Supplier encroachment strategy in the presence of retail strategic inventory: Centralization or decentralization?

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Supplier Encroachment Strategy in the Presence of Retail Strategic Inventory: Centralization or Decentralization? Jin Li , Liao Yi , Victor Shi , Xiding Chen PII: DOI: Reference:

S0305-0483(19)30257-9 https://doi.org/10.1016/j.omega.2020.102213 OME 102213

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Omega

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26 February 2019 9 February 2020

Please cite this article as: Jin Li , Liao Yi , Victor Shi , Xiding Chen , Supplier Encroachment Strategy in the Presence of Retail Strategic Inventory: Centralization or Decentralization?, Omega (2020), doi: https://doi.org/10.1016/j.omega.2020.102213

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Highlights    

We analyze the interaction between a supplier’s centralized or decentralized encroachment and a retailer’s use of strategic inventory. We explicitly analyze a supplier’s choice between centralized and decentralized encroachment. We study how a retailer can use strategic inventory to deal with supplier encroachment. We prove that decentralized encroachment outperforms centralized encroachment for both the supplier and the retailer.

Supplier Encroachment Strategy in the Presence of Retail Strategic Inventory: Centralization or Decentralization?

Jin Lia*, Liao Yib, Victor Shic, Xiding Chend a

School of Management and E-Business, Key Research Institute-Modern Business Research Center, Zhejiang

Gongshang University, Hangzhou, 310018, China; e-mail: [email protected] b

School of Business Administration, Southwestern University of Finance and Economics, Chengdu, 610074,

China; e-mail: [email protected] c

Lazaridis School of Business and Economics, Wilfrid Laurier University, Waterloo, ON N2L 3C5, Canada; e-

mail: [email protected] d

Wenzhou Business College, Wenzhou University, Wenzhou, 325035, China; e-mail: [email protected]

Abstract There has been extensive research on supplier encroachment. That is, a supplier sells to consumers through a direct retail subsidiary in addition to an independent retailer. However, most research assumes centralized encroachment, where the supplier makes decisions for the subsidiary. In this research, we make a major contribution by explicitly analyzing the option of decentralized encroachment, where the subsidiary makes its own pricing and/or quantity decisions. Furthermore, we make another major contribution by studying the retailer’s possible use of strategic inventory as a countermeasure to achieve wholesale price concessions from the supplier. In a dual channel consisting of a retailer and a supplier and its subsidiary in a two-period model, we find that when the retailer can employ strategic inventory, decentralized encroachment outperforms centralized encroachment for the supplier and the retailer. Moreover, if the supplier adopts the strategy of decentralized encroachment, the retailer’s use of strategic inventory always benefits the supplier, but it benefits the retailer only when the unit inventory holding cost is below a threshold. We also conduct numerical examples to further illustrate our analytical findings and gain more managerial insights.

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Keywords: supplier encroachment; decentralization; strategic inventory; dual channel.

1. Introduction As consumer online shopping continues to grow, it is more and more common for a supplier (e.g., a manufacturer) to sell to the end consumers through a direct channel, which can lead to competition between the supplier and her independent downstream retailers or distributors. This competition phenomenon is often referred to as supplier encroachment (Yang et al., 2018). For example, soft-drink makers (e.g. Coca-Cola and Pepsi-Cola) and electronic producers (e.g. Apple, Dell, and Samsung) have long sold their products both to their downstream subsidiaries (their own online/physical stores) and to competitive retailers (e.g., department stores). Such examples can also be found in many other industries, including apparel and fashion industry (e.g., Nike, Adidas and Coach), food industry (e.g., Campbell Soup and Budweiser beer), cereal manufacturers and gasoline refiners (Li et al., 2016). Existing research demonstrates that encroachment usually benefits the supplier because it can reach consumers across multiple channels (Tsay & Agrawal, 2004), promote brand awareness and loyalty (Blair & Lafontaine, 2005), and provide a stronger incentive to reduce production cost (Yoon, 2016). However, most existing studies assume centralized encroachment, where the supplier centrally makes all decisions (e.g., pricing) for her subsidiary direct channel. In business reality, it is also common for a supplier to adopt the strategy of decentralized encroachment. For example, in 2006, Sony Corporation established StylingLife Holdings (―SLH‖), a holding company for Sony’s group of retail businesses with the intention of enabling the independent management of these retail businesses by their own management and employees. Sony also announced that ―in order to support the independence of this retail business group, Sony plans to gradually reduce its stake in the holding company‖ (Sony Corporation, 2006). For another example, several outdoor

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companies (such as Fjällräven, Teva, Timbuktu, Merrell and GoLite) not only sell to independent retailers like Nordstrom but also open their own retail brand stores across the USA. These brand stores are endowed with many independent retail decisions especially in accounting, discounts and promotions (Tracy, 2017). Research has shown that centralized encroachment may be difficult to flexibly deal with the intracompany trade and alleviate the fierce competition with the retailer (Kalnins, 2004; Vinhas and Anderson, 2005). On the other hand, several studies have shown that decentralized encroachment with a transfer price between a supplier and her downstream subsidiary is desirable because the downstream affiliate making independent decisions can alter the nature of dual-channel interactions and address concerns of dual distribution (Alles and Datar, 1998; Goex and Schiller, 2006). In particular, Arya et al. (2008) prove that decentralized encroachment where a supplier charges a transfer price above marginal cost to her downstream subsidiary conveys a less aggressive posture in retail competition. This in turn engenders higher wholesale price, and thereby benefits the supplier. Therefore, unlike previous research, in this paper, we consider both strategies of centralized and decentralized encroachment and compare them. As for a retailer, there are two possible consequences caused by supplier encroachment. On the one hand, the encroachment may pose a threat to a retailer by reducing his market share (Li et al., 2014; Ha et al., 2016; Li et al., 2015). On the other hand, it may improve the retailer’s profit due to a lower wholesale price or spill-over effect of cost-reduction investment brought by encroachment (Arya et al., 2007; Chiang et al., 2003; Li et al., 2016; Yoon, 2016; Zhang et al., 2018). Furthermore, a retailer can employ strategic inventory to alleviate the competitive pressure caused by supplier encroachment. Different from typical inventory types such as pipeline inventory, speculative inventory and safety inventory (Zipkin, 2000; Mantin and Jiang, 2017; Yao et al., 2009), strategic inventory is purchased

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and held by a retailer with the sole purpose of enhancing bargaining power with his supplier in the future (Anand et al., 2008). The use of strategic inventory is widely observed in business practice. For example, Martínez-de-Albéniz and Simchi-Levi (2013) show that although the supply is sufficient and diversified, the steel manufacturer Celsa in Spain still stores large piles of scrap metal outside its factory as an effective negotiation tool to force the local dealers to quote lower prices. The research on strategic inventory has been growing but still limited. Anand et al. (2008) are the first to formalize the role of strategic inventory as a negotiation tool with suppliers. Building and analyzing a two-period supply chain model with a supplier and a retailer, they prove that the retailer’s optimal strategy is to carry strategic inventory to lower the secondperiod wholesale price. Since this seminar work, strategic inventory is investigated in several papers which confirm that it may be beneficial to the supplier as well (e.g., Arya and Mittendorf, 2013; Hartwig et al., 2015; Arya et al., 2015; Moon et al., 2018). In particular, Guan et al. (2019) illustrate that a retailer can employ strategic inventory to alleviate the threat from its supplier’s centralized encroachment. They note that the joint use of strategic inventory and supplier encroachment strategies may benefit both players. However, in their models and analysis, Guan et al. (2019) omit the common business practice of decentralized encroachment. As explained earlier, decentralized encroachment has been shown to be advantageous than centralized encroachment in the literature (e.g., Arya et al., 2008). Hence, it is worthwhile to study the interplay between strategic inventory and both strategies of centralized and decentralized encroachment. Furthermore, it is also worthwhile to see if and how decentralized encroachment can be better than centralized encroachment in the presence of the retailer’s use of strategic inventory. To this end, in this paper, we build a two-period supply chain model with a supplier and a retailer. The supplier can directly sell to consumers through her subsidiary, which may be centralized or decentralized

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with the supplier. The subsidiary competes with the retailer selling substitutable products. To deal with competition from the supplier, the retailer, in turn, can carry strategic inventory from the first period to the second period to achieve wholesale price concessions from the supplier. In this paper, we make the following major contributions. In terms of modelling, first, unlike most existing studies considering centralized encroachment only, our paper explicitly considers the fact that a supplier can choose between centralized and decentralized encroachment. Second, this research is a first to study a retailer’s use of strategic inventory when his supplier can choose between decentralized and centralized encroachment. In terms of our main findings, first, our research is the first to show that decentralized encroachment outperforms centralized encroachment for both the supplier and the retailer. Second, we show that with decentralized encroachment, both the supplier and the retailer will benefit from the use of strategic inventory if the unit holding cost is below a threshold. Otherwise, only the supplier can benefit. The remainder of this paper is organized as follows. Section 2 reviews the related literature and compare this paper with the most related papers in the literature. Section 3 introduces our two-period model and initial analysis. In Section 4, we identify the equilibrium outcomes under centralized and decentralized encroachment in the presence of strategic inventory. Section 5 presents the results from comparing different strategies: Subsection 5.1 compares centralized and decentralized encroachment. Subsection 5.2 compares two strategies: decentralized encroachment with and without strategic inventory. Conclusions and future research are provided in Section 6. All key proofs are provided in Appendix.

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2. Literature Review There are three streams of literature related to our study: supplier encroachment, dual channel supply chain and strategic inventory. In this section, we review those three research streams one by one. Supplier encroachment occurs when in addition to independent retailer(s), a supplier uses a direct channel to sell her products to end customers. Traditionally, since supplier encroachment usually ends up with the reduction of the retailer’s market share, it will lead to channel conflict and hurt the retailer under asymmetric information (Li et al., 2014), retailer-led supply chain (Zhang et al., 2019), endogenous product quality (Ha et al., 2016) or pay-on-delivery scheme (Liu, et al., 2019). Li et al. (2015) study supplier encroachment in competitive supply chains and find that encroachment may hurt the suppliers and the retailers. However, in some cases, the retailer can benefit from supplier encroachment (Arya et al., 2007; Chiang et al., 2003). Li et al. (2016) show that supplier encroachment may be detrimental to the supplier when the retailer has strong fairness concerns and significant marketing advantage, but the retailer may benefit from it. Yang et al. (2018) explore supplier encroachment in a supply chain with nonlinear pricing. They show that the retailer is always worse off from it, but the supplier with weaker power is more likely to be better off. However, contrasts to conventional wisdom, some studies demonstrate that both the supplier and the retailer could benefit from supplier encroachment (Dumrongsiri, et al., 2008; Tsay and Agrawal, 2004; Yoon, 2016; Zhang et al., 2018). The prevalence of supplier encroachment motivates many researchers to analyze dualchannel supply chains, where a supplier can sell her product to end consumers directly as well as through independent retailers (Chen et al., 2012; Lu et al., 2018; Feng, 2017; Qing et al, 2017; Yan et al., 2015; Yu et al., 2017). Cattani et al. (2006) investigate the pricing decisions when a manufacturer has a direct Internet channel and show that the manufacturer

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will not choose the equal-pricing policy if the direct channel is more convenient than the traditional channel. Cai et al. (2009) study the effect of price discount and pricing schemes on dual-channel supply chain competition and find that price discount contracts can outperform the non-contract scenarios. Zhou et al. (2018) examine the effects of free riding on dual-channel sales effort and propose a cost-sharing contract to coordinate the dualchannel supply chain. Chen et al. (2017) study price and quality decisions in dual-channel supply chains and demonstrate that introducing a new channel can improve product quality. Modak and Kelle (2018) analyze a dual-channel supply chain with price and delivery-time dependent stochastic customer demand. Furthermore, the effects of risk aversion on the optimal pricing strategy and coordination of a dual-channel supply chain are also investigated (Liu and Cao, 2016; Li et al, 2016). Most of the research above studies dual-channel systems assuming centralized encroachment, where a supplier sells to the final consumers and centrally make all relevant decisions in her direct channel. The downsides of centralized encroachment include in excessive supplier encroachment and the inflexibility when dealing with the direct channel (Kalnins, 2004). However, several papers (e.g., Alles and Datar, 1998; Goex and Schiller, 2006; Liu et al., 2015) demonstrate the benefits of a supplier decentralizing with its affiliates through strategic transfer pricing. In particular, our paper is related to Arya et al. (2008), where they demonstrate that a supplier can benefit from decentralization by charging transfer price above marginal cost. However, our paper differs in the following aspects. First, they employed a one-period model, while we build a two-period framework. Second, they focused on the supplier’s use of transfer price, but our focus also includes the retailer’s use of strategic inventory. Third, they only evaluate the supplier’s profit. We pay more attention on both the supplier’s and the retailer’s profits and investigate when win-win outcomes can be achieved in the presence of strategic inventory.

