Product Variety and Capacity Investments in Congested Production

Submitted to Operations Research manuscript

Product Variety and Capacity Investments in Congested Production Systems Sergio Chayet, Panos Kouvelis Olin Business School, Washington University in St. Louis, Campus Box 1133, St. Louis, Missouri 63130-4899, [email protected], [email protected]

Dennis Z. Yu School of Business, Clarkson University, P.O. Box 5790, Potsdam, New York 13699-5790, [email protected]

We investigate a firm’s product line design, and capacity investment problem for vertically differentiated products along design quality levels. Customers arrive according to a Poisson process and are heterogeneous in their marginal valuation of the quality level. Customers make product choices to maximize a linear utility function of price, quality level, and waiting cost. Resulting product demands are met through capacity investments in production processes, which are modeled as queuing systems. We consider two different types of production processes: product-focused, dedicated to the production of a single product variant; or productflexible, processing all product variants in the product line. Capacity investment and variable production costs are functions of the processed product’s quality. We develop an integrated marketing-operations model that provides insights on the factors determining the right level of product variety to offer, the relative quality positioning of the products in the line, the resulting market coverage and segmentation, and the effects on production costs and congestion levels of the processes. We show that the statistical economies of scale resulting from the congestion phenomena in the production system impose limits on the optimal product variety. For product-focused processes the optimal product variety is captured by an aggregate parameter M , which emphasizes the market size as a facilitator and the per unit capacity investment and customer waiting costs as deterrents for higher product variety. For product-flexible processes optimal product variety depends not only on the above parameter M , but also on the specific type of flexibility and the ratio of capacity investment to variable production costs. Finally, access to full information about customers’ preferences on quality increases the firm’s profitability via higher quality level(s) and increased capacity investments. Subject classifications : Inventory/production: Capacity investments, flexible or dedicated facilities, statistical scale-economies; Marketing: Product variety, product line design, vertical differentiation, segmentation. Area of review : Manufacturing, Service and Supply Chain Operations

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Chayet, Kouvelis, and Yu: Product Variety and Capacity Investments in Congested Production Systems Article submitted to Operations Research; manuscript no.

1. Introduction The clearly cross-functional nature of product line decisions amplifies their difficulty, as different functional areas bring to the table different interests and managerial measures of success. From a marketing perspective, variety in product offerings is the key to increased market share and revenue growth. From an operations and logistics perspective, variety adds complexity, reduces economies of scale, defocuses efficient lean processes, and results in longer lead-times and increased costs for matching supply with demand. While both functions have sound arguments to make for their views, their answers, in terms of an optimal product line and variety to offer, are at diametric opposite extremes and are not helping the overall business decision. The lack of integrated decision frameworks to capture and quantify all relevant tradeoffs on determining optimal product variety, and the proper differential positioning of products within the resulting product line is further contributing to the confusion on the issue. Our research attempts to offer such an integrated decision framework, and within a stylized model, outlines the factors that affect product variety for a given market and production environment. It also suggests how to position the products to achieve the necessary quality differentiation and to price the product offerings for profit maximization. Furthermore, it explicitly captures implications for capacity investments, the flexibility of the production system, and the right utilization of the production resources to match supply and demand effectively. The study of product line design and positioning has been a major research area in industrial organization and marketing science for decades. The main focus of this research stream has been the revenue optimization through careful pricing, and the effective matching of customer preferences and product design attributes of a diverse product line. However, such research models overemphasize the accuracy of depiction of customer choice in the presence of a vertically differentiated product line, mostly along dimensions of price and product quality, at the expense of an oversimplified representation of the capacity investment, variable production, and congestion-related costs of the production and supply system. On the other hand, a rich operations management literature has been building detailed cost and congestion-level models of production systems responding to uncertain and variable product mixes, but under the assumption of given prices and product lines. Recent research efforts have come closer to closing the gap between the front-end customer choice representation and the back-end operational implications of product offerings, but as we will argue, have so far remained short in effectively capturing capacity investment and congestion implications of product variety for the production system. In this paper, we investigate a firm’s product line design and capacity investment problem for vertically differentiated products along design quality levels. The customer population is heterogeneous in the marginal valuation of the quality level. Utility-maximizing customers make choices using a linear utility function of both price, quality level, and waiting cost. The customer heterogeneity of quality preferences is captured via a parameter assumed to have a uniform distribution over the range [0, 1]. Product demand has to be met through investments in production resources. We use simple queuing systems to capture the relevant congestion effects in the production process, and consider two different types of production technologies available to our firm. The productfocused (dedicated) technology, which sets up separate production facilities (or manufacturing lines within the same facility) for each product of a predetermined quality level. The flexible production technology can produce a wide range of product quality levels within the same production facility (or manufacturing line). Capacity investment costs of a production technology are functions of the product quality levels that the facility can produce. Our firm is a make-to-order producer. The variable production cost of each product is a function of its quality level. A heavily utilized production system suffers the consequences of its congestion via elongated lead-times and increased customer waiting, which erodes the firm’s revenue. Suitable congestion penalties account for such costs via the customers’ assessed waiting costs as a part of a


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full price (nominal price plus waiting costs) attributed to all waiting customers in the system. The offered product variety and pricing influence the customer arrival rate to the production facility, and proper capacity investment decisions have to be made to fully reflect tradeoffs in capacity investment costs and congestion penalties. Thus, a profit maximizing firm has to simultaneously optimize its product line design (product offerings along a quality dimension) and pricing; which are essentially demand arrival rate controls, in addition to capacity investments; which indirectly affect congestion levels, and with the variable production costs, determine its profit margins. While the existing economics literature argues that limits on product variety exist because of fixed costs in the production environment, in other words, as a result of production economies of scale, we argue that even without imposing any fixed costs, such limits still exist because of production congestion effects and the resulting queuing-type statistical economies of scale (risk pooling effects as often referred to in the operations literature). We will use product line decisions at the Hospital Equipment Corporation (HEC) as a motivating example. HEC is a leading manufacturer of hospital beds for acute care hospital and nursing homes. HEC is a monopolist in the acute care hospital market, with an over 90% market share and revenues in the $300M range. To ensure its continuing profitable growth in this market, HEC carefully plans its product line offerings in hospital beds. The company had to make relevant product line decisions in the early 1990s requiring a product line expansion that addressed the needs of specialty segments with high margins. For example, the firm studied expanding its primary electric bed line by adding specialty beds such as a birthing bed and a critical care bed with all needed furnishings (such as over-bed tables, lighting, control panels, and power columns). While these specialty beds were not as high in volume as the primary electric beds, they did add incremental sales at a high value per unit (the low end of the electric bed market sells at $1,000–$1,500 per unit, while the high end is in the $3,500–$4,500 range). The new beds had more sophisticated, value-added features, which required investments in expensive production equipment and tooling, and resulted in higher variable production—both higher material and skilled labor—costs. All beds sold to hospitals were madeto-order in response to specific hospital orders, which often required some minor customization of existing products to the specifics of the particular hospital. Hospitals chose their products based on price and quality tradeoffs, and delivery lead-times were within standard windows for that industry. Failure to meet delivery promises could lead to an order cancellation, but delivery was an order qualifier and not an order winner for this business. HEC’s production process was a low-volume batch or job shop process, with some of the higher volume primary beds handled via higher volume processes having some resource commitment and routing inflexibility resembling a production line. As the company was planning for added product variety, it had to consider the implications for both design and manufacturing costs. The relevant question from the manufacturing side was the degree and type of flexibility its production process needed to have, to more effectively handle the expanded product line while offering proper levels of quality and delivery window commitments to its customers (for additional details on the product line and operational specifics of HEC, we refer the interested reader to the Christensen (1997) HBS case). HEC’s product line decisions require a careful understanding of the revenue growth benefits of additional market segmentation of electric beds differentiated along “quality” (mostly design quality dimension reflecting added features and higher quality materials) and prices, while simultaneously handling the increased capacity investment and variable productions costs of such decisions. Furthermore, the nature of the production facilities to be used to handle it (dedicated production lines for different products versus flexible batch processes with some degree of postponed assembly) became an essential decision issue as well. Our integrated marketing-operations model introduced in Section 3 effectively fits the above described decision-making setting, and offers useful insights for how such decisions can be framed and analyzed.

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2. Literature Review Product line design and product variety issues have been extensively studied in economics/industrial organization, marketing and operations management literatures. Karmarkar (1996) pointed out the study of such decisions is an important issue for multiproduct firms that “require a simultaneous understanding of the market benefits of and manufacturing costs imposed,” and Alptekinoglu and Corbett (2008) highlight the emergence of mass customization strategies as accentuating the need for a better understanding of the implications of product variety. The implications of broader product lines on costs and firm performance were described and empirically verified in Kekre and Srinivasan (1990). Marketing and industrial organization research emphasizes product design and associated pricing decisions to extract value from a heterogeneous population of customers via effective price discrimination. In their seminal work on vertical differentiation via product quality, Mussa and Rosen (1978) derive the monopolist’s optimal price-quality schedule offered to a heterogeneous customer population with a continuous preference parameter along the quality dimension over a bounded range. They assume that the variable cost of quality is a convex increasing function of the quality level. Moorthy (1984) substantiates the benefits of market segmentation through product line design when customers have discrete types. In this work, the monopolist offers a menu of products with higher quality products priced higher. By assuming the unit variable cost is a quadratic function of the product attributes, Kim and Chajed (2002) study a monopolist’s product design problem with multiple quality-type attributes and with the market partitioned into two distinctive segments. They show that a single product offering is never optimal for their two-segment case. Choudhary et al. (2005) utilize vertical differentiation models to study the effect of personalized pricing on the firm’s choices over quality. By using personalized pricing, the firm can charge different prices to different consumers based on their willingness to pay, assuming the firm can implement a pricing policy based on complete knowledge of the customer’s willingness to pay (first-degree price discrimination). In all the above literature, there is usually no explicit consideration of relevant operational costs beyond variable production costs (even those are typically simplified to a constant or a quadratic function of quality). Such relevant operational costs, especially with respect to the study of product variety issues, will capture inherent economies of scale in production systems through fixed costs of production, capacity related costs in installing the needed capacity for an effective service level, and important non-linear congestion phenomena and associated costs (e.g., customer waiting costs due to long lead-times) because of the uncertain nature of order arrivals and processing times of the production system. In the operations management literature on product line design, the design of product attributes is commonly predetermined and customer choice and associated pricing decisions are usually not considered in its models. Emphasis is typically placed on the challenges that an extended product line with multiple stock-keeping units places on scheduling and inventory policies. The book edited by Ho and Tang (1998) is an effective compilation of papers in this area by several authors, and a key reference on product variety research until the late 1990s. An early prominent work on the operational implications of product variety is Karmarkar and Kekre(1987). They study the impact of the nature of the production system (dedicated versus flexible multi-product facilities) and its capacity on the operational costs and experienced lead-times of processing a given product line. With similar assumption of given number of products and customer demands, Benjaafar and Gupta (1998) study the effect of scheduling and batch sizing policies on the choices of product mix and capacity of flexible and dedicated production facilities. By considering an un-capacitated multi-product lot-sizing problem, de-Groote (1994) analyzes the monopolist’s problem of selecting product line breadth and production flexibility. The author shows that an increase in the product variety makes it optimal to select a more flexible technology. Furthermore, the firm chooses an optimal number of products so that the market is covered, product locations are equally spaced, and optimal variety increases with demand and decreases with changeover and holding costs. Jiang


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et al. (2006) look at horizontally differentiated product offerings in deterministic and economies of scale settings, and study the number of product variants to offer and how it interacts with the type of production system used (mass customization vs. mass production). Our paper deals explicitly with vertically differentiated product lines in a stochastic and statistical economies of scale setting, and thus has to deal with issues of design quality specification and resulting market segmentation with the use of both price and quality levels. By using a model with two distinctive customer segments, Desai et al. (2001) investigate the marketing-manufacturing tradeoff of product differentiation decisions in the presence of component commonality. With a base model of a similar assumption of two distinct consumer segments (multiple segments studied as an extension of the base model), Netessine and Taylor (2007) study the effect of production technology on the firm’s product line design strategies with an Economic Order Quantity (EOQ) type economies of scale (fixed cost vs. linear variable and holding costs) production model. Tang and Yin (2007) also analyze a two segment model with exogenous quality levels and investigate the effects of fixed costs and capacity constraints on the product selection decision. Netessine and Taylor (2007) and Tang and Yin (2007) substantially improve previous modeling attempts by using a more balanced approach in representing customer preferences (“front-end”) and operational (“back-end”) details. Our model differs from these two papers in that it captures operational details via a queuing model of the production system, thus emphasizing non-linear congestion effects and statistical economies of scale (risk pooling) considerations in capacity investments, instead of economies of scale driven by fixed costs. Furthermore, our work explicitly models capacity costs that are functions of design quality. In further contrasting our work to the above models, we note that the results of the existing models (with the exception of the Netessine and Taylor (2007) work) are contingent on the assumption of given customer segments, and therefore, the market coverage of a specific product and/or product line is known a-priori. In other words, assuming predetermined market segmentation prevents capturing the interrelatedness of the product positioning, resulting market segmentation, and generated product demand for each offering in the line. Our work allows product positioning along a continuous line, following the rich tradition of such “front-end” decisions and utility maximizing consumer choices within a capacitated production system context. The main focus of our study is to understand the factors that affect the monopolist’s choice of optimal product variety in the presence of a heterogeneous customer population of utilitymaximizers with utility function linear in price and product quality, uncertain demand, and a production system (either dedicated facilities for each product or a flexible facility for all products) subject to congestion effects as in usual queuing models. Furthermore, we are explicitly asking the question of how the offered product line partitions the market via its quality-price choices, and what is the resulting degree of vertical differentiation among products in a diverse product line and what factors affect it. Our model includes reasonable detail in its representation of both market segmentation and operational decisions. Following the rich tradition of marketing science models, we carefully capture the heterogeneous customer preferences for product quality and the way utility-maximizing customers are affected by expanded product offerings via added product variants in the product line. In addition to that, we carefully capture the implications of the expanded product variety and the resulting market segmentation and demand for the various products on the operational system that produces them. Our model of the operational structure is a queuing system, with its capacity a decision variable, which captures congestion effects of the product line decisions (capacity costs are functions of the design quality and the offered product line). Our results contribute to the product variety literature through analytical clarity on factors affecting product line variety for dedicated and flexible production systems, and insightful answers on the profit maximizing market coverage, market segmentation, and product positioning of a monopolist’s product line for uncertain and heterogeneous consumer markets. Furthermore, we

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offer insights on the needed capacity investments to service the product variety, and how these interact with the nature of the production technology used. As a quick preview of some of our results, we list a few of the important insights from our work: • Limited optimal product variety might not be only due to fixed costs associated with new product introductions (as is often emphasized in the previous literature), but also as a result of a congested production system with capacity costs that are design quality dependent. We provide clear predictions on optimal variety (closed formulas for various cases) and the factors that affect it. • The offered optimal product variety is highly dependent not only on variable production costs used (mostly affected by production technology), but also on the nature of capacitated production facilities processing the variety (dedicated versus flexible facilities). We clearly describe the differences in terms of optimal product variety and relative positioning of product variants, when the nature of the facilities that process them is different. For example, when we compare the sets of two product positioning decisions of dedicated facilities with that of tailored customization and customized standardization flexible facilities, we observe that: The market coverage of the high quality product remains the same across the three cases. The total market coverage for dedicated facilities is larger than that of tailored customization and smaller than that of customized standardization. And thus, the market coverage of the low quality product for dedicated facilities is larger than that of tailored customization and smaller than that of customized standardization. • The product positioning of the product variants, and therefore the resulting market segmentation, is affected by the presence of a congested production system, and in that aspect our results differ from the offered predictions in the marketing literature. For example, for the case of two differentiated products processed by dedicated facilities, our work predicts that a congested production system will expand the market coverage and associated profits of the high-quality product while contracting these variables for the low quality product as compared to an un-capacitated system with quality dependent variable costs (e.g., the predictions in the classic work of Moorthy (1984)). • For the same set of exogenous parameters, the profitability of a dedicated facilities system is lower than that of a “customized standardization” flexible facility, but higher than that of a “tailored customization” flexible facility. Our results highlight the two main forces at play: Statistical scale-economies (favoring the single queue in the flexible systems) vs. optimally making market coverage and positioning decisions by leveraging the convexity in the quality level of costs and coupling with customer preferences. In the case of tailored customization, the second force dominates, because it allows the dedicated facilities system to overcome the statistical scale-economies advantage of the flexible facility by offering a larger market coverage overall, through the expanded coverage of the low quality product via less expensive production capacity. Customized standardization enjoys advantages in both forces relative to dedicated facilities. A single queue offers the statistical economies of scale benefit, and the resulting advantageous product positioning offers a larger total market coverage, through the expanded coverage of the low quality product via less expensive production capacity. • Full information on customers preferences on quality leads to increased profitability via higher offered quality levels and increased capacity investments. The rest of the paper is organized as follows. We first study the case when the firm produces a single product in Section 3. In Section 4 we discuss the firm’s optimal product line design and capacity investment when offering two differentiated products with the use of dedicated facilities. Then, we move on to the multiple-product with dedicated facilities case, and characterize analytically the optimal product variety and the factors that affect it in Section 5. We also investigate the effect of using flexible facilities on producing two differentiated products in Section 6, and use these results to generalize our framework on factors that affect product variety from a single-dimension


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aggregate measure representation of relevant factors to a two-dimensional space with the addition of a new cost ratio. We discuss the results with the firm having full information of customers’ preferences on quality in Section 7. We conclude with the summary of our results and managerial insights in Section 8. All proofs are provided in the Appendix.

3. Single Product Case To study the interactions between market partitioning decisions in product line design and congestion effects in capacity planning, we formulate an integrated model with both product line and capacity variables as endogenous choices. Such a model allows us to better understand important interactions previously ignored as independent choices of these variables. Many of the insights we obtain on the effects of factors such as variable production costs, market size, and congestion levels on product variety and capacity could not have been predicted via models that treated either the marketing (product line) or operational (capacity) variables as exogenous. We first consider the case in which the firm produces a single product using a single dedicated production facility. The firm’s decisions are the product design quality, selling price, and the capacity of the production facility. Customers arrive according to a Poisson process with rate λ, and have no obligation to purchase the product. The customers are heterogeneous in their marginal valuation of the product quality level and are sensitive to product delivery lead times. Each customer has a ˜ − cˆ wσ − p, where q is the quality level, wσ the quoted lead time (with utility function u(p, q) = θq service level σ), and p the price. Every customer who has a non-negative utility of consumption is willing to buy the product. The parameter θ˜ captures the customer’s valuation of the quality level, and is uniformly distributed on [0, 1]. We assume that θ˜ is private information to each customer ˜ This assumption carries through Sections 3–6. We and the firm only knows the distribution of θ. discuss the case of the firm knowing each customer’s θ˜ in Section 7. For a given quality level, a higher θ˜ represents a higher willingness to pay. Without loss of generality we assume the quality level is bounded, i.e., q ∈ [0, qmax ]. The parameter cˆ represents the marginal cost of delay. The production system is make-to-order. There is a capacity investment cost incurred when the firm sets up the facility, and production capacity refers to the system’s processing rate. We assume the marginal capacity investment cost per unit time b(q) is a sufficiently smooth, strictly increasing and convex function of the quality level, hence b0 (·) > 0, b00 (·) > 0, and b(0) = 0. There is also a variable production cost a(q) with analogous properties, i.e., a0 (·) > 0, a00 (·) > 0, and a(0) = 0. When offered a product with a given quality level q, price p, and lead time quotation wσ , a customer with parameter θ˜ purchases it if and only if θ˜ ≥ θ, where θ := (p + cˆ wσ ) /q. Hence, the effective customer arrival rate λe = (1 − θ)λ is the arrival rate of customers who buy the product. Since λ is the effective arrival rate under full market coverage (i.e., θ = 0), we also refer to it as the market size. The firm’s revenue (in fact, revenue per unit of time) is thus πS (p, q) = [p − a(q)]λe = [θq − cˆ wσ − a(q)](1 − θ)λ.

(1)

Revenue function (1) with cˆ = 0 is widely used in the economics and marketing literature, but it ignores delay costs. Moreover, it does not include any relevant production information, such as capacity limitations and costs. In our model, we represent the production system as an M/M/1 queuing system, with both the optimal quality level q and capacity (processing rate) µ as relevant decision variables, and incorporate the economic consequences of the production system’s congestion as it affects lead times and the willingness to pay of time-sensitive customers. Use of a single-server queuing system to model production systems is common (see e.g., Karmarkar (1987), Duenyas and Hopp(1995), and Duenyas (1995)). The firm’s problem can be formulated as: max

(p,q,µ)∈R3 +

s.t.

ΠS (p, q, µ) = [p − a(q)]λe − b(q)µ P {T > wσ } ≤ 1 − σ,

(2) (3)


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where T is the actual lead time. In optimality, constraint (3) is binding because capacity is costly and strictly exceeding lead time performance is wasteful. Notice that customer behavior depends on the system’s performance and vice versa. One mechanism by which an equilibrium is reached starts with the firm quoting truthful lead times for the given service level σ, which can be considered an industry standard. Customers make ex-ante purchasing decisions based on the firm’s lead time quotation and the system reaches the anticipated equilibrium. In the long-run, the firm is truthful for fear of negative reputational effects since customers can observe long term service levels expost. Customers have no incentive to deviate either, because doing so is sub-optimal in terms of their individual utilities. For the M/M/1 system P {T > wσ } = e−(µ−λe )wσ , (3) is equivalent to wσ = − log(1 − σ)/(µ − λe ). For convenience we define c := −cˆ log(1 − σ) and w := w1−e−1 , thus, for every σ ∈ [0, 1), cˆ wσ = c w. Using (θ, q, µ) as decision variables instead of (p, q, µ), problem (2)–(3) can be rewritten as c max ΠS (θ, q, µ) = θq − − a(q) (1 − θ)λ − b(q)µ (4) µ − (1 − θ)λ (θ,q,µ)∈R3 + c ≥0 (5) s.t. θq − µ − (1 − θ)λ θ ≤ 1. (6) Recall that p = θq − c w, therefore customer waiting, caused by congestion effects, erodes the firm’s margin. With this in mind, we will sometimes refer to cw(1 − θ)λ as “congestion costs,” even though these are not costs which are directly incurred by the firm. Notice that it also follows from p = θq − c w that (5) corresponds to p ≥ 0. It is straightforward to show that (4) is strictly concave in p ∗ µ. The optimal capacity is given by µ = (1 − θ)λ + c(1 − θ)λ/b(q) . In general, the profit function (4) is not jointly concave in θ and q. We solve the optimization problem (4)–(6) sequentially and we provide the justification of sequential optimization as a remark in the Appendix. For expositional ease we assume that both b(q) and a(q) are quadratic functions of the quality level, i.e., b(q) = βq 2 and a(q) = αq 2 with α, β > 0. However, using the more general power functions b(q) = βq γ , a(q) = αq γ , γ > 2 leads to similar results and insights. For the interested reader we provide the corresponding analysis in the Appendix. The optimization problem can be formulated in terms of the single decision variable θ as max ΠS (θ) =

θ∈[0,1]

√ 2 λ θ 1−θ−M , 4(α + β)

s.t.

p √ θ 1 − θ ≥ M, with M = 2 βc/λ,

where the constraint is required to ensure q ∗ (θ) ≥ 0. We provide the optimal decision variables in the following theorem. Theorem 1. When 0 ≤ M ≤ 3√2 3 , the optimal: product positioning θ∗ , quality level q ∗ , production capacity µ∗ , price p∗ , and profit Π∗S are given by ! r √ 2 1 1 3βc λ λ(α + β) 3c ∗ ∗ ∗ √ , θ = , q = − , µ = +√ 3 α+β 3 λ 3 βλ − 3β 3c √ √ √ √ 2 p 1 2λ λ − 9λ 3βc + 27βc λ √ p∗ = , and Π∗S = λ − 3 3βc . 27(α + β) 9(α + β)λ λ √

√

βλ−3β √3c The system’s utilization level is given by ρ∗ = √βλ+3α . When M > 3c negative and the optimal choice is not to produce any product.

The solution has the following properties.

