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The impacts of manufacturers' direct channels on competitive supply chains

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The impacts of manufacturers' direct channels on competitive supply chains

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Content THE IMPACTS OF MANUFACTURERS’ DIRECT CHANNELS ON COMPETITIVE SUPPLY CHAINS by Liang Han A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (BUSINESS ADMINISTRATION) May 2012 Copyright 2012 Liang Han Dedication To my parents and my wife, for their endless love and encouragement. ii Acknowledgments I would like to express my deepest gratitude to my advisor and dissertation chair, Professor Greys So˘ si´ c, for her continuous guidance, understanding, patience and support over the years of my doctoral study. I am deeply grateful to her for the time and effort she has spent providing valuable suggestions, proofreading the manuscripts and improving my presentationskills. Withoutherpersistenthelpandencouragement, thisdissertationwould not have been possible. I would also like to express my gratefulness to my dissertation committee members, ProfessorHamidNazerzadehandProfessorHarrisonChengfortheirvaluableinput, helpful discussions and accessibility. I am also heartily grateful to Professor Sriram Dasu and Professor Leon Chu for their valuable research opportunities and advice. I also appreciate thesupportandencouragementfromProfessorYehudaBassokandProfessorGarethJames whenIfaceddifficultiesintheprogram. IamalsogratefultotheDepartmentofInformation and Operations Management and the PhD program at Marshall School of Business for providing the scholarship and the research environment during these years. Finally, and most importantly, I wish to thank my dear parents and my loving wife for their endless love and enduring support. iii Table of Contents Dedication ii Acknowledgments iii List Of Tables vi List Of Figures viii Abstract xi Chapter 1: Channel Selection and Pricing Issues 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Model Assumption Validation . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.1 Bricks-and-clicks manufacturer . . . . . . . . . . . . . . . . . . . . . 15 1.4.2 The manufacturer controls three channels . . . . . . . . . . . . . . . 19 1.4.3 Decentralized case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4.3.1 Second stage . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4.3.2 First stage . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.4.4 Analysis of the special cases . . . . . . . . . . . . . . . . . . . . . . . 27 1.4.5 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.4.5.1 The retailer dominates the market. . . . . . . . . . . . . . 31 1.4.5.2 The manufacturer enters the market to compete with the traditional retailer.. . . . . . . . . . . . . . . . . . . . . . . 39 1.4.5.3 The manufacturer gets into the market to compete with the bricks-and-clicks retailer. . . . . . . . . . . . . . . . . . . . 42 1.4.5.4 Comparisons of different industries . . . . . . . . . . . . . . 45 1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Chapter 2: Capacity Investment and Allocation 49 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2 Literature Reivew . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.3 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 iv 2.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.4.1 Convex capacity investment cost: ψ ′′ (K)≥0 . . . . . . . . . . . . . 57 2.4.1.1 Manufacturer’s problem . . . . . . . . . . . . . . . . . . . . 57 2.4.1.2 Retailer’s problem . . . . . . . . . . . . . . . . . . . . . . . 64 2.4.2 Concave capacity investment cost: ψ ′′ (K)<0 . . . . . . . . . . . . . 65 2.4.2.1 Manufacturer’s problem . . . . . . . . . . . . . . . . . . . . 66 2.4.2.2 Retailer’s problem . . . . . . . . . . . . . . . . . . . . . . . 71 2.4.3 Comparison of different capacity cost functions . . . . . . . . . . . . 72 2.5 Obtaining a first-best outcome . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.5.1 Centralized case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.5.1.1 Sensitivity analysis of the centralized system: . . . . . . . . 75 2.5.2 Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 References 85 Appendices 87 Appendix A: Long Proofs in Chapter 1. . . . . . . . . . . . . . . . . . . . . . . . 87 Appendix B: Survey Questions and Analysis . . . . . . . . . . . . . . . . . . . . . 92 Appendix C: The manufacturer controls three channels. . . . . . . . . . . . . . . 97 Appendix D: Long Proofs in Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . 101 Appendix E: Detailed Analysis in Convex Case . . . . . . . . . . . . . . . . . . . 109 Appendix F: Detailed Analysis in Concave Case. . . . . . . . . . . . . . . . . . . 115 Appendix G: Additional Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 v List Of Tables Table 1.1: Relationship between θ and α, β, or γ . . . . . . . . . . . . . . . . 10 Table 1.2: Some possible parameter choices across industries . . . . . . . . . . 13 Table 1.3: Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Table 1.4: Equilibrium prices in special cases . . . . . . . . . . . . . . . . . . . 31 Table B.1: Illustrative example: DayQuil (CVS) . . . . . . . . . . . . . . . . . 95 Table B.2: Illustrative example: DayQuil (WMT) . . . . . . . . . . . . . . . . 95 Table B.3: Illustrative example: Pantene shampoo . . . . . . . . . . . . . . . . 95 Table B.4: Illustrative example: Olay skin care cream . . . . . . . . . . . . . . 96 Table B.5: Illustrative example: Gillette razor . . . . . . . . . . . . . . . . . . 96 Table B.6: Illustrative example: HP personal printers/toners . . . . . . . . . . 96 Table B.7: Illustrative example: Canon digital cameras . . . . . . . . . . . . . 96 Table B.8: Illustrative example: PUR faucet and filter . . . . . . . . . . . . . . 96 Table E.1: More notations and definitions . . . . . . . . . . . . . . . . . . . . . 109 Table E.2: Possible Equilibria in Case 3 . . . . . . . . . . . . . . . . . . . . . . 113 Table E.3: Possible Equilibria in Case 4 . . . . . . . . . . . . . . . . . . . . . . 113 Table E.4: More Possible Equilibria in Case 4 . . . . . . . . . . . . . . . . . . 113 Table E.5: Possible Equilibria in Case 5 . . . . . . . . . . . . . . . . . . . . . . 114 vi Table E.6: More Possible Equilibria in Case 5 . . . . . . . . . . . . . . . . . . 114 Table E.7: More Possible Equilibria in Case 5 . . . . . . . . . . . . . . . . . . 114 vii List Of Figures Figure 1.1: Market structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Figure 1.2: Feasibility in vertically integration . . . . . . . . . . . . . . . . . . 16 Figure 1.3: Channel Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Figure 1.4: Feasible region 1 for Retailer . . . . . . . . . . . . . . . . . . . . . . 21 Figure 1.5: Feasible region 2 for Retailer . . . . . . . . . . . . . . . . . . . . . . 21 Figure 1.6: Feasible region for manufacturer’s problem . . . . . . . . . . . . . . 27 Figure 1.7: Price when |c r s −c r o | is small . . . . . . . . . . . . . . . . . . . . . . 33 Figure 1.8: Price when |c r s −c r o | is large . . . . . . . . . . . . . . . . . . . . . . 33 Figure 1.9: Demand when |c r s −c r o | is small . . . . . . . . . . . . . . . . . . . . 33 Figure 1.10: Demand when |c r s −c r o | is large . . . . . . . . . . . . . . . . . . . . 33 Figure 1.11: Profit when |c r s −c r o | is small . . . . . . . . . . . . . . . . . . . . . . 34 Figure 1.12: Profit when |c r s −c r o | is large . . . . . . . . . . . . . . . . . . . . . . 34 Figure 1.13: Price when |c r s −c r o | is small . . . . . . . . . . . . . . . . . . . . . . 36 Figure 1.14: Price when |c r s −c r o | is large . . . . . . . . . . . . . . . . . . . . . . 36 Figure 1.15: Demand when |c r s −c r o | is small . . . . . . . . . . . . . . . . . . . . 37 Figure 1.16: Demand when |c r s −c r o | is large . . . . . . . . . . . . . . . . . . . . 37 Figure 1.17: Profit when |c r s −c r o | is small . . . . . . . . . . . . . . . . . . . . . . 37 viii Figure 1.18: Profit when |c r s −c r o | is large . . . . . . . . . . . . . . . . . . . . . . 37 Figure 1.19: Price when |c r s −c m o | is small . . . . . . . . . . . . . . . . . . . . . . 41 Figure 1.20: Price when |c r s −c m o | is large . . . . . . . . . . . . . . . . . . . . . . 41 Figure 1.21: Demand when |c r s −c m o | is small . . . . . . . . . . . . . . . . . . . . 41 Figure 1.22: Demand when |c r s −c m o | is large . . . . . . . . . . . . . . . . . . . . 41 Figure 1.23: Profit when |c r s −c m o | is small . . . . . . . . . . . . . . . . . . . . . 41 Figure 1.24: Profit when |c r s −c m o | is large. . . . . . . . . . . . . . . . . . . . . . 41 Figure 2.1: p m =5, p r =4, a=0.005 . . . . . . . . . . . . . . . . . . . . . . . 77 Figure 2.2: p m =5, p r =4, a=0.01 . . . . . . . . . . . . . . . . . . . . . . . . 77 Figure 2.3: p m =4, p r =5, a=0.005 . . . . . . . . . . . . . . . . . . . . . . . 78 Figure 2.4: p m =4, p r =5, a=0.01 . . . . . . . . . . . . . . . . . . . . . . . . 78 Figure 2.5: p m =5, p r =4, α m =0.1 . . . . . . . . . . . . . . . . . . . . . . . . 78 Figure 2.6: p m =5, p r =4, α m =0.1 . . . . . . . . . . . . . . . . . . . . . . . . 78 Figure G.1: Demand when |c r s −c r o | small. . . . . . . . . . . . . . . . . . . . . . 119 Figure G.2: Demand when |c r s −c r o | large . . . . . . . . . . . . . . . . . . . . . . 119 Figure G.3: Price when |c r s −c r o | small . . . . . . . . . . . . . . . . . . . . . . . 119 Figure G.4: Price when |c r s −c r o | large . . . . . . . . . . . . . . . . . . . . . . . . 119 Figure G.5: Profit when |c r s −c r o | small . . . . . . . . . . . . . . . . . . . . . . . 119 Figure G.6: Profit when |c r s −c r o | large . . . . . . . . . . . . . . . . . . . . . . . 119 Figure G.7: Demand when β =γ . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Figure G.8: Demand when α=γ . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Figure G.9: Price when β =γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 ix Figure G.10: Price when α=γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Figure G.11: Profit when β =γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Figure G.12: Profit when α=γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 x Abstract We investigate a competitive dual-channel supply chain with one manufacturer and one retailer. In the first part, we study the pricing strategies made by the manufacturer and the retailer. In order to characterize the market interaction, we introduce two essential sets of parameters, absolute demand coefficients and channel differentiation coefficients. Both sets of parameters are critical for identification of the optimal channel selection and pricing strategiesinthecentralizedanddecentralizedsystems. Inaverticallyintegratedsystem,the decision maker has to make tradeoffs between cannibalization and larger potential market demand. Wealsonotethatcostdifferences, ratherthanthecostmagnitudes, areimportant in deciding optimal channel choices and pricing strategies. In the decentralized system, the manufacturercancompetewiththeretailerbyusingherownonlinestores,whiletheretailer can use physical stores, online stores, or both. We characterize the conditions under which the retailer prefers a single channel over dual channels, and under which the manufacturer chooses to enter the market. We also investigate the pricing strategies used by the two players. The manufacturer can use the wholesale price and the online price to influence the pricessetbytheretailer. Whenthecompetitionlevelissmall,themanufacturermayreduce thewholesalepriceandgivetheretailermorefreedominchoosingoptimalpricingstrategies, xi which benefits both players. When the competing channels are much more substitutable, the manufacturer faces an increased threat from the retailer’s stores. As a result, she may use both prices to compete with the retailer and even start price wars, which can hurt both parties. In the second part, we investigate the impact of capacity investment and allocation in a competitive manufacturer-retailer supply chain system. The manufacturer makes products and sells through both his own channel and the retailer. Given exogenous selling prices, we analyze the equilibrium quantity strategies made by the manufacturer and the retailer. Faced with different types of production cost structures and additional penalty costs, the manufacturer may not always allocate products to both the retailer and himself. Moreover, even when the manufacturer chooses to deliver products to the retailer, he may only fulfill the order partially. Compared to the efficient centralized system, the manufacturer would liketoover-allocateproductstobothchannelsandtheretailerismorelikelytoover-orderin many situations. In order to resolve these issues, we develop contracts that can coordinate the decentralized competitive supply chain. xii Chapter 1 Channel Selection and Pricing Issues 1.1 Introduction In 2010, Proctor & Gamble (P&G) launched a wholly-controlled online store named PGestore, attracting attention from both industry counterparts and academic researchers. Being a manufacturer of a wide range of consumer goods, P&G was the first company in that area to introduce an online store in order to compete with and complement its contracted retailers, such as Wal-Mart, Target, etc. Almost all kinds of products made by P&G were available in this online store, and consumers could see various promotions on the website. Moreover, P&G also introduced an online community for consumers where theycoulddiscussandprovideimmediatefeedbacksontheproductsandservices. Although P&Geventuallyoutsourcedtheoperationofthiswebsitetoanotherspecializedonlineretail service provider, the trial has ignited discussions about the channel selection and pricing strategies for manufacturers in many different industries. P&G is not the only leading manufacturer in its industry to test the water. Among the IT firms, Apple is famous for its dual-channel and fixed pricing strategy: consumers can purchase products from Apple’s physical stores and apple.com at exactly the same price. However, some of its competitors, 1 such as Lenovo in the PC industry, HTC in the smart-phone industry, Samsung in the tablet industry, etc., still focus on their contracted retailers. In the digital camera industry, Canon continues to operate its online stores and offers discounts in its online stores for almost every type of products, while Nikon launched a trial with limited types of products and eventually gave up the online sales strategy. In the apparel and cosmetics industries, many manufacturers start to operate both physical stores and online stores (e.g., Levi’s or Clinique). Similar examples exist in other industries as well. TheInternetasamarketplaceprovidesretailerswithadditionalmeanstoreachpotential customers. Moreandmoretraditionalbricks-and-mortarretailerspaysubstantial attention to the bricks-and-clicks model. Major retail giants, such as Wal-Mart, Target, Bestbuy, or CVS,havebeenmovingfromtraditionalphysicalstorestomultiplesaleschannels. Inorder to maximize their profits, retailers make tradeoffs between capturing more potential cus- tomer demand and possible market cannibalization. This kind of tradeoffs depends on both retailers’ market power and product attributes. Furthermore, across different industries, even the same retailer may choose different retail channels for different types of products. However, in some industries, retailersstill prefers traditional physical stores. There are also some large pure online retailers, such as Amazon.com or Alice.com, actively participating in competitions and attracting a significant fraction of consumers. One of the interesting questionsis, then, howshouldtheretailerspickspecificretailchannelsafterconsideringthe myriads of product attributes and the threats from competitors. Knowing that retailers may have multiple channels, manufacturers have to re-consider the relationships with retailers and the option of using direct channels. 2 Besides the tradeoffs faced by retailers, manufacturers must carefully examine the im- pact brought by new sales channels. As both manufacturers and retailers are faced with intense competition among traditional physical stores and novel multiple-channel stores, interesting questions that we want to address are: 1) what kind of key factors should both manufacturers and retailers take into account? 2) what are the major differences that man- ufacturers and retailers face when they choose their optimal channel and pricing strategies? Although our analysis was motivated by examples in which manufacturers and retailers useonlinechannels,ourmodelcanbeeasilyextendedtoothercasesinwhichmanufacturers use their direct channels and retailers have multiple-channel options. In Section 1.2, we explore the relevant literature. Then, in Section 1.3, we formulate the basic model and discuss an online survey used to validate some of our key assumptions. In Section1.4, wefirstinvestigatetheoptimalstrategiesinthecaseinwhichthemanufacturer operates both online and physical channels; then, we analyze the equilibrium channel selec- tion and pricing strategies made by the manufacturer and the retailer in the decentralized case. We also use numerical analysis to study the more general cases. In Section 1.5, we conclude and discuss possible directions to extend our research questions. All longer proofs are shown in Appendix A. 1.2 Literature Review We begin with the discussions on pricing issues. There are several streams of research questions related to our model. In general, we can distinguish them by the number of different parties involved and their roles in the competition. 3 Choi (1991) is one of the seminal papers discussing channel structure competitions. It studies price competition in a market with two competing manufacturers and a common retailer. Although a common retailer is often a powerful player in the market, all chan- nel members, as well as the consumers, are better off when noone dominates the market. BrynjolfssonandSmith(2000)usesthecustomeracceptanceoftheInternetchanneltochar- acterize the product or channel substitutability in the market; we model this acceptance by introducing differentiation (substitutability) parameters in our paper. McGuire and Staelin (1983) investigates the impact of product substitutability on Nash equilibrium vertical dis- tribution structures in a duopoly setting. In their model, two manufacturers contract with their exclusive retailers respectively. We follow their modeling approach and extend the structure to a multiple-channel case with different kinds of channel substitutability. The question of who introduces the new channel into the market affects the competi- tion structures. Manufacturers can introduce an online sales channel to compete with the traditional retail stores; alternatively, the retailers can launch online shops to complement theirretailstores; finallybothmanufacturersandretailersmayhandlemorethanonechan- nel, both traditional and Internet. When it is the manufacturer who introduces the online sales channel, the natural inclination is to price products on the web at the same level as their traditional retailers. An obvious benefit of maintaining consistent prices is avoidance of cannibalization; however, this reasoning overlooks the opportunity to profit from differ- ences existing between the two different channels. The question is, then, how to optimally set prices in different channels. 4 Anumber of papersinthis areaimplicitlyassume that market isfully covered, and only discuss the distribution of profit in the market. There are also several papers that focus on the strategies for enlarging the pie size. Lal and Sarvary (1999) addresses the issue of when and how is the Internet likely to decrease price competition in the setting with two vertically integrated brands. Consumers face search costs incurred when uncovering more information. The authors analyze two scenarios: in one, products are distributed through stores only, while in the other products are distributed through both stores and Internet. Although they analyze the case in which both channel exist, the authors assume that both products are equally priced. We, on the other hand, allow price differentiations in our channels. Cattani et al. (2004) uses a consumer choice model to determine the optimal prices that firms should charge when they compete in both a traditional channel and an Internet channel. They introduce a random variable, the effort to purchase the product through different channels, as a factor that influences the individual consumer’s utility. Assuming that the manufacturer opens the Internet channel and the retailerusesthe retail store, they analyze possible equilibria by using different wholesale and retail pricing strategies. Cattanietal. (2002)studiestheeffectofforwardintegrationbythemanufactureronthe equilibrium prices and profits. Unlike previous studies, they assume that the manufacturer attempts to minimize channel conflict by setting price equally. Instead of adopting single choice of channel structure, we investigate the possibility of dual channel competition. Kumar and Ruan (2006) examines the strategic forces that may influence the manufac- turer’s decision to complement the retail channel with a direct online channel. In addition 5 to the focal manufacturer’s product, the retailer carries products of competing manufactur- ers. The paper reveals that both brand loyalty and price sensitivity are important factors that the manufacturer takes into account when making the entry decisions. Chiang et al. (2003) uses a deterministic parameter to measure customers’ acceptance of products sold over Internet. An independent manufacturer has the option to implement direct sales channel, and faces the problem of double marginalization. The existence of customer acceptance affects the choice of pricing strategies made by both the manufacturer and the retailer. However, the authors assume that the retailer only opens retail stores, while the manufacturer considers the choice of launching a new direct-sales channel; in our model, the retailer is a bricks-and-clicks retailer. In another paper discussing market entry problem, Huang and Swaminathan (2009) studies the optimal pricing strategies when a product is sold through two channels. A styl- ized deterministic demand model is used to discuss price equilibrium. For a monopoly case, the authors provide theoretical bounds for the pricing strategies, while for the duopoly case they analyze the situation in which an incumbent with mixed channels faces competition from a pure bricks-and-mortar retailer. Although we start with a similar structure as the one described here, the relationship between the two parties is quite different in our case, as we study the pricing strategies between a manufacturer and a retailer, instead of looking at two retailers. Besidesthesituationinwhichsinglemanufacturerandsingleretailercompetewitheach other, there are other cases in which multiple players compete in one or more echelons. 6 Hendershott and Zhang (2006) models consumers with different product valuations who can either purchase a product from one of many traditional retailers, or directly from the manufacturer. Generally speaking, the traditional retailers here are competitors, while in our case, the dual channels of the retailer are not only competitors, but can cooperate with each other as well. Tsay and Agrawal (2004a) presents a model in which a traditional retailer and an Internet retailer compete against each other for customers based on both the price they charge and on effort they put forth. Druehl and Porteus (2010) analyzes a stylized duopoly model of price competition between an online firm and an offline firm, both selling a single physical product in a single period. Bell et al. (2009) analyzes models with three retailers, in which either all retailers are traditional or two are traditional and one is integrated (direct) retailer. Their analysis also includes the sales effort exerted by retailers. Unlike these models, we take into account both the competitions among retailers and the manufacturer’s action. 1.3 Model Formulation In this section, we introduce our model and its basic assumptions. 1.3.1 Model setup Suppose that there are one manufacturer and one retailer in the market, and their relationship is depicted in Figure 1.1. The manufacturer makes the products at the cost rate ofC m per unit, and charges the retailer the wholesale priceW. The manufacturer also has her online sales channel in which the online retail price is P m o , and incurs a per unit 7 onlinesalescostC m o ,wherethesuperscript“m”standsfor”manufacturer”andthesubscript “o” stands for “online”. Similarly, we have the retailer’s online unit sales cost, C r o and the unit store cost, C r s . The retailer charges P r o per unit online and P r s in retail stores. Figure 1.1: Market structure In order to capture the characteristics of competition among two firms and the two differentchannels,wefollowthemodelintroducedbyMcGuireandStaelin(1983)(hereafter referenced as M&S) and extend it to our framework. M&S investigate the effect of product substitutability on Nash equilibrium distribution structures in a duopoly where each manufacturer distributes the goods through a single exclusive retailer, which may be either a franchised outlet or a factory store. In their paper, the demand functions faced by the retail outlets 1 and 2, respectively, are: q ′ 1 =μd 1− s 1−θ p ′ 1 + sθ 1−θ p ′ 2 ,q ′ 2 =(1−μ)d 1− s 1−θ p ′ 2 + sθ 1−θ p ′ 1 , (1.1) where 0≤μ≤1,0≤θ≤1 and s,d are positive. The constantd is a scale factor which is equal to industry demandq ′ ≡q ′ 1 +q ′ 2 when the prices are zero. The parameter μ captures the absolute difference in demand, and θ≥0 8 representsthesubstitutabilityasreflectedbythecrosselasticities. Alargerθ meansalarger substitutability, therefore, a more intense competition in the market. The constants is also a scale factor used to ensure the non-emptiness of the feasible pricing space. After altering the term sequences, we can rewrite (1.1) as q ′ 1 =μd 1−sp ′ 1 + θ 1−θ s(p ′ 2 −p ′ 1 ) , q ′ 2 =(1−μ)d 1−sp ′ 2 + θ 1−θ s(p ′ 1 −p ′ 2 ) . (1.2) Byextendingexpressionsfrom (1.2)toathree-channelcase,wecharacterizethedemand functions in our case as follows: d r s =μ r s d[1−P r s /k+α(P r o −P r s )/k+γ(P m o −P r s )/k], d r o =μ r o d[1−P r o /k+α(P r s −P r o )/k+β(P m o −P r o )/k], d m o =μ m o d[1−P m o /k+γ(P r s −P m o )/k+β(P r o −P m o )/k]. (1.3) In the above expressions, we follow M&S and denote the total market potential by a constant d>0. The parameters μ r s , μ r o and μ m o capture the absolute difference in demand. We assume thatμ r s ,μ r o ,μ m o ≥0 andμ r s +μ r o +μ m o =1. The parameterα represents the pure channel differentiation coefficient (between the retailer’s physical store and online channel), β denotesthepurefirmdifferentiationcoefficient(betweentheonlinechannelsownedbythe retailer and the manufacturer), andγ represents the cross-firm-cross-channel differentiation coefficient(betweentheretailer’sphysicalstoreandthemanufacturer’sonlinechannel). It’s easy to see that these parameters are similar to θ 1−θ in M&S: a larger α is equivalent to a larger θ (See Table 1.1), which indicates higher substitutability and less differentiation in the two channels. The same reasoning applies to β and γ. We also assume α,β,γ ≥0. 