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1 Wokng Pape o Competton n Mult Echelon Dst butve Sup ply Cha ns wth Lnea Dem mand Demng Zhou Uday S. Kamaka Jang o Copyght 04 by Demng Zhou and Uday S. Kamaka. All ghts eseved. PHS wokng papes ae dstbuted fo dscusson andd comment puposes only.. Any addtonal epoducton fo othe puposes eques the t consent of the copyght holde.

2 Competton n Mult-Echelon Dstbutve Supply Chans wth Lnea Demand Demng Zhou HSC usness School, Pekng Unvesty Unvesty Town, anshan Dstct, Shenzhen, 58055, P.R. Chna dmzhou@phbs.pku.edu.cn Uday S. Kamaka UCLA Andeson Gaduate School of Management Los Angeles, CA uday.kamaka@andeson.ucla.edu Abstact We study competton n mult-echelon supply chans wth a dstbutve stuctue. Fms n the supply chan ae gouped nto homogeneous sectos that contan dentcal fms wth dentcal poducton capabltes that all poduce exactly one undffeentated poduct. Each secto may dstbute ts poduct to seveal dffeent downsteam sectos, and each secto s only suppled by a sngle upsteam secto. The demand cuves n fnal makets ae assumed to be lnea, as ae the vaable costs of poducton n all sectos. Competton s modeled va the Successve Counot model n whch fms choose poducton quanttes fo the downsteam maket so as to maxmze the own pofts, gven pces fo the nput. Unde these assumptons, equlbum pces, quanttes, and fm-level pofts fo any mult-echelon dstbutve netwok can be deved. We dscuss the netwok tansfomaton popetes, and by usng these popetes, we examne the effect of demand paamete changes and cost changes on any fm s equlbum pce, quantty, and poft. We also exploe the effects of enty on the equlbum soluton. Whle the effects of upsteam enty on downsteam sectos ae as expected, the effect of downsteam enty on upsteam sectos, and theefoe on sectos n paallel (lateal) paths can be qute countentutve. Keywods: Dstbuted decson makng; Successve Counot competton; Supply chan management

3 Intoducton In many ndustes t s common to see mult-stage supply chans, n whch dffeent companes occupy dffeent stages of the chan. The numbe of entants at any level of a supply chan can vay qute substantally. Supply chans also exhbt consdeable stuctual vaaton acoss ndustes. asc goods and commodty poducts ae often puchased by seveal dstnct sectos. Fo example, steel s puchased by the automotve ndusty as well as by the constucton ndusty. Goods can also be sold n moe than one geogaphcally dstnct maketplace. Such examples dsplay a dstbutve stuctue. In othe cases, the supply netwok can have an assembly stuctue whee many nputs may be equed to assemble a poduct. Of couse, both of these chaactestcs can also be pesent smultaneously. In ths pape we addess competton n mult-echelon dstbutve supply chans, a genealzaton of the seal chan model of Cobett and Kamaka (00). Fms ae gouped nto sectos whee all fms wthn a secto ae dentcal. Each secto poduces exactly one poduct and uses exactly one nput. A poduct may be puchased by many sectos. The esultng system can be pctued as a netwok whch has a mult-echelon o aboescent stuctue n whch each secto s epesented by a node. Each node s suppled by a unque node, but may be a supple to moe than one node. The netwok epesentaton of such dstbutve stuctue s sketched n Fgue. 3 4 Consumes Consumes 5 6 Consumes Consumes Fgue : An Example of Dstbutve etwok Examples of dstbutve netwok stuctues abound. Pehaps the most obvous case s that of smple geogaphcal dstbuton, whee goods fom a plant ae sold n moe than one egon, wth no cosssellng acoss egons (due to dstance and cost). Dstbutve stuctues also occu n many pocess ndustes, whee one poduct s used n moe than one dstnct downsteam poducton pocess to make dffeent poducts. Examples nclude metals (steel, alumnum, coppe), agcultual poducts (mlk, wheat, con), peto-chemcals (leadng to a vast ange of sectos ncludng polyme plastcs lke

4 PVC and HDPE, fbes lke polyeste and chemcals lke methanol, ethylene and olefns), and electonc components (lke memoy chps) gong nto dffeent boads and poducts. Cement poducton povdes a good example of a two-te dstbutve netwok. The fst stage s the poducton of clnke fom lmestone. At a second te, clnke s gound, blended wth flles lke slag, and packaged nto the fnal poduct. The gndng plants tend to be close to maket egons and so ae geogaphcally dstbuted, whle clnkeng must be done close to the souces of aw mateal (lmestone and coal). Gndng plant costs ae an ode of magntude smalle than clnke plants, so that thee can be many moe entants at the lowe te than at the uppe. In some egons, a thd te n ths secto s dstbuton to etal locatons, whch happens n countes wth many small contactos who buy cement locally. Changes n costs, concentaton, technology, netwok stuctue, o demand can have both vetcal and lateal effects n dstbutve systems. Fo example, n the cement poducton, concentaton decease n one egon due to enty of gndng plants mght affect the pce of clnke, and thus affect the poducton and poftablty of gndng plants n anothe egon, and the pce of cement n that egon. As anothe example, con s used fo poducng gasohol, as well as stach, con syup, and hundeds of othe foods, cosmetcs and ndustal poducts. An nceased demand fo gasohol mght ncease pce and poducton of con, and thus affect the behavos of fms n sectos that s not dectly lnked wth the gasohol ndusty. In a dstbutve netwok, the magntudes of the changes n any secto depend on many, pehaps all, othe sectos. Methods fo undestandng and estmatng these changes, and fo solvng lage poblems, ae not eadly avalable. If multple concuent changes occu n these systems, t s not easy to assess the composte effects of those changes. And n some cases, the changes can be counte-ntutve even n tems of decton. Ou pupose s to study system-wde equlbum behavo whle consdeng vetcal and hozontal nteactons among sectos n the whole netwok as well as the competton between fms wthn each secto (node). We develop models of competton whee each ndvdual fm n each secto of a lage mult-echelon dstbutve netwok acts as a decson make optmzng ts own poft. Ths model allows us to examne the mpact of cost stuctue, netwok (dstbuton) stuctue and secto concentaton on pces, quanttes and pofts. As n Cobett and Kamaka (00) we model competton usng the Successve Counot famewok. We povde explct expessons fo equlbum pces, quanttes and fm-level pofts. We constuct netwok tansfomaton methods that can compess netwoks to smple foms, o expand them to patcula (bnay) foms. y usng these tansfomaton methods, we ae able to analyze the mpact of cost and demand paametes wthout havng to deal wth the complexty o 3

