The method of relative differences is an example of a solution. The method of relative (percentage) differences of deterministic factor analysis. Method of chain substitutions

Types of deterministic models that use the chain substitution method. Essence and rules of its application. Algorithms for calculating the influence of factors by this method in various types of models.

One of the most important methodological issues in AHD is to determine the magnitude of the influence of individual factors on the growth of performance indicators. In deterministic factor analysis (DFA), the following methods are used for this: chain substitution, index, absolute differences, relative differences, proportional division, integral, logarithms, etc.

The first four methods are based on the elimination method. To eliminate means to eliminate, reject, exclude the influence of all factors on the value of the effective indicator, except for one. This method proceeds from the fact that all factors change independently of each other: first one changes, and all others remain unchanged, then two change, then three, etc., while the rest remain unchanged. This allows you to determine the influence of each factor on the value of the studied indicator separately.

The most versatile of these is chain substitution method. 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 the reporting period. For this purpose, a number of conditional values of the effective indicator are determined, which take into account the change in one, then two, three, etc. factors, assuming that the others do not change. Comparing the value of the effective indicator before and after changing the level of one or another factor allows you to eliminate the influence of all factors except one, and determine the impact of the latter on the growth of the effective indicator.

The procedure for applying this method will be considered in the following example (Table 6.1).

As we already know, the volume of gross output ( VP) depends on two main factors of the first level: the number of workers (CR) and average annual output (GV). We have a two-factor multiplicative model: VP = Czech Republic X GV.

The algorithm for calculating by the method of chain substitution for this model:

As you can see, the second indicator of gross output differs from the first one in that when calculating it, actual number workers instead of planned. Average annual output production by one worker in both cases is planned. This means that due to the increase in the number of workers, output increased by 32,000 million rubles. (192,000 - 160,000).

The third indicator differs from the second one in that when calculating its value, the output of workers is taken at the actual level instead of the planned one. The number of employees in both cases is actual. Hence, due to the increase in labor productivity, the volume of gross output increased by 48,000 million rubles. (240,000 - 192,000).

Thus, the overfulfillment of the plan in terms of gross output was the result of the influence of the following factors:

a) increase in the number of workers + 32,000 million rubles.

b) increasing the level of labor productivity + 48,000 million rubles.

Total +80,000 million rubles

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

The absence of such equality indicates errors in the calculations.

For clarity, the results of the analysis are given in table. 6.2.

If it is required to determine the influence of three factors, then in this case not one, but two conditional additional indicators are calculated, i.e. the number of conditional indicators is one less than the number of factors. Let's illustrate this on a four-factor model of gross output:

The initial data for solving the problem are given in Table 6.1:

The plan for the production of products as a whole was overfulfilled by 80,000 million rubles. (240,000 - 160,000), including by changing:

a) the number of workers

Using the chain substitution method, it is recommended to adhere to a certain 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 level of subordination, and then the lower one. 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 Scheme 5.2, the number of workers in this case is the factor of the first level of subordination, the number of days worked is the second level, the length of the working day and the average hourly output are factors of the third level. This determined the sequence of placement of factors in the model and, accordingly, the sequence of their study.

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.

We considered an example of calculating the influence of factors on the growth of the effective indicator in multiplicative models.

In multiple models the algorithm for calculating factors for the value of the studied indicators is as follows:

where FD- return on assets; VP- gross output; OPF - average annual cost of basic production assets.

Method for calculating the influence of factors in mixed models:

a) Multiplicatively-additive type P = VPP (C - WITH)

where P- the amount of profit from the sale of products; VPP - the volume of sales of products; C - selling price; C - unit cost of production;

In a similar way, the influence of factors is calculated for other deterministic mixed-type models.

Separately, it is necessary to dwell on the methodology for determining the influence structural factor on the increase in the effective indicator using this method. For example, sales revenue (AT) depends not only on the price (C) and quantity of products sold (VPN), but also from its structure (UDi). If the share of products of the highest quality category, which is sold at higher prices, increases, then revenue will increase due to this, and vice versa. The factorial model of this indicator can be written as follows:

In the process of analysis, it is necessary to eliminate the influence of all factors, except for the structure of the product. For this, we compare the following indicators revenue:

The difference between these indicators takes into account the change in revenue from the sale of products due to changes in its structure (Table 6.3.).

