Technical Appendix: Auxiliary Results and Proofs

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1 A Technical Appendix: Auxiliary Reult and Proof Lemma A. The following propertie hold for q (j) = F r [c + ( ( )) ] de- ned in Lemma. (i) q (j) >, 8 (; ]; (ii) R q (j)d = ( ) q (j) + R q (j)d ; (iii) R q (j)d > q (j): Proof. (i) i veri ed by inpecting the expreion of q (j). To prove (ii), note that q (j) = ( ) q (j) : Uing thi and by Leibni rule and integration by part, Z q (j)d = = = Z q (j) d = ( ) ( ) ( ) q (j) = q (j) + Z Z = Z q (j) d q (j)d q (j)d : (iii) immediately follow by combining (i) and (ii). Proof of Propoition. Note that the retriction = e me limit the range of from to e and, imilarly, may vary between and e. Both upper bound are reached if and only if =. For a xed and a realied value of, the rm optimally chooe the quantity q(j) = F r G (j) = F r (c + ( ( )) ), where we ued the relation (). Subtituting thi, the objective function in (B) become U( ; ) = r R J(q(j))d. Di erentiating thi with repect to and yield Z U = ( ) Z U = ( ) m e q(j)d q(j)d ( ) ; em : Conider U. If =, it i optimal to chooe = ince U = < for all. If >, on the other hand, lim U em! =, implying that thaximum occur either at an interior point that ati e the rt-order condition U = or at the corner, i.e., =. A imilar analyi with repect to how that thaximum occur either at an interior point that ati e 33

2 U = or at = a long a >. Otherwie, =. Hence, poible optimal outcome are characteried by: (i) = = =, (ii) U = and U = with <, (iii) U e= = and U > with =, (iv) e=e U = and em=e U > with =. Whow rt m em= that (i) i never optimal and then among the ret, (ii) with = dominate (iii) and (iv). Conider (ii). Combining the rt-order condition U = and U = with = e me allow u to rewrite the rm expected pro t a a function of : U() = r R J(q(j))d K. Di erentiating thi, U () = ( ) R q(j)d K and U () = ( ) R q(j) d >, where we ued Leibni rule. Since U() i convex and U() < U() according to (), it i optimal to chooe =. Hence, (i) i eliminated. Since = i optimal, uing the relation = e me, we can rewrite the rm expected pro t a a function of only: U( ; ) = r R J(q(j))d e. Therefore, maximiing U with the contraint = i equivalent to nding that minimie the total e ort cot e +. Tholution i e = km. Subtituting thi in em = = e yield e m = peci ed in (ii). From U. We now how that e m and e are obtained from the rt-order condition = and U = with =, we get = ( ) R q(j)d and = ( ) R q(j)d. Subtituting them in = e yield ( ) R q(j) = K, and therefore, = K = e m and = K = e. Thi con rm that the optimal value e m are e reult from the rt-order condition U (iv) a poible optimal outcome. = and U = with =, thereby eliminating (iii) and Proof of Lemma. The proof cloely follow the approache found in Ha () and Bolton and Dewatripont (5). Let ( j ) =g(g ( j ) j ), which i equal to ( ( )) under (). Since i xed at the time thanufacturer decide the contract term, we drop it in thubequent expreion for notational convenience. Let t() w()q() be the total payment to the -typupplier. Once the optimal t () and q () are found, w () i computed a w () = t ()=q (). The expected pro t are rewritten a (; b) = t(b) G ()q(b) and m = R (re[minfd; q()g] t()) d. For brevity, let () (; ). Firt, we etablih that (IR-S) can be rewritten a () =. To ee thi, oberve that (IC- S) and G () G () together imply that () = t() G ()q() t() G ()q() t() G ()q() = (). Therefore, () guarantee (IR-S). Suppoe () > at the optimum. By reducing t() by thame in niteimal amount for all, neither (IR-S) or (IC-S) i violated while m i increaed. Therefore, () = at the optimum. Next, whow that (IC-S) i equivalent to the two condition t () = G ()q (); (5) q () : (6) 34

