Price Relatives Average
When this technique is used to construct a price index, First of all price relatives are obtained for the different items included in the index and then average of these relatives are obtained by using any one of the measures of central value median, arithmetic mean, mode, geometric mean or harmonic mean. When an arithmetic mean is used for averaging the relatives the formula for computing the index is:
p01 Σ (p1 / p0 x 100) / N
Where N refers to the number of items or commodities whose price relatives are thus averaged.
Although any measure of the central value can be used to obtain, the overall index price relatives are normally average either by the arithmetic or the geometric mean. When geometric mean is used for raging the price relatives, then the formula for obtaining the index becomes.
Log p01 Σ log [p1 / p0 x 100] / N or Σ log p / N where p = p1 / p0 x 100
Or p01 = antilog [(Σ log p1/ p0 x 100) /N] = antilog Σ log p / N
another measures of central value are not in common use for averaging relatives.
Illustration:
From the following data construct an index for 2009 taking 2008 as base by the average of relative method using (a) arithmetic mean and (b) geometric mean for averaged relatives.
|
Commodity
|
Price in 2008 ($)
|
Price in 2009 ($)
|
|
A
|
50
|
70
|
|
B
|
40
|
60
|
|
C
|
80
|
90
|
|
D
|
110
|
120
|
|
E
|
20
|
20
|
Solution:
(a) Index numbers by using arithmetic mean of price relatives
|
Commodity
|
Price in 2008 ($)p0
|
Price in 2009 ($) p1
|
Price relatives p1 / p0 x 100
|
|
A
|
50
|
70
|
140.0
|
|
B
|
40
|
60
|
150.0
|
|
C
|
80
|
90
|
112.5
|
|
D
|
110
|
120
|
109.1
|
|
E
|
20
|
20
|
100.0
|
|
|
|
|
Σ p1 / p0 x 100 = 611.6
|
Σ p1 / p0 x 100 / N = 122.32
(b) Index numbers by using geometric mean of price relatives
|
Commodity
|
Price in 208 p0
|
Price in 2009 p1
|
Price relatives p
|
Log p
|
|
A
|
50
|
70
|
140.0
|
2.1461
|
|
B
|
40
|
60
|
150.0
|
2.1761
|
|
C
|
80
|
90
|
112.5
|
2.0512
|
|
D
|
110
|
120
|
109.1
|
2.0378
|
|
E
|
20
|
20
|
100.0
|
2.0000
|
|
|
|
|
|
Σ log p = 10.4112
|
p01 = antilog [Σ log P / N = antilog [10.4112/5 = antilog 2.0822=120.9
Although the arithmetic and geometric mean have both been used, the arithmetic mean is often preferred as it is easier to compute and much better known. Some economist like F.Y Edgeworth, have preferred to use the median which is not affectedly extreme values. As the argument is important only when an index is based on a very small number of commodities, it normally does not carry much weight and the median is seldom used in actual practice.