The Impact of Price Comparison Service on Pricing Strategy in a Dual-Channel Supply Chain

Abstract

Recently, price comparison service (PCS) websites are more and more popular due to its features in facilitating transparent price and promoting rational purchase decision. Motivated by the industrial practices, in this study, we examine the pricing strategies of retailers and supplier in a dual-channel supply chain influenced by the signals of PCS. We categorize and discuss three situations according to the signal availability of PCS, under which the optimal pricing strategies are derived. Finally, we conduct a numerical study and find that in fact the retailers and supplier are all more willing to avoid the existence of PCS with the objective of profit maximization. When both of retailers are affected by the PCS, the supplier is more willing to reduce the availability of price information. Important managerial insights are discussed.

1. Introduction

Price is one of the key competitive dimensions of purchasing a product. As consumers are eager to have access to the better price information in the market, price comparison service (PCS) is naturally born. The PCS provides specific product information and price differences for consumer’s reference [1]. By checking the PCS, consumers are able to compare prices with other retailers and make better decisions [2, 3]. For example, Skyscanner, an online air ticket price comparison website, helps online consumers to compare flight prices of any given route over a month period among different airline and agents. More examples of PCS are summarized in Table 1. According to Table 1, products shown in the PCS include air ticket, hotel, fashion, home product, computer, and electronics. One observation from Table 1 is noteworthy: all products shown in PCS are under fierce price competition.

Table 1 

Examples of PCS [3].

Price differences and price fluctuations exert an impact on the decision making among consumers and supply chain parties. Huang and Swaminathan [4] compare various products from Amazon and BELK and find that there exists price difference under an online duopoly environment. This observation motivates us to explore the pricing strategy with consideration of PCS. Serenko and Hayes [3] state that the PCS offers tremendous benefits for consumers who may potentially receive lower prices and for online vendors (i.e., retailer), who may not only capture more price information of their rivals but also get more exposure for their brands.

Pricing issues have been extensively studied by the scholars of supply chain; however, the impact of PCS has rarely been investigated. In this study, we first examine how the signals of PCS affect the decision making of supply chain members in a dual-channel supply chain. More specifically, we mainly develop a model with consideration of PCS and assume that supplier’s pricing decision is affected by the PCS, while the retailers’ pricing decision can be divided into three situations: (i) neither of retailers is affected by the PCS; (ii) either of retailers is affected by the PCS; and (iii) both of retailers are affected by the PCS. According to the three situations mentioned above, we derive the optimal pricing decision of supply chain parties in such a dual-channel supply chain, and our numerical results show that the best strategies for the retailers and supplier are to avoid the existence of PCS. In addition, we find if both of the retailers are affected by the PCS, the supplier should tend to reduce the availability of price information, and if either of retailers is affected by the PCS, the retailer should tend to exclusively cooperate with PCS.

The paper is organized as follows. We show the related literature in Section 2. The model is presented in Section 3, and we obtain the optimal retail prices on both channels under vertically competition in Section 4. In Section 5, we study the supplier’s pricing strategies when facing two competing retailers. We further conduct numerical studies to investigate the impact of PCS on pricing and expected profit among supply chain parties in Section 6. Finally, conclusion is presented in Section 7.

2. Literature Review

In the last decade, researchers have studied many issues related to the dual-channel supply chain with traditional and internet channel (please refer to [5, 6] for more discussions and review). More recently, Yao and Liu [7] study the pricing competition between retail and e-tail distribution channels under the Bertrand and the Stackelberg price competition models. Interestingly, they find that an optimal wholesale price exists under a different market structure in which the retailer is encouraged to accommodate the additional e-tail channel. Cai [8] examines the impact of channel structure on various supply chain parties with and without coordination under a dual-channel supply chain. Chen et al. [9] investigate the contracting strategies in a dual-channel supply chain. According to their results, the wholesale price contract could coordinate the dual-channel supply chain.

Pricing is popularly investigated in supply chain management, particularly in the dual-channel supply chain [10]. An early research, conducted by Ingene and Parry [11], discusses the case of a single manufacturer selling an identical product to two competing retailers with a linear quantity discount schedule and a two-part tariff. Ingene and Parry claim that coordination is not always in the manufacturer’s interest when retailers are competing. Chiang et al. [12] also examine a price competition game in a dual-channel supply chain. They find that a direct channel strategy could lead the manufacturer to be more profitable by posing a viable threat to draw customers away from the retailer. Huang and Swaminathan [4] investigate the pricing strategies in a dual-channel supply chain by assuming a stylized deterministic demand model. Under such demand function, the retailers tend to set a higher retail price in order to make more profit. Recently, Tang and Xing [13] compare the pricing behavior between online branches of traditional retailers and pure internet retailers. They conclude that the price charged by pure e-tailers for DVD titles is 14% lower than those charged by e-tailers with traditional channels. From the marketing perspective, price comparison can affect buyer’s perceptions of acquisition value, transaction value, and behavioral intentions [2]. The impact of PCS on pricing decision is thus inevitable.

The service is significantly important for supply chain management. Dumrongsiri et al. [14] consider the impact of retailer’s service quality on a dual-channel supply chain in which a manufacturer sells to a retailer as well as to consumers directly. They reach an interesting conclusion that a higher retailer’s service quality may lead to a higher manufacturer’s profit in a dual-channel supply chain. Dan et al. [15] examine the retail services in a centralized and a decentralized dual-channel supply chain using the two-stage optimization technique and Stackelberg game. Their results imply that retail services could strongly influence the manufacturer and the retailer’s pricing strategies. In this paper, we examine the impact of PCS on a dual-channel supply chain. PCS is new to the literature of supply chain management, although it has been largely explored in the area of electronic business [1, 3].

The signals of PCS are closely related to information availability, namely, information asymmetry or symmetry. However, information asymmetry has been largely discussed in the literature of supply chain management. Desiraju and Moorthy [16] study information asymmetry regarding a price- and service-sensitive demand curve. They show that the coordination can be achieved by requiring service performance. Cakanyildirim and Sethi [17] find that information asymmetry regarding a manufacturer’s production cost does not necessarily cause inefficiency in supply chain. Mukhopadhyay et al. [18] examine the mixed channels under information asymmetry and propose retailer to add differentiated value to the product so as to eliminate the possibility of channel conflict. In this study, we consider both information asymmetry and symmetry cases.

3. The Model

In this paper, we consider a dual-channel supply chain system in which one supplier provides the products to two retailers at the same wholesale price, and then two retailers (e.g., Wal-Mart and Amazon) sell products to consumers. Here, we consider one retailer is a physical retailer and the other is an online retailer (As a remark, the physical retailer and the online retailer may or may not be homogenous due to the signal availability of PCS. The heterogeneous structure has particularly happened in the scenario that either of retailers is affected by the PCS). To simplify, the two retailers are referred as retailer1 and retailer2. We consider the retail prices are public information for every party in this supply chain system. All parties face the risk of price competition brought by price comparison service (PCS) (The PCS is a third party who is independent with all parties in such a supply chain), which positively affect their pricing decisions. The sequence of events is as follows. First, supplier decides wholesale price. Second, based on the wholesale price offered by supplier, both retailer1 and retailer2 determine the retail price simultaneously. As a remark, all decisions made by supply chain parties might be influenced by the PCS. This game is analyzed by using backward induction technique. This dual-channel supply chain with the PCS is depicted in Figure 1.