5 specfc fom of the netwok stuctue. Impotantly, we ae also able to study the effect concentaton n any secto n the netwok on any othe secto. These effects ae n some cases qute non-ntutve. Conventonal wsdom and eale studes n the lteatue state that lowe concentaton ( a lage numbe of fms) n a maket geneally causes the total output of that maket to ncease, the consume pce to decease, and each ncumbent's poft to decease (Seade 980a). These esults ae ndeed obtaned n the seal and assembly cases, unde assumptons smla to those of ths pape. Howeve, we fnd that fo the dstbutve case, wth the consdeaton of vetcal and hozontal nteactons, fmlevel poft fo ncumbents could ncease o decease wth changes n concentaton. Ths esult s due to the combned effect of competton and upsteam esouce pce changes. Although ths pape adopts the same Successve Counot Model to analyze the dstbutve stuctue as Cobett and Kamaka (00) fo the seal case and Ca and Kamaka (005) fo the assembly case, the equlbum soluton povdes dffeent nsghts fom those exstng papes. In the dstbutve netwok, contay to tadtonal wsdom, when secto concentaton changes, the esouce pce could ncease, decease, o emans the same dependng on the elatve paametes of all the dstbutve sectos. Thus, pofts and poducton of the ncumbents could change n ethe decton, unde the combned effects of esouce pces and competton. Ths unque stuctue behavo s not seen n the seal and assembly stuctues analyzed n the pevous papes. Moeove, dffeent fom the seal case, vetcal ntegaton fo dstbutve case can make the total poft of the ntegato ncease o decease dependng on the concentaton level and cost paametes. We note that the (post-enty competton) model of ths pape s a pe-equste not only fo enty decsons, but also fo othe types of analyses. Fo example, a faclty locaton decson n the context of a supply chan could use the post-enty poducton model to undestand the consequences of altenatve locaton choces. Ho et al. (004) study compettve locaton wth a smple mbedded one-te Counot model of poducton competton afte locaton choce. ut n many sectos, fo example cement, petochemcals, o food pocessng, thee ae multple tes nvolved. Ths moe complex and ealstc locaton poblem emans to be nvestgated. As anothe example, consde the consequences of pocess mpovement at some stage n an ndusty to educe vaable costs. The esults of such changes can pple to upsteam, downsteam and lateal sectos. The tools to analyze such changes do not exst today. Vetcal ntegaton decsons can be analyzed n a manne smla to the example n Cobett and Kamaka (00) fo the seal chan case. In all these examples, the fast soluton of the compettve model s a necessay pe-equste. The dstbutve model can also be combned wth the assembly model of Ca and Kamaka (004) to study moe complex poducton netwoks. Howeve, ths extenson s not tval, and the soluton 4

6 appoach may not be geneal, snce t can depend on the specfc stuctue of the netwok beng consdeed. Lteatue Ou analyss s n the tadton of the Successve Counot olgopoly lteatue, though most pevous eseach has been lmted to the two te seal case wth only one o two entants at each te. In fact, the upsteam te s often taken to consst of a sngle (monopolst) fm. Futhemoe, most of the exstng lteatue s dected towads polcy ssues elated to vetcal ntegaton and maket foeclosue, often n a settng wth the upsteam monopolst ntegatng fowad. Ou appoach s dected at the modelng, analyss and soluton of geneal netwoks, n ode to undestand the mplcatons fo poducton decsons (quanttes), the esultng pces, and the effects of changes n netwok stuctue, vaable cost stuctue acoss the netwok, end demand, and concentaton at netwok stages. Machlup and Tabe (960) pesent an ealy dscusson of successve olgopoly and vetcal ntegaton. Geenhut and Ohta (979) and Abu (988) show that vetcal ntegaton by a monopolst n the supplyng secto, by and lage leads to hghe outputs and lowe pces. The seemng paadox hee s that a monopolst ntegatng fowad, can dve out compettos fom downsteam makets, and yet socal welfae can be nceased. Essentally, ths happens because vetcal ntegaton avods double magnalzaton. Quan and Roges (004), follow an appoach smla to Cobett and Kamaka (004) to examne a two-te netwok of a telecommuncatons fm puchasng softwae tools fom an upsteam vendo; the fm then employs the tools to poduce both a poduct (pogammed quees) and sevces. Tyag (999) studes the effects of downsteam enty n a two te seal settng when the upsteam te conssts of a monopolst and the downsteam te conssts of dentcal fms. He fnds that downsteam enty could affect the (upsteam) pce chaged to the downsteam fms. Dependng on the consume demand functons, ths change of upsteam pce could have negatve o postve effect on the pofts of downsteam ncumbents. Ths effect occus fo cetan demand condtons, and cannot occu fo lnea demand n a seal chan. In ths pape we show that t can occu n dstbutve chans even when the end maket demand s lnea, due to an entely dffeent mechansm that has to do wth the dstbutve stuctue athe than wth the shape of the end maket demand cuve. Othe elated economc lteatue ncludes Zss's (995) study of hozontal meges wthn a settng of two tes wth two entants n each te, Vckes's (995) study of egulaton n seal chan competton and Seade's (980) study on the effects of concentaton and enty. As we have mentoned, none of these papes consdes a netwok whee thee ae multple tes of dstbutos o manufactues, o a dstbutve stuctue. 5

