Application of economic and mathematical methods in economics. Tasks for independent work

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION

FEDERAL AGENCY FOR EDUCATION

State educational institution of higher professional education

RUSSIAN STATE TRADE AND ECONOMICS UNIVERSITY

TULA BRANCH

(TF GOU VPO RGTEU)

Abstract in mathematics on the topic:

"Economic and mathematical models"

Completed:

2nd year students

"Finance and Credit"

day department

Maksimova Kristina

Vitka Natalya

Checked:

Doctor of Technical Sciences,

Professor S.V. Yudin _____________

Introduction

1.Economic and mathematical modeling

1.1 Basic concepts and types of models. Their classification

1.2 Economic and mathematical methods

Development and application of economic and mathematical models

2.1 Stages of economic and mathematical modeling

2.2 Application of stochastic models in economics

Conclusion

Bibliography

Introduction

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

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

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

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

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

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

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

-variability (dynamism);

-inconsistent behavior;

-tendency to deteriorate performance;

-environmental exposure

predetermine the choice of method for their research.

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

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

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

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

1.Economic and mathematical modeling

1.1 Basic concepts and types of models. Their classification

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

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

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

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

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

By purposemodels are divided into:

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

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

Taking into account the time factormodels are divided into:

· Dynamic (describe an economic system in development);

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

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

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

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

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

According to the purpose of creation and usemodels are distinguished:

· Balance sheet;

· Econometric;

· Optimization;

·Network;

· Queuing systems;

· Imitation (expert).

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

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

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

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

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

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

Taking into account the uncertainty factormodels are divided into:

· Deterministic (with uniquely defined results);

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

By type of mathematical apparatusmodels are distinguished:

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

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

· Correlation-regression;

·Matrix;

·Network;

·Game theories;

· Queuing theories, etc.

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

modeling mathematical stochastic

1.2 Economic and mathematical methods

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2. Development and application of economic and mathematical models

2.1 Stages of economic and mathematical modeling

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

.Statement of the economic problem and its qualitative analysis;

2.Construction of a mathematical model;

.Mathematical analysis of the model;

.Preparation of background information;

.Numerical solution;

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

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

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

It is wrong to believe that the more facts a model takes into account, the better it “works” and gives better results. The same can be said about such characteristics of the complexity of the model as the forms of mathematical dependencies used (linear and nonlinear), taking into account randomness factors and uncertainty, etc.

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

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

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

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

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

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

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

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

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

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

2.2 Application of stochastic models in economics

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

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

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

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

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

Table 1

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

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

N=1+3.322 * log N;

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

Where n- number of groups;

N- the number of the population.

h=(KFUmax- KFUmin) / 10.

table 2

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

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

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

LG1 =KFUmin;

PG1 =KFUmin+h;

LG2 =PG1;

PG2 =LG2 +h;

PG10 =KFUmax.

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

Table 3

Distribution of discrete random variables X, Y, Z

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

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

Pi= ni /12;

Xi= (LGi+PGi)/2.

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

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

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

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

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

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

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

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

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

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

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

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

M(X) = x1 p1 +x2 p2 +…+xnpn

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

Table 4

Numerical characteristics of discrete random variables X, Y, Z

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

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

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

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

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

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

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

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

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

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

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

or in case of reverse probability:

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

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

jar":

Bank "C":

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

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

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

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

Conclusion

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

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

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

· intended purpose;

· taking into account the time factor;

· the duration of the period under review;

· purposes of creation and use;

· taking into account the uncertainty factor;

· type of mathematical apparatus;

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

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

· formulation of an economic problem and its qualitative analysis;

· building a mathematical model;

· mathematical analysis of the model;

· preparation of background information;

· numerical solution;

· analysis of numerical results and their application.

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

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

Bibliography

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)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 others. Mathematical modeling of economic processes. - M.: Agropromizdat, 1990.

)Ed. Fedoseeva V.V. Economic-mathematical methods and applied models: Textbook for universities. - M.: UNITY, 2001.

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

)Gmurman V.E. Theory of Probability and Mathematical Statistics. M.: Higher School, 2002

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

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

)Kochetygov A.A. Probability theory and mathematical statistics: Textbook. Manual / Tool. State Univ. Tula, 1998. 200 p.

)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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The group of economic and mathematical methods is divided into two subgroups:

· Methods of mathematical extrapolation;

· Methods of mathematical modeling.

Mathematical extrapolation is the extension of the law of change of a function from the region of its observation to a region lying outside the observation segment.

Extrapolation methods are based on the assumption of the invariability of the factors determining the development of the object under study, and consists in extending the patterns of development of the object in the past to its future.

