# Int. J. Production Economics

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4 178 P. Guo et al. / Int. J. Production Econoics 18 (1) Then, the first-order conditions with respect to w and w s are, respectively, k w w s and k w w s F Þ f Þ, F s s Þ f s s Þ ð5þ Suppose the OEM s prior belief about the CM s (supplier s) cost is instead ^X j, j or s. Denote the corresponding wholesale prices by ^w C i, i, s. We have the following proposition. Proposition 1. If ^X s Z rh X s, then ^w C s ZwC s and ^wc rwc. Siilarly, if ^X Z rh X, then ^w C ZwC and ^wc s rwc s. Thus, under consignent, if the OEM akes a larger estiation of the supplier s cost in the sense of the reversed hazard rate order, then it will offer a higher wholesale price to the supplier and a lower wholesale price to the CM. (Throughout this paper, higher and lower and increasing and decreasing are not used in the strict sense, i.e., they include equality.) 3.3. Integration Under integration, the OEM has priors X and X ~ s, and offers a contract pair (w a, q) to the integration party. The alliance then decides whether to accept or reject. Suppose that the OEM offers w a to the alliance. Then, the latter s best response is accept if w a Zc þc s and reject otherwise. Because the OEM s prior about the total cost incurred by the alliance is X þ X ~ s, its contract success probability is PfX þ X ~ s rw a g R wa F a x s Þd F ~ s ðx s Þ. The OEM chooses w a and q that axiizes its expected profit P I o ðq,w aþ, as follows Z wa OEM Max P I o ðq,w aþðk yq w a Þq F a x s Þ d F ~ s ðx s Þ ð6þ q,wa Note that R wa F a x s Þ d F ~ s ðx s Þ is the convolution of F and ~F s. According to Lea 3, it is log-concave. The overall objective function is also log-concave in (q, w a ), and the optiu is unique. For fixed w a, (6) is concave in q, and the first-order condition is q k w a ð7þ Substitute (7) back into (6), and we obtain the first-order condition for w a k w a R wa F a x s Þ d ~ F s ðx s Þ R wa f a x s Þ d ~ F s ðx s Þ ð8þ Denote the solution of (8) by ~w I a. Suppose the OEM s prior belief about the supplier s cost is instead X s. Denote the corresponding wholesale price offered to the alliance by w a I. Then, we have the following proposition. Proposition. If ~ X s Z rh X s, then ~w I a ZwI a. Proposition shows that under integration, the CM has the otivation to provide a stochastically larger prior in the sense of the reversed hazard rate order to the OEM when it is asked for. Therefore, credible inforation sharing cannot be achieved without a valid echanis Turnkey Under this structure, the sequence of events is (1) the OEM offers a contract consisting of wholesale price w and production quantity q to the CM; () if the CM accepts, then it offers wholesale price w s and production quantity q to the supplier; and ð4þ (3) the supplier decides to accept or reject. This can also be analyzed in reverse order. First, we derive the CM s best response given the OEM s offer. Then, we analyze the OEM s decision in anticipation of the CM s best response. Given the OEM s offer (w, q), the CM chooses w s to axiize its expected profit, CM Max P T sjw,qþ w s c ÞqF s s Þ ws This is log-concave in w s, and the first-order condition for w s is w c w s þ F s s Þ f s s Þ ð1þ However, the OEM cannot fully observe the best response function of the CM as its prior knowledge of the supplier s cost, X ~ s, is stochastically larger than the CM s prior X s. Hence, fro the perspective of the OEM, conditional upon X x, the CM s best response is the solution of F w x w s þ ~ s s Þ ~f s s Þ ð11þ Denote the optial solution to the above equation by ~w s,x Þ. Then, the OEM can infer that the success probability of the turnkey strategy is joint probability PfX rw, X ~ s r ~w s,x Þg. The OEM thus wishes to choose w and q to axiize its expected profit, as follows OEM Max P T o ðq,w Þ q,w ðk yq w ÞqPfX rw, X ~ s r ~w s,x Þg ð9þ ð1þ The two events fx rw g and f ~ X s r ~w s,x Þg are not independent. Note that if X 4w, then the CM s profit argin, w X w s, is negative. Therefore, the optial pricing strategy for the CM is to set ~w s,x Þ. Because the doain is positive, i.e., ~ X s 4, event ~ X s r ~w s,x Þ iplies ~w s,x Þ4, and thus X 4w is ipossible. As a result, a positive price paid to the supplier by the CM iplies that the latter accepts the contract offered by the OEM. Thus, PfX rw, ~ X s r ~w s,x Þg Pf ~ X s r ~w s,x Þg Z w ~F s ð ~w s,x ÞÞ df ðx Þ Then, (1) reduces to Z w Max P T o ðq,w Þðk yq w Þq ~F s ð ~w s,x ÞÞ df ðx Þ ð13þ q,w Again, given w, (13) is concave in q. Thus, axiization over q yields q k w ð14þ Substituting (14) back into (13) gives Max w P T o Þ ðk w Þ 4y Z w ~F s ð ~w s,x ÞÞ df ðx Þ ð15þ Lea 1. If F ~ s ðxþ= f ~ s ðxþ is convex in x, the objective function in (15) is log-concave and the optial w satisfies the first order condition of (15) k w R w R w ~F s ð ~w s,x ÞÞ df ðx Þ ~f s ð ~w s,x ~w s,x Þ df ðx ð16þ where ~w s,x Þ ~w s,x can be derived fro (11).

