Equation (7) means that it does not matter which period we regard as the base period; we obtain essentially the same answer either way. Since the choice of which period to regard as the base is essentially arbitrary, other things being equal, we would like our price index to satisfy the time reversal test. (It is worth noting that neither the Laspeyres nor the Paasche price index satisfies the time reversal test.)
Diewert (1997:138) showed that only one average satisfies all the properties listed. This is the geometric mean (the square root of the product) of the Paasche and the Laspeyres
The price index defined by (8) is known as the Fisher (1922) ideal price index. The foregoing argument provides one justification for thinking of the Fisher price index as a “best” estimator of price change between periods 0 and t.
An alternative approach to combining the Paasche and the Laspeyres is to average not the indexes themselves but the two different baskets that go into them, an approach that was originated by Walsh (1901, 1921) and Knibbs (1924). If we use a geometric mean of the two baskets, we obtain the Walsh price index, PW, written as
If we replace the geometric mean in (9) with the simple arithmetic mean, we reach yet another index in the Walsh-Knibbs family, known as the Marshall Edgeworth price index (Marshall, 1887; Edgeworth, 1925).
We have been careful so far not to distinguish individual from aggregate quantities. Paasche and Laspeyres indexes can be equally well constructed using individual baskets or aggregate (or average) baskets. In this section, we consider the relationships between these various types of Laspeyres and Paasche indexes under the assumption that each household faces the same vector of prices in each period.
Suppose that there are H households in the economy. Household h’s period t Laspeyres index can be written following (3) but with the household superscript h in the form