Horizontal Coordinating Contracts in the Semiconductor Industry

Transcription

1 Horizontal Coordinating Contracts in the Semiconductor Industry Xiaole Wu* School o Management, Fudan University, Shanghai 2433, China wuxiaole@udaneducn Panos Kouvelis Olin Business School, Washington University in St Louis, St Louis, MO 6313, USA kouvelis@wustledu Hiroumi Matsuo Graduate School o Business Administration, Kobe University, Kobe , Japan matsuoh@kobe-uacjp Hiroki Sano McCombs School o Business, University o Texas at Austin, Austin, TX 78712, USA hirokisano@phdmccombsutexasedu Integrated device manuacturers (IDMs and oundries are two types o manuacturers in the semiconductor industry IDMs integrate both design and manuacturing unctions while oundries solely ocus on manuacturing Since oundries oten have cost advantage over IDMs due to their specialization and economies o scale, IDMs have incentives to source rom oundries or the purpose o avoiding excessive capacity investment risk As the IDM is also a potential capacity source, the IDM and oundry are in a horizontal setting rather than a purely vertical setting In the absence o sophisticated contracts, the benchmark contract or the IDM and oundry is a wholesale price contract We deine coordinating contracts as those that improve both the IDM s and oundry s expected proits over the benchmark wholesale price contract and also lead to the maximum system proit This paper examines i there exist coordinating capacity reservation contracts It is ound that wholesale price contracts in the horizontal setting cannot achieve the maximum system proit due to either double marginalization eect, or misalignment o capacity-usage-priority In contrast, i the IDM s capacity investment risk is not too low, there always exist coordinating capacity reservation contracts Furthermore, under coordinating contracts, the IDM s sourcing structure, either sole sourcing rom the oundry or dual sourcing, is contingent on the irms cost structures Keywords: supply chain management, horizontal capacity coordination, reservation contract, wholesale price contract in horizontal setting, sourcing structure * Corresponding author Tel:

2 2 1 Introduction In the semiconductor industry, there are two types o manuacturers Integrated device manuacturers (IDMs both design and manuacture semiconductor devices, while oundries concentrate only on manuacturing and take orders rom IDMs and/or abless irms For both irm types, the cost o establishing production capacity is extremely high, and equipment technology becomes obsolete quickly as new technologies and demanding product requirements emerge Thereore, manuacturers have to keep a consistently high ab utilization in order to pay back the capital expenditure in a timely manner On the demand side, the customers o such manuacturers are themselves established irms that produce products such as cell phones, personal computers, and automobiles, among others To gain orders rom these customers, the semiconductor manuacturers need to be costeective, responsive, and lexible in accommodating highly luctuating and uncertain order quantities Because o this hard-to-match supply and demand situation, only a small number o manuacturers typically the larger ones in terms o capacity, such as Intel (IDM, Samsung (IDM and TSMC (oundry are consistently proitable The rest are less well on their own and are in need o horizontal coordination to justiy the needed investment to reach a competitive level o economies o scale and lexibility In 29, IDMs like Texas Instruments, Freescale Semiconductor, STMicroelectronics, and Renesas Electronics are estimated to outsource their production to oundries at percentages o 55%, 23%, 2%, and 1% to 2%, respectively (IC Insights 211 Due to the signiicant lead times required to add capacity, capacity decisions and related investment outlays are made well ahead o actual demand Once the capacity is installed, the rest o the production activities are conducted in a make-to-order ashion When an IDM and a oundry interact in a decentralized ashion, rom the IDM s perspective, it is important to secure enough capacity internally and/or externally to meet uncertain demand without incurring excessive costs Since only the IDM has access to the end market, the oundry has to work with the IDM in order to be proitable On the other hand, the oundry is concerned about over-committing capacity The IDM has to oer the oundry incentives to induce suicient capacity commitment In the absence o sophisticated contracts, the benchmark contract or the IDM and oundry is a wholesale price contract In the vertical supply chain setting, a wholesale price contract cannot coordinate the supply chain due to the well known double marginalization eect (Spengler 195 The IDM and oundry are not in a vertical supply chain relationship,

3 3 because the IDM himsel can produce the product in addition to the option o sourcing the product rom the oundry Thus, the IDM and oundry are in a horizontal setting This paper identiies new eatures o wholesale price contracts in the horizontal setting compared to the vertical setting, and investigates whether wholesale price contracts can achieve the maximum system proit (ie, the maximum sum o the IDM s and oundry s expected proits A prevailing type o contract used in the semiconductor industry is capacity reservation contract According to the structure o reservation contracts, the IDM reserves capacity R rom the oundry by paying a reservation ee per unit The reservation ee is reundable, which means the IDM pays to the oundry a wholesale price deducted by the reservation ee or a unit o product when the purchasing quantity is less than R When R =, the IDM always pays the wholesale price That is, the pathological case o R = in reservation contracts involves a single contract parameter, wholesale price, and is actually a wholesale price contract in the horizontal setting Hereater, we reer to wholesale price contracts in the horizontal setting as zero reservation contracts (ZRCs We emphasize two dierences between ZRCs and traditional wholesale price contracts in the vertical setting First, under ZRCs, the oundry may or may not build capacity, even in some cases with the wholesale price exceeding her cost That is, even though the IDM is willing to pay the wholesale price, there may be no available supply rom the oundry In contrast, in traditional wholesale price contracts, the buyer can always buy whatever quantity he needs rom the supplier or a reasonable wholesale price (eg, a price that exceeds the supplier s cost Second, under ZRCs, the IDM may himsel be a source o supply in addition to the oundry But under traditional wholesale price contracts in the vertical setting, the buyer solely relies on the supplier Due to these nuances o wholesale price contracts in the horizontal setting, we reer to them as ZRCs We reer to reservation contracts with positive capacity reservation (R > as positive reservation contracts (P RCs A P RC is coordinating i it satisies the ollowing three conditions (1 Capacity investment coordination: The resulting capacity investments are the same as in the centralized system that maximizes the system proit (2 Production coordination: The production decisions (ie, the demand allocation decisions are consistent with those in the centralized system

