OH. 8 COMPARISONS OF PURCHASING POWER 107

were not available in the first position; for in thisevent average satisfaction per unit from b 2 is likelyto be higher in proportion to average satisfaction perunit from a than is average satisfaction per unit fromb u so that we need some further hypothesis if we areto compare the purchasing powers successfully.

The method of approximation by taking as thebasis of our comparison that part of expenditure awhich is common to all the positions under considera-tion and neglecting the rest has the great advantageover more complicated formulae that we are dealingwith the same composite commodity throughout, andhave, therefore, no need to resort to the Chain Method(which will be considered later). Moreover, it wouldseem to be obviously preferable to, and not less simplethan, the method nearly always adopted by practicalstatisticians, namely, to take either a + b 1 or a + b 2 &sthough it were appropriate throughout. For this pro-cedure must, as a rule, involve a greater error than totake a throughout; inasmuch as its effect is always tooverestimate purchasing power in the position to whichour composite is strictly appropriate as compared withthe position to which it is less appropriate, wheneverthe change in the character of the expenditure is dueto a change in relative prices and represents a moveon the part of the consumer to take advantage of whatis relatively cheaper.

In those cases, therefore, where we are not entitledto assume that nothing has changed except relativeprices, this “ highest common factor ” method—as itmay be called—will probably give us as good £ resultas is obtainable in the circumstances. For example,the best index of Purchasing Power over a period often years may be obtained by taking as our com-promise composite the highest common factor ofgeneral expenditure in the successive yearly periods,and—as a check on the closeness of the approximation—setting alongside this index-number a statement