Giacomo Calzolari and Giancarlo Spagnolo*

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1 INTERNATIONAL PUBLIC PROCUREMENT CONFEREN CE PROCEEDINGS September 2006 REPUTATIONAL COMMITMENTS AND COLLUSION IN PROCUREMENT Giaomo Calzolari and Gianarlo Spagnolo* ABSTRACT. When gains from trade eist both along ontratible and nonontratible dimensions suh as with R&D for innovative projets and prourement is repeated, non-ontratible gains from trade an be realized through reputation/relational ontrating beause the buyer an threaten replaement of sellers that perform poorly along non-ontratible dimensions. In suh a dynami prourement framework, keeping the optimal number of eligible sellers and ontrat duration endogenous, we find that: a) there is a general trade-off between reputation for quality and ollusion beause shorter ontrats - more frequent re-autioning - together with redued ompetition - small number of eligible suppliers - failitate non-ontratible quality provision, but also ollusive agreements among suppliers; and b) when non-ontratible quality and variability in suppliers' effiieny are both important, short ontrat duration and a ollusive agreement between a few eligible sellers may be nevertheless desirable for the buyer and also welfare maimizing. Indeed, ollusion or ooperation among firms (e.g. firms onsortia) may boost non-ontratible dimensions. INTRODUCTION Non-ontratible dimensions are present in different measure in every eonomi ehange. 1 For eample in prourement of omple IT servies it is often impossible to fully speify all the requirements that are of value for a buyer. R&D ativities are often at the frontier of tehnologial knowledge, effort towards improvements is diffiult to * Giaomo Calzolari, Ph.D., is a faulty member, Department of Eonomis, University of Bologna, Italy. Gianarlo Spagnolo, Ph.D., is a faulty member, Stokholm Shool of Eonomis, and a member of Consip Researh Unit, and CEPR. Copyright by Giaomo Calzolari and Gianarlo Spagnolo

2 222 CALZOLARI & SPAGNOLO measure and it is often impossible, to hek whether the ontrator has behaved as promised. Similarly, in health prourement it is often the ase that the more ritial is quality of proured goods and servies, the more diffiult is to orretly speify required properties of servies to be proured and whether prourement omply with them. It is well known that when these non-ontratible dimensions are important in terms of gains from trade, letting suppliers ompete on prie - say, in an aution - may lead to a very ineffiient outome, for a buyer and in general. 2 Ehanges, however, are often regularly repeated, partiularly in prourement where buyers need to be served over time. Reputational fores may then help governing transations on nonontratible dimensions. An opportunisti supplier that overstates the non-ontratible quality of an eperiene good or that purposely redues non-verifiable but e-post observable qualitative aspets to ut osts and bust profits an be punished by its buyer(s). Clearly, this annot take plae under any infringement of ontratual terms, but the buyer an eerise some of his disretion to deliberately hinder an "unfaithful" ontrator. With this respet, the most effetive and, probably, the most natural punishment onsists in eluding the supplier from (some) future trade(s). This form of punishment is ertainly available in private ontrating where the buyer his generally free to elude any buyer from its seleting proess. In the ase of publi prourement this deision is still viable to some etent, but it may be partially limited by ruling laws whih often restrit ivil servants disretion so as to avoid orruption. However, some national legislations do provide room for disretion in elusion of dubious providers and leave some aountability in the hands of publi buyers. 3 This paper analyzes repeated prourement proesses (reurrent autions or other forms of searh) where non-ontratible dimensions are an important soure of gains from trade. In partiular, quality is nonontratible in our analysis and will be interpreted in broad sense apturing all value-enhaning deisions that a supplier is free to take during the ontrat eeution and whih the buyer observes but annot diretly enfore beause they are not verifiable. Among the most interesting instanes of this type of deisions we list many type of investments for innovation, R&d ativities and, in general, effort. Certainly, there are normally other dimensions in the relationships that are under the ontrol of the supplier but that are ontratible and an then be governed with a proper design of the ontrat by the buyer and with

3 REPUTATIONAL COMMITMENTS AND COLLUSION IN PROCUREMENT 223 an adequate prie paid for prourement. In addition, the buyer an also ondition the prourement environment. In partiular, she an deide to vary the amount of searh done before proposing a ontrat, namely it has the possibility to fi the number of suppliers in the pool admitted to partiipate at the reurrent autions. Furthermore, she an also deide the length of the prourement ontrat and onsequently also the frequeny of the seletion proedure. As previously disussed, we will also allow the buyer to punish with elusion from future prourement ontests firms that have deided to offer low levels of quality along the ontratual relationship. We show that, in a dynami prourement proess the buyer may want to restrit the number of potential trading partners at the ost of redued sreening and more epensive prourement. This is done to boost non-ontratible quality. Indeed, by restriting ompetition, the buyer leaves firms suffiient future rents so that they an find profitable to build reputational ommitments for future interations and prefer to provide aeptable levels of non-ontratible quality in the urrent relationship (and refrain from moral hazard). Duration of supply ontrats is also a ruial aspet in dynami prourement. Abstrating from (important) tehnologial aspets suh as the rate of obsolesene, a shorter duration of supply ontrats implies more frequent re-seletion or searh. We then show that with higher frequeny of interation it is easier for a buyer to obtain high non-ontratible quality levels from sellers by threatening elusion from future trade. Indeed, with more frequent ontrating the threat of elusion is loser in time and gains from "heating" are smaller so that larger implementable quality an be epeted. Furthermore, shortening ontrat length also redues the buyer s risk to be loked-in with undesirably low quality but at the same time these desirable effets for the buyer have to be ontrasted with the buyer s osts of running and organizing more frequent autions and with firms inability to spread possible fied (and relation-speifi) osts of prourement over a longer ontrat. Considering pros and ons we then show that in repeated prourement a buyer may profit from suppliers stake on future profits and indue larger non-ontratible quality by restriting the pool of firms admitted to ompete for the prourement ontrat and reduing the ontrat length or, whih is equivalent, inreasing the frequeny of reurrent autions. With respet, we also show that these two important dimensions of prourement whih would be normally independent