4 4 (3 Individual rationality: The resulting IDM s and oundry s expected proits are both greater than what they can earn under the benchmark wholesale price contract Conditions (1 and (2 are required to achieve the maximum system proit Condition (3 guarantees that the contract is individually rational or both the IDM and oundry We reer to coordinating P RCs, i they exist, as CP RCs In our paper, we pursue answers to the ollowing questions: (1 What are the centralized capacity investments o the IDM and oundry that maximize the system proit? What are the main actors aecting such capacity investments? (2 Can ZRCs maximize the system proit? I not, are the underlying reason(s the same as in the vertical setting? (3Under what circumstances do CP RCs exist? What is the resulting sourcing structure or the IDM? The answers to these questions will oer us a better understanding o the distinctions between horizontal capacity coordination and traditional vertical supply chain coordination This paper is organized as ollows Section 2 reviews the relevant literature Section 3 introduces the model assumptions and notation Section 4 derives the centralized capacity investments Sections 5 and 6 study ZRCs and P RCs, respectively, to investigate their roles in achieving the maximum system proit when the IDM and oundry interact in a decentralized scheme, and check i there exist CP RCs Section 7 compares the irms expected proits under P RCs with those under ZRCs and illustrates the impact o proit margin Section 8 provides concluding remarks 2 Literature Review Vertical supply chain coordination has received substantial attention in the operations literature Various orms o contracts are proposed to align supply chain partners decisions with those o the centralized system Among these are buy-back (Pasternack 1985, revenue-sharing (Cachon and Lariviere 25, quantity-lexibility (Tsay 1999, and salesrebate contracts (Taylor 22 Cachon (23 provides a comprehensive review o this literature beore 22 More recently, Tomlin (23 studies a vertical supply chain with a supplier selling a key component to the manuacturer who processes the component into end product Tomlin (23 proves the existence o a class o price-only contracts that arbitrarily allocate the supply chain proit between the supplier and manuacturer Chick et al (28 show that cost-sharing contract can coordinate the vertical inluenza vaccine

5 5 supply chain with yield uncertainty In contrast to the above papers, we emphasize horizontal capacity coordination between an IDM and a oundry using capacity reservation contracts There are dual capacity sources in meeting end-demand (both the oundry and IDM may build capacity or the end product, but without direct access to the market, the oundry serves as a subcontractor to the IDM Reservation contracts have been studied as part o vertical supply chain coordination in Erkoc and Wu (25 and Jin and Wu (27 Wu et al (25 provide an excellent survey o the reservation contracts literature Erkoc and Wu (25 propose two variants o capacity reservation contracts: partially deductible reservation contract whose reservation ee is partially deducted i the reserved capacity rom the supplier is used, and cost-sharing contract or which the buyer pays a portion o the capacity cost associated with her reservation Jin and Wu (27 consider deductible and take-or-pay reservation contracts or vertical supply chain coordination, and extend the model rom one customer to two or more customers Brown and Lee (1998 study pay-to-delay capacity reservation contracts and derive optimal policies or the buyer in the semiconductor industry Our paper diers rom the above work in that it has the buyer (IDM not only decide how much capacity to reserve rom the supplier (oundry, but also how much o his own capacity to build, thus injecting horizontal coordination concerns Another stream o literature combines reservation contracts and the spot market, such as Serel et al (21, Wu et al (22, Spinler et al (23, Wu and Kleindorer (25, Spinler and Huchzermeier (26, Fu et al (21, and Inderurth et al (213 In these papers, a buyer can reserve capacity rom a supplier in addition to buying rom the spot market In our paper, the IDM can reserve capacity rom the oundry in addition to his own capacity i built upront However, the supply rom the spot market is ininite while in our paper, the supply is limited and the capacities at the IDM and oundry are all that can be used ater the demand realization Recently, Peng et al (212 propose a dualmode equipment procurement ramework, with both modes using reservation contracts Both the base and lexible suppliers in their paper are in a vertical relationship with Intel, which diers rom the horizontal setting in our paper that the IDM does not need to use any contract or own capacity In addition, we ocus on the coordinating capability o reservation contracts, but they are interested in the value o the added lexibility o dual-mode or the irm

6 6 Capacity expansion is an important topic in operations management Van Mieghem (23 provides a comprehensive review o the literature Our work diers rom singleresource, multiagent capacity models (eg, Cachon and Lariviere 1999 in that both the IDM and oundry can build core capacity The paper that explicitly addresses horizontal capacity coordination and is most closely related to ours is Van Mieghem (1999 He considers horizontal capacity coordination between two irms to satisy two distinct market demands, and the two irms make capacity decisions simultaneously In our paper, both the IDM and oundry are potential supply sources to meet one market demand The IDM is the only one with market access, and he usually builds his own capacity beore subcontracting the rest o his needs to the oundry Hence, a sequential-move capacity game is more appropriate or our setting 1 Furthermore, our work studies contracting between the IDM and oundry using reservation contracts besides linear price contracts Finally, there is a stream o literature on interirm surplus inventory transshipment that addresses horizontal coordination between/among retailers by setting ex-ante transshipment prices beore demand realizations (Rudi et al 21 and Hu et al 27 A common eature o transshipment papers is that irms irst satisy their own demand using their own inventory and then transship surplus inventories to others or receive inventories rom others Our capacity coordination diers rom inventory coordination in that the IDM does not necessarily use his own capacity irst He may use the oundry s irst as long as it is economically justiied To summarize, our paper is one o the earliest to study horizontal capacity coordination in the presence o uncertain demand using capacity reservation contracts Our studied setting is unique in the asymmetric roles o the IDM and oundry: Only the IDM has access to the market while both serve as capacity sources The capacity decisions o the IDM and oundry are modeled by a sequential-move game with the IDM as the leader Our paper aims at deriving horizontal coordinating contracts as important reerences or the IDM and oundry in their choice o contractual relationship 1 Nevertheless, we have also checked the results under simultaneous-capacity-investment setting, ie, the setting with the IDM and oundry simultaneously deciding their capacity investments ater the IDM s capacity reservation decision We have ound that the insights derived in this paper about the coordinating capability o reservation contracts still hold under simultaneous-capacity-investment setting

