

7. Methods of experimental study of economic phenomena ( experimental economy):

mathematical methods of planning and analysis economic experiments;

methods machine simulation And bench experimentation;

business games.

EMM uses various branches of mathematics, mathematical statistics And mathematical logic ; a big role in machine decision economic and mathematical problems play computational mathematics, theory of algorithms and other related disciplines.

The practical application of EMM in some countries has become widespread, in a sense, routine. In thousands companies tasks are solved planning production, distribution resources using established and often standardized software ensure installed on computers. This practice is being studied in the field - surveys, surveys .. In the USA, a special magazine "Interfaces" is even published, regularly publishing information on the practical use of EMM in various sectors of the economy. For example, here is a summary of one of the articles of this magazine: “In 2005 and 2006, Coca-Cola Enterprises (CCE), the largest manufacturer and distributor of the Coca-Cola drink, software ORTEC for transport routing. Currently, over three hundred controllers use this software, planning routes for approximately 10,000 trucks daily. In addition to overcoming some non-standard limitations, the use of this technology is notable for its progressive (non-disruptive) transition from previous business practices. CCE has reduced annual costs by $45 million and improved customer service. This experience was so successful that (parent multinational company) Coca Cola expanded it beyond the CCE, to other companies for the production and distribution of this drink, as well as beer.

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION

FEDERAL AGENCY FOR EDUCATION

State educational institution 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 It 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 industries modern science brought the modeling method of the twentieth century. However, the modeling methodology long time developed independently by separate sciences. absent one system concepts, common terminology. Only gradually did the role of modeling as universal method scientific knowledge.

The term "model" is widely used in various fields 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 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 kinds of economic models.

The purpose of mathematical modeling of economic systems is the use of mathematical methods for the most effective solution tasks arising in the field of economics, with the use, as a rule, of 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 systems 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 view 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.

Model- this is 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 gives 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 economic theory, rationale for economic patterns, processing and bringing into the system of empirical data. In practical terms, economic and mathematical models are used as a tool for forecasting, planning, managing and improving various aspects economic activity 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).

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

Options econometricmodels are evaluated using methods of mathematical statistics. 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 a trend (long-term trend) of the main indicators of the modeled economic system. Econometric models are used to analyze and predict specific economic processes using real statistical information.

Optimizationmodels allow you to find from a variety of possible (alternative) options the best option production, distribution or consumption. Limited resources will be used the best way to achieve the set 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 features of their classification, the process of integration of models is carried out. different types into more complex model structures.

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.

practical tasks 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 managerial 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 high level 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 selection most important features and properties of the object being modeled and abstraction from secondary 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 more facts takes into account the model, 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 real opportunities information and mathematical support, but also to compare the costs of modeling with the effect obtained.

One of important features mathematical models - 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. Most important point- 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 certain deadlines), 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 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.On this final stage 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.

Integral coefficients used in existing methods financial stability(KFU) of commercial banks often characterize the balance of their condition, but do not allow to give complete description development trends. It should be borne in mind that the result (KFU) depends on many random reasons(endogenous and exogenous nature), which cannot be fully taken into account in advance.

In this regard, it is reasonable to consider possible results research steady state 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 test the random variable takes on one and only one possible value, we conclude that the events x1 , x2 , …, xnform full 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\u003d 1 + 3.322 * ln12 \u003d 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 to minimum value found step. 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 ?(x) \u003d 0.164 "B" M (Y) \u003d 1.124 D (Y) \u003d 0.010 ?(y) \u003d 0.101 "C" M (Z) \u003d 1.037 D (Z) \u003d 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 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 ?.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 ? 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 a smaller side and, considering the deviations from the central value both to a smaller and 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 ?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 slightly 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.

Economic and mathematical methods acquire a particularly large role as information technologies are introduced in all areas of practice. The main stages of the modeling process were also considered, namely:

· 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

1)Krass M.S. Mathematics for economic specialties: Textbook. -4th ed., rev. - M.: Delo, 2003.

)Ivanilov Yu.P., Lotov A.V. Mathematical models in economics. - M.: Nauka, 2007.