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Our work also relates to the growing stream of literature on strategic inventory. Strategic inventory as a retailer’s competitive tool is formulated first in Anand et al. (2008). In a twoperiod model, they show that strategic inventory can induce the supplier to lower the wholesale price in the second period. Since then, several papers have investigated strategic inventory in different settings. Arya and Mittendorf (2013) illustrate the role of strategic inventory in the design of manufacturer-to-consumer rebates. Hartwig et al. (2015) confirms the importance of strategic inventory via empirical laboratory experiments. Arya et al. (2015) show that in the presence of strategic inventory, decentralized procurement outperforms centralized procurement, which is counter-intuitive. Roy et al. (2017) find that even when a supplier cannot observe the inventory level, she may initiatively lower the wholesale price to respond to her retailer’s willingness to hold inventory. Mantin and Jiang (2017) study the effect of inventory deterioration on the supplier-retailer interaction and find that the carrying strategic inventory may result in higher wholesale price as well as selling price. Moon et al. (2018) show that when strategic inventory is used, retailer-investment efforts are beneficial to both the manufacturer and the retailer. However, this may not be true for manufacturerinvestment efforts. Our work differs from the above papers on strategic inventory along three important dimensions. First, we consider the setting where two firms engage in dual-channel vertical competition, rather than a single channel setting adopted by most existing studies. Second, we compare the equilibrium outcomes between centralized and decentralized encroachment, while existing studies focus only on centralized encroachment. Finally, our work is related to Guan et al. (2019) where they focus on the centralized encroachment strategy in a dualchannel supply chain with perfectly substitutable products. Our work significantly differs by studying both centralized and decentralized encroachment when products are imperfect substitutes. Note that decentralized encroachment is common in business practice but rarely-

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investigated in the literature. Furthermore, to make the analysis clearer, they focus on supplier encroachment in the second period only, while we explicitly consider supplier encroachment in both periods. Although they show that both the supplier and her buyer can benefit from the joint use of centralized encroachment and strategic inventory, we find that when the strategic inventory can be used, the supplier with decentralized encroachment is less aggressive. In particular, decentralized encroachment leads to less retail quantities and higher wholesale prices in both periods for the supplier. Furthermore, decentralized encroachment can make the retailer sell more and hold less strategic inventory. In short, in this paper, we prove a major new finding that compared with centralized encroachment, decentralized encroachment is more beneficial for the supplier and the retailer.

3 Model and Benchmark Analysis In this section, we consider strategic inventory in a two-period model. We analyze both centralized and decentralized encroachment. For comparison purpose, the results for the two-period benchmark model without strategic inventory are also outlined.

3.1. Model and Notation We study dual-channel competition and strategic inventory in a dyadic supply chain with one supplier (She) and one independent downstream retailer (He). In addition to the retailer, the supplier sells directly to consumers through a downstream (retail) subsidiary. In this paper, we focus on a two-period model so that strategic inventory can be considered. In each period t ( t  1,2 ), the upstream supplier can wholesale differentiated products to her downstream retail subsidiary and to the retailer. The two downstream parties then sell these products to end markets. For ease of exposition, we use subscripts u , s and r on variables to indicate upstream supplier, subsidiary and retailer, respectively. The inverse demand for the retail product of firm i ( i, j  s, r , i  j ) in period t is given by pti  a  qti  kqtj , where pti is the retail price paid by consumers for firm i ’s products, and qti denotes the product

quantities sold by firm i in the retail market. The value of k ( 0  k  1 ) represents the substitution degree between the two competing products. In particular, k  0 and k  1

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correspond to the cases of independent and perfectly substitutable products, respectively. Finally, the demand intercept a is assumed to be sufficiently large so that the level of strategic inventory can be positive for at least some k . As shown later, we need a  4h . In each period, the supplier and the retailer make pricing and/or quantities decisions to maximize their own profit. As downstream retail firms, the subsidiary and the retailer may procure extra products in the first period and carry excess purchases as inventory to the second period. We assume that the unit holding cost for strategic inventory, if any, is h , h  0 . The supplier has two alternative encroachment strategies: centralization and

decentralization. Under centralized encroachment, the supplier fully integrates with her downstream subsidiary. In other words, the supplier centrally makes all decisions for the subsidiary. Under decentralized encroachment, the supplier charges the subsidiary a unit transfer price but the subsidiary will decide on quantities and retail prices to maximize its own profit. As the supplier may have less experience in selling directly to consumers, the encroachment can be costly (Arya et al., 2007; Guan et al., 2019; Ha et al., 2016). Hence, we assume that the unit direct selling cost for the supplier’s retail subsidiary is b , 0  b  a . Without loss of generality, the production cost, the retailer’s selling cost and other fixed costs are normalized to zero. With centralized encroachment, the sequence of events is as follows. In the first period: Stage 1: The supplier sets her a unit wholesale price w1r for the retailer. Stage 2: Simultaneously, the supplier chooses retail quantity q1s and the retailer chooses retail quantity q1r . The retailer orders the quantities Q1r  q1r and carries strategic inventory I r  Q1r  q1r to the second period.

In the second period: Stage 1: The supplier establishes a unit wholesale price w2 r for the retailer. Stage 2: Simultaneously, the supplier and the retailer choose their selling quantities q2 s and q2r , respectively. For the retailer, his order quantity is Q2 r  q2 r  I r . With decentralized encroachment, the difference is that the supplier will decide the unit intracompany transfer price for the subsidiary, and the subsidiary has the option to carry strategic inventory. The sequence of moves is listed as follows.

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In the first period: Stage 1: The supplier sets a wholesale price w1r for the retailer and an intracompany transfer price w1s for the subsidiary. Stage 2: The subsidiary decides on her retail quantity q1s and how much to buy from the supplier Q1s  q1s . Simultaneously, the retailer decides on her retailer quantity q1r and how much to buy from the supplier Q1r  q1r . The subsidiary and the retailer then carry strategic inventories I s  Q1s  q1s and I r  Q1r  q1r , respectively, to the second period. In the second period: Stage 1: The supplier quotes the wholesale price w2 r for the retailer and the intracompany transfer price w2 s for the subsidiary. Stage 2: The subsidiary and the retailer jointly choose their retail quantities q2 s and q2r , respectively. Equivalently, they decide on order quantities Q2 s  q2 s  I s and Q2 r  q2 r  I r , respectively. Throughout this paper, we use backward induction to derive the subgame-perfect equilibrium outcomes. For clarity, the decision variables and model parameters used in this paper are listed in Table 1. Table 1. Notation List Notations Indices i

j t ˆ ~

c d

Description Index of firms: upstream supplier ( i  u ), subsidiary ( i  s ) or retailer ( i  r ) ( i ofd rival )trerreretailerSsssssssssssss Index firm in dual-channel, j  {s, r} \ {i} Index of number of periods, t  1,2 Index of centralized encroachment without strategic inventory Index of decentralized encroachment without strategic inventory Index of centralized encroachment with strategic inventory Index of decentralized encroachment with strategic inventory

Parameters

a

k h

b

a

qt wt I

Market base (the intercept of demand function) The substitutability between competing products and 0  k  1 The unit holding cost for strategic inventory and h  0 The unit direct selling cost at the encroachment channel and 0  b  a The net demand intercept of downstream cost and a  a  b The vector of retail quantities in period t  1,2 The vector of supplier’s prices charged to the retailer and subsidiary The vector of inventory levels for the retailer and subsidiary

Decision variables

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qti wti Ii Dependent variables pti Qti  ti

Retail quantity for firm i  s, r in period t  1,2 Supplier’s wholesale/transfer price for the firm i  s, r in period Strategic inventory level held by firm i  s, r

t  1,2

Retail price for firm i  s, r in period t  1,2 Ordering quantities for firm i  s, r in period t  1,2 Profit function of firm i  u, s, r in period t  1,2

3.2. Two-period Benchmark Model without Strategic Inventory To facilitate discussions in the rest of the paper, we begin with a natural benchmark model where strategic inventory is disallowed. This is possible when the retailer sticks to a just-in-time strategy. Without strategic inventory, the decisions in the first period and in the second period are independent. The profits of the supplier and the retailer in each period are respectively given by:

tu  wtrqtr  (a  qts  kqtr  b)qts , t  1,2 . tr  (a  qtr  kqts )qtr  wtr qtr , t  1,2 .

Models similar to our one-period dual-channel supply chain have been extensively studied in the literature (e.g., Arya et al. 2008). However, to lay a foundation for future comparisons, the equilibrium outcomes under centralized and decentralized encroachment are summarized in Table 2. Table 2. Equilibrium Outcomes under Centralized and Decentralized Encroachment when Strategic Inventories are Disallowed Centralized encroachment

Decentralized encroachment

wr  w1r  w2r

k 3a  4(2  k 2 )a 2(8  3k 2 )

ws  w1s  w2 s

-------

qs  q1s  q2 s

(8  k 2 )a  2ka 2(8  3k 2 )

qr  q1r  q2 r

2(a  ka ) 8  3k 2

k (a  ka ) 2(2  k 2 ) 2a  ka 2(2  k 2 ) a  ka 2(2  k 2 )

Supplier’s profit (  u )

4a 2  8ka a  (8  k 2 )a 2 2(8  3k 2 )

a 2  2ka a  2a 2 2(2  k 2 )

Retailer’s profit (  r )

8(a  ka ) 2 (8  3k 2 ) 2

(a  ka ) 2 2( 2  k 2 ) 2

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a 2

It can be verified from Table 2 that in each period, decentralized encroachment with ~  0 ) on the one hand depresses the affiliated transfer prices above marginal cost (i.e., w s

retail arm’s market (i.e., q~s  qˆ s ), and on the other hand expands the retailer’s market share (i.e., q~r  qˆ r ). The use of transfer price provides a competitive posture that the supplier is less aggressive in the downstream retail encroachment, which in turn permits higher ~  wˆ ). Although decentralization with modest transfer price wholesale prices (i.e., w r r ~ w ~ ) lowers the supplier’s retail profit, it boosts her wholesale profit even more. As (0 w s r

a result, the use of transfer price can benefit both the supplier (i.e., (i.e.,

~ ˆ ) r   r

for

~ ˆ ) u   u

and the retailer

0  k 1.

4 Equilibrium Outcomes with Strategic Inventory In this section, to examine the effects of strategic inventory as well as the transfer pricing, we use our models in Subsection 3.1 to identify the equilibrium outcomes under centralized and decentralized encroachment when strategic inventory are allowed.

4.1. Centralized Encroachment By backward induction, at stage 2 in the second period, the supplier chooses her retail quantity, q2 s , to maximize firm-wide profit given the retailer’s chosen retail quantity q2r , the strategic inventory I r he has carried, and the stated wholesale price w2 r . Denoting the vector of retail quantities in period t by q t , the supplier’s problem is: Max  2u (q 2 , I r , w2 r )  w2 r (q2 r  I r )  (a  q2 s  kq2 r  b)q2 s .

(1)

q2 s

In (1), the first term, w2r (q2 r  I r ) , reflects the supplier’s wholesale profit, whereas the second term, (a  q2s  kq2r  b)q2s , reflects her retail profit. Similarly, given the supplier’s wholesale price w2 r , her chosen quantity q2 s , and the retailer’s strategic inventory I r , the retailer chooses quantity q2r to maximize his profit: Max  2 r (q 2 , I r , w2 r )  (a  q2 r  kq2 s )q2 r  w2 r (q2 r  I r ) .

(2)

q2 r

Solving (1) and (2) simultaneously reveals the following retail quantities: q2 s ( w2 r ) 

2a  ka  kw2 r 2a  ka  2 w2 r and q2 r ( w2 r )  . 2 4k 4  k2

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(3)

It can be seen from (3) that intuitively, the retailer’s quantity decreases in the wholesale price charged to him. Due to the strategic inventory carried from the first period, his associated order quantity is q2 r ( w2 r )  I r . Given this, at stage 1 in the second period, the supplier sets her wholesale price to maximize her profit: Max  2u (q 2 ( w2r ), I r , w2r ) .

(4)

w2 r

Solving the first-order condition of (4) yields the supplier’s second-period wholesale price as follows: w2 r ( I r ) 

4(2  k 2 )a  k 3a  (16  8k 2  k 4 ) I r . 2(8  3k 2 )

(5)

As can be expected, the supplier’s wholesale price decreases in the inventory carried forward by the retailer. In particular, since the retailer’s strategic inventory can meet highdemand consumers’ needs, higher level of strategic inventory will reduce his incentive to order more, leading to a lower wholesale price in period 2. Given the best responses in period 2, we continue to solve the game in period 1. At stage 2, the supplier chooses her direct selling quantity q1s to maximize her two-period profit: Max 1u (q1 , I r , w1r )  w1r (q1r  I r )  (a  q1s  kq1r  b)q1s q1 s

  2u (q 2 ( w2 r ( I r )), I r , w2r ( I r )) .

