The Purchasing Power of Money
APPENDIX TO CHAPTER II§ I (TO CHAPTER II, § 3)

PERSONS  PERIODS 
TOTAL  PERSONS  PERIODS 
AVERAGE  

1  2  3  1  2  3  
1  _{1}q_{1}  _{2}q_{1}  _{3}q_{1}  Q_{1}  1  _{1}p_{1}  _{2}p_{1}  _{3}p_{1}  p_{1} 
2  _{1}q_{2}  _{2}q_{2}  _{3}q_{2}  Q_{2}  2  _{1}p_{2}  _{2}p_{2}  _{3}p_{2}  p_{2} 
3  _{1}q_{3}  _{2}q_{3}  _{3}q_{3}  Q_{3}  3  _{1}p_{3}  _{2}p_{3}  _{3}p_{3}  p_{3} 
—  —  —  —  —  —  —  —  —  — 
—  —  —  —  —  —  —  —  —  — 
Total  _{1}Q  _{2}Q  _{3}Q  Q  Average  _{1}p  _{2}p  _{3}p  p 
We have just stated the meaning of the letters inside these arrays. Those outside are as follows: Q_{1} is the total quantity bought by person 1 and is the sum (_{1}q_{1} + _{2}q_{1} + _{3}q_{1} +...) of all quantities purchased by him in all the different periods of time. Like definitions apply to Q_{2}, Q_{3}, etc. _{1}Q is the total quantity purchased in moment 1 and is the sum (_{1}q_{1} + _{1}q_{2} + _{1}q_{3} +...) of all quantities purchased in that moment by all the different persons. Like definitions apply to _{2}Q, _{3}Q,.... 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
PERSONS  PERIODS 
TOTAL  

1  2  3  
1  _{1}p_{1}_{1}q_{1}  _{2}p_{1}_{2}q_{1}  _{3}p_{1}_{3}q_{1}  p_{1}Q_{1} 
2  _{1}p_{2}_{1}q_{2}  _{2}p_{2}_{2}q_{2}  _{3}p_{2}_{3}q_{2}  p_{2}Q_{2} 
3  _{1}p_{3}_{1}q_{3}  _{2}p_{3}_{2}q_{3}  _{3}p_{3}_{3}q_{3}  p_{3}Q_{3} 
Total  _{1}p_{1}Q  _{2}p_{2}Q  _{3}p_{3}Q  pQ 
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 p_{1}) must conform to the type of formula:—
Hence, p is a weighted average of _{1}p_{1}, _{2}p_{1}, etc., the weights being _{1}q_{1}, _{2}q_{1}, 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, _{1}p, 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 p_{1}, p_{2}, etc., the weights being Q_{1}, Q_{2}, etc.; or (using the row instead of column), p is the like weighted arithmetical average of _{1}p, _{2}p, etc., the weights being _{1}Q, _{2}Q, etc.; or lastly, either of these two expressions for p, combined with the preceding expression for p_{1}, p_{2}, etc., or with that for _{1}p, _{2}p, 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.
In the preceding section we have seen that there exists an "array" of p's, pQ's and Q's for each commodity. These relate to the right side of the equation of exchange. Similar arrays relate to the left side.
If, as before, we assume a community of any number of persons, distinguished respectively by subscripts at the right, and if we divide the year into moments, distinguished by subscripts at the left, we may designate the amount of money expended in the first moment by the first person as _{1}e_{1}, the average amount of money he has on hand at that moment as _{1}m_{1} and his velocity of circulation at that moment (reckoned at its rate per year) as _{1}V_{1}. The expenditure in the moment being _{1}e_{1}, that moment's rate per annum is _{1}n_{1} _{1}e_{1}, there being n moments in the year, so that the velocity of circulation or rate of turnover per annum, _{1}V_{1}, is
A similar notation may be used to express the amounts expended and held and the velocity of circulation for each member of the community during each moment of the year as shown in the following three "arrays" (inside the lines).