)Ashmanov S.A. Introduction to mathematical economics. - M.: Nauka, 1984.

)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.

)Gmurman V.E. Theory of Probability and Mathematical Statistics. Moscow: Higher school, 2002

)Operations research. Tasks, principles, methodology: textbook. allowance for universities / E.S. Wentzel. - 4th ed., stereotype. - M.: Drofa, 2006. - 206, p. : ill.

)Mathematics in economics: textbook / S.V. Yudin. - M.: RGTEU Publishing House, 2009.-228 p.

)Kochetygov A.A. Probability Theory and Mathematical Statistics: Proc. Allowance / Tul. State. Univ. Tula, 1998. 200p.

)Boyko S.V., Probabilistic models in assessing the financial stability of credit institutions /S.V. Boyko // Finance and credit. - 2011. N 39. -

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Topic 1. Basic concepts of mathematical modeling of socio-economic systems.

Modeling as a method of scientific knowledge.

SES, their properties.

Stages of economic and mathematical modeling.

Classification of economic and mathematical models.

Modeling in scientific research began to be used in ancient times and gradually captured more and more new areas of scientific knowledge: technical design, construction and architecture, astronomy, physics, chemistry, biology and, finally, social sciences. The 20th century brought great success and recognition in almost all branches of modern science to modeling.

Modeling methodology has long been developed independently by individual sciences. 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.

Under modeling the process of building, studying and applying models is understood.

The modeling process necessarily includes the construction of abstractions, inferences by analogy and the construction of scientific hypotheses. The main feature of modeling is that it is a method of indirect cognition with the help of proxy objects.

A model is a conditional image, a scheme of the object of study. The model acts as a kind of tool of knowledge, which the researcher puts between himself and the object and with the help of which he studies the object of interest to him.

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 investigate or not at all, or this research requires a lot of time and money.

The modeling process includes 3 elements: the subject (researcher), the object of study, the model that mediates the relationship between the subject and the object.

Most of the objects studied by economic science can be characterized by the cybernetic concept a complex system . The most common understanding of the system as a set of elements that are in interaction and form a certain integrity, unity. Complexity a system of any nature (technical, economic, biological, social, etc.) is determined by the number of elements included in it, the connections between them, as well as the relationship between the system and the environment.

The economy has all the hallmarks of a complex system. It combines a huge number of elements that are distinguished by a variety of internal connections and connections with other systems (the natural environment, the economic activities of other entities, social relations). 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. Complex objects are of the greatest interest for modeling; it is here that modeling can provide results that cannot be obtained by other research methods.

Thus, the main method for studying systems is simulation method, those. a method of theoretical analysis and practical action aimed at the development and use of models.

System evolution modeling is based on two methodological approaches:

System Analysis, i.e. dividing the system into separate elements, studying their interrelations and patterns of development using the model.

System approach, i.e. synthesis– study of an object as a single whole based on the use of a complex of logical, informational and algorithmically interconnected systems of models and methods for their solution.

If the economic system is interpreted as a system of social production and consumption of material goods, then the social aspects of society are very multifaceted and not always accessible for detailed analysis, modeling and forecasting. At the same time, some social problems are the object of research for practitioners (analysis and forecasting of consumer demand in marketing, the distribution of workers by wage level in economics and the sociology of labor). Many of these kinds of problems can be solved using economic and mathematical methods and models.

Economics and Mathematics model is a similarity or analogue of the studied economic phenomenon or process, expressed using mathematical dependencies and relationships.

Under economic and mathematical methods it implies a cycle of scientific disciplines, the subject of which are the quantitative characteristics and patterns of economic processes, considered inextricably linked with their qualitative characteristics.

In research, methods of mathematical statistics, probability theory are used, to a large extent they use the apparatus of mathematical programming and modeling of economic processes, network planning, queuing theory, expert assessments, etc.

The use of mathematical methods in solving practical problems makes it possible to improve the system of economic information, improve the accuracy of economic calculations, deepen the quantitative analysis of economic problems, and solve fundamentally new economic problems.

practical tasks economic and mathematical modeling are:

Analysis of economic objects and processes;

Economic forecasting of the development of processes and phenomena;

Development of managerial decisions at all levels of management.

