Factor analysis
Comprehensive and systematic study and measurement of the impact of factors on the magnitude of performance indicators.
Functional (deterministic)
Stochastic (correlation)
・Forward and reverse
Statistical
· Dynamic
retrospective and prospective
Main task: selection of factors, classification and systematization, determination of the form of communication, calculation of the influence of the factor and the role of its influence on complex indicators.
Types of factor models:
1 Additive models: y=x1+x2+x3+…+xn=
2 Multiplicative models: y=x1*x2*x3*…*xn=P
3 Multiple models: y=
4 Mixed models: y=
Chain substitution method
A universal method that is used for any factorial models.
Allows ODA influence individual factors on the measure of the value of the effective indicator, the way. Gradual replacement of the base value of each factor by its actual value.
Replacement begins with the main quantitative factor and ends with a qualitative indicator.
The influence of each factor is determined by successive steps. For 1 step, you can make one replacement. The algebraic sum of the influence of factors should be equal to the total increase in the effective indicator.
Tactics of application:
y=a*b*c where y0,a0,b0,c0 are base values
y1=a1*b1*c1 – actual values
Influence on the growth of the effective indicator of the change in factor a:
∆ y’ a = y’-y0
y''=a1*b1*c0
∆ y'' b = y''-y'0
y'''=a1*b1*c1
∆ y’’’ c = y’’’-y’’0
∆y=∆y a +∆y b +∆y c
Example: TP \u003d K * C
TPpl \u003d Kpl * Cpl - base value
TPF \u003d Kf * Tsf - actual value
TPus \u003d Kf * Tspl
∆TP=TPf-TPpl
∆TPc=TPsl-Tpl
∆TPc=TPav-Tpusl
∆TP=∆TPc+∆TPc
1) TPpl \u003d 135 * 1200 \u003d 16200
2) TPF=143*1370=195910
3) ∆TP=TPf-TPpl=195910-162000=33910
4) TPusl=135*1370=184950
5) ∆TPc=184950-162000=22950
∆TPc=195910-184950=10960
∆TP=22950+10960=33910
Absolute difference method
This is a modification of the chain substitution method. Used only in multiplicative models.
The magnitude of the influence of factors is calculated by multiplying the absolute increase of the used factor by the fictitious value of the factors that are used in the model to the left of it and by the base value of the factors that are to the right.
yb=a0*b0*c0 – basic
y1=a1*b1*c1 – actual
∆у a =∆ a*b0*c0, where ∆а=а1-а0
∆ y b = a1*∆b*c0
∆ y c = a1*b1*∆c
∆TPk = (1370-1200)*135=22950
∆TPc = 1370*(143-145)=10960
∆TP = 195910-162000=33910
Relative difference method
It is desirable to use only in what models? type when you need to calculate the influence of more than 8 factors.
Step 1. We calculate the relative deviations of factor indicators:
y0=a0*b0*c0 ∆а=а1-а0 – absolute deviation
y1=a1*b1*c1 relative deviation:
Step 2. Deviation of the effective indicator due to a change in each factor:
Index Method
The method is widely used to quantify the role of individual factors. All factors change independently of each other.
Based on relative performance indicators, and distribution comparisons, what? Plan.
Defined as a level ratio relative indicator to its level in the base period.
Index methods are used in multiplicative and real models. Allocate individual and group indices. Indices expressing ratios of directly commensurate values are called individual, and are calculated according to indicators for which factor models are not compiled.
Group indices characterize the ratio of what? Phenomena (total indices). Calculated by multifactorial models, index cost marketable products.
Index of the cost of marketable products:
Index of what? What? Shows how much revenue decreased with a decrease in sales.
The price index reflects the amount of change in revenue due to price changes.
Main indicators: gross output (cost of all manufactured products, including unfinished production), marketable products (not including unfinished products), sold products (sold, 91-1 account).
The minimum allowable sales volume is the break-even point.
Max allowable sales volume - at max capacity utilization.
Optimal allowable scope of implementation - methods of research operations.
5.2.4 Relative difference method
The method of relative differences, like the previous one, is used to measure the influence of factors on the growth of the effective indicator only in multiplicative models and combined type Y = (a - b) s. It is much simpler than chain substitutions, which makes it very efficient under certain circumstances. This primarily applies to those cases where the original data contain previously determined relative deviations of factor indicators in percentages or coefficients.