7 Ou pape s closest n methodology and spt to Cobett and Kamaka (00) who study mult-te seal supply chans. Also closely elated s the pape by Ca and Kamaka (005) that analyzes multechelon assembly netwoks. The dstbutve and assembly settngs ae complementay genealzatons of the seal case. Thee ae sgnfcant dffeences n the two genealzatons that deve fom the dffeent undelyng netwok stuctues. In the assembly case, a key advance was n epesentng how quantty matchng takes place acoss supplyng sectos (coespondng to bll-of-mateals elatonshps). In the dstbutve case, an mpotant nsght s that upsteam fms have to stke a balance between multple downsteam sectos when dscmnaton s not possble. Ths compomse then has mplcatons fo the consequences of changes n paametes o stuctue. The ablty to analyze the equlbum behavo of geneal dstbutve netwoks dstngushes ou pape fom pevous wok, and as fa as we ae awae, ths s the fst pape to addess ths settng. 3 The Two-te Dstbutve Model In the two-te case, thee s a sngle upsteam secto (consstng of seveal homogenous fms) and multple sectos n the lowe te (each wth seveal fms) as shown n Fgue. 0 k Fgue : A Two-te Case Fms at each downsteam secto face an ndependent consume maket, whee the pce and the aggegate poducton quanttes ae lnealy elated by p = a b Q. Hee, a s the maket esevaton pce, the supemum pce at whch demand wll be postve. The othe paametes b s pce senstvty of the maket. We also defne : / b fo convenence. We use the followng notaton: v, the vaable cost of poducton and dstbuton n secto, =0,,, k. n, (a postve ntege), the numbe of fms n secto. Fo convenence, we also use a paamete defned by : n / ( n ). P =a -b Q P =a -b Q P k=a k-b kq k p, the common pce chaged by all fms n secto to all downsteam sectos. 6

8 q, the quantty chosen by a sngle fm n secto. As n a standad one secto Counot model, all fms wthn a secto poduce the same quantty at equlbum, so the ndvdual fms' poducton quanttes can (at equlbum) be expessed as q = Q /n. π, the poft of a sngle fm n secto. We assume that fms n each lowe te secto compete n the Counot sense, and competton n the netwok follows the Successve Counot famewok (Machlup and Tabe, 960). That s to say, each fm n each lowe te secto chooses poducton quantty to maxmze ts poft, gven a demand cuve and a pce fo the nput suppled by the upsteam secto. Fo the upsteam secto the aggegate quantty acoss all downsteam (lowe te) sectos as a functon of the esouce pce, establshes a demand cuve fo the esouce supplyng (upsteam) secto. Fms n that secto compete n the Counot sense by choosng quanttes gven ths demand cuve. Costs of esouces o nputs at ths secto ae assumed to be exogenous to the model. We adopt the equlbum ctea defned by Ca and Kamaka (005). Fo each secto:. Gven the demand cuve faced by the secto and the esouce pce chaged by the upsteam secto, no fm n the secto has an ncentve to unlateally devate fom ts poducton quantty.. The aggegate quantty poduced n evey secto s balanced wth the aggegate quantty of the equed esouce (.e., makets clea). The optmal poducton quanttes and equlbum pces fo ths system ae as follows. Poposton The equlbum pces, poducton quanttes, and pofts fo the two-te dstbutve netwok n fgue ae p ( ) a v Q ( a p ) ( a v ) ( ) Q / ( ) ( ) ( a v ) p [( ) a v ] [( ) a v ],... k Q ( a p ) [ a v v ( )( a v )],... k ( ) ( ) Q [ 0 ( 0)( 0 0)],... and 0 whee a0 : ( a v ) /.. k.. k a v v a v k :.. k. The poof s gven n the Appendx. An mpotant popety of ths poposton s that the equlbum pce fo the esouce suppled by the upsteam te s elated to the pces that would be chaged n the pefect dscmnaton case, n a specfc way. Suppose that pefect dscmnaton wee possble,.e. 7

9 that n the Counot settng the uppe te could detemne the quanttes suppled to each downsteam secto (node) ndependently. The system can then be decomposed nto k sepaate seal systems. Let p 0 be the esouce pce n the th such system. Then we have Coollay The esouce pce p 0 s a convex combnaton of the pefect dscmnaton pces p 0, =,..., k: p p /. () k.. k The poof s gven n the Appendx. The qualtatve nsght fom ths esult s that the esouce pce n the dstbutve case s a compomse between the pces that could have been obtaned fom each downsteam puchasng secto, unde pefect dscmnaton. So fo example, the enty of anothe potental buye of the esouce wll ceate a pce shft, whch could be n ethe decton, dependng on the pce and the weghtng gven to the new secto. ote that the weghts n the convex combnaton tem depend on the s whch ae measues of the concentaton of a secto, and on the s whch ae a measue of the pce senstvty of the mputed demand cuve of the secto. Ths esult eveals the undelyng foces of nteacton among downsteam sectos when affectng the esouce pce. Howeve, t deseves to be mentoned that the detaled fom of convex combnaton s hghly elated to the demand lneaty, wthout whch t would be had to deve a smple fom of decomposton. 4 The Mult-te Dstbutve etwok and Stuctual Popetes We now consde geneal dstbutve supply netwoks as exemplfed by Fgue. We estct attenton to stuatons n whch the aggegate poducton quantty along each ac of a gven netwok s stctly postve at equlbum. Ths assumpton can be fomally expessed as equlbum condton, whch, togethe wth the demand lneaty, guaantees the unqueness of the equlbum soluton. The aboescent stuctue of dstbutve netwoks mples that each netwok has a sngle oot secto. We label the sectos of netwok top down wth the oot secto as secto. Fo each secto (node ), we defne: D as the set of all the sectos downsteam of secto. Fo example, n fgue, D = {, 3, 4, 5, 6} and D 4 = {5, 6}. S as the set of sectos mmedately downsteam of secto. In fgue, S = {, 3, 4}. φ,s as the set of sectos along (, s), the path fom secto to secto s. In fgue, path (, 5) s the path fom secto, though secto 4, to secto 5; and φ,5 = {, 4, 5}. Poposton Fo a gven dstbutve netwok, the deved demand cuve fo any upsteam secto s 8

