Suppose, for instance, that the quantity of money were doubled, while its velocity of circulation and the quantities of goods exchanged remained the same. Then it would be quite impossible for prices to remain unchanged. The money side would now be $10,000,000 × 20 times a year or $200,000,000; whereas, if prices should not change, the goods would remain $100,000,000, and the equation would be violated. Since exchanges, individually and collectively, always involve an equivalent quid pro quo, the two sides must be equal. Not only must purchases and sales be equal in amount—since every article bought by one person is necessarily sold by another—but the total value of goods sold must equal the total amount of money exchanged. Therefore, under the given conditions, prices must change in such a way as to raise the goods side from $100,000,000 to $200,000,000. This doubling may be accomplished by an even or uneven rise in prices but some sort of a rise of prices there must be. If the prices rise evenly, they will evidently all be exactly doubled.... If the prices rise unevenly, the doubling must evidently be brought about by compensation; if some prices rise by less than double, others must rise by enough more than double to exactly compensate.

But whether all prices increase uniformly, each being exactly doubled, or some prices increase more and some less (so as still to double the total money value of the goods purchased), the prices are doubled on the average.... From the mere fact, therefore, that the money spent for goods must equal the quantities of those goods multiplied by their prices, it follows that the level of prices must rise or fall according to changes in the quantity of money, unless there are changes in its velocity of circulation or in the quantities of goods exchanged.

If changes in the quantity of money affect prices, so will changes in the other factors—quantities of goods and velocity of circulation—affect prices, and in a very similar manner. Thus a doubling in the velocity of circulation of money will double the level of prices, provided the quantity of money in circulation and the quantities of goods exchanged for money remain as before....

Again, a doubling in the quantities of goods exchanged will not double, but halve, the height of the price level, provided the quantity of money and its velocity of circulation remain the same....

Finally, if there is a simultaneous change in two or all of the three influences, i. e., quantity of money, velocity of circulation, and quantities of goods exchanged, the price level will be a compound or resultant of these various influences. If, for example, the quantity of money is doubled, and its velocity of circulation is halved, while the quantity of goods exchanged remains constant, the price level will be undisturbed. Likewise, it will be undisturbed if the quantity of money is doubled and the quantity of goods is doubled, while the velocity of circulation remains the same. To double the quantity of money, therefore, is not always to double prices. We must distinctly recognize that the quantity of money is only one of three factors, all equally important in determining the price level....

We now come to the strict algebraic statement of the equation of exchange.... Let us denote the total circulation of money, i. e., the amount of money expended for goods in a given community during a given year, by E (expenditure); and the average amount of money in circulation in the community during the year by M (money). M will be the simple arithmetical average of the amounts of money existing at successive instants separated from each other by equal intervals of time indefinitely small. If we divide the year's expenditures, E, by the average amount of money, M, we shall obtain what is called the average rate of turnover of money in its exchange for goods, E/M that is, the velocity of circulation of money. This velocity may be denoted by V, so that E/M = V; then E may be expressed as MV. In words: the total circulation of money in the sense of money expended is equal to the total money in circulation multiplied by its velocity of circulation or turnover. E or MV, therefore, expresses the money side of the equation of exchange. Turning to the goods side of the equation, we have to deal with the prices of goods exchanged and quantities of goods exchanged. The average price of sale of any particular good, such as bread, purchased in the given community during the given year, may be represented by p (price); and the total quantity of it purchased, by Q (quantity); likewise the average price of another good (say coal) may be represented by p´ and the total quantity of it exchanged, by Q´; the average price and the total quantity of a third good (say cloth) may be represented by p´´ and Q´´ respectively; and so on, for all other goods exchanged, however numerous. The equation of exchange may evidently be expressed as follows:

MV = pQ
+ p´Q´
+ p´´Q´´
+ etc.

The right-hand side of this equation is the sum of terms of the form pQ—a price multiplied by a quantity bought. It is customary in mathematics to abbreviate such a sum of terms (all of which are of the same form) by using "Σ" as a symbol of summation. This symbol does not signify a magnitude as do the symbols M, V, p, Q, etc. It signifies merely the operation of addition and should be read "the sum of terms of the following type." The equation of exchange may therefore be written:

MV = ΣpQ.

That is, the magnitudes E, M, V, the p's and the Q's relate to the entire community and an entire year; but they are based on and related to corresponding magnitudes for the individual persons of which the community is composed and for the individual moments of time of which the year is composed.