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1 Ingredient Branding Strategies in an Assembly Supply Chain: Models and Analysis Juan Zhang, Qinglong Gou, Liang Liang School of Management, University of Science & Technology of China, Hefei, Anhui, 230026, P.R.China [email protected], [email protected], [email protected] Xiuli He Belk College of Business, University of North Carolina at Charlotte, Charlotte, NC 28223-0001 [email protected]

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Ingredient Branding Strategies in an Assembly Supply Chain:

Models and Analysis

Juan Zhang, Qinglong Gou, Liang Liang

School of Management, University of Science & Technology of China,

Hefei, Anhui, 230026, P.R.China

[email protected], [email protected], [email protected]

Xiuli He

Belk College of Business, University of North Carolina at Charlotte,

Charlotte, NC 28223-0001

[email protected]

2

Abstract

We consider a supply chain in which an original equipment manufacturer (OEM)

procures a key component from a supplier. We consider an ingredient branding strategy under

which the supplier and the OEM form a brand alliance. Specifically, the supplier invests in

ingredient branding to build up her goodwill and additionally she shares a portion of the

OEM’s advertising cost through a cooperative advertising program. Under a differential game

framework, we obtain the equilibrium advertising efforts of the supplier and OEM, and the

supplier’s equilibrium subsidy rate for the cooperative advertising program. We further

extend the model to the case in which the OEM procures two complementary components

from two suppliers. We consider three different scenarios of supplier interaction, i.e., the

suppliers are (I) independent, (II) allied and keep two brands, and (III) allied and keep one

brand. We demonstrate how the different interactions between suppliers affect the channel

members’ advertising efforts, goodwill levels, and their profits.

Keywords: Ingredient Branding, Cooperative advertising, Goodwill, Differential game

models

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1. Introduction

Owing to the Intel’s huge success in its “Intel Inside” program, Ingredient Branding, as a

special brand alliance form which emphasizes on identification of components in the final

product, is picking up in popularity in recent years. According to Kotler and Pfoetsh (2010),

Ingredient Branding is the brand policy concerning a branded object of materials,

components, or parts (raw materials, component materials, or component parts) that

represents a brand for the respective target group. Besides Intel, a number of companies from

different industries have successfully used Ingredient Branding strategy. Examples include

DuPont, Dolby Laboratories, Tetra Pak, Microban and so on.

According to the motivation behind it, Ingredient Branding can be manufacturer- initiated

or supplier-initiated one (Norris, 1992). In manufacturer-initiated Ingredient Branding, the

Original Equipment Manufacturer (OEM) usually chooses an existing ingredient brand with

strong brand awareness and promotes the fact that this ingredient is part of his final products.

Supplier-initiated Ingredient Branding occurs when a component supplier promotes her

ingredient to final users in efforts to build up brand awareness.

Combing the two kinds of Ingredient Branding together, Luczak et al. (2007) proposed a

new concept of InBranding, which is shown in Figure 1. To create the brand awareness of her

ingredient brand, the ingredient/component supplier forms an alliance with an OEM in a

supply chain framework. Under the alliance, the component supplier leaps the OEM and

communicates to final users directly (supplier-initiated Ingredient Branding), whereas the

OEM labels the ingredient brand logos in his final products, trying to persuade final users that

his products have certain positive attributes which are related to this ingredient brand

(manufacturer-initiated Ingredient Branding).

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Figure 1. InBranding framwork

InBranding strategy could create a win-win situation (Luczak et al., 2007). For the

component suppliers, such a strategy enables them to communicate their product offerings

and performance directly to end consumers and hence increase their brand equity.

Accordingly, the component suppliers gain greater bargaining power as the brand awareness

of suppliers will result in the consumers’ request or pulling the ingredient brands from the

OEMs. Furthermore, the good brand image built in consumers makes the component

suppliers escape the anonymity and substitutability of supplying a part or component by

competitors easily. For the downstream OEMs, incorporating a reliable supplier’s component

may enhance the image of their end products and increase demand due to the superior

performance of a key component.

Doyle (1989) argues that advertising is central to the process by informing consumers of

inherent product benefits and positioning the brand in the mind of the consumer. Our model

focuses on advertising decisions for InBranding and tries to answer the following questions:

Ingredient

Supplier

OEM

Final user

Cooperative

advertising program

Manufacturer-Initiated

Ingredient Branding

Advertising efforts

Demand Supply

Supply Demand

Supplier-Initiated

Ingredient Branding

Advertising efforts

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(i) When a component supplier implements her InBranding strategy, what are her optimal

advertising efforts directly to final users? (ii) What are the OEM’s optimal advertising

efforts?

Furthermore, noting that Intel has launched one of the largest cooperative advertising

program in the world to stimulate computer OEMs to label the “Intel Inside” logos in their

computers for its InBranding strategy, we also incorporate cooperative advertising decisions

in our model, trying to illustrate: (iii) Whether and under what condition should a component

supplier offer a cooperative advertising program to her OEM when she implements an

InBranding strategy? (iv) And if so, what is the supplier’s optimal subsidy rate under a

cooperative advertising program?

To answer the above questions, we consider a supply chain in which an OEM procures a

key component from a supplier. To implement the InBranding strategy, the supplier not only

builds up her goodwill through advertising, which is just the same as the OEM, but also

shares a portion of the OEM’s advertising cost via a cooperative advertising program. Noting

that building a brand is a long-term process and must be regarded as an investment in the

future (Meenaghan, 1995), we model the impact of advertising efforts on the goodwill of the

two channel members in a Nerlove-Arrow framework. Specifically, the goodwill of the OEM

depends on his own advertising and the supplier’s goodwill level whereas the supplier’s

goodwill depends on her own ingredient branding efforts and the OEM’s advertising efforts.

The sales of the final products are expressed as a function of the OEM’s advertising efforts

and goodwill. Under a Stackelberg-Nash game framework, we calculate out the equilibrium

advertising efforts of the supplier and OEM, as well as the supplier’s optimal subsidy rate for

her cooperative advertising program.

Main findings include the following. First, the component supplier shares the OEM’s

advertising cost only when her profit margin exceeds a threshold. Second, the supplier will

not advertise directly to final users if the OEM does not label her component brand logos in

his final products. Third, the supplier’s decisions on her own advertising efforts and subsidy

rate of the cooperative advertising program can be made separately. That is to say, the

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supplier’s optimal subsidy rate of the cooperative advertising program does not depend on

whether the supplier implements an supplier-initiated Ingredient Branding strategy (i.e., the

supplier advertises directly to the final users), it is just influenced by the relative magnitude

of the OEM’s and the supplier’s profit margins.

In practice, there are usually multiple component suppliers for OEMs and the marginal

profit threshold plays a key role when a firm decides whether to offer a cooperative

advertising program to its partner (Huang and Li, 2001; Jørgensen et al., 2000; 2001; Li et al.,

2002). We extend our model to the case in which the OEM procures two complementary

components from two suppliers in three different scenarios in which the suppliers are (I)

independent, (II) allied and keep two brands, and (III) allied and keep one brand. We attempt

to address the following questions: How does a complementary component supplier affect a

supplier’s ingredient branding strategy? Whether the two suppliers should cooperatively offer

a cooperative advertising program when they implement ingredient branding strategies?

Analysis of the extended models result in the following findings: (i) Whether or not a

supplier should offer the OEM a cooperative advertising program depends on the supplier’s

own profit margin and not on the other supplier’s profit margin; (ii) Suppliers will share a

part of the OEM’s advertising costs when their profit margins exceed a threshold level; (iii) If

both share the OEM’s cost, the subsidy rate of each supplier decreases compared to that with

a single supplier, but the total subsidy rate for the OEM increases; (iv) When the suppliers

form a strategic alliance, the allied supplier system is more likely to meet the profit margin

threshold and provides the OEM a cooperative advertising program with a higher subsidy rate

compared to the single supplier case, which implies that both the OEM and the suppliers

benefit from such alliances.

The rest of the paper is organized as follows. Section 2 provides a review of related

research. Section 3 presents the model and analysis for the case of a single component

supplier. Section 4 explores the case of two component suppliers. Section 5 presents

numerical analysis. Section 6 provides concluding remarks.

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2. Literature Review

This paper is related to the streams of literature in ingredient branding and co-op

advertising. In marketing theory and industrial practice, ingredient branding is defined as the

marking or labeling of components or their industrial goods (Kotler and Pfoertsch 2010).

