Channel coordination with extended warranty when retailers compete

Abstract This paper studies coordination mechanisms for the supply chain with one manufacturer and two competing retailers, where the manufacturer directly sells the extended warranty service to the consumers. We find that the efficiency of the wholesale contract increases with retail competition, whereas it decreases with the extended warranty duration. For the symmetric competing retailers, a set of quantity discount contracts have been designed to coordinate channel with Pareto improvement. Especially, when retail competition approaches infinity, the quantity discount contract to coordinate the channel will degenerate into a simple wholesale contract. For the asymmetric competing retailers, three sets of contract structures have been identified, under which the complicated quantity discount contract combined with a two-part tariff or slotting allowance can achieve channel coordination with Pareto improvement of tripartite. The results are remarkably different from the symmetric retailer scenario.


Introduction
The extended warranty, which extends the period of warranty coverage after the expiration of the base warranty, has become the mainspring of the aftersale markets. Studies indicate that consumers are inclined to overestimate the possibility and repair cost of product breakdown, and in order to avoid the perceived risk, consumers have more willing to buy an extended warranty (Cai et al., 2020). Not only do the consumers welcome the extended warranties, but the extended warranty providers also gain benefits (Kelley & Conant, 1991). Marketing extended warranty provides an opportunity to maintain the relationship with the consumer (Hartman & Laksana, 2009) and thereby gain a large profit margin (e.g., Ma et al., 2019). In the US, an estimated 20-40% of people who purchase consumer durables also buy the extended warranty, and 40-80% of the profit on electronic products comes from the sale of the extended warranties (Week, 2005).
Typically, there are three main providers in the extended warranty market (Zhang et al., 2020). Many manufacturers offer extended warranties directly to the end consumers; for example, GE Appliances offers extended warranties on almost all the products it manufactures. Manufacturers like Ford, GM, JVC, and Apple even have devoted whole divisions solely to managing and to serving extended warranty contracts (Padmanabhan, 1995). Third-party insurers like Warranty Group and Warrantech offer extended warranties on many kinds of consumer durables, including home appliances, electronic equipment, computers, furniture, and automobile (Lutz & Padmanabhan, 1998). Some powerful retailers (e.g., Sears in North America and Suning.com) also rewrite extended warranties themselves rather than offering third-party insurers' extended warranties. Compared to the extended warranty provided by retailers and third-party insurers, manufacturers' extended warranties are usually more attractive since the lower price and the recommendations from consumer experts (Armstrong, 2004).
To ensure the profit from extended warranty sales, the manufacturer needs to solve two main difficulties. The first difficulty is how to balance its extended warranty pricing and wholesale pricing behavior. The extended warranty profit is determined by the extended warranty price and extended warranty demand. The extended warranty price is determined by the manufacturer and directly influences the extended warranty demand. But it is worth noting that the extended warranty demand is also influenced by the product demand since the extended warranties are after-sale services of products. Since the product demand is determined by the manufacturer's wholesale price and the double marginalization in the vertical structure, the manufacturer needs to balance its extended warranty pricing and wholesale pricing behavior. The second difficulty is that the retail competition in the market makes it more difficult to balance the wholesale pricing and extended warranty pricing. The second difficulty is, however, not avoidable since retail competition is common in durable goods markets (such as Bestbuy and Sears in the US; and Suning and Gome in China.), and the durable goods markets are also the main market of extended warranties. As the retail competition increases in the market, the retailer will decrease the retail price; thus, the retail competition weakens the double marginalization. The manufacturer should also consider the retail competition when setting the wholesale price and extended warranty price.
Since it is strictly better to pre-announcing the price of the extended warranty when the manufacturer directly promotes the extended warranty (Desai & Padmanabhan, 2004), we focus on the case in which the manufacturer chooses pre-announcing. The following problems have been investigated in this paper. Firstly, how does the introduction of extended warranty affect the supply chain performance under retail competition? Secondly, under the supply chain with both extended warranty selling and retail competition, can the quantity discount contract coordinate the supply chain? Thirdly, how do we design the complicated quantity discount contract to coordinate the extended warranty supply chain with asymmetric retailers?
The major contributions of our paper are as follows. Firstly, although the extended warranty pricing and the efficient modes of distribution have been discussed (e.g., Desai & Padmanabhan, 2004;Gallego et al., 2014;Heese, 2012;Li et al., 2012), none of the studies has examined how manufacturers should design the coordination contract with the extended warranty. In this paper, we explore how to design the complicated quantity discount contracts to coordinate the channel under the competing retailer scenario. Secondly, we identify three sets of contract structures under which the manufacturer with an extended warranty should offer a quantity discount contract combining a two-part tariff or slotting allowance for asymmetric retailers. The results are remarkably different from that the symmetric competing retailers can offer simple quantity discount contracts for channel coordination. Although some studies have focused on quantity discount contracts (e.g., Karray & Surti, 2016;Weng, 1995), to the best of our knowledge, there has been no investigation of asymmetric competing retailers. Thirdly, our research integrates the two streams of marketing management on extended warranty pricing strategies and operation management on channel coordination. Finally, we provide some new managerial insights based on our results, such as the academic explanation of the classical proverb "Rivalry between Mussels and Mussels." This paper is organized as follows. Section 2 briefly reviews the related literature, and in Sec. 3, the basic models are introduced. In Sec. 4, we construct decentralized supply chain decision models with an extended warranty. In Sec. 5, the centralized supply chain decision models are provided for a benchmark. In Sec. 6, we design a set of quantity discount contracts for coordination under symmetric retailers with Pareto improvement and explore quantity discount contracts structure combining two-part tariff or slotting allowance to coordinate supply chain under asymmetric retailers. In Sec. 7, we do the numerical analysis of decision behaviors and policies' robustness. A summary of findings and limitations is described in the last section. All proofs are in the Appendix (see online supplementary materials).