The bottom line is that the trajectory of an object’s development up to the moment from which it begins to predict future development can be expressed after appropriate processing of actual data by some mathematical function that adequately describes the patterns of the previous development of the object

Depending on the characteristics of changes in levels in the dynamics series, extrapolation techniques can be simple or complex.

The first group consists of forecasting methods based on the assumption of relative constancy in the future of the absolute values of levels, the average level of a series, the average absolute increase, and the average growth rate.

The second group of methods is based on identifying the main trend, that is, using statistical formulas that describe the trend. They can be divided into two main types: adaptive and analytical (growth curves). Adaptive forecasting methods are based on the fact that the process of their implementation consists in calculating time-sequential values of the predicted indicator, taking into account the degree of influence of previous levels. These include the methods of moving and exponential averages, the method of harmonic weights, and the method of autoregressive transformations.

Analytical methods (growth curves) of forecasting are based on the principle of obtaining, using the least squares method, an estimate of the deterministic component Ft, which characterizes the main trend.

The essence of the method is that the trajectory of an object’s development up to the moment from which forecasting begins can be expressed after appropriate processing of actual data by any mathematical function that adequately describes the patterns of previous development. It is carried out as follows:

1. it is necessary to obtain a sufficiently long series of indicators;

2. it is necessary to construct an empirical curve that graphically displays the dynamics of this indicator over time;

3. it is necessary to align the series using graph analysis or statistical selection of functions, which maximizes the approximation to the actual values of the time series;

4. We calculate the coefficient or parameter of this function (a,b,c...), the result is the simplest mathematical model suitable for forecasting over time, while it is assumed that the cumulative factor determining the trends of the time series in the past will, on average, retain its strength.

In economic research, the most common method of predictive extrapolation is the method based on time series smoothing.

The sequence of statistical indicators arranged in chronological order that characterize changes in an economic phenomenon over time is a time (dynamic) series. Individual values of indicators (observations) of a time series are called levels of this series.

Time series are divided into moment and interval.

The purpose of analyzing time series of economic phenomena over a certain time interval is to establish the trend of their change over the period under consideration, which will show the direction of development of the phenomenon being studied.

In order to identify the general trend of changes in economic phenomena during the studied period of time, the time series should be smoothed. The need to smooth time series is due to the fact that in addition to the influence on the levels of a number of main factors that ultimately form the specific value of the non-random component (trend), they are affected by random factors that cause deviations of the actual (observed) values of the series levels from the trend.

A trend is understood as a characteristic of the main tendency of a time series of values of a certain indicator, i.e. the basic pattern of its movement in time, free from random influences.

Thus, individual levels of the time series (y t ) represent the result of the influence of the main factors that form the specific value of the non-random (deterministic) component ( ), as well as a random component (е t), caused by the influence of random factors, the value of which is the deviation of the actual (observed) values of the series levels from the trend. To eliminate random deviations, the time series is smoothed.

Non-random components of the levels of a time series can be expressed by some approximating function, reflecting the patterns of development of the phenomenon under study.

Let's consider forecast extrapolation based on smoothing time series using the least squares method.

The essence of the least squares method is to determine the parameters of the trend model that minimize its deviation from the points of the original time series, i.e. in minimizing the sum of square deviations between observed and calculated values.

Thus, the essence of smoothing a time series of observed indicator values is that the actual (observed) levels of the series are replaced by levels calculated on the basis of a certain function that most closely matches the observed values of the time series indicators.

The graph of a linear function is a straight line.

In order to determine the parameters a and A of the straight line equation, you need to solve the system of equations:

Often time series data has a nonlinear relationship, which is expressed as a quadratic function: y = ax 2+ b x + s. The graph of a quadratic function is a parabola. In order to determine the parameters a, b, c equations of a parabola, you should solve the system of equations:

Economic and mathematical modeling involves constructing a model based on a preliminary study of an object or process, identifying its essential characteristics or features.

Economic and mathematical model is a system of formalized relationships that describe the basic relationships of the elements that form a certain economic system.

Depending on the level of management of economic and social processes, macroeconomic, intersectoral, sectoral, regional models and macro-level models (individual enterprises, firms) are distinguished.

An example of an economic-mathematical model at the macro level can be a production function model when forecasting the volume of gross domestic product (GDP) country, which looks like this:

It should be noted that the calculation of economic and mathematical models is carried out using appropriate computer programs.

Economic and mathematical models are used to develop an inter-industry balance, modeling capital investments, labor resources, etc.

