Advance Selling in the Presence of Experienced Consumers

Transcription

1 Advance Selling in the Presence of Eerienced Consumers Oksana Loginova X. Henry Wang Chenhang Zeng June 30, 011 Abstract The advance selling strategy is imlemented when a firm offers consumers the oortunity to order its roduct in advance of the regular selling season. Advance selling reduces uncertainty for both the firm and the buyer and enables the firm to udate its forecast of future demand. The distinctive feature of the resent theoretical study of advance selling is that we divide consumers into two grous, eerienced and ineerienced. Eerienced consumers know their valuations of the roduct in advance. The resence of eerienced consumers yields new insights. Secifically, re-orders from eerienced consumers lead to a more recise forecast of future demand by the firm. We show that the firm will always adot advance selling and that the otimal re-order rice may or may not be at a discount to the regular selling rice. Key words: advance selling, the Newsvendor Problem, demand uncertainty, eerienced consumers, ineerienced consumers. We thank Stanislav Kolenikov for his hel on roofs. Deartment of Economics, University of Missouri-Columbia, loginovao@missouri.edu (corresonding author). Deartment of Economics, University of Missouri-Columbia, wang@missouri.edu. Deartment of Economics, University of Missouri-Columbia, czvd7@mail.mizzou.edu. 1

2 1 Introduction Advance selling occurs when firms and retailers offer consumers the oortunity to order the roduct or service in advance of the regular selling season. Remarkable develoments in the Internet and information technology have made advance selling an economically efficient strategy in many roduct categories. Eamles include new books, movies and CDs, software, electronic games, smart hones, travel services and vacation ackages. There are several major advantages of advance selling. First, advance selling reduces uncertainty for both the firm and the buyer, because advance orders are re-committed. In situations when the firm needs to decide how much to roduce (rocure) rior to the regular selling season, advance orders reduce demand uncertainty. For the buyer, an advance order guarantees delivery of the roduct in the regular selling season, ossibly at a discount to the retail rice. Second, orders from advance selling may rovide valuable information for the firm to better forecast the future demand. In articular, the firm may be able to udate its forecast of the size of consumer ool and the distribution of consumers valuations. Finally, advance selling may increase the overall demand. Indeed, when a consumer re-orders the roduct, she commits to urchase it. In the absence of advance selling the same consumer will not urchase the roduct if she learns her valuation is low. The motivation for the resent study is based on two observations. One is that many reorders are from consumers who have revious eerience with the roduct or its earlier versions. The other is that some roducts were not made available for re-orders when they were first introduced, but re-orders became ossible for later versions. These observations oint to an imortant role layed by eerienced consumers in advance selling. Table 1 reorts the roduct release history of several well-known roducts. These roducts are also widely cited as eamles of advance selling. In the first four eamles there were no re-orders for the first one or two versions, and re-orders were offered for later versions, some with discount and some without. While ineerienced consumers learn more about their valuations of the roduct when it becomes available, eerienced consumers are likely to have a good idea about their valuations of the roduct in advance. Therefore, eerienced consumers have less incentives to wait until the regular selling season. It follows that when there are eerienced consumers, advance selling is more likely to be utilized by consumers. In addition, re-orders from eerienced consumers are more informative than those from ineerienced consumers. One can thus conclude that the resence of eerienced consumers makes the first two of the aforementioned advantages of advance selling more ronounced. A number of aers in the literature have emhasized some or all of the three advantages of advance selling (see the literature review in Section ), but none have modeled eerienced consumers and the role they lay in advance selling. This aer is the first study of advance selling with both eerienced consumers and ineerienced consumers. Our model has two eriods. The first is the advance selling season and the second is the regular selling season. In the first eriod, the firm chooses whether to make its roduct available for re-orders, and if so, the level of discount from the retail rice. There are two grous of consumers eerienced and ineerienced. Eerienced consumers know their valuations of the roduct from the outset, while ineerienced consumers learn their valuations only in the

3 Table 1: Release history and re-order availability for several roducts Product Version Release date Pre-order availability/discount Amazon Kindle Kindle Nov. 19, 007 No Kindle Feb. 3, 009 Yes/No discount Kindle 3 Aug. 7, 010 Yes/No discount Harry Potter Book 1 Se. 1, 1998 No Book Jun., 1999 No Book 3 Se. 8, 1999 Yes/40% off Book 4 Jul. 8, 000 Yes/40% off Book 5 Jun. 1, 003 Yes/40% off Book 6 Jul. 16, 005 Yes/40% off Book 7 Jul. 1, 007 Yes/49% off iphone iphone Jun. 9, 007 No iphone 3G Jul. 11, 008 No iphone 3GS Jun. 19, 009 Yes/No discount iphone 4 Jun. 4, 010 Yes/No discount ipod Touch ipod Touch 1st Se. 5, 007 No ipod Touch nd Se. 9, 008 No ipod Touch 3rd Se. 9, 009 Yes/No discount ipod Touch 4th Se. 8, 010 Yes/No discount PlayStation PlayStation 1 Se. 9, 1995 Yes/No discount PlayStation Oct. 6, 000 Yes/No discount PlayStation 3 Nov. 17, 006 Yes/No discount Nintendo Wii Wii Fit May 1, 008 Yes/$0 off Wii Fit Plus Oct. 4, 009 Yes/$10 off second eriod. All consumers decide whether to re-order the roduct (if this otion is available) or wait until the regular selling season, in which they will face a robability of not being able to get the roduct (the stock-out robability). At the conclusion of the first eriod, the firm must choose its roduction quantity, which has to be at least the size of re-orders. The roduct is delivered at the end of the second eriod. Consumers are heterogeneous in their valuations, which are assumed to follow a normal distribution. The firm does not know the mean of this distribution. The grou size of eerienced consumers is fied and known to the firm. However, the firm is uncertain about the number of 3