Literature review
This paper is primarily related to research in two main streams: extended warranty and the channel coordination mechanisms under retail competition. We first discuss the issue of the extended warranty. We will then focus on the channel coordination mechanisms under retail competition.

Literature on the extended warranty
Based on the previous research about the consumer behavior toward extended warranties (see, e.g., Chen et al., 2009;Gan et al., 2009;Hollis, 1999;Jindal, 2015;Voss & Ahmed, 1992) and the extended warranty pricing (see, e.g., Gallego et al., 2015;Hartman & Laksana, 2009;Jack & Murthy, 2007;Lam & Lam, 2001), the distribution channel design and the coordination mechanism design of supply chain with extended warranty have been the new focus of recent research. Desai and Padmanabhan (2004) analyzed different channel arrangements (independent retail channel and dual-channel) and showed that the optimal warranty policy depended on two key forces, the complementary goods, and the double marginalization effect. Given that both manufacturer and retailer can provide the extended warranty, Li et al. (2012) considered the optimal price, extended warranty length, and supply chain profits with two scenarios. Jiang and Zhang (2011) explored the supply chain conflicts in the interactions between original product warranties and extended warranties. Gallego et al. (2008) studied the supply chain where a supplier sells a decentralized product and an ancillary service such as an extended warranty through a retailer, and they investigated channel coordination mechanisms in the coexistence of decentralized product and ancillary service. Mai et al. (2017) found that the extended warranty could be applied in quality coordination for the store-brand product. The literature above is all based on the monopoly channels. Zhang et al. (2020) investigate the optimal extended warranty strategies in a supply chain with a single manufacturer and a single retailer and consider two single-channel models and a dual-channel model. It solves the problem of price decisions under different extended warranty service provision modes, and our paper focuses on the choice of supply chain cooperation mechanism under the determination of the extended warranty service provision model. Three scenarios, i.e., the manufacturer or third party or directly retailers can offer third party extended warranty, have been considered. A similar structure has been extensively studied in the product remanufacturing field (Savaskan et al., 2004), in which there is a competitive relationship between new products and remanufacturing products. Unlike this, the products and extended warranty services in our study can coexist, and the demand for extended warranty services only arises after the products are purchased.
Recent researchers extended the extended warranty problem from monopoly channel to channel competition. Under the scenario where two competing manufacturers sell their products through a common retailer, Heese (2012) investigated the retail strategies for extended warranty sales and the impact on manufacturers' base warranty. Bian et al. (2015) extended the model in Jiang and Zhang (2011) to the retail competition market and found that the existence of an extended warranty gave the manufacturer no incentive to offer its corresponding base warranty. Li et al. (2014) analyzed different extended warranty selling modes under the setting of dual-channel. The literature above introduces the extended warranty to the competitive market, but there is still no research considering the supply chain coordination with extended warranties under the retail competition. Ma et al. (2019) studied operational decisions of two competitive supply chains under retailers' extended warranties, which is different from our market structure. Cai et al. (2020) consider a supply chain with one supplier providing basic products to two retailers, which is the same as our supply chain structure. However, they focus on the retailer's risk aversion information disclosure mechanism without the supply chain cooperation mechanism.

