CHAPTER VIII

THE "EQUATION OF EXCHANGE"

In Professor Irving Fisher's _Purchasing Power of Money_[130] we have the most uncompromising and rigorous statement of the quantity theory to be found in modern economic literature. We have, too, a book which follows the logic of the quantity theory more consistently than any other work with which I am acquainted. The book deals with the theory more elaborately and with more detail than any other single volume, and sums up most of what other writers have had to say in defence of the quantity theory. Professor Fisher's book has, moreover, received such enthusiastic recognition from reviewers and others as to justify one in treating it as the "official" exposition of the quantity theory. Thus, Sir David Barbour cites Professor Fisher as the authority on whom he relies for such justification of the theory as may be needed,[131] while Professor A. C. Whitaker declares that he adopts "without qualification the whole body of general monetary theory" for which Professor Fisher stands.[132] Professor J. H. Hollander has recently referred to Professor Fisher's work on money and prices as a model of that combination of theory and inductive verification which constitutes real science.[133] The _American Economic Review_ presents as an annual feature Professor Fisher's "Equation of Exchange."

Not all, by any means, of those who would call themselves quantity theorists would concur in Professor Fisher's version of the doctrine--Professor Taussig, notably, introduces so many qualifications, and admits so many exceptions, that his doctrine seems to the present writer like Professor Fisher's chiefly in name. But there is no other one book which could be chosen which would serve nearly as well for the "platform" of present-day quantity theorists as _The Purchasing Power of Money_. Partly for that reason, and partly because the book lends itself well to critical analysis, I shall follow the outline of the book in my further statement and criticism of the quantity theory, indicating Professor Fisher's views, and indicating the points at which other expositions of the quantity theory diverge from his, setting his views in contrast with those of other writers. We shall find that this method of discussion will furnish a convenient outline on which to present our final criticisms of the quantity theory, and parts of the constructive doctrine of the present book.

First, Professor Fisher presents in the baldest possible form the dodo-bone doctrine. The quality of money is irrelevant. The sole question of importance is as to its quantity--the number of money-units.[134] I shall not here discuss this point, as a previous chapter has given it extended analysis, except to repeat that it is in fact an essential part of the quantity theory. If the _quality_ of money is a factor, a necessary factor, to consider, then obviously we have something which will disturb the mechanical certainty of the quantity theory. Professor Fisher is thoroughly consistent with the spirit of his general doctrine on this point.

Second, Professor Fisher has no absolute value in his scheme. By the value of money he means merely its purchasing power, and by its purchasing power he means nothing more than the fact that it does purchase: the purchasing power of money is defined as the reciprocal of the level of prices, "so that the study of the purchasing power of money is identical with the study of price levels." (_Loc. cit._, p. 14.) In this, again, Professor Fisher is absolutely true to the spirit and logic of the quantity theory doctrine. The equilibration of numbers of goods, and numbers of dollars, in a mechanical scheme, gives prices--an average of prices, and nothing else. Any psychological values of goods or of dollars would upset the mechanism, and mess things up. They are properly left out, if one is to be happy with the quantity theory. Fisher, in discussion of Kemmerer's _Money and Credit Instruments_, has criticised the exposition of the utility theory of value with which Kemmerer prefaces his exposition of the quantity theory, as "fifth wheel." I agree thoroughly with Fisher's view in this, and would add that the only reason that it has made Kemmerer little trouble in the development of his quantity theory is that he has made virtually no use of it there! The two bodies of doctrine, in Kemmerer's exposition, are kept, on the whole, in separate chapters, well insulated. Coupled with this purely relative conception of the value of money, however, there is, in Fisher's scheme, an effort to get an absolute out of it: the general price-level is declared to be independent of, and causally prior to,[135] the particular prices of which it is an average. I mention this remarkable doctrine here, reserving its discussion for a later chapter.[136]

