Outsourcing Service Processes to a Common Service Provider under Price and Time Competition

Transcription

1 Submtted to manuscrpt (Please, provde the mansucrpt number!) Outsourcng Servce Processes to a Common Servce Provder under Prce and Tme Competton Gad Allon Kellogg School of Management, 2001 Sherdan Road Evanston, IL , g-allon@kellogg.northwestern.edu Aw Federgruen Columba Busness School, 3022 Broadway, New York, NY 10027, af7@columba.edu In many ndustres, frms consder the opton of outsourcng an mportant servce process assocated wth the goods or servces they brng to the market. Often, competng frms outsource ths servce process to one or more common servce supplers. When they outsource to a common servce provder, ths gves rse to a servce supply chan. We develop analytcal models to characterze the benefts and dsadvantages of outsourcng n servce ndustres n whch the retalers compete wth each other n terms of the prce they charge and/or the watng tme expectatons and standards whch they adopt and sometmes advertse. We show that the benefts of outsourcng are affected by the suppler s ablty to explot the benefts of servce poolng as well as dfferences n the cost rates themselves. 1. Introducton In many ndustres, frms consder the opton of outsourcng an mportant servce process assocated wth the goods or servces they brng to the market. Often, competng frms outsource ths servce process to one of more common servce supplers. For example, nternet retalers such as Amazon, Barnes and Noble and other book dstrbuters use common carrers to delver the merchandse to ther customers. After-sales support and mantenance servces of applances and electronc equpment are often outsourced by competng frms to a common mantenance or repar servce provder. The same apples to (part of) the techncal support functon of software vendors. For make-to-order consumer goods, the potentally outsourced servce process may refer to the fnal producton or assembly stage of the product tself. Japanese automoble manufacturers, for example, have adopted make-to-order assembly systems to cut nventory costs for themselves and ther dealers, and to reduce the need for rebates for slow sellng vehcles. The outsourced servce process s often of crtcal strategc mportance to the competng frms, as the above examples ndcate: nternet book dstrbuters dfferentate themselves to a large extent n terms of the 1

2 2 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) delvery tme guarantees or expectatons they are specfyng to ther customers ( along wth prce advantages.) In many ndustres, the product tself becomes ncreasngly commodtzed and frms dfferentate themselves n terms of the qualty of the after-sales servce they provde, even though mantenance, repar and techncal support functons may fal to be part of the frms core competency and, as a consequence, are often outsourced. A recent survey by AMR, (see Bjesse et al. (2002)) estmates that after-sales servces represent, on average, 24% of the revenues and no less than 45% of the profts earned by consumer goods merchandsers. Denns and Kambl (2003) and Wse and Baumgartner (1999) report smlar results for a large verty of ndustres. (see also Cohen et al. (2006).) The attractveness of the after-sales servce process s, once agan, often determned by the watng tme customers experence before ther problem s solved, along wth the prce they pay for ther servce contract. Fnally, n the automoble ndustry, customer watng tmes become an ncreasngly mportant component of the frms compettve strateges, along wth the purchasng prce of the automoble tself, n partcular now that the qualty gap between Japanese manufacturers and ther Amercan and European compettors has shrunk sgnfcantly. For example, Toyota s goal s to reduce the average watng tme between dealer order and delvery to fourteen days; see e.g. Fay (2004). In ths paper, we develop analytcal models to characterze the benefts and dsadvantages of outsourcng n servce ndustres n whch the retalers compete wth each other n terms of the prce they charge and/or the watng tme expectatons and standards whch they adopt and sometme advertse. When some or all of the frms outsource the servce process to a common servce provder, ths gves rse to a servce supply chan. The competng retalers may represent ndependent companes or dvsons of the same corporaton. Stayng wth the example of the automoble ndustry, we have observed that most domestc, European and Japanese manufacturers have chosen to brng the market nto the frm by creatng separate and competng dvsons wth broad manageral autonomy. Perhaps surprsngly, dfferent dvsons often sell smlar cars wth dentcal chasss and engnes, often produced-to-order n a common plant. An example s Toyota whch sells, the Senna and Lexus Rx-30, wth dentcal chasss and engnes, va two ndependent dvsons, whch outsource the assembly process to common plants n West Vrgna and Canada 0. Several papers n the 0 Baye et al. (1996) document smlar practces n the md nnetes among the bg three domestc manufacturers: For example, General Motors produced the LeSabre (Buck Motor Dvson) and the Olds 88 (Oldsmoble Dvson) whch, whle dfferng n

3 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) 3 economc lterature provde theoretcal frameworks to characterze the benefts of dvsonalzaton, see e.g. Baye et al. (1996), Baker et al. (2001) and the references theren. In the general strategy lterature, several reasons are mentoned as to why frms may beneft from outsourcng: frst, the servce provder to whom the servce process s outsourced, may enjoy lower operatng cost rates, as the servce process represents hs core competency rather than that of the servce retalers. A second explanaton relates to possble economes of scale and economes of scope. The former consst of reduced per unt costs as the customer volume ncreases whle the latter ponts at the benefts of poolng demand streams nto a common servce faclty. Benson and Ieoronmo (1996) and Lacty and Hrschhem (1993), for example, dentfy cost effcences as the prncple factor drvng outsourcng decsons for mantenance functons and nformaton systems, respectvely. Gupta and Zhender (1994) emphasze economes of scale n the cost structure as a prme reason why outsourcng s often benefcal. To assess the benefts of outsourcng strateges, we need to address the followng questons: (I) When are frms better off f all of them choose to outsource rather than perform the servce n-house? Gven a unque equlbrum under n-house servce, the frms profts are unambguously specfed. Under outsourcng, these depend on () what fracton e 1 of frst best aggregate performance the servce chan acheves, and () what (mnmum) partcpaton profts P the outsde servce provder demands. For the sake of brevty and smplcty only, we present all of our results where e = 1 and P = 0, thus portrayng the outsourcng opton n the best possble lght. (In 7, we show that frst best performance can ndeed be acheved by a decentralzed servce chan, under specfc prcng schemes, the structure of whch depends on the type of competton the frms engage n.) (II) when wll a servce chan n whch all frms choose to outsource to a common provder be stable n the sense that no frm has an ncentve to unlaterally abandon the chan and provde n-house servce nstead? More generally, assumng dentcal retalers, how large a servce chan of outsourcng frms s stable? How do the answers to these questons depend on the ntensty of the competton, the number of frms n the ndustry and the sales volume of the frms? stylng, are bult on the same chasss and, comparable equpped, sell for vrtually dentcal prces. Smlarly, the Ford Motor Co. produced the Sable (Lncoln- Mercurey Dvson) and the Taurus (Ford Dvson), whch are effectvely the same car wth dfferent name plates, as are the Chrysler Corporaton s Plymouth Voyager and Dodge Caravan. Indeed, n just about every prce range, all the major domestc manufacturers have several dvsons producng competng products.

