90 Irving Fisher — Mathematical investigations 



APPENDIX I. 

 MISCELLANEOUS REMARKS ON PART L 



I. FAILURE OF EaUATIONS. 



Jevons (p. 118) discusses the failure of equations for simple ex- 

 change. It is clear that such failure must frequently occur in com- 

 plex exchanges but no one has apparently commented on it. It would 

 seem at first sight that this would introduce an indeterminate element 

 into our results. Such however is not the case unless we take account 

 of articles neither produced nor consumed ; then the highest price 

 which any consumer will pay for the first infinitesimal is less than 

 the lowest price at which any one will produce it; there is no pro- 

 duction nor consumption and the term price has no determinate 

 meaning. As soon as changes in industrial conditions, tKat is in the 

 shape of the cisterns or their number makes this inequality into an 

 equality, the article enters into our calculations. 



Suppose A is produced by 71^ people, consumed by w^, and ex- 

 changed or retailed by ?^^, where 71^^ n^ and ?i^ are each less than ?i 

 (the number of individuals.) Moreover from the nature of our 

 former suppositions if any of the three are greater than zero all 

 must be, for anything once in the system is supposed to be produced,, 

 exchanged and consumed within the given period of time. 



The number of people who do not 



produce A is n—n^, 

 exchange A is n — n^, 

 consume A is n — n^. 



The number of unknowns dropped out of the equations in Ch. VI,. 



§ 2, is 



3n-(7i^-\-n^i-n^) of the type A^,^, A^,^, A,,^, etc., 



and Sn-in^-i-n^-^-nJ of the type ^, 



or Qn — 2{7i^-\-n^-{-nJ altogether. 



The failing equations in the first set are none, 

 " " '' " second " none, 



" " " " third " 3n—{n^-\'7i^-{-n^), 



" " " " fourth " 3?i — (;i^-f-?i^4-?^J, 



" " " " fifth " none, 



or 6n—2{?ip-{-n,-{-n,,) altogether. 



From the above agreement it appears that there can be no indeter- 

 minate case under the suppositions which were first made. Let us 

 look at this somewhat, more closely. 



