Incentive provision for demand information acquisition in a dual-channel supply chain

Transportation Research Part E 116 (2018) 42–58

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Incentive provision for demand information acquisition in a dualchannel supply chain

T

⁎

Song Huanga, Xu Guanb, , Binqing Xiaoc a b c

College of Economics and Management, South China Agricultural University, Guangzhou, Guangdong 510642, China School of Economics and Management, Wuhan University, Wuhan, Hubei 430072, China School of Management and Engineering, Nanjing University, Nanjing, Jiangsu 210093, China

A R T IC LE I N F O

ABS TRA CT

Keywords: Supply chain management Information acquisition Dual-channel Endogenous adverse selection Contracts

This paper studies an endogenous adverse selection model in a dual-channel supply chain setting, in which the manufacturer can offer a menu of contracts to induce the retailer to costly acquire private demand information. We derive the manufacturer’s optimal incentive provision decision and show that although the increase of acquisition cost results in higher distortion effect on the retailer’s selling quantity, such a distortion effect can be alleviated in a dual-channel setting. The manufacturer’s incentive provision exhibits a threshold policy. When demand variation is high and information acquisition cost is low, acquiring demand information does not necessarily benefit the retailer.

1. Introduction With the rapid development of third-party logistics and information technology (IT), many manufacturers, such as Epson, Sony, and Cisco Systems, have established their own direct selling channels (Chiang et al., 2003). The establishment of a direct selling channel is normally beneficial to the manufacturer, as it endows the manufacturer with direct control of distribution and pricing flexibility (Tsay and Agrawal, 2004). However, its impact on the retailer’s side is mixed: direct channel would intensify the downstream competition but also incentivize the manufacturer to cut down the wholesale price (e.g., Liu and Zhang, 2006; Arya et al., 2007; Khouja and Wang, 2010). The establishment of direct selling channel has been extensively examined in the literature, and a common assumption is that the demand information is observable to both the manufacturer and retailer. Nonetheless, in practice, it is more likely that the manufacturer has no access of demand information, due to its remoteness to the end-market and lack of acquisition tactics. In contrast, the retailer is more professional at acquiring market information. For example, a retailer can invest substantial resources in establishing demand forecasting systems (Shin and Tunca, 2010; Ausfoodnews.com, 2012), employing personnel to gather and process the data obtained (Guo, 2009; Li et al., 2014), or hiring external experts who have professional knowledge in a specific field to carry out the forecasting (Fu and Zhu, 2010). Amazon, the world’s largest on-line retailer, is hiring scientists who are specialized in enabling machines to make demand forecasts for its research teams in New York and Berlin (Mizroch, 2015). Given the retailer’s superior knowledge of demand information, the manufacturer needs to consider how to induce the retailer to share her privately observed demand information with him, which is particularly valuable in a dual-channel setting. This is because compared to the singlechannel setting, demand information influences not only the manufacturer’s wholesale pricing decision in the retail channel, but also his selling quantity decision in the direct selling channel.

⁎

Corresponding author. E-mail addresses: [email protected] (S. Huang), [email protected] (X. Guan), [email protected] (B. Xiao).

https://doi.org/10.1016/j.tre.2018.05.012 Received 12 September 2017; Received in revised form 21 April 2018; Accepted 25 May 2018 1366-5545/ © 2018 Elsevier Ltd. All rights reserved.

Transportation Research Part E 116 (2018) 42–58

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Although the issue of information asymmetry has been investigated by some researchers in a dual-channel setting (e.g., Cao et al., 2013; Li et al., 2014; Li et al., 2015), a common assumption is that the retailer is freely endowed with superior demand information. Nonetheless, based on the above discussion, in practice, acquisition should be an endogenous decision for the retailer, given that such a process could be quite costly. According to the data, in the United States alone, the annual spending of firms on information acquisition is approximately $6.7 billion and such a number has already increased to $18.9 billion worldwide (Shin and Tunca, 2010; Guan and Chen, 2016). Therefore, whether the retailer has the incentive to costly acquire the demand information becomes questionable, especially in a dual-channel supply chain setting wherein the establishment of the direct selling channel would potentially intensify the market competition. On the other hand, from the manufacturer’s perspective, is it profitable for him to incentivize the retailer to undertake the costly acquisition under such a circumstance? If the manufacturer does so, how should he design the optimal contracts to induce the retailer to share her private demand information? To answer these questions, this paper seeks to conduct a novel approach by relaxing the assumption of costless acquisition and to investigate the manufacturer’s contract design and the retailer’s incentive of acquisition in a dual-channel context. We consider a manufacturer selling to end consumers through his own direct selling channel and a retailer in the traditional retail channel. The market demand is uncertain initially and could exhibit two possible values: a high demand and a low demand. At the beginning, the manufacturer determines whether or not to provide incentives for the retailer to costly acquire demand information. To achieve this, the manufacturer can either provide a contract menu to induce information acquisition or offer one single contract to induce nonacquisition. Then, the retailer chooses whether or not to costly acquire demand information. If she acquires information, she becomes informed of her demand type; otherwise, she remains uninformed.1 Next, the retailer either accepts and chooses one contract consistent with her demand type or just rejects the manufacturer’s contract offer. Meanwhile, the manufacturer determines the selling quantity in the direct selling quantity accordingly. Finally, the demand is realized and payments are made as stipulated. To our knowledge, this paper belongs to the few ones considering the endogenous adverse selection problem (e.g., Taylor and Xiao, 2009; Chen et al., 2016; Huang and Yang, 2016). Complementing the extant research, we extend the game context from a standard supply chain setting to a dual-channel supply chain. We further characterize the manufacturer’s optimal contracts by incorporating the substitution effect, the demand variation, and the information acquisition cost. Combining these elements, our analysis yields several interesting results speaking to the strategic interactions between the manufacturer and the retailer, which can be elaborated as follows. To incentivize the retailer, it is generally believed that the manufacturer has to distort the selling quantity, which consequently hurts his profitability. However, we show that such a distortion effect can be alleviated in the dual-channel setting, as now the manufacturer has the flexibility in adjusting his selling quantity in the direct selling channel. Similarly, to prevent the retailer from acquiring information, the manufacturer decreases the selling quantity in the retail channel to eliminate the retailer’s motivation of acquiring information, and correspondingly, increases the selling quantity in the direct selling channel. During this process, we surprisingly find that the retailer is able to obtain extra profit beyond the reservation profit, even if she has no superior demand information. From the manufacturer’s perspective, there always exists a threshold for the retailer’s information acquisition cost, below which the manufacturer prefers to induce information acquisition. The manufacturer’s optimal incentive provision decision also gives rise to some interesting payoff implications. For example, the retailer’s optimal expected profit is weakly decreasing in the information acquisition cost, whereas the expected profits of both the manufacturer and the supply chain exhibit non-monotonic relationships with respect to the retailer’s information acquisition cost. The remainder of this paper is organized as follows. Section 2 reviews the related literature. Section 3 presents the model. The manufacturer’s optimal incentive provision decision is presented in Section 4. Section 5 compares the results with the single-channel setting. Section 6 concludes the paper. All proofs are presented in the Appendix. 2. Literature This paper also falls into the vast literature that studies the optimal decisions in a dual-channel supply chain (see, e.g., Liu and Zhang, 2006; Arya et al., 2007; Hua et al., 2010; Chen and Chang, 2012; Liu et al., 2016; etc.). In this stream of literature, some studies have examined the impact of the establishment of the direct selling channel on the performance of the retailer and the manufacturer. For example, Liu and Zhang (2006) show that the establishment of the direct selling channel impairs the retailer’s profitability. However, some researchers find that the establishment of the direct selling channel does not necessarily hurt the retailer (see, e.g., Arya et al., 2007; Ha et al., 2016). Arya et al. (2007) indicate that the retailer can benefit from the establishment of the direct selling channel through a reduced wholesale price. Ha et al. (2016) further show that a higher manufacturer’s cost of quality could benefit the retailer in a dual-channel supply chain. The above literature on supplier encroachment usually assumes symmetric information. However, in practice, retailers can acquire demand information and have superior private demand information to the supplier. When information is asymmetric between the supplier and the retailer, the consequences of supplier encroachment may be varied. Li et al. (2014, 2015) study the optimal contracting with asymmetric demand information in a dual-channel supply chain by utilizing the adverse selection model and the signalling model. Cao et al. (2013) study the optimal contract design in a dual-channel supply chain when the retailer has private cost information. The above literature assumes that the retailer is freely endowed with private demand information. However, in practice, the retailer usually needs to exerting costly effort to acquire private demand information. Thus, the present paper complements an 1

Denote the retailer who finds that the demand is in the high (low) state as the High (Low) type retailer.

43

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Table 1 Comparison with the related literature.