4 4 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) (III) In what drecton do equlbrum prces, watng tme standards and demand volumes change when all frms move from n-house servce to outsourcng? (IV) How do the answers to the questons rased n (I) -(III) depend on whether the outsde suppler pools the servce processes completely or n part, and whether t s able to operate at lower cost rates than the servce retalers themselves? Each of these above questons s analyzed under each of three types of competton: () Prce competton, n whch all watng tme standards are exogenously gven and the frms compete on the bass of ther prces only, () Watng tme competton: here all prces are exogenously gven and the competton s n terms of watng tme standards, and () Smultaneous competton: all prces and watng tme standards are selected smultaneously by the varous servce retalers. We represent a frm s demand rate as a general functon of all prces and all watng tme standards n the ndustry. We focus prmarly on two mportant classes of demand models: The frst class uses a separable specfcaton whch, n addton, s lnear n the prce vector. The second class s that of the attracton models. Here, each frm s characterzed by an attracton value gven by a general functon of ts prce and watng tme standard. A frm s market share, s gven by the rato of ts attracton value and the attracton values of all frms n the ndustry, that of the no-purchase opton ncluded. These broad classes of demand models allow us to represent general tradeoffs for potental customers among the prces, watng tme standards and other servce attrbutes. Prce and watng tme are treated as truly ndependent attrbutes, n that, n general, a change n a frm s watng tme standard can not be compensated by a prce change that leaves all market effects unchanged. 1 The system of demand functons determnes the rates at whch customers arrve to the dfferent servce retalers. We assume all servce retalers face Posson demand processes whle servce tmes are exponentally dstrbuted, wth a rate to be determned by the servce provder. Under outsourcng, the common servce provder may employ general dynamc prorty schemes to prortze among customers referred by 1 We refer to Allon and Federgruen (2007) for a dscusson of why t s mportant to treat prces and watng tme standards as ndependent strategc nstruments, n contrast to the tradtonal approach n the lterature on servce competton, n whch the prce and watng tme are aggregated nto a sngle, so called, full prce measure.

5 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) 5 the dfferent frms. Whoever provdes the servce ncurs two types of costs: Frst, a cost per unt of tme proportonal to the adopted capacty level and a second, transacton/handlng cost per customer served. In 3-6, we analyze the specal case where (a) all servce retalers share dentcal characterstcs, and (b) the demand functons are affne n both the prce and servce level. Ths specal case permts analytcal characterzatons of the benefts of outsourcng. The followng represent some of our man nsghts: startng wth the case where the retal frms engage n prce competton, under a gven watng tme standard, we show that they may earn lower profts under outsourcng, even f the resultng servce chan operates at maxmum effcency. The potental for such proft reductons arses, even when the servce chan under outsourcng operates at maxmum effcency (e = 1) and even f economes of scope are exploted under outsourcng, by poolng of the servce processes, and as long as the cost rates of the external suppler are not excessvely below those faced under n-house servce. More specfcally, assumng the suppler operates under dentcal cost rates as the retalers, they beneft from outsourcng f and only f ther sales volume, when prcng at cost, falls below a gven threshold. Ths threshold s a decreasng functon of the watng tme standard,.e., the more ambtous the frms are regardng ther servce level, the more lkely t s that the frms beneft from outsourcng. Only n the extreme case, where the retalers reap all the profts of the servce chan - and the suppler merely breaks even (P = 0) - are the retalers guaranteed to beneft from outsourcng. Moreover, even under ths best case scenaro for the outsourcng opton, the servce chan may fal to be mmune to unlateral defectons by one or more retalers. Such defectons wll not occur when there are up to 3 frms n the ndustry. When the number of frms N 4, defectons can be avoded f a certan ndex whch measures the ntensty of the prce competton and vares between 0 and 1, s above a threshold, gven by a functon of N alone. Ths threshold s ncreasng n N and converges to the maxmum value of one as N goes to nfnty. If the compettve ntensty s below the threshold value, defecton can be avoded f and only f the frms demand volume, when prcng at cost, s below a crtcal value, whch ncreases wth the degree to whch benefts of servce poolng are exploted under outsourcng. In other words, n an ndustry wth a large number of frms, there s an ncentve for a sngle frm to leave the servce chan, even under ths most favorable servce chan desgn, unless the prce competton s very ntense or the frm s sales volume value suffcently small. We

