# Contract Strategies in Competing Supply Chains with Risk-Averse Suppliers

Authors: Baixun Li; Yongwu Zhou; Baozhuang Niu
Journal: Mathematical Problems in Engineering
 (2013)

---

## Abstract

This paper investigates the equilibrium contract strategies of two competing supply chains. Each chain is composed of one risk-averse supplier and one risk-neutral retailer. The two suppliers, as Stackelberg leaders, can choose either wholesale price or revenue sharing contract. We compare the outcomes obtained from different scenarios and study the impact of competition density and risk attitudes on the suppliers’ contract choices. We find that it is always optimal for the first-moving supplier to choose revenue sharing contract if the follower chooses wholesale price contract. We also find that the retail price under revenue sharing contract is lower than that under wholesale price contract. Compared to Pan et al. (2010), we find that a threshold policy with respect to the degree of product substitutability holds in a sequential competition environment. There also exists a threshold policy with respect to the degree of risk aversion for the suppliers to choose their individual contract types and the corresponding chain-to-chain contract strategies.


---

## Body

## 1. Introduction

Nowadays, with the development of information technology, many “pure” service companies have arisen, such as Expedia, Alibaba.com, Crip.com, and China Mobile. In these companies, no physical products are sold—what they can provide are intangible products, that is, services. Even some companies, who traditionally produce and sell physical products, are changing to be service providers. One typical example is IBM, who uses its LotusLive Connections to apply a new business model named “Software-as-a-Service (SaaS)”, under which customers need only “pay a subscription fee to use a software over the internet” [1]. To find more customers, these service companies rely on agents/retailers and hence form service supply chains. For example, China Mobile contracts with agents to sell its telecommunication services. As the key supply chain member, China Mobile is very powerful. They determine the service charges, the service contracts, and so on. Retailers just follow China Mobile’s decisions to earn some agent fee.Similar phenomena can be observed in nonservice supply chains. For example, in the petrol retailing industry, Caffarra and Mattei [2] find that UK oil companies hold the right of selecting the organizational structures (and hence the contract parameters such as the wholesale price) in their retail networks. It is suggested that the petrol suppliers would be significantly “rewarded or penalized” according to their contract strategies. For the other industries, such as the electronic industry [3, 4], the television industry [5], and the agriculture industry [6], the determination of contract type and/or parameters are continuously discussed and investigated. This indicates that the competing suppliers, especially those who sell their products through retailers, must be very careful at the contract design stage so as to gain some competitive advantages.In practice, different suppliers may choose different kinds of contracts with retailers. A variety of contractual relationships have been discussed in the literature, for example, wholesale-price contract [7], revenue sharing contract [8], quantity-discount contract [9], and consignment contract [10]. See Cachon [11] for the literature review on supply chain contracts. Among these studies, we note that Pan et al. [8] study the contract choice problems (revenue sharing versus wholesale price) by assuming two competing manufacturers make their decisions simultaneously, which is reasonable but ignores the more common sequential-moving supply chains. Therefore, in this paper, we relax their assumption and examine scenarios where suppliers have imbalanced power, so that suppliers can act as Stackelberg leaders. Such supply chains can be found in many industries, for example, the domestic appliance industry, tourism industry, telecommunication industry, and beverage industry. Moreover, we note that more and more suppliers are concerning the loss minimization of their companies; thus, it is important to investigate the role of risk aversion in the competing supply chains. Tsay [12] points out that the ‘‘risk sensitivity’’ has an important impact on the decisions of the supply chain members. However, few studies have addressed risk aversion in chain-to-chain competition environments. Thus, in this paper, we relax the risk-neutral assumption of Pan et al. [8] to examine scenarios where suppliers are risk-averse, and analyze the effects of the degree of risk aversion on the suppliers’ contract decisions. Readers hence can view this paper as an extension of that of Pan et al. [8] with the consideration of sequential-moving supply chains and risk-averse suppliers.Our analysis leads to the following insights. (I) Revenue sharing contract is the best option for the first-moving supplier if the follower chooses wholesale price contract, regardless of the degree of competition density and risk aversion. However, if the follower chooses revenue sharing contract, it is possible for the leader to choose wholesale price contract. (II) There exist threshold policies with respect to the degree of competition density and risk aversion for the suppliers to choose contract strategies. The retail price under revenue sharing contract is lower than that under wholesale price contract. Thus, if the follower could not afford the loss due to low retail price, he/she has incentives to switch to wholesale price contract.The remainder of this paper is organized as follows. Section2 briefly reviews related literature. Section 3 presents our basic model. Section 4 derives close form equilibrium solutions for different contract structures, including the WW, WR, RW, and RR cases. Section 5 compares the equilibrium profits under different cases and identifies the conditions under which the suppliers should choose wholesale price contract or revenue sharing contract. Finally, we summarize the paper and suggest research extensions in Section 6.


