Index numbers
- What are index numbers?
- Base-weighted indices
- Current-weighted indices
- Difference between base-weighted and current-weighted indices
- Chaining or splicing index numbers
- Effects of reweighting/out-of-date weights
- Convergence of indices
What are index numbers?
Index numbers are values expressed as a percentage of a single base figure. For example, if annual production
of a particular chemical rose by 35%, output in the second year was 135% of that in the first year. In index terms,
output in the two years was 100 and 135 respectively.
Index numbers have no units. Chemical production in the second year is referred to as 135, not 135 tonnes or 135%. The advantages are that distracting units are avoided and changes are easier to assess by eye. The arithmetic is straightforward, as shown in Table 1.
Composite indices and weighting.
Frequently two or more indices are combined to form one composite index. For example, indices of consumer
spending on food and on all other items might be combined into one index of total spending.
Base-weighted indices
The most straightforward way of combining indices is to calculate a weighted average using the same weights
throughout. This is known as a base-weighted index, or sometimes a Laspeyres index after the German economist
who developed the first one. The following is an example of a base-weighted price index for single-person
household consumption of wine and cheese each week.
Where prices are in, say, dollars and quantities are litres of wine/kilos of cheese:
Weekly expenditure in 1990
= (1990 quantity of wine * 1990 price of wine)
+ (1990 quantity of cheese * 1990 price of cheese)
= (5 * 9.00) + (2 * 5.00)
= 45.00 + 10.00 = 55.00
Weekly expenditure in 1995, based on 1990 quantities
= (1990 quantity of wine * 1995 price of wine)
+ (1990 quantity of cheese * 1995 price of cheese)
= (5 * 10.50) + (2 * 8.00)
= 52.50 + 16.00 = 68.50
Index number for 1990 = 55.00/55.00 * 100 = 100.0
Index number for 1995 = 68.50/55.00 * 100 = 124.5
Current-weighted indices
The problem with weighted averages is that weights usually need revising from time to time. With the consumer
price index, spending habits change because of variations in relative cost, quality, availability, and so on.
One way to proceed is to calculate a new set of current weights at regular intervals, and use these to derive a
single long-term index. This is known as a current-weighted index, or occasionally a Paasche index, after its
founder. The following is an example of a current-weighted price index for single-person household
consumption of wine and cheese each week.
Where prices are in, say, dollars and quantities are litres of wine/kilos of cheese:
Weekly expenditure in 1990, based on 1995 quantities
= (1995 quantity of wine * 1990 price of wine)
+ (1995 quantity of cheese * 1990 price of cheese)
= (6 * 9.00) + (3 * 5.00)
= 54.00 + 15.00 = 69.00
Weekly expenditure in 1995
= (1995 quantity of wine * 1995 price of wine)
+ (1995 quantity of cheese * 1995 price of cheese)
= (6 * 10.50) + (3 * 8.00)
= 63.00 + 24.00 = 87.00
Index number for 1990 = 69.00/69.00 * 100 = 100.0
Index number for 1995 = 87.00/69.00 * 100 = 126.1
Difference between base-weighted and current-weighted indices
Neither is perfect. Base-weighted indices are simple to calculate but they tend to overstate changes over time.
Current-weighted indices are more complex to produce and they understate long-term changes.
Current-weighted price indices reflect changes in both prices and relative volumes, while base-weighted versions record price changes only.
Mathematically, there is no ideal method for weighting indices; expediency usually rules. Most commonly used indices are a combination of base-weighted and current-weighted. A new set of weights might be introduced every five years or so and the new index then spliced or chained to the old index.
Chaining or splicing index numbers
Step 1) Identify one period when there are figures for both indices; 1992 in Table 1.
Step 2) For this period, divide the new figure by old figure; 83 / 133 = 0.62.
Step 3) Multiply all old figures by the result; each figure in column C = figure in column A * 0.62.
Step 4) Put the rebased data with the new figures to create one long run of data.
Effects of reweighting/out-of-date weights
To show the effects of reweighting, consider GDP (total output) based on 1990 weights when, say, manufacturing accounted
for half of all economic activity. If in 1990 manufacturing grew by 6% while all other activity was static, initial
1990 figures showed total GDP rising by 6 * 0.50 = 3%. By 1995 the results of a major survey were available and
GDP from 1988 was reweighted to take account of the fact that the manufacturing sector had shrunk to a mere 10%
of total GDP. As a result the revised figure for total growth in 1990 was 6 * 0.10 = 0.6%.
This is obviously an extreme example, but index numbers can easily become distorted if one item is much less or much more significant than the others. For example, demand tends to grow most rapidly for goods and services which increase least in price, and so on rebasing these items are allocated larger relative weights.
When looking at index numbers it is a good idea to check when they were last rebased and ask whether any component is increasing or decreasing in relative importance. The same approach should be taken to constant price series, because these are essentially index numbers with a base value other than 100.
Convergence of indices
Look out also for illusory convergence on the base. Two or more series will always meet at the base period because
that is where they both equal 100 (see Figure). This can be highly misleading. When you encounter indices on a
graph, the first thing to do is check where the base is located.
Related topics:
Measuring changes
