# Consumer price index calculation in practice

## Introduction

10.1 The Consumer Price Index (CPI) can be thought of as a basket of goods and services which is notionally purchased each quarter. The total cost (or price) of the basket changes from one quarter to the next as prices change. Of the various ways in which a CPI could be described, this description conforms most closely with the procedures actually followed.

10.2 The CPI can be thought of as being constructed in six major steps:

(i) subdividing the total expenditure into individual items for which price samples can be selected;

(ii) collecting price data;

(iii) estimating price movements for elementary aggregates;

(iv) calculating the current period cost of the basket;

(v) calculating the weighted average of eight capital cities; and

(vi) calculating index numbers and points contribution.

10.3 Consumer Price Index calculation in practice provides a stylised account of the steps above. It also indicates how analytical indexes are calculated, and describes the ABS rounding practices.

## Subdividing the basket

### Expenditure aggregates

10.4 Based mainly on the results of the Household Expenditure Survey (HES), estimates are obtained for the total annual expenditure of private households in each capital city for each of the 87 expenditure classes in the CPI. As these estimates are for the expenditure of households in aggregate, they are referred to as expenditure aggregates.

10.5 Expenditure aggregates are derived for well-defined categories of household expenditure (e.g. flooring), but are still too broad to be of direct use in selecting samples of products for pricing. For this purpose, expenditure aggregates need to be subdivided into as fine a level of commodity detail as possible. As the HES is generally not designed to provide such fine level estimates, it is necessary to supplement the HES data with information from other sources such as other official data collections and industry data. The processes involved are illustrated below using a hypothetical example for the Carpets and other floor coverings expenditure class of the CPI.

10.6 Suppose that, based on information reported in the HES, the annual expenditure on flooring by all private households in a particular city is estimated at $8 million. Further, suppose that some industry data exists on the market shares of various types of flooring. In combination, these two data sources can be used to derive expenditure aggregates at a much finer level of detail than that available from the HES alone. The hypothetical results are shown in Table 10.1.

10.7 The next stage in the process involves determining the types of flooring for which price samples should be constructed. This is not a simple exercise and relies on the judgement of the Prices Statisticians. In reaching decisions about precisely which items to include in price samples, a balance needs to be struck between the cost of collecting and processing the data, and the accuracy of the index. Factors taken into account include the significance of individual items, the extent to which different items are likely to exhibit similar price behaviour, and any practical problems with measuring prices to constant quality.

Type of floor covering | Market share | HES data | Derived expenditure aggregates | |
---|---|---|---|---|

% | $'000 | $'000 | ||

1 | Laminate | 17 | 1 360 | |

2 | Timber | 19 | 1 520 | |

3 | Tiles | 19 | 1 520 | |

4 | Carpets | 22 | 1 760 | |

5 | Rugs | 3 | 240 | |

6 | Vinyl | 10 | 800 | |

7 | Other flooring | 10 | 800 | |

Total Carpets and other floor coverings expenditure class | 100 | 8 000 | 8 000 |

10.8 In this example, a reasonable outcome would be to construct pricing samples for types 1, 2, 3, and 4. Separate price samples would not be constructed for types 5, 6 or 7 because of their small market share relative to the other types.

### Elementary aggregates must have a price sample

10.9 When no more information is available to disaggregate the expenditure values any further, the resulting product groupings are called elementary aggregates. Each elementary aggregate will contain its own price sample. Ideally, all the products in an elementary aggregate (and there should only be a few) would be homogeneous goods or services and would be substitutes for each other. In the Australian CPI, there are approximately 800 elementary aggregates for each of the eight capital cities and approximately 6,400 price samples nationally. The expenditure for the items that are not explicitly priced are reallocated across the elementary aggregates of closely related goods or services under the assumption that the price movements for these items are similar.

10.10 In the flooring example, the reallocation is carried out in two stages. First, the expenditure for rugs is added to carpets resulting in an elementary aggregate for soft floor coverings (as they are likely to experience similar price movements). In the second stage, the expenditure for vinyl and other flooring which have no closely matching characteristics with any of the other types of flooring, would be allocated proportionally across the remaining elementary aggregates under the assumption that the average movement in prices for all other flooring types is the most representative estimate of price change. The derived expenditure for vinyl and other flooring is 25% of the remaining elementary aggregates. Each of the remaining elementary aggregates are therefore increased by 25% in the second stage. This gives an expenditure aggregate for each elementary aggregate. The outcome of this process is presented in Table 10.2.

