96 Irving Fisher — Mathematical investigations 



just begin to buy, then his consumer's rent or gain is (see Ch. I, § 8) 



^^J^'yxdy — y^J'^^' xdy 

 yh yk 



or measuring this gain in the given commodity as valued at y^ cents 

 (say) per unit, 



This may be interpreted by a simple geometrical construction. 

 In fig. 38 OA zz: y^ and OR — y^. 



39. 



R 



IN 



7 



A / ^ 



4' \'\ ' 



A /// \, 



;b 



'/2 M 



E 



Selecting the point 3 make the evident dotted construction deter- 

 mining a point 5. 



y_ 



^.-,,75 05 07 



Evidently: - := ^ ^ ^ 



75 



<■■')■ 



Let 3 assume all positions from M to 6. Then 5 will trace a curve 



26. 



The area described by the moving line 73 is evidently / xdy. 

 The area described by the moving line 75 is evidently / ( '— x \dy. 



*/y, \y. I 



Hence area described by the moving line 35 is the difference of 

 these expressions or G/y^. 



That is the area M62 represents the gain measured in commodity. 



Thus suppose a person buys com measured by RAM6 and let corn 

 at the valuation RO be the unit of utility. It is only the last layer 

 R6 on which no gain is felt. For any preceding layer 75 the price 

 really paid is OR while the price which it is worth to him is 07. 

 The layer 73 may be considered as lengthened in his e3^es by that 



