Foundations of Models with Complementarities

We generalize standard models with complementarities used widely in economics and other disciplines. We unify and expand their foundations, increase their scope of application, and identify previously unknown structural relationships among equilibrium sets in these models. Widely used results for existence of extremal equilibrium, nonempty complete lattice structure of the equilibrium set, and monotone comparative statics of extremal equilibria are generalized, subsuming standard models as special cases and allowing for new situations. We generalize the famous results due to Tarski (1955) and Zhou (1994), without using the strong set order or subcompleteness. Defining new set orders, we formulate new theories for structural comparisons of equilibrium sets, and prove new theorems for monotone comparative statics of the infimum equilibrium set, the supremum equilibrium set, and the full equilibrium set. These results are proved to hold in the standard models in the literature. Moreover, order comparability of equilibrium sets provides a new theory of order approximation of equilibria. This holds under weak conditions that are satisfied in the standard models