Relatives Weighted Average
In weighted aggregative technique, price relative are not be computed. However like unweighted relative technique it is also possible to compute the weighted average of relative for the purposes of averaging. We may use either the arithmetic mean or the geometric mean. The steps in the computation of weighted arithmetic mean of relative index number are as follows:
(i) At first express each item of the period for which the index number is being calculated as a percentage often same item in the base period.
(ii) In the next step, multiply the percentages as obtained in step (i) for each item by the weight which has been assigned to that item.
(iii) As the results obtained from the several multiplications carried out in step (ii), add it
(iv)Now divide the sum obtained in step (iii) by the sum of the weights used. The result is the index number symbolically
p01 = Σ p v / ΣV
Where p =price relative
V = value weights p0 q0
Instead of using the arithmetic mean the geometric mean may be used for assigning relatives. The weighted geometric mean of relatives is computed in the similar manner as the unweighted geometric mean of relative's index number except that weights are introduced by applying them to the logarithms of the relatives. When this technique is used the formula for computing the index is:
p01 = antilog { ΣV . log p / Σ V }
Where p = p1 / p0 x 100 V = value weight p0 q0 for each item.
Illustration:
From the following data compute the price index by supplying weighted average of price technique using
(a) Arithmetic mean and
(b) Geometric mean.
|
Commodity
|
p0 ($)
|
q0
|
P1 ($)
|
|
Sugar
|
3.0
|
20 Kg.
|
4.0
|
|
Flour
|
1.5
|
40kg
|
1.6
|
|
Milk
|
1.0
|
10 lt.
|
1.5
|
Solution:
Weighted arithmetic mean of price relatives
|
Commodity
|
p0
|
q0
|
p1
|
p0 q0 V
|
p1/ p0x100 p
|
pV
|
|
Sugar
|
$ 3.0
|
20 kg
|
$4.0
|
60
|
4/3 x 100
|
8000
|
|
Flour
|
$1.5
|
40 kg
|
$ 1.6
|
60
|
1.6/1.5 x100
|
64000
|
|
Milk
|
$1.0
|
10lt.
|
$1.5
|
10
|
1.5/10 x100
|
1500
|
|
|
|
|
|
Σ V = 130
|
|
Σ p V = 15,9000
|
p01 Σ p v / Σ V = 15,900/130 = 122.31
This means that there has been a 22.3 percent increase in prices over the base level.
Index number using geometric mean of price relative
|
Commodity
|
p0
|
q0
|
P1
|
V
|
P
|
Log p
|
V log p
|
|
Sugar
|
$3.0
|
20 kg.
|
$4.0
|
60
|
133.3
|
2.1279
|
127.494
|
|
Flour
|
$ 1.5
|
40 kg
|
$ 1.6
|
60
|
106.7
|
2.0282
|
121.692
|
|
Milk
|
$ 1.0
|
10 lt
|
$ 1.5
|
10
|
150.0
|
2.1761
|
21.761
|
|
|
|
|
|
|
Σ V = 130
|
|
Σ V log p = 270.947
|
p01 = antilog [Σ V log / Σ V] = antilog [270.947/130] = antilog 2.084 = 120.9
The result obtained by applying the Laspeyres technique would come out to be the same as obtained by weighted arithmetic mean of price relative's technique as shown below:
Price index by Laspeyres technique
|
Commodity
|
p0
|
q0
|
p1
|
p1 q0
|
p0 q0
|
|
Sugar
|
$3.0
|
20 kg.
|
$4.0
|
80
|
60
|
|
Flour
|
$ 1.5
|
40 kg
|
$ 1.6
|
64
|
60
|
|
Milk
|
$ 1.0
|
10 lt.
|
$ 1.5
|
15
|
10
|
|
|
|
|
|
Σ p1 q0 = 159
|
Σ p0 q0 = 130
|
p01 = Σ p1 q0 / Σ p0 q0 x 100 = 159 / 130 x 100 = 122.3
The answer is the same as that obtained by the weighted arithmetic mean of price relative technique. This is because the weighted average of price relative technique can be changed to the simple aggregative technique given by Laspeyres as follows:
Σ p1/ p0 x p0 q0 / Σ p0 q0 = Σ p1 q0 / Σ p0 q0 s