## 2. Literature Review

This paper is related to the work on chain-to-chain competition. McGuire and Staelin [13] investigate the effect of product substitutability on the suppliers’ decision of integrating with an outside retailer or not. Trivedi [14] looks into two manufacturers and two common retailers and analyzes the effect of product and channel substitution. Boyaci and Gallego [15] consider a market comprising two supply chains and explore the value and limitation of channel coordination. Wu et al. [16, 17] extend the seminal work of McGuire and Staelin [13] by considering demand uncertainty. They build competitive newsvendor models to investigate the channel design problems. Allon and Federgruen [18] build multiple stage models to study competition in the service industries. Shou et al. [19] study the chain-to-chain competition problems with the consideration of supply uncertainty. Chiu et al. [20] consider price-dependent demand and study methods to avoid price war among multiple retailers. Xie et al. [21] consider the selection of supply chain structures and quality improvement strategies of two supply chains. Li et al. [22] investigate the channel choice problem in two competing supply chains with asymmetric cost information. Different from the aforementioned literature, we study the contract strategies with the consideration of sequential-moving supply chains and risk-averse suppliers.The literature on supply chain contract is also related. Cachon [11] gives an excellent review on supply chain contract and coordination. This work has extensively reviewed the one-to-one or one-to-N supply chain structures (see, e.g, [23–25]). One conclusion is that the wholesale-price contract is Pareto inferior for the supply chain, which leads to the double marginalization effect. To eliminate the double marginalization effect, manufacturers need to design sophisticated contracts, such as quantity flexibility contract, quantity discount contract, and revenue sharing contract. However, are sophisticated contracts always better for all the supply chains?Recently, some papers find that the degree of competition has a significant impact on the contract choice decision. Ha and Tong [26] investigate contracting and information sharing problems in two competing supply chains. They highlight that contract type is a key driver of the value of information sharing. Differing from Ha and Tong [26], Ha et al. [27] assume imperfect demand signals, production diseconomy, and both Cournot and Bertrand competition with differentiated products. Cachon and Kök [9] study the problem of contract choice in the supply chain structure comprising two competing manufacturers and a single retailer. They show that sophisticated contracts (e.g., quantity discount, two-part tariff contract) may make manufacturers worse off and the retailer substantially better off. Pan et al. [8] discuss different contract strategies (wholesale price contract versus revenue sharing contract) under both manufacturer dominated and retailer dominated scenarios in a supply chain with either two-one structure or one-two structure. Zhao and Shi [10] consider two supply chains, each with multiple upstream suppliers producing complementary products and selling to a single buyer. They focus on the channel structure (integration versus decentralization) issues and the contract strategies (wholesale price contract versus revenue sharing contract) issues. Ai et al. [28] demonstrate the impact of demand uncertainty and retailers’ forecasting precision on contract decisions of the manufacturers in two competing supply chains. Feng and Lu [7] analyze contract strategies in a supply chain comprising two competing manufacturers and two competing retailers. They derive the outcomes of Stackelberg games via generalized Nash bargaining scheme. Li et al. [29] explore the effect of supply chain structure and competition at both manufacturer and retailer levels on contract choices of two competing supply chains.In the literature above, most studies assume that manufacturers are risk-neutral. When facing random demand, however, the manufacturers will be more concerned with the risk associated with demand uncertainties. The inclusion of risk into decision making has gained increasing interest in supply chain studies (e.g., [30–34]). Different methods have been used to study risk aversion; for example, Hsieh and Lu [35] characterize each retailer’s risk-embedded objects and compare them with conditional value-at-risk. They study the manufacturer’s return policy and the retailers’ decisions. Caliskan-Demirag et al. [36] model risk aversion by adopting the conditional-value-at-risk (CVaR) decision criterion, and analyze the manufacturer’s rebate amount decisions and the retailer’s joint inventory and pricing decisions.Some other papers develop the risk aversion model in a mean-variance framework. Chen and Seshadri [37] consider a single period model in which multiple risk-averse retailers purchase a single product from a common distributor with a mean-variance utility approach. Choi et al. [38] carry out a mean-variance analysis of the newsvendor problems. Choi and Chow [39] study quick response program using a mean-variance approach. Xiao and Choi [40] focus on channel structure strategies in a two-echelon system consisting of two manufacturers and two retailers, where all players are risk-averse. Wei and Choi [32] explore the use of wholesale price contract and profit sharing contract to coordinate supply chains under the mean-variance decision framework. Chiu et al. [41] carry out a mean-variance analysis of supply chains under target sales rebate contracts. Chiu et al. [42] explore the performance of sales rebate contract in fashion supply chains under a mean-variance framework. Ma et al. [43] examine the impact of bargaining powers and the retailer’s risk attitudes using a CVaR framework. Li et al. [44] conduct a mean variance analysis of a fast fashion supply chain with return policy. Ma et al. [45] study the inventory control problem for a loss-averse retailer with financial constraint in a finite periodic review system. Shen et al. [46] explore how markdown money policy performs in a two-stage TC/fashion supply chain with an upstream risk-averse manufacturer and a downstream risk-neutral retailer under the mean-variance framework. See Chiu and Choi [47] for an excellent review of this line of literature.Our paper is most closely related to Cachon and Kök [9], Pan et al. [8], Feng and Lu [7], and Li et al. [29]. We extend their models in two dimensions. First, we examine scenarios where manufacturers have imbalanced power, so that manufacturers can move first. Second, we examine scenarios where manufacturers are risk-averse and investigate the impact of the degree of risk aversion on the contract choice decisions. Referring to Xiao and Choi [40], we model the risk aversion with mean-variance utility functions in this paper.


## 3. Model Settings

Consider a market in which two supply chains (denoted by SC1 and SC2) compete with each other. Each supply chain is composed of a single supplier (denoted by Mi, i=1,2) and a single exclusive retailer (denoted by Ri, i=1,2). Without loss of generalization, we assume SC1 is the market leader who produces and launches a product first. This is a common phenomenon in practice, for example, China Mobile launches its M-Zone service earlier than China Unicom, whose competing product is Up-Power (admittedly, it is possible to investigate the timing decisions of the two supply chains, named as endogenous leadership game; see Wang et al. [3, 4, 48] for further information. However, here our focus is the contract type choice and the impact of risk attitudes we leave the timing decisions for future research). We assume the two chains compete strictly in retail price, pi, which directly determines their respective customer demand, Di~. Referring to McGuire and Staelin [13], we assume the following linear demand function:
(1)Di~=Ai~-pi+θp3-i,
where i=1,2. Ai~ denotes the stochastic market potential for product i, with the mean Ai¯>0 and variance σi2. The parameter θ∈(0,1) represents the degree of product substitutability, which can be viewed as a measurement of demand competition density [26]. When θ→0, there is no cross market influence, implying that the products are independent. When θ→1, there is a perfect product substitution.Towards the contract types adopted in each supply chain, we consider wholesale price contract (denoted by W) and revenue sharing contract (denoted by R). The reasons for choosing these two contract types are as follows. First, both wholesale price contract and revenue sharing contract are widely studied and used in theory and practice; we can thus compare our results directly with previous literature such as Pan et al. [8] and Zhao and Shi [10]. Second, these contract types represent the “simple” contract and the “sophisticated” contract, respectively.Under a wholesale price contract,wi is the wholesale price of product i, pi is the retail price, and ui is the unit profit margin of retailer i, where ui=pi-wi. Under a revenue sharing contract, the retailer keeps a portion of channel revenue (ϕipi), and (1-ϕi)pi is the fraction the supplier earns. Moreover, as Pan et al. [8] have pointed out, the supplier who chooses a revenue-sharing contract should make sure that the retailer’s profit for each unit of product sold is not less than the marginal profit under a wholesale price contract, that is, ϕipi-wi≥ui. As the leader of a supply chain, service providers are authorized to decide whether wholesale price contract or revenue sharing contract should be offered to the exclusive retailer. Similar assumption has been made by Pan et al. [8]. Combing the suppliers’ choices, in the two supply chains, there will be four possible scenarios: WW, WR, RW, and RR, where WW denotes that two supply chains are restricted to choose the wholesale price contract; WR means that supplier 1 (M1) chooses the wholesale-price contract, while supplier 2 (M2) chooses the revenue sharing contract. RW and RR are similar. Our model includes a two-level game. On the one hand, there is a Stackelberg game between SC1 and SC2. The market leader SC1 determines his/her contract strategies to maximize its own profit, anticipating the follower SC2’s possible actions. On the other hand, the individual members of each supply chain maximize their own profits, and there is a Stackelberg relationship between the supplier and the retailer.In particular, similar to Xiao and Choi [40] and Choi et al. [49], we model the risk attitudes of the suppliers with mean-variance utility function as follows:
(2)UMil(πMil)=E(πMil)-λiVar(πMil),
where UMil(πMil) denotes the utility function of supplier i and E(πMil) and Var(πMil) represent the mean and the variance of supplier i’s expected profit, respectively.  l=WW, WR, RW, and RR and the parameter λi≥0 denotes the constant absolute risk aversion of supplier i. The larger λi is, the more conservative supplier i’s behavior will be.


