WASHINGTON UNIVERSITY IN ST. LOUIS Olin Business School Disseraion Examinaion Commiee: Nan Yang, Chair Fuqiang Zhang, Co-Chair Amr Faraha Jake Feldman John Nachbar Dynamic Pricing and Invenory Managemen: Theory and Applicaions by Renyu Zhang A disseraion presened o he Graduae School of Ars & Sciences of Washingon Universiy in parial fulfillmen of he requiremens for he degree of Docor of Philosophy May 2016 S. Louis, Missouri
2016, Renyu Zhang
Table of Conens Page Lis of Figures..................................... v Lis of Tables...................................... Acknowledgmens.................................... ABSTRACT OF THE DISSERTATION....................... vi vii x 1 Inroducion.................................... 1 1.1 Moivaion................................... 1 1.2 Conribuion.................................. 2 1.3 Organizaion of he Disseraion....................... 6 2 Operaions Impac of Nework Exernaliies: he Monopoly Seing. 7 2.1 Inroducion.................................. 7 2.2 Relaed Research............................... 11 2.3 Model Formulaion.............................. 12 2.4 Analysis of he Base Model......................... 17 2.4.1 Opimal Policy............................ 17 2.4.2 Sae Space Dimension Reducion.................. 19 2.4.3 Managerial Implicaions of Nework Exernaliies......... 22 2.5 Effecive Sraegies o Exploi Nework Exernaliies........... 27 2.5.1 Price Discriminaion......................... 27 2.5.2 Nework Expanding Promoion................... 30 2.6 Numerical Sudies............................... 33 2.6.1 Impac of Nework Exernaliies................... 34 2.6.2 Effecive Heurisic Policies under Nework Exernaliies...... 37 2.7 Summary................................... 38 3 Operaions Impac of Nework Exernaliies: Dynamic Compeiion Seing....................................... 41 3.1 Inroducion.................................. 41 3.2 Relaed Research............................... 45 3.3 Model..................................... 48 3.4 Simulaneous Compeiion.......................... 53 3.4.1 Equilibrium Analysis......................... 53 3.4.2 Exploiaion-Inducion Tradeoff................... 61 3.5 Promoion-Firs Compeiion........................ 66 ii
Page 3.5.1 Equilibrium Analysis......................... 66 3.5.2 Exploiaion-Inducion Tradeoff................... 73 3.6 Comparison of he Two Compeiion Models................ 77 3.7 Summary................................... 78 4 Trade-in Remanufacuring, Sraegic Cusomer Behavior and Governmen Subsidies................................ 80 4.1 Inroducion.................................. 80 4.2 Relaed Research............................... 84 4.3 Model and Equilibrium Analysis....................... 86 4.3.1 Model Seup.............................. 86 4.3.2 Equilibrium Analysis......................... 89 4.4 Impac of Trade-in Remanufacuring.................... 93 4.4.1 Impac on Firm Profi........................ 94 4.4.2 Impac on Environmen and Cusomer Surplus........... 98 4.5 Social Opimum and Governmen Inervenion............... 102 4.6 Summary................................... 108 5 Pricing and Invenory Managemen under he Scarciy Effec of Invenory....................................... 111 5.1 Inroducion.................................. 111 5.2 Relaed Research............................... 115 5.3 Model Formulaion.............................. 117 5.3.1 Discussions on Assumpion 5.3.3................... 122 5.4 Unified Model................................. 124 5.5 Addiional Resuls in Two Special Cases.................. 131 5.5.1 Wihou Invenory Wihholding................... 131 5.5.2 Wihou Invenory Disposal..................... 136 5.6 Responsive Invenory Reallocaion...................... 137 5.7 Numerical Sudies............................... 140 5.7.1 Opimal Policy Srucure wih Non-concave R(, ) Funcions... 140 5.7.2 Impac of Scarciy Effec....................... 142 5.7.3 Value of Dynamic Pricing...................... 143 5.8 Summary and Exension........................... 145 6 Comparaive Saics Analysis Mehod for Join Pricing and Invenory Managemen Models.............................. 148 6.1 Inroducion.................................. 148 6.2 Relaed Research............................... 152 6.3 A New Comparaive Saics Mehod.................... 154 6.3.1 An Illusraive Example....................... 154 6.3.2 Proof of Lemma 15 wih Our New Mehod............. 156 iii
Page 6.4 Applicaion of he New Comparaive Saics Mehod in a General Join Pricing and Invenory Managemen Model................. 160 6.4.1 Model................................. 161 6.4.2 Comparaive Saics Analysis wih Our New Mehod....... 165 6.5 Applicaion of he New Comparaive Saics Mehod in a Compeiion Model173 6.5.1 Effor-Level-Firs Compeiion.................... 174 6.5.2 Simulaneous Compeiion...................... 177 6.5.3 A Comparison of Equilibria in he Two Compeiion Models... 179 6.6 Summary................................... 180 7 Concluding Remarks.............................. 183 References........................................ 185 A Appendix for Chaper 2............................ 201 A.1 Proofs of Saemens............................. 201 A.2 More Condiions on Assumpion 2.3.1.................... 220 B Appendix for Chaper 3............................ 223 B.1 Proofs of Saemens............................. 223 B.2 Sufficien Condiions for he Monooniciy of πs, sc [π pf s, ] in βs, 1 sc [βs, 1] pf. 246 C Appendix for Chaper 4............................ 252 C.1 Equilibrium Definiions............................ 252 C.2 Proofs of Saemens............................. 253 D Appendix for Chaper 5............................ 269 D.1 Proofs of Saemens............................. 269 E Appendix for Chaper 6............................ 291 E.1 Proofs of Saemens............................. 291 E.2 Discussions on Sopping Condiion (ii) of he Ieraive Procedure.... 312 E.2.1 Comparaive Saics Analysis of y i (γ) (p + 1 i p + q)..... 312 iv
Lis of Figures Figure Page 2.1 Value of λ m : θ = 0.5, η = 0.5......................... 35 2.2 Value of λ m : k = 0.5, η = 0.5......................... 35 2.3 Value of λ m : k = 0.5, θ = 0.5......................... 36 2.4 Value of λ m and λ i h : θ = 0.5, η = 0.5..................... 36 2.5 Value of λ m and λ i h : k = 0.5, η = 0.5..................... 37 2.6 Value of λ m and λ i h : k = 0.5 θ = 0.5..................... 37 5.1 Opimal Ordering-up-o Level....................... 141 5.2 Opimal Price-induced Demand...................... 141 5.3 Value of λ scarciy : = 5............................. 142 5.4 Value of λ scarciy : = 10............................ 142 5.5 Value of λ pricing : = 5............................. 144 5.6 Value of λ pricing : = 10............................. 144 v
