Factors the influence of which is studied in the analysis economic activity are classified according to various criteria. First of all, they can be divided into two main types: internal factors that depend on the activities of a given organization, and external factors not dependent on this organization.

Internal factors, depending on the magnitude of their impact on economic indicators, can be divided into major and minor. The main factors include factors related to the use of labor resources, fixed assets and materials, as well as factors due to supply and marketing activities and some other aspects of the functioning of the organization. The main factors have a fundamental impact on the generalized economic indicators. External factors that do not depend on this organization are due to natural and climatic (geographical), socio-economic, as well as external economic conditions.

Depending on the duration of their impact on economic indicators, constant and variable factors can be distinguished. The first type of factors affects economic indicators, which is not limited in time. Variable factors affect economic performance only for a certain period of time.

Factors can be subdivided into extensive (quantitative) and intensive (qualitative) according to the nature of their influence on economic indicators. So, for example, if the influence of labor factors on the volume of output is studied, then a change in the number of workers will be an extensive factor, and a change in the productivity of one worker will be an intensive factor.

Factors affecting economic indicators, according to the degree of their dependence on the will and consciousness of employees of the organization and other persons, can be divided into objective and subjective factors. Objective factors can include weather conditions, natural disasters that do not depend on human activities. Subjective factors are entirely dependent on people. The overwhelming majority of factors should be classified as subjective.

Factors can also be subdivided, depending on the scope of their action, into factors of unlimited and factors of limited action. The first type of factors operates everywhere, in all sectors of the national economy. The second type of factors affects only within an industry or even an individual organization.

By their structure, factors are divided into simple and complex. The overwhelming majority of factors are complex, including several component parts... At the same time, there are also factors that do not lend themselves to dismemberment. For example, return on assets is an example of a complex factor. The number of days the equipment worked during a given period is a simple factor.

By the nature of the impact on the generalizing economic indicators, direct and indirect factors are distinguished. So, the change in the cost of goods sold, although it has the opposite effect on the amount of profit, should be considered direct factors, that is, a factor of the first order. A change in the value of material costs has an indirect effect on profit, i.e. affects profit not directly, but through the cost, which is a factor of the first order. Based on this, the level of material costs should be considered a factor of the second order, that is, an indirect factor.

Depending on whether it is possible to quantify the influence of a given factor on a generalizing economic indicator, there are measurable and non-measurable factors.

This classification is closely interconnected with the classification of reserves for increasing the efficiency of economic activities of organizations, or, in other words, reserves for improving the analyzed economic indicators.

Economic analysis of RAP

Economic analysis production activities of an enterprise, or situational analysis, is the first type of analysis that determines the situations in which the enterprise is located, i.e. identifying the circumstances affecting the entire course of his production, economic and financial activities.

The objectives of the analysis are to identify the place that the enterprise occupies in the general economic space, its current production capabilities, the consumed labor, material and technical and financial resources.

The task of the analysis is to reflect the main factors that determine the strategy of the enterprise, i.e. ways to achieve this goal.

Enterprise strategy should:

correspond to the real state of affairs and market requirements, for which mechanisms of its adaptation to the ongoing changes are necessary;

be reflected in the actions of all divisions of the enterprise (production, supply, finance, marketing, management, personnel, research and development) and implemented through effective actions of managers to achieve specific, pre-planned results;

to be the main goal of the enterprise as a whole and, therefore, all its divisions and each employee individually.

In the second case, a comprehensive analysis of the internal resources of the enterprise is carried out:

organizational and management analysis;

financial and economic analysis.

test

Chapter 3. INDEX METHOD FOR DETERMINING THE INFLUENCE OF FACTORS ON THE GENERAL INDICATOR

In statistics, planning and analysis of economic activity, index models are the basis for a quantitative assessment of the role of individual factors in the dynamics of changes in generalizing indicators.

So, studying the dependence of the volume of output at an enterprise on changes in the number of employees and their labor productivity, one can use the following system of interrelated indices:

In = eD1R1 / eD0R0;

In = eD0R1 / eD0R0, eD1R1 / eD0R1;

where In is the general index of the change in the volume of output,

Ir - individual (factor) index of changes in the number of employees;

Id - factorial index of changes in labor productivity of workers;

D0, D1 - average annual output of marketable (gross) output per worker, respectively, in the baseline and reporting periods;

R1, R0 - the average annual number of industrial and production personnel, respectively, in the baseline and reporting periods.

If a generalizing economic indicator is a product of quantitative (volumetric) and qualitative indicators-factors, then when determining the influence of a quantitative factor, the qualitative indicator is fixed at the basic level, and when determining the influence of a qualitative factor, the quantitative indicator is fixed at the level of the reporting period.

The index method makes it possible to decompose not only relative, but also absolute deviations of the generalizing indicator by factors.

In our example, the formula In = eD1R1 / eD0R0 allows you to calculate the value of the absolute deviation (increase) of the generalized indicator - the volume of output of the enterprise's marketable output:

пNт = еD1R1 - еD0R0,

where пNт is the absolute increase in the volume of commercial output in the analyzed period.

This deviation was formed under the influence of changes in the number of workers and their labor productivity. To determine what part of the total change in the volume of output was achieved due to the change in each of the fators separately, it is necessary to eliminate the influence of the other factor when calculating the influence of one of them.

The formula In = eD0R1 / eD0R0 `eD1R1 / eD0R1 corresponds to this condition. In the first factor, the influence of labor productivity is eliminated, in the second - the number of employees, therefore, the increase in the volume of output due to a change in the number of employees is determined as the difference between the numerator and denominator of the first factor:

pNtR = eD0R1 - eD0R0.

The increase in the volume of output due to changes in the productivity of workers is determined similarly by the second factor:

pNDT = eD1R1 - eD0R1.

The stated principle of decomposition of the absolute increase (deviation) of the generalized indicator by factors is suitable for the case when the number of factors is equal to two (one of them is quantitative, the other is qualitative), and the analyzed indicator is presented as their product.

Index theory does not give general method decomposition of the absolute deviations of the generalizing indicator by factors when the number of factors is more than two.

Analysis and evaluation of the accounting policy of the accounting policy of EKOIL LLC

Table 1 Main economic indicators of EKOIL LLC for 2012-2014. Indicators 2012 2013 2014 Deviations 2014 to 2013 2013 to 2012 +; -% +; -% Revenue, t. P. 21214 27401 16712 -10689 60.99 6187 129.16 Cost of sales, tons ...

Analysis accounting statements in OOO "MiD-Line"

Let's assess the influence of factors on the profit from sales. Table 2 Analysis of profit from sales, thousand rubles ...

Features of management accounting in an organization

For the purposes of strategic enterprise management, the management accounting system is considered as a system for recruiting and interpreting information about costs, expenses and product costs, i.e. ...

The cost of production and its reduction (on the example of the Zhemkonsky consumer society)

According to the data given in table 2.5 ...

Preparation and analysis of the financial statements of the enterprise

The effectiveness of the production, investment and financial activities of an organization is characterized by its financial results. The overall financial result is profit ...

Management audit

Consider. external factors of the macro-environment and factors of the micro-environment, factors of the internal environment using situational audit ...

Accounting for finished products and their sales

The change in the volume of production is influenced by the factors characterizing the use of labor and material resources, OPF ...

Cost accounting, cost and efficiency analysis of milk and finished product production

Gross livestock production is the total volume of the industry's products produced for a given period of time ...

Accounting and analysis of distribution costs in trade on the example of NRUTP "Krynitsa"

Different factors influence distribution costs in different ways. So, the factors contributing to cost reduction include: - overfulfillment of the turnover plan ...

Accounting intangible assets and planning of administrative expenses

Unlike direct material costs, direct costs for wages or other types of costs, management costs in the corresponding budget are not tied to sales or production volumes ...

Accounting for the formation and use net profit

Profitability indicators characterize the efficiency of the enterprise as a whole, the profitability of various areas of the enterprise as a whole, the profitability of various areas of activity, cost recovery, etc.

An index is a statistical indicator that represents the ratio of two states of a feature. With the help of indices, comparisons are made with the plan, in dynamics, in space. The index is called simple (synonyms: private ...

Factor index analysis. Methodology and problems

In the process of economic analysis, analytical processing of economic information, a number of special methods and techniques are used ...

Differential calculus method.

The theoretical basis for a quantitative assessment of the role of individual factors in the dynamics of the effective (generalizing) indicator is differentiation.

In the method of differential calculus, it is assumed that the total increment of functions (resulting indicator) differs into terms, where the value of each of them is determined as the product of the corresponding partial derivative by the increment of the variable over which this derivative is calculated. Let us consider the problem of finding the influence of factors on the change in the resulting indicator by the method of differential calculus using the example of a function of two variables. Let the function z = f (x, y) be given, then if the function is differentiable, its increment can be expressed as

where - change of functions;

Δx (x 1 - x o) - change in the first factor;

- change in the second factor;

Is an infinitely small quantity of a higher order than

The influence of factors x and y on the change in z is determined in this case as

and their sum is the main (linear with respect to the increment of factors) part of the increment of the differentiable function. It should be noted that the parameter is small for sufficiently small changes in factors and its values can differ significantly from zero for large changes in factors. Since this method gives an unambiguous decomposition of the influence of factors on the change in the resulting indicator, this decomposition can lead to significant errors in assessing the influence of factors, since it does not take into account the value of the remainder, i.e. .

