Analysis of Pre-order Strategies with the Presence of Experienced Consumers

Transcription

1 Analysis of Pre-order Strategies with the Presence of Experienced Consumers Chenhang Zeng February 19, 2014 Abstract We study pre-order strategies in a two-period, continuous-valuation model with the presence of both demand uncertainty and consumer heterogeneity. Consumers face different levels of uncertainty about their valuations in the pre-order stage: experienced consumers know their individual valuations while inexperienced consumers only know the distribution of their valuations. We find three candidates for the retailer s profit maximizing selling strategy. In general, when the proportion of experienced consumers is small, the future demand uncertainty is large, or the salvage value of unsold products is small, the retailer is more likely to offer a deep discount for pre-orders; when the proportion of experienced consumers is large, the future demand uncertainty is small, or the salvage value is large, the retailer is more likely to offer a moderate discount for pre-orders; otherwise, a premium price is most likely to be charged to pre-orders. Furthermore, in an extended model where the retail price is endogenously determined as well as the pre-order price, we find that the retailer chooses different retail prices for the above three strategies because of different demand uncertainty levels in the retail stage. Key words: pre-order strategies, advance selling, demand uncertainty, experienced consumers, endogenous retail price. This paper is based on my Ph.D. thesis at University of Missouri. I am grateful to Oksana Loginova and X. Henry Wang for many helpful discussions. Financial support from RCGEB, Shandong University is gratefully acknowledged. Research Center for Games and Economic Behavior, Shandong University, China, cz sdu@163.com 1

2 1 Introduction A pre-order is an order placed for a new product before the release date which guarantees immediate delivery on release. Many retailers take pre-orders for new to-be-released products, such as books, CDs, video games, software and technical products, and the practice has been proved to be a huge success especially for popular items which are hard to get in stores after the release. Two common features for these new products are shortselling season and high demand uncertainty. By taking pre-orders, retailers can be assured of minimum sales and are able to forecast how much demand there will be after release. In such a way retailers manage to reduce the demand uncertainty and forecast the future demand. One popular example of pre-order practice is Harry Potter book series. Amazon offered 40% off to consumers who pre-ordered the fourth book in 2000 and received more than 410,000 pre-orders. Henceforward Amazon continued this pricing strategy and shattered the pre-order record each time a new book was released. Different from advance selling at deep discounts for Harry Potter book series, Amazon offered a $10 Gift Card to consumers who purchased Gears of War 2 in advance, which is the same as a $10 moderate discount for pre-orders. Another example is the release of Miata by Mazda in Dealers charged a premium price for orders in advance (20% more than the sticker price) and then dropped the price for late orders. These three examples describe different pre-order strategies for new generations of series products. A salient feature of series products is that a portion of consumers purchased the early generation(s) while the rest did not. Then a natural question arises as to how a retailer set prices under advance selling in a market composed of consumers with product-use experience and consumers without product-use experience. The present paper studies pre-order strategies for new generations of series products when both valuation uncertainty and consumer heterogeneity are of concern. The starting point of our study is the observation that consumers with product-use experience (we call them experienced consumers) know more about their true valuations in advance than others (we call them inexperienced consumers). 1 Consumer heterogeneity is therefore reflected in the following two characteristics in our model. First, consumers are separated into two groups based on their knowledge about valuations of the product in advance. Second, consumers true valuations follow a normal distribution, and experienced consumers value the product more than inexperienced consumers on average. After observing the prices, the two groups of consumers behave differently in the pre-order period. While experienced consumers know their valuations in advance and therefore have less incentives to wait until the product release, inexperienced consumers face valuation uncertainty in advance and make homogeneous pre-order decisions on the basis of valuation expectation. Hence, retailers face a trade-off among a deep pre-order discount-high sales which takes full advantage of inexperienced consumers valuation uncertainty, a moderate pre-order discount-moderate sales which maximizes total expected profit by separating two groups to purchase in two periods, and a pre-order premium-low sales which price discriminates experienced consumers to obtain the highest possible profit. The goal of this paper is to provide retailers with a guideline for the optimal pre-order strategy before the release of new series products whose future demands are uncertain. Considering that the retail prices 1 Compared to other consumers, consumers with product-use experience are more involved in learning about new generations on average. And the product information on the new generation works more effective in shaping their valuations because most likely they know the current product better and understand what improvement are valuable for the new generation. 2