Consider the methodology for calculating the influence of factors in this way for multiplicative models of the type Y = A * B * C. First, you need to calculate the relative deviations of factor indicators:
Then the deviation of the effective indicator due to each factor is determined as follows:
According to this rule, to calculate the influence of the first factor, it is necessary to multiply the base (planned) value of the effective indicator by the relative growth of the first factor, expressed as a percentage, and divide the result by 100.
To calculate the influence of the second factor, you need to add the change due to the first factor to the planned value of the effective indicator and then multiply the resulting amount by the relative increase in the second factor in percent and divide the result by 100.
The influence of the third factor is determined in a similar way: it is necessary to add its growth due to the first and second factors to the planned value of the effective indicator and multiply the resulting amount by the relative growth of the third factor, etc.
Let's fix the considered technique on the example shown in Table 15:
As you can see, the calculation results are the same as when using the previous methods.
The method of relative differences is convenient to use in cases where it is required to calculate the influence of a large complex of factors (8-10 or more). Unlike the previous methods, the number of calculations is significantly reduced.
5.2.5 Method of proportional division and equity participation.
In some cases, to determine the magnitude of the influence of factors on the growth of the effective indicator, the method of proportional division can be used. This applies to those cases when we are dealing with additive models of the type Y = ∑Х i and mixed models of the type
In the first case, when we have a single-level model of the type Y = a + b + c, the calculation is carried out as follows:
For example, the level of profitability decreased by 8% due to an increase in the capital of the enterprise by 200 million tenge. At the same time, the cost of fixed capital increased by 250 million tenge, and the value of working capital decreased by 50 million tenge. So, due to the first factor, the level of profitability decreased, and due to the second - increased:
The calculation procedure for mixed models is somewhat more complicated.
When ∆Vd are known; ∆Вn and ∆Вm as well as ∆Yb, then to determine ∆Yd, ∆Yn, ∆Ym, you can use the method of proportional division, which is based on the proportional distribution of the increase in the effective indicator Y due to a change in the factor B between the second level factors D, N and M, respectively their size. The proportionality of this distribution is achieved by determining a coefficient that is constant for all factors, which shows the amount of change in the effective indicator Y due to a change in factor B by one.
The value of the coefficient (K) is determined as follows:
Multiplying this coefficient by the absolute deviation B due to the corresponding factor, we find the deviations of the effective indicator:
∆Yb=K*∆Bd; ∆Yn=К*∆Bn; ∆Ym=К*∆Bm
For example, the cost of 1 ton/km increased by 180 rubles due to a decrease in the average annual output of a car. At the same time, it is known that the average annual production of a car has decreased due to:
a) overscheduled downtime of machines - 5000 t/km
b) overplanned idle runs - 4000 t/km
c) incomplete use of carrying capacity - 3000 t / km
Total-12000 t/km
From here you can determine the change in cost under the influence of factors of the second level:
Table 18 - Calculation of the influence of factors on the performance indicator by the method of equity participation
To solve this type of problem, you can also use the method of equity participation. To do this, first determine the share of each factor in the total amount of their growth, which is then multiplied by the total growth of the effective indicator:
There are a lot of similar examples of the application of this method in AHD, as you can see in the process of studying the industry course of analysis. economic activity at enterprises.
5.2.6 Logarithm method in business analysis.
The logarithm method is used to measure the influence of factors in multiplicative models. AT this case the result of the calculation, as in the case of integration, does not depend on the location of the factors in the model and, in comparison with the integral method, provides more high accuracy calculations. If, in integration, the additional gain from the interaction of factors is distributed equally between them, then using the logarithm, the result of the combined action of the factors is distributed in proportion to the share of the isolated influence of each factor on the level of the effective indicator. This is its advantage, and the disadvantage is the limited scope of its application.
In contrast to the integral method, when taking logarithms, not absolute increases in indicators are used, but indices of growth (decrease).
Mathematically, this method is described as follows. Let's assume that the performance indicator can be represented as a product of three factors: F = xyz. Taking the logarithm of both sides of the equation, we get
Considering that the same dependence remains between the indexes of change in indicators as between the indicators themselves, we will replace their absolute values with indices:
It follows from the formulas that the overall 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. And it doesn't matter which logarithm is used - natural or decimal.
Comparing the results of calculating the influence of factors in different ways using this factorial model, one can be convinced of the advantage of the logarithm method. This is expressed in the relative simplicity of calculations and an increase in the accuracy of calculations.