3 Oberve that (IC-S) implie the rt- and econd-order condition b (; b) b= = and b (; b) b=. (5) follow from the rt-order condition. Thecond-order condition i written a t () G ()q (). On the other hand, di erentiating (5) yield t q () () G ()q () =. g(g ()) (6) follow from combining thee two reult. Therefore, (IC-S) implie (5) and (6). Converely, uppoe that (5) and (6) are true. Aume that there exit that violate (IC-S), i.e., () = t() G ()q() < t(b) G ()q(b) = (; b), or equivalently, R b t (x) G ()q (x) dx >. Conider < b. From (6), we have G ()q (x) G (x)q (x) for x [; b]. Then R b t (x) G ()q (x) dx R b t (x) G (x)q (x) dx = by (5). But thi contradict the earlier aertion R b t (x) G ()q (x) dx >. A imilar argument can bade for b. Summariing, we have replaced (IR-S) with () = and (IC-S) with t () = G ()q () and q (). To olve thi modi ed optimiation problem, we ignore the lat contraint q () and olve the relaxed problem intead, and then verify that the omitted contraint i indeed ati ed. By (5), we have () = obtain () = R q() or equivalently, w() = t() q() = G () + q() pro t become g(g ()). Integrating both ide of thi equation and uing () =, we g(g (x)) dx. Therefore, t() = G ()q() + () = G ()q() + R R m = Z re[minfd; q()g] Oberve that, by integration by part, Z Z Z g(g (x)) dx d = g(g (x)) dx, dx. A a reult, thanufacturer expected g(g (x)) G ()q() Z Z g(g (x)) dx + g(g (x)) dx d: q() g(g ()) d = Z ()q()d: (7) Hence, m = R re[minfd; q()g] G () + () q() d. Di erentiating thi with repect to q(), m q() = rf (q()) G () + () and m = rf(q()) <. Hence, thanufacturer q() problem i concave. Suppoe that q() = at the optimum. Then m q() at q() =, i.e., G () + () r. However, thi contradict the earlier aumption G ( ) + ( ) < r. Therefore, q() > at the optimum. Noting that lim m q()! q() = G () + () <, we conclude that the optimal q() i found from the rt-order condition m q() =, i.e., q () = F r G () + (). Finally, q () < ince G () + () i increaing, con rming the condition (6) that wa left out earlier. R Thupplier expected pro t i = R ()d = R where we ued (7). function, i m = r R J(q ())d: q (x) g(g (x)) dx d = R ()q ()d, Thanufacturer expected pro t, after ubtituting q () in her objective Proof of Propoition. Note that the retriction = e me from to e and, imilarly, from to e 35 m limit the range of. Both upper bound are reached if

4 and only if =. Di erentiating thanufacturer and thupplier expected pro t U m ( j ) = r R J(q (j))d and U ( j ) = ( ( )) R q (j)d, U m e = ( ) U = ( )( ) [q (j) + ()] ; em () ; where we ued part (ii) of Lemma A. in thecond equation. Let b be the root of (). We proved in Lemma that () > for < b and () < for > b. Suppoe b. Then () for all [; ]. From the expreion above, wee that thi implie U < for all [; ], and therefore, thupplier chooe =. Thi in turn implie Um = < for all [; ], and therefore, thanufacturer chooe =. Hence, = = = i the equilibrium outcome if (), and therefore (), for all [; ]. Thi i conitent with thtatement in the propoition for the cae () K. Next, uppoe < b <. Then, according to Lemma, () for [; b ] and () < for ( b ; ]. Let b be ( ) < and b be m <. In the interval [; b ], varie between and b while varie between and b. Similarly, (b ; ] and (b ; ] in the interval ( b ; ]. Conider ( b ; ]. Since () < in thi interval, U <, which implie that the optimal value of doe not exit in (b ; ]; it ha to be in [; b ]. In other word, an equilibrium, if it exit, hould reult in [; b ]. Therefore, a earch for an equilibrium in [; ] i reduced to a earch in [; b ], in which () and hence (). De ne a ( ) () and k b m. We conider the three cae tated in the propoition in turn. ( ) q (j)+ () (i) Suppoe () < K for all [; ]. Then () < K for all [; b ]. Thi inequality can be rewritten a a < b. From the expreion of Um U m U < and U < if if a, (b) Um b. In each de ned interval of and U < and U < if a < e we derived above, we nd that (a) < b, (c) Um and, it i optimal for either thanufacturer or thupplier to decreae hi/her e ort a much a poiblince hi/her expected pro t i monotonically decreaing. Since we encompa all poible range of e de ned under [; b ] and [; b ] (which are together equivalent to [; b ]), it implie that one of the two partie chooe a ero e ort at the optimum. Thi in turn implie that the other party chooe ero e ort a well, ince Um < for all [; b ] if = and U < for all [; b ] if =. Hence, = = in equilibrium, and a a reult, = in equilibrium if () < K for all [; b ]. (ii) The cae () > K for all [; ] i not permitted under the aumption < b <, ince () < and hence () < for ( b ; ]. We conider thi cae below when we aume b. 36