3 for some series products are relatively stable (for example, iphone and ipad), we first take the retail price as a constant (following the literature) to study how the retailers decide the pre-order price and production quantity to maximize his total expected profit. 2 Then we examine how the retailer s optimal pre-order strategy and profit change with the proportion of experienced consumers and future demand uncertainty. In an extended model, we will allow the retail price to be endogenously decided by the retailer as well as the pre-order price, and study the retailer s optimal pre-order strategy. We believe that an endogenously determined retail price will bring more interesting results to the literature. In our model, a monopolistic retailer sells a product with a fixed release date and commits to a two-period price path. Consumers can purchase either in the first period (pre-order stage), or in the second period (retail stage). There are two groups of consumers: experienced and inexperienced, depending on whether they have product-use experience of the early generation(s). Consumers valuations follow a normal distribution and we assume that experienced consumers value the product more than inexperienced consumers on average. Furthermore, the group size of experienced consumers is fixed and known to the monopolist. However, the monopolist is uncertain about the number of inexperienced consumers. In the first period, experienced consumers know their true valuations while inexperienced do not. All consumers decide whether to preorder the product or wait until the retail stage in anticipation of a future stock out risk. At the conclusion of the first period, the monopolist must choose its production quantity to satisfy the overall market demand, which has to be at least the size of pre-orders. If the production quantity is greater than the realized demand, the monopolist must dispose of the remaining units at a loss. In the second period, individual uncertainty is resolved for inexperienced consumers. For consumers who did not pre-order, they buy this product if their valuations are greater than or equal to the retail price. The product is delivered during the second period. In the benchmark model when the retail price is assumed to be exogenous, we show that the retailer has three possible strategies for pre-orders: advance selling at a deep discount, advance selling at a moderate discount, and advance selling at a premium price. Under advance selling at deep discount, the retailer s optimal pre-order price is such that no consumer waits until the retail stage. Experienced consumers whose valuations are above the pre-order price and all inexperienced consumers pre-order and get this product for sure after release. Under advance selling at a moderate discount, while experienced consumers with valuations above the pre-order price buy in the first period, all inexperienced consumers wait to make purchasing decisions based on the realization of their true valuations. Under advance selling at a premium price, the retailer is able to price discriminate experienced consumers by attracting high valuation experienced consumers to pre-order. Regarding other experienced consumers with valuations above retail price but did not buy in advance, they make purchases in the retail stage at a lower price but face a stock-out risk. All inexperienced consumers wait until the retail stage. Furthermore, we show that if the proportion of experienced consumers is small, the demand uncertainty from inexperienced consumers is large, or the salvage value of unsold products is small, the retailer is more likely to charge a deep discount for pre-orders. If the proportion of experienced consumers is large, the demand uncertainty from inexperienced consumers is small, or the salvage value is large, the retailer is more likely to charge a moderate discount for pre-orders. In the extended model where the retail price is endogenous, we show that the retailer strategically 2 The retailer will be referred as he and a consumer as she hereinafter. 3

4 chooses the retail price under different pre-order strategies. For example, if he decides to sell exclusively in advance, i.e., advance selling at a deep discount, the retailer will charge an infinite retail price so as to increase the threshold pre-order price as much as possible; if he decides to sell in advance at a premium, the retailer anticipates that the demand in the retail stage is composed of both experienced consumers (whose valuations are above the retail price but did not pre-order) and inexperienced consumers (whose valuations are above the retail price), and thus chooses the optimal retail price to maximize his profits. Our paper contains at least two contributions to the literature on advance selling. First, we build up a twoperiod, continuous-valuation model to study the retailer s optimal pre-order strategies for series products and consider two cases: exogenous and endogenous retail price. It substantially extends the two-type valuation model which is widely used in the existing pre-order literature and furthermore an endogenously determined regular selling price will bring more interesting results to the literature. Second, our model is one of the few that capture an important aspect of the economic phenomena in series products by introducing heterogeneity in consumers knowledge about their demand uncertainties (experienced vs. inexperienced consumers). Furthermore, we consider a more general situation that experienced consumers value the product more than inexperienced consumers on average. After the literature review (Section 2), the rest of the paper is organized as follows. In Section 3 we set up the model. In Section 4 we study the benchmark model where the retail price is exogenously given, and study the optimal pre-order strategy for the retailer. Comparison of the three pre-order strategy candidates is conducted to see how the retailer chooses the optimal pre-order strategy. In Section 5 we allow the retailer to determine the retail price endogenously and further study the corresponding optimal pre-order strategy. Concluding remarks are presented in Section 6 and roofs are relegated to Appendix. 2 Literature Review This paper belongs in the intertemporal pricing literature with strategic consumer behaviors. Within that literature, a number of papers focus specifically on pre-order (or advance selling) strategies, which can be classified into two strands. The first strand focuses on advance selling under limited capacity, with application to service industry. For examples, Gale and Holmes (1993) studies the pricing strategies of a monopoly airline and shows that advance purchase discounts allow the monopolist to allocate capacity more efficiently. Dana (1998) examines a competition model where consumers are heterogeneous in valuations and demand uncertainty and shows that firms may offer advance-purchase discounts to maximize profits. Xie and Shugan (2001) determine when and how the seller should sell in advance and focus on how the optimal pricing strategies change with the capacity. Möller and Watanabe (2010) compare different pricing strategies in a two-period advance selling model and further study how the relative profitability depends on price commitment, capacity constraint, rationing rule and resale. The second strand focuses on pre-order strategies in the manufacturing industry. DeGraba (1995) studies a monopoly model with consumer heterogeneity and demand uncertainty. He shows that the monopolist may 4

