Improving Supplier Yield under Knowledge Spillover - Arizona State

Submitted tomanuscript August 2011

Improving Supplier Yield under Knowledge Spillover

Yimin WangW. P. Carey School of Business, Arizona State University, Tempe, AZ 85287, yimin [email protected]

Yixuan XiaoOlin School of Business, Washington University in St. Louis, St. Louis, MO 63130, [email protected]

Nan YangOlin School of Business, Washington University in St. Louis, St. Louis, MO 63130, [email protected]

We study the effect of knowledge spillover on OEM manufacturers’ efforts to improve suppliers’ process

yield. The OEM manufacturers compete with imperfectly substitutable products, and they share a common

component supplier whose production process is subject to random output due to variations in quality or pro-

duction yield. We characterize the manufacturers’ equilibrium improvement efforts and provide managerial

insights on market characteristics that influence the manufacturers’ equilibrium improvement efforts. With

spillover effect, each manufacturer’s feasible strategy set of improvement effort is dependent on the strategy

adopted by its competitor, and the set of joint feasible strategies is not a lattice. As such, the existence

of equilibrium on improvement effort is not guaranteed. Nevertheless, we are able to characterize sufficient

conditions for the existence of the manufacturers’ equilibrium improvement efforts with a surrogate game.

From a managerial point of view, we find that, contrary to common wisdom, spillover effect often brings

favorable improvement to the manufacturers’ expected profits, regardless of whether the manufacturers are

asymmetric or symmetric. We also find that manufacturers’ improvement efforts typically decline in mar-

ket competition or market uncertainty, although, paradoxically, both manufacturers benefit from increased

spillover effect.

Key words : knowledge spillover; random yield; supplier improvement

1. Introduction

Across industries (e.g. automobile, electronics, and aero spaces), manufacturers increasingly depend

on their suppliers for competitive advantage. To ensure that suppliers are capable, manufacturers

often engage in supplier improvement initiatives, for example, to improve supplier quality or yield.

United Technologies Corporation (UTC), for example, treat supplier improvement as an essen-

tial element of their corporate strategy, and they “provide numerous resources for suppliers to

support continuous improvement and lean manufacturing.” (United Technologies 2011) Supplier

1

Wang, Xiao, and Yang: Improving Supplier Yield under Knowledge Spillover2 Article submitted to ; manuscript no. August 2011

improvement is also prevalent in the automobile industry, where major manufacturers often form

close relationship with suppliers (Liker and Choi 2004). In fact, many manufacturers, and notably

Japanese manufacturers, have long engaged in active supplier improvement, e.g. investing in tech-

nology knowhow, to enhance the supplier’s operational performance, such as delivery, average cost,

production yield, and quality.

Suppliers, however, often work with multiple manufacturers (Markoff 2001), which can create

intricate incentive conflicts in manufacturers’ supplier improvement efforts. In particular, when one

manufacturer engages in supplier improvement activities, the knowledge gained in the supplier’s

manufacturing process may spillover to other manufacturers. In this case, a manufacturer’s supplier

improvement effort may inadvertently benefit its direct competitors. Empirical evidence suggests

that manufacturing OEMs are often aware of, and also wary of, the potential knowledge spillover,

but different manufacturers seem to take the issue differently.

Farney (2000) describes the 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 plan to initiate supplier improvement with DGT despite the fact that (a) DGT has close

relationship with a P&W competitor, Rolls Royce, and (b) Rolls had engaged in “projects to

help DGT improve delivery reliability through improved manufacturing flexibility.” (p.4). More

troubling, perhaps, is the fact that, historically, “by leveraging its expertise and knowledge gained

from working with UTC, Dynamic successfully won bids from Allied Signal, Rolls Royce, and

General Electric.” (p.5).

Although being aware of the competitive concerns above, “UTC had attempted to bring process

improvements to DGT operations. DGT had received substantial assistance from P&W in the form

of workshops in Kaizen process improvement, lean manufacturing, and UTC’s ACE program for

quality improvement.”. (p.5). UTC were, however, wary of the potential spillover: “Their [supplier

improvement] goal did not include the scenario where the supply chain partner sold parts into

the aftermarket, competed with UTC for other business, and partnered with UTC’s competitors.”

Wang, Xiao, and Yang: Improving Supplier Yield under Knowledge SpilloverArticle submitted to ; manuscript no. August 2011 3

(p.8). In this case, UTC seem to reluctantly accept the existence of knowledge spillover from their

supplier improvement effort.

Toyota, on the other hand, seems to have a more open mind towards knowledge spillover: “Pro-

duction processes are simply not viewed as proprietary and Toyota accepts that some valuable

knowledge will spill over to benefit competitors.” (Dyer and Nobeoka 2000, p.358). In addition,

“Toyota provide guidance, quite consciously bringing our attention to other aspects from which

we can gain for ourselves regardless of Toyota. They tell us from time to time to direct our kaizen

effort to these aspects.” (p.291)

Yet in Honda’s case, its supplier improvement program implicitly encourages the suppliers to

work with competitors: Honda espouses suppliers’ self-reliance, i.e. “balancing responsiveness to

Honda’s needs with a sufficiently diversified customer base.” (Sako 2004, p.296). In fact, encour-

aging suppliers to work with multiple manufacturers is often a policy of many firms to avoid the

“captive supplier issue” (Lascelles and Dale 1990, p.53).

To study the effect of potential knowledge spillover on manufacturers’ supplier improvement

efforts, we consider the scenario where two manufacturers share a common component supplier,

which has a production process that is subject to random output due to variations in quality

or production yield. We adopt the commonly used stochastically proportional yield model, as

surveyed in Yano and Lee (1995), to describe the supplier’s random production process. The

stochastically proportional yield model is not as restrictive as it might appear in the context of

supplier development: lean manufacturing effort, for example, is often aimed at reducing production

waste which is directly related to yield. In addition, process quality and capability, as reflected by

Cp and Cpk indices (Montgomery 2004), can also be represented by production yield. On demand

side, we consider a class of demand functions under widely satisfied conditions (including general

attraction model, linear model and log-separable model), that allow us to incorporate service levels

(in-stock probability) offered by the two manufacturers. Both manufacturers may exert effort to

improve the reliability (e.g. expected yield) of the supplier’s production process.


We characterize the manufacturers’ equilibrium improvement efforts and provide managerial

insights on the market characteristics that influence their equilibrium improvement efforts. The

presence of spillover effect makes the manufacturers’ feasible strategy sets (in improvement effort)

dependent on each other, and the set of joint feasible strategies is not a lattice. As such, the

existence of equilibrium on improvement effort is not guaranteed. In fact, we have observed numer-

ical instances of no equilibrium, unique equilibrium and multiple equilibria in the manufacturers’

improvement efforts. Nevertheless, under various conditions, we are able to characterize the manu-

facturers’ equilibrium improvement efforts with a surrogate game. From a managerial point of view,

we find that, contrary to common wisdom, the spillover effect often brings favorable improvement

to the manufacturers’ expected profits, regardless of whether the manufacturers are asymmetric or

symmetric.

When manufacturers are asymmetric, we find that the ex ante more advantageous manufacturer

typically exerts a higher proportion of improvement efforts, especially when the spillover effect is

large. While such higher proportion of improvement efforts benefits the focal manufacturer, it may

also benefit, in terms of relative profit, its competitor more. Thus, the ex ante less advantageous

manufacturer may prefer a higher spillover effect than the ex ante more advantageous one does.

When manufacturers are symmetric, we find that both manufacturers’ improvement efforts

decline in market competition or market uncertainty, although, paradoxically, both manufacturers

benefit from increased spillover effect. In the context of our modeling framework, our findings seem

to support Toyota’s (and Honda’s) attitude on supplier improvement effort: one should engage in

supplier improvement without worrying too much about the spillover effect.

2. Literature

This research is closely related with several streams of existing literature in supplier development,

information spillover, random yield, and market competition. In what follows we examine these

four streams of literature in turn.

A large body of extant literature examines firms’ supplier development efforts, with a majority of

them empirically investigating the antecedents of successful supplier development effort as well as


key elements for success in supplier development. Some of these studies rely upon primary survey

data to isolate important common factors that influence supplier development, e.g. Krause and

Ellram (1997), Krause (1997), Humphreys et al. (2004), and Modi and Mabert (2007). Using a

sample of 215 manufacturing firms in the US, for example, Modi and Mabert (2007) find that prior

evaluation and certification are two important prerequisites for manufacturers to initiate supplier

development program. They also find that collaborative communication is one of the key factors

for the supplier development program to be successful. In a similar vein, surveying 142 electronic

manufacturers in Hong Kong, Humphreys et al. (2004) identify seven factors that influence the per-

formance of supplier improvement effort, and they find that trust, strategic objectives, and effective

communication are among the most critical antecedents for successful supplier development. Simi-

larly, using surveys of 527 firms, Krause (1997) empirically examines different approaches a buying

firm may use to engage in supplier improvement, where the top three methods used include perfor-

mance feedback, inviting supplier’s personnel, and site visits (p.14). Krause (1997) concludes that

supplier improvement is generally viewed favorably by the manufacturers but further improvements

are still possible. A concurrent study in Krause and Ellram (1997) identifies additional elements

that contribute to successful supplier development, including top management involvement and

buyer’s clout.

Some other empirical studies rely on more in-depth case studies to contrast supplier development

practices at different firms. A notable example is Sako (2004), who compares supplier development

effort using case studies of three major automobile manufacturers: Toyota, Honda, and Nissan.

Sako (2004) finds that shared practice is often more effective than standard training sessions where

tacit knowledge cannot be easily transmitted. In addition, supplier development requires a certain

level of the manufacturer’s intervention on the supplier’s internal process, which requires a proper

form of corporate governance to make the supplier development program effective. Similar studies

also include Dyer and Nobeoka (2000) as well as references in Krause and Ellram (1997) for earlier

case studies.


All of the above study (except for Dyer and Nobeoka (2000)) focus on the dyadic buyer-supplier

relationships and do not explore the knowledge spillover. In contrast, Stump and Heide (1996),

Bensaou and Anderson (1999), and Kang et al. (2007) explore general opportunism in the buyer-

supplier relationship and the related control mechanism, such as screening and monitoring, to

mitigate such opportunistic behaviors. For example, Kang et al. (2007) note that “the exchange

relationship between an OEM supplier and its buyer enables the supplier to develop capabilities that

over time enable this OEM supplier to gain economically profitable business from other buyers.”

(p.14). Note that in these studies the opportunism includes knowledge spillover but also includes

many other aspects such as direct competition and misappropriate valuable information between

the buyer and the supplier. Recently, Bernstein and Kok (2009) study suppliers’ own improvement

efforts in an assemble to order system. They consider spillover effects as an extension, but they

mainly focus on two different contracting mechanisms to motive the suppliers to exert improvement

efforts.