4 224 CALZOLARI & SPAGNOLO instruments for the buyer in ase of fully ontratible quality, turn out to be losely related (either in terms of omplementarity or substitutability) when quality is non ontratible. Non-ontratible dimensions thus indue the buyer to run frequent autions within a limited number of potential suppliers. However, it is also well known that this environment that the buyer designs to (at least partially) govern non-ontratible quality is also the most favorable environment for induing and sustaining ollusive behavior between the seleted suppliers. Hene, we illustrate a rather general and possibly disappointing trade-off between reputation for non-ontratible quality and ollusion in dynami prourement. Longer duration of supply ontrats - less frequent autions - together with larger pool of ompeting suppliers both deter ollusion among eligible suppliers but also redue non-ontratible quality levels obtainable from them. Symmetrially, shorter ontrats - more frequent autions - and a smaller pool of suppliers both failitate suppliers ollusion but also the enforement of non-ontratible quality standards. 4 This trade-off may show up being rather disappointing beause it seems to mark limits on the remedies that an be put at work for nonontratible quality provision. However, our analysis larifies that ollusion itself an diretly interat with firms inentives to provide non-ontratible quality. In fat, by inreasing the selling prie, ollusion learly inreases the epeted gains from partiipating in prourement autions whih, as usual, an also be seen as the ost of being eluded when the buyer reats to low levels of non-ontratible quality. Hene, we show that the seeming trade-off between non-ontratible quality and ollusion, that naturally seems to emerge in repeated prourement, may in fat reveal to be only apparent beause larger future rents assoiated with ollusion make firms ready to offer larger non-ontratible quality. This somehow provoative result suggests that the buyer may not be neessarily onerned by the spontaneous formation of artels and may even foster establishment of (legal) onsortia among firms that may alternatively ompete. Indeed, reinterpreting our result on ollusion and non-ontratible quality, ooperation among firms in the form of onsortia or joint-ventures are made stable by frequent interation among a limited number of potential ompetitors and, in addition, they may also be epeted to provide larger levels of quality. Hene, lessening ompetition by reduing the pool of potential suppliers and shortening ontrat duration that both indue a ollusive or legal agreement between

5 REPUTATIONAL COMMITMENTS AND COLLUSION IN PROCUREMENT 225 eligible suppliers may leave the buyer better off. For eample, this suggests that in the prourement of innovation ooperation among firms and in the limit also ollusion, do not neessarily hinder innovation when R&D ativity is not fully-ontratible. Further eploring this point, we also note that it is often the ase in repeated prourement that partiipating firms have better knowledge on their relative abilities and effiieny than the buyer. Hene, in addition to the previous effets of agreements among firms, onsortia and ollusion may also have a desirable sorting effet whereby more effiient firms may be seleted for supplying. This may not neessarily translate into lower pries for the buyer, and ertainly does not in the ase of ollusion. Rather, larger effiieny in prodution in these ases tend to inrease firms rent and then further boosts implementable non-ontratible quality. We then onlude our analysis disussing how a buyer onerned by non-ontratible quality, ost of prourement and, possibly, also diretly by effiieny in supplying should design the prourement environment (i.e. the number of ompeting firms and the elusion rule for low quality provision) together with the ontratible dimensions of the relationship (i.e. the ontrat duration and aution frequeny) to effetively ompose its objetives. Relation with the literature Our work is related to several strands of literature. Manelli and Vinent (1995) analyze the adverse effets of ompetition in prourement with non-ontratible quality. They ast their model in terms of adverse seletion (sellers are of different quality whih is refleted in the produed good), fous on single transation, and show that when gains from trade are onentrated on non-ontratible dimensions, autions deliver the worst possible outome, as they selet firms produing the good at the lowest ost but with the lowest level of non-ontratible quality. Sequential take-it-or-leave-it offers to randomly seleted sellers is then a better mehanism. We learly differentiate from this paper by onsidering an environment where quality is an effort deision of prouring firms and not an innate harateristi of suppliers and also onsidering a dynami ontet with repeated prourement. Reahing similar onlusions on the desirability of limited ompetition but for a different reason than the "bad seletion" effet in Manelli and Vinent (1995), our paper omplements their analysis. 5 In a different

6 226 CALZOLARI & SPAGNOLO ontet where unepeted ontingenies may make renegotiation neessary, Bajari and Tadelis (2001) also show that with relevant nonontratible dimensions autioning with fied-prie ontrats may be dominated by bilateral negotiation over ost-plus ontrats. The literature on optimal prourement of innovative goods and servies, where preaution non-ontratible R&D investments are ruial, inluding for eample Taylor (1995), Fullerton and MAfee (1999), and Che and Gale (2003), is also related to our work. In these analyses even though prourement is not repeated, limiting partiipation is optimal, as in our model, but for a different reason than reputational effets. Reduing the number of partiipant inreases eah partiipant s probability of winning the award, and thereby enourages pre-aution non-ontratible R&D effort-investment. Our environment on repeated purhase with non-ontratible quality relates our analysis to the industrial organization literature on reputation and ompetition whih studies whether firms reputational ommitments to high non-ontratible quality an be ompatible with a ompetitive environment and that has been initiated by the seminal works on eperiene good markets of Maaulay (1963), Klein and Leffler (1981), Shapiro (1983), and Allen (1984). These early analyses were onerned with the ompatibility of "quality-assuring" reputational equilibria - requiring rents that make the effort of maintaining reputation worthwhile also with free entry in the market - but did not analyze in detail firms ompetitive interation (firms inentives to steal business from eah other). 6 Stiglitz (1989) also raised the question how ould reputation be ompatible with perfet ompetition that should eliminate any future supraompetitive gains. More reently, Kranton (2003) offers a model that aptures this dilemma. In presene of moral hazard on quality and ompetition, high quality equilibria are unfeasible and he suggests restriting ompetition in industries where non-ontratible quality is important. In a different model Bar-Isaa (2005) onfirms Kranton and Stiglitz s view at the limit, but shows that at intermediate levels of ompetition a further inrease in ompetition (number of firms and substitutability) may well inrease equilibrium produt quality. Hoerner (2002) offers the first elegant answer to Stiglitz s question: in his model with heterogeneous onsumers, adverse seletion and moral hazard, high pries signal high quality and make ompetition ompatible with (in fat neessary for) reputation to work. 7