It can be seen from the table that due to the increase specific gravity second-class products in the total volume of its sales, revenue decreased by 10 million rubles. (655 - 665). This is the unused reserve of the enterprise.

6.2. Index Method

The essence and purpose of the index method. Algorithm for calculating the influence of factors by this method for different models.

The index method is based on relative indicators dynamics, spatial comparisons, implementation of the plan, expressing the ratio of the actual level of the analyzed indicator in the reporting period to its level in the base period (or to the planned one or for another object).

With the help of aggregate indices, it is possible to identify the influence of various factors on the change in the level of performance indicators in multiplicative and multiple models.

For example, let's take the index of the cost of marketable products:

It reflects the change in the physical volume of marketable products (q) and prices (R) and is equal to the product of these indices:

To establish how the cost of marketable products has changed due to the quantity of manufactured products and due to prices, it is necessary to calculate the index of physical volume Iq and price index 1 p:

In our example, the volume of gross output can be represented as the product of the number of workers and their average annual output. Therefore, the index of gross output 1ch will be equal to the product of the index of the number of workers lchr and index of average annual output 1gv:

If we subtract the denominator from the numerator of the above formulas, then we will obtain the absolute growth of gross output as a whole and due to each factor separately, i.e. the same results as the chain substitution method.

6.3. Absolute difference method

Essence, purpose and scope of the method of absolute differences. The procedure and algorithms for calculating the influence of factors in this way

Way absolute differences is one of the elimination modifications. Like the chain substitution method, it is used to calculate the influence of factors on the growth of the effective indicator in deterministic analysis, but only in multiplicative and multiplicative-additive models: Y= (a -b)with and Y = a(b- with). And although its use is limited, but due to its simplicity, it has been widely used in AHD. This method is especially effective if the initial data already contains absolute deviations in factorial indicators.

When using it, the value of the influence of factors is calculated by multiplying the absolute increase in 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.

Consider the calculation algorithm for multiplicative factor model of the type Y= a x b x c x d. There are planned and actual values for each factor indicator, as well as their absolute deviations:

We determine the change in the value of the effective indicator due to each factor:

As can be seen from the above diagram, the calculation is based on the successive replacement of the planned values of factor indicators with their deviations, and then with the actual level of these indicators.

Consider the methodology for calculating the influence of factors in this way for a four-factor multiplicative model of gross output:

Thus, the absolute difference method gives the same results as the chain substitution method. 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 mixed models type V = (a - b)with. For example, let's take the factorial model of profit from the sale of products, which was already used in the previous paragraph:

P = VRP(C - WITH).

The increase in the amount of profit due to changes in the volume of sales of products:

selling prices:

production cost:

Calculation of the influence of the structural factor using this method is carried out as follows:

As can be seen from Table. 6.4, due to the change in the structure of sales, the average price for 1 ton of milk decreased by 40 thousand rubles, and for the entire actual volume of sales of products, the profit was received less by 10 million rubles. (40 thousand rubles x 250 tons).

6.4. Relative difference method

The essence and purpose of the method of relative differences. Scope of its application. Algorithm for calculating the influence of factors in this way.

Relative difference method, like the previous one, it is used to measure the influence of factors on the growth of the effective indicator only in multiplicative and additive-multiplicative models of the type V= (a - b)c. It is much simpler than chain substitutions, which makes it very efficient under certain circumstances. This primarily applies to those cases where the initial data contain previously determined relative increases in factor indicators in percentages or coefficients.

Consider the methodology for calculating the influence of factors in this way for multiplicative models of the type V = BUT X AT X WITH. First, you need to calculate the relative deviations of factor indicators:

Then the change in 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 given in tab. 6.1:

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.

A variation of this method is acceptance of percentage differences. We will consider the methodology for calculating the influence of factors with its help using the same example (Table 6.1).

