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By Root 1356 0

Then the utility of the 150th loaf may be equal, say, to the utility of ²/2. 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 ² is equal to twice the utility of ²/2. 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 ² is equal to twice the utility of ²/2, 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 ², or 2n small quantities ²/2. 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 ² and the value of commodity unit y as equal to that of ²/2. And he makes the further assumption that in both valuations, that is, both in equating the value of x with that of ² and in equating the value of y with that of ²/2, 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 ² is twice as great as that of ²/2. 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 ² = 2²/2. 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 ²/2. Here again he finds that both alternatives are of equal value. Now the question arises, what is the proportion between the marginal utility of ² and that of ²/2? 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 ²/n and that of ²/2n. For this purpose let us imagine the supply B split up into 2n portions of ²/2n. 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 than this product. The value of two units is greater than, but not twice as great as, the value of one unit. [5]

Perhaps Fisher thinks that this consideration may be disposed of by supposing ² and ²/2 to be such small quantities that their utility may be reckoned infinitesimal. If this is really his opinion, then it must first of all be objected that the peculiarly mathematical conception of infinitesimal quantities is inapplicable to economic problems. The utility afforded by a given amount of commodities, is either great enough for valuation, or so small that it remains imperceptible to the valuer and cannot therefore affect his judgment. But even if the applicability of the