## 4. Equilibrium Solutions

In this section, we derive the close form solutions for different contract structures, including WW, WR, RW, and RR. We use backward induction to find the equilibriums for all of the participants.


### 4.1. Symmetric Subgame 1: WW

In the WW case, both M1 and M2 are restricted to choose wholesale price contracts. We consider SC2’s decisions first. Given wholesale price w2WW, R2 determines u2WW (note that uiWW=piWW-wiWW, so Ri’s decision of uiWW is equivalent to the decision of retail price piWW) by maximizing his/her expected profit:
(3)maxu2WW≥0πR2WW=u2WW[A2¯-(u2WW+w2WW)+θ(u1WW+w1WW)].
It can be shown that πR2WW is concave with respect to u2WW; thus, the best response u2WW can be written as
(4)u2*WW(w2WW,u1WW,w1WW)=A2¯-w2WW+θ(u1WW+w1WW)2.
Anticipating the R2’s actions, M2 determines wholesale price to maximize his/her utility:
(5)maxw2WW≥0UM2WW(πM2WW)=(w2WW-c2)×D2(u2*WW)-12λ2σ22(w2WW-c2)2,
where ci denotes the manufacturing cost of product i, i=1,2. The first-order condition of (5) leads to the best response wholesale price:
(6)w2*WW(u1WW,w1WW)=c2+A2¯-c2+θ(u1WW+w1WW)2+2λ2σ22.We now turn to SC1’s decisions. Anticipating SC2’s actions, M1 and R1 of SC1 determine their best responses sequentially. Using similar event sequence, we derive the equilibriums with respect to u1WW and w1WW. For the convenience, let Δi=1+λiσi2, i=1,2. We have
(7)u1*WW(w1WW)=4Δ2A1¯+2θΔ2A2¯+θ(2Δ2c2-c2+A2¯)4Δ2θ2+2θ2-8Δ2+(θ2+2Δ2θ2-4Δ2)w1WW4Δ2θ2+2θ2-8Δ2,w1*WW=2Δ2(2c1+2A1¯+4c1λ1σ12-c1θ2+θA2¯+θc2)8Δ1Δ2-2θ2(2Δ2+1)+θ(A-2-c2-c1θ)8Δ1Δ2-2θ2(2Δ2+1).We use superscriptWW to represent the optimal solutions in the WW case. Substituting w1*WW back to u1*WW(w1WW), w2*WW(u1WW,w1WW), u2*WW(w2WW,u1WW,w1WW), we get all the equilibriums u1*WW, w2*WW, u2*WW and the corresponding production quantities and profits.


### 4.2. Hybrid Subgame 1: WR

The WR case represents that the M1 chooses wholesale price contract while the M2 chooses revenue sharing contract. Similar to the WW case, we first analyze SC2’s decisions. As pointed out by Pan et al. [8], under a revenue sharing contract, the supplier M2 should make sure that ϕ2p2WR-w2WR≥ν2, where ν2=u2*WW; otherwise, the retailer will prefer a wholesale price contract to a revenue sharing contract. As for M2, he/she expects that the retailer shares the revenue as small as possible such that ϕ2WRp2WR-w2WR=ν2. Therefore, given ϕ2WR, the decision of p2WR and w2WR is equivalent. The profit maximization problem of M2 is
(8)maxp2WR≥0UM2WR(πM2WR)=(p2WR-c2-ν2)[A2¯-p2WR+θ(w1WR+u1WR)]-12λ2σ22(p2WR-c2-ν2)2.The first-order condition of (8) leads to the best response p2*:
(9)p2*WR(u1WR,w1WR)=c2+ν2+A2¯-c2-ν2+θ(u1WR+w1WR)2+λ2σ22.Then, M1 and R1 of SC1 determine their best response conditional on the response of SC2, p2*WR(u1WR,w1WR). It can be shown that
(10)u1*WR(w1WR)=(1+Δ2)(A1¯-w1WR)2[(1+Δ2)-θ2]+θ(A-2+c2Δ2+ν2Δ2+θw1WR)2[(1+Δ2)-θ2],w1*WR=θ[A2¯-c1θ+Δ2(c2+ν2)]2[Δ1(1+Δ2)-θ2]+(1+Δ2)(A1¯-c1+2c1Δ1)2[Δ1(1+Δ2)-θ2].We use superscriptWR to represent the optimal solutions in the WR case. Substituting w1*WR into u1*WR(w1WR) and p2*WR(u1WR,w1WR) yields all the equilibriums in the WR case.


### 4.3. Hybrid Subgame 2: RW

The RW case represents that M1 chooses revenue sharing contract while M2 chooses wholesale price contract. The detail of solution process is similar to that in WR case. The best responses u2RW, w2RW, and p1RWare given by
(11)u2*RW(w2RW,p1RW)=A2¯-w2RW+θp1RW2w2*RW(p1RW)=c2+A2¯-c2+θp1RW2+2λ2σ22,p1*RW=2Δ2(2A1¯+θA2¯)+θ(2c2Δ2-c2+A2¯)2[2Δ1Δ2-θ2(1+2Δ2)+2Δ2],-(c1+ν1)(2θ2Δ2+θ2-4Δ2+4Δ2λ1σ12)2[2Δ1Δ2-θ2(1+2Δ2)+2Δ2],
where ν1=u1*WW. Substituting p1*RW into w2*RW(p1RW) and u2*RW(w2RW,p1RW) yields all the equilibriums in the RW case.


### 4.4. Symmetric Subgame 2: RR

The RR case represents that both M1 and M2 choose revenue sharing contract. Note that M1 and M2 have to guarantee R1 and R2 to obtain at least the same profits as those they can obtain under a wholesale price contract. We derive the best response retail price of SC2 first as
(12)p2*RR(p1RR)=c2+ν2+A2¯-c2-ν2+θp1RR2+2λ2σ22.Anticipating the SC2’s actions, SC1 determines his best response p1RR as
(13)p1*RR=2Δ2A1¯-(c1+ν1)(θ2-2Δ2+4Δ2λ1σ12)2(2Δ2-θ2+Δ2λ1σ12)+θ[(2Δ2-1)(c2+ν2)+A2¯]2(2Δ2-θ2+Δ2λ1σ12).
Substituting p1*RR back yields all the equilibriums in the RR case.


