Factor analysis. Methods of factor analysis of economic indicators Method of chain substitutions

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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 should 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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Chapter 3. 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.

In = eD1R1 / eD0R0 ;

In = еD0R1 / еD0R0 ` еD1R1 / еD0R1 ;

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;

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

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, the formula In = еD1R1 / еD0R0 allows us to calculate the absolute deviation (increase) of the generalizing indicator - the volume of output of commercial products of the enterprise:

pNt = eD1R1 - eD0R0,

where pNt 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 output volume 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.

The formula In = еD0R1 / еD0R0 ` еD1R1 / еD0R1 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:

pNtR = еD0R1 - еD0R0.

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

nNDT = eD1R1 - eD0R1.

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.

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Economic analysis that studies the influence of individual factors on economic indicators is called factor analysis.
It is worth noting that the main types of factor analysis will be 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 the general economic indicator will be 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 will be probabilistic, otherwise - correlation.

In conditions of 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

Factors, the influence of which is studied when analyzing economic activities, are classified according to various criteria. First of all, they can be divided into two main types: internal factors, depending on the activities of this organization, and external factors, independent of this organization.

Internal factors, depending on the magnitude of their impact on economic indicators, can be divided into main and secondary. Among the main factors are factors related to the use of labor resources, fixed assets and materials, as well as factors determined by supply and sales activities and certain 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.

Taking into account the dependence 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 during 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. 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, which 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 exclusively within an industry or even a separate organization.

According to this 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 worked by the equipment during a given period will be 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 the cost of products sold, although it has a reverse 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.

Considering the dependence 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.

By the way, 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

In economic analysis, those signs that characterize the cause are called factorial, independent. Note that the same signs that characterize the investigation are usually called resultant, dependent.

The set of factor and resultant characteristics, which are in the same cause-and-effect relationship, is called factor system. There is also the concept of a factor system model. It is worth noting that 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) economic indicators and the factors influencing them is called economic-mathematical modeling.

In economic analysis, two types of relationships between general indicators and the factors influencing them are studied:

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

Functional connection- such a connection in which each value of a factor (factorial characteristic) has a well-defined non-random value of a generalizing indicator (resultative characteristic)

Stochastic communication— ϶ᴛᴏ such a connection, for which each value of the factor (factorial characteristic) ϲᴏᴏᴛʙᴇᴛϲᴛʙ creates a set of values of the generalizing indicator (resultative characteristic). Under these conditions, for each value of the factor x, the values of the generalizing 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 relation to the two types of relationships considered, a distinction is made between methods of deterministic factor analysis and methods of stochastic factor analysis. Let's study 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.

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.

To make it easier to study the material, we divide the article into topics:

P cr = In report * (U cr report -U cr. base)/100
At the cr.otch. and bases – columns 6 and 7.

5. Calculation of the “administrative expenses” factor

Pupr. =Watch. *(Uuro -U urb)/100
Where Uuro and U ur are, respectively, the levels of management expenses in the reporting and base periods

6. Calculation of the total influence of all factors on sales profit

The “Total” amount must be equal to the absolute deviation on line 050 of Form No. 2 (column 5). If this is not the case, then the calculations are erroneous and further analysis makes no sense.

Factor analysis can be continued to net income. The methodology for carrying it out is as follows:

1. According to the above diagram, sales profit is analyzed.
2. The influence of all other factors (operating income, expenses, etc.) is assessed in column 5 in the table above.

Factor analysis methods

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.

Factor analysis in the educational literature is interpreted as a section of multivariate statistical analysis that combines methods for assessing the dimension of many observed variables by studying the structure of covariance or correlation matrices.

Factor analysis begins its history in psychometrics and is currently widely used not only in psychology, but also in neurophysiology, sociology, political science, economics, statistics and other sciences. The basic ideas of factor analysis were laid down by the English psychologist and anthropologist F. Galton. The development and implementation of factor analysis in psychology was carried out by such scientists as: C. Spearman, L. Thurstone and R. Cattell. Mathematical factor analysis was developed by Hotelling, Harman, Kaiser, Thurstone, Tucker and other scientists.

This type of analysis allows the researcher to solve two main problems: to describe the subject of measurement compactly and at the same time comprehensively. Using factor analysis, it is possible to identify factors responsible for the presence of linear statistical relationships of correlations between observed variables.

For example, when analyzing scores obtained on several scales, a researcher notes that they are similar to each other and have a high correlation coefficient, in which case he can assume that there is some latent variable that can be used to explain the observed similarity of the scores obtained. Such a latent variable is called a factor that influences numerous indicators of other variables, which leads to the opportunity and need to mark it as the most general, higher order.

Thus, two goals of factor analysis can be distinguished:

Determination of relationships between variables, their classification, i.e. “objective R-classification”;
reducing the number of variables.

To identify the most significant factors and, as a consequence, the factor structure, it is most justified to use the principal components method. The essence of this method is to replace correlated components with uncorrelated factors. Another important characteristic of the method is the ability to limit oneself to the most informative principal components and exclude the rest from the analysis, which simplifies the interpretation of the results. The advantage of this method is also that it is the only mathematically based method of factor analysis.

Factor analysis is a technique for a comprehensive and systematic study and measurement of the impact of factors on the value of a performance indicator.

The following types of factor analysis exist:

1. Deterministic (functional) – the effective indicator is presented in the form of a product, quotient or algebraic sum of factors.
2. Stochastic (correlation) - the relationship between the effective and factor indicators is incomplete or probabilistic.
3. Direct (deductive) – from the general to the specific.
4. Reverse (inductive) – from the particular to the general.
5. Single stage and multi stage.
6. Static and dynamic.
7. Retrospective and prospective.

