Compiling the Primary Price indexes

Once a price movement for the elementary aggregate is determined, the resulting C-index price movement is used to price update the value aggregate associated with the elementary aggregate. The resulting measure is known as the price updated value aggregate (or current period value aggregate).

For a given elementary aggregate:

\(VA^t_{EA}=\frac{I^t_C}{I^{t-1}_C}\times VA^{t-1}_{EA}\)

where \(VA^t_{EA}\) is the current period value aggregate for the elementary aggregate in period \(t\), \(VA{_{EA}}^{t-1}\) is the previous period value aggregate, and \(I^t_C\) and \(I_C^{t-1}\)are respectively the current and previous period C-indexes for the elementary aggregate.

The price updated value aggregate is then used to determine the current period P-index for the elementary aggregate.

\(I^t_{P,EA}=\frac{VA^t_{EA}}{VA^{LINK}_{EA}}\times I^{LINK}_{P,EA}\)

where \(I^t{_p}\) is the current period P-index for the elementary aggregate in period \(t\), \(VA_{EA}{^{LINK}}\)is the link period value aggregate for the elementary aggregate and \(I^{LINK}{_{P,EA}}\) is the link period P-index for the elementary aggregate.

Once the current period value aggregates for all elementary aggregates are determined, the current period value aggregates for all higher level components of the index structure are calculated by summing the price updated value aggregates of their components.

Current period price indexes for any component in the aggregation structure are then calculated by price updating the link period P-index for the component. That is, for any component, the current period P-index is given by:

\(I^t_p=\frac{VA^t}{VA^{LINK}}\times I^{LINK}_{P}\)

where \(I^t{_P}\) is the current period P-index in period \(t\), \(VA^t\) is the current period value aggregate for the component, \(VA^{LINK}\) is the link period value aggregate for the component and \(I_P{^{LINK}}\) is the link period P-index for the component of the index (or aggregation) structure.

Points contributions are also calculated using the value aggregates. In any period, the points contribution of a component to the top level is calculated by multiplying the root index number for the period by the value aggregate for the component in that period and dividing by the root value aggregate for that period. This can be stated algebraically as:

\(PC^t_i=I^t{_{P,ROOT}} \times \frac{VA^t_i}{VA^t_{ROOT}}\)

where \(PC^t{_i}\) is the points contribution for component \(i\) in period \(t\), \(I^t{_{P,ROOT}}\) is the P-index for the root in period \(t,VA^t{_i}\) is the value aggregate for component \(i\) in period \(t\) and \(VA^t{_{ROOT}}\) is the value aggregate for the root of the index in period \(t.\)

Changes in points contribution for a component of a price index give an assessment of the component’s contribution to net price change. However, such a comparison is limited to periods between linking of price indexes. Comparisons of a component’s contribution to the index that cross a link period are comparing contributions on different weighting bases and therefore do not measure the contribution to net price change; any attempt at such comparison will confound change of weight with change of price.

Calculation of upper level price indexes is illustrated in Table 4.2. This table shows an input price index where products are classified by source (domestic and imported) and then by type of product. In Part 1 the P-index for period 1 is calculated. Part 2 shows the calculation of the percentage movement in the elementary aggregate C-index from period 1 to period 2. Part 3 shows how the current period value aggregates for period 2 are then derived and used to calculate the P-index for period 2.