Int. J. Production Economics

Transcription

1 Int. J. Prodution Eonomis 135 (212) Contents lists available at Sieneiret Int. J. Prodution Eonomis journal homepage: Coordinating a hannel with asymmetri ost information and the manufaturer s optimality Yuelin Shen a,n, Sean P. Willems b a Shanghai University of Finane and Eonomis, Shool of International Business Administration, Shanghai 2433, People s Republi of China b Boston University, Shool of Management, Boston, MA 2215, USA artile info Artile history: Reeived 27 November 29 Aepted 15 November 21 Available online 23 November 21 Keywords: Supply hain oordination Information asymmetry Buybak ontrat Mehanism design abstrat In a manufaturer retailer system with private retail ost information, we find that a set of inentiveompatible ontrats onsisting of wholesale and buybak pries an oordinate the hannel for any retail ost. We then design two wholesale-buybak ontrats by imposing a utoff point on the retail ost. The first ontrat maximizes the manufaturer s expeted profit while ensuring the hannel is oordinated. The seond ontrat assumes the same ontratual struture without onsidering the effet on the hannel. Both ontrats are exatly solved. We find from numerial study that the manufaturer in the first ontrat an perform losely to the seond one in many ases, and ases exist where both the manufaturer and the hannel an do better in the first ontrat versus the seond one. & 21 Elsevier B.V. All rights reserved. 1. Introdution Original equipment manufaturers (OEMs) are ontinuing to outsoure portions of their supply hains. Instead of a vertially integrated entity ontrolling all aspets of the produt as it evolves from a onept into a ommerially available produt, there are now multiple independent ompanies involved in the proess with the OEM ating as the hannel master oordinating the entire proess. When a supply hain involves independent ompanies, there is inherent diffiulty in oordinating the hannel to optimize the system s profitability and determining how to share the hannel s profits aross the ompanies. In order to align inentives between different parties in the supply hain, a wide diversity of ontrating strategies have been devised and implemented in industry. This has in turn generated a signifiant stream of aademi researh in supply ontrats; see the review artiles by Tsay et al. (1999) and Cahon (23). While most researh has foused on supply ontrats for hannels where eah party in the supply hain has omplete knowledge regarding all the parameters aross the hannel, there is more and more researh designing ontrats for supply hains where there is private information held by one party. Two primary areas where information asymmetries an our relate to ost and demand information. Works inluding Lewis and Sappington (1988), Cahon and Lariviere (21), Ozer and Wei (26), and Burnetas et al. (27) onsider asymmetri demand n Corresponding author. addresses: shen.yuelin@mail.shufe.edu.n (Y. Shen), willems@bu.edu (S.P. Willems). information. Works onsidering asymmetri ost information inlude Baron and Myerson (1982), Corbett and degroote (2), Ha (21), and Corbett et al. (24). In order to mitigate the presene of asymmetri information, these papers have designed a set of inentive-ompatible ontrats suh that the party with private information is indued to selet the ontrat whih truthfully reveals the party s private information. This is the mehanism design based on the fundamental revelation priniple (Fudenberg and Tirole 2). Beyond the researh on asymmetri ost or demand reviewed above, Chakravarty and Zhang (27) also study asymmetri apaity information based on mehanism design. Regarding the asymmetri ost information at the downstream stage, Corbett et al. (24) ompare different manufaturer retailer ontrats in the ase of deterministi but prie-dependent demand. Ha (21) onsiders an environment where the market demand is the sum of prie-dependent deterministi and stohasti omponents. When the manufaturer has omplete information regarding the retailer s ost, oordinating the hannel is possible through some well-known mehanisms. When the retail ost information is private, an inentive-ompatible, nonlinear-priing ontrat with a utoff level poliy is proposed. However, it is no longer possible to ahieve the hannel-optimal solution. A utoff poliy in this ase ditates a threshold of the retail ost, above whih the manufaturer or the retailer will reeive less than the reservation profit and therefore an effetive ontrat will not be issued. Suh a utoff poliy is shown optimal due to the fat that both the manufaturer s and retailer s profits are dereasing funtions of the retail ost. Our researh is motivated by the pratie at a teleommuniation ompany. This ompany is an OEM in the omputer peripherals industry that sells its produts through several large distributors /$ - see front matter & 21 Elsevier B.V. All rights reserved. doi:1.116/j.ijpe

2 126 Y. Shen, S.P. Willems / Int. J. Prodution Eonomis 135 (212) The OEM had outsoured the vast majority of its manufaturing operations. The OEM realized that from the distributor s perspetive, the OEM s wholesale prie was just one aspet of the ost inurred by the distributor. In addition to the wholesale prie, the distributor was surely inurring a distribution ost. While the OEM did not know the distributor s exat struture of this ost, the OEM ould develop reasonable estimates for the distributor s osts. The OEM employed a wholesale-buybak ontrat with the distributor. There are two onfliting objetives faing the OEM in its role as the hannel master in the supply hain. First, the OEM held the belief that the most effiient supply hain wins so the OEM was genuinely interested in understanding how best to oordinate the supply hain. Furthermore, sine the OEM was the hannel master, operating in its own best interest meant that the OEM had the ability to struture the ontrat terms that would be to its best advantage. This problem setting motivated us to ompare two wholesale-buybak ontrats under asymmetri ost information in a manufaturer