Factoring an index
When calculated over many periods, a price index gives a measure of the change in prices over time relative to a fixed base period. Calculating a price index over many periods poses no new challenges—once a base period is selected, one of the index-number formulas in the previous section can be directly applied to produce an index that evolves over time by simply forming price relatives that always calculate the change in price relative to the base period. For example, the series of geometric indices computed over periods 0, 1, and 2, with period 0 as the base period, is \[\begin{align*} 1, \prod_{i = 1}^{n} \left(\frac{p_{i1}}{p_{i0}}\right)^{\omega_{i}}, \prod_{i = 1}^{n} \left(\frac{p_{i2}}{p_{i0}}\right)^{\omega_{i}}. \end{align*}\] Thus the period 1 index value gives the change in prices between period 1 and period 0, and the period 2 index gives the change in prices between period 2 and period 0. The period 0 value is 1 because the ratio of period 0 prices to period 0 prices is always 1.
Most price indices are calculated frequently—usually monthly or quarterly—and it is useful to be able to calculate a price index using only the previous period’s index value and the most recent price relatives for each good covered by the index.11 That is, it is useful to be able to factor an index into two terms: the previous period’s index value, and an index that depends on only the current-period prices and previous-period prices. Such a factorization allows for an index to be calculated period-by-period, and saves from having to hold onto the price data in the base period for the life of an index. This is practically significant, as it means that an index can still be produced even if base-period price data are lost.
Factoring a geometric index is trivial; for a given set of weights, the geometric index that runs from period 0 to period \(t\) can always be written as the product of the geometric index that runs for period 0 to period \(k\) and the geometric index that runs from period \(k\) to period \(t\). That is, \[\begin{align*} I^{G}(0, t) &= \prod_{i = 1}^{n} \left(\frac{p_{it}}{p_{i0}}\right)^{\omega_i} \\ &= \prod_{i = 1}^{n} \left(\frac{p_{ik}}{p_{i0}}\right)^{\omega_i} \times \prod_{i = 1}^{n} \left(\frac{p_{it}}{p_{ik}}\right)^{\omega_i} \\ &= I^{G}(0, k) \times I^{G}(k, t). \end{align*}\]
This should be deeply intuitive. If prices increased by 20% between period 0 and period \(k\), and then increase by another 10% between period \(k\) and period \(t\), the total increase in price from period 0 to period \(t\) is 32%. This is the same as multiplying an index value of 1.2 by an index value of 1.1, as the result is 1.32.
Factoring an arithmetic index is slightly more complex, as it requires changing the weights used to aggregate price relatives. The arithmetic index that runs from period 0 to period \(t\) can be written as \[\begin{align*} I^{A}(0, t) &= \sum_{i = 1}^{n} \omega_{i} \frac{p_{it}}{p_{i0}} \\ &= I^{A}(0, k) \times \sum_{i = 1}^{n} \tilde{\omega}_{i} \frac{p_{it}}{p_{ik}}, \end{align*}\] where \[\begin{align*} \tilde{\omega}_{i} = \frac{\omega_{i} \frac{p_{ik}}{p_{i0}}}{I^{A}(0, k)}. \end{align*}\]
Unlike the geometric index, factoring an arithmetic index requires changing the weights for the index running from period \(k\) to \(t\). Sometimes this gets called price updating the weights, but this terminology is used inconsistently. Despite this wrinkle, however, the basic idea is the same as in the geometric case—an arithmetic index can always be decomposed into two arithmetic indices.
Intuitively, factoring an index simply tinkers with value of the index in the base period so that it can continue from a previous value without needing to be recalculated from the start. Rather than starting the index at the value 1 in period \(k\), it starts from the value of the index in period \(k\), essentially building on the cumulative change in price up to that point in time.
One reason is that it can make sampling price information easier, as the same units don’t need to be sampled in both the current period and base period.↩︎