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 x 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....