Literature on channel coordination under retail competition
Channel coordination under retail competition is always the main topic in supply chain coordination literature (Ingene & Parry, 2000;Jeuland & Shugan, 1983;Pan et al., 2010). Ingene and Parry (1995) examined the applicability of two-part tariffs under the retail competition and identified the limitation of two-part tariffs. The result showed, however, that implementing a specified quantity-discount schedule could gain the same profit as the integrated channel. Under the same setting with a supplier and many homogeneous retailers, Weng (1995) considered the inventory and found out that both an all-unit quantity discount policy and an incremental quantity discount policy could achieve channel coordination. Karray and Surti (2016) examine the interactive effects between quantity discounts and cooperative advertising. Chen et al. (2001) demonstrated that coordination is achieved via periodically charged, fixed fees and a nontraditional discount pricing schedule under which the discount is the sum of three components determined by the retailer's annual sales volume, order quantity, and order frequency. Considering the retail competition and demand uncertainty, Padmanabhan and Png (1997) suggested that the optimality of return policy depends on the tradeoff between the intense retail competition and the excessive stocking. The literature above presents several main contracts used in channel coordination under retail competition. The coordination mechanisms under retail competition are also discussed in supply chains with diversified settings. Savaskan and Van Wassenhove (2006) considered the reverse channel design under retail competition and found that the buy-back payment with a fixed franchise fee can coordinate the product collection effort and the retail price. When the retailers compete with retail price and service, Tsay and Agrawal (2000) found that a wholesale pricing scheme and two-part tariff can coordinate the channel under very limiting conditions. Xiao and Qi (2008) focused on the cost and demand disruptions in the supply chain and developed the conditions under which the two coordination mechanisms, an all-unit quantity discount, and an incremental quantity discount, can coordinate the supply chain. The literature above, however, has not discussed the coordination problem with extended warranties.

The basic model
We consider a supply chain consisting of a manufacturer M, who sells its product through two competing retailers, donated by R i , i ¼ 1, 2, and directly sells extended warranty to consumers. Retailer i chooses its own retail price, and the demand faced by the retailer is deterministic and dependent on the retail prices charged by two competing retailers.
The manufacturer is the Stackelberg leader and treats the retailers equally (i.e., provides the same contract for both retailers). Here we assume the manufacturer has the unlimited capability to supply products, and for the aim of simplicity, we assume the unit production cost is zero.
The notation used in this paper is summarized in Table 1.
Our model uses the duopoly static demand function (Ingene & Parry, 2007;Tsay & Agrawal, 2000), given by: The demand of retailer i is, by assumption, downward sloping in its own retail price p i and upward sloping in the competitive retailer's price p 3Ài : Then we consider the extended warranty demand as the additive function for retailer i (Li et al., 2012), which is: In order to focus on the impact of the extended warranty, we simplify the model by normalizing the base warranty to zero (Bian et al., 2015;Heese, 2012;Jiang & Zhang, 2011). Different from Li et al. (2012), we take the extended warranty duration as exogenous. It seems reasonable because products within the same category usually have the same extended warranty duration. For example, manufacturers mostly provide one year extended warranty for mobile phones, three years for white goods, and five years for automobiles.
To ensure the existence, uniqueness as well as nonnegative of the equilibrium, we assume a 1 þ a 2 > 8c 0 ht, a 1 À a 2 j j< 4c 0 ht, and 2h > bt, which assure that the manufacturer prefers to offer the extended warranty. This assumption is possible for most industrial situations since an average of 27% of new car buyers purchase an extended warranty (Lutz & Padmanabhan, 1998), and an estimated 20-40% of people who purchase consumer durables also buy an extended warranty (Huysentruyt & Read, 2010). This paper considers the most widely used extended warranty service, which extends the free replacement warranty policy after the expiration of the base warranty. A free replacement policy means the seller agrees to repair or provide replacements for failed items free of charge up to a time t from the time of the initial purchase (Blischke & Murthy, 1992;Murthy & Blischke, 2005). According to Wu et al. (2009) and Jack and Murthy (2007), when the cumulative failure distribution function FðtÞ follows a Weibull distribution with failure rate k and shape parameter b, the expected extended warranty cost with duration t is c 0 ðktÞ b , where ðktÞ b represents the expected failure time within the extended warranty duration t, and c 0 represents the cost for every replacement. Consistent with Li et al. (2012) and for simplicity, we assume k ¼ 1, b ¼ 2, c 0 t 2 , where the quadratic form indicates the diminishing return on extended warranty cost. The manufacturer's cost for providing an extended warranty with duration t is c 0 t 2 :

