The Traditional Approach

The traditional way of dealing with the index-number problem has been to design measures based on some sort of averaging process, aimed at breaking away from the obvious one-sidedness of the Laspeyres and the Paasche measures. In its simplest form this approach to the problem, if applied to the measurement of the price development of a commodity aggregate, leads one to suggest that maybe this development can best be measured by the arithmetic mean of the Laspeyres and the Paasche price index: (2.1) P ( A M ) = P L + P P 2 $$P\left( {AM} \right) = \frac{{{P_L} + {P_P}}}{2}$$ However, apart from its simplicity, there is no reason why the arithmetic mean should be preferred to the harmonic mean: (2.2) P ( H M ) = 2 1 P L + 1 P P = 2 P L P P P P + P L $$P\left( {HM} \right) = \frac{2}{{\frac{1}{{{P_L}}} + \frac{1}{{{P_P}}}}} = \frac{{2{P_L}{P_P}}}{{{P_P} + {P_L}}}$$ or to the geometric mean (2.3) P ( G M ) = P L P P $$P\left( {GM} \right) = \sqrt {{P_L}{P_P}}$$ Indeed, if one has to choose between these three types of average, one might well prefer the geometric mean to the other two, since the geometric mean is itself a mean of the other two: (2.4) P ( G M ) = P ( A M ) P ( H M ) $$P\left( {GM} \right) = \sqrt {P\left( {AM} \right)P\left( {HM} \right)}$$ But that is not all. Irving Fisher, after examining well over a hundred possible index-number formulae, reached the conclusion that P(GM) was in fact the ‘ideal’ index number (see the note on p. 99).