Publihed online ahead of print July 3, 212 MANAGEMENT SCIENCE Article in Advance, pp. 1 22 ISSN 25-199 (print ISSN 1526-551 (online http://dx.doi.org/1.1287/mnc.112.1545 212 INFORMS Selling to Conpicuou Conumer: Pricing, Production, and Sourcing eciion Necati Tereyağoğlu College of Management, Georgia Intitute of Technology, Atlanta, Georgia 338, necati.tereyagoglu@mgt.gatech.edu Senthil Veeraraghavan The Wharton School, Univerity of Pennylvania, Philadelphia, Pennylvania 1914, enthilv@wharton.upenn.edu Conumer often purchae good that are hard to find to conpicuouly diplay their excluivity and ocial tatu. Firm that produce uch conpicuouly conumed good uch a deigner apparel, fahion good, jewelry, etc., often face challenge in making optimal pricing and production deciion. Such firm are confronted with precipitou trade-off between high ale volume and high margin, becaue of the highly uncertain market demand, trategic conumer behavior, and the diplay of conpicuou conumption. In thi paper, we propoe a model that addree pricing and production deciion for a firm, uing the rational expectation framework. We how that, in equilibrium, firm may offer high availability of good depite the preence of conpicuou conumption. We how that carcity trategie are harder to adopt a demand variability increae, and we provide condition under which carcity trategie could be uccefully adopted to improve profit. Finally, to credibly commit to carcity trategy, we how that firm can adopt ourcing trategie, uch a ourcing from an expenive production location/upplier or uing expenive raw material. Key word: trategic cutomer behavior; game theory; conpicuou conumption; pricing; carcity; ourcing Hitory: Received February 25, 21; accepted Augut 31, 211, by Preya eai and Pradeep Chintagunta, marketing. Publihed online in Article in Advance. 1. Introduction Conumer looking to ignal their uniquene and excluivity have often expreed themelve by conuming good prominently to diplay their tatu. Firm that deign and ell luxury product or innovative gadget have often deired excluivity in their look and deign. The prominent diplay of logo, limited availability, and expenive deign are ome way through which firm have diplayed their excluivity. For intance, the Big Pony apparel line deigned by Ralph Lauren ha more prominent logo that can be diplayed conpicuouly by the wearer. 1 Many luxury watche with intricate deign, uch a Piaget watche, are old only through limited number of boutique tore and authorized retailer in the United State (http://www.piaget.com. Firm often face deciion on how to make production and pricing deciion when elling uch conpicuou good. We tudy the deciion of a firm when there i conpicuou conumption, i.e., when ome member of 1 A comparion between the Claic-Fit Polo and the more conpicuou Claic-Fit Big Pony Polo hirt on http://www.ralphlauren.com how the Big Pony deign being old at higher price. the population are motivated by invidiou comparion (Bagwell and Bernheim 1996. Invidiou comparion refer to ituation in which a member of a ocial cla conume conpicuouly to ditinguih himelf from other member. We examine the cae when ome conumer eek, purchae, and conume hard-to-find product to diplay their ditinction from the other conumer in the population. Conitent with the literature (Leibentein 195, Amaldo and Jain 25a, we term conumer that are driven by uch invidiou comparion a engaging in nobbih behavior. With increaingly unpredictable market demand condition, many firm elling to conpicuou conumer face difficult trade-off between profit and excluivity, which put them in a bind. Some firm adopt the trategy to compete on price and hope to increae revenue through ale volume. Recently, retailer uch a Nordtrom have attributed their increaed revenue to lahed price and increaed inventory availability. 2 On the other hand, other firm 2 Nordtrom chief executive officer avid Spatz argued for cutting price of everal product to repond effectively to the market. For intance, handmade Anyi Lu deigner hoe old at le than $4, intead of the regular retail price $595, which wa accompanied by 69% increae in tore inventory (Giacobbe 29. 1
2 Management Science, Article in Advance, pp. 1 22, 212 INFORMS have choen to limit their product availability by creating carcity (Rigby et al. 29, and uch hortage for new product have been commonly oberved (Gumbel 27. In general, a reduction in product availability lead to reduced ale, which may hurt firm profit. It i till unclear if the firm hould or hould not ue carcity trategie in elling good and, if they do, when thoe trategie hould be implemented. Thu, both from practitioner and reearch perpective, it i imperative to undertand how firm hould make interconnected deciion uch a how much of the good to produce, how to price thoe good, and when to invet in innovative deign or ue an expenive upplier, etc. In thi paper, we analyze a monopolit firm pricing and production deciion while elling a good to a market with uncertain demand from conpicuou conumer. 3 When the demand i not determinitic, it i difficult to identify if carcity occurred becaue of an unexpected high demand (a random realization or becaue of decidedly low inventory (a trategic deciion. Often, it i difficult to eparate the two effect becaue of the lack of full information on the production proce (unobervability. Thi i a key focal point of our approach. We how that carcity trategy could emerge in equilibrium in market with tochatic demand becaue of conpicuou conumer. Our model ue the rational expectation (RE framework (Muth 1961, Stokey 1981 to analyze the impact of conpicuou conumption on firm pricing, production, and ourcing deciion. The framework ha been ued in ome recent paper (ee, for intance, Jerath et al. 21, Su 27. The cheme of our paper i a follow. We firt poition our contribution with repect to extant literature. In 2, we analyze the equilibrium pricing and production deciion of a firm elling to conpicuou conumer in a homogeneou market (in 2.1 and in a heterogeneou market (in 2.2. In 3, uing our tructural reult, we conider trategic carcity deciion. In 4, we how that our tructural inight on carcity hold under a variety of alternative model and condition. In 5, we dicu how firm can commit to a carcity trategy, either by limiting clearance pricing or ourcing expenively. Finally, in 6, we conclude and dicu future reearch. 