Figure 1 

Dual-channel supply chain system with PCS.

3.1. Notations

 𝐷₁,𝐷₂: Market demand for retailer1 and retailer2 𝐷_𝑠: Market demand 𝑝_𝑤: Wholesale price offered by supplier 𝑝₁,𝑝₂: Retail prices decided by retailer1 and retailer2 𝑑: Market size base 𝛼: Demand sensitivity for retailer1’s market price 𝛽: Demand sensitivity for retailer2’s market price 𝜇: Impact of PCS on demand, which follows a normal distribution 𝜇∼𝑁(0,𝜎) 𝑥₀,𝑥₁,𝑥₂: Impact of PCS on pricing decision determined by supplier, retailer1, and retailer2, respectively 𝜃_𝑚,𝛿_𝑚: Adjusted coefficients of retail pricing, 𝑚∈1,2,3,4 𝜆₀,𝜆: Adjusted coefficients of wholesale pricing.

3.2. Basic Model

Without any loss of generality, we consider that the demand curve in the dual-channel supply chain is a linear function of the selling price, where 𝜇∼𝑁(0,𝜎²) is a random parameter to represent random uncertain information [19]. The function of market demand is expressed as follows: 𝐷_𝑖𝑝_𝑖,𝑝_𝑗,𝜇=𝑑_𝑖−𝛼𝑝_𝑖+𝛽𝑝_𝑗+𝜇, (1) where 𝑖,𝑗∈(1,2), 𝑖≠𝑗, and 𝑑,𝛼,𝛽 are constants.

Supply chain parties might enable to receive the signal of PCS. We denote 𝑥₀, 𝑥₁, and 𝑥₂ as the degree of PCS signal the supplier and two retailers received, respectively. According to the degree of signal, the parties would make price adjustment, and the adjusted price will in turn affect the market demand. Assuming 𝑥_𝑖=𝜇+𝜀_𝑖, both 𝜇 and 𝜀_𝑖 are independent random variables, where 𝜀_𝑖 follows normal distribution with mean of 𝜂, (𝜂>0) and variance of 𝜀_𝑖, 𝑖∈0,1,2. Covariance matrix of 𝜀_𝑖 can be denoted as ∑=diag(𝜎,𝑠₀,𝑠₁,𝑠₂). For example, if the price comparison exerts greater impact on retailer1, then the retailer1 will adjust its price, which will cause decrease in demand; while if the impact is much less, then the retailer1 will tend to keep the price stable, which will not affect the market demand. Prices offered by the supplier and retailers are functions of 𝑥₀,𝑥₁,𝑥₂, namely, 𝑝_𝑤=𝑓₀(𝑥₀) and 𝑝_𝑖=𝑓_𝑖(𝑥_𝑖).

Supplier adjusts its wholesale price based on the signal of PCS. However, due to asymmetry of transmission, the retailers may or may not be affected by the PCS. Here, we divide it into three situations accordingly.

Situation 1. The supplier does not share price comparison information 𝑥₀ with both of the retailers, namely, neither of retailers are not affected by the PCS information 𝑥₀ directly. Instead, they are able to adjust retail price based on respective price comparison 𝑥_𝑖 as well as wholesale price 𝑝_𝑤. The adjusted retail price hence is 𝑝_𝑖=𝜃₁+𝜃₂𝑥_𝑖+𝜃₃𝑝_𝑤, 𝑖∈1,2.

Situation 2. PCS information 𝑥₀ exerts impact on either of retailers, then the one who is affected by 𝑥₀ will adjust its price and the corresponding retail price is 𝑝_𝑖=𝜃₁+𝜃₂𝑥_𝑖+𝜃₃𝑝_𝑤+𝜃₄𝑥₀, 𝑖=1 or 𝑖=2.

Situation 3. PCS information 𝑥₀ exerts impact on both of retailers, namely, retailers make pricing decisions based on 𝑥₀ and 𝑥_𝑖. The adjusted retail price is 𝑝_𝑖=𝜃₁+𝜃₂𝑥_𝑖+𝜃₃𝑝_𝑤+𝜃₄𝑥₀, 𝑖∈1,2.

4. Retail Pricing Strategy

4.1. Neither of Retailers Is Directly Affected by 𝑥₀

We consider the supplier is affected by price comparison 𝑥₀, and its updated wholesale pricing strategy is 𝑝_𝑤=𝜆₀+𝜆𝑥₀, (2) where 𝜆₀, 𝜆 are price coefficients (see [20]). In this subsection, we consider that the supplier does not share information with the two retailers who are only affected by 𝑥₁ and 𝑥₂, respectively. The updated pricing strategies of the two retailers are hence as follows: 𝑝₁=𝑓₁𝑥₁,𝑝_𝑤=𝜃₁+𝜃₂𝑥₁+𝜃₃𝑝_𝑤,𝑝₂=𝑓₂𝑥₂,𝑝_𝑤=𝛿₁+𝛿₂𝑥₂+𝛿₃𝑝_𝑤. (3)(4) Assume that the competitive relationship between the two retailers who are well-matched in strength and the impact of PCS on each retailer are equal. Then, the expected demand of retailer1 can be expressed as: 𝐸𝜇∣𝑥₁,𝑥₀=𝜎𝑠₀𝑥₁+𝜎𝑠₁𝑥₀𝜎𝑠₁+𝜎𝑠₀+𝑠₁𝑠₀. (5) According to the above assumption, the formula can be further described as: 𝐸𝜇∣𝑥₁,𝑝_𝑤=𝐸𝑥₂∣𝑥₁,𝑝_𝑤=𝜎𝑠₀𝑥₁𝜆+𝜎𝑠₁𝑝_𝑤−𝜆₀𝜎𝑠₁+𝜎𝑠₀+𝑠₁𝑠₀𝜆, (6) and the expected demand of retailer2 can be expressed as: 𝐸𝜇∣𝑥₂,𝑝_𝑤=𝐸𝑥₁∣𝑥₂,𝑝_𝑤=𝜎𝑠₀𝑥₂𝜆+𝜎𝑠₂𝑝_𝑤−𝜆₀𝜎𝑠₂+𝜎𝑠₀+𝑠₂𝑠₀𝜆. (7) In addition, the signal of PCS is symmetric for both retailers; the expected prices of retailer1 and retailer2 are 𝐸𝑝₂∣𝑥₁,𝑝_𝑤=𝛿₁+𝛿₂𝐸𝑥₂∣𝑥₁,𝑝_𝑤+𝛿₃𝑝_𝑤,𝐸𝑝₁∣𝑥₂,𝑝_𝑤=𝜃₁+𝜃₂𝐸𝑥₁∣𝑥₂,𝑝_𝑤+𝜃₃𝑝_𝑤. (8)(9) Then, the expected profits of retailer1 and retailer2 are expressed as: 𝐸Π₁∣𝑥₁,𝑝_𝑤=𝑑₁−𝛼𝑝₁+𝛽𝐸𝑝₂∣𝑥₁,𝑝_𝑤+𝐸𝜇∣𝑥₁,𝑝_𝑤𝑝₁−𝑝_𝑤,𝐸Π₂∣𝑥₂,𝑝_𝑤=𝑑₂−𝛼𝑝₂+𝛽𝐸𝑝₁∣𝑥₂,𝑝_𝑤+𝐸𝜇∣𝑥₂,𝑝_𝑤𝑝₂−𝑝_𝑤. (10)(11)

Proposition 1. When neither of retailers is influenced by the supplier price comparison impact 𝑥₀, the profit functions 𝐸(Π₁∣𝑥₁,𝑝_𝑤) and 𝐸(Π₂∣𝑥₂,𝑝_𝑤) are strictly concave in 𝑝₁ and 𝑝₂, respectively, namely, the optimal retail prices 𝑝^∗₁ and 𝑝^∗₂ uniquely exist.

