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Managing product variety in a competitive environment

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Managing product variety in a competitive environment

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Content MANAGING PRODUCT V ARIETY IN A COMPETITIVE ENVIRONMENT by Nan Xia A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (BUSINESS ADMINISTRATION) August 2008 Copyright 2008 Nan Xia Dedication To my mom, dad and my dear husband, for their endless love and encouragement. ii Acknowledgements Looking back to the years of my doctoral life, I owe much of today’s achievements to the people that have supported me during these years. I would like to express my deepest gratitude to my advisor, Prof. Raj Rajagopalan. Thank you for your years of guidance in my academic research, your kindness and patience, and your unwavering support for me, which have helped me overcome so many difficulties in my doctoral study. I have learned that the most valuable character for succeeding in academic life is persistence. Thank you for guiding me into the won- derland of academic research. It has been a real pleasure working with you on various research problems. I would like to thank my co-advisor Prof. Yehuda Bassok. I would not have had the courage to explore so many research areas without your support and encouragement for pursuing my own ideas. Discussing research with you opened my mind and stimulated my interests in research. I also benefited greatly from your experience in inventory management and bargaining theory. I also want to express my gratefulness to the other members of my dissertation committee: Prof. Guofu Tan, whose knowledge on industrial economics benefited my research, Prof. Sriram Dasu, whose help and insightful comments facilitated my research, Prof. Greys Sosic, who has kindly offered me valuable advice on research writing and presentations, and Prof. Hao Zhang, who has helped me a lot in several iii ways. It was an honor to have you in my committee. I really appreciate all the feedback and experience you shared with me during my doctoral study. In addition, I would like to thank Mahesh Nagarajan, who has helped and supported me during these years. USC is a wonderful place to study and make friends. In the math department where I spent two years, I made many friends who have supported me thereafter. In particular I want to thank Huamei Dong for her kind support and consolation when I was low. It was sweet memories playing tennis with you in the beautiful sunshine of Southern California. I also want to thank Zheying Wu for listening and sharing. I enjoyed the years spent with OM classmates and Marshall classmates and wish them best of luck in their future study. Now I want to express my deepest thanks to my family, my mom, dad, my brother Mu and my dear husband Zheng. Without their love and encouragement, I would not have been able to go through the difficult times. My beloved parents, Lu and Guoping, who have been my role models with their great visions and hardworking, have supported me all the way through these years. My big brother Mu, who has always cared about me, reached out for me in every possible way. Last but not least, I want to thank my loving husband Zheng, whose kind understanding, enduring support and encouragement made all these years unforgettable memories. iv Table of Contents Dedication ii Acknowledgements iii List of Tables vii List of Figures ix Abstract x Chapter 1: Introduction 1 Chapter 2: Standard versus Custom Products: Variety, Lead Time and Price Competition 6 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 Standard versus Standard Products: Variety and Price Decisions 17 2.3.2 Custom versus Custom Products: Lead Time and Price Decisions 20 2.3.3 Standard versus Custom Products: Variety, Lead Time and Price Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.4 Product Strategies in Equilibrium . . . . . . . . . . . . . . . . 26 2.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.1 Impact of Variable Costs . . . . . . . . . . . . . . . . . . . . . 29 2.4.2 Impact of Fixed Costs . . . . . . . . . . . . . . . . . . . . . . 31 2.4.3 Impact of Reputation . . . . . . . . . . . . . . . . . . . . . . . 33 2.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 35 Chapter 3: Assortment Planning and Inventory Decisions Under Retail Com- petition 38 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.1 Product Variants, Prices, and Costs . . . . . . . . . . . . . . . . 44 v 3.2.2 The Customer Demand Model . . . . . . . . . . . . . . . . . . 44 3.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3.1 Structure of the Optimal Assortment . . . . . . . . . . . . . . . 49 3.3.2 Special Case: When There are No Product Substitutions . . . . 52 3.4 Effect of Market and Cost Parameters . . . . . . . . . . . . . . . . . . 53 3.4.1 General Problem with No-Purchase Options . . . . . . . . . . . 54 3.4.2 Effect of Market and Cost Parameters . . . . . . . . . . . . . . 56 3.5 Assortment Structure with Heterogeneous Variable Costs . . . . . . . . 65 3.6 Conclusions and Discussions . . . . . . . . . . . . . . . . . . . . . . . 68 Chapter 4: Retail Assortment and Price Competition with Effect of Brand Rep- utation 72 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3.1 Product Variants, Prices, and Costs . . . . . . . . . . . . . . . . 77 4.3.2 The Customer Demand Model . . . . . . . . . . . . . . . . . . 77 4.4 Analysis with Homogeneous Product Segments . . . . . . . . . . . . . 79 4.4.1 Equilibrium of Price Competition . . . . . . . . . . . . . . . . 79 4.4.2 Product Variety in Equilibrium . . . . . . . . . . . . . . . . . . 81 4.5 Analysis with Heterogeneous Product Segments . . . . . . . . . . . . . 83 4.5.1 Equilibrium of Price Competition . . . . . . . . . . . . . . . . 84 4.5.2 Product Variety in Equilibrium . . . . . . . . . . . . . . . . . . 85 4.6 Effect of Brand Reputation . . . . . . . . . . . . . . . . . . . . . . . . 88 4.6.1 The Price Equilibrium . . . . . . . . . . . . . . . . . . . . . . 88 4.6.2 Effect of Brand Reputation on Equilibrium Variety and Prices . 90 4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Chapter 5: Conclusions and Future Directions 94 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 References 98 Appendix: Proofs 102 vi List of Tables 2.1 Threshold Value ofI C Above Which the Equilibrium Shifts from(C;C) to(S;S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1 Firm 1’s Market Share for Each Segment when Customers Have No- purchase Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Comparisons of Assortment Decisions with Same Variety . . . . . . . . 55 3.3 Effect of Competitor’s OfferingS 2 on the Optimal Assortment . . . . . 57 3.4 Effect of Segment Heterogeneity on the Optimal Assortment: S 2 = [1;1;0;0] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5 Effect of Utility Difference(u¡u 0 ) on the Optimal Assortment: S 2 = [1;1;0;0] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.6 Effect of Product Misfit Cost d on the Optimal Assortments: S 2 = [1;1;0;0] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.7 Effect of Transportation Cost t on the Optimal Assortments: S 2 = [1;1;0;0] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.8 Firm 1’s Market Shares with Different Assortments Under Differentt . 59 3.9 Firm 1’s Profits Under Different Assortments . . . . . . . . . . . . . . 60 3.10 Effect of Segment Heterogeneity on the Equilibrium Assortments . . . 60 3.11 Effect of(u¡u 0 ) on the Equilibrium Assortments . . . . . . . . . . . 61 3.12 Effect oft on the Equilibrium Assortments . . . . . . . . . . . . . . . . 62 3.13 Effect ofd on the Equilibrium Assortments . . . . . . . . . . . . . . . 63 3.14 Effect of Market V olume¸ on the Equilibrium Assortments . . . . . . . 63 3.15 Effect of Retail Pricer on the Equilibrium Assortments: c=1 . . . . . 64 vii 3.16 Effect of Variable Costc on the Equilibrium Assortments . . . . . . . . 64 3.17 Effect of Fixed CostK on the Equilibrium Assortments . . . . . . . . . 65 viii List of Figures 2.1 Product and customer space where firm 1 offers standard products and firm 2 offers custom products. . . . . . . . . . . . . . . . . . . . . . . 14 3.1 Product and customer space for assortment planning under competition. 45 ix Abstract Product proliferation has been observed in many product categories over the years. It is commonly believed that increasing variety would potentially attract more customers and hence yield more profit. However, there are variety-associated costs as well, and whether firms in a competitive environment would benefit from increasing product vari- ety is unclear. Therefore, this dissertation studies how to balance the costs and ben- efits of product variety in presence of competition. More specifically, the following research questions are explored: In presence of competition, how should a firm choose the product strategies that involve production technology (mass production or mass cus- tomization), product selection or assortment planning, pricing, and inventory decisions in order to maximize its overall profits? How do these decisions interact with each other, and what is the effect of market characteristics and cost factors on these variety-related decisions? We analyze competitive variety management problems along different dimensions: in the first essay, we study the competitive production technology strategies faced by manufacturers and the trade-offs between standard products and customized products, along with variety, lead time and pricing decisions; in the second essay we examine the assortment planning and inventory decisions faced by retailers, in consideration of the competitor’s choices; the third essay studies the interaction between variety and pricing x strategies in presence of competition and the effect of firm reputation on these decisions, which may apply to both manufacturers and retailers. This dissertation makes the following contributions: 1. It proposes a modeling framework to analyze the product variety management problems in a competitive set- ting. In contrast to the models in literature, it incorporates customers’ heterogeneous preferences for firm and for product attribute, and it models customer behaviors in response to firms’ production technology, product assortment and pricing strategies. Therefore, we are able to analyze how these decisions should be made in order to maxi- mize a firm’s profit in competition. 2. We demonstrate the strategic interactions among decisions including production technology, product assortment, pricing and stocking quantities in presence of competition, and identify the effect of market characteristics and cost factors on those decisions. The modeling framework proposed and the insights obtained contribute to the literature on product variety management. xi Chapter 1 Introduction Throughout the last several decades, there has been product proliferation in many prod- uct categories, especially in consumer products, such as consumer electronics (personal computers, digital cameras and televisions), apparel, and staples such as shampoos and toothpaste. The incentives for expanding product variety are manifold. First, it is a lever- age to maintain or capture more market share as consumers demand for more variety and increased personalization. It is also facilitated by the development of flexible manufac- turing technologies that have reduced the cost of product proliferation and the availabil- ity of new channels of distribution such as the internet. However, product proliferation is not free. It increases the operational costs such as production costs as it requires more changeovers or production facilities, inventory costs, and distribution costs such as ship- ping costs. For retailers, it also increases the book-keeping cost, or manual cost, for the price labeling. Furthermore, product proliferation significantly complicates firms prod- uct strategy decisions that may involve the configuration, inventory level and pricing for each variant, and makes it more difficult to manage the supply chain activities across production, inventory and distribution. Therefore, how to strike the balance, to provide the optimal level of product variety where the benefits justify the costs, has become an important issue in the competitive strategies of both manufacturers and retailers. Product variety, which is more of a strate- gic level decision, usually affects other tactical decisions which are made later such as 1 pricing and inventory decisions, and therefore it should not be dealt with alone. Gen- erally speaking, product variety management includes the following interrelated deci- sions: product line or assortment decisions–how many and what variants to offer, or which segments to target; pricing decisions; inventory decisions. In addition, it also needs to take into account of competitors offerings and decisions. Aimed to contribute to a better decision making in product variety management, this dissertation proposes a modeling framework to investigate variety management problems when firms jointly optimize these interrelated decisions in a competitive environment and analyze how the market characteristics and cost factors affect the variety-related decisions. The frame- work also incorporates customers’ heterogeneous preferences for firm and for product attribute and model the customer behavior. Because these problems involve interaction between different entities (manufacturer, retailer and customer) each being interested in maximizing its own individual utility/profitability, they are analyzed using tools from the mathematical theory of optimization as well as the theory of games and economic behavior. This dissertation consists of three essays (organized in chapters 2,3 and 4 respec- tively) that deal with different aspects of variety management and offer different insights, with a central theme of managing product variety in a competitive environ- ment. Motivated by the emerged trend of product customization in personal computers and apparel, etc, the first essay (Chapter 2), titled “Standard versus Custom Products: Variety, Lead Time and Price Competition”, studies firms strategic choice of produc- tion technologies (mass production vs. mass customization) in a competitive setting, along with variety, lead time and price decisions. It proposes a modeling framework in which customers have heterogeneous preferences for firm and for product attribute. Firms can decide to use mass customization to offer customized products, which can match customers’ preferences exactly but incur a certain lead time that brings disutility 2 to customers, or to use mass production to offer standard products, which are avail- able immediately but may not satisfy customers’ preferences. The essay focuses on the tradeoffs between standard and custom products and ask the following questions: if two firms compete in the same product category, and each can choose to offer a variety of standard products or custom products with certain lead time, then under what condi- tions will firm choose to offer custom products over standard products, and what are the corresponding variety, lead time and pricing strategies? The essay characterizes the equilibrium outcome and shows that whether firms will choose to offer custom products over standard products depends on the cost efficiencies of the production technologies as well as the consumer sensitivity to product fit and lead time. An index is developed that signifies the relative attractiveness of the standardization and customization strate- gies and the potential outcomes. In addition, the essay identifies the strategic roles of product variety and lead time in the competition. In the extension, the essay further analyzes the impact of asymmetric variable costs, fixed costs and brand reputation on the equilibrium decisions. Although manufacturers have expanded product variety in many product categories, retail stores, faced by limited shelf space and inventory costs, often select a subset of variants to offer. They need to decide which variants to offer and how much to stock, in consideration of what the nearby store offers in this product category. Therefore, the second essay (Chapter 3) investigates the assortment planning and inventory decisions under retail competition. Two retailers who compete in a product category choose their assortments from a same set of product variants offered by the manufacturer, and then determine the inventory levels under exogenous prices. Customers have their ideal prod- uct preferences and choose which store to visit based on their location and the stores’ assortment offerings. They incur some disutility if their ideal product is not offered and they make assortment-based substitution. It is shown that the optimal assortment for 3 a retailer in response to the competitor’s offerings contains the most popular variants offered by the competitor and the most popular variants not offered by the competitor. In contrast to the monopoly results in previous literature, the optimal assortment may not cover a contiguous set of the most popular segments. The assortment structure in equilibrium is also analyzed. It is found that in equilibrium both firms offer variants for major segments but differ in product offerings for smaller segments. The essay further examines the effect of cost structures on the assortment offerings and shows that when products have different variable costs but similar segment sizes, the optimal assortment consists of the least costly variants offered by the competitor and the least costly vari- ants not offered by the competitor, which mirrors the optimal assortment structure when products have different segment sizes but identical variable costs. Based on a framework similar to the one in the second essay, the third essay (Chapter 4) titled “Retail Assortment and Price Competition with Effect of Brand Competition” focuses on the retailers’ assortment and pricing decisions in a competitive setting. Cus- tomers make the store choices based on their location, the stores’ product offerings and prices charged. Stores can attract more customers by expanding the variety or lowering the prices, but they also incur more costs. We characterize the assortment and pricing strategies in response to the competitor’s offerings and in equilibrium. The effect of market characteristics is then studied which leads to the following results. As market potential increases, the optimal variety increases at a lower rate, which shows a decreas- ing return from variety. The equilibrium variety increases when customer have more heterogeneous preferences or less willing to substitute between products. Sometimes customers may be willing to pay more if a retail store has higher reputation because of a better store environment or better service. We find that in equilibrium, the store with a higher reputation will offer a larger variety and charge higher prices, if both stores offer same quality of products. 4 Finally, in Chapter 5 we provide concluding remarks and discuss possible directions for future research. 5 Chapter 2 Standard versus Custom Products: Variety, Lead Time and Price Competition 2.1 Introduction For the past few decades, there has been an explosion of choice in almost every product category. Further, an increasing number of firms have begun making custom products to better meet customers’ needs and tastes, competing against traditional firms offer- ing standard products. Examples range from customized footwear (NikeID, Digitoe), apparel (Land’s End), to custom packaged nutritional supplements (Acumins). Mass customization has been touted as the next wave and the holy grail of manufacturing [Pin93] leading to the demise of mass production as a means to meet the growing demand for product variety. Wind and Rangaswamy [WR01] point out that many prod- ucts are being customized and offer the notion of “customerization” wherein the cus- tomers help customize the products by exploiting the power of the internet. Zipkin [Zip01] on the other hand argues that mass customization may not work for many prod- uct categories and provides several examples to substantiate his assertion. Huffman and Kahn [HK98] point out that increased customization may lead to a confusing set of choices for the consumer and has to be carefully managed. 6 Two forces seem to be at work here. One is a demand pull – the need for firms to maintain or increase their market share by better meeting customer preferences for var- ious product attributes. Second is a supply push – the ability of firms to offer increased variety without significantly increasing their costs due to more flexible manufacturing capabilities and distribution channels. As the capabilities of firms to fine tune their products to meet consumer preferences improve, the question that arises is: are we more likely to see firms meeting the consumer demand for increased variety and personaliza- tion by offering several standard products as we observe in, say, cereals or by offering custom products as claimed by the mass customization advocates? Let us consider some anecdotal evidence. In many product categories, only stan- dard products are offered, for example TVs, breakfast cereals, and cameras, while only custom products are offered in others – for example, airplanes and heavy transporta- tion equipment. The website madeforone.com provides many instances of existing and new customized products and services. There are a few industries where both custom and standard products are offered, e.g. apparel, furniture, but the proportion of custom product sales is typically small. Sales of catalog apparel retailers, e.g. Land’s End, is only 3-4% of retail store apparel sales (US Census Bureau Retail Survey 2005), even assuming catalog retailers’ entire sales represent custom products, which is not the case. Overall, one finds very few industries or markets wherein both standard and custom products enjoy substantial market shares. An exception is the PC industry, where Dell has won a large share of the market by offering customized PCs. Interestingly however, its major competitors such as Hewlett-Packard and Lenovo have also started to offer custom PCs through their websites. As we point out in the next section, since the emergence of mass customization is recent, there have been only a few academic works on this topic. In this paper, we consider a duopoly wherein both firms have the choice of offering either standard or 7 custom products and attempt to address the following questions: 1 First, under what market conditions is it profitable for firms to offer custom products versus standard products? In equilibrium, will firms always pursue the same product strategies? Second, if it is better to offer standard products, what is the optimal variety to provide and how is it related to market characteristics? Third, when it is optimal to offer custom products, what is the optimal lead time to quote? Finally, what are the effects of product variety and lead time on the market shares and prices? To address these questions, we construct a game-theoretic duopoly model, where customers are heterogeneous in firm and product preferences. Firms will first choose between providing a limited number of standard products, which do not meet product attribute preferences exactly but are available immediately, and custom products, which meet product attribute preference exactly but are available only after a certain lead time. Depending on the product type decisions, the firms then determine variety and lead time, and finally set prices. The paper offers several interesting insights. First, we find that in equilibrium the domination of a product type strategy depends on its cost efficiencies on the supply side and the attractiveness on the demand side. Specifically, we develop an index for customization and for standardization based on costs and demand sensitivity, and the product strategy that has a smaller index value is the strategic choice of both firms. Second, variety and lead time can influence consumer store choice. Increasing the variety for standard products or shortening the lead time for custom products can boost the sales and margin, with decreasing returns. As a consequence, when the pro- duction technology becomes more cost efficient, the firm will be able to provide a larger variety or shorter lead time. Further, in contrast to the previous literature, we find that increasing the variety will not intensify the price competition, if there is sufficient firm 1 Henceforth, we use “production technology”, “product type” and “product strategy” interchangeably to refer to the choice of standardization or customization. 8 differentiation. Rather, it relieves the price pressure for the firm as it satisfies consumer needs better and enables higher price premiums. We also find that brand reputation does not affect the product strategy, but a firm with higher reputation will offer a larger variety or shorter lead time to exploit the consumer surplus and enjoy higher prices and margins. The rest of the paper is organized as follows. In the next section we review the related literature. In Section 2.3 we present our model and characterize the equilibrium strategies in terms of production technology, variety, lead time and prices. In Section 2.4 we further analyze the impact of asymmetric variable costs, fixed costs and brand reputation on the equilibrium strategies. We present our conclusions in Section 2.5. 