retailer (distributor) setting. In the first ontrat, the manufaturer tries to extrat the maximum profit under the ondition that the hannel is oordinated. In the seond ontrat, the manufaturer tries to extrat the highest profit without onsidering the hannel performane. We hypothesize that oordinating the hannel may not neessarily finanially harm the OEM, while ating selfishly may not be the best hoie either. Our ontributions to the literature will be threefold: first, in ontrast to the existing literature, we demonstrate that hannel oordination under asymmetri ost information is possible via a speifi wholesale-buybak ontrat; seond, we prove that a wholesale-buybak ontrat is equivalent to a nonlinear-priing ontrat under asymmetri ost information; third, we demonstrate that the hannel-oordinated ontrat an benefit the hannel yet harm or benefit the manufaturer depending on the irumstanes. Coordinating hannels has long been of researh interest. Pasternak (1985) is the first to study this problem with stohasti demand. He shows that a buybak poliy where a manufaturer offers a retailer partial redit for all unsold goods an optimize the hannel s profit. Suh a buybak poliy allows the deentralized two-stage supply hain to ahieve the profit produed by operating the supply hain in a entralized fashion. For a detailed exposition of the many extensions to the Pasternak framework, see Pasternak (28). For example, onohue (2) extends the one-period return poliy into a two-period problem setting. Granot and Yin (25) extends the problem into the ase that the demand is prie-dependent. Very reently, Xiao et al. (21) oordinate a hannel where the retailer an return the unsold produts while the final onsumers an also return the goods under their valuation after reeiving. Our researh studies hannel oordination via buybak ontrat under asymmetri retailer ost information. The remainder of the paper is strutured as follows. Setion 2 builds the problem preliminaries by inorporating retail ost into Pasternak s buybak framework in a manufaturer-retailer system, assuming this retail ost is known to eah party. In Setion 3 we study the situation that the manufaturer only has an estimate of the retailer s ost, i.e., an asymmetri ost information ase. We first find that the inentive-ompatibility ondition from revelation priniple with the oordinating ondition between the wholesale and buybak pries derived in Pasternak (1985) an warrant hannel optimality in the asymmetri information ase for any realized retail ost. We then propose a ontrat to maximize the manufaturer s profit based on this type of oordinated hannel. We propose another ontrat that preserves the ontrat setting but optimizes the manufaturer s profit by allowing the manufaturer to independently set the wholesale and buybak pries. In Setion 4, we ondut numerial experiments to ompare the performane of the manufaturer, the retailer and the hannel in the two ontrats presented in Setion 3, whih partially verifies our hypothesis. We onlude and disuss the impliation of the paper in Setion Baseline results in the full information ase To introdue notation and establish the first-best results for this hannel, we first present the full information ase where the manufaturer has omplete knowledge of the retail ost. The resulting buybak ontrat extends Pasternak (1985) to the ase of known retail ost at the retailer. We introdue the notations first. We denote s as unit manufaturing ost, w as unit wholesale prie, p as unit selling prie, as unit inremental retail ost, known or unknown to the manufaturer, r as unit buybak prie. Finally we define f(x) as the probability density funtion of the demand and F(x) as the umulative distribution funtion. As usual, we assume sowop and w+op in order not to inur negative profit for any party. We also assume that the retailer must inur the retail ost for eah unit that the retailer reeives. In order to simplify notation, without loss of generality we have assumed the goodwill ost and the salvage ost are both zero. When the retail ost is known to the manufaturer, we adopt the full return with partial redit poliy proposed in Pasternak (1985) and find the hannel s optimal order quantity, Q T, satisfies F(Q T )¼(p s)/p. In addition, the retailer s optimal order quantity, Q R, satisfies F(Q R )¼(p w)/(p r). Letting r() denote the buybak prie as a funtion of the retail ost, it is easy to see that setting r() as in Eq. (1) will oordinate the hannel rðþ¼ pðw sþ ð1þ p s Namely, under (1) we have Q T ¼Q R. (1) plays a pivotal role in ahieving hannel optimality in a deentralized system; as suh we refer to it as the hannel-optimal buybak prie. When the hannel is oordinated, we denote the optimal hannel order quantity as Q F (). In this ase, the oordinated hannel profit is expressed as a type of newsvendor profit funtion P F T ðþ¼ðp s ÞQ F ðþ p Z Q F ðþ h i where Q F ðþ¼f 1 p s p. If the value of w is fixed, we state the following proposition: Proposition 1. With the wholesale prie fixed, we have (a) rðþowþ; (b) there is, suh that if 4 then r()4w; () P F TðÞ dereases with respet to. All the proofs are in the appendix. Proposition 1(a) says the buybak prie is bounded by the sum of the wholesale prie and the manufaturer ost, whih prohibits the retailer from profiting by buying produts from the manufaturer but then returning them immediately. However, Proposition 1(b) implies there are retail osts for whih the manufaturer is willing to offer a buybak prie that will generate negative margins if returned. The reason behind suh willingness is that although the manufaturer loses profit from those returned units for ertain materialized demand, he still gains from other materialized demand; therefore the profit is higher in expetation. Finally, it is not surprising that total profit dereases as retail ost inreases. It is lear that the feasible region for the retail ost is [,p s]in the entralized supply hain whereas it is [,p w] in the ð2þ