Planning methods, as an integral part of the planning methodology, are a set of calculations that are necessary for the development of individual sections and indicators of the plan and their justification. At the same time, the achievements of branch economic sciences are widely used: economic statistics; industrial economics; agricultural economics; economics of construction and others. When planning indicators, it is important not only to calculate their value in the planning period, but also to identify possible reserves for its improvement and involve them in economic turnover.

The main planning methods that are widely used in economic practice include the following: balance sheet method; normative method; program-target method; economic and statistical methods; economic and mathematical methods.

Balance sheet method- ensures the linking of needs and resources both on the scale of all social production and at the level of the industry and individual enterprise. The following types of balances are used in planning practice: 1) material balances; 2) cost balances; 3) labor resource balances.

The basic diagram of material balance in natural units of measurement is as follows:

Cost balances include: intersectoral balance of production and distribution of products, works and services; state budget, etc. As a balance of labor resources, one of the topics of the course will consider the consolidated balance of labor resources.

Normative planning method based on the development and use of norms and standards in planning. As an example, we can give the rate of consumption of various materials in physical measurement per unit of output. As an example, we can cite the standard for deduction of funds from the profit of an enterprise in the form of taxes.

Program-target planning method based on the development of socio-economic programs to solve individual socio-economic problems. This method involves defining a set of interrelated organizational, legal, financial and economic measures aimed at implementing the developed programs. Using this method involves concentrating resources on solving the most important problems.

Economic and statistical methods of planning represent a set of individual methods with the help of which individual socio-economic indicators for the planning period and their dynamics are calculated. The absolute and relative dynamics of indicators are determined, i.e. their change over time.

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

List of sources used

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

One of the directions for improving the analysis of economic activity is the introduction of economic and mathematical methods and modern computers. Their use increases the efficiency of economic analysis by expanding the factors studied, justifying management decisions, choosing the optimal option for using economic resources, identifying and mobilizing reserves for increasing production efficiency.

Mathematical methods are based on the methodology of economic-mathematical modeling and scientifically based 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 studying mass empirical data by constructing 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 is manifested in the fact that when one of them changes, the distribution law of the other changes.

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

Studying the presence and closeness of the connection between function and factors, as well as between factors;

Ranking and classification of factors of economic phenomena;

Identification of the analytical form of connection between the phenomena being studied;

Smoothing the dynamics of changes in the level of indicators;

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

Study of the dimension (complexity, versatility) of economic phenomena;

Quantitative change in informative indicators;

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

Stochastic modeling and analysis of relationships between the studied indicators begin with correlation analysis. The correlation is that the average value of one of the characteristics changes depending on the value of the other. A characteristic on which another characteristic depends is usually called factorial. The dependent characteristic is called effective. In each specific case, to establish factorial and resultant characteristics in unequal populations, an analysis of the nature of the connection is necessary. Thus, when analyzing various characteristics in one set, the wages of workers in connection with their production experience act as an effective characteristic, and in connection with indicators of living standards or cultural needs - as a factor one. Often dependencies are considered not on one factor characteristic, but on several. To do this, a set of methods and techniques is used to identify and quantify the relationships and interdependencies between characteristics.

When studying mass socio-economic phenomena, a correlation relationship appears between factor characteristics, in which the value of the resulting characteristic is influenced, in addition to the factor characteristic, by many other characteristics acting in different directions simultaneously or sequentially. Often, a correlation relationship is called incomplete statistical or partial, in contrast to a functional one, which is expressed in the fact that at a certain value of a variable (independent variable - argument), another (dependent variable - function) takes on a strict value.

A correlation can only be revealed in the form of a general trend through a massive comparison of facts. Each value of a factor characteristic will correspond not to one value of the resulting characteristic, but to their combination. In this case, to reveal the relationship, it is necessary to find the average value of the resulting characteristic for each factor value.

If the relationship is linear:

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

N is the number of observations.

In the case of a linear relationship between the studied indicators, the correlation coefficient is calculated using the formula:

If the correlation coefficient is squared, we obtain the coefficient of determination.

Discounting is the process of converting the future value of capital, cash flows or net income into the present. The rate at which discounting is carried out is called the discount rate (discount rate). The basic premise behind the concept of discounted flow of real money is that money has a time price, that is, an amount of money available today is worth more than the same amount in the future. This difference can be expressed as an interest rate that represents the relative change over a specified period (usually a year).

Many of the 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, you have to find the optimal one among the many possible ones. A significant part of such problems have been solved for a long time based on common sense and experience. At the same time, there was no certainty that the found option was the best.

In modern conditions, even minor mistakes can lead to huge losses. In this regard, the need arose to involve optimization economic-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 using 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 is the relationship of equality, less or more.

