in the theory of value and prices. 77 



for the purchase of A, B, and C, by each individual. Construct 

 three mutually perpendicular axes (OA, OB, OC,) in sj^ace. Con- 

 ceive this space to be filled with matter whose density distribution 

 is the total utility for A, B, and C, relative to a particular individual 

 I. There may be "empty" portions of space. The locus of points 

 representing combinations of A, B, and C, possessing a given utility 

 will be an indifference surface. All such loci will form a " family " 

 of concentric surfaces like the coats of an onion around one or more 

 points of maxima. 



Lay off on the A axis OA, equal to as many units of A as can be 

 bought for the sum of money disposable by I for the purchase of 

 A, Bj and C. Lay off OB and OC similarly defined. Draw the 

 plane ABC. This is the locus* of all consumption-combinations 

 of A, B, and C, purchasable with the given sum of money. It is a 

 "partial income plane." Its point of tangency with an indifference 

 surface will mark the chosen combination. A normal at this point 

 indicates the "maximum direction" and its A, B, and C components 

 are the marginal utilities, proportional to the prices of A, B, and C. 



§ 2- 



The utility distributions may be very complicated. If the three 

 articles are substitutes like oats, corn, and rye, the indifference sur- 

 faces may be almost plane and will allow but little change in the 

 orientation of the partial income plane, while each slight change 

 shifts the point of tangency greatly (cf. fig. 20 for two dimensions). 

 If they are completing articles as cuffs, collars, and ties the indiffer- 

 ence surfaces are arranged like concentric cocoons directed toward 

 the origin (cf. fig. 22 for two dimensions). 



But the three articles may be more intricately related in utility. 

 Of tea, coffee and sugar, the first two are substitutes while the last 

 is completing to both. If this triple completing and competing rela- 

 tion of articles were " perfect," the utility distribution would reduce 

 to a plane passing through the origin and cutting between the 

 " sugar " and " tea " axes, also between the " sugar " and " coffee " 

 axes. Several characteristics of such an ideal utility dependence 

 would exist. If the triple dependence is not " perfect " the plane 

 referred to swells out into a flat disk or rather a " f amilv " of con- 

 centric disks. The triple variation of prices and its effects on the 



*Forit8equa.iB— +_+_ = !, whence : A . — -h B .- -h C.^ = oO 

 or Apa + Bpb + Cpc = 50. 



