Page 245 - DMGT404 RESEARCH_METHODOLOGY

Unit 11: Index Numbers

Solution: Notes
Let p be the price in 1980 and p be the price in 1981. Thus, we have
0 1
S p 0 = 350 and pS 1 = 400

\ P = 400 ´ 100 = 114.29
01
350
Weighted Aggregative Method

This index number is defined as the ratio of the weighted arithmetic means of current to base
year prices multiplied by 100.
Using the notations, defined earlier, the weighted arithmetic mean of current year prices can be
å p w
written as = 1i i
å w i
å p w
Similarly, the weighted arithmetic mean of base year prices = 0i i
å w i
å p w i
1i
å w å p w
\ Price Index Number, P = i ´ 100 = 1i i ´ 100
å p w i å p w i
01
0i
0i
å w i
å p w
Omitting the subscript, we can also write P = å p w ´ 100
1
01
0
Nature of Weights
In case of weighted aggregative price index numbers, quantities are often taken as weights.
These quantities can be the quantities purchased in base year or in current year or an average of
base year and current year quantities or any other quantities. Depending upon the choice of
weights, some of the popular formulae for weighted index numbers can be written as follows:
1. Laspeyres' Index: Laspeyres' price index number uses base year quantities as weights.
Thus, we can write
La å p q La å p q
1i
1
P 01 = å p q 0i 0i ´ 100 or P 01 = å p q 0 0 ´ 100
0i
0
2. Paasche's Index: This index number uses current year quantities as weights. Thus, we can
write
Pa å p q Pa å p q
P = 1i 1i ´ 100 or P = 1 1 ´ 100
å p q 1i å p q 1
01 01
0i
0
3. Fisher's Ideal Index: As will be discussed later that the Laspeyres's Index has an upward
bias and the Paasche's Index has a downward bias. In view of this, Fisher suggested that an
ideal index should be the geometric mean of Laspeyres' and Paasche's indices. Thus, the
Fisher's formula can be written as follows:

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