## 4.1. Symmetric Subgame 1: WW

In the WW case, both M1 and M2 are restricted to choose wholesale price contracts. We consider SC2’s decisions first. Given wholesale price w2WW, R2 determines u2WW (note that uiWW=piWW-wiWW, so Ri’s decision of uiWW is equivalent to the decision of retail price piWW) by maximizing his/her expected profit:
(3)maxu2WW≥0πR2WW=u2WW[A2¯-(u2WW+w2WW)+θ(u1WW+w1WW)].
It can be shown that πR2WW is concave with respect to u2WW; thus, the best response u2WW can be written as
(4)u2*WW(w2WW,u1WW,w1WW)=A2¯-w2WW+θ(u1WW+w1WW)2.
Anticipating the R2’s actions, M2 determines wholesale price to maximize his/her utility:
(5)maxw2WW≥0UM2WW(πM2WW)=(w2WW-c2)×D2(u2*WW)-12λ2σ22(w2WW-c2)2,
where ci denotes the manufacturing cost of product i, i=1,2. The first-order condition of (5) leads to the best response wholesale price:
(6)w2*WW(u1WW,w1WW)=c2+A2¯-c2+θ(u1WW+w1WW)2+2λ2σ22.We now turn to SC1’s decisions. Anticipating SC2’s actions, M1 and R1 of SC1 determine their best responses sequentially. Using similar event sequence, we derive the equilibriums with respect to u1WW and w1WW. For the convenience, let Δi=1+λiσi2, i=1,2. We have
(7)u1*WW(w1WW)=4Δ2A1¯+2θΔ2A2¯+θ(2Δ2c2-c2+A2¯)4Δ2θ2+2θ2-8Δ2+(θ2+2Δ2θ2-4Δ2)w1WW4Δ2θ2+2θ2-8Δ2,w1*WW=2Δ2(2c1+2A1¯+4c1λ1σ12-c1θ2+θA2¯+θc2)8Δ1Δ2-2θ2(2Δ2+1)+θ(A-2-c2-c1θ)8Δ1Δ2-2θ2(2Δ2+1).We use superscriptWW to represent the optimal solutions in the WW case. Substituting w1*WW back to u1*WW(w1WW), w2*WW(u1WW,w1WW), u2*WW(w2WW,u1WW,w1WW), we get all the equilibriums u1*WW, w2*WW, u2*WW and the corresponding production quantities and profits.


## 4.2. Hybrid Subgame 1: WR

The WR case represents that the M1 chooses wholesale price contract while the M2 chooses revenue sharing contract. Similar to the WW case, we first analyze SC2’s decisions. As pointed out by Pan et al. [8], under a revenue sharing contract, the supplier M2 should make sure that ϕ2p2WR-w2WR≥ν2, where ν2=u2*WW; otherwise, the retailer will prefer a wholesale price contract to a revenue sharing contract. As for M2, he/she expects that the retailer shares the revenue as small as possible such that ϕ2WRp2WR-w2WR=ν2. Therefore, given ϕ2WR, the decision of p2WR and w2WR is equivalent. The profit maximization problem of M2 is
(8)maxp2WR≥0UM2WR(πM2WR)=(p2WR-c2-ν2)[A2¯-p2WR+θ(w1WR+u1WR)]-12λ2σ22(p2WR-c2-ν2)2.The first-order condition of (8) leads to the best response p2*:
(9)p2*WR(u1WR,w1WR)=c2+ν2+A2¯-c2-ν2+θ(u1WR+w1WR)2+λ2σ22.Then, M1 and R1 of SC1 determine their best response conditional on the response of SC2, p2*WR(u1WR,w1WR). It can be shown that
(10)u1*WR(w1WR)=(1+Δ2)(A1¯-w1WR)2[(1+Δ2)-θ2]+θ(A-2+c2Δ2+ν2Δ2+θw1WR)2[(1+Δ2)-θ2],w1*WR=θ[A2¯-c1θ+Δ2(c2+ν2)]2[Δ1(1+Δ2)-θ2]+(1+Δ2)(A1¯-c1+2c1Δ1)2[Δ1(1+Δ2)-θ2].We use superscriptWR to represent the optimal solutions in the WR case. Substituting w1*WR into u1*WR(w1WR) and p2*WR(u1WR,w1WR) yields all the equilibriums in the WR case.


## 4.3. Hybrid Subgame 2: RW

The RW case represents that M1 chooses revenue sharing contract while M2 chooses wholesale price contract. The detail of solution process is similar to that in WR case. The best responses u2RW, w2RW, and p1RWare given by
(11)u2*RW(w2RW,p1RW)=A2¯-w2RW+θp1RW2w2*RW(p1RW)=c2+A2¯-c2+θp1RW2+2λ2σ22,p1*RW=2Δ2(2A1¯+θA2¯)+θ(2c2Δ2-c2+A2¯)2[2Δ1Δ2-θ2(1+2Δ2)+2Δ2],-(c1+ν1)(2θ2Δ2+θ2-4Δ2+4Δ2λ1σ12)2[2Δ1Δ2-θ2(1+2Δ2)+2Δ2],
where ν1=u1*WW. Substituting p1*RW into w2*RW(p1RW) and u2*RW(w2RW,p1RW) yields all the equilibriums in the RW case.


## 4.4. Symmetric Subgame 2: RR

The RR case represents that both M1 and M2 choose revenue sharing contract. Note that M1 and M2 have to guarantee R1 and R2 to obtain at least the same profits as those they can obtain under a wholesale price contract. We derive the best response retail price of SC2 first as
(12)p2*RR(p1RR)=c2+ν2+A2¯-c2-ν2+θp1RR2+2λ2σ22.Anticipating the SC2’s actions, SC1 determines his best response p1RR as
(13)p1*RR=2Δ2A1¯-(c1+ν1)(θ2-2Δ2+4Δ2λ1σ12)2(2Δ2-θ2+Δ2λ1σ12)+θ[(2Δ2-1)(c2+ν2)+A2¯]2(2Δ2-θ2+Δ2λ1σ12).
Substituting p1*RR back yields all the equilibriums in the RR case.