5 intentionally adopt scarcity strategies to lure consumers into pre-ordering to avoid a stock-out risk. 3 Li and Zhang (2013) examine the optimal pre-order strategies for a monopolistic seller in an uncertain market with heterogeneous consumers. They assume that consumer valuations are deterministic due to sufficient information in advance and show that advance selling discounts can never be optimal. Zeng (2013) assumes that the retailer is able to commit to both prices at the beginning of the first period, and analyze the retailer s optimal pre-order strategies in a model with both experienced consumers and inexperienced consumers. In the above three papers, consumers valuation follows a two-point distribution. That is, a customer can realize either high valuation for the good or low. Our paper is closely related to the literature which propose a two-period, continuous-valuation model. Most papers in this group treat the regular selling price as exogenously given and study the optimal pre-order price for the retailer. Zhao and Stecke (2010) classify consumers into two groups according to whether they are loss averse, and assume that consumers do not consider stock-out risk in their decision making. Prasad et al. (2011) set up a model where informed consumers enter the market in the first period while uninformed consumers enter the market in the second period. Informed consumers are uncertain about their valuations in advance, and face a endogenously determined stock-out probability if they do not pre-order the product. Although price premiums can induce strictly positive demand in the advance selling period, the authors in the above two papers restrict the retailer s strategy to advance selling discounts. Loginova et al. (2012) assume that the demand uncertainty for the retailer not only comes from the market size (as stated in the literature) but also from the distribution of consumers valuations. Based on pre-orders obtained in advance, the retailer learns the consumers valuation distribution and thus updates his forecast of the future demand. The authors show that there are two candidates for the retailer s profit maximizing pricing strategy: advance selling at a discount price and advance selling at a premium price. The only two papers (in the second strand) that treat the second-period price as a decision variable in a continuous-valuation model are Chu and Zhang (2011) and Nasiry and Popescu (2012). Chu and Zhang (2011) investigate the optimal pre-order strategy for a seller who strategically releases the product information in advance. In their model, population size is normalized to 1, and each consumer has an expectation of her true valuation in the pre-order stage given the information released in advance. The seller sets prices in two periods, and chooses how much information about the product to release. They show that it is never optimal for the seller to release all information in the pre-order stage, and there is always a pre-order price discount to induce some of the consumers to place advance orders. Our paper differs from Chu and Zhang (2011) in the following aspects. First, we divide consumers into two groups depending on whether they have product-use experience with any early generation of the product to capture different consumer characteristics in the market of series products. Second, we consider aggregate demand uncertainty in the model and the retailer needs to forecast the future demand based on the number of pre-orders. Third, our model shows that the retailer may charge premium price for pre-orders in some situations because consumers face a positive stock-out risk if they buy in the retail stage. Nasiry and Popescu (2012) study the optimal pricing strategy in an advance selling context where buyers act strategically with anticipated regrets. In their model, population size is normalized to 1 and all consumers 3 Since there is no aggregate uncertainty in his model, the stock-out risk is due to capacity constraints. 5

6 are uncertain about their valuations in advance. The firm supplies the product with a capacity constraint and sets prices for pre-orders and retail orders at the beginning of the pre-order stage. Because of the scarce production capacity, consumers face a stock-out risk if they delay their purchases. The authors investigate how anticipated regret affects consumer purchasing decisions and therefore the firm s advance selling strategies. Different from Nasiry and Popescu (2012) which model consumer regret in advance selling, our paper focuses on the pre-order strategies for new generation of series products and thus introduce experienced and inexperienced consumers in the model. We show that the existence of experienced consumers provides retailers with incentives to charge pre-order price premiums, which services as a new explanation for premium advance selling other than regret heterogeneity. Other than the literature which assume strategic consumers in the setup, the following three papers treat consumers as non-strategic. Tang et al. (2004) characterize the optimal discount price for the retailer and evaluate the benefits of pre-orders. McCardle et al. (2004) present a duopoly model and focus on competition between two firms. Boyaci and Özer (2010) focus on advance selling from manufacturers to retailers so as to obtain the optimal advance selling price and optimal stopping policy for a manufacturer. 3 The Setup Consider a two-period advance selling model where the retailer takes pre-orders for a new to-be-released product. The first period ends before the release date and we call it pre-order stage; the second period starts after the product release and we call it retail stage. Each consumer wants to purchase at most one unit of the specific product in either period. Pre-orders made at the pre-order price are guaranteed to be fulfilled immediately after the product release. However, orders submitted in the retail stage face a stock-out risk. 4 A market of risk-neutral and forward looking consumers decide whether to pre-order or to wait for the retail stage. We assume that demand exhibits the following two characteristics. First, there are two types of consumers. A number m e (fixed number) are experienced consumers and a number M i are inexperienced consumers, where M i follows a lognormal distribution M i LN ( ν i, τi 2 ) with mean m i = exp { ν i + τi 2/2}. 5 The uncertainty in the size of inexperienced consumers implies aggregate demand uncertainty for the retailer. While experienced consumers learn their valuations for the product in the pre-order stage, inexperienced consumers do not and their valuation uncertainty are resolved only in the retail stage. Consumers types are private knowledge. Let m = m e + m i denote the total expected number of consumers and α denote the proportion of experienced consumers. Thus, m e = αm and m i = (1 α)m. Second, each consumer has an idiosyncratic valuation, i.e., the maximum amount of money she would like to pay for this product. The consumer valuation of this product V follows normal distribution with mean µ and variance σ 2. In particular, we assume the consumer valuation distribution for experienced consumers is V e N ( µ e, σ 2) and that for inexperienced consumers is V i N ( µ i, σ 2), where µ i µ e, which implies that experienced consumers value the product more than inexperienced consumers on average. We use v e and v i to denote the realization of V e and V i, respectively. 4 The stock-out risk, which results from the uncertainty of future demand, is endogenously determined in this paper. And a positive stock-out risk is crucial for a pre-order price premium under advance selling. 5 A lognormal distribution avoids the realization of a negative number of inexperienced consumers. 6

