Thus, the Laspeyres composite index of product volume is essentially the weighted average of the individual indexes of product volume of each item, with the weights being the revenue of each item in the base period.
If set d 0
= p 0 q 0
p 0 q 0
then the Laspeyres aggregate index of product volume
is determined by the following formula:
I q = i q d 0 (6.14)
Thus, the weight in this case is the proportion of revenue of each item in the base period.
- Paasche's composite index of product volume: is a composite index of product volume with weights determined in the research period.
Calculation formula:
1 1
pq
I q = (6.15)
p 1 q 0
Using example 6.3, the Paasche composite index of product volume reflecting the overall change in product volume of three items is defined as follows:
1 1
pq (17 x 1,650) + (22 x 1,250) + (24 x 1,000)
I = =
qp 1 q 0 (17 x 1,500) + (22 x 1,050) + (24 x 1,300)
= 0.9969 times (or 99.69%)
In case the data has determined the single index of product volume and revenue level (D) of each item in the research period, the Paasche's composite index of product volume is calculated according to the following formula:
1 1 1 1
pqpq
I = = (6.16)
qp 1 q 0
p 1 q 1
i q
Thus, Paasche's composite index of product volume is essentially the weighted average of the individual indexes of product volume of each item, with the weights being the revenue of each item in the study period.
If set d 1
= p 1 q 1
p 1 q 1
then Paasche's aggregate index of product volume is
determined by the following formula:
d 1
q
I = 1
i q
(6.17)
Thus, the weight in this case is the revenue proportion of each item in the research period.
As with the price composite index, when the Laspeyres and Paasche indices differ significantly, the Fisher volume composite index is most appropriate.
- Fisher's composite index of product volume: is the geometric mean of the two composite indexes of product volume of Laspeyres and Paasche.
Calculation formula:
= x (6.18)
Ip 0 q 1p 1 q 1
qp 0 q 0p 1 q 0
Based on Example 6.3, Fisher's composite price index is determined as follows: I p = 1.0252 x 0.9969 = 1.0109 times (or 101.09%)
6.2.2. Spatial index
Spatial index is a relative number reflecting the comparative relationship between two levels of a research phenomenon in two different spaces.
Similar to the development index, let us consider the following example to illustrate the methodology for calculating the spatial index.
Example 6.4: There is data on the consumption situation of two products X and Y in two markets A and B as follows:
Table 6.4. Consumption situation of two products X, Y in two markets A, B
Item
Market A | Market B | |||
Selling price (million VND/product) | Volume of goods consumed receiver (sp) | Selling price (million VND/product) | Volume of goods consumed receiver (sp) | |
X | 130 | 95 | 150 | 105 |
Y | 180 | 115 | 190 | 100 |
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We calculate the following types of spatial indices:
6.2.2.1. Single index
- A single price index reflects the comparative price relationship of a particular item in two different spaces.
Calculation formula:
i = p A ori = p B (6.19)
From example 6.4, we have:
p(A/B) p B p(B/A) p A
i = p A = 130 = 0.8667 times (or 86.67%)
pX(A/B)
p B 150
i = p A = 180 = 0.9474 times (or 94.74%)
pY(B/A)
p B 190
- The single index of product volume reflects the comparative relationship of product volume of a specific item in two different spaces.
Calculation formula:
i = q A ori = q B (6.20)
q(A/B) q B q(B/A) q A
From example 6.4, we have:
i
= q A = 95 = 0.9048 times (or 90.48%)
qX(A/B)
q B 105
i = q A = 115 = 1.15 times (or 115.00%)
qY(A/B)
q B 100
Like the single growth index, the limitation of the single spatial index is that it cannot be calculated for many items and does not reflect the combined effect of both price and volume of products. Therefore, it is necessary to use a composite index.
6.2.2.2. Composite index
- The composite price index reflects the comparative price relationship of a group or all items in two different spaces.
Similar to the price index in the development index, the weight of the spatial price index is the volume of output.
Calculation formula:
I p(A/B)
pq
A
= orI
p B q
p(B/A)
pq
B
= (6.21)
p A q
In which, choose the weight Q = q A + q B as the product volume of each item in both spaces A and B to ensure uniformity.
According to example 6.4, calculate the aggregate index of selling prices of 2 goods in 2 markets A and B:
A
pq 130 x (95+ 105) + 180 x (115 + 100)
I = =
p(A/B)
p B q
150 x (95+105) + 190 x (115 + 100)
= 0.9132 times (or 91.32%)
- The aggregate index of product volume reflects the comparative relationship of product volume of a group or all items in two different spaces with the weight being price.
In reality, there are many different types of prices that can be used as weights to calculate the spatial index such as fixed prices (or comparable prices - p n ), average prices, ...
+ In case of fixed price (p n ), the calculation formula is as follows:
I q(A/B)
qp
A n
= orI
q B p n
q(B/A)
qp
You
= (6.22)
q A p n
The disadvantage of fixed pricing is that when a new item is launched, there is no fixed price.
+ In case of average price of each item ( p ):
The average price in both spaces A and B of each item is calculated by the
awake:
p = p A q A + p B q B (6.23)
q A + q B
Then calculate the total index of product volume:
I q(A/B)
pq
A
= orI
pq B
q(B/A)
pq
B
= (6.24)
pq A
According to example 6.4, calculate the aggregate index of the volume of goods consumed in two markets A and B:
p X
pq + pq 130 x 95 + 150 x 105
= AABB =
= 140.50 (million VND/product)
q A + q B
95+105
p Y
pq + pq 180 x 115 + 190 x 100
= AABB =
= 184.65 (million VND/product)
q A + q B
115+100
A
pq 140.50 x 95 + 184.65 x 115
I = = = 1.0411 times (or 104.11%)
q(A/B)
pq B
140.50 x 105 +184.65 x 100
6.3. Index system
6.3.1. Concept
An index system is a series of related indicators, forming an equation. The basis for forming an index system is the actual relationship between the indicators, often having a product relationship. The composition of an index system usually includes a total index and factor indicators.
- The overall index reflects the fluctuation of a general phenomenon composed of many factors. For example, revenue is affected by selling price and consumption volume.
- Factor index reflects the fluctuations of each factor and its influence on the general phenomenon.
For example: Revenue (D) = pq i.e. I D = I p x I q
6.3.2. Effect of the index system
The index system is mainly applied to indicators that are related to each other and has two effects as follows:
- Analyze the role and level of influence of the factors that make up the general phenomenon. The level of influence of each factor can be expressed in relative or absolute numbers.
- Calculate unknown index when the remaining indexes in the system are known.
When using the index system for analysis, the indexes included in the system must have practical meaning. The index system is used to analyze the fluctuating influence of the constituent factors on the research phenomenon, thereby knowing which factor has the main effect on the general fluctuation.
6.3.3. Construction method
The index system can be built according to a number of different methods. Of these, the most commonly used is the continuous method. When building an index system according to this method, the following rules are required:
- Quality factors come first, mass factors come after in order of decreasing quality and increasing mass.
- When studying the fluctuating effect of one factor, the remaining factors must be fixed.
- The weights of the research factors are the remaining factors and are taken from the base period for the factors ranked first and the research period for the factors ranked later.
6.3.3.1. Composite index system
The composite index system is formed based on the actual relationship between indicators.
For example: Revenue = selling price x quantity of goods sold; Production cost = unit product cost x output, ...
In the system of synthetic indexes, if the general index of the research phenomenon consists of how many factors, the index system has that many factor indexes and the total index is always equal to the product (or sum) of the factor indexes.
Returning to example 6.3, set up the spreadsheet as follows:
Table 6.5. Worksheet 1
Item
p (million VND/product) | q (sp) | p 1 q 1 (million) copper) | p 0 q 0 (million) copper) | p 0 q 1 (million) copper) | |||
p 0 | p 1 | q 0 | q 1 | ||||
A | 16 | 17 | 1,500 | 1,650 | 28,050 | 24,000 | 26,400 |
B | 28 | 22 | 1,050 | 1,250 | 27,500 | 29,400 | 35,000 |
C | 20 | 24 | 1,300 | 1,000 | 24,000 | 26,000 | 20,000 |
Total | x | x | x | x | 79,550 | 79,400 | 81,400 |
We have: D = pq , in which p is the quality index, q is the quantity index. According to the convention of the continuous method, we have the index system:
ID = I p x I q
Write in full form:
p 1 q 1 =p 1 q 1 xp 0 q 1 (6.25)
Absolute change:
p 0 q 0p 0 q 1p 0 q 0
p 1 q 1 - p 0 q 0 = ( p 1 q 1 - p 0 q 1 ) + ( p 0 q 1 - p 0 q 0 )(6.26) Substituting the numbers, we have:
79,550 = 79,550 x 81,400
79,400 81,400 79,400
Absolute change:
1.0019 = 0.9773 x 1.0252
(100.19%) (97.73%) (102.52%)
(+0.19%) (-2.72%) (+2.52%)
79,550 – 79,400 = (79,550 – 81,400) + (81,400 – 79,400)
150 = -1,850 + 2,000 (million VND)
Thus, total research revenue compared to the base period increased by 0.19%, equivalent to an increase of 150 million, due to the influence of 2 factors:
- Due to the general selling price of all items decreasing by 2.27%, total revenue decreased by 1,850 million VND.
- Due to the general consumption volume of all products increasing by 2.52%, total revenue increased by 2,000 million VND.
6.3.3.2. Average indicator index system
We know that the weighted average is calculated by the formula:
ii
xf
x = (6.27)
f i
The above formula shows that the average index is affected by two factors.
𝑓
𝑖
The factors are: the quantity of the research criterion (xi) and the overall structure (fi/ f i or 𝑑 ). From
Based on that relationship, we can build an index system of the average indicator as follows:
I x= I xx I d f (6.28)
𝑥 1 𝑓 1
𝑓 1
𝑥 0 𝑓 0
𝑓 0
𝑥 1 𝑓 1
=
𝑓 1
𝑥 0 𝑓 1
𝑓 1
𝑥 0 𝑓 1
x
𝑓 1
𝑥 0 𝑓 0
𝑓 0
x
x 1 d 1 =x 1 d 1
x 0 d 1
x 0 d 0x 0 d 1
x 0 d 0
x
x 1=x 1x01
In there:
x 0x01
x 0
- The variability component index shows the change in the average indicator between two research periods, that is, it reflects the comparative relationship between the level of the average indicator of the research period and the base period. The variability component index is determined according to the following formula:
𝑥 1 𝑓 1
𝐼x 1 𝑓 1 x 1 d 1
𝑥 =x 0=𝑥 0 𝑓 0 =x 0d 0 (6.29)
𝑓 0
- The fixed component index shows the variation of the average index due to the specific influence of the research criteria (excluding the structural influence). The fixed component index is determined according to the following formula:
𝑥 1 𝑓 1
I x =
x 1=
x 01
𝑓 1
𝑥 0 𝑓 1
𝑓 1
xd
1 1
= (6.30)
x 0 d 1
- The structural influence index shows the variation of the average index due to the influence of the overall structure (while the research criteria themselves remain unchanged, fixed in the base period). The structural influence index is determined according to the following formula:
I df
=x 01=
x 0
𝑥 0 𝑓 1
𝑓 1
𝑥 0 𝑓 0
𝑓 0
xd
0 1
= (6.31)
x 0 d 0
Example 6.5: There are the following statistics of a business:
Table 6.6. Labor situation statistics at an enterprise
Factory
Labor productivity (products/person) | Number of employees (people) | |||
Base period | Research period | Base period | Research period | |
A | 100 | 110 | 10 | 40 |
B | 100 | 120 | 10 | 20 |
C | 200 | 220 | 30 | 20 |
Requirement: Analyze the fluctuations in average labor productivity of the entire enterprise due to the influence of factors.
We have the following system of indicators to analyze the fluctuations of the average index:
x
I w= I wx I dL
1
w
=
w 1w 01
w 0
w 01
w 0
With the given data, we have the following calculation table:
Table 6.7. Worksheet 2
Factory
NSLD (product/person) | Labor (people) | w 0 L 0 (product) | w 1 L 1 (product) | w 0 L 1 (product) | |||
w 0 | w 1 | L 0 | L 1 | ||||
A | 100 | 110 | 10 | 40 | 1,000 | 4,400 | 4,000 |
B | 100 | 120 | 10 | 20 | 1,000 | 2,400 | 2,000 |
C | 200 | 220 | 30 | 20 | 6,000 | 4,400 | 4,000 |
Total | x | x | 50 | 80 | 8,000 | 11,200 | 10,000 |
From that we can calculate:
𝑤
w L 11,200
1 1
= = =140 (products/person)
1L 180
𝑤
w L 8,000
0 0
= = =160 (products/person)
0L 050
𝑤
w L 10,000
0 1
= = =125 (products/person)
01L 180
Instead of the above index system, we have:
140 = 140 x 125
160 125 160
0.8750 = 1.120 x 0.7813
Absolute change:
Comment:
(87.50%) (112.00%) (78.13%)
(-12.50%) (+12.00%) (-21.87%)
(140 – 160) = (140 – 125) + (125 – 160)
-20 = 15 + (-35) (products/person)
The average labor productivity of the entire enterprise in the research period compared to the base period decreased by 12.5%, equivalent to a decrease of 20 products/person, due to the influence of two factors:
- Because the labor productivity of each workshop in general increased by 12%, the average labor productivity increased by 15 products/person.
- Due to changes in labor structure, average labor productivity decreased by 35 products/person.