1.1. Our Poition in the Literature Many new product gadget, fahion apparel, and good (deigner brand are often treated a vehicle of elf-expreion through which conumer exhibit 3 Example of uch conpicuou product might include Chritian ior watche (Amaldo and Jain 25a, Yve Saint Laurent hoe (Bruner et al. 25, and Cherry Bloom Murakami bag (Amaldo and Jain 28. their deire for excluivity or conformity. Recently, there ha been emergent interet in literature, on how production deciion are impacted by conumer deciion-making behavior (within the rational framework. Thi paper tudie the operational deciion of a firm when it ell to conumer involved in conpicuou conumption and note how operational deciion in production, alvaging, and ourcing can be employed together with pricing and carcity trategie. Economit have pointed out how conumption could be beet with poitive externalitie due to ocial conformity in the context of retaurant choice (Becker 1991, due to network effect in the context of technology (Katz and Shapiro 1985, due to market frenzie (egraba 1995, or due to herd behavior (Bikhchandani et al. 1992. However, the notion of conumer purchaing good to be conpicuou date back to Veblen (1899 who, in hi The Theory of the Leiure Cla, explained how individual conumed highly conpicuou good and ervice to advertie their wealth or ocial tatu. Leibentein (195 emphaized the ignificance of ocial factor in conumption and argued that price by itelf might enhance utility. Bagwell and Bernheim (1996 argued that the relationhip between price and demand hould emerge in equilibrium, and derive condition for uch Veblen effect to arie in equilibrium. Corneo and Jeanne (1997 etablihed that conpicuou conumption might emerge a a tool to ignal wealth. Although economic literature ha examined when Veblen effect may emerge in market equilibrium, production and pricing deciion of a firm facing conpicuou conumer have been relatively underexplored. We believe that our paper i the firt attempt on modeling the production, pricing, and ourcing deciion of a firm facing conpicuou conumption in a market with uncertain demand and trategic cutomer behavior. Along with demand uncertainty, the interet in modeling cutomer behavior and tudying the effect on firm deciion i gaining increaed attention. The impact of forward-looking or trategic cutomer on operational deciion ha been conidered in a variety of context uch a eaonal good (Aviv and Pazgal 28, commitment in upply chain performance (Su and Zhang 28, triggering early purchae (Liu and van Ryzin 28, price-match guarantee (Lai et al. 21, reervation (Çil and Lariviere 21, and quick repone trategy (Cachon and Swinney 21. See Neteine and Tang (29 and paper therein for an excellent overview of trategic conumer behavior literature. However, none of the above paper conider conpicuou conumption or tudie intentional carcity. Little ha been done to undertand the cutomeroriented reaon behind firm exceive pricing
Management Science, Article in Advance, pp. 1 22, 212 INFORMS 3 trategie ued in parallel with carce availability and addition of cotly feature to a product. In a erie of paper, Amaldo and Jain (25a, b howed that conpicuou conumer may exhibit an upward loping demand curve only in a heterogeneou market, and the firm may charge extreme price to extract that margin. Amaldo and Jain (28 howed that the addition of cotly feature to a product can increae profit in a market with reference group effect. Stock and Balachander (25 provided a ignaling reaon behind product hortage to ell hot product in a market with quality uncertainty. Alo, there ha been ome recent interet in undertanding how inventory hortage (ebo and van Ryzin 29 or long queue (Veeraraghavan and ebo 29 may ignal quality. In contrat, we believe that our work how how carcity trategie along with exceive pricing and addition of cotly feature could arie in equilibrium within a market with conpicuou conumption and uncertain demand. We build an analytical framework for a firm making operational deciion (viz. pricing, production quantity, and ourcing trategy when elling to a market with uncertain demand and when conumer exhibit trategic purchaing behavior and/or conpicuou conumption. Our equilibrium reult hold under general condition of demand uncertainty. We how that demand uncertainty coupled with conpicuou conumption can indeed lead to market condition where product are carce and the firm make higher profit. We how that when elling to market with conpicuou conumption, becaue of increaed margin, firm may overproduce good compared to it production deciion in a market without uch conpicuou conumption. Therefore, urpriingly, there may be fewer tockout in a market in which a ufficient number of conumer prefer excluivity. For intance, if the market i compoed of nob, it may be optimal for the firm to overproduce, even more than it would produce in a market in which all trategic conumer behavior i ignored. Thi finding contrat with the extant literature, which how that trategic buying lead to a reduction in production quantitie (ee Su and Zhang 28, p. 64. We how that carcity trategy i beneficial to the firm when the fraction of conumer engaging in conpicuou conumption ( nob i neither too high nor too low. When there are too few nob in the market, the firm decide to ell to everyone at lower price. When there are too many nob in the market, the attractive profit margin trigger the firm to overcommit to large production quantitie to minimize the lot ale. A a reult, the product will not be carce. Finally, firm that ell to conumer exhibiting conpicuou conumption may reort to expenive ourcing or increaed production cot. In uch cae, firm deliberately ource the good from a more expenive location, or ue a cotlier upplier, and/or ue more expenive raw material component in producing the good. Traditionally, ourcing excluively from a more expenive upplier ha been conidered an unviable trategy unle the upplier ha fater delivery time or better reliability (Tomlin 26. In thoe cae, an expenive upply ource i ued paringly a an expeditiou alternative. Often inventory commitment i not fully verifiable by conumer. A firm can credibly commit to it carcity trategy by advertiing it more expenive ourcing trategy. If product are produced through an expenive proce, it i unlikely that the firm can invet in upfront cot to produce too many unit of the good. Therefore, conumer believe that the product i likely to be carce, which drive up the valuation for nob in the market. 2. Pricing and Production eciion Our model involve a ingle monopolit firm who ha two deciion to make production quantity, Q, and the price charged per unit, p before a random demand,, i realized in a market compoed of nonatomitic cutomer. The demand i ditributed with cumulative ditribution function F, with denity function f. 4 The firm incur a contant marginal cot, c, per unit produced. If the firm produce more than the realized demand, it will be able to alvage the remaining leftover inventory at a lower price, <c, at the end of the elling period (i.e., during the alvage period. Let x + denote maxx. The firm expected profit can be written a N Q p = Ɛp min Q + Q + cq. We aume that all the cutomer have the ame valuation, v, for the product. 5 In a market without any trategic behavior or conpicuou conumption, the optimal production quantity and price are et a per newvendor deciion (ee Cachon and Terwiech 28: p = v F Q = p c/p. Henceforth, the p and Q hall be referred to a the traditional newvendor price and quantity. However, cutomer are trategic, i.e., the cutomer recognize that if the product remain unold it will be available in the alvage market at price. We term uch a cutomer a trategic cutomer (Su and Zhang 28. The deciion of the trategic cutomer i to chooe whether to buy the product in the elling period or wait to buy the product later in the alvage 4 We aume that demand ditribution F ha increaing generalized failure rate (Lariviere 25. We uppre ubcript and ue F to denote F when it i unambiguou. Furthermore, let denote 1 F. 5 To eliminate trivial outcome, we aume that the cutomer will value a product more than it cot of production, i.e., v > c.