7 7 Step 1: Under ZRCs, the IDM commits his production capacity by K s and goes to Step 2 Under P RCs, the IDM commits his production capacity by K s and reserves R(R > units o the oundry s capacity Step 2: Observing the IDM s capacity commitment decision, the oundry determines her own capacity commitment K, with K R Step 3: The demand is realized Step 4: The IDM determines the production quantity o his own, z, and the production order to the oundry, y, subject to the capacity constraints Table 1 Protocol o Capacity Investment and Production Decisions under P RCs and ZRCs 3 Model Assumptions and Notation The IDM (he designs a device and sells it or a unit price o p to customers The demand is realized in one period and is expressed as a random variable X (x is its probability density unction where (x > or x, and (x is dierentiable F (x is its cumulative distribution unction where F (x is dierentiable and invertible or x Anticipating an uncertain demand to meet, the IDM irst decides whether to do business with the oundry (she; i so, the IDM and oundry agree upon a contract to use The contracts considered in this paper are positive reservation contracts (P RCs and wholesale price contracts in the horizontal setting (ZRCs Given the contract, the IDM and oundry execute the capacity investment and production decisions ollowing the sequence speciied in Table 1 Beore the demand is realized, the IDM and oundry commit their production capacities K s and K, respectively, and expend the corresponding capacity costs K s and K Let K = (K s, K Ater the demand x is realized, the IDM internally produces z units and places an order to the oundry o y units so that the ex-post proit o the IDM, π s is maximized The corresponding ex-post proit o the oundry is denoted by π The per-unit production costs o the IDM and oundry are c s and c, respectively The IDM subcontracts manuacturing to the oundry or a unit subcontract price o w In P RCs, the IDM proposes to reserve R (R > units o the oundry s capacity, and the oundry must commit at least R units o capacity to the IDM In return, the IDM pays to the oundry rr in total as a capacity reservation ee, where r is the per-unit reservation ee with < r w When the oundry delivers y units to the IDM, she receives a unit price o w r or the quantity less than or equal to R and w or any quantity above R That is, the total revenue or the oundry is yw + (R yr or y R and yw or y > R This

8 8 pricing scheme shits part o the capacity investment risk rom the oundry to the IDM In return, the IDM secures a certain amount o capacity rom the oundry On the other hand, in ZRCs there is no capacity reservation decision In this paper, we assume the IDM and oundry hold symmetric inormation about p, the cost structure,, c s, c, and the demand distribution These assumptions are reasonable in the semiconductor industry The selling price p is veriiable in a business-to-business setting Cost structure can be eectively guessed by each other since most players in this industry are long-term partners with repeated dealings, and they have a good understanding o the other s acility and technology; urthermore, some third party (eg, isuppli provides products such as integrated circuit (IC cost evaluator, which estimates the cost and price involved in IC procurement The symmetric inormation about the demand distribution can be viewed as a result o joint orecasting based on common economic indicators Throughout this paper, in order to avoid trivial cases, we assume: p > c s +, w > c +, p > w, < r w, and c s,, c, > As oundries oten have the cost advantage over IDMs due to their dedicated ocus on manuacturing, we assume c + < c s + Let K s and K denote the equilibrium capacity or the IDM and oundry, respectively, R denote the IDM s equilibrium reservation capacity out o the capacity game speciied in Table 1, and K = (Ks, K All the notation is summarized in Table 2 4 Centralized Capacity Investments As assumed in the protocol o Table 1, the IDM and oundry are part o a decentralized system and take actions sequentially, with each trying to maximize its own expected proit In contrast, when the IDM and oundry collaborate, with their capacity investment decisions dictated by a central planner to maximize the sum o their expected proits, we reer to the IDM and oundry orming a centralized system (CS Note that the CS has the choice between two production technologies: producing at a cost o c s i using the IDM s capacity, and c i using the oundry s capacity Based on the relationship between c s and c, the CS gives priority o usage to the one with lower production cost I c s = c, the CS is indierent on the priority o use o the IDM s or oundry s capacity I c s > c (c s < c, then ater the demand is realized, the oundry s (IDM s capacity is used irst In contrast to the purely vertical coordination problem (only one irm can build capacity and serves as the supplier o the other irm, the horizontal capacity coordination problem between

9 9 x X realized demand demand random variable F ( cumulative distribution unction (cd o X; F ( = 1 F ( ( p w c s, c, K s, K r R y z MOF ZRC ZRC P RC CCP RC CP RC π s, π Π s, Π Π i s Π cs probability density unction (pd o X per-unit price o product per-unit subcontract price o product variable production cost or the IDM and oundry, respectively variable capacity cost or the IDM and oundry, respectively capacity o the IDM and oundry, respectively per-unit capacity reservation ee reserved capacity production quantity or the oundry production quantity or the IDM p c v, ie, proit margin over ixed cost v zero reservation contract ZRC with the oundry building positive capacity positive reservation contract candidate coordinating positive reservation contract coordinating positive reservation contract ex-post proits or the IDM and oundry, respectively expected proits or the IDM and oundry, respectively IDM s expected proit under the IDM independent case centralized system proit N T (µ, σ 2 truncated normal distribution with mean µ and variance σ 2 over (, Table 2 Notation the IDM and oundry leads to three objective unctions depending on the values o c s and c Let Π cs be the centralized system proit The three cases are summarized in Lemma 1 Lemma 1 K Ks +K For c s > c, Π cs = (p c (1 F (x dx + (p c s (1 F (x dx K s K K (1