3 Y. Shen, S.P. Willems / Int. J. Prodution Eonomis 135 (212) deentralized supply hain. When the deentralized supply hain is oordinated, the hannel profit is divided into P F w s MðÞ¼ h i p s PF T ðþ for the manufaturer and P F w s RðÞ¼ 1 p s P F TðÞ for the retailer (Cahon 23). With suh a profit split, setting a wholesale prie w¼p allows the manufaturer to absorb all the hannel profit. However, suh a ase is not rational beause the retailer will refuse to partiipate. In order for the manufaturer to determine the optimal wholesale prie, we define w as w() in general. Therefore we have a pair of deision variables, w() and r(). If we rephrase (1) as rðþ¼ pðwðþ sþ ð1 Þ p s the hannel is oordinated for any feasible. In order to determine the feasible, we propose the individual rationality onditions: Assumption 1. The retailer will not partiipate in the transation without earning a reservation profit P R ; meanwhile, the manufaturer will not partiipate without earning a reservation profit P M. In general, P R and P M will of ourse be nonzero. Assumption 1 implies the hannel profit must be at least P M þp R in order for a transation to our. In order to have a non-trivial problem, we assume P F T ðþ4p M þp R throughout this paper. Sine PF TðÞ is dereasing and ðp sþ¼, we an define the following threshold of the retail ost: P F T efinition 1. efine a utoff point op s for by P F T ð Þ¼ P M þp R. The feasible region A[, * ] defines the maximal region where a feasible transation an be onsummated from the hannel perspetive. In the full-information setting, the manufaturer will set the wholesale prie suh that the retailer s profitability onstraint is binding. The assoiated wholesale prie, w ðþ, is found by w ðþ¼ðp Þ ðp sþ P R P F T ðþ From the above we an dedue that the retailer always earns P R while the manufaturer extrats the rest of the hannel profit. Under full information, employing an individual rationality onstraint on eah party, we have proposed a mehanism to determine the wholesale prie and buybak prie suh that the hannel is oordinated while maximizing the manufaturer s profit. 3. Wholesale-buybak ontrats under asymmetri information In the full-information ase, the manufaturer knows the transation will not take plae if the retail ost is above *.Sowe assume in the asymmetri information ase the manufaturer s estimate of the retail ost is in the region [, U]C[, * ], bounded by an upper bound U and a lower bound, and he estimates the retailer ost has a probability distribution of h() in the region of [, U] with the umulative distribution funtion of H(). We also assume the retailer s true ost always falls into this region. With the two instruments of wholesale and buybak pries, the following sequene of events unfolds in the presene of asymmetri retail ost information. First, the manufaturer estimates the retailer s ost; i.e., the OEM an bound the retail ost and likely an estimate it within a relatively narrow range. Seond, onsidering the manufaturer s estimate of the retail ost, the manufaturer offers a ontrat menu onsisting of the wholesale prie and the buybak prie. Third, the retailer, with full knowledge of the retail ost, aepts a set of wholesale and buybak pries from the menu and then deides the order quantity. Fourth, the manufaturer delivers this quantity to the retailer. Fifth, the end-onsumer demand is realized and the retailer fulfills as muh demand as possible, returning any unsold units to the manufaturer Channel oordination under asymmetri information We will find that an inentive-ompatible ontrat menu onsisting of {w(),r()} with the hannel-optimal buybak prie in (1 ) an oordinate the hannel under asymmetri information, and therefore a hannel-optimal solution is ahieved in A[, * ]. If the retail ost is but the retailer aepts the ontrat {wðuþ,rðuþ} from the manufaturer, the retailer will determine the optimal order quantity by solving Z QðwðuÞ9Þ P A R ðwðuþ9þ¼max QðwðuÞ9Þ ðp wðuþ ÞQðwðuÞ9Þ ðp rðuþþ ð3þ where relation (1 ) has been inserted in Eq. (3) and therefore we only need to determine the optimal w(). The solution to the above problem will yield the optimal order quantity, Q(w( )9), as a funtion of w( ). Under this ontrat, the manufaturer s profit is Z QðwðuÞ9Þ P A M ðwðuþ9þ¼ðwðuþ sþqðwðuþ9þ rðuþ Therefore, the manufaturer needs to set the optimal wholesale prie, followed by the buybak prie given in Eq. (1 ), by solving the following: Z QðwðuÞ9Þ max wðuþ P A M ðwðuþ9þ¼ðwðuþ sþqðwðuþ9þ rðuþ where is estimated by the manufaturer while is known by the retailer. In this asymmetri information ase, the revelation priniple (Laffont and Tirole 1993) states that if there is an optimal ontrat for the manufaturer, then there exists an optimal ontrat under whih the retailer will truthfully reveal her ost. Thus we are only interested in ontrats based on the revelation mehanism. From Eq. (3), the manufaturer an predit how a retailer with ost will behave, and therefore what ontrat she will aept. The manufaturer an onstrut a mapping between and from Eq. (3). The revelation priniple is interpreted by the following inentiveompatible ondition P A R ðwðuþ9þopa RðwðÞ9Þ 8,u ð4þ Sine Eq. (4) will enfore the retailer to tell the truth of her ost while (1 ) an oordinate the hannel, we have: Theorem 1. Solving Eq. (4) with (1 ) will oordinate the hannel in the asymmetri information ase for any [, U], a subset of [, * ], whih gives the following equation in this region for w() dwðþ ¼ðwðÞ sþ p R Q F ðþ ðp sþp F T ðþ Taking into aount the individual rationality onditions, we an find if Uo *, there are a lass of solutions to Eq. (5) whih only differ in an initial ondition. If U¼ *, there is only one unique solution to Eq. (5) whereby the hannel is oordinated as in the full information ase. We will assume [, U] to be general in the following Buybak ontrat under hannel optimality From Theorem 1, a follow-up question is how muh profitability the manufaturer an extrat from a hannel-oordinated ontrat ð5þ

