Ever since Antonelli noted ([ 2], [3]) the "integrability" (symmetry) conditions necessarily obeyed by an indirect demand function derived from maximizing a utility function, and ever since Volterra emphasized ([37], [38]) their importance to Pareto's attempt [23] to construct utility from consumer purchase data these conditions have retained a technical character eluding intuitive motivation.It is our purpose here to show that an axiom of Ville ([35], [36]) provides an intuitively appealing equivalent of these symmetry conditions.In doing this, with the help of our integrability theorem of [16], we will extend Ville's result and, we hope, clarify his very important contribution to axiomatic consumer theory.lFrom the dual versions ([27], Theorems 16 and 18) of an extension of Hurwicz and Uzawa's Theorems 1 and 2 [17], we know, roughly speaking, that the following two conditions together are equivalent to utility-rationality2 of a given C 1 competitive inverse demand function satisfying the budget identity: negative semi-definiteness of the Antonelli matrix 11.5) below), and symmetry of the Antonelli matrix.From duality theorems ([27], Theorems 20 and l2(b) applied to a recent result of Kihlstrom, Mas-Colell, and Sonnenschein ([19], Theorems 1 and 2), we know that the first (negative semi-definiteness) condition is equivalent to a weak version of the intuitively appealing Weak Axiom of Revealed Demand Preference. 3 What about the second condition, the