Procurement, Cost Reduction, and Vertical Integration

Transcription

1 Procuremet, Cost Reductio, ad Vertical Itegratio Simo Loertscher Uiversity of Melboure Michael Riorda Columbia Uiversity December 9, 20 Abstract We study a two-stage model of vertical itegratio that sheds ew light o two importat questios: Does vertical itegratio reduce procuremet costs? Does it icrease ecoomic efficiecy? I our model, a buyer who wats to procure a iput of a give quality rus a first-price procuremet auctio. I the first stage, the competig suppliers make simultaeous ivestmet decisios that reduce their expected costs of productio. I stage two, each producer observes his cost realizatio ad makes his bid. Without vertical itegratio, the buyer procures from the supplier who submits the lowest bid. Therefore, abset vertical this is a stadard procuremet auctio augmeted by a cost reducig ivestmet stage. With vertical itegratio, the buyer has access to the productio techology of oe supplier ad procures from a o-itegrated supplier if ad oly if the lowest submitted bid is less tha her ow productio cost. Whether or ot the buyer is vertically itegrated affects the ivestmet decisios of all suppliers. If the problem of miimizig expected productio cost is covex the o-itegratio is the efficiet market structure. With vertical itegratio, the itegrated supplier overivests ad o-itegrated suppliers uderivest relative to first-best. If ivestmets shift the mea of the cost distributios, a vertical merger decreases (icreases) total ivestmet if the margial cost of ivestmet is covex (cocave). With a expoetial cost distributio ad quadratic ivestmet costs, o-itegratio ca be efficiet but vertical itegratio is joitly profitable. If the cost distributio is uiform ad ivestmet cost is quadratic, vertical itegratio is efficiet if abset itegratio the umber of suppliers is two. JEL-Code: D43, D44, L3 Keywords: Vertical Itegratio, Procuremet, Cost Reductio, First-Price Auctios. Ackowledgemets to be added. simol@uimelb.edu.au mhr2@columbia.edu

2 INTRODUCTION 2 Itroductio That vertical market structure matters for ivestmet icetives is uderstood. Williamso (985) argues that asset specificity, bouded ratioality, ad opportuism cospire to udermie efficiet ivestmets. Grossma ad Hart (986) echoes the setimet by modelig how icomplete cotractig causes a holdup problem that dimiishes the ivestmet icetive of the party lackig cotrol rights. Bolto ad Whisto (993) add that vertical itegratio may cause ivestmet distortios motivated by the pursuit of a bargaiig advatage. This paper revisits these issues by examiig the cosequeces of vertical itegratio for ivestmet i cost reductio i the cotext of a simple procuremet model. The model features icomplete cotracts i the sese that ay trasactio betwee a customer ad a exteral supplier is determied by a reverse auctio i which the supplier must bid the low price to wi the supply cotract. The model also features asset specificity by assumig that potetial suppliers make relatioship-specific ivestmets i cost reductio before commecemet of the auctio. Vertical itegratio is modeled as a prior acquisitio of a potetial supplier who the becomes a preferred supplier. The preferred supplier has the optio to produce after observig the bids of the exteral potetial suppliers, ad therefore elects to do so wheever its ow cost is below the low bid. While iteral sourcig eables the vertically itegrated firm to avoid payig profits to exteral suppliers, the acquirig firm must compesate the acquisitio target for the expected value of foregoe profits. Furthermore, as a istace opportuism, vertical itegratio distorts the sourcig decisio, which, i tur, also distorts ivestmets i cost reductio. I particular, vertical itegratio leads exteral suppliers to uderivest i cost reductio i aticipatio of sourcig distortios. Cosequetly, it is ot clear a priori whether vertical itegratio is o balace a attractive strategy for reducig expected procuremet costs. I this procuremet eviromet, the oitegrated market structure results i socially efficiet ivestmets i cost reductio if disecoomies of ivestmet are sufficietly proouced ad the variace of cost outcomes (give ivestmets) is sufficietly great. I such circumstaces, the distortios arisig from vertical itegratio raise expected productio costs. Nevertheless, a vertical acquisitio may be a profitable strategy because it squeezes the profits of the remaiig exteral suppliers. Ideed, for a special case of a expoetial cost distributios ad quadratic ivestmet costs, we show that a vertical acquisitio reduces expected procuremet costs eve though it compromises social welfare by icreasig expected productio cost. O the other had, if the beefits of multiple potetial suppliers are sufficietly small, the vertical itegratio may cofer the social beefit of accomplishig a asymmetric patter of ivestmet that is more closely aliged with socially optimal cost reductio. We demostrate this for the special case of a uiform cost distributio ad quadratic cost of ivestmet. Our idea that vertical is motivated by reducig the profits of exteral suppliers is remiiscet of Bolto ad Whisto (993) s idea that vertical itegratio is motivated by the creatio of bargaiig advatages. Bolto ad Whisto (993) cosiders how forward itegratio eables a upstream supplier to extract rets from dowstream customers who make relatioshipspecific ivestmets, whereas our model turs the icetives aroud to cosider how backward itegratio extracts rets from upstream suppliers who make relatioship specific ivestmets i cost reductio. While the directio of vertical itegratio is maily a matter of iterpretatio, there are other importat differeces betwee the models. First, the models make differet assumptios about iformatio ad the market mechasim. The Bolto-Whisto (BW) model assumes complete iformatio ad assumes a particular bargaiig process to allocate