Early research by Norris (1992) shows that ingredient branding, if successfully implemented,

can be very beneficial to both partner brands. Norris (1992) distinguishes between the

manufacturer-initiated and supplier-initiated ingredient branding strategies. The motivation

behind the former revolves around the host brand and usually extends or modifies an attribute

of the host brand in an effort to enhance consumer brand evaluations, whereas the motivation

behind the latter revolves around the component brand forming an alliance with an end

product manufacturer in an effort to create brand awareness for the ingredient brand and

generate pull effects through the value chain (Desai and Keller 2002, Havenstein 2004,

Pfoertsch and Müeller 2006). Hillyer and Tikoo (1995) describe the impact of ingredient

branding on consumer product evaluations. Rao and Ruekert (1994, 1999) evaluate ingredient

branding from the perspectives of multiple beneficiaries. McCarthy and Norris (1999) show

that branded ingredient could improve the competition position of host brands with moderate

quality. Bartlett et al. (2004) and Havenstein (2004) show that the brand cooperation between

the OEM and ingredient suppliers enables firms to establish and maintain their competitive

advantages and provide criteria for their customers to differentiate between competing

products. Venkatesh and Mahajan (1997), McCarthy and Norris (1999), and Pfoertsch and

Müeller (2006) show that ingredient branding can prompt suppliers to create a pull effect

from downstream customers.

Luczak et al. (2007) combine the manufacturer-initiated and supplier-initiated ingredient

branding and propose a new concept, InBranding, in which the ingredient suppliers promote

their brands to the OEMs and the end users simultaneously. While Baumgarth (2001) studies

the single stage branding in which the brand is promoted directly to the next users in the

supply chain, Luczak et al. (2007) study three kinds of relationship: (i) suppliers with OEM,

(ii) OEM with final user, and (iii) suppliers with final user. Pfoertsch et al. (2008) and Kotler

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and Pfoertsch (2010) use a similar framework as proposed by Luczak et al. (2007).

There are a few empirical studies in the ingredient branding literature. McCarthy and

Norris (1999) design two experiments to investigate the impact of branded ingredients on the

host products. They find that branded ingredients consistently and positively affect

moderate-quality host brands while branded ingredients only positively affect higher-quality

host brands occasionally. Desai and Keller (2002) conduct a laboratory experiment to analyze

the impact of two different branding strategies on consumer acceptance of brand expansions,

i.e., branding the target attribute ingredient as a self-branded ingredient versus as a

co-branded ingredient.

There is very limited analytical research on the ingredient branding except Venkatesh and

Mahajan (1997) and Erevelles et al. (2008). Venkatesh and Mahajan (1997) consider a

bundled product with two jointly consumed components. They propose an analytical

approach for the sellers of bundled product to make optimal pricing and partner selection

decisions. Erevelles et al. (2008) employ an econometric modeling approach to discuss why

the ingredient co-branding occurs. We propose an analytical model to study the dynamic

ingredient branding strategy.

Our research is also related to the co-op advertising in supply chains. Berger (1973)

proposes a static co-op advertising model. Bergen and John (1997), Huang and Li (2001) and

Li et al. (2002) extend Berger’s work in a single-manufacturer single-retailer supply chain.

Other static models in co-op advertising include Yue et al. (2006), Karray and Zaccour (2007),

Karray and Martín-Herrán (2009), Xie and Neyret (2009) and Karray (2011). In this paper,

we study the dynamic ingredient branding and co-op advertising strategy. A few papers have

studied the dynamic co-op advertising strategies. Jørgensen et al. (2000) study the co-op

advertising in a supply chain composed of a manufacturer and an exclusive retailer. They

explicitly distinguish between the long-term and short-term advertising efforts. The

short-term efforts stimulate the demand at the retailer store while the long-term advertising

efforts affect the demand through the accumulated goodwill. The manufacturer and retailer

choose their short-term and long-term advertising efforts. Jørgensen et al. (2001) assume

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decreasing marginal returns to goodwill and use a more flexible functional form for the sales

dynamics. These works are extended by Karray and Zaccour (2005) in which the retailer sells

both his own private product and the manufacturer’s product. He et al. (2009) study the co-op

advertising and pricing decisions in a decentralized supply chain with a single manufacturer

and single retailer in which the OEM decides the subsidy rate for the retailer. He et al. (2011,

2012) extend He et al. (2009) to a competitive retail market and they investigate how the

competition in retailers affects the co-op advertising policies. All of the above papers study

the manufacturer’s optimal subsidy policy for the retailer(s) while they do not model the

ingredient branding strategies as the components suppliers are absent in their models.

To our best knowledge, this is the first paper to study the joint decisions of ingredient

branding and co-op advertising in a dynamic assembly supply chain. We contribute to the

ingredient branding literature by proving an analytical model. Our results provide quantitative

guidelines for marketing brands and supply chain managers. We also contribute to the co-op

advertising literature by extending the practices into the supply chains with manufacturers

and suppliers while the previous research primarily focuses on the manufacturer-retailer

supply chains.

3. Basic Model: Single Component Supplier

In the basic model, we consider a supply chain in which an original equipment

manufacturer (OEM) procures a key component (ingredient) from a supplier and uses the

component to produce a final product. The supplier may implement an InBranding strategy.

To build up her ingredient brand awareness, the supplier not only advertises herself directly

to the consumer, but also offers a cooperative advertising program to encourage the OEM to

label the ingredient brand logos in the final products, as well as in the advertisement of the

final products. The OEM’s determines his advertising effort level to build up the final product

brand awareness.

Brand building is a long-term process (Meenaghan 1995). We model the impact of

advertising efforts in a Nerlove-Arrow framework to catch the long term influences. Let

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( )MU t and ( )SU t be the advertising effort levels of the OEM and supplier at time t ,

respectively. If the OEM does not label the supplier’s brand logo in the final product, the rate

of change in goodwill for the supplier ( )S t and that of the OEM’s goodwill ( )M t are just

given by the Nerlove-Arrow model as Equations (1) and (2), i.e.,

( ) ( ) ( )S S

dS t U t S t

dt , 0(0)S S , (1)

( ) ( ) ( )M M

dM t U t M t

dt , 0(0)M M , (2)

where M and S are positive constants which illustrate the efficiency of OEM’s and

supplier’s marketing efforts, respectively; 0M and 0S are the initial goodwill levels of the

two channel members at 0t ; 0 is the constant diminishing rate of goodwill.

If the OEM labels the supplier’s ingredient brand logos in the final product and promotes

the fact that the ingredient is a part of the end product, some changes occur. First, due to the

“brand halo” effect, consumer’s goodwill on the ingredient brand can be partly transferred to

the OEM (which is the main purpose of manufacturer-initiated Ingredient Branding), and thus

the rate of change in the OEM’s goodwill is translated to Equation (3), i.e.,

( ) ( ) ( ) ( )M M M

dM t U t S t M t

dt , 0(0)M M . (3)

where the M is a positive constant and ( )M S t represents the impact of such “brand

halo” effect. Generally, a larger M implies that the ingredient/component plays a more

important role in the final product. Second, since consumers can see the supplier’s ingredient

brand logos in the OEM’s final products and their advertisement, the OEM’s advertisements

are promoting the supplier’s brand partly. Thus, the rate of change in the supplier’s goodwill

is transferred to

( ) ( ) ( ) ( )S S S M

dS t U t U t S t

dt , 0(0)S S (4)

where S is a positive constant that captures the overflowing effect of the OEM’s

advertising effort, reflecting the fact that the OEM’s continued advertising investments

always have a positive effect on the supplier’s goodwill accumulation.

Note that the asymmetry between Equations (3) and (4) is original to the fact that the

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OEM’s advertisement labels the supplier’s brand logo while the supplier’s advertisement does

not label that of the OEM. Furthermore, letting 0M S in Equations (3) and (4), they

can be rewritten as Equations (1) and (2), implying the case that the OEM does not label the

supplier’s ingredient brand in his final products.

We assume that the sales of the final product ( )Q t are determined by the goodwill level

of the OEM and his instant advertising effort level, i.e., ( ) ( ( ), ( ))MQ t Q U t M t . While most

previous literature on Nerlove-Arrow framework assumes that the sales are only affected by

the stock of goodwill (e.g., Nerlove and Arrow 1962; Chintagunta and Jain 1992; Viscolani

and Zaccour 2009), our sales response function is more general in that it captures both the

long-term carry-over effect and the instant effect of advertising on the final product sales.