The data obtained as a result of economic and mathematical modeling can be used as "advising" means.

An important concept in EMM is the concept model adequacy , i.e. correspondence of the model to the modeled object or process in terms of those properties that are essential for the study. Checking the adequacy of economic and mathematical models is complicated by the difficulty of measuring economic values.

The scope of practical application of the modeling method is limited by the possibilities and efficiency of formalizing economic problems and situations, as well as the state of information, mathematical, and technical support of the models used.

At present, the most promising direction for the use of economic and mathematical methods is the implementation of the EMM system within the framework of automated control systems, automated workplaces of specialists, managers within local information networks (LIS).

Socio-economic system(SES) refers to complex systems. It is more complex than economic and is determined by the system of human relations with nature, society, production, and entrepreneurship. It covers the processes of production, exchange, distribution and consumption of material and other goods.

In the economic subsystem, the relationship of man to production is considered, in the social subsystem, the relationship of man to nature.

SES includes economic and social subsystems.

Within the framework of the "economic system", the concept of "production system" is singled out. This is a naturally stable connection and relationship of all industries and elements of production in a certain period of time. Models of the production system make it possible to describe a purposefully developed type of human labor activity, its dynamics.

The production system is subdivided into sub-complexes of agro-industrial complex sectors:

industries that ensure the development of the agro-industrial complex;

agriculture proper;

creation of final products (processing industry) .

Such systems can be considered at the federal, regional level, the level of inter-farm associations and enterprises, enterprises and their divisions.

Complex systems in the economy have a number of properties that must be taken into account when modeling them, otherwise it is impossible to talk about the adequacy of the constructed economic model.

The most important of these properties are:

emergence- manifestation of the integrity of the system, i.e. the presence in the economic system of such properties that are not inherent in any of its constituent elements, taken separately. Therefore, SES needs to be investigated and modeled as a whole.

Mass character of economic phenomena and processes. Patterns of economic processes are not detected on the basis of a small number of observers. Therefore, modeling in economics should be based on mass observations.

Dynamism of economic processes consists in changing the parameters and structure of economic systems under the influence of the environment (external factors).

Randomness and uncertainty in the development of economic phenomena. Therefore, economic phenomena and processes are mainly of a probabilistic nature, and their study requires the use of EMM based on probability theory and mathematical statistics.

The inability to isolate the phenomena and processes occurring in economic systemsfrom the environment to observe and explore them in their purest form.

Active response to emerging new factors, the ability of SES to take active actions, depending on the attitude of the system to these factors, ways and methods of their influence.

The selected properties of SES, of course, complicate the process of their modeling, but these properties should always be kept in mind when considering various aspects of economic and mathematical modeling, starting with the choice of the type of model and ending with the practical use of modeling results.

The development of EMM is carried out in stages, in a certain sequence :

1. Statement of the economic problem and its qualitative analysis.

An economic formulation is required, including the purpose of the decision, the establishment of a planning period, the clarification of the known parameters of the object and those whose value needs to be determined, their production and economic relations, as well as many factors and conditions that reflect the simulated process.

The goal of solving the problem is expressed quantitatively by a specific indicator, called the optimality criterion. It must correspond to the economic essence of the problem being solved. This requires a comprehensive and deep qualitative analysis of the essence of the problem and the exact formulation of the goal of its solution.

2. Building a mathematical model.

This is the stage of formalization of the economic problem, i.e. expressing it in the form of specific mathematical dependencies (functions, equations, inequalities). Model building is divided into several stages. First, the type of EMM is determined, the possibilities of application in this problem are studied, then the specific list of variables and parameters and the form of relationships are specified.

3. Mathematical analysis of the model.

The purpose of this step is to clarify the general properties of the model. This is where mathematical methods of research are applied. The most important point is the proof of the existence of solutions in the formulated model.

4. Preparation of initial information.

Mathematical modeling imposes strict requirements on the information system; At the same time, it is necessary to take into account not only the possibility of preparing information, but also the costs of its preparation. In systemic economic and mathematical modeling, the results of the functioning of some models serve as initial information for others.