10.11 In summary, the rationale for this allocation is as follows. Price behaviour of type 5 (rugs) is likely to be best represented by the price behaviour of type 4 (carpets). The price behaviour for types 6 (vinyl) and 7 (other flooring) are likely to be best represented by the average price behaviour of all other flooring types.

Type of floor covering | Initial | Stage 1 | Stage 2 | Elementary aggregate | |
---|---|---|---|---|---|

$'000 | $'000 | $'000 | |||

1 | Laminate | 1 360 | 1 360 | 1 700 | Laminate |

2 | Timber | 1 520 | 1 520 | 1 900 | Timber |

3 | Tiles | 1 520 | 1 520 | 1 900 | Tiles |

4 | Carpets | 1 760 | 2 000 | 2 500 | Soft floor coverings |

5 | Rugs | 240 | |||

6 | Vinyl | 800 | 800 | ||

7 | Other flooring | 800 | 800 | ||

Total Carpets and other floor coverings expenditure class | 8 000 | 8 000 | 8 000 |

### Determining outlet types

10.12 The next step is to determine the outlet types (respondents) from which the prices will be collected. In order to accurately reflect changes in prices paid by households for flooring, prices need to be collected from the types of outlets from which households normally purchase flooring. Data are unlikely to be available on the expenditures at the individual elementary aggregate level by type of outlet. It is more likely that data will be available for expenditure on flooring in total by type of outlet. Suppose industry data indicate that specialty stores account for about 75% of flooring sales, and department stores the remainder. A simple way to construct a pricing sample for each elementary aggregate that is representative of household shopping patterns is to have a ratio of three specialty stores for every department store.

## Collecting price data

### Selecting respondents

10.13 When the pricing samples are constructed, ABS staff decide from which individual outlets the prices will be collected. The respondents are chosen to be representative of the types of outlets (in the example above, department and specialty stores) taking into account the demographic characteristics of the city and the number of observations required for the sample. Prices are collected from any particular respondent on the same day in each collection period (e.g. the second Monday of the first month of the quarter).

### Selecting items to price

10.14 A pricing sample may contain specifications to either national standards, respondents standards or a combination of both (see Sampling of this manual). When a pricing sample contains respondent standard specifications, ABS staff will decide which specific items are most representative of the required type of product. Usually this is done by consulting with the manager of the outlet. Using the flooring example above, at one outlet they might decide that porcelain tiles are the most representative tiles, but at another outlet it might be ceramic tiles. Once selected, the same item will be priced at that respondent so long as it remains the most representative example of the product.

10.15 An important part of the price collection process is the continual monitoring of the items for quality change. In the flooring example, quality change could occur, for example, with a change in the material used to make the item. In this case, the price movement attributable to the change in material would be removed to derive a pure price movement for the flooring.

## Estimating price movements for elementary aggregates

10.16 Price relatives are calculated for each item in the sample. Price relatives are the ratio of the current period price and the reference period price (see paragraph 4.16). In samples where items are determined to be substitutable the geometric mean of these price relatives is used in the calculations. The ratio of the current period’s geometric mean of price relatives to the previous period’s geometric mean of price relatives provides the change in the average price for the elementary aggregate. The alternative is to use the relative of average prices (see Price index theory of this manual). Using the hypothetical flooring example, Table 10.3 shows price relatives being used to estimate the price movement for laminate flooring. These estimates of price movements are used to revalue the expenditure aggregates to current period prices by applying the period to period price movement to the previous period's expenditure aggregate for each elementary aggregate. The updated expenditure aggregate provides an estimate of the cost of acquiring the reference base quantity of the elementary aggregate’s products in the current period. This new aggregate can be referred to as the 'price updated' aggregate.

Price relative in | |||
---|---|---|---|

Laminate | Period 1 | Period 2 | Price movement % |

Specialty Store A | 1.025 | 1.030 | 0.5 |

Specialty Store B | 1.030 | 0.950 | -7.8 |

Specialty Store C | 0.980 | 0.965 | -1.5 |

Department Store | 1.100 | 1.250 | 13.6 |

Geometric mean | 1.033 | 1.042 | 0.9 |

## Calculating the current cost of the basket

10.17 The price updated expenditure aggregates for the elementary aggregates are then summed to derive the current cost of the basket of goods and services (or any portion of the basket). Index numbers are calculated from the expenditure aggregates at every level of the index. Table 10.4 shows the calculation of the expenditure aggregate for the total of flooring (an expenditure class in this example) from the elementary aggregates.

