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 = Σ 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 ΣpQ, 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 ΣpQ as PT. One might restate ΣpQ 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. ΣpQ 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.×10) + (20c.×3) + (35c.×1)]/14 P is then 105/14 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 105/14 cents. The T in the new equation is 13. The product of P and T is $1.349/14. 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]