Design of Extended Warranties in Supply Chains

Kunpeng LiUniversity of Illinois at Urbana−Champaign, College of Business

Dilip Chhajed Suman MallikUniversity of Illinois at Urbana−Champaign, College

of BusinessUniversity of Illinois at Urbana−Champaign, College

of Business

Abstract

Consider a supply chain involving an independent retailer and an independent manufacturer.The manufacturer produces a single product and sells it exclusively through the retailer.Using this supply chain framework, we develop a game theoretic model to study twocommonly observed practices of selling extended warranties: the manufacturer offers theextended warranty directly to the end consumer, and the retailer selling the product offersextended warranty. We show that, of the two decentralized systems, when the retailer offersan extended warranty, it is for a longer duration and generates more system profit. Wecompare and contrast the two decentralized models with a centralized system where a singleparty manufactures the product, sells to the consumer and offers the extended warranty. Weidentify the different causes of inefficiencies in each of the two decentralized models andpropose coordination mechanisms that eliminate the inefficiencies. We also provide contractsto achieve both coordination and a Pareto improvement over a wholesale price contract.

Published: September 2005URL: http://www.business.uiuc.edu/Working_Papers/papers/05−0128.pdf

Design of Extended Warranties in Supply Chains

Kunpeng Li Dilip Chhajed Suman Mallik kli@uiuc.edu chhajed@uiuc.edu mallik@uiuc.edu

Department of Business Administration University of Illinois at Urbana-Champaign

350 Wohlers Hall 1206 South Sixth Street Champaign, IL 61820

September 2005

i

Design of Extended Warranties in Supply Chains

Abstract Consider a supply chain involving an independent retailer and an independent

manufacturer. The manufacturer produces a single product and sells it exclusively

through the retailer. Using this supply chain framework, we develop a game theoretic

model to study two commonly observed practices of selling extended warranties: the

manufacturer offers the extended warranty directly to the end consumer, and the retailer

selling the product offers extended warranty. We show that, of the two decentralized

systems, when the retailer offers an extended warranty, it is for a longer duration and

generates more system profit. We compare and contrast the two decentralized models

with a centralized system where a single party manufactures the product, sells to the

consumer and offers the extended warranty. We identify the different causes of

inefficiencies in each of the two decentralized models and propose coordination

mechanisms that eliminate the inefficiencies. We also provide contracts to achieve

both coordination and a Pareto improvement over a wholesale price contract.

Key words: supply chain management, extended warranty, game theory

ii

1. INTRODUCTION Selling extended warranties on products is a rapidly expanding business. During the 1980s,

extended warranties were offered only on large, expensive items. Now, extended warranties are

offered on almost all consumer electronics and domestic appliances ranging from laptop

computers to simple sewing machines. An extended warranty is actually a service plan under

which the provider promises to repair, replace, or maintain the product for free or at a lower

price over a certain period of time after the manufacturer’s original warranty expires. The

extended warranty may also offer additional benefits (such as return and/or exchange privileges)

that are not provided by the manufacturer’s original warranty. Extended warranties are sold

separately from the products and usually cost extra money. Generally, an extended warranty is

offered by a manufacturer, a retailer, or by a third party (Publication 153, Best Business Bureau).

The terms of a typical extended warranty specify the price and the time length during which the

product is covered. The provider of the extended warranty incurs costs related to the warranty,

for example, the actual repair costs, costs associated with the administration of the claim,

extended warranty division setup and maintenance costs.

The primary focus of our current research is to analyze the design of extended warranties in a

supply chain context. We study a simple supply chain involving a single manufacturer and a

single retailer. The manufacturer produces a single product and sells exclusively through the

retailer. The extended warranty could, however, be offered either by the manufacturer or by the

retailer. The party offering the extended warranty decides the terms of the policy in its best

interest and incurs all costs associated with administering the policy. Thus, the

repair/administration costs directly influence the provider’s extended warranty decisions. Under

such a setting, we use game theoretic models to answer the following questions. Which scenario

leads to a higher total supply chain profit, a retailer offering the extended warranty or the

manufacturer? How do the optimum price and warranty length vary under the two scenarios?

How is the total supply chain profit distributed between the parties in the two scenarios? We also

compare the two scenarios with a centralized system in which a single firm manufactures the

product, sells directly to the end consumer, and offers the extended warranty.

The two scenarios, a retailer or a manufacturer offering an extended warranty are quite

common in practice. Many manufacturers offer extended warranties directly to the end

consumers. GE Appliances, a leading manufacturer of major appliances, parts, and accessories,

1

with products ranging from consumer electronics (VCRs, camcorders, CD players, etc.) to

appliances (ovens, refrigerators, washers, etc.) offers GE extended warranties on almost all the

products it produces. Firms like Ford, GM, JVC, and Apple have devoted whole divisions solely

to managing and to serving extended warranty contracts (Padmanabhan 1995). Manufacturer

offering extended warranties are also common for office machines such as copiers, fax machines,

and printers. Retail stores such as Best Buy, Circuit City, and Home Depot offer and promote

their own extended warranties on most of the items they carry. Normally, the extended warranty

policy that is offered by the retailer is called “service plan”. Often, these plans extend the

manufacturer’s original warranty to a longer period and may offer additional benefits not

provided by the manufacturer’s original warranty. For example, Best Buy offers a three-year

extended performance plan for notebook computers in the price range $1500 - $1999.99 (Carry-

In) for $299.99. An example of a centralized system selling a product as well as managing the

extended warranty is Dell Computer. A customer can choose from a menu of extended

warranties while customizing a computer at Dell’s website.

In this paper, we use game theoretic models to analyze the two scenarios: a retailer offering

the extended warranty (Model R) and a manufacturer offering the extended warranty (Model M).

We model the extended warranty as a free repair service over the length of the contract. The two

models are compared with respect to the total channel profit and the terms of the extended

warranty. Model R gives rise to higher total channel profit than Model M and offers a longer

extended warranty. We discuss how the total channel profit is distributed between the

manufacturer and the retailer in the two models. In addition, we compare and contrast the two

models with a centralized system offering extended warranty. These results further explain the

difference between Models R and M. We benchmark the performance of the decentralized

models with those of a centralized system and show that the centralized system will offer the

longest extended warranty while a system in which the manufacturer offers the extended

warranty will have the shortest extended warranty. We develop two parameters that influence

the profitability of the extended warranty business. We also identify the source of inefficiency in

Models R and M and discuss ways to coordinate our decentralized supply chains with extended

warranties. A revenue sharing contract with side payments not only achieves coordination but

also offers a Pareto improvement over the wholesale price contract for the channel described in

2

Model R. A profit-sharing-and-quantity-discount contract is shown to achieve coordination as

well as Pareto improvement for the channel described in Models M and R.

Our paper provides insights about the influence of extended warranties on supply chain

decisions and performance. It also helps to understand some of the unique features of the

extended warranty, its price and duration, as well as its demand dependency on the primary

product demand. Note that extended warranties sold by third parties, though common in

practice, is not a focus of our current work. The contract between a third party warranty provider

and a customer can easily be modeled as an insurance policy. The design, pricing, and analyses

of insurance policies have been well studied in the economics and insurance and risk

management literature (e.g., Lutz and Padmanabhan 1998, Manove 1983, Schlesinger 1983, and

Taylor 1995). Unlike insurance literature, the extended warranty provider in our paper not only

sells and administers the policy, but also influences the retail price of the product directly (in

Model R) or indirectly (through the wholesale price in Model M). Our model thus allows us to

study the interactions of the product and extended warranty decisions simultaneously in a supply

chain.

