in the theory of value and prices.

15

third utility is that of oil (say) of which let B gallons be consumedby I during the year. Let P be that infinitesimal or small incrementof 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 150thloaf will be pronounced by him equal (say) to the utility of £ P-Then the utility of the 150th loaf is said to be half the utility ofthe 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 _ P _ut. of 150th loaf Pj 2

B being total,B being total,C being total,C beirnr total.

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 definedby 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 p,and (2) ut. of 150th loaf = ut. of /J/2,also (3) ut. of 100th loaf = lit. of y,

To prove ut. of 150th loaf = ut. of y/2,where C is the quantity of another commodity (c) consumed by I inthe same period and y is such an increment of C that its utilityshall equal that of the 100th loaf.

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

ut. of 100th loaf = ut. of P — ut. of y,

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

Now, if the first total (100 loaves) be changed to 150, B and C beingunchanged, the above equation, dropping the first member, will stillbe true, viz :

ut. of p = ut. of y,

(150 loaves, B and C, totals),

for, by our preliminary assumption these utilities are independent ofthe quantity of bread.

* As a matter of fact an individual who, if consuming 100 loaves of breadwould consume B gallons of oil might, if consuming 150 loaves, use also moreoil. But this fact in no wise hinders our inquiring how he would reckon utili-ties if he used the same amount.