Weak and strong monotone comparative statics.

This paper develops necessary and sufficient conditions for the set of solutions to an optimization problem to be nondecreasing in a weak sense still strong enough to guarantee the existence of an increasing selection, and thus strong enough to guarantee monotonicity when the solution is unique, as well as necessary and sufficient conditions for the set of optimizers to be nondecreasing in a strong sense which is strong enough to rule out the possibility of a decreasing selection. These necessary and sufficient conditions are variations of quasisupermodularity and the single crossing property introduced in Milgrom-Shannon [13]. Moreover, to determine when an objective function satisfies these conditions, this paper develops several characterizations of quasisupermodularity and the single crossing property and their variants, both in terms of differential conditions and in terms of restrictions on the structure of the level sets of these functions. Several examples are given to choice theory under loss aversion and to an auction problem. [ABSTRACT FROM AUTHOR]