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Signaling New Product Reliability with After-Sales ServiceContractsNitin Bakshi, Sang-Hyun Kim, Nicos Savva
To cite this article:Nitin Bakshi, Sang-Hyun Kim, Nicos Savva (2015) Signaling New Product Reliability with After-Sales Service Contracts.Management Science 61(8):1812-1829. http://dx.doi.org/10.1287/mnsc.2014.2029
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Signaling New Product Reliability withAfter-Sales Service Contracts
Nitin BakshiLondon Business School, Regent’s Park, London NW1 4SA, United Kingdom, [email protected]
Sang-Hyun KimYale School of Management, Yale University, New Haven, Connecticut 06511, [email protected]
Nicos SavvaLondon Business School, Regent’s Park, London NW1 4SA, United Kingdom, [email protected]
Prior studies on performance-based contracting (PBC) for after-sales services have highlighted its advantagesover traditional resource-based contracting (RBC), when products are established and their reliability is known
to all parties. We develop a game theoretic model to investigate how these insights are affected when the vendor isprivately informed about the reliability of a newly developed product. A novel feature of our model is theinteraction between reliability signaling (private information) and the vendor’s discretionary investment in sparesinventory (private action), which arises naturally in the setting we consider. We find that this interaction leads tocontrasting equilibrium outcomes under the two contracts: RBC induces the vendor to focus on inventory savings,leading to underinvestment in spares, whereas PBC induces the vendor to focus on reliability signaling, achievedthrough overinvestment in inventory. As a result, neither contract is efficient. We investigate two means to mitigatethis inefficiency, but either approach has caveats: (a) making inventory verifiable removes the trade-off betweenreliability signaling and inventory investment, but results in diverging contract preferences between the vendorand the buyer; (b) pooling inventories across multiple buyers saves inventory costs but it also hinders reliabilitysignaling, potentially exacerbating inefficiency.
Keywords : signaling games; performance-based contracting; product reliability; aerospace sector; after-sales servicesHistory : Received February 1, 2012; accepted June 18, 2014, by Yossi Aviv, operations management. Published
online in Articles in Advance January 19, 2015.
1. IntroductionAfter-sales product support, namely maintenance,repair, and overhaul (MRO) operations, is a key busi-ness area in the aerospace industry. This sector alonegenerated revenue of $43 billion in 2009 (Standard &Poor’s 2011). One of the main drivers of financial per-formance in this sector is product reliability, since MROoperations revolve around preventing and respondingto flight disruptions that occur due to unanticipatedproduct malfunctions. To mitigate the impact of imper-fect reliability, it is critical for the product supportprovider to invest in spares inventory, since sparesreplace defective products and thus increase systemusage. In a typical decentralized supply chain consist-ing of buyer organizations (e.g., airline companies) andvendors (e.g., aircraft engine manufacturers), an impor-tant challenge is to establish a contractual relationshipthat enables effective management of reliability andinventory.
Prior studies have emphasized the role of relia-bility and spares inventory in determining a firm’spayoff under two types of contracts that are widelyused in practice: resource-based contracts (RBC) andperformance-based contracts (PBC). Under RBC, the
compensation for the vendor who provides productsupport is proportional to the amount of resourcesutilized (e.g., labor and spare parts consumed to repaira defective product). By contrast, PBC unties resourceusage from compensation, since the vendor underPBC is rewarded or penalized based on the realizedperformance outcome that directly impacts the buyer,such as aircraft uptime. A general consensus from theliterature on PBC (see, e.g., Hypko et al. 2010, Kimet al. 2007, Randall et al. 2011, Ward and Graves 2007)is that PBC is a superior contracting mechanism thatbetter aligns incentives between the buyer and thevendor, resulting in higher product utilization at alower overall cost.
Despite the consensus in the academic literaturetouting the advantages of PBC over RBC, practitionersappear to be ambivalent about the choice between thetwo, some preferring PBC based on the aforementionedmerits and others remain unwilling to switch from themore traditional RBC.1 Such reluctance to switch from
1 Conklin & de Decker (2014) report, “[PBC] programmes generatea lot of heated debate, with some believing they can get theirengine maintenance for a lot less money [through programmes
RBC may simply reflect the reality that some buyersare comfortable with the status quo. However, thisinertia-based argument is not entirely consistent withsome of the observed facts. In particular, it does notexplain why buyers are often willing to adopt PBCwhen they consider acquiring products with maturetechnologies, but not when they acquire productsequipped with newly developed technology; thesediffering preferences based on the maturity of theunderlying technology are reported in the survey ofbuyer organizations conducted by the U.S. GovernmentAccountability Office (GAO 2004). The report indicatesthat buyers are reluctant to switch to PBC for newlydeveloped products due to the absence of publiclyavailable baseline data on product reliability. In otherwords, buyers view RBC as offering better value thanPBC when product reliability is unknown to them.
This observation identifies an important aspect ofafter-sales service contracting that has been overlookedin the literature: buyers’ uncertainty about productreliability and its impact on contract performance. Avendor faces a challenge in this environment becausethe lack of independently verifiable data on reliabilitylimits her ability to propose the best contract termsfor herself and the supply chain. That is, the vendorwho bundles an after-sales service contract with anew product has to consider not only how her profitis maximized in the long run but also how she cancredibly convey information about product reliabilityin the short run. How does this added requirement tosignal reliability impact contract design? Do the benefitsof PBC over RBC identified in the literature continueto exist in this situation? Is the observed preference forRBC among buyers influenced by the vendor’s need tosignal? We aim to answer these questions in this paper.
We develop a stylized principal–agent model thatcaptures the dynamics arising when a vendor (principal)possesses an informational advantage concerning thereliability of a newly developed product that a buyer(agent) plans to acquire. We adopt the classical signalingframework as the model basis, but with an importantadded feature that makes the dynamics richer and moresubtle: an interaction between the vendor’s reliabilitysignaling and her discretionary inventory investment.Although investing in spares inventory compensates for
other than PBC]. Others are convinced that PBC is the best thingsince sliced bread. Still others provide a full range of opinions inbetween.” Responding to such heterogeneous preferences, vendorshave adopted different priorities for their contract offerings. Forexample, Rolls-Royce reports that in the period 2001–2009, 80% of itslarge commercial-airliner engines (the Trent 700 and its variants)were covered by PBC, and this is forecast to grow in the future. Atthe same time, at Pratt and Whitney only a third of the installedengine base in 2009 was covered by PBC and the company isreporting a slowdown in PBC agreements going into the future(Adler et al. 2009).
(§5.1) Efficient reliabilitysignaling and inventoryinvestment, with partialrent extraction
(§5.2) Efficient reliabilitysignaling and inventoryinvestment, with full rentextraction
Unverifiableinventory
(§6.1) Focus on inventorysavings over reliabilitysignaling; underinvestmentin inventory
(§6.2) Focus on reliabilitysignaling over inventorysavings; overinvestmentin inventory
imperfect reliability (the more the spares, the smallerthe chance of prolonged outage after a product failure),this mitigating effect also interferes with the vendor’sability to credibly signal product reliability throughcontract terms. This interaction forces the vendor to setcontract terms so that the dual goals of inventory costsavings and effective signaling are balanced.
Our analysis reveals how RBC and PBC differ asvehicles to signal product reliability. As it turns out,verifiability of inventory plays a crucial role. Wheninventory is verifiable, both RBC and PBC are effectiveas a signaling mechanism as they allow the vendorto credibly convey private reliability information andachieve supply chain efficiency. When inventory isunverifiable, by contrast, neither contract brings effi-ciency. In particular, the two contracts shift the balancebetween reliability signaling and inventory investmentin opposite directions: RBC shifts the focus to inventorysavings over effective signaling, whereas PBC shiftsfocus to effective signaling over inventory savings.Consequently, the supply chain is left with underin-vestment in inventory under RBC and overinvestmentin inventory under PBC. Table 1 summarizes thesefindings.
This observation suggests that the two supply chainparties may have a mutual interest in having the ven-dor invest in technology to make inventory verifiable,as doing so will decouple the interaction between relia-bility signaling and inventory investment and bringsupply chain efficiency, thereby potentially benefittingboth parties. We find, however, that allowing for suchtechnology investment results in diverging contractpreferences between the vendor and the buyer; thevendor prefers PBC, whereas the buyer prefers RBC.Thus, the prediction from our analysis is consistentwith the aforementioned reports from the aerospaceindustry. Another option for improving supply chainefficiency, which is of particular interest when inven-tory verifiability is prohibitively expensive, is for thevendor to pool inventory across multiple buyers. Wefind that this option also comes with limitations due toits interaction with signaling; although pooling savesinventory costs, it hinders reliability signaling, poten-tially exacerbating inefficiency. Overall, our findings
show that after-sales service contracting for newlydeveloped products presents a new set of challengesthat require managerial actions that are quite distinctfrom those used for established products.
This paper is organized as follows. After a literaturereview in §2, we present our model in §3 followed by abrief discussion of the benchmark first-best scenario in§4. To illustrate the structural differences between RBCand PBC, we conduct our analysis in §§5 and 6, assum-ing that the contract type is exogenously specified. In§5 we focus on the problem of signaling in isolation,assuming that inventory is verifiable and is contractedupon. In §6 we study how the vendor’s discretionaryinventory choice interacts with her reliability signaling.After evaluating how RBC and PBC perform underthese scenarios, we discuss in §7 the two means bywhich system inefficiency can be alleviated: (a) con-tract choice coupled with investment in inventoryverifiability and (b) inventory pooling. We concludein §8.
2. Literature ReviewOur work relates to three distinct streams of litera-ture: the first explicitly deals with PBC for outsourcedservices; the second, with which our work shares amethodological connection, is the literature on asym-metric information in operations management (OM);and the third is the literature on the use of warrantiesas a signaling mechanism.
PBC has been studied in a service delivery setting by,among others, Roels et al. (2010), Gümüs et al. (2012),and Jain et al. (2013). PBC in the context of after-salesservices in the aerospace industry has been studied inKim et al. (2007, 2010). Our paper is closest in spirit toKim et al. (2007), who investigate a setting in whichthe buyer proposes the terms of the PBC while thevendor exerts private effort to reduce maintenance costand invests in spare parts inventory. The paper showsthat in a setting with risk-averse players, the first-best cannot be attained and the optimal second-bestcontract involves a performance-related component.Although this stream of research has shown that PBCis a preferred contracting mechanism in settings undermoral hazard issues because of its ability to alignincentives between the vendor and buyer, it invariablyassumes that the failure characteristics of the productsare common knowledge. Our paper complements thisline of research by studying a setting with asymmetricinformation: the vendor has better knowledge aboutthe product’s reliability than the buyer does becausethe product is newly developed. In contrast to previousstudies, we identify circumstances under which PBCis not favored due to the presence of asymmetricinformation.