Expenditure aggregate | Price movement | Expenditure aggregate | |
---|---|---|---|

$'000 | % | $'000 | |

Elementary aggregate | Period 1 | Period 1 to Period 2 | Period 2 |

Laminate | 1 700 | 0.9 | 1 715 |

Timber | 1 900 | 3.3 | 1 963 |

Tiles | 1 900 | 0.0 | 1 900 |

Soft floor coverings | 2 500 | 1.8 | 2 545 |

Total Carpets and other floor coverings expenditure class | 8 000 | 1.5 | 8 123 |

## Calculating the weighted average of eight capital cities

10.18 The ABS compiles the Australian CPI on a separate basis for each capital city based on the acquisition of goods and services by the resident population of that city. The ABS also constructs the equivalent of a national index at the All groups CPI, group, sub-group and expenditure class level, which is published as the weighted average of the eight capital cities. The construction of a CPI weighted average of eight capital cities series is demonstrated below using a stylised example for the Carpets and other floor coverings expenditure class in three cities.

10.19 A base period expenditure aggregate is calculated for each city at the group, sub-group and expenditure class level, using information primarily sourced from the HES on the number of households in each of the three cities and the average weekly household expenditure for specific items. These weekly expenditure aggregates are converted to yearly expenditure aggregates by multiplying the final weekly expenditure aggregates by the number of weeks in the year. This process is demonstrated in Table 10.5 for the Carpets and other floor coverings expenditure class in three cities.

Item | City A | City B | City C | Weighted average of three cities | |
---|---|---|---|---|---|

Households(b) | no. | 8 000 | 3 000 | 2 000 | 13 000 |

Average weekly household expenditure on flooring(b) | $ | 19.18 | 15.98 | 13.42 | |

Weeks in a year | no. | 52.143 | 52.143 | 52.143 | |

Yearly expenditure aggregate for flooring(c) | $'000 | 8 000 | 2 500 | 1 400 | 11 900 |

Weight contribution to three cities(d) | % | 67 | 21 | 12 | 100 |

- Any discrepancies between totals and sums are due to rounding.
- Information sourced from a Household Expenditure Survey (HES).
- Calculated as the product of the number of households, the average weekly household expenditure and weeks in a year.
- Calculated as the ratio of each city's expenditure aggregate relative to the weighted average of three cities' expenditure aggregate.

10.20 The expenditure aggregates for Carpets and other floor coverings in each city are price updated from period 1 to period 2 by the price change in the relevant price samples of the Carpets and other floor coverings elementary aggregates for each city (such as tiles, soft floor coverings, etc. as described in paragraphs 10.16 and 10.17).

10.21 The expenditure aggregates for Carpets and other floor coverings in each city are summed to arrive at a weighted expenditure aggregate for all three cities in period 1 and period 2. The price movement of the weighted average of three cities is calculated from the change in the weighted expenditure aggregates for the three cities between period 1 and period 2.

10.22 The calculation of price change for the weighted average of three cities is demonstrated in Table 10.6. For Carpets and other floor coverings, the period 1 expenditure aggregate for the weighted average of three cities is $11,900,000. In period 2, the Carpets and other floor coverings expenditure aggregate for the weighted average of three cities is now ($8,123,000 + $2,470,000 + $1,407,000), which is equal to $12,000,000. Using the above expenditure aggregates, the price change for Carpets and other floor coverings for the weighted average of three cities is calculated to be 0.8% from period 1 to period 2:

Percentage change from period 1 to period 2 = ($12,000,000 - $11,900,000) / $11,900,000 x 100 = 0.8%.

Item | City A | City B | City C | Weighted average of three cities(b) | |
---|---|---|---|---|---|

Period 1 expenditure aggregates | |||||

Carpets and other floor coverings | $'000 | 8 000 | 2 500 | 1 400 | 11 900 |

All other items | $'000 | 42 000 | 22 500 | 13 600 | 78 100 |

All groups CPI | $'000 | 50 000 | 25 000 | 15 000 | 90 000 |

Weight contribution to three cities(c) | % | 55 | 28 | 17 | 100 |

Price change | |||||

Carpets and other floor coverings | % | 1.5 | -1.2 | 0.5 | 0.8 |

All other items | % | 2.1 | 2.4 | 3.5 | 2.4 |

All groups CPI | % | 2.0 | 2.0 | 3.2 | 2.2 |

Period 2 expenditure aggregates | |||||

Carpets and other floor coverings | $'000 | 8 123 | 2 470 | 1 407 | 12 000 |

All other items | $'000 | 42 882 | 23 040 | 14 076 | 79 998 |

All groups CPI | $'000 | 51 005 | 25 510 | 15 483 | 91 998 |

Weight contribution to three cities(c) | % | 55 | 28 | 17 | 100 |

- Any discrepancies between totals and sums are due to rounding.
- The expenditure aggregates for each city are summed at the All groups CPI, group, sub-group and expenditure class level to arrive at an expenditure aggregate for the weighted average of three cities.
- It is important to note that the weight of each city relative to the weighted average of three cities can change from period to period, depending on the city's price movement relative to the other cities.

