The influence of factors on the general indicator is determined by the method. Factor analysis

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Ministry of Agriculture of the Russian Federation

FSBEI HPE "VORONEZH STATE AGRARIAN UNIVERSITY NAMED AFTER K.D. GLINKA"

Department of Statistics and Analysis of Economic Activities of Agricultural Enterprises

Test

Subject: Theory of economic analysis

On the topic: Methods for analyzing the quantitative influence of factors on changes in the performance indicator

Pavlovsk - 2011

Methods for analyzing the quantitative influence of factors on changes in performance indicators

Method of differential calculus. The theoretical basis for 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) is divided into terms, where the value of each of them is determined as the product of the corresponding partial derivative and the increment of the variable by which this derivative is calculated. Let's consider the problem of finding the influence of factors on the change in the resulting indicator using the differential calculus method 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 is the change in functions;

Dx(x1 - xo) - change in the first factor;

Change in the second factor;

An infinitesimal 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 represents the main (linear relative to the increment of factors) part of the increment of the differentiable function. It should be noted that the parameter is small for fairly small changes in factors and its values can differ significantly from zero for large changes in factors. Because This method provides an unambiguous decomposition of the influence of factors on the change in the resulting indicator, then this decomposition can lead to significant errors in assessing the influence of factors, since it does not take into account the value of the residual term, i.e. .

Let's consider the application of the method using the example of a specific function: z = xy. Let the initial and final values of the factors and the resulting indicator (x0, y0, z0, x1, y1, z1) be known, then the influence of the factors on the change in the resulting indicator is determined accordingly by the formulas:

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

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 irreducible remainder, which is interpreted as a logical error in the differentiation method, is simply discarded. This is the “inconvenience” of differentiation for economic calculations, in which, as a rule, an exact balance of changes in the effective indicator and the algebraic sum of the influence of all factors is required.

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

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

where IN is the general index of changes in production volume;

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

ID - factor index of changes in labor productivity of workers;

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

R0, R1 - the average annual number of industrial production personnel, respectively, in the base 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 accepted in statistics for constructing factor indices, the essence of which can be formulated as follows. If a generalizing economic indicator is the product of quantitative (volume) 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 into factors not only relative, but also absolute deviations of the generalizing indicator. In our example, formula (5.2.1) allows us to calculate the magnitude of the absolute deviation (increase) of the general indicator - the volume of output of commercial products 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. In order to determine what part of the total change in the volume of output was achieved due to changes 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 output due to a change in the number of employees is determined as the difference between the numerator and the denominator of the first factor:

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

The stated principle of decomposition of the absolute increase (deviation) of a generalizing indicator into 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 provide a general method for decomposing the absolute deviations of a generalizing indicator into factors when the number of factors is more than two.

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

In general, we have the following system of calculations using the chain substitution method:

The basic value of the summary indicator;

Intermediate value;

Actual value.

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

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

due to changes in factor a

due to changes in factor b

The chain substitution method, like the index method, has disadvantages that you should be aware of when using it. Firstly, the calculation results depend on the sequential replacement of factors; secondly, the active role in changing the general indicator is unreasonably often attributed to the influence of changes in the qualitative factor.

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

where Dz is the increment of the general indicator;

Dx, Dy - increment of factors;

x0 y0 - basic values of factors;

t0 t1 are the base and reporting periods of time, respectively.

By grouping the last term in this formula with one of the first, we obtain two different variants 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 one).

This formula reveals the influence of the qualitative factor on the change in the general indicator, i.e. expressing a more active connection, it is not possible to obtain an unambiguous quantitative value of individual factors without meeting 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 second order of substitution, then the result is summed up and the average value is taken from the resulting sum, giving a single answer about the value of the factor’s influence. If more factors are involved in the calculation, then their values are calculated using all possible substitutions. Let us describe this method mathematically, using the notation adopted above.

As you can see, the weighted finite difference method takes into account all substitution options. At the same time, when averaging, it is impossible to obtain an unambiguous quantitative value of individual factors. This method is very labor-intensive and, compared to the previous method, complicates the computational procedure, because you have to go through all possible substitution options. At its core, the method of weighted finite differences is identical (only for a two-factor multiplicative model) to the method of simply adding an indecomposable remainder when dividing this remainder equally between factors. 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 therefore the number of substitutions, the described identity of the methods is not confirmed.

Logarithmic method. This method consists in achieving a logarithmically proportional distribution of the remainder over the two required factors. In this case, there is no need to establish the order of action of the factors.

Mathematically, this method is described as follows.

The factor system z = xy can be represented as log z=log x + log y, then

Dividing both sides of the formula by and multiplying by Dz, we get

Expression (*) for Dz is nothing more than its logarithmic proportional distribution over the two required factors. That is why the authors of this approach called this method “the logarithmic method of decomposing the increment Dz into factors.” The peculiarity of the logarithmic decomposition method is that it allows one to determine the 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. Khumal, who wrote: “Such a division of the increase in a product 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 the 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 Dz will be

The decomposition of growth into factors is achieved by entering the coefficient k, which, if equal to zero or mutual cancellation of factors, does not allow the use of this method. The formula for Dz can be written differently:

In this form, this formula is currently used as a classical one, describing the logarithmic method of analysis. From this formula it follows that the total increase in 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 when analyzing any type of factor system models. 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), then with the same analysis of multiple models of factor systems, obtaining exact values of the influence of factors is not possible.

Thus, if the multiple model of the factor system is represented in the form

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

If in a multiple model of a factor system, then when analyzing this model we obtain:

It should be noted that the subsequent division of the factor Dz"y by the logarithmic method into factors Dz"c and Dz"q cannot be carried out in practice, since the logarithmic method in its essence provides for obtaining logarithmic ratios, which will be approximately the same for the factors being divided. This is precisely the drawback 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 influence the change in the effective indicator. The presence of approximate calculations of the magnitude of factor changes proves the imperfection of the logarithmic method of analysis.