The decentralized supply chain game model
The game sequence in the decentralized model is as follows: In the first stage, the manufacturer sets the same wholesale price for both retailers and pre-announcing the extended warranty price p e (Desai & Padmanabhan, 2004). In the second stage, given the wholesale price and extended warranty price, each retailer determines its own retail price p i : A Stackelberg game is played between a manufacturer and the two retailers, and a Nash game is played between the two retailers.
Let p D R i denote the retailer i's profit, so the retailer i's problem is: Acting as Stackelberg leader, the manufacturer can correctly anticipate how the retailers react to the given wholesale price and extended warranty price. Let p D M denote the profit of manufacturer M in the decentralized model, so we have: Maximizing with respect to w and p e and solving the first order conditions simultaneously, we have: Proposition 1. In the decentralized supply chain system, the two-stage Stackelberg game equilibrium results of the wholesale price, retailer price, extended warranty price, and profit of manufacturer and retailers are as follows.
In order to identify the effect of extended warranty duration t on the equilibrium, we can get the following results.
@p D e @t > 0: Case 1 implies that when product competition is high, or product competition is low, and the market size is large enough, the wholesale price and retail price will decrease with the extended warranty duration. Case 2 implies that when product competition is low and the market size is moderate, the wholesale price and retail price will increase with the extended warranty duration. Case 3 implies the extended warranty price always increases with the extended warranty duration.
In order to identify the effect of competition h on the equilibrium, the results are as follows.
Proposition 3. (1) The wholesale price decreases with the competition h, i.e., @w @h < 0; however, the extended warranty price increases with the competition h, i.e., @p D e @h > 0: (2) If a i ¼ a, the retail price decreases with the competition h, i.e., @p D i @h < 0:

The centralized supply chain game model
We first model the centralized supply chain case, which is a benchmark to evaluate the efficiency of the channel-coordinating contracts. The profit of the centralized model (i.e., a manufacturer controls both retailers) with retail price p i and extended warranty price p e is: The optimal policy in the centralized model is obtained by maximizing the supply chain profit (10) with respect to variables p i and p e , so the optimal solutions are given by: Proposition 4. For the centralized supply chain case, (i) the product price is lower than that in the decentralized supply chain, i.e., p C i < p D i , and product demand is higher than that in the decentralized supply chain, i.e., q C i > q D i . (ii) the extended warranty price and the extended warranty demand are all higher than that in the decentralized supply chain, i.e., p C e > p D e , q C e > q D e : Under the setting that the manufacturer sells the extended warranty directly to the consumers, there only exists a double marginalization problem in the product sales. The double marginalization in the decentralized model leads to a higher product price and thus decreases the corresponding product demand and the extended warranty demand. Facing the profit loss of product sales, the manufacturer would have to set a lower extended warranty price to maximize the profit from extended warranty sales. The proposition we find above is consistent with the empirical study that the manufacturer authorized retailers to sell washers and warranties (Jindal, 2015), which indicated that the decentralized modes led to a 7% increase in the washer price and a 3% decrease in the warranty price.