7 The retailer produces the product at marginal cost c and charges pre-order price x and retail price p during the two periods. At the end of the retail stage, the retailer gets salvage value s for each unsold unit. We assume s < c < p, which ensures that the retailer makes positive profit and avoids infinite stock. Following the literature (Van Cayseele, 1991; Courty, 2003; Möller and Watanabe, 2010; and Nocke et al. 2011), we assume that the retailer can commit to a price schedule (x, p) in advance and there is no cost to implement advance selling. At the beginning of the pre-order stage the retailer announces the pre-order price x and the retail price p to the market. After pre-order is available, all consumers are allowed to submit pre-orders at price x which will be fulfilled by the retailer in the retail stage. At the end of this period the retailer gets the number of pre-orders, denoted by D 1. Informed by the pre-orders D 1, the retailer must decide how much to produce: Q = D 1 + q, where D 1 fulfills the pre-orders and quantity q satisfies the stochastic demand during the retail stage, which is denoted by D 2. 4 The Benchmark Model In the benchmark model, we assume that the retail price is exogenously given. This assumption will be relaxed later to study the corresponding optimal pre-order strategies. Assumption 1. The retail price is exogenously given as p from the outset. Figure 1 displays the timeline of the benchmark model. Note that the decision variables for the retailer is how much to price in the first period (x) and how much to produce (Q). 1 st Period all experienced consumers learn their valuations retailer announces x and p some consumers pre-order at x 2 nd Period all inexperienced consumers learn their valuations some consumers buy at p product delivery retailer decides on x retailer observes pre-orders and produces t Figure 1: Timeline of the Benchmark After observing the pre-order price x and retail price p in period 1, consumers make the decisions to pre-order or wait. Since experienced consumers know their valuations in period 1, each of them compares her payoff from pre-ordering, v e x, with her expected payoff from waiting, (1 η)(v e p), where η describes the stock-out probability in period 2. Mathematically, η is endogenously decided by [ (D2 q ) ] + η = E, (1) D 2 7

8 where q is the optimal quantity satisfying the stochastic demand during the retail stage (D 2 ). This expression implies that the probability of any consumer who wants to purchase the product in the second period but is unable to get it is the fraction of the excess demand (Loginova et al. 2012). It follows immediately that experienced consumers never wait when there is a discount for pre-orders and those with valuations v e x pre-order. If there is a premium for pre-orders, those experienced consumers with valuation pre-order this product to avoid a future stock-out. v e p + x p η Regarding inexperienced consumers, they are uncertain about their own valuations in period 1 and all act homogenously based on valuation expectations. An inexperienced consumer pre-orders if and only if µ i x (1 η) + p (v i p)f i (v i ) dv i, where f i ( ) denotes the density function of N ( µ i, σ 2). The left hand side is the inexperienced consumer s expected payoff from pre-ordering at price x, and the right hand side is her expected payoff from waiting to buy in the retail stage. Note that the right hand side captures the fact that an inexperienced consumer purchases in the retail stage only if her realized valuation of this product is greater than or equal to the retail price p, and she is able to get it with probability 1 η. Let ˆx µ i (1 η) + p (2) (v i p)f i (v i ) dv i (3) denote the threshold value for the pre-order price. All inexperienced consumers pre-order the product if x ˆx; and all wait to make purchasing decisions until the second period if x > ˆx. Lemma 1 (Properties of ˆx). The threshold value ˆx, given by (3), increases with η, but can not exceed µ i. For the rest of our analysis in this section, we assume that a pre-order price discount is required to attract all inexperienced consumers to pre-order. That is, ˆx < p. Most studies on advance selling also focus on the situation that consumers whose true valuations are uncertain are attracted to pre-order only when there is a discount for pre-orders, which is commonly observed in practice as well. Assumption 2. The threshold value for the pre-order price ˆx is smaller than the retail price p. This assumption supports pre-order discount price being the candidate of the retailer s optimal pricing strategies and thus keeps the market behaviors as close to the real world as possible. Without it, the retailer most likely charges pre-order premium price in the equilibrium. Therefore, we have the following three regions. Region A: x ˆx. Experienced consumers with valuations above x and all inexperienced consumers pre-order. No sales occur in the second period. Region B: ˆx < x p. Experienced consumers with valuations above x pre-order, while all inexperienced consumers wait to make decisions in the second period. 8

9 Region C: x > p. Experienced with valuations above x buy this product either in the first period or in the second period, while all inexperienced consumers wait to make decisions in the second period. 4.1 A pre-order price discount In this section we consider x p so as to focus on advance selling at a discount price. Following the above discussion, experienced consumers never wait to buy in the retail stage. Those with valuations above x pre-order at a discount. The part of the retailer s expected profit that comes from experienced consumers equals m e F e (x)(x c), where F e ( ) denotes the cumulative distribution function of N ( µ e, σ 2) and F e ( ) = 1 F e ( ). Next, we analyze inexperienced consumers optimal purchasing decisions with Assumption 2 to derive the retailer s expected profit that comes from inexperienced consumers. If x ˆx (Region A), all inexperienced consumers pre-order and the part of the retailer s expected profit that comes from inexperienced consumers equals m i (x c). Hence, the retailer s expected total profit Π as a function of the pre-order price x can be written as Π A (x) = m e F e (x)(x c) + m i (x c). (4) If ˆx < x p (Region B), all inexperienced consumers wait until the second period and make purchases if their valuations exceed p. Thus, the second-period demand is comprised of inexperienced consumers with valuations above p, D 2 = M i Prob(v i > p) = M i F i (p), where F i ( ) denotes the cumulative distribution function of N ( µ i, σ 2) and F i ( ) = 1 F i ( ). Because M i LN ( ν i, τi 2 ), it is straightforward to show that D 2 LN ( ν i + ln F i (p), τi 2 ). In addition to D 1 pre-orders, the retailer needs to produce q to satisfy the stochastic second-period demand, where the quantity q is chosen to maximize his expected profit π(q) = p E [min {q, D 2 }] + s E [ (q D 2 ) +] cq. (5) The solution is a traditional Newsvendor Problem with a lognormally distributed future demand (e.g., Silver et al and Loginova et al. 2012). 6 Following the standard solution method, the optimal order quantity q and the optimal expected profit π are q = exp{ν i + τ i z β }F i (p) (6) 6 For the lognormal distribution D 2 LN ( ν, τ 2), the optimal quantity is given by q = exp{ν + τz β } and the resulting expected profit is given by π = (p s) (1 Φ(τ z β )) exp{ν + τ 2 2 }. 9

