Rule for finding the price relative
If we let p_{o} be the price in the base period and
let p_{n} be the price in the later period, then the
price relative for the price change between
these periods is given by (p_{n}/p_{o}) 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 1990 party | The 2000 party | |||
---|---|---|---|---|
Drink | Unit price | Quantity | Unit price | Quantity |
p_{o} | q_{o} | p_{n} | q_{n} | |
wine | £2.50 | 25 | £3 | 30 |
beer | £4.50 | 10 | £6.00 | 8 |
soft drinks | £0.60 | 10 | £0.84 | 15 |
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 = p_{o}q_{o} =
(2.5 x 25) + (4.5 x 10) + (0.6 x 10) = 113.5.
The party's cost in 2000 = p_{n}q_{n} =
(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
= (p_{n}q_{n}/
p_{o}q_{o}) 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.
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
(p_{n}q_{o}/
p_{o}q_{o}) x 100.
In this particular case we have
p_{n}q_{o} =
(3 x 25) + (6 x 10) + (0.84 x 10) = 143.4 and
p_{o}q_{o} =
(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 |
p_{o} | q_{o} | p_{n} | q_{n} | |
wine | £2.50 | 25 | £3 | 30 |
beer | £4.50 | 10 | £6.00 | 8 |
soft drinks | £0.60 | 10 | £0.84 | 15 |
The 1990 party | The 2000 party | |||||
---|---|---|---|---|---|---|
Drink | Unit price | Quantity | Expenditure | Unit price | Quantity | Price relative |
p_{o} | q_{o} | p_{o} x q_{o} | p_{n} | q_{n} | (p_{n}/p_{o}) 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's how the calculation now goes for the Tastynibbles example.
Substituting in the general rule, we have
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 1990 party | The 2000 party | |||
---|---|---|---|---|
Drink | Unit price | Quantity | Unit price | Quantity |
p_{o} | q_{o} | p_{n} | q_{n} | |
wine | £2.50 | 25 | £3 | 30 |
beer | £4.50 | 10 | £6.00 | 8 |
soft drinks | £0.60 | 10 | £0.84 | 15 |
The base weighted or Laspeyre's volume index is given by (p_{n}q_{o}/ p_{o}q_{o}) x 100 using the base period prices as weights.
The end weighted or Paasche's volume index is given by (p_{n}q_{n}/ p_{o}q_{n}) 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 |
p_{o} | q_{o} | p_{n} | q_{n} | |
wine | £2.50 | 25 | £3 | 30 |
beer | £4.50 | 10 | £6.00 | 8 |
soft drinks | £0.60 | 10 | £0.84 | 15 |