The co-opetitive strategy of a closed-loop supply chain with remanufacturing

Transportation Research Part E 48 (2012) 387–400

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

The co-opetitive strategy of a closed-loop supply chain with remanufacturing Jen-Ming Chen ⇑, Chia-I Chang Institute of Industrial Management, National Central University, 300 Jhongda Road, Jhongli City, Taoyuan County 32001, Taiwan, ROC

a r t i c l e

i n f o

Keywords: Closed-loop supply chain Reversed logistics Remanufacturing Cross-channel competition Newsvendor

a b s t r a c t This paper deals with a strategic issue of closed-loop supply chains with remanufacturing by developing analytic models under cooperative and competitive settings. The primary goal behind analytic formulation is to investigate under what conditions an original equipment manufacturer (OEM) may take a cooperative approach by participating in remanufacturing. In contrast, the OEM may take a competitive approach by letting the third-party firm remanufacture the returned cores and remarket in the secondary market that competes with the new product. Our analysis reveals that the strategic decision depends critically on the costs of remanufacturing and the competition intensity between the two versions. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Remanufacturing is the process of restoring end-of-used or end-of-life products (i.e., cores), components, modules, and parts to like-new condition in a manufacturing environment. The restoring or recovering process tends to be energy-saving, less material-consuming, and often have a lower impact on the environment than its manufacturing counterpart that makes the brand-new product from virgin materials. Evidences can be found in the case study of furniture remanufacturing (Sahni et al., 2010) and the economic analysis in a variety of industries including tire (Ferrer, 1997a), personal computer (Ferrer, 1997b), and many others (Ferrer and Ayres, 2000). As a result, the cost of taking back and remanufacturing a core is typically 40–60% less than those of brand-new production (Giuntini and Gaudette, 2003). Remanufacturing incorporating with a closed-loop supply chain is therefore getting more prevalent and well adopted in practice. Examples from companies include GE Transportation, Xerox, Robert Bosch Tool, Black and Decker, and Hewlett-Packard (Atasu et al., 2010; Martin et al., 2010). It is estimated that the US annual expenditure on remanufacturing, including overhaul and rebuild, is approximately 40 billion in year 2003 (Remanufacturing Central, 2011). Given the massive market magnitude and the widespread practices of the remanufacturing industry, however, it remains unclear why some Original Equipment Manufacturers (OEMs) participate in remanufacturing aggressively but some do not. The void is often filled by the third-party independent operators (IOs) whose primary business is to remanufacture the endof-used products of the major OEMs within a given industry. Hauser and Lund (2008) found that only 6% of over 2000 remanufacturing firms were OEMs. The remanufactured product made by the IOs competes with the new product in the market and cannibalizes the sales of the OEMs. Therefore a strategic issue in closed loop supply chains is to determine whether or not the OEMs should participate in remanufacturing the take-back cores and reselling them in the market? Without the help of analytic tools, the OEMs tackle the strategic decision often by using the rule of thumb. For example, Robert Bosch Tool’s decision on what product to remanufacture is based upon the retail price and market share (Valenta, ⇑ Corresponding author. Tel.: +886 3 4258192; fax: +886 3 4258197. E-mail address: [email protected] (J.-M. Chen). 1366-5545/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2011.10.001

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2004). Ferguson (2010) was one of the very few academic researchers who provided scientific guidance on this crucial issue. Through extensive review of related literature, directives, and case studies, he shed light on the strategic issue by identifying some direct costs of remanufacturing as well as opportunity costs that should be included in a firm’s decision to remanufacture or not. Ordoobadi (2009) proposed a multi-phased decision model for strategic analysis of outsourcing reverse logistics to a third-party firm. The conceptual model provides the OEM a comprehensive tool for effective decision making by considering economic and strategic factors. In this paper, we deal with the same problem by using an analytic scheme that helps policy-makers response to such a strategic question: under what conditions an OEM should participate in remanufacturing, in addition to the manufacturing operations. By coordinating the production quantities and retail prices of both versions of the same product, the OEM may generate more profit than that by manufacturing the new product alone. In contrast, under what conditions an OEM should not be involved with the remanufacturing, and choose a head-to-head competition with the third-party remanufacturer counterpart. Taking the competition strategy, both firms may mutually beneficial by generating a larger pie for the same product. Under a game-theoretical setting, we assume the OEM acts as the dominant leader who can take cooperative or competitive strategy, and the IO acts as the follower. The problem being considered in the paper is unique in the sense that we incorporate the effects of manufacturing and remanufacturing costs and the competition intensity into account. The cost of remanufacturing a returned core is typically less than that of manufacturing a brand new product (Giuntini and Gaudette, 2003). And the cost of remanufacturing performed by a third-party IO is generally lower than that by the OEM. For examples, Arruñada and Vázquez (2006) pointed out that the production cost by the third-party contract manufacture is usually lower than that by the OEM due to the economics of scale. Cross (1995) revealed that the operating cost can reduce up to 25% by outsourcing information technology operations to a third-party service provider. In this paper, we assume the cost between manufacturing and remanufacturing differs, so as the remanufacturing cost incurred by the OEM and the IO. The co-opetitive strategy has been widely applied in the supply chain and logistics management for decades. General Mills Yogurt and Land O’Lakes butter delivered by the same truck, en route to the same supermarket, is a prominent example of co-opetitive partnership (Hammer, 2001). Such a co-opetitive relationship among upstream vendors and downstream buyers has recently been investigated using analytic model and behavioral experiment (Nair et al., 2011) and through empirical study (Li et al., 2011). This paper deals with such an issue between two parallel cross-channels in a CLSC environment. Our analysis provides scientific guidance to the OEM in deciding whether or not to participate in the remanufacturing. This study contributes to the literature by considering such a strategic issue under stochastic price-dependent demand function with substitution in a dual channel problem setting. It differs significantly from the utility function approach given by Atasu et al. (2008) and Ferguson and Toktay (2006) whose analyses were under the key assumption of a uniformly distributed willing-to-pay function, and the closed-loop decision model proposed by Shi et al. (2010, 2011a) and Shi et al. (2011b) who assumed both products were sold at the same price. In what follows, literature review is given in Section 2. The problem context, assumptions, and notations are described and defined in Section 3. Model formulation and analysis are given in Sections 4 and 5, respectively. Section 6 provides an in-depth numerical study and sensitivity analysis. Section 7 concludes by summarizing our research contributions and providing future research directions.

