D(Maximum)=P(Argmaximum)

TL;DR

This paper derives a formula for the subdifferential of the maximum functional on continuous functions over a compact metric space, linking it to probability measures on the arg-maximum set, with applications in economics and optimization.

Contribution

It provides a novel explicit characterization of the subdifferential of the maximum functional in terms of probability measures on the arg-maximum set.

Findings

Subdifferential of maximum functional equals probability measures on arg-maximum.

The result underpins key identities in microeconomics and duality theory.

Applicable to continuous functions on arbitrary compact metric spaces.

Abstract

In this note, we propose a formula for the subdifferential of the maximum functional $m(f) = \max_K f$ on the space of real-valued continuous functions $f$ defined on an arbitrary metric compact $K$. We show that, given $f$, the subdifferential of $m(f)$ always coincides with the set of all probability measures on the arg-maximum (the set of all points in $K$ at which $f$ reaches the maximal value). In fact, this relation lies in the core of several important identities in microeconomics, such as Roy's identity, Sheppard's lemma, as well as duality theory in production and linear programming.