2.2 Literature Review There is only a limited academic literature in mass customization but since our research draws upon the literature in product differentiation, specifically spatial and horizontal differentiation, we briefly review the relevant literature on product customization and differentiation which can be found in economics, marketing and management science journals. The spatial differentiation literature is primarily based on Hotelling’s model [Hot29] in which two firms compete in location and price within a linear city. The initial litera- ture on spatial competition is reviewed by Gabszewicz and Thisse [GT96]. As the loca- tional ideas in Hotelling’s model are extended into virtual spaces in product characteris- tics, the literature expands into multi-product firms in both monopolistic and oligopolis- tic structures. Lancaster [Lan90] gives a full review of product variety under differ- ent market structures. More recent surveys of product differentiation models include [MW01] and [And05]. Some key ideas from these surveys are that price competition 9 becomes more intense when transport costs become more convex and price discrimi- nation is allowed. Also, firms tend to offer less variety to reduce the intensity of price competition. For instance, Martinez-Giralt and Neven [MGN88] analyze a two-stage game where firms decide on the locations of their two outlets first and then compete in prices. In both circular and linear city models, in equilibrium both firms will collapse their two stores into a single point far apart from each other to relax the intense price competition. Shugan [Shu89] explores when and why premium quality producers may provide a smaller assortment than lower quality producers, focusing on the ice-cream industry. He finds that a super-premium producer will offer a smaller relative assortment than a regular producer when its variable costs are higher, customers are more price-sensitive or the products (lines/assortments) are more substitutable. While there are some linkages between our work and theirs and we discuss these when we present our results, our focus is on standardization versus customization and not just product variety. There is also a popular literature focusing on mass customization. As customers demand more product variety and new manufacturing technologies alter the economics of manufacturing, mass customization emerged as a new business strategy and was pop- ularized by Pine [Pin93] and others (e.g., [Kot89]). Mass customization was proposed as a means to offer unlimited variety and give customers exactly what they want, with low costs achieved through economies of scope–using a single process to produce a variety of products more cheaply and quickly [Pin93]. Piller and Tseng [PT03] provide numerous examples of mass customization and personalization strategies by companies such as Proctor&Gamble, Lego, Adidas, Land’s End and BMW. Zipkin [Zip01] points out that production technologies in many industries may not be cost effective or flexible enough to achieve mass customization. 10 Now we review the literature in product customization that is related to our model, i.e., the ones that deal with the interaction of standard products with custom products. Balasubramanian [Bal98] considers a model with a direct marketer (who could be seen as a customizer) competing with multiple retailers that are evenly distributed on a cir- cle and develop several interesting results on the price equilibrium and the impact of the direct marketer’s entry. Dewan, Jing and Seidmann [DJS03] develop a model of product customization on a circle where the firm chooses a customization scope (an arc segment) to produce custom products with two standard products at the end of the cus- tomization scope. In a duopoly, customization reduces the differentiation between the firms’ standard products and the firms are worse off compared to a monopolist. When the firms adopt customization sequentially, the early adopter achieves an advantage and may be able to deter subsequent entry by strategically choosing its customization scope. Alptekinoglu and Corbett [AC05] study the competition between a firm offering custom products and a firm offering a finite set of standard products. They show that the mass producer can survive the competition even if it has a small cost disadvantage. Our paper differs from the above papers in several respects: First, we incorporate the disutility of lead time for custom products and allow the custom product firm to choose the lead time – a shorter lead time attracts more customers, but incurs a higher cost. More important, we allow each firm to choose either strategy – mass production or mass customization, instead of letting one firm adopt mass production and the other adopt mass customiza- tion. Finally, we allow for heterogeneity in the preference of consumers for the two firms. These key differences result in important new insights, as will be clear later. Syam, Ruan and Hess [SRH05] explore the appropriate level of customization in a competitive setting. Using a two-dimensional attribute space where each firm has a standard product that is located diagonally opposite from the other, they study a two- stage game where firms choose whether and which attribute to customize in the first 11 stage and set prices in the second stage. They find that the equilibrium could be no customization or partial customization, in which both firms choose one and the same attribute to customize in an effort to avoid a price war, but not customization on both attributes. The focus of our work is very different, as we model variety and lead time decisions rather than which attribute to customize. Therefore, the insights obtained from our work are quite different. Syam and Kumar [SK06] study a duopoly model where the firms may offer cus- tomized products in addition to the standard products. They assume that each firm offers only one standard product, but can choose the degree of customization for custom products. They find that firms can profit by offering custom products and tend to offer partial rather than full customization to reduce price competition. Further, the degree of customization is lower when both firms offer customized products relative to the case where only one firm offers customized products. The differences between our paper and their work are the following: First, we allow the firm to decide the number of standard products to offer, as is generally observed in practice, for example in jeans or cosmetics. Second, we model the lead time decision for custom products, which is important as custom products almost always require long lead times and this is a potentially signif- icant drawback of custom products in the minds of consumers. Finally, we allow each firm to offer standard or custom products but not both while they assume that both firms offer one standard product and can decide whether or not to offer a custom product. Our model results in several new insights regarding the impact of variety and lead time. Syam, Krishnamurthy and Hess [SKH07] present an interesting model that blends ideas of uncertain consumer preferences about an ideal product attribute with regret to determine the optimal consumer choice when they can choose between customized and standard products. They provide several interesting analytical findings coupled with experimental results. First, when consumers are likely to have high regret levels, they are 12 likely to choose a standard product. Second, the presence of two standard products may increase the attractiveness of a custom product relative to having one standard product as an alternative. The focus of our work is quite different: while we do not model uncertain preferences or regret, we allow multiple standard products rather than just one or two. Further, our focus is on the optimal product strategy, variety, lead time and pricing decisions in a duopoly. Our work contributes to the existing literature in the following ways. We study the standardization and customization strategies of firms in a competitive setting while incorporating variety and lead time decisions and the associated costs. We also include consumer heterogeneity in store convenience or firm preference and in product prefer- ences. Therefore, we are able to offer insights not available in the literature, such as the attractiveness of each product strategy measured by an index, and strategic use of the variety and lead time to increase store or firm attraction and to reduce the competitive pressure. 2.3 Model Formulation We consider a duopoly with two symmetric firms competing with each other for theM customers. The customers are distributed uniformly in a rectangular product space with Firm 1’s products located at the left end, denoted byx = 0 and Firm 2’s products at the right end, denoted by x = 1 (Figure 2.1). Each consumer is identified by a point that represents her ideal product and has a unitary demand. The location on the horizontal axis represents the preference for a firm while the vertical axis represents differentiation along a product attribute. For example, if the product category is running shoes, the foot size may represent differentiation along the vertical axis while location on the horizontal axis may reflect relative preference for Nike versus Adidas. A firm may offer standard 13 Firm 1 Firm 2 x . . . . . Brand Differentiation Attribute Differentiation Customer Figure 2.1: Product and customer space where firm 1 offers standard products and firm 2 offers custom products. or custom products. A standard product is available immediately but does not meet the customer’s product attribute preference perfectly while a custom product meets the product attribute preference exactly but is available only after a certain lead time l i : If a firm i decides to offer standard products, it offers n i products. If it decides to offer custom products, it offers essentially an infinite number of products targeted at each customer and it has to specify a lead time l i : The utility for a consumer of getting her ideal product isU and is the same for all consumers. Suppose Firm 1 offersn 1 standard products at a pricep 1 . For a customer at location x along the horizontal axis, the net utility of purchasing a standard product from Firm 1 is denoted by (U ¡ tx¡ d=n 1 ¡ p 1 ). Note that tx is the loss of utility due to the “distance” of a consumer atx from Firm 1 andt represents the “transportation cost” or the intensity of relative preference for a firm. So, higher values of t imply greater firm loyalty and so a customer “closer” to Firm 1 will require a larger price differential to switch to Firm 2’s products. The termd=n 1 represents the loss of utility due to getting a 14 standard product that does not meet the consumer’s product preference exactly 2 . d; the product misfit cost, also represents the degree of substitution between products and as Firm 1 increases the varietyn 1 the disutility decreases since a consumer is more likely to get something close to her ideal product. In the limit, this disutility vanishes as n 1 approaches infinity, i.e. the product offerings meet the customer’s preference exactly. Suppose Firm 2 offers a custom product with a lead timel 2 . Then the net utility of buying the custom product is (U¡t(1¡x)¡kl 2 ¡p 2 ) wherep 2 is the price charged by Firm 2 andk denotes the sensitivity to lead time. kl 2 is the loss of utility due to the lead timel 2 . A higher value ofk implies greater sensitivity to lead time. A consumer buys the product from the firm that provides a higher net utility. We assume thatU is large enough so that the net utility is always greater than zero and so all consumers will buy a product from one of the two firms, i.e. the market is fully covered. Also, we assume that the price p i is the same for all standard products (or all custom products), i.e. there is no price discrimination based on the consumer’s location. Price discrimination is not common in horizontal differentiation contexts. For instance, Lands End charges the same amount for custom dress shirts of the same quality, independent of color, size, etc. Draganska and Jain [DJ06] point out that firms typically charge the same price for SKUs within a product line along characteristics such as scent, color and flavor that relate to horizontal differentiation. Syam, et al. [SRH05] and Alptekinoglu and Corbett [AC05] also provide persuasive arguments for not considering price dis- crimination in their models. The unit cost for either type of product is assumed to bec 2 We make this assumption because the expected distance between a customer and her nearest stan- dard product is inversely proportional to the variety offered under uniform distribution. Under other distributions, d=n 1 is also a good estimate of the expected disutility due to product misfit, under the assumption that standard products are evenly spread out along the attribute dimension which is shown to be the optimal strategy in a monopoly setting [GH06]. Therefore, the results will also apply to the case of non-uniform distributions. 15 initially for ease of exposition. We point out later how the results change if the unit cost is different for standard and custom products. Increasing the number of standard products is equivalent to increasing product vari- ety and there is anecdotal evidence that increased product variety results in higher pro- duction and distribution costs. However, the nature of this increase is not clear and empirical evidence about whether and how production cost increases with variety is inconclusive [KS90]. We assume that the cost of variety is linear in the number of stan- dard products offered, given by Vn i , which is a common assumption in the relevant literature ([DW96], [AC05] and [DJ06]). Each product type produced incurs a fixed costV , which may also include product development costs. Custom products require a lead time to produce as pointed out earlier, unlike stan- dard products. However, since longer lead times create greater disutility for consumers 3 , the custom firm has to carefully choose a lead time. Typically, firms can reduce lead times by incurring additional costs, for instance by expediting production, shipping by air, or acquiring technologies that can quickly produce customized products. Piller and Tseng [PT03] and Pine [Pin93] provide numerous examples of such technologies that help achieve short lead times with more expensive and flexible technologies. To capture this phenomenon, custom products incur an additional cost ¯ l in our model, which is convex, decreasing in the lead time l. Note that as l ! 0; ¯ l ! 1 and so the cost of offering infinite variety with zero lead time using custom products tends to1; the same as the cost of offering an infinite number of standard products, which have zero lead time by definition. Let F s and F c denote the fixed cost required to purchase and install the production system necessary to offer standard and custom products respectively. The production 3 There may be exceptions for certain products and services (e.g., restaurants), where waiting time may signal higher quality. 16 systems or at least some subsystems required to produce customized products may be quite different from that for standard products and these differences are reflected in the fixed costs. We allow both firms to offer either standard or custom products but not both. When the fixed cost incurred for offering custom or standard products is high, it will not be profitable to offer both product types. Also, the equipment, processes and labor required for making standard and custom products are likely to be quite different in many indus- tries. Our objective is to understand whether and when the equilibrium solution will have the firms offering standard or custom products. Further, we would like to explore the impact of competition on the firms’ variety and lead time decisions as well as prices. We consider a three-stage game. At stage 1, each firm decides whether to offer standard or custom products. We refer to each subgame at the end of stage 1 as (i, j) with i, j2 fS, Cg where for example (S, C) denotes that Firm 1 offers standard products and Firm 2 offers custom products. At stage 2, a standard product firm decides the product variety n i while a custom firm decides on the lead time l i . Finally, the firms choose prices at stage 3. Since each firm can decide to offer standard or custom products at stage 1, we con- sider each of the following scenarios first before analyzing the overall game: (1) Both firms offer only standard products. (2) Both firms offer only custom products. (3) One firm offers standard products while the other offers custom products. 2.3.1 Standard versus Standard Products: Variety and Price Deci- sions In the subgame (S,S) where both firms offer standard products, they determine the vari- ety decision at stage 2 and pricing decision at stage 3. We first analyze the pricing game at stage 3, then the equilibrium variety at stage 2. 17 Price equilibrium From the consumer utility functions, we can derive the indifferent customer’s location as: x SS = t+ d n 2 ¡ d n 1 +p 2 ¡p 1 2t (2.1) To focus on the duopoly and exclude the possibility of monopoly scenarios, we impose the conditionx SS 2(0;1) in the price equilibrium, which is satisfied if we assume3t> d, i.e. there is sufficient firm differentiation. Then the customers located between0 and x SS will purchase one of the n 1 standard products offered by Firm 1, while customers located within[x SS ;1] will purchase one of then 2 standard products offered by Firm 2. Givenn i , Firmi’s optimization problem at stage 3 is max p i ¼ i (n 1 ;n 2 ;p 1 ;p 2 )=D i (p i ¡c)¡Vn i (2.2) fori=1;2;whereD 1 =Mx SS ;D 2 =M(1¡x SS ): Using the first order conditions, the optimal pricing strategy for Firms 1 and 2 respec- tively arep 1 = 1 2 ³ p 2 +c+t+ d n 2 ¡ d n 1 ´ andp 2 = 1 2 ³ p 1 +c+t+ d n 1 ¡ d n 2 ´ : We then obtain the price equilibrium at stage 3: p ¤ 1 = c+t+ 1 3 ( d n 2 ¡ d n 1 ) (2.3) p ¤ 2 = c+t+ 1 3 ( d n 1 ¡ d n 2 ) The following proposition is obtained from equation (2.3). Proposition 1. Ifn i >n j , thenp ¤ i >p ¤ j ;i6=j: Thus, in the (S;S) subgame, the firm which offers a greater variety charges higher prices as it is more attractive to potential customers in terms of net utility to them. As 18 an example, Booz Allen [Ham03] suggests that firms often increase variety so that they can charge higher prices. Retailers demand unique packaging for home DVDs from the studios so as to reduce the intensity of price competition at the retail level. This result is unlike those in the literature where increased variety leads to more intense price competition, e.g. Martinez-Giralt and Neven [MGN88]. This is because we allow for firm-specific preferences. Variety equilibrium At stage 2, Firm 1 determines the optimal variety to maximize its profit, anticipating the subsequent equilibrium prices. Firm 1’s market share in the price equilibrium is m 1 = t+ d n 2 ¡ d n 1 +p ¤ 2 ¡p ¤ 1 2t = t+ 1 3 ( d n 2 ¡ d n 1 ) 2t (2.4) and Firm 2’s market share is m 2 = t+ 1 3 ( d n 1 ¡ d n 2 ) 2t (2.5) Therefore, the profit functions of the two firms are ¼ 1 (n 1 ;n 2 ) = M 2t · t+ 1 3 ( d n 2 ¡ d n 1 ) ¸ 2 ¡Vn 1 (2.6) ¼ 2 (n 1 ;n 2 ) = M 2t · t+ 1 3 ( d n 1 ¡ d n 2 ) ¸ 2 ¡Vn 2 (2.7) The equilibrium variety decisions are stated in the following proposition: 19 Proposition 2. If both firms choose standardization, the equilibrium variety and prices are: n SS 1 = n SS 2 = q Md 3V ; p SS 1 = p SS 2 = c + t: The equilibrium variety is con- cave increasing in the market potential and degree of product substitution, and convex decreasing in the variety cost. The equilibrium profits are¼ SS 1 =¼ SS 2 = Mt 2 ¡ q MVd 3 : Thus, in equilibrium, both firms charge a price that exceeds unit variable cost by the amount of transportation costt. Thus, greater the differentiation between the firms, higher the price they can charge. The equilibrium variety is increasing in the mar- ket potential, but at a slower rate as the market size enlarges, because of the decreasing returns from variety. The same is true for the impact on variety of the sensitivity to prod- uct fit. For instance, sensitivity to product fit is likely to be much higher for a personal item such as apparel as compared to, say, a binder. Correspondingly, one is likely to find much greater variety in apparel relative to a binder. Finally, as variety cost increases, the optimal variety would decrease as expected. For instance, the cost of producing sig- nificant variety is likely to be higher for a car as compared to a more modular and less complex product with fewer parts such as a computer and one correspondingly finds Toyota offering only a few option combinations for a Camry as compared to the number of configurations offered on a laptop. That the equilibrium variety is convex decreasing in the variety cost offers interesting predictions: as the variety cost decreases, the equi- librium variety expands in an ever faster rate. That may explain the soaring variety for some products such as yogurt, cereals, toothpastes, etc. 2.3.2 Custom versus Custom Products: Lead Time and Price Deci- sions In the subgame (C,C), both firms provide custom products that is tailored to each cus- tomer’s specification but after a lead time l i . The indifferent customer’s location is 20 x CC = t+kl 2 ¡kl 1 +p 2 ¡p 1 2t : To ensure thatx CC 2(0;1) in the price equilibrium, we assume 3t > kl, where l is an upper bound on the lead time. That is, we assume sufficient firm differentiation to focus on the case where both firms cover part of the market. The customers located between 0 and x CC purchase a custom product from Firm 1, while customers located within[x CC ;1] buy a custom product from Firm 2. Price equilibrium Given lead timesl 1 andl 2 ; Firmi’s optimization problem at stage 3 is: max p i ¼ i (l 1 ;l 2 ;p 1 ;p 2 )=D i (p i ¡c)¡ ¯ l i (2.8) fori=1;2;whereD 1 =Mx CC ;D 2 =M(1¡x CC ): Using first order conditions, the two firms’ optimal pricing strategies are, respec- tively,p 1 = 1 2 (p 2 +c+t+kl 2 ¡kl 1 ) andp 2 = 1 2 (p 1 +c+t+kl 1 ¡kl 2 ). Therefore, the price equilibrium at stage 3 is given by: p ¤ 1 = c+t+ 1 3 (kl 2 ¡kl 1 ) (2.9) p ¤ 2 = c+t+ 1 3 (kl 1 ¡kl 2 ) Proposition 3. Ifl i <l j , thenp ¤ i >p ¤ j ,i6=j: The proof follows easily from equation (2.9). So, the firm which offers a shorter lead time charges higher prices. This is because the firm with longer lead time is less attractive to customers, and so it attempts to com- pensate for this disadvantage by lowering the price. In contrast, the firm offering a shorter lead time will charge a premium to exploit the consumer surplus. This phe- nomenon is not uncommon in practice. For instance, custom drapery is available from 21 different vendors with those offering shorter lead times charging higher prices. The same is true for products such as custom windows and doors (Mullet Door) or precision mechanical components (W.M. Berg). Lead time equilibrium At stage 2, Firm 1 decides on the optimal lead time to maximize its profit. Firm 1’s market share in the price equilibrium is: m 1 = t+kl 2 ¡kl 1 +p ¤ 2 ¡p ¤ 1 2t = t+ 1 3 (kl 2 ¡kl 1 ) 2t (2.10) and Firm 2’s market share is m 2 = t+ 1 3 (kl 1 ¡kl 2 ) 2t (2.11) Thus the profit functions of the two firms are ¼ 1 (l 1 ;l 2 ) = M 2t · t+ 1 3 (kl 2 ¡kl 1 ) ¸ 2 ¡ ¯ l 1 (2.12) ¼ 2 (l 1 ;l 2 ) = M 2t · t+ 1 3 (kl 1 ¡kl 2 ) ¸ 2 ¡ ¯ l 2 (2.13) The following proposition provides the equilibrium lead time. Proposition 4. If both firms choose customization, the lead time and prices offered in equilibrium are: l CC 1 = l CC 2 = q 3¯ Mk ; p CC 1 = p CC 2 = c+t: The equilibrium lead time is convex decreasing in the market potential and sensitivity of lead time, and concave increasing in the lead time cost. The equilibrium profits are¼ CC 1 =¼ CC 2 = Mt 2 ¡ q M¯k 3 : As one might expect, the equilibrium lead time is increasing in the cost parameter ¯ relating to lead time reduction but the increase is at a decreasing rate. This is due to 22 the price competition which limits the extent of increase in lead time and so substantial increases in lead time reduction costs will result in a less than proportional increase in lead time. A similar reasoning also holds for increases in the sensitivity to lead time. Finally, the equilibrium lead time is decreasing in the market potential, but at a slower rate as the market size enlarges. As the market potential increases, the fixed lead time related costs can be spread among more customers and so the firm can afford to have shorter lead times. Observe that the prices in equilibrium are the same as in the previous subgame where both firms produce standard products. Since both firms are symmetric and both offer the same variety or lead time, depending on the subgame, their prices are identical and depend only on the unit variable cost and the transportation costt. 2.3.3 Standard versus Custom Products: Variety, Lead Time and Price Decisions In the subgame (S,C), one firm chooses standardization (assume it is Firm 1 without loss of generality) while the other firm (Firm 2) chooses customization. At stage 2, Firm 1 chooses the number of standard products to offer,n 1 ; and Firm 2 chooses the lead time l 2 for custom products simultaneously. After they both observe n 1 and l 2 , they set the prices at stage 3 which we consider next. Price equilibrium The indifferent customer is located at x SC = t+kl 2 ¡ d n 1 +p 2 ¡p 1 2t : We have x SC 2 (0;1) in the price equilibrium under the previous assumption3t>d and3t>kl. The customers located between 0 and x SC will purchase Firm 1’s standard products, while customers located within[x CC ;1] will buy Firm 2’s custom products. 23 Firm 1’s optimization problem at stage 3 is max p 1 ¼ 1 (n 1 ;l 2 ;p 1 ;p 2 ) = D 1 (p 1 ¡c)¡ Kn 1 ; while Firm 2 maximizes¼ 2 (n 1 ;l 2 ;p 1 ;p 2 )=D 2 (p 2 ¡c)¡ ¯ l 2 ; whereD 1 =Mx SC ; D 2 =M(1¡x SC ): Firm 1’s optimal price for standard products is then given by p 1 = 1 2 ³ p 2 +c+t+kl 2 ¡ d n 1 ´ ; and Firm 2’s optimal price for custom products is p 2 = 1 2 ³ p 1 +c+t+ d n 1 ¡kl 2 ´ : Therefore, the price equilibrium at stage 3 given n 1 and l 2 is, p ¤ 1 = c+t+ 1 3 (kl 2 ¡ d n 1 ) (2.14) p ¤ 2 = c+t+ 1 3 ( d n 1 ¡kl 2 ) As we would expect, Firm 1’s price is increasing in the variety it can offer to its customers. Further, as Firm 2 increases the lead time for custom products, Firm 1 can afford to charge a higher price and still be competitive. As the disutility from its limited variety decreases relative to disutility from the lead time of the custom products, Firm 1 can charge a higher price and Firm 2 has to charge a lower price. We now determine the equilibrium variety and lead time that will be offered by the two firms. Variety and lead time equilibrium In the price equilibrium, Firm 1 and Firm 2’s market shares are m 1 = t+ 1 3 (kl 2 ¡ d n 1 ) 2t (2.15) and m 2 = t+ 1 3 ( d n 1 ¡kl 2 ) 2t (2.16) 24 respectively. At stage 2, Firm 1 chooses the variety that maximizes its profit: ¼ 1 (n 1 ;l 2 )= M 2t · t+ 1 3 (kl 2 ¡ d n 1 ) ¸ 2 ¡Vn 1 (2.17) The optimal varietyn 1 satisfies the first order condition µ 3t d + kl 2 d ¡ 1 n 1 ¶ 1 n 2 1 ¡ 9Vt Md 2 =0 (2.18) At the same time, Firm 2 chooses the lead time so as to maximize its profit, ¼ 2 (n 1 ;l 2 )= M 2t · t+ 1 3 ( d n 1 ¡kl 2 ) ¸ 2 ¡ ¯ l 2 (2.19) and the corresponding first order condition is, µ 3t k + d kn 1 ¡l 2 ¶ l 2 2 ¡ 9¯t Mk 2 =0 (2.20) The following theorem characterizes the equilibrium variety and lead time in the (S,C) game relative to those in the (S,S) game and (C,C) game. It also provides insights into the prices and market shares of the two firms in equilibrium. Let m ij i denote the equilibrium market share of firmi in the subgame(i;j). Theorem 1. When one firm offers standard products while the other firm offers custom products, the equilibrium variety, lead time, prices and market share satisfy the follow- ing relationships: (i) If Vd > ¯k, then n SC < n SS ; l SC < l CC ; p SC S n SS ; l SC > l CC ; p SC S > p SS S = p CC C > p SC C ; m SC S >m SS S =m CC C >m SC C ; 25 (iii) If Vd = ¯k, then n SC = n SS ; l SC = l CC ; p SC S = p SC C = p SS S = p CC C ; m SC S =m SC C =m SS S =m CC C : wheren SS = q Md 3V ;l CC = q 3¯ Mk ;p SS S =p CC C =c+t;m SS S =m CC C = 1 2 : Let I S = Vd; which is the product of variety cost and sensitivity to product fit, denote an index for standardization, while I C = ¯k, the product of lead time cost and lead time sensitivity, denotes an index for customization. The product strategy that has a smaller index is more attractive for a firm to adopt in the sense that it will incur less costs or/and draw more demand. When I S > I C , in the (S;C) subgame, we find that the custom product firm has a larger market share than the standard product firm despite charging a higher price (m SC S <m SC C ;p SC S <p SC C ). Furthermore, a custom product firm faces less pressure when competing with a standard product firm than when competing with a custom product firm and so it can choose a shorter lead time without worrying about price competition (l SC <l CC ). The larger market share and prices of the custom firm in the (S;C) subgame imply higher revenues and so it can afford to incur the higher costs associated with shorter lead times. In contrast, the standard product firm has greater competitive pressure in the(S;C) subgame than in the(S;S) subgame due to the inherent disadvantage implied by I S > I C , hence it chooses a smaller variety in the(S;C) subgame (n SC <n SS ) and charges a lower price (p SC S <p SS S ). The situation is the opposite whenI S <I C . Only whenI S =I C and the two production technologies are equivalent, do we find that equilibrium variety, lead time, prices and market shares are the same in all the subgames. 