36 Irving Fish'-ir — Mathematical investigations 



§2. 



First of all an analysis \vi\\ serve to set the two preceding discus- 

 sions in a common point of Tiew. 



In any purchase the last infinitesimal commodity bought has a 

 utility equal to that of the money given, that is : 



ut. of dA z= ut. of dm ' 



dU ,. dV , A ^, T - X 



or : -.— dA = ^- . dm (see Ch. I, $ 3.) 



^^ d\ - '^-^ 



dA "'" dm 



. dm 



'JV 'TV 



dm 



dA dm 



' dA 



dU dT 



dA ~ dm ' 



P. 



or 



or : 



where p^ is the money price. 



That is, the marginal utility of a commo<iity (per pound, yard, 

 etc.) equals the marginal utility of money (per dollar) times the 

 ratio of exchange of money for commodity : 



This equation is fundamentaL In our first discussion (one com- 

 modity, various consumers) the marginal utility of money was sup- 

 posed constant so that 



dU 



dA^^'' 



or the marginal utility of a commodity is measured by it price. 



In the second discussion the other factor, the price, was supposed 



constant, and : 



dV 'fC 



oc 



c7A dm 



or the marofinal utilitv is measured bv the valuation of monev. 



§3. 



In the present chapter we are restricted to neither of these special 

 suppositions. For the individual I. we may write 



'/r dV 



dA^ ~~ dm^ 



du dx: 



<7B. *" dm^ 



•/'. 

 •i?» 



dV dV 

 <OI. — dm^ 



•i>. 