7 REPUTATIONAL COMMITMENTS AND COLLUSION IN PROCUREMENT 227 In the ontet of prourement Laffont and Tirole (1993, hapter 4) eplain in a two-stage model that when demand is fied or inelasti, as in the ase of many prourement situation in whih the buyer buys a fied amount, one way to stimulate quality provision is linking future business to urrent quality. Reputation an be eploited to threaten suppliers, espeially when the buyer an potentially deal with many suppliers. Kim (1998) and Doni (2004) also study a repeated prourement autions with moral hazard on non-ontratible quality, and show that it may be good to restrit partiipation and threaten elusion if the level of nonontratible quality is too low. Even if Shapiro (1983) noted that the frequeny of interation may failitate the operation of reputational mehanisms with eperiene goods, the effets of ontrat duration and aution frequeny have been largely negleted in the prourement literature. Contrary to these papers, we onsider the possibility to adjust the frequeny of prourement, its interation with the deision to restrit the pool of potential suppliers and also the potential of artels and onsortia in sorting the several trade-offs between quality, prourement osts and effiieny. 8 Ellman (2006) analyses a ontrat length trade-off assoiated to the presene of investment whih beomes speifi to the ontratual relationship but does not look at the role of threats in ase of repeated purhase. 9 Our paper also ontributes to the literature on inomplete epliit ontrats (the "Hart and Moore paradigm"), and in partiular on how dynami interation allows to omplement these by impliit/relational ontrats, from the early ontributions of Bull (1987) and MLeod and Malomson (1989, 1998), to the reent ones like Levin (2003) and Fuhs (2005). As in our paper, the fous of this literature is how relational ontrating allows parties to enfore and govern agreements on observable but not verifiable dimensions (effort, investment, quality). Within this rih and growing literature, our work is losest in spirit to the ontributions more diretly foussing on the interation between epliit and impliit ontrats, i.e. on how epliit ontrats should be strutured or modified to optimize the joint outome of epliitly ontrated dimensions and impliit effets, like Baker, Gibbons and Murphy (1994, 2002), Peare and Stahetti (1998), Halonen (2002), Che and Yoo (2001), Blonski and Spagnolo (2003), and Rayo (2004), among others. To our knowledge, none of these studies onsiders how the design of prourement proess may be influened by relational ontrating, nor deals with how repeated sreening through autions, the number of

8 228 CALZOLARI & SPAGNOLO eligible suppliers, the length of epliit ontrats, and firms ooperation interat with relational quality ommitments. 10 We onlude this literature review with notiing that there are at least two approahes to reputation in markets, as onviningly emphasized by Bar-Isaa (2003). A first approah views "reputation as beliefs" where uninformed players infer intrinsi qualities of ontrating parties by their behavior so that a firm s reputation onsists in buyer s beliefs about its quality-type. See for eample Kreps and Wilson (1982), Milgrom and Roberts (1982). A seond non-elusive view, whih is the one employed in our paper and is related to the previously disussed literature on markets with eperiene goods and impliit ontrating, onsiders reputation as a self-sustaining ommitment to provide desirable but non-ontratible quality by a redible threat on the side of the other partner in ase reputational ommitment is violated. The rest of the paper is organized as follows. Setion 2 presents the model setup. Setion 3 analyzes prourement and quality with ompeting firms. Setion 4 disusses the effet of ollusion on implementable quality. Setion 5 illustrates optimal prourement. Setion 6 etends the base model and disusses its main assumptions. Finally, Setion 7 onludes. MODEL SETUP A buyer needs to proure a unit of a good (or a servie) at any period and she ares for the quality of supply. The per-period valuation of the good V( q) is inreasing in the proured quality q 0, with 11 V (0) = 0. Amongst the N potential suppliers, any firm i an proure a unit of the good with quality by inurring in a per-period ost q i i θθ] i ( q ) ψ () is a positive real valued funtion inreasing in i The per-period (soial) value of quality Vq ( ) ψ ( q) θ + ψ i where θ Θ [, is an (in-)effiieny parameter and ψ 0 = 0. q, with ( ) is onave in q, time horizon is infinite and all the players have a onstant ommon disount fator equal to 1. A prourement ontrat determined by the buyer speifies the number of prourement periods 1 and the per-period payment p that the supplier reeives for any of the periods ontemplated by the

9 REPUTATIONAL COMMITMENTS AND COLLUSION IN PROCUREMENT 229 ontrat. Supplied quality in any ontratual relationship is publily observable but not verifiable and then it is not ontratible. Contrats that last more than one period annot be unilaterally renegotiated (i.e. reneging is ruled out by law), but bilateral renegotiation is admissible. The buyer is not fully informed on eah firm s ost θ i so that she may run an aution for any awarding proess that selets a supplier. An aution awards a prourement ontrat to the bidder that offers the lowest aeptable bid b i for the ontrat and maps the vetor of all bids b into the payment bw for ontrat eeution. In the rules of the aution, the buyer may also set a reservation prie r 0 so that aeptable bids must satisfy r b i (if r < bi for any i, the buyer does not award the ontrat) and she may also deide to limit to n N the number of bidders that are eligible and admitted to the aution proess. 12 In the following, we will ontinue to desribe the awarding proess as an aution also in the etreme ase where n = 1 so the relationship is in fat bilateral. The firm that is awarded the ontrat sets the level of quality it will provide one and for all the duration of the ontrat. 13 Quality is not ontratible so that the buyer annot laim ontrat infringement on the basis of inadequate quality. Nevertheless, the buyer may reat to a low proured quality with deisions that affet future ontratual relationships with the supplier. In partiular, if the quality provided by the aution winning firm does not satisfy the buyer, in a sense that we now larify, then the latter an disretionally elude the ontrator for some future aution rounds. 14 We model this elusion rule with a minimum quality requirement q so that if the firm proures a good of quality q < q, the buyer an disretionally deide to elude this seller for the net T 0 autions. The relationship between the buyer and any potential ontrator is thus omposed by some terms that are verifiable (and ourt-enforeable), namely ontrat duration and remuneration b w, the reservation prie r and the number of eligible bidders N, together with terms that are non-ontratible and disretionary, the proured quality q and the elusion rule σ ( qt, ).