6 6 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) also show that there s always a chan sze m o, such that a servce chan wth m o frms s stable n the sense that no frm outsde the chan has an ncentve to jon the chan and no frm n the chan has an ncentve to brng servce n-house. When the suppler s margnal cost rate per customer s lower than under n-house servce, the ncreased benefts of outsourcng manfest themselves, for example, n that the break-even sales volume (between outsourcng and n-house servce) shfts upwards as a roughly affne functon of the suppler s cost rate advantage. Smlar conclusons preval when the frms compete n terms of ther watng tme standards and under smultaneous prce and watng tme competton. In the absence of servce poolng or the suppler enjoyng lower cost rates, whether the retalers beneft from outsourcng depends on whch fee combnaton s used, but sometmes the retalers are worse off under any of these coordnatng schemes (whch avod perodc fxed transfer payments). For example, under watng tme competton, f the external suppler servces the dfferent retalers wth dedcated facltes, the servce chan s, even n the above best case scenaro for outsourcng, always stable when N 3, never stable when N 5, and under N = 4, t s stable f and only of a smlar ndex of the ntensty of the watng tme competton s above In contrast, f the servce chan fully explots the benefts of servce poolng, the chan s always stable. More generally, under partal explotaton of the benefts of servce poolng, there exsts, as n the case of prce competton, a chan sze m o, such that a chan wth m o frms s stable. We gve a general condton under whch ths chan sze s unque, and show that t only depends on the number of frms n the ndustry and the compettve ntensty. In 7, we dentfy for the general model, payment schemes for the retalers such that, under outsourcng, the servce supply chan can operate at a frst best level (e = 1), as assumed n many of the companes n 3-6. Under prce competton, t suffces to charge each retaler a constant fee per customer, but under watng tme competton and smultaneous competton, t s necessary to add a second perodc fee whch s nversely proportonal to the watng tme standard requested. (We also show how the exact fee levels can be determned.) Alternatvely, the second perodc fee may be set to be proportonal to the capacty level whch the suppler would need to adopt f he were to serve the retalers customers n a dedcated M/M/1 faclty. In the case of smultaneous competton, a unque volume based fee and capacty based fee are

7 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) 7 requred to acheve frst best profts but under watng tme competton, a contnuum of such fee pars may coordnate the servce chan. The above fee structures may be complemented wth a perodc fxed fee to be pad by the retalers to the suppler (or vce versa). Outsourcng contracts often nclude volume based and capacty based fees, see K Djah.com and Jackson (1999), and Hasja et al. (2006) and Chander (2007). In 8 we use a numercal study to demonstrate how the above fndngs carry over n asymmetrc ndustres and when the demand functons are gven by an attracton model. 9 completes the paper wth a bref outlne of how our analyss can be extended to systems wth more general servce processes, represented by general queueng systems. The lterature on outsourcng n servce ndustres s recent. Aksn et al. (2006) and Gans and Zhou (2003) consder a sngle server retaler (e.g. call center) who can outsource part of ts busness to an outsde suppler. In Aksn et al. (2006), both frms ncur costs proportonal to ther selected capacty level, but the outsde suppler enjoys a lower capacty cost rate. Demand n any gven tme nterval can be satsfed f t falls below the total avalable capacty, wth any excess beng lost. The authors state that for tractablty reasons they do not model the servce faclty as a queue. Aksn et al. (2006) consder the followng two arrangements between the outsde suppler and the servce retaler whch are closely related to the above volume- and capacty- based fees: n the frst, the retaler buys a certan capacty level from the suppler, for whch t s charged a gven fee per unt of capacty. All demand s frst drected to the suppler wth any excess overflowng to the retaler s own faclty. In the second arrangement, demand s frst drected to the retaler s own faclty whle any excess s handled by the outsde retaler, at a gven fee per customer referred. Under ths arrangement the servce retaler and outsde suppler choose ther capacty levels noncooperatvely and the authors establsh the exstence of a unque Nash equlbrum. Gans and Zhou (2003), model the servce facltes as queueng processes, but assume that both the capacty level of the retaler and that of the outsde suppler are determned by the former. (Two customer classes are consdered; outsourcng s an opton for one of them accordng to a queue-sze dependent strategy.) To our knowledge, Cachon and Harker (2002) s the only paper whch addresses an ndustry of two competng servce retalers, modeled as M/M/1 systems. In the outsourcng part of the paper, demand rates are lnear functons of the frms full prce (= prce + multple of the expected sojourn tme). Moreover, the

8 8 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) demand of a frm s equally senstve to a change n ts own prce as to that of ts compettor. Both frms may outsource to a common suppler, who faces the same cost structure as the retalers themselves, and who serves the customers of each retaler n a dedcated faclty, for a gven fee per customer. The authors show that both frms beneft when they both outsource, compared to when they servce the customers nhouse, rrespectve of the fee charged by the suppler. But, f ths fee s set at a suffcently hgh level, one of the retalers may beneft by keepng ts servce process n-house, assumng the compettor contnues to outsource. As mentoned, these results contrast wth ours, where frms do not necessarly beneft from outsourcng, even when the servce chan operates at a frst best level (e = 1), unless the suppler s proft s reduced to zero (P = 0). Hassn and Havv (2003) and Allon and Federgruen (2007) offer surveys of competton models n servce ndustres. See also the nascent lterature on competton models n whch watng tme senstve customers are segmented nto multple classes. e.g. Loch (1991), Lederer and L (1997), Armony and Havv (2001), Afeche (2004), and Allon and Federgruen (2004). 2. Model and Notaton We consder a servce ndustry wth N competng servce retalers each actng as an M/M/1 faclty when provdng n-house servce. Each frm dfferentates tself n the market by selectng a prce p, as well as a watng tme standard w. The watng tme standard s defned as the expected steady state watng tme experenced by the customers,.e. w = E(W ). ( Alternatvely, the watng tme standard may be specfed as a gven fractle of the watng tme dstrbuton. All of our results contnue to apply, snce the structural form of all proft functons remans unchanged.) When frm serves ts customers n-house, t faces two types of cost: frst, t ncurs a cost c per customer served, and, second, t ncurs a capacty cost, at a rate γ per unt of capacty nstalled. Smlarly, when the servce processes are outsourced to an outsde suppler, ths suppler ncurs the same two types of costs, at rates c 0 and γ 0 respectvely. The capacty of a servce faclty s defned as the faclty s servce rate. Let µ,µ 0 denote the capacty levels chosen by frm and the outsde suppler respectvely. Under n-house servce, the followng smple relatonshp exsts between µ, w and λ, the frm s demand rate. Assumng λ > 0, we have µ = λ + 1 w (1)