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

The described method of coefficients is captivating in its simplicity, but when substituting digital values into the formulas, the result given by I.A. Belobzhetsky turned out to be correct only by accident. When algebraic transformations are performed accurately, 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.

Method of splitting factor increments. In the analysis of economic activity, the most common problems are direct deterministic factor analysis. From an economic point of view, such tasks include analyzing 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 performance indicator is calculated. From a mathematical point of view, problems of direct deterministic factor analysis represent the study of the function of several variables.

A further development of the method of differential calculus was the method of crushing increments of factor characteristics, in which it is necessary to split the increment of each variable into sufficiently small segments and recalculate the values of partial derivatives for each (already quite small) movement 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;

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

Ayn = - change in the function z = f(x, y) due to a change in the factor y by the amount

The error e decreases as n increases.

For example, when analyzing a multiple model of a factor system of the form by crushing increments of factor characteristics, 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 crushing increments of factor characteristics 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, and is not associated with the sequence of substitutions and the choice of qualitative and quantitative indicators-factors. The fractionation method requires compliance with the conditions of differentiability of the function in the region under consideration.

Integral method for assessing factor influences. A further logical development of the method of crushing increments of factor characteristics was the integral method of factor analysis. This method is based on the summation of the increments of a function, defined as the partial derivative multiplied by the increment of the argument over infinitesimal intervals. In this case, the following conditions must be met:

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 varies along a straight line;

constancy of the ratio of the rates of change of factors

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

where Ge is a straight-line oriented segment on the plane (x, y) connecting the point (x0, y0) with the point (x1, y1).

In real economic processes, changes in factors in the area of definition of the function can occur not along a straight line segment e, but along some oriented curve. But because the change in factors is considered over an elementary period (i.e., over the minimum period of time during which at least one of the factors will receive an increase), then the trajectory of the curve is determined in the only possible way - a straight-line oriented segment of the curve connecting the starting and ending points of the elementary period.

Let us derive the formula for the general case.

The function of changing the resulting indicator from factors is specified

Y = f(x1, x2,..., xm),

where xj is the value of the factors; j = 1, 2,..., t; y is the value of the resulting indicator.

Factors change over time, and the values of each factor at n points are known, i.e. We will assume that n points are given in m-dimensional space:

where xji is the value of the j-th indicator at time i.

Points M1 and Mn correspond to the values of factors at the beginning and end of the analyzed period, respectively.

Let us assume that the indicator y received an increase Dy for the analyzed period; let the function y = f(x1, x2,..., xm) be differentiable and f"xj(x1, x2,..., xm) be the partial derivative of this function with respect to the argument xj.

Let's say Li is a straight line connecting two points Mi and Mi+1 (i=1, 2, …, n-1).

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

Let us introduce the notation

Given these two formulas, the integral over the segment Li can be written as follows:

j = 1, 2,…, m; I = 1,2,…,n-1.

Having calculated all the integrals, we obtain the matrix

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

Having summed up the values of Дyij according to the tables of the matrix, we obtain the following line:

(Dy1, Dy2,..., Dyj,..., Dym.);

differential index factor factor

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 Dy. The sum of all Дyj (j = 1, 2,..., m) is the full increment of the resulting indicator.

We can distinguish two directions for the practical use of the integral method in solving problems of factor analysis. The first direction includes problems of factor analysis, when there is no data on changes in factors within the analyzed period or they can be abstracted from, i.e. 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 factor analysis problem can be conventionally called static, because in this case, the factors participating in the analysis are characterized by the invariance of their position in relation to one factor, the constancy of the conditions for the analysis of the measured factors, regardless of their location in the factor system model. The comparison of factor increments occurs in relation to one factor selected for this purpose.

The static types of problems of the integral method of factor analysis should include calculations related to the analysis of plan implementation or dynamics (if 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 includes the tasks of factor analysis, when there is information about changes in factors within the analyzed period and it must be taken into account, i.e. the case when this period, in accordance with the available data, is divided into a number of elementary ones. In this case, calculations should be carried out along some oriented curve connecting the point (x0, y0) and the point (x1, y1) for a two-factor model. The problem is how to determine the true form of the curve along which the movement of factors x and y occurred over time. This type of factor analysis problems can be conventionally called dynamic, because in this case, the factors involved in the analysis change in each period divided into sections.

Dynamic types of problems of the integral method of factor analysis 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 over time over the entire period under consideration. In this case, in each divided elementary period an individual value can be taken that is different from the others. The integral method of factor analysis is used in the practice of deterministic economic analysis.

Unlike the chain method, the integral method has a logarithmic law of redistribution of factor loads, which indicates its great advantages. This method is objective because it excludes any assumptions about the role of factors before the analysis. Unlike other methods of factor analysis, the integral method adheres to the principle of 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 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, but the resulting final formulas for calculating 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 fully 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 factor system can be reduced to two types - multiplicative and multiple. This condition predetermines that the researcher deals with two main types of factor system models, because the rest of the models are their variations.

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 machine’s memory. In this regard, the task is reduced only to constructing 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 will 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 integrands of the elements of the structure of the factor system for any set of elements of the model of the finite factor system. Basically, the construction of integrand expressions for the elements of the structure of a factor system is an individual process, and in the case when the number of elements of the structure is measured in a large number, which is rare in economic practice, they proceed from specifically specified conditions.

An example of 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.

Bibliography

1. Bakanov M.I., Sheremet A.D. Theory of economic analysis: Textbook. - 4th ed., add. and processed - M.: Finance and Statistics, 2000. - 416 p.

2. Zenkina I.V. Theory of economic analysis, part 1: Textbook. Benefit/Growth. state econ. Univer. - Rostov n/d., - 2001. - 131 p.