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

Basically, in practice, nonlinear programming problems by linearization are reduced to a linear programming problem. Of particular practical interest among nonlinear 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 good mathematics and software are available today.

The dynamic programming method is a special mathematical technique for optimizing nonlinear mathematical programming problems, which is specially adapted to multi-step processes. A multi-step process is usually considered to be 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 break down into steps naturally (for example, the process of planning the economic activities 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 searching for an 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 we forget about all subsequent steps. Thus, dynamic programming is forward-looking planning with a perspective in mind.

The principle of choosing a solution in dynamic programming is decisive and is called the Bellman optimality principle. Let us formulate it as follows: an 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 decision made at the moment 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 maximum effect. Having optimally planned the last step, you can “attach” the penultimate one to it so that the result of these two steps is optimal, etc. It is in this way - from end to beginning - that the decision-making procedure can be developed. 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 branch of modern applied mathematics that studies methods for justifying optimal decisions in conflict situations. In the theory of statistical games, concepts such as the original strategic game and the statistical game itself are distinguished. In this theory, the first player is called “nature,” which is understood as the totality of circumstances under which the second player—“statistics”—has to make decisions. In a strategy game, both players act actively, assuming that the opponent is a "reasonable" player. A strategic game is characterized by complete uncertainty in the choice of strategy by each player, that is, the players know nothing about each other’s strategies. In a strategy game, both players act based on deterministic information defined by a loss matrix.

In a statistical game proper, 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 a statistical game strives to win the game against an imaginary opponent - nature. If in a strategic game players act 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 study these laws gradually, for example through 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 service requests arise at random (or not at random) times and there are devices (channels) for executing requests. Queuing theory explores mathematical methods for quantitative assessment of queuing processes and the quality of functioning of systems, where both the moments of appearance of requirements (applications) and the time spent on their execution can be random.

The queuing system is used in solving the following problems: for example, when applications (requirements) for service are received en masse with their subsequent satisfaction. In practice, this may be the receipt of raw materials, materials, semi-finished products, products to the warehouse and their issue from the warehouse; processing a wide range of parts using the same technological equipment; organization of adjustment and repair of equipment; transport operations; planning reserve and insurance reserves of resources; determining 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 characterizing the availability of resources (products) in kind or monetary terms and the directions of their use. At the same time, the availability of resources (products) and the need for them quantitatively coincide. The solution to such models is based on linear vector-matrix algebra methods. Therefore, balance methods and models are called matrix methods of analysis. The clarity of images of various economic processes in matrix models and the elementary methods of 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 20th century, is today increasingly used in the financial analysis of enterprise activities, including analysis and forecast of the financial position of the enterprise, analysis of changes in the working capital, free cash flows, economic risk, 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 quite cumbersome, since there is a large amount of information involved. The practical use of the theory of fuzzy sets makes it possible to develop traditional methods of financial and economic activity and adapt them to the new needs of taking into account the uncertainty in the future of the main performance indicators of enterprises.

Task 1

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

Solution:

Let's define:

1) acceptance coefficient (K pr):

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

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

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

2) coefficient for 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 disposal also decreased by 0.007 (0.108 - 0.115).

3) staff turnover rate(To 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 personnel.

4) coefficient of total labor turnover(To about):

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

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

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

Task 2

Create an initial model of production volume. Determine the type of factor model. Calculate the influence of factors on changes in production volume using all known techniques.

Solution:

The effective indicator is capital productivity.

Initial mathematical model:

FO = VP / OF.

Model type - multiple. The total number of performance indicators used to calculate 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.

The following techniques are applicable for this model: chain substitution, index and integral.

1. Let's calculate the level of influence of factors changing the performance indicator using the method of chain substitution.

Solution algorithm:

FO pl = VP pl / OF pl = 20433 / 2593 = 7.88 rub.

FO conv1 = VP f /OF pl =20193 / 2593 = 7.786 rub.

FO f = VP f /OF f =20193 / 2577 = 7.836 rub.

The calculation of the factors that influenced the change in capital productivity will be presented in the table.

No. of factors	Name of factors	Calculation of the level of influence of factors	Level of influence of factors changing the total amount of profit
	Change capital productivity by changing production volume	7,786-7,88 =-0,094
	Change capital productivity by changing fixed assets	7,836-7,786 = 0,05
	TOTAL (balance sheet linkage)

2. Let's calculate the level of influence of factors changing the performance indicator using an integral method.

VP = VP f - VP pl = 20193 - 20433 = -240;

OF = OF f - OF pl = 2577 - 2593 = -16.