4 128 Y. Shen, S.P. Willems / Int. J. Prodution Eonomis 135 (212) in the asymmetri information ase. The answer to this question depends on the hoie of the right ontrat range of the retailer ost. Rather than designing the inentive-ompatible ontrat in the full range of [, U], we will first restrit the ontrat to A½,UŠ where U ru, and then find the optimal U. The ontrat is void if A½U,UŠ. This is the utoff poliy introdued in the literature (Corbett et al., 24; Ha, 21). The optimality of suh a utoff poliy will be demonstrated after solving the ontrat. Aording to this utoff poliy, we develop a ontrat menu under asymmetri information, labeled as Problem 1. Problem 1. Contrat under hannel optimality max U A ½,UŠ max wðþ P A M ðwðþ9þhðþ Subjet to dwðþ ¼ðwðÞ sþ p R Q F ðþ ðp sþp F T ðþ A,U P A R ðwðþ9þzp R A½,UŠ ð8þ P A M ðwðþ9þzp M ð6þ ð7þ A½,UŠ ð9þ Here the term hannel optimality is a loose meaning sine the hannel profit is equal to that in the full information ase only in the region A½,UŠ instead of the full range of A[, U]. The objetive in Eq. (6) is to optimize the manufaturer s expeted profit under the ondition that the hannel is oordinated for any A½,UŠ from Eq. (7), whih is the repetition of Eq. (5). To solve Problem 1, we first solve it with fixed U. We start with solving (7) and yield wðþ¼sþðwðþ sþe LðÞ ð1þ where Z LðÞ¼ u p R Q F ðuþ ðp u sþp F T ðuþ and w() is the initial ondition to be determined. Sine a pair of {w(),r()} an oordinate the hannel with retail ost in A½,UŠ, the resulting manufaturer and retailer profits are P A wðþ s MðwðÞ9Þ¼ p s PF T ðþ ð11þ P A wðþ s RðwðÞ9Þ¼ 1 P F T p s ðþ ð12þ Proposition 2. The optimal solution to Problem 1 is to set P A R ðwðuþ9uþ¼p R. Therefore, in the region A½,UŠ, themanufaturer earns a onstant profit whih is a funtion of U and dereasing in U, while the retailer extrats the rest of hannel profit whih is dereasing in. Setting P A R ðwðuþ9uþ¼p R yields the optimal wðþ, denoted as w 1 ðþ " # R w 1ðÞ¼sþðp s UÞ 1 P P F T ðuþ ð13þ Proposition 2 has demonstrated the utoff poliy is valid and optimal, sine for 4U the retailer s individual rationality ondition is violated and thus no ontrat is signed. From Eq. (13), we have the optimal w() for any A½,UŠ, denoted by w 1ðÞ, as " # R e LðUÞ w 1ðÞ¼sþðp s UÞ 1 P P F T ðuþ ð14þ when the buybak prie is set as r1 ðþ¼pðw 1 ðþ sþ=ðp sþ. e ðlðuþ LðÞÞ Corollary 1. The ontrat menu {w(), r()} for any fixed U is independent of the retail ost distribution, h(). While it is self-evident, Corollary 1 implies the ontrat from Problem 1 is robust as it does not depend on the information of the retail ost distribution. We need to determine the optimal utoff point U. IfU ¼ U then the manufaturer s materialized profit is the least for any A[, U] from Proposition 2, when the ontrat is non-trivial for the full range of A[, U]. In this ase, the hannel ahieves the optimality as in the full information ase. If we redue U, then we have a larger value of the manufaturer s profit for any A½,UŠ. By varying U in the range of [, U], we then find the optimal U to maximize the manufaturer s profit. The optimal utoff point U is found by solving the following one-dimensional optimization problem max r U r U GðUÞ¼ max wðþ P A M ðwðþ9þhðþ " # ¼ðp s UÞ 1 P R P F T ðþe ðlðþ LðUÞÞ P F T ðuþ hðþ p s ð15þ In the above equation, (11) and (14) are applied in the seond equality. The onavity or onvexity of GðUÞ is not lear; in general, it ould be non-onave and non-onvex. Sine Eq. (15) is a onedimensional problem, it an be solved by a line searh. After finding the optimal utoff point of U, denoted as U 1, we an determine the wholesale prie and the profit alloation between the two parties for A½,U 1 Š Buybak ontrat without hannel optimality In this setion, we study a wholesale-buybak ontrat to maximize the manufaturer s profit under asymmetri ost information without onsidering the hannel performane. Informally speaking, the manufaturer will offer a ontrat menu of wholesale and buybak pries without ondition (1 ). We again employ the revelation priniple with the utoff poliy. With the manufaturer estimating the retail ost in the region [, U]C[, * ], we assume the manufaturer issues ontrats only if the retail ost is less than a threshold U A½,UŠ while the retailer will at least reeive her minimal expeted profit in this region. Suh a utoff poliy will be proved to be optimal again after the whole ontrat is solved. If the retailer aepts the ontrat of wðuþ,rðuþ with her real ost value of, the retailer solves P A R ðwðuþ,rðuþ9þ¼max QðwðuÞ,rðuÞ9Þ ðp wðuþ ÞQðwðuÞ,rðuÞÞ9Þ ðp rðuþþ Z QðwðuÞ,rðuÞ9Þ and yields the optimal QðwðuÞ,rðuÞ9Þ. The manufaturer s profit is then given by Z QðwðuÞ,rðuÞ9Þ P A M ðwðuþ,rðuþ9þ¼ðwðuþ sþqðwðuþ,rðuþ9þ rðuþ The manufaturer an now find a menu of wholesale and buybak pries by solving the following inentive-ompatible truth-telling optimization problem: Problem 2. Manufaturer-optimal ontrat max r U r U max wðþ,rðþ P A M ðwðþ,rðþ9þhðþ subjet to ð16þ P A R ðwðuþ,rðuþ9þrpa RðwðÞ,rðÞ9Þ r,uru ð17þ

5 Y. Shen, S.P. Willems / Int. J. Prodution Eonomis 135 (212) P A R ðwðþ,rðþ9þzp R rru ð18þ P A M ðwðþ,rðþ9þzp M rru ð19þ Though the above ontrat is typial in the prinipal-agent literature, we are questioning whether this must be the best that the manufaturer should offer, in partiular if the leader (manufaturer) in the hannel is also sensitive to the hannel performane. We will address this question after solving Problem 2 and then ompare it with the ontrat in Problem 1. We demonstrate in the Appendix that the wholesale-buybak ontrat in Problem 2 is equivalent to the nonlinear-priing ontrat in the literature. To solve it ompletely, we need the following mild assumption, Assumption 2. efining zðþ¼hðþ=hðþ, the retailer ost distribution satisfies _zðþ¼dzðþ=4. This assumption means the distribution of the retail ost is dereasing reversed hazard rate, whih is true for many distribution funtions (Corbett and e Groote 2). Under Assumption 2, we have Proposition 3. The optimal solutions of (w(), r()) for A½,U 2 Š in Problem 2, denoted as ðw 2 ðþ,r 2ðÞÞ, satisfy (a) r2 ðþ¼ pðw 2 ðþ zðþ sþ p zðþ s (b) dw 2 ðþ R q ðþ dr2 ðþ ¼ q ðþ where () q ðþ¼f 1 p zðþ s p (d) U 2 ¼ minðu, a, b Þ (e) a þzð a Þþs ¼ p (f) ðp b zð b Þ sþq ð b Þ p Z q ð