In order to establish how much the volume of gross output has changed due to the number of workers, it is necessary to multiply its planned value by the percentage of overfulfillment of the plan by the number of workers CR%:

To calculate the influence of the second factor, it is necessary to multiply the planned volume of gross output by the difference between the percentage of the plan fulfilled by the total number of days worked by all workers D% and the percentage of the implementation of the plan for the average number of workers CR%:

The absolute increase in gross output due to a change in the average length of the working day (intra-shift downtime) is established by multiplying the planned volume of gross output by the difference between the percentage of the plan fulfilled by the total number of hours worked by all workers t% and the total number of days they worked D%:

To calculate the impact of average hourly output on the change in the volume of gross output, the difference between the percentage of the implementation of the plan for gross output VP% and the percentage of plan fulfillment by the total number of hours worked by all workers t% multiply by the planned volume of gross output VPpl:

The advantage of this method is that when it is applied, it is not necessary to calculate the level of factor indicators. It is sufficient to have data on the percentage of fulfillment of the plan in terms of gross output, the number of workers and the number of days and hours worked by them for the analyzed period.

6.5. Method of proportional division and equity participation

Essence, purpose and scope of the method of proportional division. The procedure and algorithms for calculating the influence of factors in this way.

In some cases, to determine the magnitude of the influence of factors on the growth of the effective indicator, one can use proportional division method. This applies to those cases when we are dealing with additive models of the type V = Xi and multiply additive type

In the first case, when we have a single-level model of type V= a + b+ p. calculation is carried out as follows:

For example, the level of profitability decreased by 8% due to an increase in the company's capital by 200 million rubles. At the same time, the value of fixed capital increased by 250 million rubles, while the value of circulating capital decreased by 50 million rubles. 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. The relationship of factors in the combined model is shown in fig. 6.1.

When known ATd, Vp and W, as well as Yb, then to determine Yd, Y n, 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 AT between second level factors D, N and M according to their growth. The proportionality of this distribution is achieved by determining a coefficient constant for all factors, which shows the amount of change in the effective indicator Y due to a change in the factor AT per unit.

Coefficient value (TO) is defined as follows:

Multiplying this coefficient by the absolute deviation AT due to the corresponding factor, we find the change in the effective indicator:

For example, the cost of 1 tkm 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 tkm

b) overplanned idle runs -4000 tkm

c) incomplete use of load capacity -3000 tkm

Total-12000 tkm

From here you can determine the change in cost under the influence of factors of the second level:

To solve this type of problem, you can also use the method of equity participation. First, the share of each factor in the total amount of their growth is determined, which is then multiplied by the total growth of the effective indicator (Table 6.5):

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 enterprises.

6.6. Integral method in the analysis of economic activity

The main disadvantages of the elimination method. The problem of decomposition of additional growth from the interaction of factors between them. The essence of the integral method and the scope of its application. Algorithms for calculating the influence of factors in different models in an integral way.

Elimination as a way of deterministic factor analysis has a significant drawback. When using it, it is assumed that the factors change independently of each other. In fact, they change together, interconnectedly, and this interaction results in an additional increase in the effective indicator, which, when applying elimination methods, is added to one of the factors, usually the latter. In this regard, the magnitude of the influence of factors on the change in the effective indicator varies depending on the place that this or that factor is placed in the deterministic model.

Let's consider it on an example which is given in tab. 6.1. According to the data given in it, the number of workers at the enterprise increased by 20%, labor productivity - by 25%, and the volume of gross output - by 50%. This means that 5% (50 - 20 - 25), or 8,000 million rubles. gross output is an additional increase from the interaction of both factors.

When we calculate the conditional volume of gross output, based on the actual number of workers and the planned level of labor productivity, then the entire additional increase from the interaction of two factors refers to a qualitative factor - a change in labor productivity:

If, however, when calculating the conditional volume of gross output, we take the planned number of workers and the actual level of labor productivity, then the entire additional increase in gross output refers to the quantitative factor, which we change secondarily:

We will show a graphical solution to the problem in different versions (Fig. 6.2).

In the first version of the calculation, the conditional indicator has the form: VP cond. = ChRf X GV pl, in the second - VP conv = CH pl X GVf.

Accordingly, deviations due to each factor in the first case

in the second

On the graphs, these deviations correspond to different rectangles, since with different substitution options, the value of the additional increase in the effective indicator, equal to the rectangle ABCD, relates in the first case to the magnitude of the influence of annual output, and in the second case, to the magnitude of the influence of the number of workers. As a result, the magnitude of the influence of one factor is exaggerated, while the other is underestimated, which causes ambiguity in assessing the influence of factors, especially in cases where the additional increase is quite significant, as in our example.