6 18 P. Guo et al. / Int. J. Production Econoics 18 (1) the supplier ay pretend to have very high costs in the first period in the expectation of higher wholesale prices in the next period. In other words, they ay intentionally reject the contract in the first period even though the wholesale prices offered are high enough to cover their costs. In contrast, intentional rejection is never optial under a one-period contract scenario. As a result, our focus in this section is on coparing the significance of the ratchet effect between the two outsourcing structures, consignent and turnkey. We oit the integration structure in this section as we want to focus on the discussion of the ratchet effect aong the three parties. For any t1,, denote w it as the wholesale price offered to party i, i, s, at the beginning of tie period t. The production quantity in each period is deterined by the corresponding firstorder conditions in the one-period setting (a function of the wholesale prices). Let a it be the action of party i in period t, i, s. Then, a it r eans the action is reject; and a it a eans it is accept. We use a t ða t,a st Þ to denote the joint action of the CM and the supplier; then, a t ASð,sÞfða,aÞ,ðr,aÞ,ða,rÞ,ðr,rÞg. Let h t be the action history up to the end of period t, i.e., h 1 ða 1 Þ, h ða 1,a Þ. Denote by P i ða it Þ and P i ðh Þ the profit of party i at the end of periods 1 and, respectively, io,, s. We assue that there is a discount factor gðrgr1þ Consignent Under two-period consignent (i) at the beginning of period 1, the OEM offers take-it-or-leave-it wholesale prices w 1 and w s 1 to the CM and the supplier, respectively, based on its initial prior inforation, F ðþ and F s ðþ. (ii) Then, the CM and the supplier decide whether to accept the contract with the objective of axiizing their own discounted two-period profits. (iii) After their response, the OEM updates its knowledge about the upstrea cost structures based on its current inforation set the original prior beliefs about costs and the observed history of actions h 1 according to Bayes rule. (iv) With this updated knowledge, the OEM then akes the second-round pricing decisions, w (h 1 ) and w s (h 1 ), at the beginning of period to axiize its expected profit in that period. Let q(h 1 ) denote the resulting optial ordering quantity at the beginning of period. (v) Then, the CM and the supplier decide whether to accept or reject the contract with the objective of axiizing their own profits in period, given the other player s strategy fixed. Assuptions (i), (ii), (iv) and (v) ean perfection, naely optiization by each player for any history of the gae. An equilibriu satisfying the foregoing five conditions is said to be a perfect Bayesian equilibriu (PBE). A perfect Bayesian equilibriu represents a strategy profile and a belief syste, such that the strategies are sequentially rational given the particular belief syste and that the belief syste is consistent, wherever possible, given the strategy profile. In the reainder of this section, we show that the two-period consignent gae has a PBE. We first establish a key observation. Note that the intentional rejection has a cost the CM (supplier) sacrifices its first-period positive profit. The following lea shows that only when the CM s (supplier s) cost is larger than a certain threshold will it reject the contract. Lea. In period 1, given wholesale prices w i 1, i, s, and under the rational expectation assuption, there exists the cutoff level ^c ð^c s Þ such that if c 4 ^c ðc s 4 ^c s Þ, the CM s (supplier s) optial decision is to reject. Otherwise, it is to accept. When c ^c ðc s ^c s Þ, the CM (supplier) is indifferent between accepting and rejecting. Anticipating this threshold-type strategy, if party i rejects contract w i 1 in period 1, then the OEM knows that its cost ust be at least higher than soe value ^c i. Then, the OEM updates its belief as follows. ( f i ðx i jrþ f i ðx iþ=ð1 F i ð^c i ÞÞ for x i Z ^c i, ð17þ for x i o ^c i Thus, the posterior distribution is a left-truncation of the prior. Correspondingly, we have R xi F i ðx i jrþ f i ðx i jrþ ^c i f i ðx i Þ=ð1 F i ð^c i ÞÞ dx i f i ðx i Þ=ð1 F i ð^c i ÞÞ F iðx i Þ F i ð^c i Þ o F iðx i Þ f i ðx i Þ f i ðx i Þ ð18þ In other words, conditional updated belief X i jr satisfies X i jr Z rh X i. According to Proposition 1, we ust have w i ðh 1 Þ4w i1. Thus, this inforation-updating echanis is consistent with party i s rational expectation. If party i accepts w i 1, then the sae contract is repeated in period, according to the coitent assuption. We now derive the profit functions of the CM and the supplier assuing their costs are at the cutoff levels ð^c, ^c s Þ in Lea. Because of their indifference at these threshold points, we obtain a nuber of siultaneous equations to solve for the equilibriu. Proposition The CM s indifference between reject and accept when c ^c leads to gf s ð^c s Þðk w ðr,aþ w s1 Þ ðr,aþ ^c Þ þg½f s s ðr,rþþ F s ð^c s ÞŠðk w ðr,rþ w s ðr,rþþ ðr,rþ ^c Þ ð1þgþf s ð^c s Þðk w 1 w s1 Þ 1 ^c Þ þgðf s s ða,rþþ F s ð^c s ÞÞðk w 1 w s ða,rþþ 1 ^c Þ, ð19þ where the left-hand side is the expected profit for the CM when it chooses to reject the offer; the right-hand side is that it accepts. Siilarly, the supplier s indifference when c s ^c s leads to gf ð^c Þðk w s ðr,aþ w 1 Þ s ðr,aþ ^c s Þ þg½f ðr,rþþ F ð^c ÞŠðk w s ðr,rþ w ðr,rþþ s ðr,rþ ^c s Þ ð1þgþf ð^c Þðk w s1 w 1 Þ s1 ^c s Þ þgðf ða,rþþ F ð^c ÞÞðk w s1 w ða,rþþ s1 ^c s Þ ðþ Solving (19) and () yields the equilibriu solution to ^c and ^c s.. If condition ð1þgþp ðða,aþj^c ÞþgP ððr,rþ,ða,aþj^c Þ 4gP ðða,rþ,ða,aþj^c ÞþgP ððr,aþ,ða,aþj^c Þ ð1þ holds, then ^c is increasing with ^c s. A siilar result holds for the supplier when the condition is ð1þgþp s ðða,aþj^c s ÞþgP s ððr,rþ,ða,aþj^c s Þ 4gP s ðða,rþ,ða,aþj^c s ÞŠþgP s ððr,aþ,ða,aþj^c s Þ ðþ 3. Given that conditions (1) and () are satisfied, there exists a unique solution for ð^c, ^c s Þ. Condition (1) eans that the su of the discounted profits for the CM when it and the supplier take the sae action in the first period (both accept or both reject) should be greater than that when the CM and the supplier take different actions in the first period. Under this condition, when the supplier chooses a larger ^c s, the CM also chooses a larger ^c. Siilar interpretation applies to ().

