70 



Irv ing Fisher — JIathematical investigations 



suppose (a) and {h) are two brands of flour. If I consumes during 

 the period X units of one brand and 20 units of the other his desire 

 for a 21st unit of the latter will de2->end on how much he has of 

 the former (how large X is). If he has much of the first kind his 

 desire is small. 



A similar pair of curves may be found by moving P horizontally. 



If the supposition in Part I were true the two strange curves (viz: 

 connecting marginal utilities of A and 13 with quantities of B and 

 A, respectively), would reduce to straight lines parallel to the plane 

 of the paper. 



§ 10. 



The relations indicated by these three surfaces are really all 

 included in one of them — the primitive. Consequently, to avoid 

 troublesome transitions from one mode of representation to another 

 we shall hereafter confine ourselves to this primitive surface. 



Consider horizontal sections of this surface, that is sections j^ar- 

 allel to the plane of the A and B axes. Each section forms a curve 

 which may be called an indifference curve. It is the locus of points 

 representing all consumption-combinations of A and B which have a 

 o'iven total utilitv. In fio-. 18 the attached number to each curve 

 represents the amount of this ntility. They in general form a 

 family of concentric curves vanishing finally at the point M of max- 

 imum satisfaction. M is the point at which the individual would 

 arrange his consumption-combination of A and B if they cost noth- 

 ing. There may be two or more maxima. For competing articles 

 these maxima may lie in the axes (fig. 19), for one may prefer not to 

 consume both. 



The ordinates may of course have any units of length. Suppose 

 19. this unit to be indefinitely reduced from an 



inch to a millimeter, etc. Then our surface 

 becomes a layer. Its thickness may be fig- 

 ured as a density (rather than an ordinate), 

 distributed over the plane of the paper as 

 electricity over a conductor. Each indif- 

 ference curve is the locus of points where 

 the density (formerly ordinate), is a given 

 amount. This idea of density will be hence- 

 forth used though the necessity for its use 

 does not come till the next chapter. 



Fig. 20 shows the curves for competing articles and fig. 22 for 

 completing. For '* perfect " substitutes the curves (fig. 21) reduce 