To enhance the presentation, all proofs are relegated to the Appendices.

By solving the first-order derivative of 𝑝₁ in 𝐸(Π₁∣𝑥₁,𝑝_𝑤) and 𝑝₂ in 𝐸(Π₁∣𝑥₁,𝑝_𝑤), respectively, one can have optimal retail pricing strategies of retailer1 and retailer2 in this situation as follows: 𝑝^∗₁=1𝑑2𝛼¹+𝛼𝑝_𝑤+𝛽𝐸𝑝₂∣𝑥₁,𝑝_𝑤+𝐸𝜇∣𝑥₁,𝑝_𝑤,𝑝^∗₂=1𝑑2𝛼²+𝛼𝑝_𝑤+𝛽𝐸𝑝₁∣𝑥₂,𝑝_𝑤+𝐸𝜇∣𝑥₂,𝑝_𝑤. (12) By solving (2)–(12) simultaneously, one finds 𝜃_𝑖, 𝑖=1,2,3,4 and 𝛿_𝑖, 𝑖=1,2,3. Consider 𝜃₁=1−𝑡−2−𝑡2𝜎1−𝑡²41𝑠𝛼+𝛽−𝑑₁𝑘₂+2𝑑𝑠₂𝑘₁+𝑑𝜎𝑠₀2𝑠₁+𝑡𝑠₂×𝑉4−𝑡²⁻¹,𝜃₂=𝜎𝑠₀1−𝑡²2/𝛼+𝛽−2𝑑𝑘₂+𝑑𝜎𝑠₀𝑉,𝜃₃=1−2−𝑡2𝜎1−𝑡²41𝑠𝛼+𝛽−𝑑₁𝑘₂+2𝑑𝑠₂𝑘₁+𝑑𝜎𝑠₀2𝑠₁+𝑡𝑠₂×𝑉4−𝑡²⁻¹,𝛿₁=1−𝑡−2−𝑡2𝜎1−𝑡²41𝑠𝛼+𝛽−𝑑₂𝑘₁+2𝑑𝑠₁𝑘₂+𝑑𝜎𝑠₀2𝑠₂+𝑡𝑠₁×𝑉4−𝑡²⁻¹,𝛿₂=𝜎𝑠₀1−𝑡²2/𝛼+𝛽−2𝑑𝑘₁+𝑑𝜎𝑠₀𝑉,𝛿₃=1−2−𝑡2𝜎1−𝑡²41𝑠𝛼+𝛽−𝑑₁𝑘₁+2𝑑𝑠₂𝑘₂+𝑑𝜎𝑠₀2𝑠₂+𝑡𝑠₁×𝑉4−𝑡²⁻¹, (13) where 𝑡=𝛽/𝛼, 𝑑₁=𝑑₂=𝑑/2, 𝑘₁=(𝜎+𝑠₀)𝑠₁+𝜎𝑠₀, 𝑘₂=(𝜎+𝑠₀)𝑠₂+𝜎𝑠₀, and 𝑉=4𝑘₁𝑘₂−𝑡²𝜎²𝑠²₀.

Thus, the optimal pricing strategies of the two retailers can be obtained by substituting (13) into 𝑝^∗₁=𝜃₁+𝜃₂𝑥₁+𝜃₃𝑝_𝑤 and 𝑝^∗₂=𝛿₁+𝛿₂𝑥₂+𝛿₃𝑝_𝑤.

4.2. Either of Retailers Is Affected by 𝑥₀

In this situation, either of retailers is affected by price comparison information 𝑥₀. We assume that retailer1 is affected by 𝑥₀ while retailer2 is not affected by 𝑥₀ directly, therefore, we can have 𝑝₁=𝜃₁+𝜃₂𝑥₁+𝜃₃𝑝_𝑤+𝜃₄𝑥₀,𝑝₂=𝛿₁+𝛿₂𝑥₂+𝛿₃𝑝_𝑤. (14)(15) Retailer1 enables to speculate pricing strategy of retailer2 based on 𝑥₀ and 𝑥₁. Consider 𝐸𝑝₂∣𝑥₁,𝑥₀=𝛿₁+𝛿₂𝜎𝑠₀𝑥₁+𝜎𝑠₁𝑥₀𝜎𝑠₁+𝜎𝑠₀+𝑠₁𝑠₀+𝛿₃𝑝_𝑤. (16) But retailer2 enables to speculate pricing strategy of retailer1 based on 𝑥₂ and 𝑝_𝑤. Consider 𝐸𝑝₁∣𝑥₂,𝑝_𝑤=𝜃₁+𝜃₂𝜎𝑠₀𝑥₂+𝜎𝑠₂𝑝_𝑤−𝜎𝑠₂𝜎𝑠₂+𝜎𝑠₀+𝑠₂𝑠₀+𝜃₃𝑝_𝑤+𝜃₄𝑝_𝑤−1,𝐸𝜇∣𝑥₁,𝑥₀=𝜎𝑠₀𝑥₁+𝜎𝑠₁𝑥₀𝜎𝑠₁+𝜎𝑠₀+𝑠₁𝑠₀,𝐸𝜇∣𝑥₂,𝑝_𝑤=𝜎𝑠₀𝑥₁𝜆+𝜎𝑠₁𝑝_𝑤−𝜆𝜎𝑠₁+𝜎𝑠₀+𝑠₁𝑠₀𝜆. (17)(18)(19) In order to reduce potential risks brought by price changes with the signal of PCS, the two retailers determine the optimal retail price by maximizing the expected profit. To this end, the expected profits for retailer1 and retailer2 are 𝐸Π₁∣𝑥₁,𝑥₀𝑑=𝐸2−𝛼𝑝₁+𝛽𝑝₂+𝜇𝑝₁−𝑝_𝑤∣𝑥₁,𝑥₀,𝐸Π₂∣𝑥₂,𝑝_𝑤𝑑=𝐸2−𝛼𝑝₂+𝛽𝑝₁+𝜇𝑝₂−𝑝_𝑤∣𝑥₂,𝑝_𝑤. (20)(21)

Proposition 2. When either of retailers is influenced by the supplier price comparison impact 𝑥₀, the profit functions 𝐸(Π₁∣𝑥₁,𝑥₀) and 𝐸(Π₂∣𝑥₂,𝑝_𝑤) are strictly concave in 𝑝₁ and 𝑝₂, respectively, namely, the optimal retail price 𝑝^∗₁ and 𝑝^∗₂ uniquely exist.