3. Lysenko D.V. Economic analysis: textbook. - M.: TK Welby, Prospekt Publishing House, 2008. - 376 p.

4. Zenkina I.V. Theory of economic analysis: Textbook. - M.: Publishing and trading corporation “Dashkov and K?”, Rostov n/d: Nauka - Press, 2007. - 208 p.

5. Theory of economic analysis: Educational and methodological complex / E.A. Edalina; Ulyan. State tech. Univ. - Ulyanovsk: St. Vocational school, 2003. - 108 p.

6. Theory of economic analysis: Textbook / ed. M.I. Bakanov. - 5th ed. Reworked and additional - M.: Finance and Statistics, 2006. - 536 p.

7. Firstova S.Yu. Economic analysis in questions and answers: textbook. Benefit. - M.: KNORUS, 2006 - 184 p.

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

Method of differential calculus. The theoretical basis for quantitative assessment of the role of individual factors in the dynamics of the resulting general 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 and the increment of the variable by which this derivative is calculated. Let's consider the problem of finding the influence of factors on the change in the resulting indicator using the differential calculus method 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

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

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

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

0(f Дх +лу2) is an infinitesimal quantity of a higher order than

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

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

A, =-Ah and A, =-Ay,

and their sum represents the main, linear relative to the increment of the factor part of the increment of the differentiable

functions. It should be noted that the parameter O (АА*2 + Ау2) is small at

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

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

Let's consider the application of the method using the example of a specific function: £ = VI Let the initial and final values be known

factors and re;\ na iru yuikch o | |okch;;ie|h 1ha, )’;l, sch, X1, t o| -

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

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

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

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

0’1 - Fo) (X\-Ho> =AhDu.

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

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

Thus, when studying the dependence of the sales volume of products at an enterprise on changes in the number of employees and their labor productivity, one can “reliably” use the following system of interrelated indices: £ A>^o

(3)

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

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

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

B, Bu - average annual production per worker, respectively, in the base and reporting periods;

Nuclear weapons, nuclear facilities - the average annual number of personnel in the base and reporting periods, respectively.

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

If a generalizing economic indicator is the product of quantitative (volume) and qualitative indicators-factors, then when determining the influence of a quantitative factor, the qualitative indicator is fixed at the base 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 into factors not only relative, but also absolute deviations of the generalizing indicator.

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

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

where AJ is the absolute increase in production volume 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 production volume is

is achieved by changing 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 (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 output due to a change in the number of employees is determined as the difference between the numerator and the denominator of the first factor:

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

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

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

In general, we have the following system of calculations using the chain substitution method:

У0 =/(я0/>оСО^П ") - basic value of the generalizing indicator; factors

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

Pr intermediate value;

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

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

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

due to changes in factor a -

due to changes in factor b -

The chain substitution method, like the index method, has disadvantages that you should be aware of when using it. Firstly, the calculation results depend on the sequence of factor replacement; secondly, the active role in changing the general indicator is unreasonably often attributed to the influence of changes in the qualitative factor.

For example, if the indicator r under study has the form of a function r =/(x, y) - xy, then its change over the period A1 - ^ - Г0 is expressed by the formula

Ag -HtsAu + UoDx + y0Dx + DxDu,

where M is the increment of the general indicator;

Ah, Au - increment of factors; x, y0 - basic values of factors;

O - base and reporting periods of time, respectively.

By grouping the last term in this formula with one of the first, we obtain two different variants 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 one.

This formula reveals the influence of the qualitative factor on the change in the general indicator, i.e. the expression (y0 + Ay)Ax is more active, since its value is established by multiplying the increment of the qualitative factor by the reported value of the quantitative factor. Thus, the entire increase in the general 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 general indicator cannot be solved by the usual method of chain substitutions.

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

The task is to find a rational computational procedure (factor analysis method), in which conventions and assumptions are eliminated and an unambiguous result of the magnitude of the influence of factors is achieved.

Method of simple addition of an indecomposable remainder. Not finding a sufficiently complete justification for what to do with the remainder, in the practice of economic analysis they began to use the method of adding an indecomposable remainder to a qualitative or quantitative (basic or derivative) factor, as well as dividing this remainder equally between the factors. The last proposal is theoretically justified by S. M. Yugenburg 1104, p. 66 - 831.

Taking into account the above, we can obtain the following set of formulas.

First option

]ZtppppT/G iyapt/gyatyat

DgL - Lhu0; Mx. - Lux0 + LxLu = Au (x0 + Dx) = DuX|.

Dhuo+Luho

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, will be the method of interconnected study of the influence of factors using in the index a qualitative indicator of the weights of the reporting period, and in the index of a volumetric indicator - the weights of the base period".

The described method, although it eliminates the problem of the “irreducible remainder,” is associated with the condition for determining quantitative and qualitative factors, which complicates the task when using large factor systems. At the same time, the decomposition of the total increase in the result indicator using 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 meeting additional conditions.

Let us describe this method mathematically, using the notation adopted above.

As you can see, the weighted finite difference method takes into account all substitution options. At the same time, when averaging, it is impossible to obtain an unambiguous quantitative value of individual factors. This method is very labor-intensive and, compared to the previous method, complicates the computational procedure, since it is necessary to go through all possible substitution options. At its core, the method of weighted finite differences is identical (only for a two-factor multiplicative model) to the method of simply adding an indecomposable remainder when dividing this remainder equally between factors. This is confirmed by the following transformation of the formula:

Lx' + Uo) ^Lhyu

Likewise

It should be noted that with an increase in the number of factors, and therefore 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 achieving a logarithmically proportional distribution of the remainder over the two desired factors. In this case, there is no need to establish the order of action of the factors.

Mathematically, this method is described as follows.