Also, factor analysis can be exploratory - it is carried out when studying the latent factor structure without assumptions about the number of factors and their loads, and confirmatory, designed to test hypotheses about the number of factors and their loads. The practical implementation of factor analysis begins with checking its conditions.

Mandatory conditions for factor analysis:

All signs must be quantitative;
The number of features must be twice the number of variables;
The sample must be homogeneous;
The original variables must be distributed symmetrically;
Factor analysis is carried out on correlated variables.

During the analysis, variables that are highly correlated with each other are combined into one factor, as a result, the variance is redistributed between the components and the most simple and clear structure of factors is obtained. After combining, the correlation of components within each factor with each other will be higher than their correlation with components from other factors. This procedure also makes it possible to isolate latent variables, which is especially important when analyzing social ideas and values.

As a rule, factor analysis is carried out in several stages.

Stages of factor analysis:

Stage 1. Selection of factors.
Stage 2. Classification and systematization of factors.
Stage 3. Modeling the relationships between performance and factor indicators.
Stage 4. Calculation of the influence of factors and assessment of the role of each of them in changing the value of the performance indicator.
Stage 5. Practical use of the factor model (calculation of reserves for growth of the effective indicator).

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

Deterministic factor analysis represents the influence of factors, the connection of which with the effective indicator is functional in nature, that is, when the effective indicator of the factor model is presented in the form of a product, quotient or algebraic sum of factors.

Methods of deterministic factor analysis: Method of chain substitutions; Absolute difference method; Relative difference method; Integral method; Logarithm method.

This type of factor analysis is the most common because, being quite simple to use (compared to stochastic analysis), it allows you to understand the logic of the action of the main factors of enterprise development, quantify their influence, understand which factors and in what proportion it is possible and advisable to change for increase .

Stochastic analysis is a technique for studying factors whose connection with an effective indicator, unlike a functional one, is incomplete, probabilistic (correlation). If with a functional (complete) dependence with a change in the argument there is always a corresponding change in the function, then with a correlation connection a change in the argument can give several values of the increase in the function depending on the combination of other factors that determine this indicator.

Methods of stochastic factor analysis: - Method of pair correlation;
- Multiple correlation analysis;
- Matrix models;
- Mathematical programming;
- Operations research method;
- Game theory.

It is also necessary to distinguish between static and dynamic factor analysis. The first type is used when studying the influence of factors on performance indicators on the corresponding date. Another type is a technique for studying cause-and-effect relationships in dynamics.

And finally, factor analysis can be retrospective, which studies the reasons for the increase in performance indicators over past periods, and prospective, which examines the behavior of factors and performance indicators in the future.

Factor analysis of profitability

The main goal of any company is to find optimal solutions aimed at maximizing profits, the relative expression of which is profitability indicators. The advantages of using these indicators in the analysis lie in the possibility of comparing performance not only within one company, but also using multivariate analysis of several companies over a number of years. In addition, profitability indicators, like any relative indicators, represent important characteristics of the factor environment for the formation of profit and income of companies.

The problem with using analytical procedures in this area is that the authors propose various approaches to the formation of not only a basic system of indicators, but also profitability indicators.

To analyze profitability, use the following factor model:

R = P/N, or
R = (N - S)/N * 100
where P is profit; N - revenue; S - cost.

In this case, the influence of the factor of price changes on products is determined by the formula:

RN = (N1 - S0)/N1 - (N0 - S0)/N0
Accordingly, the influence of the cost change factor will be:
RS = (N1 - S1)/N1 - (N1 - S0)/N1
The sum of factor deviations will give the total change in profitability for the period:
R = RN + RS

Using this model, we will conduct a factor analysis of the profitability indicators for the production of hardware products by a conditional enterprise. To carry out the analysis and build a factor model, the following data is needed: on the prices of products sold, sales volumes and the cost of production or sale of one unit. product.

Deterministic factor analysis

Deterministic modeling of factor systems is limited by the length of the factor field of direct connections. With an insufficient level of knowledge about the nature of direct connections of a particular indicator of economic activity, a different approach to understanding objective reality is often necessary. The scope of quantitative changes in economic indicators can only be determined by stochastic analysis of mass empirical data.

In deterministic factor analysis, the model of the phenomenon being studied does not change across economic objects and periods (since the relationships of the corresponding main categories are stable). If it is necessary to compare the results of activities of individual farms or one farm in certain periods, the only question that may arise is about the comparability of the quantitative analytical results identified on the basis of the model.

Deterministic factor analysis is a technique for studying the influence of factors whose connection with the performance indicator is functional in nature, i.e. can be expressed by a mathematical relationship.

Deterministic models can be of different types: additive, multiplicative, multiple, mixed.

Factor analysis of the enterprise

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

Internal factors, depending on the magnitude of their impact on economic indicators, can be divided into main and secondary. 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, constant and variable factors can be distinguished. 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 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.

Based on the nature of their influence on general economic indicators, a distinction is made between direct and indirect factors. Thus, a change in the cost of products sold, although it has a reverse 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 analysis models

Let's say you conduct a (somewhat "dumb") study in which you measure the height of one hundred people in inches and centimeters. So you have two variables. If you next want to investigate, for example, the effects of different nutritional supplements on growth, would you continue to use both variables? Probably not, because Height is one characteristic of a person, regardless of the units in which it is measured.