Let's consider the application of the method using a specific function as an example: z = xy. Let the initial and final values of the factors and the resulting indicator (x 0, y 0, z 0, x 1, y 1, z 1) be known, then the influence of factors on the change in the resulting indicator is determined, respectively, by the formulas:

It is easy to show that the remainder in the linear expansion of the function z = xy is

Indeed, the total change in the function was, and the difference between the total change and is calculated by the formula

Thus, in the method of differential calculus, the so-called indecomposable remainder, which is interpreted as a logical error of the method of differentiation, is simply discarded. This is the "inconvenience" of differentiation for economic calculations, which, as a rule, require an exact balance of changes in the effective indicator and the algebraic sum of the influence of all factors.

Index method for determining the influence of factors on the generalizing indicator.

So, studying the dependence of the volume of output at an enterprise on changes in the number of employees and their labor productivity, one can use the following system of interrelated indices:

(5.2.1)

(5.2.2)

where I N is the general index of the change in the volume of production;

I R - individual (factor) index of changes in the number of employees;

I D - factorial index of changes in labor productivity of workers;

D 0, D 1 - the average annual output of marketable (gross) output per worker, respectively, in the baseline and reporting periods;

R 0, R 1 - the average annual number of industrial and production personnel, respectively, in the baseline and reporting periods.

The above formulas show that the overall relative change in the volume of output is formed as the product of relative changes in two factors: the number of workers and their labor productivity. The formulas reflect the practice of constructing factor indices adopted in statistics, the essence of which can be formulated as follows. If a generalizing economic indicator is a product of quantitative (volumetric) and qualitative indicators-factors, then when determining the influence of a quantitative factor, the qualitative indicator is fixed at the basic level, and when determining the influence of a qualitative factor, the quantitative indicator is fixed at the level of the reporting period.

The index method makes it possible to decompose not only relative, but also absolute deviations of the generalizing indicator by factors. In our example, the formula (5.2.1) allows you to calculate the value of the absolute deviation (increase) of the generalizing indicator - the volume of output of the commercial output of the enterprise:

where is the absolute increase in the volume of commercial output in the analyzed period.

This deviation was formed under the influence of changes in the number of workers and their labor productivity. To determine what part of the total change in the volume of output was achieved due to the change in each of the factors separately, it is necessary to eliminate the influence of the other factor when calculating the influence of one of them.

Formula (5.2.2) corresponds to this condition. In the first factor, the influence of labor productivity is eliminated, in the second - the number of employees, therefore, the increase in the volume of output due to a change in the number of employees is determined as the difference between the numerator and denominator of the first factor:

The increase in the volume of output due to changes in the productivity of workers is determined similarly by the second factor:

The stated principle of decomposition of the absolute increase (deviation) of the generalizing indicator by factors is suitable for the case when the number of factors is equal to two (one of them is quantitative, the other is qualitative), and the analyzed indicator is presented as their product.

The theory of indices does not provide a general method for decomposing the absolute deviations of the generalized indicator by factors when the number of factors is more than two.

Chain substitution method.

This method consists, as already proved, in obtaining a number of intermediate values of the generalizing indicator by successive replacement of the basic values of the factors with the actual ones. The difference between two intermediate values of the generalizing indicator in the chain of substitutions is equal to the change in the generalizing indicator caused by a change in the corresponding factor.

V general view we have the following system calculations by the method of chain substitutions:

- the basic value of the generalizing indicator;

- intermediate value;

………………………………………………..

…………………………………………………

Is the actual value.

The general absolute deviation of the generalizing indicator is determined by the formula

The general deviation of the generalized indicator is decomposed into factors:

due to a change in factor a

due to a change in factor b

The chain substitution method, like the index method, has drawbacks that you should be aware of when using it. First, the calculation results depend on the successive replacement of factors; secondly, an active role in changing the generalizing indicator is unreasonably often attributed to the influence of a change in a qualitative factor.

For example, if the investigated indicator z has the form of a function, then its change over the period is expressed by the formula

where Δz is the increment of the generalizing indicator;

Δx, Δy - increment of factors;

x 0 y 0 - basic values of factors;

t 0 t 1 - respectively the base and reporting periods of time.

Grouping the last term with one of the first in this formula, we obtain two different versions of chain substitutions.

First option:

Second option:

In practice, the first option is usually used (provided that x is a quantitative factor and y is a qualitative factor).

This formula reveals the influence of a qualitative factor on the change in the generalizing indicator, that is, the expression of a more active connection to obtain an unambiguous quantitative value of individual factors without observing additional conditions is not possible.

Weighted finite difference method.

This method consists in the fact that the magnitude of the influence of each factor is determined by both the first and the second order of substitution, then the result is summed up and the average value is taken from the resulting sum, which gives a single answer about the value of the influence of the factor. If more factors are involved in the calculation, then their values are calculated for all possible substitutions. Let us describe this method mathematically using the notation used above.

As you can see, the method of weighted finite differences takes into account all variants of substitutions. At the same time, when averaging, it is impossible to obtain an unambiguous quantitative value of individual factors. This method is very laborious and, in comparison with the previous method, complicates the computational procedure, since it is necessary to iterate over all possible options substitutions. Basically, the method of weighted finite differences is identical (only for the two-factor multiplicative model) to the method of simple addition of the indecomposable remainder when dividing this remainder between the factors equally. This is confirmed by the following transformation of the formula

Likewise

It should be noted that with an increase in the number of factors, and hence the number of substitutions, the described identity of the methods is not confirmed.

Logarithmic method.

This method consists in the fact that a logarithmically proportional distribution of the remainder is achieved according to the two desired factors. In this case, no prioritization of the factors is required.

Mathematically, this method is described as follows.

The factorial system z = xy can be represented in the form lg z = lg x + lg y, then

Dividing both sides of the formula by and multiplying by Δz, we obtain

(*)

where

The expression (*) for Δz is nothing more than its logarithmic proportional distribution over the two desired factors. That is why the authors of this approach called this method “the logarithmic method for decomposing the increment Δz into factors”. The peculiarity of the logarithmic decomposition method is that it allows you to determine the non-residual influence of not only two, but also many isolated factors on the change in the effective indicator, without requiring the establishment of a sequence of actions.

In a more general form, this method was described by the mathematician A. Humal, who wrote: “Such a division of the increase in the product can be called normal. The name is justified by the fact that the obtained division rule remains in force for any number of factors, namely: the increment of the product is divided between variable factors in proportion to the logarithms of their coefficients of change. " Indeed, in the case of the presence of a larger number of factors in the analyzed multiplicative model of the factor system (for example, z = xypm), the total increment of the effective indicator Δz will be

Decomposition of the increment into factors is achieved by introducing the coefficient k, which, in the case of equality to zero or mutual compensation of factors, does not allow using the specified method. The formula for Δz can be written differently:

where

In this form, this formula is currently used as a classical one, describing the logarithmic method of analysis. It follows from this formula that the total increment of the effective indicator is distributed among the factors in proportion to the ratio of the logarithms of the factor indices to the logarithm of the effective indicator. It does not matter which logarithm is used (natural ln N or decimal lg N).

The main disadvantage of the logarithmic method of analysis is that it cannot be “universal”, it cannot be used in the analysis of any kind of models of factor systems. If, when analyzing multiplicative models of factor systems using the logarithmic method, obtaining the exact values of the influence of factors (in the case when) is achieved, then with the same analysis of multiple models of factor systems, it is not possible to obtain exact values of the influence of factors.

So, if the multiple model of the factorial system is presented in the form

then ,

then a similar formula can be applied to the analysis of multiple models of factorial systems, i.e.

where

If in the multiple model of the factor system , then when analyzing this model we get:

It should be noted that the subsequent division of the factor ∆z ’y by the method of logarithm into the factors ∆z’ c and ∆z ’q cannot be carried out in practice, since the logarithmic method, in its essence, provides for obtaining logarithmic ratios, which for the decomposing factors will be approximately the same. This is precisely the disadvantage of the described method. The use of a "mixed" approach in the analysis of multiple models of factor systems does not solve the problem of obtaining an isolated value from the entire set of factors that affect the change in the effective indicator. The presence of approximate calculations of the magnitudes of factorial changes proves the imperfection of the logarithmic method of analysis.

Method of coefficients. This method, described by the Russian mathematician I. A. Belobzhetsky, is based on comparing the numerical value of the same basic economic indicators under different conditions. A. Belobzhetsky proposed to determine the magnitude of the influence of factors as follows:

The described method of coefficients captivates with its simplicity, but when substituting digital values into the formulas, the result of I. A. Belobzhetsky turned out to be correct only by chance. With the exact implementation of algebraic transformations, the result of the total influence of factors does not coincide with the magnitude of the change in the effective indicator obtained by direct calculation.

The method of splitting the increments of factors.

In the analysis of economic activity, the most common tasks are the tasks of direct deterministic factor analysis. From an economic point of view, such tasks include an analysis of the implementation of the plan or the dynamics of economic indicators, in which the quantitative value of the factors that influenced the change in the effective indicator is calculated. From a mathematical point of view, the problems of direct deterministic factor analysis represent the study of a function of several variables.