A further feature of Professor Fisher's system, to which especial attention must be given, is the large role played in it by the "equation of exchange." This device has been used by other writers before him, notably by Newcomb, Hadley, and Kemmerer, receiving at the hands of the last named an elaborate analysis. But Fisher, basing his work on Kemmerer's, has made even more extensive use of the "equation of exchange," and has given it a form which calls for special consideration.[137] The "equation of exchange," on the face of it, makes an exceedingly simple and obvious statement. Properly interpreted, it is a perfectly harmless--and, in the present writer's opinion, useless--statement. It gives rise to complications, however, as to the meaning of the algebraic terms employed, which we shall have to study with care. The starting point is a single exchange: a person buys 10 pounds of sugar at seven cents a pound. "This is an exchange transaction in which 10 pounds of sugar have been regarded as equal to 70 cents, and this fact may be expressed thus: 70 cents = 10 pounds of sugar multiplied by 7 cents a pound. Every other sale and purchase may be expressed similarly, and by adding them all together we get the equation of exchange _for a certain period in a given community_."[138] The money employed in these transactions usually serves several times, and hence the money side of the equation is greater than the total amount of money in circulation. In the preliminary statement of the equation of exchange, foreign trade, and the use of anything but money in exchanges are ignored, but later formulations of the equations are made to allow for them. "The equation of exchange is simply the sum of the equations involved in all individual exchanges in a year.... And in the grand total of all exchanges for a year, the total money paid is equal in value to the total value of the goods bought. The equation thus has a money side and a goods side. The money side is the total money paid, and may be considered as the product of the quantity of money multiplied by its rapidity of circulation. The goods side is made up of the products of quantities of goods exchanged multiplied by their respective prices."

Letting M represent quantity of money, and V its velocity or rapidity of circulation, p, p', p'', etc., the average prices for the period of different kinds of goods, and Q, Q', Q'', etc., the quantities of different kinds of goods, we get the following equation:

MV = pQ + p'Q' + p''Q'' + etc.[139]

"The right-hand side of this equation is the sum of terms of the form pQ--a price multiplied by the quantity bought."[140] The equation may then be written,

MV = [Greek: S] pQ (Sigma being the symbol of summation).

The equation is further simplified[141] by rewriting the right-hand side as PT, where P is the weighted _average_ of all the p's, and T is the _sum_ of all the Q's. "P then represents in one magnitude the level of prices, and T represents in one magnitude the volume of trade."

It may seem like captious triviality to raise questions and objections thus early in the exposition of Professor Fisher's doctrine. And yet, serious questions are to be raised. First, in what sense is there an equality between the ten pounds of sugar and the seventy cents? Equality exists only between _homogeneous_ things. In what sense are money and sugar homogeneous? From my own standpoint, the answer is easy: money and sugar are alike in that both are _valuable_, both possess the attribute of economic social value, an absolute quality and quantity. The degree in which each possesses this quality determines the exchange relation between them. And the degree in which each other good possesses this quality, taken in conjunction with the value of money, determines every other particular price. Finally, an average of these particular prices, each determined in this way, gives us the general price-level. The value of the money, on the one hand, and the values of the goods on the other hand, are both to be explained as complex social psychological forces. But when this method of approach is used, when prices are conceived of as the results of organic social psychological forces, there is no room for, or occasion for, a further explanation in terms of the mechanical equilibration of goods and money. Professor Fisher, as just shown, very carefully excludes this and all other psychological approaches to his problem of general prices, and has no place in his system for an absolute value. In what sense, then, are the sugar and the money equal? Professor Fisher says (p. 17), that the equation is an equation of values. But what does he mean by values in this connection? Perhaps a further question may show what he _must_ mean, if his equation is to be intelligible. That question is regarding the meaning of T.

