Page 218 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Unit 10: Index Number

In a similar manner the Paasche's price index can be interpreted as the change in cost of purchasing Notes
the bundle q . Out of these two, the Laspeyres's index is preferred because weights do not change
1
over different periods and hence the index numbers of various periods remain comparable.
Furthermore, Laspeyres's index requires less calculation work than the one with changing
weights in every period. The main disadvantage of Laspeyres's formula is that with passage of
time the relative importance of various items may change and the base year weights may
become outdated. Paasche's index, on the other hand, uses current year weights which truly
reflect the relative importance of the items. The main difficulty, in this case, is that index numbers
of various periods are not comparable because of changing weights. Moreover, it may be too
expensive and difficult to obtain these weights.
When both the index number formulae are applied to the same data, they will in general give
different values. However, "if prices of all the commodities change in the same ratio, then the
Laspeyres's index is equal to Paasche's index, for then the two weighing systems become
irrelevant; or, if quantities of all the commodities change in same ratio, the two index numbers
will again be equal, for then the two weighing systems are same relatively." (Karmel & Polasek)

In order to show this, let p be the price of i th commodity in current year and p be its price in
1i 0i
p 1i 105
base year. If prices of all the commodities increase by 5%, then we can write or
p 100
0i
p = 1.05 ×p , for all values of i. To generalise, we assume that p = a.p (or p = a.p , on dropping
1i 0i 1i 0i 1 0
the subscript i), where a is constant.
p q
1 0
La
We can write the Laspeyres's index as P 01 100
p q
0 0
Substituting p =ap , we have
1 0
p q p q
0 0
0 0
La
P 01 100 100 100 .... (1)
p q p q
0 0
0 0
Similarly, the Paasche's index is given by
p q p q p q
Pa 1 1 0 1 0 1
P 100 100 100
01 .... (2)
p q p q p q
0 1
0 1
0 1
La
Hence, P = P Pa
01 01
Further, when quantities of all the commodities change in same proportion, we can write,
q = b.q , for all commodities. Here b is a constant.
1 0
Thus, we can write the Paasche's index as
p q p q p q
1 0
1 0
1 1
Pa
P 01 100 100 100
p q p q p q
0 1 0 0 0 0
Pa La
Hence, P = P
01 01

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