2. Literature review In this section, we provide a review on the research of closed-loop supply chains with remanufacturing, especially for those work using management science and operations research techniques. Some of the research themes in this stream include inventory control and lot-sizing of remanufacturing systems (Teunter et al., 2004; Toktay et al., 2000; Van der Laan et al., 1999; Wang et al., 2011), acquisition and/or sorting policy (Galbreth and Blackburn, 2006, 2010; Zikopoulos and Tagaras, 2008), reverse channel/network design (Lee and Dong, 2008, 2009; Mar-Ortiz et al., 2011; Nagurney and Toyasaki, 2005; Savasßkan et al., 2004; Savasßkan and Van Wassenhove, 2006; Shulman et al., 2010; Yang et al., 2009), and location and territory design of collecting depots (De Figueiredo and Mayerle, 2008; Görmez et al., 2011; Ramos and Oliveira, 2011; Fernández et al., 2010). Some CLSCs and remanufacturing models place their focus on considering time value of product return (Guide et al., 2006), quality consideration (Dobos and Richter, 2006), technology selection (Debo et al., 2005), limited durability and finite life cycles (Geyer et al., 2007), and Nash bargaining game (Sheu, 2011). The proposed model deals with pricing and lot-sizing joint decisions with considering the effect of product substitution between new and remanufactured products, which differs from the aforementioned literature in research theme, focus, and problem setting. The literature that is most closely related to ours is to be reviewed. The models proposed by Ferrer and Swaminathan (2006, 2010) and Webster and Mitra (2007) considered a multiple period problem using a deterministic demand function. Mitra and Webster (2008) extended their own research by considering the effects of government subsidies. The models by Ferguson and Toktay (2006) and Atasu et al. (2008) use a utility function approach with a key assumption of the consumers’ willingness-to-pay being uniformly distributed over [0, 1], which are also deterministic in nature. Bayindir et al. (2005) dealt with a one-way substitution problem, i.e., the remanufactured product can be substituted by the new one under a continuous review control policy. We consider a single-period inventory problem with two-way substitution and price-dependent stochastic demand.

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We are also aware of four recently published works by Shi et al. (2010, 2011a,b) and Wang et al. (2011). The three works by Shi et al. assumed the remanufactured and new products are sold in the same market at the same price. In our model, the consumers’ perception and the selling price of the remanufactured product are different from that of the new one. Wang et al. dealt with the cost-minimization problem under stochastic demand and returned rated. We deal with the profit-maximization problem with a key assumption of two retail prices being endogenous decision variables. A broader collection and comprehensive review on CLSCs and green/reverse supply chain can be found in the review articles by Guide and Van Wassenhove (2009), Atasu et al. (2008), and Srivastava (2007). A review of more than sixty real case studies on reverse logistics is provided by De Brito et al. (2003). A survey on environmental management and sustainability using management science methodologies is given by Paucar-Caceres and Espinosa (2011). Guide and Van Wassenhove (2009) classified the activities of reverse supply chain into three broad categories: frond-end issues of product collection and acquisition, engine issues of remanufacturing operations, and back-end issues of channel development and remarketing. This paper contributes to the literature by forging the remanufacturing operations with the back-end pricing in a competing dual-channel setting. First, we bring a marketing perspective to the remanufacturing problem by investigating the effect of the cross-sensitivity coefficient of demand on the joint decisions and corresponding profit. Second, we bring operational factors to the marketing decision-making by investigating the effect of the manufacturing cost and remanufacturing cost-saving on the optimal price-setting. Finally, our analytic model is developed under a newsvendor framework which is generic and applicable to the uncertain marketing environment. 3. The problem context As stated earlier, our primary goal is to investigate under what conditions an OEM may want to remanufacture the returned cores by herself or to let the third-party independent operator close the loop. In the former case, the OEM has direct control over the two manufacturing lines and thus can take a cooperative strategy by jointly setting the price and lot-size for both versions of the same product. In the latter case, the manufacturing and remanufacturing lines are owned respectively by the OEM and the IO. The new and like-new products produced by the two firms compete with each other in the market. Therefore, the decisions are made separated by two decision-makers without considerations being given to its counterpart. We assume the OEM is the dominate leader and the IO is the follower in the two-person game setting. With bearing it in mind, we consider both centralized and decentralized systems to resemble the cooperative and competitive strategies respectively. For notational convenience, let subscript i = 1 represent the manufacturing system or the brand-new products produced by the system and let subscript i = 2 represent the remanufacturing system or the likenew products produced by the system. The objective of the firm (OEM or IO) is to jointly determine the selling price and production lot-size so as to maximize the expected profit generated by the system. To differentiate the solutions generated by the two systems, let superscripts c and d represent the realized values of the price, lot-size, and expected profit generated by the cooperative and competitive strategies respectively. The problems are formulated under a newsvendor setting with an additive price-dependent demand function. The brandnew and remanufactured products differ in production cost, quality, durability, and consumers’ perception; they have different prices in the market per se. Under such a setting, let C1 = c be the cost of manufacturing input from the virgin material and C2 = dc, 0 6 d 6 1, be the cost of remanufacturing input from the returned cores under the cooperative setting. In addition, we assume the IO has the production cost advantages over the OEM, which is reflected by letting c1 = C1 = c be the manufacturing cost incurred by the OEM and c2 = DC2 = DdC, 0 6 D 6 1, be the remanufacturing cost incurred by the IO under the competitive setting. By defining so, we can represent the cost components (C1, C2, c1, c2) in terms of c, d, and D which will facilitate subsequent analysis and numerical study. We assume the quantity of returned cores (i.e., q2) can be exogenous or endogenous. The term of ‘‘exogenous’’ implies that q2 is a given parameter and cannot be controlled by the firm; while ‘‘endogenous’’ implies that it is a decision variable and can be controlled by the firm. The model by Shi et al. (2011a) assumes the quantity of returned cores is endogenous and depends solely on the acquisition fee. This study does not consider the collection and acquisition related efforts and decisionmaking. Furthermore, we assume the total potential market size (or market magnitude) of both brand-new and remanufactured products is M when both prices are set to zero. Let M1 = aM and M2 = (1  a)M, 0 6 a61, be respectively the potential markets of the new and remanufactured products, where a represents the market share of the new product. For brevity, the holding cost per period per unit is h and the shortage cost (e.g., good-will loss) per unit is s for both products, regardless of its production cost or selling price. A more general notation can easily be used for h and s by adding subscript i that will take into account the factors of heterogeneous prices and costs. However, it will complicate the presentation and distract from our main theme of the research objective. Based on the aforementioned notations and assumptions, the price-dependent uncertain demand is represented by the additive form:

Di ðpi ; p3i ; ei Þ ¼ yi ðpi ; p3i Þ þ ei ;

for i ¼ 1; 2;

ð1Þ

where yi(pi, p3i) and ei are respectively the mean demand and the random term of demand function Di() and p1 and p2 are selling prices for the new and remanufactured products, respectively. The random term ei is defined on the range [Ai,Bi], which represents the maximal possible deviation from the mean demand yi(pi, p3i). Let the cumulative distribution function

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and the probability density function of ei be F() and f(), respectively. Likewise, let the mean and standard deviation of ei be li and ri, respectively. The mean demand yi(pi, p3i) is price-dependent and substitutable between the new and remanufactured versions of the product. It can be defined in many ways, such as the rescaled additive form used by McGuire and Staelin (2008):

Mi ð1  pi þ cp3i Þ;

i ¼ 1; 2;

ð2aÞ

or the form used by Ingene and Parry (1995):

Mi  pi þ cp3i ;

i ¼ 1; 2;

ð2bÞ

where Mi is the potential market size of product i and c is the (symmetric) cross-sensitivity coefficient of demand, 0 6 c < 1. Both forms are negatively sloped response functions in p3i, and positively response functions in pi However, the price-sensitivity coefficient of demand in (2a) is proportional to the market size Mi; while the coefficient in (2b) is fixed to 1, disregard of the market size. In a pilot study of the research, the two mean demand forms were considered in problem formulation, analysis, and numerical study, both of which generated similar results. To prevent from distraction, only the mean demand function (2a) is used and reported in the subsequent modeling and analysis. 4. The model We formulate the decision problems of a closed-loop supply chain with remanufacturing under cooperative and competitive settings. Since they are extended variations of the newsvendor model given by Petruzzi and Dada (1999), we will recapture their context to a certain extent in our problem formulation. 4.1. The competitive setting Based on our setting, the decision problem facing the OEM (i = 1) and IO (i = 2) individually is to determine the pricing and production quantity pi, qi so that the total profit generated by product i, represented by p(pi, p3i, qi, q3i), i = 1, 2, is maximized:

pi ðÞ ¼



pi Di ðpi ; p3i ; ei Þ  ci qi  h½qi  Di ðpi ; p3i ; ei Þ; DðÞ 6 qi ; pi qi  ci qi  s½Di ðpi ; p3i ; ei Þ  qi ;

DðÞ > qi :

i ¼ 1; 2:

ð3Þ

Define the lot-sizing factor: zi = qi  yi(pi, p3i), i = 1, 2, as the difference of the production quantity and the mean demand. The production quantity qi can be re-expressed as qi = zi + yi(pi, p3i). Substituting qi of Eq. (3) with zi + yi(pi, p3i), the expectation of the profit function Eq. (3) becomes:

E½pi ðÞ ¼

Z

zi

ðpi ½yi ðpi ; p3i Þ þ u  h½zi  uÞf ðuÞdu þ

Ai

Z

Bi

ðpi ½yi ðpi ; p3i Þ þ zi   s½u  zi Þf ðuÞdu  ci ½yi ðpi ; p3i Þ þ zi ; i ¼ 1; 2 zi

ð4Þ The equation above represents a sum of expected sales, overage cost, underage cost, and variable production cost. Let Rz RB Kðzi Þ ¼ Aii ðzi  uÞf ðuÞdu be the expected leftovers and let Hðzi Þ ¼ zi i ðu  zi Þf ðuÞdu be the expected shortages. Note that

Z

zi

ð½yi ðpi ; p3i ÞÞf ðuÞdu þ

Ai

Z

Bi

ð½yi ðpi ; p3i Þ þ zi Þf ðuÞdu ¼ yi ðpi ; p3i Þ þ l 

Z

zi

Bi

½u  zi f ðuÞdu;

zi

and

½yi ðpi ; p3i Þ þ zi ¼ ½yi ðpi ; p3i Þ þ l þ

Z

zi

Ai

ð½zi  lÞf ðuÞdu 

Z

Bi

½u  zi f ðuÞdu:

zi

Eq. (4) can be re-expressed as follows:

E½pi ðpi ; p3i ; zi ; z3i Þ ¼ ðpi  ci Þ½yi ðpi ; p3i Þ þ li   ½ðci þ hÞKðzi Þ þ ðpi þ s  ci ÞHðzi Þ; i ¼ 1; 2:

ð5Þ

In Eq. (5), the first term is the riskless profit denoted by Wi(pi, p3i), and the second term is the loss function denoted by Li(i, zi), which consists of the total expected holding cost and the expected shortage cost. The riskless profit is the profit for a given pair of prices in the certainty equivalent problem in which ei is replaced by li. In what follows, we present the lemma and theorem forthwith, and all the proofs are detailed in Appendix A. Lemma 1. For a fixed zi, the optimal price is determined uniquely as a function of zi:

pdi ¼

2ð1 þ ci Þ þ cð1 þ c3i Þ 2ðli  Hðzi ÞÞ cðl3i  Hðz3i ÞÞ þ þ ; 4  c2 M i ð4  c2 Þ M 3i ð4  c2 Þ

i ¼ 1; 2:

ð6Þ

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The pricing decision is determined by the production costs ci, c3i, the mean and variance of the error terms (ei, e3i), the market sizes (Mi,M3i), and the (symmetric) cross-sensitivity coefficient of demand c Substituting pdi ; i ¼ 1; 2, into Eq. (5), the optimization problem becomes a maximization over two stocking variables: zi and z3i. To maximize the function    E pi pdi ; pd3i ; zi ; z3i with two variables, an exhaustive search over all values of zi, i = 1, 2, in the region [Ai, Bi] will determine zdi . As discussed in Petruzzi and Dada (1999), if the concavity condition is met, the optimal lot-size can be determined by the following standard fractile:

zdi

¼F

1

! pdi þ s  ci ; pdi þ s þ h

i ¼ 1; 2:

ð7Þ

     Substituting pdi ; zdi into Eq. (5) generates the maximal expected profit of the system: E pi pdi ; pd3i ; zi ; z3i If one of the stocking factors is exogenous, the problem is reduced into the optimization problem with only one variable. Therefore the theorem given by Petruzzi and Dada (1999) can be applied to solve the single-variable maximization problem, which is detailed in the following theorem. Theorem 1. If q3i or z3i (i = 1, or 2) is exogenous, the optimal production quantity and pricing policy under the competitive   strategy is to produce qdi ¼ yi pdi ; pd3i þ zdi units to sell at the unit price pdi ; i ¼ 1; 2, where pdi is determined by Lemma 1, and zdi is determined by the following:

(i) If F() is an arbitrary distribution function, then an exhaustive search over all values of zi in the region [Ai, Bi] will determine zdi . (ii) If F() is a distribution function satisfying the condition 2r(zi)2 + dr(zi)/dzi > 0 for Ai 6 zi 6 Bi, where r()/[1  F()] is the hazard rate, then zdi is the largest zi in the region [Ai, Bi] that satisfies dE[pi()]/dzi = 0. (iii) If the condition for (ii) is met, and p(Ai) + s  c > 0, then zdi is the unique zi in the region [Ai, Bi] that satisfies dE[pi()]/dzi = 0. In practice, the quantity of end-of-used cores can be partially determined by using economic incentives like deposit fee, buy back, and acquisition fee, etc., which stimulate or enforce the acquisition (De Brito et al., 2003). Therefore it is practical by assuming q3i or z3i is exogenous, which can facilitate theoretic discussions in a rigid manner. However, if the supply of the end-of-used cores is endogenous and both zi and z3i are decision variables, analytic analysis seems intractable, and an exhaustive search over all values of zi, i = 1, 2, in the region [Ai, Bi] will determine the optimal production quantities. Fortunately, we can numerically show its optimality in our illustrative example. After solving the three variables (given q2 is exogenous) or solving the four variables (given q2 is endogenous), the ex  pected profit E pdi ðÞ can be obtained. The total expected profit generated by the twocompeting versions of the product  is to sum up E pdi ðÞ ; i ¼ 1; 2. Let the realized value of the sum denoted by Pd, Pd ¼ E pd1 ðÞ þ pd2 ðÞ . 4.2. The cooperative setting If the OEM decides to remanufacture the take-back cores by herself, the cooperative decision-making becomes a centralized problem by jointly determining the pricing and lot-sizing (pi, zi), i = 1, 2, so that the expected total profit generated by the two versions of the product is maximized:

E½Pðpi ; p3i; zi ; z3i Þ ¼ E

2 X

½pi ðpi ; p3i; zi ; z3i Þ;

ð8Þ

i¼1

where

E½pi ðÞ ¼ ðpi  C i Þ½yi ðpi ; p3i Þ þ li   ½ðC i þ hÞKðzi Þ þ ðpi þ s  C i ÞHðzi Þ

Phc represent thei realized value of the total expected profit generated by the centralized system, P ¼ E P pci ; pc3i; zci ; zc3i . Let c

Lemma 2. For fixed zi, i = 1, 2, the optimal price of the centralized system is determined uniquely as a function of zi and z3i:

pci ¼

2M 3i ½Mi ð1 þ C i Þ  cM 3i C 3i þ li  Hðzi Þ þ cM½M 3i ð1 þ C 3i Þ  cM i C i þ l3i  Hðz3i Þ 4M i M 3i  c2 M 2

;

i ¼ 1; 2:

ð9Þ

The pricing decision generated by the integrated firm is determined by the production costs, the mean and variance of the error terms, the market sizes (where M = Mi + M3i), and the cross-sensitivity coefficient. Substituting pci ; i ¼ 1; 2, into Eq. (8),    the optimization problem becomes a maximization over two stocking variables: E P pci ; pc3i ; zi ; z3i . If the supply of the end-of-used cores is endogenous and both zi and z3i are decision variables, seeking for analytical optimal solutions of

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(z1, z2) seems not possible. It is because the optimization problem involves with multiple decision variables. Instead, an exhaustive search over all values of zi in the region [Ai, Bi] will determine zci ; i ¼ 1; 2. As discussed previously, if the concavity condition is met, the optimal lot-size can be determined by the standard fractile. In our numerical example, we provide graphical illustration to show its optimality of the problem being studied. If one of the stocking factors is exogenous, the problem can be simplified into the single variable optimization problem, which is analogous to that under the competitive setting. Therefore Theorem 1 can be applied in an analogous manner and hence the detail is omitted. 5. Analysis and managerial implications This section reports on analytic analysis that characterizes various qualitative properties of the pricing and lot-sizing decisions, and the corresponding expected profits generated by the system with respect to major parameters: c, d, D, and c. It is worthy of noting that c1 = C1 = c is the manufacturing cost for both systems, d = C2/C1 = C2/c is the percentage of the remanufacturing to the manufacturing cost, D = c2/C2 is the percentage of remanufacturing cost if it is performed by the IO, and c is the substitution coefficient. Based on the parameters defined above, we carry out analytic analysis and address some managerial implications. In what follows, we first outline the analysis of pricing and lot-sizing decisions in Propositions 1 and 2, respectively. Since profit analysis under a random price-dependent demand is intractable, we only report the analysis of riskless profit functions: Wi(pi, p3i), i = 1, 2, in Proposition 3. Proposition 4 presents the necessary conditions under which the total expected profit generated by the cooperative strategy will be greater than that by the competitive strategy. Proposition 1. Given lot-sizing factors zi, i = 1, 2, the pricing decisions generated by the system have the following properties: (i) pd1 is increasing in c, (ii) pd2 is increasing in c and d, and (iii) pc1 is increasing in c, provided M1 > dcM2 and dM2 > cM1, and is also increasing in d. The proposition is intuitive, suggesting that the pricing decisions generated by the two systems are directly proportional to the costs. It is consistent with the pricing rule of thumb in a manufacturing system. Proposition 2. The production lot-sizes generated by the system have the following properties: (i) Given z2 ; zd1 is decreasing in c and increasing in c, and zc1 is decreasing in c, and (ii) given z1 ; zd2 is decreasing in c and increasing in c, and zc2 is decreasing in c. Part (i) suggests that the production quantity generated by the manufacturing operation in both systems is decreasing in its cost, and increasing in the substitution rate. Part (ii) indicates that the remanufacturing quantity generated by both systems has similar properties as that of the new product manufacturing. Proposition 3. Given lot-sizing factors zi, i = 1, 2, the riskless profits Wdi ðÞ generated by the decentralized system have the following properties: (i) Wd1 ðÞ is decreasing in c and increasing in d, (ii) Wd2 ðÞ is decreasing in c and d, provided (2Dd + c)/ (4  c2)  d < 0, and Dd(2 + c2) + c < 0, and (iii) Wd ðÞ ¼ Wd1 ðÞ þ Wd2 ðÞ is decreasing in c, provided (2Dd + c)/(4  c2)  d < 0 and Dd(2 + c2) + c < 0. The proposition is intuitive, i.e., the individual and the total profits will be less if the cost is high. However, the riskless profit generated by the new product will be higher when the cost of its remanufacturing counterpart is relative high, or the cost-saving of remanufacturing is less. It can be explained by the substitution effect between the two products. Proposition 4. Given lot-sizing factors zi, i = 1, 2, the total expected profit generated under the cooperative strategy is more than that generated by the competitive strategy, i.e., the cooperative strategy is more profitable, only if the following condition is met: cP

h  i                     pc1  c M 1 1  pc1 þ l1  pc2  dc M 2 1  pc2 þ l2 þ Lc1 ðÞ þ Lc2 ðÞ þ pd1  c M 1 1  pd1 þ l1 þ pd1  c M1 1  pd1 þ l2  Ld1 ðÞ  Ld2 ðÞ         ; pc2 M 1 pc1  c þ pc1 M2 pc2  dc  pd2 M 1 pd1  c þ pd1 M 2 pc2  Ddc

or equivalently,  

d6









 



