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Economic Indexes
Indexes in statistics Indexes are indicators of size comparison of any socio-economic process. Index number measures how much a variable changes over time or in space
All indexes can be classified as:
single and composite time and territorial aggregate and average with chain base and fixed base with chain weight and fixed weight with variable structure and constant
structure
Indexes in statistics Index number can be calculated as the ratio of the current value to the base value. The resulting number is usually multiplied by 100 to express the index as a percentage
Indexes in statistics To calculate an index number it is necessary to compare two identical periods of time, for example month and month, year and year
The index method is applied to solve the following problems:
study the change of the event in time carry out spatial comparisons measure plan fulfillment indicate the degree of structural changes
influence
Indexes are subdivided into individual (single) and composite (summary)
Individual (single) index
is applied to find out the change degree of a separate element of a
complexsocio-economic process
Individual (single) index
It turns out to be a result of a comparison of the one commodity process. Single index is the dynamic ratio DR, plan ratio PR, plan fulfilling PF or ratio of comparison RCom
Notations
P – price of a good, q – quantity of sold (produced) goods; pq – value (monetary worth) of sold (produced) goods, turnover; z - unit cost (себестоимость); zq – total cost, t - labor intensiveness (трудоемкость) of producing a good; w – labor productivity; T=tq - labor inputs, or number of employed persons; h = zq/pq – cost per one money unit of good production…
Notations
f – individual wage or salary; F –wages & salaries fund (compensation of employees); S – cultivated area; y - crop yield; Sy - total harvest, Q – total amount of production or total sales, in natural units or in cost units
Individual (single) index
There are many single indexes, first of all we will consider single price index, single quantity index, single value index, single unit cost index, and single labor intensiveness index
Single price index
Single price index is expressed by the ratio of price per one unit for one period to the price per one unit of the same commodity for another period. It shows by how many percent the price has been reduced or increased
Individual price index
where P1 – the price of the accounting period;
P0 – the price of the basic period
1
0
,p
PiP
Single quantity index (Individual index of physical volume of goods turnover)
where q1, q0 – quantities of sold (produced)
goods in current and base periods, respectively
,0
1
qqi
q
The Single quantity index (Individual index of physical volume of goods turnover)
shows by how many percent the quantity volume has been increased or reduced. Multiplying the single price index and the single quantity index gives us the single value index:
Individual index of goods turnover (the single value index)
1 1
0 0
,pq p q
P qi i iP q
where P1q1, P0 q0– values for current and base periods of time, respectively
.
The single value index (individual index of goods turnover) shows by how many percent the turnover (of total monetary worth) for one commodity has been increased or reduced
Price per
unit, rub.
QuantityPrice per unit,
rub.
1,58
1,25
1,00
1,27
1,25
1,5
1,25
1,0
0,67
9500
2500
1500
25
30
10
7500
2000
1000
20
30
15
А
B
C
II periodI periodGoods
0P 0q1P 1q pi qi pqi
Quantity
Find the change of prices, sales volumes and turnover for each
commodity and for all commodities
Single price index for commodity A:
1
0
251.25(*100% 100% 25%).
20
AAp A
Pi
P
Price of commodity A has increased by 25%
Single quantity index for commodity A:
1
0
95001.267(*100% 100% 26.7%).
7500
AAq A
qi
q
Sales volume of commodity A has increased by 26.7%
Single value index for commodity A:
1 1
0 0
25*95001.25*1.267 1.583(*100% 100% 58.3%).
20*7500
A AApq A A
P qi
P q
Total monetary worth (turnover) of commodity A has increased by 58.3%
We can say that the turnover of commodity A has increased by 58.3% because the sales volume has increased too by 26.7%, and the price per unit has increased by 25%
Thus, the change of turnover consists of the price change and quantity change:
Turnover, pq 58.3%
26.7% Volume, q Price, p 25%
Single cost indexes
The next indexing system characterizes the change of total cost, unit cost and quantity volume, it includes single unit cost index, single quantity index and single total cost index
Single unit cost index shows by how many percentages the cost per one unit has changed:
where z1, z0 – unit cost in current and base periods,
respectively.