7 P. Guo et al. / Int. J. Production Econoics 18 (1) The CM s and the supplier s best response in period 1 can be characterized by ð^c, ^c s Þ, and takes the following for a 1 ða,aþ if c r ^c, c s r ^c s ; a 1 ða,rþ if c r ^c, c s 4 ^c s ; a 1 ðr,aþ if c 4 ^c, c s r ^c s ; and a 1 ðr,rþ if c 4 ^c, c s 4 ^c s. (When a player is indifferent between accepting and rejecting, we assue it chooses to accept.) The action taken in period depends on the order between wholesale price w i (h 1 ) and cost c i, i,s. Ifw i ðh 1 ÞZc i, then party i chooses to accept the offer. Otherwise, it chooses to reject it. The OEM s decisions about the wholesale prices in period 1, (w 1, w s 1 ), can then be obtained by solving its two-period discounted profit function with a nuerical exclusive search. 4.. Turnkey In the two-period turnkey setting, we assue that as long as the OEM offers a wholesale price that is greater than the CM s cost, the CM will always accept the contract offer. One arguent is that the CM can always axiize its profit by deciding the price offered to the supplier. Even though the CM wants to reject the offer, it can instead offer a low price to the supplier and let the supplier reject the contact. Hence, we can concentrate on analyzing the supplier s behavior. Here, the supplier ay intentionally reject the CM s offer in anticipation of a higher price in the next period. Siilar to our analysis of the two-period consignent, there exists a cutoff level ^c s. If the supplier s cost c s falls at this point, then it is indifferent between rejecting and accepting the contract. If c s 4 ^c s, then the optial decision for the supplier is to reject it. Now consider indifferent point c s ^c s. Suppose the supplier chooses to accept in period 1; then, the CM ust accept the contract (as argued in the single-period turnkey). Thus, a 1 ða,aþ. According to our coitent assuption, the contract in the second period reains the sae. As a result, a ða,aþ and P ðða,aþj^c s ÞP ðða,aþ,ða,aþj^c s Þ ðk w 1Þ s1 c s Þ Hence, the supplier s total discounted profit is P s ðaj^c s Þð1þgÞ P ðða,aþj^c s Þ. If the supplier rejects the contract in period 1, then, at the end of period 1, both the OEM and the CM update their prior inforation about the supplier s cost. The analysis of this gae is siilar to that for the one-period turnkey, but with updated prior inforation. The OEM updates prior F ~ s according to (17) and offers updated wholesale price w (a, r) to the CM, which solves k w R w ½ ~ F s ð ~w s,x ÞÞ ~ F s ð^c s ÞŠ df ðx Þ R w ~f s ð ~w s,x ~w s,x Þ df ðx ð3þ where ~w s,x Þ is the solution to F w x w s þ ~ s s Þ F ~ s ð^c s Þ ~f s s Þ The CM also updates its belief about the supplier s cost and offers the supplier w s (a,r), which is the solution to w ða,rþ c w s þ F s s Þ F s ð^c s Þ ð4þ f s s Þ In this case, the supplier s total discounted profit is P s ðrj^c s Þg ðk w ða,rþþ s ða,rþ c s Þ Solving equation P s ðaj^c s ÞP s ðrj^c s Þ generates the solution for ^c s. We suarize the foregoing conclusions in the following proposition. Proposition 7. The supplier s best response in period 1 can be characterized by ^c s, which solves the equation gðk w ða,rþþ s ða,rþ c s Þð1þgÞðk w 1 Þ s1 c s Þ, ð5þ where w (a, r) is the solution to (3) and w s (a,r) is the solution to (4). The supplier chooses to accept in period 1 if and only if (iff) c s r ^c s s1 Þ and chooses to accept in period iff w s ðh 1 ÞZc s ; otherwise, the supplier chooses to reject. Anticipating the supplier s best response, the OEM and the CM are able to choose their wholesale prices, w 1 and w s 1, respectively, to axiize their discounted two-period expected profit. 5. Nuerical experients In this section, we copare the significance of the ratchet effect under consignent and turnkey through a nuerical study. We exaine each party s best response, profit and resulting chain perforance by varying the production costs, prior distributions and discount factors. In all of our experients, the arket deand paraeters are k 3 and y 1. Production costs c and c s are drawn fro the following set {1,3,5,7,9,11,13}. We assue the cost priors are independent rando variables with noral distributions. The eans are unbiased, that is, i c i,i,s, and the standard deviation s i Af1 i, i,3 i g. The discount factors are set at g f35,55,75,95g. In total, there are 1764 cobinations for the two-period odel. In choosing the OEM s prior on the supplier s cost X ~ s under turnkey, we consider an extree case where X ~ s is in the best interests of the CM, that is, the CM s profit is axiized with X ~ s. In this way, we can eliinate the CM s self-interested behavior under turnkey and focus on investigating the ipact of the rachet effect in the threetier supply chain. We observe that in the one-period setting, when X ~ s is ost