18 Irving Fisher — Matheraatical ioivestigations 



(5) The marginal utility of any arbitrarily chosen commodity on 

 the margin of some arbitrarily chosen quantity of that commodity 



^nay serve as the unit of utility for a given individual at a given 

 time. 



This unit may be named a util. 



Any unit in mathematics is valuable only as a divisor for a 

 second quantity and constant only in the sense that the quotient is 

 constant, that is independent of a third quantity. If we should 

 awaken to-morrow with every line in the universe doubled, we 

 should never detect the change, if indeed such can be called a 

 change, nor would it disturb our sciences or formulae. 



With these definitions it is now possible to give a meaning to 

 Jevons' utilitj^ curve, whose abscissas represent the amounts of a 

 commodity (say bread) which a given individual might consume 

 during a given period and the ordinates, the utilities of the last (i. e. 

 the least useful) loaf. For if corresponding to the abscissa 100 

 loaves an ordinate of arbitrary length (say one inch) be drawn to 

 stand for the utility of the 100th loaf, we may use this as a unit 

 {iitil.) For any other abscissa as 85 loaves whose marginal utility 

 is (say) twice the former, the ordinate must be two inches, and so 

 on. For any other commodity as oil the marginal utility of A 

 gallons being contrasted with the utility of the 100th loaf of bread 

 and this ratio being (say) three, an ordinate of three inches must be 

 drawn. In all the curves thus constructed only one ordinate is 

 arbitraril}" selected, viz: that representing the utility of the 100th 



loaf. 



§8. 



Only differentials of utilities have hitherto been accounted for. 

 To get the total utility of a given amount of bread we sum up the 

 utilities for the separate loaves. Or in general: 



(6) The total utility of a given quantity of a commodity at a 

 given time and for a given individual is the integral of the mar- 

 ginal utility times the differential of that commodity. 



That is : 



ut. of (x) = ut. {d^\) + ut. (d.\\) +....+ ut. {d?.\) 





f 



Ut. {dx) 



'■^•d\J , 

 —r- ax. 

 dx 



