Information acquisition and transparency in a supply chain with asymmetric production cost information

Int. J. Production Economics 182 (2016) 449–464

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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Information acquisition and transparency in a supply chain with asymmetric production cost information Song Huanga, Jun Yangb, a b

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College of Economics and Management, South China Agricultural University, Guangzhou 510642, China School of Management, Huazhong University of Science and Technology, Wuhan 430074, China

A R T I C L E I N F O

A BS T RAC T

Keywords: Forecasting Transparency Information acquisition Endogenous adverse selection

This paper studies a retailer outsourcing the production to a supplier who can improve the quality of the production cost information by exerting costly forecasting effort. The outcome of the supplier's information acquisition may turn out to be either successful, with the supplier becoming informed, or unsuccessful, with the supplier remaining uninformed. Once the outcome of the forecasting is resolved, the supplier knows the information status (informed or uninformed) and content (high type or low type). We consider two-layer information asymmetry and analyze three different scenarios: no forecasting, forecasting with transparent information acquisition (disclosing information status) and forecasting with nontransparent information acquisition (hiding information status). We study both the retailer's contract design and the supplier's information disclosure decision. We obtain some interesting observations. First, the retailer's incentive for the supplier's forecasting is a threshold policy: If the forecasting cost is low, then the retailer will prefer the supplier to forecast, otherwise, the retailer will prefer the supplier not to forecast. Second, when the forecasting cost is high and the production cost variance is small, under transparent information acquisition, the high cost supplier's production quantity may be either upward or downward distorted; while under nontransparent information acquisition, the uninformed supplier's production quantity is either upward or downward distorted, and the high cost supplier's production quantity is always downward distorted. At last, the supplier can benefit from transparency only under some specific conditions, and when the production cost variance is extremely large, nontransparent information acquisition is always the supplier's first choice.

1. Introduction Global outsourcing has been increasingly popular in the past a few decades with the development of economic globalization. Many giant brand retailers, such as Apple, Huawei, and Nike, are outsourcing the production of their products to contract suppliers. For example, the production of iPhone cellphone of Apple is conducted in China factories owned by the Taiwan-based company known as Foxconn Technology Group (Zhang, 2012). This outsourcing strategy makes the retailer concentrate more efficiently on the core businesses, such as designing, development and marketing, and enhances the retailer's competence in the market. Meanwhile, the supplier can efficiently focus on improving the production process control. Fierce competition among the retailer, especially in the industries of electronics and apparels, has made the production cost control extremely important for the success of pricing and marketing decisions, which will eventually impact the retailer's competitiveness. Usually, the production cost is affected by many factors, such as the prices of raw materials, the

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complexity of production process, wages of skilled workers, and sometimes even the oil price, which has been experiencing continuous price fluctuations. For example, rare earth elements, which are widely used in fluorescent lamp, LED device, mobile phone, and some other semi-conductor industries, have experienced dramatic price fluctuations since 2011. It is reported that the prices of rare earth elements increased 10 times in 2011, leading to a price increase of 37% for the energy saving lamp industry (Bradsher, 2011). Before a production process is initiated, the real production cost is usually unknown to both parties. The supplier, who is more familiar with the production environment, is in the ideal position to invest resources to acquire fresh production cost information. For example, the supplier can devote recourses to analyze the price trends of the critical materials, hire more talented experts to manage the production processes, and use more advanced software to help control the cost, to acquire accurate production cost information. The retailer is also willing to know the accurate production cost information, because accurate production cost information can benefit the ordering deci-

Corresponding author. E-mail addresses: [email protected] (S. Huang), [email protected] (J. Yang).

http://dx.doi.org/10.1016/j.ijpe.2016.10.005 Received 15 July 2015; Received in revised form 23 June 2016; Accepted 19 September 2016 Available online 05 October 2016 0925-5273/ © 2016 Elsevier B.V. All rights reserved.

Int. J. Production Economics 182 (2016) 449–464

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sions. Since information acquisition is usually not free, whether the supplier will find it attractive to exert such costly forecasting effort depends critically on the contractual terms offered by the retailer. In addition, even if the supplier decides to conduct information acquisition activities, the outcome of which is usually unobserved by the retailer, which means that the supplier's production cost information is only privately known (Çakanyıldırım et al., 2012; Lei et al., 2015). Thus, the retailer is facing an adverse selection problem but with a major modification. The retailer now should provide contracts that are effective both to encourage the supplier to forecast and to elicit the supplier's private production cost information. This problem is called the endogenous adverse selection problem. Are the traditional contracts with asymmetric information still valid in the endogenous adverse selection problem? If not, what are the optimal contracts? How does the forecasting cost impact the optimal contracts? In addition, the extant literature on information acquisition mainly assumes that information can always be acquired after forecasting. However, the supplier may obtain nothing useful after exerting costly forecasting effort. The outcome of the forecasting may turn out to be either successful or unsuccessful. Following Li et al. (2014), we define the supplier's information status as being informed or uninformed, depending on whether or not the supplier succeeds in getting useful information after forecasting. The term information status is different from the term information content. The former refers to whether the supplier gets useful information, while the latter refers to what the supplier really gets. Since both the information status and the information content are the supplier's private knowledge, the supplier can decide whether or not to disclose the information status (not the information content) to the retailer. If the supplier voluntarily discloses the information status to the retailer, we refer to this kind of information acquisition as the transparent information acquisition. Otherwise, we refer to it as the nontransparent information acquisition. Under transparent or nontransparent information acquisition, how should the retailer design the contracts? Is transparent information acquisition always beneficial to the supplier? If not, under what conditions? To answer these questions, we consider a supply chain composed of a giant retailer and a supplier. The retailer can choose to induce the supplier to forecast or not to forecast. We consider both the retailer's forecasting decision and the supplier's information status disclosure decision. The retailer should determine under what conditions to forecast, and what is the impact of the retailer's forecasting cost on the optimal contracts. The supplier should determine whether to disclose the information status to the retailer after forecasting. The extant literature usually consider these two problems separately, however, since the information acquisition decision and the information status disclosure decision are conducted individually by the retailer and the supplier, it is necessary to address these two problems in a unified framework. We have obtained some interesting observations. First, the retailer's incentive for the supplier's forecasting is a threshold policy, no matter whether the supplier chooses to share the information status or not. If the forecasting cost is low, then the retailer will prefer the supplier to forecast, otherwise, the retailer will prefer the supplier not to forecast. Second, when the forecasting cost is high and the production cost variance is small, under transparent information acquisition, the high cost supplier's production quantity may be either upward or downward distorted, but the low cost supplier's and the uninformed supplier's production quantities are not distorted. While under nontransparent information acquisition, the low cost supplier's production quantity remains unchanged, the uninformed supplier's production is either upward or downward distorted, and the high cost supplier's production quantity is always downward distorted. At last, the supplier can benefit from transparency only under specific conditions, but when the production cost variance is extremely large, nontransparency is always the supplier's first choice.

The rest of the paper is organized as follows. Section 2 reviews the relevant literature. In Section 3, we present the model setup and analyze the optimal decisions in an integrated supply chain. Section 4 derives the retailer's optimal contract when the retailer does not induce the supplier to forecast. Section 5 analyzes the scenario of transparent information acquisition, and Section 6 analyzes the scenario of nontransparent information acquisition. Section 7 concludes. All proofs are relegated to the Appendix A. 2. Literature review This paper falls into the screening models in supply chain contracting literature (Chen, 2003; Cachon and Lariviere, 2001; Ha, 2001; Özer and Wei, 2006; Ha and Tong, 2008; Ha et al., 2011; Zhang et al., 2010; Lei et al., 2012; Çakanyıldırım et al., 2012; Dai and Chao, 2016). Chen (2003) provides an excellent review of the literature. Cachon and Lariviere (2001) consider the capacity reservation contract with asymmetric demand information. Özer and Wei (2006) study both the capacity reservation contract the advance purchase contract with asymmetric forecasting information to ensure credible demand information sharing. Ha and Tong (2008) and Ha et al. (2011) study the issue of contracting and information sharing with competing supply chains. Zhang et al. (2010) investigate the contracting under asymmetric inventory information. In a paper also studying the contracting under asymmetric cost information, Çakanyıldırım et al. (2012) assume nonzero reservation profits for the manufacturer. They find that in equilibrium, the optimal production may be overproduction, efficient production or underproduction, depending on the level of reservation profit. These papers consider the supply chain contracting under various asymmetric information scenarios and assume that the agent is freely endowed with the private information. Our work differs the above literature in that we incorporates both the supplier's information acquisition decision and information status disclosure decision into the framework of supply chain contracting with asymmetric production cost information. Our work is also related to the literature on contracting to induce the downstream firm to forecast by exerting forecast effort (Taylor and Xiao, 2009; Fu and Zhu, 2010; Shin and Tunca, 2010; Chen et al., 2016). Taylor and Xiao (2009) investigate the contracting in a newsvendor model when the retailer can improve the quality of the demand information by exerting costly forecasting effort. They study two kinds of contracts, rebates and returns, and find that returns are superior to rebates. Fu and Zhu (2010) investigate contracting to induce the downstream retailer to exert costly forecasting effort and examine the performance of several commonly used contracts. Shin and Tunca (2010) study the effect of downstream competition on incentive for demand forecast investments in supply chains. Chen et al. (2016) study the contracting to induce the retailer to improve demand forecast and sales effort simultaneously by exerting costly information acquisition effort. All these papers assume that once the agent forecasts, the agent can obtain useful information and the principal knows the agent's information status. In other words, information acquisition is always successful and the retailer's information status is transparent to the manufacturer. Sometimes, the agent may be reluctant to share the information status with the principal especially when the principal also cooperates with other competing agents, because disclosing such information to the principal will make the agent lose competitive advantage. Therefore, we consider the situation of nontransparent information acquisition in this paper, that is, the principal does not know the agent's information status after forecasting. In a paper closely related to our work, to the best of our knowledge, Li et al. (2014) first consider the impact of information transparency on the optimal contacts under demand asymmetry. They find that under some specific conditions, sharing the information status with the manufacturer is superior to holding back the information. In other words, there are benefits that are foregone by following strict con450

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fidentiality but can be potentially recovered by switching to a policy of partial confidentiality. Our work is distinct from theirs in the following aspects. First, they consider the demand asymmetry while we consider the production cost asymmetry, which is also very common in the outsourcing practice. Second, we incorporate the forecasting cost into our model and find that the existence of forecasting cost significantly affects the upstream retailer's incentive decisions and the optimal contracts. Third, we analyze the conditions under which the retailer should induce the supplier to forecast, and the impact of the forecasting cost on the supplier's information status disclosure decision.

forecasting or not, but does not know which type the supplier is, then this information acquisition is referred to as the transparent information acquisition. On the contrary, if the retailer neither knows whether the supplier succeeds in forecasting nor which type the supplier is, this information acquisition is referred to as the nontransparent information acquisition. In face of two-layer information asymmetry, designing appropriate compensation plans for outsourcing products can be very important for the retailer. Under this framework, the retailer should determine whether to induce the supplier to forecast and how to devise optimal contracts. By contrast, the supplier should decide whether to disclose his information acquisition status (not the information content) to the retailer. We assume the retailer uses a quantity-payment bundle contract to extract the supply chain profit, which is very common in the principalagent literature (Tirole, 1988; Laffont and Martimort, 2002; Taylor and Xiao, 2010; Li et al., 2014). Under symmetric information scenario, it is well known that the retailer can devise a menu of contracts to extract the entire supply surplus, leaving the supplier only the reservation profit. Under asymmetric information scenario, a menu of quantitypayment bundle contracts, taking the form of nonlinear pricing, can achieve the second best outcome (Tirole, 1988; Laffont and Martimort, 2002). For ease of exposition and without loss of generality, we normalize the supplier's reservation profit to zero, which is very common in operations management literature (Taylor and Xiao, 2010; Li et al., 2014). The timeline of the model is illustrated in Fig. 1.

