Improving Reliability of a Shared Supplier with

Competition and Spillovers

Yimin Wanga, Yixuan Xiaob,∗, Nan Yangb

aW. P. Carey School of Business, Arizona State University, Tempe, AZ 85287, USAbOlin Business School, Washington University in St. Louis, St. Louis, MO 63130, USA

Abstract

Supplier reliability is a key determinant of a manufacturer’s competitiveness.It reflects a supplier’s capability of order fulfillment, which can be measuredby the percentage of order quantity delivered in a given time window. Aperfectly reliable supplier delivers an amount equal to the order placed byits customer, while an unreliable supplier may deliver an amount less thanthe amount ordered. Therefore, when suppliers are unreliable, manufacturersoften have incentives to help suppliers improve delivery reliability. Suppliers,however, often work with multiple manufacturers and the benefit of enhancedreliability may spill over to competing manufacturers. In this study, we ex-plore how potential spillover influences manufacturers’ incentives to improvesupplier’s reliability. We consider two manufacturers that compete with im-perfectly substitutable products on Type I service level (i.e., in-stock proba-bility). The manufacturers share a common supplier who, due to variationsin production quality or yield, is unreliable. Manufacturers may exert effortsto improve the supplier’s reliability in the sense that the delivered quantityis stochastically larger after improvement. We develop a two-stage modelthat encompasses supplier improvement, uncertain supply and random de-mand in a competitive setting. In this complex model, we characterize themanufacturers’ equilibrium in-stock probability. Moreover, we characterizesufficient conditions for the existence of the equilibrium of the manufactur-ers’ improvement efforts. Finally, we numerically test the impact of marketcharacteristics on the manufacturers’ equilibrium improvement efforts. We

∗Corresponding author. Tel.: +1 314 600 3532Email addresses: yimin_wang@asu.edu (Yimin Wang), xiaoy@wustl.edu (Yixuan

Xiao), yangn@wustl.edu (Nan Yang)

Preprint submitted to European Journal of Operational Research September 27, 2013

find that a manufacturer’s equilibrium improvement effort usually declinesin market competition, market uncertainty or spillover effect, although itsexpected equilibrium profit typically increases in spillover effect.

Keywords: Supply chain management, Game theory, Reliability, Spillover,Supplier improvement

1. Introduction

Supplier reliability is increasingly being recognized as a key determinantof manufacturers’ competitiveness. It reflects a supplier’s capability of or-der fulfillment, which can be measured by the percentage of order quantitydelivered in a given time window. A recent survey by Accenture finds that“supplier reliability is seen across the industry as significantly more impor-tant than cost” (Erhardt et al., 2010, p.10). Manufacturers therefore oftenhelp to improve delivery reliability when suppliers are unreliable. Suppliers,however, often work with multiple manufacturers (Markoff, 2001), and hencea manufacturer’s effort to improve supplier reliability may benefit its directcompetitors. Empirical evidence suggests that manufacturers are often awareof, and also wary of, the potential spillover effect, but different manufacturersseem to take the issue differently.

Farney (2000) describes supplier improvement effort by UTC’s Pratt &Whitney Aircraft unit (P&W) with Dynamic Gunver Technologies (DGT)(now a unit of Britain’s Smiths Group). UTC initiates improvement effortwith DGT despite the fact that (a) DGT has close relationship with a P&W’scompetitor, Rolls Royce, and (b) “by leveraging its expertise and knowledgegained from working with UTC, Dynamic successfully won bids from AlliedSignal, Rolls Royce, and General Electric.”(p.5). Although being aware ofthe competitive concerns above, “UTC had attempted to bring process im-provements to DGT operations. DGT had received substantial assistancefrom P&W in the form of workshops in Kaizen process improvement, leanmanufacturing, and UTC’s ACE program for quality improvement.”(p.5).UTC were, however, wary of the spillover effect: “Their goal did not includethe scenario where [DGT] competed with UTC for other business and part-nered with UTC’s competitors.”(p.8). In this case, UTC seems to reluctantlyaccept the existence of spillover effect.

Toyota, on the other hand, seems to have a more open mind towards thespillover effect: “Production processes are simply not viewed as proprietary

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and Toyota accepts that some valuable knowledge will spill over to benefitcompetitors.” (Dyer & Nobeoka, 2000, p.358).

Yet in Honda’s case, its supplier improvement program implicitly en-courages suppliers to work with its competitors: Honda espouses suppliers’self-reliance, i.e., “balancing responsiveness to Honda’s needs with a suffi-ciently diversified customer base.” (Sako, 2004, p.296). In fact, encouragingsuppliers to work with multiple manufacturers is often a policy of many firmsto avoid the “captive supplier issue” (Lascelles & Dale, 1990, p.53).

Given the different attitudes exhibited by different firms toward the spillovereffect, we explore in this research how potential spillover effect influencesmanufacturers’ reliability improvement efforts and the resulting expectedprofits. We consider the scenario where two manufacturers share a commonsupplier, which has a production process that is subject to random outputdue to variations in quality or production yield. Both manufacturers mayexert effort to improve the reliability (e.g., expected yield) of the supplier’sproduction process. We develop a two-stage model to investigate the impactof spillover upon the manufacturers’ improvement efforts in a competitivesetting. In the first stage, each manufacturer decides its improvement effortand incurs the corresponding improvement cost. In the second stage, as aresult of the manufacturers’ improvement efforts, the supplier has a new,more reliable production process. The manufacturers then place orders fromthe supplier, and compete on service levels in the consumer market.

In this model, we show the existence of the equilibrium in the servicelevel competition, and characterize the manufacturers’ equilibrium servicelevels and corresponding inventory decisions in the second stage. Moreover,we characterize sufficient conditions for the existence of the equilibrium ofthe manufacturers’ improvement efforts in the first stage. Finally, we nu-merically test the impact of market characteristics upon the manufacturers’equilibrium improvement efforts. We find that spillover effect often bringsfavorable improvement to the manufacturers’ expected profits, regardless ofwhether the manufacturers are asymmetric or symmetric. When manufac-turers are asymmetric, the ex ante more advantageous manufacturer typi-cally exerts a higher proportion of improvement efforts, especially when thespillover effect is large. While such higher proportion of improvement effortsbenefits the focal manufacturer, it may benefit, in terms of relative profit,its competitor more. When manufacturers are symmetric, we find that bothmanufacturers’ improvement efforts decline in market competition, marketuncertainty or spillover effect, although their expected profits increase in

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spillover effect.The rest of this paper is organized as follows: we discuss related literature

in §2, and develop the two-stage model in §3. In §4, we characterize themanufacturers’ equilibrium service levels and the corresponding inventorydecisions in the second stage. In §5, we study the manufacturers’ supplierimprovement efforts, and characterize sufficient conditions for the existenceof the equilibrium in the first stage. In §6, we numerically test the impactof market characteristics upon the manufacturers’ equilibrium improvementefforts and profit. In §7, we summarize our findings and conclude the paper.All proofs are relegated to Appendix A.

2. Literature Review

Our research is related to three streams of literature: (1)economics andindustrial organization literature on spillover effect; (2)empirical literatureon supplier improvement, and (3) operations management literature on un-certain supply. In this section, we review these three streams of literature,and discuss their relationship with this paper.

First, a large body of economic and industrial organization literature stud-ies spillover effect in the absence of supplier improvement, e.g., Spence (1984),D’Aspremont & Jacquemin (1988), Levin & Reiss (1988), Harhoff (1996),Gilbert et al. (2006) and Gupta (2008). Spence (1984) finds that spilloversincrease industry-wide efficiency while lower the individual firm’s incentiveto innovate. Harhoff (1996) suggests that strategic spillovers from upstreamto downstream enhance the upstream firm’s innovation. D’Aspremont &Jacquemin (1988) study competing manufacturers’ incentives to cooperatehorizontally in the presence of R&D spillovers, and their work has been gen-eralized by numerous studies, see Kamien et al. (1992), Suzumura (1992), etc.Gupta (2008) analyzes the role of channel structure in determining firms’ in-vestment in process innovation. D’Aspremont & Jacquemin (1988) and thefollow-up work are primarily concerned with the impact of R&D cooperationon manufacturers’ R&D investment level, industry outcome and social wel-fare, and evaluating whether the R&D investments by firms with and withoutcooperation is socially efficient under different cooperation and competitionstructures. Our work differs from this stream of research in terms of researchfocus. The existing literature studies the impact of spillover effect on costreduction, while our research investigates the impact of spillover effect onsupplier improvement efforts. Our research complements the existing litera-

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ture by considering manufacturers’ role in supplier reliability improvement,as opposed to the manufacturers’ own cost reduction effort. Compared to thecost reduction setting, our model shows that manufacturers may not neces-sarily benefit from improving supplier’s reliability even if the improvement iscostless. In addition, we numerically investigate how the manufacturers’ sup-plier improvement efforts and their expected profits are affected by marketcharacteristics such as manufacturers’ competitive advantage and demanduncertainty, which has not been examined in the existing literature thatstudies spillover effect on cost reduction.

Secondly, there is extensive empirical literature that examines manufac-turers’ role in supplier improvement effort, with a focus on antecedentsof successful supplier improvement effort, e.g., Krause & Ellram (1997),Humphreys et al. (2004), and Modi & Mabert (2007). Some common factorsemerging from this stream of research include trust, buyer involvement, andcommunication that are likely to predict successful supplier improvementeffort. Talluri et al. (2010) proposes a decision model for assisting firms inmaking resource allocation decisions in supplier development initiatives. Thisstream of literature, however, focuses on dyadic buyer-supplier relationshipsand therefore does not explore spillover effect.

Thirdly, this research is also related with the vast literature on unreliablesupply, including random yield (e.g., Anupindi & Akella, 1993; Federgruen &Yang, 2009; Dada et al., 2007; Kazaz & Webster, 2011; Yeo & Yuan, 2011; Xu& Lu, 2013; Inderfurth & Vogelgesang, 2013), random capacity (e.g., Ciaralloet al., 1994; Erdem, 1999; Wang et al., 2010; Chao et al., 2008), and randomdisruption (e.g., Parlar & Perry, 1996; Gurler & Parlar, 1997; Tomlin, 2006;Babich et al., 2007; Schmitt et al., 2010). This stream of research does nottypically consider spillover effect or supplier improvement effort.

We adopt the most commonly used stochastically proportional yield modelto describe supplier unreliability, which “applies to circumstances where yieldlosses occur because of limited capabilities of the production system to adaptto random environmental changes or variations in materials.”(Yano & Lee,1995, p.313) This modeling approach allows us to capture scenarios wheremanufacturers may improve supplier reliability in different ways, such as leanproduction assistance aimed at reducing production variability and waste, orquality assistance aimed at improving process capability and quality.

Recently, Qi et al. (2011) study supplier improvement effort under randomcapacity, where manufacturers share the same supplier but they can stipulatecapacity usage terms to partially mitigate the capacity spillover effect. They

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focus on how capacity usage restrictions affect manufacturers’ equilibriumcapacity investment, and find that rigid restrictions of capacity usage canlead to adverse market outcomes. Our research is complementary to theirstudy, as we focus on reliability improvement and manufacturers’ competitionon service level.

To sum up, existing literature has considered, separately, (a) spillover ef-fect in the absence of supplier improvement, (b) supplier improvement with-out spillover effect, (c) supply uncertainty without the combined effects ofsupplier improvement and spillover. Our research is the first to investigatemanufacturers’ supplier improvement efforts under the combined effects ofspillover, competition, random yield and uncertain demand.

3. Model

We consider a market where there are two manufacturers, each sellinga single product to the final consumers. The products offered by the twomanufacturers are imperfectly substitutable, which means that they are notfully interchangeable, and customers have to make a trade-off (either in priceor in quality) when replacing one product with the other. This assumptionreflects the reality that customers typically have different preferences overdifferent products, which can be attributed to factors such as brand loyalty.

Manufacturers outsource production to a common, unreliable supplier,and the manufacturers compete on service level against each other. Servicelevel, as measured by in-stock probability, is now recognized by many firmsas one of the most important performance metrics. Bernstein & Federgruen(2004) have noted that an increasing number of companies attempt to obtainlarger market shares by increasing the service level of their products.

