12 Irving Fisher — Mathematical investigations 



The sense in which utility is a quantity is determined by three 

 definitions: 



(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 of 

 another (^), if the individual has 7io desire for the one to the exclu- 

 sion of the other. 



A and B are here used as numbers. Thus if the first commodity 

 is sugar and the second calico and if the individual prizes 2 pounds 

 of 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 units 

 of (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 the 

 exclusion of A. In the same case the utility of B is said to be less 

 than that of A. 



The third definition will be given in § 4. 



The two preceding definitions are exactly parallel to those of any 

 other mathematical mas^nitude. 



Thus: two forces are equal if at the same time they alone act on 

 the same particle in opposite directions and no change of motion 

 results. One is greater when additional motion is produced in its 

 direction. Again : "two masses are equal which if moving with 

 equal velocities along the same straight line in opposite directions 

 and 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 and 

 inequality of weights, so is the desire of the individual, the test of 

 the equality and inequality of utilities. It is to be noted that in 

 each definition of equality the word " no " or some equivalent 

 occurs. 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 two 

 bushels of {b), then a second gallon for two more bushels and so on 



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