4 ineerienced consumers. In the second eriod the firm faces the Newsvendor Problem by analogy with the situation faced by a newsvendor who must decide how many coies of the day s aer to stock on a newsstand before observing demand, knowing that unsold coies will become worthless by the end of the day. If the roduced quantity is greater than the realized demand, the firm must disose of the remaining units at a loss (due to the salvage value being below the marginal cost). If the roduced quantity is lower than the realized demand, the firm forgoes some rofit. 1 Our main research questions are the following. Will the firm adot the advance selling strategy? If so, will an advance selling discount be offered? How do eerienced and ineerienced consumers behave in the advance selling season? What can the firm learn from re-orders? How much should the firm roduce? How are the answers to (some of) these questions affected by arameters of the model, such as the salvage value and the comosition of eerienced/ineerienced consumers in the oulation? Our main results are summarized below. The firm always adots advance selling. Advance selling may be at a discount and may be not. Eerienced consumers never wait until the regular selling season. Ineerienced consumers sometimes re-order, sometimes wait until the regular selling season. When the re-order discount is dee, ineerienced consumers re-order. When the discount is moderate, ineerienced consumers re-order if the mean of the distribution from which their valuations are drawn is high, and wait if otherwise. The firm learns from re-orders, which softens the Newsvendor Problem. It learns whether there are any consumers who have chosen to wait until the regular selling season. If nobody waits, the firm only needs to fill all re-orders. In the case when some consumers wait, the firm learns the mean of consumers valuations. However, the uncertainty about the number of ineerienced consumers remains. Our sensitivity analysis in regard to changes in some arameters of the model yields several interesting results, some intuitive and some counterintuitive. For eamle, as the salvage value decreases, the firm s eected rofit may decrease (intuitive), but may also increase (counterintuitive). Likewise, as the roortion of eerienced consumers decreases, the firm s eected rofit may decrease (intuitive), but may also increase (counterintuitive). Our aer contains several contributions to the literature on advance selling. First, as mentioned earlier, we are the first to study advance selling in a model with eerienced consumers. We believe our model catures an imortant asect of the advance selling henomena. Second, learning by the firm in our model is not only on the consumer ool but also on the distribution of consumers valuations of the roduct. Finally, the stock-out robability that consumers face when they wait until the regular selling season is endogenously determined in our model. In the 1 If there is no uncertainty about the second-eriod demand or if the salvage value equals the marginal cost, then the Newsvendor Problem disaears. 4

5 literature, the stock-out robability has been modeled as eogenously given. We think that the correct way to model the stock-out robability is through endogenous determination, since this robability affects consumers choices in the advance selling season and these choices in turn affect the stock-out robability in the regular selling season. After the literature review (Section ), the rest of the aer is organized as follows. In Section 3 we introduce the model. Section 4 is devoted to equilibrium analysis. In Section 5 sensitivity analysis results are resented. In Section 6 we consider two etensions. Concluding remarks are rovided in Section 7. Proofs of all lemmas and roositions, as well as derivations for some eressions and claims, are relegated to Aendi. Literature Review Several strands of the literature have studied advance selling. One deals with advance selling from manufactures to retailers, e.g. Cachon (004) and Taylor (006). Another is on advance selling from firms and retailers to consumers under limited caacity, with alications to the airline and hotel industries (Xie and Shugan, 001, and Liu and van Ryzin, 008). The literature that is closest to the resent study is on advance selling from firms and retailers to consumers without caacity constraints. Our review below focuses on the third strand. Two modeling aroaches have been adoted by researchers. In the first aroach consumers are non-strategic in their decisions on whether to re-order the roduct. In the second aroach consumers are strategic. Paers that model consumers as non-strategic include Weng and Parlar (1999), Tang, Rajaram, Altekinoğlu, and Ou (004), McCardle, Rajaram, and Tang (004), and Chen and Parlar (005). In all of these aers the fraction of consumers who lace advance orders is an eogenously given decreasing function of the advance selling rice. 3 In Tang, Rajaram, Altekinoğlu, and Ou (004) and McCardle, Rajaram, and Tang (004) there are two brands belonging to rivalry firms. Advance selling by a firm attracts customers of the other brand. The former aer eamines the decision on advance selling by a single firm, while the latter focuses on cometition between two firms in adoting the advance selling strategy. Several aers have treated consumers as strategic. Strategic consumers comare the otions of ordering in advance and of waiting until the regular selling season. Zhao and Stecke (010) classify consumers according to whether they are loss averse. A loss averse consumer is more averse to a negative surlus (when the realized valuation is below the advance selling rice) than is attracted to the equivalent ositive surlus. Prasad, Stecke, and Zhao (011) divide consumers into two grous. The informed grou consists of consumers who know about the otion to buy in advance, while the uninformed grou is not aware of this otion. Chu and Zhang (011) allow the firm to control the release of information about the roduct at re-order. The common issues resent in the literature are (i) the Newsvendor Problem, and (ii) learning and udating by the firm. 4 The Newsvendor Problem arises because the firm, facing uncer- The main difference between the second and the third strands is that in situations of limited caacity firms mainly choose rices, while without caacity constraints firms choose their roduction quantities as well as rices. 3 In Chen and Parlar (005) an alternative model is considered, in which the robability that each consumer orders in advance is a beta-distributed random variable. 4 An ecetion is Chu and Zhang (011) in which the Newsvendor Problem is (imlicitly) assumed away because 5

