CH. 8 COMPARISONS OF PURCHASING POWER 111

formula is simpler than those previously given, andthe proof is more rigorous. It is reached, for example,by Professor Pigou ( Economics of Welfare, part i. chap,vi.). The matter is also very well treated by Haberler (Der Sinn der Indexzahlen, pp. 83-94). The depend-ence of the argument, however, on the assumption ofuniformity of tastes, etc., is not always sufficientlyemphasised. 1 Moreover, it must be noticed that, inthe class of case where p is less than q, this conditionproves that tastes must have changed, and that anassumption to the contrary must be invalid, since it

infringes the condition that — is less than p and

n i

greater than q.

In certain circumstances a combination of theHighest Common Factor Method with the Method ofLimits may be practicable. We may know that tastesare unchanged over a considerable part of the field ofconsumption, measured in terms of expenditure, andalso that the real-incomes derived from this part of thefield are either a considerable part or a constant partof the total real-incomes. In this case our methodof approximation may consist first of all in using theHighest Common Factor Method to reduce the fieldof comparison to that part of the expenditure over theseries of positions for which it is legitimate to assumeunchanged tastes, and then applying the Method ofLimits to this part of the field.

3. The “ Crossing of Formulae ”

The method of procedure, to which Professor IrvingFisher has devoted much attention 2 and to which hehas given the name of “ Crossing the Formulae ”, 3 is,

1 Dr. Bowley in his “ Notes on Index Numbers ” published in theEconomic Journal, June 1928, may be mentioned amongst those who haveexpressly introduced this necessary condition.

2 Vide especially his The Making of Index Numbers.

8 Op. cit. chap. vii.