Comparative Statics With Linear Objectives: Normality, Complementarity, and Ranking Multi‐Prior Beliefs

Abstract

We formulate an order over constraint sets $A\subseteq {\mathbb{R}}^{\ell }$, called the parallelogram order, which guarantees that argmin{ p ⋅  x: x ∈  A} increases in the product order as A increases in the parallelogram order, for any vector $p\in {\mathbb{R}}^{\ell }$. Using this result, we characterize the utility/production functions that lead to normal demand as well as the closely related class of production functions with marginal costs that increase with factor prices. By generalizing the concept of supermodularity, we also characterize the class of production functions for which factors are complements. In the context of decision‐making under uncertainty, our new set order leads to natural generalizations of first‐order stochastic dominance in multi‐prior models.