By solving the first derivative of 𝑝₁ and 𝑝₂, one can have the optimal pricing strategies of the two retailers as follows: 𝑝^∗₁=1𝑑2𝛼2+𝛽𝐸𝑝₂∣𝑥₁,𝑥₀+𝐸𝜇∣𝑥₁,𝑥₀+𝛼𝑝_𝑤,𝑝^∗₂=1𝑑2𝛼2+𝛽𝐸𝑝₁∣𝑥₂,𝑝_𝑤+𝐸𝜇∣𝑥₂,𝑝_𝑤+𝛼𝑝_𝑤. (22) Then, solving (14)–(22) simultaneously, one finds 𝜃_𝑖, 𝑖=1,2,3,4 and 𝛿_𝑖, 𝑖=1,2,3. Consider 𝜃₁=1−𝑡−2−𝑡𝜎𝑑1−𝑡²2𝜎+𝑠₀𝑠₁𝑠₂+𝜎𝑠₀2𝑠₂+𝑡𝑠₁𝑉(),2−𝑡𝜃₂=𝜎𝑠₀1−𝑡²21/𝛼+𝛽−𝑑𝑘₂+𝑑𝜎𝑠₀𝑉,𝜃₃=1−2−𝑡𝜎𝑑1−𝑡²2𝜎+𝑠₀𝑠₁𝑠₂+𝜎𝑠₀2𝑠₂+𝑡𝑠₁(),𝛼𝑉2−𝑡𝜃₄=1−2−𝑡𝛼𝜎𝑠₁1−𝑡²2𝑘₂+𝑡𝜎𝑠₁𝑉,𝛿₁=1−𝑡−2−𝑡4𝜎²1−𝑡²1×2𝛼+𝛽−𝑑𝛼+𝛽−2𝑑𝜎+𝑠₀𝑠₁𝑠₂+𝜎𝑠₀×𝑉4−𝑡²⁻¹,𝛿₂=𝜎𝑠₀1−𝑡²2/𝛼+𝛽−2𝑑𝑘₁+𝑑𝜎𝑠₀𝑉,𝛿₃=1−2−𝑡2𝜎1−𝑡²2𝑠₂2𝑘𝛼+𝛽−2𝑑₁+𝑑𝜎𝑠₀+𝑑𝑠₁2𝑘𝛼+𝛽−2𝑑₂+𝑑𝜎𝑠₀×𝛼𝑉4−𝑡²⁻¹, (23)(24)(25)(26)(27)(28)(29) where 𝑡=𝛽/𝛼, 𝑘₁=(𝜎+𝑠₀)𝑠₁+𝜎𝑠₀, 𝑘₂=(𝜎+𝑠₀)𝑠₂+𝜎𝑠₀, and 𝑉=4𝑘₁𝑘₂−𝑡²𝜎²𝑠²₀.

The optimal pricing strategies of the two retailers can be obtained by substituting (22) to (29) into 𝑝^∗₁=𝜃₁+𝜃₂𝑥₁+𝜃₃𝑝_𝑤+𝜃₄𝑥₀ and 𝑝^∗₂=𝛿₁+𝛿₂𝑥₂+𝛿₃𝑝_𝑤.

4.3. Both of Retailers Are Affected by 𝑥₀

In this situation, the supplier shares price comparison information with retailers, which enables both of retailers to be affected by 𝑥₀ directly. Hence, price functions of retailer1 and retailer2 are as follows: 𝑝₁=𝜃₁+𝜃₂𝑥₁+𝜃₃𝑝_𝑤+𝜃₄𝑥₀,𝑝₂=𝛿₁+𝛿₂𝑥₂+𝛿₃𝑝_𝑤+𝛿₄𝑥₀. (30)(31) In this situation, due to price competition, the retailer enables to speculate pricing strategy of its rival according to 𝑥₁ or 𝑥₂. The new pricing strategies of retailer1 and retailer2 are 𝐸𝑝₂∣𝑥₁,𝑥₀=𝛿₁+𝛿₂𝜎𝑠₀𝑥₁+𝜎𝑠₁𝑥₀𝜎𝑠₁+𝜎𝑠₀+𝑠₁𝑠₀+𝛿₃𝑝_𝑤+𝛿₄𝑥₀,𝐸𝑝₁∣𝑥₂,𝑥₀=𝜃₁+𝜃₂𝜎𝑠₀𝑥₁+𝜎𝑠₁𝑥₀𝜎𝑠₁+𝜎𝑠₀+𝑠₁𝑠₀+𝜃₃𝑝_𝑤+𝜃₄𝑥₀,𝐸𝜇∣𝑥₁,𝑥₀=𝜎𝑠₀𝑥₁+𝜎𝑠₁𝑥₀𝜎𝑠₁+𝜎𝑠₀+𝑠₁𝑠₀,𝐸𝜇∣𝑥₂,𝑥₀=𝜎𝑠₀𝑥₂+𝜎𝑠₁𝑥₀𝜎𝑠₁+𝜎𝑠₀+𝑠₁𝑠₀. (32)(33)(34)(35) The expected profits of retailer1 and retailer2 are expressed as: 𝐸Π₁∣𝑥₁,𝑥₀𝑑=𝐸2−𝛼𝑝₁+𝛽𝑝₂+𝜇𝑝₁−𝑝_𝑤∣𝑥₁,𝑥₀,𝐸Π₂∣𝑥₂,𝑥₀𝑑=𝐸2−𝛼𝑝₂+𝛽𝑝₁+𝜇𝑝₂−𝑝_𝑤∣𝑥₂,𝑥₀. (36)(37)

Proposition 3. When both of retailers are influenced by the supplier price comparison impact 𝑥₀, the profit functions 𝐸(Π₁∣𝑥₁,𝑥₀) and 𝐸(Π₂∣𝑥₂,𝑥₀) are strictly concave in 𝑝₁ and 𝑝₂, respectively, namely, the optimal retail prices 𝑝^∗₁ and 𝑝^∗₂ uniquely exist.

By solving the first derivative of 𝑝₁ and 𝑝₂, one can have the optimal pricing strategies of the two retailers as follows: 𝑝^∗₁=1𝑑2𝛼2+𝛽𝐸𝑝₂∣𝑥₁,𝑥₀+𝐸𝜇∣𝑥₁,𝑥₀+𝛼𝑝_𝑤,𝑝^∗₂=1𝑑2𝛼2+𝛽𝐸𝑝₁∣𝑥₂,𝑥₀+𝐸𝜇∣𝑥₂,𝑥₀+𝛼𝑝_𝑤. (38) Solving (30)–(38) simultaneously, one finds 𝜃_𝑚, 𝑚=1,2,3,4 and 𝛿_𝑚, 𝑚=1,2,3,4. Consider 𝜃₁=1−𝑡−2−𝑡𝜎𝑑1−𝑡²4𝜎+𝑑𝑠₀𝑘₁+𝜎𝑠₀2𝑠₁+𝑡𝑠₂𝑈2−𝑡²,𝜃₂=𝜎𝑠₀1−𝑡²21/𝛼+𝛽−𝑑𝑘₂+𝑑𝜎𝑠₀𝑈,𝜃₃=1−2−𝑡𝜎𝑑1−𝑡²4𝜎+𝑑𝑠₀𝑘₂+𝜎𝑠₀2𝑠₂+𝑡𝑠₁𝛼𝑉4−𝑡²,𝜃₄=𝛼𝜎𝑠1−𝑡²₀2𝑘₂+𝑡𝜎𝑠₀𝑈,𝛿₁=1−𝑡−2−𝑡4𝜎²1−𝑡²1×2𝛼+𝛽−𝑑𝛼+𝛽−2𝑑𝜎+𝑠₀𝑠₁𝑠₂+𝜎𝑠₀(2+𝑡)𝑠₁+𝑠₂×𝑈2−𝑡²⁻¹,𝛿₂=𝜎𝑠₀1−𝑡²2/𝛼+𝛽−2𝑑𝑘₁+𝑑𝜎𝑠₀𝑈,𝛿₃=1−2−𝑡𝜎𝑑1−𝑡²4𝜎+𝑑𝑠₀𝑘₁+𝜎𝑠₀2𝑠₁+𝑡𝑠₂𝑉4−𝑡²,𝛿₄=𝛼𝜎𝑠₀1−𝑡²2𝑘₁+𝑡𝜎𝑠₀𝑈, (39) where 𝑡=𝛽/𝛼, 𝑘₁=(𝜎+𝑠₀)𝑠₁+𝜎𝑠₀, 𝑘₂=(𝜎+𝑠₀)𝑠₂+𝜎𝑠₀, 𝑉=4𝑘₁𝑘₂−𝑡²𝜎²𝑠₀², and 𝑈=2𝑘₁𝑘₂−𝑡²𝜎²𝑠²₀.

