Economic Indexes Indexes in statistics Indexes are indicators of size comparison of any...

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

qq

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

qq

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 %