To overcome this shortcoming, deterministic factor analysis uses integral method, which is used to measure the influence of factors in multiplicative, multiple and mixed models of a multiple-additive type

Using this method allows you to get more accurate results of calculating the influence of factors compared to the methods of chain substitution, absolute and relative differences and avoid an ambiguous assessment of the influence of factors because in this case the results do not depend on the location of the factors in the model, and an additional increase in the effective indicator, which formed from the interaction of factors, is decomposed between them equally.

At first glance, it may seem that in order to distribute an additional increase, it is enough to take half of it or a part corresponding to the number of factors. But this is most often difficult to do, since factors can act in different directions. Therefore, certain formulas are used in the integral method. Here are the main ones for different models.

The logarithm method is used to measure the influence of factors in multiplicative models. In 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, an even higher accuracy of calculations is provided. If, when integrating, 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 its limited scope.

Unlike the integral method, the logarithm uses not absolute increases in indicators, but indices of their growth (decrease).

Mathematically, this method is described as follows. Suppose that the performance indicator can be represented as a product of three factors: f = xz. 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 index of the effective indicator. And it doesn't matter which logarithm is used - natural or decimal.

Using the data in Table. 6.1, we calculate the increase in gross output due to the number of workers (CR), number of days worked by one worker per year (D) and average daily output (DV) according to the factor model:

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 main methods of deterministic factor analysis and the scope of their application, the results can be systematized in the form of the following matrix:

Knowledge of the essence of these techniques, their scope, calculation procedures - necessary condition qualified quantitative research.

The essence and purpose of the method of relative differences. Scope of its application. Algorithm for calculating the influence of factors in this way.

Relative difference method, like the previous one, it is used to measure the influence of factors on the growth of the effective indicator only in multiplicative and additive-multiplicative models of the type V = (a - b)c. It is much simpler than chain substitutions, which makes it very efficient under certain circumstances. This primarily applies to those cases where the initial data contain previously determined relative increases in factor indicators in percentages or coefficients.

Then the change in the effective indicator due to each factor is determined as follows:

Let's fix the considered technique on the example given in tab. 6.1:

As you can see, the calculation results are the same as when using the previous methods.

A variation of this method is acceptance of percentage differences. We will consider the methodology for calculating the influence of factors with its help using the same example (Table 6.1).

See also:

(to the table of contents)

Example 1 Create a factor system of the volume of gross output, which is in functional dependence on the following indicators:

· the number of days worked by one employee per year (D);

· average hourly output of products by one worker (CV);

· average working day (P);

· average daily output of products by one worker (DV);

· average annual output of products by one worker (HW);

· average annual number of workers (HR).

Decision:

Factor model of gross output volume:

VP \u003d CR * GV or VP \u003d CR * D * DV or VP \u003d CR * D * P * CV.

Example 2 Based on the initial data of Table 14 (highlighted in italics), determine the absolute and relative change in sales proceeds and the magnitude of the influence of the volume and price of products sold on this indicator using the following methods:

· chain substitutions;

· absolute differences;

· relative differences;

· integral;

· logarithms

based on the model:

B =VRP * C,

where B is the proceeds from the sale of products,

VRP - the volume of products sold,

C - the price of sold products.

Table 14

Indicators	Base	Report	Changes
Indicators	Base	Report	abs.	rel.
1	2	3	4=3-2	5=4/2*100%
1. The volume of products sold, thousand pieces.	10	12
2. Price of sold products, thousand rubles.	7	10		42,8
3. Revenue (2*3), million rubles		120		71,4

Decision:

1. Method of chain substitutions

We calculate the value of revenue by successively replacing the basic values of factor indicators with the values of the reporting period:

B 0 =VRP 0 * C 0 \u003d 10 * 7 \u003d 70 million rubles.

Condition1 =VRP 1 * C 0 \u003d 12 * 7 \u003d 84 million rubles.

B 1 =VRP 1 * C 1 \u003d 12 * 10 \u003d 120 million rubles.