Now suppose you want to measure people's satisfaction with life, for which you create a questionnaire with various items; Among other questions, you ask the following: are people satisfied with their hobby (point 1) and how intensively do they engage in it (point 2). The results are transformed so that the average responses (for example, for satisfaction) correspond to a value of 100, while below and above the average responses there are lower and higher values, respectively. Two variables (responses to two different items) are correlated with each other. (If you are not familiar with the concept of a correlation coefficient, we recommend that you refer to the section Basic Statistics and Tables - Correlations). From the high correlation of these two variables, we can conclude that the two questionnaire items are redundant.

Combining two variables into one factor. The relationship between variables can be detected using a scatterplot. The line obtained by fitting gives a graphical representation of the relationship. If you define a new variable based on the regression line shown in this diagram, then that variable will include the most significant features of both variables. So, in effect, you have reduced the number of variables and replaced two with one. Note that the new factor (variable) is actually a linear combination of the two original variables.

Principal component analysis. An example in which two correlated variables are combined into a single factor shows the main idea of the factor analysis model, or more precisely, principal components analysis (this distinction will be discussed later). If the example with two variables is extended to a larger number of variables, the calculations become more complex, but the basic principle of representing two or more dependent variables as one factor remains valid.

Isolation of main components. Basically, the principal component extraction procedure is similar to rotation that maximizes the variance (varimax) of the original variable space. For example, in a scatterplot, you can treat the regression line as the x-axis, rotating it so that it coincides with the regression line. This type of rotation is called a variance-maximizing rotation because the criterion (goal) of the rotation is to maximize the variance (variability) of the "new" variable (factor) and minimize the variance around it (see Rotation Strategies).

Generalization to the case of many variables. When there are more than two variables, they can be considered to define a three-dimensional "space" in the same way that two variables define a plane. If you have three variables, you can create a 3D scatterplot.

For the case of more than three variables, it becomes impossible to represent points on a scatterplot, but the logic of rotating the axes to maximize the variance of the new factor remains the same.

Several orthogonal factors. Once you have found the line for which the variance is maximum, there remains some scatter in the data around it. And it’s natural to repeat the procedure. In principal component analysis, this is exactly what is done: after the first factor has been isolated, that is, after the first line has been drawn, the next line is determined that maximizes the residual variation (the spread of data around the first line), etc. Thus, the factors are sequentially identified one after another. Since each subsequent factor is determined in such a way as to maximize the variability remaining from the previous ones, the factors turn out to be independent of each other. In other words, uncorrelated or orthogonal.

How many factors should be identified? Let us recall that principal component analysis is a method of data reduction or reduction, i.e. by reducing the number of variables. A natural question arises: how many factors should be identified? Note that in the process of sequential identification of factors, they include less and less variability. The decision about when to stop the factor selection procedure depends largely on one's view of what constitutes small "random" variability.

Review of principal component analysis results. Let's now look at some standard results from principal component analysis. With repeated iterations, you identify factors with less and less variance. For simplicity of presentation, we assume that work usually begins with a matrix in which the variances of all variables are equal to 1.0. Therefore, the total variance is equal to the number of variables. For example, if you have 10 variables, each of which has a variance of 1, then the most variance that can potentially be extracted is 10 times 1. Suppose that in a study of life satisfaction you included 10 items to measure different aspects of satisfaction with home life and work.

Eigenvalues. In the second column (Eigenvalues) of the results table, you can find the variance of the new factor you just identified. The third column for each factor gives the percentage of the total variance (in this example it is 10) for each factor. As you can see, the first factor (value 1) explains 61 percent of the total variance, factor 2 (value 2) explains 18 percent, and so on. The fourth column contains the accumulated or cumulative variance. The variances extracted by the factors are called eigenvalues. This name comes from the calculation method used.

Eigenvalues and the problem of the number of factors. Once you know how much variance each factor contributed, you can return to the question of how many factors should be retained. As stated above, this decision is arbitrary in nature. However, there are some generally accepted recommendations, and in practice, following them gives the best results.

Kaiser criterion. First, you can select only factors with eigenvalues greater than 1. Essentially, this means that if a factor does not emit variance equivalent to at least the variance of one variable, then it is omitted. This criterion was proposed by Kaiser (1960) and is probably the most widely used. In the example above, based on this criterion, you should only retain 2 factors (two principal components).

Scree criterion. The scree criterion is a graphical method first proposed by Cattell (1966). You can plot the eigenvalues presented in the table earlier as a simple graph.

Cattel suggested finding a place on the graph where the decrease in eigenvalues from left to right slows down as much as possible. It is assumed that to the right of this point there is only a "factorial scree" - "slide" is a geological term for rock fragments that accumulate at the bottom of a rocky slope. In accordance with this criterion, you can leave 2 or 3 factors in this example.

What criterion should be used? Both criteria have been studied in detail by Browne (1968), Cattell and Jaspers (1967), Hakstian, Rogers and Cattell (1982), Lynn (1968), Tucker, Koopman and Lynn (Tucker, Koopman, Linn, 1969). Theoretically, it is possible to calculate their characteristics by generating random data for a specific number of factors. Then you can see whether the criterion used has detected a sufficiently accurate number of significant factors or not. Using this general method, the first criterion (Kaiser criterion) sometimes retains too many factors, while the second criterion (Scree criterion) sometimes retains too few factors; however, both criteria are quite good under normal conditions, when there are a relatively small number of factors and many variables. In practice, an important additional question arises, namely: when the resulting solution can be meaningfully interpreted. Therefore, several solutions with more or less factors are usually examined, and then the one that makes the most sense is selected. This issue will be further discussed within the framework of factor rotations.