There is a large stream of literature that studies pure information exchange and information

spillover in different settings, e.g. information sharing to reduce cost (Bonte and Wiethaus 2005),

information misappropriation (Baiman and Rajan 2002), demand information sharing (Lee et al.

2000, Li 2002, Zhang 2002), and information sharing in an R&D setting (Harhoff 1996). None

of these studies consider supplier improvement, nor do they consider supplier process reliability

issues, such as random yield or quality.

Our research is related with the vast literature in random yield, e.g. Gerchak and Parlar (1990),

Anupindi and Akella (1993), Federgruen and Yang (2009a) and Dada et al. (2007). None of these

papers consider supplier improvement effort. A recent paper by Wang et al. (2010) contrasts sup-

plier improvement strategy with diversification strategy under random capacity (yield). They do

not, however, consider knowledge spillover, market competition, or service level effects considered

in this paper. We refer interested readers to Yano and Lee (1995) for an excellent review of the

literature in random yield.


Our research is also related with the stream of literature that considers market competition on

the basis of strategic choices other than price (alone). Bernstein and Federgruen (2004, 2007) model

an industry in which make-to-stock suppliers of goods that are (perfect or imperfect) substitutes,

compete by selecting both a price and a service level, defined as the item’s fill rate, i.e. the fraction

of the demand which can be met immediately from stock. In their infinite horizon models, the

expected per period demand volume of each supplier is given by a general function of all prices

and all service levels offered in the industry. As an alternative to suppliers committing themselves

to a given fill rate, Bernstein and de Vericourt (2008) assume the firms offer a buyer-specific direct

penalty for each unit and each unit of time a buyer needs to wait for its demand to be filled.

De Vany and Saving (1983), So (2000), Cachon and Harker (2002) and Chayet and Hopp (2005)

model industries with suppliers of make-to-order goods or services, who compete by selecting a

price as well as either a capacity level or a waiting time guarantee. These authors assume that all

buyers or consumers aggregate the offered price and waiting time into a single full price measure

selecting the supplier whose full price is lowest. Allon and Federgruen (2008, 2009, 2007) model

settings where the suppliers’ goods or services are imperfect substitutes and the demand volumes

faced by each depend on all prices and waiting time standards in the industry, according to a

general set of demand functions.

A few recent papers analyze game theoretic models for industries in which the suppliers face

uncertain yields. Corbett and Deo (2009) assume that an arbitrary number of suppliers, offering a

homogenous good, engage in Cournot competition, where the (common) per unit price is given by

a linear function of the total actual supply offered to the market. (A firm’s actual supply is given

by its intended production volume multiplied with a random yield factor, generated from a com-

mon Normal distribution.) The firms compete by selecting their intended production volumes. As

standard Cournot competition models, this model does not capture demand uncertainty. Corbett

and Deo (2009) use their model to explain the number of flu vaccine suppliers in the United States.

Also motivated by the flu vaccine supply problem, Chick et al. (2008) consider a supply chain with

a single buyer and a single supplier, whose random yield factor follows a general distribution. The


buyer derives a benefit from its order, the magnitude of which grows as a concave function of the

order size. The authors characterize how the buyer and the supplier sequentially determine their

order size and intended production volume, respectively. Federgruen and Yang (2009b) analyze a

supply chain in which suppliers compete by targeting specific key characteristics of their uncertain

yield processes. Their analysis of the Stackelberg game starts with a characterization of the optimal

procurement policies of the M buyers, in response to chosen yield characteristics by the N suppli-

ers. To the best of our knowledge, our research is the first to consider supplier improvement under

random yield, knowledge spillover, stochastic demand and imperfectly substitutable products.

To sum up, existing literature has either adopted the case or survey based approach to iden-

tify important antecedents for supplier development, or adopted the modeling approach to study

the random yield and/or market competition without consideration of supplier improvement. Our

research provides the first model to study supplier improvement effort under both knowledge

spillover and market competition.

The rest of this paper is organized as follows. We describe the model in §3 and present some

preliminary analysis in §4. The manufacturers’ equilibrium improvement efforts are characterized

in §5, and managerial insights are discussed in §6. We then conclude in §7. All proofs are contained

in the Appendix.

3. Model

We consider a newsvendor model with two manufacturers that source a common component from

an unreliable supplier, i.e. the supplier’s delivery may be less than the quantity ordered. The man-

ufacturers compete against each other with imperfectly substitutable products, and their market

demand is determined by a general demand model described below.

3.1. Demand

Each manufacturer’s market demand is stochastic, imperfectly substitutable, and influenced by

both manufacturers’ product service levels. The service level here refers to in-stock probability

(Type-1 service level), as market demand is stochastic. If a manufacturer has a low service level, it


may experience a product shortage which can have a negative impact on its demand. Conversely,

a high service level often has a positive impact on market demand.

Let Di(f) denote the random demand of manufacturer i, given the vector of the service levels

f provided by both manufacturers. We assume that demand uncertainty is of the multiplicative

form, that is,

Di(f) = di(f)ϵ, (1)

where ϵ is a continuous random variable on the support [0,+∞). We assume the distribution and

density of ϵ exist and are given by Gϵ(·) and gϵ(·), respectively. The multiplicative form of demand

uncertainty in (1) implies that the market uncertainty (measured by coefficient of variation) is not

affected by the service levels f , but the expected demand is. Without loss of generality, we can

normalize ϵ such that E[ϵ] = 1, and interpret di(f) as the expected market demand of manufacturer

i. We require the following two assumptions on demand.

Assumption 1. Gϵ(ϵ) is log-concave.

Assumption 2.

∂di(f)

∂fi≥ 0,

∂di(f)

∂fj≤ 0,

∂2 logdi(f)

∂fi∂fj≥ 0, and

∂2 logdi(f)

∂f2i

+∂2 logdi(f)

∂fi∂fj≤ 0.

Assumption 1 is satisfied by many common distributions, including uniform, normal, exponential,

and gamma. In addition, any concave function is also log-concave. 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), Federgruen

and Yang (2009b)) that is widely used in the operations and marketing literature. Next, we prove,

under mild conditions, that these three types of demand functions satisfy Assumption 2.

Lemma 1 (Attraction Model). Suppose di(f) is given by the following attraction model

di(f) =Mxi(fi)∑2

j=0 xj(fj), (2)


where M is the average total market size. Here, we assume that x0 is a constant, which is the value

of the no-purchase option; and xi(fi) (i= 1,2) is manufacturer i’s product attraction value which

is increasing and log-concave in its service level, i.e. dxi(fi)/dfi ≥ 0, d2 logxi(fi)/df2i ≤ 0. If

2∑j=1

∂di(f)

∂fj≥ 0 for all i, (3)

then di(f) satisfies Assumption 2.

The technical condition (3) implies that a uniform increase in service levels by both manufacturers

will not decrease any manufacturer’s demand.

Lemma 2 (Linear Model). Suppose di(f) is given by the following linear model

di(f) = ai + bifi − cijfj, (4)

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

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

all i, then di(f) satisfies Assumption 2.

Lemma 3 (Log-Separable Model). Suppose di(f) is given by the following log-separable model

di(f) =ψi(fi)hi(fj) (5)

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.

A special form of log-separable function that satisfies Assumption 2 is the Cobb-Douglas function:

di(f) = γifβii f

−βijj (6)

with γi > 0, βi > 0 and βij > 0 for all i and j.

3.2. Supply

The supplier is not perfectly reliable, and its production process can be characterized by the

standard proportional random yield process. In this setting, for any given order Qi placed with

manufacturer i, the delivered quantity is qi = YiQi, where Yi is the random yield factor.


The distribution of the random yield factor Yi is influenced by the supplier’s reliability index pi,

an aggregate index reflecting factors (e.g., equipment, technical knowhow, and training) that may

influence supplier yield. Let ΦYi(·, pi) denote the distribution of Yi. We adopt the convention that

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.

As common in the operations literature, we assume that the manufacturer pays only for quantity

delivered, and manufacturer i offers a unit payment of wi to the supplier.

3.3. Improvement Effort and Knowledge Spillover

Both manufacturers may exert efforts (e.g., knowledge transfer, technical assistance, equipment

investment) to improve the supplier’s reliability. If manufacturer i exerts improvement effort level

at zi and let z= (z1, z2), then the supplier’s production reliability index for manufacturer i increases

from p0i to

pi(z) = p0i + zi +αzj, i= 1,2, j = 3− i, (7)

where 0≤ α≤ 1 capturers the knowledge spillover effect in the supplier’s production process.

The extent of the knowledge spillover effect depends on the configurations of the supplier’s pro-

duction process. Sometimes the supplier may use the same production line for both manufacturers

if the components are identical. In this case, pi = pj and α= 1. In contrast, the supplier oftentimes

may use somewhat different production lines if the components are customized for each manu-

facturer. For example, the supplier may use a shared production line for the early steps of the

production process, and then perform additional customization steps to satisfy each manufacturer’s

specific requirement. In this setting, the knowledge gained from improvement in the production

process for manufacturer i can partially spillover to the production process for manufacturer j. In

this case, pi = pj and α is in general less than 1. It is also possible that the supplier maintains

completely separate production lines (focused factory within factory). In this case, there is no

knowledge spillover effect, and hence α equals zero.


It is natural to assume that manufacturer i’s improvement cost mi(zi) is increasing in its effort

level zi, i.e. dmi(zi)/dzi ≥ 0. Note that we do not consider other benefits associated with the

manufacturer’s improvement effort, such as improved efficiency or improved contract terms.

3.4. Problem Formulation

Sequence of events. First, each manufacturer decides its level of supplier improvement effort.

Second, after exerting supplier improvement effort but before yield and demand uncertainty is

resolved, each manufacturer decides its service level fi and its order quantity Qi from the supplier

to meet the service level. Last, after observing realized production yield and demand uncertainty,

each manufacturer satisfies its demand as much as possible.

Given the above sequence of events, the manufacturers’ decision problem can be naturally for-

mulated as a two-stage stochastic program. Let ri, si, and πi denote manufacturer i’s unit price

(charged in the market), unit salvage value (for leftovers), and unit penalty cost (for shortages),

respectively. Define Hi(u, v) = rimin{u, v}+ si(u− v)+ −πi(v−u)+.

The second stage problem. Because the supplier’s production process is random, manufac-

turer i will choose an order quantity Qi such that the manufacturer will achieve a service level

fi. Let Y= (Y1, Y2) and p= (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,ϵ [Hi (YiQi,Di(f))] , (8)

s.t. Prob (Di(f)≤ YiQi)≥ fi. (9)

The service level fi defined in (9) is the in-stock probability, commonly known as Type-1 service

level in the literature. With random yield, the service level varies with different yield realizations,

and therefore the in-stock probability needs to be conditioned and unconditioned over all possible

yield realizations. The service level defined in (9) implies that manufacturers can credibly commit

on their service levels, which is true in most business to business markets. Such service level

constraint is also applicable in consumer markets with short life cycle products, where commitment

to service levels can “attract potential customers” (Sethi et al. 2007, p.322-323).