3 2 GENERAL MODEL 3 scarce supplies. I cotrast, our model features icomplete iformatio about cost reductio ad assumes source selectio via a first-price auctio. Secod, the logic of the distortios arisig from vertical itegratio is differet. I the BW model, the itegrated dowstream firm overivests to create a more powerful outside optio whe bargaiig with a idepedet custormer, ad this ivestmet distortio leads to distortios i the allocatio of scarce supplies. I our model, vertical itegratio leads to sourcig distortios, which i tur lead to ivestmet distortios. Thus the causal relatioships betwee allocatio ad ivestmet distortios are differet. Vertical itegratio i our model effectively establishes a preferred supplier, who serves to limit the market power of o-itegrated suppliers. The itegrated firm avoids givig away rets by allocatig productio to its upstream divisio wheever its cost is below the low bid. These allocatio distortios from a preferred suppler are similar to those aalyzed by Burguet ad Perry (2009). Our model goes further by aalyzig the cosequeces for ivestmet i cost reductio. As result of edogeous ivestmets, the preferred supplier has a more favorable cost distributio tha the idepedet suppliers i our model, i cotrast to the Burguet ad Perry (BP) model which assumes idetical cost distributios. Obviously, edogeous ivestmets are a additioal dimesio alog which to cosider the cosequeces of a preferred supplier. We show quite geerally that the itegrated supplier overivests i cost reductio ad idepedet suppliers uderivest. For the special case i which ivestmet shifts mea cost, we also provide coditios uder which total ivestmet is o larger with vertical itegratio. Uder auch coditios, a vertical merger reduces ad shifts ivestmet away from oitegrated suppliers toward the vertically itegrated supplier, ad may also reduce total ivestmet if the margial cost of ivestmet icreases too quickly. These ivestmet distortios, i additio to the sourcig distortio from preferred supplier status, accout for the social iefficiecy of vertical itegratio whe more symmetric ivestmets by potetial suppliers are cost miimizig. 2 Geeral model 2. Basic setup A dowstream buyer procures a fixed iput from a upstream idustry cosistig of 2 potetial suppliers. The value of the fial good to the buyer gross of the procuremet cost is V. Supplier i makes a costly relatioship-specific o-cotractible ivestmet x i, which radomly determies the cost of productio c i accordig to a cumulative distributio fuctio F (c i ; x i ) with positive desity f(c i ; x i ) o its support. We assume the support of F (c; x) is bouded below ad let µ(x) deote the ifium of the support; that is, F (c; x) 0 if c µ(x). For expositioal coveiece, we assume for ow that that V ad the support is ubouded F (c;x) x above for all x, ad assume that lim c cf x (c; x) 0, where F x (c; x). The cost of the ivestmet is Ψ(x i ). The buyer selects oe of the potetial suppliers to produce the iput. 2 The product techology for the iput is described by F (c i ; x i ) ad Ψ(x i ). Higher ivestmet is assumed to shift the cost distributio smoothly accordig to first-order stochastic domiace, The support of F (c; x) is exteded o the real lie i the usual maer, i.e. the support of F (c) is [c o, c o ] the let F (c; x) 0 for c < c o ad F (c; x) for c > c o. 2 The implicit assumptio justifyig a first-price auctio is V. Alteratively, if the support of support of F (c; x) is bouded above, the V is above the supremum of the support for the relevat rage of x.

4 2 GENERAL MODEL 4 ad the cost of effort is assumed to be covex icreasig ad differetiable: F x (c; x) > 0 for all c i the iterior of the support of F (c; x) ad Ψ (x) ψ(x) is strictly positive ad strictly icreasig for all strictly positive x. The aalysis compares two modes of the procuremet. I both modes, cost realizatios are the private iformatio of the suppliers. I the o-itegrated mode the buyer is idepedet of the suppliers, ad procures the iput i a first-price reverse auctio i which the suppliers bid a price ad the buyer selects the low price supplier. I the vertically-itegrated mode, the buyer is itegrated with oe of the suppliers, who becomes a preferred supplier, ad obtais bids from each of the remaiig suppliers i a reverse auctio with a secret reserve price equal to the realized cost of the preferred supplier. Vertical itegratio ca be iterpreted as forward itegratio by a upstream supplier to acquire the property rights of the dowstream buyer. Suppose that the buyer has property rights over the techology to produce the dowstream product with a value V gross of procuremet costs for a required iput. Each of the upstream suppliers has property over a productio techology for the required iput. Vertical itegratio occurs whe oe of the upstream firms acquires the dowstream productio rights. Alteratively, ad for our purposes equivaletly, the buyer ca be viewed as itegratig backwards to acquire oe of the upstream suppliers. O this iterpretatio, the buyer acquires to cotrol rights to direct the ivestmet of the acquired supplier ad to observe its realized cost. There is a obvious questio of why the buyer stops at oly oe acquisitio. A possible aswer is that competitio laws prevet a cosolidatio of the upstream idustry. 3 A systematic ivestigatio of this competitio policy issue, however, is beyod the scope of this paper. 2.2 Noitegratio The timig of the game uder oitegratio is as follows: Suppliers simultaeously choose ivestmets x i ad observe costs c i. Suppliers simultaeously submit bids b i. The low-bid supplier, say i, produces the iput ad icurs cost c i. The payoff of the buyer is V b i, the payoff of the low-bid supplier is b i c i Ψ(x i ), ad the payoff of the others is Ψ(x i ). This might be thought of as a extesive form game i which suppliers choose ivestmets i the first stage, ad submit bids i the secod stage. The appropriate equilibrium cocept is subgame perfectio. Sice the ivestmets are uobserved, the ormal formal of the game has firms simultaeously choosig a ivestmet ad biddig strategy. We focus o symmetric equilibria, by which we mea Nash equilibria i which all firms choose the same ivestmet level x, so that all firms draw their costs idepedetly from the same distributio F (c) F (c; x ) ad accordigly employ the same biddig fuctio b(c). The structure of equilibrium biddig is well uderstood from auctio theory. Cosider the biddig icetives of a represetative firm with cost realizatio c whe rival bidders use a ivertible bid strategy b(c). A represetative bidder chooses β to maximize (β c)[ F (b (β))]. Therefore, a symmetric equilibrium biddig strategy b(c) is such that { c arg max [b(z) c] [ F (c)] }, () z 3 Aother possible explaatio is that first-best may be achieved by vertically itegratig with oly oe supplier, which may, for example, occur i the model with uiform distributios.

