A Comment on: “Monotone Comparative Statics”

Milgrom and Shannon (1994) provide necessary and sufficient conditions on parameterized optimization problems for their solution sets to be globally monotone in the parameter. We establish that their conditions may be significantly relaxed when focusing on discrete, binary comparisons between solution sets. Such binary comparisons are ubiquitous in economics and may involve comparing the same decision maker across two distinct regimes or two distinct decision makers with related objectives (e.g., a monopolist firm versus a social planner). While the single-crossing property remains prominent in the theory, quasisupermodularity of the objective functions of interest is not needed. Our approach relies upon a novel method of embedding a new optimization problem with a quasisupermodular objective function “between” the two original problems of interest. In smooth problems, sufficient conditions for our new assumptions may be verified by elementary differential comparisons, making them well suited for applied work. We illustrate the relevance of this novel approach with several economic applications.