Having considered the basic techniques of deterministic factor analysis and their scope, the results can be systematized in the form of the following matrix:
Table 19 - Deterministic factor methods and models
|
| Multiplicative | Additive | Multiples | mixed |
| Chain substitution | + | + | + | + |
| index | + | - | + | - |
| Absolute difference | + | - | - | Y=a (b-c) |
| Relative differences | + | - | - | - |
| Proportional division (equity) | - | + | - | Y=a/Sxi |
| Integral | + | - | + | Y= a/Sxi |
| Logarithms | + | - | - | - |
ModelsBibliography
1. Bakanov M.I., Sheremet A.D., Theory economic analysis. - M.: Finance and statistics, 2000.
2. Savitskaya G.V. Analysis of the economic activity of the enterprise: Tutorial. - Mn.: IP "Ekoperspektiva", 2000. - 498 p.
3. Methodology of economic analysis industrial enterprise(Associations) / Ed. A.I. Buzhinsky, A.D. Sheremet. - M.: Finance and statistics, 1988
4. Muravieva A.I. Theory of economic analysis. - M.: Finance and statistics, 1988.
Chain substitution method
Determining the magnitude of the influence of individual factors on the growth of performance indicators is one of the most important methodological tasks in AHD. In deterministic analysis, the following methods are used for this: chain substitution, absolute differences, relative differences, proportional division, integral, logarithms, balance, etc.
The most universal of them is the method of chain substitution. 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 the change in the value of the effective indicator by gradually replacing the base value of each factor indicator in the volume of the effective indicator with the actual value in reporting period. For this purpose, a number of conditional values of the performance indicator are determined, which take into account the change in one, then two, three and subsequent factors, assuming that the rest do not change. Comparison of the values of the performance indicator before and after the change in the level of one or another factor makes it possible to eliminate the influence of all factors except one, and to determine the impact of the latter on the growth of the performance indicator. The procedure for applying this method will be considered using the example given in Table. 4.1.
As we already know, the volume of gross output (GRP) depends on two main factors of the first order: the number of workers (HR) and the average annual output (GW). We have a two-factor multiplicative model:
VP \u003d CR GW.
The algorithm for calculating by the method of chain substitution for this model:
VP 0 = CR 0 GV 0 = 100 4 = 400 million rubles;
VP cond. = CR ■ GV 0 = 120 -4 = 480 million rubles; VP 2 = CR, TBj = 120 5 = 600 million rubles.
| Table 4.1
|
As you can see, the second indicator of output differs from the first one in that when calculating it, the number of workers in the current period is taken instead of the base one. The average annual output of products by one worker in both cases is basic. This means that due to the growth in the number of workers, output increased by 80 million rubles. (480-400).
The third indicator of output differs from the second one in that when calculating its value, the output of workers is taken at the actual level instead of the base one. The number of employees in both cases - the reporting period. Hence, due to the increase in labor productivity, output increased by 120 million rubles. (600-480).
Thus, the increase in output is caused by the following factors:
a) increase in the number of workers + 80 million rubles;
b) increased productivity
labor +120 million rubles.
Total + 200 million rubles.
The algebraic sum of the influence of factors must necessarily be equal to the total increase in the effective indicator:
WUA chr + WUA gv = WUA total
The absence of such equality indicates errors in the calculations.
If it is required to determine the influence of four factors, then in this case not one, but three conditional values of the effective indicator are calculated, i.e. the number of conditional values of the effective indicator is one less than the number of factors. Schematically, this can be represented as follows.
Overall change in performance indicator:
AY o6ui =Y,-Y 0 ,
including through:
l y \u003d v - Y ■ AY \u003d Y -Y
A condition1 I 0" ziI B condition2 uel 1"
AY=Y-Y AY=Y-Y
С ^slZ conv2> ziI D M conv"
Let's illustrate this with a four-factor model of output:
VP \u003d CR d p chv.
The initial data for solving the problem are given in Table. 4.1: VP 0 = PR 0 ■ D 0 P 0 PV 0 = 100 200 8 2.5 = 400 million rubles;
VP conv1 = PR, Up to n 0 PV 0 = 120,200 8 ■ 2.5 = 480 million rubles;
VG1 conditional2 - PR, D 1 P 0 CV 0 = 120 208.3 ■ 8 2.5 = 500 million rubles;
VP conv3 = CR, D; P, PV 0 = 120,208.3 7.5 ■ 2.5 = = 468.75 million rubles;
VP, \u003d PR, D, P, CV, \u003d 120 208.3 7.5 3.2 \u003d 600 million rubles.