Our paper is related methodologically to the OM lit-erature on games of asymmetric information. Examples
of screening games, where the uninformed principaloffers contracts designed to elicit information fromthe informed agent, include Corbett et al. (2004) andLi and Debo (2009). Our work is closer to papersthat adopt a signaling framework (e.g., Anand andGoyal 2009, Cachon and Lariviere 2001, Özer andWei 2006), in which an informed principal signals hersuperior information through the contracts offered toan uninformed agent. We contribute to this literatureby studying how PBC and RBC perform as signal-ing devices. Unlike the aforementioned works, whichimplicitly assume that the contracting parties somehowcommit not to renegotiate, in our analysis we employthe signaling-with-renegotiation framework developedby Beaudry and Poitevin (1993), where the agentsmay renegotiate after a contract has been signed butbefore it is implemented. (It can be shown using thisframework that the classic (and inefficient) second-bestequilibrium outcome of a one-shot signaling game thatignores renegotiation (e.g., Theorem 7 in Cachon andLariviere 2001) is never renegotiation-proof and thatthe renegotiation-proof outcome is, in general, moreefficient.) Correspondingly, our work also complementsthe recent OM literature on the impact of renegotiationon equilibrium outcomes (e.g., Plambeck and Taylor2007a, b; Xiao and Xu 2012). Whereas in these articlesrenegotiation is triggered when some of the uncertaintyis naturally resolved after the contract is implemented,in our setting renegotiation is triggered by the potentialfor Pareto improving offers that arise at the time acontract is signed but before the contract is actuallyimplemented.
The risk-sharing and signaling capabilities of RBCand PBC result in characteristics similar to those inproduct warranties. Research on signaling productquality through warranties has a long tradition in eco-nomics (Gal-Or 1989, Lutz 1989, Riley 2001), marketing(Boulding and Kirmani 1993, Moorthy and Srinivasan1995), and OM (Courville and Hausman 1979, Gümüset al. 2012). See Kirmani and Rao (2000) for an extensivesurvey of the literature. The underlying premise ofthis literature is that firms selling low-quality productswill face higher costs for the same level of warrantythan will high-quality firms because low-quality firms’products are likely to require more frequent repair.A major difference between our work and those inthe quality-signaling literature is that our results aredriven by the interaction between the principal’s pri-vate information (about product reliability) and privateaction (represented by inventory investment), a featurethat gives rise to new dynamics.2 Another departure
2 Although many papers in the signaling literature have investi-gated the interaction between a principal’s private informationand an agent’s moral hazard, the interaction between a principal’sprivate information and her own private action has received little
from the extant literature is that, in addition to identi-fying the scope for signaling, our model allows us toevaluate the performances of two widely used contractsin the after-sales setting: RBC and PBC.
3. ModelWe consider a supply chain that consists of two risk-neutral parties: a vendor (“she”) and a buyer (“he”).The buyer faces constant demand for product usagethat he plans to satisfy by purchasing and deployingN identical copies of a newly developed product fromthe vendor. The vendor provides after-sales repair andmaintenance services to the buyer. The vendor setsthe terms of the service contract and proposes themto the buyer. Hence, the vendor serves as a principalin the principal–agent framework. The contract takesone of two forms: RBC or PBC. The duration of thecontracting period is normalized to one.
3.1. Repair Process and Inventory PolicyWe adopt the standard modeling framework establishedin the literature to represent the repair process andspares inventory management, as elaborated by Kimet al. (2007, 2010), among others. During the course ofdeployment, the products fail occasionally due to mal-function. These products are repairable items that arenot discarded upon failure but repaired and restoredto working order. We focus on the types of failurethat require unscheduled repairs, which incur largeand unanticipated costs. For instance, unscheduledmaintenance on a wide-bodied aircraft could cost asmuch as 10 times the capital outlay over the operatinglife of the aircraft (see Hopper 1998). We do not explic-itly model preventive and scheduled maintenance, thecost of which is either fixed or regulated and is notsignificantly affected by contractual incentives.
A one-for-one base stock policy is employed forspares inventory control (Feeney and Sherbrooke 1966).A failed unit immediately enters a repair facility, whichis modeled as a GI/G/� queue with expected repairlead time l. We assume that the distribution for repairlead time is exogenously specified. The expected num-ber of product failures during the contracting periodis denoted by �. We assume that the arrival processfor failures is exogenous and state independent: acommon convention found in the spare parts inven-tory management literature (e.g., Sherbrooke 1968).Although in practice there are situations where this
attention. To the best of our knowledge, the only other paper that hasstudied this type of interaction is Jost (1996). In this paper, however,the authors focus on a completely different setting in which theprincipal chooses from a binary action set and the agent’s payoffis independent of this action. As a consequence of these differingassumptions, they obtain results that are very different from ours:they predict that only a pooling equilibrium will emerge.
condition is violated, it is not overly restrictive as theinfrequent nature of failures makes this an excellentapproximation.
A product failure may affect system availability,which is defined as the fraction of system uptime overthe length of contract duration. System availability isunaffected if a spare product can be pulled from theinventory to replace the defective unit immediately. Ifthe inventory is empty at the time of failure, however,system availability is reduced until a repaired unitbecomes available from the repair facility; this causesinventory backorder, denoted by B. It is clear fromthis description that backorders are reduced as theinventory s increases. In our analysis we approximateall discrete variables (including B and s) as continuousvariables in order to facilitate game-theoretic analysis.Let F be the stationary distribution function of theinventory on order O, that is, the number of repairsbeing performed at the repair facility at a given point intime. When a one-for-one base stock policy is followed,O can be thought of as the number of busy serversin a GI/G/� queue, the distribution for which isstationary for any finite repair lead time l (Kaplan1975). We assume that F 4x5 = 0 for x < 0 and thecorresponding density function f 4x5 > 0 for x > 0. Wealso assume that the on-order distribution has theincreasing hazard rate property, i.e., f 4x5/41 − F 4x55 ismonotone increasing in x. This property is satisfiedby a wide range of distributions including Gamma,Weibull, Poisson, and truncated normal (Barlow et al.1963, Gupta et al. 1997). For a given level of sparesinventory s, the expected backorders in steady state isthen equal to E6B � s7=
∫ �
s41− F 4x55 dx. Moreover, a one-
to-one correspondence can be made between systemavailability and the expected backorders: availabilityis equal to 1 − E6B � s7/N . For this reason, we use theterms “availability” and “backorders” interchangeablythroughout the paper.
3.2. Cost StructureGiven the assumption that the buyer faces constantdemand for product usage, the maximum revenue thatshe can generate (in case there is no product outage) isa constant. Without loss of generality, we normalizethis value to zero. Each time a product failure occurs,the buyer incurs a fixed cost r ≥ 0. In commercialairline operations, r represents, among other things,the cost associated with rescheduling flights that resultfrom delays due to an engine coming off-wing forrepair (Adamides et al. 2004). In addition, if the vendordoes not have inventory on hand to replace the failedproduct, the buyer incurs a variable cost � > 0 perunit time until the system function is restored. Thecost � represents the direct revenue loss to the buyerresulting from system inoperability due to the failedproduct (e.g., the cost of grounding an aircraft). Thebuyer’s expected cost is equal to �r +�E6B � s7.
The vendor expects to incur a repair cost (such asthe cost of labor and component parts) equal to M aftereach failure, plus the cost of acquiring and maintainingspares inventory, each unit costing c per unit time.Note that since we have normalized the duration of thecontract to 1, the costs per unit for the whole durationof the contract are also c. The vendor’s expected costis equal to �M + cs. To rule out trivial results, weassume � ≥ c.
3.3. Information StructureAs the developer of a product equipped with new tech-nology, the vendor possesses superior knowledge aboutthe product’s characteristics, including an estimate ofits failure distribution, at the time she introduces theproduct to the market (GAO 2004, Boito et al. 2009).Although this gives the vendor an informational advan-tage, it also creates an incentive to misrepresent theinformation: a vendor whose product has low reliabilitymay claim otherwise. This presents a challenge for thevendor whose product has high reliability, since anysuch claim may not be viewed by the buyer as credible.This problem is compounded by the fact that the buyerlacks the ability to independently verify the vendor’sclaim. For example, Kappas (2002) reports that OEMs’databases describing the material properties of alloysin aircraft engines are not publicly available. Moreimportantly, inferring the reliability of products fromactual failures of deployed units is challenging becauselittle historical data exist for the new technology andfailures occur infrequently; for instance, the mediantime between failures for an aircraft engine reported inGuajardo et al. (2012) is five years.
To represent information asymmetry in a succinct andanalytically tractable way, we assume that the productoffered by the vendor is one of two possible types, L orH , where L denotes low reliability and H denotes highreliability. We use the terms “unreliable vendor” and“reliable vendor” to refer to the vendor having a lowreliability product type and high reliability producttype, respectively. Whereas the vendor observes hertype perfectly, the buyer believes ex ante that thevendor is of type L with probability p > 0. This belief iscommon knowledge. The expected number of productfailures occurring during the contract period satisfies�L >�H , i.e., more failures are expected to occur whenproduct reliability is low. Moreover, the distributions FLand FH of the on-order inventory OL and OH areassumed to have the following two properties:
fL4x5
1 − FL4x5>
fH 4x5
1 − FH 4x51 (1)
∫ �
F −1H 4p5
41 − FH 4x55 dx ≤
∫ �
F −1L 4p5
41 − FL4x55 dx1
∀p ∈ 601170 (2)
In other words, the distribution functions follow thehazard rate ordering (1) and the excess wealth (EW)ordering (2) (see Shaked and Shanthikumar 2007 fordetails). It can be shown that these properties aresatisfied if the failures occur according to a Poissonprocess. We note that (1) and (2) play important rolesin our analysis in identifying the equilibria.
To focus on the dynamics arising from signaling,we sidestep the issue of double moral hazard, i.e., thepossibility that the buyer exerts an insufficient levelof care for the product he has acquired (Lutz 1989,Jain et al. 2013). This modeling choice is reasonablegiven that our research motivation comes primarilyfrom the aerospace industry, where vendors typicallyhave real-time access to information about the usagepatterns and performance of their deployed products,thereby alleviating concerns about buyer moral hazard.Furthermore, we do not model risk aversion explicitlyas it would add another layer of complexity, divertingattention away from the main trade-offs we investigate.
3.4. Contract Types and PayoffsIn this paper we restrict attention to two predominanttypes of maintenance contracts used in the aerospaceindustry: resource-based contracts and performance-based contracts. As evidenced by numerous industryreports (e.g., Adler et al. 2009), practitioners focus onthe contrasts between these two types especially due tothe growing adoption of PBC in recent years. The twocontract types differ on the basis of compensation formaintenance activities performed by the vendor. RBCis the older and more traditional transaction-basedapproach, which includes the popular time and mate-rial (T&M) contract. It is based on the simple idea thatcompensation for the vendor is proportional to theamount of resources utilized to repair a defective prod-uct, such as labor and spare parts consumption. In itsgeneral form, RBC also includes warranty coverage thatis designed to protect the buyer from any unexpectedout-of-pocket expenses incurred after a product failure.In this paper we generalize the definition of RBC toinclude warranty coverage in addition to compensationfor utilized resource. PBC is a fundamentally differentconcept in that the vendor is compensated based onthe realized performance outcome (e.g., aircraft uptime)instead of the amount of resources utilized for repairs.A typical PBC specifies the rate at which a vendorwill be paid per unit of time the product is functionalor, equivalently, the rate at which the vendor will bepenalized for each unit of product downtime.