3. Model setup and baseline case 3.1. Model setup We consider a supply chain composed of one supplier (he) and one retailer (she). The retailer relies on the supplier to produce the product to serve the end customers. Assume that the market demand, by means of analyzing historical sales data, is deterministic and known. Following the operations management literature with asymmetric information (Guo, 2009; Shin and Tunca, 2010; Li et al., 2014; Bian et al., 2016), we assume that the inverse demand function is p=a−q, where a is the market potential, p is the market clearing price, and q is the quantity the retailer sets to sell. Assume that the production cost C is a nonnegative random variable, which is unknown before the production starts. However, the distribution of C is C=cH with possibility λ and C=cL with possibility 1–λ. Assume that λ is common knowledge. The expectation of C is denoted by cN=λcH+(1–λ)cL. Obviously, cH > cN > cL. Both parties do not know the real production cost, but know the distribution of the production cost. Before the production, the supplier can obtain an forecast of C, by exerting a forecasting cost κ (Taylor and Xiao, 2009). Since the outcome of the forecast may turn out to be either successful or unsuccessful, following Li et al. (2014), we assume that after forecasting, the information status of the supplier has two possible values: becoming informed of his true production cost type with possibility μ when the forecasting turns out to be successful, or remaining uninformed of his true production cost type with possibility 1–μ when the forecasting turns out to be unsuccessful. The value of μ is the probability that the forecasting turns out to be successful. The larger μ is, the more likely that the supplier succeeds in forecasting. Similar to Li et al. (2014), we assume that μ is common knowledge and is known by the two parties as a prior, and it is exogenously determined by the nature and has nothing to do with the forecasting cost. Given that the supplier acquires useful information after forecasting, he becomes informed of his real production cost, either cH or cL. However, since the forecast is unobserved by the retailer, the retailer does not know the supplier's real production cost, but he knows that the supplier's expected production cost is cN. We consider information acquisition and two-layer information asymmetry (information status and information content) in this paper. Specifically, if the retailer knows whether the supplier succeeds in

1. The retailer decides whether to induce the supplier to forecast or induce the supplier not to forecast. 2. The supplier commits to the supplier whether to disclose his information status to the retailer if he forecasts. 3. The retailer either offers a single contract to induce the supplier not to forecast, or offers a menu of contracts to induce the supplier to forecast. 4. If the retailer offers the contract that induces no forecasting, then the supplier chooses to accept it or reject it. If the retailer offers the contract that induces forecasting, then the supplier determines whether to accept it or reject it. If he accepts it, then he forecasts. He acquires a private forecast which is unobservable to the retailer. The forecasting is either successful and makes the supplier informed of his true type with possibility μ, or unsuccessful and makes the supplier remain uninformed with possibility 1–μ. Under successful forecasting, the supplier knows his true production cost type (either high type or low type). 5. The supplier produces the required quantity to meet the retailer's requirement. 6. Production is proceeded and the true production cost is realized. The demand is then realized. The retailer pays the supplier. We assume that once the supplier commits to disclosing the information status, this commitment is valid and can be verified (Laffont and Martimort, 2002). Our main research questions are as follows. (1) Will the retailer prefer the supplier to forecast? (2) And

Fig. 1. The timeline of the model.

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s.t.

when the retailer determines to induce the supplier to forecast, how would she design the contracts to maximize her expected profit? (3) Is the supplier willing to disclose his information status (not the information content) to the retailer? Our analysis is done from two distinct perspectives. First, from the perspective of the retailer, we characterize the optimal contracts that maximizing her expected profit while ensuring that the supplier obtains reservation profits. Second, from the perspective of the supplier, we characterize the supplier's information status disclosure decision: whether to make his information status transparent or nontransparent to the retailer.

(4)

tN − cN qN ≥ μλ max {tN − cH qN , 0} + μ (1 − λ )max {tN − cL qN , 0} + (1 − μ)max {tN − cN qN , 0} − κ

(5)

The objective function in (3) is the retailer's profit if she offers the contract (qN,tN). The first term in (3) is the retailer's sales revenue and the second term in (3) is her transfer payment to the supplier for producing qN products. The individual rationality constraint in (4) ensures that the contract satisfies the supplier's participation constraint, so that the nonforecasting supplier will accept the contract. The forecasting constraint in (5) ensures that it is in the supplier's interest not to forecast. The left hand side of (5) is the supplier's profit if he does not forecast, and the right side of (5) is his expected profit after forecasting: with probability μ he becomes informed (The real cost is either cH or cL), and with probability 1–μ he remains informed. Note that the supplier may reject the contract and obtain zero profit if he finds that the contract is unprofitable. For ease of exposition, it is 1 convenient to define Γ (c ) ≡ 2 μλ (1 − λ )(cH − cL )(a − c ), which is decreasing in c.

3.2. Baseline case: integrated supply chain We first consider the optimal solution under symmetric information in the integrated supply chain as the benchmark. In the integrated supply chain when the decision-maker chooses not to forecast, she does not know the real production cost and only knows that the expected production cost is cN. The supply chain's expected total payoff is given by

πSC (qN ) = EC [(a − qN ) qN − CqN ]00=(a − qN ) qN − [λcH + (1 − λ ) cL ] qN = (a − qN ) qN − cN qN ,

tN − cN qN ≥ 0

(1)

Proposition 1. An optimal contract (qNnf , tNnf ) that does not induce the supplier to forecast is.

where the subscript SC denotes the supply chain and the qN denotes uninformed supplier's ordering quantity. The optimal production quan1 1 I (qNI ) = 4 (a − cN )2 . tity qNI = 2 (a − cN ), and the optimal profit πSC If the decision-maker chooses to forecast, then she incurs forecasting cost κ, acquiring a signal, and chooses the optimal production quantity. Conditional on the signal she obtained, the supply chain's expected payoff is given by

⎧⎛ ⎞ ⎪ ⎜qHI , cH qHI − κ ⎟ μλ ⎠ ⎪⎝ ⎪ (qNnf , tNnf ) = ⎨ ⎛ κ ⎪ ⎜ μλ (1 − λ)(cH − cL ) , cN ⎪⎝ ⎪ (q I , cN q I ) ⎩ N N

if κ < Γ (cH ), ⎞ κ ⎟ μλ (1 − λ )(cH − cL ) ⎠

if Γ (cH ) ≤ κ < Γ (cN ), if κ ≥ Γ (cN ),

πSC (qH , qL , qN ) = μ {λ [(a − qH ) qH − cH qH ] + (1 − λ )[(a − qL ) qL − cL qL ]} + (1 − μ)[(a − qN ) qN − cN qN ] − κ .

(2)

and the retailer's optimal profit is

The subscript H, L, and N denote high cost type, low cost type and the uniformed supplier who does not know her production cost type after forecasting. The first term in (2) denotes the supply chain's profit if the forecasting is successful, and the second term is the supply chain's profit if the forecasting is unsuccessful, and the third term is the forecasting cost. Note that if the decision-maker acquires no useful information after forecasting, then she still does not know her real production cost, but only knows that the expected production cost is cN, thus, the optimal production quantity is still qNI = 12 (a − cN ). While if the decision-maker acquires useful information, then the optimal pro1 duction quantity qSI = 2 (a − cS ), S ∈ {H , L}. The integrated supply chain's expected profit after forecasting is 1 1 1 I πSC (qHI , qLI , qNI ) = 4 μλ (a − cH )2 + 4 μ (1 − λ )(a − cL )2 + 4 (1 − μ)(a − cN )2 − κ . Intuitively, acquiring accurate production cost information is beneficial to the supply chain. Thus, there exists a threshold 1 κ I = 4 μλ (1 − λ )(cH − cL )2 for the forecasting cost κ such that I I πSC (qHI , qLI , qNI ) = πSC (qNI ), and below this threshold forecasting is beneficial to the supply chain.

πRnf

We now turn to the decentralized supply chain. In the decentralized supply chain, the retailer has the option to induce the supplier to forecast or not to forecast. This section concentrates on the optimal contract that induces the retailer not to forecast, maximizing her expected profit and ensuring that the supplier will accept the contract and does not forecast. In this case, the retailer can devise a contract (qN,tN), under which the retailer pays the supplier tN for the production quantity qN. The optimal contract that induces the retailer not to forecast is the solution to

max [(a − qN ) qN − tN ]

if κ < Γ (cH ), ⎞ κ ⎟ if Γ (cH ) ≤ κ < Γ (cN ), μλ (1 − λ )(cH − cL ) ⎠ if κ ≥ Γ (cN ).

The superscript nf denotes the scenario of no forecasting. The proof of Proposition 1, as well as other propositions and corollaries, is relegated in the Appendix A. Since the retailer is in the dominant position to offer the contract, we next show that she will always sell the quantity q = qNnf defined in the Proposition 1 and selling any quantity q < qNnf is always suboptimal. Assume that the retailer offers the contract (qN,tN), promising to transfer tN to the supplier for production quantity qN. Since the transfer payment has been (will be) given to the supplier, this cost is sunk to some extent. Now, the retailer will determine how many products to sell. Obviously, she can only sell q ≤ qNnf because of production quantity limit. The retailer needs to solve the problem max q≥0 [(a − q ) q], which is maximized at a . The transfer 2 payment tN is absent from the objective function because it is predetermined and is a constant. a is the maximum quantity that the 2 retailer is willing to sell anytime. Thus, the optimal selling quantity is nf a min {qN , 2 }. Without loss of generality, we can impose a constraint a q ≤ 2 on the problem defined in (3)–(5) to eliminate the possibility that

4. No forecasting

(qN , tN )

⎧ 1 (a − c ) 2 + κ H μλ ⎪4 ⎪ ⎪ ⎛ κ =⎨ ⎜a − cN − ⎪ μλ (1 − λ)(cH − cL ) ⎝ ⎪1 ⎪ (a − cN )2 ⎩4

the retailer will sell q < qNnf . The above Proposition 1 indicates that a qNnf < 2 always hold, which means that the retailer will always sell out all the products. So the motivation that the retailer will sell q < qNnf is completely eliminated. Under the scenario of no forecasting, when the forecasting cost κ is high, i.e., κ ≥ Γ (cN ), the supplier obtains only the reservation profit, and the optimal production quantity, which is irrelevant to the forecasting cost κ, is the same as that in the integrated supply chain. When the forecasting cost κ is medium, i.e., Γ (cH ) ≤ κ < Γ (cN ), the supplier also only gets the reservation profit. However, at this time, the

(3) 452

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shutdown policy of not cooperating with the high cost supplier. This is because when the supplier is more likely to be in the high cost type, on the one hand, the production quantity for the high cost supplier may be distorted severely, and on the other hand, the retailer will pay more information rent to the low cost type supplier because of his informational advantage and efficiency advantage. These two effects will eventually hurt the retailer. We analyze the case of contracts without and with shutdown of the high cost supplier, respectively. In both cases, we focus on characterizing the optimal contracts, the expected profits for the retailer and the supplier, and examining whether the retailer should induce the supplier to forecast, as well as under what conditions will the retailer induce the supplier to forecast.

optimal production quantity is downward distorted and is increasing in κ. When the forecasting cost κ is low, i.e., κ < Γ (cH ), the optimal production quantity is further downward distorted to qHI , which is also irrelevant to κ. In equilibrium, the supplier does not forecast, and the retailer knows that the supplier will not have accurate production cost information. Note that, when the forecasting cost is low, the supplier can get information rent (1 − λ )(cH − cL ) qHI − κ /(μλ ), which is decreasing in κ and increasing in the production cost variance cH − cL . The larger the production cost variance is, the more information rent the supplier gets, which is consistent with the classical adverse selection results (Salanié, 2005). The intuition is as follows. When the forecasting cost is low, the supplier is more willing to forecast, in order to induce the supplier not to forecast, the retailer will have to cede some information rent to the supplier. When the forecasting cost becomes larger, the supplier's motivation to forecast becomes weaker, thus, the information rent is becoming smaller. Meanwhile, if the production cost variance cH − cL is large, the supplier will be more willing to forecast, thus, in order to weaken the supplier's motivation to forecast, the retailer also needs to cede information rent to the supplier. Here it is not because the supplier has superior information that leads to the information rent, but the threat to acquire it. In other words, the low cost supplier has the efficiency advantage, thus, he can threaten to acquire accurate production cost information to require the retailer to cede some information to him. This result is similar to that in Taylor and Xiao (2009), in a different setting with asymmetric demand information and the manufacturer can induce the retailer not to forecast. They find that the efficient retailer can obtain information rent when the forecasting cost is low because of efficiency advantage. Our result enriches the extant literature by showing that when the principal chooses not to induce the agent who has information advantage to forecast, the efficient agent can always acquire positive information rent.

5.1. Contracts without shutdown of the high cost supplier Under transparent information acquisition, the retailer knows the supplier's information status, either informed or uninformed, after forecasting, but does not know the true type of informed supplier if the supplier succeeds in forecasting, so she can offer a menu of contracts {{(qH , tH ), (qL , tL )}, {(qN , tN )}}. The first part of the contracts is a menu of two contracts {(qH , tH ), (qL , tL )}, which is intended for the supplier who knows her true type after successful forecasting, and the second part is a single contract {(qN , tN )}, which is intended for the supplier who does not know her true types after unsuccessful forecasting. Because under transparent information acquisition the retailer knows exactly whether the supplier is informed or not after forecasting, the informed supplier does not have the option to choose the contract {(qN , tN )}, similarly, the uninformed supplier also does not the option to choose from the contracts {(qH , tH ), (qL , tL )}. The optimal contract that induces the supplier to forecast is the solution to

max

nf Corollary 1. (1) qNnf is weakly increasing in κ ; (2) πRnf and πSC are both weakly increasing in κ . Corollary 1 indicates that if the retailer induces the supplier not to forecast, then the optimal production quantity is weakly increasing in κ, and the retailer's profit and the supply chain's profit are both weakly increasing in κ. In other words, larger forecasting cost is beneficial to the retailer and the supply chain. The intuition is as follows. When the forecasting cost becomes lower, the supplier will be more willing to forecast, in order to deter the supplier from forecasting, the retailer has to distort the production quantity downward, which will have a negative impact on the profits for the retailer and the entire supply chain. On the contrary, when the forecasting cost becomes higher, the supplier will be less willing to forecast, therefore, it is increasingly unnecessary for the retailer to distort the production quantity downward, which is beneficial to the retailer and the supply chain.

{{(qH , tH ),(qL , tL )},{(qN , tN )}}

πR = μ {λ [(a − qH ) qH − tH ] + (1 − λ )[(a − qL ) qL − tL ]} + (1 − μ)[(a − qN ) qN − tN ]

s.t.

tH − cH qH ≥ 0

(6) (7)

tL − cL qL ≥ 0

(8)

tH − cH qH ≥ tL − cH qL

(9)

tL − cL qL ≥ tH − cL qH

(10)

tN − cN qN ≥ 0

(11)

μ [λ (tH − cH qH ) + (1 − λ )(tL − cL qL )] + (1 − μ)(tN − cN qN ) − κ ≥ max {tN − cN qN , 0}

(12)

The objective in (6) is to maximize the retailer's expected profit, with the first part denoting the profit from the informed supplier and the second part the profit from the uninformed supplier. (7) and (8) are the individual rationality constraints for the high cost supplier and low cost supplier, which ensures that the contract offer satisfies the informed supplier's participation constraint, so the supplier will accept the contract offer. The incentive compatibility constraints (9) and (10) ensure that each type of informed supplier is better off choosing the contract consistent with his true type rather than choosing the opposite contract. The individual rationality constraint (11) ensures that the uninformed supplier is willing to accept the contract (qN , tN ). (12) is the forecasting constraint. The left hand side of (12) is the supplier's expected profit if he forecasts, and the right hand side of (12) is the profit when he chooses not to forecast. Upon forecasting, given the supplier is informed with probability μ, he may either earn a surplus of tH − cH qH with possibility λ or a surplus of tL − cL qL with possibility 1–λ. And given the supplier is unformed with probability 1–μ, he may earn a surplus of tN − cN qN . For the supplier who does not forecast, he may either accept the contract (qN , tN ), earning a surplus of tN − cN qN , or just

5. Forecasting with transparent information acquisition In subsequent, we analyze the scenario when the retailer induces the supplier to forecast. After forecasting, there is still a possibility that the supplier acquires nothing useful. Thus, after forecasting, the supplier may either acquire useful information and become informed of his true type, or acquire nothing useful and remain uninformed of his true type. The acquired information is privately known to the supplier, and is unobservable to the retailer. Under transparent information acquisition, the supplier discloses his information status to the retailer and withholds the information content. Under nontransparent information acquisition, the retailer discloses neither his information status nor his information content. We analyze the scenario of transparent information acquisition in Section 5 and nontransparent information acquisition in Section 6, respectively. Note that, when the probability that a supplier is in the high cost type is high, or the cost variance is large, the retailer may prefer to a 453

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reject it, earning zero profit. In order to make the forecasting attractive to the supplier, the expected profit from forecasting minus the forecasting cost, should be no less than that from choosing the contract tN − cN qN or from just rejecting it. With the constraints (9) and (10), an informed supplier will always choose the contract consistent with his true type. To see this, assume that a supplier becomes informed and knows that he is the high type, from (9) we know that if he pretends to be the low type by choosing the contract (qL , tL ), then he will obtain less profit, compared with the situation when he chooses the contract (qH , tH ). Therefore, the incentive compatibility constraints (9) and (10) are sufficient to guarantee truthtelling. What's more, an uninformed supplier will never have the chance to pretend to be an informed supplier, this is because the retailer knows exactly whether a supplier is informed or not. When facing an uninformed supplier, the retailer only offers him {(qN , tN )}, so the uninformed supplier does not have the chance to pretend to be an informed supplier by choosing a contract from {(qH , tH ), (qL , tL )}. c −c

πStr

λ

1⎛ (1 − λ )(cH − cL ) ⎞ ⎟, ⎜a − cH − ⎠ λ 2⎝ ⎞ ⎛ 1 (1 − λ )( c − c H L) tHtr = cH ⎜a − cH − ⎟, ⎠ 2 ⎝ λ

qHtr =

1 (a − cL ), 2 ⎛ 1 1 (1 − λ )(cH − cL ) ⎞ tLtr = cL (a − cL ) + (cH − cL ) ⎜a − cH − ⎟, ⎠ ⎝ λ 2 2 qLtr = qLI =

(2) When κ > κ tr ,

κ κ , tHtr = cH , μ (1 − λ )(cH − cL ) μ (1 − λ )(cH − cL ) 1 1 κ (a − cL ), tLtr = cL (a − cL ) + , 2 2 μ (1 − λ )

qNtr = qNI =

1 1 (a − cN ), tNtr = cN (a − cN ). 2 2

where the superscript tr denotes transparency 1 (1 − λ )(cH − cL ) . κ tr = 2 μ (1 − λ )(cH − cL )(a − cH − ) λ

if κ > κ tr .

Corollary 2. (1) If 12 μ (1 − λ)(cH − cL )(a − cH ) ≤ κ < 12 μ (1 − λ)(cH − cL )(a − cL ) , then qHtr |κ > κ tr ≥ qHI > qHtr |κ ≤ κ tr . If κ tr ≤ κ < 12 μ (1 − λ)(cH − cL )(a − cH ) , then tr qHtr |κ ≤ κ tr ≤ qHtr |κ > κ tr < qHI ; (2) πR is weakly decreasing in κ . In the classic adverse selection models, there are distortions for all types of agents except for the most efficient type (Laffont and Martimort, 2002; Salanié, 2005). However, in our framework, the production quantities for the low cost supplier and the uninformed supplier do not suffer from distortions. In other words, both the most efficient and secondary efficient agents’ efficiency are guaranteed. The main reason is that we consider transparent information acquisition, that is, the retailer knows exactly the supplier's information status after forecasting. The asymmetry between the principal and agent is weakened under transparency. Transparency efficiently diminishes the efficiency loss in asymmetric information environment. In a recent paper addressing the contracting and coordination problem under asymmetric production cost, Çakanyıldırım et al. (2012) find that in equilibrium, the production quantity for the high cost supplier may be downward distorted. Proposition 2 indicates that the production quantity for the high cost supplier can be either downward or upward distorted when the forecasting cost is high. This is because we consider the endogenous adverse selection model and supplier can only get it by exerting costly forecasting effort. Our result offers another explanation for why the high cost supplier produces more in equilibrium when the forecasting cost is high. When the forecasting cost is high, both types of suppliers are not willing to forecast. In order to induce the supplier to forecast, the retailer has to cede more information rent to the supplier to cover the forecasting cost. To this end, given fixed production cost variance, the retailer has to raise the quantity for the high cost supplier to increase the information rent ceded to the low cost supplier, which can also be verified in the proof of Proposition 2.

1 1 (a − cN ), tNtr = cN (a − cN ). 2 2

qLtr = qLI =

⎛ cH − cL ⎞ ⎤ ⎜ a − c ⎟ ⎥ − κ if κ ≤ κ tr , H ⎠⎦ ⎝ (14)

(1) When κ ≤ κ tr ,

qHtr =

1−λ λ

From Proposition 2, we get some interesting observations. First, under transparent information acquisition, no matter the forecasting cost is high or low, the production quantities for the low cost supplier and uninformed supplier are not distorted. Only when the forecasting cost exceeds the threshold κtr, is the production quantity for the high cost supplier either downward or upward distorted. Second, when the forecasting cost κ≤κtr, the supplier can obtain information rent that is enough to cover the forecasting cost κ, while when the forecasting cost κ > κtr, the supplier can only get the information rent that is equivalent to the forecasting cost. The intuition is that when the forecasting cost is low, the information rent the supplier obtains is higher than the forecasting cost. While when the forecasting cost is high, the production quantity is severely distorted, the supplier will obtain less information rent that can only cover the forecasting cost. Notably, the retailer's profit is dependent on the forecasting cost κ if the forecasting cost κ is low. This is in contrast with that in Taylor and Xiao (2009). In the setting of asymmetric demand information, they find that when the supplier uses a menu of rebate contracts to induce the retailer to forecast, in equilibrium, the manufacturer bears all the forecasting cost. In other words, the forecasting cost is fully borne by the principal who does not own the information. The reason for this difference is as follows. In their model, the optimal production quantities for the two types of retailers are independent of the forecasting cost. While in our model, the forecasting cost impacts the high cost supplier's production quantity.

Proposition 2. Suppose aH− c L ≤ 1 − λ .With transparent information H acquisition, an optimal contract that induces the supplier to forecast is given by:

qNtr = qNI =

⎧ ⎡ ⎪ 1 μ (1 − λ )(cH − cL )(a − cH ) ⎢1 − = ⎨2 ⎣ ⎪ ⎩0

and

1

Note that κ < 2 μ (1 − λ )(cH − cL )(a − cL ) must hold to ensure that ≤ qLtr , which is implicitly implied by (9) and (10). Under transparent information acquisition, given the optimal contract parameters defined in Proposition 2, the retailer's optimal expected profit is

qHtr

⎧ ⎡ ⎛ ⎞ ⎤2 ⎪ 1 μλ (a − cH )2 ⎢1 − (1 − λ) ⎜ cH − cL ⎟ ⎥ + 1 μ (1 − λ )(a − cL )2 if κ ≤ κ tr , 4 λ a − c 4 H ⎠⎦ ⎪ ⎝ ⎣ ⎪ ⎪ + 1 (1 − μ)(a − cN )2 4 πRtr = ⎨ ⎡ ⎤ 1 ⎪ λκ κ 2 tr ⎪ (1 − λ)(cH − cL ) ⎢⎣a − cH − μ (1 − λ)(cH − cL ) ⎥⎦ + 4 μ (1 − λ )(a − cL ) if κ > κ . ⎪ ⎪ + 1 (1 − μ)(a − c )2 − κ N ⎩ 4

κ 1tr ≥ 0 such that πRtr ≥ πRnf if and Corollary 3. There exists a threshold ⌢ κ 1tr . only if κ ≤ ⌢ Corollary 3 shows that, when the forecasting cost is low, the retailer

(13) The supplier's optimal expected profit is 454

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this assumption is deviated, the optimization problem defined in (6)– (12) will reach a boundary solution, and the retailer will not design a contract for the high cost supplier. c −c λ When aH− c L > 1 − λ , the retailer will offer a menu of two contracts H {{(qL , tL )}, {(qN , tN )}}. After forecasting, the low cost supplier will choose the contract (qL , tL ), and the uninformed supplier will choose the contract (qN , tN ). Note that under transparent information acquisition, the low cost supplier does not have the chance to choose the contract (qN , tN ) and the uninformed supplier also does not have the chance to choose the contract (qL , tL ). The contract offer is unacceptable to the high cost supplier and he also does not have the chance to choose the contract (qN , tN ). The optimal contract is the solution to

should offer a menu of contracts to induce the supplier to forecast, while when the forecasting cost is high, the retailer should offer a single contract to induce the supplier not to forecast. The intuition is as follows. When the forecasting cost is low, from constraint (12) we know that the supplier is more willing to forecast, and from Corollary 2 we know that the retailer will benefit from forecasting cost reduction. Meanwhile, as more accurate cost information is beneficial to the supply chain, it will eventually increase the retailer's profit. However, when the forecasting cost is high, the supplier is less willing to forecast. To encourage the supplier to forecast, the retailer should cede more information to the supplier, which hurts the retailer. Thus, the retailer will prefer the supplier not to forecast only when the forecasting cost is high. The following Fig. 2 depicts the retailer's optimal expected profits under no forecasting and transparent information acquisition when offering contracts without shutdown of the high cost supplier. The parameters are as follows: a=50, cH=8, cL=2. In (a) and (b), we fix λ and change μ. In (c) and (d), we fix μ and change λ. It can be found that when the retailer does not induce the supplier to forecast, the retailer's expected profit is increasing in the forecasting cost κ. When the forecasting cost is low, the retailer's expected profit under transparent information acquisition is constant, but when the forecasting cost becomes larger, the retailer's expected profit is decreasing in κ. Under κ 1tr for the different parameters, there always exists a critical threshold ⌢ forecasting cost. If the forecasting cost is below this threshold, the retailer would prefer the supplier to forecast, otherwise, the retailer would offer a single contract to induce the supplier not to forecast.

max {{(qL , tL )},{(qN , tN )}}

πR = μ (1 − λ )[(a − qL ) qL − tL ] + (1 − μ)[(a − qN ) qN − tN ]

(15)

s.t.

tL − cH qL < 0

(16)

tL − cL qL ≥ 0

(17)

tN − cN qN ≥ 0

(18)

μ (1 − λ )(tL − cL qL ) + (1 − μ)(tN − cN qN ) − κ ≥ max {tN − cN qN , 0} (19) The objective function in (15) is to maximize the retailer's expected profit from cooperating with the low cost supplier and the uninformed supplier. (16) ensures that the contract offer is unacceptable to the high cost supplier. The individual rationality constraints (17) and (18) indicate that both the low cost supplier and uninformed supplier will choose the contract intended for their types. The forecasting constraint (19) shows that the expected profit from forecasting exceeds the profit the supplier will obtain when he does not forecast and remains uninformed.

5.2. Contracts with shutdown of the high cost supplier We now turn to the scenario of contracts with shutdown of the high cost supplier. From Proposition 2 we know that in order to ensure that the production quantity for the high cost supplier to be positive when κ ≤ κ tr , the underlying assumption is that caH−−ccL ≤ 1 −λ λ . However, when

Proposition 3. Suppose

H

cH − cL a − cH

>

λ . 1−λ

An optimal contract that

Fig. 2. Retailer's expected profits under no forecasting and transparent information acquisition without shutdown of the high cost supplier.

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induces the supplier to forecast is:

6. Forecasting with nontransparent information acquisition

(1) When κ ≤ κ tr ,

We now turn to the scenario of nontransparent information acquisition. To the retailer, nontransparent information acquisition prevents her from knowing the information status of the supplier (Li et al., 2014), i.e., whether the supplier is informed or not after forecasting is unknown to the retailer. When the supplier gets no useful information and remains uninformed after forecasting, it is the same as that he does not forecast with the only exception that he incurs the forecasting cost. Therefore, under nontransparent information acquisition, the retailer can offer a menu of contracts {(qH , tH ), (qL , tL ), (qN , tN )}, which are intended for the high cost supplier, the low cost supplier, and the uninformed supplier, respectively. Similar to the analysis in Section 5, we consider the problem in two different cases.

qLtr = qLI =

1 1 κ (a − cL ), t Ltr = cL (a − cL ) + , 2 2 μ (1 − λ )

qNtr = qNI =

1 1 (a − cN ), t Ntr = cN (a − cN ). 2 2

(2) When κ > κ tr ,

qLtr =

κ κ , t Ltr = cH , μ (1 − λ )(cH − cL ) μ (1 − λ )(cH − cL )

qNtr = qNI =

6.1. Contracts without shutdown of the high cost supplier

1 1 (a − cN ), t Ntr = cN (a − cN ), 2 2

Forecasting will make the supplier informed or uninformed about his true type. If the retailer offers contracts without shutdown of the high cost supplier, then the optimal contract is the solution to

1

where κ tr = 2 μ (1 − λ )(cH − cL )(a − cL ).

max

Proposition 3 shows there exists a threshold κ tr for the forecasting cost κ, below which both the low cost supplier's and the uninformed supplier's production quantities are not distorted, and above which the production quantity for the low cost supplier is upward distorted and the production quantity for the uninformed supplier is not affected. Comparing Proposition 2 with Proposition 3, we find that when offering contracts without shutdown of the high cost supplier, the production quantities for the low cost supplier and the uninformed supplier are not distorted, while when offering contracts with shutdown of the high cost supplier, the production quantity for the uninformed supplier remains unaffected, but the production quantity for the low cost supplier is upward distorted when the forecasting cost κ > κ tr . In Çakanyıldırım et al. (2012), the production quantity for the low cost supplier may be upward distorted. Proposition 3 obtains a similar result, but in a different framework. Here the reason why there is upward distortion for the low cost supplier's production quantity is that we consider the forecasting cost and transparent information acquisition. Under the optimal contract defined in Proposition 3, the retailer's expected profit is

{(qH , tH ),(qL , tL ),(qN , tN )}

+ (1 − μ)[(a − qN ) qN − tN ]

s.t.

tL − cL qL ≥ 0

(23)

tN − cN qN ≥ 0

(24)

tH − cH qH ≥ tL − cH qL

(25)

tH − cH qH ≥ tN − cH qN

(26)

tL − cL qL ≥ tH − cL qH

(27)

tL − cL qL ≥ tN − cL qN

(28)

tN − cN qN ≥ tH − cN qH

(29)

tN − cN qN ≥ tL − cN qL

(30)

(31)

The objective function in (21) is to maximize the retailer's expected profit when she offers a menu of contracts {(qH , tH ), (qL , tL ), (qN , tN )}. The individual rationality constraints (22)–(24) indicate that each supplier will accept the contract intended for his true type. The incentive compatibility constraints (25)–(30) ensure that it is in the supplier's interest to choose the contract intended for him. (31) is the forecasting constraint. It shows that the supplier's expected profit when he forecasts is no less than that when he does not forecast and chooses a contract randomly or just rejects it.