9 Therefore, our characterization is consistent with the case analyzed in M&S. The positive parameter k has to satisfy k≥C m +max              C r s +α(C r s −C r o )+γ(C r s −C m o ), C r o +α(C r o −C r s )+β(C r o −C m o ), C m o +β(C m o −C r o )+γ(C m o −C r s )              . It plays the role of 1 s in M&S and is used to adjust the magnitude of prices and make the feasible region non-empty. We assume that the industry demand should not increase with θ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0.999 α,β, or γ 0 0.11 0.25 0.43 0.67 1 1.5 2.33 4 9 99 999 Table 1.1: Relationship between θ and α, β, or γ an increase in price in any store. While this may seem restrictive, it is rather intuitive in most cases—for common goods, the increase in price should discourage customers from making a purchase. In general, the price increase in one channel should lead to a smaller total industry demand. Thus, when we add up the total industry demand, the coefficients of price variables should be non-positive. Then, we have (1+α+γ)μ r s − αμ r o − γμ m o ≥0, (1+α+β)μ r o − βμ m o − αμ r s ≥0, (1+β+γ)μ m o − γμ r s − βμ r o ≥0. (1.4) We know that the three inequalities in condition (1.4) cannot be binding at the same time. Whenever two of the three inequalities are binding, we get a vertex (extreme point) of the feasible region: 10 V 1 = (μ r s ,μ r o ,μ m o )= γ+Δ 1 Δ , β+Δ 1 Δ , 1+2α+β+γ+Δ 1 Δ , V 2 = (μ r s ,μ r o ,μ m o )= 1+α+2β+γ+Δ 1 Δ , α+Δ 1 Δ , γ+Δ 1 Δ , V 3 = (μ r s ,μ r o ,μ m o )= α+Δ 1 Δ , 1+α+β+2γ+Δ 1 Δ , β+Δ 1 Δ , (1.5) where Δ=1+2α+2β+2γ+3αβ+3βγ+3γα, Δ 1 =αβ+βγ+γα. Thus, any feasible point (μ r s ,μ r o ,μ m o ) f is a convex combination of these three extreme points: (μ r s ,μ r o ,μ m o ) f =ξ 1 V 1 +ξ 2 V 2 +ξ 3 V 3 , whereξ 1 ,ξ 2 ,ξ 3 ≥0 andξ 1 +ξ 2 +ξ 3 =1. We now want to simplify the model in (1.2) and make our analysis more tractable. Using simple algebra, we can rewrite the demand functions as follows: D r s = 1−(1+α+γ)p r s +αp r o +γp m o , D r o = 1−(1+α+β)p r o +αp r s +βp m o , D m o = 1−(1+γ+β)p m o +γp r s +βp r o , (1.6) where we have D r s = d r s μ r s d ,D r o = d r o μ r o d ,D m o = d m o μ m o d and p r s =P r s /k,p r o =P r o /k,p m o =P m o /k. In this formulation, we can easily observe the “effect” that every price variable con- tributes to the final demand quantities. Similarly, we let c r s = C r s /k,c r o = C r o /k,c m o = C m o /k,c m =C m /k,w =W/k. To make this problem reasonable, all demands, (i.e.,D r s ,D r o ,D m o ) must be nonnegative. This defines a feasible region for the price space: P = {(p r s ,p r o ,p m o ,w)|p r s ≥ w +c r s ,p r o ≥ w+c r o ,p m o ≥c m +c m o ,w≥c m ; D r s ,D r o ,D m o ≥0}. Weacknowledgethatdemandisuncertaininreallife,whichmayelicitargumentsagainst the assumption of the deterministic demand. While in operations literature the demand randomness is one of the key issues, in most marketing literature deterministic models are 11 commonly used for study of the market equilibria. Since our main interest is in identifying potential market equilibrium structures, we focus on the deterministic case and investigate the impact of absolute market differences and relative channel differentiations. We also assume that the manufacturer has infinite production capacity, so she can satisfy all the demand realized in the market. This assumption can be extended to the case in which the manufacturer has finite production capacity and considers how to allocate the quantities. However, this analysis is beyond the scope of this work. We analyze capacity issuesinanotherpaper,whereweinvestigatetheimpactofdifferentcapacitycoststructures. 1.3.2 Model Assumption Validation We notice that in real life, different industries may have different sets of values of parameters used in our model (α,β,γ, and μ’s), and our formulation does capture the characteristics of various market situations. For instance, in the comic book industry, most publishers only focus on managing copyrights and advertisements, and leave most of sales to their retailers. Therefore, it is quite reasonable to assume that the retailers fulfill most of the demand,μ r s +μ r o ≈1. For those retailers, the online traffic is usually small compared to the traffic in the physical stores. Therefore, one possible choice of parameters μ in the comicbookindustryis(μ m o ,μ r o ,μ r s )=(0.1,0.3,0.6). Moreover, sincetheretailercaresmore about the consumers’ valuation, the differentiation impact between manufacturer and the retailer is large and the price difference sensitivity is then very small, e.g., β = γ = 0.11 (which corresponds toθ =0.1 in M&S); the price sensitivity is relatively large between the retailer’s two stores, e.g., α=1 (θ =0.5). Some more possible choices of parameter values 12 across different industries are illustrated in Table 1.2; we shall discuss them further in later sections. Products (μ m o ,μ r o ,μ r s ) α β γ Comic books (0.1, 0.3, 0.6) 1 0.11 0.11 Electronics (0.3, 0.35, 0.35) 2.33 2.33 2.33 (0.4, 0.3, 0.3) 10 0.2 0.2 Large appliances (0.1, 0.38, 0.52) 2.33 0.11 0.11 Easy-to-assemble furniture (0.25, 0.25, 0.5) 1 1 1 Scooters (0.4, 0.1, 0.5) 0.11 0.11 0.11 Grocery (0.15, 0.1, 0.75) 0.11 1 0.25 K-J wines (0.4, 0.1, 0.5) 0.11 0.11 1 Cosmetics (0.35, 0.4, 0.25) 1 9 1 Apparel (0.3, 0.3, 0.4) 0.2 1 0.2 Table 1.2: Some possible parameter choices across industries Fordifferentindustries,itseemstobeapparentthatwemayhavedifferentsetsofα,β,γ. But do we really need to introduce different sets of μ’s as well? Furthermore, we assume thatμ’sareindependentofprices; i.e.,μ’scapturetheabsolutedemanddifference. Inorder to obtain more solid evidence that our modeling assumptions are reasonable, we conduct surveys and experiments in two different scenarios. The questionnaires used in our surveys are provided in Appendix B; we discuss the surveys in more detail below. We carefully examine both assumptions using a survey on the M-Turk system from Amazon.com. The survey consists of two parts. The first part considers possible reasons why consumers may prefer manufacturers or retailers, and the reasons why one channel is morepopularthantheotherchannel. Wetrytocaptureconsumerpreferencesoverdifferent possible channels using the knowledge in consumer behavior studies. Questions 1 and 2 in our survey capture these elements. In the second part (question 3 in our survey), we focus on the independence assumption. From a set of different industries, we collect data for 13 8 products of different types (home care, health care, personal care, small and large elec- tronics) from different manufacturers (P&G, HP, and Canon) and sold through different retailers (Walmart, Target, Bestbuy, CVS). Each subject goes through five scenarios: we change the prices in each of the scenarios, but keep the prices across all channels equal. For example, we initially set the price of shampoo to be $6.00 on Walmart physical store, Walmart.com, and P&G online store, and in the second scenario reduced it to $5.70 across all channels. In each scenario, we give subjects four options for purchase—manufacturers’ online stores, retailers’ online store, retailers’ physical store, and do not purchase. In order to see the possible changes in μ’s, we study the percentage of customers who prefer one channel. We randomly pick 150 participants each time and find that more that 75% of them remain consistent in the preference over channels, while 15% feel indifferent among all channels. For different types of products, we also see different types of μ’s (please see Appendix B for more details). These results give us the confidence that our assumptions on μ’s are reasonable. Some may argue that M-Turk is an online system, hence participants are already prone to shop online and the result may be biased. However, it is shown in the marketing literature that it is rather easy to get access to the Internet nowadays. It is also very common that consumers get product information and suggestions online, and then decide to purchase from online or physical stores. In most marketing analyses, this kind of bias can be ignored. The results we get in our experiments are consistent with our intuition: among all products we analyzed, 35%−55% of participants would like to pur- chase from retailers’ physical stores, even though these participants completed the survey online. 14 1.4 Analysis We start our analysis with a special case in which the manufacturer owns both the physical store and the online store. In other words, we first look at the vertically integrated system. We then proceed by looking at the decentralized case, in which the manufacturer may sell products through his online store, while the retailer can use the physical store and/or the online store. 1.4.1 Bricks-and-clicks manufacturer Suppose the manufacturer owns two channels: one online store and one physical store, so thatμ s ,μ o ≥0 andμ s +μ o =1. In addition, we need the following restrictions onα and μ’s: μ s (1+α)−μ o ≥0, −μ s +μ o (1+α)≥0. (1.7) Now, the manufacturer chooses optimal prices to maximize the total profits, Π(p s ,p o )=(p s −c s )D m s +(p o −c o )D m o , (1.8) where the demand functions are D m s =μ s [1−(1+α)p s +αp o ], D m o =μ o [1−(1+α)p o +αp s ], (1.9) and c s =c m +c m s ,c o =c m +c m o . We first introduce a technical result. Proposition 1. The objective function is concave with respect to (w.r.t.) all price parameters. 15 Figure 1.2: Feasibility in vertically integration Figure 1.3: Channel Decisions WeuseProposition1toshowourmainresultsforthissubsection,whichdescribechannel decisions in vertically integrated supply chain and are illustrated in Figure 1.3. We define μ s = α(1−co) 2(1+α)(1−cs) , μ s = 2(1−co)+α(1−2co+cs) 2(1+α)(1−co) , ˆ μ s = (1−co) 2 (1−co) 2 +(1−cs) 2 , ˆ α= 2(1−co−cs+cocs) (co−cs) 2 . Wecancharacterizetheoptimalchannelselectionandpricingstrategiesinthefollowing theorem. Theorem 2. I) When α< ˆ α, there always exists an interval [μ s ,μ s ]⊂[0,1], such that: a) if μ s ∈[μ s ,μ s ], the manufacturer chooses bricks-and-clicks; b) if μ s <μ s , then D m s =0; c) if μ s >μ s , then D m o =0. II) When α≥ ˆ α, then μ s >μ s . b) if μ s < ˆ μ s , then D m s =0; c) if μ s > ˆ μ s , then D m o =0. This result is consistent with some existing literature, which consider the centralized system with a manufacturer and a retailer. Most of the papers in this area focus on the marketentryissuesandinvestigatetheentrydecisionmadebythemanufacturer. Therefore, they ignore the role of the critical threshold ˆ α. 16 (I) When the substitution effect is small (i.e., α< ˆ α), at least one store is profitable. a) When the potential market share of the physical store is in a moderate range (i.e., μ s ∈ [μ s ,μ s ]), the manufacturer faces positive demand in both channels. The interior solutions are p s = 2c ′ 1 (1+α)μ o +c ′ 2 α 4μ s μ o (1+α) 2 ,p o = 2c ′ 2 (1+α)μ s +c ′ 1 α 4μ s μ o (1+α) 2 , where c ′ 1 =μ s [1+(1+α)c s ]−μ o αc o , c ′ 2 =μ o [1+(1+α)c o ]−μ s αc s . b) Whenμ s <μ s , the market share in the physical channel is small. The manufacturer maximizes the total profits by focusing on the online stores and adjusting the physical store prices. The optimal prices are then {p s ,p o }= n 2+α(1+c m +c m o ) 2(1+α) , 1+c m +c m o 2 o . c) When μ s > μ s , the manufacturer maximizes the total profit by focusing on the physical store. The optimal prices are {p s ,p o }= n 1+c m +c m s 2 , 2+α(1+c m +c m s ) 2(1+α) o . (II) When α ≥ ˆ α, the two channels have a high substitutability level, and the manu- facturer can obtain positive demand from at most one channel. b) When μ s < ˆ μ s , the manufacturer maximizes the total profits by focusing on the online stores and the optimal prices are {p s ,p o }={ 2+α(1+c m +c m o ) 2(1+α) , 1+c m +c m o 2 }. c) When μ s > ˆ μ s , the manufacturer maximizes the total profit by focusing on the physical store and the optimal prices are {p s ,p o }={ 1+c m +c m s 2 , 2+α(1+c m +c m s ) 2(1+α) }. Thresholds μ s , μ s and ˆ μ s depend on both the channel differentiation coefficient α and the cost difference |c m s −c m o |, which is described in the following corollary. 17 Corollary 3. As |c m s −c m o | increases, the interval [μ s ,μ s ] shrinks. The term |c m s −c m o | captures the cost difference between the two channels. A larger difference in costs makes the manufacturer focus on the more cost-efficient channel. We can furthershowthat,ifthereisnocostdifferencebetweenchannels,theinterval[μ s ,μ s ]⊂[0,1] always exists, and all feasible μ r s must satisfy μ s = α 2(1+α) ≤μ s ≤μ s = 2+α 2(1+α) . Corollary 4. If c m s =c m o , then ˆ α=+∞. As α increases, the interval [μ s ,μ s ] shrinks. When α approaches +∞, the interval shrinks to a single point, 1 2 . In other words, as the substitution effect increases, the decision maker is more likely to obtain profits from one channel. Whentwochannelsarefunctional,thedecisionmakermarksuptheonlinepriceasfewer customers are interested in the online channel. When only one channel is in business, the decision maker creates a quasi-monopoly problem, in which the non-profitable channel is used to drive all possible customers to the profitable channel, and the profitable channel price is optimized by solving the monopoly problem. This can be summarized in the following result. Corollary 5. When the manufacturer opens two channels, the optimal online price p m o increases with respect to (w.r.t.) μ r s . When only one channel is profitable, the optimal price for this channel is independent of μ r s and α, while the price for the other channel is decreasing in α. In the above analysis, we use the constraints {p s ≥c m +c m s ,p o ≥c m +c m o } to ensure a non-negative profit margin for each product sold. However, in some cases the sellers may choose to attract more customers by charging a lower-than-cost price for some promotional 18 products. Therefore, the feasible region may change toP c ′ ={(p s ,p o )|1−(1+α)p o +αp s ≥ 0,1−(1+α)p s +αp o ≥0}. However, the analysis of optimal solutions remains the same even if we enlarge the feasible region P c to P c ′ in the vertically integrated case. In our setting, the reason is obvious: we only have one category of products, and price reduction in one channel leads to lower profit from this channel and less demand from the otherchannel. Wewillusethisresultintheanalysisofretailer’sprobleminthedecentralized scenario. 1.4.2 The manufacturer controls three channels In the above analysis, we assume that the manufacturer owns one physical store and oneonlinestore. Wealsoconsiderthecaseinwhichthemanufactureroperatesonephysical store and two different online stores in the same regional market, which corresponds to the centralized version of our decentralized model. Consider, for example, iPod Touch sold in Greater China Region by Apple. It is quite easy for consumers in mainland China to purchase iPod Touch from Apple’s physical stores in Shanghai or Beijing. It is also not difficult for them to get the product from two different websitesoperatedbyApple: apple.com.cn andapple.com/hk. Thepricesinthesetwoonline stores are different not only in currency, but also in their real value. To simplify the analysis, we use the same notation introduced in our model setup. In other words, p r s , p r o , p m o represent the prices in the manufacturer’s physical store, the first online store and the second online store, respectively. In this system, the manufacturer 19 maximizes the total profit by choosing optimal prices in three channels: Π c (p r s ,p r o ,p m o ) = (p r s −c r s −c m )d r s +(p r o −c r o −c m )d r o +(p m o −c m o −c m )d m o , where d r s = μ r s [1−(1+α+γ)p r s +αp r o +γp m o ], d r o = μ r o [1−(1+α+β)p r o +αp r s +βp m o ], d m o = μ m o [1−(1+γ+β)p m o +γp r s +βp r o ]. The feasible region is P c ={(p r s ,p r o ,p m o )|d r s ≥0,d r o ≥0,d m o ≥0,p r s ,p r o ,p m o ≥0}. We can show that this objective function is concave with respect to p r s ,p r o ,p m o . There are eight possible scenarios and the analysis is quite complicated for general cases. For a more detailed analysis, please see our discussions in Appendix C. 1.4.3 Decentralized case We next consider the case in which the manufacturer and the retailer compete against each other. Different game sequences can affect the choice of optimal pricing strategies. For example, the manufacturer may refuse to announce the online sales price when signing the wholesale contract, but commits to publish his online sales price simultaneously with the retailer. In our model, we assume that the manufacturer announces the online sales price together with the wholesale price; this online sales price can be viewed as one kind of manufacturer suggested retail price. The game is played in the following sequence: 1 st stage: the manufacturer announces the wholesale price she’s going to charge the retailer and the price she will charge online; 20 2 nd stage: based on the pricing information from the manufacturer, the retailer chooses the prices in his retail store and his online store; 3 rd stage: all the sales occur. We can determine equilibrium results by using the backward approach. 1.4.3.1 Second stage First, weconsiderthepricingstrategieschosenbytheretailer. Afterlearningthewhole- sale price w and manufacturer’s online price p m o , the retailer decides his optimal price pair (p r o , p r s ). The feasible region for the prices is depicted by the shadowed area in Figure 1.4. However, the manufacturer may charge a high wholesale price w, such that the retailer incurs a loss if he opens two channels, or even if he opens one channel only. If the man- ufacturer offers a contract which is not profitable to the retailer, we obtain a trivial case, so we assume that the wholesale price still gives the retailer a non-empty region of feasible prices, as illustrated in Figure 1.5. Figure 1.4: Feasible region 1 for Retailer Figure 1.5: Feasible region 2 for Retailer The total profit for the retailer is a function of p r o and p r s : Π r (p r s ,p r o |w,p m o ) = Π r o (p r s ,p r o |w,p m o )+Π r s (p r s ,p r o |w,p m o ), where we have 21 Π r o (p r s ,p r o |w,p m o )= μ r o (p r o −w−c r o )[1−(1+α+β)p r o +αp r s +βp m o ], Π r s (p r s ,p r o |w,p m o )= μ r s (p r s −w−c r s )[1−(1+α+γ)p r s +αp r o +γp m o ]. (1.10) We first have a technical result. Proposition 6. The retailer’s objective function is concave w.r.t. p r o and p r s . Based on the results in Proposition 6, we can now further identify conditions under which the retailer operates in both channels. Proposition 7. For any given positive μ r s , μ r o and α, β, γ satisfying the inequality (1.4) and any feasible p m o , there exists ˆ w such that i) when w< ˆ w, the retailer operates two channels; ii) the demand in each channel decreases as w increases; iii) the optimal prices are increasing w.r.t. both w and p m o . When the wholesale price w is sufficiently small, the retailer will keep both channels. Notethatthemanufacturermaysetahighwholesalepricew andthentheretailermayonly getpositivedemandfromthepopularchannel. Inreallife,itisnotnecessaryfortheretailer to shut down a channel even if no positive profit can be earned through that channel. The choice of maintaining a channel is beneficial if it can keep the competitors from seizing a larger market share and/or provide more freedom on the pricing strategies. While the second result is quite intuitive, in general, we are not sure if the increase in p m o leads to a larger retailer demand. However, we can show that the demand from each channel is monotone w.r.t. p m o , and the sign depends on the actual value of μ’s and parameters α,β,γ. In addition, when β (resp., γ) is extremely large compared to α and γ (resp., α and β), the demand in the retailer’s physical (resp., online) store increases as p m o increases. 22 This is consistent with our intuition — when the two channels are very substitutable, consumers can be very sensitive to the price change; consumers may move to the other channel when one channel increases the price. The price increase in the manufacturer’s store may drive customers to retailer’s stores. When the potential market share of the retailer’s two channels are close to each other, the additional demand may be allocated to both channels. However, when one channel is extremely popular, not only will the additional demand be directed to the popular channel, but the customers from the less favored channel may also be guided to the more popular channel. Itisalsopossiblethattheretaileronlygetsprofitfromonepopularchannelandnothing from the other one when the manufacturer starts a price war and sets a very small online price, p o m . In addition, the manufacturer may actually want to attract more customers to her own channel by setting a very large wholesale price w such that the retailer has little freedom to choose prices in his two sales channels, and only obtains positive demand from at most one channel. Thus, based on the price pair (w, p m o ), when two channels are both profitable, we denote the interior solutions byp rI o ,p rI s ; when only one channel is profitable, we denote the boundary solutions by p rB o ,p rB s . We may have three possible solutions. I. When both channels are profitable, we have the unique pair of maximizers p rI o (w,p m o )= [α(μ r s +μ r o )c 2 +2(1+α+γ)μ r s c 1 ] det(H r ) , p rI s (w,p m o )= [α(μ r s +μ r o )c 1 +2(1+α+β)μ r o c 2 ] det(H r ) . 23 And we have c 1 =μ r o [βp m o +1+(1+α+β)(w+c r o )]−μ r s α(w+c r s ), c 2 =μ r s [γp m o +1+(1+α+γ)(w+c r s )]−μ r o α(w+c r o ). In addition, we can have two boundary solutions. II. When D r o =0, then we have p rBs s (w,p m o )= w+c r s 2 + 1+2α+β 2δ 0 + 1 2 − 1+2α+β 2δ 0 p m o , p rBs o (w,p m o )= 1+αp rBs s +βp m o 1+α+β , where δ 0 =(1+α+β)(1+α+γ)−α 2 >0. Both optimal prices are increasing in w, but the physical store price increases much faster: ∂p rBs s ∂w = 1 2 , ∂p rBs o ∂w = α 2(1+α+β) . Both optimal prices are also increasing in p m o , ∂p rBs s ∂p m o = 1 2 − 1+2α+β 2δ 0 >0, ∂p rBs o ∂p m o = β 1+α+β + α 1+α+β 1 2 − 1+2α+β 2δ 0 >0. When β >γ, we have ∂p rBs s ∂p m o < ∂p rBs o ∂p m o ; and when β <γ, we have ∂p rBs s ∂p m o > ∂p rBs o ∂p m o . In other words, larger substitution level leads to stronger price responses. III. When D r s =0, the optimal prices are p rBo o (w,p m o )= w+c r o 2 + 1+2α+γ 2δ 0 + 1 2 − 1+2α+γ 2δ 0 p m o , p rBo s (w,p m o )= 1+αp rBo o +γp m o 1+α+γ . Both optimal prices are increasing inw, but the online channel price increases much faster, ∂p rBo o ∂w = 1 2 , ∂p rBo s ∂w = α 2(1+α+γ) . 24 Both optimal prices are also increasing in p m o , ∂p rBo o ∂p m o = 1 2 − 1+2α+γ 2δ 0 , ∂p rBo s ∂p m o = γ 1+α+γ + α 1+α+γ 1 2 − 1+2α+γ 2δ 0 . When β > γ, we have ∂p rBo s ∂p m o < ∂p rBo o ∂p m o ; for β < γ, we have ∂p rBo s ∂p m o > ∂p rBo o ∂p m o . Once again, larger substitution level leads to larger price responses. We can summarize the above results in the following proposition. Proposition 8. When only one channel is profitable to the retailer, both prices increase in the wholesale price w, but the profitable channel’s price increases much faster. The substitution threat (β, γ) from the manufacturer’s online stores affects both prices in retailer’s channels, with higher substitution leading to a faster price increase in the retailer’s corresponding channel. As the wholesale price w increases, we expect the retailer to set higher prices in the sales channels. Interestingly, we find that the price in the profitable channel increases much faster. More precisely, when the substitution effect between the retailer’s two channels is small(i.e.,α is very small), he faces insignificant cannibalization from the two channels, and theprofitablechannel’spriceincreasesfaster; whenαisverylarge,thecannibalizationissue is serious, and the retailer increases both prices at approximately the same rate, 1 2 , w.r.t. w. Whenthemanufacturerraisestheonlinepricep m o , shegivestheretailermorefreedomof choosing optimal prices in both channels, and the retailer does increase both prices. When the substitution effect between the manufacturer’s online channel and the retailer’s online store is larger, the retailer increases the price in his online channel much faster to compete 25 with the manufacturer and saves profits in his other channel. When the substitution ef- fect between the manufacturer’s online channel and the retailer’s physical store is larger, the retailer increases the price in his physical channel much faster to compete with the manufacturer. Given the possible best responses from the retailer, the total demand faced by the retailer and the manufacturer can be rewritten as D rI (w,p m o )=μ r s [1−(1+α+γ)p rI s +αp rI o +γp m o ] +μ r o [1−(1+α+β)p rI o +αp rI s +βp m o ], D rBs (w,p m o )=μ r s [1−(1+α+γ)p rBs s (w,p m o )+αp rBs o (w,p m o )+γp m o ], D rBo (w,p m o )=μ r o [1−(1+α+β)p rBo o (w,p m o )+αp rBo s (w,p m o )+βp m o ], D mi o (w,p m o )=μ m o [1−(1+γ+β)p m o +γp ri s (w,p m o )+βp ri o (w,p m o )], (1.11) where i=I,Bs,Bo. 1.4.3.2 First stage By anticipating the actions taken by the retailer, the manufacturer decides her optimal pricingstrategiesforhertwosourcesofprofits,itsowndirectonlinesalesandretailer’sorder contract. Some may argue that ifp m o ≤w, the retailer may have arbitrage opportunities to purchase from manufacturer’s online stores. The shadowed area in Figure 1.6 is the feasible region in the case p m o ≥ w, while the dotted area represents the additional feasible region without the constraint p m o ≥ w. Therefore, we can summarize the manufacturer’s total profits as follows: Π mi (w,p m o )=(w−c m )D ri (w,p m o )+(p m o −c m −c m o )D mi o (w,p m o ). 26 Figure 1.6: Feasible region for manufacturer’s problem We again start with a technical result about the convexity of the problem (i.e., the concavity of the objective function). Proposition 9. When i=I, we have positive demand in both channels, and the manufacturer’s objective function is concave w.r.t. w and p m o . However, we can show that it is not easy to obtain elegant results for the boundary cases in which the retailer operates in a single channel. Therefore, we start with analyses of some special cases, and then use numerical analysis to investigate the more general cases. 1.4.4 Analysis of the special cases In this section, we consider some special cases in which some of the channels may not be used, while all channels actually being used have equal market share. In total, we have 7 special cases (See Table 1.3). Moreover, we assume that the inequality (1.4) holds, which leads to the following result. 27 ① ② ③ ④ ⑤ ⑥ ⑦ μ m o 0 0 1 0 1 2 1 2 1 3 μ r o 0 1 0 1 2 1 2 0 1 3 μ r s 1 0 0 1 2 0 1 2 1 3 Table 1.3: Special cases 1) If only one of the μ’s is zero, we have μ m o =0=⇒β =γ =0, μ r o =0=⇒α=β =0, μ r s =0=⇒α=γ =0. (1.12) 2) If two of the μ’s are zero, we have μ m o =μ r o =0=⇒α=γ =0, μ r o =μ r s =0=⇒β =γ =0, μ r s =μ m o =0=⇒α=β =0. (1.13) These relationships substantially simplify our analyses of special cases. 1. μ m o =0, μ r o =0, μ r s =1 =⇒ α=γ =0. It is straightforward that when the retailer’s physical store dominates the market, the substitutioneffectfromtheothertwoonlinechannelsisnegligibleregardlessofthesubstitu- tion effect between the two online channels. The retailer as a monopolist only gains profits from physical stores and set the optimal physical store pricep r s = 1+w+c r s 2 . Then, the manu- facturer only obtains profits from wholesale contract and sets wholesale pricew = 1+c m −c r s 2 . 2. μ m o =0, μ r o =1, μ r s =0 =⇒ α=β =0. Similarlyasbefore, weconcludethattheretailerasamonopolistonlygainsprofitsfrom onlinestores,andsetstheoptimalphysicalstorepricep r o = 1+w+c r o 2 . Then,themanufacturer only obtains profits from wholesale contract and sets wholesale price w = 1+c m −c r o 2 . 28 3. μ m o =1, μ r o =0, μ r s =0 =⇒ γ =β =0. The retailer in this case gets zero profit and the manufacturer as a monopolist only obtains profits from online stores and sets p m o = 1+c m +c m o 2 . 4. μ m o =0, μ r o = 1 2 , μ r s = 1 2 =⇒ γ =β =0. When the retailer’s two channels dominate the market, the substitution effect from the third channel is negligible. The retailer gains profits from both online and physical stores if the degree of competition between these two channels is small (i.e., α< ¯ α), where ¯ α=        2−2c m −3c r o +c r s 4(c r o −c r s ) , c r o >c r s , 2−2c m −3c r s +c r o 4(c r s −c r o ) , c r s >c r o . The optimal physical store price is p r o = 1+w+c r 0 2 , the optimal online store price is p r s = 1+w+c r s 2 , while the manufacturer only obtains profits from wholesale contract in which she chooses wholesale pricew = 1+c m −(c r o +c r s )/2 2 . When the competition is intense, (i.e.,α≥ ¯ α), the retailer only obtains profits from the channel with the smaller operating cost and the analysis corresponds to that for cases 1 and 2. 5. μ m o = 1 2 , μ r o = 1 2 , μ r s =0 =⇒ γ =α=0. When the competition between the two online channels is small (i.e., β < 1 c m +c r o −c m o ), the retailer can get positive profit. The optimal physical store price is p r o = w+c r o 2 + 1+βp m o 2(1+β) , and the manufacturer obtains profits from both the wholesale contract and online stores by setting wholesale price w = 1+c m −c r o 2 and online price p m o = 1+c m +c m o 2 . It is obvious that p m o > w, so there is no arbitrage opportunities for the retailer by ordering from the manufacturer’s online stores. 29 Whenthesubstitutioneffectbetweenthetwoonlinechannelsislarge(i.e.,β ≥ 1 c m +c r o −c m o ), the retailer charges p r o = 1+βp m o 1+β and gets zero profit from this channel, while the manufac- turer still charges p m o = 1+c m +c m o 2 and gets positive profits from his own online channel. 6. μ m o = 1 2 , μ r o =0, μ r s = 1 2 =⇒ β =α=0. When the competition between the two channels is small (i.e., γ < 1 c m +c r s −c m o ), the retailer can get positive profit from physical stores, and the optimal physical store price is p r s = w+c r s 2 + 1+γp m o 2(1+γ) . The manufacturer obtains profits from both the wholesale contract and online stores by setting wholesale price w = 1+c m −c r s 2 and online price p m o = 1+c m +c m o 2 . It is obvious thatp m o >w, so there is no arbitrage opportunities for the retailer by ordering from the manufacturer’s online stores. When the substitution effect between the two channels is large (i.e.,γ ≥ 1 c m +c r s −c m o ), the retailer chargesp r s = 1+γp m o 1+γ and gets zero profits from this channel, while the manufacturer still charges p m o = 1+c m +c m o 2 and gets positive profits from his own online channel. 7. μ m o =μ r o =μ r s = 1 3 . In this case, all non-negative α,β,γ are feasible and optimal prices are given by p mI o = 1 2 (c m +c m o )+ A 2 2A 1 (c r o −c r s ), w I = 1 2 c m − A 3 2A 1 c r s −( 1 2 − A 3 2A 1 )c r o , p rI o = w I +c r o 2 + (1+2α+γ)+p mI o (β+αβ+αγ+βγ) 2(1+2α+γ+β+αβ+αγ+βγ) , p rI s = w I +c r s 2 + (1+2α+β)+p mI o (γ+αβ+αγ+βγ) 2(1+2α+β+γ+αβ+αγ+βγ) . (1.14) 30 We define A 1 =4+10γ+5γ 2 +β 2 (5+6γ)+α(2+β+γ)(4+6β+6γ) +β(10+18γ+6γ 2 )), A 2 =(β−γ)((1+β)(1+γ)+α(2+β+γ)), A 3 =α(4+β 2 +10γ+5γ 2 +6β(1+γ))+(1+γ)(2+β 2 +4γ+β(4+5γ)). (1.15) Then, we can show p mI o > w I as well. Our main results for these special cases are summarized in Table 1.4. (μ r s ,μ r o ,μ m o ) p r s p r o p m o w ( 1, 0, 0 ) 3+c m +c r s 4 − − 1+c m −c r s 2 ( 0, 1, 0 ) − 3+c m +c r o 4 − 1+c m −c r o 2 ( 0, 0, 1 ) − − 1+c m +c m o 2 − ( 1 2 , 1 2 , 0 ) 6+2c m +3c r s −c r o 8 6+2c m +3c r o −c r s 8 − 2+2c m −c r o −c r s 4 ( 0, 1 2 , 1 2 ) − 1+c m +c r o 4 + 2+β(1+c m +c m o ) 4(1+β) 1+c m +c m o 2 1+c m −c r o 2 ( 1 2 , 0, 1 2 ) 1+c m +c r s 4 + 2+γ(1+c m +c m o ) 4(1+γ) − 1+c m +c m o 2 1+c m −c r s 2 ( 1 3 , 1 3 , 1 3 ) p rI s p rI o p mI o w I Table 1.4: Equilibrium prices in special cases 1.4.5 Numerical Analysis Duetoalargenumberofparametersinourmodel, itisnotpossibletoobtainanalytical results for general cases. Therefore, we conduct some numerical analysis to investigate transitions between special cases analyzed in Section 1.4.4. 1.4.5.1 The retailer dominates the market. Consider, for instance, the comic book industry in Japan. Many publishers focus on managing copyrights and advertisements, and leave most of sales to the retailers (i.e.,μ r s + μ r o ≈1). Many large retailers build up theme stores to attract comic-lovers and help to set 31 up online communities for those consumers where they can share their ideas and organize activities (e.g., Cosplay). Consumers can purchase items online, but most of them choose to visit physical stores (i.e.,μ r s >0.5). Consumers value the experience both in the physical stores and from online communities. Therefore, the substitution between the retailer’s two channels is moderate (e.g., α = 1). However, the physical stores usually require more resources to take care of the books, so the operating cost c r s is much larger compared to c r o . We are most interested in the impact of the channel differentiation coefficient, α, and the the absolute demand differences,μ r s andμ r o . It is very difficult to investigate the impact of these parameters simultaneously, so we focus on one of the two sets of parameters each time. 1) First, we study the impact of α. Suppose μ r s is fixed and μ r s >0.5 (i.e., the retailer’s physical channel is more popular). • Price. ∗ |c r s −c r o | small. The price in the less costly channel decreases as α increases. When α is small, the price in the more costly channel increases in α; when α is large, the price in the costly channel decreases in α (see Figure 1.7). ∗ |c r s −c r o | large. Whenα is small, we have similar price patterns compared to the caseinwhich|c r s −c r o |issmall;whenαislarge,onlythelow-costchannelgenerates positive demand. The retailer’s prices are constant, while the manufacturer’s wholesale price decreases in α (see Figure 1.8). 32 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 α Price c m =0.5 c r s =0.15 c r o =0.12 u r s =0.55 p r o p r s w Figure 1.7: Price when |c r s −c r o | is small 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 α Price c m =0.5 c r s =0.25 c r o =0.05 u r s =0.55 p r o p r s w Figure 1.8: Price when |c r s −c r o | is large • Demand. ∗ |c r s −c r o | small. The retailer may obtain positive demand from both channels even if α is large (see Figure 1.9). ∗ |c r s −c r o | large. The retailer obtains positive demand from both channels only if α is very small; demand in the high-cost channel always decreases in α (see Figure 1.10). 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.01 0.02 0.03 0.04 0.05 0.06 α Demand c m =0.5 c r s =0.15 c r o =0.12 u r s =0.55 d r o d r s 0 Figure 1.9: Demand when |c r s −c r o | is small 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.02 0.04 0.06 0.08 0.1 α Demand c m =0.5 c r s =0.25 c r o =0.05 u r s =0.55 d r o d r s 0 Figure 1.10: Demand when |c r s −c r o | is large • Profit. ∗ |c r s −c r o | small. As α increases, the retailer’s profit increases and the manufac- turer’s profit decreases (see Figure 1.11). 33 ∗ |c r s −c r o | large. When α is small, we have similar profit patterns compared to the case in which |c r s −c r o | is small; when α reaches certain critical level, the retailer’s profit drops suddenly as only one channel remains profitable. As α increases further, the retailer’s profit increases again and the manufacturer’s profit increases as well (see Figure 1.12). 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.008 0.01 0.012 0.014 0.016 0.018 0.02 α Profit c m =0.5 c r s =0.15 c r o =0.12 u r s =0.55 Π r Π m Figure 1.11: Profit when |c r s −c r o | is small 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 α Profit c m =0.5 c r s =0.25 c r o =0.05 u r s =0.55 Π r Π m Figure 1.12: Profit when |c r s −c r o | is large In the comic book industry, |c r s −c r o | is quite large and α is not small. In physical stores, retailers usually face heavy traffic (μ r s ), set higher prices (p r s ), and sell fewer items. Retailers fail to generate significant realized demand in the physical stores, and the profit from physical stores is smaller compared to that from the online stores. A natural way to increase the total profit for retailers is to differentiate the channels so that they face less cannibalization between the two channels. Our analysis suggests that this may not be the best strategy for retailers. Since keeping both channels in operation means operating costs from both channels, retailers may incur large costs from the physical stores (recall that in our case, c r s is large compared to c r o ). Even when they obtain positive profit from physical stores, retailers may be better off by stocking and selling fewer items through the physical channel. 34 Aswecanseefromthefiguresabove,itmayactuallybeagoodideaforretailerstomake two channels more substitutable (i.e., increase α). Since consumers visiting physical stores may not actually purchase anything there, the retailer may want more consumers to move from physical stores to online stores, where consumers may actually make the purchase. To achieve this goal, retailers can make two channels similarly attractive to consumers. On onehand, retailerscanprovidefewerin-storeservices(e.g., reducethefrequencyofCosplay program). On the other hand, retailers should introduce more online services (e.g., provide more useful comic-related information in online stores, even online Cosplay shows). We also notice that retailers can be better-off by reducing operating costs in physical stores. When c r s is close to c r o , retailers do not incur large costs from both channels. By increasing the cannibalization effect (i.e., a larger α) within an allowable region, retailers can still survive price wars and generate higher profit. This might be obtained by enriching user-generated content in online communities. 2) Now, suppose α is fixed. We are interested in how the μ r s influences the equilibrium strategies. Here, we assume thatc r s >c r o (that is, the physical store incurs higher operation costs). • Price. ∗ |c r s −c r o | small. Whenμ r s issmall,boththeretailer’spricesandthemanufacturer’s wholesalepriceareconstant. Whenμ r s increases,thepriceinthelow-costchannel (c r o ) first decreases then increases w.r.t. μ r s . The price in the high-cost (c r s ) channel and the wholesale price always decrease in μ r s (see Figure 1.13). 35 ∗ |c r s −c r o | large. Both the retailer’s prices and the manufacturer’s wholesale price are constant (see Figure 1.14). 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 μ r s Price c m =0.5 c r s =0.2 c r o =0.1 α=1 p r o p r s w Figure 1.13: Price when |c r s −c r o | is small 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 μ r s Price c m =0.5 c r s =0.3 c r o =0.1 α=1 p r o p r s w Figure 1.14: Price when |c r s −c r o | is large • Demand. ∗ |c r s −c r o | small. When μ r s is small, the retailer only generates positive demand in the low-cost channel (c r o ), and the demand decreases in absolute share (μ r s ) in the other channel. When μ r s is large, the retailer may obtain positive demand from the high-cost (c r s ) channel. The demand in the high-cost channel increases as the absolute share (μ r s ) increases (see Figure 1.15). ∗ |c r s − c r o | large. The retailer may only generate positive demand in the low- cost channel (c r o ), and the demand decreases in absolute share (μ r s ) in the other channel (see Figure 1.16). • Profit. ∗ |c r s −c r o | small. The manufacturer’s profit always decreases. The retailer’s profit first decreases until μ r s reaches certain level. Then the retailer gets positive demandfrombothchannels, hisprofitjumpssuddenlybutdecreasesinμ r s again. (See Figure 1.17.) 36 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0 0.02 0.04 0.06 0.08 0.1 μ r s Demand c m =0.5 c r s =0.2 c r o =0.1 α=1 d r o d r s 0 Figure 1.15: Demand when |c r s −c r o | is small 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0 0.02 0.04 0.06 0.08 0.1 μ r s Demand c m =0.5 c r s =0.3 c r o =0.1 α=1 d r o d r s 0 Figure 1.16: Demand when |c r s −c r o | is large ∗ |c r s −c r o | large. Both the retailer’s profit and the manufacturer’s profit decrease in μ r s . (See Figure 1.18.) 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.008 0.01 0.012 0.014 0.016 0.018 0.02 μ r s Profit c m =0.5 c r s =0.2 c r o =0.1 α=1 Π r Π m Figure 1.17: Profit when |c r s −c r o | is small 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.005 0.01 0.015 0.02 μ r s Profit c m =0.5 c r s =0.3 c r o =0.1 α=1 Π r Π m Figure 1.18: Profit when |c r s −c r o | is large As we discussed earlier,|c r s −c r o | in the comic book industry is relatively large. Retailers may incur large costs from physical stores and have to set high prices, p r s . Therefore, consumers will not purchase products through the physical channel, and retailers incur costs but get low (if any) benefits from this high-cost channel. Naturally, retailers should attractmoreconsumerstoonlinestoresandreducethesizesofphysicalstores(i.e.,asmaller μ r s ). For example, retailers can provide free digital samples for most comic books and make the online purchase, subscription, and return experiencemore customer-friendly. Moreover, 37 retailers can reduce the number of physical stores and decrease the selection of comic books in physical stores. It is also common for retailers to try to reduce the operating costs in physical stores. When c r s is close to c r o , retailers do not incur large costs in both channels. Therefore, retailers enjoy more freedom in setting prices in both channels and do not have to transfer more consumers from physical stores to online stores. It might be a sensible option for retailers to maintain moderate potential market shares in both channels (i.e., moderate μ r s and μ r o ). Recall that in Section 1.4.1 we discuss the case in which the manufacturer operates two channels: one physical store and one online store. The numerical simulations obtained here are closely related to the results from Section 1.4.1. In Section 1.4.1, we show that when the cost difference |c s −c o | is small, ˆ α is large, and the manufacturer is able to obtain positive demand from both channels when μ s is in a moderate range, h μ s ,μ s i . When the cost difference |c s −c o | is large, ˆ α is small, and the manufacturer is less likely to obtain positive demand from both channels. Whenμ is fixed in a moderate range, we find similar patternsinFigure1.21and1.22: when|c r s −c r o |issmall,theretailerobtainspositivedemand from both channels for all feasibleα; when|c r s −c r o | is large, the high cost channel (i.e., the retailer’s physical store) loses demand asα increases, and whenα reaches critical level, the retailer’s physical channel faces zero demand. When α is fixed, the cost difference |c r s −c r o | influences the size of interval h μ r s ,μ r s i in which both channels get positive demand: when|c r s −c r o | is small, h μ r s ,μ r s i is large, and the retailer obtains positive demand from both channels; when |c r s −c r o | is large, h μ r s ,μ r s i is 38 small, and the high cost channel (i.e., the retailer’s physical store) loses demand as μ r s decreases; when μ r s <μ r s , the retailer’s physical channel faces zero demand. 1.4.5.2 The manufacturer enters the market to compete with the traditional retailer. Kendall-Jackson (K-J) is a local wine producer in California. Traditionally, all K-J wines were sold in retail physical stores. Now, with the help of the Internet, consumers can also purchase wines from K-J’s online stores directly. Therefore, this represents the case in which the manufacturer enters the market with a direct sales channel, and we have μ r s +μ m o ≈ 1 and μ r s > 0.5. The substitution between K-J’s online store and the retailer’s physical store is moderate (e.g., γ =1). 1) Following an approach similar to that used in Section 1.4.5.1, we first fix μ r s and investigate the impact of γ. • Price. Both the retailer’s price (p r s ) and the wholesale price (w) decrease, and then stabilize as γ increases. ∗ |c r s −c r o | small. p m o increases and then stabilizes asγ increases (see Figure 1.19). ∗ |c r s −c r o | large. p m o first increases, then decreases, and finally stabilizes (see Fig- ure 1.20). • Demand. ∗ |c r s − c m o | small. The retailer may always observe positive demand from his physical store, even if γ is large (see Figure 1.21). 39 ∗ |c r s −c r o | large. The retailer observes positive demand only if γ is very small. As γ increases, demand in the manufacturer’s channel always increases, but the retailer gets less demand (see Figure 1.22). • Profit. Asγ increases, the manufacturer’s profit always increases, while the retailer’s profit always decreases. When γ is very large, the retailer may see no demand what- soever (see Figure 1.23, 1.24). IntheK-Jwine’sscenario,themanufacturerhasasmalleroperatingcost(c m o )compared to the retailer. The magnitude of cost difference does matter when two players choose their pricingstrategies,butitdoesnotaffectthewayinwhichγ influencestheprofit. Toincrease the profit, the retailer always prefers to differentiate his channel from the manufacturer’s online store, while the manufacturer prefers to make the channels more substitutable. For this particular type of products (i.e., wine), there are not many approaches that retailercanusetodistinguishhimself,andhehastoworkhardertodifferentiatehischannel whenthecostdifferenceislarger. Infact,itisratherconvenientforK-Jtomakewineprices the same in both channels—when the retailer launches special events, K-J can do the same thing online. 2) We now fix γ and study the impact of μ r s . • Price. Asμ r s increases, bothp r s andw decrease, andp m o increases. When|c r s −c m o | is large, all prices are constant ifμ r s is very small (see Figure G.3, G.4 in Appendix G). • Demand. In general, as μ r s increases, the demand in the manufacturer’s channel decreases and the retailer faces increasing demand. When |c r s −c m o | is large, the retailer may get zero demand ifμ r s is very small (see Figure G.1, G.2 in Appendix G). 40 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 γ Price c m =0.5 c r s =0.25 c m o =0.22 u r s =0.55 p m o p r s w Figure 1.19: Price when |c r s −c m o | is small 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 γ Price c m =0.5 c r s =0.25 c m o =0.05 u r s =0.55 p m o p r s w Figure 1.20: Price when |c r s −c m o | is large 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.02 0.04 0.06 0.08 0.1 γ Demand c m =0.5 c r s =0.25 c m o =0.22 u r s =0.55 d m o d r s 0 Figure 1.21: Demand when |c r s −c m o | is small 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.05 0.1 0.15 0.2 γ Demand c m =0.5 c r s =0.25 c m o =0.05 u r s =0.55 d m o d r s 0 Figure 1.22: Demand when |c r s −c m o | is large 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.005 0.01 0.015 0.02 γ Profit c m =0.5 c r s =0.25 c m o =0.22 u r s =0.55 Π r Π m Figure 1.23: Profit when |c r s −c m o | is small 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.01 0.02 0.03 0.04 0.05 γ Profit c m =0.5 c r s =0.25 c m o =0.05 u r s =0.55 Π r Π m Figure 1.24: Profit when |c r s −c m o | is large 41 • Profit. As μ r s increases, the manufacturer’s total profit decreases and the retailer’s profitincreases. When|c r s −c m o |islarge,theretailergetsnothingifμ r s istoosmall.(see Figure G.5, G.6 in Appendix G). In this scenario, the retailer can reduce c r s to increase his profit. In some sense, the manufacturer also benefits from this strategy, because she may be able to increase her profit through the wholesale contract when the retailer orders more. In general, as the retailer’s potential market share (i.e., μ r s ) increases, the retailer can be better-off, but the manufacturer will be worse-off. Therefore, there would be a contest between the two to attract more consumers to their respective sales channels. The retailer can enlarge the potential market demand by improving shopping experience (e.g., open stores in convenient locations, introduce face-to- face interactions with wine experts in stores) and providing more shopping opportunities (e.g., put complementary products along with the K-J wines), while K-J can provide more convenient and safe shipping options. K-J can also attract more consumers by announcing special warrantee options for wines sold in her online store. 1.4.5.3 The manufacturer gets into the market to compete with the bricks- and-clicks retailer. The apparel industry and the electronics industry are two of the most commonly seen industries fitting this scenario. However, these two industries are different in terms of the two sets of parameters used in our model. In the apparel industry, consumers usually do not distinguish very much the products from the two online channels. Therefore, the 42 substitutability between two online channels is moderate (e.g.,β =1). However, due to the specialattributesofclothing(e.g.,thefit,truecolor,patterns),thesubstitutabilitybetween onlinestoresandphysicalstoresismuchsmaller(e.g.,α=γ =0.2). Ontheotherhand, for some electronic products (e.g., laptops), the manufacturer usually provides more different configurations and adds more after-sale services for the similar products than the retailer. Therefore, the products sold in the manufacturer’s online store are highly differentiated from those sold by the retailer, so we may set, e.g.,β =γ =0.2 andα=10. Therefore, we investigate two special cases: 1) α=γ and 2) β =γ. 1) When α = γ, the substitution effect between the physical store and either online store is the same. We set α=γ =0.2, β =1, which means the substitution effect between online stores is quite high, while that between physical store and either online store is very small. If μ r s = μ r o , the result is trivial, so we assume that the two channels have different potential demand (e.g.,μ r s −μ r o =0.02). If we suppose that the retailer’s two channels lose demand at the same magnitude when the manufacturer enters the market, then the feasible region for μ m o is [0.27,0.4]. 2) When β = γ, the substitution effect between the manufacturer’s online store and either of two retailer’s channels is the same. We set β = γ = 0.2, α = 10, which means the substitution effect between the retailer’s two channels is quite high, while that between the manufacturer’s store and either retailer’s store is very small. We also assume that μ r s −μ r o =0.02. Again, we suppose that the retailer’s two channels lose demand at the same magnitude when the manufacturer enters the market, and obtain the feasible region for μ r s as [0.27,0.44]. From our numerical analysis we have the following observations: 43 • Price. In general, as μ m o increases, p m o decreases. When γ = β is small compared to α, all other prices are decreasing in μ m o . When γ =α is small compared to β, all other prices are increasing in μ m o (see Figure G.9, G.10 in Appendix G). • Demand. In general, as μ m o increases, the demand in the manufacturer’s channel increases and the retailer faces decreasing demand in both channels. When γ =β is small compared toα, the retailer may abandon the high-cost channel even if the high cost channel has higher potential demand (see Figure G.7, G.8 in Appendix G). • Profit. In general, asμ m o increases, the manufacturer’s total profit increases and the retailer’s profit decreases (see Figure G.11, G.12 in Appendix G). In both apparel and electronics industries, as μ m o increases, the manufacturer can act more like a monopolist and enhance the profit by marking down p m o . When γ = β is small compared to α, the cannibalization threat from the retailer is quite small, and the manufacturer can benefit from the retailer’s sales by setting a smaller wholesale price. Then, the retailer has more freedom to choose the best prices, which leads to lower overall prices. When γ = α is small compared to β, the cannibalization threat from the retailer is quite high, and the manufacturer does not see high benefits from the retailer’s sales. Therefore, the manufacturer may set a higher wholesale price and force the retailer to increasehisprices. Inbothindustries,themanufacturercanincreasetheprofitbyattracting more potential demand to her online store. For instance, in the electronics industry, the manufacturer can introduce special financing offers (such that consumers can pay over a longer time period). She can also make the repair and return process more convenient, so that consumers rely more on the manufacturer’s online store. 44 In the apparel industry, the manufacturer can standardize certain types of clothing and introduce limited version online only. Then, the realized demand in the manufacturer’s channel increases and the retailer faces decreasing demand in both channels. As a result, in the electronics industry the retailer may have to display fewer types of products in the physical stores(i.e., the high-cost channel), and focus more on online stores. In the apparel industry, although the retailer can still get positive profit from both channels, it is much easier and cheaper for him to display more articles in online stores and attract more consumers to the low cost channel. The retailer also wants to attract more consumers to his channels. As we mentioned earlier, the retailer’s online store is relatively easier to operate in terms of virtual product availability. Similartothemanufacturer,theretailer’sonlinestorehastodealwiththesizeandship- ping issues in the apparel industry. By providing a more convenient shopping environment and experience in physical stores, the retailer may increase his foot traffic. Introducing new arrivals and displaying more popular articles as soon as the manufacturer delivers them is also important for retailers to attract consumers in both apparel and electronics industries. 1.4.5.4 Comparisons of different industries As can be seen from the preceding analysis, different combinations of parameters can lead to completely different equilibrium channel selection and pricing strategies for both the manufacturer and the retailer. For different industries and different types of products, 45 themanufacturerandtheretailershouldcarefullyestimatethevaluesofsubstitutabilityand marketshareparametersacrossdifferentchannelsandchooseappropriateoptimalstrategies. 1.5 Conclusion In this paper, we investigate a competitive dual-channel supply chain with one manu- facturer and one retailer. We introduce two sets of parameters to characterize the market demand. By using surveys and behavior experiments, we show that both absolute demand coefficients and channel differentiation coefficients are important to describe consumer preferences for dif- ferent products through different channels in various industries. Therefore, both sets of parameters are essential to model the demand functions. Based on the assumptions on the two sets of parameters, we identify the optimal chan- nel selection and pricing strategies for the vertically integrated system. A higher level of potential demand in a given channel does not necessarily guarantee high realized demand, or rather, the choice of this channel. The channel differentiation coefficients can have a huge impact on optimal channel selections. The decision maker has to make a tradeoff between cannibalization and larger potential market demand. We note that cost differences are also important in deciding optimal channel choices and pricing strategies. When the associated costs in online and physical channels are very close, the decision maker may prefer dual channels and obtain positive profits from both channels. 46 In the decentralized system, the manufacturer can compete with the retailer directly, while the retailer has the option to utilize the dual channel strategy. Although many retail- ers use dual channels, the profitability of a second channel is always questionable. Many manufacturers introduce online shopping regardless of the industry and product attributes. However, based on different industry characteristics and cost parameters, their final market equilibrium structures can be very different. For several special cases, we analyt- ically identify the equilibrium channel selection and pricing strategies under different sets of parameters. We find the conditions under which the retailer prefers one channel over dual channels and the manufacturer chooses to enter the market. For more general cases, we use numerical simulations to obtain interesting insights. We observe that both sets of parameters and the cost difference are important. When the channel differentiation coef- ficients are fixed, a larger cost difference can lead to smaller total profits and even single channelstrategyfortheretailer. Whenthepotentialabsolutedemandcoefficientsarefixed, sometimes a higher substitution level between retailer’s channels may actually increase the retailer’s total profits. In fact, this counter-intuitive result comes from the impact of the absolute demand coefficients. Whenmanufacturerentersthemarket,shecanusewholesalepricetoinfluencetheprices setbytheretailer. Weobservethatwhenthecompetitionbetweenthemanufacturerandthe retailer is small, the manufacturer may reduce the wholesale price to give the retailer more freedom in choosing optimal pricing strategies, which in return gives the manufacturer a higherprofit. However,whenthetwoonlinechannelsaremuchmoresubstitutablecompared to the other channels, the manufacturer faces larger threat from the retailer’s online stores. 47 Evenwhenthepotentialdemandinthemanufacturer’schannelincreases, themanufacturer still increases the wholesale price and forces the retailer to increase prices in both channels. This action leads to a higher profit for the manufacturer and hurts the retailer. Manufacturers can also use online prices to compete with retailers, and even start price wars. When the manufacturer’s potential demand is quite small, she benefits more if the retailer orders more. Therefore, the manufacturer purposely marks up the online price and forces consumers to purchase from the retailer, which in return provides higher profit to the manufacturer. There are many possible directions to extend our analysis. We assume that the manu- facturer has infinite production capacity and all demand can be fulfilled. If the production capacity is limited, it would be interesting to see how is the capacity allocated among dif- ferent channels. The capacity level can also be the manufacturer’s private information. We also assume that all cost information is common knowledge, but in practice, both the manufacturer and the retailer may have their own hidden information. Thus, moving our model to the asymmetric information setting may provide valuable new insights. Moreover, the manufacturer in our research is only capable to use online channels; the option of dual channels may affect the manufacturer’s optimal pricing strategies as well. 48 Chapter 2 Capacity Investment and Allocation 2.1 Introduction Another interesting issue is related to quantity strategies made by manufacturers and retailers. After receiving the retailer’s order size, the manufacturer has to select his ca- pacity and determine the final delivery amounts to different channels. Since the capacity investment decisions are closely related to the actual cost structures faced by the manufac- turer, we look at two general types of capacity cost functions; namely, convex-increasing capacity cost functions and concave-increasing capacity cost functions. When the produc- tion materials and/or resources are limited, the manufacturer faces an increasing marginal cost (i.e., the capacity cost function is convex in production quantity). For instance, in the semiconductor industry, the resources (e.g., crystalline solids) are usually limited. It is rather costly to obtain additional raw materials for manufacturer. When the raw materials are abundant, the manufacturer benefits from economy of scale (i.e., the capacity cost func- tion is concave in production quantity). For instance, in the fast-moving consumer goods industry, the raw materials are usually not a problem, and the production is often planned based on economy of scale. Based on different cost structures, the manufacturer may face 49 threats from scarce capacity and possible benefits from economy of scale. These potential differences may affect the manufacturer’s decisions on the delivery amounts. The retailer makes her ordering decisions in line with her own interest. Consequently, she may under-order or over-order compared to the optimal centralized system’s quantity decision. The manufacturer offers penalty cost for each unit of unsatisfied order from the retailer, as a signal that he will not sell items in his channel at the retailer’s expense in the case of scarce capacity. However, this may induce the retailer to over-order, regardless of the manufacturer’s capacity investment decisions, and this can hurt the manufacturer. Therefore, it is very important to devise effective and easy-to-use contracts to resolve the issue. During the global economy downturn, it is also of great interest to point out how to maintain good and sustainable relationships between competitive manufacturers and retailers. In this chapter, we investigate the quantity decisions made by the manufacturer and the retailer. Section 2.2 provides a related literature review. In Section 2.3, we present our model and analyze the decentralized competitive supply chain under different cost scenarios in Section 2.4. We then investigate the centralized system and propose contracts to coordinate the supply chain in Section 2.5. Section 2.6 provides concluding remarks. All longer technical proofs are given in Appendix D. 2.2 Literature Reivew Competitive supply chain management issues have been carefully studied in the recent operations management literature. Here, we provide a review of the papers that we believe 50 are most closely related to ours, and refer the reader to Tsay and Agrawal (2004a, 2004b) and Boyaci (2005) for a recent overview of this vast literature. Our work is closely related to the field of inventory competition. Inventory competition has been studied extensively in newsvendor context (see, for example, Parlar and Goyal, 1984; Parlar, 1988; Lippman and McCardle, 1997; Ernst and Kouvelis, 1999; Smith and Agrawal, 2000; Netessine and Rudi, 2003; etc.) Recently,vanRyzinandKe(2011)considersasingleretailertradingwithanoligopolyof manufacturers whose products are substitutable. They show that competitive overstocking due to substitution counteracts the understocking induced by double marginalization. Netessine et al. (2006) considers the infinite-horizon repeated inventory competition between two retailers. They show that, under appropriate conditions, a stationary base- stock policy is the unique Nash equilibrium of the game. Mahajan and van Ryzin (2001) analyzes a model of inventory competition among n firms that provide competing, substi- tutable goods. Each firm chooses initial inventory levels for their good in a single period (newsvendor-like) inventory model. Customers choose dynamically based on current avail- ability, so the inventory levels at one firm affect the demand of all competing firms. They characterize the Nash equilibrium of the resulting stocking game and prove it is unique in the symmetric case. Boyaci (2005) is one of the first papers studying a multi-channel distribution system in which a manufacturer sells its product through an independent retailer, as well as through hiswholly-ownedchannel. Eachchannelhasalocalstochasticdemand,buttheproductsare substitutable. It explores the channel inefficiencies induced by the presence of simultaneous 51 verticalcompetition(double-marginalization)andhorizontalcompetition(substitutability), and shows an overall tendency for both channels to overstock due to substitution, which intensifies under increasing substitution rates. Following a similar setting with one manufacturer and one retailer, Geng and Mallik (2007) considers the case in which the manufacturer faces capacity constraints. They es- tablish the necessary condition for a manufacturer to undercut a retailer’s order and show that a manufacturer may deny the retailer of inventory even when the capacity is ample. In this paper, we also investigate the manufacturer-retailer multi-channel distribution system with substitution rates. Unlike the models described above, we consider a manufacturer whohascapacityinvestment choicesaswell. The manufacturer facesdifferent capacitycost structures and has to make a decision on the capacity size when he receives orders from the retailer. This combines both the capacity investment decisions and substitution effects. We show that cost structure has a huge impact on the channel equilibrium results. Another issue of interest in our analysis is capacity allocation. Cachon and Lariviere (1999b) considers a single supplier with limited capacity selling to several retailers who are privately informed of their optimal stocking levels. It evaluates several allocation mecha- nismsfortruth-elicitingproperties,andfindsthatsupplychainmightbebetteroffunderan allocation mechanism that induces retailers to over-order. Deshpande and Schwartz (2005) considersageneralizationoftheabovemodelusingbothpricingandallocationmechanisms. Unlikeourwork,allthesepapersonlystudytheallocationproblemsamongretailers. While Boyaci(2005)doesnotaddresscapacityallocationissues, GengandMallik(2007)considers capacity allocation along with inventory competition. They allow the manufacturer to 52 undercut the order size from the retailer. We also allow the manufacturer to underdeliver, but in our model we incorporate a penalty on unfulfilled orders. Furthermore, our work also relates to the supply chain coordination literature. Van Ryzin and Ke (2011) combines elements of both vertical and horizontal competition. How- ever, the manufacturer-retailer relation is a strict supplier-buyer relationship and, there- fore, does not involve multi-channel distribution. In such a setting, they show that retail- managed-inventory achieves coordination, and vendor-managed-inventory achieves coordi- nation in the limit as the number of competing manufacturers becomes large. A similar structure is studied in Anupindi and Bassok (1999). They consider a one-manufacturer- two-retailer supply chain without multi-channel distribution and explore the impact of cen- tralization of retail stocks. In the manufacturer-retailer distribution framework, Boyaci (2005) also investigates co- ordination mechanisms, and shows that most of the well-known, simple contracts fail to achieve coordination in this setting. An exception to this is an appropriately designed penalty contract which can, indeed, coordinate the supply chain, but is hard to implement. In search of practically more appealing coordination mechanisms, it proposes a novel two- part compensation-commission contract, whose terms depend on the retail channel sales. Geng and Mallik (2007) develops a simple contract, called the reverse revenue sharing contract, and show that along with a fixed franchise fee this contract can coordinate the decentralized supply chain. However, the suggested contract is not seen in real life and can be very difficult to implement. We propose an alternative, simpler contract with penalties and order limits to coordinate the supply chain. 53 2.3 Model Formulation We consider a model with one manufacturer and one independent retailer. The manu- facturer makes products at per unit cost c> 0 and distributes them through two possible channels: hisownonlinestoresandtheretailer’ssaleschannel. Throughthemanufacturer’s own online channel, a quantity of Q m is sold at a price p m > c . The retailer places an order of sizeQ o r , obtains the actual quantityQ r at a wholesale pricew, and sells at a retail price p r . The manufacturer pays a penalty cost b to the retailer for each undelivered unit. We assume that both selling prices, p m and p r , are exogenous. That is, the market prices are in equilibrium, and we can focus on the impact of other strategies. Let D m and D r denote the total demand faced by the manufacturer and the retailer, respectively. Consisting of the primary demand and the potential shifted demand from the other channel, the total respective demand functions are defined as follows: D m = ˜ D m +α r ( ˜ D r −Q r ) + , D r = ˜ D r +α m ( ˜ D m −Q m ) + , where ˜ D m and ˜ D r denote the primary demand for the manufacturer and the retailer, respectively. Both ˜ D m and ˜ D r are random, and we let f(x) and g(y) denote their respective density functions. For simplicity, we assume that ˜ D m and ˜ D r are independent and their properties are common knowledge. When consumers find shortage in one channel, certain fraction will choosethepurchasetheproductfromtheotherchannel. Letα r andα m denotethefractions of potential demand shift, where 0 < α r , α m ≤ 1. This type of demand model has been usedtostudyinventoryissuesinoperationsmanagementarea(see, e.g., Boyaci, 2005; Geng and Mallik, 2007; etc.). 54 Wenextassumethatthemanufacturercaninvestincapacityandobtainthecapacitysize ofK atcostψ(K)≥0,whereψ ′ (K)≥0forall0≤K <∞. Thiscoststructurecanbevery general; we will focus on two specific types,ψ ′′ (K)≥0 (convex) andψ ′′ (K)<0 (concave). Although the retailer does not know the choice of K, she knows the cost structure, ψ(·). We will discuss the two different scenarios in detail in later subsections. We start with the decentralized system and we use the following game sequence: The manufactureroffersacontract(w,b)withwholesalepricewandunitpenaltybonunfulfilled order quantities. After observing the contract, the retailer decides and reports her order size Q o r to the manufacturer. After obtaining this information, the manufacturer considers whether to accept the contract or reject it. If the manufacturer accepts the contract, he decides the size of production capacity K, and allocates Q m and Q r to the two different channels. The manufacturer cannot deliver more than the retailer asks for, i.e., Q r ≤ Q o r . For each unfulfilled unit, the manufacturer pays b to the retailer. Then, the selling season begins. When determining capacity allocation, the manufacturer may keep extra products for hisownchannelandonlyallocateapartialordersizetotheretailer. Inordertodemonstrate his willingness to cooperate with the retailer, the manufacturer offers the unit penaltyb for each unfulfilled unit. This penalty not only discourages the manufacturer from allocating less than the order size, but also encourages the retailer to order more, which can help to reduce the “double-marginalization effect” in traditional supply chain. However, we show thatthiscompetitivesystemstillsuffersfrominefficienciesinlateranalysis;thisisconsistent with Boyaci (2005). 55 Using backward approach, we can find the equilibrium of the game. Based on Q r o , the manufacturer solves the following optimization problem: max Qm,Qr,K Π m (Q m ,Q r ,K) = p m E[min{Q m ,D m }]+wQ r −c(Q m +Q r ) −ψ(K)−b(Q o r −Q r ) s.t. 0≤Q r ≤Q o r , Q m +Q r ≤K. (2.1) When ψ ′ (K) > 0 for all K > 0, it is easy to see that Q m +Q r ≤ K is always binding at optimality, so we can remove this constraint and replaceK byQ m +Q r in the objective function. We note that there may be cases in which ψ ′ (K) = 0 for K ∈ [K,K] and Q m +Q r <K at optimality. Under this circumstance, we assume that the manufacturer does not set up excess ca- pacity. This assumption is very natural when there is one type of products and one selling season, and it makes the binding constraint hold again. Hence, we can rewrite the opti- mization problem for the manufacturer as max Qm,Qr Π m (Q m ,Q r ) = p m E[min{Q m ,D m }]+wQ r −c(Q m +Q r ) −ψ(Q m +Q r )−b(Q o r −Q r ) s.t. 0≤Q r ≤Q o r . (2.2) To distinguish between the order size from the retailer and the actual delivery size to the retailer, we use Q o r and Q d r to denote the original order size and the actual quantity delivered, respectively. We show that different cost structures will have large impact on the equilibrium results. Thus, we analyze separately two specific cases, 1) ψ ′′ (K)≥0, and 2) ψ ′′ (K)<0. That is, we study the cases when the capacity investment cost is convex and when it is concave. Taking into account the actions taken by the manufacturer, the retailer responds by solving max Q o r Π r (Q o r )=p r E[min{Q r ,D r }]−wQ r +b(Q o r −Q r ). 56 Finally, the manufacturer chooses appropriate wholesale price w and penalty cost b to maximize his profit. 2.4 Analysis 2.4.1 Convex capacity investment cost: ψ ′′ (K)≥0 We begin with the case in which the manufacturer faces scarce capacity investment resources; i.e.,themanufacturerconstructthecapacityataconvexincreasingcostψ ′′ (K)≥ 0. Note that we treat the resource purchasing cost as part of the total capacity setup cost, and the variable cost c is simply the variable production cost. For example, in the semiconductor industry, the setup cost includes the money used to purchase crystalline solids. The setup cost is convex increasing with respect to production quantity due to the limited amount of raw materials. First,wesolvethemanufacturer’soptimizationproblem,andthenconsidertheretailer’s decision. 2.4.1.1 Manufacturer’s problem If ψ ′′ (Q m +Q r ) ≥ 0 (i.e., the marginal capacity cost is increasing in the production quantity), then we have ∂ 2 Πm ∂Q 2 m ≤ 0 for all (Q m ,Q r ). This enables us to show the following result, which is very useful in our subsequent analysis. Proposition 1.When capacity investment cost is convex, the unique relationship between Q m and Q r is guaranteed. 57 We will describe the relationship betweenQ m andQ r by using the threshold ˆ Q r ,deter- mined by ˆ Q r =            +∞, if p m −c>ψ ′ (Q r ), ∀Q r ≥0; 0, if p m −c<ψ ′ (Q r ), ∀Q r ≥0; ψ ′−1 (p m −c), otherwise. (2.3) If Q o r ≤ ˆ Q r , the relationship between Q m and Q r can be fully determined by the first- order conditions (FOC), (p m −c)−p m Z Qm 0 Z Qr+(Qm−x)/αr 0 f(x)g(y)dydx−ψ ′ (Q r +Q m )=0 for all Q r ≤Q o r . If Q o r > ˆ Q r , then we have the following result:        Q m solves FOC, Q r ≤ ˆ Q r , Q m =0, ˆ Q r <Q r ≤Q o r . Therefore, for any Q r ≤ ˆ Q r , the best response Q m (Q r ) can be determined by ∂Πm ∂Qm = 0. In other words, Q m and Q r satisfy the condition R Qm 0 R Qr+(Qm−x)/αr 0 f(x)g(y)dydx+ ∂ψ(Qm+Qr) ∂Qm 1 pm = pm−c pm . By the implicit function theorem, we get dQ m (Q r ) dQ r =− α r h R Qm 0 f(x)g(Q r + Qm−x αr )dx+ψ ′′ (Q m +Q r )/p m i α r f(Q m )G(Q r )+ R Qm 0 f(x)g(Q r + Qm−x αr )dx+α r ψ ′′ (Q m +Q r )/p m . Both the denominator and the numerator are positive in the above expression, and the denominator is larger than the numerator. Consequently, we can get a bound on dQm(Qr) dQr ; that is, −1≤ dQm(Qr) dQr <0. 58 Since the relationship between Q m and Q r is uniquely determined, we can rewrite the manufacturer’s objective function as a function of Q r . Given the order size Q o r from the retailer, the manufacturer can solve his optimization problem. There are two possible cases we need to consider separately, Q o r ≤ ˆ Q r and Q o r > ˆ Q r . Case 1. Q o r ≤ ˆ Q r . IfQ o r ≤ ˆ Q r ,theretailerorderslessthanthethresholdquantity,andtheFOCissufficient to determine the relationship betweenQ m andQ r at optimality. By the envelope theorem, dΠm dQr = (w−c+b)−p m ∂A ∂Qr + ∂B ∂Qr −ψ ′ (Q m (Q r )+Q r ) = (w+b)−p m [1−(1−α r )L 1 −α r F(Q m (Q r ))G(Q r )], where L 1 = Z Qm 0 Z Qr+(Qm−x)/αr 0 f(x)g(y)dydx. The question now is, will the manufacturer always deliver full order size to the retailer, or is it possible that the manufacturer allocates less than the amount requested by the retailer, maybe even nothing? If (w+b)≥p m , then dΠm dQr >0. Therefore, the manufacturer delivers whatever the retailer orders (i.e., Q d r = Q o r ). In other words, when the marginal benefitfromwholesalecontractislargerthanthepotentialmarginallossinhisownchannel, the manufacturer will always deliver the amount required by the retailer. Note that, when the penalty cost b is large (e.g., w = p m ), the above inequality always holds, and the manufacturer will always deliver the full-size order. However, if (w +b) < p m , dΠm dQr may not be positive and monotone with respect to (w.r.t.) Q r for all Q r ≤ Q o r . Thus, the manufacturer may or may not deliver the entire order to the retailer. As we can see, the sign of d 2 Πm dQrdQr is unclear given dQm dQr <0. 59 For many common distributions (such as normal, gamma, or uniform), both L 1 and F(Q m )G(Q r ) are concave and not monotone w.r.t. Q r . Note that this is quite different from the results obtained by Geng and Mallik (2007). One of the reasons is the impact of a convexcapacityinvestmentcost. InGengandMallik(2007),F(Q m )G(Q r )ismorelikelyto be an increasing function of Q r without the influence of the capacity investment decisions. The detailed analysis of different scenarios is provided in Appendix D. For brevity and better flow, we summarize our results in Propositions 2–4. Each of the propositions identifies conditions for different delivery decisions for the manufacturer. Before we state the propositions, we define ˜ Q m and ˜ Q i r as follows: Q m = ˜ Q m solves ∂Π m ∂Q m Qr=0 =(p m −c)−p m Z Qm 0 Z (Qm−x)/αr 0 f(x)g(y)dydx−ψ ′ (Q m )=0; (2.4) Q r = ˜ Q i r solves dΠ m dQ r =(w+b)−p m [1−(1−α r )L 1 −α r F(Q m (Q r ))G(Q r )], (2.5) and dΠm dQr | Qr> ˜ Q i r <0. First, we can characterize the conditions under which the manufacturer always fulfills the entire order. Proposition 2. Suppose that the retailer orders Q o r ≤ ˆ Q r . The manufacturer delivers full order size to the retailer if one of the following holds: 1. (w+b)≥p m ; 2. (w+b)≥α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )), and (i) d 2 Πm dQrdQr >0, or (ii) dΠm dQr first increases then decreases w.r.t. Q r , and Q r o ≤ ˜ Q i r ; 60 3. (w+b)<α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )) and Π m (Q m (Q o r ),Q o r )>max{Π m ( ˜ Q m ,0),0}, and (i) d 2 Πm dQrdQr >0, or (ii) dΠm dQr first increases then decreases w.r.t. Q r , and Q r o ≤ ˜ Q i r ; where ˜ Q m and ˜ Q i r are defined in Equations (2.4) and (2.5). We also show the cases in which the manufacturer may deliver nothing in the following proposition. Proposition 3. Suppose that the retailer orders Q o r ≤ ˆ Q r . The manufacturer delivers nothing to the retailer if (w+b)<α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )) and one of the following is true: 1. d 2 Πm dQrdQr >0 and Π m ( ˜ Q m ,0)≥max{Π m (Q m (Q o r ),Q o r ),0}; 2. dΠm dQr first increases then decreases, and Π m ( ˜ Q m ,0)≥max{Π m (Q m (Q o r ),Q o r ),Π m (Q m ( ˜ Q i r ), ˜ Q i r ),0}; where ˜ Q m and ˜ Q i r are defined in Equations (2.4) and (2.5). Moreover, the manufacturer may fulfill the order partially. Proposition 4. Suppose that the retailer orders Q o r ≤ ˆ Q r . The manufacturer delivers partial order size to the retailer if dΠm dQr first increases then decreases w.r.t. Q r and one of the following holds: 1. (w+b)≥α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )) and Q r o > ˜ Q i r ; 2. (w+b)<α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )), Q r o > ˜ Q i r , and Π m (Q m ( ˜ Q i r ), ˜ Q i r )>max{Π m ( ˜ Q m ,0),0}; where ˜ Q m and ˜ Q i r are defined in equations (2.4) and (2.5). 61 Case 2. Q o r > ˆ Q r . Another possible caseisQ o r > ˆ Q r ; that is, the retailerordersmorethan the thresholdvalue. Then, we have ∂Πm ∂Qm <0 and Q d m =0 for Q r ∈( ˆ Q r , Q o r ]. Moreover, for Q r ∈( ˆ Q r , Q o r ], we get dΠ m dQ r =(w−c+b)−ψ ′ (Q r ), d 2 Π m dQ r dQ r =−ψ ′′ (Q r )≤0. This leads to the following result. Proposition 5. If (w−c+b)>ψ ′ (Q o r ), then dΠm dQr >0 for Q r ∈( ˆ Q r , Q o r ]. The manufacturer delivers Q o r , when one of the following is satisfied: 1. (w+b)≥p m ; 2. (w+b)<p m and (w+b)≥α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )); 3. (w+b)<α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )) and Π m (0,Q o r )≥0. In real life, we may find situations in which the first condition holds. For example, the manufacturermaychooseb=p m toprotectgoodrelationshipwiththeretailerbyremoving the incentive for selling in her channel while not delivering to the retailer. In this case, (w+b)≥p m is always true as long asw≥0, which is quite natural, and the manufacturer allocates the entire order size requested by the retailer. When the marginal capacity cost is large, the manufacturer may not benefit from allo- cating full order to the retailer; when the cost is in the moderate range, the manufacturer may prefer to deliver partial order size instead; and finally, when the cost is very large, the manufacturer may just deliver nothing. These results are summarized in the following proposition. 62 Proposition 6. If (w−c+b)<ψ ′ ( ˆ Q r ), then dΠm dQr <0 for Q r ∈( ˆ Q r , Q o r ]. 1. When dΠm dQr increases for Q r ∈(0, ˆ Q r ], then the manufacturer delivers nothing. 2. When dΠm dQr first increases then decreases for Q r ∈(0, ˆ Q r ], the manufacturer delivers partial size ˜ Q i r (which solves equation (2.5)) when one of the following conditions holds: a) (w+b)≥α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )); b) (w+b)<α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )) and Π m (0, ˜ Q i r )≥0. Even if the benefit from the contract is substantial for the manufacturer, there are still some cases in which he allocates partial orders. Interestingly, there is another possible case inwhichthemanufacturerdeliverspartialsizes. Wecharacterizetheresultsinthefollowing proposition. Proposition 7. Suppose there exists some ˆ Q i r ∈( ˆ Q r , Q o r ] such that (w−c+b)=ψ ′ ( ˆ Q i r ). (2.6) When one of the following conditions holds, the manufacturer allocates ˆ Q i r to the retailer: 1. (w+b)≥p m ; 2. (w+b)<p m and (w+b)≥α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )); 3. (w+b)<α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )) and Π m (0, ˆ Q i r )≥0. It is quite interesting to note that, even when the marginal benefit from allocating more products to the retailer is larger than selling the products in his own channel, the manufacturer may not do so. However, as we will show later, this can not be a good option in terms of profits. 63 2.4.1.2 Retailer’s problem The retailer’s objective is to choose the optimal order size, Q o r . The retailer’s objective function can be rewritten as Π r (Q o r )=(p r −w)Q r −p r ( ¯ A+ ¯ B)+b(Q o r −Q r ), where ¯ A = R Qm 0 R Qr 0 (Q r −y)f(x)g(y)dydx, ¯ B = R Qm+Qr/αm Qm R Qr−αm(x−Qm) 0 [Q r −y−α m (x−Q m )]f(x)g(y)dydx. Considering that Q r ∈{0, Q i r , Q o r }, the retailer decides the actual order quantity, Q o r . Basedondifferentprice-costpairs, themanufacturerwillrespondtotheretailer’sordersize differently. Beforeproceedingwithmoredetailedanalysis,wedefineseveralpossiblesolutionstothe retailer’s problem (for more details, please see Table E.1 in Appendix E). Q d m and Q d r are theamountsallocatedtothemanufacturer’schannelandtheretailer’schannel,respectively, and Q o r is the order size from the retailer. We use “hat” ( ˆ Q r ) when Q d m = 0 (that is, the manufacturerdoesnotdeliveranythingtohischannel),and“tilde”( ˜ Q r )whenQ d m >0(that is, the manufacturer delivers a positive quantity to his channel). Superscriptod denotes the retailer’s order size when the manufacturer delivers full quantity, superscript odi denotes the retailer’s order size when the manufacturer delivers partial (but positive) quantity, and superscript od0 denotes the retailer’s order size when the manufacturer delivers nothing. The analysis of existence and uniqueness of each solution is very tedious and is omitted here due to limited space and better content flow. We provide more detailed results of 64 possible solutions in Appendix E. Based on the possible reactions from the manufacturer, the retailer derives the optimal ordering strategies. We summarize the equilibrium results as Q d r , Q o r , the manufacturer’s delivery size and the retailer’s optimal order size, in the following Theorem. Recall that ˆ Q r is defined by (2.3). Theorem 1. When one of the following two conditions hold: 1. (w+b)≥p m ; 2. (w+b)<p m , (w+b)≥α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )) and dΠm dQr increases, then the equilibria are given as ˜ Q od r ≤ ˆ Q r ˜ Q od r > ˆ Q r (w−c+b)≥ψ ′ ( ˆ Q od r ) (w−c+b)<ψ ′ ( ˆ Q od r ) ˜ Q od r , ˜ Q od r ˆ Q od r , ˆ Q od r ˆ Q i r , ˆ Q odi r When the manufacturer sets the penalty b =p m , the first condition in this proposition always holds, and we obtain very straightforward equilibrium results. As discussed earlier, this might be a viable option when the manufacturer wants to signal to the retailer that he willnotcoverhisdemandattheexpenseoftheretailerinthecaseofscarceresources. If, on theotherhand,thetwoconditionsfailtohold,theequilibriumbecomesrathercomplicated; fordetails,pleaseseetheanalysisinAppendixE.Wenowanalyzeanotherplausiblecapacity cost structure, with concave capacity investment cost. 2.4.2 Concave capacity investment cost: ψ ′′ (K)<0 When ψ ′′ (Q m +Q r ) < 0, the manufacturer faces diminishing marginal capacity cost. This type of capacity setup cost is commonly seen in various industries; for instance, in 65 the fast-moving consumer goods industry, the raw materials are abundant, and the man- ufacturers usually enjoy the economy of scale in production. Similarly as before, we first consider manufacturer’s decision. 2.4.2.1 Manufacturer’s problem Before proceeding with the detailed analysis, we use the following theorem to identify the limiting structure of ψ ′ and ψ ′′ . Theorem 2. Suppose φ(x) is twice-differentiable. If φ(x)≥0 and φ ′ (x)<0 for all 0<x<∞, then lim x→∞ φ(x)=C 0 ≥0 and lim x→∞ φ ′ (x)=0. Recall that it follows from the manufacturer’s problem introduced by (2.2) that ∂Π m ∂Q m =(p m −c)−p m Z Qm 0 Z Qr+(Qm−x)/αr 0 f(x)g(y)dydx−ψ ′ (Q m +Q r ). Givenψ ′ (K)≥0andψ ′′ (K)<0for0≤K <∞,weknowthatlim K→∞ ψ ′ (K)=C o ≥0 and lim K→∞ ψ ′′ (K) = 0. Hence, ∂ 2 Πm ∂Q 2 m ≤ 0 when Q m goes to ∞. More precisely, if this is the case, we know that ∂Π m ∂Q m =        (p m −c)−ψ ′ (Q r ), Q m =0 −c−C o , Q m =∞ , and ∂ 2 Π m ∂Q 2 m =        −p m f(0)G(Q r )−ψ ′′ (Q r ), Q m =0 −p m [f(∞)G(Q r )+g(∞)], Q m =∞ . Since ∂Πm ∂Qm | Qm=∞ < 0, the manufacturer never allocates infinite number of products to his own channel, regardless of the order size from the retailer. 66 If we assumeψ ′ (0)≤(p m −c), then the inequalityψ ′ (K)≤(p m −c) holds for allK >0. Therefore, (p m −c)−ψ ′ (Q r )>0 holds for allQ r ≤Q o r , and the optimal allocated quantity shouldbepositive(i.e.,Q m >0). Inotherwords, whenp m >c+ψ ′ (Q r )holds, themarginal benefit from the production is larger than the marginal cost, and the manufacturer prefers to make some products for his own channel. What if the opposite is true; i.e., ψ ′ (0)>(p m −c)? Compared to the convex capacity cost scenario, it is quite natural to believe that we should have a threshold ˆ Q r , similar to that defined by (2.3), such that whenever Q r ≤ ˆ Q r , the manufacturer never allocates products to his own channel. However, this is not always truewhenthecostisconcave. Themanufacturermayactuallyproducealargequantityand allocate products to his own channel if he can benefit from economies of scale. We again define the threshold, ˆ Q r , such that the manufacturer allocates products to both channels when Q r > ˆ Q r . For certain scenarios in which the manufacturer always allocates products to both channels, we let ˆ Q r =0. Although we may get the relationship between the two allocation sizes,Q m andQ r , by solving the FOC, it is not quite clear whether the allocation size in one channel increases when that in the other channel increases. In fact, the sign of ∂ 2 Πm ∂Q 2 m is not determined, hence the uniqueness of best response Q m (Q r ) characterized by FOC is not guaranteed, which complicates the analysis. In order to make the analysis more tractable, we assume that ∂ 2 Πm ∂Q 2 m is negative for all (Q r ,Q m ) (we will relax this assumption later). 67 Thus, dQ m (Q r ) dQ r =− α r h R Qm 0 f(x)g(Q r + Qm−x αr )dx+ψ ′′ (Q m +Q r )/p m i α r f(Q m )G(Q r )+ R Qm 0 f(x)g(Q r + Qm−x αr )dx+α r ψ ′′ (Q m +Q r )/p m . When dQm(Qr) dQr <0, we have a bound−α r ≤ dQm(Qr) dQr <0, and the following proposition characterizes the sign of dQm(Qr) dQr . Proposition 8. Suppose ∂ 2 Πm ∂Q 2 m <0 for all (Q r ,Q m ). Then, the relationship between Q m and Q r is uniquely determined by the FOC if the manufacturer allocates products to both channels. For most cost functions and most unimodal distributions (with slight restrictions), there exists unique Q r such that dQm(Qr) dQr ≤0 for Q r <Q r , and dQm(Qr) dQr ≥0 for Q r >Q r . Oneinterestingobservationisthatthemanufacturermayactuallyincreasetheallocation sizes in both channels, which is quite different from the convex capacity investment cost case. Since for most unimodal distribution the density function approaches zero as the random variable goes to infinity, we can further show that dQm(Qr) dQr goes to zero and Q m stabilizes as Q r becomes larger. Let us first look at the possible reactions of the manufacturer given the retailer’s order size Q o r . Case 1. Q o r ≤ ˆ Q r . If Q o r ≤ ˆ Q r , then Q m =0. So, for Q r ≤Q o r , we get Q d m =0 and dΠ m dQ r =(w−c+b)−ψ ′ (Q r ), d 2 Π m dQ r dQ r =−ψ ′′ (Q r )>0. 68 Proposition 9. When the retailer’s order size is small, Q o r ≤ ˆ Q r , the manufacturer makes the following decisions: 1. If (w−c+b)−ψ ′ (0)≥0, then the manufacturer delivers the full order size, Q d r =Q o r . 2. If (w−c+b)−ψ ′ (Q o r )<0, then the manufacturer allocates nothing to the retailer. 3. If there exists ˆ Q i r ∈(0, Q o r ] solving equation (2.6), then the manufacturer delivers full order Q o r only when Π m (0,Q o r )≥0. However, the retailer’s order size can exceed ˆ Q r , in which case the manufacturer has to get prepared for a large order quantity. Case 2. Q o r > ˆ Q r . If Q o r > ˆ Q r , then ∂Πm ∂Qm = 0 at optimality for Q r ∈ ( ˆ Q r , Q o r ]. By the envelope theorem, for Q r ∈( ˆ Q r , Q o r ], we get dΠm dQr = (w−c+b)−p m ∂A ∂Qr + ∂B ∂Qr −ψ ′ (Q m (Q r )+Q r ) = (w+b)−p m [1−(1−α r )L 1 −α r F(Q m (Q r ))G(Q r )]. We define L 1 = Z Qm 0 Z Qr+(Qm−x)/αr 0 f(x)g(y)dydx. Proposition 10. When the retailer orders more than the critical quantity (i.e., Q o r > ˆ Q r ), the manufacturer delivers full order size if (w+b)≥p m , given Π m (0,Q o r )≥0. Proposition 11. When dΠm dQr first decreases and then increases w.r.t. Q r for Q r ∈( ˆ Q r , Q o r ], 69 1. If (w−c+b)−ψ ′ (0)≥0, the manufacturer only chooses Q d r from { ˜ Q i1 r , Q o r }; 2. If (w−c+b)−ψ ′ (0)<0, the manufacturer chooses Q d r from {0, ˜ Q i1 r , Q o r }; where ˜ Q i1 r is defined by the following equation (w+b)=p m [1−(1−α r )L 1 −α r F(Q m ( ˜ Q i r ))G( ˜ Q i r )]. (2.7) Note that if such ˜ Q i r which satisfies (2.7) does not exist, then this will not be chosen by themanufacturerasanoptimalallocationstrategy; pleaseseeAppendixFformoredetailed analysis. Themanufacturerbehaviorwhenthecapacitycostisconcaveisalmosttheoppositefrom his decisions when the capacity cost is convex. That is, whenQ o r is small, the manufacturer may only deliver products to satisfy the demand in the retailer’s channel. We also show that the manufacturer now benefits from economies of scale and always allocate positive quantities to both channels when the retailer’s order size Q o r is large. Although the manufacturer faces a different cost structure, the partial delivery (or even zero delivery) can still be optimal for him; we identify above the conditions under which the manufacturer chooses full delivery as the optimal strategy. Similarlytothecasewithconvexcost, (w+b)≥p m isoneoftheconditionsthatensures full delivery—when the penalty b is large, the manufacturer suffers a significant loss from anyunfulfilledunit. Forinstance,whenb=p m ,thelossisequivalenttothehighestpossible revenue the manufacturer can obtain from selling this item himself. We next analyze the retailer’s problem. 70 2.4.2.2 Retailer’s problem The retailer’s objective is to choose the optimal order size, Q o r ; her objective function can be rewritten as Π r (Q o r )=(p r −w)Q r −p r ( ¯ A+ ¯ B)+b(Q o r −Q r ), where ¯ A = R Qm 0 R Qr 0 (Q r −y)f(x)g(y)dydx, ¯ B = R Qm+Qr/αm Qm R Qr−αm(x−Qm) 0 [Q r −y−α m (x−Q m )]f(x)g(y)dydx. Case 1. If Q o r ≤ ˆ Q r , then Q m =0, and for Q r ≤Q o r , we have Q d m =0 and dΠ m dQ r =(w−c+b)−ψ ′ (Q r ), d 2 Π m dQ r dQ r =−ψ ′′ (Q r )>0. This leads to the following result. Theorem 3. The retailer may order ˆ Q od r if ˆ Q od r < ˆ Q r . The manufacturer delivers full order ˆ Q od r if one of the following conditions holds: 1. (w−c+b)−ψ ′ (0)>0; 2. (w−c+b)−ψ ′ (0)≤0 but ˆ Q i r < ˆ Q od r and Π m (0, ˆ Q od r )≥0. Case 2. If Q o r > ˆ Q r , then ∂Πm ∂Qm =0 at optimality for Q r ∈( ˆ Q r , Q o r ]. By the envelope theorem, for Q r ∈( ˆ Q r , Q o r ], we have dΠm dQr =(w+b)−p m [1−(1−α r )L 1 −α r F(Q m (Q r ))G(Q r )], where L 1 = Z Qm 0 Z Qr+(Qm−x)/αr 0 f(x)g(y)dydx. If (w+b)≥p m , then dΠm dQr >0, and the result is similar to that in Theorem 3. However, if (w+b)<p m , the sign of dΠm dQr is not negative for allQ r ∈( ˆ Q r , Q o r ]. In Proposition 8, we identify critical point Q r such that dQm dQr ≥0 for Q r >Q r , and dQm dQr ≤0 for Q r <Q r . 71 If Q r < ˆ Q r , then dQm dQr ≥ 0 and d 2 Πm dQrdQr > 0 for Q r > ˆ Q r , and the result is similar to that in Theorem 3. In other words, when selling to the retailer is more profitable, the manufacturerallocateshertheentireordersize; whensellingtotheretailerisnotprofitable enough, but the benefit from economies of scale from the two channels is substantial, the manufacturer delivers full order to the retailer as well. Corollary 12. The retailer may order ˜ Q od r if ˜ Q od r > ˆ Q r . The manufacturer delivers ˜ Q od r if at least one condition from set A and one condition from set B are true: A: 1. (w+b)≥p m ; or 2. Q r < ˆ Q r ; B: 1. (w−c+b)−ψ ′ (0)>0; or 2. (w−c+b)−ψ ′ (0)≤0 and Π m ( ˜ Q od r , ˜ Q od r )≥0. In real life, we may find situations in which the first condition, (w +b) ≥ p m , holds. For example, the manufacturer may choose b = p m to protect good relationship with the retailer by removing the incentive for selling in his own channel at the expense of shorting theretailer. Therefore,(w+b)≥p m istrueaslongasw≥0,whichisanaturalassumption, andthemanufacturerallocatestheentireorderrequestedbytheretailer. Forcompleteness, we provide more detailed analysis for the case (w+b)<p m in Appendix F. Notice that, if ˆ Q r =0,themanufactureralwaysallocatesproductstohisownchannel; underthisscenario, the analysis can be simplified substantially. 2.4.3 Comparison of different capacity cost functions As we can see from the above analysis, under different capacity cost structures (i.e., convex and concave), both the manufacturer and the retailer in general choose different quantity and allocation strategies. 72 Overall, when Q o r is small, the manufacturer may deliver products only to the retailer if the cost is concave; but he may deliver products only to his own channel if the cost is convex. When the retailer’s order size Q o r is large, the manufacturer always allocates positive quantities to both channels when the cost is concave; but he may focus on the retailer’ channel when the cost is convex. Although the manufacturer faces a different cost structure,weshowthatthepartialdelivery(orevenzerodelivery)canbeoptimalforhimby either cost structure. Based on the actions taken by the manufacturer, the retailer changes her optimal order quantities. Regardless of the cost function properties, (w+b)≥p m is one simple conditions under which the full order and full delivery strategy can be implemented. In more general situ- ations, the actual equilibrium results depend on the actual form of the cost function and demand types. Once we have obtained possible equilibrium order and allocation strategies for the de- centralizedsupplychainunderdifferentdemanddistributionandpriceparameters,wewant to study how they compare to the decisions made in the centralized system. More impor- tantly, we want to investigate if there exists any contracts that can be used to coordinate the competitive supply chain. This is the subject of our next section. 2.5 Obtaining a first-best outcome We first analyze, as a benchmark, the centralized model. Then, we develop a contact that can be used to coordinate the decentralized case. 73 2.5.1 Centralized case Suppose that the manufacturer owns the entire supply chain and acts as a central decision-maker. The manufacturer solves the following maximization problem: max Qm,Qr Π c (Q m ,Q r ) = p m E[min{Q m ,D m }]+p r E[min{Q r ,D r }] −c(Q m +Q r )−ψ(Q m +Q r ). FOCs give us ∂Πc ∂Qm = (p m −c)−p m R Qm 0 R Qr+(Qm−x)/αr 0 f(x)g(y)dydx−ψ ′ (Q m +Q r ) −p r R Qm+Qr/αm Qm R Qr+αm(Qm−x) 0 α m f(x)g(y)dydx, ∂Πc ∂Qr = (p r −c)−p r R Qr 0 R Qm+(Qr−y)/αm 0 f(x)g(y)dxdy −p m R Qr+Qm/αr Qr R Qm+αr(Qr−y) 0 α r f(x)g(y)dxdy−ψ ′ (Q m +Q r ). If we compare them with the FOCs in the decentralized case, we observe that the allocation size in the manufacturer’s online channel is now smaller, Q c m <Q d m , given fixed Q r . One possible reason is that the manufacturer, now as a central decision-maker, tries to avoid the unnecessary cannibalization between the two channels. However, the size change in the retailer’s channel is not straightforward. We can characterize further the conditions under which only one channel is profitable and when both channels are profitable. We show the results in the following proposition. Proposition 13. If pm pr <1−(1−α m ) R Q c r 0 R (Q c r −y)/αm 0 f(x)g(y)dxdy, the manufacturer never allocates products to his own channel; the manufacturer never allocates products to the retailer’s channel, if pr pm <1−(1−α r ) R Q c m 0 R (Q c m −x)/αr 0 f(x)g(y)dydx, where Q c r and Q c m solve 74 (p r −c)=p r Z Q c r 0 Z (Q c r −y)/αm 0 f(x)g(y)dxdy+ψ ′ (Q c r ), (p m −c)=p m Z Q c m 0 Z (Q c m −x)/αr 0 f(x)g(y)dydx+ψ ′ (Q c m ). Therefore, when both conditions above are not ture, both channels will be profitable. Then, the manufacturer allocates products to both channels. In addition, we obtain the relationship between Q c m and Q c r in the following proposition. Proposition 14. Given ψ ′′ >0, when p m ≥α m p r and p r ≥α r p m hold, the manufacturer allocates products to both channels but never increases the allocation sizes in both channels at the same time (compared to the decentralized system). It is interesting to see that two simple conditions can easily help to determine the sign of dQ c m dQ c r given the convex capacity setup cost. However, if ψ ′′ < 0, we may not have a straightforward relationship between Q c m and Q c r ; in fact, the manufacturer may increase the allocation sizes due to economies of scale. We next analyze the impact of different parameters in the model on the allocation decisions, and identify scenarios in which manufacturer allocate the same amount to both channels under different demand distributions. 2.5.1.1 Sensitivity analysis of the centralized system: 1. The impact of α m and α r on Q c m and Q c r . Given general distributions and arbitrary capacity cost functions, it is not possible to identify general impacts of the demand shift parameters, α m and α r . Therefore, we run 75 numerical simulations to analyze the results for more general cases. We also use specific distributions and cost functions to show analytical results. From the analysis in the centralized system, we note that the prices andα’s have signif- icant impact on the optimal allocation sizes. Therefore, we consider different relationships between selling prices in our analysis. If we assume ψ(K)= a 2 K 2 , we can derive the critical value of ˆ a satisfying the FOCs: (p m −c)−p m I 1 −p r [I 2 −F(Q)G(Q)]·α m = 2a·Q, (2.8a) (p r −c)−p r I 2 −p m [I 1 −F(Q)G(Q)]·α r = 2a·Q, (2.8b) where I 1 = Z Q 0 Z Q+(Q−x)/αr 0 f(x)g(y)dydx, I 2 = Z Q 0 Z Q+(Q−y)/αm 0 f(x)g(y)dxdy. a) p m >p r . In general, the optimal Q c m increases and Q c r decreases as α r increases. When α r is large enough, the manufacturer may allocate nothing to the retailer, Q c r = 0. When α r is small, both channels are allocated positive amount of products, and the total quantity Q T decreases as α r increases. In other words, as the potential demand coverage from the two channels increases, the total quantity gets smaller. However, when α r exceeds some threshold ˆ α r , the manufacturer may only deliver products to his own channel. Under this scenario, the higher the demand shift from the retailer’s channel, the more profits the manufacturer can obtain. In order to meet the demand shift,Q c m increases asα r increases, and thus the total Q T increases. These observations are illustrated in Figure 2.1 and 2.2. 76 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 100 200 300 400 500 α r Q m Q r Q T Figure 2.1: pm = 5, pr = 4, a = 0.005 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 50 100 150 200 250 300 350 α r Q m Q r Q T Figure 2.2: pm = 5, pr = 4, a = 0.01 Moreover, when a (in ψ(K) = a 2 K 2 ) gets larger, Q c m , Q c r and Q T become smaller for anyα r , while the threshold ˆ α r decreases asa increases. Ifa is very large, the manufacturer only allocates products to his own channel. Whenp m ≥α m p r ,wehave ∂Q c m ∂a ≤0;whenp r ≥α r p m ,wehave ∂Q c r ∂a ≤0. Notethatboth of these conditions are satisfied here, so both ∂Q c m ∂a and ∂Q c r ∂a are negative. That is, as the marginal setup cost decreases, the inventory allocated to each channel increases. Moreover, from the analysis in the centralized case, we can see that as long as either p m ≥ α m p r or p r ≥α r p m is satisfied (which is true in this case), ∂Q c m ∂Q c r ≤0 holds. 2) p m <p r . In general, the optimalQ c m increases andQ c r decreases asα r increases. In addition, the total quantity Q T gets larger when α r becomes larger, which is quite different from the results whenp m >p r . Whena gets larger,Q c m ,Q c r andQ T become smaller for anyα r (see Figure 2.3 and 2.4). 2. Equal allocation. Another interesting issue is when will the manufacturer allocate the same amount to both channels, even if they have different demand distributions. 77 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100 150 200 250 300 350 400 450 α r Q m Q r Q T Figure 2.3: pm = 4, pr = 5, a = 0.005 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 50 100 150 200 250 300 α r Q m Q r Q T Figure 2.4: pm = 4, pr = 5, a = 0.01 Note that the solution to (2.8), ˆ a, is not necessarily positive. When ˆ a is negative, the equal allocation is impossible. When ˆ a > 0, the manufacturer may find it interesting to allocate the same amount to each channel. The relationship between ˆ a and α m and that betweenQ andα m are depicted in Figure 2.5 and 2.6. Given the parameters, we have both p m ≥ α m p r and p r ≥ α r p m , which implies ∂Q c m ∂a ≤ 0 and ∂Q c r ∂a ≤ 0. In other words, as the marginal setup cost decreases, the inventory allocated to each channel increases. Moreover, ∂Q c m ∂Q c r ≤0 also holds in this case. 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.002 0.004 0.006 0.008 0.01 α r a Figure 2.5: pm = 5, pr = 4, αm = 0.1 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 160 180 200 220 240 260 280 300 320 α r Q Figure 2.6: pm = 5, pr = 4, αm = 0.1 When ˆ a>0, themanufacturercanbalancetheallocations. Asα r increases, higherlevel of potential demand shift from the retailer’s channel may lead to more inventory allocated to the manufacturer’s channel. To balance the allocation, the manufacturer needs a smaller 78 a and makes more products; i.e.,a decreases andQ c m =Q c r increases asα r becomes larger. We can also show that there never exists the case in which Q c r = Q c m for certain rage of price parameters. 2.5.2 Coordination One of the most important issues in supply chain management is how to coordinate the decentralized system. Suppose that, before signing the contract, both the manufacturer and the retailer agree on the constraint on order size; i.e., Q o r < ¯ Q r . When ¯ Q r = ∞, the analysis from Section 3 can be applied. However, when ¯ Q r < ∞, the retailer may not be abletoextracttoomuchbenefitfromthemanufacturerbyoverordering; ontheotherhand, the manufacturer may have incentives to allocate too many units to his own channel. In order to coordinate the system, we need more levers than can be obtained by simply using one of a well-known contracts, such as a buyback contract. Geng and Mallik (2007) proposes a reverse revenue sharing contract to coordinate a competitive supply chain with- out capacity constraints. In their coordination contract, they introduce three parameters: a fixed franchise fee, a penalty cost and a reverse revenue sharing fraction. This contract may be appropriate for an intra-firm coordination, but it may be impractical to use for a multi-channel supply chain system. A major difficulty in implementing this contract is to ask the manufacturer to announce the revenue truthfully and share it with the retailer. In practice, announcing the true revenue for specific products is not easy to verify (not en- forceable in court). The manufacturer does not have incentives to report the actual sales data to the retailer. In some sense, the retailer can play as a free rider, and it may be 79 beneficialforthemanufacturertowithholdmoreprofit. Tocopewiththeseimplementation difficulties, we propose a contract with quantity “caps” and penalty costs. The FOCs for the decentralized model with full order delivery can be written as ∂Πm ∂Qm = (p m −c)−p m R Qm 0 R Qr+(Qm−x)/αr 0 f(x)g(y)dydx−ψ ′ (Q m +Q r ), ∂Πm ∂Qr = (w−c+b)−p m R Qm 0 R Qr+(Qm−x)/αr Qr α r f(x)g(y)dydx−ψ ′ (Q m +Q r ), dΠr dQ o r = p r α m RQm+ Q o r αm Qm R Q o r −αm(x−Qm) 0 f(x)g(y)dydx· dQm(Q o r ) dQ o r +(p r −w)−p r R Q o r 0 RQm+ Q o r −y αm 0 f(x)g(y)dxdy. If we look at the centralized case, the FOCs are given as: ∂Πc ∂Qm = (p m −c)−p m R Qm 0 R Qr+(Qm−x)/αr 0 f(x)g(y)dydx−ψ ′ (Q m +Q r ) −p r R Qm+Qr/αm Qm R Qr+αm(Qm−x) 0 α m f(x)g(y)dydx, ∂Πc ∂Qr = (p r −c)−p r R Qr 0 R Qm+(Qr−y)/αm 0 f(x)g(y)dxdy −p m R Qr+Qm/αr Qr R Qm+αr(Qr−y) 0 α r f(x)g(y)dxdy−ψ ′ (Q m +Q r ). The manufacturer in the decentralized supply chain prefers a larger quantity allocated to his own channel. Thus, in order to coordinate this supply chain, we need another cap, ¯ Q m on Q m . When the order size Q o r exceeds the cap ¯ Q r , the retailer will pay u r for each unit over-ordered. When the allocation sizeQ m exceeds the cap ¯ Q m , the manufacturer will pay u r for each unit over-allocated to his own channel. While we acknowledge that, in practice, the last clause may not be easy to verify, we also observe that in practice we will not need to implement any of the two penalties, as neither party would want to overorder. However, introduction of these penalties enables as to use { ¯ Q r , ¯ Q m ,u r ,u m ,b} to coordinate the supply chain. 80 We rewrite the optimization problems for the manufacturer and the retailer as follows: max Qm,Qr Π m (Q m ,Q r ) = p m E[min{Q m ,D m }]+wQ r −c(Q m +Q r )−ψ(Q m +Q r ) −b(Q o r −Q r )−u m (Q m − ¯ Q m ) + +u r (Q o r − ¯ Q r ) + s.t. 0≤Q r ≤Q o r , max Q o r Π r (Q o r ) = p r E[min{Q r ,D r }]−wQ r +b(Q o r −Q r ) +u m (Q m − ¯ Q m ) + −u r (Q o r − ¯ Q r ) + . Let ¯ Q m =Q c m , ¯ Q r =Q c r , and u m =p r Z Q c m +Q c r /αm Q c m Z Q c r +αm(Q c m −x) 0 α m f(x)g(y)dydx, (2.9) b =(p r −w)−p r Z Q c r 0 Z Q c m +(Q c r −y)/αm 0 f(x)g(y)dxdy =(p r −w)− u m α m −p r F(Q c m )G(Q c r ), (2.10) u r =p m Z Q c r +Q c m /αr Q c r Z Q c m +αr(Q c r −y) 0 α r f(x)g(y)dxdy+ψ ′ (Q c m +Q c r ) +p r α m Z Q c m + Q c r αm Q c m Z Q c r −αm(x−Q c m ) 0 f(x)g(y)dydx· dQ c m (Q c r ) dQ c r −(w−c) =b+ dQ c m (Q c r ) dQ c r u m . (2.11) The existence of positive b and u r is not always assured for arbitrarily chosen price pa- rameters. Whatifbobtainedin(2.10)isnegative? Thiswouldimplythatthemanufacturer obtains |b| by not delivering one unit of product. However, this benefit may not be able to cover the loss of the margin, w−c. Note that, if b is negative, then ∂Πm ∂Qr b=0 > 0; i.e., the manufacturer is willing to deliver more. However, the order size Q o r from the retailer prevents the manufacturer from delivering more than requested. Therefore, in the contract, we may choose b=max n 0,(p r −w)− um αm o . 81 Wecanalsomanipulatethesignofu r bychoosingappropriatew;thekeyissueiswhether such w≥c exists. Note that the largest u r we can obtain is (p r −c)− 1 αm − dQ c m dQ c r u m . If this quantity is nonnegative, then we can always find a w such that u r is non-negative. As a matter of fact, this describes the tradeoffs when the retailer changes Q o r . Although for any given w the existence of positive penalties is not guaranteed, we can show that the manufacturer can always choose appropriate w such that nonnegative penalties exist, and the supply chain can be coordinated. The result is described in the following theorem. Theorem 4: Given α r p m ≤α m p r ≤p m , there always exists some w∈[c,p r ] such that b,u r ≥0. Using such wholesale contracts with penalties and order limits, { ¯ Q r , ¯ Q m ,u r ,u m ,b}, we can coordinate the decentralized system easily and implement the first-best optimal order quantities. 2.6 Conclusion In this paper, we investigate the impact of capacity investment and allocation in a competitive manufacturer-retailer supply chain system. Given exogenous selling prices, we focus on the capacity issues from the manufacturer’s side. Based on different capacity cost structures, the manufacturer alters the capacity choices and allocation quantities in both channels. We show that in both convex and concave capacity cost scenarios, the manufacturer may prefer to deliver partial order size, or even nothing, to the retailer, even when there is a penalty for each unfulfilled unit. When the capacity cost is convex, the 82 manufacturer may only allocate products to the retailers channel when he faces scarce production restrictions and the capacity cost increases dramatically. When the capacity cost is concave increasing, the manufacturer may only deliver products to the retailer’s channel when the entire quantities from both channels are small. As the retailer’s order size gets larger, the manufacturer can benefit from selling to his retailer and then allocate less to his own channel. When the order size continues to increase, the benefits from the economy of scale can induce the manufacturer to increase the allocations to both channels. When the retailer’s order size is very large, the manufacturer cannot obtain excess demand from the retailer, and chooses to deliver as many items as requested by the retailer, but allocates an almost constant amount (expected demand) to his own channel. The retailer also changes the equilibrium order sizes according to the manufacturer’s different capacity choices. Foreseeing the optimal capacity allocation strategies by the manufacturer, the retailer can cleverly choose to overorder in several cases, due to possi- ble penalty payments from the manufacturer. When the manufacturer suffers from large marginal capacity costs, the retailer does overorder and extracts all possible profits from the manufacturer. Although this does not benefit the manufacturer in terms of profit, the two players may sign the contract to maintain sustainable relationships. Moreover, even when obtaining zero profits, the manufacturer may still want to keep machines running and skilled workers employed during economic downturns. We also analyze the centralized supply chain, and show that the central decision maker may allocate equal quantities to both channels even if the demands and profit margins are quite different in the two channels. The allocation size in one channel increases as the 83 possible demand shift parameter for this channel increases. Compared to the decentralized system,wealsoshowthatthemyopicoverallocationissuescanberesolvedinthecentralized system. To coordinate the decentralized system, we propose contracts with penalty costs and order limits, which can be easily implemented in real life when there is a sustainable relationship between the manufacturer and the retailer. There are several interesting research questions that can further extend our work. Our model assumes that all selling prices are exogenous. However, when these prices can be manipulated by both players (which alters the demand), we may obtain additional insights closely related to real-life situations. In terms of demand sharing between channels, some researchers considered the cases in which the substitution rates between the manufacturer’s and the retailer’s channels are stochastic, which combines more practical issues but also complicates the analysis. 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Proof: The first order condition gives: ∂Π/∂p s =−2(1+α)μ s p s +αp o +c ′ 1 , ∂Π/∂p o =αp s −2(1+α)μ o p m o +c ′ 2 , withc ′ 1 =μ s [1+(1+α)c s ]−μ o αc o , c ′ 2 =μ o [1+(1+α)c o ]−μ s αc s , and the Hessian matrix is H = −2(1+α)μ s α α −2(1+α)μ o . Givenα≥0,μ s ,μ o ≥0,μ s +μ o =1,it’seasytoseethat−2(1+α)μ s ≤0,−2(1+α)μ o < 0. Under the assumption (1.7), we havedet(H)≥0. Therefore,H is negative semi-definite, and the objective function is concave. Proposition 6. The retailer’s objective function is concave with respect to (w.r.t.) the decision variables p r o and p r s . Proof: The total profit for the retailer is a function of p r o and p r s : Π r (p r s ,p r o |w,p m o )=Π r o (p r s ,p r o |w,p m o )+Π r s (p r s ,p r o |w,p m o ) where Π r o (p r s ,p r o |w,p m o ) = μ r o (p r o −w−c r o )[1−(1+α+β)p r o +αp r s +βp m o ] Π r s (p r s ,p r o |w,p m o ) = μ r s (p r s −w−c r s )[1−(1+α+γ)p r s +αp r o +γp m o ] We can write down the first order condition: ∂Π r /∂p r o = [−2μ r o (1+α+β)p r o +(μ r s +μ r o )αp r s ] +μ r o [βp m o +1+(1+α+β)(w+c r o )]−μ r s α(w+c r s ) ∂Π r /∂p r s = [(μ r s +μ r o )αp r o −2μ r s (1+α+γ)p r s ] +μ r s [γp m o +1+(1+α+γ)(w+c r s )]−μ r o α(w+c r o ) as well as the Hessian matrix: H r = ∂ 2 Π r ∂p r o ∂p r o ∂ 2 Π r ∂p r o ∂p r s ∂ 2 Π r ∂p r s ∂p r o ∂ 2 Π r ∂p r s ∂p r s ! = −2μ r o (1+α+β) (μ r s +μ r o )α (μ r s +μ r o )α −2μ r s (1+α+γ) 87 Given α>0, β >0 and γ >0, it’s apparent that ∂ 2 Π r ∂p r o ∂p r o =−2μ r o (1+α+β)<0, ∂ 2 Π r ∂p r s ∂p r s =−2μ r s (1+α+γ)<0, anddet(H r )=4μ r o μ r s (1+α+β)(1+α+γ)−(μ r s +μ r o ) 2 α 2 . Under the additional restrictions on μ’s we mentioned above, we have 2(1+α+γ)μ r s ≥α(μ r s +μ r o )+μ r s +γ(1−μ r o )≥α(μ r s +μ r o ) 2(1+α+β)μ r o ≥α(μ r s +μ r o )+μ r o +γ(1−μ r s )≥α(μ r s +μ r o ) Therefore, the determinant is positive and the problem is again concave w.r.t. the decision variables p r o and p r s . Proposition 7. For any given positive μ r s , μ r o and α, β, γ satisfying the non-increasing industry demand assumption and any feasible p m o , there exists ˆ w s.t. 1) when w< ˆ w, the retailer operates two channels; ii) the demand in each channel decreases as w increases; iii) the optimal prices are increasing w.r.t. both w and p m o . Proof: The critical magnitude of ˆ w depends on value of parametersμ r o ,μ r s ,α,β,γ and the man- ufacturer’s online price p m o . First, we can show that demand from the two channels are both decreasing in w. It’s easy to verify that ∂D r o ∂w = −((1+β)(1+γ)+α(2+β+γ))μ r s ((2+α+2β)μ r o −αμ r s ) det(H r ) , ∂D r s ∂w = −((1+β)(1+γ)+α(2+β+γ))μ r o ((2+α+2γ)μ r s −αμ r o ) det(H r ) . By using the constraints on μ’s, we have μ r s ≥ α−γ 1+α+2γ ·μ r o + γ 1+α+2γ ≥ α 1+α+2γ ·μ r o μ r o ≥ α−β 1+α+2β ·μ r s + β 1+α+2β ≥ α 1+α+2β ·μ r s which gives us ∂D r o ∂w < 0 and ∂D r s ∂w < 0. Therefore, we can always find ˆ w such that when w< ˆ w both demand functions are positive. Moreover, we have ∂p r o ∂w ·det(H r )=2(1+α+β)μ r s ((1+α+β)μ r o −αμ r s )+α(μ r o +μ r s )(−αμ r o +(1+α+γ)μ r s ), 88 ∂p r s ∂w ·det(H r )=2(1+α+γ)μ r o ((1+α+γ)μ r s −αμ r o )+α(μ r o +μ r s )(−αμ r s +(1+α+β)μ r o ). Since we know (1+α+γ)μ r s ≥αμ r o and (1+α+β)μ r o ≥αμ r s , then ∂p r o ∂w >0 and ∂p r s ∂w >0 must hold, and we have ∂p r o ∂p m o ·det(H r )=2β(1+α+γ)μ r o μ r s +αγμ r s (μ r o +μ r s )>0, ∂p r s ∂p m o ·det(H r )=2γ(1+α+β)μ r s μ r o +αβμ r o (μ r o +μ r s )>0. Proposition 9. When i=I, we have positive demand in both channels, the manufacturer’s objective function is concave w.r.t. w and p m o . Proof: The first order conditions are: ∂Π mI /∂w = D rI (w,p m o )+ ∂D rI (w,p m o ) ∂w (w−c m )+ ∂D mI o (w,p m o ) ∂w (p m o −c m −c m o ) ∂Π mI /∂p m o = ∂D rI (w,p m o ) ∂p m o (w−c m )+ ∂D mI o (w,p m o ) ∂p m o (p m o −c m −c m o )+D mI o (w,p m o ) And the Hessian matrix is H m = ∂ 2 Π m ∂w∂w ∂ 2 Π m ∂w∂p m o ∂ 2 Π m ∂p m o ∂w ∂ 2 Π m ∂p m o ∂p m o ! where we have ∂ 2 Π m ∂w∂w = 2[αμ r o −(1+α+γ)μ r s ] ∂p rI s ∂w +2[αμ r s −(1+α+β)μ r o ] ∂p rI o ∂w , ∂ 2 Π m ∂p m o ∂w = (μ r o β+μ r s γ)+μ m o γ ∂p rI s ∂w +μ m o β ∂p rI o ∂w +[μ r o α−μ r s (1+α+γ)] ∂p rI s ∂p m o +[μ r s α−μ r s (1+α+β)] ∂p rI o ∂p m o , ∂ 2 Π m ∂w∂p m o = (μ r o β+μ r s γ)+μ m o γ ∂p rI s ∂w +μ m o β ∂p rI o ∂w +[μ r o α−μ r s (1+α+γ)] ∂p rI s ∂p m o +[μ r s α−μ r s (1+α+β)] ∂p rI o ∂p m o , ∂ 2 Π m ∂p m o ∂p m o = 2μ m o h −(1+β+γ)+γ ∂p rI s ∂p m o +β ∂p rI o ∂p m o i . It’s easy to see that the objective function is concave w.r.t. w and p m o . When i = Bs, the retailer only faces positive demand in the retail store. And the first-order conditions are ∂Π mB1 /∂w = D rBs (w,p m o )+ ∂D rBs (w,p m o ) ∂w (w−c m )+ ∂D mB1 o (w,p m o ) ∂w (p m o −c m −c m o ), ∂Π mB1 /∂p m o = ∂D rBs (w,p m o ) ∂p m o (w−c m )+ ∂D mB1 o (w,p m o ) ∂p m o (p m o −c m −c m o )+D mB1 o (w,p m o ). 89 And the elements of the Hessian matrix are ∂ 2 Π m ∂w∂w = 2μ r s [−(1+α+γ) ∂p rBs s ∂w +α ∂p rBs o ∂w ], ∂ 2 Π m ∂p m o ∂w = μ m o (γ ∂p rBs s ∂w +β ∂p rBs o ∂w )+μ r s [−(1+α+γ) ∂p rBs s ∂p m o +α ∂p rBs o ∂p m o +γ], ∂ 2 Π m ∂w∂p m o = μ m o (γ ∂p rBs s ∂w +β ∂p rBs o ∂w )+μ r s [−(1+α+γ) ∂p rBs s ∂p m o +α ∂p rBs o ∂p m o +γ], ∂ 2 Π m ∂p m o ∂p m o = 2μ m o h −(1+β+γ)+γ ∂p rBs s ∂p m o +β ∂p rBs o ∂p m o i . Or more precisely, ∂ 2 Π m ∂w∂w = −μ r s · δ 0 (1+α+β) <0, ∂ 2 Π m ∂p m o ∂w = (μ m o +μ r s ) δ 0 −(1+2α+β) 2(1+α+β) >0, ∂ 2 Π m ∂w∂p m o = (μ m o +μ r s ) δ 0 −(1+2α+β) 2(1+α+β) >0, ∂ 2 Π m ∂p m o ∂p m o = −μ m o · 1 (1+α+β) [2δ 0 −2(α−β)+ (δ 0 −(1+2α+β)) 2 δ 0 ]<0. The determinant is 1 (1+α+β) 2 μ m o μ r s [δ 2 0 −2(α−β)δ 0 +s 2 1 ]− (μ m o +μ r s ) 2 4 ·s 2 1 , where s 1 = δ 0 −(1+2α+β) < 0. We know (μ m o +μ r s ) 2 4 ≥ μ m o μ r s > 0, but the sign of the determinant is not clear. When i = Bo, the retailer only faces positive demand in the online store. And the first-order conditions are ∂Π mB2 /∂w = D rBo (w,p m o )+ ∂D rBo (w,p m o ) ∂w (w−c m )+ ∂D mB2 o (w,p m o ) ∂w (p m o −c m −c m o ), ∂Π mB2 /∂p m o = ∂D rBo (w,p m o ) ∂p m o (w−c m )+ ∂D mB2 o (w,p m o ) ∂p m o (p m o −c m −c m o )+D mB2 o (w,p m o ). And the components of the Hessian matrix are ∂ 2 Π m ∂w∂w = 2μ r o [−(1+α+β) ∂p rBo o ∂w +α ∂p rBo s ∂w ], ∂ 2 Π m ∂p m o ∂w = μ m o (γ ∂p rBo s ∂w +β ∂p rBo o ∂w )+μ r o [−(1+α+β) ∂p rBo o ∂p m o +α ∂p rBo s ∂p m o +β], ∂ 2 Π m ∂w∂p m o = μ m o (γ ∂p rBo s ∂w +β ∂p rBo o ∂w )+μ r o [−(1+α+β) ∂p rBo o ∂p m o +α ∂p rBo s ∂p m o +β], ∂ 2 Π m ∂p m o ∂p m o = 2μ m o h −(1+β+γ)+γ ∂p rBo s ∂p m o +β ∂p rBo o ∂p m o i . Equivalently, ∂ 2 Π m ∂w∂w = −μ r s · δ 0 (1+α+γ) <0, ∂ 2 Π m ∂p m o ∂w = (μ m o +μ r s ) δ 0 −(1+2α+γ) 2(1+α+γ) >0, ∂ 2 Π m ∂w∂p m o = (μ m o +μ r s ) δ 0 −(1+2α+γ) 2(1+α+γ) >0, ∂ 2 Π m ∂p m o ∂p m o = −μ m o · 1 (1+α+γ) [2δ 0 −2(α−γ)+ (δ 0 +1+2α+γ) 2 δ 0 ]<0. 90 The determinant is 1 (1+α+γ) 2 μ m o μ r s [δ 2 0 −2(α−γ)δ 0 +s 2 2 ]− (μ m o +μ r s ) 2 4 ·s 2 2 , where s 2 = δ 0 −(1+2α+γ) < 0. We know (μ m o +μ r s ) 2 4 ≥ μ m o μ r s > 0, but the sign of the determinant is not clear as well. 91 Appendix B Survey Questions and Analysis Survey Questions Experiments to investigate the general assumptions on μ’s and α, β, γ. 1. Demographic background Gender, age, education level, household income, household size, online shopping fre- quency 2. Opinions on shopping methods (Five-level classification: strongly agree, somewhat agree, neutral, somewhat disagree, strongly disagree.) 1) I can shop in privacy of my home 2) I believe manufacturers often provide more varieties of the same brand 3) I am cautious in trying new products 4) I find shopping over the Internet to be a pleasant experience 5) I would rather obtain the product at store than wait for delivery 6) I appreciate a store with a pleasant atmosphere 7) I believe manufacturers often charge lower prices 8) I like to buy the same brand 9) I believe manufacturers often have fewer stock-outs 10) I like to have a great deal of information before I buy 11) I like to plan my purchases carefully 12) I can shop whenever I want 13) I can save a lot of time by shopping online 14) Shopping over the Internet is not a pleasant experience 15) While shopping online, I don’t have to wait to be served 16) While shopping online, I miss the experience of interacting with people 17) While shopping online, I may purchase something by accident 18) While shopping online, I may not find personal information to be safe 19) While shopping online, I find most items are available 20) While shopping online, I don’t feel embarrassed if I don’t buy 21) While shopping online, I can’t examine the actual product 22) While shopping online, I have broader selections of products 23) After shopping online, I may find the product to be damaged on the way 24) After shopping online, I may not get the product I purchased 92 3. Suppose the prices in different stores are the same, through which channel would you prefer to purchase the item? 1) Pantene Pro-V Shampoo 25.4 floz PGestore.com is the P&G official website selling products from P&G. Walmat.com is the Wal-Mart official website selling products for Wal-Mart. Pantene is a brand of P&G. Options: Wal-Mart, Walmart.com, PGestore.com, Not buy Prices: $6.00, $5.70 (5% off), $5.40 (10% off), $6.30 (5% up), $6.60 (10% up) 2) PUR 3-Stage Horizontal Faucet Stainless Water Filters PGestore.com is the P&G official website selling products from P&G. PUR is a brand of P&G. Options: Wal-Mart, Walmart.com, PGestore.com, Not buy Prices: $46.00, $43.70 (5% off), $42.32 (10% off), $48.30 (5% up), $49.68 (10% up) 3) DayQuil Cold&Flu 10 floz CVS.com is the CVS official website selling products for CVS. DayQuil is produced by P&G. Option a): CVS, CVS.com, PGestore.com, Not buy Prices: $6.00, $5.70 (5% off), $5.40 (10% off), $6.30 (5% up), $6.60 (10% up) Option b): Wal-Mart, Walmart.com, PGestore.com, Not buy Prices: $5.60, $5.32 (5% off), $5.04 (10% off), $5.88 (5% up), $6.16 (10% up) 4) Olay Professional Pro-X Eye Restoration Complex Target.com is the Target official website selling products for Target. Olay is a brand of P&G. Options: Target, Target.com, PGestore.com, Not buy Prices: $41.50, $39.43 (5% off), $37.35 (10% off), $43.58 (5% up), $45.65 (10% up) 5) Gillette Fusion Proglide Manual Gillette is a brand of P&G. Options: Target, Target.com, PGestore.com, Not buy Prices: $10.00, $9.50 (5% off), $9.00 (10% off), $10.50 (5% up), $11.00 (10% up) 6) Canon Digital Camera, Rebel T1i with EF 18-55mm IS Lens Canon.com is the Canon official website selling products for Canon. Bestbuy.com is the BestBuy official website selling products for BestBuy. Options: BestBuy, Bestbuy.com, Canon.com, Not buy Prices: $699.00,$664.05(5%off),$629.10(10%off),$733.95(5%up),$768.90(10%up) 7) HP Deskjet F4480 All-in-one Printer, Scanner, Copier HP.com is the HP offical website selling products for HP. Options: BestBuy, Bestbuy.com, HP.com, Not buy 93 Prices: $69.99, $64.99 (7% off), $59.99 (14% off), $74.99 (7% up), $79.99 (14% up) Analysis of μ’s For each group discussed in previous section, we calculate the percentages of total qual- ified responses for all relevant options (i.e., the μ’s in different cases). Then, we obtain the mean and standard deviation (std) in each case. In order to investigate the consistency, we also calculate the coefficient of variation (CV); a smaller CV means a more consistent result. We notice that CV is less than 10% in most cases. This implies that the μ’s estimated by our analysis are quite consistent with the assumption that μ’s are price independent. Moreover, for different types of products, we have different sets of μ’s. In Table B.1 and B.2, over the counter medicine (e.g., Dayquil) gives us largeμ r s ’s, with the highest values of all cases considered. Consumers may tend to purchase the medicine only when they need them immediately. In order to get the cure immediately and con- veniently, they may prefer the nearest retail store and pick the necessary products. We also notice that μ r s in the CVS case is larger than that in the Walmart case. This is also reasonable because CVS pharmacy is more specialized in providing medicines, and more conveniently located in neighborhoods than Walmart stores. InTableB.3, B.4and B.5,personalcareproducts(e.g.,Panteneshampoo,Olaycream, and Gillette razor) provide relatively smaller μ r s ’s compared to medicines, but their values are still larger than those of the remaining product categories. Because these products are daily necessities, many consumers still buy them along other items when they shop for regular daily/weekly needs in physical stores. Consumers can plan regularly and purchase these products before they run out of them, so some consumers may move from traditional physical stores to online shops. In Table B.6 and B.7, we notice that consumers tend to purchase electronics (e.g., HP personal printers/toners and Canon digital cameras) through online channels. Compared to Canon digital cameras, HP personal printers are usually less expensive and the toners need replacement more frequently. Therefore, the percentage of consumers who purchase HP personal printers and the toners through physical channels is relatively larger than the percentage of consumers purchasing Canon digital cameras in physical stores. Because the branded electronics are rather standardized, and we can see some exclusive financing offers providedonlineforthoseexpensiveproducts(e.g., Canondigitalcameras), moreconsumers may prefer to choose both online stores. 94 DayQuil -10% -5% - +5% +10% mean std CV CVS 53.70% 52.53% 55.13% 52.27% 51.43% 53.01% 1.28% 2.42% CVS.com 27.78% 29.29% 25.64% 27.27% 25.71% 27.14% 1.37% 5.04% PGestore.com 18.52% 18.18% 19.23% 20.45% 22.86% 19.85% 1.69% 8.53% Table B.1: Illustrative example: DayQuil (CVS) Dayquil -10% -5% - +5% +10% mean std CV Walmart 46.30% 48.45% 48.19% 45.83% 43.24% 46.40% 1.88% 4.06% Walmart.com 24.07% 22.68% 24.10% 25.00% 27.03% 24.58% 1.43% 5.83% PGestore.com 29.63% 28.87% 27.71% 29.17% 29.73% 29.02% 0.73% 2.50% Table B.2: Illustrative example: DayQuil (WMT) Pantene -10% -5% - +5% +10% mean std CV Walmart 45.45% 41.49% 42.86% 42.11% 44.12% 43.20% 1.43% 3.30% Walmart.com 27.27% 29.79% 31.17% 31.58% 32.35% 30.43% 1.79% 5.87% PGestore.com 27.27% 28.72% 25.97% 26.32% 23.53% 26.36% 1.71% 6.48% Table B.3: Illustrative example: Pantene shampoo 95 Olay -10% -5% - +5% +10% mean std CV Target 45.28% 46.74% 43.42% 44.74% 40.00% 44.04% 2.28% 5.18% Target.com 28.30% 27.17% 30.26% 34.21% 33.33% 30.66% 2.74% 8.95% PGestore.com 26.42% 26.09% 26.32% 21.05% 26.67% 25.31% 2.14% 8.44% Table B.4: Illustrative example: Olay skin care cream Gillette -10% -5% - +5% +10% mean std CV Target 44.76% 46.74% 44.00% 40.00% 41.38% 43.38% 2.41% 5.55% Target.com 22.86% 20.65% 22.67% 25.71% 27.59% 23.90% 2.45% 10.26% PGestore.com 32.38% 32.61% 33.33% 34.29% 31.03% 32.73% 1.08% 3.29% Table B.5: Illustrative example: Gillette razor HP -10% -5% - +5% +10% mean std CV Bestbuy 37.27% 37.11% 37.04% 35.00% 33.33% 35.95% 1.55% 4.32% Bestbuy.com 26.36% 23.71% 24.69% 22.50% 27.27% 24.91% 1.73% 6.95% HP.com 36.36% 39.18% 38.27% 42.50% 39.39% 39.14% 1.99% 5.09% Table B.6: Illustrative example: HP personal printers/toners Canon -10% -5% - +5% +10% mean std CV Bestbuy 28.85% 28.74% 28.57% 26.47% 25.00% 27.52% 1.54% 5.58% Bestbuy.com 36.54% 36.78% 35.71% 38.24% 35.71% 36.60% 0.93% 2.53% Canon.com 34.62% 34.48% 35.71% 35.29% 39.29% 35.88% 1.76% 4.91% Table B.7: Illustrative example: Canon digital cameras PUR -10% -5% - +5% +10% mean std CV Walmart 35.14% 34.34% 36.84% 35.00% 33.33% 34.93% 1.15% 3.29% Walmart.com 27.93% 29.29% 26.32% 25.00% 27.78% 27.26% 1.47% 5.40% PGestore.com 36.94% 36.36% 36.84% 40.00% 38.89% 37.81% 1.40% 3.69% Table B.8: Illustrative example: PUR faucet and filter 96 Appendix C The manufacturer controls three channels In this system, the manufacturer maximizes the total profit by choosing optimal prices in three channels, Π c (p r s ,p r o ,p m o )=(p r s −c r s −c m )d r s +(p r o −c r o −c m )d r o +(p m o −c m o −c m )d m o , where d r s = μ r s [1−(1+α+γ)p r s +αp r o +γp m o ], d r o = μ r o [1−(1+α+β)p r o +αp r s +βp m o ], d m o = μ m o [1−(1+γ+β)p m o +γp r s +βp r o ]. ThefeasibleregionisP c ={(p r s ,p r o ,p m o )|d r s ≥0,d r o ≥0,d m o ≥0,p r s ,p r o ,p m o ≥0}. Wefirst derive the first order conditions, ∂Π c ∂p r s = −2(1+α+γ)μ r s p r s +α(μ r o +μ r s )p r o +γ(μ m o +μ r s )p m o +e 1 , ∂Π c ∂p r o = α(μ r o +μ r s )p r s −2(1+α+β)μ r o p r o +β(μ m o +μ r o )p m o +e 2 , ∂Π c ∂p m o = γ(μ m o +μ r s )p r s +β(μ m o +μ r o )p r o −2(1+β+γ)μ m o p m o +e 3 , where e 1 = (1+α+γ)μ r s (c m +c r s )−αμ r o (c m +c r o )−γμ m o (c m +c m o ), e 2 = −αμ r s (c m +c r s )+(1+α+β)μ r o (c m +c r o )−βμ m o (c m +c m o ), e 3 = −γμ r s (c m +c r s )−βμ r o (c m +c r o )+(1+β+γ)μ m o (c m +c m o ). Furthermore, we can get the Hessian matrix as follows H c =   h 11 h 12 h 13 h 21 h 22 h 23 h 31 h 32 h 33   =   −2(1+α+γ)μ r s α(μ r o +μ r s ) γ(μ m o +μ r s ) α(μ r o +μ r s ) −2(1+α+β)μ r o β(μ m o +μ r o ) γ(μ m o +μ r s ) β(μ m o +μ r o ) −2(1+β+γ)μ m o   . If H c is negative semi-definite, then the objective function is concave with respect to p r s ,p r o ,p m o . Intuitively, this should be correct given the assumption that total demand de- creases when any of the selling prices increases, but we need to prove this result. First-orderprincipalminors,−2(1+α+γ)μ r s ,−2(1+α+β)μ r o and−2(1+β+γ)μ m o , are allnegative. Itisalsoeasytocheckthatsecond-orderprincipalminorsareallnon-negative. However, the sign of the third-order principal minor is not easy to verify. 97 But we know that det(H) = −8(1+α+γ)(1+α+β)(1+β+γ)μ r s μ r o μ m o +2αβγ(μ r o +μ m o )(μ m o +μ r s )(μ r o +μ r s ) +2(1+α+γ)β 2 (μ m o +μ r o ) 2 μ r s +2(1+α+β)γ 2 (μ m o +μ r s ) 2 μ r o +2(1+β+γ)α 2 (μ r o +μ r s ) 2 μ m o . Only one μ is zero Without loss of generality, we assume thatμ r s =0. Using condition (1.4), we know that αμ r o =γμ m o =0. Therefore, we have (1+β)μ r o ≥βμ m o , (1+β)μ m o ≥βμ r o , μ m o +μ r o =1, μ m o ≥0, μ m o ≥0. To satisfy all these conditions, we must have α = γ = 0 and μ m o ,μ r o > 0, which implies det(H c )=0. Similarly, when μ r o =0 or μ m o =0, we always have det(H c )=0. Two μ’s are zero Without loss of generality, we assume thatμ r s =μ r o =0. Using condition (1.4), we know that β =γ =0 and μ m o =1. Therefore, we have det(H c )=0. All μ’s are positive First, we can expand and re-write det(H c ) as follows: det(H c )=μ r s (μ r o −μ m o ) 2 ·Z 1 +μ r o (μ r s −μ m o ) 2 ·Z 2 +μ m o (μ r s −μ r o ) 2 ·Z 3 −μ r s μ r o μ m o ·Z 4 , where Z 1 = 2β[β(1+α+γ)+αγ], Z 2 = 2γ[γ(1+α+β)+αβ], Z 3 = 2α[α(1+β+γ)+βγ], Z 4 = 8(1+2α+2β+2γ). Whenμ r s =μ r o =μ m o = 1 3 , we have the minimumdet(H c )=− 8(1+2α+2β+2γ) 27 . Due to the symmetry of the problem, when the differences among μ r s ,μ r o ,μ m o become larger, the sum of the first three terms becomes larger, while the last term becomes smaller. Therefore, when the differences among μ r s ,μ r o ,μ m o get larger, det(H c ) becomes larger. According to the extreme cases discussed in previous subsections, we know that det(H c ) = 0 when at least one of theμ’s are zero. So,det(H c )≤0 always holds given the condition (1.4), which implies thatH c is negative semi-definite, and the objective function is concave with respect to p r s ,p r o ,p m o . 98 We get three vertices (extreme points) of the feasible region: V 1 = (μ r s ,μ r o ,μ m o ) = γ+Δ 1 Δ , β+Δ 1 Δ , 1+2α+β+γ+Δ 1 Δ , V 2 = (μ r s ,μ r o ,μ m o ) = 1+α+2β+γ+Δ 1 Δ , α+Δ 1 Δ , γ+Δ 1 Δ , V 3 = (μ r s ,μ r o ,μ m o ) = α+Δ 1 Δ , 1+α+β+2γ+Δ 1 Δ , β+Δ 1 Δ , where Δ=1+2α+2β+2γ+3αβ+3βγ+3γα, Δ 1 =αβ+βγ+γα. Any feasible point (μ r s ,μ r o ,μ m o ) f is a convex combination of these three extreme points: (μ r s ,μ r o ,μ m o ) f =ξ 1 V 1 +ξ 2 V 2 +ξ 3 V 3 , where ξ 1 ,ξ 2 ,ξ 3 ≥0 and ξ 1 +ξ 2 +ξ 3 =1. We can show that det(H c ) ≤ 0 at these three vertices and the convex combination of any two extreme points. It is very challenging to verify the general cases, so we implement numericalanalysisbasedonthechoicesof(ξ 1 ,ξ 2 ,ξ 3 ). Sincetheproblemisquitesymmetric, we assume thatξ 1 ≥ξ 2 ≥ξ 3 . At a step size of 0.01, we can show that the sign of each term of det(H c ) containing (α,β,γ) is negative at any feasible point (μ r s ,μ r o ,μ m o ) f . Therefore, given (α,β,γ), it is reasonable to assume that det(H c ) for all feasible (μ r s ,μ r o ,μ m o ) f . Supposetheobjectivefunctionisconcavewithrespecttop r s ,p r o ,p m o . Wecouldhaveeight possible solutions: • 1 three-channel (d r s , d r o , d m o >0), • 3 two-channel (only one of d r s , d r o , d m o is 0), • 3 one-channel (only one of d r s , d r o , d m o is positive), • 1 close-all (d r s = d r o = d m o =0). 1. Three-channel case. After solving the FOCs, we get p r s = [(h 32 h 23 −h 22 h 33 )e 1 +(h 12 h 33 −h 32 h 13 )e 2 +(h 22 h 13 −h 12 h 23 )e 3 ]/det(H c ), p r o = [(h 21 h 33 −h 31 h 23 )e 1 +(h 13 h 31 −h 11 h 33 )e 2 +(h 11 h 23 −h 21 h 13 )e 3 ]/det(H c ), p m o = [(h 31 h 22 −h 21 h 32 )e 1 +(h 11 h 32 −h 31 h 12 )e 2 +(h 21 h 12 −h 11 h 22 )e 3 ]/det(H c ). 2. Two-channel case. Suppose, for instance, that d r s = 0. We have the optimal physical store price p r s = 1+αp r o +γp m o 1+α+γ . Therefore, the demand from the other two channels are as follows: d r o = μ r o h 1+2α+γ 1+α+γ − (1+α+β)(1+α+γ)−α 2 1+α+γ p r o +βp m o i , d m o = μ m o h 1+α+2γ 1+α+γ − (1+β+γ)(1+α+γ)−γ 2 1+α+γ p m o +βp r o i , 99 and the total profit is (p r o −c r o −c m )d r o +(p m o −c m o −c m )d m o . Again, this objective is concave with respect to p r o and p m o . We have the FOCs ∂Π c ∂p r o =f 11 p r o +f 12 p m o +g 1 , ∂Π c ∂p m o =f 21 p r o +f 22 p m o +g 2 , where f 11 = − 2(1+α+β)(1+α+γ)−2α 2 1+α+γ μ r o , f 12 = f 21 = β(μ r o +μ r o ), f 22 = − 2(1+β+γ)(1+α+γ)−2γ 2 1+α+γ μ m o , g 1 = μ r o h 1+2α+γ 1+α+γ + (1+α+β)(1+α+γ)−α 2 1+α+γ (c r o +c m ) i −μ m o β(c m o +c m ), g 2 = −μ r o β(c r o +c m )+μ m o h 1+α+2γ 1+α+γ + (1+β+γ)(1+α+γ)−γ 2 1+α+γ (c m o +c m ) i . The optimal prices are p r s = 1+αp r o +γp m o 1+α+γ , p r o = [f 12 g 2 −f 22 g 1 ]/(f 11 f 22 −f 12 f 21 ), p m o = [f 21 g 1 −f 11 g 2 ]/(f 11 f 22 −f 12 f 21 ). 3. One-channel case. Suppose, for instance, that d r s =d r o =0. Then, we have p r s = (αβ+αγ+βγ+γ)p m o +(1+2α+β) (1+α+β)(1+α+γ)−α 2 , p r o = (αβ+αγ+βγ+β)p m o +(1+2α+γ) (1+α+β)(1+α+γ)−α 2 . The total profit is (p m o −c m o −c m )d m o . Therefore, it is easy to get p r s = (αβ+αγ+βγ+γ)p m o +(1+2α+β) (1+α+β)(1+α+γ)−α 2 , p r o = (αβ+αγ+βγ+β)p m o +(1+2α+γ) (1+α+β)(1+α+γ)−α 2 , p m o = c m +c m o 2 − A+(2+β+γ) 2A , where A=−(1+β+γ)+ γ(1+2α+β)+β(1+2α+γ) (1+α+β)(1+α+γ)−α 2 . 100 Appendix D Long Proofs in Chapter 2 Proposition 1.Unique relationship between Q m and Q r is guaranteed. Proof: It is easy to see that the first-order derivative w.r.t. Q m gives us (p m −c)−p m Z Qm 0 Z Qr+(Qm−x)/αr 0 f(x)g(y)dydx−ψ ′ (Q r +Q m ) Apparently, the second term ( i.e., the integral ) is non-negative, and it is equal to zero iff Q m =0. Therefore, there is a threshold ˆ Q r determined by ˆ Q r =            +∞, if p m −c>ψ ′ (Q r ), ∀Q r ≥0; 0, if p m −c<ψ ′ (Q r ), ∀Q r ≥0; ψ ′−1 (p m −c), otherwise. (D.1) When p m < c +ψ ′ (0), the marginal production cost is larger than the selling price. Given that the marginal cost is increasing (i.e., ψ ′′ (.) ≥ 0), the manufacturer will give up the production for himself regardless of the order size from the retailer. When p m ≥c+ψ ′ (0), the marginal production cost is less than or equal to the selling price. When ˆ Q r is finite (i.e., when (p m −c)=ψ ′ ( ˆ Q r )), we have Q m (Q r )=0 for Q r > ˆ Q r . That is, the manufacturer never allocates products to his own channel if the retailer’s order size is too large. If there does not exist finite Q r such that (p m −c) =ψ ′ (Q r ), then ˆ Q r is defined to be +∞. In this case, the FOC is sufficient to determine the best response ofQ m to Q r . Note that the manufacturer never allocates infinite amount of products to his own channel, i.e., Q m is always finite. Knowing that the maximum Q m is Q m (0), we only need to check if Q m (0) < ∞. Suppose Q m (0) = ∞, then the LHS of the FOC becomes −c−ψ ′ (∞) < 0, which implies the manufacturer will decrease the production size and Q m (0)=∞ is not optimal. Proofs of Propositions 2-4. If (w +b) ≥ p m , we have dΠm dQr > 0. Therefore, in this situation, the manufacturer delivers whatever the retailer asks for, i.e., Q d r = Q o r . In other words, when the marginal benefitfromwholesalecontractislargerthanthepotentialmarginallossinhisownchannel, 101 the manufacturer will always deliver the maximum amount required by the retailer. From another aspect, when the penalty cost b is extremely large, the above inequality always holds, and the manufacturer will always deliver the full-order size. However, if (w +b) < p m , dΠm dQr may not be positive and monotone w.r.t. Q r for all Q r ≤Q o r . Thus, the manufacturer may or may not deliver the full order size to the retailer. As we can see, the sign of d 2 Πm dQrdQr is unclear given dQm dQr < 0. For many common dis- tributions, such as normal distribution, gamma distribution, uniform distribution, bothL 1 andF(Q m )G(Q r ) are concave and not monotone w.r.t. Q r . Note that this is quite different from the results obtained by Geng and Mallik (2007). One of the reasons is the impact from a convex capacity investment cost. In their paper, F(Q m )G(Q r ) is more likely to be a increasing function of Q r without the influence of the capacity investment decisions. Furthermore, we know that dΠm dQr | Qr=0 = (w +b)−p m +(1−α r )(p m −c−ψ ′ ( ˜ Q m )), where ˜ Q m solves dΠm dQm | Qr=0 = 0. We also have ψ ′ ( ˜ Q m )) ≤ (p m −c). Therefore, we get dΠm dQr | Qr=0 ≥(w+b)−p m . Since the general shape of dΠm dQr can be increasing or an inverse-v curve for Q r , we further divide it into two situations. Situation 1. When dΠm dQr is increasing in Q r for all Q r ≤Q o r , then Q d r ∈{0, Q o r }. When (w+b)≥α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )), then dΠm dQr ≥ 0 and Q d r =Q o r . Full order size will be delivered. When (w+b)<α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )). (1)If dΠm dQr | Qr=Q o r <0,theoptimalQ d r =0. Thatis,themanufacturerdeliversnothingto the retailer. The penalty cost may hurt a lot, and the manufacturer will reject the contract at the very beginning when Π m ( ˜ Q m ,0)<0. (2) If there exists some Q 1 r ≤ Q o r such that dΠm dQr | Qr=Q 1 r = 0, then the manufacturer allocates either 0 or Q o r to the retailer. When Π m (Q m (Q o r ),Q o r ) ≤ Π m ( ˜ Q m ,0), we have Q d r = 0. Moreover, if Π m ( ˜ Q m ,0) < 0, the manufacturer simply rejects the contract at the very beginning. When Π m (Q m (Q o r ),Q o r )>Π m ( ˜ Q m ,0), the manufacturer prefers to deliver the exactQ o r requested by the retailer, i.e., Q d r =Q o r . Situation 2. When dΠm dQr first increases and then decreases w.r.t. Q r for Q r ≤Q o r . Therefore,wecouldhavethreecandidatesolutions,Q d r ∈{0, ˜ Q i r , Q o r },where ˜ Q i r satisfies (w+b) =p m [1−(1−α r )L 1 −α r F(Q m ( ˜ Q i r ))G( ˜ Q i r )] and dΠm dQr | Qr> ˜ Q i r < 0. If ˜ Q i r ≥Q o r , ˜ Q i r will not be the optimal choice. We are more interested in the case when ˜ Q i r <Q o r . The shape of d 2 Πm dQrdQr plays a critical role in choice of optimal allocation strategies. We see in many numerical simulations, the shape can be inverse-v curve, which actually sheds some light on the basic tradeoff from the manufacturer’s perspective. Theorem 2. Suppose φ(x) is twice-differentiable. If φ(x)≥0 and φ ′ (x)<0 for all 0<x<∞, then we have lim x→∞ φ(x)=C 0 and lim x→∞ φ ′ (x)=0. 102 Proof: First, we need to show that lim x→∞ φ(x) exists. If not, we have two ordered sequences {x i 1 } ∞ i=1 and {x j 2 } ∞ j=1 such that lim x i 1 →∞ φ(x i 1 ) =C 1 and lim x j 2 →∞ φ(x j 2 ) =C 2 , where C 1 > C 2 . Therefore, we know that ∀ǫ > 0, ∃N < ∞ s.t. ∀i,j > N, |φ(x j 2 )−C 2 | < ǫ and |φ(x i 1 )−C 1 | < ǫ . Now we can choose ǫ 0 = C 1 −C 2 4 > 0, and ∃N 0 < ∞ s.t. ∀i,j > N 0 , |φ(x j 2 )−C 2 | < ǫ 0 and |φ(x i 1 )−C 1 | < ǫ 0 . That is ∀i,j > N 0 , φ(x j 2 ) < φ(x i 1 ). We can choose two sequences of {i l } ∞ l=1 and {j m } ∞ m=1 , such that N 0 < i 1 < j 1 < i 2 < j 2 < ... and φ(x jm 2 )<φ(x i l 1 ). However, by definition of φ ′ (x)< 0, we know that φ(x j n+1 2 )<φ(x i n+1 1 )< φ(x jn 2 )<φ(x in 1 ). We obtain a contradiction. Therefore, lim x→∞ φ(x) exists, we denote the limit by C 0 ≥0. Suppose that φ ′ (x) is continuous and differentiable. Since φ ′ (x)< 0 for all 0<x<∞, by the continuity of φ ′ (x), we know that lim x→∞ φ ′ (x) ≤ 0. Obviously if lim x→∞ φ ′ (x) exists, it can not be negative, otherwise, lim x→∞ φ(x) < 0. Therefore, if the limit exists, lim x→∞ φ ′ (x)=0. If lim x→∞ φ ′ (x) does not exists, then we could have two sequences{x i 1 } ∞ i=1 and{x j 2 } ∞ j=1 such that lim x i 1 →∞ φ ′ (x i 1 ) = d 1 and lim x j 2 →∞ φ ′ (x j 2 ) = d 2 , where d 1 > d 2 . It is easy to see that 0≥d 1 >d 2 , however, this implies that lim x j 2 →∞ φ ′ (x j 2 )=d 2 <0 and lim x j 2 →∞ φ(x j 2 )< 0. By continuity of φ(x), we know that ∃N 2 <∞ such that, ∀j >N 2 , φ(x j 2 )< 0 and this contradicts the assumption φ(x)≥0 for all 0<x<∞. Proposition 8. Suppose ∂ 2 Πm ∂Q 2 m <0 for all (Q r ,Q m ), then the relationship between Q m and Q r is uniquely determined by the FOC if the manufacturer allocates products to both channels. For most cost functions and slight restrictions on most unimodal distributions, there exists unique Q r such that dQm(Qr) dQr ≤0 for Q r <Q r , and dQm(Qr) dQr ≥0 for Q r >Q r . Proof: We look at the following set of equations: p m Z Qm 0 f(x)g Q r + Q m −x α r dx+ψ ′′ (Q m +Q r )=0 (p m −c)−p m Z Qm 0 Z Qr+(Qm−x)/αr 0 f(x)g(y)dydx−ψ ′ (Q m +Q r )=0 If the solution does not exist, then the sign the LHS of the first equation is always the same, either positive or negative. Let Q m be close to zero, we then know the LHS must be negative, which implies that the LHS is always negative. Therefore, if we let Q r be the solution solving the second equation, we must have dQm(Qr) dQr ≥ 0 for Q r > Q r and dQm(Qr) dQr =0 for Q r <Q r . 