Appendix to Chapter X, § 1. Each Form of Index Number for Prices implies a Correlative Form of Index Number for Quantiti

We have seen that the number of possible forms of averages is infinite. Since an index number, as P1, is an average, it follows that there exists an infinite number of possible forms of index numbers. Forty-four of the simplest and most important forms are given in the table which follows. In this table the subscript "1" relates to any specified year called for convenience "year 1," while the subscript "0" likewise relates to "year 0," called the "base" year. The headings of the columns in the table give the formula for the index number, P1, for the year 1 relatively to the base year 0. By substituting "2" for "1," each formula could be made to refer to a second year, 2, considered relatively to the base year 0. Likewise, substituting "3," "4," etc., for "1," we have an entire series of index numbers, P1, P2, P3, P4, etc., for different years, all relative to the same base year 0. Since the formulæ are all alike and differ only in subscripts, it is unnecessary to waste space by expressing P2, P3, etc., in the headings. Consequently, in each column heading, only the formula for P1 is expressed.

Also to save space, the column headings omit the formulæ for T1, etc., correlative to those for P1. Each form of price index, P1, applicable to the equation of exchange, implies a correlative trade index, T1, such that the product of the two is equal to Sp1Q1, the right side of the equation of exchange.

Since

Hence, given a particular formula for P1, we have a resultant and particular formula for T1. For instance, if P1 is a simple arithmetic average of, i.e. if, (formula 3 of the table), where n is the number of commodities of which the price ratios are included, then the correlative formula for T1 will evidently be

Again, if P1 is the geometric average, (formula 7 of the table), then T1 has the correlative form

Conversely any particular formula for T1 implies a correlative particular formula for P1. For, since

it follows that

By means of this equation, if we have given any particular formula for T1, we may obtain a resultant particular formula for P1.

The examples already given of P1 (the arithmetical and geometric average) illustrate how to obtain the correlative formula for T1. If we work backward from these somewhat complicated formulæ for T1, we may in turn derive as the correlative formulæ for P1 the arithmetic and geometric averages.

As a third example illustrating the derivation of the formula for P1 from a given formula for T1, let T1 be defined as Sp0Q1; then

(Formula 11 of the table.)

We may consider, then, that each column heading, though stating only the formula for P1, implies also a corresponding formula for T1; that is, P1 and T1 occur in correlative pairs. P1 and T1 are such that if one of them (say P1) is given independently of the equation, Sp1Q1 = P1T1, the other is then defined by means of that equation.

The two magnitudes P1 and T1 are not, however, absolutely symmetrical. There is this important distinction between them: that, while P1 is an abstract number, T1 is concrete, being expressible in dollars and cents.

It thus appears that, although the p's and Q's enter symmetrically into the expression Sp1Q1, yet, when this expression is replaced by P1T1, the first factor, P1, represents the p's in a somewhat different manner from that in which the second factor, T1, represents the Q's. P1 is a pure number, an average of pure numbers—the ratios the p's bear to the base prices, p0's, whereas T1, being

is a concrete number, being a value found by dividing the value Sp1Q1 by the pure number P1.

Thus, while the p's and Q's occur symmetrically in the original formula Sp1Q1, the process by which we convert Sp1Q1 into P1T1 treats them asymmetrically. But evidently we can reverse the asymmetry in their treatment; for, instead of putting Sp1Q1 equal to P1T1, we may put it equal to A1Q1, in which Q1 is now a quantity index, that is, an average of the ratios which the Q1's bear to the Q0's or base quantities (i.e. an average of ...), and A1, being therefore is the "aggregate price," that is, the value found by dividing the value Sp1Q1 by the pure number Q1. Here, if the form of Q1 is given independently of the equation Sp1Q1 = A1Q1, the form of A1 is defined by means of this equation, and conversely.

Thus we may convert Sp1Q1 into either P1T1 or A1Q1. In the first, the p's are represented by a ratio, P1; in the second, by a value, A1; in the first, the Q's are represented by a value, T1; in the second, by a ratio, Q1. The asymmetry of each of the two formulæ P1T1 and A1Q1 is the reverse of the other.

Finally we may, if we wish, treat both the p's and Q's alike by putting Sp1Q1 equal to (Sp0Q0)P1Q1, where P1 and Q1 are, both of them, index numbers for the p1's and Q1's respectively. That is (as we shall prove), P1 and Q1 are averages respectively of price ratios like p1/p0, and of quantity ratios like Q1/Q0. The equation Sp1Q1 = (Sp0Q0) P1Q1 may be said to define either one of the two averages (P1 and Q1) in terms of the other. One or the other must be defined irrespective of the equation.

Thus there are three ways of resolving Sp1Q1, as follows:—

Sp1Q1 = P1T1 = A1Q1 = (Sp0Q0) P1Q1.

The third form becomes, dividing the equation through by Sp0Q0,

We wish now to prove that if either P1 or Q1 is first determined in any way conformably to the definition of an average, leaving the other to be determined by the above equation, then the latter also necessarily conforms to the definition of an average. We have to prove that if Q1 is taken as an average of ..., then the correlative expression for P1 derived from (1), viz.:— is an average of....

It is, therefore, only necessary to show (in accordance with the most general definition of an average as given in the Appendix to Chapter II) that expression (2) shall be equal to k when

by the definition of an average.

Hence the expression (2) may now be written

which is evidently equal to k.

Therefore expression (2) is, by definition, an average of....By the same reasoning we may show conversely

that

is a true average of....

We conclude that if either P1 or Q1 in the formula

is an average of the p or Q ratios respectively, then the other is also an average respectively of the Q or p ratios.