10.23 This process is carried out at the All groups CPI, group, sub-group and expenditure class level in the index. The relative contribution of any city to the price change for the weighted average of three cities will be determined by the ratio of the individual city expenditure aggregate to the weighted expenditure aggregate for all three cities.

## Calculating index numbers and points contributions

10.24 Table 10.7 shows the calculation of index numbers and points contribution. It is assumed that index numbers already exist for the link period (September quarter 2017 for the 17th series CPI) and period 1. Assume the expenditure aggregates for Furniture have been calculated using the same method as that for Carpets and other floor coverings so that they can be added and a movement calculated for the Furniture and furnishings subgroup. Similarly, assume the expenditure aggregates for period 2 have been calculated for other Furnishings, household equipment and services groups and Non-Furnishings, household equipment and services groups so that expenditure aggregates can be calculated for the Furnishings, household equipment and services group and the All groups CPI.

10.25 When a price index has not been linked, indexes for any component can be calculated simply by dividing the current period expenditure aggregate by its expenditure aggregate in the reference period (when the index is set to 100.0). The index numbers must be calculated from

\(I_{LP}\times {\frac {V_{CP}}{V_{LP}}}\space \space \space \space \space (10.1)\)

where \(I_{LP}\) is the index number in the link period (September quarter 2017 for the 17th series CPI), and \(V_{CP}\) and \(V_{LP}\) are the expenditure aggregates in the current period and link periods respectively. Using the example in Table 10.7,

\(I_{LP}\) = 108.0 (Index number for Carpets and other floor coverings in the link period)

\(V_{CP}\) = $8,123,000 (Expenditure aggregate for Carpets and other floor coverings in the current period)

\(V_{LP}\) = $6,500,000 (Expenditure aggregate for Carpets and other floor coverings in the link period)

Thus the index number for Carpets and other floor coverings in period 2 is 108.0 x $8,123,000 / $6,500,000 = 135.0.

10.26 Points contributions allow users to understand how much each component contributes to the overall price movement. Points contributions are calculated using the expenditure aggregates. In any period, the points contribution of a component to the All groups CPI index number is calculated by multiplying the All groups CPI index number for the period by the expenditure aggregate for the component in that period, and dividing by the All groups CPI expenditure aggregate for that period. This can be stated algebraically as

\(I^{AG}_t\times {\frac {V^i_t}{V^{AG}_t}}\space \space \space \space \space (10.2)\)

where \(I^{AG}_t\) is the index for the All groups CPI in period \(t\), \(V^i_t\) is the expenditure aggregate for component \(i\) in period \(t\) and \(V^{AG}_t\) is the expenditure aggregate for the All groups CPI in period \(t\).

10.27 In the example in Table 10.7, the points contribution for Carpets and other floor coverings in period 2 is calculated as 144.3 x ($8,123,000 / $146,066,000) = 8.02.

10.28 The change in index points contribution for a component between any two periods is found by simply subtracting the points contribution for the previous period from the points contribution for the current period. For example, the change in index points contribution for Carpets and other floor coverings between periods 1 and 2 is 8.02 - 7.90 = 0.12. This means that between periods 1 and 2, Carpets and other floor coverings contributed 0.12 index points to the overall increase in the All Groups CPI of 5.2 (144.3 - 139.1) index points.

10.29 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. Index numbers and points contributions are a better way to present the information.

Link period | Period 1 | Period 2 | |
---|---|---|---|

Expenditure aggregates ($'000) | |||

ALL GROUPS CPI | 124 500 | 140 800 | 146 066 |

Furnishings, household equipment and services | 34 500 | 42 800 | 44 066 |

Furniture and furnishings | 14 500 | 17 700 | 18 213 |

Furniture | 8 000 | 9 700 | 10 090 |

Carpets and other floor coverings | 6 500 | 8 000 | 8 123 |

Other Furnishings, household equipment and services sub-groups | 20 000 | 25 100 | 25 853 |

Non-Furnishings, household equipment and services groups | 90 000 | 98 000 | 102 000 |