Lis of Tables Table Page 4.1 Summary Saisics: Firm Profi (%)................... 97 4.2 Summary Saisics: Environmenal Impac (%)............ 100 vi
Acknowledgmens Firs and foremos, I would like o hank and praise he Lord Jesus Chris for graning me he wisdom, he perseverance, and he necessary suppor and resources o navigae he PhD sudy and finish he disseraion. I is my hope ha his disseraion could glorify His name. I am exremely blessed o have Professors Nan Yang and Fuqiang Zhang as my docoral advisors. Nan and Fuqiang have always been incredibly caring and helpful hroughou our collaboraions on various research projecs. More imporanly, hey have spen significan effor on encouraging and faciliaing my scholarly growh. I owe my sincere graiude o Nan and Fuqiang, and I believe his disseraion represens he beginning of a life-long journey of academic collaboraions beween us. I would also like o hank Professors Amr Faraha, Jake Feldman, and John Nachbar for serving on my disseraion commiee, and giving me insrumenal feedbacks a differen sages of he hesis. Thanks o heir insighful commens and suggesions, he qualiy of his disseraion has improved subsanially. I is my grea forune and honor o collaborae wih various scholars oher han my advisors hroughou my PhD sudy. Here, I would like o express my graiude o hem: Professor Long Gao from UC Riverside, Ting Luo from UT Dallas, and Guang Xiao from WashU. I am graeful for he suppor and help from he suden communiies a WashU o which I belong. I enjoy my ime wih my fellow PhD sudens a Olin Business School hroughou hese years. I am indebed o he kind prayers of he brohers and sisers from he Faih Hope Love suden fellowship. My disseraion research has been funded by he Docoral Fellowship of Olin Business School. I would like o ake his opporuniy o acknowledge Olin Business School s generous financial suppor. The saff members of Olin Business School s docoral program, Dona Cerame, Sarah Graham and Erin Murdock, are also graefully acknowledged for heir kind adminisraive suppor. vii
Las bu no leas, I would like o hank my wife Sicong Zhu, my daugher Naalie Y. Zhang, and my parens Xuwei Zhang and Renwu Lu. I is my parens who raised me and provided me good educaion opporuniies, and my wife and daugher who accompanied me hrough he ups and downs of my disseraion research. Finishing his disseraion would simply be impossible wihou heir coninuous love and suppor viii
Dedicaed o Jesus Chris and my family. ix
ABSTRACT OF THE DISSERTATION Dynamic Pricing and Invenory Managemen: Theory and Applicaions by Renyu Zhang Docor of Philosophy in Business Adminisraion Washingon Universiy in S. Louis, 2016 Professor Nan Yang, Chair Professor Fuqiang Zhang, Co-Chair We develop he models and mehods o sudy he impac of some emerging rends in echnology, markeplace, and sociey upon he pricing and invenory policy of a firm. We focus on he siuaion where he firm is in a dynamic, uncerain, and (possibly) compeiive marke environmen. The marke rends of paricular ineres o us are: (a) social neworks, (b) susainabiliy concerns, and (c) cusomer behaviors. The wo main running quesions his disseraion aims o address are: (a) How hese emerging marke rends would influence he operaions decisions and profiabiliy of a firm; and (b) Wha pricing and invenory sraegies a firm could use o leverage hese rends. We also develop an effecive comparaive saics analysis mehod o address hese wo quesions under differen marke rends. Overall, our resuls sugges ha he curren marke rends of social neworks, susainabiliy concerns, and cusomer behaviors have significan and ineresing impac upon he operaions policy of a firm, and ha he firm could adop some innovaive pricing and invenory sraegies o exploi hese rends and subsanially improve is profi. Our main findings are: (a) Nework exernaliies (he monopoly seing). We find ha nework exernaliies promp a firm o face he radeoff beween generaing curren profis and inducing fuure demands when making he price and invenory decisions, so ha i should increase he base-sock level, and o decrease [increase] he sales price when he x
nework size is small [large]. Our exensive numerical experimens also demonsrae he effeciveness of he heurisic policies ha leverage nework exernaliies by balancing generaing curren profis and inducing demands in he near fuure. (Chaper 2.) (b) Nework exernaliies (he dynamic compeiion seing). In a compeiive marke wih nework exernaliies, he compeing firms face he radeoff beween generaing curren profis and winning fuure marke shares (i.e., he exploiaioninducion radeoff). We characerize he pure sraegy Markov perfec equilibrium in boh he simulaneous compeiion and he promoion-firs compeiion. We show ha, o balance he exploiaion-inducion radeoff, he compeing firms should increase promoional effors, offer price discouns, and improve service levels. The exploiaion-inducion radeoff could be a new driving force for he fa-ca effec (i.e., he equilibrium promoional effors are higher under he promoion-firs compeiion han hose under he simulaneous compeiion). (Chaper 3.) (c) Trade-in remanufacuring. We show ha, wih he adopion of he very commonly used rade-in remanufacuring program, he firm may enjoy a higher profi wih sraegic cusomers han wih myopic cusomers. Moreover, rade-in remanufacuring creaes a ension beween firm profiabiliy and environmenal susainabiliy wih sraegic cusomers, bu benefis boh he firm and he environmen wih myopic cusomers. We also find ha, wih eiher sraegic or myopic cusomers, he socially opimal oucome can be achieved by using a simple linear subsidy and ax scheme. The commonly used governmen policy o subsidize for remanufacuring alone, however, does no induce he social opimum in general. (Chaper 4.) (d) Scarciy effec of invenory. We show ha he scarciy effec drives boh opimal prices and order-up-o levels down, whereas increased operaional flexibiliies (e.g., he invenory disposal and invenory wihholding opporuniies) miigae he demand loss caused by high excess invenory and increase he opimal order-up-o levels and sales prices. Our exensive numerical sudies also demonsrae ha dynamic pricing leads o a much more significan profi improvemen wih he scarciy effec of invenory han wihou. (Chaper 5.) xi
