in the theory of value and prices.

11

for the purchase of A, B, and C, by each individual. Construct-three mutually perpendicular axes (OA, OB, OC,) in space. Con-ceive this space to be filled with matter whose density distributionis the total utility for A, B, and C, relative to a particular individualI. There may be “ empty ” portions of space. The locus of pointsrepresenting combinations of A, B, and C, possessing a given utilitywill be an indifference surface. All such loci will form a “family”of concentric surfaces like the coats of an onion around one or morepoints of maxima.

Lay off on the A axis OA, equal to as many units of A as can bebought for the sum of money disposable by I for the purchase ofA, B, and C. Lay off OB and OC similarly defined. Draw theplane ABC. This is the locus* of all consumption-combinationsof A, B, and C, purchasable with the given sum of money. It is a“ partial income plane.” Its point of tangency with an indifferencesurface will mark the chosen combination. A normal at this pointindicates the “maximum direction” and its A, B, and C componentsare the marginal utilities, proportional to the prices of A, B, and C.

§ 2 -

The utility distributions may be very complicated. If the threearticles are substitutes like oats, corn, and rye, the indifference sur-faces may be almost plane and will allow but little change in theorientation of the partial income plane, while each slight changeshifts the point of tangency greatly (cf. fig. 20 fpr two dimensions).If they are completing articles as cuffs, collars, and ties the indiffer-ence surfaces are arranged like concentric cocoons directed towardthe 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 lastis completing to both. If this triple completing and competing rela-tion of articles were “ perfect,” the utility distribution would reduceto a plane passing through the origin and cutting between the“ sugar ” and “ tea ” axes, also between the “ sugar ” and “ coffeu ”axes. Several characteristics of such an ideal utility dependencewould exist. If the triple dependence is not “ perfect ” the planereferred to swells out into a flat disk or rather a “ family ” of con-centric disks. The triple variation of prices and its effects on the

*For its equa. is ^ ^ = 1, whence : A . “■ + B

or A pa + Bps + Cpc = 50.

— + C . — = 50OB OC