(6)

Similarly, the retailer chooses period 1 retail quantity q1r and strategic inventory level I r to maximize his two-period profit: Max 1r (q1 , I r , w1r )  (a  q1r  kq1s )q1r  w1r (q1r  I r )  hI r q1r , I r

  2r (q 2 ( w2r ( I r )), I r , w2r ( I r )) .

(7)

Solving (6) and (7) jointly yields their retail quantities and the retailer’s strategic inventory level as follows: q1s ( w1r )  I rc ( w1r ) 

2a  ka  kw1r 2a  ka  2 w1r , q1r ( w1r )  , and 2 4k 4  k2

4(3k 4  16k 2  24)a  (3k 4  16k 2  32)ka  (128  96k 2  18k 4 )(h  w1r ) . 6(2  k 2 )(4  k 2 )2

(8)

Back to stage 1 in the first period, the supplier chooses the wholesale price to solve: Max 1u ( w1r )  w1r [q1r ( w1r )  I r ( w1r )]  [a  q1s ( w1r )  kq1r ( w1r )  b]q1s ( w1r ) c

w1 r

 2u (q2 (( w2r ( I rc (w1r )))), I rc (w1r ), w2r ( I rc (w1r ))) .

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(9)

Solving (9) yields supplier’s period 1 wholesale price. We then substitute it back to obtain the equilibrium outcome under centralized encroachment, which is stated in the following lemma. LEMMA 1: Under centralized encroachment, the equilibrium wholesale prices, quantities 2 and strategic inventory level are as follows. When h  f (k , b)  4(a  ka )(42 2 k ) ,

(8  3k )

4(576  832 k  396 k  63k )a  (128  400 k  288k  63k )ka  2(4  3k 2 )(8  3k 2 ) 2 h , 2(8  3k 2 )(272  264 k 2  63k 4 ) 4(48  68k 2  21k 4 )a  (80  8k 2  21k 4 )ka  2(20  9k 2 )(8  3k 2 )h , w2cr  2(272  264 k 2  63k 4 ) (k 2  4)[2k (256  252 k 2  63k 4 )a  (2176  2416 k 2  792 k 4  63k 6 )a ]  2k (4  3k 2 )(8  3k 2 ) 2 h , q1cs  2(4  k 2 )(8  3k 2 )(272  264 k 2  63k 4 ) 2

w1cr 

4

6

2

4

6

2(4  k 2 )(256  252 k 2  63k 4 )(a  ka )  2(4  3k 2 )(8  3k 2 ) 2 h (4  k 2 )(8  3k 2 )(272  264k 2  63k 4 ) (4  k 2 )[2k (21k 2  44)a  (272  176 k 2  21k 4 )a ]  2k (20  9k 2 )(8  3k 2 )h q2c s  2(4  k 2 )(272  264 k 2  63k 4 ) q1cr 

q2cr 

2(4  k 2 )(44  21k 2 )(a  ka )  2(20  9k 2 )(8  3k 2 )h (4  k 2 )(272  264k 2  63k 4 )

I rc 

2(20  9k 2 )[4(4  k 2 )(a  ka )  (8  3k 2 ) 2 h] . (4  k 2 ) 2 (272  264 k 2  63k 4 )

, , ,

Otherwise, they are identical to those under centralized encroachment without strategic inventory. From Lemma 1, we can see that only when the holding cost is below a threshold, f ( k , b) , the retailer will carry strategic inventory. Furthermore, only when the direct selling cost is less than F ( k , h) , the supplier can use the strategy of centralized encroachment in both periods, where  (4  k 2 ) A1a  2k (20  9k 2 )(8  3k 2 ) 2 h ,   (4  k 2 )(272  176 k 2  21k 4 ) F ( k , h)   2 2 2 2  (2  k ) A2 a  2k (4  3k )(8  3k ) h , 2 2 4 6   (4  k )(2176  2416 k  792 k  63k )

4(4  k 2 )a (8  k 2 )(8  3k 2 ) 4(4  k 2 )a h (8  k 2 )(8  3k 2 ) h

,

A1  272  88k  176k 2  42k 3  21k 4 , A2  2176  512k  2416 k 2  504k 3  792k 4  126k 5  63k 6 .

Since

df (k , b) 4k (4  k 2 )   0 , f ( k , b) always increases in b . As b increases, the db (8  3k 2 ) 2

supplier’s strategy of centralized encroachment becomes less effective and she relies more on the wholesale channel. Thus, the retailer’s use strategic inventory becomes more effective. On the other hand, we can verify that f ( k , b) increases in k if:

16

bb 

32  32k  12k 2  6k 3  3k 4 a. 32  12k 2  3k 4

Otherwise, f ( k , b) decreases in k . Therefore, when the supplier’s encroachment cost is sufficiently large, the retailer is more willing to carry strategic inventory with more substitutable or competitive products. Otherwise, the retailer is less willing to adopt strategic inventory. Based on the solutions summarized in Lemma 1, the profits of the supplier and the retailer can be derived, which is stated below. LEMMA 2: Under centralized encroachment, the equilibrium profits of the supplier and the retailer are:  C1  32(4  k 2 )(8  3k 2 ) 2 (a  ka )h  4(8  3k 2 ) 4 h 2 , h  f ( k , b)   2(4  k 2 ) 2 (8  3k 2 )(272  264 k 2  63k 4 ) c , u   2 2 2 4 a  8 k a a  ( 8  k ) a  , h  f ( k , b)  2(8  3k 2 )   2[C2 (a  ka ) 2  C3 (a  ka )h  (8  3k 2 )5 (304  321k 2  81k 4 )h 2 ] ,   [(4  k 2 )(8  3k 2 )(272  264 k 2  63k 4 )]2 c r   8(a  ka ) 2  ,  (8  3k 2 ) 2 

(10)

h  f ( k , b)

,

(11)

h  f ( k , b)

where C1  (4  k 2 )2[12(96  88k 2  21k 4 )(a  2ka )a  (2176 1776k 2  240k 4  63k 6 )a 2 ] , C2  4(4  k 2 )2 (79360 149760 k 2  106128 k 4  33480 k 6  3969 k 8 ) , C3  4(4  k 2 )(8  3k 2 ) 2 (3776  4944 k 2  2124 k 4  297 k 6 ) .

4.2. Decentralized Encroachment In contrast to centralized encroachment, the supplier under decentralized encroachment will charge a transfer price to its downstream retail subsidiary. Then the subsidiary decides on retail quantities and strategic inventory levels to maximize its own profit. In this case, the retailer’s period 2 decision is the same as in (2). Given transfer price w2 s , the subsidiary’s inventory levels I s , and the retailer’s chosen quantity q2r , the subsidiary chooses its retail quantity q2 s to solve: Max  2 s (q 2 , I s , w2 s )  (a  q2 s  kq2 r  b)q2 s  w2 s (q2 s  I s ) . q2 s

Solving (2) and (12) simultaneously reveals the retail quantities as follows:

17

(12)

q2 s ( w 2 ) 

2a  ka  2 w2 s  kw2 r 2a  ka  kw2 s  2 w2 r and q2 r (w 2 )  . 2 4k 4  k2

(13)

Then the supplier’s period 2 pricing problem is: Max 2u (q 2 (w 2 ), I r , w2r ) .

(14)

w2 s , w2 r

Solving (14) yields the period 2 transfer price and wholesale price in the equilibrium: w2 s ( I r ) 

k[a  ka  (2  k 2 ) I r ] and w2 r ( I r )  a  I r . 2 2 2(2  k )

(15)

Note that both the transfer price and the wholesale price are k (a  ka2 ) and a , 2(2  k )

2

respectively, identical to those under the case of no strategic inventory, minus an adjustment for the retailer’s inventory levels. This is similar to the effect of strategic inventory on the supplier’s wholesale price under centralization. Hence, higher level of strategic inventory will reduce both the transfer price and the wholesale price in period 2. In period 1, the subsidiary chooses its selling quantity q1s and inventory levels I s to maximize its two-period total profit. Max 1s (q1 , w1s , I)  (a  q1s  kq1r  b)q1s  w1s (q1s  I s )  hI s q1 s , I s

  2 s (q 2 (w 2 ( I r )), I s , w2 s ( I r )) .

(16)

Similarly, the retailer decides his period 1 retail quantity q1r and inventory levels I r to maximize his two-period total profit. Max 1r (q1 , w1r , I r )  (a  q1r  kq1s )q1r  w1r (q1r  I r )  hI r q1r , I r

  2 r (q 2 (w 2 ( I r )), I r , w2 r ( I r )) .

(17)

Solving the first-order conditions of (16) and (17) jointly yield their retail quantities and strategic inventory levels as below: 2a  ka  kw1s  2 w1r 2a  ka  2w1s  kw1r , q1r (w1 )  , 2 4k 4  k2 (3  k 2 )a  ka  2(2  k 2 )( h  w1r ) . I sd  0 , and I rd ( w1r )  3(2  k 2 )

q1s (w1 ) 

(18)

Interestingly, from (18), we can see that the subsidiary’s equilibrium strategic inventory is zero. The reason is as follows. The supplier makes pricing decisions to maximize her total profit. And her total profit consists of her wholesale profit from the retailer and her direct selling profit, which is independent of the subsidiary’s inventory I s . This can be seen from the supplier’s period 2 profit function  2u (q 2 (w 2 ), I r , w2 r ) in (14). Hence, the subsidiary’s

18

strategic inventory level has no effect on the period 2 transfer price and wholesale price as reflected in (15). Consequently, there is no incentive for the subsidiary to carry any strategic inventory. For clarity, when we refer to strategic inventory in the following discussions, we shall focus on the retailer’s strategic inventory. It can be seen from (3), (8), (13) and (18) that when the transfer price equals to the marginal cost (i.e., w1s  w2 s  0 ), two-period retail quantities are identical under centralized and decentralized encroachment, reflecting the fact that they are equivalent to the limiting case of marginal-cost transfer pricing. Moreover, in each period, as the transfer price increases, the subsidiary’s retail quantity decreases while the retailer’s retail quantity increases. This is consistent with the typical view that when the subsidiary is imposed with a higher transfer price, the competing retailer can benefit from it. Given the induced demand and inventory levels in (18), the supplier’s period 1 pricing problem is as follows: d d Max 1u (w1 )  w1r [q1r (w1 )  I r ( w1r )]  [a  q1s (w1 )  kq1r (w1 )  b]q1s (w1 )  hI s

w1 s , w1 r

 2u (q2 (w 2 (I rd (w1r )), I rd (w1r ), w2r ( I rd (w1r ))) .

(19)

Solving the first-order condition of (19) will give us the equilibrium wholesale prices. Substituting them back, we can then obtain the equilibrium outcomes under decentralized encroachment, as summarized below. LEMMA 3: Under decentralized encroachment, the equilibrium prices, quantities and

inventory levels are as follows. When h  g (k , b)  a  ka2 , 2(2  k )

w1ds 

k[9(a  ka )  (2  k )h] 17(2  k 2 )

w2ds 

2a  ka k[6(a  ka )  5(2  k 2 )h] , d (24  17 k 2 )a  10ka  20(2  k 2 )h , d , q1s  q2ds  w  2r 2 2 2(2  k 2 ) 34(2  k ) 17(2  k )

q1dr 

8(a  ka )  (2  k 2 )h 11(a  ka )  5(2  k 2 )h 5(a  ka )  10(2  k 2 )h . d d d , , , I  0 q  I  s 2r r 17(2  k 2 ) 17(2  k 2 ) 17(2  k 2 )

2

,

w1dr 

(36a  17 k 2 )a  2ka  4(2  k 2 )h 34(2  k 2 )

,

Otherwise, they are identical to those under decentralized encroachment without strategic inventory.