9 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) 9 (When λ = 0,µ = 0 as well). The frst term n ths expresson s the base capacty level, ensurng stablty of the system, whle the second term s the servce based capacty requred to guarantee a gven watng tme standard. If an outsde suppler servces the customers of the dfferent frms n separate (dedcated) facltes, the requred capacty s, therefore gven by: µ B 0 = ) N =1 (λ + 1w. Allon and Federgruen (2004) have shown that, under pooled servce, the mnmum requred capacty level, consderng all non-antcpatng dynamc prorty schemes, s gven by: µ P 0 = max S {1,...,N} { λ + S S λ S λ w } = max S {1,...,N} λ + S 1 λ w S j S λ j (2) (Note, under non-dentcal watng tmes, t s no longer feasble to serve all customers on a FIFO bass) Proposton 1(a) bd show that µ P 0 µb 0, reflectng economes of scope. (More generally, the poolng of any two groups results n cost savngs.) Ths observaton, n tself, bodes well for the ncreased benefts of outsourcng under pooled servce versus servce wth dedcated facltes. The presence of economes of scope do not mply that the cost structure (2) exhbts economes of scale as well: as demonstrated n Allon and Federgruen (2004), both margnal and average costs per customer may ncrease f the demand volume of a sngle frm s ncreased. The prce p s chosen from an nterval [p mn,p max ], = 1,...,N. Clearly, frm selects a prce p whch results n a non-negatve gross proft margn p c γ. (By (1), c + γ s the margnal cost per unt of demand.) Thus, wthout loss of generalty, we select p mn to be suffcently large as to have no mpact on the equlbrum behavor. = c +γ, = 1,...,N. As to p max, t s chosen The demand rate λ may depend on all of the ndustry s prces and watng tme standards, accordng to a general set of twce dfferentable functons:λ = λ (p,w), = 1,...,N, wth λ p 0, λ w 0 and λ varyng concavely wth w,.e. 2 λ w 2 0. In our base models, we focus on the followng class of demand functons whch are lnear n the prces: (In 8, when addressng the general model, we do consder demand functons gven by attracton models.)

10 10 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) [ λ = a (w ) α j (w j ) b p + ] + β j p j, = 1,...,N (3) =j =j where x + = max(x,0). a s a decreasng concave functon, reflectng the fact that reductons of a frm s watng tme standard result n ncreases of ts demand volume; however, these ncreases become progressvely smaller as the watng tme standard contnues to be cut. The functons α j are general decreasng functons, snce a reducton of a compettor s watng tme standard results n a decrease of the frm s demand volume. The prce coeffcents b > 0, β j > 0, j satsfy the well known domnant dagonal condtons (D) b > j β j, (D ) b > j β j, = 1,...,N These condtons are generally satsfed; they merely stpulate that a unform prce ncrease by all N frms cannot result n an ncrease n any frm s demand volume, and that a prce ncrease by a gven frm cannot result n an ncrease of the ndustry s aggregate demand. 2 Let b = b β j j > 0 denote the total prce senstvty of frm s demand,.e., the (absolute value of the) margnal change n frm s demand volume due a to unform prce ncrease by all frms. In contrast, (3), n addton to enjoyng analytcal smplfcatons, specfes a frm s demand to be zero under such extreme choces. Allon and Federgruen (2007) show that under (3) the frms equlbrum choces nduce a postve market share for each. To guarantee that ths s the case, t suffces n the Prce Competton model to assume, λ (c +γ,w) > 0, = 1,...,N, w [0,w max ] N, (4).e., any frm can acheve a postve market share at least when wllng to operate wth zero varable proft margn,.e. when p = p mn = c + γ. (4) guarantees that under ths prce, λ > 0, regardless of the compettors choces. 3 However, other prce-servce level combnatons may result n zero demand. In the 2 As s well known from the lterature on olgopoly models wth product dfferentaton, systems of demand equatons need not, but often can be obtaned from one of several underlyng consumer utlty models, n partcular the representatve consumer model, the random utlty model and the address model. Smlarly, (3) may, e.g. be derved from a representatve consumer model wth utlty functon U(λ,θ) C λt B 1 λ + λ T B 1 a(w) where the N N matrx B has B = b and B j = β j, j, a(w) a (w ) j αj(wj) and C > 0. ( (D) ensures that B 1 exsts and s negatve sem-defnte, gvng rse to a jontly concave utlty functon). The demand functons (3) arse by optmzng the utlty functon subject to a budget constrant. 3 (4) reduces to lower bounds for the ntercept values: a (0) > j αj(w)+b(c + γ) j βj(cj + γj).

11 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) 11 two remanng competton models (Watng Tme Competton and Smultaneous Competton),Allon and Federgruen (2007) show that a somewhat stronger condton s needed, namely λ (p,w) > 0, p N =1 [ p mn ],p max,w [0,w max ] N 4 (5) 3. Identcal Retalers: Prce Competton We ntally focus on a basc class of models wth dentcal servce retalers, and demand functons of the quas-separable type (3). Ths allows for analytcal comparsons of the equlbra and coordnatng prcng schemes under each of the three types of competton and each of the varous outsourcng optons. Thus, assume the frms face dentcal cost parameters {c,γ } N =1 = (c,γ). Consder frm ndependent demand functons wth prce lnearty as n (3), and assume that the functons a ( ) and α j ( ) are affne as well. Thus, for postve constants a 0,a,α,b and β: [ λ = a 0 aw +α w j bp +β ] + p j, = 1,...,N. (6) j j Smlar to condton (D), we assume wthout practcal loss of generalty, that a > (N 1)α (7),.e., no frm experences an ncrease n ts demand volume when all frms ncrease ther watng tme standards by the same amount. Defne the ntensty of the prce competton by ρ (N 1)β b and the ntensty of the watng tme competton by θ (N 1)α. Note that both ρ and θ are dmensonless ndces, wth a 0 < ρ < 1 and 0 < θ < 1, by (D) and (7). In the symmetrc model, condton (5) can be replaced by a consderably weaker requrement, namely: λ (p,...,p;w,...,w) > 0 for all c +γ p p max and w mn w w max (8) In other words, when the frms offer dentcal terms to the customers, each faces an dentcal and postve 4 (5) reduces to smlar lower bounds for the ntercept values a (0).

