in the theory of value and prices.

79

could be found whatever the position of the plane. But the planemust assume such an orientation that the center of gravity of thesepoints shall be the origin. That is the algebraic sum of all the Acoordinates consumed must equal the sum produced. Likewise thealgebraic sum of the B and C coordinates must each be zero.

Hence with the geometrical analysis just described the equilib-rium for a market of three commodities is determined when :

(1) All individuals’ combinations lie in a common plane throughthe origin (each individual’s sales and purchases cancel).

(2) Each individual’s combination is at the point where this planeis tangent to an indifference surface for that individual (the point ofmaximum net utility).

(3) The points in the plane are so distributed as to make the origintheir centre of gravity (the production and consumption of each com-modity balance).

Whence it follows geometrically that the “ maximum directions ”are parallel, their components (marginal utilities) proportional asbetween different individuals and that this proportion is that of theorientation of the plane (the ratio of prices).

§ 4 .

When this equilibrium is attained, let us, through the point oftangency I, representing the consumption combination for I, pass asection parallel to the plane of the A and B axes. The section ofthis plane with the total income plane gives a straight line which isnone other than the partial income line of Ch. I, § 11 and its sectionwith the indifference surfaces gives back the indifference curves ofCh. I, § 10.

§5.

We have temporarily assumed only three commodities for we haveonly three dimensions wherewith to represent them. A completepresentation of the interdependence of utilities would require mdimensions, for the utility of any one commodity A, is subject tom independent variations according to a change in any one of them commodities, though (in general) the change of the quantity Aitself is most important.

There is a curious glamour over “the fourth dimension.” Thepopular interest is all to prove that it “ exists.” Its origin histor-ically and its present usefulness is in the interpretation of a fourthindependent variation, i. e. in representing just such relations as now