3.1. Demand

Market demand is stochastic, imperfectly substitutable, and influencedby both manufacturers’ in-stock probabilities, commonly known as Type-1 service levels. Let Di(f) denote the random demand of manufacturer i,given the vector of the in-stock probabilities f provided by both manufac-turers. Following Bernstein & Federgruen (2004), we assume that demanduncertainty is of the multiplicative form, that is,

Di(f) = di(f)ǫ, (1)

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where ǫ denotes market uncertainty. The multiplicative demand form in (1)implies that the market uncertainty (measured by coefficient of variation) isnot affected by the in-stock probabilities f , but the expected demand is. LetGǫ(·) and gǫ(·) denote the distribution and density of ǫ, respectively. Withoutloss of generality, we normalize ǫ such that E[ǫ] = 1 and interpret di(f) asthe expected demand of manufacturer i.

Assumption 1. Gǫ(ǫ) is log-concave.

Assumption 2.

∂di(f)

∂fi≥ 0,

∂di(f)

∂fj≤ 0,

∂2 log di(f)

∂fi∂fj≥ 0, and

∂2 log di(f)

∂f 2i

+∂2 log di(f)

∂fi∂fj≤ 0.

Assumption 1 is satisfied by many common distributions, including uni-form, normal, exponential, and gamma distributions (see Bernstein & Fed-ergruen, 2004). In addition, any concave cumulative distribution function isalso log-concave. Although Assumption 2 is made for technical tractability,it is satisfied by many different forms of demand functions under mild condi-tions, including linear demand function, log-separable demand function, andthe general class of attraction models. See Appendix B for details.

3.2. Supply

The supplier’s production process follows the standard stochastically pro-portional yield model, reflecting variations in supplier’s production process,limited production capabilities, or quality variations. In this setting, for anygiven order Qi placed by manufacturer i, the delivered quantity is qi = YiQi,where Yi is the random yield factor with support on [0,1].

The distribution of the random yield factor Yi is influenced by the sup-plier’s reliability index pi, an aggregate index reflecting factors (e.g., equip-ment, technical knowhow, and training) that may influence the supplier’syield. Let ΦYi

(·, pi) denote the distribution of Yi. We adopt the conventionthat a higher pi is associated with a more reliable production process, i.e.,Yi is first-order stochastically increasing in pi. Thus, for any given pi > p′i,ΦYi

(y, pi) ≤ ΦYi(y, p′i) for any y.

3.3. Improvement Efforts and Spillovers

Both manufacturers may exert efforts, such as knowledge transfer or cap-ital investment, to improve the supplier’s reliability. If manufacturer i exerts

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improvement effort level at zi and let z = (z1, z2), then the supplier’s pro-duction reliability index for manufacturer i increases from p0i to

pi(z) = p0i + zi + αzj, i = 1, 2, j = 3− i, (2)

where 0 ≤ α ≤ 1 capturers the spillover effect in the supplier’s productionprocess.

We do not restrict the level of the spillover effect. Therefore, α can takeany value on [0, 1], and the extent of the spillover effect depends on the con-figurations of the supplier’s production process. For example, if the supplieruses the same production line for both manufacturers’ components, then itis possible that full spillover might occur, i.e., α = 1. In contrast, if thesupplier uses a shared production line for the early steps of the productionprocess and performs additional customization steps to satisfy each manufac-turer’s specific requirement, then the knowledge gained from manufacturer ican partially spillover to manufacturer j, i.e., 0 < α < 1. It is also possiblethat the supplier maintains completely separate production lines such thatthere is no spillover and hence α = 0.

It is natural to assume that manufacturer i’s improvement cost mi(zi)(weakly) increases in its effort level zi, i.e., dmi(zi)/dzi ≥ 0. We do notconsider other benefits associated with the manufacturer’s improvement ef-fort, such as improved efficiency or improved contract terms. As is commonin game theory literature, we assume manufacturers’ improvement costs arecommon knowledge. This typically occurs when the manufacturers have beenworking with the supplier for several years, and therefore each manufacturerhas a good idea of the other’s supplier reliability improvement capabilities.This assumption also reflects the reality that manufacturers can resort tobusiness intelligence or industry association, or develop estimates based ontheir own experience to assess their competitors’ improvement cost structure.

3.4. Problem Formulation

Sequence of events. The competing manufacturers play a two-stagegame with complete information. In the first stage, each manufacturer si-multaneously decides its supplier improvement effort. In the second stage,after exerting supplier improvement effort but before yield and demand un-certainty is resolved, each manufacturer decides its in-stock probability fiand its order quantity Qi from the supplier. In the end, after observing real-ized production yield and demand, each manufacturer pays the supplier forquantities delivered and satisfies its demand as much as possible.

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Given the above sequence of events, the manufacturers’ decision problemcan be formulated as a two-stage stochastic program. Let wi, ri, si, and πidenote the wholesale price manufacturer i pays to the supplier, manufactureri’s unit price charged in the market, unit salvage value for leftovers, and unitpenalty cost for shortages, respectively. Define Hi(u, v) = ri min{u, v} +si(u− v)+ − πi(v − u)+.

The second stage problem. Since the supplier’s production processis random, manufacturer i will choose an order quantity Qi such that it willachieve the expected in-stock probability fi ex-ante. Let Y = (Y1, Y2) andp = (p1, p2), manufacturer i’s second stage problem is to find the optimal{f ∗

i , Q∗i } that maximizes

vi(fi, Qi|p, fj) = −wi EYi[YiQi] + EYi,ǫ [Hi (YiQi, Di(f))] , (3)

s.t. Prob (Di(f) ≤ YiQi) ≥ fi. (4)

The in-stock probability fi defined in (4) is commonly known as Type-1service level in the literature. With random yield, the in-stock probabilityvaries with different yield realizations, and therefore it needs to be condi-tioned and unconditioned over all possible yield realizations. The in-stockprobability defined in (4) implies that manufacturers can credibly commit ontheir product availabilities.

The first stage problem. Let the supplier’s initial production reliabil-ity index be p0 = (p01, p

02), the first stage problem for manufacturer i can be

formulated as maximizing

Ji(zi) = −mi(zi) + vi(f∗i , Q

∗i |p(z), f ∗

j ), (5)

where p(z) = (p1(z), p2(z)) and pi(z) is given by (2).We note that it is fairly straightforward to incorporate probabilistic sup-

plier improvement results, e.g., an improvement effort may not always befully successful. In this case, (5) can be modified by imposing an expectationoperator on vi(f

∗i , Q

∗i |p(z), f ∗

j ) over all possible realizations of p(z).In the following sections, we study the competition between the two man-

ufacturers, and focus on characterizing pure strategy equilibrium of the two-stage game. Henceforth, the Nash equilibrium in the following context refersto pure strategy equilibrium.

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4. Second Stage: Service Level Competition

In this section, we analyze the manufacturer’s second stage problem,where manufacturer i chooses in-stock probability fi and order quantity Qi tomaximize its expected profit (given by (3)) in the service level competition.

4.1. Stocking Factor

For analytical convenience and expositional ease, define qi(f) = Qi/di(f).Hereafter we refer to qi(f) as the stocking factor, which is the ratio of man-ufacturer i’s order quantity and its expected demand for any given vectorof in-stock probability f . Using the stocking factor qi(f), we can rewrite (3)and (4) as

vi(fi, qi(f)|p, fj) = −widi(f)EYi[Yiqi(f)] + EYi,ǫ [Hi (Yiqi(f)di(f), di(f)ǫ)] ,

(6)

s.t. EYi[Gǫ (Yiqi(f))] ≥ fi, (7)

where the service level constraint (7) follows from the fact that

Prob (Di(f) ≤ YiQi) ≥ fi ⇔ Prob (ǫ ≤ Yiqi(f)) ≥ fi ⇔ EYi[Gǫ (Yiqi(f))] ≥ fi.

Note that for any given fi, vi(·) is concave in qi(f); and for any given qi(f),vi(·) is log-concave in fi. We adopt a two-stage approach by first character-izing the optimal q∗i as a function of fi, and then characterizing the optimalservice level f ∗

i . Simplifying (6), we have

vi(fi, qi(f)|p, fj) = di(f)

{qi(f)

((ri − wi + πi)EYi

[Yi]− (ri − si + πi)EYi[YiGǫ(Yiqi(f))]

)

+ (ri − si + πi)EYi

[ ∫ Yiqi(f)

0

ǫgǫ(ǫ)dǫ

]− πi

}. (8)

Let ui(fi, qi(f)|p, fj) denote the terms in the curly bracket in (8), i.e., man-ufacturer i’s expected profit per unit of demand. The following lemma char-acterizes the optimal stocking factor.

Lemma 1. For any given vector of in-stock probability f , (a) ui(fi, qi(f)|p, fj)is concave in the stocking factor qi(f). (b) The optimal stocking factor q∗i (f)satisfies one of the following conditions:

EYi[YiGǫ (Yiq

∗i (f))] =

ri − wi + πiri − si + πi

EYi[Yi]; (9)

EYi[Gǫ (Yiq

∗i (f))] = fi. (10)

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Notice that (9) gives an interior solution to ∂ui(fi, qi(f)|p, fj)/∂qi(f) = 0,and (10) gives a corner solution constrained by (7). Leveraging Lemma 1, thefollowing lemma completely characterizes the properties of optimal stockingfactor for any given in-stock probability fi.

Lemma 2. (a) Manufacturer i’s optimal stocking factor q∗i (f) is independentof its competitor’s in-stock probability fj and is a function of fi only. (b)There exists a unique threshold of in-stock probability f i ∈ (0, 1) such that iffi ≤ f i, q

∗i (fi) uniquely solves (9) and dq∗i (fi)/dfi = 0; and if fi > f i, q

∗i (fi)

uniquely solves (10) and dq∗i (fi)/dfi > 0. (c) f i can be uniquely obtained bysolving the system of equations specified by (9) and (10) simultaneously.

Lemma 2 shows that if the in-stock probability is less than a criticalthreshold f i, the optimal stocking factor, q∗i (fi), is a constant and indepen-dent of the chosen in-stock probability.1 In contrast, if the in-stock proba-bility fi > f i, the optimal stocking factor q∗i (fi) increases in fi, i.e., a largerinflation of the expected demand is required to satisfy a higher service level.

4.2. Equilibrium In-stock Probabilities

We now turn our attention to manufacturer i’s in-stock probability deci-sion, given its competitor’s in-stock probability. It is immediate from Lemma2(a) that ui is independent of fj , so we can rewrite (3) as a univariate func-tion, i.e.,

vi(fi|p, fj) = di(fi|p, fj)ui(fi|p), where (11)

ui(fi|p) = q∗i (fi)((ri − wi + πi)EYi

[Yi]− (ri − si + πi)EYi[YiGǫ(Yiq

∗i (fi))]

)

+ (ri − si + πi)EYi

[ ∫ Yiq∗

i (fi)

0

ǫgǫ(ǫ)dǫ

]− πi. (12)

The following proposition characterizes the necessary and sufficient conditionfor each manufacturer to enter the market.

1One should not confuse the stocking factor with the order quantity: even thoughthe optimal stocking factor is constant in fi and independent of fj, the optimal orderquantity Q∗

i = q∗i (fi)di(f) increases in fi and decreases in fj, because ∂di(f)/∂fi ≥ 0 and∂di(f)/∂fj ≤ 0. The rate of increase (decrease) in Q∗

i is exactly proportional to that ofthe expected demand as fi increases (decreases).

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Proposition 1. (a) The best response f ∗i |fj ≥ f i for all fj. (b) A necessary

and sufficient condition for manufacturer i to enter the market is

(ri − si + πi)EYi

[ ∫ Yiq∗

i (f i)

0

ǫgǫ(ǫ)dǫ

]> πi. (13)

Proposition 1 shows that the best response f ∗i |fj can never be lower than

the threshold service level f i. One can therefore interpret f i as the minimumservice level required for manufacturer i to compete in the market place. Themanufacturer either enters the market and achieves a service level at least ashigh as f i, or exits the market altogether. Hereafter, we focus on the casewhere the market entry condition (13) holds for both manufacturers, anddo not study the degenerate case where only one manufacturer enters themarket.