Liu and Zhang (2006), Arya et al. (2007), Hua et al. (2010), Chen and Chang (2012), Liu et al. (2016), Ha et al. (2016) Li et al. (2014, 2015), Cao et al. (2013) Zhang et al. (2010), Xu et al. (2010), Feng et al. (2015), Zhu (2016) Guo (2009), Li et al. (2014) Shin and Tunca (2010), Fu and Zhu (2010) Taylor and Xiao (2009), Chen et al. (2016), Huang and Yang (2016) This paper

Channel Structure

Information Structure

Information

Endogenous

Single

Dual

Symmetric

Acquisition

Adverse Selection

√

√

√ √ √ √

√ √

√ √ √ √ √

Asymmetric

√ √ √ √

√

√ √

emerging area that explicitly studies motivating the downstream retailer to acquire demand information in a single-channel setting (see, e.g., Guo, 2009; Shin and Tunca, 2010; Fu and Zhu, 2010; Guo, 2009; Taylor and Xiao, 2009). This stream of literature studies how the retailer’s information decision affects the manufacturer’s contract decisions in a single channel. Our work is also related to a large body of literature on supply chain contracting with asymmetric information by adverse selection models (see, e.g., Zhang et al., 2010; Xu et al., 2010; Feng et al., 2015; Zhu, 2016; etc.). In this stream of literature, some papers study the endogenous adverse selection in the single-channel setting. That is, the manufacturer should provide contracts not only to elicit the retailer’s private demand information, but also to induce the retailer to costly acquire such demand information in the first place. For example, Taylor and Xiao (2009) compare the performance of the returns and the rebates when the retailer can obtain a signal about the demand by exerting a fixed forecasting cost and demonstrate that returns are superior. Chen et al. (2016) investigate the retailer’s costly information acquisition effort and sales effort simultaneously in a sales force contracting setting under asymmetric demand information. In addition, Huang and Yang (2016) study the endogenous cost information acquisition in a singlechannel setting when the manufacturer can costly obtain private cost information. Different from the above literature that considers the single-channel setting, we focus on deriving the manufacturer’s optimal contracts in a dual-channel supply chain setting. Most importantly, we focus on characterizing how the retailer’s information acquisition cost and channel competition intensity affect the manufacturer’s optimal incentive provision decision. This paper complements the previous literature by investigating an endogenous adverse selection problem in a dual-channel supply chain setting, and characterizing the manufacturer’s optimal contracts under given market conditions. A detailed comparison with the related literature is summarized in the following Table 1. 3. The model Consider a supply chain where a manufacturer (he) sells products to end consumers through the retailer (she) in the traditional retail channel. In addition to the retail channel, the manufacturer can also sell a substitute product directly to end consumers in his own direct selling channel. In the presence of dual-channel, the manufacturer and retailer engage in the quantity competition in the market. The manufacturer must produce the product before the selling season starts at a constant unit (marginal) cost c, which is normalized to zero. This assumption is in line with Arya et al. (2007), Li et al. (2014), Li et al. (2015), Yoon (2016). Note that if the unit production cost c > 0 , we can interpret the market potential as Θ−c . Thus, the assumption of zero unit production cost will not analytically change the analysis and results. The nonnegative random variable, denoted by Θ, captures the potential size of the market and can take two values: θ H with probability μ or θ L with probability 1−μ , where θ H (θ L ) denotes that the demand is in the High (Low) state, θ H > θ L and μ ∈ (0, 1) . Denote that the retailer who finds that the demand is in the high (low) state as the High (Low) type retailer. The two demand states assumption is very common in the operations literature (see, e.g., Cachon and Lariviere, 2001; Taylor and Xiao, 2009; Shamir and Shin, 2016). Note that when the retailer does not acquire information, both firms only hold the same prior belief of the demand distribution. Thus, the expected demand can be represented as θ N = μθ H + (1−μ) θ L , where the superscript N denotes the scenario of non-acquisition. Following Lus and Muriel (2009), Yang et al. (2015), and Yoon (2016), the inverse demand functions in the retail channel and in the direct selling channel are given by

pr (qr , qm) = Θ−qr −bqm and pm (qr , qm) = Θ−qm−bqr .

(1)

The parameter b ∈ (0, 1] represents the substitution rate, which can be interpreted as a manufacturer’s offering of different brands or products through its own direct selling channel (Webb, 2002). In practice, a manufacturer usually sell differentiated products in different channels (Yoon, 2016). For example, Kendall-Jackson, a wine producer, sold a rarely carried wine through his own online channel while selling other wines through the retail channel (Lee et al., 2003). When the value of b increases, the two products become more identical and the competition between the two channels becomes more intense. Specifically, when b = 1, the products sold in these two channels are completely substitutable. The relation that b ⩽ 1 is required to insure that the demand for a channel should be more sensitive to changes in its price than to changes in the price of the other channel (Lus and Muriel, 2009). Although the exact value of the market potential Θis unknown to the manufacturer, the retailer must endogenously determine 44

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whether or not to costly acquire such a market demand information before the selling season starts. If she does so, she incurs a fixed non-negative information acquisition cost k and becomes privately informed of the realized value of Θ; otherwise, she remains uninformed of Θ. Anticipating that the retailer has the option of costly acquiring information and becoming informed of Θ, the manufacturer now faces the fundamental question on whether or not to induce the retailer to costly acquire such demand information. The effect of inducing information acquisition is twofold. On the one hand, information acquisition help the retailer better craft her selling quantity decision, thus benefiting the manufacturer not only in producing more revenue in the retail channel but also in finetuning his selling quantity decision in the direct selling channel. On the other hand, the informed retailer puts the manufacturer in an informational disadvantage position, making it less efficient for the latter to extract the surplus. Therefore, the manufacturer’s incentive provision decision arises from trading off the benefit brought by more efficient decisions and the loss induced by informational disadvantage. Without loss of generality, we further normalize the retailer’s reservation profit to zero. This assumption has been widely used in operations literature on asymmetric information (see, e.g., Taylor and Xiao, 2009; Shin and Tunca, 2010; Li et al., 2015; Chen et al., 2016; Huang and Yang, 2016). Actually, the managerial insights will remain unchanged if we assume that the retailer’s reservation profit is a nonnegative constant, as we can add a constant reservation profit to the retailer’s optimal profit. The timing of the game is as follows. First, the manufacturer determines either to offer a menu of contracts to induce information acquisition, or to just offer a single contract to induce no information acquisition. Second, the retailer determines whether or not to acquire the demand information. If she does so, she incurs a fixed information acquisition cost k and privately observes the realized value {θ H , θ L} ; otherwise, she remains uninformed. The retailer then chooses the contract consistent with her type, meanwhile, the manufacturer determines the selling quantity in the direct selling channel. Finally, demand is realized and payments are made according to the retailer’s contract choice. 4. The optimal contracts This section characterizes the manufacturer’s optimal incentive provision decision on whether to induce the retailer to costly acquire demand information in a dual-channel supply chain. To this end, we derive the manufacturer’s optimal menu of contracts that induces costly information acquisition in §4.1, and the manufacturer’s optimal single contract that induces no information acquisition in §4.2. Comparing the manufacturer’s expected profits under these two options, in §4.3, we characterize the manufacturer’s optimal incentive provision. 4.1. Inducing information acquisition If the manufacturer chooses to incentivize the retailer to costly acquire information, by revelation principle and without loss of generality, we can restrict our attention to the menu of quantity-payment contracts {(qrH , trH ), (qrL , trL)} , where qrS and trS denote the retailer’s selling quantity and transfer payment to the manufacturer when the observed demand state S ∈ {H , L} , respectively. Under this menu of contracts, the manufacturer’s selling quantities are {qmH , qmL} correspondingly. We denote π (I , S ) = (θ I −qrS −bqmS) qrS −trS as the retailer’s profit under the demand type I and the contract (qrS , trS ) and qmS , where I , S ∈ {H , L} , and the manufacturer’s profit function is represented by Π(I , S ) = (θ I −qmS−bqrS ) qmS + trS . By contrast, if the retailer does not acquire information, we denote π (N , S ) = (θ N −qrS −bqmS) qrS −trS as her profit function under the contract (qrS , trS ) and qmS for S ∈ {H , L} . Acquiring information allows the retailer to better finetune her selling quantity in the retail channel. However, since information acquisition is costly, the retailer may give up acquiring information and choose a contract randomly. In this case, we assume that the manufacturer will choose the selling quantity in the direct selling channel following the retailer’s contract choice. Similar approach is also implicitly implied in Cao et al. (2013). The optimal menu of contracts that induces information acquisition is the solution to

max

{μ [(θ H −qmH −bqrH ) qmH + trH ] + (1−μ)[(θ L−qmL−bqrL) qmL + trL]}

H , q L} {(qrH , trH ),(qrL, trL), qm m

(OBJ1)

s. t. π (H , H ) ⩾ 0

(IR1)

π (L, L) ⩾ 0

(IR2)

π (H , H ) ⩾ π (H , L)

(IC1)

π (L, L) ⩾ π (L, H )

(IC2)

μπ (H , H ) + (1−μ) π (L, L)−k ⩾ π (N , H )

(IC3)

μπ (H , H ) + (1−μ) π (L, L)−k ⩾ π (N , L)

(IC4)

μπ (H , H ) + (1−μ) π (L, L)−k ⩾ 0.

