Product Launches and Buying Frenzies: A Dynamic Perspective

Transcription

1 Product Launches and Buying Frenzies: A Dynaic Perspective Pascal Courty Departent of Econoics, University of Victoria, PO Box 700, STN CSC, Victoria, BC, V8W Y, Canada, pcourty@uvic.ca Javad Nasiry School of Business and Manageent, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, nasiry@ust.edu Buying frenzies in which a fir intentionally undersupplies a product during its initial launch phase are a coon practice within several industries such as electronics (cell phones, video gaes, gae consoles), luxury cars, and fashion goods. We develop a dynaic odel of buying frenzies that captures the production and sales of a product over tie by the fir and then characterize the conditions under which frenzies are an optial policy for the fir. We show that buying frenzies occur when custoers are sufficiently uncertain about their product valuations and when custoers discount the future but not excessively. Further, we propose a easure of custoer desperation to easure the agnitude of frenzies and deonstrate that buying frenzies can have a significant ipact on the fir s profit. This paper provides anagerial insights on how firs can affect the arket response to a new product through their pricing, production, and inventory decisions to induce profitable frenzies. Key words : Advance Selling, Buying Frenzy, Custoer Desperation, Strategic Custoer Behavior. Introduction Long queues of enthusiastic custoers were coon when Apple s ipad hit the Hong Kong arkets on April 9, 00. Given the incleent weather, Apple stores were glad to provide ubrellas and raincoats (bearing Apple s logo) to waiting custoers. Each store received a liited nuber of ipads and distributed the on a first-coe, first-served basis. Soe custoers were willing to pay considerably higher prices to obtain the product in the gray arket. Such shortage of supply is not confined to Apple s ipad. The shortage of Nintendo s gae console Wii, for exaple, lasted fro its initial introduction in 006 until 009 (Liu and Schiraldi 0). Sony s PlayStation faced the sae probles after entering the US arket eight onths later than Japan s arkets (Stock and Balachander 005).

2 A buying frenzy is induced by a fir that intentionally undersupplies a arket and excluded custoers are strictly worse off. This practice is coon in several industries such as luxury cars, fashion, and especially electronics (cell phones, video gaes, gae consoles). Although shortages ight be attributed to deand forecasting errors, issues in coponent supplies or production probles, their repeated occurrence particularly during the launch phase of innovative products suggests that a conscious arketing strategy is the true reason. The few studies investigating this issue have considered ainly static odels. These odels capture situations where firs sell the good only once; exaples include tickets for sporting or usic events, liited edition products, and one-off auctions. Yet, the predictions of static odels collapse when the fir produces repeatedly over tie, as is the case for ost anufactured products. The reason is that the fir wants to serve the custoers who were excluded fro the early sales. But why should custoers be desperate to buy early when products are available in the future? An iportant gap in the literature is the absence of a dynaic analysis that supports an initial buying frenzy followed by a period of regular sales without frenzies. We develop a odel in which production and sales occur in two periods and then characterize the dynaics of sales, prices, and scarcity. The two-period odel that we eploy is a stylized representation that features the classical dichotoy between the launch phase of a new product and its subsequent ature phase. We show that the fir s gains fro inducing a buying frenzy (relative to atching supply and deand) can be econoically substantial. We also investigate the conditions under which buying frenzies are optial. Finally, we copute custoers loss fro being excluded during the initial launch phase, which is a proxy for custoer desperation, and show that this loss can be significant. This explains why custoers ay invest resources to obtain the good early (e.g., wait in queues) and why prices can be significantly higher in resale arkets. There are several ingredients to our analysis. We assue that custoers are initially uncertain about their preferences for the product (see e.g., Xie and Shugan 00, Gallego and Şahin 00 and Yu et al. 0). This assuption applies to the innovative and fashion products that have been the object of buying frenzies. As DeGraba (995) and Denicolò and Garella (999), we assue that the fir cannot coit to future prices and quantities. This iplies that the fir discounts prices when inventories build up and is consistent with the response of car and electronics anufacturers when products do not sell as expected. For exaple, HP had to slash prices to clear the unsold inventory of TouchPad only two onths after its launch (The Wall Street Journal 0). Finally, we use Pareto doinance as the selection criterion aong the equilibria of the gae (Cachon and Netessine 004). In other words, given the fir policy, the equilibriu selected gives custoers the highest payoff. The choice of the Pareto-doinant equilibriu is consistent with the interpretation

3 3 that custoers coordinate on the equilibriu that gives the the highest surplus which reflects the social diension of buying frenzies. In a dynaic odel, how does one account for uninfored custoers rushing to buy early? Custoers can always obtain the good upon waiting, and waiting benefits the because they can then incorporate the inforation they learn about their preferences in the decision to purchase the product. When a fir produces a large quantity in the first period, custoers anticipate that they can get the product at a lower price in the second period in the equilibriu where custoers wait. Thus, for a product to sell in the first period, its price ust be sufficiently low to prevent such strategic custoer waiting. This deterines the axiu price a fir can charge in the first period. At that price, custoers expected utility fro buying early is equal to the utility they obtain in the waiting equilibriu. But an individual custoer is strictly worse-off waiting because the price the fir charges in the second period (on the equilibriu path) is higher than the price the custoer could have obtained in the waiting equilibriu (off the equilibriu path). This explains why custoers strictly prefer obtaining the good early. Buying frenzies are ore likely to happen when custoers are initially uncertain about their preferences for the product. An increase in preference uncertainty increases custoers utility in the waiting equilibriu. Custoer valuation uncertainty is likely to play a greater role in the case of an innovative product that will atch the needs of soe custoers but not others. This ay explain why frenzies are coon for new electronic products (e.g., the iphone) and not for siilar e too products (e.g., Android phones) that are released later. It is reasonable to assue that custoers have ore uncertainty about new products than about knock offs produced later. There are two ain theories of buying frenzies. The first is based, as in our odel, on interteporal price discriination. Denicolò and Garella (999) show that a onopolist ay ration heterogenous custoers to prevent strategic waiting. DeGraba (995) considers a static odel with individual preference uncertainty. In both of these odels, buying frenzies exist only for specific rationing rules. Our odel is dynaic and the existence of frenzies does not depend on the rationing rule used. Moreover, our analysis delivers a tractable odel to study how preference uncertainty influences the existence of buying frenzies and to forally easure custoer desperation. Other papers on inter-teporal price discriination and scarcity policies that coe close to our paper include Liu and van Ryzin (008), Cachon and Swinney (009), and Liu and Schiraldi (0). These papers, however, focus on different issues than we do. In particular, Liu and van Ryzin (008) show that a stock-out possibility in the second period can prevent custoers with known Coordination transpires through press reports and social edia such as online networking sites (Facebook, Twitter, etc.). Through these channels, custoers develop a sense of whether other custoers are buying the product and whether they should do the sae.

