Mathematical methods in economics and management. Mathematical methods and models in the economy

24.09.2019

Read also

Why did Sofya preferred to Chapkom Mollylly?

Symbolic meaning of the name of the Store Eery A

Methods of economic theory

The study of the economic life of a person was part of the interests of scientists since ancient times. The gradual complication of economic relations demanded the development of economic thought. Jumps in science have always been accompanied by tasks arising before humanity at various stages of evolution. Initially, people mined food, then began to exchange it. Over time, agriculture arose, which contributed to the division of labor and the emergence of the first handicraft professions. An important stage in the economic life of humanity was the industrial revolution, which gave impetus for the rapid growth in production volume, and also influenced social changes in society.

Modern economic science has been formed relatively recently when scientists moved from solving problems that arise before the primary class, to the study of the processes occurring in systems, regardless of the interests of society.

The subject of economic theory is to optimize the ratio of increasing demand in conditions when the amount of the proposal is limited due to the limitation of resources.

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

Economic systems are quite complex formations in which each entity simultaneously enters many connections. They can be considered from the point of view of macroeconomic cumulative indicators, as well as the result of the work of a separate economic agent. In the science of the economy, various methods are used to facilitate the processes of research and analysis of economic phenomena. Most often in practice apply:

the abstraction method (selection of an object from its relations and existing factors);
synthesis method (combining elements in general);
analysis method (crushing the general system into components);
deduction (study from private to general) and induction (study of the subject from common to private);
systematic approach (allows us to consider the object being studied as a structure);
mathematical modeling (building models of processes and phenomena on the mathematical language).

Modeling in economics

The nature of modeling is that the real model of the process, phenomenon or system, replace another model that can simplify its research and analysis. It is important to observe the approach of the original model to its scientific analogue. Simulation is used to simplify. Often in practice there are such phenomena that cannot be studied without the use of visual scientific generalizations.

The following modeling objectives can be distinguished:

Search and describing the causes of the behavior of the original model.
Forecasting future behavior of the model.
Drawing up projects, plans for systems.
Automation of processes.
Search for ways to optimize the original model.
For training professionals, students and others.

In essence, the model can also be different types. The verbal model is built on the verbal description of any system or process. The graphic model is a visual image of various dependencies from each other. It can also describe the behavior of the original model in the dynamics. Modeling is a natural lies in creating a layout that can partially or completely display the behavior of the original. Mathematical modeling is most widely used. It makes it possible to use the fullness of mathematical tools and language. In mathematics, statistical models, dynamic and information models are applied. Each of their species is used to achieve specific goals arising from experts.

Note 1.

The separation of the economy on the macro- and micro levels led to the fact that the simulation simulates the systems of various levels of the organization. Econometrics, which applies statistics and probability theory, is most often used to study economic structures. It is worth noting that it is the mathematical modeling that allows you to consider an important time factor in the dynamic development of systems.

Mathematical models in the economy

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

Setting goals and objectives.
Conduct formalization of the studied process or phenomenon.
Search for the necessary solution.
Check the solution obtained and the model for adequacy.
If the results of the check are satisfactory, these models can be applied in practice.

Mathematical models are distinguished by the application of the mathematics language at the stage of their construction, as well as for further calculations. This language allows you to most accurately describe communication, dependencies and patterns. When the transition to the solution of models is performed, there may be different types of solutions. For example, accurate or analytical gives the final calculation rate. Approximate value has a specific calculation error, often used to build graphic models. The solution expressed by the number gives the final result that is often displayed using computer computing. It should be remembered that the accuracy of solutions does not mean the accuracy of the calculated model.

An important step in mathematical modeling is to verify the results obtained and the simulation model for adequacy. Usually, the verification work is based on comparing the data of the real model with the data constructed. However, in mathematical and economic modeling it is rather difficult to make this action. Usually the adequacy of calculations is determined subsequently in practice.

Note 2.

Mathematical modeling in the economy allows you to simplify the phenomena and processes in economic systems, produce calculations and obtain relatively correct results of calculations. It is important to remember that this approach is also not universal, as it has a number of the flaws listed above. The adequacy of the simulation is often achieved by tested hypotheses and calculated formulas.

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.

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 activity.

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:

LG.1 \u003d Kf.mIN.;

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

LG.2 \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. -

Tutoring

Need help to study what language themes?