Because manufacturer i decides its service level fi and ordering quantity Qi jointly, one can

view fi as a surrogate for the classic quantity competition. Introducing the service level fi allows

us to generalize the classic quantity competition considerably, i.e. incorporate stochastic demand,

random supply, and imperfectly substitutable products.

The first stage problem. Let the supplier’s initial production reliability index be p0 = (p01, p02),

the first stage problem for manufacturer i can be formulated as

Ji(zi) =−mi(zi)+ vi(f∗i ,Q

∗i |p(z), f∗

j ), (10)

where p(z) = (p1(z), p2(z)) and pi(z) is given by (7).

We note that it is fairly straightforward to incorporate probabilistic supplier improvement results,

e.g. an improvement effort may not always be fully successful. In this case (10) can be modified by

imposing an expectation operator on vi(f∗i ,Q

∗i |p(z), f∗

j ) over all possible realizations of p(z).

4. Preliminaries

In this section, we characterize the manufacturer’s second stage problem, where manufacturer i

chooses service level fi and order quantity Qi to maximize its expected profit given by (8).

4.1. Stocking Factor

For analytical convenience and expositional ease, define qi(f) =Qi/di(f). Hereafter we refer to qi(f)

the stocking factor, which captures the extent by which the manufacturer inflates its order quantity

relative to the expected demand for any given vector of service level f . Using the stocking factor

qi(f), we can rewrite (8) and (9) as

vi(fi, qi(f)|p, fj) =−widi(f)EYi[Yiqi(f)]+EYi,ϵ [Hi (Yiqi(f)di(f), di(f)ϵ)] , (11)

s.t. EYi[Gϵ (Yiqi(f))]≥ fi, (12)

where the service level constraint (12) follows from the fact that

Prob (Di(f)≤ YiQi)≥ fi ⇔Prob(ϵ≤ Yiqi(f))≥ fi ⇔ EYi[Gϵ (Yiqi(f))]≥ fi.


Note that constraint (12) is in general not a convex set, and therefore one cannot use the standard

Lagrangian approach in constrained optimization to solve (11). However, for any given fi, vi(·) is

concave in qi(f); and for any given qi(f), vi(·) is log-concave in fi. We therefore adopt a two-stage

approach by first characterizing the optimal q∗i as a function of fi, and then characterizing the

optimal service level f∗i . Simplifying (11), we have

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

{qi(f)

((ri −wi +πi)EYi

[Yi]− (ri − si +πi)EYi[YiGϵ(Yiqi(f))]

)+(ri − si +πi)EYi

[∫ Yiqi(f)

0

ϵgϵ(ϵ)dϵ

]−πi

}. (13)

Let ui(fi, qi(f)|p, fj) denote the terms in the curly bracket in (13), i.e. manufacturer i’s expected

profit per unit of demand. The following proposition characterizes the optimal stocking factor.

Proposition 1. For any given vector of service level 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[YiGϵ (Yiq

∗i (f))] =

ri −wi +πi

ri − si +πi

EYi[Yi]; (14)

EYi[Gϵ (Yiq

∗i (f))] = fi. (15)

Notice that (14) gives an interior solution to ∂ui(fi, qi(f)|p, fj)/∂qi(f) = 0, and (15) gives a corner

solution constrained by (12). Leveraging Proposition 1, the following theorem completely charac-

terizes the properties of optimal stocking factor for any given service level fi.

Theorem 1. (a) Manufacturer i’s optimal stocking factor q∗i (f) is independent of its com-

petitor’s service level fj and non-decreasing in its own service level fi, i.e., q∗i is a function of fi

only, and dq∗i (fi)/dfi ≥ 0.

(b) There exists a unique threshold service level f i such that if fi ≤ f i, q∗i (fi) uniquely solves

(14), and if fi > f i, q∗i (fi) uniquely solves (15).

(c) (ri −wi +πi)/(ri − si +πi)EYi[Yi]≤ f i ≤ (ri −wi +πi)/(ri − si +πi).

(d) f i can be uniquely obtained by solving the system of equations specified by (14) and (15)

simultaneously.


(e) 0 = dq∗i (fi)/dfi∣∣fi≤fi

<dq∗i (fi)/dfi∣∣fi>fi

.

Theorem 1 shows that if the service level is less than a critical service threshold f i, the optimal

stocking factor, q∗i (fi), is a constant that is independent of the service level chosen.1 In contrast, if

the service level fi > f i, the optimal stocking factor is no longer constant in service level. Instead,

q∗i (fi) increases in fi, i.e., a larger inflation of the expected demand is required to satisfy a higher

service level. The service level threshold f i plays an important role in characterizing the optimal

service level, and, as we will see shortly, it also has an important practical interpretation.

4.2. Equilibrium Service Level

We now turn our attention to manufacturer i’s optimal service level decision, given its competitor’s

service level. It is immediate from Theorem 1(a) that ui is independent of fj, so we can rewrite (8)

as a univariate function, i.e.,

vi(fi|p, fj) = di(fi|p, fj)ui(fi|p), where (16)

ui(fi|p) = q∗i (fi)((ri −wi +πi)EYi

[Yi]− (ri − si +πi)EYi[YiGϵ(Yiq

∗i (fi))]

)+(ri − si +πi)EYi

[∫ Yiq∗i (fi)

0

ϵgϵ(ϵ)dϵ

]−πi. (17)

The following theorem establishes a lower bound on the optimal service level.

Theorem 2. The optimal f∗i ≥ f i.

Theorem 2 shows that the optimal service level f∗i can never be lower than the threshold service

level f i. One can therefore interpret f i as the minimum service level required for manufacturer i

to compete in the market place. The manufacturer either enters the market and achieves a service

level at least as high as f i, or exits the market altogether. The following theorem ensues.

Theorem 3. A necessary and sufficient condition for manufacturer i to enter the market is

(ri − si +πi)EYi

[∫ Yiq∗i (fi|fi=fi)

0

ϵgϵ(ϵ)dϵ

]>πi. (18)

Hereafter we focus on the case where the market entry condition (18) holds for both manufac-

turers, and do not study the degenerate case where only one manufacturer enters in the market.


Now we are ready to study the second stage problem on manufacturers’ equilibrium service

levels. Given the reliability index p for both manufacturers, manufacturer i’s second stage expected

profit vi depends on both manufacturers’ service levels. For the ease of exposition, we rewrite

vi as a function 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 in fi, hence vi(f |p) is not necessarily

log-concave in fi. Nevertheless, by Assumption 2, vi(f |p) is log-supermodular in f , and thus the

following theorem ensues.

Theorem 4. The service level equilibrium f∗ exists.

While Theorem 4 shows the existence of service level equilibrium, it is challenging to prove the

uniqueness of the service level equilibrium f∗ under general yield distributions. With the following

assumption, however, we are able to prove the uniqueness of service level equilibrium. We thus

make the following assumption for the rest of the analysis.

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

Theorem 5. There exists a unique Nash equilibrium on the service levels f∗ under Bernoulli

yield distribution. Furthermore, the equilibrium f∗i satisfies

q∗i (fi)∂di(f |p)/∂fi + di(f |p)q∗′

i (fi)

di(f |p)ui(fi|p)

{(ri −wi +πi)EYi


∗i (fi))]

}+

∂di(f |p)/∂fidi(f |p)ui(fi|p)

{(ri − si +πi)EYi

[∫ Yiq∗i (fi)

0

ϵgϵ(ϵ)dϵ

]−πi

}= 0. (19)

Theorem 5 demonstrates that although each manufacturer’s objective function is constrained by

a non-convex set (in service level and stocking factor), there exists nevertheless a unique Nash

equilibrium such that both firms achieve the unique, stable equilibrium service levels. It is worth

noting that all analysis so far directly extends to the N -manufacturer case.

In closing, we note that Propositions A1 and A2 in Appendix A provide additional sensitivity

analysis on the minimum service level f i and the equilibrium service level f∗.


5. Supplier Improvement

In this section, we characterize the manufacturers’ first stage problem, where each manufacturer

chooses its improvement effort to maximize its objective function given by (10).

5.1. Do Manufacturers Benefit From Higher Supplier Reliability?

In order to characterize the improvement effort z, it is helpful to first understand the effect of

target reliability index p (after improvement effort) on each manufacturer’s expected profit. While

one might expect that a higher pi is always beneficial to manufacturer i, it is unclear whether

such conjecture holds in the presence of spillover effects. In what follows, we first study cross effect

and direct effect of target reliability index p, and then explore the role of spillover index α on the

combined effect of p.

Note that under Bernoulli yield distribution, the second-stage expected profit at the equilibrium

service levels is given by vi(f∗|p) = di(f

∗|p)ui(f∗i |p), where f∗ = (f∗

i , f∗j ) = (f∗

i (pi, pj), f∗j (pi, pj)) are

implicitly given by (19). We can therefore treat vi(f∗|p) as a function of target reliability index

p only, where pi explicitly appears in ui(·) and implicitly appears in f∗i and f∗

j (see (17) under

Bernoulli distribution), but pj only implicitly appears in f∗i and f∗

j . For expositional ease, we

denote the second-stage expected profit at the equilibrium service levels as a function of p, i.e.,

v∗i (p)≡ vi(f∗|p).

The cross effect. The following proposition proves that, as one might expect, the focal manu-

facturer’s expected profit declines as its competitor’s target reliability increases.

Proposition 2. Manufacturer i’s expected profit is (weakly) decreasing in supplier’s reliability

index pj, i.e. ∂v∗i (p)/∂pj ≤ 0.

The intuition for Proposition 2 is as follows. An increase in pj affects manufacturer i’s expected

profit v∗i by affecting both manufacturers’ equilibrium service levels f∗i and f∗

j . However, v∗i is not

affected by its own equilibrium service level at f∗. Hence, pj influences v∗i through its impact on f∗

j

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

i’s expected profit. Leveraging Proposition 2, the following corollary (proof omitted) is intuitive.


Corollary 1. If supplier improvement cost is fixed, then manufacturer i’s first stage objective

Ji is (weakly) decreasing in pj regardless of the spillover effect α, i.e. ∂Ji/∂pj ≤ 0.

