Index Numbers

• Fro measuring changes in a variable or a group of related variables with respect to time, geographical location, or other characteristics is INDEX Numbers

• NEWS PAERS headlines the fact that prices are going up or down , that industrial production is rising or falling, that imports are increasing or decreasing, that crimes are rising in a particular period compared to previous period are disclosed by index numbers.

• An Index Number (I.N.) is a number which is used as a device for comparison between the prices, quantities or values of a group of articles (related variables) in different situations, e.g. at a certain place or a period of time and that at another place or period of time. When the comparison in respect of prices, it is called index number of prices, when in respect of physical quantities, it is termed as Index number of quantities, other index no. are defined in the same manner.

Selection of Base

• Usually base is selected in three different ways and according to these three types of base periods, the following are the three methods of constructing index numbers:

(a) Fixed Base Method (b) Chain Base Method (C) Average Base Method.

Fixed Base Method

1. Simple Index Numbers2. Composite Index Numbers

In fixed base method, a year is fixed as a base period and the prices during the base year are represented by 100. The price relatives of other years are the required index numbers.

Simple Index Numbers eg.

Year Price of rice per quintal (Rs.)

Index Number with 1960 as the base year

Index Number with 1957 as the base year

1957 40 (40/50)*100=80 100

1958 36 (36/50)*100=72 (36/40)*100=90

1959 48 96 (48/40)*100=120

1960 50 100 125

1961 44 88 110

1962 52 104 130

1963 46 92 115

The price index of a single commodity (rice) w.r.t. the base year is shown in the table

Composite Index Numbers

In case of more than one item, their price relatives w.r.t. a selected base are determined separately. The statistical average of these relatives is called a composite Index Number

Composite Index Numbers eg.

Year Price of rice (Rs./qt)

Price of wheat (Rs./qt)

Price of Pulse (Rs./qt)

Index Number Composite Index Number

Rice Wheat Pulse

1957 40 25 20 100 100 100 300/3=100

1958 36 21 24 90 84 120 98

1959 48 27 21 120 108 105 111

1960 50 26 22 125 104 110 113

1961 44 23 19 110 92 95 99

1962 52 28 23 130 112 115 119

1963 46 24 17 115 96 85 98.67

The price index of a single commodity (rice) w.r.t. the base year is shown in the table

Chain Base Methodsor Link Index Numbers

When year to year comparison is desired Index numbers are determined by Chain Base Method. In this method each year is taken as the base for the immediately next year, the first year itself is being its own index number.

Chain Base Method eg.Year Price Index of Rice (Chain Base Method)

Price of rice (Rs./qt)

Chain Base Index Number

1957 40 (40/40)*100=100

1958 36 (36/40)*100=90

1959 48 (48/36)*100=133.3

1960 50 (50/48)*100=104.2

1961 44 (44/50)*100=88

1962 52 (52/44)*100=118.2

1963 55 (55/52)*100=105.7

• One of the great advantage of Chain Base Index Numbers are that new items may be readily included or old one dropped in their calculations; also such indices are more free from seasonal variations than the fixed base indices.

(C) Average Base Method

• The average of a number of years’ prices may be used as base price in determining index numbers. This has the effect of minimizing the abnormalities of any particular year.

Avg. Base Method eg.Year Price Index of Rice (Chain Base Method)

Price of rice (Rs./qt)

Avg. Price (Rs./qt) Base Price

Index Number

1957 40

(40+25+32+28.57+39.43)/5= 33

(40/33)*100=121.2

1958 25 (25/33)*100=75.8

1959 32 (32/33)*100=97.0

1960 28.57 (28.57/33)*100=86.6

1961 39.43 (39.43/33)*100=119.5

Method of Constructing Index Number (Prices)

(1) Fixed Base Method

Method of Aggregates

a) Simple Aggregate of Pricesb) Weighted Aggregate of

Pricesc) Simple Arithmetic Mean of

Prices Relativesd) Simple Geometric Mean of

Price Relativese) Weighted Arithmetic Mean

of Price Relativesf) Geometric Mean of two

weighted aggregates of Prices (special case)

A) Laspeyres’ IndexB) Paasche’s IndexC) Fisher’s Index

NumberD) Dorbish & Bowley

MethodE) Marshall &

Edgeworth Method

F) Walsche’s MethodG) Kelly’s Method

Method of Constructing Index Number (Prices)…

(2) Chain Base Method

Simple Arithmetic Mean Or Geometric Mean of Link Relatives (Chain Index)

Notations: For the purpose of showing the above modes of construction by mathematical formulae, the following symbols will be used.