(20)

= 0 . This is because even if the And the supplier's expected profit retailer offers two contracts, these contracts are unacceptable to the high cost supplier, and the low cost supplier and the uninformed supplier both have only one contract to choose from under transparent information acquisition. So the retailer does not need to cede information rent to differentiate these two types of suppliers. Therefore, the retailer extracts the entire supply chain surplus except reimbursing the supplier for the forecasting cost, leaving the supplier only the reservation profit.

c −c

μλ

Proposition 4. Suppose aH− c L < (1 − λ)(1 − μλ) . Under nontransparent H information acquisition, an optimal contract that induces the supplier to forecast is

c −c λ Corollary 4. When aH− c L > 1 − λ , (1) π Rtr is decreasing in κ ; (2) There H κ tr > 0 such that π tr ≥ π nf if and only if κ ≤ ⌢ κ tr . exists a threshold ⌢

R

(22)

≥ max {tH − cN qH , tL − cN qL , tN − cN qN , 0}

πStr

R

tH − cH qH ≥ 0

(21)

μλ (tH − cH qH ) + μ (1 − λ )(tL − cL qL ) + (1 − μ)(tN − cN qN ) − κ

⎧ 1 μ (1 − λ )(a − c )2 + 1 (1 − μ)(a − c )2 − κ if κ ≤ κ tr , L N 4 ⎪4 tr πR = ⎨ κ ⎡ ⎤ κ 1 ⎪ ⎢a − cL − μ (1 − λ)(cH − cL ) ⎥ + 4 (1 − μ)(a − cN )2 − κ if κ > κ tr . ⎦ ⎩ cH − cL ⎣

2

πR = μ {λ [(a − qH ) qH − tH ] + (1 − λ )[(a − qL ) qL − tL ]}

(1) If κ ≤ κ nt , then.

2

Corollary 4 indicates that the retailer will prefer the supplier to forecast when the forecasting cost is below a threshold. Integrating the results in Corollary 3 and Corollary 4, we know that the retailer prefers the supplier to forecast when the forecasting cost is low, and prefers the supplier not to forecast when the forecasting cost is high. In other words, the existence of forecasting cost has significant impact on the retailer's incentive decision for forecasting.

qHnt =

(1 − μλ )(1 − λ ) 1 (a − cH ) − (cH − cL ), tHnt = cH qHnt , 2 2μλ

qNnt =

μλ (1 − λ ) 1 (a − cN ) − (cH − cL ), tNnt = cN qNnt + (cH − cN ) qHnt , 2 2(1 − μ)

qLnt = qLI = 456

1 (a − cL ), tLnt = cL qLnt + (cN − cL ) qNnt + (cH − cN ) qHnt . 2

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(2) If κ > κ nt , then

qHnt =

Similar to the results in section subsection 5.1, the retailer's expected profit is independent of the forecasting cost κ when the forecasting cost κ is low. While when the forecasting cost κ is high, the retailer's expected profit depends on the forecasting cost κ. However, there is a significant difference. If the forecasting cost κ is high, then the supplier can only obtain the reservation profit under transparent information acquisition; while under nontransparent information acquisition, the supplier can obtain positive profit beyond the reservation profit. The reason why the retailer can always get nonnegative profit is that the supplier can make full of his another information advantage: he knows his information status.

μ (1 − λ )(cH − cL ) (1 − μ) κ 1 (a − cH ) − − 2 2(1 − μ + μλ ) μλ (1 − λ )(1 − μ + μλ )(cH − cL ) , tHnt = cH qHnt ,

κ , tNnt = cN qNnt + (cH − cN ) qHnt , μλ (1 − λ )(cH − cL )

qNnt = qHnt + qLnt = qLI =

1 (a − cL ), tLnt = cL qLnt + (cN − cL ) qNnt + (cH − cN ) qHnt , 2

Corollary 5. (1) When κ > κ nt , qHnt is decreasing and qNnt is increasing in κ , and both πRnt and πSnt are decreasing in κ ; (2) There exists a threshold κl1nt ≥ 0 , below which forecasting is beneficial to the retailer, and above which forecasting is detrimental to the retailer. When the forecasting cost is low, qHnt and qNnt are both independent of κ . While when the forecasting cost is high, qHnt is decreasing and qNnt is increasing in κ . In order to differentiate the high cost supplier from the uninformed supplier, the retailer has to increase the production quantity variance, which is linearly increasing in the forecasting cost. This will make the production quantity for the uninformed supplier increasingly upward distorted with the increase of the forecasting cost. The second part of Corollary 5 is intuitive. Forecasting is beneficial to the retailer only when the forecasting cost is low. From Corollary 5 we know that retailer will induce the supplier to forecast if and only if κ ≤ κl1nt . The following Fig. 3 depicts the expected profits of the retailer, the supplier and the supply chain when the retailer chooses the optimal incentive decisions. The parameters are as follows: a=50, cH=8, cL=2, λ=0.4 and μ=0.8. The threshold κl1nt = 3.2 . The retailer prefers the supplier to forecast if and only if κ ≤ 3.2 , which is depicted in (a). When κ ∈ (0, 3.2), the retailer's expected profit is constant and the supplier's profit is decreasing in κ, thus, the supply chain's expected profit is decreasing, too. When κ ∈ [3.2, 24.2), the retailer's expected profit is increasing in κ while the supplier's expected profit is decreasing in κ, and the supply chain's expected profit is constant. When κ ∈ [24.2, 26.3), the supplier's expected profit is zero, and the retailer's expected profit is increasing in κ. Thus, the supply chain's expected profit is increasing in κ. When κ ∈ [26.3, 28.2) , the supplier's expected profit, the retailer's expected profit and the supply chain's expected profit are all constant. From Fig. 3 we also find that the retailer always benefits from the increase of the forecasting cost, while the supplier's expected profit is not monotonous. The reason why the retailer's expected profit is increasing in the forecasting cost κ is that the retailer is in the dominant position to offer the contracts. When it is profitable for the retailer to induce the supplier not to forecast, i.e.,

where the superscript nt denotes nontransparent information acquisition and κ nt =

(1 − μ − μ2 λ2)(1 − λ )2 (cH − cL )2 . 2(1 − μ)

To avoid triviality and to guarantee that the production quantity for the high cost supplier is nonnegative, μλ (1 − λ )(1 − μ + μλ )(cH − cL ) κ≤ [(1 − μ + μλ )(a − cH ) − μ (1 − λ )(cH − cL )] 2(1 − μ) must hold. Under the optimal contract defined in Proposition 4, the retailer's optimal expected profit is

πRnt ⎧ ⎡ ⎤2 ⎪ 1 μλ ⎢a − cH − (1 − μλ)(1 − λ)(cH − cL ) ⎥ + 1 μ (1 − λ )(a − cL )2 if κ ≤ κ nt , 4 4 μλ ⎣ ⎦ ⎪ ⎪ 2 ⎪ + 1 (1 − μ) ⎡a − c − μλ (1 − λ)(cH − cL ) ⎤ ⎢ ⎥ N 4 1 − μ ⎪ ⎣ ⎦ ⎪ = ⎨ μλ (a − cH − qHnt ) qHnt + 1 μ (1 − λ )(a − cL )2 4 ⎪ ⎡ ⎤ ⎪ κ nt ⎪ + (1 − μ) ⎢⎣a − cN − μλ (1 − λ)(cH − cL ) − qH ⎥⎦ if κ > κ nt . ⎪ ⎪⎡ ⎤ κ nt nt ⎪ ⋅⎢ μλ (1 − λ)(c − c ) + qH ⎥ − (1 − λ )(cH − cL ) qH − κ H L ⎦ ⎩⎣ (32) And the supplier's optimal expected profit is

⎧ ⎪ 1 (1 − λ )(cH − cL )(a − cH ) if κ ≤ κ nt , ⎪2 ⎪ ⎧ ⎡ ⎤ (μλ − 1 + μ)(1 − λ ) ⎛ cH − cL ⎞ ⎪ ⎜ a − c ⎟ ⎥ + (1 − μλ ) ⎪ ⎨μλ ⎢⎣1 − 1−μ H ⎠⎦ ⎝ nt ⎩ πS = ⎨ ⎪ ⎡ ⎫ ⎪ ⎢1 − (1 − μλ)(1 − λ) ⎛⎜ cH − cL ⎞⎟ ⎤⎥ ⎬ − κ − μλ a c ⎪ ⎣ H ⎠⎦ ⎝ ⎭ ⎪ ⎪ (1 − λ )(cH − cL ) q nt if κ > κ nt . ⎩ H ⎪ ⎪

⎪ ⎪

(33)

Fig. 3. The expected profits of the retailer, supplier and supply chain when the retailer chooses the optimal incentive decision.

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κ > 3.2 , the retailer's expected profit is increasing in the forecasting cost κ , because he can cede less information rent to the supplier, which is indicated in Corollary 1. Similar results also appear in Taylor and Xiao (2009).

that each type of supplier is better off choosing the contract intended for his type rather than choosing the other contract. The forecasting constraint (41) means that the supplier's expected profit after forecasting is no less than that when the supplier does not forecast.

6.2. Contracts with shutdown of the high cost supplier

Proposition 5. Suppose aH− c L > (1 − λ)(1 − μλ) . An optimal contract that H induces the supplier to forecast and exclude the high cost supplier is:

c −c

We next analyze the scenario when the retailer chooses to offer contracts with shutdown of the high cost supplier. From Proposition 4 we get that, in order to ensure that the production quantity for the high cost supplier is positive, the underlying assumption is μλ cH − cL ≤ (1 − λ)(1 − μλ) . However, if the differential between the high cost a − cH and the low cost is large, the optimization problem defined in (21)– (31) will reach a boundary solution, where the retailer will offer two contracts and exclude the high cost supplier. μλ c −c When aH− c L > (1 − λ)(1 − μλ) , the retailer will offer contracts unaccepH table by the high cost supplier. The optimal contract is solution to the following problem

max {(qL , tL ),(qN , tN )}

(1) If κ ≤ κ nt , then.

1 (a − cL ), 2 ⎛ μλ (1 − λ )(cH − cL ) ⎞ 1 1 = (cN − cL ) ⎜a − cN − ⎟ + cL (a − cL ), 2 1−μ 2 ⎠ ⎝

qLnt = qLI = t Lnt

μλ (1 − λ ) 1 (a − cN ) − (cH − cL ), 2 2(1 − μ) ⎞ μλ (1 − λ ) 1 ⎛ = cN ⎜a − cN − (cH − cL ) ⎟ . 2 ⎝ 1−μ ⎠

qNnt = t Nnt

πR = μ (1 − λ )[(a − qL ) qL − tL ] + (1 − μ)[(a − qN ) qN − tN ] (2) If κ > κ nt , then transparency of information acquisition is

(34)

s.t.

μλ

tL − cL qL ≥ 0

(35)

tN − cN qN ≥ 0

(36)

tN − cH qN < 0

(37)

tL − cH qL < 0

(38)

tL − cL qL ≥ tN − cL qN

(39)

tN − cN qN ≥ tL − cN qL

(40)

qLnt = qLI = qNnt =

tN − cN qN , 0}

κ κ , t Nnt = cN , μλ (1 − λ )(cH − cL ) μλ (1 − λ )(cH − cL )

where

κ nt =

μ (1 − λ )(tL − cL qL ) + (1 − μ)(tN − cN qN ) − κ ≥ max {tL − cN qL ,

1 1 κ (a − cL ), t Lnt = cL (a − cL ) + , 2 2 μ (1 − λ )

the

superscript

μλ (1 − λ )(cH − cL ) ⎛ ⎜a 2 ⎝

nt

− cN −

denotes

nontransparency

μλ (1 − λ )(cH − cL ) ⎞ ⎟. (1 − μ) ⎠

1

and

Note that to ensure qLnt ≥ qNnt , κ ≤ 2 μλ (1 − λ )(cH − cL )(a − cL ) must hold. Proposition 5 shows that there exists a threshold κ nt for the forecasting cost, below which the uninformed supplier's production quantity is not affected by the forecasting cost, and above which the uninformed supplier's production quantity is increasing in the forecasting cost κ. The production quantity for the low cost supplier is not distorted. And the production quantity for the uninformed supplier is downward distorted when κ ≤ κ nt and is either downward distorted or upward distorted, depending on the forecasting cost κ.