4 Management Science, Article in Advance, pp. 1 22, 212 INFORMS period. The production quantity remain unoberved. Hence, each trategic cutomer ha to form rational expectation of not being able to find the product, i.e., the tockout probability, during the elling period. Baed on thee expectation, the cutomer expected urplu if he face an actual regular price p i U trategic = maxv p 1 v. We apply rational expectation (Muth 1961 to olve for the equilibrium price and production quantity choen by the firm in thi environment. 2.1. Modeling Conpicuou Conumption in a Homogeneou Market Cutomer, in addition to being trategic (or forward looking, may alo exhibit conpicuou conumption. A per Leibentein (195 and Amaldo and Jain (25a, we addre thee cutomer a nob. In thi ection, we begin the analyi with a market compoed olely of trategic cutomer who exhibit conpicuou conumption (the aumption i relaxed in 2.2. Snob have a higher utility for conuming a product when they figure that other conumer are unable to conume the ame product. Suppoe that a firm produce a good in mall quantitie. If nob acquire the product and conume it, they will be een a the elect few member in the market who conume uch a carce good (i.e., their conumption i conpicuouly oberved, which in turn increae their utility for uch product (Brown 21. A before, we aume that the actual quantitie produced by the firm for the market remain unobervable to the nob. 6 Thu, belief on product availability i one important factor that nob can ue to exhibit their conpicuou conumption. Baed on their rational expectation, they eek out hard-to-find product and derive a higher utility in their excluivene. A conumer might bae her belief on availability baed on the obervation that inventory/helf pace dedicated to the product at a retailer i low. In general, the nonavailability of the product increae her utility for the product, although it might be equally hard for her to get the product. Mathematically, we integrate thi nobbihne to her utility function baed on the tockout belief a U nob = maxv + k p, 1 v, where k repreent the enitivity to tockout. 7 It meaure a conumer reponivene to the product carcity. 8 For a nob, the higher 6 Many firm produce excluive good to ell to nob. Sometime, firm announce the exact quantitie (Liverpool FC commemorative phone; Sung 29. Often being proprietary, inventory and hipment quantitie are often not eaily verifiable information, becaue the production proce remain unoberved by cutomer. 7 We keep the dependency on tockout linear only for analytical and expoitory eae. A hown in 4.3, our reult extend to utility function that have nonlinear dependencie on the tockout belief. 8 There i evidence that ome nob wait for a good deal. Firm uch a bluefly.com and Gilt Groupe pecialize in alvage market for the value of k, the higher the utility he get from purchaing the product on the obervation of a tockout. 9 There i ubtantial evidence from literature regarding how tockout may improve a cutomer utility or enhance her preference for the product (ee Lynn 1991 and reference therein. The firm ha to develop belief on the cutomer reervation price for the product. Thi belief may be accrued from ale/conumer data from a imilar portfolio of product from the firm. 1 We denote the firm (eller belief on the reervation price a r. Baed on r, it chooe the price optimally and will produce the correponding optimal quantity to maximize it profit. A cutomer problem i to decide whether he hould buy the product in the elling period or in the alvage period. She buy in the elling period if and only if v + k p 1 v. Thi lead to the nob reervation price, r = k + v +. In our cae, given every agent rational expectation, the game between the firm and the conumer decompoe into two eparate deciion problem: for the conumer, a binary choice problem regarding whether to buy in the elling period or wait (baed on tockout, and for the firm, newvendor profit maximization under demand uncertainty. Both of thee problem have unique olution. The firm need to know the nature of the deciion problem that cutomer are olving and vice vera. Therefore, once all player have rational belief, given market parameter, their optimal action will ettle into the unique rational equilibrium outcome. We are now ready to define the rational expectation equilibrium (RE equilibrium condition for our model. efinition 1. An RE equilibrium (p Q r r atifie the following condition: (i p = r ; (ii Q = arg max q N q p; (iii r = k + v + ; (iv = Q; (v r = r. Condition (i (iii aert that, under expectation r and, the firm and all conumer will rationally act to maximize their utilitie. Condition (iv pecifie luxury good (Rice 21. In any cae, our reult hold even if nob are myopic and not forward looking in their purchae behavior (ee 4.1. 9 We ue one variable to connect both demand realization and production deciion (Q, i.e., the probability of tockout, Prbuyer > Q Q = (baed on rational expectation. Our reult continue to hold even with alternate formulation in which the belief could be baed on fill rate, expected ale, or the production quantity (ee 4.4 4.6. 1 Often the development proce of thee belief i unpecified. In contrat to evolutive jutification baed on the learning offered by the repetition of game, Guenerie (1992 offered an eductive approach forecating forecating of other that alo align well with the game we tudy.
