vertical cost information sharing in a supply chain with value-adding retailers

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Omega 36 (2008) 838 – 851 www.elsevier.com/locate/omega Vertical cost information sharing in a supply chain with value-adding retailers Dong-Qing Yao a , , Xiaohang Yue b , John Liu c a Department of Management, College of Business and Economics,Towson University, MD 21252, USA b School of Business Administration, University of Wisconsin-Milwaukee, WI 53201, USA c Department of Logistics, Hong Kong Polytechnic University, Hung Hom, Kowloon, China Received 12 August 2005; accepted 19 April 2006 Available online 9 June 2006 Abstract We consider a supply chain consisting of one supplier and two value-adding heterogeneous retailers. Each retailer has full knowledge about his own value-added cost structure that is unknown to the supplier and the other retailer. Assuming there is no horizontal information sharing between two retailers, we model the supply chain with a three-stage game-theoretic framework. In the first stage each retailer decides if he is willing to vertically disclose his private cost information to the supplier. In the second stage, given the information he has about the retailers, the supplier announces the wholesale price to the retailers. In response to the wholesale price, in the third stage, the retailers optimize their own retail prices and the values added to the product, respectively. Under certain conditions, we prove the existence of equilibrium prices and added values. Furthermore, we obtain the condition under which both retailers are unwilling to vertically share their private information with the supplier, as well as the conditions under which both retailers have incentives to reveal their cost information to the supplier, thus leading to a win–win situation for the whole supply chain. 2006 Elsevier Ltd. All rights reserved. Keywords: Value-added; Supply chain; Information sharing; Distribution channel; Game theory 1. Introduction The concept of “value-added”, the idea of adding service or components to a product to increase its value or price [1], has been a buzzword in recent years. In order not to be disintermediated in this competitive E-business era, each stage in a supply chain needs to add appropriate value to a product. There are numerous ways to add value to a product. A retailer can bundle the product with value-added service/delivery or provide desirable value-added packaging [2]. In the IT industry, value can be added through services, free software, technical training or maintenance. For example, software companies (e.g., Systemax) may bundle the PC with a package of free internet access such as basic AOL service [3]. In hi-tech industries (e.g., electronics industry), value can be added through simple components labeling and kitting to complex supply chain management service [4]. This manuscript was processed by Associated Editor Richard Metters. Corresponding author. Tel.: +1 410 704 6185; fax: +1 410 704 3236. E-mail addresses: [email protected] (D.-Q. Yao), [email protected] (X. Yue), [email protected] (J. Liu). 0305-0483/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.omega.2006.04.003

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Omega 36 (2008) 838–851www.elsevier.com/locate/omega

Vertical cost information sharing in a supply chain withvalue-adding retailers�

Dong-Qing Yaoa,∗, Xiaohang Yueb, John Liuc

aDepartment of Management, College of Business and Economics, Towson University, MD 21252, USAbSchool of Business Administration, University of Wisconsin-Milwaukee, WI 53201, USA

cDepartment of Logistics, Hong Kong Polytechnic University, Hung Hom, Kowloon, China

Received 12 August 2005; accepted 19 April 2006Available online 9 June 2006

Abstract

We consider a supply chain consisting of one supplier and two value-adding heterogeneous retailers. Each retailer has fullknowledge about his own value-added cost structure that is unknown to the supplier and the other retailer. Assuming there is nohorizontal information sharing between two retailers, we model the supply chain with a three-stage game-theoretic framework. Inthe first stage each retailer decides if he is willing to vertically disclose his private cost information to the supplier. In the secondstage, given the information he has about the retailers, the supplier announces the wholesale price to the retailers. In response to thewholesale price, in the third stage, the retailers optimize their own retail prices and the values added to the product, respectively.Under certain conditions, we prove the existence of equilibrium prices and added values. Furthermore, we obtain the condition underwhich both retailers are unwilling to vertically share their private information with the supplier, as well as the conditions underwhich both retailers have incentives to reveal their cost information to the supplier, thus leading to a win–win situation for the wholesupply chain.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Value-added; Supply chain; Information sharing; Distribution channel; Game theory

1. Introduction

The concept of “value-added”, the idea of adding service or components to a product to increase its value or price[1], has been a buzzword in recent years. In order not to be disintermediated in this competitive E-business era, eachstage in a supply chain needs to add appropriate value to a product. There are numerous ways to add value to a product.A retailer can bundle the product with value-added service/delivery or provide desirable value-added packaging [2].In the IT industry, value can be added through services, free software, technical training or maintenance. For example,software companies (e.g., Systemax) may bundle the PC with a package of free internet access such as basic AOLservice [3]. In hi-tech industries (e.g., electronics industry), value can be added through simple components labelingand kitting to complex supply chain management service [4].

� This manuscript was processed by Associated Editor Richard Metters.∗ Corresponding author. Tel.: +1 410 704 6185; fax: +1 410 704 3236.

E-mail addresses: [email protected] (D.-Q. Yao), [email protected] (X. Yue), [email protected] (J. Liu).

0305-0483/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.omega.2006.04.003

D.-Q. Yao et al. / Omega 36 (2008) 838–851 839

Since there are always operating costs incurred related to corresponding value-added service, one natural researchquestion arises: how much value should be added, especially under a competitive market structure? In this paper, wewill examine a distribution channel in a supply chain made up of one supplier and two heterogeneous value-addingretailers/distributors. Both retailers order generic products from a supplier, and add certain value to the product, thensell the product to customers. We are interested in studying how much appropriate value should be added to the genericproduct by each retailer.