10 1 Ks +K Ks For c > c s, Π cs = (p c (1 F (x dx + (p c s (1 F (x dx K s K K s (2 Ks+K For c = c s, Π cs = (p c s (1 F (x dx K s K (3 In Proposition 1, we derive the centralized capacities or the IDM and oundry, denoted by K c = (K cs, K c, to maximize the system proit, where the subscript c denotes centralized decision Let us call p c v as the proit margin over ixed cost (MOF MOF = and MOF = Proposition 1 The centralized capacity investments K c = (K cs, K c to maximize the system proit are summarized in the ollowing table: MOF MOF s MOF > MOF s ( ( c s > c F 1 F 1 cs+vs c c s c, F 1 cs+vs c c s c, F 1 p c p c ( ( c > c s c + < c s + implies this case is impossible, F 1 p c p c ( ( c = c s c + < c s + implies this case is impossible, F 1 p c Proposition 1 indicates that the IDM s (oundry s capacity should be positive i MOF s MOF (MOF s < MOF MOF is an important measure or the optimization o the centralized capacity investments This is because when using a single capacity source, either the IDM s or oundry s capacity to meet the demand, the newsvendor critical ractile is = MOF s or the IDM and MOF s +1 = MOF MOF or the oundry (please reer to Porteus 22 or the deinition o newsvendor critical ractile Thus, a larger M OF +1 implies a larger newsvendor critical ractile The centralized total capacity is always determined by the larger newsvendor critical ractile out o p cs vs and Dual sourcing occurs (ie, both irms commit positive capacity when the IDM has a larger proit margin over ixed cost (MOF s MOF In this case, although the total capacity is determined by the IDM s newsvendor critical ractile p cs vs, it is optimal or the centralized system to have the oundry build part o the capacity since the oundry s per-unit total cost and production cost are both smaller than the IDM s Given the total capacity is targeted at F 1 (, i the oundry builds one less unit o capacity, then the IDM has to build an additional unit Thus, the underage cost o the oundry s capacity or the CS is c s + c and the corresponding overage cost is ; it is optimal or the CS to have the oundry build the capacity o F 1 ( cs+ c build the rest c s c, and have the IDM

11 11 To summarize, there are two types o contracting relationship: I MOF > MOF s, then ( the resulting centralized capacity is, F 1 p c, which is complete subcontracting, or outsourcing to the oundry case I MOF s MOF, then the resulting centralized capacity is (F 1 F 1 cs+vs c c s c, F 1 cs+vs c c s c, which is dual sourcing rom both the IDM and oundry The centralized capacity and production decisions lead to the maximum system proit However, in the absence o a central planner, the IDM and oundry are part o a decentralized system and in need o contracts to coordinate their decisions In the ollowing, we discuss two types o contracts in the decentralized horizontal setting, wholesale price contracts (ZRCs and P RCs, with the ocus on the latter to investigate whether they are coordinating 5 Wholesale Price Contract in the Horizontal Setting (ZRC In this section, we investigate whether wholesale price contracts in the horizontal setting (ie, ZRCs can achieve the maximum system proit First, we derive irms expected proits under ZRCs There are two cases: w c s and w > c s When w c s, the IDM always allocates the demand to the oundry irst, because buying rom the oundry is less expensive than producing by his own capacity We call this the oundry-irst case w > c s corresponds to the IDM-irst case That is, the IDM uses his own capacity irst to produce Hereater, i necessary, we emphasize the expected proits under ZRCs using the superscript z ; that is, Π z s and Π z ZRCs They are expressed in Lemma 2 are expected proits or the IDM and oundry under Lemma 2 I w c s, Π z = (w c K (1 F (x dx v K and Π z s = (p w K (1 F (x dx + (p c s K s+k K (1 F (xdx K s I w > c s, Π z = (w c K s +K K s (1 F (x dx K and Π z s = (p w K s +K K s (1 F (x dx + (p c s K s (1 F (xdx v s K s The capacities as a result o the IDM and oundry maximizing Π z s and Π z, respectively, are expressed in Proposition 2 Proposition 2 The equilibrium capacities or the IDM and oundry (K s, K under ZRCs can be expressed as ollows: For w c s, there are two cases:

12 12 ( F 1 F 1 w c, F 1 w c ( <, F 1 ( w c For c s < w c s +, there are two cases: ( < p cs, F 1 w c or (, F 1 ( w c For w > c s +, there are three cases: ( w c s F 1, w c s ( F 1 w c s w c s < ( < p cs F 1, (F 1, (, F 1 w c or F 1 w c s w c s (F 1 w c s w c s, F 1 w c F 1 w c s w c s Basically, there are our possible capacity equilibria under ZRCs, depending on the relative magnitude o the IDM s two possible critical ractiles, and w c s w c s, and the oundry s critical ractile The unique equilibrium can be determined except in two cases in Proposition 2 with In the scenario o < p cs (, the IDM s ideal total capacity level is F 1 < p cs (ie, < One option is to build by himsel the whole capacity, which leads to the equilibrium (F 1,, and the other option is to also use the oundry s capacity In the latter option, the oundry knows that the IDM will use his own capacity irst in the production stage by w > c s, so the oundry will only bring the total capacity to her critical ractile F 1 w c w c by building the dierence between F 1 w c and the IDM s capacity K s Thus, ( i K s F 1 w c (F 1,, the oundry will not build any capacity, in which case, again is the IDM s optimal outcome I K s < F 1 w c, the oundry will bring the total capacity to F 1 w c, in which case, the demand exceeding the IDM s capacity will be satisied rom the oundry s capacity, and thus the IDM s underage capacity investment cost is (w c s + and overage cost is When w c s +, the underage cost is, which means the IDM should not build any capacity and solely rely on the oundry to build F 1 w c When w > c s +, the underage cost is w c s, leading to the IDM s critical ractile w c s w c s ; This explains the possible equilibrium (F 1 w c s w c s, F 1 w c F 1 w c s w c s To summarize, in the scenario ( o c s < w < c s + and < p cs, the IDM s tradeo in choosing, F 1 w c or (F 1, is between (i achieving a suboptimal total capacity level but bearing