A further development of the method of differential calculus was the method of splitting the increments of factor signs, in which one should split the increments of each of the variables into sufficiently small segments and recalculate the values of the partial derivatives for each (already sufficiently small) displacement in space. The degree of fragmentation is taken such that the total error does not affect the accuracy of economic calculations.

Hence, the increment of the function z = f (x, y) can be represented in general form as follows:

where n is the number of segments into which the increment of each factor is divided;

A x n = - a change in the function z = f (x, y) due to a change in the factor x by the value;

A y n = - a change in the function z = f (x, y) due to a change in the factor y by the value

The error ε decreases with increasing n.

For example, when analyzing a multiple model of a factorial system of the form by the method of splitting increments of factorial features, we obtain the following formulas calculation of quantitative values of the influence of factors on the resulting indicator:

ε can be neglected if n is large enough.

The method of splitting increments of factor signs has advantages over the method of chain substitutions. It allows you to determine unambiguously the magnitude of the influence of factors with a predetermined accuracy of calculations, is not associated with the sequence of substitutions and the choice of qualitative and quantitative indicators-factors. The splitting method requires compliance with the differentiability conditions of the function in the considered region.

An integral method for assessing factor influences.

The further logical development of the method of splitting the increments of factor signs was the integral method of factor analysis. This method is based on summing the increments of a function defined as a partial derivative multiplied by the increment of the argument over infinitesimal intervals. In this case, the following conditions must be observed:

continuous differentiability of a function, where an economic indicator is used as an argument;

the function between the start and end points of the elementary period changes in a straight line;

the constancy of the ratio of the rates of change of factors

In general form, the formulas for calculating the quantitative values of the influence of factors on the change in the resulting indicator (for the function z = f (x, y) - of any kind) are derived as follows, which corresponds to the limiting case when:

where Гe is a straight-line oriented segment on the plane (x, y) connecting the point (x 0, y 0) with the point (x 1, y 1).

In real economic processes, the change in factors in the domain of the function definition can occur not along a straight line segment e, but along a certain oriented curve. But since the change in factors is considered for an elementary period (that is, for the minimum period of time during which at least one of the factors will gain an increment), then the trajectory of the curve is determined in the only possible way - a straight-line oriented segment of the curve connecting the initial and final points of the elementary period.

Let us derive a formula for the general case.

The function of changing the resulting indicator from factors is set

Y = f (x 1, x 2, ..., x t),

where x j is the value of factors; j = 1, 2, ..., m; y - the value of the resulting indicator.

The factors change in time, and the values of each factor at n points are known, i.e., we will assume that n points are given in an m-dimensional space:

where x ji is the value of the j-th indicator at the moment i.

Points M 1 and M p correspond to the values of the factors at the beginning and end of the analyzed period, respectively.

Suppose that the indicator y received an increment of Δy for the analyzed period; let the function y = f (x 1, x 2, ..., x m) be differentiable and f "xj (x 1, x 2, ..., x m) is the partial derivative of this function with respect to the argument x j.

Suppose Li is a line segment connecting two points M i and M i + 1 (i = 1, 2,…, n-1).

Then the parametric equation of this straight line can be written in the form

Let us introduce the notation

Taking into account these two formulas, the integral over the segment Li can be written as follows:

j = 1, 2, ..., m; I = 1,2, ..., n-1.

Calculating all the integrals, we obtain the matrix

The element of this matrix y ij characterizes the contribution of the j-th indicator to the change in the resulting indicator for the period i.

Summing up the values of Δy ij according to the matrix tables, we get the following row:

(Δy 1, Δy 2, ..., Δy j, ..., Δy m.);

The value of any j-th element of this line characterizes the contribution of the j-th factor to the change in the resulting indicator Δy. The sum of all Δy j (j = 1, 2, ..., m) is the full increment of the resulting indicator.

There are two areas of practical use of the integral method in solving problems of factor analysis. The first direction can be attributed to the tasks of factor analysis, when there is no data on changes in factors within the analyzed period, or it is possible to abstract from them, that is, there is a case when this period should be considered as elementary. In this case, calculations should be carried out along an oriented straight line. This type of factorial analysis problems can be conventionally called static, since the factors involved in the analysis are characterized by the invariability of position in relation to one factor, the constancy of the conditions for the analysis of the measured factors, regardless of their location in the model of the factor system. The increments of factors are measured in relation to one factor selected for this purpose.

The static types of tasks of the integral method of factor analysis should include calculations related to the analysis of the implementation of the plan or dynamics (if the comparison is made with the previous period) of indicators. In this case, there is no data on changes in factors within the analyzed period.

The second direction can be attributed to the tasks of factor analysis, when there is information about changes in factors within the analyzed period and it must be taken into account, that is, the case when this period, in accordance with the available data, is divided into a number of elementary ones. In this case, the calculations should be carried out along some oriented curve connecting the point (x 0, y 0) and the point (x 1, y 1) for the two-factor model. The problem is how to determine the true form of the curve along which the movement of factors x and y took place in time. This type of factorial analysis problems can be conventionally called dynamic, since the factors involved in the analysis change in each period divided into sections.

The dynamic types of problems of the integral method of factor analysis should include calculations related to the analysis of time series of economic indicators. In this case, it is possible to choose, albeit approximately, an equation that describes the behavior of the analyzed factors in time for the entire period under consideration. In this case, in each elementary period being divided, an individual value, different from others, can be taken. The integral method of factor analysis finds application in the practice of deterministic economic analysis.

In contrast to the chain method, the integral method has a logarithmic law of redistribution of factor loads, which indicates its great advantages. This method is objective, since it excludes any assumptions about the role of factors prior to analysis. Unlike other methods of factor analysis, the integral method observes the provision on the independence of factors.

An important feature of the integral method of factor analysis is that it provides a general approach to solving problems of the very different kind regardless of the number of elements included in the model of the factor system, and the form of communication between them. At the same time, in order to simplify the computational procedure for decomposing the increment of the resulting indicator into factors, one should adhere to two groups (types of factor models: multiplicative and multiple.)

The computational procedure for integration is the same, and the resulting final formulas for calculating the factors are different. Formation of working formulas of the integral method for multi-plication models. The use of the integral method of factor analysis in deterministic economic analysis most completely solves the problem of obtaining uniquely determined values of the influence of factors.

There is a need for formulas for calculating the influence of factors for many types of models of factor systems (functions). It was established above that any model of a finite factorial system can be reduced to two types - multiplicative and multiple. This condition predetermines that the researcher deals with two main types of models of factor systems, since the rest of the models are their varieties.

Calculation operation definite integral according to a given integrand and a given interval of integration, it is performed according to a standard program stored in the memory of the machine. In this regard, the task is reduced only to the construction of integrands that depend on the type of function or model of the factor system.

To facilitate the solution of the problem of constructing integrands, depending on the type of model of the factor system (multiplicative or multiple), we propose matrices of initial values for - constructing integrands of the elements of the structure of the factor system. The principle inherent in the matrices makes it possible to construct the integrands of the elements of the structure of the factorial system for any set of elements of the model of the finite factorial system. Basically, the construction of integrands for the elements of the structure of a factor system is an individual process, and in the case when the number of structural elements is measured in a large number, which is a rarity in economic practice, they proceed from specifically specified conditions.

When forming working formulas for calculating the influence of factors in the conditions of using a computer, use following rules, reflecting the mechanics of working with matrices: the integrands of the elements of the structure of the factor system for multiplicative models are constructed by multiplying the complete set of elements of the values taken for each row of the matrix, referred to a specific element of the structure of the factor system, with subsequent decoding of the values given on the right and at the bottom of the matrix of initial values ( Table 5.1).

Table 5.1

The matrix of initial values for constructing the integrands of the structure elements of multiplicative models of factor systems

Elements of the structure of the factor system	Elements of the multiplicative model of the factor system							The integrand formula








The integrand formula		y / x = (y 0 + kx) dx	z / x = (z 0 + lx) dx	q / x = (q 0 + mx) dx	p / x = (p 0 + nx) dx	m / x = (m 0 + ox) dx	n / x = (n 0 + px) dx

Let us give an example of constructing integrands.

Example:

Type of models of the factorial system f = x y zq (multiplicative model).

Factor system structure

Construction of integrands

where

Formation of working formulas of the integral method for multiple models. The integrands of the elements of the structure of the factor system for multiple models are constructed by entering under the integral sign the initial value obtained at the intersection of the lines, depending on the type of the model and the elements of the structure of the factor system, with subsequent decoding of the values given to the right and down from the matrix of initial values.

The subsequent calculation of a definite integral over a given integrand and a given integration interval is performed using a computer according to a standard program in which the Simpson formula is used, or manually in accordance with the general rules of integration.

In the absence of universal computing facilities Let us propose a set of formulas for calculating structure elements for multiplicative and multiple models of factor systems, which are most often found in economic analysis, which were derived as a result of the integration process. Taking into account the need to simplify them as much as possible, a computational procedure was performed to compress the formulas obtained after calculating certain integrals (integration operations).

Let us give an example of constructing working formulas for calculating the elements of the structure of a factor system.

Example:

The type of the model of the factorial system f = xyzq (multiplicative model).