Analysis of the main factors. Before we continue to look at the various aspects of the output of principal component analysis, let us introduce principal factor analysis. Let's return to the example of the life satisfaction questionnaire to formulate another “thought model.” You can imagine that the subjects' responses depend on two components. First, we select some relevant general factors, such as, for example, “satisfaction with one's hobby,” discussed earlier. Each item measures some portion of this overall aspect of satisfaction. In addition, each item includes a unique aspect of satisfaction not shared by any other item.

Commonalities. If this model is correct, then you cannot expect the factors to contain all the variance in the variables; they will contain only that part that belongs to common factors and is distributed over several variables. In factor analysis model language, the proportion of variance in a particular variable that is attributed to common factors (and shared with other variables) is called communality. Therefore, the additional work facing the researcher when applying this model is to estimate the commonalities for each variable, i.e. the proportion of variance that is common to all items. The proportion of variance accounted for by each item is then equal to the total variance associated with all variables minus the communality. From a general point of view, the multiple correlation coefficient of the selected variable with all others should be used as an assessment of generality (for information about the theory of multiple regression, refer to the section Multiple Regression). Some authors propose various iterative "post-solution improvements" to the initial communality estimate obtained using multiple regression; for example, the so-called MINRES method (the method of minimum factor residuals; Harman and Jones (Harman and Jones, 1966)), which tests various modifications of factor loadings in order to minimize the residual (unexplained) sums of squares.

Principal factors versus principal components. Principal factors versus principal components. The main difference between the two factor analysis models is that in principal component analysis, you assume that all of the variability in the variables should be used, whereas in principal factor analysis, you only use the variability in a variable that is common to other variables. A detailed discussion of the pros and cons of each approach is beyond the scope of this introduction. In most cases, these two methods lead to very similar results. However, principal component analysis is often preferred as a method of data reduction, while principal factor analysis is better used to determine the structure of the data (see the next section).

Factor analysis of sales

In a similar way, we will derive models for factor analysis of profitability of sales.

The initial indicator looks like:

RPr = Prp/RP = SRP - Srp)/RP.

Changes in sales profitability under the influence of relevant factors:

Lrpr = Prp1 /RP1- PrpO /RP0= (RP1 - Srp1)/RP1 - (RP0 - Srp0)/RL0 = - CpnJ/RSh + Srp0/RP0 = (Crp0/RSh - Srp1/RP1) + (Cpn0/RP0 - Srp0/RP1) = LrsPRS + A/V.

Here, the component Ap prS characterizes the impact of changes in the cost of goods sold on the dynamics of profitability of sales. And component A//PPR is the impact of changes in sales volume. They are determined accordingly: ArsPRs = Srp0/RP1 - Srp1/RP1; A/pPr = Srp0/RP0 - Srp0/RP1.

Using the method of chain substitutions, the factor analysis of profitability of sales can be continued by studying the influence on the component Ar prS of the dynamics of such factors as:

A) cost of sales of goods, products, works, services:
ArsPrr = (Ср0 - Ср1)/РП1,
where СРО, Cpl - cost of sales of goods, products, works, services, respectively, in the base and reporting periods (line 020 of form 2), rub.;

B) administrative expenses:

Ar „, y = (SuO - Su1)/RP1, where SuO, Su1 are administrative expenses, respectively, in the base and reporting periods (line 030 of form 2), rub.,

B) business expenses:

LrsPrk = (SkO - Sk1)/RP1, where SkO, Sk1 are commercial expenses, respectively, in the base and reporting periods (line 040 of form 2), rub.

If an enterprise keeps records of cost and revenue for certain types of products, then in the analysis process it is necessary to assess the impact of the sales structure on changes in product profitability. However, such a study is possible only based on operational data, that is, it is performed in the process of in-house analysis. Let's demonstrate it with the following example.

Example: Assess the impact of the sales structure on changes in the profitability of products sold.

Products Share of j-th Profitability of j-th product in product volume, Pj sales, %, dj Last reporting year Last reporting year A 30 40 0.25 0.245 B 70 60 0.125 0.128

Profitability of products sold:

Last year p»t = ^podo = 0.25*0.3 + 0.125*0.7 = 0.1625,
reporting YEAR ^ = = 0.245*0.4 + 0.128*0.6 = 0.1748,
LrRP = r\n - r\n = 0.1748 - 0.1625 = 0.0123.

This change in profitability is the result of two factors:

Change in profitability of individual products:
ршР1 =ip>jd)-ipw =
P 1=1
= 0,1748 - (0,25*0,4 + 0,125*0,6) = 0,1748 - 0,1750 = -0,0002.
Implementation structure changes:
PMd. = Z P°Jd) ~ Z P°JdJ = °"1750 " °"1625 = +0"0125 "" M M

Conclusion: The increase in the level of profitability of sold products occurred due to changes in the sales structure. Increasing the share of more profitable products (product A) from 30% to 40% in sales volume led to an increase in the profitability of products sold by 1.25%. However, the decrease in the profitability of product A caused a decrease in the profitability of products sold by 0.02%. Therefore, the overall increase in product profitability was 1.23%.

Problems of factor analysis

1. Selection of factors for the analysis of the studied performance indicators and their classification.
2. Determination of the form of dependence between factor and performance indicators, construction of a factor model.
3. Calculation of the influence of factors and assessment of the role of each of them in changing the value of the effective indicator.