2.3.4 Product Strategies in Equilibrium We finally consider the decisions at the first stage, where the firms choose between standardization and customization. The following theorem characterizes the equilibrium product type and profits. 26 Theorem 2. (The Equilibrium Product Type Theorem) (i) IfI S >I C ; then¼ SC S <¼ CC C ;¼ SS S <¼ SC C , and(C;C) is the equilibrium; (ii) IfI S ¼ CC C ;¼ SS S >¼ SC C , and(S;S) is the equilibrium; (iii) IfI S = I C , then¼ SC S = ¼ CC C =¼ SS S = ¼ SC C , and(S;S);(S;C);(C;S);(C;C) are all possible equilibria. We find that when the sensitivity to product fit or/and the variety cost is small rel- ative to the sensitivity to lead time and/or lead time reduction cost, i.e. I S I C and both firms offer custom products. Whether it is attractive to offer standard or custom products depends on many factors and is a function of product characteristics, consumer prefer- ences, and production technologies. For example, the sensitivity to product fitd is likely to be higher for apparel or cosmetics as compared to say a VCR. So, ceteris paribus, one is more likely to see apparel being customized as compared to a VCR but other factors also play a role. If the cost of lead time reduction ¯ or the sensitivity to lead time k is high for an apparel item, then customization may not be an attractive strategy even for an apparel item. Levis found the cost and lead time for customized jeans to be high and abandoned the idea [Wag02]. But some firms have been successful, such as Land’s End even though their lead time for custom apparel is 3-4 weeks because they provide a perfect or close-to-perfect fit. However, this has not been the case for a majority of apparel manufacturers and custom apparel sales is still a tiny proportion of total apparel sales. For example, Piller [Pil06] points out that some apparel firms are unable to truly achieve a custom fit for apparel despite the promise of customization and this leads to consumer disappointment and poor sales. Even if manufacturing technologies become flexible, transportation lead times are likely to be large unless shipment is by air in which case the cost of reducing lead time (¯) is large. Thus, General Mills test-marketed custom cereals but never followed 27 through due to the long lead times [Sch04]. Thus, transportation lead time plays a significant role in determining the lead time for many products and if the cost of reducing transportation lead time is high (say shipping by air versus road or ship) relative to the cost of the item, then customization is never an optimal strategy independent of whether sensitivity to product fit is high or it is cost-effective to customize, etc. Since we allow for each firm to offer a standard or custom product, our results pro- vide different and broader insights than those in [AC05] where one firm pursues a stan- dard and the other a custom strategy. Also, because we allow for firm differentiation through the parameter t, we find that both firms pursuing a custom product strategy need not result in Bertrand competition and zero profits. When we consider a product such as a PC, it is not likely that the sensitivity to product fitd is high for many consumers. But the modular nature of the product implies that the cost of variety and customization is not high and achieving short lead times does not lead to very high costs. Further, the sensitivity to lead timek may not be high since a substantial proportion of the demand is replacement demand. Hence, one is more likely to see firms potentially producing custom products and in fact, we do observe more firms allowing consumers to customize their PCs. So, this is a good example of a scenario where the equilibrium has been shifting from(S;S) to(C;C) due to the lower cost of customization. Theorem 2 also suggests that the market size does not affect the equilibrium product strategies. However, from Theorem 1, the equilibrium variety increases and the lead time decreases as the market grows larger. We note that (S;C) is not an equilibrium unless I S = I C , which is a point rather than a range of parameters. Furthermore, as we point out later,(S;C) is not an equilib- rium under most scenarios even if the variable costs of the two products are different. 28 However, we show later that if custom products and standard products have different fixed costs, then(S;C) may be an equilibrium. 2.4 Extensions In the previous section, we find that when standard and custom products have the same fixed and variable costs, the equilibrium outcome depends on the index values. Next, we look at the impact of asymmetric variable and fixed costs on the equilibrium. Then we study the effect of brand reputation on the equilibrium strategies. 2.4.1 Impact of Variable Costs Let c S and c C denote the unit cost of standard and custom products respectively. We assume that both products have the same fixed costs to understand the impact of variable costs. Theorem 3. When standard and custom products incur different variable costs, the equilibrium outcome is the following: (i) IfI S ·I C andc S <c C ; then(S;S) is the equilibrium; (ii)IfI S ¸I C andc S >c C ; then(C;C) is the equilibrium. In particular, when I S = I C ; then both firms will choose the type of products with smaller variable costs. Theorem 2 says that when standard and custom products have same variable costs, both firms choose the products that have a smaller index value. In contrast, Theorem 3 says that when the index value is the same, the equilibrium product type is the one with smaller variable costs. If a product type with smaller variable cost also has a smaller index, then that product type is the one chosen by both firms. 29 Table 2.1: Threshold Value of I C Above Which the Equilibrium Shifts from (C;C) to (S;S) c S c C I S I C ¯ ¯ ¯ c C ¡c S c S ¯ ¯ ¯ ¯ ¯ ¯ I C ¡I S I S ¯ ¯ ¯ p I S ¡ p I C c C ¡c S 1 1.1 10 8.22 10% 17.8% 2.96 1 1.2 10 6.61 20% 33.9% 2.96 1 1.3 10 5.18 30% 48.2% 2.96 1.1 1 10 11.96 9.1% 19.6% 2.96 1.2 1 10 14.09 16.7% 40.9% 2.96 1.3 1 10 16.40 23.1% 64.0% 2.96 However, if I S > I C and c S < c C ; then the equilibrium could be (S;S) or (C;C) depending on the relative magnitude of the parameters. We illustrate the nature of the equilibrium as a function of these parameter values with some numerical results. Table 2.1 shows the threshold value of I C for the given values of c S , c C and I S ; such that the equilibrium shifts from (C;C) to (S;S) when I C exceeds the threshold. We let M =100;t=2: We observe that variable costs have a bigger impact on the equilibrium than the index values. For instance, if the variable cost of custom products is 20% higher than for standard products (c S = 1; c C = 1:2), then (S,S) will be the equilibrium unless I C is 33.9% less than I S . More specifically, if d = 1; k = 1; V = 10; then (S,S) will be the equilibrium unless ¯ < 6:61: Interestingly, we find that the ratio of p I S ¡ p I C to c C ¡c S is a constant, which also indicates that the effect of index values is less than that of variable costs. In summary, the equilibrium product type will be the one with smaller variable costs as long as its index value is not far bigger than the other’s index value. Therefore, if the custom products have significantly higher variable costs as is true for many product categories [Sch04], it is unlikely that firms will produce custom products in equilibrium. 30 2.4.2 Impact of Fixed Costs We now consider the impact of fixed costs on the equilibrium strategies. First, fixed costs do not have any impact on the equilibrium pricing, variety and lead time decisions and only impact whether each firm will offer standard or custom products. We first list some observations that follow directly from our analysis in the previous sections but take into account the impact of fixed costs. We assume that the variable costs are the same for both product types to isolate the impact of fixed costs. Theorem 4. When the fixed costs of mass production and mass customization are dif- ferent, the equilibrium outcome is the following: (i) IfI S >I C andF S ¸F C then(C;C) is the equilibrium; (ii) IfI S >I C andF S <<F C then(S;S) is the equilibrium; (iii) IfI S >F C then(C;C) is the equilibrium. Thus, even ifI S >I C ,(S;S) can be the equilibrium if the fixed costs of customiza- tion are substantially larger than the fixed costs of standardization. Conversely, even if I S < I C , the equilibrium outcome may be for both firms to produce custom products if the fixed costs of standardization are substantially larger than the fixed costs of cus- tomization. In reality, it is far more likely that the fixed cost of customization is higher. For instance, Archetype Solutions, the firm that has helped Lands End, J.C.Penney, etc. to launch customized products, points out that it takes a lot of effort and cost to cre- ate the necessary customization infrastructure for each new retailer [Wag02]. Hence, it appears that the(S;S) equilibrium is more likely than the(C;C) equilibrium. The result also tells us that whenI S >I C andF S <F C butF S is not much smaller thanF C , then depending on the relative values of the various costs and other parameters, 31 the equilibrium outcome could be (S;C) or (C;C) or (S;S). In particular, the (S;C) equilibrium will arise only under the following condition: (F C ¡F S )<¼ SC C ¡¼ SS S and(F C ¡F S )>¼ CC C ¡¼ SC S (2.21) where ¼ here represents profits without considering fixed costs. So, the (S;C) equi- librium arises only when the fixed cost of customization is larger than the fixed cost of standardization but not substantially larger. Furthermore, the above condition also implies that the sum of profits of both firms in the(S;C) equilibrium is larger than the sum of profits of following the(S;S) and(C;C) strategies. Hence, the range of possible parameter values wherein(S;C) is an equilibrium is narrower than the ranges for(S;S) and(C;C) strategies. From Theorem 2,¼ SC C ¡¼ SS S and¼ CC C ¡¼ SC S have the same sign asI S ¡I C , therefore condition (2.21) impliesF C ¡F S must have the same sign asI S ¡I C , i.e. a necessary condition for(S;C) to be the equilibrium is (F C ¡F S )(I S ¡I C )>0 (2.22) The product type that has a smaller index needs to have a larger fixed cost, which is unlikely but not impossible. 4 4 The necessary and sufficient condition for(S;C) to be the equilibrium is 9¯ 2 t 2Mk 2 1 l SC 4 ¡ ¯ l SC >F C ¡F S + Mt 2 ¡ r MVd 3 and 9V 2 t 2Md 2 n SC 4 ¡Vn SC >F S ¡F C + Mt 2 ¡ r M¯k 3 wheren SC andl SC are solutions of Equation (A.1) and (A.2).) 32 2.4.3 Impact of Reputation We now analyze how differences in reputation or brand equity will affect the equilibrium outcome. Here the reputation refers to the brand image as perceived by all consumers. For example, the retail stores that offer better store environment and product quality will have a higher reputation in the customers’ minds. Nordstrom has a higher reputation than J.C. Penney and Nike has a better reputation than L.A. Gear. We model the higher reputation with a willingness to pay more for that firm’s products. Specifically, we assume that the utility for a consumer of getting her ideal product from Firm i is U i , i=1;2. We analyze the equilibrium structure whenU 1 6=U 2 : To investigate the effect of brand reputation, we assume that both products have the same fixed and variable costs for both firms. However, we comment on how the results would change if the costs are different, specifically if the firm with a higher reputation has a higher variable cost perhaps due to higher quality inputs. We find that the result of Theorem 2 on equilibrium product types is robust under differences in brand reputation. That is, both firms will choose the products with smaller index value in the equilibrium, regardless of their brand reputation. In particular, when I S =I C , the equilibrium could be(S;S);(S;C) or(C;C): 5 However, the brand reputation does affect the variety, lead time, prices and market shares in equilibrium, as stated in the following proposition. Proposition 5. When the two firms have different reputations, the equilibrium strategies have the following properties: (i) If the equilibrium is (S;S), the firm with higher reputation will provide a larger variety than the competitor. (ii) If the equilibrium is (C;C), the firm with higher reputation will provide a shorter 5 We ignore the lengthy proof here due to the length constraint. 33 lead time. In either case, the firm with higher reputation will enjoy a larger market share, higher retail prices (and unit margin) and greater profitability than the competitor. Thus, a firm with higher reputation in effect provides better product features (greater variety or shorter lead times) but charges correspondingly higher prices. We observe that the Ralphs grocery chain, which offers a better store environment, provides a larger assortment and charges higher prices for products such as yogurt and shampoo, as com- pared to a lower-tier store such as Food for Less. There may be examples where a higher reputation firm offers less variety. Shugan [Shu89] points out that in some industries producers of premium quality goods offer a smaller assortment, for example, Haagen-Dazs offers fewer flavors of ice cream than Baskin Robins. However, note that in these cases, the quality of the ice-cream is differ- ent and so are the corresponding variable costs. In fact, higher variable cost of the higher quality firm is a key condition required for the result in Shugan (p.316 of [Shu89]). While our model does not explicitly model differences in quality, it can be shown that higher variable costs (signifying higher quality) can result in a smaller assortment in our model. Using an analysis similar to that used in section 2.4.1, it can be shown that if two firms providing standard products have different variable costs, then the one with higher variable cost will have a smaller variety in equilibrium. However, our model and conclusions are more conditional. Specifically, if the firm with higher quality (and variable costs) also has a higher reputation, then depending on the relative differences in reputations and variable costs, the higher reputation firm may offer a higher or smaller variety. 34 2.5 Discussion and Conclusions In this paper we have studied the competition between two firms to investigate the rela- tive strengths of standardization and customization and conditions under which firms are likely to follow a particular strategy. The two firms compete for customers through their choices of product technology (product type), variety, lead time, and price. Consumers have heterogeneous tastes for the stores or firms and for product attributes. First, we have shown that in equilibrium, whether customization dominates stan- dardization or not hangs on its relative cost efficiencies and attractiveness to the con- sumers, as consumers base their store choice on product variety and lead time in addi- tion to the price offered. The firm that chooses a superior product strategy, represented by a smaller index value and/or smaller variable costs, can gain a larger market share and charge higher prices, and thus achieve higher profitability. Hence, we find that (S, S) or (C, C) are the equilibrium strategies for a wide range of parameter values and (S, C) is the equilibrium only over a relatively narrow range. This is true when variable or fixed costs are different for the two product types and even when reputations of the two firms are different. Further, our analysis showed that if the fixed or variable costs for custom products are likely to be higher as one might expect, then the (S, S) equilibrium is far more likely. This perhaps explains why we tend to see only standard products in many industries such as consumer electronics, office supplies, etc., while custom prod- ucts are thriving in a few industries such as PCs. Further, the proportion of customized product sales is small even in product categories such as apparel where several firms offer customization. The cost of achieving flexibility and consumer’s sensitivity to lead time play an important role and depending on the product category, it is sometimes a cost issue (e.g. consumer electronics) while in others it may be due to sensitivity to lead time (e.g., toothpaste). In general, as car manufacturers have discovered [Sch04], the cost of achieving customization with lead times acceptable to consumers is quite high 35 for many product categories. Customers have to wait several weeks or months to get a customized car, which reduces the potential demand substantially. Therefore, most cars sold are not customized. For many impulse purchase products where sensitivity to lead time, k, is likely to be high, customization is not a feasible strategy. In general, items with a high value density (such as PCs or furniture) are more likely to have a smaller k but whether such products are customized depends on the cost parameter ¯: As time goes by and the costs of customization decrease, customization may become a dominant strategy in more product categories but this is not imminent. Second, we have shown the strategic roles of product variety and lead time in the competition. Increasing variety or decreasing lead time can both increase market share and margin, but has diminishing returns and incurs the costs. Therefore, the firm should carefully choose the variety/lead time that balances the costs and benefits. Third, we find that with sufficient firm differentiation, increasing variety will NOT intensify the price competition. Previous literature found that increasing the product variety would force the firm itself to lower the price due to intensified price competition (e.g., [MGN88] and [AC05]), where there is no firm differentiation. In contrast, we find that when there is sufficient firm differentiation (or equivalently, high enough trans- portation cost), increasing the variety would allow the firm to increase the market share as well as charge a higher price, while the competitor would lower the price. Therefore, increasing the variety actually relieves the price pressure for itself as it better caters to consumer needs and enables higher price premiums. Furthermore, we find that the reputation of a firm does not impact its product strat- egy and so it is not more likely that a firm with a better brand or quality image will necessarily offer customization for instance. However, such firms do offer greater prod- uct variety and shorter lead time, depending on whether it offers standard or custom products respectively, and charges higher prices. Thus, the product type offered appears 36 to be largely a function of the costs and sensitivity of consumers to product fit and lead time and not a function of a firm’s brand image. Further, it appears that symmetric product strategies are the most likely ones in most scenarios. Our model has some limitations that suggest future research ideas. First, we assumed that market is fully covered. Relaxing this assumption may generate more insights regarding the role of variety and lead time under competition. For example, the equi- librium variety may increase and lead time decrease as they can attract more customers to purchase the product. Second, we could have heterogeneity in the sensitivity of con- sumers to the product fit and lead time. Finally, we could allow one firm to be an incumbent and study entry-deterrent strategies by analyzing sequential move games. 37 Chapter 3 Assortment Planning and Inventory Decisions Under Retail Competition 3.1 Introduction Over the past few decades, there has been product proliferation in many product cat- egories, especially in consumer products such as shampoos, toothpastes, cereals, etc. This is because manufacturers have tried to offer more variety to capture market share, and economies of scope, enabled by flexible technologies, have allowed manufacturers to produce a greater variety of products without much increase of the costs. However, many retailers may not be able to carry all of the variants offered by the manufacturer due to operational constraints or costs such as limited shelf space or stocking costs. Therefore, how many number variants to select, which ones to select, and how much to stock have become prominent problems faced by many retailers. In addition, the ever- intense retail competition forces retailer firms to consider what nearby stores offer in the same product categories. The previous assortment planning literature mostly focuses on the assortment and inventory decisions of monopoly firms. In particular, van Ryzin and Mahajan [vRM99] find that the optimal assortment for a monopoly firm contains the most popular variants in the product category, using the multinomial logit (MNL) model for the consumer behavior. In this paper, we consider a duopoly wherein both retailers select the assort- ment from a set of variants offered by the manufacturer and make inventory decisions. 38 We attempt to answer the following questions. Given the competitor’s product offer- ings, what is the optimal assortment for the retailer to maximize its profit? How does it differ from the optimal solution under monopoly settings? Furthermore and finally, what are the effects of market characteristics on the optimal and equilibrium assortment offerings? To address these questions, we propose a consumer demand model that is differ- ent from the MNL model and allows us to study assortment planning and inventory decisions under retail competition. Consider a single product category with horizontally differentiated products, such as yogurt with different flavors, or shirts of different colors, etc. The manufacturer offers retailers a set of variants from which to select, and each of the two retailers decides which variants to offer and how much to stock. Consumers, who may have different ideal variants, make the store choice based on the product offer- ings of the two stores. We assume a two-stage game: first, the two retailers make their assortment decisions simultaneously–i.e. which variants to offer; then they determine the stocking quantities simultaneously. Offering more variety may attract more cus- tomers, but it also incurs more operational costs. We characterize the structure of the optimal assortment and contrast it with those under monopoly settings obtained from previous research to identify the effect of competition on the assortment selections. We also analyze the effects of market conditions on the assortment decisions. Our work is related to the rapidly growing literature in assortment planning and inventory management. Kok et al. [KFV06] provide a thorough review of the assortment planning literature as well as industry approaches. Here, we focus only on the research that is closely related to our work. The majority of the literature studies assortment planning under monopoly settings. Van Ryzin and Mahajan [vRM99] were the first to study retail assortment decisions with inventory costs for the case of static substitution. They use a multinomial logit model to 39 describe consumer choice behavior and a newsboy model to represent inventory costs. Their main finding is that the optimal assortment consists of the most popular variants in the potential product set to offer. Our results in the competitive setting of assortment planning contrast with theirs in an interesting way, as we will show later. Subsequent literature explores the dynamic substitution problem. For example, Smith and Agrawal [SA00] consider a probabilistic demand model with dynamic substitution and construct lower and upper bounds to the problem solution. They point out that with substitution effects the assortment size is reduced if fixed costs are present or if items have differ- ent profit margins. Mahajan and van Ryzin [MvR01b] who study the retail assortment stocking decisions under dynamic consumer substitution, propose a sample path gradi- ent algorithm to determine the optimal assortment and inventory levels and show that under substitution the retailer should stock relatively more of popular variants and less of unpopular variants than a traditional newsvendor analysis would suggest. Gaur and Honhon [GH06] study an assortment planning and inventory management problem using a locational choice model, with infinite potential variants and nonuniform customer distribution. They show that in the optimal assortment under static substitu- tion, products are equally spaced-out such that there is no substitution among them, regardless of the distribution of customer preferences. They also construct bounds on profit under dynamic substitution and evaluate the impact of dynamic substitution on the retailer’s profit. Honhon et al. [HGS06] find that when customers have homo- geneous ranking of preferences, an efficient dynamic programming algorithm can be constructed to obtain the optimal assortment under dynamic substitution. They find that the most profitable product or the most preferred product might not be included in the optimal solution because they are dominated by combinations of other products. Caro and Gallien [CG07] study the dynamic