# Index Numbers

### BUILDING UP AN INDEX NUMBER

#### Introduction

In commercial and economic affairs (and in some others, too) there are some very important quantities which are too complex to be measured directly; such things, for example, as the “level of industrial production” or the “cost of living”. These are real enough things, but they are made up of a very large number of component parts, all affecting the main issue in different ways or to different extents. The index number notion is especially suited to dealing with such matters.

#### Simple Index

Index numbers make a comparison between a value (quantity or price) in the current period and the corresponding value in a base period.  All calculations are given in percentages without the % sign.

#### Price Relatives

We can get round this problem by using the ratio of prices of a given item rather than the actual prices themselves. The price of a pint of milk in Year 10 as a percentage of its price in Year 1 is 420.0 and is called the price relative for milk in Year 10 (Year 1 = 100).

Similarly, we can work out price relatives for the other items.

(Remember, all we are doing is making the current price into a percentage of the base year price.)

#### WEIGHTED INDEX NUMBERS (LASPEYRES AND PAASCHE INDICES)

You may think that the mean of relatives index is still not very satisfactory, in that all items are treated as of equal importance and no account has been taken of the different quantities of the items consumed. For instance, the average family is much more concerned about a 5c increase in the price of a loaf of bread than a 10c increase in the price of a drum of pepper, as far more bread is consumed than pepper.

If you look back at Table 9.2, you will see that we are, in fact, given the average weekly consumption of each item in Year 1 and Year 10. You can see that the consumption pattern, as well as the prices, has changed over the 10-year period. We are interested in calculating an index for prices, so we have to be careful not to over-emphasise the increase in prices by incorporating the changes in consumption.

##### Weighted Aggregative Index Numbers

We can adopt either of two approaches:

1. a) We can consider the consumption pattern in Year 1 as “typical” and:
• work out the total expenditure on the four items in Year 1; then,
• work out what the total expenditure would have been in Year 10 if the family had consumed at Year 1 levels; and finally,

express the sum in

as a percentage of

to form an index number.

##### Weighted Price-Relative Index Numbers

We can also form base-weighted or current-weighted price-relative index numbers. As before, we work out the price relatives for each commodity and as we now want to take into account the relative importance of each item in the family budget, we use as weight the actual expenditure on each item. The expenditure is used rather than the quantities consumed, to avoid exaggeration of variations arising from the change in consumption pattern rather than the change in price

#### QUANTITY OR VOLUME INDEX NUMBERS

You must not think that we are always concerned with price indices. Often we are interested in volume or quantity indices as, for instance, in the Index of Industrial production which seeks to measure the changes in volume of output in a whole range of industries over a period of time. We can calculate such quantity index numbers in exactly the same sort of way as we dealt with the price indices

#### THE CHAIN-BASE METHOD

In the chain-base method, the index for the current period is based on the last (i.e. the immediately preceding) period.

For example, if we are calculating an index for 2003, we use 2002 as the base year; then, when we come to calculate the index for 2004, we use 2003 as the base year; and so on. This system has the advantage that it is always up-to-date and it is easy to introduce new items or delete old ones gradually without much upset to the reliability of the index. Its disadvantage is that it cannot be used for making comparisons over long periods of time, as we are simply comparing each year with the immediately preceding year.

If we do need to make long-term comparisons when a chain-base index number is in use, then it is necessary to convert the indices from a chain base to a fixed base

#### DEFLATION OF TIME SERIES

These days we are all very familiar with the term “inflation”, as it impinges directly on our own lives to a greater or lesser extent, depending upon the country we live in.

We make such remarks as, “Things cost twice as much now as they did ten years ago!”, “We will need at least a 10% wage rise to keep up with inflation!”, and, “I don’t know where my money is going, everything costs so much nowadays”.

We are, of course, referring to the effects of a positive cost of living index. I am sure you will not remember a time when the cost of living actually fell.

A static index from one year to the next indicates no inflation and although in Rwanda we managed a 6% index figure not too long ago, we have not had a 0% index for many many years. There has therefore always, to all intents and purposes, been some degree of inflation in the economy.

As all workers naturally want at least to maintain their standard of living, they look for annual wage rises at least equal to the cost of living rise, measured by the index. Very often, they negotiate clauses into their wage agreements specifying a cost of living increase, or inflation-proof clause, without being specific or trying to anticipate the figure.

For the purposes of the examination, we are interested in how these vague statements can be measured. What we do is relate values backwards in the same manner as with indices.

##### Changing the Index Base-Year

To convert indices from an earlier to a later base year, divide all the indices by the index for the new base year. This is really a variation on the technique of chain-based indices except that we relate to one particular year rather than continuing to roll forward.

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