68 



Irving Fisher — Matheyaatical investigations 



18. 



as follows. He is directed to alter this consumption combination 

 by arranging his quantities A and B of the two selected commodities 

 («)• and [IS) in all j^ossible ways, but Avithout changing the quantities 

 C, D, etc. of other commodities. The marginal utility of each will 

 vary not only in relation to its own quantit}^ but also the quantity 

 of the other commodity. Thus, 



g; = F(A..B.) 



f^ = F(B.,A, 



These may be regarded as derivatives with respect to A and B of 



r. = <p(A„B,) 



where U^ is the total utility to I of the consumption combination 

 A, and B^. 



In fig. 18 let the abscissa OX represent the quantities B^ of {h) and 



the ordinates (OY) the quantities A^ of («). 

 Any point P by its co-ordinates represents 

 a possible combination of quantities A^ 

 and Bj consumed by I. B}^ varying point 

 P all possible combinations of A^ and B^ 

 are represented. At P erect a perpen- 

 dicular to the plane of tlie page whose 

 length shall represent the marginal utility 

 of A^ for the combination, that is, the 

 degree of utility of a small addition of 

 Aj, (B, remaining the same). If P as- 

 sumes all possible positions, the locus of the extremity of this per- 

 pendicular will be a surface. 



Again at P erect a different perpendicular for the marginal utility 

 of Bj its extremity will generate another surface. The first sur- 

 face takes the place of a utility curve for [a), the second for (b). 

 These two surfaces may be regarded as the derivative surfaces 

 (with respect to the variation of A^ and of B^), from a primitive 

 whose ordinate (perpendicular at P), is the total utility of the com- 

 bination of Aj and B^ represented by the point P. This surface is 

 usually convex like a dome with a single maximum part, but it need 

 not always be. There may be two maxima as will presently appear. 

 In such a case it cannot be everywhere convex. 



If a plane be drawn tangent to this last surface at a point over P, 

 the slope of the plane parallel to the A direction will be the ordinate 