Factor system structure

Working formulas for calculating the elements of the structure of the factor system:

The use of working formulas is significantly expanded in deterministic chain analysis, in which the identified factor can be stepwise decomposed into components, as it were, in another plane of analysis.

An example of a deterministic chain factor analysis can be an on-farm analysis of a production association, in which the role of each production unit in achieving better result in general for the union.

Rating analysis- one of the options for a comprehensive assessment of the financial condition of the enterprise. Rating analysis is a method of comparative assessment of the activities of several enterprises. The essence of the rating assessment is as follows: businesses line up(grouped) according to certain characteristics or criteria.

Signs or criteria reflect either individual aspects of the enterprise (profitability, solvency, etc.) or characterize the enterprise as a whole (sales volume, market volume, reliability).

When conducting rating analysis there are two main methods: expert and analytical. The expert method is based on the experience and qualifications of experts. Experts on the basis of available information, according to their methods, analyze the enterprise. The analysis takes into account both quantitative and qualitative characteristics of the enterprise.

Unlike the expert method, the analytical method is based only on quantitative indicators... The analysis is carried out according to formalized calculation methods. When applying analytical method there are three main stages:

primary "filtration" of enterprises. At this stage, enterprises are eliminated, about which, with a high degree of probability, we can say that their reporting causes great suspicion;

calculation of coefficients, according to a previously approved method;

There are several disadvantages that reduce the effectiveness of rating analysis when determining the financial condition of an enterprise:

The reliability of the information on which the rating is based. The rating analysis is carried out by independent agencies on the basis of public, official statements of the enterprise. The official reporting that enterprises publish in the media is the balance sheet. The imperfection of the Russian accounting system, gaps in Russian financial legislation, the large volume of the shadow economy - all this does not allow one to fully trust the official reporting of enterprises. This problem can be partially solved by conducting an audit of the company's reporting.

Late rating analysis. As a rule, the rating is calculated based on the balance for the year. Annual balances are due until March 31 of the year following the reporting year. Then it takes some time to compile a rating. Thus, the rating appears on the basis of information that was relevant 3-4 months ago. During this time, the state of the enterprise could change significantly.

Subjectivity of expert opinion (for expert method rating analysis). It is difficult to formalize the opinions of experts, and the position of an enterprise in the rating largely depends on them.

The most complete and detailed study of the company's activities for assigning a rating assessment can be carried out by the company's employees. Since, in addition to official information, they can use inside information... However, enterprise employees may be subjective in assessing performance and are not always competent enough to conduct such an analysis.

5.3. methods of quantitative analysis of the influence of factors on the change in the final indicator

In the analysis of economic activity, which is sometimes called accounting analysis, methods of deterministic modeling of factor systems prevail, which give an accurate (and not with some probability characteristic of stochastic modeling), balanced characterization of the influence of factors on the change in the result indicator. But this balance is achieved different methods... Let's consider the main methods of deterministic factor analysis.

Differential calculus method. Theoretical basis for a quantitative assessment of the role of individual factors in the dynamics of the resultant generalizing indicator is differentiation.

In the method of differential calculus, it is assumed that the total increment of a function (resulting indicator) is decomposed into terms, where the value of each of them is determined as the product of the corresponding partial derivative by the increment of the variable over which the given derivative is calculated. Let us consider the problem of finding the influence of factors on the change in the resulting indicator by the method of differential calculus using the example of a function of two variables.

Let the function z -fix, y) be given; then, if the function is differentiable, its increment can be expressed as

Dg = - Dx4 - Du + 0 (h / dx2 + D ;; 2), 5x 8y Y

where Az = (zi -Zo) change of function; Ax = (*! X0) change of the first factor; Ay = (yi -y0) change in the second factor;

0 (- / Dx + & y2) is an infinitely small quantity of a higher order than

This value is discarded in the calculations (it is often denoted r - epsilon).

The influence of the factor x and y on the change in z is determined in this case as

AZx = -Ax and AZv = -yAy "

and their sum is the main, linear with respect to the increment of the factor, part of the increment of the differentiable

functions. It should be noted that the parameter O (VA * 2 + Ay2) is small at

sufficiently small changes in factors and its values can differ significantly from zero with large changes in factors. Since this method gives an unambiguous decomposition of the influence of factors on the change in the resulting indicator, then this time

position can lead to significant errors in assessing the influence of factors, since it does not take into account the value of the

Consider the application of the method using a specific example

functions: Let the initial and final values

factors and the resulting indicator, then the influence of factors on the change in the resulting indicator is determined, respectively, by the formulas

It is easy to show that the remainder in the linear expansion of the function z xy is equal to AxAy. Indeed, the total change in the function was - and the difference between the total change (Azx + Azy) and Az is calculated by the formula

Δz Azx Azy = (xlyi XaYv) y0Ax x ^ Ay =

UM) - (* oYi - * oyo) = * i (Y. Yo) -ho (Yi ~ Yo) =

"(* Yi ~ JCqVo)" ki ~ xo) Ui (Yi "U = = (x #) y ^)) (x0yi Xou0) ~ ui ~ y0) x0 (yi Yo) ~~ = (Yi Y0) ^ xz) Ahay.

Thus, in the method of differential calculus, the so-called indecomposable remainder, which is interpreted as a logical error of the method of differentiation, is simply discarded. This is the "inconvenience" of differentiation for economic calculations, which, as a rule, require an exact balance of changes in the final indicator and the algebraic sum of the influence of all factors.

Index method for determining factors for a generalizing indicator. In statistics, planning and analysis of economic activity, index models are the basis for a quantitative assessment of the role of individual factors in the dynamics of changes in generalizing indicators.

So, studying the dependence of the volume of sales of products at the enterprise on changes in the number of employees and their productivity, you can use the following system

interrelated indices:

where ./* is the general index of the change in the volume of product sales;

Г - individual (factorial) index of changes in the number of employees;

1 ° factorial index of changes in labor productivity of workers;

D, Dy - average annual output per worker, respectively, in the baseline and reporting periods; RQ, RX average annual headcount, respectively, in baseline

and reporting periods.

The above formulas show that the overall relative change in the volume of production is formed as the product of relative changes in two factors: the number of workers and their labor productivity. The formulas reflect the practice of constructing factor indices adopted in statistics, the essence of which can be formulated as follows.

If the generalizing economic indicator is a product of quantitative (volumetric) and qualitative indicators-factors, then when determining the influence of a quantitative factor, the qualitative indicator is fixed at the basic level, and when determining the influence of a qualitative factor, the quantitative indicator is fixed at the level of the reporting period.

The index method makes it possible to decompose not only relative, but also absolute deviations of the generalizing indicator by factors.

In our example, the formula (1) allows you to calculate the value of the absolute deviation (increase) of the generalizing indicator - the volume of production of the enterprise:

dlg = id, * i-ІЗД).

where AN is the absolute increase in the volume of production in the analyzed period.

This deviation was formed under the influence of changes in the number of workers and their labor productivity. To determine what part of the total change in the volume of production dos

tied by changing each of the factors separately, it is necessary when calculating the influence of one of them to eliminate the influence of the other factor.

The increase in the volume of production due to changes in the productivity of workers is determined similarly by the second factor:

Formula (2) corresponds to this condition. In the first factor, the influence of labor productivity is eliminated, in the second - the number of workers, therefore, the increase in the volume of production due to a change in the number of workers is determined as the difference between the numerator and the denominator of the first factor:

The theory of indices does not provide a general method for decomposing the absolute deviations of the generalizing indicator by factors when the number of factors is more than two and if their relationship is not multiplicative.

Chain substitution method (difference method). This method consists in obtaining a number of intermediate values of the generalizing indicator by sequentially replacing the basic values of the factors with the actual ones. The difference between two intermediate values of the generalizing indicator in the substitution chain is equal to the change in the generalizing indicator caused by the change in the corresponding factor.

In general, we have the following system of calculations by the method of chain substitutions:

= / (af $ ya ...) - the basic value of the summarizing indicator; factors

Yo = / (in | A () C () D? D ...) - intermediate value; - intermediate value;

intermediate value;

actual value.

The general absolute deviation of the generalizing indicator is determined by the formula

The general deviation of the generalized indicator is decomposed into factors:

due to a change in factor a -

W ^ ya-yo - / (eoVo4> ->;

due to a change in factor b -

FiafactftQ ...)

The chain substitution method, like the index method, has drawbacks that you should be aware of when using it. First, the calculation results depend on the sequence of replacing factors; secondly, an active role in changing the generalizing indicator is unreasonably often attributed to the influence of a change in a qualitative factor.

For example, if the investigated indicator z has the form of a function, then its change over the period is expressed by the formula

where Az is the increment of the generalizing indicator; Ah, ay increment of factors; х№ у0 - basic values of factors;

the baseline and reporting periods, respectively.

Grouping in this formula the last term with one of the first, we obtain two different options chain substitutions. First option:

Second option:

Az = x ^ y + (y0 + Ay) Ax = XdAy + y) AX.

In practice, the first option is usually used, provided that x is a qualitative factor and y is a quantitative factor.

This formula reveals the influence of the qualitative factor on the change in the generalizing indicator, that is, the expression (y0 + Ay) Ax is more active, since its value is set by multiplying the increment of the qualitative factor by the reported value of the quantitative factor. Thus, the entire increase in the generalizing indicator due to the joint change in factors is attributed to the influence of only the qualitative factor.