Specifically, we assume a linear sales response function:

( ) ( ) ( )MQ t aU t bM t , (5)

where a is a non-negative constant that captures the instant effect of the OEM’s advertising

on the final product sales and b is a positive constant that captures the long-term effect.

When the OEM’s advertising effort does not stimulate any instant sales, we have 0a and

Equation (5) reduces to the sales response function in Chintagunta and Jain (1992) and

Jørgensen et al. (2000). We assume that one unit of final product requires one unit of

component, so the sales of the component ( )SQ t are equal to the sales of the final product

( )Q t , i.e.,

( ) ( )SQ t Q t . (6)

Let ( ( ))MC U t and ( ( ))SC U t denote the OEM’s and the supplier’s advertising cost functions.

We assume that the advertising cost functions of the OEM and supplier are quadratic in their

advertising efforts:

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( ( )) ( ( ))2

M MC U t U t , 21

( ( )) ( ( ))2

S SC U t U t . (7)

The quadratic cost assumption is common in the literature (Deal 1979; Sorger 1989;

Chintagunta and Jain 1992; Prasad and Sethi 2004; Bass, Krishnamoorthy, Prasad, and Sethi

12

2005; He et al.2009; He et al. 2011). It implies diminishing returns to advertising costs. We

assume that the firms have certain advertising budget constraints so that the efforts MU and

SU are limited by upper bounds:

0 ,0M SU K U K , (8)

where K is a positive constant which is large enough.

Let , [0,1] , denote the supplier’s subsidy rate of her cooperative advertising

program, the profit rate functions of the OEM, ( )M t , and supplier, ( )S t , are then given

by

( ) ( ( ), ( )) (1 ) ( ( ))M M M Mt Q U t M t C U t , (9)

( ) ( ( ), ( )) ( ( )) ( ( ))S S M M St Q U t M t C U t C U t , (10)

where M and S are the OEM’s and the supplier’s profit margins, respectively. We

assume that the objective functions for the OEM and supplier MJ and SJ are the

discounted values of their profit streams over an infinite horizon with a common discount

factor r :

0

max ( , , ) ( )M

rt

M M S MU

J U U e t dt

, (11)

,

0

max ( , , ) ( )S

rt

S M S SU

J U U e t dt

, (12)

subject to the state equations (3) and (4); ( )M t and ( )S t are given by (9) and (10),

respectively.

The sequence of events is as follows. First the supplier offers the OEM a co-op

advertising program and announces her subsidy rate . Then the supplier chooses her

ingredient branding effort and the OEM chooses his advertising effort for the final product

simultaneously. Sales are then realized. The supplier and OEM maximize their individual

profits while taking each other’s best response into consideration. The result for the subgame

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is a Nash equilibrium. There are two types of equilibrium strategies for this dynamic

optimization problem: open-loop and closed-loop strategy. A closed loop strategy is a

function of time and the supplier’s and OEM’s current goodwill levels while an open-loop

strategy is a function of time only. Closed-loop strategies are generally more complex

mathematically and more desirable as they are time-consistent. However, for our particular

model formulation, closed-loop strategies are mathematically intractable. Therefore, we focus

on deriving the open-loop strategies. Since our supply chain operates in deterministic

situations, open-loop strategies are reasonable for those occasions (Eliashberg and Chatterjee

1985). To derive the dynamic equilibrium strategies for the channel members, we first keep

the subsidy rate fixed and derive the best response advertising effort levels of the OEM

and supplier.

We follow the standard procedure of dynamic optimization. Substituting Equations (9)

and (10) for ( )M t and ( )S t in (11) and (12) respectively and taking into account the

state Equations (3) and (4), we obtain the present value Hamiltonian equation for the OEM as

2

1 2

1( ) (1 )

2

( ) ( )

M M M M

M S S S M M M M M

H aU bM U

U U S U S M

, (13)

where 1M and 2M are the co-state variables (shadow prices) in the OEM’s problem

corresponding to OEM and supplier goodwill levels, respectively. Similarly, the supplier’s

present value Hamiltonian equation is:

2 2

1 2

1 1( ) ( ) ( )

2 2

( ) ( )

S S M S M

S S S S M S M M M

H aU bM U U

U U S U S M

, (14)

where 1S and 2S represent the co-state variables in the supplier’s problem corresponding

to the supplier and OEM’s goodwill levels, respectively.

Let ( )MU and ( )SU denote the OEM’s and the supplier’s best response advertising

effort levels for a given subsidy policy , respectively. We obtain the following result:

Proposition 1. When the single supplier offers the OEM a cooperative advertising program,

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for a given subsidy rate , the supplier’s best response advertising effort level for the

ingredient brand is given by

2

( )( )

S S MS

bU

r

, (15)

and the OEM’s best response advertising effort level for the final product is given by

2( )

1

S MM MM

bbU a

r r

. (16)

We find that for an announced subsidy rate , the best response advertising effort of the

OEM, ( )MU , and best response ingredient branding effort, ( )SU , are both constant over

time. The strategies of maintaining the constant advertising levels are easy to implement from

the managerial perspective. The OEM’s best response advertising effort is increasing in the

supplier’s subsidy rate , which makes intuitive sense as the OEM has more incentive to

promote the end product if his promotional expenditure is shared by the supplier. Although

we allow the supplier to decide on her advertising effort after she announces her subsidy rate,

her best response advertising effort ( )SU does not depend on the announced subsidy rate

. In other words, the supplier can make separate decisions on the ingredient branding effort

and a co-op subsidy rate. Furthermore, the supplier’s best response advertising effort ( )SU

is increasing in her own profit margin S .

With a slight abuse of notation, we sometimes omit the time argument t for the state

equations and replace it with to denote that the corresponding variables are the best

response functions. We substitute the best response functions (15)-(16) into the state

equations (3)-(4) to get the response state equations as:

0( ) ( ) ( ) ( ), (0)M M MM U S M M M ,

0( ) ( ) ( ) ( ), (0)S S S MS U U S S S .

We substitute the best response functions (15)-(16) into (9)-(10) and express the OEM’s

and the supplier’s profits as functions of :

( ) ( ( ), ( )) (1 ) ( ( ))M M M MQ U M C U ,

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( ) ( ( ), ( )) ( ( )) ( ( ))S S M M SQ U M C U C U .

We then solve for the supplier’s equilibrium subsidy rate for the OEM and the

equilibrium advertising efforts of the channel members. Results are summarized in the

following proposition and the proofs are in the Appendix.

Proposition 2. When the single supplier offers the OEM a cooperative advertising program,

her equilibrium subsidy rate * is

*

2

2 2

02

S M MS

S M

MS

if

if

. (17)

the supplier’s equilibrium advertising effort *

SU for her ingredient brand is

*

2( )

S S MS

bU

r

, (18)

and the OEM’s equilibrium advertising effort *

MU for the final product is

2

*

2

( )2 2

2

S MM M MS S

M

S MM MM S

bba if

r rU

bba if

r r

. (19)

The comparative statics are: * / 0S , * / 0M , * / 0M MU , * / 0M SU ,

* / 0M MU , * / 0M SU , * / 0S SU , * / 0S MU .

The supplier will subsidize the OEM’s advertising expenditure only when her profit

margin S exceeds a threshold

/ 2M . Below the threshold, the supplier does not share the

OEM’s cost of advertising because the OEM is so profitable that he himself will invest

significantly in promoting the final product. Such threshold could be prohibitive for some

component suppliers which only have lower profit margins.

16

The supplier’s equilibrium subsidy rate is non-decreasing in her profit margin and

non-increasing in the OEM’s profit margin. The equilibrium subsidy rate is independent of

parameters such as M and S , implying that the supplier will share the OEM’s

advertising costs even if he does not label the supplier’s logos in his final products. Our result

partly justifies why Intel offered plenty of preferential conditions to Apple even when Steve

Jobs refused to label the “Intel Inside” logo in Apple’s products in 2005. The supplier’s

equilibrium advertising efforts are linearly increasing in the component’s importance degree

in the final products, i.e., M , which is consistent with the observations in practice. For

example, companies producing key components, such as Intel (computer microprocessor),

Microban (anti-bacterial protection), and DuPont (Teflon, Lycra, etc.) have successfully

implemented InBranding strategies. In contrast, the suppliers producing non-critical

components rarely advertise directly to final users to build up brand awareness.