Let's evaluate the influence of each factor separately:

∆V V RP \u003d In condition 1 - B 0 \u003d 84 - 70 \u003d 14 million rubles.

∆V C \u003d V 1 - V condition 1 \u003d 120 - 84 \u003d 36 million rubles.

Examination:

∆V= V 1 -V 0 =∆V V RP +∆V C \u003d 120-70 \u003d 14 + 36 \u003d 50 million rubles.

2. Method of absolute differences

∆V V RP = ∆ VRP * C 0 \u003d 2 * 7 \u003d 14 million rubles.

∆V C =VRP 1 * ∆C \u003d 12 * 3 \u003d 36 million rubles.

Examination:

3. Method of relative differences

∆V V RP = В 0 *(∆VRP/VRP 0) = 70 * (2/10) = 14 million rubles.

∆V C \u003d (V 0 + ∆V V RP ) * (∆C / C 0) \u003d 84 * (3/7) \u003d 36 million rubles.

Examination:

∆В= 120-70=14+36=50 million rubles

4. Integral method

∆V V RP = 0,5*∆ VRP * (C 0 + C 1) \u003d 0.5 * 2 * (7 + 10) \u003d 17 million rubles.

∆V C = 0.5*∆C*(VRP 0+VRP 1) \u003d 0.5 * 3 * (10 + 12) \u003d 33 million rubles.

Examination:

5. Logarithm method

∆V V RP = ∆V*lg( VRP 1 /VRP 0)/lg(B 1 / B 0) \u003d 50 * (0.079 / 0.23) \u003d 17 million rubles.

∆V C =∆V*lg(C 1 /C 0)/lg(B 1 / B 0) \u003d 50 * (0.15 / 0.23) \u003d 33 million rubles.

Examination:

∆В= 120-70=17+33=50 million rubles

Conclusion: calculations showed that the greatest impact on the increase in sales proceeds was the increase in the price of products. Three out of five methods gave the same results for the values of the factorial influence on the performance indicator. The use of the integral method and the logarithm method made it possible to take into account the interaction of factor indicators with each other and, as a result, to more accurately determine their impact on the effective indicator, in particular, to identify more strong influence volume factor.

Example 3 Based on the initial data (highlighted in italics) given in Table 15, determine the absolute and relative change in gross profit from product sales and the magnitude of the influence of factors on gross profit by the method of proportional division and the equity method, using the model:

where Pr - gross profit from product sales,

B - proceeds from the sale of products,

C is the cost of goods sold.

Table 15

Indicators	Basic year	Reporting year	Changes
Indicators	Basic year	Reporting year	abs.	rel.
			4=3-2	5=4/2*100%
1. Revenue, thousand rubles	56 377	62 849	6472	11,48
2. Cost price, thousand rubles.	46 496	57 738	11242	24,18
3. Gross profit (1-2), thousand rubles	9881	5111	4770	48,27

Decision:

1. Method of proportional division

thousand. rub.

Examination :

thousand. rub.

2. Method equity participation

thousand. rub.

Examination :

thousand. rub.

Conclusion: gross profit from product sales in the reporting period decreased by 4,770 thousand rubles. or by 48.27% compared to the base period due to the outstripping growth of production costs over the growth of sales proceeds. The share of the negative impact of cost growth on the decrease in gross profit amounted to 63.46% (3027.23/4770*100%).

Example 4 Based on the data in Table 16, determine the relationship between sales revenue and advertising costs, calculate the correlation coefficients, determinations and determine the correlation ratio.

Table 16

Decision: Calculate the derivatives for analysis in Table 17:

Table 17

			X*Y	x2	Y2	Y x
			2800	1600	4900
			3024	1764	5184	71,2
			2584	1444	4624	68,8
			2990	2116	4225	73,6
			3520	1936	6400	72,4
			3600	2304	5625	74,8
			3900	2500	6084
Total	308	508	22418	13664	37042	506,8

Based on the table, we build a system of equations

from here

The relationship equation describing the dependence of sales revenue on advertising costs has received the following expression:

Y x =46+ 0,6 x

Calculate the correlation coefficient:

Calculatecoefficientdeterminations:

Conclusion: in this case, the relationship between the indicators is insignificant, the value of the coefficient of determination indicates that the revenue from product sales is 22% dependent on advertising costs, and other factors account for 78% of the change in its level.