## 5. Analysis of Contract Strategies

In this section, we compare the suppliers’ equilibrium profits under different cases and identify the conditions under which they choose wholesale price contract or revenue sharing contract. We also would like to find the equilibrium contract structure. According to previous literatures such as Choi et al. [38], Xiao and Choi [40], and Pan et al. [8], the degree of product substitutability and risk attitudes are the critical factors; therefore, we assume a symmetric scenario in the sense that A1¯=A2¯=A¯, σ1=σ2=σ, c1=c2=c. Then their impacts are ruled out, leaving M1 and M2 differing from each other via their risk attitudes: λ1 and λ2. We find that under this assumption, all of the optimal UMil, l=WW,WR,RW,RR, i=1,2, can be expected as a function of a common factor (A--c+cθ)2. Then, the comparison of the utilities depends only on three parameters: the degree of product substitutability (θ), the degree of risk aversion (λi), and the variance of stochastic market base (σ). Moreover, the parameters λi and σ always arise together with the form of λiσ2, therefore, to focus on the impact of risk aversion, we first set σ as a given positive constant. Next we investigate how the degree of product substitutability and the risk attitudes of the suppliers influence the supply chain contract decisions and compare our results with the previous literature. We analyze symmetric and asymmetric risk attitudes of the two suppliers.


### 5.1. Impact of Product Substitutability (Risk-Neutral Suppliers)

To obtain analytical comparison results with respect to the degree of product substitutability, we assumeλ1=λ2=λ=0, in other words, both suppliers are risk-neutral. We analyze the incentive of a supplier to switch from wholesale price contract (revenue sharing contract) to revenue sharing contract (wholesale price contract). Lemmas 1 and 2 describe M1’s (M2’s) preference given M2’s (M1’s) choice.Lemma 1.
From M1’s perspective, (i) if M2 chooses wholesale price contract, then it is optimal for M1 to choose revenue sharing contract. θ has no impact. (ii) If M2 chooses revenue sharing contract, then M1 is better to choose revenue sharing contract when 0<θ≤0.69, otherwise, wholesale price contract.Lemma 2.
From M2’s perspective, (i) if M1 chooses wholesale price contract, then M2 is better off with revenue sharing contract under 0<θ≤0.55, otherwise, wholesale price contract. (ii) If M1 chooses revenue sharing contract, then M2 is better off with revenue sharing contract under 0<θ≤0.51, otherwise, wholesale price contract.Based on Lemmas1 and 2, Table 1 summarizes the suppliers’ incentives to switch from one structure to another one.Table 1
Stability analysis of risk-neutral suppliers’ perspectives.
Original structure
M
1
M
2
New structure
Change in profits
New structure
Change in profits
WW
RW
0
<
θ
<
1, ↑*
WR
0
<
θ
≤
0.55, ↑
0.55
<
θ
<
1, ↓
RW
WW
0
<
θ
<
1, ↓
RR
0
<
θ
≤
0.51, ↑
0.51
<
θ
<
1, ↓
WR
RR
0
<
θ
≤
0.69, ↑
WW
0
<
θ
≤
0.55, ↓
0.69
<
θ
<
1, ↓
0.55
<
θ
<
1, ↑
RR
WR
0
<
θ
≤
0.69, ↓
RW
0
<
θ
≤
0.51, ↓
0.69
<
θ
<
1, ↑
0.51
<
θ
<
1, ↑
*When the utility of new structure is more than the original structure, we denote it by “↑”, otherwise, “↓”.This leads to the equilibrium contract strategies summarized in Theorem3.Theorem 3.
Assuming risk-neutral suppliers, the equilibrium contract strategies for the two competing supply chains is RR ifθ is smaller than θ¯^, and RW otherwise.Thus, there exists a unique threshold value ofθ¯^ which guides M2’s choice of contract type. When competition is tense, that is, θ is large, it is better for M2 to adopt a different contract type. As the degree of product substitutability decreases, competition becomes not so tense, then it is optimal for the market follower to copy the leader’s strategy: adopting a revenue sharing contract. Note that, with simultaneous competition, Pan et al. [8] show that it is optimal for the suppliers to choose revenue-sharing contract when there is no tense competition. Here, we verify this finding in a sequential competition environment. The underlying reason lies in their different preferences towards the “low-price-high-volume” strategy. To further explain our findings, we compare the retail prices in different scenarios and have the following lemma.Lemma 4.
Assuming risk-neutral suppliers, the following orders always hold:(14)p1*RR-p1*WW<0;p2*RR-p2*WW<0;p2*RR<p1*RR.As we show in Lemma4, the retail price under revenue sharing contract is lower than that under wholesale price contract. Market leader has to decide a low retail price to snatch a large market share, while market follower has the right of cutting down the leader’s retail price and hence enjoys a second-mover advantage in the price competition (p2*RR<p1*RR). This behavior will not occur when there is no information interaction under simultaneous competition, as Pan et al. [8] have studied. On the other hand, facing tense competition, the follower’s profits with “low-price-high-volume” strategy may not compensate its loss due to low retail price. Thus, the follower has to switch to the wholesale price contract so as to enjoy a higher profit margin. This is consistent with Pan et al. [8].