The volume of output as a whole increased by 200 million rubles. (600 - 400), including by changing:
a) the number of workers
DVP chr \u003d VP conv. - VP 0 \u003d 480 - 400 \u003d +80 million rubles;
b) the number of days worked by one worker per year
WUA D = VP cond.2 - VP cond.1 = 500 - 480 = +20 million rubles;
c) average working hours
WUA n \u003d VP cond3 - VP conv2 = 468.75 - 500 = -31.25 million rubles;
d) average hourly output
DVP cv \u003d VP, - VP cond3 \u003d 600 - 468.75 \u003d +131.25 million rubles.
Total +200 million rubles.
Using the chain substitution method, you need to know the rules for the sequence of calculations: first of all, you need to take into account the change in quantitative, and then qualitative indicators. If there are several quantitative and several qualitative indicators, then first you should change the value of the factors of the first order, and then the lower ones. In the above example, the volume of production depends on four factors: the number of workers, the number of days worked by one worker, the length of the working day and the average hourly output. According to fig. 2.3 the number of workers in relation to gross output - a factor of the first level, the number of days worked - the second level, the length of the working day and the average hourly output - the factors of the third level: This determined the sequence of placement of factors in the model and, accordingly, the order in which their influence was determined.
Thus, the application of the method of chain substitution requires knowledge of the relationship of factors, their subordination, the ability to correctly classify and systematize them.
Absolute difference method
The method of absolute differences is used to calculate the influence of factors on the growth of the effective indicator in deterministic analysis, but only in multiplicative models (Y = x, x
x x 2 x 3 ..... x n) and multiplicative-additive type models:
Y= (a - b)c and Y = a(b - c). And although its use is limited, but due to its simplicity, it has been widely used in AHD.
When using it, the value of the influence of factors is calculated by multiplying the absolute increase in the value of the factor under study by the base (planned) value of the factors that are to the right of it, and by the actual value of the factors located to the left of it in the model.
Calculation algorithm for a multiplicative four-factor model gross output is as follows:
VP \u003d CR D P CV.
DVP chr \u003d FHR Up to n 0 CV 0 \u003d (+20) ■ 200 8.0 2.5 \u003d +80,000;
DVPd \u003d 4Pj DD P 0 FO 0 \u003d 120 (+8.33) 8.0 2.5 \u003d +20,000;
DVP n \u003d CR, ■ D, DP ■ CV 0 \u003d 120 208.33 ■ (-0.5) 2.5 \u003d -31 250;
DVP chv \u003d 4Pj D x P] DCHV \u003d 120 208.33 7.5 (+0.7) \u003d +131 250
Total +200 000
Thus, using the method of absolute differences, the same results are obtained as with the method of chain substitution. Here it is also necessary to ensure that the algebraic sum of the increase in the effective indicator due to individual factors is equal to its total increase.
Consider the algorithm for calculating factors in this way in multiplicative-additive models. For example, let's take a factorial model of profit from the sale of products:
P \u003d URP (C-S), where P - profit from the sale of products;
URP - the volume of sales of products;
C - the price of a unit of production;
C is the unit cost of production.
The increase in the amount of profit due to changes in:
the volume of sales of products DP urp \u003d DURP (C 0 - C 0);
Relative difference method
The method of relative differences is used to measure the influence of factors on the growth of the effective indicator only in multiplicative models. Here, relative increases in factor indicators are used, expressed as coefficients or percentages. Consider the methodology for calculating the influence of factors in this way for multiplicative models of the Y= abc type.
AY c \u003d (Y 0 + AY a + AY b) ^
According to this algorithm, to calculate the influence of the first factor, it is necessary to multiply the base value of the effective indicator by the relative growth of the first factor, expressed as a decimal fraction.
To calculate the influence of the second factor, you need to add the change due to the first factor to the base value of the effective indicator and then multiply the resulting amount by the relative increase in the second factor.
The influence of the third factor is determined similarly: it is necessary to add its growth due to the first and second factors to the base value of the effective indicator and multiply the resulting amount by the relative growth of the third factor, etc.
Let's fix the considered technique on the example given in tab. 4.1:
DVP chv \u003d (vp 0 + DVP CR + DVPd + DVPd) ■
\u003d (400 + 80 + 20-31.25) \u003d + 131.25 million rubles.