13 13 no capacity investment risk, and (ii achieving the optimal total capacity level but meanwhile bearing high capacity risk Intuitively, (i will outperorm (ii when is suiciently high Similar tradeo presents in the scenario o w > c s + and w c s < < All capacity equilibria in Proposition 2 are candidates to replicate the centralized capacity investments or the coordination purpose In the vertical supply chain setting, the wholesale price contract cannot coordinate the supply chain due to the well known double marginalization eect (Spengler 195 In the horizontal setting, we ind the wholesale price contracts cannot achieve the maximum system proit or two possible reasons: One is still the double marginalization eect, and the other is misalignment o capacity-usagepriority In the case o MOF > MOF s, the centralized system has only the oundry build capacity F 1 ( p c In this case, the horizontal supply chain setting reduces to the traditional vertical supply chain setting because in both cases the IDM does not build any capacity As in the vertical setting, the maximum capacity the oundry is willing to build acing a wholesale price w in the horizontal setting is F 1 w c, which is always strictly less than F 1 ( p c or w < p In such cases, no ZRC can coordinate capacity investment decisions, also due to the double marginalization eect In the other case with MOF MOF s and c s > c (Proposition 1, it is optimal or both the IDM and oundry to build capacity in the centralized system and the total capacity ( is F 1 The ZRCs with w c s cannot coordinate capacity investment decisions also due to double marginalization eect, because the oundry s capacity F 1 w c under ZRCs with w c s is always less than her capacity in the centralized system, F 1 ( cs + c c s c On the other hand, ZRCs with w > c s cannot coordinate the production decision because under the condition o c s > c, the centralized system uses the oundry s capacity irst, but w > c s implies the IDM will use his own capacity irst under ZRCs This is called the misalignment o capacity-usage-priority 6 Positive Reservation Contracts (P RCs In this section, we are interested in whether there exist coordinating P RCs (CP RCs We irst derive the expected proits and optimal capacities or the IDM and oundry when they are part o a decentralized system and ollow the decision protocol o Table 1 under a P RC Then we derive in Section 61 candidates or coordinating P RCs that achieve the maximum system proit but may not be individually rational or both the IDM and

14 14 oundry Finally, we check in Section 62 the IDM s and oundry s individual rationality, respectively We derive the objective unctions used in Steps 1 and 2 o Table 1 backward The oundry maximizes Π in Step 2 and determines her optimal capacity K In Step 1, anticipating the oundry s optimal action in Step 2, the IDM determines K s and R > to maximize Π s The IDM s ex-post production decision in Step 4 depends on the cost structure The unctional orms o Π and Π s are derived in Lemma 3 Lemma 3(a corresponds to the oundry-irst case, where in Step 4, the IDM uses the oundry s capacity irst and then his own capacity This is because the per-unit subcontract price w is less than or equal to the IDM s per-unit production cost c s Lemma 3(b corresponds to the IDM-irst case where the IDM s capacity is used irst, and then the oundry s This is because the IDM s per-unit production cost is less than the per-unit sourcing cost rom the oundry even i the IDM can obtain the reservation ee r back Lemma 3(c corresponds to the oundry-irst up-to-r case where the oundry s capacity is used irst up to R, ollowed by use o the IDM s capacity, and inally by the oundry s remaining capacity, i any This is because under the condition o c s + r > w > c s, it is more expensive or the IDM to produce by himsel i he can get the reservation ee back by sourcing rom the oundry; however, i there is no urther reservation ee to get back, the IDM preers to produce on his own Recall the production decisions under ZRCs Since the capacity investments have been sunk ater the demand gets realized, the IDM only needs to compare sourcing rom the oundry at per-unit cost o w and producing by himsel at per-unit cost o c s, and chooses the less expensive option However, under P RCs the oundry s capacity investment has not been ully sunk due to the reservation ee that the IDM pays ex ante to the oundry In order to get the reservation ee back, the IDM has more incentives to use the oundry s capacity, which results in the oundry-irst up-to-r case in Lemma 3(c Lemma 3 The expected proit unctions or the oundry and IDM under P RCs are as ollows: (a I w c s, then Π = (w c Π s = (p w K K (1 F (x dx K + r (1 F (x dx + (p c s R Ks+K K F (x dx, (4 (1 F (xdx K s r R F (x dx, (5

