Optimal Preorder Strategy with Endogenous Information Control

MANAGEMENT SCIENCE

Vol. 57, No. 6, June 2011, pp. 1055–1077 issn 0025-1909 eissn 1526-5501 11 5706 1055

doi 10.1287/mnsc.1110.1335 © 2011 INFORMS

INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/.

Optimal Preorder Strategy with Endogenous Information Control Leon Yang Chu, Hao Zhang Marshall School of Business, University of Southern California, Los Angeles, California 90089 {[email protected], [email protected]}

I

n this paper, we investigate the integrated information and pricing strategy for a seller who can take customer preorders before the release of a product. The preorder option enables the seller to sell a product at an early stage when consumers are less certain about their valuations. We find that the optimal pricing strategy may be highly dependent on the amount of information available at preorder and that a small change in the latter may cause a dramatic change in the proportion of consumers who preorder under optimal pricing. Furthermore, the seller’s optimal information strategy depends on a key measure, the normalized margin, which is the ratio between the expected profit margin and the standard deviation of consumer valuation. Although the seller may want to release some information or none, she should never release all information. Finally, under the optimal information and pricing strategy, the benefit of preorder is most pronounced when the normalized margin is in a medium range. Key words: preorder; advance selling; information release; consumer valuation control History: Received August 18, 2009; accepted January 23, 2011, by Martin Lariviere, operations management. Published online in Articles in Advance April 21, 2011.

1.

Introduction

product is mediocre on those aspects. In the equilibrium, consumers can form a rational expectation (Muth 1961) of the product value. In other words, the mean of the consumer valuation is dictated by the design of the product, and the seller cannot systematically inflate consumers’ expectation through an information strategy.1 In contrast, the variance (or dispersion) of the consumer valuation is largely caused by product characteristics that split the consumer population—e.g., the taste of a yogurt product, the efficacy of a medicine, the screen finish (glossy versus matte) of a notebook computer, and the user interface of a smartphone. Such characteristics are assessed according to consumers’ idiosyncratic needs and preferences. If better informed, consumers can form more accurate personal valuations of the product, resulting in a larger valuation dispersion from the seller’s point of view. If the information is instead concealed, the dispersion would be smaller. Thus, the seller can control the dispersion of consumer valuation by limiting the amount of taste-related information in advertisements, demonstrations, announcements, etc., or by restricting the extent of customer tryouts. In this

With rapid development in information technologies, preorder has gained popularity in recent years, especially for books, CDs, video games, and software items, and the practice has attracted increasing attention from both practitioners and academics. Although various aspects of the preorder practice have been scrutinized by researchers, one important dimension has been largely neglected thus far. When a new product is available for preorder, consumers often do not have complete product information or clear personal preferences. A natural question arises as to whether a manufacturer (or seller) should release information to help consumers make better preorder decisions. Information plays a crucial role in shaping consumers’ valuation of the product, which can be represented by a distribution parameterized by its mean and variance. The mean valuation of a product is driven by the product’s characteristics or features that are universally valued—e.g., the memory space of a cellular phone, the battery life of an electronic reader, and the safety ratings of an automobile. Information control over such characteristics is straightforward, as suggested by the large literature on voluntary information provision (e.g., Grossman 1981, Milgrom 1981, Okuno-Fujiwara et al. 1990). Intuitively, if the product excels on those characteristics, the seller would disclose such information without reservation; otherwise, a rational consumer would assume that the

1

As discussed in Guo (2009), disclosure costs, information acquisition costs, and other factors may lead to partial disclosure of quality information. Information-related costs are beyond the scope of this paper. We thank an anonymous referee for this point. 1055

Chu and Zhang: Optimal Preorder Strategy with Endogenous Information Control

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Management Science 57(6), pp. 1055–1077, © 2011 INFORMS

Table 1

Preorder Information-Release Methods and Discounts for Top Video Games in 2010 Game


Name Star Trek Online BioShock 2 Heavy Rain Final Fantasy XIII God of War III Crackdown 2 Starcraft II Halo Reach

Platform

Information release Online mode

Windows Xbox 360/PS3/Windows PS3 Xbox 360/PS3 PS3 Xbox 360 OS X/Windows Xbox 360

paper, we focus on the control of consumer valuation dispersion at the preorder stage to maximize the seller’s total profit. To maximize profit, the seller should integrate information control with strategic pricing. As an example, Table 1 lists the top video games in 2010 with their information-release methods and preorder discounts, ordered by their release dates.2 Game publishers release game information through three common forms: (1) trailers, (2) playable demos, and (3) open beta versions (for online network play if the game has an online mode). Similar to the release of movies, trailers are provided for all video games to impress consumers with visual and sound effects. A playable demo enables consumers to experience the user interface and explore a small portion of the content. It can be distributed in various ways—e.g., the God of War III demo was preloaded on the Bluray disc of District 9. Finally, for games with a multiplayer online mode, an open beta version offers a comprehensive (sometimes nearly complete) preview of a game and familiarizes consumers with not only the user interface but also team dynamics of the game. For the eight games listed in Table 1, the retail price is $49.99 for Star Trek Online and $59.99 for all others. An interesting observation from Table 1 is that the preorder discount decreases as more information is released, i.e., as the information-release method changes from trailer to demo and to online beta. Preorder discount is commonly used to stimulate the preorder demand and compensate consumers for their possible losses due to valuation uncertainty. Intuitively, a small discount would be sufficient if consumers are well informed, whereas a large discount may be needed to attract consumers if the seller withholds information. 2

The list consists of the top three 2010 games on Xbox 360, PS3, and PC platforms, respectively, according to the video game website 1UP.com (the game Star Wars: The Old Republic has been delayed to 2011, and no preorder information is available). The information-release methods and preorder discounts are obtained through gaming website http://www.gamingbits.com, deal website http://www.dealcatcher.com, etc.

Yes Yes

Yes Yes Yes

Trailer Yes Yes Yes Yes Yes Yes Yes Yes

Demo

Beta (online)

Preorder discount ($)

Yes

0 15 5 10 10 10 0 20

Yes Yes Yes Yes Yes

Given numerous options for releasing information and setting prices, what is the seller’s optimal strategy? More information at preorder can help match consumers with the product better and improve social welfare but may also yield more surplus to the consumers, which poses an intriguing trade-off for the seller. In this paper, we investigate the seller’s optimal preorder strategy with endogenous information control. We are particularly interested in the following questions: How much information should the seller release at the preorder stage? How do the optimal preorder and retail prices depend on the amount of information released? And when will preorder be most beneficial to the seller? Our main findings and contributions are summarized below. • We propose a two-period, continuous-valuation, dispersion-control model based on solid microfoundations, with applications in advertising and consumer experiencing. It substantially extends the two-type valuation model predominant in the existing advance-selling literature and seamlessly integrates the seller’s information and pricing strategies. • We find that the seller’s optimal pricing strategy may be highly dependent on the amount of information available at preorder. A small perturbation of the latter may trigger a sudden switch of the former between two competing strategies at preorder: one aims at the high end of the market and the other aims for high market participation. • The seller’s optimal information strategy at preorder is determined by the nature of the product and the consumer population, represented by the ratio between the expected profit margin and the standard deviation of consumer valuation (referred to as the “normalized margin”). When the ratio is large, the seller should withhold information and offer a large preorder discount to capture a large portion of consumers at preorder; when the ratio is small, the seller should release a great deal of information and offer a small preorder discount to attract only high valuation consumers; and when the ratio is in between, the optimal strategy also depends on the amount of information consumers initially have. In any case, the




seller should never release all information, in contrast to the extreme information strategy in the literature. • We further find that the benefit of preorder to the seller is most pronounced when the normalized margin lies in a medium range. The remainder of this paper is organized as follows. In §2, we review the literature related to preorder. In §3, we discuss the model, its applications, and microfoundations. We study the seller’s optimal pricing strategy in §4 and optimal information strategy in §5. Section 6 provides some extensions of the results, and §7 concludes.

2.

Literature Review

Though not precisely defined, preorder is commonly regarded as a special form of advance selling, and the two bodies of literature are intertwined. As defined by Xie and Shugan (2009), advance selling refers to the general practice that a seller induces buyers to commit to purchasing a good before the time of consumption, which can take many different forms. In contrast, preorder usually refers to the practice that the seller allows buyers to purchase a product at a particular price until a specified time prior to its release. For example, Hui et al. (2008) analyze the preorder and sales pattern for DVDs through a consumer behavior model with optimal stopping. Because preorder is often associated with the introduction of new products, the study of preorder pricing is related to the literature on strategic pricing of new products and services, as in Chaterjee (2009). Sellers engage in advance selling (or preorder) for many plausible reasons. It may help them update demand forecasts and manage inventory, as discussed by Chen (2001), Moe and Fader (2002), Tang et al. (2004), and Li and Zhang (2010). It may also help the sellers effectively segment the market, especially under limited capacity. Capacity control is crucial to the field of revenue management, which has wide applications in the hospitality industry where late customers are often willing to pay higher prices; see Boyd and Bilegan (2003) and Talluri and van Ryzin (2004) for comprehensive reviews of this literature. Xie and Shugan (2009) demonstrate that if capacity is limited and rationing is possible, the optimal price scheme may entail shutdowns or preorder premiums rather than discounts. DeGraba (1995) shows that a seller may even benefit from purposely restricting its capacity and generating buying frenzies. Liu and Xiao (2008) study a seller’s optimal return policy and inventory rationing policy when inventory is endogenously determined through a newsvendor model. Zhao and Stecke (2009) and Li and Zhang (2010) both study the advance selling (preorder) strategy of a newsvendor retailer who must procure all

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units before the regular selling season starts. Liu and van Ryzin (2008) and Su and Zhang (2008) show that capacity and inventory considerations may enhance the seller’s credibility in charging a high retail price. In contrast to the above studies, we focus on situations such as the Windows 7 upgrade, a Harry Potter book release, and a video game release, in which the seller has ample capacity. If the seller can match the demand through continuous production and is not restricted by capacity or inventory, the problem can be framed as a sequential screening problem in which consumer valuation is private information at both the preorder and release dates. In this setting, Courty and Li (2000) show that the seller’s optimal pricing policy consists of a menu of refund contracts, pairing different preorder prices with different cancellation fees. Akan et al. (2009) show that a menu of refund contracts with various expiration dates is optimal when each consumer learns his or her valuation at some random time. Xie and Shugan (2001) also note that a well-designed refund option can benefit the seller, and the topic is investigated further by Gallego and S¸ ahin (2010). However, optimal refund contracts differ noticeably from the simple transactions prevalent in the preorder practice, in which a single preorder price and a single retail price are announced at the preorder date. We restrict our attention to the latter form of price mechanism in this paper. All of the above papers assume exogenous information structures and focus on the seller’s pricing decision. In reality, the seller can often influence consumer valuation through information control. Few papers in the literature have studied the seller’s information strategy. Among the few exceptions, Lewis and Sappington (1991, 1994) examine an extension of the standard screening model such that the seller can alter the probability that consumers receive perfect private information. They find that the seller will always set the probability at zero or one, corresponding to zero or full information release. Johnson and Myatt (2006) find that the seller’s profit is typically a U-shaped function of the dispersion of consumer valuation, which again leads to extreme information release. However, these papers only consider the static setting in which the seller interacts with consumers once, unlike the two-period setting when preorder is allowed. Recently, extreme information release has also been studied in the operations management literature. For instance, Shulman et al. (2009) investigate whether a seller should inform customers of their match with the product before their purchase given that they can return or exchange the product later, subject to a restocking fee. In this paper, we adopt a two-period continuous-valuation model and investigate the optimal (possibly intermediate) level of information release.