The direct effect. A more interesting question is whether manufacturer i always benefits from

an increase in its own target reliability index pi, and how knowledge spillover index α affects such

benefits. Applying 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

=∂di(f

∗|p)∂fj︸︷︷︸−

∂f∗j

∂pi︸︷︷︸+

ui + di∂ui(f

∗|p)∂pi︸︷︷︸+

,

(20)

where ∂di(f∗|p)/∂fj ≤ 0 follows from Assumption 2, ∂f∗

j /∂pi ≥ 0 follows from part (a) of Proposi-

tion A2 in Appendix A, and ∂vi(f∗|p)/∂fi = 0 follows from (19). In general, (20) can be positive

or negative, so manufacturer i may not necessarily benefit from an improvement in the supplier’s

reliability index pi. Under certain conditions, however, we are able to unambiguously sign (20).

Proposition 3. If demand function is log-separable and satisfies Assumption 2, then each

manufacturer’s second stage expected profit is strictly increasing in its own reliability index, i.e.,

∂v∗i (p)/∂pi > 0 (i= 1,2).

Proposition 3 shows that, with log-separable demand, manufacturer i’s expected profit v∗i (p)

increases in its own target reliability index pi, and the rate of increase ∂v∗i (p)/∂pi = di · (∂ui/∂pi)

depends on its own demand.

The combined effect. Combining the cross effect (Proposition 2) and direct effect (Proposition

3), we have

∂Ji(p)

∂zi=∂v∗i (p)

∂pi︸︷︷︸+

+α∂v∗i (p)

∂pj︸︷︷︸−

− dmi(zi)

dzi︸︷︷︸+

. (21)

Since the spillover index α links the cross effect (a negative impact) with the direct effect (a positive

impact) of an increase in p, the net effect (i.e., whether the focal manufacturer is better off with

higher supplier reliability) can be ambiguous. In fact, we have observed numerical examples in

which the focal manufacturer is better off (or worse off) with a higher reliability index of the


supplier. We thus conclude that the manufacturers may not necessarily benefit from higher supplier

reliability due to the spillover effect.

It is natural to conjecture, however, that if there is no spillover effect (i.e., α = 0) the manu-

facturers should benefit from higher supplier reliability and therefore have incentives to improve

supplier reliability p.

5.2. The Case of No Spillover Effect (but with Demand Competition)

The following proposition proves that, when there is no spillover effect and the improvement cost

is fixed, each manufacturer’s improvement effort is of the all-or-nothing type.

Proposition 4. If there is no spillover effect (i.e., α = 0) and supplier improvement cost is

fixed, then it is optimal for each manufacturer to either target a perfect reliability or not to improve

the supplier at all.

Proposition 4 shows that if the gain from improving supplier reliability (to perfect yield) offsets the

fixed improvement cost, then the manufacturer will target a perfect yield; otherwise it will exert

no improvement effort. Note that if the improvement cost varies with improvement effort, then the

manufacturer’s improvement effort no longer exhibits this all-or-nothing type of behavior.

Next we study the equilibrium behavior of the manufacturers’ improvement efforts. Since each

manufacturer will not exert more effort than necessary to bring the target reliability to 1 (under

Bernoulli yield distribution), the set of feasible joint strategies is given by the following nonempty

convex and compact set: zi ∈ [0,1− p0i ], i= 1,2.

Theorem 6. If there is no spillover effect (i.e., α = 0), demand function is log-separable and

satisfies Assumption 2, then each manufacturer’s expected profit Ji(·) is submodular in improvement

efforts z, and there exists a Nash equilibrium on the optimal improvement efforts z∗.

The setting of α = 0 can be loosely interpreted as a market setting where two manufacturers

compete on demand but with distinct technologies such that there is no “collaboration” on supplier

improvement. In this special setting, Theorem 6 shows that the two manufacturers settle in a

stable equilibrium in their improvement efforts. When α > 0, however, since each manufacturer’s


improvement effort inevitably affects the other’s reliability, the spillover effect may disturb the

above equilibrium or even lead to non-existence of Nash equilibrium.

5.3. The Case of Positive Spillover Effect (but without Demand Competition)

For the more general case with positive spillover effect, the set of feasible joint strategies is

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

In this case, each manufacturer’s feasible strategy set is dependent on the strategy adopted by

its competitor, and the set of joint feasible strategies is not a lattice. To tackle this challenge, we

consider the surrogate 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 existence of equilibrium improvement

efforts in this subsection and next (§5.3 and §5.4).

A special case is when each manufacturer’s demand depends on its own service level only. This can

be the case, for example, when the two manufacturers serve two geographically separate markets.

This special case can be modeled by setting hi(fj) = 1 in the log-separable demand model defined

in (5). In this case, Assumption 2 is reduced to d′i(fi)≥ 0 and d2 logdi(fi)/df2i ≤ 0. The following

corollary (proof omitted) is immediate from Proposition 3 and (21).

Corollary 2. If each manufacturer’s demand is a weakly increasing and log-concave func-

tion of its own service level only, then each manufacturer’s second stage expected profit is strictly

increasing in its own reliability index, i.e. ∂v∗i (p)/∂pi > 0 (i= 1,2).

If supplier improvement cost is fixed, we can further characterize each manufacturer’s best

response function in optimal improvement effort.

Proposition 5. If supplier improvement cost is fixed, and each manufacturer’s demand is a

weakly increasing and log-concave function of its own service level only, then both manufacturers

would either target perfect reliability or exert no improvement effort at all.


(a) If the fixed improvement cost is greater than the maximum gain from improved supply reli-

ability, i.e., mi > v∗i (pi = 1)− v∗i (pi = p0i ), then manufacturer i exerts no improvement effort.

(b) Otherwise, there exists a unique threshold zj ∈ [0, (1−p0i )/α] such that manufacturer i would

target perfect reliability if zj < zj, exert no improvement effort if zj > zj, and be indifferent between

the two options if zj = zj.

(c) Moreover, if zj > 1− p0j , then manufacturer i will always target perfect reliability under any

feasible strategy of manufacturer j.

Leveraging Proposition 5, we can now characterize the equilibrium behavior of the manufacturers’

improvement efforts.

Theorem 7. If supplier improvement cost is fixed, and each manufacturer’s demand is a weakly

increasing and log-concave function of its own service level only, then each manufacturer’s optimal

improvement effort in the surrogate game is increasing in its competitor’s improvement effort.

There exists at least one Nash equilibrium in the surrogate game. The set of equilibria is a lattice;

there exists a componentwise smallest equilibrium z∗ and a componentwise largest equilibrium z∗.

Theorem 7 shows that, if there are (unintentional) “collaboration” on supplier improvement

(i.e., positive spillover effect) but no direct competition on demand (i.e., demand depends on each

manufacturer’s own service level only), then again, in the original game, the best response of

one manufacturer’s supply improvement effort is decreasing in the other manufacturer’s supplier

improvement effort and there exists one or more equilibria. This is the case when two manufacturers

share a common supplier with similar technologies but they serve geographically separate markets.

Next we study the most general setting with positive spillover effect and direct market compe-

tition between two manufacturers.

5.4. The Case of Positive Spillover Effect With Demand Competition

When two manufacturers engage in general demand competition under positive spillover effect

in improvement efforts, it is very challenging to characterize the equilibrium behavior. We there-

fore assume the Cobb-Douglas demand function, see (6). The following theorem gives sufficient

conditions for the existence of Nash equilibrium in the manufacturers’ improvement efforts.


Theorem 8. Assume that there is no shortage penalty cost (i.e., πi = 0) and each manufac-

turer’s demand function is a Cobb-Douglas function (see (6)).

(a) The equilibrium service level satisfies f∗i =Kipi, where Ki depends on cost parameters ri,

wi, si and the distribution of ϵ only.

(b) Define

Mi =max

{βip

0j

βij

+βij +1

(βi +1)p0j,βi

βijp0i+

(βij +1)p0iβi +1

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

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

and 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 has at least one equilib-

rium. The set of equilibria is a lattice; there exists a componentwise smallest equilibrium z∗ and a

componentwise largest equilibrium z∗.

Theorem 8 shows that under certain technical conditions, the best response of one manufacturer’s

supplier improvement effort is again decreasing in his competitor’s supplier improvement effort

in the original game, and there exist one or more equilibria in the manufacturers’ improvement

efforts. An important implication of this result is that a stable balance (in improvement effort)

can be achieved, despite a significant presence of spillover effect (α) and a fairly intense market

competition (Cobb-Douglas demand).

5.5. Summary

In this section, we first show that manufacturer i may not necessarily benefit from its supplier

improvement effort, even if the effort is costless, because, although it benefits from a higher pi, it

suffers from a higher pj due to spillover effect. The net effect can therefore be ambiguous (§5.1).

This is a static view of the spillover effect for a given vector of manufacturers’ improvement efforts.

When manufacturers optimally respond to each other’s improvement effort, however, we will see

in §6 that the spillover effect has a quite surprising impact on manufacturers’ expected profits.

We then characterize the sufficient conditions for the existence of manufacturers’ equilibrium

improvement efforts in three different cases: (i) the manufacturers use distinct or proprietary


technologies (i.e., no spillover effect) and compete on demand, (ii) the manufacturers use similar

technologies (i.e., positive spillover effect) but they serve different markets, and (iii) the manu-

facturers use similar technologies (i.e., positive spillover effect) and they compete in the same

market. In all three cases (§5.2-§5.4), the best response of one manufacturer’s supplier improvement

effort is decreasing in his competitor’s supplier improvement effort (in the original game), i.e., the

manufacturers’ improvement efforts are substitutes.

In closing, we note that under general demand functions we numerically observed instances

where there exists no equilibrium, unique equilibrium or multiple equilibria in the manufacturers’

improvement efforts. The no-equilibrium case typically arises when spillover effect is large, demand

is highly sensitive to service levels and the manufacturers differ significantly in their market char-

acteristics.

6. Managerial Implications

In §5, we characterize manufacturers’ equilibrium improvement efforts under certain conditions.

It is of interest to further explore the impact of market characteristics on manufacturers’ optimal

improvement efforts and the resulting implications on manufacturers’ expected profits.

6.1. Endowed Market Advantage

Oftentimes manufacturers differ from one another in the expected market demand even if they

set identical service levels. Such persistent difference in market demand can often be attributed

to exogenous factors such as brand image, product design, and idiosyncratic consumer tastes. We

refer to a manufacturer that all else being equal enjoys a higher market demand as the one that has

endowed market advantage (EMA). In the context of our demand model, for example, manufacturer

i has EMA if (all else being equal) xi(f)>xj(f) for any f in attraction model (see (2)), or ai >aj

in linear model (see (4)), or γi >γj in log-separable model (see (6)).

It is unclear a priori whether the manufacturer with EMA would exert a higher or lower improve-

ment effort relative to its competitor. On one hand, intuition suggests that it should exert less

effort: since it already enjoys a market advantage, it does not have to compete as aggressively.