Now we are ready to study the second stage problem on manufacturers’equilibrium service levels. Given the reliability index p for both manufac-turers, manufacturer i’s second stage expected profit vi depends on bothmanufacturers’ service levels. For the ease of exposition, we rewrite vi as afunction of f , i.e., vi(f |p) = di(f |p)ui(fi|p). For general yield distributions,the expected revenue per unit demand, ui(·), is not necessarily concave infi, and hence vi(f |p) is not necessarily log-concave in fi. Nevertheless, byAssumption 2, vi(f |p) is log-supermodular in f , and thus the following propo-sition ensues.

Proposition 2. The service level equilibrium f∗ exists.

While Proposition 2 shows the existence of service level equilibrium, it ischallenging to prove the uniqueness of the service level equilibrium f∗ undergeneral yield distributions. The following condition ensures the uniquenessof the equilibrium in-stock probabilities f∗, and we assume it holds for therest of the analysis.

Assumption 3. ΦYi(·, pi) follows a Bernoulli distribution with parameter pi.

Such an assumption is fairly common in the random yield literature(see Anupindi & Akella, 1993; Swaminathan & Shanthikumar, 1999, etc).Bernoulli random yield factors represent settings of a complete shutdown ofthe supplier’s production process, as in the case of machine breakdown orcontamination of raw materials. We focus on Bernoulli yield distribution

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for analytical tractability, but relax this assumption in our numerical exper-iment. In our extensive numerical study with general yield distribution, weobtain unique equilibrium in most cases, and therefore we believe the primaryinsight of our paper can be carried over to more general yield distributions.

Proposition 3. Assume Bernoulli yield distribution. (a) There exists aunique Nash equilibrium on the in-stock probability f∗, which satisfies

q∗i (fi)∂di(f |p)/∂fi + di(f |p)q∗′i (fi)di(f |p)ui(fi|p)

{(ri − wi + πi)EYi

[Yi]− (ri − si + πi)EYi[YiGǫ(Yiq

∗i (fi))]

}

+∂di(f |p)/∂fidi(f |p)ui(fi|p)

{(ri − si + πi)EYi

[ ∫ Yiq∗i (fi)

0

ǫgǫ(ǫ)dǫ

]− πi

}= 0. (14)

(b) Each manufacturer’s equilibrium in-stock probability is increasing in itsown process reliability, and nondecreasing its competitor’s process reliability.Moreover, an increase in a manufacturer’s process reliability leads to a largerincrease in its own equilibrium in-stock probability than its competitor’s, i.e.,∂f ∗

i /∂pi − ∂f ∗j /∂pi > 0.

5. First Stage: Supplier Improvement Competition

In this section, we characterize the manufacturers’ equilibrium behaviorof the first stage game, where each manufacturer chooses its improvementeffort to maximize its objective function given by (5).

5.1. Do Manufacturers Benefit From Higher Supplier Reliability?

In order to characterize the improvement effort z, it is helpful to first un-derstand the effect of target reliability index p (after improvement effort) oneach manufacturer’s expected profit. While a higher pi benefits manufactureri, it may also benefit manufacturer j through spillovers. In what follows weexplore the role of spillover index α on the combined effect of p.

Recall that the manufacturer i’s second-stage expected profit at the equi-librium in-stock probabilities is given by vi(f

∗|p) = di(f∗|p)ui(f ∗

i |p), wheref∗ = (f ∗

i , f∗j ) = (f ∗

i (pi, pj), f∗j (pi, pj)) are implicitly given by (14). One can

therefore treat vi(f∗|p) as a function of target reliability index p only, and,

for expositional ease we define v∗i (p) ≡ vi(f∗|p).

The indirect effect. The following proposition proves that, as one mightexpect, the focal manufacturer’s expected profit declines as its competitor’starget reliability increases.

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Proposition 4. Manufacturer i’s expected profit is (weakly) decreasing in itscompetitor’s reliability index, i.e., ∂v∗i (p)/∂pj ≤ 0.

The intuition for Proposition 4 is as follows. An increase in pj affectsmanufacturer i’s expected profit v∗i by affecting both manufacturers’ equilib-rium in-stock probabilities f ∗

i and f ∗j . However, v

∗i is not affected by its own

equilibrium in-stock probability at f∗. Hence, pj influences v∗i through itsimpact on f ∗

j only. Since manufacturer i’s demand is decreasing in f ∗j , the

increase in pj reduces manufacturer i’s expected profit.The direct effect. A more interesting question is whether manufacturer ialways benefits from an increase in its own target reliability index pi. Ap-plying the envelope theorem to v∗i , we have

∂v∗i (p)

∂pi=dvi(f

∗|p)dpi

=∂vi(f

∗|p)∂fi︸ ︷︷ ︸=0

∂f ∗i

∂pi+∂vi(f

∗|p)∂fj︸ ︷︷ ︸−

∂f ∗j

∂pi︸︷︷︸+

+∂vi(f

∗|p)∂pi︸ ︷︷ ︸+

, (15)

where ∂vi(f∗|p)/∂fj =

∂di(f∗|p)

∂fj︸ ︷︷ ︸−

ui follows from Assumption 2, ∂vi(f∗|p)/∂pi =

di∂ui(f

∗|p)∂pi︸ ︷︷ ︸+

follows from the proof of Proposition 3 (b), and ∂vi(f∗|p)/∂fi =

0 follows from (14). It is evident from (15), the reliability index pi affectsv∗i (p) both directly and indirectly (through in-stock probabilities). The directeffect, ∂vi(f

∗|p)/∂pi, is always positive. The indirect effect, (∂vi(f∗|p)/∂fi) ·

(∂f ∗i /∂pi) + (∂vi(f

∗|p)/∂fj) · (∂f ∗j /∂pi), is negative at equilibrium in-stock

probabilities, since a manufacturer is always hurt by an increase in its com-petitor’s in-stock probability. In general, (15) can be positive or negative,so manufacturer i may not necessarily benefit from an improvement in thesupplier’s reliability index pi. Under certain conditions, however, we are ableto unambiguously sign (15).

Proposition 5. If demand function is log-separable with

di(f) = ψi(fi)hi(fj), (16)

where dψi(fi)/dfi ≥ 0, dhi(fj)/dfj ≤ 0 and d2 logψi(fi)/df2i ≤ 0, then each

manufacturer’s second stage expected profit is strictly increasing in its ownreliability index, i.e., ∂v∗i (p)/∂pi > 0 (i = 1, 2).

14

Proposition 5 and its proof show that, with log-separable demand, manu-facturer i’s expected profit v∗i (p) increases in its own target reliability indexpi, and the rate of increase ∂v∗i (p)/∂pi = di · (∂ui/∂pi) depends on its owndemand only.The combined effect. Combining the indirect effect (Proposition 4) anddirect effect (Proposition 5), we have

∂Ji(p)

∂zi=∂v∗i (p)

∂pi︸ ︷︷ ︸+

+α∂v∗i (p)

∂pj︸ ︷︷ ︸−

− dmi(zi)

dzi︸ ︷︷ ︸+

. (17)

Since the spillover index α links the indirect effect (a negative impact) withthe direct effect (a positive impact) of an increase in zi, the net effect, i.e.,whether the focal manufacturer is better off by exerting improvement effortscan be ambiguous. In fact, we have observed numerical examples in whichthe focal manufacturer is better off (or worse off) with a higher reliabilityindex of the supplier. We thus conclude that the manufacturers may not nec-essarily benefit from improving supplier’s reliability, even if the improvementis costless, due to the spillover effect.

5.2. Equilibrium Improvement Efforts

5.2.1. The Case of No Spillover Effect (But with Demand Competition)

The case of no spillover effect with α = 0 can be loosely interpretedas a market setting where two manufacturers compete on demand but withdistinct technologies such that there is no “collaboration” on supplier im-provement effort. In this case, if demand function is log-separable and satis-fies Assumption 2, it can be shown that each manufacturer’s expected profit,Ji(·), is submodular in improvement efforts z. Since the set of manufacturers’feasible joint strategies is given by the nonempty convex and compact set:zi ∈ [0, 1− p0i ] (i = 1, 2), there exists a Nash equilibrium on the improvementefforts.

5.2.2. The Case of Positive Spillover Effect (But without Demand Competi-tion)

With positive spillovers, the manufacturers’ set of feasible joint strategiesis

zi ∈ [0, 1− p0i − αzj], i = 1, 2, j = 3− i.

15

We will show the existence of equilibrium using supermodular game ap-proach, which requires that the set of joint feasible strategies is a lattice, seeTopkis (1998). In the case of positive spillover effect, each manufacturer’sfeasible strategy set is dependent on the strategy adopted by its competitor,and the set of joint feasible strategies is not a lattice. We therefore consider asurrogate game with the following change of variables: (z1, z2) = (1− z1, z2).The set of feasible joint strategies in the surrogate game is the following:

z1 ∈ [p01 + αz2, 1], z2 ∈ [0, 1− α− p02 + αz1],

which is a lattice. We will utilize this surrogate game to show existenceof equilibrium improvement efforts in this subsection and next (§5.2.2 and§5.2.3).

The special case of positive spillovers without demand competition cap-tures settings where two manufacturers share a common supplier with sim-ilar technologies but they serve geographically separate markets. This casecan be modeled by setting hi(fj) = 1 in the log-separable demand modeldefined in (16). In this case, Assumption 2 is reduced to d′i(fi) ≥ 0 andd2 log di(fi)/df

2i ≤ 0. It follows from Proposition 5 that each manufacturer’s

second stage expected profit is independent of its competitor’s reliability in-dex and strictly increasing in its own reliability index.

If supplier improvement cost is fixed, we can further characterize eachmanufacturer’s best response function in optimal improvement effort.

Proposition 6. If supplier improvement cost is fixed, and each manufac-turer’s demand is a weakly increasing and log-concave function of its ownin-stock probability only, then both manufacturers would either target perfectreliability or exert no improvement effort at all.

(a) If its fixed improvement cost is greater than the maximum gain in itssecond stage expected profit from improved supply reliability, i.e., mi >v∗i (pi = 1) − v∗i (pi = p0i ), then manufacturer i exerts no improvementeffort.

(b) Otherwise, there exists a unique threshold zj ∈ [0, (1 − p0i )/α] suchthat manufacturer i would target perfect reliability if zj < zj, exertno improvement effort if zj > zj, and be indifferent between the twooptions if zj = zj.

(c) Moreover, if zj > 1− p0j , then manufacturer i will always target perfectreliability under any feasible strategy of manufacturer j.

16

Leveraging Proposition 6, we can now characterize the equilibrium be-havior of the manufacturers’ improvement efforts.

Proposition 7. Assume that each manufacturer’s demand is a weakly in-creasing and log-concave function of its own in-stock probability only.

(a) If supplier improvement cost is fixed, then each manufacturer’s optimalimprovement effort in the surrogate game is increasing in its competi-tor’s improvement effort. There exists at least one Nash equilibriumin the surrogate game. The set of equilibria is a lattice; there existsa componentwise smallest equilibrium z∗ and a componentwise largestequilibrium z∗.

(b) If the equilibrium improvement efforts exist and manufacturer i’s equi-librium effort z∗i is an interior solution2, then manufacturer i’s expectedprofit at the equilibrium weakly increases in the spillover effect.

Proposition 7 shows that, if manufacturers “collaborate” on supplier im-provement but have no direct competition on demand, then the manufac-turers’ improvement effort is decreasing in each other and there exists oneor more equilibria. Interestingly, despite the decrease in improvement effort,the spillover effect often leads to an increase in the expected profits.

5.2.3. The Case of Positive Spillover Effect with Demand Competition

When two manufacturers engage in general demand competition underpositive spillover effect in improvement efforts, it is very challenging to char-acterize the equilibrium behavior. We therefore assume the Cobb-Douglasdemand function, i.e.,

di(f) = γifβi

i f−βij

j (18)

with γi > 0, βi > 0 and βij > 0 for all i and j. The Cobb-Douglas demandfunction is commonly used in the literature (e.g., Bernstein & Federgruen,2004; Aydin & Porteus, 2008). By Lemma 4 in Appendix B, it is easilyverified that the Cobb-Douglas function (i.e., constant elasticity) is a specialform of log-separable function that satisfies Assumption 2. The followingproposition gives sufficient conditions for the existence of Nash equilibriumin the manufacturers’ improvement efforts.