2 √ , 3 3

the firm’s profit is


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Proposition 1. (i.) The optimal quality level q ∗ increases in λ and decreases in β, α and c; (ii.) the optimal price p∗ decreases in α, β and c and increases in λ; (iii.) the optimal profit function Π∗S increases in λ and decreases in β, α and c. Observe that because the market coverage 1 − θ∗ is constant, the effective arrival rate λe = (1 − θ∗ )λ is only sensitive to changes in the arrival rate λ. Therefore, when production costs α or capacity costs β increase, the firm has no other option but to lower the quality level to control capacity investment costs. We establish the monotonic effect on the optimal quality level of changes in either lead time costs c or the arrival rate λ. But even more insightful, and counterintuitive in their nature, are the effects of such factors on the behavior of the optimal capacity level µ∗ , which we present in the following comparative statics. For interested readers, the extensive second order comparative statics of µ∗ are provided in the Appendix. √ √ Proposition 2. (i.) We define Mr (ω) := 2[1 − ω + 1 + ω + ω 2 ]/9 3, where ω = 27αc/λ. There ˆ S that solves M = Mr (ω), and the optimal production capacity µ∗ has the following exists a unique λ properties ( ˆS decreases in λ, 0 < λ < λ ∗ µ : ˆS . increases in λ, λ ≥ λ (ii.) (a.) If α = 0, µ∗ is increasing in β. (b.) For α > 0 : If β ≥ α, µ∗ increases in β; if β < α, ( decreases in β, 0 < β < βˆS µ∗ : increases in β, β ≥ βˆS , √ √ 2 . where βˆS = α 27αc + λ − 3 3αc λ. ∗ (iii.) The optimal capacity µ is increasing in both α and c. The optimal capacity µ∗ increases in c because with no change in the effective arrival rate λe , the firm can only control uncaptured revenue due to congestion by reducing it through increased capacity. Hence, because production and capacity costs are convex in the quality level, lowering quality allows the firm to control both the increased capacity costs as well as margins, by reducing variable costs to compensate for the price reduction from p = θq − cw. Analogously, when α increases, the resulting lower quality level decreases the cost of capacity, which the firm increases in order to relieve congestion and its associated costs. At first glance, one might expect the optimal capacity µ∗ to be decreasing in the marginal capacity investment cost β. However, as Proposition 2 shows, the response of µ∗ to changes in β (though λ∗e is independent from it) depends on the relative magnitudes of α and β. If β dominates α (i.e., β > α or α = 0), µ∗ is increasing in β. To understand this result it is sufficient to consider the effect of β on q ∗ , and the convexity in q ∗ of capacity costs: For relatively large β, q ∗ is low and marginal capacity investment cost are low and relatively insensitive to changes in β, therefore, as β increases, the firm resorts to increasing its capacity investments in order to reduce congestion costs. The following result shows that for this case (α = 0 or β ≥ α), the total operations costs (congestion plus capacity investment) decrease in β, supporting the assertion that µ∗ increasing in β is an effective way to manage the system. Proposition 3. If α = 0 or β ≥ α, the sum of congestion and capacity investment costs decreases in β. If variable production costs dominate (i.e., α > β), the above result no longer holds. For low levels of β (β < βˆS ), q ∗ is high, and the marginal capacity investment cost is high and highly sensitive to changes in β, therefore as β increases the firm tightens its capacity. When β increases past a

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critical value (β ≥ βˆS ), q ∗ becomes sufficiently low for the marginal capacity cost to be relatively low and insensitive to changes in β. Hence capacity investment costs are dominated by congestion costs, which the firm controls by increasing capacity. Even though θ∗ is constant, the effective arrival rate λe is increasing in λ. Therefore, an increase in λ generates higher congestion, which can be relieved through capacity investments whose cost can in turn be controlled by reducing the quality level. However, the directional change in margins cannot be unequivocally determined at first glance because profits are also affected by the increase in λe . But from Proposition 1 it follows that as λ increases, optimal profits increase in q ∗ and outweigh the associated increase in marginal capacity costs and costlier congestion management. The lack of monotonicity of µ∗ in λ can be explained similarly to that in β. Notice p that Mr (ω) is decreasing in ω, which in turn is linearly increasing in α, and recall that M = 2 βc/λ. When β is relatively low, M is small, and the firm handles an increase in customer arrival rate by raising production capacity to reduce congestion. In contrast, when capacity investment costs are high, M is large, and the high quality level makes adding capacity expensive, therefore, the savings from reducing capacity outweigh the increase in congestion costs and µ∗ becomes decreasing in λ. Because θ∗ is constant, the utilization level ρ∗ exhibits similar properties to those of µ∗ . Utilization ρ∗ is increasing in λ and decreasing in α and c. If α is zero or less than β, ρ∗ decreases in β. When α is greater than β, ρ∗ is no longer monotonic with respect to β; it first increases before β reaches a threshold value and decreases thereafter. The complete comparative statics of ρ∗ can be found in the Appendix. As we have shown in this section, even if the product variety issue is not in play, the correct product positioning is a challenging decision with unanticipated operational implications. According to our stylized model, such product positioning targets the top 1/3 of the market, with product quality that is increasing in market size, and decreasing in capacity investment, variable production, and congestion (customer waiting) costs per unit of met demand. However, the total capacity and its resulting utilization for meeting the demand of the optimal product positioning have far from immediately intuitive comparative statics, especially with respect to market size and capacity investment cost per unit. The invested optimal capacity is non-monotonic to increased market size, first decreasing and then increasing, while the utilization level stays increasing in it. When the per unit capacity costs are dominant relative to the variable production costs, the capacity and its utilization are increasing in these costs, however, this effect becomes non-monotonic when such dominance is not present. These results emphasize the importance of capturing the queuing related implications of product positioning decisions on the operational processes that will meet the resulting demand, and how such effects lead to counter-intuitive insights.

4. Two Differentiated Products in Dedicated Facilities We now assume the firm produces two products with quality levels q1 and q2 respectively, where q1 > q2 . Let p1 , p2 denote the prices of the two products. The lead time quotations are represented ˜ 1 − cˆwσ − p1 for purchasing by wσ1 and wσ2 . A customer with θ˜ has utility function u(p1 , q1 , wσ1 ) = θq 1 ˜ 2 − cˆwσ − p2 for buying the low quality product. the high quality product and u(p2 , q2 , wσ2 ) = θq 2 There exist θˆ1 and θ2 such that p1 = θˆ1 q1 − (θˆ1 − θ2 )q2 − cˆwσ1 and p2 = θ2 q2 − cˆwσ2 , where θˆ1 , θ2 ∈ (0, 1), and θˆ1 q1 − cˆwσ1 − p1 = θˆ1 q2 − cˆwσ2 − p2 . We illustrate these quantities in Figure 1, which shows plots of the customer’s utility functions for each product. By making utility maximizing choices, arriving customers generate appropriately split Poisson processes for the two products. To ensure that every product serves a positive market segment, one must further impose the participation constraints θ2 < θˆ1 < 1. Customers purchase the high quality product if θˆ1 ≤ θ˜ ≤ 1, and the low quality product if θ2 ≤ θ˜ < θˆ1 . The effective arrival rates of the customers who purchase products q1 and q2 are λe1 = (1 − θˆ1 )λ and λe2 = (θˆ1 − θ2 )λ respectively. The firm uses dedicated production facilities with capacities µ1 and µ2 , each one producing a single quality level and


11

u (θ ) ˆ 1 − p1 u (θ ) = θq1 − cw ˆ 2 − p2 u (θ ) = θq2 − cw

θ2

0

ˆ 2 − p2 −cw

θˆ1

1

θ

ˆ 1 − p1 −cw

Figure 1

Customer utility functions when two quality levels (q1 and q2 ) are available.

modeled as a separate M/M/1 queuing system. Let T1 , T2 be the customer’s actual waiting time in the system. The service level constraint in this case is P{Ti > wσi } ≤ 1 − σ, which is equivalent to wσi ≥ − ln(1 − σ)/(µi − λei ), i = 1, 2. The firm’s profit maximization problem is X max ΠD (p, q, µ) = {[pi − a(qi )] λei − b(qi )µi } (7) p,q,µ

s.t.

i=1,2

λe1 = λ(1 − θˆ1 ) λe = λ(θˆ1 − θ2 )

(8) (9)

2

p1 + cˆwσ1 − (p2 + cˆwσ2 ) , θˆ1 = q1 − q2 − ln(1 − σ) wσi ≥ , i = 1, 2 µ i − λ ei θ2 < θˆ1 < 1

θ2 =

p2 + cˆwσ2 q2

q1 > q2 > 0

(10) (11) (12) (13)

where p = (p1 , p2 )0 , q = (q1 , q2 )0 , and µ = (µ1 , µ2 )0 are vectors. As in the single product case, the service constraint (11) is always binding. Therefore, wσi = − ln(1 − σ)/(µi − λei ) for i = 1, 2. We define cˆwσi = cwi where wi = 1/(µi − λei ), i = 1, 2. Notice that even though the products are made in dedicated facilities, the decisions are coupled by the market partitioning constraints (8)–(10). The prices of the two products p1 and p2 are fully determined by the market partitioning variables θˆ1 , θ2 and capacity variables µ1 , µ2 according to p1 = θˆ1 q1 − (θˆ1 − θ2 )q2 − c/(µ1 − λe1 ) and p2 = θ2 q2 − c/(µ2 − λe2 ). Thus, problem (7)–(13) can be recast in terms of q, µ, and θˆ1 , θ2 , with p2 ≥ 0 being equivalent to θ2 q2 − c/(µ2 − λe2 ) ≥ 0. Therefore, the constraints reduce to θ2 q2 − c/(µ2 − λe2 ) ≥ 0, 0 ≤ θ2 < θˆ1 < 1 and q1 > q2 > 0. The firm’s optimization problem becomes max ΠD (q, µ, θˆ1 , θ2 ) = [θˆ1 q1 − a(q1 )]λe1 + [(θˆ1 + θ2 − 1)q2 − a(q2 )]λe2 X cλe i − + b(qi )µi µ i − λ ei i=1,2

q,µ,θˆ1 ,θ2

s.t.

θ2 q2 −

c >0 µ2 − λe2

θ2 < θˆ1 < 1

(14) (15) (16)


12

q1 > q2 > 0

(17)

As in the single product case we assume quadratic cost functions, i.e., b(qi ) = βqi2 and a(qi ) = αqi2 , i = 1, 2 (please see the Appendix for the corresponding analysis for general power cost functions, which leads to similar results). We are able to recast the problem exclusively in terms of the optimal market partitioning variables: ( q 2 2 ) q max ΠD (θˆ1 , θ2 ) = A θˆ1 1 − θˆ1 − M + (θˆ1 + θ2 − 1) θˆ1 − θ2 − M (18) θˆ1 ,θ2

q q ˆ ˆ 1 − θ2 > M 1/ 1 − θ1 − 1/ θ1 − θ2 q (θˆ1 + θ2 − 1) θˆ1 − θ2 > M 0 ≤ θ2 < θˆ1 < 1

s.t.

(19) (20)

(21) q λ , M = 2 βc and (19) and (20) correspond to q1 > q2 and q2 > 0 respectively. where A = 4(α+β) λ Solving optimization problem (18)–(21) we characterize the firm’s optimal product variety deci2 , the firm is better off producing two differentiated sions by a single factor M . When M is below 27 2 2 √ products. If M falls in the range of ( 27 , 3 3 ), the firm’s optimal choice is to produce a single product. When M is higher than 3√2 3 the firm cannot produce any products profitably. Thus, the p optimal product variety is completely determined by the aggregate measure M = 2 βc/λ. Offering differentiated products is optimal when capacity investment or customer waiting costs are relatively low, or the total market size, represented by the arrival rate, is sufficiently large. In the next result, we provide the optimal product positioning and capacity investment decisions for a firm offering two products. 2 , Theorem 2. For two products in dedicated facilities with quadratic cost functions, if 0 ≤ M < 27 ∗ ∗ ∗ ∗ ˆ the firm’s optimal product positioning decisions θ1 and θ2 , optimal quality levels q1 and q2 , optimal prices p∗1 and p∗2 , and optimal capacities µ∗1 and µ∗2 √ are given by: ˆ1 +5θˆ2 ) 1−θˆ1 (8−14 θ 1+θˆ1∗ √ 1 (i.) θˆ∗ is the unique solution of M = in the range ( 2 , 4 ), and θ∗ (θˆ∗ ) = . 1

18−27θˆ1 +12

2

3 5

3(1−θˆ1 )(2θˆ1 −1)

1

3

(ii.) # " s ˆ∗ 1 βc θ 1 , q1∗ (θˆ1∗ ) = − α+β 2 (1 − θˆ1∗ )λ

# " s ˆ∗ − 1 1 3βc 2 θ 1 q2∗ (θˆ1∗ ) = . − α+β 3 (2θˆ1∗ − 1)λ

(iii.) "

p∗1 (θˆ1∗ , q1∗ , q2∗ )

=

q1∗

θˆ1∗

s −

# βc 2θˆ∗ − 1 ∗ q2 , − 1 3 (1 − θˆ1∗ )λ

"

p∗2 (θˆ1∗ , q2∗ )

=

q2∗

s

θˆ1∗

−

βc

#

(θˆ1∗ − θ2∗ )λ

(iv.) 1 µ∗1 (θˆ1∗ , q1∗ ) = (1 − θˆ1∗ )λ + ∗ q1

s

c(1 − θˆ1∗ )λ , β

2θˆ∗ − 1 1 µ∗2 (θˆ1∗ , q2∗ ) = 1 λ+ ∗ 3 q2

s

c(2θˆ1∗ − 1)λ . 3β

The utilization levels of the dedicated facilities are #−1 " #−1 " s s 1 c 1 3c ∗ ˆ∗ ∗ ∗ ˆ∗ ∗ ρ1 (θ1 , q1 ) = 1 + ∗ , ρ2 (θ1 , q2 ) = 1 + ∗ . q1 β(1 − θˆ1∗ )λ q2 β(2θˆ1∗ − 1)λ

.


13

The market positioning and quality components of the optimal solution have the following properties. Proposition 4. For the case of two differentiated products in dedicated facilities with quadratic 2 cost functions and 0 ≤ M < 27 : 4 5 3 2 ∗ ∗ ˆ (i.) θ1 ∈ ( 3 , 5 ), θ2 ∈ ( 9 , 5 ). (ii.) Both θˆ1∗ and θ2∗ increase in λ and decrease in β and c. They do not change with α. (iii.) Both q1∗ and q2∗ increase in θˆ1∗ and λ, and decrease in α, β, and c. (iv.) Both p∗1 and p∗2 increase in θˆ1∗ and λ, and decrease in α, β, and c. Furthermore, p∗1 > p∗2 . In contrast to the single product case, the market partitioning decisions θˆ1∗ and θ2∗ are no longer constant. The following result about market coverage follows directly from Proposition 4. Corollary 1. The total market coverage 1 − θ2∗ and the market coverage by the high quality product 1 − θˆ1∗ increase in M . The market coverage by the low quality product θˆ1∗ − θ2∗ decreases in M.

0.8

^"1

0.75 0.75

^"1

0.7

"$

0.7

"i

"i

"#

0.65

0.65 0.6 0.6

"#

"2 0.55

0.55 0

0.2

0.4

0.6

!

0.8

1

1.2

(a)θˆ1∗ and θ2∗ as functions of β Figure 2

1.4

200

400

600

!

800

1000

(b)θˆ1∗ and θ2∗ as functions of λ

Illustration of comparative statics of market decisions in the two-product, quadratic cost functions case.

To illustrate these results we include plots for a specific example of θˆ1∗ and θ2∗ as functions of β and λ in Figure 2, and the corresponding plots for q1∗ and q2∗ in Figure 3 (c = 1, α = 0.5, λ = 1000 in Figures 2(a) and 3(a), and c = 1, α = 0.1, β = 0.2 in Figures 2(b) and 3(b)). Both figures also include plots of the optimal single-product market decision θ∗ and quality level q ∗ . Observe in Figure 3 how as β increases and λ decreases, the high quality product behaves as the single product, and that the quality difference between the two products is decreasing in both β and λ. When β = 0 or c = 0 the market partitioning decisions are θˆ1∗ = 4/5 and θ2∗ = 3/5, which coincide with the results by Moorthy (1984). As the customer arrival rate increases, the firm raises both quality levels, and compensates the increase in unit capacity costs by reducing market coverage. The firm responds to an increase in β by expanding the market coverage and associated profits of the high quality product, while simultaneously contracting the market coverage and associated profits of the low quality product. To obtain comparative statics for the optimal capacities we exploit the property that the first order derivatives of µ∗1 and µ∗2 with respect to c, λ, and β can be cast as pure functions of θˆ1 . We


14

0.8

1.2

q

Quality

0.6

q*

0.9

q*

qi

0.4

q1

0.6

q 0.2

0.3

q2 0

0 0

0.2

0.4

0.6

0.8

1

E

(a)q1∗ and q2∗ as functions of β Figure 3

1.2

1.4

200

400

600

800

1000

O

(b)q1∗ and q2∗ as functions of λ

Illustration of comparative statics of quality levels in the two-product, quadratic cost functions case.

summarize the comparative statics of optimal capacities µ∗1 and µ∗2 in the following Proposition (we provide second order comparative statics in the Appendix): Proposition 5. (i.) Both optimal capacities µ∗1 and µ∗2 are increasing in α and c. ˆ D such that (ii.) The optimal capacity µ∗1 is increasing in λ. There exists a unique λ ( ˆD is decreasing in λ, 0 < λ < λ µ∗2 : ˆD . is increasing in λ, λ ≥ λ (iii.) If α = 0, both µ∗1 and µ∗2 are increasing in β. If α > 0, there exists a unique βˆ1D (respectively D ˆ β2 ) such that ( is decreasing in β, 0 < β < βˆ1D (βˆ2D ) ∗ ∗ µ1 (µ2 ) : is increasing in β, β ≥ βˆ1D (βˆ2D ). The optimal utilization levels ρ∗1 and ρ∗2 have similar properties as those of the optimal capacities µ∗1 and µ∗2 . Both ρ∗1 and ρ∗2 are increasing in λ decreasing in α and c. When β is greater than α, ρ∗1 (ρ∗2 ) is decreasing in β; otherwise, there exists a threshold value of β which separates the increasing and decreasing cases of ρ∗1 (ρ∗2 ) with respect to β. Interested readers can find rigorous comparative statics results of ρ∗1 and ρ∗2 in the Appendix. Like in the single product case, as capacity investment costs grow, the firm increases the total market coverage and lowers the quality level of its offerings to afford capacity increases for both facilities. The lower utilization levels reflect the reduction in the system’s congestion and associated cost savings. To illustrate the comparative statics for optimal capacities, in Figure 4 we show plots of a particular example (c = 1, and α = 0.5, λ = 1000 in Figure 4(a); α = 0.1, β = 0.2 in Figure 4(b)) and include the corresponding values µ∗ for the single product case. Figure 4(b) also illustrates that ˆ S ) is smaller than the smallest λ for any β, c > 0, the critical λ where µ∗ changes direction (i.e., λ p supporting two products (i.e., λ = 729βc = 145.8; the unique solution to 2 βc/λ = 2/27), which is always the case (see Appendix). Notice that as λ approaches 145.8 from the right, even though the capacity for the low quality product becomes increasingly higher, since q2 approaches zero, both the price the firm can charge as well the cost of capacity approach zero, and so does the revenue contribution from the low quality product. In the limit, the firm is indifferent between offering one


1200

15

800

1050 600 900

μi

Pi 750

400

μ2

P 600 200

P

450

μ∗

P 300

0 0

0.25

0.5

0.75

1

1.25

E

(a)µ∗i as function of β Figure 4

μ1

100 145.8 200

λ

300

400

500

(b)µ∗i as function of λ

Illustration of comparative statics of capacity decisions in the two-product, quadratic cost functions case.

or two products, since the lower quality product is produced with costless infinite capacity, and it is worthless to both the firm (it commands a zero price) and the customers (its utility is zero because its value is zero and the firm’s infinite capacity results in zero waiting). Recall that 1 − θ2∗ is the total market coverage by both products. Then ω1 = (1 − θˆ1∗ )/(1 − θ2∗ ) and ω2 = (θˆ1∗ − θ2∗ )/(1 − θ2∗ ) respectively are the relative market coverages by the high and low quality products. Proposition 6. The total market coverage 1 − θ2∗ and the relative coverage of high quality product ω1 decrease in λ and increase in β and c. The relative coverage of low quality product ω2 increases in λ and decreases in β and c. To understand better the role of each product we analyze the relative profits ∆Π∗i = Π∗i /Π∗D , where Π∗i is the optimal profit contributed by product i = 1, 2, and Π∗D is the firm’s total optimal profit. Proposition 7. ∆Π∗1 is decreasing in θˆ1 and increasing in M . ∆Π∗2 is increasing in θˆ1 and decreasing in M . In Figure 5 we show plots for a particular (but representative) example (c = 1, α = 0.5, λ = 2000) of behavior observed over a wide range of parameter values. Figure 5(a) illustrates the large relative contribution to profits from the high quality product (more than 80%), and Figure 5(b) illustrates the increase in benefits from vertical product differentiation as M decreases.

5. Multiple Products with Dedicated Facilities We now generalize the model to the case of a firm offering n > 2 differentiated products with quality levels q1 , . . . , qn , where q1 > q2 > · · · > qn . There exist 0 < θn < 1 such that pn = θn qn − cˆwσn and pi +ˆ cwσi −(pi+1 +ˆ cwσi+1 ) , such that θî qi − cˆwσi − pi = θî qi+1 − cˆwσi+1 − market partitioning variables θî = qi −qi+1 pi+1 , i = 1, . . . , n − 1. Figure 6 depicts a plot of the customers’ utility functions and market partition variables when the firm offers n differentiated products. The effective arrival rates λei for products i = 1, . . . , n are λe1 = (1 − θˆ1 )λ, λei = (θî−1 − θî )λ, i = 2, . . . , n − 1, λen = (θˆn−1 − θn )λ. Defining


16

160

100

ΔΠ1

140

3 D

120

Profit

ΔΠi (%)

80

3S

≈ 40 60

100

ΔΠ2

20

80

0 0.01

0.02

0.03

0.04

0.005

0.05

0.01

0.015

0.025

0.03

(b)Optimal two-product Π∗D and single-product Π∗S profits as functions of M

(a)Relative profit contributions with M Figure 5

0.02

M

M

Total profits and relative profit contribution for the two-product, quadratic cost functions case.

u (θ ) ˆ 1 − p1 u (θ ) = θq1 − cw

ˆ 2 − p2 u (θ ) = θq2 − cw ˆ n −1 − pn −1 u (θ ) = θqn −1 − cw ˆ −p u (θ ) = θ q − cw n

0

ˆ n − pn −cw

n

n

θn

θˆn −1

θˆ2

θˆ1

1

θ

ˆ n −1 − pn −1 −cw

Figure 6

Customer utility functions and market partition variables for n quality level offerings.

vectors θˆ = (θˆ1 , . . . , θˆn−1 , θn )0 , p = (p1 , . . . , pn )0 , and q = (q1 , . . . , qn )0 we formulate the problem with n dedicated production facilities with capacities µi , i = 1, . . . , n, as max ΠM (p, q, µ) = p,q,µ

s.t.

n X

{[pi − a(qi )] − b(qi )µi }

(22)

i=1

λe1 = λ(1 − θˆ1 ) λe = λ(θî−1 − θî ),

i = 2, . . . , n − 1 ˆ λen = λ(θn−1 − θn−1 ) pi + cˆwσi − (pi+1 + cˆwσi+1 ) pn + cˆwσn θî = , θn = , qi − qi+1 qn − ln(1 − σ) wσi ≥ , i = 1, . . . , n µ i − λ ei

(23) (24)

i

(25) i = 1, . . . , n − 1

(26) (27)


θn < θˆn−1 < · · · θî < θî−1 < q1 > q2 > · · · > qn > 0,

···

θˆ1 < 1

17

(28) (29)

whose decisions are coupled by the marketing partitioning constraints (23)–(26). The participation constraints (28) ensure every product has a positive market share. As with one and two products, the service constraint (27) is always binding. Let cˆwσi = cwi and ˆ q and µ, i.e., pi = θî qi − wi = 1/(µi − λei ) for i = 1, . . . , n. The prices can be characterized by θ, ˆ ˆ ˆ ˆ (θi − θi+1 )qi+1 − c/(µi − λei ) (i = 1, . . . , n − 2.), pn−1 = θn−1 qn−1 − (θn−1 − θn )qn − c/(µn−1 − λen−1 ) and pn = θn qn − c/(µn − λen ). We subsequently solve for optimal capacities in terms of θˆ and qˆ, ˆ which allows us to write the optimization problem followed by optimal quality levels in terms of θ, ˆ exclusively in terms of θ (see Appendix). The following result characterizes the optimal product variety when n > 2 as we did for n = 2. Proposition 8. Let N be the maximum number of product variants supported by the firm’s dedicated technology. Define M N√as the largest M yielding a positive profit when producing N products. From Section 4, M 1 = 2/(3 3) and M 2 = 2/27. For N ≥ 3, M N is uniquely determined by the solution of the following N − 1 equation system with N − 1 variables θˆ1 , . . . , θˆN −2 , and M N :   q q ˆ θ1  θˆ1 1 − θˆ1 − M N  1 − θˆ1 − q 2 1 − θˆ1  q q ˆ1 + θˆ2 − 1 θ =0 + (θˆ1 + θˆ2 − 1) θˆ1 − θˆ2 − M N  θˆ1 − θˆ2 + q ˆ ˆ 2 θ1 − θ2 .. .  q q ˆ ˆ θi−1 + θi − 1  (θî−1 + θî − 1) θî−1 − θî − M N  θî−1 − θî − q 2 θî−1 − θî   q q ˆ ˆ θi + θi+1 − 1  =0 + (θî + θî+1 − 1) θî − θî+1 − M N  θî − θî+1 + q î − θî+1 θ 2 .. .   q q ˆ ˆ θN −3 + θN −2 − 1  (θˆN −3 + θˆN −2 − 1) θˆN −3 − θˆN −2 − M N  θˆN −3 − θˆN −2 − q 2 θˆN −3 − θˆN −2 s   ! 32 2θˆN −2 − 1  2θˆN −2 − 1 +2 2 − MN = 0 3 3 √ 23 3 323MN + 1 θˆN −2 = . 2 Furthermore, N is the optimal number of variants to produce when M N +1 ≤ M < M N . In Table 1 we show M N values for up to ten variants. Additional values are readily available from the above equation system via standard numerical techniques. Note that according to the above result, when vertical differentiation in the marketplace is operationally supported by dedicated facilities, the value of the single aggregate parameter M relative to the predetermined threshold M N decides the optimal variety to be offered. And as before, we can conclude that capacity investment and congestion costs of the production system act as deterrents to the offering of variety, while increased market size promotes variety. The following result for the market partitioning variables follows directly from Proposition 8.