Single quantity index has been considered above
1
0
*100%,z
zi
z
Single total cost index is calculated by the following equation:
where z1q1, z0 q0 – - total cost for current
and base periods, respectively
1 1
0 0
*100% *100%,zq z q
z qi i i
z q
Single labor intensiveness index (индекс трудоемкости) and single quantity index are joined by the single labor inputs index
Single labor intensiveness (трудоемкости) index shows by how many percent the labor intensiveness increased or reduced
This index is an exception to the rule because for constructing it, the labor intensiveness for base period has to be divided by the labor intensiveness for the reporting period:
0
1
*100%t
ti
t
Single quantity index
1
0q
qiq
Single labor inputs index could be calculated by dividing the single quantity index by the single labor intensiveness index:
1 1
0 0
*100% *100%qtq
t
it qi
t q i
The above indexes are used for analysis of one commodity process. However, to analyze commodity set the composite (общие) indexes are required. Composite index combines the individual indicators such as price, quantity and value. Composite index can be expressed by aggregate or average formulas
General (composite) indexis the relative indicator characterising change of the complex phenomenon,
consisting of the elements which can not be added directly
The simplest form of a composite index is an aggregate
index. An aggregate index is calculated by adding the
elements in the composite for the given time period and then
dividing this result by the sum of the same elements during the
base period
Aggregate value index can be calculated using the following
equation:
1 1
0 0
*100%,i ipq
i i
P qI
P q
0 0i iP q
,
where - total turnover or total monetary worth of all commodities for base and current periods of time, respectively
1 1,i iP q
The aggregate quantity index and the aggregate price index are named weighted aggregate indexes
,
The aggregate quantity index is calculated by adding the volume of all elements in
comparable prices for the given time period and then dividing this result by the sum of the
same elements during the base period:
0 1
0 0
*100%,i iq
i i
P qI
P q
,
where - comparable price (prices for the same period, usually base period)
0iP
The General index of physical volume of goods turnover
The index shows how the total receipts change as the result of change in quantity of goods sold
1 0
0 0q
q PIq P
weighting The weighting allows us to improve the
accuracy of the general quantity estimate based on our data set. For constructing the aggregate quantity index, the influence of price should be excluded by means of price fixing during the base period
The idea of constructing the general price index
The general price index shows how the prices on all considered commodity groups
vary on average. It is impossible to add directly prices of various goods, it is
necessary to choose a certain indicator to make economic sense in summation action. We use goods turnover or a profit in the
role of such an indicator
The goods turnover size is influenced by two factors:
price level; quantity of goods sold.
As we are interested to measure only the change of prices, so influence of the second factor should be eliminated. For this purpose the quantity of goods sold is fixed at constant level:
qp
qpIp
0
1
Two variants are possible:
1. The quantity of the goods sold is fixed at level of the accounting period:
1 1
0 1
,Pa Pap
p qI P Pp q
where - Paashe indexPap
I
2. The quantity of the goods sold is fixed at level of the basic period:
Lp
I
1 0
0 0
,L Lp
p qI P Lp q
where - Laspeyres’ price index
Aggregate price index in Russian statistics is calculated using the Paashe formula:
1iq
1 1
0 1
*100%,i i
i i
P qPap P q
I P
where - quantity volume of each element for the current period and weighting factor
For constructing the aggregate price index the influence of volume should be excluded by quantity fixation during the current or reporting period. The numerator of the aggregate quantity index and the denominator of the aggregate price index are the same. It expresses the possible total
monetary worth without price changing
To get a uniform result Fisher's index is used which is calculated as
simple geometric mean from indexes of Paashe and Laspeyres:
F P L
These indexes represent a system:
Pa L
pq p qI I I
or
L Pa
pq p qI I I
Quant-ity
Price per
unit, rub.
Quan-tity
Price per
unit, rub.
257500287500327500225000--- - Total
187500
60000
10000
190000
75000
22500
237500
75000
15000
9500
2500
1500
25
30
10
7500
2000
1000
20
30
15
А
B
C
II periodI period
Goods
0P 0q1P 1q 00qp
11qp 10qp 01qp
150000
60000
15000
The aggregate value index:
1 1
0 0
25*9500 30* 2500 10*1500*100%
20*7500 30* 2000 15*1000i i
pqi i
P qI
P q
The total monetary worth (total turnover) has increased by 45.6%. Why? The prices and sales volume have changed
3275001.456(*100% 100% 45.6%).
