Irving Fisher—Mathematical investigations

(Unit of commodity is dollar’s worth.)

j <flT d\J _ dU )m — 1 independent equation.

I dA dB ' dM \ no new unknowns.

Number of equations = m + 1 + wt — 1 = 2wi.

“ “ unknowns = 2m + 0 + 0 = 2m.

Hence the system is determinate.

§5.

AGGREGATE INCOME.

Let I, fig. 6, be the average curve* of all the separate commoditycurves A, B, C, . . . M, and let the new cistern have a thicknessequal to the sum of the thicknesses of the original cisterns. Thenthe water in the resultant cistern equals the sum of that in the com-ponents.*

The liquid in the new cistern represents the money collectivelyconsidered and the ordinate the utility of the last dollar.

If this income increases, its marginal utility decreases and de-creases in a law whose relation to the laws of utility for the separatecommodities is shown by the relation of the resultant cistern to thecomponents.

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* In this case the average is not a simple arithmetical mean but a weightedaverage. Select points of like utility on the component curves, that is, points ofequal ordinates. Average their abscissas, multiplying each by the ratio of thethickness of its cistern to that of the resultant cistern (viz : the sum of the thick-nesses of the original cisterns). Thus if the thicknesses are p a , p^, . . . p m andthe abscissas x a , x b , . . . x m , the resulting thickness and abscissa (P and X) are :

P =P a +Pi + ■ ■ ■ ■ +P m

X gPg + X l,P / , + • • • +x m P„.

Pa+Pi+ ■ ■ • +P m

If in a cistern thus formed liquid enters to the level of the component cisterns,the liquid in the resultant cistern equals the total in the component. For thesum of the free surfaces in the component cisterns is

X aPa + X tPi + ‘ ■ • +X mPmand the free surface in the resultant is

( P a +P b +• • -+P„)

/ * a P. + *iPi + - ■ ■+ x m P m \\ P.+P t + ---+P m j

Since these two expressions are equal and this equality holds of infinitesimallayers at the free surface and so successively at all levels it must hold of thesums of these layers.