12 12 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) demand volume. ((8) s dentcal to the followng lower bound for a 0 : a 0 b(1 ρ)p max +a(1 θ)w max.) When all N frms provde n-house servce, the proft functon for frm s, by 1, gven by ) π = λ (p c) γµ = λ (p c) γ (λ + 1w. (9) Assume frst that the frms engage n prce competton, under dentcal watng tme standards w = w. Theorem 1 n Allon and Federgruen (2007) establshes that, lke wth all prce-lnear demand functons, for dentcal as well as non-dentcal frms, the prce competton model has a unque prce equlbrum p whch satsfes the set of lnear equatons λ = b(p c γ), = 1...,N (10) where λ = λ (p,w) > 0. It s easly verfed, by substtuton n (6) and (9), that p = p a0 w(a (N 1)α) +b(c +γ) ; π = b (p c γ) 2 γ 2b (N 1)β w. (11) In partcular, by (10), the model has a unque symmetrc equlbrum, wth p = p > c +γ. We now compare the decentralzed system wth n-house servce at the servce retalers, wth varous outsourcng scenaros. We start wth a base model n whch a common suppler faces the same cost rates as the retalers encounter when provdng n-house servce. To put the outsourcng opton n the best possble lght, we assume the resultng servce chan, whle contnung to be decentralzed, operates at maxmum effcency. The profts n a centralzed chan, under outsourcng, are unaffected by transfer payments between the retalers and the suppler, and are gven by Π(p) = N λ =1 p c ( N λ N ) =1 γ λ =1 + ν. Here w 1 ν N represents an neffcency ndex. The extreme values ν = 1 and ν = N arse when the supplers explot economes of scope maxmally, or not at all. When servng the customers of dfferent retalers n dedcated facltes, ν = N, see (1); when poolng all servce processes nto a sngle faclty, ν = 1. (When the same watng tme standards and the servce tme dstrbutons apply to all customers, there s no advantage to gve prorty to some frms over others..e., customers are optmally served on a FIFO bass and the

13 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) 13 suppler s servce process s an M/M/1 system wth arrval rate N λ =1. The capacty related cost s thus ( N ) gven by γ λ =1 + 1.) More generally, the potental for economes of scale may only be exercsed n w part, gvng rse to an ntermedate value for the neffcency ndex. The functon Π s a strctly jontly concave quadratc functon of p. (In ts Hessan, the dagonal elements equal 2b, whle the off-dagonal elements equal β. Ths matrx s negatve sem-defnte,snce by (D), the absolute value of the dagonal element domnates the sum of the absolute values of the off dagonal elements n each row.) Thus, the unque optmal prce vector p CB s the one that satsfes the set of frst order condtons: b(p c γ) +λ +β j (p j c γ) = 0, (12) Ths system of equatons has a symmetrc soluton, p CB, whch s ndependent for p CB = p CB a0 w(a (N 1)α) 2(b (N 1)β) + c +γ 2 = a0 w(a (N 1)α) 2b + c +γ 2. (13) Let λ CB = λ(p CB,w) and Π CB = Π(p CB ) denotes the optmal chan-wde profts n the servce chan. Note that p,p CB,λ and λ CB are all ndependent of the neffcency ndex. Proposton 3.1 shows that each frm s optmal prce level s hgher and ts demand volume lower than under n-sourcng. Proposton 3.1(Comparng decentralzed system vs centralzed system) Assume (4). (a) p < p CB ; (b) λ > λ CB, (c) For a fxed value of b, the total prce senstvty of demand, Π CB s ndependent of ρ, whle, under n-house servce, each retaler s profts decreases wth ρ: π = ( 1 b [a 0 w(a (N 1)α)] (c +γ)b ) 2 1 ρ. (2 ρ) 2 As shown n 7 the servce chan, under outsourcng, may acheve frst best level aggregate profts (e = 1), whenever a specfc per customer fee c WB s charged to each retaler. The retalers share of these profts depends on whether a fxed per perod payment or subsdy K s added to the volume based fee. Hasja et al. (2006) s survey shows that such fxed payments are added to some of the contracts, but not to others. In the next theorem, we therefore consder three potental values for a fxed payment K: () K=0, () K s set to compensate the suppler for the cost of the servce based capacty, and () K s set to ensure that

14 14 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) the suppler merely breaks even. We are partcularly nterested n () as t reflects a best case scenaro for outsourcng: a maxmally effcent servce chan n whch the retalers share of the profts s maxmzed. Theorem 3.1(Proft comparsons between n-house servce and outsourcng) Assume the servce chan under outsourcng acheves frst best aggregate profts. For suffcently large w max : (a) N =0 π OB N =1 π regardless of what fxed transfer payments are used. Under any fxed payment K, π OB s ndependent of the suppler s neffcency ndex ν, = 1,...,N. π OB 0 decreases wth ν. (b) In the absence of fxed transfer payments (K = 0) () π OB π ff λ(c +γ,w) λ0 2 ρ = 4γb 1 ρ 4ρ ρ 2 w () π OP 0 0 for w suffcently small,.e., w < w(ν), wth w(ν) ncreasng n ν. (c) Assume K = γ,.e., each retaler compensates the suppler perodcally for the servce based capac- w ty he mposes (f served by herself): π OB 0 > 0, π OB π rrespectve of the watng tme standard w. (d) Assume the fxed payment K s set to ensure that the suppler breaks even. Then 0 π OB (ν) π = λ 2 (c+γ,w) b ρ 2 + (N ν)γ 4(2 ρ) 2 Nw Thus, wthout fxed transfer payments, whether or not the retal frms beneft from outsourcng depends on the gven watng tme standard. For any gven value of w, outsourcng s benefcal f and only f the frms demand volume, when prcng at cost (p = c + γ), s less than a gven break even value, and ths break-even volume s nversely proportonal wth the square-root of the watng tme standard. Outsourcng s benefcal only when the demand volume falls below a gven break-even pont, snce the frm s cost functon, under n house servce, s affne wth a per unt cost gven by c + γ + ( γ 1 w). In other words, the λ average cost functon exhbts economes of scale, even though the margnal cost s constant. In addton, the smaller the watng tme standard w, the more pronounced the economes of scale are, and the larger the break even volume. In the absence of fxed transfer payments, the break even volume s ndependent of the suppler s neffcency ndex ν. If the coordnatng volume based fee c WB s complemented wth a perodc fee K = γ,.e., the cost of w the servce based capacty whch the retaler mposes on the suppler, f served by hmself, the retalers are always worse off under outsourcng, rrespectve of ther demand volume. However, Theorem 3.1(d) shows