6 tain demand, has to choose its roduction quantity rior to the regular selling season. Obviously, learning from re-orders benefits the firm because it hels to better forecast the demand in the regular selling season. Both issues are also central in our aer. Because we assume that the mean of the distribution of consumers valuations is unknown to the retailer, learning in our model is not only on the consumer ool, but also on the distribution of consumers valuations. Like Zhao and Stecke (010), Prasad, Stecke, and Zhao (011), and Chu and Zhang (011), consumers in our model are strategic. The key difference between our aer and eistent literature is the introduction of eerienced consumers into the model. As stated before, eerienced consumers make the strategy of advance selling more attractive to the firm. 3 Model Setu Consider a firm or a retailer who sells a roduct over two eriods. 5 The first eriod is the advance selling season and the second eriod is the regular selling season. Any consumer who re-orders in the first eriod is guaranteed delivery of the roduct in the second eriod. Those who do not re-order can buy in the regular selling season, but there is a risk that the roduct will be out of stock. There are two tyes of consumers eerienced and ineerienced. Eerienced consumers know their valuations (i.e., their willingness to ay for the roduct) from the outset, whereas ineerienced consumers learn their valuations only in the second eriod. Each consumer is willing to buy at most one unit of the roduct. The number of eerienced consumers is m e. The number of ineerienced consumers, M i, is a random variable; the distribution of M i is lognormal LN ( ν i, τi ) with the mean mi = e ν i + τi /. 6 Both m e and the distribution of M i are common knowledge. Consumers valuations of the roduct are normally distributed with mean µ and variance σ (i.e., v N ( µ, σ ) ). While all consumers know the distribution from which their valuations are drawn, the retailer does not. We model the retailer s uncertainty by assuming that µ is high, µ H, with robability γ and low, µ L, with robability 1 γ. The marginal roduction cost is c and the rice during the regular selling season is. For each unsold unit of the roduct at the end of the regular selling season the retailer gets its salvage value s. We assume s < c <. The retailer decides on advance selling rice in the beginning of the advance selling season. After the conclusion of the advance selling season the retailer must decide how much to roduce. Let D 1 denote the number of consumers who buy in the advance selling season. Then the retailer s quantity choice is Q = D 1 + q, where D 1 fulfills the re-orders. Quantity q satisfies the (stochastic) demand during the regular selling season, denoted by D. Table lists the notation introduced above and also some of the notation introduced later. Figure 1 dislays the timeline of the model. In the beginning of the first eriod, all consumers learn µ and all eerienced consumers learn their valuations. During the first eriod, the retailer announces advance selling rice, then each consumer decides whether to re-order. At the end of the first eriod, the retailer observes the number of re-orders D 1, udates his forecast of the salvage value equals the marginal cost. 5 From now on we shall seak of the firm as the retailer. 6 We use a lognormal distribution to avoid negative realizations of the number of ineerienced consumers. 6

7 Table : Notation Parameters c s m e marginal cost salvage value rice in the regular selling season number of eerienced consumers Random variables v N ( µ, σ ) distribution of consumers valuations µ µ H, µ L two-oint mass distribution, Prob(µ H ) = γ and Prob(µ L ) = 1 γ M i LN ( ) ν i, τi number of ineerienced consumers, mean m i = e ν i + τ i Decision variables q Q quantity roduced for the regular selling season total quantity roduced (includes re-orders) advance selling rice Distribution and density functions F ( ) cdf of N ( µ, σ ), F (y) = Φ ( ) y µ σ F (y) = 1 F (y) f( ) density function of N ( µ, σ ), f(y) = 1 e πσ G( ) cdf of LN ( ν, τ ) ( ), G(y) = Φ ln y ν τ g( ) density function of LN ( ν, τ ) 1, g(y) = y e πτ Other notation (y µ) σ (ln y ν) τ D 1, D π Π η β demands in the advance and regular selling seasons retailer s eected rofit from the regular selling season retailer s total eected rofit (includes re-orders) stock-out robability = c s the second-eriod demand D and chooses roduction quantity Q. During the second eriod, all ineerienced consumers learn their valuations and those consumers who did not re-order then decide whether to urchase the roduct at rice. The roduct is delivered at the end of the second eriod. 7

8 1st eriod: retailer announces some consumers re-order at rice nd eriod: all ineerienced consumers learn their valuations some consumers urchase at rice time all consumers learn µ all eerienced consumers learn their valuations retailer observes re-orders D 1 and udates his forecast of D retailer roducers Q = D 1 + q roduct delivery Figure 1: Timeline of the model 4 Equilibrium Analysis Our goal in this section is to find the otimal advance selling rice. To do this, we first derive consumers otimal resonses to any advance selling rice, and how the retailer learns from the re-orders and chooses his outut. We then determine the endogenous stock-out robability and resent the retailer s eected rofit function. 4.1 Consumers otimal urchasing decisions Since eerienced consumers know their valuations from the outset, they never wait until the regular selling season. Eerienced consumers with valuations above re-order the roduct and ay discounted rice. Ineerienced consumers do not know their valuations in the advance selling season. An ineerienced consumer has two otions. The first is to re-order and ay. In this case the consumer s eected ayoff is µ. The other otion is to wait until the regular selling season. The consumer learns her valuation v and urchases the roduct (rovided it is in stock) if v. Her eected ayoff is (1 η) (v )f(v) dv, where η is the stock-out robability and f( ) is the density function of N ( µ, σ ), from which the consumer s valuation is drawn. The stock-out robability is the robability that the consumer will not be able to get the roduct when she actually wants to urchase it. Thus, ineerienced consumers re-order if and only if µ (1 η) 8 (v )f(v) dv,

