500 



lotka: an objective standard of value 



perfect, economic activities and transactions, such as exchange 

 upon the market, would be conducted in such manner as to 

 make the rate of increase per head, r, of the community a 

 maximum. 



In these circumstances, if mi, m^, . . . fi, fi, . . . are the 

 quantities of the commodities ^i, A^, . . . Fi, F2, . . . con- 

 sumed by a representative individual per unit of time, we must 

 have 



diUi + dmo + 



drrii ' ' dm-z 



for all values of dmi, dnh, 

 the equations of exchange 



dnii = dix2 + djxz -f- . . 

 V2 dnii = — Vi djU2 

 V3 drrii - - Vi c?M3 



v'l dfi = - Vi dipi 

 v'2 dfi = — Vi d<p'i 



■ +^df, + ^df,+ . . . = (I) 

 . . dfi, dfi, . . . compatible with 



+ d,pi + d<p2 + 



(ID 



where dfxo is the quantity of Ai taken in exchange for A^ 

 dixz is the quantity of Ai taken in exchange for As 



dipi is the quantity of Ai taken in exchange for Fi 

 etc., etc. 

 By (I) and (II) we then have 



dr 

 bnii 



(dp2 + f/yUs + . . . + d<pi + d(p2 + . . .) 



dr Vi , ^ 



bnii V2 



dr Vi , 

 Ofis — . 



bms V3 



dr Vi , dr Wi , 

 dfi v'l d/2 v'2 



= 



(III) 



This relation must hold for all arbitrary values of the infini- 

 tesimals C?/X2, C?M3, . . . dipi, d(p2, ... 



It follows that 



. br . dr dr ^ dr ^ , 



dm, dm2 dfi d/2 



Vj : V2 : . . : v'l : v'o 



