Appendix S1 1. ( z) α βc. dβ β δ β

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1 Appendix S1 1 Proof of Lemma 1. Taking first and second partial derivatives of the expected profit function, as expressed in Eq. (7), with respect to l: Π Π ( z, λ, l) l θ + s ( s + h ) g ( t) dt λ Ω( z) l l l λ z β ( z, λ, l ) z ( s h ) g ( l ) λ ( z) l + l λ Ω <. B B Note that Ω ( z ) ( x z ) f ( x ) dx ( x A ) f ( x ) dx A z. Therefore λ ( z) λ A A Ω >, and thus Π(z, λ, l) is concave in l. The optimal solution follows directly from the first-order condition. Proof of Lemma. From Eq. (9), it is easy to verify that Π(z, λ, l(λ))/ λ {() (1 + θδ + βφ)[λ Ω(z)]}/β and Π(z, λ, l(λ))/ λ (1 + θδ + βφ)/β. Therefore the concavity of Π(z, λ, l(λ)) in λ is established. From the first-order condition, we have * 1 λ λ + Ω ( z ), where λ, which is the optimal mean demand when there is no demand (1 + θδ + βϕ) uncertainty. It is obtained by substituting Eq. (6) and l * l(λ) into Eq. (4) and then optimizing over λ. p is the price corresponding to this λ. The optimal price p * is then obtained by substituting λ * and l(λ) into Eq. (6). Proof of comparative statics in table 1. We only present an example here to show how price sensitivity β affects the average demand. The other results can be derived in a similar fashion. We first note that Also, δ ϕ h φ ( y) dy + s φ ( y) dy ( h + s ) Φ ( δ ) s θ / β. δ l l l l l δ d(1 + θδ + βϕ) δ ϕ δ θ + β + ϕ ϕ >. dβ β δ β 1 Supplemental information for: Wu, Z., B. Kazaz, S. Webster, K.K. Yang. 11. Ordering, pricing, and lead-time quotation under lead-time and demand uncertainty. Production and Operations Management, to appear. 1

2 Therefore, it is easy to see λ decreases in β. Proof of Theorem 1. The proof is similar to that of Theorem 1 in Petruzzi and Dada (1999). It is produced here for completeness. Recall that the expected profit function is given by Eq. (1). Taking the first-order derivative with respect to z, we have R(z) (, λ, ) dπ z z l z dz 1 F( z) + βs (1 + θδ + βϕ) Ω( z) ( h + c) F( z) β [ ] (1 + θδ + βϕ) Ω( z) h + c + F z h + c + s + β [ 1 ] In order to identify values of z that satisfy the first-order necessary condition, we characterize the shape of R(z) through the following: dr ( z ) f ( z ) 1 c ( h c s) (1 ) ( z) F z α β + β θδ + βϕ dz β Ω + r( z), where r( ) f( )/[1 F( )] denotes the hazard rate (or failure rate). Also, d R z dr z dz df z dz f ( z) dz / (1 + θδ + βϕ) f ( z) f ( z) [ dr( z) / dz][1 F( z)] 1 F( z) + + β r( z) r ( z) d R ( z ) (1 + θδ + βϕ) f ( z )[1 F ( z )] dr( z)/ dz r ( z) dr z dz βr ( z) + dz. If F( ) is a distribution satisfying the condition r (z) + dr(z)/dz >, then it follows that R(z) is either monotone or unimodal, implying that R(z) has at most two roots. Further, R(A) s + [ + A(1 + θδ + βφ)]/(β) > according to condition (14), and R(B) (h + c) <. Therefore, R(z) can only have one root, indicating a change of sign for R(z) from positive to negative, and thus it corresponds to a local maximum of Π(z, λ(z), l(z)). As a result, the unique root of R(z) is the optimal stocking factor z * that maximizes Π(z, λ(z), l(z)). Proof of Theorem. Part (a) and (c) are immediate from results in Lemma. Part (b) follows directly from part (a), because the optimal lead-time to quote is always proportional to the average demand. To prove part (d), note that Π max Ψ ( p, l) Ψ ( p, l ), and ( p, l )

3 [ p l L z p l ] * Π Ψ ( z, p, l) max (, ) (,, ) Ψ < Ψ Ψ * ( p, l ) L( z, p, l ) ( p, l ) Π ( p, l ). The last inequality follows from the fact that p and l are the global maximizers of Ψ(p, l). Proof of Theorem 3. Note that p 1 β θτλ + ( 1+ θτρ ) > η 1+ θτρ + βηρ λ ( θτ + βη ) ( 1+ θτρ + βηρ ) λ < The associated optimal expected profit is given by Π ( p c ηρλ ) λ λ β. 4β 1+ θτρ + βηρ,. It is straightforward to see that Π / <. Proof of Lemma 3. Consider two arbitrary values of ρ, with ρ H > ρ L. It is straightforward to see that the derivative of the expected profit function with respect to z, R(z), is decreasing in ρ, i.e., R(z, ρ L ) > R(z, ρ H ). Therefore we have R(z * (ρ H ), ρ L ) > R(z * (ρ H ), ρ H ), where z * (ρ H ) is the root of R(z, ρ H ). In other words, at z * (ρ H ), the curve R(z, ρ L ) is still positive. Recall that we have established in proof of Theorem 1 that under conditions (13) and (14), R(z) is either monotone or unimodal with a change of sign from positive to negative. Thus we conclude that R(z, ρ L ) will cross the x-axis to the right of z * (ρ H ), namely, z * (ρ L ) > z * (ρ H ). See the figure below for a sketch of the proof. A simpler way to show z * / < is to make use of the concept of supermodularity (Topkis 1998). It can be shown that 3

4 Π λ ( z, ( z), l( z)) R( z) 1 F( z) ( θτ + βη ) Ω ( z ) <, z β which implies that the expected profit function Π ( z, λ( z), l( z)) is submodular in z and ρ, or supermodular in -z and ρ. The fact that z * / < then directly follows from Theorem.8. of Topkis (1998). R(z,ρ) R(z,ρ L ) R(z,ρ H ) Z * (ρ H ) Z * (ρ L ) ρ Taking derivative of l * with respect to ρ yields ( c) ( + θτρ + βηρ ) l α β τ τ τρ z + Ω ( z) + [ F( z) 1] >. 1 Proof of Theorem 4. Note that Eq. (1), the expected profit as a function of the single variable z, can be simplified as follows θτρ + βηρ * 1 * Π ( z ) ( z ) ( z ) β Ω Ω ( 1+ θτρ + βηρ ) + Ω From the envelope theorem, ( h c) ( z ) s ( z ) * { 1 ( z ) } ( θτ βη ) ( α βc) ( θτρ βηρ ) Ω Π ( z ) <, ρ 4β 1+ θτρ + βηρ where the inequality follows directly from our assumption (17). 4

5 l ω kτρλ + (1 k) ρλ ω + 1 λ ω 1 + kθτρ + kβηρ + (1 k) ω + 1 ω α 1 + kθτρ + (1 k) λ p ω 1 + β (A1) (A) (A3) ω 1+ θ ηρ p 1 / k ω + ω 1 + kθτρ + kβηρ + (1 k) ω + 1 References >. (A4) Petruzzi, C. N., M. Dada Pricing and the newsvendor problem: A review with extensions. Operations Research 47() Topkis, M. D Supermodularity and Complementarity. Princeton University Press, Princeton, NJ. 5