Lemma B.1, p. 28 — u(x) − xu′(x) is strictly increasing for increasing, strictly concave, differentiable u on ℝ₊

Let $u : \mathbb{R}_+ \to \mathbb{R}$ be increasing, strictly concave and differentiable. Then the function

is strictly increasing on $\mathbb{R}_+$.

Geometrically, $u(x) - x u'(x)$ is the intercept at $0$ of the tangent line to $u$ at $x$; for a strictly concave function this intercept rises as the point of tangency moves right. In the paper it is the marginal value of an additional (fractional) platform, and the lemma drives the concavity statements of Lemmas B.2 and B.3.

Formalization Note. $u$ is a function on $\mathbb{R}$ of which only the restriction to $[0,\infty)$ matters; monotonicity, strict concavity and differentiability are required on $[0,\infty)$, and $u'(x)$ is the derivative within $[0,\infty)$ (one-sided at $0$).