The factor system z - xy can be represented in the form ^ = !yah + !yay, then

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

gas 1^, = 18Л-, +18^!/ ^ = 1в^о + 1ВУ0-

(4)

Expression (4) for L1 is nothing more than its logarithmic proportional distribution over the two required factors. That is why the authors of this approach called this method “the logarithmic method of decomposing the L1 increment into factors.” The peculiarity of the logarithmic decomposition method is that it allows one to determine the 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. Khumal, who wrote: “Such a division of the increase in a product 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 the variable factors in proportion to the log

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 = khurt), the total increment of the effective indicator Dg will be:

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

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

The main disadvantage of the logarithmic method of analysis is that it cannot be “universal”; it cannot be used when analyzing any type of factor system models. 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 Dg = 0), then with the same analysis of multiple models of factor systems, obtaining exact values of the influence of factors is not possible.

Thus, if a brief model of the factor system is presented in the form

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

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

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

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

Having not received an accurate result in the subsequent analysis, D. I. Vainshenker and V. M. Ivanchenko limited themselves to using the logarithmic method only at the first stage (when determining the factor Lg"). They obtained subsequent values of the influence of factors using the proportional (structural) coefficient b, which is nothing more than the share 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 a brief factor system model

* = -, U=s+d,

then when analyzing this model we get:

It should be noted that the subsequent division of the factor At!y by the logarithm method into factors A1C and Ar\ cannot be carried out in practice, since the logarithmic method in its essence provides for obtaining logarithmic deviations, which will be approximately the same for the dismembered factors. 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 influence changes in the result indicator. The presence of approximate calculations of the magnitudes of factor changes proves the imperfection of the logarithmic method of analysis.

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

I. A. Belobzhetsky proposed to determine the magnitude of the influence of factors as follows;

The described method of coefficients is captivating in its simplicity, but when substituting digital values into the formulas, I. A. Belobzhetsky’s result turned out to be correct only by chance. When algebraic transformations are carried out accurately, 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.

Method of splitting factor increments. In the analysis of economic activity, the most common problems are direct deterministic factor analysis. From an economic point of view, such tasks include analyzing 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 result indicator is calculated. From a mathematical point of view, problems of direct deterministic factor analysis represent the study of the function of several variables.

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

АІ - А"х^Т, Л(х0 +і^"х>Уо +‘&У) - change of function r =/(x, y)

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

Apu =D >Ё/;(x0 +іA"x,y0 +іA"y) + є, - change of function

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

For example, when analyzing a multiple factor system model

type - by the method of crushing increments of factor recognition

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 crushing increments of factor characteristics 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, and is not associated with the sequence of substitutions and the choice of qualitative and quantitative indicators-factors. The fractionation method requires compliance with the conditions of differentiability of the function in the region under consideration.

Integral method for assessing factor influences. A further logical development of the method of crushing increments of factor characteristics 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 the summation of the increments of a function, defined as the partial derivative multiplied by the increment of the argument over infinitesimal intervals. In this case, the following conditions must be met:

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

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

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

In general terms, formulas for calculating quantitative values of the influence of factors on changes in the resulting indicator

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

A” = lim A" = lim £ L"(*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 (x1yy().

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

Let us derive the formula for the general case.

The function of changing the resulting indicator from factors is specified

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

y is the value of the resulting indicator.

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

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

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

Points Mx and M2 correspond to the values of factors at the beginning and end of the analyzed period, respectively.

Let us assume that the indicator y has received an increment Ay for the analyzed period; let the function y =/(x1, x2,..., xm) be differentiable and y -/x] (xb x, x) be the partial derivative of this function with respect to the argument xy.

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

Let us introduce the notation

Given these two formulas, the integral over 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.

We can distinguish two directions for the practical use of the integral method in solving problems of factor analysis.

The first direction includes problems of factor analysis when there is no data on changes in factors within the analyzed period or they can be abstracted from, i.e., there is a case when this period should be considered as elementary. In this case, calculations should be carried out along the oriented straight line Ge. This type of factor analysis problem can be conventionally called static, since in this case the factors involved in the analysis are characterized by an unchanged position in relation to one factor, the constancy of the conditions for the analysis of the measured factors, regardless of their location in the factor system model. The comparison of factor increments occurs in relation to one factor selected for this purpose.

The second direction includes the tasks of factor analysis, when there is information about changes in factors within the analyzed period and it should 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, calculations should be carried out along some oriented curve Г connecting the point (x0, y) and the point (xy y) for a two-factor model. The problem is how to determine the true form of the curve G along which the movement of factors x and y occurred over time. This type of factor analysis problem can be conventionally called dynamic, since in this case the factors involved in the analysis change in each period divided into sections.

The integral method of factor analysis is used 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 deterministic economic analysis of the economic activities of managed 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 allowed us to obtain the most accurate result. The results of calculations using the integral method differ significantly from those obtained by the method of chain substitutions or modifications of the latter. The greater the magnitude of changes in factors, the more significant the difference.

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

Unlike the chain method, the integral method has a logarithmic law of redistribution of factor loads, which indicates its great advantages. This method is objective because it excludes any suggestions about the role of factors before the analysis is carried out. Unlike other methods of factor analysis, the integral method adheres to the principle of independence of factors.

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

most fully 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 factor system can be reduced to two types - multiplicative and multiple. This condition predetermines that the researcher deals with two main types of factor system models, since the remaining models are their varieties.

To facilitate the solution of the problem of constructing integrands, depending on the type of model of the factor system (multiplicative or multiple), we will 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 integrands of the elements of the structure of the factor system for any set of elements of the model of the finite factor system. Basically, the construction of integrand expressions for the elements of the structure of a factor system is an individual process, and in the case when the number of elements of the structure is measured in a large number, which is rare 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, reflecting the mechanics of working with matrices: integrands of the elements of the structure of the factor system for multiplicative models are constructed by multiplying the complete set of elements of values taken for each row of the matrix, assigned to a specific element of the factor structure system with subsequent decoding of the values given to the right and bottom of the matrix of initial values (Table 5.2).