10 230 CALZOLARI & SPAGNOLO Finally, running any prourement ontrat implies a ontrat-speifi investment with a total (for the buyer and the supplier) osts K 0. For eample, K may be a fied set-up ost that eah ontrator inurs to proure the good of whatever quality and that has to be paid anew in any ontratual relationship. Alternatively, it may also be the buyer s ost for organizing any aution or bargaining with a new supplier. For the sake of onreteness we will often use the latter interpretation. In this set-up, a strategy for the buyer thus onsists in, one and for all, (i) setting the awarding rules and the ontrat length, whih we will indiate as the ontratible elements of the prourement relationship ( nrb,, w, ), and (ii) publily announing the elusion rule σ whih may help the buyer governing non-ontratible behavior of suppliers. 15 A strategy for any firm i is omposed by a partiipation deision and a bid for any awarding proess and a deision on proured quality for any ontrat it is awarded. The timing of the game is as follows. t = 2 : The buyer sets the ontrat length, the number of eligible bidders n, the reservation prie r and announes the elusion rule σ = ( qt, ). t = 1: The buyer randomly selets the n potential sellers. firms amongst the t = 0 : An infinite repetition of the following stage game (or aution game) takes plae. Stage Game At time t1 0, an aution publily awards the prourement ontrat, the winning ontrator i obtains the payment b w and sets proured quality ; q i N At any period quality ; q i t [ t, t + 1] 1 1 the winner proures the good of At time t1 +, if qi < q the supplier i is eluded for net T awarding proesses and possibly replaed by a new firm,

11 REPUTATIONAL COMMITMENTS AND COLLUSION IN PROCUREMENT 231 otherwise the pool n starts. remains unhanged and a new stage game Although relatively simple, the dynami desribed game may turn to be intratable in its most general form. We will temporarily limit our attention to a simplified version based on some (strong) working assumptions. Then, having emphasized the main ideas and results in this lear-ut environment, in Setion 6 we will disuss these assumptions and alternative model speifiations. - Assumption 1 (i) In eah period, the ost parameter θ i of any firm i is drawn anew from a time-invariant and independent distribution f ( θ i )( > 0 for any θi Θ ) whih is ommon knowledge; (ii) Firms are fully informed. - Assumption 2 The number of potential suppliers N is infinite. - Assumption 3 The buyer needs to be served at any period, otherwise she obtains a per-period payoff equal to k with k >> 0. Assumption 1 (i) ould be substituted by onsidering a θ i whih is drawn anew at any aution stage but that remains the same for the ontrat duration. This would not qualitatively alter our results. On the ontrary, a model where eah firm is haraterized by a permanent level of effiieny θ i forever would be muh less tratable beause the buyer would then learn firms effiieny aution after aution and firms would antiipate that their ations signal information. Also assumption 1 (ii) is ertainly not without loss of generality. It greatly simplifies our analysis, but as we will further disuss in Setion 6, our results and the underlying trade-offs seem to be robust to asymmetrially informed firms. Assumption 2 will be relaed and disussed in Setion 6 as well as assumption 3 whih simplifies the analysis of the reservation prie beause interrupting the flow of goods is etremely ostly for the buyer. Note also that to simplify the eposition we have not epliitly modelled the many other elements of the prourement relationship that are ontratible. However, a more general interpretation of our model is that the buyer evaluates all these ontratible elements with a soring rule that is here simply represented by firms bids, sores for all ontratible elements. In Setion 6 we will also eplore the possibility

12 232 CALZOLARI & SPAGNOLO for the buyer to use soring rules on non-ontratible quality, whih in the urrent model setup are irrelevant. IMPLEMENTABLE QUALITY WITH COMPETING SUPPLIERS Consider a buyer strategy omprising a pool n of eligible firms, a ontrat length, and a given elusion rule σ. For a given awarded ontrat in whih the supplier offers a quality q and reeives the payment b w, the buyer surplus is 1 1 V ( q ) b K. w Hene, in ase at any aution any winning firm reeives the same offers the same quality q, the buyer s total surplus is b w and t S( q,, n) = t= 0 V( q) bw K V( q) bw K 1 = 1 1 where the dependene on n impliitly takes plae through the prie b w, as it will be lear in the sequel. For any vetor of period, let θ ( n) min { θ θ } n firms ost-effiieny θ ( θ,..., θ ) in a given 1 n 1 n,..., be the ost parameter of the most effiient firm in the pool of n eligible suppliers (i.e. the first order θ n learly dereasing in n. Then, maimal statistis of θ ), with ( ) surplus that an be generated by a ontrat of length and proured quality q is 1 V( q) E[ θ ( n) ] ψ ( q) K 1 where [] E is the epetation operator over the realization of θ and, assuming a repetition of the same ontrat with the same quality, overall welfare is