### 5.2. Impact of Product Substitutability and Risk Attitudes (Identical Risk-Averse Suppliers)

We now turn to the caseλ1=λ2=λ>0, that is, both suppliers are risk-averse, but they are identical. With this assumption, suppliers’ choices are influenced by the degree of product substitutability and risk aversion. We find that given M2’s choice of wholesale price contract, M1 always prefers revenue sharing contract. This is consistent with our previous finding shown in Lemma 1. However, it is not easy to analytically find M1’s preference when M2 chooses revenue sharing contract. We conduct numerical studies and observe similar curves; see Figure 1 for illustration. The parameters are A-=1, σ=1, c=0.5, λ∈(0,1), and θ∈(0,1).Figure 1
M1’s preference given M2 choosing revenue sharing contract (λ>0).The rectangular area{(θ,λ)∣0<θ<1,0<λ<1} is divided into two distinct regions by the curve fa=UM1RR-UM1WR|λ>0=0, where “W” denotes that M1 prefers wholesale price contract and “R” denotes that M1 prefers revenue sharing contract. From Figure 1 we can obtain the following observations. (i) If λ=0, that is, both suppliers are risk-neutral, given M2 choosing revenue sharing contract, M1 prefers revenue sharing contract when 0<θ≤0.69, otherwise, wholesale price contract. This is consistent with Lemma 1. (ii) As the degree of risk aversion increases (e.g., λ>0.21), given M2 choosing revenue sharing contract, the wholesale price contract is attractive for M1 no matter what the value of θ is. This observation can be explained as follows. When M1 becomes more and more conservative (λ becomes larger and larger), it can be observed that p1*RR is decreasing (see Figure 2). Thus, to avoid the retail price dropping too much, M1 tends to replace the revenue sharing contract by the wholesale price contract, owing to the fact that the retail price under the revenue sharing contract is always lower than that under the wholesale price contract (see Lemma 4 and Figure 3).Figure 2
Impact ofλ on p1*RR.Comparison of the retail prices withλ1=λ2=λ>0.
(a)
(b)We now also illustrate the follower M2’s preference given M1’s choice, by assuming the same numerical parameters.Similar observations can be found in Figures4(a) and 4(b). “W” denotes that M2 prefers wholesale price contract, and “R” denotes that M2 prefers revenue sharing contract. We can obtain the following observations. (i) In Figure 4(a), if λ=0, given M1 choosing wholesale price contract, M2 prefers revenue sharing contract, when 0<θ≤0.55, otherwise, wholesale price contract. In Figure 4(b), if λ=0, given M1 choosing revenue sharing contract, M2 prefers revenue sharing contract, when 0<θ≤0.51, otherwise, wholesale price contract. This is consistent with Lemma 2. (ii) As the degree of risk aversion increases (e.g., λ>0.19), regardless of M1 choosing wholesale price contract or revenue sharing contract, it is optimal for M2 to choose the wholesale price contract. A possible reason is that as M2 becomes more and more conservative, he/she tends to avoid the retail price dropping too much (p2*RR is decreasing in λ). We obtain equilibrium contract strategies illustrated by Figure 5.(a) M2’s preference given M1’s choice of wholesale price contract (λ>0). (b) M2’s preference given M1’s choice of revenue sharing contract (λ>0).
(a)
(b)Figure 5
Equilibrium contract strategies of suppliers (λ>0).In Figure5, “RW” denotes that M1 chooses revenue sharing contract and M2 chooses wholesale price contract. “RR” denotes that both M1 and M2 choose revenue sharing contract. We first verify the findings summarized in Theorem 3, that is, “RR” is the equilibrium contract strategy when 0.51<θ<1, and “RW” is the equilibrium contract strategy when 0.51<θ<1. Then, we find that if the degree of risk aversion and product substitutability are both low, “RR” sustains as the equilibrium contract strategy. Recall that when the competition intensity is low, suppliers tend to keep a relatively low retail price to stimulate demand by adopting revenue sharing contract. However, when suppliers are very conservative, as we have observed, the retail price will become too low. Consequently, M2 may switch to the wholesale price contract, which guarantees a larger profit margin. This differentiates our work from that of Pan et al. [8], in which the impact of risk attitude is not studied.


### 5.3. Verification with Asymmetric Risk Attitudes (Heterogeneous Risk-Averse Suppliers)

In this subsection, we assume that the risk aversion between two suppliers is asymmetric, that is,λ1≠λ2. Interestingly, given M2’s choice of wholesale price contract, it can be shown that M1 always prefers revenue sharing contract. Combining our previous findings with symmetric risk attitudes, this phenomenon is rather robust.Consider the following:(15)UM1RW-UM1WW|λ1≠λ2=(A--c+cθ)2(4+4λ2σ2-2λ2θ2σ2-3θ2)2×(2θλ2σ2+3θ+4+4λ2σ2)2×(256Γ(1+λ2σ2)(Γ+2λ1λ2σ4+2λ1σ2)2)-1>0,
where Γ=4-2λ2θ2σ2+2λ1λ2σ4-3θ2+4λ2σ2+2λ1σ2.Clearly,UM1RW-UM1WW|λ1≠λ2>0 always holds regardless of the value of λ1, λ2, and θ.Now, we analyze M1’s preference given M2’s choice of revenue sharing contract. The parameters are A-=1, σ=1, c=0.5, θ∈(0,1), λ2∈(0,20), and λ1=0,0.1.As observed from Figures6(a) and 6(b), when λ1 increases, it becomes more likely for M1 to choose wholesale price contract. A possible reason is that M1 tend to avoid sharply lowered retail price. Another observation is when λ2 is large, it is better for M1 to choose revenue sharing contract. The reason lies in M2’s possible choice of wholesale price contract when he/she is very conservative. It has been shown that M1’s optimal strategy is revenue sharing, given M2’s choice wholesale price contract.(a) M1’s preference given M2’s choice of revenue sharing contract (λ1=0). (b) M1’s preference given M2’s choice of revenue sharing contract (λ1=0.1).
(a)
(b)We now also illustrate M2’s preference given M1’s choice under asymmetric risk attitudes. First of all, assuming M1 chooses wholesale price contract, we use the following parameters: A-=1, σ=1,  c=0.5, θ∈(0,1), λ1∈(0,20), and λ2=0,0.1. As seen from Figures 7(a) and 7(b), the outcomes can be classified into two distinct regions by the critical curves fg=0, fh=0, fi=0, respectively. Again, with larger λ2, M2 tends to choose wholesale price contract. Assuming M1 chooses revenue sharing contract, the observation is similar.(a) M2’s preference given M1’s choice of wholesale price contract (λ2=0). (b) M2’s preference given M1’s choice of wholesale price contract under (λ2=0.1).
(a)
(b)


### 5.4. Verification with Stochastic Market Variance

In this sub-section, we investigate the role that stochastic market variance (σ) plays in the suppliers’ contract choices. First, we examine M1’s preference. Given M2’s choice of wholesale price contract, it can be shown that M1 always prefers revenue sharing contract. Second, we illustrate M1’s preference given M2’s choice of revenue sharing contract and M2’s preference given M1’s choice of wholesale price contract or revenue sharing contract. The parameters are A-=1, c=0.5, θ∈(0,1), σ∈(0,2), and λ=0.1,0.5. As seen from Figure 8(a), the rectangular area {(θ,σ)∣0≤θ<1,0≤σ<2} is divided into two parts by the curves fj(θ,σ)=0, fk(θ,σ)=0, fl(θ,σ)=0. In the upper right part (i.e., region I) of the area, all functions fj(θ,σ), fk(θ,σ), and fl(θ,σ) are less than zero. In contrast, in the lower left part (i.e., region II) of the area, all functions fj(θ,σ), fk(θ,σ), and fl(θ,σ) are greater than zero. These observations imply that if the competition density is small and the market variance is also small, revenue sharing contract will be the best option for the two suppliers. This indicates a threshold policy with respect to σ for the suppliers to select the contract type. Figure 8(b) shows similar curves. Possible explanation is as follows. Recall that the comparison of the suppliers’ utilities depends only on θ, λi, and σ, and the parameters λi and σ always arise together with the form of λiσ2. Thus, keeping λi as a constant, the impact of σ is similar to that of λi shown previously (threshold type).(a) M1’s and M2’s preference given (λ=0.1). (b) M1’s and M2’s preference given (λ=0.5).
(a)
(b)