Table 52

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

Elements			multiplicative model >actor system						Podyntefal formula
		X	U	G	I	R	T	P	Podyntefal formula
I I	Oh	-	Wow	UH	iGH	R"x	TO		-
s- 35 £6 Р1 5	AU		-	Wow	bgcolor=white>P"x	t"x	-	Ux=p(xo+x)yoh
s- 35 £6 Р1 5	Podyntefalnaya	St.	1	3	3	8	3	3	3	bx
Where		1	£13	313	£\|3	£13	3\|z	313

Let us give examples of constructing a subset of intephal expressions.

Example 1 (see Table 5.2).

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

Structure of the factor system

Construction of subscript expressions

LH = \ Ux^xdx ~ \ (l + kx)i+bc)(d0+tx)s_x- o o

AU = 1 Xx 1xYax - \ *(*0 +*)(go +bc)(4 0 +tx)ex- o o

	Type of multiple model
Elements of the factor system structure		X	X	X	X
		X	U + 1	y+y+h	y+g+h+r
	Oh	eh	Oh	eh	eh
	Oh	Uo + kh	Uo + go + bg	Uo+a+cho	Uo +*o+Cho + Po+kh
	Ay	-k(x^ + x)ex	-/(x0 + x)ex	-/(ho +x)yoh	-1(x0 +x)ex
	Ay	(Uo + kx)2	(Uo + io + kx)2	(Uo + + Cho + kh)*	(Uo + %0 + Cho + Po + kh)2
	A,	-	-t(ho + x)yoh	-t(x0 + x)ex	-t(x0 +x)ex
	A,	-	(Yo + ^o + kx)2	(Yo + th + ^o + ^x)2	(Uo + io + Cho + Po + kh)2
	Ah		-	-n(x0 + x)ex	-n(x$ + x)ex
	Ah		-	(Uo + io + Cho + kx)2	(Uo+Ts+Cha + Po+kh)2
	A,	-	-	-	-o(ho + x)yoh
	A,	-	-	-	(Uo + 1o+Cho + Po+kh)2
		X	X	X	X
		X	Y + Z	y + 1 + H	U+I+H+R
	At	-		-	-
	Up	-	-	-	-
	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
Uy+^+%+Ry+t0+kh	Uo +£o+Yo+Po+to+po +^c
-1(Ho +x)(1x	-/(Ho +x)s!x	Oh
(Uy+Ъl+%+Po+Sh+kh)2	(Uo + £y+(1o+ Ry+Sh + Sh+k*)2	Oh
-t(ho+x)yoh	-t(x o + x)yoh
(Z"o + th +bgcolor=white>
(Uo+go +?o +#) +у+кх)2	(UO +go+?o +Ro+Sh + Po+kh)2
	-r(x0+ x)ex	Up
	(UO + ^ +?0 +Po+pChUpo +kh)2	Oh
. Du+Dg+D? +Ar+At	o Ау +Az +Ag + Ar +At +An	Oh
Oh	Oh	0

Type of factor system model

Structure of the factor system

Formula for calculating structure elements

/=xy

S = x1y1 -XoYo =AX+A

■- Ах =ТДх(3"0+ Уі)

Lu=-Du(x0 + *,)

And

/ -khushch

^=Х\У1ы\ - ХУо^о =

Ах= ^дх(3^0у0г0+ Уія о(гі + Дг)+

DxDuDgThe integral method requires knowledge of the basics of differential calculus, integration techniques and the ability to find derivatives of various functions. At the same time, in the theory of business analysis, for practical applications, final working formulas of the integral method have been developed for the most common types of factor dependencies, which makes this method accessible to every analyst. Let's list some of them.

1. Factor model of type u = xy:

a Ah i D their 1p

Ai = Ai + Aig.

4. Factor model type

The use of these models allows you to select factors, the targeted change of which allows you to obtain the desired value of the result indicator.

All phenomena and processes of economic activity of enterprises are interconnected and interdependent. Some of them are directly related to each other, others indirectly. Hence, an important methodological issue in economic analysis is the study and measurement of the influence of factors on the value of the economic indicators under study.

Under economic factor analysis is understood as a gradual transition from the initial factor system to the final factor system, the disclosure of a full set of direct, quantitatively measurable factors that influence the change in the performance indicator.

Based on the nature of the relationship between indicators, methods of deterministic and stochastic factor analysis are distinguished.

Deterministic factor analysis is a methodology for studying the influence of factors whose connection with the performance indicator is functional in nature.

The main properties of the deterministic approach to analysis:
· construction of a deterministic model through logical analysis;
· the presence of a complete (hard) connection between indicators;
· the impossibility of separating the results of the influence of simultaneously acting factors that cannot be combined in one model;
· study of relationships in the short term.

There are four types of deterministic models:

Additive Models represent an algebraic sum of indicators and have the form

Such models, for example, include cost indicators in relation to elements of production costs and cost items; an indicator of the volume of production in its relationship with the volume of output of individual products or the volume of output in individual departments.

Multiplicative models in generalized form can be represented by the formula

An example of a multiplicative model is a two-factor model of sales volume

Where H- average number of employees;

C.B.- average output per employee.

Multiple models:

An example of a multiple model is the indicator of the turnover period of goods (in days). T OB.T:

Where Z T- average stock of goods; O R- one-day sales volume.

Mixed models are a combination of the above models and can be described using special expressions:

Examples of such models are cost indicators per 1 ruble. commercial products, profitability indicators, etc.

To study the relationship between indicators and quantitatively measure the many factors that influenced the effective indicator, we present general model transformation rules in order to include new factor indicators.

To detail the generalizing factor indicator into its components, which are of interest for analytical calculations, the technique of lengthening the factor system is used.

If the original factor model is , a , then the model will take the form .

To identify a certain number of new factors and construct the factor indicators necessary for calculations, the technique of expanding factor models is used. In this case, the numerator and denominator are multiplied by the same number:

To construct new factor indicators, the technique of reducing factor models is used. When using this technique, the numerator and denominator are divided by the same number.