Thus, the problem of accurately determining the role of each factor in changing the generalizing indicator is not solved by the usual method of chain substitutions.

In this regard, the search for ways to improve the precise unambiguous definition of the role of individual factors in the context of introducing complex economic and mathematical models of factor systems in economic analysis is of particular relevance.

The task is to find a rational computational procedure (method of factor analysis), which eliminates conventions and assumptions and achieves an unambiguous result of the magnitudes of the influence of factors.

Method of simple addition of indecomposable remainder. Not finding a sufficiently complete justification of what to do with the remainder, in the practice of economic analysis, they began to use the method of adding an irreducible remainder to a qualitative or quantitative (main or derived) factor, and also to divide this remainder among the factors equally. The last proposal is theoretically substantiated by S. M. Yugenburg 1104, p. 66 - 831.

Taking into account the above, you can get the following set of formulas.

First option

& ZX ^ & xy0 + AxAy + Yes "O" o + Ay) = Axy ^;

Wtppg> ™ ISYAPYANT

D? L = AxyQ; Azv = Auh $ + AxAy - Ay (xQ + Ax) = Auh ^.

The third option

There are other proposals that are rarely used in the practice of economic analysis. For example, refer AXAy to the second term with a coefficient equal to

Ahuo + Auchts

And add the remainder to the first

term. This technique was defended by V.E. Adamov. He believed that “despite all the objections, the only practically unacceptable, although based on certain agreements on the choice of index weights, would be the method of interrelated study of the influence of factors using the qualitative indicator of the weights of the reporting period in the index, and the basis weights in the volume index. period ".

The described method, although it removes the problem of "indecomposable residue", is associated with the condition of determining quantitative and qualitative factors, which complicates the task when using large factor systems. At the same time, the expansion of the total increase in the result indicator by the chain method depends on the sequence of substitution. In this regard, it is not possible to obtain an unambiguous quantitative value of individual factors without observing additional conditions.

Weighted finite difference method. This method consists in the fact that the magnitude of the influence of each factor is determined by both the first and the second order of substitution, then the result is summed up and the average value is taken from the resulting sum, which gives a single answer about the value of the influence of the factor. If more factors are involved in the calculation, then their values are calculated for all possible substitutions.

Let us describe this method mathematically using the notation used above.

As you can see, the method of weighted finite differences takes into account all variants of substitutions. At the same time, when averaging, it is impossible to obtain an unambiguous quantitative value of individual factors. This method is very laborious and, in comparison with the previous method, complicates the computational procedure, since it is necessary to sort out all possible variants of substitutions. Basically, the method of weighted finite differences is identical (only for the two-factor multiplicative model) to the method of simple addition of the indecomposable remainder when dividing this remainder between the factors equally. This is confirmed by the following transformation of the formula:

Likewise

It should be noted that with an increase in the number of factors, and hence the number of substitutions, the described identity of the methods is not confirmed.

Logarithmic method. This method, described by V. Fedorova and Yu. Egorov, consists in the fact that a logarithmically proportional distribution of the remainder is achieved by the two desired factors. In this case, no prioritization of the factors is required.

Mathematically, this method is described as follows.

The factorial system z - xy can be represented as Igz = lgx + lgy, then

where U = logx (+] g jv Igzo = IgXQ + 1

Expression (4) for Az is nothing more than its logarithmic proportional distribution over the two desired factors. That is why the authors of this approach called this method “the logarithmic method for decomposing the increment Az into factors”. The peculiarity of the logarithmic decomposition method is that it allows one to determine the non-residual influence of not only two, but also many isolated factors on the change in the result indicator, without requiring the establishment of a sequence of actions.

In a more general form, this method was described by A. Humal, who wrote: “Such a division of the increase in the work can be called normal. The name is justified by the fact that the resulting division rule remains in force for any number of factors, namely: the increase in the product is divided between variable factors in proportion to the log-

rhymes of their coefficients of change. " Indeed, if there is more factors in the analyzed multiplicative model of the factor system (for example, z the total increment of the effective indicator will be:

M = & + Mu = ■ Mkx + (5)

In this form, this formula (5) is currently used as a classical one, describing the logarithmic method of analysis. It follows from this formula that the total increment of the final indicator is distributed among the factors in proportion to the ratio of the logarithms of the factor indices to the logarithm of the final indicator. It does not matter which logarithm is used (natural mN or decimal IgN).

The main disadvantage of the logarithmic method of analysis is that it cannot be “universal”, it cannot be used in the analysis of any kind of models of factor systems. If, when analyzing multiplicative models of factor systems using the logarithmic method, it is possible to obtain exact values of the influence of factors (in the case when Δr = 0), then with the same analysis of multiple models of factor systems, it is not possible to obtain exact values of the influence of factors.

So, if a short model of the factorial system is presented in the form

then a similar formula (5) can be applied to the analysis of multiple models of factorial systems, i.e.

Az = U + My + Aztx + Dg * yy

gae $ --k; th

This approach was used by D.I. Vainshenker and V.M. Ivanchenko when analyzing the implementation of the plan for profitability. When determining the magnitude of the increase in profitability due to the increase in profit, they used the coefficient k "x.

Having failed to obtain an exact result in the subsequent analysis, D.I. Vainschenker and V.M. Ivanchenko limited themselves to using the logarithmic method only at the first stage (when determining the factor Az "J. which is nothing more than specific gravity an increase in one of the factors in the total increase in the constituent factors. The mathematical content of the coefficient L is identical to the “equity method” described below.

If in the short model of the factorial system Y

then, when analyzing this model, we get:

& Z = Z C = Azx + Azy = Azx + AZtAZql

Azx ~ Azkx = Az-Dyu = & z-Azxi

It should be noted that the subsequent division of the factor Az "y by the method of logarithm into the factors Az" c and Az "q cannot be carried out in practice, since the logarithmic method, in its essence, provides for obtaining logarithmic deviations, which for the separating factors will be approximately the same. and the disadvantage of the described method.Application of the "mixed" approach in the analysis of multiple models of factorial systems does not solve the problem of obtaining an isolated value from the entire set of factors influencing the change in the final indicator.The presence of approximate calculations of the values of factorial changes proves the imperfection of the logarithmic method of analysis.

Method of coefficients. This method, described by I.A. different conditions.

IA Belobzhetskiy proposed to determine the magnitude of the influence of factors as follows;

The described method of coefficients impresses with its simplicity, but when substituted digital values Belobzhetskii's result in the formulas was correct only by accident. With the exact implementation of algebraic transformations, the result of the total influence of factors does not coincide with the magnitude of the change in the result indicator obtained by direct calculation.

The method of splitting the increments of factors. In the analysis of economic activity, the most common tasks are the tasks of direct deterministic factor analysis. From an economic point of view, such tasks include an analysis of the implementation of the plan or the dynamics of economic indicators, in which the quantitative value of the factors that influenced the change in the final indicator is calculated. From a mathematical point of view, the problems of direct deterministic factor analysis represent the study of a function of several variables.

Hence, the increment of the function z -f (x, y) can be represented in general form as follows:

function change

due to the change in the factor x by the value Ax xx xih

due to the change in the factor y by the value of the Error e decreases with increasing n.

For example, when analyzing a multiple model of the factor system

of the form z = - by the method of splitting the increments of factorial recognition

kov we obtain the following formulas for calculating the quantitative values of the influence of factors on the resulting indicator:

e can be neglected if n is large enough. The method of splitting increments of factor signs has advantages over the method of chain substitutions. It allows you to determine unambiguously the magnitude of the influence of factors with a predetermined accuracy of calculations, is not associated with the sequence of substitutions and the choice of qualitative and quantitative indicators-factors. The splitting method requires compliance with the conditions for the differentiability of the function in the region under consideration.

An integral method for assessing factor influences. Further

logical development of the method of splitting increments of factor

features became an integral method of factor analysis. This

the method, like the previous one, was developed and justified by A.D. Sheremet and his students.It is based on summation

increments of a function defined as a partial derivative,

multiplied by the increment of the argument at infinitesimal intervals. In this case, the following conditions must be observed:

continuous differentiability of a function, where an economic indicator is used as an argument;

the function between the starting and ending points of the elementary period changes along the straight line Ge;

the constancy of the ratio of the rates of change of factors

In general form of the formula for calculating the quantitative values of the influence of factors on the change in the resulting indicator

where Ge is a straight-line oriented segment on the plane (x, y) connecting the point (xa, y) with the point (x1r y ().

In real economic processes, a change in factors in the field of definition of a function can occur not along a straight line segment Ge, but along a certain oriented curve G. But since the change in factors is considered for an elementary period (i.e., for a minimum period of time during which at least one of the factors will receive an increment), then the trajectory Г is determined uniquely possible way- a straight-line oriented segment Ge connecting the start and end points of the elementary period.

Let us derive a formula for the general case.

The function of changing the resulting indicator from factors is set

y = f (xx, x2, ..., хт),

where Xj is the value of factors; j - 1, 2, ..., t;

y - the value of the resulting indicator.

The factors change in time, and the values of each factor at n points are known, that is, we will assume that n points are given in space:

Mx = (x, x, ..., X1m), M2 = * m)> Mi = (A> Ar- ^

where x | the value of the th indicator at the moment /.

