Title: [Notes] - Weighted versus Simple Index Post by: sperl on Oct 8, 2012 [html]
Index numbersWhat 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:
Rule for finding the price relative 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
Now we'll use this data to show how to work out various index numbers.
The expenditure index 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
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. 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 .
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.
Here is the general rule for working out the base weighted or Laspeyre's price index using price relatives. (pnqo/ poqo) x 100 as before.
Here's how the calculation now goes for the Tastynibbles example. 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 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 poqnPowered by SMF | SMF © 2015, Simple Machines |