18
Irving Fisher—Mathematical investigations
(5) the marginal utility of any arbitrarily chosen commodity onthe margin of some arbitrarily chosen quantity of that commoditymay serve as the unit of utility for a given individual at a giventime.
This unit may be named a util.
Any unit in mathematics is valuable only as a divisor for asecond quantity and constant only in the sense that the quotient isconstant, that is independent of a third quantity. If we shouldawaken to-morrow with every line in the universe doubled, weshould never detect the change, if indeed such can be called achange, nor would it disturb our soiences or formulae.
§*•
With these definitions it is now possible to give a meaning toJevons ’ utility curve, whose abscissas represent the amounts of acommodity (say bread) which a given individual might consumeduring a given period and the ordinates, the utilities of the last (i. e.the least useful) loaf. For if corresponding to the abscissa 100loaves an ordinate of arbitrary length (say one inch) be drawn tostand for the utility of the 100th loaf, we may use this as a unit{util.) For any other abscissa as 85 loaves whose marginal utilityis (say) twice the former, the ordinate must be two inches, and soon. For any other commodity as oil the marginal utility of Agallons being contrasted with the utility of the 100th loaf of breadand this ratio being (say) three, an ordinate of three inches must bedrawn. In all the curves thus constructed only one ordinate isarbitrarily selected, viz: that representing the utility of the 100thloaf.
§ 8 .
Only differentials of utilities have 'hitherto been accounted for.To get the total utility of a given amount of bread we sum up theutilities for the separate loaves. Or in general:
(6) The total utility of a given quantity of a commodity at agiven 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.. (<fe,) + ut. (dxj + ... ■ + ut. (dx„)= J** ut. (dx)
= f
Jo
* <ZU
dx
dx.