T, in Professor Fisher's equation, is defined as the sum of all the Q's. But how does one sum up _pounds_ of _sugar_, _loaves_ of _bread_, _tons_ of _coal_, _yards_ of _cloth_, etc.? I find at only one place in Professor Fisher's book an effort to answer that question, and there it is not clear that he means to give a general answer. He needs units of Q which shall be homogeneous when he undertakes to put concrete figures into his equation for the purpose of comparing index numbers and equations for successive years. "If we now add together these tons, pounds, bushels, etc., and call this grand total so many 'units' of commodity, we shall have a very arbitrary summation. It will make a difference, for instance, whether we measure coal by tons or hundred-weights. The system becomes less arbitrary if we use, as the unit for measuring any goods, not the unit in which it is commonly sold, but the amount which constitutes a 'dollar's worth' at some particular year called the base year" (p. 196). If this be merely a device for the purpose of handling index numbers, a convention to aid mensuration, we need not, perhaps, challenge it. The unit chosen is, in that case, after all a fixed physical quantity of goods, the amount bought with a dollar in a given year, and remains fixed as the prices vary in subsequent years. That it is more "philosophical" or less "arbitrary" than the more common units is not clear, but, if it be an answer, designed merely for the particular purpose, and not a general answer, it is aside from my purpose to criticise it here. If, however, this is Professor Fisher's _general_ answer to the question of the method of summing up T, if it is to be employed in his equation when the question of _causation_, as distinguished from _mensuration_, is involved, then it represents a vicious circle. If T involves the price-level in its definition, then T cannot be used as a causal factor to explain the price-level. I shall not undertake to give an answer, where Professor Fisher himself fails to give one, as to his meaning. I simply point out that he himself recognizes that the summation of the Q's is arbitrary without a common unit, and that the only common unit suggested in his book, if applied generally, involves a vicious circle.

What, then, is T? Perhaps another question will aid us in answering this. What does it mean to _multiply_ ten pounds of sugar by seven cents? What sort of product results? Is the answer seventy pounds of sugar, or seventy cents, or some new two-dimensional hybrid? One multiplies feet by feet to get _square_ feet, and square feet by feet to get cubic feet. But in general, the multiplication of _concrete_ quantities by _concrete_ quantities is meaningless.[142] One of the generalizations of elementary arithmetic is that concrete quantities may usually be multiplied, not by other concrete quantities, but rather by _abstract_ quantities, pure numbers. Then the product has meaning: it is a concrete quantity of the same denomination as the multiplicand. If the Q's, then, are to be multiplied by their respective p's, the Q's must be interpreted, not as bushels or pounds or yards of concrete goods, but merely as abstract numbers. And T must be, not a sum of concrete goods, but a sum of abstract numbers, and so itself an abstract number. Thus interpreted, T is equally increased by adding a hundred papers of pins,[143] a hundred diamonds, a hundred tons of copper, or a hundred newspapers. This is not Professor Fisher's rendering of T, but it is the only rendering which makes an intelligible equation.

We return, then, to the question with which we set out: in what sense is there an equality between the two sides of Professor Fisher's equation? The answer is as follows: on one side of the equation we have M, a quantity of money, multiplied by V, an abstract number; on the other side of the equation, we have P, a quantity of money, multiplied by T, an abstract number. The product, on each side, is a _sum of money_. These sums are equal. They are equal because they are _identical_. The equation asserts merely that what is _paid_ is equal to what is _received_. This proposition may require algebraic formulation, but to the present writer it does not seem to require any formulation at all. The contrast between the "money side" and the "goods side" of the equation is a false one. There is no goods side. Both sides of the equation are money sides. I repeat that this is not Professor Fisher's interpretation of his equation. But it seems the only interpretation which is defensible.