The detail of factor analysis is largely determined by the number of factors whose influence can be quantitatively assessed, therefore multifactorial multiplicative models are of great importance in the analysis. Their construction is based on the following principles:
· the place of each factor in the model must correspond to its role in the formation of the effective indicator;
· the model should be built from a two-factor complete model by sequentially dividing factors, usually qualitative, into components;
· when writing a formula for a multifactor model, factors should be arranged from left to right in the order of their replacement.

Building a factor model is the first stage of deterministic analysis. Next, determine the method for assessing the influence of factors.

Chain substitution method consists in determining a number of intermediate values of the generalizing indicator by sequentially replacing the basic values of the factors with the reporting ones. This method is based on elimination. Eliminate- means to eliminate, exclude the influence of all factors on the value of the effective indicator, except one. Moreover, based on the fact that all factors change independently of each other, i.e. First, one factor changes, and all the others remain unchanged. then two change while the others remain unchanged, etc.

In general, the application of the chain production method can be described as follows:

where a 0, b 0, c 0 are the basic values of factors influencing the general indicator y;

a 1, b 1, c 1 - actual values of factors;

y a, y b, are intermediate changes in the resulting indicator associated with changes in factors a, b, respectively.

The total change D у = у 1 – у 0 consists of the sum of changes in the resulting indicator due to changes in each factor with fixed values of the remaining factors:

Let's look at an example:

table 2

Initial data for factor analysis

Indicators	Legend	Basic values	Actual values	Change
Indicators	Legend	Basic values	Actual values	Absolute (+,-)	Relative (%)
Volume of commercial products, thousand rubles.
Number of employees, people
Output per worker,

We will analyze the impact of the number of workers and their output on the volume of marketable output using the method described above based on the data in Table 2. The dependence of the volume of commercial products on these factors can be described using a multiplicative model:

Then the effect of a change in the number of employees on the general indicator can be calculated using the formula:

Thus, the change in the volume of marketable products was positively influenced by a change in the number of employees by 5 people, which caused an increase in production volume by 730 thousand rubles. and a negative impact was had by a decrease in output by 10 thousand rubles, which caused a decrease in volume by 250 thousand rubles. The combined influence of two factors led to an increase in production volume by 480 thousand rubles.

The advantages of this method: versatility of application, ease of calculations.

The disadvantage of the method is that, depending on the chosen order of factor replacement, the results of factor decomposition have different meanings. This is due to the fact that as a result of applying this method, a certain indecomposable residue is formed, which is added to the magnitude of the influence of the last factor. In practice, the accuracy of factor assessment is neglected, highlighting the relative importance of the influence of one or another factor. However, there are certain rules that determine the substitution sequence:
· if there are quantitative and qualitative indicators in the factor model, the change in quantitative factors is considered first;
· if the model is represented by several quantitative and qualitative indicators, the substitution sequence is determined by logical analysis.

Under quantitative factors in analysis they understand those that express the quantitative certainty of phenomena and can be obtained by direct accounting (number of workers, machines, raw materials, etc.).

Qualitative factors determine the internal qualities, signs and characteristics of the phenomena being studied (labor productivity, product quality, average working hours, etc.).

Absolute difference method is a modification of the chain substitution method. The change in the effective indicator due to each factor using the method of differences is defined as the product of the deviation of the factor being studied by the basic or reporting value of another factor, depending on the selected substitution sequence:

Relative difference method used to measure the influence of factors on the growth of a performance indicator in multiplicative and mixed models of the form y = (a – c) . With. It is used in cases where the source data contains previously determined relative deviations of factor indicators in percentages.

For multiplicative models like y = a . V . The analysis technique is as follows:

· find the relative deviation of each factor indicator:

· determine the deviation of the performance indicator at due to each factor

Example. Using the data in table. 2, we will analyze using the method of relative differences. The relative deviations of the factors under consideration will be:

Let's calculate the impact of each factor on the volume of commercial output:

The calculation results are the same as when using the previous method.

Integral method allows you to avoid the disadvantages inherent in the chain substitution method, and does not require the use of techniques for distributing the indecomposable remainder among factors, because it has a logarithmic law of redistribution of factor loads. The integral method makes it possible to achieve a complete decomposition of the effective indicator into factors and is universal in nature, i.e. applicable to multiplicative, multiple and mixed models. The operation of calculating a definite integral is solved using a PC and is reduced to constructing integrand expressions that depend on the type of function or model of the factor system.
1. What management problems are solved through economic analysis?
2. Describe the subject of economic analysis.
3. What distinctive features characterize the method of economic analysis?
4. What principles underlie the classification of techniques and methods of analysis?
5. What role does the method of comparison play in economic analysis?
6. Explain how to construct deterministic factor models.
7. Describe the algorithm for using the simplest methods of deterministic factor analysis: the method of chain substitutions, the method of differences.
8. Characterize the advantages and describe the algorithm for using the integral method.
9. Give examples of problems and factor models to which each of the methods of deterministic factor analysis is applied.

This may be of interest (selected paragraphs):

Called factor analysis. The main types of factor analysis are deterministic analysis and stochastic analysis.

Deterministic factor analysis is based on a methodology for studying the influence of such factors, the relationship of which with a general economic indicator is functional. The latter means that the generalizing indicator is either a product, a quotient of division, or an algebraic sum of individual factors.

Stochastic factor analysis is based on a methodology for studying the influence of such factors, the relationship of which with a general economic indicator is probabilistic, otherwise - correlation.

In the presence of a functional relationship with a change in the argument, there is always a corresponding change in the function. If there is a probabilistic relationship, a change in the argument can be combined with several values of the change in the function.

Factor analysis is also divided into straight, otherwise deductive analysis and back(inductive) analysis.

First type of analysis carries out the study of the influence of factors by a deductive method, that is, in the direction from the general to the specific. In reverse factor analysis the influence of factors is studied inductively - in the direction from particular factors to general economic indicators.

