The Le Chatelier Principle in lattices

The standard Le Chatelier Principle states that the long-run demand for a good (in which by definition there are fewer restraints on the variables) is more elastic than short-run demand. The fundamental insight above goes well beyond demand theory, and proofs of this basic idea have been found in various settings. Nearly all of these have been continuous optimization problems requiring assumptions on the continuity of the objective function and on the convexity of the choice set. However, the statement and intuition for the original principle do not seem to rely on any such `technical' assumptions. Work by Milgrom & Shannon on monotone [ordinal] comparative statics provides an obvious framework to pursue a broader result in a discrete environment. The present paper therefore formulates and proves a very general Le Chatelier Principle in the context of lattices. A further generalization is that we allow the choice set to vary (potentially as a function of the underlying parameter).