An Observation on Gross Substitutability and the Weak Axiom of Revealed Preference

Suppose that the consumption side of a constant returns production economy is described by an excess demand function f: R’,, -+ R’ satisfying the usual properties of continuity, homogeneity of degree zero, and p. f( p) = 0 for all p. A property of central importance is the Weak Axiom of Revealed Preference (WA). The excess demand function f satisfies WA iff(p)#f(q) and q.f(p)O. It is not difficult to verify that the WA property is necessary and sufficient if we are interested in guaranteeing that the equilibrium price set be convex (hence there is a unique equilibrium in the regular case) for any production technology. This was pointed out to us years ago by H. Scarf. The sufficiency part was originally demonstrated by Wald (1951). A familiar property in applications is Gross Substitutability (GS). The excess demand function f satisfies GS if q = p + ae,, a > 0, implies f’(q)fJ(p) for all j#i (here e, is the ith unit vector). There is an extensive literature on the GS property [see Arrow and Hahn (1971, chs. 8 and 9)]. Note that, in contrast to the WA, the GS