(IR3)

The model in (OBJ1)–(IR3) is the endogenous adverse selection model, since that the manufacturer needs to provide incentive for the retailer to costly acquire private demand information. (IR1) and (IR2) are individual rationality constraints, which indicate that both types of retailers are willing to accept the contracts. (IC1) and (IC2) are incentive compatibility constraints, which indicate that it is in the retailer’s interest to choose the contract intended for her demand type rather than choose the contract intended for the other 45

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demand type. Note that the forecasting cost k is excluded because it is sunk when the retailer makes her decision on accepting or rejecting the manufacturer’s contract offer (Taylor and Xiao, 2009). Constraints (IC3) and (IC4) ensure that the retailer is better off acquiring information than not acquiring information. Constraint (IR3) satisfies the retailer’s ex ante participation constraint, i.e., the retailer is better off acquiring demand information and choosing the contract consistent with her demand type rather than not acquiring information and rejecting the contract offer. first

We

Ω2 =

define

the

following

2k (1 − b2) + (1 − μ)(1 − b + bμ)(θ H − θ L)2 2k (1 − b2) + bμ (1 − μ)(θ H − θ L)2

useful

notations:

μ (1 − μ) θ H (θ H − θ L) μ (1 − b + bμ)(θ H − θ L)2 1 − b + bμ , k 2A = , Ω1 = , and 2(1 + b) μ 2(1 − b2) H 1 − b + bμ θ and only if L ⩾ , and Ω1 > Ω2 if and only if k > k 2A . As μ θ

k1A =

. It can be verified that k1A ⩽ k 2A if H

information acquisition cost k and the demand ratio θ L are critical in devising the optimal menu of contracts that induces information θ acquisition, we then define the following four regions:

{ (k , R = { (k , R = { (k , R = { (k , R1 =

θH θL

) ) ) )

2

θH θL

3

θH θL

4

θH θL

θH θL

0 < k ⩽ k 2A and 1 <

⩽ Ω1⎫; ⎬ ⎭

k > k 2A and 1 <

θH θL

⩽ Ω 2 ⎫; ⎬ ⎭

0 < k ⩽ k1A and

θH θL

> Ω1⎫; ⎬ ⎭

k1A < k ⩽ k 2A and

θH θL

> Ω1, or k > k 2A and

θH θL

> Ω 2 ⎫. ⎬ ⎭

Proposition 1. The optimal menu of contracts {(qrH , trH ), (qrL , trL)} , which induces the retailer to acquire demand information, is given by(1)

(

if k ,

θH θL

) ∈ R , then

qrL

1

=

qrH =

μ (θ H − θ L) θL − , tL 2(1 + b) 2(1 − μ)(1 − b2) r θH , tH 2(1 + b) r

under which qmH =

=

θH , 2(1 + b)

qrH =

θH 2(1 + b)

qrL qrH

= 0, =

trL

qrH

= 0, =

λ1∗ (θ H − θ L) 2μ (1 − b2)

trL

trL =

, trH =

bλ∗ (θ H − θ L) θH − 1 , 2(1 + b) 2μ (1 − b2)

bμ (θ H − θ L)

+

2(1 − μ)(1 − b2)

(1 − μ) θ L + (μ + λ1∗)(θ H − θ L) 2(1 − μ)

μθ H − λ1∗ (θ H − θ L) 2μ θL 2(1 + b)

qmL =

θH ) θL

;(2) if (k ,

∈ R2 , then

qrL , k

qrH −(θ H −θ L) qrL− μ , b (μ + λ1∗)(θ H − θ L)

+

and λ1∗ =

2(1 − μ)(1 − b2)

2k (1 − b2) −μ (1−b (θ H − θ L)2

+ bμ) ;(3) if (k ,

θH ) θL

∈ R3, then

= 0, =

(θ H )2 k − , 4(1 + b) μ

θH , 2(1 + b)

and qmL =

θH , tH 2(1 + b) r

under which qmH =

qrL

θL 2(1 + b)

and qmL =

qrL =

under which qmH =

(1 − μ) θ L + μ (θ H − θ L) L qr , 2(1 − μ)

(θ H )2 k −(θ H −θ L) qrL− μ , 4(1 + b)

(μ + λ1∗)(θ H − θ L) θL − , 2(1 + b) 2(1 − μ)(1 − b2)

+

=

θL ;(4) 2

if (k ,

θH ) θL

∈ R 4 , then

= 0,

θH 2(1 + b)

under which qmH =

+

λ2∗ (θ H − θ L) 2μ (1 − b2)

, trH =

bλ∗ (θ H − θ L) θH − 2 , 2(1 + b) 2μ (1 − b2)

μθ H − λ2∗ (θ H − θ L) 2μ

qmL =

θL 2

k

qrH − μ ,

and λ 2∗ =

(1 − b) μθ H 2k (1 − b2) − H L . (1 − μ)(θ H − θ L)2 θ −θ

Note that the thresholds k1A and k 2A follow directly from letting λ 2∗ = 0 and λ1∗ = 0 , respectively. Moreover, the retailer’s selling quantities qrH in regions R1 and R3 are constant, whereas qrH in regions R2 and R 4 are strictly increasing with k, and the retailer’s selling quantity qrH in region R2 (R 4 ) is larger than that in region R1 (R3) . Thus, the four regions can then be defined according to whether qrH is upward distorted and whether qrL is zero (see Fig. 1). Proposition 1 implies that when the retailer’s information acquisition cost is low (regions R1 and R3), if Θ = θ H , the selling quantities in both channels are not distorted; otherwise, the retailer’s selling quantity is downward distorted, whereas the manufacturer’s selling quantity is upward distorted. However, when the retailer’s information acquisition cost is high (regions R2 and R 4 ), if Θ = θ H , the retailer’s selling quantity is upward distorted, whereas the manufacturer’s selling quantity is downward distorted; otherwise, the retailer’s selling quantity is downward distorted and the manufacturer’s selling quantity is upward distorted. Note that in regions R3 and R 4 , the Low type retailer sells no products in the market, leading to the dualchannel degenerating to a single direct selling channel. The scenario under which qrL = 0 is referred to as shutdown of the Low type retailer (Laffont and Martimort, 2002). Corollary 1. In regions R2 and R 4 ,

∂ (qrH − qrL) ∂k

> 0 and

H − q L) ∂ (qm m

∂k

< 0.

Corollary 1 indicates that when the retailer’s information acquisition cost is high, the selling quantity differential between the High type retailer and the Low type retailer increases in the information acquisition cost, whereas the differential between the 46

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S. Huang et al.

H L

R3

R4

qrH not distorted

qrH distorted

qrL

qrL

0

0

1 2

R2

R1 H r

q distorted

H r

q not distorted L r

q

qrL

0

0

1

0

k1A k2A

k

Fig. 1. The graphical description of the four regions.

manufacturer’s selling quantity when Θ = θ H and that when Θ = θ L decreases in the information acquisition cost. The retailer has less incentive to acquire information with the increase of the information acquisition cost; thus, in order to encourage the retailer to acquire information, the manufacturer must provide stronger incentives. To this end, the manufacturer raises the High type retailer’s quantity qrH and decreases the Low type retailer’s quantity qrL . The reason is as follows. Given that the retailer does not acquire information, if a High (Low) type retailer incorrectly chooses the quantity for the Low (High) retailer, she will suffer great losses. To elaborate, a High type retailer cannot maximize her profit because she orders fewer products than its optimal level; while a Low type retailer cannot maximize her profit because she orders more products than that she can sell. Under both scenarios, the retailer suffers a lot from not knowing her exact demand type. To avoid this potential profit loss, the retailer is better off acquiring information and choosing the quantity consistent with her demand type. ∂ (q H − q L)

Moreover, the retailer’s quantity differential increases with the information acquisition cost, i.e., r ∂k r > 0 . It indicates that the manufacturer should enlarge the retailer’s quantity differential to provide stronger incentives for the retailer to acquire information. Note that in the setting of dual-channel supply chain, when the retailer sells more products, the manufacturer has to sell less products, and vice versa. Thus, the manufacturer’s quantity differential decrease with the information acquisition cost, i.e., H − q L) ∂ (qm m

∂k

< 0 . In other words, when the manufacturer chooses to induce information acquisition and the acquisition is very costly, he

should increase qrH and decrease qmH when facing the High type retailer, while decrease qrL and increase qmL when facing the Low type retailer. This selling quantity adjustment strategy mitigates the detrimental impact of the retailer’s selling quantity distortions on the manufacturer’s expected profit. The adverse selection literature has widely documented that the manufacturer’s expected profit in the single-channel setting decreases with the information acquisition cost under the scenario of inducing acquisition (Taylor and Xiao, 2009). However, how the manufacturer’s expected profit in the direct selling channel varies to k has been paid little attention to. To this end, we denote ΠdA = μ (θ H −qmH −bqrH ) qmH + (1−μ)(θ L−qmL−bqrL) qmL as the manufacturer’s expected profit in the direct selling channel under inducing information acquisition.

(

Corollary 2. ΠdA (k ) decreases in k if k ,

θH θL

) ∈ R or (k, ) ∈ R . 2

θH θL

4

Corollary 2 suggests that when the retailer’s information acquisition cost k is low, the manufacturer’s expected profit in the direct selling channel is independent of the acquisition cost. However, when the information acquisition cost is high, the manufacturer expected profit in the direct selling channel decreases with k. The following Fig. 2 depicts the manufacturer’s expected profits in the retail channel, the direct channel, and the dual-channel. The parameters are b = 0.5, μ = 0.2, θ H = 2, and θ L = 1. It indicates that the manufacturer’s expected profit in the direct selling channel is decreasing in k only when k ⩾ 0.08; whereas the manufacturer’s expected profit in the retail channel is strictly decreasing in k. Although the retailer’s selling quantity distortions undoubtedly hurt the manufacturer’s expected profit in the retail channel, the manufacturer can correspondingly adjust his own selling quantity in the direct selling channel to alleviate such detrimental distortion effect. The optimal expected profits for the retailer and the manufacturer are respectively given by

47

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S. Huang et al.