The OEM’s equilibrium advertising effort is non-decreasing in his own profit margin and

the supplier’s profit margin. We illustrate the impacts of the three terms in

2/ ( ) /M S Ma b r b r in (19). The term a shows that instant advertising

effort increases the immediate sales. Even when the other two terms are equal to zero, the

OEM still keeps certain level of advertising to stimulate instant sales. The term

/ ( )Mb r captures the long-term carry-over effect of the OEM’s advertising on the

product sales. This term is large if the OEM puts more weight on future profits (a small r )

and/or the consumers have a long memory (small ). The term 2

/S Mb r represents

the benefit of manufacturer-initiated ingredient branding. When the OEM promotes the

supplier’s brand, the component brand image is improved and the improved component brand

image in turn enhances the OEM’s brand image. Note that this “feedback effect” gets less

significant with a large discount factor r or decay factor .

When the OEM does not promote the supplier’s brand, we get the equilibrium advertising

efforts of the two channel members by setting 0M S in Equations (18) and (19):

* 0SU , *

*1

M MM

bU a

r

, (20)

17

where * is given by (17). The above result shows that the supplier will not advertise

directly to the final users if the OEM does not inform the customers that the ingredient is

included in his final product. In other words, the OEM’s cooperation is necessary for the

supplier to initiate an ingredient branding strategy. Furthermore, comparing *

MU in (16) with

that in (20), we find that the OEM exerts less advertising effort when the supplier does not

initiate an ingredient branding strategy.

Proposition 3. The equilibrium accumulated goodwill levels of the OEM and supplier along

with time t are given by

*( ) ( ) t t

M SSM t X te Ye M , *( ) t

SSS t Xe S ,

respectively, where *

0 SSX S S , *

0 SSY M M , *

SSM and *

SSS are given by

* * *1( )SS M M M SSM U S

,

* * *1( )SS S S S MS U U

.

When t , *( ) SSS S and *( ) SSM M , therefore *

SSM and *

SSS are steady

state goodwill levels of the OEM and the supplier, respectively. The implication of this result

is that after a sufficiently long period of time, the supply chain members’ goodwill levels will

not change dramatically. The supplier’s stable goodwill *

SSS is linearly increasing in *

SU

and *

MU , indicating that advertising efforts exerted by the supplier and the OEM have

positive effects on building up the long-term supplier brand awareness. The OEM’s stable

goodwill * * *( ) /SS M M M SSM U S indicates that the OEM’s brand reputation is enhanced

by the reputation of the supplier. This explains why the OEM would rather select an existing

ingredient brand with high brand awareness for his manufacturer-initiated Ingredient

Branding strategy.

4. Two Complementary Component Suppliers

In this section, we consider the case in which the OEM procures two complementary

components (e.g., monitors and processors for computers) from component suppliers and

incorporates them in the final product. We assume that the OEM is independent of component

suppliers while the two suppliers are independent of each other or may form a strategic

18

alliance. We consider three scenarios in which the two suppliers are: (I) independent, (II)

allied and keep two distinct brands, and (III) allied and keep one brand. We derive the OEM’s

advertising effort and suppliers’ equilibrium subsidy rates for the OEM as well the suppliers’

branding efforts under each scenario.

Let ( )iS t and ( )iU t denote the goodwill level and advertising effort of Supplier i ,

1,2i , respectively. The rate of change in goodwill for the OEM, ( )M t and for Supplier i ,

( )iS t , are given by

1 1 2 2( ) ( ) ( ) ( ) ( )M M M M

dM t U t S t S t M t

dt , 0(0)M M , (21)

( ) ( ) ( ) ( )i i i i M i

dS t U t U t S t

dt , 0(0)i iS S , (22)

where the parameters i , Mi , i and 0 ,iS 1,2,i have the same meanings as in Section

3. When the OEM does not label the brand logo of Supplier i , 1,2i , in the final product,

we set 0Mi i in Equations (21) and (22).

We assume that one unit of final product requires one unit of components 1 and 2.

Accordingly, we have ( ) ( )iQ t Q t 1,2i . The cost function of Supplier i is 21

( ) ( )2

i iC U U ,

1,2i .

4.1 Independent Suppliers (Scenario I)

In Scenario I, the OEM and two complementary suppliers make independent decisions by

maximizing their own profits over an infinite horizon. Let i , [0,1]i , 1,2i , denote

Supplier i ‘s subsidy rate for the OEM under a co-op advertising program. The OEM’s

portion of the advertising expenditure is 1 2(1 ) ( ( ))MC U t .

Assume that the two suppliers are the Stackelberg leaders and the OEM the follower. The

sequence of the events is as follows. First, the two component suppliers simultaneously

announce their subsidy rates i , 1,2i , to the OEM. Second, the OEM and the two

suppliers decide their advertising efforts for their own brands independently. Sales are then

realized.

The OEM’s profit function ( )M t is

19

1 2( ) ( ( ), ( )) (1 ) ( ( ))M M M Mt Q U t M t C U t , (23)

and Supplier i ‘s profit function ( )i t is

( ) ( ( ), ( )) ( ( )) ( ( ))i i M i i Mt Q U t M t C U t C U t , 1,2i , (24)

where i is Supplier i ‘s profit margin. The OEM and suppliers maximize the present

values of their profits, i.e.,

1 2 1 2

0

max ( , , , , ) ( )M

rt

M M MU

J U U U e t dt

,

1 2 1 2,

0

max ( , , , , ) ( )i i

rt

M M iU

J U U U e t dt

,

subject to Equations (21) and (22); M and i are given by Equations (23) and (24),

respectively. We use a similar procedure to derive the open-loop equilibrium strategies.

Proposition 4. When the OEM and two suppliers are independent, Supplier i ‘s equilibrium

subsidy rate for the OEM under the co-op advertising program is

(3 )

1 2

*

(3 )

1

2,

2 2 2 2

2,

2 2 2

0,2

i M M Mi i

I i M M Mi i i

M

Mi

if and

if and

if

, 1,2i . (25)

The sum of the two equilibrium subsidy rates is

1 2

11 2

* * 11 2

22 1

2

1 ,2

2,

2 2 2

2,

2 2 2

0,2

M Mi

M M M

I I M

M M M

M

Mi

if

if and

if and

if

. (26)

We have a few observations from Proposition 4. First, when the suppliers and the OEM

20

are independent, each supplier offers the OEM a co-op advertising program if and only if her

profit margin is larger than a threshold of / 2M .

Second, whether or not a supplier offers a co-op advertising program to an OEM does not

depend on the other supplier’s profit margin. When only one supplier subsidizes the OEM,

her equilibrium subsidy rate depends only on her own margin and the subsidy rate is equal to

that when she is a single supplier. When both suppliers offer co-op advertising programs to

the OEM, the subsidy rate of each supplier decreases in the other supplier’s profit margin.

Thus a component supplier may benefit when complementary component suppliers are

present. Because the complementary components are incorporated into the same end product,

the interests of the complementary suppliers are aligned in encouraging the OEM to advertise

more to stimulate the end product sales.

Third, the OEM benefits from having multiple complementary component suppliers as it

is more likely for him to get support from suppliers and the total subsidies for him increase

with more suppliers supporting him.

Proposition 5. When the OEM and two suppliers are independent, Supplier i ‘s equilibrium

advertising efforts for her ingredient brand is

*

2( )

I i i Mii

bU

r

, 1,2.i (27)

The OEM’s equilibrium advertising effort is

* 1 1 2 2

2* *

1 2

2

1 2 1 1 2 2 1 22

2

1 1 1 2 2 1 22

2

2 1 1 22

( )[ ]

1

1( )[ ( )], ,

2

1( )[ ( )],

2 2 2

1( )[ (

2

I M M M MM I I

MM M M

M M MM M M

MM M

b bU a

r r

a r b r b ifr

a r b r b if andr

a r b r br

2 2 1

2

1 1 2 2 1 22

)],2 2

1[ ( )], , .

2

M MM

MM M M M

if and

a r b r b ifr

.

(28)

Proposition 5 shows the channel members’ equilibrium advertising efforts are constant

21

over time. Such constant equilibrium strategies are robust and easy to implement. We have a

few more observations. First, the supplier’s ingredient branding effort does not depend on

whether or not she offers the OEM a subsidy. Second, each supplier’s decision on the

ingredient branding effort depends on her own system parameters while not on the other

supplier’s parameters. Third, the OEM invests more in advertising in Scenario I than in the

case of a single supplier.

