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CALCULATING THE CPI
4.5 The next stage in the process involves determining the types of bread for which price samples should be constructed. This is not as simple an exercise as might be imagined and relies heavily on the judgement of the prices statisticians. In reaching decisions about precisely which items to include in price samples, the prices statisticians need to strike a balance between the cost of data collection (and processing) and the accuracy of the index. Factors taken into account include the relative significance of individual items, the extent to which different items are likely to exhibit similar price behaviour, and any practical problems associated with measuring prices to constant quality.
4.6 In this example, a reasonable outcome would be to decide to construct price samples for items 1, 3, 5 and 6. Separate price samples would not be constructed for items 2 and 4 due to their relatively small market share. Price samples would also not be constructed for items 7 and 8 (bread rolls and specialty breads) as they would prove difficult to price to constant quality due to the tendency for these items to be sold by number rather than weight.
Elementary aggregates must have a price sample
4.7 The items for which it is decided to construct specific price samples are referred to as ‘elementary aggregates’. (There are approximately 1,000 elementary aggregates for each of the eight capital cities, or approximately 8,000 price samples at the national level.) The expenditure aggregates for the items that are not to be explicitly priced are reallocated across the elementary aggregates in such a way as to best preserve the representativeness of the price samples. In this example, this would be done in two stages. First, the expenditure aggregate for item 2 would be allocated to item 1 and the expenditure aggregate for item 4 would be allocated to item 3. In the second stage, the expenditure aggregates for items 7 and 8 would be allocated, on a proportional basis, across the four elementary aggregates. This process is illustrated in the following table.
Determining outlet types
4.9 Having settled on the items for which price samples are to be constructed, the next step is to determine the outlet types (respondents) from which prices will be collected. In order to accurately reflect changes in prices paid by households for bread, prices need to be collected from the various types of outlets from which households purchase bread. Data are unlikely to be available on the expenditure at the individual elementary aggregate level by type of outlet. It is more likely that data will be available for expenditure on bread in total by type of outlet. Suppose industry data indicates that supermarkets accounted for about 80% of bread sales and specialist bakeries the remainder. A simple way to construct the price sample for each elementary aggregate that is representative of household shopping patterns is to have a ratio of four prices from supermarkets to every specialist bakery price.
COLLECTING PRICE DATA
4.10 When price samples have been determined, ABS field staff determine from which individual supermarkets and specialist bakeries the prices will be collected. The individual outlets are chosen to be representative of the two types of outlets taking into account many perspectives. For example, the outlets should be representative of the socio–economic characteristics of the city. The prices are collected each quarter from the same respondents for the same items.
Selecting items to price
4.11 When a respondent is first enrolled in the collection process the field staff will determine, in conjunction with the outlet management, which specific items are best representative of each elementary aggregate. For example, at one outlet it might be decided that the 680gm sliced white sandwich loaf is best representative of white sandwich bread; at another outlet it might be a 700gm white toast sliced sandwich loaf.
4.12 An important part of the ongoing price collection process is the monitoring of the items for quality change. In the stylised bread example quality change could occur in various ways. A possible quality change would be a change in the size (weight) of the loaf of bread. In this case prices would be adjusted to derive a pure price for the item along the lines illustrated in the example in 2.51 above. Individual item prices are also compared with prices collected in the previous period to check their accuracy and to verify any large movements.
ESTIMATION OF PRICE MOVEMENTS FOR ELEMENTARY AGGREGATES
4.13 Price samples are constructed for the sole purpose of estimating price movements for each elementary aggregate. These estimates of price movements are required to revalue the expenditure aggregates to current period prices in much the same way as illustrated in the example on using price indexes (see 3.16-3.18 above). This is achieved by applying the period to period price movement to the previous period’s expenditure aggregate for each elementary aggregate. It provides an estimate of the cost of acquiring the base period quantity of the elementary aggregate in the current quarter.
Four options for calculating price movement
4.14 There is no single correct method for calculating the price movement for a sample of observations. Four commonly used methods are described below, using as an example, the price observations from two periods for multigrain bread.
4.15 The differences between the four methods involve choices as to:
4.16 The two commonly used forms of average are the arithmetic mean and the geometric mean. For a sample of n price observations, the arithmetic mean is the sum of the individual prices divided by the number of observations, while the geometric mean is the nth root of the product of the prices. For example, the arithmetic mean of 4 and 9 is 6.5, while the geometric mean is 6 (the geometric mean is always less than or equal to the arithmetic mean).
Relative of arithmetic mean of prices
4.17 Based on these options, one method is to construct a ratio of the arithmetic mean prices in the two periods. In the example below the arithmetic mean of prices in period 1 is $4.14 and in period 2 it is $4.17, giving a relative of 1.007 (4.17/4.14) or a percentage change of 0.7%. This method is called the ‘relative of arithmetic mean prices’ (RAP), sometimes referred to as the ‘Dutot’ index formula.
