APPENDIX TO CHAPTER VI
§ 1 (TO CHAPTER VI, § 1)
Modification of Equation of Exchange required by International Trade
We have already seen that there are two equations of exchange, one for purchases, and the other for sales. In a closed community, these two are necessarily identical, for every purchase by one member of the community is a sale by another member. But in a community with international trade they will be slightly different. The equation of exchange as developed in this book relates to the expenditure of money for the purchase of goods, and not to the receipt of money for the sale of goods. This equation of exchange at its last stage of elaboration was
MV + M'V' + E'' - E''' = SpQ
where the letters have the meanings previously given to them, E'' relating to the debts contracted in the given period in book accounts and promissory notes used in purchase of goods, and E''' relating to the extinguishment of such debts during the same period. Since MV was developed from E, and M'V' from E', this equation may be written as follows:—
E + E' + E'' - E''' = SpbQb
where the letters E on the money side of the equations are used to indicate that the money is expended money and the subscripts b on the goods side are used to indicate that the goods are goods bought; likewise, if the letter R is used to indicate received money, and the subscript s to indicate that the goods are goods sold, the following equation expresses the receipt of money, etc., in exchange for the sale of goods:—
R + R' + R'' - R''' = SpsQs.
If there is no external trade, the several magnitudes in these two equations will evidently be identical on each side. If external trade exists, each equation may be resolved into an equation in which are distinguished the home trade and the outside trade. Thus, for the first equation, relating to expenditures, the E,E', etc., may be replaced by H + O,H' + O', etc., where the H's relate to the purchases at home and the O's to money spent outward. On the other side of the equation the SpbQb may be replaced by SphQh + SpiQi where the subscripts h relate to the goods purchased at home and the subscripts i to those coming inward. The equation will then become:—
(H + H' + H'' - H''') + (O + O' + O'' - O''') = SphQh + SpiQi
which, for brevity, we may write SH + SO = SphQh + SpiQi. Similarly, the second equation, relating to sales, may be written:—
SH + SI = SphQh + SpoQo.
That is, the net sum of the receipts at home (of money, bank credit, and book credit) plus the sum of payments for goods coming inward, is equal to the sum of the value of the goods sold at home plus the value of those sent out of the country. The last two equations, one relating to purchases and the other to sales, may be added together so as to give in a common equation the total trade in which the given community is concerned, that is, the total sales and purchases within itself and the sales and purchases with respect to the outside world. The combined equation will be:—
2 SH + SO + SI = 2 SphQh + SpiQi + SpoQo.
Here the internal trade is counted twice, because every transaction occurs both as a sale and as a purchase. This expresses the equation of exchange for the total trade (domestic and foreign) in which the country under consideration engages. If, instead of adding, we subtract one equation from the other, we obtain the following:—
SO - SI = SpiQi - SpoQo.
which is the equation of the balance of trade in its most general form, taking account, as it does, of credit as well as of money. The flow of money, as to or from a nation, depends upon this last equation.
The right-hand side of the penultimate equation, depends on three sets of prices,—the home prices (the ph's), the prices of goods which come into the country (the pi's), and the prices of goods which go out (the po's).
For a large country like the United States, the outside trade is so small, compared with the internal trade, as to be negligible. As we shall see in Chapter XII, the foreign trade of the United States is only a fraction of one per cent of the internal trade. And, because the export and import sides of the various magnitudes (O's, I's, Qo's and Qi's) nearly cancel each other, the net balance remaining on either side of the equation of exchange seldom amounts to more than one eighth of one per cent of the internal trade of the United States.
Almost equally insignificant is the difference, E'' - E''', between debts annually contracted and liquidated, if we may judge from estimates of that indebtedness, such as Holmes's. We are at any rate safe in saying that the corrections to our equation of exchange which have been discussed in this and the previous section are needless complications so far as the United States s concerned. We may therefore consider the equation MV + M'V' = SpQ as practically a precise form of the equation.