1'''

1''

1'1

1'''

1''

1'1

0'''

0''0

'0

0'''

0''0

'0

Ysay year base in thely respective ,... 3 2, 1, items of quantities ........ ,q ,q ,q

Ysay year base in thely respective ,... 3 2, 1, items of prices ........ ,p ,p ,p

Ysay year base in thely respective ,... 3 2, 1, items of quantities ........ ,q ,q ,q

Ysay year base in thely respective ,... 3 2, 1, items of prices ........ ,p ,p ,p

Similar Notations for prices and quantities of items 1,2,3… resp. in the year Y1,Y2,Y3,… etc. Will be used. However, p0, q0 and p1, q1 will generally refer to the prices & quantities at the base year Y0 and at the current year Y1 respectively without any specific mention about the different items.I01 = Index no. for the year Y1 with the year Y0 as baseI12= Index no. for the year Y2 with the year Y1 as base ……

1(a): Method of Aggregates: Simple Aggregate of Prices

Simple Aggregate Price Index = I01=

Eg. Determination of Simple Aggregative Index Numbers.

Simple Aggregate Index Number = I01 =

100 * p

p 100*

......ppp

.......ppp

0

1

'''0

''0

'0

'''1

''1

'1

Commodity Price (Rs/qt) =p0 (Base 2001)

Price (Rs/qt) =p1 (current 2011)

Rice 32 50

Wheat 25 25

Oil (edible) 90 100

Fish 120 140

Potato 35 40

Total = 302 = 355 0p 1p

117.5 100 * 302

355 100 *

p

p

0

1

1(b): Weighted aggregate of prices

Weighted Aggregate of prices= I01=

Eg. Find by the Weighted Aggregate Method, the Index Numbers

Weighted Index Number = I01 =

100 * p

p 100*

......ppp

.......ppp

0

1

''''''0

''''0

''0

''''''1

''''1

''1

w

w

www

www

Commodity Price (Rs/qt) =p0 (Base 2001)

Price (Rs/qt) =p1 (current 2011)

Weight

Rice 32 50 8

Wheat 25 25 6

Oil (edible) 90 100 7

Fish 120 140 3

Potato 35 40 3

119.03 100 * 1571

1870 100 *

p

p

0

1

w

w

A. Laspeyres’ Index. Named after the name of German economist Etienne Laspeyres who formulated it in 1871, we have an Index Number known as Laspeyres Index which is equal to

100* prices baseat quantities period base of prices of Sum

pricescurrent at quantities period base of prices of Sum

100 * p

p 100*

......ppp

.......ppp

00

01

'''0

'''0

''0

''0

'0

'0

'''0

'''1

''0

''1

'0

'1

q

q

qqq

qqq

B. Paasche’s Index: Named after German statistician Paasche who formulated it in 1874. We have,

100* prices baseat quantities periodcurrent of prices of Sum

pricescurrent at quantities periodcurrent of prices of Sum

100 * p

p 100*

......ppp

.......ppp

10

11

'''1

'''0

''1

''0

'1

'0

'''1

'''1

''1

''1

'1

'1

q

q

qqq

qqq

Example

From the following data, construct the index number for 1988 with 1985 as base using Laspeyre’s and Paasche’s formula.

Commodity Prices Quantity

1985 1988 1985 1988

A 20 25 10 12B 18 32 16 10C 35 48 8 8D 28 40 12 10

Example….Commodity

Base Year (1985)

Current Year (1988) p0q0 p1q0 P0q1 p1q1

Price (p0)

Quantity (q0)

Prices (p1)

Quantity (q1)

A 20 10 25 12 200 250 240 300

B 18 16 32 10 288 512 180 320

C 35 8 48 8 280 384 280 384

D 28 12 40 10 336 480 280 400

Total 1104 1626 980 1404

143.26980

1404100 *

qp

qp IIndex Price sPaasche'

147.281104

1626100 *

qp

qp IIndex Price sLaspeyre'

10

11Pa01

00

01La01

C. Fisher’s Index Number: It is obtained by the (GM) of Laspeyres’ Index and Paasche’s Index. It is named after Prof Irving Fisher.

This is also called Fisher’s Ideal Index.Because:1.Geometric Mean is useful in averaging % ’s and

ratios. Index no. indicates % changes and Fisher’s Index no is a G.M. between Paspeyres’ & Paasche’s Index nos.

100 X qp

qpX

qp

qp I

10

11

00

01F01

Because……..:2. It takes into account both current year and base

year quantities.3. It satisfies Time Reversal test and Factor Reversal

test.4. It is free bias upward as well as downward.

D. Dorbish and Bowley Method: To take into account the influence of both the base as well as current periods, Dorbish and Bowley suggested the arithmetic average of the Laspeyre’s and Paasche’s indices.