(e) Comparaive saics analysis mehod. We develop a comparaive saics mehod o sudy a general join pricing and invenory managemen model wih muliple demand segmens, muliple suppliers, and sochasically evolving marke condiions. Our new mehod makes componenwise comparisons beween he focal decision variables under differen parameer values, so i is capable of performing comparaive saics analysis in a model where par of he decision variables are non-monoone, and i is well scalable. Hence, our new mehod is promising for comparaive saics analysis in oher operaions managemen models. (Chaper 6.) xii
1. Inroducion 1.1 Moivaion Price and invenory are definiely wo key operaions decisions of any firm ha delivers (physical) producs o cusomers. The developmen of advanced informaion echnologies faciliaes he sellers o plan, implemen and ake advanage of he dynamic pricing sraegies. Thanks o he IT decision suppor applicaions, sellers are now able o opimize sales prices and invenory conrol policies based on complex analyics and opimizaion mehods. Therefore, he join dynamic pricing and invenory managemen sraegies have been exensively sudied in he lieraure, and widely used in pracice. For example, Amazon no only dynamically adjuss he sales prices of housands of is iems everyday, bu also adops a complex procuremen and delivery sysem o manage is invenories. The emerging rends in echnology, markeplace, and sociey have led o unprecedened challenges o opimize heir pricing and invenory conrol policy. The primary goal of his disseraion is o develop he models and mehods o undersand he impac of some emerging marke rends upon a firm s pricing and invenory policy. Specifically, we consider hree ypes of curren marke rends: (a) social neworks, (b) susainabiliy concerns, and (c) cusomer behaviors. ˆ Social neworks. The recen fas developmen of online social media has significanly inensified he ineracions beween cusomers. The social neworks make cusomers easily know and follow heir friends purchasing decisions, hus giving rise o (posiive) nework exernaliies for almos all producs. Tha is, cusomers are more likely o purchase a produc if here are more oher cusomers who purchase he same produc. Nework exernaliies enable firms o use curren cusomers o arac fuure cusomers and, hus, may have ineresing implicaions on he pricing and invenory policy of a firm. ˆ Susainabiliy concerns. rend of susainabiliy/environmenal concerns. In he recen years, he sociey embraces an increasing Remanufacuring, and he associaed rade-in program o collec used producs for remanufacuring, have been 1
increasingly used for he sake of is environmenal benefi. We are especially ineresed in characerizing how rade-in remanufacuring would influence he pricing and producion policy of a firm, and he economic and environmenal values of his business pracice. From he governmen s perspecive, i is also ineresing o sudy he public policy ha could improve he social welfare when aking ino accoun firm profi, cusomer surplus, and environmenal impac. ˆ Cusomer behaviors. We sudy wo cusomer behaviors in his disseraion. The firs is he sraegic waiing behavior of cusomers. Wih his behavior, cusomers will sraegically seek for fuure discoun and rade-in opporuniies. We are curious abou he impac of sraegic cusomer behavior upon he economic and environmenal values of rade-in remanufacuring. The second cusomer behavior sudied in his disseraion is he scarciy effec of invenory, which refers o he phenomenon ha cusomers are discouraged by high invenory and encouraged by low invenory available o hem. The operaional implicaions of he scarciy effec of invenory have also been analyzed in his disseraion. 1.2 Conribuion In his disseraion, we esablish dynamic programming and game heoreic models o sudy he dynamic pricing and invenory conrol issues under he presence of hese new marke rends. Our focus is o address wo main quesions: (a) How hese emerging marke rends would influence he operaions decisions and profiabiliy of a firm; and (b) Wha pricing and invenory sraegies a firm could use o leverage hese rends. Our analysis reveals ha he curren marke rends of social neworks, susainabiliy concerns, and cusomer behaviors give rise o some new radeoffs he firm has o balance and, hus, have significan and ineresing impac upon he operaions policy of a firm. On he oher hand, he firm could adop some innovaive pricing and invenory sraegies o exploi hese rends and subsanially improve is profi. To faciliae he analysis of he wo main quesions, we also develop an effecive comparaive saics analysis mehod for a general class of join pricing and invenory managemen models. Nework exernaliies (he monopoly seing, Chaper 2). We sudy he impac of nework exernaliies upon a firms pricing and invenory policy under demand 2
uncerainy. The firm sells a produc associaed wih an online service or communicaion nework, which is formed by (par of) he cusomers who have purchased he produc. The produc exhibis nework exernaliies, i.e., a cusomer s willingness-o-pay and, hus, he poenial demand are increasing in he size of he associaed nework. We show ha a nework-size-dependen base-sock/lis-price policy is opimal. Moreover, he invenory dynamics of he firm do no influence he opimal policy as long as he iniial invenory is below he iniial base-sock level. Hence, we can reduce he dynamic program o characerize he opimal policy o one wih a single-dimensional sae-space (he nework size). Nework exernaliies give rise o he radeoff beween generaing curren profis and inducing fuure demands, hus having several imporan implicaions upon he firm s operaions decisions. Compared wih he benchmark case wihou nework exernaliies, he firm under nework exernaliies ses a higher base-sock level, and charges a lower [higher] sales price when he nework size is small [large]. When he marke is saionary, he firm adops he inroducory price sraegy, i.e., i charges a lower price a he beginning of he sales season o induce higher fuure demands. The price discriminaion and nework expanding promoion sraegies can effecively leverage nework exernaliies and improve he firm s profi. Boh sraegies faciliae he firm o (parially) separae generaing curren profis and inducing fuure