Economic-mathematical methods and models of analysis. An example of building a multiplicative model

They are used in cases where the performance indicator is an algebraic sum of several factorial indicators.

2. Multiplicative models

Y=
.

This type of model is used when the performance indicator is the product of several factors.

3. Multiples models

Y= .

They are used when the effective indicator is obtained by dividing one factorial indicator by the value of another.

4. Mixed (combined) models are a combination in various combinations of previous models:

Y= ; Y= ; Y=(a+b)c .

transformation factor systems

1. Transformation multiplicative factor systems is carried out by successive partitioning of the factors of the original system into factor factors.

For example, when studying the process of forming the volume of production (see Figure 6.1), you can use such deterministic models as

VP=KR GV; VP=KR D LW, VP=CR D P ST.

These models reflect the process of detailing the original factor system of a multiplicative type and expanding it by dividing complex factors into factors. The degree of detail and expansion of the model depends on the purpose of the study, as well as on the possibility of detailing and formalizing indicators within the established rules.

2. Simulation is carried out in a similar way additive factor systems due to dismemberment of one of the factor indicators into its constituent elements-terms.

Example. As is known, the sales volume

VRP \u003d VVP - VI,

where GDP is the volume of production;

VI - the volume of on-farm use of products.

In an agricultural enterprise, grain products were used as seeds (S) and feed (K). Then the given initial model can be written as follows: VП = VVP - (С + К).

3. To class multiples models, the following methods of their transformation are used:

elongation;

formal decomposition;

extensions;

abbreviations.

The first the method involves lengthening the numerator of the original model by replacement of one or more factors by the sum of homogeneous indicators.

For example, the unit cost of production can be represented as a function of two factors: the change in the amount of costs (3) and the volume of output (VVP). The initial model of this factorial system will have the form

C= .

If the total amount of costs (3) is replaced by their individual elements, such as wages (OT), raw materials (CM), depreciation of fixed assets (A), overhead costs (NC), etc., then the deterministic factor model will have kind of additive model with a new set of factors

C= +++=X + X + X + X ,

where X - the complexity of products; X - material consumption of products; X - capital intensity of products; X – overhead level

Formal decomposition method factor system provides lengthening the denominator of the original factor model by replacing one or more factors with the sum or product of homogeneous indicators.

If a b=l+m+n+p, then

Y=
.

As a result, a final model of the same type as the original factorial system (multiple model) was obtained. In practice, such a decomposition occurs quite often. For example, when analyzing the indicator of profitability of production (P):

P= ,

where /7 - the amount of profit from the sale of products;

3 - the amount of costs for the production and sale of products.

If the sum of costs is replaced by its individual elements, the final model as a result of the transformation will take the following form:

P=
.

The cost of one ton-kilometer (C
) depends on the amount of costs for the maintenance and operation of the car (3) and on its average annual output (GW). The initial model of this system will look like

FROM
=.

Given that the average annual output of a car, in turn, depends on the number of days worked by one car per year (D), the duration of the shift (P) and the average hourly output (AM), we can significantly lengthen this model and decompose the increase in cost into more factors:

FROM
=
.

Extension method provides for the expansion of the original factorial model due to multiplying the numerator and denominator of a fraction by one or more new indicators. For example, if the original model

introduce a new indicator c, then the model will take the form

.

The result is a final multiplicative model in the form of a product of a new set of factors.

This method of modeling is very widely used in analysis. For example, the average annual production of products by one employee (an indicator of labor productivity) can be written as follows: GV = VP / KR. If we introduce such an indicator as the number of days worked by all employees (D), we get the following model of annual output:

HW=
,

where DV is the average daily output; D - the number of days worked by one employee.

After the introduction of the indicator of the number of hours worked by all employees (T), we will obtain a model with a new set of factors: average hourly output (AM), the number of days worked by one employee (D) and the length of the working day (P):

Reduction method is the creation of a new factorial model by dividing the numerator and denominator of a fraction by the same factor:

.

In this case, we get the final model of the same type as the original one, but with a different set of factors.

Another example. The economic profitability of the company's assets (ROA) is calculated by dividing the amount of profit (P) by the average annual cost of the company's fixed and working capital (A): ROA=P/A.

If we divide the numerator and denominator by the volume of sales of products (S), then we get a multiple model, but with a new set of factors: the profitability of products sold and the capital intensity of products:

 Performance indicators can be decomposed into constituent elements (factors) in various ways and presented in the form of various types of deterministic models. The choice of modeling method depends on the object of study, the goal, as well as on the professional knowledge and skills of the researcher. The process of modeling factor systems is a very complex and crucial moment in economic analysis. The final results of the analysis depend on how realistically and accurately the created models reflect the relationship between the studied indicators..

