On a System of Inequalities in Demand Analysis: An Extension of the Classical Method

AN EXPENDITURE CORRESPONDENCE being defined as a correspondence between budget constraints and admitted commodity bundles, a demand system, or its associated expenditure system, appears as one special case, and another case is that of a finite correspondence. A theorem will be proved on the existence of a positive solution for a certain system of homogeneous linear inequalities. Such a system can be associated with any finite expenditure correspondence, together with a number e between 0 and 1 interpreted as a level of cost-efficiency. The existence of a solution is equivalent to the admissibility of the hypothesis that the consumer, whose behavior is represented by the correspondence, (i) has a definite structure of wants, represented by an order in the commodity space, and (ii) programs at a level of cost-efficiency e. Any solution permits the immediate construction of a semi-increasing polyhedral concave utility function which realizes the hypothesis. The method for this construction is shown. When e = 1 the utility function fits the data exactly, in the usual sense that its maximum under any budget constraint is at the corresponding commodity point, and when e < 1 it can be considered to fit it approximately, to an extent indicated by e. A formula is given for the critical cost-efficiency, defined as the upper limit of possible e. Standard demand analysis which involves a strict maximum under the budget constraint, expressed also by the revealed preference idea, is put in perspective with this approach.