The Fisher index
One important price index that is neither a geometric index nor an arithmetic index is the Fisher index, which is the geometric average of the (arithmetic) Laspeyres index and the (arithmetic) Paasche index, \[\begin{align*} I_{F} &= \sqrt{\frac{\sum_{i = 1}^{n} p_{i1} q_{i0}}{\sum_{i = 1}^{n} p_{i0} q_{i0}} \times \frac{\sum_{i = 1}^{n} p_{i1} q_{i1}}{\sum_{i = 1}^{n} p_{i0} q_{i1}}} \\ &= \sqrt{I^{A}_{L} \times I^{A}_{P}}. \end{align*}\]
The Fisher index is often seen as an ideal index because it treats information in period 0 and period 1 symmetrically.10 In practice, however, the Fisher index is not frequently used by national statistical agencies because it is not timely to calculate. This is because it depends on the Paasche index, which requires information on period 1 quantities in addition to period 1 prices, something that has historically been impractical for national statistical agencies to collect.
The Törnqvist index is also seen as ideal for a similar reason, and these types of indices often get called superlative.↩︎