Theorem 14.1 (first part): under the wholesale price contract $Q^*_r = Q^*_s = Q_0$ iff $w = c_s - \frac{c - v}{r - v + p}p_s$

Theorem 14.1, first part. Under the wholesale price contract with price $w$, the retailer's and supplier's optimal order quantities both coincide with the supply-chain-optimal quantity $Q_0$ if and only if

Coordination is stated as the equality of the sets of maximizers: every maximizer of $\pi_r(\cdot, w)$ is a maximizer of $\Pi$ and conversely, and the same for $\pi_s$. The proof substitutes $w$ into the first-order conditions (14.11)-(14.12) and uses that $S'(Q) = \bar F(Q)$ is strictly decreasing and continuous, so the fractile equations have a common unique solution. This is double marginalization (Spengler 1950): only a price below the supplier's cost aligns the retailer's fractile with the chain's.

Formalization Note $F$ is assumed continuous (an atomless demand law), $p_s > 0$ (the book's proof divides by it), and the chain optimum positive, $\bar F(0) > (c - v)/(r - v + p)$. Strict monotonicity of $F$ is not assumed. On all of $\mathbb{R}$ it would exclude every nonnegative demand law, since $F = 0$ on $(-\infty, 0)$, and so the book's own setting. It is not needed either: with the maximizer sets compared as sets, at $w = w^*$ all three sets are $\{Q : \bar F(Q) = (c - v)/(r - v + p)\}$, and for $w \ne w^*$ the retailer's set is disjoint from the chain's, which is nonempty.