The most important task of deterministic factor analysis is to calculate the influence of factors on the value of performance indicators, for which the analysis uses a whole arsenal of methods, the essence, purpose, and scope of which are discussed below.

It is important to distinguish factors according to their content: extensive (quantitative), intensive (qualitative); and by level of subordination.

Some factors have a direct impact on the performance indicator, others have an indirect impact. Based on the level of subordination (hierarchy), factors of the first, second, third and subsequent levels of subordination are distinguished.

Currently, when analyzing the actual cost of manufactured goods, identifying reserves and the economic effect of reducing it, factor analysis is used.

Since cost is a complex resulting indicator, and knowledge of the conditions for its formation is important for the effective management of an organization, it is of interest to assess the influence on this indicator of various factors or reasons when they change during the production process, in particular, deviations from planned values, values in the base period, etc. P.

Economic factors most fully cover all elements of the production process - means, objects of labor and labor itself. They reflect the main directions of work of enterprise teams to reduce costs: increasing labor productivity, introducing advanced equipment and technology, better use of equipment, cheaper procurement and better use of labor items, reduction of administrative, managerial and other expenses, reduction of defects and elimination of unproductive expenses and losses.

The most important groups of factors that have a significant impact on cost include the following:

1) Increasing the technical level of production: introduction of new, progressive technology; mechanization and automation of production processes; improving the use and application of new types of raw materials and materials; changes in the design and technical characteristics of products. They are also reduced as a result of the integrated use of raw materials, the use of economical substitutes, and the complete use of waste in production. A large reserve also conceals the improvement of products, a reduction in their material and labor intensity, a reduction in the weight of machinery and equipment, a reduction in overall dimensions, etc.

For this group of factors, the economic effect is calculated for each event, which is expressed in a reduction in production costs. Savings from implementing measures are determined by comparing the cost per unit of production before and after implementing the measures and multiplying the resulting difference by the volume of production in the planned year:

EC = (Z0 – Z1) * Q, (7.8)
where EK is savings in direct current costs;
Z0 - direct current costs per unit of production before the implementation of the event;
Z1 - direct current costs per unit of production after the implementation of the event;
Q is the volume of production of goods in natural units from the beginning of the implementation of the event to the end of the planned period.

2) Improving the organization of production and labor: changes in the organization of production, forms and methods of labor with the development of production specialization; improving production management and reducing production costs; improved use; improvement of logistics; reducing transport costs; other factors that increase the level of organization of production. With the simultaneous improvement of technology and production organization, it is necessary to establish savings for each factor separately and include them in the appropriate groups. If such a division is difficult to make, then savings can be calculated based on the targeted nature of the activities or by groups of factors.

A reduction in current costs occurs as a result of improving the maintenance of the main production (for example, developing continuous production, increasing the shift ratio, streamlining auxiliary technological work, improving the tool economy, improving the organization of quality control of work and goods). A significant reduction in human labor costs can occur with an increase in standards and service areas, a reduction in losses, and a decrease in the number of workers who do not meet production standards. These savings can be calculated by multiplying the number of redundant workers by the average in the previous year (with social insurance charges and taking into account the costs of special clothing, food, etc.). Additional savings arise when improving the management structure of the organization as a whole. It is expressed in a reduction in management costs and in savings in wages and salaries due to the release of management personnel.

When improving the use of fixed assets, savings are calculated as the product of the absolute reduction in costs (except depreciation) per unit of equipment (or other fixed assets) by the average amount of equipment (or other fixed assets).

Improving the logistics supply and use of material resources is reflected in a reduction in the consumption rates of raw materials and supplies, reducing their cost by reducing procurement and storage costs. Transport costs are reduced as a result of reduced costs for the delivery of raw materials and materials from the supplier to the organization’s warehouses, from factory warehouses to places of consumption; reducing the cost of transporting finished products.

3) Change in the volume and structure of goods: changing the nomenclature and increasing the quality and volume of production of goods. Changes in this group of factors can lead to a relative decrease in semi-fixed expenses (except for depreciation), a relative decrease. Conditionally fixed costs do not depend directly on the quantity of goods produced; with an increase in production volume, their quantity per unit of goods decreases, which leads to a decrease in its cost.

Relative savings on semi-fixed costs are determined by the formula

EKP = (TV * ZUP0) / 100, (7.9)
where EKP is the saving of semi-fixed costs;
ZUP0 - the amount of conditionally fixed expenses in the base period;
TV is the growth rate of production compared to the base period.

The relative change in depreciation charges is calculated separately. Part of the depreciation charges (as well as other production costs) is not included in the cost price, but is reimbursed from other sources (special funds, payments for external services that are not included in commercial products, etc.), so the total amount of depreciation may decrease. The decrease is determined based on actual data for the reporting period. The total savings on depreciation charges are calculated using the formula

EKA = (AOK / QO - A1K / Q1) * Q1, (7.10)
where ECA is savings due to a relative reduction in depreciation charges;
A0, A1 - the amount of depreciation charges in the base and reporting periods;
K is a coefficient that takes into account the amount of depreciation charges attributed to the base period;
Q0, Q1 - volume of production of goods in natural units of the base and reporting period.

To avoid double billing, the total amount of savings is reduced (increased) by the part that is taken into account by other factors.