The optimal pricing strategies of the two retailers can be obtained by substituting (39) into 𝑝^∗₁=𝜃₁+𝜃₂𝑥₁+𝜃₃𝑝_𝑤+𝜃₄𝑥₀ and 𝑝^∗₂=𝛿₁+𝛿₂𝑥₂+𝛿₃𝑝_𝑤+𝛿₄𝑥₀.

5. Wholesale Pricing Strategy

In this section, we investigate the supplier’s pricing decision. The supplier is affected by the signal of PCS. We denote the impact as 𝑥₀. Then, it can be divided into three situations according to previous section, namely, according to whether or not the retailers are affected by 𝑥₀. We consider the wholesale pricing is 𝑝_𝑤=𝜆₀+𝜆𝑥₀, and the supplier aims to maximize its expected profit 𝐸Π₃∣𝑥₀=𝐸𝐷₁+𝐷₂𝑝_𝑤∣𝑥₀, (40) where 𝐷₁(𝑝₁,𝑝₂,𝜇)=𝑑₁−𝛼𝑝₁+𝛽𝑝₂+𝜇, 𝐷₂(𝑝₁,𝑝₂,𝜇)=𝑑₂−𝛼𝑝₂+𝛽𝑝₁+𝜇𝑑₁=𝑑₂=𝑑/2; that is, 𝐸Π₃∣𝑥₀=𝐸𝑑−𝛼−𝛽𝑝₁+𝑝₂+2𝜇𝑝_𝑤∣𝑥₀=𝑑−𝛼−𝛽𝐸𝑝₁+𝑝₂∣𝑥₀+2𝐸𝜇∣𝑥₀𝑝_𝑤,𝐸𝑥₁∣𝑥₀=𝐸𝑥₂∣𝑥₀=𝐸𝜇∣𝑥₀=𝜎𝑥₀𝜎+𝑠₀. (41) Denoting 𝜎/(𝑥₀(𝜎+𝑠₀)) as Δ, optimal wholesale pricing is discussed in the following parts.

5.1. Neither of Retailers Is Affected by 𝑥₀

In this subsection, we consider that 𝑥₀ is not correlated with the two retailers, but retailers are affected indirectly by 𝑥₀ of 𝑝_𝑤. According to (3), (4), and (41), we can get the expected profit of supplier as follows: 𝐸Π₃∣𝑥₀=𝑑−𝛼−𝛽𝜃₁+𝛿₁−𝜃₂+𝛿₂Δ+𝜃₃+𝛿₃𝐸𝑝_𝑤∣𝑥₀+2Δ𝑝_𝑤. (42)

Proposition 4. When 𝑥₀ exerts impact to neither of retailers, the profit function 𝐸(Π₃∣𝑥₀) is strictly concave in 𝑝_𝑤1, and the optimal wholesale price 𝑝^∗_𝑤1 uniquely exists.

Solving the first-order derivative of 𝐸(Π₃∣𝑥₀) with respect to 𝑝_𝑤, one can get 𝑝^∗_𝑤1=8𝛼𝑑𝜎²𝑠₀𝑠₂+𝑠₁()×()2−𝑡𝛼−𝑡𝛽+12−𝑡²1−𝑡²𝑘₁+𝑘₂×1𝑠𝛼+𝛽−𝑑₁+2𝑑𝑠₂1+𝑘₁+𝜎²𝑠₀𝜎+𝑠₀(2−𝑡)𝛼−𝛽1−𝑡²𝑘₁+𝑘₂×2𝛼+𝛽−2𝑑+𝑑𝜎𝑠₀−(2−𝑡)𝑑𝜎+𝑠₀+2𝜎(1−𝑡)𝜎+𝑠₀, (43) where 𝑡=𝛼/𝛽, 𝜃₁, 𝛿₁, 𝜃₂, 𝛿₂, 𝜃₃, 𝛿₃ are the same as the results mentioned in Section 4.1.

5.2. Either of Retailers Is Affected by 𝑥₀

In this subsection, we consider either of retailers is affected by 𝑥₀. According to (14), (15), and (41), we can get the expected profit of supplier as follows: 𝐸Π₃∣𝑥₀=𝑑−𝛼−𝛽𝜃₁+𝛿₁+𝜃₂+𝛿₂Δ+𝜃₃+𝛿₃×𝐸𝑝_𝑤∣𝑥₀+𝛿₄𝐸𝑥₀∣𝑥₀+2Δ𝑝_𝑤. (44)

Proposition 5. When 𝑥₀ exerts impact on either of the retailers, the supplier’s profit function 𝐸(Π₃∣𝑥₀) is strictly concave in 𝑝_𝑤2, and the optimal wholesale price 𝑝^∗_𝑤2 uniquely exists.

Solving the first-order derivative of 𝐸(Π₃∣𝑥₀) with respect to 𝑝_𝑤, and let 𝑑𝐸𝑝_𝑤/𝑑𝑝_𝑤=0, one can get 𝑝^∗_𝑤2𝑝^∗_𝑤2=4𝑘₁𝑘₂−𝑡²𝜎²𝑠²₀−𝑑4𝑘₁𝑘₂−𝑡²𝜎²𝑠²₀4−𝑡²×(1−𝑡)4𝑘₁𝑘₂−𝑡²𝜎²𝑠²₀()×21−4𝜎2−𝑡𝑠𝛼+𝛽−𝑑₁𝑘₂+𝑑𝑠₂𝑘₁⁻¹⋅𝑥₀+4𝜎𝛼(2−𝑡)𝛼−𝑡𝛽1−𝑡²𝑘₁+𝑘₂×2𝑠₁1𝛼+𝛽−𝑑+𝑑𝑠₂+𝑑𝜎𝑠₀(2+𝑡)𝑠₁+𝑠₂+2𝑠₀𝜎²𝜎+𝑠₀(2−𝑡)𝛼−𝛽1−𝑡²𝑘₁+𝑘₂×1𝛼+𝛽−𝑑+𝑑𝜎𝑠₀−𝜎𝑠₁2𝑘₂+𝑡𝜎𝑠₁()×1+𝑡4𝑘₁𝑘₂−𝑡²𝜎²𝑠²₀4−𝑡²()×21−4𝜎2−𝑡)(1+𝑡𝑠𝛼+𝛽−𝑑₁𝑘₂+𝑑𝑠₂𝑘₁⁻¹, (45) where 𝑡=𝛼/𝛽, 𝜃₁, 𝛿₁, 𝜃₂, 𝛿₂, 𝜃₃, 𝛿₃ are the same as the results in Section 4.2.