It is clear from Lemma 3 that the retailer can use strategic inventory when h  g (k , b) , and the

supplier

b  G (k ) 

can

employ

the

strategy

of

decentralized

encroachment

when

(2  k )a . It can be verified that g (k , b) is a function increasing in b . Therefore, as 2

b increases, the supplier’s decentralized encroachment strategy become less effective but

19

the retailer’s use of strategic inventory becomes more effective. On the other hand, similar to the effect of product substitution on f ( k , b) , we note that g (k , b) increases in k when ~ ~ 2  2k  k 2 bb  a and decreases in k when b  b , which reflects the strategic interaction 2 2k

between the supplier and the retailer. In particular, when the encroachment cost exceeds a threshold, more competition will make the retailer more willing to adopt strategic inventory. Interestingly, Lemma 3 shows that the supplier’s retail quantities

q1ds

and

q2ds

in each

period under decentralization just replicates the outcomes under the benchmark decentralization without strategic inventory. In other words, the strategic inventory held by the retailer has no impact on the supplier’s retail quantities. The reasons are as follow. The supplier, as the first mover, can anticipate the retailer’s response and set the transfer price and wholesale price accordingly to offset the effect on her retail sales caused by the retailer’s use of strategic inventory. Mathematically, plugging w2 s ( I r ) and w2 r ( I r ) from (15) into q2 s (w 2 ) from (13), I r will disappear in the supplier’s period 2 equilibrium retail quantity. Similarly, substituting w1s and w1r into q1s ( w1 ) from (18), q1s will become independent of h . This implies that q1s will not be affected by the strategic inventory as well. Using the equilibrium outcomes presented in Lemma 3, the profits of the supplier and the retailer can be obtained, which is stated in the following lemma. LEMMA 4: Under decentralized encroachment, the equilibrium profits of the supplier and the retailer are respectively:  (36  17 k 2 )(a  2ka )a  4(17  8k 2 )a 2  8(2  k 2 )(a  ka )h  8(2  k 2 ) 2 h 2 , h  g ( k , b)   34(2  k 2 ) 2 , (20)  ud   2 2 a  2 k a a  2 a  , h  g ( k , b)  2(2  k 2 ) 

155(a  ka ) 2  59(2  k 2 )(a  ka )h  76(2  k 2 ) 2 h 2 ,   289 (2  k 2 ) 2 d r   (a  ka ) 2  ,  2(2  k 2 ) 2 

h  g ( k , b)

.

(21)

h  g ( k , b)

5 Comparing Strategies and Results We use this section to compare the supplier’s equilibrium outcomes under centralized and decentralized encroachment, and to obtain managerial insights into the interplay between the

20

supplier’s encroachment strategy and the retailer’s use of strategic inventory. Furthermore, to show the effect of strategic inventory more clearly, we compare decentralized encroachment with and without the possibility of strategic inventory.

5.1. Centralized vs. Decentralized Encroachment with Strategic Inventory As explained in the benchmark analysis earlier, the use of transfer price above marginal cost allows the supplier to convey to the retailer that she is less aggressive in the retail competition. On the one hand, this reduces her own retail demand. On the other hand, this enhances the demand from the retailer, her wholesale customer. In this sense, decentralization generates higher wholesale profit at the expense of less retail profit. However, the retailer can use strategic inventory to influence the supplier’s decisions. Moreover, we can verify that f (k , b)  g (k , b) for 0  k  1 . Thus, when h  f (k , b) , the retailer will carry the strategic inventory under both centralized and decentralized encroachment. By comparing the wholesale prices, quantities and inventory levels under centralization and decentralization, the following proposition is confirmed. PROPOSITION 1: When the retailer carries strategic inventory under both centralized and decentralized encroachment, i.e., h  f (k , b) , we have: (i) The transfer prices of both periods under decentralization are set above marginal cost, i.e.,

w1ds  0

and

w2ds  0 .

(ii) The supplier’s retail quantities of both periods are lower under decentralization, i.e., q1ds  q1cs

and

q2ds  q2cs .

(iii) The retailer’s first-period retail quantity is higher under decentralization, i.e., q1dr  q1cr ;

In the second period, when 0  k  0.9759 or h  u(k , b) , q2dr  q2cr ; when

and h  u(k , b) ,

q2dr  q2cr ,

where u (k , b) 

(a  ka )( 4  k 2 )( 21k 2  20 ) (2  k 2 )( 2152  1662 k 2  315 k 4 )

0.9759 k  1

is a small value.

(iv) The wholesale prices of both periods are higher under decentralization, i.e.,

w1dr  w1cr ,

w2dr  w2cr . 2 2 4 (v) When h  v(k , b)  (a  k2a )( 4  k )(1472 2 1356 k 4 315 k )6 , the retailer’s strategic inventory

2(2  k )(5888  6556 k  2463 k  315 k )

level is lower under decentralization, i.e., I rd  I rc ; otherwise, Ird  Irc . For the benchmark case without strategic inventory, part (i), (ii) and (iii) of Proposition 1 indicate the downside of decentralization as it lowers the market share of the supplier’s direct selling. More specifically, in each period, transfer price above marginal cost in part (i)

21

depresses the retail demand in part (ii). As expected, this boosts the retailer’s selling quantities, as confirmed in part (iii). However, in contrast to the benchmark results where the retailer’s retail quantities are always higher under decentralization, when the product are close to perfectly substitutable and the unit inventory holding cost is very low (i.e, 0.9759 k  1 and h  u (k , b) ), the retailer’s period 2 retail quantity is lower under

decentralization. There are two possible reasons for this result. First, for sufficiently large k , the cost of softened competition is so high that the supplier prefers to reduce the transfer

price and increase the second-period wholesale price, which can intensify the double marginalization problem in the wholesale market. Second, as shown in part (v), the retailer’s strategic inventory level is lower under decentralization when the inventory cost is not too high. Consequently, the intensified double marginalization and lower level of strategic inventory lead to a lower second-period retail quantity. In general, the reduced retail demand under decentralization imposes a strict loss in the supplier’s retail profit. However, this less aggressive competitive posture from decentralization increases the retailer’s wholesale prices of both periods, as shown in part (iv), which together with his higher order quantity in each period is helpful to generate higher wholesale profit. As indicated in Subsection 4.2, marginal-cost transfer pricing yields equilibrium outcomes as those under centralization. Therefore, we can infer that a higher transfer price from the supplier can better balance the potential gain in the wholesale market and the potential loss in the retail market. We also note that 0  w1ds  w1dr and 0  w2ds  w2dr , which indicates that the optimal transfer price in each period is still a preferential pricing because it is lower than the corresponding wholesale price. Moreover, part (v) shows that the retailer will carry a modest level of strategic inventory depending on the holding cost. In particular, when h  v(k , b) , I rd  I rc ; when v(k , b)  h  f (k , b) , I rd  I rc ; when f (k , b)  h  g (k , b) , I rd  I rc (since I rc  0 for f (k , b)  h  g (k , b) ). This implies that

the chosen level of strategic inventory under decentralization has the potential to better balance between the costs in the period 1 induced by the higher wholesale price w1r and holding cost h and the benefit in the period 2 from a lower wholesale price w2 r . To see this clearly, we compare (5) and (15) and can confirm that when the transfer price is employed, the supplier’s period 2 wholesale price for the retailer becomes more sensitive to the strategic inventory level. This is because w2 r ( I r ) decreases in I r at the rate of 1 under

22

2 4 decentralization and at a lower rate 16  8k 2k  1 under centralization. Therefore, when

2(8  3k )

strategic inventory is less costly to hold (i.e., h  v(k , b) ), the retailer under decentralized encroachment can reduce the wholesale price more substantially via lower inventory level (i.e., I rd  I rc ) and vice versa. We summarize our results as below. PROPOSITION 2: When strategic inventories can be used, decentralized encroachment outperforms centralized encroachment for both the supplier and the retailer. Proposition 2 demonstrated bright side of the supplier’s decentralized encroachment with a moderate transfer price. This is different from Guan et al. (2019) and Arya et al. (2007), where they showed the bright side of the supplier’s centralized encroachment relative to no encroachment only. Furthermore, our results show that the supplier prefers decentralized encroachment regardless the retailer’s final choice of holding strategic inventory or not. At the extreme case of k  1 , Arya et al. (2008) shows that decentralized and centralized encroachment are equivalent in the absence of strategic inventory. As an extension to this result, we prove that decentralized encroachment can still be beneficial to the supplier and the retailer even for perfectly substitutable products. Comparing with the benchmark results in Arya et al. (2008), we can conclude that strategic inventory plays an important role on the profit improvement of decentralized vs. centralized encroachments, as confirmed in the following proposition. PROPOSITION 3: When strategic inventories can be used, c d c d (i) d (q1s  q1s )  0 , d (q2 s  q2 s )  0 .

dh

dh

(ii) when 0  k  0.7779 ,

d ( q1dr  q1cr ) 0 dh

, otherwise,

d ( q1dr  q1cr ) 0 dh

; when 0  k  0.9759 or

d (q2dr  q2cr ) d (q2cr  q2dr )  0 , when 0.9759 k  1 and h  u (k , b) , 0. dh dh d c d c (iii) d ( w1r  w1r )  0 , d ( w2 r  w2 r )  0 . dh dh c d d c (iv) when h  v(k , b) , d ( I r  I r )  0 , otherwise, d ( I r  I r )  0 . dh dh

h  u ( k , b) ,

(v) when h  α(k , b) 

(a  ka )( 4  k 2 )(376  312 k 2  63k 4 ) (2  k 2 )(3008  3112 k 2  1077 k 4  126 k 6 )

d ( ud   uc ) d ( dr   cr ) 0;  0. dh dh

23

d c , d ( u   u )  0 , otherwise

dh

Proposition 3 shows the effect of inventory holding cost on outcomes of decentralized vs. centralized encroachment. In particular, as h increases, the retailer’s use of strategic inventory becomes less effective and he will carry more inventory for h  (k , b) and less inventory for h  (k , b) in order to force the supplier to charge lower wholesale prices in both periods. This increases the supplier’s retail demand in the first period and the wholesale demand in the second period. Consequently, as h increases, the supplier’s profit improvement will increase for h  (k , b) but decrease for h  (k , b) while the retailer’s profit improvement always increases. Therefore, the benefit of decentralized encroachment still exists when the retailer’s strategic inventory is considered. More interestingly, both the supplier and the retailer can benefit more even for higher inventory holding cost. This complements a finding in Arya et al. (2008) that decentralized encroachment could outperform centralized encroachment when the inventory holding cost is zero. To better illustrate our analytic results and further discuss the effect of product substitutability and encroachment cost, we now conduct numerical examples. Without loss of generality, a is normalized to be 1. Following the thresholds on h used by Anand et al. (2008), we choose h =0, 0.11 and 0.22 to represent the low, medium and high inventory costs, respectively. The benchmark problem where strategic inventory can’t be used can be regarded as the special case of h   . We define the notations u  ud  uc and  r   dr  cr to represent the profit improvements of the supplier and the retailer from

decentralized encroachment, respectively. Moreover, based on the joint effect of h and b , we shall identify the optimal strategies for both the retailer and the supplier. For simplicity, we denote the retailer’s strategy choices as HI (Holding Inventory) and NI (No Holding Inventory) and the supplier’s as D (Decentralized Encroachment), C (Centralized Encroachment) and NE (No Encroachment). We also denote the strategy pairs for the retailer and the supplier by (α, β ) , where α =HI or NI and β =D, C or NE. Based on Lemma 1 and Lemma 3, when b  min{ F (k , h), G(k )} , the direct channel has no effect on the supply chain under centralized or decentralized encroachment. Therefore, to focus on comparing centralized and decentralized encroachment, we choose b = 0, 0.3, 0.6 to represent low, medium and high levels of direct selling cost, respectively. With b = 0, 0.3, 0.6, we plot the levels of strategic inventory and profit improvement of decentralization and centralization in Figures 1-3 for h  0, 0.11 and 0.22, respectively.

24

5.1.1. Low Inventory Holding Cost (h=0) From Figure 1(a), we can see that low holding cost allows the retailer to carry strategic inventory under either decentralization or centralization for all 0  k  1 . Moreover, we d c observe that I r  I r for any b , which is consistent with Proposition 1(v). As b increases,

the supplier’s direct selling ability becomes less effective and the retailer’s strategy of strategic inventory becomes more competitive due to low holding cost. Consequently, strategic inventory under either centralized or decentralized encroachment will increase with b . However, when b =0.6 and k  0.8 , the direct selling cost under decentralized

encroachment is so high, i.e. b  G (k ) , that the supplier only relies on the retailer to sell to the customers. ×10-3

 ud

I rc (b  0.6)

 r

uc

I rd (b  0.6)

 u

I rc (b  0.3) I rd (b  0.3) I rc (b  0) I rd (b  0)

 cr

k

 dr

k (HI, D)

(a) The retailer’s inventory Ir

(b) Player’s profit improvement (b=0) ×10-3

×10-2

 r

 ud uc

 ud

uc

 r

 u

 u

 dr

 cr  dr

 cr

k

k (HI, D)

(HI, D)

(HI, NE)

(d) Player’s profit improvement (b=0.6)

(c) Player’s profit improvement (b=0.3)

Figure 1. The levels of strategic inventory and profit improvement from decentralized (vs. centralized) encroachment when h=0 and b=0, 0.3, 0.6.