15 15 where K = R, or R > F 1 w c, F 1 w c, or < R F 1 w c (b I w > c s + r, then Π = (w c Π s = (p w Ks+K K s Ks +K Ks+R (1 F (x dx K + r F (x dx, (6 K s K s (1 F (x dx + (p c s Ks (1 F (xdx K s r Ks +R K s F (x dx, (7 where K = R, or R + K s F 1 w c, F 1 w c K s, or R + K s < F 1 w c (c I c s < w c s + r, then ( R Π = (w c (1 F (xdx + ( R Π s = (p w (1 F (xdx + where K s r K = R F (x dx, Ks+K K s+r Ks +K K s +R (1 F (xdx K + r (1 F (xdx + (p c s R, or R + K s F 1 w c, F 1 w c K s, or R + K s < F 1 w c R Ks +R R F (x dx, (8 (1 F (xdx In a positive reservation contract, the oundry has the obligation to build at least the capacity reserved by the IDM, ie, K R However, < R < K will never emerge in the capacity equilibrium, because i the oundry is willing to build a capacity level above the reservation quantity, then it is optimal or the IDM not to reserve any capacity to save the capacity reservation ee Thus, in a P RC, R = K (9 must hold in the equilibrium Lemma 4 in the appendix characterizes the possible capacity equilibria or the oundry-irst, IDMirst, and oundry-irst up-to-r cases, respectively This lemma serves as an intermediate result or deriving positive reservation contracts that achieve the maximum system proit Recall that a contract is coordinating i it satisies three conditions: capacity coordination,

16 16 production coordination, and individual rationality A contract that satisies the irst two conditions maximizes the system proit To check these two conditions, we examine i there exist P RCs so that 1 the centralized capacity investments characterized in Proposition 1 can be replicated by the decentralized capacity investments under the P RCs (see Lemma 4, and 2 the production decisions are also consistent with that in the centralized system 61 Candidate Coordinating Positive Reservation Contracts The ollowing proposition derives P RCs that replicate the centralized capacity investment and production decisions under both cases o MOF s MOF and MOF s < MOF, and characterizes the corresponding proit allocation between the IDM and oundry Proposition 3 There exist the ollowing P RCs (w, r that coordinate the IDM s and oundry s capacity investment and production decisions (a Assume MOF s MOF holds Then set w = c + + α and r = α ( c s+ c or (c s c + < α < c s + c The expected proits or the oundry and IDM are ( Π c = α c s+ c (c s c K c (1 F (x dx (v K c, Π c s = Π cs Π c, where Π cs = (p c K c (1 F (xdx + (p c s K cs +K c K c (1 F (xdx K c K cs The corresponding capacity investments are (K cs, K c = (F 1 ( F 1 ( cs + c c s c (b Assume MOF > MOF s holds Then set, F 1 cs + c c s c w = c + + α and r = α or < α c s + c The expected proits or the oundry and IDM are Π c = α Π cs, Π c s = 1 α Π cs, where Π cs = (p c K c (1 F (xdx v K c ( The corresponding capacity investments are (K cs, K c =, F 1 p c In Proposition 3, each α (deined as α = w c corresponds to a P RC(w, r, and α takes values in a continuous range So there are an ininite number o P RCs that can replicate the centralized capacity investment and production decisions and thus achieve the maximum system proit The case o MOF s MOF leads to horizontal dual sources under P RCs that maximize the system proit Thereore, the coordination problem in this case is clearly dierent rom the coordination o a vertical supply chain Contract parameters have to be set so

17 17 that the capacities built at each irm and the capacity usage priority are consistent with that in the centralized system In the vertical supply chain, there is no need or coordinating capacity usage priority (or production decision, as only supplier builds capacity The coordination problem or the case o MOF > MOF s in the horizontal setting looks similar to the coordination problem in the purely vertical supply chain setting because in the resulting capacity investment decision, it is optimal that only the oundry builds capacity, which essentially reduces the horizontal setting to a vertical supply chain Despite this similarity, there is a key dierence Under a vertical supply chain ramework, several dierent contract types are shown to maximize and arbitrarily divide system proit, such as buy-back contracts, revenue-sharing contracts, quantity-lexibility contracts, sales-rebate contracts and quantity-discount contracts (Cachon 23 Arbitrary proit allocation is not possible in the horizontal setting where the IDM himsel is able to build capacity and earns at least a proit o Π i s when he acts independently (note that the upper bound o α is c s + c, instead o p c, whereas in the vertical supply chain, the buyer has to rely on the supplier or a key input or product that the buyer cannot produce by itsel Thereore, both parties will earn zero proit without collaboration The P RCs derived in Proposition 3 maximize the system proit However, they may not be coordinating i they do not satisy the IDM s or oundry s individual rationality condition Thus, all P RCs in Proposition 3 are reerred to as candidate coordinating P RCs (CCP RCs hereater Proposition 3 also presents the expected proits o the IDM and oundry under the CCP RCs In the ollowing subsection, these proits are compared with the IDM s and oundry s expected proits when they use the benchmark wholesale price contracts (ZRCs to check the individual rationality condition 62 Individual Rationality Condition Among ZRCs, there exists an optimal contract or the IDM to maximize his expected proit Let Π z s denote the IDM s expected proit under his optimal ZRC, and Π z denote the oundry s expected proit under this ZRC That is, under the benchmark wholesale price contract, the oundry has a business opportunity to earn Π z The IDM s individual rationality condition can be checked by comparing Π c s under the CCP RCs in Proposition 3 with Π z s I the IDM has incentives to improve his expected proit over the wholesale price contract by adopting a P RC with the oundry, although theoretically he can switch unilaterally, in most practical settings especially in the semiconductor industry, due