assortment problem using a finite horizon multi- armed bandit model with Bayesian learning. Using Lagrangian relaxation, they obtain a 40 closed-form dynamic index policy that captures the key exploration versus exploitation trade-off. A number of papers deal with implementations of assortment planning models in dif- ferent industries. Chong et al. [CHT01] present an empirically based modeling frame- work for managers to use in assessing the revenue and lost sales implications of alterna- tive category assortments. Rajaram [Raj01] develops an assortment planning model for fashion retailing and implements the heuristics with a catalog retailer. Kok and Fisher [KF04] present a methodology for estimating the substitution rate and demand using data from multiple stores, and they propose and implement an optimization heuristic in a supermarket chain. Among the inventory management studies that consider retail competition, Mahajan and van Ryzin [MvR01a] and Netessine and Rudi [NR03] investigate inventory compe- tition among multiple firms, with each firm stocking inventory of a single good. Under dynamic consumer choice, Mahajan and van Ryzin [MvR01a] characterize the Nash equilibrium of the stocking game and show that competition results in overstocking rel- ative to the solution that maximizes system profits. With a deterministic substitution matrix, Netessine and Rudi [NR03] obtain analytically tractable solutions that facilitate comparisons between centralized and competitive inventory management solutions. In contrast, we study duopoly competition in which each firm carries multiple products and needs to decide the assortment selections as well as stocking levels. Retail assortment planning in a competitive setting has been studied in several research works (e.g., [Shu89], [CTX05], [CTX06], [CK07]), but they do not consider inventory decisions. Shugan [Shu89] focuses on the variety and quality of product offerings in a triopoly setting. He finds that a premium quality producer will offer a smaller assortment than a lower quality producer when its variable costs or assortment costs are higher or when customers are more price-sensitive. Cachon et al. [CTX05] 41 study the effects of consumer search on assortment planning decisions. They show that search costs can induce retailers to carry a broader assortment to prevent consumers from searching other retailers. In particular, it may be optimal to include an unprof- itable product just to reduce consumer search. Ignoring consumer search may result in significantly lower expected profits compared with the optimal solution. Cachon et al. [CTX06] show that easier consumer search exhibits two effects: the competition inten- sifying effect, which forces firms to lower their prices and reduce assortments, and the market expansion effect, which encourages firms to expand their assortments. Under certain circumstances the latter effect may dominate and therefore increase the over- all consumer welfare. Cachon and Kok [CK07] study the multiple category assortment planning problem in the presence of basket-shopping consumers. Consumers make their store choice based on the prices and variety level that retailers set for each category. The authors show that the common practice in category management as a decentralized regime yields less variety and higher prices than the optimal centralized solution and that the profit loss can be significant. They propose a decentralized regime using basket profits as a metric and show that it can produce near-optimal solutions. Our contribution to the assortment literature is as follows. We propose a different modeling framework that incorporates heterogeneous consumer preferences for firms and for products, with stochastic demand. This consumer demand model allows us to study the assortment planning and inventory decisions under competitive conditions, and to obtain insights not available from traditional models. Our paper yields the following main insights: ² When faced with competition, the retailer’s optimal assortment contains the most popular variants offered by the competitor and the most popular variants not offered by the competitor, though it may not cover a contiguous set of the most 42 popular segments. This, in contrast to the optimal assortment structure under monopoly settings, shows the effect of competition. ² Although in general the optimal assortment may not contain the most popular variants, we find that when retail price is high enough or the market potential is sufficiently large the optimal assortment does contain a continuous set of most popular variants. ² In equilibrium, both firms offer variants for major segments but differ in product offerings for smaller segments, in order to reduce the competition while capturing enough market share. ² In equilibrium, firms are more likely to offer different product variants and more variety when customers have more heterogeneous tastes, in order to capture dif- ferent customer groups while alleviating competition. ² The assortment structure is affected by the competitive intensity in a nonmono- tonic way: firms offer the most popular variants when the competitive intensity is considerably high or low, but differ in product offerings when competitive pres- sure is moderate. ² When product variants have different variable costs but comparable segment sizes, the optimal assortment consists of the least costly variants offered by the competi- tor and the least costly ones not offered by the competitor. It may not contain a continuous set of least costly variants. This chapter is organized as follows: We present our model in Section 3.2, and in Section 3.3 we analyze the model and characterize the structure of the optimal assort- ment. Inx3.4, we illustrate the effects of market characteristics and cost parameters on 43 assortment offerings, and inx3.5 we show the structural results when variable costs are different. Finally, we conclude inx3.6 and point out future research directions. 3.2 Model Formulation 3.2.1 Product Variants, Prices, and Costs Within a single product category, there aren customer segments, and each firm can pro- duce a product variant j that is targeted at segment j. Let S =f1;2;:::;ng denote the set of possible variants,S i ½S denote the subset of variants stocked by storei,i=1;2, and n i denote the variety, i.e., n i = jS i j. To focus on the effect of competition on assortment decisions, we assume that each variant is offered at an identical exogenous price,r, and has an identical cost,c. Similar assumptions are applied by van Ryzin and Mahajan [vRM99]. In practice, retailers tend to set a uniform retail price for horizon- tally differentiated products, such as yogurt with different flavors. We assume that each variant incurs a fixed costK associated with ordering and stocking a product. The same assumption has been made in [GH06]. 3.2.2 The Customer Demand Model The total number of customers choosing this product category within the selling season is assumed to be Poisson distributed with mean ¸. For each customer, the probability that his or her ideal segment is j is given by m j , j = 1;2;:::n; where P n j=1 m j = 1. Each segment is presented by a line segment of unit length. Assume Firm 1’s products are located at the left ends of the segments, while Firm 2’s products are at the right ends, in terms of store locations or brand images (see Figure 3.1). Customers are uniformly distributed on the line in terms of their physical locations or preferences for the brands. 44 Figure 3.1: Product and customer space for assortment planning under competition. If firm i offers a product for this segment, then for a customer located at x, her utility of purchasing a product from this firm is u¡ tjx¡ X i j¡ r, where u represents the customer’s reservation price for her ideal product, t represents the transportation cost, andX 1 = 0;X 2 = 1. If, however, firmi does not offer a product for this segment, then the customer may substitute to other products offered by this firm, and obtain utility u¡ d n i ¡tjx¡X i j¡r, where d n i represents the disutility of substitution. With a larger variety, customers are more likely to find a variant close to their ideal product and hence incur a smaller disutility, and so they are more likely to substitute. Here,d represents the cost of product misfit, which also indicates the degree of substitution between products, while t measures the degree of substitution between firms. A lower value of d or t implies a higher degree of substitution between products or between firms. We assume that the reservation price is high enough (a sufficient condition is u > t+d+r) so that every customer can get positive utility by making a purchase. This may apply to basic product categories such as milk, bread, cereal, and shampoo, which must 45 be purchased regularly. 1 Each consumer selects the store that gives her the highest net utility to make the purchase. We can derive the indifferent customer location for each segment using the consumer utility functions, given the two firms’ offeringsS 1 andS 2 . The indifferent customer for segmentj is: x j = 8 > > > > > > > < > > > > > > > : 1 2 if j2S 1 \S 2 t+ d n 2 2t if j2S 1 nS 2 t¡ d n 1 2t if j2S 2 nS 1 t+ d n 2 ¡ d n 1 2t if j2S 1 \S 2 (3.1) Customers located between0 andx j will visit store 1, while customers within[x j ;1] will visit store 2. To focus on duopoly and the related insights, we make a technical assumption of t > d; to ensure x j 2 (0;1). That is, we assume that there is more differentiation between the two firms than between the products so that no firm can capture all the customers in a segment. This assumption will be relaxed in Section 3.4. The timing is as follows. Customers first decide which store to visit based on their net utilities, which take into account the stores’ product offerings. They may decide to buy their ideal product or second ideal product depending on the assortment offerings. Then they will visit that store and choose their intended product. However, if the store has run out of their intended product by the time of their arrival, they do not substitute to other products. This assumption may apply when an unexpected substitution would dra- matically reduce their utility so that they are unwilling to buy. That is, customers make assortment-based substitutions, but not inventory-based substitutions. Similar assump- tions are made in [vRM99] and the first part of [GH06]. This assumption provides a reasonable starting point and lets us focus on the effect of competition on assortment 1 This assumption helps to limit the number of cases to discuss, and it will be relaxed later in Section 3.4. 46 decisions and obtain managerial insights without the distraction of a complex demand process. For the sake of simplicity, we assume that each product is equally likely to be chosen as the second ideal product. Then the probability that a customer chooses product j offered by firm 1 is p 1 j =m j x j + 1 n 1 2 4 X k2S 2 nS 1 m k t¡ d n 1 2t + X k2S 1 \S 2 m k t+ d n 2 ¡ d n 1 2t 3 5 ; (3.2) wherex j = 8 < : 1 2 if j2S 1 \S 2 t+ d n 2 2t if j2S 1 nS 2 : The first part of (3.2) is the probability that the customer’s ideal product is j and that she chooses firm 1, while the second part is the probability that the customer’s ideal product is not offered by firm 1 but she chooses variantj of firm 1 as a substitute. Similarly we can obtain the probability that a customer selects productj of firm 2: p 2 j =m j (1¡x j )+ 1 n 2 2 4 X k2S 1 nS 2 m k t¡ d n 2 2t + X k2S 1 \S 2 m k t+ d n 1 ¡ d n 2 2t 3 5 ; (3.3) wherex j = 8 < : 1 2 if j2S 1 \S 2 t¡ d n 1 2t if j2S 2 nS 1 : Then the total demand for variantj offered by firmi,D i j , is Poisson distributed with mean ¸p i j . To facilitate the analysis, we use a normal approximation of the Poisson distribution to represent the demand for each variant, i.e. D i j is normally distributed with mean ¸p i j and standard deviation q ¸p i j . This allows us to obtain closed-form expressions for profit functions, as we show in the next section. 2 2 Normal approximation of the Poisson distribution has been used in the literature, such as in [GH06]. 47 3.3 Analysis In this section we aim to obtain the structural properties of the optimal assortment and inventory policies for retailers when facing competition. For simplicity, we assume that the sales season lasts for one period. At the begin- ning of the sales season, each retailer selects the subset of variants to stock, S i . Then, after becoming aware of each other’s assortment, they decide on the stocking quan- tities for each variant offered. Let vector q i denote the ordering quantities by store i; q i =(q i 1 ;:::;q i n ), whereq i j denotes its ordering quantity for variantj. Finally, the demand is realized during the sales season. The expected profit made by retaileri on variantj isE £ rmin © q i j ;D i j ª ¡cq i j ¡K ¤ , where the three parts represent the revenue, variable cost, and fixed cost for variant j, respectively. Then the total maximum expected profit of firmi givenS 1 andS 2 is max q i ¸0 X j2S i E £ rmin © q i j ;D i j ª ¡cq i j ¡K ¤ : Under the normal demand distribution, the optimal stocking quantity for variant j is given by e q i j = ¸p i j + z q ¸p i j , where z = © ¡1 ¡ r¡c r ¢ is the service level and ©(¢) denotes the cumulative distribution function of the standard normal distribution. Define e q i =( e q i 1 ;:::; e q i n ) as the vector of optimal stocking quantities, and let e q i j = 0 if j = 2 S i : The optimal profit of firmi givenS 1 andS 2 is then ¼ i (S 1 ;S 2 )= X j2S i h (r¡c)¸p i j ¡rÁ(z) q ¸p i j ¡K i ; whereÁ(¢) denotes the density function of the standard normal distribution. 48 3.3.1 Structure of the Optimal Assortment For convenience, let the variants inS be ranked in decreasing order of segment sizem j ; so thatm 1 ¸m 2 :::¸m n : At stage 1, each firm chooses the optimal assortmentS i to maximize its own profit, as represented above. Its optimization problem is max S i ½S ¼ i (S 1 ;S 2 )= X j2S i h (r¡c)¸p i j ¡rÁ(z) q ¸p i j ¡K i wherep i j is given by Equations (3.2) and (3.3). As the first step, we analyze the profit associated with adding a variantk toS 1 given the currentS 1 andS 2 : The following lemma will help us to obtain the optimal assortment structure. Lemma 1. Let0· ±· maxfm l : l2 Sg; and define the profit associated with adding a variantk2S 2 nS 1 withm k =± toS 1 asf(±), and the profit associated with adding a variantk2S 1 \S 2 withm k =± toS 1 asg(±). Thenf(±) andg(±) are convex in±: Functionf(±) is convex in±, which means that the maximum off(±) on the closed interval [0;maxfm k :k2S 2 nS 1 g] is achieved at the end points of the interval. There- fore, if firm 1 adds a variant k 2 S 2 nS 1 , the profit is maximized either by adding no variants or by adding the variant with the largest segment size available in the setS 2 nS 1 . Using the same logic,g(±) is convex in±, so if firm 1 adds a variantk 2 S 1 \S 2 , then it should either add no variants or add the variant with largest segment size available in the set S 1 \S 2 . These properties allow us to show the structure of firm 1’s optimal assortment, as stated in the following theorem. Theorem 5. Let the variants inS 2 be ranked ask 1 ;k 2 ;:::;k n 2 ; such thatm k 1 ¸m k 2 :::¸ m k n 2 ; and let the variants inS 2 bek n 2 +1 ;:::;k n , such thatm k n 2 +1 ¸ m k n 2 +2 :::¸ m k n : 49 DefineA i =fk 1 ;:::;k i g for1· i· n 2 , andB i =fk n 2 +1 ;:::;k i g forn 2 +1· i· n; andA 0 =;;B n 2 =;: Then there existx2f0;1;:::;n 2 g;y2fn 2 ;n 2 +1;:::;ng; such that the assortmentS ¤ 1 =A x [B y maximizes firm 1’s profit. Proof of Theorem 5 The proof is by construction. LetS ¤ 1 be an optimal assortment set, and letx;y be the cardinality ofS ¤ 1 \S 2 andS ¤ 1 \S 2 ; respectively. Let the variants inS ¤ 1 \S 2 be denoted as k ¤ 1 ;:::;k ¤ x ; such that m k ¤ 1 ¸ ::: ¸ m k ¤ x ; and let the variants in S ¤ 1 \S 2 be denoted as k ¤ n 2 +1 ;:::;k ¤ y ; so thatm k ¤ n 2 +1 ¸ :::¸ m k ¤ y : IfS ¤ 1 = A x [B y ; the theorem holds trivially. IfS ¤ 1 6=A x [B y ; there are two possible cases: Case 1. There exists a k j 2 S 2 ; and k j = 2 S 1 ; such that m k j > m k ¤ x : Due to the convexity of function f(±), we can either remove k ¤ x or exchange it with k j without decreasing profits. RedefineS ¤ 1 to be this new set. Case 2. There exists a k j = 2 S 2 ; and k j = 2 S 1 ; such that m k j > m k ¤ y : Due to the convexity of function g(±), we can either remove k ¤ y or exchange it with k j without decreasing profits. RedefineS ¤ 1 to be this new set. Repeat the procedure. Eventually, we will arrive at an optimal setS ¤ 1 =A x [B y for somex;y:¦ In words, the optimal assortment for firm 1 contains the most populari variants that firm 2 offers and the most popularj variants that firm 2 does not offer, where0·i·n 2 and0· j · n¡n 2 . This simple structure allows us to restrict the optimal assortment to(n 2 +1)(n¡n 2 +1) sets, which means that we can obtain the optimal assortment in O(n 2 ) time using a simple algorithm. In a monopoly setting, van Ryzin and Mahajan [vRM99] find that the firm should choose the best variants in the product category to maximize its profit. Our results show the effect of competition on the assortment decisions: the optimal assortment depends on the competitor’s offerings, and it is the joint set of the most popular variants that the 50 competitor offers and the most popular variants that the competitor does not offer. The joint set may be discontinuous and not cover all of the popular variants. Examples will be provided in the numerical studies in Section 3.4. Theorem 5 points out that the optimal assortment depends on the competitor’s offer- ings. This raises interesting questions about when to select a variant offered by the competitor and when to select a variant not offered by the competitor. The following theorem delineates the conditions under which it is better to add to S 1 a variant in S 2 versus inS 2 . Theorem 6. For all1·i<n 2 ;n 2 +1·l·n; we have a)¼(A i+1 [B l )>¼(A i [B l+1 ) ifm i+1 >m l+1 b)¼(A i+1 [B l )<¼(A i [B l+1 ) ifm i+1 <m l+1 when the selling price r is sufficiently high or the market volume ¸ is sufficiently large. Theorem 6 implies that, when the retail margin or market potential is large, it is optimal to add the most popular variant available, regardless of whether the competitor offers it or not. Thus, although Theorem 5 says that the optimal assortment may not cover a contiguous set of the most popular variants, Theorem 6 tells us that when the retail margin is sufficiently high, or the market volume is large enough, the optimal assortment does contain a contiguous set of the most popular variants. However, we need to point out that the variety, and therefore the assortment, still depend on what and how many variants the competitor offers since the marginal benefits of additional variants depend on the competitor’s assortment through consumer demand. For regularr and¸, we may have¼(A i+1 [B l )>¼(A i [B l+1 ) even ifm i+1 <m l : This sounds counter-intuitive: why should the firm choose a variant already offered by the competitor that has a smaller segment size than the variant not offered by the competitor? The reason is as follows: adding a variant targeted at a niche segment 51 may attract more customers, but it will also cannibalize the sales of existing products and result in a defragmentation of inventory, which in turn reduces the benefit of scale economies. Adding a variant already offered by the competitor may not bring much more sales, but it does not significantly affect the sales of existing products. Therefore, the firm faces the trade-offs between generating more revenue and keeping the cost sav- ings of inventory pooling, and sometimes one may dominate the other. Only when the retail margin is high or the market volume is large enough that the revenue generated by capturing a bigger segment dominates the reduced economies of scale due to inventory defragmentation, and the firm chooses the variant that is more popular. 3.3.2 Special Case: When There are No Product Substitutions Now let us consider an extreme scenario in which customers are not willing to substitute between products, because of the large disutility it brings, but they are willing to substi- tute between firms. That is, the product misfit costd is large but the transportation costt is small. For example, if a customer could not find her size of clothes in a store, she may go to another store to buy, but she will not substitute to other sizes. The same applies to shoes such as sneakers. We still assume everybody makes a purchase, but there is no substitution between products. Since there is no product substitution, the demands for different products are inde- pendent of each other. The net profit brought by each variant is then given by ¼ j (m j )= 8 < : (r¡c) ¸m j 2 ¡rÁ(z) q ¸m j 2 ¡K if j2S 1 \S 2 (r¡c)¸m j ¡rÁ(z) p ¸m j ¡K if j2S 1 nS 2 : The optimal assortment consists of all of the variants that have positive net profit. Theorem 7. When there is substitution between firms but no substitution between prod- ucts, the optimal assortment contains the most popular variants offered by firm 2 and 52 the most popular variants not offered by firm 2. The threshold segment sizes are m ¤ = ³ Á(z)+ p Á 2 (z)+4K(1¡c=r) ´ 2 2¸(1¡c=r) 2 andm ¤¤ = ³ Á(z)+ p Á 2 (z)+4K(1¡c=r) ´ 2 4¸(1¡c=r) 2 , respectively. Proof: Rewrite ¼ j (m j ) ase ¼ j (x) = (r¡c)x 2 ¡rÁ(z)x¡K, where x = q ¸m j 2 if j 2 S 1 \S 2 , and x = p ¸m j if j 2 S 1 nS 2 . If the variant can bring positive benefit (i.e.,e ¼ j (x) > 0) then (r¡c)x 2 ¡rÁ(z)x > 0, i.e., x > rÁ(z) (r¡c) : Then de ¼ j (x) dx = 2(r¡ c)x¡rÁ(z)>0: Therefore,¼ j (m j ) is increasing inm j when¼ j (m j )>0: That means that the optimal assortment contains all of the variants in S 2 with segment sizes larger than m ¤ , and all of the variants in S 2 with segment sizes larger than m ¤¤ , where m ¤ and m ¤¤ are the solutions of (r¡c) ¸m j 2 ¡rÁ(z) q ¸m j 2 ¡K = 0 and (r¡c)¸m j ¡ rÁ(z) p ¸m j ¡K = 0, respectively. We have m ¤ = ³ Á(z)+ p Á 2 (z)+4K(1¡c=r) ´ 2 2¸(1¡c=r) 2 ; m ¤¤ = 1 2 m ¤ = ³ Á(z)+ p Á 2 (z)+4K(1¡c=r) ´ 2 4¸(1¡c=r) 2 :¦ We can see that even if customers do not substitute between products, Theorem 5 still holds, and that because the thresholdsm ¤ andm ¤¤ are decreasing in market volume ¸ and retail pricer, Theorem 6 also holds. Example: n=4;¸=100;r =2;c=1;K =10;m=[0:5;0:25;0:15;0:1]: The thresholds are m ¤ = 0:2572 and m ¤¤ = 0:1286: Therefore, if S 2 = [1;1;0;0], thenS ¤ 1 =[1;0;1;0]: That is, the firm does not offer the second-most-popular variant in an effort to avoid competition and target a niche market. GivenS 1 =[1;0;1;0],S ¤ 2 =[1;1;0;0]: Therefore([1;0;1;0];[1;1;0;0]) is an equi- librium. Another equilibrium is([1;1;1;0];[1;0;0;0]): 3.4 Effect of Market and Cost Parameters In earlier sections, we assume that every customer makes a purchase, possibly through substitution between products or between firms, which may apply to the basic product 53 category. In this section, we relax this assumption to include the no-purchase options. We first present a general model that incorporates the no-purchase option. Then we provide the results of a numerical study to understand the effect of market characteristics and cost parameters on assortment decisions. 3.4.1 General Problem with No-Purchase Options We consider the more general problem in which consumers may choose not to purchase and get an outside utility, u 0 , which is also called no-purchase utility. Therefore, cus- tomers have three options: purchasing from firm 1, or firm 2 or not purchasing at all, which yield utility u¡ tx¡ r¡ d n 1 I(j = 2 S 1 ); u¡ tx¡ r¡ d n 2 I(j = 2 S 2 ); and u 0 , respectively 3 . To exclude the trivial case in which demand becomes zero, we assume that u¡ r¡u 0 > 0; so that someone will purchase something. We can derive the market share of each firm in each segment. In Table 3.1, x j represents the fraction of customers in segmentj who will purchase from firm 1. Then the probability that a customer chooses productj offered by firm 1 is p 1 j =m j x j + 1 n 1 2 4 X k2S 2 nS 1 m k x k + X k2S 1 \S 2 m k x k 3 5 ; (3.4) wherex j andx k are given by the above table. We are not able to show the property of f(±), since the different cases disrupt the differentiability off(±), but from the numerical studies we believe that Theorem 5 still holds in the general case. Example 1 below illustrates when and why the optimal assortment could be discon- tinuous. 3 I(¢) represents the indicator function. 54 Table 3.1: Firm 1’s Market Share for Each Segment when Customers Have No-purchase Options Set Conditions x j j2S 1 \S 2 u¡r¡ t 2 >u 0 u¡r¡ t 2 u 0 u¡r¡ t+ d n 2 2 u 0 u¡r¡ t+ d n 1 2 u 0 u¡r¡ t+ d n 1 + d n 2 2 <u 0 middlef t¡ d n 1 + d n 2 2t ;0;1g middlef u¡r¡u 0 ¡ d n 1 t ;0;1g Table 3.2: Comparisons of Assortment Decisions with Same Variety S 1 x p q Itemized Profits Total Profit [1 1 0 0] [.50 .50 .12 .12] [.19 0 .17 0] [19.03 16.81 ] [5.55 0 3.53 0] 9.08 [1 0 1 0] [.50 .12 .62 .12] [.19 0 .19 0] [19.17 0 19.17 0] [5.67 0 5.67 0] 11.35 [0 0 1 1] [.12 .12 .62 .62] [0 0 .20 .11] [0 0 20.56 10.83] [0 0 6.94 -1.79] 5.14 Example 1: n = 4; ¸ = 100; r = 2; c = 1; t = 2; d = 2; u = 3:25; u 0 = 0; K = 10; m=[0:3;0:26;0:24;0:1];S 2 =[1;1;0;0]: Table 3.2 shows the comparisons along different dimensions when the firm chooses to offer two variants. If the firm offers the two most popular variants, then it gets half of the market share for those two segments, but just a small fraction of the other two segments. By offer- ing the first- and the third-most-popular variants, it captures half of the first segment and more than half of the third segment. Because the second and third segments have comparable sizes, the latter choice attracts more customers. If the firm offers the third- 55 and fourth-most-popular variants, even though it reduces the competition, the market segments are not large enough to increase the sales, so the middle choice is the best. Among all of the possible selections with different assortment sizes, the optimal assortment isS ¤ 1 = [1;0;1;0], which does NOT cover a contiguous set of most popular variants. The motives behind these decisions are the following: the firm tries to avoid the competition and attract more customers by offering products that are different from its competitor’s, but if the size of the segment covered by the competitor is large enough, then it is worthwhile competing with the rival store by offering the same variant. In fact, we can show that under the above parameter values, the equilibrium assort- ment is S ¤ 1 = [1;1;0;0]; S ¤ 2 = [1;0;1;0], where firm 2 does not offer a continuous set of popular products. In summary, when choosing the assortment to maximize its profit, the firm faces the decision of whether to offer competitive products for major market segments or to target niche markets. It is optimal to add the variant that gives the largest marginal benefit. However, keep in mind that the possible candidates could only be the most popular variant (MPV) in S 2 and MPV in S 2 : The exact assortment selection depends on all of the market and cost parameters, as illustrated in the following numerical studies. 