4 4 preferences fro waiting. Cachon and Swinney (009) suggest that by liiting the initial stocking level and offering optial arkdowns, the fir ay constrain the strategic purchase behavior of its custoers. Liu and Schiraldi (0) show that the existence of resale arkets can induce the fir to under-stock the product and increase its equilibriu price. The second theory is based on asyetric inforation. Stock and Balachnder (005) argue that a high-quality fir eploys scarcity to signal quality to uninfored custoers. Rationing as a signal of quality has also been studied in Debo and van Ryzin (009), and in Allen and Faulhaber (99), aong others. Papanastasiou et al. (0) develop a odel of boundedly-rational social learning and show that a fir ay restrict the availability of a product to elicit favorable reviews fro early adopters which positively affect the preferences of other custoers. Our odel is not based on inforation asyetry and does not rely on irrationalities to explain frenzies. Neither do we assue that custoers are yopic or the fir has to charge a fixed price over tie. Other odels of buying frenzies are based on deand externality (Becker 99) and psychological drive (Verhallen and Robben 994). Becker s odel accounts for social interactions and shows that an upward-sloping deand curve ight result in an unstable equilibriu with arbitrarily sall frenzies. Verhallen and Robben (994) argue that scarcity itself can increase custoers willingness to pay provided custoers attribute that scarcity to deand-side variables (such as popularity) and not supply-side variables (such as the fir intentionally liiting supply). Finally, buying frenzies are distinct fro herding (Debo and Veeraraghavan 009), in which custoers ignore their private noisy inforation about a product and erely follow what previous custoers did. In contrast, iperfect inforation is absent fro our odel of frenzies. Not uch epirical work has been done on buying frenzies. An iportant exception is Balachander et al. (009) who present a thorough analysis of buying frenzies in the context of the autoobile arket. Although their findings are not consistent with DeGraba s (995) static odel of buying frenzies, we argue that they are consistent with a dynaic odel. This deonstrates the iportance of distinguishing static and dynaic odels of buying frenzies. The rest of the paper is organized as follows. Section outlines the basic odel. Section 3 analyzes the dynaic odel of buying frenzies when the fir can produce over tie; and also defines easures of custoer desperation and frenzy intensity. Section 4, extends the odel to include second-period arrivals. Section 5 concludes the paper.. A Model of Buying Frenzies A onopolist sells to a population of N custoers who arrive in the first period of a two-period horizon. Custoer valuations are identically and independently distributed with density f(v) and survival function F (v). We assue that f and F are continuous functions with support [v, v]

5 5 [0, ] and E[v] = µ. We also assue that the function f(x) > 0 is log-concave. This deand specification approxiates a large arket in which custoers are infinitesial 3 and have idiosyncratic preferences that are discovered over tie (Lewis and Sappington 994). The two-period horizon is a stylized representation of a product launch phase followed by a ature phase of sales. It is therefore unnecessary for the two periods to have equal length; the second period can be arbitrarily longer than the first. What atters is that individual custoer uncertainty is resolved by the end of the first period. Custoers discount future utility by δ c, and the onopolist discounts future profits by δ. Although we do not restrict the paraeter space for (δ c, δ ), it is plausible that custoers have a lower discount factor than the fir (Liu and van Ryzin 008). The onopolist can produce in both periods. We denote q and q as the production quantities in the first and second periods, respectively. To siplify the exposition, we assue that the arginal cost of production is zero. The fir announces (q, p ) at the start of the first period. Custoers are strategic and for expectations on what will happen in period. They buy or wait depending on which option axiizes their discounted utility given these expectations. As DeGraba (995) and Denicolò and Garella (999), we assue that all inventory available in the second period (period- production plus left-over inventory fro period ) is sold at the arket-clearing price. In other words, the fir cannot coit to withhold inventory in the second period. It is a realistic assuption because product launches are rare and isolated in tie, which helps keep the fir fro developing a reputation for withholding or destroying excess inventory. As discussed in the Introduction, this assuption is consistent with practices observed for anufactured products. In contrast, firs selling perishable products (e.g., in the airline or hotel industry) soeties coit to policies that do not necessarily discount excess inventories when sales are low; see Liu and van Ryzin (008) for an extended discussion on this assuption. We solve for the syetric rational expectations equilibria (REE) once the fir announces (q, p ). 4 Denote a custoer s decision to buy or wait in an REE by the probability x [0, ]. A custoer waits if x = 0, buys if x =, and her strategy is ixed if x (0, ). We focus on syetric equilibria that is, we assue x is the sae for all infinitesial custoers. The price that a custoer expects to face in period depends on the fraction of custoers who have attepted to buy, x, and on the fir s initial production quantity q. Denote that expected period- price p e (x q ). The expected period- surplus is then T (p e (x q )) = δ c E[v p e (x q )] +. 5 This is a standard technical assuption to ensure the uniqueness of the equilibriu. However, the results do not hinge on this assuption. Widely applied paraetric failies including the unifor, exponential, noral, and logistic have log-concave density (Bagnoli and Bergstro 005). 3 This eans that whether a custoer decides to purchase or wait does not affect the payoff of other custoers. 4 Rational expectations equilibriu is a coon concept applied in operations anageent; see e.g., Cachon and Swinney (009), Liu and van Ryzin (008), and Su and Zhang (008); see the latter for a description. 5 We use subscript to denote that the expectation is with respect to valuation distribution of custoers arriving in period. This ephasis will be iportant in Section 4 where we extend the odel to include period- arrivals.