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Figure 1

Sequence of Events

Preorder stage


Seller decides information-release and pricing strategies.

Retail stage

Seller releases product information and announces prices p0 and p . 1

Seller releases the product.

Each buyer forms Each buyer decides her initial type 0 whether to preorder information and prices. at price p0.

3.

The Model

In this section, we introduce the model, with an emphasis on the evolution of the consumer valuation and the seller’s control over it. We discuss two applications of the model, in advertising and consumer experiencing, and two microfoundations of the model. 3.1.

Sequence of Events, Consumer Valuation Evolution, and Seller’s Information Control In the model, a monopolistic seller sells a product to a set of consumers (the seller will be referred to as “she” and a consumer as “he” hereinafter). A consumer has a unit demand and an idiosyncratic utility from consumption. The seller’s marginal production cost is constant, denoted by c > 0. The seller releases the product at time (or stage) 1 but can take preorders at time (stage) 0. The preorder price p0 and retail price p1 are announced in advance and all consumers are informed.3 We assume that the seller can commit to the retail price at the preorder stage because she can earn a higher expected profit in doing so.4 Consumers are risk-neutral and maximize their expected utilities. Each consumer learns his utility in two steps: at time (stage) 0, he learns the initial valuation 0 ; then a random shock occurs, and he learns his final valuation 1 = 0 + at time (stage) 1. The final valuation is the consumer’s true utility from consuming the product. Without loss of generality, we 3

The model can be easily extended to accommodate the situation in which only a subset of consumers is aware of the preorder opportunity. The optimality of adopting preorder will continue to hold, and the discontinuity in the optimal pricing and information-release strategies will continue to exist. 4

To establish credibility, the seller may (1) announce the preorder discount and offer a refund to preorder consumers if the retail price is later reduced (e.g., Amazon offers a preorder price guarantee), (2) build the reputation for a high spot price (Xie and Shugan 2009), or (3) artificially set capacity and inventory limitations.

Each buyer discovers her final type 1.

Each remaining buyer decides whether to purchase at price p1.

assume that has a zero mean. Thus, a consumer’s initial valuation 0 can be viewed as an expectation of his true valuation 1 , given the available information at the preorder stage. This dynamic valuation process describes a common phenomenon that consumers learn their valuations of a product gradually over time. Both 0 and 1 are consumers’ private information and will also be referred to as their initial type and final type, respectively. The distributions of 0 and 1 0 (or ) are common knowledge, denoted by F 40 5 and G41 0 5, respectively. The sequence of events is illustrated in Figure 1. A consumer’s initial valuation of a product, 0 , before he actually owns or even sees the product, can be influenced by many factors, one of which (perhaps the most important) is how much information the consumer has about the product. The seller can play a crucial role in this by selectively releasing product information, through advertisements, exhibitions, professional reviews, etc. However, a consumer’s intrinsic valuation of a product, represented by his final valuation 1 , can hardly be altered by the seller’s information handling. To capture these important facts, we assume that the distribution of the final valuation 1 is exogenously determined but the seller can alter the distributions of the initial valuation 0 and the valuation shock . For analytical tractability and consistency with the tradition of the literature (to be discussed soon), we focus on the normal-normal setting described below. 1. The unconditional distribution of a consumer’s intrinsic valuation 1 follows a normal distribution N 41 2 5, with > 0. The variance 2 measures the dispersion of the true consumer valuation (we will use “dispersion,” “variance,” and “heterogeneity” interchangeably hereinafter). This distribution is determined by the characteristics of the product and the consumer population and is independent of the information available at the preorder stage.




2. The seller can control the variances of 0 and via a variable ∈ 401 15 such that 0 ∼ N 41 2 5 and ∼ N 401 41 − 52 5. The variable measures the amount of information available at the preorder stage, and we call the preorder information intensity parameter. By changing , the seller can allocate valuation uncertainty between 0 and 1 0 . Because < 1, the distribution of 0 is more concentrated around the mean than that of 1 , and consumers are more homogeneous at stage 0 than at stage 1. 3.2. Consumer Valuation Control Applications We discuss two applications of the valuation control model, to search goods and experience goods. In economics, a “search good” refers to a product the characteristics of which can be evaluated through information acquisition prior to its purchase (Stigler 1961)—e.g., the style of a dress and the hardware specifications of a computer. In contrast, an “experience good” is dominated by characteristics that can only be evaluated through consumption (Nelson 1970, 1974)—e.g., the taste of a yogurt product, the user experience of a software product, and the effectiveness of a medicine. Advertising is a common and effective way for releasing information on a search good. The seller can influence the dispersion of consumer valuation at the preorder stage through advertisements. As modeled by Johnson and Myatt (2006), the seller may send an advertisement signal to consumers at time 0; conditional on a consumer’s intrinsic valuation 1 ∼ N 41 2 5, the distribution of is given by N 41 1 2 5. Consumers form their expectation of 1 at time 0 based on the advertisement signal; that is, a consumer’s initial type 0 is his conditional expectation of 1 given . Notice that the same advertisement can send distinct signals to different consumers, depending on their intrinsic preferences. Standard calculation yields 0 = E41 5 = 4 2 + 2 5/4 2 + 2 5, and hence 0 ∼ N 41 4 /4 2 + 2 55 and 1 0 ∼ N 40 1 2 2 /4 2 + 2 55. The variance 2 measures the noise in the advertisement signal, which can be controlled by the seller. At one extreme, a perfectly informative advertisement helps consumers learn their true valuations of the product—i.e., sending a perfect signal = 1 to a consumer with intrinsic valuation 1 ; at the other extreme, a totally uninformative advertisement sends a completely noisy signal ∼ N 41 1 5 to the same consumer. By defining = 2 /4 2 + 2 5, we have 0 ∼ N 41 2 5 and 1 0 ∼ N 40 1 41−52 5, which is consistent with the valuation control model. A preorder setting differs from a typical experiencegood situation because consumers do not actually own the product at the preorder stage. However,

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as argued by Klein (1998), the Internet can “virtually” transform certain experience characteristics into search characteristics and experience goods into search goods. Such transformations may be done at the preorder stage, especially for software products that can be easily distributed through the Internet. As in the video game example discussed in the introduction, the seller can grant consumers limited access to the product prior to its launch and enable them to gain partial experience. Dynamic consumer valuation is central to the experience-good literature, and a key building block in many of the consumer learning models (e.g., Ackerberg 2003, Crawford and Shum 2005, Osborne 2011) is the following: the prior belief (actual distribution) of the consumer valuation of the product is given by 1 ∼ N 41 2 5, a signal ∼ N 41 1 2 5 is generated from the trial of the product, and consumers update their beliefs subsequently. This is a special hierarchical Bayesian model, and normal distributions are chosen according to the theory of conjugate distributions (DeGroot 1970). Because this model is mathematically identical to the advertising model discussed above, the discussion following that model is also applicable here. The variable = 2 /4 2 + 2 5, negatively related to 2 , measures the effectiveness of the experience in revealing the idiosyncratic consumer valuations (or resolving the valuation uncertainty). 3.3.

Consumer Valuation Control Microfoundations In this subsection, we provide two microfoundations (mechanisms) for controlling the preorder information intensity in the valuation control model (or signal noise 2 in the advertising and consumer experiencing models). They justify the proposed continuous-valuation model; a simple two-type model as commonly seen in the preorder literature would be too coarse to fully describe the seller’s information control capability. As commonly assumed in the marketing literature, a product can be described by a set of characteristics and consumers form their valuations based on their preferences for these characteristics. As discussed in the introduction, we focus on the dispersion of consumer valuation and on characteristics that may appeal to some consumers but displease others. These characteristics can be either advertised (e.g., the shape, color, keyboard type, and carrier of a cellular phone) or experienced (e.g., subtle features of the user interface of an operating system or a video game). Thus, the following model can serve as a microfoundation for both the advertising and consumer experiencing models. Assume that a product has n characteristics and that a consumer’s true valuation of the product is


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P V = + ni=1 Xi , where is the mean valuation and each Xi ∈ 811 −19 captures the variation in the valuation of characteristic i. If the consumer learns the specification of characteristic i, his actual Xi is realized; otherwise, he only knows its expectation. For simplicity, assume that each characteristic of the product polarizes the consumer population independently—i.e., P 4Xi = 15 = P 4Xi = −15 = 005. At stage 0, the seller may release information on (or limit a consumer’s access to) n0 ≤ n characteristics. Then a consumer’s valuation of the product at stage 0 is Pn0 givenP by V0 = + i=1 Xi and that at stage 1 is V V0 = V0 + ni=n0 +1 Xi , contingent on V0 . The valuations V0 and V V0 follow the distributions + 2B4n0 1 0055 − n0 and V0 + 2B4n − n0 1 0055 − 4n − n0 5, respectively, where B4n1 p5 represents the binomial distribution with n trials and success probability p. When n is large enough and p is not near 0 or 1 (e.g., both np and n41 − p5 are greater than 5), B4n1 p5 can be closely approximated by the normal distribution N 4np1 np41 − p55 (Box et al. 2005). Therefore, V0 and V V0 can be approximated by the normally distributed 0 and 1 0 in the basic model. When the number of characteristics n is fixed, the distribution of a consumer’s total valuation V (approximated by 1 ) is fixed. The amount of information available at preorder is determined by the number of characteristics n0 revealed at preorder, and n0 /n corresponds to exactly. The above model is essentially a simple random walk model with +1 and −1 steps and is a special case of the additive martingale model of forecast evolution (Graves et al. 1986, 1998; Heath and Jackson 1994). The next model provides an alternative microfoundation for controlling consumer experience, in the spirit of the hierarchical Bayesian random-effects model (e.g., Lenk et al. 1996). Assume that the seller can control the extent of consumer experience by limiting the time of experience or the contents to be experienced. Suppose that the distribution of the intrinsic consumer valuation is 1 ∼ N 41 2 5 and the ith unit of “experience time” or “experience sample” yields a signal xi ∼ N 41 1 2 5 (xi ’s are independent of each other given 1 ). If the seller limits the total experience time (or contents) at n units per consumer, then a consumer’s posterior valuation given 4x1 1 0 0 0 1 xn 5 follows the dis2 2 2 2 tribution 1 xP 1 1 0 0 0 1 xn ∼ N 40 1 /4 + n 55, with n 2 2 2 2 0 = 4P+ 4 i=1 xi 5 5/4 P + n 5 (DeGroot 1970). Because ni=1 xi = n1 + ni=1 i for i ∼ N 401 2 5, 0 is distributed according to 0 ∼ N 41 4n4 /42 + n2 55. Define = n2 /42 + n2 5, and we again arrive at the basic model, with 0 ∼ N 41 2 5 and 1 0 ∼ N 40 1 41 − 52 5. The seller can control through n: the more experience is allowed at the preorder stage (n increases), the more valuation uncertainty is resolved early ( increases). The basic notion behind


this model is that more thorough experience creates more heterogeneous valuations following the experience, which is backed by empirical studies. For instance, Osborne (2011) finds that consumers in a laundry detergent market have similar expectations prior to experience but become very heterogeneous afterward. In the video game example presented in the introduction, after selecting a basic method of information release (trailer, demo, or beta version, corresponding to different ranges of ), the seller can further fine tune the extent of consumer experience— e.g., by carefully choosing the size and contents of the demo (the demos of Heavy Rain and Crackdown 2 are 1.3 GB and 0.9 GB large, respectively).