15 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) 15 that there are payment schemes under whch all retalers beneft from outsourcng and the suppler breaks at least even. The expresson, there, denotes the maxmum beneft retalers can reap when outsourcng,.e., assumng the servce chan, under outsourcng, s maxmally effcent and the retalers are able to squeeze the suppler s profts down to zero. The expresson for these maxmum benefts conssts of two parts. The frst term represents the benefts, f the suppler makes no use of the potental for servce poolng, whle the second term denotes the addtonal benefts that accrue when the suppler apples full or partal servce poolng. The benefts ncrease as the compettve ntensty ρ ncreases, for a fxed smple prce senstvty b, and even, for a fxed total prce senstvty b and hence for a fxed demand volume, when prcng at cost. They grow n proporton to the square of ths demand volume. Not surprsngly, the more effcent the suppler, the larger the maxmum benefts assocated wuth outsourcng: ndeed the maxmum benefts depend lnearly on the neffcency ndex ν. As mentoned n the ntroducton, beyond the outsde suppler s ablty to explot economes of scope by poolng the servce processes for the dfferent retal frms, a second drver behnd the benefts of outsourcng often results from the suppler s ablty to operate wth lower cost rates than the retalers. (The servce process often represents the suppler s core competency rather than that of the retalers.) We refer to the on-lne Appendx for Theorem A.1, whch generalzes Theorem 4.1 to allow for arbtrary cost dfferentals between the suppler and the retalers. (To smplfy the exposton, we confne ourselves, there, to the case where the suppler uses a dedcated faclty for each of the retalers. All results can be extended to the general case wth a general effcency ndex ν. ) Theorem 4.1 showed that n the base outsourcng model wthout fxed transfer payments, the retalng frms are better off outsourcng ff ther demand volume, when prcng at cost, s below a crtcal value. The same characterzaton apples to the case where the suppler enjoys lower cost rates. The break even value for ths demand volume ncreases wth, the dfferental n the total cost rate enjoyed by the suppler. The break even value s also decreasng n the watng tme standard w. Thus, the hgher the servce level, the hgher the mnmum demand under whch a retaler prefers preformng servce n-house. Whle ths phenomenon occurs n the base model (see Theorem 1(b)), t s all the more pronounced when the suppler operates under lower cost rates.

16 16 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) 4. Prce Competton: Stablty of Servce Chan Even f all retalers beneft from collectve outsourcng compared to all of them provdng n-house servce, t s not clear whether the servce chan, under outsourcng, s mmune to defectons by ndvdual retalers, or, whether collectve outsourcng arses as an equlbrum, when each frm has an upfront choce whether to outsource or to keep servce n-house. Ths apples even when outsourcng s portrayed n the best possble lght,.e., the servce chan acheves frst best level profts, all of whch are earned by the retalers. Thus, consder the followng two-stage game: n the frst stage each retaler decdes whether to outsource the servce process or to perform t n-house. For the frms who opt for outsourcng, the resultng chan s assumed to guarantee maxmum possble profts for ts partcpatng retalers,.e., t operates under a perfectly coordnatng prcng scheme wth fxed transfer payments, reducng the suppler s profts to zero. In the second stage game, the servce chan competes wth each of the retalers that have chosen to perform the servce n-house, once agan ensurng the frms nsde the chan of frst-best level profts, gven the competton of the frms outsde the chan. Theorem 4.1 characterzes under what condtons the soluton where all retalers decde to outsource n the frst stage, s a Sub game Perfect Nash Equlbrum (SPNE). Theorem 4.1(Sustanablty of servce chans under prce competton) (a) Let 0 ρ (N) 2(N 2 4N +5)+2 (N 2 4N +5) 2 +(N 1)(N 3) 1, and λ(ρ,n,ν) ( ) ( ) γ w 1 ν 2b+ N 1 b N+1 ρ 1. The soluton where all frms decde to outsource n stage one N 4 b2 1 ρ ρ 2 1 ρ N 1 4b and adopt the prce vector p CB n stage two s a SPNE f and only f one of the followng three condtons s satsfed: () N 3, () N 4, ρ ρ (N) () N 4,ρ ρ (N) and λ(c +γ,w) λ(ρ,n,ν) (b) ρ (N) s non-decreasng n N wth lm N ρ (N) = 1. As mentoned, the proft values n Theorem 4.1(d) s arguably the best a partcpatng servce retaler can hope for, as t maxmzes chan-wde profts whle preservng all such profts for the servce retalers and reducng those of the outsde suppler to zero. When N 3, the servce chan, under outsourcng, s mmune to defectons. However, when N 4 and the suppler fals to explot economes of scope (ν = N),