The remainder of the paper is organized as follows. The next section reviews the related

literature. We present our models and discuss the solution procedures in Section 3. In Section 4,

we analyze the results and develop insights. Channel coordination is discussed in section 5, while

Section 6 summarizes and concludes the paper.

2. LITERATURE REVIEW The research on design and analyses of extended warranty policies is limited. However,

theories of product failure and warranties have received extensive attention in both economics

and operations research literature.

2.1 Economic literature

Three distinct theories have been proposed in the economics literature to explain the

existence of product warranties. The insurance theory (first addressed by Heal, 1977) assumes

that consumers are more risk-averse than sellers, so that warranties are provided to consumers as

a form of insurance against product failure. That is, economic literature typically treats

3

warranties as compensation paid to consumers in case of product failure. The concept may be the

result of consumer heterogeneity, as mentioned by Hollis (1999) and as observed by Emons

(1989) and Padmanabhan (1995). The signaling theory states that a product warranty is a signal

of product quality. A longer and more comprehensive warranty usually indicates better product

quality. Spence (1977) was the first to show the signaling relationship between the product and

its warranty. Finally, the incentive theory is the indication of double moral hazards, i.e.,

producer’s and consumer’s moral hazards, in which product reliability could be affected by

actions not observable by the other party. The terms of the warranty contract describe liabilities

for both parties. Therefore, the manufacture has incentives to produce and maintain a certain

level of product quality, while the consumers have incentives for proper use and care of the

product. Examples of this research are Cooper and Ross (1985), Emons (1988), Mann and

Wissink (1988), and Dybvig and Lutz (1993).

The sparse literature on extended warranties is dominated by the insurance theory. The

extended warranty is usually modeled as a cash payment, and the self-selection method is

adopted to deal with the consumer heterogeneity. Padmanabhan (1995) addresses consumer

moral hazard and heterogeneity in product usage in the optimal choices of product price and

warranty payment. Lutz and Padmanabhan (1998) study the influence of extended warranties on

manufacturer’s warranty policy under producer moral hazard. In their model, warranties are

modeled as cash payments, and two products with different qualities are offered to

heterogeneous consumers in the market. Padmanabhan and Rao (1993) empirically demonstrate

that the risk aversion influence in choosing an extended warranty can be reduced by increasing

the length of the base warranty. Again, the warranties are considered as cash payments in the

model. Hollis (1999) also uses the standard self-selection method trying to distinguish the heavy-

user and the light-user in the market. However, his work differs from the previous researches in

the economic literature by modeling warranty as duration, instead of cash payment. He argues

that although extended warranties are a form of insurance, warranty contracts usually vary by

duration rather than by amount of payment.

2.2 Operations Research literature

The operations research literature on warranties focuses on mathematical models with

considerable scope and details in the description of warranty costs. The product quality is usually

4

modeled in conjunction with the product failure time. Product life-time distribution and failure

rate function are very important components of the formulation of the warranty cost model.

Sahin and Polatogu (1998) provide an excellent review of various warranty policies and product

failure models. Wamer (1987) analyzes the trade-off between warranty and quality, and

illustrates the sensitivity of warranty costs to environmental variables. Anderson (1977) develops

an optimization decision model for the optimal choices of the warranty period and the product

price. Balcer and Sahin (1986) derive total product replacement cost under both “pro rata” and

“free replacement” warranty policies by assuming that the successive failure times form a

renewal process. Opp et al. (2003) consider a cost minimization problem of outsourcing warranty

services to repair vendors under static allocation. They develop an efficient optimal algorithm

and demonstrate that the optimal algorithm can handle industry-size problems and also performs

much better than common static allocation heuristics.

Another unique work related to warranties by Cohen and Whang (1997) develop a product

life-cycle model in which warranty cost is incorporated in the profit function of a firm seeking to

maximize total lifecycle profit from a product. They assume that the warranty will run for a fixed

interval and that the warranty cost is linear with the manufacturer’s quality of after-sales service.

However, the design or the management of the warranty is not the focus of their work.

As the reader will see in the next Section, our model is an implementation of the signaling

theory, which is reflected in the choice of our extended warranty demand function. The

following features distinguish our model from the literature cited.

• Consistent with our observation in practice, we model the extended warranty as duration of

time, instead of cash payment to the customer when the product fails.

• We study the design of extended warranties in a supply chain setting, which allows us to

study product pricing decisions and extended warranty decisions within a unified context.

Our model, thus, allows a party offering an extended warranty to incur loses from the product

sale if the profit from the warranty sale can compensate the loss. To the best of our

knowledge, the design of the extended warranties in a supply chain context has not been

studied before.

3. THE MODELS

Consider a supply chain consisting of a single manufacturer and a single retailer. The

manufacturer produces a single product (e.g., TV, microwave, computer, or oven) and sells it

5

exclusively through the retailer at a unit wholesale price x. The retailer, in turn, sells the product

to the end consumer at a unit retail price p. The manufacturer offers the original product

warranty, whose length, without loss of generality, is normalized to zero. This assumption

enables us to focus exclusively on the analysis of the extended warranty. In line with our

observations in practice, two separate models for extended warranties are considered. In Model

M, the manufacturer offers the extended warranty, while in model R, the retailer offers the

extended warranty. In either model, the manufacturer sets the wholesale price while the retailer

sets the retail price. The specification of the extended warranty has two components, which are

the decision variables for the party offering it: the length of the warranty in units of time, denoted

by ; and the price, denoted by . During the lifetime of the extended warranty, the provider

commits to offering free repair service for a failed product and incurs the associated repair and

administration costs. We will compare and contrast the two decentralized models (R and M)

with a centralized system, Model C, in which the manufacturer makes the product, sells directly

to the end consumer and offers the extended warranty. Figure 1 schematically describes these

three models along with the relevant decision variables.

ew ep

Manufacturer Manufacturer Manufacturer

(x) (x)

Retailer

(p)

Customers

Retailer

(p)

Customers

(p)

Customers

(we, pe) (we, pe) (we, pe)

(a) Model C (b) Model R (c) Model M

Figure 1: Models When Supply Chain Members Offer Extended Warranty

Demand functions

Two demand functions are involved in our model, the product demand and the extended

warranty demand. First, consider the product demand. We assume that the demand is linearly

decreasing in price and is given by:

6

bpq −= 1 , (1)

where, is the retail price of the product, q is its demand, and b∈ (0, 1] is the price sensitivity

of the consumers.

p

When an extended warranty is offered, its demand depends on three factors: product

demand , extended warranty price , and length of the extended warranty . If no extended

warranty policy is offered, , by definition, equals to zero. We use the following demand

function to model the demand for the extended warranty.

eq

q ep ew

eq

eee ewpbbp +−− )1( , if > 0 (2) ew=eq

where, > and

warranty length sen

candidates for buyi

and (2), and we g

warranty from (2)

made because the

changes, than that o

generally lower tha

same amount of c

eb b

influenced more tha

The costs

We assume tha

that the wholesale

models. Two comp

repair and adminis

warranty. Here

, that depen

rc

2/2ecw

0 , if = 0, ew

∈ (0, 1] is the extended warranty price sensitivity, is the extended

sitivity. Note that only the consumers who bought the product are potential

ng the extended warranty, i.e. ≤ . Simplifying this inequality using (1)

et . Thus, the maximum allowable demand for an extended

is simply , the demand for the product. The assumption > is

extended warranty demand is more sensitive to extended warranty price

f the retail demand for the retail price change. The extended warranty price is

n the product retail price. Thus, if the extended warranty price has the

hange as the retail price

eb 0>e

eq q

eee ewpb ≥

)1( bp− eb b

ep

p , the extended warranty demand will be

n the retail demand .

eq

q

t the production cost for the product is constant and is normalized to zero, so

price is also the manufacturer’s product profit margin in the decentralized

onents of the extended warranty cost are considered. First, the average unit

tration cost, , which is linearly proportional to the length of extended

is the average repair cost per unit of time. Second, a quadratic cost,

ds on the length of the warranty but is independent of the demand for the

er wc

0>

7

extended warranty. This component captures other exogenously determined cost components

involved in managing the extended warranty process, such as the cost of setup and maintenance

of a repair division. Here is the cost parameter. A similar quadratic cost assumption can be

found in Balasubramanian and Bhardwaj (2004). Note that, while it is intuitively appealing to

interpret as the length of the extended warranty, our modeling framework is general enough

to allow other interpretations of . For example, might also be interpreted as the quality or

the amount of coverage included in the warranty (e.g. drive train vs. bumper-to-bumper coverage

in an auto). Clearly, the higher the included coverage in a warranty, the more extensive is the

required support facility. The quadratic cost component is also consistent with this interpretation.

