The use of mathematical methods in the economy is an example. Mathematical modeling in the economy

Ministry of Education and Science of the Russian Federation

Federal Agency for Education

State Educational Institution of Higher Professional Education

Russian State Trade - Economic University

Tula branch

(TF GOU VPO RGTEU)

Abstract for mathematics on the topic:

"Economic and Mathematical Models"

Performed:

2 student students

"Finance and Credit"

day separation

Maksimova Kristina

Top 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 the economy

Conclusion

Bibliography

Introduction

Relevance. Modeling in scientific research began to be applied in deep antiquity and gradually excited all new areas of scientific knowledge: technical design, construction and architecture, astronomy, physics, chemistry, biology and, finally, social sciences. Big successes and recognition in almost all branches of modern science brought the method of modeling the XX century. However, the modeling methodology has long developed independently individual sciences. There was no uniform system of concepts, single terminology. Only gradually began to be aware of the role of modeling as a universal method of scientific knowledge.

The term "model" is widely used in various spheres of human activity and has many semantic values. Consider only such "models", which are tools for obtaining knowledge.

The model is such a material or mentally represented object, which in the process of study replaces the original object so that its direct study gives new knowledge about the original object.

Under the modeling is understood as the process of building, studying and using models. It is closely related to such categories as abstraction, analogy, hypothesis, etc. The simulation process necessarily includes the construction of abstractions, and conclusions by analogy, and the design of scientific hypotheses.

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

The purpose of mathematical modeling of economic systems is the use of mathematics methods for the most effective solution to the tasks arising in the field of economy, with the use of, as a rule, modern computing technology.

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

-variability (dynamism);

-conflicting behavior;

-tendency to deteriorate characteristics;

-exposure to environmental exposure

the choice of the method of their research is predetermined.

Penetration of mathematics in economic science is associated with overcoming significant difficulties. This was partly a "guy" mathematics, developing over several centuries, mainly due to the needs of physics and technology. But the main reasons are still in the nature of economic processes, in the specifics of economic science.

The complexity of the economy was sometimes viewed as a substantiation of the impossibility of its modeling, studying mathematics. But this point of view is in principle incorrect. You can simulate an object of any nature and any complexity. And just complex objects are the greatest interest for modeling; It is here that modeling can give results that cannot be obtained by other research methods.

The purpose of this work - disclose the concept of economic and mathematical models and explore their classification and methods on which they are based, and also consider their use 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 research, the object is often inexpedient or even impossible to deal directly with this object. It is more convenient to replace it with another object similar to this in those aspects that are important in this study. In general modelyou can define as a conditional image of a real object (processes), which is created for a deeper study of reality. The study method based on the development and use of models is called modeling. The need for modeling is due to complexity, and sometimes the impossibility of directly studying the actual object (processes). It is much more accessible to create and explore the prototype of real objects (processes), i.e. Models. It can be said that theoretical knowledge of anything, as a rule, is a combination of various models. These models reflect the essential properties of the real object (processes), although in fact the reality is significantly infident and richer.

Model - This is a mentally represented or financially implemented system, which, displaying or reproducing an object of the study, is able to replace it in such a way that its study gives new information about this object.

To date, the generally accepted single classification of models does not exist. However, from a variety of models, verbal, graphic, physical, economic and mathematical and some other types of models can be distinguished.

Economic and Mathematical Models- These are models of economic objects or processes, which are used by mathematical means. The goals of their creation are varied: they are built to analyze certain prerequisites and provisions of economic theory, the logical substantiation of economic patterns, processing and bringing the empirical data system. In practical terms, economic and mathematical models are used as a tool for the forecast, planning, management and improvement of various aspects of the Company's economic activities.

Economic and mathematical models reflect the most significant properties of a real object or process using a system of equations. There is no uniform classification of economic and mathematical models, although you can select the most significant groups of their groups depending on the sign of classification.

By intended purpose Models are divided into:

· Analytical theoretical (used in the study of general properties and patterns of economic processes);

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

According to the time of time Models are divided into:

· Dynamic (describe the economic system in development);

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

The duration of the time period under considerationdistinguish models:

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

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

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

For the purpose of creating and use distinguish models:

· Balance;

· Econometric;

· Optimization;

· Network;

· Mass maintenance systems;

· Imitation (expert).

IN balance Models reflect the requirement for the availability of resources and their use.

Parameters econometric Models are estimated using mathematical statistics methods. The most common models that are systems of regression equations. These equations reflect the dependence of endogenous (dependent) variables from exogenous (independent) variables. This dependence is mainly expressed through a trend (long-term tendency) of the main indicators of the simulated economic system. Econometric models are used to analyze and predict specific economic processes using real statistical information.

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

Network Models are most widely used in project management. The network model displays a complex of work (operations) and events, and their relationship in time. Usually the network model is designed to perform work in such a sequence so that the timing of the project is minimal. In this case, the task of finding a critical path. However, there are also network models that are not focused on time criteria, but, for example, to minimize the cost of work.

Models mass maintenance systems Created to minimize the cost of time to wait in the queue and time of downtime of service channels.

Imitation The model, along with machine solutions, contains blocks where solutions are made by a person (expert). Instead of the direct involvement of a person in making decisions, a knowledge base can be. In this case, the personal computer, specialized software, database and knowledge base form an expert system. Expert The system is designed to solve one or a number of tasks by imitating a person's action, an expert in this area.

According to the uncertainty factor Models are divided into:

· Deterministic (with uniquely defined results);

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

By type of mathematical apparatus distinguish models:

· Linear programming (the optimal plan is achieved at the extreme point of the area of \u200b\u200bchange in variable values \u200b\u200bof the limit system);

· Nonlinear programming (optimal values \u200b\u200bof the target function may be several);

· Correlation-regression;

· Matrix;

· Network;

· Game theory;

· Mass maintenance theory, etc.

With the development of economic and mathematical research, the problem of classification of the models used is complicated. Along with the advent of new types of models and new signs of their classification, the process of integrating models of different types in more complex model structures is carried out.

modeling mathematical stochastic

1.2 Economic and Mathematical Methods

Like any modeling, economic and mathematical modeling is based on the principle of analogy, i.e. Opportunities to study the object by building and considering another, similar to it, but a simpler and accessible object, its model.

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

The essence of economic and mathematical modeling is to describe 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 are like a tool.

Consider the issues of the classification of economic and mathematical methods. These methods are a complex of economic and mathematical disciplines that are an alloy of the economy, mathematics and cybernetics. Therefore, the classification of economic and mathematical methods is reduced to the classification of scientific disciplines included in their composition.

With a known propulsion, the classification of these methods can be represented as follows.

· Economic cybernetics: systemic analysis of the economy, theory of economic information and the theory of control systems.

· Mathematical statistics: Economic applications of this discipline - selective method, dispersion analysis, correlation analysis, regression analysis, multidimensional statistical analysis, index theory, etc.

· Mathematical savings and studying the same questions from the quantitative side of Econometrics: the theory of economic growth, the theory of production functions, inter-sectoral balance sheets, national accounts, analysis of demand and consumption, regional and spatial analysis, global modeling.

· Methods for making optimal solutions, including the study of operations in the economy. This is the most surround section comprising the following disciplines and methods: optimal (mathematical) programming, network planning and management methods, theory and methods of stock management, mass maintenance theory, game theory, theory and decision-making methods.

In addition, the optimal programming includes linear and nonlinear programming, dynamic programming, discrete (integer) programming, stochastic programming, etc.

· Methods and disciplines specifically separately both for a centralized planned economy and for a market (competitive) economy. To the first one can be attributed to 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. To the second - methods allowing to develop models of free competition, model of the capitalist cycle, model of monopoly, firm theory model, etc. . Many of the methods designed for centrally planned economies may be useful and in economic and mathematical modeling in a market economy.

· Methods of experimental study of economic phenomena. These include, as a rule, mathematical methods of analyzing and planning economic experiments, machine simulation methods (simulation), business games. Methods of expert estimates, designed to estimate phenomena, not directly measurable, can also be attributed.

In economic and mathematical methods, various sections of mathematics, mathematical statistics, mathematical logic are used. A large 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 brought tangible results when solving problems of analyzing the processes of extended production, determining the optimal growth rates of investment, optimal placement, specialization and concentration of production, the tasks of selecting optimal production methods, determining the optimal sequence of launch in production, the task of preparing production by network planning methods and many other .