In the first table, E_{1} at the right of the first line is the sum expended by the first person, being the sum of _{1}e_{1}, _{2}e_{1}, _{3}e_{1},...in the first line representing the amounts expended by him at successive moments during the year. Likewise, E_{2} is the sum expended by the second person during the year, and E_{3} is the sum expended by the third. _{1}E at the foot of the first column is the amount expended by all persons in the first moment; that is, it is the sum of all the amounts in the column above it; _{2}E is likewise the amount expended by all persons in the second moment; _{3}E, the amount expended in the third, etc. Finally, E, in the lower righthand corner, is, as employed in the text, the grand total expended by all persons in all moments of the year. Evidently E can be obtained by adding the row to the left of it, or by adding the column above it. It is also the sum of all the elements inside the lines, i.e. E = S_{1}E = SE_{1} = S_{1}e_{1}.
In the second table, M in the lower right corner is a sum of the average amounts held by the different members of the community during the year, i.e. it is the sum of the elements in the column above it, m_{1}, m_{2}, m_{3}, etc., each of which is by hypothesis a simple average of the row to its left.
Or, again, M is a simple average of the row to its left, _{1}M, _{2}M, _{3}M, etc., the average amounts of money in the community, in the successive moments of the year, each of which averages is in turn the sum of the column above it, i.e.
Thus M is both the sum of averages and the average of sums. That the two are equal follows by expressing both in terms of the elementary quantities _{1}m_{1} by means of the equations
and the equations _{1}M = _{1}m_{1} + _{1}m_{2} + _{1}m_{3} +....It is, of course, easy also to express M directly in terms of _{1}m_{1}, etc., within the table. Thus expressed, it is
The third table (that for velocities) is derived from the other two. As just explained, _{1}V_{1} is the velocity of circulation (considered as a per annum rate) for the first person in the community in the first moment.
There remain to be shown the relations of the elements in the V table.
Form (1) shows that V is a weighted average of the yearly velocities of the different persons, the velocity of each person being for the entire year and weighted according to his average amount of money on hand.
Following an analogous but slightly different sequence, we have
Form (2) shows that V is also the weighted average of the yearly velocities of the successive moments into which the year is divided, the velocity of each moment being for the entire community and weighted according to its average amount of money then in circulation.
Thus form (1) gives V in terms of the column above it, while form (2) gives V in terms of the row at its left. A formula similar to (1) may be constructed to express each of the magnitudes _{1}V, _{2}V, _{3}V, etc., in terms of the column above it, while a formula similar to (2) may be constructed to express each of the magnitudes V_{1}, V_{2}, V_{3}, etc., in terms of the row at its left. That is, the velocity in the entire community at any particular moment is a specific form of average of the velocities of different persons at that moment; and the velocity for the entire year of any particular person is a specific form of average of the velocities at different moments for that person.
Finally, V may be expressed, not only as an average of its column and row as in formula (1) and (2), but also as an average of the magnitudes in the interior of the table. This last result may be obtained in several ways, of which the most direct may briefly be expressed as follows: We know that E is the sum of the interior of the first or E table, that is, E = S_{1}e_{1}; and that M is equal to
Hence, we have
That is, V is the weighted arithmetical average of the yearly velocities pertaining to different persons in different moments, each velocity being weighted by the amount of money on hand in that instance. The mathematical reader will perceive that an alternative treatment would derive the result in terms of an harmonic average.
We now turn to the cointransfer concept of velocity of circulation. To show what kind of an average V is of the velocity of circulation of individual coins, or rather of individual pieces of money in general, let us denote the values of the individual pieces of money circulating in the community by the letters a, b, c, d, etc., and let us denote the net velocity of circulation of these (the number of times exchanged against goods minus the number of times exchanged with goods or "in change") by h, i, j, k, respectively, etc. Then E, the total amount expended, is denoted by ha + ib + jc + kd +...; and the amount of money, M, in the community is a + b + c + d.