desirable for the supplier, the OEM always prefers consignent to turnkey, and the CM always prefers turnkey to consignent. This is not surprising. However, when the outsourcing contract lasts for two periods, in contrast, the OEM ay prefer turnkey to consignent, and the CM ay prefer consignent to turnkey. Table 1 illustrates the profits of each player under certain prior inforation structures, where the bold nuber eans either that the OEM prefers turnkey or that the CM prefers consignent. Also, the outcoe for each player under two-period renegotiable outsourcing ay be worse than that under one-period outsourcing when intentional rejection (by the CM or the supplier) occurs in the first period. We then find that the profits of the OEM and the entire supply chain decrease when intentional rejection occurs. Table copares the profits of each player under one- and two-period consignent, where intentional rejection occurs, and Table 3 copares siilar behavior under one- and two-period turnkey. We observe that when the CM (supplier) intentionally rejects the offer while the supplier (CM) does not, then the supplier s (CM s) expected profits are lower. Therefore, the ratchet effect brings inefficiency to two-period outsourcing. By coparing the behavior of the CM and the supplier in 1764 cobinations, we find that, as the variances of the prior distributions increase, ore intentional rejections occur, either by the CM, or the supplier or by both. Thus, the ratchet effect is reinforced by prior inforation uncertainty. In addition, we find that the ratchet effect is ore severe under consignent than under turnkey. Intentional rejection occurs ore frequently under consignent than under turnkey. Aong

8 18 P. Guo et al. / Int. J. Production Econoics 18 (1) Table 1 Two-period outsourcing perforance ðg 75Þ. Input paraeters Consignent Turnkey 1 s 1 s ða 1,a Þ P oðh Þ P ðh Þ P sðh Þ ða 1,a Þ P oðh Þ P ðh Þ P sðh Þ a a, a a a a, a a a a, a a a a, a a a a, a a a r, a a a a, a a a a, a a a a, a a a r, a a a r, a a a r, a a a a, a a a a, a a a a, a a a a, a a a a, a a a r, a a r a, a a a a, a a a a, a a a r, a a a r, a a a r, a a a a, a a a a, a a a a, a a a a, a a a a, a a a a, a a a a, a a a a, a a r a, a a a a, a a a a, a a a a, a a r a, a a a a, a a r a, a a a a, a a r a, a a a r, a a r a, a a a a, a a r a, a a a r, a a r r, a a a r, a a Table Ratchet effect on inhouse consignent ðg 75Þ. Input paraeters One period consignent Two period consignent 1 s 1 s a 1 P C o P C P C s ða 1,a Þ P oðh Þ P ðh Þ P sðh Þ a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a a r, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a a a r a, a a the 1741 cases, we observed 114 cases of intentional rejection under two-period consignent and 99 cases of it under twoperiod turnkey. This is because of the non-cooperative gae between the CM and the supplier under consignent that is, consignent s success depends on the participation of both the CM and the supplier. If one player accepts the contract and the other rejects it, then the forer gets zero in the current period and the sae price in the next period; however, if instead, they both reject it, then that player is better off with a higher price in the next period. Thus, the non-cooperative gae between the CM

9 P. Guo et al. / Int. J. Production Econoics 18 (1) Table 3 Ratchet effect on turnkey ðg 75Þ. Input paraeters One period turnkey Two period turnkey 1 s 1 s a 1 P T o P T P T s ða 1,a Þ P oðh Þ P ðh Þ P sðh Þ a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a a a a r, a a Table 4 Ipact of discount factor on ratchet effect. Input paraeters Two-period inhouse consignent Two-period turnkey g Failure Intentional rejection Failure Intentional rejection and the supplier ay diinish the profits of both, as well as those of the entire supply chain. In contrast, the CM has no incentives to intentionally reject the contract under turnkey, which itigates the non-cooperative gae s inefficiency. Lastly, we explore the effect of the discount factor on the ratchet effect. We observe that, as this factor increases, the failure of the outsourcing contract in the first period occurs ore frequently. That is, as g increases, the profits in the second period becoe ore iportant, which stiulates rejection in the first period. Hence, increasing the discount factor enhances the ratchet effect. Table 4 lists the total incidence of outsourcing failure as well as that of intentional rejection (fro the CM, the supplier and both) in the first period under each discount factor. It also shows that consignent has ore outsourcing failure and intentional rejection than does turnkey. 