To solve the standard problems, the clarity of the target is characterized, the ability to develop procedures and rules for making calculations in advance.

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

Before proceeding to developing models, it is necessary to carefully analyze the situation, identify the goals and relationships, problems that require solutions, and the initial data for solving them, to keep the system of designations and only then describe the situation in the form of mathematical relations.

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 essential features associated with both modeling object and apparatus used and simulation tools. Therefore, it is advisable to analyze in more detail the sequence and maintenance of the stages of economic and mathematical modeling, highlight the following six steps:

.Statement of economic problem and its qualitative analysis;

2.Construction of a mathematical model;

.Mathematical analysis of the model;

.Preparation of source information;

.Numerical solution;

Consider each of the stages in more detail.

1.Statement of economic problem and its qualitative analysis. The main thing here is to clearly formulate the essence of the problem, the assumptions and the questions you want to get answers. This stage includes the allocation of the most important features and properties of the simulated object and abstraction from the secondary; study of the structure of the object and the main dependences connecting its elements; Formulation of hypotheses (at least preliminary), explaining the behavior and development of the object.

2.Construction of a mathematical model. This is the stage of formalization of the economic problem, expressing it in the form of concrete mathematical dependencies and relations (functions, equations, inequalities, etc.). Usually, the main design (type) of the mathematical model is determined, and then the details of this design (a specific list of variables and parameters, a form of links) is specified. In this way, the construction of the model is divided in turn into several stages.

It is wrong to believe that the more facts take into account the model, the better "works" better and gives the best results. The same can be said about such characteristics of the complexity of the model, as used forms of mathematical dependencies (linear and nonlinear), accounting for the factors of accidentability of uncertainty, etc.

Excessive complexity and bulkiness of the model make it difficult to research the process. It is necessary to take into account not only the real possibilities of information and mathematical support, but also to compare the costs of modeling with the resulting effect.

One of the most important features of mathematical models is the potential for their use for solving disabilities. Therefore, even facing a new economic task, it is not necessary 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 find out the general properties of the model. Here are applied purely mathematical methods of research. 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 task has no solution, then the need for subsequent work on the initial version of the model disappears and should be adjusted or the formulation of an economic task, or the methods of its mathematical formalization. With an analytical study of the model, issues such as, for example, is the only solution, which variables (frenitive) can be included in the decision, what are the relationship between them, in what limits and depending on the initial conditions they change, what are the trends of their change, etc. d. Analytical study of the model compared with empirical (numerical) has the advantage that the resulting conclusions retain their strength at various specific values \u200b\u200bof the external and internal parameters of the model.

4.Preparation of source information. Modeling places strict information system requirements. At the same time, the real possibilities for obtaining information limit the choice of models intended for practical use. At the same time, not only the principal possibility of preparing information (for certain time) is taken into account, but also the costs of preparing relevant information arrays.

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

In the process of preparing information, methods of the theory of probabilities, theoretical and mathematical statistics are widely used. With systemic economic and mathematical modeling, the initial information used in some models is the result of the operation of other models.

5.Numerical solution. This stage includes the development of algorithms for the numerical solution of the problem, drawing up programs for the computer and direct settlement. The difficulties of this stage are primarily due, above all, the large dimension of economic tasks, the need to process significant arrays of information.

The study conducted by numerical methods can significantly add the results of an analytical study, and for many models it is the only feasible. The class of economic tasks that can be solved with numerical methods is much wider than the class of tasks available to analytical research.

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

Mathematical test methods can identify incorrect construction of the model and thereby narrow out the class potentially correct models. An informal analysis of theoretical conclusions and numerical results obtained through the model, comparing them with the existing knowledge and facts of reality, also make it possible to detect the shortcomings of the economic task of the designed mathematical model, its information and mathematical support.

2.2 Application of stochastic models in the economy

The basis for the effectiveness of bank management is a systematic control over the optimality, balance and resistance of functioning in the context of all elements forming the resource potential and defining the prospects for the dynamic development of the credit institution. Its methods and tools require modernization, taking into account the 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 search.

The integral coefficients of financial stability (CFCs) of commercial banks used in existing methodologies often characterize the balance of their state, but do not allow to fully characterize the development trend. It should be borne in mind that the result (CFU) depends on many random causes (endogenous and exogenous nature), which cannot be fully taken into account in advance.

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

Taking into account that in one test, a random value takes one and only one possible value, conclude that events x.1 , X.2 ..., xn.form a complete group, therefore, the sum of their probabilities will be equal to 1: p.1 + P.2 + ... + pn.=1 .

Discrete random variability X. - the coefficient of financial sustainability of the bank "A", Y. - Bank "B", Z. - Bank "C" for a given period. In order to obtain a result, which gives reason to conclude the sustainability of bank development, the assessment was carried out on the basis of a 12-year retrospective period (Table 1).

Table 1

Sequence number of Letabank "A" Bank "B" Bank "C"11,3141,2011,09820,8150,9050,81131,0430,9940,83941,2111,0051,013,11,0981,1541,0981,11,11,151,11,11,1151,11,11,111,02981,1111,3281,06591 2451,1911,1451,19611,2041,1261,084121,1431,1511,028MIN0,8150,9050,811MAX1,5701,3281,296SHA0,07550,04230,0485

For each sample, on a specific bank, the values \u200b\u200bare divided into N. Intervals, the minimum and maximum value are defined. The procedure for determining the optimal number of groups is based on the use of Formula Sterges:

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 aggregate.

h \u003d (kfmax- KFU.mIN.) / 10.

table 2

The boundaries of the intervals of the values \u200b\u200bof discrete random variables X, Y, Z (financial stability coefficients) and the frequencies of these values \u200b\u200bin the indicated boundaries

Intervalist number intervalism appearances (n. ) Xyzxyz.10,815-0,8910,905-0,9470,811-0,86011220,891-0,9660,947-0,9900,860-0,90800030,966-1,0420,990-1,0320,908-0,95702041,042-1,1171,032-1,0740,957-1,00540051,117-1,1931,074-1,1171,005-1,05412561,193-1,2681,117-1,1591,054-1,10223371,268-1,3441,159-1,2011,102-1,15131181,344-1,4191,201-1,2431,151-1,19902091,419-1,4951,243-1,2861,199-1,248000101,495-1,5701,286-1,3281,248-1,296111

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

LG1 \u003d Kf.mIN.;

Pg.1 \u003d Kf.mIN.+ H;

LG2 \u003d Pg.1;

Pg.2 \u003d LG.2 + H;

Pg.10 \u003d Kf.max.

Data on the frequency of focus of financial stability (discrete random variables x, y, z) is grouped into the intervals, and the likelihood of their values \u200b\u200bto the specified boundaries is determined. At the same time, the left value of the border is included in the interval, and the right - no (Table 3).

Table 3.

Distribution of discrete random variables x, y, z

Indicator indicators "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

In the frequency of appearance of values n.their probabilities are found (the frequency of appearance is divided into 12, based on the number of units of the aggregate), as well as the meanings of the discrete random variables, the mid-intervals were used. Laws of their distribution:

P.i.\u003d N.i. /12;

X.i.\u003d (LG.i.+ Pg.i.)/2.

Based on the distribution, one can judge the likelihood of the unstable 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 values \u200b\u200bof the financial stability coefficient, equal to 0.853. In other words, the likelihood that its costs exceed income is 8.3%. By the bank "B" the probability of falling the coefficient below the unit also amounted to 0.083, however, taking into account the dynamic development of the organization, this decrease will be insignificant - up to 0.926. Finally, a high probability (16.7%) is high, that the activities of the Bank "C", with other things being equal, is characterized by the value of financial stability equal to 0.835.

At the same time, on distribution tables, you can see the probability of sustainable development of banks, i.e. The amount of probabilities, where the options for coefficients are important, greater than 1:

P (x\u003e 1) \u003d 1 - p (x<1) = 1 - 0,083 = 0,917

P (Y\u003e 1) \u003d 1 - P (y<1) = 1 - 0,083 = 0,917

P (Z\u003e 1) \u003d 1 - P (z<1) = 1 - 0,167 = 0,833.

It can be observed that the least sustainable development is expected to be in the bank "C".

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

The mathematical expectation of the discrete random variable is called the amount of works of all possible values \u200b\u200bon its probability:

M (x) \u003d x1 p.1 + X.2 p.2 + ... + xn.p.n.

The results of calculations of the values \u200b\u200bof mathematical expectations of random variables are presented in Table 4.

Table 4.

Numeric characteristics of discrete random variables x, y, z

BankMathematical Explanation External Quadratic Deviation"A" m (x) \u003d 1,187d (x) \u003d 0.027 ?(x) \u003d 0.164 "in" m (y) \u003d 1,124d (y) \u003d 0.010 ?(y) \u003d 0.101 "C" m (z) \u003d 1,037d (z) \u003d 0.012? (z) \u003d 0,112

The resulting mathematical expectations make it possible to estimate the average values \u200b\u200bof the expected probable values \u200b\u200bof the financial stability coefficient in the future.

So according to the calculations, it can be judged that the mathematical expectation of the sustainable development of the bank "A" is 1.187. The mathematical expectation of banks "B" and "C" is 1,124 and 1.037, respectively, reflecting the estimated profitability of their work.

However, knowing only the mathematical expectation showing the "center" of the estimated possible values \u200b\u200bof the random variable - KFU, it is also impossible to judge its possible levels or the degree of their scattering around the resulting mathematical expectation.

In other words, the mathematical expectation due to its nature is not fully sustainable the development of the bank. For this reason, there is a need to calculate other numeric characteristics: dispersion and rms deviation. Which make it possible to estimate the degree of absentness of possible values \u200b\u200bof the financial stability coefficient. Mathematical expectations and average quadratic deviations allow you to assess the interval in which the possible values \u200b\u200bof the financial sustainability of credit organizations will be.

With a relatively high characteristic value of the mathematical expectation of sustainability by the Bank "A", the average quadratic deviation was 0.164, which indicates that the stability of the bank can either increase by this value or decrease. With a negative change in stability (which is still unlikely, considering the obtained probability of unprofitable activity, equal to 0.083) The coefficient of financial stability of the bank will remain positive - 1, 023 (see Table 3)

The activity of the Bank "B" with a mathematical expectation in 1.124 is characterized by a smaller difference of the ratio of the coefficient. So, even with an unfavorable coincidence, the Bank will remain stable, since the average quadratic deviation from the predicted value was 0, 101, which will allow it to remain in the positive zone of profitability. Consequently, it can be concluded about the stability of the development of this bank.

Bank "C", on the contrary, with a low mathematical expectation of its reliability (1, 037), will face with other things being equal to an unacceptable deviation for it equal to 0.112. With an unfavorable situation, as well as given the high percentage of the probability of unprofitable activity (16.7%), this credit organization is likely to reduce its financial stability to 0.925.

It is important to note that by making conclusions about the sustainability of bank development, it is impossible to pre-confidently anticipate which of the possible values \u200b\u200bwill receive a financial stability coefficient as a result of testing; It depends on many reasons to take into account that is impossible. From this position, we have very modest information about each accidental value. In connection with which it is hardly possible to establish the 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 is almost lost and becomes natural.

Evaluating the stability of bank development, it remains to estimate the likelihood that the deviation of a random variable from its mathematical expectation does not exceed the absolute value of the positive number ?. To give an estimate that interests us allows the inequality of P.L. Chebyshev. The likelihood 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 the inverse probability:

Given the risk associated with the loss of sustainability, we will evaluate the likelihood of deviating a discrete random variable from the mathematical expectation in a smaller side and, considering the equally accurate deviations from the central value both to smaller and on the major sides, rewrite inequality again:

Next, based on the task, it is necessary to estimate the likelihood that the future value of the financial stability coefficient will not be below 1 of the proposed mathematical expectation (for the Bank "A" ? We will take equal to 0.187, for the bank "B" - 0.124, for "C" - 0.037) and make calculation of this probability:

bank "A":

bank "C":

According to the inequality P.L. Chebyshev, the most sustainable in its development is the Bank "B", since the probability of deviating the expected values \u200b\u200bof the random variable from its mathematical expectation is low (0.325), while it is relatively fewer than by other banks. In second place on comparative stability of development, the bank "A" is located, where the coefficient of this deflection is somewhat higher than in the first case (0.386). In the third bank, the likelihood that the value of the financial sustainability coefficient to deviate on the left side of the mathematical expectation of more than 0, 037 is a practically reliable event. Especially, if we consider that the probability cannot be greater than 1, exceeding values, according to the proof of L.P. Chebyshev, must be taken over 1. In other words, the fact that the development of the bank can move to an unstable zone characterized by a financial stability coefficient less than 1 is a reliable event.

Thus, describing the financial development of commercial banks, the following conclusions can be drawn: the mathematical expectation of the discrete random variable (the average expected value of the financial stability coefficient) of the bank "A" is 1.187. The average quadratic deviation of this discrete value is 0.164, which objectively characterizes a small variation of the values \u200b\u200bof the coefficient of the average. However, the degree of instability of this series is confirmed by a sufficiently high probability of negative deviation of the coefficient of financial stability from 1, equal to 0.386.

Analysis of the activities of the second bank showed that the mathematical expectation of the KFU is 1.124 with an average quadratic deviation of 0.101. Thus, the activities of the credit institution is characterized by a small variation of the values \u200b\u200bof the financial stability coefficient, i.e. It is more concentrated and stable, which is confirmed by a relatively low probability (0.325) of the bank transition to the unprofitability zone.

The stability of the bank "C" is characterized by a low meaning of mathematical expectation (1.037) and also a small variation of values \u200b\u200b(the 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 1, i.e. Waiting for the positive dynamics of its development, with other things being equal, it will look very unreasonable. Thus, the proposed model based on the determination of the existing distribution of discrete random variables (values \u200b\u200bof the financial stability coefficients of commercial banks) and confirmed by the assessment of their equilibrium positive or negative deviation from the resulting mathematical expectation, it allows its current and promising level.

Conclusion

The use of mathematics in economic science, gave impetus in the development of both the most economics and applied mathematics, in terms of the methods of an economic and mathematical model. The proverb says: "Some seven times - a rejection once." The use of models has time, strength, material means. In addition, the calculations on the models are opposed to volitional solutions, since they allow pre-evaluating the consequences of each decision, to discard invalid options and recommend the most successful. Economic and mathematical modeling is based on the principle of analogy, i.e. Opportunities to study the object by building and considering another, similar to it, but a simpler and accessible object, its model.

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

In the work it was found that economic and mathematical models can be divided by signs:

· target;

· accounting of the time factor;

· the duration of the period under consideration;

· goals of creating and use;

· accounting of the uncertainty factor;

· such as a mathematical apparatus;

A 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 apply at all levels of management.

Economic and mathematical methods are especially important, as information technologies are implemented in all fields of practice. The main stages of the modeling process, namely:

· statement of economic problem and its qualitative analysis;

· construction of a mathematical model;

· mathematical analysis of the model;

· preparation of source information;

· numerical solution;

· analysis of numerical results and their application.

The article contains an article of a candidate of economic sciences, associate professor of the Department of Finance and Credit S.V. Boyko, in which it is noted that the domestic credit institutions subject to the influence of the external environment are the task of finding management instruments involving 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 adequate determination of financial stability through various methods and models, one of whose species is stochastic (probabilistic) models, allowing not only to identify the proposed growth factors or reduce sustainability, but also to form a complex 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 at a given level of economic and mathematical knowledge that has specific information and computational technology. And although it is impossible to indicate the absolute boundaries of 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)Crass MS Mathematics for economic specialties: tutorial. -4-E ed., Act. - M.: Case, 2003.

)Ivanilov Yu.P., Lotov A.V. Mathematical models in the economy. - M.: Science, 2007.

)Ashmanov S.A. Introduction to the mathematical economy. - M.: Science, 1984.

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

)Ed. Fedoseeva V.V. Economic and Mathematical Methods and Applied Models: a tutorial for universities. - M.: Uniti, 2001.

)Savitskaya G.V. Economic analysis: textbook. - 10th ed., Act. - M.: New Knowledge, 2004.

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

)Operations research. Tasks, principles, methodology: studies. Handbook for universities / E.S. Ventcel. - 4th ed., Stereotype. - M.: Drop, 2006. - 206, p. : IL.

) Mathematics in the economy: Tutorial / S.V. Yudin. - M.: Publishing House of RGTEU, 2009.-228 p.

)Kochetkov A.A. Probability theory and mathematical statistics: studies. Benefit / Tul. State Un-t. Tula, 1998. 200c.

)Boyko S.V., probabilistic models in assessing the financial sustainability of credit organizations. Boyko // Finance and Credit. - 2011. N 39. -

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Introduction

1. Mathematical Modeling in Economics

1.1 Development of methods modeling

1.2 Simulation of the catalog of scientific knowledge

1.3 Economic and mathematics and models

Conclusion

Literature

Introduction

The doctrine of the similarity and modeling began to create more than 400 years ago. In the middle of the XV century. The substantiation of the methods was simulated by Leonardo da Vinci: he made an attempt to withdraw common patterns, used a mechanical and geometric similarity when analyzing situations in the examples considered by them. He used the concept of an analogy and addictive to the need for experimental verification of the results of similar examinations, the importance of experience, the ratio of experience and theory, their role in knowledge.

The ideas of Leonardo da Vinci about the mechanical similarity in the XVII century developed Galilee, they used to build a gallery in Venice.

In 1679, Mariott used the theory of mechanical similarity of lifestyment about the disadvantaged bodies.

The first strict scientific formulations of the conditions of similarity and the clarification concept of similarity were given at the end of the 18th century I. Newton in the "mathematical fines of natural philosophy".

In 1775-76 I.P. Kulibin used a static appliance in experiments with the model of the bridge through the Neva Poly 300 m. Models Ilover, in 1/10 of the natural value and weighing over 5 tons. The calculations of the grinding were checked and approved by L. Euler.

1. Mathematical modeling in the economy

1.1 Development of modeling methods

The successes of mathematics stimulated the use of formalized methods and in unconventional spheres of science and practice. So, O. Kourno (1801-1877) introduced the concept of demand and suggestions, and earlier German economist I.G. Tunene (1783-1850) began to apply mathematical methods in the economy and proposed the theory of production, anticipating the theory of labor productivity. For pionerars using the modeling method, F. Kene (1694-1774), the author of the "Economic Table" (Zigzag Kene) - one Of the first modified reproduction, the three-sector macroeconomic model of simplicity.

In 1871, Hyams Wenley Jevons (1835-1882) published the "theoria-absorbing economy", where he outlined the theory of utmost utility. The ability to satisfy the needs of a person underlying goods and prices is understood. Jevonsov distinguished:

- Abstract utility that is deprived of a specific form;

- usefulness in general as the pleasure of man consistent with the consignment;

- Maximum utility - the smallest utility among all the whole.

Almost at the same time (1874) with the work of the javelized work "Elements of pure political economy" Leon Valras (1834-1910), in which he set the task of finding such a price system, with several demand for all goods and markets would be equal to the cumulative proposal. Valrasa pricing factors are:

Costs produce;

Faolescence of good;

Ask the proposal of the goods;

Exposure to the price of this product of the entire price system
The rest of the goods.

End of the XIX - the beginning of the 20th century was marked by widespread use in the economy. In the XX century Mathematical methods are simulating used so much widely that almost all works received by the Nobel premium economy are associated with their use (D. Hicks, R. Solow, V. Leontiev, P. Samuelson, L. Kantorovich, etc.). The development of substantive disciplines of science and practice is due to the increasing level of formalization, intellectualization and use of computers. Not complete list of academic disciplines and their partitions include: functions and graphs of functions, differential and integral calculation, functions of many variables, analyticalgeometry, linear spaces, multidimensional spaces, linear algebra, statistical methods, matrix calculus, logic, graph theory, game theory, theory Usefulness, optimization methods, theory of schedules, research projects, theory of mass service, mathematical programming, dynamic, nonlinear, integer and stochastic programming, network methods, Monte Carlo method (statistical test method), reliability methods, random processes, Markov chains, modeling theory HAPPY.

Formalized simplified descriptions of economic phenomena are foreign models. The models are used to detect the messenger factors of phenomena and the processes of the functioning of economic objects, to prepare the forecast of the possible effects of the impact of the accuracy objects and systems, for various estimates and the use of ectogencies in management.

The construction of the model is carried out as the implementation of the following steps:

a) formulating the purpose of the study;

b) a description of the subject of research in generally accepted terms;

c) analysis of the structure of famous objects and connections;

d) a description of the properties of objects and the nature and quality of ties;

e) evaluation of relative weights of objects and a bonding method;

(e) Building a system of the most important elements in verbal, graphic or symbolic form;

g) collecting the necessary data and verify the accuracy of modeling results;

and) analysis of the structure of the model for the adequacy of the present phenomenon and the introduction of the adjustment; Analysis of the provision of initial information and planning or additional studies for possible replacement data by other, or special experiments to obtain missing.

Mathematical models used in the economy can be divided classes depending on the characteristics of the simulated objects, targets and methods.

Macroeconomic models are designed to describe the economy of the Cycle. The main characteristics used in the analysis, are the consumption, consumption, investment, employment, amount of money, etc.

Microeconomic models describe the interaction of structural ideological components of the economy or behavior of one of the components of the rest. The main objects of the simulation application in microeconomics are etopling, demand, elasticity, cost, production, competition, consumer choice, pricing, monopoly theory, firm theory, etc.

According to the nature of the model, there may be theoretical (abstract), applied, static, dynamic, deterministic, stochastic, equilibrium, optimization, natural, physical.

Theoretical models Allow the general properties of the economy based on formal prerequisites using the deduction method.

Applied models Allow the parameters of the functioning-economic object. They operate with numerical knowledge of economic permanent. Most often in these models use statistical or actual-stuck data.

Equilibrium models Describe such a state of the economy as a system in which the sum of all forces acting on it is zero.

Optimization models Operated with the concept of maximizing utility, the result of which is the choice of behavior in which the state of the micro level is preserved.

Static models Describe an instantaneous state of an economic object or phenomenon.

Dynamic model Describes the state of the object as a function of time.

Stochastic models Take into account the random effects on economic accomitics and use the apparatus of probability theory.

Deterministic models It is assumed between the studied accelerations of the functional communication and, as a rule, use apparathydifferential equations.

Washing modeling It is carried out on the actual objects of anciently selected conditions, for example, an experiment conducted at the time of production process at the current enterprise that is responsible for the production of the industry itself. The method of one-time study arose as an absreteability of material production when science did not exist. It coexists on a par with the natural science experiment and at present, demonstrating the unity of theory and practice. A variation of attendant simulation modeling by generalizing production experience. The difference between the state that, instead of a specially educated in production conditions, the existing material is used, processing it in the corresponding crirological ratios using the similarity theory.

The concept of the model always requires the introduction of the concept of similarity, which is determined as mutually unambiguous correspondence between objects. Function-input from the parameters characterizing one of the objects to the parameters characterizing another object is known.

The model ensures the similarity of only those processes that cause the similarity criteria.

The theory of similarity is applied at:

a) finding analytical dependencies, the ratios of the inflation of specific tasks;

b) processing the results of experimental studies in cases where the results are presented in the form of generalized criteria and dependentations;

c) creating models that reproduce objects or phenomena in smaller saves, or by complexity differing from the source.

In physical modeling, the study is carried out by the login on the physical selection, i.e. When mainly preserved the phenomenon. For example, communications in economic systems simulated electrical circuit / network. Physical modeling can be temporary, the phenomena flowing only in time; the spatial-temporal - when non-stationary phenomena, distributed over time and space, are studied; Spatial, or object - there are once equilibrium states, independent of other objects or time.

The processes are considered similar if there are compliance values \u200b\u200bof the systems under consideration: sizes, parameters, IDR positions.

The patterns of similarities are formulated in the form of two theorems that establish relations between the parameters of such phenomena, not specifying the implementation of the sale of similarity when building models. The third, or inverse referred to defines the necessary and sufficient conditions for the similarity of phenomena, requiring the help of the definition conditions (allocating this process from the diversity of diversity) and such a selection of parameters in which the criteria for similarity containing initial and boundary conditions become the same.

First Theorem.

Similar in the same sense of the phenomenon have the same combinations of parameters.

Indexed combinations of parameters, numerically the same for all-like processes, are called similarity criteria.

Second theorem.

Any complete equation of the process recorded in the definition of units may be represented by the dependence between the similarity criteria, i.e. the escape that binds the dimensionless values \u200b\u200bderived from the parameters participating.

Dependence is complete, if you take into account all the links by the values \u200b\u200bthat are beyond it. Such a dependence cannot change with the change in the measurement of physical quantities.

Third Theorem.

For the similarity of phenomena should be accordingly the same identifying criteria for similarity and similar conditions of unambiguity.

Under the defining parameters, it is understood as criteria containing the parameters of processes and systems that in this problem can be considered independent (time, capital, resources, etc.); Under the conditions, the parameter group is unambiguously. The values \u200b\u200bof which specified in the form of functional and numbers or numbers are allocated from the possible diversity of phenomena specifies.

The similarity of complex systems consisting of several subsystems, similarity, is ensured by the similarity of all similar elements that are for subsystems.

The similarity of nonlinear systems is preserved if the calculation of the relative characteristics of the similar parameters, which are nonlinearly variable.

Similarity of inhomogeneous systems. The approach to establishing the conditions of similarity systems is the same as an approach to nonlinear systems.

Similarity at the probabilistic nature of studied phenomena. All similarity theorems relating to deterministic systems are permanent, subject to the coincidence of the probability densities of the similar parameters presented in the form of relative characteristics. In this case, the dispersion of immature expectations of all parameters, taking into account the scale, should be likely to be the same. An additional condition for the similarity is to fulfill the requirements of the physical realizability of similar correlation and interrestorally specified parameters included in the definition condition.

There are two ways to identify similarity criteria:

a) bringing the process equations to a dimensionless form;

b) Using the parameters describing the process when the process equation is unknown.

In practice, also enjoy another way of relatively orders that are modifications of the first two. In this case, all parameters are expressed by shares from certainly selected basic values. The most important parameters expressed in the shares of the basic can be considered as criteria of similarity acting in specific conditions.

Thus, the economic and mathematical models and methods are not enough to obtain economic patterns, but also a widely used toolkit of practical solving problems in managing, predicting, business, banking and other sections of the economy.

1.2 Modeling as a method of scientific knowledge

Scientific research is the process of developing new knowledge, one of the types of cognitive activity. For research exploits, various methods are used, one of which is the current, i.e. The study of any phenomenon, process or system objects by building and examining its models. Modeling means also use of models to determine or clarify the characteristics of the methods of constructing newly designed objects.

"Modeling is one of the main categories of the theory of knowledge; The naided simulation is essentially any method of scientific knowledge of both both the experimental cognition. " Modeling began to apply some scientific studies in ancient times and gradually covered the greatest and new areas of scientific knowledge: technical design, construction, architecture, astronomy, physics, chemistry, biology and finally social sciences. It should be noted that the modeling methodology long ago developed in relation to specific sciences, regardless of the other. In these conditions, there was no uniform system of knowledge, terminology. Then began to be the role of modeling as a universal method of scientific knowledge, in a bit of a gnoseological category. However, it is necessary to clearly understand what formulation is a method of indirect knowledge with a certain tool - a model that is placed between the researcher and object examination. Modeling is used either when the object cannot be impossible directly (land core, solar system, etc.), or when the object does not yet exist (the future state of the economy, future demand, expected proposal, etc.), or when the study requires A lot of disruption time, or finally, to test various hypotheses. The simulation is only part of the general process of knowledge. Currently there are many different definitions and classifications of models as applied by the pile of different sciences. We accept the definition given by the economist VS Nemchinov, famous, in particular, by the development of models of planned economy: "The model is a means of allocating any objectively active system of monomer bonds and relations that occur in the study of the realization."

The main requirement for models is the identity of real reality, although the model and reproduces the studied object or the process in a simplified form. When building any model, a complex task is to build a complication: on the one hand, simplifying, throwing all the minor to focus on the inheritance features of the object, on the other hand, not to simplify to the such level to weaken the connection of the model with real reality. AmericanMatimatik R. Bellman figuratively described such a task as "WestureProves and Marsh Rooting".

In the process of scientific research, the model can work in two-plating: from the observations of the real world to theory and back; Those., from one meward, the construction of the model is an important step to the creation of theory, on the other, one of the means of experimental research. Depending on modeling elections, model and abstract (iconic) models are allocated. Material (physical) models are widely used in the technique, architecture of the Idrogiy regions. They are based on obtaining a physical image of the study paper or process. Abstract models are not associated with the construction of physical formations. They are some intermediate link between abstractteoretic thinking and real reality. The abstract models (their names) include numerical (mathematical expressions by cycling numerical characteristics), logical (block diagrams of algorithm for computers on computer, graphics, diagrams, drawings). Models, with the constructions of which the purpose of determining this: the state of the object, which is the best from the point of view of a certain criterion, are called regulatory. Models intended for explaining the observed facts or forecasting the object of the object are called descriptive.

The effectiveness of the use of models is determined by the scientific relations of their prerequisites, the ability of the researcher to highlight the material specifications of the modeling object, select the source information, interpret in relation to the system obtained the results of numerical accounts.

1.3 Economic and Mathematical Methods and Models

Like any modeling, an economic and mathematical is simulated at the principle of analogy, i.e. The possibilities of studying the facility of the construction and consideration of another, similar to it, but more simple an affordable object, its model.

The practical tasks of economic and mathematical modeling, firstly, the analysis of economic objects; secondly, economic planning, the foresight of the development of economic processes and behavior of the information indicators; Thirdly, the development of management solutions on all levels of management.

A description of economic processes and phenomena in video economic and mathematical models is based on the use of one isetheromic and mathematical methods. The generalizing name of a complex of economic imathematic disciplines - economic and mathematical methods - introduced at the beginning of 60-hodors Academician V.S. Nemchinov. With a known proportion of the classification of methods, methods can be represented as follows.

1. Economic and statistical methods:

· Economicalstabe;

· math statistics;

· Multifactorial analyze.

2. Econometrics:

· Macroeconomic model;

· The theory of productive functions

· Interdisciplinarybalances;

· National;

· Analysis and consumption;

· Globiculture.

3. Study of operations (methods for making optimal solutions):

· Mathematical programming;

· Network management planning;

· Theoryamas maintenance;

· game theory;

· Theory;

· Methodification of economic processes in industries and enterprises.

4. Economic cybernetics:

· Systemanalysis of the economy;

· Theoryaeconomic information.

5. Methods of experimental study of economic phenomena:

· MethodsMisty imitation;

· Business game;

· Methodologic Economic Experiment.

In economic and mathematical methods, various sections are applied, mathematical statistics, mathematical logic. Computational mathematics, theoriyalgorithms and other disciplines are played by a major role of the loss of economic and mathematical problems. Using the mathematical apparatus Briginal results in solving problems of analyzing the processes of expanded production, matrix modeling, determining the optimal rates of rostapplies, optimal placement, specialization and concentrations of production, the tasks of selecting optimal production methods, determining the production sequence of launch in production, optimal options for industrial materials and compilation of mixtures, preparation tasks by methods Network planning and many others.

To solve the standard problems, the clarity of the target is characterized, the ability to develop procedures and rules for making calculations in advance.

There are the following prerequisites for using methodopicomico-mathematical modeling.

The most important of them are, firstly, a high level of knowledge of the economic theory, economic processes and phenomena, methodology of the sophisticated analysis; Secondly, a high level of mathematical training, ownership of economic and mathematical methods.

Before proceeding to developing models, it is necessary to thoroughly analyze the situation, identify goals and relationships, problems, demanding, and initial data for solving them, enter the system of designations, and only it is possible to describe the situation in the form of mathematical relationships.

Conclusion

A characteristic feature of scientific and technological progress in developed by the development of the role of economic science is. The economy is put forward by a vicinary plan precisely because it determines the effectiveness of the priority of the directions of scientific and technological progress reveals the wiper of the implementation of cost-effective achievements.

The use of mathematics in economic science, gave the impetus to the body of both the most economics and applied mathematics, in the parties of the economic and mathematical model. The proverb says: "Seven times will die - graduate". The use of models there is time, strength, material means. In addition, the calculations on models are opposed to volitional solutions, since the consequences of each solution are pre-assessing the consequences of each decision, to discard unacceptablely analysts and recommend the most successful.

At all levels of management, all sectors use methods of monomic and mathematical modeling. We highlight the following areas of practical application, for which a large economic effect is obtained.

The first direction is forecasting and promising planning. The rates and proportions of the economic development of the economy are carried out, on their basis, the definitions and factors of the growth of national income, its distribution to the consumption of costs and so on. An important point is the use ofconomic and mathematical methods not only in drawing up plans, but also in the dealerical guidelines for their implementation.

The second direction is the development of models that are used to make the coordination and optimization of planned solutions, in particular, the modest and interregional balance sheets of production and distribution of products. According to the economic content and nature of the information, the balancerupt and natural-product products, each of which may be reportable iplanov.

The third direction is the use of economic and mathematical models on the sectoral level (the implementation of the calculations of the optimal plans of the industry, analysis with the help of production functions, predicting the main production proportions of the industry). To solve the problem of placing the use of enterprise, optimal attachment to suppliers or DRIVERs, etc. The models of optimization of two types are used: in some of the production volume of production, it is required to find an option to be implemented with the smallest costs ", in others it is necessary to determine the scale and product structure in other products in order to obtain a maximum effect. It includes the calculation of the calculations from the statistical models of kdynamic and from statistical models to the dynamic and modeling industries to optimize multi-sectoral complexes. If I used to create a unified model of the industry, then the use of models complexes, mutually related both vertically and horizontally, is now the most promising.

The fourth direction is an economic and mathematical modeling and operational planning of industrial, construction, transportation ideologies, enterprises and firms. The field of practical applications is also included by the units of agriculture, trade, communications, health care, the protection of nature, etc. In mechanical engineering, the bulky of various models is used, the most "debugged" of which are optimizing, allowing to determine the production programs and the most pricing options for the use of resources, distribute the production program in time and efficiently organize the work of intra-water supply, significantly improve the loading of the equipment and reasonably organize product controls, etc.

The fifth direction is territorial modeling, the beginning of which was the development of reporting inter-sectoral balances of some regions in the late 50s.

As the sixth direction, it is possible to exclude economic and mathematical modeling of logistics, including optimization of transport and economic bonds and levels of stocks.

The seventh direction includes models of the functional block economic system: the movement of the population, training of personnel, formation of themselves and demand for consumer benefits, etc.

Economic and mathematical methods are especially important to introduce information technologies in all areas of practice.

Literature

1. Ventcel E.S. Operations research. - M: Soviet radio, 1972.

2. Syrilov A.A. Create the best solution in real conditions. - M.: Radio and Communication, 1991.

3. Kantorovich L.V. Economic and the best use of resources. - M.: Science, Academy of Sciences of the USSR, 1960.

4. Cofman A., debases of network planning methods and their application. - M.: Progress, 1968.

5. Cofman A., Form R. I will deal with the study of operations. - M.: Mir, 1966.

The model is, first of all, a simplified representation of a real object or phenomenon that maintains its main, essential features. The process of developing a model itself, i.e. Modeling can be implemented in various ways, of which physical and mathematical modeling is most common. However, each of these methods can be obtained by various models, since their specific implementation depends on which features of the real object The creator of the model considers the main, the main. Therefore, in engineering practice and scientific research, various models of the same object can be applied, since their diversity allows you to carefully study the most different aspects of a real object or phenomenon.

In engineering practices and natural sciences, physical models are widespread, which differ from the object being studied, as a rule, smaller than the sizes, and serve to conduct experiments, the results of which are used to study the source object and for the conclusions about the choice of one or another development of its development or design, If we are talking about the project of an engineering structure. The path of physical modeling turns out to be unproductive for analyzing economic objects and phenomena. Concerning the main method of modeling in the economy is the method of mathematical modeling . Description of the main features of the real process using the system of mathematical formulas.

How do we act, creating a mathematical model? What are the mathematical models? What features occur when modeling economic phenomena? We will try to clarify these questions.

When creating a mathematical model, proceed from the real task. Initially, the situation understands, important and secondary characteristics, parameters, properties, quality, communications, etc. are detected. Then one of the existing mathematical models is selected or a new mathematical model is created to describe the object being studied.

Designations are introduced. Restricted restrictions to which variables must satisfy. The target is determined - the target function is selected (if possible). Not always the choice of the target function is unequivocal. There are situations when I want it, and this, and more than many other things ... But various goals lead to various solutions. In this case, the task refers to the class of multicriterial tasks.

The economy is one of the most complicated areas of activity. Economic objects can be described by hundreds, thousands of parameters, many of which are random. In addition, the economy has a human factor.

The person's behavior is difficult to predict, it is sometimes impossible.

The complexity of the system of any nature (technical, biological, social, economic) is determined by the number of elements included in it, connections between

these elements, as well as relationships between the system and the medium. The economy has all the signs of a very complex system. It combines a huge number of elements, is distinguished by the diversity of internal connections and connections with other systems (natural environment, economic activities of other subjects, social relations, etc.). Natural, technological, social processes, objective and subjective factors interact in the national economy. The economy depends on the social structure of society, from politics and from many and many factors.

The complexity of economic relations was often justified by the impossibility of modeling the economy, studying its means of mathematics. And yet the modeling of economic phenomena, objects, processes is possible. You can simulate an object of any nature and any complexity. For the modeling of the economy, not one model is used, but the system of models. This system has models describing different parties to the economy. There are models of the country's economy (they are called macroeconomic), there are models of economic models on a separate enterprise or even a model of one economic event (they are called microeconomic). When drafting the model of the economy of a complex object, the so-called aggregation is produced. In this case, a number of related parameters are combined into one parameter, thereby the total number of parameters is reduced. At this stage, experience and intuition play an important role. As parameters, you can choose not all characteristics, but the most important.

After the mathematical task is drawn up, the method of solving it is selected. At this stage, as a rule, a computer is used. After obtaining the solution, it is compared with reality. If the results obtained are confirmed by practice, the model can be used and to build forecasts. If the answers received on the basis of the model do not correspond to reality, the model is not suitable. It is necessary to create a more complex model that is better consistent with the object being studied.

Which model is better: simple or complicated? The answer to this question cannot be unequivocal.

If the model is too simple, it does not correspond to the real object. If the model is too complicated, it may be so that with the existence of a good model, we are not able to receive an answer based on it. There may be a good model and there is an algorithm for solving the corresponding task. But the decision time will be so big that all other advantages of the model will be crossed out. Therefore, when choosing a model, the Golden Middle is needed.

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Introduction

Modeling in scientific research began to be applied in deep antiquity and gradually excited all new areas of scientific knowledge: technical design, construction and architecture, astronomy, physics, chemistry, biology and, finally, social sciences. Big successes and recognition in almost all branches of modern science brought the method of modeling the XX century. However, the modeling methodology has long developed independently by individual sciences. There was no uniform system of concepts, single terminology. Only gradually began to be aware of the role of modeling as a universal method of scientific knowledge.

The term "model" is widely used in various spheres of human activity and has many semantic values. Consider only such "models", which are tools for obtaining knowledge.

The model is such a material or mentally represented object, which in the process of study replaces the original object so that its direct study gives new knowledge about the original object.

Under the simulation means the process of building, studying and applying models. It is closely related to such categories as abstraction, analogy, hypothesis, etc. The simulation process necessarily includes the construction of abstractions, and conclusions by analogy, and the design of scientific hypotheses.

The main feature of modeling is that it is a method of mediated knowledge with the help of deputy objects. The model acts as a peculiar instrument of knowledge that the researcher puts each other and the object and with which it is exploring the object of interest. It is this feature of the modeling method that determines the specific forms of the use of abstractions, analogies, hypotheses, other categories and methods of knowledge.

The need to use the modeling method is determined by the fact that many objects (or problems relating to these objects) directly investigate or not at all, or this study requires a lot of time and means.

The simulation process includes three elements: 1) the subject (researcher), 2) the object of study, 3) a model that mediate the relationship of a learning entity and a knowledgeable object.

Let it be either necessary to create some object A. We design (materially or mentally) or we find another object in the real world in - the model of the object A. Stage construction of the model involves the presence of some knowledge about the original object. Cognitive capabilities of the model are determined by the fact that the model reflects any essential features of the original object. The question of the need and sufficiently similarity of the original and the model requires a specific analysis. Obviously, the model loses its meaning as in the case of identity with the original (then it ceases to be the original) and in the case of excessive differences from the original in all significant relationships.

Thus, the study of the same sides of the simulated object is carried out by the price of refusal to reflect other parties. Therefore, any model replaces the original only in a strictly limited sense. It follows from this that several "specialized" models can be built for one object, focusing on certain sides of the object under study or the object characterizing the object with different degrees of detail.

At the second stage of the modeling process, the model acts as an independent object of research. One of the forms of such a study is the conduct of "model" experiments, in which the conditions for the functioning of the model consciously change and data on its "behavior" is systematized. The end result of this stage is a lot of knowledge about the model R.

In the third stage, knowledge is transferred from the model to the original - the formation of a multitude of knowledge S about the object. This knowledge transfer process is carried out according to certain rules. Knowledge of the model must be adjusted taking into account those properties of the original object that have not found reflections or have been changed when building a model. We can carry out any result from the model to the original, if this result is required to be related to the signs of the similarity of the original and the model. If a certain result of the model study is associated with the difference between the model from the original, then this result is wrong with it.

The fourth stage is a practical verification of the knowledge models and their use to build a generalizing theory of the object, its transformation or management of them.

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

Modeling is a cyclic process. This means that the second, third, etc. may follow the first four-stage cycle. At the same time, the knowledge of the test object is expanded and refined, and the original model is gradually improved. Disadvantages found after the first modeling cycle, due to small knowledge of the object and errors in the construction of the model, can be corrected in subsequent cycles. In the modeling methodology, therefore, large capabilities of self-development are laid.

1. Features of the application of the Mathematic methodskogo modeling in the economy

Penetration of mathematics in economic science is associated with overcoming significant difficulties. This was partly a "guy" mathematics, developing over several centuries, mainly due to the needs of physics and technology. But the main reasons are still in the nature of economic processes, in the specifics of economic science.

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

The most common understanding of the system as a set of elements in the interaction and forming some integrity, unity is. An important quality of any system is Emergenity - the presence of such properties that are not inherent in any of the elements included in the system. Therefore, when studying systems, it is not enough to use the method of their dismemberment to elements, followed by the study of these elements separately. One of the difficulties of economic research is that there are almost no economic objects that could be considered as separate (non-system) elements.

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

The complexity of the economy was sometimes viewed as a substantiation of the impossibility of its modeling, studying mathematics. But this point of view is in principle incorrect. You can simulate an object of any nature and any complexity. And just complex objects are the greatest interest for modeling; It is here that modeling can give results that cannot be obtained by other research methods.

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

2. E. classificationcondomico-mathematical models

Mathematical models of economic processes and phenomena can more briefly be called economic and mathematical models. To classify these models, different bases are used.

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

Economic and mathematical models can be intended to study different parties to the national economy (in particular, its production and technological, social, territorial structures) and its individual parts. When classifying models on the studied economic processes and meaningful issues, models of the national economy as a whole and its subsystems - industries, regions, etc., complexes of production models, consumption, formation and distribution of income, labor resources, pricing, financial ties, and T ..

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

In accordance with the general classification of mathematical models, they are divided into functional and structural, and also include intermediate forms (structural-functional). In research at the national economic level, structural models are more often used, since the relationship between the subsystems have great importance for planning and management. Typical structural models are models of intersectoral ties. Functional models are widely used in economic regulation, when the behavior of the object ("output") is affected by changing the "login". An example is the model of consumer behavior in the conditions of commodity-monetary relations. The same object can be described simultaneously both by the structure, and the functional model. For example, for planning a separate sectoral system, a structural model is used, and at the national economic level, each industry can be represented by the functional model.

Above the differences between models are descriptive and regulatory. Discipriptive models answer the question: how does this happen? Or how does it most likely can further develop?, i.e. They only explain the observed facts or give a likely forecast. Regulatory models are answering the question: how should it be?, I.e. suggest targeted activities. A typical example of regulatory models are models of optimal planning, formalizing in one way or another method of economic development, opportunities and means of their achievement.

The use of a descriptive approach in the modeling of the economy is explained by the need to empirically identify various dependencies in the economy, the establishment of statistical patterns of economic behavior of social groups, studying the likely ways to develop any processes under immutable conditions or external influences. Examples of descriptive models are production functions and customer demand functions based on the processing of statistical data.

Whether an economic and mathematical model is descriptive or regulatory, depends not only on its mathematical structure, but on the nature of the use of this model. For example, the model of intersectoral balance is descriptive if it is used to analyze the proportions of the last period. But the same mathematical model becomes regulatory when it is used for calculating the balanced options for the development of national economy, satisfying the ultimate needs of society under planned standards of production costs.

Many economic and mathematical models combine signs of descriptive and regulatory models. A typical situation where the regulatory model of a complex structure combines individual blocks that are private descriptive models. For example, an inter-sectoral model may include customer demand functions that describe consumer behavior when income changes. Such examples characterize the tendency of an effective combination of descriptive and regulatory approaches to modeling economic processes. A descriptive approach is widely used in imitation modeling.

By the nature of the reflection of causal relationships, the models are distinguished by rigidly deterministic and models that take into account the accident and uncertainty. It is necessary to distinguish the uncertainty described by probabilistic laws, and uncertainty, to describe which the laws of probability theory are not applicable. The second type of uncertainty is much more complicated for modeling.

According to methods of reflection of the time factor, economic and mathematical models are divided into static and dynamic. In static models, all dependencies refer to one point or time period. Dynamic models characterize changes in economic processes in time. According to the duration of the time period in question, models of short-term (up to year), medium-term (up to 5 years), long-term (10-15 or more years) forecasting and planning are distinguished. Self-time in economic and mathematical models may vary either continuously or discretely.

Models of economic processes are extremely diverse in the form of mathematical dependencies. It is especially important to allocate the class of linear models that are most convenient for analysis and computing and received a large distribution due to this. The differences between linear and nonlinear models are significant not only from a mathematical point of view, but also in economic economic relations, since many dependencies in the economy are fundamentally non-linear in nature: the efficiency of resource use with increasing production, the change in the demand and consumption of the population with increasing production, the change in demand and population consumption with income growth, etc. The theory of "linear economy" is significantly different from the theory of "nonlinear economy". From whether many production capabilities are assumed to subsystems (industries, enterprises) by convex or non-deputies, conclusions are substantially dependent on the possibility of a combination of centralized planning and economic independence of economic subsystems.

By the ratio of exogenous and endogenous variables included in the model, they can be divided into open and closed. Fully open models do not exist; The model must contain at least one endogenous variable. Fully closed economic and mathematical models, i.e. not including exogenous variables, extremely rare; Their construction requires full abstraction from the "medium", i.e. Serious degradation of real economic systems, always having external communications. The overwhelming majority of economic and mathematical models occupies an intermediate position and differ in the degree of openness (closedness).

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

Depending on whether national economic models include spatial factors and conditions or do not include, distinguish between spatial and point models.

Thus, the general classification of economic and mathematical models includes more than ten basic signs. With the development of economic and mathematical research, the problem of classification of the models used is complicated. Along with the advent of new types of models (especially mixed types) and new signs of their classification, the process of integrating models of different types in more complex model structures is carried out.

3 . Stages of economicso-mathematical modeling

The main stages of the simulation process were already considered above. In various branches, including in the economy, they acquire their specific features. We analyze the sequence and content of the stages of one cycle of economic and mathematical modeling.

1. Statement of the economic problem and its qualitative analysis. The main thing here is to clearly formulate the essence of the problem, the assumptions and the questions you want to get answers. This stage includes the allocation of the most important features and properties of the simulated object and abstraction from the secondary; study of the structure of the object and the main dependences 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 formalization of the economic problem, expressing it in the form of concrete mathematical dependencies and relations (functions, equations, inequalities, etc.). Usually, the main design (type) of the mathematical model is determined, and then the details of this design (a specific list of variables and parameters, a form of links) is specified. Thus, the construction of the model is divided in turn into several stages.

It is wrong to assume that the more facts take into account the model, the better "works" and gives the best results. The same can be said about such characteristics of the complexity of the model, as used forms of mathematical dependencies (linear and nonlinear), accounting for accidentability and uncertainty, etc. Excessive complexity and bulkiness of the model make it difficult to research the process. It is necessary to take into account not only the real possibilities of information and mathematical support, but also compare the costs of modeling with the resulting effect (with an increase in the complexity of the model, cost increases may exceed the effect of the effect).

One of the important features of mathematical models is the potential for their use for solving disabilities. Therefore, even facing a new economic task, you do not need to strive to "invent" the model; First you need to try to apply already known models to solve this problem.

In the process of building a model, the interconnection of two systems of scientific knowledge - economic and mathematical systems is carried out. Naturally strive to get a model belonging to a well-studied class of mathematical tasks. Often it can be done by some simplification of the original prerequisites of the model, not distorting essential features of the simulated object. However, this situation is also possible when the formalization of an economic problem leads to an unknown meathematic structure. The needs of economic science and practice in the middle of the twentieth century. Protected the development of mathematical programming, game theory, functional analysis, computational mathematics. It is likely that in the future the development of economic science will become an important stimulus to create new sections of mathematics.

3. Mathematical analysis of the model. The purpose of this stage is to find out the general properties of the model. It uses purely purely mathematical research techniques. The most important point is proof of the existence of solutions in a formulated model (existence theorem). If it is possible to prove that the mathematical task has no solution, then the need for subsequent work on the initial version of the model disappears; It should be adjusted either formulation of an economic task or methods for its mathematical formalization. In analytical research, the model, such issues, such as, for example, are the only solution, which variables (unknown) may be in solution, what will the relationship between them, in what limits and depending on what initial conditions they change, what are the trends in their changes and etc. Analytical study of the model compared with empirical (numerical) has the advantage that the resulting conclusions retain their strength at various specific values \u200b\u200bof the external and internal parameters of the model.

Knowledge of the general properties of the model has so important, often for the sake of evidence of such properties, the researchers consciously go to the idealization of the initial model. Nevertheless, the models of complex economic objects with great difficulty are amenable to analytical research. In cases where the analytical methods cannot find out the general properties of the model, and model simplifications lead to unacceptable results, switch to numerical research methods.

4. Preparation of source information. Modeling places strict information system requirements. At the same time, the real possibilities for obtaining information limit the choice of models intended for practical use. At the same time, not only the principal possibility of preparing information (for certain time) is taken into account, but also the costs of preparing relevant information arrays. These costs should not exceed the effect of using additional information.

In the process of preparing information, methods of the theory of probabilities, theoretical and mathematical statistics are widely used. With systemic economic and mathematical modeling, the initial information used in some models is the result of the operation of other models.

5. Numerical solution. This stage includes the development of algorithms for a numerical solution of the problem, drawing up programs for the computer and direct settlement. The difficulties of this stage are primarily due to the large dimension of econnomic problems, the need to process significant information arrays.

Typically, calculations on an economic and mathematical model are a multivariate character. Due to the high speed of modern computer, it is possible to carry out numerous "model" experiments, studying the "behavior" of the model with various changes in some conditions. The study conducted by numerical methods can significantly add the results of an analytical study, and for many models it is the only feasible. The class of economic tasks that can be solved with numerical methods is much wider than the class of tasks available to analytical research.

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

Mathematical testing methods can detect the incorrect construction of the model and thereby narrow out the class of potentially correct models. An informal analysis of theoretical conclusions and numerical results obtained through the model, comparing them with the existing knowledge and facts of reality, also make it possible to detect the shortcomings of the economic task of the designed mathematical model, its information and mathematical support.

The relationships of the stages. We will pay attention to the return links of the stages arising from the fact that in the process of research, the disadvantages of the preceding stages of modeling are found.

Already at the stage of building a model, it may be found out that the setting of the problem of contradictory or leads to a too complicated mathematical model. In accordance with this, the initial setting of the problem is adjusted. Next, a mathematical analysis of the model (step 3) may show that a small modification of the setting of the problem or its formalization gives an interesting analytical result.

Most often, the need to return to the preceding stages of modeling occurs in the preparation of the original inforiation (step 4). It may be found that the necessary information is missing or the cost of its preparation is too large. Then you have to return to the formulation of the problem and its formalization, changing them so as to adapt to the available information.

Since economic and mathematical tasks can be complex in their structure, have a greater dimension, it often happens that well-known algorithms and computer programs do not allow to solve the problem in its original form. If it is impossible to develop new algorithms and programs in a short time, the initial setting of the problem and the model simplifies: remove and combine conditions, reduce the number of factors, nonlinear relations are replaced with linear, strengthen the determinism of the model, etc.

Disadvantages that cannot be corrected at intermediate stages of modeling are eliminated in subsequent cycles. But the results of each cycle also have a completely independent value. Starting a study with the construction of a simple model, you can quickly get useful results, and then proceed to the creation of a more advanced model, complemented by new conditions, including refined mathematical dependencies.

With the development and complications of economic and mathematical modeling, its individual stages are isolated into specialized areas of studies, differences between theoretical and applied models increase, the models are defrastically based on abstraction levels and idealization.

The theory of mathematical analysis of models of the economy developed into a special branch of modern mathematics - a mathematical economy. Models studied within the framework of the mathematical economy lose direct relationship with economic reality; They deal with exclusively idealized economic objects and situations. When building such models, the main principle is not so much approach to reality, how much receipt is possible greater than the analytical results by means of mathematical evidence. The value of these models for economic theory and practice is that they serve as a theoretical basis for applied type models.

Pretty independent areas of research are the preparation and processing of economic information and the development of mathematical support for economic problems (creating databases and banks of information, automated models and software service programs for user economists). At the stage of practical use of models, specialists should play a leading role in the relevant area of \u200b\u200beconomic analysis, planning, management. The main plot of work of economists-mathematicians remains the formulation and formalization of economic problems and synthesis of the process of economic and mathematical modeling.

economic mathematical modeling

List of used literature

1. Fed-seas, economic methods

2. I.L.Akulich, mathematical programming in examples and objectives, Moscow, "Higher School", 1986;

3. S.A. Abramov, Mathematical Construction and Programming, Moscow, Science, 1978;

4. J. Littlewood, Mathematical Mathematics, Moscow, "Science", 1978;

5. Recommends the Academy of Sciences. Theory and Management Systems, 1999, No. 5, p. 127-134.

7. http://exsolver.narod.ru/books/mathematic/gametheory/c8.html

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There is a significant variety of species, types of economic and mathematical models necessary for use in managing economic objects and processes. Economic and mathematical models are divided into: macroeconomic and microeconomic depending on the level of the simulated control object, dynamic, which characterize changes in time control object, and static, which describe the relationship between different parameters, indicators of the object at that time. Discrete models display the status of the control object into separate, fixed time points. Simulation is called economic and mathematical models used to simulate managed economic objects and processes using the means of information and computing. According to the type of mathematical apparatus used in models, economic and statistical, models of linear and nonlinear programming, matrix models, network models are allocated.

Factor models. The group of economic and mathematical factor models includes models that, on the one hand, include economic factors on which the state of the managed economic object depends, and the parameters of the state of the object dependent on these factors. If the factors are known, the model allows you to define the desired parameters. Factor models are most commonly provided in mathematical terms with linear or static functions that characterize the relationship between factors and dependent parameters of the economic object.

Balance models. Balance models such as statistical and dynamic are widely used in economic and mathematical modeling. The creation of these models is the balance sheet method - the method of mutual comparison of material, labor and financial resources and needs of them. Describing the economic system as a whole, the system of equations is understood under its balance model, each of which expresses the need for a balance between the amounts of products manufactured by individual economic objects and the cumulative need for this product. With this approach, the economic system consists of economic objects, each of which produces some product. If instead of the concept of "product" to introduce the concept of "resource", then under the balance sheet model it is necessary to understand the system of equations that satisfy the requirements between a certain resource and its use.

The most important types of balance models:

• · Material, labor and financial balances for the economy as a whole and individual industries;
• · Inter-sectoral balances;
• · Matrix balance of enterprises and firms.

Optimization models. The large class of economic and mathematical models form optimization models that allow you to choose from all solutions the best optimal option. In mathematical content, optimality is understood as the achievement of an extremum of optimality criterion, also called target function. Optimization models are most often used in the tasks of finding a better way to use economic resources, which makes it possible to achieve the maximum target effect. Mathematical programming was formed on the basis of solving the problem about the optimal discovery of plywood sheets, which ensures the most complete use of the material. Putting such a task, famous Russian mathematician and economist Academician L.V. Kantorovich was recognized as worthy of the Nobel Prize in the economy.