25

In Figure 1(b), when b =0, encroachment and strategic inventory are effective for the supplier and the retailer respectively. Therefore, we observe that as the product competition increases, both players’ profits will decrease under centralized or decentralized encroachment. On the other hand, lower level of strategic inventory under decentralization would reduce the retailer’s first-period costs due to the less need for procuring at higher wholesale price and withholding excess inventory. This improves the retailer’s profit, as illustrated in Figure 1(b). At the same time, the supplier will charge a higher wholesale price in second period when the retailer holds less strategic inventory. As a result, the supplier can also benefit because of the increased wholesale profit. That’s why the strategy pairs are (HI, D) for all 0  k  1 . We also note that  r   u . The reason is that decentralization brings a strong boost to the wholesale market but lowers the retail demand. In other words, the retailer benefits more when the supplier chooses her preferred decentralization strategy. When b increases to 0.3, compared with b =0, the retailer can hold more inventories to achieve better bargaining position. We can see from Figure 1(c) that the profit improvements of both the supplier and the retailer increase with k . Therefore, the retailer can benefit from his low-cost strategic inventory, and the supplier can also benefit from the increased wholesale market driven by the retailer. This win-win situation leads to (HI, D) strategy pairs for 0  k  1 . Next, when b =0.6, the retailer’s strategy of strategic inventory is the most effective compared with b =0 and 0.3 and he will carry the highest level of strategic inventory. Figure 1(d) shows that the retailer’s profits under centralized or decentralized encroachment tend to first decrease and then increase as k increases from 0 to 0.8. In addition, both players’ profit improvements always increase with k . However, when k  0.8 , fierce product competition and high direct selling cost will make encroachment cost exceeds the threshold where the supplier does not actually sell through the direct channel, i.e. q1s  q2 s  0 , and encroachment will not be an option for the supplier. Therefore, the strategy pairs are (HI, NE) for k  0.8 . The decentralization problem is actually reduced to the no-encroachment case but with strategic inventory, which leads to lower profit improvements of both players when k increases from 0.8 to 1. Finally, we investigate the impact of b on both players’ profits. For any given k , h and b , we denote the profits of the retailer and the supplier by  r (k , h, b) and  u (k , h, b) , respectively. Moreover, let  r (k , h, b) be the retailer’s profit improvement and similarly

26

 u (k , h, b) be the supplier’s profit improvement. For all 0  k  1 in Figures 1(b)-(d),

iu (k ,0,0)  iu (k ,0,0.3)  iu (k ,0,0.6) and ir (k ,0,0)  ir (k ,0,0.3)  ir (k ,0,0.6) , i  c, d . These

findings indicates that the supplier’s profit decreases in her direct selling cost while the retailer’s profit increases in this direct selling cost. Therefore, higher direct selling cost would benefit the retailer especially when his inventory holding cost is low. On the other hand, note that  u (k ,0,0)   u (k ,0,0.3)   u (k ,0,0.6) , for all 0  k  1 . Hence, the supplier’s profit improvement increases with b . In addition, we also observe that the retailer’s profit improvement increases with b , i.e.,  r (k ,0,0)   r (k ,0,0.3)   r (k ,0,0.6) for 0  k  0.8 . However, when 0.8  k  1 , decentralized encroachment can’t come into play for b =0.6, and it is possible  r (k ,0,0.3)   r (k ,0,0)   r (k ,0,0.6) . Therefore, when the decentralized encroachment channel can be used, higher direct selling cost would increase the profit improvements of both players. 5.1.2. Medium Inventory Holding Cost (h=0.11)

27

×10-3

 ud

 r

uc

I rc (b  0.6)

 u

I rd (b  0.6) I rc (b  0.3) I rc (b  0)

I rd (b  0)

I rd (b  0.3)

 dr

 cr

0.65

k

k

0.65

(NI, D)

(HI, D)

(b) Player’s profit improvement (b=0)

(a) The retailer’s inventory Ir ×10-2

uc

×10-2

 ud

 ud

uc  r

 r

 u

 dr

 cr

 dr

 cr

 u

k

k (HI, D)

(HI, D)

(HI, NE)

(d) Player’s profit improvement (b=0.6)

(c) Player’s profit improvement (b=0.3)

Figure 2. The levels of strategic inventory and profit improvement from decentralized (vs. centralized) encroachment when h =0.11 and b=0, 0.3, 0.6.

For the case with medium inventory holding cost, we see a similar pattern in Figure 2 as that in Figure 1. However, there are several main differences. First, Figure 2(a) shows that the retailer will hold less inventories under centralized or decentralized encroachment for each b =0, 0.3 and 0.6 as the unit holding cost increases. In particular, for b =0, the retailer’s strategic inventory is zero when k is larger than a threshold (  0.65). Intuitively, this implies that when the unit holding cost increases, the retailer’s inventory strategy becomes cost inefficient. Therefore, it would be easier for the supplier to use the channel of decentralized encroachment and our model converges to the benchmark one when k  0.65 . This also explains why the strategy pairs of both players are (NI, D) for k  0.65 .

28

Second, we look at the impact of direct selling cost on the players’ profits in a way similar to the case for low inventory holding cost. The supplier still suffers from the higher direct selling cost while the retailer can benefit from it, i.e., for all 0  k  1 in Figure 2(b)-(d), iu (k ,0.11,0)  iu (k ,0.11,0.3)  iu (k ,0.11,0.6) and ir (k ,0.11,0)  ir (k ,0.11,0.3)  ir (k ,0.11,0.6) ,

i  c, d . In general, the profit improvement of both players’ increases with the direct selling

costs. The difference lies in the retailer’s profit improvement when 0.8  k  1 , where  r (k ,0.11,0.3)   r (k ,0.11,0.6)   r (k ,0.11,0) can be true. To explain this, recall that

decentralized encroachment can benefit the retailer more than centralized one by increasing the wholesale demand for higher direct selling cost. However, higher inventory holding cost limits this effect and reduces the retailer’s profit improvement, which eventually leads to the retailer’s profit improvement with encroachment is even worse than the one with no encroachment, i.e.,  r (k ,0.11,0)   r (k ,0.11,0.6) for 0.8  k  1 . Finally, we can see the impact of higher inventory holding cost on profits of the retailer and the supplier. For any 0  k  1 and b = 0, 0.3 and 0.6, we find that cr (k ,0, b)  cr (k ,0.11, b) and dr (k ,0, b)  dr (k ,0.11, b) . An observation is that the retailer’s profit decreases in his holding cost. We also find the negative correlation between the supplier’s profit and the holding cost under either centralization or decentralization, i.e., uc (k ,0, b)  uc (k ,0.11, b) and ud (k ,0, b)  ud (k ,0.11, b) . This shows that the supplier can’t benefit from a cost-inefficient

retailer. However, when it comes to players’ profit improvement, a different picture is found. For any 0  k  1 and b = 0, 0.3 and 0.6 in Figures 1 and 2,  r (k ,0, b)   r (k ,0.11, b) and  u (k ,0, b)   u (k ,0.11, b) . Therefore, under decentralization, higher holding cost can

increase the players’ profit improvement. Although the retailer’s higher holding cost limits the supplier from charging a higher wholesale price, the transfer price is the main driver to alleviate this limitation and thus generates more players’ profit improvement as mentioned above.

5.1.3. High Inventory Holding Cost (h=0.22)

29

×10-3

 ud

 r

uc

 u

I rc (b  0.6)

I rc (b  0) I rd (b  0) I rd (b  0.3)

I rc (b  0.3)

I rd

 cr

(b  0.6)

k

 dr

k

0.13

0.13

(HI, D)

(a) The retailer’s inventory Ir

(NI, D)

(b) Player’s profit improvement (b=0)

×10-2

 ud

×10-2

 r

 ud

uc

uc  u

 r

 dr

 cr

(HI, D)

 cr

 dr

 u

k

k (HI, D)

(NI, D)

(HI, NE)

(d) Player’s profit improvement (b=0.6)

(c) Player’s profit improvement (b=0.3)

Figure 3. The levels of strategic inventory and profit improvement from decentralization (vs. centralization) when h= 0.22, b = 0, 0.3, 0.6.

Turning to the high inventory holding cost case ( h  0.22 ), we illustrate the equilibrium outcomes in Figure 3. First, we note that when b =0, the threshold with no inventory is shifted from k  0.65 to k  0.13 . This is because the retailer’s higher holding cost will make the use of strategic inventory less efficient for large k . That’s why the strategy pairs of both players are (HI, D) for 0  k  0.13 and (NI, D) for k  0.13 in Figure 3(b). Second, in Figure 3(a), when b =0.3, the inventory holding cost is so high that the retailer will hold no inventory for k  0.2 . Therefore, higher holding inventory cost for the retailer will make the strategy pairs become (NI, D) for k  0.2 as shown in Figure 3(c). Moreover, since the strategic inventory level decreases in its holding cost, higher holding cost leads to less difference between I rd and I rc for b =0, 0.3 and 0.6. However, we still see I rd  I rc when the strategic inventory can be used. Consequently, as shown in Figure 3(b)-(d), the profit

30

improvements of both the retailer and the supplier are positive. Similar to the observation with h  0.11, when strategic inventory option is available, for any 0  k  1 and b =0, 0.3 and 0.6,  r (k ,0.11, b)   r (k ,0.22, b) and  u (k ,0.11, b)   u (k ,0.22, b) (see Figures 2(b)-(d) and 3(b)-(d)). As such, even though holding inventory is costly, using transfer prices under decentralization allow both the supplier and the retailer to improve their profit.

5.2. Decentralized Encroachment with and without Strategic Inventory So far, we have shown that compared with centralized encroachment, decentralized encroachment is always preferred by the supplier regardless the retailer’s final choice of carrying strategic inventory or not. One of the attendant problems is how strategic inventory impacts supply chain performance when the supplier adopts the strategy of decentralized encroachment. It is often believed that inventory should be eliminated as much as possible. As a matter of fact, several zero-inventory strategies, such as just-in-time ordering systems, are widely used in business practice. Recall that the pronounced merit of decentralized encroachment is that it helps the retailer keep modest inventory levels rather than eradicate them. Therefore, to better show the effect of strategic inventory, we shall compare between two possible strategies: decentralized encroachment with and without the possibility of strategic inventory. For clarity, we sometimes call the former inventory-holding decentralization and the latter zero-inventory decentralization. Benchmark results show that compared with zero-inventory centralization, zero-inventory decentralization benefits both the supplier and the retailer. Thus, comparing decentralized encroachment with and without the use of strategic inventory is sufficient when we aim to examine the effect of strategic inventory. Following the logic of inventory moderation, one can conjecture that decentralized encroachment with strategic inventory has the potential to create the right mix of sales and inventories. This conjecture is confirmed below. PROPOSITION 4: When the retailer carries strategic inventory under decentralization, i.e., h  g (k , b) ,

we have:

(i) The first-period transfer price is higher, i.e., price is lower, i.e.,

~ w1ds  w s

while the second-period transfer

~ . w2ds  w s

(ii) The supplier’s retail quantities for both periods are the same under inventory-holding decentralization and zero-inventory decentralization, i.e.,

31

~ . q1ds  q2ds  q s

q1dr  q~r ,

(iii) The retailer’s first-period retail quantity is lower, i.e., period retail quantity is higher, i.e.,

~ . q2dr  q r

(iv) The first-period wholesale price is higher, i.e., wholesale price is lower, i.e.,

while his second-

~ , w1dr  w r

while the second-period

~ . w2dr  w r

Proposition 4 shows the effect of strategic inventory on equilibrium prices and quantities when the supplier purses decentralized encroachment. In the first period, the supplier will ~ ) to reduce the retailer’s set higher transfer price ( w1ds  w~s ) and wholesale price ( w1dr  w r

incentive to hold excess inventory. Indeed, increasing

w1ds

and

keeps the supplier’s retail

w1dr

demand fixed ( q1ds  q~s ) and lowers the retailer’s selling quantity ( q1dr  q~r ). This increases the retail price and thus benefits the supplier’s retail profit. One the other hand, holding inventory boosts the retailer’s ordering quantities ( q1dr  I rd

~ ), q r

and is expected to improve

the supplier’s wholesale profit. In particular, the supplier’s period 1 retail profits under inventory-holding and no-inventory decentralization are respectively: ~ ~ ~  kq ~  b)q ~. q1ds , ur1  ur2  (a  q s r s

d d d ur 1  (a  q1s  kq1r  b)

Comparing these retail profits yields:

k (2a  ka)[ h  g (k , b)] ~ d ur  0. 1   ur1   34(2  k 2 )

