Index Numbers I

Introduction:

  1. An index number is a device which shows by its variations the change in a magnitude which is not capable of accurate measurement in itself or of direct valuation over time.
  2. To measure changes in a situation we combine the prices and qualities and find a single number.  This single number which shows overall changes in a phenomenon is known as ‘Index Number’.
  3. It is used to compare changes in a complex phenomenon like the cost of living, total industrial production, wages, etc.
  4. It is very useful in measuring changes in prices and quantities of commodities with different measuring units, for example, wheat per maund, cloth per yard, etc., which cannot be compared directly.

Types of Index Number:

(a)   Price Index Number: It compares changes in prices, from one period to another.  Wholesale price index and cost of living index are the examples.

(b)   Quantity Index Number: It measures how much the quantity of a variable changes over time.  Index of industrial production and business activity index are examples.

(c)    Value Index Number: It measures changes in total monetary worth.  It combines price and quantity changes to present a more informative index.  Index of GNP and index of retail sales are the examples.

Uses of Index Numbers:

  1. An index number is a device for measuring changes in a variable or a group of related variables.
  2. It can be used to compare changes in one or more variables in one period with those of others or in one region with those in the others.
  3. The index number of industrial activity enables us to study the progress of industrialisation in the country.
  4. The quantity index numbers show rise or fall in the volume of production, volume of exports and imports, etc.
  5. The cost of living index numbers are, in fact, the retail price indices.  They show changes in the prices of goods generally consumed by the people.  Therefore, they can help the government to formulate the suitable price policy.
  6. The cost of living index number can be made a basis for regulation of wage rates and can be used by industrial and commercial organisations to grant dearness allowance and bonus to their employees in order to meet the increased cost of living.
  7. Index numbers are also used for forecasting business activity and in discovering seasonal fluctuations and business cycles.

Steps in the Construction of Index Numbers of Prices:

(i)                  Defining the purpose and scope of index number, i.e., the general-purpose or special purpose,

(ii)                Selecting commodities to be included,

(iii)               Collection of prices, i.e., (a) considering the prices to be used like average price, retail price or wholesale price, etc; and (b) the sources of price data like from representative markets, price lists or trade journals.

(iv)              Selecting base period, (a) fixed-base method, and (b) chain-base method.

(v)                Choice of average to be used, i.e., AM, median or GM.

(vi)              Selecting suitable weights: (a) implicit weighting, and (b) explicit weighting.

Notations:

Pn   =          Price in current year

Po   =          Price in base year

Qn  =          Quantity in current year

Qo  =          Quantity in base year

Pon =          Price for the nth year to the base year

Qon =          Quantity for the nth year to the base year

Construction of Price Index Numbers:

(a)    Simple Relatives or Simple Index Numbers,

(b)   Unweighted Index Numbers, and

(c)    Weighted Index Numbers.

(a)   Simple Relatives: are further classified into two categories:

                       (i)      Price Relatives: are obtained by dividing the price in a given year by the base year price and expressed as percentage.  Thus:

Example:

The prices of sugar for 2001 and 2005 are given as below:

Year

Price / Kg

2001

11

2005

30

Required:

(a)    Taking 2001 as base year, find price relative for 2005.

(b)   Taking 2005 as base year, find price relative for 2001.

Solution:

(a)   Base year: 2001

Year

Price

Price Relative (v)

2005

30

(b)   Base year: 2005

Year

Price

Price Relative (V)

2001

11

                       (ii)      Link Relatives: are obtained by dividing the price in a given year by the price in the preceding year and expressed as percentage:

Link relatives are not directly comparable, therefore, they are converted to a fixed based index number.  The process of conversion is called the ‘chaining process’, and the index numbers so obtained are chain indices:

= (L.R. × C.I. of preceding year) ÷ 100

Example:

The price of rice for the 6 years is as follows:

Year

Price / Kg

2000

21

2001

20

2002

20

2003

22

2004

25

2005

28

Required:

Taking 2000 as base year, find price relatives for the years 2001 to 2005.

Solution:

Year

Price

Price Relative (V)

Chain Indices

2000

21

100%

2001

20

2002

20

2003

22

2004

25

2005

28

(b)   Unweighted Index Numbers: There are two methods of constructing this type of index:

                       (i)      Simple Aggregative Method: In this method, the total of the prices of commodities in a given year is expressed as percentage of the total of the prices of commodities in the base year:

This method has two disadvantages which make it unsatisfactory:

·        It does not take into account the relative importance of various commodities.

·        The units in which prices are given, e.g., maunds, yards, gallons, etc., affect the value of index very much.

Example:

The prices of 3 commodities for the 5 years are as follows:

Commodity

Prices (per kg)

2001

2002

2003

2004

2005

Rice

20

20

22

25

28

Sugar

11

12

14

27

30

Tea

178

176

174

180

180

Required:

Simple aggregative index numbers for the years 2001-05, with 2001 as base year.