Management Science, Article in Advance, pp. 1 22, 212 INFORMS 5 Figure 1 (a Equilibrium quantity choice 68 66 64 62 6 58 Equilibrium Quantitie (a and Stockout Probabilitie P > Q (b are Plotted with Repect to the Senitivity to Stockout, k, for the Three Market: Newvendor, Strategic Conumer, and Conpicuou Conumer Q * 56 2 4 6 8 1 Senitivity to tockout Conpicuou conumption Q * > Q Regular newvendor Q * < Q (Q (Q c * Abence of conpicuou con. (b Stockout probability.6.5.4.3.2 Abence of conpicuou con. P( > Q c * Regular newvendor P( > Q P( > Q * Conpicuou conumption 2 4 6 8 1 Senitivity to tockout Note. For illutrative purpoe, the demand ditribution i N = 6, 2 = 1, and the parameter are v = 2, c = 1, and = 5. that, in equilibrium, the tockout expectation mut match with the actual probability of not being able to find the product (conitency condition. Conider a cutomer who i indifferent between buying in the firt period and waiting to buy in the alvage period. Becaue he know that every other cutomer i alo trategic and nobbih, he develop a belief on the availability of the product. She rationalize that other cutomer who are trying to buy the product face the ame tockout probabilitie a he doe. We aume nonatomitic deciion maker, hence the ma of remaining cutomer i if the realization of the demand i. Thu, he face a poibility of tockout, P Q, which mut be conitent with her belief, a tated by (iv. Finally, condition (v require that the firm correctly predict the nob reervation price. Condition in efinition 1 can be reduced to condition in p and Q only: p = Qk + v + and Q = arg max q N q p. With the aforementioned condition, we are ready to decribe the RE equilibrium in Propoition 1. Propoition 1. In the RE equilibrium, all cutomer can buy immediately, and the firm price and quantity choice are characterized by p = + k + v c and Q = c /k + v, repectively. All proof are deferred to the appendix. We ue p and Q (and ubcript in general to denote the equilibrium price and quantity deciion the firm make when it chooe to ell the product baed on nob reervation price. For the purpoe of benchmarking, we compare the optimal deciion when conpicuou conumption i preent in the market to the deciion (p and Q when there i none (k =. 1. The equilibrium price when faced with nob,, turn out to be higher than the equilibrium price p choice when faced with jut trategic cutomer, pc. Thi reaffirm our intuition. 2. Intriguingly, the equilibrium production quantity, when conpicuou conumption i preent, Q, i higher than the equilibrium production, Qc (when there i no conpicuou conumption. 11 The firm overproduce becaue of higher margin (underage cot. Jut becaue conumer exhibit conpicuou conumption doe not imply that the conequent production quantitie will be low. In fact, a cutomer become more nobbih (i.e., their valuation increae ignificantly due to tockout, the equilibrium tockout probability fall. Thi i illutrated in Figure 1(b, where equilibrium tockout probability decreae teadily with enitivity to tockout. 3. Note from Figure 1(a that the production quantity when cutomer are jut trategic i lower than in the newvendor cae, i.e., Qc < Q. However, becaue Q > Qc, it i unclear whether, under conpicuou conumption, the production quantitie are lower or higher than the newvendor quantity. Comparing our equilibrium deciion with the newvendor deciion, we find that Q < Q and p < p when k < v v c/c, and Q Q otherwie (proof are deferred to Lemma A1 in the appendix. If k i high, the firm produce more than it would in the traditional newvendor etting. Thu, even though conumer exhibit trategic buying behavior and conpicuou conumption, we find that the firm may not necearily produce le inventory. The higher margin that can be accrued from conpicuou conumer make the firm overcommit to higher production volume, more o than it would if thoe conumer were not conpicuou conumer. Thu, 11 Notice that Q > Q c becaue Q = c c /v > c /k + v = Q.