On the other hand, intuition also suggests that it should exert more effort to exploit its market

advantage more effectively. Figure 1 illustrates the effect of EMA on manufacturers’ equilibrium

improvement efforts. (In Figure 1, yield is uniform Yi ∼ [1− 0.6/pi,1], where p0i = 1.0; demand is

linear (as in (4)), where the base values are ai = 0.5, bi = 1.0, and cij = 0.5; demand uncertainty ϵ is

normal with µ= 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 varying α and a1 while fixing all other

parameter values. The rest of figures are obtained using similar settings.)

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 endowed market advantage on manufacturers equilibrium improvement efforts

Figure 1 indicates that the manufacturer with EMA tends to exert a higher effort level relative to

its competitor, and this phenomenon becomes more pronounced as the spillover index α increases.

We note that, as the spillover effect increases, both manufacturers improvement effort declines, but

the declining speed is slower for the manufacturer with EMA. As a result, its relative improvement

effort increases in α.


One naturally wonders whether the fact that the manufacturer with EMA exerts an increasingly

higher proportion of improvement efforts also translates into a relative profit advantage, i.e. whether

its relative financial performance is also enhanced by the spillover effect. Figure 2 illustrates the

effect of spillover α and the EMA on manufacturers’ expected profits.

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 endowed market advantage on equilibrium expected profits

Figure 2(a) shows that both manufacturers’ expected profits increase in the spillover effect α,

but the relative profit advantage of the manufacturer with EMA (manufacturer 1) declines as the

spillover effect increases (Figure 2(b)). Thus, the manufacturer with EMA may view the spillover

effect with some sort of dilemma: in absolute profit terms, its increasingly higher proportion of

improvement effort pays off because its expected profit increases in α; in relative profit terms, how-

ever, its profit advantage (over its competitor) declines, suggesting that its ever larger proportion

of improvement effort benefits its competitor more than itself.

Back to the UTC’s example, its Pratt & Whitney’s ambivalence about the spillover effect is


somewhat understandable. Although Pratt &Whitney trails GE and Rolls Royce in terms of overall

market share (Malone 2010), it dominates “the commercial aircraft market with its jet engines”

(Hinton 2007). Thus, although its supplier improvement effort increases its own expected profit,

such improvement effort may benefit its competitors more than itself. The latter concern may be

the reason for Pratt & Whitney’s hesitation. Nevertheless, the expected benefit seems to outweigh

the latter concern, with a result that Pratt & Whitney reluctantly engages in improvement effort.

In the Toyota’s example, existing evidence suggests that, despite significant spillover effects,

Toyota expends much more improvement effort than its major competitors (Liker and Choi 2004).

Based on our results, one may conjecture that all else being equal Toyota enjoys an endowed market

advantage, at least before the recent quality woes. Toyota’s high level of improvement effort also

suggests that Toyota is concerned perhaps more about its expected profit than its relative profit

advantages over its competitors. We acknowledge, of course, that Pratt & Whitney’s and Toyota’s

strategic choices of improvement efforts are more complex than the stylized model studied here,

and our above observation is not enough for decision support. Instead, our research serves as the

initial step towards the answer of this important and challenging problem.

6.2. Manufacturer’s Pricing Power Over Supplier

Manufacturers often can negotiate different purchasing prices wi, depending on their relative power

and their relationship structure with the supplier. We refer to the manufacturer that can negotiate

a lower wi as the one with more pricing power. A lower wi itself does not necessarily imply a

combative or cooperative relationship. Depending on the volume and profit margins, for example,

a lower wi may be necessary for both the supplier and the manufacturer to stay in the market.

Similar to our earlier discussion in §6.1, it is unclear whether the manufacturer with more pricing

power would exert more or less improvement efforts relative to its competitors. Nevertheless, we

can form some intuitions from the following proposition.

Proposition 6. Each manufacturer’s equilibrium service level is decreasing in its unit procure-

ment cost, as well as its competitor’s unit procurement cost; increasing in its unit price, unit


salvage value and unit penalty cost, as well as its competitor’s unit price, unit salvage value and

unit penalty cost; i.e. ∂f∗i /∂wi < 0 and ∂f∗

i /∂wj ≤ 0; ∂f∗i /∂ri > 0 and ∂f∗

i /∂rj ≥ 0; ∂f∗i /∂si > 0

and ∂f∗i /∂sj ≥ 0; ∂f∗

i /∂πi > 0 and ∂f∗i /∂πj ≥ 0; for i= 1,2 and j = i.

Proposition 6 demonstrates that (among others) manufacturer i’s equilibrium service level f∗i

increases if the unit procurement cost wi decreases. This suggests that if manufacturer i has more

pricing power, it will in equilibrium set a higher service level and hence face a higher supply yield

risk (of not being able to fulfill the service level). Intuitively, the manufacturer would benefit more

from an improvement in supplier yield, and exert a higher level of improvement efforts relative to

its competitor.

It turns out that our numerical observation (not illustrated here) is consistent with the above

intuition, i.e., the manufacturer with more pricing power exerts a higher proportion of improvement

effort relative to its competitor. The implications on the expected profit is also qualitatively similar

to our earlier observation in §6.1, that is, both manufacturers’ expected profits increase in spillover

effect, but the relative profit advantage (of the manufacturer with more pricing power) declines as

spillover effect increases, suggesting that the increasingly higher proportion of improvement effort

exerted by the focal manufacturer benefits its competitor more than itself.

In the Toyota and Honda’s case, both can often negotiate a lower purchasing price wi relative

to their competitors, because they both practice the so called target pricing policy where wi is

set based on the supplier’s production cost (and any lower wi would typically leave the supplier

with such anaemic profit margins that the supplier may bankrupt). While other manufacturers

may also try to squeeze suppliers on wi, it is often a hit-and-miss activity because the arms

length relationship typically practiced by these manufacturers prevents them from obtaining precise

information about the supplier’s cost structures (Li et al. 2011). It is not surprising, then, that

Toyota and Honda engage in higher level of supplier improvement efforts than their competitors. In

practice, manufacturers’ supplier improvement effort is often motivated by a multitude of reasons.

Nevertheless, our above observation offers some insights on one aspect of such complex activities.


6.3. Competition Intensity

As analyzed in §5.3, the spillover effect can be viewed as an “unintentional collaboration” between

the manufacturers if they do not compete in the same market. One may therefore conjecture

that, as market competition intensifies, the spillover effect may have a negative impact on the

manufacturers’ improvement efforts and the resulting expected profits.

To capture the market competition effect, we use the ratio of ci/bi in the linear demand model

(4) to measure the competition intensity (where we set cij = cji = ci). If ci/bi = 0 then each man-

ufacturer’s demand depends on its own service level only (i.e., no direct competition), whereas if

ci/bi = 1 then each manufacturer’s demand depends on its own service level as much as it depends

on its competitor’s service level (i.e., perfect substitution/competition). Figure 3 illustrates the

effect of competition intensity on the manufacturers’ equilibrium service levels.

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

Consistent with our earlier conjecture, Figure 3 suggests that the manufacturers’ improvement

efforts decline as competition intensifies. This is somewhat as expected, because any improvement


effort only serves to further intensify the already fierce demand competition. Figure 3 also indicates

that, under intense competition, both manufacturers significantly reduce their improvement efforts

as spillover effect increases. The intuition is similar to the above observation: improvement efforts

will worsen the already intense competition.

This seems to suggest that both manufacturers should prefer less spillover effect under intense

market competition. It turns out, surprisingly, that this intuition is largely incorrect, at least in

terms of the manufacturers’ expected profits. Figure 4 illustrates the effect of market competition

on the manufacturers’ equilibrium profits (where the two manufacturers are symmetric).

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

Figure 4(a) illustrates the percentage change in the manufacturers’ expected profits as spillover

effect increases, where we use the expected profits under α= 0 as the base case. Interestingly, the

spillover effect improves the manufacturers’ expected profits, and the relative benefit in expected

profit is increasing in market competition. The latter observation is partially caused by the fact


that, at a higher level of market competition, the expected profit is much smaller and hence even a

small improvement (due to spillover effect) translates into a rather significant percentage increase

in expected profits.

Figure 4(b) directly illustrates the effect of market competition on the expected profits, and

it is clear that it has a detrimental effect on the manufacturers’ expected profits. Nevertheless,

Figure 4(b) indicates that even with intense market competition, spillover effect improves the

manufacturers’ expected profits. Perhaps, then, manufacturers should not view spillover effect with

trepidation, at least when both manufacturers are similar in their characteristics.

In summary, intense market competition is often a characteristic of commodity products, where

manufacturers use similar technologies and hence are more likely to experience spillover effects.

The presence of spillover effect in this type market perhaps should be viewed more favorably than

what we have previously anticipated. One important implication is that the manufacturers in this

type of market should not try to curb the supplier’s spillover effect, should they decide to exert

supplier improvement efforts.

6.4. Market Uncertainty

Different industries often exhibit different levels of market uncertainty: some industries (e.g., fash-

ion) exhibits considerably higher market uncertainty than others (e.g., aerospace). It is therefore of

interest to understand how market uncertainty influences the manufacturers’ equilibrium improve-

ment efforts and the resulting expected profits. While it is fairly straightforward to conjecture

that higher market uncertainty would lead to lower expected profit, it is not clear how market

uncertainty influences optimal improvement efforts and what a role the spillover effect plays in this

environment.

It turns out that the effect of market uncertainty on the manufacturers’ improvement efforts (not

illustrated here) is very similar to the effect of market competition discussed in §6.3. That is, higher

market uncertainty reduces the manufacturers’ equilibrium improvement efforts, and the spillover

effect further exacerbates the declining improvement efforts. Part of the intuition is that higher


market uncertainty renders the “reward” of improvement effort much less certain: an improvement

effort may be entirely unnecessary if market condition turns out to be “lower than expected”.

Figure 5 illustrates the effect of market uncertainty on the manufacturers’ expected profits.

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

Figure 5(a) indicates that the effect of market uncertainty on the relative changes in the man-

ufacturers’ expected profits differs from the effect of market competition (Figure 4(a)). Here, as

one might expect, market uncertainty unambiguously reduces the manufacturers’ expected profits,

both in relative and absolute terms. Nevertheless, at any level of market uncertainty, the spillover

effect improves the manufacturers’ expected profits. This suggests that the presence of spillover

effect may not necessarily be a bad thing for the manufacturers, especially when manufacturers

are similar in their characteristics (e.g., oligopoly markets).

The above observation is consistent with empirical observations that “market uncertainty pro-

vides an unfavourable environment for specific investments [supplier improvement] ...” (Lee et al.


2009, p.190). From a strategic management perspective, Lee et al. (2009) note that “when faced

with greater environmental uncertainty, firms may need to avoid long-term entanglements that

could later prove to be disadvantageous, and will instead favour more flexible, arm’s-length rela-

tionships” (p.191). While our model does not capture strategic relationships, it does point to the

fact that, regardless of the types of relationships, high demand uncertainty does not seem to be

conducive to supplier improvement efforts.

