§ 3 (to Chapter II, § 5) "Arrays" of p's, Q's, and pQ's

Let us assume that the year is divided into an indefinite number of periods, or moments, and distinguish the prices and quantities relating to those successive periods by the subscripts 1, 2, 3, etc., at the left; and that we are dealing with a community of an indefinite number of persons, distinguished likewise by subscripts at the right. Thus the quantity of a particular kind of goods purchased by individual No. 1 in moment No. 3 is represented by 3q1 and the price of the sale by 3p1. The entire system of quantities and prices is represented by the two following "arrays."

We have just stated the meaning of the letters inside these arrays. Those outside are as follows: Q1 is the total quantity bought by person 1 and is the sum (1q1 + 2q1 + 3q1 +...) of all quantities purchased by him in all the different periods of time. Like definitions apply to Q2, Q3, etc. 1Q is the total quantity purchased in moment 1 and is the sum (1q1 + 1q2 + 1q3 +...) of all quantities purchased in that moment by all the different persons. Like definitions apply to 2Q, 3Q,.... Finally Q is (as already employed in the text) the grand total of quantities bought by all persons in all periods of time. Evidently,

Like definitions apply to the letters outside the p array, but the relations to the letters inside are here averages instead of sums. We may best derive the form of these averages from a third or intermediate array for pQ indicating the money values of the purchases.

This last named array is

In this array the same relations must evidently hold as in the Q array. That is, pQ, the entire sum spent on the given commodity by all persons in the community during all periods of the year, must be equal to (1) the sum of the column above it, (2) the sum of the row at its left, and (3) the sum of the interior terms of the array. In other words, it must be equal to (1) the sum of the total yearly amounts spent by the many different persons, (2) the sum of the total amounts spent in the community at the many different periods of the year, and (3) the sum of the purchases of all the individuals in all the periods.

The nature of the p array is now determined by the Q and the pQ arrays. It must namely be such as to permit the summation just described for the pQ array. That is, each of the average prices (such as p1) must conform to the type of formula:—

Hence, p is a weighted average of 1p1, 2p1, etc., the weights being 1q1, 2q1, etc. That is, the average price paid by person No. 1 is the weighted arithmetical average of the prices paid by him at different moments through the year, the weights being the quantities bought. The same principle obtains for all other persons.

Similarly, the average price, 1p, may be shown to be

That is, the average price in period No. 1 is the weighted arithmetical average of all prices paid by different persons at moment No. 1, the weights being the quantities bought by each. The same principles obtain at all other moments.

Finally, the average price, p, in the lower right corner of the p array, is either

(that is, p is a weighted arithmetical average of p1, p2, etc., the weights being Q1, Q2, etc.; or (using the row instead of column), p is the like weighted arithmetical average of 1p, 2p, etc., the weights being 1Q, 2Q, etc.; or lastly, either of these two expressions for p, combined with the preceding expression for p1, p2, etc., or with that for 1p, 2p, etc., may be used to show that p is a weighted arithmetical average of all the p's within the array, the weights being the corresponding q's. In short, the price of each commodity for the year is its average at all times and for all purchases in the year weighted according to the quantities bought.

This principle covers the method of averaging prices in different localities. Thus the average price of sugar in 1909 in the United States is the weighted arithmetical average of all prices of sales by all individuals throughout the United States, and at all moments throughout the year, the weights being the quantities bought. Thus, if there are large local or temporal variations in price, it is important to give chief weight to the largest purchases.

What has been said as to Q and p arrays relates only to one commodity. But the same principles apply to each commodity yielding separate arrays corresponding to each of the total quantities, Q, Q', Q'', etc., as well as corresponding to each of the average prices, p, p', p'', etc.