82 Irving Fisher — Mathematical investigations 



§ 10- 

 It is seen that analytically the treatment of interdependent com- 

 modities differ from that of independent commodities only in this, 

 that the equations which represent the functions have more letters ; 

 i. e. we have 



^ = F(A, , B. . . . N,) iustead of = F(A,). 



All other equations are just as in Part I. In fact these function 

 equations are, so to speak, the residuary formulae ; they contain all 

 the unanalyzed conditions of the problem. 



The marginal utilities are (as in Part I) in a continuous ratio 

 which is the ratio of prices. Yet there are some peculiar cases 

 which could not occur under the suppositions of Part I, viz : those 

 cases arising when the marginal utility of one or more articles has 

 no meaning. 



If two articles are perfect completing articles, as gun and trigger, 

 there is no such quantity as the marginal utility of triggers alone. 

 There is, however, a marginal utility of a combined gun and trigger. 

 Now there are separate maririnal disutilities for producing the o-un 

 and trigger. How are all these quantities to be introduced into our 

 continuous proportion of marginal utilities V 



Suppose for a moment thei-e were no difficulty of this sort. The 

 proportion for each indiAddual would be just as before (Part I, 

 Ch. IV, § 10) and might be expressed as follows [G&</ for gun T&^ 

 for trigger] : 



r p, + Pi 1 _ p&p^ 



j ^dX_ ^dV \ dV 



I ,K\ ^ </T. J <7(G&T). 



Finally: It is clear that 



- ^. c?U cm f'U 



I = Aa -f- B^ T- . . . + M/<( and v Ui = — — a + — — 6 + . . . + -r^ m . 



dKi (?Bi c/Mi 



Substituting these values in I . v Ui = we have after perfonning the multipli- 

 cation 'and remembering that a . (i = \ and <^f .?;=«. c =...= 6 . c— . . . r=0, 



A, —- + B, ---+... +Mi -—= 

 f'Ai r/B, (^M, 



or since prices are proportional to marginal utilities: 



Ai p„ + B, pi + . . . M, p;„ = 0. 

 Likewise for II, III, etc. making n equations. 



Conversely we could derive the vector equations from the scalar. 