5.3. Both of Retailers Are Affected by 𝑥₀

In this subsection, 𝑥₀ exerts impact on both of retailers. According to (30), (31), and (41), we can get the expected profit of supplier as follows: 𝐸Π₃∣𝑥₀=𝑑−𝛼−𝛽𝜃₁+𝛿₁+𝜃₂+𝛿₂Δ+𝜃₃+𝛿₃×𝐸𝑝_𝑤∣𝑥₀+𝜃₄+𝛿₄𝐸𝑥₀∣𝑥₀+2Δ𝑝_𝑤. (46)

Proposition 6. When 𝑥₀ exerts impact on both of retailers, the profit function 𝐸(Π₃∣𝑥₀) is strictly concave in 𝑝_𝑤3, and the optimal wholesale price 𝑝^∗_𝑤3 uniquely exists.

Solving the first-order derivative of 𝐸(Π₃∣𝑥₀) with respect to 𝑝_𝑤, one obtains the optimal wholesale price 𝑝^∗_𝑤3 as follows: 𝑝^∗_𝑤3=𝛼𝜎𝑠₀1−𝑡²2𝑘₁+𝑘₂+𝑡𝜎𝑠₀2𝑘₁𝑘₂−𝑡²𝜎²𝑠²₀⋅𝑥₀+1𝛼−𝑡𝛽−𝑑×4𝛼𝜎²1−𝑡²1×2𝛼+𝛽−𝑑+()𝛼+𝛽−2𝑑2+𝑡𝜎𝑠₀𝑠₁+𝑠₂+𝛼𝜎²𝑑1−𝑡²×4𝜎+𝑑𝑠₀𝑘₁+𝜎𝑠₀2𝑠₁+𝑠₂×𝛼2𝑘₁𝑘₂−𝑡²𝜎²𝑠²₀2−𝑡²×4𝑘₁𝑘₂−𝑡²𝜎²𝑠²₀4−𝑡²⁻¹−𝜎𝜎+𝑠₀⋅1−𝑡²𝜎𝑠₀2/𝛼+𝛽−2𝑑𝑘₁+𝑘₂+𝑑𝑠₀2𝑘₁𝑘₂−𝑡²𝜎²𝑠²₀, (47) where 𝑡=𝛼/𝛽, 𝜃₁, 𝛿₁, 𝜃₂, 𝛿₂, 𝜃₃, 𝛿₃ are the same as the results in Section 4.3.

6. Simulation Analysis

Assuming that 𝑑=200, 𝛼=0.5, 𝛽=0.3, diag(𝜎,𝑠₀,𝑠₁,𝑠₂)=diag(10,2,4,3), and 𝜆=10, 𝜆₀=40 based on the three situations discussed above, we make an analysis of how 𝑥₀,𝑥₁,𝑥₂ affects optimal pricing 𝑝^∗₁,𝑝^∗₂,𝑝^∗_𝑤. Based on the analysis of the three situations shown above, we are able to obtain optimal wholesale prices in different situations. The numerical results are shown in Tables 2, 3, 4, and 5.

Table 2 

Optimal retail pricing strategies in different situations.

Table 3 

Optimal profits in different situations.

Table 4 

Optimal wholesale price offered by supplier under different 𝑥₀.

Table 5 

Optimal profits of supplier under different 𝑥₀.

6.1. Simulation Analysis of Retail Pricing Model

Recall that 𝑥₀,𝑥₁,𝑥₂ represent the degree of PCS signal on price, we take the values of 𝑥₀,𝑥₁,𝑥₂ from 0 to 1, where 𝑥₀,𝑥₁,𝑥₂=0 implies that the corresponding parties are not affected by the signal of PCS, whereas 𝑥₀,𝑥₁,𝑥₂=1 implies that the corresponding party are fully affected by the signal of PCS. As for the retailers, from Figure 2, we can see when 𝑥₀,𝑥₁,𝑥₂ take values from 0 to 1, the line of Situation 1𝑝^∗₁ is higher than those in Situations 2 and 3; the line of Situation 2𝑝^∗₂ is higher than the line of Situation 2𝑝^∗₁; the line of Situation 3𝑝^∗₁ is the lowest one among all situations. Besides, optimal prices in every situation are increasing with 𝑥₀,𝑥₁,𝑥₂. From Figure 3, we can find that the line of Situation 1𝐸(𝑝^∗₁) is the highest among all situations. When 𝑥₀,𝑥₁,𝑥₂ are from 0.4 to 0.7, the line of Situation 2𝐸(𝑝^∗₁), 𝐸(𝑝^∗₂) and the line of Situation 3𝐸(𝑝^∗₁) are getting closer. Besides, optimal profits of every situation are increasing with 𝑥₀,𝑥₁,𝑥₂.

Figure 2 

Optimal retail prices for 4.1, 4.2, and 4.3.

Figure 3 

Optimal profits of retailers for 4.1, 4.2, and 4.3.

According to Figures 2 and 3, three observations for the retailers are noteworthy.

Observation 1. The optimal retail price of Situation 1 and its corresponding optimal profit are higher than those of Situations 2 and 3. In other words, in order to obtain a higher profit, the retailers are more willing to avoid the existence of PCS.

Observation 2. Although the retailers tend to avoid the existence of PCS, they are also willing to seek more availability of price information because the optimal retail price and their corresponding profit would be higher if the availability of pricing information is more sufficient.

Observation 3. In Situation 2, obtaining the information from the PCS could lead retailer1 to set a higher retail price and gain more profits than those of retailer2. Hence, the retailer should tend to exclusively cooperate with the PCS. This observation enhances the value of the existence of PCS.

Based on Observations 1, 2, and 3, the best strategy for the retailers is to avoid the PCS and meanwhile to increase the availability of price information. In reality, the PCS might be a third party service who is out of the control of the retailers. However, our results provide an important implication about how the retailers react if the PCS existed.

6.2. Simulation Analysis of Wholesale Pricing Model

As for the supplier, from Figure 4, we can see that the wholesale prices in every situation are linearly correlated with 𝑥₀. For certain 𝑥₀, the optimal wholesale prices in Situation 3 is higher than those in Situations 1 and 2. And the optimal wholesale prices in Situations 2 and 3 are increasing with 𝑥₀. From Figure 5, we can find that profits in Situations 2 and 3 are increasing with 𝑥₀; however, the one in Situation 1 is linearly negatively correlated with 𝑥₀.

Figure 4 

Optimal wholesale prices for 5.1, 5.2, and 5.3.

Figure 5 

Optimal profits of supplier for 5.1, 5.2, and 5.3.

According to Figures 4 and 5, three observations are noteworthy.

Observation 4. The optimal wholesale price of Situation 1 is lower than those of Situations 1 and 2, while the corresponding optimal profit is higher than those of Situations 1 and 2. This observation implies that the supplier tends to set a higher wholesale price if the PCS affects the decisions of retail price made by both of retailers. Similar to the results of retailers, the supplier is also more willing to avoid the existence of PCS.

