in the theory of value and prices. 13 



until finally lie has given A gallons and received B bushels. At 



what point does he stop ? 



Although the " exchange values " of A gallons of (ci) and B 



bushels of {h) are equal, their utilities (to I) are not. He prefers B 



to 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 [h) ? He sold exactly A gallons for 

 B bushels. By stopping here he has shown his preference to buy 

 no more (postulate)o Ergo the utility of a small increment, say 

 another bushel of {b) is less than the utility of the corresponding 

 number 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 greater 

 than the gallons for buying it. Now by the mathematical principle 

 of continuity, if the small increment or decrement be made infinites- 

 imal <^B, the two above inequalities become indistinguishable, and 

 vanish in a common equation^ viz: 



ut. of c?B = ut. of dA. 



<:?B and c^A are here exchangeable increments. But the last incre- 

 ment c?B is exchanged for dK at the same rate as A was exchanged 



for B; that is 



A _ ^ 



B ~ t^ 



where each ratio is the ratio of exchange or the price of B in terms 

 of A. 



_?. - A 



^^ ^ ~ ^A 



multiplying this by the first equation, we have: 



ut. of c?B ^^ ut. of c?A . 

 f/B dK 



which may be written :* 



<^U d\] 



d^ ' ~~ dA ' 



The differential coefficients here employed are called by Jevons 

 " final degree of utility/'f and by Marshall " marginal utility.";|; 

 Hence the equation just obtained may be expressed: Eor a given 



* Of. Jevons, Pol. Econ., p. 99. f Jevons, Ibid., p. 51. 



X Marshall, Prin. of Econ., Preface, p. xiv. 



