34 Irving Fisher — Mathematical investigations 



(3) 



(Unit of commodity is dollar's worth.) 

 j fZU d\] d\5 I m — 1 independent equation. 



dA. c^B cl'Sl \ no new unknowns 



Number of equations = m 4- 1 + m — \-=: 2m. 

 " " unknowns = 2m + + = 2m. 



Hence the system is determinate. 



AGGREGATE INCOME. 



Let I, fig. 6, be the average curve* of all the separate commodity 

 curves A, B, C, . . . M, and let the new cistern have a thickness 

 equal to the sum of the thicknesses of the original cisterns. Then 

 the water in the resultant cistern equals the sum of that in the com- 

 ponents.* 



The liquid in the new cistern represents the money collectively 

 considered 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 separate 

 commodities is shown by the relation of the resultant cistern to the 

 components. 



* In this case the average is not a simple arithmetical mean but a weighted 

 average. Select points of like utility on the component curves, that is, points of 

 equal ordinates. Average their abscissas, multiplying each by the ratio of the 

 thickness 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 , p^, . . . p and 

 the abscissas x , x^, . . . x , the resulting thickness and abscissa (P and X) are : 



P =Pa+lh+ ■ ■ ■ ■ +Pm 



Pa+Pl,+ ' ■ ■ +P„, 



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 the 

 sum of the free surfaces in the component cisterns is 



X p +«?.». + . . . +x p 

 and the free surface in the resultant is 



iPn^Ph + - ■ -^Pj- 



X.P„ + ^kP>.+ - • •+^ P^ 



P„+P, + - ■ •+P^ 



Since these two expressions are equal and this equality holds of infinitesimal 

 layers at the free surface and so successively^ at all levels it must hold of the 

 sums of these layers. 