6 6 Fir announces (q,p ) Custoers for expectations about period- price p e Custoers buy with probability x N in(xn,q ) reaining custoers observe v Fir produces q and sets p to clear q +q in(xn,q ) Period Period Figure The sequence of events Definition Assue the fir announces (q, p ) in period. (a) x = 0 is a pure strategy REE if and only if µ p δ c T (p e (0 q )). (b) x = is a pure strategy REE if and only if µ p δ c T (p e ( q )). (c) x (0, ) is a ixed strategy REE if and only if µ p = δ c T (p e (x q )). There are typically ultiple syetric REE for a given announceent (q, p ). To resolve the proble of equilibriu ultiplicity, we select the Pareto-doinant REE and denote it PDREE. If there are ultiple PDREE that is, custoers expected surplus is the sae, we select the PDREE that axiizes the fir s profits. This is consistent with other works in the literature (see Cachon and Netessine (004) and references therein) and with the notion of social coordination although we do not forally odel the echanis behind coordination. At the beginning of period, the fir observes sales in period, Q = in(xn, q ), and sets q. The second-period price p is such that the second-period supply q + q in(xn, q ) equals second-period deand. The rational expectations assuption iplies that p = p e (x q ). In period, custoers buy if their valuation is greater than the arket-clearing price. Figure shows the sequence of events. 3. Model Analysis Profit axiization in period subject to arket clearing iplies that p e (x q ) = argax p F (p) () s.t. N in(xn, q ) F (p) q in(xn, q ). This price is unique for a given announceent (q, p ) because f is log-concave. Define p argax p p F (p) as the unconstrained period- profit-axiizing price. Our first result establishes the properties of p e (x q ). Lea Assue x [0, ] is part of an REE. (a) If q N F (p ), then p e (x q ) = p for all x [0, ]. (b) If q > N F (p ), then p e (x q ) = F ( q xn ) for x [0, q N F (p ) N ( x) N ] and is strictly F (p ) increasing in x and p e (x q ) = p for x [ q N F (p ) N, ] and is constant in x. F (p )

7 7 Define p b (q ) µ δ c T (p e ( q )) and p w (q ) µ δ c T (p e (0 q )). Buying, x =, is an REE for any price p p b (q ) and waiting, x = 0, is an REE for any price p p w (q ). Because T (p) is decreasing in p, Lea iplies that p w (q ) p b (q ). This establishes that an REE exists for any announceent (q, p ). Our next result shows that whenever p e (x q ) is strictly increasing at x = 0, we can ignore all ixed strategy equilibria. Lea Any x (0, ) such that p e (x q ) > p e (0 q ) cannot be part of a PDREE. To see why, assue x (0, ) is a ixed strategy PDREE for a fir announceent (q, p ). Therefore, custoers are indifferent between buying and waiting, i.e., µ p = δ c T (p e (x q )). Because µ p = δ c T (p e (x q )) < δ c T (p e (0 q )), we conclude that the pure strategy waiting REE (x = 0) exists and gives higher expected surplus than PDREE x; a contradiction. For any announceent (q, p ), we show in the Appendix that any ixed strategy REE is either strictly Pareto-doinated by the waiting pure strategy equilibriu x = 0 (by Lea ) or Paretoequivalent to the pure strategy buying REE (x = ) which is weakly preferred by the fir. We prove the latter by coparing the fir s profits in ixed strategy REE x and REE x =. We subsequently ignore all ixed strategy REE and derive the set of pure strategy PDREE which could be x = 0, x = or both. We then fix q and solve for the price p that axiizes fir profits. That is, we consider all possible prices p, copute the profits in all PDREE associated with (q, p ), and select the price associated with the highest profits. Denote that price p (q ). Proposition Assue the period- production is q. The profit axiizing period- price is p (q ) = p w (q ) and the associated PDREE is x =. The fir cannot produce q and charge p b (q ) because waiting is the Pareto-doinant equilibriu. In order to sell q, the fir can charge at ost p w (q ). At that price, the two REE (buying and waiting) are Pareto-equivalent. Proposition iplies that the period- price has to be less than custoers expected valuation, µ, which is the profit per custoer when the fir can credibly coit not to produce and sell in period. Hence, δ c T (p e (0 q )) is a easure of the cost to the fir associated with not being able to coit. Further, Proposition shows that the fir axiization proble is well-defined. In particular, the fir chooses q to axiize π(q ) = q p (q ) + δ (N q )p F (p ). () This expression covers the case where the fir sells only in period. This happens when q = 0 and custoers have to wait. A frenzy occurs if custoers are worse off when they do not obtain