10 p a bq, o Q ( a p ), () whee the paametes, a, and b ae computed teatvely te by te as ss ( ) ss as vs ss, b, and a. ss ss s s ss s s The poof s smla to the poof fo Poposton and s omtted hee. The expessons n () povde some nteestng popetes of the deved demand cuves seen by upsteam sectos. As n the seal supply chan case (Cobett and Kamaka, 00), deceases gong upsteam. That s to say, the quanttes demanded at upsteam sectos ae less senstve to upsteam pce than sectos whch ae close to the makets. ote also that the esevaton pce a seen by a te s equal to the weghted aveage of the esevaton pces along 's downsteam acs, a v, s S. In the exteme case when a o s vss the same along each banch, a equals as v s s s. In such stuatons, a does not change wth s. Ths popety s cucal fo ou late dscusson of the effect of downsteam enty to the pces and pofts n upsteam sectos. Usng Poposton, we can teatvely deve the demand cuve fo each secto, statng fom the sectos facng fnal consumes. Fom these cuves, the equlbum pce condton fo each secto can be deved. All the pce condtons togethe compse a system of ndependent lnea equatons, whch s solvable. The equlbum quanttes ae then deved by substtutng the pces back nto demand cuves. Ths soluton method can be convenently expessed n matx notaton, as gven n the followng poposton. Poposton 3 Equlbum pces of sectos n a dstbutve netwok ae the soluton to T p R whee p s the vecto of pces (one element pe secto), R s a column vecto (one element pe secto) wth each element R ( ) a v, and T s a lowe tangula matx, populated wth element f T f S. 0 othewse The poof s omtted as t follows fom the above dscusson. ote that T can be nveted to gve T - wth elements s 9

11 whee, s defned as, : s, f T, / f D 0 othewse. s. Applyng T - gves the followng esult. Poposton 4 At equlbum, the pce of secto n a dstbutve netwok s p ( / )[( ) a v ], (3),, whee, s defned as, :. s, s T s a stuctue matx that captues the elatonshp between the stuctual featues of a dstbutve netwok (.e., the secto connectons, concentatons, and demand functons) and the equlbum competton pces. Ths poposton povdes an explct fom fo the equlbum pces fo a gven netwok. Equlbum quanttes and pofts can then be deved accodngly. ote that although the pce of secto, p n (3) depends only on paametes of sectos on the upsteam of secto, those paametes ae deved fom paametes of all the downsteam sectos, as shown n Poposton. Theefoe, the equlbum pce of one secto, p, s the esult of nteacton of all sectos n the whole netwok. The man pupose of the cuent eseach s to analyze the equlbum changes as a esult of cost paametes and secto concentaton changes fo any gven dstbutve netwok. Ths goal, howeve, cannot be easly eached by dectly takng fst ode dffeentaton of the equlbum esults deved fom Poposton 3 and 4, due to the ntetwne of paametes acoss a geneal complex dstbutve netwok. We now constuct netwok tansfomaton methods usng some stuctual popetes of the dstbutve netwok. Though these tansfomaton methods, we can dscuss the compaatve statcs wth cost and secto concentaton changes wthout havng to deal wth the geneal stuctue any moe. The stuctual popetes ae qute smla to the assembly netwok dscussed n Ca and Kamaka (005). As n Ca and Kamaka (005), we egad two sub-netwoks o two sets of nodes as equvalent wth espect to the est of the netwok f, afte substtutng one netwok (set of nodes) fo the othe, the est of the netwok has the same equlbum pces, quanttes, and fm-level pofts. otng that a subnetwok essentally communcates to the est of the 0

12 netwok though the demand cuve that s povded to the subnetwok's oot node, ths means that two sub-netwoks ae equvalent when they show the same demand cuve to the est of the netwok. Usng ths concept, dstbutve netwok has the featues of expandable and compessble, smla to the assemble netwok shown n Ca and Kamaka (005). As llustated by Fgue 3, a netwok (a) can be expanded to a netwok (b) wthout affect any exstng sectos equlbum esults. Secto D s a dummy secto wth d and vd 0. Fgue 4 shows that secto fms ae unaffected f secto and 3 ae compessed nto a smple demand cuve. ( a ) ( b ) 3 4 D Consumes Consumes Consumes Consumes 3 4 Consumes Consumes Fgue 3: etwok Expandablty (a) 3 (b) Consumes Consumes Consumes Fgue 4: etwok Compessblty

13 Tunng to compaatve statcs, we now consde the senstvty of a secto's equlbum pces, pofts, and quanttes to othe sectos' paametes whee wll be the secto at whch a change occus, and we consde how ths affects anothe secto s. Poposton 5 Suppose s s any secto not upsteam of secto, and let u be the fst node encounteed that s upsteam of both and s. Then, a paamete change at s completely communcated to secto s though p u. That s, at equlbum, p s nceases ff p u nceases (as a esult of the change at ) and, equvalently, π s deceases ff p u nceases. The poof s omtted. Usng ths poposton togethe wth compessblty and expansblty allows us to analyze the effect of paametc changes vey smply. We only need to examne the sectos along the path (, ) to dscuss the effects of paamete changes n secto. Usng the compessblty popety, the est of the netwok can be compessed to sngle sectos wthout changng the equlbum soluton to the netwok. Theefoe, the geneal mult-te dstbutve netwok can be compessed to a smple bnay tee as shown n Fgue 5(b). In the followng dscusson, we focus on the equlbum pces, quanttes, and pofts of the sectos along the path (, ) n the bnay stuctue shown n Fgue 5(b). The othe sectos ae dstngushed by a pme ( '). In the fgue, we assume the maket paametes (a s and s) of the leaf sectos (secto, -,..., and secto ) ae gven. (a) (b) k k s Fgue 5: Convet a Dstbutve etwok nto a nay Tee The followng poposton shows how v nfluences equlbum pces and pofts n each secto along the path (, ).

14 Poposton 6 (Illustated by Fgue 5(b).) Suppose that v, the vaable poducton cost n secto, nceases. At equlbum: ) p nceases, Q and π decease. ) If secto s upsteam of (, ), then p, Q, and π decease. 3) If secto s downsteam of then p nceases; Q and π decease. 4) Othewse (.e., s a lateal secto that lnks wth secto though the same esouce secto) p deceases; Q and π ncease. The effects of changes n v ae ntutve n that nceased poducton cost fo fms n a secto nceases sellng pces and lowes pofts. It also deceases the pces and fm-level pofts of upsteam sectos n the whole supply chan, whch means the cost ncease s passed along the supply chan eventually esultng n lowe pces, lowe poducton, and lowe poft magns fo all upsteam sectos. Inteestngly, howeve, fo the lateal sectos n the netwok that ae not upsteam no downsteam of secto, cost ncease n secto causes the esouce pce of the connectng node to decease, and thus causes the fms n those sectos to be moe poftable. Poposton 7 Suppose that a nceases (at a secto supplyng a consume maket). At equlbum: ) Q, p, and π all ncease. ) If secto s upsteam of then Q, p, and π all ncease. 3) If secto s not upsteam of then p nceases, Q and π decease. So, nceased poft magns can be passed upsteam along the chan, causng pces to ncease n esouce makets and pofts to ncease n upsteam fms. Howeve, to fms n the "substtute channels" of the netwok, an ncease n a has negatve effects: the pces ncease and pofts decease as a esult of the change. 5. The Effects of Secto Concentaton The dependence of equlbum outputs, pces and pofts on ndusty concentaton s a fundamental ssue n economc analyss. Conventonal wsdom holds that wth lowe concentaton (moe fms), ndusty pce ought to declne and pe fm output and poft ought to decease. In sngle te Counot competton, ths clam holds fo most geneal 3