Our specialists will advise or have tutoring services for the subject of interest.
Send a request With the topic right now, to learn about the possibility of receiving consultation.

Theory

1.

Model - This is a simplified representation of a real device and processes occurring in it, phenomena. . Modeling - This is the process of creating and researching models. Modeling facilitates the study of the object in order to create it, further transformation and development. It is used to study the existing system when the real experiment is inexpedient due to significant financial and labor costs, as well as if necessary, analyzing the designed system, i.e. which still does not physically exist in this organization.

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

The model has the following functions:

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

During the modeling of knowledge of the test object expand and specify, and the original model is gradually improved. The disadvantages found after the first modeling cycle are corrected, and the simulation is carried out again. In the modeling methodology, therefore, large capabilities of self-development are laid.

2.

Modeling in economics - This is an explanation of socio-economic systems with sign mathematics. The practical tasks of economic and mathematical modeling are: analysis of economic objects and processes, economic forecasting, prediction of economic processes, preparation of management decisions at all levels of economic activity.

The characteristics of the economy as an object of modeling are:

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

2) Emergenism, meaning that economic objects, processes and phenomena have such properties, which none of the elements of their generators do not possess;

3) probabilistic, indefinite, accidental nature of economic processes and phenomena;

4) the inertial nature of the development of the economy, in accordance with which laws, patterns, trends, relationships, dependencies that took place last period continue to act for some time in the future.

All of the above and other properties of the economy complicate its study, identifying patterns, dynamic trends, connections and dependencies. Mathematical modeling is the tools, the skillful use of which allows you to successfully solve the problems of studying complex systems, including such complex as economic objects, processes, phenomena.

3.

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

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

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

Methodology: Branch of knowledge, exploring the conditions, principles, structure, logical organization, methods and methods of activity.

Mechanism: The system of methods of practical orientation aimed at ensuring the practical use of methods and models to solve economic management problems.

Method: A combination of tools aimed at solving a certain problem.

Mathematical method: A study method aimed at analyzing, synthesis, optimization or prediction of the state, structure, functions or behavior of the economic system, consequences and prospects for its functioning, management or development, using formal methods and apparatus of mathematical research.

Mathematical model: The mathematical description of the object (process or system) is used in the study instead of the original object, in order to analyze, identify quantitative or logical connections between its parts.

Complex of mathematical models: The set of shared applying mathematical models that use or exchange shared data and are aimed at achieving a common goal or solving a common problem.

4.

There are two basic Approach to the modeling of the economy: microeconomic and macroeconomic. Microeconomic approachreflects the functioning and structure of individual elements of the system being studied (for example, in the study of the banking sector, this element is a commercial bank) or the state and development of certain socio-economic processes occurring in it, and is primarily implemented by developing applied methods for analyzing the results of activities. So, for example, in relation to the bank is an analysis of the liquidity of the bank, the assessment of bank risks, etc. The tasks within the microeconomic approach are also implemented by developing special economic and mathematical models. Macroeconomic approach Inappropriate an analysis of the specifics of the functioning of the system under study in relationships with the main macroeconomic indicators of the National Economic Development. In relation to the analysis of the activities of the banking sector, this approach consisions in its interaction with various segments of the financial market and, accordingly, in the relationship of the banking sector indicators with the macroeconomic indicators of the economy as a whole. In this case, the macroeconomic approach can practically be implemented through the construction of models of factor analysis, such as the factor model of the market of state short-term obligations, the model of the loan capital market, as well as when building and evaluating the projected values \u200b\u200bof the dynamics of individual banking sector indicators.

A number of directions in modeling relies on microeconomics, a row - on macroeconomic. There are no clear faces, for example, it can be said that the economy of the industrial enterprise, the labor economy, the communal economy relate to microeconomics, the monetary economy, the investment of the consumption of consumption is macroeconomics, and the financial market, international trade economic development is the area of \u200b\u200boverlapping.

5.

In the most general form, the equilibrium in the economy is balanced and the proportionality of its main parameters, in other words, the situation when there are no incentives for the participants in economic activity to change the existing situation.

Market balance is the market situation, when the demand for the product is equal to its proposal. Usually, equilibrium is achieved through either the restriction of needs (they always perform in the market in the form of solvent demand), or increase and optimize the use of resources.