As you can see, the calculation results are the same as when using the previous methods.
The method of relative differences is convenient to use in cases where it is required to calculate the influence of a large complex of factors (8-10 or more). Unlike the previous methods, the number of computational procedures is significantly reduced here, which determines its advantage.
19. Method of relative differences
used in deterministic factor analysis to assess the impact of each individual factor on the growth of the effective indicator. The advantage of this method is its simplicity. The relative difference method can only be used for multiplicative and multiplicative-additive factor models.
This method is based on the elimination method. Elimination (from English. eliminate) means the elimination of the influence of all other factors (except one), that is, all other factors remain static. The method proceeds from the fact that all factors change independently of each other. First, the base value changes to the reporting value for one factor with the other factors unchanged, static, then for two, three, and so on.
To calculate the magnitude of the impact of the first factor on the performance indicator, multiply the base value of the performance indicator by the relative growth of the first factor in percent and divide by 100.
To calculate the influence of the second factor, multiply the sum of the base value of the effective indicator and its growth due to the first factor by the relative growth of the second factor.
To calculate the influence of the third factor, multiply the sum of the base value of the effective indicator, the influence of the first and second factors by the relative deviation of the third factor. And so on.
When using this method, the order in which the factors are arranged in the factorial model and, accordingly, the sequence of changing the values of the factors is of great importance, since the quantitative assessment of the influence of each factor depends on this.
For method of relative differences, a correctly constructed deterministic factorial model should be used, it is necessary to observe a certain order in the arrangement of factors.
If the factor model contains quantitative and qualitative factors, then the replacement of factors should begin with a quantitative factor.
Quantitative factors reflect the quantitative certainty of phenomena. Quantitative factors can be expressed both in value and in physical terms. For example, quantitative factors characterize the volume of production and sales of products, and the value of these factors can be expressed both in rubles and in pieces, meters, etc.
Qualitative Factors characterize the internal properties, features and characteristics of the objects under study. For example, a qualitative factor is the fat content of milk, labor productivity, product quality, etc.
If there are several quantitative and several qualitative indicators, then first you should change the value of the factors of the first level of subordination, and then the lower one.
Hierarchically, the factors are divided into factors of the first, second, third level etc. The factors of the first level are the factors that directly affect the performance indicator. Factors that affect the performance indicator indirectly, through factors of the first level, are factors of a lower level (second, third, etc.).
The algorithm for calculating the relative difference method for a two-factor multiplicative model is as follows:
X = A* B;
Δ rel A-((A 1 -BUT 0 )/BUT 0 *100;
Δ rel B-((B 1 -B 0 )/B 0 *100;
Δ XA= X plan* Δ rel BUT;
ΔX B = (X plan +ΔХ(а)) Δ rel B.
The sum of these quantities (ΔXa and ΔXb) must be identical to the difference between X 1 and X 0
Consider the calculation algorithm on a specific example.
The annual production of the enterprise depends on the average annual number of workers (H) and the average annual output of one worker (AT). A two-factor multiplicative model is compiled, where the number of workers is a quantitative factor, and therefore it comes first in the model, and production is a qualitative factor, and it is behind the quantitative one.
OP=H*V.
The data we will use is entered in tab. 6.
Table 6 Data for factor analysis
So on first tag we need to calculate the relative increments of the factors.
Δ rel H \u003d ((H fact - H plan) / H plan) * 100 \u003d ((27 - 25) / 25) 100 \u003d 8;
Δ rel B \u003d ((In fact - In plan) / In plan) * 100 \u003d ((230-200) / 200) * 100 \u003d 15.
The relative change in the average annual number of workers was 8%, and the relative change in the average annual output was 15 %.
Second step. We find the influence of the first factor on the value of the effective indicator. In our case, how will the volume of production change if the number of workers increases by two people. We must multiply the planned output by the relative increase in the number of workers and divide the resulting number by 100.
ΔOP(H) = OP plan * Δ rel H;
Δ OP (H) \u003d 5000 8/100 \u003d 400.
Conclusion: an increase in the average annual number of workers by 2 people led to the fact that the volume of production increased by 400 thousand rubles.
Third step. We continue to consistently consider the factors in our model. Now we find the influence of the second factor on the value of the effective indicator. In our example, how will the volume of production change if the average annual output of one worker increases (by 30 thousand rubles). We must multiply the sum of the planned value of the effective indicator (production volume) and the influence of the first factor (average annual number of workers) by the relative growth of the second factor (average annual output per worker) and divide the resulting figure by 100:
ΔOP (V)= ((OP plan + ΔOP(H)) * Δ rel B)/100;
ΔOP (V)= ((5000+400) 15)/100 = 810.