Changes in the product range and range are one of the important factors influencing the level of production costs. With different profitability of individual products (relative to cost), shifts in the composition of goods associated with improving the structure and increasing production efficiency can lead to both a decrease and an increase in production costs. The impact of changes in the structure of goods on cost is analyzed based on variable costs for costing items of the standard nomenclature. Calculation of the impact of the structure of goods on cost must be linked to indicators of increasing labor productivity.

4) Improving the use of natural resources: changing the composition and quality of raw materials; changes in the productivity of deposits, the volume of preparatory work during extraction, methods of extraction of natural raw materials; changes in other natural conditions. These factors reflect the influence of natural conditions on the amount of variable costs. An analysis of their impact on reducing production costs is carried out on the basis of industry methods in the extractive industries.

5) Industry and other factors: commissioning and development of new workshops, production units and production facilities, preparation and development of production; other factors.

Significant reserves are included in reducing the costs of preparing and mastering new types of production of goods and new technological processes, in reducing the costs of the start-up period for newly commissioned workshops and facilities.

The amount of changes in expenses is calculated using the formula:

EKP = (З1/Q1 - З0/Q0) * Q1, (7.11)
where EKP is the change in costs for preparation and development of production;
Z0, Z1 - the amount of costs of the base and reporting period;
Q0, Q1 - volume of production of goods of the base and reporting period.

If changes in the amount of costs during the analyzed period are not reflected in the above factors, then they are classified as other. These include, for example, changing the size or termination of mandatory payments, changing the amount of costs included in the cost of production, etc.

The cost reduction factors and reserves identified as a result of the analysis must be summarized in the final conclusions, and the total impact of all factors on reducing the total cost per unit of goods must be determined.

In order to conduct a factor analysis of labor productivity, i.e. determine how one or another technical and economic factor influences changes in this indicator, and calculate the relative savings (increase) in the number of employees. Calculations are carried out in the following sequence.

First, the relative release of industrial production personnel is determined in comparison with the reporting period as a result of the influence of all factors:

L = L sp 0 qQ t 0 .

Then, using any of the factor analysis methods, the influence of changes in the value of the corresponding factor is determined: the output of marketable products, which can be achieved through an increase in production volume (extensive factor), and the increase in average annual output per payroll worker, which can be achieved as a result of measures to increase the technical level of production (intensive factor).

One of the important aspects of assessing a company's performance is to study its effectiveness from the owner's point of view. Efficiency in this case, as in many others, can be assessed by determining the profitability indicator. However, a simple calculation may not be enough and will need to be supplemented with analysis. The most popular method is, perhaps, factor analysis of return on equity. Let us dwell in more detail on the methodology of its implementation and the main features.

Factor analysis of return on equity is usually associated with DuPont formulas, which allow you to quickly make all the necessary calculations. It is important to understand how these formulas were obtained, and besides, there is nothing complicated about it. The return on the owner's capital is obviously determined by the ratio of what is received to the amount of this capital. The factor model is obtained from this relationship through elementary transformations. Their essence is to multiply the numerator and denominator by revenue and assets. After this, it is easy to notice that the efficiency of using this part of the capital, its profitability, is determined by the product of the indicator of the degree of financial dependence by the turnover of property (assets) and the level of profitability of sales. After compiling a mathematical model, it is directly analyzed. It can be carried out in any way suitable for deterministic models. Factor analysis of return on equity using DuPont formulas is a variation of the absolute difference method. It, in turn, is also a special case of the chain substitution method. The main principle of this method lies in the sequential determination of the impact of each factor in isolation, regardless of the others.

It is worth noting that factor analysis of economic profitability is carried out in a similar way. It is the ratio of profit to assets. After minor transformations, this indicator can be represented by the product of the company’s property turnover times the profitability of sales. The subsequent analysis proceeds in the same way.

It is necessary to pay special attention to what indicators should be used in the calculations. Obviously, it is necessary to use information for at least two periods in order to be able to observe changes. Data taken from the income statement are cumulative in nature, since they represent a certain value for a particular period. In the balance sheet, the data is presented for a specific date, so it is best to calculate their average value.

The above methods, that is, the method of chain substitutions and its modifications, can be used to analyze almost any deterministic factor model. For example, factor analysis of the current ratio can be carried out extremely simply. For greater detail, it is advisable to reveal the formula of this coefficient, reflecting the components of current assets in the numerator, and short-term liabilities in the denominator. Then it is necessary to calculate the influence of each of the identified factors. It should be noted that absolute differences and the method of the same name cannot be used for this model, since it is multiple in nature.

The value of any type of analysis is difficult to overestimate, and factor analysis of return on equity and other indicators is one of the best methods for making the right management decisions. The identification of a strong negative impact of a particular factor clearly indicates where the impact should be directed. On the other hand, a positive impact may indicate, for example, the presence of certain reserves for profit growth.

Stochastic factor analysis

Stochastic modeling of factor systems of interrelations between individual aspects of economic activity is based on a generalization of patterns of variation in the values of economic indicators - quantitative characteristics of factors and results of economic activity. Quantitative parameters of the relationship are identified based on a comparison of the values of the studied indicators in a set of economic objects or periods.

Thus, the first prerequisite for stochastic modeling is the ability to compose a set of observations, that is, the ability to repeatedly measure the parameters of the same phenomenon under different conditions.

In stochastic analysis, where the model itself is compiled on the basis of a set of empirical data, a prerequisite for obtaining a real model is the coincidence of the quantitative characteristics of connections in the context of all initial observations. This means that variation in the values of indicators should occur within the limits of unambiguous determination of the qualitative side of the phenomena, the characteristics of which are the modeled economic indicators (within the range of variation there should not be a qualitative leap in the nature of the reflected phenomenon).

