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

Economic models may include the following models:

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

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

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

The nature of the models may vary. Models are divided into: real, symbolic, verbal and tabular description, etc.

Economic and mathematical model

In managing business processes, the greatest importance is, 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 studying and managing them. This is a mathematical notation of the economic problem being solved.

• Extrapolation models
• Factor econometric models
• Optimization models
• Balance models, Inter-Industry Balance (IOB) model
• Expert assessments
• Game theory
• Network models
• Models of queuing systems

Economic and mathematical models and methods used in economic analysis

R a = PE / VA + OA,

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

So, first you should build an economic and mathematical model that describes the influence of individual factors on the general economic indicators of the organization’s activities. Widespread in the analysis of economic activity multifactor multiplicative models, since they make it possible to study the influence of a significant number of factors on general indicators and thereby achieve greater depth and accuracy of analysis.

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

When using the integral method, it is possible to obtain more substantiated results for calculating the influence of individual factors than when using the method of chain substitutions and its variants. The method of chain substitutions and its variants, as well as the index method, have significant disadvantages: 1) the results of calculations of the influence of factors depend on the accepted sequence of replacing the basic values ​​of individual factors with actual ones; 2) the additional increase in the general 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 a given model, as well as regardless of the form of connection between these elements.

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

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

d y / d x = const

When using the integral method, the calculation of a definite integral for a given integrand and a given integration interval is carried out using an existing standard program using modern computer technology.

If we solve a multiplicative model, then to calculate the influence of individual factors on the general economic indicator, we can use the following formulas:

ΔZ(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 a given period. Examples of such tasks include analysis of the implementation of business plans or analysis of changes in economic indicators compared to the previous period. The dynamic type of tasks occurs in the presence of information about changes in the analyzed factors during a given period. This type of task includes calculations related to the study of time series of economic indicators.

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

Logarithm method

In addition to this method, the logarithm method (method) is also used in analysis. It is used in factor analysis when solving multiplicative models. The essence of the method under consideration is that when it is used, there is a logarithmically proportional distribution of the magnitude of the joint action of factors between the latter, that is, this value is distributed among 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 equally among the factors. Therefore, the logarithm method makes calculations of the influence of factors more reasonable compared to the integral method.

In the process of logarithmization, not absolute values ​​of growth in 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 general 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 general economic indicator. Thus, the influence of the first factor can be determined by the following formula:

Δf x = Δf log(x 1 / x 0) / log(f 1 / f 0)

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

Δf y = Δf log(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 = Δf log(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.

Differential calculus method

When carrying out 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 individual 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 us determine the influence of individual factors on the general indicator, using a function of two variables as an example.

Function specified 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)- magnitude of change in function;

Δx = (x 1 - x 0)— the magnitude of change in one factor;

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

- an infinitesimal quantity of a higher order than

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

ΔZ x = δZ / δx Δx; ΔZ y = δZ / δy · Δy.

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

Participation method

In terms of solving additive, as well as multiple-additive models, the equity method is also used to calculate the influence of individual factors on changes 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. This proportion is then multiplied by the total change in the summary indicator.

Suppose we determine the influence of three factors − A,b And With to a general indicator y. Then for the factor, and determining its share and multiplying it by the total amount of change in the generalizing indicator can be done using the following formula:

Δy a = Δa/Δa + Δb + Δc*Δy

For factor b, the formula under consideration will have the following form:

Δy b =Δb/Δa + Δb +Δc*Δy

Finally, for factor c we have:

Δy c =Δc/Δa +Δb +Δc*Δy

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

Linear programming method

Queuing theory

Game theory

Game theory is also used. Just like queuing theory, game theory is one of the branches of applied mathematics. Game theory studies the optimal solutions possible in gaming situations. This includes 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 activities of organizations is the so-called sensitivity analysis. This method is often used in the process of analyzing investment projects, as well as for the purpose of predicting the amount of profit remaining at the disposal of a given organization.

In order to optimally plan and forecast the activities of an organization, it is necessary to provide in advance for those changes that may occur in the future with the analyzed economic indicators.