(41)

The objective function in (34) is to maximize the retailer's expected profit, containing the profit from cooperating with the low cost supplier and the uninformed supplier after forecasting. The individual rationality constraints (35) and (36) ensure that the low cost supplier and the uninformed supplier are both willing to accept the contract offer. (37) and (38) show that these two contacts are unacceptable to the high cost supplier. The incentive compatibility constraints (39) and (40) imply

Fig. 4. The expected profits of the retailer and the supplier under nontransparent information acquisition with shutdown of the high cost supplier vs. no forecasting.

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(2A) If 0 ≤ κ ≤ κ tr ≤ κ nt or 0 ≤ κ ≤ κ nt ≤ κ tr , then πStr ≥ πSnt . (2B) If κ nt < κ < κ tr , then πStr ≥ πSnt if and only if

The retailer's optimal expected profit is

π Rnt

⎧ ⎪ 1 μ (1 − λ )(a − cL )2 + 1 (1 − μ) if κ ≤ κ nt , 4 ⎪4 2 ⎪ ⎡ μλ (1 − λ )(cH − cL ) ⎤ a − cN − ⎪ ⎥ 1−μ ⎪ ⎢⎣ ⎦ =⎨ ⎪1 (1 − μ) κ nt 2 ⎪ 4 μ (1 − λ )(a − cL ) + μλ (1 − λ)(cH − cL ) if κ > κ . ⎪ ⎤ κ ⎪ ⎡a − c − −κ N ⎪ ⎢⎣ μλ (1 − λ )(cH − cL ) ⎥ ⎦ ⎩

(1 − μ) − μλ (1 − μ + μλ ) 1 κ ≥ (1 − μ)(a − cH ) 2 μλ (1 − λ )(1 − μ + μλ )(cH − cL ) μ (1 − λ )[λ − (1 − μ + μλ )](cH − cL ) − . 2λ (1 − μ + μλ ) (2C) If κ > κ tr , then πSnt > πStr . (42)

⎧ 1 ⎫ (3) If 0 < μ < min ⎨ 1 + λ , 1⎬, ⎭ ⎩

The supplier's optimal expected profit is

⎧1 ⎡ ⎪ μλ (1 − λ )(cH − cL ) ⎢a − cN − πSnt = ⎨ 2 ⎣ ⎪ ⎩0

μλ (1 − λ ) (cH 1−μ

⎤ − cL ) ⎥ − κ if κ ≤ κ nt , ⎦ if κ > κ nt . (43)

(4) If

We have analyzed the retailer's optimal contract under different scenarios, as well as under what conditions should the retailer induce the supplier to forecast. In subsequent, we would like to analyze, from the perspective of the supplier, whether he would like to disclose his information status to the retailer, i.e., whether he is willing to make his information status transparent or nontransparent to the retailer. The supplier's information sharing decision depends critically on the comparison of his profits between transparency and nontransparency. The following Proposition 6 characterizes the conditions under which the supplier is willing to share his information status with the retailer. Proposition 6. Whether transparency of information acquisition is beneficial to the supplier depends on the ratio of cH − cL , μ ,λ and the a − cH forecasting cost κ . Specifically,

≤ μ < 1 and

cH − cL a − cH

≤

λ , 1−λ

then.

(1A) If 0 ≤ κ ≤ κ tr ≤ κ nt or 0 ≤ κ ≤ κ nt ≤ κ tr , then πStr ≥ πSnt if and only if.

⎡ μλ (μλ − 1 + μ) μ2 − (1 − μλ )2 ⎤ ⎛ cH − cL ⎞ 1 − μ . + ⎟≥ ⎢ ⎥⎜ ⎣ ⎦ ⎝ a − cH ⎠ 1 − λ 1−μ μλ (1B) If κ nt < κ < κ tr , then πStr ≥ πSnt if and only if (1 − μ) − μλ (1 − μ + μλ ) μ (1 − μ)(1 − λ )2 1 (cH − cL ). κ ≥ (1 − μ)(a − cH ) + μλ (1 − λ )(1 − μ + μλ )(cH − cL ) 2 2λ (1 − μ + μλ )

(1C) If κ > κ tr , then πSnt > πStr .

⎧ 1 ⎫ (2) If 0 < μ < min ⎨ 1 + λ , 1⎬ and ⎭ ⎩

cH − cL a − cH

<

μλ , (1 − λ )(1 − μλ )

cH − cL a − cH

<

λ , 1−λ

then

cH − cL a − cH

>

λ , 1−λ

then πSnt > πStr or πSnt > πStr .

In case (1) and (2), the retailer offers contracts without shutdown of the high cost supplier. In case (3), the retailer offers contracts without shutdown of the high cost supplier under transparent information acquisition, and offers contracts with shutdown of the high cost supplier under nontransparent information acquisition. In case (4), the retailer offers contracts with shutdown of the high cost supplier under transparent information acquisition, and may offer contracts with or without shutdown of the high cost supplier under nontransparent information acquisition. Proposition 6 indicates that transparency is not always superior to nontransparency. Transparency dominates nontransparency only under some specific conditions. First, from the part (4) of Proposition 6 we know that if the production cost variance is large enough, then transparency is always inferior to nontransparency. The intuition is as follows. When the production cost variance is large, under transparent information acquisition, since the retailer knows the supplier's information status, the retailer will offer contracts which are unacceptable to the high cost supplier, thus, the supplier can only get zero reservation profit because the low cost supplier does not have the chance to mimic the high cost supplier's contract choice to get the information rent. However, under nontransparent information acquisition, the supplier can always get nonnegative profit, thus when the production cost variance is large, transparency is always inferior to nontransparency. However, even if the production cost variance is not too large, transparency can also be inferior to nontransparency, see the case (3B). In this case, when the forecasting cost satisfies κ tr < κ < κ nt , under transparent information acquisition, the supplier can only get zero profit. Under nontransparent information acquisition, the supplier can always get nonnegative expected profit. Therefore, transparency can be inferior to nontransparency. Under some specific conditions, transparency dominates nontransparency; see case (2A) and (3D). In these cases, transparency is always superior to nontransparency. Li et al. (2014) find that transparency harms the retailer (the agent) for sufficient small demand variance. In the scenario of asymmetric production cost asymmetry, our result is to some extent different from theirs. Proposition 6 indicates that transparency may harm the supplier (the agent) for sufficient large production cost variance. This is because we consider the endogenously information acquisition, and especially the impact of forecasting cost on the retailer's contract design. We explicitly characterize the conditions under which transparency dominates nontransparency in Proposition 6. Traditional operations management literature shows that sharing production cost information can create a competitive advantage for the supply chain and a win-win can also be achieved (Chopra and Meindl, 2015). But Proposition 6

6.3. Transparency vs. nontransparency

1 1+λ

≤

(3A) If 0 ≤ κ ≤ κ tr ≤ κ nt or 0 ≤ κ ≤ κ nt ≤ κ tr , then πStr ≥ πSnt if and 1 + λ2 − (1 + λ2 + λ3) μ ⎛ cH − cL ⎞ only if ⎜ a − c ⎟ ≤ 1. λ (1 − μ) H ⎠ ⎝ (3B) If κ tr < κ < κ nt , then πSnt > πStr = 0 . (3C) If κ tr ≤ κ nt ≤ κ , then πStr = πSnt = 0 . (3D) If κ nt < κ tr and κ > κ nt , then πStr > πSnt = 0 .

From (43) we find that when the forecasting cost exceeds the threshold κ nt , the supplier gets zero profit. In other words, the retailer only reimburses the supplier for the forecasting cost. The following Fig. 4 depicts the expected profits of the retailer and the supplier under no forecasting and nontransparent information acquisition when offering contracts with shutdown of the high cost supplier. The parameters are as follows: a=50, cH=10, cL=2, λ=0.1 and μ=0.6. Fig. 4(a) shows the retailer's expected profits under no forecasting and under nontransparent information acquisition. We find that when κ ∈ [0, 7.43], the retailer will induce the supplier to forecast, while when κ ∈ (7.43, 10.37], the retailer will induce the supplier not to forecast. Fig. 4(b) shows the supplier's expected profit under no forecasting and nontransparent information acquisition. We find that κ nt = 9.96 , and when κ ∈ [0, 9.96], the supplier can always get nonnegative expected profit.

(1) If

μλ (1 − λ )(1 − μλ )

then

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S. Huang, J. Yang

always beneficial to the supplier. Under certain conditions, sharing the information status will increase the supplier's profit. However, there are still some limitations in our work. First, to simplify the analysis, we assume deterministic demand function in this paper. Though this assumption is very commonly used in operations management and marketing literature and it helps analyze and obtain the closed form results, whether the results obtained still hold under stochastic demand model are still unknown. Second, we assume that the supplier, once he forecasts, may be either completely informed or uninformed of his cost type. However, in practice, the supplier may be partially informed of his cost type. Li et al. (2014) analyze the scenario when the acquired information can partially help inform the forecaster's type. However, in our paper, we do not consider this scenario. Because we consider the supplier's forecasting cost is this paper, and it would be somewhat complicated to obtain analytical results if both the forecasting cost and the partial information acquisition are considered together. These constitute important directions for future work.

indicates that it is not always in the supplier's interest to disclose his information status to the retailer. However, under some specific conditions sharing the information status can create a win-win situation. Proposition 6 enriches the extant literature by addressing when the supplier should disclose or hide his information status. This will help the firms devise their information sharing agreement more efficiently. 7. Concluding remarks In this paper, we consider the optimal contract design problem when the retailer has the option to induce the supplier to forecast to obtain accurate production cost information. After forecasting, the supplier may become informed of his true type if the forecasting turns out to be successful or the supplier may remain uninformed if the forecasting turns out to unsuccessful. We consider two kinds of information disclosure strategy for the supplier: he can share his information status but not the information content he obtained with the retailer, which is the transparent information acquisition scenario, or he neither share his information status nor the information content with the retailer, which is the nontransparent information acquisition scenario. We obtain the optimal contracts under both scenarios, and obtain the conditions under which the retailer should induce the supplier to forecast. We find that, there exists a threshold for the forecasting cost, below which the retailer will prefer the supplier to forecast and above which discouraging the supplier from forecasting is the retailer's best choice. We further study the conditions under which the supplier should disclose his information status to the retailer, and characterize the conditions under which nontransparency dominates transparency. We find that strict information confidentiality is not

Acknowledgements The authors are grateful to the editor and referees for their constructive comments that led to this improved version. The first author is supported in part by the National Natural Science Foundation of China under Grant 71301055 and 71333004, and Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT_14R17. The corresponding author's research is supported in part by the National Natural Sciences Foundation of China under Grant 71672065, 71172093, 71320107001 and 71332001, and Wuhan Yellow Crane Talents Special Program.