2z∗i is an interior solution means that z∗i ∈ (0, 1− p0i − αz∗j ), i = 1, 2, j = 3− i.

17

Proposition 8. Assume that there is no shortage penalty cost (i.e., πi = 0)and each manufacturer’s demand function is a Cobb-Douglas function (see(18)).

(a) The equilibrium in-stock probability satisfies f ∗i = Kipi, where Ki de-

pends on cost parameters ri, wi, si and the distribution of ǫ only.

(b) Define

Mi = max

{βip

0j

βij+

βij + 1

(βi + 1)p0j,βiβijp

0i

+(βij + 1)p0iβi + 1

}, i = 1, 2, and M = max{M1,M2}.

If M ≤ 2, the first stage surrogate game is a supermodular game forall α ∈ [0, 1]; otherwise (M > 2), the first stage surrogate game is a

supermodular game for α ∈[0, 2

M+√M2−4

].

(c) When the first stage surrogate game is a supermodular game, it hasat least one equilibrium. The set of equilibria is a lattice; there existsa componentwise smallest equilibrium z∗ and a componentwise largestequilibrium z∗.

Proposition 8 shows that under certain conditions, the best response ofone manufacturer’s supplier improvement effort is again decreasing in itscompetitor’s supplier improvement effort in the original game, and there ex-ist one or more equilibria in the manufacturers’ improvement efforts. Animportant implication of this result is that despite a fairly intense marketcompetition as modeled by Cobb-Douglas demand function, equilibrium im-provement efforts exist, as long as the spillover effect, α, is bounded fromabove by 2/(M +

√M2 − 4), a constant that depends on demand parame-

ters and initial reliability index.In closing, we note that, under general demand functions, we have ob-

served numerical instances with no equilibrium, unique equilibrium or multi-ple equilibria in the manufacturers’ improvement efforts. The no-equilibriumcase typically arises when spillover effect is large, demand is highly sensitiveto in-stock probabilities (i.e., Cobb-Douglas demand function) and the man-ufacturers differ significantly in their market characteristics (i.e., wholesaleprices charged by the supplier). Typically, the best response of a manufac-turer is discontinuous in these settings, leading to the no equilibrium out-come. Mixed-strategy equilibrium might exist, but it is not the focus of ourpaper.

18

6. Managerial Implications

Since the existence or uniqueness of manufacturers’ equilibrium (in purestrategies) in general cannot be guaranteed, we do not analytically study thecomparative statics of the equilibrium. Instead, we conduct a comprehensivenumerical study to investigate market characteristics that influence manufac-turers’ equilibrium improvement efforts. In this numerical study, we employuniform yield distributions and linear demand functions.

6.1. Endowed Market Advantage

Oftentimes manufacturers differ from one another in the expected marketdemand even if they set identical service levels. Such persistent differences inmarket demand can often be attributed to exogenous factors such as brandimage, product design, and idiosyncratic consumer tastes. We refer to amanufacturer that enjoys a higher market demand with all else being equalas the one that has endowed market advantage (EMA). In our numericalstudy, EMA is measured by the difference in the base market demand of thetwo manufacturers.

It is unclear a priori whether the manufacturer with EMA would exerta higher or lower improvement effort relative to its competitor. On onehand, intuition suggests that it should exert less effort: it does not have tocompete as aggressively since it already enjoys a market advantage. On theother hand, intuition also suggests that it should exert more effort to exploitits market advantage more effectively.

Figure 1 illustrates the effect of EMA on manufacturers’ equilibrium im-provement efforts. (In Figure 1, yield is uniform Yi ∼ [1 − 0.6/pi, 1], wherep0i = 1.0; demand is linear (di(f) = ai + bifi − cijfj), where the base valuesare ai = 0.5, bi = 1.0, and cij = 0.5; demand uncertainty ǫ is normal withµ = 1.0 and σ = 0.3; the other base values are ri = 1.0, wi = 0.5, si = 0.2,and πi = 0.3; improvement cost is mi(zi) = z2i . Figure 1 is obtained by vary-ing α and a1 while fixing all other parameter values.) Figure 1(a) shows howthe ratio of equilibrium improvement effort (z1/z2−1) varies by the spillovereffect, and the arrow shows the direction when the EMA (as measured bythe value of a1− a2) increases or decreases. Figure 1(b) shows how the ratioof equilibrium improvement effort (z1/z2− 1) varies by the EMA (a1− a2),and the arrow shows the direction when the spillover effect α increases.

Figure 1 indicates that the manufacturer with EMA tends to exert ahigher effort level relative to its competitor, and this phenomenon becomes

19

0 0.2 0.4 0.6 0.8 1−30%

−20%

−10%

0%

10%

20%

30%

40%

Spillover α

Rat

io o

f Opt

imal

Impr

ovem

ent E

ffort

(z 1/z

2−1)

x100

%

(a) Effect of Knowledge Spillover (α)on Equilibrium Improvement Effort

−1.0 −0.6 −0.2 0.2 0.6 1.0−30%

−20%

−10%

0%

10%

20%

30%

40%

Endowed Market Advantage (a1−a

2)

(b) Effect of Endowed Market Advantage (a1−a

2)

on Equilibrium Improvement Effort

Manufacturer 1 increasinglysuffers from market disadvantage

Manufacturer 1 enjoys increasingendowed market advantage

Increasing Spillover α

Increasing Spillover α

Figure 1: Effect of EMA on manufacturers’ equilibrium improvement efforts

more pronounced as the spillover index α increases. As the spillover effectincreases, both manufacturers’ improvement efforts decline, but the speed ofdecline is slower for the manufacturer with EMA. As a result, its relativeimprovement effort increases in α.

One naturally wonders whether the fact that the manufacturer with EMAexerts an increasingly higher proportion of improvement efforts also trans-lates into a relative profit advantage. Figure 2(a) shows that both manu-facturers’ expected profits increase in the spillover effect α, but the relativeprofit advantage of the manufacturer with EMA (manufacturer 1) declinesas the spillover effect increases (Figure 2(b)). Thus, the manufacturer withEMA may view the spillover effect with some sort of dilemma: in absoluteprofit terms, its increasingly higher proportion of improvement effort pays offbecause its expected profit increases in α; in relative profit terms, however,its profit advantage (over its competitor) declines, suggesting that its everlarger proportion of improvement effort benefits its competitor more thanitself.

Back to the UTC’s example, its Pratt & Whitney’s ambivalence aboutthe spillover effect is somewhat understandable. Although Pratt & Whitneytrails GE and Rolls Royce in terms of overall market share (Malone, 2010),

20

0 0.2 0.4 0.6 0.8 110

15

20

25

30

35

40

45

50

Spillover α

Opt

imal

Exp

ecte

d P

rofit

(a) Effect of Knowledge Spillover (α)on Optimal Expected Profit

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

3.5

4

Spillover α

Rat

io o

f Opt

imal

Exp

ecte

d P

rofit

(J 1/J

2)

(b) Effect of Knowledge Spillover (α)on Relative Expected Profit

Manufacturer 2 with market disadvantage

Manufacturer 1 with endowed market advantage

Ratio of expected profit when manufacturer 1 has

endowed market advantage

Manufacturer 1 & 2 with identical market advantage

Ratio of expected profit whenManufacturer 1 & 2 have

identical market advantage

Figure 2: Effect of EMA on equilibrium expected profits

it dominates “the commercial aircraft market with its jet engines” (Hinton,2007). Thus, although its supplier improvement effort increases its own ex-pected profit, such improvement effort may benefit its competitors more thanitself. The latter concern may be the reason for Pratt &Whitney’s hesitation.Nevertheless, the expected benefit seems to outweigh the latter concern, witha result that Pratt & Whitney reluctantly engages in improvement effort.

In the Toyota’s example, existing evidence suggests that, despite signif-icant spillover effects, Toyota expends much more improvement effort thanits major competitors (Liker & Choi, 2004). Based on our results, one mayconjecture that, all else being equal, Toyota enjoys an endowed market ad-vantage, at least before the recent quality woes. Toyota’s high level of im-provement effort also suggests that Toyota is concerned perhaps more aboutits expected profit than its relative profit advantage over its competitors.

6.2. Manufacturer’s Pricing Power with Supplier

Manufacturers often can negotiate different wholesale prices wi, depend-ing on their relative power and their relationship structure with the sup-plier. We refer to the manufacturer that can negotiate a lower wi as the onewith more pricing power. It is unclear whether the manufacturer with morepricing power would exert more or less improvement efforts relative to its

21

competitors. Nevertheless, we can form some intuitions from the followingproposition.

Proposition 9. Assume Assumption 1-3. Each manufacturer’s equilibriumin-stock probability is (a) decreasing in its wholesale price, and weakly de-creasing in its competitor’s wholesale price; (b) increasing in its unit price,unit salvage value and unit penalty cost, and weakly increasing in its com-petitor’s unit price, unit salvage value and unit penalty cost.

Proposition 9 proves that manufacturer i’s equilibrium in-stock proba-bility f ∗

i increases if its wholesale price wi decreases. This suggests that ifmanufacturer i has more pricing power, it will set a higher in-stock proba-bility in equilibrium. Intuition suggests that the manufacturer would benefitmore from an improvement in supplier yield, and exert a higher level ofimprovement efforts relative to its competitor.

Our numerical observation (not illustrated here) confirms the above intu-ition, i.e., the manufacturer with more pricing power exerts a higher propor-tion of improvement effort relative to its competitor. The implication on theexpected profit is also qualitatively similar to our earlier observation in §6.1,that is, both manufacturers’ expected profits increase in spillover effect, butthe relative profit advantage (of the manufacturer with more pricing power)declines as spillover effect increases, suggesting that the increasingly higherproportion of improvement effort exerted by the focal manufacturer benefitsits competitor more than itself.

6.3. Competition Intensity

To study the impact of competition intensity, we consider the settingwhere two manufacturers are symmetric, i.e., cij = cji = ci = c, ai = a, andbi = b. To capture the market competition effect, we use the ratio of c/b inthe linear demand model (recall that di(f) = ai+ bifi− cijfj) to measure thecompetition intensity, and we change the value of the ratio c/b by fixing b andvarying c. If c/b = 0, then each manufacturer’s demand depends on its ownin-stock probability only (i.e., no direct competition), whereas if c/b = 1,then each manufacturer’s demand depends on its own in-stock probabilityas much as it depends on its competitor’s in-stock probability (i.e., perfectsubstitution/competition).

Figure 3 illustrates the effect of competition intensity on the manufactur-ers’ equilibrium improvement efforts. Figure 3 suggests that the manufactur-ers’ improvement efforts decline as competition intensifies. Moreover, under

22

0 0.2 0.4 0.6 0.8 10.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Spillover α

Opt

imal

Impr

ovem

ent E

ffort

(a) Effect of Knowledge Spillover (α)on Equilibrium Improvement Effort

0% 20% 40% 60% 80% 100%0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Market Competition (ci/b

i)

(b) Effect of Market Competition (ci/b

i)

on Equilibrium Improvement Effort

Increasing market competition

Decreasing spillover

Figure 3: Effect of market competition on manufacturers’ equilibrium improvement efforts

intense competition, both manufacturers significantly reduce their improve-ment efforts as spillover effect increases.

Figure 4 illustrates the effect of market competition on the manufactur-ers’ equilibrium profits. Figure 4(a) illustrates the percentage change in themanufacturers’ expected profits as spillover effect increases, where we use theexpected profits under α = 0 as the base case. Interestingly, the spillover ef-fect improves the manufacturers’ expected profits, and the relative benefit inexpected profit is increasing in market competition. Figure 4(b) shows thatmarket competition has a detrimental effect on the manufacturers’ expectedprofits, whereas spillover effect improves the manufacturers’ expected profits.

In summary, intense market competition is often a characteristic of com-modity products, where manufacturers use similar technologies and henceare more likely to experience spillover effects. Our numerical results suggestthat spillover effect in this type of market should be viewed favorably, andmanufacturers should not try to curb the supplier’s spillover effect in thistype of market.

