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Irving Fisher—Mathematical investigations

purchaser at the time of purchase the quantity of the commoditypurchased multiplied by its marginal utility equals the like productfor the commodity sold. Or again: for a given purchaser the utili-ties of A and B, though actually unequal would be equal if everyportion of A (and also of B) were rated at the same degree of utilityas the last infinitesimal. This hypothetical equality underlies, aswill subsequently appear, the notion of the equality of values of Aand B.

§ 4 -

But the two definitions (1) and (2) do not fully determine thesense in which utility is a quantity. To define when the “grades”of two parts of a highway are equal or unequal (viz : when theymake equal or unequal angles with a horizontal), does not inform uswhen one shall be twice as steep as the other. It does not oblige usto measure the “grade” by the sine of the angle of elevation, orby the tangent, or by the angle itself. If the two highways wereinclined at 10° and 20° respectively, the “grades” have a ratio of1-97 if measured by sines, of 2-07 by tangents, and exactly 2 byangles. For a long time philosophers could define and determinewhen two bodies were equally or unequally hot. But not till themiddle of this century* did physicists attach a meaning to the phrase“ twice as hot.”

It is here especially that exactitude has been hitherto lacking inmathematical economics. Jevons freely confesses that “We canseldom or never affirm that one pleasure is an exact multiple ofanother.”f

Now throughout Part I the assumption is made that the utility ofany one commodity (or service) depends on the quantity of thatcommodity or service, but is' independent of the quantities of othercommodities and services. This assumption is preliminary to thedefinition we seek.

Our first problem is to find the ratio of two infinitesimal utilities.If an individual I consumes 100 loaves of bread in a year the utilityof the last infinitesimal, or to fix our ideas, the utility of the lastloaf is (presumably) greater than what it would be if he consumed150 loaves. What is their ratio f It is found by contrasting theutilities of the 100th and 150th loaves with a third utility. This

* The first thermodynamic definition of one temperature as a multiple ofanother was made by W. Thomson in 1848. See Maxwell, Theory of Heat, p.155.

+ p. 13.