225000pqI
The aggregate quantity index:
0 1
0 0
20*9500 30* 2500 15*1500*100%
225000i i
qi i
P qI
P q
2875001.278(*100% 100% 27.8%).
225000qI
Thus, the total monetary worth (total turnover) has increased by 45.6% while the sales volume has increased by 27.8%
The aggregate price index:
1 1
0 1
327500*100% 1.139(*100% 100% 13.9%).
287500i i
pi i
P qI
P q
Thus, prices on three commodities have increased by 13.9%
Weighted aggregate quantity index and weighted price index turn into aggregate value index:
* 1.139*1.278 1.456.pq p qI I I
Thus, the total monetary worth (total turnover) has increased by 45.6% while prices increased by 13,9%, and the sales volume has increased by 27.8%
To construct the aggregate indexes it is recommended to apply the following rule:
Constructing the index quantitative indicator, the weights of base period should be used. The example of this index is aggregate quantity index. Constructing the index of qualitative indicator, the weights of reporting period should be used. The examples of this index are aggregate price index, aggregate unit cost index, and aggregate labor intensiveness index
The factorial analysis
For the analysis of influence of separate factors on a goods turnover rate of growth we take a difference between numerator and a denominator of a corresponding general index.
1. Absolute change of goods turnover (numerator minus denominator from the value index of goods turnover):
0011 qpqppq
Factorial analysis
The goods turnover rate of growth occurs under the influence of two factors: changes of quantity of the goods sold and price change for a commodity unit. The sum of the rates of growth under the influence of these factors should be equal to the general increase in value of goods turnover .
Factorial analysis
For reception of comparable results it is recommended to observe such sequence of inclusion of factors in the analysis: in the beginning there are quantitative factors (in our case q), then qualitative (P).
2. Absolute change in goods turnover at the expense of quantity change of the goods sold (numerator minus denominator of the general physical volume of goods turnover index according to Laspeyres):
0010 qpqppqq
3. Absolute change of goods turnover at the expense change of the prices (numerator minus price index denominator according to Paashe):
Here two cases are possible: economy or buyers over-expenditure at the expense of the
price change
1011 qpqppqp
Interrelation of absolute values
q ppq pq pq
Average harmonious index
p
p
iqpqpI
11
11
pp i
ppp
pi 10
0
1
;10
11
qp
qpIp
In this case the general price index is calculated as average harmonious value from individual indexes, where
as scales act the values of goods turnover of the accounting period.
x
wwx
h
71877--73000Total
22115
20528
29234
1,040
1,023
0,992
+4,0
+2,3
-0,8
23000
21000
29000
А
Б
В
Price change
%
Realization in the current period, rub.
Goods
11qppi
pi
qp 11
Prices of given commdity group in the current period in comparison with the basic period
have grown by 1,6%
1 1
1 1
730001,016( 1,6%)
1 71877p
p
p qI
p qi
Average arithmetic index
0 0
0 0
i p qI
p q
;
00
10
qp
qpIq
010
1 qiqq
qiqq
In this case the general index of physical volume of goods
turnover is calculated as the average arithmetic value from
individual indexes of physical volume of goods turnover where as scales act values of goods turnover of the
basic period:
ffxx
119505--124000Total
43056
24786
51663
0,936
0,918
1,013
-6,4
-8,2
+1,3
46000
27000
51000
А
Б
В
Change of the physical volume of
realization, %
Realization during the
current period, rub.
Goods
00 pq
qi 00 pqiq
0 0
0 0
1195050,964( 3,6%)
124000q
q
i q pI
q p
Physical volume of realization of given goods on average decreases by 3,6%
Indexes of average levels (indexes of variable structure,
constant structure and structural shifts)
Sale of goods by several firms is considered. Each firm has certain volume of sale and the price. It is required to analyse,
how the average price of the goods changes.
The average price index (the index of variable structure)
0 01 1 1
0 1 0p
p qp p qIp q q
From the formula of an index of variable structure changes it is visible that the
average price changes as a result of action of two factors:
change of the prices in different firms; change of relative density of firms in total
amount of realisation of the goods.
Hence, the index of variable structure can be spread out on two subindexes, each of them characterises actions of one of these factors.
1. A subindex - an index of constant structure. It shows, how
the average price changes as a result of price changes in
different firms.