6.4. Market Uncertainty

Different industries often exhibit different levels of market uncertainty:some industries (e.g., fashion) exhibits considerably higher market uncer-

23

0 0.2 0.4 0.6 0.8 10%

1%

2%

3%

4%

5%

6%

7%

8%

Spillover α

Per

cent

age

Cha

nge

in O

ptim

al E

xpec

ted

Pro

fit

(a) Effect of Knowledge Spillover (α) on%Change in Optimal Expected Profit

0% 20% 40% 60% 80% 100%15

20

25

30

35

40

45

50

Market Competition (ci/b

i)

Opt

imal

Exp

ecte

d P

rofit

(b) Effect of Market Competition (ci/b

i)

on Optimal Expected Profit

Increasing spillover

Increasing market competition

Figure 4: Effect of market competition on equilibrium expected profits

tainty than others (e.g., aerospace). It is therefore of interest to understandhow market uncertainty influences the manufacturers’ equilibrium improve-ment efforts and the resulting expected profits.

In this section, we study the impact of market uncertainty by varying thestandard deviation of the random term of the demand (ǫ) from 0.1 to 0.5,while keeping its mean constant at 1. It turns out that the effect of marketuncertainty on the manufacturers’ improvement efforts (not illustrated here)is very similar to the effect of market competition discussed in §6.3. Thatis, as in the case of market competition, both higher market uncertainty andspillover effect reduce the manufacturers’ equilibrium improvement efforts.

Figure 5 illustrates the effect of market uncertainty on the manufactur-ers’ expected profits. Figure 5 shows that market uncertainty reduces themanufacturers’ expected profits, both in relative and absolute terms. How-ever, at any level of market uncertainty, the spillover effect improves themanufacturers’ expected profits. This suggests that the spillover effect maybenefit the manufacturers, especially when manufacturers are similar in theircharacteristics.

24

0 0.2 0.4 0.6 0.8 10%

1%

2%

3%

4%

5%

6%

7%

Spillover α

Per

cent

age

Cha

nge

in O

ptim

al E

xpec

ted

Pro

fit

(a) Effect of Knowledge Spillover (α) on%Change in Optimal Expected Profit

0.1 0.2 0.3 0.4 0.522

24

26

28

30

32

34

36

38

40

Market Uncertainty (σ)

Opt

imal

Exp

ecte

d P

rofit

(b) Effect of Market Uncertainty (σ)on Optimal Expected Profit

Increasing market uncertainty

Increasing spillover

Figure 5: Effect of market uncertainty on equilibrium expected profits

7. Conclusion

Supplier reliability is one of the key determinant of manufacturers’ com-petitiveness. To ensure reliable supply delivery, more and more companiestake a proactive role and invest directly in their suppliers’ reliability im-provement. In this research, we investigate how potential spillover influencesmanufacturers’ supplier improvement efforts in a competitive setting.

We analyze a supply chain with two manufacturers, who compete on ser-vice levels (measured by in-stock probability) in their end product market.The manufacturers share a common supplier whose production process is sub-ject to random output. Each manufacturer may exert effort to improve thesupplier’s delivery reliability, but the enhanced reliability may spill over toits competitor. We develop a two-stage model that encompasses supplier im-provement, spillover effect and random yield in a competitive setting. In thefirst stage, each manufacturer simultaneously decides its supplier improve-ment effort. In the second stage, after exerting supplier improvement effortbut before yield and demand uncertainty is resolved, each manufacturer de-cides its in-stock probability and its order quantity from the supplier. In thiscomplex model, we analytically characterize each manufacturer’s equilibriumin-stock probability and order quantity in the second stage. Moreover, we

25

characterize sufficient conditions for the existence of the equilibrium of themanufacturers’ improvement efforts in the first stage. In addition, we conductextensive numerical experiments and provide managerial insights on marketcharacteristics that influence the manufacturers’ equilibrium improvementefforts.

When there is spillover effect, we analytically show that the manufacturersmay not necessarily benefit from increased supplier reliability, even if sup-plier improvement is costless. Under certain conditions, we prove that eachmanufacturer’s expected equilibrium profit weakly increases in the spillovereffect. Our extensive numerical study confirms that our analytical findingscan be carried over to the more general settings. In particular, in all ournumerical experiments, both manufacturers’ expected equilibrium profits al-ways increase in the spillover effect. Moreover, our numerical study alsobrings about additional insights on the impact of spillover effect and mar-ket characteristics. We find that manufacturers typically exert less effortsin the equilibrium as the spillover effect increases. When manufacturers areasymmetric, the ex ante more advantageous manufacturer typically exertsa higher proportion of improvement efforts compared to its competitor, es-pecially when the spillover effect is large. While such higher proportion ofimprovement efforts benefits the focal manufacturer, it may benefit, in rela-tive profit terms, its competitor more. When manufacturers are symmetric,both manufacturers’ equilibrium improvement efforts and expected equilib-rium profits decline in market competition or market uncertainty.

In the context of our modeling framework and numerical study, our resultssuggest that the presence of spillover effect may not necessarily be viewed asa negative thing. Manufacturers should have a more open-minded attitudeand engage in supplier improvement without worrying too much about thespillover effect.

We hope this work encourages future research in the area of reliabilityimprovement. A number of interesting extensions are worth further explo-ration. First, the supplier may initiate improvement effort as well. In thiscase, who is in a better position to improve process reliability, the supplieror the manufacturers? Second, manufacturers may procure from multiplesuppliers, and it is of interest to contrast the effect of supplier competitionwith manufacturers’ improvement efforts. We hope future studies by us andothers will bring additional insights to this important area of research.

26

References

Anderson, S. P., de Palma, A., & Thisse, J.-F. (1992). Discrete Choice Theoryof Product Differentiation. Cambridge, MA: MIT Press.

Anupindi, R., & Akella, R. (1993). Diversification under supply uncertainty.Management Science, 39 , 944–963.

Aydin, G., & Porteus, E. L. (2008). Joint inventory and pricing decisions foran assortment. Operations Research, 56 , 1247–1255.

Babich, V., Burnetas, A. N., & Ritchken, P. H. (2007). Competition anddiversification effects in supply chains with supplier default risk. Manu-facturing & Service Operations Management , 9 , 123–146.

Bernstein, F., & Federgruen, A. (2004). A general equilibrium model forindustries with price and service competition. Operations Research, 52 ,868–886.

Chao, X., Chen, H., & Zheng, S. (2008). Joint replenishment and pricing de-cisions in inventory systems with stochastically dependent supply capacity.European Journal of Operational Research, 191 , 142–155.

Ciarallo, F. W., Akella, R., & Morton, T. E. (1994). A periodic review,production planning model with uncertain capacity and uncertain demand:Optimality of extended myopic policies. Management Science, 40 , 320–332.

Dada, M., Petruzzi, N. C., & Schwarz, L. B. (2007). A newsvendor’s pro-curement problem when suppliers are unreliable. Manufacturing & ServiceOperations Management , 9 , 9–32.

D’Aspremont, C., & Jacquemin, A. (1988). Cooperative and noncooperativeR&D in duopoly with spillovers. The American Economic Review , 78 ,1133–1137.

Dyer, J. H., & Nobeoka, K. (2000). Creating and managing a high-performance knowledge-sharing network: The Toyota case. Strategic Man-agement Journal , 21 , 345–367.

Erdem, A. (1999). Inventory models with random yield. Chapter 4, PhDdissertation, .

27

Erhardt, G., Langlinais, T., & Ratta, V. (2010). The call to become customersmart . Technical Report Accenture.

Farney, S. (2000). United Technologies Corporation: Supplier developmentinitiative. Darden Business Publishing, University of Virginia. UVA-M-0646.

Federgruen, A., & Yang, N. (2009). Optimal supply diversification undergeneral supply risks. Operations Research, 57 , 1451–1468.

Federgruen, A., & Yang, N. (2009b). Competition under generalized attrac-tion models: Applications to quality competition under yield uncertainty.Management Science, 55 , 2028–2043.

Gilbert, S. M., Xia, Y., & Yu, G. (2006). Strategic outsourcing for competingOEMs that face cost reduction opportunities. IIE Transactions , 38 , 903–915.

Gupta, S. (2008). Channel structure with knowledge spillovers. MarketingScience, 27 , 247–261.

Gurler, U., & Parlar, M. (1997). An inventory problem with two randomlyavailable suppliers. Operations Research, 45 , 904–918.

Harhoff, D. (1996). Strategic spillovers and incentives for research and de-velopment. Management Science, 42 , 907–925.

Hinton, C. (2007). Pratt & Whitney aims high for big-jet engine sales. Mar-ketWatch, Nov 29, 2007.

Horn, M. (1979). Some Inequalities for The Expectation of A Product ofFunctions of A Random Variable and for The Multivariate DistributionFunction at A Random Point. Biometrical Journal , 21 , 243–245.

Humphreys, P. K., Li, W. L., & Chan, L. Y. (2004). The impact of supplierdevelopment on buyer-supplier performance. Omega, 32 , 131–143.

Inderfurth, K., & Vogelgesang, S. (2013). Concepts for safety stock determi-nation under stochastic demand and different types of random productionyield. European Journal of Operational Research, 224 , 293301.

28

Kamien, M. I., Muller, E., & Zang, I. (1992). Research joint ventures andR&D cartels. The American Economic Review , 82 , 1293–1306.

Kazaz, B., & Webster, S. (2011). The impact of yield-dependent tradingcosts on pricing and production planning under supply uncertainty. Man-ufacturing & Service Operations Management , 13 , 404–417.

Krause, D. R., & Ellram, L. M. (1997). Critical elements of supplier devel-opment: The buying-firm perspective. European Journal of Purchasing &Supply Management , 3 , 21–31.

Lascelles, D. M., & Dale, B. G. (1990). Examining the barriers to supplierdevelopment. International Journal of Quality & Reliability Management ,7 , 46–56.

Levin, R. C., & Reiss, P. C. (1988). Cost-reducing and demand-creatingR&D with spillovers. The RAND Journal of Economics , 19 , 538–556.

Liker, J. K., & Choi, T. Y. (2004). Building deep supplier relationships.Harvard Business Review , 82 , 104–113.

Malone, S. (2010). Analysis: Rolls Royce woes could boost GE, Pratt &Whitney. Reuters, Nov 5, 2010.

Markoff, J. (2001). Ignore the label, it’s flextronics inside; outsourcing’s newcachet in silicon valley. The New York Times.

Milgrom, P., & Roberts, J. (1990). Rationalizability, learning, and equi-librium in games with strategic complementarities. Econometrica, 58 ,1255–1277.

Modi, S. B., & Mabert, V. A. (2007). Supplier development: Improvingsupplier performance through knowledge transfer. Journal of OperationsManagement , 25 , 42–64.

Parlar, M., & Perry, D. (1996). Inventory models of future supply uncertaintywith single and multiple suppliers. Naval Research Logistics , 43 , 191–210.

Qi, A., Ahn, H.-S., & Sinha, A. (2011). Investing in a shared supplier in acompetitive market. Working paper, University of Michigan, .

29

Sako, M. (2004). Supplier development at Honda, Nissan and Toyota: com-parative case studies of organizational capability enhancement. Industrialand Corporate Change, 13 , 281–308.

Schmitt, A. J., Snyder, L. V., & Shen, Z.-J. M. (2010). Inventory systemswith stochastic demand and supply: Properties and approximations. Eu-ropean Journal of Operational Research, 206 , 313328.

Spence, M. (1984). Cost reduction, competition, and industry performance.Econometrica, 52 , 101–122.

Suzumura, K. (1992). Cooperative and noncooperative r&d in an oligopolywith spillovers. The American Economic Review , 82 , 1307–1320.

Swaminathan, J. M., & Shanthikumar, J. G. (1999). Supplier diversification:effect of discrete demand. Operations Research Letters , 24 , 213–221.

Talluri, S., Narasimhan, R., & Chung, W. (2010). Manufacturer coopera-tion in supplier development under risk. European Journal of OperationalResearch, 207 , 165173.