4.

Seller’s Optimal Pricing Strategy

In this section, we study the seller’s optimal pricing strategy when the valuation distributions F 40 5 and G41 0 5 are exogenously given. We first formulate the seller’s problem in terms of threshold consumer types and then streamline the model around two key parameters—information intensity at preorder, , and normalized margin of the product, 4 − c5/. After that, we discuss properties of the seller’s profit function and show that it is optimal for the seller to exercise both preorder and retail. Finally, we characterize the optimal solution through numerical analysis. In §5, we will allow the seller to control the amount of information at the preorder stage, i.e., altering distributions F and G, and study the seller’s informationrelease strategy. 4.1.

Seller’s Pricing Problem, Consumer Valuation Thresholds, and Normalized Margin The seller tries to maximize her profit, whereas consumers try to maximize their expected utility given the preorder price p0 and retail price p1 . If the distributions F and G are given, the seller solves the following optimal pricing problem: Z max 4p0 −c5 x40 5 dF 40 5 p0 1 p1 1 x4·51 y4·5 0 Z Z +4p1 −c5 y41 5 dG41 0 5 41−x40 55 dF 40 5 (1) 0

1

( 1 1 ≥ p1 1 s.t. y41 5 = 0 1 < p1 3 Z   1 0 −p0 ≥ y41 541 −p1 5dG41 0 51 Z1 x40 5 =  0 0 −p0 < y41 541 −p1 5dG41 0 50

(2)

(3)

1

Constraint (2) describes the purchase behavior of a type 1 consumer at stage 1, and constraint (3) describes that of a type 0 consumer at stage 0. To


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focus on the joint effect of pricing and information release instead of capacity or inventory planning, we assume that the seller has ample capacity and can perfectly match supply with demand. In addition, we assume without loss of generality that a consumer will purchase the product if he is indifferent between buying and not buying. Constraint (2) states that consumers’ purchase decision at the retail stage follows a simple threshold policy. The theorem below shows that their purchase decision at the preorder stage also follows a threshold policy, which implies that preorder enables the seller to capture high valuation consumers at an early stage. Theorem 1. Given the preorder price p0 and retail price p1 , a consumer with initial type 0 would preorder if and only if 0 ≥ 0 , and he would purchase at the retail stage (given that he has not preordered) if and only if his final type 1 ≥ 1 , where 0 and 1 are determined by 1 = p1 1 p0 =

Z

p1 −

(4)

° 1 0 5 1 dG41 0 5 + p1 G4p

= p1 −

Z

p1 −

G41 0 5 d1 0

(5)

A sketch of the proof is presented here for completeness, and more details are provided in Appendix A. Because a consumer purchases the product at stage 1 if and only if 1 − p1 ≥ 0, it can be shown thatR he should preorder at stage 0 if and only p1 if p1 −p0 ≥ − 4p1 −1 5 dG41 0 5. That is, he preorders if and only if the preorder discount p1 − p0 exceeds R p1 his potential regret, 4p − 1 5 dG41 0 5, or equiv− 1 R p1 alently, − G41 0R5 d1 .5 Because G41 0 5 decreases p1 with 0 for any 1 , − G41 0 5 d1 decreases with 0 and hence a consumer’s optimal preorder strategy is a thresholdRpolicy. p1 Because − G41 0 5 d1 is monotone in 0 , 0 is uniquely determined by p0 and p1 from expression (5). Thus, the price pair 4p0 1 p1 5 and threshold-type pair 40 1 1 5 have a one-to-one correspondence, and the seller’s pricing problem (1)–(3) can be reformulated based on the threshold types: max ç40 1 1 5 = 4p0 40 1 1 5 − c5F°40 5 0 1 1

° 1 0 < 0 5F 40 51 (6) + 41 − c5G4 5

We assume that a consumer derives negative utility from the product if his final type 1 is negative. If the consumer can R p dispose of the product costlessly, his potential regret would be 0 1 G41 0 5 d1 . In our model, due to the normal distributions of the types, when > 3, the probability of a negative final valuation and the difference between the two regret expressions are negligible. We thank Guillermo Gallego for this comment.

where p0 40 1 1 5 = 1 −

Z

1

−

G41 0 5 d1

° 1 0 5 + = 1 G4

Z

1

−

1 dG41 0 50

(7)

° 1 0 < 0 5F 40 5 is a shortIn the objective function, G4 R 0 R + hand notation for − 1 dG41 0 5 dF 40 5. Notice that setting 0 = + (or p0 ≥ p1 ) is equivalent to a pure-retail strategy. By the assumption of the valuation control model that 0 ∼ N 41 2 5 and 1 √0 ∼ N 40 1 41 − 52 5, we have F 4 √0 5 = ê440 − 5/ 5 and G41 0 5 = ê441 − 0 5/ 1 − 5, where ê4 · 5 and later 4 · 5 denote the cumulative distribution function (c.d.f.) and probability density function (p.d.f.) of the standard normal distribution. Clearly, the seller’s problem (1)–(3) or (6)–(7) is fully determined by four parameters 41 c1 1 5. The next result shows that the problem is essentially determined by only two measures: the normalized margin, defined as z = 4 − c5/, and the preorder information intensity, ∈ 401 15. The measure z captures the intrinsic profitability of the product, whereas captures the amount of information available at the preorder stage. Theorem 2. Given the normalized margin z = 4 − c5/ and preorder information intensity , the seller solves the following normalized model without loss of generality: Z 0 − − z 0 ° 1 ° 0 − z + 1 ê d ê 0 1 max p0 ê 0 1 1 0 1 0 − (8) where 1 − 0 1 − 0 ° p0 = 0 − 1 + 41 − 0 5ê 1 1 1 √ √ 0 = 1 and 1 = 1 − 0 If the solution to the normalized model is 40∗ 1 1∗ 5, the solution to the original model (6)–(7) is 40∗ + c1 1∗ + c5 and the optimal profit under the original model is times the normalized one. The normalized model can be viewed as a special instance of the original model with marginal cost c = 0, standard deviation = 1, and mean valuation = z. Because of the simple relationship between the normalized model and the original one, we will focus on the normalized model in the rest of this paper. 4.2.

Seller’s Profit Function and Optimality of Preorder The seller’s total profit ç40 1 1 5 consists of the preorder profit ç0 40 1 1 5 from stage 0 and the retail profit


1062 Figure 2


Profit Functions ç0 40 1 1 5, ç1 40 1 1 5, and ç40 1 1 5, from Left to Right, for 4z1 5 = 411 0025


Preorder profit

Retail profit

ç1 40 1 1 5 from stage 1. By the normalized model (8), we have ° 0 −z ç0 40 11 5 = p0 ê 0 1 −0 1 −0 ° = 0 −1 +41 −0 5ê 1 1 ° 0 −z 1 (9) ·ê 0 Z 0 − −z ° 1 0 d ê 0 ê ç1 40 11 5 = 1 1 0 − Z 0 ° 1 −0 0 −z d0 0 ê = 1 (10) 0 − 1 0 The three profit functions, ç0 40 1 1 5, ç1 40 1 1 5, and ç40 1 1 5, are illustrated in Figure 2, from left to right, for the instance 4z1 5 = 411 0025.6 As shown in the figure, the graphs of ç0 40 1 1 5 and ç1 40 1 1 5 resemble two mountain ridges, roughly parallel to the two axes, and the sum of the two, ç40 1 1 5, resembles an L-shaped ridge. This non-concavity is an important feature of the seller’s total profit function. One implication of this fact, as will soon be seen, is the possible existence of multiple optimal solutions, which is rarely mentioned in the existing preorder (advanceselling) literature. To facilitate subsequent analysis, we examine the first-order conditions next. Proposition 1. The first-order condition ¡ç40 1 1 5 =0 ¡0 is equivalent to 0 − 6

° 0 − z5/0 5 1 4ã/1 5 0 ê44 = 1 440 − z5/0 5 ê4ã/1 5

(11)

The attentive reader may have noticed that the preorder and total profits are negative at 40 1 1 5 = 401 05. This is because 0 = 1 = 0 corresponds to p0 < p1 = 0.

Total profit

where ã = 1 − 0 . For any ã ∈ 4−1 +5, Equation (11) is satisfied by a unique 0 . The first-order condition ¡ç40 1 1 5/¡1 = 0 is equivalent to Z 0 − −ã −z ° ã ê ° 0 −z + ê 0 0 d ê 0 ê 1 0 1 0 − 2 2 2 − 0 ã − 1 z = 1 41 − z5ê 1 0 0 (12) 0 1 Equation (11) enables us to express 0 as a function of ã and the total profit function ç40 1 1 5 as ç40 4ã51 ã + 0 4ã55, which reduces the twovariable optimization problem to a one-variable problem that can be reliably solved by standard computing software such as MATLAB. This lays the foundation for the numerical analysis in the remainder of this paper. Next, we show that it is always optimal for the seller to sell at both stages. Theorem 3. Assume ∈ 401 15 or neither F nor G is a one-point distribution. Then the optimal solution to the seller’s problem (6)–(7) or (8) must satisfy 0 < + and 1 < +. That is, it is optimal for the seller to adopt both preorder and retail. The proof of this result builds on the monotone hazard-ratio properties of the normal distribution: ° ° (1) ê45/45 decreases with and lim→+ ê45/45 = 0, and (2) ê45/45 increases with and lim→− ê45/45 = 0. In other words, the result holds for any distributions with such properties. For a fixed 1 (or p1 ), choosing 0 < + has two effects. On one hand, it leaves a positive surplus to consumers with high initial valuations. On the other hand, it increases market participation, because a consumer with an initially high but finally low valuation would purchase the product through preorder but not through retail. The proof of the theorem suggests that the second effect dominates the first. Therefore, for


1063

a fixed p1 , both the seller and the consumers benefit from the preorder option: the seller gets the chance to segment the market twice and meet different consumers in different stages, and consumers receive an offer of price discount. As a result, the seller’s profit, aggregate consumer surplus, and social welfare all improve. This differs from the result of Shugan and Xie (2004) that in a discrete-valuation model the optimality of preorder depends on model parameters in general. 4.3.

Optimal Solution Through Numerical Analysis In this subsection, we investigate the seller’s optimal pricing strategy through representative examples, with the same normalized margin z = 005 but different preorder information intensity . The discussion also prepares us for the detailed analysis of the impact of in the next section. Benchmark Case: Pure Retail 4 = 15. We start with the case in which the seller uses retail exclusively, which is equivalent to choosing = 1 so that consumers are fully informed at the preorder stage. In this situation, the seller should charge the optimal pure-retail price 009220. At this price, 33065% of the consumers purchase the product, and the seller’s profit, aggregate consumer surplus, and social welfare are 003103, 002229, and 005332, respectively. Case I: = 005. The seller’s profit function ç40 1 1 5 in this scenario is illustrated in Figure 3 through a contour map, with contour interval 000025 (the profit difference between consecutive contour lines). For the sake of clarity, the profit function is truncated after 20 contour levels. The map demonstrates a unique local (and global) maximizer 40 1 1 5 = 41014121 1000475, with seller profit 003147. This solution corresponds to preorder and retail prices 4p0 1 p1 5 = 40078561 1000475. The seller captures high valuation consumers early by setting the preorder (retail) price lower (higher) than the pure-retail price 009220. Compared with the benchmark case without preorder, the change in consumer surplus given the preorder option depends on the initial valuation 0 : consumers with low initial valuations will see their surplus decline due to the higher retail price, whereas those with high initial valuations will enjoy higher surplus due to the preorder discount and relatively low risk of regret. We depict consumer surplus for various initial types in Figure 4(a). The solid and dotted lines represent consumer surplus with and without the preorder option, respectively. Figure 4(b) illustrates the (normal) probability density function of the initial type 0 in the given range. The proportion √ of consumers who preorder is given ° 0 − z5/ 5, as shown in expression (8). In this by ê44 example, the proportion is 1802%. That is, the seller

Figure 3

Contour Map of the Seller’s Total Profit ç40 1 1 5, for 4z1 5 = 40051 0055

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1



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focuses on the high end of the market at the preorder stage, and we say that the seller adopts a niche preorder pricing strategy. Because consumers are reasonably well informed about the product when = 005, the total market participation with the preorder option is similar to that without it (the benchmark case). The seller’s profit increases by 104% (to 003147), the aggregate consumer surplus decreases by 303% (to 002156), and the social welfare declines by less than 1% (to 0.5303). Case II: = 000193. The contour map of the seller’s total profit in this case is displayed in Figure 5, with contour interval 0000075. For clarity, the profit function is truncated after 10 levels. The map reveals two optimal solutions 40 1 1 5 = 40070301 0096705 and 40042421 1090595, with the same seller profit 003112 (the map also reveals a saddle point at 40052761 1028765). The two optimal solutions represent two distinctive pricing strategies. (1) The valuation thresholds 40 1 1 5 = 40070301 0096705 correspond to prices 4p0 1 p1 5 = 40042601 0096705. At this optimal solution, the total market participation, the seller’s profit, aggregate consumer surplus, and social welfare are all similar to those in Case I. Even though the seller offers a larger preorder discount than in Case I, the preorder proportion is actually lower (at 702%), because consumers are less certain about their valuations now, as is much smaller. The seller again adopts a niche preorder pricing strategy at this solution. (2) The valuation thresholds 40 1 1 5 = 40042421 1090595 correspond to prices 4p0 1 p1 5 = 40039491 1090595. Instead of capturing only high valuation consumers at preorder, the seller offers a deep discount to induce


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Figure 4

Consumer Surplus (a) and p.d.f. (b) of the Initial Type 0 , for 4z1 5 = 40051 0055

Figure 5

Contour Map of the Seller’s Total Profit ç40 1 1 5, for 4z1 5 = 40051 0001935

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With preorder

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the majority (7007%) of consumers to preorder, which can be called a mass preorder pricing strategy. Compared with the benchmark case, although the participation rate increases dramatically, the social welfare in fact decreases by 20% due to the inefficiency in allocating the good to ill-informed consumers who will likely regret later (the relatively low normalized margin z = 005 makes this situation particularly relevant). The difference in consumer surplus with and without the preorder option also becomes larger, as can be seen from Figure 6(a). The aggregate consumer surplus declines by more than 40% because consumers are almost homogeneous at the preorder stage and the seller can extract their surplus (or information rent) relatively easily. The combination of consumer surplus reduction and social welfare loss improves the seller’s profit by less than half a percent. This example contrasts with the result of Xie and Shugan (2001) that advance selling benefits the seller through the increase of market participation, not the reduction of buyer surplus. Comparing Case II with Case I, we observe that the seller can improve her profit by increasing from 000193 to 005, i.e., releasing more information at the

preorder stage. Although the social welfare is lower with the preorder option in both cases, we remark that one can easily find examples in which market participation, social welfare, and seller’s profit all increase significantly with the preorder option, especially when z is relatively large. The scenario z = 005 is chosen in this subsection for its connection with a later discussion in §5.3. General Results. Extensive numerical analysis shows that the seller’s pricing problem (8) given z and may have a unique local optimum, as in Case I above, or two local optima, as in Case II. The two optima situation occurs only for z ∈ 40022671 0058625, the “medium margin” scenario in §5.3. When it happens, one of the local optima corresponds to a niche preorder pricing strategy and the other a mass preorder one; the latter dominates the former when is small but is overtaken by the former as increases. When the normalized mean z is smaller, the niche preorder optimum is the unique local optimum; when z is large enough, the mass preorder optimum is the only one. Another general result is that for any given z, the optimal preorder discount p1 − p0 decreases with and approaches 0 as approaches 1, as illustrated in Figure 7. This is consistent with the video game example in the introduction, in which smaller discounts are coupled with more information. Notice that the curve for z = 005 is discontinuous at = 000193 because the optimal preorder and retail prices are discontinuous at this , as evident from the gap between the two optimal solutions in Figure 5. A thorough investigation of the impact of the preorder information intensity is the subject of the next section. If the seller releases more information, social


1065


Consumer Surplus (a) and p.d.f. (b) of the Initial Type 0 , for 4z1 5 = 40051 0001935 and 40 1 1 5 = 40042421 1090595

Figure 6

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welfare can be improved through more efficient allocations, but consumers can obtain higher information rent as well. How should the seller strike a balance between information rent extraction and social welfare improvement? This will be a key question as we study the seller’s information-release strategy. Figure 7

Seller’s Optimal Information Strategy

In the above analysis of the seller’s pricing strategy, the preorder information intensity was exogenously given. In the real world, however, the seller can often control the amount of product information at the preorder stage. The more information the seller releases (the larger the ), the better a consumer understands the characteristics of the product and his own preferences at the preorder stage. The existing literature on information release is restricted to the static environment in which the seller and consumers interact only once (e.g., Lewis and Sappington 1991, Johnson and Myatt 2006), and hence the finding that it is optimal to release all or nothing may not apply in the dynamic preorder setting. When the seller releases all information, a consumer’s initial valuation 0 fully reveals his final valuation 1 and the maximum profit attainable by the seller is the same as that under pure retail, which is suboptimal by Theorem 3. Therefore, full information release is never optimal in our model, as summarized below. Corollary 1. If the seller can choose the preorder information intensity in problem (8), the optimal must be strictly less than 1. That is, full information release is suboptimal.

1.5

0.5

Optimal Preorder Discount for z ∈ 801 0051 1059 and ∈ 401 15

6 z = 0.0 z = 0.5 z = 1.5

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4 3 2 1 0 0.1

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The departure from the existing literature indicates that the trade-off between consumer surplus and social welfare is more complex in a dynamic environment such as ours. To find the seller’s optimal information strategy, we start with the following benchmark case. 5.1.

Benchmark Case: No Information at Preorder 4 = 05 In the extreme case in which consumers have no initial information about the product and the seller does not release any information at all, the preorder information intensity is given by = 0, and consumers’ initial valuations reduce to a single point, 0 = z. Then the seller should choose either preorder or retail instead of both because either all consumers or none will preorder. Notice that this situation differs from the one in the previous section, in which it is assumed that > 0, or 0 is normally distributed, and hence Theorem 3 applies (nevertheless, it is shown in Appendix B that the seller’s profit function and optimal pricing strategy are continuous at = 0). We consider the seller’s two extreme strategies, pure retail and pure preorder, next. Pure Retail and Pure Preorder. When the seller chooses pure retail, consumers can only learn their (true) valuations at stage 1 and would purchase if and only if 1 ≥ p1 . (This situation is equivalent to setting = 1, where consumers learn their true valuations


Figure 8

p ′′

7

With a unit of consumers, the demand curve can be obtained from the c.d.f. of the valuation through a 90 counterclockwise rotation and proper relabeling of the axes, as can be seen in Figure 8.

z

p′

0.5

Quantity (b)

Proposition 2. Given = 0, the seller should choose either preorder or retail, and the former is better than the latter if and only if z > z† = 002267. In other words, the seller prefers completely homogeneous consumers to extremely heterogeneous ones if and only if z > z† = 002267. Rotation of the Demand Curve. Proposition 2 implies that if the seller only sells at a single stage, the right choice between pure preorder (homogeneous consumers) and pure retail (heterogeneous ones) depends on the normalized margin z. This can be explained through the rotation of the demand curve, introduced by Johnson and Myatt (2006). If all consumers have the same valuation z, the demand curve is flat; when consumers become more and more heterogeneous, the demand curve rotates clockwise.7 When z is large enough, the optimal static price p should be smaller than z to capture the majority of consumers—i.e., the seller would adopt a mass-market pricing strategy; a clockwise rotation of the demand curve would result in fewer sales at the price p, and hence the seller would prefer more homogeneous consumers and withholding information. This scenario is illustrated in Figure 8(a) for p0 < z. On the contrary, when z is small or even negative, the optimal static price p ought to be larger than z to capture high valuation consumers only— i.e., the seller would opt for a niche-market pricing strategy; a clockwise rotation of the demand curve (by releasing more information) would bring in more sales and benefit the seller, as illustrated in Figure 8(a) for p00 > z.

Demand Curve Rotation in the (a) One-Period Setting and (b) First Period of a Two-Period Setting

(a)

Price

at stage 0 and the preorder and retail stages essentially collapse into one.) Because the distribution of 1 is ê41 − z5, the seller’s pricing problem reduces to a static problem, çR 4z5 = maxp pê4z − p5, and the optimal price satisfies the first-order condition ê4z − pR 5 − pR 4z−pR 5 = 0, or pR = ê4z − pR 5/4z − pR 5 (the superscript “R” stands for pure “r”etail). It is straightforward to show that both pR 4z5 and çR 4z5 increase with z and 4çR 50 4z5 = ê4z − pR 5 ∈ 401 15, which approaches 1 as z goes to infinity. If the seller wants all consumers to preorder, she can set the preorder price p0 = z and retail price p1 = +, generating profit çP 4z5 = z (“P ” stands for pure “p”re-order). Because 4çP 50 4z5 = 1 > 4çR 50 4z5, there exists a single threshold z† such that çP 4z5 ≥ çR 4z5 if and only if z ≥ z† , which can be found through straightforward computation. We obtain the following result immediately:


0′′ Price


1066

p0′′ z

0′ p0′

0.5

Quantity

When the seller adopts both preorder and retail, the initial valuation threshold 0 is brought into the picture, and Figure 8(b) demonstrates the rotation effect of the preorder demand curve. We will see that the above effect remains a driving factor, but the result is more intricate, especially for intermediate z’s. Information Release: Impact of on Seller’s Profit In this subsection, we examine how the seller’s profit is affected by a small change of . Previously, the seller’s profit function was expressed as ç40 1 1 5, in terms of the valuation thresholds 40 1 1 5. It can be written in terms of the prices 4p0 1 p1 5 as well, which turns out to be more convenient in explaining the impact of . By the envelope theorem, if p0 45 and p1 45 are the optimal prices given , the derivative of the seller’s profit with respect to , dç41 p0 451 p1 455/d, equals ¡ç41 p0 451 p1 455/¡, and it suffices to investigate the impact of given p0 and p1 , at their optimal values. When the seller sells at both stages, the purchasing threshold at stage 0 is given by 0 and the pre° order profit is given by ç0 41 p0 1 p1 5 = p0 ê4u5, where 5.2.


1067



° u = 40 − z5/0 and ê4u5 gives the proportion of consumers who preorder. Because ¡ ° ¡u 1 d ¡ ê4u5 = −4u5 = 2 4u5 40 − z5 0 − 0 0 1 ¡ ¡ 0 d ¡ the change in the proportion of preorder consumers is driven by two effects simultaneously: (1) a demandrotation effect on marginal (preorder) consumers, measured by 40 − z5d0 /d, that may increase or decrease the preorder demand depending on the sign of 0 − z; and (2) a regret effect on marginal consumers, measured by −0 ¡0 /¡, that always increases the preorder demand, as shown below: Lemma 1. The partial derivative of the preorder threshold 0 with respect to the preorder information intensity is given by ¡0 41 p0 1 p1 5 1 44p1 − 0 5/1 5 =− < 0. ¡ 21 ê44p1 − 0 5/1 5

(13)

When the prices p0 and p1 are fixed, as increases, consumers become more heterogeneous at the preorder stage, and hence the preorder demand curve rotates clockwise as in Figure 8(b). In the meantime, consumers become more assertive about their true valuations at the preorder stage and as a result are more willing to purchase early to secure the preorder discount, which pushes 0 downward and drives the demand up. When z < 0 (which implies z < 0 < p0 < 0 ), as more information is available at preorder, both the demandrotation effect (the scenario z < p000 < 000 in Figure 8(b)) and the regret effect propel the preorder demand, so the aggregate preorder proportion increases. Proposition 3. When z < 0, given p0 and p1 , the ° proportion of consumers who preorder, ê4u41 p0 1 p1 55, increases with , or equivalently, ¡u41 p0 1 p1 5/¡ < 0. In the other direction, we can show that as z approaches infinity, the proportion of preorder consumers approaches 1 and the demand rotation effect suppresses the preorder demand: Proposition 4. For any < 1, there exists a z0 such ° that for all z > z0 , ê4u5 > under the optimal pricing policy. ° Because ê4u5 > 005 is equivalent to z > 0 , the proposition implies that the demand-rotation effect reduces the preorder demand when z is large enough (the scenario z > 00 > p00 in Figure 8(b)), in opposite to the regret effect. Further numerical analysis verifies that when z is large enough the demand° rotation effect dominates the regret effect and ê4u5 decreases with (or ¡u/¡ > 0) under the optimal pricing policy.

The change of also impacts the seller’s retail profit, Z 0 p − − z 0 ° 1 ç1 41 p0 1 p1 5 = p1 ê d ê 0 0 1 0 − Because 0 − z ¡ Z 0 ° p1 − 0 ê d ê ¡ − 1 0 −ã ¡u 1 ã =ê 4u5 − 4u5 1 ¡ 20 1 1 (derived in the proof of Theorem 4), for fixed p0 and p1 , the change in retail demand also depends on two effects: (1) a demand-substitution effect on marginal consumers, captured by the first term above, that influences consumer participation at the retail stage depending on the sign of ¡u/¡, the oppo° site of 4¡/¡5ê4u5; and (2) a regret effect on low valuation consumers, captured by the second term, which always dampens the retail demand because with more accurate initial valuations, the lower ê4u5 quantile of consumers is less likely to buy at the retail stage. The next theorem aggregates the profit changes at both stages. The functions u41 p0 451 p1 455 and ã41 p0 451 p1 455 are simply denoted by u and ã in the theorem. Theorem 4. The derivative of the seller’s total profit with respect to is given by dç41 p0 451 p1 455 d ¡u p1 45 ã ° = −0 ê4u5 − 4u5 1 ¡ 20 1 1 √ √ where 0 = , and 1 = 1 − .

(14)

The first term on the right-hand side of Equation (14) aggregates the profit changes from marginal consumers whose initial valuations are near 0 . The second term captures the change from the lower ê4u5 quantile of consumers who do not preorder. The theorem implies that if more information results in lower preorder demand (¡u/¡ > 0), which occurs when the normalized margin z is large enough as mentioned above, the seller should withhold information at the preorder stage. This theorem facilitates the investigation of the seller’s information-release strategy at preorder. 5.3. Optimal Information Strategy To reflect the fact that the seller may be forced to disclose certain product information at the preorder stage and consumers may have certain initial knowledge about the product beyond the seller’s control, we extend the basic model so that consumers


1068 Figure 9


Seller’s Optimal Information-Release Strategy Map 1.0 Upper threshold *(z) Lower threshold *(z)

0.8

0


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Withhold information

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can possess some initial information at stage 0, represented by the initial information intensity 0 ∈ 601 15. The total amount of information that can be released by the seller at preorder is therefore limited to ∈ 60 1 17. The seller’s optimal information strategy clearly depends on 0 and the normalized margin z. In Figure 9, we draw a strategy map for the seller on how much additional information to release for any given z and 0 . The map can be best explained in three sections, according to the value of z. High-Margin Section. When z is relatively large (z > 005862), the seller’s optimal profit ç4z1 5 decreases with in the entire interval 601 17. In this region of the strategy map, the seller should not disclose any information in addition to the initial information already gained by consumers. Because the expected margin is large enough, the seller should adopt a mass preorder strategy and use preorder effectively to capture average consumers. Consequently, she should withhold information. As an example, the optimal profit function ç4z1 5 for z = 1 is illustrated in Figure 10(a). The two extreme scenarios are ç4z1 05 = çP 4z5 = 1 (via pure preorder, at price 1) and ç4z1 15 = çR 4z5 = 005066 (via pure retail, at price pR 4z5 = 101317). The profit function is strictly decreasing in ∈ 601 17. Thus, the seller should not release any additional information regardless of 0 . Low-Margin Section. When z is small (z < z† = 002267), the seller’s optimal profit function ç4z1 5 is quasi-concave in and has a unique maximizer ∗ 4z5 ∈ 401 15. In this region of the strategy map, the seller should release information up to ∗ 4z5 if consumers have less information to begin with—i.e., 0 < ∗ 4z5—and should withhold information otherwise. Because the expected margin is low, consumers are skeptical at stage 0. The seller should adopt a niche preorder strategy and release a great deal of information to stimulate demand from high valuation

consumers; on the other hand, she should also withhold some information so as to benefit from both preorder and retail (Corollary 1). The case z = 0 is illustrated in Figure 10(c), with extreme scenarios ç4z1 05 = ç4z1 15 = çR 4z5 = 001700 (via pure retail, at price 007518). The case z = −002 is illustrated in Figure 10(d), with ç4z1 05 = ç4z1 15 = çR 4z5 = 001288 (via pure retail, at price 006940). We observe that both profit functions have a single peak and that the optimal ∗ 4z5 increases as z decreases because more information is needed to stir up the preorder demand when z is smaller. Medium-Margin Section. When z is in the middle (002267 < z < 005862), the seller’s optimal information strategy is more intriguing because the profit function ç4z1 5, for ∈ 601 17, has two local maxima, corresponding to two preorder strategies: a mass preorder strategy and a niche preorder one. By choosing a tiny (≥0 ), the seller may effectively capture the majority of consumers at the preorder stage when consumers are almost homogeneous; by choosing a larger and injecting more heterogeneity into the consumer valuation, the seller may attract high valuation consumers at the preorder stage, which can be particularly useful when z is relatively small. Which strategy is better for the seller is affected by how much information consumers initially possess. We find that when z < 002288 the seller should continue to adopt the niche preorder strategy as she does in the low-margin situation, by releasing a large amount of information up to the threshold ∗ 4z5. When z > 002288 and 0 is smaller than a certain threshold ∗ 4z5, the seller should adopt the mass preorder strategy and withhold information as she does in the high-margin situation. However, when z > 002288 but 0 ∈ 4∗ 4z51 ∗ 4z55, the seller should adopt the niche preorder strategy and release information up to ∗ 4z5. Intuitively, when the information about a somewhat niche product is leaked to the extent


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Figure 10

Optimal Profit ç4z1 5 as a Function of , When (a) z = 1, (b) z = 005, (c) z = 0, or (d) z = −002

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that the mass market becomes too costly to attract, it would be wise for the seller to primarily focus on high valuation consumers. Finally, we note that the upper threshold ∗ 4z5 decreases with z whereas the lower threshold ∗ 4z5 increases with z. At z = 005862, the two thresholds meet at ∗ 4z5 = ∗ 4z5 = 001320. The case z = 005 is illustrated in Figure 10(b). The two extreme scenarios are ç4z1 05 = çP 4z5 = 005 (via pure preorder, at price 005) and ç4z1 15 = çR 4z5 = 003103 (via pure retail, at price 009220). The seller’s profit function ç4z1 5 has two local optima at = 0 and 003330, one interior local minimum at = 000193, and the global optimum is located at = 0. The function is nonsmooth at = 000193 because the seller is indifferent between the niche and mass preorder pricing strategies at this and the optimal pricing strategy jumps from mass preorder to niche preorder as passes through 000193 from below (Case II, §4.3). According to the information strategy map (Figure 9), when z = 005, if 0 < ∗ 4z5 = 000171, the seller should choose the mass preorder strategy and withhold information; if ∗ 4z5 < 0 < ∗ 4z5 = 003330, he should adopt the niche preorder strategy and release information up to ∗ 4z5; if 0 > ∗ 4z5, the seller should withhold further information to maximally benefit from the preorder option (Corollary 1).

6.

Further Discussions

6.1. Relative Gain from Preorder When does preorder benefit the seller the most? To answer this question, we compute the seller’s relative gain from the preorder option under the optimal information and pricing strategies against the benchmark case of pure retail without preorder. The ratio 4ç4z1 5 − çR 4z55/çR 4z5 is depicted in Figure 11, where is optimally chosen for each z. The line 0 = 0 describes the scenario in which consumers have no initial information. In this scenario, the preorder option has only minor impact on the Figure 11

The Seller’s Profit Gain from Preorder Under the Optimal Strategy, Relative to the Pure-Retail Profit

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4.0


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seller’s profit when z is small, because the seller needs to release a great deal of information to stir up demand for a low-margin product, which at the same time blurs the distinction between preorder and retail and limits the gain from preorder. The gain becomes significant when z increases above 002288 and the seller switches from the niche preorder strategy to the mass one, as discussed in §5.3. When z becomes too large, however, pure retail becomes a compelling choice for the seller—she can capture almost the entire market without sacrificing much relative profit—so the room for further improvement through preorder shrinks. To describe the more realistic situation with nonzero 0 , additional lines for 0 = 001, 002, and 005 are drawn in Figure 11. As 0 increases, the gain from the preorder option declines because the seller can only operate in a smaller information strategy domain. Nevertheless, the gain is still substantial for medium z’s when a mass preorder strategy is viable (e.g., more than 10% at z = 3, given 0 = 005). 6.2. Risk-Averse Consumers We have focused on risk-neutral consumers throughout the paper. Here we show that it is still optimal for the seller to adopt preorder if consumers are risk averse; interestingly, her profit is even higher in this case. For simplicity, we assume that a consumer derives total utility u4−p5 + g from the good purchased at price p or u405 if no purchase is made, where u0 > 0 and u00 < 0. At stage 1, the consumer knows exactly his utility derived directly from the good, g (which varies across consumers). To facilitate the comparison between the risk-neutral and risk-averse cases, we define the final type 1 as the maximum money the consumer is willing to pay for the good—i.e., satisfying u4−1 5 + g = u405. At stage 0, the consumer knows his initial type 0 and the conditional distribution of 1 0 , while the seller only knows the distributions of 0 and 1 0 . We assume that 1 = 0 + and both 0 and follow independent normal distributions as in §3.8 We have the following result: Theorem 5. Given any preorder and retail prices, consumers’ purchase decisions at both stages are threshold policies. The proof is analogous to the proof of Theorem 1 and is omitted. Then, we can show the following: Theorem 6. It is optimal for the seller to practice preorder when consumers are risk averse. 8

In §3, the initial type 0 coincides with the expected valuation of the good for each consumer and has a natural explanation. When consumers are risk averse, 0 is no longer the expected valuation (or the maximum money a consumer is willing to pay) at the preorder stage. Nevertheless, similar analysis still carries through. The essence of 0 is that it partially reveals a consumer’s final (true) type 1 .


The preorder option enables consumers with high initial valuations to secure the product at a relatively low price early. These consumers likely have high true valuations as well, and preorder provides them insurance against possible high future payments. If they are averse to monetary risk, they would be willing to pay a premium at the preorder stage, as compared to risk-neutral consumers.9

7.

Conclusion

In this paper, we investigate the seller’s preorder strategy with endogenous information control and identify the seller’s optimal information and pricing strategies. The seller’s optimal preorder strategy generally entails selling at both the preorder and retail stages with a preorder discount, and it may aim at the mass market or the high valuation niche depending on the normalized margin of the product (which combines the profitability of the product and the dispersion of consumer valuation into a single measure) and the amount of information at the preorder stage. When the margin is large, the seller should adopt a mass preorder strategy by withholding information and offering a large preorder discount; when the margin is small, the seller should adopt a niche preorder strategy by releasing a great deal of information and offering a small preorder discount; and when the margin is in between, the optimal strategy depends on the amount of information consumers initially possess. We also find that although preorder benefits the seller across the board, the relative gain is most pronounced when a mass preorder strategy is viable (the normalized margin is not too low) and retail alone is inadequate to realize the profit potential (the margin is not too high either). In the future, we would like to investigate other factors that influence the preorder practice, including return policies, inventory and capacity considerations, heterogeneity in the information noise level, logistic costs at different stages, etc. Given the existence of alternative theories on the preorder practice, empirical studies present another promising research direction. 9

As an example, assume that a consumer’s true valuation of the product (final type) 1 is either $20 or $0 with equal probability, and the preorder and retail prices are 0 < p0 < p1 = $10. If the true valuation turns out to be $20, the consumer’s utility would be u4−p0 5 + u405 − u4−205 through preorder (where g = u405 − u4−205 is the utility derived from the good itself) or u4−105 + u405 − u4−205 through retail purchase, and his net gain from preorder would be u4−p0 5 − u4−105. If the true valuation is $0, the consumer’s utility would be u4−p0 5 through preorder or u405 otherwise, and his net gain from preorder would be u4−p0 5−u405 (which is negative). Thus, the indifferent preorder price for this consumer satisfies 0054u4−p0 5 − u4−1055 + 0054u4−p0 5 − u4055 = 0, or u4−p0 5 = 005u405 + 005u4−105. If the consumer is risk neutral (where u is linear), we obtain p0 = $5; if he is risk averse (where u is concave), we have p0 > $5.


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Acknowledgments

and 425 for any h > 0,

The authors are grateful to Martin Lariviere (the department editor), the anonymous associate editor, and three reviewers, whose comments and suggestions have improved the paper significantly. The authors also thank David Sappington; Guillermo Gallego; conference participants at the Overseas Chinese Scholars Association in Management Science and Engineering annual conference and the Manufacturing and Service Operations Management annual conference; and seminar participants at University of California Riverside, University of California Davis, University of Michigan, Columbia University, Northwestern University, Duke University, the Hong Kong Polytechnic University, and the Chinese University of Hong Kong for their valuable feedback.

Appendix A. Proofs Proof of Theorem 1. If a consumer waits until time 1, he will buy if and only if 1 −p1 ≥ 0, which is a threshold policy with the threshold type 1 = p1 . At time 0, the consumer buys the product if and only if his expected net utility from buying is no less than that from waiting—i.e., Z + 0 − p0 ≥ 41 − p1 5 dG41 0 50 p1

Recall that 0 = E6 R 1p1 0 7. The above expression can be ° 1 0 5, where simplified to p0 ≤ − 1 dG41 0 5 + p1 G4p ° G4p1 0 5 = well defined, R p11 − G4p1 0 5. Because E61 0 7R is p1 we have − 1 dG41 0 5 = p1 G4p1 0 5 − − G41 0 5 d1 , by integration by parts. Therefore, Rthe consumer preorders p1 the product if and only if p1 − p0 ≥ − G41 0 5 d1 . Because G41 0 5 = G41 − 0 05 is decreasing in 0 , the consumer’s optimal preorder strategy is also a threshold policy with the threshold type 0 determined by expression (5). √ √ Proof of Theorem 2. We define 0 = and 1 = 1 − throughout this proof. By expression (7) and the fact that 0 4x5 = −x4x5, we have Z 1 ° 1 0 5 p0 = 1 dG41 0 5 + 1 G4 − Z 1 − 0 ° 1 − 0 = 1 dê 1 + 1 ê 1 1 − Z 1 − − 1 0 0 = 1 d1 1 1 − Z 1 − 0 0 ° 1 − 0 + 1 d1 + 1 ê 1 1 − 1 Z 41 −0 5/1 Z 41 −0 5/1 = 1 x1 4x1 5 dx1 + 0 4x1 5 dx1 − − ° 1 − 0 + 1 ê 1 − 0 − 0 ° 1 − 0 0 = −1 1 + 0 ê 1 + 1 ê 1 1 1 Therefore, given any 0 1 1 ∈ and 1 > 0, we have 415 for any h ∈ , p0 40 + h1 1 + h1 1 5 = p0 40 1 1 1 1 5 + h,

(A1)

p0 4h0 1 h1 1 h1 5 = hp0 40 1 1 1 1 5.

(A2)

The above expressions indicate a linear relationship between the preorder price p0 and the threshold types 0 and 1 (1) shifting the threshold types by h (keeping 1 constant) shifts the preorder price by h as well, and (2) scaling the threshold types (along with the standard deviation 1 ) by a factor of h scales the preorder price by h as well. This relationship provides the basis for normalizing the model parameters below. Define z = 4 − c5/, i0 = 4i − c5/, and i0 = i /, for i = 01 1. Then Equations (A1) and (A2) imply p0 40 1 1 1 1 5 − c = p0 440 − c5/1 41 − c5/1 1 /5 = p0 400 1 10 1 10 5, and the seller’s objective function (6) can be rewritten as 0 ° 0 − z ç40 1 1 5 = p0 400 1 10 1 10 5ê 00 0 0 Z 0 0 0 ° 1 − 0 d ê 0 − z 0 + 10 ê 10 00 − √ √ Notice that 00 = and 10 = 1 − , which depend only on , and that the scaled profit function ç40 1 1 5/ depends on , c, and only through z = 4 − c5/. Therefore, the seller’s problem can be simplified by using parameters 4z1 5 and decision variables 400 1 10 5, which is the normalized model. To simplify expressions, we use the same notation as in the original model—i.e., 0 , 1 , 0 , 1 , etc. Proof of Proposition 1. (1) From expression (7), the partial derivatives of p0 with respect to 0 and 1 , respectively, are given by ° 1 0 5 ¡p0 40 1 1 5 Z 1 ¡ G4 = d1 1 ¡0 ¡0 −

(A3)

¡p0 40 1 1 5 ° 1 0 50 = G4 ¡1

(A4)

From Equation (A3), it is straightforward to show that ¡p0 40 1 1 5 ã =ê 0 (A5) ¡0 1 (Note that the notation ã is used only to simplify the expression. When taking the partial derivative with respect to 0 , we hold 1 constant, not ã. The same is true for the expressions below.) From (9), (10), and (A5), we obtain ¡ç40 11 5 ã ° 0 −z =ê ê ¡0 1 0 1 ã −z ° ã − 0 −1 +ã ê 0 0 1 1 0 −z ° ã + 1ê 0 0 1 0 ã ° 0 −z =ê ê 1 0 1 ã ã −z − 0 ê −1 0 0 0 1 1 0 Therefore, the first-order condition ¡ç40 1 1 5/¡0 = 0 is equivalent to Equation (11).


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It can be verified that the right-hand side of (11) is strictly decreasing in ã and that the left-hand side is strictly increasing in 0 because of the monotone hazard-ratio properties of the normal distribution. As ã varies from − to +, the right-hand side varies from + to 0; as 0 varies from − to +, the left-hand side varies from − to + as well. Therefore, for any ã ∈ 4−1 +5, there is a unique 0 satisfying Equation (11). (2) By Equation (A4), it is straightforward to show that ° ¡p0 40 1 1 5/¡1 = ê4ã/ 1 5, where ã = 1 −0 . By definition, ¡ç0 40 1 1 5 ° ã F°40 5 = ê ° ã ê ° 0 − z 1 =ê ¡1 1 1 0 Z 0 − − z − z ¡ç1 40 1 1 5 1 = ê 0 0 d 0 ¡1 1 0 0 − Z 0 − 1 −z −z − 1 0 0 d 0 0 1 − 1 0 0 The last term can be simplified as follows: Z 0 − − z − z 1 0 d 0 0 1 0 0 − Z 0 1 1 −z 2 2 2 2 = √ e−40 −1 5 /421 5 √ e−40 −z5 /420 5 d 0 0 − 2 2 Z 0 1 −z 1 2 2 2 2 2 2 = √ e−40 −0 1 −1 z5 /420 1 5 √ e−41 −z5 /2 d 0 0 − 2 2 Z 0 1 2 −0 1 −12 z 2 2 2 2 2 = 1 41 −z5 √ e−40 −0 1 −1 z5 /420 1 5 d 0 0 1 − 2 2 2 − 0 1 − 1 z = 1 41 − z5ê 0 0 0 1 Therefore, we have ¡ç40 11 5 ° 0 −z ° ã ê =ê ¡1 1 0 Z 0 − −z + ê 0 1 d ê 0 1 0 − 2 −0 1 −12 z −1 41 −z5ê 0 0 1 ° ã ê ° 0 −z =ê 1 0 Z 0 − −ã −z + ê 0 0 d ê 0 1 0 − 2 1 0 −02 ã −12 z −40 +ã540 +ã −z5ê 1 0 1 and ã satisfies the following equation at optimality: Z 0 − 0 − ã −z ° ã ê ° 0 − z + ê ê 0 d ê 0 1 0 1 0 − 2 2 2 − 0 ã − 1 z = 40 + ã540 + ã − z5ê 1 0 0 0 1 To prove Theorem 3, we show the following lemma first. The proof relies on the monotone hazard ratio properties of the distribution G rather than the specific normal distribution or the normalization of the parameters.


Lemma A1. Given any 1 < +, the ratio R 1 G41 0 5 d1 − G41 0 5 is decreasing in 0 and approaches 0 as 0 approaches infinity. Proof. Notice that ¡ ¡0

R 1

−

G41 0 5 d1 G41 0 5

has the same sign as Z 1 ¡ Z 1 ¡ G41 0 5 G41 0 5 d1 − G41 0 5 G41 0 5 d1 0 ¡0 − ¡0 − Because ¡ ¡ G41 0 5 = − G41 0 5 ¡0 ¡1

and

¡ G41 0 51 ¡1 R 1 the aboveR expression equals −G41 0 5 − g41 0 5 d1 + 1 g41 0 5 − G41 0 5 d1 . Because G41 0 5/g41 0 5 ≤ G4 R 1 1 0 5/g41 0 5 for any R11 < 1 , we have −G41 0 5 · g41 0 5 d1 + g41 0 5 − G41 0 5 d1 ≤ 0. Therefore, R − 1 G41 0 5 d1 /G41 0 5 is decreasing in 0 . − Because R 1 R 1 G41 0 5 d1 G41 0 5 d1 G41 0 5 − R = − ≥ 1 ≥0 g41 0 5 G4 g4 5 d 1 0 5 1 0 1 − g41 0 5 =

and G41 0 5/g4 R 1 1 0 5 approaches 0 as 0 approaches infinity, the ratio − G41 0 5 d1 /G41 0 5 approaches 0 as 0 approaches infinity by the sandwich theorem. Proof of Theorem 3. Because problems (6)–(7) and (8) are equivalent, we consider the former problem in this proof. (1) We show the first part, 0 < +. If the seller excludes the preorder option, she must charge 1 > c at time 1 to obtain a positive payoff. It suffices to show that for any 1 > c, the partial derivative of ç40 1 1 5 with respect to 0 is negative as 0 → +. Expressions (6) and (7) imply ¡ç40 1 1 5 ¡0 Z 1 ¡G4 5 1 0 = − d1 F°40 5 ¡0 − Z 1 + G41 0 5 d1 f 40 5 + 4c − 1 5G41 0 5f 40 50

(A6)

−

Because Z 1 ¡G4 5 1 0 − d1 F°40 5 ¡0 −

and

Z

1 −

G41 0 5d1 f 40 5

are positive and 4c − 1 5G41 0 5f 40 5 is negative, it suffices to show that the ratios between the positive terms and the negative term approach 0 as 0 approaches infinity. Notice that Z Z 1 1 ¡G4 5 1 0 − d1 F°40 5 = g41 0 5 d1 F°40 5 ¡0 − − = G41 0 5F°40 50

Chu and Zhang: Optimal Preorder Strategy with Endogenous Information Control Management Science 57(6), pp. 1055–1077, © 2011 INFORMS

The ratio between G41 0 5F°40 5 and 4c − 1 5G41 0 5f 40 5 is


1 F°40 5 1 4c − 1 5 f 40 5 which approaches 0 as 0 approaches infinity. The ratio R 1 between 4 − G41 0 5 d1 5f 40 5 and 4c − 1 5G41 0 5f 40 5 is R 1 G41 0 5 d1 1 − 1 4c − 1 5 G41 0 5 which approaches 0 as 0 approaches infinity, by Lemma A1 above. Therefore, given any 1 , it is optimal for the seller to choose 0 < + and exercise preorder. (2) We show the second part that 1 < +. It suffices to show that for any 0 , the partial derivative with respect to 1 is negative as 1 → +. The first-order condition with respect to 1 implies ¡p 4 1 5 ¡ç40 1 1 5 ° 1 0 < 0 5F 40 5 = 0 0 1 F°40 5 + G4 ¡1 ¡1 + 4c − 1 5g41 0 < 0 5F 40 50

(A7)

° 1 0 < 0 5F 40 5 are Because 4¡p0 40 1 1 5/¡1 5F°40 5 and G4 positive and 4c − 1 5g41 0 < 0 5F 40 5 is negative for large 1 , it suffices to show that the ratios between the positive terms and the negative term approach 0 as 1 approaches infinity. Notice that Z 0 ° 1 0 < 0 5 = ° 1 0 5d F 40 5 G4 and G4 F 40 5 − Z 0 F 40 5 g41 0 < 0 5 = g41 0 5d 0 F 40 5 − ° 1 0 5/g41 0 5 is increasing in 0 , the ratio Because G4 ° 1 0 < 0 5 and g41 0 < 0 5 is bounded by between G4 ° 1 0 5/g41 0 5, which approaches 0 as 1 approaches G4 infinity. Thus, the ratio between the second term and the third term in expression (A7) approaches 0 as 1 approaches infinity. ° 1 0 5 by (A4), we can rewrite Because ¡p0 40 1 1 5/¡1 = G4 ° the first term in (A7) as G41 0 5F°40 5. ° 1 0 5 is decreasing in 0 . If we Notice that g41 0 5/G4 pick a small ã > 0, we also have g41 0 5 g41 0 5 < ° 1 0 5 G4 ° 1 0 5 G4

for 1 ∈ 41 − ã1 1 70

Therefore, g41 0 5 g41 − ã 0 5 ≥ ° 1 0 5 G4 ° 1 − ã 0 5 G4

for 1 ∈ 61 − ã1 1 70

° 1 0 5 with respect to 1 over Integrating g41 0 5/G4 ° 1 − ã 0 5 − G4 ° 1 0 55/G4 ° 1 0 5 ≥ 61 − ã1 1 7 yields 4G4 ° 1 − ã 0 55. The right-hand side goes to ã4g41 − ã 0 5/G4 infinity as 1 goes to infinity. Therefore, the ratio between ° 1 − ã 0 5 and G4 ° 1 0 5 goes to infinity as 1 goes G4 to infinity. It is equivalent to say that the ratio between ° 1 + ã 0 5 and G4 ° 1 0 5 goes to zero as 1 goes to infinity. G4 Pick a small ã > 0 and define k = min8f 40 5/F 40 5 0 −ã ≤ 0 ≤ 0 9. We have k > 0, as f 40 5 is continuous and f 40 5 > 0

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R 0 for all . Because g41 0 < 0 5 = − g41 0 5d4F 40 5/F 40 55 > R 0 ° 1 0 5− G4 ° 1 +ã 0 55, the ratio bek 0 −ã g41 0 5d0 = k4G4 ° 1 0 5 and g41 0 < 0 5 is no greater than 1/k as 1 tween G4 approaches infinity. ° 1 0 5F°40 5 and As a result, the ratio between G4 4c − 1 5g41 0 < 0 5F 40 5 approaches 0 as 1 approaches infinity because 1/4c − 1 5 approaches 0 and the ratio ° 1 0 5F°40 5 and g41 0 < 0 5F 40 5 is bounded. between G4 Therefore, it is optimal to select a 1 < + and sell at both stages. Proof of Lemma 1. Because 1 = p1 , we can rewrite p0 as p − 0 ° p1 − 0 1 + 4p1 − 0 5ê (A8) p 0 = 0 − 1 1 1 1 √ where 1 = 1 − . Thus, 0 can be expressed as a function of , p0 , and p1 . By Equation (A8) and the fact that 0 4x5 = −x4x5, we have d p − p − ¡ p1 −0 ¡ 0 = 0 − 1 1 0 +4p1 −0 5 1 0 ¡ d 1 1 ¡ 1 ¡ ° p1 −0 p − ¡ p1 −0 − 0ê −4p1 −0 5 1 0 ¡ 1 1 ¡ 1 ¡ p − d p − = 0ê 1 0 − 1 1 0 0 ¡ 1 d 1 Therefore, 1 44p1 − 0 5/1 5 ¡0 d1 44p1 − 0 5/1 5 = =− < 00 ¡ d ê44p1 − 0 5/1 5 21 ê44p1 − 0 5/1 5

Proof of Proposition 3. The partial derivative ¡ ¡ ° ¡u 1 d ê4u5 = −4u5 = − 2 4u5 0 0 − 40 − z5 0 0 ¡ ¡ ¡ d 0 Because z < 0 and 0 > 0, the term 40 − z54d0 /d5 > 0, and hence 0 4¡0 /¡5 − 40 − z54d0 /d5 < 0 by Lemma 1. There° fore, 4¡/¡5ê4u5 > 0, or ¡u/¡ < 0. The following bounds about the standard normal distribution will be used in the proof of Proposition 4. Lemma A2. For any x > 0, 4x/41 + x2 554x5 < 1 − ê4x5 < ° 41/x54x5. Let a 4x5 = 4x5/ê4x5 − x and b 4x5 = 1 − ° x4ê4x5/4x55. Then a 4x5 > 0, b 4x5 > 0, and both approach 0 as x approaches infinity. Proof. Assume x > 0. The upper bound is derived from Z Z u 1 − ê4x5 = 4u5 du < 4u5 du x x x 2 2 2 1 Z e−u /2 u e−u /2 4x5 = d = − √ = 0 √ x x 2 x 2 x 2 x The lower bound follows from the fact that 0 4u5 = −u4u5 and Z 1 1 1+ 2 1 − ê4x5 = 1 + 2 4u5 du x x x Z 1 > 1 + 2 4u5 du u x 4u5 4x5 =− = 0 u x x



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Thus, a 4x5 ∈ 401 1/x5, b 4x5 ∈ 401 1/41 + x2 55, and both approach 0 as x approaches infinity.

° ê4u5 → 1) uniformly as z → under an optimal pricing policy.

Proof of Proposition 4. We first show that when z > 3022, u < 0 under the optimal pricing policy. If this is not true, there must exist z > 3022 and u ≥ 0 under an optimal pricing policy. By Equation (11) and the ¯ monotonicity of ê4u5/4u5, we have

Proof of Theorem 4. In the following analysis, the prices 4p0 1 p1 5 will always be fixed at their optimal levels 4p0 451 p1 455, and hence will be dropped from the expressions when it is clear from the context. The envelope theorem implies that 4d/d5ç41 p0 1 p1 5 = 4¡/¡5ç41 p0 1 p1 5, and hence it suffices to evaluate 4¡/¡5ç41 p0 1 p1 5. Recall that ç41 p0 1 p1 5 = ç0 41 p0 1 p1 5 + ç1 41 p0 1 p1 5, where ° 0 − z and ç0 41 p0 1 p1 5 = p0 ê 0 Z 0 − p − z 1 ê 0 ç1 41 p0 1 p1 5 = p1 d ê 0 0 1 0 −

° 0 − z5/0 5 1 4ã/1 5 ê44 = 0 − 0 ê4ã/1 5 440 − z5/0 5 = z + 0 u −

° ° 0 ê4u5 ê405 ≥z− 0 0 4u5 405

Because the right-hand side is at least 10966 (as z > 3022 and 0 ≤ 1) and the left-hand side is decreasing in ã/1 with 4−105325/ê4−105325 ≈ 10966, we must have ã/1 < −10532. On the other hand, because ê4x5 = 1 − ê4−x5 > 4−x/41 + x2 554x5 for any x < 0 (by Lemma A2 above), we have 0 −

° 0 − z/0 5 1 4ã/1 5 0 ê4 12 = < 0 − 1 + 1 40 − z/0 5 ê4ã/1 5 0 − 1

and hence 1
2055 > 00764 even when z = 3022). Therefore, the above pricing policy is not optimal, which is a contradiction. ° Now, we show that u → − (or, ê4u5 → 1) uniformly as z → under the optimal pricing policy. If this is not true, there must exist u¯ < 0 such that we can find a u > u¯ and an arbitrarily large z under an optimal pricing policy. From Equation (11), we have ° 0 − z5/0 5 4ã/1 5 4ã/1 5 ê44 ≥ 1 = 0 − 0 ê4ã/1 5 ê4ã/1 5 440 − z5/0 5 ° ° u5 ¯ ê4 ê4u5 = z + 0 u − > z + 0 u¯ − 0 ¯ 4u5 4u5 0 ° The last inequality holds because 4u − ê4u5/4u55 =2− ° uê4u5/4u5 > 0 (as shown above, u < 0 when z is large enough). Thus, due to the monotonicity of 4x5/ê4x5, as z approaches infinity, ã/1 must approach negative infinity uniformly. By Equation (11) and the definition of a 4 · 5 in Lemma A2, we obtain ° ° ¯ ã ê4u5 ã ê4u5 1 = 1 a − + 0 + < a − 0 ¯ 1 4u5 1 4u5

