Laspeyres and Paasche Index

They differ because they use different basket weights: Laspeyres holds base-period quantities fixed, Paasche holds current-period quantities fixed, and consumers substitute between the two periods toward goods that became relatively cheaper. Neither is exactly right — for a cost-of-living comparison the true index lies between them, with Laspeyres biased upward and Paasche downward. When you have both periods' data, a superlative index such as Fisher (their geometric mean) or Törnqvist is preferred because it sits between them and approximates the true index closely.

Substitution bias is the error a fixed-weight index makes by ignoring that consumers shift away from goods whose relative prices rise. A Laspeyres index, freezing the old basket, therefore overstates the cost of maintaining welfare; a Paasche index understates it. The size depends on how much relative prices move and how readily consumers substitute; for consumer price indices the upper-level substitution bias is typically estimated at a few tenths of a percentage point per year, enough to matter cumulatively for indexed pensions, taxes, and contracts, which is why statistical agencies adopt chained or superlative indices to reduce it.

Fisher called the geometric mean of the Laspeyres and Paasche indices 'ideal' because, of all the formulas he examined axiomatically, it best satisfied his battery of tests — notably time reversal (reversing the periods inverts the index) and factor reversal (price and quantity indices multiply to the value ratio) — which Laspeyres and Paasche each fail. It is 'superlative' in Diewert's later economic sense: it is exact for a flexible (second-order) aggregator function, so it approximates the true cost-of-living or output index to second order regardless of the underlying preferences or technology, justifying its use when both periods' data are available.