ABSTRACT
In considering the behavior of the consumer, a market is assumed which offer some n goods for purchase at certain prices and in whatever quantities. Apurchase requires an expenditure of money
e = π1ξ1 + · · · + πnξn = p′x which is determined as the scalar product of the vector x = {ξ1, . . . , ξn} of quantities, which shows the composition of the purchase, and the vectorp = {π1, . . . ,πn} of prevailing prices, where braces { } denote a column vector, and a prime its transposition. The classical assumption about the consumer is that any purchase is such as to give a maximum of utility for the money spent. The consumer is supposed to attach a number φ(x) to any purchase, according to its composition x, which is the measure of the utility, to the effect that a purchase with composition x made at price p and, therefore, requiring an expenditure, e = p′x, is such as to satisfy the maximum utility condition
φ(x) = max{φ(y) :p′y e}. An equivalent statement of this condition is
φ(x) = max{φ(y) : u′y 1}, where u = p/e is the vector of prices divided by expenditure, that is with expenditure taken as the unit ofmoney and is to be called the balance vector, corresponding to those prices and that expenditure. The fundamental property required for a utility function φ(x) is that, given a balance u, any composition xwhich is determined by the condition of maximum utility satisfies u′x = 1, so that
u′y 1 ⇒ φ(y) φ(x) and
φ(y) φ(x) ⇒ u′y 1.