When neither of the suppliers’ brands is promoted/labeled by the OEM, the equilibrium

advertising levels of the three channel members can be obtained by setting 0Mi i ,

1,2i , in Equations (27) and (28):

*

* *

1 2

[ ]1

I M MM I I

bU a

r

, * 0I

iU , 1,2i ,

where the equilibrium subsidy rate *I

i , 1,2i , is given by (25). Similar to that in the single

supplier case, the two suppliers will not advertise directly to consumers if the OEM does not

label the suppliers’ logos in the final product.

We obtain the trajectories of equilibrium cumulative goodwill levels for the OEM and two

suppliers.

Proposition 6. When the OEM and two suppliers are independent, the equilibrium accumulated

goodwill levels of the OEM and Supplier i along with time t are given by

*

1 1 2 2 1( ) ( ) t t I

M M SSM t X X te Y e M ,

*( ) t I

i i iSSS t X e S , 1,2i ,

where*

0

I

i i iSSX S S , 1,2i , *

1 0

I

SSY M M , and *I

SSM and *I

iSSS are given by

* * * *

1 1 2 2

1( )I I I I

SS M M M SS M SSM U S S

, * * *

2

1( )I I I

iSS i i i MS U U

, 1,2i .

Note that *I

SSM and *I

iSSS are the steady state goodwill levels of the OEM and Supplier ,i

2,1i . The goodwill levels will not change after a sufficiently long period of time. Proposition 6

illustrates that the component brands’ goodwill levels positively affect the OEM’s goodwill level

while each ingredient brand benefits from the OEM’s and the other supplier’s advertising efforts. This

22

result implies that a strategic alliance between suppliers may be beneficial to all the supply chain

members, which will be investigated subsequently.

4.2 Supplier Alliance with Two Brands (Scenario II)

We consider the case where the two component suppliers form a strategic supplier

alliance and the alliance keeps both component brands. The alliance determines its

advertising effort for each component/brand and proposes a single subsidy rate for the

OEM. In this subsection, we use “component” and “brand” interchangeably. Let S denote

the supplier alliance’s profit function which is given by

1 2 1 2( ) ( ) ( ( ), ( )) ( ( )) ( ( )) ( ( ))S M Mt Q U t M t C U t C U t C U t , (29)

where 1 2 is the total profit margin from selling two components; ( ( ))iC U t is the

advertising cost function for Brand i , 1,2i . The supplier alliance solves the following

problem:

1 2

1 2, ,

0

max ( , , , ) ( ) ,rt

S M SU U

J U U U e t dt

subject to Equations (22) and (22), where ( )SJ is the supplier alliance’s objective function

and ( )S t is given by (29).

Proposition 7. When two suppliers form an alliance and keep two distinct component brands,

the supplier alliance’s equilibrium subsidy rate for the OEM is given by

1 21 2

* 1 2

1 2

2 2,

2 2 2

0,2

M M

II M

M

if

if

. (30)

Similar to the cases of a single supplier (Scenario I) and independent suppliers (Scenario

II), the supplier alliance’s equilibrium subsidy rate is a threshold policy, i.e., it supports the

OEM’s advertising only if the combined profit margin from two components exceeds / 2M .

23

Recall that in Scenario I, we derive a challenging threshold, / 2S M , above which the

single supplier subsidizes the OEM’s advertising effort. Since the alliance sells two

components to the OEM and obtains a larger profit, it is more likely to reach the threshold

profit margin and support the OEM. The alliance’s equilibrium subsidy rate does not depend

on whether or not the OEM labels the component brand logos in the final product.

Proposition 8. When the two suppliers form an alliance and keep two distinct component

brands, the supplier alliance’s ingredient branding efforts for the two components are given

by

* 1 2

2

( )

( )

II i Mii

bU

r

, 1,2i , (31)

and the OEM’s equilibrium advertising effort for the final product is given by

* 1 1 2 2

2*

2

1 2 1 1 2 2 1 22

2

1 1 2 2 1 22

( )[ ]

1

1( )[ ( )],

2 2

1[ ( )],

2

II M M M MM II

M MM M M

MM M M M

b bU a

r r

a r b r b ifr

a r b r b ifr

(32)

Recall that in Scenario I, the equilibrium advertising effort of each supplier depends on her

own profit margin only. In contrast, under an alliance, the equilibrium advertising effort for

each brand depends on the sum of the profit margins from two components shown in

Equation (31). Therefore, the supplier alliance invests more in building the component brands

than each independent supplier, i.e., * *II I

i iU U , 1,2.i

When only one ingredient brand is labeled in the final product, we get the equilibrium

advertising efforts of the channel members by setting 0Mi ( 1 2i or i ) in (31) and (32):

*

2*( )

1

II i MiM MM II

bbU a

r r

,

* 1 2

2

( )

( )

II i Mii

bU

r

, *

(3 ) 0II

iU , 1,2,i

where*II is given by (30).

24

When neither of the two ingredient brands is labeled in the final product, we set 0Mi ,

1,2i , in (31) and (32) to get the equilibrium advertising efforts of the channel members:

*

*( )

1

II M MM II

bU a

r

, * 0II

iU , 1,2i .

We next obtain the accumulated goodwill levels of the OEM and two component brands.

Proposition 9. When the two suppliers form an alliance and keep two brands, the equilibrium

accumulated goodwill levels for the OEM and for the two components brands along with time

t are given by

*

1 1 2 2 2( ) ( ) t t II

M M SSM t Z Z te Y e M ,

*( ) t II

i i iSSS t Z e S , 1,2i ,

where *

0

II

i i iSSZ S S , 1,2i , *

2 0

II

SSY M M , and *II

SSM and *II

iSSS are given by

* * *1( )II II II

iSS i i i MS U U

,

* * * *

1 1 2 2

1( )II II II II

SS M M M SS M SSM U S S

.

The stable goodwill levels of channel members, *II

iSSS and *II

SSM , have the same

structures as those in Scenario I. Since * *II I

i iU U and * *II I

M MU U , we can easily show

that * *II I

iSS iSSS S and * *II I

SS SSM M , implying that all the channel members have higher levels of

brand awareness when suppliers form an alliance than when suppliers are independent

(Scenario I).

4.3 Supplier Alliance with One Brand (Scenario III)

In this scenario, the two suppliers form a strategic alliance but the allied supplier system

keeps one brand rather than two distinct brands. Without loss of generality, we assume the

system keeps Supplier 1’s brand. The supplier alliance decides the advertising effort for the

component brand and the subsidy rate for the OEM. The profit function of the OEM is the

same as that in Equation (9). The supplier alliance’s profit function is given by

1 2 1( ) ( ) ( ( ), ( )) ( ( )) ( ( ))S M Mt Q U t M t C U t C U t . (33)

25

Note that Scenario III is a special case of the single supplier case with 1 2S .

Accordingly, the allied system’s equilibrium subsidy rate *III is

1 21 2

* 1 2

1 2

2( ),

2 2 2

0,2

M M

III

M

if

if

. (34)

Since 1 2 1 , it is more likely for the supplier alliance to provide a co-op

advertising program to the OEM. Furthermore, when the OEM is actually subsidized by the

supplier alliance, he gets a higher subsidy rate than that in a single supplier case.

The equilibrium advertising efforts for the OEM and the allied supplier in Scenario III

are given by:

* 1 1

2*1

III M M MM III

b bU a

r r

, * 1 2 1 1

1 2

( )

( )

III M bU

r

. (35)

It can be easily verified that *

1

III

SU U , where SU is given by Proposition 1. The

alliance invests more in advertising due to a higher profit margin. The result suggests that the

suppliers pool the financial resources together to build up one strong brand if each alone does

not have enough resources to do so. In addition, the OEM invests more in advertising in

Scenario III than in the single supplier case due to a higher subsidy rate.

5. Numerical Analysis

In this section, we present numerical analysis to demonstrate the equilibrium subsidy

rates, goodwill levels, profits to the channel members by varying Supplier 2’s profit margin

and ingredient branding strategy. Most of the following results were proven in earlier sections.

