An Incentive Effect of Multiple Sourcing *

Transcription

1 cetive Effect of Multiple Sourcig * Staley aima Serguei Netessie 1 The Wharto School Uiversity of Pesylvaia Philadelphia, P July 004 bstract: We cosider a supply chai with oe maufacturer who assembles a ed-product cosistig of two compoets ( ad, where each compoet is purchased from multiple outside suppliers. The maufacturer s decisio as to the cotract to offer each supplier is complicated by two factors: (1 each potetial supplier is self-iterested ad privately-iformed as to his margial cost of productio; ad ( ay parts delivered i excess of parts (or vice versa caot be ivetoried but must be disposed of at o value. Thus, the maufacturer must choose the cotract to offer each so as to coordiate or balace their productio, while takig ito cosideratio their selfiterested behavior. We demostrate that icreasig the umber of suppliers mitigates (but does ot etirely elimiate the productio distortios arisig from these two factors. Hece, we show that icreasig the supplier base ca have efficiecy ehacig effects that have ot bee previously described i the literature. *We gratefully ackowledge the helpful commets of Madhav Raja, Stefa Reichelstei, Gerard Cacho, Mau Goyal, Marcelo Olivares ad uqiag Zhag. We also thak Paul ischer ad the participats of the ccoutig Workshop at Columbia Uiversity, especially Tim aldeius ad Nahum Melumad, for very helpful commets o a earlier draft of this paper. 1 Correspodig author, etessi@wharto.upe.edu

2 cetive Effect of Multiple Sourcig page 1. troductio Two major treds i maufacturig withi the last two decades have bee: the icreased use of outsourcig of parts ad subassemblies ad the reductio i the umber of first-tier suppliers to whom these parts ad subassemblies are outsourced (see, for example, (McMilla, The efficiecy effect of the umber of first-tier suppliers is usually aalyzed i terms of maufacturig (disecoomies of scale, risk of supply disruptio ad coordiatio costs. While these three effects have bee studied before, we demostrate that there is aother, icetiverelated, efficiecy effect associated with the umber of suppliers which has ot bee studied i the literature. 3 This icetive-related efficiecy effect arises whe the suppliers are privatelyiformed ad self-iterested ad there is a eed to balace their productio (as is the case i assembly systems. 4 To demostrate this icetive efficiecy effect, we model a simple assembly process with a maufacturer (referred to as the buyer ad she whose ed-product is made up of two compoets (parts ad. The buyer outsources the productio of both parts to suppliers (each referred to as him ad merely assembles the ed-product from the parts delivered. Each supplier s cost to maufacture his parts is ucertai at the time of cotractig. efore each supplier produces his parts, he lears his cost realizatio ad commuicates it to the buyer. The latter the iforms each supplier of the desired productio schedule. The icetive efficiecy effect associated with the umber of suppliers arises from the followig two assumptios. irst, the suppliers are privately-iformed about their cost realizatios ad are self-iterested; each acts so as to maximize his ow expected profit. s a result, the privately-iformed suppliers will ot ecessarily truthfully commuicate their cost observatios to the buyer (this is referred to as the adverse selectio or supplier-icetive problem. Secod, each of the buyer s ed-products requires oe part ad oe part ad, because of spoilage, fashio cocers, or other costraits, the buyer caot ivetory ay uused parts for use at some later date. Thus the Coordiatio costs typically iclude the cost of commuicatig with ad keepig track of the activities of suppliers. 3 or the effects of supply chai disruptios see, for example, (Reitma, 1997, (Latour, 001 ad (Hedricks ad Sighal, 003. or a discussio of coordiatio costs see, for example, (McMilla, See (aima ad Raja, 00 for a survey of the research literature which addresses the icetive implicatios of other sourcig decisios.

3 cetive Effect of Multiple Sourcig page 3 buyer has to balace or coordiate the suppliers productio schedules because ay parts delivered i excess of parts (or vice versa are valueless ad must be discarded (this is referred to as the productio-balacig problem. The presece of these two assumptios iduces the buyer to distort the optimal productio schedules offered to the suppliers (relative to the optimal productio schedules i the absece of a supplier-icetive problem. This productio distortio reduces the buyer s efficiecy ad expected profit. We show that icreasig the umber of suppliers reduces the relative size of each supplier s productio distortio, thereby icreasig efficiecy. 5 The reaso for our result is that icreasig the umber of suppliers reduces the effect that ay oe supplier ca have by misrepresetig his private iformatio, thereby reducig the productio distortio required to elicit the truth ad improvig coordiatio. This result relies o both of our earlier stated assumptios. With o supplier-icetive problem there is o eed to distort productio i order to truthfully elicit each supplier s cost realizatio. With o productio-balacig problem, the optimal cotracts treat each supplier as idepedet; hece there is o relative productio distortio reductio effect associated with the umber of suppliers. The paper is orgaized as follows. Sectio reviews the relevat literature. sectio 3 we discuss i more detail the basic assumptios uderlyig the model. We also solve for the optimal supplier cotracts ad supplier productio schedules assumig the productio-balacig problem but assumig away the supplier-icetive problem. That is, we assume that the suppliers hoestly report their iformatio ad implemet whatever productio schedules the buyer gives them. This is referred to as the irst-est solutio ad provides a bechmark for our subsequet aalysis. We eed this bechmark because, eve without a supplier-icetive problem, varyig the umber of suppliers affects the optimal productio. sectio 4 we derive the optimal supplier cotracts ad productio schedules whe there is the supplier-icetive problem (this is referred to as the Secod-est case. We the compare the optimal solutios derived i sectios 3 ad 4 to isolate the icetive-efficiecy effect of the umber of suppliers. Sectio 5 provides a coclusio. 5 creasig the umber of suppliers ca also lead to improved cotractig via relative performace evaluatio whe the agets are subject to correlated shocks (see (Holmstrom, 198. This effect does ot arise i our model because our suppliers are subject to ucorrelated shocks.

4 cetive Effect of Multiple Sourcig page 4. Literature Review The preset paper lies at the itersectio of the procuremet/assembly literature i supply chai maagemet ad the cotractig/icetives literature i ecoomics/accoutig. the assembly literature, as i our paper, the maufacturer (assembler costructs the ed-product from parts provided by suppliers. Oe subset of this literature looks at the buyer s problem of coordiatig the suppliers productio schedules i a cetralized settig (i.e., there are o icetive problems i motivatig the suppliers see, for example, (Sog et al., 000 ad citatios therei. More closely related to our work is the subset of the literature that looks at the buyer s problem of coordiatig the suppliers withi a decetralized settig the buyer ad suppliers make their respective decisios i their ow best iterests based o their cotracts. much of this literature, ulike i the preset paper, there is o uderlyig icetive problem ay coflict betwee the iterests of suppliers ad the buyer is caused solely by the assumed cotracts betwee them (e.g., (Wag ad Gerchak, 003 ad (erstei ad DeCroix, 003. That is, it is the use of o-optimal cotracts which create the supplier-icetive problem. Closest to our work is (Gurai ad Gerchak, 003 which does aalyze a settig i which there is a uderlyig icetive problem ivolvig the suppliers. that paper each supplier s actios are subject to moral hazard ad the paper aalyzes the efficiecy effects of two differet (ooptimal cotracts betwee the buyer ad suppliers. cotrast, we study a decetralized decisio situatio i which each supplier is privately iformed about his cost of productio (i.e., a adverse selectio problem ad aalyze the efficiecy effects associated with the optimal (screeig cotract. Much of the procuremet literature (see the survey by (Elmaghraby, 000 aalyzes the allocatio, to oe or more privately-iformed suppliers, of the right to provide a sigle good or service to the buyer. Whe, istead, the right is to supply multiple uits, it is typically assumed that they are idetical ad that there are o exteralities amog uits or the suppliers. 6 Our paper is differet from this literature i a umber of ways. irst, while much of this literature restricts 6 See, for example, (Stole, 1994, (Rob, 1986, (Laffot, 1993, (Riorda ad Sappigto, 1989, (Seshadri, 1991 ad (Klotz, 1995.

5 cetive Effect of Multiple Sourcig page 5 itself to a auctio approach, we take a more geeral optimal cotractig approach. 7 Secod, i our model, the goods beig supplied are ot idetical ad are ot used idepedetly. particular the buyer must cofrot productio-balacig issues amog her suppliers. The efficiecy effect of varyig the umber of suppliers i the preset paper is therefore differet from that i the procuremet literature. 8 s oted above, our work also fits withi the ecoomics/accoutig literature dealig with icetives ad orgaizatioal desig i that a assembly system ca be iterpreted as a orgaizatio i which a pricipal combies iputs from subordiate departmets. (Demski ad Sappigto, 1984 ad (Ma et al., 1988 study the implicit coordiatio amog the agets that may arise as a result of their cotracts. Uder our equilibrium cocept implicit collusio does ot arise. (Harris et al., 198 studies the iteral resource allocatio process i which the pricipal assembles a ed-product made up of parts produced by the privately iformed divisios. However, it assumes that each divisio is a sole source for the part it supplies, while our focus is o the value of multi-sourcig. 9 (Ziv, 1993, (Ziv, 000 ad (alakrisha et al., 1998 study the effect of a firm s moitorig techology o its orgaizatioal desig. (alakrisha et al., 1998 focuses o differet ways of orgaizig a fixed umber of workers (team vs. idividual productio. Our paper ad (Ziv, 1993 ad (Ziv, 000 study the effect of varyig the umber of workers. There are two major differeces i the assumptios which uderlie the works of Ziv ad the preset paper. irst, differet assumptios are made with respect to productio exteralities amog the workers/suppliers. Secod, the models aalyzed by Ziv do ot icorporate adverse selectio o the part of the workers/suppliers, while the model i the preset paper does. While the above icetives literature assumes complete cotractig, aother subset allows for icomplete cotractig. 10 With icomplete cotracts, the umber of suppliers with whom a buyer chooses to deal, as well as the way i which the buyer orgaizes them, ca affect 7 (Tuca ad Zeios, 004 study the relative efficiecy of procuremet auctios versus log-term relatioal cotractig. 8 (Cacho ad Zhag, 003 studies the effect of creatig exteralities amog the suppliers through competitio. 9 lso, see (aima et al., 004 for a assembly model with multiple parts but sigle-sourcig of each part. 10 cotract is icomplete if it does ot specify the desired behavior by the cotractig parties for each joitly observable future evet. Whe a uspecified future cotigecy does occur, the parties must reegotiate the cotract.

6 cetive Effect of Multiple Sourcig page 6 the rage of self-eforcig behaviors of both the buyer ad the suppliers. or example, see (akos ad ryjolfsso, 1993 ad (Levi, The cotracts aalyzed i the preset paper are complete ad, therefore, these issues regardig the umber of suppliers do ot arise. ially, there is the literature which studies the strategic effects of outsourcig, (see, e.g., (Shy ad Stebacka, 003 ad (Cacho ad Harker, 00. No such strategic effects exist i our model because the buyer sells her uits to a competitive market. 3. ssumptios, otatio ad first-best aalysis a. ssumptios ad otatio Our model cosists of a oe-period world with oe risk-eutral buyer ad risk-eutral suppliers. The buyer maufactures a ed-product made from oe part ad oe part. Part is supplied by a set of suppliers (referred to as part suppliers ad part is supplied by a o-overlappig set of suppliers (referred to as part suppliers. 11 The buyer cotracts with the suppliers ad assembles the ed-products from the parts delivered by them. The suppliers productio fuctios have a radom compoet (that might be due to ucertaity i equipmet quality/yield, radom disruptios i the productio process, reliability of the trasportatio, currecy fluctuatios etc. that is modeled as follows. The cost to supplier i (the ith supplier of part of supplyig x parts is ( ( i C + c x i, where C( is the fixed cost which the ij supplier must icur i order to produce ay uits of ad c ij is supplier i s jth cost realizatio. 1 The same holds true for the part suppliers. We further assume that: C( is the same kow costat for both the ad suppliers; the suppliers cost realizatios are draw from idetical but idepedet probability distributios; there are oly two cost realizatios, c1 ad c, where c1 < c < occur with probabilities 0< p < 1 ad 1- p, respectively; ad the 11 The assumptio that the umber of ad suppliers is the same ca be easily relaxed. 1 We allow the cost icurred by each supplier to be a fuctio of the umber of other suppliers of that part. Oe reaso for this is that a equivalet iterpretatio of C( is that it represets the additioal commuicatio/coordiatio cost which the buyer icurs for each supplier.

7 cetive Effect of Multiple Sourcig page 7 iformatio about the cost distributio is commo kowledge. 13 c 1 is referred to as the low-cost realizatio ad c is referred to as the high-cost realizatio. t the start of the period, o oe has yet observed the cost realizatios, but everyoe shares commo beliefs about those cost realizatios. The game proceeds as follows: Step 1. The buyer offers a cotract to each supplier which stipulates how may parts each will be asked to produce ad how much each will be paid for doig so, depedig upo the iformatio subsequetly commuicated to the buyer by all suppliers. 14 Step. Each supplier privately observes his ow cost realizatio ad commuicates iformatio about it to the buyer. Each supplier chooses his message so as to maximize his expected profit give the cotract offered i Step 1. The suppliers messages are joitly observable ad, hece, cotractible. Step 3. fter receivig the suppliers messages, the buyer commuicates to each supplier the umber of parts to be delivered ad the paymet for doig so, cosistet with the cotract terms egotiated i Step 1. Step 4. t this poit, each supplier has the right to abrogate the cotract egotiated i Step 1 ad leave the supply chai uless the umber of parts ordered ad the promised paymet at least satisfy his best outside opportuity, assumed to be zero profit. f ay supplier does abrogate his cotract, the maufacturer caot reegotiate the cotract terms for the remaiig suppliers. 15 f the supplier hoors the cotract he first ivests C( ad the produces the required umber of uits. Step 5. Give that the cotract satisfies each supplier s outside opportuity, each supplier delivers the umber of parts specified by the buyer. Step 6. The buyer receives the parts ad assembles oe fiished ed-product from oe part ad oe part ad sells each fiished ed-product for $R per uit. f more tha 13 We assume that the suppliers are ex ate idetical ad that there are oly two potetial cost realizatios i order to more clearly focus o the icetive-efficiecy effects of the umber of suppliers. The qualitative results would still hold eve if the ad suppliers were ot ex ate idetical ad if there were more tha two potetial cost realizatios. 14 This is referred to as a screeig cotract i ecoomics literature. the above settig, this cotract is optimal for the maufacturer (see (Kreps, brogatio of the cotract by a supplier is a off-equilibrium evet ad, thus, will ever occur. However, we eed this assumptio (or equivaletly that the maufacturer caot desig the cotract to icorporate a two stage process i case a supplier abrogates i order to have a o-trivial productio-balacig problem.

8 cetive Effect of Multiple Sourcig page 8 parts are delivered (or vice versa, the excess parts caot be used ad are costlessly disposed. 16 b. irst-est alysis s stated above, we are iterested i studyig the icetive-efficiecy effect of the umber of suppliers. To do so, we must hold costat the other efficiecy effects that the umber of suppliers might have. or example, varyig the umber of suppliers also affects optimal productio eve without the supplier-icetive problem because of its effect o the margial cost of productio. Therefore i order to separate the icetive from the o-icetive efficiecy effects of varyig the umber of suppliers, i this sub-sectio we derive the solutio to the problem described i the previous sectio, but with the supplier-icetive problem assumed away. This is referred to as the irst-est solutio ad is comparable to the solutio derived i the cetralized assembly literature. 17 That is, we first study the case i which each supplier hoestly commuicates his cost realizatio to the buyer ad produces whatever the buyer asks, as log as doig so leaves him o worse off tha his outside opportuity wage of zero expected profit. We restrict our aalysis to symmetric solutios. ssume that the cost realizatio is ( k, j, i.e., there are k low-cost part suppliers, -k high-cost part suppliers, j low-cost part suppliers, ad -j high-cost part suppliers. Let: x = the umber of uits a part supplier with a cost realizatio of give the ( k, j cost realizatio. T = the compesatio to a part supplier for producig x. c is directed to provide The order sizes ad compesatios for suppliers with c 1 cost realizatios ad for suppliers are defied aalogously. 16 The aalysis would be qualitatively the same as log as there is some cost to havig a excess of part or. 17 See, for example, (Sog et al., 000.

9 cetive Effect of Multiple Sourcig page 9 The total umber of parts produced ad delivered to the buyer is thus ( k x kx 1 +, the total umber of parts produced ad delivered to the buyer is ( j x jx 1 +, ad the total umber of ed-uits which the buyer ca produce is (( k x 1 ( kx j x jx1 mi + ; +. The mi(..;... fuctio captures the fact that ay parts i excess of parts (or vice versa have o value (i.e., the productio-balacig problem. This effect arises i may maufacturig/supply chai settigs (e.g., JT productio i which it is costly to have a excess or shortage of ay oe part. t is the usual objective fuctio i the assembly literature (e.g., (Wag ad Gerchak, 003. Give that uder irst-est the buyer observes the cost realizatios before decidig o productio, she will choose ( x, x1, x, x 1 such that ( ( k x + kx = j x + jx =, so as to avoid payig for ay uused 1 1 productio. urther, give that each supplier s outside opportuity is assumed to be zero, each is reimbursed oly for his out-of-pocket productio cost T C( c ( x = +, = ( + (, etc. The buyer solves for the optimal (, 1,, 1 T C c x realizatio i two steps: 1. or each (, x x x x for each ( k, j k j realizatio, choose the cost miimizig vector (, 1,, 1 satisfies ( ( k x + kx = j x + jx =, for some is used to idicate irst-est or each ( k, j realizatio, choose the expected profit maximizig x x x x that, where the superscript. Lemma 1: The optimal productio orders for each ( k, j realizatio are: Lemma 1 is straight-forward ad offered without proof. Other proofs are icluded i the ppedix.

10 cetive Effect of Multiple Sourcig page 10 x 1 x x 1 x ( 1+ ( 1 1+ ( 1 cc 1 c1+ ( j/ + k/ ( c c1 R( c1+ ( c c1 j/ ( + ( + ( R( c1+ ( c c1 j/ ( + ( + ( R( c1+ ( c c1 k/ ( + ( + ( R( c1+ ( c c1 k/ + ( + ( / ( / ( R c c c k c c c j = = c c j/ k/ c c = c c j/ k/ c c 1 1 = c c j/ k/ c c = c c j/ k/ c ( c 1 1 (1 The effect of the productio-balacig problem is maifested i (1 by the fact that the required productio by each supplier is a fuctio of the umber of low-cost ad suppliers, k ad j. 19 The same is true for each supplier. Deote the buyer s profit for the cost realizatio ( k, j by ( 1 c1+ ( k + j( c c1 ( c1+ k ( c c1 ( c1+ j ( c c1 π calculated as follows: π = R cc C fter substitutig i the optimal, we obtai R ( c1+ k( c c1 ( c1+ j( c c 1 R π = C ( = C (. 4cc 1 ( c1+ ( j+ k( c c1 or each ( k, j realizatio, total expected profit is liear i total productio so that, except for the fixed cost charge, the effect of o total expected profit ad o total expected productio is j,, 1! ( 1! k k P k j = p p p p represet the j! j! k! k! proportioal. 0 j Let ( ( ( ( probability of the (k,j cost realizatio give that there are suppliers for each part. The buyer s total expected profit is the: (. 19 the absece of the productio-balacig problem, each idividual supplier s productio would be idepedet of the umber of the other suppliers ad their cost realizatios. See the later discussio. 0 This effect is due to liearity of the reveue fuctio.

11 cetive Effect of Multiple Sourcig page 11 R P( k, j, C(. ( j= 0 k= 0 4. cetive problem a. Secod est aalysis We ext aalyze the Secod-est case, i which the buyer faces self-iterested suppliers ad must iduce them to reveal their privately-observed cost realizatios ad follow the productio schedule give to them. oth the irst-est ad Secod-est formulatios icorporate the productio-balacig problem. Thus, the oly differece betwee the irst-est solutio derived i sectio 3 ad the Secod-est solutio derived i this sectio is the existece of the supplier-icetive problem. Hece, ay differece betwee the solutios, ad ay differece i the effect of the umber of suppliers o those solutios, is due totally to the supplier-icetive problem. Give the Revelatio Priciple (see (Kreps, 1990, we ca, without loss of geerality, restrict the space of cotracts to those which iduce each supplier to hoestly reveal his cost realizatio to the buyer ad implemet the specified productio schedule. 1 The Secod-est problem ca the be stated as: 1 The usual formulatio of the Truth-Tellig costrait uses the ayesia Nash equilibrium cocept which states that each supplier s expected profit, give his cost realizatio ad expectig over the other suppliers messages, must be at least as great from tellig the truth as from lyig. problem with the ayesia Nash equilibrium i a situatio with multiple suppliers is that it ca result i cotracts for which, i additio to the truth-tellig equilibrium, there may exist a o-truth-tellig equilibrium which all of the suppliers prefer but which makes the buyer strictly worse off (see (Demski ad Sappigto, that case, the suppliers ca implicitly collude to implemet their preferred o-truth-tellig equilibrium, rather tha the buyer s preferred truth-tellig equilibrium. To avoid that situatio, we use the Domiat Strategy equilibrium cocept which is usually more restrictive. t states that each supplier s expected profit, give his cost realizatio, must be at least as great from tellig the truth as from lyig, regardless of the other suppliers messages. With the Domiat Strategy equilibrium, there is o possibility of implicit collusio, because each supplier prefers truth-tellig regardless of what the other suppliers decide to do (see (Kreps, ortuately, give our model, the use of the more restrictive Domiat Strategy equilibrium is costless because the optimal solutio usig the ayesia Nash equilibrium ca be equivaletly implemeted i domiat strategies. or sufficiet coditios for this to be true, see (Mookherjee, 199.

12 cetive Effect of Multiple Sourcig page 1 { (( ( } R mi k x + kx1 ; j x + jx 1 max P( k, j, C(. { x, x1, x, x1} j= 0 k= 0 kt1 ( k T jt1 ( j T subject to, for each feasible ( k, j cost realizatio: ( ( ( ( ( ( ( ( T c x C 0 T c x C 0 T c x C 0 T c x C 0 (LL ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( T c x C T c x C ( k 1 j 1 ( k 1 j T c x C T c x C ( k 1 j ( k 1 j 1 1 T c x C T c x C k ( j 1 1 k ( j 1 T c x C T c x C k( j 1 k( j (TT where ow: x = the umber of uits a part supplier with a reported cost realizatio of to provide give that there were a total of k part suppliers who reported c 1 c is directed realizatios, -k part suppliers who reported c realizatios, j part suppliers who reported c 1 realizatios, ad -j part suppliers who reported c realizatios. T = the compesatio to a part supplier for reportig c ad producig x. The order sizes ad compesatios for the part suppliers with c 1 cost realizatios ad for suppliers are defied aalogously. The Limited Liability costraits (LL idicate that uless the productio ad compesatio plas offered to each supplier assure at least zero profit for each cost realizatio, he will leave the supply chai. The Truth-Tellig costraits (TT restrict (without loss of geerality our attetio to truth-tellig equilibria. or example, cosider the viewpoit of a supplier with a low-cost realizatio. ssume that that the other -1 part suppliers ad the

13 cetive Effect of Multiple Sourcig page 13 part suppliers report that there are k-1 low-cost part suppliers, -k high-cost part suppliers, j low-cost part suppliers ad -j high-cost part suppliers. The the TT costrait esures that low-cost part supplier should prefer truthfully revealig that he is low-cost ad havig the buyer believe that she is facig the (k,j cost realizatio rather tha lyig ad havig the buyer believe that she is facig the (k-1,j cost realizatio. Notice that a part supplier s reportig decisio affects both what he reports his cost realizatio to be ad the total umber of reported part low-cost realizatios. urther, the messages reported by the other suppliers affect each supplier s productio ad compesatio pla, but ot that particular supplier s cost of fulfillig the pla. That is, the other suppliers cost realizatios oly have a idirect effect o each supplier s utility (i.e., through the offered pla. The followig Lemma uses the dividual Ratioality ad Truth-Tellig costraits to provide a iitial characterizatio of the optimal solutio to the Secod-est problem formulatio: Lemma : The optimal solutio to the Secod-est problem has the followig characteristics: ( ( ( ( T c x C > 0 k, j where k 0 T c x C > 0 k, j where j k 1 j k 1 j ( ( k j 1 1 ( ( ( k j ( T c x C = 0 j, k where k 0 T c x C = 0 k, j where j 0 ( (, ad 0 ad x x j k x x k, ad j 0 1 ( k 1 j 1 k( j 1 ( ( ( ( ( ( ( ( ( k 1 j 1 ( k 1 j T = C + c x + c x c x j, k where k 0. T = C + c x + c x c x k, j where j k ( j 1 1 k ( j 1 Our subsequet aalysis igores the costrait embodied i Result 3 of Lemma. However, the resultig optimal solutio satisfies it as a strict iequality, so igorig the costrait is without loss of geerality.

14 cetive Effect of Multiple Sourcig page 14 The characterizatio i Lemma is typical of adverse selectio problems such as ours, i which the agets (i.e., suppliers have superior iformatio to that possessed by the pricipal (i.e., buyer. The low-cost suppliers productio is at least as great as the high-cost suppliers (Result 3, the high-cost suppliers just ear their outside reservatio wage (Result, ad the low-cost suppliers ear more tha their outside reservatio wage (Result 1. This extra amout eared by the low-cost supplier is referred to as a iformatioal ret. 3 This ret is the cost to the buyer associated with the supplier-icetive problem. order to properly coordiate productio, the buyer must elicit the true cost realizatios from the ad suppliers. ut to do so, the buyer must share some of her profit with the low-cost suppliers (i.e., bribe the supplier with the iformatioal ret. rom Result 4 of Lemma, the buyer ca reduce the low-cost supplier s ( iformatioal ret by decreasig the high-cost supplier s productio x ( k 1 but this will j distort the high-cost suppliers productio away from irst-est ad result i ubalaced ad parts productio. Of course, the resultig productio mismatch ca be mitigated by adjustig the low-cost supplier s productio ad/or both suppliers productio. ut ievitably, the presece of the supplier-icetive problem forces the buyer to trade-off iformatioal rets ad productio efficiecy. Propositio 1, below, further characterizes the Secod-est Problem. Propositio 1: t optimality, ( ( Secod-est problem ca be restated as max max { 1, 1} { } x x j= 0 k= 0 jx 1 jc1 x1 j c where pc ( c/ ( 1 p 1 (,, C( k x + kx = j x + jx = j k ad the 1 1, kx 1 R kc1( x1 ( k c + ( k ( ( ( + ( j P k j ( = ad the superscript refers to the optimal Secod-est (or cetive solutio. (3 ( j 3 or the low-cost supplier the iformatioal ret is ( c c1 x( k 1, see Result 4.

15 cetive Effect of Multiple Sourcig page 15 Propositio 1 demostrates that the Secod-est problem is exactly the same as the irst- est, except that, i the Secod-est case, c is everywhere replaced by c +. Hece, the same is true for the Secod-est solutio. c + is referred to as the buyer s virtual cost (see (Laffot ad Martimort, 00. The represets the additioal cost which the buyer must bear as a result of the supplier-icetive problems. Thus, the optimal solutio for the Secod-est problem is: 4 ( 1( + ( 1 1( + ( 1 = c1( c pc1 c1( 1 p + ( j/ + k/ ( c c1 R( c1( 1 p + ( c c1 j/ x1 = c1( c1( 1 p + ( j/ + k/ ( c c1 R( 1 p ( c1( 1 p + ( c c1 j/ x = ( c pc1 ( c1( 1 p + ( j/ + k/ ( c c1 R( c1( 1 p + ( c c1 k/ x1 = c1( c1( 1 p + ( j/ + k/ ( c c1 R( 1 p ( c1( 1 p + ( c c1 k/ x = ( c pc c ( 1 p + ( j/ + k/ ( c c ( ( R c 1 p c c k/ c 1 p c c j/ ( (4 The differece betwee the optimal solutio to the irst-est problem (1 ad the Secod-est problem (4 is due solely to the supplier-icetive problem. alogous to the irst- est case, the buyer s profit for the cost realizatio ( k, j is: ( ( c + k + j c + c π = R c c + C. fter substitutig i the optimal 1 1 ( 1( ( c1+ k ( c + c1 ( c1+ j ( c + c1, we obtai ( 1+ ( + 1 ( 1+ ( + 1 4c1( c + ( c1+ ( j+ k( c + c1 R c k c c c j c c R ( ( C ( π = C =. The buyer s total expected profit is: 4 Recall that we iitially igored Result 3 from Lemma i our subsequet aalysis. However, it is straightforward to verify that the optimal solutio i (4 satisfies it as a strict iequality for all k, j.

16 cetive Effect of Multiple Sourcig page 16 R P( k, j, C(. (5 j= 0 k= 0 Thus, for each ( k, j realizatio the icetive efficiecy effect of the supplier-icetive problem is captured by the distortio i total productio - vs.. Therefore, we ext aalyze the productio distortios, begiig with the distortios i the idividual supplier s orders. b. Productio distortios or the high-cost part supplier (the same is true for the part supplier it is straightforward to show that: x / x > 1 ( x x / < 0 ( x x lim / > 1 The supplier-icetive problem causes each high-cost supplier to always produce less tha irst- est, regardless of the cost realizatio. or ay give (k,j cost realizatio, this relative distortio i idividual productio is mitigated by a icrease i the umber of suppliers, but is ever elimiated. Thus, icreasig the umber of suppliers mitigates the distortio caused by the supplier-icetive problem for the high-cost supplier ad, hece, has a positive icetive effect o idividual high-cost supplier productio. (6 or the low-cost (ad suppliers oe ca also show that: whe k > j, x / x > 1, x / x < 1; whe k = j, x / x = x / x = 1; whe k < j, x / x < 1, x / x > 1; ( x1 x1 ( x1 x1 lim / = lim / = 1. (7

17 cetive Effect of Multiple Sourcig page 17 The Secod-est solutio is such that the low-cost part ( supplier will produce efficietly oly whe k = j, whe the proportio of low-cost part ad suppliers is the same. However, give the supplier-icetive problem, the low-cost ( supplier will produce more (less tha the efficiet amout whe k < j ad less (more tha the efficiet amout whe k > j. s with the high-cost supplier, the relative productio distortio for the low-cost supplier varies i. However there are two differeces betwee the high ad low-cost suppliers. irst, for ay fixed (k,j realizatio, this distortio disappears as gets large for the low-cost supplier but ot for the high-cost supplier. Secod, the effect o the low-cost relative productio distortio depeds o the specific parameters, while it is uambiguous for the high-cost supplier. To put (6 ad (7 i cotext, recall that that our two mai assumptios are: a suppliericetive or adverse selectio problem (i.e., privately-iformed ad self-iterested suppliers ad a productio-balacig problem (i.e., the buyer pays for but obtais o value from ay parts delivered i excess of parts (or vice versa. With o supplier-icetive problem, irst-est ca always be achieved ad there is o distortio i productio. With a supplier-icetive problem but o productio-balacig problem, which is the stadard adverse selectio case examied i the literature, the low-cost supplier always produces the irst-est quatity, while the high-cost supplier always produces less tha irst-est, ad the idividual supplier productio distortios are ivariat to the umber of suppliers. 5 Hece, i this case the umber of suppliers has o icetive-related effect o idividual relative productio distortio. With both the supplier-icetive problem ad the productio-balacig problem, both low-cost ad high-cost suppliers produce iefficietly ad each idividual supplier s relative productio distortio varies i the umber of suppliers. Thus, it is the existece of the supplier-icetive problem that gives rise to productio distortios, but it is the presece of the productiobalacig problem, ormally associated with assembly operatios, which results i idividual supplier relative productio distortios that vary i the umber of suppliers. 5 model comparable to ours, but without the productio-balacig problem is: there is a ed-product, ad a ed-product; each ed-product is made up of oe part; each ed-product is made up of oe part; each edproduct is sold for $R/ per uit. this case, irst-est low ad high-cost productios are ( R /4 c1, R/4c ad Secod-est low ad high-cost productio are ( R/4 c1, R/ ( 4c +. Relative productio distortio is thus idepedet of. Like us, (Levi, 003 fids iefficiet productio for all cost realizatios i a model of adverse selectio but with icomplete cotractig ad repeated play.

18 cetive Effect of Multiple Sourcig page 18 The reaso our productio distortio results are differet from the usual is that, as with all supplier-icetive problems, the buyer trades off the cost of iefficiet productio agaist iformatioal ret by distortig the high-cost productio schedule. the absece of the productio-balacig problem, the buyer ca distort the high-cost suppliers productio without icurrig ay productio-balacig cost (e.g., payig for excess parts without beig able to use them i productio ad thus without the eed to distort the low-cost suppliers productio to mitigate that balacig cost. 6 However, with the productio-balacig problem, distortig the high-cost suppliers productio icurs the additioal cost of ubalaced productio. To avoid this, the buyer fids it optimal for most ( k, j cost realizatios to distort the low-cost suppliers productio as well. summary, we have foud that the high-cost supplier always produces less tha irst- est while the low-cost supplier s productio distortio depeds upo the cost realizatio. urther, while icreasig mitigates the high-cost relative productio distortio caused by the supplier-icetive problem; it ca either mitigate or exacerbate, depedig o the problem parameters, the same effect o the low-cost relative productio distortio. Whether the overall effect of icreasig the umber of suppliers is to ehace or reduce the overall relative productio distortio depeds o its aggregate effect o relative to. order to address this issue we must first determie the relative sizes of ad. Propositio below shows that the uderproductio by high-cost suppliers more tha compesates for ay overproductio by low-cost suppliers ad hece, that Secod-est total productio for ay give ( k, j (, realizatio is always less tha irst-est. Propositio 1. / 1 ( k, j (, < ad / = 1 6 This is why, i the absece of the productio-balacig problem, the cost of elicitig the truth from ay supplier is idepedet of the umber of other suppliers.

19 cetive Effect of Multiple Sourcig page 19. ( lim / < 1 k, j Thus, the supplier-icetive problem results i total productio for each ( k, j (, realizatio which is less tha irst-est. particular, the productio distortio for ay fixed ( k, j cost realizatio does ot disappear as the umber of suppliers gets large. Propositio 3 below establishes how this total productio distortio is affected by the umber of suppliers. Propositio ( k, j. ( k, j. rom Propositio we kow that the Secod-est total productio for each (k,j cost realizatio is less tha or equal to the irst-est, resultig i a productio distortio ad a loss of efficiecy. Result 1 i Propositio 3 shows that a icrease i the umber of suppliers reduces the relative distortio i total productio for each ( k, j realizatio. t should be oted that icreasig the umber of suppliers results i a direct productio effect, the icrease i total productio for both Secod ad irst-est for each ( k, j realizatio. However, Secod-est total productio icreases relatively faster tha irst-est productio ad, thus, the relative distortio for each ( k, j realizatio caused by the supplier-icetive problem is decreasig i. Therefore, for a buyer facig privately-iformed ad self-iterested suppliers, the choice of has a additioal icetive effect beyod the direct productio effect. or such a buyer choosig is a way of mitigatig the cost of dealig with privately-iformed ad self-iterested suppliers. 7 or this part of the aalysis we allow to take o-iteger values. While clearly ot the case, this assumptio still captures the essece of the problem.

20 cetive Effect of Multiple Sourcig page 0 Note that, from Part of Propositio, for ay fixed ( k, reduces this productio distortio, it does ot elimiate it. j realizatio, while icreasig Result of Propositio 3 represets the distortio somewhat differetly. Result we are cocered with the chage i total productio caused by a chage i relative to the average total productio per supplier, that is, the elasticity of productio for each ( k, j realizatio as a fuctio of the umber of suppliers. gai, the effect of a icrease i is greater o Secod- est total productio tha o irst-est total productio. Propositio 3 thus shows that, i the presece of a supplier-icetive problem, there is a positive icetive efficiecy effect associated with icreasig the umber of suppliers for each ( k, distortio i total productio. j realizatio it reduces the relative The basic ituitio for Propositio 3 ca be explaied as follows. ecause of the productio-balacig problem, each supplier s optimal productio will deped upo the other suppliers cost realizatios. s the umber of suppliers of each part icreases, what ay oe supplier reports about his cost realizatio becomes relatively less importat to the buyer i determiig the optimal productio schedules for the other suppliers. This, i tur, reduces his iformatioal ret ad the relative distortio i total productio required to iduce him to hoestly reveal his cost realizatio (Propositio 3. s oted earlier, without the productiobalacig problem, each supplier s productio is idepedet of the umber of the other suppliers ad their cost realizatios, ad thus, icreasig the umber of suppliers has o effect o the suppliers cotracts or productio. Hece, supply chais which are subject to the productiobalacig problem will exhibit this efficiecy ehacig icetive-related effect associated with the umber of suppliers. c. mplicatios for the optimal umber of suppliers summary, Propositios ad 3 establish that, for each ( k, j cost realizatio, total Secod-est productio is less tha or equal to irst-est but that icreasig the umber of

21 cetive Effect of Multiple Sourcig page 1 suppliers reduces this relative productio distortio. This would seem to suggest that, give the productio-balacig problem, the buyer facig a supplier-icetive problem has a greater demad for suppliers tha does the same buyer ot facig a supplier-icetive problem. That is, these results suggest that the optimal umber of suppliers uder Secod-est should be at least as great as uder irst-est. The difficulty i establishig this aalytically is that varyig has three effects: 1. it chages the optimal irst ad Secod-est total productios for every ( k, j realizatio,. it fills i the distributio by icreasig the set of ( k, j realizatios, 3. it chages the probabilities associated with every ( k, j outcome. Propositio 3 addressed the first effect. We are uable to characterize aalytically the secod ad third effects, ad thus caot aalytically establish this cojecture. Thus, we ext tur to umerical aalysis. To best test this cojecture, cosider the irst ad Secod-est problems aalyzed so far, but with two modificatios. irst, we assume that the buyer has a fixed amout of output that she eeds to produce regardless of the suppliers cost realizatios,. This is a special case of the edogeous productio sceario which we have studied so far. The reaso for this modificatio is that the optimal irst ad Secod-est productios will, i geeral, be differet. Differet productio will imply differet optimal umbers of suppliers. order to isolate the demad for suppliers strictly arisig from the supplier-icetive problem, we keep the productio for irst ad Secod-est the same,. Secod, we assume that C(=, so that there is a fixed cost to productio for each supplier, or alteratively, the buyer icurs a commuicatio/coordiatio cost which is liear i the umber of suppliers. Extesive umerical aalysis produces cosistet results, the optimal umber of suppliers for Secod-est is always greater tha or equal to that for irst- est. igure below illustrates this for the followig parameter values: ( c1.3, c 1, p.5, R 1,.0, 1 = = = = = =. The figure idicates that for the give parameter values the optimal umber of suppliers uder irst-est is 7, while uder Secod-est it is 8. This further demostrates that the umber of suppliers has a icetive-efficiecy effect. Give a productio-balacig problem, the presece of privately-iformed ad self-iterested suppliers make it optimal for the buyer to have

22 cetive Effect of Multiple Sourcig page a larger supplier base tha he would have if the suppliers were ot privately-iformed ad selfiterested. 0.8 irst est Profit Secod est Profit uyer Expected Profit Coclusio this paper we study the effect that multi-sourcig has o a assembler who eeds to coordiate her privately-iformed ad self-iterested suppliers. Much of the literature has bee devoted to studyig the effect that the umber of supplier s has o: (disecoomies of scale of productio, risk of disruptio, ad cost of commuicatio. this paper we poit out that there is a additioal effect that must be cosidered whe the buyer chooses her umber of suppliers. The umber of suppliers affects the relative size of productio distortios caused by the suppliericetive problem. creasig the umber of suppliers mitigates this productio distortio. The reaso is that as the umber of suppliers of each part icreases, what ay oe supplier reports has a relatively smaller effect i determiig the productio schedules. This, i tur, reduces the amout of iformatioal ret which ay supplier ca extract from the buyer ad hece reduces the amout by which the buyer has to distort productio i order to reduce the iformatioal ret.

23 cetive Effect of Multiple Sourcig page 3 Simply stated, our fidig is that by icreasig the umber of suppliers the buyer is decreasig their importace, thereby decreasig their iformatioal rets ad improvig efficiecy. a very differet model, oe with icomplete cotractig ad buyer hold-up behavior, (akos ad ryjolfsso, 1993 foud that decreasig the umber of suppliers icreased their relative bargaiig power, thereby icreasig their icetive to ivest i relatioship-specific capital, thereby improvig efficiecy. Our work, together with (akos ad ryjolfsso, 1993, idicates that the umber of suppliers i a supply chai ca have subtle efficiecy effects ad that the appropriate umber of suppliers depeds o the uderlyig icetive problems. Moreover, our work demostrates the importace of cosiderig fuctioal exteralities amog product parts that ofte exist but are typically igored i much of the procuremet literature. Our lie of iquiry is a first step towards a deeper uderstadig of icetive effects i multi-sourcig arragemets. While we cosidered oly oe (but perhaps the most atural exterality that ca exist amog a buyer ad her suppliers, other types of exteralities ca be studied. our work the assembly system had oly two tiers while i practice there are ofte multi-tiered assembly systems (see, for example, (erstei ad DeCroix, orthcomig. Hece, oe iterestig questio is: what icetive-related effect does the umber of suppliers i each tier have o overall efficiecy? alysis of icetive effects uder relatioal (multi-period ad icomplete cotracts could also prove a fruitful directio.

24 cetive Effect of Multiple Sourcig page 4 ppedix Proof of Lemma : We oly prove these results for the part suppliers sice the proof for the part suppliers is aalogous. Result 1. Note that ( ( ( ( ( T c x C T c x C > T c x C. The ( k 1 j 1 ( k 1 j ( k 1 j ( k 1 j 0 first iequality follows from s (TT Costrait, the secod from c1 < c ad the third from s (LL Costrait. Result. or high-cost suppliers assume otherwise (i.e., the high-cost (LL Costrait is ot bidig ad suppose that we start with a optimal solutio so that all of the costraits are satisfied. Reduce both T1 ad T( 1 by ε. Noe of the costraits will be violated by the k j chage ad the maufacturer icreases his objective fuctio. This is a cotradictio. Result 3. rom the (TT costraits for the part suppliers, ( k j ( k j ( k j ( c x c x T T c x c x. The result the follows from the 1 ( ( 1 1 ( 1 1 fact that c1 < c. Result 4. The (TT costraits for the part suppliers ad Result above imply that ( ( ( ( ( k 1 j 1 ( k 1 j T C + c x + c x c x. The maufacturer is iterested i miimizig 1 follows. T, hece, ( ( ( T ( 1 C c1 x1 c x( k 1 j c1 x( k 1 j = + + ad the result Proof of Propositio 1: We first eed to establish that i the Secod-est problem, ( ( k x + kx = j x + jx = j k. 1 1, Proof: ssume the cotrary so that for some ( k, j realizatio ( k x + kx 1 > ( j x + jx 1. this case, oly ( j x jx 1 + ed-products

25 cetive Effect of Multiple Sourcig page 5 k x + kx j x jx uits of part will be throw will be produced ad ( ( 1 1 away. Usig the results of Lemma, the Secod-est problem ca the be restated as: max (( ( Rmi k x + kx1; j x + jx 1 k( c1( x1 c( x( k 1 j c1( x + ( k 1 j (,, P k j j j k= 0 ( k c( x j( c1( x1 + c( x k ( j 1 c1( x k ( j 1 ( j c( x Rmi (( k x + kx1 ;( j x + jx 1 k( c1( x1 + c( x( k 1 j c1( x ( 1 k j + { x, x1, x, x1} ( k c( x j( c1( x1 + c( x k( j 1 c1( x k( j 1 j= 0 k k ( j c( x P( k, j, R( ( j x + jx 1k j k ( c1( x1 k j c( x( k 1 j c1( x( k 1 j + + P( k, j, C( ( k c( x j( c1( x1 c( x k ( j 1 c1( x k ( j 1 + ( j c( x Notice that x ad x 1 appear i the objective fuctio oly with egative coefficiets. Hece, their optimal values will be zero. O the other had, x ad x 1 appear with both positive ad egative coefficiets. Hece their optimal values will be o-egative. ut this k x + kx > j x + jx cotradicts the assumptio that ( ( 1 1. Give this result, the Secod-est problem ca be restated as: max max {, 1,, 1} { } 1( 1 ( 1( ( 1 ( ( ( ( ( k j ( ( jc x j c c x j c x P k j R kc x k c c x k j k c x x x x x j= 0 k= ( 1 (,, C( (8

26 cetive Effect of Multiple Sourcig page 6 or a give ( k, j cost realizatio, the buyer s Secod-est objective fuctio is: 1( 1 ( 1( ( 1 ( ( ( ( ( k j ( ( R kc x k c c x k j k c x π = jc x j c c x j c x ( 1 Give that the Secod-est objective fuctio sums over j ad k, we ca re-express each term i (8 by movig out the terms ( 1 kc с x( 1 ad j( c с1 x ( 1 to π( 1 ad π ( 1, k j k j π k j k j ad movig i the terms ( 1( 1 k+ c с x ad ( j+ 1( c с1 x from π( k+ 1 j ad π k( j+ 1. doig so we have to adjust for the differeces i probabilities associated with these terms. The probability mass P( k 1, j, ( 1, j, P k + = Hece, we obtai + associated with the term ( k+ 1( c с1 x ca be writte as 8 k p! k k! j j k p p ( 1 p p ( 1 p = Pr( k, j, k+ 11 p k! k! j! j! k+ 11 p ( ( p π = 1 p p jc x j c c x j c x 1 p ( ( ( ( ( ( R kc1 x1 k c c1 x k c x ( ( ( ( ( ( ( ( ( ( ( ( ( ( = R kc x k x c + jc x j x c where p( c c /( 1 p =. The Propositio follows immediately. 1. (9 Proof of Propositio : Result 1: Recall that the Secod-est objective fuctio is the same as the irst-est, except with c +, > 0 replacig c. Hece = at = 0. Therefore, if is everywhere decreasig i, the >. t is straightforward to show that ( j( k ( j k + ( c + (( k j c1+ ( j+ k ( c + ( k j = < 0 k, js.t. j k 8 The term ( j+ 1( c с1 x is treated aalogously.

27 cetive Effect of Multiple Sourcig page 7 Hece ( j, k (, >. Result follows immediately from (1 ad (4. Proof of Propositio 3: oth Results 1 ad ca be established together. ssume that there are suppliers ad suppliers. t is the straightforward to show that: = R( c 1+ k( c c1 ( c 1+ j( c c1. cc 1 ( ( + c1+ ( j+ k( c c1 1+ ( ( + ( + + ( + ( + ( ( ( 1 ( 1 1 R c k c c c j c c = c c c j k c c. We first wat to demostrate that / / = 0. ( (10 s log as / 0, / 0 (10 this is equivalet to / / 0. (11 k( j+ k + = / / jc c1 ( j k (( j c1+ jc ( j c1+ j( c + 1 ( We will study the shape of this expressio as a fuctio of. Oe ca verify that c1 ( j k ( ( k( j+ k + j c + j c + = / / jc1 ( 1 1 ( ( ( c j k 0. = 3 / / ( j c1+ j c + (1 (13 That is, / / is a cocave fuctio with a maximum i.

28 cetive Effect of Multiple Sourcig page 8 ecause we are iterested i showig that (11 is a positive fuctio, we oly eed to check boudary cases =0 ad =. Clearly, at =0 (11 is zero because at =0 irst-est ad Secod-est solutios are equivalet. p( c c1 Next we check =. Recall that = so p 1 { k, j }. 1 p R. = = Hece, = 1 ad 0. c + 1 ( = To summarize, we have show that for ay problem parameters it is the case that 0. alogously, oe ca verify that 0. Hece, if = =, we ca vary ad sequetially ad Results 1 ad follow. =