6. Suary In this paper, we investigate three outsourcing structures in a three-tier supply chain consisting of an OEM, a CM and a supplier consignent, integration and turnkey. We first obtain the equilibriu wholesale prices under the one-period setting and exaine the ipact of the OEM s prior inforation about other parties costs on those wholesale prices. We then extend the study to a two-period scenario. We discuss how to update the cost inforation between the periods and deonstrate how to derive the corresponding perfect Bayesian equilibria. Finally, we conduct a nuerical study to explore the ratchet effects under different structures. Without inforation difference, and with an exaple of the power function distributions of costs, we illustrate that the wholesale price offered by the OEM for each sei-product is the sae under integration and turnkey. When the OEM has a low estiation of the CM s and supplier s costs, the wholesale price it offers for each sei-product is also the sae under consignent and integration. However, when this estiation is high, the OEM tends to offer a lower wholesale price under integration than under consignent. We show that the OEM is always better off under integration than under consignent or turnkey. This is because integration itigates double arginalization. We then characterize the sensitivity of the prior inforation on wholesale prices. We find that, under integration, as long as

10 184 P. Guo et al. / Int. J. Production Econoics 18 (1) the OEM has stochastically larger prior in the reversed hazard rate order, the wholesale price it offers will be higher. However, under turnkey, the conditions propting the OEM to offer a higher price are stronger. Fro this analysis, we conclude that credible inforation sharing between the OEM and the CM is not possible without a suitable echanis. We note that there exist several research work along this direction such as Kaya and Özer (9) and Kayis et al. (7). We find that when outsourcing contracts last ore than one period, even though the OEM s indirect prior about the supplier s cost under turnkey is in alignent with the CM s best interests, it ay be better off under this structure than under consignent even though it possesses ore inforation under the latter. One possible reason for this is the inefficiency that arises fro the ratchet effect. Through our nuerical studies, we find that consignent results in a larger ratchet effect than does turnkey, a difference that is caused by the non-cooperative gae between the CM and the supplier. Under turnkey, the CM has no incentives to reject an offer intentionally, and the syste s success depends solely on the supplier s optial decision. Under consignent, the contract s success depends on the participation of both the CM and the supplier. As one player s rejection tends to induce the other s rejection, this strengthens the ratchet effect and renders supply chain coordination ore difficult. We also find that the ratchet effect is strengthened by a high degree of variance in the cost priors and a larger discount factor. Our study shows that the choice between turnkey and consignent depends on such situational factors as the priors about costs and the length of contracts. Our research also deonstrates the iportance of coordination under consignent and turnkey to achieve contract success, an issue worth further exploration in the future. Acknowledgents We would like to thank the editor and two anonyous referees for their helpful coents and good suggestions. Guo s research was supported in part by Hong Kong RGC Research Grant (no. B-Q11F). Song s research was supported in part by National Natural Science Foundation of China (no ). Wang s research was supported in part by the Hong Kong Polytechnic University Departental Grant (G-U639). Appendix To prove the propositions in this paper, we first introduce the following leas and definitions. Lea 3 (Bagnoli and Bergstro, 5). Suppose rando variable Y i has a log-concave distribution function G i and pdf g i, i1,. Then, (1) the reversed hazard rate g i (x)/g i (x) is decreasing in x; () if t(x) is twice differentiable, strictly increasing and concave, then G i (t(x)) is a log-concave function of x; and (3) the convolution GðxÞ R x G 1ðx x ÞdG ðx Þ is log-concave. These properties, along with the following reversed hazard rate order and its properties, are useful in later sections for coparative static analysis of the optial wholesale prices. Lea 4 (Shaked and Shanthikuar, 1994). If rando variables Y 1 and Y are such that Y 1 r rh Y, and if Z is a rando variable independent of Y 1 and Y with a decreasing reversed hazard rate, then Y 1 þz r rh Y þz. Proof for Proposition 1. Condition ^X s Z rh X s iplies that ^F s ðxþ= ^f s ðxþrf s ðxþ=f s ðxþ Fro (5), we can argue by contradiction that ^w s Zw s ; and fro (4), we can further derive that ^w rw. The proof for the second part is siilar. & Proof for Proposition. As X has a log-concave