18 18 to considerations such as long-term business relationship and mutual beneit both the IDM and oundry have a say in the contracting stage In order to make a smooth switch rom the wholesale price contract to a more sophisticated one, both parties should have incentives or the switch That is, the oundry should also earn a larger expected proit in the alternative contract than Π z To check the oundry s individual rationality, we compare Π c under the CCP RCs in Proposition 3 with Πz The CCP RCs with Πc s and Π c greater than Π z s are CP RCs and Π z respectively satisy both irms individual rationality condition, and thus Under the IDM s optimal ZRC, the oundry may or may not build capacity in the capacity equilibrium For urther discussion purpose, ZRC denotes a ZRC with the oundry building positive capacity Denote the IDM s expected proit under a ZRC Π z s I the oundry does not build capacity ZRC reduces to IDM independent case In this case, Π z s Thus, Π z s = Π i s, where Π i s is the IDM s expected proit when he acts independently = Π i s or Π z s = Π z s As a irst step to compare Π c s and Π c under the CCP RCs with Πz s and Π z Propositions 4 and 5 derive some properties o irms proits under CCP RCs as respectively, Proposition 4 Consider all Π c s under the CCP RCs speciied in Proposition 3 (i I MOF s MOF, Π c s is greater than Π i s As w converges to c s +, Π c s converges to Π i s (ii I MOF s < MOF, Π c s is strictly greater than Π i s Proposition 4 shows that by using any CCP RC, the IDM earns at least what he can earn by acting independently Note that Π c s under the CCP RCs deined in Proposition 3 decreases in w The upper bound o w is c s +, which is the IDM s per-unit total cost and the corresponding reservation ee r = or MOF s MOF and r < or MOF s < MOF In act, when w = c s +, r =, and the total capacity is ixed, the IDM is indierent between using the positive reservation contract (w, r and relying on his own capacity This is because securing one unit o capacity by either building his own capacity or reserving rom the oundry costs the same, and producing one unit o product costs the same c s by using either the oundry s capacity or his own capacity The act that the IDM s least expected proit under CCP RCs is greater than or equal to Π i s leads to Proposition 4 I the IDM preers to act independently (ie, Π z s = Π i s, Π z =, all the CCP RCs satisy both the IDM s and the oundry s individual rationality condition Otherwise, i the oundry builds positive capacity (ie, Π z s = Π z s, then we compare Π z s and Π z with

19 19 Π c s and Π c Π z under the CCP RCs respectively to check the individual rationality Πz s can be derived by substituting the capacity equilibria with positive oundry s capacity speciied in Proposition 2 to Lemma 2 Due to the complexity o the newsvendor ormula, Π z s cannot be directly compared with Π c s However, Proposition 5 below shows that the oundry s expected proit under any CCP RC is always greater than that under the ZRC or the same w This is partially because under any CCP RC the demand is allocated to the oundry irst, and urther the oundry receives the reservation ee under CCP RCs, but not under ZRCs Proposition 5 The oundry s expected proit under any CCP RC(w, r speciied in Proposition 3 is greater than that under the ZRC or the same w Based on Propositions 4, 5, and urther analysis, we derive suicient but not necessary conditions or CP RCs to exist in Proposition 6 and Proposition 6 Coordinating P RCs (CP RCs exist when (i MOF s MOF c s c +, or (ii MOF > MOF s and the IDM s optimal ZRC has w c s + and The condition (i in Proposition 6 implies the IDM s cost structure is more lexible and his unit production cost is not very high In this case, or w converging to c + the IDM can achieve the maximum system proit, which must be strictly greater than his proit under the optimal ZRC, Π z s As Π c s decreases in w, there exists a w (c + < w c s + so that Π c s = Π z s at w and all CCP RCs with w < w is strictly preerred by the IDM to his optimal ZRC On the other hand, or w converging to c s +, the oundry must obtain a strictly greater proit than her proit under the IDM s optimal ZRC, because at c s +, the IDM s expected proit under the CCP RC is exactly the same as when he acts independently, which is less than or equal to his proit under the optimal ZRC, Π z s However, the total proit under CCP RC must be greater than that under ZRC, so the oundry s proit under the CCP RC with w converging to c s + must be greater than Π z Since Πc increases in w, there exists a lower bound w so that Πc = Πz at w, and all the CCP RCs with w > w are preerred by the oundry to the IDM s optimal ZRC Based on the above discussion, i w > w, CCP RCs with w (w, w satisy both irms individual rationality condition, and hence are coordinating In act, w > w always holds: At w, Π c s = Π z s, which implies the oundry s proit at the CCP RC must be greater than Π z ; at w, the oundry s proit at the CCP RC equals Πz Since the oundry s proit under CCP RCs increases in w, we have w > w

20 w w : s, or w c s under ZRCs : s, or c s w c s under ZRCs : s, under CCPRCs in Proposition 3 a s Figure 1 An example o the IDM preerring his optimal ZRC to any CCP RC in Proposition 3 under MOF s > MOF and c s > c +, where c s = 5, = 5, c = = 23, p = 575, D N T ( 5, 2 2 I MOF > MOF s, w can also be arbitrarily close to c + by the observation rom Proposition 3 Following the same argument above, there exists a w so that CCP RCs with w < w satisy the IDM s individual rationality condition Suppose the IDM s optimal ZRC has w c s + ; then by Proposition 5, the oundry s proit under the CCP RC with w is strictly greater than her proit under the IDM s optimal ZRC Then there exists a w < w so that Π c = Πz at w, and all the CCP RCs with w > w are preerred by the oundry to the IDM s ZRC However, i the IDM s optimal ZRC has w > c s +, then the oundry s proit under the IDM s optimal ZRC might be so large that it is greater than her largest proit under the CCP RC with w = c s + In this case, the oundry s individual rationality condition is not guaranteed Similarly, it can be shown w > w holds, and CCP RCs with w (w, w satisy both irms individual rationality condition, and hence are CP RCs This explains the suicient conditions in Proposition 6 (ii I nether (i nor (ii in Proposition 6 holds, irms individual rationality condition is not guaranteed Because under MOF s MOF and c s > c +, the oundry s proit is strictly greater than a positive value under CCP RCs (since α is strictly greater than a positive value This implies that the upper bound o Π c s under all the CCP RCs is less than the maximum system proit In this case, it is not guaranteed that the largest Π c s under CCP RCs is greater than Π z s To give an example, Figure 1 shows under the case o MOF s MOF and c s > c +, there exist ZRCs (to the right o the vertical dashed line leading to larger IDM s expected proits compared to all the CCP RCs Then, the IDM preers to use his optimal ZRC, and no CCP RC satisies the IDM s individual rationality condition, ie, no P RC is coordinating