demands hrough nework exernaliies. Finally, we perform exensive numerical sudies o demonsrae he significan profi loss of ignoring nework exernaliies. We also propose near-opimal heurisic policies ha leverage nework exernaliies by balancing generaing curren profis and inducing demands in he near fuure. Nework exernaliies (he dynamic compeiion seing, Chaper 3). We sudy a dynamic compeiion model, in which reail firms periodically compee on promoional effor, sales price, and service level over a finie planning horizon. The key feaure of our model is ha he curren decisions influence he fuure marke sizes hrough he service effec and he nework effec, i.e., he firm wih a higher curren service level and a higher curren demand is more likely o have larger fuure marke sizes and vice versa. Hence, he compeing firms face he radeoff beween generaing curren profis and inducing fuure demands (i.e., he exploiaion-inducion radeoff). Using he linear separabiliy approach, we characerize he pure sraegy Markov perfec equilibrium in boh he simulaneous compeiion and he promoion-firs compeiion. The exploiaion- 3
inducion radeoff has several imporan managerial implicaions under boh compeiions. Firs, o balance he exploiaion-inducion radeoff, he compeing firms should increase promoional effors, offer price discouns, and improve service levels under he service effec and he nework effec. Second, he exploiaion-inducion radeoff is more inensive a an earlier sage of he sales season han a laer sages, so he equilibrium sales prices are increasing, whereas he equilibrium promoional effors and service levels are decreasing, over he planning horizon. Third, he compeing firms need o balance he exploiaion-inducion radeoff iner-emporally under he simulaneous compeiion, whereas hey need o balance his radeoff boh iner-emporally and inra-emporally under he promoion-firs compeiion. Finally, we show ha, in he dynamic game wih marke size dynamics, he exploiaion-inducion radeoff could be a new driving force for he fa-ca effec (i.e., he equilibrium promoional effors are higher under he promoion-firs compeiion han hose under he simulaneous compeiion). Trade-in remanufacuring (Chaper 4). We invesigae he impac of sraegic cusomer behavior on he economic and environmenal values of he rade-in remanufacuring pracice. There are several major findings. Firs, under rade-in remanufacuring, a firm may earn a higher profi wih sraegic cusomers han wih myopic cusomers, which differs from he common belief ha firms dislike forward-looking cusomer behavior due o is derimenal effec on profi. This is because sraegic cusomers can anicipae he fuure price discoun brough by he rade-in opion, so when he revenuegeneraing effec of remanufacuring is srong enough, hey migh be willing o pay a higher firs-period price han he myopic cusomers. Second, we show ha sraegic cusomer behavior may creae a ension beween profiabiliy and susainabiliy: On one hand, by exploiing he forward-looking cusomer behavior, rade-in remanufacuring is more valuable o he firm wih sraegic cusomers han wih myopic cusomers; on he oher hand, wih sraegic cusomers, rade-in remanufacuring may have a negaive impac on he environmen and also on social welfare, since i may give rise o a significanly higher producion quaniy wihou improving cusomer surplus. Therefore, our research demonsraes ha i is imporan o undersand he ineracion beween rade-in remanufacuring and sraegic cusomer behavior. Finally, o resolve he above ension, we sudy how a social planner (e.g., he governmen) should design a public policy o maximize social welfare. I has been shown ha subsidizing remanufacured producs alone 4
may lead o undesired oucomes; however, he social opimum can be achieved by using a simple linear subsidy and ax scheme for all produc versions. Scarciy effec of invenory (Chaper 5). We analyze a finie horizon periodic review join pricing and invenory managemen model for a firm ha replenishes and sells a produc under he scarciy effec of invenory. The demand disribuion in each period depends negaively on he sales price and cusomer-accessible invenory level a he beginning of he period. The firm can wihhold or dispose of is on-hand invenory o deal wih he scarciy effec. We show ha a cusomer-accessible-invenory-dependen orderupo/dispose-down-o/display-up-o lis-price policy is opimal. Moreover, he opimal order-up-o/display-up-o and lis-price levels are decreasing in he cusomer-accessible invenory level. When he scarciy effec of invenory is sufficienly srong, he firm should display no posiive invenory and deliberaely make every cusomer wai. The analysis of wo imporan special cases wherein he firm canno wihhold (or dispose of) invenory delivers sharper insighs showing ha he invenory-dependen demand drives boh opimal prices and order-up-o levels down. In addiion, we demonsrae ha an increase in he operaional flexibiliy (e.g., a higher salvage value or he invenory wihholding opporuniy) miigaes he demand loss caused by high excess invenory and increases he opimal order-up-o levels and sales prices. We also generalize our model by incorporaing responsive invenory reallocaion afer demand realizes. Finally, we perform exensive numerical sudies o demonsrae ha boh he profi loss of ignoring he scarciy effec and he value of dynamic pricing under he scarciy effec are significan Comparaive saics analysis mehod (Chaper 6). We consider a general join pricing and invenory managemen model, in which a firm sources from muliple supply channels o serve a marke wih muliple demand segmens. Moreover, boh he marke size of each demand segmen and he reference procuremen cos of each supply channel are flucuaing over he planning horizon according o an exogenous Markov process. Comparaive saics analysis is essenial in his model, bu he commonly used implici funcion heorem (IFT) approach and monoone comparaive saics (MCS) approach are no amenable. Hence, we develop a new comparaive saics mehod for his model. We uilize he mehod o characerize he srucure of he opimal policy and he impac of marke flucuaion, demand segmenaion, and supply diversificaion upon he opimal policy in each period. The new mehod esablishes he desired comparaive saics resuls 5