This means that the second prerequisite for the applicability of the stochastic approach to modeling connections is the qualitative homogeneity of the population (relative to the connections being studied).

The studied pattern of changes in economic indicators (modeled connection) appears in a hidden form. It is intertwined with random (from the point of view of research) components of variation and covariation of indicators. The law of large numbers states that only in a large population does a regular relationship appear more stable than a random coincidence of the direction of variation (random variation).

From this follows the third prerequisite of stochastic analysis - a sufficient dimension (number) of the set of observations, which allows one to identify the studied patterns (modeled connections) with sufficient reliability and accuracy.

The fourth prerequisite of the stochastic approach is the availability of methods that allow one to identify quantitative parameters of economic indicators from mass data on variations in the level of indicators. The mathematical apparatus of the methods used sometimes imposes specific requirements on the empirical material being modeled. Fulfillment of these requirements is an important prerequisite for the applicability of methods and the reliability of the results obtained.

The main feature of stochastic factor analysis is that in stochastic analysis it is impossible to create a model through qualitative (theoretical) analysis; a quantitative analysis of empirical data is necessary.

Methods of stochastic factor analysis:

Pair correlation method. The method of correlation and regression (stochastic) analysis is widely used to determine the closeness of the relationship between indicators that are not functionally dependent, i.e. connection does not manifest itself in each individual case, but in a certain dependence. With the help of pair correlation, two main problems are solved: a model of the operating factors is left (regression equation); a quantitative assessment of the closeness of connections is given (correlation coefficient).

Matrix models. Matrix models are a schematic representation of an economic phenomenon or process using scientific abstraction. The most widely used method here is the “input-output” analysis, which is built according to a checkerboard pattern and makes it possible to present the relationship between costs and production results in the most compact form.

Mathematical programming is the main means of solving problems to optimize production and economic activities.

The operations research method is aimed at studying, including the production and economic activities of enterprises, in order to determine such a combination of structural interconnected elements of systems that will best determine the best economic indicator from a number of possible ones.

Game theory as a branch of operations research is the theory of mathematical models for making optimal decisions under conditions of uncertainty or conflict of several parties with different interests.

Integral method of factor analysis

Elimination as a method of deterministic factor analysis has an important drawback. When using it, it is assumed that the factors change independently of each other, but in fact they change interconnectedly, as a result, some indecomposable remainder is formed, which is added to the magnitude of the influence of one of the factors (usually the last one). In this regard, the magnitude of the influence of factors on the change in the performance indicator fluctuates depending on the place of the factor in the deterministic model. To get rid of this drawback, deterministic factor analysis uses an integral method, which is used to determine the influence of factors in multiplicative, multiple and mixed models of multiple additive form.

Using this method makes it possible to obtain more accurate results for calculating the influence of factors compared to methods of chain substitution, absolute and relative differences, and to avoid ambiguous assessment of the influence: in this case, the results do not depend on the location of factors in the model, but an additional increase in the effective indicator arising from interaction of factors is distributed equally between them.

To distribute additional growth, it is not enough to take its part corresponding to the number of factors, since factors can act in different directions. Therefore, the change in the effective indicator is measured over infinitely small periods of time, i.e., the increment of the result is summed up, defined as partial products multiplied by the increments of factors over infinitely small intervals. 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. Due to the complexity of calculating some definite integrals and additional difficulties associated with the possible action of factors in opposite directions.

Factor analysis of net profit

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Net profit is an indicator of a company’s performance that, on the one hand, is influenced by the largest number of factors compared to other types of profit, and on the other hand, is the most accurate and “honest” indicator. It is for these reasons that this value requires close attention and should be subject to detailed study. One of the most popular and frequently used methods is factor analysis of net profit. As the name implies, studying profit in this way involves determining those factors that most affect it, as well as determining the specific magnitude of this impact.

Before considering factor analysis of net profit, it is necessary to study how it is formed. Analysis of the formation of net profit is carried out according to the profit and loss statement. This is understandable, since it is this form of reporting that reflects the order in which the financial result of the company is formed. When studying profit generation, it is useful to conduct a vertical analysis of the specified reporting form. It involves finding the specific weight of each of the indicators included in the report, as well as the subsequent study of its dynamics. As a rule, revenue is chosen as the basis of comparison, which is considered equal to one hundred percent.

It is also advisable to carry out factor analysis of net profit on the income statement. This is explained by the fact that this form of reporting allows you to easily and simply create a mathematical model that will include factors affecting profit margins. Factors that have the greatest influence should be placed in the model before factors whose influence is less significant. The profit and loss statement reflects the amount of revenue, but does not allow one to judge its changes under the influence of price and sales volume. These factors are extremely important, so they must be further taken into account in the model by dividing the impact on revenue revenue into two appropriate parts. After compiling a mathematical model, it is necessary to directly subject it to analysis using a certain technique. Most often they resort to using the method of chain substitutions or its modifications, for example, the method of absolute differences. This choice is due to ease of use and accuracy of results.

After studying the formation process and dynamics, it is necessary to analyze the use of net profit. The most logical and easiest way to study this process would be to conduct a vertical analysis, which was already mentioned above. Obviously, in this case, it is necessary to take net profit as the base. Then you need to determine the shares of each direction of spending this profit: on, in reserve funds, on investments, and so on. Naturally, it is necessary to study changes in this structure over time.