6.5. Summary

In summary, we find that, contrary to common wisdom, the spillover effect often brings favor-

able improvement to manufacturers’ expected profits, regardless of whether the manufacturers are

asymmetric (§6.1 and §6.2) or symmetric (§6.3 and §6.4).

When manufacturers are asymmetric, we find that the ex ante more advantageous manufacturer

typically exerts a higher proportion of improvement efforts, especially when the spillover effect

is large. While such higher proportion of improvement efforts benefits the focal manufacturer, it

may benefit (in relative profit terms) its competitor more. Thus, the ex ante less advantageous

manufacturer may prefer a higher spillover effect more than the ex ante more advantageous one

does.

When manufacturers are symmetric, we find that both manufacturers’ improvement efforts


benefit from increased spillover effect. In the context of our modeling framework, our findings seem

to support Toyota’s (and Honda’s) attitude on supplier improvement effort: one should engage in

supplier improvement without worrying too much about the spillover effect.

7. Conclusion

In this research, we study the effect of knowledge spillover on OEM manufacturers’ supplier

improvement efforts. We characterize the structure of the manufacturers’ equilibrium improvement

efforts as well as their equilibrium service levels. In addition, we provide managerial insights on

market characteristics that influence the manufacturers’ equilibrium improvement efforts.


In the special case of no knowledge spillover, e.g., the manufacturers adopt distinct technologies

but they compete in the same market place, we prove that if improvement effort is at a fixed cost,

then it is optimal for each manufacturer to either target a perfect supplier reliability or not to

improve the supplier at all. When there is knowledge spillover, the manufacturers’ improvement

efforts are dependent on each other, and the set of joint feasible strategies of improvement efforts

is not a lattice. To the best of our knowledge, there is no known results in the structure or even the

existence of Nash equilibriums when the action set is not a lattice. We have numerically observed

that all forms of equilibrium can happen, e.g. no equilibrium, one equilibrium, or multiple equilibria.

Nevertheless, by reformulating the manufacturers’ strategy set into a surrogate game, we are able

to characterize sufficient conditions for the existence of manufacturers’ equilibrium improvement

efforts under positive spillover effect with demand competition.

From a static point of view, we prove that the manufacturers may not necessarily benefit from

increased supplier reliability due to the spillover effect. From a “dynamic”, equilibrium point of view

(e.g., each manufacturer decides optimal improvement effort in response to the other), however,

we find that, contrary to common wisdom, the spillover effect often brings favorable improvement

to the manufacturers’ expected profits, regardless of whether the manufacturers are asymmetric

or symmetric. When manufacturers are asymmetric, we find that the ex ante more advantageous

manufacturer typically exerts a higher proportion of improvement efforts, but such higher propor-

tion of improvement efforts may benefit its competitor more. Thus, the ex ante less advantageous

manufacturer may prefer a higher spillover effect more than the ex ante more advantageous one

does. When manufacturers are symmetric, we find that both manufacturers’ improvement efforts


benefit from increased spillover effect.

A number of interesting extensions are possible for future research. First, the supplier may

initiate improvement effort as well. In this case, who is in a better position to improve process

reliability, the supplier or the manufacturers? Second, manufacturers may procure from multiple

suppliers, and it is of interest to contrast the effect of supplier competition with manufacturers’


improvement efforts. Third, one may study the cumulative effect of supplier improvement effort in

a multi-period setting. In this case, it is of interest to understand the equilibrium path of supplier

improvement efforts. We hope future studies by us and others will bring additional insights to this

important area of research.

Appendix A: Auxiliary Results

Proposition A1. (a) The minimum market entry service level f i is (weakly) decreasing in the manu-

facturer’s unit procurement cost and (weakly) increasing in the manufacturer’s unit retail price, unit salvage

value and unit shortage penalty, i.e., ∂f i/∂wi ≤ 0, ∂f i/∂ri ≥ 0, ∂f i/∂si ≥ 0, ∂f i/∂πi ≥ 0. (b) For any given

random yield Yi and minimum service level f i, there exists a Yi that first-order stochastically dominates Yi,

such that f i|Yi ≥ f i|Yi for any Yi ≥st Yi.

The sensitivity results in part (a) are intuitive. Part (b) shows that in general a stochastic increase in the

supplier’s yield increases the minimum service level. Note that, however, depending on the particular yield

distribution, there is no guarantee that f i is monotone in random yield Yi - although a sufficiently large

stochastic increase in Yi always raises the minimum service level f i.

Proposition A2. For i = 1,2 and j = i, we have: (a) Each manufacturer’s equilibrium service level is

increasing in its process reliability, and nondecreasing its competitor’s process reliability, i.e. ∂f∗i /∂pi > 0

and ∂f∗j /∂pi ≥ 0.

(b) An increase in a manufacturer’s process reliability will lead to a larger increase in its own equilibrium

service level than its competitor’s, i.e. (∂f∗i /∂pi − ∂f∗

j /∂pi)> 0.

Part (a) of Proposition A2 states that an improvement in supplier’s process reliability index pi increases

its own equilibrium service level and weakly increases its competitor’s equilibrium service level regardless

of the spillover. Hence, supplier improvement may not necessarily benefit the manufacturer in terms of the

equilibrium expected profit. Part (b) complements part (a) by examining the relative change in service

levels between the two manufacturers: the increase in service level for manufacturer i is greater than that

for manufacturer j, i.e., ∂f∗i /∂pi > ∂f∗

j /∂pi regardless of the spillover effect. This shows that the improved

process reliability pi allows manufacturer i to compete more effectively in service levels.

The proofs of Propositions A1 and A2 are delegated to Appendix B, after the proof of Theorem 5.


Appendix B: Proofs

Proof of Lemma 1 Using (2) (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)

∂f2i

=(x0 +xj) (x

′′i (x0 +xi +xj)− 2x′2

i )

(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 transform and relevant derivatives, we have

∂2 logdi(f)

∂f2i

=

∂2di(f)

∂f2idi(f)−

(∂di(f)

∂fi

)2d2i (f)

=(x0 +xj) (x

′′i xi(x0 +xi +xj)−x′2

i (x0 +2xi +xj))

x2i (x0 +xi +xj)2

,

∂2 logdi(f)

∂fi∂fj=

∂2di(f)

∂fi∂fjdi(f)− ∂di(f)

∂fi

∂di(f)

∂fj

d2i (f)=

x′ix

′j

(x0 +xi +xj)2≥ 0.

Note that

∂2 logdi(f)

∂f2i

=(x0 +xj) (x

′′i xi(x0 +xi +xj)−x′2

i (x0 +2xi +xj))

x2i (x0 +xi +xj)2

<(x0 +xj)(x

′′i xi −x′2

i )

x2i (x0 +xi +xj)

=x0 +xj

x0 +xi +xj

d2 logxi

df2i

< 0.

The last inequality follows from the assumption that xi is log-concave. Therefore, di(f) is log-concave in fi.

We will further show that ∂2 log di(f)

∂f2i

+ ∂2 log di(f)

∂fi∂fj≤ 0. Note that

∂2 logdi(f)

∂f2i

+∂2 logdi(f)

∂fi∂fj=

(x0 +xj)(x0 +xi +xj)(x′′i xi −x′2

i )−xix′i

((x0 +xj)x

′i −xix

′j

)x2i (x0 +xi +xj)2

.

By condition (3), we have

∂di(f)

∂fi+∂di(f)

∂fj=x′i(x0 +xj)−xix

′j

(x0 +xi +xj)2≥ 0.

Therefore,

∂2 logdi(f)

∂f2i

+∂2 logdi(f)

∂fi∂fj=

(x0 +xj)(x0 +xi +xj)(x′′i xi −x′2

i )−xix′i

((x0 +xj)x

′i −xix

′j

)x2i (x0 +xi +xj)2

≤ (x0 +xj)(x0 +xi +xj)(x′′i xi −x′2

i )

x2i (x0 +xi +xj)2

=x0 +xj

x0 +xi +xj

d2 logxi

df2i

≤ 0.

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

Proof of Lemma 2 Using (4), we have

∂di(f)

∂fi= bi ≥ 0,

∂di(f)

∂fj=−cij ≤ 0,

∂2di(f)

∂f2i

= 0, and∂2di(f)

∂fi∂fj= 0.


Taking the logarithm transform and relevant derivatives, we have

∂2 logdi(f)

∂f2i

=

∂2di(f)

∂f2idi(f)−

(∂di(f)

∂fi

)2d2i (f)

=−

(∂di(f)

∂fi

)2d2i (f)

≤ 0,

∂2 logdi(f)

∂fi∂fj=

∂2di(f)


∂fi

∂di(f)

∂fj

d2i (f)=−

∂di(f)

∂fi

∂di(f)

∂fj

d2i (f)≥ 0.

Note that

∂2 logdi(f)

∂f2i

+∂2 logdi(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.

Proof of Lemma 3 Using (5), 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 derivative of ψi(fi) with respect to fi (or hi(fj) with respect to fj). Note that

logdi(f) = logψi(fi) + loghi(fj). Taking the logarithm transform and using ψi(fi) is log-concave in fi, we

have

∂2 logdi(f)

∂f2i

=d2 logψi(fi)

df2i

≤ 0,∂2 logdi(f)

∂fi∂fj= 0,

∂2 logdi(f)

∂f2i

+∂2 logdi(f)

∂fi∂fj=∂2 logdi(f)

∂f2i

=d2 logψi(fi)

df2i

≤ 0.

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

Proof of Proposition 1 By definition of ui(fi, qi(f)|p, fj) (see (13)), we have

∂ui(fi, qi(f)|p, fj)∂qi(f)

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

[YiGϵ(Yiqi(f))]. (22)

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

i gϵ(Yiqi(f))]< 0. Thus, ui(fi, qi(f)|p, fj) is

concave in qi(f). The proposition statement follows from the fact that q∗i (f) is either an interior solution

satisfying (22) or a corner solution constrained by (12).

Proof of Theorem 1 (a) It is immediate from (14) and (15) that q∗i is independent of fj , therefore q∗i is a

function of fi only, i.e. q∗i = q∗i (fi). If it satisfies (14), q

∗i (fi) is independent of fi, and hence dq∗i (fi)/dfi = 0.

In contrast, if q∗i (fi) satisfies (15), by the Implicit Function Theorem, dq∗i (fi)/dfi = 1/EYi[Yigϵ(Yiq

∗i (fi))]> 0.