18

Table 1

Critical M N values and number of variants N .

MN Number of variants N √ 2/ 3 3 1 2/27 2 0.027196064 3 0.013281966 4 0.007606549 5 0.007204386 6 0.004749820 7 0.003293344 8 0.002781853 9 0.001785259 10 ∗ Corollary 2. For N product variants, all the optimal product positioning decisions θˆ1∗ , . . . , θˆN −1 , ∗ ∗ and θN are decreasing in M . The highest-quality product market coverage 1 − θˆ1 is increasing in ∗ ∗ M and the lowest-quality product market coverage θˆN −1 − θN is decreasing in M .

0.9

θ4,1 θ3,1

0.8

θ4,2

θi

θ2,1

0.7

θ∗ θ3,2

θ4,3 0.6

θ2,2 θ3,3 θ4,4

0.5 0.015

0.03

0.045

0.06

0.075

0.09

M

Figure 7

Optimal market partitioning as a function of M for a firm producing up to 4 variants.

We will now present useful insights on the management of product variety via focused production facilities from our numerical studies. Figure 7 illustrates the evolution of optimal market partitioning decisions as functions of M for the particular example of α = 0.01. We denote the market partitioning variable for product i when the firm produces j variants by θj,i , and the optimal choice for the single product case by θ∗ . In other words, for N variants this notation relates ∗ ∗ to the one used in Corollary 2 as follows: θN,1 = θˆ1∗ , θN,2 = θˆ2∗ , . . . , θN,N −1 = θˆN −1 , θN,N = θN . Using analogous notation, Figure 8 depicts the corresponding behavior of optimal quality levels for the same example. The results graphically depicted in Figures 7 and 8 portray the role of increased product variety on the relative product positioning within the product line and the resulting market segmentation. As we increase product variety from N − 1 to N variants, a new bottom-line


19

q4,1 8

q3,1 q2,1

6

q*

q4,2

qi q3,2

4

q4,3 q2,2

2

q4,4

q3,3

0 0.015

0.03

0.045

0.06

0.075

0.09

M

Figure 8

Optimal quality levels as functions of M for a firm producing up to 4 variants.

110 1

μ2,2

"&

%$0.98

"&$&

μ2,1

≈

0.94

"i

μ3,3

80

"#$#

0.96

100

μi

"#$% "#$&

"'

0.92

μ3,1

μ3,2

0.9 0.88

70 0

0.005

0.01

0.015

β

0.02

0.025

(a)Optimal capacities as functions of β Figure 9

0

0.005

0.01

0.015

!

0.02

0.025

(b)Optimal utilizations as functions of β

Optimal capacities and utilization levels as functions of β for a firm producing 3 variants.

product with lower quality than any previous product in the line is introduced, while the quality of the top-line product is increased. Similarly, the qualities of products 2, . . . , N − 1 in the new product line are improved relative to the same rank products in the previous product line. From a market segmentation perspective, the total market coverage is increased through increased product variety. However, this expanded market coverage is as a result of expanded coverage at the low end of the quality spectrum, while the high-end product fills the needs of a smaller market segment than before. Figure 9 includes plots of optimal capacities and utilization levels as functions of β for a specific example (c = 0.01, α = 0.04, λ = 500), using similar notation as before. It is straightforward to see that, because of the previously explained quality positioning and market segmentation of the expanded product line, as product variants increase from N − 1 to N the utilization levels of all existing N − 1 dedicated facilities decrease as a new variant is introduced. The newly added bottom-of-the-line product facility picks up the expanded market demand while remaining the

20


lowest utilized among all facilities. This last effect is owed to the reduced capacity investment costs per unit for this facility because of the lower product quality requirements. In summary, when product-focused (dedicated to each product variant) facilities are used, all relevant cost and market factors to describe thepanswer to the optimal product variety question are captured in the aggregate parameter M (= 2 βc/λ), which within a square root type formula emphasizes the market size as the facilitator, and the capacity investment (but not the variable production cost) and congestion costs per unit as the deterrents for higher product variety. The optimal product variety is determined by comparing the value of M to predetermined threshold levels. The lower the value of M , the more product variants to be offered as part of the product line, but with the feasible ranges for a particular level of product variety declining in size as the number of variants increase. Furthermore, from a profitability perspective, the marginal benefits of increased product variety are decreasing in both the number of product variants and in M .

6. Two Differentiated Products in a Flexible Facility We now incorporate flexible production technology into the model so that a single make-to-order facility is capable of producing two products with quality levels q1 > q2 . An M/M/1 queue with flexible capacity (service rate) µ handles jobs on a first-come first-served basis. For simplicity, we neglect switching times, but these can be incorporated into the service rate (see e.g., Karmarkar (1987)). However, the applicability of the M/M/1 model is limited whenever setup or changeover times are sufficiently high, as would be the case for traditional batch processes processing a wide product range. The types of flexible production systems we intend to represent in our study are those that can switch between product variants within a reasonable range without noticeable setup or changeover times. This is true for lean batch processes, where flexible production cells are capable of effortlessly switching between production of variants within a given product family. The per unit capacity cost in this case is assumed to be bF (q1 , q2 ) = β[δq1 + (1 − δ)q2 ]2 with δ between 0 and 1. In this formulation we assume flexibility is attained by a mixture of two production paradigms: “tailored customization” (δ = 1), and “customized standardization” (δ = 0). Tailored customization refers to examples where products specific to the customer applications are designed and produced, with the obvious examples of tailors making to measure high quality suits for wealthy clients, architects, and building contractors designing and building custom ordered homes; and semiconductor producers specializing their technological products to the needs of sophisticated defense contractors and state-of-the-art consumer electronic firms. The production equipments in this case are such that they can meet the most demanding request of the high quality customers, and resulting capacity costs end up being dependent on the highest quality product. In the second case, in order to obtain a product line, the firm makes additional efforts to reduce the range of a wide spectrum of initial quality levels created by natural variation. There are several types of processes represented by this paradigm. Consider for example the case of oil refineries, where early stages produce a wide range of intermediate products and any effort to restrict the range for quality purposes requires further processing. Similarly, in agribusiness, natural variation resulting from many uncontrollable factors generates a wide spectrum of quality levels, with the higher end occurring naturally and restricting the lower end products requires additional sorting and inspection processes. Thus, in this case capacity costs for producing the range (qL , qH ) will be lower than those for producing the range (qM , qH ), where qL < qM < qH , due to this added processing requirement for eliminating (sorting out) naturally occurring low quality variants and imperfections. Notice that because in this paradigm the cost of capacity depends exclusively on the low quality product, a pathological scenario could arise in which the low quality level may attain an infinitesimal value in order to offer a premium quality product with an arbitrarily low cost of capacity. However, this scenario would rarely happen in practice because there always exists a minimum industry quality level, e.g., the lowest admissible configuration of a product. This pathological situation can be


21

easily ruled out by imposing the constraint q1 ≥ qmin , where qmin represents the minimum industry standard quality level. But we will not include such constraint in the model because it would only add unnecessary complexity without providing any additional insights. Consistent with our queuing modeling assumptions (first-come first-served M/M/1), all customers must be quoted the same lead time wσA . Given qi , pi , i = 1, 2 and wσA , there exist θˆ1 and θ2 such that p1 = θˆ1 q1 − (θˆ1 − θ2 )q2 − cˆwσA and p2 = θ2 q2 − cˆwσA , where θˆ1 , θ2 ∈ (0, 1), and θˆ1 q1 − p1 = θˆ1 q2 − p2 . The firm’s profit maximization problem is X max ΠδF (p, q, µ) = [pi − a(qi )]λei − bF (q1 , q2 )µ (30) p,q,µ

i=1,2

λe1 = λ(1 − θˆ1 ) λe = λ(θˆ1 − θ2 )

s.t.

2

p2 + cˆwσA p1 − p2 θˆ1 = , θ2 = q1 − q2 q2 λ A = λ p1 + λ p2 − ln(1 − σ) wσA ≥ µ − λA ˆ θ2 < θ1 < 1 p1 > p2 ≥ 0, q1 > q2 > 0,

(31) (32) (33) (34) (35) (36) (37)

which includes interactions via the aggregate demand rate λA = (1 − θ2 )λ and the capacity investment cost bF (q1 , q2 ) in addition to the coupling by the market partition constraints (31)–(33). Using cˆwσA = cwA , wA = 1/(µ − λA ), p2 = θ2 q2 − c/(µ − λA ), and p1 = p2 + θˆ1 (q1 − q2 ), the firm’s optimization problem can be recast as max ΠδF (θˆ1 , θ2 , q1 , q2 , µ) = θˆ1 q1 − αq12 (1 − θˆ1 )λ + (θˆ1 + θ2 − 1)q2 − αq22 (θˆ1 − θ2 )λ θˆ1 ,θ2 ;q1 ,q2 ;µ

c(1 − θ2 )λ − β[δq1 + (1 − δ)q2 ]2 µ µ − (1 − θ2 )λ µ > (1 − θ2 )λ c >0 θ2 q2 − µ − (1 − θ2 )λ 0 < θ2 < θˆ1 < 1 q1 > q2 > 0. −

s.t.

(38) (39) (40) (41) (42)

As with dedicated facilities, we characterize the firm’s optimal product variety decisions by solving optimization problem (38)–(42). Theorem 3. If 0 ≤ δ ≤ 1, a feasible pair of M and r can be characterized by M (r|δ) and rδ , where M (r|δ) is an envelope function and rδ is the cutoff value defined by M (rδ |δ) = 0. When 0 ≤ M < M (r|δ) and 0 ≤ r < rδ , there exists a one to one correspondence between the firm’s optimal product positioning decisions (θˆ1∗ , θ2∗ ) and (M, r). Furthermore, the optimal solution of (38)–(42) can be uniquely determined by θˆ1∗ and θ2∗ . In particular: (i.) For δ = 0, r0 = ∞ and θˆ1∗ and θ2∗ are uniquely determined solving (θˆ1 − θ2 )[(1 − θˆ1 ) − (θˆ1 − θ2 )] − r(1 − θ2 )(3θˆ1 − 2) √ 1 − θ2 q 2 ˆ ˆ 5 − 24θ1 − 48θ1 + 25 θ2 = . 6 M=

22


(ii.) For δ = 1, r1 = 1 and θˆ1∗ and θ2∗ can be uniquely determined by solving (1 − θˆ1 )[(1 − θˆ1 ) − (θˆ1 − θ2 )] − r(1 − θ2 )(3θˆ1 − θ2 − 1) √ 1 − θ2 6θˆ13 − 10θˆ12 − 6θˆ12 θ2 + 6θˆ1 + 5θˆ1 θ2 + 3θˆ1 θ22 + 3θ2 − 8θ22 + 3θ23 − 2 = 0. M=

The detailed definition and properties of the envelope function M (r|δ) are in the Appendix (the proof of Theorem 3). Figure 10 shows plots of M (r|δ) as functions of r for several values of δ and is used to generate Figure 11.

Figure 10

M (r|δ) as functions of r for δ = 0, 0.05, 0.1, . . . , 0.9, 0.95, 1.

In contrast to the p case of dedicated facilities, when capacity is flexible the value of the single parameter M = 2 βc/λ (a measure of congestion and capacity costs relative to market size) no longer suffices to determine whether it is optimal to produce two products or fewer. Here the parameter r = β/α (capacity investment to production costs ratio) also plays an important role: For any given δ between 0 and 1, it is optimal to offer two variants only in a specific region of the two-dimensional parameter space r × M , namely, the set of points (r, M ) for which M < M (r|δ) (the Appendix includes plots of M (·|δ) for several values of δ). We generated Figure 11 using the envelope to the set of M (r|δ) curves across all δ. It depicts the different regions in the two-dimensional parameter space. In region A it is unprofitable to produce even a single variant. For any point in region C it is optimal to offer two products provided δ is sufficiently close to 1 (tailored customization), and similarly for region F provided δ is sufficiently close to 0 (customized standardization). In region E it is optimal to offer two products provided δ is sufficiently close to either 0 or 1. In regions B and D, it is optimal to produce one and two products respectively, regardless of the value of δ. As Figure 11 illustrates, besides M and r, the type of flexibility also plays a key role in determining optimal variety. In what follows, we provide further details on this issue. Theorem 3 fully characterizes θ2∗ as a function of θˆ1∗ . We plot these functions for several values of δ as 2/3 ≤ θˆ1∗ ≤ 4/5 in Figure 12. Consider the top portion of region C in Figure 11. When congestion is relatively expensive (high M ) the firm offers two products only when capacity is relatively inexpensive (low r) and the


0 variants (A)

2 √ 3 3

0.2

0.15

1 variant for any δ (B)

M 0.1 2 variants if δ close to 1 (C)

0.05

! √ 4/9 63 + 11 33

(D) 2 var any δ

2 variants if δ close to 0 or 1 (E)

0.2

0.4

0.6

0.8

2 variants if δ close to 0 (F)

1

1.2

1.4

r Figure 11

Two-product optimality space for flexible capacity.

2 3

δ 1

0.65

.95 .9 .85 .8 .75 .7 .65 .6 .55 .5 .45 .4 .35

0.60 θ2∗

.3 .25 .2 .15

0.55

.1 .05

0

√ 15− 33 18 2 3

0.68

0.70

0.72

0.74

0.76

0.78

θ1∗

Figure 12

θ2∗ as functions of θˆ1∗ for multiple values of δ.

0.8

23

24


production system’s flexibility must be close to pure tailored customization (δ = 1). The following result illustrates how a flexible-capacity firm with δ = 1 offers variety. Proposition 9. When δ = 1, θ2∗ spans the interval 53 , 23 as θˆ1∗ ∈ 23 , 45 . θˆ1∗ decreases in M while θ2∗ increases in M . Both optimal quality levels q1∗ and q2∗ decrease in M . Moreover, the entire market coverage 1 − θ2∗ and the market coverage of the low quality product θˆ1∗ − θ2∗ decrease in M while the market coverage of the high quality product 1 − θˆ1∗ increases in M . Recall from Proposition 4 that for two products in dedicated facilities θˆ1∗ spans the interval ( 32 , 45 ) and θ2∗ spans ( 95 , 35 ). Therefore, comparing a tailored customization flexible facility with a system of two dedicated facilities across the sets of all product positioning decisions we observe that: The total market coverage is smaller for tailored customization, the coverage of the high quality product remains the same across both cases, and thus the market coverage of the low quality product is smaller for the tailored customization. Moreover, the properties of θˆ1∗ and θ2∗ indicate that θˆ1∗ is increasing in α and λ, and decreasing in β and c, whereas θ2∗ decreases in θˆ1∗ . The optimal quality levels q1∗ and q2∗ increase in the market size and decrease in capacity investment, variable production, and congestion costs per unit. The optimal prices p∗1 and p∗2 share the exact same properties as those of q1∗ and q2∗ . Here, the optimal quality levels behave similarly to the case of two products in dedicated facilities, but the product positioning decision of the low quality product moves in the opposite direction. This happens because the firm controls increasing congestion costs by reducing total market coverage, which depresses revenues. This in turn prompts the firm to both control capacity costs by reducing the high quality level (the only one affecting them), and to improve revenues by expanding coverage of the high quality product. Reducing the low quality level can be interpreted as a market segmentation adjustment. When operating in region C of Figure 11, efforts to reduce congestion costs strongly support the offering of product variety. As congestion costs decrease, the firm can offer two products over a wider variety of capacity costs and production flexibility options (i.e., as M decreases, the ranges of possible r and δ expand to include higher and lower values respectively), and similarly as M decreases further into regions D and E of the figure. Firms with decreasing degrees of tailored customization, operating in regions B through E of Figure 11, must be conservative in q1∗ to control capacity investment costs, which limits revenues, and requires controlling congestion costs by limiting total market coverage. Even when congestion costs are relatively high (large M ) and production costs dominate capacity costs (small r), utilization levels tend to be high for these firms, which highlights the delicate balance between quality levels, market coverage, and capacity decisions. This is also made apparent by the sensitivity to changes in r of quality levels, market partitioning decisions, and relative revenue contributions of each product. A decrease in congestion costs eases this situation, enabling these firms to increase quality levels, market coverage, and the proportion of revenues from the lower quality product. Regions D through F of Figure 11 correspond to relatively low congestion costs, and the resulting optimal policies for these regions further validate our previous statement that reduction of congestion costs is a catalyst for offering product variety. When capacity costs are relatively low, any flexible system regardless of type offers variety. As relative capacity costs increase, only facilities with specific flexibility types will offer two products in regions E and F. The only firms offering two products in all three bottom regions (D through F) are those with flexibility close to pure customized standardization (δ = 0). The following result shows how these firms offer variety. √ √ √ 11 , 3 Proposition 10. When δ = 0, θ2∗ spans the interval 5 63− as θˆ1∗ ∈ 23 , 45 . Both θˆ1∗ and θ2∗ 5 3 as well as the optimal quality levels q1∗ and q2∗ decrease in M . Moreover, the entire market coverage 1 − θ2∗ , the market coverage of the high quality product 1 − θˆ1∗ , and the market coverage of the low quality product θˆ1∗ − θ2∗ increase in M .


25

Recall from Proposition 4 that for two products in dedicated facilities θˆ1∗ spans the interval and θ2∗ spans ( 59 , 35 ). Therefore, comparing a customized standardization flexible facility with a system of two dedicated facilities across the sets of all product positioning decisions we observe that: The total market coverage is larger for customized standardization, the coverage of the high quality product remains the same across both cases, and thus the market coverage of the low quality product is larger for the customized standardization. Remember also that in the region in which δ = 0 firms offer two products, congestion costs are relatively low. Moreover, these firms can control capacity costs via the low quality level and use the high quality level to attain the best margin to volume tradeoff for that product. These firms deal with increasing congestion costs by reducing the low quality level, and counteract the resulting lower margins by increasing the low-quality coverage. The reduced high quality level and increased market coverage of the high can be interpreted as a market segmentation √ product √ √ quality adjustment. Finally, notice that 5 3 − 11 /6 3 < 3/5, therefore, the total market coverage for δ = 0 is larger than the coverage for δ = 1 firms. Figure 12 shows that this is also true for moderately customized standardization firms relative to moderately tailored customization firms. For moderately customized standardization firms, operating in regions B, C, and F of Figure 11, q1∗ is relatively high because it has little effect on capacity costs, and its level is tempered only by production costs and market partitioning decisions. Market coverage is relatively high, and high quality product sales account for a large percentage of revenues. Given M , as r increases, capacity costs become more important and are controlled by lowering q2∗ , resulting in lower utilization levels. But quality levels, market partitioning decisions, and relative revenue contributions are not very sensitive to changes in r and become less so as r grows beyond 1. As congestion costs decrease, quality levels and utilization increase, and so does the relative revenue of the low quality product. Evidently, the optimal product variety answer gets more complex when product-flexible facilities are used. First, one has to understand the capacity investment cost implications of processing the multiple product variants through the same facility. In our stylized model, this is effected through the δ parameter, which describes a spectrum between “customized standardization” (δ = 0) and “tailored customization” (δ = 1). Customized standardization settings rely heavily on the processes to restrict the quality range from a wide spectrum by natural variation, which are not heavily dependent on the nature of the particular product variants in the range. The processes become more expensive to obtain narrower range of product variants. Close to this end of the spectrum are the operational processes at oil refineries and agribusiness. However, tailored customization relies on production tasks that are heavily dependent on the customer products and applications, thus becoming very costly from a capacity investment perspective. For such cases, the capacity investment costs are determined by the most difficult tasks that can be performed by such facilities. Flexible facilities in continuous processing industries, like active ingredient production in chemicals and pharmaceuticals, fall closer to this end of the spectrum. Second, beyond the aggregate parameter M, it is necessary to actively consider the ratio r (= β/α) of capacity investment cost per unit to variable production cost per unit. Notice that most elements from the preceding discussion are depicted in Figure 11, which is based entirely on analytical results and covers all instances of the two product flexible facility model. To illustrate the effect of flexibility and flexibility type on profitability, we compare the optimal profits of the dedicated and flexible capacity models for tailored customization and customized standardization. ( 32 , 54 )

Proposition 11. For any set of parameters λ, α, β and c that is feasible for the two-product dedicated capacity and flexible capacity cases δ = 1 and 0, let Π∗D (λ, α, c; β), Π1∗ F (λ, α, c; β), and Π0∗ (λ, α, c; β) represent the optimal profits of the dedicated and the two flexible capacity cases F ∗ 1∗ respectively. Then Π0∗ (λ, α, c; β) > Π (λ, α, c; β) > Π (λ, α, c; β). F D F

26


This result highlights the interaction between two major factors at play in our model. One is the statistical economies of scale advantage of the single queuing system of the flexible capacity models relative to the dedicated facilities model with two separate queues. The other is the convexity of production and investment costs in the quality level coupled with the optimal market coverage and positioning decisions stemming from customers’ preferences. This second effect clearly favors customized standardization over tailored customization because in the former case the firm can cover the high end of the market relatively inexpensively since the high quality product is only affected by variable production costs and is independent of capacity investments; whereas in tailored customization, the ability of the firm to effectively use market segmentation to counteract the convexity of capacity investment costs borne by the high quality variant by profiting via high volume with the low quality product is severely limited by the relatively high costs incurred by the low quality product. The fact that the customized standardization system is more profitable than the dedicated model confirms intuition because the former enjoys advantages in the two factors we described above. The result that the dedicated facilities model is more profitable than tailored customization is more noteworthy because it shows that when each system is endowed with one of the two factors, convexity coupled with market preferences dominates, because it allows the dedicated facilities system to overcome the statistical economies of scale advantage of the flexible facility by offering a larger market coverage overall, through the expanded coverage of the low quality product via less expensive production capacity. Notice that the above result is inconclusive towards making technology choice decisions (which are beyond the scope of this study) because the final decision will hinge on the particular details of each production system under consideration (e.g., the exogenous parameters and additional fixed costs will vary across candidate systems). To complete our analysis, we study the case of a flexible facility whose marginal capacity investment costs are independent of the quality levels of either product. These assumptions reflect the conditions present in simple assembly lines (e.g., the Dell assemble-to-order system). The following proposition summarizes our results. Proposition 12. Consider a two-product flexible production system whose capacity investment costs are independent of the quality levels of either product. p Let κ denote the marginal capacity investment cost. When α(κ + 10κc/λ) < 1/10, θ2∗ ∈ 53 , 1 and is uniquely determined by solving p p p 1 − θ2 (14θ2 − 15θ22 − 3) + 16α[ κ(1 − θ2 ) + κc/λ] = 0 . ∗

∗

∗

θˆ θˆ +θ −1 θˆ1∗ = (1 + θ2∗ )/2. q1∗ = 2α1 and q1∗ = 1 2α2 . Both optimal quality levels q1∗ and q2∗ are decreasing in κ, c and α, and increasing in λ. The entire market coverage 1 − θ2∗ , the market coverage of the high quality product 1 − θˆ1∗ , and the market coverage of the low quality product θˆ1∗ − θ2∗ are decreasing in κ, c and α, and increasing in λ. Moreover, 1 − θˆ1∗ = (1 − θ2∗ )/2.

Hence, the unit capacity cost κ and the customer’s delay cost c have the same impact on the firm’s product positioning and quality design decisions. The overall market coverage is reduced due to the congested production facility and customer’s waiting while both the high quality and low quality products have the same share of the covered market (1 − θ2∗ )/2. Both quality levels are decreasing in capacity and customer’s delay costs. In summary, from a managerial perspective, it is important to understand the clear identification of five separate regions in the two-dimensional space r × M (see Figure 11), with each region having different product positioning guidelines. With the exception of the uninteresting case of region A (unprofitable business across all r when M is large enough), we will describe in more detail the optimal product variety and positioning implications for all others. Region B occurs for reasonably large values of M and r which do allow profitable single product production, but discourage further variety across a wide range of δ values. The optimal product variety insights for this case are


27

captured in Section 3, and have been already summarized. Regions D and E, for small values of M and small to medium values of r (medium values of r corresponding to region E), are the regions that fully support variety across all values of δ. The insights on product positioning in these cases are heavily dependent on the value of δ, with the extreme cases of δ = 1 and δ = 0 captured in the results of propositions 9 and 10 respectively. For tailored customization, which is fully represented at the higher M end of region D and the whole region C (high M , small to medium r), increases in M are followed by increased market coverage at the high-end and reduction of market coverage at the low end. The expensive capacity investment costs per unit are justified for meeting highend demand, but not so for the low-end. However, for the standardized customization extreme, which is represented at the very low M but the higher end of r values in region D and in the whole region F (medium to large r values for relatively small values of M ), the insensitivity of the capacity investment costs per unit to the actual quality levels of the product variants results in increased market coverage across the board to counteract increases in M . Furthermore, apparently in tailored customization the expensive capacity decisions force a very delicate balance between quality levels offered, market coverage, and installed capacity, with these facilities heavily utilized. For standardized customization there is the propensity to increase the quality of the high-end products, while relying on the low-end product quality to moderate capacity costs, and frequently such facilities are operating at lower levels of utilization.