For example, one should predict in advance changes in the values ​​of those factors that affect the profit margin: the level of purchase prices for purchased material resources, the level of sales prices for the products of a given organization, changes in customer demand for these products.

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

For example, they establish by what amount profit will change in the future, subject to a change in the quantity of products sold per unit. By doing this, we analyze the sensitivity of net profit to changes in one of the factors influencing it, that is, in this case, the sales volume factor. The remaining factors influencing the amount of profit remain unchanged. It is also possible to determine the amount of profit if the influence of several factors changes simultaneously in the future. Thus, sensitivity analysis makes it possible to establish the strength of the response of a general economic indicator to changes in individual factors influencing this indicator.

Matrix method

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

Network planning method

Extrapolation Analysis

In addition to the methods discussed, extrapolation analysis is also used. It includes consideration of changes in the state of the analyzed system and extrapolation, that is, 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; choosing the type of empirical functions; determination of the main parameters of these functions; extrapolation; establishing the degree of reliability of the analysis performed.

Economic analysis also uses the principal component method. They are used for the purpose of comparative analysis of individual components, that is, the parameters of the analysis of the organization’s activities. The principal components represent the most important characteristics of linear combinations of components, that is, the parameters of the analysis that have the most significant dispersion values, namely, the largest absolute deviations from the average values.

Multiplicative model.

Example 2. Revenue from sales of products (product volume - V) can be expressed as the product of a set of factors: number of personnel (nr), the share of workers in the total number of personnel (dр); average annual output per worker (Vr)

V = Chp * dр * Вр

Ø Transformations of deterministic factor models

To model various situations in factor analysis, special methods for transforming standard factor models are used. They are all based on reception detail. Detailing– decomposition of more general factors into less general ones. Detailing allows, based on knowledge of economic theory, to streamline the analysis, promotes a comprehensive consideration of factors, and indicates the significance of each of them.

The development of a deterministic factor system is achieved, as a rule, by detailing complex factors. Elemental (simple) factors are not decomposed.

Example 1. Factors

Most of the traditional (special) techniques of deterministic factor analysis are based on elimination. Reception elimination used to identify an isolated factor by excluding the effects of all others. The starting premise of this technique is as follows: All factors change independently of each other: first one changes, and all the others remain unchanged, then two, three, etc. change. with the rest remaining unchanged. The elimination technique is, in turn, the basis for other techniques of deterministic factor analysis, chain substitutions, index, absolute and relative (percentage) differences.

Ø Acceptance of chain substitutions

Target.

Application area. All types of deterministic factor models.

Restricted use.

Application procedure. A number of adjusted values ​​of the performance indicator are calculated by sequentially replacing the basic values ​​of the factors with the actual ones.

It is advisable to calculate the influence of factors in an analytical table.

Original model: P = A x B x C x D

Ø Acceptance of absolute differences

Target. Measuring the isolated influence of factors on changes in performance indicators.

Application area. Deterministic factor models; including:

1. Multiplicative

2. Mixed (combined)

type Y = (A-B)C and Y = A(B-C)

Restrictions on use.Factors in the model should be sequentially arranged: from quantitative to qualitative, from more general to more specific.

Application procedure. The magnitude of the influence of an individual factor on the change in the performance indicator is determined by multiplying the absolute increase in the factor under study by the basic (planned) value of the factors that are located to the right of it in the model, and by the actual value of the factors located to the left.

In the case of the original multiplicative model P = A x B x C x D we obtain: change in the effective indicator

1. Due to factor A:

DP A = (A 1 – A 0) x B 0 x C 0 x D 0

2. Due to factor B:

DP B = A 1 x (B 1 - B 0) x C 0 x D 0

3. Due to factor C:

DP C = A 1 x B 1 x (C 1 - C 0) x D 0

4. Due to factor D:

DP D = A 1 x B 1 x C 1 x (D 1 - D 0)

5. General change (deviation) of the performance indicator (balance of deviations)

D P = D P a + D P in + D P c + D P d

The balance of deviations must be maintained (just as in the reception of chain substitutions).