Channel coordination mechanism design
Firstly, we will design the quantity discount contract to coordinate the tripartite of the supply chain under symmetric retailers with Pareto improvement. Secondly, we will explore the complicated contract based on quantity discount contract to coordinate the tripartite of the supply chain under asymmetric retailers. Especially, we will try to identify the conditions under which the quantity discount contract combing with franchises fee and the quantity discount contract combing with slotting allowance can coordinate the tripartite of the supply chain under asymmetric retailers.

Channel coordination with symmetric retailers
Consistent with Ingene and Parry (1995), we focus on a linear quantity-discount schedule of form ðW À wq i Þq i , q i < W w , where W means the vertical intercept of the per-unit wholesale price schedule, and w is the slope of the per-unit wholesale price schedule. We assume that the manufacturer provides an identical quantity-discount schedule to both retailers (Xiao & Qi, 2008), i.e., a i ¼ a: 6.1.1. The retailer's problem We begin by solving for retailers' decisions in the second stage of the game. Let p q R i denote the retailer i's profit, and according to the definition of the quantity discount contract above, retailer i's profit function is: The first order condition equilibriums of retailers' pricing are as follows: 6.1.2. The manufacturer's problem Knowing the response function of retailers, that manufacturer simultaneously chooses p i ¼ p c i , p e ¼ p c e to solve W; we can get the result: Let p q M denote manufacturer M's profit, then: In order to achieve channel coordination, the decentralized decisions are not only consistent with centralized decisions, but also the participants get Pareto performance improvement compared with the decentralized supply chain system.
Proposition 5. If quantity discount unit price W q satisfies (15), and the quantity discount rate w satisfies w 1 < w < w 2 , as shown in (18) and (19), then the quantity discount contract in the form of ðW q À wq i Þq i , where q i < W q w , can achieve not only the centralized supply chain performance but also the tripartite of supply chain get the Pareto improvement compared with the decentralized supply chain system.
Especially when the competition intensity h approaches infinity, we will get an interesting finding.
Corollary 1. When the competition intensity approaches infinity, i.e., h ! 1, the feasible discount rate will approach zero, i.e., w ! 0, then the quantity discount contract will degenerate into the simple wholesale contract, i.e., W q ¼ ðbc 0 hÞt 2 þatð2hÀbÞ ð4bhÀb 2 tÞ , which can achieve the supply chain coordination.
This corollary implies that the manufacturer has no motivation to implement the complicated coordination contract when competition intensity is very high, which stems from the fact that the wholesale contract is effective enough. In a way, this is the academic explanation of the classical proverb "Rivalry between Mussels and Mussels." In other words, the manufacturer can introduce retail competition as another effective coordination strategy.
In order to characterize this coordinated quantity discount contract deeply, we can get the following proposition.
Proposition 6. (1) The first part unit price W increases with the discount rate w and the competition intensity h, i.e., @W q @w > 0 and @W q @h > 0.

Channel coordination with asymmetric retailers
In Sec. 6.1, we have identified the conditions under which the quantity discount can achieve channel coordination with homogenous retailers. So in this sector, we will explore the complicated quantity discount policies to coordinate the channel with heterogeneous retailers. We will focus on contracts combining linear quantity-discount with slotting allowance in form ðW À wq i Þq i ÀK i , q i < W w , where K i represents the fixed fee paid by the manufacturer to retailer i: 6.2.1. The retailer's problem We begin by solving for retailers' decisions in the second stage of the game. Let p Q R i denote retailer i's profit, and according to the definition of the quantity discount contract above, retailer i's profit function is: Solving the first order conditions of Eq. (20), we can get the subgame equilibrium: (21)

The manufacturer's problem
Knowing the response function of retailers, the manufacturer simultaneously chooses w and p e to maximize the supply chain profit. Let p Q M denote the manufacturer's profit, then the manufacturer's decision problem is: In order to coordinate the supply chain system, we should design the decentralized system decision the same as the centralized system; the results are summarized as the following proposition.
Proposition 7. There exists the quantity discount schedule ðW Ã À w Ã q i Þq i , where q i < W Ã w Ã , which enables the supply chain to achieve the centralized performance, the quantity discount parameters are designed as follows: Proposition 8.
(1) The first-part unit price W Ã decreases with the price-sensitive factor b; however, it increases with the potential total market size a 1 þ a 2 .
(2) The quantity discount rate w Ã decreases with the retail competition h and also decreases with the price-sensitive factor b, i.e., @w Ã @h < 0, @w Ã @b < 0:

The complicated contract under symmetric retailers
Firstly, we consider the complicated contract under symmetric retailers, i.e., a i ¼ a, then there exists the following Lemma 1.
As the symmetric retailers under the quantity discount contract combined with slotting allowance, we can get the following result.
Proposition 9. Under the symmetric retailers with quantity discount contract as (27)-(29), the following results hold, The linear quantity discount contract combined with slotting allowance in the form of ð a b À 2bþh 2bðbþhÞ q i Þq i À K , where q i < 2aðbþhÞ 2bþh ; and the slotting allowance satisfies K R i j j < K < K M 2 , can achieve the channel coordination with Pareto improvement compared with the decentralized supply chain system.
(3) When minða 21 , a 2 þ 4Þ > a 1 > maxða 21 , 8Àa 2 , a 2 À4Þ (area III in Figure 1), then p Q M > p D M , p Q R 1 < p D R 1 , and p Q R 2 < p D R 2 hold. The quantity discount contract combined with slotting allowance in the form of ð ða 1 þa 2 Þ 2 À q 1 Þq 1 ÀK 1 for retailer 1, where K R 1 j j < K 1 < K M À K R 2 j j, and another in the form of ð ða 1 þa 2 Þ 2 À q 2 Þq 2 ÀK 2 for retailer 2, where K R 2 j j < K 2 < K M À K R 1 j j, K R 1 j jþ K R 2 j j < K 1 þ K 2 < K M , and q i < W Ã w Ã , can achieve the channel coordination with Pareto improvement. Contract's efficiency is one of the most important factors for contract benchmarking (Cachon, 2003). Contract's efficiency was defined as the ratio of supply chain profit with the contract to the centralized supply chain profit (Cachon & Lariviere, 2005), i.e.
To see how the price sensitivity, retail competition, and extended warranty duration affects the product price, extended warranty price, and contract's efficiency in the centralized solution and decentralized model, we use the numerical illustrations.
Setting parameters as a 1 ¼ 10, a 2 ¼ 9, h ¼ 1, c 0 ¼ 1, b ¼ 1, t ¼ f0:5, 1g, then we can get the relationship between product price and competition as shown in Figure 2, the relationship between the extended price and competition as shown in Figure  3, and the relationship between wholesale contract efficiency and competition, as shown in Figure 4.
From Figure 2, we can find that the product price both in the decentralized system and the centralized system all decrease with retail competition, and the product price in the decentralized system are higher  than that in the centralized system because of double marginalization. Furthermore, the product price in both systems all decrease with the extended warranty duration because of the complementarity of the product and the extended warranty. Figure 3 implies that the extended warranty price in both systems all increases with the warranty duration, and the extended warranty price in the decentralized system is lower than that in the centralized system because of double marginalization. The extended warranty price in the decentralized system increases with the retail competition; however, the extended warranty price in the centralized system is invariant with retail competition. It is reasonable that the manufacturer charges a higher extended warranty price when providing a longer extended warranty duration since the manufacturer has to bear more extended warranty costs under the longer extended warranty duration. From the view of the manufacturer, there are two sources of profits, profit of product sales and profit of extended warranty sales, and the manufacturer must tradeoff between the two profits under specific settings. With the higher extended warranty price, the sales of extended warranty are more profitable than the product sales, and thus the manufacturer would rather lower the wholesale price so as to ensure greater demand for the extended warranty. Figure 4 indicates that the efficiency of the wholesale contract increases with the retail competition; however, it decreases with the extended warranty duration. This is because the competition between two retailers can lead to a lower product price and improve the contract efficiency. Due to the complementarity of the product and the extended warranty, coordination with the complicated contract is necessary for the decentralized supply chain.