Solution:

Commodity

Prices (per kg)

2001

2002

2003

2004

2005

Rice

20

20

22

25

28

Sugar

11

12

14

27

30

Tea

178

176

174

180

180

Total

209

208

210

232

238

Simple

Aggregative

Index

     (ii)          Average of Relatives’ Method: In this method, we use the average (mean, median, GM, etc.) of the price relatives or link relatives.  It does not affect the value of index numbers.  The only disadvantage of this method is that it gives equal weight to all commodities.

Example:

The prices of 3 commodities for the 5 years are as follows:

Commodity

Prices (per kg)

2001

2002

2003

2004

2005

Rice

20

20

22

25

28

Sugar

11

12

14

27

30

Tea

178

176

174

180

180

Required:

Construct price index numbers using average of relatives’ method, taking 2001 as base year.

Solution:

Commodity

Prices (per kg)

2001

2002

2003

2004

2005

Rice

Sugar

Tea

Total

300

307.97

335.02

471.52

513.85

Mean

(Index)

100

102.66

111.67

157.17

171.28

(c)   Weighted Index Numbers: This type of index can be further classified into two categories:

                       (i)      Weighted Aggregative Index Numbers: In these index numbers, the quantities produced, sold or bought or consumed during the base year or current year are used as weights.  These weights indicate the importance of the particular commodity.  Some well-known weighted index numbers are given below:*

(1)   Lespeyre’s Index: This index uses base year quantities as weights.  For this reason, it is also known as ‘Base Year Weighted Index’:

Here W = Qo

(2)   Paasche’s Index: This index uses current years quantity as weights.  For this reason, it is known as ‘Current Year Weighted Index’:

Here W = Qn

(3)   Fisher’s Ideal Index: This index number is the GM of the Lespeyre’s and Paasche’s index numbers.  It is called ‘ideal’ because it satisfies two tests (Time Reversal and Factor Reversal Tests):

(4)   Marshall-Edgeworth’s Index: This index number uses the average of the base year and current quantities as weights:

Example:

Commodities

2001

2005

Price (Rs. / kg)

Qty. (kgs)

Price (Rs. / kg)

Qty. (kgs)

Rice

20

100

28

160

Sugar

11

18

30

37

Salt

1

1

5

1.1

Milk

18

57

32

149

Required:

Construct the following price index numbers using 2001 as base year:

(a)    Lespeyre’s

(b)   Paasche’s

(c)    Fisher’s

(d)   Marshall-Edgeworth’s

Solution:

 

2001

2005

PoQo

PnQo

PnQn

PoQn

Qo+Qn

Po(Qo+Qn)

Pn(Qo+Qn)

Po

Qo

Pn

Qn

Rice

20

100

28

160

2000

2800

4480

3200

260

5200

7280

Sugar

11

18

30

37

198

540

1110

407

55

605

1650

Salt

1

1

5

1.1

1

5

5.5

1.1

2.1

2.1

10.5

Milk

18

57

32

149

1026

1824

4768

2682

206

3708

6592

Total

 

 

 

 

3225

5169

10363.5

6290.1

 

9515.1

15532.5

(a)   Lespeyre’s:

(b)   Paasche’s:

(c)    Fisher’s:

(d)   Marshall-Edgeworth’s:

                     (ii)      Weighted Average of Relatives: The formula of weighted average of relatives is:

or

 (Arithmetic Mean taken as average); where

or

 (Geometric Mean taken as average)

The total value of the commodity is used as weights.  If the base year value (PoQo) is used as base, then the formula becomes:

or

If the current year value (PnQn) is used as base, then the formula becomes:

Example:

Commodity

Prices

Weights

2001

2005

Rice

20

28

35

Tea

178

180

5

Sugar

11

30

24

Required:

Weighted index for 2005, taking 2001 as base year using:

(a)    Arithmetic Mean

(b)   Geometric Mean

Solution:

(a)   Arithmetic Mean:

Commodity

V

W

VW

Rice

35

4900

Tea

5

505.6

Sugar

24

6545.52

 

 

64

11951.12

(b)   Geometric Mean:

Commodity

V

W

log V

W.logV

Rice

35

2.146

75.11

Tea

5

2.005

10.025

Sugar

24

2.436

58.464

 

 

64

 

143.599

Quantity Index Number: The formula described for obtaining price indices can be easily used to obtain quantity indices or volume indices simply by interchanging the Ps and Qs, for example:

and:

The Lespeyre’s index number can be converted as follows:

and so on.

Value Index Numbers: Like price or quantity index numbers, we can obtain formulae for value index numbers.  The simplest value index number is defined as below:

This is the ‘Simple Aggregative Index’ because the values have not been obtained.


* W.A.I.N. is equal to

Continued

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