Points Мх and М2 correspond to the values of the factors at the beginning and end of the analyzed period, respectively.

Suppose that the exponent y received an increment of Ay for

analyzed period; let the function y = f (xl, x2, ..., xm) be differentiable and y -fxj (xl xj is the partial derivative of

this function by argument xy.

Suppose L "is a segment of a straight line connecting two points M" and M * 1 (/ "= 1,2, n - D). Then the parametric equation of this straight line can be written in the form

Xj = x "j + Xі) f.j = 1, 2, m; 0< і < I.

Let us introduce the notation

AUi, = J / v (^ i ^ 2, ..., xm) (i> c (; Y = 1,2, ..., m.

Considering these two formulas, the integral over the segment i can be written as follows:

The element of this matrix characterizes the contribution of the th indicator to the change in the resulting indicator for the period

Summing up the values to the matrix tables, we get

the following line:

The value of any i-th element of this line characterizes the contribution of the y-th factor to the change in the resulting indicator Ay. The sum of all Ay, (/ = 1,2, ..., t) is the full increment of the resulting indicator.

There are two areas of practical use of the integral method in solving problems of factor analysis.

The first direction can be attributed to the tasks of factor analysis, when there is no data on changes in factors within the analyzed period, or it is possible to abstract from them, that is, there is a case when this period should be considered as elementary. In this case, the calculations should be carried out along the oriented straight line Ge. This type of factor analysis tasks can be conventionally called static, since the factors participating in the analysis are characterized by the invariability of position in relation to one factor, the constancy of the conditions for the analysis of the measured factors, regardless of their location in the model of the factor system. The increments of factors are measured in relation to one factor selected for this purpose.

The second direction can be attributed to the tasks of factor analysis, when there is information about changes in factors within the analyzed period and it must be taken into account, that is, the case when this period, in accordance with the available data, is divided into a number of elementary ones. In this case, the calculations should be carried out along some oriented curve Г connecting the point (x0, y) and the point (x, y) for the two-factor model. The problem is how to determine the true form of the curve Г, along which the movement of factors x y took place in time. the type of factor analysis problems can be conventionally called dynamic, since the factors involved in the analysis change in each period divided into sections.

The integral method of factor analysis finds application in the practice of computer deterministic economic analysis.

The static type of problems of the integral method of factor analysis is the most developed and widespread type of problems in the deterministic economic analysis of the economic activity of controlled objects.

In comparison with other methods of a rational computational procedure, the integral method of factor analysis eliminated the ambiguity in assessing the influence of factors and made it possible to obtain the most accurate result. The results of calculations by the integral method differ significantly from those given by the method of chain substitutions or modifications of the latter. The greater the magnitude of changes in factors, the greater the difference.

The method of chain substitutions (its modifications), in its essence, takes into account the ratio of the values of the measured factors weaker. The greater the gap between the values of the increments of factors included in the model of the factor system, the more the integral method of factor analysis reacts to this.

In contrast to the chain method, the integral method has a logarithmic law of redistribution of factor loads, which indicates its great advantages. This method is objective as it excludes any suggestions about the role of factors prior to analysis. Unlike other methods of factor analysis, the integral method observes the provision on the independence of factors.

An important feature of the integral method of factor analysis is that it provides a general approach to solving problems of various types, regardless of the number of elements included in the model of the factor system and the form of connection between them. At the same time, in order to simplify the computational procedure for decomposing the increment of the resulting indicator into factors, one should adhere to two groups (types) of factor models: multiplicative and multiple. The computational procedure for integration is the same, and the resulting final formulas for calculating the factors are different.

Formation of working formulas of the integral method for multiplicative models. The use of the integral method of factor analysis in deterministic economic analysis most completely solves the problem of obtaining uniquely determined values of the influence of factors.

There is a need for formulas for calculating the influence of factors for many types of models of factor systems (functions).

It was established above that any model of a finite factorial system can be reduced to two types - multiplicative and multiple. This condition predetermines that the researcher deals with two main types of models of factor systems, since the rest of the models are their varieties.

The operation of calculating a definite integral for a given integrand and a given integration interval is performed according to a standard program stored in the memory of the machine. In this regard, the task is reduced only to the construction of integrands that depend on the type of function or model of the factor system.

To facilitate the solution of the problem of constructing integrands, depending on the type of model of the factor system (multiplicative or multiple), we propose matrices of initial values for constructing integrands of the elements of the structure of the factor system. The principle inherent in the matrices makes it possible to construct the integrands of the elements of the structure of the factorial system for any set of elements of the model of the finite factorial system. Basically, the construction of integrands for the elements of the structure of a factor system is an individual process, and in the case when the number of structural elements is measured in a large number, which is a rarity in economic practice, they proceed from specifically specified conditions.

When forming working formulas for calculating the influence of factors in the conditions of using a computer, the following rules are used that reflect the mechanics of working with matrices: the integrands of the elements of the structure of the factorial system for multiplicative models are constructed by the product of a complete set of elements of the values taken for each row of the matrix, referred to a certain element of the structure of the factorial system with the subsequent decoding of the values given to the right and at the bottom of the matrix of initial values (Table 5.2).

Let us give examples of constructing subintephalic expressions.

Example 1 (see table 5.2).

Type of models of the factor system / = xyzq (multiplicative model).

Factor system structure

Formation of working formulas of the integral method for multiple models. The integrand expression of the elements of the structure of the factor system for multiple models is constructed by entering under the integral sign the initial value obtained at the intersection of the lines, depending on the type of model and the elements of the structure of the factor system, with subsequent decoding of the values shown on the right and at the bottom of the matrix of initial values.

Example 2 (Table 5.3).

Du + Dg + d # +

■ A * + ^ + Az + ^ + Ap

4 o (y0 + zu +? O + kx) z

Lou + Az + Hell, & Az has Hell

- -; / = -; t = -; n = -H

Dx lx ah ah

The subsequent calculation of a definite integral over a given integrand and a given integration interval is performed using a computer according to a standard program in which Simpson's formula is used, or manually in accordance with general rules integration.

In the absence of universal computational tools, we will offer a set of formulas for calculating structural elements for multiplicative (Table 5.4) and multiple (Table 5.3) models of factor systems that are most often found in economic analysis, which were derived as a result of the integration process. Taking into account the need to simplify them as much as possible, a computational procedure was performed to compress the formulas obtained after calculating certain integrals (integration operations).

Let us give examples of constructing working formulas for calculating the elements of the structure of the factor system.

Example 1 (see table 5.4).

The type of the model of the factorial system f = xyzq (multiplicative model).

Factor system structure

a / = shtt shrt = A * + 4 + 4 + 4 Working formulas for calculating the elements of the structure of the factor system:

Factor system model view

Working formulas for calculating the elements of the structure of the factor system

An example of a deterministic chain factor analysis can be an on-farm analysis of a production association, in which the role of each production unit in achieving the best result for the association as a whole is assessed.

The integral method gives accurate estimates of factor influences. The calculation results do not depend on the sequence of substitutions and the sequence of calculating factorial influences. The method is applicable for all types of continuously differentiable functions; it does not require prior knowledge of which factors are quantitative and which are qualitative.

The application of the integral method requires knowledge of the basics of differential calculus, integration techniques and the ability to find the derivatives of various functions. At the same time, in the theory of the analysis of economic activity for practical applications, the final working formulas of the integral method for the most common types of factor dependencies have been developed, which makes this method available to every analyst. Here are some of them.

1. Factor model of the type u = xy: Au = Aih + Aig

Ax-Ay, Aih = y0Ax + ---;

Auy = x0Ay + -; Au = Au + Aih.

2, Dm = Aih + Diu + Dmg;

Dm = l: 0 -ts -Ay + -l0 -Ay-Az + -Zq ■ Ax -Ay + -Ay ■ Az ■ Dx;

4. Factorial type model

The use of these models makes it possible to select factors, the purposeful change of which makes it possible to obtain the desired value of the result indicator.

In the analysis of economic activity, which is sometimes called accounting analysis, methods of deterministic modeling of factor systems prevail, which give an accurate (and not with some probability characteristic of stochastic modeling), balanced characterization of the influence of factors on the change in the result indicator. But this balance is achieved by different methods. Let's consider the main methods of deterministic factor analysis.

Differential calculus method. Differentiation is the theoretical basis for a quantitative assessment of the role of individual factors in the dynamics of the resultant generalizing indicator.

Let the function z -fix, y) be given; then, if the function is differentiable, its increment can be expressed as

where Az = (zj - th) - function change;

Ax = (*! - x0) - change in the first factor;

Du - (yi -y0) - change in the second factor;

0 (f Dx + by2) is an infinitely small quantity of a higher order than

This value is discarded in the calculations (it is often denoted r - epsilon).