Formally, a contract C is a vector of real-valuedparameters that specifies the total transfer paymentmade by the buyer to the vendor. Depending onwhether or not the inventory s is verifiable, C mayinclude s, in addition to other contract parameters. RBCis characterized by the fixed fee w ≥ 0 and the warranty
coverage �, which assumes a value between 0 and1 that represents the buyer’s share of the repair costincurred after a product failure. Thus, �= 0 denotesfull warranty coverage (i.e., the vendor bears the entirerepair cost), whereas �= 1 represents no warranty (i.e.,the buyer is responsible for the entire cost). RBC is thenrepresented as either C = 4w1�1 s5 or C = 4w1�5, and theexpected transfer payment is equal to T 4C5=w+��M .On the other hand, PBC is characterized by the fixedfee w ≥ 0 and the performance penalty v ≥ 0 chargedto the vendor for each unit of product downtime.Therefore, PBC is represented as either C = 4w1v1 s5 orC = 4w1v5, and the expected transfer payment is equalto T 4C5=w− vE6B � s703
With the transfer payments defined as above we canwrite the buyer’s expected payoff as
U4C5= −T 4C5−�E6B � s7−�r3
similarly, the vendor’s expected payoff is given by
V 4C5= T 4C5− cs −�M0
3.5. Signaling GameConsistent with the majority of signaling models foundin the literature, we assume that the vendor, whopossesses private information about the reliability of anewly developed product, chooses contract terms andproposes them to the buyer. The buyer then acceptsthe contract terms if his expected payoff exceeds anoutside option valued at �. In §§5 and 6 we presentour analysis under the premise that the contract typeis exogenously given (either RBC or PBC). This isan assumption we relax in §7.1, where we considerendogenous contract choice.
We seek a perfect Bayesian equilibrium (PBE) thatis also renegotiation-proof. When a PBE is inefficient(e.g., second-best) it is potentially vulnerable to Pareto-improving renegotiations after it is signed but beforeit is executed; for such cases we adopt the signaling-with-renegotiation framework developed by Beaudryand Poitevin (1993). As is common in signaling games,we are confronted with the issue of multiple equilibria,which hinders the ability to make sharp predictionsabout the equilibrium outcome. To circumvent thismultiplicity problem, we also require the candidatePBEs to satisfy the extended divinity (XD) criterion.This refinement captures the notions behind boththe divinity criterion (Banks and Sobel 1987) andthe intuitive criterion (Cho and Kreps 1987), suitably
3 Notice that neither contract penalizes the vendor based on thenumber of product failures. Contracts that penalize the vendorbased on the number of failures are rarely observed in practice, asvendor firms may invoke force majeure claiming that random failuresare beyond their control. Both RBC and PBC provide contractualremedies for post-failure damages rather than preventive protections.
adapted to a multiperiod treatment to accommodaterenegotiation-proofness (see Beaudry and Poitevin 1993for more details). As it turns out, the XD criterionidentifies a set of equilibria that are payoff equivalent;i.e., the surviving equilibria are unique with respect tothe payoff, even though the combinations of contractterms that lead to the outcomes may differ. With theequilibrium identification strategy set, we outline thesequence of events as follows.
Step 1. Nature reveals to the vendor her type � ∈
8L1H9.Step 2. The vendor offers contract terms (possibly
type contingent) to the buyer.Step 3. The buyer updates his beliefs about the ven-
dor’s type and accepts or rejects the contract. If thebuyer rejects the contract, the game ends. If he acceptsand the resulting outcome is efficient, the negotiationprocess is terminated (since Pareto-improving rene-gotiations do not exist) and we proceed directly toStep 5.
Step 4. The outcome in Step 3 is renegotiated as perBeaudry and Poitevin (1993).
Step 5. The vendor decides the inventory of sparesto be maintained (if not specified in the contract);products are deployed, failures occur, and repair andmaintenance takes place; transfer payment is madeby the buyer and final payoffs are realized by bothplayers.
4. Benchmark: First-Best UnderComplete Information
We first establish the first-best benchmark against whichthe performances of different contractual agreementsare to be compared. The first-best treatment requiresthat all attributes, decisions, and actions are completelyobservable and verifiable to both parties. Under such acondition, it is sufficient to devise a contract consistingonly of the fixed fee w and the inventory s; no warranty(as under RBC) or performance penalty (as underPBC) term is needed. Given the vendor type � ∈
8L1H9, the first-best outcome is obtained by solving theoptimization problem
maxw1s≥0
V� =w− cs −��M1
subject to U� = −w−�E�4B � s5−��r ≥ �0(IR)
The solution is summarized in the following prop-osition.
Proposition 1. When the vendor’s type � ∈ 8L1H9 andinventory are verifiable, the optimal contract specifies thefirst-best contract parameters s� = F −1
� 41 − c/�5 and w� =
−�E�4B � s�5−��r −�. Furthermore, the reliable vendorachieves a higher payoff and maintains a lower inventorythan the unreliable vendor: VH > VL; sH < sL.
All proofs are provided in the appendix. The re-sult VH > VL confirms our intuition that the reliablevendor is better off than the unreliable vendor when noinformation asymmetry exists. The next result, sH < sL,states that the optimal inventory of the reliable ven-dor is lower than that of the unreliable vendor. Thisis a direct consequence of the substitutable relation-ship between reliability and inventory: the higher thereliability, the less frequently product failures occur,and therefore the required level of spares inventory islower. As we discuss in the subsequent sections, thisinherent dependence between reliability and inventoryhas significant implications for signaling effectivenesswhen information is asymmetric.
5. Signaling with Verifiable InventoryWe now consider the setting in which the vendorpossesses private information about product reliability.Note that the reliable vendor’s lower inventory andhigher payoff in the benchmark case together imply thatthe first-best contracts are not incentive compatible; i.e.,the unreliable vendor has an incentive to misrepresenther type. We first examine the case in which thevendor’s inventory choice can be verified and henceincluded as part of the contract terms. This analysisallows us to isolate the effect of signaling under RBCand PBC and establish a baseline for the next casepresented in §6: a more complex scenario in which thevendor makes a discretionary choice of inventory inaddition to her signaling effort.
5.1. Resource-Based ContractWhen inventory is verifiable the vendor includes it inher RBC contract along with other parameters. Thus,the contract specifies the inventory s� , the fixed fee w� ,and the warranty coverage �� ∈ 60117 that defines thebuyer’s share of the repair cost. These terms dependon the vendor type � ∈ 8L1H9. Under the assumptionslaid out in §3, the payoff of the vendor of type � isequal to V� =w� − 41 −��5��M − cs� , and the payoffof the buyer, when the vendor is of type � , is U� =
−w� −����M −�E�4B � s�5−��r .Since the warranty coverage offers to protect the
buyer from unanticipated out-of-pocket expensesincurred after a product failure, the value of �� set bythe vendor in her contract may relay information aboutwhether or not the product is reliable. The question isCan the reliable vendor credibly signal her type usingthe warranty term included in RBC? The followingproposition answers this question. Throughout thepaper we use the superscript asterisk (∗) to denote theequilibrium outcomes.
Proposition 2. When inventory is verifiable, the PBEsof a signaling game under RBC that satisfy the conditionsin §305 are payoff equivalent, separating, and efficient. In
each equilibrium, sL = s∗L > s∗
H = sH , w∗L <w∗
H < wH , and�∗L ≥ �∗
H = 0. As a result, the unreliable vendor recovers herfirst-best payoff but the reliable vendor does not: VL = V ∗
L <V ∗
H < VH . The buyer is left with a positive expected rentabove his outside option �.
Although we find that multiple equilibria exist,they are all payoff equivalent; i.e., they only differin the amount of warranty and fixed fee offered bythe unreliable vendor without altering the net payoff.In all equilibria the reliable vendor signals her type(i.e., they are separating equilibria) by setting a lowerinventory than the unreliable type (s∗
L > s∗H ) and offering
a full warranty (�∗H = 0) in return for a higher (but
discounted from first-best) fixed fee (w∗L <w∗
H < wH ).This ability to differentiate herself from the unreliabletype comes at a cost, as she fails to earn the sameamount of payoff that she would have earned underthe first-best condition (V ∗
H < VH ). It is worth notingthat this reduction in payoff for the reliable vendordoes not in fact result in system inefficiency; at thesupply chain level no distortion exists, as evidenced bythe same inventory investments as under the first-best(s∗
L = sL, s∗H = sH ). Instead the reduction in payoff of the
vendor is captured entirely by the buyer.Two aspects in Proposition 2 warrant further exami-
nation. First, to understand how RBC allows the vendorto signal, it is instructive to examine the trade-offsthat the reliable vendor faces. She has a dual objec-tive: maximize her payoff while signaling her typeby differentiating herself from the unreliable type.Maximizing profit can be further decomposed into“enlarging the pie size” and “dividing the pie.” Amongthe three levers at the vendor’s disposal—the inventorysH , warranty coverage �H , and fixed fee wH—only theinventory has an impact on the pie size; i.e., on sup-ply chain efficiency. This is because higher inventoryincreases product availability and hence determinessystem performance. The fixed fee and warranty do notimpact efficiency since they represent transfers withinthe supply chain. The former is used by the vendor toextract rents from the buyer. The reason the warrantyterm is used as a primary signaling device is becausethe marginal impact of increasing warranty differsbetween the two vendor types, whereas the marginalimpact of altering the fixed fee or inventory is exactlythe same for the two. Specifically, since high warrantycoverage requires a vendor to internalize more repaircosts, it is more costly for the unreliable vendor tomatch the same coverage offered by the reliable vendor,who enjoys less frequent product failures.
A second feature of the equilibrium outcome withRBC is that it leaves the buyer with surplus withoutsacrificing efficiency. This feature distinguishes thisequilibrium from the classic second-best separatingequilibrium—for example, in the job market signaling
analysis (Spence 1973)—in which economic value isdestroyed without leaving any surplus to the agent.The reason for this distinction is that, unlike Spence’sexample, in which the employee’s education does notimpact the employer’s payoff, the warranty offeredunder RBC directly increases the buyer’s payoff. Thus,the vendor’s investment in the signal, represented byhigher warranty coverage, is captured by the buyer.Moreover, the requirement that the equilibrium contractshould be immune to Pareto-improving renegotiationspushes the outcome toward efficiency.
In summary, successful signaling by the reliablevendor is possible under RBC when her inventorychoice is verifiable, but it comes at a cost since thevendor has to leave positive surplus to the buyer.
5.2. Performance-Based ContractWith inventory verifiable, a PBC contract specifiesthe inventory s� , fixed fee w� , and penalty rate v�
applied to each unit of downtime (or equivalently,backorder). The contract terms may differ by vendortype � ∈ 8L1H9. The payoff for the vendor of type � isV� =w� − v�E�4B � s�5− cs� −��M , while the payoff forthe buyer is U� = −w� + 4v� −�5E�4B � s�5−��r0 Thenext result parallels Proposition 2 obtained for the RBCcase.