9 or, equivalently, Let and µ (1 η) L µ L (1 η) H µ H (1 η) (v )f(v) dv. (v )f L (v) dv (1) (v )f H (v) dv () denote the threshold values for µ = µ L and µ = µ H, resectively. Here f L ( ) is the density function of N ( µ L, σ ) and f H ( ) is the density function of N ( µ H, σ ). In Subsection 4.4 we show that the endogenously determined η is the same in (1) and (). The elicit eressions for L and H (derived in Aendi) are L = µ L (1 η) ( (µ L )F L () + σ f L () ) and H = µ H (1 η) ( (µ H )F H () + σ f H () ), where F L ( ) and F H ( ) are the cumulative distribution functions of N ( µ L, σ ) and N ( µ H, σ ), resectively, and F ( ) = 1 F ( ). Lemma 1 (Proerties of L and H ). The threshold values L (η, σ) and H (η, σ) ossess the following roerties: (i) L (η, σ) < H (η, σ) for all η and σ; (ii) L / σ < 0 and H / σ < 0; (iii) L / η > 0 and H / η > 0. Since L < H always holds, we consider the following three regions for advance selling rice. Region A: L. All ineerienced consumers re-order. Region B: L < H. Ineerienced consumers re-order if µ = µ H. Region C: > H. All ineerienced consumers wait until the second eriod. Proerties (ii) and (iii) will be useful for our sensitivity analysis in Section 5. 9

10 4. Learning by the retailer from re-orders When advance selling rice is in region A, eerienced consumers with valuations above and all ineerienced consumers re-order. No one will wait until the regular selling season. As the retailer s forecast of the second-eriod demand is D = 0, he roduces Q = D 1. When is in region B, the retailer learns µ through observing D 1. If D 1 = m e F L (), then the retailer infers that µ = µ L. The retailer roduces Q = D 1 + q, where q satisfies the second-eriod demand that comrises of ineerienced consumers with valuations above, D = M i Prob(v > ) = M i F L (). Because M i LN ( ν i, τi ), it is straightforward to show that D LN ( ν i + ln F L (), τi ). If D 1 m e F L (), the retailer infers µ = µ H and D = 0, hence roduces Q = D 1. (Like in region A, no one waits until the regular selling season eerienced consumers with valuations above and all ineerienced consumers re-order.) In region C the retailer also learns µ. If D 1 = m e F L (), then the retailer infers that µ = µ L. The retailer roduces Q = D 1 + q, where q is for the second-eriod demand D = M i F L () LN ( ν i + ln F L (), τi ). If D 1 = m e F H (), the retailer knows µ = µ H. The retailer roduces Q = D 1 + q, where q is for the second-eriod demand D = M i F H () LN ( ν i + ln F H (), τi ). 4.3 Otimal value of q In regard to the retailer s choice of q, it remains to find the otimal q when D follows the distribution LN ( ν i + ln F L (), τi ) ( and when it follows LN νi + ln F H (), τi ). For any D, if q units are roduced, then min q, D units are sold and (q D ) + = ma q D, 0 are salvaged. The retailer s eected rofit from the second eriod, denoted by π, is π(q) = E [min q, D ] + s E [ (q D ) +] cq. (3) The roblem of maimizing (3) is the Newsvendor Problem, well-known in the oerations management literature. Using the fact that min q, D = D (D q) +, we can rewrite the retailer s eected rofit as π(q) = E[D ]( c) E [ (D q) +] ( c) E [ (q D ) +] (c s). The otimal value of q, therefore, minimizes the eected underage and overage cost E [ (D q) +] ( c) + E [ (q D ) +] (c s). The first-order condition is Prob(D q ) = β, 10

11 where β c s. It is clear that q selected this way increases in β and therefore increases in the er unit underage cost c and decreases in the er unit overage cost c s. For the lognormal distribution D LN ( ν, τ ) the otimal roduction quantity is given by and q = eν + τz β (4) π(q ) = ( s) (1 Φ(τ z β )) e ν + τ, (5) where z β is the β-th ercentile of the standard normal distribution, z β Φ 1 (β). See Aendi for derivations of (4) and (5). Alying (4) and (5) to D LN ( ν i + ln F L (), τi ), the otimal q, denoted by q L, and the resulting eected rofit π L are q L = eν i + τ i z β F L () and π L = ( s) (1 Φ(τ i z β )) m i F L (). Similarly, under D LN ( ν i + ln F H (), τi ) the otimal q, denoted by q H, and the resulting eected rofit π H are qh = eν i + τ i z β F H () and 4.4 Stock-out robability π H = ( s) (1 Φ(τ i z β )) m i F H (). Given D, the (conditional) robability of any consumer who wants to urchase the roduct in the regular selling season but is unable to get it is the fraction of ecess demand, ( D q ) + 0, D q, Prob(stock-out D ) = = D D q D, D > q. Hence, the stock-out robability is the eected value of this eression over the distribution of D, [ (D q ) ] + η = E. (6) D For D LN ( ν, τ ), η = q D q D g(d ) dd, 11

12 where g( ) is the density function of LN ( ν, τ ) and q is given in (4). The elicit eression for the stock-out robability is obtained in Aendi, η = 1 β e τz β + τ (1 Φ(z β + τ)). (7) Note that the eression in (7) is indeendent of ν. Accordingly, the same η results from D LN ( ν i + ln F L (), τi ) and D LN ( ν i + ln F H (), τi ). This finding is resented in Lemma. Lemma (Stock-out robability η). The stock-out robability under the second-eriod demand D LN ( ν i + ln F L (), τi ) is equal to that under D LN ( ν i + ln F H (), τi ) and is given by η = 1 β e τ i z β + τ i (1 Φ(z β + τ i )). It is imortant to note that the stock-out robability in our model is endogenously determined, because q is the otimal choice. It follows that η < 1 β, which is eected, as [ (D q ) ] + η = E < [1I(D > q )] = Prob(D > q ) D where 1I(D > q ) is the indicator of the corresonding event, and ( D q ) + < 1I(D > q ) = 1 Prob(D < q ) = 1 β, for all D. 7 D Lemma 3 (Proerties of η). The stock-out robability η = η(β, τ i ) ossesses the following roerties: (i) η/ τ i > 0, η(β, 0) = 0, and lim τi + η(β, τ i ) = 1 β; (ii) η/ β < 0, η(0, τ i ) = 1, and η(1, τ i ) = 0. Combining the results of Lemma 1(iii) and Lemma 3 we can conclude that the threshold values L and H increase in τ i and decrease in β. 7 Another theoretical study that models elicitly the risk of stock out that consumers face is Prasad, Stecke, and Zhao (011). In their study the stocking out robability (defined on age 5) has the same meaning as η in our model: a consumer who waited until the regular selling season will not be able to get the roduct with robability η. However, instead (6), they use the eression η = Prob(D > q ), which yields η = 1 β (age 7). Their formula, however, does not account for the fact that when D > q, the consumer might still get the roduct. 1

