in the theory of value and prices.
13
until finally lie has given A gallons and received B bushels. Atwhat point does he stop ?
Although the “exchange values” of A gallons of (a) and Bbushels of ( b) are equal, their utilities (to I) are not. He prefers Bto the exclusion of A, for his act proves his preference (postulate).Therefore by definition (2) the utility of B exceeds that of A.
We may write:
ut. of B ]> ut. of A.
Why then did he cease to buy (b) ? He sold exactly A gallons forB bushels. By stopping here he has shown his preference to buyno more (postulate) Ergo the utility of a small increment, sayanother bushel of (b) is less than the utility of the correspondingnumber of gallons of (a) (Def. 2). Likewise he prefers to buy no less.Ergo the utility of a small decrement, say one less bushel is greaterthan the gallons for buying it. N ow by the mathematical principleof continuity, if the small increment or decrement be made infinites-imal dB, the two above inequalities become indistinguishable, andvanish in a common equation , viz:
ut. of dB = ut. of dA
dB and dA are here exchangeable increments. But the last incre-ment dB is exchanged for dA at the same rate as A was exchangedfor B; that is
A _ dAB dB
where each ratio is the ratio of exchange or the price of B in termsof A.
A - A : J ■ '
dB ~ dA
or
io/. • - • 4,
'V'
multiplying this by the first equation, we have:
ut. of c?B ut. of dA .
. B= --. A
dB
dA
wit
»■ -- (-
which may be written :*
tfU „ d U A
sb • B = jx ■ A -
The differential coefficients here employed are called by Jevons “final degree of utility,”! and by Marshall “marginal utility.”^Hence the equation just obtained may be expressed: For a given
* ‘A
! _
.yH-'
JJ
V //