Introduction:
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.
Steps in the Construction of
Index Numbers of Prices:
(i) Defining the purpose and scope of index number, i.e., the generalpurpose 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) fixedbase method, and (b) chainbase 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:
P_{n} = Price in current year
P_{o} = Price in base year
Q_{n} = Quantity in current year
Q_{o} = Quantity in base year
P_{on} = Price for the n^{th} year to the base year
Q_{on} = Quantity for the n^{th} 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 200105, 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 wellknown 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 = Q_{o}
(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 = Q_{n}
(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) MarshallEdgeworth’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) MarshallEdgeworth’s
Solution:

2001 
2005 
P_{o}Q_{o} 
P_{n}Q_{o} 
P_{n}Q_{n} 
P_{o}Q_{n} 
Q_{o}+Q_{n} 
P_{o}(Q_{o}+Q_{n}) 
P_{n}(Q_{o}+Q_{n}) 

P_{o } 
Q_{o } 
P_{n } 
Q_{n } 

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)
MarshallEdgeworth’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 (P_{o}Q_{o}) is used as base, then the formula becomes:
or
If the current year value (P_{n}Q_{n}) 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