Tomlin, B. (2006). On the value of mitigation and contingency strategies formanaging supply-chain disruption risks. Management Science, 52 , 639–657.

Topkis, D. M. (1998). Supermodularity and Complementarity . (1st ed.).Princeton University Press.

Wang, Y., Gilland, W., & Tomlin, B. (2010). Mitigating supply risk: Dualsourcing or process improvement? Manufacturing & Service OperationsManagement , 12 , 489–510.

Xu, M., & Lu, Y. (2013). The effect of supply uncertainty in price-settingnewsvendor models. European Journal of Operational Research, 227 ,423433.

Yano, C. A., & Lee, H. L. (1995). Lot sizing with random yields: A review.Operations Research, 43 , 311–334.

Yeo, W. M., & Yuan, X.-M. (2011). Optimal inventory policy with supplyuncertainty and demand cancellation. European Journal of OperationalResearch, 211 , 2634.

30

Appendix A. Proofs of “Improving Reliability of a Shared Sup-

plier with Competition and Spillovers”

Proof of Lemma 1. By definition of ui(fi, qi(f)|p, fj) (see (8)), we have

∂ui(fi, qi(f)|p, fj)∂qi(f)

= (ri − wi + πi)EYi[Yi]− (ri − si + πi)EYi

[YiGǫ(Yiqi(f))].

(A.1)

It follows that ∂2ui(fi, qi(f)|p, fj)/∂qi(f)2 = −(ri−si+πi)EYi[Y 2

i gǫ(Yiqi(f))] <0. Thus, ui(fi, qi(f)|p, fj) is concave in qi(f). The lemma statement followsfrom the fact that q∗i (f) is either an interior solution satisfying (A.1) or acorner solution constrained by (7).

Proof of Lemma 2. (a) It is immediate from (9) and (10) that q∗i is indepen-dent of fj, therefore q

∗i is a function of fi only, i.e., q

∗i = q∗i (fi). (b) If q∗i

satisfies (9), q∗i (fi) is independent of fi, and hence dq∗i (fi)/dfi = 0. In con-trast, if q∗i (fi) satisfies (10), by the Implicit Function Theorem, dq∗i (fi)/dfi =1/EYi

[Yigǫ(Yiq∗i (fi))] > 0. Let qci (fi) and qii(fi) denote the stocking factor

implied by (10) and (9), respectively. As shown above, qci (fi) strictly in-creases in fi, whereas q

ii(fi) remains constant in fi. By the interior point

theorem, there exists a unique point f i at which qci (f i) = qii(f i). By (9)and (10), we have f i = EYi

[Gǫ(Yiqci (f i))] ≥ EYi

[YiGǫ(Yiqii(f i))] = (ri − wi +

πi)/(ri − si + πi)EYi[Yi] > 0. Note that (ri −wi + πi)/(ri − si + πi)EYi

[Yi] =EYi

[YiGǫ(Yiqii(f i))] ≥ EYi

[Yi]EYi[Gǫ(Yiq

ii(f i))] = EYi

[Yi]EYi[Gǫ(Yiq

ci (f i))] =

EYi[Yi]f i, where the first inequality follows from Lemma 1 in Horn (1979).

We have f i ≤ (ri−wi+πi)/(ri− si+πi) < 1. (c) is immediate from (b).

Proof of Proposition 1. Part (a). Using (11), we have

dvi(fi|p, fj)dfi

=ddi(fi|p, fj)

dfiui(fi|p) + di(fi|p, fj)

dui(fi|p)dfi

. (A.2)

By (12), we have

dui(fi|p)dfi

=dq∗i (fi)

dfi

{(ri − wi + πi)EYi

[Yi]− (ri − si + πi)EYi[YiGǫ(Yiq

∗i (fi))]

},

(A.3)

d2ui(fi|p)df 2

i

=d2q∗i (fi)

df 2i

{(ri − wi + πi)EYi

[Yi]− (ri − si + πi)EYi[YiGǫ(Yiq

∗i (fi))]

}

−(dq∗i (fi)

dfi

)2

(ri − si + πi)EYi

[Y 2i gǫ(Yiq

∗i (fi))

]. (A.4)

1

By (A.3) and Lemma 2, dui(fi|p)/dfi = 0 when fi ≤ f i. In addition, becauseq∗i (fi) satisfies (9) when fi ≤ f i, (A.2) can be simplified to

dvi(fi|p, fj)dfi

∣∣∣∣fi≤f i

=ddi(fi|p, fj)

dfiui(fi|p)

=ddi(fi|p, fj)

dfi

{(ri − si + πi)EYi

[ ∫ Yiq∗

i (fi)

0

ǫgǫ(ǫ)dǫ

]− πi

}.

(A.5)

We can show that when fi ≤ f i, either vi(fi|p, fj) < 0 , or vi(fi|p, fj) > 0and increases in fi. It follows that it is never optimal for manufacturer i toset fi ≤ f i, given its competitor’s service level fj . Therefore, manufactureri’s best response f ∗

i |fj must satisfy f ∗i |fj ≥ f i.

Part (b). Manufacturer i enters the market iff there exists an fi such thatvi(fi|p, fj) > 0. Since di(fi|p, fj) > 0 always holds, manufacturer i enters themarket iff there exists an fi such that ui(fi|p) > 0. By (A.3) and Lemma 2,dui(fi|p)/dfi = 0 when fi ≤ f i, and dui(fi|p)/dfi < 0 when fi > fi. Hence,manufacturer i participates if and only if ui(f i|p) > 0, which is equivalent

to (ri − si + πi)EYi

[∫ Yiq∗

i (fi)

0ǫgǫ(ǫ)dǫ

]> πi.

Proof of Proposition 2. Note that vi(f |p) = di(f |p)ui(fi|p), where di(f |p) isa function of fi and fj, while ui(fi|p) is a function of fi only. Therefore,∂2 log vi(f |p)

∂fi∂fj= ∂2 log di(f |p)

∂fi∂fj≥ 0 by Assumption 2, vi(f |p) is log-supermodular

in f . Since each manufacturer’s set of feasible in-stock probability values isa compact lattice, it follows that the game is log-supermodular. Hence, theequilibrium in-stock probabilities f∗ exist, see Topkis (1998) or Milgrom &Roberts (1990).

Proof of Proposition 3. Part (a). A unique Nash equilibrium (f ∗i , f

∗j ) is guar-

anteed, by Milgrom & Roberts (1990), if

∂2 log vi(f |p)/∂f 2i + ∂2 log vi(f |p)/∂fi∂fj < 0.

By (11), vi(f |p) = di(f |p)ui(fi|p), where di(f |p) is a function of fi and fj ,while ui(fi|p) is a function of fi only. Note that

∂2 log vi(f |p)∂f 2

i

+∂2 log vi(f |p)

∂fi∂fj=∂2 log di(f |p)

∂f 2i

+∂2 log di(f |p)

∂fi∂fj+d2 log ui(fi|p)

df 2i

=∂2 log di(f |p)

∂f 2i

+∂2 log di(f |p)

∂fi∂fj+u′′i (fi|p)ui(fi|p)− u′2i (fi|p)

u2i (fi|p),

2

where ∂2 log di(f |p)/∂f 2i + ∂2 log di(f |p)/∂fi∂fj ≤ 0 follows from Assump-

tion 2. We will show that under Bernoulli yield distribution, ui is log-concave in fi, i.e., u′′i (fi|p)ui(fi|p) − u′2i (fi|p) < 0. Then uniqueness ofequilibrium will be guaranteed. Under Bernoulli yield distribution, we have

q∗i = G−1ǫ (fi/pi),

dq∗idfi

= 1pigǫ(q∗i )

,d2q∗idf2

i

=−g′ǫ(q

∗

i )

p2i g3ǫ (q

∗

i ). We therefore can simplify

(A.4) to

d2ui(fi|p)df 2

i

=−g′ǫ(q∗i )p2i g

3ǫ (q

∗i )

{(ri − wi + πi)pi − (ri − si + πi)fi

}− ri − si + πi

pigǫ(q∗i )

.

(A.6)

Note that (ri−wi+πi)− (ri−si+πi)fi/pi ≤ 0 (by Lemma 1 under Bernoulli

distribution). Therefore, if g′ǫ(q∗i ) ≤ 0, we can get d2ui(fi|p)

df2

i

< 0. Otherwise,

if g′ǫ(q∗i ) > 0, we can rewrite (A.6) as

d2ui(fi|p)df 2

i

=−g′ǫ(q∗i )(ri − wi + πi)

pig3ǫ (q∗i )

+(ri − si + πi)

(fipig′ǫ(q

∗i )− g2ǫ (q

∗i ))

pig3ǫ (q∗i )

=−g′ǫ(q∗i )(ri − wi + πi)

pig3ǫ (q∗i )

+(ri − si + πi) (Gǫ(q

∗i )g

′ǫ(q

∗i )− g2ǫ (q

∗i ))

pig3ǫ (q∗i )

< 0.

The inequality follows from Assumption 1. Therefore, in either case, we

have d2ui(fi|p)/df 2i < 0. So d2 log ui(fi|p)

df2

i

=u′′

i (fi|p)ui(fi|p)−u′2

i (fi|p)u2

i

< 0, and

∂2 log vi(f |p)/∂f 2i + ∂2 log vi(f |p)/∂fi∂fj < 0. The uniqueness of equilibrium

is guaranteed.Note that under Bernoulli distribution, ui is log-concave in fi, hence vi is

log-concave in fi. Therefore, the equilibrium in-stock probability f ∗i either is

an interior solution that satisfies the first order condition specified in (14),or a corner solution of either fi (lower bound of fi) or pi (upper bound offi under Bernoulli distribution). We will show as follows that the cornersolution cannot be optimal.

At fi = fi, dui(fi|p)/dfi = 0 by (A.2), (A.3) and the definition of fi.

Sodvi(fi|p,fj)

dfi|fi=fi =

ddi(fi|p,fj)dfi

ui(fi|p) > 0, which means vi is increasing at

fi = fi. Therefore, fi cannot be optimal. Bernoulli distribution gives anupper bound of fi = pi. At fi = pi, q

∗i = G−1

ǫ (1) = +∞. And we can derivethat vi|fi=pi = di

(− pi(wi − si)q

∗i + pi(ri − si + πi)− πi

)< 0. Hence, fi = pi

cannot be optimal as well. Therefore, the equilibrium in-stock probabilitycan only be the interior solution that satisfies the first order condition.

3

Part (b). Define, for i = 1, 2, Li(f) = ∂ log vi(f |p)/∂fi = (∂di(f |p)/∂fi)/di(f |p)+u′i(fi|p)/ui(fi|p), where u′i(fi|p) is given by (A.3). We refer di(f) as di(f |p)and ui(fi) as ui(fi|p) in the following proof for the simplicity. By the ImplicitFunction Theorem, we have

∂f ∗1

∂p1= − det

∣∣∣∣∣∂L1

∂p1

∂L1

∂f2∂L2

∂p1∂L2

∂f2

∣∣∣∣∣

/det

∣∣∣∣∣∂L1

∂f1

∂L1

∂f2∂L2

∂f1∂L2

∂f2

∣∣∣∣∣ ,∂f ∗

2

∂p1= − det

∣∣∣∣∣∂L1

∂f1

∂L1

∂p1∂L2

∂f1∂L2

∂p1

∣∣∣∣∣

/det

∣∣∣∣∣∂L1

∂f1

∂L1

∂f2∂L2

∂f1∂L2

∂f2

∣∣∣∣∣ .

(A.7)

First we will show that the determinant of denominating matrix in (A.7) ispositive at f ∗

i . Note that

det(L) ≡ det

∣∣∣∣∣∂L1

∂f1

∂L1

∂f2∂L2

∂f1∂L2

∂f2

∣∣∣∣∣ =∂2 log v1∂f 2

1

∂2 log v2∂f 2

2

− ∂2 log v1∂f1∂f2

∂2 log v2∂f1∂f2

. (A.8)

As we showed in proof of Proposition 3 (a), ∂2 log vi/∂f2i +∂

2 log vi/∂fi∂fj <0. And note that ∂2 log vi/∂fi∂fj = ∂2 log di/∂fi∂fj ≥ 0 by Assumption2. We can derive that ∂2 log vi/∂f

2i < 0, |∂2 log vi/∂f 2

i | > |∂2 log vi/∂fi∂fj |.Therefore, (A.8) is positive. Hence, the key to determine the sign of ∂f ∗

1 /∂p1and ∂f ∗

2 /∂p1 is to find the sign of the determinant of the numerator matricesin (A.7).