15 demand condtons. In mult-te netwoks, we mght expect that lowe concentaton n a lowe te would lead to hghe maket powe and hghe pofts n an uppe (supplyng) te. Howeve, these expectatons may be volated n dstbutve netwoks. The followng dscusson shows that even wth lnea demand thee exsts a ange of demand condtons fo whch an ncease n the numbe of fms n a secto (lowe concentaton) nceases the pofts of the secto's exstng fms and deceases the pofts of upsteam fms. Ths supsng esult occus when enty deceases the potental upsteam maket, and causes the upsteam pce to decease; ths then pemts the poft magn of the downsteam fms to ncease. If the effect of upsteam pce outweghs the effect of nceased competton due to the enty, the ncumbents' pofts n the secto wth enty can go up. We note that ths seemngly pevese effect s not necessaly a common phenomenon. It occus fo cetan paamete anges, and can dsappea wth futhe enty. Howeve, what t undelnes s that the dstbutve stuctue has chaactestcs whch ae specfc to that stuctue, and whch lead to consequences not seen n the pue seal and assembly cases. In the followng dscusson of enty effects, we stat wth two-te case to deve explct esults. Then, we extend the analyss to the mult-te case and show that cetan popetes genealze smply whle othe effects can be moe complex. The followng poposton summazes the effects of secto concentaton on equlbum pces, outputs, and pofts fo two-te netwoks. p p p 0 v a v a v * (- )(a -v ) ( a v)( ) n a a v v whee 3 3 /( ) Fgue 6: Effect of Inceases n on p and π Poposton 8 The effects of concentaton changes (enty o ext) n the two-te netwok of Fgue 4(a) ae: ) An ncease (decease) n n and causes p, p, π to decease (ncease), and causes Q, Q, π to ncease (decease). 4

16 ) An ncease (decease) of due to enty (ext) n that downsteam secto causes p to decease (ncease). 3) Howeve, wth an ncease of, p and π could ncease, decease o eman constant dependng on the elatve demand paametes (as shown n Fgue 6): Fo p : f a v > a (.e., a v > a 3 v 3 ), p nceases wth ; f a v = a (.e., a v = a 3 v 3 ), p emans constant; f a v < a (.e., a v < a 3 v 3 ), p deceases wth. Fo π : f a v v >(a v )/, π nceases wth ; f a v v =(a v )/, π emans constant; f a v v <(a v )/, π deceases wth. a v v s always geate than (- )(a v ) and smalle than (a v )(+ 3 3 /( )) due to the egulaty condton. 4) Moeove, π could ncease, decease, o eman constant wth the ncease of. Specfcally ( ) f a v v ( )( a v ), ( ) ( ) π nceases wth. Othewse, π deceases wth (o emans constant when the above expesson holds wth equalty). Pats ) and ) of Poposton 8 state that lowe concentaton o an nceased numbe of fms n a secto, leads to lowe pces n the secto. Moe nteestng and qute countentutve fndngs ae the effects of on p, π, and π as stated n pats 3) and 4) of Poposton 8. Contay to conventonal wsdom, educed concentaton (enty) n a downsteam secto can cause the esouce pce chaged by the upsteam supples to go up unde cetan demand and cost condtons. Moeove, fo a cetan ange of demand paametes, the poft of upsteam supples could go down as a esult of the downsteam enty. Smlaly, ncumbents n the same secto (secto n ou analyss) could see pofts ncease. The last phenomenon s not pevasve. It only happens when the effect of esouce pce decease ovewhelms the effect of competton. Futhemoe, notce that the ght hand sde of the nequalty condton (4) monotoncally deceases n. Wth contnued enty n secto, the anomalous effect goes away. Howeve, t llustates a chaactestc of dstbutve systems. The undelyng eason fo these effects s the natue of upsteam esouce pce, whch as shown n Poposton, s a compomse between the esouce pces that would be seen wth pefect dscmnaton. The weght of the balance s contolled by the concentaton (). Fo a secto n whch the esouce pce unde pefect dscmnaton s hghe, enty ntensfes the weght, and makes the equlbum esouce pce hghe, and vce vesa. (4) 5

17 To examne the effect of concentaton changes at a secto n a mult-te netwok, we only need to focus on the behavo of sectos along the path (, ). We note that, the esevaton pces fo upsteam sectos a ( ) play a key ole n the changes of pces, outputs and fm-level pofts. In the followng poposton, we summaze the decton of change of a wth espect to. Poposton 9 The decton of change of the maket esevaton pce a at secto,,, wth espect to can be captued by ts fst devatve as a, ( a v, a ) whee, s (f =, s, ) and v, svs. Thus the esevaton pce fo an upsteam secto nceases, deceases, o emans the same dependng on the elatve value of the esevaton pces along the channel whee enty occus and the esevaton pce of the maket, whch s the weghted aveage value of each downsteam channel as dscussed eale. Ths esult, whch s dffeent fom the cases n seal chans (Cobett and Kamaka, 00) o assembly netwoks (Ca and Kamaka, 005), enables us to examne the pce changes of any secto wth the change of. Poposton 0 In a mult-echelon dstbutve netwok, the pce at te, p, deceases monotoncally wth. Usng (3), p, the pce chaged by the upsteam sectos, can be expessed as p [( ) a v ] k, k k k k k.. Snce each ndvdual "a" s a functon of, the dectve of p to can be expessed as p k.. ak ( ) k, k ( ) ( ). ext, poft of secto can be expessed as and the fst dffeentaton then gves k, k, a vk, ak.. k ( ) ( a v p ), 6