Proposition 3. When inventory is verifiable, the PBEsof a signaling game under PBC that satisfy the conditions in§305 are payoff-equivalent, separating, and efficient. In eachequilibrium, sL = s∗
L > s∗H = sH and w∗
L <w∗H . The penalty
rates v∗H and v∗
L satisfy the relation v∗L6EL4B � sL5− EH 4B �
sL57≤ �6EL4B � sL5−EH 4B � sH 57+ 4�L−�H 5r+c4sL− sH 5≤
v∗H 6EL4B � sH 5− EH 4B � sH 570 Moreover, both vendor types
recover their first-best payoffs: VL = V ∗L <V ∗
H = VH . Thebuyer is left with his outside option �.
Recall that the warranty term �� served as theprimary signaling device under RBC. Under PBC, thesame role is played by the penalty term v� . Similarto the RBC case, the multiple equilibria identified inProposition 3 for the PBC case differ only in theirfixed fee/penalty combinations and do not affect thevendor’s net payoff. Also similar to the RBC case is theresult that the first-best inventory levels are maintained(s∗
L = sL1 s∗H = sH ).
Despite the similarities, there are a few significantdepartures from the RBC case. First, unlike under RBC,the reliable vendor under PBC signals her type withoutceding rents to the buyer. Instead, she retains the entiresurplus and attains her first-best payoff (V ∗
H = VH ).Second, despite the seemingly conflicting goals ofsuccessfully signaling her type while maximizing profit,the reliable vendor is able to achieve separation fromthe unreliable vendor with a wide range of penalty rates.This is evidenced by the condition for v∗
L and v∗H given
in Proposition 3, which imposes loose constraints on the
two values, even including the case in which the reliablevendor subjects herself to a lower penalty rate than theunreliable type would.4 Combined, these observationssuggest that, from the vendor’s perspective, PBC issuperior to RBC as a signaling device since it permitsthe reliable vendor to retain all of her rents.
What is different about PBC that makes it a superiorsignaling mechanism? To answer this, it is important torecognize that the types of risk that are shared betweenthe vendor and the buyer under RBC and PBC arenot the same. Under RBC, it is the repair risk that isshared through the warranty coverage; the vendorperforms a costly repair each time a random productfailure occurs, and the contract specifies how muchof this cost is reimbursed by the buyer. By contrast,under PBC, it is the outage risk that is shared throughthe penalty term; the buyer incurs a cost proportionalto the outage duration after each failure (e.g., lostrevenue), and the contract specifies to what degree thiscost is compensated for by the vendor. Whereas therepair risk is a function of reliability only, the outagerisk is a function of reliability as well as inventorychoice, since higher inventory lowers the expectedoutage duration. The limitation of RBC then becomesclear: even if warranty is used to its maximum extent(i.e., full coverage is offered) it still exposes the buyerto the risk of prolonged outage because the warrantydoes not cover this risk. PBC relaxes this constraintbecause it ties the vendor’s compensation directly to theoutage risk that the buyer cares about. This risk-sharingarrangement allows the vendor to signal her type moreefficiently. As noted in §2, PBC is known to align theincentives better than RBC when efficiency loss due todecentralization is caused by private actions. In thecontext of our problem, in which private informationplays the central role, the same characteristic of PBCmanifests itself as a signaling advantage.
Therefore, successful signaling by the reliable vendorcan be done under both RBC and PBC when theinventory is verifiable. However, PBC presents anadvantage because its risk sharing mechanism is moreamenable to signaling than that of RBC, thus allowingthe reliable vendor to extract all buyer surplus withoutcompromising efficiency.
6. Signaling with UnverifiableInventory
In this section we relax the assumption that inven-tory is verifiable. We examine how the two sources of
4 The reason for such a wide range of permissible penalty rates isthat, under PBC with verifiable inventory, the incentive compatibilityconstraints required for separation are not binding in equilibrium,even though the first-best payoffs are attained. This is one of the threequalitatively different equilibrium outcomes possible in signalinggames, the other two being the second-best separating equilibriumand the pooling equilibrium (Spence 2002).
inefficiency (absence of verifiability and private infor-mation) interact with each other and whether RBC andPBC lead to qualitatively different outcomes. In thissetting, inventory cannot be included in a contract, andtherefore the vendor cannot use it to relay informationabout product reliability. As a result, the vendor ismore limited in her ability to signal, but at the sametime she has more freedom to set the inventory to alevel that would improve her payoff. The vendor isthen faced with the challenge of balancing the potentialbenefit of payoff increase with the potential downsideof signaling ineffectiveness. We study these issues inthis section.
6.1. Resource-Based ContractEven with inventory no longer verifiable, the pay-off functions for the vendor and the buyer underRBC remain the same as those of the verifiable inven-tory case. The difference now is that the vendor oftype � ∈ 8L1H9 makes a discretionary choice about theinventory s� that will indirectly influence the choices ofthe other two variables that are included in the RBCcontract: the fixed fee w� and the warranty coverage �� .We summarize the equilibrium of the signaling gamefor this case in the next proposition.
Proposition 4. When inventory is unverifiable, thePBE of the signaling game with RBC, which satisfy theconditions in §305, are payoff equivalent and inefficient,with s∗
L = s∗H = 0, �∗
H = 0 and w∗H =w∗
L +�∗L�LM . The PBE
can be either pooling or separating, but the reliable vendorcannot recover her first-best payoff: V ∗
H < VH . The buyer’spayoff is equal to his outside option �.
Recall from Proposition 2 that when inventory isverifiable and hence is included in a contract, thereliable vendor using RBC succeeds in signaling hertype by offering full warranty coverage combinedwith a discounted fixed fee, maintaining the first-bestinventory. This characterization stands in direct contrastto that of the equilibrium identified in Proposition 4.With contracting on inventory no longer an option,even though a separation from the unreliable vendorcan be achieved, separation does not bring any benefitover the pooling equilibrium, in which both types ofvendor offer full warranty (�∗
L =�∗H = 0) and set the
inventory level to zero (s∗L = s∗
H = 0). In other words,none of the combinations of contractual levers at herdisposal allow the vendor to generate a payoff greaterthan that of the pooling outcome.5
5 Maintaining zero inventory is a by-product of the simplifyingassumptions of our model that allow us to focus on the maintrade-offs. In reality, the vendor may choose to keep some minimumacceptable level of inventory s > 0 for reasons that are not capturedin the model, such as the reputation effect. Our analysis indicatesthat, as long as this minimum level s is no greater than a thresholds∗ < sH , none of the qualitative insights of this paper are impacted.
Intuitively, this happens because the vendor’s inabil-ity to make inventory verifiable removes any benefitthat it might have brought the vendor. Inventory iscostly to keep, so it is best for the vendor to minimizeits level. Given that inventory cannot be used as asignaling device and that RBC does not incentivizethe vendor to mitigate the outage risk with inven-tory, setting s� to its minimum value is indeed theoptimal course of action for the vendor. The vendorthen adjusts other contract parameters (w� and �� )to accommodate this decision, thereby shifting thepriority from signaling to cost savings. It turns out thatthe vendor maximizes her payoff either in a poolingequilibrium with full warranty or in a payoff-equivalentseparating equilibrium. Furthermore, any pooling con-tract comprising a partial warranty is not offered inequilibrium, because the reliable vendor always doesbetter by deviating to an alternate contract (whichmay itself not survive in equilibrium) with slightlyhigher warranty coupled with a slightly higher fixedfee, which the unreliable vendor finds too costly tomimic.
In summary, the vendor who adopts RBC is able tosignal product reliability without affecting supply chainefficiency only if the inventory is verifiable. If not, thevendor’s discretionary inventory choice exacerbates theproblem of incentive misalignment that already existswith the signaling challenge, leading to an outcomewhere the vendor is not better off than in a poolingequilibrium with an underinvestment in inventory thatcauses economic inefficiency.
6.2. Performance-Based ContractThe players’ payoff functions under PBC remain thesame as those of the verifiable inventory case. Withinventory no longer included in the contract, however,the vendor’s discretionary choice of s� will influenceher choices for the remaining contract terms: the fixedfee w� , and the performance penalty rate v� .
Recall from §5.2 that when inventory is verifiableand is included in the contract, PBC allows the reliablevendor to successfully signal her type and achieveeconomic efficiency. A key reason why this happens isthat the penalty rate vH offered by the reliable vendoris used only as a signaling device and serves no otherpurpose. Once inventory is not verifiable, however, vH
assumes another role in addition to that of signaling:providing a financial incentive to hold inventory. Thus,when inventory is unverifiable, the penalty rate servesa dual role. On one hand, the reliable vendor prefers tosignal by setting a high penalty rate in order to determimicking by the unreliable type. On the other hand,her temptation to forgo investment in noncontractibleinventory exerts a downward pressure on the penaltyrate. Our next result reveals the equilibrium outcomethat emerges from this trade-off.
� Backorder cost $2,000,000 per month Estimated based on revenue per aircraft figures reported byairlinefinancials.com: an independent airline industry consulting firm.
c Cost per spare $70,000 per month Estimated based on engine lease-rental rates (Canaday 2010) and cost of aspare engine (Adler et al. 2009).
1/�H Expected time between failures for H-type vendor 0.5–5 years Within range reported in Guajardo et al. (2012).1/�L Expected time between failures for L-type vendor 0.5 years Within range reported in Guajardo et al. (2012).r Inconvenience cost $175,000 Engine replacement time ∼ two days; linear extrapolation at rate of
$61 per minute of delay (Ramdas et al. 2013).N Number of engines 150 Order size for large airline (Kim et al. 2007).M Cost per unscheduled maintenance $800,000 Estimated based on industry expert input and consistent with figures
reported in Hopper (1998).l Average repair lead time for engine overhaul 3 months Based on numbers in Adamides et al. (2004) (accounting for
operational improvements).Fi 4x5 Failure distribution, i ∈ 8L1H9 — Normal approximation to Poisson distribution.p Buyer’s prior belief that the vendor is L-type 50%
Proposition 5. When inventory is unverifiable, thePBE of the signaling game under PBC that satisfies theconditions in §305 is unique, separating, and inefficient. Inthis equilibrium, sL = s∗
L1 s∗H > sH , w∗
L <w∗H , and � = v∗
L <v∗H . As a result, it is not possible for the reliable vendor to
recover her first-best payoff: V ∗H < VH . The buyer’s payoff is
equal to his outside option �.
As the proposition describes, the resulting equilib-rium is separating but fails to achieve efficiency; thereliable vendor destroys economic value in order to sig-nal her type. Moreover, the reliable vendor overinvestsin inventory in equilibrium (s∗
H > sH ). This is in contrastto the verifiable inventory case, in which we found thatthe reliable vendor is able to achieve both separationand efficiency with the first-best inventory under PBC.Therefore, when inventory cannot be contracted upon,the two competing forces exerted upon the penalty ratevH do not prevent the reliable vendor from signalingher type: she eventually does so at the expense of theincreased cost of maintaining inventory. Note also thatoverinvestment in inventory is exactly the oppositeoutcome of the equilibrium under RBC, where it wasfound that underinvestment arises.