by ieraively linking he comparisons beween opimizers and hose beween he parial derivaives of he objecive funcions. The mehod makes componenwise comparisons beween he opimizers wih differen parameer values, so i applies o he models where no all of he opimal decision variables are monoone in he parameer, and i is well scalable. The mehod does no require he objecive funcion o be wice coninuously differeniable or joinly supermodular. We also employ his comparaive saics mehod o sudy a join price and effor compeiion model. 1.3 Organizaion of he Disseraion The remainder of his disseraion is organized as follows. Chapers 2 and 3 examine he impac of nework exernaliies upon he pricing and invenory managemen policy in he monopoly and dynamic compeiion seings, respecively. In Chaper 4, we sudy how sraegic cusomer behavior would influence he economic and environmenal values of rade-in remanufacuring. Chaper 5 presens he analysis of he combined pricing and invenory conrol issue under he scarciy effec of invenory. Chaper 6 is devoed o he developmen of a new comparaive saics analysis mehod for a general class of join pricing and invenory managemen models. We conclude he disseraion in Chaper 7, where we also discuss poenial direcions for fuure research. All proofs are relegaed o he Appendices. For Chapers 2 o 6, he noaions wihin each chaper are self-conained, so he same noaion may have differen meanings in differen chapers. 6
2. Operaions Impac of Nework Exernaliies: he Monopoly 2.1 Inroducion Seing 1 Nework exernaliies refer o he general phenomenon ha a cusomer s uiliy of purchasing a produc is increasing in he number of oher cusomers who buy he same produc. See, e.g., [66]. Wih he fas developmen of informaion echnology, nework exernaliies have become a key driver of profiabiliy for a high-ech firm. Take Apple as an example. Around year 2000, Apple compuers were beer, by all accouns, han he PCs wih he Windows sysem. However, he vas majoriy of deskop and lapop compuers ran Windows as heir operaing sysems because of nework exernaliies (see, e.g., [107]). Due o Windows dominaing role in he operaing sysem marke, sofware developers made only one sixh as many applicaions for Macinosh as hey did for Windows by he ime of Microsof s anirus rial. This, in urn, made Apple compuers unaracive o new consumers, despie is funcional advanages (see [65]). A he era of smarphones, however, Apple becomes he winning side of he nework exernaliies game. Since he launch of App Sore in 2008, here have been more han 1.4 million mobile apps wih more han 75 billion downloads on his digial disribuion plaform. The App Sore no only generaes huge revenues (Apple akes 30% of all revenues generaed hrough apps), bu also creaes large availabiliy of apps for iphones, hus enabling Apple o exploi nework exernaliies o a large exen. As a consequence, iphones have a marke share of 47.4% among all smarphones in November 2014 (see, e.g., [101]). The example of Apple clearly demonsraes he imporance of nework exernaliies upon a firm s success in he marke. In paricular, he online mobile sofware disribuing plaform App Sore plays an imporan role in srenghening he nework exernaliies of Apple producs, and in boosing he sales of iphones. As an analogous example, Xbox Live, he online muliplayer gaming nework for Xbox game consoles, significanly inensifies he nework exernaliies of Xbox consoles. This is because he value of an 1 This chaper is based on he auhor s earlier work [190] 7
Xbox o an user increases if she has more opporuniies o play games wih her friends on Xbox Live (see, also, [127]). Thus, he size of he online gaming nework Xbox Live is crucial o Microsof s game console business, and he firm should manage he size of his nework carefully. Being aware of his, Microsof offered a discoun of $50 for Xbox One cusomers who guaraneed o sign up for Xbox Live Gold membership for a leas one year ([85]). This sraegy helps Microsof price discriminae in favor of he cusomers who would join Xbox Live. In anoher promoion, he 12-monh Xbox Live Gold membership was discouned by 33% in February 2015 o arac Xbox cusomers ino he online gaming nework ([153]). Firms like Apple and Microsof naurally face he quesion of how o opimally coordinae he price and invenory policy of heir producs (iphone and Xbox One). To address his quesion, we sudy a periodic-review single-iem dynamic pricing and invenory managemen model under nework exernaliies. The firm may launch an online service nework associaed wih he produc (e.g., App Sore and Xbox Live). Wih he recen rends of online social media, he associaed nework can also be in he form of a social communicaion nework (e.g., Facebook), where cusomers share heir purchasing and consumpion experiences of he produc. To model nework exernaliies, we assume ha a cusomer s willingness-o-pay is increasing in he size of he associaed nework. Moreover, in each period, a fracion of he cusomers who make a purchase would join he nework, whereas he res direcly leave he marke. We call he former cusomers he social cusomers, and he laer ones he individual cusomers. The firm may generae revenues from he nework via, e.g., service fees. This model enables us o characerize he opimal pricing and invenory policy of a profi-maximizing firm under nework exernaliies. Our analysis highlighs he impac of nework exernaliies upon he firm s opimal price and invenory policy, and idenifies effecive sraegies o exploi nework exernaliies. To he bes of our knowledge, we are he firs in he lieraure o sudy he dynamic pricing and invenory managemen problem under nework exernaliies. We show ha a nework-size-dependen base-sock/lis-price policy is opimal. Moreover, we make an ineresing echnical conribuion in his chaper: The invenory dynamics of he firm would no affec is opimal policy. As a consequence, he opimal policy can be characerized by a dynamic program wih a single-dimensional sae space (he nework 8