Condition: to determine the impact of the number of personnel, the number of shifts worked and the output per shift per employee on the change in output (N p).

Make a conclusion.

Solution algorithm:

The factor model describing the relationship of indicators has the form: N = h * cm * v

Initial data - factors and the resulting indicator are presented in the analytical table:

Deterministic factor analysis methods used to solve three-factor models:

- chain substitution;

- absolute differences;

 weighted final differences;

- logarithmic;

- integrated.

Application various methods to solve a typical problem:

1. Chain substitution method. The application of this method involves the allocation of quantitative and qualitative factor characteristics: here, the quantitative factors are the number of personnel and the number of shifts worked; quality sign - production.

a) N 1 = h 0 * Cm 0 * AT 0 =5184 thousand pieces;

b) N 2 = h 1 * Cm 0 * AT 0 \u003d 25 * 144 * 1500 \u003d 5400 thousand pieces;

c) N (h) \u003d 5400 - 5184 \u003d 216 thousand pieces;

N 3 = h 1 * Cm 1 * AT 0 \u003d 25 * 146 * 1500 \u003d 5475 thousand pieces;

N (cm) \u003d 5475 - 5400 \u003d 75 thousand pieces;

N 4 = h 1 * Cm 1 * AT 1 \u003d 25 * 146 * 1505 \u003d 5493.25 thousand pieces;

N (B) \u003d 5493.25 - 5475 \u003d 18.25 thousand pieces;

N= N(h) + N(cm) + N (B) \u003d 216 + 75 + 18.25 \u003d 309.25 thousand pieces.

4.2 . Absolute difference method also involves the allocation of quantitative and qualitative factors that determine the sequence of substitution:

a) N(h) = h*cm 0 * AT 0 \u003d 1 * 14 * 1500 \u003d 216 thousand pieces;

b) N(cm) = cm*h 1 * AT 0 = +2 * 25 * 1500 = 75 thousand pieces;

in) N(B)= b*h 1 * Cm 1 = +5 * 25 * 146 = 18.25 thousand pieces;

N= N(h) + N(cm) + N (B) = 309.25 thousand pieces

Relative difference method

a) N(h) =
thousand pieces;

b) N(cm) = thousand PCS.;

in) N(B) thous. PCS.;

General influence of factors: N= N(h) + N(cm) + N (B) = 309.3 thousand pieces

4.4 . Weighted Finite Difference Method involves the use of all possible settings based on the method of absolute differences.

Substitution 1 is made in the sequence
the results are determined in the previous calculations:

N(h) = 216 thousand pieces;

N(cm) = 75 thousand pieces;

N (B) = 18.25 thousand pieces

Substitution 2 is made in the sequence
:

a) + 1 * 1500 * 144 \u003d 216 thousand pieces;

b) +5 * 25 * 11 \u003d 18 thousand pieces;

c) +2 * 25 * 1505 = 75.5 thousand pieces;

Substitution 3 is made in the sequence
:

a) 2 * 24 * 1500 = 72 thousand pieces;

b) 1 * 146 * 1500 = 219 thousand pieces;

c) + 5 * 25 * 146 = 18.25 thousand pieces.

Substitution 4 is made in the sequence
:

a) 2 * 1500 * 5 * 146 * 24 = 17.52 thousand pieces;

b) 5 * 146 * 24 = 17.52 thousand pieces;

c) 1 * 146 * 1515 = 219.73 thousand pieces;

Substitution 5 is made in the sequence
:

a) 5 * 144 * 24 = 17.28 thousand pieces;

b) 2 * 1505 * 24 = 72.27 thousand pieces;

c) 1 * 146 * 1505 = 219.73 thousand pieces.

Substitution 6 is made in the sequence
:

a) 5 * 24 * 144 = 17.28 thousand pieces;

b) 1 * 1505 * 144 = 216.72 thousand pieces;

c) 2 * 1505 * 25 = 75.25 thousand pieces.

Influence of factors on the resulting indicator

4.5. logarithmic method assumes the distribution of the deviation of the resulting indicator in proportion to the share of each factor in the sum of the deviation of the result

a) the share of influence of each factor is measured by the corresponding coefficients:

b) the influence of each factor on the resulting indicator is calculated as the product of the deviation of the result by the corresponding coefficient:

309,25*0,706 = 218,33;

309,25*0,2438 = 73,60;

309,25* 0,056 = 17,32.