5 2 GENERAL MODEL 5 or b(c) c + c [ F (z)] dz [ F (c)]. (2) Note that b(c) is a icreasig fuctio ad is ideed ivertible o the support F ( ). 4 Next cosider equilibrium ivestmet icetives. Eve if a deviat firm had a differet distributio of costs, the deviat would still follow the equilibrium biddig strategy if it expects its rivals to do so; similarly, rivals also would follow the equilibrium biddig strategy because the deviatio is uobserved. Cosequetly, i cosiderig coditios for a symmetric Nash equilibrium of the ormal form game, it is eough to cosider a isolated ivestmet deviatio. Suppose that a represetative firm were to deviate ad choose x + ε istead of x. The deviat would have cost distributio G(c; ε) F (c; x +ε), but, as oted above, would cotiue to follow the same biddig strategy give by (2). Let U(c) (b(c) c)( F (c)) be the expected payoff of a firm with cost c whe placig the equilibrium bid b(c). By the evelope theorem, U (c) [ F (c)]. Therefore, usig itegratio by parts, the deviat s expected profit gross of ivestmet cost is Π(ε) µ(x +ε) µ(x +ε) U(c)dG(c; ε) (3) [ F (c)] G(c; ε)dc ad the derivative of Π(ε) is Π (ε) µ(x +ε) [ F (c)] G ε (c; ε)dc [ F (µ(x + ε))] µ (x + ε)g(µ(x + ε); ε) (4) where G ε (c; ε) G(c;ε) ε ad g(c; ε) G(c;ε) c. A ecessary coditio for a symmetric equilibrium is ψ(x ) Π (0), or, equivaletly, ψ(x ) µ(x ) [ F (c; x )] F x (c; x )dc f(µ(x ); x )µ (x ) (5) sice F (µ(x )) 0. Therefore, for c µ(x ), the equilibrium bid fuctio is b(c) c + c [ F (z; x )] dz [ F (c; x )] (6) For the rest of this sectio, we maitai the assumptio that these equilibrium coditios are ot oly ecessary but also sufficiet for a equilibrium of the ormal form game. I a symmetric equilibrium, the low-cost firm wis the procuremet auctio. Thus, if all firms have the same ivestmet x, the the realized productio cost is determied by the distributio of the miimum order statistic for idepedet draws from F (c; x). The distributio of this miimum order statistic is L(c; x, ) [ F (c; x)] 4 While this defiitio is strictly correct if the support of F ( ) has o upper boud, it readily exteds to the case of bouded support. If the supremum of the support of F ( ) is ν, the it is coveiet to defie b(c) c o the exteded support where c ν.

6 2 GENERAL MODEL 6 ad the expected productio cost is C(x) µ(x) cdl(c; x, ). (7) Cosequetly, usig itegratio by parts ad lim c cf x (c; x) 0, the cost reductio from a symmetric margial icrease i ivestmet is C (x) µ(x) µ(x) L x (c; x, )dc µ (x)l(µ(x); x, ) (8) [ F (c; x)] F x (c; x)dc f(µ(x); x)µ (x) where L x (c; x, ) L(c;x,) x ad l(c; x, ) L(c;x,) c. It follows from (5) ad (8) that ψ(x ) C (x ) (9) at a symmetric equilibrium. I other words, each firm fully iteralizes the expected cost reductio from a margial icrease i its ivestmet. The result is summarized as follows. Propositio I equilibrium uder o-itegratio, dowstream ivestmets miimize expected productio plus effort costs, assumig that this miimum is achieved with symmetric ivestmets. 2.3 Vertical Itegratio 2.3. Model Uder vertical itegratio, the dowstream buyer is vertically itegrated with upstream firm, ad the idepedet suppliers are labeled i 2,... The timig of the game is the same as for oitegratio, except at the last stage the low-bid idepedet firm (i ) is selected to produce oly if b i < c. Otherwise the vertically-itegrated firm produces ad icurs c. The payoff of the itegrated firm is V mi{b i, c } Ψ(x ), the payoff of the low-bid idepedet firm is b i c i Ψ(x i ) if selected ad Ψ(x i ) otherwise, ad the payoff of the others is Ψ(x i ). Sice the itegrated firm ad idepedet firms are positioed asymmetrically, the aalysis focuses o equilibria i which the itegrated firm ivests x I ad the idepedet firms symmetrically ivest x N. Itis useful to distiguish the cocepts of productio cost ad procuremet cost. Productio cost is the cost of actually producig the iput, while procuremt cost is the expese icurred by the buyer which may iclude a profi margi for the supplier. Uder o-itegratio, the distictio is simple. The distributio of miimum productio cost is L(c; x, ), ad the distributio of the price icurred by the buyer is L(b (b); x, ). Thus the probability that procuremet cost is o more tha b(c) is also L(c; x, ). Uder vertical itegratio, the distributios of productio ad of procuremet costs are illustrated i Figure. O the vertical axis is the cost of the itegrated firm c I, ad o the horizotal axis is the miimum cost draw of a oitegrated firm c N. The lie above the 45- degree lie is b(c N ). The itegrated firm procures from the lowest-cost idepedet supplier if

7 2 GENERAL MODEL 7 Figure : Distributio of productio ad of procuremet costs ad oly if c I > b(c N ). Cosequetly, the probability that realized procuremet cost (excludig its ivestmet cost Ψ(x I )) is at least b(ĉ) is give by the probability mass i the rectagle to the ortheast of the poit (ĉ, b(ĉ)). This probability is [ F (b(ĉ); x I )][ L(ĉ; x N, )] ad cosequetly the probability that this cost is ot more tha b(ĉ) is P (b(ĉ); x N, x I ) [ F (b(ĉ), x I )][ L(ĉ; x N, ). O the other had, the probability that actual productio cost is less tha ĉ is give by the probability mass over the shaded area, which is give by R(c; x N, x I ). The actual productio cost geerally is higher tha the miimum productio cost because of the sourcig distortio. The fist-order approach to equilibrium aalysis proceeds similarly to the oitegratio case, except (a) there is oe fewer o-itegrated firm, (b) the upstream divisio of the itegrated firm is a preferred supplier, ad (c) there are differet equilibrium ivestmets for the itegrated ad oitegrated suppliers Noitegrated suppliers Let b(c) ad x N deote the symmetric equilibrium bid strategy ad ivestmet for oitegrated suppliers, ad x I the ivestmet of the itegrated supplier. I a symmetric equilibrium, the expected profit of a oitegrated firm biddig b(c) with cost c is U N (c) (b(c) c)( F (c; x N )) 2 ( F (b(c); x I )). (0) This equilibrium profit for a o-itegrated firm reflects that the itegrated firm will self supply if its cost is below the low bid b(c). Ivokig the revelatio priciple ad the evelope theorem, ad imposig the boudary coditio U N(c) ( F (c; x N )) 2 ( F (b(c)); x I ). () lim U N(c) 0, (2) c