Movement in expenditure aggregates (period 1 to period 2) | |||

ALL GROUPS CPI | 1.131 | 1.037 | |

Furnishings, household equipment and services | 1.241 | 1.030 | |

Furniture and furnishings | 1.221 | 1.029 | |

Furniture | 1.213 | 1.040 | |

Carpets and other floor coverings | 1.231 | 1.015 | |

Other Furnishings, household equipment and services sub-groups | 1.255 | 1.030 | |

Non-Furnishings, household equipment and services groups | 1.089 | 1.041 | |

Index numbers | |||

ALL GROUPS CPI | 123.0 | 139.1 | 144.3 |

Furnishings, household equipment and services | 115.0 | 142.7 | 146.9 |

Furniture and furnishings | 110.0 | 134.3 | 138.2 |

Furniture | 113.0 | 137.0 | 142.5 |

Carpets and other floor coverings | 108.0 | 132.9 | 135.0 |

Other Furnishings, household equipment and services sub-groups | 117.0 | 146.8 | 151.2 |

Non-Furnishings, household equipment and services groups | 125.0 | 136.1 | 141.7 |

Points contribution | |||

ALL GROUPS CPI | 123.0 | 139.1 | 144.3 |

Furnishings, household equipment and services | 34.08 | 42.28 | 43.53 |

Furniture and furnishings | 14.33 | 17.49 | 17.99 |

Furniture | 7.90 | 9.58 | 9.97 |

Carpets and other floor coverings | 6.42 | 7.90 | 8.02 |

Other Furnishings, household equipment and services sub-groups | 19.76 | 24.80 | 25.54 |

Non-Furnishings, household equipment and services groups | 88.92 | 96.82 | 100.77 |

Note: It is assumed the link period precedes period 1.

## Secondary indexes

10.30 The following separate inflation series are currently published to assist users to analyse the CPI:

- All groups CPI, seasonally adjusted;
- Underlying trend series, 'Trimmed mean' and 'Weighted median';
- International trade exposure series, Tradables component;
- International trade exposure series, Non-tradables component;
- All groups CPI, goods component;
- All groups CPI, services component;
- All groups CPI including Deposit and loan charges (indirect charges);
- Market goods and services excluding 'volatile items';
- All groups CPI excluding (various).

10.31 These are called secondary indexes as they use the same weights (or expenditure aggregates) as the CPI, and are compiled by summing the appropriate expenditure aggregates. For example, in Table 10.7, the starting point for compiling an index for All groups CPI excluding Furniture and furnishings would be to add up the expenditure aggregates for the Other Furnishings, household equipment and services sub-groups, and Non-Furnishings, household equipment and services groups and then calculate index values as described previously. For more information on the analytical series published, see Appendix 2.

## Tertiary indexes

10.32 A further range of analytical indexes are compiled from the price samples collected for the CPI. Price indexes compiled under the outlays approach (see Purposes and uses of consumer price indexes of this manual) are published quarterly for four household types: employee households; age pensioner households; other government transfer recipient households; and self-funded retiree households. The ABS also publishes the Pensioner and Beneficiary Living Cost Index (PBLCI), which combines the age pensioner and other government transfer recipient household types. These indexes, unlike the secondary indexes, have their own weighting patterns. For each component in the household type indexes, the movement in the corresponding CPI index is used to update the expenditure aggregate and index number for the population sub-group. The purpose of the population sub-group indexes is to show any differences in the aggregated price changes faced by each of the four demographic groups arising from their differing expenditure patterns. For further information, see The system of price statistics of this manual or the explanatory notes for the publication Selected Living Cost Indexes, Australia (cat. no. 6467.0).

## Consumer Price Index rounding conventions

10.33 To ensure consistency from one publication to the next, the ABS uses a set of rounding conventions or rules for calculating and presenting the results. These conventions strike a balance between maximising the usefulness of the information for analytical purposes, and retaining the underlying precision of the estimates. These conventions need to be taken into account when CPI data are used for analytical or any other purpose.

10.34 Index numbers are always published relative to a base of 100.0. Index numbers and percentage changes are always published to one decimal place, and the percentage changes are calculated from the rounded index numbers. An exception to this are the Underlying trend series 'Trimmed mean' and 'Weighted median', which have index numbers published to four decimal places. Index numbers for periods longer than a single quarter (e.g. for financial years) are calculated as the simple arithmetic average of the rounded quarterly index numbers in that period.

10.35 Points contributions are published to two decimal places, except the All groups CPI which is published to one decimal place. Change in points contributions is calculated from the rounded points contributions. Rounding differences can arise in the points contributions where different levels of precision are used.