Appendix A Proof of Proposition 1. We begin with simplifying (5). (4) implies tN − cL qN ≥ 0 because cN > cL . Thus, (5) can be transformed as.

tN − cN qN ≥ λ max {tN − cH qN , 0} + (1 − λ )(tN − cL qN ) − κ / μ.

(A1)

Next we analyze the problem in three cases. Case 1: If κ is very large, then (A1) always holds. Since the objective function is decreasing in tN , at the optimal solution, (4) must bind. That is, qNnf = qNI and tNnf = cN qNI when κ ≥ Γ (cN ). Case 2: If κ becomes smaller, then (A1) becomes binding, which means tN = cH qN − κ /(μλ ). Plugging tN = cH qN − κ /(μλ ) into (3), we get qNnf = qHI and tNnf = cH qHI − κ /(μλ ) when κ < Γ (cH ). Case 3: If both (4) and (A1) κ c κ are binding, then qN = μλ (1 − λ)(c − c ) and tN = μλ (1 − λN)(c − c ) when Γ (cH ) ≤ κ < Γ (cN ). Summarizing the three cases above, we obtain the results. H

L

H

L

nf = qHI is constant, πRnf is linearly increasing in κ , and πSC Proof of Corollary 1. (1) When κ < Γ (cH ), nf nf nf which is constant. (2) When Γ (cH ) ≤ κ < Γ (cN ), qN is linearly increasing in κ , and πSC = πR are both nf nf = πRnf are constant. Therefore, qNnf , πRnf and πSC κ ≥ Γ (cN ), qNnf and πSC are all weakly increasing in κ .

qNnf

1

1

= 4 (a − cH )2 + 2 (1 − λ )(a − cH )(cH − cL ), concave and increasing in κ . (3) When

Proof of Proposition 2. From (11) and (12) is equivalent to the following constraint:

λ (tH − cH qH ) + (1 − λ )(tL − cL qL ) − κ / μ ≥ tN − cN qN .

(A2)

The first step to characterize the optimal solution is to show that some constraints are redundant or binding. First, because cH > cL, (7′) and (10′) imply (8). Second, at the optimal solution, (11) must bind. Otherwise the retailer can improve the objective value by decreasing tN by an infinitesimally small amount without violating the constraints, that is, tN = cN qN . Removing the redundant (8) and using the binding constraint (11), as well as the equivalent constraint (A2), the original problem simplifies to

max {{(qH , tH ),(qL , tL )}, qN }

s.t.

πR = μλ [(a − qH ) qH − tH ] + μ (1 − λ )[(a − qL ) qL − tL ] + (1 − μ)[(a − qN ) qN − cN qN ]

tH − cH qH ≥ 0

(7′)

tH − cH qH ≥ tL − cH qL

(9′)

tL − cL qL ≥ tH − cL qH

(10′)

λ (tH − cH qH ) + (1 − λ )(tL − cL qL ) − κ /μ ≥ 0

(A3)

Adding (9′) and (10′), we get qL > qH . The second step to characterize the optimal solution is to ignore some constraints first and solve the relaxed constrained optimization problem, and then verify that the solution of the relaxed problem satisfies the constraints ignored before. To this end, we first ignore (9′) and (A3) and solve the relaxed optimization problem with the rest constraints (7′) and (10′). For the relaxed optimization 460

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problem, (7′) and (10′) must bind together. From the two binding constraints (7′) and (10′), we get

tH = cH qH

(A4)

tL = cL qL + (cH − cL ) qH

(A5)

Given the transfer payment defined in (A4) and (A5), (9′) is always satisfied. Then the retailer's optimization problem can be rewritten as

max {{qH , qL}, qN }

s.t.

⎡ ⎤ (1 − λ ) πR = μλ ⎢ (a − qH ) qH − cH qH − (cH − cL ) qH ⎥ + μ (1 − λ )[(a − qL ) qL − cL qL ] + (1 − μ)[(a − qN ) qN − cN qN ] ⎣ ⎦ λ

(1 − λ )(cH − cL ) qH ≥ κ /μ

(A6)

Solving the above problem by using the KKT conditions, we obtain the results presented in Proposition 2, and we can easily verify that (9′) and (A3) are always satisfied. Proof of Corollary 2. (1) The results can be easily obtained by comparing the production quantities under different scenarios. (2) Since ⎡ ⎤ ∂πRtr |κ > κ tr λ (1 − λ )(cH − cL ) 2κ = (1 − λ)(c − c ) ⎢a − cH − − μ (1 − λ)(c − c ) ⎥ < 0 , where the inequality follows directly from κ > κ tr . Since πRtr |κ ≤ κ tr is a constant, thus, ∂κ λ H L ⎣ H L ⎦ πRtr is weakly decreasing in κ .

κ 1tr ≥ 0 , we only need to prove Proof of Corollary 3. Since πRtr is weakly decreasing and πRnf is weakly increasing in κ , to show the existence of ⌢ πRtr ≥ πRnf at κ = 0 . From the proof of Proposition 2, we have. πRtr |κ =0 = μ {λ [(a − qHtr ) qHtr − cH qHtr ] + (1 − λ )[(a − qLtr ) qLtr − cL qLtr − (cH − cL ) qHtr ]} + (1 − μ)[(a − qNtr ) qNtr − cN qNtr ] ≥ μ maxq {λ [(a − q ) q − cH q] + (1 − λ )[(a − q ) q − cL q − (cH − cL ) q]} + (1 − μ)maxq [(a − q ) q − cN q] ≥ μ max q {λ [(a − q ) q − cH q] + (1 − λ )[(a − q ) q − cL q − (cH − cL ) q]} + (1 − μ)max q [(a − q ) q − cH q] = max q [(a − q ) q − cH q] = πRnf |κ =0 where the first inequality follows directly from the definition of qHtr , qLtr and qNtr , and the second inequality follows from the fact that cH > cN . Thus, κ 1tr ≥ 0 such that πRtr ≥ πRnf if and only if κ ≤ ⌢ κ 1tr . there exists a threshold ⌢ Proof of Proposition 3. From (18) and (19) can be rewritten as the following equivalent constraint.

(1 − λ )(tL − cL qL ) − κ / μ ≥ tN − cN qN

(A7)

(A7) and (18) imply (17). At the optimal solution, (18) must bind. Otherwise, the retailer can decrease tN by a small enough ε to improve the profit while still satisfying all the constraints. (A7) must bind, too. Otherwise, the retailer can decrease tL by a small amount to improve the profit while without violating the constraints. From the binding constraints (18) and (A7), we obtain

tN = cN qN

(A8)

κ tL = cL qL + μ (1 − λ )

(A9)

Deleting the redundant constraint (17) and using the optimal transfer payment obtained in (A8) and (A9), the retailer's optimization problem can be rewritten as

max πR = μ (1 − λ )[(a − qL ) qL − cL qL ] + (1 − μ)[(a − qN ) qN − cN qN ] − κ

(A10)

{qL , qN }

s.t.

κ ≤ μ (1 − λ )(cH − cL ) qL

(A11)

where constraint (A11) is equivalent to (16). Solving the problem above without considering (A11), we get the unconstrained production a−c a−c quantity qL = 2 L and qN = 2 N . These production quantities are the optimal solution when they satisfy (A11), or equivalently, κ ≤ κ tr . When a−c κ κ > κ tr , (A11) must bind. Solving the constrained optimization problem defined in (A10) and (A11), we get qL = μ (1 − λ)(c − c ) and qN = 2 N . H L Substituting these production quantities into (A8) and (A9), we get the transfer payment.□ Proof of Corollary 4. (1) When κ ≤ κ tr , π Rtr is linearly decreasing in κ , while when κ > κ tr ,

∂π Rtr ∂κ

=

⎛ 1 ⎜a (cH − cL ) ⎝

− cL −

⎞ 2κ ⎟ μ (1 − λ )(cH − cL ) ⎠

− 1 < 0 , where the

inequality follows directly from κ > κ tr . Therefore, π Rtr is weakly decreasing in κ . (2) Since πRnf is weakly increasing in κ and π Rtr is weakly decreasing in 1 κ , in order to prove the existence of ⌢ κ 2tr > 0 , we only need to prove π Rtr |κ =0 > πRnf |κ =0 . Since πRnf |κ =0 = 4 (a − cH )2 , in order to prove ⎛ a − c ⎞2 ⎛ a − c ⎞2 ⎛a − c ⎞ c −c λ (a − cH )2 ≤ (1 − λ )(a − cL )2 , we only need to prove λ ≤ 1 − ⎜ a − cH ⎟ . From the assumption aH− c L > 1 − λ , we get λ ≤ 1 − ⎜ a − cH ⎟ < 1 − ⎜ a − cH ⎟ . L⎠ L⎠ L⎠ H ⎝ ⎝ ⎝ 1 1 1 1 1 1 1 That is 4 (a − cH )2 < 4 (1 − λ )(a − cL )2 . What is more, 4 (a − cH )2 < 4 (a − cN )2 , thus, we have 4 (a − cH )2 < 4 μ (1 − λ )(a − cL )2 + 4 (1 − μ)(a − cN )2 , or equivalently, π Rtr |κ =0 > πRnf |κ =0 .□ Proof of Proposition 4. The constraint (31) is equivalent to the following constraint.

λ (tH − cH qH ) + (1 − λ )(tL − cL qL ) − κ / μ ≥ tN − cN qN

(A12)

The first step to characterize the optimal solution is to show that some constraints are tL − cL qL ≥ tH − cL qH ≥ tH − cH qH ≥ 0 , which means that (23) is redundant. (22) and (29) imply means that (24) is redundant. From (26) and (29), we have cN (qH − qN ) ≥ cH (qH − qN ), that cL (qN − qL ) ≥ cN (qN − qL ), that is, qN ≤ qL . From (29) we have tN − tH ≥ cN (qN − qH ) ≥ cL (qN − qH ), 461

redundant. From (22) and (27), we have tN − cN qN ≥ tH − cN qH ≥ tH − cH qH ≥ 0 , which is, qH ≤ qN . From (28) and (30), we have and from (28) we have tL − tN ≥ cL (qL − qN ),

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S. Huang, J. Yang

thus, tL − tH ≥ cL (qL − qH ) holds. In other words, (28) and (29) imply (27). (30) implies tN − tL ≥ cN (qN − qL ) ≥ cH (qN − qL ), and from (26) we get tH − tN ≥ cH (qH − qN ). Combining these two inequalities, we get tH − tL ≥ cH (qH − qL ). In short, (25) is redundant if (26) and (30) both hold. The second step to characterize the optimal solution is to show that some constraints are binding. At the optimal solution, (22) must bind, otherwise, the retailer can decrease tH , tL and tN simultaneously by an infinitesimally small amount without violating the constraints, but improve the objective function. The third step is to ignore some constraints first and then solve the relaxed constrained optimization problem. We then verify that the obtained solutions also satisfy the ignored constraints. To this end, we first ignore (26) and (30), and then we find that at the optimal solution, (28) must bind, otherwise, the retailer can simultaneously decrease tL and tN by ε (where ε is positive and small enough) to improve profit while still satisfying all the constraints. Similarly, (29) must bind, too. Otherwise, the retailer can decrease tN by a small amount to improve profit while still satisfying all constraints. Not that if (29) binds, then (26) is redundant. If (28) binds, then (30) is redundant. Thus, we have verified that the ignored (26) and (30) are satisfied if (28) and (29) are binding. From the binding constraints (22), (28) and (29), we get

tH = cH qH

(A13)

tN = cN qN + (cH − cN ) qH

(A14)

tL = cL qL + (cN − cL ) qN + (cH − cN ) qH

(A15)

Substituting (A13)–(A15) into the retailer's problem and after some simplification, the retailer's optimization problem becomes to

max πR = μ {λ [(a − qH ) qH − cH qH ] + (1 − λ )[(a − qL ) qL

− cL qL − (cN − cL ) qN − (cH − cN ) qH ]}

{qH , qL , qN }

+ (1 − μ)[(a − qN ) qN − cN qN − (cH − cN ) qH ]

(A16)

κ qN ≥ qH + μλ (1 − λ )(cH − cL )

s.t.