5 (iii) The only remaining poibility i min b f ()g K max b f ()g. Since () i continuou in [; b ], tholution of () = K exit. Thame equation i obtained by combining the rt-order condition Um = and U three equation alo yield the expreion () = ( = with = e me. Thi ytem of ) [q (j) + ()] and () = ( ) (), from which the equilibrium e ort level are identi ed once the optimal i found from the optimality condition () = K. Since tholution exit, the equilibrium alo exit. Finally, uppoe b. Then by Lemma, () for all [; ]. We conider the three cae tated in the propoition a we did for < b <. Cae (i) and (iii) proceed imilarly a above, with b!, b!, and b!. Hence, we only conider cae (ii): (ii) Suppoe () > K for all [; ]. Thi can be rewritten a b < a. Then (a) Um and U U if > if b, (b) Um > and U > if b < e < a, and (c) Um > and a. In each cae it i optimal for either party to increae hi/her e ort a much a poiblince hi/her expected pro t i monotonically increaing. Thi lead to the corner olution, i.e., either = or =, at which =. From the inequalitie in (a)- (c) it i clear that the equilibrium i reached when the pro t of the other party i maximied, i.e., when either Um e= = or U em= =. If =, then U em= =, which i equivalent to e = a. From thi we get = ( ) em=e () and = = m e = ( ) (). Similarly, if =, = ( ) [q (j) + ()] and = = em = ( ) [q (j) + ()]. We have exhauted all poibilitie, and the concluion i ummaried in the propoition. Proof of Propoition 3. From Propoition, wee that S (; ) i determined from the equation () = K, which implie ( S ) > ince K >., e B = km and e B = km e B m ( Note that, from Propoition. It wa hown in the proof of Propoition that e S m = ) S q (j S ) + ( S ) and e S = ( ) S ( S ) if < S <. Then e S = ( ) S ( S ) < ( ) = S ( S ) = S K = S km S q (j S ) + ( S ) ( S ) = S e B < e B : Alo, e S e S m = km ( S ) q (j S ) + ( S ) < km = e B e B m : 37

6 Proof of Propoition 4. (i) Under price commitment, the event unfold a follow: () thanufacturer commit to w, () thanufacturer and thupplier decide and imultaneouly, and (3) thanufacturer o er q. In the lat tep, thanufacturer chooe q(w) = F w r to maximie her pro t re[minfd; qg] wq. Anticipating thi, thupplier chooe for a given value of to maximie hi expected pro t R (w G (j))q(w)d = w c ( ( )) q(w). It i traightforward to how that thi function i concave and i maximied at = ( ) q(w). At thame time, thanufacturer chooe to maximie her expected pro t re[minfd; q(w)g] wq(w) = rj(q(w)). Tholution i =, and therefore, = and = in equilibrium. (ii) Under quantity commitment, thequence of event i: () thanufacturer commit to the quantity q, () thanufacturer and thupplier decide the e ort and imultaneouly, and (3) thanufacturer o er the price w. At the time of the price o er, thanufacturer face the problem max w re[minfd; qg] wq ubject to the participation contraint (w G (j))q, 8 [; ]. Tholution i w = G (j) = c + ( ( )), i.e., thanufacturer chooe a price that leave ero pro t to thupplier with the highet cot. Anticipating thi pricing, thupplier chooe hi e ort for a given value of to maximie hi expected pro t R (w G (j))qd = ( ( )) q. Thi function i decreaing in, and therefore, thupplier chooe =. At thame time, thanufacturer chooe her e ort to maximie her expected pro t re[minfd; qg] wq = re[minfd; qg] [c + ( ( ))] q. It i traightforward to how that thi function i concave and maximied at = ( ). From thi expreion wee that it i optimal for the manufacturer to chooe = ince thupplier chooe =, and a a reult, =. (iii) Under price-quantity commitment, thanufacturer o er w and q, and then thanufacturer and thupplier decide and imultaneouly. Thupplier chooe that maximie hi expected pro t R (w G (j))qd = w c = ( )( )q(w) ( ( )) q. Tholution i. At thame time, thanufacturer chooe to maximie her expected pro t re[minfd; qg] wq. Tholution i =, and therefore, = and = in equilibrium. 38