13 4.5 The retailer s eected rofit We can now write the retailer s eected total rofit Π as a function of advance selling rice. The art of the retailer s eected rofit that comes from eerienced consumers equals m e ( γf H () + (1 γ)f L () ) ( c), as eerienced consumers never wait until the regular selling season and only those with valuations above (fraction F H () in the case µ = µ H and fraction F L () in the case µ = µ L ) urchase the roduct in the advance selling season. The urchasing behavior of ineerienced consumers deends on the region that belongs to. If L (region A), then all ineerienced consumers re-order. Hence, Π A () = m e ( γf H () + (1 γ)f L () ) ( c) + m i ( c). (8) Net, consider L < H (region B). In the case µ = µ H all ineerienced re-order, yielding m i ( c) to the retailer. In the case µ = µ L all ineerienced consumers wait until the second eriod, yielding π L (calculated in Section 4.3) to the retailer. Hence, Π B () = m e ( γf H () + (1 γ)f L () ) ( c) + γm i ( c) + (1 γ)π L. (9) Finally, if > H (region C), then all ineerienced consumers wait until the second eriod. Since the retailer learns µ in the first eriod, his eected ayoff from ineerienced consumers is γπ H + (1 γ)π L. Hence, Π C () = m e ( γf H () + (1 γ)f L () ) ( c) + γπ H + (1 γ)π L. (10) The retailer s total eected rofit as a function of advance selling rice is, therefore, Π A (), L, Π() = Π B (), L < H, Π C (), > H. 4.6 Advance selling vs. no advance selling Before moving onto the otimal advance selling rice for the retailer, we elore net whether there always is an incentive for the retailer to imlement the advance selling strategy. Without advance selling, the retailer sells in the regular selling season at rice. Let Π H (Q) and Π L (Q) denote the eected rofits as functions of roduction quantity Q for µ = µ H and µ = µ L, resectively. Let Q 0 denote the otimal quantity. Then, without advance selling, the retailer s maimum eected rofit is Π 0 = γπ H (Q 0 ) + (1 γ)π L (Q 0 ). We will show that advance selling at rice leads to a higher eected rofit than no advance selling, that is, Π() > Π 0. It follows that the retailer s eected rofit under advance selling 13

14 (at the otimal rice which may or may not be equal to ) must be greater than that under no advance selling. Advance selling brings two benefits for the retailer: learning the true value of µ before choosing roduction quantity and receiving recommitted orders. We show net that the first benefit alone imroves the retailer s eected rofit over that under no advance selling. Let Q H and Q L denote the otimal quantities for µ = µ H and µ = µ L, resectively. It follows that Π H (Q H ) Π H (Q 0 ) and Π L (Q L ) Π L (Q 0 ) with at least one in strict inequality. Accordingly, γπ H (Q H ) + (1 γ)π L (Q L ) > Π 0. This inequality indicates that knowing the true value of µ before choosing Q is suerior to choosing Q without knowing the true value of µ. Since Π() incororates the benefit from ossible re-orders, we must have Π() γπ H (Q H ) + (1 γ)π L (Q L ). Hence, Π() > Π 0. Therefore, we have the following roosition. 8 Proosition 1 (Advance selling vs. no advance selling). Advance selling is always suerior to no advance selling for the retailer. 4.7 Otimal advance selling rice For the rest of our analysis, we will assume that c < L < H < holds. 9 Furthermore, we make the following simlifying assumtion. Assumtion 1. The function ( γf H () + (1 γ)f L () ) ( c) increases in on [c, ]. This assumtion imlies that the eected rofit from eerienced consumers, m e ( γf H () + (1 γ)f L () ) ( c), is an increasing function of for all [c, ]. It follows that, as far as eerienced consumers are concerned, the retailer has no incentives to offer an advance selling discount. Accordingly, if discounting for re-orders is offered it must due to the resence of ineerienced consumers. It is easy to see that under Assumtion 1 the retailer s eected rofit Π() increases in in each of the three regions A, B, and C. This does not imly, however, that Π() increases in 8 We do not consider the cost of adoting advance selling in this aer. Proosition 1 holds as long as the adotion cost is lower than Π() Π 0. 9 There are si ossible relationshis between c,, L, and H: c < < L < H, c < L < < H, c < L < H <, L < c < < H, L < c < H <, and L < H < c <. 14

15 on [c, ], as Π() can jum down at = L and/or at = H. A jum down at L occurs if and only if Π A ( L ) > Π B ( L ). 10 That is, or equivalently, (1 γ)m i ( L c) > (1 γ)π L, L c > ( s) (1 Φ(τ i z β )) F L (). Similarly, a jum down at H occurs if and only if Π B ( H ) > Π C ( H ). 11 That is, or equivalently, γm i ( H c) > γπ H, H c > ( s) (1 Φ(τ i z β )) F H (). Hence, we have the following four atterns for Π() (see Figure ). Pattern 1: Π() jums down at both L and H. Pattern : Π() jums u at both L and H. Pattern 3: Π() jums u at L, jums down at H. Pattern 4: Π() jums down at L, jums u at H. Regarding the otimal advance selling rice, under attern 1 it is either L, H, or ; under attern it is ; under attern 3, it is either H or ; under attern 4, it is either L or. Our etensive numerical simulations indicate that all four atterns may arise and under each attern the otimal advance selling rice can take any of the ossible values mentioned above. It follows that the retailer s otimal advance selling rice can be any one of the three values: L, H, or. Proosition (Otimal advance selling rice). Under Assumtion 1, the otimal advance selling rice is either L, H, or. The three likely otimal advance selling rices reflect two different tradeoffs for the retailer: between low rice-high sales and high rice-low sales, and between low rice-low eected overage and underage cost and high rice-high eected overage and underage cost. Under all three rices, eerienced consumers only buy in the advance selling season. Hence, the tradeoff for the retailer from this grou of consumers is only between a lower rice and therefore higher eected sales and a higher rice and lower eected sales. For the ineerienced grou of consumers, let us first focus on the comarison between L and. At L all ineerienced consumers re-order while at none re-order and only those with realized values above will buy in the regular selling season. Hence, both tradeoffs described above are resent here. First, L corresonds to higher sales and a lower rice, while corresonds to much lower sales and a higher rice. Second, L means zero overage and underage cost, while leads to a ositive eected overage and underage cost. 10 Since Π B( ) is defined on ( L, H], Π B ( L) is the limiting value. 11 Since Π C( ) is defined on ( H, ], Π C ( H) is the limiting value. 15

16 Π() Π() c L H c L H (a) Pattern 1 (b) Pattern Π() Π() c L H c L H (c) Pattern 3 (d) Pattern 4 Figure : The retailer s eected rofit as a function of Adding the intermediate rice H to the mi, the same two tradeoffs are involved between each air of these rices. For eamle, moving from L to H, the eected sales decrease while the eected overage and underage cost increases, both are due to the fact that there is a ositive robability that at H all ineerienced consumers will wait until the regular selling season. Comaring all three ossible advance selling rices, we conclude that, as the advance selling rice changes from L to H to, the eected sales decrease and the eected overage and underage cost increases. 5 Sensitivity Analysis In this section we consider how the retailer s otimal advance selling rice and the eected rofit Π( ) are affected by some imortant arameters of the model. Let Π Π( ). Intuitively, we should eect that a decrease in the salvage value s or an increase in demand uncertainty τ i would result in lower Π. Surrisingly, we find that Π might actually increase 16

17 (Subsection 5.1). We are also interested in how consumer characteristics the relative number of eerienced consumers (measured by arameter α introduced below) and valuation uncertainty of ineerienced consumers (catured by σ) affect the retailer. Will a decrease in α and/or an increase in σ lead to lower Π? As we show in Subsections 5. and 5.3, both lower and higher Π are ossible. In all of our sensitivity analysis in this section, we focus on small changes in the arameter values so that the retailer s otimal choice stays the same in that it does not jum to one of the other two oints. Although we work with τ i and σ, the same results aly to small changes in τ i and σ. In Table 3 all of the directional changes in the arameters (the first row) are chosen to hurt the retailer on an intuitive basis. Therefore, all cases in which Π increases (Π ) reresent counterintuitive results. 5.1 Sensitivity analysis s and τ i In this subsection we consider how a decrease in the salvage value s and an increase in demand uncertainty τ i affects the retailer. We first focus on s. Suose s decreases. Then β = ( c)/( s) decreases. By Lemma 3(ii), η increases. An increase in η ositively affects L and H. As to π L and π H, they go down as s decreases. Indeed, alying the Enveloe Theorem to (3) yields π s = E [ (q D ) +] > 0. q=q Geometrically, a decrease in s shifts the thresholds L and H to the right and the curves Π B () and Π C () defined in (9) and (10) down. Π() Π() c L L H H c L L H H (a) Pattern 1 (b) Pattern Figure 3: The effects of a decrease in s (increase in τ i ) on Π() 17

18 Consider, for eamle, attern 1 and suose = L. In Figure 3(a) the black and gray curves reresent, resectively, Π() before and after a decrease in s. It is easy to see that both and Π increase. If = H, then a decrease in s leads to higher ; Π can increase or decrease. If =, then a decrease in s leads to lower Π. Consider net attern. The otimal advance selling rice is. As can be seen from Figure 3(b), a decrease in s leads to a lower Π. Similar reasoning alies to atterns 3 and 4. It is imortant to oint out that the same results on the directional changes of and Π hold across all four atterns as long as the same otimal choice is obtained. The second column of Table 3 reorts the sensitivity results on and Π in terms of s. Table 3: Sensitivity analysis results otimal choice L H arameter change s τ i α σ Π Π unchanged Π Π Π unchanged Π unchanged Π unchanged Π unchanged Π Π Π unchanged Π What is the effect of an increase in τ i on and Π? By Lemma 3(i), η increases. As a result, L and H increase. Clearly, an increase in τ i negatively affects π L and π H. It follows that an increase in τ i affects and Π in similar ways as a decrease in s. (See the third column of Table 3.) The intuition for the above results in regard to a change in the salvage value s is as follows (similar for an increase in τ i ). When s becomes lower, the er unit overage cost becomes higher and therefore the retailer reduces his outut to avoid too much unsold roduct at the end of the regular selling season. This raises the stock-out robability η for consumers who wait until the regular selling season. As a result, ineerienced consumers are willing to ay a higher advance selling rice (i.e., higher L and H ) to secure delivery of the roduct. Consider the case in which the retailer s otimal advance selling rice = L. Since the retailer s eected rofit from the grou of eerienced consumers is an increasing function of (Assumtion 1), it becomes higher. Obviously, the retailer s eected rofit from the grou of ineerienced consumers rises as well since all of them re-order at a higher rice. Hence, the total eected rofit of the retailer becomes greater. In this case, we obtain the counterintuitive result that the retailer can benefit from a decrease in s. The increased stock-out robability does bite the retailer if the otimal advance selling rice =. In this case, the retailer s eected rofit from eerienced consumers remains the same as there is no change for this grou both in rice and in the number of buyers. However, the retailer s eected rofit from the grou of ineerienced consumers, who all wait until the regular selling season, becomes lower due to the increased er unit overage cost. It follows that 18

19 the retailer is hurt by a decrease in s. Finally, consider = H. With robability γ the retailer will benefit from a decrease in s (similar to the case = L ) and with robability 1 γ the retailer will be hurt by a decrease in s (similar to the case = ). As a result, deending on these robabilities and the resective gain and loss, both directions of change are ossible for the retailer s total eected rofit. 5. Sensitivity analysis α Let m = m e + m i denote the total eected number of (eerienced and ineerienced) consumers, and let α denote the roortion of eerienced consumers in the market. Thus, m e = αm and m i = (1 α)m. In this subsection we consider how a decrease in α affects the retailer. Note that η, L and H are indeendent of α. Hence, stays the same. To see how the retailer s eected rofit is affected, we substitute m e = αm and m i = (1 α)m into the eressions (8) through (10), remembering that π L and π H deend on m i. In Aendi we calculate the derivatives of Π A ( L ), Π B ( H ), and Π C () with resect to α, and show that the first derivative is negative, the second can be ositive or negative, and the third is ositive. Therefore, if = L then a decrease in α leads to higher Π, and if = to lower Π. If = H, Π can increase or decrease. The fourth column of Table 3 reorts the results of our sensitivity analysis with resect to α. The intuition is straightforward. A decrease in α changes the comosition of eerienced and ineerienced consumers in the total consumer oulation by decreasing the grou size of eerienced consumers and increasing the grou size of ineerienced consumers. Such a change does not alter the incentive for the retailer to roduce to satisfy the demand in the regular selling season and does not affect each individual consumer s incentive to re-order in the advance selling season. That is, η, L and H all remain unchanged. It does, however, affect the retailer s eected rofit. In the case in which the retailer s otimal advance selling rice = L, no one buys in the regular selling season. The size of re-orders laced by eerienced consumers decreases while the size of re-orders laced by ineerienced consumers increases. The latter change is greater than the former, because only a fraction of eerienced consumers urchase the roduct. Hence, the retailer s eected rofit increases. If the otimal advance selling rice =, the oosite occurs. In this case all eerienced consumers whose valuations are above re-order but only a fraction of those ineerienced consumers whose valuations are above get to buy the roduct in the regular selling season due to the ositive stock-out robability. This leads to a net loss in the eected sales for the retailer and therefore to a lower eected rofit. In the case in which the otimal advance selling rice = H, the retailer s eected rofit rises with robability γ (corresonding to all ineerienced consumers re-ordering like in the case = L ) and falls with robability 1 γ (corresonding to all ineerienced consumers waiting to buy in the regular selling season like in the case = ). As a result, deending on these robabilities and the resective gain and loss, both directions of change are ossible for the retailer s total eected rofit. 19

20 5.3 Sensitivity analysis σ Parameter σ catures variation in consumer valuations. For ineerienced consumers an increase in σ also means they become more uncertain about their valuations. By Lemma 1(ii), both L and H decrease as σ increases. In Aendi we show that whenever = L, an increase in σ leads to lower eected rofit for the retailer. If = H or, Π can increase or decrease (we constructed numerical eamles that show that both directions are ossible). The fifth column of Table 3 reorts the results. The intuition is as follows. When σ increases, ineerienced consumers become less certain about their valuations of the roduct. As a result, they require better incentives than before to be willing to re-order in the advance selling season (i.e., lower L and H ). In the case in which the otimal advance selling rice = L, all consumers who buy re-order, now at a lower rice. By Assumtion 1, the retailer s eected rofit from eerienced consumers becomes lower. The retailer s eected rofit from ineerienced consumers falls as well since all of them re-order at a lower rice. Hence, the total eected rofit of the retailer becomes smaller. Consider now the case in which the otimal advance selling rice =. As σ increases the valuation distribution functions become more disersed in that more consumers have valuations farther away from the mean value than before. As a result, if > µ then more consumers have valuations above and if < µ then less consumers have valuations above. Suose > µ H, which also imlies > µ L. The retailer gets more re-orders from eerienced consumers in the advance selling season and a larger demand from ineerienced consumers in the regular selling season, both yielding greater rofit. On the other hand, for < µ L (hence, < µ H ), the oosite occurs and the retailer s total eected rofit decreases. Obviously, either an increase or a decrease in the retailer s eected rofit is ossible if µ L < < µ H. Thus, in the case = the directional change in the retailer s eected rofit deends on the value of in relation to µ L and µ H, all of which are eogenously given in our model. Finally, consider the case in which the otimal advance selling rice = H. The retailer s total eected rofit falls with robability γ (corresonding to all ineerienced consumers reordering like in the case = L ) and falls or rises with robability 1 γ (corresonding to all ineerienced consumers waiting to buy in the regular selling season like in the case = ). As a result, deending on these robabilities (γ and 1 γ) and the resective gain and loss, both directions of change are ossible for the retailer s total eected rofit. 6 Etensions In this section, the equilibrium analysis in Section 4 is etended in two directions. We first rela the assumtion that c < L < H < so as to elore the ossibility that the retailer sets an advance selling rice that is below cost. Then, we discuss the retailer s otimal ricing strategy without Assumtion Advance selling below cost ( < c) Based on our analysis in Section 4, only if the assumtion that c < L < H < is relaed can it become ossible that the otimal advance selling rice is less than c. Accordingly, we 0