(22)

Accordingly, the supplier’s period 1 wholesale profits under inventory-holding and noinventory decentralizations are respectively:

d d d d uw 1  w1r (q1r  I r )

,

~ ~ ~q ~ uw1  uw2  w r r

. The

wholesale profit difference between inventory case and no inventory case is: 18[h  α(k , b)][ h  g (k , b)] ~ d  uw 0, 1   uw1  289

(23)

where α(k , b)  358 a  52 ka 2153 ak  g (k , b) . 2

36 (2  k )

It can be seen from (22) and (23) that decentralized encroachment with strategic inventory improves the supplier’s retail profit as well as her wholesale profit in period 1. This is different from the results of the benchmark model, where higher transfer price without the use of strategic inventory increases the supplier’s wholesale profit but lower’s her retail profit compared with centralization. In the second period, if the retailer holds strategic inventory, it will reduce his willingness to procure from the supplier. As a counterattack, the supplier would offer a lower wholesale price ( w2dr  w~r ) to encourage the retailer to order more ( q2dr

32

~ ). q r

In addition, the supplier is

also willing to set a lower transfer price ( w2ds  w~s ) to maintain identical retail demand ( q2ds  q~s ). In the presence of strategic inventory, in the second period, the retailer benefits from a lower wholesale price, but the supplier will suffer in both the retail market and the wholesale market. This is confirmed by the followings: 5k (2a  ka)[ h  g (k , b)] ~ d ur  0, 2   ur2  34(2  k 2 )

50[h  β (k , b)][ h  g (k , b)] ~ d  uw 0, 2   uw 2  289

(24)

(25)

where β (k , b)   58 a  24 ka 2 17 ak  0 . 2

20 (2  k )

The results above show that strategic inventory can force the supplier to reduce period 2 wholesale price and engender the retailer’s profit margin exceeding the inventory holding cost. At the same time, the retailer’s withholding inventory also leads to higher demand and higher wholesale price for the supplier in the period 1. Hence, this may benefit the supplier as well. Comparing of the profit under decentralized encroachment with and without the possibility of strategic inventory, we have the following result. PROPOSITION 5: When strategic inventory can be used, i.e. h  g (k , b) , comparing with zero-inventory decentralization, (i) the supplier’s profit is higher under inventory-holding decentralization, (ii) when h  h~(k , b)  21(a  ka2 ) , the retailer’s profit is higher under 76(2  k )

inventory-holding decentralization; otherwise the retailer’s profit is lower. Proposition 5(i) confirms that the supplier always benefits from the use of strategic inventory. However, Proposition 5(ii) shows that the retailer benefits from withholding ~

strategic inventory if and only if h  h (k , b) . Therefore, a win-win situation can achieved when the retailer’s holding cost is sufficiently low. Moreover, note that the threshold ~ h (k , b) decreases in k but increases in b . Thus, the higher the product substitutability (the

more the direct selling cost), the less (the more) likely the retailer benefits from the use of the strategic inventory. The role of strategic inventory has also been studied in Guan et al. (2019). Their key results are that compared with zero-inventory case, inventory-holding centralized

33

encroachment can sometimes benefit both the supplier and the retailer. However, Guan et al. (2019) mainly consider centralized encroachment for completely substitutable products in the second period only. To make a more meaningful comparison between our study and Guan et al. (2019), we simplify our model to examine decentralized encroachment for completely substitutable products in the second period as well. The following proposition summarizes how decentralized encroachment can benefit the supplier and the retailer when the strategic inventory can be used. PROPOSITION 6: If encroachment can only occur in the second period, compared with centralized encroachment, (i) the supplier’s profit is higher under inventory-holding decentralized encroachment, (ii) when 0  b  0.1526a , the retailer’s profit is lower under inventory-holding h h

decentralized

encroachment;

when

0.1526a  b  0.2264a

and

1113b 1113 b , or 0.2264a  b  0.5a and 0  h  , the retailer’s profit is higher under 2600 2600

inventory-holding decentralized encroachment, where h 

1 (4279563 a  29995959 b 106516294

 17187 62001 a 2  4466896 ab  26610156 b 2 ) .

The first part of Proposition 6 states that the supplier still prefers decentralized encroachment when the encroachment is considered only in the second period with completely substitutable products. Therefore, the joint presence of decentralized encroachment and strategic inventory can help the supplier. The second part of Proposition 6 shows that when the direct selling cost is small, the retailer will prefer centralized encroachment. Otherwise, the retailer will prefer decentralized encroachment in most cases. Moreover, Proposition 6 confirms the robustness of our previous work in Proposition 2. We now illustrate and discuss the players’ profit improvement under inventory-holding decentralization and zero-inventory decentralization. Similar to Subsection 5.1, we consider three scenarios where h  0, 0.11, 0.22 represent low, medium and high levels of inventory holding cost, respectively. We assume the profit improvements of the supplier and retailer ~ ~ ~ ~ d d are  u   u   u and  r   r   r , respectively. Given each h , we plot the players’

profit improvement in Figures 4-6 by fixing b =0, 0.3, 0.6 and varying k from 0 to 1. 5.2.1. Low Inventory Holding Cost (h=0)

34

×10-2

×10-2

 ud ~ u ~ u

 ud

~  u

~  u

~  r

~ r

~  r

 dr

~ r

 dr

k

k (HI, D)

(HI, D)

(b) b=0.3

(a) b=0 ×10-1

×10-2

 ud

~  u

~ u

~  r

 dr

~ r

k (HI, D)

(HI, NE)

(c) b=0.6 Figure 4. The players’ profit improvement under inventory-holding decentralization (vs. zero-inventory decentralization) when h=0.

We start to discuss the case of low inventory holding cost ( h  0 ) and direct selling cost (b=0) in Figure 4(a), where each player can use their own strategies at zero costs. First, we can see that similar to Figure 1(b), as k increases, stronger competition will reduce both players’ profits under either inventory-holding or zero-inventory decentralization. Second, ~ ~ Figure 4(a) indicates that u and  r are always positive for all k  [0,1) . This means that

both the supplier and the retailer can benefit from the use of strategic inventory except for the case of perfect substitutability, which leads to the strategy pairs (HI, D) for k [0,1) . As such, the retailer can strategically withhold inventory to influence the supplier’s pricing ~ ~ strategy and improve the channel performance. Third, note that both  u and  r decrease in k . This intuition is as follows. As k increases, retail competition becomes

35

fiercer and the retailer tends to carry lower inventory level (as shown in Figure 1(a)) to save the inventory holding cost. As a result, both players’ profit under inventory-holding ~ decentralization close to those without inventory for higher k . Finally, we can see that  u ~ is greater than  r , contrary to the case in Figure 1(b). This implies that the use of strategic

inventory benefits the supplier more when the product substitution k is lower. This is because the supplier gains more profit in the first period both in the retail market and the wholesale market via higher transfer price and wholesale price. Those will discourage the retailer from holding strategic inventory. When b increases to 0.3 in Figure 4(b), the supplier’s strategy of decentralized encroachment becomes worse compared to b =0. First, similar to Figures 1(b)-(c), Figures 4(a)-(b) shows that higher direct selling cost reduces the supplier’s profit but increase the retailer’s

profit

under

inventory-holding or zero-inventory decentralization, i.e., ~ ~ ~ ud (k ,0,0)  ud (k ,0,0.3) , u (k ,0,0)  u (k ,0,0.3) ,  dr (k ,0,0)   dr (k ,0,0.3) , and  r (k ,0,0) 

~  r (k ,0,0.3) , for all 0  k  1 . Second, the profit improvements of the supplier and the retailer

~ ~ still decrease with k . However, for all 0  k  1 , i (k ,0,0)  i (k ,0,0.3) , i  u, r . Hence,

higher direct selling cost makes the retailer’s inventory strategy relatively more effective. This increases the retailer’s profit improvements. At the same time, the supplier’s profit improvement can also benefit from increased wholesale profits despite her retail profit is reduced. Consequently, the strategy pairs are still (HI, D) for both players. In Figure 4(c), when b =0.6, compared with b =0 and 0.3, the retailer can use the strategic inventory most effectively in competing with the supplier. This can be seen from the players’ profit curves under each regime, where the supplier’s profit still decreases while the retailer’s profit tends to first decrease and then increase as k increases from 0 to 0.8. However, when k  0.8 , strong competition and high direct selling cost will make the decentralized encroachment with or without inventory not an option for the supplier. That’s why the strategy pairs are (HI, NE) for k  0.8 . On the other hand, we observe that for all ~ ~ 0  k  1 , i (k ,0,0.3)  i (k ,0,0.6) , i  u, r . Therefore, both players’ profit improvements still increase with higher direct selling cost. 5.2.2. Medium Inventory Holding Cost (h=0.11)

36

×10-2

×10-2

 ud ~ u

 ud ~ u

~  u

~  u

~  r

~  r

 dr ~ r

~ r 0.22

0.65

 dr

0.37

k

(HI, D)

(NI, D)

(HI, D)

k

(b) b=0.3

(a) b=0

×10-3

×10-1

 ud

~  u ~ u

 dr

~  r

~ r

k (HI, D)

(HI, NE)

(c) b=0.6 Figure 5. The players’ profit improvement under inventory-holding decentralization (vs. zero-inventory decentralization) when h=0.11.

Figure 5 shows the results for medium holding inventory cost ( h  0.11 ). In Figure 5(a), when b  0 , we find that there is a threshold k  0.22 , where the retailer’s profit under ~ inventory-holding decentralization is not improved, i.e.  dr   r . In particular, when k  0.22 ,

~ ~ ~ ~ both  u and  r are always positive; when 0.22  k  0.65 ,  u is positive while  r ~ ~ becomes negative; when k  0.65 ,  u   r  0 . The reason is as follows. As stated in

Proposition 5(ii), higher holding cost ( h  h~(k , b) ) will cause the retailer’s loss from ordering and holding strategic inventory in the first period to exceed his benefit from lowering wholesale price in the second period. Since h~(k , b) decreases in k for given b , as k ~

increases from 0 to 0.22, we have h (k ,0)  0.11 . Hence, both the supplier and the retailer

37

can be better off when strategic inventory is used. However, as k increases from 0.22 to ~

0.65, we have h (k ,0)  0.11 . In line with Proposition 4, the supplier still benefits while the retailer becomes worse off. Finally, as k exceeds 0.65, Figure 2(a) shows that the retailer will employ zero-inventory strategy. Therefore, both players’ profit improvements remain unchanged, which leads to strategy pairs (NI, D) for k  0.65 . When b reaches to 0.3, we note that the threshold of retailer’s profit improvement ~

increases from k  0.22 to k  0.37 . To explain this, recall that the threshold h (k , b) increases ~ in b . Therefore, higher direct selling cost permits larger threshold value of  r . This can ~ be seen more prominently when b increases to 0.6, where  r is always positive for all 0  k  1 . Meanwhile, compared with the case of low holding inventory cost ( h  0 ), we find ~ ~ the supplier’s profit improvement increases with b , i.e., u (k ,0.11,0)  u (k ,0.11,0.3)

~  u (k ,0.11,0.6) , for 0  k  1 . Moreover, for retailer’s positive profit improvement, ~ ~ ~ r (k ,0.11,0)  r (k ,0.11,0.3)  r (k ,0.11,0.6) . Accordingly, even for higher inventory

holding cost, both players’ profit improvement can still benefit from less aggressive posture exhibited through higher direct selling cost. Finally, we discuss the effect of inventory holding cost on the players’ profit ~ improvements. For all 0  k  1 and b =0, 0.3, 0.6 in Figures 4 and 5, u (k ,0, b)  ~ ~ ~ u (k ,0.11, b) and  r (k ,0, b)   r (k ,0.11, b) . Therefore, similar to the negative correlation

between the players’ profits and the holding cost as shown in Subsection 5.1, lower holding cost can help the retailer achieve more competitive edge, which leads to more profit improvements for both the supplier and the retailer. 5.2.3. High Inventory Holding Cost (h=0.22) When the inventory holding cost is high ( h  0.22 ), we illustrate the equilibrium profits in Figure 6. When b =0 in Figure 6(a), the threshold of zero-inventory under each regime starts ~ ~ at k  0.13. In particular, when k  0.13 ,  u is always positive while  r is always negative;

~

~

when k  0.13 , u  r  0 . Similar to the observation in Figure 5(a), in this case, the ~

effect of higher holding cost yields h (k , b)  0.22 for k  0.13 . Therefore, the retailer will suffer from the use of strategic inventory under higher holding cost. Even so, the supplier still benefits from the use of strategic inventory. An explanation is that once the retailer

38

withholds strategic inventory, his supplier will seek to increase the first-period pricing (transfer price and wholesale price) to counter the retailer’s threat as indicated in Proposition 4. This in turn on the one hand increases the retailer’s holding costs and on the other hand boosts the supplier’s profits in the first period. However, as reported in Figure 3(a), the retailer will not withhold strategic inventory when k  0.13 . As a result, both player’s profits remain unchanged for k  0.13 . ×10-1

×10-4

×10-1

 ud

 ud ~  u

×10-4

~  u

~ u

~ u

~  r

~  r

~ r

 dr

 dr

~ r

0.13

(HI, D)

k

k (NI, D)

(HI, D)

(NI, D)

(b) b=0.3

(a) b=0

×10-4

×10-1

~  u ~ u

 ud

~  r

 dr

~ r (HI, D)

k

(HI, NE)

(c) b=0.6 Figure 6. The players’ profit improvement under inventory-holding decentralization (vs. zero-inventory decentralization) when h = 0.22.