8 8 the good early. Because custoers can always buy at p in period if they wait, this condition is satisfied only if p e (0 q ) < p e ( q ) = p. Therefore, custoers expected loss fro being rationed out is L(q ) = p b (q ) p w (q ) = δ c (T (p e (0 q )) T (p )). (3) We distinguish two scarcity strategies. A frenzy takes place when custoers are strictly worse off being rationed out, i.e., L(q ) > 0, and there is excess deand, i.e., N > q. If, on the other hand, L(q ) = 0 and N > q, we say the fir eploys a rationing policy. Note that our definitions of frenzy and rationing policy differ fro previous works that have not ade a clear distinction between the two concepts. Here, L(q ) is a easure of the intensity of the frenzy (custoer desperation). Conditional on this easure being positive, a easure of the extent of the frenzy is N q. Denote the axiu profit per custoer obtained by selling in period, π = p F (p ). The following proposition characterizes the conditions under which a buying frenzy occurs. Proposition The fir s optial policy induces a unique buying frenzy if F (p ) < and δ c < δ c < δ, c where δ c µ δ π = T (p ) + [ F and δ c µ δ π (p )] =. µ v + f(p ) f(v) If custoers are sufficiently patient (δ c > δ c ), the fir sells in both periods. If custoers are excessively patient (δ c δ c ), then the fir rations supply in the early period without frenzy. In this case, the optial production quantity is such that custoers are indifferent between buying early and waiting as individuals. Finally, if custoers are sufficiently but not excessively patient (δ c < δ c < δ c ), the fir s optial policy is to induce a buying frenzy in the early arket. The equilibriu selection rule says that custoers are indifferent between buying early and waiting as a group. But an individual custoer who waits is strictly worse-off (L (q ) > 0). This is because p > p e (0 q ) and so µ p (q ) > δ c T (p ). To deonstrate the relevance of our dynaic fraework, we revisit the finding by Balachander et al. (009) that the introductory price of a new car is positively correlated with its scarcity (H3B, p. 67). The concept of scarcity corresponds to N q in our fraework. Fro Proposition, it follows that the introductory price decreases with q. Yet the extent of a frenzy, N q, also decreases with q. Therefore, a decrease in q (as ight be caused by a shift in one of the odel s priitives) will increase both the extent of a frenzy and the introductory price. Corollary (a) The fir s optial profit is increasing in δ and decreasing in δ c. (b) The optial period- production q is decreasing in both δ c and δ. Figure shows the optial policy for unifor valuation distributions with ean µ and standard deviation σ. A unifor distribution is indexed by its coefficient of variation, σ/µ. The optial

9 Rationing without frenzy δ c Buying frenzy Rationing without frenzy δ c δ c δ c δ c Buying frenzy Sell only in period (No rationing) C.V δ c Sell only in period (No rationing) C.V. (a) δ = 0.6 (b) δ = Figure The fir s optial policy when the fir can produce in both periods and when custoers valuation is unifor with coefficient of variation of C.V. C.V. 3/3 to ensure that the support of v is non-negative. policy does not induce a buying frenzy if σ/µ 3/9. The reason is that if σ/µ 3/9, then p = v and Proposition iplies that the optial policy is q = N, i.e., to serve the entire arket early. Therefore, a buying frenzy can occur only when custoers are sufficiently uncertain about their valuations, which is a typical state with respect to new or innovative products. As δ increases, the range of custoer ipatience that supports a buying frenzy shrinks. 6 In particular, it is never optial to induce a buying frenzy if δ c = δ =. Our odel distinguishes rationing fro frenzy policies. Although for δ c > δ c it is not optial to induce a frenzy, the fir does ration custoers (i.e., q < N ). Custoers are indifferent between buying early and waiting and custoer desperation, as defined in (3), is zero. Figure 3(a) shows custoer desperation, noralized as L (q)/(δ c T (p )) and Figure 3(b) plots the extent of the frenzy under the optial policy. Custoer desperation can be econoically significant. The loss fro not obtaining a good early is 0% of the expected value of consuption when, for exaple, δ = and δ c = 0.5. For a given value of δ, custoers relative desperation decreases as they becoe ore patient (Figure 3(a)) whereas the extent of the frenzy N q increases (Figure 3(b)) because q is decreasing in δ c (see Corollary ). Moreover, for a given δ c, custoer desperation also decreases in δ whereas the extent of the frenzy increases because q is decreasing in δ (Corollary ). 6 This is because δ c δ c is decreasing in δ (note that T (p) + [ F (p)] /f(p) is decreasing in p and that p > v).

10 Custoer desperation (%) δ = δ = 0.6 N q δ = δ = δ c δ c (a) Custoer desperation (b) Frenzy extent Figure 3 Custoer desperation L(q) L(q (noralized as ) δ c T (p 00) and the frenzy intensity ) N q in a buying frenzy when custoers valuation is unifor with ean µ = and standard deviation σ = 4 (see Appendix B for details). In this figure, N =. For δ c δ c and δ c δ c, we have L(q ) = 0. Figure 4 copares the fir s optial profits with two benchark cases. One corresponds to the fir s profit under period- sales, N F (p )p, and the other corresponds to the coitent profit that is charging µ in period to N custoers (and coitting, credibly, not to sell in period ). Figure 4(b) shows that, at σ/µ = 0.5, the loss due to noncoitent is alost 00% whereas inducing a frenzy recovers 66.7% to 36.% of the profits (the ore patient custoers are the lower the percentage recovered). The equivalent profit recovered when σ/µ = 0.5 is 66.9% to 55.%; that is, the lower the coefficient of variation the higher the percentage recovered. Furtherore, the percentage of profits recovered by inducing a frenzy decreases with increasing fir ipatience (lower δ ). 4. Second-Period Arrivals A liitation of the analysis in Section 3 is that custoers arrive only in period. A natural way to generalize the odel is to distinguish early adopters, who arrive in period and face valuation uncertainty, and followers, who arrive in period and know their valuations. In this section, we deonstrate that the results are robust to the arrival of followers. Assue that N new custoers arrive in the second period. These custoers have independently and identically distributed valuations with density f and survival function F. The period- deand is a regular downward-sloping deand N F (p). This setup in consistent with other odels in the literature (see e.g., Gallego and Şahin 00 and Xie and Shugan 00).