(A17)

By using the KKT conditions, we can obtain the optimal solutions presented in Proposition 4. Proof of Corollary 5. (1) Taking the first-order derivatives of qHnt and qNnt , respectively, we get.

∂qHnt ∂κ

=−

∂q nt (1 − μ) μλ < 0 and N = >0 μλ (1 − λ )(1 − μ + μλ )(cH − cL ) ∂κ μλ (1 − λ )(1 − μ + μλ )(cH − cL )

Taking the first-order derivative of πSnt , we get

∂πSnt ∂κ

(1 − μ)

= − μλ (1 − μ + μλ) < 0 . From envelope theorem, we obtain

∂πRnt 1 − μ − μ2 λ 2 1 − μ − μ2 λ 2 2κ 2κ nt = (1 − λ )(cH − cL ) − ≤ (1 − λ )(cH − cL ) − ∂κ (1 − μ)(1 − μ + μλ ) (1 − λ )(1 − μ + μλ )(cH − cL ) (1 − μ)(1 − μ + μλ ) (1 − λ )(1 − μ + μλ )(cH − cL ) = 0, where the second inequality follows directly from κ > κ nt , the last equality follows from the definition κ nt . (2) Since πRnt is decreasing in κ , and πRnf is increasing in κ , in order to prove the existence of κl1nt ≥ 0 , we only need to prove πRnt ≥ πRnf at κ = 0 .

πRnt |κ =0 = μ {λ [(a − qHnt ) qHnt − cH qHnt ] + (1 − λ )[(a − qL ) qLnt − cL qLnt − (cN − cL ) qNnt − (cH − cN ) qHnt ]} + (1 − μ)[(a − qNnt ) qNnt − cN qNnt − (cH − cN ) qHnt ] ≥ maxq {μ {λ [(a − q ) q − cH q] + (1 − λ )[(a − q ) q − cL q − (cN − cL ) q − (cH − cN ) q]} + (1 − μ)[(a − q ) q − cN q − (cH − cN ) q]} 1 = maxq [(a − q ) q − cH q] = (a − cH )2 = πRnf |κ =0 4 where the second inequality follows directly from the definitions of qHnt ,qLnt , and qNnt , and the third equality follows simplification manipulation. Thus, there exists a threshold κl1nt ≥ 0 such that πRnt ≥ πRnf when κ ≤ κl1nt , and πRnt < πRnf when κ > κl1nt .□ Proof of Proposition 5. From (39) and (40), we have qL ≥ qN . And (41) can be rewritten as the following equivalent constraint.

(1 − λ )(tL − cL qL ) − κ / μ ≥ tN − cN qN

(A18)

The first step to characterize the optimal solution is to show that some constraints are redundant or binding. From (36) and (39), we get tL − cL qL ≥ tN − cL qN ≥ tN − cN qN ≥ 0 , thus, (35) is redundant. At the optimal solution, (36) must bind. Otherwise, the retailer can decrease tN and tL simultaneously by a small amount, improving the objective value without deviating all the constraints. When (36) binds, (37) is redundant. From (38) and (40), we have tL − cH qL ≤ tL − cN qL ≤ tN − cN qN = 0 , thus, (38) is redundant. The second step to characterize the optimal solution is to ignore (A18) first and obtain a solution, can then verify the solution obtained satisfy (A18). After we ignore (A18), (39) must bind. If (39) is binding, then (40) is redundant if and only if qL ≥ qN . From the binding constraints (36) and (39), we have tN = cN qN and tL = cL qL + (cN − cL ) qN . Putting the transfer payments into the original problem, the retailer's optimization problem can be simplified as

max πR = μ (1 − λ )[(a − qL ) qL − cL qL − (cN − cL ) qN ] + (1 − μ)[(a − qN ) qN − cN qN ]

{qL , qN }

s.t.

(1 − λ )(cN − cL ) qN ≥ κ /μ

(A19) (A20)

By using the KKT conditions, we can obtain the optimal solutions presented in Proposition 5.□ λ

μλ

1

Proof of Proposition 6. Note that 1 − λ < (1 − λ)(1 − μλ) is equivalent to μ > 1 + λ . We consider the following cases. (1A) In this case, the supplier's expected profit under transparent information acquisition is

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S. Huang, J. Yang

πStr =

⎡ 1 (1 − λ ) ⎛ cH − cL ⎞ ⎤ μ (1 − λ )(cH − cL )(a − cH ) ⎢1 − ⎟⎥ − κ ⎜ 2 λ ⎝ a − cH ⎠ ⎦ ⎣

and the supplier's expected profit under nontransparent information acquisition is

πSnt =

⎡ (μλ − 1 + μ)(1 − λ ) ⎛ cH − cL ⎞ ⎤ (1 − μλ )(1 − λ ) ⎛ cH − cL ⎞ ⎤ ⎫ (1 − λ )(cH − cL )(a − cH ) ⎧ ⎡ ⎨μλ ⎢1 − ⎟⎥⎬ − κ ⎜ ⎟ ⎥ + (1 − μλ ) ⎢1 − ⎜ ⎝ a − cH ⎠ ⎦ ⎭ ⎠ ⎝ 2 1 − μ a − c μλ ⎣ ⎦ ⎣ H ⎩

πStr ≥ πSnt is equivalent to

⎪

⎪

⎪

⎪

cH − cL a − cH

≥

μλ (1 − μ)2 (1 − λ ){μ2 λ2 (μλ − 1 + μ) + (1 − μ)[μ2 − (1 − μλ )2]}

.

(1B) In this case, the supplier's expected profit under transparent information acquisition is

πStr =

⎡ 1 (1 − λ ) ⎛ cH − cL ⎞ ⎤ μ (1 − λ )(cH − cL )(a − cH ) ⎢1 − ⎟⎥ − κ ⎜ 2 λ ⎝ a − cH ⎠ ⎦ ⎣

and the supplier's expected profit under nontransparent information acquisition is

⎡1 μ (1 − λ )(cH − cL ) πSnt = (1 − λ )(cH − cL ) ⎢ (a − cH ) − ⎣2 2(1 − μ + μλ ) πStr ≥ πSnt is equivalent to

(1 − μ) − μλ (1 − μ + μλ ) κ μλ (1 − λ )(1 − μ + μλ )(cH − cL )

−

⎤ (1 − μ) κ ⎥ μλ (1 − λ )(1 − μ + μλ )(cH − cL ) ⎦

1

≥ 2 (1 − μ)(a − cH ) +

μ (1 − μ)(1 − λ )2 (cH 2λ (1 − μ + μλ )

− cL ).

(1C) In this case, the supplier's expected profit under transparent information acquisition πStr = 0 , and the supplier's expected profit under nontransparent information acquisition πSnt > 0 , thus, πSnt > πStr . (2) (2A) In this case, the supplier's expected profit under transparent information acquisition is

πStr =

⎡ 1 (1 − λ ) ⎛ cH − cL ⎞ ⎤ μ (1 − λ )(cH − cL )(a − cH ) ⎢1 − ⎟⎥ − κ ⎜ 2 λ ⎝ a − cH ⎠ ⎦ ⎣

and the supplier's expected profit under nontransparent information acquisition is

πSnt =

⎡ (μλ − 1 + μ)(1 − λ ) ⎛ cH − cL ⎞ ⎤ (1 − μλ )(1 − λ ) ⎛ cH − cL ⎞ ⎤ ⎫ (1 − λ )(cH − cL )(a − cH ) ⎧ ⎡ ⎨μλ ⎢1 − ⎟⎥⎬ − κ ⎜ ⎟ ⎥ + (1 − μλ ) ⎢1 − ⎜ ⎝ a − cH ⎠ ⎦ ⎭ ⎝ a − cH ⎠ ⎦ 2 1−μ μλ ⎣ ⎩ ⎣

πStr ≥ πSnt is equivalent to

⎪

⎪

⎪

⎪

cH − cL a − cH

≥

μλ (1 − μ)2 (1 − λ ){μ2 λ2 (μλ − 1 + μ) + (1 − μ)[μ2 − (1 − μλ )2]}

. Note that 1 − μλ > μ, the above inequality always holds.

(2B) The proof is similar to that in (1B). (2C) The proof is similar to that in (1C). (3)(3A) In this case, the supplier's expected profit under transparent information acquisition is

πStr =

⎡ 1 (1 − λ ) ⎛ cH − cL ⎞ ⎤ μ (1 − λ )(cH − cL )(a − cH ) ⎢1 − ⎟⎥ − κ ⎜ 2 λ ⎝ a − cH ⎠ ⎦ ⎣

and the supplier's expected profit under nontransparent information acquisition is

πSnt =

⎡ (μλ − 1 + μ)(1 − λ ) ⎛ cH − cL ⎞ ⎤ 1 μλ (1 − λ )(cH − cL )(a − cH ) ⎢1 − ⎟⎥ − κ ⎜ ⎝ a − cH ⎠ ⎦ 2 1−μ ⎣

⎛ cH − cL ⎞ ⎜ a − c ⎟ ≤ 1. H ⎠ ⎝ (3B) In this case, the supplier's expected profit under transparent information acquisition πStr = 0 , and the supplier's expected profit under nontransparent information acquisition πSnt > 0 , thus, πSnt > πStr . (3C) In this case, the supplier's expected profit under transparent information acquisition πStr = 0 , and the supplier's expected profit under nontransparent information acquisition πSnt = 0 , thus, πStr = πSnt . (3D) In this case, the supplier's expected profit under transparent information acquisition is nonnegative, and the supplier's expected profit under nontransparent information acquisition πSnt = 0 , thus, πStr > πSnt = 0 . (4) In this case, the supplier's expected profit under transparent information acquisition πStr = 0 , and the supplier's expected profit under nontransparent information acquisition is nonnegative, thus, πSnt > πStr or πSnt > πStr .

πStr ≥ πSnt is equivalent to

1 + λ2 − μ (1 + λ2 + λ3) λ (1 − μ)

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