6 Management Science, Article in Advance, pp. 1 22, 212 INFORMS accounting for cutomer behavior uch a pricing under conpicuou conumption impact other area of the firm and lead to ditinct operational deciion. We now analyze the heterogeneou market that form the bae model for all extenion analyzed in thi paper. 2.2. Conpicuou Conumption in a Heterogeneou Market (Snob and Commoner The market i compoed of two different type of cutomer, which we term nob and commoner (ee Leibentein 195. We ue to denote the fraction of the cutomer population who are nob. The ret of the population 1 i compoed of commoner. A commoner i ditinguihed from a nob in the following ene: A commoner doe not exhibit any inclination for conpicuou conumption, but he may till be trategic in her deciion making. Both type of cutomer (nob and commoner are willing to buy the product in the elling period a long a the firm doe not charge a price higher than their own reervation price. Becaue there are two poible reervation price within the market, the firm will have two poible conitent quantity choice, and thi will in turn affect the equilibrium availability and belief (. Thu, there are two poible candidate for the RE equilibrium. The firm charge one of the reervation price baed on the percentage of nob and produce an optimal quantity that will make the expectation of the cutomer conitent. Thereafter, the cutomer oberve the price and decide whether to buy the product in the elling period. efinition 2. When the firm charge the nob (commoner reervation price, an RE equilibrium (p Q r r atifie the following condition: (i p = r ; (ii Q = arg max q N q p; (iii r = k + v + r = v + ; (iv = Q = Q; (v r = r. The condition impoed in efinition 2 are the ame a thoe impoed in efinition 1 except for condition (iii and (iv. Thoe condition relate to the belief on the reervation price and product availability. If the firm charge the nob reervation price, then only nob are preent in the market to purchae the product (becaue the high price rule out commoner buying the product. Thu, the random variable i recaled from to, and tockout probability become P Q, or imply Q. The correponding equilibrium production quantity i given by Propoition 2(i. On the other hand, if the firm charge the commoner reervation price, the demand remain identical to the initial demand ditribution, becaue the offered price i lower than everyone reervation price. In thi cae, a poibility of tockout tay the ame a in efinition 1, F Q. Thi i indicated in Propoition 2(ii. Propoition 2. (i (Limited Production In the RE equilibrium under limited production, only nob can buy, and the firm price and quantity choice are characterized by P > Q = c /k + v and p = c k + v +. (ii (Regular Production In the RE equilibrium, all cutomer (nob and commoner can buy, and the firm price and quantity deciion are characterized by P > Q c = c /v and p c = c v +. Note that given a pecific market condition (fraction of nob and the trategie of other player, only one et of rational belief will emerge. epending on the market parameter, the profit-maximizing firm would end up with one of the aforementioned trategie. Given a, becaue conumer are rational and can correctly form expectation about firm trategie, the correct RE equilibrium would emerge. We invetigate the two candidate trategie to ee when limited production or regular production would be preferred by the firm. We ue N and N c to denote the firm optimal profit obtained under the limited production trategy (elling only to nob and regular production trategy (elling to nob and commoner, repectively. Propoition 3. There exit a unique threhold 12 uch that for in the RE equilibrium, N N c, the firm adopt the regular production trategy, and all cutomer can buy. If >, then N > N c, the firm adopt the limited production trategy, and only nob can buy. Propoition 3 point out that for a given, there i a unique rational expectation equilibrium. If the number of nob in the market i low (i.e.,, the firm will price the product at the commoner reervation price, thu making it poible for all conumer to buy it. The additional profit accrued from the higher price premium can be compenated by elling to a ignificantly larger market at a lower price. However, if there i a ufficient preence of nob in the market ( >, the firm adopt the limited production trategy by attempting to ell only to the nob. On average, the firm can afford to ell to nob at high price, even though the volume of ale ha been puhed down becaue of reduced market coverage. We note that erving only to the nob might alo be perceived a a carcity trategy. However, thi i not necearily true. In thi cae, the firm deciion are dictated by two counteracting factor. Firt, elling 12 = v /k+v / F c /v F uf udu c /k+v u f u du.
Management Science, Article in Advance, pp. 1 22, 212 INFORMS 7 only to nob mean that the average demand in the market i reduced thi mean the production quantity will tend to reduce on average. However, elling only to nob increae the underage cot or the product margin, becaue the product i now marketed to nob at higher price. Thi mean that the production quantity will increae. Becaue of the higher underage cot, more unit of the product are produced to avoid the opportunity cot of miing thoe high margin ale (lot ale. Thee two effect counteract each other. Thu, the reultant production quantity may be higher or lower than the production quantity when the firm ell to everyone in the market. In fact, we find that if the fraction of nob in the market i below a certain threhold, the product availability might be low compared to the cae in which the firm ell the product to all conumer type. Thi i captured in Propoition 4. Propoition 4. There exit a unique fraction of percentage of nob, Q, where Q < Q c when < Q and Q > Q c when > Q. Thi threhold level i given by Q = c /v / c /k + v. Propoition 4 aert that the trategy of retricting the ale only to nob doe not alway imply low availability of the product. In fact, the product might be commonly available even though the firm cover only the nob in the market. Conequent to Propoition 4, the product i carce in the market only if (i the product i limited to nob (limited production and (ii the production quantitie are lower than the quantitie the firm produce when it ell to the whole market (i.e., Q < Q c. Note that Q c i the quantity available when erving the whole market, becaue all conumer (nob and commoner are trategic in our model. Thu carcity exit only when Q. We elaborate thi intereting finding further in 3. 