Theorem 14.2: under the wholesale price contract with $w > c_s$ the retailer under-orders, $Q^*_r < Q_0$

Theorem 14.2. Under the wholesale price contract, if $w > c_s$ then $Q^*_r < Q_0$: any maximizer of the retailer's profit is strictly smaller than any maximizer of the chain profit.

The retailer absorbs all of the overage risk but only part of the underage risk, since the supplier also pays a stockout penalty, so he orders less than the chain wants. The book omits the proof (Problem 14.6): the retailer's fractile $(w + c_r - v)/(r - v + p_r)$ exceeds the chain's $(c - v)/(r - v + p)$ when $w > c_s$, and $\bar F$ is nonincreasing. Only a continuous distribution function and a finite mean are needed.