0.5 Dual channel

Manufacturer's profit

0.4 0.3

Direct channel

0.2 Retail channel

0.1 0 0

0.02

0.04 0.06 0.08 Acquisition cost k

0.1

0.12

Fig. 2. The manufacturer’s expected profits in different channels. L

H

L

2

H

L 2

⎧ μθ (θ − θ ) − μ (θ − θ ) 2 2(1 − μ)(1 − b ) ⎪ 2(1 + b) π A (k )

=

μθ L (θ H − θ L) μ (μ + λ1∗)(θ H − θ L)2 ⎨ 2(1 + b) − 2(1 − μ)(1 − b2)

⎪ ⎩0

θH θL

1

θH θL

2

otherwise, H 2

ΠA (k ) =

( )∈R, if (k , ) ∈ R , if k ,

L 2

L

H

L

⎧ μ (θ ) + (1 − μ)(θ ) − μθ (θ − θ ) + 2(1 + b) 2(1 + b) ⎪ ⎪ μ (θ H )2 + (1 − μ)(θ L)2 μθ L (θ H − θ L) − 2(1 + b) + ⎪ 2(1 + b) ⎨ μ (θ H )2 ⎪ 2(1 + b) + ⎪ H2 ⎪ μ (θ ) + ⎩ 2(1 + b)

μ2 (θ H − θ L)2 4(1 − μ)(1 − b2)

(2)

−k

(μ3 − λ1*2)(θ H − θ L)2 4μ (1 − μ)(1 − b2)

(1 − μ)(θ L)2 −k 4 (1 − μ)(θ L)2 λ2∗2 (θ H − θ L)2 − −k 4 4μ (1 − b2)

( if (k , if (k , if (k , if k ,

−k

θH θL θH θL θH θL θH θL

)∈R, )∈R, )∈R, )∈R. 1

2

3

4

(3)

It is worth mentioning that when the manufacturer provides contracts with the shutdown of the Low type retailer (in regions R3 and R 4 ), the retailer can only obtain the reservation profit, which is normalized to zero in this paper. This is because under asymmetric information, the retailer’s expected profit beyond the reservation profit comes from a more efficient retailer’s ability to mimic the contract choice of a less efficient retailer. The High type retailer is now unable to mimic the Low type retailer’s contract choice as qrL = 0 . In addition, without shutdown of the Low type retailer, the retailer’s expected profit is decreasing in the information acquisition cost, i.e., dπ A (k )/ dk ⩽ 0 . This is intuitively reasonable as higher information acquisition cost impairs the benefit of information acquisition. 4.2. Inducing no information acquisition Following Crémer et al. (1998), Taylor and Xiao (2009), Huang and Yang (2016), when the manufacturer chooses not to induce the retailer to acquire information, it suffices to offer a single contract (qrN , trN ) , under which the manufacturer’s selling quantity is represented by qmN . This is because the retailer only has the prior belief of the demand distribution when the manufacturer motivates no information acquisition. In this case, the manufacturer only needs to provide a single contract to the retailer who has no accurate demand information. This single contract (qrN , trN ) should meet the retailer’s participation constraint and ensure that the retailer has no incentive to acquire demand information. Note that when acquisition is cheap, the retailer may still have the potential motivation to acquire information. This is because, when the observed demand is in the High state, the retailer can obtain more profit than the reservation profit due to the efficiency advantage. In comparison, if the observed demand is in the Low state, the retailer can just reject the contract offer and earn zero profit. Hence, information acquisition endows the retailer with the flexibility of deciding whether or not to accept the contract offer so as to evaluate her expected profit more efficiently. For ease of exposition, denote π (N , N ) = (θ N −qrN −bqmN ) qrN −trN as the retailer’s expected profit function under the contract (qrN , trN ) when she does not acquire information, and π (S, N ) = (θ S−qrN −bqmN ) qrN −trN as the retailer’s expected profit under the contract (qrN , trN ) when she acquires information and the demand is S ∈ {H , L} . The optimal contract that induces the retailer not to acquire information is the solution to

max

N} {(qrN , trN ), qm

[(θ N −qmN −bqrN ) qmN + trN ]

(OBJ2) 48

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(IR4)

s. t. π (N , N ) ⩾ 0 π (N , N ) ⩾ μmax{π (H , N ), 0} + (1−μ)max{π (L, N ), 0}−k.

(IC5)

(1 − μ)(θ N − θ L)(θ L − bθ N )

(1 − μ)(θ N − θ L) θ N . 2(1 + b)

= and The following proposition charWe first define the following notations: = 2(1 − b2) acterizes the manufacturer’s optimal menu of contract, which induces the retailer not to acquire the demand information. k1N

k 2N

Proposition 2. The optimal contract, which induces the retailer not to acquire information, is given by

(qrN , trN

( )= ( ⎨ ⎪ ⎪( ⎩ ⎧ ⎪ ⎪

under which qmN = N dqm

dk

θ L − bθ N θ L (θ L − bθ N ) , 2(1 − b2) 4(1 − b2)

+

k 1−μ

)

k (2 − b) θ N k (1 − b2) k2 , − (1 − μ)(θ N − θ L) 2(1 − μ)(θ N − θ L) (1 − μ)2 (θ N − θ L)2 θN (θ N )2 , 2(1 + b) 4(1 + b)

θ N − bθ L 2(1 − b2)

if 0 ⩽ k ⩽ k1N ,

)

)

if k1N < k ⩽ k 2N , if k > k 2N ,

if 0 ⩽ k ⩽ k1N , qmN =

θN bk − 2 (1 − μ)(θ N − θ L)

if k1N < k ⩽ k 2N , and qmN =

θN 2(1 + b)

if k > k 2N . In addition,

dqrN dk

⩾ 0 and

⩽ 0. Proposition 2 shows that, in order to induce no information acquisition, the manufacturer should offer three different contracts

corresponding to the value of the information acquisition cost. Note that qrN ⩾ 0 is equivalent to

θH θL

⩽

1 − b + bμ , bμ

indicating that the

qrN

is weakly increasing in k. Moreover, retailer prefers not to sell products if the demand variation is very high. It can be shown that the manufacturer’s selling quantity is weakly decreasing in k. The intuition is that lower information acquisition cost attracts the retailer to acquire information. To eliminate the retailer’s incentive to acquire information, the manufacturer must downward suppress the retailer’s selling quantities so as to discourage the retailer from acquiring information. Meanwhile, as distorting the retailer’s selling quantity hurts the manufacturer’s profitability, the latter can adjust his selling quantity to alleviate such distortion effect. The retailer’s and the manufacturer’s optimal expected profits are respectively given by

π N (k ) =

⎧ (θ

N − θ L)(θ L − bθ N ) k −1 − μ 2(1 − b2)

⎨0 ⎩

if 0 ⩽ k ⩽ k1N , ifk > k1N ,

N 2

N L

(4)

L 2

⎧ (θ ) − 2bθ θ2 + (θ ) + k if 0 ⩽ k ⩽ k1N , 1−μ 4(1 − b ) ⎪ ⎪ (θ N )2 (1 − b) kθ N (1 − b2) k2 ΠN (k ) = + − if k1N < k ⩽ k 2N , (1 − μ)(θ N − θ L) (1 − μ)2 (θ N − θ L)2 ⎨ 4 ⎪ (θ N )2 if k > k 2N . ⎪ 2(1 + b) ⎩

(5)

In the adverse selection model, the retailer is able to acquire extra profit beyond her reservation profit only if she has informational advantage. However, (4) suggests that the retailer can also earn positive profit beyond her reservation profit, which is normalized to zero in this paper, even if she has no informational advantage over the manufacturer. This result is consistent with Taylor and Xiao (2009) and Huang and Yang (2016). This is because the retailer’s extra profit is not driven by her informational advantage, but the threat of acquiring information when the acquisition cost is low. Comparing (2) and (4), we notice that the retailer can obtain a strictly positive profit beyond the reservation profit if she does not acquire information when 0 ⩽ k ⩽ k1N . However, under the condition of inducing information acquisition, she can only obtain the reservation profit, which is normalized to zero in this paper, when k < k1A and

θH θL

>

1 − b + bμ . In other words, when the information acquisition cost is low (i.e., k ⩽ min(k1A, k1N )) and the μ H 1 − b + bμ θ > ), acquiring private demand information does not necessarily benefit the retailer, even if μ θL

demand variation is high (i.e., her information acquisition cost k = 0 . 4.3. Optimal incentive provision

In this section, we investigate the manufacturer’s optimal incentive provision decision, i.e., whether or not he should incentivize the retailer to costly acquire demand information. Proposition 3. (1) The manufacturer’s expected profit ΠA (k ) is strictly decreasing in the retailer’s information acquisition cost k; and (2) the manufacturer’s expected profit ΠN (k ) is weakly increasing in the retailer’s information acquisition cost k. Proposition 3 shows that a higher information acquisition cost reduces the retailer’s motivation of information acquisition, because the benefit of acquiring private demand information may be insufficient to cover her information acquisition cost. Thus, in order to induce the retailer to acquire demand information, the manufacturer needs to reimburse the retailer for her information acquisition cost, thus hurting his profitability. Hence, the manufacturer’s expected profit decreases with the retailer’s acquisition cost k. 49