b Þ ¼ P M þp R The relation between w 2 ðþ and r 2ðÞ in Proposition 3(a) resembles that in Eq. (1). In the proof of Proposition 3, we also show that both the manufaturer and the retailer s profits from the ontrat menu derease with respet to, and therefore the utoff poliy is optimal. Moreover, these profit funtions are only funtions of the optimal order quantity, q * (). We therefore first derive the optimal q * () and U 2, and then the profits for the manufaturer and the retailer. We are also interested in the ontrat instruments for Problem 2. To this end, we first define R q ðþ gðþ¼ q ðþ 1þ _zðþ AðÞ¼ pð1 gðþþ zðþ s p_zðþ BðÞ¼ pð1 gðþþ zðþ s Then, we have: Corollary 2. The optimal solutions to ðw 2 ðþ,r 2ðÞÞ are solved as R R u r2 ðþ¼ e Að Þ BðuÞuþb e R AðuÞu ð2þ w 2 p zðþ s ðþ¼zðþþsþ r2 p ðþ ð21þ where the initial ondition, b, ould be given arbitrarily as long as r2 ðþ is non-negative. With the relation of (21), this onstant will not affet the order quantity and the profits in the system. 4. Numerial results In this numerial experiment, we will ompare the performane of the ontrat under hannel optimality (denoted as COC) from Setion 3.2, and the ontrat without hannel optimality (denoted as MOC) from Setion 3.3. In partiular, we want to see the differene between the utoff points and the profits of eah party in the two ontrats. We will fous on the effets of reservation profits as well as the manufaturer s estimate of the retailer s retail ost. Throughout this setion, the demand is assumed to be a normal distribution Normal (2, 5) and the retail ost is uniformly distributed in [, U]. The end-ustomer selling prie is fixed as p¼$21 throughout this setion, while the manufaturing ost is first hosen as s¼$5 but will be varied later. Therefore, if the hannel is entralized, the hannel profit is simply given by Eq. (2) and the feasible region of is $[, 16]. If the hannel is deentralized with full information, the utoff point * for the feasible region of is determined by the retailer and manufaturer reservation profits. In the asymmetri information ase with the manufaturer s estimate of the retail ost as [, U], we solve COC and MOC. In Fig. 1, we depit the profits of the manufaturer, the retailer and the hannel under both ontrats when P M ¼ $5, P R ¼ $3 and [, U]¼$[,9]. It is found the utoff point * ¼$12.5 in the full information ase, U 1 ¼ $68 in COC, and U 2 ¼ $51 in MOC. From Fig. 1, the regions of the retail ost for non-trivial COC and MOC are less than [, * ], while the feasible region in MOC is smaller than that in COC. It is also observed that the retailer s realized profit for a given in COC is always higher than that in MOC while the manufaturer s realized profit for a given in MOC is higher than that in COC, exept that in the region ½U 1,U 2Š where the MOC ontrat does not exist. The hannel profit performs very losely in the two ontrats in ½,U 1 Š. In Fig. 2, we depit the ontrat instruments of both COC and MOC. From Fig. 2, the wholesale pries in the COC and MOC are quite lose; however, the buybak pries differ dramatially with the retail ost inreasing. For COC, the manufaturer has to pay higher buybak prie in order to ahieve hannel oordination; the return is a wider region of the retailer ost that warrants an effetive ontrat. In the MOC, the manufaturer is overall onservative sine the hannel performane is beyond his onern. As a note, the onstant, b, in Eq. (2), is hosen suh that the buybak prie at the utoff point, U 2, is zero. Indeed, the buybak prie in Eq. (2) is dereasing in in this ase. With all the other problem parameters unhanged, we will now vary ðp M,P R Þ and [, U]. In Fig. 3, we have hanged [, U] to a muh narrower region, [, U]¼ $ [1,4]. In this ase, the utoff points in COC and MOC are the same, namely, U 1 ¼ U 2 ¼ $4; the profits of the hannel, manufaturer and retailer in MOC are very lose to their ounterparts in the COC for almost all of A[, U]. In order to further understand the effets of the manufaturer s estimate of the retail ost, we hoose another set of [, U]as$[6,8] while keeping the reservation profits of P M ¼ $5,P R ¼ $3. In this ase, the optimal utoff point U 1 in COC is solved to be $ 7, while the optimal utoff point in MOC is U 2 ¼ $8. The profits of the manufaturer, the retailer, and the hannel are depited in Fig. 4. From Fig. 4, the hannel profits in COC and MOC are almost idential in the region of A½,U 1 Š, while in A½U 1,U 2Š the hannel only ahieves profitability in MOC. The manufaturer s realized profit in A½,U 1 Š in COC is larger than that in MOC; however, he earns a profit in the region

6 13 Y. Shen, S.P. Willems / Int. J. Prodution Eonomis 135 (212) Profit ($) Retailer Cost ($) Manufaturer profit Manufaturer profit Fig. 1. Profit alloation with P M ¼ $5,P R ¼ $3, [, U]¼$[,9], s¼$5. Contrating Pries ($) Retailer Cost ($) Order Quantity Wholesale prie Wholesale prie Buybak prie Buybak prie Order quantity Order quantity Fig. 2. Contrat instruments for P M ¼ $5, P R ¼ $3, [, U]¼$[, 9], s¼$5. Profit ($) Retailer Cost ($) Manufaturer profit Manufaturer profit Fig. 3. Profit alloation with P M ¼ $5, P R ¼ $3, [, U]¼$[1, 4], s¼$5. of ½U 1,U 2Š in MOC. On the other hand, the retailer s realized profit for a given in MOC is larger than the ounterpart in COC. With Figs. 1 4 as illustrative examples, we want to broaden our understanding of COC and MOC through a larger number of examples. We have hosen three ases of (P M,P R ), orresponding to high, medium, and low individual rationalities for the manufaturer and the retailer. For eah pair of (P M,P R ), we find * and then hoose four segments in the range of [, * ], whih represent the ases that the retail ost spans in broad, low, medium, and high regions, respetively. For eah ombination of (P M,P R )and[, U], we find the utoff points and expeted profits of the hannel, the manufaturer and the retailer for both ontrats. In Table 1, wehavesummarized theresults for the 2 tested ases. The last olumn in Table 1 is for the perentage hanges of the profits from the MOC to the COC for different entities.