Index sPaasche'Index sLaspeyre'2

1 IDB

01

E. Marshall and Edgeworth Method: In this method, both the current year and base year prices and quantities are considered.

F. Walsche’s Method:

010 X

qpqp

qpqp100 X

pqq

pqq I

1000

1101

010

110ME01

100 X p

I100

101Wa01

qq

qqp

G. Kelly’s Method: This method is also known as fixed weight aggregative index and is currently in great favor of index number series. The formula is as

Here weights are the quantities which may refer to some period and anre kept constant for all periods.

The AM or GM of the quantities of 2, 3 or more years can be used as weights.

The important advantage of Kelly’s method over Laspeyres’ , index is that in this index, the cahnge in the base period does not necessitate a corresponding change in the weights which can be kept constant until new data become available for revising the index.

100 Xqp

qp I

0

1Wa01

Example: Construct Index number of Price from the following data by applying

Commodity Base Year (1983) (1984)

Price Quantity Prices Quantity

A 2 8 4 6

B 5 10 6 5

C 4 14 5 10

D 2 19 2 13

143.26980

1404100 *

qp

qp IIndex Price sPaasche'

147.281104

1626100 *

qp

qp IIndex Price sLaspeyre'

10

11Pa01

00

01La01

Example…Commodity

Base Year (1983)

(1984)p1q0 P0q0 p1q1 P0q1

(p0) (q0) (p1) (q1)

A 2 8 4 6 32 16 24 12

B 5 10 6 5 60 50 30 25

C 4 14 5 10 70 56 50 40

D 2 19 2 13 38 38 26 26

Total 200 160 130 103

Fixed Based Method: Methods of Relatives

1(C). Simple Arithmetic Mean of Price RelativesExpressed in symbols

The Index number calculated by this method is given by

0

1'''

0

'''1

''0

''1

'0

'1

'''0

'''1

''0

''1

'0

'1

01

100.....

100

included. items of no.n where.....1001001001

I

p

p

np

p

p

p

p

p

n

p

p

p

p

p

p

n

1,2,3... items of relatives price theare ....... ,p

p100 ,

p

p100 ,

p

p100

'''0

'''1

''0

''1

'0

'1

1(d). Simple Geometric Mean of Price RelativesThe Index number calculated by this method is given by

1(e). Weighted Arithmetic Mean of Price Relatives

But for all practical purposes, the weights adopted in this method are the values (=price X quantity) of items.

.....p

p X

p

p X

p

p 100 I n

'''0

'''1

''0

''1

'0

'1

01

0

1'''

0

3'''

1''0

2''

1'0

1'1

01 p

wp

w

100.....

p

wp

p

wp

p

wp

w

100 I

1(f). Geometric Mean of Two Aggregative Price Index Numbers (Special case)

NumberIndex sPaasche' is which 100 Xqp

qp

100 X pp

qp

qp I

10

11

0

1

11

1101

Test of Adequacy of Index numbers

1. Unit Test2. Time Reversal Test3. Factor Reversal Test4. Circular Test

Unit Test

This test requires that the formula should be independent of the unit in which or for which prices and quantities are quoted. Except for the simple (unweighted) aggregative index, all other formula satisfy this test.

2. Time Reversal Test

An Index no formula satisfies this test if works both ways, forward and

backward with respect to time. In other words, an index no I01 for

the year Y1 with base year Y0, Symbolically I01 X I10 =1 omitting

the factor 100 from both the indices. This test satisfied by –

(i) Simple aggregative Index (ii) Marshall-Edgeworth’s Index

(iii) Fisher’s Ideal Index (iv) Simple GM of price relatives

(v) Walsch Formula (vi) Kelly’s fixed weight formula

(vii) Weighted GM of price relative formula with fixed weights

3. Factor Reversal TestAn Index no formula satisfies this test if the product of the price

index and the quantity index gives the TRUE VALUE RATIO,

omitting the factor 100 from both indices. Symbolically an index

no formula satisfies this test if

Where,

I01 = Price index for Y1 with base year Y0

Q01 = Quantity ideal index for Y1 with base year Y0

Fisher’s Ideal Index is the only formula which satisfies this test.

TVR ratio value trueThe qp

qp Q X I

10

110101

4. Circular TestThis test is based on shifting the base in a circular fashion. It

may be considered as an extension of Time Reversal Test.

An index number is said to satisfy the circular test if it

satisfies

This test is concerned with the measurement of price changes

over a period of years when the shifting of base is desirable.

