Unified gross substitutes and inverse isotonicity for equilibrium problems

We introduce a notion of substitutability for correspondences and establish a monotone comparative static result. More precisely, we introduce the notions of unified gross substitutes and nonreversingness and show that if Q: P ⇉ Q is a supply correspondence defined on a set of prices P , which is a sublattice of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mrow> <a:mi mathvariant="double-struck">R</a:mi> </a:mrow> <a:mrow> <a:mi>N</a:mi> </a:mrow> </a:msup> </a:math>, and Q satisfies these two properties, then the set of equilibrium prices Q −1 ( q ) associated with a vector of quantities q ∈ Q is a sublattice of P and is increasing (in the strong set order) in q . We establish connections between our notion of substitutes and existing notions, and examine applications such as the structure of inverse demand, profit maximization, the structure of competitive equilibria, matching games, hedonic pricing, and routing problems.