density, it has a decreasing reversed hazard rate. Then, according to Lea 4, X þ X ~ s Z rh X þx s, which iplies that R wa F a x s Þ d F ~ s ðx s Þ R wa f a x s Þ d F ~ s ðx s Þ r According to (8), ~w a Zw a. R wa F a x s Þ df s ðx s Þ R wa f a x s Þ df s ðx s Þ & Proof for Lea 1. If F ~ s ðxþ= f ~ s ðxþ is convex in x, Lea 3 and (11) iply that ~w s,x Þ is an increasing and concave function of w x. Also, according to Lea 3, F ~ s ð ~w s,x ÞÞ is a logconcave function of w x. Thus the convolution R w ~F s ð ~w s,x ÞÞ df ðx Þ is log-concave in w. Consequently, (15) is log-concave and has a unique optiu. & Proof of Proposition 3. X ~ s Z rh X s iplies that F ~ s ðxþ= f ~ s ðxþr F s ðxþ=f s ðxþ. Fro (11), we can derive that ~w s,x ÞZ w s,x Þ and F ~ s ð ~w s,x ÞÞ= f ~ s ð ~w s,x ÞÞrF s s,x ÞÞ= f s s,x ÞÞ, for all (w, x ). Also fro (11), we can derive F 1þ ~ s ð ~w s Þ ~w s ~f s ð ~w s Þ Thus, condition ð F ~ s ðxþ= f ~ s ðxþþurðf s ðxþ=f s ðxþþu iplies ~w s s ~w s s Hence, we have, for all (w, x ), ~F s ð ~w s,x ÞÞ ~f s ð ~w s,x ~w r s,x F s s,x ÞÞ f s s,x s,x Because the reversed hazard rate order is closed under convolution (see Lea 4), we can further derive that R w R w ~F s ð ~w s,x ÞÞ df ðx Þ ~f s ð ~w s,x ~w s,x Þ df ðx R w F s s,x ÞÞ df ðx Þ r R w f s s,x ð6þ s,x Þ df ðx Then, according to (16), ~w Zw. Proof of Proposition 4. For consignent, (w C, w s C ) are decided by first-order conditions (4) and 5. It can be shown that w C kb þb þb s, w C s kb s þb þb s For integration, solving (8) yields, k w I a R w I a I a x Þ b s b x b 1 R w I a b s I a x Þ b s 1 b x b ð7þ 1 I As the doain of x is (,1], we need to classify w a into two cases w I a r1and1owi a. For the first case, the foregoing can be written as k w I a R w I a I a x Þ b s x b 1 R w I a b s I a x Þ b s 1 x b 1 wi a Bðb,b s þ1þ b s Bðb,b s Þ w I a b s b s ðb þb s Þ wi a, b þb s &

11 P. Guo et al. / Int. J. Production Econoics 18 (1) where Bða,bÞ R 1 ta 1 ð1 tþ b 1 dt is the beta function. Solving this equation yields w I a kðb þb s Þ ð8þ þb þb s This solution is valid if kðb þb s Þ=ðþb þb s Þr1 orb þb s r= ðk 1Þ. If, instead, b þb s 4=ðk 1Þ, then it becoes k w I a R 1 I a x Þ b s x b 1 R 1 b s I a x Þ b s 1 x b 1 wi a Bð1=wI a,b,b s þ1þ b s Bð1=w I a,b,b s Þ, where Bðz,a,bÞ R z ta 1 ð1 tþ b 1 dt is the incoplete beta function. For the incoplete beta function, it has the property B(z,a,b+1)b/(a+b)B(z,a,b)+z a (1 z) b /(a+b) (see the coplete suary of the incoplete beta function on functions.wolfra. co). Using this, we can derive that k w I a wi a Bð1=wI a,b,b s þ1þ b s Bð1=w I a,b,b s Þ 4 wi a b s b s ðb þb s Þ wi a b þb s The above inequality yields w I a o kðb þb s Þ þb þb s Note that w C þwc s ðkðb þb s ÞÞ=ðþb þb s Þ. Under turnkey, fro (11) we obtain w T x ~w T s þ ~wt s b s Thus, ~w T s T,x Þ b s T x Þ b s þ1 Fro this we can ~w T s T,x T b s b s þ1 ð9þ Substitute ~w T s T,x Þ ~w T s T,x T into (16), and we obtain k w T b s T x bs Þ b b s þ1 x b 1 bs 1 b s b s þ1 b s þ1 b xb 1 R w T R w T b s b s T x Þ R w T T x Þ b s x b 1 R w T b s T x Þ b s 1 x b 1 This equation is the sae as (7). Thus, w T w a I. Proof of Proposition 5. Assue the price for one unit of a seiproduct is the sae across the three structures, i.e., w C þwc s wi a wt. Note that PfwI a ZX þx s gzpfw C ZX g Pfw C s ZX sg always holds. Thus, integration yields a higher chance of contract success than does consignent. Denote the optial solutions to Max w,wsp C o,w s Þ as (w C, w s C ). Then, Max wa P I o aþzp I o C þwc s ÞZPC o C,wC s ÞMax P C o,w s Þ, w,ws thus proving (a). & ð3þ Eq. (11) iplies that w T x 4w T s, and thus Z w T F s T s Þ df ðx Þo Z w T F s T x Þ df ðx Þ Pfw T ZX þx s g Hence, turnkey yields a saller chance of contract success than does integration. Denote the optial solutions to Max w P T o Þ by w *. Then, Max wa P I o aþzp I o ÞZPT o ÞMax P T o Þ, w ð31þ thus proving (b). There is no certain order relationship between success probability Pfw C ZX gpfw C s ZX sg under consignent and R w T F s T s Þ df ðx Þ under turnkey, given w C +w C s w T. Thus, (c) holds. & Proof of Lea. We take the CM as an exaple. We fix the supplier s first period decision a s1, and denote it by. We first assue a s1 to be to accept. The proof for the general case is siilar. Suppose a 1 a; then, the CM obtains the wholesale price w 1 in both periods 1 and. Thus, its total profit in the two periods is P ða,þ ð1þgþ 1 c Þqða,Þ. If a 1 r, then the CM s firstperiod profit is zero, but in period it obtains a higher price, w ðr,þ, than w 1. Thus, its total profit is P ðr,þ g ðr,þ c Þqðr,Þ. Because the optial ordering quantity satisfies q ðk w w s Þ=ðÞ, condition w ðr,þ4w 1 iplies qðr,þoqða,þ. Assue P ða,þ P ðr,þ at ^c.ifc 4 ^c, then we have P ða,þ ð1þgþ 1 ða,þ c Þqða,Þ ð1þgþ 1 ða,þ ^c Þqða,Þþð1þgÞð^c c Þqða,Þ g ðr,þ ^c