Ø Acceptance of relative (percentage) differences

Target. Measuring the isolated influence of factors on changes in performance indicators.

Application area. Deterministic factor models including:

1) multiplicative;

2) combined type Y = (A – B) C,

It is advisable to use when the previously determined relative deviations of factor indicators in percentages or coefficients are known.

There are no requirements for the sequence of arrangement of factors in the model.

Original package. The resultant characteristic changes in proportion to the change in the factor characteristic.

Application procedure. The magnitude of the influence of an individual factor on the change in the effective indicator is determined by multiplying the basic (planned) value of the effective indicator by the relative increase in the factor characteristic.

Original model:

Change in performance indicator:

1. Due to factor A:

2. Due to factor C:

Balance of deviations. The total deviation of the performance indicator consists of deviations by factors:

D Y = Y 1 - Y 0 = D Y A + D Y B + D Y C

Ø Index method

Target. Measuring relative and absolute changes in economic indicators and the influence of various factors on it.

Application area.

1. Analysis of the dynamics of indicators, including aggregated (added) indicators.

2. Deterministic factor models; including multiplicative and multiple ones.

Application procedure. Absolute and relative changes in economic phenomena.

Aggregate index of product cost (turnover)

I pq – characterizes the relative change in the cost of products in current prices (prices of the corresponding period)

The difference between the numerator and denominator (åp 1 q 1 - åp o q 0) – characterizes the absolute change in the cost of products in the reporting period compared to the base one.

Aggregate price index:

I p – characterizes the relative change in the average price for a set of types of products (goods).

The difference between the numerator and denominator (åp 1 q 1 - åp o q 1) – characterizes the absolute change in the cost of products due to changes in prices for certain types of products.

characterizes the relative change in production volume at fixed (comparable) prices.

åq 1 p 0 - åq 0 p 0 – the difference between the numerator and denominator characterizes the absolute change in the cost of products due to changes in the physical volumes of its various types.

Based on index models, it is carried out factor analysis.

Thus, a classic analytical task is to determine the influence of quantity factors (physical volume) and prices on the cost of products:

In absolute terms

å p 1 q 1 - å p 0 q 0 = (å q 1 p 0 - å q 0 p 0) + (å p 1 q 1 - å p 0 q 1).

Similarly, using the index model, it is possible to determine the influence on the total cost of production (zq) of the factors of its physical volume (q) and the cost of a unit of production of various types (z)

In absolute terms

å z 1 q 1 - å z 0 q 0 = (å q 1 z 0 - å q 0 z 0) + (å z 1 q 1 - å z 0 q 1)

Ø Integral method

Target. Measuring the isolated influence of factors on changes in performance indicators.

Application area. Deterministic factor models, including

· Multiplicative

· Multiples

Mixed type

Advantages. Compared to methods based on elimination, it gives more accurate results, since the additional increase in the effective indicator due to the interaction of factors is distributed in proportion to their isolated impact on the effective indicator.

Application procedure. The magnitude of the influence of an individual factor on the change in the performance indicator is determined on the basis of formulas for different factor models, derived using differentiation and integration in factor analysis.

Change in performance indicator due to factor x

D¦ x = D xy 0 + DxDу / 2

due to factor y

D¦ y = Dух 0 +DуDх / 2

Overall change in the effective indicator: D¦ = D¦ x + D¦ y

Balance of deviations

D¦ = ¦ 1 - ¦ 0 = D¦ x + D¦ y

The simplest approach to modeling seasonal fluctuations is to calculate the values ​​of the seasonal component using the moving average method and construct an additive or.
The general appearance of the multiplicative model looks like this:

Where T is the trend component, S is the seasonal component and E is the random component.
Purpose. Using this service, a multiplicative time series model is built.