Many business practices around such types of distribution channels can be found in real life. For example, ITsolution providers may order white box PCs from suppliers, then add appropriate value (e.g. offer antivirus software,technical training, provide extended warranty). Actually, CRN research showed that white boxes are their best-sellingdesktops, compared to branded name PCs [5]. In the electronics industry, distributors may offer value-added servicessuch as connector and cable assemblies, customized integration, supply chain management services such as a vendormanaged inventory (VMI) program, or even design service [6]. In the franchising industry, each franchisee acquiresthe franchisor’s generic product (the rights to use franchisor’s names, trademarks, etc.), then adds value and sells to thecustomers [7]. In the retailing industry, retailers may add value to electronic products by providing an extended warrantyor offering lenient returns policy. For example, Best Buy’s warranty policy would cover defects the manufacturer doesnot cover, while Wal-Mart’s warranty policy begins after the manufacturer’s policy expires [8].

Researchers have examined distribution channels with one player working with two other players. For example, Choi[9] studied price competition in a distribution channel with two suppliers and one retailer. Under a different marketstructure, he analytically obtained channel decisions for three non-cooperative games between the manufacturers andthe retailer. Tsay and Agrawal [10] investigated a supply chain with one supplier supplying a common product totwo retailers, and both retailers competing with each other along both price and service policies. Tsay and Agrawalexamined each party’s pricing strategy, total sales, market share and profitability.

However, existing studies of distribution channels assume supplier and retailers compete with complete informa-tion. In other words, the research assumes that information such as production cost, operating cost and other marketparameters about supplier or retailers are common knowledge to all parties in a supply chain. This assumption maynot hold in real life. In practice, each party in a supply chain usually possesses private information that is unknown tooutsiders, and these firms are likely to protect their sales strategies by hiding their private cost or demand information[11], thus leading to a competition under incomplete information. For example, warranty cost is confidential informa-tion for retailers such as Best Buy and Circuit City because significant portions of their operating profits are from thewarranty service [8]. Franchisees also hold their private cost information of sale and decide if they will reveal their costinformation to the franchisor in the contract [7]. Therefore, what information to share and how to share informationbecomes an interesting research issue.

Some researchers have studied information (e.g., demand, cost, and inventory) sharing/asymmetry in a supply chain.For example, Li [12] studied the incentives a firm would need to share its private information (demand and costs) withits competitors; Gavirneni [13] analyzed a supply chain where inventory information is shared between the supplierand retailer. Cachon and Fisher [14] investigated the sharing of demand and inventory data in a supply chain with onesupplier and multiple identical retailers. Agrell et al. [15] examined information sharing in telecom supply chains wherea supplier has private cost information and investment opportunities. Corbett et al. [16] analyzed a supply chain withone supplier and one buyer, and studied the value to the supplier of offering more general contracts and acquiring moreaccurate information about the buyer’s cost structure. Assuming that information asymmetry between a manufacturerand a retailer, i.e., the retailer’s knowledge of the manufacturer’s cost is incomplete, Lau et al. [17] also studied how toset up the wholesale price and retail price in a supply chain.

In this paper, we will examine a supply chain consisting of one supplier and two heterogeneous retailers. There ishorizontal competition between the two retailers, each of whom orders common generic products from the supplier,adds value to the products, then sells to customers. In addition, the supplier has no complete knowledge of the retailers’cost structure regarding value-added information. Each retailer needs to decide if he is willing to vertically share hiscost information with the supplier.

Previously, Li [18] and Zhang [19] studied a vertical information exchange in a supply chain where demand in-formation is uncertain to the supplier. Their focus was on the effects of horizontal information leakage for verticalinformation sharing in a supply chain, i.e., if one retailer vertically discloses his information to the supplier, the latterwill react to the revealed information, and make decisions accordingly. Li and Zhang discovered that the other retailercan infer his competitor’s private information through the supplier’s decision.

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However, the research assumes that retailers are homogeneous, that is, the retailers sell the same product withoutdifferentiating the product through valued-added practices. Because adding value is crucial in the supply chain practiceas illustrated in previous examples, in this paper we do not explicitly consider horizontal information leakage, butmainly focus on examining the appropriate value to be added to a generic product. We are interested in investigatingthe equilibriums in our supply chain structure with value-adding retailers and under what conditions the retailers haveincentives to share their information vertically with the supplier.

The main contributions of our research to the extant literature are threefold: first we find that equilibrium points existin a supply chain with value-adding (heterogeneous) retailers under certain conditions; second, we show the upstreamplayer always benefits from more accurate information about the retailers. Third, we prove information sharing couldbenefit both upstream and downstream players, thus leading to a win–win situation for the whole supply chain.

2. Model framework

We consider a supply chain consisting of one supplier and two heterogeneous retailers. We assume there is nowholesale price discrimination. The supplier sells a common generic product to two retailers at the same wholesaleprice. Afterwards, the retailers add their own values to the product and decide on the retail prices to the customers.Here each retailer has his own private information about the cost of the value added to the product, which is unknownto the other players. For example, each retailer has his private warranty cost, or each franchisee has his private costinformation of the sale. Considering the retailers are competitors, we assume there is no horizontal information sharingbetween retailers; however, each retailer can make his own decision if he is willing to disclose his cost informationto the supplier vertically. Furthermore we assume each party only takes into account the direct effects of informationsharing between different players. As a result, there is no horizontal information leakage considered in this paper. Infact, in the real situation, one player (i.e., the supplier) makes decision based on many considerations (e.g., forecast,cost structure, market promotional strategy, brand consideration, etc.). Thus, it is very hard for a retailer to infer theother retailer’s private information based on the supplier’s decision.

We model our supply chain as a three-stage game. The sequences of moves are as follows:

(1) Each retailer has his own private cost information about value-added service, and decides if he is willing to sharethe cost information with the supplier vertically.

(2) Informed of the retailers’ cost information, if any, the supplier announces the wholesale price to optimize his(expected) profit. In this stage, the supplier may have either, both, or neither retailer’s cost information. Therefore,three different scenarios exist for the wholesale price.

(3) In response to the wholesale price, two retailers decide on the values added to the product and the retail pricessimultaneously to maximize their own expected profits, respectively.

Next, we describe three key system components in our supply chain framework: demand functions, value-addedfunctions and payoff functions. Before that, we will summarize our notations as follows:

w: The supplier’s wholesale price to the retailers (decision variable)p1: Retailer 1’s retail price to the customers (decision variable)p2: Retailer 2’s retail price to the customers (decision variable)v1: Value added to the retailer 1’s product (decision variable)v2: Value added to the retailer 2’s product (decision variable)d1: Demand via retailer 1’s channeld2: Demand via retailer 2’s channel

2.1. Demand functions

In this paper, we adopt linear demand functions mainly because linear demand functions are widely used in supplychain and marketing literature due to their tractability. Specifically we use demand functions similar to those proposedby Tsay and Agrawal [10]. It is worth noting that this kind of linear model structure has been empirically tested byother researchers. For example, Raju et al. [20] employed a similar demand structure to study market characteristicswhen retailers introduced store brands to compete against national brand products. Sayman et al. [21] used the same

D.-Q. Yao et al. / Omega 36 (2008) 838–851 841

linear model structure to study the position of store brands against national brands. In essence, the structure of ourdemand function is similar to theirs. For other demand functions, if they can be approximated or locally approximatedby linear functions, our managerial insights still hold.

For retailer 1:

d1 = �1 − �1p1 + �2v1 + �1(p2 − p1) + �2(v1 − v2). (1)

For retailer 2:

d2 = �2 − �1p2 + �2v2 + �1(p1 − p2) + �2(v2 − v1). (2)

�i (i = 1, 2) are market bases for two retailers. �1 measures the marginal demand change with channel price. �2measures the marginal demand change with value-added service. �i (i =1, 2) are migration rates for the perceived pricedifference and the perceived service difference, which means if customers perceive there is price difference (value-added service) between two retailers, they will switch to the other retailer at the rate of �1 (�2). Therefore, horizontalcompetition exists between the two retailers. We also make the following assumptions about the parameters in thedemand functions:

(1) Market base �i (i = 1, 2) is large enough compared to other market parameters. Typically this assumption holdsfor real life cases.

(2) We assume �i > �i (i = 1, 2), which means that the effect of a direct price/value change is greater than the effectof a price/value difference between the two channels (cross-price/value change).

2.2. Value-added functions

In practice, we can use a strictly convex service function c(v) to depict the relationship between the added value andits related cost. That is, the cost of value-added service have the properties of dc(v)/dv > 0 and d2c(v)/dv2 > 0. Oneform commonly adopted in previous literature (e.g., [10,22]) is given Eq. (3)

ci(v) = �i

v2

2, i = 1, 2. (3)

�i is the efficiency parameters of retailer i’s cost structure. The smaller the �i , the more efficient retailer i is. In thispaper, we also assume that each retailer knows his own �i but does not know the other retailer’s efficiency parameter.However, each retailer knows the probability distribution of the other retailer’s cost information (�i) (e.g. [23]). Therationality is as follows: each player has no complete information about a specific retailer’s cost structure; but hecan estimate the probability distribution of the parameter from the whole industry’s information. For simplicity, inthis paper, we assume �i is uniformly distributed, i.e., �i ∼ U [�̄ − ε, �̄ + ε]. Each player can estimate the averagevalue-added cost efficiency �̄ and the deviation ε. The larger the ε, the more dispersed the industry is (usually in hi-techindustries). In our model, �1 is known only to retailer 1, �2 is known only to retailer 2, and neither �1 nor �2 is knownto the supplier.

2.3. Profit functions

Without loss of generality, we assume marginal production cost for the supplier is 0. Thus the supplier’s profit canbe expressed as

�M = (d1 + d2)w. (4)

For the retailers, the profit functions are given in Eq. (5)

�Ri= [pi − w − ci(v)]di, i = 1, 2. (5)

As we mentioned before, this is a three-stage game. We analyze this dynamic game with the backward inductionapproach. First we assume the wholesale price is given and find the equilibrium retail prices and added values instage 3.

842 D.-Q. Yao et al. / Omega 36 (2008) 838–851

3. Stage 3: retailers’ equilibriums

Because we assume there are no horizontal information sharing and no information leakage, each retailer has no fullinformation about the other competitor’s cost information. Therefore for each retailer, the objective is to find optimalretail price pi and the optimal added value vi given the wholesale price w, i.e.

Maxpi,vi

E[�Ri|w] = Max

pi,vi

E[(pi − w − ci)di |w], i = 1, 2. (6)

Also we assume two retailers decide on the retail prices and added value simultaneously. Therefore, we have thefollowing proposition.

Proposition 1. There exists equilibrium retail price and added value for each retailer.

Proof. See Appendix. �

The managerial insight of Proposition 1 is that both retailers can have stable retail prices and added values even ifthey are heterogeneous, i.e., the retailers should have stable marketing strategies. No one can be better off by deviatingfrom these pricing and value-added strategies. Also the optimal value added to each retail channel is independent ofthe wholesale price w, which is decided only by the retailer’s own cost efficiency (�i). This result will not be affectedeven if horizontal information leakage occurs. The more efficient the retailer is, the more value he is willing to add tothe product, which is consistent with intuition.

On the other hand, the retail price is related to the supplier’s wholesale price, the cost efficiency and the dispersionof the industry’s efficiency. Depending on the different wholesale prices (Stage 2 in next section shows there are threedifferent wholesale prices for information sharing cases), the retail prices vary accordingly. The retailer tends to chargemore to the customers if the wholesale price is higher, and the retail price is higher if the retailer is more efficient dueto more added values. Furthermore we have the following corollary regarding the retail prices.

Corollary 1.⎧⎪⎨⎪⎩

�p∗i

�ε> 0 if 3�2�1 − (4�1 + �1)�2 > 0,

�p∗i

�ε< 0 if 3�2�1 − (4�1 + �1)�2 < 0.

Proof. See Appendix. �

In order for 3�2�1 − (4�1 + �1)�2 > 0, one possibility is that �2is larger and �1 is smaller, which indicates thatcustomers are more value-oriented than price-oriented. The managerial insight of Corollary 1 is that the retail priceswill go up if customers are more sensitive to the added value and the industry is more dispersed. In the hi-tech industry(computer, electronics products), for example, if customers need more added value (after sale service, maintenance,technical training, etc.), and the industry is highly dispersed (as in the introduction stage of every new technology), theretail price is usually higher. Afterwards when there is not much efficiency difference between retailers, the retail pricewill go down. On the other hand, if customers are more sensitive to the price rather than the added value, the retailershave lower retail prices if the whole retail industry is highly dispersed.

Next we will find the supplier’s optimal decision given the best response functions obtained above.

4. Stage 2: supplier’ equilibrium

In the second stage, the supplier needs to decide the wholesale price w to maximize his own anticipated profit basedon the information available on hand about the retailers’ cost structure. Eq. (7) is the supplier’s objective function.

Maxw

E[�M ] = Maxw

E[(d1 + d2)w]. (7)

D.-Q. Yao et al. / Omega 36 (2008) 838–851 843

Because each retailer will decide if he is willing to reveal his private information to the supplier in the first stage, wehave three different cases in the second stage, as follows:

(1) Both of the retailers will share information with the supplier.(2) One of the retailers will share information with the supplier.(3) No any retailer will share information with the supplier.

In what follows, we will discuss each of these cases, respectively.

4.1. Case 1: vertical information sharing between the supplier and both retailers

If both retailers decide to share information with the supplier in the first stage, the supplier has full knowledge of theretailers’ value-added cost information. Hence the supplier’s objective function is

Maxw

�M = (d1 + d2)w. (8)

Proposition 2 (The supplier’s equilibrium wholesale price under complete information sharing).

w∗CI = (�1 + �2)

4�1+ [2�2�1 + �1(�2 − �2)](�2 + �2)

8�1(�1 + �1)2�1

+ [2�2�1 + �1(�2 − �2)](�2 + �2)

8�1(�1 + �1)2�2

.

Proof. See Appendix. �

Since we assume �2 > �2, it is obvious that the wholesale price decreases in �i (i = 1, 2). The managerial insight ofProposition 2 is that if both retailers are willing to disclose their private information to the supplier, or if a manufacturerlike IBM, Sony, etc., has monopoly power in the supply chain, and he can coerce the retailers to share their costinformation with the manufacturer, then the wholesale price is the function of [1/�1 + 1/�2], which indicates thatthe more efficiently the retailer can add value to the product, the higher the wholesale price is. In other words, themanufacturer will be better off by acquiring information from efficient retailers. By contrast, the retailers will be worseoff since they always prefer to have a lower wholesale price.

4.2. Case 2: no vertical information sharing between the supplier and any retailer

Suppose there is no information sharing between the supplier and any of the retailers. The supplier’s objective is tofind the optimal wholesale price w to maximize his expected profit, i.e.,

Maxw

E[�M ] =∫ �̄+�

�̄−�

∫ �̄+�

�̄−�(d1 + d2)wf (�1)f (�2) d�1 d�2. (9)

Accordingly, we have Proposition 3.

Proposition 3. The supplier’s optimal wholesale price is given as

w∗NI = (�1 + �2)

4�1+

[2�2�1 + �1(�2 − �2)](�2 + �2) ln�̄ + ε

�̄ − ε

8�1(�1 + �1)2ε

.

Proof. See Appendix. �

Proposition 3 provides us the optimal wholesale price if the manufacture has no complete information about theretailers’ cost structure. The managerial insight of Proposition 3 is that if no retailer is willing to reveal his informationto the upstream supplier, the wholesale price is decided by the average industry’s cost efficiency. Also it can be proved�w∗

NI/�ε > 0, which means that the more dispersed the value-added cost structure of the retailing industry (or more

844 D.-Q. Yao et al. / Omega 36 (2008) 838–851

differentiated the retailers are), the higher the wholesale price is. In other words, the supplier could be better off inmore heterogeneous retailing markets.