0 11 1 1 1
0 1 1 1pp
p qp d p qIp d q q
2. A subindex - an index of structural shifts. It shows, how the average price changes as a result of
relative density change of firms in total amount sale of the goods (as a result of structural shifts):
0 1 0 0
1 0pd
p q p qIq q
The Listed indexes form system
dppppIII
1. Absolute change of the average price
estimated as a difference of a dividend and a divider of an index of variable structure.
0
00
1
11
q
qp
q
qpp
2. Change of the average price at the expense of price changes in different firms
is estimated as a difference of a dividend and a divider of an index of the fixed structure:
1
10
1
11
q
qp
q
qppp
3. Change of the average price at the expense of structural shifts
is estimated as a difference of a dividend and a divider of an index of structural shifts:
0
00
1
10
q
qp
q
qppd
The listed absolute values form a system:
dpppp
Three-factorial indexes
Cost of material inputs on production depends from:
q -quantities of issued production;
m -specific expenses of raw materials and materials;
p -the prices for raw materials and materials.
where z – material inputs on manufacture.
qmpz
Index of material inputs on manufacture
000
111
qmp
qmpIpmq
Index of the output volume
000
100
qmp
qmpIq
Index of specific expenses
100
110
qmp
qmpIm
Index of the raw materals’ prices
110
111
qmp
qmpIp
These indexes form a system
qmppmqIIII
Territorial indexes
During construction of territorial indexes there are questions on base of comparison and object at which level it is necessary to fix index weight.
These questions can be answered, proceeding from research specific targets.
For example, it is necessary to compare price levels of two regions
(regions A and B).
As scales we take quantity of the goods sold in region A.
АБ
ААp qp
qpI
As scales we take quantity of the goods sold in region B.
БА
ББp qp
qpI
The given indexes are NOT interconnected among themselves:
pp I
I
1
For reception of uniform result the total sales volume of two regions act as scales.
БАqqq
qp
qpIБ
Аp
It is possible the construction of a price index on the basis of
method called indirect standardization.
,
Б
ББ
А
ААp qp
qpqpqpI
where -average price for two regions.
БА
ББАА
qpqpp
p
An index of physical volume of goods turnover.
,
Б
Аq qp
qpI
где - веса.p
Goods turnover index of two regions.
ББ
ААpq qp
qpI
pqpqIII
Chain and basic indexes
If indexes are calculated for the value which is more than two numbers of the periods of time, so they can be received in the basic and chain way. We will consider construction of basic and chain indexes on an example of physical volume of goods turnover index.
Individual indexes
Let's consider realisation of any goods during the different
periods of time.
t tq
Бqi
Цqi
0 0q 1 -
1 1q 01 qq 01 qq
2 2q 02 qq 12 qq
3 3q 03 qq 23 qq
4 4q 04 qq 34 qq
-Quantity of the goods sold in the basic period;
-Quantity of the goods sold in the first period and so on.
0q
1q
Product of chain indexes gives a basic index of last period of time.
0
4
3
4
2
3
1
2
0
1
q
q
q
q
q
q
q
q
q
q
General indexes
The interrelation noted above is unconditional only for individual indexes. For the general indexes this interrelation will be fair only when the general indexes will be calculated with so-called constant scales.
Let there is data about realisation of the
several goods for four periods of time.
I II III IV
1p
2p 3
p 4
p
1q
2q 3
q 4
q
System of basic indexes
First period is considered as a basic.
11
21
12 qp
qpIq
11
4114 qp
qpIq
11
3113 qp
qpIq
System of chain indexes with constant scales
11
2112 qp
qpIq
21
3123 qp
qpIq
31
4134 qp
qpIq
11
41
31
41
21
31
11
21
qp
qp
qp
qp
qp
qp
qp
qp
System of chain indexes with variable scales
11
2112 qp
qpIq
22
3223 qp
qpIq
33
4334 qp
qpIq
Goods
Sales in the basic peiod
q0 p0
Change in the physical
volume of sales,% iq
iq * q0 p0
A 46 000 - 6,4 0,936 43 056
Б 27 000 - 8,2 0,918 24 786
В 51 000 + 1,3 1,013 51 663
Итого 124 000 - - 119 505
Iq = iq q0 p0 / q0 p0 = =119 505 / 124 000 =
0, 964 or 96,4 %Physical volume of sales of given goods
decreased on average by 3,6 %