3. Analyi of Scarcity Strategie To dicu carcity trategie in the market, we firt define the notion of carcity. We define a product to be carce when the optimal quantity available in the market when elling only to nob i lower than the optimal quantity available when elling to all trategic conumer in the market with an identical demand ditribution (i.e., Q < Q c.13 From Propoition 3, when, regular production i adopted, and Qc unit are available in the market. When the fraction of nob in the population exceed, the firm witche to elling only to 13 Figure 2 demontrate that our notion of carcity i tricter than comparing the equilibrium production quantity to the tandard newvendor production quantity. nob (i.e., limited production. A increae, elling only to nob continue to remain the optimal elling trategy. However, note that the production volume increae becaue the mean demand (i.e., the fraction of nob in the market i increaing. A a reult, if the fraction of nob in the market i higher than Q (from Propoition 4, the total production volume and the availability of product (in-tock probability are both higher than in the cae when cutomer are jut trategic. Thu, the in-tock probability for the product i lower (i.e., the product i carcer to find in the intermediate region between and Q. Furthermore, Figure 2 alo reveal that the extent of carcity i the tronget when the fraction of nob i jut higher than. 3.1. Increaed Repone to Stockout We now tudy the prevalence of carcity a the nob enitivity to tockout varie. In Figure 2, we tudy how carcity deciion vary with the fraction of nob in the market, a the enitivity to tockout increae (from k = 1 in Figure 2(a to k = 45 in Figure 2(b. We find that when the market i concentrated with nob, who are highly enitive to tockout (high k, the firm might produce more quantity than the regular newvendor quantity in equilibrium (even though the cutomer are trategic. Note that thee reult extend the obervation from 2.1 that howed that the equilibrium production may exceed the newvendor production quantity when the enitivity to tockout i high. Furthermore, a the nob become more enitive to tockout, we make two key obervation: 1. The threhold decreae with enitivity to tockout (k. If carcity become more deirable to nob, the firm i more likely to offer limited production. 2. On the other hand, the optimal equilibrium production quantity under the limited production trategy increae more teeply with the fraction of nob in the market a k increae (i.e., the lope of the line under limited production trategy increae with k. If the nob repond trongly to tockout, their reervation price will be even higher, which reult in a higher price (and an increaed underage cot. A a reult, the production quantitie increae teeply depite the firm adopting a trategy of elling only to nob. Thi ha the effect of reducing the degree of carcity (fewer tockout. Lemma 1. For higher k, the equilibrium production quantity Q increae more teeply in. Lemma 1 demontrate that the optimal production quantity increae fater in a the enitivity to tockout increae. A nob become more enitive to tockout, the firm increae it production
8 Management Science, Article in Advance, pp. 1 22, 212 INFORMS Figure 2 Equilibrium Production over ifferent (a Equilibrium quantity choice 7 6 5 4 3 2 1 (b 7 Q Q Q * c 6 Q * c Whole market can buy Scarcity Q * Only nob can buy Equilibrium quantity choice.2.4 β *.6.8 β Q 1..2 β *.4 β *.6.8 β Q 1. Fraction of nob Fraction of nob Note. For repreentative illutration, v = 2, c = 1, and = 5, and the demand ditribution i N = 6 2 = 1. In (a, k = 1, and in (b, k = 45. The dotted line above repreent the production quantity when cutomer behavior i entirely ignored. Note that in the region between and Q, the optimal production capacity i lower than the production quantity when all cutomer are trategic (Q c. Thi define the carcity region. The difference between Q and c Q, pointed out in the figure, denote the extent or the degree of carcity in the market. An increae in reponivene to tockout k (panel (a to panel (b lead to a decreae in and an increae in the lope of Q. quantitie even further becaue the margin from the ale to nob ha alo increaed. Even though nob are enitive to tockout, their willingne to pay more for excluivity caue the firm to produce more good, becaue the opportunity cot of loing a ale to uch a cutomer would be very high. In other word, the firm i avere to loing a high margin ale (on thoe rare tockout and tock up on inventory, even though it run the rik of reduced excluivity among the nob. Lemma 2 ummarize the behavior of the threhold with repect to enitivity of nob to tockout. Lemma 2. The threhold level, and Q, decreae with increae in enitivity to tockout, k. Recall that the firm adopt the limited production trategy when the number of nob in the market i more than. Lemma 2 indicate that the firm would adopt the limited production trategy more often a the enitivity to tockout increae in the market for the nobbih cutomer. Converely, Lemma 2 alo tate that Q decreae in k. The more enitive the nob are to tockout, the more likely the trategy of elling to nob lead to overproduction (i.e., more than the equilibrium quantity produced when the good i available to the whole market. A een in Lemma 1, the increaed opportunity cot drive the firm to produce more good. In other word, the cot of tockout are high when the enitivity of tockout for nob i high. A a reult, the firm produce more good, even though it i limiting it market to a fraction of cutomer (nob. Aided with the reult of Lemma 1 and 2, we can now analyze the region of carcity. It i unclear if the carcity region that exit in the region Q expand a nob become more 5 4 3 2 1 Scarcity receptive to tockout (i.e., a k increae. Propoition 5 provide condition under which the region of carcity (i.e., Q expand a nob become more enitive to tockout. Propoition 5. The carcity region expand if and only if the generalized failure rate of the ditribution i greater than a threhold, i.e., gq / Q/ MQ /, where M i a contant dependent on Q and. Propoition 5 provide a condition baed on the demand ditribution for the prevalence of the tockout trategy. When the nob are very enitive to tockout, the carcity trategy i often in equilibrium if the ditribution of the uncertain demand ha a high generalized failure rate (ee Lariviere 25. In 3.2, we focu on how the carcity region varie with increaing demand variability for a given enitivity to tockout parameter. 3.2. Scarcity Strategy: The Effect of Uncertainty In Figure 3, we hold the mean of the demand ditribution contant and increae it variance. We hold k contant and explore how the carcity trategy region change with increaing variability in demand. Figure 3 demontrate that the following occur a the demand uncertainty increae: (i decreae. Applying the reult on Propoition 3, the firm i le likely to adopt the regular production trategy a demand uncertainty increae. (ii Q decreae. Applying the reult from Propoition 4, the firm i more likely to overproduce if there are a ufficient number of nob in the market a the demand uncertainty increae. (iii Q decreae. Applying Propoition 5, the market proportion of conumer for which the Q *
Management Science, Article in Advance, pp. 1 22, 212 INFORMS 9 Figure 3 Equilibrium quantity choice 12 1 8 6 4 2 A the Coefficient of Variation of the emand Increae, Q c ecreae Q * c Whole market can buy Only nob can buy.4 β *.8 Fraction of nob Q Scarcity Note. Alo note that both and Q alo decreae. The carcity region i further reduced by the increae of the lope of the line Q. Parameter are ame a in Figure 2(a. carcity trategy i employed decreae. If demand uncertainty increae, the carcity trategy i le likely. (iv Qc decreae (note that regular newvendor quantity Q increae. Applying Propoition 2(ii, a the demand uncertainty increae, the regular production quantity decreae. (v The lope of Q increae. Applying Propoition 2(i for the ame fraction of conumer in the market, more quantity i produced under the limited production trategy. (vi The degree of carcity (i.e., Qc Q i reduced. A demand uncertainty increae, applying Propoition 2, the number of tockout conumer face due to the carcity trategy i reduced. In ummary, if the demand uncertainty increae, the firm commit to the carcity trategy le often, and the degree of carcity when it o commit i alo reduced. 