4.3. Case 3: vertical information sharing between one retailer and the supplier

Without loss of generality, we assume retailer 1 decides to share information with the manufacture due to symmetriccharacteristics, i.e., the supplier has full knowledge of the retailer 1’s cost structure �1. Thus the objective of the supplieris to find an optimal wholesale price to maximize his expected profit.

Maxw

E[�M ] =∫ �̄+ε

�̄−ε

(d1 + d2)wf (�2) d�2. (10)

Proposition 4. If one retailer discloses his cost information with the supplier, the optimal wholesale price is given as

w∗PI = �1 + �2

4�1+ 2(�2 + �2)[3�2�

21 + �2

1(�2 − 3�2) + �1�1(5�2 − 2�2)]8�1(�1 + �1)

2(2�1 + 3�1)�i

+ (�2 + �2)[3�2�21 + �2

1(�2 + �2) + �1�1(2�2 − �2)]8�1(�1 + �1)

2(2�1 + 3�1)�ln

�̄ + �

�̄ − �(i = 1, 2).

Proof. Same as Proposition 3. �

Also it can be proved that �w∗PI/�ε > 0, which suggests that the wholesale price increases in the value-added cost

dispersion if only one retailer is willing to disclose his cost information.

5. Equilibrium study in the first stage

In this section, we will study the retailers’ information revelation mechanism in the first stage, i.e., under whatconditions both retailers are willing to share their cost information with the supplier; under what conditions neitherretailer is willing to share his cost information; and under what conditions one of the retailers is willing to share hiscost information while the other is not. Also we will investigate if equilibrium exists in the first stage. First we willcompare the supplier’s profits under different vertical information sharing cases.

5.1. Supplier’s profits

We define the supplier’s profit notations according to the different cases:�M(0): The supplier’s profit under the case of no vertical information sharing with any retailer.�M(1): The supplier’s profit under the case of vertical information sharing with one retailer.�M(2): The supplier’s profit under the case of full vertical information with both retailers.

Proposition 5. �∗M(2)��∗

M(1) and �∗M(2)��∗

M(0)

Proof. See Appendix. �

This proposition indicates that the supplier is always better off with full vertical information sharing than with noinformation sharing or with partial information sharing. This result is consistent with other research (e.g. [18,19]).

Furthermore we are interested in comparing the partial information case and the no information sharing case, i.e.,is the supplier always better off by obtaining partial information than no information at all? Towards this, we have thefollowing proposition.

D.-Q. Yao et al. / Omega 36 (2008) 838–851 845

Proposition 6. Suppose retailer i (i = 1, 2) reveals his information to the supplier, and if

�3−i < �i < �̃ or �3−i > �i > �̃ then �∗M(1) > �∗

M(0) where �̃ = 2ε

ln

(�̄ + ε

�̄ − ε

) .

Proof. See Appendix. �

This proposition suggests that if the supplier shares information with one retailer, for example with retailer 1,his profit will be better off under the following two situations: (1) retailer 1’s value-added cost is efficient enough(�1 < �̃) and the other retailer is more efficient enough; (2) retailer 1’s value-added cost is not efficient enough (�1 > �̃)and the other retailer is even worse. To sum up, the supplier is better off by accessing one retailer’s informationonly if the other retailer has higher (smaller) cost parameter than this retailer when this retailer’s cost is higher(smaller) than the threshold value �̃. Therefore even if one retailer is willing to disclose his private informationvertically, the supplier may not accommodate that information into his decision process because the supplier may not bebetter off.

Also it can be proved d�̃/dε < 0, that is, the more dispersed the retailing industry is, the smaller the thresholdshould be.

5.2. Retailer’s profits

In this section, we will establish conditions under which both retailers have incentives to disclose their privateinformation, or neither retailer would share information vertically.

Similar to Zhang [19], we first define the following notations for the retailers’ profits:�Ri

(1s , 2n): Retailer i’s (i =1, 2) profit under the condition that retailer 1 shares information with the supplier whileretailer 2 does not.

�Ri(1n, 2n): Retailer i’s profit under the condition that neither retailer shares information with the supplier.

�Ri(1s , 2s): Retailer i’s profit under the condition that both retailers share information with the supplier.

�Ri(1n, 2s): Retailer i’s profit under the condition that retailer 2 shares information with the supplier while retailer

1 does not.Correspondingly we have the following conclusions:(1) For retailer 1: �R1(1s , 2n) < �R1(1n, 2n) and �R1(1s , 2s) < �Ri

(1n, 2s),For retailer 2: �R2(1n, 2s) < �R2(1n, 2n) and �R2(1s , 2s) < �R2(1s , 2n),which means that each retailer, if he will be worse off by sharing information with the supplier regardless of the other

retailer’s decision, will not be willing to reveal his private information vertically. As a result, no vertical informationsharing is a unique equilibrium in the first stage of game. In the franchising industry, each franchisee holds private costinformation. The franchisee will receive a higher margin if he has greater advertising effectiveness. Therefore he is notwilling to reveal his private information.

(2) For retailer 1: �R1(1s , 2n) > �R1(1n, 2n) and �R1(1s , 2s) > �R1(1n, 2s),For retailer 2: �R2(1n, 2s) > �R2(1n, 2n) and �R2(1s , 2s) > �R2(1s , 2n),which means if each retailer will be better off by sharing information with the supplier regardless of the other retailer’s

decision, both retailers are willing to reveal their private information vertically. Therefore vertical information sharingis a unique equilibrium in the first stage of game. This situation is desirable for the whole supply chain because everyplayer will be better off by information sharing.