11 π (C.V. = 0.5) 3.5 π (C.V. = 0.5) π (C.V. = 0.5) π (C.V. = 0.5) δ c δ c (a) δ = 0.6 (b) δ = Figure 4 The optial profit is bounded fro above by coitent profits and fro below by profits fro selling only in period. Dashed and solid lines show the corresponding profits for C.V. = 0.5 and C.V. = 0.5, respectively. In panel (a), frenzy profits when C.V. = 0.5 correspond to the interval δ c (0.7, 0.93); when C.V. = 0.5 they correspond to the interval δ c (0.50, 0.7). In panel (b), frenzy profits when C.V. = 0.5 correspond to the interval δ c (0.9, 0.66); when C.V. = 0.5 they correspond to the interval δ c (0.3, 0.46). In this figure, N =. We need additional notation to facilitate the exposition in this section. We use subscripts to denote period and superscripts to denote the arrival cohort. Thus, p i, t denotes the onopoly price in period t for custoer cohort i, i.e., p, argax p N p F (p) and p, argax p N p F (p). We assue that these two optiization probles are strictly concave (e.g., f and f are both logconcave). When Q custoers buy in period, we let Q b (p, Q ) (N Q ) F (p)+n F (p) denote the deand in period. Consistent with Section 3, we let p price when the fir sells only in period, and Q argax p pq b (p, 0) be the onopoly = Q b (p, 0) be the corresponding quantity. Suppose the fir offers q units for sale in period at a price p. The period- price p e (x q ) axiizes the period- profits subject to arket clearing constraint. In other words, p e (x q ) = argax p p Q b (p, in(xn, q )) (4) s.t. Q b (p, in(xn, q )) q in(xn, q ) Define the unconstrained axiizer of (4) as p b (y) = argax p p Q b (p, y). Lea 3 p b (y) is increasing in y if p, < p,, constant if p, = p,, and decreasing otherwise.

12 Observe that the constraint in (4) is not binding for x [ q N, ]. For x < q N, define ˆp (x, q ) as the solution to N ( x) F (p) + N F (p) = q xn. (5) There is a unique solution to (5) because its left-hand side is decreasing in p, takes a axiu of N ( x) + N q xn at p = 0 and a iniu of 0 q xn at p = vax. The following result characterizes p e (x q ). Lea 4 Assue x [0, ] is part of an REE. If x [0, q N ], p e (x q ) = in(p b (xn ), ˆp (x, q )). Otherwise, p e (x q ) = p b (q ). then We can show that p e (x q ) is (weakly) increasing only when p, p,. Thus, we distinguish this case and the case p, > p,. 4.. Buying frenzies when p, p, We first show that, siilar to Section 3, p e (x q ) is onotone in x. To do so, we define ˆx(q ) as the solution to Q b (p b (xn ), xn ) = q xn. (6) We show in the Appendix that (6) has a unique solution when Q q N + N F (p, ). Lea 5 (a) Assue p, < p,. It follows that p e (x q ) is continuous and weakly increasing for x [0, ] with x pe (x q ) > 0 at x = 0. (b) Assue p, = p,. (b) If q Q then p e (x q ) = p. (b) If Q q N + N F (p, ), then p e (x q ) = p for x ˆx(q ), and p e (x q ) = ˆp (x, q ) otherwise. (b3) If q > N + N F (p, ), then p e (x q ) = ˆp (x, q ). (b4) p e (x q ) is weakly increasing in x. Lea 5 shows that p e (x q ) is weakly increasing in x. We leverage this property and apply Lea to eliinate all ixed strategies such that p e (x q ) > p e (0 q ). We then show that a siilar result to that in Proposition holds, i.e., the profit axiizing price for quantity q is p (q ) = p w (q ) (see Appendix for proofs). Moreover, an early production quantity larger than the period- arket size is never strictly profitable for the fir. This is because excess capacity in period lowers the period- expected price (p e (x q ) is decreasing in q ) and therefore lowers the price the fir can charge in the first period while not increasing sales. In suary, the onopolist chooses the early production quantity q N to axiize π(q ) = q p (q ) + δ p e ( q )Q b (p e ( q ), q ). (7) The easure of custoer desperation derived in Section 3 generalizes as L(q ) = δ c (T (p e (0 q )) T (p e ( q ))) +. The inequality p e (0 q ) p e ( q ) holds as p e is increasing in x when p, p,.

13 3 We next investigate the conditions under which a buying frenzy equilibriu exists and is an optial policy for the fir. Lea 6 There is no buying frenzy equilibriu if N N F (p ) F (p ). One iplication of Lea 6 is that buying frenzies are less likely to be optial when the custoer cohort arriving in period is large relative to the one arriving in period (i.e., when N N ). In this case, the waiting equilibriu involves a relatively sall nuber of period- custoers who do not have uch of an ipact on the second-period price. For the rest of the section, we assue that N N < F (p ) F (p ); this assuption is equivalent to Q < N, which iplies that the period- price can change when the waiting equilibriu is selected. 7 The following proposition characterizes when a frenzy occurs. Proposition 3 Assue that N ). Sufficient conditions for the fir s optial policy to induce a buying frenzy are: (i) p, < p, and δ c > δ; c or (ii) p, = p, and δ c < δ c < δ c where N < F (p ) F (p δ c µ δ p = F (p ) T (p Q ) + F, δ c (p ) = N f (p )+N f T (p ) (p w (N )) + µ δ p, F (p, ) N F (p w (N )) N f (p w (N ))+N f (p w (N )). (8) Proposition 3 establishes sufficient conditions under which q < N and L(q ) > 0 whenever p, < p, or p, = p,. Figure 5 generalizes Figure (b) when new custoers arrive in period. Overall, the ain insights fro Section 3 carry over. The shape of the paraeter space where frenzies occur does not change uch relative to Figure (b) with one exception. When p, < p,, Figure 5(a) reveals such a paraeter space that is larger than its counterpart in Figures (b) and 5(b). The areas labeled rationing without frenzy, becoe buying frenzy in Figure 5(a). The area rationing without frenzy corresponds to the corner solution at Q with p e ( Q ) p e (0 Q ). In Figure 5(a), this area disappears because we have p e ( Q ) > p e (0 Q ). Figure 5(b) considers the special case where f ( ) = f ( ) = f( ). In that case, we can derive a closed-for solution for the paraeter space that generates frenzies. Moreover, the frenzy equilibriu is unique (see the proof of Proposition 3 in Appendix A). 4.. Buying frenzies when p, > p, We first characterize the behavior of p e (x q ) which is significantly different in this case. 7 This assuption is siilar to the assuption in Section 3 that p b > v (see Proposition ). The reason is that if p b = v then the size of the period arket when the fir sells only in period (i.e., N F (v) = N) is as large as the period- arket size.