Classification of factors influencing the efficiency of an organization

The factors whose influence is studied during the study are classified according to various criteria. First of all, they can be divided into two main types: internal factors, depending on the activity of this, and external factors, independent of this organization.

Internal factors, depending on the magnitude of their impact on, can be divided into major and minor. The main ones include factors related to the use of materials and materials, as well as factors determined by supply and sales activities and some other aspects of the functioning of the organization. The main factors have a fundamental impact on general economic indicators. External factors beyond the control of a given organization are determined by natural-climatic (geographical), socio-economic, and foreign economic conditions.

Depending on the duration of their impact on economic indicators, we can distinguish constant and variable factors. The first type of factors has an impact on economic indicators that is not limited in time. Variable factors affect economic indicators only over a certain period of time.

Factors can be divided into extensive (quantitative) and intensive (qualitative) based on the essence 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 labor productivity of one worker will be an intensive factor.

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

Factors can also be divided 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 influences only within an industry or even a separate organization.

According to their structure, factors are divided into simple and complex. The overwhelming majority of factors are complex, including several components. At the same time, there are also factors that cannot be separated. For example, capital productivity can serve as an example of a complex factor. The number of days the equipment was used during a given period is a simple factor.

According to the nature of the influence on general economic indicators, they are distinguished direct and indirect factors. Thus, a change in products sold, although it has an inverse effect on the amount of profit, should be considered direct factors, that is, a first-order factor. A change in the amount of material costs has an indirect effect on profit, i.e. affects profit not directly, but through cost, which is a first-order factor. Based on this, the level of material costs should be considered a second-order factor, that is, an indirect factor.

Depending on whether it is possible to quantify the influence of a given factor on a general economic indicator, a distinction is made between measurable and unmeasurable 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.

Factor economic analysis

Those signs that characterize the cause are called factorial, independent. The same signs that characterize the investigation are usually called resultant, dependent.

The set of factor and resultant characteristics that are in the same cause-and-effect relationship is called factor system. There is also the concept of a factor system model. It characterizes the relationship between the resultant characteristic, denoted as y, and the factor characteristics, denoted as . In other words, the factor system model expresses the relationship between general economic indicators and individual factors influencing this indicator. In this case, other economic indicators act as factors, representing the reasons for changes in the general indicator.

Factor system model can be expressed mathematically using the following formula:

Establishing dependencies between generalizing (resulting) and influencing factors is called economic-mathematical modeling.

We study two types of relationships between generalizing indicators and the factors influencing them:

functional (otherwise - functionally determined, or strictly determined connection.)
stochastic (probabilistic) connection.

Functional connection- this is a relationship in which each value of a factor (factorial characteristic) corresponds to a completely definite non-random value of a generalizing indicator (resultative characteristic).

Stochastic communication- this is a relationship in which each value of a factor (factor characteristic) corresponds to a set of values of a general indicator (resultative characteristic). Under these conditions, for each value of factor x, the values of the general indicator y form a conditional statistical distribution. As a result, a change in the value of factor x only on average causes a change in the general indicator y.

In accordance with the two types of relationships considered, a distinction is made between methods of deterministic factor analysis and methods of stochastic factor analysis. Consider the following diagram:

Methods used in factor analysis. Scheme No. 2

The greatest completeness and depth of analytical research, the greatest accuracy of analysis results is ensured by the use of economic and mathematical research methods.

These methods have a number of advantages over traditional and statistical methods of analysis.

Thus, they provide a more accurate and detailed calculation of the influence of individual factors on changes in the values of economic indicators and also make it possible to solve a number of analytical problems that cannot be done without the use of economic and mathematical methods.

Method of differential calculus.

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

Where – change of functions;

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

– change in the second factor;

– an infinitesimal 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 represents the main (linear relative 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 provides 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 residual term, i.e. .

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

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

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

Index method for determining the influence of factors on a general indicator.

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

(5.2.1)

(5.2.2)

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

I R – individual (factorial) index of changes in the number of employees;

I D – factor index of changes in labor productivity of workers;

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

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

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

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

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

Index theory does not provide a general method for decomposing the absolute deviations of a generalizing indicator into factors when the number of factors is more than two.

Chain substitution method.

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

In general, we have the following system of calculations using the chain substitution method:

– basic value of the general indicator;

– intermediate value;

………………………………………………..

…………………………………………………

– actual value.

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

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

due to changes in factor a

due to changes in factor b

For example, if the indicator z under study 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 – base and reporting periods of time, respectively.

By grouping the last term in this formula with one of the first, we obtain two different variants 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 one).

This formula reveals the influence of a qualitative factor on the change in the general indicator, i.e., by expressing the connection more actively, it is not possible to obtain an unambiguous quantitative value of individual factors without meeting 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 second order of substitution, then the result is summed up and the average value is taken from the resulting sum, giving a single answer about the value of the factor’s influence. If more factors are involved in the calculation, then their values are calculated using all possible substitutions. Let us describe this method mathematically, using the notation adopted above.

As you can see, the weighted finite difference method takes into account all substitution options. At the same time, when averaging, it is impossible to obtain an unambiguous quantitative value of individual factors. This method is very labor-intensive and, compared to the previous method, complicates the computational procedure, since it is necessary to go through all possible substitution options. At its core, the method of weighted finite differences is identical (only for a two-factor multiplicative model) to the method of simply adding an indecomposable remainder when dividing this remainder equally between factors. 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 therefore the number of substitutions, the described identity of the methods is not confirmed.

Logarithmic method.

This method consists in achieving a logarithmically proportional distribution of the remainder over the two required factors. In this case, there is no need to establish the order of action of the factors.

Mathematically, this method is described as follows.

