in the theory of value and prices. 15 



third utility is that of oil (say) of which let B gallons be consumed 

 by I during the year. Let /5 be that infinitesimal or small increment 

 of B whose utility shall equal that of the 100th loaf. Now in sub- 

 stituting the hypothesis of 150 loaves let us not permit our individ- 

 ual to alter B, his consumption of oil.* The utility of the 150th 

 loaf will be pronounced by him equal (say) to the utility of ^ /^. 

 Then the utility of the 150th loaf is said to be half the utility of 

 the 100th. 

 That is, if : 



ut. of 100th loaf = ut. of /?, B being the total, 



and ut. of 150th loaf =: ut. of /?/2, B being the total again, 



the ratio is defined : 



ut. of 100th loaf _ /5 _ 



ut. of 150th loaf ~ y^ ~ ■ 



It is essential to observe that if the 100th loaf is twice as useful 

 as the 150th when their ratio is defined as above in terms of incre- 

 ments of oil, it will also be twice as useful when the ratio is defined 

 by any other commodity ; also that it matters not what total quan- 

 tity (B) of oil or other commodity is employed. 



This theorem may be thus stated: 



Given (1) ut. of 100th loaf = ut. of /?, B being total, 



and (2) ut. of 15 0th loaf = ut. of /?/2, B being total, 



also (3) ut. of 100th loaf = ut. of y, C being total. 



To prove ut. of 150th loaf = ut. of 7/2, C being total, 



where C is the quantity of another commodity (c) consumed b}^ I in 

 the same period and y is such an increment of C that its utility 

 shall equal that of the 100th loaf. 

 We may write from (1) and (3): 



ut. of 100th loaf = ut. of /5 — ut. of y, 

 (100 loaves, B and C, being totals). 



Now, if the first total (100 loaves) be changed to 150, B and C being 



unchanged, the above equation, dropping the first member, will still 



be true, viz : 



ut. of I'd = ut. of y^ 



(150 loaves, B and C, totals), 



for, by our preliminary assumption these utilities are independent of 

 the quantity of bread. 



* As a matter of fact an individual who, if consuming 100 loaves of bread 

 would consume B gallons of oil might, if consuming 150 loaves, use also more 

 oil. But this fact in no wise hinders our inquiring liow he would reckon utili- 

 ties if he used the same amount. 