21 21 The conditions o MOF s MOF and c s > c + imply the IDM s per-unit capacity investment cost, is very low and so is the capacity investment risk Under this case, the IDM may be better o by using a ZRC with a low subcontract price and mainly relying on his own capacity, so P RC cannot be coordinating However, it is typical in most practical settings that the oundry has a more lexible cost structure than the IDM (ie, MOF s < MOF, and one party s total cost is greater than the other s production cost 2 Thus, it is rather extreme or both conditions MOF s MOF and c s > c + to hold In almost all practical situations, we have either MOF > MOF s or c s < c +, which guarantees the existence o a continuous range o CCP RCs that satisy the IDM s individual rationality condition In our numerical study, the IDM s optimal ZRC has w c s + in most cases w > c s + in the IDM s optimal ZRC does not occur requently, but is possible; even or such less requently encountered cases, we ind that Π c at w = c s + is much greater than Π z, which implies the oundry s individual rationality condition is also satisied or a range o CCP RCs Overall, these results imply that the conditions or CP RCs to exist can easily be met in practice 7 Numerical Study In the extensive numerical study conducted, we present two cases whose results are representative or those under MOF s MOF and MOF > MOF s, respectively For the case o MOF s MOF, one example o cost structure is {c s = 45, = 15, c = = 27} Although in the semiconductor industry, the per-unit total production cost is around 5% o the revenue, ie, the reasonable retail price is around 12 or the considered cost structure, we consider our dierent retail prices {p = 65, 7, 8, 13} to study the impact o proit margin In Figure 2, we plot or each case the IDM s and oundry s expected proits under both CCP RCs and ZRCs (Π s, Π under ZRCs with w (c +, c s + and w [c s +, p], respectively, are marked with dierent symbols Note that or the considered cost structure, w < c s is impossible because w > c + > c s The triangle signs denote (Π s, Π under the CCP RCs with w (c +, c s + as speciied in Proposition 3 The arrow in each plot indicates the direction o increasing w By representing (Π s, Π under both CCP RCs and ZRCs in the same plot, we are able to show the eiciency loss 2 In the semiconductor industry, both the equipment procurement and the device manuacturing present economies o scale Due to oundries specialization and economies o scale, they oten have cost advantage over IDMs; that is, < and c < c s are likely to hold, both contributing to > p cs

22 22 :HP s,p Lunder CCPRCsinProposition 3HaL æ:hp s,p Lor c s <w<c s + under ZRCs :HP s,p Lor w c s + under ZRCs P 15 æ æ æ 1 æ æ æ ææ 5 ææ ææææææ HP s z*,p z* L p=68 CPRC w æ w P 15 æ 1 æ æ 5 æ ææææ æ HP s z*,p z* L CPRC p=7 w P s P s P 15 1 CPRC w HP z* æ s,p z* L p=8 P 15 1 CPRC w HP z* s,p z* L æ p=13 P s P s Figure 2 (Π s, Π under CP RCs and ZRCs (MOF s > MOF, c s = 45, = 15, c = = 27, D N T ( 5, 2 2 o the system by using ZRCs and the proit gaps under the two types o contracts or both irms There are two interesting observations rom Figure 2 First, when the product price is large (eg, p = 8, 13, the IDM s optimal choice under ZRCs is the IDM independent case, and all the CCP RCs are indeed coordinating (all the triangle signs are on the upper-right side o the round dot Second, when the product price is low (p = 68, 7, the IDM s optimal ZRC leads to positive Π z s and Π z as indicated in the plot In this case, only the CCP RCs with the corresponding triangle signs on the upper-right side o Π z s, Π z are coordinating These observations hold in all the extensive examples we have studied Figure 2 also shows that the proit allocations under all CP RCs constitute the complete eicient rontier Figure 3 shows the results or the case o MOF > MOF s Dierent rom the case o MOF s > MOF shown in Figure 2, all the proit allocations under the

23 23 :HP s,p Lunder CCPRCsinProposition 3HbL æ:hp s,p Lor c + <w<c s + under ZRCs :HP s,p Lor w c s + under ZRCs P p=65 15 æ æ æ æ æ æ 1 æ CPRC w æ æ æ æ 5 æ ææææ w HP z* s,p z* L æ æ æ æ ææææ w ç P p=7 15 CPRC æ æ æ 1 w æ æ æ ææ HP æ ææææ æ æææææ z* s,p z* L 5 w w æç P s P s P p= CPRC 1 5 w HP z* s,p z* L æç P p= CPRC w HPz* æç s,pz* L P s P s Figure 3 (Π s, Π under CP RCs and ZRCs (MOF > MOF s, c s = = 3, c = = 27, D N T ( 5, 2 2 CP RCs in Figure 3 may not constitute the complete eicient rontier, as in the case o p = 8, 13 This is because the lower bound o Π c s is strictly greater than Π i s when MOF > MOF s, as shown in Proposition 4 Nevertheless, there exists a range o CP RCs The identiication o CP RCs is important or the IDM and oundry to understand i they can improve their expected proits over the wholesale price contract and the possible range o surplus allocation 8 Conclusion Motivated by the capacity expansion practice in the semiconductor industry, this paper studies horizontal capacity coordination in a one-market setting that dierentiates rom the purely vertical supply chain setting and the two-market horizontal setting The requent and expensive nature o capacity investment in the semiconductor industry oten motivates IDMs to seek collaboration with oundries In a decentralized decision-making scheme, the search or contracts that can achieve the maximum system proit is a worthwhile topic o study We deine coordinating contracts as those that can replicate the centralized