That is, E/M is a weighted average of the net velocities of circulation of the different pieces of money, the velocity of each piece being weighted according to its denomination. But E/M is also V, which we have already seen is the velocity of circulation in the personturnover sense.
It is clear, therefore, that the cointransfer method of averaging is the same in results as the personturnover method, if all the pieces of money in the community are included.
Finally, we come to the concept of "time of turnover."
If velocity of circulation is represented as V, then 1/V represents the time of turnover. Similarly, the reciprocals of _{1}V, _{2}V,..., V_{1}, V_{2},..., _{1}V_{1}, _{1}V_{2},..., _{2}V_{1},..., are corresponding times of turnover. Using W for the reciprocal of V and applying the appropriate subscripts, we may write an array of W's analogous to the previous array of V's, and we may show that W is an average of W_{1}, W_{2}, or of _{1}W, _{2}W,... or of _{1}W_{1}, _{1}W_{2},...,_{2}W_{1},...
But these averages are all harmonic averages. To see this, we need only remember that V has already been analyzed*3 as a weighted average of the elementary V's, and that W has been defined as the reciprocal of V. That is, W is the reciprocal of this weighted average of elementary V's. But the elementary W's are reciprocals of the elementary V's. In other words, W is the reciprocal of the weighted arithmetical average of the reciprocals of elementary W's. This makes W, by definition, a weighted harmonic average of these elementary magnitudes.
It is clear that the equation of exchange, MV = SpQ, is derived from elementary equations expressing the equivalence of purchase money and goods bought. The money expended by any particular person at any particular moment is, by the very concept of price, equal to the quantities of all commodities bought in that moment by that person multiplied by the prices, i.e.
From this equation and others like it, for every person in the community and for every moment in the year, simply by adding them together, we obtain, for the left side of the equation, the sum of the e's which we call E; and for the right side the sum of all the pq's. We have already seen in the text how the left side, E, may be converted (by multiplying and dividing by M) into MV, and we have also just seen (§ 3 of this Appendix) how the sum of all the terms relating to each particular commodity represented on the right side may be converted (by similar simple algebraic operations) into one term of the form pQ so that the whole sum becomes SpQ. The final result is, therefore, MV = SpQ. This reasoning constitutes, therefore, a demonstration of the truth of this formula, based on the simple elementary truth that in every exchange the money expended equals the quantity bought multiplied by the price of sale.
Let us assume that V and the Q's remain invariable while M changes to M_{o} and p, p', p', etc. to p_{o}, p'_{o}, p'_{o}, etc. (The subscripts "0" refer to a year called the base year other than the original year.) We have for the two years respectively the two equations:—
whence by division, we obtain
The last expression is evidently a weighted arithmetical average of
etc., the weights being p_{o}Q, p'_{o}Q', etc. We conclude that, if the velocity of circulation and the quantities of goods exchanged remain unaltered, while the quantity of money is altered in a given ratio, then prices will change in this same ratio "on the average," the average being exactly defined as a weighted arithmetical average, in which the weights are the values of goods sold, reckoned at the prices of the base year. The ratio may evidently also be written:—
which is a weighted harmonic average of
etc., in which the weights are pQ, p'Q', etc., that is, the values, not in the base year, but the other year.
If M and the Q's remain invariable, while V changes from V to V_{1}, evidently the ratio V/V_{1} will be expressed by precisely the same formulæ as above.
If the Q's remain invariable, while M and V both change, evidently the ratio
will be expressed by the same formulæ.
Again the same formulæ apply if M and V remain invariable while the Q's all vary in a given ratio, or if the Q's all vary in a given ratio in combination with any variation in M or V or both. In short, the formulae apply perfectly in all cases of variation, except when the Q's vary relatively to each other.
These formulæ, it should be noted, are those later discussed as the formulae numbered (11) in the large table of formulæ in the Appendix to Chapter X.
Notes for Appendix to Chapter V
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