Combining the above together, we have dq∗i (fi)/dfi ≥ 0. (b) Let qci (fi) and qii(fi) denote the stocking factor


implied by (15) and (14), respectively. By part (a), qci (fi) strictly increases in fi, whereas qii(fi) remains

constant in fi. By the interior point theorem, there exists a unique point f i at which qci (fi|f i) = qii(fi|f i).

(c) By (14) and (15), if fi ≤ (ri − wi + πi)/(ri − si + πi)EYi[Yi], EYi

[Gϵ(Yiqi(fi))] ≥ EYi[YiGϵ(Yiqi(fi))] =

(ri − wi + πi)/(ri − si + πi)EYi[Yi] ≥ fi ⇒ (14) is binding. In contrast, if fi ≥ (ri − wi + πi)/(ri − si +

πi)EYi[Yi], EYi

[YiGϵ(Yiqi(fi))] ≥ EYi[Yi]EYi

[Gϵ(Yiqi(fi))] = EYi[Yi]fi ≥ EYi

[Yi](ri − wi + πi)/(ri − si + πi) ⇒

(15) is binding. (d) and (e) are immediate from (b) and (c).

Proof of Theorem 2 Using (16), we have

dvi(fi|p, fj)dfi

∣∣∣∣fi≤fi

=∂di(fi|p, fj)

∂fiui(fi|p)+ di(fi|p, fj)

dui(fi|p)dfi

. (23)

By (17), we have

dui(fi|p)dfi

=dq∗i (fi)

dfi

{(ri −wi +πi)EYi


∗i (fi))]

}, (24)

d2ui(fi|p)df2

i

=d2q∗i (fi)

df2i

{(ri −wi +πi)EYi


∗i (fi))]

}−(dq∗i (fi)

dfi

)2

(ri − si +πi)EYi

[Y 2i gϵ(Yiq

∗i (fi))

]. (25)

By (24) and Theorem 1, dui(fi|p)/dfi = 0 when fi ≤ f i. In addition, because q∗i (fi) satisfies (14) when fi ≤ f i,

(23) can be simplified to

dvi(fi|p, fj)dfi

∣∣∣∣fi≤fi

=∂di(fi|p, fj)

∂fiui(fi|p) =

∂di(fi|p, fj)∂fi

{(ri − si +πi)EYi

[∫ Yiq∗i (fi)

0

ϵgϵ(ϵ)dϵ

]−πi

}. (26)

We can show that either ui(fi|p)< 0 when fi ≤ f i, or ui(fi|p)> 0 and increases in fi when fi ≤ f i. It follows

that neither case can be optimal for any fi ≤ f i, and therefore the optimal f∗i must satisfy f∗

i ≥ f i.

Proof of Theorem 3 By Proposition 1, (ri − wi + πi)EYi[Yi] − (ri − si + πi)EYi

[YiGϵ(Yiq∗i (fi))] ≤ 0. It

follows (see (17)) that ui(fi|p) ≤ 0 (and hence vi(fi|p, fj) ≤ 0) if (ri − si + πi)EYi

[∫ Yiq∗i (fi)

0ϵgϵ(ϵ)dϵ

]≤

πi. Further notice that dui(fi|p)dfi

|fi≥fi ≤ 0. Hence, the manufacturer participates if and only if (ri − si +

πi)EYi

[∫ Yiq∗i (fi|fi=fi)

0ϵgϵ(ϵ)dϵ

]>πi.

Proof of Theorem 4 Note that 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. 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 . The equilibrium on the optimal service levels f exists.

Proof of Theorem 5 A unique Nash equilibrium (f∗i , f

∗j ) is guaranteed if

∂2 log vi(f |p)∂f2

i

+∂2 log vi(f |p)

∂fi∂fj< 0.


By (16), 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


i

+∂2 log vi(f |p)

∂fi∂fj=∂2 logdi(f |p)

∂f2i

+∂2 logdi(f |p)

∂fi∂fj+d2 logui(fi|p)

df2i

=∂2 logdi(f |p)

∂f2i

+∂2 logdi(f |p)

∂fi∂fj+u′′i (fi|p)ui(fi|p)−u′2

i (fi|p)u2i (fi|p)

,

where ∂2 log di(f |p)∂f2

i+ ∂2 log di(f |p)

∂fi∂fj≤ 0 follows from Assumption 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′2

i (fi|p)< 0. Then uniqueness of equilibrium will

be guaranteed. Under Bernoulli yield distribution, we have

q∗i =G−1ϵ (fi/pi),

dq∗idfi

=1

pigϵ(q∗i ),d2q∗idf2

i

=−g′ϵ(q∗i )p2i g

3ϵ (q

∗i ).

So we can simplify (25) to

d2ui(fi|p)df2

i

=−g′ϵ(q∗i )p2i g

3ϵ (q

∗i )

{(ri −wi +πi)pi − (ri − si +πi)fi

}− ri − si +πi

pigϵ(q∗i ). (27)

Note that (ri − wi + πi) − (ri − si + πi)fi/pi ≤ 0 (see Theorem 3). Therefore, if g′ϵ(q∗i ) ≤ 0, we can get

d2ui(fi|p)df2

i< 0. Otherwise, if g′ϵ(q

∗i )> 0, we can rewrite (27) as

d2ui(fi|p)df2

i

=−g′ϵ(q∗i )(ri −wi +πi)

pig3ϵ (q∗i )

+(ri − si +πi)

(fipig′ϵ(q

∗i )− g2ϵ (q

∗i ))

pig3ϵ (q∗i )

=−g′ϵ(q∗i )(ri −wi +πi)

pig3ϵ (q∗i )

+(ri − si +πi) (Gϵ(q

∗i )g

′ϵ(q

∗i )− g2ϵ (q

∗i ))

pig3ϵ (q∗i )

< 0.

The inequality follows from Assumption 1.

Therefore, in either case, we have d2ui(fi|p)df2

i< 0. So d2 logui(fi|p)

df2i

=u′′i (fi|p)ui(fi|p)−u′2

i (fi|p)u2i

< 0, and then


i+ ∂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

optimal service level f∗i either is an interior solution that satisfies the first order condition specified in (19),

or a corner solution of either fi (lower bound of fi) or pi (upper bound of fi under Bernoulli distribution).

We will show as follows that the corner solution cannot be optimal.

At fi = fi, dui/dfi = 0 by the definition of fi. Sodvi(fi|p,fj)

dfi|fi=fi =

∂di∂fiui > 0, which means vi is increasing

at fi = fi. Therefore, fi cannot be optimal. Bernoulli distribution gives an upper bound of fi = pi. At fi = pi,

q∗i =G−1ϵ (1) = +∞. And we can derive that 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 optimal service level can only be the interior solution that

satisfies the first order condition.


Proof of Proposition A1 Part (a). Recall that f i is at the intersection of (14) and (15). In addition,

by part (e) of Theorem 1, q∗i (fi) is invariant to fi for fi ≤ f i and q∗i (fi) monotonically increases in fi for

fi > f i. Thus, the sign of, say, ∂f i/∂ri, is the same as the sign of ∂q∗i (fi)/∂ri. Hence, the sensitivity of f i

over ri,wi, si, and πi can be directly obtained by applying the Implicit Function Theorem to (14). Part (b).

First note that if Yi first-order stochastically dominates Yi, then by definition E[Yi]≥ E[Yi]. Define Yi as a

first-order stochastic increase over Yi such that E[Yi] = (ri − si +πi)/(ri −wi +πi)f i > E[Yi]. Note that such

Yi exists because by part (c) of Theorem 1 (ri−si+πi)/(ri−wi+πi)f i ≤ 1. The proposition statement then

follows by recognizing that f i|Yi ≥ (ri −wi +πi)/(ri − si +πi)E[Yi] = f i|Yi, which also holds for any Yi that

is (first-order) stochastically greater than Yi.

Proof of Proposition A2 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 (24). We refer di(f) as di(f |p) and ui(fi) as ui(fi|p) in the

following proof for the simplicity. By the Implicit Function Theorem, we have∣∣∣∣ ∂L1

∂f1

∂L1

∂f2∂L2

∂f1

∂L2

∂f2

∣∣∣∣ · ∣∣∣∣ ∂f1∂p1∂f2∂p1

∣∣∣∣=−∣∣∣∣ ∂L1

∂p1∂L2

∂p1

∣∣∣∣ .By the Cramer’s rule,

∂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

∣∣∣∣ . (28)

First we will show that the determinant of denominating matrix in (28) is positive at f∗i . Note that

det(L)≡ det

∣∣∣∣ ∂L1

∂f1

∂L1

∂f2∂L2

∂f1

∂L2

∂f2

∣∣∣∣= ∂2 log v1∂f2

1

∂2 log v2∂f2

2

− ∂2 log v1∂f1∂f2

∂2 log v2∂f1∂f2

. (29)

As we showed in proof of Theorem 5, ∂2 log vi∂f2

i+ ∂2 log vi

∂fi∂fj< 0. And note that ∂2 log vi

∂fi∂fj= ∂2 log di

∂fi∂fj≥ 0 by Assumption

2. We can derive that

∂2 log vi∂f2

i

< 0,

∣∣∣∣∂2 log vi∂f2

i

∣∣∣∣> ∣∣∣∣∂2 log vi∂fi∂fj

∣∣∣∣ .Therefore, (29) is positive.

Hence, the key to determine the sign of ∂f∗1/∂p1 and ∂f∗

2/∂p1 is to find the sign of the determinant of the

numerator matrices in (28).

