Summary: A On-line Appendix. Proofs
Proof of Lemma 1. The supplier's objective function (3) is piecewise linear. We focus on the
corner-point solutions: z = 0 or q. The result follows.
Proof of Lemma 2. Omitted for brevity. The proof is similar to that of Proposition 3.
Proof of Proposition 1. The proof of this proposition is parallel to that of Proposition 3, except
that the feasible region is pruned by the additional constraints q1(t1, t2) = 0 or q2(t1, t2) = 0.
Specifically, when determining the optimal (q1, q2)(t1, t2), instead of four corner-point solutions,
there are only three feasible corner-point solutions: (q1, q2)(t1, t2) = (0, 0), (0, D) or (D, 0).
Proof of Proposition 2. The procedure for deriving the optimal order quantities of the buyer is
similar to the proof of Proposition 3, except that for each of the four maximization problems in
(A.10), there is only one feasible corner-point solution: (q1, q2)(t1, t2) = (D, D). This is the optimal
quantity pair for this model.
Proof of Proposition 3. At optimality, from the supplier's optimal profit (3), we have
Xn(t1, t2) + pn(t1, t2)E min
qn(t1, t2), tn
n [(qn, pn)(t1, t2)]