An On-line Addendum to: “Strategic Commitments for an Optimal Capacity Decision Under Asymmetric Forecast Information”

Proof of Theorem 4. For wa ∈ [c + ck, rck w−c ], we have the boundary equilibrium. From its definition, yb(ξ) is decreasing in wa (that is, dyb(ξ) dwa < 0) and yb(ξ) = μ + ξ + G−1( r−c−ck r ) when wa = c + ck. The total supply chain profit at the boundary equilibrium is Πtot(yb(ξ), yb(ξ), ξ) = rEmin(μ + ξ + , yb(ξ)) − (c + ck)y(ξ) from Equations (16) and (17). This function is concave and increasing in yb(ξ) for all yb(ξ) < μ + ξ + G−1( r−c−ck r ) (that is, dΠtot(yb(ξ),yb(ξ),ξ) dyb(ξ) > 0). Hence dΠtot(yb(ξ),yb(ξ),ξ) dwa < 0. For wa ∈ [ rck w−c , w + (r−w)ck w−c ], we have the interior equilibrium. From its definition, y i(ξ) is also decreasing in wa ( dyi(ξ) dwa < 0). From Lemma 2 the optimal capacity Kap(ξ) is independent of wa. The total supply chain profit in the interior equilibrium is Πtot(yi(ξ),Kap(ξ), ξ) = E[rmin(μ + ξ + ,Kap(ξ)) − cmax(yi(ξ),min(μ + ξ + ,Kap(ξ)))] − ckK(ξ) and it is decreasing in yi(ξ) ( dΠtot(yi(ξ),Kap(ξ),ξ) dyi(ξ) < 0). Hence dΠ tot(yi(ξ),Kap(ξ),ξ) dwa > 0. For Part 2, note that the manufacturer’s profit function in Equation (16) decreases uniformly in wa, hence it decreases at the equilibrium as well. Part 3 follows from Parts 1 and 2 immediately since supplier’s profit is the difference between the total profit and the manufacturer’s profit. For Part 4, we differentiate the equilibrium profit functions with respect to ξ. At the interior equilibrium, we have ∂Π s(yi(ξ),Kap(ξ),ξ) ∂ξ = wa − c − ck ≥ 0 and ∂Πm(yi(ξ),Kap(ξ),ξ) ∂ξ = r − wa ≥ 0. Similarly, at the boundary equilibrium, we have ∂Π s(yb(ξ),yb(ξ,ξ)) ∂ξ = wa−c−ck ≥ 0, ∂Πm(yb(ξ),yb(ξ),ξ) ∂ξ = r − wa ≥ 0. For Part 5, from Theorem 3 we know that an advance purchase contract with wa ∈ [w,w+ r−w w−cck] leads to the interior equilibrium and credible information sharing. The equilibrium profit under such a contract is Πap(ξ) = (w − c)E(min(μ + ξ + ,Kws(ξ)) − yi(ξ))+ + (wa − c)yi(ξ) − ckK(ξ) ≥