The quadratic cost of quality or service level is well documented ( e. g., Moorthy, 1988 and Iyer,

1998).

0>c

ew

ew ew

We assume that there is no information asymmetry in the channel and that the manufacturer

acts as a Stackelberg leader in the game. The manufacturer, therefore, can look ahead and

anticipate the retailer’s pricing and extended warranty decisions.

We will use subscripts and superscripts to facilitate the comparison of the optimal values of

the decision variables across the models. The superscripts C, R, or M will denote Models C, R,

or M, respectively; while the subscripts r , , , m .prod .warr , or .sys will denote the retailer,

manufacturer, product, extended warranty, and system, respectively. We will also use superscript

“*” to denote optimal quantities. Thus, is the optimal total system profit in Model R, while

is the profit from the extended warranty in model C.

∗Rsys.π

Cwarr .π

We next define two parameters to simplify the exposition of our paper. Define

cbcbe

e

re2)( −

=α as the extended warranty desirability index. It is easy to check that 0<∂∂

cα ,

0<∂∂

rcα , 0<

∂∂

ebα , and 0>

∂∂

eα . All else being equal, a party offering an extended warranty will

make a lower profit from the extended warranty for a lower value of α than for a higher one.

Therefore, for a supply chain member, a small value of α represents relatively little desire to

offer an extended warranty; while a large value of α represents a relatively greater desire to

offer one.

8

Next, definebbe=β as the relative price sensitivity index. Based on our assumption , bbe ≥

1≥β . A higher value of β indicates that customers are much more sensitive to extended

warranty price than to product retail price, and vise versa. We will use the parameters α and β

extensively throughout the remainder of the paper as well as in the Appendix.

3.1 Model C: The centralized system

A centralized system maximizes the total supply chain profit by simultaneously considering

the product and extended warranty demands.

..2

])1[()()1(2

,,

ts

cwewpbbpwcpbppMax e

eeeereC

wpp ee

−+−−⋅−+−=Π (3)

eee pbew ≤ (4)

0≥p (5)

0>ew (6)

The objective function in (3) incorporates the two types of costs of extended warranties, while

constraint (4) ensures that the demand for an extended warranty cannot exceed the demand for

the product. The non-negativity condition on price in constraint (5) ensures the product demand

will not exceed 1. Constraint (6) guarantees that the model will indeed offer an extended

warranty. If , we must have ,0=ew 0=eq indicating that we no longer have the extended

warranty in our model.

Model C can be solved easily by forming the Lagrangian and using standard optimization

techniques. Table 1 presents the optimal solution to Model C, along with the optimal values of

all decision variables.

3.2 Model R: Retailer offers extended warranty

In Model R, besides the retail price, the retailer also decides the extended warranty policy by

specifying and . In the first stage of the game, the manufacturer chooses the wholesale

price that maximizes her profit. The optimization problem of the manufacturer is given by:

ep ew

x

9

)1( bpxMax Rmx

−=Π . (7)

In the second stage, taking the wholesale price as given, the retailer maximizes his own profit.

The retailer’s problem is as follows.

..2

])1[()()1)((2

,,

ts

cwewpbbpwcpbpxpMax e

eeeereRrwpp ee

−+−−⋅−+−−=Π (8)

eee pbew ≤ (9)

0≥p (10)

0>ew . (11)

The interpretations of the constraints (9) - (11) are similar to those of (4) - (6), respectively.

There is no equilibrium solution to the problem (7) - (11) for 0=p . To see this, note, if we let

, then the product demand becomes 1. Consequently, the manufacturer’s optimization

problem becomes . Obviously, the wholesale price x can go to infinity,

which will bring the retailer’s profit down to negative infinity. So the retailer will never set the

retail price to zero, and the constraint on

0=p

xbpxMax Rmx

=−=Π )1(

p becomes strictly positive. However, this does not

preclude the possibility that the retailer will set the retail price below the wholesale price.

We solve the model starting with the retailer’s problem and working backwards. The

Lagrangian for the retailer’s problem is as follows:

L )(2

])1[()()1)((2

eeee

eeeere ewpbcw

ewpbbpwcpbpxp −+−+−−⋅−+−−= λ . (12)

We can find the retailer’s best response function using standard optimization procedures. Two

cases are possible: 0=λ and 0>λ . Consider the case of 0=λ first. Solving for the retailer’s

best responses as a function of the wholesale price , we get, x

]/1)2(2[/1)1)(2()(βαβα

−−⋅−+−

=b

bxxp , ]/1)2(2[

)()1()(

βα −−⋅−⋅−

=cb

cbebxxw

e

ree ,

]/1)2(2[])[()1(

)(βα −−⋅+−⋅−

=cb

cccbebxxp

e

rree .

(13)

Substituting βα

α/1)2(2

)1)(2()(1)(−−−−

=−=bxxbxq into the manufacturer’s problem, we get,

βαα

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

−−−−

⋅=−=ΠbxxbpxMax R

mx . (14)

10

The optimization problem in (14) can be solved to yield b

x R

21

=∗ . Optimal values of all other

decision variables can be obtained by substituting the value of in (13). The case ∗Rx 0>λ can

be solved similarly. Table 2 describes the optimal solutions for Model R.

3.3 Model M: Manufacturer offers extended warranty

Being the Stackelberg leader in the game, the manufacturer anticipates the retailer’s pricing

decision and maximizes her profit accordingly. In the first stage of the game, the manufacturer

chooses the wholesale price x , and the extended warranty policy ( , ). In the second stage,

the retailer takes the manufacturer’s decisions as given and decides the retail price p . The

manufacturer’s optimization problem is as follows.

ep ew

..2

])1[()()1(2

,,

ts

cwewpbbpwcpbpxMax e

eeeereMmwpx ee

−+−−⋅−+−=Π (15)

eee pbew ≤ (16)

0>ew (17)

In the second stage, given the manufacturer’s decision, the retailer solves for by solving p

)1)(max(arg bpxpp −−∈ subject to . Again, we can show that is not an

equilibrium solution.

0≥p 0=p

The solution procedure for Model M is similar to that of Model R. We start with the

retailer’s problem and work backwards. Table 3 describes the optimal solution.

4. RESULTS AND ANALYSIS

Our analysis is based on the results summarized in Tables 1-3. All the extended warranty

models are valid only if the constraint is satisfied, which ensures that extended warranty

is, indeed, offered. As we have shown in the footnotes in Tables 1-3, the necessary condition for

satisfying the constraint is . We state this result in the following proposition.

0>ew

0>ew recbe >

Proposition 1: It is never optimal for any party to offer an extended warranty to the market

unless the extended warranty sensitivity e is at least recb .