7 Y. Shen, S.P. Willems / Int. J. Prodution Eonomis 135 (212) Profit ($) Manufaturer profit Manufaturer profit Retailer Cost ($) Fig. 4. Profit alloation with P M ¼ $5,P R ¼ $3, [, U]¼$[6, 8], s¼$5. Table 1 Summary of tested ases (p¼$21, s¼$5). Case P M,P R $ * $ (, U) $ U 1 $ COC U 2 $ MOC % hange of profit Manuf Retailer System Manuf Retailer System Manuf Retailer System 1 (,14) , ,882 6.% 3.% 6.% 2 (5, 3) (1,5) 5 17, , , ,879.9% 8.2%.7% 3 (6, 1) , % 64.3% 36.3% 4 (11, 14) 11 NA NA NA NA NA NA 5 (, 14) , , % 33.3% 7.3% 6 (5, 3) (1,5) 5 17, , , ,879.9% 7.7%.7% 7 (6, 1) , % 72.7% -38.8% 8 (11, 14) 11 NA NA NA NA NA NA 9 (,12) , ,48 4.6% 23.% 7.% 1 (5, 3) 13.9 (1,4) 4 16, , , ,16.5% 3.%.4% 11 (5,8) , , % 45.8% 31.7% 12 (9, 12) 8.5 NA NA NA NA NA NA 13 (, 1) 8 1, , , , % 93.7% 23.8% 14 (5, 3) 119 (1,4) 4 19, , , ,16.5% 5.5%.4% 15 (4, 7) 7 13, , , ,74.7% 6.%.6% 16 (7, 1) % 75.% 5.% 17 (, 9) , ,955.7% 67.6% 24.% 18 (5, 3) 12.5 (1,4) 4 16, , , ,16.5% 3.%.4% 19 (4, 7) 7 1, , , ,74.8% 3.2%.6% 2 (7,9) NA NA NA NA ,2 NA NA NA Table 2 More tested ases (p¼$21, s¼$1). Case P M,P R $ * $ (, U) $ COC MOC % hange of profit Manuf Retailer System Manuf Retailer System Manuf Retailer System 21 (,18) % 16.1% 5.1% 22 (5, 3) 18.6 (1,4) 1, ,954 1, , % 18.4%.8% 23 (4,7) % 54.2% 54.6% 24 (7, 1) NA NA NA NA NA NA 25 (,1) % 2.4% 7.3% 26 (5, 3) 11. (1,4) 1, , , % 16.%.8% 27 (4,7) % 87.4% 43.9% 28 (7, 1) NA NA NA NA NA NA 29 (,5) , % 112.6% 54.5% 3 (5, 3) 52.5 (1,3) 12, ,813 12, ,83 1.3% 3.9%.1% 31 (3,5) NA NA NA NA NA NA

8 132 Y. Shen, S.P. Willems / Int. J. Prodution Eonomis 135 (212) From Table 1, we an draw a number of observations. First, higher retail ost will lead to lower hannel, manufaturer and retailer profits in both COC and MOC. Moreover, when this ost is in a very high region, suh as in ases 4, 8, 12, and 2, COC beomes void, whereas MOC is still valid;wehaveused NA todenotetheasesthatnoeffetiveontrat is signed. This indiates that high retail ost needs high wholesaler prie to ahieve hannel optimality; however, this may break the manufaturer s rationality ondition. Seond, if the retail ost is in a region with relatively low value, the hannel s profit in COC is higher than that in MOC, while the manufaturer s profit in COC is very lose to that in MOC. Interestingly, in ase 17 the manufaturer s profit in COC even ould be higher than that in MOC, indiating that the hannel optimality and individual optimality ould be onsistent. This might look surprising at the first glane as we may regard the COC as a speial ase of MOC with an additional onstraint. As a matter of fat, with the utoff point poliy in both ontrats this is not true anymore. We have observed in Case 17 the utoff point in COC is higher than MOC. While Case 17 looks unique in Table 1, itmayhappenmore frequently if we vary the produt prie, p, anost, s. Infat, the differene between p and s automatially restrit the region for the retailer ost from Proposition 1; therefore, we fix p but hange s. In Table 2, we list the profits of the manufaturer, retailer, and system, as well as their perentage relations by letting s¼$1 while varying P M, P R and (, U) in the meantime. From this table, in ases 21, 22, 25, 26, 29, 3, both the manufaturer and the system perform better in the COC than in the MOC; in ases 23, 24, 27, 28, 31, in whih the retailer osts fall into high segments of the feasible regions, the COC is a hoie from eah party and the system. Comparing Table 2 with Table 1, it indiates that if the produt margin is low, using COC annot only oordinate the hannel but also yield high profit for the manufaturer if the retailer ost is not estimated in a high-value region. In pratie, the retailer ost should not be too high so the COC should be valuable in many ases. Therefore, it pays for the OEM to understand the trade-off between the two ontrats under various onditions. There are also several situations where the manufaturer and the hannel are in the win-win situation with speifi data, whih supports the hypothesis raised in the introdution. This should also happen to more senarios if we hange other problem onditions in the problem, for example, the demand profile. Overall, there is no simple way to determine a threshold where the retailer ost ditates the use of COC versus MOC. That said, solving the two ontrats in this paper will tell whih one is more suitable from the perspetive of the manufaturer and the hannel sine the manufaturer is the Stakelberg leader. 5. Conlusions We have examined the performane of wholesale-buybak ontrats in the presene of retail ost. When this ost information is known to the manufaturer, a buybak ontrat an fully oordinate the hannel. We deide the wholesale prie taking into aount the individual rationality onstraints for both the manufaturer and the retailer. When the retail ost is private information, it is not easy to ahieve hannel optimality. However, in this paper, we propose an inentive-ompatible wholesale-buybak ontrat that an ahieve hannel optimality. Starting with suh a oordinated hannel and asymmetri information, we propose a ontrat to maximize the manufaturer s profit. We then propose a new ontrat to maximize the manufaturer s profit, and solve it by mapping it onto a nonlinear-priing ontrat. The differene between these two ontrats lies in