D(Maximum)=P(Argmaximum)

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.