98 Irving Fisher — 3IathematiGal investigations 



attained and specified in Part I yields the maximum total utility, 

 for, since (Ch. lY, § 2) : 



?U dJJ dJJ 





d\j dJj dJS dJJ 

 dA^ dm^ • i a 5 ^^g^ ^^jy^^ • 



ret ore : 



dJJ dJJ dJJ 





dA^ __ dB^ _ _ dM^ 



Pa ~ Ph ' ~" Pr. 



(1) 



The numerators are the marginal utilities per unit of commodity. 

 To divide by the price is to make the unit of commodity the dollar's 

 worth. Each fraction is thu s the marginal utility per dollar's worth. 

 The equation expresses the fact that the rate of increase of utility 

 from spending more money on any one commodity equals the rate 

 of increase for any other. Hence by a familiar theorem of the cal- 

 culus the total utility must be the maximum attainable by any dis- 

 tribution of a fixed income. In like manner the individual dis- 

 tributes his production so that the marginal disutilities in all modes 

 of producing dollar's v/orth of commodity are equal so that his total 

 disutility is a minimum. Hence the difference between his total 

 utility and total disutility or his economic gain is a maximum . 



§2. 

 In the distribution of a single commodity over manj" individuals 

 since : 



dJS c?U c?U dJS dM dXS 



dA, 



— 



dm^ ' ^" ' dA^ ~ dm^ 



' Pa] ' ' 



' c?A„ dm^ 



therefore, 





dJJ dJJ 





cZU 







dA^ dA^ 





^A„ 





dU dV 





d\J' 



(2) 

 dm^ dm^ dm^ 



that is, the marginal utilities (when the unit of utility is the marginal 

 utility of money for each individual) are equal and the total utility 

 is maximum. In like manner the total disutilit}^ is a minimum and 

 srain therefore a maximum. 



§3. 



dJJ 



The first continuous equation may be divided by -' — and the 



