Design of Extended Warranties in Supply Chains Kunpeng Li University of Illinois at Urbana-Champaign, College of Business Dilip Chhajed Suman Mallik University of Illinois at Urbana-Champaign, College of Business University 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 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. Published: September 2005 URL: http://www.business.uiuc.edu/Working_Papers/papers/05-0128.pdf
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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
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
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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.