4.6. integral method involves the use of standard formulas to calculate the influence of each factor:

5. The results of the calculations of each of the listed methods are combined in the table of the cumulative influence of factors.

The combined influence of factors:

Comparison of the results of calculations obtained by various methods (logarithmic, integral and weighted final differences) shows their equality. It is convenient to replace cumbersome calculations by the method of weighted finite differences by using the logarithmic and integral methods, which give more accurate results compared to the methods of chain substitution and absolute differences.

5. Conclusion: The output volume increased by 309.25 thousand pieces.

Positive impact in the amount of 217.86 thousand units. had an increase in the number of staff.

As a result of the increase in the number of shifts, the output increased by 73.6 thousand units.

Due to the increase in output, the volume of output increased by 17.76 thousand units.

Extensive factors had the strongest impact on the volume of output: an increase in the number of personnel and the number of shifts worked. The combined effect of these factors was 94.26% (70.45 +23.81). The influence of the production factor accounts for 5.74% of the growth in output.

Note: The application of the considered techniques is similar in relation to multiplicative models of any number of factors. However, the use of the method of weighted finite differences to multifactorial models is limited by the need to perform a large number of calculations, and this is inappropriate in the presence of other, simpler and more rational methods, for example, the logarithmic one.

Deterministic factor analysis as a goal puts forward the study of the influence of factors on the effective indicator in cases of its functional dependence on a number of factor characteristics.

Functional dependence can be expressed by various models - additive; multiplicative; multiple; combined (mixed).

additive the relationship can be represented as a mathematical control, reflecting the case when the effective indicator (y) is algebraic sum several factors:

Multiplicative the relationship reflects the direct proportional dependence of the studied generalizing indicator on the following factors:

where P is the generally accepted sign of the product of several factors.

Multiple the dependence of the effective indicator (y) on the factors is mathematically reflected as a quotient of their division:

Combined (mixed) the relationship between the effective and factor indicators is a combination in various combinations of additive, multiplicative and multiple dependencies:

where a, b, c etc. - variables.

A number of techniques for modeling factor systems are known: the technique of dismemberment; extension reception; reception of expansion and reception of reduction of initial multiple two-factor systems of type: -. As a result of the modeling process, from a two-factor multiple model, additive-multiple, multiplicative and multiplicatively-multiple multifactorial systems of the type are formed:

Ways to measure the influence of factors in deterministic models

Widespread in analytical calculations received chain substitution method due to the possibility of using it in deterministic models of all types. The essence of this technique is that in order to measure the influence of one of the factors, its base value is replaced by the actual one, while the values ​​of all other factors remain unchanged. The subsequent comparison of performance indicators before and after the replacement of the analyzed factor makes it possible to calculate its impact on the change in the performance indicator. The mathematical description of the method of chain substitutions when used, for example, in three-factor multiplicative models is as follows.

Three-factor multiplicative system:

Sequential substitutions:

Then, to calculate the influence of each of the factors, you must perform the following actions:

Deviation balance:

We will consider the sequence of calculations by the method of chain substitutions using a specific numerical example, when the dependence of the effective indicator on the factorial ones can be represented by a four-factor multiplicative model.

The cost of goods sold was chosen as the performance indicator. The goal is to investigate the change in this indicator under the influence of deviations from the comparison base of a number of labor factors - the number of workers, full-day and intra-shift losses of working time and average hourly output. The initial information is given in table. 15.1.

Table 15.1

Information for factor analysis of changes in the cost of sales

products

The original four-factor multiplicative model:

Chain substitutions:

Calculations of the impact of changes in factor indicators are given below.

1. Change in the average annual number of workers:

2. Change in the number of days worked by one worker:

3. Change in the average working day:

4. Change in average hourly output:

Deviation balance:

The results of calculations by the method of chain substitutions depend on the correctness of determining the subordination of factors, on their classification into quantitative and qualitative ones. Changes in quantitative multipliers should be carried out earlier than qualitative ones.

In multiplicative and combined (mixed) models, it is widely used absolute difference method also based on the elimination technique and distinguished by the simplicity of analytical calculations. The rule for calculating this method in multiplicative models is that the deviation (delta) for the analyzed factor indicator must be multiplied by the actual values ​​of the multipliers (factors) located to the left of it, and by the base values ​​of those located to the right of the analyzed factor.

The order of factor analysis by the method of absolute differences for combined (mixed) models will be considered using a mathematical description. Initial basic and actual models:

The algorithm for calculating the influence of factors by the method of absolute differences:

Deviation balance:

Relative difference method is used, as well as the method of absolute differences, only in multiplicative and combined (mixed) models.

For multiplicative models, the mathematical description of the named technique will be as follows. Initial basic and actual four-factor multiplicative systems:

For factor analysis by the method of relative differences, it is first necessary to determine the relative deviations for each factor indicator. For example, for the first factor, this will be the percentage of its change to the base:

Then, to determine the impact of changing each factor, such calculations are made.

Consider the sequence of actions on a numerical example, the initial information for which is contained in Table. 15.1.

In gr. 7 tab. 15.1 shows the relative deviations for each factor indicator.

The results of the influence of a change in each of the factors on the deviation of the effective indicator from the comparison will be as follows:

Balance of deviations: RP, -RP 0 \u003d 432,012-417,000 \u003d +15,012 thousand rubles. (-9811.76) + 3854.62+ (-10,673.21) + 31,642.36 = 15,012.01 thousand rubles. Indexes are generalizing indicators of comparison in time and space. They reflect the percentage change in the phenomenon under study over a period of time compared to the base period. Such information makes it possible to compare changes in various factors and analyze their behavior.

In factor analysis index method used in multiplicative and multiple models.

Let us turn to its use for the analysis of multiple models. Thus, the aggregate index of the physical volume of sales (Jg) looks like:

where q- indexed value of quantity; p 0- co-meter (weight), price fixed at the level of the base period.

The difference between the numerator and denominator in this index reflects the change in trade due to changes in its physical volume.

The Paasche aggregate price index (formula) is written as follows:

Using the information contained in Table. 15.1, we calculate the impact of changes in the index of the average number of workers and the index of the average annual output of one worker on the growth rate of sales.

Labor productivity (PT) of one worker in the base year is 245.29 million rubles, and in the reporting year - 260.25 million rubles. The growth index (/pt) will be 1.0610 (260.25: 245.29).

Indices of growth of sold products (/ rp) and the average annual number of workers (/ sch) according to Table. 15.1- respectively:

The relationship of the three indicated indices can be represented as a two-factor multiplicative model:

Factor analysis by the method of absolute differences gives such results.

1. The impact of changing the index of the average number of workers:

2. The impact of changes in the labor productivity index:

Deviation balance: 1.0360 - 1.0 = +0.0360 or (-0.0235) + 0.0596= + 0.0361 100 = 3.61%.

Integral method used in deterministic factor analysis in multiplicative, multiple and combined models.

This method allows you to decompose the additional increase in the effective indicator due to the interaction of factors between them.

The practical use of the integral method is based on specially developed working algorithms for the corresponding factorial models. For example, for a two-factor multiplicative model (at = a in) the algorithm will be like this:

As an example, we use a two-factor dependence of sold products (RP) on changes in the average annual number of workers (AC) and their average annual output (PT):

Initial information is available in Table. 15.1.

The impact of the change in the average annual population:

The impact of changes in labor productivity (average annual output per worker):

Deviation balance:

In factor analysis in additive models of the combined (mixed) type, one can use proportional division method. Algorithm for calculating the influence of factors on the change in the effective indicator for an additive system of the type y = a + b + c will be like this:

In combined models, the calculation of the influence of factors of the second level can be performed share method. First, the share of each factor in the total amount of their changes is calculated, and then this share is multiplied by the total deviation of the effective indicator. The calculation algorithm is as follows:

We systematize the considered methods for calculating the influence of individual factors in deterministic factor analysis using the scheme (Fig. 15.4).

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

where h - average headcount workers;

CB - average output per worker.

Multiple Models:

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

,

where ST is the average stock of goods; RR - one-day sales volume.

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

Examples of such models are cost indicators for 1 ruble. marketable products, profitability indicators, etc.

To study the relationship between indicators and quantitative measurement of many factors that influenced the performance indicator, we present general rules transformation of models to include new factor indicators.

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

If the original factorial model

then the model will take the form

.

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

.

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

.

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

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

The method of chain substitutions consists in determining a number of intermediate values ​​of the generalizing indicator by successively replacing the basic values ​​of the factors with the reporting ones. This method is based on elimination. To eliminate means to eliminate, exclude the influence of all factors on the value of the effective indicator, except for one. At the same time, based on the fact that all factors change independently of each other, i.e. first one factor changes, and all the others remain unchanged. then two change while the rest remain unchanged, and so on.

AT general view The application of the chain setting method can be described as follows:

where a0, b0, c0 are the basic values ​​of the factors influencing the general indicator y;

a1 , b1, c1 - actual values ​​of the factors;

ya, yb, - intermediate changes in the resulting indicator associated with a change in factors a, b, respectively.

The total change Dу=у1–у0 is the sum of the changes in the resulting indicator due to the change in each factor with fixed values ​​of the other factors:

Consider an example:

table 2

Initial data for factor analysis

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

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

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

Advantages this method: universality of application, simplicity of calculations.

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

Instruction. To solve such problems, select the number of rows. The resulting solution is saved in a MS Word file.

Index- this is a relative indicator of comparison of two states of a simple or complex phenomenon, consisting of commensurate or incommensurable elements, in time or space.
The main tasks of the index method are:

• assessment of the dynamics of generalizing indicators characterizing complex, directly incommensurable aggregates;
• analysis of the influence of individual factors on the change in effective generalizing indicators;
• analysis of the impact of structural shifts on the change in the average indicators of a homogeneous population;
• evaluation of territorial, including international, comparisons.

General (composite) indices there are only group; dynamic indexes there are basic and chain; indexes with constant weights– standard, base period, reporting period; weighted average indices- arithmetic and harmonic.

Conventions used in the theory of the index method:
R - price per unit of goods (services);
q- the quantity (volume) of any product (goods) in physical terms;
pq- the total cost of products of this type (trade);
z- unit cost of production (product);
zq- the total cost of production of this type (cash costs for its production);
T - total time spent on production or total strength workers;
w=q/T- production of products of this type per unit of time (or output per employee, i.e. labor productivity);
t=T/q- the cost of working time per unit of production (labor intensity of a unit of production);
1 - subscript symbol of the indicator of the current (reporting) period;
0 - subscript symbol of the indicator of the previous (base) period

Individual index ( i) characterizes the dynamics of the level of the phenomenon under study in time for two compared periods or expresses the ratio of individual elements of the population.
The main element of the index ratio is the indexed value. Indexed value is a sign, the change of which characterizes the index.
Basic formulas for calculating individual indices:
Index of physical volume (quantity) of products

Price index

Production value index

Unit cost index

Production cost index

Labor intensity index

Index of the number of products produced per unit of time

Labor productivity index (by labor intensity)

Index relationship

Types of multiplicative index two-factor models

1. Multiplicative index two-factor model of turnover: Q 1 = Q 0 i p i q
   From an analytical point of view, i q shows how many times the total amount of revenue increased (or decreased) under the influence of changes in sales volume in natural units.
   Similarly, i p shows how many times the total amount of revenue has changed under the influence of a change in the price of goods. It's obvious that
   i Q = i q i p , or Q 1 = Q 0 i q i p
   The formula Q 1 = Q 0 i q i p represents a two-factor index multiplicative model of the final indicator. By means of such a model, the increase in the total is found under the influence of each factor separately.
   So, if the proceeds from the sale of a certain product increased from 8 million rubles. in the previous period to 12.180 million rubles. subsequently, and it is known that this is due to an increase in the quantity of goods sold by 5% at a price 45% higher than in the previous period, we can write the following ratio:
   12.180 = 8 × 1.05 × 1.45 (million rubles).
   Distributions of total growth by factors in a two-factor index multiplicative model
   The total increase in revenue in the amount of 12.180-8 = 4.180 million rubles. due to changes in volume and price. The increase in revenue due to changes in sales volume (in physical terms) will be
   ΔQ(q) \u003d Q 0 (i q -1)
   For our example: ΔQ(q) = 8(1.05-1)=+0.4 million rubles.
   Then, due to a change in the price of this product, the amount of revenue changed by
   ΔQ(p) \u003d Q 0 i q (i p -1) or ΔQ (p) \u003d 8 * 1.05 (1.45-1) \u003d +3.78 million rubles.
   The total increase in trade turnover is made up of increases explained by each factor separately, i.e. ΔQ \u003d Q 1 - Q 0 \u003d ΔQ (q) + ΔQ (p)
   or ΔQ \u003d 12.18-8 \u003d 0.4 + 3.78 \u003d 4.18 million rubles.
2. Multiplicative index two-factor model of prime cost (costs, distribution costs): Q 1 = Q 0 i z i q