Editing General equilibrium theory (section)

===Uniqueness===

{{See also|Sonnenschein–Mantel–Debreu theorem}}

Although generally (assuming convexity) an equilibrium will exist and will be efficient, the conditions under which it will be unique are much stronger. The [[Sonnenschein–Mantel–Debreu theorem]], proven in the 1970s, states that the aggregate [[excess demand function]] inherits only certain properties of individual's demand functions, and that these ([[Continuous function|continuity]], [[Homogeneous function|homogeneity of degree zero]], [[Walras' law]] and boundary behavior when prices are near zero) are the only real restriction one can expect from an aggregate excess demand function. Any such function can represent the excess demand of an economy populated with rational utility-maximizing individuals.

There has been much research on conditions when the equilibrium will be unique, or which at least will limit the number of equilibria. One result states that under mild assumptions the number of equilibria will be finite (see [[regular economy]]) and odd (see [[Poincaré–Hopf theorem|index theorem]]). Furthermore, if an economy as a whole, as characterized by an aggregate excess demand function, has the revealed preference property (which is a much stronger condition than [[revealed preference]]s for a single individual) or the [[substitute good|gross substitute property]] then likewise the equilibrium will be unique. All methods of establishing uniqueness can be thought of as establishing that each equilibrium has the same positive local index, in which case by the index theorem there can be but one such equilibrium.