The influence of the factor x and y on the change in r is determined in this case as

A, = -Ax and A, = -Ay,

and their sum is the main, linear with respect to the increment of the factor, part of the increment of the differentiable

functions. It should be noted that the parameter O (YA * 2 + Ay2) is small at

position can lead to significant errors in assessing the influence of factors, since it does not take into account the value of the remainder of the term, I e C | (\ || Dx? + dy ~ W

Let us consider the application of the method on the example of a specific function: £ = VI Let the initial and final values

factors and re; \ na iru yuikch o | | okch ;; ue | h 1ha,) '; l, u, X1, t about | -

yes, the influence of factors on the change in the resulting indicator is determined, respectively, by the formulas

It is easy to show that the remainder in the linear expansion of the function z - xy is equal to DxDy. Indeed, the overall change in function was XpY! - X ^ Yo, and the difference between the total change (Δ ^ + Δz>,) and Δz is calculated by the formula

= (x, y, - XuYo) - y0 (x, -x0) - X0 (y, - y0) =

FL) - (Xo, -X (Y0) = X, (y, -y0) -x0 (y, -y0) =

0'1 - Fo) (X \ -Ho> = AxDy.

Thus, in the method of differential calculus, the so-called indecomposable remainder, which is interpreted as a logical error of the method of differentiation, is simply discarded. This is the "inconvenience" of differentiation for economic calculations, which, as a rule, require an exact balance of changes in the final indicator and the algebraic sum of the influence of all factors.

So, studying the dependence of the volume of sales of products at an enterprise on changes in the number of employees and their labor productivity, one can "■ restore" the following system of interrelated indices: £ A> ^ o

(3)

where ./* is the general index of the change in the volume of product sales;

Г - individual (factorial) index of changes in the number of employees;

1 ° - factorial index of changes in labor productivity of workers;

B, Bu - average annual output per worker, respectively, in the baseline and reporting periods;

YaO, YaKh - the average annual number of personnel, respectively, in the baseline and reporting periods.

The index method makes it possible to decompose not only relative, but also absolute deviations of the generalizing indicator by factors.

In our example, the formula (1) allows you to calculate the value of the absolute deviation (increase) of the generalizing indicator - the volume of production of the enterprise:

AN - X A A -X A) A)>

where АЖ is the absolute increase in the volume of production in the analyzed period.

This deviation was formed under the influence of changes in the number of workers and their labor productivity. To determine what part of the total change in the volume of production was

tied by changing each of the factors separately, it is necessary when calculating the influence of one of them to eliminate the influence of the other factor.

The increase in the volume of production due to changes in the productivity of workers is determined similarly by the second factor:

In general, we have the following system of calculations by the method of chain substitutions:

Y0 = / (n0 /> oCo ^ P ") - the basic value of the generalizing indicator; factors

y0 = / (a, A (> Co ^ () ...) is an intermediate value;

Intermediate value;

G;; = / ("LrLU; ...) - fairies and reading.

The general absolute deviation of the generalizing indicator is determined by the formula

The general deviation of the generalized indicator is decomposed into factors:

due to a change in factor a -

due to a change in factor b -

For example, if the investigated exponent r has the form of a function r = f (x, y) - xy, then its change over the period A1 - ^ - T0 is expressed by the formula

Ag -HtsAu + UoDx + y0Dx + DxDy,

where M is the increment of the generalizing indicator;

Ah, Ay - increment of factors; х, у0 - basic values of factors;

О - respectively base and reporting periods of time.

Grouping the last term with one of the first in this formula, we obtain two different versions of chain substitutions. First option:

In practice, the first option is usually used, provided that x is a qualitative factor and y is a quantitative factor.

Thus, the problem of accurately determining the role of each factor in changing the generalizing indicator is not solved by the usual method of chain substitutions.

In this regard, the search for ways to improve the precise unambiguous definition of the role of individual factors in the context of the introduction of complex economic and mathematical models of factor systems in economic analysis is of particular relevance.

Method of simple addition of indecomposable remainder. Not finding a sufficiently complete justification of what to do with the remainder, in the practice of economic analysis they began to use the method of adding an irreducible remainder to a qualitative or quantitative (main or derived) factor, and also to divide this remainder between the factors equally. The last proposal is theoretically substantiated by S. M. Yugenburg 1104, p. 66 - 831.

Taking into account the above, you can get the following set of formulas.

First option

] ZtpppT / G iyapt / gyat

DgL - Lhu0; Mx. - Luh0 + LxLy = Ay (x0 + Dx) = DuX |.

Dhuo + Luho

and add the remainder to the first

Let us describe this method mathematically using the notation used above.

As you can see, the method of weighted finite differences takes into account all variants of substitutions. At the same time, when averaging, it is impossible to obtain an unambiguous quantitative value of individual factors. This method is very laborious and, in comparison with the previous method, complicates the computational procedure, since it is necessary to sort out all possible variants of substitutions. Basically, the method of weighted finite differences is identical (only for the two-factor multiplicative model) to the method of simple addition of the indecomposable remainder when dividing this remainder between the factors equally. This is confirmed by the following transformation of the formula:

Lx ’+ Yo) ^ Lhyu

Likewise

It should be noted that with an increase in the number of factors, and hence the number of substitutions, the described identity of the methods is not confirmed.

Mathematically, this method is described as follows.

The factorial system z - xy can be represented in the form ^ =! Yah +! Yy, then

Dg = 1 ^ 1 -1826 - (1in, - 1 & x0) + (1 & y, - 1 & y0)

rac 1 ^, = 18A-, +18 ^! / ^ = 1b ^ o + 1BY0-

(4)

Expression (4) for L1 is nothing more than its logarithmic proportional distribution over the two desired factors. That is why the authors of this approach called this method "the logarithmic method of decomposing the increment Л1 into factors". The peculiarity of the logarithmic decomposition method is that it allows one to determine the non-stop influence of not only two, but also many isolated factors on the change in the result indicator, without requiring the establishment of a sequence of actions.

rhymes of their coefficients of change. " Indeed, in the case of the presence of a larger number of factors in the analyzed multiplicative model of the factor system (for example, r = hurt), the total increment of the effective indicator Dg will be:

Dg = Dg * + Dg * = DgA * + Dg A

The main disadvantage of the logarithmic method of analysis is that it cannot be “universal”, it cannot be used in the analysis of any kind of models of factor systems. If, when analyzing multiplicative models of factor systems using the logarithmic method, it is possible to obtain exact values of the influence of factors (in the case when Δr = 0), then with the same analysis of multiple models of factor systems, it is not possible to obtain exact values of the influence of factors.

So, if a short model of the factorial system is presented in the form

then a similar formula (5) can be applied to the analysis of multiple models of factorial systems, i.e.

D * = Dx ", + b * y + D + d

where k "x d-; k" y ---.

Having failed to obtain an exact result in the subsequent analysis, D. I. Vainschenker and V. M. Ivanchenko limited themselves to using the logarithmic method only at the first stage (when determining the factor λ). which is nothing more than the proportion of the increase in one of the factors in the total increase in the constituent factors. The mathematical content of the coefficient b is identical to the "method of equity participation" described below.

If in the short model of the factor system

* = -, Y = c + d,

then, when analyzing this model, we get:

It should be noted that the subsequent subdivision of the factor Amy by the method of logarithm into the factors A1C and Ar1 cannot be carried out in practice, since the logarithmic method in its essence provides for obtaining logarithmic deviations, which for the separating factors will be approximately the same. This is precisely the disadvantage of the described method. The use of the "mixed" approach in the analysis of multiple models of factor systems does not solve the problem of obtaining an isolated value from the entire set of factors that influence the change in the result indicator. The presence of approximate calculations of the magnitudes of factorial changes proves the imperfection of the logarithmic method of analysis.

Method of coefficients. This method, described by I. A. Belobzhetsky, is based on comparing the numerical value of the same basic economic indicators under different conditions.

IA Belobzhetskiy proposed to determine the magnitude of the influence of factors as follows;

Hence the increment of the function r - f (x, y) can be represented in general form as follows:

AI - A "x ^ T, A (x0 + i ^" x> Yo + ‘& Y) - change in the function r = f (x, y)

due to the change in the factor x by the value Ax == x, - x (b

Apy = D> E /; (x0 + iA "x, y0 + iA" y) + є, - change of function

due to the change in the factor y by the amount Lu ~ y. - \\ y Error e decreases with increasing n.

For example, when analyzing a multiple model of the factor system

type - by the method of splitting increments of factorial signs

kov we obtain the following formulas for calculating the quantitative values of the influence of factors on the resulting indicator:

e can be neglected if n is large enough. The method of splitting increments of factor signs has advantages over the method of chain substitutions. It allows you to unambiguously determine the magnitude of the influence of factors with a predetermined accuracy of calculations, is not associated with the sequence of substitutions and the choice of qualitative and quantitative indicators-factors. The splitting method requires compliance with the differentiability conditions of the function in the considered region.

An integral method for assessing factor influences. The further logical development of the method of splitting increments of factorial features was the integral method of factor analysis. This method, like the previous one, was developed and substantiated by A.D. Sheremet and his students. It is based on summing the increments of a function defined as a partial derivative multiplied by the increment of the argument over infinitesimal intervals. In this case, the following conditions must be observed:

1) continuous differentiability of the function, where the economic indicator is used as an argument;

2) the function between the starting and ending points of the elementary period changes along the straight line Ge;

3) the constancy of the ratio of the rates of change of factors

In general form of the formula for calculating the quantitative values of the influence of factors on the change in the resulting indicator

(for a function z f (x, y) of any kind) are derived as follows, which corresponds to the limiting case when n - »oo:

A "= lim A" = lim £ A "(* o +" A "x, y0 + iA" y) A "x =) f ± dx \

where Ge is a straight-line oriented segment on the plane (x, y) connecting the point (x, y) with the point (x1r y ().