This test is satisfied by

(i) Simple aggregate index (ii) Simple GM of price relatives

(iii) Weighted aggregative formula with fixed weights

1 I* I*...........*I* I * I 0n 1)n-(n231201

Example:Using following data, show that the Fisher’s Ideal formula

satisfies the Factor Reversal Test.Commodity Price per unit (in Rs.) Number of Units

Base Period Current period Base Period Current periodA 6 10 50 56

B 2 2 100 120

C 4 6 60 60

D 10 12 30 24

E 8 12 40 36

Solution:Omitting to factor 100, Fisher’s price Index I01 is given by

This shows that Fisher’s Ideal Index satisfies Factor Reversal Test

Ratio) Value (True 1360

1880

1360*1360

1880*1880

1900

1880*

1360

1344*

1344

1880*

1360

1900Qx I

1900

1880X

1360

1344

qp

qpX

qp

qp Q

1344

1880X

1360

1900

qp

qpX

qp

qp I

0101

10

11

00

0101

10

11

00

0101

Chain Index Numbers

Chain Index NumberIn fixed base Index no., the index no of a given year on a given

fixed base was not affected by changes in the relevant values of

any other year.

But in the chain base method, the value of each period is related

with that of the immediately preceding period and not with any

fixed period.

For constructing Index no by chain base method, a series of Index

nos. are computed for each year with preceding year as the base

year. These index nos. are known as Link Index Number or

Link relatives.

Chain Index Number….The link relatives I01, I12, I23, I34,……I(n-1)n when multiplied

successively known as the chaining process gives the

relatives of a common base.

Thus I01 = First Link

I02 = I01 * I12

I03 = (I01 * I12)*I23

……………………………………

…………………………………….

I0n = I0(n-1) * I(n-1)n

Construction of Chain IndiciesStep (1): Express the figures for each period as a % of the

preceding period to obtain Link Relatives (L.R.)

Step (2): Chain base indices (CBI) are obtained by multiplying

successively the link relatives as explained above.

Chain Base Index (CBI) =

NOTE:

1. Chain relatives differ from fixed base relatives in computation,

chain relatives are computed from link relatives whereas fixed

base relatives are computed directly from the original data.

100

C.I.year Preceding X L.R.year Current

Construction of Chain IndiciesNOTE:

2. Link Relative

Price Relative

3. Conversion of Chain base Index no to Fixed base Index no

Current year FBI

The FBI for the Ist period being same as the CBI for the first period

100 X year previous for the relative Price

yearcurrent for the relative Price

100

relative price syear' Previous * relativelink yearsCurrent

100

FBIyear Previous X CBIYear Current

Example:

From the following data of wholesale prices of a certain commodity, construct Index Numbers by Chain Base Method.

Year 1979 80 81 82 83 84 85 86 87 88

Price 750 500 650 600 720 700 690 750 840 800

Solution:Year Price Link Relatives Chain Base Index No. Fixed base Index no.

(1979=100)1979 750 100 100 100

1980 500 (500/750)*100=66.67

1981 650 (650/500)*100=130

1982 600 (600/650)*100=92.31

1983 720 120

1984 700 97.22

1985 690 98.57

1986 750 108.69

1987 840 112

1988 800 95.24106.67

112

100

29

.3339 100

96 X 97.22

69 100

80 X 120

80 100

86.67 X 92.31

86.67 100

66.67 X 130

66.67 100

100 X 66.67

106.67

112

100

92

.3339 750

100 X 700

69 750

100 X 720

80 750

100 X 600

86.67 750

100 X 650

66.67 750

100 X 500

It may be noted that chain base index nos are the same as the fixed base index nos.

Base Shifting

Base shifting refers to the preparing of a new series with a new

or more recent base period than the original one. This

method requires the taking of a new base year as 100 and

express the given series of index nos as a % of the index no

of the time period selected. The series of index no with a

new base is obtained by the formula

100 X year base new theof No.Index Old

year theof No.Index Oldyearany of No.Index Recast

Base Shifting Eg.

The following are the index nos of prices based on 1977. Shift the base from 1977 to 1982

Year Index Nos

1977 100

1978 110

1979 120

1980 200

1981 320

1982 400

1983 410

1984 400

1985 380

1986 370

1987 350

1988 366

Base Shifting Eg.

The following are the index nos of prices based on 1977. Shift the base from 1977 to 1982

Old Base Year Year Index Nos (old)1977=100

Index Nos (new)1982 =100

1977 100 (100/400)*100=25

1978 110 (110/400)*100=27.5

1979 120 (125/400)*100=30

1980 200 (200/400)*100=50

1981 320 (320/400)*100=80

New Base Year 1982 400 100

1983 410 (410/400)*100=102.5

1984 400 100

1985 380 95

1986 370 92.5

1987 350 87.5

1988 366 91.5