10 π = π(q ) = m i (p s) (1 Φ(τ i z β )) F i (p), (7) where β = (p c)/(p s), z β is the β-th percentile of the standard normal distribution, i.e., z β Φ 1 (β), and Φ( ) is the distribution of the standard normal distribution. Hence, the retailer s expected total profit is Π B (x) = m e F e (x)(x c) + π. (8) Therefore, in the case of a discount for pre-orders, the retailer s expected total profit is Π A (x), x ˆx, Π(x) = Π B (x), ˆx < x p. The optimal pre-order discount price depends on the shape of unimodal function F e (x)(x c). Denote x as the solution of max x F e (x)(x c). We have F e (x)(x c) x x = 0. (9) The unimodal function F e (x)(x c) is monotonically increasing for x x and monotonically decreasing for x > x. We note that x may locate on the left of ˆx, on the right of p, or between these two values. From now on we focus on the interesting case that ˆx < x < p to study the retailer s optimal pre-order strategy. Similar analysis can be applied in the other two cases and we will see limited candidates for the retailer s profit maximizing selling strategy. 7 Assumption 3. The price that maximizes the profit from experienced consumers, x, is located between ˆx and p. With Assumption 3, the expected total profit function in Region A, Π A (x), increases with x and the retailer chooses ˆx; in Region B, all inexperienced consumers wait to make purchasing decisions in period 2 at price p and thus Π B (x) is maximized at x as well as F e (x)(x c), then the retailer chooses x. Although Π A (x) and Π B (x) are continuous in each region, it is important to note that the expected total profit Π(x) is not continuous on the interval (c, p). There is a jump at ˆx. Accordingly, we have the following two patterns for Π(x) (see Figure 2). Pattern 1: Π(x) jumps down at ˆx. Pattern 2: Π(x) jumps up at ˆx. Regarding the optimal pre-order discount price x, under Pattern 1 it is either ˆx or x; and under Pattern 2 it is x. 7 If x ˆx, the optimal choice in Region A is located between x and ˆx, and its value depends on the ratio of experienced consumers to inexperienced consumers; however, any price in Region B be can not be optimal because it can be cut down to increase the profit from experienced consumers without affecting the expected profit from inexperienced consumers. If x p, the optimal choice in Region A and Region B are ˆx and p respectively because both expected total profit functions increase with x; however, advance selling at p is dominated by advance selling at a price premium as we can see in (12). 10

11 c p c p (a) Pattern 1 (b) Pattern 2 Figure 2: The retailer s expected profit as a function of x for x p Lemma 2 (Optimal pre-order discount price). The optimal pre-order discount price x is ˆx or x. With a deeper discount for pre-orders, x = ˆx, the retailer is able to achieve a sales increase because more consumers are attracted to pre-order. 8 Furthermore, since all sales happen in the first period, the overage and underage costs under x = ˆx is zero. Therefore, the two potential optimal pre-order prices reflect two different tradeoffs for the retailer: between low price-high sales and high price-low sales, and between low price-low expected overage and underage costs and high price-high expected overage and underage costs. The optimal pre-order price is thus chosen to maximize the retailer s expected total profit after weighing the potential tradeoffs. Note that the threshold value ˆx is defined in (3) as a function of η. Under advance selling at a discount, we can calculate η as (see the derivation in Appendix) which possesses the following properties. { η = 1 β exp τ i z β + τ i 2 } (1 Φ(z β + τ i )), (10) 2 Lemma 3 (Properties of η). The stock-out probability η under advance selling at a discount has the following properties: (i) η/ s < 0, and η = 0 when s = c; (ii) η/ τ i > 0, and η = 0 when τ i = 0. with s. By combining the results of Lemma 1 and Lemma 3, we obtain that ˆx increases with τ i and decreases 8 With a deeper discount, more experienced pre-order and secondly all inexperienced consumers buy in advance. 11