13 REPUTATIONAL COMMITMENTS AND COLLUSION IN PROCUREMENT 233 t W( q,, n) = t= 0 V( q) E θ ( n) ψ ( q) K V( q) E θ ( n) ψ ( q) K 1 = 1. 1 Our simple framework is stationary for the buyer in the sense that, eept for the supplier s deision to omply or not with quality, firms effiieny is not persistent and learning on effiieny does not our. Hene, the buyer s optimal strategy is time invariant and she sets the ontratible elements of the prourement proess and the rule σ with quality requirement one and for all stages of the game. Consider now the elusion rule σ set by the buyer. We will onsider the toughest elusion rule available to the buyer that punishes disloyal and heating firms that provide quality q < q with elusion forever from future autions. This elusion rule is redible in the urrent framework beause the (infinite) firms are e-ante idential, so that replaing a firm omes at no ost for the buyer and it learly onstitutes an optimal penal ode in the sense of Abreu (1988). 16 Being quality not-ontratible, any firm i that wins an aution may hoose to satisfy the quality requirement q, i.e. to provide a quality q q or not. Clearly, none of the firms has inentive to provide a quality larger than the minimum requirement beause it will not be redited for etra quality so that in equilibrium quality will be either q = q when the firm s deision is to satisfy the quality requirement, or q = 0 if the suppliers deides to heat on quality. If the winning firm deides to omply with the buyer s quality requirement, it obtains an epeted profit equal to bw θ C( q) + E Π q q 1 where Cq ( ) E[ ] θ + ψ ( q) 1 1 is the sum of epeted ost of prodution for the 1 periods of supply after the first one (reall that firm i knows only the urrent realization of

14 234 CALZOLARI & SPAGNOLO θ i ) and the ost of prouring quality q along the ontrat, whilst the term E Π q q represents the epeted profits from any future aution of a firm always omplying with the quality requirement. 17 Alternatively, then the winning firm may deide to shirk on quality, thus providing q = 0. By so doing the firm saves the quality prodution osts 1 / 1 ψ q but it inurs in elusion from future autions, ( ) ( ) with an overall profit equal to bw 1 1 θ + E[ θ]. 1 It is then lear that at any aution, any winning firm will be ready to satisfy the quality requirement if epeted future profits are larger than the immediate ost saving it an obtain in the urrent ontratual relationship, i.e. ( ) 1 E Π q q ψ q. 1 1 Hene, in the following we will say that a minimum quality requirement q is implementable if it satisfies ondition (1).This reasoning an be further eploited notiing that, by standard arguments on prie ompetition, at any aution taking plae at stage t the winning firm will be the most effiient firm, i.e. the one with ost θ ( n) for the vetor θ ourring in t, who obtains a ontratual prie whih is equal to the minimum between the reservation prie r and the seond most effiient firm s ost. Furthermore, the risk of being left with no provision of the required good fores the buyer to set a high reservation prie that guarantees prourement for any realization of θ, i.e. r = θ + C( q). Hene, the awarded prie an be written as b w bw = θ ( n) + C( q) where θ ( n) = min{ θ/ θ ( n )} if the buyer admits at least two firms at the aution stage (i.e. 1) and θ n = θ if she prefers n > ( ) (1)

15 REPUTATIONAL COMMITMENTS AND COLLUSION IN PROCUREMENT 235 ontrating with a single firm (who an ask the reservation prie). We then obtain the following Lemma. Lemma 1. Maimal implementable quality There eists a maimal q, n suh that a quality requirement q is implementable quality ( ) implementable if q(, n) q, with q( n) if either, dereasing in n and, nil n = or = and, if ψ is onave, then the negative effet of ( n ) on q redues with n ( ). Proof of Lemma 1. As stated in the tet, the winning firm prefers to provide the required quality if ( ) 1 E Π q q ψ q. 1 1 Now, the probability that a firm i with ost θ i is the most effiient firm will be denoted as Pr( θ θ, j i θ ) and, from independene, we i j i have [ ] 1 Pr( θ ) 1 ( ) n i θ j, j i θi = F θi. Hene, the e-ante probability of being the lowest ost firm at any stage game is 1 Pr( ) 1 ( ) n θ θ, j i = F θ f θ dθ = 1/ n. [ ] ( ) i j i i i Consider now a (possible) equilibrium where all firms are ready to supply the required minimum quality q. The most effiient firm the wins and, applying the optimal reservation prie, in any future aution with Δ θ n = θ n θ n, so that ost θ obtains a rent ( ) ( ) ( ) Let (impliitly) define ˆq(, n) by ( n) E Δθ E q q Π =. n ( ) ψ ( ˆ) 1 E Δ θ n = q n(1 ), (2) 1

16 236 CALZOLARI & SPAGNOLO It is thus immediate that for any quality requirement q qˆ ( n) >,, any supplier would prefer to shirk on quality so that a quality requirement is implementable only if q qˆ (, n). We now also need to show that by offering a prie bw = θ ( n) + C( q), the most effiient firm indeed wins the aution. In priniple a less effiient firm may be able to underut by offering a lower prie and planning to offer a lower quality (alternatively it would be never able to profitably underut the offer b w ). For this it suffies to onsider the seond most effiient firm who will not underut the most effiient one if θ ( n) + C( q) θ ( n) E[ θ] ψ ( 0) E q q 1 1 Π. 1 This is learly equivalent to ( ) 1 E Π q q ψ q 1 1 whih is the same ondition analyzed for the quality provision by the q qˆ, n most effiient firm so that underutting is not profitable if ( ) Now note that, even if the buyer sets q qˆ ( n) =,, an equilibrium may prevail where all firms supply nil quality. Indeed, suppose all n 1 firms plan to offer q = 0, then the most effiient firm an win by asking a prie bw = θ ( n) + C(0) (reall the seond most effiient firm here proures q = 0 ) but still providing the required quality q. In this ase, the future epeted profits if it wins and abides the quality requirement 1 1 E Π q q = E Δθ( n) ψ ( q) n 1 so that it will prefer to provide the required quality if