In real economic processes, a change in factors in the field of definition of a function can occur not along a straight line segment Ge, but along a certain oriented curve G. But since the change in factors is considered for an elementary period (i.e., for a minimum period of time during which at least one of the factors will receive an increment), then the trajectory Г is determined in the only possible way - a straight-line oriented segment Ge, connecting the initial and final points of the elementary period.

Let us derive a formula for the general case.

The function of changing the resulting indicator from factors is set

where Xj is the value of factors; j = 1, 2, ..., m;

y - the value of the resulting indicator.

The factors change in time, and the values of each factor at n points are known, i.e., we will assume that n points are given in the / m-dimensional space:

My = (*), x \, ..., xxm), M2 = (x (, y% T .., Xm), Mn = (x "j, x £ r ..,

where x | the value of the th indicator at the moment i.

Points Мх and М2 correspond to the values of the factors at the beginning and end of the analyzed period, respectively.

Suppose that the indicator y received an increment of Ay for the analyzed period; let the function y = f (x1, x2, ..., xt) be differentiable and y - / x] (xl x, x) be the partial derivative of this function with respect to the argument xy.

Suppose 1_ "is a segment of a straight line connecting two points M 'and M + (/" = 1,2, ..., n - Г). Then the parametric equation of this straight line can be written in the form

Let us introduce the notation

Considering these two formulas, the integral over the segment I can be written as follows:

The value of any i-th element of this line characterizes the contribution of the y-th factor to the change in the resulting indicator Ay. The sum of all Ay, - (/ = 1,2, ..., t) is the full increment of the resulting indicator.

There are two areas of practical use of the integral method in solving problems of factor analysis.

The first direction can be attributed to the tasks of factor analysis, when there is no data on changes in factors within the analyzed period, or it is possible to abstract from them, that is, there is a case when this period should be considered as elementary. In this case, the calculations should be carried out along the oriented straight line Ge. This type of factorial analysis problems can be conventionally called static, since the factors involved in the analysis are characterized by the invariability of position in relation to one factor, the constancy of the conditions for the analysis of the measured factors, regardless of their location in the model of the factor system. The increments of factors are measured in relation to one factor selected for this purpose.

The second direction can be attributed to the tasks of factor analysis, when there is information about changes in factors within the analyzed period and it must be taken into account, that is, the case when this period, in accordance with the available data, is divided into a number of elementary ones. In this case, the calculations should be carried out along some oriented curve Г connecting the point (x0, y) and the point (x, y) for the two-factor model. The problem is how to determine the true form of the curve Г, along which the movement of factors x and y took place in time. This type of factor analysis problem can be conventionally called dynamic, since the factors involved in the analysis change in each period divided into sections.

The dynamic types of problems of the integral method of factor analysis should include calculations related to the analysis of time series of economic indicators. In this case, it is possible to select, albeit approximately, an equation that describes the behavior of the analyzed factors in time for the entire period under consideration. In this case, in each elementary period being divided, an individual value, different from others, can be taken.

The integral method of factor analysis finds application in the practice of computer deterministic economic analysis.

In comparison with other methods of a rational computational procedure, the integral method of factor analysis eliminated the ambiguity of assessing the influence of factors and made it possible to obtain the most accurate result. The results of calculations by the integral method differ significantly from those given by the method of chain substitutions or modifications of the latter. The greater the magnitude of changes in factors, the greater the difference.

An important feature of the integral method of factor analysis is that it provides a general approach to solving problems of various types, regardless of the number of elements included in the factor system model and the form of connection between them. At the same time, in order to simplify the computational procedure for decomposing the increment of the resulting indicator into factors, one should adhere to two groups (types) of factor models: multiplicative and multiple. The computational procedure for integration is the same, and the resulting final formulas for calculating the factors are different.

Formation of working formulas of the integral method for multiplicative models. Application of the integral method of factor analysis in deterministic economic analysis

most completely solves the problem of obtaining unambiguously determined values of the influence of factors.

There is a need for formulas for calculating the influence of factors for many types of models of factor systems (functions).

To facilitate the solution of the problem of constructing integrands, depending on the type of model of the factor system (multiplicative or multiple), we propose matrices of initial values for constructing integrands of the elements of the structure of the factor system. The principle inherent in the matrices makes it possible to construct the integrands of the elements of the structure of the factorial system for any set of elements of the model of the finite factorial system. Basically, the construction of integrands for the elements of the structure of a factor system is an individual process, and in the case when the number of structural elements is measured in a large number, which is a rarity in economic practice, they proceed from specifically specified conditions.

Table 52

The matrix of initial values for constructing the integrands of the structure elements of multiplicative models of factor systems

The elements			multiplicative model> actor system						Subintephalic formula
		X	Have	G	I am	R	T	NS	Subintephalic formula
I am	Oh	-	Uh	Uh	yah	P "x	TO		-
s- 35 £ 6 R1 5	AU		-	Uh	bgcolor = white> P "x	t "x	-	Yx = p (xo + x) ex
s- 35 £ 6 R1 5	Subintephalic	sv	1	3	3	8	3	3	3	Bx
Where		1	£ 13	313	£ \| 3	£ 13	3 \| s	313

Let us give examples of constructing subintephalic expressions.

Example 1 (see table 5.2).

Type of models of the factorial SYSTEM / = lgu # 7 (multiplicative model).

Factor system structure

Construction of subintephalic expressions

LX = \ Yx ^ xdx ~ \ (l + kx) u + bc) (q0 + tx) cix- o o

AY \ u003d 1 Xx 1xYax - \ * (* 0 + *) (ro + bc) (4 0 + mx) ex- o o

	Multiple model view
Elements of the structure of the factor system		X	X	X	X
		X	U + 1	y + g + h	y + r + h + p
	Oh	eh	Oh	eh	eh
	Oh	Wo + kx	Yo + go + br	Uo + aa + cho	Whoo + * o + Cho + Ro + kx
	Ay	-k (x ^ + x) ex	- / (x0 + x) ex	- / (xo + x) ex	-1 (x0 + x) ex
	Ay	(Whoo + kx) 2	(Uo + їo + kx) 2	(Whoo + + Cho + kx) *	(Whoo +% 0 + Cho + Ro + kx) 2
	A,	-	-t (xo + x) ex	-t (x0 + x) ex	-t (x0 + x) ex
	A,	-	(Yo + ^ o + kx) 2	(Yo + ho + ^ o + ^ x) 2	(Uo + їo + Cho + Ro + kx) 2
	Ah		-	-n (x0 + x) ex	-n (x $ + x) ex
	Ah		-	(Yo + io + Cho + kx) 2	(Yo + C + Cha + Ro + kx) 2
	A,	-	-	-	-o (ho + x) eh
	A,	-	-	-	(Whoo + 1o + Cho + Ro + kx) 2
		X	X	X	X
		X	Y + Z	y + 1 + h	Y + I + H + R
	At	-		-	-
	An	-	-	-	-
	Where	*-	, Du + Dg Dx	Lu + Dg + Dd Dx	Du + Dg + Dd + Dr Dx

factor system
X	X
■ y + z + g + p + m	y + z + g + p + m + n	Where
eh	eh
Yy + ^ +% + Pd + m0 + kx	Yo + £ o + Ro + Po + to + po + ^ c
-1 (Xo + x) (1x	- / (xo + x) c! X	Oh
(Yy + bl +% + Po + Sh + kx) 2	(Yo + £ d + (1o + Pd + U + U + U + k *) 2	Oh
-t (xo + x) ex	-t (x o + x) ex
(Z "o + go + bgcolor = white>
(Yo + go +? O + #) + u + kx) 2	(UO + go +? O + Po + U + Po + kx) 2
	-g (x0 + x) ex	An
	(UO + ^ +? 0 + Po + PCHUno + kx) 2	Oh
... Du + Dg + D? + Ar + At	o Ay + Az + Ag + Ap + Am + Ap	Oh
Oh	Oh	0

Factor system model view

Factor system structure

Formula for calculating structure elements

/ = xy

Y = x1y1 -HoYo = AX + A

■ - Ax = TDx (3 "0+ Yi)

Lu = -Du (x0 + *,)

and

/ -hush

^ = X \ Y1Y \ - XoYo ^ o =

Ax = ^ dx (3 ^ 0y0r0 + YiRo (ri + Ar) +

DxDuDgIntegral method requires knowledge of the basics of differential calculus, integration techniques and the ability to find the derivatives of various functions. At the same time, in the theory of the analysis of economic activity for practical applications, the final working formulas of the integral method for the most common types of factor dependencies have been developed, which makes this method available to every analyst. Here are some of them.

1. Factor model of type u = xy:

a Ax i D their 1p

Au = Au + Aig.

4. Factorial type model

The use of these models makes it possible to select factors, the purposeful change of which makes it possible to obtain the desired value of the result indicator.

Chapter 3. INDEX METHOD FOR DETERMINING THE INFLUENCE OF FACTORS ON THE GENERAL INDICATOR

5.3. methods of quantitative analysis of the influence of factors on the change in the final indicator

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