3.4.2 Effect of Market and Cost Parameters In this subsection, we explore the effect of market characteristics and cost parameters on the optimal and equilibrium assortments using numerical studies. Effect of parameters on the optimal assortment To better illustrate the insights, let us consider a scenario in which the competitor’s offerings cover the biggest segments. The firm then needs to decide whether to offer the same variant for the major segment, or to target a smaller but different segment. The 56 Table 3.3: Effect of Competitor’s OfferingS 2 on the Optimal Assortment S 2 S ¤ 1 q Itemized Profits Total Profit [1 0 0 0] [0 1 1 1] [0 18.97 17.72 16.47] [0 5.50 4.36 3.23] 13.09 [1 1 0 0] [1 0 1 1] [16.53 0 17.53 16.28] [3.28 0 4.19 3.06] 10.53 [1 1 1 0] [1 1 0 1] [16.33 15.33 0 16.08] [3.11 2.21 0 2.88] 8.20 [1 1 1 1] [1 1 0 0] [16.88 15.88 0 0] [3.60 2.70 0 0] 6.29 Table 3.4: Effect of Segment Heterogeneity on the Optimal Assortment: S 2 =[1;1;0;0] m S ¤ 1 q Itemized Profits Total Profit [.40 .30 .20 .10] [1 1 0 0] [21.88 16.88 0 0] [8.14 3.60 0 0] 11.74 [.34 .28 .22 .16] [1 1 1 0] [18.56 15.56 15.30 0] [5.12 2.41 2.18 0] 9.71 [.28 .26 .24 .22] [1 0 1 1] [16.52 0 17.53 16.28] [3.28 0 4.19 3.06] 10.53 [.25 .25 .25 .25] [1 0 1 1] [14.93 0 18.06 18.06] [1.85 0 4.66 4.66] 11.18 following examples illustrate this trade-off and show how the assortment decisions vary with the market conditions and cost parameters. The common parameter values are set to be the following, unless specified in the examples. n = 4; ¸ = 100; r = 2; c = 1; t = 2; d = 2; u = 3:25; u 0 = 0; K =10;m=[0:28;0:26;0:24;0:22]: When segment sizes are comparable, the firm will first target customer segments dif- ferent from the competitor’s in order to reduce the head-to-head competition, as shown in Table 3.3. However, if the competitor covers most major segments, then the firm also has an incentive to compete in major customer segments besides the niche segment. For instance, when firm 2 offers variants for the biggest three segments out of four, firm 1 would offer variants 1 and 2 in addition to variant 4. The stocking quantities and itemized profits help us to understand the logic behind these results. In addition, we find that the optimal assortment size decreases as the competitor offers more variety, which implies that the marginal benefit of adding a variant decreases as the competitor offers more variety. 57 Table 3.5: Effect of Utility Difference (u¡ u 0 ) on the Optimal Assortment: S 2 = [1;1;0;0] (u¡u 0 ) S ¤ 1 q Itemized Profits Total Profit 4 [1 1 0 0] [25.5 24.5 0 0] [11.47 10.55 0 0] 22.02 3.75 [0 0 1 1] [0 0 24.75 23.25] [0 0 10.78 9.40] 20.18 3.5 [0 0 1 1] [0 0 24.75 23.25] [0 0 10.78 9.40] 20.18 3.25 [1 0 1 1] [16.53 0 17.53 16.28] [3.28 0 4.19 3.06] 10.53 3 [1 1 1 0] [15.22 14.22 13.22,0] [2.11 1.21 0.32 0] 3.64 Table 3.6: Effect of Product Misfit Costd on the Optimal Assortments: S 2 =[1;1;0;0] d S ¤ 1 .5 [0 0 1 0] 1 [0 0 1 1] 2 [1 0 1 1] 4 [1 1 1 1] When segment heterogeneity is high, as shown in the first column of Table 3.4, it is optimal to target the biggest segments, even if the competitor does so as well. When segment sizes are more homogeneous (i.e. customers’ preferences are more spread out), the firm starts to offer different products and provide more variety, in an effort to gain market share while reducing competition. As shown in Table 3.5, when the reservation utility is high or the no-purchase utility is low, it is optimal to target the biggest segments because customers in other segments will substitute. When (u¡u 0 ) decreases, customers are less likely to substitute, and therefore it is better to target a niche segment when the segment sizes are comparable. However, if (u¡ u 0 ) becomes so low that there is little substitution, then it is better to offer more products to keep from losing customers, and the firm may also target the biggest segments even if the competitor does so as well. Table 3.6 shows that when customers are likely to substitute between products–i.e.,d is low–the firm should offer a small variety. When customers are unwilling to substitute 58 Table 3.7: Effect of Transportation Costt on the Optimal Assortments: S 2 =[1;1;0;0] t S ¤ 1 .5 [1 1 1 0] 1 [1 0 1 1] 1.5 [0 0 1 1] 2 [1 0 1 1] 2.5 [1 1 0 0] Table 3.8: Firm 1’s Market Shares with Different Assortments Under Differentt t Assortment .5 1 1.5 2 2.5 [1 1 1 0] [.5 .5 1 .83] [.5 .5 1 .58] [.5 .5 .83 .39] [.5 .5 .62 .29] [.5 .5 .5 .23] [1 0 1 1] [.5 0 1 1] [.5 .17 1 1] [.5 .28 .83 .83] [.5 .29 .62, 0.62] [.5 .5 .23 .5] [1 1 0 0] [.5 .5 .5 .5] [.5 .5 .25 .25] [.5 .5 .17 .17] [.5 .5 .12 .12] [.5 .5 .1 .1] [0 0 1 1] [0 0 1 1] [0 0 1 1] [.17 .17 .83 .83] [.12 .12 .62 .62] [.1 .1 .5 .5] between products–i.e., d is high–it is better to offer a large variety to cater to different customers’ needs. From Table 3.7, when transportation cost is high, customers are less likely to switch between firms, and therefore it is optimal to target the largest segments. Ast decreases, the competitive pressure increases, and it is better to offer different products to reduce direct competition. However, whent is too low it is optimal to target the largest segments again, as well as to offer different products. This observation is counter-intuitive, and we will explore this in detail. From Table 3.8 and Table 3.9 above, we see that, given the same variety, [1;1;1;0] gains greater market shares and revenue than [1;0;1;1] when t = 0:5 and 2:5, while [1;0;1;1] excels whent is in the middle range. Similarly, [1;1;0;0] dominates [0;0;1;1] if and only if transportation cost is very large or very small. 59 Table 3.9: Firm 1’s Profits Under Different Assortments t Assortment .5 1 1.5 2 2.5 [1 1 1 0] 27.89 22.86 15.29 8.80 4.96 [1 0 1 1] 19.36 23.31 18.91 10.53 4.00 [1 1 0 0] 22.02 11.50 8.02 6.29 5.26 [0 0 1 1] 18.35 18.35 19.57 8.78 2.39 Table 3.10: Effect of Segment Heterogeneity on the Equilibrium Assortments m Equilibrium Assortments [.40 .30 .20 .10] [1 1 0 0] [1 1 0 0] [.34 .28 .22 .16] [1 1 0 0] [1 1 1 0] [.28 .26 .24 .22] [1 1 0 0] [1 0 1 1] [.25 .25 .25 .25] [1 1 1 0] [1 1 0 1] Combine this with the variety cost, we can see why the effect of transportation cost on the optimal assortment is so irregular, i.e., the optimal variety and assortment struc- ture are non-monotonic in t. Effect of parameters on the equilibrium assortments Although the equilibrium assortment structures are difficult to obtain analytically, numerical studies reveal some interesting properties of the equilibrium assortments, as shown in the following examples. One common property, as can be seen in the exam- ples, is that the equilibrium assortments could be asymmetric even if the firms have symmetric market and cost structures. The parameter values are the same as in the last subsection, which is listed on page 57. 60 Table 3.11: Effect of(u¡u 0 ) on the Equilibrium Assortments (u¡u 0 ) Equilibrium Assortments 4 [1 1 0 0] [1 1 0 0] 3.75 [1 1 0 0] [0 0 1 1] 3.5 [1 1 0 0] [0 0 1 1] 3.25 [1 1 0 0] [1 0 1 1] 3 [1 1 1 0] [1 1 1 0] In Table 3.10, when segment sizes significantly differ from each other, both firms tar- get the largest segments, but when segment sizes are more comparable (i.e., customers’ preferences are more heterogeneous), firms tend to target different segments, if possible, to reduce direct competition and gain market share. In addition, firms offer more variety in equilibrium as segment sizes are more homo- geneous, in order to satisfy the needs of different customers. Note that two firms may offer different assortments in equilibrium, even if they incur the same variable cost and fixed cost. In Table 3.11, both firms target the largest segments when the utility difference is high or low, but they differentiate in offerings when the utility difference is moderate. When the reservation utility u is high or the no-purchase utility u 0 is low, both firms target the largest segments because customers in other segments will substitute. As u decreases oru 0 increases, customers are less likely to substitute, and firms offer differ- ent products to capture different customer segments and reduce competition. However, when the reservation utility is too low or outside utility is too high, firms offer more variety and target the major segments. There are two possible reasons. First, under low 61 Table 3.12: Effect oft on the Equilibrium Assortments t Equilibrium Assortments 1 [1 1 1 0] [1 1 1 0] 1.5 [1 1 0 0] [0 0 1 1] 2 [1 1 0 0] [1 0 1 1] 2.5 [1 1 0 0] [1 1 0 0] reservation utility, when the ideal product is not available, customers are more likely to take the no-purchase option than to substitute to other products, so firms have incen- tives to expand the assortments. Second, the lower reservation utility indirectly reduces the competition between firms, and firms may consequently move back to the major segments to keep up the sales. In Table 3.12, if there is a high transportation cost that increases the difficulty of product comparisons between stores, the competitive pressure between firms is low. As a result, both firms offer the most popular variants as if they are a local monopoly. As t decreases, firms start to offer different variants to relieve the competition, but when t is too low–i.e., it is easy for customers to switch between retailers–then firms target the largest segments again, which is somewhat counter-intuitive. It is quite surprising to notice that the effect of increasing transportation cost mirrors the effect of decreasing utility difference. However, they share a common thread in that they both reduce competitive pressure and allow the firms to gain some monopoly power. Therefore, the observed pattern may apply to more general cases: firms offer a contiguous set of the most popular variants when the competitive pressure is either low or very high, but they differ in product offerings when the competition is moderate. 62 Table 3.13: Effect ofd on the Equilibrium Assortments d Equilibrium Assortments .5 [1 0 0 0] [0 1 0 0] 1 [1 0 0 1] [0 1 1 0] 1.5 [1 0 0 1] [0 1 1 0] 2 [1 1 0 0] [1 0 1 1] 2.5 [1 1 1 0] [1 1 1 0] Table 3.14: Effect of Market V olume¸ on the Equilibrium Assortments ¸ Equilibrium Assortments 100 [1 1 0 0] [1 0 1 1] 200 [1 1 1 0] [1 1 0 1] 300 [1 1 1 1] [1 1 1 1] Recall that the product misfit costd also indicates the degree of product substitution, with a larger d implying a lower degree of product substitution. We observe in Table 3.13 that firms offer more variety under a low degree of product substitution in order to attract customers with different needs who are unwilling to substitute. By comparing the effects of market volume and retail price in Table 3.14 and 3.15, we find that as market volume or retail price increases, retailers offer more variety in equilibrium, and eventually they both offer a contiguous set of the most popular variants. 63 Table 3.15: Effect of Retail Pricer on the Equilibrium Assortments: c=1 r Equilibrium Assortments 1.8 [1 0 0 1] [0 1 1 0] 1.9 [1 1 0 0] [0 0 1 1] 2.0 [1 1 0 0] [1 0 1 1] 2.1 [1 1 1 0] [1 1 0 1] 2.2 [1 1 1 0] [1 1 1 0] Table 3.16: Effect of Variable Costc on the Equilibrium Assortments c Equilibrium Assortments whenr¡c=1 Equilibrium Assortments whenr =2 0.8 [1 1 0 0] [0 0 1 1] [1 1 1 0] [1 1 0 1] 0.9 [1 1 0 0] [1 0 1 1] [1 1 1 0] [1 1 0 1] 1.0 [1 1 0 0] [1 0 1 1] [1 1 0 0] [1 0 1 1] 1.1 [1 1 1 0] [1 1 0 1] [1 1 0 0] [1 0 1 1] 1.2 [1 1 1 0] [1 1 1 0] [1 0 0 1] [0 1 1 0] However, we can see that the equilibrium assortments are more sensitive to the price changes than to the market size differences. Table 3.15 shows the effect of retail price, which is also the effect of lost sales or understocking cost because the variable cost is fixed. The second column of Table 3.16 shows the effect of overstocking cost on the equilibrium assortment. We see that as the overstocking or understocking cost increases, firms provide more variety and target the biggest segments. This result shows the effect of inventory costs. Because without 64 Table 3.17: Effect of Fixed CostK on the Equilibrium Assortments K Equilibrium Assortments 10 [1 1 0 0] [1 0 1 1] 5 [1 1 1 0] [1 1 0 1] 1 [1 1 1 1] [1 1 1 1] inventory costs, the assortments should not depend on the retail price or variable cost, as they are the same for all the variants in the same category. The third column of Table 3.16 shows the effect of variable cost. It is the combina- tion of an increasing overstocking cost and decreasing understocking cost. The results indicate that the latter effect dominates the former one, and therefore firms tend to dif- ferentiate their products and offer less variety as variable cost decreases. The effect of fixed cost on the equilibrium assortments in Table 3.17 is intuitive: the equilibrium variety increases when the fixed cost becomes smaller. For instance, with comparable segment sizes, firms tend to offer different products in equilibrium when the fixed cost is large. AsK diminishes, firms are able to offer more variants to compete in the major segments. 3.5 Assortment Structure with Heterogeneous Variable Costs In the above model, we assumed that all of the variants have the same variable costs. In this section, we relax this assumption and investigate the optimal assortment structure when there exists heterogeneity in the variable costs. We assume identical segment 65 sizes to focus on the effect of different variable costs. That is, for each customer, the probability that her ideal segment isj ism j = 1=n; for allj: The indifferent customer location is same as in equation (3.1). Using (3.2), (3.3), andm j = 1=n, the probability that a customer chooses productj offered by firm 1 becomes p 1 j = x j n + 1 n 1 " 1 n jS 2 nS 1 j t¡ d n 1 2t + 1 n ¯ ¯ S 1 \S 2 ¯ ¯ t+ d n 2 ¡ d n 1 2t # ; (3.5) wherex j = 8 < : 1 2 if j2S 1 \S 2 t+ d n 2 2t if j2S 1 nS 2 : while the probability that a customer selects productj of Firm 2 is p 2 j = 1¡x j n + 1 n 2 " 1 n jS 1 nS 2 j t¡ d n 2 2t + 1 n ¯ ¯ S 1 \S 2 ¯ ¯ t+ d n 1 ¡ d n 2 2t # ; (3.6) wherex j = 8 < : 1 2 if j2S 1 \S 2 t¡ d n 1 2t if j2S 2 nS 1 : The expected profit of firmi givenS 1 andS 2 is max q i ¸0 X j2S i E £ rmin © q i j ;D i j ª ¡c j q i j ¡K ¤ : Using optimal stocking quantity e q i j =¸p i j +z q ¸p i j forj2S i , the optimal profit of firmi givenS 1 andS 2 is ¼ i (S 1 ;S 2 )= X j2S i h (r¡c j )¸p i j ¡rÁ(z) q ¸p i j ¡K i : Next, we analyze the optimal assortment decisions when one firm maximizes its profit in response to the other firm’s offerings. 66 Let the variants in S be ordered in increasing value of unit cost c j ; so that c 1 · c 2 :::·c n : At stage 1, firm i’s optimization problem is max S i ½S ¼ i (S 1 ;S 2 ) = P j2S i h (r¡c j )¸p i j ¡rÁ(z) q ¸p i j ¡K i ; where p i j is given by Equations (3.5) and (3.6). The following lemma will help us to obtain the optimal assortment structure. Lemma 2. Let0· ¾· maxfc l : l2 Sg; and define the profit associated with adding a variant k 2 S 2 nS 1 with c k = ¾ to S 1 as v(¾), and the profit associated with adding a variant k 2 S 1 \S 2 with c k = ¾ to S 1 as w(¾). Then v(¾) achieves the maximum at ¾ = 0 or minfc l : l 2 S 2 nS 1 g, and w(¾) achieves the maximum at ¾ = 0 or minfc l :l2S 1 \S 2 g: Lemma 2 implies that, if firm 1 wants to add a variant, the profit is maximized by either adding no variant or adding the variant with the smallest variable cost inS 2 nS 1 or inS 1 \S 2 . These properties allow us to characterize the structure of the optimal assort- ment and provide an efficient algorithm to derive the optimal assortment. Theorem 8. Let the variants inS 2 be ranked ask 1 ;k 2 ;:::;k n 2 ; such thatc k 1 · c k 2 :::· c kn 2 ; and let the variants in S 2 be k n 2 +1 ;:::;k n , such that c k n 2 +1 · ::: · c kn : Define A i =fk 1 ;:::;k i g for1·i·n 2 ,B i =fk n 2 +1 ;:::;k i g forn 2 +1·i·n; andA 0 =;; B n 2 = ;: Then there exist x 2 f0;1;:::;n 2 g; y 2 fn 2 ;n 2 + 1;:::;ng; such that the assortmentS ¤ 1 =A x [B y maximizes firm 1’s profit. In words, the optimal assortment for the firm contains the least costly i variants offered by the competitor and the least costly j variants not offered by the competitor, where 0 · i · n 2 , 0 · j · n¡ n 2 . This simple structure allows us to obtain the optimal assortment through an algorithm inO(n 2 ) time. 67 Now let us compare Theorem 8 with Theorem 5. Both show the effect of competition on the assortment decisions: the optimal assortment depends on the competitor’s offer- ings. When the variants have heterogeneous segment sizes and identical variable costs, the optimal assortment contains the most popular variants that the competitor offers and the most popular variants that the competitor does not offer, and when the variants have heterogeneous variable costs and the same segment sizes, the optimal assortment con- sists of the least costly variants offered the competitor and the least costly variants not offered by the competitor. In either case, the joint set may be discontinuous and not cover all of the best variants (which could be either the most popular variants or the least costly variants). In the case that variants have different segment sizes and variable costs, the assort- ment decisions will depend on the relative magnitude of cost heterogeneity and segment heterogeneity, which need further investigations in future. 3.6 Conclusions and Discussions Our paper investigates the assortment and inventory decisions under retail competi- tion with a consumer model that captures heterogeneous customer preferences for retail stores and for products. We have characterized the structure of the optimal assortment in response to a competitor’s offerings and identified the effect of competition by com- paring it with that obtained under monopoly settings. We have also obtained insights into the market equilibrium assortment structure and the effects on equilibrium assort- ments of customer preference heterogeneity, degree of product substitutability, degree of firm substitutability, the no-purchase option, variable costs, and fixed costs on the equilibrium assortments. 68 Our paper yields some key managerial insights that may apply more generally in spite of the stylized model. We find that when a firm is faced with competition the opti- mal assortment contains the most popular products offered by the competitor together with the most popular products not offered by the competitor. The joint set may not cover all the top segments, but when retail price is high or the market volume is large the optimal assortment does contain a contiguous set of the most popular variants, sim- ilar to the structural property in monopoly settings obtained in previous studies. In addition, we find that the size of the optimal assortment decreases when the com- petitor increases the number of product offerings, which implies that the marginal ben- efit of a new variant shrinks when the competitor offers more variety. The trade-offs that a firm faces when selecting the optimal assortment are threefold: it has an incentive to fight for market share by competing in major segments and offering more variants, but it also has an incentive to avoid direct competition by targeting niche segments and an incentive to save operational costs by limiting product offerings. This complex incentive structure results in the various assortment offerings under different market conditions. We find that in equilibrium, both firms offer variants for major segments but differ in product offerings for smaller segments. As segments sizes become more comparable, which indicates that customers’ tastes are more spread out, firms are more likely to offer different products and more variety because of an incentive to capture market share while reducing competition. Furthermore, we find that firms offer more variety in equilibrium when market potential or retail price is high, or when product misfit cost is high (degree of prod- uct substitutability is low) or fixed cost is small. The more surprising result is the effect of competitive pressure on the equilibrium assortments. When the competitive pressure is low, as indicated by a low reservation utility or a large transportation cost, firms offer 69 the most popular variants as if they are in a monopoly. When the competitive pressure is moderate, they differ in product offerings as much as possible in order to reduce com- petition. However, under extremely high competitive pressure, they move back to target the biggest segments again, because offering different products can no longer prevent customers from switching to the other firm and therefore they would rather capture the major segments. Finally, we characterize the assortment structure when the variants have different variable costs. When the segment sizes are comparable, the optimal assortment is com- posed of the least costly variants offered by the competitor and the least costly variants not offered by the competitor. In equilibrium, both firms offer the least costly vari- ants but differ in other product offerings, which mirrors the assortment structure when products have different segment sizes but identical variable costs. Our paper points to some important directions to explore in future research based on the modeling framework we have proposed. First, we may generalize the model to incorporate price as a decision variable and investigate how the assortments change when there is price competition. Firms may use price as a tool to differentiate their products, so it is possible that they offer different assortment and prices in equilibrium. Second, the model may be extended to incorporate inventory-based substitution in which customers may substitute to other products when there is a stockout. The assort- ment offerings may be affected in a way that firms are more willing to offer the most popular variants due to the dynamic substitution. Further investigations may provide more insights regarding the assortment structure and inventory levels. Finally, we may extend the model to a multi-period problem. There are two possible directions. Retailers may select the assortments and then dynamically change the inven- tory levels, or they may change the assortment and stocking quantities every period. For 70 the former scenario, if the demand distribution is static, the insights obtained in this work regarding the assortments may still hold, while the inventory policies may change. The second model, involving dynamic assortment competition with updated demand distribution, would be a more challenging problem that could yield insights about the trade-offs between exploration and exploitation under competition. 71 Chapter 4 Retail Assortment and Price Competition with Effect of Brand Reputation 4.1 Introduction Product proliferation has become prevalent in many product categories, especially in consumer products such as shampoos (e.g. Head & Shoulder), toothpastes (Crest and Colgate), cereals (Cheerios), and yogurt (Yoplait), etc. Prompted by customers’ demand for increased variety and personalization and facilitated by flexible technologies that have reduced the cost of product proliferation, manufacturers keep up offering more and more variety to capture market share. Although internet sale or direct sale is more common nowadays, most of the man- ufactures still sell goods through traditional (brick-and-mortar) retailers, who have lim- ited shelf space and incur inventory costs and hence may not be able to carry all of the variants offered by the manufacturers. On the other hand, retailers may attract customers by leveraging the prices. That is, there are two ways to gain the market share: increas- ing the number of variants offered, which increases the associated costs, or lowering the prices, which hurts the margin. Therefore, how many variants to select, which variants to offer, and how much to charge for the price are interrelated decisions that retailers constantly face. In addition, due to the nature of retail industry, competition is intense. 72 Thus these decisions should be made in consideration of what the nearby stores offer and the prices they charge. To address above issues, in this paper we propose a game-theoretic modeling frame- work to study the assortment and pricing strategies in retail competition. In our model, customers have heterogeneous preferences for retail store and for product. Two retailers choose which variants to carry from the same set of variants offered by a manufacturer. After they know each other’s product assortment, they determine the prices to charge, and finally customers make their purchasing decisions based on their preferences and the stores’ product and price offerings. Although each customer has an ideal product, she may substitute to other products if her ideal product is not offered. We ask the following research questions: In presence of retail competition, what are the optimal assortment and pricing strategies that balances the cost and benefit of product variety? In equilibrium, how are the variety and pricing strategies related to the firm and market characteristics? Especially, what is the effect of brand reputation on the assortment and pricing decisions? We obtain the following main results and insights: 1. Assortment structure: When there is sufficient firm differentiation, the optimal strategy is to target on the biggest segments, regardless of what the competitor offers. 2. Effect of market characteristics on variety strategy: A) Firms offer more variety when customer preferences are more heteroge- neous, or less willing to substitute between products. B) The firm with a higher brand reputation (e.g. better store environment or better service) offers a larger variety/assortment and charges higher prices, if they offer same product qualities. 