size). We perform a sample pah analysis of he invenory sysem and show ha, if he firm adops he opimal policy and he iniial invenory is below he iniial base-sock level, he invenory level of he firm will say below he opimal base-sock level in each period hroughou he planning horizon wih probabiliy 1. Under he base-sock/lisprice policy, invenory will no affec he opimal policy if i is below he base-sock level. Therefore, alhough he firm carries invenory, he opimal policy does no depend on he invenory dynamics once i falls below he base-sock level of any decision period. Wih a simple ransformaion o normalize he value of curren invenory, we can reduce he dynamic program ha characerizes he opimal policy o one wih a single-dimensional sae space (he nework size). This dimensionaliy reducion resul significanly simplifies he analysis, and enables us o deliver sharper insighs on he managerial implicaions of nework exernaliies. Our analysis reveals ha nework exernaliies drive he firm o balance he radeoff beween generaing curren profis and inducing fuure demands. Under nework exernaliies, since cusomers have a higher willingness-o-pay wih a larger nework size, he opimal lis-price are increasing in he curren nework size. The opimal expeced demand and base-sock level, however, may be eiher increasing or decreasing in he curren nework size. Moreover, nework exernaliies give rise o higher poenial demand, hus driving he firm o increase he base-sock level in each period. The opimal sales price, however, may be higher or lower under nework exernaliies, because he firm should decrease he sales price o induce higher fuure demands when he nework size is small, and increase he sales price o exploi he beer marke condiion when he nework size is big. From he ineremporal perspecive, he firm should pu more weigh on inducing fuure demands a he early sage of a sales season. Thus, when he marke is saionary, he firm charges lower prices a he beginning of he planning horizon. Hence, he widely-adoped inroducory price sraegy (offering price discouns when saring he sales season of a produc) may sem from nework exernaliies. We demonsrae he effeciveness of wo commonly adoped sraegies in he presence of nework exernaliies: (a) price discriminaion and (b) nework expanding promoion. The key uniform idea of boh sraegies is ha, he firm employs an addiional leverage (price or promoion) o (parially) separae generaing curren profis and inducing fuure demands hrough nework exernaliies. Under he price discriminaion sraegy, he firm 9
ailors (poenially) differen prices o differen cusomer segmens based on heir social influences. The prices for boh he social and individual cusomers help generae curren profis, bu he price for he social cusomers has he addiional role of inducing fuure demands via nework exernaliies. Therefore, i is opimal for he firm o offer discouns o social cusomers o induce fuure demands, and compensae for he reduced margin in he social segmen wih an increased margin in he individual segmen. Our model validaes he use of (cosly) nework expanding promoions (e.g., offering discouns for he service fee of he associaed nework or invesing in social media markeing sraegies). When nework exernaliies are sufficienly srong or he marginal profi of he associaed nework is sufficienly high, i is opimal for he firm o offer nework expanding promoion, regardless of is invenory level. The opimal sales price in each period is higher wih nework expanding promoion han wihou. In oher words, he firm employs nework expanding promoions o induce fuure demands via nework exernaliies, while charging a premium produc price o generae higher curren profis from selling he produc. We perform exensive numerical sudies o demonsrae ha (a) he profi loss of ignoring nework exernaliies is significan, and (b) some easy-o-implemen heurisic policies can effecively exploi nework exernaliies and achieve low opimaliy gaps. Our numerical resuls show ha ignoring he demand-inducion effec of nework exernaliies leads o a significan profi loss, especially when he nework exernaliies inensiy, he social cusomer proporion, or he nework size carry-hrough rae is high. In his case, he firm faces a srong radeoff beween generaing curren profis and inducing fuure demands, so ignoring nework exernaliies yields a misleading myopic policy. On he oher hand, he heurisic policies ha dynamically maximize he profi in a moving ime window of no more han 5 periods enable he firm o leverage nework exernaliies o a large exen, and achieve low profi losses relaive o he opimal policy. Alhough compleely ignoring nework exernaliies gives rise o significan profi losses, he firm can effecively exploi nework exernaliies by balancing he curren profis and he near fuure demands. The res of his chaper is organized as follows. In Secion 2.2, we posiion his chaper in he relaed lieraure. Secion 2.3 presens he basic formulaion, noaions and assumpions of our model. Secion 2.4 analyzes he base model. We discuss how 10
price discriminaion and nework expanding promoion sraegies help exploi nework exernaliies in Secion 2.5. The numerical sudies are repored in Secion 2.6. In Secion 2.7, we conclude his chaper by summarizing our main findings. All proofs are relegaed o Appendix A.1. 2.2 Relaed Research This chaper is buil upon wo sreams of research in he lieraure: (a) nework exernaliies and (b) join pricing and invenory managemen. Nework exernaliies have been exensively sudied in he economics lieraure. In heir seminal papers, [102, 103] characerize he impac of nework exernaliies upon marke compeiion, produc compaibiliy, and echnology adopion. [62, 67] sudy he nework exernaliy in financial markes. Several papers also sudy dynamic pricing under nework exernaliies. For example, [61] characerize he opimal nonlinear pricing sraegy for a nework produc wih heerogenous cusomers. [19] consider he opimal dynamic monopoly pricing under nework exernaliies and show ha he equilibrium prices increase as ime passes. [38] show ha, for a monopolis, he inroducory price sraegy is opimal under demand informaion incompleion or asymmery. [36] sudy he opimal pricing sraegy in a nework wih given nework srucure, and characerize he relaionship beween opimal prices and consumers cenraliy. Recenly, he operaions managemen (OM) lieraure sars o ake ino accoun he impac of nework exernaliies upon a firm s operaions sraegy. For example, [185] propose and analyze he consumer choice models ha endogenize nework exernaliies. The lieraure on he join pricing and invenory managemen problem under sochasic demand is rich. [137] give a comprehensive review on he single period join pricing and invenory conrol problem, and exend he resuls in he newsvendor problem wih pricing. [70] show ha a lis-price/order-up-o policy is opimal for a general periodicreview join pricing and invenory managemen model. When he demand disribuion is unknown, [138] address he join pricing and invenory managemen problem under demand learning. [47, 48, 49] analyze he join pricing and invenory conrol problem wih fixed ordering cos. They show ha (s, S, p) policy is opimal for finie horizon, infinie horizon and coninuous review models. [52] and [96], among ohers, sudy he join pricing and invenory conrol problem under los sales. In he case of a single unreliable 11