11

Proposition 1 implies that higher warranty sensitivity e is required for offering the extended

warranty when the unit repair cost increases. Because of the corresponding increasing repair

costs and the shrinking profit margins on extended warranty selling, the provider is less willing

to offer extended warranties unless the customers have a strong desire for it. Similarly, higher

warranty sensitivity e is required for offering an extended warranty when the extended warranty

price sensitivity increases. A higher value of implies a more dramatic change in extended

warranty demand when the extended warranty price changes. Consequently, a higher extended

warranty sensitivity, e , is required to compensate for this effect.

rc

eb eb

Proposition 2: The following statements hold for each of the three models (Models C, R and M).

(a) The system profit, the extended warranty price, the extended warranty length, the product

demand, and the extended warranty demand are non-decreasing in α ; and are non-increasing

in β ; while the product retail price has a reversed relationship in α and β .

(b) For any party offering the extended warranty, as α increases, the profit from selling the

product decreases while the profit from selling the extended warranty increases. Furthermore,

the rate of increase in extended warranty profit is higher than the rate of decrease in product

profit, implying the provider’s total profit increases with α.

The proof of Proposition 2(a) follows from observing the signs of the corresponding first

derivatives with respect to α . For example, in Model C, when βα <≤1 , 2)2( αββ

α −−

=∂∂ ∗

bp ,

2)2(2 αββ

απ

−=

∂∂ ∗

bs , 2)2( αβ

αβ −

=∂∂ ∗

bp and 2)2(2 αβ

αβπ

−−

=∂∂ ∗

bs . The proof of proposition

2(b), as well as the proofs for all other results, is included in the Appendix.

According to Proposition 2(a), when the value of α becomes smaller, the length of the

extended warranty should be shorter. A smaller value of α indicates a less favorable condition

for offering an extended warranty, which, for example, may be due to higher repair costs. A

recent article in the Wall Street Journal reports that DaimlerChrysler is shortening the extended

warranty on its vehicles beginning with the 2006 models because of increased repair costs

12

resulting from higher labor costs and more complicated technology (Saranow 2005). The

findings of our model directly support this action.

As α increases, the market becomes more favorable for a firm to offer extended warranties.

A more favorable market means that the provider will realize higher profits from the extended

warranty market. As α increases, the product retail price decreases (Proposition 2a). This results

in a higher product demand but might reduce the profit from the product market. However, the

extended warranty provider benefits from a lower retail price as a higher product demand implies

a higher potential demand for the extended warranty. This, together with the strategic choice of

extended warranty price and length, enables the extended warranty market to generate more

profit, which compensates for lower profit from product sales and, hence, the system profit

increases with α. As α continues to increase, the profits from selling extended warranties will

surpass the profits from selling the product. To further improve total profit, the provider needs to

lower the product retail price further. In Model R, the retailer may set the retail price so low that

he will sustain a loss on the product in order to gain more from the extended warranty. As α

increases, the manufacturer in Model M reduces the wholesale price so that she can benefit from

a lower retail price set by the retailer.

When the relative price sensitivity index β increases, customers become more sensitive to

extended warranty price change. Small changes in the extended warranty price could result in

relatively large fluctuations in the extended warranty market. Thus, the market becomes less

stable for the extended warranty business. Similarly, a decreasing β implies a more stable

demand that allows easier demand manipulation in the product market through the retail price.

This provides an intuitive explanation for the sensitivity analysis results with respect to β in

Proposition 2(a).

Comparing the retail price and the wholesale price in Model R, we note that when ∗Rp ∗Rx

)2

22,1( +∈β and βα (∈ , )

212β

− , . In this scenario, the retailer chooses to

lose money on the product in order to gain higher profit from extended warranties. Note that a

small value of

∗∗ << RR xp0

β indicates that the product demand is more responsive to retail price changes,

and a higher α represents a very favorable extended warranty market. When β is low, the

retailer is easily able to increase product demand by setting a low retail price. This may reduce

13

the profit coming from product sales. However, a higher demand for the product also contributes

to a higher demand for the extended warranties. When α is high, a higher profit could be

derived from the extended warranty market. The range of values for α and β provides a sufficient

condition for extended warranty profits to dominate the product profit. Note that 1≥> βα in

this range.

However, a similar scenario of the manufacturer incurring a loss on the product (i.e.

) does not arise in Model M. The condition for in Model M are 0<∗Mx 0<∗Mx ∈β (0.146,

0.853) and βα 2(∈ , )412β

− . However, the assumptions in our model dictate that 1≥β . This

implies that the manufacturer in Model M is less effective in influencing product demand and

can only do so through the choice of the wholesale price.

4.1 Profit analysis

In this sub-section, we compare the total channel profit for the three models, as well as look

at the division of total profits between the manufacturer and the retailer within the models.

Theorem 1 states our first result.

Theorem 1:

(a) When β

α2120 −≤≤ , all three models are feasible and . ∗∗∗ >> M

sysRsys

Csys ... πππ

(b) When β

αβ 4

12212 −≤<− , only Model M has feasible solutions.

(c) No feasible solution exists for any model whenβ

α412 −> .

Theorem 1 (a) shows that, as expected, for a low value of the desirability index α , a centralized

system offers the highest system profit. It is interesting to note that Model R always offers a

higher system profit than Model M. Why? In Model R, the retailer decides , and , while

the manufacturer decides the wholesale price x. Thus, the retailer can manipulate all three

variables under his control simultaneously to maximize profits. These three variables allow him

to influence the product as well as the extended warranty demand directly. On the other hand, in

Model M the manufacturer can directly choose and , and can only indirectly influence

p ep ew

ep ew p

14

through the choice of the wholesale price x . Thus the manufacturer in Model M can neither

influence the product demand directly, nor manipulate the variables , and

simultaneously, resulting in a lower system profit.

p ep ew

Theorem 1(b) and 1(c) are derived from the second-order conditions of the respective

optimization problems. For intermediate value of α , only Model M is feasible, which might

explain why it is more common for the manufacturer to offer extended warranties than the

retailer in certain industries, such as the automobile industry1. Generally, the value of α is

relatively high in the auto industry due to consumers’ strong desire (high value of e ) for the

extended warranty. Compared with other products, extended warranties for autos are also more

expensive. Therefore, consumers are less sensitive to unit price changes for the extended

warranty (small value of ), which contributes to higher value of eb α .

We next look at the division of profit between the manufacturer and the retailer within a

model, as well as between the two models. Figure 2 plots the profits as a function of the

desirability index α. The solid lines in Figure 2 represent Model R, while the dashed lines

represent Model M.

Profit

∗Rmπ

∗Rrπ

∗Mmπ

∗Mrπ

α 2β2-1/(2β) 2-1/(4β)

Figure 2: Profits of the retailer and the manufacturer in Model R and Model M

1 Admittedly, third party extended warranties are also common in the automobile industry. The website www.carbuyingtips.com/warranty.htm discusses various reasons for this. However, extended warranties provided by third parties are beyond the scope of our current work.

15

The following proposition is derived from the profit comparison of the retailer and the

manufacturer within and between Model R and Model M.

Proposition 3

(a) Comparison between Model R and Model M in their common feasible region:

Model R has both higher retailer profits and higher manufacturer profits than Model M, i.e.,

and . ∗∗ > Mr

Rr ππ ∗∗ > M

mRm ππ

(b) Comparison within Model R and Model M:

(i) In Model R, the optimal profit of the manufacturer is twice as much as that of the retailer, i.e.,

. ∗∗ = Rr

Rm ππ 2

(ii) In Model M, the optimal profit of the manufacturer is greater than that of the retailer, but is

less than twice the optimal the profit of the retailer, i.e., . Moreover, the

difference between the two profits ( ) is decreasing with

∗∗∗ >> Mr

Mm

Mr πππ2

∗∗ − Mr

Mm ππ α .