To understand why overinvestment arises in equilib-rium, we revisit the observation that PBC supportssignaling through the degree of outage risk shared bythe vendor. Note that separation cannot be achieved byagreeing to a penalty that induces first-best inventoryinvestment (i.e., v∗
H = �), since the first-best outcomedescribed in Proposition 1 is not incentive compatible.Similarly, any attempt to separate with a lower penaltywill also not be incentive compatible—the unreliablevendor will find it even easier to mimic. Therefore,the only option remaining for the reliable vendor is toincrease the penalty rate sufficiently to deter mimickingby the unreliable vendor.
A direct consequence of this overinvestment in inven-tory is that the payoff appropriated by the reliable
vendor is less than the first-best level. However, this isnot the only factor at play. Another reason for this gapin payoff stems from the renegotiation-proof require-ment that forces the reliable vendor to surrender rentsto the unreliable vendor. Although this characteristicis typical of pooling equilibria in which the unreliablevendor benefits from the fact that the buyer cannotidentify her type, it also manifests itself in renegotiation-proof separating equilibria, as a mechanism to ensurethat no other Pareto-improving proposal exists afterthe contract is signed. This point is numerically illus-trated in Figures 1 and 2 using the parameter valuesin Table 2, which are typical in the civil aerospaceindustry. In this example, the inefficiency associatedwith overinvestment in inventory is substantial (seeFigure 1), but smaller than the overall loss of rentsthat asymmetric information inflicts on the reliablevendor (see Figure 2). The additional loss is due to therents surrendered to the unreliable vendor to ensurerenegotiation proofness. We refer interested readers toBeaudry and Poitevin (1993) for further details.
Overall, we find that the interaction between thevendor’s discretionary inventory choice and her sig-naling incentive creates subtle dynamics that impact
Figure 1 Increase in Reliable Vendor’s Inventory Cost Under PBC WhenInventory Is Unverifiable Compared to First-Best
Figure 2 Vendor’s Payoffs Under PBC When Inventory Is Unverifiable
�L �H
HHLL
Notes. Buyer’s revenues are normalized to zero. Parameter values shown inTable 2.
the relative efficiency of after-sales service contracts innontrivial ways. Although system efficiency cannotbe attained under either RBC or PBC, they shouldbe distinguished in that the equilibrium outcomesunder the two contract types are qualitatively different:whereas RBC leads to underinvestment in inventory,the opposite is true under PBC. Moreover, the buyerunder RBC may find himself unable to obtain informa-tion about product reliability (i.e., pooling equilibriacannot be ruled out) but the buyer under PBC receivesan unambiguous signal about reliability.
In closing, we ask the following question: Giventhat neither RBC nor PBC gives the reliable vendor aclear advantage when inventory cannot be included ina contract, which contract type will she prefer? Theanswer is given below.
Corollary 1. The reliable vendor’s equilibrium payoffunder PBC with unverifiable inventory (Proposition 5) isgreater than that under RBC with unverifiable inventory(Proposition 4).
Thus, the reliable vendor finds that the ability tosignal her type with PBC more than compensates forthe cost associated with overinvestment in inventoryand any benefit presented by RBC, which does notimprove upon a pooling outcome. Note that, althoughno strict ordering of payoffs such as in Corollary 1exists for the unreliable vendor, a similar conclusion ismade under reasonable parameter value combinations(such as those in Table 2): the unreliable vendor’spayoff is greater under PBC than under RBC, except inextreme cases (e.g., p → 0 and �L ��H ).
7. Extensions: Efficiency EnhancementStrategies
Thus far, our focus has been on the properties of theequilibrium outcomes that arise when the contract type(RBC or PBC) and verifiability of inventory have beenexogenously specified. We found that although systemefficiency is ensured when inventory is verifiable, nei-ther PBC nor RBC leads to efficiency when inventory
is unverifiable. Therefore, a natural course of actionfor reducing inefficiency is to invest in verifiability ofinventory. We investigate this option in the contextof either the vendor or the buyer making a choiceof contract type, in order to highlight how inventoryverifiability shapes their preferences. In case the ven-dor manages inventories across multiple buyers, analternative option for reducing inefficiency is to utilizeinventory pooling. In this section we investigate thesetwo efficiency enhancement strategies and discuss theirimplications.
7.1. Endogenizing Inventory VerifiabilityWe study an extended game in which either the vendoror the buyer is empowered to decide whether inventoryshould be made verifiable and select the contract typebefore actual contract terms are determined. Giventhat the vendor is the principal in our setting, it isnatural to consider the case in which she makes thesechoices. However, motivated by the practices in the civilaerospace sector, in addition to the vendor’s choice wealso consider the case in which the buyer wields somebargaining power (e.g., a large airline company) bydeciding the contract type and inventory verifiability.6
A key addition in this setting is that inventoryverifiability does not come for free. An irreversible fixedcost K is incurred by the vendor to make the inventorydecision observable by the buyer and verifiable by athird party. In practice, verifiability is often achievedby purchasing and implementing information systems;e.g., CD Aviation Services offers the SilverSky programthat enables customers to track the progress of theirrepair job with detailed status reports and picturesvia secure Internet access anytime and from anywherein the world (Adler et al. 2009, p. 18). Alternatively,inventory can be made verifiable by maintaining it onthe customer’s site (Adams 2008, Canaday 2010). Werefer to K as the “cost of investing in verifiability.” Tofocus on nontrivial cases, we make a mild assumptionthat the fixed cost is lower than the systemwide surplusgenerated, irrespective of the vendor type (i.e., K < VL).
Endogenizing inventory verifiability along with con-tract choice alters the sequence of events in the game.Note that, in principle, such a choice itself could revealinformation and it may therefore generate equilibriumoutcomes that are different from those we obtained inthe previous sections. The altered sequence of events isas follows.
1. Nature reveals to the vendor her reliability (type):� ∈ 8L1H9.
6 Our modeling assumption is similar to that in Aksin et al. (2008),who study a call center setting in which the contractor (principal)first offers the service provider (agent) a choice between two contracttypes, and then, depending upon the contract type chosen by theagent, the principal specifies the contract parameters.
2. Player i ∈ 8vendor, buyer9 announces whether ornot inventory should be made verifiable and chooses acontract type (RBC or PBC), but not the specific contractterms. Player i’s identity is exogenously specified.
3. The vendor incurs an irreversible fixed cost K ifverifiable inventory is chosen; if player i is the vendor,then the buyer updates his beliefs about the vendor’stype.
4. The vendor offers the buyer specific contract terms(possibly type contingent).
5. Based on the proposed contract terms, the buyerupdates his beliefs about the vendor’s reliability (type),and accepts or rejects the contract after a possiblyinfinite round of renegotiations.
6. Products are deployed, failures occur, and repairand maintenance takes place; transfer payment is madeby the buyer and final payoffs are realized by bothplayers.
The analysis of this extended game builds on theresults obtained in the previous two sections. The keyresult is summarized below.
Proposition 6. In the game featuring choices on inven-tory verifiability and contract type:
(a) Suppose that it is the vendor who makes the choices.A threshold value Kv > 0 exists, such that if K > Kv,both vendor types select PBC and choose not to investin verifiability. The resulting equilibrium is separating asdescribed in Proposition 5. If K <Kv, on the other hand,the reliable vendor signals her type by choosing to investin verifiability coupled with either RBC or PBC, whereasthe unreliable vendor signals by choosing not to invest inverifiability in conjunction with PBC. In any of these cases,the buyer’s payoff is equal to his outside option �.
(b) Suppose that it is the buyer who makes the choices.The buyer then opts for RBC with verifiable inventory, andthe payoffs are determined as per the separating equilibriumin Proposition 2, except that the vendor’s payoff is reducedby K.
The proposition shows that, despite the efficiencygain that inventory verifiability brings, it may creatediverging preferences between the vendor and thebuyer about contract types: the vendor prefers PBC,whereas the buyer prefers RBC. From the vendor’sperspective, PBC dominates RBC because the formerallows the reliable vendor to signal her type moreefficiently. Moreover, endowing the vendor with theability to choose inventory verifiability and the contracttype gives her an extra lever, besides contract parame-ters, with which she can signal her reliability. This isbecause the verifiability investment acts as a crediblesignal of reliability when the cost of verifiability Kis relatively low compared to the value lost becauseof inventory overinvestment, which we identified in§6. In contrast to the vendor who prefers PBC, thebuyer prefers RBC because the combination of RBC
with verifiable inventory allows the buyer to retainpositive rents in equilibrium, whereas PBC does not,as we found in §5.
Our predictions are consistent with what is observedin practice. As has been reported by GAO (2004) andothers, when product technology is new and infor-mation on product reliability is not readily available,buyer organizations tend to opt for RBC over PBC.Moreover, the above finding also matches with thegrowing trend in the aerospace industry where manyvendor organizations (e.g., Rolls-Royce, InternationalAero Engines, etc.) are moving toward offering PBCexclusively (Adler et al. 2009). Although there arefactors other than signaling that may explain theseobservations, our analysis provides theoretical supportfor these industry practices when reliability signalingis a significant contributor to the firms’ decisions.
7.2. Pooling Inventory Across Multiple BuyersWhen inventory verifiability investment is prohibitivelyexpensive but the vendor transacts with multiple buy-ers, she may consider adopting the inventory poolingstrategy. Indeed, this practice is widely used amongvendors of products with mature technology. Althoughpooling is relevant for both RBC and PBC in principle,in our model context its impact under RBC is limitedbecause, as demonstrated in §6.1, the vendor has aminimal incentive to maintain a high level of inventoryunder RBC. On the other hand, pooling effect maybe magnified under PBC because it encourages inven-tory overinvestment. For this reason, we restrict ourattention to PBC in the following discussion.7
We modify our model to illustrate the key trade-offs associated with inventory pooling by makingtwo simplifying assumptions that enable analyticaltractability. First, we assume that the vendor tradessimultaneously with k identical buyers, each purchasingN product units. Hence, the total sales for the vendorare equal to Q = kN units. The number k is given andis common knowledge. The sequence of events andequilibrium identification strategy remain the sameas in §3.5. Since all buyers are identical, we focus onsymmetric equilibria in which all buyers receive exactlythe same (potentially type-dependent) contract terms.Second, to quantify the benefits of pooling, we focus
7 In addition, we do not study the pooling strategy when inventoryis verifiable because the typical means for achieving verifiability, i.e.,maintaining inventory on the customer’s site, makes it impracticalto pool inventory across buyers. Besides, when reliability is notcommon knowledge and inventory is contractible, pooling wouldrequire intricate considerations around multiparty contracting andcontentious inventory sharing rules. The above factors effectivelymake pooling infeasible, as is reflected in the aerospace industrywhere, for engines with new technology, buyers opt for tangibleon-site assets (dedicated spares), as opposed to participating in apool (Canaday 2010).