17 REPUTATIONAL COMMITMENTS AND COLLUSION IN PROCUREMENT E Δθ( n) ψ ( q) 1 + n 1 1 n 1 Now let define (. q%, n) the q that satisfies the ondition (3) written as an equality. On the other hand, it is also lear that this behavior by firms an be onsistent with an equilibrium only if the winning firm asks a prie bw = θ ( n) + C(0) whih is assoiated with nil quality beause for any higher prie the buyer, antiipating a nil quality, would prefer to set q = 0. %. Furthermore, for a given minimum quality requirement q, we have that the level of quality q effetively proured in equilibrium is It is immediate that ˆq(, n) q(, n) q qˆ ( n) ( ) %( ) if %(, ). 0 if >,, q = {0, q} if qˆ, n q > q, n, q q n q whih shows that if q qˆ ( n) on quality, if the buyer asks for a lower quality ˆq(, n) q > q% (, n) >, in any event all firm will prefer to heat then two types of equilibria may prevail so that the buyer ends up either with nil proured quality or with quality q. Finally, setting a suffiiently low (but positive) quality requirement q% (, n) q, firms systematially prefer to abide the requirement and the buyer does not risk to end up with nil quality. %,. That Consider now the properties of ˆq(, n) and q( n) ( ) for n qˆ, n = 0 = and any is immediate from (2). Treating n and as ontinuous variables for simpliity, from impliit differentiation we obtain (3)

18 238 CALZOLARI & SPAGNOLO Also note that ( n) F ( ) = 1 (1 F( ))n θ ( n) F F F n θ both for n 1 ( ) = 1 (1 ( )) (1 + ( )( 1)) ( q) ( qˆ ) qˆ 1 + Log[ ] ψ ˆ = 0. 1 ψ with with q F ( ) ( ) θ n n 0 and F ( ) ( ) θ n n 0, thus θ ( n) and θ ( n), inreasing n amounts to a first order θ ( ) θ ( ) E[ n n ] stohasti dominane effet. We then also have n 0 beause the larger is n the smaller is the epeted differene in term of effiieny between the most and the seond most effiient firms and also lim n E[ θ ( n) θ ( n) ] = 0. Hene, the limit for n of the left hand side in (2) is zero so that also ( ) qˆ, n = 0 with n = for any and ( 1 ) E Δθ( n) E Δθ( n) qˆ 1 1 = n (1 ) n n n ψ ( ˆ q q). Finally, differentiating ˆ ˆq with respet to n, we obtain ( ) 1 + Log[ ] ψ ( ˆ) ( ˆ q q ψqq q ) = 2 1 ψ ( qˆ ) 2 q, n n where the seond term in the right hand side is negative when ψ qq 0 so that 2 qˆ( n) n 0 and a larger n ( ) makes, larger makes ˆq n same properties when =, n = and also ˆq ( ˆq n less negative.18 Similarly, for q(, n) q qˆ n ) less negative and a % we have the

19 REPUTATIONAL COMMITMENTS AND COLLUSION IN PROCUREMENT 239 ( 1 ) ( 1 ) (%) (%) ( q% ) 2 q% n+ ( n 1) Log[ ] ψ q = 0 ψ q ( ) ( ) E Δ θ ( n) q% 1 E θ n n 1 n Δ + n = 0 2 n ψ q q 1 n ( n 1) with ( ) 2 q%, n n 0 if ψ qq 0. This analysis shows that the properties of the two boundaries ˆq n and q%, n are qualitatively the same in terms of n and. (, ) ( ) Hene, to simplify the analysis in the sequel we will disregard the presene of multiple equilibria for intermediate values of q and make as if the equilibrium with quality prouring firms would systematially prevail. Hene, setting q(, n) { qˆ (, n), q% (, n) } we have that a quality requirement q is stritly implementable if ( ) ( ) ( ) ˆ ( ) q, n = q%, n q and weakly implementable if q, n = q, n q. When all potential suppliers are ready to abide the minimum quality requirement q, the most effiient firm wins the aution at a prie bw = θ ( n) + C( q ). In this ase the epeted profit for any future aution whih is relevant for the deision on quality prourement is E Δθ E q q Π = n where E Δθ ( n) ( n) is the informational rent that firms an epet when winning any of the future autions and 1/ n is the probability of being the most effiient firm out of the n eligible firms. As in any repeated game, the framework we are disussing is haraterized by multiple equilibria with different levels of proured quality. Hene, there eists another equilibrium where the most effiient firm wins the aution, offers the required quality b = θ n + C q, but laims a lower prie ( ) (0) w

20 240 CALZOLARI & SPAGNOLO beause all rival firms do not abide the quality requirement. In this ase the epeted rent from any aution is smaller and equal to 1 1 E Π q q = E Δθ( n) ψ ( q). n 1 However, independently of the partiular type of equilibrium, with unontratible quality firms may be ready to provide quality in return of future profits so that if epeted profits are small then the implementable quality is also small. It then follows that if the buyer sets a too high quality requirement, the winning firms will prefer to save on quality osts and shirk on quality. 19 Hene, setting ondition (1) as an equality q, n whose eat value determines a maimal implementable quality ( ) depends on the partiular firm s profit E Π q q and, in any event, the larger is the pool of eligible firms, the smaller is the probability to be the winning firm and the smaller is the epeted ost differene between the most and the seond most effiient firm, i.e. the informative rent E Δθ ( n) is small. Note that this is also true if firms heterogeneity is small for the properties of the distribution f (), thus independently of the dimension n of the pool. In addition, the longer is the ontrat length the smaller is the maimal implementable quality. The ause of this effet does not rely on the dimension of future epeted profits, rather on the firms possibility to retard punishment for quality shirking when is large. In the limit, if the buyer sets a one-and-for-all ontrat (i.e. = ) the unique implementable quality is the nil quality beause q = Although Lemma 1 learly illustrates the negative effet of and n on implementable quality, and n have also other effets on the buyer s payoff that we now illustrate. First, a longer ontrat determines smaller osts for organizing the autions whih beome less frequent (or a better management of eonomies of sale in prourement). On the other hand, a larger also implies that along the ontrat the buyer is stuk with a firm that may no longer be the most effiient firm in the pool of n eligible suppliers. Seond, a larger pool of n firms implies that the prie asked by the most effiient firm b = θ n + C q dereases beause w ( ) ( )

