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Untitled 60
Index numbers
What are index numbers?
Index numbers are designed to measure the magnitude of economic changes over
time. Because they work in a similar way to percentages they make such changes
easier to compare. Briefly, this works in the following way.
Suppose that a cup of coffee in a particular café cost 75p in 1995. In 2002, an
identical cup of coffee cost 99p. How has the price changed between 1995 and
2002?
The particular time period of 1995 which we've chosen to compare against, is
called the base
period.
The variable for that period, in this case the 75p, is then given a value of
100, corresponding to 100%.
The index can
then be calculated for the later period of 2002 as a proportionate change as
follows:
The index number shows us that there has been a price increase of 32% since the
base period. An index number for a single price change like this is called a price
relative.
Rule for finding the price relative
If we let po be the price
in the base period and let pn be
the price in the later period, then the price relative for the price change
between these periods is given by (pn/po) x 100.
The Index
of Retail Prices is
probably the most generally known of all index numbers. Its aim is to measure
the change in price over time of a whole range of widely bought goods and
services and so give a measurement of the cost of living. This measurement can
then be used to alter the amounts of the payments in index-linked pensions, for
example.
Calculating Index Numbers
There are a number of different ways of calculating index numbers. Again, the
easiest way of explaining what these are and how to work them out is to look at
an example.
The small firm of Tastynibbles Ltd likes to give all its workers a Christmas
party. The snacks are provided from Tastynibbles stock (a useful way of running
down any extra stock before the Christmas/New year holiday), but they have to
buy the drinks. They buy bottles of red and white wine (all at one price),
6-packs of beer also all at one price and litre bottles of soft drinks which are
also priced the same as each other.
The first Christmas party was in 1990. It was a great success and so has been
held every year since. We'll now look at how we can use index numbers to compare
the cost for the base period of 1990 against the cost for a later period, taking
the particular year of 2000 for our example.
The table below shows the details of the purchases for these two parties.
The Tastynibbles Christmas parties of 1990 and 2000
|
The 1990 party |
The 2000 party |
| Drink |
Unit price |
Quantity |
Unit price |
Quantity |
|
po |
qo |
pn |
qn |
| wine |
£2.50 |
25 |
£3 |
30 |
| beer |
£4.50 |
10 |
£6.00 |
8 |
| soft drinks |
£0.60 |
10 |
£0.84 |
15 |
Now we'll use this data to show how to work out various index numbers.
The expenditure index
In a simple situation like my example, where we are comparing only a few
purchases made on just two occasions, we can obtain the most exact information
about the changing cost of the party by working out the expenditure index.
To do this, we first calculate the total cost of the party in 1990 and then the
total cost of the party in 2000.
Also, to help us make a general rule for finding the index number, we'll make
use of the capital Greek S which is called sigma and written 
.
Mathematicians use to
mean "the sum of everything like..."
Now, the party's cost in 1990 = poqo =
(2.5 x 25) + (4.5 x 10) + (0.6 x 10) = 113.5.
The party's cost in 2000 = pnqn =
(3 x 30) + (6 x 8) + (0.84 x 15) = 150.6.
(I have left out the £ signs here since the method is the same for all
currencies and the index number is independent of the currency.)
Now, we work in a similar way to when we found the price relative for the cup of
coffee.
The expenditure index = (party's cost in 2000)/(party's cost in 1990) x 100
= (pnqn/ poqo)
x 100 = (150.6/113.5) x 100 = 132.7 to 1 d.p.
Notice that we have taken account of the different quantities for wine, beer and
soft drinks by multiplying the unit prices by the corresponding quantities. This
process is called weighting.
We could have worked out what is called a simple
aggregative index by just
taking account of the unit prices as follows:
The simple aggregative index = ((3 + 6 + 0.84)/(2.5 + 4.5 +0.6)) x 100 = 129.5
to 1 d.p.
but it is not very useful for two reasons. Firstly, the quantities for different
drinks differ so much from each other and, secondly, the unit prices are
themselves for different quantities. We have single bottles of wine and soft
drinks but 6-packs of beer. If we had calculated the index using the two prices
for single cans of beer we would have got a different answer.