Arithmetic mean of price relatives
4.18 A second method is to calculate the price movement between periods for each individual item and then take the arithmetic average of these movements. The price movement for each item must be expressed in relative terms (i.e. period 2 price divided by period 1 price as shown in the 'Price relative' column). In the example below the arithmetic average of the price relatives is 1.021, a price change of 2.1%. This method is called the ‘arithmetic mean of price relatives’ (APR), sometimes referred to as the ‘Carli’ index formula.
4.19 A third method is to construct a ratio of the geometric mean of prices in each period. The geometric mean of the sample prices in period 1 is $4.14 and in period 2 it is $4.070 giving a relative of 1.017 (4.070/4.14) or a percentage change of 1.7%.
4.20 The fourth method is to calculate the geometric mean of the price movements for each individual item. Again, the price movements must be in the form of price relatives. In the example below, the geometric mean of the price relatives is 1.017, indicating a price increase of 1.7%, the same as using the ratio of the geometric mean of prices in each period.
4.21 In fact the geometric mean will always produce the same result whether the relative of mean prices or the mean of relative prices is used. These methods are simply referred to as the geometric mean (GM), sometimes called the ‘Jevons’ index formula.
Geometric mean is the preferred method
4.22 The method of calculating price change at the elementary aggregate level is important to the accuracy of the price index. The arithmetic average of price relatives (APR) approach has been shown to be more prone to (upward) bias than the other two methods. In line with various overseas countries, the ABS is using the geometric mean formula for calculating elementary aggregate index numbers where practical in the 16th series of the CPI. Where the geometric mean is not appropriate the relative of arithmetic mean prices (RAP) is used. The reasoning behind using geometric means is outlined below.
Geometric mean allows for substitution
4.23 At the elementary aggregate level of the index it is usually impractical to assign a specific weight to each individual price observation. The three formulas described above implicitly apply equal weights to each observation, although the bases of the weights differ. The geometric mean applies weights such that the expenditure shares of each observation are the same in each period. In other words the geometric mean formula implicitly assumes households buy less (more) of items that become more (less) expensive relative to the other items in the sample. The other formulas assume equal quantities in both periods (RAP) or equal expenditures in the first period (APR), with quantities being inversely proportional to first period prices. The geometric mean therefore appears to provide a better representation of household purchasing behaviour than the alternative formula in those elementary aggregates where there is likely to be high substitutability in consumption within the price sample.
Geometric mean not appropriate for all elementary aggregates
4.24 The geometric mean cannot be used to calculate the average price in all elementary aggregates. It cannot be used in cases where the price could be zero (i.e. the cost of a good or service is fully subsidised by the government). It is also not appropriate to use geometric means in elementary aggregates covering items between which consumers are unable to substitute. An example of this is local government rates where it is not possible to switch from a high rate area to a low rate area without physically moving location.
CALCULATING THE CURRENT COST OF THE BASKET
4.25 Once price movements are calculated for each elementary aggregate, they can be used to derive the expenditure aggregates that are then summed to derive the current cost of the basket. It is from the expenditure aggregates that index numbers are calculated at any level of the index. The stylised example above is continued to show the process for the Bread expenditure class.
4.26 The expenditure aggregates are revalued to period 2 prices by applying the movements between period 1 and period 2. The expenditure aggregate for the expenditure class Bread is the sum of the expenditure aggregates for the elementary aggregates comprising the expenditure class. Summing the elementary aggregates says that in period 2 it would cost $754.8m to buy the volume of Bread in period 1 that cost $740m. The price change for Bread between period 1 and 2 is simply the ratio of these expenditure aggregates, that is, a price increase of 2.0% (754.8/740). Thus if the price index for Bread was 100.0 in period 1, it would be 102.0 in period 2.
4.27 The derivation of the expenditure class movement as shown above is mathematically equivalent to a weighted average of the price movements for the individual elementary aggregates, that is, a weighted version of the mean of price relatives formula discussed above. In this case period 1 expenditure aggregates are the weights. The same formula is used at higher levels of the index.
4.28 Similar procedures are used to derive price movements at higher levels of the CPI. For example, the current period cost of purchasing items in the Bread and cereal products sub–group of the CPI is obtained by summing the current period expenditure aggregates of the expenditure classes Bread, Cakes and biscuits, Breakfast cereals and Other cereal products. The ratio of the current and previous period expenditure aggregates for the Bread and cereal products sub–group gives the price movement for the sub–group.
4.29 Points contributions (see 3.10–3.12 above) are also calculated using the expenditure aggregates. The current period points contribution of a CPI component, for example the expenditure class Bread, is the current period expenditure aggregate for Bread relative to the expenditure aggregate for the All groups CPI multiplied by the current period All Groups index number.
4.30 The CPI publication does not show the expenditure aggregates, but rather the index numbers derived from the expenditure aggregates. Expenditure aggregates vary considerably in size and showing them would make the publication difficult to read and interpret. The published index numbers and points contributions are a convenient presentation of the information.
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