The quantity discount contract coordination for symmetric retailers
Setting parameters as a 1 ¼ 10, a 2 ¼ 9, h ¼ 1, c 0 ¼ 1, b ¼ 1, t ¼ 1, then we can get the relationship between the coordination area of discount rate and retail competition, as shown in Figure 5. Figure 5 shows that the feasible coordination area of discount rate decreases with retail competition, which implies that the manufacturer has no motivation to implement the coordination contract when retailer competition is fierce. This result is also in line with Corollary 1.
Setting parameters as a 1 ¼ 10, a 2 ¼ 9, h ¼ 1, c 0 ¼ 1, b ¼ 1, h ¼ 0:5, then we can get the relationship between the coordination area of discount rate and the extended warranty duration, as shown in Figure 6. Figure 6 suggests that the feasible coordination area of discount rate increases with the extended warranty duration. Due to the complementarity of the product and extended warranty, the manufacturer can reduce the product price and can obtain profit from the increasing price of warranty service.
Setting parameters as a 1 ¼ 10, a 2 ¼ 9, h ¼ 1, c 0 ¼ 1, t ¼ 1, h ¼ 0:2, then we can get the relationship between the feasible coordination area of discount rate varies non-monotonously with price sensitiveness, as shown in Figure 7. Especially, the manufacturer will have no positive motivation to implement the discount contract when price sensitiveness is very high, and he can implement market    segmentation to obtain the extra profit through the extended warranty service.
7.3. The combining quantity discount with a two-part tariff for asymmetric retailers We choose different samples in three areas and get the numerical results in Table 2; then, the concrete quantity discount contract structures are as follows.

Conclusions and future research
This paper studies the strategic interaction between the manufacturer and the retailer as well as the interaction between the product sales and the extended warranty sales. The main results of our models are able to qualitatively explain some observed practices and provide some insights into contract design. Specifically, we find: 1. The product price of the decentralized system is higher than the centralized system, whereas the extended warranty price and the extended warranty demand of the decentralized system are all lower than the centralized system, which results from the complementarity of the product and the extended warranty service. 2. The wholesale contract efficiency increases with the retail competition, whereas it decreases with the extended warranty duration. The extended warranty price increases with the extended warranty duration. 3. Under the situation of symmetric retailers, there exists a set of quantity discount contracts in the form of ðW q À wq i Þq i , which can not only achieve the centralized supply chain performance but also the tripartite of supply chain get the Pareto improvement compared with the decentralized supply chain system. Especially, when retail competition approaches infinity, the quantity discount contract to coordinate the channel will degenerate into a simple wholesale contract. In a way, this is the academic explanation of the classical proverb "Rivalry between Mussels and Mussels." 4. Under the situation of asymmetric retailers, the quantity discount contract combined with a two-part tariff in the form of ð ða 1 þa 2 Þ 2 À q i Þq i þ K i , or the quantity discount contract combined with slotting allowance in the form of ð ða 1 þa 2 Þ 2 À q i Þq i À K i j j, can not only achieve the centralized supply chain performance but also the tripartite of supply chain get the Pareto improvement compared with the decentralized supply chain system. The quantity discount contract combined with the form of a two-part tariff, or slotting allowance, is dependent on the market size and other parameters, which are the novel finding different from the symmetric situation.
Our paper differs from the previous studies in two aspects. First, we explore how to design the complicated quantity discount contracts to coordinate the channel with the extended warranty instead of focusing only on who provides extended warranty services or price decisions (Cai et al., 2020;Mai et al., 2017;Zhang et al., 2020). Second, we consider that a quantity discount contract is offered by the manufacturer with an extended warranty for asymmetric retailers. The results are remarkably different from that the symmetric competing retailers can offer simple quantity discount contracts for channel coordination. Although some studies have focused on quantity discount contracts (e.g., Karray & Surti, 2016;Weng, 1995), to the best of our knowledge, there has been no investigation of asymmetric competing retailers. Though abundant conclusions are derived from this paper, the findings are facilitated by the simplified assumption, such as the zero base warranty, the unlimited production capacity, and no inventory cost. Our model also neglects some factors, such as the competition between manufacturers and the retailer's dominance in the supply chain, which may change the supply chain structure and equilibrium. One extension of this work would be to examine and compares the equilibrium when the extended warranties are provided by retailers or a third party. Furthermore, besides the quantity discount contract, other coordination contracts such as return policy and revenue sharing contract may also be applied in the coordination of the supply chain with extended warranties under retail competition.

Disclosure statement
No potential conflict of interest was reported by the authors.