In Theorem 1 we have shown that Model R generates more system profit compared with Model

M. Proposition 3(a) further states that both the retailer and the manufacturer are better off with

Model R than Model M. Model R, therefore, makes a Pareto improvement over Model M. This

insight can serve as a useful qualitative guideline for the practitioners responsible for setting up

or running extended warranty businesses.

Proposition 3(b)(i) states that the manufacturer gets twice the profit of the retailer in Model

R. Interestingly, the same relationship holds for a decentralized supply chain involving a single

manufacturer and a single retailer that offers no extended warranty. As described in the

discussion following Theorem 1, the retailer has full control over the extended warranty market

in Model R. The retailer’s only interaction with the manufacturer is through the wholesale price,

which is virtually the same as that in the model without an extended warranty.

However, the manufacturer in Model M does not have full control over extended warranty

demand, giving rise to the relationship described in Proposition 3(b)(ii). What’s more, in Model

M, 0)(<

∂−∂ ∗∗

αππ M

rMm , i.e., the profit difference between the manufacturer and the retailer is

decreasing as α increases. As α increases, offering the extended warranty becomes more

attractive. The manufacturer takes this opportunity to reduce the wholesale price in hopes that

16

the retailer will reduce the retail price, creating a higher demand for the product. The retailer

obliges the manufacturer. However, the retailer uses this as an opportunity to improve his own

profit as well, thus resulting in a greater profit increase than that of the manufacturer. As a

result, decreases with ∗∗ − Mr

Mm ππ α .

4.2 Extended warranty design

Tables 1-3 describe the optimal values for the design variables of extended warranties for

the three models discussed. In this sub-section, we use these Tables to develop insights about the

design of the extended warranty as well as product pricing. The following proposition describes

our result.

Proposition 4: Among the three models discussed, Model C offers the longest extended

warranty, while Model M offers the shortest, i.e., > > . A similar relationship

holds for extended warranty price, demand for the extended warranty, and profit from the

extended warranty, i.e.: > > , > > , and > > .

∗Cew ∗R

ew ∗Mew

∗Cep ∗R

ep ∗Mep ∗C

eq ∗Req ∗M

eq ∗Cwarr .π ∗R

warr .π ∗Mwarr .π

Proposition 4 shows that the length of the extended warranty is different for each of the three

models, even though the unit extended warranty price is same for all the three models. Model C,

benefiting from a centralized structure, is able to offer the longest warranty and captures the most

extended warranty demand and profit. Compared with Model M, Model R offers a longer

extended warranty since the retailer can influence the extended warranty demand by directly

choosing the retail price. Extended warranty demand and profit show a similar relationship. The

intuitive explanations are similar to that of the warranty length. The prices, on the other hand,

have a reverse relationship in the three models as described by Proposition 4.

We next compare the optimal values of the wholesale price , the retail price , and

the product demand . We summarize the results in the following proposition:

∗x ∗p∗q

Proposition 5:

(a) The wholesale price in Model M is lower than that in Model R, i.e., > . ∗Rx ∗Mx

17

(b) The optimal values of product price and product demand in the three models have the

following relationship: , and ∗Cp < ∗Rp < ∗Mp ∗Cq > ∗Rq > ∗Mq .

The decisions facing the manufacturer in Model R are identical to those in a single-

manufacturer-single-retailer supply chain with no extended warranty. Not surprisingly, we find

, which is also the optimal wholesale price in the no-extended-warranty model. In

Model M, in order to create higher extended warranty demand from a lower retail price, the

manufacturer has to set the wholesale price lower than that in Model R, i.e., . This

explains Proposition 5(a).

)2/(1 bx R =∗

)2/(1 bx M <∗

Compared with Model M, the retail price in Model R is lower despite the higher wholesale

price. The retailer in Model R has more decision variables under his control. As a result, he can

jointly decide on all of the variables, resulting in a higher product demand and lower product

price. Higher product demand, in turn, also gives rise to a higher demand for the extended

warranty.

5. CHANNEL COORDINATION AND IMPROVEMENT

Our discussion so far has focused on a wholesale price contract between the manufacturer

and the retailer in models R and M. In this section, we explore alternative contracts to

coordinate the decentralized channels in these two models. We say that channel coordination is

achieved when the system profit from a decentralized channel equals that of the centralized

channel. In addition to achieving coordination, we also explore the issue of making a Pareto

improvement over the wholesale price contract. A new contract achieves a Pareto improvement

over the wholesale price contract when both the manufacturer and the retailer earn profits at least

equal to that of the wholesale price contract. Indeed, as we show, it is possible to design a

contract that simultaneously achieves coordination and Pareto improvement.

5.1 The Revenue Sharing Contract and Side Payments

In Section 4, we discussed how in Model R the retailer can manipulate the variables p ,

and simultaneously. Note that in our model these are the only three variables that affect the

profit from the extended warranty. The manufacturer can, of course, influence the retail price

ep

ew

18

through her choice of the wholesale price. In addition, the decision variable for the manufacturer

in Model R is only the wholesale price. Therefore, we can think of Model R as decentralized in

product sales and somewhat like a centralized system for the extended warranty sales. The main

source of inefficiency in model R, thus, is the double marginalization in the product market.

This, however, no longer holds true in Model M. In model M, the manufacturer sells the

extended warranty while the retailer controls its demand directly through the retail price (recall

that the demand for extended warranties depends on product demand). Therefore, Model M can

be thought of as being decentralized in the product as well as in the extended warranty market.

Thus the source of inefficiency in Model M comes from the double marginalization of the

product market, as well as from the decentralization in the extended warranty market. This

implies that any contract that remedies the double marginalization in a single-manufacturer-

single-retailer decentralized supply chain should coordinate the channel described in Model R,

but not the channel described in Model M.

Observation 1: A revenue sharing contract in which only the product revenue is shared

between the manufacturer and the retailer can coordinate the channel in Model R, but no such

contract exists for the channel in Model M.

We next ask the question of whether it is possible to achieve both coordination and

Pareto improvement in Model R using a revenue-sharing contract. Consider a revenue-sharing

contract with side payments. Such a contract has two parameters: φ , the fraction of product

sales revenue allocated to the retailer (the manufacture thus gets φ−1 ); and , the side payment

the manufacturer offers to the retailer.

F

Proposition 6: A product revenue sharing contract with side payment (φ, F) simultaneously

achieves coordination and Pareto improvement for the decentralized channel described in Model

R, where, F ∈ [ , ], and ∗Rrπ ∗R

mπ

⎪⎪⎩

⎪⎪⎨

⎧

−≤<

≤≤−

=)2/(121,

2

10,)2(2

1

βαβα

ααβ

φ .

19

Recall that we have defined and as the optimal profit of the retailer and the

manufacturer, respectively, in Model R under the wholesale price contract. As described in

Observation 1, a side payment is not required for coordination. A revenue-sharing contract (with

φ as shown in Proposition 6 and F=0) alone is sufficient to coordinate the chain. Under this pure

coordination framework, the manufacturer gets the whole system profit, and the retailer gets zero

profit. That is possible since the retailer’s reservation profit is assumed to be zero, and being the

Stackelberg leader, the manufacturer offers a contract that extracts the entire surplus from the

retailer.

∗Rrπ ∗R

mπ

Under the revenue sharing scheme with side payments, the manufacturer’s profit is ,

while that of the retailer is simply the fixed side payment F. Any arbitrary distribution of the

profit is possible between the manufacturer and the retailer using the side payments. In

particular, the side payment described in Proposition 6 allows us to achieve Pareto improvement.