For simplicity of exposition, we assume �1 = �2 = �, �1 = �2 = � and �1 = �2 = � to analyze the retailers’ profits.Accordingly we have the following propositions:

Proposition 7. (1) If the market base � is large enough, �1 < �̃ and �2 < �̃, no information sharing is the uniqueequilibrium in the first stage of the game.

Proof. See Appendix. �

846 D.-Q. Yao et al. / Omega 36 (2008) 838–851

The managerial insight of Proposition 7 is that both retailers are unwilling to disclose their cost information ifthey are efficient. The intuition underlying the impact of cost structure information sharing on retailers’ profits is asfollows. When retailers are cost efficient, i.e., �1 < �̃ and �2 < �̃, the supplier sets a higher wholesale price in thecase of information sharing than it does in the case of no information sharing. This is because the wholesale price,from Proposition 2, is a decreasing function of �1 and �2. Naturally, retailers would have incentive to share their costinformation so as to motivate a lower wholesale price. However, when �1 < �̃ and �2 < �̃, the wholesale price in thecase of information sharing is higher than the price in the case of no information sharing, so retailers would haveno interest in revealing their information. This underscores the importance of the threshold value �̃ for the retailers’decision-making process. Thus these results give us important insights into the channel dynamics under asymmetricinformation.

Proposition 8. If the market base � is large enough, �1 > �̃ and �2 > �̃, vertical information sharing is the uniqueequilibrium in the first stage of the game.

Proof. Same as Proposition 7. �

The managerial insight of Proposition 8 is that both retailers have incentive to share cost information, which alsobenefits the supplier because the supplier does not need to pay either retailer for his private cost information. At thesame time the retailer’s profit can be improved as well. Therefore this is a win–win situation for all players, and thereis no information asymmetry in this case.

Proposition 9. If the market base � is large enough, �1 > �̃ and �2 < �̃, or vice versa, then one retailer is willing todisclose information to the supplier while the other is not.

Proof. Same as Proposition 7. �

Note Proposition 9 is not necessary an equilibrium due to the following reason: suppose �1 > �̃ and �2 < �̃, thenretailer 1 is willing to share information with the supplier while retailer 2 is not. From Proposition 6, it is possiblethat the manufacturer may not accept the retailer 1’s information because he may not be better off by only acquiringone retailer’s information. The managerial insight of Proposition 9 is that equilibrium may not exist if one retaileris efficient while the other is not. The supplier’s decision about the wholesale price may be based on both retail-ers’ or only one retailer’s information. Therefore, it is impossible for one retailer to infer the other’s private costinformation through information leakage. This also can justify our assumption of no information leakage in thisresearch.

However, since the supplier is always better off by sharing information with two retailers rather than one, one naturalquestion arises: If only one retailer is willing to reveal his private information to the supplier, can the supplier inducethe other retailer to disclose his information by some side payment? For example, retailer 1 is willing to disclose hiscost information while retailer 2 is not. From Proposition 9, we know that n1 < �̃ and �2 < �̃. In order for the supplierto benefit from information sharing by side payment, the following condition must be satisfied:

�M(1s , 2s) − �M(1s , 2n) > �R2(1s , 2n) − �R2(1s , 2s).

Here the left side is the upper bound the supplier is willing to pay, and the right side is the loss retailer 2 incurs due todisclosing his cost information.

Towards this, we have the following Proposition.

Proposition 10. If one retailer is willing to disclose his cost information while the other is unwilling to, the supplieralways can not afford to pay the latter to obtain his private cost information.

Proof. See Appendix. �

The managerial insight of Proposition 10 is that although the supplier is always better off by accessing two retailers’cost information, he cannot gain enough to cover the loss of the retailer who is unwilling to disclose his information.

D.-Q. Yao et al. / Omega 36 (2008) 838–851 847

Simply put, side payment is not a strategy to extract more accurate information from a downstream player. Actually, ifa retailer is cost efficient, he can gain more by not sharing information with the supplier. For example, a manufacturersupplies a certain product (e.g., Haier washer) to two retailers/distributors. If one is a very efficient retailer/distributor,like Wal-Mart, it will not share its private information with the supplier. Even if the supplier is willing to pay some sidepayment to Wal-Mart for its private information, the extra payoff the supplier gains by obtaining Wal-Mart’s privateinformation can not compensate for Wal-Mart’s loss.

6. Concluding remarks

In this paper, we study a supply chain system with one supplier and two value-adding retailers. Both retailers haveprivate information about their own cost structure that is unknown to the supplier and the other retailer. We model thesupply chain problem as a three-stage game and prove that under certain conditions, equilibrium exists. We also findthat the supplier is always better off by obtaining private information from both retailers. Furthermore we find that bothretailers may have incentive to share their private information with the supplier under certain conditions, thus leadingto a win–win situation for the whole supply chain.

This research can be extended in several directions. First, because the supplier is better off sharing information withboth retailers, if neither of the retailers reveals their information, we may study if it is feasible for the supplier to useside payment to pay just one, then which one, or both of the retailers to acquire information. Second in this paper, weassume cost efficiency is uniformly distributed; we may relax this to other probability distributions and investigate ifthe managerial insights obtained here are applicable to other situations. Another limitation of this research is we donot consider the effect of horizontal information leakage for vertical information sharing in a supply chain [24], i.e.,each retailer may infer the other’s cost structure through the wholesale price announced by the supplier, which couldbe a very interesting topic in the future research. However, if there are numerous heterogeneous retailers in the market,it is very hard for one retailer to infer the others’ private cost information.