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By contrast, under the condition of inducing no information acquisition, lower information acquisition attracts the retailer to acquire information, as accurate demand information help her make more efficient selling quantity decision. Thus, in order to deter the retailer from acquiring information, the manufacturer has to distort the retailer’s selling quantity to make information acquisition less attractive. In particular, with the decrease of the retailer’s information acquisition cost, the manufacturer needs to distort the retailer’s selling quantity more severely to provide sufficient incentive for the retailer not to acquire information, which eventually hurts the manufacturer’s profitability. In contrast, when the acquisition cost is very high, the retailer inherently has no motivation to acquire information. In this case, there is no need for the manufacturer to distort the selling quantity, which consequently improves his profit. Hence, the manufacturer’s expected profit increases with the retailer’s information acquisition cost k. Proposition 4. In the dual-channel supply chain, there always exists a threshold k > 0 , below which the manufacturer should induce information acquisition; otherwise, the manufacturer should induce no information acquisition. Regardless of whether the manufacturer provides a menu of contracts with or without shutdown of the Low type retailer, the manufacturer’s optimal incentive provision decision always exhibits a threshold policy. The result in Proposition 4 extends the result in the newsvendor situation (Taylor and Xiao, 2009) and that in the endogenous cost information asymmetry situation (Crémer et al., 1998; Huang and Yang, 2016) in a single-channel setting to a dual-channel setting. Generally, there are two effects of keeping the Low type retailer away from the market. On the one hand, it saves information rent ceded to the High type retailer as the High type retailer cannot threaten to mimic the Low type retailer’s contract choice to obtain extra profit. On the other hand, it undoubtedly decreases the manufacturer’s profit since the Low type retailer cannot sell any products in the market. These two conflicting effects jointly determine the manufacturer’s optimal incentive provision decision, and Proposition 4 suggests that the former effect dominates the latter effect. The managerial insights are as follows. When information acquisition is inexpensive, the manufacturer should offer the optimal menu of contracts (See Proposition 1 to encourage the retailer to acquire information. Otherwise, the manufacturer should offer a single contract (see Proposition 2 to discourage the retailer from acquiring information. With the increase of the retailer’s information acquisition cost, if the manufacturer still intends to motivate the retailer to acquire information, he has to distort the retailer’s quantity more severely, which will hurt the manufacturer’s profitability significantly. By contrast, with the increase of the information acquisition cost, the retailer has less incentives to acquire information and the quantity distortion becomes alleviated, which improves the manufacturer’s profitability. Thus, when it is too costly (k > k ) for the retailer to acquire information, the manufacturer would prevent the retailer from acquiring information instead of motivating her to do so. Recall that the manufacturer’s optimal profit first decreases and then increases with the retailer’s acquisition cost. Under the situation of inducing information acquisition, the improvement of the retailer’s ability of information acquisition (i.e., k decreases) increases the manufacturer’s expected profit. By contrast, under the situation of inducing no information acquisition, the improvement of the retailer’s ability of information acquisition decreases his expected profit. In other words, the manufacturer benefits from a retailer’s improvement in her ability of information acquisition if and only if the retailer is already good at acquiring information (i.e., k < k ). Moreover, information advantage does not necessarily benefit the retailer. When the demand variation is high and the acquisition cost is low, foregoing information acquisition may bring extra profit to the retailer. Thus, when the retailer is able to improve her ability of information acquisition, she needs to take into account the related market conditions. We next conduct some numerical analyses to further illustrate the results in Proposition 4. The parameters are H = 2, L = 1, and μ = 0.2 . Following Proposition 1, the manufacturer provides a menu of contracts without shutdown of the Low type retailer when b = 0.50 and with the shutdown of Low type retailer when b = 0.80 , when inducing information acquisition. Fig. 3 shows that, when the substitution rate is mild (i.e., b = 0.50 ), the manufacturer prefers inducing information acquisition if and only if k ⩽ k = 0.0311. While when the substitution rate is high (i.e., b = 0.80 ), the Low type retailer prefers not to sell any products, and the manufacturer

Fig. 3. The manufacturer’s optimal incentive provision decision with respect to b. 50

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0.2

0.15

0.1

0.05

0 1

1.5

2

2.5

3

3.5

0.044

0.08

0.042

0.07

0.04

0.06

0.038

0.05

0.036

0.04

0.034

0.03

0.032

0.02

0.03

0.01

0.028 0

0.2

0.4

0.6

0.8

0 0

0.2

0.4

0.6

0.8

1

Fig. 4. The threshold k with respect to model key parameters.

prefers inducing information acquisition if and only if k ⩽ k = 0.0299. Proposition 4 characterizes the manufacturer’s optimal incentive provision decision in terms of the retailer’s information acquisition cost k. The characterization of the threshold k appears analytical challenging. We next numerically examine the impacts of model key parameters on the manufacturer’s optimal incentive provision, which is shown in Fig. 4. The left subplot (i) shows that the threshold k increases in θ H / θ L , as higher ratio of θ H / θ L implies that the private demand information is more valuable to the retailer. The middle subplot (ii) shows that the threshold k first decreases and then increases in b. This is because when the two products are more substitute, information acquisition becomes less attractive to the retailer. However, when the substitution rate exceeds a threshold, the Low type retailer will not sell any products in the market. In this case, information acquisition helps the retailer make more efficient order decision, improving her expected profit. The right subplot (iii) suggests that the threshold k also exhibits a non-monotonic relationship with the parameter μ . This is because when the ratio of High type retailer increases in the market, both firms benefit. However, when μ exceeds a threshold, the Low type retailer will not sell any products in the market. This excessive screening hurts both firms’ profits. Thus, the threshold k first increases and then decreases in μ . Following Proposition 4, the manufacturer’s optimal profit is Π∗ (k ) = max{ΠN (k ), ΠA (k )} . We next examine how the retailer’s information acquisition cost k affects the optimal profits of the retailer, the manufacturer, and the supply chain. To do so, we assume that the parameters are H = 2, L = 1, and μ = 0.2 . The results are illustrated in Fig. 5. Note that when the channel substitution rate is high (i.e., b = 0.80 ), the manufacturer should offer a menu of contracts with shutdown of the Low type of retailer. Thus, the retailer can only obtain the reservation profit, which is normalized to zero in this paper, irrespective of her information acquisition cost k. This is because, under the scenario of inducing information acquisition (i.e., k ⩽ 0.0299), even if the High type retailer has the efficiency advantage over the Low type retailer, the manufacturer knows that a Low type retailer would not sell any products in the market. Thus, a retailer who orders a positive quantity must be the High type retailer. In this case, the manufacturer’s optimal profit decreases with k. By contrast, when k > 0.0299 , the manufacturer prefers inducing no information acquisition, thus, the manufacturer’s expected profit is weakly increasing in k. When the substitution rate is mild (i.e., b = 0.50 ), the Low type retailer’s selling quantity is strictly positive under the condition of information acquisition. If the retailer’s information acquisition cost is low (i.e., k < 0.0311), the retailer can always obtain a constant extra profit beyond the reservation profit, as the manufacturer prefers inducing information acquisition. Moreover, the manufacturer’s expected profit is linearly decreasing in k, because he needs to reimburse the retailer for her information acquisition cost. Thus, the supply chain’s expected profit is also linearly decreasing in k.

Fig. 5. The optimal profits of the retailer, the manufacturer, and the supply chain. 51

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However, when k ∈ (0.0311, 0, 0427], the manufacturer prefers inducing no information acquisition, whereas the retailer may still have the motivation to acquire information because the information acquisition cost is not too high. Therefore, the manufacturer has to cede profit to the retailer to discourage information acquisition. Given that higher information acquisition cost weakens the attractiveness of acquiring information, the manufacturer’s expected profit is linearly increasing in k, and the retailer’s expected profit is linearly decreasing in k. In sum, the decentralized supply chain’s optimal profit is constant. When k ∈ (0.0427, 0.0640], the retailer can only obtain the reservation profit. However, as the selling quantities in both channels are still downward distorted to discourage information acquisition, the increase of k alleviates the distortions and increases the manufacturer’s expected profit. When k is relatively high, i.e., k > 0.0640 , the retailer prefers not to acquire information; thus, both the retailer’s and the manufacturer’s selling quantities are not distorted, and the manufacturer’s expected profit is constant. 5. Implication of channel structure The above analysis focuses on characterizing the manufacturer’s optimal incentive provision decision in a dual-channel setting, which is prevalent in a wide range of industries. A natural question then arises on how the establishment of the direct selling channel affects firms’ expected profits under endogenous adverse selection. In this section, we extend to the situation of a single-channel setting and compare it with the dual-channel situation. This extension analysis is helpful in shedding more insights on the impact of endogenous information acquisition. Our objective in this section is to examine how the manufacturer’s incentive provision decision, as well as the firms’ profits, vary in relation to the model key parameters. To this end, we first derive the manufacturer’s optimal contracts that induce information acquisition and no information acquisition, respectively, in the single-channel setting, and then examine the manufacturer’s incentive provision motivation. Note that the inverse demand function in the single channel is pr = Θ−qr . The optimal menu of contract that induces information acquisition is the solution to

max

[μtrH + (1−μ) trL]

{(qrH , trH ),(qrL, trL)}

(OBJ3)

s. t. (θi−qri ) qri−tri ⩾ 0, i ∈ {H , L}

(IR5)

(θi−qri ) qri−tri ⩾ (θi−qrj ) qrj−trj, i, j ∈ {H , L}, i ≠ j

(IC6)

μ [(θ H −qrH ) qrH −trH ] + (1−μ)[(θ L−qrL ) qrL−trL]−k ⩾ max{(θ N −qrH ) qrH −trH , (θ N −qrL) qrL−trL, 0}.