Expenditure is made up of two different elements, prices and quantities bought.
We'll suppose first that we are particularly interested in price changes over
time. In complicated situations, where we need to compare the prices of many
items over many different time intervals (such as for the Retail Price Index),
we work with the different prices, and use the quantities to weight them in
different ways for different index numbers.
Here is how we would calculate two more index numbers using the Tastynibbles
party example in each case.
The base
weighted price index or Laspeyre's
price index .
This index concentrates on measuring price changes from a base year. It is
called a base weighted index because we use the quantities purchased in the base
year (here 1990) to weight the unit prices in both years. Keeping the quantities
constant in this way means that any change in the calculated expenditure is due
solely to price changes.
The Laspeyre's price index is given by (pnqo/ poqo)
x 100.
In this particular case we have
pnqo =
(3 x 25) + (6 x 10) + (0.84 x 10) = 143.4 and
poqo =
(2.5 x 25) + (4.5 x 10) + (0.6 x 10) = 113.5
so Laspeyre's price index = (143.4/113.5) x 100 = 126.3 to 1 d.p.
Here's the table again so that you can check this.
|
The 1990 party |
The 2000 party |
| Drink |
Unit price |
Quantity |
Unit price |
Quantity |
|
po |
qo |
pn |
qn |
| wine |
£2.50 |
25 |
£3 |
30 |
| beer |
£4.50 |
10 |
£6.00 |
8 |
| soft drinks |
£0.60 |
10 |
£0.84 |
15 |
In practice, the Laspeyre's price index is usually calculated using price
relatives. For this method, we have to use the expenditures in the base year as
weights. This sounds more complicated but the reason we do this is that it is
easier to obtain data on expenditure than on actual quantities bought when we
are dealing with a large complicated index. For example, cost of living weights
are obtained by using sampling in the Survey of Household Expenditure. Indeed
for some elements of the cost of living expenses, 'quantities' don't even make
sense. You can't really talk about 'quantities' of public transport, for
example.
I've shown the table again below, this time including the base year expenditures
and the price relatives.
|
The 1990 party |
The 2000 party |
| Drink |
Unit price |
Quantity |
Expenditure |
Unit price |
Quantity |
Price relative |
|
po |
qo |
po x qo |
pn |
qn |
(pn/po) x 100 |
| wine |
£2.50 |
25 |
62.5 |
£3 |
30 |
120 |
| beer |
£4.50 |
10 |
45 |
£6.00 |
8 |
133.3 |
| soft drinks |
£0.60 |
10 |
6 |
£0.84 |
15 |
140 |
Here is the general rule for working out the base
weighted or Laspeyre's price index using price relatives.
Notice that cancelling the po above
and below on the top line and taking out the factor of 100 gives us
(pnqo/ poqo)
x 100 as before.
Here's how the calculation now goes for the Tastynibbles example.
Substituting in the general rule, we have
giving the same answer as before.
The base weighted index has the advantage that we only have to work out the base
year expenditures once. We can then use these in the calculation of the index in
any subsequent period. However, this index can be misleading in telling us what
is actually going on. For example, the fluctuations in fashion might have a
considerable impact on an index. Suppose that skirts were considered as a
separate item in a women's clothing manufacturer's index. The greatly increased
relative popularity of trousers would dramatically affect the quantities sold
and any index which used base year quantities from some time back would be
misleading.The next index that we consider avoids this particular problem.
The end
year weighted price index or Paasche's
price index
This uses the end year quantities as weights. We'll now calculate this for the
Tastynibbles parties. I've shown the table again below.
|
The 1990 party |
The 2000 party |
| Drink |
Unit price |
Quantity |
Unit price |
Quantity |
|
po |
qo |
pn |
qn |
| wine |
£2.50 |
25 |
£3 |
30 |
| beer |
£4.50 |
10 |
£6.00 |
8 |
| soft drinks |
£0.60 |
10 |
£0.84 |
15 |
The end
weighted or Paasche's price index is
given by (pnqn/ poqn)
x 100.
In this particular case we have
pnqn =
(3 x 30) + (6 x 8) + (0.84 x 15) = 150.6 and
poqn
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