21 REPUTATIONAL COMMITMENTS AND COLLUSION IN PROCUREMENT 241 θ ( n) is a dereasing funtion of n. Hene, learly if the buyer ould diretly ontrol quality, that is if quality ould be ontratible, she would set n at its largest value n = N and the ontrat length at %where ( ) 1 if K E[ θ] E θ n % otherwise. In fat, after the first period of prourement, the buyer has to ompare the ost K of organizing a new aution (or non optimizing over the eonomies of sale) with the epeted effiieny gain E[ θ] E θ n that an be obtained if she disards the urrent ( ) supplier that has epeted ost E[ θ] ψ ( ) reverts to the most effiient firm that will laim a ost θ ( n) + ψ ( q) for the first period. + q for the net period and Coming bak to non-ontratible quality, it is also important to notie that if the ost of restriting n and or is too large, the buyer may then prefer a nil quality and in this ase she would then set n and eatly as she would do were quality ontratible, i.e. again n= N and =.% However, when quality is valuable then at the optimum the buyer may be indued to restrit ompetition through n and redue ontrat duration taking into aount that these deisions help relaing the onstraint on implementable quality q(, n). Hene, the buyer would aept the ost of reduing n and so as to inrease q(, n) only if by so doing she is interested to inrease q and the atual proured quality q. A stritly larger than zero minimum quality requirement q is then desirable if the (net) per period value of quality Vq ( ) ψ ( q) suffiiently large at least for ertain values of q. is Proposition 1. Optimal Prourement with Competing Firms If the (marginal) value of quality is large, then the buyer optimally sets n and suh that q(, n) > 0 : omparing with ontratible quality or with

22 242 CALZOLARI & SPAGNOLO q = 0, the buyer optimally restrits the pool of eligible firms n< N and the ontrat duration %. Proof. Step 1. First reall from the proof of Lemma 1 that if the most effiient firm prefers to meet the minimum quality standard, then none of the less effiient firm is able to profitably underut the winning offer by planning to heat on proured quality. The buyer s optimization program ( P ) then onsists in maimizing 1 E θ n E θ 1 S( q,, n) V( q) ψ ( q) K ( ) 1 1 [ ] 1 ( 1 ) with respet to n, ( ) q, n q. and q subjet to the implementability onstraint We now show that for any maimal implementable quality q(, n), the buyer always prefers to have the quality-implementability onstraint binding, i.e. q(, n) = q. Indeed, suppose to the ontrary that q(, n) > q, it is immediate that she an inrease n (and possibly also ) thus paying a lower prie b w still obtaining the same level of proured quality q. Hene, inreasing n (and ) is ertainly optimal up to the point where q( n) q is q = 0, then it must be that n and are suh that q( n) 0, =. Furthermore, if atual proured quality, =, beause otherwise the buyer an inrease n (and possibly also ) thus reduing the prie for prourement with no effet on quality. In other terms, it is impossible that an equilibrium an emerge where, even if the buyer sets a stritly positive minimum quality requirement q > 0, still all the firms offer nil quality beause in this ase the buyer would deviate setting q = 0 obtaining the same level of quality but at a smaller prie.

23 REPUTATIONAL COMMITMENTS AND COLLUSION IN PROCUREMENT 243 Step 2. From step1, for any level of n, and q, the implemented quality q is q = q = q(, n) 0 and the program beomes ma { n, } ( (, ),, n) S q n Differentiating we obtain ( ( ) ) ( ( ) ) θ ( ) S q n,, n, 1 1 E n q = Vq ψ q n 1 n 1 n S q n,, n, 1 q = Vq ψq log ( ) E θ 2 ( n) [ θ] + 1 (1 ) { E K} q q E θ ( n) where n 0, 0 and n 0. From whih we an see that if K E[ θ] E θ ( n) then the optimal ontrat length is = 1. If ψ is small, then the optimal n is large as well as the optimal if Vq q ( ) [ θ ] E θ n + K > E. On the other hand, if Vq ψ q is large then the optimal n and are suh that n< N and %. Reduing firms profits, ompetition in terms of a large n also redues inentives to maintain their ommitment for reputation by omplying with the buyer s quality requirement. In addition, the buyer has another instrument to improve quality, namely ontrat length, so that when quality is a real onern, she may want to inrease quality by reduing the length of the ontrat thus inreasing the frequeny of autions. As we have disussed above, this is not without osts for running autions more frequently is ostly both in terms of organizing autions and in terms of lost eonomies of sale, but it also has the advantage of avoiding to end up stuk with ineffiient firms for long periods. Hene, the Proposition shows that when non-ontratible quality matters, the buyer prefers eluding some firms thus reduing ompetition and also possibly limiting the length of the prourement ontrat, as ompared with what she would prefer were quality ontratible or were a nil quality preferable. 21 In fat, it is only by

24 244 CALZOLARI & SPAGNOLO restriting n and that she an obtain a stritly positive maimal implementable quality q(, n) and then also set a stritly positive minimum quality requirement q. An interesting orollary of the previous proposition is the following. Corollary 1. Contrary to the ase of fully-ontratible quality, if the buyer wants to implement a stritly positive non-ontratible quality, then the optimal n and are not independent. If ψ is (weakly) onave, then n and are omplement: the buyer s (marginal) benefit of n inreases with and vieversa. Proof of Corollary 1. We are here interested in analyzing the relationship between optimal and n. Differentiating with respet to n S( q( n, ), n, ) the first order ondition for, = 0 we obtain S( q( n, ), n, ) S( q( n, ), n, ) S ( q( n, ), n, ) n = n / where 0 neessary for the 2 ( ( ) ) seond order ondition. The epression S q n,, n, n an be deomposed as follows ( ( ) ) ( ) 2,,, 2 1, = Vq( q(, n) ) ψ q q(, n) S q n n q n ( ) n 1 n 1 q(, n) q(, n) + Vqq ( q(, n) ) ψ qq ( q(, n) ) 1 n log( ) E θ ( n) 1 n ( ) 2 q, n n 2 where 0 if ψ is onave. as shown in the Lemma 1. Then, if 2 S( q( n, ), n, ) Vq ψ q is large enough and ψ 0, then n 0 and n 0, i.e. n and are omplements n 0, otherwise, if the seond and third 2 ( ( ) ) term in S q n n 2,,, S( q( n, ), n, ) n may prevail, so that n 0 and n 0. +