(IC7)

In addition, the optimal contract that induces no information acquisition is the solution to

max trN

(qrN , trN )

s. t.

(OBJ4)

(θ N −qrN ) qrN −trN ⩾ 0

(IR6)

(θ N −qrN ) qrN −trN ⩾ μmax{(θ H −qrN ) qrN −trN , 0} + (1−μ)max{(θ L−qrN ) qrN −trN , 0}−k.

(IC8)

The following proposition characterizes the optimal contracts and the manufacturer’s optimal incentive provision in the singlechannel setting. Proposition 5. In the single-channel setting, (1) the optimal menu of contracts that induces information acquisition is H L {(qrĤ , ^tr ), (qrL̂ , ^tr )} = {(qrH , trH ), (qrL , trL)}|b = 0 , and the optimal contract that induces no information acquisition is N (qrN̂ , t^r ) = (qrN , trN )|b = 0 ; (2) there exists a threshold k ̂ > 0 such that the manufacturer should induce information acquisition if and only if k ⩽ k .̂ The intuitions behind Proposition 5 can be explained as follows. It shows that the manufacturer’s optimal contract in the singlechannel setting is a special case of that under the dual-channel setting. This is because when these two products are non-substituted, it looks like that the manufacturer manages these two channels separately. Thus, the manufacturer can optimize the optimal decisions over the two channels separately. The manufacturer’s optimal incentive provision decision also exhibits a threshold policy, in which he induces information acquisition only when the acquisition cost is below a threshold. Compared with the setting of the dualchannel supply chain, we then investigate how the establishment affects the manufacturer’s optimal incentive provision decision. Fig. 6 demonstrates how the establishment of the direct selling channel affects the manufacturer’s optimal incentive provision decision. We change the parameter b ∈ (0.05, 0, 95) . When the parameter μ = 0.4 (left subplot (i)), the threshold in the single-channel setting k ̂ = 0.0625, while when the parameter μ = 0.6 (right subplot (ii)), the manufacturer always provides the contracts with qrL̂ = 0 , and the threshold in the single-channel setting k ̂ = 0.1. Fig. 6 shows that when b is small, the establishment of the direct selling channel enhances the manufacturer’s incentive to induce information acquisition, which is illustrated by k > k .̂ However, when b becomes large, the establishment of the direct selling channel weakens the manufacturer’s incentive for inducing information acquisition, which is illustrated by k < k .̂ We next numerically investigate how the establishment of the direct selling channel affects the optimal profits of the retailer and ∗) as the retailer’s (manufacturer’s) expected profit differential between ∗ (ΔΠ = Π∗−Π the manufacturer in Fig. 7. Denote Δπ = π ∗−π 52

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(ii) =0.6 0.12

0.08

0.11

0.075

0.1

Threshold

Threshold

(i) =0.4 0.085

0.07

0.09

0.065

0.08

0.06

0.07

0.055 0

0.2

0.4 0.6 Substitution rate b

0.8

0.06 0

1

0.2

0.4 0.6 0.8 Substitution rate b

1

Fig. 6. The comparisons between the thresholds k and k .̂

(i) Retailer

(ii) Manufacturer

0

0.18 0.17

-0.01

Profit differential

Profit differential

0.16 -0.02 -0.03 -0.04

0.15 0.14 0.13 0.12

-0.05 -0.06 0

0.11 0.022

0.04 0.06 0.088 Acquisition cost k

0.1

0.12

0.1 0

0.02 02

0.04 0.06 0.08 Acquisition cost k

0.1

0.12

Fig. 7. Profit differentials of the retailer and the manufacturer.

the dual-channel setting and the single-channel setting. Assume that the parameters are H = 2, L = 1, μ = 0.2 and b = 0.5, then after N

N

some calculation, we can obtain the thresholds k = 0.0311 and k ̂ = 0.0278, k1N = 0.0427, k 2N = 0.064, k1̂ = 0.08 , and k 2̂ = 0.096. In particular, when k < 0.0278, both the retailer’s profit and the manufacturer’s profit are irrespective of the acquisition cost. Thus, the profit differentials for both firms are constant. When k ∈ [0.0278, 0.0311) , the retailer’s profit is constant in the dual-channel setting, and decreases with k in the single-channel setting; thus, her profit differential increases with k. However, the manufacturer’s profit decreases with k in the dual-channel setting, and increases with k in the single-channel setting; thus, his profit differential decreases with k. When k ∈ [0.0311, 0.0427) , the profit differentials for both firms are constant. When k ∈ [0.0427, 0.08) , the retailer obtains extra profit in the single-channel setting and only the reservation profit in the dual-channel setting. The profit differential for the retailer is then increasing in k. Note that when k > 0.08, the retailer only obtains the reservation profit under both settings. For the manufacturer, when k ∈ [0.0427, 0.096) , the manufacturer always induces non-acquisition, and the profit differential decreases with k. When k ∈ [0.0427, 0.096) , the manufacturer induces non-acquisition and the profit differential is constant, as the manufacturer’s profit under both settings are irrespective of the acquisition cost. Fig. 7 shows that the establishment of the direct selling channel always hurts the retailer’s profit, as the retailer will face the quantity competition from the manufacturer. However, the establishment of the direct selling channel always benefits the manufacturer, as adding a selling channel indirectly increases the market demand, thus increasing the manufacturer’s profit. Therefore, the manufacturer will establish the direct selling channel irrespective of the retailer’s ability of information acquisition. 6. Conclusions The retailer is generally more skilled in acquiring market demand information than the manufacturer does. Given that information acquisition is usually costly for the retailer, whether or not the retailer is willing to acquire information is remarkably 53

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affected by the manufacturer’s incentive contract provision. This paper investigates the manufacturer’s incentive provision decision in a dual-channel supply chain setting using the endogenous adverse selection model. Two core issues are discussed: how to elicit the retailer’s private demand information, and how to incentivize the retailer to acquire such information first. Our results can provide efficient guidance on the manufacturer’s incentive provision and retailer’s information acquisition strategies in a context with uncertain demand information. First, it suggests that the manufacturer’s optimal incentive provision decision should exhibit a threshold policy, in which inducing information acquisition is preferable only when the information acquisition cost is below a threshold. Second, for the retailer, it also indicates that acquiring superior demand information does not necessarily improve the retailer’s expected profit, even if the information acquisition cost is zero. The underlying rational is that when the demand variation is high and the information acquisition cost is low, the retailer can always obtain extra profit beyond the reservation profit under the scenario of not acquiring information. These results are driven by the two conflicting effects of acquisition uncovered in the dual-channel supply chain. On the one hand, the retailer’s information acquisition enables the two firms to determine their selling quantities more efficiently, thus maximizing the expected profits for both firms. On the other hand, it also weakens the manufacturer’s ability to extract more profit from the retailer who has superior demand information. Our study has certain limitations. We assume that the demand follows a two-point distribution. Though our framework can be easily extend to the case with continuous distribution, the derivation of the optimal contracts may be more complicated. We believe that the structural results and main managerial insights will remain unchanged. In this paper, we focus on characterizing how the retailer’s ability of information acquisition affects the manufacturer’s incentive provision in a dual-channel setting, it will be interesting to examine how the retailer’s ability of information acquisition influences the manufacturer’s decision on establishing a direct selling channel, and some more fruitful results and insights may be obtained. Acknowledgements The authors sincerely thank the Editor-in-Chief and three anonymous reviewers for their constructive comments and suggestions that improve the paper. The authors have also benefited from the support of National Natural Science Foundation of China (71772070, 71402126, 71301055, 71671083, 71633002). Appendix A Proof of Proposition 1. The proof proceeds mainly in three steps. The first step is to identify the redundant constraints. After removing the redundant constraints, the second step is to identify the binding constraints at the optimum. Following the binding constraints, the third step is to solve the simplified equivalent optimization problem. To this end, we note that (IR2) and (IC1) imply (IR1), and (IR2) and (IC4) imply (IR3). From (IC1) and (IC2), we have qrH ⩾ qrL if and only if θ H −θ L ⩾ b (qmH −qmL) , which must be satisfied to ensure the single-crossing condition. This condition is essential to guarantee the validity of the following procedure. The constraints (IC3) and (IC4) can be simplified as

(θ L−qrL−bqmL) qrL−trL−k /(1−μ) ⩾ (θ L−qrH −bqmH ) qrH −trH

(IC3’)

(θ H −qrH −bqmH ) qrH −trH −k / μ

(IC4’)

⩾

(θ H −qrL−bqmL) qrL−trL

We can easily show that (IC3’) implies (IC2) and (IC4’) implies (IC1). Removing the redundant constraints, the original problem can be readily simplified as

max

{μ [(θ H −qmH −bqrH ) qmH + trH ] + (1−μ)[(θ L−qmL−bqrL) qmL + trL]}

H , q L} {(qrH , trH ),(qrL, trL), qm m

s. t.