The factor system z = xy can be represented as log z=log x + log y, then

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

(*)

Where

Expression (*) for Δz is nothing more than its logarithmic proportional distribution over the two required factors. That is why the authors of this approach called this method “the logarithmic method of decomposing the increment Δz into factors.” The peculiarity of the logarithmic decomposition method is that it allows one to determine the 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. Khumal, who wrote: “Such a division of the increase in a product 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 the 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

Where

The main disadvantage of the logarithmic method of analysis is that it cannot be “universal”; it cannot be used when analyzing any type of factor system models. 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 ), then with the same analysis of multiple models of factor systems, obtaining exact values of the influence of factors is not possible.

Thus, if the multiple model of the factor system is represented in the form

That ,

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

Where

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

It should be noted that the subsequent division of the factor Δz' y by the logarithmic method into factors Δz' c and Δz' q cannot be carried out in practice, since the logarithmic method in its essence provides for obtaining logarithmic ratios, which will be approximately the same for the factors being divided. 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 influence changes in the performance indicator. The presence of approximate calculations of the magnitudes of factor changes proves the imperfection of the logarithmic method of analysis.

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

The described method of coefficients is captivating in its simplicity, but when substituting digital values into the formulas, I. A. Belobzhetsky’s result turned out to be correct only by chance. When algebraic transformations are performed accurately, 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.

Method of splitting factor increments.

In the analysis of economic activity, the most common problems are direct deterministic factor analysis. From an economic point of view, such tasks include analyzing 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 performance indicator is calculated. From a mathematical point of view, problems of direct deterministic factor analysis represent the study of the function of several variables.

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 = – change in the function z = f(x, y) due to a change in the factor x by the amount ;

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

The error ε decreases as n increases.

ε can be neglected if n is large enough.

Integral method for assessing factor influences.

A further logical development of the method of crushing increments of factor characteristics was the integral method of factor analysis. This method is based on the summation of the increments of a function, defined as the partial derivative multiplied by the increment of the argument over infinitesimal intervals. In this case, the following conditions must be met:

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 varies along a straight line;

constancy of the ratio of the rates of change of factors

In general, the formulas 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 type) are derived as follows, which corresponds to the limiting case when:

where Ge 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, changes in factors in the area of definition of the function can occur not along a straight line segment e, but along some oriented curve. But since the change in factors is considered over an elementary period (i.e., over the minimum period of time during which at least one of the factors will receive an increase), the trajectory of the curve is determined in the only possible way - a straight oriented segment of the curve connecting the initial and final points of the elementary period.

Let us derive the formula for the general case.

The function of changing the resulting indicator from factors is specified

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

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

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

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

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

Let us assume that the indicator y received an increment 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.

Let's say Li is a straight line connecting two points M i and M i+1 (i=1, 2, ..., n-1).

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

Let us introduce the notation

Given these two formulas, the integral over the segment Li can be written as follows:

j = 1, 2,…, m; I = 1,2,…,n-1.

Having calculated 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 period i.

Having summed up the values of Δy ij according to the matrix tables, we obtain the following line:

(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.

We can distinguish two directions for the practical use of the integral method in solving problems of factor analysis. The first direction includes problems of factor analysis when there is no data on changes in factors within the analyzed period or they can be abstracted from, i.e., 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 factor analysis problems can be conventionally called static, since in this case the factors involved in the analysis are characterized by the immutability of their position in relation to one factor, the constancy of the conditions for the analysis of the measured factors, regardless of their location in the factor system model. The comparison of factor increments occurs in relation to one factor selected for this purpose.

The second direction includes the tasks of factor analysis, when there is information about changes in factors within the analyzed period and it should 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, calculations should be carried out along some oriented curve connecting the point (x 0, y 0) and the point (x 1, y 1) for a two-factor model. The problem is how to determine the true form of the curve along which the movement of factors x and y occurred over time. This type of factor analysis problem can be conventionally called dynamic, since in this case the factors involved in the analysis change in each period divided into sections.

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

Table 5.1

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

Elements of the factor system structure	Elements of a multiplicative model of a factor system							Integrand formula








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 integrand expressions.

Example:

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

Structure of the factor system

Construction of integrands

Where

Formation of working formulas of the integral method for multiple models. Integrand expressions for the elements of the factor system structure 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 model and the elements of the factor system structure, followed by deciphering the values given to the right and down the matrix of the initial values.

The subsequent calculation of a definite integral for a given integrand and a given integration interval is performed using a computer using a standard program that uses the Simpson formula, or manually in accordance with the general rules of integration.

In the absence of universal computing tools, we will propose a set of formulas most often found in economic analysis for calculating structural elements for multiplicative and multiple models of factor systems, 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 elements of the structure of a factor system.

Example:

Type of factor system model f = xyzq (multiplicative model).

Structure of the factor system

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

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

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

Signs or criteria reflect either individual aspects of the enterprise’s activities (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 basis of the expert method is the experience and qualifications of experts. Experts, based on available information and using their own methods, conduct an analysis of 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 using formalized calculation methods. When applying the analytical method, three main stages can be distinguished:

primary “filtration” of enterprises. At this stage, enterprises are eliminated, about which with a high degree of probability it can be said that their reporting is highly suspicious;

calculation of coefficients according to a pre-approved methodology;

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

Reliability of the information on which the rating is based. Rating analysis is carried out by independent agencies on the basis of public, official reporting of the enterprise. The official reporting published by enterprises in the media is the balance sheet. The imperfections of the Russian accounting system, gaps in Russian financial legislation, and the large volume of the shadow economy - all this does not allow one to fully trust the official reporting of enterprises. An audit of the enterprise’s reporting can partially solve this problem.

Lack of timeliness of rating analysis. Typically, the rating is calculated based on the balance sheet for the year. Annual balance sheets are submitted by March 31 of the year following the reporting year. Then it takes some time to compile the rating. Thus, the rating appears based on information that was relevant 3-4 months ago. During this time, the state of the enterprise could change significantly.

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

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