Existence and uniqueness of price equilibria. Theory and application to discrete choice models

Continuous excess demand systems which do not obey homogeneity of degree zero or Walras's Law are proved to have equilibria if they satisfy certain mild regularity conditions when prices tend to the extremes of a price domain which need not be closed or bounded. A straightforward generalization of Brouwer's theorem is used. Systems also obeying a weak balance condition (of which Walras's Law is a special case) and homogeneity are treated as corollaries to the main theorem. Sufficient conditions for differentiable excess demand systems to have unique equilibria are developed in three separate theorems. The usefulness of these general existence and uniqueness theorems is demonstrated by applying them to three specific models constructed from discrete choice theory: (1) a competitive rental housing market, (2) a regulated rental housing market with fixed rents and rationing and (3) an interregional labor market in which laborers can choose among regions for employment (or voluntary unemployment) as well as the work hours they will supply.