## 5.1. Impact of Product Substitutability (Risk-Neutral Suppliers)

To obtain analytical comparison results with respect to the degree of product substitutability, we assumeλ1=λ2=λ=0, in other words, both suppliers are risk-neutral. We analyze the incentive of a supplier to switch from wholesale price contract (revenue sharing contract) to revenue sharing contract (wholesale price contract). Lemmas 1 and 2 describe M1’s (M2’s) preference given M2’s (M1’s) choice.Lemma 1.
From M1’s perspective, (i) if M2 chooses wholesale price contract, then it is optimal for M1 to choose revenue sharing contract. θ has no impact. (ii) If M2 chooses revenue sharing contract, then M1 is better to choose revenue sharing contract when 0<θ≤0.69, otherwise, wholesale price contract.Lemma 2.
From M2’s perspective, (i) if M1 chooses wholesale price contract, then M2 is better off with revenue sharing contract under 0<θ≤0.55, otherwise, wholesale price contract. (ii) If M1 chooses revenue sharing contract, then M2 is better off with revenue sharing contract under 0<θ≤0.51, otherwise, wholesale price contract.Based on Lemmas1 and 2, Table 1 summarizes the suppliers’ incentives to switch from one structure to another one.Table 1
Stability analysis of risk-neutral suppliers’ perspectives.
Original structure
M
1
M
2
New structure
Change in profits
New structure
Change in profits
WW
RW
0
<
θ
<
1, ↑*
WR
0
<
θ
≤
0.55, ↑
0.55
<
θ
<
1, ↓
RW
WW
0
<
θ
<
1, ↓
RR
0
<
θ
≤
0.51, ↑
0.51
<
θ
<
1, ↓
WR
RR
0
<
θ
≤
0.69, ↑
WW
0
<
θ
≤
0.55, ↓
0.69
<
θ
<
1, ↓
0.55
<
θ
<
1, ↑
RR
WR
0
<
θ
≤
0.69, ↓
RW
0
<
θ
≤
0.51, ↓
0.69
<
θ
<
1, ↑
0.51
<
θ
<
1, ↑
*When the utility of new structure is more than the original structure, we denote it by “↑”, otherwise, “↓”.This leads to the equilibrium contract strategies summarized in Theorem3.Theorem 3.
Assuming risk-neutral suppliers, the equilibrium contract strategies for the two competing supply chains is RR ifθ is smaller than θ¯^, and RW otherwise.Thus, there exists a unique threshold value ofθ¯^ which guides M2’s choice of contract type. When competition is tense, that is, θ is large, it is better for M2 to adopt a different contract type. As the degree of product substitutability decreases, competition becomes not so tense, then it is optimal for the market follower to copy the leader’s strategy: adopting a revenue sharing contract. Note that, with simultaneous competition, Pan et al. [8] show that it is optimal for the suppliers to choose revenue-sharing contract when there is no tense competition. Here, we verify this finding in a sequential competition environment. The underlying reason lies in their different preferences towards the “low-price-high-volume” strategy. To further explain our findings, we compare the retail prices in different scenarios and have the following lemma.Lemma 4.
Assuming risk-neutral suppliers, the following orders always hold:(14)p1*RR-p1*WW<0;p2*RR-p2*WW<0;p2*RR<p1*RR.As we show in Lemma4, the retail price under revenue sharing contract is lower than that under wholesale price contract. Market leader has to decide a low retail price to snatch a large market share, while market follower has the right of cutting down the leader’s retail price and hence enjoys a second-mover advantage in the price competition (p2*RR<p1*RR). This behavior will not occur when there is no information interaction under simultaneous competition, as Pan et al. [8] have studied. On the other hand, facing tense competition, the follower’s profits with “low-price-high-volume” strategy may not compensate its loss due to low retail price. Thus, the follower has to switch to the wholesale price contract so as to enjoy a higher profit margin. This is consistent with Pan et al. [8].


## 5.2. Impact of Product Substitutability and Risk Attitudes (Identical Risk-Averse Suppliers)

We now turn to the caseλ1=λ2=λ>0, that is, both suppliers are risk-averse, but they are identical. With this assumption, suppliers’ choices are influenced by the degree of product substitutability and risk aversion. We find that given M2’s choice of wholesale price contract, M1 always prefers revenue sharing contract. This is consistent with our previous finding shown in Lemma 1. However, it is not easy to analytically find M1’s preference when M2 chooses revenue sharing contract. We conduct numerical studies and observe similar curves; see Figure 1 for illustration. The parameters are A-=1, σ=1, c=0.5, λ∈(0,1), and θ∈(0,1).Figure 1
M1’s preference given M2 choosing revenue sharing contract (λ>0).The rectangular area{(θ,λ)∣0<θ<1,0<λ<1} is divided into two distinct regions by the curve fa=UM1RR-UM1WR|λ>0=0, where “W” denotes that M1 prefers wholesale price contract and “R” denotes that M1 prefers revenue sharing contract. From Figure 1 we can obtain the following observations. (i) If λ=0, that is, both suppliers are risk-neutral, given M2 choosing revenue sharing contract, M1 prefers revenue sharing contract when 0<θ≤0.69, otherwise, wholesale price contract. This is consistent with Lemma 1. (ii) As the degree of risk aversion increases (e.g., λ>0.21), given M2 choosing revenue sharing contract, the wholesale price contract is attractive for M1 no matter what the value of θ is. This observation can be explained as follows. When M1 becomes more and more conservative (λ becomes larger and larger), it can be observed that p1*RR is decreasing (see Figure 2). Thus, to avoid the retail price dropping too much, M1 tends to replace the revenue sharing contract by the wholesale price contract, owing to the fact that the retail price under the revenue sharing contract is always lower than that under the wholesale price contract (see Lemma 4 and Figure 3).Figure 2
Impact ofλ on p1*RR.Comparison of the retail prices withλ1=λ2=λ>0.
(a)
(b)We now also illustrate the follower M2’s preference given M1’s choice, by assuming the same numerical parameters.Similar observations can be found in Figures4(a) and 4(b). “W” denotes that M2 prefers wholesale price contract, and “R” denotes that M2 prefers revenue sharing contract. We can obtain the following observations. (i) In Figure 4(a), if λ=0, given M1 choosing wholesale price contract, M2 prefers revenue sharing contract, when 0<θ≤0.55, otherwise, wholesale price contract. In Figure 4(b), if λ=0, given M1 choosing revenue sharing contract, M2 prefers revenue sharing contract, when 0<θ≤0.51, otherwise, wholesale price contract. This is consistent with Lemma 2. (ii) As the degree of risk aversion increases (e.g., λ>0.19), regardless of M1 choosing wholesale price contract or revenue sharing contract, it is optimal for M2 to choose the wholesale price contract. A possible reason is that as M2 becomes more and more conservative, he/she tends to avoid the retail price dropping too much (p2*RR is decreasing in λ). We obtain equilibrium contract strategies illustrated by Figure 5.(a) M2’s preference given M1’s choice of wholesale price contract (λ>0). (b) M2’s preference given M1’s choice of revenue sharing contract (λ>0).
(a)
(b)Figure 5
Equilibrium contract strategies of suppliers (λ>0).In Figure5, “RW” denotes that M1 chooses revenue sharing contract and M2 chooses wholesale price contract. “RR” denotes that both M1 and M2 choose revenue sharing contract. We first verify the findings summarized in Theorem 3, that is, “RR” is the equilibrium contract strategy when 0.51<θ<1, and “RW” is the equilibrium contract strategy when 0.51<θ<1. Then, we find that if the degree of risk aversion and product substitutability are both low, “RR” sustains as the equilibrium contract strategy. Recall that when the competition intensity is low, suppliers tend to keep a relatively low retail price to stimulate demand by adopting revenue sharing contract. However, when suppliers are very conservative, as we have observed, the retail price will become too low. Consequently, M2 may switch to the wholesale price contract, which guarantees a larger profit margin. This differentiates our work from that of Pan et al. [8], in which the impact of risk attitude is not studied.