Now consider each term in the numerator matrices in (28). For notational ease, in what follows we suppress

the dependence of di and qi on the vector of f . We have

∂Li(f)

∂fi=

∂2di(f)

∂f2idi(f)−

(∂di(f)

∂fi

)2d2i (f)

+u′′i (fi)ui(fi)−u′2

i (fi)

u2i (fi)

, (30)


∂Li(f)

∂fj=

∂2di(f)


∂fi

∂di(f)

∂fj

d2i (f), (31)

∂Li(f)

∂pi=

1

u2i (fi)

(∂u′

i(fi)

∂piui(fi)−

∂ui(fi)

∂piu′i(fi)

). (32)

∂Li(f)

∂pj= 0. (33)

By Theorem 5, (30) is negative. It is straightforward to prove that (31) is nonnegative. Next we prove that

(32) is positive. Note that

∂u′i(fi)

∂pi=∂q∗

′

i

∂pi

{(ri −wi +πi)EYi


∗i )]

}+ q∗

′

i

{(ri −wi +πi)

∂ EYi[Yi]

∂pi− (ri − si +πi)

∂ EYi[YiGϵ(Yiq

∗i )]

∂pi

},

∂ui(fi)

∂pi=∂q∗i∂pi

{(ri −wi +πi)EYi


∗i )]

}+ q∗i

{(ri −wi +πi)

∂ EYi[Yi]

∂pi− (ri − si +πi)

∂ EYi[YiGϵ(Yiq

∗i )]

∂pi

}

+(ri − si +πi)∂ EYi

[∫ Yiq∗i

0ϵgϵ(ϵ)dϵ

]∂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), (34)

∂ui(fi)

∂pi=∂q∗i∂pi

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

{∫ q∗i

0

ϵgϵ(ϵ)dϵ− q∗iGϵ(q∗i )

}, (35)

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

∗i )< 0. Next we prove that both (34) and (35) are positive. With

Bernoulli yield distribution, we have q∗i =G−1ϵ (fi/pi), q

∗′i = 1/(pigϵ(q

∗i ))> 0, ∂q∗i /∂pi =−Gϵ(q

∗i )/(pigϵ(q

∗i ))<

0, and ∂q∗′

i /∂pi = −1/(pigϵ(q∗i ))

2 [gϵ(q∗i )− g′ϵ(q

∗i )Gϵ(q

∗i )/gϵ(q

∗i )] < 0, where the last inequality follows from

Assumption 1. Using these expressions, it is immediate that (34) is positive. Now consider (35), which can

be simplified to

∂ui(fi)

∂pi=−Gϵ(q

∗i )

gϵ(q∗i )A′

i + q∗i (ri −wi +πi)+ (ri − si +πi)

{∫ q∗i

0

ϵgϵ(ϵ)dϵ− q∗iGϵ(q∗i )

}

=−Gϵ(q∗i )

gϵ(q∗i )A′

i + q∗iA′i +(ri − si +πi)

∫ q∗i

0

ϵgϵ(ϵ)dϵ > 0, (36)

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

∫ q∗i0ϵgϵ(ϵ)dϵ > 0 by the market entry con-

dition in Theorem 3 (and recognizing that Yi follows a Bernoulli distribution with parameter pi). Therefore,

(32) is positive (because u′i(fi)< 0). We have now proved that ∂Li(a, f)/∂pi > 0.


Part (a). The theorem statement ∂f∗i /∂pi > 0 follows from the fact that the determinant of the numerator

matrix is negative and the determinant of the denominating matrix is positive. The statement for ∂f∗j /∂pi ≥ 0

can be analogously shown.

Part(b). Using (28), we have ∂f∗1/∂p1 =− ∂L1/∂p1·∂L2/∂f2

det(L)and ∂f∗

2/∂p1 =∂L1/∂p1·∂L2/∂f1

det(L). By (30) and (31),

and Theorem 5, we have |∂L2/∂f2|> |∂L2/∂f1|. Hence the theorem statement follows.

Proof of Proposition 2 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 the first-order condition (19).

Proof of Proposition 3 Substituting (5) in lemma 3 into (31), we have ∂Li

∂fj= 0. By (28), ∂f∗

i /∂pj = 0.

Then (20) can be simplified to

∂v∗i (p)

∂pi=dvi(f

∗|p)dpi

= di(f∗|p)∂ui(f

∗i |p)

∂pi> 0, (37)

where the last inequality follows from (36).

Proof of Proposition 4 Follows from Proposition 3 and (21).

Proof of Theorem 6 As we showed in the proof of Proposition 3, f∗i is a function of pi only, i.e. f

∗i = f∗

i (pi).

We have vi(f∗|p) = di

(f∗i (pi), f

∗j (pj)|p)

)ui(f

∗i (pi)|p). Taking the first order derivative of v∗i (p) with respect

to pi,

∂v∗i (p)

∂pi=dvi(f

∗|p)dpi

=∂vi(f

∗|p)∂pi

+∂vi(f

∗|p)∂fi

∂f∗i

∂pi

= di (f∗|p) ∂ui(f

∗i |p)

∂pi(38)

where the last equality comes from the first order condition on optimal service level f∗. Further taking

derivative of (38) with respect to pj ,

∂2v∗i (p)

∂pi∂pj=∂di∂fj

∂ui

∂pi

∂f∗j

∂pj≤ 0.

where ∂di∂fj

≤ 0 comes from Assumption 2, ∂ui

∂pi> 0 comes from (36), and

∂f∗j

∂pj> 0 follows from Proposition A2.

When there is no spillover effect, i.e. α= 0

∂2Ji

∂zi∂zj=

∂2v∗i∂zi∂zj

=∂2v∗i∂pi∂pj

≤ 0.

which means Ji is submodular in z. Both players have decreasing best responses in the first stage game,

therefore the equilibrium exists.


Proof of Proposition 5 First, note that when manufacturer j’s improvement effort zj ≥ (1− p0i )/α, man-

ufacturer 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 effort zj ≤ (1−p0)/α. When the improve-

ment 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 the resulting expected

profit is Ji(pi = 1) = v∗i (pi = 1) −mi which is a constant that only depends on primitive parameters. If

manufacturer i does not exert improvement effort, its reliability index is pi = p0i +αzj and its expected profit

is Ji(pi = p0i +αzj) = v∗i (pi = p0i +αzj). Hence, given manufacturer j’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

reliability index pi that maximizes its expected profit.

Note that Ji(pi = 1) = v∗i (pi = 1) −mi is independent of zj . On the 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 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 improvement effort on the supplier. Otherwise, Ji(pi = 1)≥ v∗i (pi = p0i +αzj) when zj = 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

threshold zj ∈ [0, (1− p0i )/α], which only depends on primitive parameters, such that Ji(pi = 1) > Ji(pi =

p0i +αzj) for zj < zj , Ji(pi = 1) = Ji(pi = p0i +αzj) for zj = 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 always target perfect reliability under any feasible strategy of manufacturer

j.

Proof of Theorem 7 We can explicitly write out each manufacturer’s best response using Proposition 5.

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, each manufacturer’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 interval 0≤ z2 < z2, because z2 ≤ (1− p01)/α by Proposition 5. Thus, z1(z2)


is a (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

mentioned in the discussion of the surrogate game, which is after Theorem 6, the set of feasible joint strategies

in the surrogate game is a lattice. The rest of the proof follows from Tarski’s fixed point theorem in Tarski

(1955).

Proof of Theorem 8 Part(a). First, as discussed after Lemma 3, Cobb-Douglas demand function satisfies

Assumption 2. When there is no penalty cost (πi = 0) and the demand function is a Cobb-Douglas function,

we can simplify (19) to the following form:

βi

[G−1

ϵ (fipi)

((ri −wi)

pifi

− (ri − si)

)+(ri − si)

pifi

∫ G−1ϵ (

fipi

)

0

ϵgϵ(ϵ)dϵ

]+

(ri −wi)− (ri − si)fipi

gϵ(G−1ϵ ( fi

pi))

= 0.

Hence, Ki ≡ f∗i

piis the solution that satisfies the above equation which only depends on primitive cost

parameters ri, wi, si and the distribution of ϵ.

Part(b). Substitute f∗i =Kipi to (17) and let π= 0, we have:

ui = pi

[G−1

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

∫ G−1ϵ (Ki)

0

ϵgϵ(ϵ)dϵ

].

Note that the part in the square bracket is dependent on the cost parameters ri, wi, si and the distribution

of ϵ only, and independent of reliability index pi and pj . Therefore, ui is a linear function of pi. Denote Bi ≡[G−1

ϵ (Ki) (ri −wi − (ri − si)Ki)+ (ri − si)∫ G−1

ϵ (Ki)

0ϵgϵ(ϵ)dϵ

]for convenience. We can rewrite v∗i as follows:

v∗i = γi(Kipi)βi(Kjpj)

−βijpiBi = γiBiKβii K

−βij

j pβi+1i p

−βij

j ,

∂2v∗i∂p2i

= γiBiKβii K

−βij

j βi(βi +1)pβi−1i p

−βij

j ,

∂2v∗i∂p2j

= γiBiKβii K

−βij

j βij(βij +1)pβi+1i p

−βij−2j ,

∂2v∗i∂pi∂pj

=−γiBiKβii K

−βij

j (βi +1)βijpβii p

−βij−1j .

As mentioned in the discussion of the surrogate game (after Theorem 6), the set of feasible joint strategies

in the surrogate game is a lattice. In the surrogate game, we have

∂2Ji

∂zi∂zj=

∂2v∗i∂zi∂zj

=−α(∂2v∗i∂p2i

+∂2v∗i∂p2j

)− (α2 +1)

∂2v∗i∂pi∂pj

= γiBiKβii K

−βij

j pβii p

−βij−1j (βi +1)βijα

[−(βi

βij

pjpi

+βij +1

βi +1

pipj

)+

(α+

1

α

)].

Note that p0i ≤ pi ≤ 1 and p0j ≤ pj ≤ 1, therefore p0j ≤pj

pi≤ 1

p0i.

βi

βij

pjpi

+βij +1

βi +1

pipj

≤max

{βi

βij

p0j +βij +1

βi +1

1

p0j,βi

βij

1

p0i+βij +1

βi +1p0i

}≡Mi.


Note that M =max{M1,M2}. If M ≤ 2, then α+ 1α≥ 2≥M ≥ βi

βij

pj

pi+

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 supermodular game.

If M > 2, then α+ 1α≥M for 0 ≤ α ≤ 2

M+√

M2−4< 1. Consequently, when 0 ≤ α ≤ 2

M+√

M2−4, α+ 1

α≥

M ≥ βi

βij

pj

pi+

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

].

Part (c). These results follow directly from standard results of supermodular games, e.g. Theorem 4.2.1

in Topkis (1998).

Proof of Proposition 6 We use a similar proof approach to that developed in the proof of Proposition

A2, and, for brevity, we omit many intermediate steps below but the details can be easily gleaned from the

proof of Proposition A2. First consider wi. Similar to (28), 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

∣∣∣∣ . (39)

One can verify that ∂L2/∂w1 = 0, and hence the key to determine the sign of ∂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)

). (40)

We now prove that the term in the bracket in (40) is negative. By (17) and (24), we have

∂u′i(fi)

∂wi

ui(fi)−∂ui(fi)

∂wi

u′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)

0

ϵgϵ(ϵ)dϵ−πi

)< 0,

where the inequality follows from the fact that dq∗i (fi)/dfi > 0 (part (a) of Theorem 1) and (ri − si +

πi)EYi

∫ Yiq∗i (fi)

0ϵgϵ(ϵ)dϵ − πi > 0 (Theorem 3). It follows from (39) that ∂f1/∂w1 < 0. The statement for

∂f2/∂w1 ≤ 0 can be analogously proved. The proposition statements with respect to ri, si and πi can be

analogously proved.

Endnotes

1. One should not confuse the stocking factor with the order quantity: even though the optimal

stocking factor is constant in fi and independent of fj, the optimal order quantity 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 the demand as fi increases (decreases).


Acknowledgments

The authors thank Awi Federgruen for many helpful discussions.

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Improving Supplier Yield under Knowledge Spillover - Arizona State

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