Observation 5. We notice that, in Situation 3, both of the retailers are affected by the PCS; more availability of price information would lead to a decrease in the supplier’s profit. Hence, the supplier in Situation 3 should keep the availability of price information as much as possible.

Observations 4 and 5 indicate that the supplier is more willing to avoid the impact of PCS and also increase the availability of price information; however, if both of the retailers are affected by the PCS, he should tend to reduce the availability of price information.

7. Conclusions

With the rapid development of e-commerce, the PCS enables to provide transparent price and promote rational consumption. However, the PCS would also affect the pricing decision made by supply chain parties. In this study, we derived the optimal pricing strategies with respective to three situations of signal availability of PCS. Our numerical study implied that when both of retailers are affected by the PCS, the supplier is more willing to reduce the availability of price information, and when either of retailers is affected by the PCS, the retailer should tend to exclusively cooperate with the PCS. In addition, the retailers and supplier are all more willing to avoid the existence of PCS and increase the availability of price information. This result implied that (1) if the entire supply chain belongs to an identical corporate, the corporate is better to merge the PCS; (2) more transparency in price information actually hurts the supply chain parties’ and supply chain’s profit. This is consistent with the industrial practice in which price war could lead to a big drop in both market and price [21].

For the research limitations, we admit that the findings are mainly based on the analytical models and numerical studies we have developed under various assumptions, such as a linear demand function with respect to the PCS. In addition, the analysis is conducted on a relatively simple dual-channel supply chain of one supplier and two competing retailers, which is an abstract version of the more complex real world supply chain. In this paper, we consider that the PCS is independent from the parties in supply chain; however, according to the practices in China, we observe that Baidu has developed new business model as a reaction to price comparison impact. It would be interesting to investigate the mechanism of how the supply chain parties should cooperate with the PCS. It is also promising to extend the model and the analysis to more complex supply chain systems (such as the ones that are risk averse agents [22–24]). It is also meaningful to investigate whether it is beneficial for the supplier to always share information from contracting perspective among supply chain parties [25, 26]. This would address another interesting topic from the perspective of social influence on the PCS in supply chain management [27].

Appendices

Proof of Proposition 1. According to (2), (4), (6), (8), and (10), we have 𝐸Π₁∣𝑥₁,𝑝_𝑤=𝑑₁−𝛼𝑝₁+𝛽𝐸𝑝₂∣𝑥₁,𝑝_𝑤+𝐸𝜇∣𝑥₁,𝑝_𝑤×𝑝₁−𝑝_𝑤=𝑑₁−𝛼𝑝₁+𝛽𝛿₁+𝛿₂𝐸𝑥₂∣𝑥₁,𝑝_𝑤+𝛿₃𝑝_𝑤×𝑝₁−𝑝_𝑤=𝑑₁−𝛼𝑝₁+𝛽𝛿₁+𝛿₂⋅𝜎𝑠₀𝑥₂𝜆+𝜎𝑠₂𝑝_𝑤−𝜆₀𝜎𝑠₂+𝜎𝑠₀+𝑠₂𝑠₀𝜆+𝛿₃𝑝_𝑤×𝑝₁−𝑝_𝑤=𝑑₁−𝛼𝑝₁+𝛽𝛿₁+𝛿₂⋅𝜎𝑠₀𝑥₂+𝜎𝑠₂𝑥₀𝜎𝑠₂+𝜎𝑠₀+𝑠₂𝑠₀+𝛿₃𝜆₀+𝜆𝑥₀𝑝₁−𝜆₀+𝜆𝑥₀=−𝛼𝑝²₁+𝑊₁𝑝₁+𝑌₁, (A.1) where 𝛼>0, 𝑊₁=𝑑₁+𝛽(𝛿₁+𝛿₂⋅(𝜎𝑠₀𝑥₂+𝜎𝑠₂𝑥₀)/(𝜎𝑠₂+𝜎𝑠₀+𝑠₂𝑠₀)+𝛿₃(𝜆₀+𝜆𝑥₀))+𝛼(𝜆₀+𝜆𝑥₀) and 𝑌₁=−(𝜆₀+𝜆𝑥₀)(𝑑₁+𝛽(𝛿₁+𝛿₂⋅(𝜎𝑠₀𝑥₂+𝜎𝑠₂𝑥₀)/(𝜎𝑠₂+𝜎𝑠₀+𝑠₂𝑠₀)+𝛿₃(𝜆₀+𝜆𝑥₀))).
According to (2), (3), (7), (9), and (11), we have 𝐸Π₂∣𝑥₂,𝑝_𝑤=𝑑₂−𝛼𝑝₂+𝛽𝐸𝑝₁∣𝑥₂,𝑝_𝑤+𝐸𝜇∣𝑥₂,𝑝_𝑤𝑝₂−𝑝_𝑤=𝑑₂−𝛼𝑝₂+𝛽𝜆𝜎𝑠₀𝑥₂+𝜎𝑠₂𝑝_𝑤−𝜆₀𝜎𝑠₂+𝜎𝑠₀+𝑠₂𝑠₀𝜆+𝜆𝜎𝑠₀𝑥₂+𝜎𝑠₂𝑝_𝑤−𝜆₀𝜎𝑠₂+𝜎𝑠₀+𝑠₂𝑠₀𝜆𝑝₂−𝑝_𝑤=𝑑₂−𝛼𝑝₂+𝛽𝜎𝑠₀𝑥₂+𝜎𝑠₂𝜎𝑠₂+𝜎𝑠₀+𝑠₂𝑠₀+𝜎𝑠₀𝑥₂+𝜎𝑠₂𝜎𝑠₂+𝜎𝑠₀+𝑠₂𝑠₀𝑝₂−𝜆₀+𝜆𝑥₀=−𝛼𝑝²₂+𝑊₂𝑝₂+𝑌₂, (A.2) where 𝛼>0, 𝑊₂=𝛼(𝜆₀+𝜆𝑥₀)(𝑑₂+𝛽(𝜎𝑠₀𝑥₂+𝜎𝑠₂)/(𝜎𝑠₂+𝜎𝑠₀+𝑠₂𝑠₀)+(𝜎𝑠₀𝑥₂+𝜎𝑠₂)/ (𝜎𝑠₂+𝜎𝑠₀+𝑠₂𝑠₀)), 𝑌₂=−(𝜆₀+𝜆𝑥₀)(𝛽((𝜎𝑠₀𝑥₂+𝜎𝑠₂)/(𝜎𝑠₂+𝜎𝑠₀+𝑠₂𝑠₀))+(𝜎𝑠₀𝑥₂+𝜎𝑠₂)/ (𝜎𝑠₂+𝜎𝑠₀+𝑠₂𝑠₀)).
According to the nature of the quadratic function and 𝑌₂<0, the profit functions 𝐸(Π₁∣𝑥₁,𝑝_𝑤) and 𝐸(Π₂∣𝑥₂,𝑝_𝑤) are strictly concave in 𝑝₁ and 𝑝₂, respectively.

