Geometric price indices

A geometric price index is entirely analogous to an arithmetic one, except that price relatives are aggregated with a geometric average instead of an arithmetic average. That is, a general geometric price index is given by5 \[\begin{align*} I^{G} = \prod_{i = 1}^{n} \left(\frac{p_{i1}}{p_{i0}}\right)^{\omega_{i}}. \end{align*}\]

As with the arithmetic indices, different geometric indices correspond to different choices for the weights.6

Jevons index. Setting \(\omega_{i} = 1 / n\) results in the Jevons index \[\begin{align*} I^{G}_{J} = \prod_{i = 1}^{n} \left(\frac{p_{i1}}{p_{i0}}\right)^{1 / n}, \end{align*}\] or, equivalently, \[\begin{align*} I^{G}_{J} = \frac{\prod_{i = 1}^{n} p_{i1}^{1 / n}}{\prod_{i = 1}^{n} p_{i0}^{1 / n}}. \end{align*}\] The Jevons index is the geometric analogue to the Carli or Dutot index.

Geometric Laspeyres index. Setting \(\omega_{i} = p_{i0} q_{i0} / \sum_{j = 1}^{n} p_{j0} q_{j0}\) results in the geometric Laspeyres index \[\begin{align*} I^{G}_{L} = \prod_{i = 1}^{n} \left(\frac{p_{i1}}{p_{i0}}\right)^{\frac{p_{i0} q_{i0}}{\sum_{j = 1}^{n} p_{j0} q_{j0}}}. \end{align*}\] Similar to the Jevons index, this is the geometric analogue of the Laspeyres index. It is trivial to define a geometric Young index as well by using period-\(b\) rather than period-0 expenditure/revenue shares.7

Törnqvist index. Setting \[\begin{align*} \omega_{i} = \frac{1}{2} \frac{p_{i0} q_{i0}}{\sum_{j = 1}^{n} p_{j0} q_{j0}} + \frac{1}{2} \frac{p_{i1} q_{i1}}{\sum_{j = 1}^{n} p_{j1} q_{j1}} \end{align*}\] results in the Törnqvist index, which is usually expressed as \[\begin{align*} \log(I^{G}_{T}) = \sum_{i = 1}^{n} \left(\frac{1}{2} \frac{p_{i0} q_{i0}}{\sum_{j = 1}^{n} p_{j0} q_{j0}} + \frac{1}{2} \frac{p_{i1} q_{i1}}{\sum_{j = 1}^{n} p_{j1} q_{j1}}\right) \log\left(\frac{p_{i1}}{p_{i0}}\right). \end{align*}\] The Törnqvist index expands on the geometric Laspeyres index by using expenditure shares in both period 0 and period 1 to form weights (i.e., the average expenditure share between period 0 and period 1).

The Jevons index is usually synonymous with a geometric index, and it finds application in situations where there is no quantity information to form weights (as opposed to using a Carli or Dutot index). Sometimes a weighted Jevons index is used as a shorthand for the general geometric index.

One point to note about the geometric indices is that they are always smaller than their arithmetic counterparts. For any given weights, it can be shown that \(I^{G} \leq I^{A}\), with equality only when all price relatives are equal or all price relatives except one have zero weight. Consequently, a geometric index always shows a smaller increase in prices over time (or a larger decrease) than the corresponding arithmetic index.8 This is an important downside to having a menu of index numbers to choose from, as the choice of index-number formula has an impact on the resulting measure of inflation.9

  1. The capital letter pi \(\Pi\) is the product operator. For a collection of numbers \(x_{1}, x_{2},\ldots,x_{n}\), \(\prod_{i=1}^{n} x_{i}\) means \(x_{1} \times x_{2} \times \ldots \times x_{n}\).↩︎

  2. It is interesting to note that the weights can also be chosen so that any geometric index is also an arithmetic index (and vice versa). That is, for any \(\omega_{i}\), there is an \(\tilde{\omega}_{i}\) such that \(\prod_{i = 1}^{n} (p_{i1} / p_{i0})^{\omega_{i}} = \sum_{i = 1}^{n} \tilde{\omega}_{i} p_{i1} / p_{i0}\). See Balk (2008, sec. 4.2) for what these weights look like. Framing an index as arithmetic or geometric is really just a case of framing the weights. The same holds true for a number of other types of indices, such as harmonic indices—see Martin (2020).↩︎

  3. Defining a geometric Lowe index is less obvious. For example, the geometric Paasche index uses period-1 expenditure/revenue shares as weights, instead of the hybrid weights used for the arithmetic Paasche index. Perhaps a better name for this index would be the geometric Palgrave index.↩︎

  4. The difference between an arithmetic index and geometric index tends to be larger when price relatives are more dispersed, although this is not always the case. It is possible that the difference becomes larger when the variance between prices relatives becomes smaller—see Lord (2002).↩︎

  5. The same reasoning can be used to show that a harmonic price index is always smaller than the corresponding geometric index—see Bullen (2003, II 2.1 Corollary 2).↩︎