In order to compare results from different scenarios, we include the equilibrium results with

the single supplier in each figure. To be consistent with previous sections, we assume that

Supplier 1’s parameter values are the same as that in the single supplier case. Our analysis

uses following parameter values: 0.2a , 4b , 0.3r , 0.2 , 1 3.6 , 2 3 ,

0.05M , 10 10S , 20 12S , 0 15M , 8M , and 1 4.5S . For Figures 2-5, we

use 1 , 1M , 2 , 2M . For Figures 6-8, we use 1 , 1M ,

26

2 , 2M if ingredient branding strategies are implemented.

Figure 2. Subsidy rates vs. Supplier 2’s Profit Margin.

Figure 2 shows the impact of changing Supplier 2’s profit margin on Supplier 1’s and

Supplier 2’s equilibrium subsidy rates and on the total subsidy rates. In Scenario I where

Supplier 1 and Supplier 2 are independent, when Supplier 2’s profit margin is low (less than

4), Supplier 2 will not support the OEM and her profit margin does not affect Supplier 1’s

subsidy policy for the OEM. When Supplier 2’s profit margin is high (greater than 4),

Supplier 2 supports the OEM and her subsidy rate increases in her profit margin. Because

Supplier 2 supports the OEM, Supplier 1’s subsidy rate decreases. The higher Supplier 2’s

profit margin, the more reduction in Supplier 1’s support. In Scenario II, the two suppliers

form an alliance which offers one subsidy rate to the OEM, the subsidy rate increases in 2 .

We find that * * * *

1 2

II I I , i.e., the total subsidy for the OEM is the highest when two

suppliers form an alliance while the total subsidy rate with two independent suppliers is

higher than that in the single supplier case.

27

Figure 3. Advertising Efforts vs. Supplier 2’s Profit Margin.

Figure 3 depicts the OEM’s equilibrium advertising effort level and the suppliers’

equilibrium advertising effort levels as Supplier 2’s profit margin varies. When the two

suppliers are independent, Supplier 1’s advertising effort is not affected by Supplier 2’s profit

margin. Supplier 2’s advertising effort increases by her own profit margin. When Supplier 2

does not subsidize the OEM, the OEM’s advertising effort is not affected by Supplier 2’s

profit margin. When Supplier 2’s profit margin is high enough to subsidize the OEM’s

advertising, the OEM’s advertising effort increases by Supplier 2’s profit margin. When the

two suppliers form an alliance, the OEM increases his advertising effort and both suppliers

increase their advertising efforts as Supplier 2’s profit margin increases. Figure 3 shows that

the OEM and Supplier 1 both benefit from having Supplier 2 and they benefit more when the

suppliers form an alliance than when the suppliers are independent: * * *II I

M M MU U U and

* * *

1 1

II I

SU U U .

28

Figure 4. Goodwill levels vs. Supplier 2’s Profit Margin.

Figure 5. Profits vs. Supplier 2’s Profit Margin.

Figures 4 and 5 show the goodwill levels and profits of the OEM and suppliers as 2

varies. Under the independent supplier system (Scenario I), the OEM’s and suppliers’ steady

29

state goodwill levels increase in 2 when Supplier 2’s profit margin is above the threshold

while they do not change with 2 when 2 is below the threshold. In contrast, under a

supplier alliance (Scenario II), the OEM’s steady state goodwill level *II

SSM is strictly

increasing in 2 and so are the suppliers’ goodwill level *

1

II

SSS and *

2

II

SSS . Not surprisingly,

the OEM’s and Supplier 1’s goodwill levels are highest when the suppliers form a strategic

alliance, i.e., * * *II I

SS SS SSM M M and * * *

1 1

II I

SS SS SSS S S . Their profits are

* * * *

1 2

II I I

S SJ J J J and * * *II I

M M MJ J J (Figure 5). Furthermore, they benefit more when

suppliers are allied than when they are independent. The horizontal supplier alliance creates a

win-win situation for all channel members.

Figure 6. Advertising Efforts vs. Supplier 2’s Ingredient Branding.

30

Figure 7. Profits vs. Supplier 2’s Ingredient strategy.

Figures 6 and 7 show the OEM’s advertising efforts and profits with and without an

ingredient branding strategy by Supplier 2. Figure 6 shows that in Scenario I, Supplier 2’s

advertising efforts affect the OEM only when she subsidizes the OEM’s advertising efforts

while in Scenario II, the OEM’s profits and advertising efforts are strictly increasing in 2 .

The OEM’s advertising efforts and profits are higher when Supplier 2 implements an

ingredient branding strategy ( 2 , 2M ) than without such a strategy

( 2 2 0M ). Figure 8 shows that in Scenario I, Supplier 1’s profit (advertising effort) is

higher when Supplier 2 implements an ingredient branding strategy; Supplier 1’s profit is

strictly increasing in 2 when Supplier 2 implements an ingredient branding strategy. When

Supplier 2 does not invest in ingredient branding, Supplier 1’s profit is not affected when 2

is below the subsidy threshold and it strictly increases when 2 exceeds the threshold.

31

Figure 8. Supplier 1’s Profit vs. Supplier 2’s Ingredient Branding

6. Conclusion

In this paper, we consider the cooperative advertising decisions for InBranding strategies

from a supply chain perspective. To create the brand awareness for her ingredient, the

ingredient/component supplier not only communicates with final product users directly via

advertising, but also offers the OEM a cooperative advertising program to stimulate the OEM

labeling her ingredient brand logo in the final products and promoting the fact that the

ingredients/components are a part of the final products. Under the Nerlove-Arrow model

framework, we calculate the equilibrium advertising efforts of the supplier and the OEM, as

well as the supplier’s optimal subsidy rate for her cooperative advertising program. Also, we

have extended our models for a supply chain with two complementary component suppliers.

Analytical results of the basic single-supplier single-OEM model provide an ingredient

supplier the following justifications for her implementing an ingredient branding strategy.

First, when a component supplier implements her ingredient branding strategy, she needs the

cooperation from the OEMs. If the OEMs do not promote the fact that they utilize the

components from her, her brand awareness among consumers has no effect on the sales of the

32

final products and thus the advertising on such ingredient brand will be in vain. Under such

conditions, the supplier will not invest in advertising for her ingredient brand. Second,

whether a supplier should provide the OEM a cooperative advertising program is only

dependent on the relative magnitude of the OEM’s and the supplier’s profit margins. That is

to say, once the supplier’s profit margin exceeds a threshold, she will share a part of the

OEM’s advertising cost, no matter whether she implies an ingredient branding strategy or not.

Third, when the supplier implements her ingredient branding strategy, her equilibrium

advertising efforts for her ingredient brand is independent on her subsidy rate offered to the

OEM.

The extended model for the three scenarios of two complementary component suppliers

illustrates the following managerial implications. First, whether a supplier should offer the

OEM a cooperative advertising program is not influenced by the other supplier. Second, once

a supplier provides the OEM a cooperative program, her optimal subsidy rate is influenced by

the other supplier’s decisions. If the two suppliers offer the OEM cooperative advertising

program simultaneously and independently, the subsidy rate of each supplier decreases

compared to that in the case of a single supplier, but the total subsidy rate for the OEM

increases. Third, when the two suppliers form a strategic alliance, the allied supplier system

is more likely to meet the profit margin threshold and provides the OEM a cooperative

advertising program with a higher subsidy rate, which implies that both the OEM and the

suppliers benefit from such alliances.

A few limitations and a future extension of the model should be noted. In our paper, we

have limited ourselves to the open-loop equilibrium due to the tractability concern.

Closed-loop strategies are more desirable as they are time-consistent. Future research may

use alternative models to explore the closed-loop equilibrium ingredient branding strategy.

We have focused on the advertising decisions while we do not explicitly study the channel

member’s pricing decisions. Incorporating price factor into the ingredient branding strategy is

a potential research direction. In practice, suppliers produce components for a number of

OEMs. Thus it would be a natural extension to incorporate OEM-level competition and

examine how the competition affects the ingredient branding strategy. Lastly, we study the

33

interactions between OEM and suppliers. Future research may bring the retailers into the

picture as they deal directly with end customers.