Acknowledgements

We thank anonymous reviewers and the associate editor for their helpful comments on the revision of the paper.

Appendix A

A.1. Proofs of All Results

Proof of Proposition 1. For the retailer 1:

Maxp1,v1

E[�R1 ] = E[(p1−w−c1)d1] =∫ �̄+�

�̄−�(p1−w−c1)d1f (�2) d�2 = 1

∫ �̄+ε

�̄−ε

(p1−w−c1) d1 d�2. (A.1)

For the retailer 2:

Maxp2,v2

E[�R2 ] = E[(p2−w−c2)d2] =∫ �̄+�

�̄−�(p2−w−c2)d2f (�1) d�1 = 1

∫ �̄+ε

�̄−ε

(p2−w−c2) d2 d�1. (A.2)

From the first-order condition (FOC), we have the optimal response function for retailer 1:

�E[�R1 ]�p1

= 0,�E[�R1 ]

�v1= 0,

p∗1 = 1

4�1 + 4�1

[2�1 + 2�1w + 2�1p2 + 2�1w − 2�2v2 + 3(�2 + �2)

2

(�1 + �1)�1

], (A.3)

v∗1 = �2 + �2

(�1 + �1)�1. (A.4)

848 D.-Q. Yao et al. / Omega 36 (2008) 838–851

For retailer 2, the optimal response function is obtained from

�E[�R2

]�p2

= 0,�E

[�R2

]�v2

= 0,

p∗2 = 1

4�1 + 4�1

[2�2 + 2�1w + 2�1p1 + 2�1w − 2�2v1 + 3(�2 + �2)

2

(�1 + �1)�2

], (A.5)

v∗2 = �2 + �2

(�1 + �1)�2. (A.6)

Eq. (A.3) is the best response function of retailer 1. Since retailer 1 has incomplete information of retailer 2, he has nocomplete knowledge about retailer 2’s price p2. Thereby retailer 1 will find the expected retail price for retailer 2.

p2 =∫ �̄+ε

�̄−ε

{1

4�1 + 4�1

[2�2 + 2�1w + 2�1p1 + 2�1w − 2�2v1 + 3(�2 + �2)

2

(�1 + �1)�2

]}f (�2) d�2. (A.7)

Similarly the retailer 2 will find the expected retail price for retailer 1.

p1 =∫ �̄+ε

�̄−ε

{1

4�1 + 4�1

[2�1 + 2�1w + 2�1p2 + 2�1w − 2�2v2 + 3(�2 + �2)

2

(�1 + �1)�1

]}f (�1) d�1. (A.8)

Substituting (A.7) into (A.3), (A.8) into (A.5), and solving (A.3), (A.5) simultaneously, we obtain the equilibriums forthe retailers:⎧⎪⎨

⎪⎩p∗

1 = A

�1+ B1 + C

εln

�̄ + ε

�̄ − ε,

v∗1 = �2 + �2

(�1 + �1)�1

(A.9)

and ⎧⎪⎨⎪⎩

p∗2 = A

�2+ B2 + C

εln

�̄ + ε

�̄ − ε,

v∗2 = �2 + �2

(�1 + �1)�2,

(A.10)

where

A = (�2 + �2)[3�1(�2 + �2) + �1(3�2 + 2�2)](�1 + �1)(2�1 + �1)(2�1 + 3�1)

,

B1 = �2�1 + 2�1(�1 + �1) + w(�1 + �1)(2�1 + 3�1)

(2�1 + �1)(2�1 + 3�1),

B2 = �1�1 + 2�2(�1 + �1) + w(�1 + �1)(2�1 + 3�1)

(2�1 + �1)(2�1 + 3�1),

C = (�2 + �2)[3�2�1 − (4�1 + �1)�2]4(�1 + �1)(2�1 + �1)(2�1 + 3�1)

. �

Proof of Corollary 1. We only need to prove that

d

⎡⎢⎢⎣

ln�̄ + ε

�̄ − ε

ε

⎤⎥⎥⎦

dε> 0.

D.-Q. Yao et al. / Omega 36 (2008) 838–851 849

It can be derived that

d

⎡⎢⎢⎣

ln�̄ + ε

�̄ − ε

ε

⎤⎥⎥⎦

dε=

1 + �̄ + ε

�̄ − ε

ε(�̄ + ε)−

ln�̄ + ε

�̄ − ε

ε2

when ε = 0,

1 + �̄ + ε

�̄ − ε

ε(�̄ + ε)>

ln�̄ + ε

�̄ − ε

ε2,

when ε > 0,⎛⎜⎜⎝

1 + �̄ + ε

�̄ − ε

ε(�̄ + ε)

⎞⎟⎟⎠

>

⎛⎜⎜⎝

ln�̄ + ε

�̄ − ε

ε2

⎞⎟⎟⎠

.