Given that both parties will be better off under this scheme, the contract seems quite practical

and easy to implement. The actual choice of F probably depends on the two parties’ relative

bargaining power. We have shown in Section 4 that under a wholesale price contract, the retailer

gets only half of the manufacturer’s profit in Model R. Under the scheme described in

Proposition 6, a powerful retailer can certainly negotiate a better deal, while making a Pareto

improvement for the manufacturer as well.

FCsys −∗.π

5.2 Profit Sharing and Quantity Discounts

We have shown that a revenue sharing contract can coordinate the channel described in

Model R. However, such a contract is unable to coordinate the channel described in Model M.

We now describe a contract involving profit sharing from the sales of extended warranties and a

quantity discount scheme that achieves coordination and Pareto improvement simultaneously for

both Models M and R. In this scheme, the party offering an extended warranty shares the profit

from extended warranty with the other party. Thus, the manufacturer will share the extended

warranty profit with the retailer in Model M. Let φ denote the fraction of the extended

warranty profit allocated to the retailer, irrespective of the party offering the warranty. The

manufacturer, thus, gets φ−1 of the extended warranty profit. In addition, depending upon the

retailer’s order quantity q, and the value of the parameter φ, the manufacturer offers a quantity

20

discount scheme for the wholesale price. As shown in Proposition 7, such a scheme achieves

both coordination and Pareto improvement for Models M and R.

Proposition 7: An extended warranty profit sharing and quantity discount contract (φ, x(φ,q))

achieves both coordination and Pareto improvement for both Models M and Model R, where,

bqqx )1()1(),( −

⋅−= φφ and

in Model R

]21,

41[∈φ , and

in Model M

]/1)2(4

)2(2,]/1)2(4[

]/1)2(2[)2(2[ 2 βαα

βαβαα

−−−

−−−−⋅− , when 10 <≤α

∈φ

The schem

centralize

models be

optimizati

improvem

This s

extended

implies th

though th

value and

manufactu

contract,

sharing. F

achieve bo

]4

2,)4(

)2(2[ 2 αββ

αβαββ

−−−⋅ , otherwise.

e in Proposition 7 allows the manufacturer to align the retailer’s profit with that of the

d channel ( cr φππ = ). The scheme is able to coordinate the supply chain for both

cause it links the product and the extended warranty markets, allowing simultaneous

on of both markets. The specific choices of the contract parameter φ ensure Pareto

ent.

cheme can achieve coordination only with 0=φ . In Model M, 0=φ means that no

warranty profit sharing is required to achieve channel coordination. In Model R, 0=φ

at the manufacturer gets the entire profit from the extended warranty business even

e retailer offers the warranty. Similarly to Proposition 6, the retailer’s zero reservation

the manufacturer’s leader position ensures that the entire system profit goes to the

rer under this pure coordinating contract. In Model M’s improvement-coordination

the manufacturer distributes the efficiency gains through extended warranty profit

or Model R, the retailer shares only part of the extended warranty profits in order to

th coordination and Pareto improvement.

21

Theoretically, our contract improves and coordinates both Models R and M perfectly.

However, the profit sharing contracts are often hard to implement in practice, as it requires

monitoring of sales as well as associated cost parameters. Nevertheless, the pure coordination

framework using a quantity discount scheme in Model M is practical and easy to implement.

6. SUMMARY AND CONCLUSION

In this paper, we have developed a game-theoretical model to compare different schemes for

extended warranties in a supply chain context. Specifically, we considered a supply chain

involving a single manufacturer and a single retailer. We studied two models with the retailer

and the manufacturer, respectively, being the provider of the extended warranty. We compared

the performances of the two models in terms of system profits, retailer profits, manufacturer

profits, product pricing decisions, and extended warranty decisions. We also benchmarked the

performances of the two models with that of a centralized system. Our results show that offering

extended warranties through a retailer generates more system profit than offering it through the

manufacturer. This is because the retailer can simultaneously optimize all relevant decision

variables and, thus, acts as a central planner in the extended warranty market. However, a

similar situation does not occur when the manufacturer offers the extended warranty. We have

also analyzed and compared the optimal values of the decision variables in the two models. We

show that a centralized system will offer the longest extended warranty while a system in which

the manufacturer offers the extended warranty will have the shortest extended warranty. We

developed two parameters that influence the profitability of the extended warranty business.

We propose channel coordination mechanisms to eliminate the causes of inefficiency in each

of the two models. Double marginalization in the product market is shown to be the primary

source of inefficiency in the channel described in Model R. On the other hand, the inefficiency

in Model M arises because of the double marginalization in the product market and because of

the decentralization of the extended warranty market. We proposed a product revenue sharing

contract with side payments that not only coordinates the supply chain but also achieves Pareto

improvement over the wholesale price contract in Model R. For Model M, an extended warranty

profit sharing contract along with a quantity discount scheme is shown to achieve both

coordination and Pareto improvement.

22

Our paper makes the following contribution to the operations management literature. First, it

provides a unique demand function for the extended warranties that explicitly takes into account

product demand. In line with our observation in practice, we model the extended warranty as a

“service product” with price and time duration. Second, while a majority of the literature in

extended warranties focuses on consumer moral hazard and heterogeneity issues, our work

addresses its role in the channel competition, performance and coordination. By adopting a

game-theoretic approach, we study the strategic interaction between the manufacturer and the

retailer as well as the interaction between the product sales and extended warranty sales. The

channel coordination and improvement mechanisms we present address the different causes of

channel inefficiency in different models. To the best of our knowledge, our paper is the first to

address these issues relating to extended warranties in a supply chain context. Third, our model

implies that extended warranties are not merely a source of revenue. Because of the unique

character of its demand, which is dependent on product demand, extended warranties could be

used strategically in channel choices to improve system profits and product pricing decisions.

These results provide useful insights as well as qualitative guidance to practicing managers

involved in designing, setting up, and managing an extended warranty business.

Like any other model in the operations management literature, our model is based on a set of

assumptions. For simplicity, we studied the extended warranty business using a monopolistic

setting. This allowed us to isolate the impact of extended warranties on total supply chain profits.

An important extension of our model would be to include competition in the model. Such

competition can be either between manufacturers or between retailers. Incorporating demand

uncertainties might be another useful extension and might require a new set of coordinating

contracts. Our model assumes no information asymmetry. A retailer might have private

information about consumer demand. In such a scenario, the retailer might use this information

strategically to improve his profit. Another way to extend our model could be to incorporate

product design elements into our model. For example, product quality can be a decision variable

in the model, which affects both product cost and the cost of the extended warranty.