Figure 3 The Value of Pooling: Difference in the Reliable Vendor’s Payoffs (Monthly) When Pooling Inventory Across k ∈ 811213149 Buyers
10
100
200
300
400
Payo
ff d
iffe
renc
e($
thou
sand
per
mon
th) Difference between k = 2 and k =1
Difference between k = 3 and k = 2
Difference between k = 4 and k = 3
2 3 4 5
�L/�H
6 7 8 9 10
–6
–2
2
6
10
Payo
ff d
iffe
renc
e($
thou
sand
per
mon
th)
2 3 4 5
�L/�H
1 6 7 8 9 10
Notes. Parameters shown in Table 1. Left-hand (right-hand) panel represents the expensive (inexpensive) backorders case with � = $2,000,000 (� = $72,500).
on a scenario in which product failures for each buyeroccur as a Poisson process with the common failurerate �� . To facilitate the game-theoretic analysis of ourpaper, we approximate the Poisson distribution withthe normal distribution.
As contract offers are made to each of the k buyerssimultaneously and independently, the dynamics ofthe signaling game are unaffected by the presence ofmultiple buyers. The only departure from the analy-sis of the previous sections is a possible cost savingthrough inventory pooling and its impact on signalingeffectiveness. The question naturally arises, Is poolingalways beneficial from the vendor’s perspective? Weaddress this question with our next result.
Proposition 7. When trading with multiple identicalbuyers, the reliable vendor is better off by pooling inventoryif and only if
VH −V ∗
H >ã1 (3)
where V ∗H is the equilibrium payoff for the reliable vendor
facing a single buyer (specified in Proposition 5), VH is thecorresponding first-best payoff (specified in Proposition 1),and ã= p4�L −�H 5r + 2c�H l−�EH 4B � sH 5.
The proposition shows that when inventory is unver-ifiable, inventory pooling is beneficial for the reliablevendor if and only if the efficiency gap from contractingwith an individual buyer is greater than a thresholdamount; otherwise, the reliable vendor ends up worseoff by pooling inventory across multiple buyers. Tounderstand the intuition behind this result, we need toexamine the two competing effects inventory poolinghas on payoffs. The first effect, which benefits both thereliable vendor and the unreliable vendor, comes fromthe fact that a larger pool of products brings morepredictability of unplanned failure events. The costsaving that follows is inherently lower for the reliablevendor than for the unreliable vendor because thevariability that the former faces is smaller to begin with.The second effect, which benefits the unreliable vendorat the expense of the reliable vendor, is due to thechange in signaling costs caused by inventory pooling.
Note that the unreliable vendor will find it relativelycheaper to pretend she is reliable if she pools inventoryacross multiple buyers: the larger pool lowers her costof matching the high penalty rate that the reliablevendor proposes. This in turn makes it more difficultfor the reliable vendor to enjoy the benefit of pooling,because lowering the penalty rate to take advantage ofsuch a benefit exposes her to possible mimicking by theunreliable vendor. Thus, pooling will actually increasethe cost of signaling for the reliable type. Which ofthese two effects of pooling—uncertainty reductionand more costly signaling—prevails depends on thesize of the efficiency gap as specified in (3).
The value of pooling is demonstrated in Figure 3.Pooling is beneficial when inventory is relatively inex-pensive compared to the cost of backorders (left-handpanel in Figure 3), i.e., c/� and hence ã in (3) is small.(Furthermore, as also evident from Figure 3, poolingexhibits the usual diminishing returns to scale withrespect to the number of buyers, k.) Nevertheless, asthe problem of asymmetric information becomes morepronounced, this benefit declines and can even becomenegative; e.g., when inventory is relatively expensivecompared to the cost of backorders (right-hand panel inFigure 3). Hence, we conclude that although inventorypooling is generally beneficial to the vendor, there aresituations in which pooling backfires. This observationserves as a reminder that the unique challenges arisingfrom asymmetric information about new product relia-bility require actions that may deviate from the usualefficiency enhancement strategies.
8. ConclusionsIn this paper we study how two widely used after-sales service contracts, resource-based contracts andperformance-based contracts, can be used to signalreliability of a newly developed product. Our analysisuncovers new dynamics that arise from the interactionbetween the vendor’s desire to signal reliability usingcontract terms, and her incentive to minimize costs(equivalently maximize profit) through investment inspares inventory.
We find that RBC and PBC differ significantly intheir ability to signal reliability, and the transparencyof inventory plays a crucial role. When inventory isverifiable, and thus signaling private information aboutnew product reliability is considered in isolation, bothRBC and PBC can be used as an efficient signalingmechanism. However, one difference exists betweenthe two contracts: it is possible for the vendor to fullyextract rent from the buyer under PBC, whereas thesame is not true under RBC. Thus, from the vendor’sperspective, PBC turns out to be a more effective sig-naling mechanism than RBC. The driver behind thisoutcome is the structural difference between the twocontract types: RBC allows signaling via the vendor’sshare of the repair risk (determined entirely by relia-bility), whereas PBC permits signaling through thevendor’s share of the outage risk (determined jointly byreliability and inventory choice). Despite the differences,both RBC and PBC give rise to efficient separatingequilibria.
The property of efficient separating equilibria nolonger holds when the vendors inventory choice isdiscretionary, because this private action interacts withher ability to signal private information about reliability.In this setting, PBC creates an incentive for the vendorto overinvest in spares inventory; a large penalty rateassociated with prolonged product outage is offered tothe buyer so as to assure him of high product reliability,but as a consequence, the vendor needs ample amountof spares in order to increase the product uptime andreduce the impact of heavy penalties that may follow.This is in contrast to what happens with RBC, underwhich the vendor is provided with little incentiveto carry spares inventory because the contract termsdo not hold the vendor accountable for prolongedoutages. The net outcome is that neither contractresults in efficiency when inventory is unverifiable.Whereas PBC puts a disproportionate emphasis onreliability signaling over inventory savings, RBC doesthe opposite. Nevertheless, we find that the vendor’spayoffs are in general higher under PBC rather thanRBC, suggesting that in this case too PBC is a moreeffective signaling mechanism that RBC.
Our analysis suggests that the key to bring efficiencyto the supply chain is inventory transparency; by mak-ing inventory verifiable, contracts can be structuredsuch that incentive distortions disappear and effec-tive signaling is enabled. However, this improvementoption may create diverging preferences for contracts.Specifically, when inventory is made verifiable the ven-dor prefers PBC since it allows her to extract more rentsfrom the buyer, whereas the buyer prefers RBC sinceit leaves him with a positive surplus. This finding isconsistent with reports of aerospace industry practices,which indicate that buyer organizations typically optfor RBC when they acquire newly developed products.
An alternative way to bring efficiency is to utilizeinventory pooling, in case the vendor manages inven-tories across multiple buyers. Although the benefit ofpooling is well known, we demonstrate that this strat-egy can backfire in our setting: it may hinder reliabilitysignaling and exacerbate inefficiency. This happensbecause the pooling effect benefits the unreliable ven-dor disproportionately, thus making it easier for her tomisrepresent her product reliability. Therefore, inven-tory pooling should be adopted with caution whenfirms operate in an environment in which products arenewly developed.
Our analysis adds a new dimension to the theoryand practice of after-sales product support contracting.By shifting the focus from the management of matureproducts to the management of newly developedproducts, we identify a new set of challenges thathave been overlooked in the existing literature. Withthe after-sales service business continually growingand new products being introduced at faster rates, webelieve that the managerial insights generated fromour analysis not only serve as useful guidelines forpractitioners, but open up new possibilities for futurework.
Appendix. ProofsBefore proceeding with the proofs, we find it useful to firstprove the following lemma.
Lemma 1. For any value of u≥ c, when OL and OH satisfy thehazard rate ordering of (1), then �4u5 2= u6EH 4B � sH 4u55− EL4B �
arg maxs6−uE� 4B � s5− cs7= F −1� 41 − c/u50 Furthermore, when
OL and OH also satisfy the EW order of (2), then �4u5 isnonincreasing.
Proof of Lemma 1. Since OL and OH follow a hazardrate ordering that implies first-order stochastic dominance(FOSD), for any u> 0, we therefore have sH 4u5≤ sL4u5. Fur-thermore, �4u5= u6EH 4B � sH 4u55− EL4B � sL4u557+ c4sH 4u5−sL4u55 = u
∫ sL4u5
sH 4u541 − FH 4x55dx + u∫ �
sL4u54FL4x5 − FH 4x55dx +
c4sH 4u5− sL4u55 < u∫ sL4u5
sH 4u541 − FH 4x55dx + c4sH 4u5− sL4u55 ≤
u41 − FH 4sH 4u5554sL4u5− sH 4u55+ c4sH 4u5− sL4u55= 0, whereto get to the second line we use the stochastic dominanceproperty, which implies that
∫ �
sL4FL4x5− FH 4x55 dx < 0. For the
third line we use the fact that 1 − FH 4x5 is a nonincreasingfunction; therefore, the area under this curve in the interval4sH 4u51 sL4u55 is no greater than 41 − FH 4sH 4u5554sL4u5− sH 4u55.Finally, we substitute sH 4u5 to show that the last line iszero. Now assume that OL and OH follow the EW order.Given the definitions of sH 4u5 and sL4u5, it follows that∫ �
sH 4u541 − FH 4x55dx ≤∫ �
sL4u541 − FL4x55dx, or EH 4B � sH 4u55 ≤
EL4B � sL4u550 Taking the derivative of �4u5 with respect to ugives �′4u5= 6EH 4B � sH 4u55− EL4B � sL4u557≤ 0. �
Proof of Proposition 1. The objective function is increas-ing in w� ; therefore, the (IR) constraint must be binding:w� = −4�E�4B � s�5+��r + �5. Using this and maximizingthe objective with respect to inventory s� , we obtain s� =
F −1� 41− c/�5 when � ≥ c and zero otherwise. The solution
is unique as the objective is concave in s� . From (1), whichimplies FOSD, we conclude that sL > sH . The vendor’s pay-off is V� = −�−�E�4B � s�5− cs� −��4r +M5. Furthermore,VL − VH = �6EH 4B � sH 5−EL4B � sL57+ c4sH − sL5+ 4�H −�L54r +
M5=�4�5+ 4�H −�L54r +M51 using Lemma 1 for u= � and�H <�L, we conclude that VL − VH < 0. �
Proof of Proposition 2. In this proof we adopt theclassic one-shot signaling treatment (without Step 4 in §3.5)to determine the equilibrium outcome, as opposed to thesignaling-with-renegotiation framework proposed by Beaudryand Poitevin (1993). This is because, as per the sequenceof events in §3.5, the accepted proposal in Step 3 turnsout to be efficient and, therefore, renegotiation-proof. Ina separating equilibrium, if it exists, the vendor is able tocredibly signal her type. Given that the L-type vendor hascredibly communicated her type to the buyer, the best that shecan do is to extract all the surplus by solving the followingproblem: maxwL1 sL1�L≥06wL − 41 −�L5�LM − csL7, subject to(IRL52 −wL−�L�LM−�EL4B � sL5−�Lr ≥ �. The constraint willbe binding at optimum (otherwise the vendor can increasethe fixed fee wL and improve her payoff); therefore, w∗
L =
−�L�LM −�EL4B � sL5−�Lr − �, and the objective becomesmaxsL1�L≥06−�EL4B � sL5−�L4r +M5− �− csL7. This functiondoes not depend on �L and is concave in sL. Therefore, theoptimal inventory to keep is equal to the first-best level, i.e.,s∗L = sL. The unreliable vendor makes her first-best payoff, VL.
