Less common arithmetic indices

There are a variety of arithmetic indices that are rarely used in practice, but can be useful to know about. As these index-number formulas are seldom used, this material can be safely be skipped.

Palgrave index. Setting \(\omega_{i} = p_{i1} q_{i1} / \sum_{j = 1}^{n} p_{j1} q_{j1}\) results in the Palgrave index \[\begin{align*} I^{A}_{p} = \sum_{i = 1}^{n} \frac{p_{i1} q_{i1}}{\sum_{j = 1}^{n} p_{j1} q_{j1}} \frac{p_{i1}}{p_{i0}}. \end{align*}\] The Palgrave index uses period-1 expenditure shares as weights, and is a special case of the Young index.

Unnamed 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 an index-number formula without a name, \[\begin{align*} I^{A}_{U} = \sum_{i = 1}^{n} \left(\frac{1}{2} \frac{p_{i0} q_{i0}}{\sum_{j = 1}^{n} p_{j0} q_{j0}} + \frac{1}{2} \frac{p_{i1} q_{i1}}{\sum_{j = 1}^{n} p_{j1} q_{j1}}\right) \frac{p_{i1}}{p_{i0}}. \end{align*}\] This index is a mixture of the Laspeyres and Palgrave indices, with weights given by the average expenditure/revenue share between period 0 and period 1.

Drobisch index. Setting \[\begin{align*} \omega_{i} = \frac{1}{2} \frac{p_{i0} q_{i0}}{\sum_{j = 1}^{n} p_{j0} q_{j0}} + \frac{1}{2} \frac{p_{i0} q_{i1}}{\sum_{j = 1}^{n} p_{j0} q_{j1}} \end{align*}\] results in the Drobisch index \[\begin{align*} I^{A}_{d} = \frac{1}{2} \frac{\sum_{i = 1}^{n} p_{i1} q_{i0}}{\sum_{i = 1}^{n} p_{i0} q_{i0}} + \frac{1}{2} \frac{\sum_{i = 1}^{n} p_{i1} q_{i1}}{\sum_{i = 1}^{n} p_{i0} q_{i1}}. \end{align*}\] This index is a mixture of the Laspeyres and Paasche indices.

Walsh index. Setting \(\omega_{i} = p_{i0} \sqrt{q_{i0} q_{i1}} / \sum_{j = 1}^{n} p_{j0} \sqrt{q_{j0} q_{j1}}\) results in the Walsh index \[\begin{align*} I^{A}_{W} = \frac{\sum_{i = 1}^{n} p_{i1} \sqrt{q_{i0} q_{i1}}}{\sum_{i = 1}^{n} p_{i0} \sqrt{q_{i0} q_{i1}}}. \end{align*}\] This index uses a basket that contains the geometric average of the period-0 and period-1 quantities.

Marshall-Edgeworth index. Setting \(\omega_{i} = p_{i0} (q_{i0} + q_{i1}) / \sum_{j = 1}^{n} p_{j0} (q_{j0} + q_{j1})\) results in the Marshall-Edgeworth index \[\begin{align*} I^{A}_{M} = \frac{\sum_{i = 1}^{n} p_{i1} (q_{i0} + q_{i1}) / 2}{\sum_{i = 1}^{n} p_{i0} (q_{i0} + q_{i1}) / 2}. \end{align*}\] Like the Walsh index, this index uses a basket that takes an average of the period-0 and period-1 quantities.

Geary-Khamis index. Setting \(\omega_{i} = p_{i0} / (1 / q_{i0} + 1 / q_{i1}) / \sum_{j = 1}^{n} p_{j0} / (1 / q_{j0} + 1 / q_{j1})\) results in the Geary-Khamis index \[\begin{align*} I^{A}_{G} = \frac{\sum_{i = 1}^{n} 2 p_{i1} / (1 / q_{i0} + 1 / q_{i1})}{\sum_{i = 1}^{n} 2 p_{i0} / (1 / q_{i0} + 1 / q_{i1})}. \end{align*}\] Like the Walsh and Marshall-Edgeworth indices, this index uses a basket that takes an average of the period-0 and period-1 quantities.