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25

Table 1: Optimal solutions for Model C

Model C: Centralized System Offers Extended Warranty

Optimal solution When

2221 +

<< β When 2

22 +≥β

Condition 10 <<α βα <≤1 β

α2121 −≤≤ 10 <<α

βα

2121 −≤≤

∗Cp ]/1)2(2[/12

βαβα−−⋅⋅

−−b

)2( αβ

αβ−⋅

−b

0 ]/1)2(2[/12

βαβα−−⋅⋅

−−b

)2( αβ

αβ−⋅

−b

∗Cq βαα

/1)2(22

−−⋅−

αββ−2

1 βα

α/1)2(2

2−−⋅

− αβ

β−2

∗Csys.π ]/1)2(2[2

2βα

α−−⋅⋅

−b

)2(2 αβ

β−⋅b

eb2

α

]/1)2(2[22

βαα

−−⋅⋅−

b

)2(2 αββ

−⋅b

∗Cprod .π

2]/1)2(2[)/12()2(

βαβαα

−−⋅⋅−−⋅−

b

2)2()(

αβαββ

−⋅−⋅

b 0 2]/1)2(2[

)/12()2(βαβαα

−−⋅⋅−−⋅−

b

2)2()(

αβαββ

−⋅−⋅

b

∗Cwarr .π 2]/1)2(2[2

)/1()2(βα

βα−−⋅⋅

⋅−b 2]/1)2(2[2

)/1()2(βα

βα−−⋅⋅

⋅−b

2)2(2 αβ

αβ−⋅

⋅b

eb2

α

2)2(2 αβ

αβ−⋅

⋅b

∗Cep ]/1)2(2[

)(βα −−⋅⋅

+−⋅cb

ccbec

e

rer

)2()/(

αββ−⋅

−⋅cb

cbee

e

r cb

cbee

e

re2

)( −⋅ ]/1)2(2[)(

βα −−⋅⋅+−⋅

cbccbec

e

rer

)2()/(

αββ−⋅

−⋅cb

cbee

e

r

∗Cew ]/1)2(2[ βα −−⋅⋅

−cb

cbe

e

re

)2(

/αβ

β−⋅

−c

cbe r cbcbe

e

re− ]/1)2(2[ βα −−⋅⋅−

cbcbe

e

re

)2(

/αβ

β−⋅

−c

cbe r

∗

∗

Ce

Ce

wp

re

rer

cbeccbec

−+−⋅ )(

eb

e eb

e re

rer

cbeccbec

−+−⋅ )(

eb

e

∗Ceq

βα /1)2(21

−−⋅

αββ−2

1 βα /1)2(2

1−−⋅

αβ

β−2

Note: (1)cbcbe

e

re2)( −

=α is the Extended Warranty Desirability Index; bbe=β is the Relative Price Sensitivity

Index. (2) Model C’s second order condition is satisfied when

βα

212 −≤ .

(3) requires condition . 0>ew recbe >

26

Table 2: Optimal solutions for Model R

Optimal Solution Model R: Retailer offers extended warranty

Condition 10 <<α β

α2121 −≤≤

∗Rp ]/1)2(2[/1)2(2/3βαβα

−−⋅⋅−−⋅

b

)2(2/3

αβαβ

−⋅−⋅

b

∗Rq ]/1)2(2[22

βαα−−⋅⋅

− )2(2 αβ

β−⋅

∗Rx b2

1

b21

∗Rrπ ]/1)2(2[8

2βα

α−−⋅⋅

−b

)2(8 αβ

β−⋅b

∗Rprodr ./π

2]/1)2(2[4]/1)2[()2(

βαβαα

−−⋅⋅−−⋅−

b

2)2(4)(

αβαββ

−⋅−⋅

b Retailer’s

Profit ∗Rwarrr ./π

2]/1)2(2[8)/1()2(βα

βα−−⋅⋅

⋅−b

2)2(8 αβ

αβ−⋅

⋅b

∗Rmπ ]/1)2(2[4

2βα

α−−⋅⋅

−b

)2(4 αβ

β−⋅b

∗Rsys.π

]/1)2(2[8)2(3

βαα−−⋅⋅

−⋅b

)2(8

3αβ

β−⋅b

∗Rprod .π

2]/1)2(2[2]/1)2(2/3[)2(

βαβαα

−−⋅⋅−−⋅⋅−

b

2)2(2)2/3(

αβαββ

−⋅−⋅

b System

Profit ∗R

warr .π 2]/1)2(2[8

)/1()2(βα

βα−−⋅⋅

⋅−b

2)2(8 αβ

αβ−⋅

⋅b

∗Rep

]/1)2(2[2)(

βα −−⋅⋅+−⋅

cbccbec

e

rer )2(2)/(

αββ−⋅

−⋅cb

cbee

e

r

∗Rew

]/1)2(2[2 βα −−⋅⋅−

cbcbe

e

re )2(2

/αβ

β−⋅

−c

cbe r

∗

∗

Re

Re

wp

re

rer

cbeccbec

−+−⋅ )(

eb

e

∗Req ]/1)2(2[2

1βα −−⋅⋅

)2(2 αβ

β−⋅

Note: (1) cbcbe

e

re2)( −

=α and bbe=β .

(2) Model R’s second order condition is satisfied when β

α212 −≤ .

(3) requires condition . 0>ew recbe >

27

Table 3: Optimal solutions for Model M

Optimal Solution Model M: Manufacturer offers extended warranty

Condition 10 <<α β

α4121 −≤≤

∗Mp ]/1)2(4[/1)2(3βα

βα−−⋅⋅

−−⋅b

)4(

3αβ

αβ−⋅

−b

∗Mq βαα

/1)2(42

−−⋅−

αββ−4

∗Mx ]/1)2(4[/1)2(2βαβα

−−⋅⋅−−⋅

b

)4(2

αβαβ−⋅−

b

∗Mrπ 2

2

]/1)2(4[)2(

βαα−−⋅⋅

−b

2

2

)4( αββ−⋅b

∗Mmπ ]/1)2(4[2

2βα

α−−⋅⋅

−b

)4(2 αβ

β−⋅b

∗Mprodm ./π

2]/1)2(4[]/1)2(2[)2(

βαβαα

−−⋅⋅−−⋅⋅−

b

2)4()2(

αβαββ

−⋅−⋅

b Manufacturer’s

Profit ∗Mwarrm ./π

2]/1)2(4[2)/1()2(βα

βα−−⋅⋅

⋅−b

2)4(2 αβ

αβ−⋅

⋅b

∗Msys.π 2]/1)2(4[2

]/1)2(6[)2(βα

βαα−−⋅⋅

−−⋅⋅−b

2)4(2

)6(αβαββ

−⋅−⋅

b

∗Mprod.π

2]/1)2(4[]/1)2(3[)2(

βαβαα

−−⋅⋅−−⋅⋅−

b

2)4()3(

αβαββ

−⋅−⋅

b System Profit

∗Mwarr .π

2]/1)2(4[2)/1()2(βα

βα−−⋅⋅

⋅−b

2)4(2 αβ

αβ−⋅

⋅b

∗Mep

]/1)2(4[)(

βα −−⋅⋅+−⋅

cbccbec

e

rer )4()/(

αββ−⋅

−⋅cb

cbee

e

r

∗Mew

]/1)2(4[ βα −−⋅⋅−

cbcbe

e

re )4(

/αβ

β−⋅

−c

cbe r

∗

∗

Me

Me

wp

re

rer

cbeccbec

−+−⋅ )(

eb

e

∗Meq

βα /1)2(41

−−⋅

αββ−4

Note: (1)cbcbe

e

re2)( −

=α and bbe=β .

(2) Model M’s second order condition is satisfied when β

α412 −≤ .

(3) requires condition . 0>ew recbe >

28

Appendix: Proof of the Results

Proof of Proposition 2

We present the proofs for Models R and M. The proofs for Model C are similar and can be

derived from Table 1.