Proof of Proposition 2. According to (2), (15), (18), and (20), we have 𝐸Π₁∣𝑥₁,𝑥₀𝑑=𝐸2−𝛼𝑝₁+𝛽𝑝₂+𝜇𝑝₁−𝑝_𝑤∣𝑥₁,𝑥₀𝑑=𝐸2−𝛼𝑝₁+𝛽𝛿₁+𝛿₂𝑥₂+𝛿₃𝜆₀+𝜆𝑥₀+𝛿₄𝑥₀+𝜇𝑝₁−𝜆₀−𝜆𝑥₀∣𝑥₁,𝑥₀=−𝛼𝑝²₁+𝑊₃𝑝₁+𝑌₃, (A.3) where 𝛼>0, 𝑊₃ represents a coefficient, and 𝑌₃ represents constant in this quadratic function.
According to (2), (14), (17), and (19), we have 𝐸Π₂∣𝑥₂,𝑝_𝑤𝑑=𝐸2−𝛼𝑝₂+𝛽𝑝₁+𝜇𝑝₂−𝑝_𝑤∣𝑥₂,𝑝_𝑤𝑑=𝐸2−𝛼𝑝₂+𝛽𝜃₁+𝜃₂𝑥₁+𝜃₃𝑝_𝑤+𝜃₄𝑥₀×𝑝+𝜇₂−𝜆₀−𝜆𝑥₀∣𝑥₂,𝑝_𝑤=−𝛼𝑝²₂+𝑊₄𝑝₂+𝑌₄, (A.4) where 𝛼>0, 𝑊₄ represents a coefficient, and 𝑌₄ represents constant in this quadratic function.
Similar to Proposition 1, according to the nature of the quadratic function, both 𝐸(Π₁∣𝑥₁,𝑥₀) and 𝐸(Π₂∣𝑥₂,𝑝_𝑤) are strictly concave in 𝑝₁ and 𝑝₂, respectively.

Proof of Proposition 3. According to (2), (31), (32), (34), and (36), we have 𝐸Π₁∣𝑥₁,𝑥₀𝑑=𝐸2−𝛼𝑝₁+𝛽𝑝₂+𝜇𝑝₁−𝑝_𝑤∣𝑥₁,𝑥₀𝑑=𝐸2−𝛼𝑝₁+𝛽𝛿₁+𝛿₂𝑥₂+𝛿₃𝑝_𝑤+𝛿₄𝑥₀×𝑝+𝜇₁−𝜆₀−𝜆𝑥₀∣𝑥₁,𝑥₀=−𝛼𝑝²₁+𝑊₅𝑝₁+𝑌₅, (A.5) where 𝛼>0, 𝑊₅ represents a coefficient, and 𝑌₅ represents constant in this quadratic function.
According to (2), (30), (33), and (35),we have 𝐸Π₂∣𝑥₂,𝑥₀𝑑=𝐸2−𝛼𝑝₂+𝛽𝑝₁+𝜇𝑝₂−𝑝_𝑤∣𝑥₂,𝑥₀𝑑=𝐸2−𝛼𝑝₂+𝛽𝜃₁+𝜃₂𝑥₁+𝜃₃𝑝_𝑤+𝜃₄𝑥₀×𝑝+𝜇₂−𝜆₀−𝜆𝑥₀∣𝑥₂,𝑥₀=−𝛼𝑝²₂+𝑊₆𝑝₂+𝑌₆, (A.6) where 𝛼>0, 𝑊₆ represents a coefficient, and 𝑌₆ represents constant in this quadratic function.
Similar to Proposition 1, the profit functions 𝐸(Π₁∣𝑥₁,𝑥₀) and 𝐸(Π₂∣𝑥₂,𝑥₀) are strictly concave in 𝑝₁ and 𝑝₂, respectively.

Proof of Proposition 4. According to (3), (4), and (41), we have 𝐸Π₃∣𝑥₀=𝐸𝑑−𝛼−𝛽𝑝₁+𝑝₂+2𝜇𝑝_𝑤∣𝑥₀=𝑑−𝛼−𝛽𝐸𝑝₁+𝑝₂∣𝑥₀+2𝐸𝜇∣𝑥₀𝑝_𝑤=𝑑−𝛼−𝛽𝜃₁+𝛿₁−𝜃₂+𝛿₂Δ+𝜃₃+𝛿₃𝐸𝑝_𝑤∣𝑥₀+2Δ𝑝_𝑤=𝛽−𝛼𝜃₃+𝛿₃𝑝²_𝑤+𝑊₇𝑝_𝑤+𝑌₇, (A.7) where 𝛼>𝛽, 𝑊₇ represents a coefficient, and 𝑌₇ represents constant in this quadratic function.
Similarly, according to the nature of the quadratic function, the profit function 𝐸(Π₃∣𝑥₀) is strictly concave in 𝑝_𝑤1.

Proof of Proposition 5. According to (14), (15), and (41), we have 𝐸Π₃∣𝑥₀=𝐸𝑑−𝛼−𝛽𝑝₁+𝑝₂+2𝜇𝑝_𝑤∣𝑥₀=𝑑−𝛼−𝛽𝐸𝑝₁+𝑝₂∣𝑥₀+2𝐸𝜇∣𝑥₀𝑝_𝑤=𝑑−𝛼−𝛽𝜃₁+𝛿₁+𝜃₂+𝛿₂Δ+𝜃₃+𝛿₃×𝐸𝑝_𝑤∣𝑥₀+𝛿₄𝐸𝑥₀∣𝑥₀+2Δ𝑝_𝑤=𝛽−𝛼𝜃₃+𝛿₃𝑝²_𝑤+𝑊₈𝑝_𝑤+𝑌₈, (A.8) where 𝛼>𝛽, 𝑊₈ represents a coefficient, and 𝑌₈ represents constant in this quadratic function.
Similarly, according to the nature of the quadratic function, the profit function 𝐸(Π₃∣𝑥₀) is strictly concave in 𝑝_𝑤2.

Proof of Proposition 6. According to (30), (31), and (41), we have 𝐸Π₃∣𝑥₀=𝐸𝑑−𝛼−𝛽𝑝₁+𝑝₂+2𝜇𝑝_𝑤∣𝑥₀=𝑑−𝛼−𝛽𝐸𝑝₁+𝑝₂∣𝑥₀+2𝐸𝜇∣𝑥₀𝑝_𝑤=𝑑−𝛼−𝛽𝜃₁+𝛿₁+𝜃₂+𝛿₂Δ+𝜃₃+𝛿₃𝐸𝑝_𝑤∣𝑥₀+𝜃₄+𝛿₄𝐸𝑥₀∣𝑥₀+2Δ𝑝_𝑤=𝛽−𝛼𝜃₃+𝛿₃𝑝²_𝑤+𝑊₉𝑝_𝑤+𝑌₉, (A.9) where 𝛼>𝛽, 𝑊₉ represents a coefficient, and 𝑌₉ represents constant in this quadratic function.
Similarly, according to the nature of the quadratic function, the profit function 𝐸(Π₃∣𝑥₀) is strictly concave in 𝑝_𝑤3.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the editor, associate editor, and two anonymous reviewers for their constructive suggestions, which have improved this paper tremendously. This work is partially sponsored by the National Natural Science Foundation of China (71172174, 70772073) and the Ph.D. Programs Foundation of the Ministry of Education of China (20110075110003), the China Scholarship Council (201206630035), the Innovation Program of Shanghai Municipal Education Commission (12ZS58), and the Shanghai Pujiang Program (12PJC020).