By Lemma A2, a 4−ã/1 5 is small (and positive) when ¯ we have z > 21 −ã/1 is large, and hence for a given u, when z is large enough. This is a contradiction because the seller’s profit is bounded by 1 under an optimal pricing policy, and she can easily secure the profit z/2 by exclusive retailing at price z. Therefore, we must have u → − (or

Thus, we have ¡ç0 /¡ = −p0 4u54¡u/¡5. The term −p0 4u5 represents the loss of preorder profit from the marginal consumers. Define t = 40 − z5/0 (i.e., 0 = z + t0 ) and v41 p1 1 t5 = 4z + 0 t − p1 5/1 . By a change of variables, ç1 can be rewritten as Z 0 − p − z 1 d ê 0 ç1 = p1 ê 0 1 0 − Z u = p1 ê4v41 p1 1 t554t5 dt1 −

and thus, ¡u ¡ç1 = p1 ê4v41 p1 1 u554u5 ¡ ¡ Z u ¡v41 p1 1 t5 + p1 4t5 dt0 4v41 p1 1 t55 ¡ − The first term on the right-hand side captures the gain of retail profit from the marginal consumers who switch from preorder to retail purchase: given p0 and p1 (at their optimal levels), a small increment ã translates into a loss of 4u54¡u/¡5ã portion of consumers at the preorder stage; ê4v41 p1 1 u55 portion of them makes a purchase at time 1, at the price p1 . The second term captures the change of retail profit from the lower ê4u5 portion of consumers with relatively low initial valuations. Now, we examine the marginal consumers who change their purchase date. By substitution, we have p1 ê4v41 p1 1 u554u5 = p1 ê4−ã/1 54u5. Thus, the overall impact of the marginal consumers on the total profit (preorder plus retail) is given by −ã 4u5 −p0 4u5 + p1 ê 1 ã ã ° = 1 − 0 ê 4u5 = −0 ê4u51 1 1 following Equations (A8) and (11). Next, we examine the demand change at time 1 for the lower ê4u5 portion of consumers who do not preorder. Because d0 1 1 1 = √ = d 2 20

and

d1 1 −1 1 = √ =− 1 d 2 1− 21


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we have ¡v41 p1 1 t5 = ¡ =

t 1 + 4z + 0 t − p1 5 20 1 21

12

t + 0 4z − p1 5 0 20 13


Furthermore, because v41 p1 1 t52 + t 2 =

t2 4z − p1 52 20 4z − p1 5t + + 2 2 2 1 1 1

= 4z − p1 52 +

4t + 0 4z − p1 552 1 12

we obtain Z u t + 0 4z − p1 5 4v41 p1 1 t55 4t5 dt 20 13 − Z u 1 2 2 = 4t + 0 4z − p1 55e−v41 p1 1 t5 /2 e−t /2 dt 3 40 1 − Z u 1 2 2 2 4t + 0 4z − p1 55e−4z−p1 5 /2 e−4t+0 4z−p1 55 /421 5 dt = 3 40 1 − Z u t + 4z − p 5 1 2 0 1 = e−4z−p1 5 /2 40 1 1 − t + 0 4z − p1 5 2 2 · e−4t+0 4z−p1 55 /421 5 d 1 Z u 1 2 2 2 = e−4z−p1 5 /2 d4−e−4t+0 4z−p1 55 /421 5 5 40 1 − =−

1 2 2 2 e−4z−p1 5 /2 e−4u+0 4z−p1 55 /421 5 40 1

=−

1 ã 1 2 2 e−u /2 e−4v41p1 1u55 /2 = − 4u5 40 1 20 1 1

< 00

This result says that for those consumers who do not preorder, the proportion of them who buy at the retail stage is decreasing in . Intuitively, the larger the , the smaller the difference between 0 and 1 , and the less likely consumers will buy later. Everything included, we arrive at expression (14): ¡u p1 ã ¡ç ° = −0 ê4u5 − 4u5 0 ¡ ¡ 20 1 1 Further, by the definition of u, expression (13), and the fact that d0 /d = 1/20 and d1 /d = −1/21 , we have ¡ 1 ¡u = 0 0 − 40 − z5 02 ¡ ¡ 20 4ã/1 5 1 = − 0 − 40 − z5 02 21 ê4ã/1 5 20 0 4ã/1 5 = − u+ 202 0 1 ê4ã/1 5 The proof of Theorem 6 requires the following lemma: Lemma A3. The seller prefers risk-averse consumers to riskneutral ones when practicing preorder.

Proof. Let 0 and 1 be the optimal valuation thresholds when consumers are risk neutral. Let p0n and p1n = 1 denote the corresponding preorder and retail prices (the superscript “n” means risk “n”eutral). Because of Theorem 5, risk-averse consumers’ optimal purchasing strategies are also threshold policies. The same thresholds 40 1 1 5 can be implemented when consumers are risk averse by charging p1a = 1 and a proper p0a ∈ 4−1 p1a 5 such that a type 0 consumer is indifferent between preordering and purchasing later (superscript “a” means risk “a”verse). Consider the seller’s profit in the risk-neutral and riskaverse worlds. At time 1, the seller obtains the same profit because the same proportion of the consumer population, ° 1 0 < 0 5F 40 5, purchases the good at the same price G4 p1n = p1a = 1 . At time 0, the same proportion of the population, F°40 5, preorders. To show that the seller can make more profit in the risk-averse world, it suffices to show that p0a > p0n . By the definition of 0 and 1 , p0a satisfies Eg 6u4−p0a 5 + g 0 = 0 7 = Eg 6max8u4051 u4−1 5 + g9 0 = 0 7. And recall that u4−1 5 + g = u405. Thus, we have Z 1 ° 1 0 5 + p0a = −u−1 4u4−1 5G4 u4−1 5 dG41 0 55 −

° 1 0 5 + > 1 G4

Z

1 −

1 dG41 0 5 = p0n 0

The inequality follows from the concavity of the utility function and that G has unbounded support, and the last equation follows from the definition of p0n . Therefore, the seller can charge a higher preorder price and will be better off in the risk-averse world. Proof of Theorem 6. By Theorem 3, it is optimal for the seller to adopt preorder when consumers are risk neutral. That is, the seller can make more profit through preorder and retail than from pure retail. By Lemma A3 above, the seller can obtain more profit when consumers are risk averse. Therefore, abandoning the preorder option and obtaining solely the pure-retail profit is suboptimal for the seller facing risk-averse consumers.

Appendix B. Continuity of the Profit Function ç4z1 5 at = 0 We prove that the seller’s total profit ç4z1 5 is continuous at = 0, for any z. Throughout this appendix, z will take a fixed value, so the optimal profit function will be simply denoted by ç45. Recall that ç405 = max8çP 4z51 çR 4z59, where çP 4z5 = z is the profit under pure preorder and çR 4z5 = maxp pê4z − p5 is the profit under pure retail. When ∈ 401 15, the seller’s problem is z − 0 ç45 = max p0 ê 0 1 1 0 Z 0 − − z 0 ° 1 + 1 ê d ê 0 1 (B1) 1 0 − where

− 0 p0 = 0 − 1 1 1 √ √ 0 = , and 1 = 1 − .

1 − 0 ° + 41 − 0 5ê 1 1

(B2)


1076


We first show the following:


Lemma B1. If pi p↑ + and limi→+ pi ê44z − pi 5/ i 5 = 0.

i ↓ 0

4i ∈ 401 155,

Proof. Because limi→+ pi ê4z − pi 5 = − limi→+ 4z − pi 5 · ê4z − pi 5 = limx→+ xê4−x5 = limx→+ x41 − ê4x55 and, by Lemma A2, limx→+ x41 − ê4x55 ≤ limx→+ 4x5 = 0, we have limi→+ pi ê4z − pi 5 = 0. When p i is sufficiently large, we have pi > z and 0 < ê44z − pi 5/ i 5 < ê4z − pi 5 because i ∈ 401 15. Therefore, z−p 0 ≤ lim pi ê p i ≤ lim pi ê4z − pi 5 = 01 i→+ i→+ i p and hence limi→+ pi ê44z − pi 5/ i 5 = 0. Now we show the following: Proposition B1. ç45 is continuous at = 0. Proof. We first show that for any > 0, there exists > 0 such that ç45 > ç405 − for any ∈ 401 5. If ç405 = çR 4z5, we can pick any ∈ 401 15 because the seller can simply forego the preorder opportunity and earn P the profit ç45 ≥ çR 4z5. √ If ç405 = ç 4z5 = z, we can pick a small so that ê4/2/ 5 > 1 − /42z5. Then z − p0 /2 ç45 > max p0 ê √ > z− ê √ p0 2 > z− 1− > z − = ç405 − for ∈ 401 5 2 2z (notice that under pure preorder, consumers preorder if and only if 0 ≥ p0 and thus 0 = p0 ). Then, to prove the proposition, it suffices to show that for any sequence of i ↓ 0 such that ç4i 5 converges, it converges to some value no more than ç405. Because Z + − − z 0 ° 1 − z5 = ° 1 d ê 0 ê4 ê 1 0 − ° 1 − 0 5/1 5 increases with 0 , the seller’s retail profit and ê44 Z 0 − − z 0 ° 1 − z5ê 0 − z 0 ° 1 1 d ê 0 ≤ 1 ê4 ê 1 0 0 − Intuitively, preorder captures those consumers with high initial valuations and leaves the remaining ê440 − z5/0 5 portion of consumers to the retail stage. Because ° 1 − z5 ≤ çR 4z5, the seller’s retail profit is no more than 1 ê4 √ çR 4z5ê440 −√ z5/ 5, and her total profit √ is no more than p0 ê44z − 0 5/ 5 + çR 4z541 − ê44z − 0 5/ 55. For any sequence i such that ç4i 5 converges, p we can find a subsequence ij such that ê44z − 0 4ij 55/ ij 5 converges to some value in 601 17. Let p0 45 and 0 45 be the optimal p0 and 0 given p . There are three cases: (1) ê44z − 0 4ij 55/ ij 5 converges to t ∈ 401 15. Then we must have p0 4ij 5 < 0 4ij 5 → z = çP 4z5, and hence lim ç4i 5 = lim ç4ij 5 ≤ çP 4z5t + çR 4z541 − t5 ≤ ç4053

i→+

j→+

p (2) ê44z − 0 4ij 55/ ij 5 converges to t = 1. Then p0 4ij 5 < 0 4ij 5 < z for sufficiently large j, and hence lim ç4i 5 = lim ç4ij 5 ≤ z = çP 4z5 ≤ ç4053

i→+

j→+

p (3) ê44z − 0 4ij 55/ ij 5 converges to t = 0. Then if p0 4ij 5 is unbounded, we have z − 4 5 z − p 4 5 0 ij 0 ij p0 4ij 5ê ≤ p0 4ij 5ê →0 p p ij ij from Lemma B1; if p0 4ij 5 is bounded, we have p0 4ij 5 · p ê44z − 0 4ij 55/ ij 5 → 0. Therefore, lim ç4i 5 = lim ç4ij 5 ≤ çR 4z5 ≤ ç4050

i→+

j→+

Because ç45 > ç405 − for any > 0 and small enough, we must have lim→0 ç45 = limi→+ ç4i 5 = ç4050

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Optimal Preorder Strategy with Endogenous Information Control

Optimal Preorder Strategy with Endogenous Information Control

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