WASHINGTON UNIVERSITY IN ST. LOUIS. Olin Business School

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1 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

2 2016, Renyu Zhang

3 Table of Conens Page Lis of Figures v Lis of Tables Acknowledgmens ABSTRACT OF THE DISSERTATION vi vii x 1 Inroducion Moivaion Conribuion Organizaion of he Disseraion Operaions Impac of Nework Exernaliies: he Monopoly Seing Inroducion Relaed Research Model Formulaion Analysis of he Base Model Opimal Policy Sae Space Dimension Reducion Managerial Implicaions of Nework Exernaliies Effecive Sraegies o Exploi Nework Exernaliies Price Discriminaion Nework Expanding Promoion Numerical Sudies Impac of Nework Exernaliies Effecive Heurisic Policies under Nework Exernaliies Summary Operaions Impac of Nework Exernaliies: Dynamic Compeiion Seing Inroducion Relaed Research Model Simulaneous Compeiion Equilibrium Analysis Exploiaion-Inducion Tradeoff Promoion-Firs Compeiion ii

4 Page Equilibrium Analysis Exploiaion-Inducion Tradeoff Comparison of he Two Compeiion Models Summary Trade-in Remanufacuring, Sraegic Cusomer Behavior and Governmen Subsidies Inroducion Relaed Research Model and Equilibrium Analysis Model Seup Equilibrium Analysis Impac of Trade-in Remanufacuring Impac on Firm Profi Impac on Environmen and Cusomer Surplus Social Opimum and Governmen Inervenion Summary Pricing and Invenory Managemen under he Scarciy Effec of Invenory Inroducion Relaed Research Model Formulaion Discussions on Assumpion Unified Model Addiional Resuls in Two Special Cases Wihou Invenory Wihholding Wihou Invenory Disposal Responsive Invenory Reallocaion Numerical Sudies Opimal Policy Srucure wih Non-concave R(, ) Funcions Impac of Scarciy Effec Value of Dynamic Pricing Summary and Exension Comparaive Saics Analysis Mehod for Join Pricing and Invenory Managemen Models Inroducion Relaed Research A New Comparaive Saics Mehod An Illusraive Example Proof of Lemma 15 wih Our New Mehod iii

5 Page 6.4 Applicaion of he New Comparaive Saics Mehod in a General Join Pricing and Invenory Managemen Model Model Comparaive Saics Analysis wih Our New Mehod Applicaion of he New Comparaive Saics Mehod in a Compeiion Model Effor-Level-Firs Compeiion Simulaneous Compeiion A Comparison of Equilibria in he Two Compeiion Models Summary Concluding Remarks References A Appendix for Chaper A.1 Proofs of Saemens A.2 More Condiions on Assumpion B Appendix for Chaper B.1 Proofs of Saemens 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 C.1 Equilibrium Definiions C.2 Proofs of Saemens D Appendix for Chaper D.1 Proofs of Saemens E Appendix for Chaper E.1 Proofs of Saemens E.2 Discussions on Sopping Condiion (ii) of he Ieraive Procedure E.2.1 Comparaive Saics Analysis of y i (γ) (p + 1 i p + q) iv

6 Lis of Figures Figure Page 2.1 Value of λ m : θ = 0.5, η = Value of λ m : k = 0.5, η = Value of λ m : k = 0.5, θ = Value of λ m and λ i h : θ = 0.5, η = Value of λ m and λ i h : k = 0.5, η = Value of λ m and λ i h : k = 0.5 θ = Opimal Ordering-up-o Level Opimal Price-induced Demand Value of λ scarciy : = Value of λ scarciy : = Value of λ pricing : = Value of λ pricing : = v

7 Lis of Tables Table Page 4.1 Summary Saisics: Firm Profi (%) Summary Saisics: Environmenal Impac (%) vi

8 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

9 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

10 Dedicaed o Jesus Chris and my family. ix

11 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

12 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

13 (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

14 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

15 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

16 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

17 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

18 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

19 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

20 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

21 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

22 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

23 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

24 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 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

25 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

26 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

27 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

28 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

29 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 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 Lemma 1 Assumpion holds for period, if and only if, for all N 0, 2(p αc b)γ (N ) (γ (N )) 2. (2.3) 16

30 Based on Lemma 1, we give more specific condiions on he nework exernaliies funcion γ( ) for Assumpion o hold in Appendix A.2. In a nushell, Assumpion 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 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

31 where F(I ) := [I, + ) [p, p] denoes he se of feasible decisions and, J (x, p, N ) = ci + E{p D (p, N ) c(x I ) h(x D (p, N )) + b(x D (p, N )) + r n (θd (p, N ) + ηn + ϵ ) +αv 1 (x D (p, N ), θd (p, N ) + ηn + ϵ ) N }, = (p αc b)( V p + γ(n )) + (b (1 α)c)x +E{r n (θ( V p + γ(n ) + ξ ) + ηn + ϵ ) (h + b)(x V + p γ(n ) ξ ) + +α[v 1 (x V + p γ(n ) ξ, θ( V p + γ(n ) + ξ ) + ηn + ϵ ) c(x V + p γ(n ) ξ )] N } = R (p, N ) + βx + Λ(x V + p γ(n )) +Ψ (x V + p γ(n ), θ( V p + γ(n )) + ηn ), (2.5) wih Ψ (x, y) := E{r n (y + θξ + ϵ ) + α[v 1 (x ξ, y + θξ + ϵ ) cx]}, Λ(x) := E{ (b + h)(x ξ ) + }, β := b (1 α)c = he moneary benefi of ordering one uni of invenory. Hence, for each period, he firm selecs (x (I, N ), p (I, N )) := argmax (x,p ) F(I )J (x, p, N ) (2.6) as he opimal price and invenory policy coningen on he sae variable (I, N ). We begin our analysis by characerizing he preliminary concaviy and differeniabiliy properies of he value and objecive funcions in he following lemma. Lemma 2 For each = T, T 1,, 1, he following saemens hold: (a) Ψ (, ) is joinly concave and coninuously differeniable in (x, y). Moreover, Ψ (x, y) is decreasing in x and increasing in y. (b) J (,, ) is joinly concave and coninuously differeniable in (x, p, N ). (c) v (, ) is joinly concave and coninuously differeniable in (I, N ). Moreover, v (I, N ) is increasing in N, and v (I, N ) ci is decreasing in I. Lemma 2 proves ha, in each period, he objecive funcion is concave and coninuously differeniable, and he value funcion is joinly concave and coninuously differeniable. Moreover, he normalized value funcion v (I, N ) ci is decreasing in he 18

32 invenory level I and increasing in he nework size N. Lemma 2 is a sandard resul in he join pricing and invenory managemen lieraure (see, e.g., Theorem 1 in [70]). The concaviy and coninuous differeniabiliy of J (,, ) ensure ha, he opimal price and invenory policy, (x (I, N ), p (I, N )), is well-defined and can be obained via firs-order condiions. Moreover, we can define he invenory-independen opimizer (x (N ), p (N )) as follows: (x (N ), p (N )) := argmax x R,p [p, p]j (x, p, N ). (2.7) In case of muliple opimizers, we selec he lexicographically smalles one. We define y (N ) := V p (N ) + γ(n ) as he opimal invenory-independen expeced demand of period. Wih Lemma 2, we characerize he opimal pricing and invenory policy in he following heorem. Theorem For any, he following saemens hold: (a) If I x (N ), (x (I, N ), p (I, N )) = (x (N ), p (N )). (b) If I > x (N ), x (I, N ) = I and p (I, N ) = argmax p [p, p]j (I, p, N ). (c) For any I R and N 0, x (I, N ) > 0. Theorem shows ha he opimal policy in he base model is a nework-sizedependen base-sock/lis-price policy. If he saring invenory level I is below he nework-size-dependen base-sock level x (N ), i is opimal o order up o his base-sock level, and charge he nework-size-dependen lis-price p (N ). If he saring invenory level is above he nework-size-dependen base-sock level, i is opimal no o order anyhing, and charge an invenory-dependen sales price p (I, N ). Moreover, as shown in Theorem 2.4.1(c), he opimal order-up-o level x (I, N ) is always posiive for any invenory level I and nework size N. This implies ha, under he opimal policy, all backlogged demand in any period will be saisfied in he nex period (i.e., period 1) Sae Space Dimension Reducion The original dynamic program o characerize he opimal pricing and invenory policy (2.4) has a sae space of wo dimensions (invenory level I and nework size N ). Hence, i is difficul o work wih (2.4) direcly. In his subsecion, we demonsrae ha he 19

33 dynamic program (2.4) can be reduced o a much simpler one wih a single-dimensional sae space (nework size N ). Moreover, wih probabiliy 1, he opimal policy in each period, (x (I, N ), p (I, N )), is independen of he dynamics of invenory level {I s : s = T, T 1,, }, as long as he iniial invenory level I T is below he opimal period-t base-sock level x T (N T ). The sae space dimension reducion, as we will show in Secion and Secion 2.5, enables us o deliver sharper insighs on he managerial implicaions of nework exernaliies and he effecive sraegies o exploi nework exernaliies. To begin wih, we employ he sample pah analysis approach o characerize he behavior of he invenory level dynamics under he opimal pricing and invenory policy. Lemma 3 For each period, he following saemens hold: (a) For all nework sizes N and N 1, we have P[x (N ) D (p (N ), N ) x 1 (N 1 )] = 1. (b) For all N 0, x (N ) = + y (N ), where := argmax {β + Λ( )}. Lemma 3(a) shows ha, if he firm adops he opimal policy and he saring invenory level in period, I, is below he period- base-sock level x (N ), he saring invenory level in period 1, I 1 = x (N ) D (p (N ), N ), is below he period-( 1) base-sock level, x 1 (N 1 ), wih probabiliy 1. Lemma 3(a) also implies ha once he saring invenory level falls below he opimal base-sock level of one period, he firm should replenish in each period hereafer hroughou he planning horizon wih probabiliy 1. Since our model bes fis he nework produc ha is eiher a new produc (e.g., he firs-generaion iphone) or a new generaion of an exising produc (e.g., Xbox One), zero invenory is socked a he beginning of he sales season, i.e., I T = 0. Therefore, Theorem 2.4.1(c) and Lemma 3(a) imply ha I x (N ) wih probabiliy 1 for each period. As a corollary of Lemma 3(a), Lemma 3(b) shows ha, if he saring invenory is below he opimal base-sock level (i.e., I x (N )), he opimal safey-sock is invarian wih respec o he period and he nework size N, and can be obained by solving a one-dimensional convex opimizaion. Based on Lemma 3, we now show ha he bivariae value funcions of he dynamic program (2.4), {v (, ) : = T, T 1,, 1}, can be ransformed ino a univariae funcion π ( ) of he curren nework size N by normalizing he value of he saring invenory ci. 20

34 Moreover, he normalized value funcion π ( ) is concavely increasing and coninuously differeniable in N. Lemma 4 There exiss a sequence of funcions {π ( ) : 1 T }, such ha, (i) π (N ) = max{j (x, p, N ) : x 0,, p [p, p]} for all N 0; (ii) for each, π ( ) is concavely increasing and coninuously differeniable in N ; (iii) v (I, N ) = ci + π (N ) for all N 0 and I x (N ); (iv) for all N 0, J (x, p, N ) = R (p, N )+βx +Λ(x V +p γ(n ))+G (θ( V p +γ(n ))+ηn ), (2.8) where G (y) := E[r n (y + θξ + ϵ ) + απ 1 (y + θξ + ϵ )] and x V + p γ(n ) ; and (v) (x (N ), p (N )) maximizes he righ-hand side of equaion (2.8). Lemma 4 enables us o reduce he original dynamic program (2.4), which has a wodimension sae-space, o one wih a single-dimension sae space. More specifically, Lemma 4 implies ha he opimal nework-size-dependen base-sock level and lis-price in each period, (x (N ), p (N )), can be recursively deermined by solving he following dynamic program wih a single dimensional sae-space of nework size N : π (N ) = max J (x, p, N ), (2.9) x 0,p [p, p] where J (x, p, N ) = R (p, N ) + βx + Λ(x V + p γ(n )) +G (θ( V p + γ(n )) + ηn ), wih G (y) := E{r n (y + θξ + ϵ ) + απ 1 (y + θξ + ϵ )}, and π 0 ( ) 0, Summarizing Theorem 2.4.1, Lemma 3, and Lemma 4, we have he following sharper characerizaion of he opimal policy in each period. Theorem Assume ha I T x T (N T ). In each period and for each I and N, (x (I, N ), p (I, N )) = (x (N ), p (N )) wih probabiliy 1. Moreover, {(x (N ), p (N )) : = T, T 1,, 1} is he soluion o he Bellman equaion (2.9). Theorem shows ha, as long as he planning horizon sars wih an invenory level below he opimal period-t base-sock level (i.e., I T x T (N T )), he opimal pricing and invenory policy in each period, (x (I, N ), p (I, N )), is idenical o he opimal base-sock level and lis-price, (x (N ), p (N )), wih probabiliy 1. Alhough he firm holds invenory hroughou he sales horizon, he opimal policy is independen of 21

35 he invenory dynamics if he iniial invenory level I T is sufficienly low. As discussed above, in mos applicaions, he firm holds zero iniial invenory a he beginning of he sales season, i.e., I T = 0. Hence, (x (I, N ), p (I, N )) = (x (N ), p (N )) for all (I, N ) wih probabiliy 1. Therefore, we will focus on analyzing he properies of he opimal invenory-independen base-sock level and lis-price (x (N ), p (N )) for he res of his secion Managerial Implicaions of Nework Exernaliies This subsecion sudies he impac of nework exernaliies upon he firm s opimal price and invenory decisions in each period. Specifically, we srive o answer he following quesions: (a) How should he firm adjus is price and invenory policy in response o he nework size evoluion? (b) How do nework exernaliies direcly impac he opimal policy of he firm? (c) How should he firm adjus is price and invenory policy ineremporally hroughou he sales season? And (d) how o balance earning profis direcly from selling he produc and from he service fees of he nework? The answers o hese quesions shed lighs on he managerial implicaions of nework exernaliies. To begin wih, we characerize he impac of nework size upon he firm s opimal pricing and invenory policy in he following heorem. Theorem For each period, assume ha ˆN > N. We have: (a) p ( ˆN ) p (N ); (b) if I x (N ), E[N 1 ˆN ] E[N 1 N ]; (c) if γ( ˆN ) = γ(n ), hen y ( ˆN ) y (N ) and x ( ˆN ) x (N ); and (d) if η = 0, hen y ( ˆN ) y (N ) and x ( ˆN ) x (N ). Theorem characerizes how he curren nework size influences he opimal join pricing and invenory policy, he opimal expeced curren-period demand, and he opimal expeced nex-period nework size. More specifically, we show ha he opimal lis-price, p (N ), and he opimal expeced nework size in he nex period, E[N 1 N ] = θy (N ) + ηn, are increasing in he curren nework size N. The opimal expeced demand y (N ), and he opimal base-sock level x (N ), however, may no necessarily be increasing or decreasing in N (see Theorem 2.4.3(c,d)). Under nework exernaliies, a larger curren nework size N gives rise o a higher poenial demand, so he firm charges a higher price o exploi he beer marke condiion. Hence, wih 22

36 a larger curren nework size, he combinaion of a beer marke condiion and an increased sales price may drive he resuling opimal expeced demand and he opimal base-sock level eiher higher or lower. In he join pricing and invenory managemen model wihou nework exernaliies (e.g., [70]), he opimal policy in each period is independen of eiher pas demands or pas decisions. Since he curren nework size N is posiively correlaed wih pas demands, Theorem implies ha nework exernaliies creae ineremporal correlaions beween demands and opimal decisions hroughou he planning horizon. Hence, he firm can employ he curren price and invenory decisions o conrol fuure demands. Therefore, he firm needs o dynamically balance he radeoff beween generaing curren profis and inducing fuure demands hrough nework exernaliies. Theorems are silen on he properies of he opimal policy when he saring invenory exceeds he opimal base-sock level (i.e., I > x (N )). Though his scenario occurs wih probabiliy 0 as long as I T x T (N T ) (see Theorem 2.4.2), we give he following heorem ha characerizes he srucure of he opimal policy herein. Theorem Assume ha η = 0. For each period, he following saemens hold, (a) v (I, N ) is supermodular in (I, N ). (b) x (I, N ) is coninuously increasing in I and N. (c) p (I, N ) is coninuously decreasing in I, and coninuously increasing in N. (d) The opimal expeced demand y (I, N ) := V p (I, N ) + γ(n ) is coninuously increasing in I and N. Hence, E[N 1 N ] = θy (I, N ) is coninuously increasing in I and N. (e) The opimal safey-sock (I, N ) := x (I, N ) V + p (I, N ) γ(n ) is coninuously increasing in I and coninuously decreasing in N. Theorem generalizes Theorems and o he seing wih high saring invenory (i.e., I > x (N )). More specifically, Theorem 2.4.4(a) shows ha if η = 0 (i.e., all cusomers who are in he nework will leave in he nex period), he value funcion in each period, v (I, N ) is supermodular in (I, N ). This is because, a larger nework size leads o a larger poenial demand and, hus, a higher marginal value of invenory. 23

37 Analogously, he opimal expeced demand, y (I, N ), and he opimal expeced nework size in he nex period are all increasing in he nework size N for all (I, N ). As a consequence, if he nework size is larger, he firm increases he order-up-o level, x (I, N ), o mach demand wih supply, and charges a higher sales price, p (I, N ), o exploi he beer marke condiion. Since he expeced demand is higher wih a larger nework size, he opimal safey-sock (I, N ) is decreasing in N. Theorem also yields how he saring invenory level I influences he opimal policy when i is above he base-sock level. We show ha, in his case, a higher saring invenory level promps he firm o increase he safey sock and, o mach supply wih demand, charge a lower sales price. Theorem shows ha nework exernaliies impac he opimal join pricing and invenory policy of he firm hrough he curren size of he associaed nework. We proceed o direcly analyze he impac of nework exernaliies by comparing he opimal policy in an invenory sysem wih nework exernaliies wih ha in an invenory sysem wihou. Theorem Assume ha wo invenory sysems are idenical excep ha one wih nework exernaliies funcion γ( ) and he oher wih ˆγ( ), where γ(0) = ˆγ(0) = γ 0 and ˆγ(N ) γ(n ) γ 0 for all N 0, i.e., he invenory sysem wih funcion γ( ) exhibis no nework exernaliies. Moreover, le ˆr n (n) = r n (n) = rn for some consan r 0. For each period and each nework size N 0, he following saemens hold: (a) ŷ (N ) y (N ); (b) ˆx (N ) x (N ); (c) There exiss a hreshold N 0, such ha ˆp (N ) p (N ) for N N, whereas ˆp (N ) p (N ) for N N. Nework exernaliies lead o a higher poenial demand for he invenory sysem, because social cusomers in he nework can arac new poenial cusomers o buy he produc. Hence, as shown in Theorem 2.4.5(a,b), he presence of nework exernaliies gives rise o a higher expeced demand and, hus, drives he firm o increase he basesock level in each period (i.e., ŷ (N ) y (N ) and ˆx (N ) x (N )). Theorem 2.4.5(c) characerizes he impac of nework exernaliies upon he firm s pricing policy: The opimal lis-price wih nework exernaliies, ˆp (N ), may be eiher higher or lower han ha wihou, p (N ). More specifically, if he nework size is sufficienly small (i.e., below he hreshold N ), ˆp (N ) p (N ). Oherwise, he nework size is sufficienly large 24

38 (i.e., above he hreshold N ) and ˆp (N ) p (N ). Under nework exernaliies, he firm faces he radeoff beween decreasing he sales price o induce high fuure demands and increasing he sales price o exploi he beer marke condiion. When he curren nework size is small (N N ), he firm should pu higher weigh on inducing fuure demands, so he opimal price is lower wih nework exernaliies. Oherwise, N N, generaing curren profis ouweighs inducing fuure demands, and, hence, he opimal price is higher wih nework exernaliies. In shor, Theorem 2.4.5(c) reveals ha, because of he radeoff beween generaing curren profis and inducing fuure demands, nework exernaliies can have some suble implicaions on he pricing policy of he firm. We now characerize he evoluion of he opimal price and invenory decisions over he planning horizon. As shown in he following heorem, when he marke is saionary, nework exernaliies moivae he firm o se lower sales prices and higher base-sock levels a he beginning of he planning horizon. Theorem Assume ha V = V 1 for all. For each = T, T 1,, 2 and any nework size N 0, we have (a) x (N) x 1 (N), (b) y (N) y 1 (N), and (c) p (N) p 1 (N). When he willingness-o-pay of he cusomers is saionary, Theorem characerizes he evoluions of he opimal base-sock level, expeced demand, and sales price under nework exernaliies. More specifically, we show ha, wih he same nework size N (and, hus, he same poenial marke size), he opimal expeced demand, y (N ), and he opimal base-sock level, x (N ), is decreasing over he planning horizon, whereas he opimal sales price, p (N ), is increasing hroughou he planning horizon. Under nework exernaliies, he firm should pu more weigh on inducing fuure demands a he beginning of he planning horizon and urn o generaing he curren profis as he sales season approaches he end. Hence, i is opimal for he firm o offer discouns and arac more cusomers o purchase he produc and join he nework a he early sage of a sales season, and o charge a higher price o exploi he curren marke owards he end of he planning horizon. To mach demand wih supply, wih he same poenial marke size, he opimal base-sock level is decreasing over he planning horizon. Theorem is consisen wih he commonly used inroducory price sraegy under which price discouns are offered a he inroducory sage of a produc. For example, when 25

39 Microsof inroduced he 500 GB Xbox 360 ino he India video game marke, i charged a surprisingly low inroducory price of $313.9 (see, e.g., [108]). When he cusomer valuaion is no saionary (i.e., V is no equal o V 1 ), he inroducory price sraegy may no necessarily be opimal. This is because, if he cusomer valuaion is higher a he beginning of he sales season, he firm may charge a higher price o exploi he cusomer preference as opposed o offering discouns o induce fuure demands. In our join pricing and invenory managemen model wih nework exernaliies, he firm has wo sources of profis: (i) selling he produc, and (ii) he service fees colleced from he associaed nework. A naural quesion o ask is how should he firm balance hese wo profi-generaing sources? The following heorem addresses his quesion by characerizing how he marginal profi from he associaed nework influence he opimal policy. Theorem Assume ha wo invenory sysems are idenical excep ha one wih nework profi funcion ˆr n ( ), and he oher wih r n ( ), where ˆr n(n) r n(n) for all N 0. For each period and any N 0, we have: (a) ˆx (N ) x (N ), (b) ˆp (N ) p (N ), and (c) ŷ (N ) y (N ). Theorem sheds lighs on how differen firms should balance he wo profi sources. More specifically, Theorem shows ha if he associaed nework has a higher profi margin (i.e., r n( ) is larger), he nework exernaliies of he produc are sronger and, as a consequence, he firm should price down and increase he poenial demand o exploi he more inensive nework exernaliies. To mach demand wih supply, he firm also increases he base-sock level wih a higher profi margin of he associaed nework. Theorem implies ha for a produc wih high inrinsic cusomer valuaions and a low margin of he associaed nework (e.g., iphone), he firm charges a premium price for he produc; whereas if he produc has low inrinsic valuaions from he cusomers due o, e.g., fierce marke compeiion, bu he margin of he associaed nework is high (e.g., Xbox), he firm charges a price wih a low margin for he produc so as o exploi nework exernaliies. In summary, nework exernaliies have several imporan managerial implicaions upon he join pricing and invenory policy of he firm. Mos imporanly, nework exernaliies creae anoher layer of complexiy in balancing he radeoff beween generaing 26

40 curren profi and inducing fuure demands. To exploi nework exernaliies, he firm should dynamically adjus is price increasing in he curren nework size. Moreover, nework exernaliies give rise o higher expeced demand and, hence, drive he firm o increase he base-sock level in each period. Since nework exernaliies creae he ension beween generaing curren profis and inducing fuure demands, he opimal sales price wih nework exernaliies is lower han ha wihou when he nework size is small (o induce high fuure demands), and is higher han ha wihou when he nework size is big (o generae high curren profis). From he ineremporal perspecive, he firm should pu more weigh on inducing fuure demands a he early sage of a sales season han a laer sages. Thus, if he cusomer valuaion is saionary, he firm should employ he inroducory price sraegy ha offers early purchase discouns o induce high fuure demands. Finally, he firm needs o rade off beween generaing profis from he produc and from he associaed nework as well. Wih higher marginal profis of he associaed nework, he firm should decrease he sales price o exploi he more inensive nework exernaliies. 2.5 Effecive Sraegies o Exploi Nework Exernaliies In his secion, we sudy wo effecive sraegies o exploi nework exernaliies: (a) he price discriminaion sraegy and (b) he nework expanding promoion sraegy. Boh sraegies adop he uniform idea ha, he firm employs an addiional leverage (price or promoion) o separae generaing curren profis and inducing fuure demands hrough nework exernaliies Price Discriminaion In his subsecion, we sudy he price discriminaion sraegy ha is commonly used in pracice under nework exernaliies. More specifically, since only social cusomers will join he associaed nework of he produc and exer nework exernaliies over poenial buyers in he fuure, he firm can beer exploi nework exernaliies by price discriminaing differen cusomer segmens in favor of social cusomers. For example, in 2015, Microsof offered price discouns for Xbox One buyers who commi o signing up for he Xbox Live Gold membership for a leas one year (see, e.g., [85]). 27

41 In period, in sead of announcing a single price p, he firm under he price discriminaion sraegy offers a price menu o cusomers: (p s, p i ) [p, p] [p, p], where p s is he uni price of he produc wih he nework sign-up commimen, and p i is he uni price of he produc wihou any nework service subscripion commimen. If p s > p i, all cusomers will ake he price p i and he model is reduced o he based model sudied in Secion 2.4. Hence, wihou loss of generaliy, we assume ha p s p i. In his case, social cusomers will ake he price p s and, as commied, join he associaed nework, whereas individual cusomers will ake he price p i wihou joining he associaed nework. Thus, in period, he demand from he social cusomers is given by D s (p s, N ) := θ( V p s + γ(n ) + ξ ), and ha from he individual cusomers is given by D(p i i, N ) := (1 θ)( V p i + γ(n ) + ξ ). The nework size a he beginning of period 1 is, hus, given by N 1 = D s (p s, N ) + ηn + ϵ. We define v d (I, N ) := he maximum expeced discouned profis wih price discriminaion in periods,, 1, when saring period wih an invenory level I and nework size N ; and (x d (I, N ), p s (I, N ), p i (I, N )) as he opimal pricing and invenory policy. As in he base model, 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 d 0(I 0, N 0 ) = ci 0 for any (I 0, N 0 ). Employing similar dynamic programming and sample pah analysis mehods, we characerize he opimal policy in he model wih price discriminaion in he following lemma. Lemma 5 Define a sequence of funcions {π d (N ) : = T, T 1,, 1} and a sequence of pricing and invenory policies {(x d (N ), p s (N ), p i (N )) : = T, T 1,, 1} as follows: π d (N ) = max J d (x, p s, p i, N ), (2.10) (x,p s,pi ) F d where J d (x, p s, p i, N ) = θr (p s, N ) + (1 θ)r (p i, N ) +Λ(x V + θp s + (1 θ)p i γ(n )) +βx + G d (θ( V p s + γ(n )) + ηn ), wih G d (y) := E{r n (y + θξ + ϵ ) + απ d 1(y + θξ )}, π d 0( ) 0, and (x d (N ), p s (N ), p i (N )) := argmax (x,p s,pi ) F d J d (x, p s, p i, N ). 28

42 (a) π d ( ) is concave, coninuously differeniable, and increasing in N. joinly concave and coninuously differeniable in (x, p s, p i, N ). J d (,,, ) is (b) If I x d (N ), (x d (I, N ), p s (I, N ), p i (I, N )) = (x d (N ), p s (N ), p i (N )) and v d (I, N ) = ci + π d (N ); oherwise, x d (I, N ) = I. If I T x d T (N T ), (x d (I, N ), p s (I, N ), p i (I, N )) = (x d (N ), p s (N ), p i (N )) for all and (I, N ) wih probabiliy 1. Lemma 5 demonsraes ha a nework-size-dependen base-sock/lis-prices policy is opimal in he model wih price discriminaion. As in he base model, afer normalizing he value of curren invenory, he sae space dimension of he dynamic program can be reduced o 1. Moreover, wih probabiliy 1, he opimal policy is independen of he saring invenory level in each period, as long as he iniial invenory level I T he opimal period-t base-sock level x d T (N T ). is below We remark ha Theorems 2.4.3, 2.4.5, and can be generalized o he model wih price discriminaion. Hence, he impac of nework exernaliies upon he opimal pricing and invenory policy is similar in he model wih price discriminaion o ha in he base model. To characerize he impac of he price discriminaion sraegy, we direcly compare he opimal policy and profi in he model wih price discriminaion wih ha in he base model. Theorem Assume ha wo invenory sysems are idenical excep ha one wih he price discriminaion sraegy and he oher wihou. For each period, we have (a) if γ ( ) > 0 and p i (N ) > p, p s (N ) < p i (N ); (b) p (N ) p i (N ) for all N ; and (c) π d (N ) π (N ) for all N, where he inequaliy is sric if p i (N ) > p s (N ). Moreover, if γ( ) γ 0, π d (N ) = π (N ) and p i (N ) = p s (N ) = p (N ) for all N. Theorem sheds lighs on he impac of he price discriminaion sraegy upon he firm s opimal pricing policy and he opimal profi. More specifically, Theorem 2.5.1(a) shows ha, as long as nework exernaliies are prevalen in he marke (i.e., γ ( ) > 0) and he opimal price for individual cusomers is no binding from below (i.e., p i (N ) > p), he firm should charge a sricly lower price for he social cusomers han ha for he individual cusomers. Under he price discriminaion sraegy, he firm can induce high fuure demands by charging a low price for he social cusomers, and generae curren profis by a high price for he individual cusomers. Theorem 2.5.1(b) shows ha, in each 29

43 period, he opimal price for cusomers wihou price discriminaion is dominaed by ha for he individual cusomers wih price discriminaion. Wihou price discriminaion, he firm should boh generae he curren profis and induce he fuure demands wih he single price charged o all cusomers, so his price is lower han ha for individual cusomers wih price discriminaion, which has he sole role of generaing he curren profis. In Theorem 2.5.1(c), we demonsrae ha he price discriminaion sraegy is beneficial o he firm wih nework exernaliies. Wihou nework exernaliies, however, he firm should charge a single price o all cusomers in each period. An imporan implicaion of Theorem is ha, under he price discriminaion sraegy, he firm earns a higher profi because i can (parially) separae generaing curren profis and inducing fuure demands, he former wih he price for he individual cusomers and he laer wih he price for he social cusomers Nework Expanding Promoion Since he willingness-o-pay of he cusomers in each period is increasing in he size of he associaed nework, he firm may launch nework expanding promoion campaigns o enlarge he nework size and, hence, increase is profiabiliy. The nework expanding promoion sraegy is commonly used in pracice for producs wih nework exernaliies. For example, in February 2015, Microsof discouned he 12-monh Xbox Live Gold membership by 33 percen o boh expand he size of Xbox Live and promoe he sales of Xbox One (see, e.g., [153]). In he case where he associaed nework is an online communicaion nework (i.e., r n ( ) 0), nework expanding promoion is he effor and invesmen he firm makes in social media markeing o arac cusomers o creae and share he messages abou he produc in he nework (i.e., hrough he elecronic wordof-mouh). As an example, in Ocober 2014, Apple bough Twier s Promoed Trend a a daily cos of $200,000 o engage Twier users for he new ipad Air 2 launch (see, e.g., [93]). To model he nework expanding promoion of he firm, le n be he number of cusomers who join he associaed service nework in period in addiion o he social cusomers who purchase he produc. The oal cos of aracing n cusomers ino he nework is c n (n ), where c n ( ) is a coninuously differeniable and convexly increasing funcion of n wih c n (0) = 0. Noe ha he nework expanding promoion do no change 30

44 he invenory dynamics of he firm, bu hey do have some impacs on he nework size dynamics. More specifically, wih nework expanding promoion, he nework size a he beginning of period 1 is given by: N 1 = θd (p, N ) + ηn + n + ϵ. We define v p (I, N ) := he maximum expeced discouned profis wih nework expanding promoion in periods, 1,, 1, when saring period wih an invenory level I and nework size N ; and (x p (I, N ), p p (I, N ), n (I, N )) as he opimal pricing and invenory policy. As in he base model, 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 p 0(I 0, N 0 ) = ci 0 for any (I 0, N 0 ). Employing similar dynamic programming and sample pah analysis mehods, we characerize he opimal policy in he model wih nework expanding promoion in he following lemma. Lemma 6 Define a sequence of funcions {π p (N ) : = T, T 1,, 1} and a sequence of pricing and invenory policies {(x p (N ), p p (N ), n (N )) : = T, T 1,, 1} as follows: π p (N ) = max (x,p,n ) F p J p (x, p, n, N ), (2.11) where J p (x, p, n, N ) = R (p, N ) + βx + Λ(x V + p γ(n )) c n (n ) +G p (θ( V p + γ(n )) + ηn + n ), wih G p (y) := E{r n (y + θξ + ϵ ) + απ p 1(y + θξ )}, π p 0( ) 0, and (x p (N ), p p (N ), n (N )) := argmax (x,p,n ) F p J p (x, p, n, N ). (a) π p ( ) is concave, coninuously differeniable, and increasing in N. J p (,,, ) is joinly concave and coninuously differeniable in (x, p, n ). (b) If I x p (N ), (x p (I, N ), p p (I, N ), n (I, N )) = (x p (N ), p p (N ), n (N )) and v p (I, N ) = ci + π p (N ); oherwise, x p (I, N ) = I. If I T x p T (N T ), (x p (I, N ), p p (I, N ), n (I, N )) = (x p (N ), p p (N ), n (N )) for all and (I, N ) wih probabiliy 1. Lemma 6 demonsraes ha a nework-size-dependen base-sock/lis-price/promoion policy is opimal in he model wih price discriminaion. By normalizing he value of curren invenory, we can reduce he sae space dimension of he dynamic program o 1. 31

45 Wih probabiliy 1, he opimal policy is independen of he saring invenory level in each period, as long as he iniial invenory level I T is below he opimal period-t base-sock level in he firs period x p T (N T ). As in he model wih price discriminaion, Theorems 2.4.3, 2.4.5, 2.4.6, and can be generalized o he model wih nework expanding promoion. We now demonsrae he effeciveness [ineffeciveness] of nework expanding promoion in he model wih [wihou] nework exernaliies. Theorem If (a) Le 0 < ι < 1, and S(N) := sup{ : P(N 1 N = N) ι}. (1 ι)[r n( S(N)) + α(p c)γ ( S(N))] > c n(0), (2.12) hen n (I, N) > 0 for all I. Moreover, S(N) is coninuously increasing in N and, for each 0 < ι < 1, here exiss an N (ι) 0, such ha (2.12) holds for all N < N (ι). (b) If γ( ) γ 0 and ( 1 τ=0 (αη)τ )r n(0) c n(0), n (I, N ) 0 for all I and N 0. Theorem characerizes he dichoomy on when he firm should offer nework expanding promoion. More specifically, Theorem 2.5.2(a) shows ha, when eiher (i) he inensiy of nework exernaliies is sufficienly srong or (ii) he associaed service nework is sufficienly profiable (as characerized by inequaliy (2.12)), i is opimal for he firm o offer nework expanding promoion o cusomers as long as he curren nework size is sufficienly low (i.e., n (I, N ) > 0 if N N (ι)). The inuiion behind Theorem 2.5.2(a) is ha, if a lower bound of he marginal value of offering nework expanding promoion, (1 ι)[r n( S(N)) + α(p c)γ ( S(N))], dominaes is marginal cos c n(0), he firm should offer nework expanding promoion o cusomers. Here, S(N) can be inerpreed as he hreshold such ha, condiioned on N = N, he probabiliy ha he nework size in period 1 exceeds S(N) is smaller han ι, regardless of he pricing sraegy he firm employs. Hence, nework expanding promoion are effecive in exploiing nework exernaliies, especially when N and, hus, he poenial demand is low. On he oher hand, Theorem 2.5.2(b) shows ha if nework exernaliies do no exis (i.e., γ( ) 0) and he associaed service nework is no sufficienly profiable (i.e., ( 1 τ=0 (αη)τ )r n(0) c n(0)), i is opimal for he firm no o offer any nework expanding promoion. 32

46 Nex, we sudy he impac of nework expanding promoion upon he firm s opimal policy. Theorem Assume ha wo invenory sysems are idenical excep ha one wih nework expanding promoion and he oher wihou. For each period and each nework size N 0, he following saemens hold: (a) p p (N ) p (N ); (b) y p (N ) y (N ); (c) x p (N ) x (N ); and (d) π p (N ) π (N ), where he inequaliy is sric if n (N ) > 0. Theorem highlighs how he firm should adjus is price and invenory policy wih nework expanding promoion. More specifically, we show in Theorem 2.5.3(a) ha, wih he same nework size (and, hence, he same poenial marke size), he firm should charge a higher sales price wih nework expanding promoion. Since boh he sales price and he nework expanding promoion helps induce fuure demands via nework exernaliies, he adopion of nework expanding promoion allows he firm o increase he sales price o generae higher profi in he curren period. As a resul, he opimal expeced demand and he opimal base-sock level are lower wih marke expanding promoion. In Theorem 2.5.3(d), we show ha nework expanding promoion can improve he profiabiliy of he firm. To summarize, nework expanding promoion helps he firm exploi nework exernaliies by boosing he nework size in each period. In paricular, nework expanding promoion faciliaes he firm o induce fuure demands wih nework expanding promoion, while generaing higher curren profis wih a higher sales price. The firm should offer nework expanding promoion when he inensiy of nework exernaliies is sufficienly srong or he associaed service nework is sufficienly profiable. 2.6 Numerical Sudies This secion repors a se of numerical sudies ha quanify he profi loss of ignoring nework exernaliies. We also propose and quaniaively evaluae some easyo-implemen heurisics in he presence of nework exernaliies. Our numerical resuls demonsrae ha (1) ignoring nework exernaliies and, hus, employing a myopic pricing and invenory policy leads o saggering profi losses when he nework exernaliies inensiy, he social cusomer proporion, or he carry-hrough rae of nework size is high; 33

47 and (2) he firm can achieve low opimaliy gaps and effecively exploi nework exernaliies wih heurisic policies ha ake ino accoun he demand inducion opporuniies in he near fuure only. Throughou our numerical sudies, we assume ha he maximum inrinsic valuaion V is saionary and equals 30 for each period. The planning horizon lengh is T = 20. The nework exernaliies funcion is γ(n ) = kn (k 0). The parameer k measures he nework exernaliies inensiy. The larger he k, he more inensive nework exernaliies he firm faces. Hence, he demand in each period is D (p, N ) = 30 + kn p + ξ, where {ξ } T =1 follow i.i.d. normal disribuions wih mean 0 and sandard deviaion σ = 2. Noe ha wih he linear nework exernaliies funcion γ( ), Assumpion does no hold. This sligh deviaion from our analyical model, however, does no influence he insighs obained in his secion. For simpliciy, we assume he random perurbaion in he marke size dynamics ϵ is degenerae, i.e., ϵ = 0 wih probabiliy 1. We se he discoun facor α = 0.99, he uni procuremen cos c = 8, he uni holding cos h = 1, he uni backlogging cos b = 10, and he feasible price range [p, p] = [0, 34]. In he evaluaion of he expeced profis, we ake I = 0 as he reference iniial invenory level and N = 0 as he reference iniial nework size Impac of Nework Exernaliies This subsecion numerically sudies he impac of nework exernaliies upon he firm s profiabiliy under differen values of nework exernaliies inensiy k, social cusomer proporion θ, and carry-hrough rae of nework size η. We evaluae he profi of he firm which ignores he radeoff beween generaing curren profis and inducing fuure demands in he presence of nework exernaliies. More specifically, we assume ha he firm adops he myopic policy in each period, i.e., i adops he pricing and invenory policy ha maximizes he expeced curren-period profi wihou aking ino accoun fuure demand-inducing opporuniies. Equivalenly, he firm employs he opimal finalperiod policy, (x 1(, ), p 1(, )), hroughou he planning horizon. Le V m be he expeced profi under he myopic policy, and V be opimal expeced profi. Thus, he meric of ineres is λ m := V V m 100%, which evaluaes he profi loss of ignoring nework exernaliies. V 34

48 We conduc he numerical experimens under he parameers = 5, 10, 15, 20, k = 0.2, 0.5, 0.8, θ = 0.2, 0.5, 0.8, and η = 0.2, 0.5, k=0.2 k=0.5 k= θ=0.2 θ=0.5 θ=0.8 Opimaliy Loss (%) Opimaliy Loss (%) Planning Horizon Lengh Planning Horizon Lengh Figure 2.1. Value of λ m : θ = 0.5, η = 0.5 Figure 2.2. Value of λ m : k = 0.5, η = 0.5 Figures summarize he resuls of our numerical sudy on he impac of ignoring nework exernaliies upon he firm s profiabiliy. Our resuls reveal ha, when he fuure demand-inducing opporuniy of nework exernaliies is ignored, he firm incurs a significan profi loss, which is a leas 4.90% and can be as high as 36.60%, as long as he nework exernaliies inensiy k, he proporion of social cusomers θ, and he nework size carry-hrough rae η are no oo low (greaer han 0.2 in our numerical case). If k, θ, and η are higher, he curren operaions decisions have greaer impac upon fuure nework sizes, hus leading o more inensive radeoff beween generaing curren profis and inducing fuure demands. Therefore, adoping he myopic policy resuls in significan losses if k, θ, and η are no oo low. Anoher imporan implicaion of Figures is ha, if k, θ, and η are no oo low, he profi loss of ignoring nework exernaliies may be significan even when he planning horizon lengh is shor (i.e., = 5). This calls for cauion ha he firm under nework exernaliies should no overlook he radeoff beween generaing curren profis and inducing fuure demands even for a shor sales horizon. 35

49 η=0.2 η=0.5 η= Myopic Policy 1 Heurisic 3 Heurisic 5 Heurisic Opimaliy Loss (%) Opimaliy Loss (%) Planning Horizon Lengh k Figure 2.3. Value of λ m : k = 0.5, θ = 0.5 Figure 2.4. Value of λ m and λ i h : θ = 0.5, η =

50 2.6.2 Effecive Heurisic Policies under Nework Exernaliies In his subsecion, we propose some easy-o-implemen heurisic policies and explore when hese heurisics effecively leverage nework exernaliies. As shown in Secion 2.6.1, he myopic policy may have a poor performance because i ignores he opporuniy of inducing fuure demands via nework exernaliies. Thus, we consider he heurisic policies ha balance generaing curren profis and inducing demands in he near fuure (wihin 5 periods) hrough nework exernaliies. More specifically, in each period, he firm dynamically maximizes he expeced oal discouned profi in he moving ime window from period o period +i (i = 1, 3, 5). We call he heurisic policy o maximize he profi in he moving ime window of lengh i as he i heurisic (i = 1, 3, 5). Clearly, obaining he i heurisic (i = 1, 3, 5) only involves solving a dynamic program wih planing horizon lengh i + 1, and is, hus, compuaionally ligh. Hence, he i heurisic policy (i = 1, 3, 5) is easy o implemen. Le V i h be he expeced oal profi under he i heurisic policy. We have V V 5 h V 3 h V 1 h V m. The meric of ineres is λ i h := V V i h V 100% which measures he opimaliy gap of he i heurisic policy (i = 1, 3, 5). We conduc he numerical experimens under he parameers = 20, k = 0.2, 0.5, 0.8, θ = 0.2, 0.5, 0.8, and η = 0.2, 0.5, Myopic Policy 1 Heurisic 3 Heurisic 5 Heurisic Myopic Policy 1 Heurisic 3 Heurisic 5 Heurisic Opimaliy Loss (%) Opimaliy Loss (%) θ η Figure 2.5. Value of λ m and λ i h : k = 0.5, η = 0.5 Figure 2.6. Value of λ m and λ i h : k = 0.5 θ = 0.5 Figures summarize he resuls of our numerical sudy on he performance of i heuriic policies (i = 1, 3, 5). The resuls show ha, compared wih he myopic 37

51 policy ha compleely ignores he fuure demand-inducing opporuniies, he i heurisics (i = 1, 3, 5) significanly improve he profiabiliy of he firm in he presence of nework exernaliies. In paricular, he 5 heurisic leads a very low profi loss compared wih he opimal policy (no more han 2%, in conras o he more-han-30% opimaliy gap of he myopic policy). Therefore, he firm can effecively exploi nework exernaliies by slighly looking ino he fuure and balancing he radeoff beween generaing curren profis and inducing near fuure demands. Moreover, as shown in Figures , if he nework exernaliies inensiy k, he social cusomer proporion θ, or he carry-hrough rae of nework size η is higher, he i heurisic policies are more valuable relaive o he myopic policy. As k, θ, or η increases, he radeoff beween generaing curren profis and inducing fuure demands becomes more inensive, and, hus, he forward-looking i heurisics can deliver higher values o he firm compared wih he myopic policy. We have also performed numerical analysis for he i heurisic policies wih i > 5. These more forward-looking heurisic policies canno generae significanly beer performances over he 5 heurisic policy. This furher demonsraes ha, o exploi nework exernaliies, i suffices for he firm o balance generaing curren profis and inducing demands in he near fuure. Finally, we remark ha our numerical resuls are robus and coninue o hold in he seings where he planning horizon lengh T is greaer han 20 and/or he marke non-saionary (i.e., he maximum inrinsic valuaion V varies wih ime ). For concision, we only presen he resuls for he case where T = 20 and he marke is saionary in his chaper. 2.7 Summary This is he firs paper in he lieraure o sudy he join pricing and invenory managemen model under nework exernaliies. To model nework exernaliies, we assume ha here is an online service or communicaion nework associaed wih he produc, and he cusomers willingness-o-pay is increasing in he size of his nework. Moreover, in each period, a fracion of he cusomers (i.e., he social cusomers) who purchase he produc would join he nework and exer nework exernaliies over poenial cusomers in he fuure. The firm may direcly generae profis from he nework via, e.g., service subscripion fees. Therefore, in each period, he firm faces he radeoff beween generaing curren profis and inducing fuure demands via nework exernaliies. 38

52 We show ha he opimal policy is a nework-size-dependen base-sock/lis-price policy. Moreover, we demonsrae ha, wih probabiliy 1, he invenory dynamics do no influence he opimal policy of he firm. As a consequence, he sae space dimension of he dynamic program can be reduced o one by normalizing he curren invenory value. Such sae space dimension reducion grealy faciliaes he analysis and enables us o deliver sharper insighs from our model. Our analysis reveals ha he firm needs o balance he radeoff beween generaing curren profis and inducing fuure demands hrough nework exernaliies. Under nework exernaliies, since he curren demand is sochasically increasing in he nework size, he opimal base-sock level and he opimal sales price are increasing in he nework size as well. Nework exernaliies lead o higher poenial demands and, hus, higher base-sock levels. The opimal sales price, however, may no necessarily increase wih he presence of nework exernaliies. This is because, wih nework exernaliies, he firm should decrease he sales price o exploi he increased nework exernaliies when he nework size is small, and increase he sales price o exploi he beer marke condiion when he nework size is large. From he ineremporal perspecive, he firm should pu more weigh on inducing fuure demands a he early sage of a sales season han a laer sages. Thus, when he marke is saionary, he firm employs he inroducory price sraegy ha offers early purchase discouns o induce high fuure demands a he beginning of he sale season. Moreover, he firm needs o rade off beween generaing profi from he produc and from he associaed nework. Wih a higher marginal profi of he associaed nework, he firm should decrease he sales price o exploi he more inensive nework exernaliies. Our analysis demonsraes he effeciveness of he price discriminaion sraegy and he nework expanding promoion sraegy in exploiing nework exernaliies. Boh sraegies faciliae he firm o (parially) separae generaing curren profis and inducing fuure demands hrough nework exernaliies wih an addiional leverage (price or promoion). Under he price discriminaion sraegy, he firm generaes a higher curren profi wih a higher price for individual cusomers, and induces higher fuure demands wih a lower price for social cusomers. Nework expanding promoion should be employed when he inensiy of nework exernaliies is sufficienly srong or he associaed service nework is sufficienly profiable. Moreover, he firm offers nework expanding 39

53 promoion o induce fuure demands hrough nework exernaliies, while generaing a higher curren profi wih an increased price of he produc. We perform exensive numerical sudies o characerize (a) he impac of ignoring nework exernaliies, and (b) he value of some easy-o-implemen heurisic policies o exploi nework exernaliies. Our numerical resuls show ha he profi loss of ignoring nework exernaliies is significan, especially when he nework exernaliies inensiy, he social cusomer proporion, or he nework size carry-hrough rae is high. In his scenario, he radeoff beween generaing curren profis and inducing fuure demands is mos inensive, so he firm should by no means myopically opimize is curren profi. On he oher hand, he heurisic policies ha dynamically maximize he expeced profi in a moving ime window of no more han 5 periods achieve low profi losses relaive o he opimal policy. Hence, o leverage nework exernaliies, i suffices for he firm o balance generaing curren profis and inducing demands in he near fuure. 40

54 3. Operaions Impac of Nework Exernaliies: Dynamic 3.1 Inroducion Compeiion Seing 1 In oday s compeiive and unsable marke environmen, i is prevalen ha modern firms compee no only on generaing curren profis, bu also on winning fuure marke shares (see, e.g., [106]). The curren decisions of all compeing firms in he marke no only deermine heir respecive curren profis, bu also significanly influence heir fuure demands. We refer o such iner-emporal dependence of fuure demands on he curren decisions as marke size dynamics. Under marke size dynamics, myopically opimizing he curren profi may lead o significan loss of fuure demands, and hur he firm s profi in he long run. Therefore, he compeing firms face an imporan radeoff beween generaing curren profis and inducing fuure demands, which we refer o as he exploiaion-inducion radeoff. Among ohers, we focus on wo main drivers of he aforemenioned exploiaioninducion radeoff: (a) The fuure demand is posiively correlaed wih he curren service level, which we refer o as he service effec; and (b) he fuure demand is posiively correlaed wih he curren demand, which we refer o as he nework effec. The service effec is driven by he well-recognized phenomenon ha he pas service experience of a cusomer significanly impacs his/her fuure purchasing decisions (see, e.g., [29, 2]). A poor service (e.g., a low fill rae of a cusomer s orders) generally diminishes he goodwill of a cusomer, hus leading o lower fuure orders from his cusomer ([1]). Moreover, i is widely observed in pracice ha sockous can adversely impac fuure demands (see, e.g., [11, 84]). In he face of a sockou experience, a naural reacion of a cusomer is o order fewer iems and/or swich he seller in a subsequen purchasing execuion (see, e.g., [77, 131]). Therefore, good [poor] pas services of a firm are likely o induce high [low] demands in he fuure. 1 This chaper is based on he auhor s earlier work [191]. 41

55 The nework effec, also known as nework exernaliies, refers o he general phenomenon ha a cusomer s uiliy of purchasing a produc is increasing in he number of oher cusomers buying he same produc (see, e.g., [66]). Under he nework effec, a higher curren demand of a firm leads o more adopions of is produc, hus increasing he uiliy of purchasing is produc for fuure cusomers and boosing fuure demands. There are hree major mechanisms ha give rise o he nework effec: (a) he direc effec, under which an increase in he adopion of a produc leads o a direc increase in he value of his produc for oher users (see, e.g., [102]); (b) he indirec effec, under which an increase in he adopion of a produc enhances he value of is complemenary producs or services, which in urn increases he value of he original produc (see, e.g., [37]); and (c) he social effec, under which he value of a produc is influenced by he social ineracions of is cusomers wih heir peers (see, e.g., [36]). In he highly iner-correlaed and compeiive marke of he curren era, he service effec and he nework effec reinforce each oher. This is because he fas developmen of informaion echnology enables cusomers o easily learn he informaion (on, e.g., qualiy, service, populariy, ec.) of any produc hrough communicaions wih heir friends and/or he cusomer reviews on online reviewing plaforms and social media. Thus, he higher he curren demand of a firm, he more informaion abou is service qualiy will be released o he public, and, hence, he higher impac is service qualiy will have upon fuure demands. Moreover, he curren service level of a firm impacs he fuure demands of iself as well as is compeiors, because cusomers are likely o paronage he firms wih good pas service and abandon hose wih poor pas service based on eiher heir own purchasing experience or he social learning process. The primary goal of his chaper is o develop a model ha can provide insighs on how he exploiaion-inducion radeoff impacs he equilibrium marke behavior under boh he service effec and nework effec. To his end, we sudy a periodic-review dynamic compeiion model, in which firms in a reail marke compee under a Markov game over a finie planning horizon. The random demand of each firm in each period is deermined by is marke size and he curren sales prices and promoional effors of all compeing firms. The promoional effor (e.g., adverising, produc innovaion, and/or afer sales service) of a firm booss he curren demand of iself and diminishes ha of is compeiors. The key feaure of our model is ha he marke sizes of he compeing firms are sochasically 42

56 evolving hroughou he planning horizon, and heir evoluions are driven by he service effec and he nework effec. More specifically, o capure he marke size dynamics, we assume ha he fuure marke size of each firm is sochasically increasing in is curren service level and demand, and sochasically decreasing in he curren service levels of is compeiors. Taking he marke size dynamics ino consideraion, each firm chooses is promoional effor, sales price, and invenory socking quaniy in each decision period, wih an aemp o balance generaing curren profis and inducing fuure demands in he dynamic and compeiive marke. We sudy wo compeiions: (a) he simulaneous compeiion, under which he firms simulaneously make heir promoion, price, and invenory decisions in each period; and (b) he promoion-firs compeiion, under which he firms firs make heir promoional effors and, afer observing he promoion decisions in he marke, choose heir sales prices and invenory levels in each period. Conducing a dynamic game analysis, we make wo main conribuions in his chaper: (a) We sudy a dynamic compeiion model wih he iner-emporal influences of curren decisions over fuure demands, and characerize he pure sraegy Markov perfec equilibrium under boh he simulaneous compeiion and he promoion-firs compeiion; (b) we idenify several imporan managerial implicaions of he exploiaion-inducion radeoff upon he equilibrium marke behavior of he dynamic compeiion under he service effec and he nework effec. We use he Markov perfec equilibrium paradigm o analyze our dynamic compeiion model, because he compeing firms need o adapively adjus heir sraegies based on heir invenory levels and marke sizes in each period. The analyical characerizaion of Markov perfec equilibria in a dynamic oligopoly wih planning horizon lengh greaer han wo is, in general, prohibiively difficul (see, e.g., [132]). To characerize he equilibrium marke oucome in our model, we employ he linear separabiliy approach (see, e.g., [131]) and show ha, under boh he simulaneous compeiion and he promoionfirs compeiion, he equilibrium profi of each firm in each period is linearly separable in is own invenory level and marke size. Such linear separabiliy grealy faciliaes he analysis and enables us o characerize he pure sraegy Markov perfec equilibrium under boh compeiions. Moreover, under boh compeiions, he pure sraegy Markov perfec equilibrium has he nice feaure ha he equilibrium sraegy of each firm only depends on he privae informaion (i.e., invenory level and marke size) of iself, bu 43

57 no on ha of is compeiors. Under he simulaneous compeiion, he subgame played by he compeing firms in each period can be decomposed ino a wo-sage compeiion, in which he firms compee joinly on promoional effor and sales price in he firs sage, and on service level in he second. Under he promoion-firs compeiion, he subgame in each period can be decomposed ino a hree-sage compeiion, in which he firms compee on promoional effor in he firs sage, on sales price in he second, and on service level in he hird. Under boh compeiions, each sage of he subgame in each period has a pure sraegy Nash equilibrium, hus ensuring he exisence of a pure sraegy Markov perfec equilibrium in he Markov game. We also provide mild sufficien condiions under which he Markov perfec equilibrium is unique under each compeiion. Under boh he simulaneous and he promoion-firs compeiions, he marke size dynamics significanly impac he equilibrium behaviors of he compeing firms via he exploiaion-inducion radeoff. This radeoff is quanified by he linear coefficien of marke size for each firm in each period. The higher he marke size coefficien, he more inensive he exploiaion-inducion radeoff for he respecive firm in he previous period. We idenify hree effecive sraegies under he service effec and he nework effec: (a) improving promoional effors, (b) offering price discouns, and (c) elevaing service levels. These sraegies are grounded on he uniform idea ha, o balance he exploiaion-inducion radeoff, he compeing firms can induce higher fuure demands a he cos of reduced curren margins. Our analysis demonsraes how he srengh of he service effec and nework effec impacs he equilibrium marke oucome. Under sronger service and nework effecs, he exploiaion-inducion radeoff is more inensive, so he compeing firms make more promoional effors, offer heavier price discouns, and mainain higher service levels. When he marke is saionary, he inensiy of he exploiaion-inducion radeoff decreases over he sales season under boh compeiions. Hence, he equilibrium sales prices are increasing, whereas he equilibrium promoional effors and service levels are decreasing, over he planning horizon. Our analysis reveals wo ineresing differences beween he simulaneous compeiion and he promoion-firs compeiion under marke size dynamics. Firs, under he simulaneous compeiion, he compeing firms need o balance he exploiaion-inducion radeoff iner-emporally, whereas, under he promoion-firs compeiion, hey have o balance his radeoff boh iner-emporally and inra-emporally. Second, we idenify a 44

58 new driving force for he fa-ca effec (i.e., in each period, he equilibrium promoional effors may be higher under he promoion-firs compeiion han hose under he simulaneous compeiion): The exploiaion-inducion radeoff is more inensive in he promoion-firs compeiion han in he simulaneous compeiion, hus promping he firms o make more promoional effors under he promoion-firs compeiion. The res of his Chaper is organized as follows. We posiion his chaper in he relaed lieraure in Secion 3.2. Secion 3.3 inroduces he model seup. We analyze he simulaneous compeiion model in Secion 3.4, and he promoion-firs compeiion model in Secion 3.5. We compare he equilibrium oucomes in hese wo compeiions in Secion 3.6. Secion 3.7 concludes his chaper. All proofs are relegaed o Appendix B Relaed Research Our work is relaed o several sreams of research in he lieraure. The lieraure on he phenomenon ha he curren service level impacs fuure demands is rich. For example, [147, 148] firs sudies he invenory managemen model, in which fuure demands are adversely affeced by curren poor service levels. [1] consider he dynamic capaciy allocaion problem of a supplier, whose cusomers remember pas service. [2] propose a dynamic behavioral model o sudy he reenion and service relaionship managemen wih he effec of pas service experiences on fuure service qualiy expecaions. The impac of curren service on fuure demands has also been analyzed in a compeiive environmen. [92] invesigae a dynamic cusomer service compeiion, in which he duopoly firms compee by invesing in capaciy wih a fixed oal number of cusomers. [114] sudy a dynamic invenory duopoly model, in which invenory is perishable and cusomers may defec o a compeior. [131] generalize his model o he seing wih non-perishable invenory and he seing in which he firms may arac dissaisfied cusomers from he compeiion. [82] invesigaes he supplier compeiion model, in which each cusomer swiches among suppliers based on her pas service qualiy experience. [84] sudy an invenory compeiion, in which each cusomer learns abou a firm s service level from her previous shopping experience, and makes her poenial paronage decision among differen firms accordingly. The conribuion of his chaper o his lieraure is 45

59 ha we characerize he equilibrium marke behavior in he join promoional effor, sales price, and service level compeiion under he service effec. The opimal pricing sraegy under nework exernaliies has received considerable aenion in he economics and markeing lieraure. [61] characerize he opimal nonlinear pricing sraegy for a nework produc wih heerogenous cusomers. [188] examine he equilibrium dynamic pricing sraegies of an incumben and a laer enran under nework exernaliies. [19] consider he opimal dynamic monopoly pricing under nework exernaliies and show ha he equilibrium prices increase as ime passes. [28] sudy he opimal pricing sraegy in a nework wih a given nework srucure and characerize he relaionship beween opimal prices and consumers cenraliy. We conribue o his sream of lieraure by analyzing he impac of nework exernaliies upon he compeing firms operaions decisions (i.e., he invenory policies) in a dynamic compeiion. This chaper is also relaed o he exensive lieraure on dynamic pricing and invenory managemen. This lieraure diverges ino wo lines of research: (i) he monopoly model, in which a single firm maximizes is oal expeced profi over a finie or infinie planning horizon, and (ii) he compeiion model, in which muliple firms play a noncooperaive game o maximize heir respecive expeced per-period profis over an infinie planning horizon. The lieraure on he monopoly model of join pricing and invenory managemen is very rich. [70] give a general reamen of his problem and show he opimaliy of he base-sock lis-price policy. [47, 48, 49] sudy he join pricing and invenory managemen problem wih fixed ordering coss for he finie horizon, infinie horizon, and coninuous review models. [52] characerize he opimal policy in he join pricing and invenory conrol model wih fixed ordering coss and los sales. [96] idenify a general condiion under which (s, S)-ype policies are opimal for a saionary join pricing and invenory conrol model wih fixed ordering coss. [112] sudy he join pricing and invenory managemen problem wih he random yield risk, and show ha such risk drives he firm o charge a higher price in each period. The join pricing and invenory conrol problem wih periodic review and posiive leadime is exremely difficul. For his problem, [136] and [46] characerize he monooniciy properies of he opimal price and invenory policy for nonperishable and perishable producs, respecively. We refer ineresed readers o [50] for a comprehensive review on he monopoly models of join pricing and invenory managemen. 46

60 The research on he compeiion model of dynamic pricing and invenory managemen is also abundan. Under deerminisic demands, [21] sudy he EOQ model of a wo-echelon disribuion sysem, characerize he equilibrium pricing and replenishmen sraegies of he compeing reailers under boh Berrand and Courno compeiions, and idenify he perfec coordinaion mechanisms herein. [22] address infinie-horizon models for oligopolies wih compeing reailers under price-sensiive uncerain demand. [23] develop a sochasic general equilibrium invenory model, in which reailers compee on boh sales price and service level hroughou an infinie horizon. [25] generalize his model o a decenralized supply chain seing, and characerize he perfec coordinaing mechanisms under price and service compeiion. Our work differs from his line of lieraure in ha we sudy he exploiaion-inducion radeoff wih he service effec and he nework effec in a dynamic and compeiive marke. To his end, we adop he Markov perfec equilibrium (i.e., he closed-loop equilibrium) in a finie-horizon model as opposed o he commonly used saionary sraegy equilibrium (i.e., he open-loop equilibrium) in an infinie-horizon model. Finally, from he mehodological perspecive, our work is relaed o he lieraure on he analysis of Markov perfec equilibrium in dynamic compeiion models. Markov perfec equilibrium is prevalen in he economics lieraure on dynamic oligopoly models (see, e.g, [122, 69, 57]). In he operaions managemen lieraure, his equilibrium concep has been widely adoped o sudy he equilibrium behaviors in dynamic games. Employing he linear separabiliy approach, [92, 114, 131] characerize he Markov perfec equilibrium in dynamic duopoly models wih marke size dynamics, and [5] analyze he srucure of he pure sraegy Markov perfec equilibria in a dynamic invenory compeiion wih subscripions. A similar approach based on he separabiliy of player decisions and probabiliy ransiion funcions has been used by [6] o sudy a join pricing and adverising compeiion, and by [130] o sudy a muli-period invenory compeiion. Due o limied echnical racabiliy, he analysis of Markov perfec equilibrium in nonlinear and nonseparable dynamic games is scarce. [120] characerize he Markov perfec equilibrium price sraegy in a finie-horizon dynamic Berrand compeiion wih fixed capaciies. [117] numerically compue he Markov perfec equilibrium in an infinie-horizon model, in which a supplier allocaes is limied capaciy o compeing reailers. [132] give condiions under which he saionary infinie-horizon equilibrium is also a Markov perfec 47

61 equilibrium in he conex of invenory duopolies. This chaper adops he linear separabiliy approach o characerize he pure sraegy Markov perfec equilibrium of a dynamic join promoion, price, and invenory compeiion under boh he service effec and he nework effec, and analyze he exploiaion-inducion radeoff herein. 3.3 Model Consider an indusry wih N compeing reail firms, which serve he marke wih parially subsiuable producs over a T period planning horizon, labeled backwards as {T, T 1,, 1}. In each period, each firm i selecs a promoional effor γ i, [0, γ i, ], which represens he effor he firm makes in adverising, produc innovaion, and/or afer-sales service o promoe he demand of is produc in he curren period. We assume ha, in any period, he oal promoional invesmen cos of each firm i is proporional o is realized demand in period, D i,, and given by ν i, (γ i, )D i,. The per-uni demand cos rae, ν i, ( ), is a non-negaive, convexly increasing, and wice coninuously differeniable funcion of he promoional effor γ i,, wih ν i, (0) = 0. Before he demand is realized in period, each firm i selecs a sales price p i, [p i,, p i, ] and adjuss is invenory level o x i,. We assume ha he excess demand of each firm is fully backlogged. In summary, each firm i makes hree decisions a he beginning of any period : (i) he promoional effor γ i,, (ii) he sales price p i,, and (iii) he invenory level x i,. The demand of each firm i in any period depends on he enire vecor of promoional effors γ := (γ 1,, γ 2,,, γ N, ) and he enire vecor of sales prices p := (p 1,, p 2,,, p N, ) in period. We denoe he demand of firm i as D i, (γ, p ). More specifically, we base our analysis on he following muliplicaive form of D i, (, ): D i, (γ, p ) = Λ i, d i, (γ, p )ξ i,, (3.1) where Λ i, > 0 is he marke size of firm i in period, d i, (γ, p ) > 0 capures he impac of γ and p on firm i s demand in period, and ξ i, is a posiive coninuous random variable wih a conneced suppor. Le F i, ( ) be he c.d.f. and F i, ( ) be he c.c.d.f. of ξ i,. The marke size Λ i, is observable by firm i a he beginning of period hrough he pre-order sign-ups and/or subscripions before he release of is produc in period. The random perurbaion erm ξ i, is independen of he marke size vecor Λ := (Λ 1,, Λ 2,,, Λ N, ), he sales price vecor p, and he promoional effor vecor γ. Moreover, {ξ i, : = 48

62 T, T 1,, 1} are independenly disribued for each i. Wihou loss of generaliy, we normalize E[ξ i, ] = 1 for each i and any, i.e., E[D i, (γ, p )] = Λ i, d i, (γ, p ). Therefore, d i, (γ, p ) can be viewed as he normalized expeced demand of firm i in period. We assume ha d i, (, ) is wice coninuously differeniable on [0, γ 1, ] [0, γ 2, ] [0, γ N, ] [p 1,, p 1, ] [p 2,, p 2, ] [p N,, p N, ], and saisfies he following monooniciy properies: d i, (γ, p ) γ i, > 0, d i, (γ, p ) γ j, < 0, d i, (γ, p ) p i, < 0, and d i,(γ, p ) p j, > 0, for all j i. (3.2) In oher words, an increase in a firm s promoional effor increases he curren-period demand of iself, and decreases he demands of is compeiors. On he oher hand, an increase in a firm s sales price decreases he demand of iself, and increases he demands of is compeiors. Moreover, we assume ha d i, (, ) is log-separable, i.e., d i, (γ, p ) = ψ i, (γ )ρ i, (p ), where ψ i, ( ) and ρ i, ( ) are posiive and wice-coninuously differeniable. Inequaliies (3.2) imply ha ψ i, (γ ) γ i, > 0, ψ i, (γ ) γ j, < 0, ρ i, (p ) p i, < 0, and ρ i,(p ) p j, > 0, for all j i. For echnical racabiliy, we assume ha ψ i, ( ) and ρ i, ( ) saisfy he log increasing differences and he diagonal dominance condiions, i.e., for any, all i and j i, 2 log ψ i, (γ ) γ 2 i, 2 log ρ i, (p ) p 2 i, < 0, < 0, 2 log ψ i, (γ ) 0, and 2 log ψ i, (γ ) γ i, γ j, γi, 2 > j i 2 log ρ i, (p ) 0, and 2 log ρ i, (p ) p i, p j, p 2 > i, j i 2 log ψ i, (γ ) γ i, γ j, ; (3.3) 2 log ρ i, (p ) p i, p j,. (3.4) The log increasing differences and he diagonal dominance assumpions are no resricive, and can be saisfied by a large se of commonly used demand models in he economics and operaions managemen lieraure, such as he linear, logi, Cobb-Douglas, and CES demand funcions (see, e.g., [124, 22, 23]). The expeced fill rae of firm i in period, z i,, is given by z i, = E[x+ i, D i,(γ, p )] E[D i, (γ, p )] = E[(Λ i,d i, (γ, p )y i, ) + (Λ i, d i, (γ, p )ξ i, )] Λ i, d i, (γ, p ) = E(y + i, ξ i,), where y i, := x i, Λ i, d i, (γ,p ) and a b := min{a, b} for any a, b R. Thus, z i, is concavely increasing in y i, for all y i, 0. Moreover, z i, = 0 if y i, 0, and z i, 1, if y i, +. The key feaure of our model is ha curren promoion, pricing, and invenory decisions impac upon fuure demands via he service effec and he nework effec. To model 49

63 hese wo effecs, we assume ha he marke size of each firm in he nex period is given by he following funcional form: where Ξ 1 i, Λ i, 1 = η i, (z, D i,, Λ i,, Ξ i, ) = Λ i, Ξ 1 i, + α i, (z )D i, Ξ 2 i,, (3.5) is a posiive random variable represening he marke size changes driven by exogenous facors such as economic environmen. Le µ i, := E[Ξ 1 i,] > 0. The erm α i, (z )D i, Ξ 2 i, summarizes he service effec and he nework effec. Specifically, α i, ( ) 0 is a coninuously differeniable funcion wih α i, (z ) z i, 0, and α i,(z ) z j, 0, for all j i, and Ξ 2 i, is a nonnegaive random variable wih E[Ξ 2 i,] = 1. Ξ 2 i, capures he random perurbaions in he marke size changes driven by he service effec and he nework effec. We refer o {α i, ( ) : 1 i N, T 1} as he marke size evoluion funcions. Moreover, for echnical racabiliy, we assume ha α i, ( ) is addiively separable, i.e., α i, (z ) = κ ii, (z i, ) j i κ ij, (z j, ), where κ ii, ( ) > 0 is concave, increasing and coninuously differeniable in z i,, and κ ij, ( ) 0 is coninuously increasing in z j, for all j i. Since α i, ( ) 0 for all z, κ ii, (0) j i κ ij,(1) 0. Le η (,,, ) := (η 1, (,,, ), η 2, (,,, ),, η N, (,,, )) denoe he marke size vecor in he nex period. The evoluion of he marke sizes, (3.5), has several imporan implicaions. Firs, he fuure marke size of each firm depends on is curren marke size in a Markovian fashion. Thus, he dynamic compeiion model in his chaper falls ino he regime of Markov games. Second, alhough he service level of each firm does no influence he curren demand of any firm due o he unobservabiliy of he firms invenory informaion o cusomers, i will impac he firms fuure demands. This phenomenon is driven by he service effec. The higher he service level of a firm, he beer service experience he cusomers have wih his firm in he curren period, and he more cusomers will paronage his firm in he fuure. Analogously, if he service levels of a firm s compeiors increase, cusomers will be more likely o purchase from is compeiors in he fuure. Therefore, he fuure demand of each firm is sochasically increasing in he curren service level of his firm and sochasically decreasing in he curren service level of any of is 50

64 compeiors. Hence, he invenory decision of each firm has he demand-inducing value driven by he service effec. Third, he fuure demand of each firm is posiively correlaed wih he curren demand of his firm. This phenomenon is driven by he nework effec. If he realized curren demand of a firm is higher, poenial cusomers can ge higher uiliies if purchasing from his firm, hus giving rise o higher fuure demand. Because of he nework effec, he sales price and promoional effor no only affec he curren demand, bu also influence fuure demands. Fourh, he service effec and he nework effec reinforce each oher. More specifically, he impac of curren service levels upon fuure marke sizes is higher wih higher realized curren demands. Wih he explosive growh of online social media, cusomers could easily learn he service qualiies of all firms hrough social learning. As a consequence, higher curren demands lead o more inensive social ineracions among cusomers, and, hence, magnify he impac of curren service levels on fuure demands. We inroduce he following model primiives: δ i = discoun facor of firm i for revenues and coss in fuure periods, 0 < δ i 1, w i, = per-uni wholesales price paid by firm i in period, b i, = per-uni backlogging cos paid by firm i in period, h i, = per-uni holding cos paid by firm i in period. Wihou loss of generaliy, we assume he following inequaliies hold for each i and : b i, > w i, δ i w i, 1 : he backlogging penaly is higher han he saving from delaying an order o he nex period for each firm in any period, so ha no firm will backlog all of is demand, h i, > δ i w i, 1 w i, : he holding cos is sufficienly high so ha no firm will place a speculaive order. p i, > δ i w i, 1 + b i, + ν i, ( γ i, ) : posiive margin for backlogged demand wih highes price and promoional effor. We define he normalized expeced holding and backlogging cos funcion for firm i in period : L i, (y i, ) := E{h i, (y i, ξ i, ) + + b i, (y i, ξ i, ) }, where y i, R. (3.6) 51

65 The sae of he Markov game is given by: I = (I 1,, I 2,,, I N, ) = he vecor for he saring invenories of all firms in period, Λ = (Λ 1,, Λ 2,,, Λ N, ) = he vecor for he marke sizes of all firms in period. We use S := R N R N + o denoe he sae space of each firm i in he dynamic compeiion. To characerize how he marke size dynamics (i.e., he service effec and he nework effec) impac he equilibrium marke oucome, we consider he Markov perfec equilibrium (MPE) in our dynamic compeiion model. An MPE saisfies wo condiions: (a) in each period, each firm i s promoion, price, and invenory sraegy depends on he hisory of he game only hrough he curren period sae variables (I, Λ ), and (b) in each period, he sraegy profile generaes a Nash equilibrium in he associaed proper subgame. In oher words, MPE is a closed-loop equilibrium ha saisfies subgame perfecion in each period. Because of is simpliciy and consisency wih raionaliy, MPE is widely used in dynamic compeiion models in he economics (e.g., [122]) and operaions managemen (e.g., [131]) lieraure. A major echnical challenge o characerize he MPE in a dynamic invenory compeiion model is ha when he saring invenories are higher han he equilibrium order-up-o levels, he model becomes illy behaved and analyically inracable (see, e.g., [132]). This issue is worsened under endogenous pricing decisions [25]. To overcome his echnical challenge, we make he following assumpion hroughou our analysis. Assumpion A he beginning of each period, each firm i is allowed o sell (poenially par of) is onhand invenory o is supplier a he curren-period per-uni wholesale price w i,. Assumpion is imposed o circumven he aforemenioned echnical challenge. As will be clear by our subsequen analysis, wih his assumpion, he equilibrium profi of each firm i in each period is linearly separable in is saring invenory level I i, and marke size Λ i,. Assumpion enables us o eliminae he influence of curren invenory decision of any firm upon he fuure equilibrium behavior of he marke, so as o single ou and highligh he exploiaion-inducion radeoff wih he service effec and he nework effec. Assumpion applies when he reail firms have such grea marke power ha hey can reach an agreemen wih heir respecive suppliers on he 52

66 reurn policy wih full price refund. [25], among ohers, also make his assumpion o characerize he MPE in an infinie-horizon join price and service level compeiion model. Wih Assumpion 3.3.1, we can define he acion space of each firm i in each period : A i, (I i, ) := [0, γ i, ] [p i,, p i, ] [min{0, I i, }, + ). 3.4 Simulaneous Compeiion In his secion, we sudy he simulaneous compeiion (SC) model where each firm i simulaneously chooses a combined promoion, price, and invenory sraegy in any period. This model applies o he scenarios where he marke expanding effors (e.g., adverising, rade-in programs, ec.) ake effec insananeously, so, in essence, he promoional effor and sales price decisions are made simulaneously in each period. Our analysis in his secion focuses on characerizing he pure sraegy MPE and providing insighs on he impac of he exploiaion-inducion radeoff in he SC model Equilibrium Analysis In his subsecion, we show ha he simulaneous compeiion model has a pure sraegy MPE. Moreover, we characerize a sufficien condiion on he per-uni demand cos rae of promoional effor, ν i, ( ), under which he MPE is unique. Wihou loss of generaliy, we assume ha, a he end of he planning horizon, each firm i salvages all he on-hand invenory and fulfills all he backlogged demand a uni wholesale price w i,0 0. The payoff funcion of each firm i is given by: E{ T =1 δ T i [p i, D i, (γ, p ) w i, (x i, I i, ) h i, (x i, D i, (γ, p )) + b i, (x i, D i, (γ, p )) ν i, (γ i, )D i, (γ, p )] + δ T i w i,0 I i,0 I T, Λ T }, (3.7) s.. I i, 1 = x i, D i, (γ, p ) for each, and Λ i, 1 = Λ i, Ξ 1 i, + α i, (z )D i, (γ, p )Ξ 2 i, for each. 53

67 Under an MPE, each firm i should ry o maximize is expeced payoff in each subgame (i.e., in each period ) condiioned on he realized invenory levels and marke sizes in period, (I, Λ ): E{ τ=1 δ τ i [p i,τ D i,τ (γ τ, p τ ) w i,τ (x i,τ I i,τ ) h i,τ (x i,τ D i,τ (γ τ, p τ )) + b i,τ (x i,τ D i,τ (γ τ, p τ )) ν i,τ (γ i,τ )D i,τ (γ τ, p τ )] + δ iw i,0 I i,0 I, Λ }, (3.8) s.. I i,τ 1 = x i,τ D i,τ (γ τ, p τ ) for each τ, τ 1, and Λ i,τ 1 = Λ i,τ Ξ 1 i,τ + α i,τ (z τ )D i,τ (γ τ, p τ )Ξ 2 i,τ for each τ, τ 1. A (pure) Markov sraegy profile in he SC model σ sc := {σ sc i,(, ) : 1 i N, T 1} prescribes each firm i s combined promoion, price, and invenory sraegy in each period, where σ sc i,(, ) := (γ sc i,(, ), p sc i,(, ), x sc i,(, )) is a Borel measurable mapping from S o A i, (I i, ). We use σ sc := {σ sc i,(, ) : 1 i N, T 1} o denoe he pure sraegy profile in he induced subgame in period, which prescribes each firm i s (pure) sraegy from period ill he end of he planning horizon. To evaluae he expeced payoff of each firm i in each period for any given Markov sraegy profile σ sc in he simulaneous compeiion, le V i, (I, Λ σ sc ) = he oal expeced discouned profi of firm i in periods, 1,, 1, 0, when saring period wih he sae variable (I, Λ ) and he firms play sraegy σ sc in periods, 1,, 1. Thus, by backward inducion, V i, (, σ sc ) saisfies he following recursive scheme for each firm i in each period : V i, (I, Λ σ sc ) = J i, (γ sc (I, Λ ), p sc (I, Λ ), x sc (I, Λ ), I, Λ σ sc 1), where γ sc (, ) = (γ1,(, sc ), γ2,(, sc ),, γn, sc (, )) is he period promoional effor vecor prescribed by σ sc, p sc (, ) = (p sc 1,(, ), p sc 2,(, ),, p sc N, (, )) is he period sales price vecor prescribed by σ sc, x sc (, ) = (x sc 1,(, ), x sc 2,(, ),, x sc N, (, )) is he period posdelivery invenory vecor prescribed by σ sc, J i, (γ, p, x, I, Λ σ sc 1) =E{p i, D i, (γ, p ) w i, (x i, I i, ) h i, (x i, D i, (γ, p )) + b i, (x i, D i, (γ, p )) ν i, (γ i, )D i, (γ, p ) + δ i V i, 1 (x D (γ, p ), η (z, D (γ, p ), Λ, Ξ ) σ sc 1) I, Λ }, 54 (3.9)

68 and V i,0 (I, Λ ) = w i,0 I i,0. model. We now formally define he pure sraegy MPE in he SC Definiion A (pure) Markov sraegy σ sc = {(γ sc i, (, ), p sc i, (, ), x sc i, (, )) : 1 i N, T 1} is a pure sraegy MPE in he SC model if and only if, for each firm i, each period, and each sae variable (I, Λ ), (γ sc i, (I, Λ ), p sc i, (I, Λ ), x sc i, (I, Λ )) =argmax (γi,,p i,,x i, ) A i, (I i, ){J i, ([γ i,, γ sc i,(i, Λ )], [p i,, p sc i,(i, Λ )], [x i,, x sc i,(i, Λ )], I, Λ σ sc 1)}. (3.10) By Definiion 3.4.1, a (pure) Markov sraegy profile in he SC model is a pure sraegy MPE if i saisfies subgame perfecion in each period. Definiion does no guaranee he exisence of an MPE, σ sc, in he SC model. In Theorem 3.4.1, below, we will show a pure sraegy MPE always exiss in he SC model. Moreover, under a mild addiional assumpion on ν i, ( ), he SC model has a unique pure sraegy MPE. By Definiion 3.4.1, he equilibrium sraegy for firm i in period, (γ sc i, (, ), p sc i, (, ), x sc i, (, )), may depend on he sae vecor of is compeiors (I i,, Λ i, ). In pracice, however, each firm i s saring invenory level I i, and marke size Λ i, are generally is privae informaion ha is no accessible by is compeiors in he marke. We will show ha he equilibrium sraegy profile of each firm i in each period is only coningen on is own realized sae variables (I i,, Λ i, ), bu independen of is compeiors privae informaion (I i,, Λ i, ). The following heorem characerizes he exisence and he uniqueness of MPE in he SC model. Theorem The following saemens hold for he SC model: (a) There exiss a pure sraegy MPE σ sc = {(γ sc i, (, ), p sc i, (, ), x sc i, (, )) : 1 i N, T 1}. (b) For each pure sraegy MPE, σ sc, here exiss a series of vecors {β sc : T 1}, where β sc = (β1,, sc β2,, sc, βn, sc ) wih βsc i, > 0 for each i and, such ha V i, (I, Λ σ sc ) = w i, I i, + βi,λ sc i,, for each firm i and each period. (3.11) (c) If he following wo condiions simulaneously hold for each i and : 55

69 (i) ν i,( ) 1 for all γ i, [0, γ i, ]; and (ii) ν i,(γ i, )(p i, δw i, 1 ν i, (γ i, )+c i, )+[ν i,(γ i, )] 2 ν i,(γ i, ) for all p i, [p i,, p i, ] and γ i, [0, γ i, ], where c i, := max{(δ i w i, 1 w i, )y i, L i, (y i, ) : y i, 0}, σ sc is he unique MPE in he SC model. In paricular, if ν i, (γ i, ) = γ i,, condiions (i) and (ii) are saisfied. Theorem 3.4.1(a) demonsraes he exisence of a pure sraegy MPE in he simulaneous compeiion model. Moreover, in Theorem 3.4.1(b), we show ha, for each pure sraegy MPE σ sc, he corresponding profi funcion of each firm i in each period is linearly separable in is saring invenory level I i, and marke size Λ i,. We refer o he consan β sc i, as he SC marke size coefficien of firm i in period. As we will show laer, he SC marke size coefficien measures he inensiy of he exploiaion-inducion radeoff. The larger he β sc i,, he more inensive he exploiaion-inducion radeoff for firm i in he previous period + 1. Theorem 3.4.1(b) also implies ha he equilibrium profi of each firm i in each period only depends on he sae variables of iself (I i,, Λ i, ), bu no on hose of is compeiors (I i,, Λ i, ). Theorem 3.4.1(c) characerizes a sufficien condiion for he uniqueness of an MPE in he SC model. In paricular, if he promoional effor γ i, refers o he acual moneary paymen of promoional invesmen per-uni demand for each firm i in each period (i.e., ν i, (γ i, ) = γ i, for each i and ), here exiss a unique MPE in he SC model. For he res of his chaper, we assume ha condiions (i) and (ii) are saisfied for each i and and, hence, he SC model has a unique pure sraegy MPE σ sc. The linear separabiliy of V i, (, σ sc ) (i.e., Theorem 3.4.1(b)) enables us o characerize he MPE in he SC model. Plugging (3.11) ino he objecive funcion of firm i in period, by x i, = Λ i, d i, (γ, p )y i, and z i, = E(y + i, ξ i,), we have: 56

70 J i, (γ, p, x, I, Λ σ sc 1) =E{p i, D i, (γ, p ) w i, (x i, I i, ) h i, (x i, D i, (γ, p )) + b i, (x i, D i, (γ, p )) ν i, (γ i, )D i, (γ, p ) + δ i V i, 1 (x D (γ, p ), η (z, D (γ, p ), Λ, Ξ ) σ sc 1) I, Λ } =E{p i, Λ i, d i, (γ, p )ξ i, w i, (y i, Λ i, d i, (γ, p ) I i, ) h i, (y i, Λ i, d i, (γ, p ) Λ i, d i, (γ, p )ξ i, ) + b i, (y i, Λ i, d i, (γ, p ) Λ i, d i, (γ, p )ξ i, ) ν i, (γ i, )Λ i, d i, (γ, p )ξ i, + δ i w i, 1 (y i, Λ i, d i, (γ, p ) Λ i, d i, (γ, p )ξ i, ) + δ i β sc i, 1(Λ i, Ξ 1 i, + α i, (z )Λ i, d i, (γ, p )ξ i, Ξ 2 i,) I, Λ } =w i, I i, + Λ i, {δ i β sc i, 1µ i, + ψ i, (γ )ρ i, (p )[p i, δ i w i, 1 ν i, (γ i, ) + π sc i,(y )]}, where π sc i,(y ) =(δ i w i, 1 w i, )y i, L i, (y i, ) + δ i β sc i, 1(κ ii, (E[y + i, ξ i,]) j i κ ij, (E[y + j, ξ j,])), and β sc i,0 :=0 for each i. (3.12) We observe from (3.12) ha he payoff funcion of each firm i in he subgame of period has a nesed srucure. Hence, he subgame of period can be decomposed ino wo sages, where he firms compee joinly on promoion and price in he firs sage, and on invenory in he second sage. Since he service level of each firm i, as measured by he expeced fill rae z i,, is increasing in he invenory decision y i,, we refer o he secondsage compeiion as he service level compeiion hereafer. we firs sudy he second-sage service level compeiion. By backward inducion, Le G sc,2 be he N player noncooperaive game ha represens he second-sage service level compeiion in period, where player i has payoff funcion π sc i,( ) and feasible acion se R. proposiion characerizes he Nash equilibrium of he game G sc,2. The following 57

71 Proposiion For each period, he second-sage service level compeiion G sc,2 has a unique pure sraegy Nash equilibrium y sc. Moreover, for each i, yi, sc > 0 is he unique soluion o he following equaion: (δ i w i, 1 w i, ) L i,(y sc i, ) + δ i β sc i, 1 F i, (y sc i, )κ ii,(e(y sc i, ξ i, )) = 0. (3.13) Proposiion demonsraes he exisence and uniqueness of a pure sraegy Nash equilibrium of he second-sage service level compeiion. Moreover, y sc i, by solving he firs-order condiion yi, π sc i,(y sc ) = 0. Le π sc can be obained := (π sc 1,, π sc 2,,, π sc N, ) be he equilibrium payoff vecor of he second-sage service level compeiion in period, where π sc i, = π sc i,(y sc ). For each i and, le Π sc i,(γ, p ) := ψ i, (γ )ρ i, (p )[p i, δ i w i, 1 ν i, (γ i, ) + π sc i, ]. (3.14) We define an N player noncooperaive game G sc,1 o represen he firs-sage join promoion and price compeiion in period, where player i has payoff funcion Π sc i,(, ) and feasible acion se [0, γ i, ] [p i,, p i, ]. We characerize he Nash equilibrium of he game G sc,1 in he following proposiion. Proposiion For each period, following saemens hold: (a) The firs-sage join promoion and price compeiion, G sc,1, is a log-supermodular game. (b) The game G sc,1 has a unique pure sraegy Nash equilibrium (γ sc unique serially undominaed sraegy of G sc,1. (c) The Nash equilibrium of G sc,1 equaions: For each i γi, ψ i, (γ sc ) ψ i, (γ sc ) pi, ρ i, (p sc ) + ρ i, (p sc ) p sc i, p sc i,, p sc ), which is he is he unique soluion o he following sysem of ν i,(γ i, sc ) δ i w i, 1 ν i, (γi, sc ) + πsc i, 0, 0 0, 1 δ i w i, 1 ν i, (γi, sc ) + πsc i, 0 if γ sc i, = 0, = 0, if γ sc i, (0, γ i, ), if γ sc i, = γ i, ; if p sc i, = p i,, = 0, if p sc i, (p i,, p i, ), if p sc i, = p i,. and, (3.15) 58

72 (d) Le Π sc := (Π sc 1,, Π sc 2,,, Π sc N, ) be he equilibrium payoff vecor of he firs-sage join promoion and price compeiion in period, where Π sc i, have Π sc i, > 0 for all i. = Π sc i,(γ sc, p sc ). We Proposiion shows ha he firs-sage join promoion and price compeiion G sc,1 is a log-supermodular game, and has a unique pure sraegy Nash equilibrium (γ sc, p sc ). The unique Nash equilibrium, (γ sc, p sc ), is deermined by (i) he serial eliminaion of sricly dominaed sraegies, or (ii) he sysem of firs-order condiions (3.15). Under equilibrium, by Proposiion 3.4.2(d) and he objecive funcion of period, (3.12), each firm i earns a posiive normalized expeced oal discouned profi, Λ i, (δ i β sc i, 1µ i, + Π sc i, ), in he subgame of period. Summarizing Theorem 3.4.1, Proposiion and Proposiion 3.4.2, we have he following heorem ha sharpens he characerizaion of he MPE in he SC model. Theorem For each period, he following saemens hold: (a) For each i, β sc i, = δ i β sc i, 1µ i, + Π sc i,. (b) Under he unique (pure sraegy) MPE σ sc, he policy of firm i is given by (γ sc i, (I, Λ ), p sc i, (I, Λ ), x sc i, (I, Λ )) = (γ sc i,, p sc i,, Λ i, y sc i, ρ i, (p sc )ψ i, (γ sc )). (3.16) Theorem 3.4.2(a) recursively compues he SC marke size coefficien vecors {β sc : T 1}. Theorem 3.4.2(b) demonsraes ha, under he MPE σ sc, each firm i s join promoion, price, and invenory policy in each period only depends on is own sae variables (I i,, Λ i, ), bu no on hose of is compeiors (I i,, Λ i, ), which are no accessible o firm i in general. Thus, for each firm i in each period, is equilibrium sraegy has he aracive feaure ha he sraegy depends on is accessible informaion only. In some of our analysis below, we will consider a special case of he SC model, where he marke is symmeric, i.e., all compeing firms have idenical characerisics. We use he subscrip s o denoe he case of symmeric marke. In his case, for all i, j, and, le ρ s, ( ) := ρ i, ( ), ψ s, ( ) := ψ i, ( ), ν s, ( ) := ν i, ( ), α s, ( ) := α i, ( ), κ sa, ( ) := κ ii, ( ), κ sb, ( ) := κ ij, ( ), w s, := w i,, h s, := h i,, b s, := b i,, µ s, := µ i,, and δ s := δ i. Thus, le L s, ( ) := L i, ( ) for each i. As shown in Theorem 3.4.1, here exiss a unique pure sraegy 59

73 MPE in he symmeric SC model, which we denoe as σs sc. The following proposiion is a corollary of Theorems Proposiion The following saemens hold for he symmeric SC model: (a) For each = T, T 1,, 1, here exiss a consan β sc s, > 0, such ha V i, (I, Λ σ sc s, ) = w s, I i, + β sc s,λ i,, for all i. (b) In each period, he second-sage service level compeiion G sc,2 s, he payoff funcion for each firm i given by is symmeric, wih π sc i,(y ) =(δ s w s, 1 w s, )y i, L s, (y i, ) + δ s β sc s, 1(κ sa, (E[y + i, ξ i,]) j i κ sb, (E[y + j, ξ j,])). Moreover, G sc,2 s, we use y sc s, has a unique pure sraegy Nash equilibrium which is symmeric, so [π sc s, ] o denoe he equilibrium sraegy [payoff] of each firm in G sc,2 s,. (c) In each period, he firs-sage join promoion and price compeiion G sc,1 s, is symmeric, wih he payoff funcion for each firm i given by Moreover, G sc,1 s, Π sc i,(γ, p ) = ψ s, (γ )ρ s, (p )[p i, δ s w s, 1 ν s, (γ i, ) + π sc s, ]. has a unique pure sraegy Nash equilibrium (γ sc ss,, p sc ss,) which is symmeric (i.e., γ sc ss, = (γ sc s,, γ sc s,,, γ sc s, ) for some γ sc s, p sc ss, = (p sc s,, p sc s,,, p sc s, ) for some p sc s, ). (d) Under he unique pure sraegy MPE, σs sc, he policy of firm i in period is (γ sc i, (I, Λ ), p sc i, (I, Λ ), x sc i, (I, Λ )) = (γ sc s,, p sc s,, Λ i, y sc s, ρ s, (p sc ss,)ψ s, (γ sc ss,)), for each (I, Λ ). and Proposiion characerizes he MPE, σs sc, and he marke size coefficiens, {βs, sc : T 1}, in he symmeric SC model. Proposiion shows ha, in he symmeric SC model, all compeing firms se he same promoional effor, sales price, and service level in each period under equilibrium, whereas he equilibrium marke oucome may vary in differen periods. 60

74 3.4.2 Exploiaion-Inducion Tradeoff In his subsecion, we sudy how he marke size dynamics (i.e., he service effec and he nework effec) influence he equilibrium marke oucome in he SC model. We focus on he managerial implicaions of he exploiaion-inducion radeoff in a dynamic and compeiive marke. To begin wih, we characerize he impac of he marke size coefficien vecors {β sc : T 1} upon he equilibrium marke oucome. The following heorem serves as he building block of our subsequen analysis of he exploiaion-inducion radeoff in he SC model. Theorem For each period, he following saemens hold: (a) For each i and j i, y sc i, β sc j, 1. is coninuously increasing in β sc i, 1 and independen of (b) For each i and j i, π sc i, decreasing in β sc j, 1. is coninuously increasing in β sc i, 1 and coninuously (c) If he SC model is symmeric, γ sc s, is coninuously decreasing in π sc s,. is coninuously increasing in π sc s,, whereas p sc s, (d) If he SC model is symmeric and ψ s, ( ) and ρ s, ( ) saisfy he following monooniciy condiion N i=1 ψ s, (γ ) γ i, > 0, for all γ, and N i=1 ρ s, (p ) p i, < 0, for all p, (3.17) β sc s, is coninuously increasing in π sc s,. (e) If he SC model is symmeric and π sc s, increasing in βs, 1, sc whereas p sc s, is increasing in βs, 1, sc γs, sc is coninuously decreasing in β sc s, 1. is coninuously (f) In he symmeric SC model, if he monooniciy condiion (3.17) holds and π sc s, increasing in β sc s, 1, β sc s, is coninuously increasing in β sc s, 1. is Theorem shows ha he marke size coefficiens {β sc i, : 1 i N, T 1} quanify he inensiy of he exploiaion-inducion radeoff in he SC model. More specifically, if β sc i, 1 is larger, firm i faces sronger exploiaion-inducion radeoff in period 61