Now consider each term in the numerator matrices in (A.7). For nota-tional ease, in what follows we suppress the dependence of di and qi on thevector of f . We have

∂Li(f)

∂fi=

∂2di(f)

∂f2

i

di(f)−(

∂di(f)∂fi

)2

d2i (f)+u′′i (fi)ui(fi)− u′2i (fi)

u2i (fi), (A.9)

∂Li(f)

∂fj=

∂2di(f)∂fi∂fj

di(f)− ∂di(f)∂fi

∂di(f)∂fj

d2i (f), (A.10)

∂Li(f)

∂pi=

1

u2i (fi)

(∂u′i(fi)

∂piui(fi)−

∂ui(fi)

∂piu′i(fi)

). (A.11)

∂Li(f)

∂pj= 0. (A.12)

By Proposition 3 (a), (A.9) is negative. It is straightforward to prove that(A.10) is nonnegative. Next we prove that (A.11) is positive. Note that

4

∂u′

i(fi)

∂pi=

∂q∗′

i

∂pi

{(ri−wi+πi)EYi

[Yi]−(ri−si+πi)EYi[YiGǫ(Yiq

∗i )]

}+q∗

′

i

{(ri−

wi + πi)∂ EYi

[Yi]

∂pi− (ri − si + πi)

∂ EYi[YiGǫ(Yiq∗i )]∂pi

}, and ∂ui(fi)

∂pi=

∂q∗i∂pi

{(ri − wi +

πi)EYi[Yi]− (ri − si + πi)EYi

[YiGǫ(Yiq∗i )]

}+ q∗i

{(ri −wi + πi)

∂ EYi[Yi]

∂pi− (ri −

si+πi)∂ EYi[YiGǫ(Yiq

∗

i )]∂pi

}+(ri−si+πi)

∂ EYi

[∫ Yiq∗

i0

ǫgǫ(ǫ)dǫ

]

∂pi. Simplifying the above

two expressions with Bernoulli yield distribution, we have

∂u′i(fi)

∂pi=∂q∗

′

i

∂pipiA

′i + q∗

′

i (ri − wi + πi), (A.13)

∂ui(fi)

∂pi=∂q∗i∂pi

piA′i + q∗i (ri − wi + πi) + (ri − si + πi)

{∫ q∗i

0

ǫgǫ(ǫ)dǫ− q∗iGǫ(q∗i )

},

(A.14)

where A′i = (ri−wi+πi)− (ri− si+πi)Gǫ(q

∗i ) < 0. Next we prove that both

(A.13) and (A.14) are positive. With Bernoulli yield distribution, we haveq∗i = G−1

ǫ (fi/pi), q∗′i = 1/(pigǫ(q

∗i )) > 0, ∂q∗i /∂pi = −Gǫ(q

∗i )/(pigǫ(q

∗i )) <

0, and ∂q∗′

i /∂pi = −1/(pigǫ(q∗i ))

2[gǫ(q∗i ) − g′ǫ(q

∗i )Gǫ(q

∗i )/gǫ(q

∗i )] < 0, where

the last inequality follows from Assumption 1. Using these expressions, itis immediate that (A.13) is positive. Now consider (A.14), which can besimplified to

∂ui(fi)

∂pi= −Gǫ(q

∗i )

gǫ(q∗i )A′

i + q∗i (ri − wi + πi) + (ri − si + πi)

{∫ q∗i

0

ǫgǫ(ǫ)dǫ− q∗iGǫ(q∗i )

}

= −Gǫ(q∗i )

gǫ(q∗i )A′

i + q∗iA′i + (ri − si + πi)

∫ q∗i

0

ǫgǫ(ǫ)dǫ > 0, (A.15)

where the inequality follows from the fact that q∗iA′i+(ri−si+πi)

∫ q∗i0ǫgǫ(ǫ)dǫ >

0. Therefore, (A.11) is positive (because u′i(fi) < 0). We have now provedthat ∂Li(a, f)/∂pi > 0.

The statement ∂f ∗i /∂pi > 0 follows from the fact that the determinant of

the numerator matrix is negative and the determinant of the denominatingmatrix is positive. The statement for ∂f ∗

j /∂pi ≥ 0 can be analogously shown.We now prove the statement ∂f ∗

i /∂pi − ∂f ∗j /∂pi > 0. Using (A.7), we

have ∂f ∗1 /∂p1 = −∂L1/∂p1·∂L2/∂f2

det(L)and ∂f ∗

2 /∂p1 =∂L1/∂p1·∂L2/∂f1

det(L). By (A.9) and

5

(A.10), and Proposition 3 (a), we have |∂L2/∂f2| > |∂L2/∂f1|. Therefore,we have ∂f ∗

1 /∂p1 − ∂f ∗2 /∂p1 > 0. ∂f ∗

2 /∂p2 − ∂f ∗1 /∂p2 > 0 can be shown

analogously.

Proof of Proposition 4. Note that

∂v∗i (p)

∂pj=dvi(f

∗|p)dpj

=∂vi(f

∗|p)∂fi

∂f ∗i

∂pj+∂vi(f

∗|p)∂fj

∂f ∗j

∂pj

=∂vi(f

∗|p)∂fj

∂f ∗j

∂pj=∂di(f

∗|p)∂fj

∂f ∗j

∂pjui(f

∗i |p) ≤ 0,

where the second equality comes from the fact that f ∗i satisfies (14).

Proof of Proposition 5. Under given conditions, it can be verified that Assmp-tion 2 holds, see Lemma 4 in Appendix B for details. Substituting (16) into(A.10), we have ∂Li

∂fj= 0. By (A.7), ∂f ∗

i /∂pj = 0. Then (15) can be simplifiedto

∂v∗i (p)

∂pi=dvi(f

∗|p)dpi

= di(f∗|p)∂ui(f

∗i |p)

∂pi> 0, (A.16)

where the last inequality follows from (A.15).

Proof of Proposition 6. First, note that when manufacturer j’s improvementeffort zj ≥ (1 − p0i )/α, manufacturer i’s reliability index pi = p0i + αzj ≥ 1.In this case, manufacturer i will not exert any improvement effort.

We’ll now focus on the case when manufacturer j’s improvement effortzj ≤ (1 − p0)/α. When the improvement cost is a fixed cost (i.e., mi(zi) =mi), ∂Ji/∂zi = ∂v∗i /∂pi + α∂v∗i /∂pj − dmi/dzi = ∂v∗i /∂pi > 0. Therefore,if manufacturer i exerts improvement effort, it will target pi = 1, and theresulting expected profit is Ji(pi = 1) = v∗i (pi = 1)−mi which is a constantthat only depends on primitive parameters. If manufacturer i does not exertimprovement effort, its reliability index is pi = p0i + αzj and its expectedprofit is Ji(pi = p0i + αzj) = v∗i (pi = p0i + αzj). Hence, given manufacturerj’s improvement effort zj , manufacturer i will compare Ji(pi = 1) = v∗i (pi =1)−mi and Ji(pi = p0i +αzj) = v∗i (pi = p0i +αzj), and target on the reliabilityindex pi that maximizes its expected profit.

Note that Ji(pi = 1) = v∗i (pi = 1) − mi is independent of zj . Onthe other hand, since ∂Ji(pi = p0i + αzj)/∂zj = ∂v∗i (pi = p0i + αzj)/∂zj =α∂v∗i (pi = p0i + αzj)/∂pi > 0, Ji(pi = p0i +αzj) = v∗i (pi = p0i +αzj) is strictly

6

increasing in zj . If mi > v∗i (pi = 1) − v∗i (pi = p0i ), then Ji(pi = 1) <v∗i (pi = p0i + αzj) for all zj ≥ 0, so manufacturer i will exert no improve-ment effort on the supplier. Otherwise, Ji(pi = 1) ≥ v∗i (pi = p0i + αzj) whenzj = 0. Note that when zj = (1 − p0i )/α, Ji(pi = 1) = v∗i (pi = 1) − mi ≤v∗i (pi = 1) = v∗i (pi = p0i + αzj). Therefore, there exists a unique thresholdzj ∈ [0, (1− p0i )/α], which only depends on primitive parameters, such thatJi(pi = 1) > Ji(pi = p0i + αzj) for zj < zj , Ji(pi = 1) = Ji(pi = p0i + αzj) forzj = zj and Ji(pi = 1) < Ji(pi = p0i + αzj) for zj > zj .

Finally, note that any feasible strategy of manufacturer j satisfies: zj ≤1 − p0j − αzi ≤ 1 − p0j . Therefore, if zj > 1 − p0j , manufacturer i will alwaystarget perfect reliability under any feasible strategy of manufacturer j.

Proof of Proposition 7. Part (a). We explicitly write out each manufacturer’sbest response using Proposition 6.

z1(z2) =

{1− p01 − αz2 for 0 ≤ z2 < z20 for z2 ≤ z2

, z2(z1) =

{1− p02 − αz1 for 0 ≤ z1 < z10 for z1 ≤ z1

.

In the surrogate game where the strategy set is z1 = 1− z1 and z2 = z2, eachmanufacturer’s best response is

z1(z2) =

{p01 + αz2 for 0 ≤ z2 < z21 for z2 ≤ z2

, z2(z1) =

{0 for z1 ≤ 1− z11− p02 − α(1− z1) for 1− z1 < z1 ≤ 1

.

Clearly, z1(z2) is a (weakly) increasing function of z2 on the intervals 0 ≤z2 < z2 and z2 ≤ z2, respectively. Moreover, p01 + αz2 ≤ 1 on the interval0 ≤ z2 < z2, because z2 ≤ (1 − p01)/α by Proposition 6. Thus, z1(z2) isa (weakly) increasing function of z2. Similarly, we can show that z2(z1)is a (weakly) increasing function of z1. Therefore, each manufacturer has(weakly) increasing best response to his competitor’s strategy. As mentionedin the discussion of the surrogate game, the set of feasible joint strategies inthe surrogate game is a lattice. The rest of the proof follows from Tarski’sfixed point theorem.

Part (b). Suppose that the equilibrium improvement efforts exist and letz∗i be the equilibrium improvement efforts undertaken by manufacturer i. Ifeach manufacturer’s demand is a weakly increasing and log-concave functionof its own in-stock probability only, then ∂v∗(p)/∂pj = 0, and v∗i is a functionof pi only. Note that Ji(z

∗i , z

∗j ) = v∗i (p

0i + z∗i + αz∗j ) − mi(z

∗i ). If z∗i is the

interior solution, we havedJi(z∗i ,z

∗

j )

dα=

∂Ji(z∗i ,z∗

j )

∂zi

∂z∗i∂α

+∂Ji(z∗i ,z

∗

j )

∂α=

∂Ji(z∗i ,z∗

j )

∂α=

∂v∗i (p)

∂piz∗j ≥ 0.

7

Proof of Proposition 8. Part(a). When there is no penalty cost (πi = 0) andthe demand function is a Cobb-Douglas function, we can simplify (14) to the

following form: βi

[G−1

ǫ ( fipi)((ri − wi)

pifi− (ri − si)

)+(ri−si)pifi

∫ G−1ǫ (

fipi

)

0 ǫgǫ(ǫ)dǫ]+

(ri−wi)−(ri−si)fipi

gǫ(G−1ǫ (

fipi

))= 0. Hence, Ki ≡ f∗

i

piis the solution that satisfies the above

equation which only depends on primitive cost parameters ri, wi, si and thedistribution of ǫ.

