§ 5 (to Chapter II, § 5) The Coin-transfer Concept of Velocity and the Concept of Time of Turnover

We now turn to the coin-transfer 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 person-turnover sense.

It is clear, therefore, that the coin-transfer method of averaging is the same in results as the person-turnover 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 1V, 2V,..., V1, V2,..., 1V1, 1V2,..., 2V1,..., 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 W1, W2, or of 1W, 2W,... or of 1W1, 1W2,...,2W1,...

But these averages are all harmonic averages. To see this, we need only remember that V has already been analyzedas 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.