8 2 GENERAL MODEL 8 simple itegratio implies U N (c) c ( F (t; x N )) 2 ( F (b(t)); x I )dt. (3) c ( F (t;x N )) 2 ( F (b(t);x I ))dt ( F (c;x N )) 2 ( F (b(c);x I )). From these relatioships, b(c) c + If a oitegrated supplier deviates ad ivests x N + ɛ, the the expected profit of the deviat is µ(x N +ɛ) U N (c)df (c; x N + ɛ) (4) A represetative o-itegrated firm s ivestmet problem is therefore max ɛ µ(x N +ɛ) yieldig the equilibrium first order coditio ψ(x N ) U N (c)df (c; x N + ɛ) Ψ(x N + ɛ), (5) U N (c)df xn (c; x N ) U N ()f(; x N )µ (x N ) (6) ( F (c; x N )) 2 ( F (b(c); x I )F xn (c; x N )dc ( F (b(c); x I )L x (c; x, )dc where the secod equality follows from itegratio by parts ad substitutio of (??), ad the third iequality is defiitioal. The ivestmet icetives of o-itegrated frims ca be uderstood alteratively with referece to the distributio of procuremet cost. Sice P xn (b(c); x N, x I ) [ F (b(c); x I )]L x (c; x N, )] (7) for c >, i a symmetric equilibrium, ψ(x N ) P xn (b(c); x N, x I )dc (8) cdp xn (b(c); x N, x I ). where the secod equality follows from itegratio by parts. This alterative characterizatio leads to the coclusio that oiitegrated firms uderivest i cost reductio. The distributio of actual productio costs is R(c; x N, x I ) P (b(c); x N, x I ) b(c) c [ L(b (t)); x N, )]df (t; x I ). (9) Notice that R(c; x N, x I ) is the probability that the productio cost is ot greater tha c. For c, the distributio of productio cost is simply R(c; x N, x I ) F (c; x I ). Therefore the expected cost of productio i the itegrated case is C(x N, x I ) µ(x I ) cdr(c; x N, x I ) (20) cdr(c; x N, x I ) + µ(xn ) µ(x I ) cdf (c; x I ) (2)

9 2 GENERAL MODEL 9 with dr(c; x N, x I ) dp (b(c); x N, x I ) [ L(c; x N, )]df (b(c); x I ) + [ L(b (c); x N, )]df (c; x I ) (22) for c > ad dr(c; x N, x I ) df (c; x I ) otherwise. This characterizatio leads to the coclusio that oitegrated firms uderivest i cost reductio. Propositio 2 I equilibrium uder vertical itegratio, o-itegrated firms symmetrically ivest less effort tha if they miimized actual expected productio plus effort costs. Proof. We prove the statemet by showig that ψ(x N ) < cdp xn (b(c); x N, x I ) cdr xn (c; x N, x I ) C(x N, x I ) x N (23) The first equality follows from (8) ad the secod from (??), so we are left to establish the iequality. From (22) we get dr xn (c; x N, x I ) dp xn (b(c); x N, x I ) + L xn (c; x N, )]df (b(c); x I ) (24) L xn (b (c); x N, )df (c; x I ). Isertig this ito (23) ad cacelig terms, the iequality i (23) is equivalet to cl xn (b (c); x N, )df (c; x I ) A chage of variables reveals that the first itegral is equal to Therefore, the iequality i (23) is equivalet to cl xn (c; x N, )]df (b(c); x I ) 0. (25) b(c)l xn (c; x N, )df (b(c); x I ). (26) [b(c) c]l xn (c; x N, )df (b(c); x I ) 0. (27) which follows because b(c) > c ad L xn (c; x N, ) 0. Propositios 2 seems ituitive o the surface. No-itegrated suppliers are discouraged, at the margi, from exertig effort because they do ot ejoy the beefits from ivestmets i some of the istaces whe they are the low-cost potetial supplier. This is because the itegrated firm opportuistically sources iterally to avoid payig profit margis to the oitegrated suppliers. Note, however, that the defiitio of expected procuremet cost already accouts for the sourcig decisio. Thus o-itegrated firms uderivest takig the sourcig rule as give. The reaso is that a oitegrated supplier does ot fully iteralize the beefit of reducig the sourcig distortio (by shiftig the cost distributio dowward) because of the mootoicity, i.e eve though the idepedet supplier might beat the cost of the itegrated supplier, its success is ucertai ad the wiig price is lower.

