Monotone comparative statics

Monotone comparative statics is a sub-field of comparative statics that focuses on the conditions under which endogenous variables undergo monotone changes (that is, either increasing or decreasing) when there is a change in the exogenous parameters. Traditionally, comparative results in economics are obtained using the Implicit Function Theorem, an approach that requires the concavity and differentiability of the objective function as well as the interiority and uniqueness of the optimal solution. The methods of monotone comparative statics typically dispense with these assumptions. It focuses on the main property underpinning monotone comparative statics, which is a form of complementarity between the endogenous variable and exogenous parameter. Roughly speaking, a maximization problem displays complementarity if a higher value of the exogenous parameter increases the marginal return of the endogenous variable. This guarantees that the set of solutions to the optimization problem is increasing with respect to the exogenous parameter.

Let ${\displaystyle X\subseteq \mathbb {R} }$ and let ${\displaystyle f(\cdot ;s):X\rightarrow \mathbb {R} }$ be a family of functions parameterized by ${\displaystyle s\in S}$, where ${\displaystyle (S,\geq _{S})}$ is a partially ordered set (or poset, for short). How does the correspondence ${\displaystyle \arg \max \limits _{x\in X}f(x;s)}$ vary with ${\displaystyle s}$?

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