Index numbers

The businessman blob

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 = (2.5 x 30) + (4.5 x 8) + (0.6 x 15) = 120
so Paasche's price index = (150.6/120) x 100 = 125.5.
We have now found out how to calculate two different price indexes to give us a measure of the fluctuations in price from a base year. But suppose the prices remain relatively stable and it is the quantities of items which are changing. In such circumstances, it may be more useful to calculate an index based on quantities, using prices as weights. The working out is then very similar to the calculations for the two price indexes.

The base weighted or Laspeyre's volume index is given by (pnqo/ poqo) x 100 using the base period prices as weights.

The end weighted or Paasche's volume index is given by (pnqn/ poqn) x 100 using the end period prices as weights.

Calculate these for yourself for the Tastynibbles party data. I have put in the table again here for your convenience.

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

You should find that Laspeyre's volume index is 105.7 to 1 d.p. and Paasche's volume index is 105.0 to 1 d.p.

One problem in the construction of any index number is choosing a suitable base period. We want a base where prices (or volumes) were not unnaturally high or low. An example of this would be if bad weather had caused an extreme shortage in a particular crop which then led to a very high price for it. Also, people are not happy with a base period which is too far in the past. Furthermore, tastes and availability can change a great deal over time so such an index could be seriously misleading. One way sometimes used to avoid these problems is to use a chain-based system where, in calculating successive index numbers, the base used is the previous period. A chain-based index number is particularly suited for period by period comparisons, but a fixed-base index number makes it easier to compare the movement of prices over time.
the blob at the tastynibbles party
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