Value (Model R) Expression 10 <<α

βα

2121 −≤≤

απ∂∂ ∗R

r 0]/1)2(2[8

/12 >

−−⋅⋅ βαβ

b 0

)2(8 2 >−⋅ αβ

βb

απ∂

∂ ∗Rprodr ./ 0

]/1)2(2[4)/1(

3

2

<−−⋅⋅

−βα

βb

0)2(4 3 <−⋅

−αβ

αβb

Retailer’s

Profit Sensitivity

απ∂

∂ ∗Rwarrr ./ 0

]/1)2(2[8)/1(]/1)2(2[

3 >−−⋅⋅⋅+−⋅βαββα

b 0

)2(8)2(3 >

−⋅+⋅αβαββ

b

<∂

∂ ∗

απ R

prodr ./ ⇔∂

∂ ∗

απ R

warrr ./ )2(2/1 αβ −< βα 2<

Value (Model M)

Expression 10 <<α β

α4121 −≤≤

απ∂∂ ∗M

m 0]/1)2(4[2

/12 >

−−⋅⋅ βαβ

b 0

)4(2 2 >−⋅ αβ

βb

απ∂

∂ ∗Mprodm ./ 0

]/1)2(4[)/1(

3

2

<−−⋅⋅

−βα

βb

0)4( 3 <−⋅

−αβ

αβb

Manufacturer’s

Profit Sensitivity

απ∂

∂ ∗Mwarrm ./ 0

]/1)2(4[2)/1(]/1)2(4[

3 >−−⋅⋅⋅+−⋅βαββα

b 0

)4(2)4(

3 >−⋅+⋅αβαββ

b

<∂

∂ ∗

απ M

prodm ./ ⇔∂

∂ ∗

απ M

warrm ./ )2(4/1 αβ −< βα 4<

Proof of Theorem 1

Use the expressions from Tables 1-3. For 10 <<α , > is obvious. > ⇔ ∗Csys.π ∗R

sys.π ∗Rsys.π ∗M

sys.π

0/1)2(8 >−− βα , which is true under condition 10 <<α . So under condition 10 <<α ,

29

we have > > . Next consider ∗Csys.π ∗R

sys.π ∗Msys.π 1≥α . Consider two case: (i) )

222,1( +

∈β , and (ii)

222+

≥β

(i) )2

22,1( +∈β , here has two sets of solutions under ∗C

sys.π 1≥α .

(i.a) Consider βα <≤1 . > is obvious. > ⇔ ∗Csys.π ∗R

sys.π ∗Rsys.π ∗M

sys.π βα 8< , which is true under

condition βα <≤1 . Therefore, > > . ∗Csys.π ∗R

sys.π ∗Msys.π

(i.b) Next, consider β

αβ212−≤≤ . > ⇔ ∗C

sys.π ∗Rsys.π βαβ 5.15.0 <≤ . Since

ββ 5.1)2/(12 ≤− , it is true under current condition; > ⇔ ∗Rsys.π ∗M

sys.π βα 8< , obviously it is

true under current condition. Therefore, for β

αβ212−≤≤ , we have > > . ∗C

sys.π ∗Rsys.π ∗M

sys.π

(i.c) Finally, when β

αβ 4

12212 −≤≤− , only has feasible solution. ∗M

sys.π

(ii) 2

22+≥β , here has only one set of solution under ∗C

sys.π 1≥α .

(ii.a) Consider, β

α2121 −≤≤ . As shown in (i.a) above, > > . ∗C

sys.π ∗Rsys.π ∗M

sys.π

(ii.b) Again, when β

αβ 4

12212 −≤<− , only has feasible solution. ∗M

sys.π

Proof of Proposition 3

(a) When 10 <<α , > ⇔ , which is true. The proof for > is

straightforward. When

∗Rrπ

∗Mrπ 0)/( 2 >ebb ∗R

mπ∗M

mπ

)2/(121 βα −≤≤ , > ⇔ , which is again true. The

proof for > is straightforward.

∗Rrπ

∗Mrπ 02 >α

∗Rmπ

∗Mmπ

(b) (i) From Table 2, it is obvious that = for Model R. ∗Rmπ

∗Rrπ2

(ii) In Model M, when 10 <<α , > ⇔ ∗Mmπ

∗Mrπ βα /1)2(2 >− , relationship holds.

When )4/(121 βα −≤≤ , > ⇔ ∗Mmπ

∗Mrπ βα 2< , relationship holds. Other relationships can

be derived rather easily.

30

Proof of Proposition 4

Consider warranty length first. Observe from Tables 1-3 that the expressions for

have the same numerator. Thus, it is sufficient for us to look at the denominators

of these expressions. It is straightforward to show that:

*** ,, Me

Re

Ce www

βαβαβα /1)2(4/2)2(4/1)2(2 −−<−−<−− , for 10 <≤α ,

αβαβαβ −<−<− 4242 otherwise,

implying > > . The proofs for prices, demands and profits are similar. ∗Cew ∗R

ew ∗Mew

Proof of Proposition 5

(a) > ⇔ , which holds trivially. ∗Rx ∗Mx 12 >

(b) For all α , < ⇔ ; < ⇔ ∗Cp ∗Rp 5.11< ∗Rp ∗Mp 5.55 < , this proves < < .

The relationship implies > > .

∗Cp ∗Rp ∗Mp

bpq −= 1 ∗Cq ∗Rq ∗Mq

Proof of Proposition 6

The optimization problems for the manufacturer and the retailer are given by:

, and FbpxbppMax Smx

−−+−−=Π )1()1()1()(

φφ

Fcw

ewpbbpwcpbpxpMax eeeeere

Srwpp ee

+−+−−⋅−+−−=Π2

])1[()()1)((2

,,φ ,

s.t. , eee pbew ≤

where, the superscript S denotes a revenue sharing contract. The solution is as follows.

10 ≤<α )2/(121 βα −≤<

Smπ F

b−

−+−⋅−

])1)(2([4)2(

φφαβφαβφ F

b−

+−⋅ )1/(41

βαφ

Srπ F

b+

−+−⋅−−⋅−

2

2

])1)(2([8])2(2[)2(

φφαβφφαβφαβφ F

b+

+−⋅−

2)1/(8/2βαφβαφ

Ssys.π 2])1)(2([8

]3)21)(2(2[)2(φφαβφ

φφαβφαβφ−+−⋅

−+−⋅−b

2)1/(8

2/34+−⋅+−

βαφβαφ

b

31

It is easy to see that when 10 ≤<α , )2(2

1αβ

φ−

= yields = and when Ssys.π ∗C

sys.π

)2/(121 βα −≤< , βαφ2

= yields = . To achieve Pareto improvement, we need to

impose the following individual rationality constraints to the profit of the manufacturer and the

retailer: and . From these two constraints, we obtain the range of the side

payment that the manufacturer offers to the retailer: F ∈ [ , ].

Ssys.π ∗C

sys.π

Smπ ≥ *R

mπSrπ ≥ ∗R

rπ

∗Rrπ ∗R

mπ

Proof of Proposition 7

(1) Consider Model R. Facing the manufacturer’s pricing schedule, the retailer solves:

, where . ewwpp

Sr bpqxpMax

ee

φπφπ +−−= )1))(,((,,

2/])1[()( 2eeeeereew cwewpbbpwcp −+−−⋅−=π

The manufacturer’s profit is: . ewSm qxbp πφφπ )1(),()1( −+⋅−=

In order to coordinate the supply chain, the manufacturer needs to find a pricing schedule so that

the retailer’s profit function is aligned with the one of the centralized supply chain, which is

. ewwpp

C bppMaxee

ππ +−= )1(,,

To verify the contract in Proposition 7, substitute bqqx /)1()1(),( −⋅−= φφ into the retailer’s

profit function, to get , and . Therefore, under the contract of

Proposition 7, the supply chain’s total profit equals to the one of the centralized channel. It is

easy to verify that this contract achieves coordination with

CSr φππ = CS

m πφπ )1( −=

0=φ . To achieve Pareto

improvement, we impose the following individual rationality constraints on the profits of the

manufacturer and the retailer: and . This leads to φ ∈[0.25, 0.5]. ≥Smπ

*Rmπ ≥S

rπ∗R

rπ

(2) The proof for Model M is similar.

32