FO pl = 20433 / 2593 = 7.88 rub.

FO f = 20193 / 2577 = 7.836 rub.

FO ch = = 15 ln|0.99| = -0.09284

FO of = ?FO total - ?FO VP = (7.836-7.88) - (-0.09284) = 0.04884

3. Let's calculate the level of influence of factors changing the performance indicator using the index method.

I FO = I VP I OF.

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

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

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

I FO = I VP I OF = 0.988 1.006 = 0.99.

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

Problem 3

Determine what the average yield level will be if the amount of fertilizer applied is 20 c. Determine the closeness of the connection between the indicator “y” and the factor “x”.

Given: Regression equation

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

x is the amount of fertilizer applied, c.

The coefficient of determination is 0.92.

Solution:

The average yield level is 62 c/ha.

Regression analysis aims to derive, define (identify) the regression equation, including statistical assessment 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 using the formula:

It has been proven 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 equal to 0.9592 (). The closer it is to one, the closer the connection between the characteristics. In this case, the connection 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 effective attribute, and the share of third-party factors accounts for 8%.

The coefficient of determination shows the proportion of the spread taken into account by regression in the total spread of the resulting characteristic. This indicator, equal to the ratio of the factor variation to the total variation of the characteristic, allows one to judge how “successfully” the type of function was chosen. The larger R2, 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 an enterprise: Textbook. allowance/Under general. ed. L. L. Ermolovich. - Mn.: Interpressservice; Ecoperspective, 2001. - 576 p.

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

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

Savitskaya G.V. Economic analysis: Textbook. - 10th ed., rev. - M.: New knowledge, 2004. - 640 p.

Skamai L. G., Trubochkina M. I. Economic analysis of enterprise activity. - 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 ECONOMICS UNIVERSITY

TULA BRANCH

(TF GOU VPO RGTEU)

Abstract in mathematics on the topic:

"Economic and mathematical models"

Completed:

2nd year students

"Finance and Credit"

day department

Maksimova Kristina

Vitka Natalya

Checked:

Doctor of Technical Sciences,

Professor S.V. Yudin _____________

Introduction

1.Economic and mathematical modeling

1.1 Basic concepts and types of models. Their classification

1.2 Economic and mathematical methods

Development and application of economic and mathematical models

2.1 Stages of economic and mathematical modeling

2.2 Application of stochastic models in economics

Conclusion

Bibliography

Introduction

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

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

-variability (dynamism);

-inconsistent behavior;

-tendency to deteriorate performance;

-environmental exposure

predetermine the choice of method for their research.

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

1.Economic and mathematical modeling

1.1 Basic concepts and types of models. Their classification

By purposemodels are divided into:

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

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

Taking into account the time factormodels are divided into:

· Dynamic (describe an economic system in development);

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

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

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

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

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

According to the purpose of creation and usemodels are distinguished:

· Balance sheet;

· Econometric;

· Optimization;

· Network;

· Queuing systems;

· Imitation (expert).

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

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

Taking into account the uncertainty factormodels are divided into:

· Deterministic (with uniquely defined results);

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

By type of mathematical apparatusmodels are distinguished:

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

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

· Correlation-regression;

· Matrix;

· Network;

· Game theories;

· Queuing theories, etc.

modeling mathematical stochastic

1.2 Economic and mathematical methods

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

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

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

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

2. Development and application of economic and mathematical models

2.1 Stages of economic and mathematical modeling

.Statement of the economic problem and its qualitative analysis;

2.Construction of a mathematical model;

.Mathematical analysis of the model;

.Preparation of background information;

.Numerical solution;

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

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

2.2 Application of stochastic models in economics

Table 1

N=1+3.322 * log N;

N=1+3.322 * ln12=9.525≈10,

Where n- number of groups;

N- the number of the population.

h=(KFUmax- KFUmin) / 10.

table 2

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

LG1 =KFUmin;

PG1 =KFUmin+h;

LG2 =PG1;

PG2 =LG2 +h;

PG10 =KFUmax.

Table 3

Distribution of discrete random variables X, Y, Z

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.

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”.

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

M(X) = x1 p1 +x2 p2 +…+xnpn

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

Table 4

Numerical characteristics of discrete random variables X, Y, Z

BankExpectationDispersionMean square deviation“A”M(X) = 1.187D(X) =0.027 σ (x) = 0.164"V"M(Y) = 1.124D(Y) = 0.010 σ (y) = 0.101 "С" M(Z) = 1.037D(Z) = 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 coefficient in the future.

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

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

or in case of reverse probability:

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