(θ L−qrL−bqmL) qrL−trL

(OBJ1)

⩾0

(IR2)

(θ L−qrL−bqmL) qrL−trL−k /(1−μ) ⩾ (θ L−qrH −bqmH ) qrH −trH

(IC3’)

(θ H −qrH −bqmH ) qrH −trH −k / μ ⩾ (θ H −qrL−bqmL) qrL−trL.

(IC4’)

Further analysis shows that (IR2) and (IC4’) must bind at the optimum. If (IR2) is not binding, then increasing both and trH simultaneously by an infinitesimal amount can surely increase the manufacturer’s expected profit without violating the constraints. Similarly, if (IC4’) is not binding, then the manufacturer is able to increase trH by an infinitesimal amount to increase her expected profit, while still satisfying the constraints. The binding constraints indicate that

trL

trL=(θ L−qrL−bqmL) qrL , trH =(θ H −qrH −bqmH ) qrH −(θ H −θ L) qrL−k / μ. Substituting the above two equalities into (OBJ)–(IC4’), we have

max

{μ [(θ H −qmH −bqrH ) qmH + (θ H −qrH −bqmH ) qrH −(θ H −θ L) qrL−k / μ] + (1−μ)[(θ L−qmL−bqrL) qmL + (θ L−qrL−bqmL) qrL]}

H , q L} {qrH , qrL, qm m

(OBJ1)

54

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s. t. (θ H −θ L)(qrH −qrL) ⩾ k /[μ (1−μ)]. Introducing the Lagrangian multiplier

λ∗

(IC3’)

⩾ 0 corresponding to the constraint (IC3’), the Karush–Kuhn–Tucker conditions are given by

θ H −2bqrH −2qmH = 0, θ H −2qrH −2bqmH +

(A.1)

λ∗ (θ H −θ L)

= 0,

μ

(A.2)

θ L−2bqrL−2qmL = 0, θ L−2qrL−2bqmL−

(A.3)

(μ +

λ∗)(θ H −θ L) 1−μ

= 0,

(A.4)

k ⎞ λ∗ ⎛⎜ (θ H −θ L)(qrH −qrL)− ⎟ = 0. μ (1 −μ) ⎠ ⎝

(A.5)

From (A.1)–(A.4) we have θH

λ∗ (θ H − θ L) , 2μ (1 − b2)

θL

(μ + λ∗)(θ H − θ L)

qrH =2(1 + b) + qrL=2(1 + b) −

2(1 − μ)(1 − b2)

,

θH bλ∗ (θ H − θ L) qmH =2(1 + b) − , 2μ (1 − b2) θL

qmL=2(1 + b) +

b (μ + λ∗)(θ H − θ L) 2(1 − μ)(1 − b2)

.

Note that non-negative selling quantity should always be ensured, i.e., qrL ⩾ 0 and qmH ⩾ 0 . (1) Assuming that qrL > 0 , we then consider the following two cases: (i) If λ∗ = 0 , then following the complementary slackness condition (A.5), we have k <

λ∗ =

2k (1 − b2)

μ (1 − b + bμ)(θ H − θ L)2 2(1 − b2)

. In this case, qrL > 0 if and only if

θH θL

1 − b + bμ . μ

<

(ii) If λ∗ > 0 , then (A.5) leads to

−μ (1−b + bμ) if qrL > 0 .

(θ H − θ L)2

(2) Assuming that qrL = 0 , then the constraint (A.4) is redundant, from (A.3) we have qmL =

θL . 2

Similarly, we consider the fol-

lowing two cases: (i) If λ∗ = 0 , then following the complementary slackness condition (A.5), we have k <

λ∗

> 0 , then (A.5) leads to

λ∗

=

(1 − b) μθ H 2k (1 − b2) − H L (1 − μ)(θ H − θ L)2 θ −θ

qrL

if

Proof of Corollary 1. In region R2 , In region R 4 ,

∂ (qrH − qrL) ∂k

=

∂qrH ∂λ2∗ · ∂λ2∗ ∂k

=

∂k

=

∂ (qrH − qrL) ∂λ1∗ · ∂k ∂λ1∗

1 μ (1 − μ)(θ H − θ L)

> 0 , and

=

(ii) If

= 0.

Summarizing the above analysis, we can obtain the results in Proposition 1. ∂ (qrH − qrL)

μ (1 − μ) θ H (θ H − θ L) . 2(1 + b)

1 μ (1 − μ)(θ H − θ L)

H − q L) ∂ (qm m

∂k

=

□

> 0 , and

H − q L ) ∂λ∗ ∂ (qm m · ∂k2 ∂λ2∗

H − q L) ∂ (qm m

=−

∂k

=

H − q L ) ∂λ∗ ∂ (qm m · ∂k1 ∂λ1∗

b μ (1 − μ)(θ H − θ L)

=−

b μ (1 − μ)(θ H − θ L)

< 0.

< 0. □

Proof of Corollary 2. Following Proposition 1, in region R2 , the manufacturer’s expected profit in the direct selling channel is

ΠdA (k ) = ∂ΠdA (k ) ∂λ1∗

=

μ (θ H )2 + (1 − μ)(θ L)2 4(1 + b) b (θ H − θ L)2 − 4(1 − b2)

+

⩽ 0 . Note that

direct selling channel is ∂ΠdA (k ) ∂k

=

∂ΠdA (k ) ∂λ2∗ · ∂k ∂λ2∗

b (θ H − θ L)[μθ L − λ1∗ (θ H − θ L)]

ΠdA (k )

4(1 − b2) ∂λ1∗

∂k

=

⩾ 0 , thus,

.

Taking

∂ΠdA (k ) ∂k

=

the

∂ΠdA (k ) ∂λ1∗ · ∂k ∂λ1∗

first-order

derivative

with

respect

to

λ1∗

leads

to

⩽ 0 . In region R 4 , the manufacturer’s expected profit in the

μ (θ H )2 + (1 − μ)(1 + b)(θ L)2 bλ2∗ θ H (θ H − θ L) − . 4(1 + b) 4(1 − b2)

Note that

∂ΠdA (k ) ∂λ2∗

=−

bθ H (θ H − θ L) 4(1 − b2)

⩽ 0 and

∂λ2∗ ∂k

⩾ 0 , thus,

⩽ 0 . Moreover, following directly from Proposition 1, the manufacturer’s expected profits from the direct

selling channel in regions R1 and R3 are independent of the retailer’s acquisition cost k. Proof of Proposition 2. Clearly, when (IR4) holds,

(θ L−qrN −bqmN ) qrN −trN

⩾

(θ H −qrH −bqmH )−trH

max{(θ L−qrN −bqmN ) qrN −trN ,

□

> 0 since θ H > θ N . Then (IC5) can be simplified as

0}−k /(1−μ).

(θ L−qrH −bqmH )−trH

< 0 must hold at the optimum, otherwise, the manufacturer can charge a higher transfer payment to Note that improve his expected profit while still satisfying all the constraints. Therefore, the problem (OBJ2)-(IC5) can be simplified as max

[(θ N −qmN −bqrN ) qmN + trN ]

N} {(qrN , trN ), qm

(OBJ2)

s. t. (θ N −qrN −bqmN ) qrN −trN ⩾ 0

(IR4)

(θ L−qrN −bqmN ) qrN −trN ⩾ −k /(1−μ).

(IC5’)

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S. Huang et al.

Given that the objective function is increasing in trN , after analyzing the structure of the constraints (IR4) and (IC5’), we know that at least one of them is binding at the optimum. We then consider the following three cases: (i) Only (IR4) is binding. In this case, substituting trN = (θ N −qrN −bqmN ) qrN into (OBJ2) leads to qrN = qmN = we have k ⩾

=

Then from (IC5’),

(1 − μ)(θ N − θ L) θ N . 2(1 + b)

(ii) Only (IC5’) is binding. In this case, substituting trN = (θ L−qrN −bqmN ) qrN +

qmN

θN . 2(1 + b)

θ N − bθ L . 2(1 − b2)

Then (IR4) is reduced to k ⩽

(1 − μ)(θ N − θ L)(θ L − bθ N ) 2(1 − b2)

k 1−μ

it into (OBJ2) results in

=

and

.

(iii) Both (IR4) and (IC5’) are binding. Then from the binding constraints (IR4) and (IC5’) we have qrN =

qmN

θ L − bθ N 2(1 − b2)

into (OBJ2) leads to qrN = k . (1 − μ)(θ N − θ L)

Substituting

θN

bk − . 2 (1 − μ)(θ N − θ L)

□

Summarizing the above three cases, we can obtain the results in Proposition 1. Proof of Proposition 3.

(1) According to whether or not the manufacturer provides the contracts with qrL = 0 , we consider two different scenarios. (i) The manufacturer provides the contracts with qrL > 0 (i.e., regions R1 and R2 ). dλ1∗ dk

2(1 − b2) (θ H − θ L)2

=

> 0 , and

dΠA (k ) dλ1∗

=−

λ1∗ (θ H − θ L)2 2μ (1 − μ)(1 − b2)

the manufacturer provides the contracts

R 4 ). dλ2∗ dk

dΠA (k )

=

dk

< 0 . Hence,

with qrL

= −1 in region R3 . In region R 4 ,

2(1 − b2) (1 − μ)(θ H − θ L)2

> 0 . Therefore,

ΠA (k )

dΠA (k ) dk

=

dΠA (k ) dk

dΠA (k ) dλ1∗ · dk −1 dλ1∗

= −1 in region R1. In region R2 , note that

< 0 . Therefore, ΠA (k ) is strictly decreasing in k if

> 0 . (ii) The manufacturer provides the contracts with qrL = 0 (i.e., regions R3 and dΠA (k ) dk

=−

λ2∗ (θ H − θ L)2 dλ2∗ 2μ (1 − b2)

· dk −1 < 0 , where the inequality follows directly from

is strictly decreasing in k if the manufacturer provides the contracts with qrL = 0 .

