This paper studies conditions under which aggregate demand behavior will satisfy the usual revealed preference axioms. Assuming a fixed distribution of income and the hypothesis that individual demand is homogeneous in income, it is shown that the weak axiom of revealed preference or the congruence axiom will hold in the aggregate if each individual demand satisfies the corresponding axiom. It is also shown that the hypothesis of homogeneity in income is not necessary for the weak axiom to hold in the aggregate. 1. INTRODUCrION THE PURPOSE OF THIS PAPER is to establish conditions under which aggregate demand behavior will have properties normally associated with individual demand when the distribution of income remains fixed. The best-known result of this type is that if each individual has a homogeneous concave utility function, and the distribution of income is fixed, then the aggregate demand correspondence will be one derived from a homogeneous concave utility function. This was first established by Eisenberg [4], though not in the context of demand theory, who employed duality theory of concave programming. More recently Chipman [1] interpreted Eisenberg's results from the point of view of demand theory and gave a proof of the aggregation theorem based in part on earlier work of Chipman and Moore [2 and 3]. In this paper utility functions will not be employed; instead, a revealed preference approach is taken. Strengthened forms of the weak axiom of revealed preference, the strong axiom of revealed preference, and the congruence axiom are used which are preserved in aggregation, and it is shown that demand correspondences homogeneous of degree one in income which satisfy the regular revealed preference axioms will also satisfy the strengthened versions. One advantage of this approach is that it shows the Eisenberg-Chipman aggregation theorem is a purely algebraic problem and does not require continuity or convexity assumptions. It will also be shown that there are demand functions not homogeneous of degree one in income which satisfy the strengthened form of the

[A consistent theory of demand is possible without the transitivity axiom. Here it is shown that a class of nontransitive orderings can be represented by a continuous numerical function, in such a way that an individual's demand function may be found by solving a constrained maximum problem.]