A further point must be made: Sigma pQ, where the Q's are interpreted as abstract numbers, is a summary of concrete money payments, each of which has a causal explanation, and each of which has effected a concrete exchange. Mathematically, PT is equal to [Greek: S] pQ, just as 3 times 4 is equal to 2 times 6. But from the standpoint of the theory of causation, a vast difference is made. Three children four feet high equal in aggregate height two men six feet high. But the assertion of equality between the three children and the two men represents a high degree of abstraction, and need not be significant for any given purpose. Similarly, the restatement of [Greek: S] pQ as PT. One might restate [Greek: S] pQ as PT, defining P as the _sum_ (instead of the average) of the p's, and T as the weighted average (instead of the sum) of the Q's. Such a substitution would be equally legitimate, mathematically, and the equation, MV = PT equally true. [Greek: S] pQ might be factorized in an indefinite number of ways. But it is important to note that in PT, as defined by Professor Fisher,[144] we are at three removes from the concrete exchanges in which actual concrete causation is focused: we have first taken, for each commodity, an average, for a period, say a year, of the concrete prices paid for a unit of that commodity, and multiplied that average by the abstract number of units of that commodity sold in that year; we have then summed up all these products into a giant aggregate, in which we have mingled hopelessly a mass of concrete causes which actually affected the particular prices; then, finally, we have factorized this giant composite into two numbers which have no concrete reality, namely, an average of the averages of the prices, and a sum of the abstract numbers of the sums of the goods of each kind sold in a given year--a sum which exists only as a pure number, and which, consequently, is unlikely to be a causal factor! It may turn out that there is reason for all this, but if a _causal_ theory is the object for which the equation of exchange is designed, a strong presumption against its usefulness is raised. Both P and T are so highly abstract that it is improbable that any significant statements can be made of either of them. As concepts gain in generality and abstractness, they lose in content; as they gain in "extension" they lose (as a rule) in "intension." On the other side of the equation, we also look in vain for a truly concrete factor. V, the average velocity of money for the year, is highly abstract. It is a mathematical summary of a host of complex activities of men. Professor Fisher thinks that V obeys fairly simple laws, as we shall later see, but at least that point must be demonstrated. Even M is not concrete. At a given moment, the money in circulation is a concrete quantity, but the average for the year is abstract, and cannot claim to be a direct causal factor, with one uniform tendency. Of course Professor Fisher himself recognizes that his central problem is, not to state and justify, mathematically, his equation[145]--that is a work of supererogation, and the statistical chapters devoted to it seem to me to be largely wasted labor. Professor Fisher recognizes that his central problem is to establish _causal_ relations among the factors in his equation of exchange. It is from the standpoint of its adaptability as a tool in a theory of causation that I have been considering it. It should be noted that "volume of trade," as frequently used, means not numbers of goods sold, but the money-price of all the goods exchanged, or PT. It is in this sense of "trade" that bank-clearings are supposed to be an index of volume of trade. The sundering of the p's and Q's really is a big assumption of many of the points at issue. Indeed, it is absolutely impossible to sunder PT. It is always the p aspect of the thing that is significant, Fisher himself finally interprets T, statistically, as billions of _dollars_.[146] As a matter of mathematical necessity, either P must be defined in terms of T or T defined in terms of P. The V's and M and M' may be independently defined, and arbitrary numbers may be assigned for them limited only by the necessity that MV + M'V' be a fixed sum.[147] But P and T cannot, with respect to each other, be thus independently defined. The highly artificial character of T has been pointed out by Professor E. B. Wilson, of the Massachusetts Institute of Technology, in his review of Fisher's _Purchasing Power of Money_ in the _Bulletin of the American Mathematical Society_, April, 1914, pp. 377-381. "Various consequences are readily obtained from the equation of exchange, but the determination of the equation itself is not so easy as it might look to a careless thinker. The difficulties lie in the fact that P and T individually are quite indeterminate. An average price-level P means nothing till the rules for obtaining the average are specified, and independent rules for evaluating P and T may not satisfy [the equation.] For instance, suppose sugar is 5c. a pound, bacon 20c. a pound, coffee 35c. a pound. The average price is 20c. If a person buys 10 lbs. of sugar, 3 lbs. of bacon, and 1 lb. of coffee, the total trading is in 14 lbs. of goods. The total expenditure is $1.45; the product of the average price by the total trade is $2.80; the equation is very far from satisfied." Wilson thinks it necessary, to make the matter straight, to define T, arbitrarily as (MV + M'V')/P in which case, the equation is true, but so obviously a truism that no one would see any point in stating it. T no longer has any independent standing. Fisher has, however, an escape from this status for T, but only by reducing P to the same position. He defines P as the _weighted_ average of the p's (27), and fails, I think, to see how completely this ties it up with T. The only method of weighting the p's that will leave the equation straight is to weight the different prices by the number of units of each kind of good sold, namely, T. Thus, in Wilson's illustration, we would define P as [(5c.x10) + (20c.x3) + (35c.x1)]/14 P is then 10-5/14 c., while T is 14. PT is, then, equal to $1.45, which is the total expenditure, or MV + M'V'. Be it noted, here, that P is defined in terms of T, _i. e._, P is defined as a fraction, the denominator of which is T. No other definition of P will serve, if T is to be defined independently.