4. Extenion of the Bae Model We etablih the robutne of our reult, by howing that thi intermediate carcity profile continue to exit under a wide variety of modeling variation. Specifically, we conider the preence of the following cae: (i myopic nob, (ii nob who have a lower value for abundant product, and (iii nob who have a nonlinear utility dependence on tockout. In addition, we conider other conpicuou conumption belief that nob might poe: namely, the nob could have belief baed on (iv fill rate, (v expected ale, (vi production quantity, and (vii product price (intead of belief baed on tockout. We how that our main concluion continue Q * β Q to hold for all the variation. The firm adopt the regular production trategy when the proportion of nob in the market i low, and the limited production trategy when the fraction of nob in the market i high. In particular, the carcity trategy i adopted in the intermediate region of. Mot of the technical detail for the ubection below are omitted due to page retriction and they are available upon requet. Reult can be driven eaily by applying the ame methodology in the proof of each propoition of 2.2 after the uggeted modification on nob utility function in the ubection below. 4.1. Myopic Snob In thi ection, we analyze the cae when cutomer exhibit conpicuou conumption but they are not trategic. Such nob are willing to pay high price for carce product to ditinguih themelve from other, and they may buy myopically (The Economit 29. To account for the myopic nature of the nob, we alter the condition for the reervation price of nob (condition (iii in efinition 2, i.e., r = v + k. Invetigating the two candidate trategie for the firm, we find that the threhold imilar to that etablihed in 2.2 hold, with one exception: the threhold for the limited production trategy when conidering myopic nob i maller than the threhold when nob are trategic (i.e., myopic. 14 The firm adopt the limited production trategy in more cenario compared to the cae when nob are trategic. A before, carcity exit only in the intermediate region of. 4.2. Snob Have a Lower Value for Commonly Available Product In 2, we aumed that nob have weakly greater utility for (carce good than commoner. We now extend that to conider the cae when nob have a trictly lower valuation than commoner while the product i widely available to everyone in the market. Let nob valuation be v + k, where i the belief on tockout probability. Furthermore, commoner valuation i v. We aume v < v. Thi would imply that if the tockout probability i lower than or equal to ome threhold, the nob will have a trictly lower utility than the commoner. Therefore, nob could have a lower utility for widely available product that are conumed in large quantitie. In other word, they uffer a diutility due to the lack of excluivity, a in Balachander and Stock (29. Neverthele, they till poe nobbihne and enjoy higher utility from carcer good. Thu, a nob utility function i U nob = maxv + k p 1 v, which change the reervation 14 = myopic c /v F c /v uf u du/v + v 2 + 4kc /2 F v 2 +4kc v /2k uf u du.
1 Management Science, Article in Advance, pp. 1 22, 212 INFORMS price of the nob to r = k + v +. In light of thee modification, condition (iii in efinition 2 i uitably changed. We can how that the firm adopt the limited production trategy when the fraction of nob in the market i ufficiently large (i.e., >. Similarly, we can how that there i a unique threhold Q below which we have Q < Qc. We can how that thee threhold and Q are both higher than in the bae cae in 2.2. The effect i qualitatively imilar to Lemma 2. Thu, a a reult of nob having more diutility from the commonly available product, the firm adopt the Limited production trategy le often. One can immediately notice that the carcity trategie exit in intermediate range Q. However, to adopt carcity trategie viably, the firm alo need an increaed number of nob in the market. In addition, the firm ha to offer a higher degree of carcity (i.e., more tockout to enable the nob to overcome their low valuation for commonly available good. Finally, the decreaed valuation of nob ha ome intereting effect when the fraction of nob in the market i low ( <, depending on the enitivity of nob to tockout. When k i ufficiently large, the concluion remain identical to Propoition 2(ii. However, when k i low (i.e., k < v v, nob have a much lower valuation than the commoner becaue of the wide availability of good. Propoition 6 (Modified Regular Production. In the RE equilibrium, when k < v v, only commoner can buy, and the firm price and quantity deciion are characterized by P1 > Q c = c /v and p c = c v +. When k > v v, Propoition 2(ii hold. Propoition 6 point out that given the low fraction of nob and their lower valuation, it i optimal for the firm to price the product at commoner reervation price. Therefore, at low (and low k, nob do not buy, and only commoner buy the product at the old market price. The reult i captured in Propoition 6, which act a a modified propoition for the regular production cae in Propoition 3. Thi reult correpond to the cae wherein a product i old at a low price and in large quantitie (which keep the nob from buying the product, depite it low price. Thi reult explain the market preence of many nonexcluive brand and low-end retailer who ell inexpenive good to commoner in large quantitie. 4.3. Nonlinear Utility from Stockout In 2.2, we aumed that the utilitie of nob are linearly dependent on the tockout probability. We now generalize the utility of nob to nonlinear dependencie on tockout probability. Cae A. We conider the utility of nob in the following functional form: U nob = maxv + k n p 1 v. We find that our concluion remain unchanged. There exit a threhold uch that if the fraction of nob i below, the firm ue the regular production trategy. When ( >, the firm adopt limited production only nob can buy, and the firm equilibrium deciion are characterized by k Q n+1 + Q 2 v = c and p = k Q n + Q v +. In the cae of regular production (, all cutomer (nob and commoner can buy. The firm deciion i characterized by Q c = c /v and pc = c v + (ame a in 2.2. Again, a in 2, the carcity trategie are employed in the intermediate region when the fraction of nob Q. Although the tructure of the equilibrium remain the ame, we find that, becaue of the nonlinearity of the nob utility, their valuation for the product i higher, which caue the carcity trategy to be employed even with a (relatively lower percentage of nob in the market. In addition, the degree of carcity i alo lower than it i in the linear cae. Cae B. We alo teted another verion of nonlinearity pecified by U nob = maxv/1 n p, 1 v. Our concluion are identical to the above concluion. The firm employ carcity in the intermediate value of fraction of nob in the market. 4.4. Belief Baed on Fill Rate In thi ection, we expanded our definition of conpicuou conumption from tockout probability to one related to fill rate, i.e., the fraction of cutomer who do not find the product ( lot ale available in equilibrium. The in-tock probability correpond to the probability that all demand i fulfilled, wherea fill rate meaure the fraction of demand that i fulfilled. Following Cachon and Terwiech (28, fill rate i 1 ELot Sale/E. Correponding to belief baed on tockout probability, we could repreent the nob utility uing belief on the fraction of unfulfilled demand lot. Thu, U nob = maxv + k lot p, where lot i the fraction of demand lot. Becaue of the tochatic nature of the demand, the number of item left unold from the elling period may vary. If there are item left over, then the demand in the elling period wa lower than the production quantity, and therefore the correponding fill rate would equal one. In thi cae, nob have no utility from conuming the product in the econd period. The aumption aid tractability. However, the property can be generalized and concluion remain the ame (jut a in myopic nob cae.
