Methodology Glossary Tier 2
Index numbers
Why use indices?

To aid understanding/interpretation – rates of change from the base year
are easily gleaned from indexed data, i.e. if the base year is equal to 100
and one year later the statistic is measured as 125, this equates to 25
per cent growth over the period.
To promote appropriate use of statistics by concentrating upon relative
rather than absolute values. Whilst it would be possible to present
quarterly Gross Domestic Product (GDP) as a constant price monetary
series i.e. with all data presented at 2000 prices, this would be rather
abstract and certainly prone to misinterpretation by some users.
To aid analysis of measures with uncommon units – e.g. one series
expressed in giga-watt hours and another in pounds sterling.
To aid comparison of measures that differ widely in size – e.g. comparing
change in population of Scotland to that of change in population of
Europe.



There are two main kinds of index – price and volume. The general
construction of each index is broadly similar, though there are important
differences to bear in mind.

A volume index is “…a weighted average of the proportionate changes
in the quantities of a specified set of goods or services between two
periods of time.” These changes in quantity are determined by holding
the price constant.

A price index is “…an average of the proportionate changes in the prices
of a specified set of goods and services between two periods of time.”
Therefore, a volume index holds price constant and examines changes in
the quantity of goods and services over time and a price index uses a fixed
basket of goods and services and examines changes in price.
Volume Indices
Creating a volume index is relatively simple when you are dealing with a
single homogenous commodity - it is simply the ratio of the two quantities in
each period. For example, if the only commodity in an economy was apples
and there were 50 produced in period one and 65 in period two, the index
would simply be the ratio of these two quantities (65/50=1.3). By setting the
value of the index in period one to 100, the value in period two is 130 which
means there has been a rise in volume of 30%.
Things get more complicated when there is more than one commodity and, in
real life, growth is measured in terms of a large number of goods and
services whose prices do not change proportionately. It therefore becomes
necessary to introduce some sort of weighting system to aggregate the
Methodology Glossary Tier 2
different rates of change into a single measure, as it is not possible to simply
add the totals of the different products together because different goods are
not worth the same as one another. The most commonly used weighting
strategy is to use the market price. In order to create the volume index,
products should be valued at their prices in one of the periods concerned and
then the total value of combined production in the second period should be
divided by that in the first. There are several ways of measuring the change
in volume depending on which period’s prices (weights) are used.
Laspeyres volume index
The Laspeyres index is one of the most commonly used aggregation
methods. It measures the percentage change in the total value of production
holding prices constant at their base year levels.
Example 1
The table below shows the number of CDs and MP3s purchased per annum
and their prices. It shows that the number of CDs purchased decreases over
time, whilst the number of MP3s rises as technology improves and newer
products begin to replace older ones.
Year
2004
2005
2006
price (£)
CDs
MP3s
12
8
13
6
14
5
Number
purchased per
annum
CDs
MP3s
9
3
6
9
4
14
Amount spend
on
CDs
MP3s
108
24
78
54
56
70
In order to calculate the combined ‘music’ series, it is necessary to create a
weighted average of the two component series using the base year weights
(2004). The formula used in the calculation is :
VL = Σ (P0 . Qt) x 100
Σ (P0 . Q0)
P0 : Price in base year
Qt : Volume in period t
Q0 : Volume in base year
2004
((12x9)+(8x3)) / ((12x9)+(8x3)) x 100 = 100
2005
((12x6)+(8x9)) / ((12x9)+(8x3)) x 100 = 109.1
2006
((12x5)+(8x14)) / ((12x9)+(8x3)) x 100 = 121.2
The disadvantage of the Laspeyres index is that as we get further and further
away from the base year, the actual volume of purchases made will become
more and more divorced from the fixed basket imposed by the Laspeyres
index. This is because, in the real world, prices would have changed and as
Methodology Glossary Tier 2
they did, people would have purchased different volumes of goods and
services.
Paasche volume index
The Paasche index is very similar to the Laspeyres index - it also measures
the percentage change in the total value of production but by using the most
recent prices (weights).
Example 2
The Paasche index can be used to create a volume index for the data in the
table above. This index is arguably a better measure of true growth in cases
such as this where one product is essentially replacing another. The
calculation is shown below.
VL = Σ (Pt . Qt) x 100
Σ (Pt . Q0)
Pt : Price in year t
Qt : Volume in period t
Q0 : Volume in base year
2004
((12x9)+(8x3)) / ((12x9)+(8x3)) x 100 = 100
2005
((13x6)+(6x9)) / ((13x9)+(6x3)) x 100 = 97.8
2006
((14x4)+(5x14)) / ((14x9)+(5x3)) x 100 = 89.4
The calculation demonstrates that ‘up to date’ prices are used when
calculating the Paasche index for each year. In this ‘music’ example, the
results of the two indices show that the Laspeyres index exhibits growth over
time and the Paasche index declines over time.
Differences between the Laspeyres and Paasche indices
It is impossible to say which of the above methods is the better measure of
growth. The differences between the two indices are caused by the fact that
they use different price sets - there is no difference between the quantities
they use. The disparities between the two can become more exaggerated if
there are large price fluctuations over time or if some prices rise whilst others
fall.
It is not a random result that the Laspeyres index is greater than the Paasche
index in the example used here. Typically, when people see the price of
certain goods rising they buy less of those goods in favour of goods which
have become relatively cheaper by comparison. This is known as the
‘substitution effect’. It means that goods whose prices have been falling tend
to exhibit faster growth and goods whose prices have been rising tend to
display slower growth. By using the most recent prices, substitutions are
likely to have been made therefore the Paasche index will tend to give less
Methodology Glossary Tier 2
weight to fast growing goods and more weight to slow growing ones. The
opposite is true of the Laspeyres index.
Fisher ‘ideal’ index
The Fisher ideal index is an attempt to produce a central estimate from the
upper and lower bounds generated by the Paasche and Laspeyres indices
and is simply defined as the geometric mean of the two, i.e.
Vf = √(Vp . Vl)
The Fisher index provides good approximations for the ‘real’ time series and
is simple to use and compute. It takes into account the weights of both the
base year and the current year which means it has the ability to incorporate
the effects of substitutions. The Fisher index for the music series in the
above examples would be calculated as follows:
√ (100 x 100) = 100
√ (97.8 x 109.1) = 103.3
√ (89.4 x 121.2) = 104.1
2004
2005
2006
Price Indices
Calculating price indices, where the basket of goods is fixed and prices are
varied, is very similar to calculating volume indices. The formulae for the
Laspeyres and Paasche prices indices are given below.
Laspeyres price index
The Laspeyres price index uses the basket of goods and services of the base
year and measures the effects of annual price changes.
PL = Σ (Pt . Q0) x 100
Σ (P0 . Q0)
P0 : Price in base year
Qt : Volume in period t
Q0 : Volume in base year
Paasche price index
The Paasche index uses the most recent basket of goods and services and
measures the effects of annual price changes.
PP = Σ (Pt . Qt) x 100
Σ (P0 . Qt)
Further Information
Tier 1 Index Numbers
Tier 2 Chain Linking