Information as a set of information about an economic object or process necessary for modeling should be complete, reliable, accessible and timely.

The purpose of processing the initial information is to develop and justify a system of technical and economic characteristics of an object or process.

For any model, these characteristics are formed in the form of technical and economic coefficients, objective function coefficients and volume indicators (constant) of resources or products.

TECs can be divided into 3 groups:

Resource input or output standards

Proportionality coefficients (provide for determining the relationship between dependent variables)

Coefficients of connection (determine the dependence of the variable on the volume index).

The cost of preparing information should not exceed the effect of its use.

5. Numerical solution.

This stage includes the development of algorithms for the numerical solution of the problem, the preparation of computer programs and direct calculations, while significant difficulties are caused by the large dimensionality of economic problems.

Usually, EMM-based calculations are of a multivariate nature. Numerous model experiments, the study of the behavior of the model under various conditions, can be carried out due to the high speed of modern computers. Optimization methods are important for solving problems.

6. Analysis of numerical results and their application.

At this stage, the issue of the correctness and completeness of the simulation results and their applicability both in practice and in order to improve the model is solved.

The listed stages of economic and mathematical modeling are closely interconnected and there may be reciprocal links between the stages. So, at the stage of building a model, it may turn out that the statement of the problem is either contradictory or leads to an overly complex mathematical model; in this case, the original formulation of the problem must be corrected. Most often, the need to return to the previous stages of modeling arises at the stage of preparing the initial information.

Thus, modeling is a cyclical process. Knowledge about the studied object is expanded and refined, and the original model is gradually improved.

In the future, you can use a more general scheme of the modeling process, including:

problem statement,

EMM formation,

The solution of the problem,

Analysis of the obtained results.

The essence of economic and mathematical modeling lies in the description of SES and processes in the form of EMM.

Mathematical models can be subdivided according to a number of features:

1. For the general purpose:

Theoretical and analytical - are used in the study of general properties and patterns of economic processes;

Applied - used in solving specific economic problems (models of economic analysis, forecasting and management).

2. By the degree of aggregation of objects:

Macroeconomic (the economy as a whole);

Microeconomic (enterprise).

3. For a specific purpose(according to the purpose of creation and application):

Balance models expressing the requirement that the availability of resources correspond to their use;

Trend models, in which the development of the modeled economic system is reflected through the trend (long trend) of its main indicators;

Optimization models designed to select the best option for the functioning of the system;

Simulation models are used in the process of machine simulation of the studied systems or processes.

4. By type of information:

Analytical (experience);

Identifiable (experiment)

5. By taking into account the time factor:

Static describe the state of an economic object at a particular moment or period of time;

Dynamic describe economic systems in development.

6. By type of mathematical apparatus:

Matrix models, linear and non-linear programming, network planning, correlation-regression, game theory, etc.

Models of economic processes are very diverse in the form of mathematical dependencies. It is especially important to single out the class of linear models that are most convenient for analysis and calculations and, as a result, have become widespread. But at the same time, many dependencies in the economy are fundamentally non-linear.

7. Taking into account the uncertainty factor:

Deterministic ones imply rigid functional relationships between model variables;

Stochastic (probabilistic) allow the presence of random effects on the studied indicators.

8. By type of approach to the studied SES:

Descriptive (descriptive) are intended to describe and explain the actually observed phenomena, their forecast (balance, trend models);

Normative ones determine how the economic system develops, how it should be arranged and how it should act, taking into account certain criteria (optimization models).

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 features of their classification, the process of integrating models of different types into more complex model constructions is being carried out.

Course subject are the quantitative characteristics of economic phenomena and processes in agro-industrial production and entrepreneurship.

Course objectives:

To study the basic techniques and methods for modeling the main patterns and economic processes in the SES of the agricultural sector.

The main method is the methods of mathematical modeling, i.e. calculation of quantitative characteristics of the development of biological-technical, organizational-technological, production-industry and entrepreneurial relations of the employee's personality with nature, society, production.

To learn how to use a package of applied programs for computers to automate the formation and calculation of the EMM system.

To study the economic and mathematical analysis of optimal solutions.