Theorem 1, p. 494 — an optimal q* with q_i* = 0, q_i* = K_i, or ∇_{q_i}Π₂(q*; a) = 0

Consider the two-supplier random-capacity newsvendor with expected profit $\Pi_2(q; a)$ (Eq. (3)), and suppose that demand $X \ge 0$ has a density and finite mean.

Then there is an optimal procurement vector $q^* = (q_1^*, q_2^*)$, that is, $q^* \ge 0$ with $\Pi_2(q; a) \le \Pi_2(q^*; a)$ for every $q \ge 0$, such that for each supplier $i = 1, 2$

The theorem has two parts: a maximizer exists, although the feasible set $q \ge 0$ is unbounded, and each of its coordinates is at a boundary ($0$ or the design capacity) or at a stationary point of the profit in that coordinate. It is the starting point for the first-order condition (6).

Formalization Note. The printed statement reads "$q_i^* = K$"; there is no un-indexed $K$ in the paper, and it means the design capacity $K_i$. The partial derivative is the derivative at $q_i^*$ of $t \mapsto \Pi_2(q^*$ with $q_i^*$ replaced by $t; a)$, and the third alternative asserts that this derivative exists and equals $0$. The density of demand is the paper's standing assumption of §3.1, expressed as absolute continuity of the demand law with respect to Lebesgue measure.