18 a p ( ) ( )( ). a v p We can thus see that upsteam maket pce and poft could decease, ncease o eman constant wth downsteam enty. Futhemoe, the tends n these stategc vaables can be even moe complcated snce they depend on the changes of any uppe te's esevaton pce and equlbum pce. Fo example, a secto's pce could ncease even when the secto's esevaton pce deceases. 6. Vetcal Integaton n Dstbutve etwoks Vetcal ntegaton abounds n ndustes wth dstbutve stuctue. It s commonly seen n cement ndusty that clnke plants supply both gndng plants that ae ntegated wth them, and ndependent gndng plants. Most of the PC manufactues, such as Dell and HP, bundle montos wth the computes, as well as sellng them sepaately though etal channels. (a) (b) p p 3 3 Consumes Consumes Consumes Consumes Fgue 7: Vetcal Integaton The explct fom of equlbum soluton fo the dstbutve netwoks shown n Poposton, 3, and 4 makes t possble to examne the effect of vetcal ntegaton n the post-enty game. Cobett and Kamaka (00) have examned vetcal ntegaton n two te seal netwoks, assumng that the numbes of fms n both tes ae the same, to pemt compason of the ntegated and un-ntegated cases. They fnd that when each te has a sngle fm (monopoly), ntegaton esults n hghe pofts. Howeve, when thee ae two o moe fms n each te, then the total pofts of the netwok declne. 7

19 In the dstbutve case, thee ae many moe stuctual altenatves that mght be consdeed wth espect to ntegaton too many to eally consde all. Howeve, as n the seal case, the modelng appoach developed hee allows fo any specfc case to be analyzed. What s moe, the dstbutve stuctue leads to phenomena whch do not occu n the seal case. Consde a secto that supples two downsteam sectos (Fgue 7). We can then have a stuaton whee the upsteam secto mght ntegate fowad wth one of the downsteam sectos but not the othe. As n the seal case, we can look at what happens elatve to that downsteam maket. Howeve, hee thee wll also be lateal effects on the othe downsteam secto. Recallng the esult of poposton, one can see that afte ntegaton, the upsteam secto wll no longe have to balance the downsteam sectos n ts pcng decsons, and wll take dffeent acton wth espect to the second (unntegated) secto. Fom the pont of poftablty, the upsteam secto wll see two souces of poft changes: that fom ntegatng fowad, and that fom changng ts actons wth espect to the second downsteam secto. In tun, the second un-ntegated downsteam secto may see ethe an ncease o a decease n esouce pce and theefoe ts pofts could ethe go up o down. The latte (lateal) effect s of couse a chaactestc of dstbutve netwok stuctue. Poposton Assume thee ae same numbe of fms n secto and n the two-te, thee-secto case (as n Fgue 7). Vetcal ntegaton of secto and secto always causes Q to ncease. Moeove, f a v > a 3 v 3, the esouce pce p deceases; f a v = a 3 v 3, p emans unchanged; and f a v < a 3 v 3, p nceases. Q 3 and π 3 ncease (decease) ff p nceases (deceases). Fnally, the total poft of the ntegated fms, π +π, nceases when n =; othewse, when n >, π +π could ncease o decease, dependng on the elatve value of a v and a 3 v 3. In the case whee thee ae moe than two downsteam sectos, ntegaton of the uppe te wth one of the downsteam sectos can lead to a wde ange of possble outcomes, dependng on the specfcs of the system. 7. Conclusons In ths pape, we have analyzed competton n pue dstbutve mult-echelon supply netwoks, usng the Successve Counot model fo olgopolstc competton wth multple tes. We developed explct expessons fo equlbum pces and quanttes as the solutons to a 8

20 set of lnea equatons that can be deved fom the stuctue of the netwok. The equlbum soluton s obtaned n two steps: ) Iteatvely calculate all the upsteam makets' demand paametes; ) Solve a system of lnea equatons that nvolve the demand paametes. We demonstated cetan netwok tansfomaton pncples that allow a netwok to be compessed o to be expanded to a bnay tee stuctue. These tansfomatons make t staghtfowad to examne the effects of paametc changes on the equlbum soluton to any dstbutve netwok. Fnally, we pesent some compaatve statcs esults and dscuss the effects of enty on equlbum pces, quanttes and fm-level pofts. Changes n the vaable costs of poducton have expected effects, as do changes n demand paametes. Howeve, the effects of changes n secto concentaton ae not as obvous. If the numbe of fms n a secto nceases, the quantty poduced n the secto nceases and pce chaged by fms n the secto deceases. Downsteam effects ae also as expected. Howeve, the upsteam consequences ae moe complcated n that whethe upsteam pces ncease o decease depends on the demand condtons of the secto whee the enty occus elatve to paallel paths. If enty occus along the channel wth less than the aveage esevaton pce of the upsteam maket, the upsteam pce could decease athe than ncease. Wth cetan demand condtons, the deceased esouce pce povdes a lage poft magn fo downsteam ncumbents and ths postve mpact on pofts can outwegh the competton effect due to enty and thus cause equlbum pofts to ncease wth enty. Deceased pces n upsteam can also cause upsteam fms to poft less (although fo the two-te case wth a monopoly supple n the upsteam te, upsteam pofts always ncease wth downsteam enty). Fo those fms not along the path between the oot secto and the secto wth enty, the equlbum pces, quanttes, and pofts change accodng to the change n the pce of the connectng node.e. the secto that connects the fms wth the path n queston. Thus, we see that some exstng ntutons, lagely deved ethe fom seal supply chans o fom models wth a sngle compettve secto, do not all suvve the extenson to moe complex dstbutve netwoks. The pesent analyss not only povdes esults fo geneal dstbutve supply chans, but also suggests dectons fo the analyss of othe netwok stuctues. In ongong eseach, we 9

