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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 in the last topic of Chapter 3. This is achieved by applying 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

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 price observations from two periods for multigrain bread.

			Price observations in
			Period 1	Period 2	Price	Estimates of
			$	$	relative	sample price
			(a)	(b)	(b) / (a)	movement
	Outlet data
		Supermarket A	1.50	1.80	1.200	. .
		Supermarket B	1.60	1.90	1.188	. .
		Supermarket C	1.85	1.50	0.811	. .
		Supermarket D	1.75	1.50	0.857	. .
		Hot bake	2.00	2.20	1.100	. .
	Average prices
		Arithmetic mean	1.74	1.78	. .	. .
		Geometric mean	1.73	1.76	. .	. .
	Four methods of calculating price movement
	Relative of average prices
		Arithmetic mean	. .	. .	. .	1.023
		Geometric mean	. .	. .	. .	1.017
	Average of price relatives
		Arithmetic mean	. .	. .	. .	1.031
		Geometric mean	. .	. .	. .	1.017

• whether the price movement for the sample is calculated as the average of each period’s prices or as the average of price movements between periods for each item; and

• the type of average used.

Relative of arithmetic mean prices

Based on these options, one method is to construct a ratio of the arithmetic average prices in the two periods. In the above example the arithmetic average of prices in period 1 is $1.74 and in period 2 it is $1.78, giving a relative of 1.023 (1.78/1.74) or a percentage change of 2.3%. This method is called the ‘relative of arithmetic mean prices’ (RAP), sometimes referred to as the ‘Dutot’ index formula.

Arithmetic mean of price relatives

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 second column from the right in the above table). In the example above the arithmetic average of the price relatives is 1.031, a price change of 3.1%. This method is called the ‘arithmetic mean of price relatives’ (APR), sometimes referred to as the ‘Carli’ index formula.

Geometric mean

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 $1.73 and in period 2 it is $1.76 giving a relative of 1.017 (1.76 / 1.73) or a percentage change of 1.7%.

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 above example, 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.

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. Both approaches are simply referred to as the geometric mean (GM), sometimes called the ‘Jevons’ index formula.

Geometric mean is the preferred method

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 13th 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

At the elementary aggregate level of the index it is usually impractical to assign a specific weight to each individual price observation. The three formulae 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 formulae 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

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.