14 Rationing without frenzy 0.8 Buying frenzy δ c δ c 0.5 Buying frenzy No rationing No rationing N N Figure 5 (a) p, < p, (b) p, = p, The fir s optial policy when the valuation distribution of early arrivals is U[0, b ] and that of late arrivals is U[0, b ]. In this figure, δ =, N =. Further, (b, b ) = (0, 0) in panel (a) and (b, b ) = (0, 0) in panel (b). The dashed line in panel (a) shows the sufficient conditions characterized in Proposition 3. The frenzy paraeter space is slightly larger than the one iplied by these conditions. Panel (b) shows the necessary and sufficient conditions for f ( ) = f ( ). Lea 7 Assue p, > p,. (i) If q Q, then p e (x q ) = p b (xn ) with dpe (x q ) < 0 for x dx [0, q N ] and p e (x q ) = p b (q ) for x [ q N, ]. (ii) If Q q N + N F (p, ), then p e (x q ) = ˆp (x, q ) with dpe (x q ) dx > 0 for x [0, ˆx(q )], p e (x q ) = p b (xn ) with dpe (x q ) dx < 0 for x [ˆx(q ), q N ] and p e (x q ) = p b (q ) for x [ q N, ]. (iii) If q N + N F (p, ), then p e (x q ) = ˆp (x, q ) with dp e (x q ) dx > 0 for x [0, ]. In Lea 7, ˆx(q ) and the conditions under which it exists and is unique are as in Section 4.. Figure 6 scheatically shows p e (x q ) in the five possible cases iplied by Lea 7. In effect, p e (x q ) can take three shapes. When case (i) in Lea 7 applies, p e (x q ) is weakly decreasing as in Panels (b) and (c). Panels (a) and (d) correspond to case (ii) where p e (x q ) is increasing up to a threshold and then weakly decrease. Finally, in Panel (e), p e (x q ) is increasing in x. The iportant difference, in this section, with the our analysis in Sections 3 and 4. is that the period- expected price, p e e(x q ), is increasing in x only in Panel (e). Otherwise, the period- expected price can be decreasing in x iplying that p e (0 q ) is not always saller than p e (x q ). Thus, the waiting equilibriu does not necessarily Pareto-doinate the ixed strategy REE. We can still rule out all ixed strategy REE such that p e (x q ) > p e (0 q ) by applying Lea. We further show in the Appendix that the reaining ixed strategy REE are equivalent to a pure

15 5 p e (x q ) p b (xn ) p e (x q ) p e (x q ) p b (xn ) p b (xn ) ˆp (x, q ) p b (q ) p b (q ) ˆx q N x q N x x (a) Q q N (b) q N and q Q (c) N q Q p e (x q ) p b (xn ) p e (x q ) ˆp (x, q ) ˆp (x, q ) ˆx x (d) N q and Q q N + N F (p, ) x (e) N + N F (p, ) q Figure 6 p e (x q ) when p, > p,. strategy PDREE associated with a different fir announceent. Therefore, all ixed strategy REE can still be ignored without loss of generality. These results are suarized in the following proposition. Proposition 4 Assue the period production is q. We can restrict the analysis, without loss of generality, to the announceent p (q ) = in(p b (q ), p w (q )) and associated PDREE x =. The interpretation of p (q ) differs fro that of p (q ) in Section 4.. p (q ) is the profit axiizing price conditional on selling q while p (q ) was the profit-axiizing price. The difference is that in the latter, selling q was always optial. However, when p, > p,, not selling at all, i.e., x = 0, ay doinate selling at p (q ). 8 But the profits under x = 0 are weakly doinated by profits under alternative announceent q = 0 and p > vax iplying that such early production volues will never be the fir s choice in equilibriu. The fir optiization proble to characterize the production level in period is ax q p (q )q + δ p e ( q )Q b (p e ( q ), q ). (9) We now focus on the conditions that induce a frenzy in the arket. Figure 6(a,b) are the only equilibriu period- prices that can be part of a frenzy because in a frenzy it ust be that q < N. 8 This will be the case when the profits fro selling in(n, q ) at p (q ), i.e., p (q )q + δ p b (q )Q b (p b (q ), q ), are lower than the profits under x = 0, i.e., δ Q p if q Q and in(n, q )ˆp (q ) otherwise.

16 Rationing without frenzy δ c Buying frenzy No rationing N Figure 7 The fir s optial policy when the valuation distribution of early arrivals is U[0, b ] and that of late arrivals is U[0, b ]. In this figure, δ = and N = and (b, b ) = (0, 5). However, a buying frenzy cannot happen when p e (x q ) is as in Figure 6(b), i.e., when q N and q < Q. In this case, p e ( q ) < p e (0 q ) which iplies p (q ) = p b (q ) (Proposition 4). Therefore, an individual custoer is indifferent between buying and waiting. As a result, a buying frenzy ay happen only in Figure 6(a). If p b (q ) p e (0 q ), then an individual custoer is indifferent between buying and waiting as both yield the surplus µ p = δ c T (p b (q )). Therefore, a frenzy ay happen only if ˆp (0, q ) < p b (q ). For general distributions f i, i {, } this inequality does not yield a threshold value for q and hence we cannot provide sufficient conditions under which a frenzy happens. One needs to directly check q < N and ˆp (0, q) < p b (q). Figure 7 shows the fir optial policy when custoer valuations are unifor distributions. To deonstrate the relevance of our analysis in Section 4. and 4., we revisit one of the two key results of Balachander et al. (009). They find a positive association between buying frenzies and intrinsic preferences for a product that lasts beyond the introductory period of the product (HA, p. 66). One needs a dynaic odel of sales such as the one presented here in order to interpret this finding properly. Assue that custoers do not discount excessively. An increase in the strength of the second-period deand can be interpreted as an increase in p. When p is low, the equilibriu is likely to be a rationing without frenzy (upper left area in Figure 7). For p sufficiently large, the equilibriu becoes a buying frenzy (upper left area in Figure 5(a)). Buying frenzies are therefore associated with a high level of p, which can be interpreted as strong and persistent intrinsic preferences.

17 7 5. Conclusion This paper offers a tractable and convenient fraework to analyze buying frenzies in a dynaic context. In contrast with other works in the literature, the existence of buying frenzies in our odel does not depend on the rule applied by the fir to allocate goods under rationing, or inforation asyetry aong custoers. Buying frenzies occur when custoers are sufficiently uncertain about their product valuations and when custoers discount the future but not excessively. Overly patient custoers wait until they learn their preferences for the product while very ipatient custoers are served early. The utility loss fro not obtaining the good in a frenzy can be substantial. Siilarly, the fir s gain fro a frenzy policy can be econoically large. The analysis presented here offers a rich fraework to interpret stylized facts and epirical findings about buying frenzies. We have illustrated this point by revisiting the two key results presented in Balachander et al. (009), i.e., the introductory price positively correlated with product scarcity and a positive association between buying frenzies and intrinsic preferences for a product that lasts beyond the introductory period. These finding are difficult to rationalize within static odels, but, both are consistent with our analysis. References Allen, F., G.R. Faulhaber. 99. Rational rationing. Econoica 58(30) Bagnoli, M., T. Bergstro Log-concave probability and its applications. Econoic Theory 6() Balachander, S., Y. Liu, A. Stock An epirical analysis of scarcity strategies in the autoobile industry. Manageent Science 55(0) Balachander, S., A. Stock Liited edition products: When and when not to offer the. Marketing Science 8() Becker, G.S. 99. A note on restaurant pricing and other exaples of social influences on price. Journal of Political Econoy 99(5) Cachon, G.P., S. Netessine Gae theory in supply chain analysis. Chapter in Handbook of Quantitative Supply Chain Analysis: Modeling in the E-Business Era, Eds. D. Sichi-Levi, S.D. Wu, M. Shen. Kluwer Acadeic Publishers. Cachon, G.P., R. Swinney Purchasing, pricing, and quick response in the presence of strategic consuers. Manageent Science 55(3) Debo, L., G.J. van Ryzin Creating sales with stock-outs. Working Paper. Debo, L., S. Veeraraghavan Models of herding behavior in operations anageent. Chapter in Consuer-Driven Deand and Operations Manageent Models, pp. 8-4, (eds. S. Netessine and C. Tang). Springer Series in Operations Research and Manageent Science.

18 8 DeGraba, P Buying frenzies and seller-induces excess deand. RAND Journal of Econoics 6() Denicolò, V., P.G. Garella Rationing in a durable goods onopoly. RAND Journal of Econoics 30() Gallego, G., Ö. Şahin. 00. Revenue anageent with partially refundable fares. Operations Research 58(4) Lewis, T., and D. Sappington Supplying Inforation to Facilitate Price Discriination. International Econoic Review 35() Liu, Q., G.J. van Ryzin Strategic capacity rationing to induce early purchases. Manageent Science 54(6) 5-3. Papanastasiou, Y., N. Bakshi, N. Savva. 0. Strategic stock-outs with boundedly rational social learning. London Business Shcool Working Paper. Liu, T., P. Schiraldi. 0. Buying frenzies in durable-goods arkets. Working paper. Stock, A., S. Balachander The aking of a hot product : A signaling explanation of arketers scarcity strategy. Manageent Science 5(8) 8-9. Su, X., F. Zhang Strategic custoer Behavior, coitent, and supply chain perforance. Manageent Science 54(0) The Wall Street Journal. 0. Losing $07 a Pop, H-P Brings Back Its ipad Rival. August 3. Verhallen, T.M.M., H.S.J. Robben Scarcity and preference: An experient on unavailability and product evaluation. Journal of Econoic Psychology 5() Xie, J., S. M. Shugan. 00. Electronic tickets, sart cards, and online prepayents: When and how to advance sell. Marketing Science 0(3) Yu, M., R. Kapuscinski, H. Ahn. 0. Advance selling with interdependent custoer valuations. Working Paper. Appendix A: Proofs Proof of Lea : The function p F (p) is uniodal and argax p F (p) = p. (a) In this case, the constraint in () is non-binding. This is because N in(xn, q ) F (p) q in(xn, q ) iplies N F (p) + in(xn, q )F (p) q. This inequality holds for p = p because q N F (p ) and hence p e (x q ) = p. (b) Assue first that x [0, q N F (p ) N ]. We have F (p ) x q N F (p ), N F (p ) xn F (p ) q N F (p ),

19 9 xn q N F (p )( x), xn q. Therefore, p e (x q ) = argax p F (p) subject to the constraint (N xn ) F (p) q xn. Because (N xn ) F (p ) q xn, it follows that the constraint is binding. We conclude that p e (x q ) is the unique solution to (N xn ) F (p) = q xn (uniqueness follows because f is log-concave), i.e., p e (x q ) = F ( q xn N xn ). Finally, because q xn N xn p e (x q ) is strictly increasing in x. is strictly decreasing in x, it follows that Now assue that x q N F (p ) N or equivalently xn F (p ) q N F (p )( x). Two cases are possible: either xn q or q xn q N F (p )( x). In either case, the constraint in () is non-binding and so p e (x q ) = p which is constant in x. Proof of Proposition : We first derive the PDREE preferred by the fir for a given (q, p ) and then select the profit axiizing price p for a given q. We proceed by doing the analysis in two possible cases. CASE : Assue period- production is q > N F (p ). We characterize the PDREE in the next result. Clai (a) p w (q ) < p b (q ). (b) Any ixed strategy REE is Pareto-doinated by REE x = 0. (c) If p < p w (q ), then x = is the unique PDREE. (c)if p = p w (q ), then there are two PDREE x = 0 and x = and the latter is preferred by the fir. (c3) Finally, if p w (q ) < p, then x = 0 is the unique PDREE. Proof: (a) follows directly fro Lea. (b) The condition in Lea holds for any x (0, ). (c) If p < p w (q ), then µ p > µ p w (q ) = δ c T (p e (0 q )) δ c T (p e (x q )) for all x. Hence x = is the unique PDREE. (c) If p = p w (q ), then x = 0 and x = are REE and PDREE because µ p = δ c T (p e (0 q )). We show that the PDREE x = is preferred by the fir. The fir profit under the equilibriu x = 0 is π(x = 0) = δ N p e (0 q ) F (p e (0 q )) and that under the equilibriu x = is π(x = ) = p q + δ (N q )p e ( q ) F (p e ( q )) = (p δ p F (p ))q + δ N p F (p ). The result then follows because p p F (p ) 0 and p F (p ) > p e (0 q ) F (p e (0 q )). The forer inequality follows because p = p w (q ) and p w (q ) p F (p ) = µ (v p e (0 q ))df p df p e (0 q ) p > µ (v p e (0 q ) + p )df because p e (0 q ) > p (Lea ) = p 0 > 0. p vdf + p (p e (0 q ) p )df

20 0 (c3) x = 0 is an REE because δ c T (p e (0 q )) > µ p. Further, x = is not an REE because µ p < δ c T (p e ( q )). Therefore x = 0 is the only PDREE. CASE : Assue period- production is q N F (p ). We characterize the PDREE in the next result. Clai (a) p w (q ) = p b (q ). (b) If p < p w (q ), then x = is the unique PDREE. (b) If p = p w (q ), then all x [0, ] are Pareto-equivalent PDREE and x = is the PDREE preferred by the fir. (b3) If p > p w (q ), then x = 0 is the unique PDREE. Proof: (a) This follows fro Lea because p e (0 q ) = p e ( q ) = p for q N F (p ). (b) We have µ p > µ p w (q ) = δ c T (p e (x q )) = δ c T (p ) for all x [0, ], hence a custoer is better off buying the product and x = is the unique PDREE. (b) Because µ p = µ p w (q ) = δ c T (p e (x q )) = δ c T (p ) for all x [0, ], it follows that all x [0, ] are PDREE because the custoer s expected surplus is the sae. We show that x = in the PDREE that axiizes the fir profits. The fir profit for a given x [0, ] is π(x) = Q p w (q ) + (N Q )p F (p ) = Q (p w (q ) p F (p )) + N p F (p ) where Q = in(xn, q ) is the sales in period. The profit π(x) is increasing in x because p w (q ) = µ δ c T (p ) and p w (q ) p F (p ) = µ (v p p )df p Therefore, x = is the PDREE that axiizes the fir profits. (b3) µ p < δ c T (p e (x q )), hence x = 0 is the unique PDREE. p p df = 0 vdf > 0. We can now conclude by collecting the PDREE preferred by the fir for an announceent (q, p ) fro the two cases above. For a period- price such that p > p w (q ), custoers wait in any PDREE and hence the fir profit is π(x = 0) = δ p e (0 q ) F (p e (0 q )). On the other hand, the fir profit, if it sets p = p w (q ), is π(x = ) = p w (q )q + δ (N q )p F (p ). Because π(x = ) > π(x = 0), it follows that any price p > p w (q ) is doinated by p (q) = p w (q ). For a period- price such that p < p w (q ), custoers buy in any PDREE and the fir profit is π(x = ) = p q + δ (N q )p F (p ). This profit is increasing in p and it follows that p = p w (q ) doinates p < p w (q ). Therefore, the fir profits are axiized at p (q ) = p w (q ). Proof of Proposition : If F (p ) = then, by Lea, p e (0 q ) = p = v and p w (q ) = µ δ c T (p ). It follows fro () that π(q ) = q (µ δ c T (p )) + δ (N q )p F (p ), which is increasing in q. This is because µ δ c T (p ) δ p F (p ) µ T (p ) p F (p ) = For the particular case δ c = δ =, the profit function is independent of q and so all q [0, N ] are optial. In this case, custoers are still indifferent between buying and waiting and a frenzy does not happen.

21 Hence q = N ; in other words, the fir serves the entire arket in period and custoers are indifferent between buying and waiting. Thus, a buying frenzy cannot occur. Assue for the reainder of the proof that F (p ) <. Observe that if q = 0 then the fir s optial policy is to sell only in period. We prove next that the optial production is strictly positive, which iplies that selling only in period is never optial when the fir has the option of producing over tie. If q N F (p ) then p e (x q ) = p and p (q ) = µ δ c T (p ), i.e., p (q ) becoes independent of q (Lea and Proposition ). The profit function in () is then linear and increasing in q because dπ(q ) dq = µ δ c T (p ) δ p F (p ) µ T (p ) p F (p ) = in(v, p ) df p F (p ) = p v vdf > 0. The last inequality follows because F (p ) <. Therefore, q = N F (p ) yields greater profits than any lower q. However, the production quantity q = N F (p ) cannot induce a buying frenzy because custoers are indifferent between buying early and waiting. It follows that a necessary condition for a buying frenzy is that q > N F (p b ). We next characterize the sufficient conditions under which it is optial for the fir to induce a buying frenzy. If N F (p ) q, then p e (0 q ) = F ( q N ) (Lea ). The fir s profit function is π(q ) = q (µ δ c T (p e (0 q ))) + δ (N q )p F (p ), which is strictly concave in q because (we suppress the arguent in p e (0 q ) for brevity) and dπ(q ) dq = µ δ c T (p e ) δ c q N f(p e ) δ q p F (p ), d π(q ) dq = δ c q ( + [f(pe )] + F (p e )f (p e ) ) < 0. N f(p e ) [f(p e )] The inequality holds because f is log-concave and so [f(p)] + F (p)f (p) > 0 (Bagnoli and Bergstro 005). The first-order conditions then characterize the optial q ; that is, q solves dπ dq = µ δ c T (p e (0 q )) δ c q N f(p e (0 q )) δ p F (p ) = 0, (0) provided that N F (p ) q N. Since the left-hand side of (0) is strictly decreasing in q (because π is strictly concave), it follows that if dπ dq q =N < 0 and dπ dq q =N F (p ) > 0 then there