21 ae nvestgatng the analyss of acyclc mult-echelon netwoks that have a mx of assembly, dstbutve, and non-aboescent netwok topologes. The eventual taget of ths steam of eseach s to povde obust technques to analyze competton n geneal supply chans and netwoks wth geneal stuctues. Refeences Abu, M Vetcal ntegaton, vaable popotons and successve olgopoles. Jounal of Industal Economcs Ca, M. S., U. S. Kamaka Competton n mult-echelon assembly supply chans. Management Scence Cobett, C., U. S. Kamaka. 00. Competton and stuctue n seal supply chans wth detemnstc demand. Management Scence Geenhut, M. L., H. Ohta Vetcal ntegaton of successve olgopolsts. Amecan Economcs Revew Machlup, F., M. Tabe lateal monopoly, successve monopoly, and vetcal ntegaton. Economca Quan, V., S. Roges Modelng a vendo-telco supply netwok to delve two types of telephony sevces. Computes and Industal Engneeng 46 (4) Rhm, H., T. H. Ho, U. S. Kamaka Compettve locaton, poducton, and maket selecton. Euopean Jounal of Opeatonal Reseach Seade, J On the effect of enty. Econometca, 48 () Tyag, R.K.999. On the effects of downsteam enty. Management Scence Vckes, J Competton and egulaton and vetcally elated makets. Revew of Economc Study 6-7. Zss, S Vetcal sepaaton and hozontal meges. Jounal of Industal Economcs Appendx Poof of Poposton Statng wth secto of the downsteam te, the fst equlbum cteon means that each fm n the secto selects a poducton quantty q that maxmzes ts pofts gven that the fms n the secto 0

22 puchase poducts at a cost of p 0 and ncu a vaable cost v fo evey unt poduced. A sngle fm thus seeks to maxmze evenue of q (p -p 0 -v ), whee p equals a -b Q. Dffeentaton gves us the fm's fst ode optmalty condton whch s to select quantty that solves q ( a v p0) / Q /, =,, k whee Q - s the aggegate quantty poduced by all the othe fms n secto. The poducton decson of each fm n secto follows ths same condton as well due to the dentcal cost stuctue of all fms n the secto. Ths gves us a system of n ndependent lnea equatons fo each secto. A symmetc soluton can be calculated n whch evey fm poduces quantty s q ( a v p ) / ( n ), =,, k 0 and the ente secto poduces an aggegate quantty, Q n q ( a v p ), =,, k (A) We now substtute (A) nto secto 's demand cuve to get the secto equlbum pce condton 0 0 p ( ) a ( v p ), =,, k (A) ext we look at the supple te, secto 0. Ou second equlbum cteon eques the supply and demand quanttes to be balanced, so the aggegate quantty poduced at secto 0 should be equal to the aggegate equlbum quanttes n each of ts dstbutve downsteam secto,.e., Q0 Q... Qk. Takng n the equlbum aggegate quantty Q fom (A), we can deve the demand cuve fo secto 0 fms as ( ) ( ) ( ) / Q a v p a v p k.. k.. k.. k ( a p ) and 0.. whee a0 : ( a v ) /.. k.. k : k. ow analyzng the poducton decsons fo secto 0 fms gves the equlbum poducton quanttes as Also, we can deve the secto 0 equlbum pce as q ( ) ( a v ), Q n q ( a v ) p ( ) a v. (A3) Equatons (A) and (A3) taken togethe ae a system of ndependent lnea equatons, and can be easly solved to get the equlbum pces (p 0 s aleady the equlbum pce), p [( ) a v ] [( ) a v ],... k

23 Futhemoe, substtutng these pces nto the elevant demand cuves gves the equlbum aggegate poducton quantty fo each secto. Fm level pofts can then be deved by substtutng back the optmal pces and quanttes. Q.E.D. Poof of Poposton Fom poposton, p0 ( 0) a0 0v0, whee a : 0 a v. Thus, () can be deved. Poof of Poposton 6 We know fo the dstbutve netwok n fgue 5(b), the pces and pofts of each ndvdual fm n secto ( ) ae: p ( ) a ( v p ), Q ( a v p ), ( ) ( a v p ) Moeove, t s staghtfowad to deve the followng expesson, a / v /, fo [, ]., To pove ), note that p / v ( p / v ), Q ( p / v ), ( ) ( a v p )( p / v ), we only need to show that p - / v > -. Ths can be done by nducton. Fst, fo secto, p / v ( ) / ( ) / /( ) ( ) /.,,, Assume the condton holds fo secto -. Then, fo secto -, we have p / v ( )( a / v ) ( p / v ) ' ' ( ) / -. ' ' Thus, () s poved. ow we use nducton to establsh (). Statng fom secto, we can deve the fst devatves wth espect to v as p / v ( )( a / v ) ( ) / 0,, Q / v ( a / v ) 0, / v ( ) ( a v )( a / v ) ( ) ( a v ) 0, Assumng these popetes hold fo secto -, we have the followng expessons fo p, Q, and π,

24 p a p ( ) 0 v v v Q a p a a p ( ) [ ( ) ] v v v v v v a p [ ( ) ( )].,, v v Snce, /, /, we have Q / v 0. As to the pofts of secto, a p ( ) ( a v p )( ) 0. v v v Thus () s poved. It s staghtfowad to deve (3) and (4) fom () and poposton 5. Q.E.D. Poof of Poposton 7 The poof s analogously smla to poposton 6, and s omtted hee. Poof of Poposton 8 ) Fo the thee-secto two-te netwok of fgue 4(a), accodng to poposton, the pce and poft of each ndvdual fm n secto and ae gven by: p ( ) a v, p ( ) a ( p v ), Q ( a v ), Q ( a v p ), b q ( ) ( a v ), b q ( ) ( a v p ). Teatng as a contnuous vaable and takng the fst devatve of these expessons wth espect to, we can easly show ). ) Dffeentatng p wth espect to, we get p ( a v p ) p [ a v p ( )( a v )] ( ) [( a v v ) ( a v )] / ( )( a v v ) / ( ) ( a v ) / Due to the egulaty condton, Q > 0, Q > 0, and Q 3 > 0. Takng Q, Q, and Q 3 fom 3

25 poposton and smplfyng the expessons, we have a v 0 a v v ( )( a v ) a v v [ / ( )]( a v ) 3 3 Theefoe, p ( )( a v ) ( )( a v ) a v ( )( ) 0. 3) To show the effect of on p, we take the fst devatve of p wth espect to as p a ( ) ( ) [( a v ) ( a v )] Thus, f a v > a3 v3, p/ 0; a v < a3 v3, p/ 0; a v = a3 v3, p/ 0. To show the effect of on, we have a ( ) ( a v ) ( ) ( a v ) ( ) ( a v ) ( ) ( a v ) ( a v a ) ( ) ( a v )[( a v v ) ( a v )] Theefoe, f a v v < (a v) /, / s negatve,.e., π deceases wth. If a v v > (a v) /, π ncease wth. 4) To show the effect of on, we take the fst devatve of wth espect to, whee X s defned as p ( )( a v p ) ( ) ( a v p ) ( )( a v p ) X, 4