7 Proof of Propoition 5. Under the expected margin commitment, () thanufacturer commit to thargin v and the payment function w() = v + R G (j) d = v + c + ( ( )), () thanufacturer and thupplier decide and imultaneouly to determine, and nally (3) w() r thanufacturer o er the quantity q. In the lat tep, for a realied value of, thanufacturer chooe q y () = F to maximie her expected Stage pro t re[minfd; qg] w()q. Anticipating thi, thanufacturer chooe to maximie her Stage expected pro t U m ( j ) = re[minfd; q y ()g] w()q y () = rj(q y ()), while thupplier chooe to maximie hi Stage expected pro t U ( j ) = R (w() G (j))q y ()d = vq y (). Di erentiating, U m = ( ) q y e () ; U = v ( ) em rf(q y ( ) : ()) The optimality condition () = K for i obtained by combining the rt-order condition Um = and U = with = e me yield the optimality condition () = K for a well a the equilibrium e ort () = ( ) () < K and Proof of Propoition 8. q y () and () = v ( ) rf(q y ()) () > K, i imilar to that of Propoition and i omitted.. The ret of the proof, including the cae It i traightforward to how that = i optimal if thupply chain i integrated. Suppoe that, in a decentralied upply chain, thanufacturer doe not commit to a contract term in the beginning and intead o er a creening contract f(w(j); q(j))g after collaboration i completed. In a make-to-order environment, thi contract i o ered after thanufacturer oberve the realied demand D. Hence, at the timhe devie the contract term, thanufacturer pro t function i R (r minfd; q(j)g w(j)q(j)) d, which i free of expectation. The optimiation problem i thame a (S ) except for thi modi ed objective. Following thtandard advere election proof tep, it can bhown that the problem reduce to Z max q( j ) [r minfd; q(j)g (c + ( ( )) ) q(j)] d: It i eay to ee that the objective function of thi problem peak at q(j) = D for each value of and any. Subtituting thi back into the objective and taking an expectation, we can how that the manufacturer Stage expected pro t i equal to U m ( j ) = [r c ( ( ))]. Similarly, thupplier pro t i U ( j ) = ( ( )), which i decreaing in, and hence, decreaing in for any xed. Thi implie that thupplier et =, and a a reult, = in equilibrium regardle of thanufacturer choice of. With = thanufacturer 39

8 pro t i decreaing in, o he chooe =. The reulting pro t i U m (j) = (r c ). Next, conider EMC. With the contant margin v the payment function i w() = v + c + ( ( )), and a before, it i optimal to et q = D regardle of. Then the Stage expected pro t of the manufacturer and thupplier are, repectively, U m ( j ) = r v c and U ( j ) = v ( ( )). Since U ( j ) i decreaing in, thupplier chooe = ; hence, = in equilibrium. It follow that thanufacturer chooe = and the reulting pro t are U m (j) = r v c and U (j) = v. At =, the optimal v that enure the participation of all upplier type i v =. Hence, U m (j) = (r c ), which i identical to the value we derived under non-commitment. 4