103 Now assume that (Q m ,Q r ) is the solution. If the solution is unique, then we must have dQm(Qr) dQr ≤ 0 for Q r < Q r , and dQm(Qr) dQr ≥ 0 for Q r > Q r . However, the uniqueness of the solution is not easy to verify for general cases. If the LHS of the first equation is decreasing w.r.t. Q r , then we always have unique solution. If we take the first-order derivative of R Qm 0 f(x)g Q r + Qm−x αr dx w.r.t. Q r , we get f(Q m )g(Q r ) dQ m dQ r + dQ m dQ r 1 α r +1 Z Qm 0 f(x)g ′ Q r + Q m −x α r dx If dQm dQr < 0, we must have a lower bound −α r < dQm dQr . Therefore, the derivative is negative when g ′ Q r + Qm−x αr < 0 for x ∈ [0, Q m ]. Even if dQm dQr ≥ 0, as long as g ′ is extremely negative, the derivative is negative as well. The result is true for most types of unimodal distributions when we put mild conditions on them. For example, if the play- ers’ base demands follow truncated normal distributions on [0,∞) with mean μ r ,μ m and standard deviation σ r ,σ m . The derivative can be extremely negative when Q r >μ r . Even for many commonly seen ψ(K), the LHS is still decreasing in Q r . If the demand follows gamma distribution with shape k and scale θ. As long as the mode (k−1)θ<Q r , we also have the conditions hold. Proposition 14. Given ψ ′′ >0, when p m ≥α m p r and p r ≥α r p m hold, the manufacturer allocates products in both channels but never increases the allocation sizes in both channels at the same time compared to the decentralized system. Proof: When p m ≥α m p r and p r ≥α r p m hold, we know that the manufacturer allocates products to both channels and FOCs are sufficient to yield the solutions. Applying the implicit function theorem to the FOCs, we get dQ c m dQ c r =− ∂ 2 Π c ∂Q c r ∂Q c m ∂ 2 Π c ∂ 2 Q c m =− R 1 R 2 , where R 1 = p m R Qm 0 f(x)g(Q r +(Q m −x)/α r )dx+p r R Qr 0 g(y)f(Q m +(Q r −y)/α m )dy +ψ ′′ (Q m +Q r ), R 2 = p m f(Q m )G(Q r )+ pm αr R Qm 0 f(x)g(Q r +(Q m −x)/α r )dx +p r R Qr 0 α m g(y)[f(Q m +(Q r −y)/α m )−f(Q m )]dy+ψ ′′ (Q m +Q r ). LetM =p m R Qm 0 f(x)g(Q r +(Q m −x)/α r )dx,N =p r R Qr 0 g(y)f(Q m +(Q r −y)/α m )dy, then we have dQ c m dQ c r =− M +N +ψ ′′ M/α r +α m N +ψ ′′ +(p m −α m p r )f(Q m )G(Q r ) , 104 and dQ d m dQ d r =− M +ψ ′′ M/α r +ψ ′′ +p m f(Q m )G(Q r ) . If M/α r +α m N +ψ ′′ +(p m −α m p r )f(Q m )G(Q r )> 0, then by simple algebra, we get dQ c m dQ c r ≤ dQ d m dQ d r ≤0. However, if M/α r +α m N +ψ ′′ +(p m −α m p r )f(Q m )G(Q r )<0, then we get dQ c m dQ c r ≥0. Suppose the capacity setup cost is convex increasing, i.e., ψ ′′ (K) > 0. As long as p m −α m p r >0 is satisfied, we always have M/α r +α m N +ψ ′′ +(p m −α m p r )f(Q m )G(Q r )>0; thus, dQ c m dQ c r ≤ dQ d m dQ d r ≤0. That is to say, the manufacturer allocates less and less inventory to himself, dQ c m dQ c r ≤ dQ d m dQ d r ≤0, in the centralized case. However, it is still not clear if the manufacturer allocates less inventory to the retailer. Note that the two channels are somewhat symmetric in the centralized system. There- fore, we can also use the second FOC to characterize the relationship betweenQ c m andQ c r . And similarly, if the simple condition p r −α r p m >0 holds, we still have dQ c m dQ c r ≤ dQ d m dQ d r ≤0. It is interesting to see that two simple conditions can easily help to determine the sign of dQ c m dQ c r given the convex capacity setup cost. Theorem 4. Given α r p m ≤α m p r ≤p m , there always exists some w∈[c,p r ] such that b,u r ≥0. Proof: First, we define b(z)=(p r −z)−J(Q c m (z),Q c r (z)), u r (z)=b(z)+ dQ c m (z) dQ c r (z) u m , ˆ u r (z)=b(z)−J(Q c m (z),Q c r (z))≤u r (z), u m (z)=α m (J(Q c m (z),Q c r (z))−p r F(Q c m (z))G(Q c m (z))), where J(Q c m (z),Q c r (z))=p r Z Q c r (z) 0 Z Q c m (z)+(Q c r (z)−y)/αm 0 f(x)g(y)dxdy. We also have ∂b(z) ∂z =−1− ∂J ∂Q c r · ∂Q c r ∂z + ∂J ∂Q c m · ∂Q c m ∂z , 105 ∂J ∂Q c r = p r F(Q c m (z))g(Q c r (z))+ 1 αm p r A c r (z), ∂J ∂Q c m = p r A c r (z), A c r (z) = R Q c r (z) 0 f Q c m (z)+ Q c r (z)−y αm g(y)dy. We need that both b(z) and u r (z) are non-negative for all z ∈ [c, p r ]. Thus, such w must exist. It is also easy to see that Q c r = 0 when c = p r . Hence, b(p r ) = u m (p r ) = u r (p r ) = ˆ u r (p r ) = 0. As long as we can show that ∂b(z) ∂z ≤ 0 and ∂ur(z) ∂z ≤ 0 or ∂ˆ ur(z) ∂z ≤ 0 for all z∈[c, p r ], then we have b(z), u r (z)≥0 for all z∈[c, p r ]. Therefore, b = b(c)−(w−c) ≥ 0 for all w ∈ [c, b(c)+c] ⊂ [c, p r ] and u r = u r (c)− (w−c)≥ 0 for all w ∈ [c, u r (c)+c]⊂ [c, p r ]. Hence, both b and u r are non-negative for w∈[c, c+min{u r (c), b(c)}]⊂[c, p r ]. Case 1. Q c r =0 WhenQ c r =0,wemusthave ∂Πc ∂Qr ≤0forallQ r andQ m . Moreover,itimplies ∂Q c r (z) ∂z =0, u m (z)=0, A c r (z)=0, which further implies ∂J ∂Q c m =0. Thus, we obtain ∂b(z) ∂z =−1<0. Since u m (z)=0, we also have u r (z)=b(z), which means ∂ur(z) ∂z =−1<0 as well. Case 2. Q c m =0 WhenQ c m =0,wemusthave ∂Πc ∂Qm ≤0forallQ r andQ m . Moreover,itmeans ∂Q c m (z) ∂c =0 and ∂Q c m ∂Q c r =0. The latter also implies u m (z)=0. We know that ∂Πc(z) ∂Qr =(p r −z)−p r R Qr(z) 0 R Qr(z)−y αm 0 f(x)g(y)dxdy−ψ ′ (Q r (z)). In order to get positive Q c r (z), we implicitly assume that p r −c ≥ ψ ′ (0). To ensure unique finite solution, we assume that ∂ 2 Πc ∂Q 2 r =− 1 αm A c r (z)−ψ ′′ (Q c r (z))<0. Therefore, we obtain ∂b(z) ∂z =−1+ A c r (z) A c r (z)+αmψ ′′ (Q c r (z)) . If ψ ′′ (Q c r (z)) ≥ 0, then −1 < ∂b(z) ∂z ≤ 0. Again, we have ∂ur(z) ∂z = ∂b(z) ∂z ≤ 0 for all z∈[c, p r ]. If ψ ′′ (Q c r (z)) < 0, then ∂b(z) ∂z > 0. So ∂ur(z) ∂z = ∂b(z) ∂z > 0 and u r (z) = b(z) ≥ b(c) = ψ ′ (Q c r (c))≥0 for all z∈[c, p r ]. Case 3. 0<Q c m ,Q c r <+∞ We need to get ∂Q c m (z) ∂z and ∂Q c r (z) ∂z . Denote S(Q c m ,Q c r ,z)= ∂Π c ∂Q m =0, T(Q c m ,Q c r ,z)= ∂Π c ∂Q r =0. Therefore, we obtain ∂Q c m (z) ∂z = T Qr −S Qr S Qm T Qr −S Qr T Qm , ∂Q c r (z) ∂z = S Qm −T Qm S Qm T Qr −S Qr T Qm , 106 S Qm = (α m p r −p m )f(Q c m )G(Q c r )−p m 1 αr A c m −p r α m A c r −ψ ′′ (Q c m +Q c r ), S Qr = −p m A c m −p r A c r −ψ ′′ (Q c m +Q c r ), T Qm = −p m A c m −p r A c r −ψ ′′ (Q c m +Q c r ), T Qr = (α r p m −p r )F(Q c m )g(Q c r )−p r 1 αm A c r −p m α r A c m −ψ ′′ (Q c m +Q c r ), A c m (z) = R Q c m (z) 0 g Q c r (z)+ Q c m (z)−x αr f(x)dx. Since we assume that both Q c m (z) and Q c r (z) are unique positive finite, we implicitly assume that S Qm T Qr −S Qr T Qm < 0. And this can be achieved by assuming α r p m ≤ p r and α m p r ≤ p m , if ψ ′′ (K) ≥ 0. If ψ ′′ (K) < 0 but the magnitude is very small, the conditionswesuggestarestillsufficient. Moreover,wecouldobtainthat dQ c m dQ c r <0,therefore, u r (z)≥ ˆ u r (z)=b(z)−J. Now if we can show that ∂J ∂Q c r ·(S Qm −T Qm )+ ∂J ∂Q c m ·(T Qr −S Qr ) < 0, then we have both ∂ˆ ur(z) ∂z ≤ 0 and ∂b(z) ∂z ≤ 0. Thus, u r (z)≥ ˆ u r (z)≥ ˆ u r (p r ) = 0 and b(z)≥b(p r ) = 0 for all z∈[c, p r ]. We have T Qr −S Qr = (α r p m −p r )F(Q c m (z))g(Q c r (z)) + 1− 1 αm p r A c r (z)+(1−α r )p m A c m (z), S Qm −T Qm = (α m p r −p m )f(Q c m (z))G(Q c r (z)) + 1− 1 αr p m A c m (z)+(1−α m )p r A c r (z). Therefore, ∂J ∂Q c r ·(S Qm −T Qm )+ ∂J ∂Q c m ·(T Qr −S Qr ) = p m A c r (z)A c m (z) h (1−α r )+ 1 αm 1− 1 αr i +A c r (z) h (α r p m −α m p r )F(Q c m (z))g(Q c r (z))+ 1 αm (α m p r −p m )f(Q c m (z))G(Q c r (z)) i +A c m (z)· 1− 1 αr p m +(α m p r −p m )f(Q c m (z))G(Q c r (z))F(Q c m (z))g(Q c r (z)). Both A c r (z) and A c m (z) are non-negative, so if all the coefficients are non-positive, then the sum must be non-positive, (1−α r )+ 1 α m 1− 1 α r ≤2−α r − 1 α r ≤0. If both α r p m ≤α m p r and α m p r ≤p m hold, we have (α r p m −α m p r )F(Q c m (z))g(Q c r (z))+ 1 α m (α m p r −p m )f(Q c m (z))G(Q c r (z)) ≤0, 107 1− 1 α r ≤0, (α m p r −p m )f(Q c m (z))G(Q c r (z))F(Q c m (z))g(Q c r (z))≤0. Therefore, we have ∂J ∂z ≥0, which implies ∂b(z) ∂z ≤0 and ∂ˆ ur(z) ∂z ≤0. So b(z)≥b(p r )=0 and u r (z)≥ ˆ u r (z)≥ ˆ u r (p r )=0 for all z∈[c, p r ]. 108 Appendix E Detailed Analysis in Convex Case Retailer’s possible order sizes The retailer’s objective is to choose the optimal order size Q o r . The retailer’s objective function can be re-written as Π r (Q o r )=(p r −w)Q r −p r ( ¯ A+ ¯ B)+b(Q o r −Q r ), where ¯ A = R Qm 0 R Qr 0 (Q r −y)f(x)g(y)dydx, ¯ B = R Qm+Qr/αm Qm R Qr−αm(x−Qm) 0 [Q r −y−α m (x−Q m )]f(x)g(y)dydx . Considering that Q r ∈ {0, Q i r , Q o r }, the retailer decides the actual order quantity Q o r . Basedondifferentprice-costpairs, themanufacturerwillrespondtotheretailer’sordersize differently. Let us define several possible solutions to the retailer’s problem when ψ ′′ ≥0. Notation and Definition ˜ Q od r The retailer’s order size Q o r when Q d r =Q o r and Q d m >0 ˜ Q odi r The retailer’s order size Q o r when Q d r = ˜ Q i r and Q d m >0 ˜ Q od0 r The retailer’s order size Q o r when Q d r =0 and Q d m >0 ˆ Q od r The retailer’s order size Q o r when Q d r =Q o r and Q d m =0 ˆ Q odi r The retailer’s order size Q o r when Q d r = ˆ Q i r and Q d m =0 ˆ ˜ Q odi r The retailer’s order size Q o r when Q d r = ˜ Q i r and Q d m =0 ˜ Q i r The manufacturer’s partial delivery size Q d r to the retailer when Q d m >0 ˆ Q i r The manufacturer’s partial delivery size Q d r to the retailer when Q d m =0 Table E.1: More notations and definitions 1. Q d m >0 and Q d r =Q o r . ˜ Q od r =argmaxΠ r (Q o r )=(p r −w)Q o r −p r ( ¯ A o + ¯ B o ), where ¯ A o = R Qm 0 R Q o r 0 (Q o r −y)f(x)g(y)dydx, ¯ B o = R Qm+Q o r /αm Qm R Q o r −αm(x−Qm) 0 [Q o r −y−α m (x−Q m )]f(x)g(y)dydx. 109 Therefore, dΠr dQ o r = ∂Πr ∂Qm dQm dQ o r + ∂Πr ∂Q o r = p r α m RQm+ Q o r αm Qm R Q o r −αm(x−Qm) 0 f(x)g(y)dydx· dQm(Q o r ) dQ o r +(p r −w)−p r R Q o r 0 RQm+ Q o r −y αm 0 f(x)g(y)dxdy. 2. Q d m =0 and Q d r =Q o r . ˆ Q od r =argmaxΠ r (Q o r )=(p r −w)Q o r −p r ˆ B o , where ˆ B o = Z Q o r /αm 0 Z Q o r −αmx 0 [Q o r −y−α m x]f(x)g(y)dydx. We have the FOC dΠ r dQ o r =(p r −w)−p r Z Q o r αm 0 Z Q o r −αmx 0 f(x)g(y)dydx. In fact, we have an interesting relationship between these two possible solutions. It’s easy to see that dΠr dQ o r | Q o r =0,Qm>0 = p r −w > 0, dΠr dQ o r | Q o r = ˆ Qr,Qm>0 = dΠr dQ o r | Q o r = ˆ Qr,Qm=0 and dΠr dQ o r | Q o r =∞,Qm=0 =−w < 0. So there exists at least one finite solution. That is to say, the retailer will only order finite amount of products from the manufacturer. Comparing the FOCs, it is easy to see that ˆ Q od r ≥ ˜ Q od r . Moreover, we have the following relationships as ˜ Q od r > ˆ Q r =⇒ ˆ Q od r > ˆ Q r , ˆ Q od r < ˆ Q r =⇒ ˜ Q od r < ˆ Q r . If ˜ Q od r > ˆ Q r , we get ˆ Q od r > ˆ Q r , the retailer has to make a decision between ˆ Q r and ˆ Q od r . We know that dΠr dQ o r | Qm>0,Q o r = ˆ Qr > 0. Because dΠr dQ o r | Qm=0,Q o r = ˆ Qr = dΠr dQ o r | Qm>0,Q o r = ˆ Qr , we then get dΠr dQ o r | Qm=0,Q o r = ˆ Qr >0. Therefore, the retailer will prefer ˆ Q od r over ˆ Q r . If ˆ Q od r < ˆ Q r , we get ˜ Q od r < ˆ Q r , the retailer has to choose between ˆ Q r and ˜ Q od r . We know that dΠr dQ o r | Qm=0,Q o r = ˆ Qr < 0. Because dΠr dQ o r | Qm=0,Q o r = ˆ Qr = dΠr dQ o r | Qm>0,Q o r = ˆ Qr , we then get dΠr dQ o r | Qm=0,Q o r = ˆ Qr <0. Therefore, the retailer will prefer ˜ Q od r over ˆ Q r . In fact, if ˜ Q od r ≤ ˆ Q r , we have dΠr dQ o r | Qm=0,Q o r = ˆ Qr ≤0, which means ˆ Q od r will not be picked by the retailer. If ˜ Q od r > ˆ Q r , we have dΠr dQ o r | Qm=0,Q o r = ˆ Qr > 0, which means ˆ Q od r is preferred by the retailer. Therefore, when full order delivery is possible, the retailer picks either ˜ Q od r or ˆ Q od r . Note that the second problem is strictly concave, so the retailer at most have one solution. However, in the first case, the uniqueness of the solution is not easy to verify. We observe that (Q o r +α m Q m ) is increasing in Q o r , given 0 < α m < 1. Therefore, as long as d 2 Qm(Q o r ) dQ o2 r ≤ 0, i.e., Q m (Q o r ) is convex w.r.t. Q o r , we must have d 2 Πr dQ o2 r < 0 and the 110 uniqueness of the optimizer is guaranteed. Hence, we can solve for the optimal order size Q o r = ˜ Q od r , which is always finite. From the FOC, we also notice that Q o r decreases as the wholesale price w increases. Therefore, ∂(Qm(Q o r )+Q o r ) ∂w = (1+ dQm dQ o r ) ∂Q o r ∂w ≤ 0, i.e., the total quantity (Q m +Q o r ) decreases as w increases. The invested capacity decreases as w increases. Now we need to show under what conditions we have d 2 Qm(Q o r ) dQ o2 r <0. In this situation, we have dQ m (Q r ) dQ r =− α r h R Qm 0 f(x)g(Q r + Qm−x αr )dx+ψ ′′ (Q m +Q r )/p m i α r f(Q m )G(Q r )+ R Qm 0 f(x)g(Q r + Qm−x αr )dx+α r ψ ′′ (Q m +Q r )/p m , and −1≤ dQm(Qr) dQr <0. Note that when ψ ′′ is extremely large, the value is close to −1. When ψ ′′ is extremely small, e.g., ψ ′′ =0, we could even have a tighter lower bound, i.e., −α r < dQm(Qr) dQr <0. Denote A m = R Qm 0 f(x)g(Q r + Qm−x αr )dx, we then obtain dQ m (Q r ) dQ r =−1+ α r f(Q m )G(Q r )p m +A m p m −α r A m p m α r f(Q m )G(Q r )p m +A m p m +α r ψ ′′ (Q m +Q r ) . Moreover, dA m dQ r =f(Q m )g(Q r ) dQ m (Q r ) dQ r + 1+ dQ m (Q r ) dQ r 1 α r Z Qm 0 f(x)g ′ Q r + Q m −x α r dx, d(αrf(Qm)G(Qr)+Am) dQr = α r f ′ (Q m )G(Q r ) dQm(Qr) dQr +f(Q m )g(Q r ) dQm(Qr) dQr +α r + 1+ dQm(Qr) dQr 1 αr R Qm 0 f(x)g ′ Q r + Qm−x αr dx, d(αrf(Qm)G(Qr)+(1−αr)Am) dQr = α r f ′ (Q m )G(Q r ) dQm(Qr) dQr +f(Q m )g(Q r ) dQm(Qr) dQr (1−α r )+α r +(1−α r ) 1+ dQm(Qr) dQr 1 αr R Qm 0 f(x)g ′ Q r + Qm−x αr dx. (a)Whenψ ′′ isextremelylargecomparedtoprobability-relatedterms, dQm(Qr) dQr iscloseto −1, thus, dQm(Qr) dQr +α r ≤ 0. Furthermore, if both f ′ (Q m (Q r )) and g ′ Q r + Qm(Qr)−x αr are positive for Q r ≤ ˆ Q r and x ∈ [0,Q m (Q r )], then we have d(αrf(Qm)G(Qr)+Am) dQr < 0. If dQm(Qr) dQr < − αr 1−αr , we also have dQm(Qr) dQr (1−α r )+α r ≤ 0 and [α r f(Q m )G(Q r )p m + A m p m −α r A m p m ] is decreasing in Q r as well. When we have α r < 1 2 and extremely large ψ ′′ , this situation can actually happen. Hence, if ψ ′′′ (Q r +Q m ) 1+ dQm(Qr) dQr < − dAm dQr , 111 then we must have d 2 Qm(Qr) dQrdQr < 0. In fact, given conditions on the distributions above, we always have dAm dQr <0, therefore, d 2 Qm(Qr) dQrdQr <0 is ensured as long as ψ ′′′ (Q r +Q m )≤0. Now we need to check if 1)ψ ′′ is extremely large compared to probability-related terms; 2)ψ ′′′ (Q r +Q m )≤0; 3)bothf ′ (Q m (Q r ))andg ′ Q r + Qm(Qr)−x αr arepositiveforQ r ≤ ˆ Q r and x∈[0,Q m (Q r )] for commonly seen distributions. For example, if both f and g follow truncated normal distributions, we know that the probability-related terms are all proportional to e −Q 2 , which can be very small compared toψ ′′ =1 for largeQ. If bothf andg are gamma distributed, then the probability-related terms are all proportional to e −Q , which again can be very small compared to ψ ′′ = 1 for largeQ. Therefore, for a whole set of cost functions, the conditions 1) and 2) hold. In fact, 3) is also true for most unimodal distributions when ˆ Q r + ˆ Q m is less then the mode in the distribution. We summarize the results in the following proposition. Proposition. When ψ ′′ (K)>0 is extremely large and ψ ′′ (K) is decreasing in K, we always have d 2 Qm(Qr) dQrdQr <0 for small α r . (b)Whenψ ′′ isextremelysmall,wecouldevenhaveatighterlowerbound,i.e., dQm(Qr) dQr > −α r . But the convexity is not easily guaranteed. 3. Q d m >0 and Q d r = ˜ Q i r . When b>0, we have ˜ Q odi r ( ˜ Q i r )= 1 b ·max{0, ˜ Π m (Q m ( ˜ Q i r ), ˜ Q i r )}, where ˜ Π m (Q m ( ˜ Q i r ), ˜ Q i r ) is the manufacturer’s profit whenQ o r =0 in the objective function. 4. Q d m =0 and Q d r = ˆ Q i r . When b>0, we get ˆ Q odi r ( ˆ Q i r )= 1 b ·max{0, ˆ Π m (0, ˆ Q i r )}, where ˆ Π m (0, ˆ Q i r ) is the manufacturer’s profit when Q o r =0 in the objective function. 5. Q d m >0 and Q d r =0. When b>0, we get ˜ Q od0 r = 1 b ·max{0, Π m ( ˜ Q m ,0)}. 6. Q d m =0 and Q d r = ˜ Q i r . When b>0, we have ˆ ˜ Q odi r = ˆ Q odi r ( ˜ Q i r )= 1 b ·max{0, Π m (0, ˜ Q i r )}. 7. Q d m =0 and Q d r =0. 112 ˆ Q od0 r =0. It’s also easy to see that ˜ Q od r ≤ ˜ Q odi r and ˆ Q od r ≤ ˆ Q odi r . More equilibrium results in the convex capacity case In addition to section 2.4.1.2, we provide more detailed equilibrium results here. 3. (w+b)<p m , (w+b)≥α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )) and dΠm dQr first increases then decreases: ˜ Q i r ≤ ˆ Q r ˜ Q i r > ˆ Q r ˜ Q od r ≤ ˜ Q i r ˜ Q od r > ˜ Q i r ˜ Q od r ≤ ˆ Q r ˜ Q od r > ˆ Q r (w−c+b)≥ψ ′ ( ˆ Q od r ) Otherwise (o/w) ˜ Q od r , ˜ Q od r ˜ Q i r , ˜ Q odi r ˜ Q od r , ˜ Q od r ˆ Q od r , ˆ Q od r ˆ Q i r , ˆ Q odi r Table E.2: Possible Equilibria in Case 3 In this situation, we have 4 possible equilibria depending on the parameters in the problem. Only full order size and partial order size are the optimal strategies for the manufacturer. 4. (w+b)<p m , (w+b)<α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )) and dΠm dQr increases: ˜ Q od r ≤ ˆ Q r < ˜ Q od0 r ˜ Q od r ≤ ˜ Q od0 r ≤ ˆ Q r Π m (Q m ( ˜ Q od r ), ˜ Q od r )≥0 Π m (Q m ( ˜ Q od r ), ˜ Q od r )≥0 Π m (Q m ( ˜ Q od0 r ), ˜ Q od0 r )<0 ˜ Q od r , ˜ Q od r ˜ Q od r , ˜ Q od r 0, ˜ Q od0 r Table E.3: Possible Equilibria in Case 4 ˜ Q od r > ˆ Q r (w+b−c)≤ψ ′ ( ˆ Q r ) (w+b−c)≥ψ ′ ( ˆ Q od r ) ψ ′ ( ˆ Q r )<(w+b−c)<ψ ′ ( ˆ Q od r ) Π m (Q m ( ˆ Q od r ), ˆ Q od r )≥0 Π m (Q m ( ˆ Q i r ), ˆ Q i r )≥0 No contract ˆ Q od r , ˆ Q od r ˆ Q i r , ˆ Q odi r Table E.4: More Possible Equilibria in Case 4 113 In this scenario, the manufacturer may prefer to deliver nothing to the retailer. In some extreme cases, the manufacturer simply decides not to purse the contract at all . 5. (w+b)<p m , (w+b)<α r p m +(1−α r )(c+ψ ′ ( ˜ Q m )) and dΠm dQr first increases then decreases: ˜ Q od r ≤ ˜ Q od0 r ≤ ˜ Q i r Π m (Q m ( ˜ Q od r ), ˜ Q od r )≥0 Π m (Q m ( ˜ Q od0 r ), ˜ Q od0 r )<0 ˜ Q od r , ˜ Q od r 0, ˜ Q od0 r Table E.5: Possible Equilibria in Case 5 ˜ Q od r ≤ ˜ Q i r < ˜ Q od0 r ≤ ˆ Q r ˜ Q od r ≤ ˜ Q i r < ˆ Q r < ˜ Q od0 r Π m (Q m ( ˜ Q od r ), ˜ Q od r )≥0 Π m (Q m ( ˜ Q i r ), ˜ Q i r )<0 Π m (Q m ( ˜ Q od r ), ˜ Q od r )≥0 ˜ Q od r , ˜ Q od r 0, ˜ Q od0 r ˜ Q od r , ˜ Q od r Table E.6: More Possible Equilibria in Case 5 ˜ Q i r < ˜ Q od r < ˆ Q r ˜ Q i r < ˆ Q r < ˜ Q od r Π m (Q m ( ˜ Q od0 r ), ˜ Q od0 r )<0 Π m (Q m ( ˜ Q i r ), ˜ Q i r )≥0 Π m (0, ˜ Q i r )≥0 0, ˜ Q od0 r ˜ Q i r , ˜ Q odi r ˜ Q i r , ˆ ˜ Q odi r Table E.7: More Possible Equilibria in Case 5 If dΠm dQr | Qr= ˆ Qr ≥0,theresultisthesameasScenario 4. If dΠm dQr | Qr= ˆ Qr <0,thenwemay have ˜ Q i r < ˆ Q r . We only consider the case when ˜ Q i r exists, otherwise, the contract never exists. Therefore, we get dΠm dQr | Qr≤ ˆ Qr < 0 and the manufacturer always allocates products to his own channel. 114 Appendix F Detailed Analysis in Concave Case Manufacturer’s problem Here, we provide additional analysis for the manufacturer’s choices. If (w +b) < p m , the sign of dΠm dQr is not always the same for Q r ∈ ( ˆ Q r , Q o r ], and the manufacturer may or may not deliver the full order size to the retailer. We know that d 2 Π m dQ r dQ r =p m f(Q m )G(Q r ) dQ m dQ r +α r F(Q m )g(Q r )+A m 1+ 1 α r dQ m dQ r (1−α r ) . The sign of d 2 Πm dQrdQr is unclear if dQm dQr < 0; if dQm dQr > 0, we must have d 2 Πm dQrdQr > 0. From Proposition 8, we know that dQm dQr ≥ 0 for Q r >Q r . When Q r becomes larger, d 2 Πm dQrdQr can remain positive. In general, we have two possible shapes of dΠm dQr : one is always increasing and the other is a v-shape curve. Recall that in the convex capacity cost case, we have scenarios with an inverse-v shape curve. Due to the possible shapes of dΠm dQr , we analyze the following two situations. Situation 1. When dΠm dQr is increasing in Q r for all Q r ∈( ˆ Q r , Q o r ], then the manufacturer either delivers Q d r =Q o r or nothing. The manufacturer will deliver full order Q o r if one of the following conditions holds: 1. (w−c+b)−ψ ′ (0)≥0; 2. (w−c+b)−ψ ′ (0)<0 and Π m (Q m (Q o r ),Q o r )≥0. Situation 2. When dΠm dQr first decreases and then increases w.r.t. Q r for Q r ∈( ˆ Q r , Q o r ], we could expect three candidate solutions, Q d r ∈{0, ˜ Q i1 r , Q o r }, where both ˜ Q i1 r and ˜ Q i2 r (≥ ˜ Q i1 r ) satisfy (w+b)=p m [1−(1−α r )L 1 −α r F(Q m ( ˜ Q i r ))G( ˜ Q i r )]. (F.1) The optimal responses of the manufacturer are characterized in Proposition 11. Retailer’s possible solutions Now, we examine the retailer’s optimal order quantities when ψ ′′ <0. 1. Q d m >0 and Q d r =Q o r . ˜ Q od r =argmaxΠ r (Q o r )=(p r −w)Q o r −p r ( ¯ A o + ¯ B o ), 115 where ¯ A o = R Qm 0 R Q o r 0 (Q o r −y)f(x)g(y)dydx, ¯ B o = R Qm+Q o r /αm Qm R Q o r −αm(x−Qm) 0 [Q o r −y−α m (x−Q m )]f(x)g(y)dydx. Therefore, dΠr dQ o r = ∂Πr ∂Qm dQm dQ o r + ∂Πr ∂Q o r = p r α m RQm+ Q o r αm Qm R Q o r −αm(x−Qm) 0 f(x)g(y)dydx· dQm(Q o r ) dQ o r +(p r −w)−p r R Q o r 0 RQm+ Q o r −y αm 0 f(x)g(y)dxdy. 2. Q d m =0 and Q d r =Q o r . ˆ Q od r =argmaxΠ r (Q o r )=(p r −w)Q o r −p r ˆ B o , where ˆ B o = Z Q o r /αm 0 Z Q o r −αmx 0 [Q o r −y−α m x]f(x)g(y)dydx. We have the FOC dΠ r dQ o r =(p r −w)−p r Z Q o r αm 0 Z Q o r −αmx 0 f(x)g(y)dydx. Itiseasytoseethat dΠr dQ o r | Q o r =0,Qm=0 =p r −w>0, dΠr dQ o r | Q o r = ˆ Qr,Qm>0 = dΠr dQ o r | Q o r = ˆ Qr,Qm=0 , and dΠr dQ o r | Q o r =∞,Qm>0 ≤ −w < 0. Therefore, there exists at least one finite solution; that is to say, the retailer will only order finite amount of products from the manu- facturer in this situation. From Proposition 8, we know that dQm(Qr) dQr ≤ 0 for Q r ∈ ( ˆ Q r ,Q r ), and dQm(Qr) dQr ≥ 0 for Q r >Q r . Suppose ˆ Q od r < ˆ Q r exists; then, for all Q r ∈ ( ˆ Q r ,Q r ) we have dΠr dQ o r < 0, and there cannot exist a solution ˜ Q od r ∈( ˆ Q r ,Q r ). Moreover, under proper conditions, we can show that dΠr dQ o r < 0 for all ˜ Q od r > ˆ Q r . Now, suppose ˆ Q od r < ˆ Q r does not exist; then, we know dΠr dQ o r | Qr= ˆ Qr > 0. Since dΠr dQ o r | Q o r = ˆ Qr,Qm>0 = dΠr dQ o r | Q o r = ˆ Qr,Qm=0 and dΠr dQ o r | Q o r =∞,Qm>0 ≤−w< 0, there exists at least one solution ˜ Q od r > ˆ Q r ; therefore, we will not have the two solutions exist at the same time. Weneedtoverifytheuniquenessofthesolution. Itiseasytoseethatwhen ˆ Q od r exists, it must be unique. The uniqueness of ˜ Q od r still remains to be shown. In general, it is quite challenging to demonstrate this analytically, so we run numerical simulations andformostcommonlyuseddistributionsandcostfunctions, theresultalwaysholds. 3. Q d m >0 and Q d r = ˜ Q i1 r . 116 Whenb>0,then ˜ Q odi r ( ˜ Q i1 r )= 1 b ·max{0, ˜ Π m (Q m ( ˜ Q i1 r ), ˜ Q i1 r )},where ˜ Π m (Q m ( ˜ Q i1 r ), ˜ Q i1 r ) is the manufacturer’s profit when Q o r =0 in the objective function. 4. Q d m >0 and Q d r =0. When b>0, then ˜ Q od0 r = 1 b ·max{0, Π m ( ˜ Q m ,0)}. 5. Q d m =0 and Q d r =0. ˆ Q od0 r =0. It’s also easy to see that ˜ Q od r ≤ ˜ Q odi r and ˆ Q od r ≤ ˆ Q odi r . More equilibrium results. In this section, we state more detailed equilibrium results for more general cases. Al- though ˆ Q r is not always smaller than Q r , we are more interested in cases in which it is. Using this threshold, we may further separate the region of interest into two segments, Q o r ∈( ˆ Q r , Q r ) andQ o r >Q r . Depending on the location of the retailer’s optimal order size with respect these two different segments, the manufacturer may respond very differently. Situation 1. When dΠm dQr is increasing in Q r for all Q r ∈( ˆ Q r , Q r ), then dΠm dQr increases for all Q r > ˆ Q r . The result when the retailer orders ˜ Q od r is very similar to the Corollary 12. Moreover, we also identify conditions when the manufacturer delivers nothing in the following proposition. Proposition F.1 When ˜ Q od0 r > ˆ Q r , the retailer orders ˜ Q od0 r and the manufacturer actually delivers nothing to the retailer only if (w−c+b)−ψ ′ (0)<0 and one of the following holds: 1. dΠm dQr | Qr= ˜ Q od0 r <0; 2. dΠm dQr | Qr= ˜ Q od0 r ≥0 but Π m (Q m ( ˜ Q od0 r ), ˜ Q od0 r )<0. When the retailer’s order size is not large, the manufacturer does not enjoy economies of scale from two channels; therefore, he may only focus on his own channel. In fact, this only happens when ˆ Q r = 0, and the manufacturer does not rely on the retailer’s order at all. When b is positive, the retailer may extract enough profits from the manufacturer by overordering. Therefore, the manufacturer may not want to sign the contract with the retailer at the very beginning in this scenario. However, during economy downturns, the two players may still want to sign the contract even if one of the players does not make profits—the manufacturer may want to keep the factory running, and also maintain good relationship with the retailer. When the economy booms again, the healthy relationship can help the manufacturer to get involved in new, profitable contracts very fast. 117 Situation 2. When dΠm dQr first decreases and then increases w.r.t. Q r for Q r > ˆ Q r , we could expect three candidate solutions, Q d r ∈{0, ˜ Q i1 r , Q o r }, where both ˜ Q i1 r and ˜ Q i2 r (≥ ˜ Q i1 r ) satisfy (w+b)=p m [1−(1−α r )L 1 −α r F(Q m ( ˜ Q i r ))G( ˜ Q i r )]. Optimal strategies are characterized in the following proposition. Proposition F.2 When dΠm dQr first decreases and then increases w.r.t. Q r for Q r > ˆ Q r , the equilibrium order size and the allocation size to the retailer’s channel, (Q o r , Q d r ), are given as follows: 1. When (w−c+b)−ψ ′ (0)≥0, then • if ˜ Q i r does not exist or ˜ Q i1 r ≥ ˜ Q od r > ˆ Q r , we have ( ˜ Q od r , ˜ Q od r ); • if ˜ Q odi r > ˜ Q od r > ˜ Q i1 r > ˆ Q r , we have ( ˜ Q odi r , Q i1 r ). 2. When (w−c+b)−ψ ′ (0)<0, then • if ˜ Q od r > ˆ Q r and Π m (Q m ( ˜ Q od r ), ˜ Q od r )≥0, we have ( ˜ Q od r , ˜ Q od r ); • if ˜ Q odi r ≥ ˜ Q od r > ˜ Q i1 r > ˆ Q r and Π m (Q m ( ˜ Q odi r ), ˜ Q odi r )<0, we have ( ˜ Q odi r , ˜ Q i1 r ); • if ˜ Q i1 r ≥ ˜ Q od0 r > ˆ Q r and Π m (Q m ( ˜ Q od0 r ), ˜ Q od0 r )<0, we have ( ˜ Q od0 r , 0); • if ˜ Q i2 r > ˜ Q od0 r > ˜ Q i1 r > ˆ Q r and Π m (Q m ( ˜ Q i1 r ), ˜ Q i1 r )<0, we have ( ˜ Q od0 r , 0). 118 Appendix G Additional Figures Additional figures mentioned in Section 1.4.5 are shown in this section. 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 μ r s Demand c m =0.5 c r s =0.25 c m o =0.22 γ=1 d m o d r s 0 Figure G.1: Demand when |c r s −c r o | small 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0 0.05 0.1 0.15 0.2 0.25 μ r s Demand c m =0.5 c r s =0.25 c m o =0.05 γ=1 d m o d r s 0 Figure G.2: Demand when |c r s −c r o | large 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 μ r s Price c m =0.5 c r s =0.25 c m o =0.22 γ=1 p m o p r s w Figure G.3: Price when |c r s −c r o | small 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 μ r s Price c m =0.5 c r s =0.25 c m o =0.05 γ=1 p m o p r s w Figure G.4: Price when |c r s −c r o | large 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0 0.005 0.01 0.015 0.02 μ r s Profit c m =0.5 c r s =0.25 c m o =0.22 γ=1 Π r Π m Figure G.5: Profit when |c r s −c r o | small 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0 0.01 0.02 0.03 0.04 0.05 0.06 μ r s Profit c m =0.5 c r s =0.25 c m o =0.05 γ=1 Π r Π m Figure G.6: Profit when |c r s −c r o | large 119 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.02 0.04 0.06 0.08 0.1 0.12 μ m o Demand α=10 β=0.2 γ=0.2 μ m o ∈[0.14,0.48] d r o d r s d m o Figure G.7: Demand when β = γ 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 μ m o Demand α=0.2 β=1 γ=0.2 μ m o ∈[0.27,0.40] d r o d r s d m o Figure G.8: Demand when α = γ 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.65 0.7 0.75 0.8 0.85 0.9 μ m o Price α=10 β=0.2 γ=0.2 μ m o ∈[0.14,0.48] p r o p r s p m o w Figure G.9: Price when β = γ 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 μ m o Price α=0.2 β=1 γ=0.2 μ m o ∈[0.27,0.40] p r o p r s p m o w Figure G.10: Price when α = γ 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 μ m o Profit α=10 β=0.2 γ=0.2 μ m o ∈[0.14,0.48] Π r Π m Figure G.11: Profit when β = γ 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 μ m o Profit α=0.2 β=1 γ=0.2 μ m o ∈[0.27,0.40] Π r Π m Figure G.12: Profit when α = γ 120