supplier wih random yield, [112] show ha supply uncerainy drives he firm o charge a higher price. [88] and [43] characerize he join dynamic pricing and dual-sourcing policy of an invenory sysem facing he random yield risk and he disrupion risk, respecively. When he replenishmen leadime is posiive, he join pricing and invenory conrol problem under periodic review is exremely difficul. For his problem, [136] parially characerize he srucure of he opimal policy, whereas [26] develop a simple heurisic ha resolves he compuaional complexiy. [46] characerize he opimal join pricing and invenory conrol policy wih posiive procuremen leadime and perishable invenory. When he firm adops supply diversificaion o complemen is pricing sraegy, [195] characerize he opimal dynamic pricing/dual-sourcing sraegy, whereas [173] demonsrae how a firm should coordinae is pricing and sourcing sraegies o address procuremen cos flucuaion. We refer ineresed readers o [50] for a comprehensive survey on join pricing and invenory conrol models. This chaper conribues o he above wo sreams of research by incorporaing nework exernaliies ino he sandard join pricing and invenory managemen model, sudying he impac of nework exernaliies upon a firm s pricing and invenory policy, and idenifying effecive sraegies and heurisics o exploi nework exernaliies. Finally, from he modeling perspecive, his chaper is relaed o he lieraure on invenory sysems wih posiive ineremporal demand correlaions (see, e.g., [100, 89, 16]). The key difference beween our work and his line of research is ha we endogenize he pricing decision in ou model and, hus, he firm can parially conrol he demand process via nework exernaliies. As a consequence, our focus is on he radeoff beween generaing curren profis and inducing fuure demands, whereas ha lieraure focuses on he demand learning and invenory conrol issues wih ineremporally correlaed demands. The new perspecive and focus of our work enable us o deliver new insighs on he managerial implicaions of nework exernaliies o he lieraure on invenory managemen wih ineremporal demand correlaions. 2.3 Model Formulaion Consider a periodic-review backlog join pricing and invenory managemen model of a firm who sells a nework produc (e.g., a smarphone or a video game console) over a T -period planning horizon, labeled backwards as {T, T 1,, 1}. We assume ha 12
here is an online service nework associaed wih he produc (e.g., he App Sore or he Xbox Live) or an online social communicaion nework (e.g. Facebook), so ha (par of) he cusomers who purchase he produc can join he nework and exhibi nework exernaliies ono poenial cusomers in he fuure. More specifically, in each period, a coninuum of infiniesimal cusomers arrive a he marke. Each cusomer requess a mos one produc. Following [102], we assume ha he willingness-o-pay of a new cusomer in period is given by V + γ(n ), where V is he cusomer ype uniformly disribued on he inerval (, V ] wih densiy 1, and γ( ) is a nonnegaive, concavely increasing, and wice coninuously differeniable funcion of he nework size a he beginning of period, N. Hence, V is he ype-v cusomer s inrinsic valuaion of he produc ha is independen of nework exernaliies, whereas γ( ) capures he nework exernaliies of he produc, i.e., he larger he associaed nework, he greaer uiliies cusomers gain o purchase he produc. We call γ( ) he nework exernaliies funcion hereafer. For echnical racabiliy, we assume ha he cusomers are bounded raional so ha hey base heir purchasing decisions on he curren sales price and nework size, insead of raional expecaions on fuure prices and nework sizes. Therefore, a ype- V cusomer would make a purchase in period if and only if V + γ(n ) p, where p [p, p] is he produc price he firm charges in period. In each period, here exiss a random addiive demand shock, ξ, which capures oher uncerainies no explicily modeled (e.g., he macro-economic condiion of period ). Hence, he acual demand in period is given by: D (p, N ) := V + γ(n ) p + ξ, where ξ is independen of he price p and he nework size N wih E[ξ ] = 0. Moreover, {ξ : = T, T 1,, 1} are i.i.d. coninuously disribued random variables. Wihou loss of generaliy, we assume ha D (p, N ) 0 wih probabiliy 1, for all p [p, p] and N 0. We now inroduce he dynamics of he nework sizes {N : = T, T 1, 1}. Given he curren nework size N, he nework size of he nex period, N 1, is deermined by wo effecs. Firs, some cusomers may leave he nework. For example, a game player may lose is enhusiasm in online gaming hree years afer purchasing he Xbox console. Analogously, an iphone user may swich o Samsung for her nex smarphone. Thus, given N, le ηn be he remaining number of cusomers saying in he nework in period 13
1, where η [0, 1] is he carry-hrough rae of he nework size. Second, a fracion of new cusomers who purchase he produc in period would join he nework. No all new cusomers will join he nework and exhibi posiive exernaliies ono poenial cusomers in he fuure, because, e.g., some Xbox players only play he games off-line and, hus, are no par of he Xbox Live nework. Clearly, hese players exer few, if any, nework exernaliies ono oher cusomers. For any given (p, N ), le θd (p, N ) be he number of new cusomers who op o join he nework associaed wih he produc, which we call he social cusomers hereafer, where θ (0, 1] is he proporion of such cusomers in he marke. The oher (1 θ)d (p, N ) cusomers who exer no nework exernaliies are called individual cusomers hereafer. Alhough we implicily assume ha he uiliy funcions of he social and individual cusomers are idenical, mos of he resuls in his chaper (excep Theorem 2.5.1) coninue o hold if V and γ( ) are differen for he social and individual cusomers. To capure he marke size dynamics, we noice ha, due o demand uncerainy and limied invenory availabiliy, no all cusomers reques a produc can ge one in he curren period. We assume ha he social cusomers who purchase bu no ge he produc sill join he nework. I is commonly observed in pracice ha cusomers exer nework exernaliies upon fuure poenial buyers before receiving he produc. For example, before obaining he pre-ordered produc, a cusomer may commen on her exciemen in waiing for and expecing he produc on Facebook, hus exering nework exernaliies upon poenial buyers. Moreover, by Theorem 2.4.1(c) below, if he firm adops he opimal pricing and invenory policy, all backlogged demand will be fulfilled in he nex period, so he backlogged social cusomers will ge he produc and join he nework shorly. For simpliciy, we ignore he differences in he iming of joining he nework beween he cusomers who ge he produc upon reques and hose who are backlogged o he nex period. Therefore, given N, he nework size a he beginning of period 1 is given by: N 1 = ηn + θd (p, N ) + ϵ, (2.1) where ϵ is he addiive random shock in he nework size dynamics no explicily capured in our model. We assume ha ϵ is independen of he price p and he nework size N wih E[ϵ ] = 0. Moreover, {ϵ : = T, T 1,, 1} are i.i.d. coninuously disribued random variables. 14