Task 2.1. Convert the analytical formula by extension:

where GW is the annual output (labor productivity);

HR - average headcount,

in such a way that it reflects the dependence of labor productivity on capital productivity and capital-labor ratio.

Task 2.2. Convert the analytical formula using the reduction method:

where FO - capital productivity of fixed production assets;

VP - gross output for the year;

OPF - the average annual cost of fixed production assets,

in such a way that it reflects the relationship between the average annual output of one worker and the capital-labor ratio.

Task 2.3. Using the lengthening method, convert the analytical formula:

where ME is the material consumption of products;

MR - the cost of material resources;

B is the revenue

in such a way that it reflects the relationship between the material intensity of raw materials and materials, fuel intensity, energy intensity, material intensity of other costs.

Task 2.4. To systematize the factors that determine the amount of profit from the sale of products:

- revenue (B);

- volume of sales (VRP);

- total costs (Z);

- unit price (P);

- structureproducts ();

- unit cost (C)

and write down the factorial profit model.

Task 2.5. Transform the analytical formula by extension in such a way that it reflects the dependence of the profitability of assets on the value of the profitability of sales and asset turnover.

Problem 2.6. Create a factor model, where the factor indicators are the volume of gross output and the average annual cost of fixed assets. Using the chain substitution method, determine the quantitative influence of factors on the performance indicator if:

· gross output for the reporting period increased in comparison with the plan from 78,000 to 82,000 rubles;

· the average annual cost of fixed production assets decreased from 72,000 to 70,000 rubles.

Problem 2.7. Based on the data in Table 18, create a factorial model of profit from the sale of products and calculate the influence of factors on the change in its amount in all possible ways.

Table 18

Indicator	Base year	Reporting year
Sales volume, pcs.	8 000	8 400
Selling price, thousand rubles
The cost of the product, thousand rubles.

Problem 2.8. Based on the data in Table 19, create a factorial model of the dependence of the volume of production on the value of the average annual cost of fixed assets and capital productivity and, using the integral method and the method of absolute differences, determine the magnitude of the influenceI factor indicators on the effective.Volume of production, million rubles

21409

22287

Average annual cost of fixed assets, million rubles

23000

23447

Problem 2.9. Using the data in Table 20, create a factorial model of a multiply additive type and use the equity method to determine the impact of changes in sales profit, the average annual cost of fixed assets and the value working capital on the change in the indicator of profitability of production.

Table 20

Indicator

Base year

Reporting year

Profit, thousand rubles

55,25

65,16

Average annual cost, thousand rubles:

fixed assets

working capital

500

350

520

385

Problem 2.10. The duration of capital turnover was reduced by 25 days. Calculate the influence of factors on the change in the duration of capital turnover by the method of proportional divisiontaking into account the change in factor indicators given in table 21.

Table 21

	Change in average balances, thousand rubles
Stocks of raw materials and materials	+2700
Remains of WIP	+1300
Finished products	- 800
Receivables	+2000
cash	- 200

Problem 2.11. The relationship between the costs of production and its volume is described by a straight-line relationship . Based on the data in Table 22, determine the coefficients of the relationship equation, the coefficients of correlation and determination, explain their economic meaning.

No. p / p

Production costs, thousand rubles

Volume of production, thousand rubles

120

200

130

270

150

280

140

250

180

200

180

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 overall profitability of the enterprise decreased 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 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 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, with significant changes in 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 of the total change in the function Dr = 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 approximateness of the considered method serves as 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.

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 the 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

Indicator	Symbol	Indicator level	Change
base	current	absolute	relative, %
Gross output, million rubles	VP			+150	+50
Average number of workers	Czech Republic			+20	+20
Average annual production output by one worker, million rubles	GV			+1	+25
Number of working days for one worker per year	D		208,3	+8,3	+4,17
Average daily output of a worker, thousand rubles	DV			+4	+20
Average shift duration, h	P		7,5	-0,5	-5
Average hourly production by one worker, thousand rubles	CV	2,5	3,2	+0,7	+28

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 = PR, 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.