7. Full Information If the firm has full information about the customers’ preferences on design quality, i.e., the firm ˜ it can price the product to extract all surplus from her. In knows every arriving customer’s θ, the single product case, if the market coverage is 1 − θf (every customer with θ˜ ∈ [θf , 1] purchases ˜ the firm charges the price pf (θ) ˜ that makes that customer’s the product), for a customer with θ, f f ˜ ˜ ˜ utility zero, i.e., u(pf (θ), qf , wσ ) = θqf − cˆwσ − pf (θ) = 0. Proceeding analogously as in Section 3 we obtain: √ √ Proposition 13. Underfull information with a single product, when 0≤M ≤ 2 2/3 3,θf∗ = q q q √ 2λ(α+β) 3c 3βc 3βc 1 1 2λ 1 ∗ ˜ = 1 ˜ − 3βc for √ , and p∗ (θ) √ 1/3, qf∗ = α+β − , µ = + − θ f f 3 2λ 3 α+β 3 2λ 2λ 2βλ−3β 3c θ˜ ∈ [1/3, 1]. Let θ∗ , q ∗ and µ∗ be the optimal solution components for partial information (from Section 3). Recall that for that case the market coverage is 1 − θ∗ = 1/3, whereas for full information 1 − θf∗ = 2/3. One can also observe that qf∗ > q ∗ and µ∗f > µ∗ . Hence, with full information the firm provides larger market coverage, sells √a higher quality product and earns higher profits while investing in f more capacity. Since M 1 = 23√23 and M 1 = 3√2 3 , under full information the firm is profitable for a wider range of M values. The comparative statics of qf∗ and µ∗f are similar to those of q ∗ and µ∗ . Consider now the case where the firm sells two differentiated products to two segments of arriving customers. A customer with θ˜ ∈ [θf1 , 1] buys the high quality product with zero utility at the price ˜ = θq ˜ f − cˆwf . Similarly, a customer with θ˜ ∈ [θf , θf ) buys the low quality product with zero pf1 (θ) σ1 1 2 1 ˜ = θq ˜ f − cˆwf . Therefore, the overall market coverage is 1 − θf . Using utility at the price pf2 (θ) σ2 2 2 similar techniques to those in Section 4 we obtain: √

Proposition 14. Under full information with two products in dedicated facilities, if 0 ≤ M < 2272 , the firm’s optimal product positioning decisions θf∗1 and θf∗2 , optimal quality levels qf∗1 and qf∗2 , and optimal capacities µ∗f1 and µ∗f2 are given √ by: (i.) θf∗1 is the unique solution of M = which spans the interval of ( 19 , 15 ).

1−θf (9−18θf +5θf2 ) 1 1 √ 1 6[3−9θf +4 6θf (1−θf )] 1

1

1

in the range ( 13 , 35 ), and θf∗2 (θf∗1 ) = θf∗1 /3,


28

(ii.) 1 qf∗1 (θf∗1 ) = α+β

1 + θf∗1 − 4

s

βc (1 − θf∗1 )λ

!

,

1 qf∗2 (θf∗1 ) = α+β

θf∗1 − 3

s

3βc 2θf∗1 λ

!

.

(iii.) 1 µ∗f1 (θf∗1 , qf∗1 ) = (1 − θf∗1 )λ + ∗ qf1 f

s

c(1 − θf∗1 )λ , β

2θf∗1 1 µ∗f2 (θf∗1 , qf∗2 ) = λ+ ∗ 3 qf2

s

2cλθf∗1 . 3β

√

2 2 . For 0 ≤ M < 27 , one can show that 1 − θf∗1 > Notice that M 2 = 2272 which is greater than M 2 = 27 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 − θˆ1 , θf1 − θf2 > θˆ1 − θ2 , qf1 > q1 , qf2 > q2 , µf1 > µ1 , and µf2 > µ2 . Therefore, with full information the firm produces higher quality products for more profit and offers larger market coverage through higher capacity investments. The comparative statics of the optimal product positioning, quality, and capacity decisions are similar to those of the corresponding partial information case. To illustrate the effect of full information on flexible production systems, we analyze the tailored customization case. Using the optimization techniques introduced in Section 6 to solve the corresponding full information model yields the following results:

Proposition 15. Under √full information, a flexible facility with δ = 1 (tailored customization), ∗ ∗ can be 0 ≤ r < 1, and 0 ≤ M < 3√23 (1 − r), the optimal product positioning decisions θ1f and θ2f uniquely determined by solving 2 1 − θ1f − (3θ1f − θ2f )[(1 − θ1f ) + r(1 − θ2f )] p 2 1 − θ2f 2 3 2 2 2 3 − 3θ1f + 8θ1f − 6θ1f + θ2f − 4θ1f θ2f + 6θ1f θ2f + 4θ2f − 3θ1f θ2f − 3θ2f = 0, ∗ ∈ 31 , 35 . Furthermore, spanning the interval 51 , 13 as θ1f

M=

∗ with θ2f (i.)

qf∗1 (θf∗1 , θf∗2 ) =

q (1 − θf∗1 2 )λ − 4 βc(1 − θf∗2 )λ

4λ[α(1 − θf∗1 ) + β(1 − θf∗2 )]

,

qf∗2 (θf∗1 , θf∗2 ) =

θf∗1 + θf∗2 . 4α

(ii.) µ∗f (θf∗2 , qf∗1 )

=

(1 − θf∗2 )λ +

1 qf∗1

s

c(1 − θf∗2 )λ . β

For every M ∈ (0, 3√1 3 (1 − r)), we can show that under full information, the total market coverage is greater than that of the corresponding partial information model, and that the firm produces higher quality level products and makes larger capacity investments. The comparative statics of θfi and qfi are similar to their counterparts in the partial information case. In short, our results show that full information of customers’ preferences on quality can help the firm make products with higher quality levels, offer a larger market coverage, and earn higher profits. The increase in profitability relative to the partial information case represents the firm’s value of the incremental information on customer preferences.

8. Conclusions Decisions on how many product variants to offer as part of a product line, and how to choose their relative positioning in terms of offered quality and prices are important and challenging decisions for all firms. Frequently such decisions are made with a sales and revenue growth mindset, with


29

increased product variety allowing to better meet heterogeneous consumer preferences in terms of quality, and thus to price in a way that extracts higher rents from them. However, such decisions have serious implications for the operational investments and variable production costs in meeting customer demand. Often, such considerations become afterthoughts of an already-made product line decision, mostly based on market growth and desirable market segmentation issues. The typical outcome of this traditional sequential nature product line approach is that the bottom line implications of product variety are exaggerated on revenue growth and underestimated on capacity investments and production expenses. In this research we provided an integrated marketingoperations decision model that depicts relevant tradeoffs on both the revenue and cost sides of the product variety debate, and offers insights on the factors that determine the right level of vertical differentiation to be offered. We provide useful insights on the relative quality positioning of the various product offerings, the resulting market coverage and segmentation, and the product variety effects on production costs and the operational investments, especially on congestion measures of the supporting processes (utilization, work-in-process inventories, etc.). The novelty of our approach is its ability to integrate, within a tractable model, front-end decisions on product line design (number of products to be offered, design quality for each product, and product positioning—market segment covered and offered price) and back-end process decisions (nature of production process—dedicated lines vs. flexible process, and capacity investment). The use of such an integrated model captures relevant tradeoffs and offers important insights that more narrowly focused either front-end (marketing and economics of product variety) or back-end (process driven queuing) models fail to consider and predict. Even for the simplest of cases, the relevant product design and capacity investment decisions for a single product, our integrated model offers non-trivial and unanticipated comparative statics (e.g., capacity investment may increase with higher marginal investment costs, and as market size increases so does the quality level. However, the capacity investment is non-monotonic in market size). The emphasis of our study is on how firms should evaluate relevant market and production tradeoffs in supporting product line decisions. Our work makes it very clear that the nature of the back-end process (dedicated vs. flexible) has a major impact on the desirable level of product variety as well as how the product line will be designed and positioned (design quality levels offered, market segments covered by specific products, and prices). For multiple products with dedicated facilities our work makes it clear that the decentralized nature of the back-end process does not imply independence of front-end decisions from relevant capacity decisions, and in no way the problem degenerates to a single product case. Even though capacity is dedicated by product, the decisions are coupled by customer preferences and the resulting segmentation in response to offered design quality levels of the various products. The capacity investment decisions are sensitive to design quality levels that drive customer preferences and segmentation, and the resulting coupling of decisions is stronger than previously anticipated. The appealing feature of our results for this setting is that level of product variety to be offered is decided by a single aggregate parameter M (defined in Section 4). As the firm becomes more profitable, new variants are introduced (with the specifics of how the market gets segmented and products positioned described in Section 5), and the total market coverage increases discontinuously with each introduction. However, between new variant introductions the market coverage decreases continuously with increased profitability. When offering multiple products in flexible facilities, our work substantiates the claim that capacity investment costs are heavily dependent on what type of process flexibility exists to support the vertically differentiated product line. We used a novel approach to model a spectrum of flexible production processes from “customized standardization” (narrowed quality range from a wide spectrum by natural variation) to “tailored customization” (expensive customization via multifunctional universal designs and disabling of features later in the process). For these settings optimal product variety is no longer determined by a single profitability parameter, but will also depend on

30


the ratio of the per unit capacity investment to variable production costs and the nature of process flexibility (using a parameter to position it in the previously mentioned spectrum). The tractability of our model allows us to fully characterize even in this case the design quality levels, market positioning, and associated capacity investments for various process flexibility configurations. We specifically compare the profitability of tailored customization and customized standardization with that of the dedicated capacity by using a capacity investment related parameter. Statistical scale-economies and savings in capacity costs are the two main factors that drive the profitability of flexible facilities. However, the statistical scale-economies though they lead to reduced capacity levels may also result in significant congestion cost to customers, which cuts into the firm’s profits. Flexible capacity does not always guarantee a dominant performance when comparing profitability across integrated marketing-operations models with different production system configurations. Full information on customers’ preferences on quality allows the firm to appropriate more value from customers purchasing its product line through personalized pricing. In the presence of such information the firm offers a higher quality product, covers a larger segment of the market, and earns higher profits. Our research is a definite first step in better understanding the optimal product variety and associated capacity investments for a monopolist firm. More research is needed in understanding how different competitive environments, in both number of players and dimensions of competition, affect the above insights. Furthermore, more detailed accounting of combined economies of scale (both fixed costs and congestion phenomena) and economies of scope (non-linear complexity costs of product variety affecting both capacity investment and variable production costs) in flexible facilities will better inform us on economics of production systems supporting product variety in the marketplace.

References Alptekinoglu, A., C. J. Corbett. 2008. Mass customization versus mass production: Variety and price competition. Forthcoming in Manufacturing Service Oper. Management. Benjaafar, S., D. Gupta. 1998. Scope versus focus: Issues of flexibility, capacity, and number of production facilities. IIE Trans. 30(5) 413–425. Choudhary, V., A. Ghose, T. Mukhopadhyay, R. Uday. 2005. Personalized pricing and quality differentiation. Management Sci. 51(7) 1120–1130. Christensen, C. 1997. Hospital Equipment Corporation. Harvard Business School Publishing (HBS # 9-697086), Boston, MA. de Groote, X. 1994. The flexibility of production processes: A general framework. Management Sci. 40(7) 933–945. Desai, P., S. Kekre, S. Radhakrishnan, K. Srinivasan. 2001. Product differentiation and commonality in design: Balancing revenue and cost drivers. Management Sci. 47(1) 37–51. Duenyas, I., W. J. Hopp. 1995. Quoting customer lead times. Management Sci. 41(1) 43–57. Duenyas, I. 1995. Single facility due date selling with multiple customer classes. Management Sci. 41(4) 608–619. Ho, T.-H., C. S. Tang. 1998. Product Variety Management: Research Advances. Kluwer Academic Publishers, Boston, MA. Jiang, K., H. L. Lee, R. W. Seifert. 2006. Satisfying customer preferences via mass customization and mass production. IIE Trans. 38 25–38. Karmarkar, U. S. 1987. Lot sizes, lead times and in-process inventories. Management Sci. 33(3) 409–418. Karmarkar, U. S. 1996. Integrative research in marketing and operations management. J. Marketing Res. 33(2) 125–133.


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Karmarkar, U. S., Kekre, S. 1987. Manufacturing configuration, capacity and mix decisions considering operational costs. J. Manufacturing Sys. 6(4) 315–324. Kekre, S., K. Srinivasan. 1990. Broader product line: A necessity to achieve success? Management Sci. 36(10) 1216–1231. Kim, K., D. Chhajed. 2002. Product design with multiple quality-type attributes. Management Sci. 48(11) 1502–1511. Moorthy, K. S. 1984. Market segmentation, self-selection, and product line design. Marketing Sci. 3(4) 288– 305. Mussa, M., S. Rosen. 1978. Monopoly and product quality. J. Econom. Theory 18 301–317. Netessine, S., T. A. Taylor. 2007. Product line design and production technology. Marketing Sci. 26(1) 101–117. Rockafellar, R. T. 1997. Convex Analysis. Princeton University Press, Princeton, New Jersey. Tang, C. S., R. Yin. 2007. The implications of costs, capacity, and competition on product line selection. Working Paper. UCLA Anderson School of Management, UCLA.


1

Appendix Proof of Theorem 1. When a(q) = αq 2 and b(q) = βq 2 , substitution of µ∗ into (4) yields the following maximization problem p (43) max ΠS (θ, q) = θq − (α + β)q 2 (1 − θ)λ − 2q βc(1 − θ)λ (θ,q)∈R2 +

s.t.

p √ θ 1 − θ ≥ βc/λ θ ≤ 1,

(44) (45)

Given θ, ΠS (θ, q) is concave in q. Using the first order condition ∂ΠS (θ, q)/∂q = 0 yields the optimal quality level s ! p √ βc 1 θ q ∗ (θ) = , where θ 1 − θ > 2 βc/λ, − α+β 2 (1 − θ)λ Substitution of q ∗ (θ) into the firm’s problem (43)–(45), we have p max ΠS (θ, q ∗ (θ)) = [θq ∗ (θ) − (α + β)q ∗ 2 (θ)](1 − θ)λ − 2q ∗ (θ) βc(1 − θ)λ θ θ2 βc θ p = cβ(1 − θ)λ + (1 − θ)λ − 4(α + β) α+β α+β √ 2 λ θ 1−θ−M = 4(α + β) r √ βc s.t. θ 1 − θ > M, where M = 2 . λ √ √ Therefore, θ 1 − θ such that θ 1√− θ > M . √ the problem can be recast as that of maximizing Since θ 1 − θ is strictly concave in θ, the optimal θ∗ can be obtained by letting (θ 1 − θ)0 = √ 1 − θ − 2√θ1−θ = 0, which yields θ∗ = 32 . To ensure positive profits, the condition M < 3√2 3 must be imposed. Finally, the expressions for q ∗ (θ∗ ), µ∗ (θ∗ , q ∗ ), and Π∗S (θ∗ , q ∗ ) follow directly from using θ∗ .

Remark: In the above optimization problem (43)–(45), the profit function (43) is not generally a joint concave function of θ and q. However, the optimization problem can be solved sequentially as follows: Let f (θ, q) denote (43), and let us consider the optimization problem max f (θ, q) subject to θ ∈ [0, 1], q ∈ [0, qmax ]. For any given θ, the function gθ (q) := f (θ, q) is strictly concave in q, and the local optimum qθ∗ = q ∗ (θ) is thus the unique global optimum of gθ (q) (notice that q ∗ (θ) is not a single point but rather a function of θ ), hence given θ ∈ [0, 1] , f (θ, q ∗ (θ)) = gθ (q ∗ (θ)) ≥ gθ (θ) = f (θ, q) for all q ∈ [0, qmax ]. The function f (θ, q ∗ (θ)) , which results from substituting q ∗ (θ) back into the original objective function, is unimodal in its single decision variable θ , hence the local optimum θ∗ is also the unique global optimum of f (θ, q ∗ (θ)). Therefore, f (θ∗ , q ∗ (θ∗ )) ≥ f (θ, q ∗ (θ)) ≥ f (θ, q) for all θ ∈ [0, 1] , q ∈ [0, qmax ]. Proof of Proposition 1. By Theorem 1, it is straightforward to show that d q ∗ /d λ > 0 and d q∗ d q∗ d q∗ , d α , d c < 0. Similar properties can also be derived for p∗ . The properties of Π∗S follow directly dβ √ 2 √ once the optimal profit function is rewritten as Π∗S = λ − 3 3βc /[27(α + β)], Proposition 16. The second order comparative statics of µ∗ are as follows: ( is concave in λ, 27 βc < λ < 243 βc µ∗ : is convex in λ, λ ≥ 243 βc.

2


For α > 0 : Let βˆC =

√ λ−216αc+



(216αc+λ)2 +1323αcλ √ 27 3cλ

s

∆(βˆC ) = 27 λβˆC 

2

, and

 cβˆC 1  +α − √ λ 9 3

! q ˆC λ 8c β √ − 3 cλβˆC + √  . 3 3 27 3

If ∆(βˆC ) ≥ 0, µ∗ is convex in β; if ∆(βˆC ) < 0, there exist ηˆ1 ∈ (0, βˆC ) and ηˆ2 > βˆC such that   0 < β < ηˆ1 is convex in β, ∗ µ : is concave in β, ηˆ1 ≤ β < ηˆ2   is convex in β, β ≥ ηˆ2 . ( λ is concave increasing in c, 0 < c < 243β µ∗ : λ λ is convex increasing in c, ≤ c < 27β . 243β Proof of Propositions 2 and 16. From Theorem 1, √ √ √ √ d µ∗ β λ(2 βλ − 9β 3c) + α(3λ 3c − 54c βλ) √ = . √ √ dλ 6 βλ( βλ − 3β 3c)2 To obtain the derivative’s sign, it suffices to consider the numerator, which can be written as ( " # ) √ √ 9 3 λ2 λ2 √ M 1− M + ω[3 3 − 27M ] = √ ϕ(M ; ω), 4 h h q √ and ω = 27αc . ϕ(M ; ω) is a quadratic function of M , with ϕ(0; ω) = 3 3ω and where M = 2 βc λ λ √ ϕ 3√2 3 ; ω = −3 3ω − 3√1 3 < 0. The unique positive root of ϕ(M ; ω) = 0 is Mr (ω) := 9√2 3 [1 − ω + √ 1 + ω + ω 2 ]. Mr (ω) decreases in ω since Mr0 (ω) < 0 and then Mr (ω) ≤ 9√4 3 < 3√2 3 . Therefore, M decreases in λ while Mr (ω) increases in λ. For 0 ≤ M < 3√2 3 , there exists a unique ˆ S such that M = Mr (ω) where λ = λ ˆ S . But λ  ˆ  > 0, for λ > λS ˆS ϕ(M ; ω) : = 0, for λ = λ   ˆS . < 0, for λ < λ The second-order derivative of µ∗ with respect to λ is √ √ √ d2 µ∗ β(α + β) c( 3βλ − 27β c) √ , = √ √ d λ2 6 βλ( βλ − 3β 3c)3 which vanishes when λ = 243βc. Since M < 3√2 3 , we need λ > 27βc. Similarly, i √ h 2 √9 M − 1 λ 3c β + α ∗ dµ 3 √ = . √ √ dβ 2β βλ( λ − 3 3βc)2 ∗ When α = 0, ddµβ > 0. If α > 0, when M ≥ 3√1 3 1 − αβ , αβ + √93 M − 1 ≥ 0. Therefore, if α = 0 or √ ∗ ˆ 6 3hβˆ β ≥ α, ddµβ > 0. If α > 0 and β < α, there exists a unique βˆS such that βαS + √λ S − 1 = 0, where


3

√ √ 2 ∗ ∗ 27αc + λ − 3 3αc . Then, ddµβ > 0 when β > βˆS ; for β = βˆS , ddµβ = 0; and, when β < βˆS , βˆS = αλ d µ∗ < 0. dβ The second-order derivative of µ∗ with respect to β is q √ √ √ √ √ cβ 2 λ cβλ λβ 27 λ − 3 + 3α( 3λ − 27 cβλ + 72 3cβ) d2 µ∗ √ = . √ d β2 4(βλ)3/2 ( βλ − 3β 3c)3 √ 2 ∗ λ . If α > 0, let x = β, and When α = 0, dd βµ2 = 0 when β = 243c r √ √ √ √ c 2 φ(x) = λx 27 x − 3 + 3α( 3λ − 27 cλx + 72 3cx2 ). λ √ √ √ √ p When β = 0, φ(x) = 3 3αλ > 0. φ√0 (x) = λx 81 λc x − 2 3 + α(432 3cx − 81 cλ). The positive λ−216αc+ (216αc+λ)2 +1323αcλ √ root of φ0 (x) = 0 is x∗ = . φ0 (x) < 0 as 0 < x < x∗ , and φ0 (x) > 0 when 27 3cλ q 2 ∗ 2 ∗ x > x∗ . Let βˆC = x∗ . Then, the minimum of dd βµ2 is attained at βˆC . dd βµ2 decreases in β as 2 ∗ 2 ∗ 0 < β < βˆC , and d µ2 increases when β > βˆC . Denote ∆(βˆC ) = φ(x∗ ). Therefore, d µ2 > 0 for all β dβ

dβ

2 ∗ if ∆(βˆC ) > 0. When ∆(βˆC ) < 0, there are two positive roots of dd βµ2 = 0 denoted by ηˆ1 and ηˆ2 , in which 0 < ηˆ1 < βˆC and ηˆ2 > βˆC . √ ∗ ∗ 2 ∗ λ 3c√ It is straightforward to show that ddµα , ddµc > 0. The result follows from dd αµ2 = √βλ−3β and 3c

d2 µ∗ d c2

=

√ √ λβ(α+β)(27 cβλ− 3λ) √ √ . 3/2 4c ( βλ−3β 3c)3

Proof of Proposition 3. Let C denote the operations costs, defined as the sum of congestion and capacity investment costs. Then, p 1 2 p 1 ∗ ∗ 2 2 C = 2q βcλ/3 + β(q ) λ/3 = βλ + √ α βcλ − β c + 2αβc . (α + β)2 27 3 3 # " √ λ(α − β) α cλ(α − 3β) dC 1 √ + 2αc(α − β) = + dβ (α + β)3 27 3 3β When α = 0 or β ≥ α, d C/d β < 0.

The following corollary includes the complete comparative statics of ρ∗ . Corollary 3. (i.) The optimal utilization level ρ∗ is concave increasing in λ and convex decreasing in both α and c; (ii.) If α = 0 or β ≥ α, ρ∗ decreases in β. If α > 0 and β < α, there exists a √ √ 2 unique βˆS = αλ 27αc + λ − 3 3αc such that ( increases in β, for β ≤ βˆS ; ∗ ρ : decreases in β, for β > βˆS . ˆ Furthermore, ρ∗ isqconvex decreasing √ in β when α = 0. If α > 0, there exists a unique βA satisfying √ √ ˆ ˆ ˆ ˆ ˆ 9β c− 3βA λ+ βA (81βA c+3λ−14 3cβA λ) √ α= A . Therefore, 6 c ( is convex in β, when 0 ≤ β < βÂ ; ρ∗ : is concave in β, when β ≥ βÂ .

4


Proof of Corollary 3. Since θ∗ = 23 , ρ∗ = 3µλ∗ . It follows from Proposition 2 that ρ∗ is decreasing in α and c. It is straightforward to show that √ d ρ∗ 3 3cβ(α + β) √ = √ > 0, √ dλ 2 βλ(3 3c + βλ)2 so ρ∗ increases in λ. The proof of (ii.) follows directly from the properties of d µ∗ /d β. The second order derivatives of utilization level ρ∗ follow from Theorem 1: √ q βc √ 54c βλ 1 − 3 3 λ d2 ρ∗ √ = >0 √ d α2 (3α √3c + √βλ)3 √ d2 ρ∗ 3(α + β) βλ( 3βλ + 27α c) √ >0 = √ 3 d c2 4h3/2 (3α√ 3c +√ βλ)√ d2 ρ∗ 9β 2 (α + β) c(3α c + 3βλ) √ = − 0. If α > 0, let β βλ − 3α( βλ + α 3c − 3β 3c) = 0, which implies d β2 √ √ √ √ p √ √ √ 9β c− 3βλ± β(81βc+3λ−14 3cβλ) √ α= . Since 9β c − 3βλ − β(81βc + 3λ − 14 3cβλ) 6 c q √ √ √ 9βÂ c− 3βÂ λ+ βÂ (81βÂ c+3λ−14 3cβÂ λ) ˆ √ exists a unique βA satisfying α = . 6 c

If α = 0,

< 0, there

We now consider the case in which both capacity investment and variable production costs are general power functions of the product quality level, i.e., b(q) = βq γ and a(q) = αq γ , where γ > 2 and α, β > 0. An increase in γ models a production technology that becomes more expensive in both capacity investment and production costs for a given quality level. By applying the fact that p = θq − c/(µ − λe ), the firm’s profit function can be written as max ΠG (θ, q, µ) = (θq − αq γ )λe − θ,q,µ

cλe − βq γ µ. µ − λe

(46)

We obtain the optimal product positioning θ∗ and the optimal quality level q ∗ using the same solution approach as for the quadratic cost functions case. Theorem 4. For product offering, let b(q) = βq γ and a(q) = αq γ where α, β > 0 and γ > 2. h i q a single γ−1 1 For 0 ≤ M < 2 2γ−1 − (α + β) , the firm’s optimal product positioning decision is given by 2γ−1 γ ∗ θ = 2γ−1 . The optimal quality level q ∗ is the solution to [θ∗ − γ(α + β)q γ−1 ](1 − θ∗ )λ − γq γ/2−1

p cβ(1 − θ∗ )λ = 0,

(47)

q ∗ )λ and the optimal capacity and system’s utilization are given by µ∗ = (1 − θ∗ )λ + (q∗ )1γ/2 c(1−θ and β −1 q c ρ∗ = 1 + (q∗ )1γ/2 β(1−θ respectively. ∗ )λ h i q γ−1 1 When M ≥ 2 2γ−1 − (α + β) , the firm’s profit is non-positive and the optimal choice is 2γ−1 not to produce any product.