Obviously, to conduct any of the types of analysis described above, information is needed for several periods, at least two years. This is due to the fact that based on one period it is simply impossible to draw any conclusions about certain changes. However, it is worth keeping in mind that the indicators must be comparable, and adjustments must be made in the event of changes in accounting policies or any other.

Whether it is a factor analysis of net profit or any other, it must necessarily end with the formulation of certain conclusions and recommendations. Based on the study of profits, one can draw many conclusions about pricing policy, cost management, and much more. Conclusions and recommendations provide the basis for making management decisions that are vital to the company's activities.

Factor analysis method of chain substitutions

The chain substitution method is the most universal of the elimination methods. It is used to calculate the influence of factors in all types of deterministic factor models: additive, multiplicative, multiple and mixed (combined). This method allows you to determine the influence of individual factors on changes in the value of the performance indicator by gradually replacing the base value of each factor indicator in the scope of the performance indicator with the actual value in the reporting period. For this purpose, a number of conditional values of the performance indicator are determined, which take into account changes in one, then two, three, etc. factors, assuming that the rest do not change. Comparing the value of an effective indicator before and after changing the level of one or another factor makes it possible to eliminate the influence of all factors except one, and determine the impact of the latter on the increase in the effective indicator.

The degree of influence of one or another indicator is revealed by sequential subtraction: the first is subtracted from the second calculation, the second is subtracted from the third, etc. In the first calculation, all values are planned, in the last - actual.

In the case of a three-factor multiplicative model, the calculation algorithm is as follows:

Y 0= a 0*b 0*C 0;
Y cond.1= a 1*b 0*C 0 ; Y a= Y conditional 1 – Y 0;
Y cond.2= a 1*b 1*C 0; Y b= Y conditional 2 – Y conditional 1;
Y f= a 1*b 1*C 1; Y с= Y f – Y conditional 2, etc.

The algebraic sum of the influence of factors must necessarily be equal to the total increase in the effective indicator:

Y a+ Y b+ Y c= Y f– Y 0.

The absence of such equality indicates errors in the calculations.

This implies the rule that the number of calculations per unit is greater than the number of indicators of the calculation formula.

When using the chain substitution method, it is very important to ensure a strict substitution sequence, since changing it arbitrarily can lead to incorrect results. In the practice of analysis, the influence of quantitative indicators is first identified, and then the influence of qualitative indicators. Thus, if it is necessary to determine the degree of influence of the number of workers and labor productivity on the size of industrial output, then first establish the influence of the quantitative indicator of the number of workers, and then the qualitative indicator of labor productivity. If the influence of quantity and price factors on the volume of industrial products sold is determined, then the influence of quantity is first calculated, and then the influence of wholesale prices. Before starting calculations, it is necessary, firstly, to identify a clear relationship between the indicators being studied, secondly, to distinguish between quantitative and qualitative indicators, thirdly, to correctly determine the sequence of substitution in cases where there are several quantitative and qualitative indicators (main and derivatives, primary and secondary). Thus, the use of the chain substitution method requires knowledge of the relationship of factors, their subordination, and the ability to correctly classify and systematize them.

An arbitrary change in the substitution sequence changes the quantitative weight of a particular indicator. The greater the deviation of actual indicators from planned ones, the greater the differences in the assessment of factors calculated with different substitution sequences.

The chain substitution method has a significant drawback, the essence of which boils down to the emergence of an indecomposable remainder, which is added to the numerical value of the influence of the last factor. This explains the difference in calculations when changing the substitution sequence. This drawback is eliminated by using a more complex integral method in analytical calculations.

Factor analysis of wages

It is carried out taking into account the analysis of the use of labor resources at the enterprise and the level of labor productivity. It is known that with the growth of labor productivity, real prerequisites are created for increasing the level of labor remuneration. At the same time, funds for wages must be used in such a way that the growth rate of labor productivity outstrips the growth rate of its payment, as this creates opportunities for increasing reproduction in the enterprise.

The analysis of the use of wages begins with the calculation of absolute and relative deviations of its actual value from the planned one.

We make sequential calculations

The absolute deviation of FZPabs is determined by comparing the funds actually used for wages with the planned wage fund of FZPpl for the entire enterprise, production divisions and categories of employees:

FZPabs = FZPf - FZPpl. = 21465-20500 = +965 million rubles

However, it must be borne in mind that the absolute deviation in itself does not characterize the use of the FZP, since this indicator is determined without taking into account the degree of implementation of the production plan.

The relative deviation of the FZPotk is calculated as the difference between the actual accrued salary amount of the FZPf and the planned fund, adjusted by the coefficient of fulfillment of the plan for the production of KVP products

Initial data for the analysis of the FZP

The constant part of wages does not change with an increase or decrease in production volume (workers’ wages at tariff rates, employees’ wages at wages, all types of additional payments, wages for workers in non-industrial production and the corresponding amount of vacation pay):

FZPotn = FZPf – FZPsk = FZPag – (FZP pl..perm * Kvp + FZP pl..post) = 21465 – (13120 * 1.026 + 7380) = 21465 – 20841 = +424 million rubles
where FZPsk is the planned salary fund, adjusted to the coefficient of fulfillment of the production plan;
FZP pl..per and FZP pl..post - variable and constant amounts of the planned planned salary fund.

When calculating the FZPotn, you can use the so-called correction coefficient Kp, which reflects the share of the variable salary in the general fund. It shows by what fraction of a percent the planned wages should be increased for each percentage of exceeding the production plan (VP, %)
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