  

h

 









 



  i

Wc1 pc1 þ pc2 M 2 1  pc2 þ cpc1 þ l2  Lc1 ðÞ  hK zc2  pc2 þ s H zc2  Wd1 pd1 þ pd2 M2 1  pd2 þ cpd1 þ l2  Ld1 ðÞ  hK zd2  pd2 þ s H zd2               c M2 1  pc2 þ cpc1 þ l2 þ K zc2  H zc2  Dc M2 1  pd2 þ cpd1 þ l2 þ K zd2  H zd2

;

or

  i     h          Lc1 ðÞ þ Lc2 ðÞ  Wc1 pc1  Wc2 pc2 þ Wd1 pd1 þ pd2 M 2 1  pd2 þ cpd1 þ l2  Ld1 ðÞ  hK zd2  pd2 þ s H zd2        DP : dc M 2 1  pd2 þ cpd1 þ l2 þ K zd2  H zd2 The proposition suggests that the cooperative strategy is more profitable only when the new and remanufactured goods are highly substitutable and/or the remanufacturing cost by the OEM is relatively low and by the IO is relatively high. On the other hand, the total system-wide expected profit generated by adopting the cooperative strategy does not outperform that by facing a head-to-head competition between the OEM and the IO in a generic setting.

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6. Numerical study In the aforementioned models and analysis, it seems unlikely to derive analytical results thoroughly regarding the behavior of the decisions and the expected profit functions. We carried out an in-depth numerical study as a complement to exploit their optimality and property. In this section we report on the numerical results to quantify such behaviors and properties, and more importantly, to attend managerial insights into the decision tendencies and economic implications associated with the two strategies. The study is carried out in the following sequence: presenting the results of a 2k factorial design, illustrating the optimality of a selected base case, and conducting an in-depth sensitivity analysis with respect to major parameters. 6.1. The factorial design In what follows, we present a 2k factorial design with k = 4. The four factors under inspection are the manufacturing cost: c, the ratio between the remanufactured and the manufacturing costs: d, the ratio of remanufacturing cost incurred between the IO and the OEM: D, and the cross-sensitivity coefficient of demand: c. The low and high levels for the factors are: c = 0.25 and 0.5, d = 0.5 and 0.75, D = 0.75 and 1.0, and c = 0.25 and 0.5. Other parameter settings are as follows: the market ratio for the new products a = 0.5, the potential market size M = 1, i.e., M1 = M2 = 0.5, the shortage and holding costs are nil (s = 0 and h = 0), and the random term of the demand is uniformly distributed over [Ai, Bi] = [0.25, 0.25], i = 1, 2, such that l1 = l2 = 0. Choosing such values can prevent from the unnecessary noise distraction from our focus on investigating the effect of the four key factors on the expected profit. Moreover, the uniform distribution is IFR (having an increasing failure rate) that satisfies the concavity condition given in Theorem 1, and ensures the optimality of the solutions. Based on the settings above, the numerical results generated by the two systems are presented in Table 1a,1b, where 1(a) reports the production quantities and selling prices, and 1(b) reports the profits and percentages of profit difference between the two systems DP and Dp1, that are defined as follows:

DP ¼

Pd  Pc  100%; jPc j

ð10aÞ

Dp1 ¼

pd1  pc1 pc  100%: 1

ð10bÞ

Eqs. (10a) and (10b) represent respectively the total channel profit improvement and the profit improvement of the manufacturing alone, provided the OEM does not participate in the remanufacturing but the IO does, i.e., the OEM adopts a decentralized and competitive strategy. It is intuitive that the decentralized competition performs no better than the centralized cooperative if there is no cost savings by the IO. It is numerically illustrated in Table 1b where DP < 0 in the eight cases with D = 1. However, when D = 0.75 and c = 0.25, the competition generates a larger pie and outperforms the cooperative in the four cases being studied. The profit improvement percentage is significant especially when the manufacturing cost is high (c = 0.5), where DP = 21.95 and 42.86 for d = 0.5 and 0.75, respectively. Under such a scenario with lower D and c and higher c, both DP and Dp1 are positive or semi-positive. The OEM can benefit from a direct cost reduction incurred by the IO and a lower competitive intensity in the market, and therefore may prefer to the competitive strategy. The mutually beneficial strategy allows the OEM to concentrate on her most profitable value chain activities. It can also explain why IBM subcontracts its factory operations to Sanmina, Porsche subcontracts Boxster’s assembly to Valmet Automotive (Arruñada and

Table 1a Pricing and quantity decisions generated by the two strategies. c

d

D

c

zc1

zc2

zd1

zd2

pc1

pc2

pd1

pd2

0.25

0.50

0.75

0.25 0.50 0.25 0.50 0.25 0.50 0.25 0.50 0.25 0.50 0.25 0.50 0.25 0.50 0.25 0.50

0.086 0.137 0.086 0.137 0.085 0.137 0.085 0.137 0.060 0.038 0.060 0.038 0.064 0.036 0.064 0.036

0.162 0.190 0.162 0.190 0.123 0.163 0.123 0.163 0.080 0.140 0.080 0.140 0.004 0.083 0.004 0.083

0.063 0.091 0.063 0.091 0.064 0.092 0.064 0.093 0.103 0.039 0.101 0.036 0.101 0.035 0.099 0.032

0.175 0.187 0.152 0.170 0.141 0.157 0.107 0.130 0.108 0.132 0.066 0.098 0.046 0.081 0.015 0.032

0.761 1.106 0.761 1.106 0.758 1.103 0.758 1.103 0.806 1.182 0.807 1.182 0.795 1.170 0.795 1.170

0.714 1.050 0.714 1.050 0.736 1.075 0.736 1.075 0.735 1.080 0.735 1.079 0.763 1.120 0.764 1.120

0.668 0.784 0.700 0.788 0.671 0.790 0.671 0.795 0.708 0.866 0.712 0.873 0.713 0.876 0.716 0.886

0.625 0.739 0.637 0.753 0.642 0.760 0.657 0.778 0.662 0.796 0.680 0.820 0.688 0.831 0.707 0.862

1.00 0.75

0.75 1.00

0.50

0.50

0.75 1.00

0.75

0.75 1.00

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Table 1b Expected profits and profit improvements generated by the two strategies. c

d

D

c

pc1

pc2

pd1

pd2

Pc

Pd

Dp1

DP

0.25

0.50

0.75

0.25 0.50 0.25 0.50 0.25 0.50 0.25 0.50 0.25 0.50 0.25 0.50 0.25 0.50 0.25 0.50

0.065 0.131 0.065 0.131 0.066 0.137 0.066 0.137 0.010 0.050 0.010 0.050 0.012 0.059 0.012 0.059

0.115 0.205 0.115 0.205 0.090 0.173 0.090 0.173 0.072 0.164 0.072 0.164 0.037 0.111 0.037 0.111

0.063 0.114 0.064 0.116 0.064 0.117 0.065 0.119 0.011 0.044 0.011 0.047 0.012 0.048 0.012 0.051

0.124 0.190 0.111 0.175 0.104 0.168 0.086 0.147 0.089 0.158 0.068 0.132 0.058 0.120 0.034 0.089

0.179 0.336 0.179 0.336 0.156 0.310 0.156 0.310 0.082 0.214 0.082 0.214 0.049 0.170 0.049 0.170

0.187 0.304 0.174 0.291 0.168 0.284 0.151 0.266 0.100 0.202 0.079 0.179 0.070 0.168 0.046 0.139

3.08 12.98 1.54 11.45 3.03 14.60 1.52 13.14 10.00 12.00 10.00 6.00 0.00 18.64 0.00 13.56

4.47 9.52 2.79 13.39 7.69 8.39 3.21 14.19 21.95 5.61 3.66 16.36 42.86 1.18 6.12 18.24

1.00 0.75

0.75 1.00

0.50

0.50

0.75 1.00

0.75

0.75 1.00

Vázquez, 2006), and British Petroleum outsources its IT operations to third-parties service providers including Granada Computer Services (Cross, 1995). The realized pricing and lot-sizing decisions given in Table 1a reveal that the production quantity for the new and remanufactured products under the competitive strategy is respectively 90.51% and 6.51%, on the average, lower than that by the cooperative, and the selling price is 20.48% and 20% respectively lower. It probably can be explained by the competition effect and the lower remanufacturing cost incurred by the IO. 6.2. The optimality Since proving the optimality of the lot-sizing solutions (z1 and z2) generated by both strategies is analytically intractable, we select a base case from the aforementioned sixteen scenarios, and show its optimality numerically. The base case settings for the four key parameters are as follows: c = 0.5, d = 0.5, D = 0.75, and c = 0.5. We used an enumeration search for zi, i = 1, 2, and the corresponding expected profit functions are graphically shown in Fig. 1, which shows nearly perfect concave response surfaces. The optimal lot-sizes (zdi and zci , i = 1, 2) are identical to those generated by the standard fractile rule in Eq. (7). We also carried out numerical study for other settings, all of which demonstrated similar results. 6.3. Sensitivity analysis Based on the parameter settings above, we conduct an in-depth sensitivity analysis to illustrate the properties and tendencies of the decisions and the expected profit functions. Special emphasis is placed on quantifying the profit-improving percentages DP and Dp1 between the two strategies with respect to the four factors: c, d, D, and c. The analysis of decision and profit tendencies with respect to the four factors is summarized in Table 2a,2b, where % represents increasing in the designated parameter, & represents decreasing, ? means no change, and- means no particular tendency between the observed values and the parameter. The numerical results reconfirm our analytical findings given in Propositions 1 and 2. Further analysis on profit is summarized in Tables 3–6 The analysis results are consistent with Proposition 4, showing that the competitive strategy generates a larger pie and more profit than the cooperative strategy under higher c and d

Fig. 1. The response surfaces of the profit functions (Pc and Pd) generated by using the base case settings.

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J.-M. Chen, C.-I. Chang / Transportation Research Part E 48 (2012) 387–400 Table 2a Analysis of decision tendencies w.r.t. the key factors. Factor

zc1

zc2

zd1

zd2

pc1

pc2

pd1

pd2

c d D

& & ? %

& & ? %

& % % %

& & & %

% & ? %

% % ? %

% % % %

% % % %

c

Table 2b Analysis of expected profit and profit improvement tendencies w.r.t. the key factors. Factor

pc1

pc2

pd1

pd2

Dp1

DP

c d D

& & ? %

& & ? %

& % % %

& & & %

– & % &

% % & &

c

Table 3 Profit analysis w.r.t. the manufacturing cost of the new product (c). c

pc1

pc2

pd1

pd2

Dp1

DP

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.250 0.197 0.152 0.112 0.078 0.050 0.027 0.009 0.004 0.012

0.250 0.232 0.214 0.197 0.180 0.164 0.149 0.134 0.120 0.106

0.220 0.173 0.132 0.097 0.068 0.044 0.026 0.013 0.004 0.000

0.220 0.210 0.197 0.184 0.171 0.158 0.145 0.133 0.120 0.107

12.00 12.18 13.16 13.39 12.82 12.00 3.70 44.44 200.00 102.50

12.00 10.72 10.11 9.06 7.36 5.61 2.84 2.10 6.90 13.83

Table 4 Profit analysis w.r.t. the cost ratio between the remanufacturing and the manufacturing (d). d

pc1

pc2

pd1

pd2

Dp1

DP

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.029 0.033 0.038 0.042 0.046 0.050 0.053 0.057 0.061 0.064 0.068

0.303 0.272 0.242 0.215 0.189 0.164 0.142 0.121 0.102 0.084 0.068

0.037 0.039 0.040 0.042 0.043 0.044 0.046 0.047 0.048 0.050 0.051

0.252 0.231 0.211 0.192 0.175 0.158 0.142 0.127 0.114 0.100 0.089

27.59 18.18 5.26 0.00 6.52 12.00 13.21 17.54 21.31 21.88 25.00

12.95 11.48 10.36 8.95 7.23 5.61 3.59 2.25 0.61 1.35 2.94

Table 5 Profit analysis w.r.t. the cost ratio between the IO and the OEM (D).

