General recognition of this fact, for which we are indebted to the authors of modern value theory, was hindered for a long time by a peculiar sort of obstacle. It is not altogether a rare thing that those very pioneers who have not hesitated to clear new paths for themselves and their followers by boldly rejecting outworn traditions and ways of thinking should yet shrink sometimes from all that is involved in the rigid application of their own principles. When this is so, it remains for those who come after to endeavor to put the matter right. The present is a case in point. On the subject of the measurement of value, as on a series of further subjects that are very closely bound up with it, the founders of the subjective theory of value refrained from the consistent development of their own doctrines. This is especially true of Böhm-Bawerk. At least it is especially striking in him; for the arguments of his which we are about to consider are embodied in a system that would have provided an alternative and, in the present writer’s opinion, a better, solution of the problem, if their author had only drawn the decisive conclusion from them.
Böhm-Bawerk points out that when we have to choose in actual life between several satisfactions which cannot be had simultaneously because our means are limited, the situation is often such that the alternatives are on the one hand one big satisfaction and on the other hand a large number of homogeneous smaller satisfactions. Nobody will deny that it lies in our power to come to a rational decision in such cases. But it is equally clear that a judgment merely to the effect that a satisfaction of the one sort is greater than a satisfaction of the other sort is inadequate for such a decision; as would even be a judgment that a satisfaction of the first sort is considerably greater than one of the other sort. Böhm-Bawerk therefore concludes that the judgment must definitely affirm how many of the smaller satisfactions outweigh one of the first sort, or in other words how many times the one satisfaction exceeds one of the others in magnitude.2
The credit of having exposed the error contained in the identification of these two last propositions belongs to Cuhel. The judgment that so many small satisfactions are outweighed by a satisfaction of another kind is in fact not identical with the judgment that the one satisfaction is so many times greater than one of the others. The two would be identical only if the satisfaction afforded by a number of commodity units taken together were equal to the satisfaction afforded by a single unit on its own multiplied by the number of units. That this assumption cannot hold good follows from Gossen’s law of the satisfaction of wants. The two judgments, “I would rather have eight plums than one apple” and “I would rather have one apple than seven plums,” do not in the least justify the conclusion that Böhm-Bawerk draws from them when he states that therefore the satisfaction afforded by the consumption of an apple is more than seven times but less than eight times as great as the satisfaction afforded by the consumption of a plum. The only legitimate conclusion is that the satisfaction from one apple is greater than the total satisfaction from seven plums but less than the total satisfaction from eight plums.3
This is the only interpretation that can be harmonized with the fundamental conception expounded by the marginal-utility theorists, and especially by Böhm-Bawerk himself, that the utility (and consequently the subjective use-value also) of units of a commodity decreases as the supply of them increases. But to accept this is to reject the whole idea of measuring the subjective use-value of commodities. Subjective use-value is not susceptible of any kind of measurement.
The American economist Irving Fisher has attempted to approach the problem of value measurement by way of mathematics.4 His success with this method has been no greater than that of his predecessors with other methods. Like them, he has not been able to surmount the difficulties arising from the fact that marginal utility diminishes as supply increases, and the only use of the mathematics in which he clothes his arguments, and which is widely regarded as a particularly becoming dress for investigations in economics, is to conceal a little the defects of their clever but artificial construction.
Fisher begins by assuming that the utility of a particular good or service, though dependent on the supply of that good or service, is independent of the supply of all others. He realizes that it will not be possible to achieve his aim of discovering a unit for the measurement of utility unless he can first show how to determine the proportion between two given marginal utilities. If, for example, an individual has 100 loaves of bread at his disposal during one year, the marginal utility of a loaf to him will be greater than if he had 150 loaves. The problem is, to determine the arithmetical proportion between the two marginal utilities. Fisher attempts to do this by comparing them with a third utility. He therefore supposes the individual to have B gallons of oil annually as well, and calls that increment of B whose utility is equal to that of the 100th loaf of bread. In the second case, when not 100 but 150 loaves are available, it is assumed that the supply of B remains unchanged. Then the utility of the 150th loaf may be equal, say, to the utility of b/2. Up to this point it is unnecessary to quarrel with Fisher’s argument; but now follows a jump that neatly avoids all the difficulties of the problem. That is to say, Fisher simply continues, as if he were stating something quite self-evident: “Then the utility of the 150th loaf is said to be half the utility of the 100th.” Without any further explanation he then calmly proceeds with his problem, the solution of which (if the above proposition is accepted as correct) involves no further difficulties, and so succeeds eventually in deducing a unit which he calls a “util.” It does not seem to have occurred to him that in the particular sentence just quoted he has argued in defiance of the whole of marginal-utility theory and set himself in opposition to all the fundamental doctrines of modern economics. For obviously this conclusion of his is legitimate only if the utility of b is equal to twice the utility of b/2. But if this were really so, the problem of determining the proportion between two marginal utilities could have been solved in a quicker way, and his long process of deduction would not have been necessary. Just as justifiably as he assumes that the utility of is equal to twice the utility of b/2, he might have assumed straightaway that the utility of the 150th loaf is two-thirds of that of the 100th.
Fisher imagines a supply of B gallons that is divisible into n small quantities b, or 2n small quantities b/2. He assumes that an individual who has this supply B at his disposal regards the value of commodity unit x as equal to that of b and the value of commodity unit y as equal to that of b/2. And he makes the further assumption that in both valuations, that is, both in equating the value of x with that of b and in equating the value of y with that of b/2, the individual has the same supply of B gallons at his disposal.
He evidently thinks it possible to conclude from this that the utility of b is twice as great as that of b/2. The error here is obvious. The individual is in the one case faced with the choice between x (the value of the 100th loaf) and b = 2b/2. He finds it impossible to decide between the two, i.e., he values both equally. In the second case he has to choose between y (the value of the 150th loaf) and b/2. Here again he finds that both alternatives are of equal value. Now the question arises, what is the proportion between the marginal utility of b and that of b/2? We can determine this only by asking ourselves what the proportion is between the marginal utility of the nth part of a given supply and that of the 2nth part of the same supply, between that of b/n and that of b/2n. For this purpose let us imagine the supply B split up into 2n portions of b/2n. Then the marginal utility of the (2n-1)th portion is greater than that of the 2nth portion. If we now imagine the same supply B divided into n portions, then it clearly follows that the marginal utility of the nth portion is equal to that of the (2n-1)th portion plus that of the 2nth portion in the previous case. It is not twice as great as that of the 2nth portion, but more than twice as great. In fact, even with an unchanged supply, the marginal utility of several units taken together is not equal to the marginal utility of one unit multiplied by the number of units, but necessarily greater