Now consider the problem of the H -type vendor. If thereexist no consistent separating deviations as per the “intuitivecriterion,” then the H -type’s contract must solve the followingproblem maxwH 1 sH 1�H≥06wH − 41 −�H 5�HM − csH 71 subject to(IRH 52 −wH −�H�HM −�EH 4B � sH 5−�H r ≥ �, (ICL52 VL ≥
wH − 41 − �H 5�LM − csH . Furthermore, for a separatingequilibrium the H-type’s contract needs to be incentivecompatible, (ICH 52 wH +�H�HM − csH ≥ �L4�H −�L5M −
�EL4B � sL5−�Lr − csL − �0 Since the objective function isincreasing in wH and an increase in wH cannot violate (ICH ),we must conclude that either (IRH ) or (ICL) is binding. Weconsider these possibilities in turn.
Case A. (ICL) is binding. This implies wH = −�EL4B � sL5−�Lr − �− c4sL − sH 5−�H�LM and the optimization problembecomes maxsH 1�H≥06−�EL4B � sL5−�Lr − �− csL −�HM −
�H 4�L −�H 5M7, subject to (IRH 52 �6EL4B � sL5− EH 4B � sH 57+c4sL − sH 5+ 4�L −�H 54�HM + r5≥ 0, and (ICH 52 4�L −�H 54�L −
�H 5M ≥ 00 Note that (ICH ) implies that any feasible solutionrequires 0 ≤ �H ≤ �L. Furthermore, the objective function isindependent of sH and is decreasing in �H . Starting withany feasible solution such that �H > 0, decreasing �H cannotviolate (ICH ), and decreasing �H does not violate (IRH ) aslong as an appropriate inventory, sH , is set. Therefore, �H = 0at optimum, provided the following condition is satisfied:�6EL4B � sL5− EH 4B � sH 57+ c4sL − sH 5+ 4�L −�H 5r ≥ 00 Thiscondition implies that sH should satisfy max801 sm9≤ sH ≤
sM , where sm and sM are the two roots of the equationcs+�EH 4B � s5= 4�L −�H 5r +�EL4B � sL5+ csL. It is easy toverify that max801 sm9≤ sH < sM by setting u= � in Lemma 1,hence �∗
H = 0. Note that sm could be negative dependingon the model parameters. The payoff of the reliable vendoris given by V ∗
H = −�EL4B � sL5 − csL − �Lr − �HM − � =
VL + 4�L −�H 5M < VH .Case B. (IRH ) is binding. This case does not contain any
further solutions. Out of all the PBE identified above, only the
efficient ones (i.e., those with s∗H = sH ) are renegotiation-proof.
We therefore eliminate all others. �Proof of Proposition 3. In this proof we adopt the
classic one-shot signaling treatment (without Step 4 in §3.5)to determine the equilibrium outcome, as opposed to thesignaling-with-renegotiation framework proposed by Beaudryand Poitevin (1993). This is because, as per the sequence ofevents in §3.5, the accepted proposal in Step 3 turns out tobe efficient, and therefore renegotiation-proof. We considerthe possibility of recovering first-best rents in a separatingequilibrium: 4w∗
H 1 v∗H 1 s
∗H ) and 4w∗
L1 v∗L1 s
∗L). Since inventory is
observable and verifiable it is straightforward to contracton its first-best levels: sH and sL. However, the transferpayments need to adhere to the incentive compatibility(IC� 2 V� ≥w� ′ − v� ′ E� 4B � s� ′ 5−��M − cs� ′ ) and participationconstraints (IR� 2 − w� + 4v� − �5E�4B � s�5 − ��r ≥ �,) for�1 � ′ ∈ 8L1H91 � 6= � ′.
Consider the following contract parameters: w∗H = −�−
�H r + 4v∗H −�5EH 4B � sH 5, w∗
L = −�−�Lr + 4v∗L −�5EL4B � sL).
For these choices of fixed fee, the participation constraints arebinding, and this implies that both types of vendor recovertheir first-best outcomes, provided the incentive compatibility(IC) constraints are satisfied. We now check (ICH ). Plugging inthe appropriate values, we obtain v∗
H 6EH 4B � sH 5− EL4B � sH 57≥ �6EL4B � sL5− EH 4B �
sH 57 + 4�L − �H 5r + c4sL − sH 5. The conditions above aresatisfied for any v∗
L and v∗H such that v∗
L6EL4B � sL5− EH 4B �
sL57 ≤ �6EL4B � sL5− EH 4B � sH 57+ 4�L −�H 5r + c4sL − sH 5 ≤
v∗H 6EL4B � sH 5 − EH 4B � sH 57. Furthermore, such v∗
H and v∗L
always exist and are nonnegative. To see this, note that thehazard rate ordering between OL and OH implies FOSD;therefore, 6EL4B � sL5−EH 4B � sL57≥ 0, 6EL4B � sH 5−EH 4B � sH 57≥0 and �6EL4B � sL5− EH 4B � sH 57+ 4�L −�H 5r + c4sL − sH 5 > 0due to Lemma 1 and �L >�H .
Note that these equilibria are efficient and, therefore,renegotiation-proof. �
Proof of Proposition 4. As per the framework forsignaling-with-renegotiation (Beaudry and Poitevin 1993),when the one-shot (without Step 4 in §3.5) equilibrium is notefficient then it may not be renegotiation-proof. Since thisis the case (proof omitted for brevity), we follow Beaudryand Poitevin (1993) to identify the renegotiation-proof out-come. Note that for � ∈ 8L1H9 the vendor’s payoff is V� =
w+���M − cs� −��M , which is decreasing in inventory s.Therefore, the optimal inventory is s∗
H = s∗L = 0. The payoff of
the buyer is U� = −w� − 4��M +�l+ r5��1 where we haveused E� 4B � 05=�� l, where l is the expected repair lead timeusing Wald’s equation and the fact that each one of the�� expected failures will generate an expected period l ofdowntime.
Then, ¡U�/¡w = −1; ¡U�/¡� = −��M ; ¡V�/¡w = 1; and¡V�/¡�=��M . It is straightforward to verify that Assumption3A and Assumption 3B (Case RS) in Beaudry and Poitevin(1993) (essentially single-crossing property for the vendor’spayoff) are satisfied. We now solve for the contract parameters,following the scheme laid out in Proposition 3 of Beaudry andPoitevin (1993). Also, we define Fp4 · 5= pFL4 · 5+ 41 − p5FH 4 · 5with corresponding interpretation for �p and Ep4B � 05= pEL4B �
05+ 41 − p5EH 4B � 05. We first solve for the reliable vendor’s
contract parameters as follows: maxw1�≥06w+��HM −�HM7,subject to (IR): −w−��pM −��pl−�pr ≥ �. Clearly (IR) isbinding at optimality, otherwise an increase in w will improvethe objective. Therefore, w∗
H = −� − ��pM − ��pl − �pr .Plugging this back into the objective, we get max�6constant+�4�H −�p5M7. Therefore, �∗
H = 0 and w∗H = −�−��pl−�pr .
Next we characterize the unreliable vendor’s contractparameters by solving the following optimization problem:maxw1�6w+��LM −�LM7, subject to −w−��LM −��Ll−�Lr ≥ −w∗
H −��Ll−�Lr ; and −w−��pM −��pl−�pr ≥ �.We ignore the second constraint and verify that is satisfied atoptimality. As before, the first constraint is clearly binding atoptimality, such that w∗
L +�∗L�LM =w∗
H . This determines thevalue of the objective function entirely. Further, the secondconstraint is satisfied for �∗
L ≥ 0. Hence, �∗L = 0 will give rise
to a pooling equilibrium, but there exist a continuum ofseparating equilibria with �∗
L > 0 that are payoff equivalentto the pooling equilibrium. �
Proof of Proposition 5. We define Fp4 · 5= pFL4 · 5+ 41 −
p5FH 4 · 5, �p = p�L+41−p5�H , and Ep4B � s4v55= pEL4B � sL4v55+41 − p5EH 4B � sH 4v55. As per the framework for signaling-with-renegotiation (Beaudry and Poitevin 1993), when theone-shot (without Step 4 in §3.5) equilibrium is not efficientthen it may not be renegotiation-proof. Since this is the case(proof-omitted for brevity), we follow Beaudry and Poitevin(1993) to identify the renegotiation-proof outcome. Recallthat for � ∈ 8L1H9, V� = w − vE�4B � s�4v55− cs�4v5−��M ;and U� = −w+ 4v−�5E�4B � s�4v55−��r , where inventoryis s� 4v5= F −1
� 41 − c/v5. Then, ¡U�/¡w = −1, ¡U�/¡v = E� 4B �
s� 4v55− 4v−�54c/v2541 − F� 4s� 4v555/f� 4s� 4v55, ¡V�/¡w = 1, and¡V�/¡v = −E� 4B � s� 4v55. Under condition (2) (i.e., EL4B � sL5 >EH 4B � sH 5), it is straightforward to verify that Assumption 3B(Case RS) in Beaudry and Poitevin (1993) (essentially single-crossing property for the vendor’s payoff) is satisfied. Usingthe hazard rate order and increasing hazard rate property,Assumption 3A in Beaudry and Poitevin (1993) is alsosatisfied, provided v ≥ �. We verify that v < � is never anequilibrium outcome, hence we can explicitly restrict theparameter space to v ≥ �. Consequently, the conditions forProposition 3 in Beaudry and Poitevin (1993) are met, andwe can conclude that the renegotiation-proof contracts areunique with respect to payoff.
We now solve for the contract parameters, following thescheme laid out in Proposition 3 of Beaudry and Poitevin(1993). We first solve for the reliable vendor’s contract param-eters as follows: maxw1v6w−vEH 4B � sH 4v55− csH 4v5−�HM7,subject to (IR): −w+4v−�5Ep4B � s4v55−�pr ≥ �. Clearly (IR) isbinding at optimality, otherwise an increase in w will improvethe objective. Therefore, w∗
H = 4v−�5Ep4B � s4v55−�pr − �.Plugging this back into the objective function and takingfirst-order conditions, we obtain
4v−�5c
v2
(
41 − p51 − FH 4sH 4v55
fH 4sH 4v55+ p
1 − FL4sL4v55
fL4sL4v55
)
= p4EL4B � sL4v55− EH 4B � sH 4v5550 (4)
Since the right-hand side is always positive for finite v, forthe equation to have a finite solution it must be the case thatvH >� , thereby resulting in inefficiency.
