§ 6 (to Chapter II, § 5) Algebraic Demonstration of Equation of Exchange

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.

1e1 = 1p1 1q1 + 1p' 1 1q' 1 + 1p'1 1q' 1 +...

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.