21 eamine the ossibility of < c under the following two scenarios: L < c < H < and L < H < c <. Following the same arguments as those resented in Section 4, we have that the otimal advance selling rice in these two scenarios must take one of the three values: L, H, and. Can L be the otimal advance selling rice? Since Π A ( L ) = m e ( γf H ( L ) + (1 γ)f L ( L ) ) ( L c) + m i ( L c) < 0, setting = L would lead to a negative eected rofit under either scenario. It is therefore inferior to setting the advance selling rice at, which imlies a ositive eected rofit. Thus, the otimal advance selling rice can never be L in either of the above two scenarios. In the scenario L < c < H <, setting = H does not imly ricing under cost. So the question becomes, can H be the otimal advance selling rice in the scenario L < H < c <? We have Π B ( H ) = m e ( γf H ( H ) + (1 γ)f L () ) ( H c) + γm i ( H c) + (1 γ)π L < m e ( γf H () + (1 γ)f L () ) ( c) + γπ H + (1 γ)π L = Π C (). Thus, the otimal advance selling rice cannot be H. In conclusion, we have shown above that the retailer s otimal advance selling rice is never below the unit roduction cost c. 6. Interior otimum Our study of the retailer s otimal advance selling rice and the subsequent sensitivity analysis were based on the simlifying Assumtion 1 that ( γf H () + (1 γ)f L () ) ( c) increases in on [c, ]. In this subsection, we wish to oint out that our main results continue to hold if Assumtion 1 is not maintained. In articular, the following numerical eamles demonstrate that the retailer s otimal advance selling rice can occur in all three relevant regions, although it is in general an interior otimum in the resective region (Figure 4). In these eamles, we resent three different interior otimal choices by the retailer that are located in the three resective regions. In all three cases, c = 100, m = 00000, α = 0.5, and γ = 0.5. Eamle 1. We use = 300, s = 45, σ = 10, µ L = 0, µ H = 300, and τ i = 4.5. In this eamle, the endogenous values are η = 0.16, L = 0, and H = 96.66; the otimal advance selling rice = is located in region A. This eamle is illustrated in Figure 4(a). Eamle. We use = 300, s = 60, σ = 15, µ L = 180, µ H = 80, and τ i = 1. In this eamle, the endogenous values are η = 0.06, L = 180, and H = 79.40; the otimal advance selling rice = is located in region B. This eamle is illustrated in Figure 4(b). Eamle 3. We use = 00, s = 55, σ = 100, µ L = 115, µ H = 140, and τ i = In this eamle, the endogenous values are η = 0.10, L = , and H = 14.74; the otimal advance selling rice = is located in region C. This eamle is illustrated in Figure 4(c). 1

22 (a) c < < L (b) L < < H (c) H < < 7 Concluding Remarks Figure 4: Interior otimum This aer has studied advance selling when the firm faces ineerienced consumers as well as a grou of eerienced consumers who have rior eerience with an earlier version of the roduct. We find that it is always in the firm s best interest to adot advance selling. The otimal re-order rice may or may not be at a discount to the regular selling rice.

23 A number of issues are worthy of further investigation. Our model has no bearing on the adotion of advance selling by a firm selling a brand new roduct. An interesting follow-u study is to investigate what haens if the set of eerienced consumers is emty. We anticiate that advance selling may still be otimal for the firm in some circumstances. Obviously, an advance selling discount becomes more likely in order to rovide ineerienced consumers with an incentive to re-order. Another issue is the ossibility of a rice remium for re-orders, which has not been analyzed in the resent aer. As eerienced consumers know their valuations of the roduct, some of them may be willing to ay a rice remium so as to avoid the ossibility of stock-out in the regular selling season. However, with a rice remium for re-orders, some eerienced consumers will choose to wait until the regular selling season. As a result, both learning by the firm and the calculation of the stock-out robability will be much different and more involved. A third issue concerns the assumtion that the distribution of valuations is the same for eerienced and ineerienced consumers. One straightforward generalization of our model is to assume that the distribution of valuations for eerienced consumers is a rightward shift of that for ineerienced consumers (i.e., eerienced consumers value the roduct more than ineerienced consumers on average). We eect many of the results in our aer to continue to hold in this etension. An alternative might be to assume a more general level of correlation between the two distribution functions. Much remains to be elored about this case. Aendi Elicit eressions for L and H : Below we show that (v )f(v) dv = (µ )F () + σ f(). The elicit eressions for L and H will follow immediately from this equality. Alying the change of variable z = (v µ)/σ to the integral on the left-hand side yields (v )f(v) dv = µ σ = (µ ) (µ + σz )φ(z) dz ( 1 Φ ( µ σ )) + µ = (µ )F () + σ e π µ σ σ z σz π e d z. z dz 3

24 Alying the change of variable u = z / yields Proof of Lemma 1: (µ )F () + σ π ( µ) σ e u du = (µ )F () + σ π e = (µ )F () + σ f(). ( µ) σ It follows immediately from (1) and () that L and H increase in η, so we have art (iii) of Lemma 1. (i) Below we show that H L = µ H µ L (1 η) [ (v )f H (v) dv ] (v )f L (v) dv > 0 for all η and σ. Since f L (v) is a arallel shift of f H (v), the term in the square brackets can be rewritten as Therefore, (v )f H (v) dv = < +µh µ L +µh µ L = (µ H µ L ) +µ H µ L (v µ H + µ L )f H (v) dv (v )f H (v) dv + (µ H µ L )f H (v) dv + +µ H µ L (µ H µ L )f H (v) dv f H (v) dv < µ H µ L. +µ H µ L (µ H µ L )f H (v) dv H L > µ H µ L (1 η)(µ H µ L ) = η(µ H µ L ) 0. (ii) In order to rove that L decreases in σ, we need to show that (v )f L (v) dv increases in σ. (v )f L (v) dv = vf L (v) dv f L (v) dv = vf L (v) dv v df L (v) (1 F L ()) ) = µ L (vf L (v) F L (v) dv + F L () = µ L + F L (v) dv. 4