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

Moreover, the equilibrium of the NC system. The farms are not only a market equilibrium. Because Disorders in the field of production inevitably lead to non-equilibrium in the markets. And in real reality, the economy is influenced by other, non-market factors (war, social excitement, weather, demographic shifts).

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

As for the state of equilibrium, it, on Valras, assumes the presence of three conditions:

1) the demand and supply of factors of production are equal; They establish a constant and stable price;

2) the demand and supply of goods (and services) are also equal and implemented on the basis of constant, sustainable prices;

3) prices of goods correspond to production costs.

There are three types of market equilibrium: instant, short-term and long-term, through which the proposal consistently passes in the process of increasing its elasticity in response to an increase in demand.

6.

Closed economy - A model of a closed economic system, focused on the exclusive use of its own resources and the refusal of foreign economic relations. This model was implemented, as a rule, in the conditions of preparation for war or war. In particular, the economy of fascist Germany was approaching, a pre-war economy of the USSR.

The closed economy is an economy, a high level of customs duties and non-tariff barriers, fell apart from the global economic community. An increasing number of developing countries is becoming closed to an open economy. There is still a closed economy. Some countries of the poor south, first of all, the countries of Africa south of Sahara. The economy of these countries will not affect the increase in international economic exchanges and movement of capital. The closed nature of the economy enhances the deep backwardness, which, in turn, does not allow them to adapt to structural changes in world markets.

Open economy - The country's economy, closely related to the global market, international division of labor. It is the opposite of closed systems. The degree of openness is characterized by such indicators as: the ratio of export and import to GDP; capital movement abroad and from abroad; Currency reversibility; Participation in international economic organizations. In modern conditions, it becomes a factor in the development of the national economy, a reference point for the best world standards.

Many directions of the economic thought of the West (representatives of the open economy) developed its own model of the open economy. This topic remains relevant to this day. The open economy models discover such a spectrum of issues as interaction between national economies, a combination of macroeconomic and foreign economic policy, and in the case of its non-equilibrium level, the issue of developing its own stabilization policy.

Models of closed and open economy:

Principal non-equilibrium economy (uneven development)

State intervention (protectionism and anti-dumping policies) and globalization (struggle for resources)

Import and export - signs of an open economy

Mutual dependence of countries (international division of labor)

Transnational corporations (capital flows)

7.

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

These models directly associate the issues and costs of production with its technology, allow us to use the ratios of material and financial balance, predict, optimization and development analysis.

Technological models can be static and Dynamic .

-Static the models operate with constant values \u200b\u200bA and B, describe the existing balance of costs and issues and are intended for short-term forecasts or optimization (for example, model Mob Leontiev)

- Dynamic models include price dynamics (and possibly - autonomous technical team), make it possible to investigate economic growth and sustainability of the economy (model Nimanana, Morishima and etc.)

At the same time, a technological approach is inherent in a number of shortcomings: in technological models usually Not considered: -Gogographic position of the object; -Real technical progress; - Dynamics of prices; - Restriction of labor resources, etc.

Model Nimanana - this model expanding economy in which all issues and costs increase in the same proportion. The model is closed, that is, all issues of one period become the costs of the next period. It also does not use primary factors and consumption is considered as costs in the technological process, so all costs are reproducible, and there is no need to consider primary resources.

Model assumptions: The real salary level corresponds to the subsistence minimum and all overcome income is reinvested; The real level of salary is given and revenues have a residual nature; There are no differences between the primary factors of production and production volumes; There are no "source" factors of production, such as labor in traditional theory.

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

simulation in economy. 2.1. The concept of "model" and " modeling". With the concept " modeling economic systems "(as well as mathematical etc.) are connected ...

Economy-mathematical modeling as a method of studying and evaluating economic activities

Essay \u003e\u003e Economy

Ed. L. N. Schechevitsy - M.: Phoenix, 2003 Mathematical modeling in economy: Tutorial / Ed. E.S. Kundyysheva ... Ed. L. T. Gilyarovskaya - M.: Prospekt, 2007 Mathematical modeling in economy: Tutorial / Ed. IN AND. Masukina ...

Application economy-mathematical Methods B. economy

Examination \u003e\u003e Economic and Mathematical Modeling

... : "Economy-mathematical Methods I. modeling" 2006 Content Introduction Mathematical modeling in economy 1.1 Development of methods simulation 1.2 Modeling as a method of scientific knowledge 1.3 Economy-mathematical ...