But notice the corollary. P must be differently defined each year, for each new equation, as T changes in total magnitude, and as the elements in T are changed. The equation cannot be kept straight otherwise. Suppose that the prices remain unchanged in the next year, but that one more pound of coffee, and two less pounds of sugar are sold. P, as defined for the equation of the preceding year would no longer fit the equation. P, as previously defined, would be unaltered, since none of the prices in it had changed. P, defined as a weighted average with the weights of the first year, would, then, still be 10-5/14 cents. The T in the new equation is 13. The product of P and T is $1.34-9/14. But the total expenditure, (MV + M'V') is $1.70. The equation is not fulfilled. To fulfill the equation, it is necessary to get a new set of weights for P, in terms of the new T of the new equation. From the standpoint of a _causal_ theory, this is delightful. P is the _problem_. But you are not allowed to _define_ the problem until you know what the _explanation_ is! Then you define the problem as that which the explanation will explain!

Fisher, however, appears unaware of this. At all events, he does not mention it. And he ignores it in filling out his equation statistically, for he assigns one set of weights to the particular prices in his P throughout.[148]

The causal theory with which the equation of exchange is associated is as follows: P is passive. A change in the equation cannot be initiated by P. If P should change without a prior change in one of the other factors, forces would be set in operation which would force it back to its original magnitude. M and T are independent magnitudes. A change in one does not occasion a change in the other. An increase or decrease in M will not cause a change in V. Therefore, an increase in M must lead to a proportionate increase in P, and a decrease in M to a proportionate decrease in P, if the equation is to be kept straight. Changes in T have opposite proportional effects on P.

Before examining the validity of the causal theory, and the arguments by which it is supported, it will be best to state the more complex formula which Professor Fisher advances as expressing the facts of to-day. The original formula ignored credit, and ignored the possibility of resort to barter. It also failed to reckon with certain complications which Fisher deals with as "transitional" rather than "normal."

The formula which includes credit is as follows:

MV + M'V' = PT

Here, MV and PT have the same significance as before. M' is the average amount of bank-deposits in the given region for the given period, and V' is the velocity of circulation of those deposits. M, money, consists of all the media of exchange in circulation which are _generally_ acceptable, as distinguished from those which are acceptable under particular conditions, as by endorsement. M excludes money in bank reserves and government vaults. Money, specifically, includes gold and silver coin, minor coins, government paper money, and bank-notes; M' consists of deposits transferable by check. This version would not satisfy such a writer as Nicholson,[149] who would limit money to gold coin, and would include in M' not only deposits, but also bank-notes, and other credit instruments. I may suggest here, what I shall later emphasize, that Fisher's "money," though he doubtless is using the most common definition of money, is really a pretty heterogeneous group of things, concerning which it is possible to make few general statements safely. In economic essence, _e. g._, bank-notes are much more like deposits than like gold, and if one wishes to separate money and credit, bank-notes belong with M' rather than with M. But we must take the theory as we find it! Again, credit is by no means exhausted when bank-deposits are named. Why should not book-credits, and bills of exchange be included? Why not postal money-orders, why not deposits subject to transfer by the giro-system? M' is defined[150] as "the total deposits subject to transfer by check," and would, thus, exclude the giro-system of Germany. It is surely a very provincial equation of exchange, with which Fisher and Kemmerer seek to set forth the universal laws of money! Fisher's reason for excluding book-credits is that book-credits merely postpone, and do not dispense with, the use of money and checks.[151] Book-credits, unlike deposits, have no _direct_ effect on prices (_Ibid._, 82, n.; 370), but only an indirect effect, by increasing the velocity of money. (_Ibid._, 81-82; 370-371.) Book-credit, indeed "time-credit" in general thus has no direct effect on prices, and is properly excluded from the equation of exchange. These distinctions seem to me highly artificial. In the first place, the use of checks, in part, merely postpones the use of money: money is moved back and forth from one part of the country to another, and from one bank to another, to the extent that checks fail to offset one another, and in the case of book-credit, while there is less of this offsetting, there is a good deal of it, especially between stockbrokers in different cities, and in small towns and at country stores, and particularly in the South, where the country storekeeper and "factor" are also dealers in cotton, etc., and where they advance provisions during the year to the small farmers, receiving their pay, in considerable degree, not in money, but in cotton, which they credit on the books in terms of money to the customer--a point which Fisher mentions in an appendix. (_Ibid._, p. 371.) The difference on this point is a difference in degree merely.[152] Further, Fisher makes the same point with reference to deposits subject to check that he makes with reference to book-credits, namely, that their use increases the velocity of money. To say that one has a _direct_ effect on prices, and the other only an indirect effect is absolutely arbitrary. If buying and selling are what count, if prices are forced up by the offer of money or credit for goods, and forced down as the amount of money and credit offered for goods is reduced, then one exchange must count for as much as any other of like magnitude in fixing prices. The same is true of transactions in which bills of exchange or other credit devices serve as media of exchange. Of course these considerations do not render the equation of exchange, as presented by Fisher, untrue. The equation simply states that the money and bank-deposits used in paying for goods in a given period are equal to the amount paid for those goods in a given period. It makes no assertion concerning payments for other goods, and makes no assertion as to the amount of other transactions which are paid for in other ways. General Walker, presented with the problem of credit phenomena, simplifies the thing even more.[153] He rules out all exchanges which are effected by credit devices, counting only those performed by coin, bank-notes and government paper money, and insists that the general price-level is determined in those exchanges in which money alone (as thus defined) is employed. His equation--if he had considered it worth while to use one--would then have been simply

MV = PT

where T would be merely the number of goods exchanged by means of money. One could make a similar equation, equally true, by defining money as gold coin, and reducing T correspondingly. Is there any reason for limiting the equation at all?[154] Is there any reason for supposing that any one set of exchanges is more significant for the determination of the price-level than any other set of exchanges? Does not the logic of the quantity theory require us to include all exchanges which run in terms of money?--If one wishes a complete picture of the exchanges, some such equation as this would be necessary:

MV + M'V' + BV'' + EV''' + OV'''' = PT,

where B represents book-credit, V'' the number of times a given average amount of book-credit is used in the period, E bills of exchange, and V''' their velocity of circulation, and O all other substitutes for money, with V'''' as their velocity of circulation. Even then we have not a complete picture, if direct barter or the equivalents of barter can be shown to be important.

For the present, I waive a discussion of the comparative importance of these different methods of conducting exchanges. The situation varies greatly with different countries. Fisher's and Kemmerer's equations are at best plausible when presented as describing American conditions, are much less plausible when applied to Canada and England, and are caricatures when applied to Germany and France.

So much for the statement of the equation of exchange, except that it is important to add that the period of time chosen for the equation is one year. Just why a year, rather than a month or two years or a decade should be chosen, may await full discussion till later. I shall venture here the opinion that the yearly period is not the period that should have been chosen from the standpoint of Fisher's causal theory, and that it probably was chosen, if for any conscious reason at all, because of the fact that statistical data which Fisher wished to put into it are commonly presented as annual averages. The question now is, however, as to the use to be made of the equation in the development of a causal theory.