Management Science, Article in Advance, pp. 1 22, 212 INFORMS 11 The condition for RE equilibrium remain the ame a in efinition 2, except for condition (iii and (iv, which are now (iii r = v + k lot and (iv lot = u Qf Q u du/e, repectively. Even if we ue (1-fill rate intead of the tockout probability in nob utility function, we note that our concluion remain identical. We can how that there exit a threhold proportion of nob in the market, below which the firm adopt the regular production, and above which the firm adopt the limited production trategy. It i traightforward to how that the equilibrium deciion when the firm engage in regular production are identical to the bae cae. However, when the firm adopt limited production, only nob can buy becaue of higher price; the firm price and quantity choice are characterized by F Q = c v + k u Q Q f u du/e and p = c / Q +. If we aume that the demand i ditributed a U N, then we obtain the following explicit reult for the equilibrium price and quantity under the limited production trategy: p = c 2/3 k + v 1/3 + ( c 1/3 Q (1 = N k + v Finally, we how that there exit a unique level of percentage of nob, Q, where Q Q c if Q, and Q > Q c when > Q. Thu, carcity trategie are adopted in the intermediate range of Q. 4.5. Belief Baed on Expected Sale We teted the robutne of our model by conidering the dependency of conpicuou conumption on unit ale in which the nob utility decreae in expected number of buyer. Thi i alo an extenion of the utility model of Amaldo and Jain (25a, if we ignore the production quantity mimatch. Accordingly, we redefine the utility function baed on the belief on expected unit ale, ale. The analyi proceed a in the previou ection. Thi analyi i tractable for ditribution with finite upport. When the belief on expected ale i ale, the utility of nob i U nob = maxv + k/ ale p. Note that the utility of a nob i decreaing in the quantity of demand that i fulfilled (i.e., the number of buyer. The condition for RE equilibrium remain the ame a in efinition 2, except for condition for nob (iii and (iv, which are now (iii r = v + k/ ale and (iv ale = Q Q F u du, repectively. We note that our concluion remain unaltered in thi cae. If the fraction of nob in the market i below ome threhold, the firm adopt regular production, and above the limited production trategy i adopted. It i traightforward to how that the equilibrium deciion when the firm engage in regular production are identical to the bae cae. A in 2.2, when exceed the threhold, the firm adopt the limited production trategy, in which it price and quantity are characterized by F Q = c v + k/q Q / F u du and p = c / Q / +. There exit a unique fraction of nob, Q, where Q < Q c if < Q, and Q > Q c otherwie. Thi lead to our final concluion that the firm employ the carcity trategy in the region Q. 4.6. Belief Baed on Firm Production Quantity In line with our previou etting, we now extend our reult to the cae in which nob value the product more when they expect that the firm produce the good in low quantitie. Thi ection matche the reult for when inventory i (credibly announced to be limited. Thu, we now repreent a nob utility uing her belief about the production quantity, Q : U nob = maxv + k/ Q p. We are ready to define the RE equilibrium condition: efinition 3. When the firm charge the nob (commoner reervation price, an RE equilibrium p Q r Q r atifie the following condition: (i p = r ; (ii Q = arg max q N q p; (iii r = v + k/ Q r = v + ; (iv Q = Q; (v r = r. If the fraction of nob in the market i below ome threhold, the firm adopt regular production, and above thi threhold the limited production trategy i adopted. The pricing and quantity deciion of the firm under regular production are identical to the bae cae. A before, when exceed the threhold, the firm adopt the limited production trategy, in which it price and quantity are now characterized by Q = c /v + k/q and p = v + k/q. There exit a unique level of percentage of nob, Q, where Q < Q c if < Q, and Q > Q c otherwie. Thi lead to our final concluion that the carcity trategy i exercied in the region Q. 4.7. Price-ependent Stockout Belief It i very likely that conumer make inference about the tockout probability for a product baed on the elling price. For intance, it i quite poible for conumer to expect that le expenive item are more likely to be produced in large quantitie and tay on helve. On the other hand, even though there are fewer cutomer in the market for more expenive product, very few of uch item are produced. Such