Next we characterize the unreliable vendor’s contractparameters by solving the following optimization problem:
maxw1v6w−vEL4B � sL4v55− csL4v5−�LM7, subject to −w+
4v−�5EL4B � sL4v55−�Lr ≥ −w∗H + 4v∗
H −�5EL4B � sL4v∗H 55−�Lr
and −w+ 4v−�5Ep4B � s4v55−�pr ≥ �. We ignore the secondconstraint and verify that it is satisfied at optimality. Asbefore, the first constraint is clearly binding at optimality,which can be written as w∗
L = 4v−�5EL4B � sL4v55+ 41−p54v∗H −
�54EH 4B � sH 4v∗H 55− EL4B � sL4v
∗H 555−�pr − �. Plugging this
back into the objective function, we are left with the residualoptimization problem minv6csL4v5+�EL4B � sL4v557, whichhas a unique minimum at v∗
L = � . Finally, using (2) we haveEL4B � sL4�55 > EH 4B � sH 4�55, which implies that the othersecond constraint, −w∗
L + 4v∗L − �5Ep4B � s4v55−�pr ≥ �, is
satisfied. �Proof of Corollary 1. Let g = 4v∗
H −�5Ep4B � sH 4v∗H 55+
��pl−v∗HEH 4B � sH 4v
∗H 55− csH 4v
∗H 5 denote the difference in
payoffs for the H -type between the PBC contract of Proposi-tion 5 and the RBC contract of Proposition 4. We will showthat this is always nonnegative. First note that v∗
HEH 4B �
sH 4v∗H 55+csH 4v
∗H 5= mins6v
∗HEH 4B � s5+cs7≤ v∗
HEH 4B � 05+c0 =
v∗H�H l. Therefore, g ≥ 4v∗
H −�5Ep4B � sH 4v∗H 55+��pl−v∗
H�H l.Next, at p = 0 we have v∗
H = � and �p =�H therefore g ≥
0. Furthermore, from (4), note that ¡g/¡v∗H = 0. Therefore,
dg/dp = 4¡g/¡v∗H 54¡v
∗H/¡p5+ ¡g/¡p = ¡g/¡p = 4v∗
H −�54EL4B �
sL4v∗H 55− EH 4B � sH 4v
∗H 555+ �4�p −�H 5l ≥ 0. Since g ≥ 0 at
p = 0 and increasing in p, we can therefore conclude thatg ≥ 0 for all 0 ≤ p ≤ 1. �
Proof of Proposition 6. For part (a) the choice in Step 2is dictated by the preferences of the two types. For theH-type, RBC is dominated by PBC. In the case of verifi-able inventory, this is because the H-type can extract allrents with PBC but not with RBC, coupled with the factthat both RBC and PBC generate the same total value (i.e.,they are both efficient). In the case of unverifiable inven-tory, this follows from Corollary 1. Thus, the H-type willnever choose RBC over PBC. Comparing PBC verifiablewith PBC unverifiable, the former is preferable for theH -type if �EH 4B � s5+cs+�H r+K≤v∗
HEH 4B � s4v∗H 55+cs4v∗
H 5−4v∗
H −�5Ep4B �s4v∗H 55+�pr . Therefore, there must exist a Kv
such that when K >Kv, PBC with unverifiable inventory ispreferable to PBC unverifiable. In this case the H -type vendorwill choose PBC with unverifiable inventory in Step 2. Thebest the L-type can do in this case is to also choose PBC withunverifiable inventory as anything else would signal her typeand restrict her to her first-best payoff, which is the worstpossible scenario for her. Since both types choose PBC withunverifiable inventory in Step 2, no new information hasbeen revealed by their choice and the separating equilibriumof Proposition 5 will be played out.
If K ≤Kv, then the H -type vendor finds it optimal to paythe fee K since the costs associated with inefficiency of thePBC unverifiable separating equilibrium of Proposition 5are higher than the fixed fee. Therefore, the H -type choosesverifiable inventory in Step 2, after which she can alwayschoose to separate using the contract of Proposition 3 andreceive her first-best rents (minus the fixed fee K alreadyincurred) should the L-type try to mimic. The L-type, however,would also receive her first-best minus the fixed fee K shouldshe choose to mimic. But she can do better by choosing tosignal her type through choosing unverifiable inventory withPBC in Step 2 and avoid paying the fixed fee K. Since her type
is revealed, the best she can offer the buyer is the first-bestcontract; i.e., vL = � , wL = −�− r�L −�EL4B � sL5. Furthermore,turning to the H -type vendor, since her type has been revealedshe can always receive her first-best rents by choosing a fixedfee–inventory contract sH = sH 1 wH = −�− r�H −�EH 4B � sH 5or any combination of PBC or RBC such that the buyer is leftwith his outside option.
For part (b) the buyer will choose RBC with verifiableinventory and thereafter the RBC separating equilibrium(Proposition 2) is played out. The buyer cannot do better inany other outcome. Note that the buyer’s choice is Step 2reveals no information since he is the uninformed party. �
Before we proceed with the proof of Proposition 7, wefind it useful to state and prove the following lemma.
Lemma 2. Under the assumptions of §702 (i.e., that the fail-ures follow a Poisson distribution that we approximate with thenormal distribution), the performance-based penalty v∗
H given inProposition 5 is independent of the number of products Q acquiredby the buyer.
Proof of Lemma 2. We approximate the Poisson dis-tribution with the normal distribution, with followingthe density function for on-order inventoryf�4x3��5 =
41/√
2��� l5exp44−4x−�� l52/2√
�� l5) for � ∈ 8L1H9, where lis the expected repair lead time. The cumulative distributionfunction is F� 4x3�� 5. With k buyers, these functions are modi-fied to f� 4x3 k�� 5 and F� 4x3 k�� 5. From the definition of v∗
H of(4) and recalling that F�4s�4v55= 1 − c/v for � ∈ 8L1H9, wehave
�4v1Q5 =c24v−�5
v3
[
1 − p
fH 4sH 4v55+
p
fL4sL4v55
]
− p
[
∫ �
sL
41 − FL4x55 dx+
∫ �
sH
41 − FH 4x55 dx
]
= 00
If T� is the expected time to failure for an engine oftype � , then for Q engines, the aggregate Poisson fail-ure rate �� = Q/T� . Also, making the observation thatd�/dQ = 0, along with the implicit function theorem, at� = 0 we obtain 4¡�/¡v54dv/dQ5 + 4¡�/¡�L54d�L/dQ5 +
4¡�/¡�H 54d�H/dQ5 = 0. We can conclude that dv/dQ = 0if and only if 44¡�/¡�L5�L + 4¡�/¡�H 5�H 5/4¡�/¡v5= 0, orassuming that 4¡�/¡v5��=0 6= 0; 4¡�/¡�L5�L+ 4¡�/¡�H 5�H = 0.Since we are using the normal approximation for the Poissondistribution,
F� 4s� 4v55 =12
[
1 + erf
(
s� 4v5−�� l√
2�� l
)]
=12
+1
2√�
∫ 4s� 4v5−�� l5/√
2�� l
−4s� 4v5−�� l5/√
2�� le−t2 dt = 1 −
c
v0
We now define z� = 4s� 4v5−�� l5/√
2�� l and note thatzH = zL = z, since FH 4sH 4v55 = FL4sL4v55 = 1 − c/v = ê4z5,where ê4 · 5 is the cumulative distribution function for thestandard normal distribution. Therefore, ¡F� 4s� 4v55/¡�� =
e−44s� 4v5−�� l525/42�� l54¡/¡��544s� 4v5−�� l5/
√
2�� l5 = 4¡/¡��5 ·
41 − c/v5= 0. This gives us the result that ¡z/¡�� = 0. Fur-thermore, it is a well-known result that for the standardnormal distribution
∫
ê4z5dz= zê4z5+�4z5, where �4 · 5 isthe density function of the standard normal. After a littlealgebra, we obtain
∫ �
s� 4v541 − F� 4x55 dx = 4−4c/v5z+�4z55
√
�� l.
Now using the result that ¡z/¡�� = 0 and the fact thatf� 4s� 4v55=�4z5/
√
�� l, it is straightforward to verify that at�= 0 we have 4¡�/¡�L5�L + 4¡�/¡�H 5�H = 0, thus implyingthat dv∗
H/dQ = 0. �
Proof of Proposition 7. Under PBC, the payoffs forthe vendor (selling to k buyers) and an individual buyerbecome V k
is the inventory chosen by the vendor when the penalty rateis v and there are k buyers. The quantity E�4B � s�4v�1k55
represents the total number of expected backorders causedby all k buyers. Because of symmetry, each buyer experiencesa kth fraction of these backorders. The characterization ofthe vendor’s outcome in the symmetric equilibrium withinventory pooling across k buyers proceeds in exactly thesame way as in Proposition 5; the only difference is that thefailure rate is k�� instead of �� for � ∈ 8L1H9. Importantly,the optimal penalty vH is that given by (4). To establish thatthe H -type vendor’s payoff improves when she has the abilityto pool inventory across buyers, we must show that ∀k ≥ 1,V k∗H >kV ∗
H ; that is, the vendor achieves a higher payoff bypooling inventory across k buyers than in the absence of pool-ing. A necessary and sufficient condition for this to be true isV ∗H 4N 5 >NV ∗
H 415, where V ∗H 4i5 is the payoff of the H -type ven-
dor when contracting with i buyers, each purchasing a singleengine. The latter condition is equivalent to treating N as acontinuous variable and verifying that 4d/dN54V ∗
H 4N 5/N > 0.We now introduce an additional notation: �� 415 denotes thefailure rate of one engine of type � , and therefore �� =N�� 415.Using the result in Proposition 5 we know that V ∗
H 4N5 =
4v∗H − �5p6EL4B � sL4v
∗H1N55 − EH 4B � sH 4v
∗H1N557 − �EH 4B �
sH 4v∗H 1N 55−csH 4v
∗H 1N 5−N4�H 415M+�p415r+�5. We define a
function g4N 5 such that V ∗H 4N 5= g4N 5−N4�H 415M+�p415r+
�5. Then 4d/dN54V ∗H 4N 5/N 5 > 0 ⇔ dg4N5/dN > g4N5/N . Using
the normal approximation to the Poisson distribution, g4N5
can be expressed as g4N5= L4z544v∗H −�5p4
√
�Ll−√
�H l5+
�√
�H l5−csH 4v∗H 1N 5, where L4z5= −z41−ê4z55+�4z5; ê and
� being the distribution and density functions, respectively,for the standard normal distribution, v∗
H is the solution to (4),z= 4sH 4v
∗H1N 5−�H l5/
√
�H l, and sH 4v∗H1N5 is the solution
of FH 4sH 4v∗H1N55= 41− c/v∗
H 5. As shown in Lemma 2 (andin its proof), ¡z/¡�� = 0, d��/dN =��/N , and dv∗
H/dN = 0.Using these relationships, we can show that dg4N5/dN =
4g4N 5− 2c�H l5/42N5. Then dg4N5/dN > g4N5/N ⇔ g4N5+
2c�H l < 0. This can be rewritten as V ∗H 4N5+N4�H 415M +
�p415r + �5+ 2c�H l < 0. Now using the expression for VH
from Proposition 1, we recover the condition of Proposition 7.Finally, note that if pooling across k buyers is possible, bothtypes of vendor will always choose to pool inventory; notpooling is not credible (renegotiation-proof) as after thecontract is signed, but before it is implemented, it is Paretoimproving to pool inventory. �
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