## 5.3. Verification with Asymmetric Risk Attitudes (Heterogeneous Risk-Averse Suppliers)

In this subsection, we assume that the risk aversion between two suppliers is asymmetric, that is,λ1≠λ2. Interestingly, given M2’s choice of wholesale price contract, it can be shown that M1 always prefers revenue sharing contract. Combining our previous findings with symmetric risk attitudes, this phenomenon is rather robust.Consider the following:(15)UM1RW-UM1WW|λ1≠λ2=(A--c+cθ)2(4+4λ2σ2-2λ2θ2σ2-3θ2)2×(2θλ2σ2+3θ+4+4λ2σ2)2×(256Γ(1+λ2σ2)(Γ+2λ1λ2σ4+2λ1σ2)2)-1>0,
where Γ=4-2λ2θ2σ2+2λ1λ2σ4-3θ2+4λ2σ2+2λ1σ2.Clearly,UM1RW-UM1WW|λ1≠λ2>0 always holds regardless of the value of λ1, λ2, and θ.Now, we analyze M1’s preference given M2’s choice of revenue sharing contract. The parameters are A-=1, σ=1, c=0.5, θ∈(0,1), λ2∈(0,20), and λ1=0,0.1.As observed from Figures6(a) and 6(b), when λ1 increases, it becomes more likely for M1 to choose wholesale price contract. A possible reason is that M1 tend to avoid sharply lowered retail price. Another observation is when λ2 is large, it is better for M1 to choose revenue sharing contract. The reason lies in M2’s possible choice of wholesale price contract when he/she is very conservative. It has been shown that M1’s optimal strategy is revenue sharing, given M2’s choice wholesale price contract.(a) M1’s preference given M2’s choice of revenue sharing contract (λ1=0). (b) M1’s preference given M2’s choice of revenue sharing contract (λ1=0.1).
(a)
(b)We now also illustrate M2’s preference given M1’s choice under asymmetric risk attitudes. First of all, assuming M1 chooses wholesale price contract, we use the following parameters: A-=1, σ=1,  c=0.5, θ∈(0,1), λ1∈(0,20), and λ2=0,0.1. As seen from Figures 7(a) and 7(b), the outcomes can be classified into two distinct regions by the critical curves fg=0, fh=0, fi=0, respectively. Again, with larger λ2, M2 tends to choose wholesale price contract. Assuming M1 chooses revenue sharing contract, the observation is similar.(a) M2’s preference given M1’s choice of wholesale price contract (λ2=0). (b) M2’s preference given M1’s choice of wholesale price contract under (λ2=0.1).
(a)
(b)


## 5.4. Verification with Stochastic Market Variance

In this sub-section, we investigate the role that stochastic market variance (σ) plays in the suppliers’ contract choices. First, we examine M1’s preference. Given M2’s choice of wholesale price contract, it can be shown that M1 always prefers revenue sharing contract. Second, we illustrate M1’s preference given M2’s choice of revenue sharing contract and M2’s preference given M1’s choice of wholesale price contract or revenue sharing contract. The parameters are A-=1, c=0.5, θ∈(0,1), σ∈(0,2), and λ=0.1,0.5. As seen from Figure 8(a), the rectangular area {(θ,σ)∣0≤θ<1,0≤σ<2} is divided into two parts by the curves fj(θ,σ)=0, fk(θ,σ)=0, fl(θ,σ)=0. In the upper right part (i.e., region I) of the area, all functions fj(θ,σ), fk(θ,σ), and fl(θ,σ) are less than zero. In contrast, in the lower left part (i.e., region II) of the area, all functions fj(θ,σ), fk(θ,σ), and fl(θ,σ) are greater than zero. These observations imply that if the competition density is small and the market variance is also small, revenue sharing contract will be the best option for the two suppliers. This indicates a threshold policy with respect to σ for the suppliers to select the contract type. Figure 8(b) shows similar curves. Possible explanation is as follows. Recall that the comparison of the suppliers’ utilities depends only on θ, λi, and σ, and the parameters λi and σ always arise together with the form of λiσ2. Thus, keeping λi as a constant, the impact of σ is similar to that of λi shown previously (threshold type).(a) M1’s and M2’s preference given (λ=0.1). (b) M1’s and M2’s preference given (λ=0.5).
(a)
(b)


## 6. Conclusion Remarks and Future Research

In this paper, we consider two sequential-moving supply chains, where each supply chain is composed of one risk-averse supplier and one risk-neutral retailer. We investigate the impact of competition intensity and risk aversion on the contract choice problems. We study by deriving and comparing the equilibriums under the symmetric subgames (including WW and RR case) and hybrid subgames (including WR and RW case), respectively.Our paper differs from the previous studies of contract choice in two important ways. First, we consider contract choice in the context of two sequential-moving supply chains. Second, we assume that two competing suppliers are risk-averse and analyze how risk aversion of one supplier affects his/her and rival’s decisions. We show that the retail prices under revenue sharing contract are lower than that under wholesale price contract. Thus, when the degrees of product substitutability and risk aversion are beyond a threshold value; the second-moving supplier may suffer from too low retail price and hence switch from revenue sharing contract to wholesale price contract. These results complement the current literature that assumes players are risk-neutral and help to explain why sophisticated contracts (e.g., revenue sharing contract) have not completely eliminated the Pareto inferior wholesale-price contract in reality.We would like to mention two potential directions for future research. First, it is worthy studying the other risk aversion models (e.g., [31, 50]) to see whether our insights hold or not. Second, we have assumed that information is common knowledge. Relaxing this assumption would lead to studies with asymmetric information and would generate insights on the value of information. As Ha and Tong [26] have pointed out, the value of information may significantly influence the choice of contract type.



---

*Source: 938124-2013-11-11.xml*