Algorithm for constructing a multiplicative model

1. Alignment of the original series using the moving average method.
2. Calculation of the values ​​of the seasonal component S.
3. Removing the seasonal component from the original series levels and obtaining aligned data (T x E).
4. Analytical alignment of levels (T x E) using the resulting trend equation.
5. Calculation of values ​​obtained from the model (T x E).
6. Calculation of absolute and/or relative errors. If the obtained error values ​​do not contain autocorrelation, they can replace the original levels of the series and subsequently use the error time series E to analyze the relationship between the original series and other time series.

Example. Construct an additive and multiplicative model of a time series that characterizes the dependence of series levels on time.
Solution. Construction multiplicative time series model.
The general view of the multiplicative model is as follows:
Y = T x S x E
This model assumes that each level of a time series can be represented as the sum of trend (T), seasonal (S) and random (E) components.
Let's calculate the components of a multiplicative time series model.
Step 1. Let's align the initial levels of the series using the moving average method. For this:
1.1. Let's find moving averages (column 3 of the table). The aligned values ​​obtained in this way no longer contain a seasonal component.
1.2. Let's bring these values ​​into correspondence with actual moments of time, for which we find the average values ​​of two consecutive moving averages - centered moving averages (column 4 of the table).

They are used in cases where the effective indicator is an algebraic sum of several factor indicators.

2. Multiplicative models

Y=
.

This type of model is used when the performance indicator is a product of several factors.

3. Multiples models

Y= .

They are used when the effective indicator is obtained by dividing one factor indicator by the value of another.

4. Mixed (combined) models are a combination in various combinations of previous models:

Y= ; Y= ; Y=(a+b)c .

Conversion factor systems

1. Conversion multiplicative factor systems are carried out by sequential division of the factors of the original system into factor factors.

For example, when studying the process of formation of production volume (see Fig. 6.1), you can use such deterministic models as

VP=KR GV; VP=KR D DV, VP=KR D P NE.

These models reflect the process of detailing the original factor system of a multiplicative form and expanding it by dividing complex factors into factors. The degree of detail and expansion of the model depends on the purpose of the study, as well as on the possibilities of detailing and formalizing indicators within the established rules.

2. Simulation is carried out in a similar way additive factor systems due to dividing one of the factor indicators into its constituent elements-components.

Example. As is known, the volume of product sales

VRP = VVP – VI,

where VVP is the volume of production;

VI – volume of on-farm use of products.

In an agricultural enterprise, grain products were used as seeds (S) and feed (K). Then the given initial model can be written as follows: VP = VVP - (C + K).

3. To class multiples models, the following methods of their transformation are used:

elongation;

formal decomposition;

extensions;

abbreviations.

First the method involves lengthening the numerator of the original model by replacing one or more factors with the sum of homogeneous indicators.

For example, the cost per unit of production can be represented as a function of two factors: the change in the amount of costs (3) and the volume of output (VVP). The initial model of this factor system will have the form

C= .

If the total amount of costs (3) is replaced by their individual elements, such as wages (W), raw materials and materials (SM), depreciation of fixed assets (A), overhead costs (OC), etc., then the deterministic factor model will have type of additive model with a new set of factors

C= +++=X +X +X +X ,

where X – labor intensity of products; X – material consumption of products; X – capital intensity of production; X – level of overhead costs

Formal decomposition method factor system provides lengthening the denominator of the original factor model by replacing one or more factors with the sum or product of homogeneous indicators.

If b=l+m+n+р, That

Y=
.

As a result, we obtained a final model of the same type as the original factor system (multiple model). In practice, such decomposition occurs quite often. For example, when analyzing the production profitability indicator (P):

P= ,

where /7 is the amount of profit from sales of products;

3 - the amount of costs for production and sales of products.

If the sum of costs is replaced by its individual elements, the final model as a result of the transformation will take on the following form:

P=
.

Cost of one ton-kilometer (C
) depends on the amount of costs for maintaining and operating the car (3) and on its average annual output (AG). The initial model of this system will have the form

WITH
=.

Considering that the average annual production of a car, in turn, depends on the number of days worked by one car per year (D), the duration of the shift (P) and the average hourly output (AS), we can significantly lengthen this model and decompose the increase in cost into a larger number of factors:

WITH
=
.