12 4.2 A pre-order price premium In this section we consider x > p (Region C) so as to explore advance selling at a premium price. Since the threshold value ˆx defined in Section 4.1 is below p, it is less than any premium price. As a result, inexperienced consumers always wait to make purchasing decisions in the retail stage after valuation uncertainty are resolved. Regarding experienced consumers who already know their valuations in the first period, those with high valuations (see (2)) pre-order in advance to avoid a positive stock-out risk, and the rest of experienced consumers with valuations above p buy in the second period. It is important to note that the stock-out risk (measured by η) is crucial for positive first-period demand at a premium price. Without it (either because the aggregate market demand is deterministic, or because s = c), advance selling is always implemented at a pre-order discount to attract consumers to pre-order. See, for example, Chu and Zhang (2011) and Nocke et al. (2011). It follows from (2) that the number of consumers who buy in the pre-order stage is ( D 1 = m e F e p + x p ). η The second-period demand D 2 is composed of inexperienced consumers with valuations above p and those experienced consumers with valuations above p but did not pre-order. It is ( ( D 2 = m e F e (p) F e p + x p )) + M i F i (p), (11) η where the first term on the right hand side is fixed and represents the number of experienced consumers who buy in the retail stage; and the second term represents the stochastic demand from inexperienced consumers. Because M i F i (p) LN ( ν i + ln F i (p), τi 2 ), D2 is a shifted lognormal distribution. Note that the second-period demand uncertainty only comes from inexperienced consumers, and it is exactly the same as that under advance selling at a discount (see Section 4.1). Following the standard solution method, in addition to satisfy the fixed demand from experienced consumers in the second period, the retailer produces q to satisfy the stochastic demand from inexperienced consumers, where q is calculated in (6). That is, the retailer produces The resulting expected total profit is ( ( Q = D 1 + m e F e (p) F e p + x p )) + q. η ( Π C (x) = m e F e p + x p ) (x p) + Π B (p), (12) η(x) where Π B (p) is obtained by setting x = p in (8). The derivation of (12) is straightforward. The demand uncertainty remains the same under both advance selling at x and p. Under both x and p the retailer produces the quantity Q = m e F e (p) + q and obtains the same sales. The difference between advance selling at x 12

13 ( ) and at p is that under x a number of orders, denoted by m e F e p + x p, are placed at a premium x, while under p all orders are placed at p. Notice that we write down η(x) instead of η. It implies that the stock-out probability takes a different value from that in Section 4.1, and it is affected by the pre-order price x. We see that the demand uncertainty remains the same under both cases (either advance selling at a discount or advance selling at a premium). Thus, the excess demand does not change. However, as some experienced consumers switch to buy in the retail stage, the second-period demand increase, which is implied by the first term (a function of x) on the right hand side of (11). Thus, it is inferred that η is smaller when the retailer adopts pre-order premium price and furthermore it is a function of x. For a given x p, we calculate the endogenous stock-out probability, η(x), as the solution of η = E η(x) + M i F i (p) exp{ν i + τ i z β }F i (p) ( ( )). (13) m e F e (p) F e p + x p + M i F i (p) Lemma 4 (Properties of η(x)). The stock-out probability η(x), defined by (13), exists for all x p. Moreover, it has the following properties: } (i) η(p) = 1 β exp {τ i z β + τ i 2 2 (1 Φ(z β + τ i )); (ii) η(x) is decreasing in x for x p. The retailer s objective is to maximize his expected total profit (12) by choosing a premium price x. Observe that Π C (x) converges to Π B (p) as x approaches p from above, and it can be shown that Π C (x) converges to Π B (p) as x. Hence, there exists x p, the price that maximizes Π C (x) for x > p. Lemma 5 (Optimal advance selling premium price). The optimal advance selling premium price is x p, which is chosen to maximize (12). η 4.3 Optimality To get the retailer s optimal pre-order price, we first combine (4), (8) and (12) to obtain the retailer s total expected profit. The retailer chooses x to maximize Π A (x), c < x ˆx, Π(x) = Π B (x), ˆx < x p, (14) Π C (x), x > p. The retailer s expected total profit Π(x) is continuous at x = p. However, there is a jump at ˆx as we discussed in Section 4.1. Accordingly, we still have the two patterns for Π(x): Pattern 1 where Π(x) jumps down at ˆx; and Pattern 2 where Π(x) jumps up at ˆx. Combining the results of Lemma 2 and Lemma 5, we have immediately the following proposition. 13

14 Proposition 1 (Optimal pre-order price). When both pre-order price discounts and premiums are considered, the optimal pre-order price is either ˆx, x or x p. If Pattern 1 in Figure 2 prevails (a jump down at ˆx), the optimal pre-order price can be either ˆx, x or x p. This results is illustrated in Figure 3. If Pattern 2 in Figure 2 prevails (a jump up at ˆx), the optimal pre-order price is x or x p. This result is illustrated in Figure x x profit 4 profit x (a) x = ˆx x (b) x = x 7 x profit x (c) x = x p Figure 3: The optimal pre-order price under Pattern 1 It is easy to observe that only when Pattern 1 occurs can ˆx, which corresponds to the case of exclusively sell in advance, be the optimal pre-order price. Otherwise, the retailer prefers an pre-order price which results in positive sales in both periods. We next look at the scenario when the retailer does not adopt advance selling. The retailer sells exclusively in the retail stage at price p. All consumers realize their valuations in this period and only those with valuations above p buy this product. Compared with advance selling at price p, selling exclusively at p produces the same quantity and get the same number of buyers. Since there is no cost to implement advance selling, it is equivalent for the retailer to advance selling at p. Therefore, we obtain the following result. 9 Figure 3 and Figure 4 are obtained with different values for parameters in the model. 14

15 4.5 x x profit profit x (a) x = x x (b) x = x p Figure 4: The optimal pre-order price under Pattern 2 Corollary 1 (Advance selling versus no advance selling). The retailer is strictly better off under advance selling than under no advance selling. This result also holds when the cost to implement advance selling is relatively small. To be precise, as long as the cost is smaller than the potential gain from advance selling, i.e., Π(x ) Π B (p), the retailer will adopt advance selling. However, we should note that a seller will not take pre-orders if the potential gain from advance selling is small considering there is a nonnegligible adoption cost. 4.4 Comparison From our analysis so far it follows that there exist three candidates for the retailer s profit maximizing selling strategy. The retailer can either sell exclusively at a deep discount (in Region A) in the pre-order stage to lure all inexperienced consumers into pre-ordering before individual valuation uncertainty has been resolved, or charge a moderate pre-order discount (in Region B) trying to achieve the maximal profit from experienced consumers while keeping inexperienced consumers buy in the retail stage. Or he can price discriminate experienced consumers by implementing a pre-order premium price (in Region C) to induce high valuation experienced consumers to pre-order at a premium while the rest of experienced consumers with valuations above p purchase in the retail stage. In a market with consumer heterogeneity (two groups of consumers) and demand uncertainty (random size of inexperienced consumers), the retailer chooses the optimal pre-order pricing strategy and resulting production quantity to maximize his total expected profit, realizing that unsold units will be salvaged at a loss (salvage value is less than the cost). To further understand how the retailer chooses among ˆx, x and x p, we next examine how the values of Π(ˆx), Π( x) and Π(x p ) change with important parameters in the model, such as the proportion of experienced consumers α, demand uncertainty τ i, and salvage value s. Based on 15

16 (4), (8) and (12), we can write down the profit functions Π(ˆx), Π( x) and Π(x p ) as Π(ˆx) = m e F e (ˆx)(ˆx c) + m i (ˆx c); ( Π( x) = m e F e ( x)( x c) F e (p)(p c) ) ( ) + Π(p); Π(x p ) = m e F e p + xp p (x p) + Π(p). η(x p ) (15) It is obvious to see that both Π( x) and Π(x p ) are greater than Π(p). And note that Π( x) Π(p) is not affected by α, τ i and s. Hence, we study Π(p) instead of Π( x) and Π(x p ) to avoid complication. Lemma 6 (Some useful properties of Π(ˆx) and Π(p)). Parameters α, τ i and s have the following effects on Π(ˆx) and Π(p): (i) As α increases, Π(ˆx) decreases and Π(p) increases. (ii) As τ i increases, Π(ˆx) increases and Π(p) decreases. (iii) As s increases, Π(ˆx) decreases and Π(p) increases. Although the pre-order price is the lowest when x = ˆx, the retailer is able to attract all inexperienced consumers to pre-order, and therefore eliminates demand uncertainty and avoids a salvage loss. Especially when the proportion of inexperienced consumers is high (small α), the demand uncertainty is high (large τ i ), or the salvage loss is high (small s), the retailer is more willing to charge x = ˆx. Otherwise, the gains from maximizing the profits in the experienced consumers market is greater than the loss from decreased sales in the inexperienced consumers market, then the retailer switches to the other two pricing strategies to focus on the experienced consumers market. And it is intuitive to see that when the proportion of experienced consumers is high (large α), demand uncertainty is low (small τ i ), or salvage loss is low (high s), there are less incentives for the retailer to charge a premium price for pre-orders; instead, he prefers to set the optimal pre-order price at a moderate discount x = x so as to extract the highest profit from experienced consumers. Therefore, we have the following corollary. Corollary 2. The retailer is more likely to adopt advance selling at a deep discount price ˆx if the proportion of experienced consumers α is small, the parameter τ i in the distribution of the number of inexperienced consumers is large, or the salvage value s is small; a moderate discount price x if the proportion of experienced consumers α is large, the parameter τ i in the distribution of the number of inexperienced consumers is small, or the salvage value s is large; a premium price x p, otherwise. There are two groups in the market (experienced and inexperienced consumers), and the retailer needs to decide on which group to target on. Setting a deep discount for pre-orders enables the retailer to attract all inexperienced consumers to pre-order, and therefore results in zero overage and underage costs. However, the other two strategies (a moderate discount for pre-orders and a premium for pre-orders) focus on experienced consumers to maximizing the profits from them. Intuitively, a smaller α means the proportion 16

17 of inexperienced consumers in the population is greater and hence the retailer is more willing to focus on inexperienced consumers market. As τ i increases, there is more uncertainty about the number of inexperienced consumers, increasing the risk associated with overage and underage. And similar situation occurs as s decreases. As a result, the retailer is more willing to charge a deep discount for pre-orders. While choosing the optimal pricing strategy to obtain the highest profit from experienced consumers, there are two candidates for the retailer: a moderate pre-order discount and a pre-order price premium. The incentives for the retailer to charge a premium for pre-orders are the demand uncertainty and salvage loss, together with the coexistence of both groups. 10 In a sense, if the proportion of experienced consumers in the population is large, the demand uncertainty is low, or the salvage value is high, there are not enough incentives for the retailer to adopt a pre-order premium price. Instead, the retailer is more likely to charge a moderate pre-order discount. 5 Endogenous Retail Price In this section we allow the retailer to choose the retail price as well as pre-order price at the beginning of the first period. It refers to the situation that the retail prices of series product may change when a new generation comes out. For example, the retail prices for the third to seventh Harry Potter book are $19.95, $25.95, $29.99, $29.99 and $34.99, respectively, which is different from the pricing pattern with fixed retail price. This extended model with an endogenously determined retail price further captures the characteristics of marketing behaviors and therefore brings more interesting results to the literature. As in Section 4, we analyze two different pricing patterns: pre-order price discount and pre-order price premium, and then obtain the retailer s optimal pricing strategy by combining the analyses. 5.1 An increasing pricing pattern Experienced consumers never wait to make purchases in the second period since there is a discount for preorders, while inexperienced consumers may prefer to wait. We then use backward induction to obtain the optimal decisions on x and p. In the second period, the seller sets q to maximize his expected profit in (5). We look at two different scenarios and start with the one where all inexperienced consumers do not pre-order in the first period. If inexperienced consumers wait to make purchases in the second period, we have D 2 = M i F i (p) LN ( ν i + ln F i (p), τi 2 ). Following (6) and (7), the optimal q and the resulting expected profit are q(p) = exp{ν i + τ i z β }F i (p), (16) π(p) = π(p, q(p)) = m i (p s) (1 Φ(τ i z β )) F i (p). (17) 10 A high proportion of experienced consumers also implies that the demand uncertainty is low since the size of inexperienced consumer is small. 17

