CH. 8 COMPARISONS OF PURCHASING POWER 105

lent, i.e. that average consumers in the two positionsare similar; and we do not know how many unitsof a + b 2 in the second position are equivalent forsimilar consumers to a given number of units ofa + b 1 in the first position. The problem before us,therefore, is to discover a valid method of approxima-tion applicable in such cases. To this problem wemust now address ourselves. I shall argue that twomethods, and only two, are legitimate, the first ofwhich is the more generally applicable, and the secondof which provides limits between which the correctsolution lies in cases where we can assume that nothinghas changed between the two positions except relativeprices.

1. The “ Highest Common Factor ” Method

Let be the price of a in the first position and p 2its price in the second position. The first method ofapproximation consists in neglecting b x and b 2 and

7 )

employing ~ as an index of the change in the price-

level as between these two positions. There are twoconditions, the satisfaction of either of which willrender this a legitimate approximation. We are as-suming—it will be remembered—that the satisfactionderived from the consumption of a per unit of a isapproximately the same in the two positions for anypair of similar persons.

The first condition is that the expenditures ofevery individual consumer on a should in both posi-tions be large compared with his expenditure on b xor b 2 .

The second condition is that the real-incomesderived by any consumer from the consumption ofa and in the first position should be approximatelyin the same proportion as his expenditures on them,and similarly with a and b 2 in the second position.