1 Proof of Theorem 4. Given θ and q, the optimal capacity is µ∗ (θ, q) = (1 − θ)λ + qγ/2 We can rewrite the firm’s problem as p max ΠG (θ, q) = θq − (α + β)q γ (1 − θ)λ − 2q γ/2 cβ(1 − θ)λ.

5

q

c(1−θ)λ β

.

θ,q

The profit function ΠG (θ, q) is strictly concave in q for every θ. Then, the optimal quality level q ∗ (θ) in terms of θ is given by the solution of p {θ − γ(α + β)[q ∗ (θ)]γ−1 }(1 − θ)λ − γ[q ∗ (θ)]γ/2−1 cβ(1 − θ)λ = 0. (48) Since q ≥ qmin = 1, to ensure that (48) yields a solution with a positive quality level, we need γ p ∗ ∗ )λ γ−1 2 −1 qmin cβ(1 − θ∗ )λ + (α + β)(1 − θ∗ )λqmin < θ (1−θ . Therefore, the profitability condition becomes γ " # r γ γ −1 1 2 . M 0, ΠG (θ, q ∗ (θ)) increases in θ when θ ∈ (0, θ∗ ), and ΠG (θ, q ∗ (θ)) decreases in ∗ ∗ θ when θ ∈ (θ , 1). By direct substitution ΠG (θ, q (θ)) θ=0 = 0 and ΠG (θ, q ∗ (θ)) θ=1 = 0. Therefore ΠG (θ, q ∗ (θ)) is unimodal in θ. The maximum is attained when θ = θ∗ . The optimal quality level q ∗ follows from solving the first-order optimality condition (47). Notice that Theorem 4 holds for any α, β > 0. Similarly to the quadratic cost functions case, the optimal product positioning decision θ∗ is constant and depends only on γ. It is straightforward to show that θ∗ > 1/2 for all γ > 2 and θ∗ → 1/2 as γ → ∞. Hence, the firm responds to higher capacity costs by expanding the market coverage. The comparative statics of the optimal solution are similar to the case of quadratic cost functions. From the optimal product positioning θ∗ and the optimal quality level q ∗ in the paper, we derive the following properties of the optimal solution for the single product case with general power cost functions. Proposition 17. (i.) The optimal quality level q ∗ is increasing in λ and decreasing in α, β, and c. (ii.) The optimal capacity level µ∗ is increasing in α and c. ∗ (iii.) µ(r is decreasing in β if β < βˆG and increasing) in β otherwise, where βˆG = 2 γ−2 h i 2(γ−1) γ 2(γ−1) √ √ γ 4αλ(γ−1)2 (2γ−1) 1 c αγ(2γ−1) + 2γ−1 − c αγ(2γ − 1) . (iv.) Let G (λ; q) = q 2 −1 cβ + 2λ(γ−1) √ q γ−1 √ q γ−1 1 γ−1 ˆ G and qˆ which βq γ−1 λ 2γ−1 and H(q) = λ 2γ−1 − αq . There exist a unique pair of λ 2γ−1 q q √ q γ 3 γ−2 ˆ G ; qˆ) = H(ˆ ˆ G + (α + β)ˆ ˆ G + (α + β)ˆ solve both G (λ q ) and 2(2γ−1) qˆ 2 β λ q 2 γ−1 β λ q γ−1 c 2γ−1 = γ−1


6

√ γ

√

c

(γ−1)(2γ−1)

ˆ G and increasing in λ otherwise. (v.) The , such that µ∗ is decreasing in λ if λ < λ

optimal utilization level ρ∗ is increasing in λ and decreasing in α and c. In addition, ρ∗ is increasing ˆ G and decreasing in λ otherwise. in λ if λ < λ h i q γ−1 1 ∗ γ−1 − (α + β)(q ) = Proof of Proposition 17. The optimal quality level q ∗ is the root of 2γ−1 2γ−1 q γ (q ∗ ) 2 −1 cβ . λ i √ q γ−1 h 1 √ γ γ−1 − (α + β)q Define G(λ, α, β, c; q) = λ 2γ−1 − q 2 −1 cβ. Clearly, G(λ, α, β, c; 0) > 0. 2γ−1 For any λ1 > λ2 > 0, let q1 and q2 be roots of G(λ1 , α, β, c; q) = 0 and G(λ2 , α, β, c; q) = 0 respectively. 1 For any q > 0 such that 2γ−1 − (α + β)q γ−1 > 0, it follows that G(λ1 , α, β, c; q) > G(λ2 , α, β, c; q), and thus q1 > q2 . Therefore, the optimal quality level q ∗ is increasing in λ. It is straightforward to show that q ∗ is decreasing in α, β, and c by showing that d G(α) , d G(β) , d G(c) < 0. dα dβ hd c i q √ γ−1 1 ∗ γ−1 The optimal quality level q ∗ satisfies F (λ, α, β, c; q ∗ ) = λ 2γ−1 − (α + β)(q ) − 2γ−1 √ ∗ γ d q (c) ∂ F ∂ F (q ∗ ) 2 −1 cβ = 0, and by the implicit function theorem d c = − ∂ c / ∂ q∗ . The optimal capacity is q ∗ ∗ ∗ ∗ ∗ cλ(γ−1) ) ) d q ∗ (c) 1 + , therefore ddµc = ∂ µ (λ,β,c;q + ∂ µ (λ,β,c;q > 0, so µ∗ is µ∗ (λ, β, c; q ∗ ) = λ(γ−1) γ 2γ−1 β(2γ−1) ∂c ∂ q∗ dc ∗ (q ) 2

∗

increasing in c. Similarly, we show that µ∗ is increasing in α by showing that d µd α(α) > 0. In contrast to the above results, µ∗ does not change monotonically in β. We have i √ h 1 ∗ γ−1 λ(γ − 1) c − αγ(q ) ∗ 2γ−1 dµ q = h √ i. √ γ dβ 2(2γ − 1)(q ∗ )γ−1 β β (γ − 1)(α + β)(q ∗ ) 2 λ(γ−1) + γ − 1 cβ 2γ−1

2

p The first order condition [θ∗ − γ(αq+ β)q γ−1 ](1 − θ∗ )λ −γq γ/2−1 cβ(1 − θ∗ )λ = 0 can be written as q √ √ √ γ γ−1 γ−1 1 q 2 −1 cβ + βq γ−1 λ 2γ−1 = λ 2γ−1 − αq γ−1 . 2γ−1 q √ √ q γ−1 1 √ γ γ−1 γ−1 Let G (β; q) = q 2 −1 cβ + βq γ−1 λ 2γ−1 and H(q) = λ 2γ−1 − αq . It is straight2γ−1 forward to show that t G (β; q) is increasing in β, G (β; 0) = 0, H(q) is decreasing in q, and H(0) = √ q γ−1 λ 2γ−1 > 0. 1 h i γ−1 1 1 − αγq1γ−1 = 0. Then, q0 = α(2γ−1) Define q0 and q1 such that H(q0 ) = 0 and 2γ−1 , q1 = 1 h i γ−1 1 , and q0 > q1 . αγ(2γ−1) ∗ Let q be the root of G (β; q ∗ ) = H(q ∗ ). It follows that q ∗ (β) is decreasing in β and q ∗ ∈ (0, q0 ). ∗ ˆ ˆ Moreover, → 0 and q ∗ → 0 as β → ∞. Let βˆG solve (rq → q0 as βγ−2 )2 G (βG ; q1 ) = H(q1 ), then βG = h i 2(γ−1) γ 2(γ−1) √ 4αλ(γ−1)2 (2γ−1) 1 c + − c αγ(2γ − 1) . Therefore, d µ∗ /d β is negative 2λ(γ−1) αγ(2γ−1) 2γ−1

when β < βˆG , zero when β = βˆG and positive when β > βˆG . Similarly, √ √ √ √ q 2γ−1 γ 3 γ−1 c γ−2 γ−1 2 + (α + β) 2 √ (γ − 1) βλq βλq + c (α + β)q − 2(2γ−1) γ−1 γ (γ−1)(2γ−1) d µ∗ (λ) h i = . γ√ dλ (2γ − 1)q 2 βλ γ−2 + (α + β)q γ−1 2(2γ−1)

q γ γ−2 2 ˆ G and q2 which solve both G (λ ˆ G ; q2 ) = H(q2 ) and ˆ G + (α + There exist λ q βλ 2(2γ−1) 2 q q √ 3 γ−1 √ c ˆ G + (α + β)q2γ−1 c 2γ−1 = √ β)q22 βλ , and it follows that d µ∗ /d λ is negaγ−1 γ

(γ−1)(2γ−1)

ˆG, λ = λ ˆ G , and λ > λ ˆ G respectively. Notice that ρ∗ = tive, zero, and positive when λ < λ


h

q

c(2γ−1) βλ(γ−1)

7

i−1

. Since q ∗ is increasing in λ and decreasing in α and c, ρ∗ is increasing in λ q h i−1 ∗ h(2γ−1) γ−1 1 λ and decreasing in α and h. Finally, from ddµβ = d A(β) , where A (β) = , dβ 2γ−1 βλ(γ−1) (q ∗ )γ/2 ∗ ∗ the comparative statics for ρ follow from those of µ . 1 + (q∗ )1γ/2

The following result provides further understanding of the effect of γ on the firm’s optimal choices. Proposition 18. With general power cost functions, for any γ > 2, both the optimal design quality q ∗ and optimal utilization ρ∗ are decreasing in γ, while the optimal capacity µ∗ is increasing in γ. Proof of Proposition 18. Applying the implicit function theorem to (48), γ−1

γ−1

(α+β)q (α+β)q 4γ−5 d q ∗ (γ) − 2(γ−1)(2γ−1)2 − 2(γ−1)(2γ−1) − 2 = γ−2 (3γ−2) (α+β)q γ−2 dγ + 4

log q

log q − 2(2γ−1)

,

4(γ−1)q

and since q ≥ 1, d q ∗ (γ)/d γ < 0. h i q p γ−1 γ−1 + (q∗D)γ/2 2γ−1 Let D = cλ/β. From µ∗ = λ 2γ−1 r r d µ∗ (γ) 1 D γ −1 D γ −1 γ d q ∗ (γ) ∗ = + − log q + ∗ . dγ (2γ − 1)2 2(q ∗ )γ/2 (2γ − 1)2 2γ − 1 2(q ∗ )γ/2 2γ − 1 q dγ Straightforward arguments lead to d µ∗ (γ)/d γ > 0. q q h i−1 c(2γ−1) c(2γ−1) Notice that ρ∗ = 1 + (q∗ )1γ/2 βλ(γ−1) . Let B (γ) = (q∗ )1γ/2 βλ(γ−1) . It is straightforward to ∗ show that d B (γ)/d γ > 0, so B (γ) increases in γ, which implies ρ is decreasing in γ. Thus, as γ increases, the firm expands market coverage and lowers the quality level, thus reducing capacity costs, which leads it to increase capacity and thereby lessen congestion costs. Our analysis with generalized power cost functions shows that expensive technologies in both capacity investment and variable production costs favor higher quality products and the careful targeting of the high-end market segments. This result carries through from the single product to diverse multi-product product lines. For the two differentiated products in dedicated facilities, the following details are needed for the transition from model (14)–(17) to model (18)–(21). Given θˆ1 , θ2 , and quality levels qi , i = 1, 2, the firm’s optimal capacity decisions are s s ˆ1 )λ 1 c(1 − θ 1 c(θˆ1 − θ2 )λ µ∗1 (θˆ1 , q1 ) = (1 − θˆ1 )λ + , and µ∗2 (θˆ1 , θ2 , q2 ) = (θˆ1 − θ2 )λ + . (49) q1 β q2 β Upon substitution of the optimal production capacities, profit function (7) becomes 2 ˆ1 )λ + (θˆ1 + θ2 − 1)q2 − (α + β)q 2 (θˆ1 − θ2 )λ (α + β)q ΠD (θˆ1 , θ2 , q1 , q2 ) = θˆ1 q1 − (1 − θ 1 2 q q − 2q1 βc(1 − θˆ1 )λ − 2q2 βc(θˆ1 − θ2 )λ.

(50)

It is straightforward to show that ΠD (θˆ1 , θ2 , q1 , q2 ) is jointly concave in q1 and q2 . The optimal quality levels q1∗ (θˆ1 ) and q2∗ (θˆ1 , θ2 ) are given by s s ˆ1 ˆ1 + θ2 − 1 θ 1 βc θ 1 βc q1∗ (θˆ1 ) = − , and q2∗ (θˆ1 , θ2 ) = − . (51) 2(α + β) α + β (1 − θˆ1 )λ 2(α + β) α + β (θˆ1 − θ2 )λ

8


For any feasible θˆ1 , because of constraint (20) it follows from (18) that 1 + θˆ1 , θ2∗ (θˆ1 ) = 3 and the maximization problem can be further restated as ( q 2 2 ) 2 3 max ΠD (θˆ1 ) = A θˆ1 1 − θˆ1 − M + √ (2θˆ1 − 1) 2 − M θˆ1 3 3 q √ q ˆ ˆ ˆ s.t. 2 − θ1 > 3M 1/ 1 − θ1 − 3/ 2θ1 − 1 2 3 √ (2θˆ1 − 1) 2 > M 3 3 1 ˆ < θ1 < 1. 2

(52)

(53) (54) (55) (56)

ˆ∗ Proof of Theorem 2. The proof of the existence and quniqueness of θ1 can be found in the proof of Proposition 4. That proof relies on the condition θˆ1 1 − θˆ1 > M corresponding to q1 > 0 instead of the more restrictive (54), which corresponds to q1 > q2 . Since θˆ1 < 4/5 implies the right-handside of (54) is negative, (54) holds for any M > 0, and the sought-after solution coincides with the relaxed problem’s solution. The expression for θ2∗ follows from (52). Given θˆ1∗ and θ2∗ , we derive the optimal quality levels from (51), the optimal capacity choices from (49), and then optimal prices. Proof of Proposition 4. Before proving the proposition, we show the following results. Lemma 1. Both θˆ1 and θ2 must be greater than 1/2. Proof. Since θˆ1 + θ2 > 1 and θˆ1 > θ2 , both θˆ1 and θ2 greater than 1/2. √ Lemma 2. Let ξ1 and ξ2 be the solutions of x 1 − x = M such that ξ1 , ξ2 ∈ (0, 1) and ξ1 < ξ2 . The profit function ΠD (θˆ1 , θ2∗ (θˆ1 )), denoted by ΠD (θˆ1 ), is unimodal in θˆ1 when θˆ1 ∈ ( 23 , ξ2 ). The optimal θˆ1∗ ∈ ( 23 , 45 ). q 2 i2 h 3 Proof. Let f1 (θˆ1 ) = θˆ1 1 − θˆ1 − M and f2 (θˆ1 ) = 3√2 3 (2θˆ1 − 1) 2 − M . When x ∈ (0, 1), x∗ = 0 ∗ arg max f1 (x) = 32 , x∗ ∈ (ξ1 , ξ2 ), and 0f2 (x) > 0. It is straightforward to verify that ΠD (x) < ΠD (x ) 2 2 ∗ for any x < x . When x ∈ 3 , 1 , f1 (x) < 0. As x → ξ2 , ΠD (x) → f2 (ξ2 ) < ΠD 3 . Therefore, the optimal θˆ1∗ = arg max ΠD (θˆ1 ) > 23 and θˆ1∗ ∈ 23 , ξ2 , and    q q   ˆ ˆ d ΠD (θ1 ) 1 ˆ2 θ1 2 = 2A (5θ1 − 14θˆ1 + 8) − M  1 − θˆ1 − q +√ 2θˆ1 − 1 .  18  3 d θˆ1 2 1 − θˆ1 √ √ 1 Let ϕ1 (x) = 18 (5x2 − 14x + 8) and ϕ2 (x) = M 1 − x − 2√x1−x + √23 2x − 1 . When x ∈ 32 , 1 , ϕ01 (x) < 0 and ϕ1 54 = 0. Since ϕ001 (x) > 0, ϕ1 (x) is a convex decreasing function. We have ϕ02 (x) < 0 and ϕ002 (x) < 0 for x ∈ 32 , 1 , and thus ϕ2 (x) is a concave decreasing function. 4 1 We also have ϕ1 23 =√18 , ϕ1 (1) = − 18 , ϕ2 23 = 23 M , and ϕ2 (x) → −∞ as x → 1. For ξ3 ∈ 23 , 1 7 and ϕ2 (ξ3 ) = 0, ξ3 = 42+4 ≈ 0.8912. Then, ΠD (x) is unimodal on 32 , ξ3 . The maximum is attained 59 at θˆ1∗ such that d ΠdDx(x) x=θˆ∗ = 0, and θˆ1∗ ∈ 32 , 45 . 1


9

By (52), we have θ2∗ ∈ 95 , 35 . When M = 0, Π0D (x) = ϕ1 (x) and ΠD (x) is unimodal on Therefore, θˆ1∗ = 54 and θ2∗ = 35 . We have h i   q q ∂ d ΠD (θˆ1 )/d θˆ1 ˆ1 2 θ 2A − 1 − θˆ1 − √ = 2A  q 2θˆ1 − 1 = − ϕ2 (θˆ1 ). ∂M M 3 2 1 − θˆ1

1 ,1 2

.

ˆ By Lemma 1, ϕ2 (x) > 0 when x ∈ 32 , 45 . Then, d ΠdDθˆ(θ1 ) decreases in M . θˆ1∗ decreases in M . 1 q θˆ1∗ βc 1 1 M ∗ ˆ∗ ∗ ˆ √ . Using We have q1 (θ1 ) = α+β 2 − (1−θˆ∗ )λ = 2(α+β) θ1 − 1−θˆ1∗ 1 √ q ˆ∗ −1 5θˆ12 −14θˆ1 +8) 1−θˆ1 ( 2 θ 3βc 1 ∗ ∗ ∗ 1 √ M (θˆ1 ) = , we show d q1 (θˆ1 )/d θˆ1 > 0. But, q2 (θˆ1 ) = α+β − (2θˆ∗ −1)λ . 3 ˆ ˆ ˆ 18−27θ1 +12

3(1−θ1 )(2θ1 −1)

1

q2∗ (θˆ1 )

It is straightforward to show that increases in θˆ1 . ∗ ∗ ˆ Therefore, θ1 and θ2 increase in λ, and decrease in β and c. They do not change with α. By (51), q1∗ and q2∗ decrease in α. For fixed α and β, when M increases θˆ1∗ , θˆ1∗ − θ2∗ , and θˆ1∗ + θ2∗ − 1 decrease. Therefore q1∗ and q2∗ increase in λ and decrease in c. To understand the effect of β on qi∗ , we use the first derivative of qi∗ with respect to β. # " s s d q1∗ (β) βc c 1 1 θˆ1 − =− − ˆ dβ (α + β)2 2 2(α + β) (1 − θ1 )λ β(1 − θˆ1 )λ # " s βc d θˆ1 (β) 1 1 1 + − . α + β 2 2(1 − θˆ1 ) (1 − θˆ1 )λ dβ q 2/27 √ 4 > 0, it follows that d q1∗ (β)/d β < 0. Since d θˆ1 (β)/d β < 0 and 21 − 2(1−1 θˆ ) (1−βcθˆ )λ > 12 − 4 1

1

4(1− 5 )

1− 5

Therefore, q1∗ decreases in β. Similarly, we show that d q2∗ (β)/d β < 0, so q2∗ decreases in β. q βc ∗ ∗ ∗ ∗ ˆ∗ The comparative statics of p1 and p2 can be derived as follows. We have p1 = q1 θ1 − (1−θˆ∗ )λ − 1

(θˆ1∗ − θ2∗ )q2∗ . Then,     √ ∗ ∗ ∗ ˆ ˆ ˆ 1  θ1 M  θˆ1∗ − qM  − 2θ1 − 1  2θ1 − 1 − q 3M  . p∗1 (θˆ1∗ ) = − q α+β 2 3 3 2 1 − θˆ1∗ 2 1 − θˆ1∗ 2 2θˆ1∗ − 1 q √ √ 2θˆ1∗ −1 1+θˆ1∗ βc 3M 3M 1 ∗ ∗ ∗ ∗ ˆ∗ √ √ And, p2 = q2 θ2 − (θˆ∗ −θ∗ )λ . Therefore, p2 (θ1 ) = α+β − − . 3 3 ˆ∗ ˆ∗ 

1

∗ Apparently, both √ p1 2 ˆ ˆ ˆ (5θ1 −14θ1 +8) 1−θ1

√

18−27θˆ1 +12

2

2

2θ1 −1

2

2θ1 −1

and decrease in α. Let = B(θˆ1 , β)/(α + β). By applying M (θˆ1 ) = , one can show that ∂B(θˆ1 , β)/∂ θˆ1 > 0, which implies p∗1 increase in θˆ1∗ and p∗2

p∗1

3(1−θˆ1 )(2θˆ1 −1)

λ, and decrease in c. One can derive " # dp∗1 (β) B(θˆ1 , β) 1 ∂B(θˆ1 , β) ∂B(θˆ1 , β) dθˆ1 =− + + . dβ (α + β)2 α + β ∂β dβ ∂ θˆ1

Because ∂B(θˆ1 , β)/∂β < 0, we have dp∗1 (β)/dβ < 0, i.e., p∗1 decreases in β. By similar approach, one can show that p∗2 increases in θˆ1∗ and λ, and decrease in β and c. As θˆ1∗ spans ( 32 , 45 ), we have p∗1 (θˆ1∗ ) − p∗2 (θˆ1∗ ) > 0. Proof of Corollary 1. Both results follow from θ2∗ = (1 + θˆ1∗ )/3 and θˆ1∗ decreasing in M .


10

The complete comparative statics of µ∗1 and µ∗2 are derived from the properties of the optimal solution. Proposition 5 provides the first order comparative statics. We summarize the second order comparative statics in the following proposition. In order to characterize the comparative statics, we need to define the following pure functions of θˆ1 . Let d µ∗2 /d c = αfc2 (θˆ1 ) + βgc2 (θˆ1 ), d µ∗1 /d λ = fλ1 (θˆ1 ) + αh g (θˆ ), d µ∗1 /d β = h[fβ1 (θˆ1 ) + αβ gβ1 (θˆ1 )], and d µ∗2 /d β = h[fβ2 (θˆ1 ) + αβ gβ2 (θˆ1 )]. λ λ1 1 0 0 0 0 We also define rλ1 (θˆ1 ) = −M 2 (θˆ1 )gλ1 (θˆ1 )/fλ1 (θˆ1 ), rβ1 (θˆ1 ) = −M 2 (θˆ1 )fβ1 (θˆ1 )/gβ1 (θˆ1 ), and rβ2 (θˆ1 ) = 2 ˆ 0 0 ˆ ˆ −M (θ1 )fβ2 (θ1 )/gβ2 (θ1 ), where √ (5x2 − 14x + 8) 1 − x p M (x) = . 18 − 27x + 12 3(1 − x)(2x − 1) Proposition 19. (i.) Both optimal capacities µ∗1 and µ∗2 are linearly increasing in α and c. µ∗1 is concave increasing in c, and there exists a unique cc which solves fc2 (θˆ1 ) + αβ gc2 (θˆ1 ) = 0. Then, ( is concave increasing in c, 0 < c < cc , µ∗2 : is convex increasing in c, c ≥ cc , where cc is the unique c solving fc2 (θˆ1 ) + αβ gc2 (θˆ1 ) = 0. (ii.)   is concave in λ, 0 < λ ≤ λmin ∗ µ1 : is convex in λ, λmin < λ ≤ λmax   is concave in λ, λ > λmax , p 0 g 0 (ξ) = 0 and M (ξ) = 2 βc/λ where (ξ1 , λmin ) and (ξ2 , λmax ) are unique pairs solving fλ1 (ξ) + αh λ λ1 such that ξ1 < ξ2 and λmin < λmax . (iii.) µ∗1 is convex in β if 4 αc/λ ≤ rβ1 (4/5) and concave in β if 4 αc/λ ≥ rβ1 (2/3). For rβ1 (4/5) < 4 αc/λ < rβ1 (2/3) ( is concave in β, 0 ≤ β < βˆC1 µ∗1 : is convex in β, β ≥ βˆC1 , 1 1 1 ˆ1 where q βC is the unique value corresponding to the unique ξC such that rβ1 (ξC ) = 0 and M (ξC ) = 2 cβˆC1 /λ. Similarly, µ∗2 is concave in β if 4 αc/λ ≥ rβ2 (4/5), otherwise ( is convex in β, 0 ≤ β < βˆC2 ∗ µ2 : is concave in β, β ≥ βˆC2 , 2 2 2 ˆ2 where q βC is the unique value corresponding to the unique ξC such that rβ2 (ξC ) = 0 and M (ξC ) = 2 cβˆ2 /λ. C

Proof of Propositions 5 and 19. From Proposition 4, neither of θˆ1∗ and θ2∗ change with α, and both q1∗ and q2∗ decrease in α. Therefore both µ∗1 and µ∗2 increase in α, and µ∗1 increases in c because θˆ1∗ and q1∗ decrease in c. Let q 5θˆ12 − 14θˆ1 + 8 1 − θˆ1 q M (θˆ1 ) = and ˆ ˆ ˆ 18 − 27θ1 + 12 3(1 − θ1 )(2θ1 − 1) q q θˆ1 2 ˆ √ √ 1 − θ1 − + 3 2θˆ1 − 1 2 1−θˆ1 ˆ . W (θ 1 ) = 5θˆ1 −7 4−3θˆ1 2 ˆ √ √ − M (θ1 ) − 9 ˆ ˆ ˆ 3(2θ1 −1)

4(1−θ1 )

1−θ1


11

Applying the implicit function theorem q to the first order condition q d ΠD (θˆ1 )/d θˆ1 = 0, we get q β c d θˆ1 (λ)/d λ = − λ1 βc W (θˆ1 ), d θˆ1 (c)/d c = cλ W (θˆ1 ), and d θˆ1 (β)/d β = βλ W (θˆ1 ). λ r

From

µ∗2 (θˆ1 , c)

=

(2θˆ1 −1)λ 3

+

(α+β) 2θˆ1 −1 − 3

c(2θˆ1 −1)λ 3β

r

3βc (2θˆ1 −1)λ

we obtain 

q ˆ ˆ ∗ ˆ ∗ ˆ ∗ ˆ  (2θ1 − 1) 2θˆ1 − 1 W (θ1 ) d µ2 (c) ∂ µ2 (θ1 , c) ∂ µ2 (θ1 , c) d θ1 (c)  − = + = α √ dc ∂c dc ∂ θˆ1 3 3M (θˆ1 )V (θˆ1 )    

q ˆ (2θˆ1 − 1) 2θˆ1 − 1 W (θ1 ) +β + √ ˆ1 )V (θˆ1 )  3 3M ( θ  

2θˆ1 −1 3

√

ˆ

4(2θˆ1 −1)2 27M (θˆ1 )

√ −

V (θˆ1 )

5

2θˆ1 −1 √ 3 3

   

M (θˆ1 ) 2θˆ1 −1

√

+

V (θˆ1 )

2θˆ1 −1 √ 3 3

   

,

  

2

. Therefore, d µ∗2 (c)/d c can be written as a pure function of θˆ1 . 2 2θ1 −1 h i Let d µ∗2 (c)/d c = α fc2 (θˆ1 ) + αβ gc2 (θˆ1 ) . For any θˆ1 ∈ 32 , 45 , fc2 (θˆ1 ) > 0 and gc2 (θˆ1 ) > 0. Therefore µ∗2 increases in c. i h 0 0 ˆ 0 ˆ 0 (θˆ1 ). (θ1 )/gc2 (θ1 ) + αβ gc2 (θˆ1 ) (d θˆ1 (c)/d c). We define rc2 (θˆ1 ) = −fc2 In addition, d2 µ∗2 (c)/d c2 = α fc2 There exists a unique θˆ1c such that rc2 (θˆ1 ) + β/α = 0. When θˆ1 < θˆ1c , rc2 (θˆ1 ) + β/α < 0; if θˆ1 > θˆ1c , rc2 (θˆ1 ) + β/α > 0. Since θˆ1c corresponds to a unique cc and d θˆ1 (c)/d c < 0 it follows that µ2 is concave in c when c < cc , and convex in c if c > cc . Similarly, we √ show that d µ∗1 (c)/d c can be written as a pure function of θˆ1 . From µ∗1 (θˆ1 , c) = (α+β) c(1−θˆ1 )λ/β (1 − θˆ1 )λ + ˆ r we obtain where V (θˆ1 ) =

θ1 2 −

− √3Mˆ(θ1 )

βc (1−θˆ1 )λ



 q q  θˆ1 (1 − θˆ1 ) 1 − θˆ1 W (θˆ1 )(2 − θˆ1 ) 1 − θˆ1  d µ∗1 (c) ∂ µ∗1 (θˆ1 , c) ∂ µ∗1 (θˆ1 , c) d θˆ1 (c)   = + = α − √ 2  √ 2 ˆ dc ∂c dc   ∂ θ1 θˆ θˆ 1−θˆ 1−θˆ 2M (θˆ1 ) 1 2 1 − M2 4 1 2 1 − M2   √ q 2   ˆ ˆ ˆ ˆ (5 θ −2) 1− θ θ (1− θ )  1 1   W (θˆ1 ) − 1M (θˆ )1   θˆ1 (1 − θˆ1 ) 1 − θˆ1  4 1 +β + . √ 2 √ 2   ˆ1 1−θˆ1 ˆ ˆ   θ θ 1− θ 1 1   − M2 − M2  2M (θˆ1 )  2 2 i h Let d µ∗1 (c)/d c = α fc1 (θˆ1 ) + αβ gc1 (θˆ1 ) . It is straightforward to show that d µ∗1 (c)/d c > 0 and d2 µ∗1 (c)/d c2 < 0 for any θˆ1 ∈ 32 , 45 so µ∗1 is concave increasing in c. We can apply the same approach to derive other comparative statics of µ∗1 and µ∗2 .

We provide the complete comparative statics of the optimal utilization levels ρ∗1 and ρ∗2 in the following corollary: Corollary 4. ρ∗1 (respectively ρ∗2 ) is increasing in λ and decreasing in α and c. If α = 0 or β ≥ α, ρ∗1 (ρ∗2 ) is decreasing in β. If α > 0 and β < α, there exists a unique βˆ1D (βˆ2D ) such that ( is increasing in β, 0 ≤ β < βˆ1D (βˆ2D ) ρ∗1 (ρ∗2 ) : is decreasing in β, β ≥ βˆ1D (βˆ2D ).


12

−1

q Proof of Corollary 4. Notice that ρ∗1 = 1 + q1∗ β(1−cθˆ

1 )λ

−1 (1+ α )M = 1 + ˆ √ βˆ . Since θˆ1 does θ1

1−θ1 −M

not change with α, ρ∗1 decreases in α. decreases in θˆ1 . Therefore, ρ∗1 increases in λ and decreases in c. Let As θˆ1 ∈ 23 , 45 , ˆ √ Mˆ ψρ1 (β) =

(1+ α β )M

θˆ1

√

θ1

1−θˆ1 −M

1−θ1 −M

. Then,

d ψρ1 (β) ∂ ψρ1 (β) ∂ ψρ1 (θˆ1 ) d θˆ1 (β) = + dβ ∂β dβ ∂qθˆ1  2  ˆ ˆ 2 ˆ ˆ 2M − θ 1 − θ M 1 1 α (3θ1 − 2)M W (θ1 )  = 2 + q  β  2Vρ1 (θˆ1 ) 4 1 − θˆ1 Vρ1 (θˆ1 )  q   ˆ ˆ 2 ˆ ˆ 1 − θ M θ 1 1 1 (3θ1 − 2)M W (θ1 )  1 α = fρ1 (θˆ1 ) + gρ1 (θˆ1 ) , + + q  β β β  2Vρ1 (θˆ1 ) 4 1 − θˆ1 Vρ1 (θˆ1 ) q 2 where Vρ1 (θˆ1 ) = θˆ1 1 − θˆ1 − M and W (θˆ1 ) is defined in the Proof of Proposition 5. Let rρ1 (θˆ1 ) = −gρ1 (θˆ1 )/fρ1 (θˆ1 ). When θˆ1 ∈ 32 , 45 , fρ1 (θˆ1 ) < 0, fρ0 1 (θˆ1 ) > 0, and fρ1 54 = 0; gρ1 (θˆ1 ) > 0, gρ0 1 (θˆ1 ) < 0, and gρ1 54 = 0. And, rρ1 (θˆ1 ) > 1, rρ0 1 (θˆ1 ) < 0, and rρ1 45 = 1. If α = 0 or β ≥ α > 0, ψρ0 1 (β) > 0 and thus ρ∗1 increases in β. When 0 < β < α, there exists a unique βˆ1D such that ρ∗1 increases in β when 0 ≤ β < βˆ1D and decreases in β when β ≥ βˆ1D . −1 i−1 h q (1+ α β )M 1 3c ∗  , hence ρ∗2 decreases in α. But = 1 + „ Similarly, ρ2 = 1 + q∗ β(2θˆ −1)λ «3 ψρ0 1 (β) =

2

1

2

2θˆ1 −1 3

2

−M

−1 3 2θˆ1 −1 2 M 2 −M decreases in θˆ1 , and ρ∗2 increases in λ and decreases in c. 3

Let ψρ2 (β) =

(1+ α β )M . „ «3 2θˆ1 −1 2 2 −M 3

Hence,

  q ˆ 32 ˆ1 −1  2θ1 −1  2 2 θ 2 ˆ   M W (θ1 ) 3 d ψρ2 (β) α M − M 3 ψρ0 2 (β) = = 2 −  dβ β  Vρ2 (θˆ1 ) Vρ2 (θˆ1 )     3 q 2θˆ1 −1 2  2θˆ1 −1  2 ˆ   1 α M M W ( θ ) 1 3 1 3 + − = fρ2 (θˆ1 ) + gρ2 (θˆ1 ) .  β Vρ (θˆ1 )  Vρ (θˆ1 )  β β 2

2

Therefore, if α = 0 or β ≥ α > 0, ψρ0 2 (β) > 0 and thus ρ∗2 increases in β. If 0 < β < α, there exists a unique βˆ2D such that ρ∗2 increases in β when 0 ≤ β < βˆ2D and decreases in β when β ≥ βˆ2D . ˆ S < 729βc”. From Proposition 2, λ ˆ S solves M = Mr (ω), where ω = Proof of the fact that “λ √ √ √ √ 27αc/λ andq Mr (ω) = 2[1 − ω + 1 + ω + ω 2 ]/9 3. Since 1 − ω + 1 + ω + ω 2 > 1, Mr (ω) > 2/9 3. √ ˆ S > 2/9 3 and λ ˆ S < 243βc < 729βc. Therefore 2 βc/λ Proof of Proposition 6. The property of 1 − θ2∗ follows directly from that of θ2∗ , and ω1 = which is a decreasing function in θˆ1∗ . Similarly ω2 =

2θˆ1∗ −1 , 2−θˆ1∗

which is increasing in θˆ1∗ .

3(1−θˆ1∗ ) , 2−θˆ∗ 1


13

Proof of Proposition 7. From the firm’s profit function ( q 2 2 ) 2 3 ∗ ∗ ∗ ΠD = Π1 + Π2 = A θˆ1 1 − θˆ1 − M + √ (2θˆ1 − 1) 2 − M , 3 3 2 q h i2 3 ∗ ˆ ˆ and Π1 = A θ1 1 − θ1 − M , Π∗2 = A 3√2 3 (2θˆ1 − 1) 2 − M . −1

−1

and ∆Π∗2 = Π∗2 /Π∗D = [Π∗1 /Π∗2 + 1] . We define Notice that ∆Π∗1 = Π∗1 /Π∗D = [1 + Π∗2 /Π∗1 ] " # 2 √ √ θˆ1 +8) 1−θˆ1 (5θˆ12 −14√ θˆ1 1−θˆ1 −M Π∗ 1 ˆ ˆ R(θ1 ) = Π∗ = 2 , we show . Applying the fact that M (θ1 ) = 3 ˆ ˆ ˆ ˆ1 −1) 2 −M √ (2θ 3 3

2

18−27θ1 +12

3(1−θ1 )(2θ1 −1)

that R0 (θˆ1 ) < 0, but R(θˆ1 ) decreasing in θˆ1 implies that Π∗1 /Π∗2 increases in θˆ1 .

Consider the two product model when both capacity investment and variable production cost are general power functions of the quality level, i.e., b(q) = βq γ and a(q) = αq γ , where γ > 2 and α, β > 0. The firm’s optimization problem becomes X cλe γ γ γ i ˆ ˆ max ΠG2 = (θ1 q1 − αq1 )λe1 + [(θ1 + θ2 − 1)q2 − αq2 ]λe1 − + βqi µi . (57) µ i − λ ei q,µ,θˆ1 ,θ2 i=1,2 Solving optimization problem (57) yields the following results for two differentiated products with general power cost functions analogous to Theorems 1 and 2. Proposition 20. For two products in dedicated facilities √ with general cost functions it is optimal: γ−1 (i.) to produce two differentiated products if 0 ≤ M < 22γ−1 [(2γ − 1)−2 − (α + β)], √ √ γ−1 γ−1 (ii.) to produce a single product if 22γ−1 [(2γ − 1)−2 − (α + β)] ≤ M < 2√2γ−1 [(2γ − 1)−1 − (α + β)], √ γ−1 (iii.) not to produce any products if M ≥ 2√2γ−1 [(2γ − 1)−1 − (α + β)]. Proof of Proposition 20. From Theorem 5, producing a single product is profitable only when q γ θˆ1 ) γ−1 1 2 , the condition ∂ ΠD /∂ q2 = 0 becomes M < 2 2γ−1 − (α + β)qmin . Using θ2 = γ−(1− γ −1 2γ−1 2 (2γ−1)qmin

Since θˆ1 >

p γ −1 γ(2θ2 − 1) − γ(α + β)q2γ−1 (γ − 1)(2θ2 − 1)λ = γq22 βc(γ − 1)(2θ2 − 1)λ.

γ 2γ−1

and q2 ≥ qmin = 1, " # √ γ 2 γ −1 1 2 M< − (α + β)qmin , γ 2γ − 1 (2γ − 1)2 q 2 −1 min

which guarantees a positive feasible solution for the quality level q2 .

Theorem 5. For two products in dedicated facilities with quadratic cost functions, if 0 ≤ M < √ 2 γ−1 − 1)−2 − (α + β)] the firm’s optimal product positioning decision θˆ1∗ and optimal quality [(2γ 2γ−1 ∗ level q1 are given by the solution to the following system of equations. q q γ γ −1 θˆ1 1 − θˆ1 −γ(α + β)q1γ−1 1 − θˆ1 = M q12 (58) 2 #γ−1 " (2γ − 1)θˆ1 − γ (2θˆ1 − 1)γ −(α + β)(2γ − 1)γ q1 2γ(γ − 1) ! γ−1 " # γ2 −1 γ+1 2 (2γ − 1) 2 2θˆ1 − 1 (2γ − 1)θˆ1 − γ =M q1 . (59) 2 γ −1 2γ

14


The corresponding optimal choices for the low quality product are θ2∗ =

γ − (1 − θˆ1∗ ) , 2γ − 1

q2∗ =

(2γ − 1)[(2γ − 1)θˆ1∗ − γ] ∗ q1 , 2γ(γ − 1)(2θˆ1∗ − 1)

γ with the product positioning decisions satisfying: θˆ1∗ > 2γ−1 , θ2∗ > 2γ(γ−1) . The optimal production (2γ−1)2 capacities are s s ˆ1∗ )λ 1 c(1 − θ 1 c(θˆ1∗ − θ2∗ )λ µ∗1 = (1 − θˆ1∗ )λ + ∗ γ/2 , µ∗2 = (θˆ1∗ − θ2∗ )λ + ∗ γ/2 , (q1 ) β (q2 ) β

with optimal utilization levels "

1 ρ∗1 = 1 + ∗ γ/2 (q1 )

s

#−1

c β(1 − θˆ1∗ )λ

"

1 , ρ∗2 = 1 + ∗ γ/2 (q2 )

s

c β(θˆ1∗ − θ2∗ )λ

#−1

.

Proof of Theorem 5. By direct computation q q p γ ∂ 2 ΠD γ/2 2 −2 ˆ ˆ (1 − θ )λ (γ − 1)(α + β)q (1 − θ )λ − (γ/2 − 1) βc < 0, = − γq 1 1 1 1 ∂ q12 q q p γ ∂ 2 ΠD γ/2 2 −2 ˆ ˆ (θ1 − θ2 )λ (γ − 1)(α + β)q1 (θ1 − θ2 )λ − (γ/2 − 1) βc < 0, = −γq2 ∂ q22 and ∂ 2 ΠD /(∂ q1 ∂ q2 ) = 0. Therefore, ΠD (θˆ1 , θ2 , qi ) is jointly concave in q1 and q2 . To obtain the optimal quality levels we solve ∂ ΠD /∂ q1 = 0, ∂ ΠD /∂ q2 = 0. Using the implicit function theorem on the first order conditions results in ∂ ΠD (θˆ1 , θ2 ) 1 ˆ 1 ∗ ∗ = λ(q1 − q2 ) 1 − 2 − θ1 + λq2 (1 − θ2 ) ˆ γ γ ∂ θ1 1 1 ∂ ΠD (θˆ1 , θ2 ) = λq2∗ 1 − 2 − θ2 + (1 − θˆ1 ) . ∂ θ2 γ γ Solving the first order conditions ∂ ΠD /∂ θˆ1 = 0, ∂ ΠD /∂ θ2 = 0 we get θ2 =

γ − (1 − θˆ1 ) , 2γ − 1

and q2 =

(2γ − 1)[(2γ − 1)θˆ1 − γ] q1 , 2γ(γ − 1)(2θˆ1 − 1)

and substituting them into ∂ ΠD /∂ q1 = 0, ∂ ΠD /∂ q2 = 0 yields (58) and (59).

To explore further the effect of the power γ, we conducted an extensive numerical study. In Figure 13, we show representative results of the firm’s optimal choices as functions of γ for the case with two differentiated products. The labels θ∗ and q ∗ denote the components of the optimal single-product solution. In Figure 13(d), ΠG1 and ΠG2 denote optimal profits for the single- and two- product cases respectively. The numerical study showed that both the market coverage of the high-end product and entire market coverage are increasing in γ. As γ increases, the optimal quality levels of both products decrease, and the optimal capacity of the high-end product increases. In addition, when feasible, it is optimal to produce two products as opposed to one.


15

12 0.8

T 10

0.7

8

q*

qi

Ti

6 0.6

T

4

q

T q

2

0.5 2

3

4

5

6

7

8

2

3

4

5

J

6

7

8

J

(a)θˆ1∗ and θ2∗ as functions of γ

(b)qi∗ as functions of γ

μ∗

900

1200 800 700

Profit

1000

550

μ1

μi

3G2

500 800

3G1

450

μ2 400

600 2

3

4

5

γ

6

7

3

8

5

6

J

(c)µ∗i as functions of γ Figure 13

4

(d)Optimal profits as functions of γ

Illustration of the effect of changes in γ to the optimal solution in the two-product case, in this example α = 0.25, β = 0.4, c = 1, and λ = 400.

Similarly to the single and two-product cases, in dealing p with optimization problem (22)–(29), ∗ we first solve for the optimal capacities µi (λei ) = λei + cλei /b(qi ), i = 1, . . . , n. Substitution of the expressions for λei yields s

c(1 − θˆ1 )λ ,..., b(qs 1) c(θî−1 − θî )λ µ∗i (θî−1 , θî , qi ) = (θî−1 − θî )λ + ,..., s b(qi ) c(θˆn−1 − θn )λ µ∗n (θˆn−1 , θn , qn ) = (θˆn−1 − θn )λ + . b(qn ) µ∗1 (θˆ1 , q1 ) = (1 − θˆ1 )λ +

Objective function ΠM (θ, q) is jointly concave in qi (for b(qi ) = βqi2 , a(qi ) = αqi2 , i = 1, . . . , n) because for any product i, ∂ 2 ΠM (θ, q)/∂ qi2 < 0 and ∂ 2 ΠM (θ, q)/∂ qi ∂ q−i = 0. Thus, the Hessian of ΠM (θ, q) has only negative diagonal elements and all off-diagonal elements are zero, so it is negative definite. The optimal quality levels are

16


" # s ˆ1 1 θ βc q1∗ (θˆ1 ) = − , ˆ α+β 2 " (1 − θ1 )λ s # î−1 + θî − 1 1 θ βc qi∗ (θî−1 , θî ) = , i = 2, . . . , n − 1, − î−1 − θî )λ α+β 2 ( θ " # s 1 βc θˆn−1 + θn − 1 ∗ ˆ qn (θn−1 , θn ) = − . α+β 2 (θˆn−1 − θn )λ

ˆ Substitution into (22)–(29) allows writing the optimization problem exclusively in terms of θ: ( q 2 X 2 q n−1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ max ΠM (θ) =A + (θi−1 + θi − 1) θi−1 − θi − M θ1 1 − θ1 − M ˆ θ i=2 2 ) q ˆ ˆ + (θn−1 + θn − 1) θn−1 − θn − M (60) s.t.

q q 1 − θˆ2 > M 1/ 1 − θˆ1 − 1/ θˆ1 − θˆ2 , q q ˆ ˆ ˆ ˆ ˆ ˆ θi−1 − θi+1 > M 1/ θi−1 − θi − 1/ θi − θi+1 , i = 2, . . . , n − 2, q q ˆ ˆ ˆ ˆ θn−2 − θn > M 1/ θn−2 − θn−1 − 1/ θn−1 − θn , q ˆ (θn−1 + θn − 1) θˆn−1 − θn > M ,

0 ≤ θn < θˆn−1 < · · · < θˆ2 < θˆ1 < 1, p where A = λ/[4(α + β)] and M = 2 βc/λ. Proof of Proposition 8. From (60), we write the firm’s profit function ΠM (θ) = A Πn (θˆ1 , . . . , θˆn−1 , θn ), where q 2 ˆ ˆ ˆ ˆ Πn (θ1 , . . . , θn−1 , θn ) = θ1 1 − θ1 − M 2 2 q q n−1 X ˆ ˆ ˆ ˆ ˆ ˆ + (θi−1 + θi − 1) θi−1 − θi − M + (θn−1 + θn − 1) θn−1 − θn − M i=2

which we henceforth abbreviate as Πn . As in the proof of Theorem 2, we first consider the problem Maximize Πn (θˆ1 , . . . , θˆn−1 , θn ) (61) θˆ1 ,...,θˆn−1 ,θn q q q s.t. θˆ1 1 − θˆ1 > M, . . . , (θî−1 + θî − 1) θî−1 − θî > M, . . . , (θˆn−1 + θn − 1) θˆn−1 − θn > M, where the conditions qi > qi+1 , i = 1, . . . , n − 1 have been relaxed to qi > 0, i = 1, . . . , n − 1. For ˆ N = 1 arg max Π1 = 23 and M 1 = 3√2 3 . For N = 2, θ2∗ (θˆ1 ) = 1+3θ1 . When product 2 generates zero “ √ ”2 q 3 3 3M +1 2 2 ∗ ∗ ∗ ∗ ∗ ˆ ˆ ˆ profit, we have (θ1 + θ2 − 1) θ1 − θ2 = M 2 . Then, θ1 = , and θˆ1∗ must also solve 2 s     ! 32 q q ˆ ˆ ˆ θ1  + 2 2θ1 − 1 2 2θ1 − 1 θˆ1 1 − θˆ1 − M 2  1 − θˆ1 − q − M 2  = 0, 3 3 ˆ 2 1 − θ1


17

2 ∗ ∗ . Suppose θˆ1∗ , . . . , θˆN Therefore M 2 = 27 −1 , θN are the optimal solutions to ΠN under a particular ∗ value of M . The maximum number of possible product variants being N , means that product θN q ∗ ∗ ∗ ∗ ∗ ˆ∗ ˆ∗ generates zero profit, so M N is defined by (θˆN θˆN −1 + θN − 1) −1 − θN = M N since θ1 , . . . , θN −1 , θN are also the solutions to the equation system ∂ ΠN /∂ θˆ1 = 0, . . . , ∂ ΠN /∂ θˆN −1 = 0, ∂ ΠN /∂θN = 0. ∗ Clearly, given θˆ1 , . . . , θˆN −1 , the optimal product positioning decision θN can be obtained from q ˆ 1+ θ N −1 ∗ ˆ ∗ ∗ ∗ ∗ ˆ ˆ θ (θN −1 ) = − θ = M N and equation ∂ ΠN /∂ θˆN −1 = 0, we . From (θ + θ − 1) θ N

N −1

3 “ √

∗ have θˆN −1 =

3 3M N 2

”2

3

+1

2

N −1

N

∗ and θˆN −1 =

∗ 1+θˆN −2 . 3

∗ θˆN −2 =

3

N

Then, we can obtain

23 √ 3 3 M N 2

+1

2

.

∗ ˆ Therefore (θˆ1∗ , . . . , θˆN −2 , M N ) must solve the above equation together with ∂ ΠN /∂ θ1 = 0, . . . , ∂ ΠN /∂ θˆN −2 = 0. For any M ∈ (M N −1 , M N ), suppose that the firm produces N − 1 product vari∗ ∗ ˆ ˆ ants θˆ1∗ , . . . , θˆN −2 , θN −1 , which are the solution of ∂ ΠN −1 /∂ θ1 = 0, . . . , ∂ ΠN −1 /∂ θN −2 = 0 0 0 0 ∗ 0 ∗ 0 0, ∂ ΠN −1 /∂θN −1 = 0. By constructing θˆ1 , . . . , θˆN −1 , θN such that θˆ1 = θˆ1 , . . . , θˆN −1 = θN −1 , and θN = ∗ 1+θN −1 ∗ ∗ ∗ ∗ 0 0 0 0 , we show that ΠN −1 (θˆ1 , . . . , θˆN −2 , θN −1 ) < ΠN (θˆ1 , . . . , θˆN −1 , θN ). Finally, proceeding as in 3 the proof of Theorem 2, it is straightforward to show that the optimal solution of (61) also satisfies the constraints in (60).

Proof of Corollary 2. Given θˆ1 , . . . , θˆn−1 , the optimal θn is found by solving ∂ ΠM /(θ)∂ θn = 0.   q q ˆ θn−1 + θn  ∂ ΠM (θ) = 2 (θˆn−1 + θn − 1) θˆn−1 − θn − M  θˆn−1 − θn − q , ∂ θn ˆ 2 θn−1 − θn which yields

q θˆ +θn θˆn−1 − θn − √n−1 = 0, i.e., θn∗ = ˆ 2

θn−1 −θn

θˆn−1 +1 . 3

Substitution of θn∗ into the profit func-

tion leads to (

ΠM [θ |θn (θˆn−1 )] =A (θˆ1

2 q q n−2 X 2 ˆ ˆ ˆ ˆ ˆ 1 − θ1 − M ) + (θi−1 + θi − 1) θi−1 − θi − M

i=2  2 q + (θˆn−2 + θˆn−1 − 1) θˆn−2 − θˆn−1 − M + 2

2θˆn−1 − 1 3

! 32

2     −M .  

Using ∂ ΠM [θ |θn (θˆn−1 )]/∂ θˆn−1 = 0, we obtain the first order condition s   q i ˆn−2 + θˆn−1 − 1 ˆn−1 − 1 1 h ˆ2 θ 2 θ 2 . 5θn−1 − (18θˆn−2 − 4)θˆn−1 + 9θˆn−2 − 1 = M  θˆn−2 − θˆn−1 − q +2 18 3 ˆ ˆ 2 θn−2 − θn−1 q θˆn−2 +θˆn−1 −1 2 2 Let ψ1 (θˆn−1 ) = 5θˆn−1 − (18θˆn−2 − 4)θˆn−1 + 9θˆn−2 − 1 and ψ2 (θˆn−1 ) = θˆn−2 − θˆn−1 − √ + 2 θˆn−2 −θˆn−1 q 2θˆn−1 −1 2 . 3 We have ψ10 (θˆn−1 ) < 0 and ψ100 (θˆn−1 ) > 0, so ψ1 (θˆn−1 ) is convex decreasing in θˆn−1 . Similarly, 0 ˆ ψ (θn−1 ) > 0 and ψ 00 (θˆn−1 ) < 0, so ψ2 (θˆn−1 ) is concave increasing in θˆn−1 . Hence, both θn and θˆn−1 2

2


18

are decreasing in M . By induction, θˆ1 , . . . , θˆn−2 are decreasing in M .

To solve optimization problem (38)–(42) we use a similar approach as with the dedicated facility cases. We start with the optimal capacity level which is given by 1 µ∗ (λ) = (1 − θ2 )λ + δq1 + (1 − δ)q2

s

c(1 − θ2 )λ . β

Substitution into (38) yields 2 ˆ ˆ ΠδF (θˆ1 , θ2 , q1 , q2 ) = θˆ1 q1 − αq12 (1 − θˆ1 )λ p+ (θ1 + θ2 − 1)q2 − αq2 (θ1 − θ22 )λ − 2[δq1 + (1 − δ)q2 ] βc(1 − θ2 )λ − β[δq1 + (1 − δ)q2 ] (1 − θ2 )λ.

By direct differentiation,

∂ 2 ΠδF (θˆ1 ,θ2 ,qi ) ∂ q12

= −2λ[α(1 − θˆ1 ) + δ 2 β(1 − θ2 )] < 0,

−2λ[α(θˆ1 − θ2 ) + (1 − δ)2 β(1 − θ2 )] < 0, and

∂ 2 ΠδF (θˆ1 ,θ2 ,qi ) ∂ q1 ∂ q2

∂ 2 ΠδF (θˆ1 ,θ2 ,qi ) ∂ q22

=

= −2λβδ(1 − δ)(1 − θ2 ) < 0. In particular,

2 δ 2 ∂ 2 ΠδF ∂ 2 ΠδF ∂ ΠF (θ, q) − ∂ q12 ∂ q22 ∂ q1 ∂ q2 = 4λ2 {α2 (1 − θˆ1 )(θˆ1 − θ2 ) + αβ(1 − θ2 )[(1 − θ2 )δ 2 − 2δ(1 − θˆ1 ) + (1 − θˆ1 )]} > 4λ2 {α2 (1 − θˆ1 )(θˆ1 − θ2 ) + αβ(1 − θˆ1 )(1 − θ2 )(1 − δ)2 } > 0,

so the Hessian of ΠδF (θˆ1 , θ2 , qi ) is negative definite and thereby ΠδF (θˆ1 , θ2 , qi ) is jointly concave in q1 and q2 . The first-order conditions yield the optimal quality levels: √ (θˆ1 − θ2 )[θˆ1 (1 − θˆ1 ) − δM 1 − θ2 ] + r(1 − δ)(1 − θ2 )[θˆ1 (1 − θˆ1 ) − δθ2 (1 − θ2 )] n o = , 2α (1 − θˆ1 )(θˆ1 − θ2 ) + r(1 − θ2 )[(1 − δ)2 − (1 − 2δ)θˆ1 − δ 2 θ2 ] √ (1 − θˆ1 )[(θˆ1 − θ2 )(θˆ1 + θ2 − 1) − (1 − δ)M 1 − θ2 ] − rδ(1 − θ2 )[θˆ1 (1 − θˆ1 ) − δθ2 (1 − θ2 )] ∗ ˆ n o q2 (θ1 , θ2 ) = , 2α (1 − θˆ1 )(θˆ1 − θ2 ) + r(1 − θ2 )[(1 − δ)2 − (1 − 2δ)θˆ1 − δ 2 θ2 ]

q1∗ (θˆ1 , θ2 )

p where M = 2 βc/λ and r = β/α. Proceeding similarly as with dedicated facilities, we impose the conditions q1∗ (θˆ1 , θ2 ) > q2∗ (θˆ1 , θ2 ) > 0 and θ2 q2∗ (θˆ1 , θ2 ) − c/[µ∗ (λ) − (1 − θ2 )λ] > 0, and reduce the firm’s problem to

max ΠδF (θˆ1 , θ2 ) θˆ1 ,θ2

s.t.

p (1 − θˆ1 )[(θˆ1 − θ2 )(θˆ1 + θ2 − 1) − (1 − δ)M 1 − θ2 ] > rδ(1 − θ2 )[θˆ1 (1 − θˆ1 ) − δθ2 (1 − θ2 )], p (1 − θˆ1 )(θˆ1 − θ2 )(1 − θ2 ) + M 1 − θ2 [(1 − θˆ1 ) − δ(1 − θ2 )] + r(1 − θ2 )[θˆ1 (1 − θˆ1 ) − δθ2 (1 − θ2 )] > 0, p 2θ2 1 − θ2 {θ2 (1 − θ2 )[(1 − θˆ1 ) + rδ 2 (1 − θ2 )] − θˆ1 (1 − θˆ1 )[(1 − θˆ1 ) + rδ(1 − θ2 )]} ˆ ˆ ˆ ˆ > M (1 p− θ1 ){(1 − θ2 )[δ θ1 + 3(1 − δ)θ2 ] − θ1 (1 − θ1 )} 2 2 ˆ −M 1 − θ2 [(1 − 2δ)(1 − θ1 ) + δ (1 − θ2 )] 0 ≤ θ2 < θˆ1 < 1,

(62)

(63) (64)

(65) (66)


19

where ΠδF (θˆ1 , θ2 ) =

n λ (1 − θˆ1 )(θˆ1 − θ2 )(1 − θ2 )[θ2 (θˆ1 + θ2 − 1) + θˆ1 (1 − θˆ1 )] 4αΨ(θˆ1 , θ2 ) + r(1 − θ2 )[δθ2 (1 − θ2 )p − θˆ1 (1 − θˆ1 )]2 + M 2 (1 − θ2 )[(1 − δ)2 − (1 − 2δ)θˆ1 − δ 2 θ2 ] − 2M (1 − θˆ1 )(θˆ1 − θ2 ) 1 − θ2 [(θˆ1 + θ2 − 1) + δ(1 − θ2 )],

and Ψ(θˆ1 , θ2 ) = (1 − θˆ1 )(θˆ1 − θ2 ) + r(1 − θ2 )[(1 − δ)2 − (1 − 2δ)θˆ1 − δ 2 θ2 ]. Proof of Theorem 3. We need to establish the following result for this proof. Result: [Begin] If 0 ≤ δ ≤ 1, 0 ≤ r < rδ , and 0 ≤ M < M (r|δ), the firm’s optimal product positioning decisions θˆ1∗ and θ2∗ satisfy 23 < θˆ1∗ < 45 and θ2∗ = θ2 (θˆ1∗ ), where θ2 (θˆ1 ) is the unique root of (1 − θˆ1 )(3θˆ1 − 2)[2θˆ1 (θˆ1 − 2) + θ2 (5 − 3θ2 )] − δ(2θˆ1 − θ2 − 1)(1 − θ2 )[6θˆ1 (θˆ1 − θ2 − 1) + θ2 (8 − 3θ2 )] = 0 √ 2 ˆ ˆ 4 5− 24θ1 −48θ1 +25 2 ˆ ˆ ˆ < θ2 < ξ2 (θ1 ) , and ξ2 (θˆ1 ) is the second-highest real in region (θ1 , θ2 ) : 3 < θ1 < 5 , 6 root of the cubic equation 3θ23 + θ22 (3θˆ1 − 8) − θ2 (6θˆ12 − 5θˆ1 − 3) + 2(3θˆ13 − 5θˆ12 + 3θˆ1 − 1) = 0. Furthermore, θˆ1∗ is uniquely determined solving Sδ (θˆ1 , θ2 (θˆ1 )) + r · Tδ (θˆ1 , θ2 (θˆ1 )) = 0, where Sδ (θˆ1 , θ2 (θˆ1 )) = (1 − θˆ1 )(θˆ1 − θ2 )(1 − 2θˆ1 + θ2 ) and Tδ (θˆ1 , θ2 (θˆ1 )) = (1 − θ2 ){2 − 5θˆ1 + 3θˆ12 − δ[4 + 2θˆ1 (3θˆ1 − 5) + θ2 − θ22 ] + 2δ 2 (1 − θ2 )(1 − 2θˆ1 + θ2 )}. ˆ ˆ ˆ ˆ In addition, let M (r, θˆ1 , δ) = √Sδ (θ1 ,θ2 (θ1 ))+r·Tˆ δ (θ1 ,θˆ2 (θ1 )) . The envelope function M (r|δ) is defined as 1−θ2 [(1−δ)(1−θ1 )+δ(θ1 −θ2 )]

2 ˆ 4 ˆ M (r|δ) := sup M (r, θ1 , δ) : < θ1 < , 3 5 θˆ1

and rδ is the cutoff value defined by M (rδ |δ) = 0. [End] The first order conditions of (62) with respect to θˆ1 and θ2 can be written as: ∂ ΠδF (θˆ1 , θ2 ) Yθˆ1 = , Zθˆ1 ∂ θˆ1

∂ ΠδF (θˆ1 , θ2 ) Yθ2 = . ∂ θ2 Zθ2

For r ≥ 0, 0 < θ2 < θˆ1 < 1, and 0 ≤ δ ≤ 1, Zθˆ1 6= 0 and Zθ2 6= 0. Imposing Yθˆ1 = 0 and Yθ2 = 0 yields M 2 (1−θ2 )[1 − θˆ1 − δ(1n− θ2 )][(1 − δ)(1 − θˆ1 ) + δ(θˆ1 − θh2 )] p p + 2M 1 − θ2 (1 − θˆ1 )2 (θˆ1 − θ2 )2 + r 1 − θ2 (1 − θˆ1 )2 (2θˆ1 − 1) − δ(1 − θˆ1 )2 (4θˆ1 + θ2 − 3) io + δ 2 (1 − θ2 )[θˆ1 (8 − 5θˆ1 ) − θ2 (1 − θ2 ) − 3] + δ 3 (1 − θ2 )2 (1 − 2θˆ1 + θ2 ) n + (1 − θ2 )[rδθ2 (1 − θ2 ) − (1 − θˆ1 )(θˆ1 + rθˆ1 − θ2 )] (1 − θˆ1 )(θˆ1 − θ2 )(1 − 2θˆ1 + θ2 ) h io + r(1 − θ2 ) 2 + 3θˆ12 (1 − 2δ) − δ[4 + θ2 − 2δ − θ22 (1 − 2δ)] − θˆ1 [5 − 10δ + 4δ(1 − θ2 )] = 0, (67)


20

p ˆ ˆ M 2 (1 − θˆ1 ) 1 − θ2n[1 − θˆ1 − δ(1 h − θ2 )][(1 − δ)(1 − θ1 ) + δ(θ1 − θ2 )] + M (1 − θˆ1 ) r(1 − θ2 ) (1 − θˆ1 )(2 − θˆ1 + θˆ2 − 5θ2 + 3θ2 ) − δ 3 (1 − θ2 )2 (θˆ1 + θ2 − 2)

1 2 i 2 ˆ ˆ ˆ − δ (1 − θ2 )[6 − θ1 (5 − θ1 ) − 9θ2 + 6θ1 θ2 + θ2 ] + δ(1 − θˆ1 )[6 − θˆ1 + 2θˆ12 − θ2 (15 + θˆ1 ) + 9θ22 ] o + (1 − θˆ1 )(θˆ1 − θ2 )2 [θˆ1 − 3(1 − δ)(1 − θ2 )] n p + 1 − θ2 (1 − θˆ1 )2 (θˆ1 − θ2 )2 (1 + θˆ1 − 3θ2 )(θˆ1 + θ2 − 1) − δr2 (1 − θ2 )2 [δθ2 (1 − θ2 ) − θˆ1 (1 −hθˆ1 )][2(1 − θˆ1 )(2θ2 − 1) + δ(1 − θˆ1 )(4 + θˆ1 − δθ2 ) + δ 2 (1 − θ2 )(3θ2 − 2)] + r(1 − θˆ1 )(θˆ1 − θ2 ) (1 − θˆ1 )(θˆ1 + θ2 − 1)(2 − θˆ1 + θˆ12 − 5θ2 + 3θ22 ) + 2δ(1 − θˆ1 )(1 − θ2 ) io (2 − 3θˆ1 + 5θˆ1 θ2 − 5θ2 + 3θ22 ) − δ 2 (1 − θ2 )2 [2 + θˆ12 + θˆ1 (θ2 − 3) + θ2 (6θ2 − 5)] = 0. (68) 2

Equation (67) is quadratic in M , with roots. 1

M1 = − √

n (1 − θˆ1 )(2θˆ12 − 3θˆ1 θ2 − θˆ1 + θ2 + θ22 )

1 − θ2 [(1 − δ)(1 − θˆ1 ) + δ(θˆ1 − θ2 )] o +r(1 − θ2 ){θˆ1 [5 − 10δ + 4δ 2 (1 − θ2 )] − 2δ 2 (1 − θ22 ) − 3θˆ12 (1 − 2δ) + δ(4 + θ2 − θ22 ) − 2} , √ 1 − θ2 {(1 + r)θˆ12 − θˆ1 (1 + θ2 + r) + θ2 [1 + rδ(1 − θ2 )]} M2 = . 1 − θˆ1 − δ(1 − θ2 ) We discard the M2 root because substitution into (64) makes that equation’s left-hand side negative, which contradicts the right-hand side. Substitution of M1 into (68) yields √

Pδ1 Pδ2 = 0, 1 − θ2 [(1 − δ)(1 − θˆ1 ) + δ(θˆ1 − θ2 )]

(69)

where n h io2 Pδ1 = (1 − θˆ1 )(θˆ1 − θ2 ) + r(1 − θ2 ) (1 − δ)2 − δ 2 θ2 + θˆ1 [δ 2 − (1 − δ)2 ] and 2 P =(1 − θˆ1 )(3θˆ1 − 2)[2θˆ1 (θˆ1 − 2) + θ2 (5 − 3θ2 )] + δ(1 − θ2 )(1 − 2θˆ1 + θ2 )[6θˆ2 + θ2 (8 − 3θ2 ) − 6θˆ1 (1 + θ2 )]. δ

1

Since Pδ1 > 0, (69) is equivalent to Pδ2 = 0, explicitly (1 − θˆ1 )(3θˆ1 − 2)[2θˆ1 (θˆ1 − 2) + θ2 (5 − 3θ2 )] + δ(1 − θ2 )(1 − 2θˆ1 + θ2 )[6θˆ12 + θ2 (8 − 3θ2 ) − 6θˆ1 (1 + θ2 )] = 0. (70) Using equation (70) constraint (63) becomes δθ2 (1 − θ2 ) − (1 − θˆ1 )[2(1 − δ) + θˆ1 (4δ − 3)] > 0,

(71)

and constraint (64) becomes 2[(1 − θˆ1 )2 − δ(1 − θ2 )(1 − 2θˆ1 + θ2 )]{(1 − θˆ1 )(θˆ1 − θ2 ) + r(1 − θ2 )[1 − θˆ1 (1 − 2δ) − δ(2 − δ + δθ2 )]} > 0. [(1 − δ)(1 − θˆ1 ) + δ(θˆ1 − θ2 )] However, the conditions 0 < θ2 < θˆ1 < 1, r ≥ 0, and 0 < δ < 1 imply that both the numerator and denominator of the above expression are always positive, thus making constraint (64) redundant. A necessary condition for (71) to hold is θˆ1 > 12 . From Theorem 3, M1 can be written as M1 = √

Sδ (θˆ1 , θ2 ) + r · Tδ (θˆ1 , θ2 ) . 1 − θ2 [(1 − δ)(1 − θˆ1 ) + δ(θˆ1 − θ2 )]


Solving for θ2 in Tδ (θˆ1 , θ2 ) = 0 yields θ2 =

√ δ−4θˆ1 δ 2 ± Λ , 2δ(1−2δ)

21

where

Λ = δ 2 (1 − 4δ θˆ1 )2 − 4δ(1 − 2δ)[3θˆ12 (1 − 2δ) + 2(1 − δ)2 + θˆ1 (10δ − 4δ 2 − 5)], and solving for θˆ1 in Λ = 0 yields p 5 − 20δ + 22δ 2 − 8δ 3 ± (1 − δ)(1 − 2δ)2 ˆ . θ1 = 6 − 24δ + 24δ 2 − 8δ 3

Let θˆ1+ and θhˆ1− denote the two expressions above, and let δ0 denote the only real root of θˆ1+ = 0, i √ 1 10 − (53−6√1 78)1/3 − (53 − 6 78)1/3 ≈ 0.42135. with δ0 = 12 For 0 ≤ δ ≤ 1 − 22/3 , θˆ1+ ≥ 1, and Λ ≥ 0 iff θˆ1 ≥ θˆ1− . For 1 − 22/3 < δ ≤ δ0 , θˆ1+ ≤ 0, and Λ ≥ 0 iff θˆ1 ≥ θˆ1− . For δ0 < δ ≤ 1, θˆ1+ ≤ θˆ1− , and Λ ≥ 0 iff θˆ1 ≥ θˆ1− or θˆ1 ≤ θˆ1+ . For δ0 < δ ≤ 1, θˆ1+ ≤ 1/2. Therefore, for all 1/2 < θˆ1 < 1 and 0 ≤ δ ≤ 1, Λ ≥ 0 iff θˆ1 ≥ θˆ1− . For 0 < δ < 1/2 and 1/2 < θˆ1 < 1, Tδ (θˆ1 , θ2 ) < 0 iff θ2− < θ2 < θ2+ for θˆ1 ≥ θˆ1− . For 1/2 < δ ≤ 1 and 1/2 < θˆ1 < 1, Tδ (θˆ1 , θ2 ) < 0 iff θ2 < θ2− or θ2 > θ2+ , but θ2 > θ2+ implies θ2 > θˆ1 so this case is irrelevant for our purposes. Moreover, θ2+ > θˆ1 on the range for θˆ1 ≥ θˆ1− , and solving for θˆ1 in θ2− = θˆ1 yield < 12 , which implies θ2− > θˆ1 , so the condition θ2 < θˆ1 implies the unique roots θˆ1 = 1 and θˆ1 = 2(1−δ) 3−2δ − θ2 < θ2 . Therefore, if 0 < θ2 < θˆ1 , 0 < δ < 1, θˆ1 > 1/2, and condition (71) holds, then Tδ (θˆ1 , θ2 ) < 0. It requires Sδ (θˆ1 , θ2 ) > 0 and together with 0 < θ2 < θˆ1 < 1 implies that any solution (θˆ1 , θ2 ) satisfies θ2 > 2θˆ1 − 1. Equation (70) can be written as Pδ2 = U + δ · V where U =(1 − θˆ1 )(3θˆ1 − 2)[2θˆ1 (θˆ1 − 2) + θ2 (5 − 3θ2 )], V =(1 − θ2 )(1 − 2θˆ1 + θ2 )[6θˆ12 + θ2 (8 − 3θ2 ) − 6θˆ1 (1 + θ2 )]. q 1 2 ˆ ˆ ˆ Let ϕδ (θ1 ) = 6 5 − 25 − 48θ1 + 24θ1 which is the relevant root of the last factor of U , then   = 0, U: > 0,   < 0,

if θˆ1 = 2/3 or θ2 = ϕδ (θˆ1 ); if θˆ1 < 2/3 or θ2 < ϕδ (θˆ1 ) or θˆ1 > 2/3 or θ2 > ϕδ (θˆ1 ); otherwise. p Similarly, we define ψδ (θ2 ) = 16 3 + 3θ2 − 9 − 30θ2 + 27θ22 which is the relevant root of the last factor of V and we have  ˆ  = 0, if θ1 = ψδ (θ2 ); V: > 0, if θˆ1 < ψδ (θ2 );   < 0, otherwise. √ It is straightforward to show that ϕδ (θˆ1 ) and ψδ (θ2 ) intersect at θˆ1 = 1/2 and θ2 = (5 − 7)/6, ϕδ (θˆ1 ) intersects with θ2 = 2θˆ1 − 1 at θˆ1 = 4/5 and θ2 = 3/5, and ψδ (θ2 ) intersects with θ2 = θˆ1 at θˆ1 = 2/3 and θ2 = 2/3. 2 2 whenever n For δ > 0, we get that Pδ 6= 0 o n U · V > 0 so Pδ = o0 has no roots in regions 2 (θˆ1 , θ2 ) : θˆ1 > 3 , 2θˆ1 − 1 < θ2 < ϕδ (θˆ1 ) and (θˆ1 , θ2 ) : ψδ (θ2 ) < θˆ1 < 23 . For 0 < δ ≤ 1, we also get that Pδ2 6= 0 whenever V (U + V ) < 0. When V < 0 and U + V > 0, then U > −V ≥ −δ · V . Similarly, V > 0 and U + V < 0 imply U + δ · V < 0. Notice that U + V = Pδ2 for δ = 1 and U + V = (θˆ1 − θ2 )U1 where U1 = 3θ23 + θ22 (3θˆ1 − 8) − θ2 (6θˆ12 − 5θˆ1 − 3) + 2(3θˆ13 − 5θˆ12 + 3θˆ1 − 1).