73 C) As the market potential increases, or the total number of segments increases, the optimal variety increases at a slower rate because of the decreasing return from variety 3. Effect of product variety in price competition: If two firms are symmetric, then the firm offering a larger variety would charge higher prices to exploit the consumer surplus, which indicates that increasing the variety does not intensify the price competition, if there is sufficient firm differentiation. This result contrasts with the previous literature (e.g. [MGN88]) which argues that firms have incentive to limit the variety because variety intensifies price competition. Our main contribution is as follows. We propose a modeling framework that cap- tures customer purchasing behaviors to analyze the assortment and pricing decisions under retail competition. Customers have heterogeneous preferences for firm and for products, and make their purchasing decisions based on product assortments, prices and brand reputation of the stores. Assortment-based product substitution is incorporated. We show that variety and pricing strategies depend on the brand reputation, market char- acteristics and cost factors. In particular, the firm with a higher reputation offers a larger assortment and charges higher prices, if product quality is the same. In contrast to the previous literature [MGN88], we find that increased variety does not intensify the price competition, if there is sufficient firm differentiation. The article will proceed as follows. In the next section, we discuss existing literature related to assortment planning and product variety. In subsequent sections, we present a model in which retailers determine their assortments and prices in a duopolistic set- ting. We then analyze the assortment and pricing strategies when product segments are homogeneous and heterogeneous, respectively. Furthermore, we analyze how the brand reputation affects the stores’ product assortment and pricing strategies. Finally, we provide concluding comments and suggest opportunities for future research. 74 4.2 Literature Review This research is related to the assortment planning literature in the operations area. Most of the research works study the product assortment and inventory decisions under monopoly settings, which we will review first. The representative early works are Smith and Agrawal [SA00] and Mahajan and Van Ryzin [MvR01b] which study inventory systems where consumer choice and substitu- tion effects are modeled in great detail and substitution in out-of-stock situations is prob- abilistic. The focus of this work is on determining both optimal product assortments as well as inventory levels of the items as a function of consumer purchase behavior. Kok, Fisher and Vaidyanathan [KFV06] provide a comprehensive review of the literature on assortment planning. Most of this literature focuses on a single retailer making assort- ment and inventory decisions. There are a few papers that address competitive models such as Parlar and Goyal [PG84], Lippman and McCardle [LM97], and Netessine and Rudi [NR03] but they are focused on inventory decisions in a competitive environment rather than assortment decisions and do not address pricing decisions. We now review the literature more closely related to our work that studies assort- ment decisions under retail competition. Shugan [Shu89] explores when and why pre- mium quality producers may provide a smaller assortment than lower quality producers, focusing on the ice-cream industry. He finds that a super-premium producer will offer a smaller relative assortment than a regular producer when its variable costs are higher, customers are more price-sensitive or the products are more substitutable. Bergen, Dutta and Shugan [BDS96] point out that manufacturers offer numerous versions of a branded product through retailers. This decreases price competition across retailers and so the retailers carry more variety and provide higher service levels. Subramanian and Kapus- cinski [SK02] consider variety and pricing decisions under various supply chain struc- tures that exhibit different levels of centralizations among manufacturers and retailers. 75 They use a market share attraction model to capture the impact of price and competition on demand and prove existence of equilibria in some supply chain structures. In recent literature, Kuksov [Kuk04] considers a scenario where two firms sell a dif- ferentiated product and examines the impact of search costs on prices and variety. The paper shows that there are two effects of lower search costs – a direct effect that low- ers prices and an indirect effect that increases product differentiation and prices. They show that counter to intuition, the net impact of search costs may be higher prices and product differentiation. Cachon, Terwiesch and Yu [CTX06] study the same issue but in an oligopolistic market and use a different consumer purchase model and consider the no-purchase option. They reach similar conclusion but also provide additional insights and point out that the market expansion effect of lower search costs may result in larger assortments and higher prices. Although they model the firm competition on price and assortment, they assume there is no overlapping between each store’s assortment, and all products have the same probability to be preferred as ideal products. In contrast, we allow the firms to choose from the same set of variants, and each product may have different popularity. Cachon and Kok [CK07] study the category management problem in the presence of basket shopping consumers, using MNL to model customer purchas- ing behaviors. They compare variety and prices under decentralized regime with the optimal centralized solution and propose a decentralized regime using basket profits as a metric to produce near-optimal solutions. The model of consumer purchase and sub- stitution in this dissertation is different from the ones in the above papers and this allows us to obtain closed-form solutions and more insights regarding the strategic interactions between assortment and pricing decisions, as pointed out in Section 1. There has also been some empirical work relating to the value of variety to con- sumers. Shugan [Shu88] found that prices vary with assortment levels in a study of retail outlets. Kekre and Srinivasan [KS90] find that broader product lines significantly 76 increase market shares without increasing much of production costs, and therefore may increase firms’ profitability. Watson (2004) performs an empirical study of variety deci- sions in the eyewear market and finds that an increase in the number of competitors in a market reduces the variety offered. Hui [Hui04] finds that there is decreasing demand returns to product variety for branded multi-product firms in the personal computer industry, which is consistent with the predictions of our model. 4.3 Model Formulation 4.3.1 Product Variants, Prices, and Costs Within a single product category, there are n customer segments, and each firm can produce a product variant j that is targeted at segment j. Let S = f1;2;:::;ng denote the set of possible variants, S i ½ S denote the subset of variants stocked by store i, i = 1;2. and n i denote the variety, i.e. n i = jS i j. Firm i offers its product variants at price p i , with unit cost c. This applies to the horizontally differentiated products, such as different colors of clothes, or different flavors of yogurt, where the firm charges same price and incurs same unit cost for all the product variants. For each variant offered, we assume that the firm incurs a fixed costK, which could include the product development costs, ordering costs, or shelf-space costs. 4.3.2 The Customer Demand Model We denote the size of customer segment j as M j , j = 1;2;:::n: Each segment is pre- sented by a line segment of unit length. Assume Firm 1’s products are located at the left ends of the segments, while Firm 2’s products are at the right ends, in terms of store locations or brand images. Customers are uniformly distributed on the line, in terms of 77 their physical locations or preferences for the brands. If Firmi offers a product for this segment, then for a customer located atx, her utility of purchasing a product from Firm i isu¡tjx¡X i j¡p i , whereu represents customer’s reservation price for her ideal prod- uct,t represents the transportation cost, andX 1 = 0;X 2 = 1. If Firmi does not offer a product for this segment, then the customer may substitute to other products offered by this firm, and obtain utility u¡ d n i ¡tjx¡X i j¡p i , where d n i represents the disutility of substitution. A larger variety would make the disutility of substitution smaller, hence the customers would be more likely to substitute, and vise versa. 1 Here d measures the degree of substitution between products, whilet measures the degree of substitution between firms. A lower value ofd ort implies a higher degree of substitution. Product and customer space is shown in Figure ?? at page ??. From the consumer utility function, we can find the indifferent customer location for each segment, given the two firms’ offerings S 1 and S 2 . The indifferent customer for segmentj is: x j = 8 > > > > > > > < > > > > > > > : t+p 2 ¡p 1 2t if j2S 1 \S 2 t+ d n 2 +p 2 ¡p 1 2t if j2S 1 nS 2 t¡ d n 1 +p 2 ¡p 1 2t if j2S 2 nS 1 t+ d n 2 ¡ d n 1 +p 2 ¡p 1 2t if j2S 1 \S 2 : (4.1) We assume x j 2 [0;1]: Later we will show that in equilibrium this condition is satisfied if we assumet > d, i.e., there is more firm differentiation than product differ- entiation so that no firm can capture the entire market share. We assume thatu is high enough so that every customer makes a purchase. We make this assumption to focus on the effect of competition. In some sense, the competitor’s 1 We can think that the product space is a circle, and product variants are evenly dispersed on the product space. Then if a customer could not find her ideal product, her closest variant would give her a disutility proportional to 1 n i . Letd represents the cost of product misfit, we have above expressions. 78 offerings can be regarded as external utilities that customers may get. Besides, there are realistic cases in which customers will buy a product if she needs one immediately, such as groceries and staples. Therefore, the customers located between0 andx j will purchase from Firm 1, while customers located within[x j ;1] will buy from Firm 2. Customers will first decide which store to visit according to their product offerings, then they will visit that store and choose their ideal product if it is offered. Otherwise they will choose the second ideal product. 4.4 Analysis with Homogeneous Product Segments The two firms engage in the assortment and price competition stated as the following: At stage 1, each firm chooses the assortment to offer,S i ½S; simultaneously. At stage 2, they determine the price to charge for the product offerings,p i : If the product variants have homogeneous segment size, we assume M j = M, for anyj2S. 4.4.1 Equilibrium of Price Competition The total demand for Store 1 is: D 1 (n 1 ;n 2 ;p 1 ;p 2 ) =M[ t+p 2 ¡p 1 2t jS 1 \S 2 j+ t+ d n 2 +p 2 ¡p 1 2t jS 1 nS 2 j+ t¡ d n 1 +p 2 ¡p 1 2t jS 2 nS 1 j + t+ d n 2 ¡ d n 1 +p 2 ¡p 1 2t ¯ ¯ S 1 \S 2 ¯ ¯ ] =M[ t+p 2 ¡p 1 2t (jS 1 \S 2 j+jS 2 nS 1 j)+ t+ d n 2 +p 2 ¡p 1 2t ¡ jS 1 nS 2 j+ ¯ ¯ S 1 \S 2 ¯ ¯ ¢ ¡ d 2tn 1 (jS 2 nS 1 j+ ¯ ¯ S 1 \S 2 ¯ ¯ )] =M[ t+p 2 ¡p 1 2t n 2 + t+ d n 2 +p 2 ¡p 1 2t (n¡n 2 )¡ d 2tn 1 (n¡n 1 )] 79 =M h n 2 + nd 2t ( 1 n 2 ¡ 1 n 1 )+ n(p 2 ¡p 1 ) 2t i ; which is decreasing inn 2 ;p 1 ; and increasing inn 1 ;p 2 . Note that the demand only depends on the variety and prices, not specific product offerings. This property is unique for homogeneous products. For heterogeneous prod- ucts, the product offerings do matter. Firm 1’s optimization problem at stage 2 is max p 1 ¼ 1 (n 1 ;n 2 ;p 1 ;p 2 )=D 1 (n 1 ;n 2 ;p 1 ;p 2 )(p 1 ¡c)¡n 1 K; whereD 1 (n 1 ;n 2 ;p 1 ;p 2 ) is given by Equation (4.4.1). The optimal pricing strategy for Firm 1 is then p 1 = 1 2 µ p 2 +c+t+ d n 2 ¡ d n 1 ¶ : Similarly, optimal price offered by Firm 2 is p 2 = 1 2 µ p 1 +c+t+ d n 1 ¡ d n 2 ¶ : We then obtain the following proposition of price equilibrium. Proposition 6. With homogeneous products, the price equilibrium given the two firms assortment offerings is p ¤ 1 = c+t+ d 3 ( 1 n 2 ¡ 1 n 1 ) p ¤ 2 = c+t+ d 3 ( 1 n 1 ¡ 1 n 2 ): Proof: Obtained from the price response functions of the two firms. 80 The equilibrium prices are linearly increasing in transportation cost, because as t increases, customers are less likely to switch between firms, and so they can charge higher prices. Note that the price charged by a firm is concave increasing in its own variety, and convex decreasing in opponent’s variety, which indicates that increasing variety has a positive yet diminishing effect on its own price and a diminishing negative effect on the competitor’s price. From Proposition 6 we can easily obtain the following corollary that indicates the interaction between variety and pricing decisions. Corollary 1. Ifn i >n j , thenp ¤ i >p ¤ j ;i6=j: The firm who offers a greater variety charges higher prices. The reasoning is as follows. The firm with a lower product variety is less attractive to potential customers, and so it tends to lower the price to compensate for this deficiency. In contrast, the firm offering a larger variety will charge a higher price to exploit the surplus. This result contrasts with those in the literature where increased variety (market segmentation) leads to more intense price competition (e.g. Martinez-Giralt and Neven, 1988). This is because we allow for firm-specific preferences and thus retailers do not compete in entirely the same market. Increasing variety will enable the firm to charge higher prices to exploit the consumer surplus without intensifying price competition. 4.4.2 Product Variety in Equilibrium Working backwards, at stage 1, Firm 1 must decide on the optimal variety to maximize its profit ¼ 1 (n 1 ;n 2 ;p ¤ 1 ;p ¤ 2 )= nM 2t · t+ d 3 ( 1 n 2 ¡ 1 n 1 ) ¸ 2 ¡n 1 K: 81 The profit function is concave in n 1 under the assumption t > d: So the optimal varietyn 1 should satisfy the first order condition µ 3t d + 1 n 2 ¡ 1 n 1 ¶ 1 n 2 1 ¡ 9Kt nMd 2 =0: (4.2) Similarly, we obtain the first order condition forn 2 : µ 3t d + 1 n 1 ¡ 1 n 2 ¶ 1 n 2 2 ¡ 9Kt nMd 2 =0: (4.3) Although these are implicit solutions, we can derive the explicit solution of the equi- librium variety, as stated in the following theorem: Theorem 9. With homogeneous products, there exists a unique Nash equilibrium, where n ¤ 1 =n ¤ 2 = q nMd 3K ;p ¤ 1 =p ¤ 2 =c+t: Several insights obtained from Theorem 9 are explained as follows. First, equilib- rium prices are increasing in transportation cost. A higher transportation cost increases the firm differentiation and consequently reduces pressure of price competition. Second, we find that equilibrium varieties are concave increasing in the market size and the total number of variants available, which demonstrates the decreasing return from variety, i.e, the marginal benefit brought by additional variant decreases as variety increases. In contrast, equilibrium varieties are convex decreasing in the fixed cost. Thus, as the variety cost decreases, the equilibrium variety expands at an ever faster rate. This may help explain the soaring variety for some products such as yogurt, cereals, toothpastes, etc., as the flexible manufacturing technologies have reduced the cost of product proliferation. 82 Finally, equilibrium varieties are concave increasing in product misfit cost, which means that more variety is provided if customers are less willing to substitute. For example, customers are usually more sensitive to the sizes of pants than the colors, and we see that retail stores offer many different sizes but only a few colors for dress pants. 4.5 Analysis with Heterogeneous Product Segments The assortment and price competition game is same as in Section 3, but now we look at the products with different segment sizeM j : For ease of exposition, we letM j = ¸m j , where¸= P j2S M j . Without loss of generality, we rank the segments in decreasing order of segment sizes: m 1 ¸ m 2 ¸ ::: ¸ m n : From Equation (4.1), we obtain the total demand for Firm 1 as P j2S 1 \S 2 t+p 2 ¡p 1 2t ¸m j + P j2S 1 nS 2 t+ d n 2 +p 2 ¡p 1 2t ¸m j + P j2S 2 nS 1 t¡ d n 1 +p 2 ¡p 1 2t ¸m j + P j2S 1 \S 2 t+ d n 2 ¡ d n 1 +p 2 ¡p 1 2t ¸m j =¸f 1 (S 1 ;S 2 )+¸ p 2 ¡p 1 2t ; where f 1 (S 1 ;S 2 ) = X j2S 1 \S 2 1 2 m j + X j2S 1 nS 2 t+ d n 2 2t m j + X j2S 2 nS 1 t¡ d n 1 2t m j + X j2S 1 \S 2 t+ d n 2 ¡ d n 1 2t m j (4.4) = 1 2 ¡ d 2tn 1 X j2S 1 m j + d 2tn 2 X j2S 2 m j (4.5) is independent ofp 1 ;p 2 : In fact, f 1 (S 1 ;S 2 ) is the market share of Firm 1 when the two firms offer the same price. 83 4.5.1 Equilibrium of Price Competition The optimal price for the problem max p 1 ¼ 1 (S 1 ;S 2 ;p 1 ;p 2 )=(¸f 1 (S 1 ;S 2 )+¸ p 2 ¡p 1 2t )(p 1 ¡c)¡n 1 K is p 1 = 1 2 [p 2 +c+2tf 1 (S 1 ;S 2 )]: Similarly Firm 2 would choose the price p 2 = 1 2 [p 1 +c+2tf 2 (S 1 ;S 2 )] to maximize its profit givenS 1 ;S 2 , where f 2 (S 1 ;S 2 )= X j2S 1 \S 2 1 2 m j + X j2S 2 nS 1 t+ d n 1 2t m j + X j2S 1 nS 2 t¡ d n 2 2t m j + X j2S 1 \S 2 t+ d n 1 ¡ d n 2 2t m j (4.6) is the market share of Firm 2 when two firms offer same prices. Usingf 2 (S 1 ;S 2 ) = 1¡f 1 (S 1 ;S 2 ); we obtain the price equilibrium Proposition 7. With heterogeneous products, the price equilibrium given the two firms assortment offerings is p ¤ 1 = c+ 2t 3 (1+f 1 (S 1 ;S 2 )) p ¤ 2 = c+ 2t 3 (1+f 2 (S 1 ;S 2 )); wheref 1 (S 1 ;S 2 ) andf 2 (S 1 ;S 2 ) are given by Equation (4.4) and (4.6). Proof: Obtained from the price response functions of the two firms. Corollary 2. Iff i (S 1 ;S 2 )>f j (S 1 ;S 2 ), thenp ¤ i >p ¤ j :i6=j: 84 The corollary means that the firm whose assortment decision allows him to win a larger market share under exogenous prices would charge a higher price to exploit the surplus. 4.5.2 Product Variety in Equilibrium Firm 1’s profit at the price equilibrium is given by ¼ 1 (S 1 ;S 2 ;p ¤ 1 ;p ¤ 2 )= 2¸t 9 (1+f 1 (S 1 ;S 2 )) 2 ¡n 1 K: (4.7) Therefore, at stage 1, given the competitor’s offering S 2 , Firm 1 should choose S 1 so thatf 1 (S 1 ;S 2 ) is as large as possible. Proposition 8. GivenS 2 andn 1 , Firm 1’s optimal assortment strategy is to choose the largestn 1 segments independent ofS 2 : Proof: We will show that f 1 (S 1 ;S 2 ) is maximized by choosing the largest n 1 seg- ments. Suppose it is not true. Let S 1 be a set that maximizes f 1 (S 1 ;S 2 ); then there exists variant j = 2 S 1 , which has larger segment size than the smallest segment k in S 1 : Let S 0 1 = S 1 nfkg; S 0 1 = S 0 1 [fjg: Then from Equation (4.4), f 1 (S 0 1 ;S 2 )¡f 1 (S 1 ;S 2 ) = ¡ d 2tn 1 P j2S 0 1 m j ¡(¡ d 2tn 1 P j2S 1 m j )= d 2tn 1 (m j ¡m k )>0: RedefineS 1 to beS 1 nfkg[fjg, and repeat the procedure. Finally we will arrive at a set which contains largestn 1 segments.¦ Proposition 8 states that if Firm 1 has chosen a variety level, then it should target on the most popular segments regardless of the competitor’s offerings. However, the optimal variety level does depend on the competitor’s assortment, as shown below. 85 By selecting the largest segments1;:::;n 1 ; f 1 (S 1 ;S 2 )= 1 2 + d 2tn 2 P j2S 2 m j ¡ d 2tn 1 n P j=n 1 +1 m j ; which is an increasing function ofn 1 : Substitute it into Equation (4.7), we get ¼ 1 (n 1 ;S 2 )= 2¸t 9 ( 3 2 + d 2tn 2 X j2S 2 m j ¡ d 2tn 1 n X j=n 1 +1 m j ) 2 ¡n 1 K; (4.8) which may not be concave inn 1 : Therefore, the optimaln 1 is obtained by searching from1 ton that maximizes Equa- tion (4.8), which takesO(n) time. Although there is no closed-form solution of optimal variety, we can characterize the equilibrium variety and pricing decisions if we restrict to symmetric equilibrium. (However, asymmetric equilibrium may exist, as shown in our numerical examples: ) Theorem 10. With heterogeneous products, there exists a unique symmetric equilib- rium, where both firms offer variants for the largestn ¤ segments,n ¤ given by: n ¤ m n ¤ + n P j=n ¤ +1 m j n ¤2 = 3K ¸d (4.9) andp ¤ 1 =p ¤ 2 =c+t: The following corollary shows the effect of of market characteristics and cost factors on the equilibrium variety. Corollary 3. The equilibrium variety increases as market potential increases, degree of product substitution decreases, or fixed cost decreases. I.e., @n ¤ @¸ >0; @n ¤ @d >0; @n ¤ @K <0: Proof: LHS= m n ¤ n ¤ + n P j=n ¤ +1 m j n ¤2 ; which is decreasing inn ¤ : As¸ ord increases, orK decreases, RHS decreases, thereforen ¤ increases.¦ 86 Therefore firms offer more variety as market size increases, variety cost decreases, or customers are less willing to substitute between products. By comparing equilibrium variety in Theorem 9 and Theorem 10, we obtain the next proposition regarding the effect of segment heterogeneity on the equilibrium varieties. Proposition 9. In equilibrium, firms offer less variety under heterogeneous segments than under homogeneous segments. Proof: We can rewrite n ¤ m n ¤ + n P j=n ¤ +1 m j as n P j=1 m j ¡ n ¤ P j=1 (m j ¡ m n ¤) = 1 ¡ n ¤ P j=1 (m j ¡ m n ¤): Rewrite (4.9) as n ¤ = s ¸d 3K (n ¤ m n ¤ + n P j=n ¤ +1 m j ) = s ¸d 3K (1¡ n ¤ P j=1 (m j ¡m n ¤)) < q ¸d 3K because n P j=1 (m j ¡ m n ¤) > 0 for heterogeneous products. From Theorem 9,we have n ¤ = q ¸d 3K for homogeneous segments. Thus the equilibrium variety under heterogeneous segments is less than that under homogeneous segments.¦ In other words, firms offer more variety when the segment sizes are comparable than when the segments sizes greatly differ from each other. When segment sizes are com- parable, customer preferences are more spread out, such as different flavors of yogurt, so firms offer more variety to satisfy different customers’ needs. In contrast, for ice creams for which customers’ tastes are concentrated in a few flavors such as vanilla and chocolate, fewer variants are offered. Another example is women’s boots, for which most people like brown or black col- ors and only a small fraction of customers like other colors such as white. Therefore only a few colors are offered. In contrast, people have quite different tastes for sandals, so a lot of colors are provided for the same style (e.g. www.zappos.com). 87 4.6 Effect of Brand Reputation In reality, customers may have different reservation prices for different retailers. For instance, if both Nordstrom and Target offer the same good, customers are willing to pay more in Nordstrom than in Target because of a better shopping environment and better service. This raises the question of whether and how the firms’ reputation affects their assortment offerings and pricing decisions. In this section we extend our model to incorporate the brand or store reputation. To focus on the effect of brand reputation on variety and pricing decisions, we restrict to homogeneous product segments. The results can be easily extended to the heteroge- neous product case. We assume that two firms have different brand reputation so that customer’s reserva- tion price for an ideal product offered by Firmi isu i , whereu 1 6=u 2 :Then for a customer located atx, her utility of purchasing a product from Firmi isu i ¡tjx¡X i j¡p i , and the indifferent customer for segmentj is Segment Indifferent Customer j2S 1 \S 2 u 1 ¡u 2 +t+p 2 ¡p 1 2t j2S 1 nS 2 u 1 ¡u 2 +t+ d n 2 +p 2 ¡p 1 2t j2S 2 nS 1 u 1 ¡u 2 +t¡ d n 1 +p 2 ¡p 1 2t j2S 1 \S 2 u 1 ¡u 2 +t+ d n 2 ¡ d n 1 +p 2 ¡p 1 2t We see that the indifferent point and market shares now depend on not only the variety and prices offered but also the relative brand reputation. 4.6.1 The Price Equilibrium The total demand for Store 1 is now 88 D 1 (n 1 ;n 2 ;p 1 ;p 2 )=M h n 2 + nd 2t ( 1 n 2 ¡ 1 n 1 )+ n(u 1 ¡u 2 +p 2 ¡p 1 ) 2t i : Firm 1’s optimization problem at stage 2 is max p 1 ¼ 1 (n 1 ;n 2 ;p 1 ;p 2 )=D 1 (n 1 ;n 2 ;p 1 ;p 2 )(p 1 ¡c)¡n 1 K; whereD 1 (n 1 ;n 2 ;p 1 ;p 2 ) is given by Equation (4.6.1). The optimal pricing strategy for Firm 1 is then p 1 = 1 2 µ p 2 +u 1 ¡u 2 +c+t+ d n 2 ¡ d n 1 ¶ : Similarly, optimal price offered by Firm 2 is p 2 = 1 2 µ p 1 +u 2 ¡u 1 +c+t+ d n 1 ¡ d n 2 ¶ : We then obtain the following proposition of price equilibrium. Proposition 10. With different brand reputation, the price equilibrium given the two firms assortment offerings is p ¤ 1 = u 1 ¡u 2 +c+t+ d 3 ( 1 n 2 ¡ 1 n 1 ) p ¤ 2 = u 2 ¡u 1 +c+t+ d 3 ( 1 n 1 ¡ 1 n 2 ): Proof: Obtained from the price response functions of the two firms. The equilibrium prices are affected by the brand reputation, substitution degree between firms and between products, and the variety levels of the two firms. Propo- sition 10 implies that if Firm 1 has a higher reputation, u 1 > u 2 , then it will charge higher prices even if the two firms offer the same level of variety. Corollary 4. Ifu i >u j ,n i =n j , thenp ¤ i >p ¤ j :i6=j: 89 If firms offer the same variety, then the one with higher reputation charges higher prices to exploit the consumer surplus, which confirms our intuition. 4.6.2 Effect of Brand Reputation on Equilibrium Variety and Prices At stage 1, Firm 1 decides on the optimal variety to maximize its profit ¼ 1 (n 1 ;n 2 ;p ¤ 1 ;p ¤ 2 )= nM 2t · u 1 ¡u 2 +t+ d 3 ( 1 n 2 ¡ 1 n 1 ) ¸ 2 ¡n 1 K: The profit function is concave inn 1 ifu 1 ¡u 2 +t>d: The optimal varietyn 1 should satisfy the first order condition µ 3(u 1 ¡u 2 +t) d + 1 n 2 ¡ 1 n 1 ¶ 1 n 2 1 ¡ 9Kt nMd 2 =0: (4.10) Similarly, we obtain the first order condition forn 2 : µ 3(u 2 ¡u 1 +t) d + 1 n 1 ¡ 1 n 2 ¶ 1 n 2 2 ¡ 9Kt nMd 2 =0 (4.11) The equilibrium variety under different brand reputation is stated in the following theorem: Theorem 11. With different brand reputation, the firm with higher reputation will offer a larger variety and charge higher prices. I.e., ifu i >u j ; thenn ¤ i >n ¤ j ; andp ¤ i >p ¤ j : We need to keep in mind that here the brand reputation refers to the store reputation, such as store environment or service. We assume both stores provide same quality of products. An example is illustrated in Table ( ). Ralphs and Food4Less are both major grocery chain stores in Southern California. Table ( ) compares the number of variants of Yoplait Yogurt offered in Ralphs vs. in Food4Less, as well as the prices labeled. It 90 shows that Ralphs grocery stores, who offer better shopping environment and services, provide a larger assortment and charge higher prices for Yoplait Yogurt than Food4Less, which is consistent with our predictions. Same phenomena have also been observed in other product categories such as shampoos. Table. Comparisons of Yoplait Yogurt Offerings in Ralphs vs. in Food4Less Ralphs Food4Less Yoplait Original 18 17 Yoplait Light 15 7 Yoplait Light Thick & Creamy Custard Style 6 1 Yoplait Whips 7 0 Price ($) 0.85 0.68 If, however, the firm with a higher reputation offers higher quality of products with potentially higher variable costs, then it may provide more or less variety depending on the relative differences in reputation and variable costs. Shugan [Shu89] points out that in some industries producers of premium quality goods offer a smaller assortment, for example, Haagen-Dazs offers fewer flavors of ice cream than Baskin Robins. However, the quality of the ice-cream is different and so are the corresponding variable costs. In fact, higher variable cost of the higher quality firm is a key condition required for the result in [Shu89] (p. 316). While our model does not explicitly model differences in quality, it can be shown that higher variable costs (which signifying higher quality) can result in a smaller assortment in our model. 