18 Since π(p) converges to zero as p approaches infinity or c, according to the Mean-Value Theorem for differential calculus there exists a price, denoted by p, that maximizes π(p). We have m i (p s) (1 Φ(τ i z β )) F i (p) p p = 0. (18) Experienced consumers with valuations above x pre-order in the first period and the retailer s expected profit that comes from experienced consumers is m e F e (x)(x c). According to (9), retailer s expected profit in the first period is maximized at x. Assumption 3 implies that the optimal price in a market with only experienced consumers is smaller than that in a market with only inexperienced consumers. Therefore, the retailer will choose an increasing pricing pattern (x = x, p = p). Lemma 7 (Optimal discounted pricing strategy when all inexperienced consumers wait). The optimal discounted pricing strategy under which all inexperienced consumers wait to make purchasing decisions is (x = x, p = p). We next study the optimal discounted pricing strategy such that all inexperienced consumers pre-order. All sales occur in the pre-order stage and the resulting expected profit is Π(x, p) = m e F e (x)(x c) + m i (x c). The first term on the right hand side is the profit from experienced consumers, which is maximized at x; and the second term is the profit from inexperienced consumers, which increases with pre-order price x. Note that in this scenario the optimal pre-order price should not exceed ˆx(p), where ˆx(p) denote the threshold value for the pre-order price which attracts inexperienced consumers to pre-order. Thus, the expression for ˆx(p), described in (19), is the same as ˆx in (3), where p on the right hand side is a decision variable rather than an exogenous value. ˆx(p) µ i (1 η) + p (v i p)f i (v i ) dv i (19) From (19) we see that lim p ˆx(p) = µ i. Therefore, the retailer will set p = such that ˆx(p) µ i, and the optimal pre-order price is x = ˆx(p) µ i. Lemma 8 (Optimal discounted pricing strategy when all inexperienced consumers pre-order). The optimal discounted pricing strategy under which all inexperienced consumers pre-order is (x = µ i, p = ). Lemma 7 and 8 imply that the profit maximizing pricing strategy candidates with pre-order price discounts can be either advance selling at a deep discount with strategy (x = µ i, p = ), or advance selling at a moderate discount with strategy (x = x, p = p). 5.2 A decreasing pricing pattern Next, we focus on the pricing pattern that x > p. In this situation, inexperienced consumers will wait to purchase in the second period. Regarding experienced consumers, some of them pre-order in the first period 18

19 and some buy later. Following the analysis in Section 4.2 (see (7), (8) and (12)), we can rewrite the expected total profit for the retailer as ( Π(x, p) = m e F e p + x p ) (x p) + m e F e (p)(p c) + π(p), (20) η(x, p) where the first two terms on the right hand side represent the profits gained from experienced consumers and the last term represents the profits gained from inexperienced consumers. Denote x and p as the solution of maximizing (20) conditional on that x > p. We have x and p satisfy the following first order conditions Π(x, p) x = 0, x= x,p= p Π(x, p) p = 0. x= x,p= p Hence, the optimal pricing strategy with a pre-order price premium is (x = x, p = p). Lemma 9 (Optimal pricing strategy with a decreasing pricing pattern). The optimal advance selling pricing strategy with a pre-order price premium is (x = x, p = p). Note that the retail prices under different pricing paths do not need to be equal. The optimal retail price under advance selling at a deep discount is p = and that under advance selling at a moderate discount is p = p, where p maximizes π(p) in (20). According to (9), the second term on the right hand side of (20) is maximized at x(< p). It is most likely that the optimal retail price that maximizes the profits gained from experienced consumers is smaller than p. Therefore, we have p < p. Our numerical examples show that this is true for all the cases we simulated. In our model, p < p indicates that the retail price is higher under advance selling at a discount compared with that under advance selling at a premium. It is reasonable to have this result because some experienced consumers go to make purchases with inexperienced consumers in the retail stage and thus a reduced retail price increases the total profits gained from experienced consumers. 5.3 Optimality As we obtain in the previous section, there exist three candidates for the retailer s profit maximizing selling strategy: advance selling at a deep discount, advance selling at a moderate discount and advance selling at a premium. The following proposition is obtained by combining the results in Lemma 7-9. Proposition 2 (Optimal pricing strategy with endogenous retail price). The profit maximizing pricing strategy candidates are: advance selling at a deep discount price with strategy (x = µ i, p = ); advance selling at a moderate discount price with strategy (x = x, p = p); advance selling at a premium price with strategy (x = x, p = p). 19