17 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) 17 Fgure 1 Stablty of the Servce Chan ρ * (N) Compettve Intensty ρ Number of Frms N an ndvdual servce retaler has an ncentve to leave the chan and brng the servce process n-house f and only f the compettve ntensty s below the crtcal value ρ (N). In other words, whether the chan s stable or not depends on whether the par of ndustry characterstcs (N, ρ) les below or above the swtchng curve {ρ (N) : N = 2,3,...} depcted n Fgure 1. (Defne ρ (2) = ρ (3) = 0.) Ths phenomenon occurs because the unmtgated competton a frm faces when defectng, acts less as a deterrent when the compettve ntensty s lower. Cachon and Harker (2002), n ther duopoly model, establshed the exstence of a lnear wholesale prcng scheme under whch both servce retalers beneft from outsourcng, the outsde suppler earns a proft and the chan s mmune for defectons. Theorem 4.1 shows that stablty of the chan, whle guaranteed under a small number of compettors, N 3, becomes ncreasngly more dffcult to acheve as the number of competng frms ncreases. The reason defecton from the servce chan may be benefcal s that the chan adopts a sgnfcantly larger prce than, say, the prce p n a decentralzed system so as to drve the profts of the partcpatng retalers to ther maxmal level. A defectng retaler, may, when ρ < ρ (N) explot ths by adoptng a sgnfcantly lower prce and thus attractng a sgnfcantly larger market share. Paradoxcally, the chan can prevent defectons, by nducng the partcpatng retalers to adopt the prce p nstead of p CB, even though the resultng profts for the retalers are lower than under the coordnaton scheme of Theorem 4.1(d) (As shown n 7, the chan may nduce the retalers to adopt any desred prce c + γ, by chargng a specfc volume based fee c W.) Stablty of the servce chan under outsourcng s enhanced to the extent the suppler explots the benefts

18 18 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) of servce poolng: when ν < N, the chan remans stable, even when N 4, and ρ ρ (N) as long as the sales volume, when prcng at cost, s below a gven threshold value λ(ρ,n,ν). The more effcent the suppler, the lower the neffcency ndex ν, and the larger the threshold λ. Smlarly, the strcter the watng tme standard or the larger the cost of capacty γ, the larger the sales volume threshold to ensure stablty, when the suppler apples partal of complete servce poolng. Fnally, t s easly verfed that any cost rate advantages for the suppler, see Appendx B, reduce the potental for defectons of ndvdual frms from a servce chan wth outsourcng: when N 3 or N 4 and ρ > ρ (N), such defectons are precluded n the base model and contnue to be precluded f the suppler enjoys cost rate advantages. At the same tme, the stablty of the servce chan contnues to preval under such cost rate advantages, even when N 4 and ρ ρ provded equlbrum demands are not too large. Theorem 5.1 dentfes the necessary and suffcent condtons for unversal outsourcng,.e., for a servce chan wth all N frms, to arse as a SPNE n the two-stage game. We now address the more general queston, when a Nash equlbrum arses wth m frms mantanng n house servce, and the remanng m o = N m frms outsourcng ther servce processes, for any 0 m N 1. In other words, when s a servce chan wth m o partcpatng frms stable, n the sense that none of the frms nsde the chan has an ncentve to brng the servce process n house, whle any of the frms provdng n house servce ncurs a strct proft loss by outsourcng and jonng the servce chan. 5 The followng Theorem shows that a (pure) SPNE exsts, for at least one value of m o, and t provdes a suffcent condton for m o, the sze of the outsourced servce chan, to be unque among all Nash equlbra. (Clearly, f a Nash equlbrum exsts wth 1 m o N frms n the chan, any of the ( N o ) m nets of o mo frms among the ndustry of N, gves rse to a Nash equlbrum.) We have observed that, except when the number of frms n the ndustry s very small (N 3), or when the compettve ntensty ρ s very large, equlbra arse wth partal outsourcng,.e. 1 < m o < N; ths, even n an ndustry where all frms have dentcal characterstcs. Let π o (,m o )[π o (o,m o )] = the proft of a frm currently n a servce chan wth m o frms, when swtchng to n-house servce [ when contnung to outsource]. 5 Ths asymmetrc defnton favors the outsourcng opton, n case of a te.

19 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) 19 π (,m )[π (o,m )] = the proft of a frm, currently among m frms provdng n house servce, when contnung to do so [ when swtchng to outsource]. Note, π o (,m o ) = π (,N m o + 1); π o (o,m o ) = π (o,n m o + 1), 1 m o N. A SPNE, wth m o frms outsourcng, exsts ff π o (o,m o ) π o (,m o ) and π (,N m o ) > π(o,n m o ) (14) Fnally, let X be a lattce wth partal order and T an ordered set. Mlgrom and Shannon (1994) defne a functon f : X T R as havng the strct sngle crossng pont property f for all x x and t T, f(x,t) > f(x,t) f(x,t ) > f(x,t ) f for all t t. Theorem 4.2 (a) The two stage outsourcng game has a (pure) SPNE (b) Assume the functon π o : X N 0 R wth X = {,o} and o has the strct sngle crossng pont property. Then, a SPNE exsts for a sngle value of outsourcng frms m o, whch decreases wth the neffcency ndex ν. Thus, under the sngle crossng pont property, the more the servce supply chan s able to explot the benefts of servce poolng, the larger a servce chan arses n the two stage outsourcng game. We now show that the sze m o of a stable outsourcng chan, depends crtcally on N, the number of frms n the ndustry, and the compettve ntensty ρ, smlar to the stablty condtons for unversal outsourcng (m o = N) n Theorem 5.3. To do so, we confne ourselves to the case where the common suppler servces each of the partcpatng frms n a dedcated faclty. In ths case, the condton for a stable chan wth m o frms reduces to a par of nequaltes, merely nvolvng m o,n, and ρ. Theorem 4.3Assume the suppler n the servce chan serves each of the partcpatng frms n a dedcated faclty. In the two stage game, an SPNE wth m o outsourcng frm arses ff ( 2 ( 2 1 (N m 1) ρ N ρ N+1 ) ρ N 1 )( 2 (m 1) ρ N 1 (N m) ) (N m)m ( ) 2 ρ N 1 2 > (15)