Conclusion: an increase in the average annual output of one worker led to an increase in production by 810 thousand rubles.
Fourth step. Examination. The algebraic sum of the influence of factors when using this method must necessarily be equal to the total increase in the effective indicator. The absence of such equality indicates errors in the calculations.
OP fact - OP plan = 6210-5000=1210;
ΔOP(H) + ΔOP(V) = 400 + 810 = 1210.
Our calculations are correct.
Calculations are carried out similarly for other admissible types of models.
The disadvantage of the method is the formation of an indecomposable residue, which is added to the magnitude of the influence of the last factor. This leads to a decrease in the accuracy of calculations. This can be avoided by using the integral method of factor analysis.
Absolute difference method
It is used in multiplicative and multiplicative-additive models and consists in calculating the magnitude of the influence of factors by multiplying the absolute increase in the factor under study by the base value of the factor located to the right of it and by the actual value of the factors located to the left. For example, for a multiplicative factorial model of the type Y \u003d a-b-c-th the change in the magnitude of the influence of each factor on the performance indicator is determined from the expressions:
where /> th, sat, ¿4- values of indicators in the base period; jaf,bf, cf - the same in the reporting period (i.e. actual); Aa \u003d df - Ob, AL \u003d bf - b6, Ac \u003d sf - sb; Asi = b?f - a.
Relative difference method
The method of relative differences, as well as the method of absolute differences, is used only in multiplicative and multiplicative-additive models to measure the influence of factors on the growth of the effective indicator. It consists in calculating the relative deviations of the values of factor indicators with the subsequent calculation of the change in the effective indicator Uf due to each factor relative to the base Yf. For example, for a multiplicative factorial model of the type
Y = abs the change in the magnitude of the influence of each factor on the performance indicator is determined as follows:
The relative difference method, having a high level of clarity, provides the same results as the absolute difference method with a smaller amount of calculations, which is quite convenient when there are a large number of factors in the models.
Proportional division (equity) method
Applies to additive Y = a + b + c and multiple models of type Y= a/(b + c + d), including multilevel ones. This method consists in the proportional distribution of the increase in the effective indicator At by changing each of the factors between them. For example, for an additive model of type Y = a + b + c influence is calculated as
We will assume that Y is the cost of production; a, b, c - material, labor and depreciation costs, respectively. Let the level of the general profitability of the enterprise decrease by 10% due to an increase in the cost of production by 200 thousand rubles. At the same time, the cost of materials decreased by 60 thousand rubles, labor costs increased by 250 thousand rubles, and depreciation costs - by 10 thousand rubles. Then due to the first factor (a) the level of profitability has increased:
Due to the second (b) and third (c) factors, the level of profitability decreased:
Method of differential calculus
It assumes that the total increment of the function differs in terms, where the value of each of them is determined as the product of the corresponding partial derivative and the increment of the variable on which this derivative is calculated.
Consider a function of two variables: r=/(x, y). If this function is differentiable, then its increment can be represented as
where Ag = (2(-2o)- function change; Oh = ("Г] - ,г0) - change of the first factor; Ay = (y^ - r/()) - change of the second factor.
Sum (dg / dx) Ah + (dg / du) Ay - the main part of the increment of the differentiable function (which is taken into account in the method of differential calculus); 0ud~r ^+d7/ - indecomposable remainder, which is an infinitesimal value for sufficiently small changes in the factors x and y. This component is not taken into account in the considered method of differential calculus. However, when significant changes factors (Oh and ay) there may be significant errors in assessing the influence of factors.
Example 16.1. Function G has the form z = x-y, for which the initial and final values of the influencing factors and the resulting indicator are known (x&y0, r0, x, y, 2). Then the influence of influencing factors on the value of the resulting indicator is determined by the expressions
Let us calculate the value of the remainder term as the difference between the value general change functions Dz = X ■ y - x0 o g / o and the sum of the influences of the influencing factors r,. + Dz(/ = y0-Ax + xn■ &y:
Thus, in the method of differential calculus, the indecomposable remainder is simply discarded (the logical
differentiation method error). This approximation of the considered method is a disadvantage for economic calculations, where an exact balance of the change in the resulting indicator and the sum of the influence of influencing factors is required.