10 2 GENERAL MODEL Itegrated Supplier The itegrated supplier chooses x I to miimize its expected procuremet cost, which equals paymets to idepedet suppliers plus productio costs of self-supply plus the ivestmet cost. Assumig µ(x I ) < b(), a sufficiet coditio for which is x I x N, expected procuremet cost is give by Θ(x I, x N ) b(c)dp (b(c); x N, x I ) + b(µ(xn )) µ(x I ) ad the first order coditio for the itegrated firm is give by ψ(x I ) b(µ(xn )) µ(x I ) cdf (c; x I ) + Ψ(x I ) (28) b(c)dp xi (b(c); x N, x I ) (29) cdf xi (c, x I ) + µ (x I )µ(x I )f(µ(x I ); x I ). Propositio 3 I equilibrium uder vertical itegratio, the itegrated supplier ivests more effort tha if it miimized expected productio plus effort costs. Proof. The partial derivative of expected productio cost C(x I, x N ) as give i (??) with respect to x I is C(x I, x N ) cdr xi (c; x I, x N ) + µ (x I )µ(x I )f(µ(x I ); x I ) µ(xn ) µ(x I ) cdf xi (c; x I ) (30) Makig use of the expressio for dr(c; x I, x N ) i (22), this derivative ca be writte as C(x I, x N ) + + µ(xn ) µ(x I ) cdp xi (b(c); x I, x N ) c( L(b (c)); x N, ))df xi (c; x I ) cdf xi (c; x I ) µ (x I )µ(x I )f(µ(x I ); x I ). c( L(c; x N, ))df xi (b(c); x I ) Re-write the term o the secod lie to get b() Substitutig, C(x I,x N ) C(x I, x N ) c( L(b (c), ; x N, ))df xi (c; x I ) + ca ow be rewritte as + + b(µ(xn )) cdf xi (c; x I ). (3) cdp xi (b(c); x I, x N ) c( L(c; x N, ))df xi (b(c); x I ) b() b(µ(xn )) µ(x I ) c( L(b (c); x N, ))df xi (c; x I ) cdf xi (c; x I ) µ (x I )µ(x I )f(µ(x I ); x I ).

11 2 GENERAL MODEL Usig a chage of variables, c( L(b (c); x N, ))df xi (c; x I ) (32) b() Cosequetly, C(x I, x N ) Observe ext that Thus, Θ(x I,x N ) Sice Θ(x I, x N ) C(x I,x N ) b(c)( L(c; x N, ))df xi (b(c); x I ). + + cdp xi (b(c); x I, x N ) (33) b(µ(xn )) µ(x I ) b(µ(xn )) + µ(x I ) is equivalet to (b(c) c)dp xi (b(c); x I, x N ) (b(c) c)( L(c; x N, ))df xi (b(c); x I ) cdf xi (c; x I ) µ (x I )µ(x I )f(µ(x I ); x I ). b(c)dp xi (b(c); x N, x I ) (34) cdf xi (c; x I ) µ (x I )µ(x I )f(µ(x I ); x I ). (b(c) c)( L(c; x N, ))df xi (b(c); x I ). (35) dp xi (b(c); x I, x N ) ( L(c; x N, ))df xi (b(c); x I ) F xi (b(c); x I )dl(c; x N, ) (36) this iequality is equivalet to 0 (b(c) c)f xi (b(c); x I )dl(c; x N, ). (37) The right had side is positive because b(c) c 0 ad F xi (b(c); x I ) 0. Thus, we have established Θ(x I,x N ) C(x I,x N ), which is equivalet to Θ(x I,x N ) C(x I,x N ). Sice ψ(.) is a icreasig fuctio ad the equilibrium level of ivestmet satisfies ψ(x I ) Θ(x I,x N ), the proof is complete. Propositio 3 is quite ituitive. As the itegrated firm obtais the additioal beefit of savig procuremet costs b(c) i some istaces where it is ot the lowest cost firm, it has a additioal icetive to ivest. We ote for referece i the ext sectio that the itegrated firm ca be viewed equivaletly as maximizig the procuremet cost savigs from self-supply. The gross procuremet cost savig of a itegrated supplier who ivests x I whe o-itegrated suppliers ivest x N is b(cn ) µ(x I ) [b(c N ) c] df (c; x I )dl(c N ; x N, )dc (38) K(b(c N ); x I )dl(c N ; x N, )

12 3 SHIFTING SUPPORT MODEL 2 where K(b; x) b µ(x) F (c; x)dc. (39) The equilibrium ivestmet choice of the itegrated supplier therefore ca be viewed as balacig the margial cost of ivestmet to the margial reductio i expected procuremet cost: ψ(x I ) where K x (b; x) K x (b(c N ); x I )dl(c N ; x N, ) (40) b µ(x) F x (z; x)dz. (4) It is straightforward to show that that this alterative characterizatio of equilibrium ivestmet is equivalet to (29). This represetatio of ivestmet icetives for the itegrated supplier is useful for cosiderig the shiftig support model that follows. 3 Shiftig Support Model We ow tur to a specializatio of the the geeral model i which icreases i ivestmet effort maitai the shape of the cost distributio but shift its support dowward; that is, F (z;x) F (z; x) F (z + x; 0) ad µ(x) µ 0 x. For otatioal ease, we let f(z + x) z. Notice that uder the shiftig support assumptio we have F x (z; x) f(z + x). It follows that K x (b; x I ) b µ 0 x I F x (z; x I )dz F (b + x I ; 0). (42) Keepig the assumptio that first-order coditios are ecessary ad sufficiet, we have ψ(x N ) [ F (c; x N )] 2 [ F (b(c); x I )]F x (c; x N )dc (43) [ F (c + x N ; 0)] 2 [ F (b(c) + x I ; 0)]f(c + x N )dc [ F (b(c) + x I ; 0)]dL(c; x N, ) ad Hece ψ(x I ) F (b(c) + x I ; 0)dL(c; x N, ). (44) ( )ψ(x N ) + ψ(x I ). (45) This implies that the equilibrium aggregate effort depeds o the shape of the effort cost fuctio. Propositio 4 I the shiftig support model, aggregate ivestmet uder vertical itegratio is the same, higher or lower tha uder o-itegratio if, for all x 0, ψ (x) 0, ψ (x) < 0 or ψ (x) > 0.