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Appendix

Proof of Proposition 1

First, we calculate the OEM’s best response advertising efforts for a given subsidy rate . The

necessary conditions of the OEM’s best response advertising efforts are given by Equations (A.1) to

(A.5), i.e.,

0M

M

H

U

(A.1)

37

1

M

M

HS

(A.2)

2

M

M

HM

(A.3)

1 1M

M M

Hr

S

(A.4)

2 2M

M M

Hr

M

(A.5)

where MH , the present value Hamiltonian for the OEM’s profit, is given by Equation (13), and 1M

and 2M represent the co-state variables (shadow prices) in the OEM’s problem corresponding to

supplier and OEM goodwill levels in Equations (3) and (4), respectively.

Equation (A.1) implies that

1 2(1 ) M S M M M MU a (A.6)

Equations (A.4) and (A.5) are, explicitly,

1 1 2( )M M M Mr (A.7)

2 2( )M M Mr b (A.8)

Differentiating Equation (A.6) with respect to time t , we get

1 2(1 ) M M MS MU (A.9)

Substituting Equations (A.7) and (A.8) into Equation (A.9), we get

1 2 2(1 ) ( )( )M S M M M M S M M MU r a (A.10)

From (A.6), we get

1 2 (1 )S M M M M MU a (A.11)

Substituting Equation (A.11) into (A.10), we obtain

2 (1 ) ( )(1 ) ( )S M M M M M M MU r U r a b

(A.12)

Differentiating Equation (A.10) with respect to time t , we get

1 2 2(1 ) ( )( )M M M MS M M SU r

(A.13)

38

Substituting Equations (A.7) and (A.8) into (A.13), we have

2

1 2 2(1 ) ( ) ) 2( )

( )

M S M M M S M M

M M M S M

U r r

r b b

(

(A.14)

Substituting Equations (A.11) and (A.12) into (A.14), we have

2

2

(1 ) 2(1 )( ) (1 )( )

( ) ( )

M M M

M M M M S M

U r U r U

r a r b b

(A.15)

Solving Equation (A.15), we get

1 2 2

( )1

r t r t S MM MM

bbU e te a

r r

(A.16)

where 1 and 2 are both constants to be determined. When either 1 or 2 is not equal to zero,

the value of advertising efforts given in Equation (A.16) will be infinity when t , which is

contrary to the constraints given by Equation (8). Thus, we know that 1 2 0 . Combining this

result with Equation (A.16), we obtain the OEM’s best response advertising efforts

2( )

1

S MM MM

bbU a

r r

(A.17)

The necessary conditions of the supplier’s equilibrium advertising efforts are:

0S

S

H

U

(A.18)

1

S

S

HS

(A.19)

2

S

S

HM

(A.20)

1 1S

S S

Hr

S

(A.21)

2 2S

S S

Hr

M

(A.22)

where SH is given by Equation (12), and 1S and 2S represent the co-state variables in the

39

supplier’s problem corresponding to the supplier and OEM’s goodwill levels, respectively. Equation

(A.18) implies that

1S S SU (A.23)

Solving Equations (A.21) and (A.22), we have

1 1 2( )S S M Sr (A.24)

2 2( )S S Sr b

(A.25)

Differentiating Equation (A.23) with respect to time t , we get

1SS SU (A.26)

Substituting (A.24) into (A.26), we get

1 2( )S S S S M SU r (A.27)

Substituting Equation (A.23) into (A.27), we have

2 2( ) ( )S SS S M S S M S SU r U or r U U (A.28)

Differentiating Equation (A.27) with respect to time t , we get

1 2( )S S SS S MU r (A.29)

Substituting Equations (A.24) and (A.25) into Equation (A.29), we get

2

1 2( ) 2( )S S S S M S S M SU r r b (A.30)

Substituting Equations (A.23) and (A.28) into (A.30), we have

22( ) ( )S S S S M SU r U r U b (A.31)

Solving Equation (A.31), we have

( ) ( )

1 2 2( )

( )

r t r t S S MS

bU e te

r

(A.32)

where 1 and 2 are constants to be determined. Similar to that of 1 and 2 , we have

1 2 0 when we take into account the advertising effort constraint, i.e., Equation (8). For a fixed

40

, the supplier’s best response advertising efforts are

2

( )( )

S S MS

bU

r

(A.33)

Appendix B: proof of the proposition 2

Let 2 2( / / ) ( ) ( ) /SS M S M M S M SM U U and

( ) / ( ) /SS S S S MS U U , where ( )MU and ( )SU are given by Proposition 1. We can

rewrite SJ in Equation (12), the present value of the supplier’s profit as a function of :

2 2

2

1 1 1 1 1 1( ) ( )

( ) 2 2S S M S M S S SS S MJ aU bX bY bM U U

r r r r r r

(B.1)

where 0 SSX S S and 0 SSY M M ; ( )MU and ( )SU are given by Proposition 1.

If we derive the first order condition of ( )SJ with regard to and set ( ) / 0SdJ d , we

get

* 2

2

S M

S M

(B.2)

The second derivative of SJ with respect to is

22

22 3

3[ ] (2 )

(1 ) 1

S S MM M MS M

J bba

r r r

(B.3)

Substituting the solution of given by Equation (B.2) into Equation (B.3), we get

*

22

22 * 3

(2 )[ ] 0

2 (1 )

S M S M S MMJ bb

ar r r

(B.4)

Since the point * (2 ) / (2 )S M S M is the unique critical point, we know that SJ

reaches its maximum at * and then decreases by . Therefore, if (2 ) / (2 ) 0S M S M ,

i.e., / 2S M , the supplier’s optimal subsidy rate is * (2 ) / (2 )S M S M . Otherwise,

the optimal subsidy rate is * 0 . Substituting

* into ( )MU and ( )SU in Proposition 1, we

get the optimal advertising efforts of the OEM and supplier as given in Proposition 2, the comparative

41

statistics can be derived straightforwardly.

Proof of Proposition 3

Substituting *

SU and *

MU into Equations (3) and (4), respectively, we have

*

0( ) ( ) ( ), (0)M M MM t U S t M t M M (C.1)

* *

0( ) ( ), (0)S S S MS t U U S t S S (C.2)

Solving (C.2), we get the general solution:

*( ) t

SSS t Xe S (C.3)

where * * *( ) /SS S S S MS U U and X is a constant to be determined. Let 0t in Equation

(C.3) and use the initial conditions in (C.2) to get *

0 SSX S S .

Bringing the general solution in (C.3) into (C.1), we get

* *( ) ( ) ( ) ( )t S M S M

M S M MM t X e U U M t

(C.4)

The general solution of Equation (C.4) is

*( ) ( ) t t

M SSM t X te Ye M (C.5)

where * * *( ) /SS M M M SSM U S , and Y is a constant to be determined. Let 0t in

Equation (C.5) and use the initial conditions in Equation (C.1) to get *

0 SSY M M .

Proofs of Propositions 4-6

The OEM’s present value Hamiltonian is given by

2

1 2 1 1 1 1 1

2 2 2 2 2 3 1 1 2 2

1( ) (1 ) ( )

2

( ) ( )

I

M M M M M M

M M M M M M M

H aU bM U U U S

U U S U S S M

(D.1)

and that of Supplier i is

2 2

1 1 1 1 1

2 2 2 2 2 3 1 1 2 2

1 1( ) ( )

2 2

( ) ( )

I

i i M i i M i M

i M i M M M M

H aU bM U U U U S

U U S U S S M

, 1,2i

42

(D.2)

where 1M , 2M and 3M ( 1i , 2i and 3i ) represent the co-state variables in the OEM’s

(Supplier i ‘s) problem corresponding to the OEM and Supplier i ‘s goodwill levels in Equations (21)

and (22), respectively.

