58 



Irving Fisher — Matliematical investigations 



changes possible are effected by change in the forms of the cisterns 

 or by changing their numher, that is by changing the " cost" of pro- 

 duction or the utility of consumption, or by changing the population 

 (which changes, we may remark, go together). By making the 

 cisterns removable and replaceable the effects of varied conditions 

 can be studied as in the preceding chapter. 



However, this equilibrium is indeterminate in one respect. Unlike 

 the former it does not fix the unit of value. The sum of the 

 income-cistern-contents is arbitrary. If all duplicate income-and- 

 expenditure-pistons are simultaneously depressed so as to increase 

 all incomes proportionately, the equilibrium will not be upset nor 

 will the distribution of commodities be affected. The rear cisterns 

 will simply dilate in uniform* ratio. The money standard has alone 

 changed. 



This may be remedied by making the thicknesses of all hack cis- 

 terns for the coinmodity A equal to unity. A thus becomes the stand- 

 ard of value, and henceforth all prices are in terms of this com- 

 modity. This is what is done in the actual world. 



§ 6. Analytical. 



A,,,-l-A^,, + ... +A^,, := A^,i + A^,,-f-... +A,,„ ^1 



B^,i + B^,, + . . . +B^,„ = B^,,+B^,2+ . . . +B^,,, ! m equations. 



... . -. . . j Imn unknowns. 



M,,, + M^,,-f ... +M^,.=M,,,+M,.,+ ...+M,„ J 



^ (^^—1) inde- 



^^..x-Pa^ '"- ■^^K^■Pm=^.^^Pa+ • • • + M^,, . ;^, | pendciit equa- 



- - y tions. 



I 

 A n 4- +M » — A 7) + +M )) \ m new un- 



J knowns (prices) , 



'^^ =F(A^,,); 





dV 



d\J 

 dA^r 



= F(A„,.) ; . . 

 = F(A„.,) ; . . 





-- F(M,,.,) 



J 



1 



J J 



2m?i equationi 



2 77in new un- 

 knowns 

 (marg. ut.). 



*Cf. Ch. IV, §8, number?. 