Thus we prove

1 + �̄ + ε

�̄ − ε

ε(�̄ + ε)>

ln�̄ + ε

�̄ − ε

ε2,

therefore

d

⎡⎢⎢⎣

ln�̄ + ε

�̄ − ε

ε

⎤⎥⎥⎦

dε> 0. �

Proof of Proposition 2. Obtained by substituting Eq. (A.9) and Eq. (A.10) into Eq. (8) directly, then set ��M/�w = 0and solve for w because �1 and �2 are both known to the supplier in this case. �

Proof of Proposition 3. Eq. (9) can be simplified as

Maxw

E[�M ] =w[2(�1 + �2 − 2�1w)(�1 + �1)

2ε] + w[2�2�1 + �1(�2 − �2)](�2 + �2) ln�̄ + ε

�̄ − ε

2(�1 + �1)(2�1 + �1)ε. (A.11)

Setting �E[�M ]/�w = 0, we obtain the optimal wholesale price under no information sharing case (w∗NI)

w∗NI = (�1 + �2)

4�1+

[2�2�1 + �1(�2 − �2)](�2 + �2) ln�̄ + ε

�̄ − ε

8�1(�1 + �1)2ε

. � (A.12)

Proof of Proposition 5. We assume the supplier shares information with retailer 1. By subtracting �∗M(1) from �∗

M(2)

and simplifying the expression, we obtain

�∗M(2) − �∗

M(1) =[3�2�

21 + �2

1(�2 − 3�2) + �1�1(5�2 − 2�2)]2(�2 + �2)2[

2� − �2 ln

(�̄ + �

�̄�

)]2

32�1(�1 + �1)3(2�1 + �1)(2�1 + 3�1)

2�2�22

�0.

We can get the same result if retailer 2 shares information with the supplier.

850 D.-Q. Yao et al. / Omega 36 (2008) 838–851

With the same approach, we have

�∗M(2)−�∗

M(0) =[3�2�

21+�2

1(�2−3�2)+�1�1(5�2−2�2)]2(�2+�2)2[ε(�1+�2)−�1�2 ln

(�̄+ε

�̄−ε

)]2

8�1(�1+�1)3(2�1+�1)(2�1+3�1)

2ε2�21�

22

�0. �

Proof of Proposition 6. Without loss of generality, we assume retailer 1 reveals his cost information to the supplier.By some algebra, we obtain

�∗M(1)−�∗

M(0)

=[3�2�

21+�2

1(�2−3�2)+�1�1(5�2−2�2)]2(�2+�2)2[

2ε(2�1+�2)−3�1�2 ln

(�̄+ε

�̄−ε

)][2ε−�1 ln

(�̄+ε

�̄−ε

)]32�1(�1+�1)

3(2�1+�1)(2�1+3�1)2ε2�2

1�22

if

�2 < �1 <2ε

ln

(�̄ + ε

�̄ − ε

) ,

we can verify that

2� − �1 ln

(�̄ + �

�̄ − �

)> 0 and

2�(2�1 + �2) − 3�1�2 ln

(�̄ + ε

�̄ − ε

)> 0.

Thus �∗M(1) − �∗

M(0) > 0.Similarly if

�2 > �1 >2ε

ln

(�̄ + ε

�̄ − ε

) ,

then

2ε − �1 ln

(�̄ + ε

�̄ − ε

)< 0 and 2�(2�1 + �2) − 3�1�2 ln

(�̄ + �

�̄ − �

)< 0

thus �∗M(1) − �∗

M(0) > 0.We obtain the same results if retailer 2 reveals his information to the supplier.

Proof of Proposition 7. First we consider

�R1 (1s , 2n) − �R1 (1n, 2n)

=

−�(�+�)2

⎧⎪⎪⎨⎪⎪⎩−4�(�+�)�1+�2[7�2+6�(�+2��1)+�(15�+8��1)] − (�+�)(3�+2�)�1�2

ln

(�̄+ε

�̄ − ε

)2ε

⎫⎪⎪⎬⎪⎪⎭

⎡⎢⎢⎣1 − �1

ln

(�̄+ε

�̄ − ε

)2ε

⎤⎥⎥⎦

16(2�+�)2(2�+3�)2�21�2

.

We assume the market base � is large enough, which is a very realistic assumption. If

�1 < �̃

⎛⎜⎜⎝�̃ = 2ε

ln

(�̄+ε

�̄ − ε

)⎞⎟⎟⎠ , i.e., 1 − �1

ln

(�̄ + ε

�̄ − ε

)2ε

> 0,

then �R1(1s , 2n) < �R1(1n, 2n).

D.-Q. Yao et al. / Omega 36 (2008) 838–851 851

Similarly for �R1(1s , 2s) − �R1(1n, 2s) < 0, �1 < �̃ must hold.For retailer 2, since retailer 2 and retailer 1 are symmetric in our model setting, so the condition is �2 < �̃. �

Proof of Proposition 10. Without loss of generality, we assume retailer 1 discloses his information to the supplierwhile retailer 2 does not. We need to solve the following inequality:

�M(1s , 2s) − �M(1s , 2n) > �R2(1s , 2n)−�R2(1s , 2s).

By some algebra,[�M(1s , 2s) − �M(1s , 2n)] − [�R2(1s , 2n) − �R2(1s , 2s)

]

=

�(�+�)2

{−2(3�2+9��+4�2)��1+4��2[�2+�(3�−4��1)+2�(�−3��1)]

−3�(�+�)�1�2 ln

(�̄ + �

�̄ − �

)}[2�−�2 ln

(�̄ + �

�̄ − �

)]

64(2� + �)2(2� + 3�)2�2�1�22

.

Because we assume market base � is large enough, then for [�M(1s , 2s)−�M(1s , 2n)]−[�R2(1s , 2n)−�R2(1s , 2s)] > 0,we solve for �2 and obtain the condition that �2 > �̃, which contradicts our assumption that retailer 2 is unwilling to sharehis cost information with the supplier (�2 < �̃). Therefore it is proved that �M(1s , 2s) − �M(1s , 2n) < �R2(1s , 2n) −�R2(1s , 2s). �

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