As b increases to 0.3, Figure 6(b) shows a similar pattern with Figure 6(a). But the difference is that the threshold of withholding no inventory under each regime moves toward 0.2. This implies that higher direct selling cost allows the retailer to use the strategic inventory more efficiently, which reduces the retailer’s loss under high holding cost and then benefits the supplier. This can also be seen from the case of b =0.6 in Figure 6(c), 39

~ where  u is always positive for all 0  k  1 . Moreover, for the positive supplier’s profit

~

~

~

improvement in Figures 6(a)-(c), we find that u (k ,0.22,0)  u (k ,0.22,0.3)  u (k ,0.22,0.6) , which is in line with the above results. Finally, for all positive profit improvement when 0  k  1 and b =0, 0.3, 0.6 in Figure 5 and ~

~

~

~

Figure 6, we see that r (k ,0.11, b)  r (k ,0.22, b) and u (k ,0.11, b)  u (k ,0.22, b) . Similar to the findings in Subsection 5.2.2, this further shows that when the retailer becomes less cost-efficient, the supplier may not benefit from it as well.

6. Conclusion and Future Research Conflicting incentives between upstream suppliers and downstream retailers and their ensuing uncoordinated behaviors have garnered much attention in supply chain management. On the one hand, a supplier can encroach into her retailer’s market by reaching consumers via direct channels. On the other hand, as an effective negotiation tool, the retailer could carry strategic inventory to seek a more competitive edge. In this paper, we make a major contribution by being a first to study the interaction of these two strategies in a dual supply chain with a retailer and a supplier in a two-period setting. Furthermore, in previous studies on supplier encroachment, it is commonly assumed that a supplier adopts the strategy of centralized encroachment. That is, the supplier centrally makes retail decisions for her subsidiary. In our research, we recognize that in business practice, a supplier often adopts the strategy of decentralized encroachment. That is, the supplier sells to her subsidiary at a transfer price which is higher than marginal production cost. Then the subsidiary can make the related retail decisions. Therefore, in this paper, we make a major contribution by studying both centralized and decentralized encroachment and comparing them. Our analysis and results show that decentralized encroachment with modest transfer price can effectively balance the profitability between the retail and the supplier. Specifically, the supplier will increase the first-period wholesale price to discourage the retailer from holding more strategic inventory, and the retailer will use the strategic inventory to force the supplier to choose a less aggressive encroachment strategy. Consequently, both the supplier and the retailer can benefit from decentralized encroachment in the presence of strategic inventory.

40

We also study the effect of strategic inventory by comparing the strategies of decentralized encroachment with and without the possibility of strategic inventory. We find that the retailer can use strategic inventory to lower the second-period wholesale price. This can compensate for the first-period cost from holding strategic inventory. At the same time, the supplier will quote higher transfer price and wholesale price to limit the level of strategic inventory to reduce the loss in the second period. As a result, the supplier is always better off when the strategic inventory is used. On the other hand, the retailer becomes better off only when the unit inventory holding cost is low. Otherwise, the retailer will adopt the strategy of zero-inventory decentralized encroachment. In our model, we assume that the supplier or its subsidiary and the retailer make simultaneous decisions in the retail market. This is practical when no player has a leadership advantage over the other. However, other timing patterns are possible. For example, an important future extension to our work is a sequential encroachment setting where the retailer is the first mover to choose his order quantity and selling quantity before the supplier decides on the selling quantity through her direct channel (Arya et al., 2007; Guan et al., 2018). Our research can also be extended in the following ways. First, we limit ourselves to deterministic models. Hence future research can incorporate demand uncertainty. Second, in our model, the unit of strategic inventory carried over from the first period to the second period is perfectly substitutable to newly purchased ones in the second period. However, the quality of many products, such as perishable goods, may deteriorate over time (Mantin and Jiang, 2017). It is interesting to study how the issue of quality deterioration can affect the retailer’s incentive to use strategic inventory and the supplier’s choice between centralized and decentralized encroachment.

Appendix A PROOF I rc ( w1r )  0

OF

4 2 4 k 2  32)ka LEMMA 1. From (8), if w1r  4(3k  16k  24)a 2(3k  16  h , then 4

128  96k  18k

, otherwise

I rc (w1r )  0

. Substituting these two

I rc (w1r )

values, q1s ( w1r ) and

q1r ( w1r ) in (9), solving the first-order condition of the problem yields: if h  f (k , b) ,

w1cr 

4(576  832 k 2  396 k 4  63k 6 )a  (128  400 k 2  288k 4  63k 6 )ka  2(4  3k 2 )(8  3k 2 ) 2 h 2(8  3k 2 )(272  264 k 2  63k 4 )

,

3 2 otherwise w1cr  k a  4(2 2k )a . Thus, for h  f (k , b) , using the period 1 w1cr value and back

2(8  3k )

41

substitution, the two-period non-negative solution is obtained and presented in Lemma 1. For h  f (k , b) ,

I rc  0 .

Hence, the centralization solution under this condition is simply the

same as that in centralized encroachment without inventory noted in the second column of Table 1. ■ PROOF then

OF

LEMMA 3. From

I rd (w1r )  0 ,

otherwise

I rd ( w1r )

I rd (w1r )  0 .

2 in (18), we can see that if w1r  (3  k )a 2 ka  h ,

2(2  k )

Using these

I rd ( w1r ) , q1s (w1 )

and q1r (w1 ) from (18) in

(19), and solving the first-order condition of (19) yields the results below: if h  g (k , b) , 2 2 2 then w1ds  k[9(a  ka )  (22  k )h] , w1dr  (36a  17 k )a  2ka2  4(2  k )h , otherwise

17(2  k )

34(2  k )

w1ds 

k (a  ka ) , d a . Using back substitution, given h  g (k , b) , the equilibrium inventory w1r  2 2(2  k 2 )

levels, prices and quantities are each non-negative and presented in Lemma 3. The condition h  g (k , b) confirms I rd  0 , and the equilibrium prices and quantities are the same to

those under decentralized encroachment without inventory. ■ PROOF

OF

PROPOSITION 1. (i) Using

w1ds

d from Lemma 3, dw1s   k  0 . This indicates

dh

17

that w1ds is a linearly decreasing function of h with one root, 9(a  k2a )  f (k , b) . Since 2k

h  f (k , b) , w1ds  0 . Using w2ds from Lemma 3, as a  ka  0 , it is easy to follow that w2ds  0 .

(ii) According to q1cs from Lemma 1 and q1ds from Lemma 3, d (q1ds  q1cs ) k (8  3k 2 )( 4  3k 2 )  0, dh (4  k 2 )( 272  264 k 2  63k 4 )

and h |q

d c 1 s ( h )  q1 s ( h )  0

Hence,

q1ds  q1cs  0 .



(a  ka )( 4  k 2 )(1152  1408 k 2  540 k 4  63k 6 )  f ( k , b) . 2(8  3k 2 ) 2 (2  k 2 )( 4  3k 2 )

Similar to this proof, using q2cs from Lemma 1 and q2ds from Lemma 3, d (q2ds  q2c s ) k (8  3k 2 )( 20  9k 2 )  0, dh (4  k 2 )( 272  264 k 2  63k 4 )

and h |q Hence,

d c 2 s ( h )  q2 s ( h )  0



(a  ka )( 4  k 2 )(12  7k 2 ) 0. 2(2  k 2 )( 20  9k 2 )

q2ds  q2cs  0 .

(iii) Using

q1cr

from Lemma 1 and

q1dr

from Lemma 3,

d (q1dr  q1cr ) k 2 (104  210 k 2  63k 4 ) , and  dh 17 (4  k 2 )( 272  264 k 2  63k 4 )

42

h |q d ( h )  q c ( h )  0  1r

1r

2(a  ka )( 4  k 2 )(1208  1242 k 2  315 k 4 ) . (2  k 2 )(8  3k 2 )(104  210 k 2  63k 4 )

So, there is a threshold k  k0  35  497  0.7779 , which is the solution of 21

104  210 k 2  63k 4

.

k  k0 ,

When

q1dr  q1cr  0

;

when

0  k  k0 ,

d ( q1dr  q1cr ) 0 dh

,

d c h |q d ( h)q c ( h)0  f (k , b) ; otherwise, d (q1r  q1r )  0 , h |qd ( h )qc ( h )0  0 . Hence, we always have 1r

dh

1r

1r

1r

q1dr  q1cr  0 .

Using

q2cr

from Lemma 1 and

q2dr

from Lemma 3,

d (q2dr  q2cr ) k 2 (2152  1662 k 2  315 k 4 )  0, dh 17 (4  k 2 )( 272  264 k 2  63k 4 )

and u (k , b)  h |q Note that there is a threshold

d c 2 r ( h )  q2 r ( h )  0

k



(a  ka )( 4  k 2 )( 20  21k 2 ) . (2  k 2 )( 2152  1662 k 2  315 k 4 )

2 105  0.9759 21

, which is the solution of 20  21k 2 . When

u (k , b)  0 , and thus q2dr  q2cr  0 . When 0.9759 k  1 , 0  u(k , b)  f (k , b) ,

0  k  0.9759,

and thus when h  u(k , b) ,

q2dr  q2cr  0

; when h  u (k , b) ,

q2dr  q2cr  0 .

Furthermore, since

u (k , b) increases in b for 0.9759 k  1 and b  a , we can verify that u(k , b)  u(k , a) and

u (k , a) is a concave function. The first-order condition of u (k , a) yields the optimal solution,

k  0.9883. Thus, u(k , b)  u(k , a)  u(0.9883 , a)  2.1323 10 -5 a , which is a small value.

(iv) Using

w1cr

from Lemma 1 and

w1dr

from Lemma 3,

d ( w1dr  w1cr ) 3k 2 (28  9k 2 )   0 , and dh 17 (272  264 k 2  63k 4 ) h |wd ( h ) wc ( h )0  1r

Hence,

w1dr  w1cr  0 .

1r

(a  ka )(9920  14000 k 2  6660 k 4  1071 k 6 )  f ( k , b) . 6(2  k 2 )(8  3k 2 )( 28  9k 2 )

Similarly, using

w2cr

from Lemma 1 and

w2dr

from Lemma 3,

d ( w2dr  w2cr ) 9k 2 (44  19 k 2 )   0 , and dh 17 (272  264 k 2  63k 4 ) h | wd

c 2 r ( h )  w2 r ( h ) 0

Hence,



(a  ka )(1552  1480 k 2  357 k 4 )  f (k ) . 18(2  k 2 )( 44  19 k 2 )

w2dr  w2cr  0 .

(v) Using

I rc

from Lemma 1 and

I rd

from Lemma 3,

43

d ( I rd  I rc ) 2k 2 (5888  6556 k 2  2463 k 4  315 k 6 )   0 , and dh 17 (4  k 2 ) 2 (272  264 k 2  63k 4 ) v ( k , b)  h | I d ( h )  I c ( h )  0  r

r

(a  ka )( 4  k 2 )(1472  1356 k 2  315 k 4 )a  f ( k , b) . 2(2  k 2 )(5888  6556 k 2  2463 k 4  315 k 6 )

Hence, when h  v(k , b) , I rd  I rc  0 ; when v(k , b)  h  f (k , b) , PROOF

OF

I rd  I rc  0 .

■

PROPOSITION 2. (i) From (10) and (20), the supplier’s profits under

centralized and decentralized encroachment are related with h . Considering that f (k , b)  g (k , b) ,

we compare the supplier’s profits under

centralization and

decentralization in terms of the retailer’s inventory choices. There are three scenarios: when 0  h  f (k , b) , the retailer will carry the inventory under both regimes; when f (k , b)  h  g (k , b) , the retailer will carry the inventory only under decentralization; when

h  g (k , b) , the retailer has no inventory under both regimes.

