Substtuton Effects n Supply Chans wth Asymmetrc Informaton Dstrbuton and Upstream Competton Jochen Schlapp, Mortz Fleschmann Department of Busness, Unversty of Mannhem, 68163 Mannhem, Germany, jschlapp@bwl.un-mannhem.de, mfleschmann@bwl.un-mannhem.de 15 February 2013 Inventory management n markets wth substtutng customers s extremely challengng, not only for a downstream wholesaler, but also for upstream manufacturers. Motvated by the structures n the agrochemcal market, we analyze the optmal producton and stockng quanttes of a manufacturer and a wholesaler, respectvely, n a two-stage supply chan wth upstream competton and vertcal nformaton asymmetres. We characterze a monopolstc wholesaler s optmal stockng quanttes and show that these quanttes are not necessarly monotonc, nether n the avalable producton quanttes nor n the customers substtuton rates. We further derve the optmal producton quanttes of a monopolstc and a compettve manufacturer when they are ncompletely nformed about the wholesaler s stockng quanttes. We fnd that the ntroducton of competton may lead to decreasng producton quanttes for some products. Furthermore, a product s end-of-season nventores at the manufacturer whch arse due to nformaton asymmetres may decrease even when ntal producton levels ncrease. Key words: customer substtuton; supply chan; asymmetrc nformaton; competton; nventory management 1 Electronc copy avalable at: http://ssrn.com/abstract=2218297
1 Introducton In recent years, a dverse body of research has focused on how frms should react to customers substtuton behavor. For frms that drectly serve customers, nvestgatons range from strategc assortment plannng (Kök and Fsher, 2007; Honhon et al., 2010) over promoton strateges (Walters, 1991) to optmal stockng decsons (Netessne and Rud, 2003; Jang et al., 2011). In a supply chan settng, only the downstream stage experences the mmedate effects of customer substtuton; however, substtuton effects also dffuse across the entre supply chan. Ths paper therefore nvestgates how dfferent stages of a supply chan are affected by customer substtuton. In partcular, we examne the optmal producton and stockng decsons of dfferent supply chan members under upstream competton and vertcal nformaton asymmetres. We are nterested n stuatons when competton and substtuton effects arse smultaneously wthn the supply chan. Whle competton occurs due to the non-cooperatve behavor of ndependent frms, substtuton emerges from the compettve structures wthn the set of avalable products. Note that competton and substtuton are nether nclusve nor exclusve concepts: Competton wthout substtuton arses f multple ndependent frms offer an dentcal product (n a supply chan settng, e.g., Cachon, 2001; Adda and DeMguel, 2011), whle substtuton wthout competton occurs f a monopolstc frm offers non-dentcal, yet smlar products that serve a common customer base. In ths paper, we concentrate on markets where substtuton and competton effects exst smultaneously. Intally, our work s motvated by the agrochemcal market. Agrochemcal manufacturers sell ther products through locally monopolstc wholesalers to ther customers, mostly farmers or farmer unons. Substtuton n ths market arses from customers focus on actve ngredents, resultng n low brand loyalty. In consequence, stock-outs at the wholesaler lead to hgh substtuton rates among products. Ths effect s even further enhanced by the nherent fnteness of the sellng season for agrochemcals and the non-durablty of some chemcal components. Informaton asymmetres n ths market stem from the wholesaler s barganng power and substantal producton lead-tmes at the manufacturers whch can amount to two years (Shah, 2004). Whle producton needs to be ntated well n advance of the desred sellng season, the wholesaler cannot be forced to commt to order quanttes at ths early stage. Fnal orders are typcally released close to the sellng season when (weather-dependent) demand can be predcted suffcently well. In essence, producton and orderng decsons are based on potentally dfferent nformaton sets, and thus, vertcal nformaton asymmetres arse. To analyze the manufacturer s (wholesaler s) optmal producton (stockng) quanttes, we consder a supply chan n whch potentally multple manufacturers sell partally substtutable products for a sngle season through a monopolstc wholesaler. We focus on a sngle perod settng because () t yelds a very good approxmaton of the agrochemcal market where the sellng season s fnte and some chemcal components cannot be stored untl the next season; and () t s a necessary frst step n the analyss of substtuton effects wthn supply chans whch s n lne 2 Electronc copy avalable at: http://ssrn.com/abstract=2218297
wth the exstng lterature. To capture the effects of upstream competton, we compare two dstnct supply chan scenaros: a horzontally ntegrated (hereafter non-compettve ) supply chan wth a sngle manufacturer producng all avalable products; and a horzontally compettve (hereafter compettve ) supply chan wth multple manufacturers, each producng only one product. Whle nspred by the agrochemcal market, our framework generally suts ndustres n whch (1) products are partal substtutes, (2) products and market structures exhbt typcal newsvendor characterstcs, and (3) customers are served by a monopolstc wholesaler. Our work contrbutes to the lterature on () vertcal nformaton asymmetres n supply chans; and, most mportantly, () optmal stockng levels under customer substtuton. Informaton sharng wthn supply chans has been a prevalent research area n the last decades (L, 2002; Özer and We, 2006). Apart from the ssue of truthful nformaton sharng, lterature also nvestgates the effects that asymmetrc nformaton exert on operatonal problems. In the presence of short capacty at the manufacturer, Cachon and Larvere (1999) show that wholesalers explot ther nformatonal advantage by manpulatng the manufacturer s allocaton mechansm. Under asymmetrc nformaton, Corbett (2001) depcts that the ntroducton of consgnment stocks at the wholesaler leads to reduced cycle stocks at the expense of ncreased safety stocks. If wholesalers are allowed to share nventores, Yan and Zhao (2011) conclude that wholesalers share demand nformaton wth each other, but not wth the manufacturer. We depart from ths stream by ncorporatng an asymmetrc nformaton structure nto a supply chan prone to customer substtuton. There has been an extensve lterature on the repercussons of customer substtuton on the wholesaler s optmal stockng quanttes. As common buldng block, the sngle-stage newsvendor nventory (competton) model wth stock-out-based substtuton as poneered by McGllvray and Slver (1978), Parlar (1988), Lppman and McCardle (1997), Bassok et al. (1999), Smth and Agrawal (2000), and Netessne and Rud (2003) has evolved. In a semnal paper, Netessne and Rud (2003) extend the precedng work by characterzng the structure of the optmal stockng levels for an arbtrary number of products under both centralzaton and competton. Based on these results, recent work has nvestgated varous compettve envronments under customer substtuton. Mshra and Raghunathan (2004), Kraselburd et al. (2004), and Km (2008) explore the consequences of ntroducng Vendor Managed Inventory for the wholesaler s stockng levels and advertsement efforts. Nagarajan and Rajagopalan (2008) embed the substtuton framework nto a mult-perod settng and Jang et al. (2011) provde a robust optmzaton approach that determnes stockng levels by mnmzng absolute regret. Recently, Vulcano et al. (2012) develop an effcent procedure to emprcally estmate requred substtuton parameters. As common n the newsvendor framework, exstng models assume that the wholesaler s unconstraned n hs stockng decson,.e., any arbtrary amount of products can be ordered. Beng true n a sngle-stage settng, ths assumpton fals to hold n a supply chan settng. Here, a manufacturer s producton or capacty decson consttutes a natural upper bound on 3
the wholesaler s decson space (compare ths to the lterature on capacty choce, e.g., Cachon and Larvere, 1999; Montez, 2007). By explctly ntegratng these dependences nto our model we make a two-fold contrbuton: frst, we nvestgate a constraned wholesaler s behavor; second, to the best of our knowledge, we are the frst to examne how customer substtuton affects upstream stages. To be specfc, the contrbutons of ths paper are as follows: (1) We derve the optmal stockng quanttes of a constraned wholesaler and characterze the non-monotonc effects that a change n a manufacturer s producton quantty exerts on optmal stockng levels. (2) We formally analyze the nfluence of changng substtuton rates on the wholesaler s stockng quanttes. In contrast to an ntutve conjecture of Netessne and Rud (2003), we show that stockng levels for certan products may ncrease even f customer substtuton away from these products ncreases. (3) We characterze the optmal producton quanttes of an ncompletely nformed manufacturer both under centralzaton and competton by applyng a Bayesan (Nash-)Stackelberg game. (4) We explctly compare optmal producton levels under competton and centralzaton and fnd that competton may lead to reduced producton. (5) We show that for some products, end-of-season nventores at the manufacturer may decrease even when ntal producton levels are ncreased under competton. The remander of ths paper s organzed as follows. The structure of the supply chan under consderaton and the dstrbuton of nformaton are descrbed n 2. Furthermore, we elaborate on the propertes of the resultng supply chan game. In 3, we present our model of a constraned wholesaler and derve the optmal stockng quanttes. We proceed by analyzng the effects of changng substtuton rates on these optmal stockng levels. The manufacturer s producton quanttes are the focus of 4. We frst characterze the equlbrum producton quanttes of a manufacturer under competton, before nvestgatng the structure of a monopolstc manufacturer s optmal producton quanttes. We then compare producton levels under centralzaton and competton, and examne the manufacturer s end-of-season nventores under both scenaros. Secton 5 provdes a dscusson of our results and concludng remarks. 2 Supply Chan Structure and Informaton Dstrbuton We consder a two-stage supply chan wth possbly multple manufacturers (she) and a sngle wholesaler (he) sellng N 2 partally substtutable products for one perod. Whle competton among manufacturers at the upstream stage may arse, we restrct attenton to a monopolstc downstream wholesaler. In the non-compettve stuaton, a sngle manufacturer provdes all N products (blateral monopoly), whereas n the compettve scenaro, N ndependent manufacturers each produce a dfferent product (unlateral monopoly wth upstream competton). Fgure 1 llustrates both supply chan structures. In the agrochemcal market, a centralzed manufacturer occurs whenever a famly of patents that allows for the provson of dfferent, yet substtutable products s exclusvely held by a sngle frm. In contrast, upstream competton s ntroduced f dfferent manufacturers hold dfferent patents for smlar, but not dentcal 4
Fgure 1: Blateral monopoly (left) and unlateral monopoly wth upstream competton (rght). Manufacturer Wholesaler Manufacturers Wholesaler Product 1 Product 1 Product 1 Product 1 Product Product Customers Product Product Customers Product N Product N Product N Product N products or f patents run out. We assume that nformaton s asymmetrcally dstrbuted between manufacturers and the wholesaler. As mentoned earler, ths vertcal nformaton asymmetry between supply chan stages arses naturally n the agrochemcal market due to the wholesaler s barganng power and manufacturers lead-tmes. Besdes such natural causes for dfferng nformaton sets, lterature has also dentfed many other reasons ncludng technologcal ssues (Lee and Whang, 2000) and the fear of nformaton leakage (Anand and Goyal, 2009). Our modelng approach allows for the ncluson of any such cause for nformaton asymmetres wthn the supply chan. To be precse, n lne wth the lterature on vertcal nformaton asymmetres, e.g., L (2002), Özer and We (2006) and Yan and Zhao (2011), we assume that manufacturers are ncompletely nformed about the wholesaler s optmal stockng quanttes. In contrast, upstream nformaton are common knowledge across manufacturers,.e., no horzontal nformaton asymmetry arses, and producton quanttes are commonly verfable. Ths assumpton s reasonable n the agrochemcal market snce manufacturers produce substtutable, hence comparable products and thus, they are able to credbly estmate ther compettors cost structures. Furthermore, to analyze the change n producton quanttes under compettve effects, we need to ensure that decsons are based on dentcal nformaton sets under both supply chan structures. Followng the argument of Harsany (1968) and Myerson (2004), we assume that manufacturers hold a common pror belef about the wholesaler s optmal stockng levels. Hence, manufacturers belefs are consstent. Ths pror belef represents the manufacturers percepton about the collecton of nformaton that are not common knowledge. In summary, supply chan structure and nformaton dstrbuton mply a Bayesan (Nash-)Stackelberg Game as frst ntroduced by Gal-Or (1987). The for our work relevant case of multple-leader Stackelberg games has frst been studed by Sheral (1984) and recently by DeMguel and Xu (2009), but only for complete, non-bayesan nformaton structures. The sequence of events s as follows: In the frst stage, manufacturers maxmze expected profts and determne ther optmal producton quanttes based on ther belefs about the wholesaler s subsequent stockng quanttes. In the second stage, before the start of the sellng 5
Fgure 2: Sequence of Events. Manufacturers determne producton quanttes Wholesaler learns producton quanttes; place orders wth manufacturers Orders are shpped; sellng season starts End of season; all profts and excess nventores determned Orderng Game Tme Supply Game season, the wholesaler learns these producton quanttes and, gven hs prvate nformaton, derves hs optmal stockng levels by maxmzng expected profts. Afterwards, orders are submtted and shpped before the sellng season starts. Throughout the sellng season the wholesaler experences customer demand and realzes profts. We refer to the subgame wth gven producton quanttes as the Orderng Game, whle the entre game s denoted as the Supply Game. Hence, producton quanttes are exogenously gven n the Orderng Game, whereas they are decson varables n the Supply Game. Fgure 2 summarzes the chronology. We assume that stochastc customer demand appears exclusvely at the wholesaler and no manufacturer can pursue a drect sellng strategy. Prces are exogenously gven by the market and nether player can negotate on the prce to pay. Furthermore, we restrct attenton to pure-strategy equlbra. 3 The Orderng Game Focusng on the Stackelberg follower n ths secton, we derve the wholesaler s optmal stockng levels gven the manufacturers producton quanttes and characterze ts senstvty wth respect to () changes n a manufacturer s producton quantty, and () substtuton effects. 3.1 Optmal Stockng Quanttes For each product {1,..., N}, the wholesaler pays a unt wholesale prce w to the manufacturer and sells the product at a unt retal prce r, satsfyng r > w > 0. Addtonally, the wholesaler ncurs a unt holdng or dsposal cost of h 0 for each unsold tem. Total demand occurrence follows the standard model of stock-out-based substtuton processes as defned by Netessne and Rud (2003), Kök et al. (2009), and Jang et al. (2011). Customers arrve at the wholesaler wth an ntal product preference. Thus, the wholesaler faces random ntal demand for product gven by D, whch s assumed to have a contnuous demand dstrbuton wth postve support. Second choce (substtuton) demand stems from customers whose ntally preferred product s out of stock. If a stock-out of product occurs, an exogenously 6
gven fracton α j of unserved customers s wllng to substtute from product to j; naturally j α j 1 for all. Each ntally unserved customer makes at most one substtuton attempt, whch, f agan unserved, results n a lost sale. Total demand for product after substtuton s denoted by D s = D + j α j max{0, D j x j }, where x j s the wholesaler s stockng level for product j. For future reference, denote by x j the (N 1)-dmensonal vector of stockng levels for all products j. Let x be the vector of stockng levels and Π W (x) be the wholesaler s expected proft when choosng x. Snce the vector of producton quanttes y s common knowledge and verfable, the wholesaler faces an optmzaton problem under complete nformaton. Thus, he determnes hs optmal stockng quanttes by solvng the followng maxmzaton problem P y : [ ] max Π W (x) = E r mn{x, D s } w x h max{x D s, 0} 0 x y [ ] = E u x (u + o ) max{x D s, 0}, (1) where u = r w and o = h +w are the wholesaler s underage and overage costs, respectvely. The wholesaler s objectve s to maxmze hs expected proft under the quantty restrctons mposed by the manufacturers producton quanttes y. If there are no such restrctons, we let y = and refer to ths case as the unconstraned problem P. We start our analyss of the optmal stockng quanttes wth a bref dscusson on the propertes of Π W (x). All proofs are n the appendx. Lemma 1. For arbtrary, Π W (x) s not concave n x, n general, for gven x. Lemma 1 formalzes the numercal results n Netessne and Rud (2003) that Π W (x) s not always concave n each ndvdual stockng level x. Ths also mples that Π W (x) s not necessarly jontly concave n x, ether. Thus, there may exst multple local optma. For the unconstraned problem P, we know from Proposton 1 n Netessne and Rud (2003) that the optmal stockng quanttes ˆx must smultaneously satsfy the followng frstorder necessary optmalty condtons for all {1,..., N}: P(D < ˆx ) P(D < ˆx < D s ) + j α j u j + o j u + o P(D s j < ˆx j, D > ˆx ) = u u + o. (2) In the remander, denote by ˆx (x ) the soluton to product s optmalty condton (2) for gven fxed values of x. Analogously, let ˆx (x ) be the soluton vector of the remanng (N 1) optmalty condtons n (2) for products j f x s fxed. We further refer to product j s entry n ˆx (x ) as ˆx j (x ). By Lemma 1, t s not ensured that ˆx (x ) s unque. Therefore, for a gven problem nstance P y, we defne ˆx (x ) to be the largest soluton that s feasble n P y and for smplcty, we let ˆx (x ) f there exsts no feasble soluton. The ntroducton of ths te-breakng rule ensures unqueness of ˆx (x ) and helps us to avod ambgutes when 7
comparng two scenaros wth multple optma. The nterpretaton of (2) s appealng. It s a standard newsvendor fractle soluton, adjusted by substtuton effects. The second term on the left hand sde ncreases the optmal stockng level to account for addtonal second choce demand, whereas the thrd term reduces the optmal stockng level by consderng that a stock-out need not result n a lost sale. The optmal soluton of the constraned problem P y follows a smlar pattern. Whenever feasble, the wholesaler tres to stock the quantty that solves (2), gven the other products optmal stockng levels. quantty y. Proposton 1 formalzes ths ntuton. If ths s not possble, he procures the entre avalable producton Proposton 1. Denote the vector of the wholesaler s optmal stockng quanttes for the constraned problem P y by x (y). Further, refer to x (y) as a partally largest optmal soluton f there exsts no other optmal soluton x (y) wth x (y) = x (y) and x (y) < x (y) for any. Then, any partally largest optmal soluton smultaneously satsfes for all = 1,..., N. x (y) = mn{ˆx (x (y)), y }, (3) In the remander, we explctly restrct our analyss to partally largest optmal solutons. Thus, from now on, x (y) refers only to partally largest optmal solutons. Analogously to our te-breakng rule for ˆx (x ), we employ ths selecton crteron to avod ambgutes and to enhance the expostonal clarty of our analyss. Obvously, each optmzaton problem P y has at least one partally largest optmal soluton. In contrast, our numercal experments ndcate that optmal solutons that are not partally largest occur very rarely. Moreover, we emphasze that most of our subsequent results also hold for optmal solutons that are not partally largest. Note that x ( ) = ˆx. Therefore, the optmal stockng quanttes gven n (3) are consstent wth the soluton to the unconstraned problem P gven n Netessne and Rud (2003). Furthermore, n any Bayesan (Nash-)Stackelberg equlbrum, the wholesaler plays hs best-response aganst the manufacturers ntal decson y, whch s gven by x (y). We now nvestgate the senstvty of the wholesaler s optmal stockng quanttes wth respect to changes n a manufacturer s producton quantty. In partcular, we are nterested n the queston f the wholesaler s optmal reacton to changes n y s monotonc. From a manufacturer s perspectve, when alterng y, monotoncty of the wholesaler s best-response functon at least guarantees predctablty of the drecton of change of x (y), even n the asymmetrc nformaton case. In contrast, under nformaton asymmetres, a non-monotonc best-response functon s much harder to predct. We start our analyss by exogenously forcng one stockng level to ncrease n the unconstraned problem P. Lemma 2. Let ε > 0 and denote by e the unt vector for product. () For gven x and x = x + εe j wth j, ˆx (x ) ˆx (x ). 8
() For gven x j and x j = x j + ε, there are nstances of P for whch ˆx (x j ) < ˆx (x j ) for some j. Usng the results of Lemma 2, we can now endogenze the ncreasng stockng level by explctly consderng changes n a manufacturer s producton quantty y j. Ths s done n the frst part of Proposton 2. Buldng on ths result, the second and thrd part transfer the fndngs of Lemma 2 to the soluton of the constraned problem P y. Proposton 2. Let y = y + εe j, ε > 0, for arbtrary j. Then: () x j (y ) x j (y). () For arbtrary and j, fx x k for all k, j and solve (3) for and j. Then, there always exsts one optmal soluton for whch x (y ) x (y). () Solve (3) for k = 1,..., N. There are nstances of P y for whch x (y ) > x (y) for some j. In essence, Proposton 2 hghlghts that the wholesaler s best-response s not necessarly monotonc n a manufacturer s producton decson. The reason for ths s the multdmensonalty of substtuton whch comprses of drect and ndrect effects. If the avalable producton quantty for one product j s ncreased, () and () ndcate that the wholesaler ncreases hs stocks for product j and, consdered n solaton, reduces any other stock j. Ths s the drect effect whch s n lne wth our common understandng of economc substtutes. However, each ncrease or decrease n any one product s stockng quantty has mmedate effects on all other products optmal stockng levels. Hence, f the wholesaler optmzes hs stockng quanttes across all products, a cascade of ndrect effects arses due to all products mutual nterdependency. We fnd that n some stuatons these ndrect effects domnate the drect effects so that, n optmum, the wholesaler may ncrease stockng levels for more than one product (). Indrect effects are domnant f, e.g., the market s substtuton structure s heterogeneous n the sense that there s few drect substtuton between products j and, but frequent substtuton between products j and k, and k and. 3.2 Substtuton Effects We now nvestgate the senstvty of the wholesaler s optmal stockng quanttes and expected proft wth respect to changng substtuton rates. A change n the customers reacton to product stock-outs mples changng substtuton rates. Naturally, ths also affects the total demand for the wholesaler s products. To be specfc, ncreasng substtuton rates mply a stochastcally larger total demand at the wholesaler, or mathematcally, D s s stochastcally ncreasng n α j for all j. Intuton suggests that ths ncreased demand s always benefcal for the wholesaler snce the probablty of ncurrng lost sales decreases. Moreover, Netessne and Rud (2003) conjecture ntutvely that optmal stockng levels for a product ncrease (decrease) f substtuton rates to (from) ths product ncrease. We now test ths ntuton. 9
We start our analyss wth the senstvty of the wholesaler s expected proft. As already argued, demand s stochastcally ncreasng n any substtuton rate. Furthermore, t s well known that, on expectaton, a wholesaler benefts from ncreased demand f trade s proftable (L, 1992). Accordngly, the wholesaler s expected proft ncreases n any substtuton rate. The followng proposton formally states ths argument. Proposton 3. Suppose j α j < 1. The wholesaler s expected proft Π W (x) s ncreasng n any substtuton rate α j f stockng quanttes x are adjusted optmally to changes n substtuton rates. Note that Proposton 3 s true for the constraned and unconstraned problems P y and P, respectvely. If j α j = 1, then α j can only ncrease f at least one other substtuton rate α jk, k, smultaneously decreases. In ths case, Π W may actually decrease n α j. Whle the senstvty of the wholesaler s expected proft has a monotonc behavor, we now show that, n contrast to common ntuton, hs optmal stockng quanttes mght be nonmonotonc n substtuton rates. As a startng pont we analyze how ˆx changes n α j. Lemma 3. () For arbtrary, ˆx / α j 0 for all j. () There are nstances of P y for whch ˆx j / α j > 0 for some and j. Ceters parbus, ˆx s monotone ncreasng n the substtuton rates to product, α j, whle ˆx j may change non-monotoncally n the substtuton rates from product j. Thus, Lemma 3 partally contradcts common ntuton. In partcular, ˆx j need not decrease n α j. Thus, n a stuaton where () holds, t s optmal for the wholesaler to lmt substtuton behavor by ncreasng ntal stockng levels. Now, the results of Lemma 3 allow us to examne the total effects that α j exerts on x. Proposton 4. There are nstances of P y for whch dx j /dα j > 0 for some and j. For the constraned optmzaton problem P y, Lemma 3() remans vald even when ncludng all ndrect substtuton dynamcs and not only drect effects. A trade-off argument between sales volumes and product margns explans these non-ntutve results. Wth ncreased substtuton the wholesaler acheves a hgher total sales volume, but potentally at the cost of reduced sales for certan hgh margn products. (Note that the overall sales volume ncreases, but not necessarly each sngle product s volume.) Consder a hgh margn product j and a low margn product. To restrct substtuton from product j to, the wholesaler may rase x j even when α j ncreases. In such a stuaton, the wholesaler delberately reduces hs sales volume, because ths negatve effect s domnated by the postve effect of more expected sales of the hgh margn product. To conclude, Lemma 3 together wth Proposton 4 ndcate that the wholesaler s optmal stockng quanttes are n general non-monotonc n changng substtuton rates. 10
4 The Supply Game In ths secton, we analyze the manufacturer s optmal producton quanttes under ncomplete nformaton about the wholesaler s stockng levels. We frst focus on the compettve scenaro wth multple Stackelberg leaders and then nvestgate the stuaton wth a sngle Stackelberg leader. Afterwards, we compare the optmal producton quanttes for both scenaros and llustrate our fndngs wth a numercal example. The Orderng Game whch takes producton quanttes y as gven s the second stage of the Supply Game. In the frst stage, manufacturers choose y to maxmze ther expected profts gven ther belefs about the wholesaler s subsequent behavor. The manufacturers unt producton cost and sellng prce for product are c and w, respectvely, wth w > c > 0, {1,..., N}. We assume that manufacturers credbly and smultaneously announce ther producton quanttes y. Further, y [0, K], wth K suffcently large so that t never constrans any manufacturer. Snce the wholesaler has prvate nformaton on hs optmal stockng quanttes, manufacturers can only hold a belef about the wholesaler s equlbrum stockng levels. We explctly model ths uncertanty about the wholesaler s orders as a random varable whch depends on the chosen producton quanttes y. To be specfc, let χ X (y) wth cumulatve dstrbuton Φ (χ, y) and densty φ (χ, y) > 0. We assume Φ (χ, y) to be twce contnuously dfferentable n all arguments y and defne µ (y) X (y) χ dφ (χ, y). We restrct attenton to ratonal belefs. Defnton 1. We say that a manufacturer s belef about the wholesaler s stockng quanttes s ratonal f t satsfes the followng condtons for all products : 1. X (y) = [0, y ]; 2. 2 Φ (χ, y)/ y y j 0, j ; 3. Φ (χ, y)/ y 0 and 2 Φ (χ, y)/ y 2 0. Defnton 1 ensures three structural propertes of a manufacturer s belef. Frst, manufacturers assgn a postve probablty mass only to non-negatve stockng levels whch are naturally bounded from above by the chosen producton quantty y. Second, ceters parbus, manufacturers consder all products to be economc substtutes. Thrd, producton quanttes exert a stmulatng effect on the wholesaler s stockng decson,.e. stockng levels stochastcally ncrease wth the avalable producton quanttes, but at a decreasng rate (for a thorough dscusson on stmulatng effects of nventores, see Balakrshnan et al., 2008). We emphasze that Defnton 1 mposes very mld restrctons on a manufacturer s belef. The wholesaler, by constructon, never orders more than y. Therefore, the frst property s n lne wth the results of Proposton 1. The second property ensures that manufacturers correctly beleve that they compete n a substtuton market. Fnally, the thrd property follows mmedately from Proposton 2() whch states that x (y) ncreases n y. Irrespectve of the 11
knd of nformaton asymmetres, any ratonal manufacturer can always predct these propertes, only the magntude of these effects may be unknown to her. Note that we nether requre belefs to be correct on expectaton, nor do we make any assumpton on how the belef for product changes wth y j, snce Propostons 2() and () ndcate that x (y) can ncrease or decrease n y j. The manufacturer s decson problem structurally dffers n two ways from the wholesaler s optmzaton problem. Frst, the wholesaler s reacton to lmted producton quanttes s fundamentally dfferent from the customers reacton to stock-outs. Whle customers only try to substtute once wth a gven probablty, the wholesaler s reacton to short producton capactes s based on a non-monotonc optmzaton strategy across all products. Second, the manufacturer can nfluence the wholesaler s stockng quantty for product by changng y, whereas the wholesaler cannot nfluence customer demand for product by varyng x. 4.1 Competng Manufacturers We now establsh the equlbrum of the frst stage of the Supply Game when there are N competng manufacturers, each sellng a dfferent, yet partally substtutable product through a monopolstc wholesaler. Before the wholesaler communcates hs stockng quanttes, manufacturers smultaneously choose ther producton levels. Accordngly, manufacturers act as Bayesan Stackelberg leaders wth respect to the wholesaler, but as Nash compettors wth respect to other manufacturers. Thus, each manufacturer maxmzes her expected proft, gven the other manufacturers producton quanttes and gven her ratonal belefs about the wholesaler s subsequent reacton. Her decson problem for gven y s max Π M (y y ) = w µ (y) c y, (4) y 0 where Π M (y y ) s the th manufacturer s expected proft. For brevty, let Π M = Π M (y y ) and denote by y c = arg max y 0 Π M producton quanttes y. the th manufacturer s best-response to her compettors We start our equlbrum analyss by notng that ratonal belefs are suffcent to guarantee concavty of each manufacturer s expected proft. Lemma 4. Assume ratonal belefs. Gven y, Π M quantty y for all. s a concave functon of the producton Due to the concavty of Π M we can derve each manufacturer s best-response y c by examnng the frst-order condtons whch provde necessary and suffcent optmalty condtons. Proposton 5. Assume ratonal belefs. The followng system of necessary frst-order optmalty condtons characterzes any manufacturer Nash equlbrum: µ (y) y y=y c = c w, (5) 12
= 1,..., N. A smple trade-off argument explans the optmalty condtons (5). On expectaton, ncreasng the producton level rases the wholesaler s stockng level. Ths generates a margnal ncrease n revenue gven by w µ (y)/ y, whle smultaneously nducng a margnal cost of c. Equatng margnal revenue and margnal costs provdes the desred result. Note that y c consttutes an upper bound on the wholesaler s decson space. Hence, n any case, the wholesaler s stockng level s smaller than y c. Naturally, (5) not only determnes each manufacturer s best-response n the manufacturer Nash game,.e. n the competton among leaders, but also perssts n the entre Bayesan Nash- Stackelberg game. Here, any Bayesan Nash-Stackelberg equlbrum s gven by the wholesaler s optmal stockng levels x (y c ) and the manufacturers producton quanttes y c whch form a Nash equlbrum n the manufacturer Nash game. In a next step, we establsh exstence and unqueness of the manufacturer Nash equlbrum. Proposton 6. Assume ratonal belefs. For the compettve scenaro, a pure-strategy manufacturer Nash equlbrum exsts and s found by solvng (5). If Π M and s strctly concave n y 2 + j y c 2 µ j (y)/ y y j y j 2 µ (y)/ y 2 j = 1,..., N, for all y, then the manufacturer Nash equlbrum s unque. > 0, (6) Proposton 6 states two suffcent condtons for unqueness of the manufacturer Nash equlbrum. Each manufacturer s expected proft Π M s strctly concave n y f and only f each manufacturer s belef satsfes 2 Φ (χ, y)/ y 2 > 0. Further note that a necessary condton for (6) to hold s gven by j yc / y j < 2. Intutvely, the senstvty of each manufacturer s best-response wth respect to the other manufacturers producton decsons should be bounded. A specal case where (6) s automatcally satsfed occurs f the effects of y and y on µ (y) are addtve separable,.e. µ (y) = g (y ) + h (y ) for arbtrary dfferentable functons g and h. If g s furthermore strctly concave, then the manufacturer Nash equlbrum s unque. Whle Proposton 6 ensures unqueness of the manufacturer Nash equlbrum, the stated condtons are not suffcent to generally guarantee unqueness of the Bayesan Nash-Stackelberg equlbrum. As dscussed n 3, the wholesaler s optmal stockng levels gven the manufacturers producton quanttes are not necessarly unque. Consequently, the wholesaler mght have multple best-responses. Accordngly, to gan a unque equlbrum n the Supply Game, the wholesaler s optmal stockng quanttes must be unque. Corollary 1 states a smple condton that guarantees unqueness. Corollary 1. Let the condtons of Proposton 6 hold. Suppose Π W (x) s jontly concave n x. Then, the Supply Game has a unque Bayesan Nash-Stackelberg equlbrum n the compettve scenaro. 13
4.2 Monopolstc Manufacturer As a benchmark, we now derve the Bayesan Stackelberg equlbrum of the Supply Game wthout manufacturer competton. To be specfc, a monopolstc manufacturer smultaneously produces all N substtutable products and sells them through a monopolstc wholesaler. Therefore, the manufacturer serves as Bayesan Stackelberg leader wth respect to the wholesaler. Thus, she maxmzes her expected proft Π M across all products gven her belef about the wholesaler s subsequent stockng levels. Her decson problem s max y 0 Π M(y) = w µ (y) c y. (7) For gven ratonal belefs, denote by y nc = arg max y 0 Π M a vector of optmal producton quanttes. In contrast to the compettve scenaro, the manufacturer s expected proft Π M s not generally concave n y. Thus, frst-order optmalty condtons provde only necessary, but not suffcent condtons for the manufacturer s optmal producton quanttes. Proposton 7. Assume ratonal belefs. In any Bayesan Stackelberg equlbrum of the noncompettve scenaro, the manufacturer s producton quanttes satsfy the system of frst-order necessary optmalty condtons µ (y) y + w j µ j (y) w y j y=y nc = c w, (8) = 1,..., N. Analogously to the optmalty condtons of the compettve scenaro, the monopolstc manufacturer s optmal decson also follows a trade-off argument. Agan, the manufacturer equates margnal costs and margnal revenues. Ths tme, the shft n revenue accounts not only for the ncreased revenue for product, but also for the decreased revenue for all other products j. Intutvely, the monopolstc manufacturer consders the nfluence of her producton quanttes on the revenue for all products, whereas each compettve manufacturer only cares about her own product. Nether the manufacturer s optmal producton quanttes y nc nor the wholesaler s optmal stockng levels x (y nc ) are necessarly unque. In consequence, the Bayesan Stackelberg equlbrum of the Supply Game s not guaranteed to be unque. A suffcent condton for unqueness s gven n Corollary 2. Corollary 2. Suppose Π W (x) and Π M (y) are jontly concave n x and y, respectvely. Then, the Supply Game has a unque Bayesan Stackelberg equlbrum n the non-compettve scenaro. 14
4.3 The Consequences of Manufacturer Competton Intutvely, competng manufacturers adopt producton quanttes y c that dffer substantally from a monopolstc manufacturer s producton quanttes y nc even though they may hold dentcal belefs about the wholesaler s subsequent stockng levels. In ths context, the natural queston arses whether competton causes manufacturers to ncrease producton quanttes,.e., y c > y nc? Furthermore, vertcal nformaton asymmetres nduce supply chan neffcences that manfest n end-of-season nventores at the manufacturer. However, are these effects smaller or larger under upstream competton? We now explore these ssues. Intuton suggests that the wholesaler prefers competng manufacturers to a monopolstc manufacturer because we expect producton quanttes to ncrease under competton. Hence, the wholesaler s decson space s less restrcted under manufacturer competton and so, he can provde a more proftable servce level to hs customers. Proposton 8 shows that ths ntuton s not always true. Proposton 8. For gven ratonal belefs, the relatonshp between y c and y nc s as follows: () If j w j µ j (y) y 0 (9) for all products, then y c ync for at least one product. () There are ratonal belefs such that y c < ync for some product. It can never happen that all producton quanttes decrease under competton, f (9) holds. Ths condton ensures that each product has n total a negatve effect on the other products, whch s the nature of substtute products. A suffcent condton for (9) are ratonal belefs that addtonally satsfy Φ (χ, y)/ y j 0 for all j, or ntutvely, each product j should exert a negatve nfluence on every other product. Note that Proposton 2() ndcates that ths need not be true for all products. There exst stuatons where two products have a postve effect on each other,.e. Φ (χ, y)/ y j < 0 for some and j. Condton (9) also captures these contngences because we only requre the weghted sum over all effects to be negatve, not each sngle effect. We propose that any product that volates (9) s no longer an economc substtute, but rather an economc complement for the other products. An avalablty trade-off explans why a monopolstc manufacturer sometmes stocks more than a compettve manufacturer (). A monopolstc manufacturer can coordnate the avalablty of all products,.e. she can optmally buld large stocks of a product, whle smultaneously decreasng the avalablty for products j. Under competton, a manufacturer cannot accomplsh ths avalablty trade-off snce she cannot force her compettors to reduce producton quanttes. Ths contngency occurs for a product f, e.g., manufacturers beleve that y exerts only a lmted nfluence on the wholesaler s stockng decson for the other products x. Markets wth such a heterogeneous substtuton structure typcally nclude no-name and brand 15
products (Alawad and Keller, 2004) or heterogeneous products. If the effects of y and y on µ (y) are addtve separable for all, or f (9) holds and all products are homogeneous and symmetrc, then producton ncreases under competton for all products,.e. y c y nc. Note that the results of Proposton 8 are smlar to the fndngs of Netessne and Rud (2003) for competton among wholesalers. However, these two results are based on dfferent problem characterstcs because the wholesaler s and manufacturer s problem dffer structurally n numerous ways. In partcular, substtuton dynamcs and demand characterstcs are completely dfferent. Therefore, Proposton 8 establshes the transferablty of the prevous results to the manufacturer stage. Naturally, as manufacturers producton quanttes change under competton, the wholesaler also adjusts hs stockng quanttes. Ths mples that end-of-season nventores at the manufacturer,.e., excess nventores after tradng, change f competton s ntroduced. Note that these resdual nventores are a drect consequence of the vertcal nformaton asymmetry wthn the supply chan. If manufacturers could perfectly determne the wholesaler s bestresponse stockng quanttes, they would never produce more than ths quantty. Accordngly, manufacturers would never ncur end-of-season nventores. Therefore, we now examne the change n manufacturers end-of-season nventores under competton n the case of nformaton asymmetres. We denote the end-of-season nventory level of product at the manufacturer by I (y) = y x (y). Proposton 9. Let y y. Then, the followng relatons between I(y ) and I(y) hold: () I (y ) I (y) for at least one product. () There are nstances of the Supply Game where I (y ) < I (y) for some product. The wholesaler s always less restrcted n hs decson under y than under y. Ths reflects, e.g., a stuaton where all producton quanttes ncrease under upstream competton. Even though all producton quanttes (weakly) ncrease, end-of-season nventores for some (), but not all () products may decrease. In such a case, the wholesaler ncreases hs stockng quantty for product more than the manufacturer ncreases y. Ths behavor s closely related to the fndngs of Proposton 2. Thus, ndrect substtuton dynamcs at the wholesaler can lead to such a dsproportonate adjustment of stockng levels. 4.4 Numercal Illustraton We now provde a small numercal example to llustrate our theoretcal fndngs. Consder a market wth three substtutable products. For the sake of analytcal tractablty, suppose that each manufacturer beleves that the wholesaler s stockng quanttes follow a truncated exponental dstrbuton wth support on [0, y ] and rate parameter λ (y),.e. Φ (χ, y) = [1 exp( λ (y)χ )]/[1 exp( λ (y)y )]. Note that our framework also works for any other common dstrbuton such as truncated Normal, Gamma, or Webull dstrbutons, but at the cost of analytcal tractablty. 16
Parameters Optmal decson Scenaro w 1 w 2 w 3 k 21 k 31 k 12 k 32 k 13 k 23 y1 c y2 c y3 c y1 nc y2 nc y3 nc A 8 8 8 0.5 0.5 0.5 0.5 0.5 0.5 0.70 0.70 0.70 0.61 0.61 0.61 B 8 8 8 0 0 0.5 0.5 0.5 0.5 1.20 0.63 0.63 1.00 0.60 0.60 C 8 8 8 0.5 0.5 0 0.5 0 0.5 0.65 0.84 0.84 0.71 0.69 0.69 D 8 8 8 0 0 0.5 0 0.5 0 1.20 0.75 0.75 0.81 0.85 0.85 E 10 8 8 0.5 0.5 0.5 0.5 0.5 0.5 1.00 0.66 0.66 1.07 0.45 0.45 F 11.9 8 8 0.5 0.5 0.5 0.5 0.5 0.5 1.24 0.62 0.62 1.89 0.06 0.06 G 11.9 10 7 0.5 0.5 0.5 0.5 0.5 0.5 1.20 0.91 0.44 1.44 0.59 0.05 Table 1: Optmal producton decsons. Followng Defnton 1, belefs about the wholesaler s stockng level for product should be stochastcally ncreasng n y. Thus, each rate parameter λ (y) s a functon of the manufacturers producton quanttes whch decreases n y. To be specfc, we employ the followng smple structural form: λ (y) = y 1 + j k jy j + 1. By settng k j 0 we ensure that the other requrements of Defnton 1 are met. We work wth the nverse of y and not wth y to ensure non-negatvty of λ (y). Intutvely, each scale parameter k j reflects the magntude of nfluence that y j exerts on the wholesaler s stockng decson for product. The truncated exponental dstrbuton together wth the specfcaton of λ (y) ensures that each manufacturer holds ratonal belefs as descrbed n Defnton 1. It s readly shown that µ (y) = [1/λ (y)] [y exp( λ (y)y )/(1 exp( λ (y)y ))]. Thus, the nfluence of y and y on µ (y) s not addtve separable. For all nvestgated scenaros, we assume c = 2 for all. All other parameter values w and k j are gven n Table 1. Parameters nclude hgh and low margn cases, and hgh and low substtuton rates. Note that for all dsplayed parameter values, a unque Bayesan (Nash-)Stackelberg equlbrum exsts. For each scenaro, we dsplay the optmal producton decsons for both supply chan confguratons. Obvously, n a market wth symmetrc prce and substtuton structure, producton quanttes ncrease f manufacturer competton s ntroduced (A). In our example, ths result remans vald f there s no substtuton to one product n the assortment (B). If nstead one product does not nfluence the other products,.e., there s no substtuton away from the product, then producton levels decrease for ths product under competton (C,D). In such a scenaro, a monopolstc manufacturer optmally ncreases the avalablty of the product at the cost of decreasng the other products avalablty. In a compettve envronment, a manufacturer cannot coordnate product avalablty across multple products because her compettors are reluctant to lose market shares. In the agrochemcal market, these heterogeneous substtuton structures arse due to the coexstence of sngle- and mult-purpose products. Whle sngle-purpose products are specalzed to fght a sngle plant dsease such as mldew, mult-purpose products are effectve aganst a wder class of dseases. Naturally, substtuton from the specalzed to the more general product s lkely to occur, because the specalzed product les wthn the applcaton range of the general product. In contrast, the specalzed product need not be useful for a customer ntally desrng the general product. 17
In our example, producton quanttes for hgh margn products decrease under competton, whle producton ncreases for low and medum margn products (E,F,G). We observe ths behavor because a monopolstc manufacturer shfts as much demand as possble to the hgh margn products, thereby reducng the other products avalablty to a mnmum. In contrast, a smlar demand shft cannot be accomplshed under competton. Note that under a monopolstc manufacturer, low margn products almost dsappear from the market, whle competton ensures product dversty (F,G). Concurrent wth ntuton, overall producton ncreases wth the ntroducton of manufacturer competton. 5 Dscusson and Conclusons In ths paper, we analyzed the optmal producton and stockng decsons of a manufacturer and a wholesaler n a two-stage supply chan wth upstream competton and vertcal nformaton asymmetres. We characterze the wholesaler s equlbrum stockng levels and show that these quanttes are non-monotonc n both, avalable producton quanttes and customer substtuton rates. For the upstream stage of the supply chan, we derve the equlbrum producton levels of a monopolstc and a compettve manufacturer, respectvely. We fnd that producton levels for some products decrease f upstream competton s ntroduced. Furthermore, we hghlght the counterntutve stuaton that some end-of-season nventores at the manufacturer decrease although ntal producton levels ncrease. 5.1 Robustness We now dscuss the robustness of our results wth respect to changes n the nformaton and supply chan structure. Addtonally, we delneate opportuntes for future research. Concernng the nformaton structure, we assume that () manufacturers producton quanttes y are verfable, and () Φ (χ, y) s dfferentable n y. Verfablty of y ensures that the wholesaler determnes hs stockng quanttes under complete nformaton about the manufacturer s strategy. Consequently, we can gnore communcaton ssues between manufacturer and wholesaler. Ths s not true f y s unverfable and thus prvately observed by the manufacturer. In ths case, the manufacturer s equlbrum behavor conssts of her producton and communcaton strategy, whch ntroduces an addtonal nference problem for the wholesaler. Under strategc communcaton, the manufacturer need not pursue a truth-tellng strategy or she may not communcate any nformaton at all, whch nherently changes the tmng of the game to smultaneous moves. Whether the structure of our results remans vald under such a scenaro, or not, s an nterestng queston for future research. We further assume that a manufacturer s belef Φ (χ, y) about the wholesaler s optmal stockng quanttes x (y) s dfferentable wth respect to y. (Cachon and Larvere, 1999; Ths s a common assumpton Özer and We, 2006), but clearly, t s not ensured that, n equlbrum, x (y) s actually dfferentable. Nevertheless, t s guaranteed that x (y) s contnuous n 18
y. For such a stuaton, Cachon and Larvere (1999) show numercally that the dfferentablty assumpton provdes an excellent approxmaton. We therefore expect our results to be robust wth respect to dfferentablty of belefs. Concernng the supply chan structure, we assume that competton occurs only among manufacturers. Ths assumpton s nspred by our observatons n the agrochemcal market, but obvously, a general extenson of our framework s to allow for downstream competton as well. Such an extenson ntroduces two new ssues that need to be ncorporated nto the model. Frst, manufacturers need to decde on allocaton mechansms for ther producton quanttes n case that total orders exceed the avalable producton quanttes. Second, these allocaton schemes nduce strategc orderng behavor of the wholesalers. The nfluence of these allocaton problems on supply chans n substtuton markets should be a focal pont of future work. Addtonally, under downstream competton, the assumpton that Φ (χ, y) s dfferentable n y becomes much more problematc. At some pont, competton among heterogeneous wholesalers can nduce some compettors to leave the market. Generally, such a market ext results n dscontnutes n the stockng quanttes of the remanng compettors. Therefore, the dfferentablty assumpton provdes a less relable approxmaton. Nevertheless, we expect that such an approxmaton yelds structurally vald results, even under downstream competton. To deepen our understandng of the repercussons that substtuton exerts on the ndvdual supply chan members, more fundamental extensons should also be examned. In partcular, we beleve that future models should also ncorporate prcng decsons, but ths mght come at the expense of analytcal tractablty. Another aspect that deserves future research s the ntroducton of multple tme perods. In such a settng, ntal product demand changes dynamcally over tme because there s a probablty of a substtutng customer changng hs product preferences due to product unavalablty. 5.2 Concludng Remarks Our analyss demonstrates that substtutng customers affect the producton and stockng decsons wthn a supply chan n non-monotonc and partally counterntutve ways. Thus, ntuton may fal to capture all relevant substtuton dynamcs and ths effect becomes stronger the more heterogeneous the competng products are. Whle n completely homogeneous (symmetrc) markets ntuton correctly predcts each supply chan member s behavor, ntutve reasonng s prone to crucal msnterpretatons as soon as the market becomes heterogeneous. Reasons for such heterogenetes are wdely spread n realty and can be found n terms of proft margns, brands, and product and demand characterstcs. The agrochemcal market, e.g., s shaped by these heterogenetes. Brand manufacturers and (former) patent holders compete wth generc products, whch oftentmes dffer n prce and proft margns. Furthermore, the market s substtuton structure s skewed due to the coexstence of sngle- and mult-purpose products. Hence, n such a heterogeneous market, t s very mportant to understand the substtuton structures among products to take the rght 19
decsons. Appendx Proof of Lemma 1. For gven x, the frst-order and second-order dervatves of Π W (x) wth respect to x are Π W (x) x =u (u + o )P(D s < x ) j (u j + o j )α j P(D s j < x j, D > x ) =u (u + o )P(D s < x ) j (u j + o j )α j P(D s j < x j D > x )P(D > x ) 2 Π W (x) x 2 = (u + o )f D s (x ) + ] (u j + o j )α j [f D (x )P(Dj s < x j D > x ) α j f D s j D >x (x j )P(D > x ), j = 1,..., N, wth f Y beng the densty functon of random varable Y. By rearrangng terms, Π W (x) s concave n x f and only f (u + o )f D s (x ) + j (u j + o j )α 2 jf D s j D >x (x j )P(D > x ) j (u j + o j )α j f D (x )P(D s j < x j D > x ) (10) for all x. To prove the lemma, we construct a scenaro for whch (10) s volated for some x. Let η > 0, and for gven x, let X η (x ) be the set of stockng quanttes x such that P(Dj s < x j D > x ) 1/(N 1) and f D s j D >x (x j ) < η. Note that for any x, X η (x ) s non-empty because P(Dj s < x j D > x ) 1 and f D s j D >x (x j ) 0 for x j. For all j, let () α j = 0,.e., D s = st D ; () α j = 1/(N 1); and () (u j + o j ) = (1 + ν)(u + o )(N 1), ν > 0. Further assume that D Normal(µ, σ ) wth σ < ν/ [ (1 + ν)η 2π ]. Gven these assumptons, (u + o ) [f D (x ) + (1 + ν)η] > (u + o )f D s (x ) + j (u j + o j )α 2 jf D s j D >x (x j )P(D > x ) (11) and (u j + o j )α j f D (x )P(Dj s < x j D > x ) (u + o )(1 + ν)f D (x ). (12) j By (10)-(12), t follows that Π W (x) s not concave n x, f for some x, (1 + ν)f D (x ) > [f D (x ) + (1 + ν)η], (13) 20
or equvalently, f D (x ) > 1 + ν η. (14) ν Snce D s normally dstrbuted, we can choose x such that f D (x ) = 1/(σ 2π) and hence, (14) holds for any σ < ν/ [ (1 + ν)η 2π ]. Proof of Proposton 1. Consder the maxmzaton problem P y. Snce Π W (x) and all constrants are contnuously dfferentable n x and all constrants are lnear n x, there exsts a unque vector λ such that (x, λ) satsfes the Karush-Kuhn-Tucker (KKT) condtons: Π W (x ) x λ = 0 (15) λ (x y ) = 0 (16) = 1,..., N. Now, suppose x s a partally largest optmal soluton. x y 0 (17) x, λ 0, (18) Case 1: x < y. For (16) to hold, we need λ = 0, whch mples by (15) and (2) that x = ˆx (x ). Case 2: x = y. We need to show that y ˆx (x ). Suppose to the contrary that there exst stuatons where x = y > ˆx (x ). By (2), ˆx (x ) s the wholesaler s optmal stockng quantty f he s unrestrcted n hs stockng decson for product. Now, f ths stockng quantty s also feasble for the bounded problem P y, then t must also be optmal n P y. Thus, x = ˆx (x ) < y = x whch s a contradcton. Combnng Case 1 and 2 for all yelds x (y) = mn{ˆx (x (y)), y }. Proof of Lemma 2. Gven x, the wholesaler s optmzaton problem s now one-dmensonal n x. Thus, to analyze how ˆx (x ) changes n x j, j, we apply the Implct Functon Theorem to gan the requred dfferental ˆx (x ) x j = 2 Π W (ˆx, x )/ x x j 2 Π W (ˆx, x )/ x 2. Due to the optmalty of ˆx (x ), we know that 2 Π W (ˆx, x )/ x 2 of the cross-partal yelds 0. Furthermore, analyss 2 Π W (ˆx, x ) x x j = (u + o ) P(D s < ˆx ) (u k + o k )α k P(Dk s x j x < x k D > ˆx )P(D > x ). j k By constructon, D s k, k j, s stochastcally decreasng n x j and so, P(D s < ˆx )/ x j 0 and P(D s k < x k D > ˆx )/ x j 0 for all k, j. Addtonally, D s j does not dependent on x j and therefore P(D s j < x j D > ˆx )/ x j 0. Combnng these arguments gves 21
2 Π W (ˆx, x )/ x x j 0 and fnally Thus, t follows that ˆx (x ) ˆx (x ). ˆx (x ) x j 0. () Consder a three-product scenaro wth products denoted by, j, and k, respectvely, and suppose that the densty functons of D, D j, and D k are strctly postve on R +. Ths mples that the nequalty n Part () s strct because 2 Π W (ˆx, x )/ x x j < 0. Assume α jk > 0, α k > 0, and any other substtuton rate to be zero. Note that ˆx (x j ) depends on x j only ndrectly through ˆx k (x j ). We now prove the lemma by a sequental argument. Frst, we analyze the drect effects between the three products. By Part (), x j > x j mples ˆx k (x j ) < ˆx k(x j ), and thus ˆx (x j ) > ˆx (x j ). Second, to complete the proof, we need to show that an ncreased stockng quantty for product also leads to a decreased stockng quantty for k, but ths s agan just an applcaton of Part (). Accordngly, snce drect and ndrect substtuton effects pont n the same drecton, we can conclude that ˆx (x j ) < ˆx (x j ). Proof of Proposton 2. () Suppose x j (y ) < x j (y). Ths can never happen because x j (y ) s feasble n P y, but by assumpton, t s domnated n P y by x j (y). Ths must also be true n P y because any feasble soluton of P y s feasble n P y. Thus, x j (y ) cannot be optmal n P y. Ths s a contradcton and therefore x j (y ) x j (y). () By Part () and Lemma 2(), t s always true that ˆx (x j (y), x j ) ˆx (x j (y ), x j ). It follows mmedately that x (y) = mn{ˆx (x j (y), x j ), y } mn{ˆx (x j (y ), x j ), y } = x (y ). () Assume y large enough so that t never constrans the wholesaler. Ths assumpton ensures the applcablty of Lemma 2 because we are guaranteed to fnd an nteror soluton to the wholesaler s optmzaton problem. stuatons where ˆx (x (y)) < ˆx (x (y )) for some j. Thus, Hence, by Part () and Lemma 2(), there exst x (y) = mn{ˆx (x (y)), y } = ˆx (x (y)) < ˆx (x (y )) = mn{ˆx (x (y )), y } = x (y ) for some j. Proof of Proposton 3. The total dfferental of Π W (x) wth respect to substtuton rates s dπ W (x (α j ), α j ) dα j = Π W α j + k Π W x k x k α j. In a frst step, we show that Π W / α j 0 for all and j, j,.e. Π [ ] W = (u + o )E (D j x j )1 α {D s <x,d j >x j } 0. (19) j Ths holds true, snce the term under the expectaton n (19) s non-negatve. 22
In a second step, we nvestgate the ndrect effects of α j on Π W. If x s optmally adjusted, then, for all k, Π W / x k = 0 f x k < y k and x k / α j = 0 f x k = y k. Thus, dπ W /dα j = Π W / α j 0 for all and j, f x s adjusted optmally. Proof of Lemma 3. () Choose an arbtrary product. Theorem yelds Applcaton of the Implct Functon ˆx (α) α j = 2 Π W (ˆx(α), α)/ x α j 2 Π W (ˆx(α), α)/ x 2. (20) 0. In addton, the cross- Due to the optmalty of ˆx(α), we know that 2 Π W (ˆx(α), α)/ x 2 partal 2 Π W / x α j s explctly gven by 2 Π W = (u + o ) P(D s < ˆx ), (21) x α j α j for all j. By constructon, D s = D + k α k(d k x k ) +. Thus, D s s stochastcally ncreasng n α j. It follows that P(D s < x )/ α j 0, and hence, 2 Π W / x α j 0. Now, by (20) and (21), ˆx / α j 0 for all j due to the optmalty of ˆx(α). () Smlar to Part (), the proof proceeds by evaluatng ˆx j (α) α j = 2 Π W (ˆx(α), α)/ x j α j 2 Π W (ˆx(α), α)/ x 2. (22) j In contrast to the proof of Part (), the cross-partal can now be postve or negatve, snce 2 [ ] Π W = (u + o ) P(D s < ˆx, D j > ˆx j ) + α j P(D s < ˆx, D j > ˆx j ), (23) x j α j α j where P(D s < ˆx, D j > ˆx j )/ α j 0. We therefore prove the lemma by provdng an example. Consder a two-product portfolo wth heterogeneous ntal demands D Unform(0, 1) and D j Beta(2, 1),.e. F j (x j ) = x 2 j. Assume all other parameters to be symmetrc across products. To be concrete: u = u j = 2, o = o j = 8, and α j = α j = 0.8. In ths settng, we obtan 2 Π W / x 2 j = ] 10 [(x + x j ) 2 + x 2 j /4 0, and 2 Π W / x j α j = 125x 3 /24 0. Consequently, ˆx j/ α j = [ ] 25/48 ˆx 3 / (ˆx + ˆx j ) 2 + ˆx 2 j /4 > 0 for ˆx > 0, whch s satsfed because ˆx = 0 s not an optmum snce there exst stockng quanttes that yeld a strctly postve proft. Proof of Proposton 4. The total dfferental of the optmal stockng level for product j wth respect to substtuton rates s dx j (x j (α j), α j ) dα j = x j + x j x k α j x. k j k α j 23
To prove the clam, we make use of the followng two propertes: For all k, (a) f x k = y k, then x k / α j = 0; and (b) f x k < y k, then x k / α j = ˆx k / α j. From Lemma 3(), for some and j, j, there are nstances of P y where ˆx j / α j > 0. Combnng ths result wth property (b), we fnd that there are nstances of P y wth x j / α j > 0. Now assume that x k = ˆx k = y k for all k, yeldng dx j /dα j = x j / α j > 0 and the proposton follows. Proof of Lemma 4. To prove the desred result, we make use of the nverse dstrbuton functon Φ 1 (ρ, y), ρ [0, 1]. In partcular, Φ (χ, y) = ρ and Φ 1 (ρ, y) = χ. Note that the assumptons on ratonal belefs mply 2 Φ 1 (ρ, y)/ y 2 0. Further, Φ (0, y) = 0 and Φ (y, y) = 1. Assumng ratonal belefs and gven y, each manufacturer s expected proft can be wrtten as y y Π M (y y ) = w χ dφ (χ, y) c y = w (1 Φ (χ, y))dχ c y. (24) 0 Usng the nverse dstrbuton functon, we can rewrte (24) as Therefore, 1 1 Π M (y y ) = w (1 ρ )dφ 1 (ρ, y) c y = w 0 0 0 Φ 1 (ρ, y)dρ c y. 2 Π M (y y ) y 2 1 2 Φ 1 (ρ, y) = w dρ 0. 0 y 2 Proof of Proposton 5. Assumng ratonal belefs, each manufacturer s expected proft gven her compettors producton levels s Π M (y y ) = w µ (y) c y. Takng the frst-order dervatve and satsfyng the optmalty condton yelds Π M (y y ) y = w µ (y) y c = 0, and the result follows mmedately. Proof of Proposton 6. A pure-strategy manufacturer Nash equlbrum exsts f () each manufacturer s strategy space s a non-empty, compact and convex set, and () each manufacturer s proft functon Π M s contnuous n y and quas-concave n y (Debreu, 1952). Lemma 4 together wth our assumptons ensures that these condtons are satsfed. Thus, there exsts at least one pure-strategy manufacturer Nash equlbrum. To derve our unqueness condtons, we rely on the fundamental results of Rosen (1965). In partcular, Theorem 2 n Rosen (1965) asserts that the manufacturer Nash equlbrum defned 24
by (5) s unque f () Π M s twce contnuously dfferentable n y for all, and () σ(y, δ) = N =1 δ Π M (y y ) s dagonally strctly concave for some fxed δ > 0. Whle condton () s guaranteed by our assumptons, we need some more defntons to verfy condton (). Let g(y, δ) be the pseudogradent of σ(y, δ) for fxed δ,.e., δ 1 Π M1 / y 1 g(y, δ) =., δ N Π MN / y N and denote by G(y, δ) the Jacoban of g(y, δ) wth respect to y,.e., G(y, δ) = y g(y, δ) = ( δ 2 Π M / y y j )j. Now, Theorem 6 n Rosen (1965) states that σ(y, δ) s dagonally strctly concave f G(y, δ) s negatve defnte for all y [0, y ] [0, K] N and some fxed δ > 0. Thus, the manufacturer Nash equlbrum s unque f, for some δ > 0, G(y, δ) s negatve defnte for all y. Negatve defnteness of G(y, δ): Denote by G T (y, δ) the transposed of G(y, δ). A basc result n fundamental algebra states that G(y, δ) s negatve defnte f ts symmetrc part G sym (y, δ) = [ G(y, δ) + G T (y, δ) ] /2 s negatve defnte. G sym (y, δ) are negatve. Ths s true f all egenvalues of Note that, due to Defnton 1, all elements of G sym (y, δ) are nonpostve. Hence, by the Gershgorn Crcle Theorem (see Varga, 2004), an upper bound for the th egenvalue of G sym (y, δ) s gven by ub = δ 2 Π M y 2 [ ] 1 2 Π M 2 Π Mj δ + δ j, 2 y y j y y j j = 1,..., N. Therefore, G sym (y, δ) s negatve defnte f, for all, ub < 0. Ths s true f Π M s strctly concave n y, and 2 + j y C y j j δ j δ 2 Π Mj / y y j 2 Π M / y 2 > 0 (25) for all y, where we make use of the Implct Functon Theorem y C y j = 2 Π M / y y j 2 Π M / y 2. By choosng δ = 1/w > 0 for all, (25) reduces to (6), whch proves the proposton. Proof of Corollary 1. If Π W (x) s jontly concave n x, then the wholesaler s optmal stockng quantty x (y) s unque for any gven y. In addton, under the condtons of Proposton 6, the manufacturer Nash equlbrum y c s unque. It follows that (x (y c ), y c ) defnes the unque Bayesan Nash-Stackelberg equlbrum n the compettve scenaro of the Supply Game. 25
Proof of Proposton 7. Assumng ratonal belefs, the manufacturer s expected proft s Π M (y) = w µ (y) c y. Takng frst-order dervatves yelds Π M (y) y = w µ (y) y + j w j µ j (y) y c, = 1,..., N. Rearrangng terms and satsfyng the optmalty condtons gves (8). Proof of Corollary 2. If Π W (x) and Π M (y) are jontly concave n x and y, respectvely, then the wholesaler s optmal stockng quantty gven y, x (y), and the manufacturer s optmal producton quantty y nc are both unque. Thus, n the non-compettve scenaro of the Supply Game, (x (y nc ), y nc ) defnes the unque Bayesan Stackelberg equlbrum. Proof of Proposton 8. We start ths proof wth a prelmnary result that s useful n the remander. Let y y and note that 2 µ (y) y y j y 2 Φ (χ, y) = dχ 0 0 y y j by the defnton of ratonal belefs. It follows that for arbtrarly fxed ỹ µ (y, y ) y µ (y, y ) y =ỹ y. (26) y =ỹ () The proof proceeds by contradcton. Assume y c < y nc. Now, by comparng and equatng the optmalty condtons (5) and (8), we requre µ (y, y c ) y y =y c ) = c = µ (y, y nc w y + w j µ j (y, y nc) w y j y =y nc to be true. By assumpton (9), the second term on the rght-hand sde of (27) s always nonpostve. So, for (27) to hold, we need µ (y, y c ) y y =y c µ (y, y nc) y y =y nc By (26) and concavty of µ wth respect to y, ths can only be true f y c ync, a contradcton to our ntal assumpton. () An example provdes the proof. Assume manufacturers belefs about the wholesaler s stockng levels for products j are ndependent of the producton quantty of product,.e.. (27) 26
µ j (y, y ) = µ j (y ) for all j. Hence, µ j / y = 0 for all j. Assume further that 2 Φ (χ, y)/ y y j > 0 for all j. Then, the nequalty n (26) becomes strct. Comparng the optmalty condtons (5) and (8) for product gves µ (y, y c ) y y =y c = c = µ (y, y nc) w y y =y nc. (28) Now, assume y c ync ; otherwse the proof would already be complete. By (26) and concavty of µ wth respect to y, (28) can only be true f y c < ync. Proof of Proposton 9. () The proof proceeds by contradcton. Let y y and suppose I(y ) < I(y). Then, for arbtrary, y x (y ) < y x (y). (29) As an mmedate consequence of (29), we know that x (y ) > x (y). Now, by repeatedly applyng Lemma 2(), and recall that x (y) = mn{ˆx (x (y)), y }. ˆx (x (y )) ˆx (x (y)), (30) If ˆx (x (y)) y, then I (y) = y y = 0, and thus I (y ) I (y). If, to the contrary, ˆx (x (y)) < y, then applyng (30) yelds I (y) = y ˆx (x (y)) y ˆx (x (y )) = I (y ). Accordngly, I (y ) I (y); a contradcton to our ntal assumpton that I(y ) < I(y). () The proof s an applcaton of Proposton 2. Suppose y = y + εe j, ε > 0, for arbtrary j. Then, by Proposton 2(), there exst stuatons where x (y ) > x (y) for some j. Thus, I (y ) = y x (y ) < y x (y) = y x (y) = I (y). References Adda, E., V. DeMguel. 2011. Supply chan competton wth multple manufacturers and retalers. Operatons Research 59(1) 156 172. Alawad, K.L., K.L. Keller. 2004. Understandng retal brandng: conceptual nsghts and research prortes. Journal of Retalng 80(4) 331 342. 27
Anand, K.S., M. Goyal. 2009. Strategc nformaton management under leakage n a supply chan. Management Scence 55(3) 438 452. Balakrshnan, A., M.S. Pangburn, E. Stavrulak. 2008. Integratng the promotonal and servce roles of retal nventores. Manufacturng & Servce Operatons Management 10(2) 218 235. Bassok, Y., R. Anupnd, R. Akella. 1999. Sngle-perod multproduct nventory models wth substtuton. Operatons Research 47(4) 632 642. Cachon, G.P. 2001. Stock wars: Inventory competton n a two-echelon supply chan wth multple retalers. Operatons Research 49(5) 658 674. Cachon, G.P., M.A. Larvere. 1999. Capacty choce and allocaton: Strategc behavor and supply chan performance. Management Scence 45(8) 1091 1108. Corbett, C.J. 2001. Stochastc nventory systems n a supply chan wth asymmetrc nformaton: Cycle stocks, safety stocks, and consgnment stock. Operatons Research 49(4) 487 500. Debreu, G. 1952. A socal equlbrum exstence theorem. Proceedngs of the Natonal Academy of Scences of the Unted States of Amerca 38 886 893. DeMguel, V., H. Xu. 2009. A stochastc multple-leader stackelberg model: Analyss, computaton, and applcaton. Operatons Research 57(5) 1220 1235. Gal-Or, E. 1987. Frst mover dsadvantage wth prvate nformaton. Revew of Economc Studes 54(2) 279 292. Harsany, J.C. 1968. Games wth ncomplete nformaton played by bayesan players, part. the basc probablty dstrbuton of the game. Management Scence 14(7) 486 502. Honhon, D., V. Gaur, S. Seshadr. 2010. Assortment plannng and nventory decsons under stockout-based substtuton. Operatons Research 58(5) 1364 1379. Jang, H., S. Netessne, S. Savn. 2011. Robust newsvendor competton under asymmetrc nformaton. Operatons Research 59(1) 254 261. Km, H. 2008. Revstng retaler- vs. vendor-managed nventory and brand competton. Management Scence 54(3) 623 626. Kök, A.G., M.L. Fsher. 2007. Demand estmaton and assortment optmzaton under substtuton: Methodology and applcaton. Operatons Research 55(6) 1001 1021. Kök, A.G., M.L. Fsher, R. Vadyanathan. 2009. Retal Supply Chan Management, chap. 6: Assortment Plannng: Revew of Lterature and Industry Practce. Sprnger Scence, 99 153. Kraselburd, S., V.G. Narayanan, A. Raman. 2004. Contractng n a supply chan wth stochastc demand and substtute products. Producton and Operatons Management 13(1) 46 62. 28
Lee, H.L., S. Whang. 2000. Informaton sharng n a supply chan. Internatonal Journal of Manufacturng Technology and Management 1(1) 79 93. L, L. 1992. The role of nventory n delvery-tme competton. Management Scence 38(2) 182 197. L, L. 2002. Informaton sharng n a supply chan wth horzontal competton. Management Scence 48(9) 1196 1212. Lppman, S.A., K.F. McCardle. 1997. The compettve newsboy. Operatons Research 45(1) 54 65. McGllvray, A.R., E.A. Slver. 1978. Some concepts for nventory control under substtutable demand. INFOR 16(1) 47 63. Mshra, B.K., S. Raghunathan. 2004. Retaler- vs. vendor-managed nventory and brand competton. Management Scence 50(4) 445 457. Montez, J.V. 2007. Downstream mergers and producer s capacty choce: why bake a larger pe when gettng a smaller slce? RAND Journal of Economcs 38(4) 948 966. Myerson, R.B. 2004. Comments on games wth ncomplete nformaton played by bayesan players, -. Management Scence 50(12) 1818 1824. Nagarajan, M., S. Rajagopalan. 2008. Inventory models for substtutable products: Optmal polces and heurstcs. Management Scence 54(8) 1453 1466. Netessne, S., N. Rud. 2003. Centralzed and compettve nventory models wth demand substtuton. Operatons Research 51(2) 329 335. Özer, Ö., W. We. 2006. Strategc commtments for an optmal capacty decson under asymmetrc forecast nformaton. Management Scence 52(8) 1238 1257. Parlar, M. 1988. Game theoretc analyss of the substtutable product nventory problem wth random demands. Naval Research Logstcs 35(3) 397 409. Rosen, J.B. 1965. Exstence and unqueness of equlbrum ponts for concave n-person games. Econometrca 33(3) 520 534. Shah, N. 2004. Pharmaceutcal supply chans: key ssues and strateges for optmsaton. Computers & Chemcal Engneerng 28(6-7) 929 941. Sheral, H.D. 1984. A multple leader stackelberg model and analyss. Operatons Research 32(2) 390 404. Smth, S.A., N. Agrawal. 2000. Management of mult-tem retal nventory systems wth demand substtuton. Operatons Research 48(1) 50 64. 29
Varga, R.S. 2004. Gersgorn and Hs Crcles. Sprnger Verlag, Berln, Germany. Vulcano, G., G. van Ryzn, R. Ratlff. 2012. Estmatng prmary demand for substtutable products from sales transacton data. Operatons Research 60(2) 313 334. Walters, R.G. 1991. Assessng the mpact of retal prce promotons on product substtuton, complementary purchase, and nterstore sales dsplacement. The Journal of Marketng 55(2) 17 28. Yan, X., H. Zhao. 2011. Decentralzed nventory sharng wth asymmetrc nformaton. Operatons Research 59(6) 1528 1538. 30