Abstract (if available)

Abstract We investigate a competitive dual-channel supply chain with one manufacturer and one retailer. In the first part, we study the pricing strategies made by the manufacturer and the retailer. In order to characterize the market interaction, we introduce two essential sets of parameters, absolute demand coefficients and channel differentiation coefficients. Both sets of parameters are critical for identification of the optimal channel selection and pricing strategies in the centralized and decentralized systems. In a vertically integrated system, the decision maker has to make tradeoffs between cannibalization and larger potential market demand. We also note that cost differences, rather than the cost magnitudes, are important in deciding optimal channel choices and pricing strategies. In the decentralized system, the manufacturer can compete with the retailer by using her own online stores, while the retailer can use physical stores, online stores, or both. We characterize the conditions under which the retailer prefers a single channel over dual channels, and under which the manufacturer chooses to enter the market. We also investigate the pricing strategies used by the two players. The manufacturer can use the wholesale price and the online price to influence the prices set by the retailer. When the competition level is small, the manufacturer may reduce the wholesale price and give the retailer more freedom in choosing optimal pricing strategies, which benefits both players. When the competing channels are much more substitutable, the manufacturer faces an increased threat from the retailer's stores. As a result, she may use both prices to compete with the retailer and even start price wars, which can hurt both parties. ❧ In the second part, we investigate the impact of capacity investment and allocation in a competitive manufacturer-retailer supply chain system. The manufacturer makes products and sells through both his own channel and the retailer. Given exogenous selling prices, we analyze the equilibrium quantity strategies made by the manufacturer and the retailer. Faced with different types of production cost structures and additional penalty costs, the manufacturer may not always allocate products to both the retailer and himself. Moreover, even when the manufacturer chooses to deliver products to the retailer, he may only fulfill the order partially. Compared to the efficient centralized system, the manufacturer would like to over-allocate products to both channels and the retailer is more likely to over-order in many situations. In order to resolve these issues, we develop contracts that can coordinate the decentralized competitive supply chain.

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Han, Liang (author)

Core Title The impacts of manufacturers' direct channels on competitive supply chains

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Publication Date 04/26/2012

Defense Date 03/27/2012

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag capacity,coordination,direct channel,OAI-PMH Harvest,Pricing,supply chain

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Sosic, Greys (committee chair), Cheng, Harrison (committee member), Nazerzadeh, Hamid (committee member)

Creator Email han.miracle@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-12546

Unique identifier UC11289069

Identifier usctheses-c3-12546 (legacy record id)

Legacy Identifier etd-HanLiang-653.pdf

Dmrecord 12546

Document Type Dissertation

Rights Han, Liang

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA