66 Irving Fisher — Mathematical investigations 



^Q^T\Q. perfect substitutes as such that this ratio is absolutely constant. 

 The essential attribute of completing articles is that the ratio of 

 the quantities actually produced and consumed tends to be constant 

 (as many shoe-strings as shoes for instance, irrespective of cost). 

 We may define perfect completing articles as such that this ratio is 

 absolutely constant. 



If we suppose each set of competing and completing articles to 

 be " perfect," it is possible to arrange the cisterns so that the change 

 of form of some cisterns as due to change in the contents of other 

 cisterns shall be small or nothing. Thus if four grades of flour be 

 "perfect" competing, so that their marginal utilities are always in 

 the ratio 8, 9, 11, 17, we may form a joint cistern for individual I 

 whose contents shall be "flour," the quality unspecified. Each 

 cubic unit of liquid shall represent equivalent quantities of each 

 grade, i. e. ^ barrel of the first quality, ^ of the second, -^^ of the 

 third or -^ of the fourth, while the ordinate shall represent the com- 

 mon utility of any one of these equivalent quantities. 



If four completing articles as the parts of a coat, sleeves, pockets, 

 buttons, and coat proper are always produced and consumed in num- 

 bers proportional respectively to 2, 4, 3 and 1 , we may form a joint 

 cistern for individual I whose contents shall be " coats," parts un- 

 distinguished. 



With such combinations as these, the cistern analysis of Part I 

 will represent the economic relations fairly well and almost per- 

 fectly if the deviations from equilibrium are not followed too far. 



But few articles are absolutely perfect representatives of either 

 the competing or the completing group, and a member of one group 

 may also belong to another. Thus butter is completing to bread 

 and biscuit, and although a cheapening of bread directly increases 

 the utility of butter it indirectly increases it by decreasing the use 

 of biscuit. 



It is readily seen that the interrelations of the shapes of the cis- 

 terns — if we now treat each quality of meat, etc. and each part of a 

 utensil as a separate commodity — are too complicated even to be 

 mentally representable without some new mode of analysis. 



§5. 



The former analysis is incomplete, not incorrect. All the inter- 

 dependence described in Part I exists, but there also exist other 

 connections between the shapes of the cisterns which could not be 

 mechanicallj^ exhibited. For any one position of equilibrium the 