91 4.7 Conclusions In this article we explored the interactions between product assortment and pricing strategies under retail competition, which has not been fully studied in the literature. Using a framework that models customers’ behaviors in response to product assortment and pricing strategies, we obtained the following main results: In terms of product variety strategy, we find that increasing variety does NOT inten- sify the price competition, if there is sufficient firm differentiation. In fact, a larger variety allows the firm to charge higher prices to exploit the consumer surplus. This result contrasts with the findings of past research which point out that variety intensifies price competition. Product variety should be adjusted according to the market characteristics. When customer preferences are more heterogeneous (spread out), or they are less willing to substitute between products, a larger variety should be offered. We also identified the decreasing return from variety which implies that the optimal variety increases at a slower rate as the market expands. Furthermore, the store reputation has an impact on variety and pricing strategies: The retail store with a better reputation (in terms of environment and service) will offer a larger variety and charge higher prices even if they offer same quality of products. Observations have been made which confirmed our predictions. Future research may extend the model to retailers competing in multiple product cat- egories and characterize the optimal assortment and pricing strategies. Our results could be tested empirically by collecting data from retail stores that are nearby and compete in the same product categories. Although we focus on retail competition in this paper, the model we proposed may also apply to the manufacturers and product-line design with appropriate modifications. One direction is to consider quality differentiation in which firms may provide products of different qualities. Then the product variety and price 92 decisions depend on product qualities as well. The firm with higher reputation may not provide a larger assortment, as higher quality products incur more variable costs. 93 Chapter 5 Conclusions and Future Directions 5.1 Conclusions In this dissertation we have proposed a modeling framework to study product variety management problems that arise in a competitive environment. In this framework, cus- tomers have heterogeneous preferences for firm and for product attribute, and they max- imize their individual net utilities by making appropriate choices. Taking into account of the customer behavior, firms that compete with each other make the strategic deci- sions to maximize their own profitability in response of competitor’s decisions. There are several ways to attract more customer demand: increasing the variety, offering prod- uct customization or lowering the prices, all of which increase the cost for the firms. This dissertation addresses the issue of how to optimize the profitability by making the appropriate decisions of production technology, product assortment, pricing, and inven- tory levels. Chapter 2 investigates production technology decisions in a competitive setting and the trade-offs between standard products and customized products. It shows that the relative strength of a product strategy depends on its cost efficiencies as well as attrac- tiveness to customers. We developed an index policy for the firms: the product strategy that has a smaller index value is superior and therefore should be chosen. In addition, the subsequent decisions of product variety and lead time also play an important role in the 94 firms’ profitability: increasing variety or decreasing lead time can both attract more cus- tomers, but with diminishing returns. The equilibrium structure is characterized which indicates strategic interactions between these decisions. Chapter 3 examines assortment planning and inventory decisions when two retailers compete in the same product category. The effect of competition is reflected in trade- offs between targeting major segments to fight for market share and targeting niche segments to avoid direct competition. We showed that the optimal assortment depends on the competitor’s offerings: it should contain the most popular products offered by the competitor and the most popular products not offered by the competitor. Interestingly, and in contrast to the result under monopoly settings, the optimal assortment may not cover a contiguous set of most popular variants. In equilibrium both firms offer variants for major segments but differ in product offerings for smaller segments. The effect of market characteristics and cost factors on the optimal assortment and equilibrium assortments is analyzed and insights obtained. In particular, firms are more likely to offer different products and more variety as customers tastes are more heterogeneous. As prices can be used to influence consumer demand, Chapter 4 studies the inter- actions between product variety and pricing strategies in presence of retail competition. In contrast to the previous literature, we find that increasing variety does not intensify the price competition if there is sufficient firm differentiation. Rather, it relieves the price pressure for the firm as it satisfies consumer needs better and enables higher price premiums. The product variety depends on the customers’ preferences for the products: firms offer more variety if customers have heterogeneous preferences, or less willing to substitute between products. Furthermore, we examine the effect of store/brand reputa- tion on variety and pricing strategies and find that the firm with a higher reputation will offer a larger variety/assortment and charge higher prices to fully exploit the consumer surplus. 95 The results and insights obtained in the dissertation either complement or contrast to the literature on product variety management or assortment planning. 5.2 Future Directions More research could be done in the future based on the framework proposed in this dis- sertation. This dissertation focused on horizontally differentiated products in a single product category. Future research may study multiple product categories and consider customers with basket-shopping behaviors, which may yield insights regarding the inter- play of assortment and pricing decisions in different product categories. Alternatively, research may be extended to include quality differentiation or vertical differentiation. There are two possible directions. In the first case, competing firms may offer products of different qualities, but each firm offers a group of products with the same qualities. In the more complex case, each firm may offer a product line with quality-differentiated products. Customer preferences for quality need to be modeled and principal-agent model may be used to analyze the optimal product quality, variety and pricing decisions for the firms. Future research may model consumer purchasing behaviors deeper by considering inventory-based substitution in which customers may substitute to other products if their intended product is out of stock. The substitution may be done once or dynamically, and the probability of a product being chosen as a substitute may depend on its inventory level or popularity. Results may provide additional implications for store’s assortment and inventory policies. A more challenging direction is the dynamic assortment planning in the competitive setting, in which demand is uncertain and may be updated in each period. Assortment and pricing strategies are used to extract the demand information as well as exploit the 96 surplus each period. Research could focus on the effect of competition on the traditional ”exploration vs. exploitation” trade-off and answer the following questions: Will firms be more interested in exploring the demand or exploiting the surplus under competition? How should the assortment and pricing decisions be made dynamically to serve their purpose? Finally, a very promising area is the product variety management involving multiple entities in a supply chain network. 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Sloan Management Review, 42(3):81–87, 2001. 101 Appendix Proofs Proof of Proposition 2 At stage 2, Firm 1 maximizes its profit ¼ 1 (n 1 ;n 2 ) = M 2t h t+ 1 3 ( d n 2 ¡ d n 1 ) i 2 ¡Vn 1 . Assuming the second order condition is satisfied, (true if2t>d), the optimal varietyn 1 for Firm 1 should satisfy the first order condition µ 3t d + 1 n 2 ¡ 1 n 1 ¶ 1 n 2 1 ¡ 9Vt Md 2 =0 (A.1) Similarly, the first order condition forn 2 is: µ 3t d + 1 n 1 ¡ 1 n 2 ¶ 1 n 2 2 ¡ 9Vt Md 2 =0 (A.2) Subtract Equation (A.2) from (A.1), we have 3t d ³ 1 n 2 1 ¡ 1 n 2 2 ´ + 1 n 1 n 2 ³ 1 n 1 ¡ 1 n 2 ´ ¡ ³ 1 n 3 1 ¡ 1 n 3 2 ´ =0 , ³ 1 n 1 ¡ 1 n 2 ´h 1 n 1 ³ 3t d ¡ 1 n 1 ´ + 1 n 2 ³ 3t d ¡ 1 n 2 ´i =0 Using 3t > d, we have 1 n 1 ³ 3t d ¡ 1 n 1 ´ + 1 n 2 ³ 3t d ¡ 1 n 2 ´ > 0: Therefore, we must have 1 n 1 ¡ 1 n 2 = 0, i.e. in equilibrium n ¤ 1 = n ¤ 2 . Then from Equation (A.1) we get n SS 1 =n SS 2 = q Md 3V ; and from Equation (2.3) we getp SS 1 =p SS 2 =c+t:¦ Proof of Proposition 4 Firm 1 maximizes its profit ¼ 1 (l 1 ;l 2 ) = M 2t £ t+ 1 3 (kl 2 ¡kl 1 ) ¤ 2 ¡ ¯ l 1 . Assuming the second order condition is satisfied, the optimal lead timel 1 should satisfy the first order condition: µ 3t k +l 2 ¡l 1 ¶ l 2 1 ¡ 9¯t Mk 2 =0 (A.3) 102 Similarly, we obtain the first order condition forl 2 : µ 3t k +l 1 ¡l 2 ¶ l 2 2 ¡ 9¯t Mk 2 =0 (A.4) Subtract Equation (A.4) from (A.3), we have 3t k (l 2 1 ¡l 2 2 )+l 1 l 2 (l 1 ¡l 2 )¡(l 3 1 ¡l 3 2 )=0 ,(l 1 ¡l 2 ) £ l 1 ¡ 3t k ¡l 1 ¢ +l 2 ¡ 3t k ¡l 2 ¢¤ =0 Using 3t > kl, we have l 1 ¡ 3t k ¡l 1 ¢ +l 2 ¡ 3t k ¡l 2 ¢ > 0: Therefore, we must have l 1 ¡l 2 = 0; i.e. in equilibriuml ¤ 1 = l ¤ 2 . Then from Equation (A.3) we getl CC 1 = l CC 2 = q 3¯ Mk ; and from Equation (2.9) we getp CC 1 =p CC 2 =c+t:¦ Proof of Theorem 1 For convenience, we rewriten 1 as d kl 1 wherel 1 is a variable representing the equiva- lence lead time. Then Equation (2.18) and (2.20) become µ 3t k +l 2 ¡l 1 ¶ l 2 1 ¡ 9Vdt Mk 3 =0 (A.5) and µ 3t k +l 1 ¡l 2 ¶ l 2 2 ¡ 9¯t Mk 2 =0 (A.6) respectively. Subtract (A.6) from (A.5), we get (l 1 ¡l 2 ) · l 1 µ 3t k ¡l 1 ¶ +l 2 µ 3t k ¡l 2 ¶¸ = 9t(Vd¡¯k) Mk 3 (A.7) Using3t>kl, we havel 1 ¡ 3t k ¡l 1 ¢ +l 2 ¡ 3t k ¡l 2 ¢ >0: 103 Alternatively, we can rewritel 2 as d kn 2 wheren 2 is a variable representing the equiv- alence variety. Then Equation (2.18) and (2.20) become µ 3t d + 1 n 2 ¡ 1 n 1 ¶ 1 n 2 1 ¡ 9Vt Md 2 =0 (A.8) and µ 3t d + 1 n 1 ¡ 1 n 2 ¶ 1 n 2 2 ¡ 9¯kt Md 3 =0 (A.9) respectively. Subtract (A.9) from (A.8), we get µ 1 n 1 ¡ 1 n 2 ¶· 1 n 1 µ 3t d ¡ 1 n 1 ¶ + 1 n 2 µ 3t d ¡ 1 n 2 ¶¸ = 9t(Vd¡¯k) Md 3 (A.10) Using3t>d, we have 1 n 1 ³ 3t d ¡ 1 n 1 ´ + 1 n 2 ³ 3t d ¡ 1 n 2 ´ >0: If Vd > ¯k, then (A.7) ) l 1 ¡ l 2 > 0: Rewrite Equation (A.6) as l 2 2 = 9¯t Mk 2 ( 3t k +l 1 ¡l 2 ) < 3¯ Mk using l 1 ¡ l 2 > 0; so l SC 2 < q 3¯ Mk ; i.e. l SC < l CC : Rewrite Equation (A.5) as l 2 1 = 9Vdt Mk 3 ( 3t k +l 2 ¡l 1) > 3¯ Mk using l 1 ¡ l 2 > 0; so l SC 1 > q 3¯ Mk : From Equation (A.10) andVd > ¯k, we have 1 n 1 ¡ 1 n 2 > 0:Rewrite Equation (A.8) as n 2 1 = Md 2 9Vt ³ 3t d + 1 n 2 ¡ 1 n 1 ´ < Md 2 9Vt 3t d = Md 3V ; son SC 1 < q Md 3V , i.e. n SC <n SS : Now we compare the price and market share. From Equation (2.14),p SC 1 =c+t+ 1 3 (kl SC 2 ¡ d n SC 1 )<c+t+ q ¯k 3M ¡ q Vd 3M (usingl SC 2 < q 3¯ Mk andn SC 1 < q Md 3V )<c+t (usingVd > ¯k) = p SS S _: Similarly, p SC 2 = c+t+ 1 3 ( d n SC 1 ¡kl SC 2 ) > c+t+ q Vd 3M ¡ q ¯k 3M > c+t = p CC C : Market share m SC 1 = t+ 1 3 (kl SC 2 ¡ d n SC 1 ) 2t < t+ 1 3 ( p ¯k 3M ¡ p Vd 3M ) 2t < 1 2 : m SC C =1¡m SC S > 1 2 : The results for the caseVd<¯k can be shown through a similar way. IfVd = ¯k, then (A.7))l 1 ¡l 2 = 0: Then solving the equations (A.8) and (A.9) yields l SC 1 = l SC 2 = q 3¯ Mk ; i.e. l SC = l CC : Then n SC 1 = d kl SC 1 = q Md 2 3¯k = q Md 3V usingVd=¯k: Substituten SC 1 andl SC 2 into Equation (2.14), we getp SC 1 =p SC 2 =c+t: 104 Market sharem SC 1 = t+ 1 3 (kl SC 2 ¡ d n SC 1 ) 2t = 1 2 :m SC 2 =1¡m SC S = 1 2 :Thus we obtain theorem 1.¦ Proof of Theorem 2 Case 1. Vd>¯k: From equation (2.18), ³ 3t d + kl SC d ¡ 1 n SC ´ 1 n SC 2 ¡ 9Vt Md 2 =0)t+ 1 3 (kl SC ¡ d n SC )= 3Vt Md n SC 2 ; substitute into equation (2.17), we have ¼ SC S = 9V 2 t 2Md 2 n SC 4 ¡Vn SC =Vn SC h 9Vt 2Md 2 n SC 3 ¡1 i <V q Md 3V · 9Vt 2Md 2 q Md 3V 3 ¡1 ¸ (From Theorem 1, ifVd>¯k, thenn SC < q Md 3V ) = Mt 2 ¡ q MVd 3 < Mt 2 ¡ q M¯k 3 (usingVd>¯k) =¼ CC C So¼ SC S <¼ CC C : Similarly, from equation (2.20), ¡ 3t k + d kn SC ¡l SC ¢ l SC 2 ¡ 9¯t Mk 2 =0)t+ 1 3 ( d n SC ¡ kl SC )= 3¯t Mk 1 l SC 2 ; substitute into equation (2.19), we have ¼ SC C = 9¯ 2 t 2Mk 2 1 l SC 4 ¡ ¯ l SC = ¯ l SC h 9¯t 2Mk 2 1 l SC 3 ¡1 i > ¯ p 3¯ Mk · 9¯t 2Mk 2 1 p 3¯ Mk 3 ¡1 ¸ (From Theorem 1, ifVd>¯k, thenl SC < q 3¯ Mk ) = Mt 2 ¡ q M¯k 3 > Mt 2 ¡ q MVd 3 (usingKd>¯k) =¼ SS S From¼ SC S < ¼ CC C and¼ SC C > ¼ SS S ; we can infer that whenVd > ¯k; (C;C) is the equilibrium. The case whenVd < ¯k andVd = ¯k can be shown through a similar way. Then we have Theorem 2.¦ Proof of Theorem 3 When both firms choose standard or custom products, they incur same variable costs, so the equilibrium is same as before. Only the equilibrium of (S,C) game is affected. 105 Suppose Firm 1 chooses standard products, and Firm 2 chooses custom products. Then Firm 1’s optimal price for standard products isp 1 = 1 2 ³ p 2 +c 1 +t+kl 2 ¡ d n 1 ´ ; and Firm 2’s optimal price for custom products is p 2 = 1 2 ³ p 1 +c 2 +t+ d n 1 ¡kl 2 ´ : The price equilibrium at stage 3 is p ¤ 1 = t+ 1 3 µ 2c 1 +c 2 +kl 2 ¡ d n 1 ¶ (A.11) p ¤ 2 = t+ 1 3 µ c 1 +2c 2 + d n 1 ¡kl 2 ¶ (A.12) At stage 2, Firm 1 chooses the variety to maximize its profit ¼ 1 (n 1 ;l 2 ;p ¤ 1 ;p ¤ 2 )= M 2t · t+ 1 3 (kl 2 ¡ d n 1 +c 2 ¡c 1 ) ¸ 2 ¡Vn 1 (A.13) The profit function is concave in n 1 . The optimal variety n 1 satisfies the first order condition µ 3t+c 2 ¡c 1 d + kl 2 d ¡ 1 n 1 ¶ 1 n 2 1 ¡ 9Vt Md 2 =0 (A.14) At the same time, Firm 2 chooses the lead time to maximize its profit ¼ 2 (n 1 ;l 2 ;p ¤ 1 ;p ¤ 2 )= M 2t · t+ 1 3 ( d n 1 ¡kl 2 +c 1 ¡c 2 ) ¸ 2 ¡ ¯ l 2 (A.15) which has the first order condition µ 3t+c 1 ¡c 2 k + d kn 1 ¡l 2 ¶ l 2 2 ¡ 9¯t Mk 2 =0 (A.16) If we rewrite l 2 as d kn 2 where n 2 represents the equivalence variety, then Equation (A.14) and (A.16) become µ 3t+c 2 ¡c 1 d + 1 n 2 ¡ 1 n 1 ¶ 1 n 2 1 ¡ 9Vt Md 2 =0 (A.17) 106 and µ 3t+c 1 ¡c 2 d + 1 n 1 ¡ 1 n 2 ¶ 1 n 2 2 ¡ 9¯kt Md 3 =0 (A.18) respectively. Subtract (A.18) from (A.17), we get µ 1 n 1 ¡ 1 n 2 ¶· 1 n 1 µ 3t d ¡ 1 n 1 ¶ + 1 n 2 µ 3t d ¡ 1 n 2 ¶¸ + c 2 ¡c 1 d µ 1 n 1 + 1 n 2 ¶ = 9t(Vd¡¯k) Md 3 (A.19) Using 3t > d, we have 1 n 1 ³ 3t d ¡ 1 n 1 ´ + 1 n 2 ³ 3t d ¡ 1 n 2 ´ > 0: Therefore, if Vd · ¯k and c 1 < c 2 ; then 1 n 1 ¡ 1 n 2 < 0; i.e. n 1 > n 2 : From Equation (A.17), n 2 1 = Md 2 9Vt ³ 3t+c 2 ¡c 1 d + 1 n 2 ¡ 1 n 1 ´ > Md 2 9Vt 3t d = Md 3V ; son sc 1 > q Md 3V , i.e. n sc <n ss : From equation (A.14), ¡ 3t+c c ¡c s d + kl sc d ¡ 1 n sc ¢ 1 n sc 2 ¡ 9Vt Md 2 =0)t+ 1 3 (kl sc ¡ d n sc + c c ¡c s )= 3Vt Md n sc 2 ; substitute into equation (A.13), we have ¼ sc s = 9V 2 t 2Md 2 n sc 4 ¡Vn sc =Vn sc h 9Vt 2Md 2 n sc 3 ¡1 i >V q Md 3V · 9Vt 2Md 2 q Md 3V 3 ¡1 ¸ (usingn sc > q Md 3V ) = Mt 2 ¡ q MVd 3 ¸ Mt 2 ¡ q M¯k 3 (usingVd·¯k) =¼ cc c So¼ sc s >¼ cc c ifVd·¯k andc 1 <c 2 Similarly we can show that¼ sc c <¼ ss s : From¼ sc s >¼ cc c and¼ sc c <¼ ss s ; we can infer that whenVd·¯k andc s <c c ;(S;S) is the equilibrium. Similarly we can show that whenVd¸¯k andc s >c c ;(C;C) is the equilibrium.¦ Proof of Proposition 5 Proof of (i) If both firms choose to offer standard products, then for a customer at loca- tion x, the net utility of purchasing a standard product from Firm 1 and Firm 2 is 107 (U 1 ¡ tx¡ d=n 1 ¡ p 1 ) and (U 2 ¡ t(1¡ x)¡ d=n 2 ¡ p 2 ), respectively. The indif- ferent customer is located at x SS = t+U 1 ¡U 2 + d n 2 ¡ d n 1 +p 2 ¡p 1 2t : At stage 3, Firm i’s opti- mization problem is max p i ¼ i (n 1 ;n 2 ;p 1 ;p 2 ) = D i (p i ¡ c)¡ Vn i for i = 1;2;where D 1 = Mx SS ; D 2 = M(1¡ x SS ):The optimal pricing strategy for Firm 1 is p 1 = 1 2 ³ p 2 +c+t+U 1 ¡U 2 + d n 2 ¡ d n 1 ´ while the optimal pricing decision of Firm 2 is p 2 = 1 2 ³ p 1 +c+t+U 2 ¡U 1 + d n 1 ¡ d n 2 ´ : We then obtain the price equilibrium at stage 3: p ¤ 1 = c+t+ 1 3 (U 1 ¡U 2 + d n 2 ¡ d n 1 ) (A.20) p ¤ 2 = c+t+ 1 3 (U 2 ¡U 1 + d n 1 ¡ d n 2 ) At stage 2, Firm 1 chooses the varietyn 1 that maximizes its profit ¼ 1 (n 1 ;n 2 ;p ¤ 1 ;p ¤ 2 )= M 2t · t+ 1 3 (U 1 ¡U 2 + d n 2 ¡ d n 1 ) ¸ 2 ¡Vn 1 (A.21) while Firm 2 chooses the varietyn 2 so as to maximize its profit ¼ 2 (n 1 ;n 2 ;p ¤ 1 ;p ¤ 2 )= M 2t · t+ 1 3 (U 2 ¡U 1 + d n 1 ¡ d n 2 ) ¸ 2 ¡Vn 2 (A.22) The first order conditions yield µ 3t+U 1 ¡U 2 d + 1 n 2 ¡ 1 n 1 ¶ 1 n 2 1 ¡ 9Vt Md 2 =0 (A.23) µ 3t+U 2 ¡U 1 d + 1 n 1 ¡ 1 n 2 ¶ 1 n 2 2 ¡ 9Vt Md 2 =0 (A.24) 108 Subtract (A.24) from (A.23), we get µ 1 n 1 ¡ 1 n 2 ¶· 1 n 1 µ 3t d ¡ 1 n 1 ¶ + 1 n 2 µ 3t d ¡ 1 n 2 ¶¸ + (U 1 ¡U 2 ) d µ 1 n 1 + 1 n 2 ¶ =0 (A.25) Using 3t > d, we have 1 n 1 ³ 3t d ¡ 1 n 1 ´ + 1 n 2 ³ 3t d ¡ 1 n 2 ´ > 0: Suppose U 1 > U 2 ; then 1 n 1 ¡ 1 n 2 < 0; i.e. n ¤ 1 > n ¤ 2 : Then from Equation (A.20), we know that p ¤ 1 > p ¤ 2 : The market sharem ¤ 1 = t+U 1 ¡U 2 + d n 2 ¡ d n 1 +p ¤ 2 ¡p ¤ 1 2t = t+ 1 3 (U 1 ¡U 2 + d n 2 ¡ d n 1 ) 2t = p ¤ 1 ¡c 2t , andm ¤ 2 = p ¤ 2 ¡c 2t ; therefore we havem ¤ 1 > m ¤ 2 . From Equation (A.21) and (A.23), profit¼ ¤ 1 = 9V 2 t 2Md 2 n ¤ 4 1 ¡ Vn ¤ 1 = Vn ¤ 1 h 9Vt 2Md 2 n ¤ 3 1 ¡1 i ; similar for ¼ ¤ 2 ; so we have ¼ ¤ 1 > ¼ ¤ 2 using n ¤ 1 > n ¤ 2 : The result whenU 1 <U 2 is the opposite. Proof of (ii) is similar to the proof of (i). Then we obtain the proposition.¦ Proof of Lemma 1 First, we look at how the demand for each variant changes. Letp j andp 0 j denote the probabilities of a customer choosing variantj offered by firm 1 before and after it adds variant k, respectively. (We drop the superscript 1 for convenience. Firm 2’s optimal assortment strategy can be obtained similarly.) Case 1. Ifk2S 2 nS 1 , then for anyj2S 1 [fkg; p 0 j =m j x j + 1 n 1 +1 2 4 X l2S 2 nS 1 nfkg m l t¡ d n 1 +1 2t + X l2S 1 \S 2 m l t+ d n 2 ¡ d n 1 +1 2t 3 5 : (A.26) The demand change can be obtained by subtracting Equation (3.2) from (A.26): 109 Forj2S 1 ; p 0 j ¡p j (A.27) = X l2S 2 nS 1 nfkg m l · 1 2 µ 1 n 1 +1 ¡ 1 n 1 ¶ ¡ d 2t µ 1 (n 1 +1) 2 ¡ 1 n 2 1 ¶¸ + X l2S 1 \S 2 m l " t+ d n 2 2t µ 1 n 1 +1 ¡ 1 n 1 ¶ ¡ d 2t µ 1 (n 1 +1) 2 ¡ 1 n 2 1 ¶ # ¡m k 1 n 1 t¡ d n 1 2t = X l2S 2 nS 1 m l · 1 2 µ 1 n 1 +1 ¡ 1 n 1 ¶ ¡ d 2t µ 1 (n 1 +1) 2 ¡ 1 n 2 1 ¶¸ + X l2S 1 \S 2 m l " t+ d n 2 2t µ 1 n 1 +1 ¡ 1 n 1 ¶ ¡ d 2t µ 1 (n 1 +1) 2 ¡ 1 n 2 1 ¶ # ¡m k · 1 2 1 n 1 +1 ¡ d 2t 1 (n 1 +1) 2 ¸ , a 1 ¡b 1 m k p 0 k ¡p k = p 0 k (A.28) = m k 1 2 + 1 n 1 +1 2 4 X l2S 2 nS 1 nfkg m l t¡ d n 1 +1 2t + X l2S 1 \S 2 m l t+ d n 2 ¡ d n 1 +1 2t 3 5 = 1 n 1 +1 2 4 X l2S 2 nS 1 m l t¡ d n 1 +1 2t + X l2S 1 \S 2 m l t+ d n 2 ¡ d n 1 +1 2t 3 5 +m k · 1 2 n 1 n 1 +1 + d 2t 1 (n 1 +1) 2 ¸ , a 2 +b 2 m k whereb 1 ;b 2 >0 using assumptiont>d, anda 1 ;a 2 are constant givenS 1 ;S 2 : Then function f(±), the new profit after adding to S 1 a variant k 2 S 2 nS 1 with m k =±; is 110 f(±)= P j2S i £ (r¡c)¸p 0 j ¡rÁ(z) p ¸p 0 j ¡K ¤ +(r¡c)¸p 0 k ¡rÁ(z) p ¸p 0 k ¡KI(±) = P j2S i h (r¡c)¸(p j +a 1 ¡b 1 ±)¡rÁ(z) p ¸(p j +a 1 ¡b 1 ±)¡K i +(r¡c)¸(a 2 + b 2 ±) ¡rÁ(z) p ¸(a 2 +b 2 ±)¡KI(±); whereI(¢) is the indicator function–i.e.,I(±) = 1 if ± > 0 andI(±) = 0 if ± > 0: The part excluding¡KI(±) is strictly convex in±, while¡KI(±) is convex in±: Therefore, f(±) is strictly convex in±. Case 2. Ifk2S 1 \S 2 , then for anyj2S 1 [fkg; p 0 j =m j x j + 1 n 1 +1 2 4 X l2S 2 nS 1 m l t¡ d n 1 +1 2t + X l2S 1 \S 2 nfkg m l t+ d n 2 ¡ d n 1 +1 2t 3 5 : (A.29) Similarly to case 1, the demand change can be obtained by subtracting Equation (3.2) from (A.29): Forj2S 1 ; p 0 j ¡p j =a 1 ¡b 3 m k ; (A.30) whereb 3 = t+ d n 2 2t 1 n 1 +1 ¡ d 2t 1 (n 1 +1) 2 =b 1 + d 2t 1 n 2 (n 1 +1) >0; p 0 k ¡p k =p 0 k =a 2 +b 4 m k ; (A.31) whereb 4 = t+ d n 2 2t n 1 n 1 +1 + d 2t 1 (n 1 +1) 2 =b 2 + d 2t n 1 n 2 (n 1 +1) >0: The new profit of Firm 1 after adding a variantk2S 1 \S 2 withm k =± is g(±)= P j2S i £ (r¡c)¸p 0 j ¡rÁ(z) p ¸p 0 j ¡K ¤ +(r¡c)¸p 0 k ¡rÁ(z) p ¸p 0 k ¡KI(±) = P j2S i h (r¡c)¸(p j +a 1 ¡b 3 ±)¡rÁ(z) p ¸(p j +a 1 ¡b 3 ±)¡K i +(r¡c)¸(a 2 + b 4 ±) ¡rÁ(z) p ¸(a 2 +b 4 ±)¡KI(±): 111 Much as in Case 1, we can show thatg(±) is convex in±. So we obtain lemma 1.¦ Proof of Theorem 6: a)¼(A i+1 [B l )>¼(A i [B l+1 ) if and only if X j2S 1 [fi+1g h (r¡c)¸p 0 j ¡rÁ(z) q ¸p 0 j ¡K i > X j2S 1 [fl+1g h (r¡c)¸p 00 j ¡rÁ(z) q ¸p 00 j ¡K i (A.32) ,(r¡c)¸ 0 @ X j2S 1 [fi+1g p 0 j ¡ X j2S 1 [fl+1g p 00 j 1 A >rÁ(z) 0 @ X j2S 1 [fi+1g q ¸p 0 j ¡ X j2S 1 [fl+1g q ¸p 00 j 1 A ; wherep 0 j =p j +a 1 ¡b 1 m i+1 forj6=i+1;p 0 i+1 =a 2 +b 2 m i+1 ;p i+1 =0; from Equations (A.27) and (A.28), andp 00 j =p j +a 1 ¡b 3 m l+1 forj6=l+1;p 00 l+1 =a 2 +b 4 m l+1 ;p l+1 =0; from Equations (A.30) and (A.31). Since P j2S 1 [fi+1g p 0 j ¡ P j2S 1 [fl+1g p 00 j = P j2S 1 (p 0 j ¡ p 00 j ) + p 0 i+1 ¡ p 00 l+1 = n 1 (b 3 m l+1 ¡ b 1 m i+1 )+b 2 m i+1 ¡b 4 m l+1 =m i+1 (b 2 ¡n 1 b 1 )¡m l+1 (b 4 ¡n 1 b 3 )=m i+1 d 2t n 1 ¡1 (n 1 +1) 2 ¡ m l+1 d 2t n 1 ¡1 (n 1 +1) 2 = d 2t n 1 ¡1 (n 1 +1) 2 (m i+1 ¡m l+1 )>0 iffm i+1 >m l+1 : Therefore, ifm i+1 >m l+1 ; then ¼(A i+1 [B l )>¼(A i [B l+1 ), ³ 1¡ c r ´ p ¸>Á(z) P j2S 1 [fi+1g p p 0 j ¡ P j2S 1 [fl+1g p p j P j2S 1 [fi+1g p 0 j ¡ P j2S 1 [fl+1g p j : LHS is increasing inr and¸, while RHS is independent ofr and¸: Thus, for sufficiently highr or¸, choosing the setA i+1 [B l is more profitable thanA i [B l+1 : For b), similarly, ifm l+1 >m i+1 ; then ¼(A i [B l+1 )>¼(A i+1 [B l ), ³ 1¡ c r ´ p ¸>Á(z) P j2S 1 [fl+1g p p 0 j ¡ P j2S 1 [fi+1g p p j P j2S 1 [fl+1g p 0 j ¡ P j2S 1 [fi+1g p j ; 112 which holds ifr or¸ is sufficiently high.¦ Proof of Lemma 2 Case 1. Ifk2S 2 nS 1 , then for anyj2S 1 [fkg; p 0 j = x j n + 1 n 1 +1 " 1 n jS 1 nS 2 nfkgj t¡ d n 1 +1 2t + 1 n ¯ ¯ S 1 \S 2 ¯ ¯ t+ d n 2 ¡ d n 1 +1 2t # : (A.33) Then the new profit after adding variantk withc k =¾ is v(¾)= X j2S i h (r¡c j )¸p 0 j ¡rÁ(z) q ¸p 0 j ¡K i +I(¾) h (r¡¾)¸p 0 k ¡rÁ(z) p ¸p 0 k ¡K i : The optimum occurs at¾ = 0, not adding any variant, orminfc l : l2 S 2 nS 1 g; adding the variant with minimum variable cost. Case 2. Ifk2S 1 \S 2 , then for anyj2S 1 [fkg; p 0 j = x j n + 1 n 1 +1 " 1 n jS 1 nS 2 j t¡ d n 1 +1 2t + 1 n ¯ ¯ S 1 \S 2 nfkg ¯ ¯ t+ d n 2 ¡ d n 1 +1 2t # : (A.34) The new profit after adding variantk withc k =¾ is w(¾)= X j2S i h (r¡c j )¸p 0 j ¡rÁ(z) q ¸p 0 j ¡K i +I(¾) h (r¡¾)¸p 0 k ¡rÁ(z) p ¸p 0 k ¡K i : The optimum occurs at¾ =0 orminfc l :l2S 1 \S 2 g.¦ Proof of Theorem 8 Prove by construction. LetS ¤ 1 be an optimal assortment set, andx=jS ¤ 1 \S 2 j;y = ¯ ¯ S ¤ 1 \S 2 ¯ ¯ : Let the variants inS ¤ 1 \S 2 be denoted ask ¤ 1 ;:::;k ¤ x ; such thatc k ¤ 1 ·:::·c k ¤ x ; and let the variants inS ¤ 1 \S 2 be denoted ask ¤ n 2 +1 ;:::;k ¤ y ; so thatc k ¤ n 2 +1 · :::· c k ¤ y : If S ¤ 1 =A x [B y ; the theorem holds. IfS ¤ 1 6=A x [B y ; then there are two cases: 113 Case 1: There exists ak j 2S 2 ; andk j = 2S 1 ; such thatc k j <c k ¤ x : Due to the property ofv(¾) described in Lemma 2, we can either removek ¤ x or exchange it withk j without decreasing profits. RedefineS ¤ 1 to be this new set. Case 2: There exists ak j = 2S 2 ; andk j = 2S 1 ; such thatc k j <c k ¤ y : Due to the property ofw(¾) described in Lemma 2, we can either removek ¤ y or exchange it withk j without decreasing profits. RedefineS ¤ 1 to be this new set. Repeat the procedure. Eventually we will arrive at an optimal setS ¤ 1 =A x [B y for somex;y:¦ Proof of Theorem 9 We use the first order conditions to derive the equilibrium varieties. Subtract Equa- tion (4.3) from (4.2), we have 3t d ³ 1 n 2 1 ¡ 1 n 2 2 ´ + 1 n 1 n 2 ³ 1 n 1 ¡ 1 n 2 ´ ¡ ³ 1 n 3 1 ¡ 1 n 3 2 ´ =0 , ³ 1 n 1 ¡ 1 n 2 ´h 3t d ³ 1 n 1 + 1 n 2 ´ + 1 n 1 n 2 ¡ ³ 1 n 2 1 + 1 n 1 n 2 + 1 n 2 2 ´i =0 , ³ 1 n 1 ¡ 1 n 2 ´h 1 n 1 ³ 3t d ¡ 1 n 1 ´ + 1 n 2 ³ 3t d ¡ 1 n 2 ´i =0: Usingt > d, we have 1 n 1 ³ 3t d ¡ 1 n 1 ´ + 1 n 2 ³ 3t d ¡ 1 n 2 ´ > 0: Therefore, we must have 1 n 1 ¡ 1 n 2 = 0, i.e. in equilibriumn ¤ 1 = n ¤ 2 . Then from Equation (4.3) we getn ¤ 1 = n ¤ 2 = q nMd 3Kt ; and from Proposition 6 we getp ¤ 1 =p ¤ 2 =c+t:¦ Proof of Theorem 10: From Proposition 8, the two firms will both choose the largest n i segments, so the equilibrium assortments depend on the equilibrium variety. If we restrict to symmetric equilibrium varietyn 1 =n 2 =n ¤ , then we just need to findn ¤ : Given that Firm 2 chooses the bestn 2 products, Firm 1 needs to choosen 1 to maxi- mize ¼ 1 (n 1 ;n 2 )= 2¸t 9 ( 3 2 + d 2tn 2 n P j=n 2 +1 m j ¡ d 2tn 1 n P j=n 1 +1 m j ) 2 ¡n 1 K = ¸d 2 18t ( 3t d + 1 n 2 n P j=n 2 +1 m j ¡ 1 n 1 n P j=n 1 +1 m j ) 2 ¡n 1 K: 114 For ease of exposition, we define f(j) = m j ; F(j) = j P k=1 m k ; F(j) = n P k=j+1 m k = 1¡F(j): Then¼ 1 (n 1 ;n 2 )= ¸d 2 18t ( 3t d + F(n 2 ) n 2 ¡ F(n 1 ) n 1 ) 2 ¡n 1 K: We treat variety n 1 as a continuous variable so that we can obtain the first order condition which helps us characterize the equilibrium variety. Take the derivative of ¼ 1 (n 1 ;n 2 ) overn 1 ; and use the fact that dF(n 1 ) dn 1 =f(n 1 ); we obtain the first order condi- tion: µ 3t d + F(n 2 ) n 2 ¡ F(n 1 ) n 1 ¶ n 1 f(n 1 )+F(n 1 ) n 2 1 = 9Kt ¸d 2 : (A.35) Assume the second order condition is satisfied, then ifn ¤ 1 =n ¤ 2 =n ¤ in equilibrium, substitute into Equation (A.35), we obtain n ¤ f(n ¤ )+F(n ¤ ) n ¤2 = 3K ¸d ; i.e. n ¤ m n ¤ + n P j=n ¤ +1 m j n ¤2 = 3K ¸d : Since LHS is decreasing inn ¤ , the solution ofn ¤ is uniquely defined by above equation. SinceS ¤ 1 =S ¤ 2 ; we havef 1 (S ¤ 1 ;S ¤ 2 )=f 2 (S ¤ 1 ;S ¤ 2 )= 1 2 : Then from Proposition 7, we getp ¤ 1 =p ¤ 2 =c+t:¦ Proof of Theorem 11 We use the first order conditions to derive the equilibrium varieties. Subtract Equa- tion (4.11) from (4.10), we have 3t d ³ 1 n 2 1 ¡ 1 n 2 2 ´ + 1 n 1 n 2 ³ 1 n 1 ¡ 1 n 2 ´ ¡ ³ 1 n 3 1 ¡ 1 n 3 2 ´ + 3(u 1 ¡u 2 ) d ³ 1 n 2 1 + 1 n 2 2 ´ =0 , ³ 1 n 1 ¡ 1 n 2 ´h 3t d ³ 1 n 1 + 1 n 2 ´ + 1 n 1 n 2 ¡ ³ 1 n 2 1 + 1 n 1 n 2 + 1 n 2 2 ´i + 3(u 1 ¡u 2 ) d ³ 1 n 2 1 + 1 n 2 2 ´ = 0 , ³ 1 n 1 ¡ 1 n 2 ´h 1 n 1 ³ 3t d ¡ 1 n 1 ´ + 1 n 2 ³ 3t d ¡ 1 n 2 ´i + 3(u 1 ¡u 2 ) d ³ 1 n 2 1 + 1 n 2 2 ´ =0: Using t > d, we have 1 n 1 ³ 3t d ¡ 1 n 1 ´ + 1 n 2 ³ 3t d ¡ 1 n 2 ´ > 0: Therefore, if u 1 > u 2 , 3(u 1 ¡u 2 ) d ³ 1 n 2 1 + 1 n 2 2 ´ > 0; then we must have 1 n 1 ¡ 1 n 2 < 0, i.e. in equilibriumn ¤ 1 > n ¤ 2 . Then from Proposition 10 we getp ¤ 1 >p ¤ 2 : 115 Similarly ifu 1 n ¤ 2 andp ¤ 1 < p ¤ 2 : Thus we obtain the theorem.¦ 116