26 X : ( a v p ) ( )( ) [( a v v ) ( a v )] [ a v v ( )( a v )] ( )( ) [( a v v ) ( a v )] [ ( )( ) ]( a v v ) [ ( ) ]( )( a v ). otce that [ ( )( ) / ] s always smalle than [ ( ) / ]. Theefoe, f a v v s vey close to ts lowe bound ( )( a v ), X could be negatve, whch means / 0. Moe specfcally, ( ) f a v v ( )( a v ), ( ) ( ) deceases wth. Othewse, nceases wth. Q.E.D. Poof of Poposton 9 Applyng poposton 3 teatvely along the path (, ) gves a ( ) ( ),, s s s ( ) ( ) s ( a v ) ( ) ( ) ( a( ) v( ) ) ( a v ) ( a v v ),,, s s s s, s s ( ) ( ) ( ) ( ) s, ( a v ). s whee v, s : kk. In a, only +, and change wth the ncease of and all the othe paametes eman constant. Theefoe, a, ( a v, ), sss ( as v, s vs) ( s ( ) ( ) ( a( ) v( ) ) ) a ( ( a v )) ( ),,,, s s s ( ) ( ) s, a ( a v,,, a v, a ( ) whee +, - =, f = -. Q.E.D. ) ( ) 5

27 Poof of Poposton 0 As n (3), p [( ) ], a v. ote both +, and a change wth. Takng fst devatve of p wth espect to and applyng (4) gve p a [ ( ) a v ] ( a v ) ( ),,, [ ( ) a v ] ( a v ),, ( ) ( a v a ) Ths can be futhe smplfed as,,, ( )( ) ( ) ( ) [( ) ( )]. p,, a v, a v, a v, a v (A4), To pove that p deceases wth, we only need to show that p / s always negatve. We do ths n two steps: ) we constuct a sees of uppe bounds U, U-,, U, and show that p / s smalle than U; ) we show that U < U- <,,< U < 0. Fst, we develop a few nequaltes used n the elaxaton. Accodng to the egulaty condton, fo a feasble dstbutve netwok, Q, Q,, Q ae stctly postve. Thus we have Q ( )( a v p ) whch gves us the followng nequalty, Smlaly, ( )( a v [( ) a v ]), ( )[( a v ) ( )( a v )] 0,,, a v ( )( a v ).,,, a v ( )( a v ),,,, a v v ( )( a v ). (A5) It deseves to be mentoned that the above nequaltes hold only when a v, > 0, whch s tue fo the dstbutve netwok to be feasble. 6

28 Moeove, snce fo any secto >,, we can deve the followng nequaltes:, 3,, (A6) ). Usng (A5), we elax p / by elmnate a v, tem n (A4) to each ts uppe bound U. In geneal, the uppe bound s constucted as [, ], k, ( k ),, U, ( )( a v, ), k k whee we defne k,k+ =. Fo =, k, ( k ), U, ( )( a v, ). k k ow, we need to pove p / s bounded by U. y combnng the a v, tems n (A4), p / can be futhe expessed as The coeffcent of a v, n (A7) s p [ ( ) ]( ) ( ) ( ).,, a v,, a v, (A7) ( ), ( ) ( ) ( ),,, ( ) ( ) ( ),, ( ) 0, (A8) whee all the nequaltes come fom (A6). Theefoe, we can elax (A7) usng (A5) as p ( ) ( a v ),,, ( ( ) ) ( )( ) k, k, a v, k k [ ( ) ] ( )( ), k, k, a v, k k U. 7

29 ).We use nducton to show that U nceases as deceases. ).We pove U < U-. Sepaatng a v, tem n U povdes U [ ( ) ]( )( a v ) k, k, k k [ ( ) ] ( )( ), k, k, a v, k k Snce, ( ) k k k k fom (A8), the a v, tem of the above expesson s negatve. Thus, elaxng t usng (A5) gves U [ ( ) ]( ) [ ( )( a v )] k, k,, k k [ ( ) ] ( )( ), k, k, a v, k k U ( ) k, k, k k, ( )( a v, ) whee the equalty comes fom collectng and smplfyng the a v, tems. ). We pove U < U- fo. Sepaatng a v, tem n U povdes Snce k, ( k ), U ( )( a v, ) k k k, ( k ),,, ( )( a v, ). k k ( ) k, k, k k ( )... ( ),,,,, ( )... ( ),, 0, we have 8

30 k, ( k ), U ( ), ( )( a v, ) k k k, ( k ),,, ( )( a v, ) k k U. k, k,, k k ( ), ( )( a v, ) ow, f we can show U < 0, the poof s completed., ( ), U ( )( a v ), ( )( a v ) 0 Poof of Poposton Fst, fom poposton, befoe ntegaton, Q s [ a v v ( )( a v )]. Afte ntegaton, t s easy to show that Q s [ a v v ], and thus, Q always nceases wth ntegaton. Smlaly, p changes fom ( ) a v to ( )( a3 v3) v,whee a [ ( a v ) ( a v )]/ ( ). Theefoe, f a v > a 3 v 3, the esouce pce p deceases; f a v = a 3 v 3, p emans unchanged; and f a v < a 3 v 3, p nceases. Q 3 and π 3 ncease (decease) ff p nceases (deceases). As to the poft π + π, befoe the ntegaton, ( ) [( )( a v ) ( a v v ( )( a v )) ]. 3 3 Afte the ntegaton, t changes to ( ) [ ( a v v ) ( a v v ) ]. Defne ont ( a v v ) / ( a v v ) as x. Afte collect tems, we have 3 3 ( ) ont ( )( a v v ) ( ) ( 3 ) x ( ) 3 3 ( ) x ( 3 ) ( )( a3 v3 v ) ( x x ). ( 3 3) whee : ( ) 3 3( 3 ), 3 : 3 3( ), and : ( 3 )

31 ote that when n (. e. 0.5), we have >0 and / 4 0.Theefoe, fo any x, 0. When n (. e. / 3), we have <0 and 4 4 [ ( ) ( ) ] Theefoe could be ethe postve o negatve, dependng on the elatve value of ( a v v ) / ( a v v ). Q.E.D