If he associaed nework is a service nework, he firm can generae profis via his nework by charging service/subscripion fees. For example, Microsof charges an annual subscripion fee of $59.99 for he Xbox Live Gold membership, whereas Apple akes 30% of all revenues generaed hrough apps in he App Sore. For any nework size N 0, le r n (N) 0 denoe he per-period profi he firm earns from he nework. Wihou loss of generaliy, we assume ha r n ( ) is a concavely increasing and coninuously differeniable funcion wih r n (0) = 0. To focus on he firm s pricing and invenory policy of is produc, we do no explicily model he firm s price decision of is nework service. Hence, he per-period profi funcion of he nework, r n ( ), is assumed o be exogenously given. Wihou loss of generaliy, we assume ha he service fees are paid a he end of each period. Hence, he oal expeced profi he firm obains from he associaed nework in period is given by: E[r n (ηn + θd (p, N ) + ϵ )]. If he associaed nework is a social communicaion nework where he social cusomers share heir purchasing and consumpion experiences, he firm obains no profi from his nework, i.e., r n ( ) 0. The sae of he invenory sysem is given by (I, N ) R R +, where I =he saring invenory level before replenishmen in period, = T, T 1,, 1; N =he saring nework size of he produc in period, = T, T 1,, 1. The decisions of he firm is given by (x, p ) F(I ) := [I, + ) [p, p], where x =he invenory level afer replenishmen in period, = T, T 1,, 1; p =he sales price charged in period, = T, T 1,, 1. In each period, he sequence of evens unfolds as follows: A he beginning of period, afer observing he invenory level I and he nework size N, he firm simulaneously chooses he invenory socking level x I and he sales price p, and pays he ordering cos c(x I ). The invenory procuremen leadime is assumed o be zero, so ha he replenished invenory is received immediaely. The demand D (p, N ) hen realizes. The revenue from selling he produc, p E[D (p, N )], and he profi from he associaed nework, E[r n (ηn + θd (p, N ) + ϵ )], are colleced. Unme demand is fully backlogged. A he end of period, he holding and backlogging coss are paid, he ne invenory is carried over o he nex period, and he nework size is updaed according o he nework size dynamics (2.1). 15
We inroduce he following model primiives: α = discoun facor of revenues and coss in fuure periods, 0 < α 1; c = invenory purchasing cos per uni ordered; b = backlogging cos per uni backlogged a he end of a period; h = holding cos per uni socked a he end of a period. Wihou loss of generaliy, we make he following assumpions on he model primiives: b > (1 α)c : he backlogging penaly is higher han he saving from delaying an order o he nex period, so ha he firm will no backlog all of is demand; p > b + αc : posiive margin for backlogged demand. The above assumpions are common in he join pricing and invenory managemen lieraure (see, e.g., [189]). For echnical racabiliy, we make he following assumpion hroughou our analysis. Assumpion 2.3.1 For each period, R (, ) is joinly concave in (p, N ) [p, p] [0, + ), where R (p, N ) := (p b αc)( V p + γ(n )). (2.2) Given he sales price, p, and he nework size, N, of period, R (p, N ) is he expeced difference beween he revenue and he oal cos, which consiss of ordering and backlogging coss, o saisfy he curren demand in he nex period. Hence, he join concaviy of R (, ) implies ha such difference has decreasing marginal values wih respec o he curren sales price and nework size. We remark ha R (, N ) is sricly concave in p for any given N. Moreover, he monooniciy of γ( ) suggess ha R (, ) is supermodular in (p, N ). The following lemma gives he necessary and sufficien condiion for Assumpion 2.3.1. Lemma 1 Assumpion 2.3.1 holds for period, if and only if, for all N 0, 2(p αc b)γ (N ) (γ (N )) 2. (2.3) 16
Based on Lemma 1, we give more specific condiions on he nework exernaliies funcion γ( ) for Assumpion 2.3.1 o hold in Appendix A.2. In a nushell, Assumpion 2.3.1 holds when (a) he curvaure of he nework exernaliies funcion γ( ) is no oo small in he region nework exernaliies exis (i.e., γ ( ) > 0), and (b) he price elasiciy of demand (i.e., ( de[d (p, N )]/E[D (p, N )])/( dp /p ) ) is sufficienly big relaive o he nework size elasiciy of demand (i.e., ( de[d (p, N )]/E[D (p, N )])/( dn /N ) ). 2.4 Analysis of he Base Model In his secion, we analyze he base model suiable for he usual sales season of he nework produc, when he firm charges a single regular price for all cusomers wihou any promoional campaigns. In Secion 2.5, we inroduce price discriminaion and nework expanding promoion sraegies, and analyze heir effeciveness in leveraging nework exernaliies. We firs characerize he srucure of he opimal pricing and invenory policy in our model. Then, we show ha he sae space dimension of he dynamic program for he join pricing and invenory replenishmen problem can be reduced o 1. Finally, we sudy he managerial implicaions of nework exernaliies. 2.4.1 Opimal Policy We now formulae he planning problem as a dynamic program. Define v (I, N ) := he maximum expeced discouned profis in periods, 1,, 1, when saring period wih an invenory level I and nework size N. Wihou loss of generaliy, we assume ha, in he las period (period 1), he excess invenory is salvaged wih uni value c, and he backlogged demand is filled wih ordering cos c, i.e., v 0 (I 0, N 0 ) = ci 0 for any (I 0, N 0 ). The opimal value funcion v (I, N ) saisfies he following recursive scheme: v (I, N ) = ci + max J (x, p, N ), (2.4) (x,p ) F(I ) 17