First, when 0  h  f (k , b) , we can verify that d 2 ( ud   uc ) 4k 2 (3008  3112 k 2  1077 k 4  126 k 6 )   0, dh2 17 (4  k 2 ) 2 (272  264 k 2  63k 4 )

which indicates that ud  uc is a concave function of h with two roots, denoted by h1 and h2 listed below: h1  

h2 

(a  ka )( 4  k 2 )[ D1  2(8  3k 2 )(376  312 k 2  63k 4 )]  0 and 2(8  3k 2 )( 2  k 2 )(3008  3112 k 2  1077 k 4  126 k 6 )

(a  ka )( 4  k 2 )[ D1  2(8  3k 2 )(376  312 k 2  63k 4 )]  f ( k , b) , 2(8  3k 2 )( 2  k 2 )(3008  3112 k 2  1077 k 4  126 k 6 )

where, D1  17 (8  3k 2 )( 272  264 k 2  63k 4 )(6016  9232 k 2  5268 k 4  1329 k 6  126 k 8 ) . In terms of the concavity of

ud  uc ,

it follows that

ud  uc  0 .

2 d c Second, when f (k , b)  h  g (k , b) , we know d (u 2 u )  8  0 . Then,

dh

convex function of h without any real root. Thus,

17

ud  uc

is a

ud  uc  0 .

Finally, when h  g (k , b) , the retailer will not withhold inventory. With zero inventories, the solution under either centralized or decentralized encroachment is reduced to the benchmark one. From Subsection 3.2, it follows that

44

ud  uc  0 .

(ii) Similar to the proof of part (i), from (11) and (21), we also take into account three conditions, that is h  f (k , b) , f (k , b)  h  g (k , b) , and h  g (k , b) . First, when 0  h  f (k , b) , we have: d 2 ( dr   cr ) 4k 2 D2  0, 2 2 dh 289[( 4  k )( 272  264 k 2  63k 4 )]2

where,

D2  13038592  26784128 k 2  21956208 k 4  8985312 k 6  1838565 k 8  150822 k 10 .

This means that h3  

is a concave function of h with two roots h3 and h4 as follows:

(a  ka )( 4  k 2 )[( 4624  4488 k 2  1071 k 4 ) D3  (8  3k 2 ) D4 ] 4(2  k 2 )(8  3k 2 ) D2

h4 

where,

 dr  cr

(a  ka )( 4  k 2 )[( 4624  4488 k 2  1071 k 4 ) D3  (8  3k 2 ) D4 ] 4(2  k 2 )(8  3k 2 ) D2

0

and

 f ( k , b) ,

D3  834535424  2082250752 k 2  2161319936 k 4 1194945408 k 6  371359440 k 8 

61563456 k 10  4258737 k 12 , D4  6667264 11757888 k 2  7700400 k 4  2212596 k 6  234171 k 8 .

Given the concavity of

 dr  cr ,

it follows that

 dr   cr  0 .

Second, when f (k , b)  h  g (k , b) , we know: d 2 ( dr   cr ) 152  0, 289 dh2

which shows that

 dr  cr

is a convex function of

h

with two roots hˆ1 and hˆ2 as follows:

(a  ka )( 472  177 k 2  17 64  2480 k 2  1073 k 4 ) and hˆ1  152 (2  k 2 )(8  3k 2 ) (a  ka )( 472  177 k 2  17 64  2480 k 2  1073 k 4 ) . hˆ2  152 (2  k 2 )(8  3k 2 )

Note that k 

1240  16 5738  0.1616 is the unique solution of 64  2480 k 2  1073 k 4  0 . 1073

When k  0.1616 , hˆ1  hˆ2  f (k , b) , Then, it follows that h  f (k , b)  hˆ2  hˆ1 .

According to the convexity of

last, when k  0.1616 , the convex function of that

 dr   cr  0

 dr  cr ,

dr  cr

. When 0  k  0.1616 ,

it holds that

 dr  cr  0 .

At

has no real roots. Hence, we obtain

 dr   cr  0 .

Finally, when h  g (k , b) , similar to the proof of part (i), the solution under either centralized or decentralized encroachment is the same as that in the benchmark case. Thus, using the results of Subsection 3.2,  dr  cr  0 . ■

45

PROOF OF PROPOSITION 3. The results of part (i)-(iv) hold from the proof of Proposition 1. Next, we prove part (v). First, from the proof of Proposition 2(i), ud  uc is a concave function of h when 0  h  f (k , b) . The first-order condition of ud  uc yields the symmetry point: α ( k , b) 

(a  ka )( 4  k 2 )(376  312 k 2  63k 4 )  f ( k , b) , (2  k 2 )(3008  3112 k 2  1077 k 4  126 k 6 )

Since h1  0 and h2  f (k , b) , we have when 0  h  α(k , b) , α(k , b)  h  f (k , b) ,

d ( ud   uc )  0 ; when dh

d ( ud   uc ) 0. dh

Second, from the proof of Proposition 2(ii),

 dr  cr

is a concave function of h for

0  h  f (k , b) . The first-order condition of ud  uc yields the symmetry point: β ( k , b) 

Since

h3  0 ,

PROOF ~  w1ds  w s

it follows that

OF

(a  ka )( 4  k 2 ) D4  f ( k , b) 4(2  k 2 ) D2

d ( dr   cr )  0, dh

for 0  h  f (k , b) . ■

PROPOSITION 4. (i) Using

w1ds

~ from Table 2, from Lemma 3 and w s

k[h  g (k , b)] ~  5k[h  g (k , b)]  0 .  0 . Similarly, using w2ds from Lemma 3, w2ds  w s 17 17

(ii) The proof follows from the expressions for

q1ds

and

Table 1. (iii) Using q1dr  q~r  

q1dr

from Lemma 3 and q~r from Table 1,

2hk 2  a k  a  4h h  g (k , b)  0. 17 34(2  k 2 )

In a similar way, using q2dr  q~r 

from Lemma 3,

5(2hk 2  a k  a  4h) 5[h  g (k , b)]  0. 2 17 34 (2  k )

(iv) Using ~  w1dr  w r

q2dr

w1dr

~ from Lemma 1, from Lemma 3 and w r

2hk 2  a k  a  4h 2[h  g (k , b)]   0. 2 17 17 (2  k )

And using

w2dr

from Lemma 3,

~  w2dr  w r

5(2hk 2  a k  a  4h) 10[h  g (k , b)]  0. ■ 17 17 (2  k 2 )

46

q2ds

from Lemma 3 and q~s from

PROOF

PROPOSITION 5. (i) Using the supplier’s profit under zero-inventory

OF

decentralization in Table 2 and that under inventory-holding decentralization in (20), for ~ 4[h  g (k , b)]2 0. h  g (k , b) ,  ud  u  17

(ii) Using the retailer’s profit under zero-inventory decentralization in Table 2 and that ~

2 d under inventory-holding decentralization in (21), we can verify that d ( r 2 r )  152  0 .

dh

289

~ This means that  dr  is a convex function of h with two roots g (k , b) and r ~ 21(a  ka ) h (k , b)  76 (2  k 2 ) ~  dr   r  0 .

PROOF

. Thus, when 0  h  h~(k , b) ,

~  dr   r  0

; when

~ h (k , b)  h  g (k , b) ,

■

OF

PROPOSITION 6. From the paper in Guan et al. (2019), under inventory-

holding centralization, the equilibrium wholesale price, retailer’s inventory and quantities in each period are as follows. w1cr  q2cr 

337 a  50 h  11b 674

627 b  520 h 1011

, w2cr  337 a  520 h  290 b , I rc  1113 b  2600 h , q1cr  337 a  50 h  11b , 674

3033

1348

, q2cs  1011 a  520 h  1638 b . 2022

Then, the supplier’s and retailer’s profits are:  uc 

9099 a 2  12132 ab  13131 b 2  9300 bh  9500 h 2 , 24264

 cr 

340707 a 2  22242 ab  101100 ah  1268979 b 2  1848732 bh  2170700 h 2 . 5451312

Next,

we

will

derive

the

equilibrium

outcomes

under

inventory-holding

decentralization where the supplier chooses the decentralized encroachment for completely substitutable products in the second period only. The selling price for the retailer in period 1 is p1r  a  q1r and the selling price for firm i ( i, s, r ) in period 2 is p2i  a  q2r  q2 s . Using the similar backward induction in Subsection 4.2, in period 2, given transfer price w2 s and the retailer’s inventory I d , solving (2) and (12) with k  1 yields the retail quantities as follows: q2 s (w 2 ) 

a  2b  2w2 s  w2r a  b  w2 s  2w2r and q2r (w 2 )  . 3 3

Then, solving the supplier’s period 2 pricing problem in (14) yields the transfer price and wholesale price given by

47

w2 s ( I r ) 

b  Ir 2

and w2 r ( I r )  a  I r . 2

In period 1, the retailer chooses its retail quantity q1r and inventory levels I r to maximize its two-period total profit. Max 1r (q1 , w1r , I r )  (a  q1r  q1s )q1r  w1r (q1r  I r )  hI r q1r , I r

  2 r (q 2 (w 2 ( I r )), I r , w2 r ( I r )) .

Solving its first-order conditions jointly yield the retailer’s quantity and strategic inventory levels as below. q1r ( w1r ) 

a  w1r 2

, I rd (w1r )  a  b  2(h  w1r ) . 3

Anticipating the retailer’s best responses, the supplier’s period 1 pricing problem is: d d d d Max 1u ( w1r )  w1r [q1r ( w1r )  I r ( w1r )]   2u (q 2 (w 2 ( I r ( w1r )), I r ( w1r ), w2 r ( I r ( w1r ))) . w1r

Solving it will give us the equilibrium wholesale price in period 1. w1dr 

17 a  2b  4h 34

.

Substituting this equilibrium period 1 wholesale price back, we can then obtain the other equilibrium prices, inventory, quantities and profits as follows. w2dr 

17 a  2b  4h 34

, w2ds  6b  5h , I rd  5b  10 h , q1dr  17 a  2b  4h , q2dr  11b  5h , 17

17

q2ds 

2 2 2 a  2b ,  d  51a  68 ab  76 b  32bh  32 h u 136 2

 cr 

289 a 2  68 ab  136 ah  1460 b 2  1216 bh  1216 h 2 . 4624

68

17

,

When the strategic inventory can be used, centralized and decentralized encroachments arise under the necessary conditions of b  b  a and h  h  1113b . 2

2600

(i) Comparing the supplier’s profits under inventory-holding centralization and decentralization, we have:  ud   uc 

7281 b 2  61044 bh  64444 h 2 . 412488

d c u u Hence, u  u is a concave function of h with two roots h1 and h2 as follows:

h1u  

3796276285 768569 b 9338892547 186911 b  0 and h2u   h . 3542846367 0198272 8857115917 549568

48

d c d c According to the concavity of u  u , it follows that u  u  0 .

(ii) Similar to the proof of Proposition 6(i), using the retailer’s profits

 cr

and

dr ,

we

know:  dr   cr 

130697289 b 2  119983836 bh  29596014 ab  17118252 ah  213032588 h 2 . 1575429168

d c r r It can be seen that r  r is a concave function of h with two roots h1 and h2 listed

below: h1r 

4279563 a  29995959 b  17187 62001 a 2  4466896 ab  26610156 b 2 , 106516294

h2r 

4279563 a  29995959 b  17187 62001 a 2  4466896 ab  26610156 b 2 , 106516294

Solving 62001 a 2  4466896 ab  26610156 b 2  0 yields two roots b1 and b2 as follows. b1 

5816958912 28309 a 9083377375 4939 a  0.0153 a , b2   0.1526 a . 5952591133 40928 3809658325 3819392

d c Hence, when b1  b  b2 , r  r is a concave function without any roots and we have

 dr  cr . When b  b1 or b  b2 , we have dr  cr  0 . When 0  b  b1 or b2  b  b , there r r d c r r  exist the two roots h1 and h2 for  r   r  0 . First, when 0  b  b1 , h2  h1  h and thus

 dr  cr . Next, when b2  b  b , h2r  h  h1r , and it follows that  dr  cr for h1r  h  h . r r Moreover, when 0.1526a  b  0.2264a , h1  0 ; when 0.2264a  b  0.5a , h1  0 . r d c Overall, Let h  h1 , when 0  b  0.1526a ,  r   r ; when 0.1526a  b  0.2264a and

h h

1113b 1113 b d c , or 0.2264a  b  0.5a and 0  h  , r  r . ■ 2600 2600

CRediT Author Statement Jin Li: Methodology, Writing- Original draft preparation, Software. Liao Yi: Supervision, Data curation. Victor Shi: Conceptualization, Writing-Reviewing and Editing, Validation. Xiding Chen: Visualization, Investigation.

49

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