75 . Therefore, o balance his srenghened radeoff and o induce high fuure demands, each firm should improve service qualiy, decrease sales price, and increase promoional effor, as shown in pars (a) and (e) of Theorem Moreover, Theorem 3.4.3(f) characerizes he relaionship beween he exploiaion-inducion radeoffs in differen periods, demonsraing ha if he exploiaion-inducion radeoff is more inensive in he nex period, i is also sronger in he curren period under a mild condiion. The monooniciy condiion (3.17) implies ha a uniform increase of all N firms promoional effors leads o an increase in he demand of each firm, and a uniform price increase by all N firms gives rise o a decrease in he demand of each firm. This condiion is commonly used in he lieraure (see, e.g., [23, 9]), and ofen referred o as he dominan diagonal condiion for linear demand models. The assumpion ha π sc s, is increasing in β sc s, 1 is no resricive eiher. In Lemma 26 in Appendix B.2, we give some sufficien condiions for his assumpion. More specifically, Lemma 26 implies ha π sc s, is increasing in β sc s, 1 if one of he following condiions holds: (i) The adverse effec of a firm s compeiors service upon is fuure marke size is no srong; (ii) he nework effec is sufficienly srong; or (iii) boh he service effec and he nework effec are sufficienly srong. Now we consider a benchmark case wihou he service effec and he nework effec. We use o denoe his model. Thus, in he benchmark model, he marke size evoluion funcion α i, ( ) 0 for each firm i and each period. Wihou he service effec and he nework effec, he curren promoion, price, and service level decisions of any firm will no influence he fuure demands. Therefore, he compeing firms can focus on generaing curren profis in each period wihou considering inducing fuure demands, i.e., he exploiaion-inducion radeoff is absen in his benchmark case. To characerize he impac of he service effec and he nework effec upon he equilibrium oucome, he following heorem compares he Nash equilibria in G sc,2 and sc,2 G, and he Nash equilibria in G sc,1 and G sc,1. Theorem π sc i, π sc i,. (a) For each firm i and each period, y sc i, ỹi, sc, zi, sc z sc i,, and (b) Consider he symmeric SC model. For each period, he following saemens hold: (i) γ sc s, (ii) p sc s, γ sc s, p sc s, and, hus, γ sc i, (I, Λ ) γ sc i, (I, Λ ) for all i and all (I, Λ ). and, hus, p sc i, (I, Λ ) p sc i, (I, Λ ) for all i and all (I, Λ ). 62

76 (iii) If he monooniciy condiion (3.17) holds, we have x sc i, (I, Λ ) x sc i, (I, Λ ) for all i and all (I, Λ ). Theorem highlighs he impac of marke size dynamics upon he equilibrium marke oucome. Specifically, Theorem 3.4.4(a) shows ha, under he service effec and he nework effec, each firm i should se a higher service level in each period. In he symmeric SC model, Theorem 3.4.4(b-i) shows ha each firm should increase is promoional effor in each period under he service effec and he nework effec, in order o induce higher fuure demands. Analogously, Theorem 3.4.4(b-ii) shows ha he service effec and he nework effec give rise o lower equilibrium sales price of each firm in each period. Under he monooniciy condiion (3.17), Theorem 3.4.4(b-i,ii) implies ha he equilibrium expeced demand of each firm in each period is higher under he service effec and he nework effec. As a consequence, o mach supply wih he curren demand and o induce fuure demands wih he service effec, each firm should increase is base sock level in each period under he service effec and he nework effec, as shown in Theorem 3.4.4(b-iii). Theorem idenifies effecive sraegies for firms o balance he exploiaioninducion radeoff under boh he service effec and he nework effec. In his case, he compeing firms have o radeoff generaing curren profis and inducing fuure demands. To balance he exploiaion-inducion rade-off, he firms can employ hree sraegies o exploi he service effec and he nework effec: (a) elevaing service levels, (b) offering price discouns, and (c) improving promoional effors. Elevaing service levels does no lead o a higher curren demand, bu helps he firm induce higher fuure demands via he service effec. Offering price discouns and improving promoional effors do no increase he curren profis bu give rise o higher curren demands and, hus, induce higher fuure demands via he nework effec. In a nushell, he uniform idea of all hree sraegies is ha, o balance he exploiaion-inducion radeoff under he service effec and he nework effec, he compeing firms should induce higher fuure demands a he cos of reduced curren margins. To deliver sharper insighs on he managerial implicaions of he exploiaion-inducion radeoff, we confine ourselves o he symmeric SC model for he res of his secion. The following heorem characerizes how he inensiies of he service effec and he nework effec influence he equilibrium marke oucome in he symmeric SC model. 63

77 Theorem Le wo symmeric SC models be idenical excep ha one wih marke size evoluion funcions {ˆα s, ( )} T 1 and he oher wih {α s, ( )} T 1. Assume ha, for each period, (i) he monooniciy condiion (3.17) holds, and (ii) κ sb, ( ) κ 0 sb, for some consan κ 0 sb,. (a) If ˆα s, (z ) α s, (z ) for each period and each z, we have, for each period, ˆβ sc s, βs,, sc ˆγ s, sc γs, sc, and ˆp sc s, p sc s,. Thus, for each period, ˆγ i, sc (I, Λ ) γi, sc (I, Λ ) and ˆp sc i, (I, Λ ) p sc i, (I, Λ ) for all i and all (I, Λ ) S. (b) If, for each period, ˆα s, (z ) α s, (z ) for all z and ˆκ sa,(z i, ) κ sa,(z i, ) 0 for all z i,, we have, for each period, sc ˆβ s, βs,, sc ˆγ s, sc γs, sc, ˆp sc s, p sc s,, and ŷs, sc ys, sc. Thus, for each period, ˆγ sc i, (I, Λ ) γ sc i, (I, Λ ), ˆp sc i, (I, Λ ) p sc i, (I, Λ ), and ˆx sc i, (I, Λ ) x sc i, (I, Λ ) for all i and all (I, Λ ) S. Theorem sharpens Theorem by showing ha if he inensiies of he nework effec and he service effec (capured by he magniudes of α s, ( ) and κ sa,( ), respecively) are higher, he exploiaion-inducion radeoff becomes sronger. To balance he srenghened exploiaion-inducion radeoff, each firm should increase is promoional effor, decrease is sales price, and improve is service level in each period. More specifically, Theorem 3.4.5(a) shows ha a higher inensiy of he nework effec (i.e., larger α s, ( )) drives all he firms o make more promoional effors and charge lower sales prices. Theorem 3.4.5(b) furher suggess ha higher inensiies of boh he nework effec and he service effec (i.e., larger α s, ( ) and κ sa,( )) promp all he firms o make more promoional effors, charge lower sales prices, and mainain higher service levels. Sronger service effec and nework effec inensify he exploiaion-inducion radeoff, hus driving he firms o pu more weigh on inducing fuure demands han on exploiing he curren marke. Therefore, o effecively balance he exploiaion-inducion radeoff, all he firms should carefully examine he inensiies of he service effec and he nework effec. Nex, we analyze he exploiaion-inducion radeoff from an iner-emporal perspecive. Under he service effec and he nework effec, how should he compeing firms adjus heir promoion, price, and service sraegies hroughou he sales season o balance he exploiaion-inducion radeoff? To address his quesion, we characerize he evoluion of he equilibrium marke oucome in he saionary and symmeric SC model. In his model, he model primiives, demand funcions, and marke size evoluion func- 64

78 ions are idenical for all firms hroughou he planning horizon. In addiion, he random perurbaions in marke demands and marke size evoluion are i.i.d. hroughou he planning horizon. The following heorem characerizes he evoluion of he equilibrium promoion, price, and service sraegy in he saionary and symmeric SC model. Theorem Consider he saionary and symmeric SC model. Assume ha, for each period, (i) he monooniciy condiion (3.17) holds, and (ii) π sc s, β sc s, 1. For each period, he following saemens hold: (a) βs, sc βs, 1, sc γs, sc γs, 1, sc p sc s, p sc s, 1, and ys, sc ys, 1. sc is increasing in (b) γ sc i, (I, Λ) γ sc i, 1(I, Λ), p sc i, (I, Λ) p sc i, 1(I, Λ), and x sc i, (I, Λ) x sc i, 1(I, Λ) for each i and each (I, Λ) S. Theorem sheds ligh on how o balance he exploiaion-inducion radeoff from an iner-emporal perspecive. More specifically, we show ha, if he marke is symmeric and saionary, he exploiaion-inducion radeoff is more inensive (i.e., β sc s, is larger) a he early sage of he sales season. Moreover, he equilibrium sales price is increasing, whereas he equilibrium promoional effor and service level are decreasing, over he planning horizon. The service effec and he nework effec have greaer impacs upon fuure demands (and, hence, fuure profis) when he remaining planning horizon is longer. Therefore, o adapively balance he exploiaion-inducion radeoff hroughou he sales season, all he firms increase heir sales prices and decrease heir promoional effors and service levels owards he end of he sales season. Our analysis jusifies he widely used inroducory price and promoion sraegy wih which firms offer discouns and launch promoional campaigns a he beginning of a sales season o arac more early purchases (see, e.g., [38, 134, 65]). To summarize, under he service effec and he nework effec, he compeing firms have o rade off beween generaing curren profis and inducing fuure demands. To effecively balance he exploiaion-inducion radeoff, he firms should (a) increase promoional effors, (b) offer price discouns, and (c) improve service levels. Moreover, he exploiaion-inducion radeoff is more inensive (a) wih sronger service effec and nework effec, or (b) a he early sage of he sales season. 65

79 3.5 Promoion-Firs Compeiion In his secion, we consider he promoion-firs compeiion (PF) model, i.e., in each period, each firm i firs selecs is promoional effor and hen, afer observing he curren-period promoional effors of all firms, chooses a combined sales price and service level sraegy. This model is suiable for he scenario in which he sickiness of marke expanding choices is much higher han ha of sales price and service level choices. For example, due o he long leadime for echnology developmen, decisions on research and developmen effor are usually made well in advance of sales price and service level decisions. Employing he linear separabiliy approach, we will show ha, in he PF model, he firms engage in a hree-sage compeiion in each period, he firs sage on promoional effor, he second on sales price, and he las on service level. We will also demonsrae ha he exploiaion-inducion radeoff has more involved managerial implicaions in he PF model han is implicaions in he SC model. In he SC model, he compeing firms balance he exploiaion-inducion radeoff iner-emporally, whereas he firms in he PF model balance his radeoff boh iner-emporally and inra-emporally. For racabiliy, we make he following addiional assumpion hroughou his secion: ρ i, (p ) = ϕ i, θ ii, p i, + θ ij, p j,, for each i and, (3.18) j i where ϕ i,, θ ii, > 0 and θ ij, 0 for each i, j, and. Moreover, we assume ha he diagonal dominance condiions hold for each ρ i, ( ), i.e., for each i and, θ ii, > j i θ ij, and θ ii, > j i θ ji,. In addiion, we make he same assumpion as [9] as follows: Assumpion For each i and, he minimum [maximum] allowable price p i, [ p i, ] is sufficienly low [high] so ha i will have no impac on he equilibrium marke behavior. We will give a sufficien condiion for Assumpion in he discussion afer Proposiion Equilibrium Analysis In his subsecion, we use he linear separabiliy approach o characerize he pure sraegy MPE in he PF model. In his model, a (pure) Markov sraegy profile of firm i in 66

80 period is given by σ pf i, = (γpf i, (, ), ppf i, (,, ), xpf i, (,, )), where γpf i, (I, Λ ) prescribes he promoional effor given he sae variable (I, Λ ), and (p pf i, (I, Λ, γ ), x pf i, (I, Λ, γ )) prescribes he sales price and he pos-delivery invenory level, given he sae variable (I, Λ ) and he curren period promoional effor vecor γ. Le γ pf (, ) := (γ1,(, pf ),, γ pf N, (, )), p pf (,, ) := (p pf 1,(,, ),, p pf N, use σ pf (,, )), and xpf (,, ) := (x pf 1,(,, ),, x pf N, (,, )). We o denoe he (pure) sraegy profile of all firms in he subgame of period, which prescribes heir (pure) sraegies from period o he end of he planning horizon. To evaluae he expeced payoff of each firm i in each period for any given Markov sraegy profile σ pf in he PF model, le V i, (I, Λ σ pf ) = he oal expeced discouned profi of firm i in periods,, 0, when saring period wih he sae variable (I, Λ ) and he firms play sraegy σ pf in periods, 1,, 1. Thus, by backward inducion, V i, (, σ pf ) saisfies he following recursive scheme for each firm i and each period : V i, (I, Λ σ pf where ) = J i, (γ pf (I, Λ ), p pf (I, Λ, γ pf (I, Λ )), x pf (I, Λ, γ pf (I, Λ )), I, Λ σ 1), pf J i, (γ, p, x, I, Λ σ pf 1) =E{p i, D i, (γ, p ) w i, (x i, I i, ) h i, (x i, D i, (γ, p )) + b i, (x i, D i, (γ, p )) ν i, (γ i, )D i, (γ, p ) + δ i V i, 1 (x D (γ, p ), η (z, D (γ, p ), Λ, Ξ ) σ pf 1) I, Λ }, and V i,0 (I, Λ ) = w i,0 I i,0. We now define he pure sraegy MPE in he PF model. (3.19) 67

81 Definiion A (pure) Markov sraegy σ pf = {(γ pf i, (, ), p pf (,, ), xpf (,, )) : 1 i N, T 1} is a pure sraegy MPE in he PF model if and only if, for each firm i, period, and sae variable (I, Λ ) S, (p pf i, (I, Λ, γ ), x pf i, (I, Λ, γ )) =argmax pi, [p i,, p i, ],x i, min{0,i i, }[J i, (γ, [p i,, p pf i, (I, Λ, γ )], [x i,, x pf i, (I, Λ, γ )], I, Λ σ pf 1)], for all γ ; and i, i, γ pf i, (I, Λ ) =argmax γi, [J i, ([γ i,, γ pf i, (I, Λ )], p pf (I, Λ, [γ i,, γ pf i, (I, Λ )]), x pf (I, Λ, [γ i,, γ pf i, (I, Λ )]), I, Λ σ 1)]. pf (3.20) Definiion suggess ha a pure sraegy MPE in he PF model is a (pure) Markov sraegy profile ha saisfies subgame perfecion in each sage of he compeiion in each period. The following heorem shows ha here exiss a pure sraegy MPE in he PF model. Theorem The following saemens hold for he PF model: (a) There exiss a pure sraegy MPE σ pf = {(γ pf i, i N, T 1}. (, ), p pf (,, ), xpf (,, )) : 1 (b) For each pure sraegy MPE σ pf, here exiss a series of vecors {β pf : T 1}, where β pf = (β1,, pf β2,, pf, β pf N, ) wih βpf i, i, i, > 0 for each i and, such ha V i, (I, Λ σ pf ) = w i, I i, + β pf i, Λ i,, for each i,, and (I, Λ ) S. (3.21) (c) If ν i, (γ i, ) = γ i, for each i and, σ pf is he unique MPE in he PF model. Theorem demonsraes he exisence of a pure sraegy MPE in he PF model. As in he SC model, in Theorem 3.5.1(b), we show ha, for each pure sraegy MPE σ pf, he associaed profi funcion of each firm i in each period is linearly separable in is own saring invenory level I i, and marke size Λ i,. We refer o he consan β pf i, as he PF marke size coefficien of firm i in period, which measures he exploiaioninducion radeoff inensiy in he PF model. Theorem 3.5.1(c) shows ha he MPE in he PF model is unique if ν i, (γ i, ) = γ i,, i.e., he promoional effor γ i, is he acual 68

82 per-uni demand marke expanding expendiure of firm i in period. For he res of his secion, we assume ha ν i, (γ i, ) = γ i, for each i and, and, hence, σ pf is he unique pure sraegy MPE in he PF model. We use {β pf size coefficien associaed wih σ pf hereafer. : T 1} o denoe he PF marke The linear separabiliy of V i, (, σ pf ) enables us o have a sharper characerizaion of MPE in he PF model. As in he SC model, we can rewrie he objecive funcion of firm i in period as follows. J i, (γ, p, x, I, Λ σ pf 1) =E{p i, D i, (γ, p ) w i, (x i, I i, ) h i, (x i, D i, (γ, p )) + b i, (x i, D i, (γ, p )) ν i, (γ i, )D i, (γ, p ) + δ i V i, 1 (x D (γ, p ), η (z, D (γ, p ), Λ, Ξ ) σ pf 1) I, Λ } =E{p i, Λ i, d i, (γ, p )ξ i, w i, (y i, Λ i, d i, (γ, p ) I i, ) h i, (y i, Λ i, d i, (γ, p ) Λ i, d i, (γ, p )ξ i, ) + b i, (y i, Λ i, d i, (γ, p ) Λ i, d i, (γ, p )ξ i, ) ν i, (γ i, )Λ i, d i, (γ, p )ξ i, + δ i w i, 1 (y i, Λ i, d i, (γ, p ) Λ i, d i, (γ, p )ξ i, ) + δ i β pf i, 1 (Λ i,ξ 1 i, + α i, (z )Λ i, d i, (γ, p )ξ i, Ξ 2 i,) I, Λ } =w i, I i, + Λ i, {δ i β pf i, 1 µ i, + ψ i, (γ )ρ i, (p )[p i, δ i w i, 1 ν i, (γ i, ) + π pf i, (y )]}, where π pf i, (y ) =(δ i w i, 1 w i, )y i, L i, (y i, ) + δ i β pf i, 1 (κ ii,(e[y + i, ξ i,]) j i κ ij, (E[y + j, ξ j,])), and β pf i,0 :=0 for each i. (3.22) We observe from (3.22) ha, in he PF model, he payoff funcion of each firm i in each period has a nesed srucure. Hence, he compeiion in each period can be decomposed ino hree sages: In he firs sage, he firms compee on promoional effor; in he second sage, hey compee on sales price; in he hird sage, hey compee on service level. By backward inducion, we sar he equilibrium analysis wih he hird-sage service level compeiion. Le G pf,3 be he N player noncooperaive game ha represens he hird-sage service level compeiion in period, where player i has he payoff funcion 69

83 π pf i, ( ) and he feasible acion se R. The following proposiion characerizes he Nash equilibrium of he game G pf,3. Proposiion For each period, he hird-sage service level compeiion G pf,3 a unique pure sraegy Nash equilibrium y pf. Moreover, for each i, y pf i, > 0 is he unique soluion o he following equaion: (δ i w i, 1 w i, ) L i,(y pf i, ) + δ iβ pf F i, 1 i, (y pf i, )κ ii,(e(y pf i, ξ i, )) = 0. (3.23) Proposiion characerizes he unique pure sraegy Nash equilibrium of he hirdsage service level compeiion. Moreover, y pf i, yi, π pf i, (ypf has is he soluion o he firs-order condiion ) = 0. Le π pf := (π pf 1,, π pf 2,,, π pf ) be he equilibrium payoff vecor of he hird-sage service level compeiion in period, where π pf i, and, le Π pf,2 i, N, = π pf i, (ypf ). For each i (p γ ) := ρ i, (p )(p i, δ i w i, 1 ν i, (γ i, ) + π pf i, ). (3.24) Therefore, given he oucome of he firs-sage promoion compeiion, γ, we can define an N player noncooperaive game G pf,2 (γ ) o represen he second-sage price compeiion in period, where player i has he payoff funcion Π pf,2 i, ( γ ) and he feasible acion se [p i,, p i, ]. We define A as an N N marix wih enries defined by A ii, := 2θ ii, and A ij, := θ ij, where i j. By Lemma 24(a) in Appendix B.1, A is inverible. Le f (γ ) be an N dimensional vecor wih f i, (γ ) := ϕ i, + θ ii, (δ i w i, 1 + ν i, (γ i, ) π pf i, ). We characerize he Nash equilibrium of he game G pf,2 (γ ) in he following proposiion. Proposiion For each period and any given γ, he following saemens hold: (b) p pf (a) The second-sage price compeiion G pf,2 (γ ) has a unique pure sraegy Nash equilibrium p pf (γ ). (γ ) = A 1 i and j. f (γ ). Moreover, p pf i, (γ ) is coninuously increasing in γ j, for each (c) Le Π pf,2 (γ ) := (Π pf,2 1, (γ ), Π pf,2 2, (γ ),, Π pf,2 (γ )) be he equilibrium payoff vecor of he second-sage price compeiion in period, where Π pf,2 i, (γ ) = Π pf,2 i, (p pf ν i, (γ i, ) + π pf i, )2 > 0 for all i. N, (γ ) γ ). We have Π pf,2 i, (γ ) = θ ii, (p pf i, (γ ) δ i w i, 1 70

84 Proposiion shows ha, for any given promoional effor vecor γ, he secondsage price compeiion G pf,2 (γ ) has a unique pure sraegy Nash equilibrium p pf (γ ) = A 1 f (γ ). By Proposiion 3.5.2(b), we have p pf i, (0) ppf i, (γ ) p pf i, ( γ ) for each i and γ, where 0 is an N-dimensional vecor wih each enry equal o 0 and γ := ( γ 1,, γ 2,,, γ N, ). Thus, a sufficien condiion for Assumpion is ha p i, p pf i, (0) and p i, p pf i, ( γ ) for all i and. Now we sudy he firs-sage promoion compeiion in period. Le Π pf,1 i, (γ ) := Π pf,2 i, (γ )ψ i, (γ ) = θ ii, (p pf i, (γ ) δ i w i, 1 ν i, (γ i, ) + π pf i, )2 ψ i, (γ ). (3.25) Thus, we can define an N player noncooperaive game G pf,1 o represen he firs-sage promoion compeiion in period, where player i has he payoff funcion Π pf,1 i, ( ) and he feasible acion se [0, γ i, ]. We characerize he Nash equilibrium of he game G pf,1 he following proposiion. in Proposiion For each period, he following saemens hold: (a) The firs-sage promoion compeiion G pf,1 is a log-supermodular game. (b) There exiss a unique pure sraegy Nash equilibrium in he game G pf,1, which is he unique serially undominaed sraegy of G pf,1. (c) The unique Nash equilibrium of G pf,1 of equaions: for each i, γi, ψ i, (γ pf ) ψ i, (γ pf ) (d) Le Π pf,1 p pf i, (γpf, γ pf, is he soluion o he following sysem 2(1 θ ii, (A 1 ) ii )ν i,(γ pf i, ) ) δ i w i, 1 ν i, (γ pf ) + π pf N, i, i, 0, 0 if γ pf i, = 0, = 0, if γ pf i, (0, γ i, ), if γ pf i, = γ i,. (3.26) := (Π pf,1 1,, Π pf,1 2,,, Π pf,1 ) be he equilibrium payoff vecor associaed wih γ pf, i.e., Π pf,1 i, = Π pf,1 i, (γ pf ) for each i. We have Π pf,1 i, > 0 for all i. As shown in Proposiion 3.5.3, in he PF model, he firs-sage promoion compeiion in period is a log-supermodular game and has a unique pure sraegy Nash equilibrium. Moreover, he unique Nash equilibrium promoional effor vecor γ pf can be deermined 71

85 by (i) he serial eliminaion of sricly dominaed sraegies, or (ii) he sysem of firsorder condiions (3.26). The following heorem summarizes Theorem and Proposiions , and characerizes he MPE in he PF model. Theorem For each period, he following saemens hold: (a) For each i, β pf i, = δ iβ pf i, 1 µ i, + Π pf,1 i,. (b) Under he unique pure sraegy MPE σ pf, he policy of firm i in period is given by (γ pf i, (I, Λ ), p pf i, (I, Λ, γ ), x pf i, (I, Λ, γ )) =(γ pf i,, p pf i, (γ ), Λ i, y pf i, ρ i,(p pf (γ ))ψ i, (γ )). (3.27) In paricular, for any (I, Λ ), he associaed (pure sraegy) equilibrium price and invenory decisions of firm i are p pf ))ψ i, (γ pf ), respecively. i, (γpf ) and Λ i, y pf i, ρ i,(p pf (γ pf Theorem 3.5.2(a) recursively deermines he PF marke size coefficien vecors, {β pf : T 1}, associaed wih he unique pure sraegy MPE σ pf. Theorem 3.5.2(b) demonsraes ha, under he unique pure sraegy MPE σ pf, each firm i s promoion, price, and invenory decisions in each period depend on is privae informaion (i.e., (I i,, Λ i, )) only, bu no on ha of is compeiors (i.e., (I i,, Λ i, )). Hence, he unique pure sraegy MPE in he PF model has he aracive feaure ha he sraegy of each firm is coningen on is accessible informaion only. As in he SC model, we will perform some of our analysis below wih he symmeric PF model, where all firms have idenical characerisics. We use he subscrip s o denoe he case of symmeric marke in he PF model. In his case, ρ s, (p ) = ϕ s, θ sa, p i, + j i θ sb,p j, for some nonnegaive consans ϕ s,, θ sa,, and θ sb,, where θ sa, > (N 1)θ sb,. We use σ pf s o denoe he unique pure sraegy MPE in he symmeric PF model. The following proposiion characerizes σ pf s in he PF model. Proposiion The following saemens hold for he symmeric PF model. (a) For each = T, T 1,, 1, here exiss a consan β pf s, > 0, such ha V i, (I, Λ σ pf s, ) = w s, I i, + β pf s,λ i,, for all i. 72

86 (b) In each period, he hird-sage service level compeiion G pf,3 s, he payoff funcion for each firm i given by is symmeric, wih π pf i, (y ) =(δ s w s, 1 w s, )y i, L s, (y i, ) + δ s β pf s, 1(κ sa, (E[y + i, ξ i,]) j i κ sb, (E[y + j, ξ j,])). Moreover, G pf,3 s, so we use y pf s, has a unique pure sraegy Nash equilibrium, which is symmeric, [π pf s, ] o denoe he equilibrium sraegy [payoff] of each firm in G pf,3 s,. (c) In each period, he second-sage price compeiion G pf,2 s, (γ ) is symmeric if γ i, = γ j, for all 1 i, j N. In his case, G pf,2 s, (γ ) has a unique pure sraegy Nash equilibrium p pf ss,(γ ), which is symmeric (i.e., p pf ss,(γ ) = (p pf s, (γ ), p pf s, (γ ),, p pf s, (γ )) for some p pf s, (γ ) [p s,, p s, ]). (d) In each period, he firs-sage promoion compeiion G pf,1 s, G pf,1 s, is symmeric. Moreover, has a unique pure sraegy Nash equilibrium γ pf ss,, which is symmeric (i.e., γ pf ss, = (γ pf s,, γ pf s,,, γ pf s, ) for some γ pf s, [0, γ s, ]). (e) Under he unique pure sraegy MPE σs pf, he policy of firm i in period is (γ pf i, (I, Λ ), p pf i, (I, Λ, γ ), x pf i, (I, Λ, γ )) =(γ sc s,, p pf i, (γ ), Λ i, y pf s, ρ s, (p pf (γ ))ψ s, (γ )), for all (I, Λ ) and γ. In paricular, for each firm i and any (I, Λ ), he equilibrium price is p pf s, (γ pf ss,), and he equilibrium pos-delivery invenory level is Λ i, y pf s, ρ s, (p pf ss,(γ pf ss,))ψ s, (γ pf ss,). Proposiion shows ha, in he symmeric PF model, all compeing firms make he same promoional effor, charge he same sales price, and mainain he same service level in each period. The PF marke size coefficien is also idenical for all firms in each period Exploiaion-Inducion Tradeoff In his subsecion, we sudy how he exploiaion-inducion radeoff impacs he equilibrium marke oucome in he PF model. As in he SC model, we firs characerize he impac of he PF marke size coefficien vecors, {β pf : T 1}. 73

87 Theorem For each period, he following saemens hold: (a) For each i and j i, y pf i, β pf j, 1. is coninuously increasing in β pf i, 1 and independen of (b) For each i and j i, π pf i, decreasing in β pf j, 1. is coninuously increasing in β pf i, 1 and coninuously (c) For each i, j, and γ, p pf i, (γ ) is coninuously decreasing in π pf j,. (d) If he PF model is symmeric, γ pf s, is coninuously increasing in π pf s,. If, in addiion, he monooniciy condiion (3.17) holds, β pf s, π pf s, as well. (e) If he PF model is symmeric and π pf s, is increasing in βs, 1, pf γ pf s, is coninuously increasing in is coninuously increasing in β pf s, 1, whereas p pf i, (γ ) is coninuously decreasing in β pf s, 1. If, in addiion, he monooniciy condiion (3.17) holds, β pf s, is coninuously increasing in β pf s, 1 as well. Theorem demonsraes ha he marke size coefficiens {β pf i, : 1 i N, T 1} quanify he inensiy of he exploiaion-inducion radeoff in he PF model. More specifically, a larger β pf i, 1 implies more inensive exploiaion-inducion radeoff of firm i in period. As in he SC model, we use o denoe he benchmark case wihou he service effec and he nework effec, where he marke size evoluion funcion α i, ( ) 0 for each firm i and each period. Thus, he exploiaion-inducion radeoff is absen in his benchmark model, and i suffices for he firms o myopically maximize heir currenperiod profis. The following heorem characerizes he impac of he service effec and he nework effec in he PF model. Theorem π pf i, π pf i,. (a) For each firm i and each period, y pf i, ỹ pf i,, zpf i, z pf i,, and (b) For each firm i and each period, p pf i, (γ ) p pf i, (γ ) for all γ. Moreover, if he PF model is symmeric and (3.17) holds, x pf i, (I, Λ, γ ) x pf i, (I, Λ, γ ) for all i,, (I, Λ ) S, and γ [0, γ s, ] N. 74

88 (c) Consider he symmeric PF model. For each period, γ pf s, γ pf s,. Thus, γ pf i, (I, Λ ) γ pf i, (I, Λ ) for all i and all (I, Λ ) S. Consisen wih Theorem 3.4.4(a), Theorem 3.5.4(a) shows ha, he service effec and he nework effec drive he compeing firms o mainain higher service levels in he PF model. Theorem 3.5.4(b) reveals he impac of he exploiaion-inducion radeoff upon he compeing firms price and invenory sraegy in he PF model. Specifically, given any oucome of he firs-sage promoion compeiion γ, in he second-sage price compeiion, each firm i should charge a lower sales price under he service effec and he nework effec, so as o exploi he nework effec and induce higher fuure demands. Moreover, in each period, he equilibrium pos-delivery invenory levels coningen on any realized promoional effor vecor γ are also higher in he PF model under he service effec and he nework effec. Theorem 3.5.4(c) sheds ligh on how he exploiaioninducion radeoff influences he equilibrium promoion sraegies under he service effec and he nework effec. In he symmeric PF model, he equilibrium promoional effor of each firm i in each period is higher under he service effec and he nework effec. Noe ha, in he PF model, he equilibrium price and invenory oucomes under he service effec and he nework effec, p pf s, (γ pf ss,) and x pf i, (I, Λ, γ pf ss,), may be eiher higher or lower han hose wihou marke size dynamics, p pf ss,( γ pf s, ) and x pf i, (I, Λ, γ pf ss,). This phenomenon conrass wih he equilibrium marke oucomes in he SC model, where he equilibrium sales price [pos-delivery invenory level] of each firm in each period is lower [higher] under he service effec and he nework effec (i.e., Theorem 3.4.4(b-i,iii)). This discrepancy is driven by he fac ha, in he PF model, each firm observes he promoion decisions of is compeiors before making is pricing decision. Hence, under he service effec and he nework effec, he compeing firms may eiher decrease he sales prices o induce fuure demands or increase he sales prices o exploi he beer marke condiion from he increased promoional effors (recall ha γ pf s, γ pf s, ). In general, eiher effec may dominae, so we do no have a general monooniciy relaionship beween eiher he equilibrium price oucomes (i.e., p pf s, (γ pf ss,) and p pf s, ( γ pf ss,)) or he equilibrium invenory oucomes (i.e., x pf i, (I, Λ, γ pf ss,) and x pf i, (I, Λ, γ pf ss,)). Therefore, he exploiaion-inducion radeoff in he PF model is more involved han ha in he SC model. The compeing firms only need o rade off beween generaing curren profis 75

89 and inducing fuure demands ineremporally in he SC model, whereas hey need o balance his radeoff boh iner-emporally and inra-emporally in he PF model. To deliver sharper insighs on he managerial implicaions of he exploiaion-inducion radeoff, we confine ourselves o he symmeric PF model for he res of his secion. Theorem Le wo symmeric PF models be idenical excep ha one wih marke size evoluion funcions {ˆα s, ( )} T 1 and he oher wih {α s, ( )} T 1. Assume ha, for each period, (i) he monooniciy condiion (3.17) holds, and (ii) κ sb, ( ) κ 0 sb, for some consan κ 0 sb,. (a) If ˆα s, (z ) α s, (z ) for each period and all z, we have, for each period, βs,, pf ˆp pf i, (γ ) p pf i, (γ ) for all i and γ [0, γ s, ] N, and ˆγ pf s, ˆβ pf s, γ pf s,. Thus, for each period, ˆp pf i, (I, Λ, γ ) p pf i, (I, Λ, γ ) and ˆγ pf i, (I, Λ ) γ pf i, (I, Λ ) for all i, (I, Λ ) S, and γ [0, γ s, ] N. (b) If, for each period, ˆα s, (z ) α s, (z ) for all z 0 for all z i,, we have, for each period, ˆβpf s, and ˆκ sa,(z i, ) κ sa,(z i, ) βs,, pf ŷ pf s, y pf s,, ˆp pf i, (γ ) p pf i, (γ ), and ˆγ pf s, γ pf s,. Thus, for each period, ˆp pf i, (I, Λ, γ ) p pf i, (I, Λ, γ ), ˆx pf i, (I, Λ, γ ) x pf i, (I, Λ, γ ), ˆγ pf i, (I, Λ ) γ pf i, (I, Λ ) for all i, (I, Λ ) S, and γ [0, γ s, ] N. Theorem 3.5.5(a) shows ha, in he symmeric PF model, higher inensiy of he nework effec (i.e., larger α s, ( )) drives all he compeing firms o make more promoional effors and charge lower sales prices for each observed promoion vecor. Moreover, if he inensiies of boh he nework effec and he service effec (i.e., he magniudes of α s, ( ) and κ sa,( )) are higher, Theorem 3.5.5(b) demonsraes ha all he compeing firms are promped o mainain higher service levels as well. Therefore, in he PF model, he exploiaion-inducion radeoff is sronger wih more inensive service effec and nework effec. Theorem Consider he saionary symmeric PF model. Assume ha, for each period, (i) he monooniciy condiion (3.17) holds, and (ii) π pf s, For each period, he following saemens hold: (a) β pf s, βs, 1, pf y pf s, γ pf s, 1. is increasing in β pf s, 1. ys, 1, pf p pf s, (γ) p pf s, 1(γ) for each γ [0, γ s ] N, and γ pf s, 76

90 (b) p pf i, (I, Λ, γ) ppf i, 1 (I, Λ, γ), xpf i, (I, Λ, γ) xpf i, 1 (I, Λ, γ), and γpf i, (I, Λ) γ pf i, 1 (I, Λ) for each i, (I, Λ) S, and γ [0, γ s, ] N. Analogous o Theorem 3.4.6, Theorem jusifies he widely used inroducory price and promoion sraegy. More specifically, his resul shows ha if he marke is saionary and symmeric in he PF model, he compeing firms should decrease he promoional effors (i.e., γ pf s, ) and service levels (i.e., y pf s, ), and increase he sales prices coningen on any realized promoional effors (i.e., p pf s, (γ )), over he planning horizon. Hence, Theorem suggess ha, in he PF model, he exploiaion-inducion radeoff is more inensive a he early sage of he sales season han a laer sages. To conclude his secion, we remark ha, because of he aforemenioned inra-emporal exploiaion-inducion radeoff under he promoion-firs compeiion, Theorems canno give he monoone relaionships on he equilibrium oucomes of each firm i s sales price (i.e., p pf i, (I, Λ, γ pf ss,)) and pos-deliver invenory level (i.e., x pf i, (I, Λ, γ pf ss,)). 3.6 Comparison of he Two Compeiion Models As demonsraed above, he exploiaion-inducion radeoff is more involved in he PF model han ha in he SC model. In his secion, we compare he unique MPE in he SC model and ha in he PF model, and discuss how he exploiaion-inducion radeoff impacs he equilibrium marke oucomes under differen compeiions. Theorem Consider he symmeric SC and PF models. Assume ha, for each period, (i) he demand funcion ρ i, ( ) is linear and given by (3.18), (ii) ν i, (γ i, ) = γ i,, (iii) he monooniciy condiion (3.17) holds, (iv) Assumpion holds, (v) π sc s, increasing in βs, 1, sc and (vi) π pf s, (a) If β pf s, 1 βs, 1, sc y pf s, ys, sc is increasing in β pf s, 1. The following saemens hold: and γ pf s, γ sc s,. (b) For each period, here exiss an ϵ [0, 1 N 1 ], such ha, if θ sb, ϵ θ sa,, we have (i) β pf s, βs, sc and, hus, V i, (I, Λ σ pf ) V i, (I, Λ σ sc ) for each firm i and all (ii) y pf s, (iii) γ pf s, (I, Λ ) S; y sc s, ; γ sc s,. 77 is

91 Theorem shows ha, if he produc differeniaion is sufficienly high (as capured by he condiion ha θ sb, ϵ θ sa, ), he PF compeiion leads o sronger exploiaioninducion radeoff (i.e., β pf s, β sc s,). As a consequence, he compeing firms should se higher service levels and promoional effors in he PF model. Compared wih he simulaneous compeiion, he promoion-firs compeiion enables he firm o responsively adjus heir sales prices in accordance o he marke condiion and heir compeiors promoion sraegies. If he produc differeniaion is sufficienly high, such pricing flexibiliy gives rise o higher expeced profis of all firms and more inensive exploiaion-inducion radeoff in he PF model. Theorem also reveals he fa-ca effec in our dynamic compeiion model: When he price decisions are made afer observing he promoional effors in each period, he firms end o overinves in promoional effors. As shown in he lieraure (e.g., [78, 9]), one driving force for his phenomenon is ha, under he PF compeiion, he firms can charge higher prices in he subsequen price compeiion wih increased promoional effors in each period. Theorem idenifies a new driving force for he fa-ca effec: The firms under he PF compeiion make more promoional effors o balance he more inensive exploiaion-inducion radeoff herein. Therefore, our analysis delivers a new insigh o he lieraure ha he exploiaion-inducion radeoff may give rise o he fa-ca effec in dynamic compeiion. 3.7 Summary This chaper sudies a dynamic join promoion, price, and service compeiion model, in which curren decisions influence fuure demands hrough he service effec and he nework effec. Our model highlighs an imporan radeoff in a dynamic and compeiive marke: he radeoff beween generaing curren profis and inducing fuure demands (i.e., he exploiaion-inducion radeoff). We characerize he impac of he exploiaioninducion radeoff upon he equilibrium marke oucome under he service effec and he nework effec, and idenify he effecive sraegies o balance his radeoff under dynamic compeiion. We employ he linear separabiliy approach o characerize he pure sraegy MPE boh in he SC model and in he PF model. An imporan feaure of he MPE in boh models is ha he equilibrium sraegy of each firm in each period only depends on he 78

92 privae invenory and marke size informaion of iself, bu no on ha of is compeiors. Moreover, he exploiaion-inducion radeoff is more inensive if he service effec and he nework effec are sronger; and his rade-off decreases over he planning horizon. The exploiaion-inducion radeoff is more involved in he PF model han in he SC model. This is because he compeing firms need o balance his radeoff boh ineremporally and inra-emporally in he PF model, whereas hey only need o balance i iner-emporally in he SC model. More specifically, in he SC model, o effecively balance he exploiaion-inducion radeoff, he firms should (a) increase promoional effors, (b) offer price discouns, and (c) improve service levels. In he PF model, he firms should increase promoional effors under he service effec and he nework effec. Given he same promoional effor in he firs sage compeiion, he firms need o decrease heir sales prices under he nework effec. However, wih an increased promoional effor in he firs sage compeiion, he equilibrium sales prices in he second sage compeiion may eiher decrease o increase. Analogously, he equilibrium pos-delivery invenory levels may eiher decrease or increase in he PF model under he service effec and he nework effec. Finally, we idenify he fa-ca effec in our dynamic compeiion model: If he produc differeniaion is sufficienly high, under he MPE, he firms make more promoional effors in he PF model han in he SC model. The driving force of his phenomenon is ha he exploiaion-inducion radeoff is more inensive under he promoion-firs compeiion han under he simulaneous compeiion. 79

93 4. Trade-in Remanufacuring, Sraegic Cusomer Behavior and 4.1 Inroducion Governmen Subsidies 1 I is a common pracice for firms o offer rade-in rebaes o recycle used producs. For example, Apple offers boh in-sore and online rade-in programs, which allow cusomers o exchange heir used iphones, ipads, and Macs for credis o purchase new ones ([13]). Analogously, Amazon allows Kindle owners o rade in heir old producs for newer versions a a discoun price ([55]). More examples on he adopion of rade-in rebaes o collec cores for remanufacuring have been repored in indusries like furniure, carpes, and power ools, ec. (see [142]). Recycling used producs hrough rade-in rebaes has been lauded for is various benefis. From he economic perspecive, he reurn produc flow from rade-in rebaes serves as an imporan source for generaing revenue and reducing coss. Wih he recycled producs, firms can recover he residual values by eiher remanufacuring hem ino new ones or reusing heir componens and maerials. Following he lieraure (e.g., [142]), hroughou he chaper we use he erm remanufacuring o represen he general revenue-generaing process hrough recycling and recovering used producs. In pracice, he revenue-generaing/cos-saving effec of rade-in based remanufacuring could be significan. Xerox, which parly bases is remanufacuring on rade-in reurns, has saved several hundred million dollars each year, which accouns for 40%-65% of he company s manufacuring coss ([146]). From he sraegic perspecive, rade-in rebaes may improve firm profiabiliy by elevaing cusomer swiching coss ([105]), discouraging second-hand markes ([109]), increasing purchase frequency ([166]), and reducing inefficiencies arising from he lemon problem ([141]). From he environmenal perspecive, rade-in rebaes encourage cusomers o reurn used producs, hus generaing less wase and disposals. In paricular, he elecronics marke is feaured wih frequen produc inroducions and generaes more han one million ons of so-called e-wases each year ([140]). Using rade- 1 This chaper is based on he auhor s earlier work [193]. 80

94 in rebaes, Apple colleced more han 40,000 ons of e-wases in 2014, which accoun for more han 75 percen of he producs hey sold seven years earlier (see [12]). I has been empirically verified ha cusomers exhibi forward-looking behaviors in he elecronics marke due o frequen produc inroducions ([140]). In paricular, when he firm offers rade-in rebaes, sraegic cusomer behavior naurally arises, because cusomers can anicipae a possible price discoun in he fuure if making a purchase now ([166, 80, 141]). Moreover, advances in informaion echnology enable cusomers o easily obain produc and price informaion. For example, Kayak launched he price forecas service o help cusomers decide when o book a fligh ([68]). As a consequence, sraegic cusomer behavior has become more prevalen in oday s business world. Alhough sraegic cusomer behavior has been widely acknowledged in he lieraure, i is no clear how such behavior would affec he economic and environmenal benefis of rade-in remanufacuring. Governmens around he world have made remendous effors o promoe recycling and remanufacuring used producs. One commonly used sraegy is o provide subsidies for remanufacuring. For insance, in January 2015, he Chinese governmen released a policy o subsidize he use of remanufacured vehicle engines and ransmissions ([44]). Analogously, he Chinese governmen esablished a special fund in 2011 o provide subsidies o companies engaged in he recycling and recovering of wase elecrical and elecronic producs (e.g., [174]). As anoher example, a recen repor backed by he Scoish governmen and Zero Wase Scoland (ZWS) concluded ha Scoland was in a unique posiion o develop a circular economy and called for governmen subsidies o help boos closed loop recycling, reuse, bio-refining, and remanufacuring ([161]). In he lieraure, he effecs of governmen subsidies for remanufacuring/rade-in remanufacuring have been sudied in seings wihou explicily modeling cusomer behaviors (e.g., [128, 118]). Despie is imporance, he quesion of how he governmen should design he subsidizaion policy under sraegic cusomer behavior o induce he social opimum has no been horoughly explored. The primary goal of his chaper is o analyze how sraegic cusomer behavior influences he value of rade-in remanufacuring from he perspecives of he firm, he environmen, and he governmen. For his purpose, we develop a wo-period model in which a profi-maximizing firm sells wo generaions of a produc in an ex-ane uncerain marke. 81

95 To highligh he impac of sraegic cusomer behavior, we consider wo scenarios, one wih sraegic cusomers and he oher wih myopic cusomers. Sraegic cusomers make heir purchasing decisions based on boh curren and anicipaed fuure uiliies, whereas myopic cusomers make decisions based on curren uiliies only. In he firs period, he firm sells he firs-generaion produc in he marke. In he second period, he firm sells he second-generaion produc o new cusomers (who have no purchased in he firs period); meanwhile he firm offers rade-in rebaes hrough which repea cusomers (who have purchased in he firs period) exchange used producs for new second-generaion ones a a discouned price. The firm generaes revenue by remanufacuring he recycled producs. This remanufacuring process also reduces he (negaive) environmenal impac of he business, because i decreases energy and raw maerial consumpion, as well as wase disposal. We model he governmen as a policy-maker whose subsidy/ax policy may affec he firm s pricing and producion sraegy as well as he cusomers purchasing decisions. The objecive of he governmen is o maximize he social welfare, i.e., he sum of firm profi and cusomer surplus less environmenal impac. We find ha sraegic cusomer behavior has imporan implicaions on he pracice of rade-in remanufacuring. Firs, under rade-in remanufacuring, he firm can earn a higher profi wih sraegic cusomers han wih myopic cusomers if he revenuegeneraing effec of remanufacuring is sufficienly srong. In oher words, sraegic cusomer behavior may improve firm profi, which is in conras wih he commonly believed noion ha sraegic cusomer behavior hurs firm profi. When he firm employs radein remanufacuring, sraegic cusomers will anicipae he fuure rade-in rebae (i.e., price discoun) in he second period, which depends on he addiional value generaed by remanufacuring. Noe ha a deeper discoun in he second period will induce a higher willingness-o-pay in he firs period. Thus, sraegic cusomers may be willing o pay a higher firs-period price han myopic cusomers if he revenue-generaing effec of remanufacuring is srong enough, which allows he firm o exrac a higher profi. This implies ha when early purchases (of sraegic cusomers) can be induced by he addiional benefis (i.e., he rade-in opion and he deep discoun brough by remanufacuring), a firm may benefi from sraegic cusomer behavior. Wihou rade-in remanufacuring, however, sraegic cusomer behavior always hurs he firm s profi, as repored in he lieraure. 82

96 Second, wih sraegic cusomers, he adopion of rade-in remanufacuring may creae a ension beween firm profiabiliy and environmenal susainabiliy. Trade-in rebae essenially offers an early purchase reward and hus can deliver addiional value by exploiing he forward-looking behavior of sraegic cusomers. As a resul, rade-in remanufacuring is more valuable o he firm wih sraegic cusomers han wih myopic cusomers. However, he early-purchase inducing effec of rade-in remanufacuring also promps he firm o increase producion quaniies significanly under sraegic cusomer behavior. The increased producion quaniies may ouweigh he environmenal advanage of remanufacuring unless he uni environmenal benefi of remanufacuring is very high. Hence, rade-in remanufacuring generally hurs he environmen wih sraegic cusomers. Moreover, we find ha rade-in remanufacuring decreases cusomer surplus, and consequenly, he social welfare may decrease as well. Therefore, our resuls call for cauion on he adopion of rade-in remanufacuring under sraegic cusomer behavior, because i is likely o be severely derimenal o he environmen and he sociey. Wih myopic cusomers, however, rade-in manufacuring generally benefis he environmen. The price discriminaion effec of rade-in rebaes increases he expeced uni profi from new cusomers in he second period. This effec drives he firm o decrease he firs-period producion quaniy and hus increase he poenial second-period marke size of new cusomers. As long as he uni environmenal benefi of remanufacuring is no oo low, rade-in remanufacuring induces lower producion quaniies and, hus, benefis he environmen. Therefore, for he scenario wih myopic cusomers, he ension beween firm and environmen does no exis in general. The ension beween firm profiabiliy and environmenal susainabiliy under sraegic cusomer behavior moivaes us o sudy how governmen inervenion can achieve he socially opimal oucome. Specifically, we focus on he subsidizaion policy he governmen can use o promoe he aciviies of used producs recycling (e.g., rade-in rebaes, remanufacuring, and ake-backs; see [128, 170, 161]). An inuiive policy observed in pracice is o subsidize he firm/cusomers for selling/purchasing remanufacured producs. However, we find ha subsidizing remanufacured producs alone acually hurs he environmen and is no sufficien o achieve he social opimum. This cauions he policy-makers abou how o promoe remanufacuring hrough subsidizaion. Wih eiher sraegic or myopic cusomers, in order o induce he social opimum, i suffices for he 83

97 governmen o use a simple linear subsidy/ax scheme for he sales of boh produc generaions and remanufacuring. In addiion, if he oal uni economic and environmenal value of remanufacuring is low, he governmen should provide more subsidies o he firm wih sraegic cusomers han wih myopic cusomers, and vice versa. The res of he chaper is organized as follows. In Secion 4.2, we posiion his chaper in he relaed lieraure. The base model and he equilibrium analysis are presened in Secion 4.3. In Secion 4.4, we analyze he impac of rade-in remanufacuring upon he firm and he environmen. In Secion 4.5, we characerize he governmen policy ha can induce he social opimum. This chaper concludes wih Secion 4.6. All proofs are given in Appendix C Relaed Research This chaper builds upon wo sreams of research in he lieraure: (1) remanufacuring and closed-loop supply chain managemen, and (2) sraegic cusomer behavior. There is a rapidly growing sream of lieraure on remanufacuring and closed-loop supply chain managemen. Comprehensive reviews of his lieraure are given by [91] and [154]. Several papers sudy he opimal invenory policy wih reurn flows of used producs; see, e.g., [167, 163], and [87]. These papers focus on characerizing he cosminimizing invenory policy in a sysem wih exogenously given demand rae, price, and remanufacurabiliy. More recenly, researchers sar o explicily model some sraegic issues relaed o remanufacuring, such as used produc acquisiion, demand segmenaion, produc cannibalizaion, and compeiion. [146] sudy he opimal reverse channel srucure for he collecion of used producs from cusomers. [74] analyze he compeiion beween new and remanufacured producs (i.e., he cannibalizaion effec) and characerize he opimal recovery sraegy. When remanufacurabiliy is an endogenous decision, [59] invesigae a join pricing and producion echnology selecion problem of a manufacurer who sells a remanufacurable produc o heerogeneous cusomers. Under he cannibalizaion effec of remanufacured producs, [75] sudy he compeiion beween an original equipmen manufacurer (OEM) and an independen operaor who only sells remanufacured producs. [14] show ha remanufacuring could serve as a markeing sraegy o arge he cusomers in he green segmen and, hence, enhance he profiabiliy of he OEM. [133] characerize he opimal relicensing sraegy of an 84

98 OEM o miigae he cannibalizaion effec in he secondary marke. [81] sudy how he rae of produc innovaion affecs he firm s reuse and remanufacuring decisions. [90] invesigae he qualiy design and environmenal consequences of green consumerism wih remanufacuring. There are papers ha address behaviorial issues relaed o remanufacuring such as how he remanufacured producs affec he cusomer valuaion of new producs ([3]). Governmen regulaions on remanufacuring have also been sudied in he lieraure; see, e.g., [118]. [56] sudy he impac of demand uncerainy on governmen subsidies for green echnology adopion. The impac of rade-in rebaes has also received some aenion in he remanufacuring lieraure. For example, [142] examine he value of price discriminaion for new and repea cusomers wih differeniaed ages (and qualiies) of he reurned producs. The impac of sraegic cusomer behavior has received an increasing amoun of aenion in he operaions managemen lieraure. [149] provide a comprehensive review on cusomer behavior models in revenue managemen and aucions. [20] show ha raional cusomers drive a monopolis firm o charge a lower price for any given sae in each period. [157] characerizes he opimal pricing sraegy wih a heerogenous group of sraegic cusomers. When cusomers are forward-looking, [17] sudy he opimal single mark-down iming wih finie invenories. In a newsvendor model where cusomers anicipae he likelihood of sockou before deciding wheher o make a purchase, [58] and [159, 160] sudy he impac of sraegic cusomer behavior on newsvendor profi, supply chain performance, and he role of produc availabiliy in inducing demand, respecively. [115] propose he effecive capaciy raioning sraegy o induce early purchases wih sraegic cusomers. [40, 41] and [176] demonsrae how quick response can be employed o miigae sraegic cusomer behavior. [99] sudy opaque selling and las-minue selling wih sraegic cusomers in a revenue managemen framework. In a cheap alk framework, [8] show ha, hough nonverifiable, he availabiliy informaion improves he profi of a service firm and he expeced uiliy of is cusomers. [7] furher demonsrae ha a single reailer providing availabiliy informaion on is own canno creae any credibiliy wih homogeneous cusomers. [54] invesigae he inegraed informaion and pricing sraegy wih sraegic cusomers and he cusomer preorders before produc release. [135] demonsraes how verical produc differeniabiliy helps miigae sraegic cusomer behavior. Recenly, here are papers addressing he opimal sraegy wih muliple produc 85

99 inroducions and sraegic cusomer behavior. For example, in he presence of sraegic consumers, [113] characerize he opimal produc rollover sraegies, whereas [116] sudy he new produc launch sraegy. There are a few papers ha invesigae rade-in rebaes wih forward-looking cusomers. [166] show ha, under sraegic cusomer behavior, rade-ins can serve as a mechanism o achieve price commimen. [80] sudy he monopoly pricing of overlapping generaions of a durable good wih and wihou a second-hand marke. In an infinie-horizon model seing, [141] demonsrae ha rade-in rebaes can alleviae he inefficiencies arising from he lemon problem. This chaper conribues o he aforemenioned sreams of research by sudying he ineracion beween rade-in remanufacuring and sraegic cusomer behavior, and how such ineracion affecs he economic and environmenal values of rade-in remanufacuring. We demonsrae ha sraegic cusomer behavior may benefi he firm, bu give rise o a ension beween firm profiabiliy and environmenal susainabiliy under rade-in remanufacuring. In addiion, we characerize how he governmen can achieve he social opimum, using a simple linear subsidy/ax scheme wih eiher sraegic or myopic cusomers. 4.3 Model and Equilibrium Analysis Model Seup We consider a monopoly firm (he) in he marke who sells a produc o cusomers (she) in a wo-period sales horizon. In he firs period, he firm produces he firsgeneraion produc a a uni producion cos c 1. The poenial marke size X, which is he oal number of poenial cusomers, is ex-ane uncerain. The cusomers are infiniesimal, each requesing a mos one uni of he produc in any period. Demand uncerainy is a common feaure wih new produc inroducion, bu he firm can obain more accurae demand informaion as he marke maures. Hence, in he second period, he marke uncerainy is resolved so he realized marke size X becomes known o he firm. Wihou loss of generaliy, we assume ha X > 0, wih a disribuion funcion F ( ) and densiy funcion f( ) = F ( ). 86

100 A cusomer s valuaion V for he firs-generaion produc is independenly drawn from a coninuous disribuion wih a disribuion funcion G( ) suppored on [v, v] (0 v < v). We call he cusomer wih produc valuaion V he ype-v cusomer. A he beginning of he sales horizon, each cusomer only knows he disribuion of her own valuaion G( ), bu no he realizaion V. This assumpion capures he cusomers uncerainies abou he qualiy, and fis he siuaion where he produc is brand new. In he second period, all cusomers observe heir own ype V. For he cusomers who purchased he produc in period 1, hey learn heir ype V by consuming he produc. For he cusomers who did no purchase he produc in period 1, hey learn is qualiy and fi (hus, heir ype V ) hrough social learning plaforms (e.g., Facebook and Amazon cusomer review sysems). Hence, he cusomers are homogeneous ex ane (i.e., a he beginning of period 1), bu heerogeneous ex pos (i.e., a he beginning of period 2). This is a common seing in he models concerning sraegic cusomer behavior (see, e.g., [175, 158]). We assume ha he valuaion disribuion G( ) has an increasing failure rae, i.e., g(v)/ḡ(v) is increasing in v, where g( ) = G ( ) is he densiy funcion and Ḡ( ) = 1 G( ). This is a mild assumpion and can be saisfied by mos commonly used disribuions. Le µ := E(V ) > c 1, i.e., in expecaion, a cusomer s valuaion exceeds he producion cos. The firm offers an upgraded version of he produc in period 2. This is a cusomary pracice for produc caegories like consumer elecronics, home appliances, and furniure. A ype-v cusomer has a valuaion of (1 + α)v for he upgraded second-generaion produc, where α 0 is exogenously given and capures he innovaion level (e.g., he improved feaures) of he upgraded produc. Accordingly, le he producion cos of he second-generaion produc be c 2. To model he produc depreciaion, we ake he approach of [166]: If a ype-v cusomer has already bough he produc in period 1, her valuaion of consuming he used produc in period 2 is (1 k)v, where k [0, 1] refers o he depreciaion facor. Specifically, if k = 0, he produc is compleely durable; if k = 1, he produc is compleely useless afer he firs period (eiher he produc is worn ou or he echnology is obsolee). Therefore, he willingness-o-pay of a ype-v cusomer in period 2 is (1+α)V if she did no purchase he produc in period 1 (i.e., a new cusomer), and is (1 + α)v (1 k)v = (k + α)v if she purchased he produc in period 1 (i.e., a repea cusomer). 87

101 As widely recognized in he lieraure, he firm can generae revenue from remanufacuring by reusing he maerials and componens of he recycled producs (see [146, 142]). We now model he revenue-generaing effec of remanufacuring. There are wo ypes of remanufacuring in our model. Firs, he firm recycles he unsold firs-generaion producs a he end of period 1. The recycled lefover invenory in he firs period is remanufacured and can generae a ne per-uni revenue r 1 (r 1 < c 1 ) for he firm. Tha is, in he base model we assume no excess invenory is carried over o he second period. This assumpion applies when he invenory holding cos is sufficienly high or he firm does no wan o dilue he sales of he newer generaion produc, which is usually he case in he elecronics marke. Moreover, his assumpion faciliaes he echnical racabiliy of our model. Our resuls can be exended o he seing wherein he firm may hold lefover invenory and offer boh produc generaions in he second period. The second ype of remanufacuring is by using he reurned producs in period 2, i.e., cusomers who bough he produc in period 1 can rade he old produc for a second-generaion one a a discouned price in period 2. The ne revenue of remanufacuring from a used produc in period 2 is r 2 (r 2 < c 2 ). Following [146], we assume ha all remanufacured producs are upgraded o he qualiy sandards of new ones, so ha consumers canno disinguish hem from newly made producs. Relaxing his assumpion will no affec our qualiaive resuls. The environmenal impac of he produc is he aggregae (negaive) lifeime impac of he produc on he environmen. The oal environmenal impac is he producion quaniy of he produc muliplied by he per-uni impac (see, e.g., [162, 4]). Le κ 1 > 0 denoe he uni environmenal impac of he firs-generaion produc. Analogously, we denoe κ 2 > 0 as he uni environmenal impac of he second-generaion produc. Such impac may refer o he use of naural resources, emission of harmful gases, and generaion of solid wases. Moreover, κ 1 and κ 2 can be esimaed by he convenional life-cycle analysis (see, e.g., [4]). To model he environmenal benefi of remanufacuring, le ι 1 (ι 1 < κ 1 ) be he uni environmenal benefi of recycling he firs-period lefover invenory, and ι 2 (ι 2 < κ 2 ) be ha of recycling he used producs hrough rade-in rebaes. Here, ι 1 and ι 2 refer o he reducions in boh he producion environmenal impac in period 2 and he end-of-use and end-of-life produc disposal, by recycling and reusing he maerials and componens of he firs-generaion producs. To capure he environmenal advanage 88

102 of he second-generaion produc, we assume ha κ 1 ι κ 2 ( = 1, 2), i.e., he oal environmenal impac of he firs-generaion produc dominaes ha of he secondgeneraion produc even if he end-of-use/end-of-life firs-generaion producs are recycled and remanufacured. The sequence of evens unfolds as follows. A he beginning of period 1, he firm announces he price p 1 and decides he producion quaniy Q 1. Each cusomer observes p 1, bu no Q 1, and makes her decision wheher o order he produc or o wai unil period 2. The firs-period demand X 1 X is hen realized, he firm collecs his firsperiod revenue, and all cusomers say in he marke. Noe ha X 1 is deermined by he collecive effec of all cusomers purchasing behaviors. If X 1 Q 1, any cusomer who requess a produc can ge one in period 1. Oherwise, X 1 > Q 1, hen he Q 1 producs are randomly allocaed o he demand, and X 1 Q 1 cusomers have o wai due o he limied availabiliy. A he end of period 1, he firm recycles and remanufacures he lefover invenory. A he beginning of period 2, he firm learns he realized oal marke size X, and each individual cusomer learns her ype V. The firm hen announces he price p n 2 for new cusomers as well as he rade-in price p r 2 p n 2 (p n 2 p r 2 is he rade-in rebae); all new cusomers decide wheher o purchase he second-generaion produc, whereas all repea cusomers decide wheher o rade heir used producs in for new second-generaion ones. Finally, he firm produces he second-generaion producs, recycles and remanufacures he used producs from repea cusomers, and collecs he second-period revenue. For noaional convenience, we will use E[ ] o denoe he expecaion operaion, x y o denoe he minimum of wo numbers x and y, and ϵ d 1 = ϵ 2 o denoe ha wo random variables ϵ 1 and ϵ 2 follow he same disribuion. The scenario wih myopic cusomers will be denoed wih Equilibrium Analysis We consider wo scenarios, one wih sraegic cusomers and he oher wih myopic cusomers. Sraegic cusomers maximize heir oal expeced surplus over he wo-period horizon, whereas myopic cusomers maximize heir expeced curren-period surplus in each period. In boh scenarios, he firm seeks o maximize his oal expeced profi over he enire horizon. For exposiional convenience, we assume here is a common discoun facor for he firm and he cusomers in period 2, denoed by δ (0, 1]. To highligh 89

103 he impac of sraegic cusomer behavior upon he economic and environmenal values of rade-in remanufacuring, we assume ha he cusomers are eiher purely sraegic or compleely myopic. In realiy, he acual cusomer behavior may ake a form beween hese wo exremes. Our model can be easily adaped o capure his siuaion by fixing he discoun facor of he firm a δ, and allowing he discoun facor of he cusomers δ c o vary in he inerval [0, δ]. The higher he δ c, he greaer he cusomers concern abou fuure uiliies, and hus he more sraegic hey are. In paricular, δ c = δ (δ c = 0) corresponds o he scenario wih purely sraegic (myopic) cusomers. To characerize he game oucome, we adop he raional expecaion (RE) equilibrium concep. The RE equilibrium was proposed by [129] and has been widely used in he operaions managemen lieraure (e.g., [159, 160, 40, 41]). Using backward inducion, we sar wih he decisions of he wo paries in period 2. There are X n 2 = X (X 1 Q 1 ) new cusomers and X r 2 = X 1 Q 1 repea cusomers in he marke. Noe ha, since period 2 is he final period, sraegic and myopic cusomers exhibi he same purchasing behavior herein. Hence, regardless of cusomer behavior, he firm should adop he same pricing sraegy in period 2 as well. Given (X n 2, X r 2), le p n 2(X n 2, X r 2) and Q n 2(X n 2, X r 2) be he equilibrium price and producion quaniy for new cusomers in period 2. Analogously, p r 2(X n 2, X r 2) and Q r 2(X n 2, X r 2) are he equilibrium rade-in price and producion quaniy for repea cusomers, respecively. Correspondingly, we denoe π 2 (X n 2, X r 2) as he equilibrium second-period profi of he firm. Lemma 7 (a) For any (X2 n, X2), r p n 2(X2 n, X2) r p n 2 and p r 2(X2 n, X2) r p r 2, where ( ) p n 2 and p2 r = argmax 1 + α p r 2 0(p r 2 c 2 +r 2 )Ḡ p n 2 = argmax p n 2 0(p n 2 c 2 )Ḡ Moreover, p r 2 < p n 2 if and only if k < 1 or r 2 > 0. (b) For any (X n 2, X r 2), Q n 2(X n 2, X r 2) = Ḡ ( p n 2 1+α ) X n 2, and Q r 2(X n 2, X r 2) = Ḡ ( p r 2 k + α ( ) p r 2 X r k+α 2. ). (c) There exis wo posiive consans β n and β r, such ha π 2 (X n 2, X r 2) = β nx n 2 + β r X r 2 for all (X n 2, X r 2), where β n = max p n 2 0(pn 2 c 2 )Ḡ ( p n 2 ) 1 + α ( ) p and βr r = max p r 2 0(pr 2 c 2 + r 2 )Ḡ 2. k + α Lemma 7 characerizes he equilibrium pricing and producion sraegy of he firm in period 2. Specifically, boh he equilibrium price for new cusomers and he equilibrium rade-in price are independen of he realized marke size (X n 2, X r 2). Hence, he 90

104 equilibrium producion quaniy for new (repea) cusomers is a fixed fracion of he corresponding marke size X2 n (X2) r in period 2. As long as he used produc is no compleely useless o cusomers in period 2 (i.e., k < 1) or remanufacuring generaes a posiive revenue (i.e., r 2 > 0), he firm offers posiive rade-in rebaes o repea cusomers. Moreover, he equilibrium profi of he firm in period 2, π 2 (X2 n, X2), r is linearly separable in X2 n and X2. r We now analyze he firm s and he cusomers decisions in period 1. We begin wih he cusomers purchasing behavior. A sraegic cusomer forms beliefs abou he firsperiod produc availabiliy probabiliy a, he second-period price for new cusomers p n 2, and he second-period rade-in price p r 2, where a, p n 2, and p r 2 are all nonnegaive random variables. Based on he belief vecor (a, p n 2, p r 2) and he observed firs-period price p 1, she compues he expeced uiliy of making an immediae purchase, U p := a(e[v ] + δe[(k + α)v p r 2] + p 1 ) + (1 a)δe[(1 + α)v p n 2] +, and he expeced uiliy of waiing, U w := δe[(1+α)v p n 2] +. Hence, he firs-period reservaion price of a sraegic cusomer, ξ r, is given by ξ r := max{p 1 : U p U w }, and she will make a purchase in period 1 if and only if p 1 ξ r. The decision-making process of a myopic cusomer is simpler because she does no form beliefs abou he firs-period availabiliy and second-period prices, bu bases her purchasing decision on he curren uiliy only. Hence, he firs-period reservaion price for a myopic cusomer equals her expeced valuaion of he produc, i.e., ξ r = E[V ] = µ. Following he sandard approach in he markeing ([175]) and he sraegic cusomer behavior ([159, 41]) lieraure wih homogeneous cusomers, we assume ha all cusomers will make a purchase in period 1 if p 1 equals heir reservaion prices (ξ r for sraegic cusomers and ξ r for myopic cusomers). Thus, wih sraegic (myopic) cusomers, he firs-period demand, X 1, is given by X 1 = X 1 {p1 ξ r} (X 1 = X 1 {p1 ξ ). r} Nex, we consider he firm s problem in period 1. The firm does no know he exac reservaion price of sraegic (myopic) cusomers ξ r ( ξ r ), bu forms a belief r 1 ( r 1 ) abou i. To maximize his expeced profi, he firm ses he firs-period price p 1 ( p 1 ) equal o he expeced reservaion price r 1 ( r 1 ), which is he highes price (he firm believes) sraegic (myopic) cusomers are willing o pay in he firs period. Thus, he firm believes ha he firs-period demand X 1 = X. Thus, he second-period marke size of new cusomers is X2 n = (X Q 1 ) +, and ha of repea cusomers is X2 r = X Q 1. Moreover, he firm ses he firs-period producion quaniy Q 1 o maximize he oal expeced profi wih sraegic 91

105 (myopic) cusomers Π f (Q 1 ) ( Π f (Q 1 )), where Π f (Q 1 ) = E{p 1 (X Q 1 ) c 1 Q 1 + r 1 (Q 1 X) + + δπ 2 (X n 2, X r 2)} and Π f (Q 1 ) = E{ p 1 (X Q 1 ) c 1 Q 1 + r 1 (Q 1 X) + + δπ 2 (X n 2, X r 2)}, wih p 1 = r 1, p 1 = r 1, X n 2 = (X Q 1 ) +, and X r 2 = X Q 1. Finally, under he RE equilibrium, all beliefs are raionally formulaed and hus consisen wih he acual oucomes. Le (p 1, Q 1, ξ r, r, a, p n 2, p r 2 ) and ( p 1, Q 1, ξ r, r ) be he RE equilibria in he marke wih sraegic and myopic cusomers, respecively. For concision, he formal definiions of he RE equilibria in boh scenarios are given in Appendix C.1. To characerize he RE equilibrium, we define wo auxiliary variables m 1 := µ + δ[β r β n + E((k + α)v p r) + E((1 + α)v p n) + ] and m 1 := µ + δ(β r β n). As will be clear in our subsequen analysis, m 1 ( m 1) is he firs-period effecive marginal revenue wih sraegic (myopic) cusomers, which summarizes he impac of he second-period decisions on he firs-period firm profi. Based on Lemma 7, we can characerize he RE equilibrium marke oucome in he scenario wih eiher sraegic or myopic cusomers. Theorem (a) Wih sraegic cusomers, an RE equilibrium (p 1, Q 1, ξ r, r, p n 2, p r 2, a ) exiss wih (i) p 1 = µ + δ[e((k + α)v p r 2 ) + E((1 + α)v p n 2 ) + ]; and (ii) Q 1 = F 1 ( c 1 r 1 ). Moreover, all RE equilibria give rise o m 1 r 1 he idenical expeced oal profi of he firm, Π f r 1 )Q 1 + δβ ne(x). = (m 1 r 1 )E(X Q 1) (c 1 (b) Wih myopic cusomers, an RE equilibrium ( p 1, Q 1, ξ r, r ) exiss wih (i) p 1 = µ; and (ii) Q 1 = F 1 ( c 1 r 1 ). Moreover, all RE equilibria give rise o he idenical m 1 r 1 expeced oal profi of he firm, Π f = ( m 1 r 1 )E(X Q 1) (c 1 r 1 ) Q 1 + δβ ne(x). Theorem 4.3.1(a) and (b) characerize he RE equilibrium marke oucomes in he scenarios wih sraegic and myopic cusomers, respecively. In each scenario, he firsperiod price equals he corresponding expeced reservaion price of he cusomers, and he firs-period producion quaniy can be deermined by he soluion of a corresponding newsvendor problem. In equilibrium, he oal environmenal impac should be he difference beween he oal environmenal impac of producion/disposal and he oal environmenal benefi of remanufacuring. Hence, he equilibrium environmenal impac wih sraegic cusomers is I e = E{κ 1 Q 1 + δκ 2 (Q n 2(X n 2, X r 2 ) + Q r 2(X n 2, X r 2 )) ι 1 (Q 1 X) + δι 2 Q r 2(X n 2, X r 2 )}, where X n 2 = (X Q 1) + and X r 2 = X Q 1; whereas ha wih 92

106 myopic cusomers is Ĩ e = E{κ 1 Q 1 + δκ 2 (Q n n 2( X δι 2 Q r 2( X n 2, X r 2 )}, where 2, X n 2 = (X Q 1) + and r X 2 ) + Q r n 2( X 2, r X 2 = X Q 1. X r 2 )) ι 1 ( Q 1 X) Impac of Trade-in Remanufacuring In his secion, we analyze he impac of rade-in remanufacuring on he firm and he environmen under differen cusomer behaviors (i.e., sraegic or myopic cusomers). Our focus is on how sraegic cusomer behavior influences he economic and environmenal values of rade-in remanufacuring. To faciliae our comparison, we firs inroduce a benchmark model where he firm does no offer rade-in rebaes o cusomers. As a consequence, he firm canno recycle used producs for remanufacuring in period 2. We call his he No Trade-in Remanufacuring (NTR) model, which is denoed by he superscrip u hereafer. We use p u 2(X n 2, X r 2) o denoe he equilibrium second-period pricing sraegy of he firm in he NTR model, which does no depend on cusomer behavior. As in he base model, he firm forms a belief abou he cusomers expeced willingness-o-pay in he firs period, and bases his price and producion decisions on his belief. The cusomers, on he oher hand, form beliefs abou he produc availabiliy and he second-period price, and ime heir purchases. Again, he formal definiions of he RE equilibrium in he NTR model are given in Appendix C.1. By he same argumen in he proof of Theorem 4.3.1, we can show ha an RE equilibrium exiss wih eiher sraegic or myopic cusomers in he NTR model. Le (p u 1, Q u 1 ) denoe he equilibrium firs-period price and producion decisions of he firm wih sraegic cusomers, and ( p u 1, Q u 1 ) denoe hose wih myopic cusomers in he NTR model. Accordingly, he associaed equilibrium expeced profi of he firm (environmenal impac) is denoed by Π u f and by Π u f (Ĩu e ) in he scenario wih myopic cusomers. (Ie u ) in he scenario wih sraegic cusomers, Le Π u f (Q 1) ( Π u f (Q 1)) be he expeced profi of he firm wih sraegic (myopic) cusomers o produce Q 1 producs in he period 1 in he NTR model. We characerize he objecive funcions Π f ( ), Π f ( ), Π u f ( ), and Π u f ( ) in he following lemma. Lemma 8 The objecive funcions are given by Π f (Q 1 ) = (m 1 r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δβ2 n E(X), Π f (Q 1 ) = ( m 1 r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δβ2 n E(X), ( p Π u f(q 1 ) = (m u 1(Q 1 ) r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 +δe{(p u 2(X2 n, X2) c r u 2 )Ḡ 2 (X2 n, X2) r 1 + α 93 ) X},

107 and ( ) p Π u f(q 1 ) = ( m u 1(Q 1 ) r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 +δe{(p u 2(X2 n, X2) c r u 2 )Ḡ 2 (X2 n, X2) r X}, 1 + α where X n 2 = (X Q 1 ) +, X r 2 = X Q 1. The expressions of m u 1( ) and m u 1( ) are given in Appendix C.2. Lemma 8 implies ha, in he NTR model, he effecive firs-period marginal revenue is producion-quaniy-dependen, and given by m u 1( ) in he scenario wih sraegic cusomers and by m u 1( ) wih myopic cusomers. The economic inerpreaion of m u 1(Q 1 ) ( m u 1(Q 1 )) is ha, when he firs-period producion quaniy is Q 1, i measures he addiional expeced marginal revenue o sell he produc in period 1 over ha in period 2 wih sraegic (myopic) cusomers. Hence, he higher he m u 1(Q 1 ) and m u 1(Q 1 ), he more profiable i is for he firm o sell he firs-generaion produc in he NTR model wih sraegic and myopic cusomers, respecively. In oher words, m u 1( ) and m u 1( ) capure he willingness-o-produce of he firm in period 1. Wihou loss of generaliy, we focus on he case where m u 1( ) > 0 and m u 1( ) > 0 for all Q 1 0, i.e., he firm can gain a posiive revenue o sell he firs-generaion produc. Oherwise, he firm will no produce or sell anyhing in period Impac on Firm Profi This subsecion invesigaes he value of rade-in remanufacuring o he firm. To begin wih, we characerize he role of sraegic cusomer behavior, depending on wheher he firm adops rade-in remanufacuring or no. Theorem (a) Under rade-in remanufacuring, le e := E((k + α)v p r 2 ) + E((1+α)V p n 2 ) +. Then, we have (i) p 1 > p 1 if and only if e > 0, (ii) Q 1 > Q 1 if and only if e > 0, and (iii) Π f > Π f if and only if e > 0. Moreover, here exiss a hreshold r 1 k 1+α c 2, such ha e > 0 if and only if r 2 > r. (b) Under no rade-in remanufacuring, we have (i) p u 1 p u 1, where he inequaliy is sric if k < 1, (ii) Q u 1 k < 1 and Q u 1 > 0. u Q 1, and (iii) Π u f u Π f, where he inequaliy is sric if Under rade-in remanufacuring, Theorem 4.4.1(a) compares he equilibrium oucomes wih differen cusomer behaviors. We find ha he key o his comparison is he 94

108 difference beween he expeced surplus of a repea cusomer and ha of a new cusomer in period 2 (i.e., e ). Wih sraegic cusomers, he firm charges a higher firs-period price, ses a higher firs-period producion level, and earns a higher oal expeced profi, if and only if he expeced second-period surplus of a repea cusomer is higher han ha of a new cusomer (i.e, e > 0). In paricular, he presence of sraegic cusomer behavior benefis he firm if he revenue-generaing effec of remanufacuring is srong enough (i.e., r 2 > r). This resul is in sharp conras wih he well-esablished noion in he lieraure ha sraegic cusomer behavior hurs a firm s profi (e.g., [17, 159]). Trade-in remanufacuring leads o a price discoun for repea cusomers in period 2, which can be perceived by sraegic cusomers when deciding wheher o make a purchase in period 1. This discoun ouweighs he benefi of sraegic waiing if he revenue generaed from remanufacuring is sufficienly high (i.e., r 2 > r). In his case, he presence of forward-looking behavior will enable he firm o charge a higher price, produce more, and hus earn a higher profi. We emphasize ha boh he rade-in opion and he revenue-generaing effec of remanufacuring are essenial for he firm o benefi from sraegic cusomer behavior: The former induces sraegic cusomers o anicipae he price discoun for repea cusomers, whereas he laer brings in he addiional benefi ha guaranees a deep discoun so ha sraegic cusomers are willing o pay an even higher firs-period price han myopic cusomers. In conras, Theorem 4.4.1(b) shows ha, wihou rade-in remanufacuring, he firm always suffers from sraegic cusomer behavior, as repored in he exising lieraure. Theorem suggess ha he presence of sraegic cusomer behavior will make rade-in remanufacuring more aracive o he firm. Nex, we sudy how rade-in remanufacuring influences he profi and he pricing sraegy of he firm under differen cusomer behaviors. The following heorem compares he equilibrium prices and profis in he NTR model and hose in he base model wih eiher sraegic or myopic cusomers. Theorem (a) In period 2, p u 2(X n 2, X r 2) is increasing in X n 2 and decreasing in X r 2. Moreover, for any (X n 2, X r 2), p r 2 p u 2(X n 2, X r 2) p n 2, where he inequaliies are sric if k < 1 and X n 2, X r 2 > 0. (b) Wih sraegic cusomers, we have (i) p u 1 p 1, where he inequaliy is sric if p r 2 < p n 2 ; and (ii) Π u f Π f, where he inequaliy is sric if pr 2 < p n 2 and Q 1 > 0. 95

109 (c) Wih myopic cusomers, we have (i) p u 1 = p 1; and (ii) inequaliy is sric if p r 2 < p n 2 and Q u 1 > 0. Π u f Π f, where he Theorem shows ha he equilibrium second-period price wihou rade-in remanufacuring, p u 2(, ), is bounded from below by he equilibrium second-period rade-in price p r 2, and from above by he equilibrium second-period price for new cusomers p n 2. Hence, under rade-in remanufacuring, he expeced uiliy of sraegic cusomers o make a purchase in he firs period increases (i.e., δe[(k+α)v p r 2 ] + δe[(k+α)v p u 2(X n 2, X r 2)] + ), whereas he benefi of waiing decreases decreases (i.e., δe[(1 + α)v p n 2 ] + δe[(1 + α)v p u 2(X n 2, X r 2)] +. This implies ha rade-in remanufacuring makes sraegic cusomers more willing o purchase immediaely han o wai unil period 2. Therefore, rade-in remanufacuring enables he firm o exploi he forward-looking behavior of sraegic cusomers and hus induces early purchases from hem. Wih myopic cusomers, however, rade-in remanufacuring does no have an early-purchase inducing effec because myopic cusomers do no care abou heir fuure surplus. This resul is also consisen wih he finding in he durable produc lieraure ha he secondary marke gives rise o greaer resale value of a durable produc and hus can increase he sales of he new produc upfron (see, e.g., [94, 169]). From Theorem 4.4.2, we can see here are hree beneficial effecs of rade-in remanufacuring ha may improve firm profi: (a) he revenue-generaing effec of remanufacuring, (b) he price discriminaion effec of rade-in rebaes, i.e., he differeniaed prices for new and repea cusomers helps he firm exploi he cusomer segmenaion in period 2, and (c) he early-purchase inducing effec of rade-in rebaes, i.e., he price discoun o repea cusomers enables he firm o exploi he forward-looking behavior of sraegic cusomers by offering early-purchase rewards. The firs wo effecs benefi he firm wih eiher sraegic or myopic cusomers, whereas he hird effec improves he firm s profi wih sraegic cusomers only. In he following, we conduc exensive numerical experimens o quanify he hird effec, and deliver insighs on how sraegic cusomer behavior influences he value of rade-in remanufacuring o he firm. The design of he numerical sudy is as follows. Le he cusomer valuaion V follow a uniform disribuion on [0, 1] (µ = E(V ) = 0.5). The discoun facor is δ = 0.95, he uni environmenal impac of he firs-generaion produc is κ 1 = 1, and he uni environmenal impac of he second-generaion produc is κ 2 = To focus on he 96

110 impac of cusomer behaviors, we se r 1 = r 2 = 0 (i.e., here is no revenue-generaing effec associaed wih remanufacuring), and he uni environmenal benefis of recycling/remanufacuring o be ι 1 = 0 and ι 2 = 0.3 (hese wo values will be useful when sudying he environmenal impac in Secion 4.2). The uni producion cos of he firs-generaion produc is c 1 {0.05, 0.1, 0.15, 0.2, 0.25}. The innovaion level of he second-generaion produc is α {0, 0.05, 0.1, 0.15, 0.2}, and he uni producion cos of he second-generaion produc is c 2 = 0.25(1 + α) {0.25, , 0.275, , 0.3}. We consider he depreciaion facor k {0.3, 0.4, 0.5, 0.6, 0.7}. The demand X follows a gamma disribuion wih mean 100 and coefficien of variaion CV (X) aking values from he se {0.5, 0.6, 0.7, 0.8, 0.9}. Thus, we have a oal of 625 parameer combinaions ha cover a wide range of reasonable problem scenarios. The above problem scenarios form a subse of he exensive experimens we have conduced. Our numerical findings are very robus. For concision, we will only presen he resuls for he parameer combinaions lised above. We calculae he expeced profi for each scenario wih eiher sraegic or myopic cusomers boh in he base model, (Π f, Π f ) and in he NTR model, (Πu f wo merics of ineres are: γ s := (Π f Πu f )/Πu f 100%, and γ m := ( Π f u, Π f ). The u u Π f )/ Π f 100%, i.e., γ s (γ m ) refers o he relaive profi improvemen of rade-in remanufacuring wih sraegic (myopic) cusomers. We evaluae γ s and γ m under he 625 parameer combinaions and repor ha, under each combinaion, γ s is significanly higher han γ m. More specifically, γ s is a leas 5.8% and can be as high as 61.6%, wih an average of 30.2%; whereas γ m ranges from 0.008% o 11.7%, wih an average of 3.1%. We give he summary saisics of γ s and γ m in Table 4.1. Min 5h percenile Median 95h percenile Max Mean San. Dev. γ s γ m Table 4.1 Summary Saisics: Firm Profi (%) Our numerical resuls deliver an imporan message on he economic value of rade-in remanufacuring: Trade-in remanufacuring delivers a much higher value o he firm wih sraegic cusomers han wih myopic cusomers (γ s is significanly higher han γ m for each 97

111 problem insance). Recall ha, wih myopic cusomers, rade-in remanufacuring only has he benefis of revenue-generaing and price discriminaion, whereas, wih sraegic cusomers, his sraegy has he addiional value of inducing early purchases. Therefore, hese resuls indicae ha he value of rade-in remanufacuring o he firm mainly comes from he early-purchase inducing effec of rade-in rebaes o exploi sraegic cusomer behavior, raher han from he revenue-generaing effec of remanufacuring or he price discriminaion effec of rade-in rebaes o exploi cusomer segmenaion Impac on Environmen and Cusomer Surplus Our nex goal is o examine he environmenal value of rade-in remanufacuring under differen cusomer behaviors. We firs characerize how rade-in remanufacuring influences he effecive firs-period marginal revenue and producion quaniies of he firm. Theorem Assume k < 1. (a) Wih sraegic cusomers, we have (i) m u 1(Q 1 ) is decreasing in Q 1 ; (ii) m u 1(Q 1 ) < m 1 for all Q 1 ; (iii) Q u 1 Q 1, where he inequaliy is sric if Q 1 > 0. (b) Wih myopic cusomers, we have (i) m u 1(Q 1 ) is increasing in Q 1 ; (ii) for each r 2 < r 2 2, here exiss a hreshold Q(r 2 ) increasing in r 2, such ha m u 1(Q 1 ) m 1 for all Q 1 Q(r 2 ), and m u 1(Q 1 ) > m 1 for all Q 1 > Q(r 2 ); (iii) for each r 2 < r 2, here exiss a hreshold c 1 (r 2 ) > 0, such ha Q u 1 > Q 1 if c 1 c 1 (r 2 ). Theorem provides an ineresing comparison beween he scenarios of sraegic and myopic cusomers: Wih sraegic cusomers, rade-in remanufacuring always increases he firs-period producion quaniy of he firm, whereas i may promp he firm o produce less wih myopic cusomers. More specifically, Theorem 4.4.3(a) shows ha, under sraegic cusomer behavior, he effecive marginal revenue wih rade-in remanufacuring always dominaes ha wihou (i.e., m u 1( ) < m 1). As a resul, he firm produces more in period 1 under rade-in remanufacuring. Theorem 4.4.3(b), however, suggess ha, wih myopic cusomers, rade-in remanufacuring may give rise o a lower firs-period effecive marginal revenue if he producion quaniy is large (i.e., 2 The expression of r 2 is given in Appendix C.2. 98

112 m u 1(Q 1 ) > m 1 if Q 1 > Q(r 2 )), hus driving he firm o lower he firs-period producion quaniy if he firs-period uni producion cos is low (i.e., c 1 c 1 (r 2 )). Recall from Theorem ha rade-in remanufacuring increases he firs-period willingness-o-pay of sraegic cusomers, which, in urn, drives he firm o produce more in period 1. Such early-purchase and, hus, early-producion inducing effecs of rade-in remanufacuring, however, are absen wih myopic cusomers. In he scenario of myopic cusomers, on he oher hand, he price discriminaion effec of rade-in remanufacuring improves he uni profi generaed from he new cusomers in period 2, hus leading o a lower effecive firs-period marginal revenue if he revenue-generaing effec of remanufacuring is no oo srong (i.e., r 2 < r 2 ). As a consequence, he firm decreases he firs-period producion quaniy o increase he second-period marke size of new cusomers. Theorem demonsraes he conrasing effecs of rade-in remanufacuring on producion quaniies under differen cusomer behaviors. How does rade-in remanufacuring affec he environmen? The answer is given in he nex heorem. Theorem (a) Wih sraegic cusomers, here exiss a hreshold ῑ u 2 > 0, such ha I e I u e if ι 2 ῑ u 2. (b) Assume ha r 2 < r 2 and c 1 c 1 (r 2 ). Wih myopic cusomers, here exiss a hreshold ῑ u 2 < κ 2, such ha Ĩu e Ĩ e if ι 2 ῑ u 2. When cusomers are sraegic, rade-in rebaes encourage hem o recycle he used firs-generaion producs more frequenly, so hey also purchase he produc more frequenly. In his scenario, rade-in remanufacuring leads o a worsened oucome for he environmen if he uni environmenal benefi of remanufacuring is no high enough o jusify he early-producion inducing effec (i.e., ι 2 ῑ u 2 in Theorem 4.4.4(a)). When he cusomers are myopic and he uni producion cos is sufficienly low, rade-in remanufacuring moivaes he firm o produce less in period 1 (see Theorem 4.4.3(b)). Hence, rade-in remanufacuring helps improve he environmen as long as he uni environmenal benefi of remanufacuring is no oo low (i.e., ι 2 ῑ u 2 in Theorem 5(b)). Theorem reveals he significan impac of cusomer behavior on he environmenal value of rade-in remanufacuring. Wih sraegic cusomers, he adopion of rade-in remanufacuring is likely o be derimenal o he environmen, whereas, wih myopic cusomers, adoping rade-in remanufacuring may benefi boh he firm and he environmen. Some 99

113 papers in he lieraure (e.g., [59, 81, 90]) have also esablished ha remanufacuring may increase he producion quaniy and hus environmenal impac. Our work, however, demonsraes ha he environmenal impac of rade-in remanufacuring depends criically on cusomer behavior. We now numerically illusrae he environmenal value of rade-in remanufacuring. We employ he same numerical seup as Secion Recall ha I e (Ĩ e ) is he expeced environmenal impac for he scenario wih sraegic (myopic) cusomers in he base model, and I u e (Ĩu e ) is ha in he NTR model. We are ineresed in he following wo merics: η s := (Ie Ie u )/Ie u 100%, and η m := (Ĩ e Ĩu e )/Ĩu e 100%, i.e., η s (η m ) refers o he relaive change of he environmenal impac when adoping rade-in remanufacuring wih sraegic (myopic) cusomers. We evaluae η s and η m under he 625 parameer combinaions and obain he following numerical findings: (i) Under each parameer combinaion, η s is significanly higher han η m ; and (ii) For mos of he parameer combinaions, η s > 0 bu η m < 0. Specifically, η s akes values from -1.2% o 171.9%, wih an average of 49.2%; whereas η m ranges from 10.2% o 4.5%, wih an average of 5.0%. Moreover, η s < 0 (i.e., rade-in remanufacuring benefis he environmen wih sraegic cusomers) for 10 ou of he 625 (i.e., 1.6%) problem insances we examine, whereas η m < 0 (i.e., rade-in remanufacuring benefis he environmen wih myopic cusomers) for 585 ou of he 625 (i.e., 93.6%) problem insances. Table 4.2 summarizes he saisics of η s and η m. Min 5h percenile Median 95h percenile Max Mean San. Dev. η s η m Table 4.2 Summary Saisics: Environmenal Impac (%) Table 2 confirms ha rade-in remanufacuring generally leads o much higher environmenal impac wih sraegic cusomers han wih myopic cusomers (η s is significanly higher han η m ). Though beneficial o he firm (see Table 4.1), he early-purchase inducing effec of rade-in remanufacuring gives rise o much higher producion quaniies under sraegic cusomer behavior, and hus leads o a much worse oucome from he environmenal perspecive. Therefore, sraegic cusomer behavior has opposing effecs 100

114 on he value of rade-in remanufacuring o he firm and he environmen: I makes his sraegy more aracive o he firm, bu less desirable o he environmen. The above resuls sugges ha rade-in remanufacuring may creae a ension beween firm profiabiliy and environmenal susainabiliy wih sraegic cusomers, bu benefis boh he firm and he environmen wih myopic cusomers. Since η s is significanly larger han zero for mos of he numerical cases we examine, rade-in remanufacuring is derimenal o he environmen for a large se of reasonable problem insances under sraegic cusomer behavior. Hence, in general, he early-purchase inducing effec dominaes he environmenal benefi of remanufacuring wih sraegic cusomers. Under sraegic cusomer behavior, he firm significanly benefis from rade-in remanufacuring, bu he environmen significanly suffers from his sraegy (i.e., γ s > 0 and, in general, η s > 0). Wih myopic cusomers, however, boh he firm and he environmen would benefi from he adopion of rade-in remanufacuring (i.e., γ m > 0 and, in general, η m < 0). Alhough an increased producion quaniy means more pressure on he environmen, i also increases he consumpion level of he produc. To conclude his secion, we explore how rade-in remanufacuring impacs he oal cusomer surplus under differen cusomer behaviors. We use S c ( S c ) and S u c ( u S c ) o denoe he equilibrium oal cusomer surplus for he scenarios wih sraegic (myopic) cusomers in he base model and he NTR model, respecively. Theorem (a) In he base model, we have Sc = δe[((1 + α)v p n 2 ) + X] and S c = δe[((1 + α)v p n 2 ) + (X Q 1) + ] + δe[((k + α)v p r 2 ) + (X Q 1)]. (b) In he NTR model, we have S u c α)v p u 2 ) + (X = δe[((1 + α)v p u 2 ) + X] and Q u 1 ) + ] + δe[((k + α)v p u 2 ) + (X Q u 1 )]. S u c = δe[((1 + (c) We have he following relaionship on he cusomer surpluses of sraegic cusomers: Sc Sc u, where he inequaliy is sric if k < 1 and Q u 1 > 0. Theorem 4.4.5(a) and (b) compue he oal cusomer surpluses in he base model and he NTR model. Moreover, in Theorem 4.4.5(c), we demonsrae ha, wih sraegic cusomers, he oal cusomer surplus always decreases wih he adopion of rade-in remanufacuring. This is because, wih sraegic cusomers, he oal cusomer surplus only depends on he (perceived) price for new cusomers in period 2, which is higher 101

115 under he adopion of rade-in remanufacuring (see Theorem 4.4.2(a)). By Theorem 4.4.3(a), one may argue ha, under sraegic cusomer behavior, rade-in remanufacuring increases producion quaniies and hus increases he cusomer surplus. Theorem 4.4.5(c), however, shows ha he oal cusomer surplus acually decreases in his scenario. Hence, under sraegic cusomer behavior, rade-in remanufacuring gives rise o higher producion quaniies wihou improving he cusomer surplus. Furher, he social welfare (i.e., firm profi plus cusomer surplus less environmenal impac) is likely o decrease under rade-in remanufacuring as well. This has been confirmed in he numerical sudy we explored in his secion. To summarize, cusomer behavior plays an imporan role in he economic and environmenal values of rade-in remanufacuring. Wih myopic cusomers, rade-in remanufacuring benefis boh he firm and he environmen. Wih sraegic cusomers, rade-in remanufacuring would be even more beneficial o he firm; however, meanwhile i may hur he environmen, decrease cusomer surplus, and possibly lower social welfare. Therefore, i is imporan for firms and policy-makers o undersand cusomer behavior when making decisions relaed o rade-in remanufacuring. 4.5 Social Opimum and Governmen Inervenion As shown in Secion 4.4, adoping rade-in remanufacuring may creae a ension beween firm profiabiliy and environmenal susainabiliy under sraegic cusomer behavior. In his secion, we analyze how a policy-maker (e.g., he governmen) can design he public policy o resolve his ension and maximize he social welfare under differen cusomer behaviors. To characerize he socially opimal oucome, we assume ha he governmen can se he prices and producion levels, wih an objecive o maximize he social welfare. Le W s denoe he social welfare, which is defined by he expeced profi of he firm Π f, plus he expeced cusomer surplus S c, ne he expeced environmenal impac I e, i.e., W s = Π f + S c I e. By backward inducion, we sar wih he second-period pricing and producion problem. As in he base model, sraegic and myopic cusomers exhibi he same purchasing behavior in period 2. For any given realized marke size in period 2 (X2 n, X2), r we use 102

116 (p n s,2(x n 2, X r 2), p r s,2(x n 2, X r 2)) o denoe he equilibrium pricing sraegy, and (Q n s,2(x n 2, X r 2), Q r s,2(x n 2, X r 2)) o denoe he equilibrium producion sraegy. Correspondingly, we denoe w 2 (X n 2, X r 2) as he equilibrium second-period social welfare. Lemma 9 (a) For any (X n 2, X r 2), p n s,2(x n 2, X r 2) p n s,2 and p r s,2(x n 2, X r 2) p r s,2, where p n s,2 = c 2 + κ 2 and p r s,2 = c 2 r 2 + κ 2 ι 2. Hence, p n s,2 > p r s,2 if and only if r 2 > 0 or ι 2 > 0. (b) For any (X n 2, X r 2), Q n s,2(x n 2, X r 2) = Ḡ ( p ) n s,2 X n 1+α 2, and Q r s,2(x2 n, X2) r = Ḡ ( p ) r s,2 X r k+α 2. (c) There exis wo posiive consans β s,n and β s,r, such ha w 2 (X n 2, X r 2) = β s,nx n 2 + β s,rx r 2 for all (X n 2, X r 2), where β s,n = E[(1 + α)v p n s,2] + and β s,r = E[(k + α)v p r s,2] +. Lemma 9 implies ha, wih eiher sraegic or myopic cusomers, he socially opimal second-period pricing sraegy akes he form ha he prices for new and repea cusomers are equal o he respecive ne uni producion cos plus he ne uni environmenal impac (i.e., p n s,2 = c 2 + κ 2 and p r s,2 = c 2 r 2 + κ 2 ι 2 ). Moreover, he equilibrium social welfare is linear in he realized marke size (X n 2, X r 2). In period 1, sraegic cusomers base heir purchasing decisions on heir raional expecaions, whereas myopic cusomers decide wheher o make a purchase by comparing he curren price and he expeced valuaion. Le (p s,1, Q s,1) denoe he equilibrium firsperiod price and producion quaniy wih sraegic cusomers, and ( p s,1, Q s,1) denoe hose wih myopic cusomers. As in he base model and he NTR model, we inroduce he firs-period effecive marginal welfare wih eiher sraegic or myopic cusomers, m s,1 = m s,1 := µ+δ[β s,r β s,n]. The following lemma characerizes he social welfare maximizing equilibrium oucomes. Lemma 10 (a) Wih sraegic cusomers, we have (i) p s,1 = m s,1; (ii) Q s,1 = F 1 ( c 1+κ 1 r 1 ι 1 m s,1 r 1 ι 1 ); and (iii) he equilibrium expeced social welfare is W s = (m s,1 r 1 ι 1 )E(X Q s,1) (c 1 + κ 1 r 1 ι 1 )Q s,1 + δβ s,ne[x]. (b) Wih myopic cusomers, we have (i) p s,1 = µ; (ii) Q 1 = F 1 ( c 1+κ 1 r 1 ι 1 m s,1 r 1 ι 1 ); and (iii) he equilibrium expeced social welfare is W s = ( m s,1 r 1 ι 1 )E(X Q s,1) (c 1 + κ 1 r 1 ι 1 ) Q s,1 + δβ s,ne[x]. 103

117 (c) Le e s := β s,r β s,n. Then, we have (i) p s,1 p s,1 if and only if e s 0; (ii) Q s,1 = Q s,1; and (iii) W s = W s. Since he social planner needs o balance firm profi, cusomer surplus, and environmenal impac, whereas he firm maximizes is own profi only, he social-welfaremaximizing equilibrium oucome may be quie differen from he profi-maximizing one, as shown by comparing Lemma 10 wih Theorem In paricular, if he uni environmenal impacs, κ 1 and κ 2, are sufficienly large, he social planner will se lower producion quaniies han he firm will do o limi he oal environmenal impacs. Lemma 10(c) characerizes how differen cusomer behaviors influence he social-welfaremaximizing RE equilibrium oucome. Specifically, we show ha he expeced opimal social welfare wih sraegic cusomers is he same as ha wih myopic cusomers, and so is he opimal firs-period producion quaniy. The equilibrium firs-period price, however, depends on cusomer behavior. If he expeced surplus of a repea cusomer dominaes ha of a new cusomer (i.e., e s 0), he equilibrium firs-period price is higher wih sraegic cusomers. Oherwise, e s < 0, he equilibrium firs-period price is higher wih myopic cusomers. We noice ha e s is he counerpar of e (see Theorem 4.4.1), boh of which characerize he addiional expeced uiliy of a repea cusomer over a new one in period 2. We now analyze how he governmen, whose objecive is o maximize he expeced social welfare W s, could induce he firm, whose objecive is o maximize his expeced profi Π f, o se he socially opimal prices and producion quaniies under differen cusomer behaviors. A commonly-observed governmen subsidizaion policy is o subsidize he firm or cusomers for he remanufacured producs (see, e.g., [128, 44]). To model his subsidizaion policy, we assume ha he governmen offers he firm a per-uni subsidy s r for remanufacuring lefover invenory and used producs. The per-uni subsidy o he firm is wihou loss of generaliy, because all resuls and qualiaive insighs in his secion coninue o hold wih he per-uni subsidy o cusomers, and he proporional subsidy 3 o eiher he firm or he cusomers. For exposiional ease, we ake he approach of per-uni subsidy o he firm. 3 The proporional subsidy refers o he governmen subsidizaion scheme under which he uni subsidy is proporional o (e.g., 10% of) he sales price/rade-in price. 104

118 We firs sudy how he governmen subsidizaion policy for remanufacured producs would influence he equilibrium oucome in he following heorem. Theorem (a) For any (X n 2, X r 2), we have (i) p r 2 is decreasing in s r ; and (ii) Q r 2(X n 2, X r 2) is increasing in s r. (b) Wih sraegic cusomers, we have (i) p 1 is increasing in s r ; (ii) Q 1 is increasing in s r ; (iii) Π f is increasing in s r; and (iv) I e is increasing in s r. (c) Wih myopic cusomers, we have (i) p 1 is independen of s r ; (ii) Q 1 is increasing in s r ; (iii) Π f is increasing in s r; and (iv) Ĩ e is increasing in s r. One of he main goals for he governmen o subsidize remanufacuring is o improve he environmen (see [44, 161]). Theorem 4.5.1(b,c), however, suggess ha if he governmen only subsidizes for remanufacuring (i.e., s r > 0), he environmen acually suffers from his subsidizaion policy wih eiher sraegic or myopic cusomers (i.e., I e and Ĩ e are increasing in s r ). This resul follows from he raionale ha subsidizing remanufacured producs no only promoes he adopion of remanufacuring, bu also increases he producion levels of he firs-generaion produc, which is he leas environmenally friendly produc version. The environmen hus suffers from he increased producion levels under he subsidizaion for remanufacuring alone. Therefore, he governmen should be careful abou designing he subsidizaion policy, because haphazard subsidizaion for remanufacuring may resul in an undesired oucome. Moivaed by he discrepancy beween he inenion and oucome of a commonly used governmen subsidizaion policy for remanufacuring, we consider an alernaive more general governmen policy ha subsidizes for/axes on he producion of boh generaion producs and remanufacuring. Some oher comprehensive governmen subsidizaion policies on producion, recycling, remanufacuring, and rade-in rebaes are discussed in, e.g., [186, 118], and [170]. The goal of such governmen subsidizaion programs is o promoe he developmen of remanufacuring, curb polluion, and simulae consumpion. We assume ha governmen subsidies (axes) are provided (charged) for he sales of boh generaion producs, and recycling/remanufacuring he lefover invenory and used producs. Specifically, le s g := (s 1, s 2, s r ) denoe he subsidy/ax scheme he governmen adops. The governmen offers he firm a per-uni subsidy s 1 for sales of he 105

119 firs-generaion produc, a per-uni subsidy s 2 for sales of he second-generaion produc, and a per-uni subsidy s r for remanufacuring. If s i < 0 (i = 1, 2, r), he firm axes on he sales of he produc or remanufacuring lefover invenory and used producs. In paricular, we remark ha he aforemenioned mos common governmen subsidizaion policy for remanufacuring alone is a special case of his general subsidy/ax scheme wih s 1 = 0, s 2 = 0, and s r > 0. We now analyze how he governmen should design he linear subsidy/ax scheme o induce he socially opimal oucome under differen cusomer behaviors. Theorem (a) Wih sraegic cusomers, here exiss a unique linear subsidy/ax scheme s g = (s 1, s 2, s r), under which he socially opimal RE equilibrium oucome is achieved. Moreover, we have (i) s 2 is he unique soluion o p n s,2 = argmax p n 2 0{(p n 2 + s 2 c 2 )Ḡ( pn 2 1+α )}; (ii) s r is he unique soluion o p r s,2 = argmax p r 2 0{(p r 2 + s r + s 2 c 2 + r 2 )Ḡ( pr 2 k+α )}; (iii) s 1 is he unique soluion o c 1+κ 1 r 1 ι 1 s r m s,1 r 1 s r m s 1(s 1 ) := s 1 + m s,1 + δ[(κ 2 + s 2 + s r ι 2 = c 1 r 1 m s 1 (s 1) r 1, where pr s,2 )Ḡ( ) (κ pn s,2 k+α 2 + s2)ḡ( )]; (iv) 1+α s 1 is decreasing in κ 1, s 2 is decreasing in κ 2, and s r is increasing in ι 2 ; and (v) here exiss a hreshold vecor ( κ s 1, κ s 2, ῑ s 2), such ha s 1 0 if and only if κ 1 κ s 1, s 2 0 if and only if κ 2 κ s 2, and s r 0 if and only if ι 2 ῑ s 2. (b) Wih myopic cusomers, here exiss a unique linear subsidy/ax scheme s g = ( s 1, s 2, s r), under which he socially opimal RE equilibrium oucome is achieved. Moreover, we have (i) s 2 = s 2; (ii) s r = s r; (iii) s 1 is he unique soluion o c 1 +κ 1 r 1 ι 1 s r m s,1 r 1 s r = c 1 r 1 m s 1 (s 1) r 1, where m s 1(s 1 ) := s 1 + µ + δ[(κ 2 + s 2 + s r ι 2 )Ḡ( pr s,2 k+α ) pn s,2 (κ 2 + s2)ḡ( )]; (iv) 1+α s 1 is decreasing in κ 1 ; and (v) here exiss a hreshold κ s 1, such ha s 1 0 if and only if κ 1 κ s 1. (c) We have (i) s 1 s 1 if and only if e s 0; and (ii) κ s 1 κ s 1 if and only if e s 0, where e s is defined in Lemma 10(c). Theorem demonsraes ha he governmen can use a simple linear subsidy/ax scheme o induce he socially opimal oucome in he scenarios wih eiher sraegic or myopic cusomers. The linear subsidy/ax policy s g helps conrol he margin of he firm and he willingness-o-pay of he cusomers. Hence, he governmen can use his incenive scheme o regulae he marke and ensure he firm ses he socially opimal prices and 106

120 producion quaniies wih eiher sraegic or myopic cusomers. More specifically, in boh scenarios, he governmen should provide a combined subsidy/ax scheme for he sales of boh produc generaions and he recycle of lefover invenory and used producs. Since some componens in s g and s g may be negaive, i is possible ha he governmen axes he firm on some produc versions o discourage heir sales. This phenomenon resuls from he governmen s goal of balancing he radeoff beween firm profi, cusomer surplus, and environmenal impac. In paricular, wih eiher sraegic or myopic cusomers, he governmen subsidizes more for (axes less on) he sales of one produc version if is uni environmenal impac increases. Analogously, more subsidies (less axes) should be provided for (charged on) remanufacuring if is uni environmenal benefi is higher. Comparing he scenarios wih sraegic and myopic cusomers (i.e., Theorem 4.5.2(c)) sheds ligh on how differen cusomer behaviors influence he opimal governmen subsidy policy. We find ha he opimal subsidy/ax raes for he second-generaion produc and remanufacuring are independen of wheher he cusomers are sraegic or myopic (i.e., s 2 = s 2 and s r = s r). The opimal subsidy/ax rae for he firs-generaion produc, however, is sensiive o cusomer behavior. The governmen should provide a higher subsidy/lower ax for sales of he firs-generaion produc wih sraegic cusomers han wih myopic cusomers (i.e., s 1 s 1) if and only if, in period 2, he expeced surplus of a new cusomer dominaes ha of a repea cusomer (i.e., e s 0). If e s 0, sraegic cusomers are relucan o make an immediae purchase, so, o regulae he marke wih sraegic cusomers, he governmen should provide more subsidies for he sales of he firs-generaion produc o induce early purchases. On he oher hand, if e s > 0, a repea cusomer has higher expeced surplus in period 2, and hus sraegic cusomers are more willing o purchase he produc immediaely in period 1. In his case, o discourage sraegic cusomers from overconsumpion in period 1, he governmen offers less subsidies for he sales of he firs-generaion produc wih sraegic cusomers han i does wih myopic cusomers. The raionale behind he dichoomy in Theorem 4.5.2(c) is ha, wih he adopion of rade-in remanufacuring, sraegic cusomers anicipae boh he purchasing opion as a new cusomer and he rade-in opion as a repea cusomer. Depending on which opion has a higher expeced uiliy, a sraegic cusomer may have a higher or lower willingness-o-pay han a myopic cusomer does. Hence, he governmen 107

121 may provide higher or lower incenives in period 1 o encourage or discourage he early purchases of sraegic cusomers accordingly. Based on Theorem 4.5.2, we now compare he oal governmen coss of he opimal subsidy/ax scheme under differen cusomer behaviors. For any subsidy/ax scheme s g, we denoe C g (s g ) ( C g (s g )) as he associaed expeced oal governmen cos under RE equilibrium wih sraegic (myopic) cusomers. Define Cg := C g (s g) and C g := C g ( s g) as he social-welfare-maximizing governmen coss wih sraegic and myopic cusomers, respecively. Theorem (a) C g C g = (s 1 s 1)E(X Q s,1). (b) C g C g if and only if e s 0. Moreover, here exiss a hreshold V 2 > 0, such ha e s 0, if and only if r 2 + ι 2 V 2. Theorem compares he social-welfare-maximizing governmen coss in scenarios wih sraegic and myopic cusomers. Specifically, we show ha he oal cos o regulae a marke wih sraegic cusomers is higher han wih myopic cusomers whenever he socially opimal subsidy for he firs-generaion produc wih sraegic cusomers dominaes ha wih myopic cusomers (i.e., s 1 s 1). Equivalenly, according o Theorem 4.5.3(b), i coss he governmen more o regulae a marke wih sraegic cusomers if he expeced surplus of a new cusomer dominaes ha of a repea cusomer in period 2 (i.e., e s 0). In his case, more subsidies should be provided o incenivise he more relucan sraegic cusomers o make an early purchase in period 1. Anoher implicaion of Theorem 4.5.3(b) is ha if he oal uni economic and environmenal value of remanufacuring, r 2 + ι 2, is sufficienly low (i.e., below he hreshold V 2 ), he oal governmen cos is lower wih sraegic cusomers. Therefore, our analysis delivers he new insigh o he lieraure ha sraegic cusomer behavior has a negaive (posiive) impac upon he governmen if he oal economic and environmenal value of remanufacuring is low (high). 4.6 Summary In his chaper, we develop an analyical model o sudy how differen cusomer behaviors influence he economic and environmenal values of rade-in remanufacuring. From 108

122 he firm s perspecive, we show ha rade-in remanufacuring is generally much more valuable wih sraegic cusomers han wih myopic cusomers. This is because a radein rebae essenially offers an early purchase reward and hus can deliver addiional value by exploiing he forward-looking behavior of sraegic cusomers. In paricular, wih he adopion of rade-in remanufacuring, sraegic cusomer behavior may help increase he firm s profi, which conrass he common belief in he lieraure ha sraegic cusomer behavior hurs firm profi. In he rade-in remanufacuring seing, he price discoun in he second period increases wih he revenue generaed from remanufacuring; hus when he revenue-generaing effec is srong enough, he willingness-o-pay of sraegic cusomers in he firs period could be even higher han ha of myopic cusomers, which allows he firm o exrac more profi wih sraegic cusomers. From he environmenal perspecive, rade-in remanufacuring decreases he uni environmenal impac, bu increases he producion quaniies hrough he early-purchase inducing effec wih sraegic cusomers. Moreover, under sraegic cusomer behavior, adoping rade-in remanufacuring may decrease he cusomer surplus and social welfare. Hence, wih sraegic cusomers, cauion is needed on he adopion of rade-in remanufacuring, because i could be derimenal o he environmen and he sociey. Wih myopic cusomers, however, rade-in remanufacuring leads o a lower firs-period producion quaniy in general. Our resuls indicae ha cusomer behavior plays an imporan role in he value of rade-in remanufacuring. Specifically, wih sraegic cusomers, radein remanufacuring may creae a ension beween firm profiabiliy and environmenal susainabiliy; bu, wih myopic cusomers, i generally benefis boh he firm and he environmen. To resolve he above ension caused by rade-in remanufacuring, we also sudy how he governmen should design a regulaory policy o balance firm profi, cusomer surplus, and environmenal impac. A commonly observed policy is o subsidize he remanufacured producs. However, we find ha despie is inenion o proec he environmen, such a policy fails o achieve he social opimum and is acually harmful o he environmen. To achieve he socially opimal oucome, we show ha i suffices for he governmen o employ a simple linear incenive scheme. This scheme imposes eiher subsidy or ax on he sales of boh produc generaions as well as he remanufacured producs: A subsidy 109

123 (ax) should be applied if he environmenal impac of he produc is sufficienly low (high). 110

124 5. Pricing and Invenory Managemen under he Scarciy 5.1 Inroducion Effec of Invenory 1 In he operaions managemen lieraure, join pricing and invenory managemen has received exensive aenion. A key assumpion in he exising models in his sream of lieraure is ha demand, hough random, is independen of invenory (e.g., [70]), so ha sales and, hence, revenue link o invenory only hrough he sockou effec. In quie a few indusries (e.g., auomobile, elecronics and luxury producs, ec.), however, we have observed srong empirical and anecdoal evidence ha demand may be correlaed wih he amoun of invenory carried by reailers. A high invenory level someimes promoes sales because i creaes a srong visual impac (he billboard effec) and signals abundan poenial availabiliy, boh of which can make he iem more desirable and increase he chance of cusomer purchase. On he oher hand, i is also commonly observed in pracice ha an ample invenory conveys o he cusomers he message ha he iem is of low populariy and qualiy, hus inducing low demand. The negaive correlaions beween demand and invenory are well suppored by psychological and economic heories as well as rich anecdoal observaions and empirical daa. The phenomena ha a low invenory level may increase and a high invenory level may decrease demand are ofen referred o as he scarciy effec of invenory. Three major mechanisms drive he scarciy effec of invenory: (1) invenory level signals he qualiy and populariy of a produc; (2) invenory level implies he sockou risk of a produc; and (3) invenory level reveals he pricing sraegy he reailer will employ. We now discuss hese hree mechanisms in deail. Firs, i has been well esablished in psychological commodiy heory ha supply scarciy increases he araciveness of a produc o cusomers ([30]). This noion has been esed and refined by various experimens wih respec o a large scope of produc caegories (e.g., food, wine and book) by, e.g., [187], [182] and [178]. The desirabiliy 1 This chaper is based on he auhor s earlier work [189] 111

125 of he produc is enhanced by scarce invenory, because cusomers are likely o infer produc qualiy and populariy from is invenory level. A lower invenory level signals more consumpion by oher cusomers and, hence, he produc is more popular and of higher qualiy. On he oher hand, observing a high invenory, a cusomer naurally believes ha he iem has many unis because no one wans o buy i. Some recen markeing (e.g., [156]) and operaions managemen (e.g., [180]) papers employ game heoreic models o demonsrae ha he scarciy sraegy can effecively signal o he cusomers he high qualiy of a produc, hus creaing a ho produc. Empirical resuls regarding he scarciy effec of invenory upon demand in auomobile indusry can also be found in, e.g., [33] and [39]. Second, a low invenory level spreads a sense of urgency among cusomers ha soon he produc will be sold ou and poenial buyers will be pu on a wai-lis. Such backlogging risk moivaes cusomers o make an immediae purchase insead of searching for beer opions. A high invenory, however, grans cusomers he luxury of waiing and searching, hus lowering he curren demand. Similar mechanism also drives he search behavior ha a low invenory of one produc ype discourages a cusomer o search for beer ypes ([42]). Knowingly limiing he availabiliy of a produc, he reailer can induce buying frenzies among uninformed cusomers and se a higher price ([60]). Third, as shown in pricing and revenue managemen lieraure (e.g., [70], [83]), reailers increase heir sales prices when invenories are low. Therefore, cusomers infer from a low invenory level ha i is unreasonable o expec a lower price and would like o purchase he iem immediaely (see, e.g., [17]). On he oher hand, a high invenory level suggess ha he sales price will be more likely o decrease and, hence, encourages cusomers o wai before buying. Carefully making use of his mechanism, he reailer can enjoy he benefis of inducing cusomers o purchase early a high prices ([115]). A similar idea has also been adoped in he advance selling lieraure (e.g., [175]), which shows ha firms may limi is capaciy for advance selling o credibly signal is pricing sraegy o cusomers. Along wih he rich heoreical and empirical jusificaions of he scarciy effec of invenory, praciioners have exensively adoped his idea in heir markeing sraegies. [64] and [31] documen ha he scarciy sraegy, in which he supply of producs is deliberaely limied, has already become a basic acic for markeers o promoe heir 112

126 sales. An increasing number of auomobile manufacurers creae significan levels of scarciy and make a long lis of hard-o-ge car models over years (see [33]). Albei facing housands of cusomers who had signed up in he wai-liss, none of he manufacurers rushed o accelerae is producion ([184]). Likewise, [123] documens ha he BMW Mini Cooper promoes is line by limiing is supply and leing he poenial owners wai for, on average, wo and half monhs before hey own heir new cars. The limied disribuion sraegy has helped he demand of Mini Cooper ake off since is reinroducion in he US marke. Similar promoional sraegy also appears in he elecronics marke, especially a he inroducion sage of a new produc generaion. Fans have been excied by he long wai o ge Sony Play Saions ([184]), Ninendo Game Boys ([183]) and Apple ipads 2 ([150]). In his chaper, we sudy he dynamic pricing and invenory managemen model under he scarciy effec of invenory. The sochasic demand is modeled as a decreasing funcion of he sales price and he cusomer-accessible invenory level a he beginning of each decision epoch. Unme demand is fully backlogged o he nex period. The wailiss observed or spread hrough word-of-mouh successfully signal he high qualiy and populariy of he produc and arac more cusomers (see, e.g., [31] and [64]). From he sraegic perspecive, join pricing and invenory decisions effecively deliver he informaion regarding he qualiy and populariy of he produc. Specifically, pricing flexibiliy induces more sraegic behavior of cusomers (e.g., waiing for poenial price discoun), which furher srenghens he scarciy effec of invenory, because cusomers may anicipae he price changes based on curren invenory (see, e.g., [115]). We develop a unified join price and invenory managemen model ha incorporaes boh invenory wihholding and invenory disposal o deal wih he scarciy effec. Under he invenory wihholding policy, he firm displays only par of is invenory and wihholds he res in a warehouse no observable by cusomers, so as o induce higher poenial demand. Analogously, wih invenory disposal, he firm can dispose is unnecessary excess invenory wih some salvage value. Boh invenory wihholding and disposal may incur a cos. We show ha a cusomer-accessible-invenory-dependen order-upo/dispose-down-o/display-up-o lis-price policy is opimal. Moreover, he order-upo/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 113

127 no posiive invenory so ha every cusomer mus wai before geing he produc. In his case, he srong scarciy effec creaes more opporuniies han risks, so he firm can proacively ake advanage of i and induce more demand by making cusomers wai (e.g., he markeing sraegy of BMW). When i is oo cosly o wihhold or dispose invenory, he unified model is reduced o he model wihou invenory wihholding or he model wihou invenory disposal, boh of which deliver sharper insighs. In he model wihou invenory wihholding/disposal, we show ha he invenory-dependen demand increases he oversocking risk and, hus, lowers he opimal sales prices and order-up-o levels. Wih higher operaional flexibiliy (a higher salvage value or he invenory wihholding opporuniy), however, he firm deals wih he scarciy effec of invenory more effecively and, hence, increases is sales prices and order-up-o/display-up-o levels. In shor, invenory disposal/wihholding benefis he firm by enhancing is operaional flexibiliy and agiliy. We also generalize he unified model by incorporaing responsive invenory reallocaion, which allows he firm o reallocae (wih a cos) is invenory beween display and warehouse afer demand realizes. In his case, he firm can keep a low invenory and beer hedge agains risks of he demand uncerainy and he scarciy effec of invenory. We perform exensive numerical sudies o demonsrae (a) he robusness of our analyical resuls, (b) he impac of he scarciy effec upon he profiabiliy of he firm, and (c) he value of dynamic pricing under he scarciy effec of invenory. Our numerical resuls show ha he analyical characerizaions of he opimal policies in our model are robus and hold in all of our numerical experimens. Boh he profi loss of ignoring he scarciy effec and he value of dynamic pricing under he scarciy effec are significan, and increase in he inensiy of he scarciy effec and/or demand variabiliy. This is because: (1) he scarciy effec decreases he fuure demand and magnifies fuure demand variabiliy; and (2) dynamic pricing faciliaes he firm o induce higher fuure demand and dampen fuure demand variabiliy. In addiion, a longer planning horizon increases he impac of he scarciy effec, and decreases he value of dynamic pricing. To conclude his secion, we summarize our main conribuions as follows: (1) To he bes of our knowledge, we are he firs o sudy he join pricing and invenory managemen under he scarciy effec of invenory. We characerize he opimal policy in a general unified model and generalize our resuls o he model wih responsive invenory 114

128 reallocaion. (2) We analyze how he scarciy effec of invenory impacs he firm s opimal price and invenory policies and sudy he effec of operaional flexibiliies on he firm s opimal decisions under he scarciy effec. (3) We idenify he raionale of he phenomenon ha firms wih inense scarciy effec deliberaely make heir cusomers wai before geing he produc. (4) We numerically sudy he profi loss of ignoring he scarciy effec and he value of dynamic pricing under he scarciy effec. The res of he chaper is organized as follows. In Secion 5.2, we posiion his chaper in he relaed lieraure. Secion 5.3 presens he basic formulaion, noaions and assumpions of our model. In Secion 5.4, we propose and analyze he unified model. Secion 5.5 discusses he addiional resuls and insighs in wo imporan special cases (he model wihou invenory wihholding and he model wihou invenory disposal). Secion 5.6 generalizes he unified model o he model wih responsive reallocaion. Secion 5.7 repors our numerical findings. We conclude his chaper by summarizing our findings and discussing a possible exension in Secion 5.8. All proofs are relegaed o Appendix D Relaed Research This chaper is mainly relaed o wo lines of research in he lieraure: (1) invenory managemen wih invenory-dependen demand and (2) opimal join pricing and invenory policy. There is a large body of lieraure on invenory-dependen demand. We refer ineresed readers o [177] for a comprehensive review. The dependence of demand on invenory is usually modeled in wo ways in he lieraure: (1) poenial demand is increasing in he invenory level afer replenishmen; and (2) poenial demand is decreasing in he invenory level before replenishmen (lefover invenory from he previous period). The firs approach o model invenory-dependen demand assumes ha demand increases wih invenory (he billboard effec). [86] sudy a periodic review invenory model, in which he random demand in each period is increasing in he invenory level afer replenishmen. [58] consider a single-period newsvendor model where demand is decreasing in price and posiively correlaed wih invenory level. Several oher operaions managemen and markeing papers also assume ha demand depends on he insananeous 115

129 (afer replenishmen) invenory level, in paricular via he shelf-space effec. We refer ineresed readers o, e.g., [171, 172], [34, 35], [119], [18] and [53]. The oher effec of invenory upon demand, as discussed in Secion 5.1, is he scarciy effec. Tha is, high lefover invenory (i.e., invenory a he beginning of he period before replenishmen) negaively influences he poenial demand. In he psychological commodiy heory lieraure, [30] argues ha supply scarciy increases he araciveness of a produc, which has been esed by numerous experimens in, e.g., [187, 178]. [156, 180] use game heoreic models o show ha he firm can use he scarciy sraegy o signal he high qualiy of a produc. [17], among ohers, demonsrae ha cusomers may sraegically wai for price discouns when observing a high invenory. [115] propose an effecive pricing scheme o induce cusomers o make early purchases under a revenue managemen framework. The idea ha supply condiion can signal he poenial pricing sraegy and he produc qualiy has also been adoped in he advance selling lieraure (e.g., [175]). [33, 39] conduc empirical sudies o show ha he scarciy effec of invenory upon demand prevails in auomobile indusry. To he bes of our knowledge, [145] is he only paper in invenory managemen lieraure ha incorporaes he scarciy effec of invenory (called wai-lis effec in ha paper) and assumes ha poenial demand is a decreasing funcion of lefover invenory. They show he opimaliy of undersocking and propose he invenory wihholding sraegy. This chaper generalizes [145] in he following aspecs. (1) We inroduce a unified model ha encompasses dynamic pricing, invenory wihholding and invenory disposal, and explicily capures he ineracion beween price, invenory and demand. In paricular, we analyically show he impac of invenory-dependen demand on he firm s pricing policy, whereas [145] do no allow price adjusmen during planning horizon and numerically es he improvemen of invenory-wihholding policy under differen price elasiciies of demand. We also numerically show ha he value of dynamic pricing under he scarciy effec of invenory is significan and increases wih he scarciy effec inensiy and/or demand variabiliy. (2) Because of he endogenous pricing decision inroduced o he dynamic program, he analysis of our model is more involved and requires a differen approach. (3) Two special cases of our unified model (i.e., he model wihou invenory wihholding and he model wihou invenory disposal) demonsrae ha invenory wihholding and invenory disposal help miigae he overage risk of invenory-dependen 116

130 demand. (4) In addiion o he undersocking and invenory wihholding policy proposed in [145], our model suggess hree oher sraegies o dampen he negaive effec of invenory-dependen demand: (a) price reducion, (b) invenory disposal, and (c) responsive invenory reallocaion. (5) We show ha when he scarciy effec of invenory is sufficienly srong, he firm should display no posiive invenory and le every cusomer wai. To sum up, his chaper generalizes he model in [145] and srenghens is resuls and insighs. There is an exensive lieraure on dynamic pricing and invenory conrol under general sochasic demand. [70] sudy he invenory sysem in a periodic review model, where he firm faces price-dependen demand in each decision period and unsaisfied demand is fully backlogged. A lis-price order-up-o policy is shown o be opimal. This line of lieraure has grown rapidly since [70]. For example, [47, 48, 49] analyze he join pricing and invenory conrol problem wih fixed ordering cos and show he opimaliy of (s, S, p) policy for finie horizon, infinie horizon and coninuous review models. [52] sudy he join pricing and invenory conrol problem under los sales. In he case of a single unreliable supplier, [112, 73] show ha supply uncerainy drives he firm o charge higher prices under random yield and random capaciy, respecively. [51] ake ino consideraion cosly price adjusmens in join pricing and invenory managemen. When he replenishmen leadime is posiive, he join pricing and invenory conrol problem under periodic review is exremely difficul, and [136] parially characerize he srucure of he opimal policy. We refer ineresed readers o [50] for a comprehensive survey on join pricing and invenory conrol models. The major difference of his chaper from his sream of research is ha we ake ino accoun invenory-dependen demand and show ha he scarciy effec of invenory drives he firm o order less/dispose more/wihhold invenory and charge a lower sales price. To he bes of our knowledge, only [58, 35] have sudied he join pricing and invenory conrol problem wih invenory-dependen demand. However, boh papers consider a single period model where demand is increasing in he available invenory afer replenishmen. 5.3 Model Formulaion We specify our unified model, noaions and assumpions in his secion. Consider a firm which faces random demand and periodically makes pricing and invenory decisions 117

131 in a T period planning horizon, labeled backwards as {T, T 1,, 1}. The firm sores is on-hand invenory in wo locaions, one wih cusomer-accessible invenory o saisfy and simulae demand, and he oher as a warehouse o wihhold invenory ha is unobservable o cusomers. The firm can eiher replenish or dispose invenory, and i can also reallocae is on-hand invenory beween he cusomer-accessible sorage and he warehouse. If he firm places an order, he replenished invenory is delivered o he warehouse, afer which he firm decides how much invenory o reallocae o he cusomer-accessible sorage. On he oher hand, if he firm disposes is on-hand invenory, i firs ships invenory, if any, from he cusomer-accessible sorage o he warehouse, and hen chooses he disposal quaniy. In each period, he sequence of evens unfolds as follows: A he beginning of each period, he firm reviews is oal and cusomer-accessible lefover invenories from las period, simulaneously chooses he order/disposal and reallocaion quaniies and he sales price, pays he ordering and reallocaion coss, and receives he disposal salvages. The ordering and reallocaion lead imes are assumed o be zero so ha he replenished and reallocaed invenories are received immediaely. Invenory disposal is also execued a once. The demand hen realizes and he revenue is colleced. A he end of he decision period, he holding and backlogging coss are paid, and he oal and cusomer-accessible invenories are carried over o he beginning of he nex period. The sae of he sysem is given by: I a = he saring cusomer-accessible invenory level before replenishmen/disposal /reallocaion in period, = T, T 1,, 1, where he superscrip a refers o cusomer-accessible ; I = he saring oal invenory level before replenishmen/disposal/reallocaion in period, = T, T 1,,

132 Noe ha, he amoun of invenory he firm wihholds in he warehouse is I I a 0. We inroduce he following noaion o denoe he decisions of he firm: p = he sales price charged in period, = T, T 1,, 1; x a = he cusomer-accessible invenory level afer replenishmen/disposal/reallocaion bu before demand realizes in period, = T, T 1,, 1; x = he oal invenory level afer replenishmen/disposal/reallocaion bu before demand realizes in period, = T, T 1,, 1. We assume ha he price p is bounded from above by he maximum allowable price p and from below by he minimum allowable price p. Wihou loss of generaliy, we also assume ha he cusomer-accessible invenory sorage capaciy of he firm is K a (0 < K a + ), whereas he warehouse capaciy is infinie. In oher words, he cusomeraccessible invenory level afer replenishmen/disposal/reallocaion canno exceed K a in each period, i.e., x a K a for all = T, T 1,, 1. Following he no-arificial wailis noion (see [145]), we assume ha he firm canno decrease is cusomer-accessibleinvenory level if a wai-lis already exiss, i.e., x x a min{i a, 0}. We inroduce he following model primiives: α = discoun facor of revenues and coss in fuure periods, 0 < α 1; c = purchasing cos per uni ordered; s = salvage value per uni disposed; b = backlogging cos per uni backlogged a he end of a period; h a = holding cos per uni socked and accessible o cusomers a he end of a period; h w = holding cos per uni socked in he warehouse a he end of a period; r d = uni reallocaion fee from he warehouse o he cusomer-accessible sorage; r w = uni reallocaion fee from he cusomer-accessible sorage o he warehouse. Wihou loss of generaliy, we assume he following inequaliies hold: b > (1 α)(r d + 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; c > s : uni procuremen cos dominaes he uni salvage value; p > α(c + r d ) + b : posiive margin for backlogged demand. 119

133 Noe ha alhough we assume ha he parameers and demand are saionary hroughou he planning horizon, he srucural resuls in his chaper remain valid when he parameers and demand disribuions are ime-dependen. As discussed in Secion 5.1, we assume ha demand in period, D, depends negaively on he prevailing price and cusomer-accessible invenory level a he beginning of his period according o a general sochasic funcional form: D = δ(p, I a, ϵ ), where ϵ is a random erm wih a known coninuous disribuion and a conneced suppor. δ(,, ϵ ) is a wice coninuously differeniable funcion sricly decreasing in p and decreasing in I a for any ϵ. We base our analysis of he problem on he following demand form: δ(p, I, ϵ ) = (d(p ) + γ(i a ))ϵ m + ϵ a, where E{ϵ a } = 0 and E{ϵ m } = 1. (5.1) We assume ha ϵ s are i.i.d. random vecors wih ϵ a suppored on [a, a] and ϵ m suppored on [m, m] (m 0). A leas one of he wo random variables (ϵ a and ϵ m ) follows a coninuous disribuion (i.e., a a or m m), which ensures ha D follows a non-degenerae coninuous disribuion suppored on he inerval: [(d(p )+γ(i a ))m+a, (d(p )+γ(i a ))m+ a], for any (p, I a ). Noe ha he above demand model is quie general and includes as special cases several demand models from he exising lieraure. For example, when ϵ m = 1 wih probabiliy 1, he demand model is reduced o he addiive demand model; if ϵ a = 0 wih probabiliy 1, i is reduced o he muliplicaive demand model (as a generalized version of he one proposed in [145]); and if γ( ) 0, he demand model is reduced o he sandard price-dependen demand model (as he one proposed in [47]). The erm d(p ) summarizes he impac of price on demand in period. As assumed above, d( ) is sricly decreasing in p. In some marke where compeiion is fierce and he firm has no pricing power, he price is exogenously fixed a p 0 and he price induced demand is fixed a d 0 = d(p 0 ). The erm γ(i a ), which is a decreasing funcion of I a, capures he scarciy effec of invenory on demand. Hereafer, we refer o γ( ) as he scarciy funcion, and γ ( ) as he inensiy of scarciy effec. The dependence of demand on invenory is measured by γ ( ). i.e., he smaller he γ ( ), he more inensive he poenial demand depends on he cusomer-accessible invenory level. When demand is independen of invenory, γ(i a ) γ 0 for all cusomer-accessible invenory level I a. Noe ha our demand model generalizes he one in [145] in he sense ha our model also capures he impac of endogenous sales price on demand. 120

134 Since d( ) is sricly decreasing in p, we assume p(d ) be is sricly decreasing inverse. For he convenience of our analysis, we change he decision variable from p o d [d, d], where d = d( p) and d = d(p). To avoid he unrealisic case where demand becomes negaive, we assume ha d + γ(k a ) 0 o ensure ha E{D } = d + γ(i a ) 0 for any d [d, d] and I a analysis. K a. We impose he following hree assumpions hroughou our Assumpion p( ) is wice coninuously differeniable and concavely decreasing in d, wih p (d ) < 0 for d [d, d]. In addiion, p(d )d is concave in d. The concaviy of p(d )d in d suggess he decreasing marginal revenue wih respec o demand, which is a sandard assumpion in join pricing and invenory managemen lieraure, see, e.g., [47, 112, 136]. For a more comprehensive discussion on decreasing marginal revenue assumpions, see [196]. The concaviy of p( ) implies ha he demand is more price-sensiive when sales prices are higher. This is also a common assumpion in he lieraure, see, e.g., [70]. As [145], we also assume ha demand is concavely decreasing in he cusomeraccessible lefover invenory: Assumpion γ( ) is concavely decreasing and wice coninuously differeniable. In addiion, lim I a γ (I a ) = 0 and lim I a γ(ia ) = γ 0. The concaviy of γ( ) refers o he phenomenon ha a higher cusomer-accessible lefover invenory level has a greaer marginal effec on poenial demand. However, when he backlogged demand is very high, is value of simulaing high poenial demand is limied, because γ( ) is bounded from above. In oher words, he impac of invenory on demand is small under a large backorder volume so demand does no increase o infiniy. Therefore, he firm canno induce arbirarily high demand by creaing an arbirarily long wai-lis. The underlying inuiion of he boundedness of γ( ) is ha he high demand induced by a long wai-lis is canceled ou by he impaience i arouses. Assumpion Le R(d, I a ) := (p(d ) b α(c + r d ))(d + γ(i a )). (5.2) R(d, I a ) is joinly concave in (d, I a ) on is domain. 121

135 Assumpion is imposed mainly for echnical racabiliy, because i is required o esablish he join concaviy of he objecive and value funcions in each period (see he discussions afer Lemma 14). Noe ha R(d, I a ) is he expeced difference beween he revenue and he oal cos (i.e., he procuring, displaying and backlogging coss) o saisfy he curren demand in he nex period, when he firm holds a cusomer-accessible invenory I a and charges a sales price p(d ). The join concaviy of R(, ) implies ha he expeced difference beween he revenue and he oal cos o mee he curren demand in he nex period has decreasing marginal values wih respec o boh he expeced price-induced demand and cusomer-accessible invenory level. The join concaviy of R(, ) is sronger han he concaviy of expeced revenue (Assumpion 5.3.1), because i also capures he impac of invenory-dependen demand upon revenue, procuremen cos, reallocaion cos and backlogging cos. We discuss his assumpion in deail in he following subsecion Discussions on Assumpion Assumpion is essenial o show he analyical resuls in his chaper. We firs characerize he necessary and sufficien condiion for Assumpion 5.3.3: Lemma 11 R(d, I a ) is joinly concave in (d, I a ) on is domain if and only if (p (d )(d + γ(i a )) + 2p (d ))(p(d ) b α(c + r d ))γ (I a ) (p (d )γ (I a )) 2, (5.3) for all d [d, d] and I a K a. Condiion (5.3) is complicaed and somewha difficul o undersand. Hence, we give he following simpler necessary condiion for Assumpion o hold. Lemma 12 If R(, ) is joinly concave on is domain, hen we have: (a) For any I a such ha γ (I a ) = 0, γ (I a ) = 0 as well. Therefore, here exiss < 0, if I a hreshold I K a (I may be ), such ha γ (I a a > I, ) and = 0, oherwise, < 0, if I γ (I a a > I, ) = 0, oherwise. 122

136 (b) There exiss an 0 < M < +, such ha, for any I a K a, (γ (I a )) 2 Mγ (I a ). Lemma 12(a) shows ha, if Assumpion is saisfied, here exiss a hreshold invenory level I, such ha here is no scarciy effec for all cusomer accessible invenory level below his hreshold and he scarciy funcion is sricly decreasing and sricly concave for all cusomer accessible invenory level above his hreshold. Lemma 12(b) proves ha R(, ) is joinly concave only if, for all I a, compared wih γ (I a ), γ (I a ) is sufficienly big. In oher words, in he region where he scarciy effec exiss (i.e., γ (I a ) < 0), he curvaure of he funcion γ( ) should be sufficienly big. This condiion is no resricive and, for example, can be saisfied by he commonly used power or exponenial families of scarciy funcions. We remark ha, mahemaically, Lemma 12(a) is a corollary of Lemma 12(b). Nex, we show ha he necessary condiion characerized in Lemma 12(b) is also sufficien o some exen. Lemma 13 If here exiss an 0 < M < +, such ha, for any I a K a, (γ (I a )) 2 Mγ (I a ), he following saemens hold: (a) For any inverse demand curve p( ), here exiss a hreshold δ < +, such ha, for any δ δ, wih ˆp δ ( ) := p( ) + δ, ˆR δ (d, I a ) := (ˆp δ (d ) b α(c + r d ))(d + γ(i a )) is joinly concave in (d, I a ) for d [d, d] and I a K a. (b) Suppose ha p ( ) 0 for any d [d, d]. For any scarciy funcion γ( ), here exiss a hreshold ς < +, such ha, for any ς ς, wih ˆγ ς ( ) := γ( ) + ς, ˆR ς (d, I a ) := (p(d ) b α(c + r d ))(d + ˆγ ς (I a )) is joinly concave in (d, I a ) for d [d, d] and I a K a. Lemma 13 demonsraes ha, as long as he condiion characerized in Lemma 12(b) on he scarciy funcion, γ( ), is saisfied, R(, ) is joinly concave on is domain if (a) he sales price of he produc, p( ), is sufficienly high relaive o he inverse of price sensiiviy, p ( ) ; or (b) he price is no linear in demand, and he scarciy effec driven demand, γ( ), is sufficienly high relaive o he scarciy inensiy, γ ( ). These sufficien condiions have a clear economic inerpreaion: he price elasiciy of demand (i.e., dd/d dp /p ) is sufficienly high relaive o he invenory elasiciy of demand (defined as dγ/γ ). In pracice, his di a/ia condiion is no resricive. Compared wih he primary demand leverage (i.e., he sales price), he cusomer accessible invenory (hrough he scarciy effec) has less impac 123

137 upon he poenial demand, because no every cusomer cares abou he backlogging risk of a produc, bu everyone cares abou is price. Therefore, Assumpion can be saisfied under a mild condiion wih economic inerpreaion. Finally, when Assumpion does no hold (i.e., R(, ) is no joinly concave), we have conduced exensive numerical experimens o es he robusness of our analyical resuls. Our numerical resuls verify ha he analyical characerizaions of he opimal policies in our model are robus and hold for non-concave R(, ) s in all of our experimens. In paricular, Lemma 12 implies ha when he scarciy funcion γ( ) conains a linear and sricly decreasing piece, R(, ) is no joinly concave. We presen our numerical experimens for his case in Secion Unified Model In his secion, we propose a unified model o analyze he join pricing and invenory replenishmen/disposal/reallocaion problem when he firm faces random demand which is negaively correlaed wih he cusomer-accessible lefover invenory. We characerize he srucure of he opimal pricing and invenory policy and give sufficien condiions under which he firm does no (a) dispose is on-hand invenory, (b) wihhold any invenory, (c) reallocae is cusomer-accessible invenory o he warehouse, or (d) display any posiive invenory o cusomers. This model is suiable for he case where he firm can boh wihhold is on-hand invenory in is privae warehouse no observable by cusomers (e.g., clohing and elecronics markes) and dispose i (e.g., in he hi-ech indusry, he evoluion of produc generaion is so fas ha he reailers/manufacurers have o sell excess old versions a a significanly discouned price). When poenial demand is negaively correlaed wih he cusomer-accessible lefover invenory, he firm faces greaer overage risk, because a high cusomer-accessible lefover invenory no only incurs a high holding cos bu also suppresses poenial demand. Boh invenory wihholding and invenory disposal policies enable he firm o sraegically keep a low cusomer-accessible invenory, so as o induce high poenial demand and miigae he oversocking risk. Hence, we incorporae invenory wihholding and invenory disposal ino our unified model. The unified model is quie general and can be reduced o several specific models ha are of ineres on heir own. For example, we show ha if he warehouse holding cos h w is 124

138 sufficienly large, he unified model is reduced o he one wihou invenory wihholding, which is discussed in deail in Secion Besides, if he disposal salvage value s is sufficienly low, he unified model is reduced o he one wihou invenory disposal, which is discussed in deail in Secion To formulae he planning problem as a dynamic program, le: V (I a, I ) = he maximum expeced discouned profis in periods, 1,, 1, when saring period wih a cusomer-accessible invenory level I a and a oal invenory level I. Wihou loss of generaliy, we assume ha he excess invenory in he las period (period 1) is discarded wihou any salvage value, i.e., V 0 (I a 0, I 0 ) = 0, for any (I a 0, I 0 ). The opimal value funcions saisfy he following recursive scheme: V (I a, I ) = r d I a + ci + max J (x a (x a,x,d ) F (I a), x, d, I a, I ), (5.4) 125

139 where F (I a ) := {(x a, x, d ) : x a [min{i a, 0}, K a ], x x a, d [d, d]} denoes he se of feasible invenory and pricing decisions, and J (x a, x, d, I a, I ) = r d I a ci + p(d )E[δ(p(d ), I a, ϵ )] c(x I ) + + s(x I ) r d (x a I a ) + r w (x a I a ) h w (x x a ) E{h a (x a δ(p(d ), I a, ϵ )) + + b(x a δ(p(d ), I a, ϵ )) } +αe{v 1 (x a δ(p(d ), I a, ϵ ), x δ(p(d ), I a, ϵ ))} = p(d )(d + γ(i a )) (c s)(x I ) (h w + c)x (r d + r w )(x a I a ) + (h w r d )x a +E[(b + αr d )(x a (d + γ(i a ))ϵ m ϵ a ) +αc(x (d + γ(i a ))ϵ m ϵ a )] +E{α[V 1 (x a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ) r d (x a (d + γ(i a ))ϵ m ϵ a ) c(x (d + γ(i a ))ϵ m ϵ a )] (b + h a )(x a (d + γ(i a ))ϵ m ϵ a ) + } = (p(d ) α(c + r d ) b)(d + γ(i a )) (c s)(x I ) (r d + r w )(x a I a ) (h w + (1 α)c)x +(h w + b (1 α)r d )x a +E{ (h a + b)(x a (d + γ(i a ))ϵ m ϵ a ) + +α[v 1 (x a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ) r d (x a (d + γ(i a ))ϵ m ϵ a ) c(x (d + γ(i a ))ϵ m ϵ a )]} = R(d, I a ) θ(x I ) (r d + r w )(x a I a ) ψx + ϕx a +E{G (x a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a )},(5.5) where G (x, y) := (b + h a )x + + α(v 1 (x, y) r d x cy), θ := c s = he uni loss of invenory disposal, (5.6) ψ := h w + (1 α)c = he uni cos of replenishing and holding invenory in he warehouse, ϕ := h w + b (1 α)r d = he uni saving of reallocaing warehouse invenory o he cusomer-accessible sorage. 126

140 We use (x a (I a, I ), x (I a, I ), d (I a, I )) o denoe he maximizer in (5.4), which sands for he opimal policy in period, wih cusomer-accessible invenory level I a and oal invenory level I. To characerize he srucure of he opimal invenory replenishmen/disposal/reallocaion and pricing policies, we define he following opimizers: (x a (I a ), d (I a )) and ( x a (I a ), x (I a ), d (I a )). Le (x a (I a ), d (I a )) := argmax x a [min{i a,0},ka],d [d, d]r(d, I a ) + βx a +E[G (x a δ(p(d ), I a, ϵ ), x a δ(p(d ), I a, ϵ ))], (5.7) where β := b (1 α)(c + r d ) > 0. (5.8) x a (I a ) is he opimal order-up-o invenory level, if he firm procures posiive invenory and displays all of is on-hand invenory o cusomers, whereas d (I a ) is he opimal expeced price-induced demand in his case. Le ( x a (I a ), x (I a ), d (I a )) :=argmax (x a,x,d ) F (I a ) {R(d, I a ) + (θ ψ)x (r d + r w )(x a I a ) + ϕx a (5.9) + E{G (x a δ(p(d ), I a, ϵ ), x δ(p(d ), I a, ϵ ))}}. When he firm disposes is on-hand invenory, x a (I a ) is he opimal display-up-o invenory level and x (I a ) is he opimal dispose-down-o invenory level, whereas d (I a ) is he opimal expeced price-induced demand. The following lemma esablishes he properies of he wo opimizers: Lemma 14 For each = T, T 1,, 1, he following saemens hold: (a) J (x a, x, d, I a, I ) is joinly concave and coninuously differeniable in (x a, x, d, I a, I ) excep for a se of measure zero; for any fixed (I a, I ), J (,,, I a, I ) is sricly joinly concave in (x a, x, d ). (b) V (I a, I ) is joinly concave and coninuously differeniable in (I a, I ), whereas V (I a, I ) r d I a ci is decreasing in I a and I. Lemma 14 proves ha he objecive funcion in each period is joinly concave and almos everywhere differeniable and he value funcion is joinly concave and coninuously differeniable. Moreover, he second half of Lemma 14(b) implies ha he normalized 127

141 value funcion, V (I a, I ) r d I a ci, is decreasing in boh he cusomer-accessible invenory level I a and he oal invenory level I, which generalizes Proposiion 5.1 in [145]. We remark ha he join concaviy of R(, ) on is enire domain is necessary o prove ha he objecive funcions J (,,, I a, I ) and he value funcions V (, ) are joinly concave, which is essenial o analyically esablish oher srucural resuls in his chaper. We can easily find examples in which R(, ) fails o be joinly concave (e.g., γ( ) conains a linear and sricly decreasing piece) and leads o non-concave J (,,, I a, I ) s and V (, ) s. In his case, we are unable o analyically show he srucural resuls in his chaper (e.g., Theorem and Theorem 5.5.1). In Secion 5.7.1, we numerically es wheher he srucure of he opimal policy characerized in our heoreical model sill holds. Wih he help of Lemma 14, we characerize he srucural properies of he opimal policy in he unified model as follows: Theorem For = T, T 1,, 1, he following saemens hold: (a) x a (I a ) x (I a ). Moreover, le q (I a, I ) := x (I a, I ) I denoe he opimal order/disposal quaniy and we have: > 0 if I < x a (I a ), q (I a, I ) = 0 if x a (I a ) I x (I a ), < 0 oherwise, i.e., i is opimal o order if and only if I < x a (I a ) and o dispose if and only if I > x (I a ). (b) If I < x a (I a ), x a (I a, I ) = x (I a, I ) = x a (I a ), d (I a, I ) = d (I a ), i.e., i is opimal o order and display up o x a (I a ) and charge a lis-price p(d (I a )). (c) If I > x (I a ), (x a (I a, I ), x (I a, I ), d (I a, I )) = ( x a (I a ), x (I a ), d (I a )), i.e., i is opimal o dispose he oal invenory level down o x (I a ), display x a (I a ), and charge a lis-price p( d (I a )). (d) If I [x a (I a ), x (I a )], x (I a, I ) = I, i.e., i is opimal o keep he oal invenory level. (e) x a (I a ) is coninuously decreasing in I a, whereas d (I a ) is coninuously increasing in I a. 128

142 Theorem generalizes Proposiion 5 in [145] by characerizing he srucure of he opimal policy in our unified model. We show ha a cusomer-accessible-invenorydependen order-up-o/dispose-down-o/display-up-o lis-price policy is opimal. The opimal policy is characerized by wo hresholds: he ordering hreshold x a (I a ) and he disposal hreshold x (I a ), boh of which depend on he cusomer-accessible invenory level, I a. If he oal invenory level, I, is below he ordering hreshold, i.e., I < x a (I a ), he firm should order-up-o his hreshold, display all of is on-hand invenory o cusomers, and charge a cusomer-accessible-invenory-dependen lis-price p(d (I a )). If he oal invenory level is higher han he disposal hreshold, i.e., I > x (I a ), he firm should dispose-down-o his hreshold, display par of is on-hand invenory, x a (I a ), o cusomers, and charge a cusomer-accessible-invenory-dependen lis-price p( d (I a )). If he oal invenory level is beween he above wo hresholds, i.e., I [x a (I a ), x (I a )], he firm should keep is oal ne invenory and display par of i o cusomers. In paricular, Theorem 5.4.1(b) implies ha if i is opimal o order, he firm should no wihhold anyhing. Order-and-wihhold policy is dominaed by displaying he same amoun of invenory o cusomers bu no ordering he invenory ha will be wihheld (so no invenory will be wihheld). This is inuiive, because he marginal cos of order-and-wihhold is a leas c + h w (procuremen cos and holding cos in he warehouse), while he marginal benefi of invenory wihholding is a mos αc (saving from he purchasing cos in he nex period). Moreover, par (e) of Theorem demonsraes ha as he excess cusomeraccessible invenory level increases, lower demand is induced and he firm has a greaer incenive o urn i over, boh of which give rise o lower opimal order-up-o levels and opimal sales prices. The excess invenory of he firm generally has hree impacs on he performance of he sysem: (1) saisfying fuure demand, (2) incurring holding coss and (3) inducing/suppressing poenial demand, he firs wih posiive marginal value and he oher wo wih negaive marginal values. Hence, afer normalizing he firs effec (V (I a, I ) r d I a ci ), he value-o-go funcion of he firm is decreasing in is cusomer-accessible invenory level and oal invenory level. To beer deal wih he inerwined radeoff beween hese hree effecs, he firm can adop dynamic pricing, invenory wihholding and invenory disposal sraegies. As suggesed in Theorem 5.4.1, he firm needs o price he produc in accordance o he cusomer-accessible invenory level so as o beer conrol 129

143 he scarciy effec of demand. Theorem also shows ha when he oal invenory is high, he firm should wihhold and dispose is on-hand invenory, which saves holding coss and miigaes he risk of suppressing poenial demand. On he oher hand, he opporuniy o redisplay he wihheld invenory in he warehouse enables he firm o saisfy poenial demand wihou discouraging i. In shor, combining dynamic pricing, invenory wihholding and invenory disposal policies helps he firm beer mach supply and demand and grealy enhances is profiabiliy. We proceed o analyze how he model primiives influence he firm s opimal operaional decisions, such as invenory disposal, invenory wihholding, and invenory display. Theorem The following saemens hold: (a) If h w αc s, x (I a ) = x a (I a ) for any = T, T 1,, 1. (b) There exiss an s < c, such ha, if s s, x (I a ) = + for any I a = T, T 1,, 1. K a and (c) If inf I a <K a γ (I a ) M, for some M < +, here exiss an r < +, such ha, if r w r, x a (I a ) I a, for any I a K a and = T, T 1,, 1. On he oher hand, if inf I a <K a γ (I a ) =, for any r w > 0, here exiss a hreshold I (r w ) < K a, such ha, if I a I (r w ), x a (I a ) < I a, for any = T, T 1,, 2. (d) Le ι < 1, and D := sup{ : P (D ) ι}, i.e., he probabiliy ha he demand in period exceeds D is smaller han ι, regardless of he policy he firm employs. If α(p b α(c + r d ) + mβ)(1 ι)γ ( D) + (r d + r w + ϕ) 0, (5.10) hen x a (I a, I ) 0 for any I a K a, I, and = T, T 1,, 1. Theorem 5.4.2(a) shows ha, when he warehouse holding cos is sufficienly high (h w αc s), he firm should display all of is on-hand invenory o cusomers. Par (b) demonsraes ha, when invenory disposal is sufficienly cosly (s s ), he firm would raher no dispose any of is invenory, regardless of is oal invenory level. When he condiion in par (a) [par (b)] holds, he unified model is reduced o he model wihou invenory wihholding [invenory disposal], which generaes addiional insighs and is horoughly discussed in Secion [Secion 5.5.2]. Theorem 5.4.2(c) reveals ha he opimal invenory reallocaion balances he radeoff beween saving he curren 130

144 reallocaion cos and simulaing fuure demand. More specifically, if he inensiy of scarciy effec is bounded, he firm should no reallocae is invenory from he cusomeraccessible sorage o he warehouse, as long as he reallocaion fee is sufficienly high. Oherwise (i.e., he inensiy of scarciy effec is unbounded), he firm should always wihhold par of is invenory in he warehouse, if he excess cusomer-accessible invenory level is high enough. Theorem 5.4.2(d) shows ha when he demand-simulaing effec/scarciy effec of invenory is sufficienly srong (characerized by (5.10)), he backlogging cos incurred by he wai-lis is dominaed by he revenue generaed by he scarciy effec. Therefore, he firm should no display any posiive invenory, and every cusomer has o join a wailis before receiving he produc. This analyical resul jusifies he markeing sraegy adoped by, e.g., BMW, in which he availabiliy of Mini Cooper is inenionally limied and more cusomers are araced by is wai-lis. 5.5 Addiional Resuls in Two Special Cases In his secion, we sudy wo imporan special cases of our unified model ha are of ineres on heir own: he model wihou invenory wihholding and he model wihou invenory disposal. As shown in Theorem 5.4.2, when i is oo expensive o wihhold [dispose] invenory, i is opimal for he firm no o wihhold [dispose] any invenory. These wo special cases deliver new resuls and sharper insighs on he impac of he invenory-dependen demand upon he firm s pricing and invenory decisions. We also characerize how he operaional flexibiliies (e.g., an increase in he salvage value and he invenory wihholding opporuniy) faciliae he firm o miigae he addiional overage risk caused by invenory-dependen demand Wihou Invenory Wihholding In some circumsances, he firm canno sore is invenory in he warehouse, due o, e.g., oo cosly wihholding or oo inconvenien ransporaion. For insance, car dealers usually display all of is auomobiles in he sore, because wihholding and redisplaying he invenory is oo cosly and inconvenien. In his subsecion, we confine our analysis o he model wihou invenory wihholding. In his model, since no invenory is sored in 131

145 he warehouse, he sae space dimension is reduced o one, and such reducion offers new resuls and sharper insighs on how he invenory-dependen demand influences he firm s opimal decisions. More specifically, we demonsrae ha he scarciy effec of invenory increases he oversocking risk and, hus, drives he firm o se a lower order-up-o level and charge a lower sales price. On he oher hand, when he firm is blessed wih a higher disposal flexibiliy (i.e., a higher salvage value), i has more capaciy o miigae such overage risk by geing rid of is surplus invenory. We show ha he firm wih a higher salvage value ses higher order-up-o levels and sales prices. To formulae he planning problem as a dynamic program, le: V s (I a ) = he maximum expeced discouned profis in periods, 1,, 1, when saring period wih a cusomer-accessible invenory level I a. Since no invenory is wihheld in he warehouse in his model, I a = I, and we don need o record he oal invenory level I. Therefore, he sae space dimension is reduced o one. Similarly, we will no incur he warehouse invenory holding cos (h w ), he redisplay cos (r d ), and he wihholding cos (r w ) in his model. The superscrip s refers o single locaion sorage. Wihou loss of generaliy, we assume he excess invenory in he las period (period 1) is discarded wihou any salvage value, i.e., V s 0 (I a 0 ) = 0, for any I a 0 K a. The value funcions saisfy he following recursive scheme: V s (I a ) = ci a + max J s (x a,d ) F s (I a) (x a, d, I a ), where F s (I a ) := [min{0, I a }, K a ] [d, d] denoes he se of feasible order-up-o/disposedown-o levels and expeced price-induced demand, and J s (x a, d, I a ) =p(d )E[δ(p(d ), I a, ϵ )] + s(x a I a ) c(x a I a ) + ci a E[b(x a δ(p(d ), I a, ϵ )) + h a (x a δ(p(d ), I a, ϵ )) + ] + αe[v s 1(x a δ(p(d ), I a, ϵ ))]. Following he algebraic manipulaion similar o ha in (5.5), we obain: J s (x a, d, I a ) =R s (d, I a ) + β s x a θ(x a I a ) + E[G s (x a δ(p(d ), I a, ϵ ))], where R s (d, I a ) :=(p(d ) b αc)(d + γ(i a )), G s (y) := (b + h a )y + + α[v s 1(y) cy], β s :=b (1 α)c, (5.11) 132

146 and θ is defined in (5.6). Noe ha, under Assumpion 5.3.3, R s (d, I a ) = R(d, I a ) + αr d (d + γ(i a )) is joinly concave on is domain. As a corollary of Theorem 5.4.1, he opimal policy in he model wihou invenory wihholding is an invenory-dependen order-up-o/dispose-down-o lis-price policy, as shown below: Theorem Consider a model wihou invenory wihholding. For each = T, T 1,, 1, he following saemens hold: (a) g s (x a, d, I a ) := E[G s (x a δ(p(d ), I a, ϵ ))] is joinly concave and coninuously differeniable in (x a, d, I a ) if x a I a ; for any fixed I a, g s (,, I a ) is sricly concave. (b) V s (I a ) is concave in I a. V s (I a ) ci a in I a. is decreasing and coninuously differeniable (c) J s (,, I a ) is sricly concave for any fixed I a, and here exiss a unique (x s (I a ), d s (I a )) such ha (x s (I a ), d s (I a )) = argmax (x,d ) F s (I a ) J s (x a, d, I a ). (d) Le q s (I a ) = x s (I a ) I a denoe he opimal order/disposal quaniy. There exis wo hreshold invenory levels I H and I L (I L < I H ), such ha, > 0 if I a < I L, q s (I a ) = 0 if I L I a I H, < 0 oherwise, i.e., he firm should order if is invenory level I a is less han he lower hreshold I L, dispose if i is more han he higher hreshold I H, and no order or dispose if i is beween he wo hresholds. (e) If I a < I L or I a > I H, he opimal order-up-o/dispose-down-o level x s (I a ) is decreasing in I a. If I L I a I H, he opimal invenory afer replenishmen/disposal is increasing in I a. (f) The opimal price-induced-demand d s (I a ) is increasing in I a. 133

147 Theorem implies ha, when he firm canno wihhold is on-hand invenory, he opimal policy is o order when he cusomer-accessible invenory level is low (below I L ), o dispose when i is high (above I H ), and no o adjus when i is beween he wo hresholds. The opimal order-up-o/dispose-down-o and lis-price levels are cusomeraccessible-invenory-dependen. As shown in Theorem 5.5.1, when he cusomer-accessible invenory level is higher, boh order-up-o/dispose-down-o levels and sales prices are lower, because a high cusomer-accessible invenory level suppresses poenial demand and he firm has a srong incenive o urn i over. We proceed o analyze how he scarciy effec of invenory impacs he opimal pricing and invenory policies. Compared wih he model in which demand is independen of invenory, when poenial demand is negaively correlaed wih cusomer-accessible lefover invenory levels, he marginal value of on-hand invenory decreases and he firm suffers from he demand reducion caused by a high invenory level. As a resul, he firm should order less/dispose more o miigae he addiional oversocking risk caused by he scarciy effec of invenory. A he same ime, o beer cach he sales opporuniy, i is opimal o underprice he produc so as o arac more cusomers. Moreover, in a marke where he firm has lile power o se he sales price, we are able o prove a sharper resul ha wih a more inensive scarciy effec, he firm should keep a lower invenory level afer replenishmen/disposal. The following heorem formalizes hese inuiions. Theorem Consider a model wihou invenory wihholding. Assume D = δ(d, I a, ϵ ) and ˆD = ˆδ(d, I a, ϵ ) wih invenory dependen erm γ(i a ) and ˆγ(I a ), respecively. We also assume ha he demand is of addiive form (i.e., ϵ m = 1 wih probabiliy 1). The following saemens hold: (a) Assume ha ˆγ(I a ) = γ 0 = lim x γ(x) for all I a K a, i.e., ˆD does no depend on he cusomer-accessible invenory level. We have ha I L ÎL, I H ÎH, x s (I a ) ˆx s (I a ) and d s (I a ) ˆd s (I a ) for all I a K a. (b) Assume ha γ (I a ) ˆγ (I a ) for all I a K a and ha lim I a γ(ia ) = lim I a ˆγ(Ia ) = γ 0. Le p = p = p 0 and d 0 = d(p 0 ). We have I L ÎL, I H ÎH and x s (I a ) ˆx s (I a ) for all I a K a. 134

148 As a generalizaion of Theorem 3.2 in [145] o he model wih dynamic pricing and invenory disposal, Theorem shows ha he firm should undersock and underprice he produc under he scarciy effec of invenory. In Theorem 5.5.2, we need he addiive demand assumpion, i.e., ϵ m = 1 almos surely. The addiive demand model is widely applied in he join pricing and invenory conrol lieraure (see, e.g., [112, 73, 136]), mosly because i enhances he echnical racabiliy and faciliaes he analysis. To show Theorem and oher comparisons beween he opimizers in differen models (Theorems below), we need o ieraively esablish he comparisons beween he derivaives of value funcions. The addiive demand form is necessary o link he monooniciy relaionship beween opimizers and ha beween derivaives. All resuls in his chaper, excep Theorems , hold for he more general demand form inroduced in (5.1). Efficienly disposing surplus invenory proecs he firm from he demand-suppressing effec of invenory. As he salvage value increases, he cos of invenory disposal decreases, and he firm has greaer disposal flexibiliy. We characerize how he salvage value impacs he opimal pricing and invenory decisions in he following heorem: Theorem Consider a model wihou invenory wihholding. For any = T, T 1,, 1, assume ha he demand is of addiive form (i.e., ϵ m = 1 wih probabiliy 1), and s < ŝ. (a) I a ˆV s (I a ) I a V s (I a ). (b) ÎL I L. (c) ˆx s (I a ) x s (I a ) and, hence, ˆq s (I a ) q s (I a ) for all I a ÎH. (d) ˆd s (I a ) d s (I a ). Theorem 5.5.3(a) shows ha he marginal value of on-hand invenory increases in he salvage value. Pars (b) - (d) demonsrae ha wih a higher salvage value, he firm should se higher ordering hresholds, order-up-o levels, and sales prices. On one hand, recall from Theorem ha he invenory-dependen demand srenghens oversocking risk by suppressing poenial demand so ha boh opimal order-up-o/disposal-down-o levels and opimal sales prices are lower in he model wih invenory-dependen demand 135

149 han hose in he model wih invenory-independen demand. On he oher hand, however, Theorem demonsraes ha increased operaional flexibiliy (i.e., a higher salvage value) miigaes he demand loss driven by a high cusomer-accessible invenory level and, hence, wih higher disposal flexibiliy, he firm is able o se higher order-up-o levels and sales prices o win more profi Wihou Invenory Disposal The model wihou invenory disposal applies o he cases where he invenory is eiher oo expensive or oo inconvenien o dispose. For example, in he auomobile indusry, he unsold cars of he las year model is oo cosly o dispose. In oher indusries like chemical engineering, producs are ofen so environmenally unfriendly ha hey canno be disposed arbirarily. The model wihou invenory disposal has a simpler opimal policy srucure (cusomer-accessible-invenory-dependen order-up-o/displayup-o lis-price policy) and, like he model wihou invenory wihholding, delivers sharper insighs regarding he impacs of invenory-dependen demand and invenory wihholding policy. More specifically, we show ha invenory-dependen demand moivaes he firm o order less and charge a lower sales price, whereas he invenory wihholding policy helps miigae he overage risk and increases he opimal order-up-o levels and sales prices. As a counerpar of Theorem 5.5.2, he following heorem shows ha invenorydependen demand drives down he opimal order-up-o levels and sales prices in he model wihou invenory disposal: Theorem Consider a model wihou invenory disposal. For any = T, T 1,, 1, assume ha r d = r w = 0, and h w h a, i.e., reallocaion is cosless and i is more cosly o sore he invenory in he warehouse. In addiion, assume ha D = δ(d, I a, ϵ ) and ˆD = ˆδ(d, I a, ϵ ) wih invenory dependen erm γ(i a ) and ˆγ(I a ), respecively, where ˆγ(I a ) = γ 0 = lim x γ(x) for all I a K a, i.e., ˆD does no depend on he cusomeraccessible invenory level. ϵ m = 1 wih probabiliy 1). We have: Furher assume ha, he demand is of addiive form (i.e., (a) The firm in he sysem wih demand ˆD should no wihhold any invenory. (b) x a (I a ) ˆx s (I a ) and d (I a ) ˆd s (I a ) for all I a K a. 136

150 Invenory wihholding policy enables he firm o beer conrol demand by inenionally making par of is invenory unavailable o is cusomers. Hence, invenory wihholding policy can sabilize he demand process and increase he opimal order-up-o levels and sales prices, as shown below: Theorem Consider a model wihou invenory disposal. For any = T, T 1,, 1, assume ha he demand is of addiive form (i.e., ϵ m = 1 wih probabiliy 1), r d = r w = 0 (i.e., reallocaion is cosless). If I = I a, we have x a (I a ) x s (I a ) for I a max{i a : x a (I a ) I a }, and d (I a, I ) d s (I a ) for I a K a. Noe ha, Theorem 6 needs he assumpion ha invenory reallocaion is cosless (r d = r w = 0), because his assumpion is necessary o reduce he sae space dimension in is proof. We also assume r d = r w = 0 for Theorem 7, mainly for exposiional convenience and he resuls sill hold under he general condiion ha r d, r w 0. To summarize, invenory wihholding and invenory disposal have similar sraegic implicaions in dealing wih invenory-dependen demand. The firm employs hese sraegies o hedge agains he overage risk caused by he scarciy effec of invenory and simulae more poenial demand. 5.6 Responsive Invenory Reallocaion In our previous analysis, we assume ha he firm can wihhold and redisplay invenory only a he beginning of he decision epoch before he demand realizes. In his subsecion, we relax his assumpion by allowing he firm o responsively reallocae is on-hand invenory afer he demand realizaion. The responsive invenory reallocaion enables he firm o opimize is invenory policy afer he demand uncerainy realizes, so ha he supply and demand are beer mached and he radeoff beween meeing curren and inducing poenial demand is beer balanced. Noe ha when responsive invenory reallocaion is allowed, he firm should no reallocae is invenory before he demand realizes. A he beginning of each period, he firm chooses is invenory replenishmen/disposal quaniy and he sales price. The demand hen realizes, afer which he firm decides he invenory reallocaion quaniies beween he warehouse and he cusomer-accessible sorage. 137

151 To formulae he planning problem as a dynamic program, le V r (I a, I ) = he maximum expeced discouned profis in periods, 1,, 1, when saring period wih a cusomer-accessible invenory level I a and a oal invenory level I, where he superscrip r refers o responsive invenory reallocaion. Wihou loss of generaliy, we assume he excess invenory in he las period (period 1) is discarded wihou any salvage value, i.e., V r 0 (I a 0, I 0 ) = 0, for any (I a 0, I 0 ). We firs analyze he opimal reallocaion policy in period. Assume ha he orderup-o/dispose-down-o level se by he firm before he demand realizaion is x and he realized demand is D. The opimal display-up-o level, x ra (I a, x, D ), afer invenory reallocaion, is given by: x ra (I a, x, D ) = argmax min{0,i a D } x a x D{ r d(x a I a + D ) + r w (x a I a + D ) bx a h a x a+ h w (x x a D ) + αv r 1(x a, x D )} Hence, he opimal value funcions saisfy he following recursive scheme: V r (I a, I ) = max {p(d )E{δ(p(d ), I a (x,d ) F r (I a), ϵ )} c(x I ) + + s(x I ) + E D { max { r d (x a I a + D ) + r w (x a I a + D ) min{0,i a D } x a min{k a,x D } bx a h a x a+ h w (x x a D ) + αv 1(x r a, x D )}}}, where F r (I a ) := {(x, d ) : x min{i a, 0}, d [d, d]}. Following he algebraic manipulaion similar o ha in Equaion (5.5), we have: V r (I a, I ) =r d I a + ci + max {R(d, I a (x,d ) F r (I a) ) + r d (d + γ(i a )) θ(x I ) ψx + E D { max { (r d + r w )(y a I a ) + ϕy a min{d,i a} ya min{x,k a +D } + G r (y a D, x D )}}}, wih G r (x, y) := (h a + b)x + + α[v r 1(x, y) r d x cy]. (5.12) Comparing he value funcions (5.12) and (5.4), i is immediae ha by posponing he reallocaion decision ill afer demand realizaion, he firm achieves a higher expeced 138

152 oal profi. In he following heorem, we characerize he opimal invenory replenishmen/disposal/reallocaion and pricing policy in he model wih responsive invenory reallocaion: Theorem The following saemens hold for = T, T 1,, 1: (a) V r (I a, I ) is joinly concave and coninuously differeniable in (I a, I ), whereas he normalized value funcion V r (I a, I ) r d I a ci is decreasing in I a and I. (b) For any given x and realized D, v r (y a I a, x, D ) := (r d + r w )(y a I a ) + ϕy a + G r (y a D, x D ) is concave in y a. Therefore, he opimal cusomer-accessibleinvenory level is: x ra (I a, x, D ) = argmax min{d,i a } ya min{x,ka+d} {v r (y a I a, x, D )} D. (c) There exis wo cusomer-accessible-invenory-level-dependen hresholds, x r (I a ) and x r (I a ) (x r (I a ) x r (I a )), such ha i is opimal o order up o x r (I a ) if and only if I < x r (I a ), o dispose down o x r (I a ), if and only if I > x r (I a ), and o keep he oal invenory level oherwise. Moreover, here exis wo cusomer-accessibleinvenory-level-dependen sales prices p(d r (I a )) and p( d r (I a )), such ha i is opimal o charge a sales price p(d r (I a )) if I x r (I a ), and o charge a sales price p( d r (I a )) if I x r (I a ). Theorem 5.6.1(a) proves he join concaviy and coninuous differeniabiliy of he opimal value funcions. Par (b) shows ha, in each period, he opimal reallocaion policy is obained by solving a one-dimensional convex opimizaion afer he demand realizes. Consisen wih Theorem 5.4.1, par (c) of Theorem proves ha i is opimal o order if he oal invenory level is low (I < x r (I a )), and o dispose if i is high (I > x r (I a )), and o keep he saring invenory level oherwise. Compared wih Theorem 5.4.1, which characerizes opimal policy in he unified model, Theorem demonsraes ha i is possible ha he firm order-and-wihholds some invenory under he opimal responsive invenory reallocaion policy, because, in his case, he firm is blessed wih he flexibiliy o reallocae invenory afer he demand uncerainy is resolved. As in Theorem 5.4.2, we can show ha if he warehouse holding cos, h w, is high enough, i is opimal no o hold any invenory in he warehouse; if he salvage value, 139

153 s, is low enough, i is opimal no o dispose anyhing; and if he reallocaion fee o wihhold invenory, r w, is high enough, i is opimal o no reallocae any cusomeraccessible invenory o he warehouse. 5.7 Numerical Sudies This secion repors a se of numerical sudies ha (a) verify he robusness of our analyical resuls when Assumpion does no hold; (b) quanify he profi loss of ignoring he scarciy effec of invenory when making he pricing and invenory decisions; and (c) quaniaively evaluae he benefi of dynamic pricing in he presence of he scarciy effec. Our numerical resuls demonsrae ha (1) he srucural resuls developed in our heoreical model are robus and hold for a large se of non-concave R(, ) funcions; (2) he impac of he scarciy effec is significan and i is higher when he scarciy inensiy, demand variabiliy, and/or planning horizon lengh increase; and (3) he value of dynamic pricing under he scarciy effec is significan and i is higher under higher scarciy inensiy, demand variabiliy and/or shorer planning horizon. Throughou our numerical sudies, we assume ha he firm can neiher wihhold nor dispose is on-hand invenory for wo reasons: (a) o have a clear illusraion of he opimal policy srucure in a model where Assumpion does no hold; and (b) o single ou and highligh he impac of he focal operaional elemens (i.e., he scarciy effec of invenory and he dynamic pricing sraegy). We also assume ha he demand in each period is of he addiive form, i.e., ϵ m = 1 almos surely and D = d + γ(i a ) + ϵ a. Le {ϵ a } T =1 follow i.i.d. normal disribuions wih mean 0 and sandard deviaion σ. The inverse demand funcion is linear wih slope 1, i.e., p(d ) = p 0 d. We se he discoun facor α = 0.95, he uni holding cos h = 1, and he uni backlogging cos b = Opimal Policy Srucure wih Non-concave R(, ) Funcions In his subsecion, we numerically examine wheher he srucural resuls in our heoreical model are robus when Assumpion does no hold, i.e., R(, ) is no joinly concave. We have performed exensive numerical experimens o es he robusness of our analyical resuls. In all our numerical experimens, alhough Assumpion is violaed, he characerizaions of he opimal policy by our heoreical analysis (i.e., The- 140

154 orem 5.4.1, Theorem 5.5.1, and Theorem 5.5.2) coninue o hold. More specifically, our numerical resuls verify ha (a) he invenory-dependen order-up-o/lis-price policy is opimal and he order-up-o level is decreasing in he saring invenory level; (b) he opimal sales price [price-induced demand] is decreasing [increasing] in he saring invenory level; and (c) compared wih an invenory sysem wihou he scarciy effec, he firm wih he scarciy effec ses lower order-up-o levels and lower sales prices. Therefore, he srucural resuls of our heoreical model are robus and hold for non-concave R(, ) funcions in all our numerical experimens. Noe ha from Lemma 12(a) ha if he scarciy funcion γ( ) conains a linear and sricly decreasing piece, R(, ) is no joinly concave. Hence, we repor our numerical γ 0 exp(ηi resuls for he case where γ(i a a ), for I a 0, ) = wih η > 0. I s γ 0 1 ηi a, for 0 < I a K a, clear ha γ( ) is concavely decreasing and coninuously differeniable in I a for all I a K a, bu R(, ) is no joinly concave in he region {(d, I a ) : d [d, d], I a [0, K a ]}. We have performed exensive numerical experimens which es many combinaions of differen values of p 0, γ 0, c, η, σ, d, d, K a, and. In all he scenarios we examine, he predicions of he opimal policy by our heoreical analysis (i.e., Theorem 5.4.1, Theorem 5.5.1, and Theorem 5.5.2) coninue o hold wihou Assumpion Figures illusrae he opimal order-up-o level and price-induced demand wih he parameer values p 0 = 30, γ 0 = 9, c = 8, η = 0.5, σ = 2, [d, d] = [6, 12], K a = 18, and = Opimal Order up o Level Wih Scarciy Effec Wihou Scarciy Effec Opimal Price induced Demand Wih Scarciy Effec Wihou Scarciy Effec Cusomer Accessible Invenory Level Cusomer Accessible Invenory Level Figure 5.1. Opimal Ordering-up-o Level Figure 5.2. Opimal Price-induced Demand 141

155 5.7.2 Impac of Scarciy Effec This subsecion numerically sudies he impac of he scarciy effec of invenory upon he firm s profiabiliy by quanifying he profi loss of ignoring his effec under differen levels of scarciy effec inensiy, demand variabiliy and planning horizon lengh. As in γ 0 exp(ηi Secion 5.7.1, we assume ha γ(i a a ), for I a 0, ) = where η > 0. γ 0 1 ηi a, for 0 < I a K a, Noe ha η represens he scarciy effec inensiy of he invenory sysem: he larger he η, he more inense he scarciy effec. We need o evaluae he profi of a firm which ignores he scarciy effec, Ṽ. To compue Ṽ, we firs numerically obain he opimal policy in an invenory sysem wihou he scarciy effec and hen evaluae he oal profis of his policy in an invenory sysem wih he scarciy effec. We also evaluae he opimal profi of a firm under he scarciy effec, V. In he evaluaion of V and Ṽ, we ake I a = 0 as he reference cusomer-accessible invenory level. The meric of ineres is λ scarciy := V Ṽ V, under differen values of η, σ and. Our numerical experimens are conduced under he following values of parameers: p 0 = 21, γ 0 = 4, c = 4, η = 0.35, 0.4, 0.45, 0.5, 0.55, σ = 1, 2, 3, [d, d] = [6, 12], K a = 18, and = 5, σ=1 σ=2 σ=3 0.6 σ=1 σ=2 σ= Opimaliy Loss 0.4 Opimaliy Loss η Value η Value Figure 5.3. Value of λ scarciy : = 5 Figure 5.4. Value of λ scarciy : = 10 Figures summarize he resuls of our numerical sudy on he impac of he scarciy effec upon he firm s profiabiliy. Our resuls reveal ha, when he scarciy 142

156 effec is ignored, all numerical experimens exhibi a significan profi loss, which is a leas 16.41% and can be as high as 64.52%. Moreover, he impac of scarciy effec is increasing in he scarciy inensiy, demand variabiliy, and planning horizon lengh. The scarciy effec has wo effecs upon he firm s profiabiliy: (a) i decreases fuure demand, and (b) i increases demand variabiliy, because he variabiliy of poenial demand is inensified by ha of he pas demand via he scarciy effec. Hence, wih higher scarciy inensiy [demand variabiliy], he firs [second] effec lowers more profi of he firm. The comparison beween Figure 5.3 and Figure 5.4 implies ha he impac of he scarciy effec accumulaes over ime, so he profi loss of ignoring he scarciy effec is higher under a longer planning horizon. In shor, he scarciy effec of invenory maers significanly o he firm s profiabiliy when he scarciy effec inensiy and demand variabiliy is high, and he planning horizon is long. Our numerical finding confirms he resul in [145] ha he profi loss is increasing in he scarciy effec inensiy. On he oher hand, our numerical finding on he impac of demand variabiliy conrass ha in [145], which shows ha he profi loss of ignoring he scarciy effec is decreasing in demand variabiliy. In heir experimens, he poenial demand is convexly decreasing in he lefover invenory level, so higher demand variabiliy increases he expeced poenial demand and, hus, he firm s profiabiliy under he scarciy effec Value of Dynamic Pricing In his subsecion, we numerically explore he value of dynamic pricing under he scarciy effec of invenory wih differen levels of scarciy effec inensiy, demand variabiliy and planning horizon lengh. As in Secions , we assume ha γ(i a ) = γ 0 exp(ηi a ), for I a 0, where η > 0. We need o evaluae he profi of a γ 0 1 ηi a, for 0 < I a K a, firm, which adops he opimal saic pricing sraegy, ˆV. To compue ˆV, we firs evaluae he oal profi of an invenory sysem for any fixed price p in each, and hen maximize over p o selec he opimal saic price. Consisen wih V, ˆV is evaluaed a he reference cusomer-accessible invenory level I a λ pricing := V ˆV ˆV = 0. The meric of ineres is, under differen values of η, σ and. 143

157 Our numerical experimens are conduced under he following values of parameers: p 0 = 21, γ 0 = 4, c = 4, η = 0.35, 0.4, 0.45, 0.5, 0.55, σ = 1, 2, 3, [d, d] = [6, 12], K a = 18, and = 5, 10. Value of Dynamic Pricing σ=1 σ=2 σ=3 Value of Dynamic Pricing σ=1 σ=2 σ= η Value η Value Figure 5.5. Value of λ pricing : = 5 Figure 5.6. Value of λ pricing : = 10 Figures summarize he resuls of our numerical sudy on he value of dynamic pricing. The resuls show ha he value of dynamic pricing is significan in he presence of he scarciy effec. Federgruen and Heching (1999) documen ha he profi improvemen of dynamic pricing in a 5-period model is beween 0.46% 2.24%, when he coefficien of variaion for demand varies beween 0.7 and 1.4. The numerical experimens of Figure 5.5 repor a much higher profi improvemen (beween 0.91% 9.78%) of dynamic pricing in a 5-period model wih he coefficien of variaion of demand beween 0.11 and Thus, he scarciy effec of invenory gives rise o significanly higher value of dynamic pricing. The value of dynamic pricing is driven by he following hree effecs: (a) i achieves beer mach beween supply and demand; (b) i helps induce higher fuure demand; and (c) i dampens fuure demand variabiliy. While effec (a) also improves he performance of an invenory sysem wihou he scarciy effec, effecs (b) and (c) have heir impac only upon a firm wih he scarciy effec. Therefore, he value of dynamic pricing is significanly increased by he scarciy effec. Moreover, wih higher scarciy effec inensiy [demand variabiliy], effec (b) [(c)] enhances he firm s profiabiliy more significanly. The comparison beween Figure 5.5 and Figure 5.6 implies ha he value of dynamic pricing decreases over ime. This is consisen wih he findings in Federgruen 144

158 and Heching (1999) ha he opimal dynamic pricing policy converges o he opimal saic pricing policy, as he planning horizon lengh goes o infiniy. In shor, he value of dynamic pricing under he scarciy effec of invenory is mos significan when he inensiy of scarciy effec and demand variabiliy is high, and he planning horizon lengh is moderae. To conclude his secion, we remark ha all he numerical resuls and insighs in his secion are robus and hold for (a) he general demand form, Equaion (5.1), and (b) a large variey of differen inverse demand funcions (i.e., p( )) and scarciy funcions (i.e., γ( )) ha give rise o concave or non-concave R(, ) funcions. 5.8 Summary and Exension We conclude his chaper wih a summary of he main resuls and managerial insighs derived from our model and some houghs on a possible direcion of fuure research. This chaper is he firs in he lieraure o sudy he join pricing and invenory managemen model under he scarciy effec of invenory. Demand is modeled as a decreasing sochasic funcion of boh price and cusomer-accessible invenory level. We propose a unified model in which he firm has several operaional flexibiliies o hedge agains he risk of he sochasic invenory-dependen demand: (a) dynamic pricing, hrough which he firm can dynamically adjus is sales price; (b) invenory wihholding, hrough which he firm can wihhold par of is invenory from cusomers; and (c) invenory disposal, hrough which he firm can dispose par of is surplus invenory. We show ha a cusomer-accessible-invenory-dependen order-up-o/dispose-down-o/display-up-o lisprice policy is opimal. The order-up-o/display-up-o and lis-price levels are decreasing in he cusomer-accessible invenory level, because of he negaive dependence of demand on invenory. When he scarciy effec of invenory is sufficienly srong, he firm can sraegically benefi from he scarciy effec by displaying no posiive invenory and making every cusomer wai, because he revenue generaed by he srong scarciy effec dominaes he backlogging cos of he wai-lis. When he warehouse holding cos [salvage value] is sufficienly high [low], i is oo cosly o wihhold [dispose] invenory, and he unified model is reduced o he model wihou invenory wihholding [disposal]. The model wihou invenory wihholding [disposal] generaes addiional resuls and sharper insighs. In he model wihou invenory 145

159 wihholding/disposal, we show ha opimal sales prices and order-up-o levels are lower under he scarciy effec of invenory han hose under invenory-independen demand. Higher operaional flexibiliy (a higher salvage value or he invenory wihholding opporuniy), however, helps he firm hedge agains he oversocking risk and, hence, drives he firm o se higher order-up-o/display-up-o levels and sales prices. In addiion, responsive invenory reallocaion is anoher effecive way o deal wih he scarciy effec of invenory. The reallocaion flexibiliy afer demand realizaion enables he firm o beer hedge agains he demand uncerainy and balance he radeoff beween meeing curren demand and inducing poenial demand. In his case, since he firm can reallocae is on-hand invenory afer demand realizes, i may be opimal o order-andwihhold when he realized demand is small. We perform exensive numerical sudies o demonsrae (a) he robusness of our analyical resuls, (b) he impac of he scarciy effec upon he profi of he firm, and (c) he value of dynamic pricing under he scarciy effec of invenory. Our numerical resuls show ha he analyical characerizaions of he opimal policies in our model are robus and hold for non-concave R(, ) funcions in all our experimens. The impac of scarciy effec upon he firm s profi is wo-fold: (a) i decreases fuure demand, and (b) i increases demand variabiliy. Hence, he profi loss of ignoring he scarciy effec is higher under higher scarciy inensiy (via effec (a)), higher demand variabiliy (via effec (b)), and longer planning horizon (via boh effecs). The value of dynamic pricing under he scarciy effec is hree-fold: (a) i beer maches supply and demand; (b) i helps induce higher fuure demand; and (c) i dampens fuure demand variabiliy. Effec (b) [(c)] leads o higher value of dynamic pricing under higher scarciy inensiy [demand variabiliy]. Moreover, he opimal dynamic pricing policy converges o he opimal saic pricing policy as he planning horizon lengh goes o infiniy, so he value of dynamic pricing decreases over ime. Finally, we remark ha all he analyical resuls in his chaper can be easily exended o he infinie horizon discouned model wih he sandard argumen ha demonsraes he preservaion of he srucural properies as he planning horizon lengh goes o infiniy. In his subsecion, we propose a possible exension of our work: he analysis of he model ha encompasses boh he scarciy effec and he promoional effec of invenory. 146

160 As discussed in Secion 5.2, he displayed invenory has boh he service and he promoional effecs (see, e.g., [34, 35]), because a higher cusomer-accessible invenory level creaes a sronger visual impac and cusomers infer a greaer chance o ge he produc. In he lieraure, his phenomenon is also called he billboard effec and he shelf-space effec (e.g. [39, 18, 53]). I is ineresing o analyze he model which incorporaes boh he scarciy effec of pre-replenishmen invenory and he promoional effec of pos-replenishmen invenory. More specifically, we assume ha he demand in period, D = δ(p, I a, x a, ϵ ) = (d(p ) + γ 1 (I a )+γ 2 (x a ))ϵ m +ϵ a, where γ 1 ( ) is a decreasing funcion of pre-replenishmen cusomeraccessible invenory level I a, and γ 2 ( ) is an increasing funcion of pos-replenishmen cusomer-accessible invenory level x a. As before, assume ha d( ) is a sricly decreasing funcion of sales price p, E{ϵ m } = 1 and E{ϵ a } = 0. I is challenging o characerize he opimal join pricing and invenory managemen policy under his generalized invenory-dependen demand. In paricular, he effec of invenory on he firm s profiabiliy is more involved and i is unclear how o srike a balance beween he overage and underage risks in his model. We will explore his problem in our fuure research. 147

161 6. Comparaive Saics Analysis Mehod for Join Pricing and 6.1 Inroducion Invenory Managemen Models 1 Comparaive saics analysis is inegral o sudying an invenory managemen sysem under dynamic pricing, because i delivers imporan insighs regarding how he sysem should opimally respond o changes in he exogenous marke condiion and/or inernal sae over he planning horizon. For insance, a firm under an uncerain marke environmen ofen faces he conundrum ha wheher i should increase or decrease he sales price and order-up-o level under a higher procuremen cos. Analogously, i is also imporan o modify he price and invenory policies in accordance o firm-level sraegic changes like conracing wih an addiional supplier or expanding he arge cusomer segmens. As an essenial ool in economics, engineering and operaions managemen, comparaive saics analysis offers a sysemaic mehod o sudy hese challenges ha are boh common and essenial in invenory managemen models under dynamic pricing. More specifically, we consider he opimal pricing and replenishmen policies in a general periodic-review join pricing and invenory managemen model wih muliple cusomer segmens and supply channels under a flucuaing marke environmen. The firm replenishes is invenory from a porfolio of supply channels wih differen cos funcions. The cos funcion of each supply channel is deermined by a supply-channel-dependen reference procuremen cos (e.g., he raw maerial procuremen cos in each supply channel). The cusomer marke is segmened ino several independen classes wih differen demand funcions. The firm charges a sales price o each demand segmen in each decision period. The demand funcion of each demand segmen is deermined by a demandsegmen-dependen marke size. Boh he reference procuremen cos of each supply channel and he marke size of each demand segmen evolve according o an underlying exogenous Markov process. Hence, our join pricing and invenory managemen model capures hree imporan feaures in oday s compeiive and unsable marke: demand 1 This chaper is based on he auhor s earlier work [192]. 148

162 segmenaion, supply diversificaion, and marke environmen flucuaion. In his quie general dynamic pricing and invenory managemen model, comparaive saics analysis plays an essenial role in he characerizaion of (a) he opimal pricing and invenory policies, and (b) he impac of marke environmen flucuaion, demand segmenaion, and supply diversificaion upon he opimal sales prices and order quaniies. There are wo sandard mehods o perform comparaive saics analysis in he economics and operaions managemen lieraure: (a) he implici funcion heorem (IFT) approach, and (b) he monoone comparaive saics (MCS) approach. The IFT approach characerizes he derivaive of he opimizer wih respec o he parameers by applying he implici funcion heorem o he firs-order condiion. In order o apply his approach, i is clear from he assumpions of he implici funcion heorem ha (a) he objecive funcion needs o be wice coninuously differeniable wih respec o he complee vecor of decision variables and parameers, and (b) he Hessian of he objecive funcion wih respec o he decision variables a he opimizer needs o be nondegenerae. In our general join pricing and invenory managemen model, as we will show laer, condiion (a), in general, is no saisfied, whereas condiion (b) is very difficul o check. Moreover, he IFT approach is no scalable, i.e., he analyical characerizaion of he derivaives via he implici funcion heorem soon becomes inracable as he number of demand segmens and supply channels increases. See, e.g., [27, 168]. In shor, he IFT approach is no effecive in performing comparaive saics analysis in our model. The MCS approach sudies he impac of a parameer change on he marginal value of decision variables for objecive funcions defined on laices. The MCS approach is very powerful in comparaive saics analysis, because i does no require any regulariy assumpion regarding he objecive funcion. In order o apply he MCS approach, he objecive funcion needs o saisfy a cerain form of complemenariy condiions (e.g., join supermodulariy or, more generally, he single crossing propery). Anoher feaure of he MCS approach is ha, all of he opimal decision variables should be monoone (in srong se order) in parameers. In our join pricing and invenory managemen model, eiher he join supermodulariy or he single crossing propery is very difficul, if no impossible, o esablish in each decision epoch. Moreover, as we will show laer, i is possible in our model ha only par of he opimal decision variables (i.e., sales prices 149

163 and order quaniies) are monoone in he marke parameers. Hence, he MCS approach does no apply o our model. The limiaions of he IFT and MCS approaches moivae us o develop a new mehod for he comparaive saics analysis of our general join pricing and invenory managemen model wih demand segmenaion, supply diversificaion, and marke environmen flucuaion. The new mehod provides rigorous proofs of comparaive saics analysis and srucural properies in our model. More specifically, our mehod proves he desired comparaive saics resuls by conradicion and carefully analyzes how changes in parameer values impac he marginal value of each decision variable (i.e., he firs-order parial derivaive of he objecive funcion). We idenify a simple ye powerful lemma which ranslaes he monooniciy relaionship beween he opimizers ino ha beween he parial derivaives of he objecive funcion under differen parameer values. Our comparaive saics mehod employs his lemma and some model-specific srucural properies (e.g., he concaviy of he objecive funcion and/or he supermodulariy of he objecive funcion in one decision variable and one parameer) o consruc a conradicion by ieraively linking he monoone relaionship beween he opimizers and ha beween he parial derivaives of he objecive funcion. Noe ha he srucural properies needed by our approach are weaker han hose required by he IFT and MCS approaches (e.g., second-order coninuous differeniabiliy and complemenariy). The lemma also enables us o make componenwise comparisons beween he opimizers under differen parameer values, because he monooniciy of he objecive funcion s parial derivaive wih respec o one decision variable a he opimizer of ineres is independen of he values of oher decision variables. Hence, unlike he IFT approach, our new mehod is scalable; and unlike he MCS approach, our new mehod enables us o perform comparaive saics analysis in a model where only par of he opimal decision variables are monoone in he parameer. To perform comparaive saics analysis in each decision epoch of our general join pricing and invenory managemen model, we inegrae our new mehod wih he sandard backward inducion argumen o ieraively link he comparison beween opimizers and ha beween parial derivaives of he value funcions and objecive funcions. We characerize he opimal join pricing and ordering policy for an arbirary number of demand segmens and supply channels as a hreshold policy, under which here exiss a 150

164 marke-environmen-dependen hreshold for each demand segmen [supply channel] such ha i is opimal o sell o [order from] his segmen [channel] if and only if he saring invenory level is above [below] is corresponding hreshold. Moreover, boh he opimal sales price for each demand segmen and he opimal order quaniy hrough each supply channel are decreasing in he saring invenory level of he firm. We also show ha he opimal sales prices and order quaniies are increasing in he marke size. When he reference procuremen coss of some supply channels increase, he firm increases he sales price in each demand segmen, and he order quaniies from he supply channels wih unchanged reference procuremen coss. Each firm s opimal order quaniy may no be monoone in is own reference procuremen cos. Serving a new demand segmen drives he firm o increase is sales prices and order quaniies, whereas expanding he supply pool has he opposie effec: i promps he firm o decrease is sales prices and order quaniies. Our mehod is robus and applicable o comparaive saics analysis in some oher seings. For example, we consider join price and effor compeiion games in which an arbirary number of firms compee on price and effor level. More specifically, we sudy wo compeiion models: (a) he effor-level-firs compeiion where he firms firs compee on effor and hen on price, and (b) he simulaneous compeiion where he firms simulaneously compee on price and effor. Each firm s demand is increasing in he oal effor level of all firms. As we will show laer, he IFT approach is no scalable whereas he complemenariy condiions required by he MCS approach are no saisfied, so he sandard IFT and MCS approaches do no work in his model. Our new comparaive saics mehod enables us o prove he exisence and uniqueness of he equilibrium in he effor-level-firs compeiion. In boh compeiion models, we show ha he equilibrium oal effor level, and he equilibrium sales price and demand volume of each firm are increasing in he marke index of any firm. We also idenify he fa-ca effec in his seing, i.e., he equilibrium oal effor level and he equilibrium price and demand of each firm in he effor-level-firs compeiion are higher han heir counerpars in he simulaneous compeiion. To sum up, we propose a new mehod for comparaive saics analysis in a general join pricing and invenory managemen model wih demand segmenaion, supply diversificaion, and marke environmen flucuaion. This new comparaive saics mehod 151

165 requires less resricive assumpions han he sandard IFT and MCS approaches. Moreover, our mehod makes componenwise comparisons beween opimizers wih differen parameer values, so i is well scalabile, and is capable of performing comparaive saics analysis in a model where some of he opimal decision variables are no monoone in he parameer. The proposed mehod also applies o he comparaive saics analysis in some oher seings where he sandard approaches do no work. The res of he chaper is organized as follows. We posiion his chaper in he relaed lieraure in Secion 6.2. Secion 6.3 presens our new mehod for comparaive saics analysis in join pricing and invenory managemen models. In Secion 6.4, we apply he proposed mehod o sudy a general join pricing and invenory managemen model wih demand segmenaion, supply diversificaion, and marke environmen flucuaion. Secion 6.5 demonsraes he applicabiliy of our mehod in compeiion games. We conclude his chaper by summarizing our mehod and findings in Secion 6.6. Mos of he proofs are relegaed o Appendix E.1. Throughou his chaper, we use o denoe he derivaive operaor of a single variable funcion, and x o denoe he parial derivaive operaor of a muli-variable funcion wih respec o variable x. For any mulivariae coninuously differeniable funcion f(x 1, x 2,, x n ) and x := ( x 1, x 2,, x n ) in f( ) s domain, we use xi f( x 1, x 2,, x n ) o denoe xi f(x 1, x 2,, x n ) x= x for any i. For any wo n-dimensional vecors v = (v 1, v 2,, v n ) and ˆv = (ˆv 1, ˆv 2,, ˆv n ), we use ˆv > v o denoe ha ˆv i v i for each 1 i n, and ˆv v. 6.2 Relaed Research This chaper is buil upon wo sreams of lieraure: (a) he mehod and applicaion of comparaive saics analysis, and (b) dynamic pricing and invenory managemen. Comparaive saics analysis is formalized in he economics lieraure by [95] and [144], where he classical IFT approach is inroduced. The MCS approach is firs esablished by [164]. He shows ha he maximizer of a supermodular funcion is increasing in he parameers in he srong se order sense. [126] derive a necessary and sufficien condiion (i.e., quasi-supermodulariy and he single crossing propery) for he soluion se of an opimizaion problem o be monoone in parameers. [15] generalizes his resul o sochasic opimizaion problems and characerizes necessary and sufficien condiions, based on he properies of uiliy funcions and probabiliy disribuions, for comparaive 152

166 saics predicions o hold. [45] esablish a new preservaion propery of supermodulariy in a class of wo-dimensional parameric opimizaion problems, where he feasible ses may no be laices. Comparaive saics analysis in game heoreic models has also been exensively sudied in he lieraure (e.g., [124, 125, 24]). This lieraure mainly focuses on supermodular games. We refer ineresed readers o he monograph by [165] ha coherenly synhesizes he heory and applicaions of he MCS approach. There is exensive applicaion of comparaive saics analysis in he operaions managemen lieraure. See, e.g., [151, 152, 98, 71, 111]. The majoriy of he papers in his sream of research apply he IFT and MCS approaches o esablish comparaive saics resuls and he srucural properies of heir models. [27] is a noable excepion ha develops a novel analyical approach for he comparaive saics analysis in muli-produc, muli-resource newsvendor neworks wih responsive pricing. In heir seing, he IFT approach is prohibiively difficul, whereas heir new approach explois he relaionship beween convex sochasic orders and dual variables, and does no suffer from he curse of dimensionaliy. This chaper conribues o his line of research by developing a new analyical mehod for comparaive saics analysis in a general join pricing and invenory managemen model. The major srengh of he proposed mehod lies in he following hree aspecs: (a) i does no need he condiions required by he IFT and MCS approaches ha are resricive in dynamic pricing and invenory managemen models (e.g., second-order coninuous differeniabiliy and complemenariy); (b) i does no suffer from he curse of dimensionaliy; and (c) i is amenable for comparaive saics analysis in a model where only par of he opimal decision variables are monoone in he parameer. This work is also relaed o he growing lieraure on he dynamic pricing and invenory managemen problem under general sochasic demand. [70] provide a general reamen of his problem, and show he opimaliy of a lis-price/order-up-o policy. This line of lieraure has grown rapidly since [70]. For example, [47, 48, 49] analyze he join pricing and invenory conrol problem wih fixed se-up cos, and show ha (s, S, p) policy is opimal for finie horizon, infinie horizon and coninuous review models. [52] and [96] sudy he join pricing and invenory conrol problem under los sales. In he case of a single unreliable supplier wih random yield, [112] show ha supply uncerainy drives he firm o charge a higher price. When he replenishmen leadime is posiive, he join pricing and invenory conrol problem under periodic review is exremely difficul. 153

167 For his problem, [136] parially characerize he srucure of he opimal policy, whereas [26] develop a simple heurisic ha resolves he compuaional complexiy. Several papers in his sream of lieraure also ake ino consideraion consumer behaviors. [179] sudy he join pricing and invenory managemen model in which cusomers bid for unis of a firm s produc over an infinie horizon. [97] characerize he opimal pricing and producion policy under cusomer subscripion and reenion/ariion. [110] esablish he concaviy of he objecive funcion in he nesed logi model, and apply his model o analyze he join pricing and invenory managemen problem wih muliple producs. The lieraure on he join pricing and invenory managemen problem under a flucuaing marke environmen is scarce. To he bes of our knowledge, [173] is he only paper which sudies he dynamic pricing and invenory managemen problem under flucuaing procuremen coss. We refer ineresed readers o [50] for a comprehensive survey on join pricing and invenory conrol models. This chaper conribues o his sream of research by developing a new analyical mehod for he comparaive saics analysis in a general join pricing and invenory managemen model wih demand segmenaion, supply diversificaion, and marke environmen flucuaion. 6.3 A New Comparaive Saics Mehod In his secion, we firs give an example o illusrae why he IFT and MCS approaches are no applicable for comparaive saics analysis in our general join pricing and invenory managemen model. We hen develop a new analyical mehod for comparaive saics analysis herein An Illusraive Example In his subsecion, we give an example ha clearly illusraes why he IFT and MCS approaches do no apply o he general join pricing and invenory managemen model wih demand segmenaion, supply diversificaion, and marke environmen flucuaion. Le f i (y i ) be a firs-order coninuously differeniable and sricly concave funcion on Y i = [A i, B i ] for i = 1, 2,, p, and g i (y i γ) be coninuously differeniable and sricly concave in y i and submodular in (y i, γ), where y i Y i = [A i, B i ] and γ Γ [γ, γ], for each i = p+1, p+2,, p+q. Moreover, assume ha h(y 0 γ) is coninuously differeniable 154

168 and concave in y 0 and supermodular in (y 0, γ), where y 0 R and γ Γ. Le λ i be a posiive consan for any i = 1, 2,, p+q. Consider he following opimizaion problem: (y1(γ), y2(γ),, yp(γ), yp+1(γ),, yp+q(γ)) =argmax y Y F (y γ), p+q p+q p s.. F (y γ) = f i (y i ) + g i (y i γ) + h( λ i y i γ), i=1 i=p+1 Y =Y 1 Y 2 Y p+q, γ Γ. i=1 (6.1) The objecive funcion F ( γ) in he opimizaion problem (6.1) is sricly joinly concave in y = (y 1, y 2,, y p+q ). Hence, he opimizer y (γ) = (y 1(γ), y 2(γ),, y p+q(γ)) is well defined and unique. As we will show in Secion 6.4, several opimizaion problems in our general join pricing and invenory managemen model can be reduced o a convex program similar o (6.1). A naural quesion on (6.1) is: how does y (γ) change wih γ? We have he following lemma ha addresses his quesion. Lemma 15 For he opimizaion problem defined in (6.1), y i (γ) is increasing in γ, for all 1 i p. Lemma 15 characerizes he impac of he parameer γ upon he opimizer {y i (γ)} 1 i p. Noe ha Lemma 15 does no give any comparaive saics resul for {y i (γ)} p+1 i p+q, because i is easy o consruc funcions {f i ( ), g i ( ), h( )} and feasible se Y Γ, such ha y i ( ) is no monoone in γ for some p + 1 i p + q. See Appendix E.2 for an example. We also remark ha he assumpion ha f i ( ) s and g i ( γ) s are sricly concave is mainly for exposiional convenience. When his assumpion is relaxed o ha hey are weakly concave, he opimizers may no be unique and we selec he lexicographically smalles one. In his case, Lemma 15 sill holds and our new comparaive saics mehod is also valid. We now explain in deail why he IFT and MCS approaches canno be used o prove Lemma 15. The firs issue relaed o he IFT approach is ha F ( ) may no be wice coninuously differeniable on is domain. For example, if Γ is finie, F ( ) is no wice coninuously differeniable. Now we assume ha Γ is an inerval and F ( ) is wice coninuously differeniable. For any opimizer y (γ) ha lies in he inerior of Y Γ, he 155

169 implici funcion heorem implies ha y (γ) is coninuously differeniable in γ, and he derivaive is given by: dy dγ = Ω 1 V, (6.2) where Ω is a (p + q) (p + q) marix wih Ω i,j = λ i λ j y 2 0 h( p+q l=1 λ lyl (γ) γ) for all i j, Ω i,i = f i (y i (γ)) + (λ i ) 2 2 y 0 h( p+q l=1 λ ly l (γ) γ) for 1 i p, and Ω i,i = 2 y i g i (y i (γ) γ) + (λ i ) 2 2 y 0 h( p+q l=1 λ ly l (γ) γ) for p + 1 i p + q, and V is a (p + q)-vecor wih V i = λ i y0 γ h( p+q l=1 λ lyl (γ) γ) for 1 i p, and V i = yi γ g i (yi (γ) γ) + λ i y0 γ h( p+q l=1 λ lyl (γ) γ) for p + 1 i p + q. As p and q increase, however, compuing Ω 1 in (6.2) suffers from he curse of dimensionaliy, and, in general, we are unable o characerize he sign of dy i / dγ for each i. When y (γ) is on he boundary of Y (i.e., some of he consrains are binding), he IFT approach can be generalized o he perurbaion analysis approach (see, e.g., [76]), which, again, deermines he sign of he derivaive of y (γ) wih respec o γ by characerizing he inverse of Hessian, and, hence, suffers from he curse of dimensionaliy. Thus, i is very difficul, if no impossible, o perform comparaive saics analysis in (6.1) by he IFT approach. The MCS approach also fails o conduc he comparaive saics analysis in (6.1). More specifically, i s clear ha F ( ) is no joinly supermodular in (y, γ), nor does i saisfy he single crossing propery in (y; γ). By [126], in order o apply he MCS approach, i is necessary ha he objecive funcion should saisfy he single crossing propery wih respec o he decision vecor and he parameer. Moreover, he MCS approach, when applicable, always gives a comparaive saics predicion for all he decision variables of an opimizaion problem (see, e.g., [165]). In our opimizaion problem (6.1), however, i is easy o specify funcions {f i ( ), g i ( ), h( )} and feasible se Y Γ, such ha yi ( ) is no monoone in γ for some p + 1 i p + q, as shown by he example in Appendix E.2. Therefore, he MCS approach does no apply o he comparaive saics analysis in (6.1) Proof of Lemma 15 wih Our New Mehod Since he sandard IFT and MCS approaches are no applicable o conducing comparaive saics analysis in (6.1), we develop a new mehod o prove Lemma 15. Before 156

170 presening he proof of Lemma 15 and our new mehod in deail, we inroduce a lemma ha plays a key role herein: Lemma 16 Le F i (z, Z i ) be a firs-order differeniable funcion in (z, Z i ) for i = 1, 2, where z [z, z] (z and z migh be infinie) and Z i Z i, where Z i is he feasible se of Z i. For i = 1, 2, le (z i, Z i ) := argmax (z,zi ) [z, z] Z i F i (z, Z i ), be he opimizer of F i (, ). If z1 < z2, we have: z F 1 (z1, Z1) z F 2 (z2, Z2). = 0 if z Proof. z1 < z2, so z z1 < z2 z. Hence, z F 1 (z1, Z1) 1 > z, and 0 if z1 = z; = 0 if z z F 2 (z2, Z2) 2 < z, i.e., z F 1 (z1, Z1) 0 z F 2 (z2, Z2). Q.E.D. 0 if z2 = z, Lemma 16 is sraighforward, bu i is a powerful ool in our new comparaive saics mehod, as illusraed by he proof of Lemma 15: Proof of Lemma 15. We show by conradicion, i.e., we derive a conradicion under he assumpion ha y i (γ) > y i (ˆγ) for some 1 i p and γ < ˆγ. Wihou loss of generaliy, we choose i = 1, i.e., y 1(γ) > y 1(ˆγ). (6.3) Denoe y 0(γ) := p+q j=1 λ jy j (γ) for all γ Γ. Lemma 16 implies ha y1 F (y (γ) γ) y1 F (y (ˆγ) ˆγ), i.e., y1 f 1 (y 1(γ)) + λ 1 y0 h(y 0(γ) γ) = y1 F (y (γ) γ) y1 F (y (ˆγ) ˆγ) = y1 f 1 (y 1(ˆγ)) + λ 1 y0 h(y 0(ˆγ) ˆγ). (6.4) The sric concaviy of f 1 ( ) yields ha y1 f 1 (y1(γ)) < y1 f 1 (y1(ˆγ)). Hence, y0 h(y0(γ) γ) > y0 h(y0(ˆγ) ˆγ). (6.5) Since h( ) is supermodular in (y 0, γ) and concave in y 0, y0(γ) < y0(ˆγ). Therefore, here exiss a j, 2 j p + q, such ha yj (γ) < yj (ˆγ). (6.6) If 2 j p, we invoke Lemma 16 again, so yj (γ) < yj (ˆγ) implies ha yj F (y (γ) γ) yj F (y (ˆγ) ˆγ), i.e., yj f j (yj (γ)) + λ j y0 h(y0(γ) γ) = yj F (y (γ) γ) yj F (y (ˆγ) ˆγ) = yj f j (yj (ˆγ)) + λ j y0 h(y0(ˆγ) ˆγ). 157

171 Since y0 h(y 0(γ) γ) > y0 h(y 0(ˆγ) ˆγ) by (6.5), yj f j (y j (γ)) < yj f j (y j (ˆγ)). Since f j ( ) is sricly concave, y j (γ) < y j (ˆγ) implies ha yj f j (y j (γ)) > yj f j (y j (ˆγ)), which conradics yj f j (y j (γ)) < yj f j (y j (ˆγ)). (6.7) Analogously, if p + 1 j p + q in (6.6), Lemma 16 implies ha yj F (y (γ) γ) yj F (y (ˆγ) ˆγ), i.e., yj g j (yj (γ) γ) + λ j y0 h(y0(γ) γ) = yj F (y (γ) γ) yj F (y (ˆγ) ˆγ) = yj g j (yj (ˆγ) ˆγ) + λ j y0 h(y0(ˆγ) ˆγ). Since y0 h(y0(γ) γ) > y0 h(y0(ˆγ) ˆγ) by (6.5), yj g j (yj (γ) γ) < yj g j (yj (ˆγ) ˆγ). Since g j ( ) is submodular in (y j, γ) and sricly concave in y j, yj (γ) < yj (ˆγ) implies ha yj g j (yj (γ) γ) > yj g j (yj (ˆγ) ˆγ), which conradics yj g j (yj (γ) γ) < yj g j (yj (ˆγ) ˆγ). (6.8) Combining he conradicions of (6.7) and (6.8), we have y1(γ) y1(ˆγ). Repea he above argumen for 1 < i p, i follows ha yi (γ) yi (ˆγ) for all 1 i p. Q.E.D. As we can see from he proof of Lemma 15, our new mehod employs Lemma 16 o make componenwise comparisons beween he opimizers under differen parameer values. More specifically, he mehod consiss of five seps: Sep (a). For each of he focal decision variable wih some poenial comparaive saics resul, we firs assume, o he conrary, ha he comparaive saics predicion of his decision variable is reversed for some parameer values (e.g., inequaliy (6.3) in he proof of Lemma 15). Sep (b). We invoke Lemma 16 o characerize some monoone relaionships of he parial derivaives of he objecive funcion wih respec o his decision variable a hese parameer values (e.g., inequaliy (6.4) in he proof of Lemma 15). Sep (c). Using some model specific properies of he objecive funcion (e.g., he supermodulariy in one decision variable and he parameer, componenwise concaviy, and firs-order differeniabiliy), such monoone relaionships of he parial derivaives can be ranslaed back ino he monoone relaionship of anoher opimal decision variable a he given parameer values (e.g., inequaliy (6.6) in he proof of Lemma 15). Sep (d). Repeaing seps (b) - (c), we employ Lemma 16 o ieraively esablish he monoone relaionship of parial derivaives and ha of some oher opimal decision variables a he given parameer values. This ieraive procedure is sopped when eiher (i) he desired comparaive saics resul 158

172 for he focal decision variable is proved by conradicion (e.g., inequaliies (6.7) and (6.8) in he proof of Lemma 15), or (ii) no furher monoone relaionship can be esablished (e.g., he case in which we assume ha yi (γ) > yi (ˆγ) or yi (γ) < yi (ˆγ) for γ < ˆγ and p + 1 i p + q, see Appendix E.2). Sep (e). We repea he same ieraive procedure, i.e., seps (a) - (d), for each focal decision variable o obain is corresponding comparaive saics resul. Noe ha here are wo sopping condiions for he ieraive procedure in Sep (d). When he sopping condiion (ii) applies, by our experience, i is very likely ha here exis some model specificaions such ha he desired comparaive saics resul for he focal decision variable does no hold. For example, in he opimizaion problem (6.1), no conradicion can be reached under any monoone comparaive saics predicion of yi ( ) (p + 1 i p + q) wih respec o γ for general {f i ( ), g i ( ), h( )} funcions. In Appendix E.2, we discuss in deail on how he ieraive procedure in Sep (d) is sopped wihou reaching a conradicion in his case, and give an example in which yi (γ) (p + 1 i p + q) is no monoone in γ. Hence, our new mehod no only helps prove he comparaive saics resuls when hey exis, bu also helps idenify cases in which comparaive saics resuls do no hold for some decision variables. Our mehod proves he desired comparaive saics resul by conradicion. The essence is o consruc a conradicion by ieraively linking he monoone relaionship beween he opimizers and ha beween he parial derivaives. Though simple, Lemma 16 plays a crucial role in his process, because, in Sep (b), i ranslaes he monooniciy of he focal decision variable (in he parameer) ino ha of he parial derivaive of he objecive funcion wih respec o his decision variable a he opimizing poin. Hence, in Sep (d), Lemma 16 enables us o ieraively link he monoone relaionship of opimizers and ha of parial derivaives, which is he key o esablish a conradicion in our mehod. The main benefi of Lemma 16 is ha he monooniciy of he parial derivaives wih respec o he focal decision variable is irrelevan o he values of oher decision variables a he opimizing poin. This benefi allows us o perform comparaive saics analysis componenwisely in Sep (e). Hence, our mehod enables us o perform comparaive saics analysis in a model where only par of he opimal decision variables are monoone in he parameer, and i is scalable. The componenwise comparison beween he opimizers is also he key difference beween our mehod and he IFT and 159

173 MCS approaches, boh of which involve he analysis of some properies of he objecive funcion in erms of he whole decision vecor (e.g., he Hessian and/or he join supermodulariy of he objecive funcion). Moreover, since he objecive funcions in Lemma 16, F i (, ) (i = 1, 2), can be compleely differen, our mehod can be used o compare he opimal decisions in differen models. See, e.g., he proofs of Theorems 6.4.7, 6.4.8, and in Appendix E.1. Alhough our mehod is fundamenally differen from he IFT and MCS approaches, i shares some similariy wih hese wo sandard approaches. As he IFT approach, he proposed mehod sudies he firs-order (KKT) condiion a he opimizer of ineres. Hence, he objecive funcion needs o saisfy he firs-order coninuous differeniabiliy condiion, bu no necessarily he second-order coninuous differeniabiliy condiion. Analogous o he MCS approach, our new mehod analyzes he impac of he parameer upon he marginal value of each decision variable in deail, so ha we can ranslae he monooniciy of parial derivaives wih respec o one decision variable back ino he monooniciy of anoher opimal decision variable. Thus, in order o obain a conradicion (and a comparaive saics resul), our mehod requires he objecive funcion o be supermodular in he parameer and each of he focal decision variables (e.g., F (y γ) is supermodular in (y i, γ) for each 1 i p in our example), bu no necessarily joinly supermodular or saisfying he single crossing propery. The above wo condiion relaxaions enhance he applicabiliy of our mehod in he general join pricing and invenory managemen model wih demand segmenaion, supply diversificaion, and marke environmen flucuaion, where he second-order coninuous differeniabiliy and/or, in paricular, he join supermodulariy of he objecive funcion in each decision epoch are hard, if no impossible, o esablish. In he nex secion, we discuss in deail how he new mehod faciliaes he comparaive saics analysis in his model. We also demonsrae he applicabiliy of our mehod in game heoreic models of join price and effor compeiion in Secion Applicaion of he New Comparaive Saics Mehod in a General Join Pricing and Invenory Managemen Model In his secion, we employ our new comparaive saics mehod o sudy a general join pricing and invenory managemen model wih demand segmenaion, supply di- 160

174 versificaion, and marke environmen flucuaion. The comparaive saics analysis is essenial o sudying his model, because i enables us o characerize he opimal pricing and replenishmen policy, and he impac of demand segmenaion, supply diversificaion, and marke environmen flucuaion herein. The analysis in his secion demonsraes he applicabiliy of our new mehod in join pricing and invenory managemen models Model We consider a T -period join pricing and invenory managemen model, in which a firm replenishes invenory from a porfolio of supply channels and serves muliple demand segmens wih endogenous sales prices. The firm maximizes is oal discouned profi over he planning horizon by opimizing is join pricing and invenory policy in each period. The periods are indexed backwards as {T, T 1,, 1} and he discoun facor is denoed as α (0, 1). We assume ha he cusomer marke is compleely segmened, i.e., each cusomer in he marke unambiguously belongs o a specific demand segmen. Complee segmenaion applies o he seings where cusomers are classified based on he differences in, e.g., (a) geographic area, (b) he need for produc feaure, (c) socioeconomic aribues, and (d) business secor (in B2B marke) (see, e.g., [10, 139]). There are n demand segmens in he marke, and we denoe hem as N := {1, 2,, n}. In period, he firm selecs a vecor of prices, p = (p 1, p 2,, p n ), for differen demand segmens. More specifically, for each i N, p i [p i min, p i max] is he sales price for cusomers in segmen i, where p i min > 0 [p i max + ] is he minimum [maximum] allowable price for his segmen. We use Λ i > 0 o denoe he expeced maximum demand (i.e., he marke size) from segmen i in period. Le Λ := (Λ 1, Λ 2,, Λ n ) be he marke size vecor. Since cusomers are compleely segmened, he demand from segmen i is independen of he sales price in segmen j (i j). Specifically, we assume ha, given he sales price p i and he marke size Λ i, he demand from segmen i in period is given as follows: D(p i i, Λ i ) = Λ i d i (p i )ς + ϵ i. (6.9) In (6.9), d i (p i ) denoes he probabiliy ha an arriving cusomer in segmen i will make a purchase when facing a sales price p i, where d i ( ) is a sricly decreasing funcion of p i. A ypical example of his specificaion is he independen reservaion price model 161

175 (e.g., s[83]). ς is he demand-segmen-independen muliplicaive marke size perurbaion, which represens he common demand shock (e.g., global economic changes) on each segmen. We assume ha {ς } 1 =T are i.i.d. posiive random variables independen of Λ wih mean 1. The addiive random perurbaion erm ϵ i capures all oher uncerainies no explicily considered in his model. We assume ha {ϵ i } 1 =T are i.i.d. coninuous random variables independen of Λ and ς wih mean 0. Hence, D(p i i, Λ i ) follows a coninuous disribuion for any given (p i, Λ i ) and i N. We use D (p, Λ ) = (D 1 (p 1, Λ 1 ), D 2 (p 2, Λ 2 ),, D n (p n, Λ n )) o denoe he demand vecor for all demand segmens, wih he sales price vecor p and he marke size vecor Λ in period. Given (p, Λ ), he accumulaive demand from all segmens in period is given by: D a (p, Λ ) = n D(p i i, Λ i ) = ( Λ i d i (p i ))ς + ϵ, (6.10) i N i=1 where he superscrip a refers o accumulaive, and ϵ := n i=1 ϵi represens he accumulaive addiive perurbaion in period. For each i, since d i (p i ) is sricly decreasing, i has a sricly decreasing inverse p i ( ) ha maps from [d i min, d i max] o [p i min, p i max], where d i min = d i (p i max) = 0 and d i max = d i (p i min) 1. We view he purchasing probabiliy vecor d := (d 1, d 2,, d n ), insead of he sales price vecor p, as he decision variable in each period. Wihou loss of generaliy, we assume ha d i max = d max 1 for any i N, i.e., he maximum expeced purchasing probabiliy is he same for every demand segmen. Since d i (p i max) = 0, our model endogenizes he opion ha, for any i N, he firm can choose no o sell o demand segmen i by charging a prohibiively high sales price p i max. We impose he following assumpion hroughou our analysis: Assumpion For each demand segmen i N, R i (d i ) := p i (d i )d i is coninuously differeniable and concave in d i [0, d max ]. Noe ha he sric monooniciy of p i ( ), ogeher wih he concaviy of R i ( ), suggess ha R i ( ) is sricly concave in d i for each i N. We remark ha, when here is only one demand segmen (n = 1), our demand model is reduced o he mos commonly sudied demand model in he join pricing and invenory managemen lieraure. See, e.g., [47, 50, 189]. The firm sources from a porfolio of m supply channels, which is denoed as M = {1, 2,, m}. In period, he firm selecs a vecor of order quaniies, q = (q 1, q 2,, q m ), 162

176 from differen supply channels. More specifically, he firm orders q j 0 from supply channel j and pays a cos C j (q j c j ), where C j ( c j ) is he cos funcion of supply channel j when he reference procuremen cos is c j, and C j (0 c j ) = 0 for all j M. The reference procuremen cos c j is an index for he acual procuremen cos of supply channel j, which is independen of he firm s pricing and invenory policy. For example, c j can be viewed as he uni procuremen cos of he raw maerial in supply channel j. Since an increase in c j increases he marginal cos of sourcing from supply channel j, we assume ha C j ( ) is supermodular in (q j, c j ) for any j M. We use c := (c 1, c 2,, c m ) o denoe he reference procuremen cos vecor in period. Moreover, we assume ha here exiss diseconomy of scale o source from each supply channel, i.e., C j ( c j ) is convexly increasing in q j for each j M. In realiy, his assumpion applies when he supply channel is capaciaed, so ha orders exceeding he sandard capaciy are charged a higher rae for he addiional ousourcing coss and/or overime labor coss (see [155]). The assumpion of convex ordering cos is necessary o prove he convexiy [concaviy] of he opimal cos [profi] funcion in a muli-period model, and common in he invenory managemen lieraure (see, e.g., [181, 50]). Wihou loss of generaliy, we assume ha C j ( c j ) is coninuously differeniable in q j for any q j 0. For exposiional ease (i.e., o ensure he uniqueness of he opimizer), we assume ha C j ( c j ) is sricly convex for each j. This assumpion is made wihou loss of generaliy. If we relax his assumpion o ha C j ( c j ) s are weakly convex, all resuls in his secion coninue o hold wih more edious proofs, as long as we selec he lexicographically smalles opimizer in each decision epoch. Consisen wih mos of he join pricing and invenory managemen models in he lieraure, we assume ha he replenishmen leadime o source from any supply channel is 0. Finally, we remark ha he firm employs he supply diversificaion sraegy o hedge agains: (a) he procuremen cos flucuaion risk caused by he volailiy of c, and (b) he diseconomy of scale for each supply channel. The firm operaes under a flucuaing marke environmen wih sochasically varying marke sizes Λ and reference procuremen coss c. Le he (n + m)-vecor θ := (Λ, c ) be he sae of he marke environmen in period. We assume ha θ evolves according o an exogenous Markov process hroughou he planning horizon. Le Λ i := (Λ 1,, Λ i 1, Λ i+1,, Λ n ) and c j := (c 1,, c j 1, c j+1,, c m ). We assume ha, for any i N [j M], condiioned on Λ i [c j ], Λ i 1 [c j 1] is independen of (Λ i, c ) [(Λ, c j )], 163

177 i.e., Λ i [c j ] is a sufficien saisic for Λ i 1 [c j 1]. Hence, he dynamics of θ can be represened as Λ i 1 = ξ Λ,i (Λ i ) and c j 1 = ξ c,j We furher assume ha, if ˆΛ i > Λ i [ĉ j > c j ], ξ Λ,i (c j ), where E{ξ Λ,i (Λ i ) θ }, E{ξ c,j (c j ) θ } < +. (ˆΛ i ) s.d. ξ Λ,i (Λ i ) [ξ c,j (ĉ j ) s.d. ξ c,j (c j )], where s.d. denoes he firs-order sochasic dominance. This is an inuiive assumpion, since a higher curren marke size is more likely o give rise o a higher marke size in he nex period, and he same is rue for he reference procuremen cos. Moreover, we assume ha, for any given θ, ξ Λ,i (Λ i ) and ξ c,j (c j ) are independen of ϵ and ς. The sequence of evens in each period unfolds as follows. A he beginning of period, he firm reviews is invenory level I and he realized sae of marke environmen θ. The firm hen simulaneously decides he sales price for each demand segmen and he order quaniy from each supply channel, and pays he oal procuremen cos j M Cj (q j c j ). The orders are received immediaely, afer which he price-dependen sochasic demand vecor D (p, Λ ) realizes. The firm hen collecs revenue from he realized demand. Unme demand is fully backlogged and excess invenory is fully carried over o he nex period. Finally, he firm pays H(z) for he invenory holding and backlogging cos for z unis of ending ne invenory, where H( ) is a convex funcion wih H(0) = 0 and H( ) > 0 oherwise. Moreover, we assume ha H( ) saisfies he Lipchiz coninuiy wih he Lipchiz consan c H, i.e., for any z 1, z 2 R, H(z 1 ) H(z 2 ) c H z 1 z 2. Noe ha alhough he demand, cos, and invenory penaly funcions are assumed o be saionary for exposiional convenience, he srucural resuls in his secion remain valid when hey are ime-dependen. To formulae he planning problem as a dynamic program, le V (I θ ) = he maximum expeced discouned oal profi in periods, 1,, 0, when he saring invenory level in period is I and he realized marke environmen sae is θ. Wihou loss of generaliy, we assume ha excess invenory a he end of he planning horizon is discarded wihou any salvage value, i.e., V 0 (I 0 θ 0 ) = 0. The opimal value funcions saisfy he following recursive scheme: V (I θ ) = max J (d, q, I θ ), (6.11) (d,q ) F 164

178 where F := {(d, q ) : i N, d i [0, d max ], j M, q j 0}, and (6.12) J (d, q, I θ ) := E{ p i (d i )D(p i i (d i ), Λ i ) C j (q j c j ) i N j M H(I + q j D a (p(d ), Λ )) j M = ( i N +αv 1 (I + j M q j D a (p(d ), Λ ) θ 1 ) θ } Λ i R i (d i )) j M C j (q j c j ) +E ς {Ψ (I + q j ( Λ i d i )ς θ )}, j M i N (6.13) wih Ψ (z θ ) := E θ 1,ϵ { H(z ϵ ) + αv 1 (z ϵ θ 1 ) θ }. (6.14) Therefore, for each period, he firm s profi-maximizing problem is o selec a join pricing and replenishmen policy (d (I, θ ), q (I, θ )) F o maximize J (d, q, I θ ), wih saring invenory level I and marke environmen sae θ. We use x := I + j M qj o denoe he oal order-up-o level, and x (I, θ ) := I + j M qj (I, θ ) o denoe he opimal oal order-up-o level. d i Moreover, le N (I, θ ) := {i N : (I, θ ) > 0} and M (I, θ ) := {j N : q j (I, θ ) > 0}, i.e., N (I, θ ) is he opimal se of acive demand segmens o which he firm sells, and M (I, θ ) is he opimal se of acive supply channels from which he firm orders. To conclude his subsecion, we characerize some preliminary concaviy and differeniabiliy properies of he value and objecive funcions in he following lemma. Lemma 17 For = T, T 1,, 1 and any given (I, θ ), he following saemens hold: (a) Ψ ( θ ) is concave and coninuously differeniable in z. (b) J (,, I θ ) is sricly joinly concave and coninuously differeniable in (d, q ). (a) V ( θ ) is concave and coninuously differeniable in I. I follows immediaely from Lemma 17 ha he opimal join pricing and ordering policy (d (I, θ ), q (I, θ )) is well-defined and unique in he feasible se F Comparaive Saics Analysis wih Our New Mehod Firs we observe ha, by Equaion (6.13), he objecive funcion in each period J (,, I θ ) is of he similar form o our illusraive opimizaion problem (i.e., Equa- 165

179 ion (6.1)) in Secion 6.3. Therefore, following he same argumen as he discussion in Secion 6.3.1, he sandard IFT and MCS approaches are generally no applicable o comparaive saics analysis in our general join pricing and invenory managemen model wih demand segmenaion, supply diversificaion and flucuaing marke environmen. Therefore, we employ our new comparaive saics mehod o sudy his model. Moreover, Lemma 15 applies o he proofs of several comparaive saics resuls in his subsecion, including Theorem and Theorems To begin wih, we apply our new comparaive saics mehod o characerize he opimal policy srucure in he following heorem. Theorem (Opimal policy srucure.) given θ, he following saemens hold: For = T, T 1,, 1 and any (a) For each i N, d i (I, θ ) is coninuously increasing in I. Moreover, here exiss a hreshold I d,i (θ ) < +, such ha i is opimal o serve demand segmen i, if > 0, I > I d,i (θ ), (I, θ ) Moreover, N (I, θ ) = 0, oherwise. and only if I > I d,i (θ ), i.e., d i N (Î, θ ) for all I < Î. (b) For each j M, q j (I, θ ) is coninuously decreasing in I. Moreover, here exiss a hreshold I q,j (θ ) < +, such ha i is opimal o order from supply channel j if and > 0, I only if I < I q,j (θ ), i.e., q j < I q,j (θ ), (I, θ ) Moreover, M (Î, θ ) = 0, oherwise. M (I, θ ) for all I < Î. (c) x (I, θ ) is coninuously increasing in I. Theorem shows ha, in each period, he opimal policy is a sae-dependen hreshold policy. More specifically, for each demand segmen i N [supply channel j M], he firm should sell o his segmen [order from his channel] if and only if he saring invenory level I is above [below] he corresponding hreshold I d,i (θ ) [I q,j (θ )]. This opimal policy srucure is characerized by employing our new mehod o esablish he monooniciy of he opimal sales price/order quaniy wih respec o he saring invenory level. More specifically, boh he opimal sales price for each demand segmen, p i (d i (I, θ )), and he opimal order quaniy from each supply channel, q j (I, θ ), are 166

180 decreasing in he saring invenory level, whereas he opimal oal order-up-o level x (I, θ ) is increasing in he saring invenory level. Consequenly, he opimal se of acive demand segmens, N (I, θ ) [acive supply channels, M (I, θ )], is increasing [decreasing], in he se inclusion order, in he saring invenory level. Theorem generalizes he base-sock lis-price policy in he join pricing and invenory managemen lieraure o he general seing wih demand segmenaion, supply diversificaion, and marke environmen flucuaion. Due o he diseconomy of scale and supply diversificaion, he order-up-o level and sales prices are invenory-dependen in his general seing. Finally, we remark ha if he muliplicaive random perurbaion in marke size is demand-segmen-dependen (i.e., D i (p i, Λ i ) = Λ i d i (p i )ς i + ϵ i for i N and ς i s are independen for differen i s), pars (b) and (c) of Theorem sill hold bu par (a) doesn. I is well esablished in he invenory managemen lieraure ha when here exis muliple muliplicaive random perurbaions in he sysem, he opimal order quaniies and/or sales prices are, in general, no monoone in he saring invenory level (see [72]). A key quesion in his invenory sysem is, for a given saring invenory level and marke sae, how o deermine he opimal se of acive demand segmens, N (I, θ ), and he opimal se of acive supply channels, M (I, θ ). The following heorem parially addresses his issue by comparing he opimal purchasing probabiliies for differen demand segmens, and he opimal order quaniies from differen supply channels. Theorem For = T, T 1,, 1 and any given θ, he following saemens hold: (a) Given i, î N, if d i R i (z) d î Rî(z) for each z [0, d max ], d i (I, θ ) dî (I, θ ), and I d,i (θ ) I d,î (θ ) for any (I, θ ). In paricular, if d 1 R 1 (z) d 2 R 2 (z) d n R n (z) for each z [0, d max ], I d,1 (θ ) I d,2 (θ ) I d,n (θ ) for any θ, and N (I, θ ) = {1, 2,,, i }, where i = max{i : I > I d,i (θ )}. (b) Assume ha c is fixed. Given j, ĵ M, if q jc j (z c j ) q ĵcĵ(z cĵ ) for any z 0, q j (I, θ ) qĵ (I, θ ) and I q,j (θ ) I q,ĵ (θ ) for any I and Λ. In paricular, if q 1 C 1 (z c 1 ) q 2 C 2 (z c 2 ) q m C m (z c m ) for any z 0, I q,1 (θ ) I q,2 (θ ) I q,m (θ ) for any Λ, and M (I, θ ) = {j, j +1,,, m}, where j = min{j : I < I q,j (θ )}. 167

181 In Theorem 6.4.2, we show ha he firm sells more o a demand segmen wih higher marginal revenue wih respec o demand, and i orders more from a supply channel wih lower marginal procuremen cos. Moreover, when he marginal revenues wih respec o demand [marginal procuremen coss] for differen demand segmens [supply channels] have he same order for all purchasing probabiliies [order quaniies], he opimal se of acive demand segmens [supply channels], N (I, θ ) [M (I, θ )], is consecuive in he marginal revenue wih respec o demand [marginal procuremen cos]. Nex, we employ our new comparaive saics mehod o sudy he impac of marke flucuaion upon he firm s opimal pricing and ordering policy. In his applicaion, we inegrae our new mehod wih he sandard backward inducion argumen o perform comparaive saics analysis in a dynamic program. More specifically, by employing Lemma 16, we ieraively link he comparison beween opimizers and ha beween parial derivaives of he value funcions and objecive funcions by backward inducion. This reamen is necessary because he curren marke sae also impacs fuure marke saes and, hus, he value funcions in he fuure. For he res of his subsecion, we make he addiional assumpion ha ς = 1 wih probabiliy 1 for all, i.e., he demand process follows an addiive form. The addiive demand assumpion is commonly imposed in he join pricing and invenory managemen lieraure for racabiliy (see, e.g., [112, 136, 189]). In our model, his assumpion enables us o ieraively link he monoone relaionship beween he opimizers and ha beween he parial derivaives. For he res of his subsecion, since ς = 1 wih probabiliy 1 for all, we rewrie he objecive funcion in period as J (d, q, I θ ) = ( i N Λ i R i (d i )) j M C j (q j c j ) + Ψ (I + j M q j ( i N Λ i d i ) θ ). Moreover, we define (I, θ ) := x (I, θ ) ( i N Λi d i (I, θ )) as he opimal safey sock in period wih saring invenory level I and marke sae θ. The following heorem characerizes he impac of curren marke size on he opimal sales prices and order quaniies. Theorem (Impac of marke size.) Assume ha, for each = T, T 1,, 1, ς = 1 wih probabiliy 1. For any given, le θ = (Λ, c ) and ˆθ = (ˆΛ, c ) wih ˆΛ > Λ. For any I, he following saemens hold: (a) I V (I ˆθ ) I V (I θ ). 168

182 (b) For each i N, d i (I, ˆθ ) d i (I, θ ), I d,i (ˆθ ) I d,i (θ ), and, hus, N (I, ˆθ ) N (I, θ ). (c) For each j M, q j M (I, ˆθ ). (d) x (I, ˆθ ) x (I, θ ). (I, ˆθ ) q j (I, θ ), I q,j (ˆθ ) I q,j (θ ), and, hus, M (I, θ ) Theorem proves ha an increase in he curren marke size of any demand segmen has he following impacs: (a) i promps he firm o increase he sales price for each demand segmen; (b) i drives he firm o order more from each supply channel; and (c) i moivaes he firm o se a higher oal order-up-o level. As he marke size of one demand segmen increases, he firm should increase is order quaniies from all he supply channels o mach supply wih demand, so he opimal se of acive supply channels is enlarged. A he same ime, he firm should increase is sales prices in all demand segmens, and he opimal se of acive demand segmens is smaller. Moreover, since he poenial marke size is more likely o become larger wih a larger curren marke size, i is opimal for he firm o keep a higher oal order-up-o level. The risks and opporuniies of procuremen cos flucuaion have been exensively sudied in [173]. In a model wih one demand segmen and wo supply channels, he paper shows ha invenory becomes more valuable under a higher curren procuremen cos, and he opimal sales price is increasing in he curren procuremen cos so ha he firm should pass par of he cos flucuaion risk o is cusomers. In Theorem below, we generalize hese resuls o our join pricing and invenory managemen model wih demand segmenaion, supply diversificaion, and marke environmen flucuaion. More specifically, we show ha, wih a higher reference procuremen cos of any supply channel, he marginal value of invenory is higher, and he firm charges a higher sales price in each demand segmen. As a resul, he demand in each segmen and he opimal se of acive segmens are decreasing in he reference procuremen cos of any supply channel. On he oher hand, [173] show ha he impac of cos on he firm s replenishmen policy is more involved, because he curren procuremen cos also summarizes he informaion on fuure coss. When facing a higher curren procuremen cos, he firm faces he radeoff beween ordering less o save curren cos and ordering more o speculae 169

183 on higher fuure coss. Numerical sudies in [173] demonsrae ha he opimal order quaniies may no be monoone in he curren procuremen cos when he firm orders is invenory eiher from a spo marke or hrough a forward-buying conrac. In our model, he opimal order quaniy from a supply channel coninues o be non-monoone in is own reference procuremen cos. However, we are able o show, in he following heorem, ha as he reference procuremen coss of one or more supply channels increase, he opimal order quaniies and ordering hresholds of he supply channels wih unchanged reference procuremen coss increase as well. Theorem (Impac of curren reference procuremen cos.) Assume ha, for each = T, T 1,, 1, ς = 1 wih probabiliy 1. For any given, le θ = (Λ, c ) and ˆθ = (Λ, ĉ ) wih ĉ > c. For any I, he following saemens hold: (a) I V (I ˆθ ) I V (I θ ). (b) For each i N, d i (I, ˆθ ) d i (I, θ ), I d,i (ˆθ ) I d,i (θ ), and, hus, N (I, ˆθ ) N (I, θ ). (c) If ĉ j = c j, q j (I, ˆθ ) q j (I, θ ) and I q,j (ˆθ ) I q,j (θ ). In addiion o he curren marke condiion, he firm should also ake ino accoun he fuure marke rend o achieve he long-run opimaliy. Our new comparaive saics mehod enables us o offer insighs on he opimal responses of he firm o poenial changes in he fuure marke condiion. We firs sudy he impac of fuure marke size rend on he firm s opimal decisions. Theorem (Impac of marke size rend.) Assume ha, for each = T, T 1,, 1, ς = 1 wih probabiliy 1. Le he wo sysems be equivalen excep ha Λ,i ˆξ (Λ i ) s.d. ξ Λ,i (Λ i ) for any, i N, and Λ. For any and (I, θ ), he following saemens hold: (a) I ˆV (I θ ) I V (I θ ). (b) For each i N, i ˆd (I, θ ) d i (I, θ ), Îd,i (θ ) I d,i (θ ), and, hus, ˆN (I, θ ) N (I, θ ). (c) For each j M, ˆq j (I, θ ) q j (I, θ ), Îq,j (θ ) I q,j (θ ), and, hus, M (I, θ ) ˆM (I, θ ). 170

184 (d) ˆx (I, θ ) x (I, θ ) and ˆ (I, θ ) (I, θ ). Theorem shows ha, under a higher marke size rend for any demand segmen, i is opimal o charge higher sales prices o all demand segmens and, hus, sell o a smaller se of segmens. On he oher hand, a higher marke size rend implies higher fuure demand, so he firm should order more from all supply channels, expand he se of acive supply channels, and se a higher safey sock o hold more invenory for fuure consumpion. As shown by [173], a higher procuremen cos rend increases he marginal value of invenory and promps he firm o increase is order quaniies boh from he spo marke and hrough he forward-buying conrac so as o save he fuure cos. A higher safey sock should also be kep. In addiion, he firm should raise is sales price o consume is invenory in he mos profiable way. In our general model, we show ha, when he reference procuremen cos rend in one sysem is higher han ha in he oher, all of he comparaive saics resuls in [173] coninue o hold for each demand segmen and supply channel. In addiion, wih a higher cos rend, he opimal se of acive demand segmens [supply channels] is smaller [larger]. Theorem (Impac of cos rend.) Assume ha, for each = T, T 1,, 1, ς = 1 wih probabiliy 1. Le he wo sysems be equivalen excep ha c,j ˆξ (c j ) s.d. ξ c,j (c j ) for any, j M and c. For any and (I, θ ), he following saemens hold: (a) I ˆV (I θ ) I V (I θ ). (b) For each i N, i ˆd (I, θ ) d i (I, θ ), Îd,i (θ ) I d,i (θ ), and, hus, ˆN (I, θ ) N (I, θ ). (c) For each j M, ˆq j (I, θ ) q j (I, θ ), Îq,j (θ ) I q,j (θ ), and, hus, M (I, θ ) ˆM (I, θ ). (d) ˆx (I, θ ) x (I, θ ) and ˆ (I, θ ) (I, θ ). In addiion, our new mehod enables us o perform comparaive saics analysis for he opimal decisions in differen models wih non-parameerizable changes. More specifically, we employ our mehod o characerize he impac of sales and procuremen flexibiliies (i.e., addiional demand segmens and supply channels) upon he firm s opimal pricing 171

185 and replenishmen policy. When he firm is blessed wih he opporuniy o sell o addiional demand segmens, he marginal value of invenory increases, and he firm should charge higher prices in he original segmens. Moreover, he firm should increase is replenishmen quaniies from all supply channels and expand he se of acive supply channels, so as o mach supply wih he higher demand from a larger pool of segmens. These inuiions are formalized in he following heorem. Theorem (Impac of addiional demand segmens.) Assume ha, for each = T, T 1,, 1, ς = 1 wih probabiliy 1. Le he wo sysems be equivalen excep for N ˆN. For = T, T 1,, 1, and any (I, θ ), he following saemens hold: (a) I ˆV (I θ ) I V (I θ ). (b) For each i N, ˆd i (I, θ ) d i (I, θ ), Îd,i (θ ) I d,i (θ ), and, hus, ( ˆN (I, θ ) N ) N (I, θ ). (c) For each j M, ˆq j (I, θ ) q j (I, θ ), Îq,j (θ ) I q,j (θ ), and, hus, M (I, θ ) ˆM (I, θ ). (d) ˆx (I, θ ) x (I, θ ). On he oher hand, he supply diversificaion sraegy enables he firm o hedge agains he procuremen cos flucuaion risk and he diseconomy of scale of he supply channels. By sourcing from a larger supply pool, he firm enjoys more procuremen flexibiliy, and orders less from each of he original supply channels. Moreover, he marginal value of invenory is smaller wih a larger supply pool, and, o mach supply wih demand, he firm should se lower sales prices in all demand segmens and sell o more segmens. Theorem (Impac of addiional supply channels.) Assume ha, for each = T, T 1,, 1, ς = 1 wih probabiliy 1. Le he wo sysems be equivalen excep for M ˆM. For = T, T 1,, 1, and any (I, θ ), he following saemens hold: (a) I ˆV (I θ ) I V (I θ ). (b) For each i N, i ˆd (I, θ ) d i (I, θ ), Îd,i (θ ) I d,i (θ ), and, hus, N (I, θ ) ˆN (I, θ ). 172

186 (c) For each j M, ˆq j (I, θ ) q j M) M (I, θ ). (I, θ ), Îq,j (θ ) I q,j (θ ), and, hus, ( ˆM (I, θ ) To sum up, comparaive saics analysis is essenial in our general join pricing and invenory managemen model wih demand segmenaion, supply diversificaion, and marke environmen flucuaion. Alhough he sandard IFT and MCS approaches do no apply o his complex model, our new comparaive saics mehod enables us o characerize is opimal policy as a sae-dependen hreshold policy, and o analyze he impac of marke flucuaion and operaional flexibiliies upon he opimal policy. 6.5 Applicaion of he New Comparaive Saics Mehod in a Compeiion Model In his secion, we apply our new mehod o comparaive saics analysis in a join price and effor compeiion model, which, as we will show, canno be conduced wih he sandard IFT and MCS approaches. We remark ha he noaions in his secion are independen of hose in Secions 6.3 and 6.4. More specifically, we consider an oligopoly indusry of N compeing firms. Each firm offers a parially subsiuable produc wih uni producion cos c i > 0. Each firm i selecs a sales price p i [p min i i, p min i, p max i ] and an effor level y i 0. We assume ha, for each = c i. In addiion, we make he same assumpion as [9] ha p max i is sufficienly large so ha i has no impac on he equilibrium behavior. Each firm i can exer effor y i (on, e.g., R&D or adverising) o increase is demand. We use Y := N i=1 y i o denoe he oal effor level. Le p := (p 1, p 2,, p N ) be he vecor of sales prices and y := (y 1, y 2,, y N ) be he vecor of effor levels. For any decision vecor (p, y), he demand for firm i is given in he following quasi-separable form: λ i (p, y, θ i ) = (θ i + f(y ) b i p i + j i β ij p j ) +, (6.15) where θ i [θ min, θ max ] is he firm-dependen marke index, capuring oher impac facors on demand beyond price and effor (e.g., brand image). I is commonly assumed in he R&D and research join venure lieraure (e.g., [104]) ha he oal effor level in he indusry has an accumulaive effec upon he demand of each firm. We model his accumulaive effec by f(y ), and assume ha f( ) is a sricly increasing, coninuously 173

187 differeniable, and sricly concave funcion ha is bounded from above by M < +. As [9], we assume ha λ i (p, y, θ i ) > 0 whenever p i = p min i = c i, i.e., each firm i can have a posiive demand under zero profi margin, regardless of he compeing firms price decisions and all firms effor decisions. We remark ha alhough we do no assume λ i (p, y, θ i ) > 0 for all feasible (p, y), we will show ha he equilibrium demand of each firm is posiive. We also make he sandard assumpion ha b i, β ij > 0 for all i, j, and ha he dominan diagonal condiion holds, i.e., b i > j i β ij > 0 and b i > j i β ji > 0. The inerpreaion of he dominan diagonal condiion is ha an uniform price increase of all firms canno resul in a demand increase in any firm, and ha a price increase of any firm canno resul in an increase in he oal demand of he indusry (see, also, [9]). Consisen wih he sandard assumpion in he economics lieraure ([121]), we assume ha, for each firm i, he cos of exering effor y i is C i (y i ), where C i ( ) is an increasing, sricly convex, and coninuously differeniable funcion. Thus, he profi of firm i in his join price and effor compeiion is given by: Π i (p, y θ) = (p i c i )λ i (p, y, θ i ) C i (y i ), (6.16) where θ := (θ 1, θ 2,, θ N ) represens he marke index vecor. Depending on he indusry dynamics, wo compeiion models are considered: (a) he effor-level-firs compeiion (EF), and (b) he simulaneous compeiion (SC). In he effor-level-firs compeiion, he firms firs choose heir effor levels (on, e.g., R&D or adverising), and hen selec he sales prices in he second sage. In he simulaneous compeiion, he firms make effor and price decisions simulaneously. In he nex wo subsecions, we employ our new comparaive saics mehod o characerize he equilibrium in hese wo compeiion models, and sudy how he equilibrium prices and effor levels change wih he marke index vecor θ. Finally, in Secion 6.5.3, we compare he equilibrium decisions in hese wo compeiion models Effor-Level-Firs Compeiion In his subsecion, we sudy he effor-level-firs compeiion model. In his model, he firms engage in a wo-sage game, in which hey compee on marke expanding effor in he firs sage and on sales price in he second sage. This model is suiable for he scenario in which he sickiness of marke expanding effor choices is much higher 174

188 han ha of sales price choices. For example, due o he long leadime for echnology developmen, decisions on R&D effor are usually made well in advance of price decisions. To analyze his wo-sage game, we begin wih he price compeiion in he second sage. In his sage, he effor level in he firs sage y is observable by all firms. Le A be an N N marix wih A ii = 2b i, A ij = β ij for i j, a(y, θ) be a column vecor wih a i (Y, θ) = θ i + f(y ) = θ i + f( N j=1 y j), and κ be a column vecor wih κ i = b i c i. Given any effor level y, he equilibrium price, p (y, θ), in he second-sage compeiion is characerized by he following heorem. Theorem (Second-sage price compeiion.) For a given effor level vecor y, he following saemens hold: (a) The equilibrium in he second-sage price compeiion is unique and given by p (y, θ) = A 1 (a(y, θ)+κ), wih p i (y, θ) > p min i = c i. The unique equilibrium demand for firm i is given by λ i (y, θ) = b i (p i (y, θ) c i ) > 0. Hence, for any i, p i (y, θ) and λ i (y, θ) depend on he effor level vecor y only hrough he oal effor level Y. (b) p i (y, θ) is sricly increasing in Y, wih yj p i (y, θ) = Y p i (y, θ) = ( N l=1 (A 1 ) il )f (Y ) for each i and j. Hence, λ i (y, θ) is sricly increasing in Y and, hus, y j for any i and j. Theorem proves ha, given any effor level vecor in he firs-sage, he secondsage price compeiion has a unique equilibrium. Moreover, under he equilibrium, each firm achieves a posiive profi margin and a posiive demand. Boh he equilibrium price and demand of each firm are sricly increasing in he oal effor level. Higher effor level in he firs sage increases he marke demand, and moivaes each firm o charge a higher sales price. Based on Theorem 6.5.1, we sudy he firs-sage compeiion, in which he firms choose heir effor levels. Since p (y, θ) depends on y only hrough he oal effor level Y, we use p (Y, θ) o denoe he equilibrium price in he second-sage compeiion. As a resul, he equilibrium demand can be represened as λ (Y, θ), wih λ i (Y, θ) = b i (p i (Y, θ) c i ). Plugging p (Y, θ) and λ (Y, θ) ino (6.16), we obain he objecive funcions in he firs-sage game: π i (y θ) = b i (p i (Y, θ) c i ) 2 C i (y i ), for i = 1, 2,, N. (6.17) 175

189 Following [104], we make he following assumpion on f( ). Assumpion (f(x) f 0 ) 2 is concave in x for x 0, where f 0 := f(0). Assumpion guaranees he concaviy of he objecive funcions in he firs-sage effor compeiion and, hus, he exisence of an equilibrium. This assumpion is he counerpar of Assumpion 3 in [104]. Wih he help of Assumpion 6.5.1, we characerize he equilibrium of he firs-sage effor compeiion in he following heorem. Theorem (Effor-level-firs compeiion.) Under Assumpion 6.5.1, he following saemens hold: (a) Given any θ, he firs-sage effor compeiion has a unique equilibrium y EF (θ). (b) Le YEF (θ) := N i=1 y EF,i (θ) be he equilibrium oal effor level in he firs-sage compeiion. p (YEF (θ), θ) is he unique associaed equilibrium price vecor and λ (YEF (θ), θ) is he unique associaed equilibrium demand vecor in he second-sage compeiion, where p (, ) and λ (, ) are given in Theorem 6.5.1(a). Theorem shows ha he wo-sage effor-level-firs compeiion has a unique subgame perfec equilibrium. The proof of Theorem heavily relies on our new comparaive saics mehod, which enables us o esablish he monoone relaionship ha he equilibrium effor level of each firm, yef,i (θ), is decreasing in he equilibrium oal effor level Y EF (θ). Such monooniciy, ogeher wih he ideniy ha N i=1 y EF,i (θ) = YEF (θ), guaranees he uniqueness of he equilibrium in he firs-sage effor compeiion. A naural quesion in his compeiion is how he marke index θ influences he equilibrium price p (YEF (θ), θ) and equilibrium effor y EF (θ). Inuiion suggess ha, under a beer marke condiion, he firms should decrease heir marke expanding effors o save coss. The following heorem shows ha his inuiion is reversed in he effor-level-firs compeiion. Theorem (Impac of marke index.) Under Assumpion 6.5.1, he following saemens hold: (a) YEF (θ) is increasing in θ i for any i. (b) p i (YEF (θ), θ) and λ i (YEF (θ), θ) are increasing in θ j for any i and j. 176

190 Theorem shows ha he equilibrium oal effor level YEF (θ) is increasing in each marke index, θ i. As a resul, he equilibrium sales price and demand of each firm are increasing in he marke index of any firm. Noe ha Theorem canno be proved by he sandard IFT and MCS approaches. I is possible ha f( ), and hence, he objecive funcion π i ( ), are no wice coninuously differeniable. Even if π i ( ) is wice coninuously differeniable in (y, θ) for each i, calculaing he inverse of he Hessian in he firs-sage effor compeiion can be prohibiively difficul when N is large. Therefore, he IFT approach has poor scalabiliy, and i is very difficul, if no impossible, o prove Theorem by he IFT approach. On he oher hand, alhough π i ( ) is supermodular in y i and θ j (y, θ j ). for any i and j, i is no joinly supermodular in Hence, he complemenariy condiions required in supermodular games (see [124]) do no hold, and he MCS approach does no apply o his model. We employ our new comparaive saics mehod o prove Theorem 6.5.3(a). We assume, o he conrary, ha Y EF (θ) is decreasing in θ i for some i, and consruc a conradicion wih he ieraive procedure in Secion This approach explois he supermodulariy of π i ( ) in y i and θ j for any i and j, bu does no require he join supermodulariy of π i ( ) in (y, θ j ). Par (b) of Theorem follows direcly from par (a) Simulaneous Compeiion In some scenarios, he marke expanding effor (on, e.g., adverising) akes effec insananeously. Hence, decisions on effor can be made a he same ime as price decisions. In his scenario, he firms engage in a simulaneous price and effor compeiion. Specifically, each firm i simulaneously selecs (p i, y i ) o maximize Π i (p, y θ) defined by (6.16). The nex heorem characerizes he equilibrium and he impac of marke index upon he equilibrium in he simulaneous compeiion. Theorem (Simulaneous compeiion.) Under Assumpion 6.5.1, he following saemens hold: 177

191 (a) Given any θ, he simulaneous compeiion has a unique equilibrium (p SC (θ), y SC (θ)), which saisfies he sysem of equaions: p SC(θ) = p (y SC(θ), θ) (p SC,i(θ) c i )f (Y SC(θ)) C i(y SC,i(θ)) = A 1 (a(ysc(θ), θ) + κ), (6.18) = 0, if ysc,i (θ) > 0, (6.19) 0, oherwise, for all i = 1, 2,, N, where YSC (θ) = N i=1 y SC,i (θ). Conversely, he sysem of equaions (6.18) and (6.19) has a unique soluion, which is he equilibrium of he simulaneous compeiion. The equilibrium demand is given by λ SC (θ) = (λ SC,1 (θ), λ SC,2 (θ),, λ SC,N (θ)), where λ SC,i (θ) = b i(p SC,i (θ) c i). Moreover, for any i, p SC,i (θ) > c i and λ SC,i (θ) > 0. (b) YSC (θ) is increasing in θ i for any i. Moreover, p SC,i (θ) and λ SC,i (θ) are increasing in θ j for any i and j. Under Assumpion 6.5.1, Theorem 6.5.4(a) proves he exisence and uniqueness of he equilibrium in he simulaneous price and effor compeiion. Moreover, we show ha, under he equilibrium, each firm earns a posiive profi margin and a posiive demand in he simulaneous compeiion. Noe ha Assumpion does no guaranee he join concaviy of Π i (p, y θ) in (p i, y i ). Hence, we canno use he sandard approach o prove he exisence of an equilibrium in he simulaneous compeiion. Insead, we show ha he sysem of equaions (6.18) and (6.19) has a unique soluion, which is an equilibrium of he simulaneous compeiion. We prove he uniqueness of he equilibrium by showing ha any equilibrium of he simulaneous compeiion mus saisfy he sysem of equaions (6.18) and (6.19). In Theorem 6.5.4(b), we employ our new comparaive saics mehod o show ha, in he simulaneous compeiion, he equilibrium oal effor level, and he equilibrium sales price and demand volume of each firm are increasing in he marke index of any firm. This resul is consisen wih is counerpar in he effor-level-firs compeiion (i.e., Theorem 6.5.3). 178

192 6.5.3 A Comparison of Equilibria in he Two Compeiion Models In his subsecion, we compare he equilibrium in he effor-level-firs compeiion (characerized in Theorem 6.5.2) and ha in he simulaneous compeiion (characerized in Theorem 6.5.4). We summarize he comparison resuls in he following heorem: Theorem Under Assumpion 6.5.1, he following saemens hold: (a) p i (yef (θ), θ) p SC,i (θ) for any i and θ. (b) YEF (θ) Y SC (θ) for any θ. (c) λ i (YEF (θ), θ) λ SC,i (θ) for any i and θ. Theorem shows ha, in he effor-level-firs compeiion, he firms exer higher oal marke expanding effor o dampen he subsequen price compeiion, which resuls in higher equilibrium price and demand of each firm han heir counerpars in he simulaneous compeiion. This phenomenon (i.e., he fa-ca effec, see [78]) has also been idenified by [9] in a join price and service level compeiion seing. They show ha he equilibrium sales prices, demand volumes, and service levels are higher in he service-level-firs compeiion model han hose in he simulaneous compeiion model. To prove his resul, [9] inducively show ha, for each k, he k h ieraion of he aônnemen scheme for he service-level-firs compeiion model is higher, in price and service level, han ha for he simulaneous compeiion model. Since he join price and service compeiion games in [9] are supermodular, he aônnemen scheme can generae he minimum equilibria and, hus, heir monoone relaionship in he wo compeiions. In our model, however, neiher he effor-level-firs compeiion nor he simulaneous compeiion is supermodular, so we employ our new comparaive saics mehod o prove Theorem We firs prove par (b) by employing he ieraive procedure in Secion o consruc a conradicion under he (incorrec) assumpion ha Y EF (θ) < Y SC (θ) for some θ. Pars (a) and (c) follow direcly from par (b) by Theorem and Theorem To conclude his secion, we remark ha all of he comparaive saics resuls on he equilibrium oal effor level in Theorems canno be generalized o ones on he equilibrium effor level of each firm. This is because, in boh compeiion models, alhough he objecive funcion of each firm i is supermodular in y i and θ j 179 for any i

193 and j, i is no joinly supermodular in (y, θ j ). In oher words, some firms may free-ride on a higher effor level of heir compeiors, and hus, decrease heir own effor levels. When his effec dominaes he effor-promping effec of a beer marke condiion, he equilibrium effor levels of some firms may be decreasing in he marke indices. 6.6 Summary In his chaper, we consider a general join pricing and invenory managemen model wih demand segmenaion, supply diversificaion, and marke environmen flucuaion. In his model, comparaive saics analysis is inegral o characerizing is opimal policy and analyzing he impac of demand segmenaion, supply diversificaion, and marke environmen flucuaion upon he opimal policy. The sandard comparaive saics mehods (i.e., he IFT and MCS approaches) do no apply, because (a) he second-order coninuous differeniabiliy and complemenariy condiions are no saisfied in our model, (b) he IFT approach is no scalable, and (c) some of he opimal decision variables are no monoone in he parameer (i.e., he MCS approach does no work in his case). Therefore, we develop a new comparaive saics mehod. Our new mehod employs a simple bu powerful lemma (Lemma 16) and some model-specific properies (e.g., he supermodulariy in one decision variable and he parameer and he componenwise concaviy of he objecive funcion) o ieraively link he comparison beween opimizers and ha beween he parial derivaives of objecive funcions, so as o consruc conradicions under he assumpion ha he desired comparaive saics resuls are reversed. Lemma 16 enables us o make componenwise comparisons of he opimal decision variables a differen parameers, which is he essenial difference beween our mehod and he sandard approaches. The componenwise comparison beween opimizers faciliaes he scalabiliy of our mehod and is applicaion in a model where only par of he opimizers are monoone in he parameer. We remark ha when a conradicion canno be reached using our new mehod, a counerexample of he original comparaive saics predicion, in general, can be found. Hence, he proposed mehod can also help idenify cases in which comparaive saics resuls do no hold for some decision variables. Though fundamenally differen, our new mehod shares some similariy wih he sandard IFT and MCS approaches. Analogous o he IFT approach, he proposed mehod sudies he firs-order (KKT) condiion a he opimizer of ineres. Hence, our mehod 180

194 requires he objecive funcion be firs-order coninuously differeniable, bu no necessarily second-order coninuously differeniable. Following he idea of he MCS approach, our mehod carefully examines he impac of he parameer upon he marginal values of he decision variables, so ha we can ranslae he monooniciy of parial derivaives back ino he monooniciy of anoher decision variable a he opimizer. Thus, o reach a conradicion (and hence, a comparaive saics resul), our mehod requires he objecive funcion be supermodular in he parameer and each of he focal decision variables, bu no necessarily joinly supermodular or saisfying he single crossing propery. The above wo condiion relaxaions enhance he applicabiliy of our mehod in he general join pricing and invenory managemen model, where he second-order coninuous differeniabiliy and join supermodulariy of he objecive funcion in each decision epoch are hard, if no impossible, o esablish. We employ our new mehod o analyze he join pricing and invenory managemen model under demand segmenaion, supply diversificaion, and marke environmen flucuaion. Our new comparaive saics mehod enables us o characerize he opimal join pricing and ordering policy for an arbirary number of demand segmens and supply channels as a hreshold policy, under which here exiss a marke environmen dependen hreshold for each demand segmen [supply channel] such ha i is opimal o sell o [order from] his segmen [channel] if and only if he saring invenory level is above [below] is corresponding hreshold. The opimal sales price for each demand segmen and he opimal order quaniy from each supply channel are decreasing in he saring invenory level of he firm, and increasing in he marke size of any demand segmen. When he reference procuremen coss of some supply channels increase, he firm increases he sales price in each demand segmen, and he order quaniies from he supply channels wih unchanged reference procuremen coss. Each firm s opimal order quaniy may no be monoone in is own reference procuremen cos. Expanding he se of demand segmens drives he firm o increase is sales price in each demand segmen and order quaniy from each supply channel, whereas expanding he supply pool decreases he opimal sales prices and order quaniies. To demonsrae he applicabiliy of our new comparaive saics mehod in oher seings, we employ i o sudy join price and effor compeiion games, in which he oal effor level has a posiive impac upon each firm s demand. More specifically, we 181

195 consider wo compeiion models: (a) he firms compee on effor in he firs sage and on price in he second sage; and (b) he firms simulaneously compee on price and effor. The sandard IFT and MCS approaches are no amenable for he comparaive saics analysis in his seing, because he IFT approach has poor scalabiliy, and he complemenariy condiions required by he MCS approach are no saisfied. We apply our new mehod o show he exisence and uniqueness of equilibrium in he effor-levelfirs compeiion. We prove ha, in boh compeiion models, he equilibrium oal effor level, and he equilibrium price and demand of each firm are increasing in he marke index of any firm. We also demonsrae he fa-ca effec in his seing, i.e., he sequenial decision making gives rise o a higher oal effor level and a higher price and demand of each firm in he effor-level-firs compeiion han hose in he simulaneous compeiion. In summary, our new mehod enables us o perform comparaive saics analysis in a general join pricing and invenory managemen model and a join price and effor compeiion model. Sandard IFT and MCS approaches are no amenable for boh seings. 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 some of he decision variables are non-monoone, and i is scalable. Hence, our new mehod is promising for comparaive saics analysis in oher operaions managemen models. 182

196 7. Concluding Remarks This disseraion focuses on he impac of some new marke rends (such as social neworks, susainabiliy concerns, and cusomer behaviors) upon a firm s pricing and invenory policies. Our resuls demonsrae ha hese emerging rends lead o ineresing new radeoffs and, hence, would significanly influence a firm s operaions decisions. On he oher hand, he firm can adop innovaive pricing and invenory sraegies o exploi hese marke rends and subsanially improve is profi. To faciliae he analysis, we develop an effecive comparaive saics analysis mehod for a general class of join pricing and invenory managemen models. The combined pricing and invenory policy is inarguably a very imporan operaions decision for any firm ha delivers physical producs o cusomers. We believe here are several promising avenues for fuure research relaed o his opic. Insead of digging ino he deails, we focus on he high-level discussions of fuure research direcions. Muli-iem invenory sysems. The disseraion only sudies he pricing and invenory policy of a single-produc model. While his seing is ineresing and relevan by iself, muli-iem invenory models would beer capure he siuaion of a reailer in he e-commerce marke. For a reailer on an online e-commerce plaform like Amazon, i generally holds and sells invenories of differen producs. So he reailer needs o joinly manage he pricing, sourcing, soring, and delivery sraegies of all is producs. Wih muliple producs handled ogeher, he key issue he firm faces is how o allocae he capial, ransporaion, and human capaciies among differen producs. Among ohers, i is ineresing o sudy how he dynamic pricing flexibiliy would complemen he firm s capaciy allocaion sraegy, and alleviae is capaciy consrain pressure. Informaion asymmery. In his disseraion, we assume in our model ha informaion is symmeric o everyone. If he marke exhibis informaion asymmery beween he firm and he cusomers, we need o employ dynamic mechanism design echniques o sudy he opimal pricing and invenory conrol policy herein. From he applicaion perspecive, inroducing informaion asymmery well capures he curren marke rends of, e.g., sharing economy, social neworks, and online aucions. An enriched join pricing 183

197 and invenory managemen model wih informaion asymmery enables us o characerize he operaional impac of hese new markeplace innovaions, and sudy he role of informaion in marke design issues. Daa-driven inegraed pricing and invenory opimizaion. In his disseraion, he decision maker (i.e., he firm) have full knowledge abou he demand funcion and demand disribuion. In realiy, however, his is no necessarily he case, since he demand funcion and disribuion may no be available o he firm. In his siuaion, he firm should collec he previous demand daa and employ some daa-driven mehods o do he predicion and prescripion simulaneously. I is ineresing o develop some daa-driven algorihms o opimize he join pricing and invenory conrol policy in an online manner. The objecive is o achieve he maximum expeced profi under he full demand informaion assumpion asympoically. The key issue wihou knowing he demand disribuion is o balance he well-known exploraion-exploiaion radeoff under he inegraed pricing and invenory managemen framework. To sum up, he inegraed pricing and invenory conrol problem is of boh heoreical ineres and pracical relevance. This disseraion s main conribuion is o esablish new models and mehods o sudy he impac of new marke rends on he join dynamic pricing and invenory policy of a firm. We also hope he disseraion would help inspire fuure research on his opic. 184

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214 APPENDICES

215 A.1 Proofs of Saemens A. Appendix for Chaper 2 We use o denoe he derivaive operaor of a single variable funcion, x o denoe he parial derivaive operaor of a muli-variable funcion wih respec o variable x, and 1 { } o denoe he indicaor funcion. For any mulivariae coninuously differeniable funcion f(x 1, x 2,, x n ) and x := ( x 1, x 2,, x n ) in f( ) s domain, i, we use xi f( x 1, x 2,, x n ) o denoe xi f(x 1, x 2,, x n ) x= x. The following lemma is used hroughou our proofs. Lemma 18 Le F i (z, Z) be a coninuously differeniable and joinly concave funcion in (z, Z) for i = 1, 2, where z [z, z] (z and z migh be infinie) and Z R n. For i = 1, 2, le (z i, Z i ) := argmax (z,z) F i (z, Z) be he opimizers of F i (, ). If z 1 < z 2, we have: z F 1 (z 1, Z 1 ) z F 2 (z 2, Z 2 ). = 0 if z 1 > z, Proof: z 1 < z 2, so z z 1 < z 2 z. Hence, z F 1 (z 1, Z 1 ) and 0 if z 1 = z; = 0 if z 2 < z, z F 2 (z 2, Z 2 ) i.e., z F 1 (z 1, Z 1 ) 0 z F 2 (z 2, Z 2 ). Q.E.D. 0 if z 2 = z, Proof of Lemma 1: Since γ( ) is wice coninuously differeniable, R (, ) is wice coninuously differeniable, and joinly concave in (p, N ) if and only if he Hessian of R (, ) is negaive semi-definie, i.e., p 2 R (p, N ) 0, and p 2 R (p, N ) N 2 R (p, N ) ( p N R (p, N )) 2, where p 2 R (p, N ) = 2, N 2 R (p, N ) = (p b αc)γ (N ), and p N R (p, N ) = γ (N ). Hence, R (, ) is joinly concave on [p, p] [0, + ) if and only if 2(p b αc)γ (N ) (γ (N )) 2 for all (p, N ). Since 2(p b αc)γ (N ) 2(p b αc)γ (N ), 2(p b αc)γ (N ) (γ (N )) 2 for all (p, N ) if and only if 2(p b αc)γ (N ) (γ (N )) 2 for all N 0. This proves Lemma 1. Q.E.D. Proof of Lemma 2: We prove pars (a) - (c) ogeher by backward inducion. We firs show, by backward inducion ha if v 1 (I 1, N 1 ) ci 1 is joinly concave in (I 1, N 1 ), decreasing in I 1, and increasing in N 1, (i) Ψ (, ) is joinly concave in (x, y), decreasing in x, and increasing in y; (ii) J (,, ) is joinly concave in (x, p, N ); and (iii) v (I, N ) ci is joinly concave in (I, N ), decreasing in I, and increasing in N. I is clear ha v 0 (I 0, N 0 ) ci 0 = 0 is joinly concave, decreasing in I 0, and increasing in N 0. Hence, he iniial condiion holds. Assume ha v 1 (I 1, N 1 ) ci 1 is joinly concave in (I 1, N 1 ), decreasing in I 1, and increasing in N 1. Because r n ( ) is concavely increasing, E[r n (y + θξ + ϵ )] is concavely increasing in y. Since concaviy and monooniciy are preserved under expecaion, Ψ (, ) is joinly concave in (x, y), decreasing in x, and increasing in y. Analogously, Λ(x) is concavely decreasing in x. We now 201

216 verify ha Ψ (x V + p γ(n ), θ( V p + γ(n )) + ηn ) is joinly concave in (x, p, N ) and increasing in N. Since γ( ) is increasing in N, whereas Ψ (x, y) is decreasing in x and increasing in y, Ψ (x V + p γ(n ), θ( V p + γ(n )) + ηn ) is increasing in N. Le λ [0, 1], x = λx + (1 λ)ˆx, p = λp + (1 λ)ˆp, and N = λn + (1 λ) ˆN, we have: λψ (x V + p γ(n ), θ( V p + γ(n )) + ηn ) +(1 λ)ψ (ˆx V + ˆp γ( ˆN ), θ( V ˆp + γ( ˆN )) + η ˆN ) Ψ (x V + p λγ(n ) (1 λ)γ( ˆN ), θ( V p + λγ(n ) + (1 λ)γ( ˆN )) + ηn ) Ψ (x V + p γ(n ), θ( V p + γ(n ) + ηn ), where he firs inequaliy follows from he join concaviy of Ψ (, ), and he second from he concaviy of γ( ), and ha Ψ (, ) is decreasing in x and increasing in y. I s clear ha Λ(x) = E{ (h + b)(x ξ ) + } is concavely decreasing in x. Hence, similar argumen o he case of Ψ (x V + p γ(n ), θ( V p + γ(n )) + ηn ) implies ha Λ(x V + p γ(n )) is joinly concave in (x, p, N ) and increasing in N. By Assumpion 2.3.1, R (p, N ) is joinly concave in (p, N ). Moreover, since γ( ) is increasing in N, R (p, N ) is increasing in N as well. Hence, by (2.5), J (x, p, N ) = R (p, N ) + βx + Λ(x V + p γ(n )) is joinly concave in (x, p, N ) and increasing in N. +Ψ (x V + p γ(n ), θ( V p + γ(n )) + ηn ) Since concaviy is preserved under maximizaion (e.g., [32] Secion 3.2.5), he join concaviy of v (, ) follows direcly from ha of J (,, ). Noe ha for any Î > I, F(Î) F(I ). Thus, v (Î, N ) cî = max (x,p ) F(Î) J (x, p, N ) max J (x, p, N ) (x,p ) F(I ) = v (I, N ) ci. Hence, v (I, N ) ci is decreasing in I. Since J (x, p, N ) is increasing in N for any (x, p, N ), for any ˆN > N, Thus, v (I, N ) ci is increasing in N. v (I, ˆN ) ci = max (x,p ) F(I ) J (x, p, ˆN ) max J (x, p, N ) (x,p ) F(I ) = v (I, N ) ci. Second, we show ha by backward inducion, ha if v 1 (, ) is coninuously differeniable, Ψ (, ), J (,, ), and v (, ) are coninuously differeniable as well. For = 0, v 0 (I 0, N 0 ) = ci 0 is clearly coninuously differeniable. Thus, he iniial condiion holds. If v 1 (, ) is coninuously differeniable, Ψ (, ) is coninuously differeniable wih parial derivaives given by x Ψ (x, y) = E{α[ I v 1 (x ξ, y + θξ + ϵ ) c]}, (A.1) y Ψ (x, y) = E{r n(y + θξ + ϵ ) + α[ N 1 v 1 (x ξ, y + θξ + ϵ )}, (A.2) 202

217 where he exchangeabiliy of differeniaion and expecaion is easily jusified using he canonical argumen (see, e.g., Theorem A.5.1 in [63], he condiion of which can be easily verified observing he coninuiy of he parial derivaives of v 1 (, ), and ha he disribuions of ξ and ϵ are coninuous.). Moreover, since ξ is coninuously disribued, Λ( ) is coninuously differeniable. Since R (, ) is coninuously differeniable, by (2.5), J (,, ) is coninuously differeniable. If I x (N ), he coninuous differeniabiliy of v (, ) follows immediaely from ha of J (,, ) and he envelope heorem. To complee he proof, i suffices o check ha, for all N 0, he lef and righ parial derivaives of he firs variable a (x (N ), N ), I v (x (N ), N ) and I v (x (N )+, N ) are equal. By he envelope heorem, I v (x (N ), N ) = c, I v (x (N )+, N ) = c + β + x Λ(x (N ) y (N )) + x Ψ (x (N ) y (N ), θy (N ) + ηn ). The firs-order condiion wih respec o x implies ha β + x Λ(x (N ) y (N )) + x Ψ (x (N ) y (N ), θy (N ) + ηn ) = 0. Therefore, I v (x (N ), N ) = I v (x (N )+, N ) = c. This complees he inducion and, hus, he proof of Lemma 2. Q.E.D. Proof of Theorem 2.4.1: Pars (a)-(b) follow immediaely from he join concaviy of J (,, N ) in (x, p ) for any N 0. We now show par (c) by backward inducion. More specifically, we prove ha if x 1 (N 1 ) > 0 for all N 1 0, x (N ) > 0 for all N 0. Since v 0 (I 0, N 0 ) = ci 0, Ψ 1 (x, y) = E[r n (y +θξ 1 )]+αe{v 0 (x ξ 1, y + θξ 1 + ϵ 1 ) cx} = E[r n (y + θξ 1 )]. Since D 1 0 wih probabiliy 1, x Λ( V 1 + p 1 γ(n 1 )) = 0 for all p 1 [p, p] and N 1 0. Hence, for any p 1 [p, p] and N 1 0, x1 J 1 (0, p 1, N 1 ) = β x Λ( V 1 + p 1 γ(n 1 )) = β > 0. Hence, x 1 (N 1 ) > 0 for any N 1 0. Thus, he iniial condiion is saisfied. Now we assume ha x 1 (N 1 ) > 0 for all N 1 0 and x (Ñ) 0 for some Ñ 0. Thus, I 1 = x (Ñ) D (p (Ñ), Ñ) 0 < x 1 (Ñ 1) almos surely, where Ñ 1 = θd (p (Ñ), Ñ)+ηÑ+ϵ. Thus, by par (a), I 1 v 1 (I 1, N 1 ) = c almos surely, when condiioned on N = Ñ. Hence, condiioned on N = Ñ, x Ψ (x, y) = αe{ I 1 v 1 (I 1, Ñ 1) c N = Ñ} = c c = 0, when (x, p ) lies in he neighborhood of (x (Ñ), p (Ñ)). As discussed above, since x (Ñ) 0, x Λ( V + p γ(ñ1)) = 0 for all p [p, p]. Hence, for any p [p, p], x J (0, p, Ñ) = β x Λ( V + p γ(ñ1)) = β > 0. Hence, x (Ñ) > 0, which conradics he assumpion ha x (Ñ) 0 is he opimizer of (2.7) when N = Ñ. Therefore, x (N ) > 0 for all N 0. This complees he inducion and, hus, he proof of par (c). Q.E.D. Proof of Lemma 3: We show Pars (a)-(b) ogeher by backward inducion. We firs show ha 203

218 Pars (a)-(b) hold for = 1. Since Par (a) auomaically holds for = 1, we only need o check Par (b). Because v 0 (I 0, N 0 ) = ci 0, Ψ 1 (x, y) = E[r n (y + θξ + ϵ )]. Thus, aking he ransformaion x 1 = 1 + V 1 p 1 + γ(n 1 ), J 1 (x 1, p 1, N 1 ) =R 1 (p 1, N 1 ) + βx 1 + Λ(x 1 V 1 + p 1 γ(n 1 )) + E[r n (θ( V 1 p 1 + γ(n 1 )) + ηn 1 )] =(p 1 b αc)( V 1 p 1 + γ(n 1 )) + β( 1 + V 1 p 1 + γ(n 1 )) + β 1 + Λ( 1 ) + E[r n (θ( V 1 p 1 + γ(n 1 )) + ηn 1 )] =(p 1 c)( V 1 p 1 + γ(n 1 )) + β 1 + Λ( 1 ) + E[r n (θ( V 1 p 1 + γ(n 1 )) + ηn 1 )]. Therefore, he opimal join price and safey-sock (p 1 (N 1 ), 1 (N 1 )) can be deermined by p 1 (N 1 ) = argmax p1 [p, p]{(p 1 c)( V 1 p 1 + γ(n 1 )) + E[r n (θ( V 1 p 1 + γ(n 1 )) + ηn 1 )]}, and 1 (N 1 ) = = argmax {β + Λ( )}, respecively. Hence, x 1 (N 1 ) = 1 (N 1 ) + y 1 (N 1 ) = + y 1 (N 1 ). We have hus shown Pars (a)-(b) for = 1. We now show ha if Pars (a)-(b) hold for period 1, hey also hold for period. Firs, aking he ransformaion x = + V p + γ(n ), J (x, p, N ) =R (p, N ) + βx + Λ(x V 1 + p γ(n )) + Ψ (x V + p γ(n ), θ( V p + γ(n )) + ηn ) =R (p, N ) + β( + V p + γ(n )) + Λ( ) + Ψ (, θ( V p + γ(n )) + ηn ) =(p c)( V p + γ(n )) + β + Λ( ) + Ψ (, θ( V p + γ(n )) + ηn ). Le (p (N ), (N )) be he opimal price and safey-sock wih nework size N. We now show ha (N ). If, o he conrary, (N ) >, Lemma 18 yields ha [(p (N ) c)( V p (N ) + γ(n )) + β (N ) + Λ( (N )) +Ψ ( (N ), θ( V p (N ) + γ(n )) + ηn )] [β + Λ( )], i.e., β + Λ ( (N )) + x Ψ ( (N ), θ( V p (N ) + γ(n )) + ηn ) β + Λ ( ). The concaviy of Λ( ) implies ha Λ ( (N )) Λ ( ). Moreover, since Ψ (x, y) is decreasing in x, x Ψ ( (N ), θ( V p (N )+γ(n ))+ηn ) 0. Therefore, Λ ( (N )) = Λ ( ) and x Ψ ( (N ), θ( V p (N ) + γ(n )) + ηn ) = 0. Thus, by he firs-order condiion wih respec o, (p (N ), ) is also he opimal price and safey-sock level, which is sricly lexicographically smaller han (p (N ), (N )). This conradics he assumpion ha we selec he lexicographically smalles opimizer in each period. Hence, (N ) for all N

219 We now show ha P[x (N ) D (p (N ), N ) x 1 (N 1 )] = 1 for all N and N 1. Noe ha, wih probabiliy 1, x (N ) D (p (N ), N ) = (N ) ξ ξ = x 1 (N 1 ) y 1 (N 1 ) ξ = x 1 (N 1 ) D 1 (p 1 (N 1 ), N 1 ), where he inequaliy follows from (N ), second equaliy from he hypohesis inducion ha x 1 (N 1 ) = y 1 (N 1 ) + d for all N 1 0, and he las equaliy from ξ 1 = ξ and he ideniy ha D 1 (p 1 (N 1 ), N 1 ) = y 1 (N 1 )+ξ 1. Because D 1 (p 1 (N 1 ), N 1 ) 0 wih probabiliy 1, x (N ) D (p (N ), N ) x 1 (N 1 ) D 1 (p 1 (N 1 ), N 1 ) x 1 (N 1 ) wih probabiliy 1, i.e., par (a) follows for period. Nex, we show ha (N ) =. Observe ha P[x (N ) D (p (N ), N ) x 1 (N 1 )] = 1 implies ha x Ψ ( (N ), θ( V p (N )+γ(n ))+ηn ) = 0 and, hus, x J ( +y (N ), p (N ), N ) = 0. Since J (,, N ) is joinly concave, he firs-order condiion wih respec o x yields ha (N ) = for all N 0. This complees he inducion and, hus, he proof of Lemma 3. Q.E.D. Proof of Lemma 4:. By pars (a) and (b) of Theorem 2.4.1, if I x (N ), v (I, N ) = ci + π (N ), where π (N ) := max{j (x, p, N ) : x 0, p [p, p]}. By Lemma 2, π ( ) is concavely increasing and coninuously differeniable in N. By Lemma 3(a), for each N 0, x (N ) D (p (N ), N ) x 1 (N 1 ) wih probabiliy 1. Since v 1 (I 1, N 1 ) = ci 1 + π 1 (N 1 ) for all I 1 x 1 (N 1 ), v 1 (x (N ) D (p (N ), N ), θd (p (N ), N ) + ηn ) =c[x (N ) D (p (N ), N )] + π 1 (θd (p (N ), N ) + ηn ) wih probabiliy 1. Taking expecaion wih respec o ξ and ϵ, we have, for all N 0 and x x (N ), Ψ (x V + p (N ) γ(n ), θ( V p (N ) + γ(n )) + ηn ) =E[r n (θ( V p (N ) + γ(n )) + ηn + θξ + ϵ )] + αe[π 1 (θ( V p (N ) + γ(n )) + ηn + θξ + ϵ )]. Therefore, for all N 0, if x + y (N ), J (x, p, N ) = R (p, N ) + βx θq + Λ(x V + p (N ) γ(n )) + G (θ( V p (N ) + γ(n )) + ηn ), where G (y) := E[r n (y + θξ + ϵ )] + αe[v 1 (y + θξ + ϵ )]. Finally, i remains o show ha (x (N ), p (N )) maximizes he righ-hand side of (2.8). Noe ha Theorem 2.4.1(c) and Lemma 3(a) imply ha, if I x (N ), wih probabiliy 1, I τ x τ (N τ ) for all τ =, 1,, 1 and, hence, {(x τ (N τ ), p τ (N τ ))} τ=, 1,,1 is he opimal policy in periods, 1,,

220 In paricular, (x (N ), p (N )) maximizes he oal expeced discouned profi given ha he firm adops {(x τ (N τ ), p τ (N τ ))} for τ = 1,, 1. I s sraighforward o check ha if he firm adops he policy {(x τ (N τ ), p τ (N τ ))} for τ = 1,, 1; and orders-up-o x and charges p in period, he oal expeced discouned profi of he firm in period is given by he righ-hand side of (2.8). Since (x (N ), p (N )) maximizes he oal expeced discouned profi in period, i also maximizes he righ-hand side of (2.8) for each. This proves Lemma 4. Q.E.D. Proof of Theorem 2.4.2: By Theorem 2.4.1(c) and Lemma 3(a), if I T x T (N ), I x (N ) for all = T, T 1,, 1 wih probabiliy 1. Therefore, by Theorem 2.4.1(a), (x (I, N ), p (I, N )) = (x (N ), p (N )) wih probabiliy 1 if I T x T (N T ). The characerizaion of (x (N ), p (N )) follows immediaely from Lemma 4 and is discussions. Q.E.D. The following lemma is used hroughou he res of our proofs. Lemma 19 For each period and any nework size N 0, he following saemens hold. (a) J (x (N ), p (N ), N ) = L (p (N ), N ) + β + Λ( ), where L (p, N ) := (p c)( V p + γ(n )) + G (θ( V p + γ(n )) + ηn ), and is he opimal safey sock characerized in Lemma 3(b). Hence, p (N ) = argmax p [p, p]l (p, N ). (b) J (x (N ), p (N ), N ) = K (y (N ), N )+β +Λ( ), where K (y, N ) := ( V +γ(n ) y c)y + G (θy + ηn ). Hence, y (N ) = argmax y [y (N ),ȳ (N )]K (y, N ), where y (N ) = V + γ(n ) p and ȳ (N ) = V + γ(n ) p. (c) Le m (N ) := θy (N ) + ηn be he opimal expeced nework size in period 1, given he curren nework size N. We have J (x (N ), p (N ), N ) = M (m (N ), N ) + β + Λ( ), where M (m, N ) := ( V + γ(n ) m ηn θ c) (m ηn ) θ + G (m ). Hence, m (N ) = argmax m [m (N ), m (N )]M (m, N ), where m (N ) = θy (N ) + ηn and m (N ) = θȳ (N ) + ηn. Proof of Lemma 19: Par (a). By Lemma 3(b), x (N ) y (N ) = for all N 0. By Lemma 4, for all N, J (x (N ), p (N ), N ) =R (p (N ), N ) + βx (N ) + Λ(x (N ) V + p (N ) γ(n )) + G (θ( V p (N ) + γ(n )) + ηn ). Therefore, J (x (N ), p (N ), N ) =R (p (N ), N ) + β( + V p (N ) + γ(n )) + Λ( ) + G (θ( V p (N ) + γ(n )) + ηn ) =(p (N ) c)( V p (N ) + γ(n )) + G (θ( V p (N ) + γ(n )) + ηn ) + β + Λ( ) =L (p (N ), N ) + β + Λ( ), (A.3) 206

221 where L (p, N ) := (p c)( V p + γ(n )) + G (θ( V p + γ(n )) + ηn ). Since (x (N ), p (N )) maximizes J (,, N ), p (N ) = argmax p [p, p]l (p, N ). This proves par (a). Par (b). Since y (N ) = V p (N ) + γ(n ), p (N ) = V y (N ) + γ(n ). Plug his ino (A.3), we have J (x (N ), p (N ), N ) =(p (N ) c)( V p (N ) + γ(n )) + G (θ( V p (N ) + γ(n )) + ηn ) + β + Λ( ) =( V y (N ) + γ(n ) c)y (N ) + G (θy (N ) + ηn ) + β + Λ( ) =K (y (N ), N ) + β + Λ( ), where K (y, N ) := ( V + γ(n ) y c)y + G (θy + ηn ). Since (x (N ), p (N )) maximizes J (,, N ), y (N ) = argmax y [y (N ),ȳ (N )]K (y, N ). The expressions of y (N ) and ȳ (N ) follow direcly from he ideniy y (N ) = V p (N ) + γ(n ) and p [p, p]. This proves par (b). Par (c). Observe ha m (N ) = θy (N ) + ηn and θ > 0 imply ha p (N ) = V y (N ) + γ(n ) = V + γ(n ) m (N ) ηn θ. Plug his ino (A.3), we have J (x (N ), p (N ), N ) =(p (N ) c)( V p (N ) + γ(n )) + G (θ( V p (N ) + γ(n )) + ηn ) + β + Λ( ) =( V + γ(n ) m (N ) ηn θ + Λ( ) =M (m (N ), N ) + β + Λ( ), c) m (N ) ηn θ + G (m (N )) + β where M (m, N ) := ( V +γ(n ) m ηn θ c) (m ηn ) θ +G (m ). Since (x (N ), p (N )) maximizes J (,, N ), m (N ) = argmax m [m (N ), m (N )]M (m, N ). The expressions of m (N ) and m (N ) follow direcly from he ideniy m (N ) = θy (N ) + ηn and ha y [y (N ), ȳ (N )]. This esablishes par (c). Q.E.D. Proof of Theorem 2.4.3: Par (a). We firs show p ( ˆN ) p (N ). By Lemma 19(a) p (N ) = argmax p L (p, N ) and p ( ˆN ) = argmax p L (p, ˆN ). Hence, i suffices o show ha L (, ) is supermodular in (p, N ). Since p L (p, N ) = V + γ(n ) 2p + c θg (θ( V p + γ(n )) + ηn ), Since G ( ) is concave, p L (p, N ) is increasing in N. Hence, L (, ) is supermodular in (p, N ) and, hus, p ( ˆN ) p (N ) for all ˆN > N (See [165]). Par (b). We now show ha E[N 1 ˆN ] = m ( ˆN ) = θy ( ˆN ) + η ˆN E[N 1 N ] = m (N ) = θy (N ) + ηn. By Lemma 19(c), m ( ˆN ) = argmax m M (m ( ˆN ), ˆN ) and m (N ) = argmax m M (m (N ), N ). To show ha m ( ˆN ) m (N ), i suffices o prove ha M (, ) is supermodular in (m, N ) and he feasible se {(m, N ) : m [m(n ), m (N )]} is a laice. Direc compuaion yields ha m M (m, N ) = 1 θ ( V + γ(n ) m ηn θ 207 c) m ηn θ 2 + G (m ).

222 Since θ > 0, m M (m, N ) is increasing in N. Thus, M (, ) is supermodular in (m, N ). Since m (N ) and m (N ) are coninuously increasing in N, he feasible se {(m, N ) : m [m(n ), m (N )]} is a laice. Hence, m ( ˆN ) m (N ) for all ˆN > N. Par (c). Since γ( ˆN ) = γ(n ), p ( ˆN ) p (N ) implies ha y ( ˆN ) = V p ( ˆN ) + γ( ˆN ) V p (N ) + γ(n ) = y (N ). Moreover, by Lemma 3(b), x ( ˆN ) = + y ( ˆN ) + y (N ) = x (N ). Par (d). Since η = 0 and m ( ˆN ) m (N ), y ( ˆN ) = m( ˆN ) θ x ( ˆN ) = + y ( ˆN ) + y (N ) = x (N ). Q.E.D. m(n) θ = y (N ). By Lemma 3(b), Proof of Theorem 2.4.4: Par (a). We show par (a) by backward inducion. More specifically, we show ha if η = 0 and v 1 (, ) is supermodular in (I 1, N 1 ), v (, ) is supermodular in (I, N ). Since v 0 (I 0, N 0 ) = ci 0, he iniial condiion is saisfied. Since supermodulariy is preserved under expecaion, Ψ (x, y) = E{r n (y + θξ + ϵ ) + α[v 1 (x ξ, y + θξ + ϵ ) cx]} is supermodular in (x, y). Le y = V p + γ(n ). Observe ha J (x, p, N ) =R (p, N ) + βx + Λ(x V + p γ(n )) + Ψ (x V + p γ(n ), θ( V p + γ(n ))) =( V + γ(n ) y αc b)y + βx + Λ(x y ) + Ψ (x y, θy ). Hence, v (I, N ) = ci + max (x,y ) F (I,N){( V + γ(n ) y αc b)y + βx + Λ(x y ) + Ψ (x y, θy )}, where F (I, N ) := {(x, y ) : x I, y [ V + γ(n ) p, V + γ(n ) p]}. Because γ( ) is increasing in N, Λ( ) is concave, and Ψ (, ) is concave and supermodular, ( V + γ(n ) y αc b)y + βx + Λ(x y ) + Ψ (x y, θy ) is supermodular in (x, y, N ). Moreover, i s sraighforward o verify ha he feasible se {(x, y, I, N ) : N 0, (x, y ) F (I )} is a laice in R 4. Therefore, v (I, N ) is supermodular in (I, N ). This complees he inducion and, hus, he proof of par (a). Par (b). The coninuiy resuls in pars (b)-(e) all follow from he join concaviy and coninuous differeniabiliy of J (,, ) in (x, p, N ). Since x (I, N ) = max{i, x (N )}, x (I, N ) is increasing in I. Moreover, because he objecive funcion ( V +γ(n ) y αc b)y +βx +Λ(x y )+Ψ (x y, θy ) is supermodular in (x, y, N ), x (I, N ) is increasing in N as well. This proves par (b). Par (c). If I x (N ), p (I, N ) = p (N ), which is independen of I. If I > x (N ), x (I, N ) = I and, hus, J (x (I, N ), p, N ) = R (p, N )+βi +Λ(I V +p γ(n ))+Ψ (I V +p γ(n ), θ( V p +γ(n ))). (A.4) Since Λ( ) is concave and Ψ (, ) is concave and supermodular, J (x (I, N ), p, N ) is submodular in (I, p ). Hence, p (I, N ) is decreasing in I for all (I, N ). By Theorem 2.4.3(d), if I x (N ), p (I, N ) = p (N ) is increasing in N. If I > x (N ), we observe from (A.4) ha J (x (I, N ), p, N ) is supermodular in (p, N ). Hence, p (I, N ) is increasing in N for all (I, N ). This proves par (c). Par (d). If I x (N ), y (I, N ) = y (N ), which is independen of I. If I > x (N ), x (I, N ) = I and, hus, J (x (I, N ), p, N ) = ( V + γ(n ) y αc b)y + βi + Λ(I y ) + Ψ (I y, θy ). 208

223 Since Λ( ) is concave and Ψ (, ) is concave and supermodular, J (x (I, N ), p, N ) is supermodular in (I, y ) and is domain is a sublaice of R 2. Hence, y (I, N ) is increasing in I for all (I, N ). By Theorem 2.4.3(d), if I x (N ), y (I, N ) = y (N ) is increasing in N. If I > x (N ), x (I, N ) = I and, hus, J (x (I, N ), p, N ) = ( V + γ(n ) y αc b)y + βi + Λ(I y ) + Ψ (I y, θy ). The supermodulariy of J (x (I, N ), p, N ) in (y, N ) follows direcly from ha γ( ) is increasing in N. Moreover, he feasible se {(y, N ) : y [ V + γ(n ) p, V + γ(n ) p]} is clearly a sublaice of R 2. Therefore, y (I, N ) is increasing in N for all (I, N ). This proves par (d). Par (e). If I x (N ), by Theorem 2.4.1(c), (I, N ) = is independen of I and N. If I > x (N ), since I = y, J (x (I, N ), p, N ) = ( V + γ(n ) + I αc b)(i ) + βi + Λ( ) + Ψ (, θ(i )). Since Ψ (, ) is concave and supermodular, J (x (I, N ), p, N ) is supermodular in (I, ). Moreover, he feasible se {(I, ) : [I V γ(n ) + p, I V γ(n ) + p]} is clearly a sublaice of R 2. Hence, (I, N ) is increasing in I for all (I, N ). Moreover, since (I, N ) = I y (I, N ), by par (d), (I, N ) is decreasing in N. This proves par (e). Q.E.D. Proof of Theorem 2.4.5: Par (a). Since γ( ) γ 0 and r n(n) r, he opimal policy of he firm (p ( ), x ( )) is independen of he curren nework size N. Hence, ˆπ (N ) π (N ) 0 for all and N 0. We now show ha ŷ (N ) y (N ) for all N 0. Noe ha ˆπ 1(N 1 ) π 1(N 1 ) for all N 1 0 implies ha Ĝ (y) = E{r n(y + θξ + ϵ ) + αˆπ 1(y + θξ + ϵ )} E{r n(y + θξ + ϵ ) + απ 1(y + θξ + ϵ )} = G (y), for all y. By Lemma 3(b), ˆx (N ) = ŷ (N ) +, x (N ) = y (N ) +, ŷ (N ) = V ˆp (N ) + ˆγ(N ), and y (N ) = V p (N ) + γ(n ). By Lemma 19(b), we have Ĵ(ˆx (N ), ˆp (N ), N ) = ˆK (ŷ (N ), N ) + β + Λ( ) and J (x (N ), p (N ), N ) = K (y (N ), N ) + β + Λ( ). We now show ŷ (N ) y (N ). Assume, o he conrary, ha ŷ (N ) < y (N ). Lemma 18 yields ha y ˆK (ŷ (N ), N ) y K (y (N ), N ), i.e., 2ŷ (N ) + ˆγ(N ) + θĝ (θŷ (N ) + ηn ) 2y (N ) + γ(n ) + θg (θy (N ) + ηn ). Because Ĝ ( ) G ( ) and ŷ (N ) < y (N ), he concaviy of Ĝ( ) and G ( ) implies ha Ĝ (θŷ (N ) + ηn ) G (θy (N ) + ηn ). Since ˆγ(N ) γ(n ), we have 2ŷ (N ) 2y (N ), which conradics he assumpion ha ŷ (N ) < y (N ). Hence, ŷ (N ) y (N ). This complees he proof of par (a). Par (b). By Lemma 3(b) and par (a), ˆx (N ) = ŷ (N ) + y (N ) + = x (N ) for all N 0. This proves par (b). Par (c). We firs show ha ˆp (0) p (0). Observe ha ˆp (0) = V + ˆγ(0) ŷ (0) and p (0) = V + γ(0) y (0). By par (a), ŷ (0) y (0). Moreover, since ˆγ(0) = γ(0) = γ 0, ˆp (0) p (0). Since γ( ) γ 0, p (N ) p (0). Moreover, Theorem 2.4.3(a) implies ha ˆp (N ) is increasing in N. The join concaviy of Ĵ(,, ) implies ha ˆp (N ) is coninuously increasing in N. Thus, le N be he smalles N such ha ˆp (N ) p (N ) = p (0). The monooniciy of ˆp ( ) hen suggess ha ˆp (N ) p (N ) if 209

224 N N, and ˆp (N ) p (N ) if N N. This proves par (c). Q.E.D. Proof of Theorem 2.4.6: We show Theorem by backward inducion. More specifically, we show ha if V = V 1 and π 1(N) π 2(N) for all N 0, (i) y (N) y 1 (N) for all N 0, (ii) p (N) p 1 (N) for all N 0, (iii) x (N) x 1 (N) for all N 0, and (iv) π (N) π 1(N) for all N 0. Since π 1(N) π 0(N) 0 for all N, he iniial condiion is saisfied. Noe ha π 1(N) π 2(N) for all N 0 implies ha G (y) = E{r n(y + θξ + ϵ ) + απ 1(y + θξ + ϵ )} E{r n(y + θξ + ϵ ) + απ 2(y + θξ + ϵ )} = G 1(y), for all y. By Lemma 19(b), J (x (N), p (N), N) = K (y (N), N) + β + Λ( ) and J 1 (x 1 (N), p 1 (N), N) = K 1 (y 1 (N), N) + β + Λ( ). We firs prove ha y (N) y 1 (N) for all N. Assume, o he conrary, ha y (N) < y 1 (N) for some N. Lemma 18 implies ha y K (y (N), N) y 1 K 1 (y 1 (N), N), i.e., 2y (N) + γ(n) + θg (θy (N) + ηn) 2y 1 (N) + γ(n) + θg 1(θy 1 (N) + ηn). Because G ( ) G 1( ) for all y and y (N) < y 1 (N), he concaviy of G ( ) and G 1 ( ) implies ha G (θy (N) + ηn) G 1(θy 1 (N) + ηn). Thus, we have 2y (N) 2y 1 (N), which conradics he assumpion ha y (N) < y 1 (N). Hence, y (N) y 1 (N) for all N 0. By Theorem 2.4.1(c), i follows immediaely ha x (N) = y (N) + y 1 (N) + = x 1 (N) for all N 0. Nex, we show ha p (N) p 1 (N) for all N 0. By Lemma 19(a), J (x (N), p (N), N) = L (p (N), N) + β + Λ( ) and J 1 (x 1 (N), p 1 (N), N) = L 1 (p 1 (N), N) + β + Λ( ). Assume, o he conrary, ha p (N) > p 1 (N) for some N. Lemma 18 implies ha p L (p (N), N) p 1 L 1 (p 1 (N), N), i.e., 2p (N) + V + c + γ(n) θg (θ( V p (N) + γ(n)) + ηn) 2p 1 (N) + V 1 + c + γ(n) θg 1(θ( V 1 p 1 (N) + γ(n)) + ηn). Because G ( ) G 1( ) for all y and p (N) > p 1 (N), he concaviy of G ( ) and G 1 ( ) implies ha G (θ( V p (N) + γ(n)) + ηn) G 1(θ( V 1 p 1 (N) + γ(n)) + ηn). Since V = V 1, we have 2p (N) 2p 1 (N), which conradics he assumpion ha p (N) > p 1 (N). Hence, p (N) p 1 (N) for all N 0. Finally, o complee he inducion, we show ha π (N) π 1(N) for all N. By he envelope heorem, π (N) = (p (N) c)γ (N) + (η + θγ (N))G (θ( V p (N) + γ(n)) + ηn), and π 1(N) = (p 1 (N) c)γ (N) + (η + θγ (N))G 1(θ( V 1 p 1 (N) + γ(n)) + ηn), If p (N) = p 1 (N), π (N) π 1(N) follows immediaely from γ (N) 0 and G ( ) G 1( ). If p (N) < p 1 (N), Lemma 18 yields ha p L (p (N), N) p 1 L 1 (p 1 (N), N), i.e., 2p (N) + V + c + γ(n) θg (θ( V p (N) + γ(n)) + ηn) 2p 1 (N) + V 1 + c + γ(n) θg 1(θ( V 1 p 1 (N) + γ(n)) + ηn). 210

225 Hence, by V = V 1, p (N) + θg (θ( V p (N) + γ(n)) + ηn) p 1 (N) + θg 1(θ( V 1 p 1 (N) + γ(n)) + ηn) + (p 1 (N) p (N)). (A.5) Since θ > 0, p (N) < p 1 (N) implies ha G (θ( V p (N) + γ(n)) + ηn) G 1(θ( V 1 p 1 (N) + γ(n)) + ηn). Therefore, π (N) π 1(N) =[(p (N) p 1 (N)) + θ(g (θ( V p (N) + γ(n)) + ηn) G 1(θ( V 1 p 1 (N) + γ(n)) + ηn))]γ (N) + η(g (θ( V p (N) + γ(n)) + ηn) G 1(θ( V 1 p 1 (N) + γ(n)) + ηn)) 0. Hence, π (N) π 1(N) for all N. This complees he inducion and, hus, he proof of Theorem Q.E.D. Proof of Theorem 2.4.7: We show Theorem by backward inducion. More specifically, we show ha if ˆπ 1( ) π 1( ) for all N 1 0 and ˆr n( ) r n( ) for all N 0, (i) ŷ (N ) y (N ) for all N 0; (ii) ˆx (N ) x (N ) for all N 0; (iii) ˆp (N ) p (N ) for all N 0; and (iv) ˆπ (N ) π (N ) for all N 0. Since ˆπ 0( ) = π 0( ) 0, he iniial condiion is saisfied. Noe ha ˆπ 1(N 1 ) π 1(N 1 ) for all N 1 0 and ˆr n( ) r n( ) for all N 0 imply ha Ĝ (y) = E{ˆr n(y + θξ + ϵ ) + αˆπ 1(y + θξ + ϵ )} E{r n(y + θξ + ϵ ) + απ 1(y + θξ + ϵ )} = G (y), for all y. By Lemma 19(a), Ĵ (ˆx (N ), ˆp (N ), N ) = ˆK (ŷ (N ), N ) + β + Λ( ) and J (x (N ), p (N ), N ) = K (y, N ) + β + Λ( ). We firs show ha ŷ (N ) y (N ). Assume, o he conrary, ha ŷ (N ) < y (N ) for some N. Lemma 18 yields ha y ˆK (ŷ (N ), N ) y K (y (N ), N ), i.e., 2ŷ (N ) + γ(n ) + θĝ (θŷ (N ) + ηn ) 2y (N ) + γ(n ) + θg (θy (N ) + ηn ). Because Ĝ ( ) G ( ) and ŷ (N ) < y (N ), he concaviy of Ĝ( ) and G ( ) implies ha Ĝ (θŷ (N ) + ηn ) G (θy (N ) + ηn ). Hence, we have 2ŷ (N ) 2y (N ), which conradics he assumpion ha ŷ (N ) < y (N ). Thus, ŷ (N ) y (N ) and, hence, ˆx (N ) = ŷ (N ) + y (N ) + = x (N ). Nex, we show ha ˆp (N ) p (N ). By Lemma 19(a), Ĵ (ˆx (N ), ˆp (N ), N ) = ˆL (ˆp (N ), N ) + β + Λ( ) and J (x (N ), p (N ), N ) = L (p (N ), N ) + β + Λ( ) Assume, o he conrary, ha ˆp (N ) > p (N ) for some N. Lemma 18 implies ha p ˆL (ˆp (N ), N ) p L (p (N ), N ), i.e., 2ˆp (N ) + V + c + γ(n ) θĝ (θ( V ˆp (N ) + γ(n )) + ηn ) 2p (N ) + V + c + γ(n ) θg (θ( V p (N ) + γ(n )) + ηn ). Because Ĝ ( ) G ( ) for all y and ˆp (N ) > p (N ), he concaviy of Ĝ ( ) and G ( ) implies ha Ĝ (θ( V ˆp (N ) + γ(n )) + ηn ) G (θ( V p (N ) + γ(n )) + ηn ). We have 2ˆp (N ) 2p (N ), which conradics he assumpion ha ˆp (N ) > p (N ). Hence, ˆp (N ) p (N ) for all N

226 Finally, o complee he inducion, we show ha ˆπ (N ) π (N ) for all N. heorem, By he envelope ˆπ (N ) = (ˆp (N ) c)γ (N ) + (η + θγ (N ))Ĝ (θ( V ˆp (N ) + γ(n )) + ηn ), and π (N ) = (p (N ) c)γ (N ) + (η + θγ (N ))G (θ( V p (N ) + γ(n )) + ηn ), If ˆp (N ) = p (N ), ˆπ (N ) π (N ) follows immediaely from γ (N) 0 and Ĝ ( ) G ( ) for all y. If ˆp (N ) < p (N ), Lemma 18 yields ha p ˆL (ˆp (N ), N ) p L (p (N ), N ), i.e., 2ˆp (N ) + V + c + γ(n ) θĝ (θ( V ˆp (N ) + γ(n )) + ηn ) 2p (N ) + V + c + γ(n ) θg (θ( V p (N ) + γ(n )) + ηn ). Hence, ˆp (N ) + θĝ (θ( V ˆp (N ) + γ(n )) + ηn ) p (N ) + θg (θ( V p (N ) + γ(n )) + ηn ) + (p (N ) ˆp (N )). (A.6) Since θ > 0, ˆp (N ) < p (N ) implies ha Ĝ (θ( V ˆp (N )+γ(n ))+ηn ) G (θ( V p (N )+γ(n ))+ ηn ). Therefore, by (A.6), ˆπ (N ) π (N ) =[(ˆp (N ) p (N )) + θ(ĝ (θ( V ˆp (N ) + γ(n )) + ηn ) G (θ( V p (N ) + γ(n )) + ηn ))]γ (N ) + η(ĝ (θ( V ˆp (N ) + γ(n )) + ηn ) G (θ( V p (N ) + γ(n )) + ηn )) 0. Hence, ˆπ (N ) π (N ) for all N. This complees he inducion and, hus, he proof of Theorem Q.E.D. Proof of Lemma 5: Par (a). The concaviy, differeniabiliy, and monooniciy of π d ( ) and J d (,,, ) follow from he same backward inducion argumen as he proof of Lemma 2. Hence, we omi he proof of par (a) for breviy. Par (b). The opimal value funcion v d (I, N ) saisfies he following recursive scheme: v d (I, N ) = ci + max (x,p s,pi ) F d(i ) J d (x, p s, p i, N ), (A.7) where F d (I ) := {(x, p s, p i ) [I, + ) [p, p] [p, p] : p s p i } denoes he se of feasible decisions wih price discriminaion and J d (x, p s, p i, N ) = θr (p s, N ) + (1 θ)r (p i, N ) + βx + Λ(x V + θp s + (1 θ)p i γ(n )) +Ψ d (x V + θp s + (1 θ)p i γ(n ), θ( V p s + γ(n )) + ηn ), (A.8) wih Ψ d (x, y) := E{α[r n (y + θξ + ϵ ) + v 1(x d ξ, y + θξ + ϵ ) cx]}. 212

227 The derivaion of (A.8) is as follows: J d (x, p s, p i, N ) := ci + E{p s D s (p s, N ) + p i D(p i i, N ) c(x I ) h(x D s (p s, N ) D(p i i, N )) + b(x D s (p s, N ) D(p i i, N )) + r n (D s (p s, N ) + ηn + ϵ ) +αv 1(x d D s (p s, N ) D(p i i, N ), D s (p s, N ) + ηn + ϵ ) N }, = θ(p s αc b)( V p s + γ(n )) + (1 θ)(p i αc b)( V p i + γ(n )) +(b (1 α)c)x +E{r n (D s (p s, N ) + ηn + ϵ ) (h + b)(x V + θp s + (1 θ)p i γ(n ) ξ ) + +α[v 1(x d V + θp s + (1 θ)p i γ(n ) ξ, θ( V p s + γ(n ) + ξ ) + ηn + ϵ ) c(x V + θp s + (1 θ)p i γ(n ) ξ )] N } = θr (p s, N ) + (1 θ)r (p i, N ) + βx + Λ(x V + θp s + (1 θ)p i γ(n )) +Ψ d (x V + θp s + (1 θ)p i γ(n ), θ( V p s + γ(n )) + ηn ). We use (ˆx d (N ), ˆp s (N ), ˆp i (N )) o denoe he unconsrained maximizer of (A.8). The same argumen as he proof of Lemma 2 yields ha J d (,,, ) is joinly concave in (x, p s, p i, N ). Hence, if I ˆx d (N ), (x d (I, N ), p s (I, N ), p i (I, N )) = (ˆx d (N ), ˆp s (N ), ˆp i (N )); oherwise (I > ˆx (N )), x d (I, N ) = I. The same argumen as he proof of Lemma 3 implies ha P[ˆx d (N ) D s (ˆp s (N ), N ) D i (ˆp i (N ), N ) ˆx d 1(N 1 )] = 1. Hence, he same argumen as he proof of Lemma 4 yields ha (ˆx d (N ), ˆp s (N ), ˆp i (N )) = (x d (N ), p s (N ), p i (N )) for all N 0. Thus, if I T x d T (N T ), I x d (N ) for all wih probabiliy 1. Hence, par (b) follows. Q.E.D. The following lemma is a counerpar of Lemma 19 in he model wih price discriminaion. Lemma 20 For each period and any nework size N 0, he following saemens hold. (a) x d (N ) = y s (N ) + y i (N ) +, where is he opimal safey sock characerized in Lemma 3(b). (b) J d (x d (N ), p s (N ), p i (N ), N ) = L s (p s (N ), N ) + (1 θ)r (p i (N ), N ) + β + Λ( ), where L s (p, N ) := θ(p c)( V p +γ(n ))+G d (θ( V p +γ(n ))+ηn ), and R (p, N ) := (p c)( V p + γ(n )). Hence, p s (N ) = argmax p s [p, p]l s (p, N ) and p i (N ) = argmax p i [p, p]r (p i, N ). (c) J d (x d (N ), p s (N ), p i (N ), N ) = K s (y s (N ), N ) + (1 θ)r (p i (N ), N ) + β + Λ( ), where K s (y, N ) := ( V +γ(n ) y θ c)y +G d (y +ηn ). Hence, y s (N )argmax [y s [y s (N ),ȳ s (N )]K s (y, N ), where y s (N ) := θ( V + γ(n ) p) and ȳ s (N ) := θ( V + γ(n ) p). (d) Le m s (N ) := y s (N ) + ηn be he opimal expeced nework size in period 1, given he curren nework size N. We have J d (x d (N ), p s (N ), p i (N ), N ) = M s (m s (N ), N )+(1 θ)r (p i (N ), N )+ β + Λ( ), where M s (m, N ) := ( V + γ(n ) m ηn θ c)(m ηn ) + G d (m ). Hence, m s (N )argmax m s [m s (N ), m s (N )]M s (m s, N ), where m s (N ) := y s (N )+ηn and m s (N ) := ȳ s (N )+ ηn. 213

228 Proof of Lemma 20: Par (a). Par (a) follows from he same argumen as he proof of Lemma 3(b), so we omi is proof for breviy. Par (b). By par (a), x d (N ) y s (N ) y i (N ) = for all N 0. By he Bellman equaion 2.10, for all N, Therefore, J d (x d (N ), p s (N ), p i (N ), N ) =θr (p s (N ), N ) + (1 θ)r (p i (N ), N ) + βx d (N ) + Λ(x d (N ) y s (N ) y i (N )) + G d (θ( V p s (N ) + γ(n )) + ηn ). J d (x d (N ), p s (N ), p i (N ), N ) =θr (p s (N ), N ) + (1 θ)r (p i (N ), N ) + β( + V θp s (N ) (1 θ)p i (N ) + γ(n )) + Λ( ) + G d (θ( V p s (N ) + γ(n )) + ηn ) =θ(p s (N ) c)( V p s (N ) + γ(n )) + (1 θ)r (p i (N ), N ) + G d (θ( V p s (N ) + γ(n )) + ηn ) + β + Λ( ) =L s (p s (N ), N ) + (1 θ)r (p i (N ), N ) + β + Λ( ), (A.9) where L s (p, N ) := θ(p c)( V p + γ(n )) + G d (θ( V p + γ(n )) + ηn ), and R (p, N ) := (p c)( V p +γ(n )). Since (x d (N ), p s (N ), p i (N )) maximizes J d (,,, N ) for all N, p s (N ) = argmax p s [p, p]l s (p s, N ) and p i (N ) = argmax p i [p, p]r (p i, N ). This proves par (b). Par (c). Since y s (N ) = θ( V p s (N ) + γ(n )) and θ > 0, p s (N ) = V ys (N ) θ his ino (A.9), we have + γ(n ). Plug J d (x d (N ), p s (N ), p i (N ), N ) =θ(p s (N ) c)( V p s (N ) + γ(n )) + (1 θ)r (p i (N ), N ) + G d (θ( V p s (N ) + γ(n )) + ηn ) + β + Λ( ) =( V ys (N ) θ + γ(n ) c)y s (N ) + (1 θ)r (p i (N ), N ) + G d (y s (N ) + ηn ) + β + Λ( ) =K s (y s (N ), N ) + (1 θ)r (p i (N ), N ) + β + Λ( ), where K s (y, N ) := ( V y θ +γ(n ) c)y +G d (y +ηn ). Since (x d (N ), p s (N ), p i (N )) maximizes J d (,,, N ) for all N, y s (N ) = argmax y s [y s (N),ȳs (N)] K s (y s, N ). The expressions of y s (N ) and ȳ s (N ) follow immediaely from he ideniy y s proves par (c). = θ( V p s + γ(n )) and ha p s [p, p]. This 214

229 Par (d). Observe ha m s (N ) = y s (N ) + ηn and θ > 0 imply ha p s (N ) = V ys (N ) θ + γ(n ) = V + γ(n ) ms (N) ηn θ. Plug his ino (A.9), we have J d (x d (N ), p s (N ), p i (N ), N ) =θ(p s (N ) c)( V p s (N ) + γ(n )) + (1 θ)r (p i (N ), N ) + G d (θ( V p s (N ) + γ(n )) + ηn ) + β + Λ( ) =( V + γ(n ) ms (N ) ηn θ + β + Λ( ) c)(m s (N ) ηn ) + G d (m s (N )) =M s (m s (N ), N ) + (1 θ)r (p i (N ), N ) + β + Λ( ), where M s (m, N ) := ( V +γ(n ) m ηn θ c)(m ηn )+G d (m ). Since (x d (N ), p s (N ), p i (N )) maximizes J d (,,, N ) for all N, m s (N ) = argmax m s [m s (N ), m s (N )]M s (m s, N ). The expressions of m s (N ) and m s (N ) follow immediaely from he ideniy m s = y s + ηn and ha y s [y s (N ), ȳ s (N )]. This esablishes par (d). Q.E.D. Proof of Theorem 2.5.1: Par (a). Direc compuaion yields ha p L s (p, N ) = θ[ 2p c + V + γ(n ) y G d (θ( V p +γ(n ))+ηn )] and p R (p, N ) = 2p c+ V +γ(n ). Since p i (N ) > p, he firs order condiion wih respec o p implies ha p R (p i (N ), N ) = 0, i.e., 2p i (N ) c+ V +γ(n ) = 0. Hence, p L s (p i (N ), N ) = θg d (θ( V p i (N ) + γ(n )) + ηn ). Since γ ( ) > 0 for all N 0, G d (θ( V p i (N ) + γ(n )) + ηn ) > 0. Moreover, θ > 0 implies ha p L s (p i (N ), N ) < 0. Because L s (, N ) is concave in p and p i (N ) > p, p s (N ) = argmax p [p, p]l s (p, N ) < p i (N ). This proves par (a). Par (b). Assume, o he conrary, ha p (N ) > p i (N ). Lemma 18 yields ha p L (p (N ), N ) p R (p i (N ), N ), i.e., θ[ 2p (N ) c + V + γ(n ) G (θ( V p (N ) + γ(n )) + ηn )] θ[ 2p i (N ) c + V + γ(n )]. G ( ) 0 implies ha p (N ) p i (N ), which conradics he assumpion ha p (N ) > p i (N ). This proves par (b). Par (c). Observe ha, if p i τ ( ) = p s τ ( ) = p τ ( ) for each τ and any N τ 0, π d (N ) = π (N ) for all N 0. Hence, π ( ) is a lower bound for π d ( ). Now assume ha p i (N τ ) > p s (N ). Because p s ( ) and p i ( ) are he lexicographically smalles opimizers, we mus have π d (N ) > π (N ). Oherwise here are wo policies (one wih p s (N ) = p i (N ) and he oher wih p s (N ) < p i (N )) ha are lexicographically differen bu generae he same opimal profi, which conradics ha he opimal policy (x d (N ), p s (N ), p i (N )) is he lexicographically smalles opimizer. On he oher hand, if γ( ) γ 0 and r( ) 0 for all N 0, N π ( ) 0 for all and, hence, p i ( ) = p s ( ) for all and N 0. Moreover, since p ( ) is he opimal pricing policy if he firm charges a single price o all cusomers in each period, he opimal price discriminaion sraegy should be p i ( ) = p s ( ) = p ( ) for each. Hence, π d (N ) = π (N ) for all N 0. This proves par (c). Q.E.D. Proof of Lemma 6: Par (a). Par (a) follows from he same argumen as he proof of Lemma 2, so we omi is proof for breviy. 215

230 Par (b). The opimal value funcion v p (I, N ) saisfies he following recursive scheme: v p (I, N ) = ci + max J p (x, p, n, N ), (A.10) (x,p,n ) F p (I ) where F p (I ) := [I, + ) [p, p] [0, + ) denoes he se of feasible decisions and J p (x, p, n, N ) = R (p, N ) + βx + Λ(x V + p γ(n )) c n (n ) +Ψ p (x V + p γ(n ), θ( V p + γ(n )) + ηn + n ), (A.11) wih Ψ p (x, y) := E{r n (y + θξ + ϵ ) + αv p 1 (x ξ, y + θξ + ϵ ) cx}. The derivaion of (A.11) is given as follows: J p (x, p, n, N ) := ci + E{p D (p, N ) c(x I ) h(x D (p, N )) + b(x D (p, N )) +r n (θd (p, N ) + ηn + n + ϵ ) c n (n ) +αv p 1 (x D (p, N ), θd (p, N ) + ηn + n + ϵ ) N }, = (p αc b)( V p + γ(n )) + (b (1 α)c)x c n (n ) +E{r n (θ( V p + γ(n ) + ξ ) + ηn + n + ϵ ) (h + b)(x V + p γ(n ) ξ ) + +α[v p 1 (x V + p γ(n ) ξ, θ( V p + γ(n ) + ξ ) + ηn + n + ϵ ) c(x V + p γ(n ) ξ )] N } = R (p, N ) + βx + Λ(x V + p γ(n )) c n (n ) +Ψ p (x V + p γ(n ), θ( V p + γ(n )) + ηn + n ). We use (ˆx p (N ), ˆp p (N ), ˆn (N )) as he unconsrained opimizer of (A.11). The same argumen as he proof of Lemma 2 yields ha J d (,,, ) is joinly concave in (x, p, n, N ). Hence, if I ˆx p (N ), (x p (I, N ), p p (I, N ), n (I, N )) = (ˆx p (N ), ˆp p (N ), ˆn (N )); oherwise (I > ˆx p (N )) x p (I, N ) = I. The same argumen as he proof of Lemma 3 implies ha P[ˆx p (N ) D (ˆp p (N ), N ) ˆx p 1 (N 1)] = 1. Hence, he same argumen as he proof of Lemma 4 yields ha (ˆx p (N ), ˆp p (N ), ˆn (N )) = (x p (N ), p p (N ), n (N )) for all N 0. Thus, if I T x p T (N T ), I x p (N ) for all wih probabiliy 1. Hence, par (b) follows. Q.E.D. The following lemma is a counerpar of Lemma 19 in he model wih nework expanding promoion. Lemma 21 For each period and any nework size N 0, he following saemens hold. (a) x p (N ) = y p (N ) +, where is he opimal safey sock characerized in Lemma 3(b). (b) J p (x p (N ), p p (N ), n (N ), N ) = L p (p p (N ), n (N ), N ) + β + Λ( ), where L p (p, n, N ) := (p c)( V p + γ(n )) c n (n ) + G p (θ( V p + γ(n )) + ηn + n ). Hence, (p p (N ), n (N )) = argmax (p,n ) [p, p] [0,+ )L p (p, n, N ). (c) J p (x p (N ), p p (N ), n (N ), N ) = K p (y p (N ), n (N ), N ) + β + Λ( ), where K p (y, n, N ) := ( V + γ(n ) y θ c)y c n (n ) + G d (θy + ηn + n ). 216

231 Hence, (y p (N ), n (N )) = argmax (y,n ) {(y,n ):y [y s y (N ),ȳ (N )]}K p (y, n, N ), where y (N ) and ȳ (N ) are defined in Lemma 19(b). (d) Le m p (N ) := θy p (N ) + ηn be he opimal expeced nework size in period 1, given he curren nework size N. We have J p (x p (N ), p p (N ), n (N ), N ) = M p (m p (N ), n (N ), N ) + β + Λ( ), where M p (m, n, N ) := ( V + γ(n ) m ηn θ c) (m ηn) θ c n (n ) + G p (m + n ). Hence, (m p (N ), n (N )) = argmax (mn ) {(m,n ):m [m (N ), m (N )]}M p (m, n, N ), where m (N ) and m (N ) are defined in Lemma 19(c). Proof of Lemma 21: Par (a). Par (a) follows from he same argumen as he proof of Lemma 3(b), so we omi is proof for breviy. Par (b). By par (a), x p (N ) y p (N ) = for all N 0. By he Bellman equaion 2.11, for all N, Therefore, J p (x p (N ), p p (N ), n (N ), N ) =R (p p (N ), N ) + βx p (N ) c n (n (N )) + Λ(x p (N ) y (N )) + G p (θ( V p p (N ) + γ(n )) + n (N ) + ηn ). J p (x p (N ), p p (N ), n (N ), N ) =R (p p (N ), N ) + βx p (N ) c n (n (N )) + Λ(x p (N ) y (N )) + G p (θ( V p p (N ) + γ(n )) + n (N ) + ηn ) =(p p (N ) c)( V p s (N ) + γ(n )) c n (n (N )) + G p (θ( V p p (N ) + γ(n )) + ηn + n (N )) + β + Λ( ) =L p (p p (N ), n (N ), N ) + β + Λ( ), (A.12) where L p (p, n, N ) := (p c)( V p + γ(n )) c n (n ) + G p (θ( V p + γ(n )) + ηn + n ). Since (x p (N ), p p (N ), n (N )) maximizes J p (,,, N ) for all N, (p p (N ), n (N )) = argmax (p,n ) [p, p] [0,+ )L p (p, n, N ). This proves par (b). Par (c). Since y p (N ) = V p p (N ) + γ(n ), p p (N ) = V y p (N ) + γ(n ). Plug his ino (A.12), we have J p (x p (N ), p p (N ), n (N ), N ) =R (p p (N ), N ) + βx p (N ) c n (n (N )) + Λ(x p (N ) y (N )) + G p (θ( V p p (N ) + γ(n )) + n (N ) + ηn ) =( V y p (N ) + γ(n ) c)y p (N ) c n (n (N )) + G p (θy p (N ) + ηn + n (N )) + β + Λ( ) =K p (y p (N ), n (N ), N ) + β + Λ( ), where K p (y, N ) := ( V y +γ(n ) c)y c n (n )+G d (θy +ηn +n ). Since (x p (N ), p p (N ), n (N )) maximizes J p (,,, N ) for all N, 217

232 (y p (N ), n (N )) = argmax (y,n ) {(y,n ):y [y (N ),ȳ (N )]}K p (y, n, N ). This proves par (c). Par (d). Observe ha m p (N ) = θy p (N ) + ηn and θ > 0 imply ha p p (N ) = V y p (N ) + γ(n ) = V + γ(n ) mp (N) ηn θ. Plug his ino (A.12), we have J p (x p (N ), p p (N ), n (N ), N ) =R (p p (N ), N ) + βx p (N ) c n (n (N )) + Λ(x p (N ) y (N )) + G p (θ( V p p (N ) + γ(n )) + n (N ) + ηn ) =( V + γ(n ) mp (N ) ηn θ c) (mp (N ) ηn ) θ + G p (m p (N ) + n (N )) + β + Λ( ) =M p (m p (N ), n (N ), N ) + β + Λ( ), where M p (m, N ) := ( V + γ(n ) m ηn θ c) (m ηn ) θ c n (n ) + G p (m ). Since (x p (N ), p p (N ), n (N )) maximizes J p (,,, N ) for all N, (m p (N ), n (N )) = argmax (m,n ) {(m,n ):m [m s (N ), m s (N )]}M p (m, n, N ). This esablishes par (d). Q.E.D. Proof of Theorem 2.5.2: Par (a). We firs show ha if (2.12) holds, n (I, N) > 0 for all I. Observe ha, since y Ψ p 1 (x, y) 0, N 1 v p 1 (I 1, N 1 ) (p b αc)γ (N +1 ) γ (N 1 )Λ (w 1), where w 1 = x 1(I 1, N 1 ) y 1(I 1, N 1 ). The firs-order condiion wih respec o x 1 yields ha Λ (w 1) β. Thus, N 1 v 1 (I 1, N 1 ) (p c)γ (N 1 ). (A.13) Therefore, for any x I and p [p, p], n J p (x, p, 0, N) E{r n(n 1 ) + α N 1 v p 1 (x D (p, N), N 1 ) N = N} c n(0) αe{r n(n 1 ) + (p c)γ (N 1 ) N = N} c n(0) (1 ι)[r n( S(N)) + α(p c)γ ( S(N))] c n(0) (A.14) >0, where he second inequaliy follows from (A.13), and he fourh from he assumpion (2.12). The hird inequaliy of (A.14) follows from he following inequaliy: E[r n(n 1 ) + α(p c)γ (N 1 ) N = N] = E N 1 S(N)[r n(n 1 ) + α(p c)γ (N 1 ) N = N] +E N 1< S(N)[r n(n 1 ) + α(p c)γ (N 1 ) N = N] 0 + E N 1< S(N)[r n( S(N)) + α(p c)γ ( S(N))] (1 ι)[r n( S(N)) + α(p c)γ ( S(N))], 218

233 where he firs inequaliy follows from he concaviy of r n ( ) and γ( ), and he second from he definiion of S(N). The inequaliy (A.14) yields ha n (I, N) > 0 for all I. Since γ( ) is coninuously increasing in N, S(N) is coninuously increasing in N. The concaviy of r n ( ) and γ( ) implies ha r n( S(N)) and γ ( S(N)) are coninuously decreasing in N. Therefore, le N (ι) := max{n 0 : (1 ι)[r n( S(N)) + α(p c)γ ( S(N))] > c n(0)}. We have (2.12) holds for all N < N (ι). This complees he proof of par (a). Par (b). Since γ( ) γ 0 and r n ( ) is concavely increasing in N, N 1 v p 1 (I 1, N 1 ) N 1 v p (I 1, 0) ( 1 τ=1 (ατ 1 η τ )r n(0). Thus, if ( 1 τ=0 (αη)τ )r n(0) c n(0), n J p (x, p, n, N ) E{r n(n ) + α N 1 v p 1 (x D (p, N), N 1 + n ) N } c n(0) 1 r n(0) + α( (α τ 1 η τ ))r n(0) c n(0) 1 ( 0. τ=1 (αη) τ )r n(0) c n(0) τ=0 Hence, n (I, N ) = 0 for all (I, N ). This complees he proof of par (b). Q.E.D. Proof of Theorem 2.5.3: Pars (a)-(c). We prove pars (a)-(c) ogeher by backward inducion. More specifically, we show ha if N 1 π p 1 ( ) N 1 π 1 ( ) for all N 1 0, (i) p p (N ) p (N ), (ii) y p (N ) y (N ), (iii) x p (N ) x (N ), and (iv) N π p ( ) N π ( ) for all N 0. Since N0 π p 0 ( ) = N 0 π 0 ( ) 0, he iniial condiion is saisfied. We firs show ha y p (N ) y (N ). Noe ha N 1 π p 1 (N 1) N 1 π 1 (N 1 ) for all N 1 0 implies ha y G p (y) =E{r n(y + θξ + ϵ ) + α N 1 π p 1 (y + θξ + ϵ )} E{r n(y + θξ + ϵ ) + α N 1 π 1 (y + θξ + ϵ )} = y G (y), for all y. By Lemma 19(b) and Lemma 21(c), J p (x p (N ), p p (N ), n (N ), N ) = K p (y p (N ), n (N ), N )+ β + Λ( ) and J (x (N ), p (N ), N ) = K (y (N ), N ) + β + Λ( ). Assume, o he conrary, ha y p (N ) > y (N ) for some N. Lemma 18 yields ha y K p (y p (N ), n (N ), N ) y K (y (N ), N ), i.e., 2y p (N ) + γ(n ) + θ y G p (θy p (N ) + ηn + n (N )) 2y (N ) + γ(n ) + θ y G (θy (N ) + ηn ). Since y p (N ) > y (N ), y G p (θy p (N ) + ηn + n (N )) > y G (θy (N ) + ηn ). (A.15) Because y G ( ) y G p ( ) and y p (N ) > y (N ), he concaviy of G p ( ) and G ( ) implies ha θy p (N )+ ηn + n (N ) < θy (N ) + ηn. However, n (N ) 0 and y p (N ) > y (N ) imply ha θy p (N ) + ηn + n (N ) > θy (N ) + ηn, which forms a conradicion. Thus, y p (N ) y (N ) for all N 0. Hence, 219

234 x p (N ) = y p (N )+ y (N )+ = x (N ) and p p (N ) = V y p (N )+γ(n ) V y (N )+γ(n ) = p (N ). Finally, o complee he inducion, we show ha N π p (N ) N π (N ) for all N 0. By he envelope heorem, N π p (N ) = (p p (N ) c)γ (N ) + (η + θγ (N )) y G p (θ( V p p (N ) + γ(n )) + ηn + n (N )), and N π (N ) = (p (N ) c)γ (N ) + (η + θγ (N )) y G (θ( V p (N ) + γ(n )) + ηn ). If p p (N ) = p (N ), N π p (N ) N π (N ) follows immediaely from γ (N) 0 and y G p ( ) y G ( ) for all y. If p p (N ) > p (N ), Lemma 18 yields ha p L p (p p (N ), n (N ), N ) p L (p (N ), N ), i.e., 2p p (N ) + V + c + γ(n ) θ y G p (θ( V p p (N ) + γ(n )) + ηn + n (N )) 2p (N ) + V + c + γ(n ) θ y G (θ( V p (N ) + γ(n )) + ηn ). Hence, p p (N ) c + θ y G p (θ( V p p (N ) + γ(n )) + ηn + n (N )) p (N ) c + θ y G (θ( V p (N ) + γ(n )) + ηn ) + (p (N ) p p (N )). (A.16) Since θ > 0, p p (N ) > p (N ) implies ha y G p (θ( V p p (N ) + γ(n )) + ηn + n (N )) y G (θ( V p (N ) + γ(n )) + ηn ). Therefore, N π p (N ) N π (N ) =[(p p (N ) p (N )) + θ( y G p (θ( V p p (N ) + γ(n )) + ηn + n (N )) y G (θ( V p (N ) + γ(n )) + ηn ))]γ (N ) + η( y G p (θ( V p p (N ) + γ(n )) + ηn + n (N )) y G (θ( V p (N ) + γ(n )) + ηn )) 0. Hence, N π p (N ) N π (N ) for all N 0. This complees he inducion and, hus, he proof of pars (a)-(c). Par (d). Noe ha π ( ) is he normalized opimal profi wih he Bellman equaion (2.9) and feasible decision se {(x, p, n ) : x 0, p [p, p], n = 0} F p, which is he feasible decision se associaed wih he profi π p ( ). Thus, π p (N ) π (N ) for all and any N 0. If n (N ) > 0, we mus have π p (N ) > π (N ). Oherwise here are wo lexicographically differen policies (one wih n (N ) = 0 and he oher wih n (N ) > 0) ha generae he same opimal normalized profi π (N ). This conradics he assumpion ha he lexicographically smalles policy is seleced. Thus, π p (N ) > π (N ), which esablishes par (d). Q.E.D. A.2 More Condiions on Assumpion Assumpion is essenial o show he analyical resuls in his paper. Thus, we characerize he condiions under which his assumpion is saisfied in he following lemma. 220

235 Lemma 22 The following saemens hold: (a) If R (, ) is joinly concave on is domain, hen we have: (i) For any N such ha γ (N ) = 0, γ (N ) = 0 as well. Thus, here exiss a hreshold N 0, > 0, if N < N, such ha γ (N ) and γ l (N < 0, if N < N, ) = 0, oherwise, = 0, oherwise. (ii) There exiss a consan 0 < M < + such ha, for any N 0, (γ (N )) 2 Mγ l (N ). (b) If here exiss a consan 0 < M < + such ha, for any N 0, (γ (N )) 2 Mγ l (N ), hen we have: (i) There exiss a hreshold δ < + such ha, for any δ δ, wih V δ := V + δ, p δ = p + δ, and p δ = p + δ, R δ (p, N ) := (p b αc)( V δ p [p δ, p δ ] and N 0. p + γ(n )) is joinly concave in (p, N ) for (ii) For any nework exernaliies funcion γ( ), here exiss an hreshold 0 < ς < + such ha, for any ς ς, wih γ ς ( ) := γ ς ( )/ς, R ς (p, N ) := (p b αc)( V p + γ ς (N )) is joinly concave in (p, N ) for p [p, p] and N 0. Par (a) characerizes a simpler necessary condiion for he join concaviy of R (, ). I implies ha R (, ) is joinly concave only if, for all N, γ (N ) is sufficienly big compared wih γ (N ). In oher words, in he region where nework exernaliies exis (i.e., γ (N ) > 0), he curvaure of γ( ) should be sufficienly big. Par (b) shows ha if he necessary condiion characerized by par (a) is saisfied, R (, ) is joinly concave if (i) p is sufficienly big relaive o he expeced demand V p +γ(n ); or (ii) γ ( ) is sufficienly small. Hence, he necessary condiions characerized in par (a) are also sufficien o some exen. The sufficien condiions in par (b) have a clear economic inerpreaion: 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 ) ). This condiion is generally saisfied in pracice, because, compared wih he primary demand leverage (i.e., sales price), nework exernaliies have less impac upon demand in general. Proof of Lemma 22: Par (a-i). If γ (N ) = 0, he lef-hand-side of (2.3) equals 0. Moreover, he righ-hand-side of (2.3) is greaer han or equal o 0 o ensure he join concaviy of R (, ) (see Lemma 1). Hence, he righ-hand-side of (2.3) has o be 0. Thus, γ (N ) = 0 for his case. For he second half of par (a-i), i suffices o show ha if γ (N 0 ) = 0 hen γ (N ) = 0 for all N N 0. Since γ (N ) 0 for all N 0, γ (N ) γ (N 0 ) = 0. On he oher hand, γ (N ) 0 for all N 0. Thus, γ (N ) = 0 for all N N 0. Par (a-ii). By par (a-i), for any N, γ (N ) = 0, γ (N ) = 0 as well. Thus, (γ (N )) 2 Mγ (N ) for any 0 < M < +. We now consider he case γ (N ) < 0. By (2.3), define M := 2(p αc b) > 0, he join concaviy of R (, ) implies ha Mγ (N ) (γ (N )) 2. This esablishes par (a). 221

236 Par (b-i). By Lemma 1, R δ (, ) is joinly concave if and only if 2(p δ αc b)γ (N ) (γ (N )) 2 for all N 0. We define δ := M 2 p + αc + b. Hence, if δ δ, 2(p δ αc b) M. Therefore, 2(p δ αc b)γ (N ) Mγ (N ) (γ (N )) 2 for all N 0, where he las inequaliy follows from he assumpion ha Mγ (N ) (γ (N )) 2 for all N 0. Par (b-i) follows. Par (b-ii). Noe ha N γ ς (N ) = γ (N )/ς and 2 N γ ς (N ) = γ (N )/ς for any ς > 0 and N 0. Thus, by Lemma 1, R ς (, ) is joinly concave if and only if Define ς = 2(p αc b) γ (N ) ς M 2(p αc b) > 0. We have, if ς ς, (γ (N )) 2 ς 2 2ς(p αc b)γ (N ) (γ (N )) 2. 2ς(p αc b)γ (N ) Mγ (N ) (γ (N )) 2, where he las inequaliy follows from he assumpion ha Mγ (N ) (γ (N )) 2 for all N. Hence, R ς (, ) is joinly concave if ς ς. Q.E.D. 222

237 B.1 Proofs of Saemens B. Appendix for Chaper 3 We use o denoe he derivaive operaor of a single variable funcion, and x o denoe he parial derivaive operaor of a muli-variable funcion wih respec o variable x. For any mulivariae coninuously differeniable funcion f(x 1, x 2,, x n ) and x := ( x 1, x 2,, x n ) in f( ) s domain, i, we use xi f( x 1, x 2,, x n ) o denoe xi f(x 1, x 2,, x n ) x= x. The following lemma is used hroughou our proof. Lemma 23 Le G i (z, Z) be a coninuously differeniable funcion in (z, Z), where z [z, z] (z and z migh be infinie) and Z R n i for i = 1, 2. For i = 1, 2, le (z i, Z i ) := argmax (z,z) G i (z, Z) be he opimizers of G i (, ). If z 1 < z 2, we have: z G 1 (z 1, Z 1 ) z G 2 (z 2, Z 2 ). = 0 if z 1 > z, Proof: z 1 < z 2, so z z 1 < z 2 z. Hence, z G 1 (z 1, Z 1 ) 0 if z 1 = z; = 0 if z 2 < z, and z G 2 (z 2, Z 2 ) i.e., z G 1 (z 1, Z 1 ) 0 z G 2 (z 2, Z 2 ). Q.E.D. 0 if z 2 = z, Proof of Theorems and Proposiions : We show Theorem 3.4.1, Proposiion 3.4.1, Proposiion 3.4.2, and Theorem ogeher by backward inducion. More specifically, we show ha, if V i, 1 (I 1, Λ 1 σ sc 1) = w i, 1 I i, 1 + β sc i, 1 Λ i, 1 for all i, (a) Proposiion holds for period, (b) Proposiion holds for period, (c) here exiss a Markov sraegy profile {(γ sc i, (, ), psc i, (, ), xsc i, (, )) : 1 i N} which forms a Nash equilibrium in he subgame of period, (d) under condiions (i) and (ii) in Theorem 3.4.1(c), he Nash equilibrium in he subgame of period, {(γ sc i, (, ), psc i, (, ), xsc i, (, )) : 1 i N}, is unique, and (e) here exiss a posiive vecor βsc, such ha V i, (I, Λ σ sc ) = w i, I i, + βi, scλ i, for all i. Because V i,0 (I 0, Λ 0 ) = w i,0 I i,0 for all i, he iniial condiion is saisfied. Since V i, 1 (I 1, Λ 1 σ sc 1) = w i, 1 I i, 1 + β sc i, 1 Λ i, 1 for all i, Equaion (3.12) implies ha he objecive funcion of player i in G sc,2 is π sc i,(y ) = (δ i w i, 1 w i, )y i, L i, (y i, ) + δ i β sc i, 1(κ ii, (E[y + i, ξ i,]) j i κ ij, (E[y + j, ξ j,])). Thus, for any given sraegy of oher players y i,, player i maximizes he following univariae funcion: ζ sc i,(y i, ) := (δ i w i, 1 w i, )y i, L i, (y i, ) + δ i β sc i, 1κ ii, (E[y + i, ξ i,]). 223

238 If y i, < 0, (y i, ξ i, ) + = 0, (y i, ξ i, ) = ξ i, y i,, and, hus, L i, (y i, ) = b i, E(ξ i, y i, ) = b i, + b i, y i,. Moreover, y i, < 0 implies ha δ i β sc i, 1 κ ii,(e[y + i, ξ i,]) δ i β sc i, 1 κ ii,(0). Hence, if y i, < 0, ζ sc i,(y i, ) = b i, + (δ i w i, 1 w i, + b i, )y i, + δ i β sc i, 1κ ii, (0). Because b i, > w i, δ i w i, 1, ζ sc i, ( ) is sricly increasing in y i, for y i, 0. Observe ha L i, ( ) is concave and coninuously differeniable in y i,. Since E(y + i, ξ i,) is concavely increasing and coninuously differeniable in y i, for y i, 0, and κ ii, ( ) is concavely increasing and coninuously differeniable, κ ii, (E[y + i, ξ i,]) is concavely increasing and coninuously differeniable in y i, for y i, 0. Hence, ζ sc i, ( ) is concave and coninuously differeniable in y i, for y i, 0. Observe ha yi, ζ sc i,(0+) = δ i w i, 1 w i, +b i, +δ i β sc i, 1 F i, (0)κ ii,(e(0 ξ i, )) = δ i w i, 1 w i, +b i, +δ i β sc i, 1κ ii,(0) > 0, where he inequaliy follows from δ i w i, 1 w i, + b i, > 0 and κ ii, (0) 0. Therefore, he opimizer of ζi, sc ( ), ysc i,, is he soluion o he firs-order condiion: y i, ζ sc i, (ysc i, ) = 0, or, equivalenly, (δ i w i, 1 w i, ) L i,(y sc i, ) + δ i β sc i, 1 F i, (y sc i, )κ ii,(e(y sc i, ξ i, )) = 0. Because ξ i, is coninuously disribued, y sc i, ζ sc i, (0) = b i, + δ i β sc i, 1 κ ii,(0) for each i. We now show ha Proposiion holds for period. is unique for each i. Moreover, y sc i, Since ζ sc i, (ysc δ i βi, 1 sc κ ii,(0) and α i, (z ) κ ii, (0) j i κ ij,(1) 0, we have πi, sc > ζi, sc(0) δ iβi, 1 sc b i,. Observe ha p i, δ i w i, 1 ν i, (γ i, ) + π sc i, > p i, δ i w i, 1 ν i, ( γ i, ) b i, > 0. Thus, if p i, = p i,, p i, δ i w i, 1 ν i, (γ i, ) + π sc i, > 0 and ζ sc i, (ysc i, ) > i, ) > ζsc i, (0) = b i, + j i κ ij,(1) > 0. Therefore, each firm i could a leas earn a posiive payoff of ( p i, δ i w i, 1 ν i, ( γ i, ) b i, )ϵ i, by charging he maximum allowable price p i,, where Le ϵ i, := min{ψ i, (γ )ρ i, (p ) : γ [0, γ 1, ] [0, γ N, ] [p 1,, p 1, ] [p N,, p N, ]} > 0. ϵ i, := max{ψ i, (γ )ρ i, (p ) : γ [0, γ 1, ] [0, γ N, ] [p 1,, p 1, ] [p N,, p N, ]} ϵ i,. Hence, we can resric he feasible acion se of firm i in G sc,1 A sc,1 i, := {(γ i,, p i, ) [0, γ i, ] [p i,, p i, ] : p i, δ i w i, 1 ν i, (γ i, ) + π sc i, ( p i, δ i w i, 1 ν i, ( γ i, ) b i, )ϵ i, ϵ i, > 0}, which is a nonempy and complee sublaice of R 2. Thus, Π sc i, (γ, p ) > 0 and o log(π sc i,(γ, p )) = log(p i, δ i w i, 1 ν i, (γ i, ) + π sc i, ) + log(ψ i, (γ )) + log(ρ i, (p )) (B.1) is well-defined on A sc,1 i,. Because ρ i,( ) and ψ i, ( ) saisfy (3.3) and (3.4), for each i and j i, we have 2 log(π sc i, (γ, p )) γ i, p i, = 2 log(p i, δ i w i, 1 ν i, (γ i, ) + π sc i, ) γ i, p i, = 224 ν i, (γ i,) (p i, δ i w i, 1 ν i, (γ i, ) + πi, sc 0, )2

239 Hence, G sc,1 2 log(π sc i, (γ, p )) γ i, p j, = 0, 2 log(π sc i, (γ, p )) p i, γ j, 2 log(π sc i, (γ, p )) γ i, γ j, = 2 log(ψ i, (γ )) γ i, γ j, 0, = 0, and 2 log(π sc i, (γ, p )) p i, p j, = 2 log(ρ i, (p )) p i, p j, 0. is a log-supermodular game and, hus, has pure sraegy Nash equilibria which are he smalles and larges undominaed sraegies (see Theorem 5 in [124]). Nex, we show ha if condiions (i) and (ii) in Theorem 3.4.1(c) hold, he Nash equilibrium of G sc,1 is unique. Firs, we show ha under condiions (i) and (ii) in Theorem 3.4.1(c), 2 log Π sc i, (γ, p ) p 2 i, 2 log Π sc i, (γ, p ) γ 2 i, Noe ha, by (B.1) and (3.4), and < 0, 2 log Π sc i, (γ, p ) p 2 > i, j i < 0, and 2 log Π sc i, (γ, p ) γi, 2 > j i 2 log Π sc i, (γ, p ) p 2 = 2 log ρ i, (p ) i, p 2 i, 2 log Π sc i, (γ, p ) p 2 i, Since 2 log(π sc i, (γ,p)) p i, γ j, = 0 for j i, and = 2 log ρ i, (p ) p 2 + i, 2 log(π sc i, (γ, p )) p i, p j, + N j=1 2 log(π sc i, (γ, p )) γ i, γ j, + 2 log(π sc i, (γ, p )) p i, γ j,, (B.2) N 2 log(π sc i, (γ, p )). (B.3) γ i, p j, j=1 1 (p i, δ i w i, 1 ν i, (γ i, ) + πi, sc )2 < 0, 1 (p i, δ i w i, 1 ν i, (γ i, ) + π sc i, )2. we have 2 log(π sc i, (γ, p )) p i, γ i, = ν i, (γ i,) (p i, δ i w i, 1 ν i, (γ i, ) + πi, sc, )2 2 log Π sc i, (γ, p ) p 2 i, = 2 log ρ i, (p ) p 2 + i, > j i = j i 2 log(π sc i, (γ, p )) p i, p j, + 2 log(π sc i, (γ, p )) p i, p j, + 1 (p i, δ i w i, 1 ν i, (γ i, ) + π sc i, )2 ν i, (γ i,) (p i, δ i w i, 1 ν i, (γ i, ) + π sc i, )2 N j=1 2 log(π sc i, (γ, p )) p i, γ j,, where he inequaliy follows from (3.4) and condiion (i). Hence, (B.2) holds for all i and all (γ, p ). and Since ν i, ( ) 0 and (3.3), we have 2 log Π sc i, (γ, p ) γi, 2 = 2 log ψ i, (γ ) γi, 2 ν i, (γ )(p i, δ i w i, 1 ν i, (γ i, ) + πi, sc ) + (ν i, (γ )) 2 (p i, δ i w i, 1 ν i, (γ i, ) + πi, sc < 0, )2 2 log Π sc i, (γ, p ) γi, 2 = 2 log ψ i, (γ ) γi, 2 + ν i, (γ )(p i, δ i w i, 1 ν i, (γ i, ) + πi, sc ) + (ν i, (γ )) 2 (p i, δ i w i, 1 ν i, (γ i, ) + πi, sc. )2 Since 2 log(π sc i, (γ,p)) γ i, p j, = 0 for j i, and 2 log(π sc i, (γ, p )) γ i, p i, = ν i, (γ i,) (p i, δ i w i, 1 ν i, (γ i, ) + πi, sc, )2 225

240 we have 2 log Π sc i, (γ, p ) γi, 2 = 2 log ψ i, (γ ) γi, 2 + ν i, (γ )(p i, δ i w i, 1 ν i, (γ i, ) + πi, sc ) + (ν i, (γ )) 2 (p i, δ i w i, 1 ν i, (γ i, ) + πi, sc )2 > 2 log(π sc i, (γ, p )) + ν i, (γ )(p i, δ i w i, 1 ν i, (γ i, ) + c i, ) + (ν i, (γ )) 2 γ i, γ j, (p i, δ i w i, 1 ν i, (γ i, ) + π sc j i i, )2 j i = j i 2 log(π sc i, (γ, p )) γ i, γ j, + 2 log(π sc i, (γ, p )) γ i, γ j, + where he firs inequaliy follows from (3.4) and π sc i, (B.3) holds for all i and all (γ, p ). We now show ha if (B.2) and (B.3) hold, G sc,1 of Nash equilibria in G sc,1 ν i, (γ i,) (p i, δ i w i, 1 ν i, (γ i, ) + π sc i, )2 N j=1 2 log(π sc i, (γ, p )) γ i, p j,, c i,, and he second from condiion (ii). Hence, has a unique Nash equilibrium. Recall ha he se forms a complee laice (see Theorem 2 in [194]). If, o he conrary, here exis wo disinc equilibria (γ, p ) and (ˆγ, ˆp ), where ˆp i, p i, for all i and ˆγ j, γ j, for all j, wih he inequaliy being sric for some i or j. If, for some i, ˆp i, > p i,, ˆp i, p i, ˆp l, p l, for all l, and ˆp i, p i, ˆγ l, γ l, for all l, wihou loss of generaliy, we assume ha i = 1. Lemma 23 suggess ha p1, log(π sc 1,(ˆγ, ˆp )) p1, log(π sc 1,(γ, p )). (B.4) On he oher hand, by Newon-Leibniz formula, we have = p1, log(π sc 1,(ˆγ, ˆp )) p1, log(π sc 1,(γ, p )) 1 N [ (ˆp j, p j,) 2 log(π sc 1,((1 s)γ + sˆγ, (1 s)p + sˆp )) p 1, p j, s=0 j=1 + 1 N (ˆγ j, γj,) 2 log(π sc 1,((1 s)γ + sˆγ, (1 s)p + sˆp )) ] ds p 1, γ j, j=1 N [ (ˆp 1, p 1,) 2 log(π sc 1,((1 s)γ + sˆγ, (1 s)p + sˆp )) p 1, p j, s=0 j=1 + N (ˆp 1, p 1,) 2 log(π sc 1,((1 s)γ + sˆγ, (1 s)p + sˆp )) ] ds p 1, γ j, j=1 < 0, where he firs inequaliy follows from ˆp 1, p 1, ˆp l, p l, for all l and ˆp 1, p 1, ˆγ l, γ l, for all l, and he second from ˆp 1, p 1, > 0 and (B.2). This conradics (B.4). If, for some j, ˆγ j, > γ j,, ˆγ j, γ j, ˆp l, p l, for all l, and ˆγ j, γ j, ˆγ l, γ l, for all l, wihou loss of generaliy, we assume ha j = 1. Lemma 23 suggess ha γ1, log(π sc 1,(ˆγ, ˆp )) γ1, log(π sc 1,(γ, p )). (B.5) 226

241 On he oher hand, by Newon-Leibniz formula, we have = γ1, log(π sc 1,(ˆγ, ˆp )) γ1, log(π sc 1,(γ, p )) 1 N [ (ˆγ j, γj,) 2 log(π sc 1,((1 s)γ + sˆγ, (1 s)p + sˆp )) γ 1, γ j, s=0 j=1 + 1 N (ˆp j, p j,) 2 log(π sc 1,((1 s)γ + sˆγ, (1 s)p + sˆp )) ] ds γ 1, p j, j=1 N [ (ˆγ 1, γ1,) 2 log(π sc 1,((1 s)γ + sˆγ, (1 s)p + sˆp )) γ 1, γ j, s=0 j=1 + < 0, N (ˆγ 1, γ1,) 2 log(π sc 1,((1 s)γ + sˆγ, (1 s)p + sˆp )) ] ds γ 1, p j, j=1 where he firs inequaliy follows from ˆγ 1, γ1, ˆp l, p l, for all l and ˆγ 1, γ1, ˆγ l, γ l, for all l, and he second from ˆγ 1, γ 1, > 0 and (B.3). This conradics (B.5). Therefore, he Nash equilibrium in G sc,1 is unique, if condiions (i) and (ii) in Theorem 3.4.1(c) hold. If ν i, (γ i, ) = γ i,, we have ν i, (γ i,) = 1 and ν i, (γ i,) = 0 for all γ i, [0, γ i, ]. Thus, if ν i, (γ i, ) = γ i,, condiions (i) and (ii) in Theorem 3.4.1(c) hold. Noe ha for any λ [0, 1] and (γ i,, p i, ), (ˆγ i,, ˆp i, ) [0, γ 1, ] [0, γ 2, ] [0, γ N, ] [p 1,, p 1, ] [p 2,, p 2, ] [p N,, p N, ], λ log(ˆp i, δ i w i, ν i, (ˆγ i, ) + π sc i, ) + (1 λ) log(p i, δ i w i, ν i, (γ i, ) + π sc i, ) log(λˆp i, + (1 λ)p i, δ i w i, λν i, (ˆγ i, ) (1 λ)ν i, (γ i, ) + π sc i, ) log(λˆp i, + (1 λ)p i, δ i w i, ν i, (λˆγ i, + (1 λ)γ i, ) + π sc i, ), where he firs inequaliy follows from he concaviy of log( ), and he second from ha log( ) is an increasing funcion and ν i, ( ) is a convex funcion. Thus, log(p i, δ i w i, ν i, (γ i, )+πi, sc ) is joinly concave in (γ i,, p i, ). Hence, he diagonal dominance condiion (3.3) and (3.4) implies ha log(π sc i, (γ, p )) is joinly concave in (γ i,, p i, ) for any given (γ i,, p i, ). Therefore, he firs-order condiions wih respec o γ i, and p i, is he necessary and sufficien condiion for (γ sc in G sc,1. Since and he Nash equilibrium of G sc,1 γi, log(π sc i,(γ, p )) = γ i, ψ i, (γ ) ψ i, (γ ) pi, log(π sc i,(γ, p )) = p i, ρ i, (p ) + ρ i, (p ), p sc ) o be he unique Nash equilibrium ν i, (γ ) p i, δw i, ν i, (γ i, ) + π sc i, 1 p i, δw i, ν i, (γ i, ) + πi, sc is a soluion o he sysem of equaions (3.15). Since G sc,1,, has a unique equilibrium, (3.15) has a unique soluion, which coincides wih he unique pure sraegy Nash equilibrium of G sc,1. As shown above, for all i, Hence, Π sc i, = Π sc i, (γsc Π sc i,(γ sc, p sc ) ( p i, δ i w i, 1 ν i, ( γ i, ) b i, )ϵ i, > 0., p sc ) > 0 for all i. 227

242 Nex, we show ha {(γ sc i,, psc i,, Λ i,y sc he subgame of period. Since y sc i, i, ρ i,(p sc > 0, Λ i, y sc )ψ i, (γ sc )) : 1 i N} is an equilibrium in i, ρ i,(p sc )ψ i, (γ sc ) > 0 for all i. Therefore, regardless of he saring invenory in period, I i,, firm i could adjus is invenory o x sc i, (I, Λ ) = Λ i, y sc i, ρ i,(p sc )ψ i, (γ sc ). Thus, by Proposiions , {(γi, sc, psc i,, Λ i,y sc i, ρ i,(p sc )ψ i, (γ sc )) : 1 i N} forms an equilibrium in he subgame of period. In paricular, if condiions (i) and (ii) hold, {(γ sc i,, psc i,, Λ i,y sc period. i, ρ i,(p sc Nex, we show ha here exiss a posiive vecor β sc V i, (I, Λ σ sc ) = w i, I i, + βi, scλ i,. By (3.12), we have ha V i, (I, Λ σ sc ) = J i, (γ sc, p sc, Λ i, y sc )ψ i, (γ sc )) : 1 i N} is he unique equilibrium in he subgame of i, ρ i,(p sc = (β1,, sc β2,, sc, βn, sc ), such ha )ψ i, (γ sc ), I, Λ σ 1) sc = w i, I i, +(σ i βi, 1 sc µ i,+π sc i, )Λ i,. Since βi, 1 sc 0 and Πsc i, > 0, βi, sc = δ iβi, 1 sc µ i, + Π sc i, > 0. This complees he inducion and, hus, he proof of Theorem 3.4.1, Proposiion 3.4.1, Proposiion 3.4.2, and Theorem Q.E.D. Proof of Proposiion 3.4.3: By Theorems , and Proposiions , i suffices o show ha, if here exiss a consan β sc s, 1 0, such ha V i, 1 (I 1, Λ 1 σ sc 1) = w s, I i, 1 + β sc s, 1Λ i, 1 for all i, we have: (a) he unique Nash equilibrium in G sc,2 (b) he unique Nash equilibrium in G sc,1 is symmeric, i.e., y sc i, is symmeric, i.e., (γ sc i,, psc i, ) = (γsc j,, psc j, = y sc j, for all i, j; ) for all i j, and (c) here exiss a consan β sc s, > 0, such ha V i, (I, Λ σ sc s, ) = w s, I i, + β sc s,λ i, for all i. Since V i,0 (I, Λ ) = w s,0 I i,0 for all i, he iniial condiion is saisfied wih β sc s,0 = 0. Since V i, 1 (I 1, Λ 1 σ sc 1) = w s, I i, 1 + β sc s,λ i, 1 for all i, by (3.12), π sc i,(y ) = (δ s w s, 1 w s, )y i, L s, (y i, ) + δ s β sc s, 1(κ sa, (E(y + i, ξ i,)) j i κ sb, (E(y + j, ξ j,))). Hence, ζi, sc(y i,) = (δ s w s, 1 w s, )y i, L s, (y i, ) + δ s βs, 1κ sc sa, (E(y + i, ξ i,)). Thus, ζi, sc ( ) ζsc j, ( ) for all i and j. Therefore, for all i and j, y sc i, = argmax y ζi,(y) sc = argmax y ζj,(y) sc = yj, sc and, hence, π sc i, = π sc i,(y sc ) = π sc j,(y sc ) = π sc j,. We denoe y sc s, G sc,1, = y sc i, for each i, and π sc s, = π sc i, for each i. Observe ha, he objecive funcions of {Π sc i,(γ, p ) = ρ s, (p )ψ s, (γ )[p i, δ s w s, 1 ν s, (γ i, ) + π sc s, ] : 1 i N} are symmeric. Hence, if here exiss an asymmeric Nash equilibrium (γ sc Nash equilibrium (γ sc, p sc ) (γ sc, p sc ), here exiss anoher, p sc ), where γ sc is a permuaion of γ sc and p sc is a permuaion of p sc. This conradics he uniqueness of he Nash equilibrium in G sc,1. Thus, he unique Nash equilibrium in G sc,1 is symmeric. Hence, Π sc i, = Π sc i, (γsc ss,, p sc ss,) = ρ s, (p sc ss,)ψ s, (γss,)[p sc sc s, δ s w s, 1 ν s, (γs, sc ) + πs, sc ] = Π sc j, (γsc ss,, p sc ss,) = Π sc j,, which is posiive. Thus, we denoe he payoff of each firm i as Π sc s,. By Theorem 3.4.2(a), βi, sc = δ s βs, 1µ sc s, + Π sc i, = δ s βs, 1µ sc s, + Π sc j, = βj, sc >

243 Thus, we denoe he SC marke size coefficien of each firm i as β sc s,. This complees he inducion and, hus, he proof of Proposiion Q.E.D. Proof of Theorem 3.4.3: Par (a). Clearly, by (3.13), y sc i, Moreover, because 2 ζi, sc(y i,) y i, βi, 1 sc = is independen of β sc j, 1 δ i Fi, (y i, )κ ii, (E(y i, ξ i, )) 0, if y i, 0; 0, oherwise, ζ sc i, (y i,) is supermodular in (y i,, β sc i, 1 ). β sc i, 1. The coninuiy of ysc i, in β sc i, 1 in (y i,, βi, 1 sc ). This complees he proof of par (a). Therefore, ysc i, for all j i. = argmax yi, Rζ sc i, (y i,) is increasing in follows direcly from he coninuous differeniabiliy of ζsc i, ( ) Par (b). Noe ha, by par (a), l i κ il,(e((yl, sc )+ ξ l, )) is independen of βi, 1 sc and coninuously increasing in βj, 1 sc for j i. Moreover, ζ sc i,(y i, ) = (δ i w i, 1 w i, )y i, L i, (y i, ) + δ i β sc i, 1κ ii, (E[y + i, ξ i,]) is coninuously increasing in βi, 1 sc and independen of βsc j, 1 for all j i. Thus, π sc i, = [ max y i, 0 ζsc i,(y i, )] κ ij, (E(yj, sc ξ j, )) j i is coninuously increasing in βi, 1 sc and coninuously decreasing in βsc j, 1 for all j i. This complees he proof of par (b). Par (c). We denoe he objecive funcion of each firm i in G sc,1 s, as Π sc i, (, πsc s, ) o capure he dependence of he objecive funcions on πs, sc. The unique symmeric Nash equilibrium in G sc,1 s, is denoed as (γ sc ss,(π sc s, ), p sc ss,(π sc s, )), where γ sc ss,(π sc s, ) = (γ sc s, (π sc s, ), γ sc s, (π sc s, ),, γ sc s, (π sc s, )) and p sc ss,(π sc s, ) = (p sc s, (πs, sc ), p sc s, (πs, sc ),, p sc s, (πs, sc )). I suffices o show ha, if π s, sc > πs, sc, γs, sc ( π s, sc ) γs, sc (πs, sc ), and p sc s, ( π sc s, ) p sc s, (π sc s, ). We firs show ha p sc s, ( π s, sc ) p sc s, (πs, sc ) for all π s, sc p sc s, ( π sc s, ) > p sc s, (π sc s, ). Lemma 23 implies ha > π sc s,. Assume, o he conrary, ha p1, log(π sc 1,(γ sc ss,( π sc s, ), p sc ss,( π sc s, ) π sc s, )) p1, log(π sc 1,(γ sc ss,(π sc s, ), p sc ss,(π sc s, ) π sc s, )), i.e., p1, log ρ s, (p sc ss,( π sc s, )) + p1, log ρ s, (p sc ss,(π sc s, )) + By (3.4) and Newon-Leibniz formula, we have = < 0. p1, log ρ s, (p sc ss,( π sc 1 N [ (p sc s, ( π sc s=0 j=1 1 p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc 1 p sc s, (π sc s, ) δ s w s, 1 ν s, (γ sc s, (π sc s, )) p1, log ρ s, (p sc s, ) p sc Hence, inequaliy (B.6) suggess ha s, (π sc ss,(πs, sc )) s, )) 2 log ρ s, ((1 s)p sc ss,(π sc p 1, p j, s, )) + πs, sc. s, ) + sp sc ss,( π s, sc )) ] ds (B.6) p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc < p sc s, (πs, sc ) δ s w s, 1 ν s, (γs, sc (πs, sc )) + πs, sc. (B.7) 229

244 Since p sc s, ( π s, sc ) > p sc s, (πs, sc ) and π s, sc γ sc s, (π sc s, ). Lemma 23 yields ha > π sc s,, ν s, (γ sc s, ( π sc s, )) > ν s, (γ sc s, (π sc s, )). Thus, γ sc s, ( π sc s, ) > γ1, log(π sc 1,(γ sc ss,( π sc s, ), p sc ss,( π sc s, ) π sc s, )) γ1, log(π sc 1,(γ sc ss,(π sc s, ), p sc ss,(π sc s, ) π sc s, )), i.e., γ1, log ψ s, (γ sc ss,( π sc s, )) γ1, log ψ s, (γ sc ss,(π sc s, )) ν s,(γ sc s, ( π sc s, )) p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc ν s,(γ sc s, (π sc s, )) p sc s, (πs, sc ) δ s w s, 1 ν s, (γs, sc (πs, sc )) + πs, sc (B.8) Since ν s, ( ) is convexly increasing, ν s,(γ sc s, ( π sc s, ) ν s,(γ sc s, (π sc s, )). Thus, inequaliy (B.7) implies ha ν s,(γ s, sc ( π s, sc )) p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc Hence, (B.8) suggess ha ν s,(γ s, sc (πs, sc )) < p sc s, (πs, sc ) δ s w s, 1 ν s, (γs, sc (πs, sc )) + πs, sc. γ1, log ψ s, (γ sc ss,( π sc s, )) > γ1, log ψ s, (p sc ss,(π sc s, )). (B.9) By (3.3) and Newon-Leibniz formula, we have = < 0, γ1, log ψ s, (γss,( π sc sc 1 N [ (γs, sc ( π sc s=0 j=1 s, )) γ1, log ψ s, (p sc s, ) γ sc s, (π sc which conradics (B.9). Therefore, for all π sc s, ss,(πs, sc )) s, )) 2 log ψ s, ((1 s)γ sc We now show ha γs, sc ( π s, sc ) γs, sc (πs, sc ) for all π s, sc γ sc s, ( π sc s, ) < γ sc s, (π sc s, ). Lemma 23 implies ha ss,(π sc γ 1, γ j, s, ) + sγ sc ss,( π s, sc )) > π sc s,, we have p sc s, ( π sc s, ) p sc s, (π sc s, ). > π sc s,. ] ds Assume, o he conrary, ha γ1, log(π sc 1,(γ sc ss,( π sc s, ), p sc ss,( π sc s, ) π sc s, )) γ1, log(π sc 1,(γ sc ss,(π sc s, ), p sc ss,(π sc s, ) π sc s, )), i.e., γ1, log ψ s, (γ sc ss,( π sc s, )) γ1, log ψ s, (γ sc ss,(π sc s, )) ν s,(γ sc s, ( π sc s, )) p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc ν s,(γ s, sc (πs, sc )) p sc s, (πs, sc ) δ s w s, 1 ν s, (γs, sc (π sc s, )) + πs, sc. (B.10) By (3.3) and Newon-Leibniz formula, we have = γ1, log ψ s, (γss,(π sc sc 1 N [ (γs, sc (π sc s=0 j=1 s, )) γ1, log ψ s, (γ sc s, ) γ sc s, ( π sc ss,( π s, sc )) s, )) 2 log ψ s, (sγss,(π sc s, sc ) + (1 s)γss,( π sc s, sc )) ] ds < 0. γ 1, γ j, Hence, inequaliy (B.10) implies ha ν s,(γ s, sc ( π s, sc )) p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc ν s,(γ s, sc (πs, sc )) < p sc s, (πs, sc ) δ s w s, 1 ν s, (γs, sc (πs, sc )) + πs, sc. Since ν s, ( ) is convexly increasing, ν s,(γ sc s, ( π sc s, )) ν s,(γ sc s, (π sc s, )). Hence, p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc < p sc s, (πs, sc ) δ s w s, 1 ν s, (γs, sc (πs, sc )) + πs, sc. 230

245 Since ν s, (γs, sc ( π s, sc )) ν s, (γs, sc (πs, sc )) and π s, sc > πs, sc, p sc s, ( π s, sc ) < p sc s, (πs, sc ). Lemma 23 implies ha p1, log(π sc 1,(γ sc ss,( π sc s, ), p sc ss,( π sc s, ) π sc s, )) p1, log(π sc 1,(γ sc ss,(π sc s, ), p sc ss,(π sc s, ) π sc s, )), i.e., Because p1, log ρ s, (p sc ss,( π sc s, )) + p1, log ρ s, (p sc ss,(π sc s, )) + 1 p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc > we have ha By (3.4) and Newon-Leibniz formula, we have = < 0, p1, log ρ s, (p sc ss,(π sc 1 N [ (p sc s, (π sc s=0 j=1 1 p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc 1 p sc s, (π sc s, ) δ s w s, 1 ν s, (γ sc s, (π sc s, )) + πs, sc. 1 p sc s, (πs, sc ) δ s w s, 1 ν s, (γs, sc (π sc p1, log ρ s, (p sc ss,( π sc s, )) < p1, log ρ s, (p sc ss,(π sc s, )). s, )) p1, log ρ s, (p sc s, ) p sc s, ( π sc which conradics (B.12). Therefore, for all π sc s, of γs, sc (πs, sc ) and p sc s, (πs, sc ) in πs, sc ss,( π s, sc )) s, )) 2 log ρ s, (sp sc ss,(π sc s, ) + (1 s)p sc p 1, p j, ss,( π s, sc )) s, )) + πs, sc, ] ds (B.11) (B.12) > π sc s,, we have γ sc s, ( π sc s, ) γ sc s, (π sc s, ). The coninuiy follows direcly from ha Π sc i, (γ, p π sc s, ) is wice coninuously differeniable and he implici funcion heorem. This complees he proof of par (c). Par (d). By Theorem 3.4.2(a), β sc s, = δ s β sc s,µ s, + Π sc s,, i suffices o show ha Π sc s, (π sc s, ) is coninuously increasing in πs, sc, where Π sc s, (πs, sc ) := Π sc i, (γsc ss,(πs, sc ), p sc ss,(πs, sc )). Assume ha π sc s, > π sc s,. γ sc s, (π sc s, ), he monooniciy condiion (3.17) implies ha Since par (c) implies ha p sc s, ( π sc s, ) p sc s, (π sc s, ) and γ sc s, ( π sc s, ) ρ s, (p sc ss,( π sc s, )) ρ s, (p sc ss,(π sc s, )) and ψ s, (γ sc ss,( π sc s, )) ψ s, (γ sc ss,(π sc s, )). (B.13) Thus, If p sc s, ( π s, sc ) = p sc s, (πs, sc ) and γs, sc ( π s, sc ) = γs, sc (πs, sc ), by π s, sc > πs, sc, we have p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc > p sc s, (πs, sc ) δ s w s, 1 ν s, (γs, sc (πs, sc )) + πs, sc. Π sc s, ( π sc s, ) = Π sc i,(γ sc ss,( π sc s, ), p sc ss,( π sc s, ) π sc s, ) = (p sc s, ( π sc s, ) δ s w s, 1 ν s, (γ sc s, ( π sc s, )) + π sc s, )ρ s, (p sc ss,( π sc s, ))ψ s, (γ sc ss,( π sc s, )) > (p sc s, (π sc s, ) δ s w s, 1 ν s, (γ sc s, (π sc s, )) + π sc s, )ρ s, (p sc ss,(π sc s, ))ψ s, (γ sc ss,(π sc s, )) = Π sc i,(γ sc ss,(π sc s, ), p sc ss,(π sc s, ) π sc s, ) = Π sc s, (π sc s, ). If p sc s, ( π sc s, ) < p sc s, (π sc s, ), Lemma 23 yields ha p1, log(π sc 1,(p sc ss,( π sc s, ), γ sc ss,( π sc s, ) π sc s, )) p1, log(π sc 1,(p sc ss,(π sc s, ), γ sc ss,(π sc s, ) π sc s, )), i.e., p1, log ρ s, (p sc ss,( π sc s, )) + p1, log ρ s, (p sc ss,(π sc s, )) + 1 p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc 1 p sc s, (π sc s, ) δ s w s, 1 ν s, (γ sc s, (π sc s, )) + πs, sc. (B.14) 231

246 By (3.4) and Newon-Leibniz formula, we have = p1, log ρ s, (p sc ss,(π sc 1 N [ (p sc s, (π sc s=0 j=1 s, )) p1, log ρ s, (p sc s, ) p sc s, ( π sc ss,( π s, sc )) s, )) 2 log ρ s, ((1 s)p sc ss,( π s, sc ) + sp sc ss,(πs, sc )) ] ds < 0. p 1, p j, Hence, (B.14) implies ha p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc > p sc s, (πs, sc ) δ s w s, 1 ν s, (γs, sc (πs, sc )) + πs, sc. Therefore, Π sc s, ( π sc s, ) = Π sc i,(γ sc ss,( π sc s, ), p sc ss,( π sc s, ) π sc s, ) = (p sc s, ( π sc s, ) δ s w s, 1 ν s, (γ sc s, ( π sc s, )) + π sc s, )ρ s, (p sc ss,( π sc s, ))ψ s, (γ sc ss,( π sc s, )) > (p sc s, (π sc s, ) δ s w s, 1 ν s, (γ sc s, (π sc s, )) + π sc s, )ρ s, (p sc ss,(π sc s, ))ψ s, (γ sc ss,(π sc s, )) = Π sc i,(γ sc ss,(π sc s, ), p sc ss,(π sc s, ) π sc s, ) = Π sc s, (π sc s, ). If p sc s, ( π sc s, ) = p sc s, (π sc s, ) and γ sc s, ( π sc s, ) > γ sc s, (π sc s, ), Lemma 23 yields ha γ1, log(π sc 1,(p sc ss,( π sc s, ), γ sc ss,( π sc s, ) π sc s, )) γ1, log(π sc 1,(p sc ss,(π sc s, ), γ sc ss,(π sc s, ) π sc s, )), i.e., γ1, log ψ s, (γ sc ss,( π sc s, )) γ1, log ψ s, (γ sc ss,(π sc s, )) By (3.4) and Newon-Leibniz formula, we have ν s,(γ sc s, ( π sc s, )) p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc ν s,(γ s, sc (πs, sc )) p sc s, (πs, sc ) δ s w s, 1 ν s, (γs, sc (π sc s, )) + πs, sc. (B.15) = γ1, log ψ s, (γss,( π sc sc 1 N [ (γs, sc ( π sc s=0 j=1 s, )) γ1, log ψ s, (γ sc s, ) γ sc s, (π sc ss,(πs, sc )) s, )) 2 log ψ s, (sγss,( π sc s, sc ) + (1 s)γss,(π sc s, sc )) ] ds < 0. γ 1, γ j, Hence, (B.15) implies ha ν s,(γ s, sc ( π s, sc )) p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc > ν s,(γ s, sc (πs, sc )) p sc s, ((πs, sc )) δ s w s, 1 ν s, (γs, sc (πs, sc )) + πs, sc. (B.16) Since ν s, ( ) is convexly increasing, ν s,(γ sc s, ( π sc s, )) ν s,(γ sc s, (π sc s, )). Hence, (B.16) implies ha p sc s, ( π s, sc ) δ s w s, 1 ν s, (γs, sc ( π s, sc )) + π s, sc > p sc s, ((πs, sc )) δ s w s, 1 ν s, (γs, sc (πs, sc )) + πs, sc. Therefore, Π sc s, ( π sc s, ) = Π sc i,(γ sc ss,( π sc s, ), p sc ss,( π sc s, ) π sc s, ) = (p sc s, ( π sc s, ) δ s w s, 1 ν s, (γ sc s, ( π sc s, )) + π sc s, )ρ s, (p sc ss,( π sc s, ))ψ s, (γ sc ss,( π sc s, )) > (p sc s, (π sc s, ) δ s w s, 1 ν s, (γ sc s, (π sc s, )) + π sc s, )ρ s, (p sc ss,(π sc s, ))ψ s, (γ sc ss,(π sc s, )) = Π sc i,(γ sc ss,(π sc s, ), p sc ss,(π sc s, ) π sc s, ) = Π sc s, (π sc s, ). 232

247 Thus, we have shown ha, if π sc s, > π sc s,, Π sc s, ( π sc s, ) > Π sc s, (π sc s, ) and, hence, by Theorem 3.4.2(a), βs,( π sc s, sc ) > βs,(π sc s, sc ). The coninuiy of βs, sc in πs, sc follows direcly from he coninuous differeniabiliy of Π sc i, (γ, p π sc s, ) in (γ, p, π sc s, ) and he coninuiy of (γ sc ss,, p sc ss,) in π sc s,. This complees he proof of par (d). Par (e). By par (c), i suffices o show ha, π sc s, is coninuously increasing in β cs s, 1. monooniciy follows from he assumpion, whereas he coninuiy follows direcly from par (a) and ha he compound funcion is coninuous if each individual funcion is coninuous. This complees he proof of par (e). Par (f). By he proof of par (e), π sc s, coninuously increasing in β sc s, 1. Q.E.D. The is coninuously increasing in β cs s, 1. By par (d), β sc s, is Proof of Theorem 3.4.4: Par (a). Because βi, 1 sc sc β i, 1 = 0 for each i and, Theorem 3.4.3(a) implies ha y sc i, Moreover, since ỹ sc i, z sc i, for all i and. Thus, = E[(y sc i, ) + ξ i, ] E[(ỹ sc i, ) + ξ i, ] = z sc i,, for all i and. sc β i, 1 = 0, πsc i, (y ) = (δ i w i, 1 w i, )y i, L i, (y i, ). Moreover, if y i, 0, π i, sc(y ) is sricly increasing in y i,. Hence, π sc i, π sc i, = max{(δ i w i, 1 w i, )y i, L i, (y i, ) : y i, 0}. Thus, = max{(δ i w i, 1 w i, )y i, L i, (y i, ) + δ i β sc i, 1(κ ii, (E[y i, ξ i, ]) j i κ ij, (E[y j, ξ j, ])) : y i, 0} max{(δ i w i, 1 w i, )y i, L i, (y i, ) + δ i β sc i, 1(κ ii, (0) j i κ ij, (1)) : y i, 0} max{(δ i w i, 1 w i, )y i, L i, (y i, ) : y i, 0} = π sc i,, where he firs inequaliy follows from ha κ ii, ( ) is increasing in y i, and κ ij, ( ) is increasing in y j,, and he second from ha α i, ( ) 0 for all i,, and z. This proves par (a). Par (b-i). Par (a) suggess ha π sc s, π sc s, for all. Thus, by Theorem 3.4.3(c), γ sc s, γ sc s, for all. By Theorem 3.4.2(b), γi, sc (I, Λ ) = γs, sc γ s, sc = γ i, sc (I, Λ ) for all and (I, Λ ) S. This proves par (b-i). Par (b-ii). Par (a) suggess ha π sc s, π sc s, for all. Thus, by Theorem 3.4.3(c), p sc s, p sc s, for all. By Theorem 3.4.2(b), p sc i, (I, Λ ) = p sc s, p sc s, = p sc i, (I, Λ ) for all and (I, Λ ) S. This proves par (b-ii). Par (b-iii). By Proposiion 3.4.3(d), x sc i, (I, Λ ) = y sc s, ρ s, (p sc ss,)ψ s, (γ sc ss,)λ i, and x sc i, (I, Λ ) = ỹs, sc ρ s, ( p sc ss,)ψ s, ( γ ss,)λ sc i,. Par (a) implies ha ys, sc ỹs, sc. Since, by pars (b-i) and (b-ii), p sc s, p sc s, and γ sc s, γ sc s,, he monooniciy condiion (3.17) yields ha ρ s, (p sc ss,) ρ s, ( p sc ss,), and ψ s, (γ sc ss,) ψ s, ( γ sc ss,). Therefore, for each (I, Λ ) S, x sc i, (I, Λ ) = y sc s, ρ s, (p sc ss,)ψ s, (γ sc ss,)λ i, ỹ sc s, ρ s, ( p sc ss,)ψ s, ( γ sc ss,)λ i, = x sc i, (I, Λ ). This complees he proof of par (b-iii). Q.E.D. 233

248 Proof of Theorem 3.4.5: Par (a). We show par (a) by backward inducion. More specifically, we show ha if ˆα s, (z ) α s, (z ) for all z and sc ˆβ s, 1 βs, 1, sc (i) ˆπ s, sc πs, sc, (ii) ˆγ s, sc γs, sc, (iii) ˆγ i, sc (I, Λ ) γs, sc (I, Λ ) for each i and (I, Λ ) S, (iv) ˆp sc s, p sc s,, (v) ˆp sc i, (I, Λ ) p sc i, (I, Λ ) for each i and (I, Λ ) S, and (vi) Since ˆα s, (z ) α s, (z ) for all z, ˆβ sc s, β sc s,. Since ˆβ sc s,0 = β sc s,0 = 0, he iniial condiion is saisfied. ˆκ sa, (y i, ) (N 1)ˆκ 0 sb, κ sa, (y i, ) (N 1)κ 0 sb, 0, for all y i, 0. Therefore, ˆπ sc s, = max{(δ s w s, 1 w s, )y i, L s, (y i, ) + δ s ˆβsc s, 1(ˆκ sa, (E[y i, ξ i, ]) (N 1)ˆκ 0 sb,) : y i, 0} ˆγ sc s, max{(δ s w s, 1 w s, )y i, L s, (y i, ) + δ s β sc s, 1(κ sa, (E[y i, ξ i, ]) (N 1)κ 0 sb,) : y i, 0} = π sc s,. Since ˆπ sc s, γ sc s, πs, sc, Theorem 3.4.3(c) implies ha ˆγ s, sc γs, sc and ˆp sc s, p sc s,. Thus, ˆγ i, sc (I, Λ ) = = γi, sc (I, Λ ) for each i and all (I, Λ ) S. Analogously, ˆp sc i, (I, Λ ) = ˆp sc s, p sc i, (I, Λ ) for each i and all (I, Λ ) S. By Theorem 3.4.3(d), ˆπ s, sc πs, sc implies ha This complees he inducion and, hus, he proof of par (a). p sc s, = sc ˆβ s, βs,. sc Par (b). By par (a), i suffices o show ha, if ˆα s, (z ) α s, (z ) for all z, ˆκ sa,(z i, ) κ sa,(z i, ) for all z i,, and (I, Λ ) S. sc ˆβ s, 1 βs, 1, sc we have (i) ŷs, sc ys, sc and (ii) ˆx sc i, (I, Λ ) x sc i, (I, Λ ) for each i and Firs, we show ha ŷ sc s, ys, sc. If, o he conrary, ŷs, sc < ys, sc, Lemma 23 yields ha yi, [(δ s w s, 1 w s, )ŷ sc s, yi, [(δ s w s, 1 w s, )y sc s, L s, (ŷs, sc ) + δ ˆβsc s s, 1(ˆκ sa, (E[ŷs, sc ξ i, ]) (N 1)ˆκ 0 sb,)] L s, (ys, sc ) + δ s βs, 1(κ sc sa, (E[ys, sc ξ i, ]) (N 1)κ 0 sb,)], i.e., (δ s w s, 1 w s, ) L s,(ŷ sc s, ) + δ s ˆβsc s, 1 F s, (ŷ sc s, )ˆκ sa,(e[ŷ sc s, ξ i, ]) (δ s w s, 1 w s, ) L s,(y sc s, ) + δ s β sc s, 1 F s, (y sc s, )κ sa,(e[y sc s, ξ i, ]). (B.17) Since L s, ( ) is sricly concave in y i, and ŷ sc s, < y sc s,, (B.17) implies ha δ ˆβsc F s s, 1 s, (ŷs, sc )ˆκ sa,(e[ŷs, sc ξ i, ]) < δ s βs, 1 sc F s, (ys, sc )κ sa,(e[ys, sc ξ i, ]). (B.18) However, since ˆκ sa,(z i, ) κ sa,(z i, ) for all z i, κ sa,(e[y sc s, ξ i, ]) and F s, (ŷ sc s, ) F s, (y sc s, ). Because and ŷ sc s, < ys, sc, we have ˆκ sa,(e[ŷs, sc ξ i, ]) ˆβ sc s, 1 β sc s, 1, δ ˆβsc F s s, 1 s, (ŷs, sc )ˆκ sa,(e[ŷs, sc ξ i, ]) δ s βs, 1 sc F s, (ys, sc )κ sa,(e[ys, sc ξ i, ]), which conradics (B.18). The inequaliy ŷ sc s, y sc s, hen follows immediaely. Now we show ha ˆx sc i, (I, Λ ) x sc i, (I, Λ ) for each i and (I, Λ ) S. By Proposiion 3.4.3(d), ˆx sc i, (I, Λ ) = ŷ sc s, ρ s, (ˆp sc ss,)ψ s, (ˆγ sc ss,)λ i, and x sc i, (I, Λ ) = y sc s, ρ s, (p sc ss,)ψ s, (γ sc ss,)λ i,. We have shown 234

249 ha ŷs, sc ys, sc. Since (3.17) holds for period, ρ s, (ˆp sc ss,) ρ s, (p sc ss,), and ψ s, (ˆγ ss,) sc ψ s, (γss,). sc Therefore, for each i and (I, Λ ) S, ˆx sc i, (I, Λ ) = ŷ sc s, ρ s, (ˆp sc ss,)ψ s, (ˆγ sc ss,)λ i, y sc s, ρ s, (p sc ss,)ψ s, (γ sc ss,)λ i, = x sc i, (I, Λ ). This complees he proof of par (b). Q.E.D. Proof of Theorem 3.4.6: We show pars (a)-(b) ogeher by backward inducion. More specifically, we show ha if βs, 1 sc βs, 2, sc (i) ys, sc ys, 1, sc (ii) γs, sc γ sc for each i and (I, Λ) S, (iv) p sc s, p sc s, 1, (v) p sc i, s, 1, (iii) γ sc i, (I, Λ) γsc i, 1 (I, Λ) (I, Λ) psc i, 1 (I, Λ) for each i and (I, Λ) S, (vi) x sc i, (I, Λ) xsc i, 1 (I, Λ) for each i and (I, Λ) S, and (vii) βsc s, βs, 1. sc Since, by Proposiion 3.4.3(a), β sc s,1 β sc s,0 = 0. Thus, he iniial condiion is saisfied. Since he model is saionary, by Theorem 3.4.3(a), βs, 1 sc βs, 2 sc suggess ha ys, sc Analogously, Theorem 3.4.3(e) yields ha γ sc s, γ sc s, 1 = γi, 1 sc (I, Λ) and psc i, (I, Λ) = psc s, γs, 1 sc and p sc s, p sc p sc s, 1 s, 1. Hence, γ sc i, y sc s, 1. (I, Λ) = γsc s, = p sc i, 1 (I, Λ) for each i and (I, Λ) S. Because he monooniciy condiion (3.17) holds, we have ρ s, (p sc ss,) ρ s, 1 (p sc ss, 1), and ψ s, (γ sc ss,) ψ s, 1 (γ sc ss, 1). Therefore, for each i and (I, Λ) S, x sc i, (I, Λ) = y sc s, ρ s, (p sc ss,)ψ s, (γ sc ss,)λ i y sc s, 1ρ s, 1 (p sc ss, 1)ψ s, 1 (γ sc ss, 1)Λ i = x sc i, 1(I, Λ). Finally, β sc s, β sc s, 1 follows immediaely from Theorem 3.4.3(f) and β sc s, 1 β sc s, 2. This complees he inducion and, hus, he proof of Theorem Q.E.D. Before presening he proofs of he resuls in he PF model, we give he following lemma ha is used hroughou he res of our proofs. Lemma 24 Le A be an N N marix wih enries defined by A ii, = 2θ ii, and A ij, = θ ij, where i j. The following saemens hold: (a) A is inverible. Moreover, (A 1 ) ij 0 for all 1 i, j N. (b) 1 2 θ ii,(a 1 ) ii < 1. (c) 1 2 N j=1 θ jj,(a 1 ) ij < 1. Proof: Par (a) follows from Lemma 2(a) in [24] and Par (b) follows from Lemma 2(c) in [24]. Par (c). Le I be he N N ideniy marix, B be he N N marix wih 0 if i = j, (B ) ij = θ ij, θ ii, if i j; and C be he N N diagonal marix wih 2θ ii, if i = j, (C ) ij = 0 if i j. 235

250 Because θ ii, > j i θ ij,, B is a subsochasic marix. Observe ha, A = C (I 1 2 B ) and, hence, A 1 be he N dimensinal vecor. Thus, N A 1 θ = (I 1 2 B ) 1 C 1 j=1 θ jj,(a 1 where he las equaliy follows from C 1 θ = 1 2I. Therefore, N j=1 θ jj, (A 1 ) ij = 1 2 = (I 1 2 B ) 1 C 1. Le θ = (θ 11,, θ 22,, θ NN, ) ) ij = (A 1 θ ) i. Moreover, θ = (I 1 2 B ) 1 (C 1 θ ) = 1 2 (I 1 2 B ) 1, N [(I 1 2 B ) 1 ] ij = 1 2 j=1 N [I + j=1 + l=1 ( ) l 1 (B ) l ] ij, 2 where he second equaliy follows from he fac ha I 1 2 B is a diagonal dominan marix. Thus, for all i, N j=1 θ jj,(a 1 ) ij 1 N 2 j=1 I ij = 1 2. On he oher hand, for all i, 1 2 N [I + j=1 + l=1 ( ) l 1 (B ) l ] ij = N [ + j=1 l=0 ( ) l 1 (B ) l ] ij = 1 + [ 2 2 l=0 ( ) l 1 N (B ) l 2 ij] < 1 2 j=1 + l=0 ( ) l 1 = 1, 2 where he inequaliy follows from ha B is a sub-sochasic marix. This complees he proof of par (c). Q.E.D. Proof of Theorems and Proposiions : We show Theorem 3.5.1, Proposiion 3.5.1, Proposiion 3.5.2, Proposiion 3.5.3, and Theorem ogeher by backward inducion. More specifically, we show ha, if V i, 1 (I 1, Λ 1 σ pf 1 ) = w i, 1I i, 1 +β pf i, 1 Λ i, 1 for all i, (a) Proposiion holds for period, (b) Proposiion holds for period, (c) Proposiion holds for period, (d) here exiss a Markov sraegy profile {(γ pf i, (, ), ppf i, (,, ), xpf i, (,, )) : 1 i N}, which forms an equilibrium in he subgame of period, (e) if ν i, (γ i, ) = γ i, for all i and γ i,, he equilibrium in he subgame of period, {(γ pf i, exiss a posiive vecor β pf (, ), ppf i, = (β pf 1,, βpf 2,,, βpf (,, ), xpf i, (,, )) : 1 i N}, is unique, and (f) here N, ), such ha V i,(i, Λ σ pf i. Because V i,0 (I 0, Λ 0 ) = w i,0 I i,0 for all i, he iniial condiion is saisfied. ) = w i, I i, + β pf i, Λ i, for all Firs, we observe ha Proposiion follows direcly from he same argumen as he proof of Proposiion We now show Proposiion holds in period. Because 2 p i, Π pf,2 i, (p γ ) = 2θ ii, < 0, Π pf,2 i, (, p i, γ ) is sricly concave in p i, for any given p i,. Hence, by Theorem 1.2 in [79], G pf,2 has a pure sraegy Nash equilibrium p pf (γ ). Since, for each i and, p i, is sufficienly low whereas p i, is sufficienly high so ha hey will no affec he equilibrium behaviors of all firms, p pf (γ ) can be characerized by firs-order condiions pi, Π pf,2 i, (p pf (γ ) γ ) = 0 for each i, i.e., θ ii, (p pf i, (γ ) δ i w i, 1 ν i, (γ i, ) + π pf i, (γ )) + ρ i, (p pf (γ )) = 2θ ii, p pf i, (γ ) + θ ij, p pf j, (γ ) + f i, (γ ) = 0, for all i. j i (B.19) In erms of he marix language, we have A p pf (γ ) = f (γ ). By Lemma 24(a), A is inverible and, hus, p pf (γ ) is uniquely deermined by p pf coninuously increasing in γ j,, we observe ha (γ ) = A 1 f (γ ). To show ha p pf p pf i, (γ ) = (A 1 ) ij θ jj, ν γ j,(γ j, ). j, 236 i, (γ ) = j (A 1 ) ij f j, (γ ) is

251 Since, by Lemma 24(a), (A 1 ) ij 0 for all i and j, we have γj, p pf i, (γ ) 0 and, hus, p pf i, (γ ) is coninuously increasing in γ j, for each j. Now, we compue Π pf,2 i, (γ ). Π pf,2 i, (γ ) = ρ i, (p pf (γ ))(p pf i, (γ ) δ i w i, 1 ν i, (γ i, ) + π pf i, ) = (ϕ i, θ ii, p pf i, (γ ) + θ ij, p pf j, (γ ))(p pf i, (γ ) δ i w i, 1 ν i, (γ i, ) + π pf i, ) j i = (θ ii, p pf i, (γ ) f i, (γ ) + ϕ i, )(p pf i, (γ ) δ i w i, 1 ν i, (γ i, ) + π pf i, ) = θ ii, (p pf i, (γ ) δ i w i, 1 ν i, (γ i, ) + π pf i, )2, where he hird equaliy follows from (B.19) and he las from f i, (γ ) = ϕ i, + θ ii, (δ i w i, 1 + ν i, (γ i, ) π pf i, ). The above compuaion also implies ha ρ i,(p pf (γ )) = θ ii, (p pf i, (γ ) δ i w i, 1 ν i, (γ i, )+π pf i, ). We now show ha Π pf,2 i, (γ ) > 0. Noe ha Π pf,2 i, (γ ) = 1 θ ii, [ρ i, (p pf (γ ))] 2 > 0, where he inequaliy follows from he assumpion ha ρ i, ( ) > 0 for all p. This complees he proof of Proposiion Nex, we show Proposiion Since Π pf,2 i, and, hence, log(π pf,1 i, ( )) is well defined. Therefore, Since (γ ) > 0 for all γ, Π pf,1 i, (γ ) = Π pf,2 i, (γ )ψ i, (γ ) > 0 log(π pf,1 i, (γ )) = log(θ ii, ) + 2 log(p pf i, (γ ) δ i w i, 1 ν i, (γ i, ) + π pf i, ) + log(ψ i,(γ )). (B.20) p pf j, (γ ) = N l=1 by direc compuaion, 2 log(π pf,1 i, (γ )) γ i, γ j, (A 1 ) jl f l, (γ ) = = 2(1 θ ii,(a 1 By Lemma 24(a,b), 1 θ ii, (A 1 Because ψ i, ( ) saisfies (3.3), and, hus, G pf,1 N l=1 [(A 1 (p pf i, (γ ) δ i w i, 1 ν i, (γ i, ) + π pf ) jl (ϕ l, + θ ll, (δ l w l, 1 + ν l, (γ l, ) π pf ))], for all j, ) ii )θ jj, (A 1 ) ij ν i, (γ i,)ν j, (γ j,) + 2 log(ψ i,(γ )), for all j i. i, )2 γ i, γ j, l, (B.21) ) ii > 0 and (A 1 ) ij 0. Thus, he firs erm of (B.21) is non-negaive. 2 log(π pf,1 i, (γ )) γ i, γ j, 2 log(ψ i, (γ )) γ i, γ j, 0, for all j i. is a log-supermodular game. The feasible acion se of player i, [0, γ i, ], is a compac subse of R. Therefore, by Theorem 2 in [194], he pure sraegy Nash equilibria of G pf,1 complee sublaice of R N We now show ha if ν i, (γ i, ) = γ i,, he Nash equilibrium of G pf,1 2 log(π pf,1 i, (γ )) γ 2 i, < 0, and 2 log(π pf,1 i, (γ )) γi, 2 > j i is a nonempy is unique. We firs show ha 2 log(π pf,1 i, (γ )) γ i, γ j,, for all i and γ. (B.22) Since ν l, (γ l, ) = γ l, for all l (i.e., ν l, ( ) 1 for all l), direc compuaion yields ha 2 log(π pf,1 i, (γ )) γi, 2 = 2 log(ψ i, (γ )) γi, 2 2(1 θ ii, (A 1 ) ii ) 2 (p pf i, (γ ) δ i w i, 1 ν i, (γ i, ) + π pf i, )2. 237

252 Inequaliy (3.3) implies ha γ 2 i, log(ψ i, (γ )) < 0 and, hus, γ 2 i, log(π pf,1 i, (γ )) < 0. Moreover, and j i 2 log(π pf,1 i, (γ )) γ 2 i, = 2 log(ψ i, (γ )) γi, log(π pf,1 i, (γ )) = 2 log(ψ i, (γ )) + γ i, γ j, γ i, γ j, j i j i 2(1 θ ii, (A 1 ) ii ) 2 (p pf i, (γ ) δ i w i, 1 ν i, (γ i, ) + π pf i, )2 2(1 θ ii, (A 1 ) ii )θ jj, (A 1 ) ij (p pf i, (γ ) δ i w i, 1 ν i, (γ i, ) + π pf i, )2. Inequaliy (3.3) implies ha 2 log(ψ i, (γ )) γi, 2 > 2 log(ψ i, (γ )). γ i, γ j, j i Lemma 24(b) implies ha 1 θ ii, (A 1 ) ii > 0. Moreover, Lemma 24(c) suggess ha 1 (A 1 ) ii θ ii, > j i (A 1 ) ij θ jj, and, hence, 2(1 θ ii, (A 1 ) ii ) 2 (p pf i, (γ ) δ i w i, 1 ν i, (γ i, ) + π pf > i, )2 Therefore, inequaliy (B.22) holds for all γ. Because G pf,1 sraegy Nash equilibria ˆγ pf j i 2(1 θ ii, (A 1 ) ii )θ jj, (A 1 ) ij (p pf i, (γ ) δ i w i, 1 ν i, (γ i, ) + π pf. i, )2 is a log-supermodular game, by Theorem 5 in [124], if here are wo disinc pure γ pf, we mus have ˆγ pf i, γ pf i, for each i, wih he inequaliy being sric for some i. Wihou loss of generaliy, we assume ha ˆγ pf 1, > γ pf 1, and ˆγ pf 1, γ pf for each i. Lemma 23 yields ha 1, ˆγ pf i, γ pf i, log(π pf,1 1, (ˆγpf )) γ 1, log(πpf,1 1, (γpf )) γ 1, (B.23) Since γ1, γi, log(π pf,1 1, (γ )) is Lebesgue inegrable for all i 1 and γ, Newon-Leibniz formula implies ha log(π pf,1 1, (ˆγpf )) γ 1, log(πpf,1 1, (γpf )) 1 = γ 1, 1 <0, N (ˆγ pf j, s=0 j=1 N (ˆγ pf s=0 j=1 γ pf j, log(π pf,1 ) 2 1, ((1 s)γpf + sˆγ pf )) ds γ 1, γ j, 1, γ pf log(π pf,1 1, ) 2 1, ((1 s)γpf γ 1, γ j, + sˆγ pf )) where he firs inequaliy follows from ˆγ pf 1, γ pf 1, ˆγ pf i, γ pf i, for all i, and he second from (B.22), and ˆγ pf 1, γ pf 1, > 0. This conradics (B.23). Thus, G pf,1 has a unique pure sraegy Nash equilibrium γ pf. We now show ha he unique pure sraegy Nash equilibrium γ pf ds can be characerized by he sysem of firs-order condiions (3.26). Firs, (B.22) implies ha log(π pf,1 i, (, γ i, )) is sricly concave in γ i, for any i and any fixed γ i,. i.e., for each i, γi, log(π pf,1 i, γi, log(π pf,1 i, (γ pf )) 0 if γ pf i, Hence, γ pf (γ pf )) 0 if γ pf i, mus saisfy he sysem of firs-order condiions, = 0; γi, log(π pf,1 i, = γ i,. Differeniae (B.20), and we have γi, log(π pf,1 i, (γ )) = γ i, ψ i, (γ ) ψ i, (γ ) (γ pf )) = 0 if γ pf i, (0, γ i, ); and 2(1 θ ii, (A 1 ) ii )ν i, (γ i,) p pf i, (γ ) δ i w i, 1 ν i, (γ i, ) + π pf i, 238.

253 Thus, γ pf saisfies he sysem of firs-order condiions (3.26). Since, by Proposiion 3.5.2(c), Π pf,2 i, (γ pf ) > 0 and ψ i, (γ pf ) > 0, we have Π pf,1 i, Proposiion Nex, we show ha {(γ pf i,, ppf = Π pf,2 i, (γ pf i, (γ ), Λ i, y pf i, in he subgame of period. By Proposiion 3.5.1, y pf i, )ψ i, (γ pf ) > 0 for all i. This complees he proof of ρ i,(p pf (γ ))ψ i, (γ )) : 1 i N} is an equilibrium > 0, Λ i, y pf i, ρ i,(p pf (γ ))ψ i, (γ ) > 0 for all i. Therefore, regardless of he saring invenory level in period, I i,, firm i could adjus is invenory o x pf i, (I, Λ, γ ) = Λ i, y pf i, ρ i,(p pf (γ ))ψ i, (γ ). Thus, {(γ pf i,, ppf i, (γ ), Λ i, y pf i, ρ i,(p pf (γ ))ψ i, (γ )) : 1 i N} forms an equilibrium in he subgame of period. In paricular, his equilibrium is he unique one, if ν i, (γ i, ) = γ i, for all i. Finally, we show ha here exiss a posiive vecor β pf V i, (I, Λ σ pf ) = w i, I i, + β pf i, Λ i,. By (3.22), we have ha V i, (I, Λ σ pf ) =J i, (γ pf i,, ppf i, (γpf ), Λ i, y pf i, = (β pf 1,, βpf 2,,, βpf N, ), such ha ρ i,(p pf =w i, I i, + (σ i β pf i, 1 µ i, + Π pf,1 i, )Λ i,. (γ pf ))ψ i, (γ pf ), I, Λ σ pf 1 ) Since β pf i, 1 > 0, βpf i, = δ iβ pf i, 1 µ i, + Π pf,1 i, > 0. This complees he inducion and, hus, he proof of Theorem 3.5.1, Proposiion 3.5.1, Proposiion 3.5.2, Proposiion 3.5.3, and Theorem Q.E.D. Proof of Proposiion 3.5.4: By Theorems , and Proposiions , i suffices o show ha, if here exiss a consan β pf s, 1 0, such ha V i, 1(I 1, Λ 1 σ pf 1 ) = w s,i i, 1 + β pf s, 1 Λ i, 1 for all i, we have: (a) he unique Nash equilibrium in G pf,3 is symmeric, i.e., y pf i, = y pf j, for all i, j; (b) he unique Nash equilibrium in G pf,2 (γ ) is symmeric if γ i, = γ j, for all i and j, (c), he unique Nash equilibrium in G pf,1, γ pf is symmeric, and (d) here exiss a consan β pf s, > 0, such ha V i, (I, Λ σ pf s, ) = w s, I i, + β pf s,λ i, for all i. Since V i,0 (I, Λ ) = w i,0 I i,0 for all i, he iniial condiion is saisfied wih β pf s,0 = 0. Firs, we observe ha y pf i, = y pf j, and π pf i, = π pf j, of Proposiion Thus, we omi heir proofs for breviy, and denoe y pf s, for each firm i in G pf,3. for all i and j follow direcly from he same proof := y pf i, and π pf s, = π pf i, Nex, we show ha if γ i, = γ j, for all i and j, p pf i, (γ ) = p pf j, (γ ). Direc compuaion yields ha, for he symmeric PF model, N of j, j=1 (A 1 ) ij is independen of i. Thus, if he value of γ j, is independen p pf i, (γ ) = N j=1 (A 1 ) ij f j, (γ ) = N [(A 1 ) ij (ϕ s, + θ sa, (δ s w s, 1 + ν s, (γ j, ) π pf s, ))] j=1 =(ϕ s, + θ sa, (δ s w s, 1 + ν s, (γ j, ) π pf s, )) N j=1 (A 1 ) ij, (B.24) which is independen of firm i, which we denoe as p pf s, (γ ). Noe ha he objecive funcions of G pf,1, {Π pf,1 i, (γ ) = θ sa, (p pf i, (γ ) δ s w s, 1 ν s, (γ i, ) + π pf s, )ψ s, (γ ) : 1 i N} 239

254 are symmeric. Thus, if here exiss an asymmeric Nash equilibrium γ pf, here exiss anoher Nash equilibrium γ pf γ pf, where γ pf is a permuaion of γ pf. This conradics he uniqueness of he Nash equilibrium in G pf,1. Thus, he unique Nash equilibrium in G pf,1 is symmeric, which we denoe as γ pf ss, = (γ pf s,, γ pf s,,, γ pf s, ). Hence, Π pf,1 i, = Π pf,1 i, (γ pf Thus, we denoe he payoff of each firm i in G pf,1 ss,) = Π pf,1 j, (γ pf ss,) = Π pf,1 j, > 0. as Π pf,1 s,. By Theorem 3.5.2(a), β pf i, = δ sβ pf s, 1 µ s, + Π pf,1 i, = δ s β pf s, 1 µ s, + Π pf,1 j, = β pf j, > 0. Thus, we denoe he PF marke size coefficien of each firm i as β pf s,. This complees he inducion and, hus, he proof of Proposiion Q.E.D. Proof of Theorem 3.5.3: Pars (a)-(b). The proof of pars (a)-(b) follows from he same argumen as ha of Theorem 3.4.3(a)-(b) and is, hence, omied. we have Par (c). Because p pf i, (γ ) = N j=1 (A 1 ) ij f j, (γ ) = N [(A 1 ) ij (ϕ j, + θ jj, (δ j w j, 1 + ν j, (γ j, ) π pf j, ))], j=1 π pf p pf i, (γ ) = θ jj, (A 1 ) ij 0, j, where he inequaliy follows from Lemma 24(a). Thus, p pf i, (γ ) is coninuously decreasing in π pf j, for each j. Par (c) follows. Par (d). We denoe he objecive funcion of each firm i in G pf,1 s, as Π pf,1 i, ( π pf s, ) o capure is dependence on π pf s,. The unique symmeric pure sraegy Nash equilibrium in G pf,1 s, is denoed as γ pf ss,(π pf s, ) o capure he dependence of he equilibrium on π pf s,, where γss,(π pf pf s, ) = (γ pf s, (π pf s, ), γ pf s, (π pf s, ),, γ pf s, (π pf s, )). We firs show ha, if π pf s, > π pf s,, γ pf s, ( π pf s, ) γ pf s, (π pf s, ). If, o he conrary, γ pf s, ( π pf s, ) < γ pf s, (π pf s, ), Lemma 23 yields ha γ1, log(π pf,1 1, (γpf s, ( π pf s, ) π pf s, )) γ1, log(π pf,1 1, (γpf s, (π pf s, ) π pf s, )), i.e., Noe ha γ1, log(ψ s, (γ pf s, ( π pf s, )) γ1, log(ψ s, (γ pf s, (π pf s, )) 2(1 θ sa, (A 1 ) ii )ν s,(γ pf s, ( π pf s, )) p pf s, (γss,( π pf pf s, )) δ s w s, 1 ν s, (γ pf s, ( π pf s, )) + π pf s, 2(1 θ sa, (A 1 ) ii )ν s,(γ pf s, (π pf s, )) p pf s, (γss,(π pf pf s, )) δ s w s, 1 ν s, (γ pf s, (π pf s, )) + π pf s, [p pf s, (γ pf ss,( π pf s, )) δ s w s, 1 ν s, (γ pf s, ( π pf s, )) + π pf s, ] [p pf s, (γ pf ss,(π pf s, )) δ s w s, 1 ν s, (γ pf s, (π pf s, )) + π pf s, ] N =(1 (A 1 ) 1j θ sa, )(ν s, (γ pf s, (π pf s, )) ν s, (γ pf s, ( π pf s, ))) + (1 >0 j=1 240 N (A 1 j=1 ) 1j θ sa, )( π pf s,. π pf s, ) (B.25)

255 where he inequaliy follows from Lemma 24(c). Thus, p pf s, (γss,( π pf pf s, )) δ s w s, 1 ν s, (γ pf s, ( π pf s, )) + π pf s, >p pf s, (γ pf ss,(π pf s, )) δ s w s, 1 ν s, (γ pf s, (π pf s, )) + π pf s, > 0. Lemma 24(b) implies ha 1 θ sa, (A 1 ) ii > 0. Hence, 2(1 θ sa, (A 1 ) ii )ν s,(γ pf s, ( π pf s, )) p pf s, (γss,( π pf pf s, )) δ s w s, 1 ν s, (γ pf s, ( π pf s, )) + π pf s, 2(1 θ sa, (A 1 ) ii )ν s,(γ pf s, (π pf s, )) p pf s, (γss,(π pf pf s, )) δ s w s, 1 ν s, (γ pf s, (π pf s, )) + π pf. s, Thus, we have γ1, log(ψ s, (γ pf s, ( π pf s, )) γ1, log(ψ s, (γ pf s, (π pf s, )). (B.26) By (3.3) and Newon-Leibniz formula, = < 0, γ1, log(ψ 1, (γ pf s, (π pf s, ))) γ1, log(ψ 1, (γ pf s, ( π pf 1 N (γ pf s, (π pf s, ) γ pf s, ( π pf s, ))[ 2 log(ψ s, (sγ pf s=0 j=1 s, ))) s, (π pf s, ) + (1 s)γ pf γ 1, γ j, s, ( π pf s, ))) which conradics (B.26). Therefore, γ pf s, (π pf s, ) is increasing in π pf s,. The coninuiy of γ pf s, (π pf s, ) in π pf s, follows direcly from ha Π pf,1 i, (γ π pf s, ) is wice coninuously differeniable in (γ, π pf s, ) and he implici funcion heorem. Nex we show ha if (3.17) holds, β pf s,(π pf s, ) is increasing in π pf s,. By Theorem 3.5.2(a), i suffices o show ha Π pf,1 s, (π pf s, ) := Π pf,1 s, (γss,(π pf pf s, ) π pf s, ) is increasing in π pf s,. Assume ha π pf s, > π pf s,. Since we have jus shown γ pf s, ( π pf s, ) γ pf s, (π pf s, ), (3.17) implies ha ψ s, (γ pf ss,( π pf s, )) ψ s, (γ pf ss,(π pf s, )). If γ pf s, ( π pf s, ) = γ pf s, (π pf s, ), p pf s, (γss,( π pf pf s, )) δ s w s, 1 ν s, (γ pf s, ( π pf s, ))+ π pf s, > p pf s, (γss,(π pf pf s, )) δ s w s, 1 ν s, (γ pf s, (π pf s, ))+π pf s,, and, hence, Π pf,1 s, ( π pf s, ) = θ sa, (p pf s, (γss,( π pf pf s, )) δ s w s, 1 ν s, (γ pf s, ( π pf s, )) + π pf s, ) 2 ψ s, (γss,( π pf pf s, )) > θ sa, (p pf s, (γ pf ss,(π pf s, )) δ s w s, 1 ν s, (γ pf s, (π pf s, )) + π pf s, ) 2 ψ s, (γ pf ss,(π pf s, )) = Π pf,1 s, (π pf s, ). If γ pf ss,( π pf s, ) > γ pf ss,(π pf s, ), Lemma 23 implies ha γ1, log(π pf,1 1, (γpf s, ( π pf s, ) π pf s, )) γ1, log(π pf,1 1, (γpf s, (π pf s, ) π pf s, )), i.e., ] ds γ1, log(ψ s, (γ pf s, ( π pf s, )) γ1, log(ψ s, (γ pf s, (π pf s, )) 2(1 θ sa, (A 1 ) ii )ν s,(γ pf s, ( π pf s, )) p pf s, (γss,( π pf pf s, )) δ s w s, 1 ν s, (γ pf s, ( π pf s, )) + π pf s, 2(1 θ sa, (A 1 ) ii )ν s,(γ pf s, (π pf s, )) p pf s, (γss,(π pf pf s, )) δ s w s, 1 ν s, (γ pf s, (π pf s, )) + π pf s,. 241

256 By (3.3) and Newon-Leibniz formula, Hence, = < 0, γ1, log(ψ 1, (γ pf s, ( π pf s, ))) γ1, log(ψ 1, (γ pf s, (π pf s, ))) 1 N (γ pf s, ( π pf s, ) γ pf s, (π pf s, ))[ 2 log(ψ s, ((1 s)γ pf s, (π pf s, ) + sγ pf γ 1, γ j, s=0 j=1 > 2(1 θ sa, (A 1 ) ii )ν s,(γ pf s, ( π pf s, )) p pf s, (γss,( π pf pf s, )) δ s w s, 1 ν s, (γ pf s, ( π pf s, )) + π pf s, 2(1 θ sa, (A 1 ) ii )ν s,(γ pf s, (π pf s, )) p pf s, (γss,(π pf pf s, )) δ s w s, 1 ν s, (γ pf s, (π pf s, )) + π pf. s, s, ( π pf s, ))) Because, by Lemma 24(b) and he convexiy of ν s, ( ), 1 θ sa, (A 1 ) ii > 0 and ν s,(γ pf s, ( π pf s, )) ν s,(γ pf s, (π pf s, )), we have p pf s, (γss,( π pf pf s, )) δ s w s, ν s, (γ pf s, ( π pf s, ))+ π pf s, > p pf s, (γss,(π pf pf s, )) δ s w s, 1 ν s, (γ pf s, (π pf s, ))+π pf s,. Therefore, Π pf,1 s, ( π pf s, ) = θ sa, (p pf s, (γss,( π pf pf s, )) δ s w s, 1 ν s, (γ pf s, ( π pf s, )) + π pf s, ) 2 ψ s, (γss,( π pf pf s, )) > θ sa, (p pf s, (γ pf ss,(π pf s, )) δ s w s, 1 ν s, (γ pf s, (π pf s, )) + π pf s, ) 2 ψ s, (γ pf ss,(π pf s, )) = Π pf,1 s, (π pf s, ). We have, hus, shown ha βs,(π pf pf s, ) is increasing in π pf s,. The coninuiy of βs,(π pf pf s, ) in π pf s, follows direcly from ha of γ pf s, (π pf proof of par (d). s, ) and ha Π pf,1 Par (e). By par (d), we have ha γ pf s, i, ] ds (γ π pf s, ) is coninuous in (γ, π pf s, ). This concludes he is coninuously increasing in π pf s, and, hus, β pf s, 1. By par (c), we have ha p pf i, (γ ) is coninuously decreasing in π pf s, and, hus, β pf s, 1. Moreover, if (3.17) holds, par (d) yields ha β pf s, is coninuously increasing in π pf s, and, hus, β pf s, 1 as well. This complees he proof of par (e). Q.E.D. Proof of Theorem 3.5.4: Par (a). Par (a) follows from he same argumen as he proof of Theorem 3.4.4(a) and is, hence, omied. Par (b). By par (a), π pf i, p pf i, (γ ) for each firm i and each γ. ha π pf i, When he PF model is symmeric, N p pf i, (γ ) p pf i, (γ ) = ( for each i. Hence, Theorem 3.5.3(c) yields ha p pf i, (γ ) j=1 θ jj,(a 1 N j=1 ) ij is independen of i. Direc compuaion yields θ jj, (A 1 ) ij )(π pf s, π pf s, ) 0, for all γ, which is independen of i. Thus, (3.17) and Newon-Leibniz formula imply ha 1 ρ s, ( p pf (γ )) ρ s, (p pf (γ )) = s=0 N ( p pf i, (γ ) p pf i, (γ )) ρ s,((1 s)p pf (γ ) + s p pf (γ )) ds 0. p i, i=1 242

257 Hence, ρ s, (p pf (γ )) ρ s, ( p pf (γ )). Since y pf s, (I, Λ ) S and γ [0, γ s, ] N, ỹ pf s,, Theorem 3.5.2(b) implies ha, for any x pf i, (I, Λ, γ ) = y pf s, ρ s, (p pf (γ ))ψ s, (γ ) ỹ pf s, ρ s, ( p pf (γ ))ψ s, (γ ) = x pf i, (I, Λ, γ ). This complees he proof of par (b). γ pf s, Par (c). Because π pf s, γ pf s, π pf s,, Theorem 3.5.3(d) yields ha γ pf s, γ pf s, and, hence, γ pf i, (I, Λ ) = = γ pf s, (I, Λ ) for each i and (I, Λ ) S. This complees he proof of par (c). Q.E.D. Proof of Theorem 3.5.5: Par (a). We show par (a) by backward inducion. More specifically, we show ha if ˆα s, (z ) α s, (z ) for all z and pf ˆβ s, 1 βpf s, 1, (i) ˆπpf s, π pf s,, (ii) ˆp pf s, (γ ) p pf s, (γ ), (iii) ˆp pf i, (I, Λ, γ ) p pf i, (I, Λ, γ ) for each i, (I, Λ ) S, and γ [0, γ s, ] N, (iv) ˆγ pf s, γ pf s,, (v) ˆγ pf i, (I, Λ ) γ pf s, (I, Λ ) for each i and (I, Λ ) S, and (vi) iniial condiion is saisfied. ˆβ pf s, β pf s,. Since The same argumen as he proof of Theorem 3.4.5(a) implies ha ˆπ pf s, pf ˆβ s,0 = βpf s,0 = 0, he π pf s,. Hence, Theorem 3.5.3(c) implies ha ˆp pf i, (γ ) p pf i, (γ ) for all i and γ. Thus, ˆp pf i, (I, Λ, γ ) = ˆp pf i, (γ ) p pf i, (γ ) = p pf i, (I, Λ, γ ) for each i, (I, Λ ) S, and γ [0, γ s, ] N. Analogously, Theorem 3.5.3(d) implies ha ˆγ pf s, γ pf s,. Hence, ˆγ pf i, (I, Λ ) = ˆγ pf s, γ pf s, Theorem 3.5.3(d), under inequaliy (3.17), ˆπ pf s, inducion and, hus, he proof of par (a). = γ pf i, (I, Λ ) for each i and all (I, Λ ) S. By π pf pf s, implies ha ˆβ s, βs,. pf This complees he Par (b). By par (a), i suffices o show ha, if ˆα s, (z ) α s, (z ) for all z, ˆκ sa,(z i, ) κ sa,(z i, ) for all z i,, and pf ˆβ s, 1 βpf s, 1, we have (i) ŷpf s, y pf s, and (ii) ˆx pf i, (I, Λ, γ ) x pf i, (I, Λ, γ ) for each i, (I, Λ ) S, and γ [0, γ s, ] N. The same argumen as he proof of Theorem 3.4.5(b) suggess ha ŷ pf s, y pf s,. We now show ha ˆx pf i, (I, Λ, γ ) x pf i, (I, Λ, γ ) for each i, (I, Λ ) S, and γ [0, γ s, ] N. Because he PF model is symmeric, N j=1 θ jj,(a 1 ) ij is independen of i. Direc compuaion yields ha p pf i, (γ ) ˆp pf i, (γ ) = ( N j=1 θ jj, (A 1 ) ij )(ˆπ pf s, π pf s, ) 0, for all γ, which is independen of i. Thus, (3.17) and Newon-Leibniz formula implies ha 1 ρ s, (p pf (γ )) ρ s, (ˆp pf (γ )) = s=0 Hence, ρ s, (ˆp pf N (p pf i, (γ ) ˆp pf i, (γ )) ρ s,((1 s)ˆp pf (γ ) + sp pf (γ )) ds 0. p i, i=1 (γ )) ρ s, (p pf (γ )) for all γ. Since ŷ pf s, y pf s,, Theorem 3.5.2(b) implies ha, for any (I, Λ ) S and γ [0, γ s, ] N, ˆx pf i, (I, Λ, γ ) = ŷ pf s, ρ s, (ˆp pf (γ ))ψ s, (γ ) y pf s, ρ s, (p pf (γ ))ψ s, (γ ) = x pf i, (I, Λ, γ ). This complees he proof of par (b). Q.E.D. Proof of Theorem 3.5.6: We show pars (a)-(b) ogeher by backward inducion. More specifically, we show ha if β pf s, 1 βpf s, 2, (i) ypf s, y pf s, 1, (ii) ppf i, (γ) ppf i, 1 (γ) for all γ [0, γ s,] N, 243

258 (iii) p pf i, (I, Λ, γ) ppf i, 1 (I, Λ, γ) for each i, (I, Λ) S, and γ [0, γ s,] N, (iv) γ pf s, γ pf i, (I, Λ) γpf i, 1 γ pf s, 1, (v) (I, Λ) for each i and (I, Λ) S, (vi) xpf i, (I, Λ, γ) xpf i, 1 (I, Λ, γ) for each i, (I, Λ) S, and γ [0, γ s, ] N, and (vii) β pf s, β pf s, 1. Since, by Theorem 3.5.2(a), βpf s,1 βpf s,0 = 0. Thus, he iniial condiion is saisfied. Since he model is saionary, by Theorem 3.5.3(a), β pf s, 1 βpf s, 2 Since π pf s, suggess ha ypf s, y pf s, 1. is increasing in β pf s, 1, βpf s, 1 βpf s, 2 implies ha πpf s, π pf s, 1. Theorem 3.5.3(c) yields ha p pf s, (γ) p pf s, 1 (γ) for all γ [0, γ s,] N. p pf i, γ pf i, Theorem 3.5.3(e) implies ha γ pf s, γ pf s, 1. Hence, (I, Λ, γ) = ppf i, (γ) ppf i, 1 (γ) = ppf i, 1 (I, Λ, γ) for each i, (I, Λ) S, and γ [0, γ s,] N, and (I, Λ) = γpf s, γ pf s, 1 = γpf i, 1 (I, Λ) for each i and (I, Λ) S. We now show ha xpf i, (I, Λ, γ) x pf i, 1 (I, Λ, γ) for each i, (I, Λ) S, and γ [0, γ]n. Because he PF model is symmeric, N j=1 θ jj,(a 1 ) ij is independen of i. Direc compuaion yields ha p pf i, 1 (γ) ppf i, (γ) = ( N j=1 θ jj (A 1 ) ij )(π pf s, π pf s, 1 ) 0, for all γ, which is independen of i. Thus, (3.17) and Newon-Leibniz formula implies ha 1 ρ s (p pf 1 (γ)) ρ s(p pf (γ)) = s=0 Hence, ρ s (p pf (I, Λ) S and γ [0, γ s, ] N, N i=1 (p pf i, 1 (γ) ppf i, (γ)) ρ s((1 s)p pf (γ) + sp pf 1 (γ)) ds 0. p i (γ)) ρ s (p pf 1 (γ)) for all γ. Since ypf s, y pf s, 1, Theorem 3.5.2(b) implies ha, for any x pf i, (I, Λ, γ) = ypf s, ρ s (p pf (γ))ψ s (γ ) y pf s, 1 ρ s(p pf 1 (γ ))ψ s, (γ ) = x pf i, 1 (I, Λ, γ). Finally, we show ha β pf s, β pf s, 1. Since he model is saionary and πpf s, π pf s, 1, βpf s, β pf s, 1 follows from Theorem 3.5.3(d) immediaely. This complees he inducion and, hus, he proof of Theorem Q.E.D. Proof of Theorem 3.6.1: Par (a). Because β pf s, 1 βsc s, 1, π pf s, as he proof of Theorem 3.4.3(a) implies ha y pf s, We now show ha, if π pf s, πs, sc, γ pf s, y sc s,. γ sc s,. π sc s,. The same argumen Proposiion implies ha p pf (γss,) pf = A 1 f (γss,). pf By Proposiion 3.4.2, he equilibrium sales prices, p sc ss,, saisfy he sysem of firs-order equaions (3.15). Equivalenly, p sc ss, = A 1 f (γss,). sc We assume, o he conrary, ha γ pf s, γ1, log(π sc 1,(γ sc ss,, p sc ss,)), i.e., < γ sc s,. Lemma 23 implies ha γ1, log(π pf,1 1, (γpf ss,)) 2(1 θ sa, (A 1 ) 11 )ν s,(γ pf s, ) N j=1 (A 1 ) 1j [ϕ sa, + θ sa, (δ s w s, 1 + ν s, (γ pf s, ) π pf s, )] δ s w s, 1 ν s, (γ pf s, ) + π pf s, + γ1, log(ψ s, (γ pf ss,)) ν s,(γ s, sc ) N j=1 (A 1 ) 1j [ϕ sa, + θ sa, (δ s w s, 1 + ν s, (γs, sc ) πs, sc )] δ s w s, 1 ν s, (γs, sc ) + πs, sc + γ1, log(ψ s, (γ sc ss,)). (B.27) 244

259 Inequaliy (3.3) and he Newon-Leibniz formula imply ha γ1, log(ψ s, (γ sc ss,)) γ1, log(ψ s, (γ pf ss,)) = By (B.27), < < 0. 1 N (γs, sc s=0 j=1 γ pf s, )[ 2 log(ψ s, ((1 s)γ pf s, + sγ sc γ 1, γ j, 2(1 θ sa, (A 1 ) 11 )ν s,(γ pf s, ) N j=1 (A 1 ) 1j [ϕ sa, + θ sa, (δ s w s, 1 + ν s, (γ pf s, ) π pf s, )] δ s w s, 1 ν s, (γ pf s, ) + π pf s, ν s,(γ s, sc ) N. j=1 (A 1 ) 1j [ϕ sa, + θ sa, (δ s w s, 1 + ν s, (γs, sc ) πs, sc )] δ s w s, 1 ν s, (γs, sc ) + πs, sc Lemma 24(b) suggess ha 0 2(1 θ sa, (A 1 ) 11 )ν s,(γ pf s, ) ν s,(γ s, sc ). Hence, s, )) ] ds < N (A 1 j=1 N (A 1 j=1 ) 1j [ϕ sa, + θ sa, (δ s w s, 1 + ν s, (γ pf s, ) π pf s, )] δ s w s, 1 ν s, (γ pf s, ) + π pf s, ) 1j [ϕ sa, + θ sa, (δ s w s, 1 + ν s, (γ sc s, ) π sc s, )] δ s w s, 1 ν s, (γ sc s, ) + π sc s,. (B.28) Since π pf s, π sc s, implies ha 1 N = = N (A 1 j=1 N (A 1 j=1 N (A 1 j=1 N (A 1 j=1 and ν s, (γ pf s, ) ν s, (γs, sc ), π pf s, ν s, (γ pf s, ) πs, sc ν s, (γs, sc ). Lemma 24(c) j=1 (A 1 ) 1j θ sa, > 0. Therefore, ) 1j [ϕ sa, + θ sa, (δ s w s, 1 + ν s, (γ pf s, ) π pf s, )] δ s w s, 1 ν s, (γ pf s, ) + π pf s, ) 1j (ϕ sa, + θ sa, δ s w s, 1 ) δ s w s, 1 + (1 ) 1j (ϕ sa, + θ sa, δ s w s, 1 ) δ s w s, 1 + (1 N (A 1 j=1 N (A 1 j=1 ) 1j θ sa, )(π pf s, ) 1j θ sa, )(π sc s, ν s, (γ pf s, )) ν s, (γ sc s, )) ) 1j [ϕ sa, + θ sa, (δ s w s, 1 + ν s, (γ sc s, ) π sc s, )] δ s w s, 1 ν s, (γ sc s, ) + π sc s,, which conradics he inequaliy (B.28). Therefore, γ pf s, γ sc s,. This complees he proof of par (a). Par (b). We firs show, by backward inducion, ha, if θ sb, = 0 for each, β pf s, β sc s, for each. Since β pf s,0 = βsc s,0 = 0, he iniial condiion is saisfied. Now we prove ha if β pf s, 1 βsc s, 1 and θ sb, = 0, we have β pf s, β sc s,. Firs, we observe ha if θ sb, = 0, (A 1 shows ha γ pf s, γs, sc. If γ pf s, = γs, sc, Π pf,1 s, ) 11 θ sa, = 1 2 and, hus, 2(1 θ sa,(a 1 ) 11 ) = 1. Par (a) =θ sa, ((A 1 f (γss,)) pf i δ s w s, 1 ν s, (γ pf s, ) + π pf s, ) 2 ψ s, (γss,) pf θ sa, ((A 1 f (γss,)) sc i δ s w s, 1 ν s, (γs, sc ) + πs, sc ) 2 ψ s, (γss,) sc =Π sc s,, where he inequaliy follows from π pf s, π sc s,. 245

260 If γ pf s, > γs, sc, Lemma 23 implies ha γ1, log(π pf,1 1, (γpf ss,)) γ1, log(π sc 1,(γss,, sc p sc ss,)), i.e., 2(1 θ sa, (A 1 ) 11 )ν s,(γ pf s, ) (A 1 f (γss,)) pf 1 δ s w s, 1 ν s, (γ pf s, ) + π pf s, ν s,(γ sc s, ) (A 1 f (γss,)) sc 1 δ s w s, 1 ν s, (γs, sc ) + πs, sc Inequaliy (3.3) and he Newon-Leibniz formula imply ha + γ1, log(ψ s, (γ pf ss,)) + γ1, log(ψ s, (γ sc ss,)). (B.29) 1 γ1, log(ψ s, (γss,)) pf γ1, log(ψ s, (γss,)) sc = By (B.29), we have 2(1 θ sa, (A 1 ) 11 )ν s,(γ pf s, ) (A 1 f (γss,)) pf 1 δ s w s, 1 ν s, (γ pf s, ) + π pf s, Because 2(1 θ sa, (A 1 ) 11 ) = 1 and γ pf s, > γ sc (A 1 f (γss,)) pf 1 δ s w s, 1 ν s, (γ pf s, ) + π pf s, By inequaliy (3.17), γ pf s, Π pf,1 s, > γ sc s, N (γ pf s, s=0 j=1 γs, sc )[ 2 log(ψ s, ((1 s)γs, sc + sγ pf s, )) ] ds < 0. γ 1, γ j, ν s,(γ s, sc ) > (A 1. f (γss,)) sc 1 δ s w s, 1 ν s, (γs, sc ) + πs, sc s,, 2(1 θ sa, (A 1 ) 11 )ν s,(γ pf s, ) ν s,(γ sc s, ). Therefore, > (A 1 f (γss,)) sc 1 δ s w s, 1 ν s, (γs, sc ) + πs, sc > 0. implies ha ψ s, (γ pf ss,) > ψ s, (γ sc ss,). Thus, we have =θ sa, ((A 1 f (γss,)) pf 1 δ s w s, 1 ν s, (γ pf s, ) + π pf s, ) 2 ψ s, (γss,) pf >θ sa, ((A 1 f (γss,)) sc 1 δ s w s, 1 ν s, (γs, sc ) + πs, sc ) 2 ψ s, (γss,) sc =Π sc s,. We have hus shown ha if β pf s, 1 βsc s, 1, Π pf,1 s, 3.5.2(a), β pf s, = δ s β pf s, 1 µ s, + Π pf,1 s, δ s βs, 1µ sc s, + Π sc s, = βs,. sc Π sc s,. By Theorem 3.4.2(a) and Theorem This complees he inducion and, by par (a), he proof of par (b) for he case θ sb, = 0. For any fixed θ sa,, boh β pf s, and β sc s, are coninuous in θ sb,. Thus, for each period, here exiss a ϵ 0, such ha, if θ sb, ϵ θ sa,, β pf s, βs,. sc I remains o show ha ϵ 1 N 1. This inequaliy follows from he diagonal dominance condiion ha θ sa, > (N 1)θ sb,. This complees he proof of par (b). Q.E.D. B.2 Sufficien Condiions for he Monooniciy of π sc s, [π pf s, ] in β sc s, 1 [β pf s, 1 ] In his secion, we give some sufficien condiions under which π sc s, [β pf s, 1 ]. Observe ha, if = 1, βsc s, 1 = β pf s, 1 = 0. So we only consider he case 2. [π pf s, ] is increasing in β sc s, 1 We define he N player noncooperaive game, G s,, as he symmeric game wih each player i s payoff funcion given by π i, (y ) = (δ s w s, 1 w s, )y i, L s, (y i, ) + δ s β(κ sa, (E[y + i, ξ i,]) j i κ sb, (E[y + j, ξ j,])), and feasible se given by R +. Hence, G sc,2 s, [G pf,3 s, ] can be viewed as G s, wih β = βs, 1 sc [β = β pf s, 1 ]. By Proposiions and 3.5.4, G s, has a unique symmeric pure sraegy Nash equilibrium. Thus, we use 246

261 ys,(β) and πs,(β) o denoe he equilibrium sraegy and payoff of each player in he game G s, wih parameer β. Le ys,(β; λ, 1) and πs,(β; λ, 1) (λ > 0) be he equilibrium sraegy and payoff of each firm in G s, (λ, 1), where G s, (λ, 1) is idenical o G s, excep ha α s, (z ) is replaced wih κ sa, (z i, ) 1 λ ( j i κ sb,(z j, )) in he objecive funcion π i, ( ), i.e., π i, (y ) = (δ s w s, 1 w s, )y i, L s, (y i, ) + δ s β(κ sa, (E[y + i, ξ i,]) 1 λ ( j i κ sb, (E[y + j, ξ j,]))). Analogously, le y s,(β; λ, 2) and π s,(β; λ, 2) (λ 0) be he equilibrium sraegy and payoff of each firm in G s, (λ, 2), where G s, (λ, 2) is idenical o G s, excep ha wih α s, (z ) is replaced wih α s, (z ) + λ in he objecive funcion π i, ( ), i.e., π i, (y ) = (δ s w s, 1 w s, )y i, L s, (y i, ) + δ s β(κ sa, (E[y + i, ξ i,]) j i κ sb, (E[y + j, ξ j,]) + λ). Finally, le y s,(β; λ, 3) and π s,(β; λ, 3) (λ > 0) be he equilibrium sraegy and payoff of each firm in G s, (λ, 3), where G s, (λ, 3) is idenical o G s, excep ha α s, (z ) is replaced wih λα s, (z ) in he objecive funcion π s, ( ), i.e., π i, (y ) = (δ s w s, 1 w s, )y i, L s, (y i, ) + δ s βλ(κ sa, (E[y + i, ξ i,]) j i κ sb, (E[y + j, ξ j,])). (3.17), In some of our analysis below, we assume ha α s, ( ) saisfies he monooniciy condiion similar o N i=1 α s, (z ) z i, > 0. (B.30) i.e., a uniform increase in he curren expeced fill raes gives rise o a higher expeced marke size of each firm in he nex period. Firs, we give a lower bound for he value of βs, 1 sc and β pf s, 1. By Theorem 3.4.2(a) and Theorem 3.5.2(a), β sc s, 1 β s, 1 and β pf s, 1 β s, 1, where 1 β s, 1 := Π s,1 (δ s µ s,τ ), wih Π s,1 := min{π sc s,1, Π pf,1 s,1 } > 0. Thus, we assume in his secion ha β β s, 1 > 0. τ=1 Le he densiy of ξ s, be defined as q s, ( ) = F s,( ) and is failure rae defined as r s, ( ) := q s, ( )/ F s, ( ). We have he following lemma on he Lipschiz coninuiy of y s,(β) and y s,(β; λ, i) (i = 1, 2, 3). Lemma 25 If κ sa, ( ) is wice coninuously differeniable and he failure rae of ξ s, is bounded from below by r s, > 0 on is suppor, here exiss a consan K s, > 0, independen of λ, i, and β, such ha ys,( ˆβ) ys,(β) K s, ˆβ β and ys,( ˆβ; λ, i) ys,(β; λ, i) K s, ˆβ β for all λ > 0, i = 1, 2, 3, and ˆβ, β

262 Proof: Since κ sa, ( ) is wice coninuously differeniable, by he implici funcion heorem, y s,(β) and y s,(β; λ, i) (i = 1, 2, 3) are coninuously differeniable in β wih he derivaives given by: and y s,(β) = y s,(β; λ, 1) β = y s,(β; λ, 2) β β δ s Fs, (ys,(β))κ sa,(e[ys,(β) ξ s, ]) = L (ys,(β)) + δ s βq s, (ys,(β))κ sa,(e[ys,(β) ξ s, ]) δ s β F s,(y 2 s,(β))κ sa,(e[ys,(β) ξ s, ]), y s,(β; λ, 3) β λδ s Fs, (ys,(β))κ sa,(e[ys,(β) ξ s, ]) = L (ys,(β)) + λδ s βq s, (ys,(β))κ sa,(e[ys,(β) ξ s, ]) λδ s β F s,(y 2 s,(β))κ sa,(e[ys,(β) ξ s, ]). Observe ha δ s Fs, (y s,(β))κ sa,(e[y s,(β) ξ s, ]) L (y s,(β)) + δ s βq s, (y s,(β))κ sa,(e[y s,(β) ξ s, ]) δ s β F 2 s,(y s,(β))κ sa,(e[y s,(β) ξ s, ]) δ s F s, (y s,(β))κ sa,(e[y s,(β) ξ s, ]) δ s βq s, (y s,(β))κ sa,(e[y s,(β) ξ s, ]) 1 β s, 1 r s, (y s,(β)) 1 β s, 1 r s,, where he firs inequaliy follows from he convexiy of L s, ( ) and he concaviy of κ sa, ( ), he second from κ sa,( ) 0, and he las from r s, ( ) r s,. Analogously, we have λδ s Fs, (y s,(β))κ sa,(e[y s,(β) ξ s, ]) L (y s,(β)) + λδ s βq s, (y s,(β))κ sa,(e[y s,(β) ξ s, ]) λδ s β F 2 s,(y s,(β))κ sa,(e[y s,(β) ξ s, ]) λδ s F s, (y s,(β))κ sa,(e[y s,(β) ξ s, ]) λδ s βq s, (y s,(β))κ sa,(e[y s,(β) ξ s, ]) 1 β s, 1 r s, (y s,(β)) 1 β s, 1 r s,. By he mean value heorem, ys,( ˆβ) ys,(β) = ˆβ β y s,( β) K s, β ˆβ β, where β is a real number ha lies beween β and ˆβ, and K s, := 1 β s, 1 r s,. The inequaliy y s,( ˆβ; λ, i) y s,(β; λ, i) K s, ˆβ β for all λ > 0 and i = 1, 2, 3 follows from exacly he same argumen. Q.E.D. We remark ha he assumpion ha he failure rae r s, ( ) is uniformly bounded away from 0 is no a resricive assumpion, and can be saisfied by, e.g., all he disribuions ha saisfy (i) he increasing failure rae propery, and (ii) he densiy q s, ( ) being posiive on he lower bound of is suppor. The same argumen as he proof of Theorem 3.4.3(a) and Theorem 3.5.3(a) imply ha, for all ˆβ > β, y s,( ˆβ) y s,(β) and y s,( ˆβ; λ, i) y s,(β; λ, i) (i = 1, 2, 3). We now characerize sufficien condiions for π s,(β) and π s,(β; λ, i) (i = 1, 2, 3) o be increasing in β. Lemma 26 The following saemens hold: (a) If κ sb, ( ) κ 0 sb, for some consan κ0 sb,, π s,(β) is increasing in β. (b) Assume ha α s, ( ) > 0 for all z and ha he condiions of Lemma 25 hold, we have: (i) If κ sb, ( ) is Lipschiz coninuous, here exiss an Ms, 1 < +, such ha for all λ Ms,, 1 πs,(β; λ, 1) is increasing in β. 248

263 (ii) If he monooniciy condiion (B.30) holds, here exiss an Ms, 2 < +, such ha for all λ M 2 s,, π s,(β; λ, 2) is increasing in β. (iii) If he monooniciy condiion (B.30) holds, here exiss an M 3 s, < +, such ha for all λ Ms,, 3 πs,(β; λ, 3) is increasing in β. Proof: Par (a). Observe ha, δ s βκ sa, (E[y + i, ξ i,]) is increasing in β for any y i,. Therefore, πs,(β) = max{(δ s w s, 1 w s, )y i, L s, (y i, ) + δ s βκ sa, (E[y + i, ξ i,]) (N 1)κ 0 sb, : y i, 0} is increasing in β. This complees he proof of par (a). Par (b-i). Le ˆβ > β, and k < + be he Lipschiz consan for κ sb, ( ). Since α s, ( ) is a coninuous funcion on a compac suppor, α s, ( ) > 0 for all z implies ha α s, ( ) α s, > 0 for some consan α s,. We define ζ i, (y i, ) := (δ s w s, 1 w s, )y i, L s, (y i, ) + δ s βκ sa, (E[y i, ξ i, ]). By he envelope heorem, ζ i, (ys,(β; λ, 1)) = δ s κ sa, (E[y β s,(β; λ, 1) ξ i, ]) δ s α s, > 0, where he firs inequaliy follows from κ sa, (z i, ) α s, (z ) α s,. By he mean value heorem and ˆβ > β, ζ i, (y s,( ˆβ; λ, 1)) ζ i, (y s,(β; λ, 1)) δ s α s, ( ˆβ β). (B.31) A he same ime, since α s,τ ( ), ρ s,τ ( ), and ψ s,τ ( ) are all uniformly bounded from above for τ 1, β sc s, 1 and β pf s, 1 have a uniform upper bound, which we denoe as β s, 1 < +. On he oher hand, δ s λ (N 1)[ ˆβκ sb, (E[y s,( ˆβ; λ, 1) ξ s, ]) βκ sb, (E[y s,(β; λ, 1) ξ s, ])] = δ s λ (N 1)[ ˆβκ sb, (E[y s,( ˆβ; λ, 1) ξ s, ]) ˆβκ sb, (E[y s,(β; λ, 1) ξ s, ]) + ˆβκ sb, (E[y s,(β; λ, 1) ξ s, ]) βκ sb, (E[y s,(β; λ, 1) ξ s, ])] (B.32) δ s λ (N 1)[ β s, 1 k (y s,( ˆβ; λ, 1) y s,(β; λ, 1)) + ( ˆβ β) κ sb, ] δ s λ (N 1)( β s, 1 k K s, + κ sb, )( ˆβ β), where he firs inequaliy follows from he Lipschiz coninuiy of κ sb, ( ), y s,( ˆβ; λ, 1) y s,(β; λ, 1), and E[y s,( ˆβ; λ, 1) ξ s, ] E[y s,(β; λ, 1) ξ s, ] y s,( ˆβ; λ, 1) y s,(β; λ, 1), wih κ sb, := max{κ sb, (z i, ) : z i, [0, 1]} < +, and he second from Lemma 25. Define If λ M 1 s,, M 1 s, := (N 1)( β s, 1 k K s, + κ sb, ) α s, < +. π s,( ˆβ; λ, 1) π s,(β; λ, 1) = ζ i, (y s,( ˆβ; λ, 1)) ζ i, (y s,(β; λ, 1)) (N 1)δ s [ λ ˆβκ sb, (ys,( ˆβ; λ, 1)) βκ sb, (ys,(β; λ, 1))] (δ s α s, δ s λ (N 1)( β s, 1 k K s, + κ sb, ))( ˆβ β) (δ s α s, δ s α s, )( ˆβ β) = 0, 249

264 where he firs inequaliy follows from (B.31) and (B.32), and he second from λ Ms,. 1 This esablishes par (b-i). Par (b-ii). Le H s, (y i, ) := (δ s w s, 1 w s, )y i, L s, (y i, ). Since δ s w s, 1 w s, h s, H s,(y i, ) b s, + δ s w s, 1 w s,, H s, ( ) is Lipschiz coninuous wih he Lipschiz consan equal o l := max{ δ s w s, 1 w s, h s,, b s, + δ s w s, 1 w s, } < +. Thus, H s, (y s,(β; λ, 2)) H s, (y s,( ˆβ; λ, 2)) l (y s,( ˆβ; λ, 2) y s,(β; λ, 2)) l K s, ( ˆβ β), (B.33) where he second inequaliy follows from Lemma 25 and y s,( ˆβ; λ, 2) y s,(β; λ, 2). On he oher hand, δ s ˆβ(κsa, (E[y s,( ˆβ; λ, 2) ξ s, ]) (N 1)κ sb, (E[y s,( ˆβ; λ, 2) ξ s, ]) + λ) δ s β(κ sa, (E[y s,(β; λ, 2) ξ s, ]) (N 1)κ sb, (E[y s,(β; λ, 2) ξ s, ]) + λ) δ s ˆβ(κsa, (E[y s,( ˆβ; λ, 2) ξ s, ]) (N 1)κ sb, (E[y s,( ˆβ; λ, 2) ξ s, ]) + λ) δ s β(κ sa, (E[y s,( ˆβ; λ, 2) ξ s, ]) (N 1)κ sb, (E[y s,( ˆβ; λ, 2) ξ s, ]) + λ) (B.34) δ s λ( ˆβ β) + δ s α s, ( ˆβ β) =δ s (λ + α s, )( ˆβ β), where he firs inequaliy follows from (B.30) and he second from he definiion of α s,. Define If λ M 2 s,, M 2 s, := l K s, δ s α s, < +. π s,( ˆβ; λ, 2) π s,(β; λ, 2) = δ s ˆβ(κsa, (E[y s,( ˆβ; λ, 2) ξ s, ]) (N 1)κ sb, (E[y s,( ˆβ; λ, 2) ξ s, ]) + λ) δ s β(κ sa, (E[y s,(β; λ, 2) ξ s, ]) (N 1)κ sb, (E[y s,(β; λ, 2) ξ s, ]) + λ) (H s, (y s,(β; λ, 2)) H s, (y s,( ˆβ; λ, 2))) (δ s λ + δ s α s, l K s, )( ˆβ β) (l K s, δ s α s, + δ s α s, l K s, )( ˆβ β) = 0, where he firs inequaliy follows from (B.33) and (B.34), and he second from λ M 2 s,. This esablishes par (b-ii). Thus, Par (b-iii). As shown in par (b-ii), H s, ( ) is a Lipschiz funcion wih he Lipschiz consan l. H s, (y s,(β; λ, 3)) H s, (y s,( ˆβ; λ, 3)) l (y s,( ˆβ; λ, 3) y s,(β; λ, 3)) l K s, ( ˆβ β), (B.35) where he second inequaliy follows from Lemma 25 and ys,( ˆβ; λ, 3) ys,(β; λ, 3). The monooniciy condiion (B.30) and ys,( ˆβ; λ, 3) ys,(β; λ, 3) implies ha κ sa, (E[ys,( ˆβ; λ, 3) ξ s, ]) (N 1)κ sb, (E[ys,( ˆβ; λ, 3) ξ s, ]) κ sa, (E[ys,(β; λ, 3) ξ s, ]) (N 1)κ sb, (E[ys,(β; λ, 3) ξ s, ]). 250

265 Therefore, δ s ˆβλ(κsa, (E[y s,( ˆβ; λ, 3) ξ s, ]) (N 1)κ sb, (E[y s,( ˆβ; λ, 3) ξ s, ])) δ s βλ(κ sa, (E[y s,(β; λ, 3) ξ s, ]) (N 1)κ sb, (E[y s,(β; λ, 3) ξ s, ])) δ s ˆβλ(κsa, (E[y s,( ˆβ; λ, 3) ξ s, ]) (N 1)κ sb, (E[y s,( ˆβ; λ, 3) ξ s, ])) δ s βλ(κ sa, (E[y s,( ˆβ; λ, 3) ξ s, ]) (N 1)κ sb, (E[y s,( ˆβ; λ, 3) ξ s, ])) (B.36) δ s λ(κ sa, (E[y s,( ˆβ; λ, 3) ξ s, ]) (N 1)κ sb, (E[y s,( ˆβ; λ, 3) ξ s, ]))( ˆβ β) δ s λα s, ( ˆβ β), where he las inequaliy follows from he definiion of α s,. Define If λ M 3 s,, M 3 s, := l K s, δ s α s, < +. πs,( ˆβ; λ, 3) πs,(β; λ, 3) = δ s ˆβλ(κsa, (E[ys,( ˆβ; λ, 3) ξ s, ]) (N 1)κ sb, (E[ys,( ˆβ; λ, 3) ξ s, ])) δ s βλ(κ sa, (E[ys,(β; λ, 3) ξ s, ]) (N 1)κ sb, (E[ys,(β; λ, 3) ξ s, ])) (H s, (ys,(β; λ, 3)) H s, (ys,( ˆβ; λ, 3))) (δ s λα s, l K s, )( ˆβ β) (l K s, l K s, )( ˆβ β) = 0, where he firs inequaliy follows from (B.35) and (B.36), and he second from λ M 3 s,. This esablishes par (b-iii). Q.E.D. Lemma 26 has several economical inerpreaions. Pars (a) and (b-i) imply ha, if he adverse effec of a firm s compeiors service level upon is fuure marke size is no srong, π sc s, [π pf s, ] is increasing in βs, 1 sc [β pf s, 1 ]. Par (b-ii) implies ha if he nework effec is sufficienly srong, πsc s, [π pf s, ] is increasing in βs, 1 sc [β pf s, 1 ]. Finally, par (b-iii) implies ha if he boh he service effec and he nework effec are sufficienly srong, π sc s, [π pf s, ] is increasing in β sc s, 1 [β pf s, 1 ]. 251

266 C.1 Equilibrium Definiions C. Appendix for Chaper 4 We now give he definiions of he RE equilibria in he four scenarios considered in his paper: (a) he base model wih sraegic cusomers, (b) he base model wih myopic cusomers, (c) he NTR model wih sraegic cusomers, and (d) he NTR model wih myopic cusomers. Le A(Q 1 ) := E[X Q 1 ]/E[X] (Q 1 0) be he availabiliy funcion given he firs-period producion quaniy Q 1 (see [160]). Definiion C.1.1 (Base model wih sraegic cusomers.) An RE equilibrium in he base model wih sraegic cusomers consiss of (p 1, Q 1, ξ r, r, a, p n 2, p r 2 ) saisfying (a) p 1 = r ; Q 1 = argmax Q1 0Π f (Q 1 ) where Π f ( ) is given in Lemma 8; (b) ξ r = µ + δe[(k + α)v p r 2 ] + δe[(1 + α)v p n 2 ] + ; (c) r = ξ r ; (d) a = A(Q 1); (p n 2, p r 2 ) d = (p n 2, p r 2 ). Definiion C.1.2 (Base model wih myopic cusomers.) An RE equilibrium in he base model wih myopic cusomers consiss of ( p 1, Q 1, ξ r, r ) saisfying (a) p 1 = r ; Q 1 = argmax Π Q1 0 f (Q 1 ) where Π f ( ) is given in Lemma 8; (b) ξ r = µ; (c) r = ξ r. Definiion C.1.3 (NTR model wih sraegic cusomers.) An RE equilibrium in he NTR model wih sraegic cusomers consiss of (p u 1, Q u 1, ξr u, r u, a u, p u 2 ) saisfying (a) p u 1 = r u ; Q u 1 = argmax Q1 0Π u f (Q 1), where Π u f ( ) is given in Lemma 8; (b) ξ u r (c) r u = ξ u r ; = µ + δe[(k + α)v p u 2 ] + δe[(1 + α)v p u 2 ] + ; (d) a u = A(Q u 1 ); p u d 2 = p u 2((X Q u 1 ) +, X Q u 1 ), where p u 2(, ) is characerized in Theorem 4.4.2(a). Definiion C.1.4 (NTR model wih myopic cusomers.) An RE equilibrium in he NTR model wih myopic cusomers consiss of ( p u u 1, Q 1, (a) p u 1 = r u ; u ξ r, r u ) saisfying u Q 1 = argmax Π u Q1 0 f (Q 1) where Π u f ( ) is given in Lemma 8; (b) ξ u r = µ; (c) r u = ξ u r. 252

267 In Definiions C.1.1-C.1.4, condiions (a) and (b) follow from ha he decisions are opimal given he raional beliefs, and condiions (c) and (d) follow from ha he raional beliefs are consisen wih acual oucomes. C.2 Proofs of Saemens We use h 1( ) o denoe he derivaive operaor of a single variable funcion h 1 ( ), x h 2 ( ) o denoe he parial derivaive operaor of a muli-variable funcion, h 2 ( ), wih respec o variable x, and 1 { } o denoe he indicaor funcion. For any mulivariae coninuously differeniable funcion h 2 (x 1, x 2,, x n ) and x := (x 1, x 2,, x n) in h 2 ( ) s domain, i, we use xi h 2 (x 1, x 2,, x n) o denoe xi h 2 (x 1, x 2,, x n ) x=x. Proof of Lemma 7: Par (a). Given (p n 2, p r 2) (p r 2 p n 2 ), a new cusomer will make a purchase if and only if (1+α)V p n 2, whereas a repea cusomer will make a purchase if and only if (k+α)v p r 2. Thus, ) he ex ane probabiliy ha a new cusomer will purchase he second-generaion produc is Ḡ, whereas he probabiliy ha a repea cusomer will join he rade-in program is Ḡ ( p r 2 k+α ( p n 2 1+α ). Therefore, condiioned on he realized marke size (X2 n, X2), r he expeced profi of he firm in period 2 is given by: ( ) ( ) p Π 2 (p n 2, p r 2 X2 n, X2) r :=X2 n (p n n 2 c 2 )Ḡ 2 p + X r 1 + α 2(p r r 2 c 2 + r 2 )Ḡ 2 k + α (C.1) =X n 2 v n 2 (p n 2 ) + X r 2v r 2(p r 2), where v n 2 (p n 2 ) := (p n 2 c 2 )Ḡ( pn 2 1+α ) and vr 2(p r 2) := (p r 2 c 2 + r 2 )Ḡ( pr 2 k+α ). quasiconcave in p n 2, and v2( ) r is quasiconcave in p r 2. Noe ha ( ) ( p p n 2 v2 n (p n n 2 ) = 2 c 2 p n g α 1 + α and p r 2 v r 2(p r 2) = ) + Ḡ ( p n 2 ) 1 + α ( ) ( ) ( ) p r 2 c 2 + r 2 p r g 2 p r + k + α k + α Ḡ 2. k + α We now show ha vn 2 ( ) is Because g(v)/ḡ(v) is coninuously increasing in v, g( pn 2 1+α )/Ḡ( pn 2 1+α ) is coninuously increasing in pn 2 and g( pr 2 k+α )/Ḡ( pr 2 k+α ) is coninuously increasing in pr 2. Hence, p n 2 v n 2 (p n 2 ) = 0 has a unique soluion p n 2 and p r 2 v r 2(p r 2) = 0 has a unique soluion p r 2, where v n 2 ( ) [v r 2( )] is sricly increasing on [0, p n 2 ) [[0, p r 2 )] and sricly decreasing on (p n 2, + ) [(p r 2, + )]. Therefore, for any realized (X n 2, X r 2), X n 2 v n 2 ( ) is quasiconcave in p n 2, and X r 2v r 2( ) is quasiconcave in p r 2. Thus, for any realized (X n 2, X r 2), (p n 2 (X n 2, X r 2), p r 2(X n 2, X r 2)) = (p n 2, p r 2 ) maximizes Π 2 (, X n 2, X r 2). I remains o show ha p n 2 > p r 2 if and only if k < 1 or r 2 > 0. Noe ha p n 2 saisfies ( ) ( ) p n p n 2 c g α ) = 1, (C.2) 1 + α Ḡ ( p n 2 1+α and p r 2 saisfies ( p r 2 c 2 + r 2 k + α ( ) p r g 2 Ḡ ) k+α ( p r 2 k+α ) = 1. (C.3) 253

268 If k < 1 or r 2 > 0, pn 2 c 2+r 2 ) k+α g ( p n 2 k+α ) /Ḡ ( p n 2 k+α g ( p n 2 1+α ( p n > pn 2 c 2 1+α ) /Ḡ ( p n 2 1+α 2 c 2 + r 2 k + α, and he increasing failure rae condiion implies ha ). Thus, ( ) p n g 2 Ḡ k+α ( p n 2 k+α ) ( p n ) > 2 c α ( ) p n g 2 Ḡ ) 1+α ( p n 2 1+α ) = 1, and, hence, p r 2 v r 2(p n 2 ) < 0. Since v r 2( ) is quasiconcave, p r 2 < p n 2. On he oher hand, if k = 1 and r 2 = 0, v n 2 ( ) v r 2( ) and hus p n 2 = p r 2. This complees he proof of Par (a). Par (b). Because all new cusomers wih willingness-o-pay (1+α)V greaer han p n 2 (X n 2, X r 2) p n 2 would make a purchase. Hence, ( ) p Q n 2 (X2 n, X2) r = E[X2 n 1 {(1+α)V p n 2 } X2 n n ] = Ḡ 2 X2 n. 1 + α Analogously, all repea cusomers wih willingness-o-pay (k +α)v greaer han p r 2(X n 2, X r 2) p r 2 would make a purchase. Hence, ( ) p Q r 2(X2 n, X2) r = E[X21 r {(k+α)v p r 2 } X2] r r = Ḡ 2 X r k + α 2. This proves Par (b). Par (c). Since π 2 (X n 2, X r 2) := max{π 2 (p n 2, p r 2 X n 2, X r 2) : 0 p r 2 p n 2 }, i follows immediaely ha π 2 (X n 2, X r 2) = [max v n 2 (p n 2 )]X n 2 + [max v r 2(p r 2)]X r 2. To complee he proof, i remains o show ha βn = [max v2 n (p n 2 )] > 0 and βr = [max v2(p r r 2)] > 0. By ( ) ( ) equaions (C.2) and (C.3), we have p n p n 2 2 c 2 > 0, Ḡ 1+α > 0, p r 2 c 2 + r 2 > 0, and Ḡ p r 2 k+α > 0. ) ) Hence, βn = (p n 2 c 2 )Ḡ > 0 and βr = (p r 2 c 2 + r 2 )Ḡ > 0. This complees he proof of Par (c). Q.E.D. ( p n 2 1+α ( p r 2 k+α Proof of Theorem 4.3.1: Par (a). Since ξ r saisfies ha U p = U w, we have a (E[V ] + δe[(k + α)v p r 2 ] + ξ r ) + (1 a )δe[(1 + α)v p n 2 ] + = δe[(1 + α)v p n 2 ] +. Direc algebraic manipulaion yields ha ξ r = µ + δe[(k + α)v p r 2 ] + δe[(1 + α)v p n 2 ] +. Hence, by Definiion C.1.1 and Lemma 7(a), Hence, p 1 = r 1 = ξ r = µ + δe[(k + α)v p r 2 ] + δe[(1 + α)v p n 2 ] + = µ + δe[(k + α)v p r 2 ] + δe[(1 + α)v p n 2 ] +. Π f (Q 1 ) = p 1E(X Q 1 ) c 1 Q 1 + r 1 E(Q 1 X) + + δe{π 2 (X (X Q 1 ), X Q 1 )} = (p 1 r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δe[βn(x (X Q 1 )) + βr (X Q 1 )] = (p 1 + δ(βr βn) r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δβne(x) = (m 1 r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δβne(x), 254

269 where he second equaliy follows from (Q 1 X) + = Q 1 (X Q 1 ), and he las from he ideniy m 1 = µ + δe[(k + α)v p r 2 ] + δe[(1 + α)v p n 2 ] + + δ(β r β n). Therefore, Q 1 is he soluion o a newsvendor problem wih marginal revenue m 1 r 1, marginal cos c 1 r 1, and demand disribuion F ( ). Hence, Q 1 = F 1 ( c 1 r 1 m 1 r 1 ) and Π f = Π f (Q 1) = (m 1 r 1 )E(X Q 1) (c 1 r 1 )Q 1 + δβ ne(x). This proves Par (a). Par (b). Since myopic cusomers will make a purchase if and only if p 1 µ, p 1 = ξ 1 = µ. Hence, Π f (Q 1 ) = p 1E(X Q 1 ) c 1 Q 1 + r 1 E(Q 1 X) + + δe{π 2 (X (X Q 1 ), X Q 1 )} = ( p 1 r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δe[β n(x (X Q 1 )) + β r (X Q 1 )] = ( p 1 + δ(β r β n) r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δβ ne(x) = ( m 1 r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δβ ne(x), where he second equaliy follows from (Q 1 X) + = Q 1 (X Q 1 ), and he las from he ideniy m 1 = µ + δ(β r β n). Therefore, Q 1 is he soluion o a newsvendor problem wih marginal revenue m 1 r 1, marginal cos c 1 r 1, and demand disribuion F ( ). Hence, Q 1 = F 1 ( c1 r1 m ) and 1 r1 Π f = Π f ( Q 1) = (m 1 r 1 )E(X Q 1) (c 1 r 1 ) Q 1 + δβne(x). This proves Par (b). Q.E.D. Proof of Lemma 8: The expressions for Π f ( ) and Π f ( ) have already been given in he proof of Theorem 4.3.1(a) and Theorem 4.3.1(b), respecively. We now compue Π u f (Q 1). Following he same argumen as he proof of Theorem 4.3.1(a), given he firs-period producion quaniy Q 1, he firs-period equilibrium price is p u 1(Q 1 ) =E[V ] + δ[e((k + α)v p u 2 ) + E((1 + α)v p u 2 ) + ] =µ + δ[e((k + α)v p u 2(X n 2, X r 2)) + E((1 + α)v p u 2(X n 2, X r 2)) + ], where X n 2 = (X Q 1 ) + and X r 2 = X Q 1. Le π u 2 (X n 2, X r 2) := max p u 2 Π u 2(p u 2 X n 2, X r 2). Hence, ( p π2 u (X2 n, X2) r = max p u 2 0{Xn 2 (p u u 2 c 2 )Ḡ α ( p = X2 n (p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r 1 + α where X n 2 = (X Q 1 ) + and X r 2 = X Q 1. Therefore, ) + X r 2(p u 2 c 2 )Ḡ( p u 2 k + α )} ) + X2(p r u 2(X2 n, X2) r c 2 )Ḡ ( p u 2 (X n 2, X r 2) k + α ), Π u f (Q 1 ) = p u 1(Q 1 )E(X Q 1 ) c 1 Q 1 + r 1 E(X Q 1 ) + + δe[π2 u (X2 n, X2)] r ( p = (p u 1(Q 1 ) r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δe[(x Q 1 ) + (p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) 1 + α ( p +(X Q 1 )(p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) ] k + α ( p = (p u 1(Q 1 ) + E[(p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) ( p (p u k + α 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) ] 1 + α ( p r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δe[(p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) X] 1 + α ( p = (m u 1(Q 1 ) r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δe[(p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) X], 1 + α 255

270 where ( p m u 1(Q 1 ) : = µ + δ{e[(p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) ] + E((k + α)v p u k + α 2(X2 n, X2)) r + ( p E[(p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) ] E((1 + α)v p u 1 + α 2(X2 n, X2)) r + }, wih X n 2 = (X Q 1 ) + and X r 2 = X Q 1. Analogously, since p u 1 = E[V ] = µ, Π u f (Q 1 ) = p u 1 E(X Q 1 ) c 1 Q 1 + r 1 E(X Q 1 ) + + δe[π2 u (X2 n, X2)] r ( p = (µ r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δe[(x Q 1 ) + (p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) 1 + α ( p +(X Q 1 )(p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) ] k + α ( p = (µ + E[(p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) ( p (p u k + α 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) ] 1 + α ( p r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δe[(p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) X] 1 + α ( p = ( m u 1(Q 1 ) r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δe[(p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) X], 1 + α where m u 1(Q 1 ) : = ( p µ + δ{e[(p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) ] E[(p u k + α 2(X2 n, X2) r c 2 )Ḡ wih X n 2 = (X Q 1 ) + and X r 2 = X Q 1. Q.E.D. ( p u 2 (X n 2, X r 2) 1 + α ) ]}, Before giving he proof of Theorem 4.4.1, we firs prove Theorem Proof of Theorem 4.4.2: Par (a). If he firm charges a single price p u 2 in period 2, all new (repea) cusomers wih willingness-o-pay (1 + α)v ((k + α)v ) greaer han p u 2 will make a purchase (join he rade-in program). Hence, he second-period profi funcion of he firm Π u 2(p u 2 X2 n, X2) r is given by ( ) ( ) p Π u 2(p u 2 X2 n, X2) r = X2 n (p u u 2 c 2 )Ḡ 2 p + X r 1 + α 2(p u u 2 c 2 )Ḡ 2 k + α = X2 n v2 n (p u 2) + X2 r ˆv 2(p r u 2), ( ) where ˆv 2(p r 2 ) := (p 2 c 2 )Ḡ p2 k+α. Clearly, ˆv 2( ) r has a unique maximizer ˆp r 2, where ˆp r 2 p r 2 wih he inequaliy being sric if r 2 > 0. Moreover, Π u 2(p u 2 X n 2, X r 2) = ˆΠ 2 (p u 2, p u 2 X n 2, X r 2), where, by he proof of Lemma 7(a), ˆΠ 2 (p n 2, p r 2 X n 2, X r 2) := X n 2 v n 2 (p n 2 ) + X r 2 ˆv r 2(p r 2) is quasiconcave funcion of (p n 2, p r 2). Thus, he equilibrium second-period pricing sraegy p u 2(X n 2, X r 2) is he maximizer of he second-period profi funcion, i.e., p u 2(X n 2, X r 2) = argmax p u 2 0Π u 2(p u 2 X n 2, X r 2). Noe ha since ˆΠ 2 (, X n 2, X r 2) is quasiconcave in (p n 2, p r 2), Π u 2(p u 2 X n 2, X r 2) = ˆΠ 2 (p u 2, p u 2 X n 2, X r 2) is also quasiconcave in p u 2. Observe ha p u 2 Π u 2(p u 2 X n 2, X r 2) = X n 2 ( ) p u 2 [Ḡ 1+α ( ) p u 2 c 2 1+α g ( )] p u 2 1+α + X2 r ( ) p u 2 [Ḡ k+α ( ) p u 2 c 2 k+α g ( )] p u 2 k+α. Since g(v)/ḡ(v) is increasing in v, p u 2 Πu 2(p u 2 X n 2, X r 2) < 0 if p u 2 > p n 2, and p u 2 Π u 2(p u 2 X n 2, X r 2) > 0 if p u 2 < ˆp r 2. Thus, p u 2(X n 2, X r 2) [ˆp r 2, p n 2 ] [p r 2, p n 2 ] = [p r 2(X n 2, X r 2), p n 2 (X n 2, X r 2)]. 256

271 If k < 1, by he proof of Lemma 7(a), ˆp r 2 < p n 2. Since X2 n, X2 r > 0, ( ) ( ) ( )] p u 2 Π u 2(ˆp r 2 X2 n, X2) r = X2 [Ḡ n ˆp r 2 ˆp r 2 1+α c2 ˆp r 2 1+α g 1+α > 0 and ( ) ( ) ( )] p u 2 Π u 2(p n 2 X2 n, X2) r = X2 [Ḡ r p n 2 p n 2 k+α c2 p n 2 k+α g k+α < 0. Therefore, p r 2 ˆp r 2 < p u 2(X2 n, X2) r < p n 2 for all X n 2, X r 2 > 0. When p u 2 [ˆp r 2, p n 2 ], Ḡ( pu 2 1+α ) ( pu 2 c 2 1+α )g( pu 2 1+α ) 0 and Ḡ( pu 2 k+α ) ( pu 2 c 2 k+α )g( pu 2 k+α ) 0. Thus, Π u 2(p u 2 X n 2, X r 2) is increasing in X n 2 and decreasing in X r 2 if p u 2 [ˆp r 2, p n 2 ], i.e., Π u 2(p u 2 X n 2, X r 2) is supermodular in (p u 2, X n 2 ) on he laice [ˆp r 2, p n 2 ] [0, + ), and submodular in (p u 2, X r 2) on he laice [ˆp r 2, p n 2 ] [0, + ). Therefore, p u 2(X n 2, X r 2) is coninuously increasing in X n 2 and coninuously decreasing in X r 2. This proves Par (a). Par (b). Noe ha p u 1 = µ + δ[e((k + α)v p u 2 ) + E((1 + α)v p u 2 ) + ], where p u d 2 = p u 2((X Q u 1 ) +, X Q u 1 ) [p r 2, p n 2 ]. Therefore, δ[e((k + α)v p u 2 ) + E((k + α)v p r 2 ) + ] 0, δ[e((1 + α)v p u 2 ) + E((1 + α)v p n 2 ) + ] 0, and hus p u 1 p 1 = δ[e((k +α)v p u 2 ) + E((k +α)v p r 2 ) + ] δ[e((1+α)v p u 2 ) + E((1+α)V p n 2 ) + ] 0. If p r 2 < p n 2, a leas one of he following wo inequaliies are sric: δ[e((k + α)v p u 2 ) + E((k + α)v p r 2 ) + ] 0 and δ[e((1 + α)v p u 2 ) + E((1 + α)v p n 2 ) + ] 0. Hence, p u 1 < p 1 if p r 2 < p n 2. I s sraighforward o compue ha, for any Q 1 0, Π f (Q 1 ) Π u f (Q 1 ) =(p 1 p u 1(Q 1 ))E(X Q 1 ) + δe[(β n v n 2 (p u 2((X Q 1 ) +, X Q 1 )))(X Q 1 ) + + (β r ˆv n 2 (p u 2((X Q 1 ) +, X Q 1 )))(X Q 1 )], where p u 1(Q 1 ) = µ+δ[e((k+α)v p u 2 ) + E((1+α)V p u 2 ) + ] p u 1 wih p u d 2 = p u 2((X Q 1 ) +, X Q 1 ). Since β n v n 2 (p 2 ) and β r v r 2(p 2 ) ˆv r 2(p 2 ) for any p 2 0, Π f (Q 1 ) Π u f (Q 1) for all Q 1 0, and hus Π f = max Q 1 Π f (Q 1 ) max Q1 Π u f (Q 1) = Π u f. If pr 2 < p n 2, by he proof of par (a), p 1 > p u 1(Q 1 ) and, hence, Π f (Q 1 ) > Π u f (Q 1) for all Q 1 > 0. Therefore, Π f = Π f (Q 1) Π f (Q u 1 ) > Π u f (Qu 1 ) = Π u. This proves par (b). Par (c). p u 1 = p 1 = µ follows immediaely from ha µ is he willingness-o-pay of myopic cusomers. Moreover, direc compuaion yields ha, for any Q 1 0, Π f (Q 1 ) Π u f (Q 1 ) =δe[(β n v n 2 (p u 2((X Q 1 ) +, X Q 1 )))(X Q 1 ) + 0, + (β r ˆv n 2 (p u 2((X Q 1 ) +, X Q 1 )))(X Q 1 )] where he inequaliy follows from he proof of par (b). If p r 2 < p n 2, a leas one of E[(β n v n 2 (p u 2((X Q 1 ) +, X Q 1 )))(X Q 1 ) + ] and E[(β r ˆv n 2 (p u 2((X Q 1 ) +, X Q 1 )))(X Q 1 )] is posiive for Q 1 > 0. Hence, he same argumen as he proof of par (b) yields ha Π f (c). Q.E.D. > Π u f if Q u 1 > 0. This proves par f Proof of Theorem 4.4.1: Par (a). Since p 1 p 1 = m 1 m 1 = e, i follows immediaely ha p 1 > p 1 and Q 1 = F 1 ( c 1 r 1 m 1 r 1 ) > F 1 ( c 1 r 1 m 1 r 1 ) = Q 1 if and only if e > 0. Moreover, for any Q 1, Π f (Q 1 ) Π f (Q 1 ) = e E(X Q 1 ) > 0 if and only if e > 0. Therefore, Π f = max Π f (Q 1 ) > max Π f (Q 1 ) = Π f if and only if e > 0 and Q 1 >

272 Nex, we show ha e > 0 if and only if r 2 > r. Observe ha v r 2(p r 2) is submodular in (p r 2, r 2 ), so p r 2 is decreasing in r 2. Moreover, e is decreasing in p r 2. Hence, e is increasing in r 2 and e > 0 if and only if r 2 > r for some r. We now show ha r 1 k 1+α c 2. I suffices o show ha if r 2 = 1 k 1+α c 2, e 0. If r 2 = 1 k 1+α c 2, v2(p r r 2) = (p r 2 k+α 1+α c2)ḡ( pr 2 k+α ). I s sraighforward o check ha pr 2 = k+α 1+α pn 2. Hence, This proves Par (a). Par (b). Observe ha, where p u 2 e =E[(k + α)v p r 2 ] + E[(1 + α)v p n 2 ] + =E[(k + α)v k + α 1 + α pn 2 ] + E[(1 + α)v p n 2 ] + = 1 k E[(1 + α)v pn 2 ] α p u 1 p u 1 = δ[e((k + α)v p u 2 ) + E((1 + α)v p u 2 ) + ], d = p u 2((X Q u 1 ) +, X Q u 1 ). Since k 1, p u 1 p u 1 and he inequaliy is sric if k < 1. This esablishes par (b-i). We now show par (b-ii). Direc compuaion yields ha m u 1(Q 1 ) m u 1(Q 1 ) = E((1 + α)v p u 2(X n 2, X r 2)) + E((k + α)v p u 2(X n 2, X r 2)) +, where X n 2 = (X Q 1 ) + and X r 2 = X Q 1. Since k < 1, we have E((1 + α)v p u 2(X n 2, X r 2)) + E((k + α)v p u 2(X n 2, X r 2)) + > 0. Le Π(Q 1, 1) = Π u f (Q 1) and Π(Q 1, 0) = Π u f (Q 1), Π(Q 1, 1) Π(Q 1, 0) = Π u f (Q 1 ) Π u f (Q 1 ) = (µ p u 1(Q 1 ))E(X Q 1 ) = [E((1 + α)v p u 2(X n 2, X r 2)) + E((k + α)v p u 2(X n 2, X r 2)) + ]E(X Q 1 ). Since [E((1+α)V p) + E((k+α)V p) + ] = P( p 1+α V p k+α ) 0 and pu 2(X n 2, X r 2) is decreasing in Q 1, Π(Q 1, 1) Π(Q 1, 0) = (µ p u 1(Q 1 ))(X Q 1 ) is increasing in Q 1, and, hence, Π(, ) is a supermodular funcion on he laice [0, + ) {0, 1}. Thus, Qu 1 = argmax Q1 0Π(Q 1, 1) argmax Q1 0Π(Q 1, 0) = Q u 1. This proves par (b-ii). Finally, since Π u f (Q 1) Π u f (Q 1) = (µ p u 1(Q 1 ))(X Q 1 ) 0 where he inequaliy is sric if p r 2 < p n 2. Hence, Π u f he proof of Theorem (b-ii) implies ha = max Π u Q1 0 f (Q 1) max Q1 0 Π u f (Q 1) = Π u f. Moreover, he same argumen as Π u f > Π u f if p r 2 < p n 2. This esablishes par (b). Q.E.D. Proof of Theorem 4.4.3: Par (a). We firs show ha m u 1(Q 1 ) is decreasing in Q 1. Observe ha m u 1(Q 1 ) = µ + δ[u r (Q 1 ) U n (Q 1 )], where ( p U r (Q 1 ) := E[(p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) ] + E((k + α)v p u k + α 2(X2 n, X2)) r +, and ( p U n (Q 1 ) := E[(p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) ] + E((1 + α)v p u 1 + α 2(X2 n, X2)) r +. Le u r (p) := (p c 2 )Ḡ( p k+α ) + E((k + α)v p)+ = E[(k + α)v c 2 ]1 {(k+α)v p} and u n (p) := (p c 2 )Ḡ( p 1+α ) + E((1 + α)v p)+ = E[(1 + α)v c 2 ]1 {(1+α)V p}. I s clear ha u r ( ) and u p ( ) are 258

273 coninuously decreasing in p. Moreover, U r (Q 1 ) = E[u r (p u 2(X n 2, X r 2))] and U n (Q 1 ) = E[u n (p u 2(X n 2, X r 2))], where X n 2 = (X Q 1 ) + and X r 2 = X Q 1. Since p u 2(X n 2, X r 2) is increasing in X n 2 and decreasing in X r 2, i is sochasically decreasing in Q 1. Hence, i suffices o show ha u r (p) u n (p) is increasing in p. Observe ha u r (p) u n (p) = [ = [ p/(k+α) p/(1+α) v p/(1+α) ((1 + α)v p)g(v ) dv + v p/(k+α) ((1 + α)v max(p, (k + α)v ))g(v ) dv ], which is coninuously increasing in p. This esablishes par (a-i). We now show ha m u 1(Q 1 ) < m 1 for all Q 1. Observe ha (1 k)v g(v ) dv ] m u 1(Q 1 ) m 1 = E[u r (p u 2(X n 2, X r 2)) u r (p r 2 )] E[u n (p u 2(X n 2, X r 2)) u n (p n 2 )]. Because p r 2 p u 2(X n 2, X r 2) p n 2, E[u r (p u 2(X n 2, X r 2)) u r (p r 2 )] 0 and E[u n (p u 2(X n 2, X r 2)) u n (p n 2 )] 0. Hence, m u 1(Q 1 ) m 1. If k < 1, p r 2 < p n 2, one of he inequaliies E[u r (p u 2(X n 2, X r 2)) u r (p r 2 )] 0 and E[u n (p u 2(X n 2, X r 2)) u n (p n 2 )] 0 mus be sric. Therefore, m u 1(Q 1 ) < m 1 for all Q 1 0. This proves par (a-ii). Nex, we show ha Q u 1 Q 1. Observe ha ( p Π u f (Q 1 ) Π f (Q 1 ) = (m u 1(Q 1 ) m 1)(X Q 1 ) + δe[(p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r 1 + α ) ] δβ ne(x). Le Π(Q 1, 1) = Π f (Q 1 ) and Π(Q 1, 0) = Π u f (Q 1). Then, ( p Π(Q 1, 1) Π(Q 1, 0) = (m 1 m u 1(Q 1 ))(X Q 1 ) + δex[βn (p u 2(X2 n, X2) r u c 2 )Ḡ 2 (X2 n, X2) r ) ] 1 + α Noe ha for any realizaion of X, p u 2(X n 2, X r 2) and hus (p u 2(X n 2, X r 2) c 2 )Ḡ( pu 2 (Xn 2,Xr 2 ) 1+α ) is decreasing in Q 1. Therefore, by par (a-ii), Π(Q 1, 1) Π(Q 1, 0) is increasing in Q 1. Hence, Π(, ) is supermodular on he laice [0, + ) {0, 1}. Hence, Q u 1 = argmax Q1 0Π u f (Q 1) argmax Q1 0Π f (Q 1 ) = Q 1. If Q u 1 > 0, since m 1 > m u 1(Q u 1 ) Π f (Qu 1 ) > Q1 Π u f (Qu 1 ) = 0. Since Π f ( ) is concave in Q 1, Q 1 > Q u 1. This proves par (a-iii). Par (b). We firs show ha m u 1(Q 1 ) is increasing in Q 1. Noe ha m u 1(Q 1 ) = µ+δe[ˆv r 2(p u 2(X n 2, X r 2)) v n 2 (p u 2(X n 2, X r 2))], where X n 2 = (X Q 1 ) + and X r 2 = X Q 1. Because ˆp r 2 p u 2(X n 2, X r 2) p n 2 and p u 2(X n 2, X r 2) is increasing in X n 2 and decreasing in X r 2. Thus, ˆv r 2(p u 2(X n 2, X r 2)) is sochasically increasing in Q 1 and v n 2 (p u 2(X n 2, X r 2)) is sochasically decreasing in Q 1. Therefore, m u 1(Q 1 ) = µ+δe[ˆv r 2(p u 2(X n 2, X r 2)) v n 2 (p u 2(X n 2, X r 2))] is increasing in Q 1. This proves par (b-i). We now show par (b-ii). Le ˆβ r = max p 0 ˆv r 2(p). I s clear ha β r ˆβ r is increasing in r 2, wih β r = ˆβ r if r 2 = 0. Moreover, since k < 1, ˆβ n := v n 2 (ˆp r 2 ) < β n. Therefore, le r 2 > 0 be he hreshold such ha β r ˆβ r = β n ˆβ n. Hence, for all r 2 < r 2, β r ˆβ r < β n ˆβ n. Moreover, by he monoone convergence heorem, lim Q 1 + mu 1(Q 1 ) = µ + δ[v2(ˆp r r 2 ) v2 n (ˆp r 2 )] = µ + δ[ ˆβ r ˆβ n] > µ + δ[βr βn] = m 1. Par (b-i) shows ha m u 1(Q 1 ) is increasing in Q 1. Hence, here exiss a hreshold Q(r 2 ) such ha m u 1(Q 1 ) m 1 if and only if Q 1 Q(r 2 ). To show ha Q(r 2 ) is increasing in r 2, we observe ha m 1 is increasing in r 2. Hence, Q(r2 ) := min{q 1 : m u 1(Q 1 ) m 1} is increasing in r 2. This proves par (b-ii). 259

274 Par (b-iii). Wihou loss of generaliy, assume ha Q u 1 > 0. Oherwise, he resul holds rivially. I s clear ha Q u 1 X and Q 1 X as c 1 0, where X is he upper bound of he suppor of X. Hence, here exiss a hreshold c(r 2 ) > 0, dependen on r 2, such ha if c 1 < c(r 2 ), Q u 1, Q 1 > Q(r 2 ). Le ˆπ 2 (Q 1 ) := δe[v n 2 (p u 2(X n 2, X r 2))X], where X n 2 differeniable and, by he chain rule = (X Q 1 ) + and X r 2 = X Q 1. I s clear ha ˆπ 2 ( ) is ˆπ 2(Q 1 ) = δe[ p v n 2 (p u 2(X n 2, X r 2))( X n 2 p u 2(X n 2, X r 2) + X r 2 p u 2(X n 2, X r 2))1 {X Q1 }X]. As Q 1 X, for any realizaion of X X, X n 2 p u 2(X n 2, X r 2) and X r 2 p u 2(X n 2, X r 2) converges o 0. Hence, by he dominaed convergence heorem, here exis a hreshold ˆQ [ Q(r 2 ), X), such ha ˆπ 2(Q 1 ) [ ϵp(x Q 1 ), 0] for all Q 1 ˆQ, where ϵ := ( m u 1( ˆQ) m 1)/2 > 0. hreshold such ha, if c 1 < c 1 (r 1 ), we have Q u 1, Q 1 > ˆQ Q(r 2 ). Therefore, Π f (Q u 1 ) = ( m 1 r 1 )P(X Q u 1 ) (c 1 r 1 ) < ( m u 1(Q u 1 ) r 1 )P(X Q u 1 ) ϵp(x Q u 1 ) (c 1 r 1 ) = 0, ( m u 1(Q u 1 ) r 1 )P(X Q u 1 ) + ˆπ 2(Q u 1 ) (c 1 r 1 ) Q1 Πu f (Q u 1 ) Le c 1 (r 2 ) (0, c(r 2 )] be he where he firs inequaliy follows from m u 1(Q u 1 ) m 1 ( m u 1( ˆQ) m 1) = 2ϵ > ϵ, he second from ˆπ 2(Q u 1 ) [ ϵp(x Q u 1 ), 0], and he las from he monooniciy ha m u 1( ) is increasing in Q 1. Because Π f ( ) is concave in Q 1, Q 1 = argmax Q1 Πf (Q 1 ) < (b-iii) and hus Theorem Q.E.D. Q u 1 follows immediaely. This esablishes par Before presening he proof Theorem 4.4.4, we give he following lemma ha compues he equilibrium environmenal impacs I e and Ĩ e. Lemma 27 (a) Wih sraegic cusomers, he oal expeced environmenal impac of he RE equilibrium is I e = I e (Q 1), where I e (Q 1 ) := (κ 1 ι 1 )Q 1 + (ι 1 + δ(κ 2 ι 2 )Ḡ ( ) ( ) p r 2 p n 2 )E(Q 1 X) + δκ 2 Ḡ E(X Q 1 ) +. k + α 1 + α (b) Wih myopic cusomers, he oal expeced environmenal impac of he RE equilibrium is Ĩ e = I e ( Q 1). (c) The funcion I e ( ) is sricly increasing in Q 1. Hence, I e Ĩ e if and only if Q 1 Q

275 Proof of Lemma 27: Pars (a) and (b). Direc compuaion yields ha I e =E{κ 1 Q 1 + δκ 2 (Q n 2 (X n 2, X r 2 ) + Q r 2(X n 2, X r 2 )) ι 1 (Q 1 X) + δι 2 Q r 2(X n 2, X r 2 )} =E{κ 1 Q 1 + δκ 2 ((X Q 1) + Ḡ ( ) p δι 2 (X Q r 2 1)Ḡ } k + α =(κ 1 ι 1 )Q 1 + (ι 1 + δ(κ 2 ι 2 )Ḡ =I e (Q 1), ( ) ( ) p n 2 p + (X Q r α 1)Ḡ ) ι 1 (Q 1 X) + k + α ( ) ( ) p r 2 p )E(X Q n 2 k + α 1) + δκ 2 Ḡ E(X Q 1 + α 1) + where he second inequaliy follows from X n 2 = (X Q 1) + and X r 2 = X Q 1, he hird from (Q 1 X) + = Q 1 (X Q 1), and he las from he definiion of he funcion I e ( ). Analogously, Ĩ e =E{κ 1 Q 1 + δκ 2 (Q n 2 ( X n 2, X r =E{κ 1 Q 1 + δκ 2 ((X Q 1) + Ḡ ( ) δι 2 (X Q p r 2 1)Ḡ } k + α 2 ) + Q r 2( ( p n α =(κ 1 ι 1 ) Q 1 + (ι 1 + δ(κ 2 ι 2 )Ḡ =I e ( Q 1). This complees he proof of Pars (a) and (b). n r X 2, X 2 )) ι 1 ( Q 1 X) + δι 2 Q r 2( ( p r 2 k + α ) + (X Q 1)Ḡ X n 2, ) ) ι 1 ( Q 1 X) + X r 2 )} ( ) ( ) p r 2 )E(X k + α Q p n 2 1) + δκ 2 Ḡ E(X 1 + α Q 1) + Par (c). To esablish he monooniciy of I e ( ), observe ha ( ) ( ) I e(q p r 2 p n 2 1 ) = κ 1 ι 1 +(ι 1 +δ(κ 2 ι 2 )Ḡ k+α )P(X > Q 1 ) δκ 2 Ḡ 1+α P(X > Q 1 ) > κ 1 ι 1 δκ 2 > 0, ) where he firs inequaliy follows from Ḡ 1 and P(X > Q 1 ) 1, and he second from ( p n 2 1+α he assumpion ha κ 1 > ι 1 + κ 2. Hence, I e ( ) is sricly increasing in Q 1. This proves Par (c). Q.E.D. Proof of Theorem 4.4.4: Par (a). Firs, we compue Ie u. Given he marke sizes (X2 n, X2), r he equilibrium oal second-period producion quaniy, Q u 2(X2 n, X2), r is given by ( p Q u 2(X2 n, X2) r = X2 n u Ḡ 2 (X2 n, X2) r ) ( p + X r u 2 (X2 n, X r ) 2) 1 + α 2Ḡ. k + α Therefore, I u e = E{κ 1 Q u 1 ι 1 (Q u = (κ 1 ι 1 )Q u 1 + E{[ι 1 + δκ 2 Ḡ ( p u +δκ 2 E[Ḡ 2 (X2 n, X2 r ) 1 + α 1 X) + + δκ 2 Q u 2(X2 n, X2 r )} ( p u 2 (X2 n, X2 r ) k + α ) (X Q u 1 ) + ], ) ](Q u 1 X)} where X n 2 = (X Q u 1 ) + and X r 2 = X Q u 1. If Q 1 = 0, Q u 1 = 0 as well by Theorem 4.4.3(a). Hence, I e = I u e regardless of he value of ι 2. In his case, par (a) rivially holds. On he oher hand, if Q 1 > 0, 261

276 Ie is sricly linearly decreasing in ι 2. Thus, le ῑ u 2 := max{ι 2 : Ie Ie u }. We have Ie I2 u if and only if ι 2 ῑ u 2. In paricular, if ι 2 = 0, Q 1 > Q u 1 implies ha ( p (κ 1 ι 1 )Q n δκ 2 Ḡ 1 + α and ) E(X Q 1) + > (κ 1 ι 1 )Q u 1 + δκ 2 E[Ḡ ( p u 2 (X2 n, X r 1 + α ( ) ( p r (ι 1 + δ(κ 2 ι 2 )Ḡ 2 p )E(X Q u k + α 1) E[(ι 1 + δκ 2 Ḡ 2 (X2 n, X r k + α Thus, ῑ u 2 > 0. This esablishes par (a). Par (b). As in he proof of par (a), we firs compue Ĩu e : Ĩe u = E{κ Qu u 1 1 ι 1 ( Q 1 X) + + δκ 2 Q u ( 2( p u 2( u = (κ 1 ι 1 ) Q 1 + E{[ι 1 + δκ 2 Ḡ +δκ 2 E[Ḡ ( p u n 2( X 2, 1 + α r X 2 ) ) (X n r X 2, X n r X 2, X 2 ) k + α Q u 1 ) + ], 2 )} ) 2 ) 2 ) ) (X Q u 1 ) + ], ) )(X Q u 1 )]. u ]( Q 1 X)} where n u X 2 = (X Q (κ 1 ι 1 ) Q 1 + δκ 2 Ḡ 1 ) + and ( p n α r u X 2 = X Q 1. By Theorem 4.4.4(b), Qu 1 Q 1. Hence, ( ) p u n r 2( X X ) E(X Q 1) + u (κ 1 ι 1 ) Q 1 + δκ 2 E[Ḡ 2, 1 + α 2 ) (X Q u 1 ) + ]. Le ῑ u 2 := pr 2 (Ḡ( k+α E{[ι 1 + δκ 2 Ḡ pn 2 ) Ḡ( k+α ( p u n 2( X pr 2 ))κ2/ḡ( ) 2, k + α r X 2 ) k+α ) < κ 2. If ι 2 ῑ u 2, since Therefore, if ι 2 ῑ u 2, ( Ĩe u u p = (κ 1 ι 1 ) Q u n 1 + E{[ι 1 + δκ 2 Ḡ 2( X 2, = Ĩ e, +δκ 2 E[Ḡ ( p u n 2( X 2, 1 + α (κ 1 ι 1 ) Q 1 + δκ 2 Ḡ which proves par (b). Q.E.D. r X 2 ) Q u 1 Q 1, u ]( Q 1 X)} E{[ι 1 + δ(κ 2 ι 2 )Ḡ ) (X k + α Q u 1 ) + ] ( p n 2 k + α ( p n 2 E{[ι 1 + δ(κ 2 ι 2 )Ḡ k + α r X 2 ) ) u ]( Q 1 X)} ) u ]( Q 1 X)} ) ](Q u 1 X)}. ( ) ( ) p n 2 E(X 1 + α Q p 1) + n 2 + E{[ι 1 + δ(κ 2 ι 2 )Ḡ ](Q u 1 X)} k + α Proof of Theorem 4.4.5: Par (a). We firs compue he equilibrium oal cusomer surplus in he scenario of sraegic cusomers, S c. If a cusomer is a new cusomer in period 2, her expeced oal surplus is δe((1 + α)v p n 2 ) + (since, by Lemma 7, p n 2 (X n 2, X r 2) = p n 2 ). Hence, he expeced surplus of a sraegic cusomer in he base model is given by: a (µ p 1 + δe((k + α)v p r 2 ) + ) + (1 a )δe((1 + α)v p n 2 ) + =a (µ µ + δe((1 + α)v p n 2 ) + δe((k + α)v p r 2 ) + + δe((k + α)v p r 2 ) + ) + (1 a )δe((1 + α)v p n 2 ) + =δe((1 + α)v p n 2 )

277 Therefore, he equilibrium oal cusomer surplus is given by S c = E[δE((1 + α)v p n 2 ) + X] = δe[((1 + α)v p n 2 ) + X]. We now compue he equilibrium oal cusomer surplus in he scenario of myopic cusomers, S c. Since he cusomers are myopic, hey ge zero uiliy in period 1. Hence, in period 2, he expeced surplus of a new cusomer is δe((1 + α)v p n 2 ) +, whereas ha of a repea cusomer is δe((k + α)v p r 2 ) +. Therefore, he oal cusomer surplus is given by This proves par (a). S c =E[δE((1 + α)v p n 2 ) + (X Q 1) + ] + E[δE((k + α)v p r 2 ) + (X Q 1)] =δe[((1 + α)v p n 2 ) + (X Q 1) + ] + δe[((k + α)v p r 2 ) + (X Q 1)]. Par (b). We firs compue Sc u. If a cusomer is a new cusomer in period 2, her expeced oal surplus is δe((1 + α)v p u 2 ) +. Hence, he expeced surplus of a sraegic cusomer in he NTR model is given by: a u (µ p u 1 + δe((k + α)v p u 2 ) + ) + (1 a u )δe((1 + α)v p u 2 ) + =a u (µ µ + δe((1 + α)v p u 2 ) + δe((k + α)v p u 2 ) + + δe((k + α)v p u 2 ) + ) + (1 a u )δe((1 + α)v p u 2 ) + =δe((1 + α)v p u 2 ) +. Therefore, he equilibrium oal cusomer surplus is given by S u c α)v p u 2 ) + X]. We now compue = E[δE((1+α)V p u 2 ) + X] = δe[((1+ u S c. Since he cusomers are myopic, hey ge zero uiliy in period 1. Hence, in period 2, he expeced surplus of a new cusomer is δe((1 + α)v p u 2 ) +, whereas ha of a repea cusomer is δe((k + α)v p u 2 ) +. Therefore, he oal cusomer surplus is given by S u c This proves par (b). =E[δE((1 + α)v p u 2 ) + u (X Q 1 ) + ] + E[δE((k + α)v p u 2 ) + u (X Q 1 )] =δe[((1 + α)v p u 2 ) + u (X Q 1 ) + ] + δe[((k + α)v p u 2 ) + u (X Q 1 )]. Par (c). Noe ha, by Theorem 4.4.2(a), p r 2 p u 2 p n 2 wih probabiliy 1. I follows immediaely ha S u c = δe[((1 + α)v p u 2 ) + X] δe[((1 + α)v p n 2 ) + X] = S c. In paricular, if k < 1 and Q u 1 > 0, p u 2 < p n 2 wih probabiliy 1 and hus Sc u > Sc. This proves par (c). Q.E.D. Proof of Lemma 9: Par (a). Le W 2 (p n 2, p r 2 X n 2, X r 2) be he expeced social welfare in period 2 when he price for new cusomers is p n 2, and ha for repea cusomers is p r 2. Since all new (repea) cusomers wih valuaion (1 + α)v p n 2 ((k + α)v p r 2) will make a purchase (rade he used producs in), he firm profi equals (p n 2 c 2 )Ḡ( pn 2 1+α )Xn 2 + (p r 2 c 2 + r 2 )Ḡ( pr 2 k+α )Xr 2, he expeced cusomer surplus equals X n 2 E((1 + α)v p n 2 ) + + X r 2E((k + α)v p r 2) +, and he environmenal impac equals κ 2 X n 2 Ḡ( pn 2 1+α ) + (κ 2 ι 2 )X r 2Ḡ( pr 2 k+α ). Therefore, W 2(p n 2, p r 2 X n 2, X r 2) = X n 2 w n (p n 2 ) + X r 2w r (p r 2), where w n (p n 2 ) := (p n 2 c 2 κ 2 )Ḡ( p n α ) + E((1 + α)v pn 2 ) + = E((1 + α)v c 2 κ 2 )1 {(1+α)V p2 }, 263

278 and w r (p r 2) := (p r 2 c 2 +r 2 κ 2 +ι 2 )Ḡ( p r 2 k + α )+E((k+α)V pr 2) + = E((k+α)V c 2 +r 2 κ 2 +ι 2 )1 {(k+α)v p r 2 }. Thus, w n(p n 2 ) = pn 2 c2 κ2 1+α g( pn 2 1+α ) and w r(p r 2) = pr 2 c2+r2 κ2+ι2 k+α g( pr 2 k+α ). Thus, w n(p n 2 ) > 0 if p n 2 < c 2 + κ 2 and w n(p n 2 ) < 0 if p n 2 > c 2 + κ 2. Analogously, w r(p r 2) > 0 if p r 2 < c 2 + κ 2 r 2 ι 2 and w r(p r 2) < 0 if p r 2 > c 2 + κ 2 r 2 ι 2. Hence, he unique maximizer of w n ( ) is c 2 + κ 2, and he unique maximizer of w r ( ) is c 2 + κ 2 r 2 ι 2. Finally, i is sraighforward o check ha c 2 + κ 2 r 2 ι 2 c 2 + κ 2, wih he inequaliy being sric if and only if r 2 > 0 or ι 2 > 0. Therefore, p n s,2(x n 2, X r 2) p n s,2 = c 2 + κ 2 and p r s,2(x n 2, X r 2) p r s,2 = c 2 + κ 2 r 2 ι 2 for any realized (X n 2, X r 2). This proves par (a). Par (b). Under he equilibrium prices (p n s,2, p r s,2), a new cusomer will make a purchase if and only if her valuaion (1 + α)v p n s,2, whereas a repea cusomer will make a purchase (and join he rade-in program) if and only if her valuaion (k + α)v p r s,2. Therefore, ( p n ) Q n s,2(x2 n, X2) r = E[X2 n 1 {(1+α)V p n s,2 } X2 n ] = X2 n Ḡ s,2, 1 + α and which proves par (b). ( p r ) Q r s,2(x2 n, X2) r = E[X21 r {(k+α)v p r s,2 } X2] r = X2Ḡ r s,2, k + α Par (c). Plugging p n s,2 and p r s,2 ino w n 2 ( ) and w r 2( ), respecively, we have w n 2 (p n s,2) = E[(1+α)V p n s,2] + and w r 2(p r s,2) = E[(1 + α)v p r s,2] +. Therefore, w 2 (X n 2, X r 2) = X n 2 E[(1 + α)v p n s,2] + + X r 2E[(1 + α)v p r s,2] +. This complees he proof of par (c). Q.E.D. Proof of Lemma 10: Par (a). Le W s (Q 1 ) be he expeced social welfare wih firs-period producion quaniy Q 1 under sraegic cusomer behavior. Following he same argumen as he proof of Theorem 4.3.1(a), we have p s,1 = µ + δe[(k + α)v p r s,2] + δe[(1 + α)v p n s,2] + = µ + δe[(k + α)v p r s,2] + δe[(1 + α)v p n s,2] + = µ + δ(β s,r β s,n) = m s,1, which proves par (a-i). We now compue W s (Q 1 ). By Lemma 9(c), w 2 (X2 n, X2) r = βs,nx 2 n + βs,rx 2, r so W s (Q 1 ) = p s,1e(x Q 1 ) + (µ p s,1)(x Q 1 ) (c 1 + κ 1 )Q 1 + (r 1 + ι 1 )E(Q 1 X) + +δe{w 2 (X (X Q 1 ), X Q 1 )} = (µ r 1 ι 1 )E(X Q 1 ) (c 1 r 1 + κ 1 ι 1 )Q 1 + δe{βs,n(x (X Q 1 )) + βs,r(x Q 1 )} = (m s,1 r 1 ι 1 )E(X Q 1 ) (c 1 r 1 + κ 1 ι 1 )Q 1 + δβs,ne(x). 264

279 Therefore, Q s,1 is he soluion o a newsvendor problem wih marginal revenue m s,1 r 1 ι 1, marginal cos c 1 +κ 1 r 1 ι 1, and demand disribuion F ( ). Hence, Q s,1 = F 1 ( c 1+κ 1 r 1 ι 1 m s,1 r 1 ι 1 ), and he equilibrium social welfare is W s = W s (Q s,1) = (m s,1 r 1 ι 1 )E(X Q s,1) (c 1 r 1 + κ 1 ι 1 )Q 1 + δβ s,ne(x). This proves par (a-ii,iii). Par (b). Le W s (Q 1 ) be he expeced social welfare wih myopic cusomers, if he firs-period producion quaniy is Q 1. The willingness-o-pay of myopic cusomers is heir expeced valuaion of he firs-generaion produc µ. Thus, p s,1 = µ. This proves par (b-i). We now compue W s (Q 1 ). By Lemma 9(c), w 2 (X2 n, X2) r = βs,nx 2 n + βs,rx 2, r so W s (Q 1 ) = p s,1e(x Q 1 ) + (µ p s,1)(x Q 1 ) (c 1 + κ 1 )Q 1 + (r 1 + ι 1 )E(Q 1 X) + +δe{w 2 (X (X Q 1 ), X Q 1 )} = (µ r 1 ι 1 )E(X Q 1 ) (c 1 r 1 + κ 1 ι 1 )Q 1 + δe{βs,n(x (X Q 1 )) + βs,r(x Q 1 )} = ( m s,1 r 1 ι 1 )E(X Q 1 ) (c r 1 + κ 1 ι 1 )Q 1 + δβs,ne(x). Therefore, Q s,1 is he soluion o a newsvendor problem wih marginal revenue m s,1 r 1 ι 1, marginal cos c 1 +κ 1 r 1 ι 1, and demand disribuion F ( ). Hence, Q s,1 = F 1 ( c1+κ1 r1 ι1 m ), and he equilibrium s,1 r1 ι1 social welfare is W s = W s ( Q s,1) = ( m s,1 r 1 ι 1 )E(X Q s,1) (c r 1 + κ 1 ι 1 ) Q 1 + δβ s,ne(x). This proves par (b-ii,iii). Par (c). Since p s,1 p s,1 = βs,r βs,n = e s, p s,1 p s,1 if and only if e s 0. The equaliies Q s,1 = Q s,1 and Ws = W s follow from he fac ha m s,1 = m s,1. This esablishes par (c). Q.E.D. Proof of Theorem 4.5.1: Par (a). Wih he uni subsidy rae s r for remanufacured producs, he expeced per demand profi from repea cusomers v2(p r r 2) = (p r 2 + s r + s 2 c 2 + r 2 )Ḡ( pr 2 k+α ). Since p r 2 sr v r 2(p r 2) = 1 1+α g( pr 2 1+α ) 0, vr 2(p r 2) is submodular in (p r 2, s r ). Hence, p r 2 = argmax p r 2 0v r 2(p r 2) is coninuously decreasing in s r. This complees he proof of par (a-i). Because Q r 2(X2 n, X2) r = ) X2Ḡ r and p r 2 is decreasing in s r, Q r 2(X2 n, X2) r is increasing in s r, which proves par (a-ii). ( p r 2 k+α Par (b). By Theorem 4.3.1(a), p 1 = µ + δ[e((k + α)v p r 2 ) + E((1 + α)v p n 2 ) + ], which is decreasing in p r 2. Since p r 2 is decreasing in s r, p 1 is increasing in s r. Wih he uni subsidy rae s r for remanufacured produc, Π f (Q 1 ) = (p 1 r 1 s r )E(X Q 1 ) (c 1 r 1 s r )Q 1 + δβ ne(x), Hence, Q 1 = F ( ) 1 c1 r 1 s r p 1 r 1 s r. The criical fracile c 1 r 1 s r p 1 r 1 s r is decreasing in p 1 and s r. Therefore, Q 1 is increasing in s r. For each Q 1, Π f (Q 1 ) is increasing in s r. Thus, Π f = max Q 1 0 Π f (Q 1 ) is increasing in s r. By Lemma 27(a), I e = I e (Q 1), which is increasing in Q 1. Thus, I e is increasing in s r as well. This esablishes par (b). 265

280 Par (c). By Theorem 4.3.1(b), p 1 = µ, which is independen of s r. Wih he uni subsidy rae s r for remanufacured produc, Π f (Q 1 ) = ( p 1 r 1 s r )E(X Q 1 ) (c 1 r 1 s r )Q 1 + δβ ne(x), Hence, Q 1 = F ( ) 1 c1 r 1 s r p 1 r 1 s r. The criical fracile c 1 r 1 s r p 1 r 1 s r is decreasing in s r. Therefore, Q 1 is increasing in s r. For each Q 1, Π f (Q 1 ) is increasing in s r. Thus, Π f = max Q 1 0 Π f (Q 1 ) is increasing in s r. By Lemma 27(b), Ĩ e = Ĩe(Q 1), which is increasing in Q 1. Thus, Ĩ e is increasing in s r as well. This esablishes par (c). Q.E.D. ( ) Proof of Theorem 4.5.2: Par (a). If s 2 is he soluion o p n s,2 = argmax p n 2 0(p n p n s 2 c 2 )Ḡ 1+α, i is clear ha he subsidy/ax scheme wih s 2 = s 2 can induce he equilibrium price for new cusomers p n s,2. We now show ha s 2 exiss. Since v n 2 (p n 2 ) is quasiconcave in p n 2 for any s 2, he firs-order condiion p n 2 v n 2 (p n 2 ) = 0 guaranees he opimal price for new cusomers. Moreover, which is sricly decreasing in s 2. ( p n ) p n 2 v2 n (p n s,2) = Ḡ s,2 pn 1 + α s,2 + s 2 c 2 g 1 + α ( p n ) s,2, 1 + α Hence, here exiss a unique s 2, such ha p n 2 v n 2 (p n s,2) = 0, hus inducing he socially opimal equilibrium price for new cusomers p n s,2. This proves par (a-i). If s r is he soluion o p r s,2 = argmax p r 2 0(p r 2 + s 2 + s r c 2 + r 2 )Ḡ ( p r 2 k+α ), he subsidy/ax scheme wih s r = s r can induce he equilibrium rade-in price for repea cusomers p r s,2. We now show ha s r exiss. Since v r 2(p r 2) is quasiconcave in p r 2 for any (s 2, s r ), he firs-order condiion p r 2 v r 2(p r 2) = 0 guaranees he opimal price for new cusomers. Moreover, if s 2 = s 2, ( p r ) p r 2 v2(p r r s,2) = Ḡ s,2 pr s,2 + s ( 2 + s r c 2 + r 2 p r ) s,2 g, k + α k + α k + α which is sricly decreasing in s r. Hence, here exiss a unique s r, such ha p r 2 v r 2(p r s,2) = 0 if s 2 = s 2, hus inducing he socially opimal equilibrium rade-in price for repea cusomers p r s,2. This proves par (a-ii). Given he subsidy/ax scheme (s 1, s 2, s r), as shown above, he firm adops he same second-period pricing sraegy as he social welfare maximizing one: (p n s,2, p r s,2). Hence, he firs-period price should also be he same as he one which is socially opimal and characerized by Lemma 10(a): p s,1 = µ + δ[β s,r β s,n]. Thus, he expeced profi of he firm in period 1 is Π s f (Q 1 ) = (p s,1 + s 1 r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δe[(x X Q 1 )(p n ( p r ) +(X Q 1 )(p r s,2 + s 2 + s s,2 r c 2 + r 2 )Ḡ ] k + α = (m s 1(s 1 ) r 1 )E(X Q 1 ) (c 1 r 1 )Q 1 + δ(p n s,2 + s 2 c 2 )Ḡ where m s 1(s 1 ) = s 1 + m s,1 + δ[(κ 2 + s 2 + s r ι 2 c 1 r 1 ( p n ) s,2 + s s,2 2 c 2 )Ḡ 1 + α ( p n ) s,2 E(X), 1 + α pr s,2 )Ḡ( k+α ) (κ pn s,2 2 + s2)ḡ( 1+α )]. Thus, Πs f (Q 1) has a unique opimizer F 1 ( m s 1 (s 1) r 1 ). Moreover, as shown in Lemma 10, Q s,1 = F 1 ( c1+κ1 r1 ι1 s r ). Therefore, if s 1 is he unique soluion o c 1 r 1 m s 1 (s1) r1 m s,1 r 1 s r = c 1+κ 1 r 1 ι 1 s r m, he opimal producion quaniy wih he s,1 r1 s r 266

281 linear subsidy/ax scheme s g = (s 1, s 2, s r) is Q s,1, which is he socially opimal firs-period producion quaniy. This proves par (a-iii). Ḡ We now show ha s 2 is increasing in κ 2. As shown in par (a-i), s 2 saisfies ) ( pn s,2 +s 2 c 2 p n ) s,2 1+α g 1+α = 0, i.e., ( p n s,2 1+α (1 + α)ḡ s 2 = g ( p n s,2 1+α ( p n s,2 1+α ) ( ) (1 + α)ḡ c2 +κ 2 ) p n 1+α s,2 + c 2 = ( ) κ 2. g c 2+κ 2 1+α Because g(v)/ḡ(v) is increasing in v, s 2 is sricly decreasing in κ 2. ) ) saisfies Ḡ pr g = 0, i.e., ( p r s,2 k+α (k + α)ḡ s r = g s,2 +s 2 +sr c2+r2 k+α ( p r s,2 k+α ( p r s,2 k+α ) ( p r s,2 k+α ) p r s,2 s 2 + c 2 r 2 = ( (k + α)ḡ c 2 r 2+κ 2 ι 2 k+α g ( c2 r 2 +κ 2 ι 2 k+α Because g(v)/ḡ(v) is increasing in v, s r is sricly increasing in ι 2. c 1 r 1 m s 1 (s 1 ) r1 Analogously, by par (a-ii), s r ) ) s 2 κ 2 + ι 2. By par (a-iii), s 1 saisfies = c 1+κ 1 r 1 ι 1 s r m, he lef-hand-side of which is sricly decreasing in s 1, whereas he righ-hand-side of which is sricly increasing in κ 1. Therefore, s 1 is s,1 r1 s r sricly decreasing in κ 1. This proves par (a-iv). Define κ s 2 as he soluion o (1+α)Ḡ( c 2+κ 2 κ 2 + ι 2 = 0, and κ s 1 as he soluion o 1+α ) g( c 2 +κ 2 1+α ) c 1 r 1 m s 1 (0) r1 = κ 2, ῑ s 2 as he soluion o (k+α)ḡ( c 2 r 2+κ 2 ι 2 = c 1+κ 1 r 1 ι 1 s r m. Since s,1 r1 s r k+α ) g( c 2 r 2 +κ 2 ι 2 k+α ) s 2 g(v)/ḡ(v) is increasing in v, κ s 2, ῑ s 2, and κ s 1 are well-defined and unique. By he proof of par (a-iv), s 2 is sricly decreasing in κ 2, s r is sricly increasing in ι 2, and s 1 is sricly decreasing in κ 1. Therefore, s 1 0 if and only if κ 1 κ s 1, s 2 0 if and only if κ 2 κ s 2, and s r 0 if and only if ι 2 ῑ s 2. This proves par (a-v). Par (b). By Lemma 9 and Lemma 10 (c), he social-welfare-maximizing equilibrium oucome is he same wih sraegic cusomers and wih myopic cusomers, excep ha p s,1 = m s,1 and p s,1 = µ. Therefore, exacly he same argumen as he proof of par (a) proves par (b) as well. In paricular, since he second-period decisions should be independen of wheher he cusomers are sraegic or myopic, s 2 = s 2 and s r = s r. Par (c). Since m s,1 = m s,1, pars (a) and (b) imply ha c 1 r 1 m s 1 (s 1 ) r 1 = c 1 r 1 m s 1 ( s 1 ) r 1. Thus, ms 1(s 1) = m s 1( s 1) and, hence, s 1+m s,1 = s 1+µ, i.e., s 1 s 1 = µ m s,1 = e s. Therefore, s 1 s 1 if and only if e s c 1 r 1 m s 1 (0) r1 c 1 r 1 m s 1 (0) r1 0. Moreover, since κ s 1 saisfies = c 1+ κ s 1 r 1 ι 1 s r m and κ s s,1 r1 s 1 saisfies = c 1+ κ s 1 r 1 ι 1 s r r m. s,1 r1 s r Because m s,1 = m s,1, κ s 1 κ s 1 if and only if m s 1(0) m s 1(0), i.e., e s 0. This proves par (c). Q.E.D. Proof of Theorem 4.5.3: Par (a). We firs compue Cg = C g (s g) and C g = C g ( s g), observe ha C g (s g) =E{s 1(X Q s,1) + s r(q s,1 X) + + δ[s 2Q n s,2((x Q s,1) +, X Q s,1) + (s r + s 2)Q r s,2((x Q s,1) +, X Q s,1)]}, and C g ( s g) =E{ s 1(X Q s,1) + s r( Q s,1 X) + + δ[ s Q 2 n s,2((x Q s,1) +, X Q s,1) + (s r + s 2) Q r s,2((x Q s,1) +, X Q s,1)]}. 267

282 By Lemma 10(c) and Theorem 4.5.2(c), Q s,1 = Q s,1, s 2 = s 2, and s r = s r, i follows immediaely ha C g (s g) C g ( s g) = (s 1 s 1)E(X Q s,1), which proves par (a). Par (b). By par (a), Cg C g if and only if s 1 s 1. By Theorem 4.5.2(c), s 1 s 1 if and only if e s 0. Since e s = E((k + α)v c 2 κ 2 + r 2 + ι 2 ) + E((1 + α)v c 2 κ 2 ) + is sricly increasing in r 2 + ι 2. Hence, le V := min{r 2 + ι 2 : e s 0}. I follows immediaely ha e s 0 if and only if r 2 + ι 2 V 2. We observe ha E((k + α)v c 2 κ 2 ) + E((1 + α)v c 2 κ 2 ) + < 0. Thus, V 2 > 0. This esablishes par (b). Q.E.D. 268

283 D.1 Proofs of Saemens D. Appendix for Chaper 5 We use o denoe he derivaive operaor of a single variable funcion, x o denoe he parial derivaive operaor of a muli-variable funcion wih respec o variable x, and 1 { } o denoe he indicaor funcion. The following lemma is used hroughou our proof. Lemma 28 Le F i (z, Z) be a coninuously differeniable and joinly concave funcion in (z, Z) for i = 1, 2, where z [z, z] (z and z migh be infinie) and Z R n. For i = 1, 2, le (z i, Z i ) := argmax (z,z) F i (z, Z), be he opimizers of F i (, ). If z 1 < z 2, we have: z F 1 (z 1, Z 1 ) z F 2 (z 2, Z 2 ). = 0 if z 1 > z, Proof: z 1 < z 2, so z z 1 < z 2 z. Hence, z F 1 (z 1, Z 1 ) 0 if z 1 = z; = 0 if z 2 < z, and z F 2 (z 2, Z 2 ) i.e., z F 1 (z 1, Z 1 ) 0 z F 2 (z 2, Z 2 ). Q.E.D. 0 if z 2 = z. Proof of Lemma 11: Since p( ) and γ( ) are wice coninuously differeniable, R(, ) is wice coninuously differeniable, and joinly concave in (d, I a ) if and only if he Hessian of R(d, I a ) is negaive semi-definie, i.e., d 2 R(d, I a ) 0, and d 2 R(d, I a ) I 2 R(d, I a ) ( d I a R(d, I a )) 2, where d 2 R(d, I a ) = p (d )(d + γ(i a )) + 2p (d ), d I a R(d, I a ) = p (d )γ (I a ), and I 2 R(d, I a a ) = (p(d ) b α(c + r d ))γ (I a ). I is easily verified ha he Hessian of R(d, I a ) is negaive semi-definie if and only if (p (d )(d + γ(i a )) + 2p (d ))(p(d ) b α(c + r d ))γ (I a ) (p (d )γ (I a )) 2. Q.E.D. Proof of Lemma 12: For par (a), if γ (I a ) = 0, he lef hand side of (5.3) equals o 0. Since he righ hand side of (5.3) is greaer han or equal o 0 and (p (d )) 2 > 0, he (5.3) holds only if γ (I a ) = 0. For he second half of par (a), i suffices o show ha if γ (I 0 ) = 0, γ (I a ) = 0 for any I a I 0. Since γ (I a ) 0 for all I a K a, γ (I a ) γ (I 0 ) = 0 for any I a I 0. On he oher hand, γ (I a ) 0 for all I a K a, so γ (I a ) = 0 and, hus, γ (I a ) = 0 for all I a I 0. Par (b): By par (a), for any I a such ha γ (I a ) = 0, γ (I a ) = 0. (γ (I a )) 2 Mγ (I a ) for any 0 < M < +. Now we suppose γ (I a ) 0. Since p( ), p ( ) and p ( ) are coninuous funcions defined on a compac se [d, d] wih p ( ) < 0 and γ(k a ) γ(i a ) γ 0, (p (d )(d + γ(i a )) + 2p (d ))(p(d ) b α(c + r d ))/(p (d )) 2 is uniformly bounded from below by a consan number, and we define his number 269

284 o be M. Hence, by (5.3), (γ (I a )) 2 Mγ (I a ). Q.E.D. Proof of Lemma 13: Par (a). Observe ha ˆp δ ( ) p ( ) and ˆp δ ( ) p ( ) for any δ > 0. Thus, le m := and max { ˆp δ (d )(d + γ(i a )) + 2ˆp δ (d ) d [d, d],i a (ˆp Ka δ (d )) 2 } = max { p (d )(d + γ(i a )) + 2p (d ) d [d, d],i a (p (d Ka )) 2 } < 0, k := Therefore, for any δ δ, d [d, d], I a K a, min {p(d ) b α(c + r d )} 0, d [d, d] δ := M m k < +. (ˆp δ (d )(d + γ(i a )) + 2ˆp δ (d ))(ˆp δ (d ) b α(c + r d )) (ˆp δ (d )) 2 γ (I a ) = p (d )(d + γ(i a )) + 2p (d ) (p (d )) 2 (p(d ) + δ b α(c + r d ))γ (I a ) p (d )(d + γ(i a )) + 2p (d ) (p (d )) 2 ( M m k + p(d ) b α(c + r d ))γ (I a ) p (d )(d + γ(i a )) + 2p (d ) (p (d )) 2 ( M m )γ (I a ) Mγ (I a ) (γ (I a )) 2, where he firs inequaliy follows from δ δ, he second from p(d ) b α(c + r d ) k, he hird from he definiion of m and he las from he assumpion ha Mγ (I a ) (γ (I a )) 2 for any I a K a. Hence, by (5.3), for any δ δ, ˆR δ (, ) is joinly concave on d [d, d], I a K a. and Par (b). Observe ha ˆγ ς( ) γ ( ) and ˆγ ς ( ) γ ( ) for any ς > 0. Since p (d ) 0, le n := max { (p(d ) b α(c + r d ))p (d ) d [d, d] (p (d )) 2 } < 0, l := min {γ(i d [d, d],i a K a ) + d + 2p (d ) a p (d ) } > 0, Therefore, for any ς ς, d [d, d], I a K a, ς := M n l < +. (p (d )(d + ˆγ ς (I a )) + 2p (d ))(p(d ) b α(c + r d )) (p (d )) 2 ˆγ ς (I a ) = (p(d ) b α(c + r d ))(p (d )(d + γ(i a ) + ς) + 2p (d )) (p (d )) 2 γ (I a ) = (p(d ) b α(c + r d ))p (d ) (p (d )) 2 (ς + γ(i a ) + d + 2p (d ) p (d ) )γ (I a ) (p(d ) b α(c + r d ))p (d ) (p (d )) 2 ( M n l + γ(ia ) + d + 2p (d ) p (d ) )γ (I a ) (p(d ) b α(c + r d ))p (d ) (p (d )) 2 ( M n )γ (I a ) Mγ (I a ) (γ (I a )) 2 = (ˆγ ς(i a )) 2, 270

285 where he firs inequaliy follows from ς ς, he second from γ(i a ) + d + 2p (d ) p (d ) l, he hird from he definiion of n and he las from he assumpion ha Mγ (I a ) (γ (I a )) 2 for any I a by (5.3), for any ς ς, ˆR ς (, ) is joinly concave on d [d, d], I a K a. Q.E.D. K a. Hence, Proof of Lemma 14: We prove pars (a) - (b) ogeher by backward inducion. We firs show, by backward inducion, ha if V 1 (I a 1, I 1 ) r d I a 1 ci 1 is concavely decreasing in boh I a 1 and I 1, boh g (x a, x, d, I a ) := E{G (x a δ(p(d ), I a, ϵ ), x δ(p(d ), I a, ϵ ))} and J (x a, x, d, I a, I ) are joinly concave, g (,,, I a ) and J (,,, I a, I ) are sricly concave for any fixed I a and I, and V (I a, I ) r d I a ci is joinly concave and decreasing in I a and I. I is clear ha V 0 (I a 0, I 0 ) r d I a 0 ci 0 = r d I a 0 ci 0 is joinly concave, and decreasing in I a 0 and I 0. Hence, he iniial condiion holds. Assume ha V 1 (I a 1, I 1 ) r d I a 1 ci 1 is concavely decreasing in boh I a 1 and I 1. Therefore, G (x, y) is joinly concave and decreasing in x and y. For every realizaion of ϵ = (ϵ a, ϵ m ), we verify ha G (x a δ(p(d ), I a, ϵ ), x δ(p(d ), I a, ϵ )) is joinly concave in (x a, x, d, I a )as follows: le 0 λ 1, x a := λx a 1 + (1 λ)x a 2, x := λx 1 + (1 λ)x 2, d := λd 1 + (1 λ)d 2 and I a := λi a 1 + (1 λ)i a 2, we have: λg (x a 1 (d 1 + γ(i 1 ))ϵ m ϵ a, x 1 (d 1 + γ(i 1 ))ϵ m ϵ a ) + (1 λ)g (x a 2 (d 2 + γ(i 2 ))ϵ m ϵ a, x 2 (d 2 + γ(i 2 ))ϵ m ϵ a ) G (x a (d + λγ(i 1 ) + (1 λ)γ(i 2 ))ϵ m ϵ a, x (d + λγ(i 1 ) + (1 λ)γ(i 2 ))ϵ m ϵ a ) G (x a (d + γ(i ))ϵ m ϵ a, x (d + γ(i ))ϵ m ϵ a ), where he firs inequaliy follows from he join concaviy of G (, ), he second from he concaviy of γ( ), he monooniciy ha G (, ) is decreasing in boh of is argumens, and ϵ m 0. Since concaviy is preserved under expecaion, g (x a, x, d, I a ) = E{G (x a δ(p(d ), I a, ϵ ), x δ(p(d ), I a, ϵ ))} is joinly concave in (x a, x, d, I a ). Noe ha R(d, I a ) is joinly concave in (d, I a ), θ(x I ) is joinly concave in (x, I ), and (r d +r w )(x a I a ) is joinly concave in (x a, I a ). Therefore, J (x a, x, d, I a, I ) is joinly concave in (x a, x, d, I a, I ). The sric concaviy of g (,,, I a ) follows direcly from he coninuous disribuion of D and ha is suppor is an inerval. Since g (,,, I a ) is sricly concave and R(, I a ) is concave for any fixed I a, J (,,, I a, I ) is sricly joinly concave for any fixed I a and I. Concaviy is preserved under maximizaion (see, e.g., Secion of [32]), so he join concaviy of V (I a, I ) follows immediaely from he join concaviy of J (,,,, ). We now verify ha V (I a, I ) is decreasing in boh I a and I. Observe ha γ(i a ), (r d + r w )(x a I a ), and G (x a δ(p(d ), I a, ϵ ), x δ(p(d ), I a, ϵ )) are decreasing in I a, and θ(x I ) is decreasing in I. Hence, J (x a, x, d, I a, I ) is decreasing in I a any I, and I for any fixed (x a, x, d ). Assume I a 1 > I a 2, we have F (I a 1 ) F (I a 2 ). Hence, for V (I a 1, I ) r d I a 1 ci = max (x a,x,d) F (Ia 1 ) J (x a, x, d, I a 1, I ) max (x a,x,d ) F (I a 2 ) J (x a, x, d, I a 2, I ) = V (I a 2, I ) r d I a 2 ci, 271

286 where he inequaliy follows from he monooniciy ha J (x a, x, d, I a, I ) is decreasing in I a, and F (I a 1 ) F (I a 2 ), hus verifying V (I a, I ) is decreasing in I a. Analogously, if I 1 > I 2, for any I a, V (I a, I 1 ) r d I a ci 1 = max J (x a (x a,x,d ) F (I a), x, d, I a, I 1 ) max J (x a (x a,x,d ) F (I a), x, d, I a, I 2 ) = V (I a, I 2 ) r d I a ci 2, where he inequaliy follows from he monooniciy ha J (x a, x, d, I a, I ) is decreasing in I. Second, we show, again by backward inducion, ha if V 1 (, ) is coninuously differeniable, g (,,, ) and V (, ) are coninuously differeniable on he inerior of heir domains. For = 0, V (I a, I ) = 0 is clearly coninuously differeniable. The iniial condiion holds. Assume V 1 (I a 1, I 1 ) is coninuously differeniable, g (x a, x, d, I ) =E{ (b + h a )(x a (d + γ(i ))ϵ m ϵ a ) + + α[v 1 (x a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ) r d (x a (d + γ(i a ))ϵ m ϵ a ) c(x (d + γ(i a ))ϵ m ϵ a )]}. Since ϵ a and ϵ m are coninuous, i is easy o compue he parial derivaives of g (,,, ) as follows: x a g (x a, x, d, I a ) =E{ (b + h a )1 {x a (d +γ(i ))ϵ m +ϵa } + α I a 1 V 1 (x a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a )} αr d, x g (x a, x, d, I a ) =E{α I 1 V 1 (x a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a )} αc, d g (x a, x, d, I a ) =E{(b + h a )ϵ m 1 {x a (d +γ(i a ))ϵm +ϵa } αϵ m I a 1 V 1 (x a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ) αϵ m I 1 V 1 (x a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a )} + α(r d + c), I a g (x a, x, d, I a ) =E{(b + h a )γ (I a )ϵ m 1 {x a (d +γ(i a ))ϵm +ϵa } αγ (I a )ϵ m I a 1 V 1 (x a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ) αγ (I a )ϵ m I 1 V 1 (x a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a )} + α(r d + c)γ (I a ), (D.0) where he exchangeabiliy of differeniaion and expecaion is easily jusified using he canonical argumen (see, for example, Theorem A.5.1 of [63], he condiion of which can be easily checked observing he coninuiy of parial derivaives of V 1 (, ), and ha he disribuion of D is coninuous.). Since a leas one of ϵ a and ϵ m follows a coninuous disribuion, x a g (x a, x, d, I a ), x g (x a, x, d, I a ), d g (x a, x, d, I a ) and I a g (x a, x, d, I a ) are coninuous. Therefore, g (,,, ) is coninuously differeniable. Since g (,,, I a ) is sricly concave and coninuously differeniable, J (,,, I a, I ) is sricly concave and coninuously differeniable. Moreover, J (,,,, ) is coninuously differeniable if x a I a x I, i.e., i is coninuously differeniable almos everywhere. By envelope heorem, V (, ) is also differeniable on he inerior of he feasible se F (I a ) for x a (I a, I ) I a and x (I a, I ) I. For he and 272

287 case x a (I a, I ) = I a or x (I a, I ) = I, we show he coninuous differeniabiliy of V (, ) in he proof of Theorem This complees he inducion and, hence, he proof of Lemma 14. Q.E.D. Proof of Theorem 5.4.1: Pars (a) - (d) and he differeniabiliy of V (I a, I ). We firs show pars (a) - (d) and he coninuous differeniabiliy of V (I a, I ). Observe ha if x > I (i.e., he firm orders), x J (x a, x, d, I a, I ) = ψ + x g (x a, x, d, I a ) < 0. Hence, if x (I a, I ) > I, x a (I a, I ) = x (I a, I ) > I I a and he opimal policy is given by Equaion (5.7). i.e., if x a (I a ) > I, (x a (I a, I ), x (I a, I ), d (I a, I )) = (x a (I a ), x a (I a ), d (I a )). This complees he proof of par (b). If x < I (i.e., he firm disposes), θ(x I ) = θ(x I ). Hence, he objecive funcion J (x a, x, d, I a, I ) = θi + R(d, I a ) + (θ ψ)x (r d + r w )(x a I a ) + ϕx a + E{G (x a δ(p(d ), I a, ϵ ), x δ(p(d ), I a, ϵ )). Hence, if x (I a, I ) < I, he opimizer prescribed in Equaion (5.9) is he opimal policy. i.e., if x (I a ) < I, (x a (I a, I ), x (I a, I ), d (I a, I )) = ( x a (I a ), x (I a ), d (I a )). Par (c) follows. Nex we show ha x a (I a ) x (I a ). If x a (I a ) > x (I a ), suppose I ( x (I a ), x a (I a )). We have ha: J (x a (I a ), x a (I a ), d (I a ), I a, I ) > sup x a I d]{j,d [d, (x a, I, d, I a, I )}, J ( x a (I a ), x (I a ), d (D.1) (I a ), I a, I ) > sup x a I d]{j,d [d, (x a, I, d, I a, I )}. By he concaviy of J (,,, I a, I ), sup x a I,d [d, d] {J (x a, I, d, I a, I )} λj (x a (I a ), x a (I a ), d (I a ), I a, I )+(1 λ)j ( x a (I a ), x (I a ), d (I a ), I a, I ), where λx a (I a ) + (1 λ) x (I a ) = I. The above inequaliy conradics inequaliy (D.1). Hence, x a (I a ) x (I a ). Par (d) hus follows from par (b), par (c), x a (I a ) x (I a ), and he concaviy of J (,,, I a, I ). The second par of par (a) summarizes pars (b) - (d). Since he proof of Lemma 14 already shows ha J (,,,, ) is coninuously differeniable, i suffices o show ha V (I a, I ) is coninuously differeniable when x a (I a, I ) = I a or x (I a, I ) = I, given ha J (,,,, ) is coninuously differeniable. We only show ha I V (I a, I ) is coninuous a he poins where x (I a, I ) = I, because he coninuiy of I a V (I a, I ) a he poins where x a (I a, I ) = I a follows from he same approach. By he proof of Lemma 14, i suffices o check ha he lef and righ parial derivaives, I V (I a, I ) and I V (I a, I +), are equal when I = x a (I a ) and I = x (I a ). heorem, I V (I a, x a (I a ) ) = c For I = x a (I a ), by he envelope I V (I a, x a (I a )+) = c + β + x a g(x a (I a ), x a (I a ), d (I a ), I a ) + x g(x a (I a ), x a (I a ), d (I a ), I a ). The firs order condiion wih respec o x a and x implies ha β + x a g(x a (I a ), x a (I a ), d (I a ), I a ) + x g(x a (I a ), x a (I a ), d (I a ), I a ) =

288 Therefore, I V (I a, x a (I a ) ) = I V (I a, x a (I a )+). For I = x (I a ), by he envelop heorem, I V (I a, x (I a ) ) = c θ I V (I a, x (I a )+) = c ψ + x g( x a (I a ), x (I a ), d (I a ), I a ). The firs order condiion wih respec o x a I = x (I a ) implies ha x g( x a (I a ), x (I a ), d (I a ), I a ) + θ ψ = 0. Hence, I V (I a, x (I a ) ) = I V (I a, x (I a )+) and he parial derivaive I V (I a, I ) is coninuous. Par (e): Le J a (x a, d, I a ) := R(d, I a ) + βx a + g a (x a, d, I a ), where g a (x a, d, I a ) = E[G a (x a δ(p(d ), I a, ϵ ))], wih G a (x) = G a (x, x). We firs show ha x a (I a ) is decreasing in I a. Le γ := γ(i a ) and y := d + γ. Then, we have J a (x a, d, I a ) = Ĵ a (x a, y, γ ), where Ĵ a (x a, y, γ ) = R (y, γ ) + βx a + E{G a (x a y ϵ m ϵ a )}, wih R (y, γ ) := R(y γ, I a ). We need he following lemma ha esablishes he supermodulariy of R (, ) and R(, ): Lemma 29 (a) R (y, γ ) is sricly supermodular in (y, γ ), where y γ = d [d, d] and y 0. In addiion, R (y, γ ) is sricly concave in y, for any fixed γ ; (b) R(d, I a ) is supermodular in (d, I a ), where d [d, d] and I a K a. In addiion, R(d, I a ) is sricly concave in d, for any fixed I a. Proof of Lemma 29: R (y, γ ) = (p(y γ ) b α(c+r d ))y is wice coninuously differeniable when y γ = d [d, d] and y 0. To prove he supermodulariy of R (, ), i suffices o show ha y γ R (y, γ ) 0. Direc compuaion yields ha: y γ R (y, γ ) = (p (y γ )y + p (y γ )). Since p ( ) < 0 and p ( ) 0, (p (y γ )y + p (y γ )) > 0. Hence, R (y, γ ) is sricly supermodular. Moreover, y 2 R (y, γ ) = p (y γ )y + 2p (y γ ) < 0, since p ( ) 0 and p ( ) < 0. Hence, R (y, γ ) is sricly concave in y, for any fixed γ. This esablishes par (a). R(, ) is wice coninuously differeniable, d I a R(d, I a ) = p (d )γ (I a ) 0. Hence, R(, ) is supermodular. In addiion, d 2 R(d, I a ) = p (d )(d + γ(i a )) + 2p (d ) < 0, so R(d, I a ) is sricly concave in d for any fixed I a.q.e.d. As shown in he proof of Lemma 14, G (, ) and, hus, G a ( ), is concave. Noe ha ϵ m 0, so, for any realizaion of (ϵ a, ϵ m ), i is easily verified ha G a (x y ϵ m ϵ a ) is supermodular in (x, y ). Hence, E{G a (x y ϵ m ϵ a )} is supermodular in (x, y ), since supermodulariy is preserved under expecaion. By Lemma 29, R (y, γ ) is supermodular and, hus, Ĵ a (x, y, γ ) is supermodular in 274

289 (x, y, γ ). Therefore, he opimal order-up-o level, x a (I a ), and opimal expeced demand y (I a ) := d (I a ) + γ are increasing in γ, and, since γ( ) is decreasing in I a, decreasing in I a. We now proceed o show ha he opimal expeced price-induced demand d (I a ) is increasing in I a. Le I a 1 > I a 2, x a 1 := x a (I a 1 ), x a 2 := x a (I a 2 ), d 1 := d (I a 1 ), d 2 := d (I a 2 ) y 1 := d 1 +γ(i a 1 ), and y 2 := d 2 +γ(i a 2 ). We prove ha d 1 d 2 by conradicion. Assume ha d 1 < d 2. By Lemma 28, d 1 < d 2 implies ha d J a (x a 1, d 1, I a 1 ) d J a (x a 2, d 2, I a 2 ). d R(d 1, I a 1 ) d R(d 1, I a 2 ) > d R(d 2, I a 2 ), where he firs inequaliy follows from he supermodulariy of R(, ) and he second inequaliy follows from he sric concaviy of R(, I a ). Hence, d g a (x a 1, d 1, I a 1 ) = d J a (x a 1, d 1, I a 1 ) d R(d 1, I a 1 ) < d J a (x a 2, d 2, I a 2 ) d R(d 2, I a 2 ) = d g a (x a 2, d 2, I a 2 ). Le f(x) := (b + h a )1 {X 0} + α[ I a 1 V 1 (X, X) + I 1 V a 1(X, X) r d c] 0, which is decreasing in X. We have: x a g a (x a i, d i, I a i ) = E{f(x a i y i ϵ m ϵ a )} and d g a (x a i, d i, I a i ) = E{ ϵ m f(x a i y i ϵ m ϵ a )} for i = 1, 2. Recall ha we have proved x a 2 x a 1 and y a 2 y a 1. If x a 1 = x a 2, x a 1 y a 1ϵ m ϵ a x a 2 y a 2ϵ m ϵ a for any realizaion of (ϵ a, ϵ m ). Hence, x a g a (x a 1, d 1, I a 1 ) = E{f(x a 1 y 1 ϵ m ϵ a )} E{f(x a 2 y 2 ϵ m ϵ a )} = x a g a (x a 2, d 2, I a 2 ), where he inequaliy follows from ha f( ) is decreasing. If x a 2 > x a 1, by Lemma 28, x a J a (x a 1, d 1, I a 1 ) x a J a (x a 2, d 2, I a 2 ) and, hence, x a g a (x a 1, d 1, I a 1 ) = x a J a (x a 1, d 1, I a 1 ) β x a J a (x a 2, d 2, I a 2 ) β = x a g a (x a 2, d 2, I a 2 ). (D.2) Noe ha here exiss an ϵ, such ha x a 1 y 1 ϵ m x a 2 y 2 ϵ m if ϵ m ϵ and x a 1 y a 1ϵ m > x a 2 y a 2ϵ m if ϵ m > ϵ (ϵ may equal m or m.). Since f( ) is decreasing, f(x a 1 y 1 ϵ m ϵ a ) f(x a 2 y 2 ϵ m ϵ a ) 0 for any ϵ m [m, ϵ ] and any realizaion of ϵ a. So ϵ m (f(x a 1 y 1 ϵ m ϵ a ) f(x a 2 y 2 ϵ m ϵ a )) ϵ (f(x a 1 y 1 ϵ m ϵ a ) f(x a 2 y 2 ϵ m ϵ a )), (D.3) for any ϵ m [m, ϵ ] and any realizaion of ϵ a. Analogously, for ϵ m [ϵ, m], f(x a 1 y 1 ϵ m ϵ a ) f(x a 2 y 2 ϵ m ϵ a ) 0, and (D.3) holds for ϵ m [ϵ, m] as well. Therefore, (D.3) holds for all ϵ m [m, m] and any realizaion of ϵ a. Taking expecaion, we have: d g a (x a 1, d 1, I a 1 ) d g a (x a 2, d 2, I a 2 ) = E{ ϵ m (f(x a 1 y 1 ϵ m ϵ a ) f(x a 2 y 2 ϵ m ϵ a ))} E{ ϵ (f(x a 1 y 1 ϵ m ϵ a ) f(x a 2 y 2 ϵ m ϵ a ))} = ϵ ( x a g a (x a 1, d 1, I a 1 ) x a g a (x a 2, d 2, I a 2 )) (D.4) 0, 275

290 where he las inequaliy follows from x a g a (x a 1, d 1, I1 a ) x a g a (x a 2, d 2, I2 a ). (D.4) conradics (D.2) and, hence, d 1 d 2, i.e., d (I a ) is increasing in I a. The coninuiy of x a (I a ) and d (I a ) follows direcly from ha he objecive funcion J a (,, I a ) is sricly concave for any given I a. The proof of par (e) follows. Q.E.D. Remark D.1.1 The supermodulariy of R (y, γ ) implies ha o beer ake advanage of he high demand induced by low invenory level, he firm should adjus is price o a level such ha he expeced demand will increase. Proof of Theorem 5.4.2: If h w αc s, θ ψ = c s h w (1 α)c = αc s h w 0. Since g (x a, x, d, I a, I ) is also decreasing in x, Equaion (5.9) implies ha x (I a ) = x a (I a ), for any and I a, which proves par (a). Observe ha for any (x a, x, d, I a, I ), x g (x a, x, d, I a, I ) ( T α j )h w ( α j )h w, = T, T 1, 1, j=1 where he firs inequaliy holds as an equaliy if x j (Ia j, I j) = I j, for all j 1. Hence, x g (x a, x, d, I a, I ) is uniformly bounded from below by ( T j=1 αj )h w, for any. Thus, if θ ψ = αc h w s ( T j=1 αj )h w, x (I a ) = + for any and I a. Hence, s = αc ( T j=0 αj )h w. This proves par (b). If inf I a <K a γ (I a ) M, for any (x a, x, d, I a, I ), x a g (x a, x, d, I a, I ) M( j=1 T α j )( p + h a ) M( α j )( p + h a ), = T, T 1, 1, j=1 where p is he maximum marginal revenue and h a is he maximum marginal holding cos. Hence, x a g (x a, x, d, I a, I ) is bounded from below by M( T j=1 αj )( p + h a ), for any. Thus, if r d + r w + ϕ M( T j=1 αj )( p + h a ), x a (I a ) I a, for any I a K a. If inf I a <K a γ (I a ) =, lim I a K a γ (I a ) =. Hence, for any x, d, and I, lim I a K a j=1 x a g (I a, x, d, I a, I ) α(p b (1 α)(c + r d )) Hence, for any r w, and any x, d and I, lim I a K a γ (I a ) =. x a J (I a, x, d, I a, I ) = r d + r w + ϕ + x a g (I a, x, d, I a, I ), as I a K a. The above limi complees he proof of Par (c). For noaional simpliciy, we denoe x a := x a (I 1, a I 1 ), x := x (I 1, a I 1 ) and d := d (I 1, a I 1 ). Observe ha I a 1 V 1 (I a 1, I 1 ) (p b α(c + r d ))γ (I a 1) + I a 1 g 1 (x a, x, d, I a 1). (D.5) By Equaion (D.0), x a 1 g 1 (x a, x, d, I 1) a = x 1 g 1 (x a, x, d, I 1) a = I a 1 g 1 (x a, x, d, I 1) a = E{f 1 (ϵ m 1)}, E{f 2 (ϵ m 1)}, γ (I a 1)E{ϵ m 1[f 1 (ϵ m 1) + f 2 (ϵ m 1)]}, 276

291 where f 1 (ϵ m 1) = E ϵ a 1 { (b + h a )1 {x a (d +γ(i ))ϵ m 1 +ϵa 1 } f 2 (ϵ m 1) = +α I a 2 V 2 (x a (d + γ(i a 1))ϵ m 1 ϵ a 1, x (d + γ(i a 1))ϵ m 1 ϵ a 1)} αr d E ϵ a 1 {α I 2 V 2 (x a (d + γ(i a 1))ϵ m 1 ϵ a 1, x (d + γ(i a 1))ϵ m 1 ϵ a 1)} αc. The firs order condiions wih respec o x a 1 and x 1 sugges ha E{f 1 (ϵ m 1) + f 2 (ϵ m 1)} (ϕ ψ) = β. Since f 1 ( ) 0 and f( ) 0, we have: E{ϵ m 1[f 1 (ϵ m 1) + f 2 (ϵ m 1)]} E{m[f 1 (ϵ m 1) + f 2 (ϵ m 1)]} = me{f 1 (ϵ m 1) + f 2 (ϵ m 1)} mβ. Therefore, by inequaliy (D.5), I a 1 V 1 (I a 1, I ) (p b α(c + r d ) + mβ)γ (I a 1). (D.6) So for any d [d, d] and any x, x a g (0, x, d, I a ) αe[ I a 1 V 1 ( (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a )] αe[(p b α(c + r d ) + mβ)γ ( (d + γ(i ))ϵ m ϵ a )] α(p b α(c + r d ) + mβ)(1 ι)γ ( D) (D.7) (r d + r w + ϕ), where he firs inequaliy follows from equaion (D.0), he second from (D.6), and he las from he assumpion ha α(p b α(c + r d ) + mβ)(1 ι)γ ( D) + (r d + r w + ϕ) 0. The hird inequaliy of (D.7) follows from he following inequaliy: E[γ ( D )] = E D[γ D ( D )] + E D[γ D ( D )] 0 + E D[γ D ( D)] (1 ι)γ ( D), where he firs inequaliy follows from he concaviy of γ( ) and he second inequaliy follows from he definiion of D. (D.7) implies ha x a (I a, I ) = 0 for all I a K a and all I, which complees he proof of par (d). Q.E.D. Before we proceed o prove he resuls in Secion 5.5, we remark ha R s (d, I a ) shares he same properies as R(d, I a ). i.e., we have he following counerpar of Lemma 29 in he model wihou invenory wihholding: Lemma 30 (a) R s (y, γ ) is sricly supermodular in (y, γ ), where R s (y, γ ) := R s (y γ, I a ), y γ = d [d, d] and y 0. In addiion, R s (y, γ ) is sricly concave in y, for any fixed γ ; (b) R s (d, I a ) is supermodular in (d, I a ), where d [d, d] and I a sricly concave in d, for any fixed I a. K a. In addiion, R s (d, I a ) is Proof of Lemma 30: The proof is idenical o ha of Lemma 29, and hence omied. Q.E.D. 277

292 Proof of Theorem 5.5.1: The proof is very similar o ha of Lemma 14 and Theorem 5.4.1, so we only skech i. For pars (a) - (c), he proof is exacly he same as ha of Lemma 14, and hence omied. To show pars (d) - (f), we define he following unconsrained opimizers: (x L (I a ), d L (I a )) := argmax x a K d]{r s a,d [d, (d, I a ) + β s x a + E[G s (x a δ(p(d ), I a, ϵ ))]}, and (x H (I a ), d H (I a )) := argmax x a K a,d [d, d]{r s (d, I a ) + (β s + θ)x a + E[G s (x a δ(p(d ), I a, ϵ ))]}. We need he following lemma: Lemma 31 Le γ := γ(i a ), Ψ(x a, y, µ γ ) := R s (y, γ )+µx a +E{G s (x a y ϵ m in (x a, y, µ) for any given γ. ϵ a )} is supermodular Proof of Lemma 31: Since G s ( ) is concave and ϵ m 0, E{G s (x a y ϵ m ϵ a )} is supermodular in (x a, y ). I s also clear ha µx a is sricly supermodular in (x a, µ). Therefore, Ψ(x a, y, µ γ ) is supermodular in (x a, y, µ) for any given γ. Q.E.D. Lemma 31 and is proof imply ha x L (I a ) < x H (I a ) since β s + θ > β s. Exacly he same argumen as in he proof of Theorem 5.4.1(e) implies ha x L (I a ) and x H (I a ) are coninuously decreasing in I a d L (I a ) and d H (I a ) are coninuously increasing in I a. I L := sup{i a : I a < x L (I a )} and I H := inf{i a : I a > x H (I a )}. I s clear ha I L and I H are he hresholds in par (d). Therefore, x s (I a ) = x L (I a ) if I a < I L ; I a if I L I a I H ; x H (I a ) if I a > I H. I s clear ha x s (I a ) saisfies he saemen in par (e). Therefore, we have d L (I a ) if I a < I L ; d s (I a ) = argmax d [d, d]j s (I a, d, I a ) if I L I a I H ; d H (I a ) oherwise. To prove par (f), i remains o show ha d s (I a ) is increasing in I a for I L I a I H. Le U s (d, I a ) := J s (I a, d, I a ) and i is easily verified ha U s (d, I a ) is supermodular in (d, I a ). Thus, d s (I a ) is increasing in I a, which complees he proof of Theorem Q.E.D. and Proof of Theorem 5.5.2: We show boh pars by backward inducion. For par (a), we use backward inducion o recursively show his resul. For = 0, V s 0 ( ) = ˆV s 0 ( ) = 0 and, hence, I a 0 V s 0 (I a 0 ) = I a 0 ˆV s 0 (I a 0 ) for all I a 0. We show ha: if I a 1 V s 1(I a 1) I a 1 ˆV s 1 (I a 1) for all 278

293 I a 1 K a, (a) I L ÎL, (b) I H ÎH, (c) x s I a (I a ) ˆx s (I a ), (d) d s (I a ) s ˆd (I a ) and (e) I a V s (I a ) ˆV s (I a ) for all I a K a. To prove hese inequaliies, we define (ˆx L (I a ), ˆd L (I a )) and (ˆx H (I a ), ˆd H (I a )) as he unconsrained opimizers in he model wih demand ˆD, corresponding o (x L (I a ), d L (I a )) and (x H (I a ), d H (I a )), respecively. Le y L (I a ) := d L (I a ) + γ(i a ), ŷ L (I a ) := ˆd L (I a ) + ˆγ(I a ) = ˆd L (I a ) + γ 0, ˆR s (d, I a ) := R s (d, ), and Ĝs (y) := (h a +b)y + +α[ ˆV s 1(y) cy]. We define he objecive funcions J L (x a, d, I a ) := R s (d, I a ) + β s x a + g s (x a, d, I a ), Ĵ L (x a, d, I a ) := ˆR s (d, I a ) + β s x a + ĝ s (x a, d, I a ), where ĝ s (x a, d, I a ) := E{Ĝs (x a ˆδ(p(d ), I a, ϵ ))}. Since ϵ m = 1 wih probabiliy 1, g s (x a, d, I a ) = H s (x a d γ(i a )) and ĝ s (x a, d, I a ) = Ĥ s (X) := E{Ĝs (X ϵ a )}. Ĥs (x a d γ 0 ), where H s (X) := E{G s (X ϵ a )} and Firs, we show ha, if I a 1 V s 1(I a 1) I a 1 ˆV s 1 (I a 1) for all I a 1 K a, x L (I a ) ˆx L (I a ), d L (I a ) ˆd L (I a ), x H (I a ) ˆx H (I a ), and d H (I a ) ˆd H (I a ). Since I a 1 V s 1(I a 1) I a 1 ˆV s 1 (I a 1), X H s (X) X Ĥ s (X) for any X. We only show ha x L (I a ) ˆx L (I a ) and d L (I a ) ˆd L (I a ), while x H (I a ) ˆx H (I a ) and d H (I a ) ˆd H (I a ) follow from he same argumen. We show by conradicion ha x L (I a ) ˆx L (I a ) and d L (I a ) ˆd L (I a ). Noe ha, for he model wih invenory-independen demand (i.e., he firm faces ˆD ), i is reduced o he classical join pricing and invenory managemen problem wih sochasic demand inroduced in Federgruen and Heching (1999). Hence, ˆx L (I a ) and ˆd L (I a ) are consans independen of I a. Assume ha x L (I a ) > ˆx L (I a ). Lemma 28 yields ha x a J L (x L (I a ), d L (I a ), I a ) x a Ĵ L (ˆx L (I a ), ˆd L (I a ), I a ). Hence, X H s (x L (I a ) y L (I a )) = x a J L (x L (I a ), d L (I a ), I a ) β s x a Ĵ L (ˆx L (I a ), ˆd L (I a ), I a ) β s = X Ĥ s (ˆx L (I a ) ŷ L (I a )). Since X H s (X) X Ĥ s (X) for any X and boh of hem are sricly decreasing, y L (I a ) > ŷ L (I a ). Thus, d L (I a ) = y L (I a ) γ(i a ) > ŷ L (I a ) γ 0 = ˆd L (I a ). Invoking Lemma 28, we have d J L (x L (I a ), d L (I a ), I a ) d Ĵ L (ˆx L (I a ), ˆd L (I a ), I a ), and d R s (d L (I a ), I a ) = d J L (x L (I a ), d L (I a ), I a ) + X H s (x L (I a ) y L (I a )) d Ĵ L (ˆx L (I a ), ˆd L (I a ), I a ) + X Ĥ s (ˆx L (I a ) ŷ L (I a )) = d ˆRs ( ˆd L (I a ), I a ) Since d R s (d, I a ) = y R s (d + γ(i a ), γ(i a )), y R s (y L (I a ), γ(i a )) y R s (ŷ L (I a ), γ 0 ). However, he sric concaviy of R s (, γ ) and he supermodulariy of R s (, ) yield ha y R s (y L (I a ), γ(i a )) < y R s (ŷ L (I a ), γ(i a )) y R s (ŷ L (I a ), γ 0 ), which leads o a conradicion. Therefore, we have x L (I a ) ˆx L (I a ). Assume ha d L (I a ) < ˆd L (I a ), so y L (I a ) = d L (I a ) + γ(i a ) < ˆd L (I a ) + γ 0 = ŷ L (I a ). Lemma 28 yields ha d J L (x L (I a ), d L (I a ), I a ) d Ĵ L (ˆx L (I a ), ˆd L (I a ), I a ). The sric concaviy of R s (, I a ) and he supermodulariy of R s (, ) imply ha d R s (d L (I a ), I a ) > d R s ( ˆd L (I a ), I a ) d R s ( ˆd L (I a ), ) = d ˆRs ( ˆd L (I a ), I a ). 279

294 Hence, we have: X H s (x L (I a ) y L (I a )) = d R s (d L (I a ), I a ) d J L (x L (I a ), d L (I a ), I a ) > d ˆRs ( ˆd L (I a ), I a ) d Ĵ L (ˆx L (I a ), ˆd L (I a ), I a ) = X Ĥ s (ˆx L (I a ) ŷ L (I a )). The firs order condiion wih respec o x a implies ha X H s (x L (I a ) y L (I a )) = X Ĥ s (ˆx L (I a ) ŷ L (I a )) = β s, which leads o a conradicion. Hence, d L (I a ) ˆd L (I a ). We have hus proved ha, if I a 1 V s 1(I a 1) I a 1 ˆV s 1 (I a 1) for all I a 1 K a, x L (I a ) ˆx L (I a ), d L (I a ) ˆd L (I a ), x H (I a ) ˆx H (I a ), and d H (I a ) ˆd H (I a ). I L ÎL and I H ÎH follow immediaely from x L (I a ) ˆx L (I a ) and x H (I a ) ˆx H (I a ). ˆd s Nex, we show ha d s (I a ) (I a ), for all I a K a. Since d L (I a ) ˆd L (I a ), d s (I a ) = d L (I a ) ˆd L (I a ), for all I a I L. If I a [I L, ÎL ], d s (I a ) d s (I L ) = d L (I L ) ˆd L (I L ) = ˆd L (I a ), where he firs inequaliy follows from Theorem 5.5.1, he second from d L (I a ) ˆd L (I a ), and he las equaliy from Federgruen and Heching (1999) Theorem 1. If I a [ÎL, I H ] (i migh be an empy se), x s (I a ) = ˆx s (I a ) = I a. The supermodulariy of R s (d, I a ) implies ha d R s ( s ˆd (I a ), I a ) d R s s ( ˆd (I a ), ) = ˆRs s d ( ˆd (I a ), I a ). Since γ 0 γ(i a ), boh H s ( ) and Ĥs ( ) are concave, and X H s (X) X Ĥ s (X) for all X, so X H s (I a ˆd s (I a ) γ(i a )) X Ĥ s (I a i.e., d s (I a ) d J s (I a, ˆd s s ˆd (I a ) γ 0 ). Hence, s ˆd (I a ), I a ) = d R s s ( ˆd (I a ), I a ) X H s (I a s ˆd (I a ) γ(i a )) d ˆRs ( = d Ĵ s (I a, s ˆd (I a ), I a ) X Ĥ s (I a s ˆd (I a ) γ 0 ) ˆd s (I a ), I a ), (I a ). If I a [I H, ÎH ], x s (I a ) ˆx s (I a ) = I a. The firs order condiion wih respec o x a implies ha X H s (x s d s (I a ) < ˆd s (I a ), Lemma 28 implies ha d J s (x s (I a ) d s (I ) γ(i a )) = (β s + θ) X Ĥ s (I a (I a ), d s (I a ), I a ) d Ĵ s (I a, s ˆd (I a ) γ 0 ). If s ˆd (I a ), I a ). Hence, d R s (d s (I a ), I a ) = d J s (x s d Ĵ s (I a, (I a ), d s = ˆRs s d ( ˆd (I a ), I a ). (I a ), I a ) + X H s (x s (I a ) d s (I ) γ(i a )) s ˆd (I a ), I a ) + X Ĥ s (I a s ˆd (I a ) γ 0 ) (D.8) The sric concaviy of R s (, I a ) and he supermodulariy of R s (, ) imply ha d R s (d s (I a ), I a ) > d R s ( s ˆd (I a ), I a ) d R s ( which conradics inequaliy (D.8). Hence, d s (I a ) ˆd H (I a ) = ˆd s ˆd s (I a ). We have compleed he proof of d s (I a ) s ˆd (I a ), ) = ˆRs s d ( ˆd (I a ), I a ), (I a ). Finally, if I a ÎH, d s (I a ) = d H (I a ) s ˆd (I a ) for all I a K a. To complee he inducion, i suffices o show ha if I a 1 V 1(I s 1) a s I a 1 ˆV 1 (I 1) a for all I 1 a K a, I a V s (I a s ) I a ˆV (I a ), for all I a K a. Noe ha ˆx d s (I ) and ˆd (I a ) are consan if I a ÎL and 280

295 , by Theorem 1 in Federgruen and Heching (1999). Hence, I a V s (I a s ) I a ˆV (I a ) for all I a I ÎH and I a ÎH, since I a V s (I a s ) I a ˆV (I a ) = c, if I a ÎL ÎL, and I a V s (I a s ) I a ˆV (I a ) = c θ = s, if I a ÎH. If ÎL I ÎH, here are wo possible cases: Î L I H ÎH and I H ÎL ÎH. If I H ÎL, I a V s (I a s ) I a ˆV (I a ) for all I a K a follows immediaely. Now assume ha I H [ÎL, ÎH ]. If I [ÎL, I H ], x s (I a ) = ˆx s (I a ) = I a. Hence, I a V s (I a ) = I a c + β s + I a R s (d s (I a ), I a ) + X H s (I a y s (I a )) γ (I a ) X H s (I a y s (I a )), ˆV s (I a ) = c + β s + X Ĥ s (I a ŷ s (I a )), where y s (I a ) = d s (I a ) + γ(i a ) and ŷ s (I a ) = X Ĥ s (I a ŷ s (I a ). We use he following lemma o prove his inequaliy: s ˆd (I a ) + γ 0. I suffices o show ha X H s (I a y s (I a )) Lemma 32 Le y 1 = argmax y {R s (y, γ 0 ) + Ĥs (I a y )}, y 2 = argmax y {R s (y, γ 0 ) + H s (I a y )} and y 3 = argmax y {R s (y, γ(i a )) + H s (I a y )}, for I a [ÎL, I H ]. We have X Ĥ s (I a y 1 ) X H s (I a y 2 ) X H s (I a y 3 ). Proof of Lemma 32: Since X Ĥ s (X) X H s (X), y R s (y 1, γ 0 ) X H s (I a y 1 ) y R s (y 1, γ 0 ) X Ĥ s (I a y 1 ), i.e., y 1 y 2. If y 1 = y 2, X Ĥ s (I a y 1 ) X H s (I a y 2 ) follows from X Ĥ s (X) X H s (X) for any X. If y 1 < y 2, y R s (y 1, γ 0 ) > y R s (y 2, γ 0 ) by he sric concaviy of R s (, ), and y R s (y 1, γ 0 ) X Ĥ s (I a y 1 ) y R s (y 2, γ 0 ) X H s (I a y 2 ) by Lemma 28. Hence, X Ĥ s (I a y 1 ) > X H s (I a y 2 ). For he second inequaliy, he supermodulariy of R s (, ) yields ha y 2 y 3 and, hus, X H s (I a y 2 ) X H s (I a y 3 ). Q.E.D. Invoking Lemma 32, X H s (I a y s (I a )) = X H s (I a y 3 ) X Ĥ s (I a y 1 ) = X Ĥ s (I a ŷ s (I a )). Hence, I a V s (I a s ) I a ˆV (I a ) for all I a [ÎL, I H ]. If I a [I H, ÎH ], I a V s (I a s ) c θ = I a ˆV (ÎH s ) I a ˆV (I a ), where he firs inequaliy follows from he firs order condiion wih respec o x a. This complees he inducion and he proof of par (a). To prove par (b), i suffices o show ha if I a 1 V 1(I s 1) a s I a 1 ˆV 1 (I 1) a for all I 1 a K a, (a) x L (I a ) ˆx L (I a ), (b) x H (I a ) ˆx H (I a ), and (c) I a V s (I a s ) I a ˆV (I a ), for all I a K a. For = 0, I a 0 V s 0 (I a 0 ) = I a 0 ˆV s 0 (I a 0 ) = 0 for I a 0 K a. Firs, we show ha x L (I a ) ˆx L (I a ), and he proof of x H (I a ) ˆx H (I a ) follows from he same argumen. If x L (I a ) > ˆx L (I a ), Lemma 28 yields ha x a J L (x L (I a ), d 0, I a ) x a Ĵ L (ˆx L (I a ), d 0, I a ). Hence, X H s (x L (I a ) y L (I a )) = x a J L (x L (I a ), d 0, I a ) β s x a Ĵ L (ˆx L (I a ), d 0, I a ) β s = X Ĥ s (ˆx L (I a ) ŷ L (I a )). 281

296 Since X H s (X) X Ĥ s (X) for any X and boh of hem are sricly decreasing, y L (I a ) > ŷ L (I a ). However, y L (I a ) = d 0 + γ(i a ) d 0 + ˆγ(I a ) = ŷ L (I a ). This conradicion shows ha x L (I a ) ˆx L (I a ). x H (I a ) ˆx H (I a ) follows analogously. To complee he proof, we need o show I a V s (I a s ) I a ˆV (I a ) for all I a K a. For he case I a [ÎL, ÎH ], he proof is idenical o ha of par (a), and, hence, omied. If I a I L, I a V s (I a ) = c + (p 0 b αc)γ (I a ) γ (I a ) X H s (x s (I a ) y s (I a )), I a Since x s (I a ) ˆx s If x s (I a ) = ˆx s ˆV s (I a ) = c + (p 0 b αc)ˆγ (I a ) ˆγ (I a ) X Ĥ s (ˆx s (I a ) ŷ s (I a )). (I a ), here are wo cases: (a) x s (I a ), x s (I a ) y s (I a ) ˆx s (I a ) = ˆx s (I a ) and (b) x s (I a ) < ˆx s (I a ). (I a ) ŷ s (I a ) and, hence, X H s (x s (I a ) y s (I a )) X Ĥ s (ˆx s (I a ) ŷ s (I a )), since X H s (X) X Ĥ s (X) for any X. If x s (I a ) < ˆx s (I a ), Lemma 28 yields ha x a J s (x s (I a ), d 0, I a ) x a Ĵ s (ˆx s (I a ), d 0, I a ). Hence, X H s (x s (I a ) y s (I a )) = x a J s (x s We have hus showed ha X H s (x s (I a ), d 0, I a ) β s x a Ĵ s (ˆx s (I a ), d 0, I a ) β s = X Ĥ s (ˆx s (I a ) ŷ s (I a )). (I a ) y s (I a )) X Ĥ s (ˆx s (I a ) ŷ s (I a )) in boh cases. Therefore, I a V s (I a ) =c + γ (I a )(p 0 b αc X H s (x s (I a ) y s (I a ))) c + ˆγ (I a )(p 0 b αc X Ĥ s (ˆx s (I a ) ŷ s (I a )) = I a ˆV s (I a ), where he inequaliy follows from γ (I a ) ˆγ (I a ) 0 and p 0 b αc X H s (x s (I a ) y s (I a )) p 0 b αc X Ĥ s (ˆx s (I a ) ŷ s (I a )) > 0. The proof of he case I a ÎH follows from he idenical argumen of he case I a I L, and is, hence, omied. If I a [I L, ÎL ], I a V s (I a ) = s ˆV (I a ) = I a Noe ha X H s (x s c + β s + (p 0 b αc)γ (I a ) + X H s (x s (I a ) y s (I a )) γ (I a ) X H s (x s (I a ) y s (I a )), c + (p 0 b αc)ˆγ (I a ) ˆγ (I a ) X Ĥ s (ˆx s (I a ) ŷ s (I a )). (I a ) y s (I a )) β s = X Ĥ s (ˆx s (I a ) ŷ s (I a )). Therefore, I a V s (I a ) =c + β s + X H s (x s (I a ) y s (I a )) + γ (I a )(p 0 b αc X H s (x s (I a ) y s (I a ))) c + ˆγ (I a )(p 0 b αc X Ĥ s (ˆx s (I a ) ŷ s (I a ))) = I a ˆV s (I a ), (D.9) where he inequaliy follows from γ (I a ) ˆγ (I a ) 0, X H s (x s (I a ) y s (I a )) X Ĥ s (ˆx s (I a ) ŷ s (I a )), and p 0 b αc X H s (x s (I a ) y s (I a )) p 0 b αc X Ĥ s (ˆx s (I a ) ŷ s (I a )) > 0. We have hus showed I a V s (I a s ) I a ˆV (I a ) for all I a Q.E.D. 282 K a, which complees he proof of par (b).

297 Proof of Theorem 5.5.3: We employ backward inducion o prove pars (a) - (d) ogeher. We define H s (X) := E ϵ a { (b + h a )(X ϵ a ) + + α(v s 1(X) cx)} and Ĥs (X) := E ϵ a { (b + h a )(X ϵ a ) + + α( ˆV s 1(X) cx)}, so ha g s (x a, d, I a ) := H s (x a d γ(i a )) and ĝ s (x a, d, I a ) := Ĥs (x a d γ(i a )). We define he objecive funcions J L (x a, d, I a ) := R s (d, I a ) + β s x a + g s (x a, d, I a ), Ĵ L (x a, d, I a ) := ˆR s (d, I a )+β s x a +ĝ s (x a, d, I a ), J H (x a, d, I a ) := R s (d, I a )+(β s +θ)x a +g s (x a, d, I a ), and Ĵ H (x a, d, I a ) := ˆR s (d, I a )+(β s + ˆθ)x a +ĝ s (x a, d, I a ), where ˆθ = c ŝ c s = θ. Le γ := γ(i a ), y s (I a ) := d s (I a ) + γ and ŷ s (I a ) := ˆd s (I a ) + γ. s I suffices o show ha if I a 1 ˆV 1 (I 1) a I a 1 V 1(I s 1) a for all I 1 a K a, (1) ÎL I L, (2) ˆx s (I a ) x s (I a ) for all I a ÎH, (3) s ˆd (I a ) d s (I a s ), and (4) I a ˆV (I a ) I a V s (I a ). Since I a 1 ˆV s 1 (I a 1) I a 1 V s 1(I a 1), X Ĥ s (X) X H s (X). For = 0, I a 0 ˆV s 0 (I a 0 ) = I a 0 V s 0 (I a 0 ) = 0, so he iniial condiion is saisfied. We firs show ha if I a 1 ˆV s 1 (I a 1) I a 1 V s 1(I a 1), ˆx L (I a ) x L (I a ), ˆdL (I a ) d L (I a ), and ˆd H (I a ) d H (I a ). ˆx L (I a ) x L (I a ) and ˆd L (I a ) d L (I a ) follows from he same argumen as he proof of Theorem We show by conradicion ha ˆd H (I a ) d H (I a ). Assume ha d H (I a ) < ˆd H (I a ), so y H (I a ) = d H (I a ) + γ < ˆd H (I a ) + γ = ŷ H (I a ). Lemma 28 yields ha d J H (x H (I a ), d H (I a ), I a ) d Ĵ H (ˆx H (I a ), ˆd H (I a ), I a ). The sric concaviy of R s (, I a ) imply ha d R s (d H (I a ), I a ) > d R s ( ˆd H (I a ), I a ). Hence, we have: X H s (x H (I a ) y H (I a )) = d R s (d H (I a ), I a ) d J H (x H (I a ), d H (I a ), I a ) > d ˆRs ( ˆd H (I a ), I a ) d Ĵ H (ˆx H (I a ), ˆd H (I a ), I a ) = X Ĥ s (ˆx H (I a ) ŷ H (I a )). The firs order condiion wih respec o x a implies ha X H s (x H (I a ) y H (I a )) = (β s + θ) < (β s + ˆθ) = X Ĥ s (ˆx H (I a ) ŷ H (I a )), which leads o a conradicion. Hence, d H (I a ) ˆd H (I a ). We have hus proved ha, if I a 1 ˆV s 1 (I a 1) I a 1 V 1(I s 1), a ˆx L (I a ) x L (I a ), ˆd L (I a ) d L (I a ), and ˆd H (I a ) d H (I a ). s Nex, we show ha ˆd (I a ) d s (I a ) for all I a K a. If I a I L or I a max{i H, ÎH }, ˆd s (I a ) d s (I a ) follows from ˆd L (I a ) d L (I a ) and ˆd H (I a ) d H (I a ). Now we assume ha I a [I L, max{i H, ÎH }]. If I a [I L, ÎL ], x s (I a ) = I a ˆx s d J s (x s (I a ), d s implies ha X H s (x s However, ˆds shows ha if I a (I a ), I a ) d Ĵ s (ˆx s (I a ), d R s (d s (I a ) > d s [I L, ÎL ], (I a ) > d s (I a ), by Lemma 28, (I a ). If ˆds s ˆd (I a ), I a ). The firs order condiion wih respec o x a (I a ) y s (I a )) β s = X Ĥ s (ˆx s (I a ) ŷ s (I a )). Therefore, (I a ), I a ) = d J s (x s d Ĵ s (ˆx s (I a ), = d R s s ( ˆd (I a ), I a ). (I a ), d s (I a ), I a ) + X H s (x s (I a ) y s (I a )) s ˆd (I a ), I a ) + X Ĥ s (ˆx s (I a ) ŷ s (I a )) (I a ) implies ha d R s (d s (I a ), I a ) > d R s s ( ˆd s ˆd (I a ) d s (I a ). If I a [ÎL, I H ], x s (I a ) = I a ˆx s (I a ). If d J s (x s (I a ), d s (I a ), I a ) d Ĵ s (ˆx s (I a ), ˆd s (I a ), I a ). The conradicion (I a ) > d s (I a ), Lemma 28 implies ha s ˆd (I a ), I a ). Since X H s (X) X Ĥ s (X) for any X and 283

298 ˆd s (I a ) > d s in he case I a ha If (I a ), X H s (x s (I a ) y s (I a )) X Ĥ s (ˆx s (I a ) ŷ s (I a )). We apply he same argumen as [I L, ÎL ] and he conradicion shows ha s ˆd (I a ) d s (I a ) for all I a [ÎL, I H ]. If I a [I H, ÎH ] (which migh be an empy se), he firs order condiion wih respec o x a implies ˆd s X H s (x s (I a ) > d s (I a ) y s (I a )) = (β s + θ) < (β s + ˆθ) X Ĥ s (ˆx s (I a ) ŷ s (I a )). (D.10) (I a ), Lemma 28 implies ha d J s (x s (I a ), d s The same argumen as in he case I a [ÎL, I H ] proves ha Hence, I a s ˆd (I a ) d s (I a ) for all I a K a. (I a ), I a ) d Ĵ s (ˆx s (I a s ), ˆd (I a ), I a ). s ˆd (I a ) d s (I a ) for all I a [I H, ÎH ]. s To complee he inducion, we nex show ha if I a 1 ˆV 1 (I 1) a I a 1 V 1(I s 1) a for all I 1 a K a, s ˆV (I a ) I a V s (I a ) for all I a K a. If I a d Ĵ d (ˆx s (I a ), I L, noe ha d J s (x s s ˆd (I a ), I a ) = d R s ( wih respec o x a, X H s (x s argumen leads o ha d s (I a ) = I a V s (I a ) s ˆV (I a ) = c + (p( Hence, I a I a (I a ), d s ˆd s (I a ), I a ) = d R s (d s (I a ), I a ) X H s (x s (I a ) y s (I a )) and (I a ), I a ) X Ĥ s (ˆx s (I a ) ŷ s (I a )). By he firs order condiion (I a ) y s (I a )) = X Ĥ s (ˆx s (I a ) ŷ s (I a )) = β s. A simple conradicion ˆd s (I a ), for I a I L. Therefore: = c + (p(d s (I a )) b αc)γ (I a ) γ (I a ) X H s (x s (I a ) y s (I a )) ˆV s (I a ) = I a V s (I a ), for I a I L. If I a [I L, ÎL ], I a V s (I a ) I a ˆV s (I a ) s ˆd (I a )) b αc)γ (I a ) γ (I a ) X Ĥ s (ˆx s (I a ) ŷ s (I a )). = c + (p(d s (I a )) b αc)γ (I a ) + β s + (1 γ (I a )) X H s (x s (I a ) y s (I a )) = c + (p( Noe ha he firs order condiion wih respec o x a X Ĥ s (ˆx s (I a ) ŷ s (I a )). If d s (I a s ) = ˆd (I a ), I a ˆV s (I a ) I a V s (I a ) = γ (I a )( X Ĥ s (ˆx s If d s (I a ) > d R s (d s We have: ˆd s s ˆd (I a )) b αc)γ (I a ) γ (I a ) X Ĥ s (ˆx s (I a ) ŷ s (I a )). implies ha X H s (x s (I a ) y s (I a )) β s = (I a ) ŷ s (I a )) X H s (x s (I a ) y s (I a ))) + β s + X Ĥ s (ˆx s (I a ) ŷ s (I a )) 0. (I a ), Lemma 28 yields ha d J s (x s (I a ), I a ) X H s (x s (I a ) y s (I a )) d R s ( (I a ), d s (I a ), I a ) d Ĵ s (ˆx s (I a ), s ˆd (I a ), I a ), i.e., s ˆd (I a ), I a ) X Ĥ s (ˆx s (I a ) ŷ s (I a )). (D.11) I a ˆV s (I a ) I a V s (I a ) =[(p( ˆd s (I a )) p(d s (I a ))) ( X Ĥ s (ˆx s (I a ) ŷ s (I a )) X H s (x s (I a ) y s (I a )))]γ (I a ) (β s + X H s (x s (I a ) y s (I a ))) [(p( ˆd s (I a )) p(d s (I a ))) ( X Ĥ s (ˆx s (I a ) ŷ s (I a )) X H s (x s (I a ) y s (I a )))]γ (I a ) [(p( ˆd s (I a )) p(d s (I a ))) ( d R s ( =[p (d s (I a ))y s (I a ) p s ( ˆd (I a ))ŷ s (I a )]γ (I a ) 0, s ˆd (I a ), I a ) d R s (d s (I a ), I a ))]γ (I a ) (D.12) 284

299 where he firs inequaliy follows from X H s (x s (I a ) y s (I a )) + β s 0, he second inequaliy from (D.11), and he las from he concaviy of p( ) and d s (I a ) > ˆd s (I a ). If I a [ÎL, I H ], x s (I a ) = I a ˆx s (I a ), I a V s (I a ) = c + (p(d s (I a )) b αc)γ (I a ) + β s + (1 γ (I a )) X H s (x s (I a ) y s (I a )) I a If d s (I a ) = If d s (I a ) > ˆV s (I a ) ˆd s = c + (p( s ˆd (I a )) b αc)γ (I a ) + β s + (1 γ (I a )) X Ĥ s (ˆx s (I a ) ŷ s (I a )). (I a ), X Ĥ s (ˆx s (I a ) ŷ s (I a )) X H s (x s (I a ) y s (I a s )), and I a ˆV (I a ) I a V s (I a ). s ˆd (I a ), as in (D.12), I a ˆV s (I a ) I a V s (I a ) [p (d s >0, + ( d R s ( (I a ))y s (I a ) p s ( ˆd (I a ))ŷ s (I a )]γ (I a ) s ˆd (I a ), I a ) d R s (d s (I a ), I a )) (D.13) where he second inequaliy follows from d s (I a ) > If I a [I H, ÎH ] (which migh be an empy se), I a V s (I a ) = I a ˆV s (I a ) = c + (p(d s c + (p( (D.10) implies ha X Ĥ s (ˆx s ˆd s (I a ). (I a )) b αc)γ (I a ) γ (I a ) X H s (x s (I a ) y s (I a )) θ s ˆd (I a )) b αc)γ (I a ) + β s + (1 γ (I a )) X Ĥ s (ˆx s (I a ) ŷ s (I a )). (I a ) ŷ s (I a )) > X H s (x s (I a ) y s (I a )) and X Ĥ s (ˆx s (I a ) ŷ s (I a ))+ β s + θ 0. The same argumen as in he case I a [I L, ÎL ] implies ha I a ˆV s (I a ) I a V s (I a ) for I a [I H, ÎH ]. If I a max{i H, ÎH }, I a V s (I a ) = s ˆV (I a ) = c + (p( Noe ha If d s (I a ) = I a c + (p(d s ˆd s (I a )) b αc)γ (I a ) γ (I a ) X H s (x s (I a ) y s (I a )) θ (I a )) b αc)γ (I a ) γ (I a ) X Ĥ s (ˆx s (I a ) ŷ s (I a )) ˆθ. X Ĥ s (ˆx s (I a ) ŷ s (I a )) = (β s + ˆθ) (β s + θ) = X H s (x s (I a ) y s (I a )). ˆd s (I a ), ˆV s (I a ) I a V s If d s (I a ) > (I a ) = γ (I a )( X Ĥ s (ˆx s s ˆd (I a ), he same argumen as (D.12) implies ha ˆV s (I a ) I a V s (I a ) [p (d s (I a ) ŷ s (I a )) X H s (x s (I a ) y s (I a ))) + θ ˆθ > 0. (I a ))y s (I a ) p s ( ˆd (I a ))ŷ s (I a )]γ (I a ) + θ ˆθ > 0. s We have hus showed ha, if I a 1 ˆV 1 (I 1) a I a 1 V 1(I s 1) a for all I 1 a s K a, I a ˆV (I a ) I a V s (I a ) for all I a K a, which complees he proof of Theorem Q.E.D. Proof of Theorem 5.5.4: We firs show par (a). Observe ha if h w h a and ˆγ(I a ) γ 0 for all I a K a, wihholding posiive invenory is dominaed by displaying his par of invenory o cusomers, because he holding cos a he cusomer-accessible sorage is smaller han ha a he warehouse, 285

300 and here is no demand-suppressing effec of cusomer-accessible invenory. Therefore, he firm should no wihhold any invenory if h w h a and ˆγ(I a ) γ 0 for all I a K a. Nex, we show par (b) by backward inducion. Since i is opimal for he firm no o wihhold any invenory in he model wih demand ˆD, his model is reduced o he one discussed in Secion 5.5.1, i.e., he model wihou invenory wihholding. Le K (I a ) := V (I a, I a ). I suffices o show ha if s I a 1 ˆV 1 (I 1) a I a 1 K 1 (I 1), a for all I 1 a K a, (a) x a (I a ) ˆx s (I a ), (b) d (I a s ) ˆd (I a ), and s (c) I a ˆV (I a ) I a K (I a ), for all I a K a. For = 0, ˆV 0 s (I0 a ) = K 0 (I0 a ) = 0, so he iniial condiion is saisfied. If I a 1 ˆV s 1 (I a 1) I a 1 K (I a 1) for I a 1 K a, X Ĥ s (X) X L (X, Y ) + Y L (X, Y ) for X = Y, where Ĥs (X) is defined in he proof of Theorem 5.5.2, and L (X, Y ) := E ϵ a { (h a + b)(x ϵ a ) + + α[v 1 (X ϵ a, Y ϵ a ) cy ]}. Therefore, he same argumen as in he proof of Theorem 5.5.2(a) shows ha x a (I a ) ˆx s (I a ) and d (I a s ) ˆd (I a ). I a s To complee he inducion, we show ha if I a 1 ˆV 1 (I 1) a I a 1 K 1 (I 1) a for all I 1 a K a, s ˆV (I a ) I a K (I a ), for I a K a. Since x a (I a ) ˆx s (I a ), x a (I L ) x s (I L ) = I L. If I a I L, I a K (I a ) c + (p b αc)γ (I a ) c = I a ˆV s (I a ). For he case I a skech i. The key sep is o show ha I L, he argumen is very similar o ha in he proof of Theorem 5.5.2, so we only X Ĥ s (I a ŷ s (I a )) X L (x a (I a, I a ) y (I a ), I a y (I a )) + Y L (x a (I a, I a ) y (I a ), I a y (I a )), where ŷ s (I a ) is defined in he proof of Theorem and y (I a ) := d a (I a, I a ) + γ(i a ). To show he above inequaliy, le y (I a ) be he opimal expeced demand in he sysem wih demand ˆD such ha he firm is forced o display x a when he curren cusomer-accessible invenory level is I a (I a, I a ) o cusomers and wihhold I a x a (I a, I a ) > 0 in he warehouse, > I L. Le ˆL s (X, Y ) = E ϵ a { (h a + b)(x ϵ a ) + + α[ ˆV s 1(Y ϵ a ) cy ]}, Following he same argumen as he proof of Lemma 32, we have: X Ĥ s (I a ŷ s (I a )) X ˆLs (x a (I a, I a ) y (I a ), I a y (I a )) + Y ˆL (x a (I a, I a ) y (I a ), I a y (I a )) X L (x a (I a, I a ) y (I a ), I a y (I a )) + Y L (x a (I a, I a ) y (I a ), I a y (I a )). Based on (D.14), he same argumen as he proof of Theorem 5.5.2(a) yields ha I a for all I a K a. This complees he inducion and he proof of Theorem 5.5.4(b). Q.E.D. (D.14) ˆV s (I a ) I a K (I a ), Proof of Theorem 5.5.5: We prove Theorem by backward inducion. Le L (X, Y ) := E ϵ a {G (X ϵ a, Y ϵ a )} and H (X) := L (X, X), hen g a (x a, d, I a ) = H (x a d γ(i a )). Le K (I a ) = V (I a, I a ). 286

301 I suffices o show ha if I a 1 K 1 (I a 1) I a 1 V s 1(I a 1), for any I a 1 K a, (a) x a (I a ) x s (I a ), (b) d (I a, I ) d s (I a ), and (c) I a K (I a ) I a V s (I a ), for any I a K a. For = 0, V s 0 (I a 0 ) = K 0 (I a 0 ) = 0, so he iniial condiion is saisfied. Because I a 1 K 1 (I a 1) I a 1 V s 1(I a 1), for any I a 1 K a, X H (X) X H s (X) for any X. Following he same argumen as he proof of Theorem 5.5.3, we have ha if X H (X) X H s (X) for any X, x a (I a ) x L (I a ) and d (I a ) d L (I a ). Hence, I L I := sup{i a : x a (I a ) > I a }. Therefore, we have ha d (I a, I ) = d (I a ) d L (I a ) d s (I a ), if I a I, where he las inequaliy follows from he supermodulariy of J s (x a, d, I a ) in (x a, d ) for any fixed I a. since If I = I a > I, x a (I a, I ) < x (I a, I ) = x s (I a ) = I a = I. Therefore, d s (I a ) =argmax d [d, d]{r(d, I a ) + H s (I a d γ(i a ))} ˆd (I a ) :=argmax d [d, d]{r(d, I a ) + H (I a d γ(i a ))}, d R( ˆd (I a ), I a ) X H s (I a ˆd (I a ) γ(i a )) d R( ˆd (I a ), I a ) X H (I a ˆd (I a ) γ(i a )), where he inequaliy follows from X H (X) X H s (X) for all X. Similar argumen yields ha: d (I a, I ) = argmax d [d, d]{r(d, I a ) + L (x a (I a, I ) d γ(i a ), I a d γ(i a ))} ˆd (I a ) = argmax d [d, d]{r(d, I a ) + L (I a d γ(i a ), I a d γ(i a ))}, because L (, Y ) is concave for any fixed Y. Hence, d (I a, I ) ˆd (I a ) d s (I a ) for any I = I a I. To complee he inducion, we need o show ha if I a 1 K 1 (I a 1) I a 1 V s 1(I a 1), for any I 1 a K a, I a K (I a ) I a V s (I a ), for any I a K a. For I a I, x a (I a, I ) = x (I a, I ). Same argumen as in he proof of Theorem implies ha I a K (I a ) I a V s (I a ), if I a I. If I a > I, he proof is based on he following lemma: Lemma 33 Assume ha I a > I. Le We have: ˆV s (I a ) = ci a + max {R(d, I a d [d, d] ) + βi a + L (I a d γ(i a ), I a d γ(i a ))}. I a V s (I a s ) I a ˆV (I a ) I a K (I a ). (D.15) Proof of Lemma 33: The firs inequaliy follows from he same argumen as he proof of Theorem For he second inequaliy, observe ha I a ˆV (I a ) =c + β + (p( ˆd (I a )) b αc)γ (I a ) + (1 γ (I a )) X L (I a ˆd (I a ) γ(i a ), I a ˆd (I a ) γ(i a )) + (1 γ (I a )) Y L (I a ˆd (I a ) γ(i a ), I a ˆd (I a ) γ(i a )), and I a K (I a ) =c + β + (p(d (I a, I )) b αc)γ (I a ) + (1 γ (I a )) X L (x a (I a, I ) d (I a, I ) γ(i a ), I a d (I a, I ) γ(i a )) + (1 γ (I a )) Y L (x a (I a, I ) d (I a, I ) γ(i a ), I a d (I a, I ) γ(i a ))). 287

302 Thus, I a K (I a ) I a ˆV (I a ) = (p(d (I a, I )) p( ˆd (I a )))γ (I a ) γ (I a )[ X L (x a (I a, I ) d (I a, I ) γ(i a ), I a d (I a, I ) γ(i a )) X L (I a ˆd (I a ) γ(i a ), I a ˆd (I a ) γ(i a )) + Y L (x a (I a, I ) d (I a, I ) γ(i a ), I a d (I a, I ) γ(i a )) Y L (I a ˆd (I a ) γ(i a ), I a ˆd (I a ) γ(i a ))] + X L (x a (I a, I ) d (I a, I ) γ(i a ), I a d (I a, I ) γ(i a )) X L (I a ˆd (I a ) γ(i a ), I a ˆd (I a ) γ(i a )) + Y L (x a (I a, I ) d (I a, I ) γ(i a ), I a d (I a, I ) γ(i a )) Y L (I a ˆd (I a ) γ(i a ), I a ˆd (I a ) γ(i a )). Based on he firs order condiion wih respec o d (D.16) and Lemma 28, he same argumen as inequaliy (D.12) yields ha I a K (I a ) I a ˆV (I a ) 0, and hence (D.15) holds. Q.E.D. By Lemma 33, I a K (I a ) I a in he proof of Theorem Q.E.D. ˆV s (I a ) I a V s (I a ) for all I a K a. This complees he inducion Proof of Theorem 5.6.1: The proof, based on backward inducion, is very similar o ha of Lemma 14 and Theorem 5.4.1, so we only skech i. In paricular, he coninuous differeniabiliy of V r (I a, I ) follows from he same argumen as in he proof of Lemma 14 and is, hence, omied. Noe ha V r 0 (I r 0, I 0 ) ci 0 r d I a 0 = ci 0 r d I a 0 is joinly concave, coninuously differeniable, and decreasing in boh of is argumens. If V r 1(I a 1, I 1 ) r d I a 1 ci 1 is joinly concave and decreasing in I a 1 and I 1, G r (x, y) is decreasing in boh x and y. Hence, he same argumen as in he proof of Lemma 14 shows ha, for any realizaion of (ϵ a, ϵ m ), (r d + r w )(y a I a ) + ϕy a + G r (y a D, x D ) = (r d + r w )(y a I a ) + ϕy a + G r (y a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ) is joinly concave in (y a, x, d, I a ). Concaviy is preserved under maximizaion and expecaion, so E D { max { (r min{d,i a} ya min{k d + r w )(y a I a ) + ϕy a + G r (y a D, x D )}} a+d,x } is joinly concave in (x, d, I a ). θ(x I ) is joinly concave in (x, I ), Since R(d, I a ) + r d (d + γ(i a )) is joinly concave in (d, I a ), and R(d, I a ) + r d (d + γ(i a )) θ(x I ) ψx + E D { max min{d,i a } ya {Ka+D,x}{ (r d + r w )(y a I a ) + ϕy a + G r (y a D, x D )}} is joinly concave in (x, d, I a ). Since concaviy is preserved under maximizaion, V r (I a, I ) is joinly concave in (I a, I ). 288

303 Nex, we show ha V r (I a, I ) r d I a ci is decreasing in I a and I. Since all of erms in (r d + r w )(y a I a ) + ϕy a + G r (y a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ) is decreasing in I a, i is decreasing in I a If y a = I a, iself, if he consrains min{i a, D } y a min{k a + D, x } is no binding. (r d + r w )(y a I a ) + ϕy a + G r (y a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ) =ϕi a + G r (I a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ). If ϕi a + G r (I a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ) is sricly increasing in I a, (r d +r w )(y a I a ) +ϕy a +G r (y a (d +γ(i a ))ϵ m ϵ a, x (d +γ(i a ))ϵ m ϵ a ) is sricly increasing in y a in a small righ-neighborhood of I a : [I a, I a + ξ), for a small enough ξ > 0. Under his condiion, y a = I a is no an opimizer. Hence, (r d + r w )(y a I a ) + ϕy a + G r (y a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ) is decreasing in I a, if i is opimal o choose y a = I a. If y a = D, (r d + r w )(y a I a ) + ϕy a + G r (y a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ) = (r d + r w )((d + γ(i a ))ϵ m + ϵ a I a ) + ϕ((d + γ(i a ))ϵ m + ϵ a ) + G r (0, x (d + γ(i a ))ϵ m ϵ a ) is decreasing I a. Analogously, if y a = K a + D, (r d + r w )(y a I a ) + ϕy a + G r (y a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ) = (r d + r w )(K a + (d + γ(i a ))ϵ m + ϵ a I a ) + ϕ(k a + (d + γ(i a ))ϵ m + ϵ a ) is decreasing in I a. If y a = x, + G r (K a, x (d + γ(i a ))ϵ m ϵ a ) (r d + r w )(y a I a ) + ϕy a + G r (y a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ) = (r d + r w )(x I a ) + ϕx + G r (x (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a ) is decreasing in I a. Hence, max min{d,i a } ya min{x,d +K a } { (r d + r w )(y a I a ) + ϕy a + G r (y a (d + γ(i a ))ϵ m ϵ a, x (d + γ(i a ))ϵ m ϵ a )} is decreasing in I a. Because, θ(x I ) is decreasing in I and F r (I a 1 ) F r (I a 2 ) for any I a 1 I a 2, V r (I a, I ) r d I a ci = max (x,d ) F r (I a){r(d, I a ) + r d (d + γ(i a )) θ(x I ) ψx + E D { max min{d,i a } ya min{x,ka+d}{ (r d + r w )(y a I a ) + ϕy a + G r (y a D, x D )}}} 289

304 is decreasing in I a and I. This concludes he proof of par (a). Par (b) follows direcly from he concaviy of V r 1(, y) for any y and (5.12), while par (c) follows from he same argumen as he proof of Theorem Q.E.D. 290

305 E.1 Proofs of Saemens E. Appendix for Chaper 6 Proof of Lemma 17: We prove pars (a) - (c) ogeher, using backward inducion. Since V 0 ( θ 0 ) 0 is concave and coninuously differeniable in I 0 for any θ 0, i suffices o show ha if V 1 ( θ 1 ) is concave and coninuously differeniable in I 1 for any θ 1, hen, for any θ, (i) Ψ ( θ ) is concave and coninuously differeniable in z, (ii) J (,, I θ ) is sricly joinly concave and coninuously differeniable in (d, q ), and (iii) V ( θ ) is concave and coninuously differeniable in I. Since H( ) and V 1 ( θ 1 ) are concave and concaviy is preserved under expecaion, by Equaion (6.14), Ψ (z θ ) is concave in z for any θ. Since ϵ follows a coninuous disribuion, Ψ (z θ ) is coninuously differeniable in z. By Assumpion 6.4.1, ( i N Λi R i (d i )) is sricly joinly concave in d. The sric convexiy of C j ( c j ) for each j, implies ha j M Cj (q j c j ) is sricly joinly concave in q. Moreover, by he concaviy of Ψ ( θ ), for any realizaion of ς, Ψ (I + j M qj ( i N Λi d i )ς θ ) is joinly concave in (d, q, I ). Therefore, by Equaion (6.13), J (d, q, I θ ) = ( i N Λ i R i (d i )) C j (q j c j ) + E ς {Ψ (I + j M j M q j ( i N Λ i d i )ς θ )} is joinly concave in (d, q, I ) and sricly joinly concave in (d, q ). Since R i ( ) is coninuously differeniable in d i for any i, C j ( c j ) is coninuously differeniable in q j for any j and c j, and Ψ ( θ ) is coninuously differeniable in z for any θ, J (,, I θ ) is coninuously differeniable in (d, q ) for any θ. Since concaviy is preserved under maximizaion, by Equaion (6.11), V ( θ ) is concave in I for any θ. The coninuous differeniabiliy of V ( θ ) follows from he envelope heorem and is derivaive is given by I V (I θ ) = I E ς {Ψ (I + j M = E ς { z Ψ (I + j M q j (I, θ ) ( Λ i d i (I, θ ))ς θ )} i N q j (I, θ ) ( Λ i d i (I, θ ))ς θ )}, i N where he firs equaliy follows from he envelope heorem and he second from Theorem A.5.1 of [63] and he coninuous differeniabiliy of Ψ ( θ ). Q.E.D. (E.1) Proof of Theorem 6.4.1: We prove par (b) firs, par (c) second, and par (a) las. Par (b). Le Φ (y θ ) := max d [0,d max ] n{( Λ i R i (d i )) + E ς {Ψ (y ( Λ i d i )ς θ )}. (E.2) i N i N 291

306 I s clear ha Φ ( θ ) is concave and coninuously differeniable in y, and (q 1 (I, θ ), q 2 (I, θ ),, q m (I, θ )) = argmax q 0{ C j (q j c j ) + Φ (I + q j θ )}. j M j M (E.3) Invoke Lemma 15 wih p = m, q = 0, γ = I, y j = q j (1 j m), λ j = 1 (1 j m), f j (y j γ) = C j (q j c j ), h(y 0 γ) = Φ (I + y 0 θ ), and Y j = [0, + ) for all 1 j m. Since Φ (I + y 0 θ ) is supermodular in ( I, y 0 ), h(y 0 γ) is supermodular in (y 0, γ). Hence, Lemma 15 implies ha q j (I, θ ) is decreasing in I for any j and θ. The sric concaviy of J (,, I θ ) yields ha q j (I, θ ) is coninuous in I for any j and θ. Hence, I q,j is decreasing in I and Î > I, q j M (I, θ ) follows immediaely. (θ ) = min{i : q j (I, θ ) = 0}. If j M (Î, θ ), since q j (I, θ ) (I, θ ) q j (Î, θ ) > 0. Thus, j M (I, θ ), and M (Î, θ ) I remains o be shown ha I q,j (θ ) < +. Firs observe ha V (I θ ) is uniformly bounded from above by E[ s=1 α s ( j N Λj s) θ ] R < +, where R := max i N,d i [0,d max ] R i (d i ). Hence, By he envelope heorem, we have lim zψ (z θ ) lim z + z + H (z+) < 0. lim yφ (y θ ) lim y + y + H (y+) < 0. Thus, here exiss a hreshold ȳ < + such ha y Φ (y θ ) < 0 for all y ȳ. Therefore, for any j M, q j C j (q j c j ) + y Φ (I + q j θ ) < 0 for all I ȳ and q 0. j M Hence, q j (I, θ ) = 0 for all I ȳ and any j M. Thus, I q,j (θ ) < + for all θ and any j M. Par (c). The coninuiy of x (I, θ ) follows from ha of q j (I, θ ) for each j M. Assume, o he conrary, ha Î > I and x (Î, θ ) < x (I, θ ). Hence, here exiss a j 0 M, such ha q j0 (Î, θ ) < q j0 (I, θ ). Wihou loss of generaliy, le j 0 = 1. The sric convexiy of C 1 ( c 1 ) implies ha q 1 C 1 (q 1 (I, θ ) c 1 ) > q 1 C 1 (q 1 (Î, θ ) c 1 ). On he oher hand, Lemma 16 yields ha q 1 C 1 (q 1 (I, θ ) c 1 ) + y Φ (x (I, θ ) θ ) q 1 C 1 (q 1 (Î, θ ) c 1 ) + y Φ (x (Î, θ ) θ ). Therefore, y Φ (x (Î, θ ) θ ) < y Φ (x (I, θ ) θ ), which conradics he concaviy of Φ ( θ ). Hence, x (I, θ ) is coninuously increasing in I. Par (a). The coninuiy of d i (I, θ ) follows from he concaviy of J (,, I θ ). For any given Î and I (Î > I ), assume, o he conrary, ha d 1 (I, θ ) > d 1 (Î, θ ). Thus, d 1 R 1 (d 1 (I, θ )) < d 1 R 1 (d 1 (Î, θ )) by he sric concaviy of R 1 ( ). On he oher hand, Lemma 16 yields ha d 1 J (d (I, θ ), q (I, θ ), I θ ) d 1 J (d (Î, θ ), q (Î, θ ), I θ ). Thus, E ς {ς y Ψ (x (I, θ ) ( Λ i d i (I, θ ))ς θ )} = d 1 R 1 (d 1 (I, θ )) 1 Λ 1 d 1 J (d (I, θ ), q (I, θ ), I θ ) i N < d 1 R 1 (d 1 (Î, θ )) 1 Λ 1 d 1 J (d (Î, θ ), q (Î, θ ), I θ ) =E ς {ς y Ψ (x (Î, θ ) ( i N Λ i d i (Î, θ ))ς θ )}. (E.4) 292

307 For any l = 2, 3,, n, furher assume ha d l (I, θ ) < d l (Î, θ ). d l R l (d l (Î, θ )). On he oher hand, Lemma 16 yields ha d l J (d (I, θ ), q (I, θ ), I θ ) d l J (d (Î, θ ), q (Î, θ ), I θ ). Thus, Hence, d l R l (d l (I, θ )) > E ς {ς y Ψ (x (I, θ ) ( i N Since (E.4) conradics (E.5), d l (I, θ ) d l Hence, if d 1 x (I, θ ). Thus, Λ i d i (I, θ ))ς θ )} = d l R l (d l (I, θ )) 1 Λ l d l J (d (I, θ ), q (I, θ ), I θ ) > d l R l (d l (Î, θ )) 1 Λ l d l J (d (Î, θ ), q (Î, θ ), I θ ) =E ς {ς y Ψ (x (Î, θ ) ( i N (Î, θ ) for all l = 2, 3,, n, if d 1 Λ i d i (Î, θ ))ς θ )}. (I, θ ) > d 1 (Î, θ ). (E.5) (I, θ ) > d 1 (Î, θ ), d i (I, θ ) d i (Î, θ ) for all i N. By par (c), x (Î, θ ) x (Î, θ ) ( i N Λ i d i (Î, θ ))ς x (I, θ ) ( Λ i d i (I, θ ))ς i N for any realizaion of ς. Thus, he concaviy of Ψ ( θ ) implies ha ς y Ψ (x (Î, θ ) ( i N Λ i d i (Î, θ ))ς θ ) ς y Ψ (x (I, θ ) ( Λ i d i (I, θ ))ς θ ) i N for any realizaion of ς. By aking expecaion on boh sides, we have E ς {ς y Ψ (x (Î, θ ) ( i N Λ i d i (Î, θ ))ς θ )} E ς {ς y Ψ (x (I, θ ) ( Λ i d i (I, θ ))ς θ )}, i N which conradics (E.4). Therefore, d 1 (I, θ ) d 1 (Î, θ ). The same argumen implies ha d i (I, θ ) is increasing in I for all i N. Hence, I d,i (θ ) = max{i : d i (I, θ ) = 0}. If i N (I, θ ), since d i (I, θ ) is increasing in I and Î > I, d i (Î, θ ) d i (I, θ ) > 0. N (Î, θ ) follows immediaely. Thus, i N (Î, θ ), and N (I, θ ) To complee he proof, i remains o be shown ha I d,i (θ ) < + for all i N. By he proof of par (b), lim z + z Ψ (z θ ) < 0. Moreover, observe ha x (I, θ ) + as I +. Thus, by he monoone convergence heorem, Therefore, for any i and d i d i lim E ς {ς y Ψ (x (I, θ ) ( Λ i I d i )ς )} < 0 for any d [0, d max ] n. + i N d i R i (0) lim E ς {ς y Ψ (x (I, θ ) ( I + [0, d max ] n 1, where d i l N,l i (I, θ ) > 0 for sufficienly large I, i.e., I d,i (θ ) < +. Q.E.D. Λ l d l )ς θ )} > d i R i (0) = p i max > 0 := (d 1, d 2,, d i 1, d i+1,, d n ). Hence, for all i N, Proof of Theorem 6.4.2: Par (a). Firs, we show ha if d i R i (z) d î Rî(z) for all z [0, d max ], d i (I, θ ) dî (I, θ ) for any (I, θ ). Assume, o he conrary, ha d i (I, θ ) < dî (I, θ ). By he inequaliy d i R i (z) d î Rî(z) 293

308 for all z [0, d max ] and he sric concaviy of R i ( ), d i R i (d i (I, θ )) > d î Rî(dî (I, θ )). On he oher hand, Lemma 16 yields ha d i J (d (I, θ ), q (I, θ ), I θ ) d î J (d (I, θ ), q (I, θ ), I θ ). Thus, E ς {ς y Ψ (x (I, θ ) ( l N Λ l d l (I, θ ))ς θ )} = d i R i (d i (I, θ )) 1 Λ i d i J (d (I, θ ), q (I, θ ), I θ ) > d î Rî(dî (I, θ )) 1 Λî d î J (d (I, θ ), q (I, θ ), I θ ) =E ς {ς y Ψ (x (I, θ ) ( l N Λ l d l (I, θ ))ς θ )}, which forms a conradicion. Thus, d i (I, θ ) dî (I, θ ), and I d,i (θ ) = max{i : d i (I, θ ) = 0} max{i : dî (I, θ ) = 0} = I d,î (θ ). For he second half of par (a), he inequaliy I d,1 (θ ) I d,2 (θ ) I d,n (θ ) follows direcly from he firs half. I remains o be shown ha N (I, θ ) = {1, 2,, i }, where i = max{i : I > I d,i (θ )}. Observe ha I > I d,i (θ ) I d,i 1 (θ ) I d,1 (θ ). Thus, by he definiion of I d,i (θ ), {1, 2,, i } N (I, θ ). Moreover, by he definiion of i, I I d,i +1 (θ ) I d,i +2 (θ ) I d,n (θ ). Thus, i N (I, θ ) for all i i + 1 and, hence, N (I, θ ) = {1, 2,, i }. Par (b). We firs show ha if q j for any (I, θ ). Assume, o he conrary, ha q j q ĵ Cĵ(z cĵ q ĵ q ĵ C j (z c j ) q ĵ Cĵ(z cĵ (I, θ ) > qĵ ) for any z 0, q j (I, θ ) qĵ (I, θ ) (I, θ ). The inequaliy q j C j (z c j ) ) for any z 0, ogeher wih he sric convexiy of C j ( c j ), implies ha q j Cĵ(qĵ (I, θ ) cĵ ). J (d (I, θ ), q (I, θ ), I θ ). Thus, C j (q j (I, θ ) c j ) > On he oher hand, Lemma 16 implies ha q j J (d (I, θ ), q (I, θ ), I θ ) E ς { y Ψ (x (I, θ ) ( Λ l d l (I, θ ))ς θ )} = q j l N C j (q j (I, θ ) c j ) + q j J (d (I, θ ), q (I, θ ), I θ ) > q ĵ Cĵ(qĵ (I, θ ) cĵ ) + q ĵ J (d (I, θ ), q (I, θ ), I θ ) =E ς { y Ψ (x (I, θ ) ( l N Λ l d l (I, θ ))ς θ )}, which forms a conradicion. Thus, q j I q,j (θ ) = min{i : q j (I, θ ) qĵ (I, θ ), and (I, θ ) = 0} min{i : qĵ (I, θ ) = 0} = I q,ĵ (θ ). For he second half of par (b), he inequaliy I q,1 (θ ) I q,2 (θ ) I q,m (θ ) follows direcly from he firs half. I remains o be shown ha M (I, θ ) = {j, j +1,, m}, where j = min{j : I < I q,j (θ )}. Observe ha I < I q,j (θ ) I q,j +1 (θ ) I q,m (θ ). Thus, by he definiion of I q,j (θ ), {j, j + 1,, m} M (I, θ ). Moreover, by he definiion of j, I I q,j 1 (θ ) I q,j 2 (θ ) I q,1 (θ ). Thus, j M (I, θ ) for all j j 1 and, hence, M (I, θ ) = {j, j +1,, m}. Q.E.D. Proof of Theorem 6.4.3: We show all pars ogeher by backward inducion. More specifically, we prove ha if I 1 V 1 (I 1 ˆθ 1 ) I 1 V 1 (I 1 θ 1 ) for all I 1 and ˆΛ 1 > Λ 1, hen we have (i) d i (I, ˆθ ) d i (I, θ ) for all i N, (ii) q j (I, ˆθ ) q j (I, θ ) for all j M, (iii) x (I, ˆθ ) x (I, θ ), and (vi) I V (I ˆθ ) I V (I θ ) for all I and ˆΛ > Λ. Since I0 V 0 (I 0 ˆθ 0 ) = I0 V 0 (I 0 θ 0 ) = 0 for all 294

309 I 0 and ˆΛ 0 > Λ 0, he iniial condiion is saisfied. Since I 1 V 1 (I 1 ˆθ 1 ) I 1 V 1 (I 1 θ 1 ) and ξ Λ,i (ˆΛ i ) s.d. ξ Λ,i (Λ i ) for any i N, z Ψ (z ˆθ ) z Ψ (z θ ) for any z. Firs, we show ha d i (I, ˆθ ) d i (I, θ ) for all i N. Wihou loss of generaliy, we assume, o he conrary, ha d 1 (I, ˆθ ) > d 1 (I, θ ). The sric concaviy of R 1 ( ) implies ha d 1 R 1 (d 1 (I, ˆθ )) < d 1 R 1 (d 1 (I, θ )). On he oher hand, Lemma 16 yields ha d 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) 0 d 1 J (d (I, θ ), q (I, θ ), I θ ), i.e., d 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ )/ˆΛ 1 0 d 1 J (d (I, θ ), q (I, θ ), I θ )/Λ 1. Thus, z Ψ ( (I, ˆθ ) ˆθ ) = d 1 R 1 (d 1 (I, ˆθ )) 1ˆΛ1 d 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) < d 1 R 1 (d 1 (I, θ )) 1 Λ 1 d 1 J (d (I, θ ), q (I, θ ), I θ ) (E.6) = z Ψ ( (I, θ ) θ ). Since z Ψ (z ˆθ ) > z Ψ (z θ ) for all z and Ψ ( θ ) is concave in z, (I, ˆθ ) > (I, θ ), i.e., I + q j (I, ˆθ ) ( ˆΛ i d i (I, ˆθ )) = (I, ˆθ ) > (I, θ ) = I + q j (I, θ ) ( Λ i d i (I, θ )). j M i N j M i N Since ˆΛ > Λ and d 1 q j (I, ˆθ ) > q j (I, θ ) for some 1 j m. (I, ˆθ ) > d 1 (I, θ ), eiher (a) d i (I, ˆθ ) < d i (I, θ ) for some 2 i n, or (b) In case (a), wihou loss of generaliy, we assume ha d 2 d 2 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) 0 d 2 J (d (I, θ ), q (I, θ ), I θ ), i.e., (I, ˆθ ) < d 2 (I, θ ). Lemma 16 yields ha d 2 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ )/ˆΛ 2 d 2 J (d (I, θ ), q (I, θ ), I θ )/Λ 2. Thus, by (E.6), d 2 R 2 (d 2 (I, ˆθ )) = 1ˆΛ2 d 2 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) + z Ψ ( (I, ˆθ ) ˆθ ) < 1 Λ 2 d 2 J (d (I, θ ), q (I, θ ), I θ ) + z Ψ ( (I, θ ) θ ) = d 2 R 2 (d 2 (I, θ )), which conradics he sric concaviy of R 2 ( ). Hence, d 2 (I, ˆθ ) d 2 (I, θ ) under he condiion ha d 1 (I, ˆθ ) > d 1 (I, θ ). I follows from he same argumen ha d i (I, ˆθ ) d i (I, θ ) for all i = 2, 3,, n, under he condiion ha d 1 (I, ˆθ ) > d 1 (I, θ ). In case (b), wihou loss of generaliy, we assume ha q 1 q 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) q 1 J (d (I, θ ), q (I, θ ), I θ ). Thus, by (E.6), (I, ˆθ ) > q 1 (I, θ ). Lemma 16 yields ha q 1 C 1 (q 1 (I, ˆθ ) c 1 ) = z Ψ ( (I, ˆθ ) ˆθ ) q 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) < z Ψ ( (I, θ ) θ ) q 1 J (d (I, θ ), q (I, θ ), I θ ) = q 1 C 1 (q 1 (I, θ ) c 1 ), which conradics he sric convexiy of C 1 ( c 1 ) in q 1. Hence, q 1 ha d 1 (I, ˆθ ) > d 1 (I, θ ). I follows from he same argumen ha q j j = 1, 2,, m, under he condiion ha d 1 ha he iniial assumpion d 1 (I, ˆθ ) > d 1 (I, ˆθ ) q 1 (I, θ ) under he condiion (I, ˆθ ) q j (I, θ ) for all (I, ˆθ ) > d 1 (I, θ ). Combining cases (a) and (b), i follows (I, θ ) is incorrec. Therefore, d 1 (I, ˆθ ) d 1 (I, θ ). The same argumen yields ha d i (I, ˆθ ) d i (I, θ ) for any i = 1, 2,, n. Hence, for each i N, I d,i (ˆθ ) = max{i : d i (I, ˆθ ) = 0} max{i : d i (I, θ ) = 0} = I d,i (θ ). 295

310 For any i N (I, ˆθ ), d i (I, θ ) d i (I, ˆθ ) > 0. Thus, i N (I, θ ), and N (I, ˆθ ) N (I, θ ) follows immediaely. Nex, we show ha q j (I, ˆθ ) q j (I, θ ) for all j M. We assume, o he conrary, ha q 1 (I, ˆθ ) < q 1 (I, θ ). The sric convexiy of C 1 ( c 1 ) in q 1 implies ha q 1 C 1 (q 1 (I, ˆθ ) c 1 ) < q 1 C 1 (q 1 (I, θ ) c 1 ). On he oher hand, Lemma 16 yields ha q 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) q 1 J (d (I, θ ), q (I, θ ), I θ ). Thus, z Ψ ( (I, ˆθ ) ˆθ ) = q 1 C 1 (q 1 (I, ˆθ ) c 1 ) + q 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) < q 1 C 1 (q 1 (I, θ ) c 1 ) + q 1 J (d (I, θ ), q (I, θ ), I θ ) (E.7) I + = z Ψ ( (I, θ ) θ ). Since z Ψ (z ˆθ ) > z Ψ (z θ ) for all z and Ψ ( θ ) is concave in z, (I, ˆθ ) > (I, θ ), i.e., j M q j (I, ˆθ ) ( ˆΛ i d i (I, ˆθ )) = (I, ˆθ ) > (I, θ ) = I + q j (I, θ ) ( Λ i d i (I, θ )). i N j M i N Since ˆΛ > Λ, and q 1 q j (I, ˆθ ) > q j (I, θ ) for some 2 j m. (I, ˆθ ) < q 1 (I, θ ), eiher (a) d i (I, ˆθ ) < d i (I, θ ) for some 1 i n, or (b) In case (a), wihou loss of generaliy, we assume ha d 1 d 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) 0 d 1 J (d (I, θ ), q (I, θ ), I θ ), i.e., (I, ˆθ ) < d 1 (I, θ ). Lemma 16 yields ha d 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ )/ˆΛ 1 d 1 J (d (I, θ ), q (I, θ ), I θ )/Λ 1. Thus, by (E.7), d 1 R 1 (d 1 (I, ˆθ )) = 1ˆΛ1 d 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) + z Ψ ( (I, ˆθ ) ˆθ ) < 1 Λ 1 d 1 J (d (I, θ ), q (I, θ ), I θ ) + z Ψ ( (I, θ ) θ ) = d 1 R 1 (d 1 (I, θ )), which conradics he sric concaviy of R 1 ( ). Hence, d 1 (I, ˆθ ) d 1 (I, θ ) under he condiion ha q 1 (I, ˆθ ) < q 1 (I, θ ). I follows from he same argumen ha d i (I, ˆθ ) d i (I, θ ) for all i = 1, 2,, n, under he condiion ha q 1 d i (I, θ ) for all i = 1, 2,, n. In case (b), wihou loss of generaliy, we assume ha q 2 (I, ˆθ ) < q 1 (I, θ ). Thus, under his condiion, d i (I, ˆθ ) = q 2 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) q 2 J (d (I, θ ), q (I, θ ), I θ ). Thus, by (E.7), (I, ˆθ ) > q 2 (I, θ ). Lemma 16 yields ha q 2 C 2 (q 2 (I, ˆθ ) c 2 ) = z Ψ ( (I, ˆθ ) ˆθ ) q 2 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) < z Ψ ( (I, θ ) θ ) q 2 J (d (I, θ ), q (I, θ ), I θ ) = q 2 C 2 (q 2 (I, θ ) c 2 ), which conradics he sric convexiy of C 2 ( c 2 ) in q 2. Hence, q 2 ha q 1 (I, ˆθ ) < q 1 (I, θ ). I follows from he same argumen ha q j j = 2,, m, under he condiion ha q 1 ha he iniial assumpion q 1 (I, ˆθ ) < q 1 (I, ˆθ ) q 2 (I, θ ) under he condiion (I, ˆθ ) q j (I, θ ) for all (I, ˆθ ) < q 1 (I, θ ). Combining cases (a) and (b), i follows (I, θ ) is incorrec. Therefore, q 1 (I, ˆθ ) q 1 (I, θ ). The same argumen yields ha q j (I, ˆθ ) q j (I, θ ) for any j = 1, 2,, m. Hence, for each j M, I q,j (ˆθ ) = min{i : q j (I, ˆθ ) = 0} min{i : q j (I, θ ) = 0} = I q,j (θ ). 296

311 For any j M (I, θ ), q j (I, ˆθ ) q j (I, θ ) > 0. Thus, j M (I, ˆθ ), and M (I, θ ) M (I, ˆθ ) follows immediaely. d i Finally, o complee he inducion, we show ha I V (I ˆθ ) I V (I θ ). Recall ha d i (I, ˆθ ) (I, θ ) for any i N. If d 1 (I, ˆθ ) < d 1 (I, θ ), he sric concaviy of R 1 ( ) implies ha d 1 R 1 (d 1 (I, ˆθ )) > d 1 R 1 (d 1 (I, θ )). On he oher hand, Lemma 16 yields ha d 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) 0 d 1 J (d (I, θ ), q (I, θ ), I θ ), i.e., d 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ )/ˆΛ 1 d 1 J (d (I, θ ), q (I, θ ), I θ )/Λ 1. Thus, By Equaion (E.1), z Ψ ( (I, ˆθ ) ˆθ ) = d 1 R 1 (d 1 (I, ˆθ )) 1ˆΛ1 d 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) > d 1 R 1 (d 1 (I, θ )) 1 Λ 1 d 1 J (d (I, θ ), q (I, θ ), I θ ) = z Ψ ( (I, θ ) θ ). I V (I ˆθ ) = z Ψ ( (I, ˆθ ) ˆθ ) > z Ψ ( (I, θ ) θ ) = I V (I θ ). The same argumen implies ha, if here exiss an i N, such ha d i (I, ˆθ ) < d i (I, θ ), we have I V (I ˆθ ) > I V (I θ ). Recall ha q j C 1 ( c 1 ) in q 1 (I, ˆθ ) q j (I, θ ) for any j M. If q 1 (I, ˆθ ) > q 1 (I, θ ), he sric convexiy of implies ha q 1 C 1 (q 1 (I, ˆθ ) c 1 ) > q 1 C 1 (q 1 (I, θ ) c 1 ). On he oher hand, Lemma 16 yields ha q 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) q 1 J (d (I, θ ), q (I, θ ), I θ ). Thus, By Equaion (E.1), z Ψ ( (I, ˆθ ) ˆθ ) = q 1 C 1 (q 1 (I, ˆθ ) c 1 ) + q 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) > q 1 C 1 (d 1 (I, θ ) c 1 ) + q 1 J (d (I, θ ), q (I, θ ), I θ ) = z Ψ ( (I, θ ) θ ). I V (I ˆθ ) = z Ψ ( (I, ˆθ ) ˆθ ) > z Ψ ( (I, θ ) θ ) = I V (I θ ). The same argumen implies ha, if here exiss a j M, such ha q j I V (I ˆθ ) > I V (I θ ). Now we assume ha for any i N and j M, d i (I, ˆθ ) = d i Since ˆΛ > Λ, (I, ˆθ ) = I + j M (I, ˆθ ) < q j (I, θ ), we have (I, θ ) and q j (I, ˆθ ) = q j (I, θ ). q j (I, ˆθ ) ( ˆΛ i d i (I, ˆθ )) I + q j (I, θ ) ( Λ i d i (I, θ )) = (I, θ ). i N j M i N Since z Ψ (z ˆθ ) z Ψ (z θ ) for any z, i follows ha z Ψ ( (I, ˆθ ) ˆθ ) z Ψ ( (I, θ ) θ ) by he concaviy of Ψ ( θ ) in z. Thus, by Equaion (E.1), I V (I ˆθ ) = z Ψ ( (I, ˆθ ) ˆθ ) z Ψ ( (I, θ ) θ ) = I V (I θ ). This complees he inducion and, hus, he proof of Theorem Q.E.D. 297

312 Proof of Theorem 6.4.4: We show all pars ogeher by backward inducion. More specifically, we prove ha if I 1 V 1 (I 1 ˆθ 1 ) I 1 V 1 (I 1 θ 1 ) for all I 1 and ĉ 1 > c 1, hen we have (i) d i (I, ˆθ ) d i (I, θ ) for all i N, (ii) q j (I, ˆθ ) q j (I, θ ) for all {j M : ĉ j = c j }, and (iii) I V (I ˆθ ) I V (I θ ) for all I and ĉ > c. Since I0 V 0 (I 0 ˆθ 0 ) = I0 V 0 (I 0 θ 0 ) = 0 for all I 0 and ĉ 0 > c 0, he iniial condiion is saisfied. ξ c,j (ĉ j ) s.d. ξ c,j (c j ) for any j M, z Ψ (z ˆθ ) z Ψ (z θ ) for any z. Since I 1 V 1 (I 1 ˆθ 1 ) I 1 V 1 (I 1 θ 1 ) and Firs, we show (i) and (ii). Wihou loss of generaliy, we assume ha ĉ j > c j for j = 1, 2,, m 1 (1 m 1 m) and ĉ j = c j oherwise. Invoke Lemma 15 wih p = n + m m 1, q = m 1, Γ = {0, 1}, y i = d i (1 i n), y n+j = q m 1+j (1 j m m 1 ), y n+m m1 +j = q j (1 j m 1 ), λ i = Λ i (1 i n), λ i = 1 (n + 1 i n + m), f i (y i ) = Λ i R i (d i ) (1 i n), f j+n (y j+n ) = C j (q j C j+m 1 (q j+m 1 c j+m c j ), if γ = 0, 1 ) (1 j m m 1 ), g j+n+m m1 (y j+n+m m1 γ) = (1 C j (q j ĉ j ), if γ = 1, Ψ (I + y 0 θ ), if γ = 0, [ d max, 0], if 1 i n, j m 1 ), h(y 0 γ) = and Y i = Since Ψ (I + y 0 ˆθ ), if γ = 1, [0, + ), if n + 1 i n + m. C j (q j c j ) is supermodular in (q j, c j ) for any 1 j m, and Ψ (I + y 0 θ ) is supermodular in (y 0, c j ), g j+p (y j γ) (1 j q) is submodular in (y j, γ), and h(y 0 γ) is supermodular in (y 0, γ). Lemma 15 implies ha d i (I, ˆθ ) d i (I, θ ) for all i N, and q j (I, ˆθ ) q j (I, θ ) for all {j M : ĉ j = c j }. For any i N (I, ˆθ ), d i (I, θ ) d i (I, ˆθ ) > 0. Thus, i N (I, θ ), and N (I, ˆθ ) N (I, θ ) follows immediaely. To complee he inducion, we show ha I V (I ˆθ ) I V (I θ ). If d 1 sric concaviy of R 1 ( ) implies ha d 1 R 1 (d 1 16 yields ha d 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) d 1 J (d (I, θ ), q (I, θ ), I θ ). Thus, By Equaion (E.1), (I, ˆθ ) < d 1 (I, θ ), he (I, ˆθ )) > d 1 R 1 (d 1 (I, θ )). On he oher hand, Lemma z Ψ ( (I, ˆθ ) ˆθ ) = d 1 R 1 (d 1 (I, ˆθ )) 1 Λ 1 d 1 J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) > d 1 R 1 (d 1 (I, θ )) 1 Λ 1 d 1 J (d (I, θ ), q (I, θ ), I θ ) = z Ψ ( (I, θ ) θ ). I V (I ˆθ ) = z Ψ ( (I, ˆθ ) ˆθ ) > z Ψ ( (I, θ ) θ ) = I V (I θ ). The same argumen implies ha, if here exiss an i N, such ha d i (I, ˆθ ) < d i (I, θ ), hen we have I V (I ˆθ ) > I V (I θ ). Now we assume ha for all i N, d i (I, ˆθ ) = d i (I, θ ). If q j (I, ˆθ ) q j (I, θ ) for all j M, (I, ˆθ ) = I + j M q j (I, ˆθ ) ( Λ i d i (I, ˆθ )) I + i N j M q j (I, θ ) ( Λ i d i (I, θ )) = (I, θ ). i N Since z Ψ (z ˆθ ) z Ψ (z θ ) for any z, he concaviy of Ψ ( θ ) in z implies ha z Ψ ( (I, ˆθ ) ˆθ ) z Ψ ( (I, θ ) θ ). Thus, I V (I ˆθ ) = z Ψ ( (I, ˆθ ) ˆθ ) z Ψ ( (I, θ ) θ ) = I V (I θ ). 298

313 In he remaining case, q j (I, ˆθ ) > q j (I, θ ) for some 1 j m, Wihou loss of generaliy, assume ha q l (I, ˆθ ) > q l (I, θ ). In his case, he supermodulariy of C l ( ) in (q l, c l ) and he sric convexiy of C l ( c l ) in q l imply ha q l C l (q l (I, ˆθ ) ĉ l ) > q l C l (q l (I, θ ) c l ). On he oher hand, Lemma 16 implies ha q l J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) q l J (d (I, θ ), q (I, θ ), I θ ). Thus, Thus, by Equaion (E.1), z Ψ ( (I, ˆθ ) ˆθ ) = q l C 1 (q l (I, ˆθ ) ĉ l ) + q l J (d (I, ˆθ ), q (I, ˆθ ), I ˆθ ) > q l C l (q l (I, θ ) c l ) + q l J (d (I, θ ), q (I, θ ), I θ ) = z Ψ ( (I, θ ) θ ). I V (I ˆθ ) = z Ψ ( (I, ˆθ ) ˆθ ) > z Ψ ( (I, θ ) θ ) = I V (I θ ). This complees he inducion and, hus, he proof of Theorem Q.E.D. Proof of Theorem 6.4.5: We show all pars ogeher by backward inducion. More specifically, we prove ha if I 1 ˆV 1 (I 1 θ 1 ) I 1 V 1 (I 1 θ 1 ) for all I 1, hen we have (i) ˆd i (I, θ ) d i (I, θ ) for all i N, (ii) ˆq j (I, θ ) q j (I, θ ) for all j M, (iii) ˆx (I, θ ) x (I, θ ), (iv) ˆ (I, θ ) (I, θ ), and (v) I ˆV (I θ ) I V (I θ ) for all I. Noe ha I0 ˆV0 (I 0 θ 0 ) = I0 V 0 (I 0 θ 0 ) for all I 0, so he iniial condiion is saisfied. Since I 1 ˆV 1 (I 1 θ 1 ) I 1 V 1 (I 1 θ 1 ) and ˆξ Λ,i (Λ i ) s.d. ξ Λ,i (Λ i ) for any i and Λ, by Theorem 6.4.3(a), z ˆΨ (z θ ) z Ψ (z θ ) for any z. Firs, we show ha ˆd i (I, θ ) d i (I, θ ) for any i N and ˆq j (I, θ ) q j (I, θ ) for any j M. We apply Lemma 15 o prove hese resuls. Le p = n + m, q = 0, Γ = {0, 1}, y i = d i (1 i n), y j+n = q j (1 j m), λ i = Λ i (1 i n), λ i = 1 (n + 1 i n + m), f i (y i ) = Λ i R i (d i ) (1 i n), f j+n (y j+n ) = C j (q j c j ) (1 j m), h(y 0 0) = Ψ (I +y 0 θ ), h(y 0 1) = ˆΨ (I +y 0 θ ), and [ d max, 0], 1 i n, Y i = Since z ˆΨ (z θ ) z Ψ (z θ ) for any z, h(y 0 γ) is supermodular [0, + ), n + 1 i n + m. in (y 0, γ). Lemma 15 implies ha ˆd i (I, θ ) d i any j M. For any i ˆN (I, θ ), d i (I, θ ) (I, θ ) q j (I, θ ) for (I, θ ) for any i N and ˆq j i ˆd (I, θ ) > 0. Thus, i N (I, θ ), and ˆN (I, θ ) N (I, θ ) follows immediaely. For any j M (I, θ ), ˆq j (I, θ ) q j (I, θ ) > 0. Thus, j ˆM (I, θ ), and M (I, θ ) ˆM (I, θ ) follows immediaely. and Moreover, ˆx (I, θ ) = I + j M ˆq j (I, θ ) I + q j (I, θ ) = x (I, θ ), j M ˆ (I, θ ) = ˆx (I, θ ) ( Λ l l ˆd (I, θ )) x (I, θ ) ( Λ l d l (I, θ )) = (I, θ ). l N l N To complee he inducion, we show ha I ˆV (I θ ) I V (I θ ). d i (I, θ ) for any i N. If ˆd 1 Recall ha (I, θ ) < d 1 (I, θ ), he sric concaviy of R 1 ( ) implies ha ˆd i (I, θ ) 299

314 d 1 R 1 ( ˆd 1 (I, θ )) > d 1 R 1 (d 1 (I, θ )). On he oher hand, Lemma 16 yields ha d 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) d 1 J (d (I, θ ), q (I, θ ), I θ ). Thus, By Equaion (E.1), z ˆΨ ( ˆ (I, θ ) θ ) = d 1 R 1 ( 1 ˆd (I, θ )) 1 Λ 1 d 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) > d 1 R 1 (d 1 (I, θ )) 1 Λ 1 d 1 J (d (I, θ ), q (I, θ ), I θ ) = z Ψ ( (I, θ ) θ ). I ˆV (I θ ) = z ˆΨ ( (I, θ ) θ ) > z Ψ ( (I, θ ) θ ) = I V (I θ ). The same argumen implies ha, if here exiss an i N, such ha I ˆV (I θ ) > I V (I θ ). Recall ha ˆq j C 1 ( c 1 ) in q 1 (I, θ ) q j (I, θ ) for any j M. If ˆq 1 ˆd i (I, θ ) < d i (I, θ ), we have (I, θ ) > q 1 (I, θ ), he sric convexiy of implies ha q 1 C 1 (ˆq 1 (I, θ ) c 1 ) > q 1 C 1 (q 1 (I, θ ) c 1 ). On he oher hand, Lemma 16 yields ha q 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) q 1 J (d (I, θ ), q (I, θ ), I θ ). Thus, By Equaion (E.1), z ˆΨ ( ˆ (I, θ ) θ ) = q 1 C 1 (ˆq 1 (I, θ ) c 1 ) + q 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) > q 1 C 1 (d 1 (I, θ ) c 1 ) + q 1 J (d (I, θ ), q (I, θ ), I θ ) = z Ψ ( (I, θ ) θ ). I ˆV (I θ ) = z ˆΨ ( ˆ (I, θ ) θ ) > z Ψ ( (I, θ ) θ ) = I V (I θ ). The same argumen implies ha, if here exiss a j M such ha ˆq j I ˆV (I θ ) > I V (I θ ). In he remaining case, We have ˆ (I, θ ) = I + j M ˆd i (I, θ ) = d i (I, θ ) and ˆq j (I, θ ) > q j (I, θ ), we have (I, θ ) = q j (I, θ ) for any i N and j M. ˆq j (I, θ ) ( Λ i i ˆd (I, θ )) = I + q j (I, θ ) ( Λ i d i (I, θ )) = (I, θ ). i N j M i N Since z ˆΨ (z θ ) z Ψ (z θ ) for any z, z ˆΨ ( ˆ (I, θ ) θ ) z Ψ ( (I, θ ) θ ). Thus, by Equaion (E.1), I ˆV (I θ ) = z ˆΨ ( ˆ (I, θ ) θ ) z Ψ ( (I, θ ) θ ) = I V (I θ ). This complees he inducion and, hus, he proof of Theorem Q.E.D. Proof of Theorem 6.4.6: The proof of Theorem follows from similar argumen o ha of Theorem 6.4.5, so we only skech i. We show all pars ogeher by backward inducion. More specifically, we prove ha if I 1 ˆV 1 (I 1 θ 1 ) I 1 V 1 (I 1 θ 1 ) for all I 1, hen we have (i) ˆd i (I, θ ) d i (I, θ ) for all i N, (ii) ˆq j (I, θ ) q j (I, θ ) for all j M, (iii) ˆx (I, θ ) x (I, θ ), (iv) ˆ (I, θ ) (I, θ ), 300

315 and (v) I ˆV (I θ ) I V (I θ ) for all I. Noe ha I0 ˆV0 (I 0 θ 0 ) = I0 V 0 (I 0 θ 0 ) for all I 0, so he iniial condiion is saisfied. Since I 1 ˆV 1 (I 1 θ 1 ) I 1 V 1 (I 1 θ 1 ) and for any j and c, by Theorem 6.4.4(a), z ˆΨ (z θ ) z Ψ (z θ ) for any z. c,j ˆξ (c j ) s.d. ξ c,j (c j ) i Firs, we employ Lemma 15 o show ha ˆd (I, θ ) d i (I, θ ) for any i N and ˆq j (I, θ ) q j (I, θ ) for any j M. Le p = n + m, q = 0, Γ = {0, 1}, y i = d i (1 i n), y j+n = q j (1 j m), λ i = Λ i (1 i n), λ i = 1 (n + 1 i n + m), f i (y i ) = Λ i R i (d i ) (1 i n), f j+n (y j+n ) = C j (q j c j ) (1 j m), h(y 0 0) = Ψ (I + y 0 θ ), h(y 0 1) = ˆΨ (I + y 0 θ ), and [ d max, 0], 1 i n, i Y i = Invoking Lemma 15, we have ha ˆd (I, θ ) d i (I, θ ) for any [0, + ), n + 1 i n + m. i N and ˆq j immediaely from (I, θ ) q j ˆd i (I, θ ) d i (I, θ ) for any j M. Î d,i (I, θ ) for any i N, whereas Îq,j (θ ) I d,i (θ ) and ˆN (I, θ ) N (I, θ ) follow (θ ) I q,j (θ ) and M (I, θ ) ˆM (I, θ ) follow immediaely from ˆq j (I, θ ) q j (I, θ ) for any j M. ˆx (I, θ ) x (I, θ ) and ˆ (I, θ ) (I, θ ) also follow direcly. To complee he inducion, we show ha I ˆV (I θ ) I V (I θ ). Following he same argumen as he proof of Theorem 6.4.5, we have ha if here exiss an i N, such ha ˆd i (I, θ ) < d i (I, θ ), or here exiss a j M, such ha ˆq j (I, θ ) > q j (I, θ ), hen I ˆV (I θ ) = z ˆΨ ( ˆ (I, θ ) θ ) > z Ψ ( (I, θ ) θ ) = I V (I θ ). In he remaining case, ˆdi (I, θ ) = d i (I, θ ) and ˆq j (I, θ ) = q j (I, θ ) for any i N and j M. Thus, ˆ (I, θ ) = (I, θ ). Since z ˆΨ (z θ ) z Ψ (z θ ) for any z, z ˆΨ ( ˆ (I, θ ) θ ) z Ψ ( (I, θ ) θ ). Thus, by Equaion (E.1), I ˆV (I θ ) = z ˆΨ ( ˆ (I, θ ) θ ) z Ψ ( (I, θ ) θ ) = I V (I θ ). This complees he inducion and, hus, he proof of Theorem Q.E.D. Proof of Theorem 6.4.7: We show all pars ogeher by backward inducion. More specifically, we prove ha if I 1 ˆV 1 (I 1 θ 1 ) I 1 V 1 (I 1 θ 1 ) for all I 1, hen we have (i) d i ˆd i (I, θ ) (I, θ ) for all i N, (ii) ˆq j (I, θ ) q j (I, θ ) for all j M, (iii) ˆx (I, θ ) x (I, θ ), and (iv) I ˆV (I θ ) I V (I θ ) for all I. Since I0 ˆV0 (I 0 θ 0 ) = I0 V 0 (I 0 θ 0 ) for all I 0, he iniial condiion is saisfied. Since I 1 ˆV 1 (I 1 θ 1 ) I 1 V 1 (I 1 θ 1 ), z ˆΨ (z θ ) z Ψ (z θ ) for any z. Denoe N = {1, 2,, n} and ˆN = {1, 2,, ˆn}, where ˆn > n. Firs, we show ha i ˆd (I, θ ) d i (I, θ ) for all i N. We assume, o he conrary, ha d 1 (I, θ ). The sric concaviy of R 1 ( ) implies ha d 1 R 1 ( ˆd 1 ˆd 1 (I, θ ) > (I, θ )) < d 1 R 1 (d 1 (I, θ )). On he oher hand, Lemma 16 yields ha d 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) d 1 J (d (I, θ ), q (I, θ ), I θ ). Thus, z ˆΨ ( ˆ (I, θ ) θ ) = d 1 R 1 ( 1 ˆd (I, θ )) 1 Λ 1 d 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) < d 1 R 1 (d 1 (I, θ )) 1 Λ 1 d 1 J (d (I, θ ), q (I, θ ), I θ ) (E.8) = z Ψ ( (I, θ ) θ ). Since z ˆΨ (z θ ) z Ψ (z θ ) for all z, ˆ (I, θ ) > (I, θ ), i.e., I + j M ˆq j (I, θ ) ( Λ i i ˆd (I, θ )) = ˆ (I, θ ) > (I, θ ) = I + q j (I, θ ) ( Λ i d i (I, θ )). i ˆN j M i N 301

316 Since N ˆN and ˆd 1 (I, θ ) > d 1 (I, θ ), eiher (a) (b) ˆq j (I, θ ) > q j (I, θ ) for some j = 1, 2,, m. In case (a), wihou loss of generaliy, we assume ha ˆd i (I, θ ) < d i (I, θ ) for some i = 2, 3,, n, or ˆd 2 (I, θ ) < d 2 (I, θ ). By Lemma 16, we have d 2 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) d 2 J (d (I, θ ), q (I, θ ), I θ ). Thus, by (E.8), d 2 R 2 ( 2 ˆd (I, θ )) = z ˆΨ ( ˆ (I, θ ) θ ) + 1 Λ 2 d 2 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) < z Ψ ( (I, θ ) θ ) + 1 Λ 2 d 2 J (d (I, θ ), q (I, θ ), I θ ) = d 2 R 2 (d 2 (I, θ )), which conradics he sric concaviy of R 2 ( ). Hence, ˆd2 1 ha ˆd (I, θ ) > d 1 (I, θ ). I follows from he same argumen ha 1 i = 2, 3,, n, under he condiion ha ˆd (I, θ ) > d 1 (I, θ ). In case (b), wihou loss of generaliy, we assume ha ˆq 1 (I, θ ) d 2 (I, θ ) under he condiion ˆd i (I, θ ) d i (I, θ ) for all (I, θ ) > q 1 (I, θ ). By Lemma 16, we have q 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) q 1 J (d (I, θ ), q (I, θ ), I θ ). Thus, by (E.8), q 1 C 1 (ˆq 1 (I, θ ) c 1 ) = z ˆΨ ( ˆ (I, θ ) θ ) q 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) < z Ψ ( (I, θ ) θ ) q 1 J (d (I, θ ), q (I, θ ), I θ ) = q 1 C 1 (q 1 (I, θ ) c 1 ), which conradics he sric convexiy of C 1 ( c 1 ) in q 1. Hence, ˆq 1 ha 1 ˆd (I, θ ) > d 1 (I, θ ). I follows from he same argumen ha ˆq j j = 1, 2,, m, under he condiion ha ha he iniial assumpion same argumen yields ha ˆd 1 Î d,i (θ ) = max{i : If i ( ˆN (I, θ ) N ), d i (I, θ ) follows immediaely. ˆd 1 (I, θ ) q 1 (I, θ ) under he condiion (I, θ ) q j (I, θ ) for all (I, θ ) > d 1 (I, θ ). Combining cases (a) and (b), i follows (I, θ ) > d 1 (I, θ ) is incorrec. Therefore, ˆd 1 (I, θ ) d 1 (I, θ ). The ˆd i (I, θ ) d i (I, θ ) for any i = 1, 2, n. Hence, for each i N, ˆd i (I, θ ) = 0} max{i : d i (I, θ ) = 0} = I d,i (θ ). ˆd i (I, θ ) > 0. Thus, i N (I, θ ), and ( ˆN (I, θ ) N ) N (I, θ ) Nex, we show ha ˆq j (I, θ ) q j (I, θ ) for all j M. We assume, o he conrary, ha ˆq 1 (I, θ ) < q 1 (I, θ ). The sric convexiy of C 1 ( c 1 ) in q 1 implies ha q 1 C 1 (ˆq 1 (I, θ ) c 1 ) < q 1 C 1 (q 1 (I, θ ) c 1 ). On he oher hand, i follows from Lemma 16 ha q 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) q 1 J (d (I, θ ), q (I, θ ), I θ ). Thus, z ˆΨ ( ˆ (I, θ ) θ ) = q 1 C 1 (ˆq 1 (I, θ ) c 1 ) + q 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) < q 1 C 1 (q 1 (I, θ ) c 1 ) + q 1 J (d (I, θ ), q (I, θ ), I θ ) = z Ψ ( (I, θ ) θ ). (E.9) I + Since z ˆΨ (z θ ) z Ψ (z θ ) for each z and Ψ ( θ ) is concave in z, ˆ (I, θ ) > (I, θ ), i.e., j M ˆq j (I, θ ) ( Λ i i ˆd (I, θ )) = ˆ (I, θ ) > (I, θ ) = I + q j (I, θ ) ( Λ i d i (I, θ )). i ˆN j M i N 302

317 Since N ˆN and ˆq 1 (I, θ ) < q 1 (I, θ ), eiher (a) (b) ˆq j (I, θ ) > q j (I, θ ) for some j = 2, 3,, m. In case (a), wihou loss of generaliy, we assume ha ˆd i (I, θ ) < d i (I, θ ) for some i = 1, 2,, n, or ˆd 1 (I, θ ) < d 1 (I, θ ). By Lemma 16, we have d 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) d 1 J (d (I, θ ), q (I, θ ), I θ ). Thus, by (E.9), d 1 R 1 ( 1 ˆd (I, θ )) = z ˆΨ ( ˆ (I, θ ) θ ) + 1 Λ 1 d 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) < z Ψ ( (I, θ ) θ ) + 1 Λ 1 d 1 J (d (I, θ ), q (I, θ ), I θ ) = d 1 R 1 (d 1 (I, θ )), which conradics he sric concaviy of R 1 ( ). Hence, ˆd1 ha ˆq 1 (I, θ ) < q 1 (I, θ ). I follows from he same argumen ha i = 1, 2,, n, under he condiion ha ˆq 1 d i (I, θ ) for all i = 1, 2,, n. In case (b), wihou loss of generaliy, we assume ha ˆq 2 (I, θ ) d 1 (I, θ ) under he condiion ˆd i (I, θ ) d i (I, θ ) for all (I, θ ) < q 1 (I, θ ). Thus, under his condiion, ˆd i (I, θ ) = (I, θ ) > q 2 (I, θ ). By Lemma 16, we have q 2 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) q 2 J (d (I, θ ), q (I, θ ), I θ ). Thus, by (E.9), q 2 C 2 (ˆq 2 (I, θ ) c 2 ) = z ˆΨ ( ˆ (I, θ ) θ ) q 2 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) < z Ψ ( (I, θ ) θ ) q 2 J (d (I, θ ), q (I, θ ), I θ ) = q 2 C 2 (q 2 (I, θ ) c 2 ), which conradics he sric convexiy of C 2 ( c 2 ) in q 2. Hence, ˆq 2 ha ˆq 1 (I, θ ) > q 1 (I, θ ). I follows from he same argumen ha ˆq j j = 2, 3,, m, under he condiion ha ˆq 1 ha he iniial assumpion ˆq 1 (I, θ ) < q 1 (I, θ ) q 2 (I, θ ) under he condiion (I, θ ) q j (I, θ ) for all (I, θ ) < q 1 (I, θ ). Combining cases (a) and (b), i follows (I, θ ) is incorrec. Therefore, ˆq 1 (I, θ ) q 1 (I, θ ). The same argumen yields ha ˆq j (I, θ ) q j (I, θ ) for any j = 1, 2,, m. Hence, for each j M, Î q,j (θ ) = min{i : ˆq j (I, θ ) = 0} min{i : q j (I, θ ) = 0} = I q,j (θ ). If j M (I, θ ), ˆq j (I, θ ) q j (I, θ ) > 0. Thus, j ˆM (I, θ ), and M (I, θ ) ˆM (I, θ ) follows immediaely. In addiion, d i ˆx (I, θ ) = I + j M ˆq j (I, θ ) I + q j (I, θ ) = x (I, θ ). j M Finally, o complee he inducion, we show ha I ˆV (I θ ) I V (I θ ). Recall ha (I, θ ) for any i N. If d 1 R 1 ( ˆd 1 ˆd 1 (I, θ ) < d 1 (I, θ ), he sric concaviy of R 1 ( ) implies ha (I, θ )) > d 1 R 1 (d 1 (I, θ )). On he oher hand, Lemma 16 implies ha d 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) d 1 J (d (I, θ ), q (I, θ ), I θ ). Thus, z ˆΨ ( ˆ (I, θ ) θ ) = d 1 R 1 ( 1 ˆd (I, θ )) 1 Λ 1 d 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) > d 1 R 1 (d 1 (I, θ )) 1 Λ 1 d 1 J (d (I, θ ), q (I, θ ), I θ ) = z Ψ ( (I, θ ) θ ). 303 ˆd i (I, θ )

318 By Equaion (E.1), I ˆV (I θ ) = z ˆΨ ( ˆ (I, θ ) θ ) > z Ψ ( (I, θ ) θ ) = I V (I θ ). The same argumen implies ha, if here exiss an i N, such ha I ˆV (I θ ) > I V (I θ ). Recall ha ˆq j of C 1 ( c 1 ) in q 1 (I, θ ) q j implies ha q 1 C 1 (ˆq 1 (I, θ ) for any j M. If ˆq 1 ˆd i (I, θ ) < d i (I, θ ), we have (I, θ ) > q 1 (I, θ ), he sric convexiy (I, θ ) c 1 ) > q 1 C 1 (q 1 (I, θ ) c 1 ). On he oher hand, i follows from Lemma 16 ha q 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) q 1 J (d (I, θ ), q (I, θ ), I θ ). Thus, By Equaion (E.1), z ˆΨ ( ˆ (I, θ ) θ ) = q 1 C 1 (ˆq 1 (I, θ ) c 1 ) + q 1 Ĵ ( ˆd (I, θ ), ˆq (I, θ ), I θ ) > q 1 C 1 (q 1 (I, θ ) c 1 ) + q 1 J (d (I, θ ), q (I, θ ), I θ ) = z Ψ ( (I, θ ) θ ). I ˆV (I θ ) = z ˆΨ ( ˆ (I, θ ) θ ) > z Ψ ( (I, θ ) θ ) = I V (I θ ). The same argumen implies ha, if here exiss a j M, such ha ˆq j I ˆV (I θ ) > I V (I θ ). In he remaining case, ˆd i (I, θ ) = d i Since ˆn > n (or equivalenly, N ˆN ), ˆ (I, θ ) = I + j M (I, θ ) and ˆq j (I, θ ) > q j (I, θ ), we have (I, θ ) = q j (I, θ ) for any i N and j M. ˆq j (I, θ ) ( Λ i i ˆd (I, θ )) I + q j (I, θ ) ( Λ i d i (I, θ )) = (I, θ ). i ˆN j M i N Since z ˆΨ (z θ ) z Ψ (z θ ) for each z, i follows ha z ˆΨ ( ˆ (I, θ ) θ ) z Ψ ( (I, θ ) θ ) by he concaviy of Ψ ( θ ) in z. Thus, by Equaion (E.1), I ˆV (I θ ) = z ˆΨ ( ˆ (I, θ ) θ ) z Ψ ( (I, θ ) θ ) = I V (I θ ). This complees he inducion and, hus, he proof of Theorem Q.E.D. Proof of Theorem 6.4.8: The proof of Theorem follows from similar argumen o ha of Theorem 6.4.7, so we only skech i. We show all pars ogeher by backward inducion. More specifically, we prove ha if I 1 ˆV 1 (I 1 θ 1 ) I 1 V 1 (I 1 θ 1 ) for all I 1, hen we have (i) i N, (ii) ˆq j ˆd i (I, θ ) d i (I, θ ) for all (I, θ ) q j (I, θ ) for all j M, and (iii) I ˆV (I θ ) I V (I θ ) for all I. Since I0 ˆV0 (I 0 θ 0 ) = I0 V 0 (I 0 θ 0 ) = 0 for all I 0, he iniial condiion is saisfied. Since I 1 ˆV 1 (I 1 θ 1 ) I 1 V 1 (I 1 θ 1 ), z ˆΨ (z θ ) z Ψ (z θ ) for any z. Denoe M = {1, 2,, m} and ˆM = {1, 2,, ˆm}, where ˆm > m. ˆd 1 Firs, we show ha ˆd i (I, θ ) d i (I, θ ) for all i N. We assume, o he conrary, ha (I, θ ) < d 1 (I, θ ). Following he same argumen as ha in he proof of Theorem 6.4.7, we have z ˆΨ ( ˆ (I, θ ) θ ) > z Ψ ( (I, θ ) θ ), so ˆ (I, θ ) < (I, θ ), i.e., I + j ˆM ˆq j (I, θ ) ( Λ i i ˆd (I, θ )) = ˆ (I, θ ) < (I, θ ) = I + q j (I, θ ) ( Λ i d i (I, θ )). i N j M i N 304

319 Since M (b) ˆq j ˆM and ˆd 1 (I, θ ) < d 1 (I, θ ), eiher (a) ˆd i (I, θ ) > d i (I, θ ) for some i = 2, 3,, n, or (I, θ ) < q j (I, θ ) for some j = 1, 2,, m. We follow he same argumen as ha in he proof of Theorem o reach a conradicion in eiher case (a) or case (b). Thus, 1 ˆd (I, θ ) d 1 (I, θ ). i The same argumen applies o show ha ˆd (I, θ ) d i (I, θ ) for any i = 1, 2, n. Îd,i (θ ) I d,i (θ ) and N (I, θ ) ˆN (I, θ ) follows immediaely. ˆq 1 Nex, we show ha ˆq j (I, θ ) q j (I, θ ) for all j M. We assume, o he conrary, ha (I, θ ) > q 1 (I, θ ). Following he same argumen as ha in he proof of Theorem 6.4.7, we have z ˆΨ ( ˆ (I, θ ) θ ) > z Ψ ( (I, θ ) θ ), so ˆ (I, θ ) < (I, θ ), i.e., I + j ˆM Since M (b) ˆq j ˆq j (I, θ ) ( Λ i i ˆd (I, θ )) = ˆ (I, θ ) < (I, θ ) = I + q j (I, θ ) ( Λ i d i (I, θ )). i N j M i N ˆM and ˆq 1 (I, θ ) > q 1 (I, θ ), eiher (a) ˆd i (I, θ ) > d i (I, θ ) for some i = 1, 2,, n, or (I, θ ) < q j (I, θ ) for some j = 2, 3,, m. We follow he same argumen as ha in he proof of Theorem o reach a conradicion in eiher case (a) or case (b). Thus, ˆq 1 (I, θ ) q 1 (I, θ ). The same argumen applies o show ha ˆq j (I, θ ) q j I q,j (θ ) for any j M. ( ˆM (I, θ ) M) M (I, θ ) follows immediaely. (I, θ ) for any j = 1, 2, m. Thus, Îq,j (θ ) To complee he inducion, we show ha I ˆV (I θ ) I V (I θ ). Following he same argumen as ha in he proof of Theorem 6.4.7, we have ha if q j ˆd i (I, θ ) > d i (I, θ ) for some i N or ˆq j (I, θ ) < (I, θ ) for some j M, I ˆV (I θ ) = z ˆΨ ( ˆ (I, θ ) θ ) < z Ψ ( (I, θ ) θ ) = I V (I θ ). i In he remaining case, ˆd (I, θ ) = d i (I, θ ) and ˆq j (I, θ ) = q j (I, θ ) for any i N and j M. Since ˆm > m (or equivalenly, M ˆM), ˆ (I, θ ) = I + j ˆM ˆq j (I, θ ) ( Λ i i ˆd (I, θ )) I + q j (I, θ ) ( Λ i d i (I, θ )) = (I, θ ). i N j M i N Since z ˆΨ (z θ ) z Ψ (z θ ) for each z, i follows ha z ˆΨ ( ˆ (I, θ ) θ ) z Ψ ( (I, θ ) θ ) by he concaviy of Ψ ( θ ) in z. Thus, by Equaion (E.1), I ˆV (I θ ) = z ˆΨ ( ˆ (I, θ ) θ ) z Ψ ( (I, θ ) θ ) = I V (I θ ). This complees he inducion and, hus, he proof of Theorem Q.E.D. Proof of Theorem 6.5.1: Par (a). We consider an auxiliary game of N players Ĝ, in which he objecive funcion of player i is ˆΠ i (p y, θ) = (p i c i )(θ i + f(y ) b i p i + j i β ijp j ) C i (y i ), wih decision variable p i [p min i, p max i ]. We firs prove ha Ĝ has a unique equilibrium given by A 1 (a(y, θ) + κ). I is clear ha, given any (y, θ), he objecive funcion of player i in Ĝ, ˆΠ i (p y, θ), is concave in p i. Therefore, here exiss an equilibrium in Ĝ. p max i We now show ha he equilibrium in Ĝ, ˆp (y, θ), is an inerior vecor in he feasible se. Since is sufficienly large so ha i will no affec he equilibrium behavior, i remains o be shown ha ˆp i (y, θ) > pmin i for each i. Taking he firs order derivaive of he funcion ˆΠ i (p y, θ) wih respec o p i, we have: pi ˆΠi (p y, θ) = b i (p i c i ) + θ i + f(y ) b i p i + j i β ij p j. 305

320 Evaluaing he above derivaive a p i = c i, we have: pi ˆΠi (p y, θ) pi =c i = [θ i + f(y ) b i p i + j i β ij p j ] pi =c i. Following he assumpion ha λ i (p, y, θ i ) > 0 when p i = c i, we have pi ˆΠi (p y, θ) pi=c i = λ i (p, y, θ i ) pi=c i > 0 for any (p i, y, θ), and, hus, ˆp i (y, θ) > c i = p min i. Hence, ˆp (y, θ) saisfies he firs-order condiion: pi ˆΠi (ˆp (y, θ) y, θ) = b i (ˆp i (y, θ) c i ) + a i (Y, θ) b i ˆp i (y, θ) + j i β ij ˆp j (y, θ) = 0. Equivalenly, Aˆp (y, θ) = a(y, θ) + κ, where he (N N)-marix A and N-vecors a(y, θ) and κ are defined in Secion Since A saisfies he diagonal-dominance condiion, A 1 exiss. Hence, ˆp (y, θ) = A 1 (a(y, θ) + κ) (E.10) is he unique equilibrium in Ĝ. Noe ha ˆp (y, θ) coninues o be an equilibrium in he original second-sage price compeiion, as long as i generaes posiive demand for each firm. If he firms selec he price vecor ˆp (y, θ) in he second-sage price compeiion, by (6.15), he associaed demand for firm i is given by ˆλ i (y, θ) =(a i (Y, θ) (Aˆp (y, θ)) i + b i ˆp i (y, θ)) + =b i ( a i(y, θ) b i =b i ( a i(y, θ) b i =b i (ˆp i (y, θ) κ i b i ) + =b i (ˆp i (y, θ) c i ) > 0, (Aˆp (y, θ)) i b i + ˆp i (y, θ)) + (AA 1 (a(y, θ) + κ)) i b i + ˆp i (y, θ)) + where he hird equaliy follows from (E.10), and he las from κ i = b i c i. Hence, ˆp (y, θ) generaes posiive demand for each firm and, hus, forms an equilibrium in he second-sage price compeiion. I remains o be shown ha he original second-sage price compeiion does no have oher equilibria. We assume, o he conrary, ha here exiss anoher equilibrium price vecor p (y, θ), wih he associaed equilibrium demand vecor λ (y, θ). Since pi Π i ([p i, p i(y, θ)], y θ) pi =c i = λ i ([p i, p i(y, θ)], y, θ i ) pi =c i > 0, p i (y, θ) > c i for all i. If λ i (y, θ) > 0 for all i, p (y, θ) mus saisfy he firs-order condiion given by (E.10), so p (y, θ) = ˆp (y, θ). In he remaining case, λ i (y, θ) = 0 for some i. Wihou loss of generaliy, we assume ha λ 1(y, θ) = 0. Since λ 1 ([p min 1, p 1(y, θ)], y, θ) > 0, here exiss a price p 1 > p min 1 = c 1 such ha λ 1 ([ p 1, p 1(y, θ)], y, θ) > 0. Hence, Π 1 ([ p 1, p 1(y, θ)], y θ) = ( p 1 c 1 )λ 1 ([ p 1, p 1(y, θ)], y, θ) C 1 (y 1 ) > C 1 (y 1 ) = Π 1 ( p (y, θ), y θ), 306

321 which conradics he assumpion ha p (y, θ) is an equilibrium. Therefore, given any (y, θ), p (y, θ) = ˆp (y, θ) = A 1 (a(y, θ) + κ) is he unique equilibrium in he second-sage price compeiion. Thus, for any i, λ i (y, θ) = ˆλ i (y, θ) = b i(p i (y, θ) c i) > 0. Par (b). By par (a), p i (y, θ) = N l=1 (A 1 ) il (θ l + f(y ) + κ l ). Therefore, yj p i (y, θ) = Y p i (y, θ) = N (A 1 ) il f (Y ). By Lemma 2 in [24], every enry of A 1 is nonnegaive, so, ogeher wih he non-singulariy of A 1 and he sric monooniciy of f( ), N l=1 (A 1 ) il f (Y ) > 0. Thus, yj p i (y, θ) > 0 for any i and j, and p i (y, θ) is sricly increasing in Y for each i. Hence, λ i (y, θ) = b i(p i (y, θ) c i), which is sricly increasing in p i (y, θ), is sricly increasing in Y and y j for any i and j. Q.E.D. l=1 Proof of Theorem 6.5.2: Par (a). By (6.17), we have ha π i (y θ) = b i ( N l=1 (A 1 ) il (θ l + f(y ) + κ l ) c i ) 2 C i (y i ). By Theorem 6.5.1(a), p i (Y, θ) > c i for any Y. Le Y = 0, we have ψ i := p i (0, θ) c i = N l=1 (A 1 ) il (θ l + f 0 + κ l ) c i > 0. Therefore, Since ψ i N π i (y θ) =b i ( (A 1 ) il (f(y ) f 0 ) + ψ i ) 2 C i (y i ) l=1 N N =b i ( (A 1 ) il ) 2 (f(y ) f 0 ) 2 + 2b i ψ i ( (A 1 ) il )(f(y ) f 0 ) + b i (ψ i ) 2 C i (y i ). l=1 l=1 > 0, f( ) is concavely increasing in Y, and C i ( ) is convexly increasing in y i, π i (y θ) is joinly concave in y under Assumpion Since f( ) is bounded from above by M, we have ha lim yi + yi π i (y θ) < 0 for any θ. Hence, here exiss an upper bound y max <, such ha he equilibrium of he firs sage game is he same as ha of a game wih he same payoff funcions, bu he feasible se is consrained o [0, y max ] N. Since, for any given θ, π i (y θ) is concave in y i for any i, and [0, y max ] N is compac, he firs-sage game has an equilibrium y EF (θ) [0, ymax ] N. For any equilibrium y EF (θ), we denoe Y EF (θ) := N i=1 y EF,i (θ). Now, we show ha y EF (θ) is unique. Le F i(y θ) := b i ( N l=1 (A 1 ) il (θ l + f(y ) + κ l ) c i ) 2. Thus, π i (y θ) = F i (Y θ) C i (y i ), where Y := N i=1 y i. By our argumen above, F i ( θ) is concave and coninuously differeniable in Y for any i and θ. Assume, o he conrary, ha here exis wo equilibria ŷef (θ) and y EF (θ) (ŷ EF (θ) y EF (θ)). Wihou loss of generaliy, assume Ŷ EF (θ) Y EF (θ). Hence, here exiss an i such ha ŷef,i (θ) > y EF,i (θ). Wihou loss of generaliy, we ake i = 1. Lemma 16 yields ha y1 π 1 (y EF (θ) θ) y 1 π 1 (ŷ EF (θ) θ). Since C i( ) is sricly convex, C 1(ŷ EF,1 (θ)) > C 1(y EF,1 (θ)). Thus, Y F 1 (Ŷ EF (θ) θ) = y1 π 1 (ŷ EF (θ) θ)+c 1(ŷ EF,1(θ)) > y1 π 1 (y EF (θ) θ)+c 1(y EF,1(θ)) = Y F 1 (Y EF (θ) θ), which conradics he concaviy of F i ( θ). Hence, ŷef,1 (θ) y EF,1 (θ). The same argumen shows ha, for each i, ŷ EF,i (θ) y EF,i (θ). Hence, Ŷ EF (θ) = N i he equaliy holds only when ŷef,i (θ) = y EF,i (θ) for all i. Since ŷ EF,i (θ) N i yef,i (θ) = Y EF (θ), where Ŷ EF (θ) Y EF (θ) by assumpion, 307

322 Ŷ EF (θ) = Y EF (θ) and ŷ EF (θ) = y EF (θ), which conradics he iniial assumpion ha ŷ EF (θ) y EF (θ). Therefore, he equilibrium in he firs-sage effor level compeiion is unique. Par (b). Par (b) follows immediaely from par (a) and Theorem 6.5.1(a). Q.E.D. Proof of Theorem 6.5.3: Par (a). Since every enry of A 1 is nonnegaive and A 1 is non-singular, θj Y F i (Y θ) = 2b i (A 1 ) ij ( N l=1 (A 1 ) il )f (Y ) 0. Hence F i (Y θ) is supermodular in (Y, θ j ) for any 1 i, j N. Assume ha ˆθ > θ and Y EF (θ) > Y EF (ˆθ). Hence, he concaviy and supermodulariy of F i ( θ) implies ha, for each 1 i N, Y F i (Y EF (θ) θ) Y F i (Y EF (ˆθ) ˆθ). (E.11) If y EF,1 (θ) > y EF,1 (ˆθ), he sric convexiy of C 1 ( ) yields ha C 1(y EF,1 (θ)) > C 1(y EF,1 (ˆθ)). On he oher hand, Lemma 16 implies ha y1 π 1 (y EF (θ) θ) y 1 π 1 (y EF (ˆθ) ˆθ). Thus, Y F 1 (Y EF (θ) θ) = y1 π 1 (y EF (θ) θ)+c 1(y EF,1(θ)) > y1 π 1 (y EF (ˆθ) ˆθ)+C 1(y EF,1(ˆθ)) = Y F 1 (Y EF (ˆθ) ˆθ), which conradics (E.11). Thus, under he condiion Y EF (θ) > Y EF (ˆθ), y EF,1 (θ) y EF,1 (ˆθ). Similar argumen implies ha, under he condiion Y EF (θ) > Y EF (ˆθ), y EF,i (θ) y EF,i (ˆθ) for any i. Therefore, if Y EF (θ) > Y EF (ˆθ), Y EF (ˆθ) = N i Thus, Y EF (θ) is increasing in θ i for any i. y EF,i (ˆθ) N i yef,i (θ) = Y EF (θ), which forms a conradicion. Par (b). Since every enry of A 1 is nonnegaive, by par (a) and Theorem 6.5.1, p i (Y EF (θ), θ) = N l=1 (A 1 ) il (θ l +f(yef (θ))+κ l) is increasing in θ j for any i and j. Thus, λ EF,i (θ) = b i(p i (Y EF (θ), θ) c i) is increasing in p i (Y EF (θ), θ) and, hence, θ j for any i and j. Q.E.D. Proof of Theorem 6.5.4: Par (a). We consider an auxiliary game of N players G, in which he objecive funcion of player i is Π i (p, y θ) = (p i c i )(θ i + f(y ) b i p i + j i β ijp j ) C i (y i ), wih decision variable p i [p min i, p max i ] and y i 0. We firs prove ha G has a unique equilibrium characerized by he unique soluion of he sysem of equaions (6.18) and (6.19). Noe ha (6.18) and (6.19) characerize he firs-order condiion: = 0, if y i > 0, for any i, pi Πi = 0 and yi Πi 0, oherwise. We firs show ha (6.18) and (6.19) have a unique soluion on (p min 1, p max 1 ) (p min 2, p max 2 ) (p min N, pmax N ) [0, + )N 1. Le G i (Y θ) := 1 2b N F i l=1 (A 1 ) i (Y θ) = il 2 N ( N l=1 (A 1 ) il l=1 (A 1 ) il (θ l +f(y )+ κ l ) c i ) 2 for any i. By he proof of Theorem 6.5.2, G i (Y θ) is concave and coninuously differeniable in Y for each i. Plugging (6.18) ino (6.19), he lef-hand-side of (6.19) becomes: Y G i (Y SC (θ) θ) C i (y SC,i (θ)), and (6.19) is reduced o: = 0, if y Y G i (YSC(θ) θ) C i(y SC,i(θ)) SC,i (θ) > 0, for all i = 1, 2,, N, 0, oherwise, (E.12) 308

323 where YSC (θ) = N i=1 y SC,i (θ). We define an auxiliary sysem of equaions on (Y SC (θ), y SC,1 (θ), y SC,2 (θ),, y SC,N (θ)): = 0, if y Y G i (Y SC (θ) θ) C i(y SC,i (θ) > 0, SC,i (θ)) for all i = 1, 2,, N. 0, oherwise, (E.13) Noe ha he difference beween (E.12) and (E.13) is ha he ideniy Y SC (θ) = N i=1 y SC,i (θ) [Y SC(θ) = N i=1 y SC,i(θ)] always holds [may no hold] in (E.12) [(E.13)]. Hence, for any soluion of (E.13) (Y SC (θ), y SC,1 (θ), y SC,2 (θ),, y SC,N (θ)), if i also saisfies he ideniy Y SC (θ) = N i=1 y SC,i(θ), i is also a soluion o (E.12). Since C i ( ) is sricly convex in y i for any i, here exiss a unique vecor y SC (θ) ha saisfies (E.13) for any fixed Y SC (θ). Thus, we use A : R + R N o denoe he mapping from Y SC (θ) o y SC (θ), such ha y SC (θ) = A(Y SC (θ)) saisfies (E.13) for any given Y SC (θ). Moreover, le B : R + R denoe he following funcion: B(Y SC (θ)) = N i=1 y SC,i(θ), where y SC (θ) = A(Y SC (θ)). Now, we show ha B( ) has a unique fixed poin on R +. I follows from he concaviy of G i ( θ) and he sric convexiy of C i ( ) ha (A(Y SC (θ))) i is coninuously decreasing in Y SC (θ) for any i. Hence, B(Y SC (θ)) is coninuously decreasing in Y SC (θ). By (E.13), (A(0)) i 0 for each i. Thus, B(0) 0. Le C(Y ) := B(Y ) Y. Thus, C( ) is sricly decreasing on R + wih C(0) 0 and lim Y + C(Y ) lim Y + (B(0) Y ) =. Therefore, C( ) has a unique roo on R +. Hence, B( ) has a unique fixed poin on R + and, hus, (E.12) has a unique soluion ysc (θ). As shown by he proof of Theorem 6.5.1, given ysc (θ), here exiss a unique p SC (θ) ha saisfies (6.18), and p SC (θ) (pmin 1, p max 1 ) (p min 2, p max 2 ) (p min N, pmax N (p min 1, p max 1 ) (p min 2, p max 2 ) (p min ). Therefore, (6.18) and (6.19) has a unique soluion (p SC (θ), y SC (θ)) on N, pmax N ) [0, + )N. We now show ha he equilibrium in G, ( p, ỹ ), if exiss, mus have an inerior price vecor p (p min 1, p max 1 ) (p min 2, p max 2 ) (p min shown ha p i > pmin i N, pmax N ). Since pmax is sufficienly large for any i, i remains o be = c i for all i. Assume, o he conrary, ha p i = pmin i i = c i for some i. Wihou loss of generaliy, we ake i = 1. Since λ 1 ([p min 1, p 1], ỹ, θ) > 0, here exiss a price p 1 > p min 1 = c 1 such ha λ 1 ([ p 1, p 1], ỹ, θ) > 0. Hence, Π 1 ([ p 1, p 1], ỹ, θ) = ( p 1 c 1 )λ 1 ([ p 1, p 1], ỹ, θ) C 1 (ỹ 1) > C 1 (ỹ 1) = Π 1 ( p, ỹ, θ), which conradics he assumpion ha ( p, ỹ ) is an equilibrium. Therefore, he equilibrium in G, if exiss, mus have an inerior price vecor, and by he KKT necessary condiion, mus saisfy he firsorder condiion characerized by he sysem of equaions (6.18) and (6.19). I remains o be shown ha he unique soluion o (6.18) and (6.19), (p SC (θ), Y SC (θ)), is an equilibrium in G. (p SC,i (θ), y SC,i (θ)) maximizes I suffices o prove ha, for any i, given oher firms decisions (p SC, i (θ), y SC, i (θ)), Π i (p i, y i p SC, i(θ), y SC, i(θ), θ) := (p i c i )(θ i + f(y i + j i y SC,j(θ)) b i p i + j i β ij p SC,j(θ)) C i (y i ). 309

324 Following he same argumen as ha in he characerizaion of (p SC (θ), y SC (θ)), we have (p SC,i (θ), y SC,i (θ)) is he unique vecor ha saisfies he firs-order condiion: pi Πi (p i, y i p SC, i(θ), ysc, i(θ), θ) = 0 (E.14) = 0, if y yi Πi (p i, y i p SC, i(θ), ysc, i(θ), SC,i (θ) > 0, θ) (E.15) 0, oherwise. Since Π i (p i, y i p SC, i (θ), y SC, i (θ), θ) is a coninuously differeniable funcion on a compac domain [p min i i, p max ] [0, y max ], where y max is defined in he proof of Theorem 6.5.2, i has a maximizer characerized by he firs-order condiion (E.14) and (E.15) for any given (p SC, i (θ), y SC, i (θ)). Therefore, given (p SC, i (θ), y SC, i (θ)), he unique soluion o (E.14) and (E.15), (p SC,i (θ), y SC,i (θ)), is he unique maximizer of Π i (p i, y i p SC, i (θ), y SC, i (θ), θ) for any i. Therefore, he unique soluion o (6.18) and (6.19), (p SC (θ), y SC (θ)), is he unique equilibrium in G. Noe ha (p SC (θ), y SC (θ)) coninues o be an equilibrium in he original simulaneous compeiion, as long as i generaes posiive demand for each firm. If he firms selec he price vecor p SC (θ) and he effor vecor ysc (θ) in he simulaneous compeiion, by (6.15) and (6.18), he associaed demand for firm i is given by λ i (θ) = (a i(y SC (θ), θ) (Ap SC (θ)) i + b i p SC,i (θ))+ = b i (p SC,i (θ) c i) > 0, where he inequaliy follows from p SC,i (θ) > c i for any i. Hence, (p SC (θ), y SC (θ)) generaes posiive demand for each firm and, hus, forms an equilibrium in he simulaneous compeiion. I remains o be shown ha he original simulaneous compeiion does no have oher equilibria. We assume, o he conrary, ha here exiss anoher equilibrium (p (θ), y (θ)), wih he associaed equilibrium demand vecor λ (θ). Since pi Π i ([p i, p i (θ)], y (θ) θ) pi =c i = λ i ([p i, p i (θ)], y (θ), θ i ) pi =c i > 0, p i > c i for all i. If λ i (θ) > 0 for all i, (p (θ), y (θ)) mus saisfy he firs-order condiion (6.18) and (6.19), i.e., (p (θ), y (θ)) = (p SC (θ), y SC (θ)). In he remaining case, λ i (θ) = 0 for some i. Wihou loss of generaliy, we assume ha λ 1(θ) = 0. Since λ 1 ([p min 1, p 1 (θ)], y (θ), θ) > 0, here exiss a price p 1 > p min 1 = c 1 such ha λ 1 ([p 1, p 1 (θ)], y (θ), θ) > 0. Hence, Π 1 ([p 1, p 1 (θ)], y (θ) θ) = (p 1 c 1 )λ 1 ([p 1, p 1 (θ)], y (θ), θ) C 1 (y 1 (θ)) > C 1(y 1 (θ)) = Π 1(p (θ), y (θ) θ), which conradics ha (p (θ), y (θ)) is an equilibrium in he simulaneous compeiion. Therefore, given any θ, (p SC (θ), y SC (θ)), which is he unique soluion of (6.18) and (6.19), is he unique equilibrium in he simulaneous compeiion. Thus, for any i, λ SC,i (θ) = b i(p SC,i (θ) c i) > 0. Par (b). By (E.12), y SC (θ) is he unique equilibrium of an N-player game, in which he ih player has he payoff funcion ˆπ i (y θ) := G i (Y θ) C i (y i ) and feasible se R +, where Y = N i=1 y i. By he proof of Theorem 6.5.2(a) and Theorem 6.5.3(a), F i (Y θ), and hence G i (Y θ) = 1 2b i N l=1 (A 1 ) il F i (Y θ), are coninuously differeniable and concave in Y and supermodular in (y i, θ j ) for any (i, j). Assume ˆθ > θ and Y SC (θ) > Y SC (ˆθ). Hence, he concaviy and supermodulariy of G i ( θ) imply ha, for each 1 i N, Y G i (Y SC(θ) θ) Y G i (Y SC(ˆθ) ˆθ). (E.16) 310

325 If y SC,1 (θ) > y SC,1 (ˆθ), he sric convexiy of C 1 ( ) yields ha C 1(y SC,1 (θ)) > C 1(y SC,1 (ˆθ)). On he oher hand, Lemma 16 implies ha y1 ˆπ 1 (y SC (θ) θ) y 1 ˆπ 1 (y SC (ˆθ) ˆθ). Thus, Y G 1 (Y SC(θ) θ) = y1 ˆπ 1 (y SC(θ) θ)+c 1(y SC,1(θ)) > y1 ˆπ 1 (y SC(ˆθ) ˆθ)+C 1(y SC,1(ˆθ)) = Y G 1 (Y SC(ˆθ) ˆθ), which conradics (E.16). Thus, under he condiion ha Y SC (θ) > Y SC (ˆθ), y SC,1 (θ) y SC,1 (ˆθ). Similar argumen implies ha, under he condiion ha Y SC (θ) > Y SC (ˆθ), y SC,i (θ) y SC,i (ˆθ) for any i. Therefore, if Y SC (θ) > Y SC (ˆθ), Y SC (ˆθ) = N i y SC,i (ˆθ) N i ysc,i (θ) = Y SC (θ), which forms a conradicion. Thus, Y SC (θ) is increasing in θ i for any i. Since every enry of A 1 is nonnegaive and A 1 is non-singular, by (6.18), p SC,i (θ) = N l=1 (A 1 ) il (θ l + f(y SC (θ)) + κ l) is increasing in θ j for any i and j. Thus, λ SC,i (θ) = b i(p SC,i (θ) c i) is increasing in p SC,i (θ) and, hence, θ j for any i and j. Q.E.D. Proof of Theorem 6.5.5: We prove par (b) firs, and pars (a) and (c) second. Par (b). As shown in he proofs of Theorem and Theorem 6.5.4, yef (θ) is he equilibrium of an N-player game wih he concave objecive funcion π i (y θ) = F i (Y θ) C i (y i ) and feasible se R + for player i, and y CS (θ) is he equilibrium of an N-player game wih he concave objecive funcion ˆπ i(y θ) = G i (Y θ) C i (y i ) and feasible se R + for player i. Noe ha Y F i (Y θ) = 2b i (p i (Y, θ) c i) Y p i (Y, θ) > 0, where he inequaliy follows from Theorem Recall ha F i (Y θ) = 2b i n l=1 (A 1 ) il G i (Y θ). By Lemma 2 in [24], 2b i n l=1 (A 1 ) il 2b i (A 1 ) ii 1. Thus, Y F i (Y θ) Y G i (Y θ) 0 for each i and θ. We assume, o he conrary, ha YSC (θ) > Y EF (θ). Hence, for each i, Y F i (Y EF (θ) θ) Y G i (Y SC(θ) θ). (E.17) If ysc,1 (θ) > y EF,1 (θ), he sric convexiy of C 1( ) implies ha C 1(y SC,1 (θ)) > C 1(yEF,1 (θ)). On he oher hand, Lemma 16 yields ha y1 ˆπ 1 (ysc (θ) θ) y 1 π 1 (yef (θ) θ). Thus, Y G 1 (Y SC(θ) θ) = y1 ˆπ 1 (y SC(θ) θ) + C 1(y SC,i(θ)) > y1 π 1 (y EF (θ) θ) + C 1(y EF,i(θ)) = Y F 1 (Y EF (θ) θ), which conradics (E.17). Thus, under he condiion ha Y SC (θ) > Y EF (θ), y SC,1 (θ) y EF,1 (θ). Similar argumen implies ha, under he condiion ha YSC (θ) > Y EF (θ), y SC,i (θ) y EF,i (θ) for any i. Therefore, if Y SC (θ) > Y EF (θ), Y EF (θ) = N i conradicion. Thus, YSC (θ) Y EF (θ) for any θ. y EF,i (θ) N i ysc,i (θ) = Y SC (θ), which forms a Par (a). Since YSC (θ) Y EF (θ) for any θ, by Theorems and and ha every enry of A 1 is nonnegaive, N N p EF,i(yEF (θ), θ) = ( (A 1 ) il )(θ l + f(yef (θ)) + κ l ) ( (A 1 ) il )(θ l + f(ysc(θ)) + κ l ) = p SC,i(θ), for any i and θ. l=1 Par (c). Since p i (Y EF (θ), θ) p SC,i (θ), i follow immediaely from Theorems and ha l=1 λ i (Y EF (θ), θ) = b i (p i (Y EF (θ), θ) c i ) b i (p SC,i(θ) c i ) = λ SC,i(θ), for any i and θ. Q.E.D. 311

326 E.2 Discussions on Sopping Condiion (ii) of he Ieraive Procedure Lemma 15 is silen abou how yi (γ) changes wih γ for p + 1 i p + q in he convex program (6.1). In his secion, we firs presen in deail how our comparaive saics mehod sops in Sep (d) wihou leading o a conradicion for yi (γ) (p + 1 i p + q). We hen give an example o illusrae ha, in he convex program (6.1), yi (γ) (p + 1 i p + q) may no be monoone in γ. E.2.1 Comparaive Saics Analysis of yi (γ) (p + 1 i p + q) We consider wo (hypoheical and incorrec) scenarios: (a) yi (γ) is increasing in γ for all γ Γ and some p + 1 i p + q; and (b) yi (γ) is decreasing in γ for all γ Γ and some p + 1 i p + q. For scenario (a), we assume, o he conrary, ha yi (ˆγ) < y i (γ) for some ˆγ > γ. By Lemma 16, yi F (y (γ) γ) yi F (y (ˆγ) ˆγ), i.e., yi g i (y i (γ) γ)+λ i y0 h(y 0(γ) γ) yi g i (y i (ˆγ) ˆγ)+λ i y0 h(y 0(ˆγ) ˆγ). Since g i ( ) is sricly concave in y i and submodular in (y i, γ), i may be possible ha (i) yi g i (y i (γ) γ) yi g i (yi (ˆγ) ˆγ) or (ii) y i g i (yi (γ) γ) < y i g i (yi (ˆγ) ˆγ). In case (i), he argumen sops because we canno obain any monoone relaionship beween y0 h(y 0(γ) γ) and y0 h(y 0(ˆγ) ˆγ). Hence, no conradicion can be reached for his scenario. For scenario (b), we assume, o he conrary, ha yi (ˆγ) > y i (γ) for some ˆγ > γ. By Lemma 16, yi F (y (γ) γ) yi F (y (ˆγ) ˆγ), i.e., yi g i (y i (γ) γ)+λ i y0 h(y 0(γ) γ) yi g i (y i (ˆγ) ˆγ)+λ i y0 h(y 0(ˆγ) ˆγ). Since g i ( ) is submodular in (y i, γ) and sricly concave in y i, yi g i (yi (γ) γ) > y i g i (yi (ˆγ) ˆγ). Thus, we have ha y0 h(y 0(γ) γ) < y0 h(y 0(ˆγ) ˆγ). Since h( ) is supermodular in (y 0, γ), we canno obain any monoone relaionship beween y 0(γ) and y 0(ˆγ). Hence, he argumen sops and no conradicion can be reached for his scenario. Since he ieraive procedure is sopped wihou reaching a conradicion, we suspec ha, in he convex program (6.1), yi (γ) (p + 1 i p + q) may no be monoone in γ, and consruc he following example. Example E.2.1 In he convex program (6.1), le p = q = 1, λ 1 = λ 2 = 1, Γ = R, and Y 1 = Y 2 = R. Le (y f 1 (y 1 ) = (y 1 ) 2 2 ) 2, if γ 0, (y 0 γ) 2, if γ 0, ; g 2 (y 2 γ) = and h(y 0 γ) = (y 2 + γ) 2, oherwise; (y 0 ) 2, oherwise. Clearly, f 1 ( ), g 2 ( ), and h( ) saisfy he condiions of (6.1). I s easy o obain ha ( γ (y1(γ), y2(γ)) 3 =, γ 3 ), if γ 0, ( γ 3, 2γ 3 ), oherwise. Therefore, in his example, y 2(γ) is sricly increasing in γ for γ 0, and sricly decreasing in γ for γ > 0. Example E.2.1 implies ha, in he convex opimizaion problem (6.1), yi (γ) (p + 1 i p + q) may no be monoone in γ for generally specified {f i ( ), g i ( ), h( )} 1 i p+q. 312