A pre-recorded version of Leonid Hurwicz' Prize Lecture was presented on 8 December 2007 at Aula Magna, Stockholm University. The lecture was introduced by Professor Jorgen Weibull, Chairman of the Economics Prize Committee.

Recent formulations of social decisions problems focus on rules by which a group of individuals (the society) arrives at a choice among available alternatives. Such a rule, which we shall call the outcome function (Gibbard's "game form, " see [ 3 J), specifies the

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

This paper is a sequel to On the Stability of the Competitive Equilibrium, I, by K. J. Arrow and L. Hurwicz. It extends the results of I in several directions. In particular, it provides a proof of stability in the large (and not merely locally) when all goods are gross substitutes; this result is found to be valid for processes where the price adjustment rate is a continuous sign-preserving, but not necessarily proportionate, function of excess demand. The paper deals both with systems where one of the commodities plays the role of numeraire and with systems where all commodities are treated symmetrically.