(2) When the manufacturer induces no information acquisition, if 0 ⩽ k ⩽ k1N , then dΠN (k ) dk

=

(1 − b) θ N (1 − μ)(θ N − θ L)

(θ − N

2(1 + b) k (1 − μ)(θ N − θ L)

dΠN (k ) dk

=

) ⩽ 0, where the inequality follows directly from k ⩽ k

1 1−μ

N 2 ;

> 0 ; if k1N < k ⩽ k 2N , then

and if k > k 2N , then ΠN is a □

constant. Recall that ΠN (k ) is continuous with respect to k, thus, ΠN (k ) is weakly increasing in k.

Proof of Proposition 4. Following the results in Proposition 3, we only need to compare the manufacturer’s expected profits when the retailer’s information acquisition cost k = 0 . (1) When the manufacturer provides the contracts with qrL > 0 to induce the retailer to acquire information,

ΠA (0)−ΠN (0) =

μ [μ2 + 2(1 − b)(1 − μ)](θ H − θ L)2 4(1 − μ)(1 − b2)

> 0 . Therefore, there exists a threshold k > 0 , below which the manufacturer should

induce the retailer to acquire information, where k is the unique solution to ΠA (k ) = ΠN (k ) . (2) When the manufacture provides the contracts with qrL = 0 to induce the retailer to acquire information,

ΠA (0)−ΠN (0) =

(θ L)2Γ(μ, b, Δ) 4(1 − b2)2

, where Δ =

θH θL

and

Γ(μ, b, Δ) = μ (2−2b−μ)Δ2 + 2μ (b + μ−1)Δ + (1−μ)(μ + 2b−b2)−1. We need to show that Γ(μ, b, Δ) ⩾ 0 when respect to Δ , we obtain ∂Γ ∂Δ ∂Γ ∂Δ

∂Γ ∂Δ

1 − b + bμ μ

⩽Δ⩽

1 − b + bμ . bμ

To this end, taking the first-order derivative of Γ(μ, b, Δ) with

= 2μ [(2−2b−μ)Δ−(1−b−μ)]. We then consider the following two cases: (i) If 2−2b−μ ⩾ 0 , then

⩾ 2[(2−2b−μ)(1−b + bμ)−μ (1−b−μ)] = 2(1−b)[(1−μ)(2−2b−μ) + μ] > 0 .

2 ⩾ b [(2−2b−μ)(1−b 1 − b + bμ Γ(μ, b, ) ⩾ 0. μ

+ bμ)−bμ (1−b−μ)] =

Define F (μ, b) ≡ Γ(μ, b, ∂F ∂b ∂2F ∂b2

1 − b + bμ ) μ

=

2(1 − b)2 (2 − μ) b

1 [(2−2b−μ)(1−b μ

> 0.

In

summary,

∂Γ ∂Δ

(ii)

If

> 0.

Thus,

2−2b−μ < 0 , it

suffices

to

then show

that

+ bμ)2 + 2μ (b + μ−1)(1−b + bμ) + μ (1−μ)(μ + 2b−b2)−μ]. Given that

= 2(1−μ)[(−3 + 3μ−μ2 ) + (6−6μ + μ2 ) b−3(1−μ) b2], = 2(1−μ)[6(1−μ)(1−b) + μ2 ] > 0,

it indicates that

∂F ∂b

is strictly increasing in b. Recall that b ∈ (0, 1) , hence,

∂F ∂b

<

∂F | ∂b b = 1

decreasing in b. Hence, F (μ, b) > F (μ, 1) = 0 . In other words, ΠA (0) > ΠN (0) for any Δ ⩾

= 0 , which implies that F (μ, b) is strictly 1 − b + bμ . μ

Therefore, there exists a threshold

k > 0 , below which the manufacturer should induce the retailer to acquire information, where k is the unique solution to ΠA (k ) = ΠN (k ) . □ Proof of Proposition 5. (1) We first derive the manufacturer’s optimal contracts. Under the scenario of inducing information acquisition. Compare the problem (OBJ3)–(IC7) with the problem (OBJ1)–(IR3), we find that the constraints (IR5)–(IC7) are a special case of (IR1)–(IR3) when b = 0 . Following the similar procedure in proof of Proposition 1, the problem (OBJ3)-(IC7) is then reduced 56

Transportation Research Part E 116 (2018) 42–58

S. Huang et al.

to

max {μ [(θ H −qrH ) qrH −(θ H −θ L) qrL−k / μ] + (1−μ)[(θ L−qrL) qrL]}

{qrH , qrL}

s. t. (θ H −θ L)(qrH −qrL) ⩾ k /[μ (1−μ)]. A

A

Define k ̂ = k A|b = 0 , k1̂ = k1A |b = 0 and k 2̂ = k 2A |b = 0 . Solving the above optimization problem, A

k < k̂ , A

(1) if

qrL̂ =

then

k ̂ = μ (1−μ)(θ H −θ L)(qrĤ −qrL̂ ), A

(2)

k ⩾ k̂ , A

if

(

qrL̂ =

then

+

L r

) , t^

H (1 − μ) θ L + μ (θ H − θ L) L θH (θ H )2 k qr̂ , qrĤ = 2 , and ^tr = 4 −(θ H −θ L) qrL̂ − μ , 2(1 − μ) A A > 0 and k ̂ = k 2̂ if qrL̂ = 0 ; + ∗ ∗̂ ∗ H L L ^t L = (1 − μ) θ L + (μ + λ ̂ )(θ H − θ L) q L̂ , q Ĥ = θ H + λ ̂ (θ H − θ L) , ⎛ θ − (μ + λ )(θ − θ ) ⎞ , r r r 2(1 − μ) 2 2μ 2 2(1 − μ)

θ L μ (θ H − θ L) − 2(1 − μ) 2 A A k ̂ = k1̂ if qrL̂

⎝ ^t H = μθ H − λ (θ H − θ L) q Ĥ −(θ H −θ L) q L̂ − k , where λ ∗̂ = r r r 2μ μ ∗̂

=

⎠

2k −μ (θ H − θ L)2

if qrL̂ > 0 , or λ ̂ = ∗

μθ H 2k − (1 − μ)(θ H − θ L)2 θ H − θ L

where

and

if qrL̂ = 0 .

Under the scenario of inducing no information acquisition, following the similar procedure in Proof of Proposition 1, the problem (OBJ4)–(IC8) is reduced to

max trN

(qrN , trN )

s. t. trN ⩽ min{(θ N −qrN ) qrN , (θ L−qrN ) qrN + k /(1−μ)}. N

Define k1̂ =

(1 − μ) θ L (θ N − θ L) 2

( (q ̂ , t^ ) = ( ⎨ ⎪ ⎪( ⎩ N r

N r

⎧ ⎪ ⎪

θL 2

,

(1 − μ) θ N (θ N − θ L) . 2

N

and k 2̂ =

(θ L)2

+

4

k 1−μ

)

N

if 0 ⩽ k ⩽ k1̂ ,

k [(1 − μ) θ N (θ N − θ L) − k ] k , (1 − μ)(θ N − θ L) (1 − μ)2 (θ N − θ L)2 θ N (θ N )2 , 4 2

The optimal contract is

)

)

N

N

if k1̂ < k < k 2̂ , N

if k ⩾ k 2̂ .

i as region Ri when b = 0 for i ∈ {1, 2, 3, 4} . Under the optimal contracts, the (2) For exposition simplicity, we define region R manufacturer’s optimal expected profits under inducing information acquisition and inducing no information acquisition are respectively given by H 2

 A (k ) = Π

H L

L 2

⎧ μ (θ ) − 2μθ θ + (1 + μ)(θ ) + 4 ⎪ ⎪ μ (θ H )2 − 2μθ H θ L + (1 + μ)(θ L)2 + ⎪ 4 ⎨ μ (θ H )2 ⎪ 4 −k ⎪ H 2 *̂ 2 H L 2 ⎪ μ (θ ) − λ (θ − θ ) −k 4μ ⎩ 4

μ2 (θ H − θ L)2 −k 4(1 − μ) 2 (μ3 − λ *̂ )(θ H − θ L)2

4μ (1 − μ)

(

if k , −k if(k ,

θH θL θH θL

) ∈ R , 1

2 , )∈R

if(k ,

θH ) θL

3, ∈R

if(k ,

θH ) θL

4 , ∈R

N  A (k ) is strictly decreasing in k in regions R  N (k ) = ^tr . We can easily to verify that Π 1-R 2 and regions R 4 , respectively, and 3 -R and Π H − θ L)2 N A N μ ( θ  1, Π  (0)−Π  (0) = Π (k ) is weakly increasing in k. Note that in region R > 0 , and in region 4(1 − μ) H 2

L 2

3, Π  A (0)−Π  N (0) = μ (θ ) − (θ ) > 0 , where the inequality follows directly from R 4 which the manufacturer prefers inducing information acquisition. □

θH θL

> μ . Thus, there exists a threshold k ̂ > 0 , below 1

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