Part(b). Substitute f ∗i = Kipi to (12) and let πi = 0, we have: ui =

pi[G−1ǫ (Ki)(ri −wi − (ri − si)Ki) + (ri − si)

∫ G−1ǫ (Ki)

0ǫgǫ(ǫ)dǫ]. Note that the

part in the square bracket is dependent on the cost parameters ri, wi, siand the distribution of ǫ only, and independent of reliability index pi and pj .Therefore, ui is a linear function of pi. Denote

Bi ≡[G−1

ǫ (Ki) (ri − wi − (ri − si)Ki) + (ri − si)

∫ G−1ǫ (Ki)

0

ǫgǫ(ǫ)dǫ

]

for convenience. We can rewrite v∗i as follows: v∗i = γi(Kipi)

βi(Kjpj)−βijpiBi =

γiBiKβi

i K−βij

j pβi+1i p

−βij

j ,∂2v∗i∂p2i

= γiBiKβi

i K−βij

j βi(βi + 1)pβi−1i p

−βij

j ,∂2v∗i∂p2j

=

γiBiKβi

i K−βij

j βij(βij+1)pβi+1i p

−βij−2j , and

∂2v∗i∂pi∂pj

= −γiBiKβi

i K−βij

j (βi+1)βijpβi

i p−βij−1j .

As mentioned in the discussion of the surrogate game, the set of feasible jointstrategies in the surrogate game is a lattice. In the surrogate game, we have∂2Ji∂zi∂zj

=∂2v∗i∂zi∂zj

= −α(

∂2v∗i∂p2

i

+∂2v∗i∂p2

j

)−(α2+1)

∂2v∗i∂pi∂pj

= γiBiKβi

i K−βij

j pβi

i p−βij−1j (βi+

1)βijα[−(

βi

βij

pjpi+

βij+1

βi+1pipj

)+(α + 1

α

)]. Note that p0i ≤ pi ≤ 1 and p0j ≤

pj ≤ 1, therefore p0j ≤ pj/pi ≤ 1/p0i .βi

βij

pjpi

+βij+1

βi+1pipj

≤ max{

βi

βijp0j +

βij+1

βi+11p0j, βi

βij

1p0i

+βij+1

βi+1p0i}≡ Mi. Note that M = max{M1,M2}. If M ≤ 2,

then α+ 1α≥ 2 ≥M ≥ βi

βij

pjpi+

βij+1

βi+1pipj

for both i = 1, 2. Therefore, ∂2Ji∂zi∂zj

≥ 0

for all α ∈ [0, 1] and both i = 1, 2. The surrogate game is a supermodulargame.

If M > 2, then α + 1α≥ M for 0 ≤ α ≤ 2

M+√M2−4

< 1. Consequently,

when 0 ≤ α ≤ 2M+

√M2−4

, α + 1α≥ M ≥ βi

βij

pjpi

+βij+1

βi+1pipj

for both i = 1, 2.

Therefore, ∂2Ji∂zi∂zj

≥ 0 for α ∈[0, 2

M+√M2−4

]and both i = 1, 2. The surrogate

game is a supermodular game when α ∈[0, 2

M+√M2−4

].

8

Part (c). These results follow directly from standard results of supermod-ular games, e.g., Theorem 4.2.1 in Topkis (1998).

Proof of Proposition 9. We use a similar proof approach to that developed inthe proof of Proposition 3 (b), and, for brevity, we omit many intermediatesteps below but the details can be easily gleaned from the proof of Proposition3 (b). First consider wi. Similar to (A.7), we have

∂f1∂w1

= − det

∣∣∣∣∣∂L1

∂w1

∂L1

∂f2∂L2

∂w1

∂L2

∂f2

∣∣∣∣∣

/det

∣∣∣∣∣∂L1

∂f1∂L1

∂f2∂L2

∂f1

∂L2

∂f2

∣∣∣∣∣ ,∂f2∂w1

= − det

∣∣∣∣∣∂L1

∂f1∂L1

∂w1

∂L2

∂f1

∂L2

∂w1

∣∣∣∣∣

/det

∣∣∣∣∣∂L1

∂f1∂L1

∂f2∂L2

∂f1

∂L2

∂f2

∣∣∣∣∣ .

(A.17)

One can verify that ∂L2/∂w1 = 0, and hence the key to determine the signof ∂f1/∂w1 is determining the sign of ∂L1/∂w1. Note that

∂Li(f)

∂wi

=1

u2i (fi)

(∂u′i(fi)

∂wi

ui(fi)−∂ui(fi)

∂wi

u′i(fi)

). (A.18)

We now prove that the term in the bracket in (A.18) is negative. By

(12) and (A.3), we have∂u′

i(fi)

∂wiui(fi) − ∂ui(fi)

∂wiu′i(fi) = −dq∗i (fi)

dfiEYi

[Yi]ui(fi) +

q∗i (fi)EYi[Yi]u

′i(fi) = EYi

[Yi](−dq∗i (fi)

dfiui(fi)+q

∗i (fi)u

′i(fi)

)= −EYi

[Yi]dq∗i (fi)

dfi

((ri−

si + πi)EYi

∫ Yiq∗

i (fi)

0ǫgǫ(ǫ)dǫ − πi

)< 0, where the inequality follows from the

fact that dq∗i (fi)/dfi > 0 (part (b) of Lemma 2) and (ri−si+πi)EYi

∫ Yiq∗

i (fi)

0ǫgǫ(ǫ)dǫ−

πi > 0. It follows from (A.17) that ∂f1/∂w1 < 0. The statement for∂f2/∂w1 ≤ 0 can be analogously proved. The proposition statements withrespect to ri, si and πi can be analogously proved.

Appendix B. Discussion on Assumption 2

Assumption 2 is satisfied by many different forms of demand functions,including the linear demand function, the log-separable demand function,and the general class of attraction models (e.g., Anderson et al., 1992; Fed-ergruen & Yang, 2009b) that is widely used in the operations and marketingliterature. Next, we prove, under mild conditions, that these three types ofdemand functions satisfy Assumption 2.

Lemma 3 (Linear Model). Assume di(f) is given by the following linearmodel

di(f) = ai + bifi − cijfj (B.1)

9

with ai, bi, and cij being nonnegative constants. If∑2

j=1 ∂di(f)/∂fj ≥ 0 forall i, i.e., bi ≥ cij for all i, then di(f) satisfies Assumption 2.

Proof of Lemma 3. Using (B.1), we have ∂di(f)/∂fi = bi ≥ 0, ∂di(f)/∂fj =−cij ≤ 0, ∂2di(f)/∂f

2i = 0, and ∂2di(f)/∂fi∂fj = 0. Taking the logarithm

transformation and relevant derivatives, we have

∂2 log di(f)

∂f 2i

=

∂2di(f)∂f2

i

di(f)−(

∂di(f)∂fi

)2

d2i (f)= −

(∂di(f)∂fi

)2

d2i (f)≤ 0,

∂2 log di(f)

∂fi∂fj=

∂2di(f)∂fi∂fj

di(f)− ∂di(f)∂fi

∂di(f)∂fj

d2i (f)= −

∂di(f)∂fi

∂di(f)∂fj

d2i (f)≥ 0.

Note that∂2 log di(f)

∂f 2i

+∂2 log di(f)

∂fi∂fj= −

∂di(f)∂fi

(∂di(f)∂fi

+ ∂di(f)∂fj

)

d2i (f)≤ 0.

The last inequality comes from ∂di(f)/∂fi+∂di(f)/∂fj ≥ 0. Therefore, di(f)satisfies all the conditions in Assumption 2.

Lemma 4 (Log-Separable Model). Assume di(f) is given by the followinglog-separable model

di(f) = ψi(fi)hi(fj) (B.2)

with dψi(fi)/dfi ≥ 0 and dhi(fj)/dfj ≤ 0. If ψi(fi) is log-concave in fi, i.e.,d2 logψi(fi)/df

2i ≤ 0, then di(f) satisfies Assumption 2.

Proof of Lemma 4. Using (B.2), we have ∂di(f)/∂fi = ψ′ihi ≥ 0, ∂di(f)/∂fj =

ψih′i ≤ 0, where for notational ease we have suppressed the dependence

of ψi(·) on fi and hi(·) on fj, and used ′ to denote the first order deriva-tive of ψi(fi) with respect to fi (or hi(fj) with respect to fj). Note thatlog di(f) = logψi(fi) + log hi(fj). Taking the logarithm transformation andusing ψi(fi) is log-concave in fi, we have

∂2 log di(f)

∂f 2i

=d2 logψi(fi)

df 2i

≤ 0,∂2 log di(f)

∂fi∂fj= 0,

∂2 log di(f)

∂f 2i

+∂2 log di(f)

∂fi∂fj=∂2 log di(f)

∂f 2i

=d2 logψi(fi)

df 2i

≤ 0.

Hence, di(f) satisfies all the conditions in Assumption 2.

10

Lemma 5 (Attraction Model). Assume di(f) is given by the following at-traction model

di(f) =Mxi(fi)∑2j=0 xj(fj)

, (B.3)

where M denotes the average total market size, x0 is a constant denoting thevalue of the no-purchase option, and xi(fi) (i = 1, 2) denotes manufactureri’s product attraction value which is increasing and log-concave in its servicelevel, i.e., dxi(fi)/dfi ≥ 0 and d2 log xi(fi)/df

2i ≤ 0. If

2∑

j=1

∂di(f)

∂fj≥ 0 for all i, (B.4)

then di(f) satisfies Assumption 2.

The technical condition (B.4) implies that a uniform increase in servicelevels by both manufacturers will not decrease any manufacturer’s demand.

Proof of Lemma 5. Using (B.3) (and scaling M = 1), we have

∂di(f)

∂fi=

x′i(x0 + xj)

(x0 + xi + xj)2≥ 0,

∂di(f)

∂fj= −

xix′j

(x0 + xi + xj)2≤ 0,

∂2di(f)

∂f 2i

=(x0 + xj) (x

′′i (x0 + xi + xj)− 2x′2i )

(x0 + xi + xj)3,

∂2di(f)

∂fi∂fj=x′ix

′j(xi − x0 − xj)

(x0 + xi + xj)3,

where for notational ease we have suppressed the dependence of xi(·) on fi,and used ′ and ′′ to denote the first and second order derivative of xi(fi).Taking the logarithm transformation and relevant derivatives, we have

∂2 log di(f)

∂f 2i

=

∂2di(f)

∂f2

i

di(f)−(

∂di(f)∂fi

)2

d2i (f)=

(x0 + xj) (x′′i xi(x0 + xi + xj)− x′2i (x0 + 2xi + xj))

x2i (x0 + xi + xj)2,

∂2 log di(f)

∂fi∂fj=

∂2di(f)∂fi∂fj

di(f)− ∂di(f)∂fi

∂di(f)∂fj

d2i (f)=

x′ix′j

(x0 + xi + xj)2≥ 0.

Note that

∂2 log di(f)

∂f 2i

=(x0 + xj) (x

′′i xi(x0 + xi + xj)− x′2i (x0 + 2xi + xj))

x2i (x0 + xi + xj)2

<(x0 + xj)(x

′′i xi − x′2i )

x2i (x0 + xi + xj)=

x0 + xjx0 + xi + xj

d2 log xidf 2

i

< 0.

11

The last inequality follows from the assumption that xi is log-concave. There-

fore, di(f) is log-concave in fi. We will further show that ∂2 log di(f)

∂f2

i

+∂2 log di(f)∂fi∂fj

≤0. Note that

∂2 log di(f)

∂f 2i

+∂2 log di(f)

∂fi∂fj=

(x0 + xj)(x0 + xi + xj)(x′′i xi − x′2i )− xix

′i

((x0 + xj)x

′i − xix

′j

)

x2i (x0 + xi + xj)2.

By condition (B.4), we have

∂di(f)

∂fi+∂di(f)

∂fj=x′i(x0 + xj)− xix

′j

(x0 + xi + xj)2≥ 0.

Therefore,

∂2 log di(f)

∂f 2i

+∂2 log di(f)

∂fi∂fj=

(x0 + xj)(x0 + xi + xj)(x′′i xi − x′2i )− xix

′i

((x0 + xj)x

′i − xix

′j

)

x2i (x0 + xi + xj)2

≤ (x0 + xj)(x0 + xi + xj)(x′′i xi − x′2i )

x2i (x0 + xi + xj)2=

x0 + xjx0 + xi + xj

d2 log xidf 2

i

≤ 0.

Hence, di(f) satisfies all the conditions in Assumption 2.

12