20 20 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) Fgure 2 Stablty of the Servce Chan Equlbrum Chan Sze Compettve ntensty ( ) ρ 2 + ρ N 1 1 (N m) ( )( ) ( N (N m) ρ 2 (m 2) ρ N 1 N 1 2 ρ N 1) (m 1)(N m + 1) 2 ( 2 ( 2 1 (N m 2) ρ N ρ N+1 ) ρ N 1 )( 2 m ρ N 1 (N m 1) ) (N m 1)(m + 1) ( ) ρ 2 + ρ N 1 1 (N m 1) ( )( ) ( N (N m 1) ρ 2 (m 1) ρ N 1 N 1 ( ρ N 1 ) ρ N 1) m(n m) < (16) We have observed that for almost all values of N and ρ, the functon π o has the sngle crossng pont property so that the equlbrum sze of the servce chan s unque. However, for some very hgh compettve ntenstes, the property may fal to hold, and multple equlbrum chan szes may arse. Fgure 2 shows for an ndustry wth N = 10 frms, how the equlbrum value(s) of m o vary as a functon of ρ; even under a low compettve ntensty, m o = 3; as ρ ncreases so does m o. When ρ ρ (N), a chan wth m o = N frms s stable, but so s one wth m o = 6 frms Identcal Retalers: Watng Tme Competton In ths secton we assess the benefts of outsourcng when the retalers compete by selectng ther watng tme standards, under a gven, exogenously specfed, prce level p. Corollary 2 n Allon and Federgruen (2007) establshes for general systems wth possbly non-dentcal retalers, that a unque equlbrum vector w exsts. In our case, ths equlbrum satsfes the frst order

21 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) 21 condtons π w = a(p c γ) + γ w 2 = 0,.e. w1 = = w N = γ. Allon and Federgruen (2007) a(p c γ) show, n fact, that ths equlbrum s a domnant soluton,.e. t s optmal for each frm to adopt ths watng tme standard regardless of the choces made by ts compettors. In comparng the case of n-house servce wth the equlbrum under outsourcng we portray the latter agan n the best possble lght,.e., we assume the servce chan under outsourcng operates n the most effcent possble way. We frst need to followng Lemma: Lemma 5.1A servce chan acheves frst-best performance, when all frms adopt a common watng tme standard w CB, both when the suppler serves the dfferent frms n dedcated facltes and when t pools the servce process. In the former case w CB = 1 1 θ w 1 γ =, n the latter case 1 θ a(p c γ) wcb = 1 1 N 1 θ w 1 γ = N(1 θ) a(p c γ) Thus, aggregate profts n the servce chan, under an optmal (common) watng tme standard w, are gven by Π CB (w) = (p c γ) N =1 λ γ N w when dedcated facltes are used, and by ΠCB (w) = (p c γ) N λ =1 γ, under servce poolng. As n the case of prce competton, we gve a unfed w treatment to both settngs, as well as ntermedate ones wth partal poolng, gvng rse to proft functon Π CB (w) = (p c γ) N λ =1 γ ν, for a general neffcency ndex 1 ν(n) N wth a correspondng w optmal watng tme standard w CB γν(n) =. Thus, N(1 θ)a(p c γ) wcb w ff θ 1 ν and N wcb ncreases n proporton to the square root of the neffcency ndex. In 7, we show that the servce chan can acheve frst best level aggregate profts under a contnuum of volume based and capacty based fee pars. We also show how the retalers and the suppler s profts are affected by the choce of a coordnatng fee par. We now show that, even when the retalers reap maxmum benefts from outsourcng the servce chan may fal to be mmune to defectons. Even more pronounced than n the case of prce competton, the ablty of the suppler n a servce chan to explot the benefts of servce poolng, has fundamental mplcatons for the stablty of the chan: wthout servce poolng the chan s stable only when N 4 (and for N = 4 only when the compettve ntensty θ 0.96). Under full servce poolng, the chan s always stable. Theorem 5.2(Sustanablty of servce chans under watng tme competton)

22 22 Artcle submtted to ; manuscrpt no. (Please, provde the mansucrpt number!) (a) The servce chan s stable,.e., the soluton where all frms decde to outsource n stage one of the outsourcng game, and adopt the watng tme standard w CB n stage 2 s an SPNE ff θ ν(n 2) (1 θ)ν(n) +2 2 (17) (N 1) (N 2)θ N In partcular, stablty s enhanced whenever the neffcency ndex ν( ) s decreased. (b) Wth dedcated facltes (ν = N), the chan s stable f N 3 or N = 4 and θ The chan s never stable when N 5. (c) Wth full servce poolng (ν = 1), the chan s always stable. Fnally, t s easly verfed, along the lnes of the above proof, that for example wth dedcated facltes, an SPNE exsts wth m o outsourcng frms ff θ N mo 1 N 1 1 θ N mo θ N mo > 2 and 2 > N 1 N 1 θ N mo θ N mo 2 N 1 N θ N mo 2 N 1 In general, Theorem 4.2 contnues to apply,.e. a pure SPNE always exsts n the two-stage outsourcng game. It nduces a unque chan sze m o f the functon π o has the sngle pont crossng property. 6. Identcal Retalers: Smultaneous Competton When the retalers compete smultaneously n terms of ther prces and watng tmes, the benefts of outsourcng are somewhat more dffcult to assess. For one, a suffcent condton s requred to ensure that a decentralzed system has a unque equlbrum whch s symmetrc and that the centralzed system has a symmetrc optmal soluton as well. We start wth the case where an outsde suppler serves the customers of each frm n a dedcated faclty. For = 1,...,N, let π (p,w) denote the equlbrum proft for retaler, under the prce vector p and the vector of watng tme standards w, and let Π CB (p,w) denote the optmal aggregate proft of the servce chan under outsourcng. Theorem (Comparson of prce and watng tme choces, wth and wthout outsourcng, under smultaneous competton) There exsts a value γ such that for all γ γ: 6 The parameter condtons are overly restrctve. They provde guarantees under the smplest verfable condton that the Nash equlbrum under n-house servce and the optmal soluton under outsourcng are symmetrc