12 Irving Fisher—Mathematical investigations

§2.

The sense in which utility is a quantity is determined by threedefinitions:

(1) For a given individual at a given time , the utility of A units,of one commodity or service (a) is equal to the utility of B units ofanother ( b ), if the individual has no desire for the one to the exclu-sion of the other.

A and B are here used as numbers. Thus if the first commodityis sugar and the second calico and if the individual prizes 2 poundsof sugar as much as 10 yards of calico, A is 2 and B is 10.

(2) For a given individual, at a given time, the utility of A unitsof (a) exceeds the utility of B units of ( b ) if the individual prefers(has a desire for) A to the exclusion of B rather than for B to theexclusion of A. In the same case the utility of B is said to be lessthan that of A.

The third definition will be given in §4.

The- two preceding definitions are exactly parallel to those of anyother mathematical magnitude.

Thus: two forces are equal if at the same time they alone act onthe same particle in opposite directions and no change of motionresults. One is greater when additional motion is produced in itsdirection. Again : “two masses are equal which if moving withequal velocities along the same straight line in opposite directionsand impinging on each other are reduced to rest by the collision.”*Two geometrical magnitudes are equal if they can be made to coin-cide, etc., etc.

Just as coincidence is the test of equality and inequality of geo-metrical figures, and the tip of the scales the test of equality andinequality of weights, so is the desire of the individual, the test ofthe equality and inequality of utilities. It is to be noted that ineach definition of equality the word “ no ” or some equivalentoccurs. A standard mode of cancellation is thus designated.

§3-

Let us see how these definitions of utility apply to an act of pur-chase. An individual I enters a market with fixed prices to ex-change some of a commodity (a) for another ( b). We may sup-pose prices to be such that he gives one gallon of (a) and receives twobushels of (6), then a second gallon for two more bushels and so on

* Price, Calculus, vol. iii, p. 316.