Extension method provides for the expansion of the original factor model due to multiplying the numerator and denominator of a fraction by one or more new indicators. For example, if the original model

introduce a new indicator c, then the model will take the form

.

The result was a final multiplicative model in the form of a product of a new set of factors.

This modeling method is very widely used in analysis. For example, the average annual production by one worker (labor productivity indicator) can be written as follows: GV = VP / KR. If we introduce such an indicator as the number of days worked by all employees (D), we obtain the following model of annual output:

GV=
,

where DV is the average daily output; D – number of days worked by one employee.

After introducing the indicator of the number of hours worked by all employees (T), we obtain a model with a new set of factors: average hourly output (AS), number of days worked by one employee (D) and length of the working day (P):

Reduction method represents the creation of a new factor model by dividing the numerator and denominator of a fraction by the same exponent:

.

In this case, the final model is of the same type as the original one, but with a different set of factors.

Another example. The economic return on assets of an enterprise (ROA) is calculated by dividing the amount of profit (P) by the average annual cost of the enterprise's fixed and working capital (A): ROA=P/A.

If we divide the numerator and denominator by the sales volume of products (S), we obtain a multiple model, but with a new set of factors: profitability of products sold and capital intensity of products:

 Performance indicators can be decomposed into their constituent elements (factors) in various ways and presented in the form of various types of deterministic models. The choice of modeling method depends on the object of study, the goal, as well as the professional knowledge and skills of the researcher. The process of modeling factor systems is a very complex and important moment in economic analysis. The final results of the analysis depend on how realistically and accurately the created models reflect the relationship between the indicators being studied..

To identify the structure of a time series, i.e. To determine the quantitative values ​​of the components that make up the levels of a series, additive or multiplicative models of time series are most often used.

Multiplicative model. Y=T*S*E

T-trend component

S-seasonal component

E-random component

The multiplicative model is used if the amplitude of seasonal fluctuations increases or decreases.

Algorithm for building a model. The model building process includes the following steps:

Aligning the levels of the original series using the moving average method.

Calculation of the values ​​of the seasonal component S

Removing the seasonal component from the original series level and obtaining aligned data without S

Analytical alignment of series levels and calculation of T factor values

Calculation of the obtained values ​​(T* S) for each level of the series

Calculation of absolute or relative model errors.

(or 4. Determination of the trend of the time series and the trend equation; 5. Calculation of absolute or relative errors of the model.)

26 Highlighting the seasonal component

An estimate of the seasonal component can be found as the quotient of the actual levels of the series divided by the centered moving averages.

To start it is necessary to find the average estimates of the seasonal component Si for the period (quarter, month). Seasonal component models typically assume that seasonal interactions cancel out over a period.

In the multiplicative model, the mutual absorption of seasonal impacts is expressed in the fact that the sum of the values ​​of the seasonal component for all quarters should be equal to the number of periods in the cycle.

Aligning initial levels using a moving average: A) The levels of the series are summed up sequentially for each period of time for every 4 quarters with a shift by 1 point in time and the conditional annual volumes of consumption are determined b) Divide the resulting amounts by 4, we get moving averages. The resulting adjusted values ​​do not contain a seasonal component. c) We bring these values ​​into accordance with the actual moments of time, for which we find the average value of 2 moving averages - centered moving averages.

27. Correlation coefficient.

To determine the degree linear connection, the correlation coefficient is calculated.

To determine the nonlinear relationship, the correlation index is determined

, 0 1

Determination coefficient: R 2 = 2 - for line. communications. R 2 = 2 - for nonlinear. communications.

Shows how much of a change in the indicator y from its average value depends on the change in the factor x from its average value. The closer the R² value is to 1, the more accurate the model.

Of all the regression equations obtained, the best one is the one with the largest coefficient of determination.

If several factors are studied (more than 2), then in this case the multiple correlation coefficient is calculated. R Y, X 1, X 2.. XN - multiple correlation coefficient.

When analyzing the influence of several factors on each other, a correlation matrix is ​​determined, which consists of all possible paired linear correlation coefficients.

Correlation matrix: