§ 2. Index Numbers for Prices occur in Antithetical Pairs as also do Index Numbers for Quantities

We have seen that, given any special form of average for P1, there results therefrom a correlative form for Q1 and vice versa. Thus if P1 is the simple arithmetical average

then Q1 is.

Again, if we start with Q1 as the simple arithmetical average

then we discover in its correlate a new formula for P1, viz.

In this way any given formula for P1 leads to another formula for P1 which may be called its antithesis. This second formula for P1 is identical in form with the formula for Q1 correlative to the first formula for P1, differing merely in the fact that the p1's and Q1's are interchanged.

The four forms and their relations are best seen by placing them in a square, as in the following example:—

The pair of formulæ in the left vertical column express correlative formulæ, one for P1 and the other for Q1. The formula diagonally opposite each of these has the p's and Q's interchanged. The right pair thus formed are evidently also correlates of each other and each is the antithesis of that in the same horizontal line at its left. Since any formula for P1 involves an antithetical formula for P1, we have here a means of devising new formulæ from old and also of noting certain unsuspected relationships between formulæ already in use.

In the table of index numbers for P1 which follows, the antithetical formulæ are in each case placed side by side and joined by a bracket. The two together represent, in each case, the upper half of a square like the above. The omitted lower half, giving the correlative forms for Q1, can be readily supplied in every case; but, to save space, each column heading in the table only includes the formula for P1, omitting its correlate for Q1 as well as the corresponding formulæ for T1 and A1. The formula for Q1 is, as above explained, easily written by interchanging p's and Q's in the formula given in the neighboring (antithetical) columns; that for T1 is found by dividing Sp1Q1 by P1; that for A1 by dividing Sp1Q1 by Q1. Practically P1 and A1 serve the same purpose, namely, that of indicating price changes; likewise Q1 and T1 serve the same purpose, namely, that of indicating quantity changes. For any series of years the numbers for P1 and for A1 will be proportional, the only difference being that P1 is expressed in percentages, the figure for the base year being 100 per cent, whereas A1 is expressed in dollars, the figure for the base year being the actual exchanges Sp0Q0 in that year. Likewise Q1 and T1 differ merely as percentage and dollar measurements, the base figure being 100 per cent and Sp0Q0 dollars respectively.