OH. 8 COMPARISONS OF PURCHASING POWER 99

of the population, are arranged in order of magnitude.In the numerical example given above, that is to say,we can only conclude that the increase in purchasingpower lies between 2 and 4.

This difficulty in making precise quantitative com-parisons is the same as arises in the case of many otherfamous concepts, namely of all those which are com-plex or manifold in the sense that they are capable ofvariations of degree in more than one mutually in-commensurable direction at the same time. The con-cept of Purchasing Power, averaged over populationswhich are not homogeneous in respect of their real-incomes, is complex in this sense. The same difficultyarises whenever we ask whether one thing is superiorin degree to another on the whole, the superiority de-pending on the resultant of several attributes whichare each variable in degree but in ways not commen-surable with one another. 1

In what follows we shall assume, for the sake ofsimplicity and precision, that we are dealing withthe non-complex cases where the purchasing powerof money has changed equally for all relevant levelsof real-incomes.

(ii.) Methods oe Approximation

We have seen that the right way to compare thepurchasing powers of money in two positions is tocompare the total money-incomes of two “ similar ”persons in the two positions. But there is a difficultyin applying this method of comparison in practiceowing to the want of an objective test by which wecan select our “ similar ” pair for comparison. Con-sequently, the general practice hitherto has been, notto make any attempt to find a pair of similar personsand then compare their money-incomes, but to find

1 The difficulty is the same as that which I have discussed in my Treatiseon Probability, chap, iii., especially §§ 7-16.