25 REPUTATIONAL COMMITMENTS AND COLLUSION IN PROCUREMENT 245 If the buyer ould diretly ontrol quality or when she prefers a nil non-ontratible quality, then the optimal ontrat length is simply % either ditated by fied-osts or by the possibility of limiting the lok in effet of a long ontrat. In any ase, it is independent of n. Interestingly, the Corollary 1 shows that this independene property of the optimal and n breaks down when quality is a onern and the buyer prefers to obtain a stritly positive quality. In this ase the two instruments an be both omplements or substitutes. If quality is suffiiently important for the buyer and quality ost is onave, they turn out to be omplements so that any event whih auses an optimal redution of firms admitted at any aution also fosters a redution of the ontrat length and vieversa. The result in the previous Corollary thus sheds some light on a relationship whih is often negleted in the literature, namely that between ompetition and ontrat length in providing rents when the buyer needs to indue the suppliers to provide non-ontratible quality. The results in Proposition 1 have emphasized a trade-off between non-ontratible quality, ompetition and frequeny of autions. However, it is well known that in a ontet with repeated ompetition, reduing the number of ompeting firms and inreasing the frequeny of interations are (among) the most effetive onditions that foster and strengthen ollusion between potential suppliers. It is important also to realize that ollusion in prourement is far from being a simple theoretial uriosity and, on the ontrary, it is a pervasive phenomenon whih has been identified in several ases. Hene, our disussion may point to a possibly muh more disappointing trade-off, namely that between non-ontratible quality and ollusion: if the buyer really ares for quality and annot epliitly ontrat on it, she may be fored to aept ollusion among suppliers. QUALITY AND COLLUSION To aount for the possibility that firms may ollude we let firms deide whether to ollude in the aution supergame or not at date t = 1. If a ollusive agreement is reahed, the most effiient firm is awarded the ontrat, proures the good for periods and reeives the payment by the buyer whih the firm sets at the highest admissible prie, i.e. the reservation prie r. All the other firms abstain from bidding or submit not aeptable / winning bids so that ollusion takes plae with

26 246 CALZOLARI & SPAGNOLO bid rotation. Deviation from a ollusive agreement is punished in the harshest way. If a defetion is observed, firms ompete forever in all the following autions (i.e. we employ grim trigger strategy). All the other features of the game are unhanged and desribed in Setion Consider the optimal prourement strategies desribed in the q, n previous setion, inluding the maimum implementable quality ( ) and elusion rule σ disussed therein. We now study firms inentives to ollude under these strategies. For the ollusive agreement to be sustainable at any aution stage, the seond most effiient firm with ost θ is the one with the highest inentives to deviate and should prefer not to underut the most effiient one. If it does not deviate from the ollusive agreement, this firm as well as any firm other than the most effiient one, an epet a ollusive payoff that we will indiate with C Π. If it does deviate, then it has now two possible ations: either deviating from the artel but abiding to the minimum quality requirement, or deviate and also heat on quality. We now investigate this firm s deision. Eah firm s effiieny is drawn anew at any aution stage so that, ontrary to standard models of ollusion, inentives to deviate are not fied one and for all but depend on the period-per-period realization of effiieny. Hene, we need to onsider a more sophistiated ollusive agreement that also ontemplates temporary phases of ompetitive priing when firms osts make too strong inentives to deviation. More preisely, olluding firms observing the realization of osts know that with a ollusive priing r the seond most effiient firm may bee too prone to deviation and, to avoid a break down of the artel, the agreement presribes a temporary reversion to ompetition, until in subsequent aution stages osts allow to sustain high ollusive priing. Clearly, all this does not affet firms ability to detet deviations when osts status would onede ollusive priing and to punish suh deviation.

27 REPUTATIONAL COMMITMENTS AND COLLUSION IN PROCUREMENT 247 The epeted future ollusive profit C Π for any firm is { E θ r θ ( n) C( q) Pr( bw r) Eθ θ ( n) θ ( n) [ 1 Pr( bw r) ]} C Π = = + = = n(1 ) { θ E[ θ ( n) ] Pr( bw r) E θ( n) [ 1 Pr( bw r) ]} = = + Δ = n(1 ) where Pr( b = r) w allows ollusive priing and represents the probability of a ost realization that [ 1 Pr( b r) ] = is the probability assoiated to ompetitive priing. Note that we are assuming that olluding firms agree to omply with the quality requirement and in the following we will study whether this is indeed the ase or not. 23 By deviating, the seond most effiient firm (the one with the most inentives to deviate) an ask a prie slightly smaller than r thus winning the aution. This firm may deide to deviate from the ollusive agreement but to omply with quality so that it obtains the following payoff, ( n) E ( n) θ θ + Δθ n(1 ) where again ( ) E Δ θ n / n(1 ) is the epeted payoff for future autions upon deviation. Alternatively, the deviating firm an also deide to shirk on quality so that its profit turns out to be w ( n) ( q) 1 1 θ θ + ψ Hene, the seond most effiient firm at any aution does not deviate if the following inentive ompatibility onstraint is satisfied, 24 ( n) E θ ( ) ma ( ) 1 C Δ Π θ θ n +, ψ q n(1 ) 1 (4)

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