a speifi wholesale-buybak ondition whih has been derived in the full information ase that an oordinate the hannel. From the pratial perspetive, we are partiularly interested in implementing the results in this paper in real world suh as the motivating problem in the introdution. There are several logial areas to extend this researh. First, we are interested in a more ompliated ontrating mehanism suh that not only the hannel is oordinated but also the manufaturer is always better off than in the MOC. Seond, a natural evolution of this researh will be to integrate both asymmetri ost information and asymmetri demand information into the problem setting. This question arises naturally as we have observed substantial researh towardseitherofthetwo,butnotmuhtowardsboth. Aknowledgements The authors would like to thank the two anonymous reviewers for their valuable omments. The researh of the first author is supported by the National Siene Foundation of China(Grant No ) and by Leading Aademi isipline Program, 211 Projet for Shanghai University of Finane and Eonomis (the 3rd phase). Appendix Proof of Proposition 1. (1) From Eq. (1) in the text, we have rðþ w ¼ pðw sþ w ¼ ðp w ÞðþsÞ o: p s p s (2) It is easy to hek if p w44 ¼ sðp wþ w where sow by assumption, then we have r()4w. F T ðþ ¼ Q F Proof of Theorem 1. We prove ondition (5) is neessary and suffiient to indue the retailer to tell her ost truthfully. (1) Neessary ondition We first prove that if the extreme value for w( ) to maximize P A RðwðuÞ9Þ is w(), then w() satisfies Eq. (5). This is proved starting with Eq. (3), from whih we an derive the optimal value of Q(w( )9) satisfies ðp wðuþ Þ ðp rðuþþfðqðwðuþ9þþ ¼ : ða1þ Therefore we an derive from some straightforward algebra A R 9 u ¼ ¼ dwðþ QðwðÞ9Þþðp 9 u Z QðwðÞ9Þ 9 u ¼

9 Y. Shen, S.P. Willems / Int. J. Prodution Eonomis 135 (212) where we have used (A1) at ¼ u ¼ ¼ p p s dwðþ þ pðwðþ sþ ðp sþ 2 9 u ¼ ¼ dwðþ QðwðÞ9Þþ The inentive-ompatible ondition Eq. (4) R 9 u ¼ ¼ and yields dwðþ ¼ pðwðþ sþ R Q F ðþ ðp sþ½ðp sþq F ðþ p R Q F ðþ Š ¼ pðwðþ sþ R Q F ðþ ðp sþp F T ðþ " # Z p dwðþ þ pðwðþ sþ QðwðÞ9Þ p s ðp sþ 2 where we have replaed the solution to (A1) at ¼ by Q F () from Setion 2. It onludes that Eq. (5) ditates the extreme path for w(). (2) Suffiient ondition The proof of the suffiient ondition follows Laffont and Tirole (1993), pp121. We prove Eq. (5) is suffiient by showing P A R ðwðuþ9þopa R ðwðþ9þ for any a under this ondition. If this is not true, there is a suh that P A R ðwðuþ9þ4pa RðwðÞ9Þ. We first prove this auses ontradition assuming 4. From the proof of the neessary ondition, Eq. (5) is from 1 P A R ðwðuþ9þ9 u ¼ ¼, where 1 is the differentiation with respet to the first variable,. That said, 1 P A R ðwðxþ9xþ¼,8x under (5). If PA R ðwðuþ9þ4pa R ðwðþ9þ, then R u 1P A RðwðxÞ9Þdx4, whih further implies Z u Z u Z ð 1 P A R ðwðxþ9þ 1P A R ðwðxþ9xþþdx ¼ dx 12 P A R ðwðxþ9uþdu4 x where 12 P A R ðwðxþ9uþ¼ð@2 P A RðwðxÞ9uÞ=@x@uÞ and the seond differentiation is with respet to u. However " Z # QðwðxÞ9uÞ 12 P ðp wðxþ uþqðwðxþ9uþþðp ðp 2 QðwðxÞ9uÞ þ ðp wðxþ uþ ðp @x þfðqðwðxþ9uþþ ðp where we have employed ðp wðxþ uþ ðp rðxþþfðqðwðxþ9uþþ ¼ : whih repeats Eq. (A1). ifferentiating Eq. (A2) with respet to u yields 1þðp rðxþþf ða2þ ða3þ Therefore 12 P @u FðQðwðxÞ9uÞÞ We further differentiate Eq. (A2) with respet to @QðwðxÞ9uÞ FðQðwðxÞ9uÞÞ ¼ ðp ða4þ Hene 12 P ðwðxþ9uþ¼ðp Sine F(Q(w(x)9u))¼(p w(x) u)/(p r(x)), we an 1/f(Q(w(x)9u))(p r(x))) resulting In the next, we whih is ahieved by due (@Q(w(x)9u)/@x). In other words, we need to prove that (@/@x)[(p w(x) u)/(p r(x))]4. However, we an prove by some alulations p wðxþ u p rðxþ " # 1 pðx uþ dwðxþ þ pðp wðxþ uþ ðwðxþ sþ ðp rðxþþ 2 p x s dx ðp x sþ 2 Sine xa[, ], xz, i.e., uox, anddw(x)/dx4 from Eq. (5), we see the first term in the braket of the right side in the above equation is positive. The seond term in the braket is positive too beause of w(x)4s and p4w(x)+u from the assumptions in Setion 2. Therefore,

10 134 Y. Shen, S.P. Willems / Int. J. Prodution Eonomis 135 (212) qq(w(x)9u)/qx4, resulting in 12 P A RðwðxÞ9uÞo and Z u Z x 12 P A R ðwðxþ9uþduo However, this onlusion is in ontradition to the assumption. If we assume o, we have the same onlusion. Therefore, ondition (5) is suffiient for the inentive ompatibility. Proof of Proposition 2. When the retailer truthfully reveals the retail ost information, the hannel is oordinated. The manufaturer s profit is P A M ðwðþ9þ¼ðwðþ sþq F ðþ rðþ Z Q F ðþ where Q F ðþ¼f 1 ½ðp wðþ Þ=ðp rðþþš while w() and r() satisfy Eqs. (1) and (5). We then A M ¼ dwðþ Q F ðþ drðþ Z Q F ðþ Z Q F ðþ þ ðwðþ sþ rðþfðq F ðþ ¼ dwðþ Q F ðþ drðþ þ ðwðþ sþ rðþ p F ðþ ¼ dwðþ Q F ðþ drðþ Z Q F ðþ p where we have plugged (1 ) into the seond identity. We further use the expressions for (dw()/) and (dr()/) derived in the proof for Theorem 1 and then have " A Z M ðwðþ9þ ¼ dwðþ Q F p dwðþ ðþ þ pðwðþ s Q F ðþ ¼ dwðþ P F T ðþ Z pðwðþ sþ Q F p s ðp sþ 2 p s ðp sþ 2 A M ðwðþ9þ=@ ¼ from Eq. (5), or, PA M ðwðþ9þ is onstant with respet to. Moreover, for A½,UŠ, PA R ðwðþ9þ¼ P F T ðþ PA M ðwðþ9þ, sopa RðwðÞ9Þ is dereasing of. The manufaturer will offer the w() to maximize his profit by binding the retailer s profit at ¼ U. From this ondition, the less U is, the higher