13 4 EXPONENTIAL-QUADRATIC MODEL 3 Proof. Uder oitegratio, equilibrium effort is give by ψ(x ). O the other had, rewritig the cosolidated equilibrium coditio with vertical itegratio, (45), as ψ(x N)+ ψ(x I), it follows from Jese s iequality that ( )x N + x I x if ψ 0 ad ( )x N + x I > (<)x if ψ < (>)0. Expected productio costs are miimized uder o-itegratio, assumig a symmetric solutio to the cost miimizatio problem. Outcomes uder vertical itegratio depart from this bechmark i three importat ways. First, if ψ (x) 0, the equilibrium aggregate effort is either too high or too low uder vertical itegratio. Secod, eve assumig ψ (x) 0 so that aggregate ivestmet is fixed, vertical itegratio equilibrium iefficietly shifts ivestmet toward the itegrated supplier. This misallocatio ot oly icreases expected productio cost, but also the cost of effort because the margial cost of effort is icreasig. Third, the sourcig decisio is distorted i favor of the vertically itegrated firm. This sourcig biases icreases expected productio cost, eve though the itegrated firm is motivated to reduce procuremet cost. 4 Expoetial-quadratic model 4. Cost miimizatio There are potetial suppliers whose costs are idepedet ad idetically distributed draws from a expoetial distributio that shifts with ivestmet: F (c; x) e λ(c+x k) (46) Assumig symmetric ivestmets, the miimum cost of productio is distributed accordig to the miimum order statistic: The expected miimum productio cost is therefore L(c, x, ) e λ(c+x k) (47) C(x, ) λ k x λ + k x If i additio ivestmet cost is quadratic, i.e. the total expected cost is ce λ(c+x k) dc (48) Ψ(x) 2 x2 (49) C(x, ) + Ψ(x) λ + k x + 2 x2 (50) ad is miimized at x. The more geeral statemet of the cost miimizatio problem allows for asymmetric ivestmets. Assume without loss of geerality that x x 2... x. The expected miimum

14 4 EXPONENTIAL-QUADRATIC MODEL 4 productio cost is: ( j C(x,...x ) λ j je λ h h) x k x j + ce jλ(c k) dc (5) k x j + λe λ h x h k x ce λ(c k) dc. For λ <, C(x,...x ) is miimized at the symmetric solutio x. This is easiest to see for 2, i which case the two first order coditios for a miimum are C(x,x 2 ) x + 2 e λ(x x 2 ) + x 0 ad C(x,x 2 ) x 2 e λ(x x 2 ) + x 2 0. Subtractig the first from the secod yields, the differece equatio e λ, (52) where x x 2. Sice e λ is cocave i ad its slope is λ at 0, it follows that 0 is the uique solutio for λ <. Pluggig 0 back ito the first order coditios the gives the result. Though the argumet is somewhat more complicated for > 2, the basic idea geeralizes directly to arbitrary. 4.2 Noitegratio The equilibrium bid for the expoetial model with symmetric bidders is a fixed markup o cost: b(c) c e λ( )(t+x k) dt c + e λ( )(c+x k) (53) c + λ( ) If x is a cadidate symmetric equilibrium ivestmet ad µ k x, the a deviat bidder who icreases ivestmet by choosig x + ε with ε > 0 ears a expected profit: 5 Π(ε, x) e λ(c+x k) λe λ(c+x+ε k) dc (54) λ( ) k x k x [ ] + λ( ) + k x c λe λ(c+x+ε k) dc e λε λ k x ε (x + ε)2 2 5 For followig formulas are useful for this derivatio: µ µ µ ε µ µ ε ce λ(c µ) dc 2 λ + ε 2 (x + ε)2. λ ; e λ(c µ) dc λ + λ eλε ; ce λ(c µ) dc µ λ + µ ε λ eλε λ 2 + λ 2 eλε.

15 4 EXPONENTIAL-QUADRATIC MODEL 5 The first partial derivative is Π(ε, x) ε e λε + (x + ε), (55) which at ε 0 if x. Therefore, i a symmetric equilibrium with suppliers, total ivestmet i the expoetial-quadratic model is equal to uity. It remais to cosider secod-order coditios for profit maximizatio to show that a symmetric equilibrium exists. The secod partial derivative of the deviat s profit fuctio (for ay value of x) is 2 Π(ε, x) ε 2 λe λε. (56) Sice e λε is a decreasig fuctio of ε, 2 Π(ε,x) λ ε 2. This is opositive if ad oly if λ. (57) Thus, coditio (57) is ecessary ad sufficiet for the profit fuctio to be cocave i a icrease i effort ε startig from ay cadidate, symmetric equilibrium effort level. Alteratively, cosider a decrease i ivestmet of by ε > 0 from the cojectured symmetric equilibrium level x /; the profit fuctio is Π( ε, ) k x+ε λ ( ) e λ( )ε 2 e λ[(c+x ε k)] dc (x ε) 2 /2 ( ) 2 ε, (58) which follows usig similar argumets as above. The first partial derivative is Π( ε, ) ( ) ε e λ( )ε + ε, (59) which is ideed 0 at ε 0, ad the secod partial derivative is 2 Π( ε, ) λ( ) ε 2 e λ( )ε, (60) which is o-positive for all ε 0 if ad oly if (57) is satisfied. Thus, (57) is ecessary ad sufficiet for the existece of a uique symmetric equilibrium. I this equilibrium, x /. 6 Propositio 5 I the expoetial-quadratic model, the socially efficiet ad uique symmetric equilibrium outcome uder oitegratio is for each supplier to ivest. This equilibrium exists if ad oly if λ. The equilibrium expected procuremet cost to the buyer uder oitegratio equals the expected low bid: P b(c)dl(c, x, ) λ ce λ(c+x k) dc + (6) k x k x λ( ) k + λ + λ( ) k + 2 λ ( ) 6 Notice that x does ot appear i ay of the secod partial derivatives; thus the profit fuctio is globally cocave i ay symmetric equilibrium. Cosequetly, the symmetric equilibrium is uique, provided it exists.

