Törnqvist Index

An index is called superlative, in Diewert's 1976 terminology, when it is exact for a flexible functional form — one that can provide a second-order approximation to an arbitrary aggregator function. The Törnqvist index is exact for the translog function, meaning that if the underlying cost or production function is translog, the Törnqvist index recovers the true index exactly. Because the translog flexibly approximates any smooth technology to second order, superlative indices like Törnqvist closely track the true cost-of-living or productivity index without assuming a restrictive functional form, which is why they are preferred over fixed-basket alternatives.

Both are superlative indices that use information from both periods and closely approximate a true economic index, and they typically give very similar numbers in practice. The Törnqvist index is a geometric mean of price relatives weighted by average value shares, exact for the translog aggregator; the Fisher index is the geometric mean of the Laspeyres and Paasche indices, exact for a quadratic aggregator. The key practical difference is that Fisher satisfies the factor-reversal and time-reversal tests exactly — its price and quantity indices multiply to the value ratio — whereas Törnqvist does not satisfy factor reversal exactly. Törnqvist, by contrast, decomposes additively in logs, which is convenient for growth accounting.

Using the average of the base-period and current-period value shares is precisely what makes the index superlative and lets it capture substitution behaviour. A Laspeyres index fixes weights at the base period and overstates inflation because it ignores that consumers shift away from goods that become relatively expensive; a Paasche index fixes weights at the current period and understates it. Averaging the two shares splits the difference symmetrically and corresponds to the exact aggregation of a flexible translog function, so the resulting index neither systematically over- nor under-states the true change.