P A M ðwðþ9þ¼pa MðwðUÞ9UÞ is. Therefore, Proposition 2 is proved. Proof of Proposition 3. In order to solve Problem 2, we R 9 u ¼ ¼ and yield a weak form, or a neessary ondition, for (17) _wðþf 1 p wðþ Z F 1 p wðþ p rðþ þ _rðþ ¼ ða5þ p rðþ where _wðþ¼ðdwðþ=þ, _rðþ¼ðdrðþ=þ. With Eq. (A5), we an A R ðwðþ,rðþ9þ ¼ F 1 p wðþ ¼ p rðþ for whih we have the integration form P A R ðwðþ,rðþ9þ¼p R þ QðwðÞ,rðÞ9Þ where we have set P A R ðwðuþ,rðuþ9uþ¼p R. The fat that PA RðwðÞ,rðÞ9Þ dereases with respet to from (A6) validates the effetiveness of the utoff poliy from the retailer perspetive. Moreover, Eq. (A6) implies that the first-best solution in the full information ase an no longer be ahieved in this ontrat. Now solving Problem 2 is equivalent to solving (16) with (19), (A5) and (A6). Sine we are mainly interested in the manufaturer s profits, retailer s profits, and the hannel profits, we an irumvent the diffiulty in the following way. Given a pair of (w(), r()) for any, we have Q(w(),r()9)¼F 1 [p w() /p r()], and the hannel profit is ða6þ Z QðwðÞ,rðÞ9Þ P A T ðwðþ,rðþ9þ¼ðp sþqðwðþ,rðþ9þ p Sine P A M ðwðþ,rðþ9þ¼pa T ðwðþ,rðþ9þ PA RðwðÞ,rðÞ9Þ, from (A6) and (A7) we an re-write the manufaturer s profit as ða7þ Z QðwðÞ,rðÞ9Þ P A M ðwðþ,rðþ9þ¼ðp sþqðwðþ,rðþ9þ p QðwðÞ,rðÞ9Þ P R Observing that in (A6), (A7) and (A8) the profit funtions are funtions of Q(w(),r()9) rather than w(),r() independently, we write q()¼q(w(),r()9). We will solve (w(), r()) in Problem 2 by first solving q() in the following problem: ða8þ Problem 3. Solving for q() max r U r U max qðþ P A M ðqðþ9þhðþ ¼ max r U r U Subjet to P A M ðqðþ9þzp M P A R ðqðþ9þ¼p R þ qðxþdx max qðþ " # ðp sþqðþ p Z qðþ qðxþdx P R hðþ ða9þ ða1þ ða11þ

11 Y. Shen, S.P. Willems / Int. J. Prodution Eonomis 135 (212) The solution to Problem 3 is dependent on the distribution of the retail ost. With Assumption 2, the manufaturer s profit in Eq. (A9) ould be proved to derease with respet to, validating the utoff poliy from the manufaturer perspetive; in other words, the utoff poliy is optimal. We an then derive the optimal value for q() and Uas q ðþ¼f 1 p zðþ s p h i A,U 2 ða12þ U 2 ¼ minðu, a, b Þ ða13þ where z()¼(h()/h()), a and b satisfy a þzð a Þþs ¼ p ða14þ Z q ð b Þ ðp b zð b Þ sþq ð b Þ p ¼ P M þp R ða15þ Proof of Corollary 2. From Proposition 3(a), we take derivative of r2 ðþ with respet to and then insert _w 2ðÞ¼gðÞ_r 2 ðþ from Proposition 3(b), whih gives us a first-order differential equation dr2 ðþ þaðþr2 ðþþbðþ¼ where g(), A(), and B() are all defined in the text. The solution to the above equation is R R u r2 ðþ¼ e Að Þ BðuÞuþb e R AðuÞu whih is Eq. (2) in the text. As a result, Eq. (21) provides the solution to w 2ðÞ. The only undetermined fator is b; this is due to the fat that there is an arbitrary fator between the wholesale prie and the buybak prie for Problem 2 while their joint funtion, q ðþ¼qðw 2 ðþ,r 2ðÞ9Þ, finally determines the result of the ontrat. We selet b in suh a way that the buybak prie will not be negative in the feasible region of. Referenes Baron,., Myerson, R., Regulating a monopolist with unknown osts. Eonometria 5, Burnetas, A., Gilbert, S., Smith, C., 27. Quantity disount in single period supply ontrats with asymmetri demand information. IIE Transations 39, Cahon, G., 23. Supply Chain Coordination with Contrats. In: de Kok, A.G., Graves, S. (Eds.), Handbooks in Operations Researh and Management Siene: Supply Chain Management. North-Holland, Amsterdam. Cahon, G., Lariviere, M., 21. Contrating to assure supply: how to share demand foreasts in supply hain. Management Siene 47, Chakravarty, A., Zhang, J., 27. Collaboration in ontingent apaities with information asymmetry. Naval Researh Logistis 54, Corbett, C., e Groote, X., 2. A supplier s optimal quantity disount poliy under asymmetri information. Management Siene 46, Corbett, C., Zhou,., Tang, C., 24. esigning supply ontrats: ontrat type and information asymmetry. Management Siene 5, onohue, K., 2. Effiient supply ontrats for fashion goods with foreast updating and two prodution modes. Management Siene 46, Fudenberg,., Tirole, J., 2. Game Theory. The MIT Press, Cambridge, MA. Grannot,., Yin, S., 25. On the effetiveness of return poliies in the priedependent newsvendor model. Naval Researh Logistis 52, Ha, A., 21. Supplier-buyer ontrating: asymmetri ost information and utoff poliy for buyer partiipation. Naval Researh Logistis 48, Laffont, J., Tirole, J., A theory of inentives in prourement and regulations. The MIT Press, Cambridge, MA. Lewis, T., Sappington,., Regulating a monopolist with unknown demand. The Amerian Eonomi Review 78, Ozer, O., Wei, W., 26. Strategi ommitment for optimal apaity deision under asymmetri foreast information. Management Siene 52, Pasternak, B., Optimal priing and return poliies for perishable ommodities. Marketing Siene 4, Pasternak, B., 28. Commentary: optimal priing and return poliies for perishable ommodities. Marketing Siene 27, Tsay, A., Nahmias, S., Agrawal, N., Modeling supply hain ontrats: a review. In: Tayur, S., Magazine, M. (Eds.), Quantitative Models for Supply Chain Management. Kluwer Aademi Publishers. Xiao, T., Shi, K., Yang,., 21. Coordination of a supply hain with onsumer return under demand unertainty. International Journal of Prodution Eonomis 124,