The calculation of Fisher indices requires both Laspeyres and Paasche indices to be calculated. The Fisher index is defined as the geometric mean of the Laspeyres index and the Paasche index. Because Laspeyres and Paasche are the upper and the lower boundary of a theoretical index, Fisher tends to be a closer approximation to a theoretical ideal index.
The Fisher price index is defined as:
Being a geometric mean of Paasche and Laspeyres the Fisher index uses weight information from two periods – period 0 and t – thus, it assumes product substitution and is therefor a ‘better’ measure of true inflation. The Fisher index is a superlative index in the sense that it has the property of being a closer approximation to the ideal consumer utility function than the Paasche and the Laspeyres.
The Fisher index passes all tests with the exception of the circularity test.
It can be shown that:
Note! The ideal Fisher index is not the geometric mean between a harmonic average of the price ratios using current weights and a harmonic mean using base period weights. But the Fisher is defined as the geometric mean between a Laspeyres and a Paasche. Doing the mistake of just changing the weights, and not the formula in what should have been a Paasche index will give a quasi index – called a Palgrave index. The Palgrave will generally be upward biased compared to a Paasche. Unfortunately, making a Palgrave (instead of a Paasche)in order to construct the ideal index is a rather common mistake, and the result will be an upward biased index compared to the ideal Fisher index.
Since:
then:
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