For the announced subsidy rates ( 1 , 2 ), we get the best response advertising efforts for the OEM

using the necessary conditions as follows:

0I

M

M

H

U

(D.3)

1

1

I

M

M

HS

(D.4)

2

2

I

M

M

HS

(D.5)

3

I

M

M

HM

(D.6)

1 1

1

I

MM M

Hr

S

(D.7)

2 2

2

I

MM M

Hr

S

(D.8)

3 3

I

MM M

Hr

M

(D.9)

Equation (D.3) implies

1 2 1 1 2 2 3(1 ) M M M M M MU a (D.10)

Solving Equations (D.7) – (D.9), we get

1 1 3 1( )M M M Mr (D.11)

2 2 3 2( )M M M Mr (D.12)

3 3( )M M Mr b (D.13)

Differentiating (D.10) with respect to time t , we get

43

1 2 1 1 2 2 3(1 ) M M M M MU

(D.14)

Substituting (D.11)-(D.13) into in (D.14), we get

1 2 1 1 2 2 3 3 1 1 2 2(1 ) ( )( ) ( )M M M M M M M M M MU r b

(D.15)

After rewriting (D.10), we get

1 1 2 2 3 1 2(1 )M M M M M MU a (D.16)

Substituting (D.16) into (D.15), we obtain

1 2 1 2 3 1 1 2 2(1 ) (1 )( ) ( ) ( )M M M M M M M MU r U r a b

(D.17)

After rewriting (D.17), we have

3 1 1 2 2 1 2 1 2( ) (1 ) (1 )( ) ( )M M M M M M M MU r U r a b

(D.18)

Differentiating (D.15) with respect to time t , we get

1 2 3 31 2 1 2 1 1 2 2(1 ) ( )( ) ( )M M M M MM M MU r (D.19)

Substituting (D.11)-(D.13) into (D.19), we get

2

1 2 1 1 2 2 3

1 1 2 2 3

1 1 2 2

(1 ) ( ) ( )

2( )( )

( ) ( )

M M M M M

M M M

M M M M M

U r

r

b r b

(D.20)

Substituting (D.16) and (D.18) into (D.20), we have

2

1 2 1 2 1 2

2

1 1 2 2

(1 ) 2(1 )( ) (1 )( )

( ) ( ) ( )

M M M

M M M M M M

U r U r U

a r b r b

(D.21)

Solve (D.21) to get the OEM’s best response effort trajectory MU as

1 2 1 2

1 1 2 2

2

1 2

( , )

( )

1

r t r t

M

M M M M

U e te

b ba

r r

(D.22)

44

where 1 and 2 are both constants to be determined. Incorporating the advertising effort

constraint in (8), we have 1 2 0 . Combining this result with Equation (D.22), we get the best

response advertising efforts of the OEM as:

1 1 2 21 2 2

1 2

( )( , )

1

M M M MM

b bU a

r r

(D.23)

Similarly, we derive the best response for Supplier 1 by using the necessary conditions:

1

1

0IH

U

(D.24)

11

11

IHS

(D.25)

12

21

IHS

(D.26)

1

31

IHM

(D.27)

111 11

1

IHr

S

(D.28)

121 21

2

IHr

S

(D.29)

131 31

IHr

M

(D.30)

Equation (D.24) implies

1 11 1U (D.31)

Solving Equations (D.28)- (D.30) simultaneously, we get

11 11 31 1( ) Mr

(D.32)

21 21 31 2( ) Mr

(D.33)

31 31 1( )r b

(D.34)

45

Differentiating (D.31) with regard to time t , we get

1 11 1U (D.35)

Substituting (D.32) into (D.35), we get

1 1 11 31 1 1( ) MU r (D.36)

Substituting (D.31) into (D.36), we get

1 1 31 1 1( ) MU r U

(D.37)

Rewrite (D.37) as

31 1 1 1 1( )M r U U

(D.38)

Differentiating (D.36) with respect to time t , we get

1 1 11 31 1 1( ) MU r

(D.39)

Substituting (D.32) and (D.34) into (D.39), we get

21 1 11 1 31 1 1 1 1( ) 2( ) M MU r r b (D.40)

Substituting (D.31) and (D.38) into (D.40) gives us

21 1 1 1 1 12( ) ( ) MU r U r U b (D.41)

Solving Equation (D.41) to get 1U , we get

( ) ( ) 1 1 11 1 2 1 1 2( , )

( )

r t r t M bU e te

r

(D.42)

In a similar manner, we can get the best response effort level for Supplier 2:

( ) ( ) 1 2 22 1 2 1 2 2( , )

( )

r t r t M bU e te

r

(D.43)

where 1 , 2 , 1 and 2 are all constants to be determined. Incorporating the advertising effort

constraint, i.e., Equation (8), we have 1 2 0 and 1 2 0 . Thus the best response

advertising efforts of two suppliers are given by

46

1 1 11 1 2 2( , )

( )

M bU

r

(D.44)

1 2 22 1 2 2( , )

( )

M bU

r

(D.45)

Substituting 1 2( , )iU and 1 2( , )MU into (21) and (22), we have

1 2 1 1 2 2 0( ) ( , ) , (0)M i M MM t U S S M M M (D.46)

1 2 1 2 0( ) ( , ) ( , ) , (0) , 1,2i i i i M i i iS t U U S S S i (D.47)

Solving Equation (D.47), we get the general solution by

( ) t

i i iSSS t X e S , 1,2i (D.48)

where 1 2 1 2[ ( , ) ( , )] /iSS i i i MS U U and iX , 1,2i , are constants to be determined.

Let 0t in Equation (D.48) and use the initial conditions in (D.47) to get 0 ( 1,2)i i iSSX S S i .

Substituting (D.48) into (D.46), we get

1 1 2 2 1 1 1 1 2 2 2 2 1 2

1 1 2 2 1 2

1( ) [ ( , ) ( , )]

1( ) ( , )

t

M M M M

M M M M

M X X te U U

U M

(D.49)

The general solution of Equation (D.49) is

1 1 2 2 1( ) ( ) t t

M M SSM t X X te Y e M (D.50)

where 2

1 1 2 2 1 2( ) ( , ) /SS M M M MM U and 1Y is a constant to be determined. Let

0t in Equation (D.50) and use the initial conditions in Equation (D.46) to get 1 0 SSY M M .

We rewrite the present value of Supplier i ‘s profit as a function of 1 and 2 in Scenario I as

1 2 1 1 2 2 12

2 2

1 2 1 2

1 1 1 1( , ) ( )

( )

1 1( ( , )) ( ( , ))

2 2

i i M i M M i i SS

i i M

J aU b X X bY bMr r r r

U Ur r

, 1,2i

(D.51)

Derive the first order condition of iJ with regard to i and set / 0i idJ d , 1,2i , to get

47

1 1 1 2(2 ) (2 )(1 )M M (D.52)

1 1 1 2(2 ) (2 )(1 )M M (D.53)

Solving Equations (D.52) and (D.53) simultaneously, we find that if / 2i M , then the

unique solution of i is

*

1 2

2

2 2

I i Mi

(D.54)

and if only Supplier i ‘s marginal profit / 2i M , the unique solution of i is

*

1

2

2

I i Mi

M

(D.55)

The second derivative of iJ with respect to i is

221 1 2 2

22 3

1 2 1 2

1 2 ( )(2 )[ ]

(1 ) 1

S iM M M Mi M M

i

J b ba

r r r

(D.56)

Substituting the solutions of i given by Equations (D.54) and (D.55) into Equation (D.56), we

get respectively

*

221 2 1 1 2 2

22 * * 3

1 2

( ) ( )[ ] 0

(1 )Ii i

i M M M M

I I

i

J b ba

r r r

,

*

1 2

2

2 2

I i Mi

(D.57)

and

*

221 1 2 2

22 * * 3

1 2

(2 ) ( )[ ] 0

2 (1 )Ii i

i M i M M M M

I I

i

J b ba

r r r

,

* 2

2

I i Mi

i M

(D.58)

Similar to the proof of Proposition 3, iJ reaches its maximum at its unique critical point,

1 2(2 ) / (2 2 )i M when / 2i M and (3 ) / 2i M ; at the point

(2 ) / (2 )i M i M when / 2i M and (3 ) / 2i M , and then both decreases by i .

48

Therefore, if / 2i M and (3 ) / 2i M , Supplier i ‘s equilibrium subsidy rate is

*

1 2(2 ) / (2 2 )I

i i M and if / 2i M and (3 ) / 2i M ,

* (2 ) / (2 )I

i i M i M . Otherwise, its optimal subsidy rate is * 0I

i .

Substituting *I

i , 1,2i , into Equations (D.23), (D.42) and (D.43), we get the equilibrium

advertising efforts of the OEM and two suppliers, *I

MU and *I

iU , 1,2i , in Proposition 5.

Substituting *I

i , 1,2i , into Equations (D.48) and (D.50), we get the equilibrium accumulated

goodwill levels of the OEM and Supplier i , *I

SSM and *I

iSSS , 1,2i , as provided in Proposition 6.

Proofs of Propositions 7-9

The proof of Proposition 7 is similar to that of Proposition 4. The proof of Proposition 8 is similar to

that of Proposition 5. The proof of Proposition 9 is similar to that of Proposition 6. Thus we omit the

detailed proofs.