Þqðr,Þþð1þgÞð^c c Þqða,Þ og ðr,þ ^c Þqðr,Þþð1þgÞð^c c Þqðr,Þ og ðr,þ ^c Þqðr,Þþgð^c c Þqðr,Þ g ðr,þ c Þqðr,Þ P ðr,þ Thus, the optial decision is to reject. Siilarly, we can show that when c o ^c, P ða,þ4p ðr,þ, the optial decision is to accept. & Proof of Proposition 6. We focus on the CM s profit functions. The results for the supplier can be derived siilarly. Assue c ^c ; then, the CM can take one of the following two actions. Case 1 The CM rejects the contract in period 1. In this case, the CM s profit in the first period is zero, i.e., P ððr,þj^c Þ, and the OEM updates its belief about the CM s cost according to (17). If the supplier s action a s1 a, then the OEM offers the sae price to the supplier in period, i.e., w s (r,a)w s 1, and offers a higher price, w (r,a), to the CM, which solves the following firstorder condition (siilar to (4)) k w w s1 F jrþ f jrþ F Þ F ð^c Þ f Þ (According to (18), this indeed guarantees w ðr,aþ4w 1.) The corresponding optial order quantity is qððr,aþ k w ðr,aþ w s1 The behavior of each player in period resebles that under oneperiod consignent. In particular, according to the coitent assuption, because the supplier obtains the sae price as in the

12 186 P. Guo et al. / Int. J. Production Econoics 18 (1) first period, its action in period is again a. The expected secondperiod profit for the CM can then be expressed as ð1þgþf s ð^c s Þðk w 1 w s1 Þ 1 ^c Þ þgðf s s ða,rþþ F s ð^c s ÞÞðk w 1 w s ða,rþþ 1 ^c Þ ð38þ P ððr,aþ,ða,aþj^c Þ k w ðr,aþ w s1 ðr,aþ ^c Þ ð3þ If the supplier chooses a s1 r, then the OEM updates its belief about the supplier s cost according to (17), and the optial wholesale prices offered by the OEM in period, (w (r,r), w s (r,r)), solve k w w s F Þ F ð^c Þ f Þ F s s Þ F s ð^c s Þ, ð33þ f s s Þ which are siilar to (4) and (5). The expected second-period profit for the CM under this condition is then P ððr,rþ,ða,aþj^c Þ Pfw s ðr,rþzx s g k w ðr,rþ w s ðr,rþ ðr,rþ ^c Þ F s s ðr,rþþ F s ð^c s Þ 1 F s ð^c s Þ k w ðr,rþ w s ðr,rþ ðr,rþ ^c Þ ð34þ In period 1, anticipating the threshold-type policy of the supplier, the CM holds the belief that the supplier will chooses to accept with probability F s ð^c s Þ and to reject with probability 1 F s ð^c s Þ. Thus, if the CM chooses to reject in period 1, then its expected total discounted profit in the two periods can be expressed as P ðrj^c Þg½F s ð^c s ÞP ððr,aþ,ða,aþj^c Þ þð1 F s ð^c s ÞÞP ððr,rþ,ða,aþj^c ÞŠ ð35þ Case The CM accepts the contract in period 1. In this case, if the supplier chooses a s1 a, then the OEM ends the negotiation processes with both the CM and the supplier and offers the the sae prices in period. Thus, P ðða,aþj^c ÞP ðða,aþ,ða,aþj^c Þ k w 1 w s1 1 ^c Þ If the supplier chooses a s1 r, then P ðða,rþj^c Þ. The CM gets the sae price in the second period, i.e., w (a,r)w 1. The supplier gets a higher price, w s (a,r), which solves k w 1 w s F s s jrþ f s s jrþ F s s Þ F s ð^c s Þ f s s Þ (According to (18), this indeed guarantees w s ða,rþ4w s1.) In period, the contract is successful if and only if w s ða,rþzx s. Fro (17), we have PfX s rw s ða,rþg ðf s s ða,rþþ F s ð^c s ÞÞ=ð1 F s ð^c s ÞÞ Hence, P ðða,rþ,ða,aþj^c Þ F s s ða,rþþ F s ð^c s Þ 1 F s ð^c s Þ k w 1 w s ða,rþ 1 ^c Þ ð36þ Here again, the CM holds the belief that the supplier will choose to accept with probability F s ð^c s Þ and to reject with probability 1 F s ð^c s Þ. Thus, if the CM chooses to accept in period 1, then its expected total discounted profit in the two periods is P ðaj^c Þð1þgÞF s ð^c s ÞP ðða,aþj^c Þ þgð1 F s ð^c s ÞÞP ðða,rþ,ða,aþj^c Þ ð37þ Because the CM is indifferent between rejecting and accepting when c ^c, P ðrj^c ÞP ðaj^c Þ. This leads to gf s ð^c s Þðk w ðr,aþ w s1 Þ ðr,aþ ^c Þ þg½f s s ðr,rþþ F s ð^c s ÞŠðk w ðr,rþ w s ðr,rþþ ðr,rþ ^c Þ In addition, the supplier is indifferent between rejecting and accepting when c s ^c s, i.e., P s ðaj^c s ÞP s ðrj^c s Þ. Analogous to our analysis of the CM s perforance when c ^c, we can derive a siilar equation to (19) for the supplier when c s ^c s gf ð^c Þðk w s ðr,aþ w 1 Þ s ðr,aþ ^c s Þ þg½f ðr,rþþ F ð^c ÞŠðk w s ðr,rþ w ðr,rþþ s ðr,rþ ^c s Þ ð1þgþf ð^c Þðk w s1 w 1 Þ s1 ^c s Þ þgðf ða,rþþ F ð^c ÞÞðk w s1 w ða,rþþ s1 ^c s Þ ð39þ For part, let us take the CM as an exaple. Consider the supplier to deviate fro ^c s to ^c u s ^c s þe, e4. To prove onotonicity, we need to show that ^c u 4 ^c. For the CM, P ðrj^c, ^c s Þ gf s ð^c s Þ k w ðr,aþ w s1 ðr,aþ ^c Þ þg½f s s ðr,rþþ F s ð^c s ÞŠ k w ðr,rþ w s ðr,rþ ðr,rþ ^c Þ Assue that, as the supplier deviates fro ^c s by a very sall e, the prices offered by the OEM reains the sae. Then. for the CM, li½p ðrj^c, ^cu s Þ P ðrj^c, ^c s ÞŠ=e e- d½p ðrj^c, ^c s ÞŠ=d^c s gf s ð^c s Þ k w ðr,aþ w s1 ðr,aþ ^c Þ gf s ð^c s Þ k w ðr,rþ w s ðr,rþ ðr,rþ ^c Þ f s ð^c s