16 4 EXPONENTIAL-QUADRATIC MODEL 6 Expected productio cost, o the other had, is as for the plaig problem. 4.3 Vertical Itegratio 4.3. Noitegrated suppliers C(, ) λ λ + k (62) The expected profit of a represetative supplier is Π λ( ) 2 2. (63) Suppose ow there is oe itegrated bidder I who ivests x I, ad ad oitegrated suppliers who ivest x N. Cojecture that the oitegrated suppliers use a fixed markup bid fuctio b(c) c + α, where α > 0 is the markup. Substitutig the cojecture ito the right-had side of the equilibrium bid fuctio cofirms that α b(c) c + c + c + λ( ) : c ( F (t; x N )) 2 ( F (b(t); x I ))dt ( F (c; x N )) 2 ( F (b(c); x I )) c e λ( )z dx e λ( )c λ( ). Thus, idepedet suppliers uder vertical itegratio use the same bid fuctio as with symmetric suppliers uder oitegratio. Cosider first a idepedet supplier s ivestmet problem give this fixed markup bid strategy. Let k x N ad µ I k x I. Followig a decrease of effort to x N ε, the represetative idepedet supplier s profit is Π N (ε,, µ I ) λ( ) +ε +ε (64) e λ(c )( 2) e λ[c+ λ( ) µ I] λe λ(c ε) dc (k µ I ε) 2 /2 e λ[(c )( ) ε] λ[c+ λ( ) µ I] dc (k µn ε) 2 /2. (65) Lettig µ µ I (which is presumably positive), the partial derivative with respect to is Π N ε e λ[ε( )+ µ] λ + e λ[(c ) ε+ µ ] dc + (k µn ε) +ε Lettig ε 0, the first order coditio becomes x N k e λ µ e λ µ ( ) e λ µ e λ µ λ e λ[(c )+ µ] dc (66) Thus, the ivestmet of a oitegrated supplier is a fuctio of µ, which remais to be determied i equilibrium.

17 4 EXPONENTIAL-QUADRATIC MODEL Itegrated supplier Cosider ow the itegrated firm s ivestmet problem. The itegrated firm s beefit B I (ε) from icreasig effort to x I + ε with ε 0 is B I (ε, ) b(cn ) µ I ε [b(c N ) c I ]df (c I ; x I ε)dl(c N ; x N, ) (k µ I + ε) 2 /2. (67) Neglectig the cost of effort for the momet, the expected profit of the buyer from icreasig effort to x I + ε with ε 0 is b(cn ) λ 2 ( ) λ( ) µ I ε λ( ) µ N [b(c N ) c I ]df (c I ; x I ε)dl(c N ; x N, ) (68) µ N cn + λ( ) µ I ε [ c N + λ( ) [ c N + ] λ( ) c I e λ[c I µ I +ε] e λ(c N )( ) dc I dc N ] e λ( )(c N ) dc N [ µ I ε + λ λ e λ[c N + λ( ) µ I+ε] ] e λ(c N )( ) dc N Notice that the term i the secod to last lie is idepedet of ε. Thus, the itegrated firm ca be viewed as maximizig ˆB I (ε,, µ I ) + µ +ε λ +( ) The partial derivative with respect ot ε is ˆB I (ε,, µ I ) ε λ( ) The first-order coditio ˆB I (0,,µ I ) ε e λ[(c N )+ µ+ e λ[(c N )+ µ + e λ µ (k µi + ε) 0 therefore implies λ( ) +ε] dc N (k µ I +ε) 2 /2. (69) λ( ) +ε] dc N (k µ I + ε) (70) x I e λ µ, (7) i.e. the itegrated firm s ivestmet also depeds o equilibrium µ Equilibrium Combiig (66) ad (7), we get µ e λ µ (72) as the equilibrium differece i effort levels by the itegrated ad o-itegrated firms. The left had side is trivially liear while the right had side is icreasig ad cocave i µ. At µ 0 the left had side is smaller tha the right had side while the coverse is true at µ. Thus, there is a uique µ (λ, ) solvig equatio (72). Moreover, 0 < µ (λ, ) < will hold for ay fiite 2. Summarizig, we have:

18 4 EXPONENTIAL-QUADRATIC MODEL 8 Propositio 6 For λ < i the expoetial-quadratic model, there is a uique equilibrium that is symmetric i the decisios of the o-itegrated firms. The effort levels i this equilibrium are give by (66) ad (7) with µ determied by (72). Uiqueess of the symmetric equilibrium follows from the uiqueess of µ (λ, ). Notice also that the aggregate effort level x I + ( )x N is costat, cosistet with the more geeral the shiftig support model with quadratic effort cost. Furthermore, a higher value of µ shifts ivestmet toward the vertically itegrated. This occurs for higher values of λ ad lower values of : ad µ λ µ λ( µ ) > 0 (73) µ µ ( ) 2 λ < 0. (74) These partial derivatives are readily established. Their sig depeds o the fact that λ( µ ) > 0 which is equivalet to λ < µ. Sice we assume λ <, the left had side is ot bigger tha while the right had side is o less tha e/( ) sice µ 0. To see that < e/( ) for ay fiite 2 otice first that this holds at 2. Sice i the limit as the two expressios are both while the derivative of e /( ) is always less tha the derivative of the result follows. The expected procuremet cost of the vertically itegrated firm equals the expected price paid to the idepedet suppliers, plus the expected productio cost of self supply, plus the ivestmet cost of the itegrated supplier. The expected price paid to idepedet suppliers is µ N b(c N ) b(c N )df (c I ; x I )dl(c N ; x N, ) (75) b(c N )[ F (b(c N ); x I )]dl(c N ; x N, ) {[ ] } c N + e λ(c N +/(λ( )) µ I ) dl(c N ; x N, ) λ( ) ad the expected productio cost of self supply is b(cn ) µ N c I df (c I ; x I )dl(c N ; x N, ) (76) µ I {µ I + λ [ c N + λ( ) + ] } e λ(c N +/(λ( )) µ I ) dl(c N ; x N, ) λ Addig these two expressios ad cacellatio of terms yields a simplified expressio: [ ] b(cn ) b(c N )df (c I ; x I ) + c I df (c I ; x I ) dl(c N ; x N, ) b(c N ) µ I { µ I + λ } λ e λ(c N +/(λ( )) µ I ) λ( )e λ(c N )( ) dc N µ I + λ λ e λ µ /( ) k + λ λ x I.

