78 



Irving Fisher — 3Iathematical investigations 



relative amounts of the three articles (that is on the position of 

 the point of contact) can readily be discerned by its aid. Far more 

 complicated cases are supposable and exist in reality. 



§ 3. 



If we suppose for an instant that there are but three commodities 

 in the market, the preceding analysis yields a complete account of 

 the equilibrium in that market. 



To sketch this briefly let us suppose the space to be filled with a 

 utility density for I, another superposed but different distribution 

 for II, and so on. Let us include production. If one man should 

 be both a consumer and producer of the same article, the 7iet con- 

 sumption or production is now to be taken, and the total utility or dis- 

 utility of this net amount is the density. The planes before referred 

 to as j^artial income j)lanes may now be called "total income and 

 expenditure planes," and they must each 2^ass through the origin* 



27. 



(01, fig. 27 for two dimensions). Since the "maximum directions" 

 (normal to their planes) are parallel, these planes must all coincide. 

 The point in this plane selected by I will be that of tangency to an 

 indifference surface for I. Likewise for II, HI, etc. Such points 



*For since income balances expenditure, if Ai, Bi, Ci, represent the (net) 

 amounts consumed or produced by I, those consumed being treated as positive, 

 and those produced as negative, the whole money value must be zero : i. e. 



A . i?„ + B . 2J^ + C . p^ = 0, 

 which is the equation of a plane passing through the origin. 