Þ½gP ððr,aþ,ða,aþj^c Þ gp ððr,rþ,ða,aþj^c ÞŠ Siilarly, P ðaj^c, ^c s Þð1þgÞF s ð^c s Þ k w 1 w s1 1 ^c Þ þgðf s s ða,rþþ F s ð^c s ÞÞ k w 1 w s ða,rþ 1 ^c Þ Thus, as ^c s changes to ^cu s ^c s þe, li½p ðaj^c, ^cu s Þ P ðaj^c, ^c s ÞŠ=e e- d½p ðaj^c, ^c s ÞŠ=d^c s ð1þgþf s ð^c s Þ k w 1 w s1 1 ^c Þ gf s ð^c s Þ k w 1 w s ða,rþ 1 ^c Þ f s ð^c s Þ½ð1þgÞP ðða,aþj^c Þ gp ðða,rþ,ða,aþj^c ÞŠ Hence, condition (1) iplies that P ðaj^c, ^cu s Þ P ðaj^c, ^c s Þ 4P ðrj^c, ^cu s Þ P ðrj^c, ^c s Þ. Because P ðaj^c, ^c s Þ P ðrj^c, ^c s Þ, we obtain P ðaj^c, ^cu s Þ4P ðrj^c, ^cu s Þ. Besides, (19) iplies that if ^c increases to ^c u ^c þz, and Z4, then ½P ðrj^c u, ^c sþ P ðrj^c, ^c s ÞŠ ½P ðaj^cu, ^c s Þ P ðaj^c, ^c s ÞŠ gf s ð^c s Þ k w ðr,aþ w s1 þg½f s s ðr,rþþ F s ð^c s ÞŠ k w ðr,rþ w s ðr,rþ Z n þ ð1þgþf s ð^c s Þ k w 1 w s1 þgðf s s ða,rþþ F s ð^c s ÞÞ k w 1 w s ða,rþ Z4

13 P. Guo et al. / Int. J. Production Econoics 18 (1) The above inequality follows fro the fact that w ðr,aþ ^c 4w 1 ^c and w ðr,rþ ^c 4w ^c, along with (19). Hence, P ðrj^c u, ^c sþ4p ðaj^c u, ^c sþ. Thus, as ^c s increases to ^c s u, ^c should increase to a larger value ^cu, such that P ðrj^cu, ^cu s ÞP ðaj^cu, ^cu s Þ. Thus, part is proved. & Part 3 follows fro Tarski s fixed point theore. References Aaral, J., Billington, C., Tsay, A., 6. Safeguarding the proise of production outsourcing. Interfaces 36, 33. Bagnoli, M., Bergstro, T., 5. Log-concave probability and its applications. Econoic Theory 6, Blair, B., Lewis, T., Optial retail contracts with asyetric inforation and oral hazard. Rand Journal of Econoics 13, Burnetas, B., Gilbert, S., Sith, C., 7. Quantity discounts in a single period supply contract with asyetric deand inforation. IIE Transactions 39, Cachon, G., 3. Supply chain coordination with contracts. In de Kok, A., Graves, S. (Eds.), Handbooks in Operations Research and Manageent Science Supply Chain Manageent. Elsevier, North-Holland, Asterda. Cachon, G., Lariviere, M., 1. Contracting to assure supply how to share deand forecasts in a supply chain. Manageent Science 47, Carbone, J., 4. Hewlett Packard wins for the nd tie. Purchasing, Septeber. Chen, F., 3. Inforation sharing and supply chain coordination. In de Kok, A., Graves, S. (Eds.), Handbooks in Operations Research and Manageent Science Supply Chain Manageent. Elsevier, Asterda, The Netherlands. Chen, F., 7. Auctioning supply contracts. Manageent Science 53, Chen, Y., Deng, M., Huang, K., 6. Hierarchical screening for capacity allocation in distribution systes. Working Paper, New York University, New York, NY. Corbett, C., 1. Stochastic inventory systes in a supply chain with asyetric inforation cycle stocks, safety stocks, and consignent stocks. Operations Research 49, Corbett, C., de Groote, X.,. A supplier s optial quantity discount policy under asyetric inforation. Manageent Science 46, Corbett, C., Tang, C., Designing supply contracts contract type and inforation asyetry. In Tayur, S., Ganeshan, R., Magazine, M. (Eds.), Quantitative Models for Supply Chain Manageent. Kluwer Acadeic Publishers, Massachusetts. Freixas, X., Guesnerie, R., Tirole, J., Planning under incoplete inforation and the ratchet effect. Review of Econoic Studies 5, Ha, A., 1. Supplier buyer contracting asyetric cost inforation and cutoff level policy for buyer participation. Naval Research Logistics 48, Huckan, R., Pisano, G., 4. Harvard Business School Cases. Flextronics International Ltd. Kaya, M., Özer, Ö., 9. Quality risk in outsourcing noncontractible product quality and private quality cost inforation. Naval Research Logistics 56, Kayis, E., Erhun, F., Plabeck, E., 7. Delegation vs. control of coponent procureent. Working Paper, Stanford University, Stanford, CA. Lutze, H., Özer, Ö., 8. Proised lead-tie contracts under asyetric inforation. Operations Research 56, Margonelli, L.,. How Ikea designs its sexy price tags. Business., October, Özer, Ö., Wei, W., 6. Strategic coitent for optial capacity decisions under asyetric forecast inforation. Manageent Science 5, Plabeck, E., Taylor, T., 7. Iplications of breach reedy and renegotiation design for innovation and capacity. Manageent Science 53, Porteus, E., Whang, S., Supply chain contracting non-recurring engineering charge, iniu order quantity and boilerplate contracts. Technical Report, Stanford University, Research Paper no Shaked, M., Shanthikuar, J., Stochastic Orders and their Applications. Acadeic Press, Boston, USA. Taylor, T., Plabeck, E., 7. Supply chain relationship and contracts the ipact of repeated interaction on capacity investent and procureent. Manageent Science 53, Ülkü, S., Toktay, L., Yücesan, E., 7. Risk ownership in contract anufacturing. Manufacturing & Service Operations Manageent 9, 5 41.

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