19 4 EXPONENTIAL-QUADRATIC MODEL 9 Fially, addig i the ivestmet cost, the total expected procuremet cost uder vertical itegratio is P INT k + λ λ x I + 2 x2 I (77) where x I is determied accordig to Propositio Icetive for vertical itegratio Fially, we tur to aalyzig the icetive for vertical itegratio i the expoetial-quadratic model. Toward that ed, we iterpret the expected costs ad profits uder alterative market structures as determiig the reduced form payoff of a acquisitio game. I the the acquisitio game, the dowstream firm (buyer) sequetially makes take-it-or-leave-it lump-sum offers to the dowstream firms (suppliers). After receivig a offer, a supplier accepts or rejects. If ay supplier accepts a offer, the the acquisitio game eds ad the resultig market structure is vertical itegratio. If all suppliers reject, the the resultig market structure is oitegratio. Each supplier s objective is to maximize expected profits. The buyer s objective is to maximize procuremet cost savigs (relative to the oitegratio equilibrium) mius the acquisitio price. It is straightforward that i equilibrium the buyer offers a supplier a acquisitio price equal to the expected profit uder oitegratio ad the supplier accepts if ad oly if this outcome is profitable for the buyer; otherwise, the buyer makes oserious offers that all suppliers reject. Therefore, the equilibrium outcome is vertical itegratio if ad oly if procuremet cost savigs from vertical itegratio exceed supplier profit uder oitegratio. Formally, vertical itegratio is a equilibrium outcome of the acquisitio game if Π P P INT. Substitutig the outcomes for the expoetial quadratic model, this coditio is equivalet to ad vertical itegratio is the equilibrium outcome of the acquisitio game if 2 Φ(λ, ) [ λ( ) ] [ λ λ x I + 2 x2 I] [ λ( ) 2 2 ] 0 (78) where x I is determied by (7) ad (72) as a fuctio of λ ad. Note that the parameter k cacels o the left-had side of this iequality. Sice there is o closed form solutio for x I, the coditio is evaluated umerically imposig the parameter restrictio λ <. Figure 2 evaluates Φ(λ, ) for differet values of λ ad. It shows that vertical itegratio geerally is profitable. The complete ituitio for the result that vertical itegratio is always profitable i the expoetial case remais to be developed. The followig factors seem importat. I the shiftig support model with quadratic effort cost aggregate ivestmet is the same with ad without vertical itegratio (Propositio 4). Sice the expoetial distributio has a costat hazard rate, this implies that the expected cost of productio is the same o the commo support. 7 7 Deote by L ε (c) ( F (c + e N ε/( ))) ( F (c + e I + ε)) the distributio of the miimum order statistic. So dl ε (c)/dε ε0 f(c + e I )[ F (c + e N )] f(c + e N )( F (c + e N )) 2 ( F (c + e I )) > 0 f(c + e N ) f(c + ei) < F (c + e N ) F (c + e I ), which is a mootoe hazard rate coditio, i.e. if the hazard rate is icreasig, the L ε (c) > L 0 (c) for ε > 0. So this redistributio will always reduce the expected lowest cost. By the same toke, the expected lowest cost will ot be affected if the distributio is expoetial because it has a costat hazard rate of /λ.

20 5 UNIFORM-QUADRATIC MODEL (IN PREPARATION) 20 Figure 2: Φ(λ, ) for differet values of λ ad. Because the additioal ivestmet of the itegrated firm shifts the support dowwards, expected productio cost falls. O top of that, the itegrated firm self-sources (iefficietly) i some istaces, thereby reducig its procuremet cost compared to the case without vertical itegratio. The dowside to vertical itegratio for the vertically itegrated firm is that its effort cost icreases. Notice that revealed prefereces argumets caot be applied directly here: Though it is true that it could keep its ivestmet at the pre-itegratio level but chooses ot to do so, the other firms reduce their ivestmets, ad so all we ca coclude is that, give that the other firms reduce their ivestmets, the itegrated buyer prefers slightly more to less ivestmet, but this does ot allow us to coclude that it is better off with itegratio. 5 Uiform-quadratic model (i preparatio) TBD: Uiform couter example: 2, quadratic cost with a : The socially efficiet ivestmet is x ad x 2 0. Without itegratio, there is o equilibrium i which firstbest is achieved. However, with vertical itegratio such a equilibrium exists. Thus, vertical itegratio ca be beeficial ad a equilibrium outcome of the acquisitio game. 6 Coclusio I a procuremet eviromet i which cost miimizatio requires equal ivestmets by symmetric suppliers, a vertical acquisitio raises expected costs by distortig sourcig ad ivestmet i favor of the itegrated supplier. Additioally, i a eviromet i which ivestmet shifts expected costs ad the margial cost of ivestmet rises quickly eough, such vertical itegratio also reduces total ivestmet i cost reductio. Nevertheless, despite the cost iefficiecy, there are strog private icetives for vertical itegratio to reduce expected procuremet cost. I cotrast, whe cost miimizatio requires asymmetric ivestmet levels, vertical itegratio ca be more efficiet tha o-itegratio ad may arise edogeously as the outcome of a acquisitio game.

21 7 REFERENCES 2 7 Refereces Patrick Bolto ad Michael Whisto (993), Icomplete Cotracts, Vertical Itegratio, ad Supply Assurace, Review of Ecoomic Studies, 60, Roberto Burguet ad Marti Perry (2009), Preferred Suppliers i Auctio Markets,: RAND Joural of Ecoomics, 40, Saford Grossma ad Oliver Hart (986), The Costs ad Beefits of Owership: A Theory of Vertical ad Lateral Itegratio, Joural of Political Ecoomy, 94: Oliver Williamso (985), The Ecoomic Istitutios of Capitalism, The Free Press.