Abstract (if available)

Abstract Product proliferation has been observed in many product categories over the years. It is commonly believed that increasing variety would potentially attract more customers and hence yield more profit. However, there are variety-associated costs as well, and whether firms in a competitive environment would benefit from increasing product variety is unclear. Therefore, this dissertation studies how to balance the costs and benefits of product variety in presence of competition. More specifically, the following research questions are explored: In presence of competition, how should a firm choose the product strategies that involve production technology (mass production or mass customization), product selection or assortment planning, pricing, and inventory decisions in order to maximize its overall profits? How do these decisions interact with each other, and what is the effect of market characteristics and cost factors on these variety-related decisions?

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Asset Metadata

Creator Xia, Nan (author)

Core Title Managing product variety in a competitive environment

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Publication Date 06/26/2010

Defense Date 05/01/2008

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

Tag assortment planning,competition,customization,game theory model,OAI-PMH Harvest,Pricing,product variety

Language English

Advisor Bassok, Yehuda (committee chair), Rajagopalan, Sampath (committee chair), Dasu, Sriram (committee member), Sosic, Greys (committee member), Tan, Guofu (committee member)

Creator Email nan.xia.2008@marshall.usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1296

Unique identifier UC1104190

Identifier etd-Xia-20080626 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-80655 (legacy record id),usctheses-m1296 (legacy record id)

Legacy Identifier etd-Xia-20080626.pdf

Dmrecord 80655

Document Type Dissertation

Rights Xia, Nan

Type texts

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

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu