The concept of a system in economic and mathematical analysis. Economic models may include models

Consider a number of basic concepts related to system analysis and
modeling of socio-economic systems, so that with their help more
fully reveal the essence of such a key concept as
economic and mathematical methods. The term economic and mathematical methods
is understood in turn as a generalized name for the complex
economic and mathematical scientific disciplines united for
study of socio-economic systems and processes.

Under the socio-economic system we mean a complex
probabilistic dynamic system covering production processes,
exchange, distribution and consumption of material and other goods. She is
belongs to the class of cybernetic systems, i.e., controlled systems.
Let us first consider the concepts associated with such systems and methods.
their research.

The central concept of cybernetics is the concept of "system". One
there is no definition of this concept; the following formulation is possible:
called a complex of interrelated elements together with the relationships between
elements and between their attributes. The set of elements under study can be
be considered as a system if the following four features are identified:

The integrity of the system, i.e. the fundamental irreducibility of the properties of the system
to the sum of the properties of its constituent elements;

The presence of a goal and criterion for the study of a given set of elements,

The presence of a larger, external in relation to this, system,
called "environment";

The possibility of selecting interconnected parts in this system
(subsystems).

The main method for studying systems is the modeling method, i.e.
method of theoretical analysis and practical action, aimed at
development and use of models. In this case, by model we mean
image of a real object (process) in a material or ideal form
(i.e., described by sign means in any language), reflecting
essential properties of the modeled object (process) and its replacement
during research and management. The modeling method is based on
the principle of analogy, i.e., the possibility of studying a real object is not
directly, but through consideration of a similar and more accessible
object, its model. In what follows, we will only talk about
economic and mathematical modeling, i.e. about the description by symbolic
mathematical means of socio-economic systems.

The practical tasks of economic and mathematical modeling are:

Analysis of economic objects and processes;

Economic forecasting, foreseeing the development of economic
processes;

Development of management decisions at all levels

economic hierarchy.

However, it should be borne in mind that not in all cases the data
obtained as a result of economic and mathematical modeling, can
be used directly as ready-made management solutions. They are
rather, they can be considered as "advising" means. Adoption
managerial decisions remain with the individual. Thus,
economic and mathematical modeling is only one of the
components (albeit very important) in man-machine systems
planning and management of economic systems.

The most important concept in economic and mathematical modeling, as in
any modeling, is the concept of model adequacy, i.e.
correspondence of the model to the modeled object or process. Adequacy
models - to some extent a conditional concept, since full compliance
there can be no model for a real object, which is also typical for
economic and mathematical modeling. When modeling, there is
mind, not just adequacy, but correspondence in those properties that
considered essential for the study. Adequacy check
economic and mathematical models is a very serious problem,
especially since it is complicated by the difficulty of measuring economic quantities.
However, without such verification, the application of simulation results in
management decisions can not only be of little use, but also
cause significant harm.

Socio-economic systems usually belong to the so-called
complex systems. Complex systems in the economy have a number of properties,
which must be taken into account when modeling them, otherwise it is impossible
talk about the adequacy of the constructed economic model. The most important of
these properties:

Emergence as a manifestation in the most vivid form of a property
system integrity, i.e. the presence of such properties in the economic system,
which are not inherent in any of the elements that make up the system, taken
separately. outside the system. Emergence is the result of the emergence
between the elements of the system of so-called synergistic links, which
provide an increase in the overall effect to a value greater than the sum
effects of system elements acting independently. So
socio-economic systems need to be researched and modeled in
in general;

Mass nature of economic phenomena and processes. patterns
economic processes are not detected on the basis of a small number
observations. Therefore, modeling in the economy should be based on
mass observations;

The dynamism of economic processes, which consists in changing
parameters and structure of economic systems under the influence of the environment (external
factors);

Randomness and uncertainty in the development of economic phenomena.
Therefore, economic phenomena and processes are mainly probabilistic
character, and for their study it is necessary to apply
economic and mathematical models based on probability theory and
mathematical statistics;

The inability to isolate the phenomena occurring in economic systems
and processes from the environment to observe and explore them in
pure form;

Active response to emerging new factors, the ability to
socio-economic systems to active, not always predictable
actions depending on the attitude of the system to these factors, methods and
their methods of influence.

Selected properties of socio-economic systems. naturally,
complicate the process of their modeling, but these properties should be
keep in mind when considering various aspects
economic and mathematical modeling, starting with the choice of the type of model and
ending with questions of practical use of simulation results.

1.2. Stages of economic and mathematical modeling

The process of modeling, including economic and mathematical, includes
three structural elements: the object of study; subject
(researcher); a model that mediates the relationship between the knower
subject and known object. Consider the general scheme of the process
modeling, which consists of four stages.

Let there be some object that we want to explore by the method
modeling. At the first stage, we construct (or find in
real world) another object is a model of the original original object. Stage
building a model requires certain information about
original object. The cognitive capabilities of the model are determined by the fact that
that the model reflects only some of the essential features of the original
object, so any model replaces the original in a strictly limited
sense. It follows from this that for one object one can construct
several models reflecting certain aspects of the object under study
or characterizing it with varying degrees of detail.

At the second stage of the modeling process, the model acts as
independent object of study. For example, one of the forms
research is the conduct of model experiments, in which
purposefully change the conditions for the functioning of the model and
data about its "behavior" are systematized. The end result of this
stage is the body of knowledge about the model in relation to essential
sides of the original object, which are reflected in this model.

The third stage is to transfer knowledge from the model to the original, in
as a result, we form a lot of knowledge about the original object and, when
In this case, we pass from the language of the model to the language of the original. With sufficient
reason to transfer any result from the model to the original can be
only if this result corresponds to the signs of similarity
original and model (in other words, signs of adequacy).

At the fourth stage, a practical verification of the received
using the knowledge model and their use both for building a generalizing
theory of a real object, and for its purposeful transformation
or their management. Finally, we return to the problem
original object.

Modeling is a cyclical process, i.e. after the first
a four-stage cycle may be followed by a second, a third, etc. At the same time
knowledge about the object under study is expanded and refined, and initially
The constructed model is gradually improved. Thus, in
modeling methodology has great potential
self-improvement.

Let us now proceed directly to the process of economic and mathematical
modeling, i.e. descriptions of economic and social systems and
processes in the form of economic and mathematical models. This variety
modeling has a number of significant features associated with both
the object of modeling, and with the apparatus and means used
modeling. Therefore, it is advisable to analyze in more detail
the sequence and content of the stages of economic and mathematical
modeling, highlighting the following six stages: formulation of the economic
problems, its qualitative analysis; building a mathematical model;
mathematical analysis of the model; preparation of initial information; numerical
decision; analysis of numerical results and their application. Consider each
of the steps in more detail.

1. Statement of the economic problem and its qualitative analysis. On this
stage, it is required to formulate the essence of the problem, accepted
background and assumptions. It is necessary to highlight the most important features and properties
modeled object, to study its structure and

The relationship of its elements, at least preliminarily formulate
hypotheses explaining the behavior and development of the object.

2. Building a mathematical model. This is the stage of formalization of economic
problem, i.e., its expression in the form of specific mathematical
dependencies (functions, equations, inequalities, etc.). Model building
is subdivided into several stages. First determined
type of economic and mathematical model, the possibilities of its application are being studied
in this task, a specific list of variables and parameters is specified
and form of connections. For some complex objects, it is advisable to build
several multidimensional models; while each model is allocated only
some sides of the object, while other sides are taken into account in aggregate and
approximately. Justified is the desire to build a model related to good
studied class of mathematical problems, which may require some
simplification of the initial assumptions of the model, without distorting the main features
modeled object. However, it is also possible that
formalization of the problem leads to a previously unknown mathematical
structure.

3. Mathematical analysis of the model. At this stage, purely mathematical
research methods reveal the general properties of the model and its solutions. AT
In particular, an important point is the proof of the existence of a solution
formulated task. Analytical research reveals that
whether the solution is unique, what variables can be included in the solution, in
what limits do they change, what are the trends in their change, etc.
However, models of complex economic objects are very difficult to
analytical research; in such cases, go to the numerical
research methods.

4. Preparation of initial information. In economic problems, this is how
as a rule, the most time-consuming stage of modeling, since it is not
reduced to passive data collection. Mathematical modeling
imposes stringent requirements on the information system; at the same time it is necessary
take into account not only the fundamental possibility of preparing
information of the required quality, but also the cost of preparing
information arrays. In the process of preparing information, we use
methods of probability theory, theoretical and mathematical statistics
for organizing sample surveys, assessing the reliability of data and
etc. With systemic economic and mathematical modeling, the results
functioning of some models serve as initial information for others.

5. Numerical solution. This stage includes the development of algorithms
numerical solution of the problem, preparation of computer programs and direct
carrying out calculations;

At the same time, significant difficulties are caused by the large dimension
economic tasks. Usually calculations based on economic and mathematical
models are multivariate. Numerous model
experiments, studying the behavior of the model under various conditions is possible
due to the high speed of modern computers. numerical
solution significantly complements the results of the analytical study, and
for many models is the only possible one.

6. Analysis of numerical results and their application. At this stage before
of all, the most important question of the correctness and completeness of the results is solved.
modeling and their applicability both in practice and in
to improve the model. Therefore, first of all, there must be
the adequacy of the model was checked for those properties that are selected
as material (in other words, must be produced
model verification and validation). Application of numerical results
modeling in economics is aimed at solving practical problems
(analysis of economic objects, economic forecasting of development
economic and social processes, development of managerial decisions
at all levels of the economic hierarchy).

The listed stages of economic and mathematical modeling are in
close relationship, in particular, there may be reciprocal relationships
stages. So, at the stage of building a model, it may turn out that setting
problem is either inconsistent, or leads to too complicated mathematical
models; In this case, the initial statement of the problem should be
adjusted. Most often, the need to return to previous
stages of modeling arises at the stage of preparation of the initial information.
If the necessary information is not available or the cost of its preparation
are too large, we have to return to the stages of setting the problem and its
formalizations in order to adapt to the information available to the researcher.

It has already been said above about the cyclic nature of the modeling process.
Deficiencies that cannot be corrected at certain stages
simulations are eliminated in subsequent cycles. However, the results
each cycle have a completely independent meaning. Having started
study with the construction of a simple model, you can get useful
results, and then move on to creating more complex and better
model, which includes new conditions and more accurate mathematical
dependencies.

1.3. Classification of economic and mathematical methods and models

The essence of economic and mathematical modeling lies in the description
socio-economic systems and processes in the form
economic and mathematical models. § 1.1 briefly discusses the meaning
concepts of "method of modeling" and "model". Based on this
economic and mathematical methods should be understood as a tool, and
economic and mathematical models - as a product of the process
economic and mathematical modeling.

Let's consider questions of classification of economic and mathematical methods. These
methods, as noted above, are a complex
economic and mathematical disciplines, which are an alloy of economics,
mathematics and cybernetics. Therefore, the classification of economic and mathematical
methods is reduced to the classification of scientific disciplines included in their
compound. Although the generally accepted classification of these disciplines is not yet
developed, with a known degree of approximation in the composition
economic and mathematical methods can be divided into the following sections:

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

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

Mathematical economy and studying the same questions with quantitative
sides of econometrics: the theory of economic growth, theory
production functions, intersectoral balance sheets, national accounts,
demand and consumption analysis, regional and spatial analysis,
global modeling, etc.;

Optimal decision-making techniques, including operations research
in economics. This is the most extensive section, including the following
disciplines and methods: optimal (mathematical) programming, in
including branch and bound methods, network scheduling methods, and
management, program-target methods of planning and management, theory
and inventory management methods, queuing theory, game theory.
theory and methods of decision making. scheduling theory. To the optimum
(mathematical) programming enter into turn linear
programming, non-linear programming, dynamic
programming, discrete (integer) programming,
fractional linear programming, parametric programming,
separable programming, stochastic programming,
geometric programming;

Methods and disciplines specific both for centralized
planned economy, and for. market (competitive) economy. To
the first can be attributed to the theory of optimal functioning of the economy,
optimal planning, optimal pricing theory, models
logistics, etc. To the second - methods that allow
develop models of free competition, models of capitalist
cycle, monopoly models, indicative planning models, models
theories of the firm, etc. Many of the methods developed for
of a centrally planned economy, can also be useful in
economic and mathematical modeling in a market economy;

Methods of experimental study of economic phenomena. To them
include, as a rule, mathematical methods of analysis and planning
economic experiments, methods of machine simulation (simulation
modeling), business games. This also includes methods
expert assessments designed to assess phenomena that are not amenable to
direct measurement. Let's move on to classification issues.
economic and mathematical models, in other words, mathematical
models of socio-economic systems and processes. unified system
classification of such models currently does not exist either,
however, more than ten main features of their classification are usually distinguished,
or classification headings. Let's take a look at some of these sections.

According to the general purpose, economic and mathematical models are divided into
theoretical-analytical, used in the study of general properties and
patterns of economic processes, and applied ones used in
solving specific economic problems of analysis, forecasting and
management. Various types of applied economic and mathematical models
just discussed in this tutorial.

According to the degree of aggregation of modeling objects, models are divided into
macroeconomic and microeconomic. Although there is no clear cut between them
distinctions, the first of them include models that reflect
the functioning of the economy as a whole, while
microeconomic models are usually associated with such links
economy as businesses and firms.

For a specific purpose, i.e. for the purpose of creation and use,
allocate balance models that express the requirement of compliance with the availability
resources and their use; trend models in which development
of the simulated economic system is reflected through a trend (long-term
trend) of its main indicators; optimization models,
designed to select the best option from a certain number
options for production, distribution or consumption; imitation
models intended for use in the process of machine simulation
studied systems or processes, etc.

According to the type of information used in the model, economic and mathematical
models are divided into analytical, built on a priori information, and
identifiable, built on a posteriori information.

By taking into account the time factor, the models are divided into static models, in which
all dependencies are related to one point in time, and dynamic,
describing economic systems in development.

By taking into account the uncertainty factor, the models are divided into
deterministic if their output results are unambiguous
are determined by control actions, and stochastic
(probabilistic) if, when specifying a certain
set of values ​​at its output can produce different results
depending on the action of a random factor.

Economic and mathematical models can also be classified according to
characterization of mathematical objects included in the model, other
words. by the type of mathematical apparatus used in the model. By
matrix models, models of linear and
nonlinear programming, correlation-regression models, models
queuing theory, network planning models and
control, game theory models, etc.

Finally, according to the type of approach to the studied socio-economic systems
distinguish descriptive and normative models. With descriptive
(descriptive) approach, models are obtained that are intended to describe and
explanations of actually observed phenomena or for the forecast of these phenomena;
As an example of descriptive models, we can cite the previously mentioned
balance and trend models. In the normative approach, one is not interested in
how the economic system is organized and develops, and how
it must be arranged and how it must act in the sense of certain
criteria. In particular, all optimization models are of the type
regulatory; another example would be normative level models
life.

Consider, as an example, the economic-mathematical model
intersectoral balance (EMM MOB). Considering the above
classification headings are applied, macroeconomic,
analytical, descriptive, deterministic, balance, matrix
model; in this case, there are both static and dynamic EMM MOBs.

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

Economic models may include models:

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

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

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

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

Economic and mathematical model

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

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

Main types of models

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

Economic and mathematical models and methods used in economic analysis

R a \u003d PE / VA + OA,

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

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

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

Integral method of economic analysis

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

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

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

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

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

dy / dx = const

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

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

∆Z(x) = y 0 * Δ x + 1/2Δ x *Δ y

Z(y)=x 0 * Δ y +1/2 Δ x* Δ y

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

Z=x/y;

Δ Z(x)= Δ x/Δ y Lny1/y0

Δ Z(y)=Δ Z- Δ Z(x)

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

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

Log method

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

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

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

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

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

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

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

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

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

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

Method of differential calculus

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

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

Let us explain the individual elements of this formula:

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

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

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

is an infinitesimal value of a higher order than

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

ΔZx = δZ / δx Δx; ΔZy = δZ / δy Δy.

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

Equity method

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

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

Δy a = Δa/Δa + Δb + Δc*Δy

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

Δyb =Δb/Δa + Δb +Δc*Δy

Finally, for the factor c we have:

∆y c =∆c/∆a +∆b +∆c*∆y

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

Linear programming method

See further:

Queuing Theory

See further:

Game theory

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

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

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

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

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

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

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

Matrix method

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

Network planning method

See further:

Extrapolation Analysis

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

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

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

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

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

0. Principles of economic and mathematical methods:

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

theory optimization of economic processes.

1. Mathematical statistics (its economic applications):

sampling method;

dispersion analysis;

correlation analysis;

regression analysis;

multivariate statistical analysis;

factor analysis;

index theory, etc.

2. Mathematical economy and econometrics:

theory of economic growth (models of macroeconomic dynamics);

theory of production functions;

intersectoral balances (static and dynamic);

national accounts, integrated material and financial balances;

analysis of demand and consumption;

regional and spatial analysis;

global modeling, etc.

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

optimal (mathematical) programming;

linear programming;

non-linear programming;

dynamic programming;

discrete (integer) programming;

block programming;

fractional linear programming;

parametric programming;

separable programming;

stochastic programming;

geometric programming;

branch and bound methods;

network methods of planning and management;

program-target methods of planning and management;

theory and methods of inventory management;

queuing theory;

game theory;

decision theory;

scheduling theory.

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

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

optimal planning:

economic;

perspective and current;

sectoral and regional;

theory of optimal pricing;

5. EMM specific to competitive economy:

models of the market and free competition;

business cycle models;

models of monopoly, duopoly, oligopoly;

indicative planning models;

models of international economic relations;

models of the theory of the firm.

6. Economic cybernetics:

system analysis of the economy;

economic Information Theory, including economic semiotics;

control systems theory, including theory of automated control systems.

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

mathematical methods of planning and analysis economic experiments;

methods machine simulation and bench experimentation;

business games.

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

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

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

List of sources used

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

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

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

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

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

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

Ranking and classification of factors of economic phenomena;

Revealing the analytical form of connection between the studied phenomena;

Smoothing the dynamics of changes in the level of indicators;

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

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

Quantitative change of informative indicators;

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

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

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

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

If the relationship is linear:

.

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

, n - number of observations.

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

.

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

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

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

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

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

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

under restrictions

f i (x) Rb i , i =

,

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

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

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

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

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

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

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

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

LECTURES

By discipline:

Economic and mathematical

methods and models

TEACHER MATSNEV A.P.

Moscow2004 year

1. Modeling of economic systems.

Basic concepts and definitions

1.1. The emergence and development of system representations

1.2. Models and modeling. Model classification

1.3. Types of model similarity

1.4. Adequacy of models

2. MATHEMATICAL MODELS AND METHODS FOR THEIR CALCULATION

2.1. The concept of operational research

2.2. Classification and principles of constructing mathematical models

3. Some information from mathematics

3.1. Convex sets

3.2. Linear inequalities

3.3. Linear form values ​​on a convex set

4. EXAMPLES OF LINEAR PROGRAMMING PROBLEMS

4.1. Transport task

4.2. General formulation of the linear programming problem

4.3. Graphical interpretation of the solution of linear programming problems

5. METHODS FOR SOLVING LINEAR PROGRAMMING PROBLEMS

5.1. General and basic problems of linear programming

5.2. Geometric Method for Solving Linear Programming Problems

5.3. Graphical solution of the resource allocation problem

5.4. Simplex method

5.5. Simplex table analysis

5.6. Solution of transport problems

6. NONLINEAR PROGRAMMING METHODS

AND MULTI-CRITERIA OPTIMIZATION

6.1. Problem Statement of Nonlinear Programming

6.2. Statement of the dynamic programming problem

Basic conditions and scope

6.3. Multiobjective optimization

1. Modeling of economic systems.

Basic concepts and definitions.

1.1. The emergence and development of system representations

The scientific and technological revolution has led to the emergence of such concepts as large and complex economic systems with specific problems for them. The need to solve such problems led to the emergence of special approaches and methods that were gradually accumulated and generalized, eventually forming a special science - system analysis.

In the early 1980s, consistency became not only a theoretical category, but also a conscious aspect of practical activity. There is a widespread notion that our successes are related to how systematically we approach solving problems that arise, and our failures are caused by a lack of systematicity in our actions. A signal of insufficient consistency in our approach to solving a problem is the appearance of a problem, while the resolution of the problem that has arisen occurs, as a rule, when moving to a new, higher, level of systematicity of our activity. Therefore, consistency is not only a state, but also a process.

In various spheres of human activity, various approaches and corresponding methods for solving specific problems have arisen, which have received various names: in military and economic issues - "operations research", in political and administrative management - "systems approach", in philosophy "dialectical materialism", in applied scientific research - "cybernetics". Later it became clear that all these theoretical and applied disciplines form, as it were, a single stream, a “system movement”, which gradually took shape in a science called “system analysis”. At present, system analysis is an independent discipline that has its own object of activity, its rather powerful arsenal of tools, and its own application area. Being essentially applied dialectics, system analysis uses all the means of modern scientific research - mathematics, modeling, computer technology and natural experiments.

 The most interesting and difficult part of system analysis - this is "pulling out" a problem from a real practical problem, separating the important from the unimportant, finding the right wording for each of the problems that arise, i.e. what is called "problem setting".

Many often underestimate the work involved in formulating a problem. However, many experts believe that “setting a problem well means solving it halfway”. Although in most cases it seems to the customer that he has already formulated his problem, the system analyst knows that the problem statement proposed by the client is a model of his real problem situation and inevitably has a target character, remaining approximate and simplified. Therefore, it is necessary to check this model for adequacy, which leads to the development and refinement of the original model. Very often, the initial formulation is stated in terms of languages ​​that are not necessary for building the model.

1.2. Models and modeling. Model classification

Initially, a model was called a kind of auxiliary tool, an object that, in certain situations, replaced another object. For example, a mannequin in a certain sense replaces a person, being a model of a human figure. Ancient philosophers believed that nature could be displayed only with the help of logic and correct reasoning, i.e. according to modern terminology with the help of language models. A few centuries later, the motto of the English Scientific Society became the slogan: “Nothing with words!”, Only conclusions supported by experimental or mathematical calculations were recognized.

Currently, there are 3 ways to comprehend the truth:

 theoretical research;

 experiment;

 modeling.

 Model a substitute object is called, which, under certain conditions, can replace the original object, reproducing the properties and characteristics of the original that are of interest to us, and has significant advantages:

- cheapness;

- visibility;

- ease of operation, etc.

 In model theory modeling called the result of mapping one abstract mathematical structure to another - also abstract, or as a result of the interpretation of the first model in terms and images of the second.

The development of the concept of a model went beyond mathematical models and began to refer to any knowledge and ideas about the world. Since models play an extremely important role in the organization of any human activity, they can be divided into cognitive (cognitive) and pragmatic, which corresponds to the division of goals into theoretical and practical.

 cognitive model focused on the approximation of the model to the reality that this model displays. Cognitive models are a form of organization and presentation of knowledge, a means of connecting new knowledge with existing ones. Therefore, when a discrepancy between the model and reality is detected, the task of eliminating this discrepancy by changing the model arises.

 Pragmatic Models they are a means of management, a means of organizing practical actions, a way of presenting exemplary correct actions or their results, i.e. are a working representation of the goals. Therefore, when a discrepancy is found between the model and reality, efforts must be directed to changing reality in such a way as to bring reality closer to the model. Thus, pragmatic models are of a normative nature, they play the role of a model, under which reality is adjusted. Examples of pragmatic models are plans, codes of laws, shop drawings, and so on.

Another principle for classifying the goals of modeling can be the division of models into static and dynamic.

 For some purposes, we may need a model of a specific state of an object at a certain point in time, a kind of “snapshot” of an object. Such models are called static . An example is the structural models of systems.

 In those cases when it becomes necessary to display the process of changing states, it is required dynamic models systems.

At the disposal of man there are two types of materials for building models - the means of consciousness itself and the means of the surrounding material world. Accordingly, the models are divided into abstract (ideal) and material.

 It is obvious that abstract models include language constructs and mathematical models. Mathematical models have the highest accuracy, but in order to reach their use in this area, it is necessary to obtain a sufficient amount of knowledge. According to Kant, any branch of knowledge can be called a science the more it uses mathematics to a greater extent.

1.3. Types of model similarity

So that some material structure can be a model, i.e. replaced the original in some respect, a relationship of similarity must be established between the original and the model. There are different ways to establish this similarity, which gives the models features that are specific to each method.

 First of all, this is the similarity established in the process of creating a model. Let's call it similarity to direct . An example of such similarity is photographs, scale models of aircraft, ships, building models, patterns, dolls, etc.

It should be remembered that no matter how good the model is, it is still only a substitute for the original, only in a certain respect. Even when the model of direct similarity is made of the same material as the original, i.e. similar to it substratively, there are problems of transferring the simulation results to the original. For example, when testing a reduced model of an aircraft in a wind tunnel, the problem of recalculating the data of a model experiment becomes nontrivial and a branched, meaningful similarity theory arises, which makes it possible to bring the scale and conditions of the experiment, flow velocity, viscosity and air density into line. It is difficult to achieve the interchangeability of the model and the original in photocopies of works of art, holographic images of works of art.

 The second type of similarity between the model and the original is called indirect . Indirect similarity between the original and the model objectively exists in nature and is found in the form of sufficient closeness or coincidence of their abstract mathematical models and, as a result, is widely used in the practice of real modeling. The most characteristic example is the electromechanical analogy between a pendulum and an electric circuit.

It turned out that many patterns of electrical and mechanical processes are described by the same equations, the difference lies in the different physical interpretation of the variables included in this equation. The role of models with indirect similarity is very great and the role of analogies (models of indirect similarity) in science and practice can hardly be overestimated. Analog computers make it possible to find a solution to almost any differential equation, thus representing a model, an analogue of the process described by this equation. The use of electronic analogues in practice is determined by the fact that electrical signals are easy to measure and fix, which gives the well-known advantages of the model.

 The third, special class of models is made up of models whose similarity to the original is neither direct nor indirect, but established by agreement . Such similarity is called conditional. Models of conditional similarity have to be dealt with very often, since they are a way of material embodiment of abstract models. Examples of conditional similarity are money (value model), identity card (owner model), all kinds of signals (message models).