D

pc1

pc2

pd1

pd2

Dp1

DP

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.164 0.164 0.164 0.164 0.164 0.164 0.164 0.164 0.164 0.164

0.037 0.038 0.039 0.040 0.041 0.042 0.043 0.044 0.045 0.046

0.252 0.238 0.224 0.211 0.198 0.186 0.175 0.163 0.153 0.142

26.00 24.00 22.00 20.00 18.00 16.00 14.00 12.00 10.00 8.00

35.05 28.97 22.90 17.29 11.68 6.54 1.87 3.27 7.48 12.15

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Table 6 Profit analysis w.r.t. the substitution coefficient (c).

c

pc1

pc2

pd1

pd2

Dp 1

DP

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.000 0.002 0.007 0.015 0.028 0.050 0.086 0.153 0.296 0.751

0.034 0.045 0.061 0.084 0.117 0.164 0.238 0.365 0.622 1.400

0.000 0.002 0.007 0.015 0.027 0.044 0.069 0.102 0.149 0.215

0.051 0.063 0.079 0.099 0.125 0.158 0.200 0.253 0.324 0.417

– 0.00 0.00 0.00 3.57 12.00 19.77 33.33 49.66 71.37

50.00 38.30 26.47 15.15 4.83 5.61 16.98 31.47 48.47 70.62

and lower D and c. In brief, if the manufacturing cost incurred by the OEM is high, the remanufacturing cost incurred by the IO is low, and the new and the like-new products have a low substitutability, the competitive strategy is preferred. Under such a circumstance, however, the OEM may suffer from a lower profit from giving up and letting the IO take over the remanufacturing, i.e., DP > 0 and Dp1 < 0. It can be rectified by adopting such coordination mechanisms as revenue-sharing agreements with or without consignments (Chen et al., 2010). 7. Conclusions This paper dealt with an emerging research of investigating the co-opetitive strategy of a closed-loop supply chain incorporating with remanufacturing using an analytic approach. We formulated the decision-making of the system as a multi-variable optimization problem under a newsvendor setting. As we have shown in our analysis, the competitive strategy can generate higher expected profit than the cooperative strategy under such conditions of a lower degree of product substitutability and/or a lower remanufacturing cost incurred by the IO. In additions, the numerical study suggests that the production quantity and selling price are lower in a competitive setting. It is probably due to the production cost reduction incurred by the IO and the substitutable effect. Our research contributes to the literature (e.g., Ferguson and Toktay, 2006; Atasu et al., 2008; Ferrer and Swaminathan, 2006, 2010; Shi et al., 2010, 2011a,b) by considering a stochastic price-dependent demand function with substitution in a dual channel problem setting. It differs significantly from the utility function approach given by Atasu et al. (2008) and Ferguson and Toktay (2006) and the exogenous-pricing decision models by Shi et al. (2010) and Shi et al. (2011b). Shi et al. (2011a) assumed both new and remanufactured products are sold at the same price. Future direction may be aimed at dealing with Pareto improvement and reverse channel coordination, conducting empirical research for a variety of industries (e.g., Martin et al., 2010), and developing environmental performance measures for CLSCs (e.g., Paksoy et al., 2011). Acknowledgments The authors would like to thank the editor-in-chief and two anonymous reviewers for their insightful comments and suggestions that have significantly improved the paper. This research was partially supported by the National Science Council (Taiwan) under Grant NSC98-2410-H-008-009-MY3. Appendix A Proof of lemma 1. To prove the price given in Eq. (6) is optimal and unique for the decentralized system, we take the first partial derivative of Eq. (5) with respect to pi and set the result equal to zero:

@E½pi ðpi ; p3i; zi ; z3i Þ ¼ M i ð1  2pi þ cp3i þ ci Þ þ li  Hðzi Þ ¼ 0: @pi Solving the necessary condition above yields the price given in (6). To prove its optimality, we can check the second order sufficient condition:

@ 2 E½pi ðpi ; p3i; zi ; z3i Þ ¼ 2M i ; @p2i

i ¼ 1; 2:

which is negative. Therefore the profit function is concave in pi for a given zi, and the solution of the first order necessary condition (FONC) is the unique optimal solution. h

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Proof of Theorem 1. Assuming z2 is given, and the distribution F() of the error term has a hazard rate r() = f()/[1  F()]. Such that 2r(x)2 + dr(x)/dx > 0 for all x 2 [Ai, Bi]. Karakul (2008) has shown that there is a unique solution that satisfies the first order necessary condition. Define PZ 1 ¼ dP1 ðz@z1 ;p1 1 ;p2 Þ ¼ ðc þ hÞ þ ðp1 ðz1 Þ þ s þ hÞ½1  Fðz1 Þ being the FONC of P1(z1, p1, p2). Solving the FONC and the second order sufficient condition (SOSC) of PZ1, respectively, yields

   dPZ 1 f ðz1 Þ 2½1  Fðz1 Þ 2 ¼ f ðz1 Þðp1 ðz1 Þ þ s þ hÞ þ ½1  Fðz1 Þ p01 ðz1 Þ ¼ ð4  c Þðp ðz Þ þ s þ hÞ  M ; 1 1 1 M 1 ð4  c2 Þ rðz1 Þ dz1

and 2

d PZ 1 2

dz1

¼

" #

 df ðz1 Þ=dz1 2½1  Fðz1 Þ 2f ðz1 Þ  ½1  Fðz1 Þ 2rðz1 Þ2 þ r 0 ðz1 Þ 2 ð4  c Þðp ðz Þ þ s þ hÞ  M  : 1 1 1 rðz1 Þ M1 ð4  c2 Þ M 1 ð4  c2 Þ rðz1 Þ2

Since any stationary point of PZ1 needs to satisfy the FONC, and hence

2 d PZ 1 2 dz 1

¼ dPZ 1 =dz1 ¼0

2f ðz1 Þ  ½1  Fðz1 Þ M1 ð4  c2 Þrðz1 Þ2

½2rðz1 Þ2 þ drðz1 Þ=dz1 ;

2

if 2r(z1) + dr(z1)/dz1 > 0 for all z1 2 [Ai, Bi]. This suggests that all stationary points of PZ1 are local maximum. Hence, PZ1 = 0 can have at most two roots over [Ai, Bi], and consequently, P[z1, p1] might have two stationary points, with the larger one being the local maximum over this range. However, PZ1(A) = p(A) + s  c > 0 and hence PZ1 equals zero at most once in (Ai, Bi], proving the unimodality of P[z1, p1]. h Proof of lemma 2. To prove the optimal price given in Eq. (9) is unique for the centralized system, we take the first partial derivative of Eq. (8) with respect to pi and set the result equal to zero:

@E½Pðp1 ; p2; z1 ; z2 Þ ¼ M i ð1  2pi þ cp3i þ C i Þ þ li  Hðzi Þ þ M 3i cðp3i  C 3i Þ ¼ 0; @pi

i ¼ 1; 2:

We can obtain the optimal price by solving the equation above. The sufficient condition is

@ 2 E½Pðp1 ; p2; z1 ; z2 Þ ¼ 2Mi ; i ¼ 1; 2; @p2i

and

! 2 ! ! 2 ! @ 2 E½Pðp1 ; p2; z1 ; z2 Þ @ E½Pðp1 ; p2; z1 ; z2 Þ @ 2 E½Pðp1 ; p2; z1 ; z2 Þ @ E½Pðp1 ; p2; z1 ; z2 Þ  ¼ 4M i M3i  c2 M2 > 0; @p1 @p2 @p1 @p2 @p21 @p22 where

@ 2 E½Pðp1 ;p2; z1 ;z2 Þ @pi @p3i

¼ cM; i ¼ 1; 2. Since the expected profit function is jointly concave in p1 and p2 for a given pair

of zi, i = 1, 2, the solution of the first order necessary condition is the unique optimal solution. h Proof of Proposition 1. We should prove part (i) of the proposition by taking the first partial derivative of Eq. (6) w.r.t. c, and showing the result is positive, that is @pd1 =@c ¼ ð2 þ DcdÞ=ð4  c2 Þ > 0. In a like, proofs of parts (ii) and (iii) can be done by showing

@pd2 ð2Dd þ cÞ @pd2 ð2DcÞ ¼ > 0; ¼ > 0; 2 ð4  c Þ ð4  c2 Þ @c @d @pc1 ½2M 2 ðM 1  dcM2 Þ þ cMðdM 2  cM 1 Þ ¼ > 0; @c ½4M 1 M 2  c2 M2  provided M1 > dcM2 and dM2 > cM1, and

@pc1 ccM1 M2 > 0: ¼ @d 4M 1 M 2  c2 M 2



Proof of Proposition 2. It can be done by taking the first partial derivative of the corresponding function with respect to the parameter, and checking the result being positive or negative. For examples,

h     i 2cðDdcÞ D dc d h  2þ c @Fðzd1 Þ  p1 þ s þ 4c2 4c2 < 0; ¼  d 2 @c ð4  c2 Þ p þ s þ h 1

  @F zd1 ðh þ cÞð4  c2 Þð1 þ dcÞ þ 2c½2ð1 þ cÞ þ cð1 þ dcÞ ¼ > 0;  2 @c ð4  c2 Þ pd þ s þ h 1

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and

 c  h  c i @p @p    pc1 þ s þ @c1  1 h  @c1 c @F zc1 < 0: ¼  2 @c ð4  c2 Þ pd þ s þ h 1

The proof of part (i) is completed. The proof of part (ii) can be conducted in a similar fashion. h Proof of Proposition 3. Differentiating the expected riskless profits Wi() with respect to c and d generates, respectively,

@ Wd1 ðp1 ; p2 Þ ¼ @c





    d 2 þ Ddc cdD þ c2  2 d d 6 0;  1 M 1  p þ c p þ l  c M þ p 1 1 1 1 2 1 4  c2 4  c2

and



    d @ Wd1 ðp1 ; p2 Þ Dc c Dc c d d > 0: ¼ M 1  p þ c p þ l  c M þ p 1 1 1 2 1 1 @d 4  c2 4  c2 Therefore, @ Wd1 ðÞ is decreasing inc, and is increasing in d. Likewise, differentiating the expected riskless profits W2() with respect to c and d, respectively, generates

@ Wd2 ðp1 ; p2 Þ ¼ @c





    d 2Dd þ c cð1 þ cdDÞ  2dD d d < 0;  d M 1  p þ c p þ l  dc M þ p 2 21 2 2 1 2 4  c2 4  c2

provided (2Dd + c)/(4  c2)  d < 0, and Dd(2 + c2) + c < 0, and

@ Wd2 ðp1 ; p2 Þ ¼ @d





    d 2 Dc ð2 þ cÞDc d d < 0:  c M 1  p þ c p þ l  dc M þ p 2 21 2 2 1 2 4  c2 4  c2

To prove part (iii), we have the first derivative:

@ WðÞd ¼ @c





    d 2Dd þ c cð1 þ cdDÞ  2dD d d  d M 1  p þ c p þ l  dc M þ p 2 21 2 2 1 2 4  c2 4  c2



    2Dc ð2 þ cÞDc d d d < 0;  c M 1  p þ c p þ l  dc M þ þ p 2 21 2 1 2 2 4  c2 4  c2

provided (2Dd + c)/(4  c2)  d < 0, and Dd(2 + c2) + c < 0.

h

Proof of Proposition 4. It can be shown by proving the profit difference: E[Pc  Pd] being positive, that is 2 X 

 



Wci pci  Lci pci ; zci



h    i  Wdi pdi  Ldi pdi ; zdi > 0:

i¼1

Manipulating some algebraic operations generates the following inequalities: cP

h  i                     pc1  c M 1 1  pc1 þ l1  pc2  dc M 2 1  pc2 þ l2 þ Lc1 ðÞ þ Lc2 ðÞ þ pd1  c M 1 1  pd1 þ l1 þ pd1  c M1 1  pd1 þ l2  Ld1 ðÞ  Ld2 ðÞ         ; pc2 M 1 pc1  c þ pc1 M2 pc2  dc  pd2 M 1 pd1  c þ pd1 M 2 pc2  Ddc

or equivalently,  

d6









 



  

h

 









 



  i

Wc1 pc1 þ pc2 M 2 1  pc2 þ cpc1 þ l2  Lc1 ðÞ  hK zc2  pc2 þ s H zc2  Wd1 pd1 þ pd2 M2 1  pd2 þ cpd1 þ l2  Ld1 ðÞ  hK zd2  pd2 þ s H zd2 



 c

c M2 1  pc2 þ cp1

           þ l2 þ K zc2  H zc2  Dc M2 1  pd2 þ cpd1 þ l2 þ K zd2  H zd2

;

or

  i     h          Lc1 ðÞ þ Lc2 ðÞ  Wc1 pc1  Wc2 pc2 þ Wd1 pd1 þ pd2 M 2 1  pd2 þ cpd1 þ l2  Ld1 ðÞ  hK zd2  pd2 þ s H zd2        : DP dc M 2 1  pd2 þ cpd1 þ l2 þ K zd2  H zd2



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