22


The polynomial U1 is cubic in θ2 , and for θˆ1 ∈ (1/2, 1) has three real roots 0 < ξ1 (θˆ1 ) < ξ2 (θˆ1 ) < ˆ θ1 < ξ3 (θˆ1 ) with ( > 0, if 0 < θ2 < ξ1 (θˆ1 ) or ξ2 (θˆ1 ) < θ2 < ξ3 (θˆ1 ); U1 : < 0, if ξ1 (θˆ1 ) < θ2 < ξ2 (θˆ1 ) or θ2 > ξ3 (θˆ1 ). 2 The additional regions where n condition V (U + V ) < 0 yields theofollowing n o Pδ 6= n 0: o ˆ ˆ ˆ ˆ ˆ ˆ ˆ (θ1 , θ2 ) : 2θ1 − 1 < θ2 < θ1 , θ2 > ξ2 (θ1 ) , (θ1 , θ2 ) : 2θ1 − 1 < θ2 < ξ1 (θ1 ) , and (θˆ1 , θ2 ) : θ2 < θˆ1 < ψδ (θ2 ) . Therefore, any solution to Pδ2 = 0 is confined to one of the two regions 1 ˆ 2 ˆ ˆ ˆ B0 = (θ1 , θ2 ) : < θ1 < , ξ1 (θ1 ) < θ2 < ϕδ (θ1 ) and 2 3 2 4 B = (θˆ1 , θ2 ) : < θˆ1 < , ϕδ (θˆ1 ) < θ2 < ξ2 (θˆ1 ) . 3 5

However, any (θˆ1 , θ2 ) in B0 violates the constraint from (71). We have thus characterized the solution (θˆ1 , θ2 ) as the unique root of Pδ2 = 0 in B . The feasibility conditions of M and r must also satisfy the existence condition of solution (θˆ1 , θ2 ) in region B , i.e., 0 ≤ r < rδ and 0 ≤ M < M (r|δ), where 2 ˆ 4 ˆ M (r|δ) := sup M (r, θ1 , δ) : < θ1 < 3 5 θˆ1 and rδ is the cutoff value defined by M (rδ |δ) = 0. When δ = 0, (θˆ1 − θ2 )[(1 − θˆ1 ) − (θˆ1 − θ2 )] − r(1 − θ2 )(3θˆ1 − 2) √ M1 = , 1 − θ2 and Equation (70) simplifies to 2θˆ1 (4 − 12θˆ1 + 11θˆ12 − 3θˆ13 ) − 5θ2 (2 − 5θˆ1 + 3θˆ12 ) + θ22 (6 − 15θˆ1 + 9θˆ12 ) = 0, √ √ 5− 24θˆ12 −48θˆ1 +25 5+ 24θˆ12 −48θˆ1 +25 + with roots θ2− = = and θ . We discard the root θ2+ because θ2+ > 1 2 6 6 for all θˆ1 ∈ (0, 1). Substitution of θ2− into M1 yields M0 (r, θˆ1 ) = s0 (θˆ1 ) + r · t0 (θˆ1 ) where q q ˆ ˆ ˆ ˆ ˆ ˆ 8 25 + 24θ1 (θ1 − 2) − 5 − 3θ1 16θ1 + 3 25 + 24θ1 (θ1 − 2) − 19 ˆ r s0 (θ1 ) = q √ 3 6 1 + 25 + 24θˆ1 (θˆ1 − 2) r q ˆ (2 − 3θ1 ) 1 + 25 + 24θˆ1 (θˆ1 − 2) ˆ √ t0 (θ1 ) = . 6

It is straightforward to show that t0 (θˆ1 ) is negative decreasing for θˆ1 ∈ (2/3, 1), and that s0 (θˆ1 ) is decreasing in θˆ1 with 4 s0 (2/3) = p √ 9 63 + 11 33


23

and s0 (4/5) = 0, which implies θˆ1 ∈ (2/3, 4/5). Let 2 ˆ 4 ˆ M (r|0) = sup M0 (r, θ1 ) : < θ1 < . 3 5 θˆ1 Since t0 (2/3) = 0, M0 (r, θˆ1 ) is decreasing in θˆ1 for any r > 0. Therefore, M (r|0) = limθˆ1 →2/3 M0 (r, θˆ1 ) = s0 (2/3) for all r > 0. By definition of rδ , r0 = ∞. Similarly, for δ = 1 M1 =

(1 − θˆ1 )[(1 − θˆ1 ) − (θˆ1 − θ2 )] − r(1 − θ2 )(3θˆ1 − θ2 − 1) √ 1 − θ2

and (70) yields 6θˆ13 − 10θˆ12 − 6θˆ12 θ2 + 6θˆ1 + 5θˆ1 θ2 + 3θˆ1 θ22 + 3θ2 − 8θ22 + 3θ23 − 2 = 0. Using θ2 (θˆ1 ), we can write M1 (r, θˆ1 ) = s1 (θˆ1 ) + r · t1 (θˆ1 ) q h i−1/2 where s1 (θˆ1 ) = (1 − θˆ1 )[1 − 2θˆ1 + θ2 (θˆ1 )] 1 − θ2 (θˆ1 ) and t1 (θˆ1 ) = [1 − 3θˆ1 + θ2 (θˆ1 )] 1 − θ2 (θˆ1 ). Taking derivatives of s1 (θˆ1 ) and t1 (θˆ1 ) with respect to θˆ1 , we show that both s1 (θˆ1 ) and t1 (θˆ1 ) are decreasing in θˆ1 for all θˆ1 ∈ (2/3, 4/5). Therefore 2 ˆ 4 1−r ˆ M (r|1) = sup M1 (r, θ1 ) : < θ1 < = s1 (2/3) + r · t1 (2/3) = √ , 3 5 3 3 θˆ1 and r1 = 1.

The envelope function M (r|δ) defined in the Proof of Theorem 3 has the following properties. Corollary 5. For any δ ∈ (0, 1), M (r|δ) is strictly decreasing and convex in r as r ∈ [0, rδ ], where rδ solves M (rδ |δ) = 0. In addition, M (r|0) = √ 4 √ and M (r|1) = 3√1 3 (1 − r). 9

63+11 33

Proof of Corollary 5. From the Proof of Theorem 3 above 4 M (r|0) = p √ 9 63 + 11 33

and

1 M (r|1) = √ (1 − r). 3 3

Using θ2 = θ2 (θˆ1 ) which solves Pδ2 = 0, we can write M (r, θˆ1 , δ) = sδ (θˆ1 ) + tδ (θˆ1 )r. For convenience we use the slightly modified definition 2 4 ˆ ˆ M ε (r|δ) := max M (r, θ1 , δ) : + ε ≤ θ1 ≤ , 3 5 θˆ1 where ε > 0 is arbitrarily small. Let ra and rb be such that 0 < ra < rb ≤ rδ , and let 2 4 i ˆ ˆ ˆ θ1 = arg max M (ri , θ1 , δ) : + ε ≤ θ1 ≤ , i = a, b. 3 5 θˆ1 We want to show that M ε (rb |δ) < M ε (ra |δ). By definition of θˆ1a , sδ (θˆ1a ) + tδ (θˆ1a )ra ≥ sδ (θˆ1b ) + tδ (θˆ1b )ra . Suppose M ε (rb |δ) ≥ M ε (ra |δ), then sδ (θˆ1b ) + tδ (θˆ1b )rb ≥ sδ (θˆ1a ) + tδ (θˆ1a )ra , which implies sδ (θˆ1b ) + tδ (θˆ1b )rb ≥ sδ (θˆ1b ) + tδ (θˆ1b )ra and tδ (θˆ1b )(rb − ra ) ≥ 0. Then, tδ (θˆ1b ) ≥ 0, which contradicts the fact that Tδ (θˆ1 , θ2 ) < 0 (see Proof of Theorem 3). Therefore M ε (rb |δ) < M ε (ra |δ).


24

To show that M (r|δ) is convex, notice that it is defined as the pointwise supremum of a collection of linear functions which are convex, and the pointwise supremum of an arbitrary collection of convex functions is convex (Theorem 5.5 of Rockafellar 1997). Proof of Proposition 9. From the proof of Theorem 3, M1 (r, θˆ1 ) is decreasing in θˆ1 and θˆ1∗ ∈ (2/3, 4/5). For δ = 1, from the properties of θ2 (θˆ1 ), θ2∗ ∈ (3/5, 2/3) and θ2 (θˆ1 ) is decreasing in θˆ1 , therefore θ2∗ is increasing in M . The optimal quality levels can be written as functions of θˆ1 q   ˆ ˆ ˆ ˆ ˆ θ (1 − θ ) − M 1 − θ ( θ ) 1 1 2 1 1   and q2∗ (θˆ1 ) = θ1 + θ2 (θ1 ) − 1 . q1∗ (θˆ1 ) = 2α 2α 1 − θˆ1 + r(1 − θ2 (θˆ1 )) Using the implicit function theorem it is straightforward to show that d q1∗ /d θˆ1 > 0 and d q2∗ /d θˆ1 > 0 for θˆ1 ∈ (2/3, 4/5). Therefore, both q1∗ and q2∗ are decreasing in M . The market coverage results follow directly from the properties of θˆ1∗ and θ2∗ In addition, we state the comparative statics of the optimal solutions in the following proposition. Proposition 21. θˆ1∗ increases in α and λ, and decreases in β and c. θ2∗ increases in β and c, and decreases in α and λ. Both optimal quality levels q1∗ and q2∗ increase in λ and decrease in α, β and c. Both optimal prices p∗1 and p∗2 increase in λ and decrease in α, β and c. Moreover, the entire market coverage 1 − θ2∗ and the market coverage of the low quality product θˆ1∗ − θ2∗ increases in α and λ, and decreases in β and c. The market coverage of the high quality product 1 − θˆ1∗ increases in β and c, and decreases in α and λ. Proof of Proposition 21. By Theorem 3, θ2∗ can be fully characterized by θˆ1 . θ2∗ decreases in θˆ1 . θˆ1∗ is the solution of M=

(1 − θˆ1 )[(1 − θˆ1 ) − (θˆ1 − θ2∗ (θˆ1 ))] − r(1 − θ2∗ (θˆ1 ))(3θˆ1 − θ2∗ (θˆ1 ) − 1) q . ∗ ˆ 1 − θ2 (θ1 )

Let M = s1 (θˆ1 ) + t1 (θˆ1 )r. We have s1 (θˆ1 ) > 0, s01q (θˆ1 ) < 0, t1 (θˆ1 ) < 0 and t01 (θˆ1 ) < 0. By applying implicit function theorem to s1 (θˆ1 ) + αβ t1 (θˆ1 ) − 2 βc = 0, one can derive the following first order λ derivatives. q ˆ c β − t1 (αθ1 ) ˆ ˆ ˆ βλ t ( θ ) dθ1 dθ1 α2 1 1 >0 0 and thus

2

q1∗

1

increases in λ and decreases in β and

β t (θˆ ) dq1∗ (θˆ1 ) 1 1 α2 1 1 = − 2 (3θˆ1 − θ2∗ (θˆ1 ) − 1) + (3 − θ2∗ 0 (θˆ1 )) < 0, dα 2α 2α s01 (θˆ1 ) + β t01 (θˆ1 ) α

which implies

q1∗

decreases in α.


Similarly, by q2∗ (θˆ1 ) =

1 ˆ (θ1 2α

25

+ θ2 (θˆ1 ) − 1), we have dq2∗ (θˆ1 )/dθˆ1 > 0 and

β t (θˆ ) dq2∗ (θˆ1 ) 1 1 α2 1 1 = − 2 (θˆ1 + θ2∗ (θˆ1 ) − 1) + (1 + θ2∗ 0 (θˆ1 )) < 0. dα 2α 2α s01 (θˆ1 ) + β t01 (θˆ1 ) α

The optimal prices are given by p∗1 (θˆ1 ) = p∗2 (θˆ1 ) + θˆ1 (q1∗ (θˆ1 ) − q2∗ (θˆ1 )) and q ∗ (θˆ1 )M . p∗2 (θˆ1 ) = θ2∗ (θˆ1 )q2∗ (θˆ1 ) − q1 ∗ ˆ 2 1 − θ 2 (θ 1 ) By applying M , we have " 1 1 − 6θˆ13 + 5θ2∗ 2 (θˆ1 ) − 2θ2∗ 3 (θˆ1 ) + θˆ12 (15 + θ2∗ (θˆ1 )) − θˆ1 (6 + 9θ2∗ (θˆ1 ) − θ2∗ 2 (θˆ1 )) ∗ ˆ p1 (θ1 ) = 4α 1 − θ2∗ (θˆ1 ) β + (1 − 3θˆ1 + θ2∗ (θˆ1 ))2 α" 1 1 − 6θˆ13 + 5θ2∗ 2 (θˆ1 ) − 2θ2∗ 3 (θˆ1 ) + θˆ12 (11 + 5θ2∗ (θˆ1 )) − θˆ1 (6 + 5θ2∗ (θˆ1 ) + 3θ2∗ 2 (θˆ1 )) p∗2 (θˆ1 ) = 4α 1 − θ2∗ (θˆ1 ) β + (1 − 3θˆ1 + θ2∗ (θˆ1 ))2 α i i h h Let p∗ (θˆ1 ) = 1 B1 (θˆ1 ) + β (1 − 3θˆ1 + θ∗ (θˆ1 ))2 and p∗ (θˆ1 ) = 1 B2 (θˆ1 ) + β (1 − 3θˆ1 + θ∗ (θˆ1 ))2 . 1

4α

2

α

2

4α

α

2

Both B1 (θˆ1 ) and B2 (θˆ1 ) are increasing in θˆ1 . Since (1 − 3θˆ1 + θ2∗ (θˆ1 ))2 is also increasing in θˆ1 , p∗1 (θˆ1 ) and p∗2 (θˆ1 ) increase in θˆ1 . Therefore, both p∗1 and p∗1 increase in λ and decrease in α, β, and c. Proof of Proposition 10. As in the proof of Proposition 9, M0 (r, θˆ1 ) is decreasing in θˆ1 and θˆ1∗ √ ∈ (2/3, from the of θ2 (θˆ1 ), for δ = 0, θ2∗ ∈ √ 4/5) √ proof of Theorem 3. From the properties ∗ [5 3 − 11]/6 3, 3/5 and θ2 (θˆ1 ) is increasing in θˆ1 , therefore θ2 is decreasing in M . Writing the optimal quality levels as functions of θˆ1 q   ˆ ˆ ˆ ˆ ˆ ˆ ( θ − θ ( θ ))( θ + θ ( θ ) − 1) − M 1 − θ ( θ ) 2 1 1 2 1 2 1 θ1 1  1 , and q2∗ (θˆ1 ) = q1∗ (θˆ1 ) = ˆ ˆ ˆ 2α 2α θ1 − θ2 (θ1 ) + r(1 − θ2 (θ1 )) it is straightforward to show that d q1∗ /d θˆ1 > 0 and d q2∗ /d θˆ1 > 0 for θˆ1 ∈ (2/3, 4/5). Therefore, ∗ ∗ ∗ ˆ∗ qh1∗ and q q2 are decreasing in i.M . The properties of 1 − θ1 and 1 − θ2 follow directly. From θ2 = 5 − 24 (θˆ1∗ )2 − 48 θˆ1∗ + 25 6, θˆ∗ − θ∗ is decreasing in θˆ∗ and therefore increasing in M . 1

2

1

Proof of Proposition 11. Because Π0F (θˆ1 , θ2 , q1 , q2 ) > ΠD (θˆ1 , θ2 , q1 , q2 ) for any feasible θˆ1 , θ2 , q1 , ∗ and q2 , we have Π0∗ F (λ, α, c; β) > ΠD (λ, α, c; β). By using r2 and θ˜2 defined in the proof of Proposition 21, we can parameterize Π1F (θˆ1 , θ2 ) by θˆ1 and M . For any 0 < M < 3√1 3 , we have λ Π1F (θˆ1 , M ) = 4α

(

p ) 2 θˆ1 (1 − θˆ1 ) − M 1 − θ˜2 + (θˆ1 + θ˜2 − 1)2 (θˆ1 − θ˜2 ) (1 − θˆ1 ) + r2 (1 − θ˜2 )


26

“

and Π∗S (M ) ∗ Π1∗ F > ΠS .

=

λ 4α

” 2 −M 2 √ 3 3 1+r2

. One can show that Π1F (θˆ1 , M ) > Π∗S (M ) for any θˆ1 ∈

2 4 , 3 5

. Then,

Similarly, we have Π∗D (M ) parameterized by M since the optimal θˆ1∗ and θ2∗ can be uniquely determined by M for dedicated capacity case, i.e., ( q 2 2 ) q λ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ˆ ˆ ˆ ˆ ΠD (M ) = θ1 1 − θ1 − M + (θ1 + θ2 − 1) θ1 − θ2 − M . 4α(1 + r2 )

2 , it can be shown that Π∗D (M ) > Π1F (θˆ1 , M ). Therefore, Π∗D (λ, α, c; β) > For any 0 < M < 27 1∗ ΠF (λ, α, c; β). For any feasible θˆ1 , θ2 , q1 , and q2 , ΠD (θˆ1 , θ2 , q1 , q2 ), Π1F (θˆ1 , θ2 , q1 , q2 ) and Π0F (θˆ1 , θ2 , q1 , q2 ) mono0∗ tonically decrease in β so do Π∗D (λ, α, c; β), Π1∗ F (λ, α, c; β) and ΠF (λ, α, c; β).

Proof of Proposition 12. We apply the techniques used in the previous flexible capacity models to formulate the case with fixed per unit capacity cost. Let κ denote the per unit capacity cost. The firm’s problem is to maximize the following profit function ΠcF (θˆ1 , θ2 , q1 , q2 , µ) = θˆ1 q1 − αq12 (1 − θˆ1 )λ + (θˆ1 + θ2 − 1)q2 − αq22 (θˆ1 − θ2 )λ c(1 − θ2 )λ − − κµ µ − (1 − θ2 )λ c > 0 and 1 > θˆ1 > θ2 > 0. ΠcF (θˆ1 , θ2 , q1 , q2 , µ) is strictly concave subject to q1 > q2 > 0, θ2 q2 − µ−(1−θ )λ 2p in µ. We have µ∗ (θ2 ) = (1 − θ2 )λ + c(1 − θ2 )λ/κ. The profit function becomes p ΠcF (θˆ1 , θ2 , q1 , q2 ) = θˆ1 q1 − αq12 − κ (1 − θˆ1 )λ + (θˆ1 + θ2 − 1)q2 − αq22 − κ (θˆ1 − θ2 )λ − 2 κc(1 − θ2 )λ. θˆ1 One can show that ΠcF (θˆ1 , θ2 , q1 , q2 ) is jointly concave in q1 and q2 . Then, q1∗ (θˆ1 ) = 2α and q2∗ (θˆ1 , θ2 ) = θˆ1 +θ2 −1 . Therefore, the firm’s profit maximization problem can be written as 2α h io p λ nˆ [θ1 (1 + θ2 ) − θ2 (1 − θ2 ) − θˆ12 ](1 − θ2 ) − 4α κ(1 − θ2 ) + 2 κc(1 − θ2 )/λ max ΠcF (θˆ1 , θ2 ) = 4α θˆ1 ,θ2 p √ subject to 1 > θˆ1 > θ2 > 1/2 and θ2 (θˆ1 + θ2 − 1) 1 − θ2 > 2α κc/λ. ΠcF (θˆ1 , θ2 ) is strictly concave in θˆ1 . We can derive θˆ1∗ (θ2 ) = (1 + θ2 )/2. Eventually, the firm wants to maximize i p λ h ΠcF (θ2 ) = (1 − 2θ2 + 5θ22 )(1 − θ2 ) − 16ακ(1 − θ2 ) − 32α κc(1 − θ2 )/λ 16α p √ 1), (1 − 2θ2 + 5θ22 )(1 − θ2 ) subject to 1 > θ2 > 1/2 and θ2 (3θ2 − 1) 1 − θ2 > 4α κc/λ. When θ2 ∈ ( 12 , p is unimodal and achieves maximal as θ2 = 3/5. Since 16ακ(1 − θ2 ) + 32α κc(1 − θ2 )/λ is strictly decreasing in θ2 , θ2∗ which maximizes ΠcF (θ2 ) should be [ 53 , 1). One can show p located in the interval c ∗ that ΠF (θ2 √ ) is unimodal as 3/5 ≤ θ2 < 1 and < 1/10. θ2 is uniquely determined p √ α(κ + 10κc/λ) 2 by solving 1 − θ2 (14θ2 − 15θ2 − 3) + 16ακ 1 − θ2 + 16α κc/λ = 0. According to the properties of ΠcF (θ2 ), θ2∗ is increasing in κ, c and α, and decreasing in λ.

Proof of Proposition 13. Using the revenue function πSf to replace πS (p, q) in Section 3, we can follow the same approach to get the firm’s optimization problem in terms of θf . " #2 p (1 + θ ) 1 − θ λ f f max ΠfS (θf ) = −M (72) θf 4(α + β) 2 p p (1 + θf ) 1 − θf s.t. > M, 2θf 1 − θf > M. (73) 2


27

The objective function at θf∗ = 1/3. The constraints (73) yield feasibility √ (72) √ attains its maximum ˜ can be derived condition 0 ≤ M ≤ 2 2/3 3, which implies θf∗ ∈ [1/3, 1]. Then, qf∗ , µ∗f , and p∗f (θ) accordingly. Direct differentiation gives the comparative statics of the optimal solution. By comparing the formulation of qf∗ and µ∗f (in Proposition 13) with q ∗ and µ∗ (in Theorem 1), one can show that qf∗ > q ∗ , µ∗f > µ∗ and ΠfS∗ > Π∗S . f Proof of Proposition 14. The firm’s revenue function πD in this case is Z 1 Z θf 1 f f 2 ˜ ˜ ˜ f − cˆwf − αq 2 ]λdθ. ˜ πD = [θqf1 − cˆwσ1 − αqf1 ]λdθ + [θq σ2 f2 2 θf

1

θf

2

Proceeding as in Section 4 yields the firm’s optimization problem with θf1 and θf2 as decision variables. " #2 " #2  p p   (1 + θ ) 1 − θ (θ + θ ) θ − θ λ f1 f1 f1 f2 f1 f2 −M + −M max ΠfD (θfi ) = (74) θf ,i=1,2  4(α + β)  2 2 i s.t.

M M 1 − θf2 −p , >p 2 1 − θ θ − θ f f f 1 1 2 p (θf1 p + θf2 ) θf1 − θf2 > 2M, p 2θf1 1 − θf1 > M, 2θf2 θf1 − θf2 > M, 0 < θf2 < θf1 < 1.

(75) (76) (77) (78)

By the proof of Theorem 2, the optimization problem (74)–(78) can be characterized by a single decision variable θf1 with θf∗2 (θf1 ) = θf1 /3. Furthermore, the unique optimal solution of θf∗1 is determined by p 1 − θf1 (9 − 18θf1 + 5θf21 ) p M= . 6[3 − 9θf1 + 4 6θf1 (1 − θf1 )] √

As 0 ≤ M ≤ 2272 , θf∗1 spans the interval ( 13 , 35 ) and is decreasing in M . The optimal quality levels qf∗1 and qf∗2 and optimal capacities µ∗f1 and µ∗f2 can be derived based on the fact that all of them can be represented as pure functions of θf∗1 . We can characterize the optimal solution of the dedicated case by θˆ1∗ according to the proof 2 of Theorem 2. For any M ∈ (0, 27 ), one can derive 1 − θf∗1 > 1 − θˆ1∗ , θf∗1 − θf∗2 (θf∗1 ) > θˆ1∗ − θ2∗ (θˆ1∗ ), ∗ ∗ ∗ ˆ∗ ∗ ∗ ∗ ˆ∗ qf1 (θf1 ) > q1 (θ1 ), qf2 (θf1 ) > q2 (θ1 ), µ∗f1 (θf∗1 ) > µ∗1 (θˆ1∗ ), µ∗f2 (θf∗1 ) > µ∗2 (θˆ1∗ ), and ΠfD∗ (θf∗1 ) > Π∗D (θˆ1∗ ). Proof of Proposition 15. With full information, the firm’s revenue is identical to that of the dedf icated capacity case, i.e., πD = πFf . When bF (q1 , q2 ) = βq12 , the firm’s profit maximization problem can be written as h  i2 p (1+θ1f )(1−θ1f )    − M 1 − θ 2 2f 2 (θ1f − θ2f )(θ1f + θ2f )  λ f + (79) max ΠF (θfi ) = θf ,i=1,2  4α  (1 − θ1f ) + r(1 − θ2f ) 4 i   s.t.

p (1 + θ1f )(1 − θ1f ) > M 1 − θ2f , 2 p 2 (1 − θ1f ) − 2M 1 − θ2f > (θ1f + θ2f )[(1 − θ1f ) + r(1 −p θ2f )], 2 M [(1 − θ1f ) − 2M 1 − θ2f ] p (θ1f + θ2f )[(1 − θ1f ) + r(1 − θ2f )] > , 2θ2f 1 − θ2f 0 < θ2f < θ1f < 1.

(80) (81) (82) (83)

28


To solve the problem (79)–(83) we use the same approach as in the proof of Theorem 3. The √ 2 ∗ envelope function is given by M f (r|1) = 3√3 (1 − r), where 0 ≤ r < 1. For any feasible θ1f , θ2f (θ1f ) can be uniquely determined by solving 2 3 2 2 2 3 −3θ1f + 8θ1f − 6θ1f + θ2f − 4θ1f θ2f + 6θ1f θ2f + 4θ2f − 3θ1f θ2f − 3θ2f = 0. ∗ θ1f is determined by solving

M (θ1f ) =

2 ∗ ∗ 1 − θ1f − [3θ1f − θ2f (θ1f )]{(1 − θ1f ) + r[1 − θ2f (θ1f )]} q . ∗ 2 1 − θ2f (θ1f )

∗ ∗ For any θ1f ∈ 13 , 35 , θ2f spans the interval of 51 , 13 . The optimal quality levels qf∗1 and qf∗2 and the optimal capacity µ∗f can be derived according to the result that all of them can eventually be ∗ characterized as pure functions of θ1f . Since M f (r|1) > M (r|1) for any r ∈ (0, 1), we have to use M (r|1) as the envelope function to compare the optimal solutions of the full information case and partial information case. As 0 < M < M (r|1), we use θˆ1∗ to describe the optimal solution (such as ∗ qi∗ and µ∗ ) of the case with partial information, and θ1f for the full information case. Therefore, we ∗ ∗ ∗ ˆ∗ ∗ ∗ ∗ ˆ∗ can show that 1 − θ2f (θ1f ) > 1 − θ2 (θ1 ), qf1 (θf1 ) > q1 (θ1 ), qf∗2 (θf∗1 ) > q2∗ (θˆ1∗ ), µ∗f (θf∗1 ) > µ∗ (θˆ1∗ ), and ΠfF∗ (θf∗1 ) > Π∗F (θˆ1∗ ).