The Dynamis of Bidding Markets with Finanial Constraints Pablo F. Beker University of Warwik Ángel Hernando-Veiana Universidad Carlos III de Madrid July 1, 2014 Abstrat We develop a model of bidding markets with finanial onstraints à la Che and Gale (1998b) in whih two firms hoose their budgets optimally and we extend it to a dynami setting over an infinite horizon. We provide three main results for the ase in whih the opportunity ost of budgets is arbitrary low, the finanial onstraint is su iently severe and the exogenous ash-flow is not too large. First, firms keep small budgets and markups are high most of the time. Seond, the dispersion of markups and money left on the table aross prourement autions hinges on di erenes, both endogenous and exogenous, in the availability of finanial resoures rather than on signifiant private information. Third, we explain why the empirial analysis of the size of markups based on the standard aution model may have a bias, downwards or upwards, positively orrelated with the availability of finanial resoures. A numerial example illustrates that our model is able to generate a rih set of values for markups, bid dispersion and onentration. JEL Classifiation Numbers: L13, D43, D44. Keywords: bidding markets, finanial onstraints, markups, money left on the table, market shares, industry dynamis. We are grateful to Luis Cabral, Dan Kovenok, Gilat Levy and Matthew Shum for their omments. We also thank an anonymous referee for very detailed omments that greatly improved the paper. We thank Fundaión Ramón Arees and the Spanish Ministry of Eonomis and Competitiveness (projet ECO2012-38863) for their finanial support. Department of Eonomis. University of Warwik, Coventry, CV4 7AL, UK. Email: Pablo.Beker@warwik.a.uk URL: http://www2.warwik.a.uk/fa/so/eonomis/sta /faulty/beker Department of Eonomis. Universidad Carlos III de Madrid. / Madrid, 126. 28903 Getafe (Madrid) SPAIN. Email: angel.hernando@u3m.es URL: http://www.eo.u3m.es/ahernando/
1 Introdution An impliit assumption of the standard model of bidding is that the size of the projet is relatively small ompared to the finanial resoures of the firm. That this assumption is key to derive the main preditions of the standard model is known sine the analysis of Che and Gale (1998b). In their model, the extent to whih a firm is finanially onstrained depends on its budget, working apital hereafter, whih is assumed exogenous. In our paper, as it happens in reality, the firm s working apital is not exogenous but hosen out of the firm s internal finanial resoures, the ash hereafter, whih in turn depends on the past performane of the firm. Our first main result, stated in Theorem 1, hallenges the view that autions [still] work well if raising ash for bids is easy (Aghion, Hart, and Moore (1992, p. 527)). 1 Although the standard model arises in our infinite horizon setup when working apitals are su iently abundant, firms tend to keep too little of it and markups are high if the opportunity ost of working apital is arbitrarily low, the finanial onstraint is su severe and the exogenous ash-flow is not too large in a sense we formalise later. iently Besides, our model displays sensible features regarding the behaviour of markups, money left on the table and market shares that suggests that we should be more autious in the empirial analysis of bidding markets. Our seond main result, see Corollaries 2 and 4, provides a new explanation for the dispersion of markups and money left on the table 2 observed aross prourement autions. Interestingly, this explanation, disussed immediately below the aforementioned orollaries, does not hinge on signifiant private information about working apitals and osts, but on di erenes in the availability of finanial resoures aross autions in a sense that we formalise later. This asts doubts about the usual interpretation for the dispersion of markups and money left on the table observed in prourement as indiative of inomplete information and large heterogeneity in prodution ost. 3 Our third main result, see Corollaries 3 and 4, explains why the empirial 1 This onjeture has been reently questioned by Rhodes-Kropf and Viswanathan (2005) under the assumption that firms finane their bids by borrowing in a ompetitive finanial market. 2 Money left on the table is the di erene between the two lowest bids in prourement autions. 3 Indeed, as Weber (1981) pointed out: Some authors have ited the substantial unertainty onerning the extratable resoures present on a trat, as a fator whih makes large bid spreads [i.e. money left on the table ] unavoidable. More reently, Krasnokutskaya (2011) noted that The magnitude of the money 1
analysis of the size of markups may be biased downwards or upwards with a bias positively orrelated with the availability of finanial resoures when the researher assumes that the data are generated by the standard model. We also use a numerial example to illustrate that the model is able to generate a rih set of values for key variables like markups, bid dispersion and onentration. We are interested in markets in whih only bids that have seured finaning an be submitted, i.e. are aeptable, 4 as when surety bonds are required. 5 We also follow Che and Gale s (1998b) insight that the set of aeptable bids inreases with the working apital. This feature is present in a number of settings in whih firms have limited aess to external finanial resoures. One example is an aution in whih the prie must be paid upfront, and hene the maximum aeptable bid inreases with the firm s working apital. Another example is a prourement ontest in whih the firm must be able to finane the di erene between its working apital and the ost of prodution. If the external funds that are available to the firm inrease with its bid or its profitability, it follows that the firm s minimum aeptable bid dereases in the firm s working apital. The latter property arises when the sponsor pays in advane a fration of the prie, 6 a feature of the ommon pratie of progress payments, or when the amount banks are willing to lend depends on the profitability of the projet, as it is usually the ase. 7 A representative example of the institutional details of the bidding markets we are interested in is highway maintenane prourement. As Hong and Shum (2002) pointed out many of the ontrators in these autions bid on many ontrats over time, and likely left on the table variable [...] indiates that ost unertainty may be substantial. 4 Alternatively, we ould have assumed that it was ostly for the firm to submit a bid and not omplying, e.g. the firm may bear a diret ost in ase of default. 5 In the U.S., the Miller At and Little Miller Ats regulate the provision of surety bonds for federal and state onstrution projets, respetively. A surety bond plays two roles: first, it ertifies that the proposed bid is not jeopardized by the tehnologial and finanial onditions of the firm, and seond, it insures against the losses in ase of non-ompliane. Indeed, the Surety Information O e highlights that Before issuing a bond the surety ompany must be fully satised that the ontrator has [...] the finanial strength to support the desired work program. See http://suretyinfo.org/?wpfb_dl=149. 6 A numerial illustration an be found in Beker and Hernando-Veiana (2011). 7 We show in Appendix II that this is also the theoretial predition of a model inspired by the observation of Tirole (2006), page 114, that The borrower must [...] keep a su the projet in order to have an inentive not to waste the money. ient stake in the outome of 2
derive a large part of their revenues from doing ontrat work for the state. Besides, Porter and Zona (1993) explain that The set of firms submitting bids on large projets was small and fairly stable[...] There may have been signifiant barriers to entry, and there was little entry in a growing market. 8 Motivated by these observations, we build a stati model in whih two firms endowed with some ash hoose working apitals to ompete in a first prie aution for a prourement ontrat. The ost of omplying is known and idential aross firms, the minimum aeptable bid inreases with the firm s working apital and only ash is publily observable. 9 Sine using ash as working apital means postponing onsumption, it is ostly. 10 Firms hoose their working apitals and bids optimally. The stati model provides a simple setting with a unique equilibrium that illustrates the strategi fores that shape our results. The dynami model onsists of the infinite repetition of the stati model. The ash at the beginning of eah period is equal to the last period unspent working apital plus the earnings in previous prourement ontrat and some exogenous ash-flow. In our stati model, to arry more working apital than stritly neessary to make the bid aeptable is stritly dominated beause of its ost. Thus, the firm that arries more working apital wins the ontrat 11 and both firms inur the ost of their working apital. The strategi onsiderations that shape the equilibrium working apitals are the same as in the all pay aution with omplete information. 12 Not surprisingly, in a version of our game with unlimited ash, there is a unique symmetri equilibrium in whih firms randomize in a bounded interval with an atomless distribution. This is also the unique equilibrium in our game when the firms ash is larger than the upper bound of the support of the equilibrium randomization. We all the senario symmetri if this is the ase, and laggard-leader otherwise. In this latter ase, firms also randomize in a bounded interval, 8 Moreover, it an be shown that in a model with many firms and entry the natural extension of the equilibrium we study has the feature that only the two firms with more ash enter the market. 9 Our first main result and the part of our seond main result regarding markups also hold in a version of our model with observable working apital, see Beker and Hernando-Veiana (2011). 10 Any other motivation for the ost of working apital would deliver similar results. 11 This feature seems realisti in many prourement ontrats: It is thought that Siemens superior finanial firepower was a signifiant fator in it beating Canada s Bombardier to preferred bidder status on Thameslink, in Minister bloks..., The Guardian, 11/De/2011. 12 It resembles Che and Gale s (1998a) model of an all pay aution with aps in that working apitals are bounded by ash. Our model is more general in that they assume exogenous aps ommon to all agents. 3
though the firm with less ash, the laggard hereafter, puts an atom at zero and the other firm, the leader, at the laggard s ash. In our dynami model, we haraterize a lass of equilibria that ontains the limit of the sequene of the unique equilibrium of models with an inreasing number of periods. Remarkably, the marginal ontinuation value of ash is equal to its marginal onsumption value under a mild assumption about the minimum aeptable bid. Thus, as in the stati model, firms do not arry more working apital than stritly neessary to make the bid aeptable and the strategi interation eah period is, again, similar to an all pay aution. On the equilibrium path, the frequeny of eah senario depends on the severity of the finanial onstraint, that we define as the ratio between the working apital for whih the minimum aeptable bid equals the ost of the prourement ontrat and the exogenous ash flow. If this ratio is large, the laggard-leader senario ours most of the time, as the ost of working apital beomes negligible. This implies our first main result. Another onsequene is that a firm tends to win onseutive prourement ontrats. 13 In ontrast, when the ratio is so small that the symmetri senario ours eah period, the probability that a firm wins a ontrat is onstant aross periods. To understand the seond main result, note that the dispersion of markups and money left on the table is due to heterogeneity aross autions in the availability of finanial resoures, i.e. either the firms ash in the laggard-leader senario of the stati model, the minimum aeptable bids or the exogenous ash flow in the dynami model. Either of these variables a et the equilibrium working apitals whih determine the bids, and hene the markups and money left on the table. To understand the third main result, note that biases in the strutural estimation of markups an also arise if, as it is often the ase, the researher does not observe osts. Imagine bid data from several autions with idential finanial onditions and suppose the data are generated by our stati model. On the one hand, there are large markups and little money left on the table if the laggard has little ash. However, a researher who assumed the standard model would onlude that there is little ost heterogeneity and, onsequently, small markups, i.e. the estimation would be biased downwards. On the other hand, if the laggard has relatively large ash, 13 To the extent that joint profits are larger in the laggard-leader senario than in the symmetri senario, our result is related to the literature on inreasing dominane due to e ieny e ets (see Budd, Harris, and Vikers (1993), Cabral and Riordan (1994) and Athey and Shmutzler (2001).) 4
but not too large, there is sizable money left on the table and relatively low markups. However, a researher that assumed the standard model would onlude that there is large ost heterogeneity and, as a onsequene, large markups, i.e. the estimation would be biased upwards. Che and Gale (1998b) and Zheng (2001) already showed that the dispersion of markups an reflet heterogeneity of working apital if this is su iently sare. 14 We show that sarity is the typial situation if firms hoose their working apital. Whereas they assume that the distribution of working apitals is onstant aross firms, our results show that this distribution is seldom onstant aross firms. This di erene is important beause the lak of asymmetries in the distribution of working apitals preludes the possibility of large expeted money left on the table when private information is small. Firms also hoose working apitals in Galenianos and Kirher s (2008) model of monetary poliy and in Burkett s (2014) prinipal-agent model of bidding. Whereas the all pay aution struture only arises in the former, the laggard-leader senario does not our beause working apital is not bounded by ash. Our paper ontributes to a reent literature that explains how asymmetries in market shares arise and persist in otherwise symmetri models. In partiular, Besanko and Doraszelski (2004), and Besanko, Doraszelski, Kryukov, and Satterthwaite (2010) show that firm-speifi shoks an give rise to a dynami of market shares similar to ours. The di erene, though, is that the dynami in our model arises beause firms randomize their working apital due to the all pay aution struture. Our haraterization of the dynamis resembles that of Kandori, Mailath, and Rob (1993) in that we study a Markov proess in whih two persistent senarios our infinitely often and we analyse their frequenies as the randomness vanishes. While the transition funtion of their proess is exogenous, ours stems from the equilibrium strategies. Setion 2 explains how we model finanial onstraints. Setions 3 and 4 analyse the stati and the dynami model, respetively. Setion 5 onludes. Appendies I and II ontain the proofs and an extension that endogenizes finanial onstraints, respetively. 14 See also Che and Gale (1996, 2000), and DeMarzo, Kremer, and Skrzypaz (2005). Pithik and Shotter (1988), Maskin (2000), Benoit and Krishna (2001) and Pithik (2009) study how bidders distribute a fixed budget in a sequene of autions. This is not an issue in our setup. 5
2 A Redued Form Model of Prourement with Finanial Constraints In this setion, we desribe a model of prourement that we later embed in the models of Setions 3 and 4. Two firms 15 ompete for a prourement ontrat of ommon and known ost in a first prie aution: eah firm submits a bid, and the firm who submits the lowest bid gets the ontrat at a prie equal to its bid. 16 Only bids in a restrited set, the aeptable bids, are allowed. In partiular, we assume that the minimum aeptable bid of a firm with working apital w 0 is given by 17 b (w) (w) + where is stritly dereasing, satisfies (0) > 0 and lim w!1 (w) < 0 and is ontinuously di erentiable. As we disuss in the Introdution, our assumption that firms an submit only aeptable bids aptures a wide range of institutional arrangements whose aim is to prelude firms from submitting bids they annot omply suh as bids that annot be finaned. 18 Alternatively, the sponsor may provide inentives to guarantee that firms submit only aeptable bids by making them bear some of the ost of default. The monotoniity of the set of aeptable bids arises naturally in markets in whih firms have limited aess to external finanial resoures, as we disussed in the Introdution and Appendix II. For any given bids b 1 and b 2,weusemarkup to denote min{b 1,b 2 } left on the table to denote b 1 b 2. and we use money Definition 1. is the working apital for whih the minimum aeptable bid is equal to the ost of the prourement ontrat so that ( ) = 0 or, equivalently, = 1 (0). Our assumptions on imply that there exists a unique 2 (0, 1). 15 As in all pay autions, see Baye, Kovenok, and de Vries (1996), if there are more than two firms then there are multiple equilibria. One suh equilibrium is that in whih two firms hoose the equilibrium strategies of the two-firm model and the other firms hoose zero working apital. 16 Asaleautionofagoodwithommonandknownvaluev an be easily enompassed in our analysis assuming that = v < 0 and bids are negative numbers. 17 Thus, the model of autions with budget onstraints analysed by Che and Gale (1998b) in Setion 3.2 orresponds in our framework with b (w) = w and (w) =v w, and the interpretation in Footnote 16. 18 For instane, Meaney (2012) says that As well as onsidering the finanial aspets of bids, the DfT [the sponsor] assesses the deliverability and quality of the bidders proposals so as to be onfident that the suessful bidder is able to deliver on the ommitments made in the bidding proess. 6
3 The Stati Model Eah firm i 2{1, 2} starts with some ash m i 0. We assume the firm s ash to be publily observable. Eah firm i hooses simultaneously and independently (I) how muh of its ash to keep as working apital w i 2 [0,m i ] and (II) an aeptable bid b i b (w i ) for a market as desribed in Setion 2. A pure strategy is thus denoted by the vetor (b i,w i ) 2{(b, w) :b b (w),w 2 [0,m i ]}. Firmi s expeted 19 profit in the market against another firm with ash m j that bids b j is equal to: 8 b >< i if b i = b j and m i >m j or if b i <b j, V (b i,b j,m i,m j ) 1 2 (b i ) if b i = b j and m i = m j, >: 0 otherwise, (1) where we are applying the usual uniformly random tie breaking rule exept in the ase in whih one firm has stritly more ash than the other. In this ase, we assume that the firm with stritly more ash wins. 20 We assume that the firm maximises m i w i + (w i + V (b i,b j,m i,m j )), that is, m i w i,itsonsumption hereafter, plus the disounted sum, at rate 2 (0, 1), of the working apital and the expeted profit in the market. Note that a unit inrease in working apital is ostly in the sense that it redues the urrent utility in one unit and inreases the future utility in. Thus, the ost of working apital beomes negligible when inreases to 1. We start by simplifying the strategy spae. First, any strategy (b, w)inwhihb>b (w) is stritly dominated by the strategy (b, w) where w satisfies b = b ( w) so that it is never optimal to arry more working apital than is stritly neessary. 21 Thus, we restrit to the set of pure strategies {(b, w) :b = b (w),w 2 [0,m]} where m denotes the firm s ash. In our seond simplifiation of the strategy spae, we use the following definition: 19 We take expetations with respet to the tie breaking rule in the ase b i = b j and m i = m j. 20 We deviate from the more natural uniformly random tie-breaking rule that is usual in Bertrand games and all pay autions in order to guarantee the existene of an equilibrium. In our game, a su iently fine disretisation of the ation spae would overome the existene problem and yield our results with the usual uniformly random tie-breaking rule at the ost of a more umbersome notation. 21 The probability that a firm wins the ontrat is una eted but the ost of working apital inreases. 7
Definition 2. 2 [0, ) is the unique solution 22 in w to (w) =(1 )w. (2) Sine w = solves Equation (2) for = 1, the Impliit Funtion Theorem implies: lim "1 = (3) Thus, denotes the working apital for whih ( ), the disounted prourement profits assoiated with the minimum aeptable bid orresponding to working apital, equals (1 ), the impliit osts of seleting working apital that are assoiated with postponing onsumption. Any pure strategy (b (w),w)inwhihw> is stritly dominated by (b ( ), ). As a onsequene, we further restrit the set of pure strategies to {(b, w) :b = b (w),w 2 [0, min{m, }]} where m denotes the firm s ash. One we eliminate the above stritly dominated strategies, the resulting redued game has a unidimensional strategy spae as an all pay aution. Eah firm hooses a working apital and its orresponding minimum aeptable bid. The firm with the higher working apital wins the prourement ontrat and arrying working apital is ostly for eah firm. As in all pay autions, there is no pure strategy equilibrium. This an be easily understood when eah of the two firms ash is weakly larger than. If both firms hoose di erent working apitals, the one with more working apital has a stritly profitable deviation: to derease marginally its working apital. 23 If both firms hoose the same working apital w, there is also a stritly profitable deviation: to inrease marginally its working apital if w<, and to hoose zero working apital if w =. 24 A mixed strategy over the set of stritly undominated strategies is desribed by 22 Note that this equation is equivalent to m w + w + (w) =m. 23 It saves on the ost of working apital without a eting to the ases in whih the firm wins and inreases the profits from the prourement ontrat beause it inreases the prie. 24 In the former ase, the deviation is profitable beause winning the prourement ontrat at w< gives stritly positive profits and the deviation breaks the tie in favor of the deviating firm with an arbitrarily small inrease in the ost of working apital and an arbitrarily small derease in the profits from the prourement ontrat. In the latter ase, w = implies that one of the firms is winning with a probability stritly less than one, and hene the definition of, see Footnote 22, means that this firm makes stritly lower expeted payo s than with zero working apital. 8
a distribution funtion with support 25 ontained in the set {(b, w) : b = b (w),w 2 [0, min{m, }]} where m denotes the firm s ash. This randomization an be desribed by the marginal distribution over working apitals F. With a slight abuse of notation, we denote by (b,f) the mixed strategy where the firm randomises its working apital w aording to F and submits a bid b (w). If a firm uses (b,f)wheref is di erentiable and has support [w, w], then the expeted payo to the other firm with ash m w from hoosing w 2 (w, w) is m w + w + (w)f (w) (4) so that indi erene aross the support results only if F satisfies the di erential equation 1 = F 0 (w) (w)+f (w) 0 (w) (5) for any w 2 (w, w). Thus, (1 ), the inrease in the ost of working apital w(1 ), must equal F 0 (w) (w) +F (w) 0 (w), the hange in the expeted disounted profits (w)f (w). There is both a positive e et and a negative e et of an inrease in w on the hange in expeted disounted profits. The former arises due to the higher probability of winning a ontrat and the latter due to the lower profits assoiated with a win. We distinguish two senarios: Definition 3. Let m l min{m 1,m 2 }.Thesymmetri senario denotes the ase in whih m l. The laggard-leader senario denotes the omplementary ase. Let y denote the degenerate distribution that puts weight 1 on y 2 R. Proposition 1. If m l, then the unique equilibrium is symmetri and denoted by the single (mixed) strategy b,f where F (w) (1 )w (w) (6) with support [0, ] solves the di erential equation (5) with initial ondition F (0) = 0. Besides: (i) the equilibrium probability of winning the ontrat is ommon aross firms; (ii) the equilibrium is una eted by any hange in ash that leaves m l F (w) onverges to (w) as inreases to 1. ; and (iii) 25 We use the definition of support of a probability measure in Stokey and Luas (1999). Aording to their definition, the support is the smallest losed set with probability one. 9
This equilibrium satisfies the usual property of all pay autions that bidders without ompetitive advantage get their outside opportunity, i.e. the payo of arrying zero working apital and losing the prourement ontrat. Besides, one an dedue the following orollary from Equation (6) using Equation (3). Corollary 1. If m l,theninequilibrium, (w 1 ) (w 2 ) and (max {w 1,w 2 }) onverge in distribution to 0 as inreases to 1. In the standard aution model, ost heterogeneity vanishes as the distribution of osts onverges to the degenerate distribution that puts all the weight on one value. As ost heterogeneity vanishes, the markup and money left on the table vanish (Krishna (2002), Chapter 2). Corollary 1 says that this limit outome also arises as inreases to one in the symmetri senario, see Definition 3, sine the markup min{b 1,b 2 } is equal to (max{w 1,w 2 }) and money left on the table b 1 b 2 is equal to (w 1) (w 2 ).Inthissense, finanial onstraints beome irrelevant as inreases to one. We next onsider the laggard-leader senario, see Definition 3. In what follows, the leader refers to the firm that starts with more ash and the laggard to the other firm. Proposition 2. If m l < and m 1 6= m 2,thentheuniqueequilibrium 26 is denoted by the laggard and leader strategies b,f l and b,f L,respetively,where: F l (w) F (w) + (m l) (1 )m l (w) 8 < F (w) if w 2 [0,m l ), F L (w) : 1 if w = m l, if w 2 [0,m l ], (7) (8) have both support [0,m l ] and solve the di erential equation (5) with boundary onditions F (m l ) = 1 and F (0) = 0, respetively. Besides: (i) the leader is more likely to win the ontrat in equilibrium, (ii) F L (w) onverges to ml (w) as inreases to 1, and (iii) the equilibrium probability that the winner is the leader onverges to 1 as inreases to 1. Both firms put their atom of probability at points that do not upset the inentives of the rival to play its equilibrium randomization. There is only one suh point for the 26 Interestingly, this equilibrium has similar qualitative features as the equilibrium of an all pay aution in whih both agents have the same ap but the tie-breaking rule alloates to one of the agents only. The latter model has been studied in an independent and simultaneous work by Sze (2010). 10
laggard, whereas the leader s atom is at the minimum working apital whih ensures that it wins the prourement ontrat. Interestingly, it an be shown that the laggard gets its outside opportunity, as in the symmetri senario, whereas the leader gets an additive positive premium. The latter is a onsequene of the leader s ability to underut any aeptable bid of the laggard and the fat that any suh bid is stritly profitable. Corollary 2. If m l < and m 1 6= m 2, (i) an inrease in m l for whih m l < inreases (in the sense of first order stohasti dominane) both equilibrium distributions of working apitals and hene, dereases the equilibrium expetation of (max{w 1,w 2 }), and (ii) the equilibrium probability that the laggard s hooses working apital 0 and the leader m l is: (m l ) (0) 1 (1 )m 2 l. (9) (m l ) Corollary 2 is diret from Equations (7) and (8) and it is the starting point for our seond main result. Point (i) shows that the dispersion of markups, min{b 1,b 2 } (max{w 1,w 2 }) = observed aross autions an be explained by variations in the laggard s ash and it suggests that the same an apply to the dispersion of money left on the table, b 1 b 2 = (w 1) (w 2 ). Note that a similar argument also applies with respet to hanges in. Point (ii) also asts doubts about the usual interpretation of money left on the table as indiative of inomplete information. To see why, onsider the linear example 27 (w) = w. In this ase, the probability that both firms play at their atoms, 0 and m l, tends to one as diverges to infinity, whih also means that the money left on the table tends to (0) (m l) = m l. Thus, a su iently large implies almost no unertainty together with sizable money left on the table. Note that the impliations about money left on the table that are only suggested by Corollary 2, are proved in Corollary 4 for the dynami model under the assumptions that the finanial onstraint is su iently severe, in a sense we formalise later, and is su iently lose to one. Here, the laggard s ash is exogenous but in the model of Setion 4 we show in a numerial example that the endogenous distribution of the laggard s ash has su ient 27 If (w) = w, then the probability that both firms play at their atoms is: (m l ) (0) 1 2 (1 )m l = (m l ) ml 1 2 (1 )m l. ( m l ) 11
variability to generate signifiant dispersion of markups and money left on the table aross otherwise idential autions. Interestingly, these results are provided for parameter values for whih there is little unertainty. Corollary 3. If m l < and m 1 6= m 2, then as inreases to 1: (i) in equilibrium, (max{w 1,w 2 }) onverges in distribution to of (w 1 ) (w 2 ) onverges to 28 (m l ) ln (ml ), and (ii) the equilibrium expetation. (0) (m l ) The orollary follows by inspetion of Equations (7) and (8). Intuitively, (i) an be explained beause the leader inreases its probability of winning by shifting all its probability mass to m l as inreases to one. Sine working apital is ostless in the limit, the laggard s randomization guarantees the indi erene of the leader by balaning the positive and negative e ets of an inrease in working apital on the expeted disounted profits, whih explains (ii). Corollary 3 implies that when is lose to one and m l < ˆm, where ˆm 1 ( (0) e ), the markup, min{b 1,b 2 } = (max{w 1,w 2 }), dereases 29 and money left on the table, b 1 b 2 = (w 1 ) (w 2 ), inreases as the laggard s ash m l inreases. This is the basis for our third main result. Suppose that is lose to one and that the bid data from several autions with idential finanial onstraints are generated by the model with onstant prourement ost. If m l <, then Corollary 3 states that there will be little money left on the table and there will be large markups when m l is lose to zero but that there will be substantial money left on the table and small markups when m l = ˆm. In what follows we assume that m l <. The bid data reveals the money left on the table but osts and, therefore, markups are not observable. If there is little money left on the table, as would happen if m l is lose to zero, an interpretation of the bid data using the standard model would onlude that there was little ost heterogeneity and small markups even though there were large markups in the generated data. That is, the results would be 28 Proving (ii) requires some non-trivial omputations. F l onverges to a distribution with an atom of probability (m l) at zero and density 0 (w) (m l) in (0,m (0) (w) 2 l ]. This together with the onvergene of F L (w) to ml (w) implies that the expetation of b 1 b 2 = (min{w 1,w 2}) (max{w 1,w 2}) onverges to: (0) (m Z ml l) (0) + (w) 0 (w) (m Z ml l) 0 (w) (0) dw (m (w) 2 l )= (m l ) dw = (m l ) ln. (w) (m l ) 0 29 Sine @ (m) ln (0) > 0ifm< ˆm. @m (m) 0 12
biased downward. If there is a substantial amount of money left on the table, as would happen if m l =ˆm, then an interpretation of the bid data using the standard model would onlude that there was large ost heterogeneity and therefore large markups even though there were small markups in the generated data. That is, the results on markups would be biased upwards. Finally, in Proposition 3 we desribe the equilibrium strategies when both firms have idential ash but smaller than.weuse 2 [0, ) defined as the unique m that solves: (m) (1 )m =0. (10) 2 The left hand side of Equation (10) is equal to the di erene in a firm s expeted payo s between hoosing working apital m and zero working apital when the other firm hooses working apital m. If m 2 (, ), we let (m) 2 [0,m]beimpliitlydefinedby: 30 F ( (m)) + 1 F ( (m)) (m) (1 )m =0, (11) 2 where F is defined in Equation (6). The left hand side of Equation (11) is equal to the di erene in a firm s expeted payo s between hoosing working apital m and zero working apital when the other firm hooses a working apital in (0, F ( (m)) and a working apital equal to m with probability 1 Proposition 3. (m)) with probability F ( (m)). If m 1 = m 2 = m 2 (0, ], then the unique equilibrium is symmetri and denoted by the single pure strategy (b (m),m). If m 1 = m 2 = m 2 (, ), then the unique equilibrium is symmetri and denoted by the single mixed strategy (b,f ), where 8 F (w) >< if w 2 [0, (m)] F (w) F ( (m)) if w 2 ( (m),m) >: 1 if w m, and F is defined in Equation (6). 30 Existene and uniqueness of the solution follow from the properties of the left hand side of the equation. This is inreasing in (m), it is negative at (m) = 0 and it is stritly positive at (m) = m. The first one is diret, the seond an be dedued from Equation (10) using that m>, and the third from the definition of,inequation(2),usingthatm<,andthedefinitionoff in Equation (6). 13
The equilibrium in the first ase is explained by the fat that m apple implies that the left hand side of Equation (10) is weakly positive and hene the best response to a working apital equal to the ommon amount of ash m is to also hoose a working apital m. This is not the ase when m> as the left hand side of Equation (10) is stritly negative. Instead, the equilibrium in this ase is onstruted by shifting probability away from the ommon amount of ash and plaing it at the bottom of the spae of working apitals aording to a distribution that solves the di erential equation (5). We shall not disuss the impliations of Proposition 3 as in our dynami model the ase in whih both firms ash is less than does not arise along the game tree. See our disussion after introduing Assumption 1. 4 The Dynami Model In this setion, we endogenise the distribution of ash by assuming that it is derived from the past market outomes. This approah provides a natural framework to analyse the onventional wisdom in eonomis that autions [still] work well if raising ash for bids is easy. In Theorem 1, we provide onditions under whih the laggard-leader senario ours most of the time. This is the basis for our first main result. Besides, we provide formal results in Corollary 4 and a numerial example that, on the one hand, omplement the previous setion analysis of the seond and third main results and, on the other hand, shed some light on the onentration and asymmetries of market shares. 4.1 The Game We onsider the infinite repetition of the time struture of the game in the last setion. We assume that both firms have the same amount of ash in the first period. Afterwards eah firm s ash is equal to its working apital in the previous period plus the profits in the prourement ontrat and some exogenous ash flow 31 m > 0. We assume that m is onstant aross time and firms, and interpret it as derived from other ativities of the firm. Hene, in any period t in whih firms start with ash (m 1,t,m 2,t ), hoose working 31 All our results also hold true for the ase m = 0. However, the analysis in Setion 4.3 di ers, as explained in Footnote 42. 14
apitals (w 1,t,w 2,t ) and bids (b 1,t,b 2,t ), and Firm 1 wins the prourement ontrat with profits b 1,t, the next period distribution of ash is equal to: (m 1,t+1,m 2,t+1 )=(w 1,t + b 1,t + m,w 2,t + m). (12) Firm i 2{1, 2} wins in period t with probability one if b i,t <b j,t or if b i,t = b j,t and m i,t >m j,t, with probability 1/2 ifb i,t = b j,t and m i,t = m j,t, and loses otherwise. The payo in period t of a firm with ash m t that hooses working apital w t is equal to its onsumption m t w t. The firm s lifetime payo in a subgame beginning at period if its ash and working apital in the subsequent periods are given by {m t,w t } 1 t= is equal to: 1X t (m t w t ). t= We assume that the firm maximises its expeted lifetime payo at any period. The following assumption 32 is used in the proof of Proposition 4. Assumption 1. (w) m w for any w 2 [0, 1). Sine (w) is the minimum profit that a firm with working apital w an make when it wins the prourement ontrat, Assumption 1 and Equation (12) imply that the firm that wins the prourement ontrat one period, starts next period with ash at least. As we explain after Proposition 4, this assumption guarantees that firms do not want to arry more working apital than stritly neessary to make the bid aeptable. Assumption 1 also implies that must be less than any ommon amount of ash held by the firms in any information set after the first period. We show in Proposition 3, for the ase of the stati model, that a tedious ase di erentiation is neessary if one allows firms to have idential ash less than. For the same reason, we assume that both firms start in the first period with ash greater than. 33 We denote by the set of ash vetors that may arise in the information sets of the game tree. A Markov mixed strategy onsists of a randomization over the set of working apitals and aeptable bids for eah point (m, m 0 )in,wherem denotes the firm s ash and m 0 the rival s. We shall restrit to equilibria in Markov mixed strategies 32 A large lass of funtions satisfy this assumption, for instane the linear funtion (w) = w. 33 In this sense, our result that firms arry too little ash in the long term arises even when firms start with su iently large amounts of ash. 15
with support ontained in the set {(b, w) :b = b(w m, m 0 ),w 2 [0,m]} for some funtion b( m, m 0 ):[0,m]! R that satisfies that b(w m, m 0 ) (w)+ for any w 2 [0,m]. This Markov mixed strategy an be desribed by its marginal distribution funtion ( m, m 0 ) over working apitals and the bid funtion b ( m, m 0 ). We let W (m, m 0 ) denote the lifetime expeted payo of a firm that has ash m when its rival has m 0. In Definition 4 below, we denote the expeted ontinuation payo of a firm who bids b with working apital w, ash m and fae a rival who bids b 0, has working apital w 0 and ash m 0 by W (b, w, m, b 0,w 0,m 0 ) whih is equal to: (b, m, b 0,m 0 )W w+ m+b, w 0 + m +(1 (b, m, b 0,m 0 ))W w+m,w 0 +m+b 0, (13) where: 8 1 if either b<b >< 0, or if b = b 0 and m>m 0 (b, m, b 0,m 0 )= 0 if either b>b 0, or if b = b 0 and m<m 0 >: 1 2 if b = b 0 and m = m 0. desribes the alloation rule of the prourement ontrat. Definition 4. A (symmetri) Bidding and Investment (BI) equilibrium 34 is a value funtion W, a working apital distribution and a bid funtion b suh that for every (m, m 0 ) 2, W is the value funtion and ( m, m 0 ) and b( m, m 0 ) are the optimisers of the right hand side of the following Bellman equation: ZZ h W (m, m 0 ) = max m w + W ( b(w), w, m, b(w 0 m 0,m),w 0,m)i 0 (w) 2 (m) b(w) (w) + dw 0 m 0,m (dw), where Equation (13). (m) denotes the set of distributions with support in [0,m] and W is defined in 4.2 The Equilibrium Strategies In what follows, we define a value funtion, a bid funtion and a working apital distribution and show that they are a BI equilibrium. Our proposed strategies generalize 34 In a version of our model with finitely many periods studied in the supplementary material there is a unique equilibrium that is symmetri. We also show that as the horizon inreases to infinity, the limit of that equilibrium is a BI equilibrium. 16
the equilibrium strategies in Setion 3. The bid funtion is, as in the stati model, the minimum aeptable bid (with a slight abuse of notation): b (w m, m 0 ) (w)+. We find our equilibrium distribution of working apital by setting up a fixed point problem over a set of funtions and then use the solution of this problem to desribe the equilibrium distribution. We set up the fixed point problem as follows. We start with a non-empty, losed, bounded and onvex subset P of the spae of all bounded ontinuous funtions. For eah funtion in this lass P we set up a di erential equation that depends on. We then onsider the unique ontinuous solution, F m, to this di erential equation with initial ondition F (m) = 1. Lastly, we seek in P that is a fixed point of an operator T : P!P where T ( ) is desribed in terms of F m.onewehavethisfixed point, say, we then use F m to define the equilibrium distribution of working apital. Let P be defined as: apple :[0, 1)! 0, 1 (0) is ontinuous, dereasing and (m) =08m. (14) Definition 5. For any 2P and m 2 [0, ), we denote by F m :[0,m]! R the unique ontinuous solution to the first order di erential equation: 35 1 = F 0 (w)( (w)+ (w + m)) + F (w) 0 (w) and F (m) =1. (15) The funtional form of F m an be found in Equation (A5) in the Appendix. 36 Note that Equation (15) is analogous to Equation (5) and that Equation (15) is idential to Equation (5) when is the zero funtion. Definition 6. We denote by ˆ the unique value of m2 [0, ) for whih F m(0) = 0. By Definition 2, Equation (6) and Definition 6, we see that = F 1 (1) and ˆ = when is the zero funtion. We undersore that, for any m apple ˆ, F m(w) is a distribution of w (given the pair (,m)) with support in [0,m] that is ontinuous for w 2 (0,m) but it has an atom of size F m(0) at w =0whenm<ˆ. Reall that F ˆ (0) = 0 by Definition 6. Consider the following funtional equation: T ( ) =, (16) 35 The uniqueness of the solution follows from Theorem 7.1 in Coddington and Levinson (1984), pag. 22. 36 We thank an anonymous referee for pointing out that Equation (15) has an expliit solution. 17
where T : P!P is defined as: 37 8 < F m(0) ( (0) + (m)) if 0 apple m apple ˆ, T ( )(m) : 0 if m>ˆ. (17) Definition 7. For any 2 (0, 1), we denote by b P P the set of fixed points of T,by an element of b P and by ˆ the upper end of the support of the distribution F. Lemma A4 in the Appendix shows that the set of fixed points b P is not empty. Let: F l,m (w) =F L,m (w) = F (w) if w apple apple m, (18) F l,m (w) = F m (w) if w apple m<, (19) 8 < F (w) if w<m<, F L,m (w) (20) : 1 if w = m<. For any (m, m 0 ) 2, let: 8 < (w m, m 0 F ) l,m (w) if m apple m 0, : F L,m 0(w) if m>m 0, and: 8 m + >< 1 m if m apple m 0, W m, m 0 >: m + 1 m + (m 0 ) if m>m 0. (21) (22) Thus, (m 0 ) is an additive premium assoiated to being leader. Note that Assumption 1 implies that the ase in whih both firms have the same ash m = m 0 and (m, m 0 ) 2 an only arise if m = m 0. By Definitions 6 and 7, <. Thus, Equation (18) implies that F l,m = F L,m 0 = F, and Equations (16) and (17) imply that (m 0 ) = 0. Thus, neither nor W hange disontinuously at any of these points. Proposition 4. For eah 2 P b, Equations (18)-(22) define a W and a. Then, (W,,b )isabiequilibrium. 38 37 That T ( ) 2P follows from heking the onditions in (14). Sine F m dereases in m, byequation (A5), Equation (17) implies that T ( )(m) dereases ontinuously from F 0 (0)( (0) + (m)) to Fˆ (0)( (0) + (m)) as m inreases from 0 to ˆ, and it is then equal to zero. Besides, T ( )(m) =0 for m sine >ˆ, by Definition 6, Fˆ (0)( (0) + (m)) = 0 sine Fˆ (0) = 0, by Definition 6, and F 0 (0)( (0) + (m)) apple (0) sine F (1 ) 0 (0) = 1, by Definition 5, and (m) apple (0) sine 2P. 38 (1 ) The limit of the unique equilibrium of the finite horizon model is one of the equilibria desribed in Proposition 4, see the supplementary material. 18
The intuition behind the proposition is based on our results in the stati model. There, we use the property that the game has the all pay aution struture: after deleting stritly dominated strategies, the firm that arries more working apital wins but arrying working apital is ostly for both firms. This argument also applies here beause this property is inherited from one period to the previous one in the following sense: if the payo s of the redued game in period t satisfy the property, so do the payo s of the redued game in period t 1. To see why, note that the usual result of all pay autions that bidders without ompetitive advantage get their outside opportunity implies here that the laggard s equilibrium payo s in the redued game of period t are equal to the payo s of onsuming all its ash and starting period t + 1 as a laggard with ash m. The leader s equilibrium payo s in the redued game in period t have an additive premium whih is a onsequene of the leader s ability to arry su ient working apital to underut any aeptable bid of the laggard. This ability is independent of the amount of ash the leader has and so it is the premium. Consequently, the value of a marginal inrease in the ash with whih the firm starts period t is equal to its onsumption value plus the value of swithing from laggard to leader. The value of swithing from laggard to leader is zero beause a marginal inrease in ash swithes the leadership only when the ash of the firms is the same and, by Assumption 1, no less than, whih means that the premium is zero beause none of the firms is onstrained by ash to bid above ost. We an thus onlude that, in period t 1, a unit inrease in working apital, keeping onstant the bid, is ostly in the sense that it redues the urrent onsumption in one unit but only inreases the future utility in its disounted value. This means, as in the stati model, that it is not profitable to arry more working apital than neessary to make the bid aeptable. Thus, in period t 1, after deleting stritly dominated strategies, the firm that arries more working apital wins but arrying working apital is ostly for both firms. 39 We an also distinguish here between a symmetri and a laggard-leader senarios and it may be shown that an analogous version of points (i)-(iii) in Propositions 1 and 2 and properly adapted versions of Corollaries 1-3 hold true as well. 39 Note that the property that firms do not want to arry more working apital than stritly neessary to make the bid aeptable is also a property of the unique equilibrium of the finite version of our model. This is beause the reursive argument in the previous paragraph an be applied starting from the last period sine the last period is the same game as the stati model. See the supplementary material. 19
4.3 The Equilibrium Dynamis To study the frequeny of the symmetri and the laggard-leader senarios, we study the stohasti proess of the laggard s ash indued by our equilibrium. Its state spae is equal to [m, + m] beause the prourement profits are non negative and none of the firms working apitals is larger than. The pair of ash holdings (m 1,t+1,m 2,t+1 )inperiodt+1 and, therefore, the laggard s ash in period t+1, denoted by m t+1 min {m 1,t+1,m 2,t+1 }, are determined by the distribution over working apitals (w 1,t,w 2,t ) and bids (b 1,t,b 2,t )in period t whih is ompletely determined by the laggard s ash m t in period t. Thus, the laggard s ash follows a Markov proess. Let B denote the Borel sets of [m, + m]. The probability that m t+1 lies in a Borel set given that m t = m is given by a transition funtion Q :[m, + m] B![0, 1] that an be easily dedued from the equilibrium. In partiular, it is defined by: 40 8 < 1 1 F l,m (x m) 1 F L,m (x m) if x m<m,, Q (m, [m, x]) = : 1 o.w. This expression is equal to one minus the probability that both the laggard s and the leader s working apitals are stritly larger than x m. Definition 8. A distribution µ : B![0, 1] is invariant if it satisfies: Z µ (M) = Q (m, M) µ (dm) for all M2B. (24) Standard arguments 41 an be used to show that there exists a unique invariant distribution, it is globally stable and its support is equal to 42 [m, (23) + m]. A suitable law of large numbers an be applied to show that the fration of time that the Markov proess spends on any set M2Bonverges (almost surely) to µ(m). Typially, the frequeny of eah senario depends on a non trivial way on the transition probabilities. An exeption is when the transition probabilities do not depend on the 40 As a onvention, we denote by [m,m]thesingleton{m}. 41 See Hopenhayn and Presott (1992). 42 Here is where the assumption m > 0 makes a di erene as the support of the invariant distribution would be equal to {0} if m = 0. This is beause zero beomes an absorbing state of the dynamis of the laggard s ash when m = 0. To see why, note that a feature of the equilibrium is that a laggard that hooses zero working apital in any given period loses with probability one in the aution of that period. Thus, the laggard starts next period with zero ash if m = 0 and its only feasible working apital is zero. 20
state. 43 Equations (18) and (23) imply that this independene ours in our model when m, this is when the exogenous ash flow m is so large that only the symmetri senario an our. Sine m also means that (m) = 0 for any m m, by Equation (17), then F = F and thus, F L,m = F l,m = F, for m m, by Equations (6), (18) and (A5), and both firms play eah period as in the symmetri senario of the stati model. Thus, the following proposition is a onsequene (and no proof is neessary) of Equations (23) and (24), Proposition 1 and Corollary 1. Proposition 5. If m < 1, then: (i) the equilibrium probability of winning the ontrat at any date t is ommon aross firms, (ii) lim "1 µ ({ + m}) = 1 and (iii) the fration of time that both firms hoose working apital arbitrarily lose to, and (max{w 1,t,w 2,t }) and (w 1,t ) inreases to one. The ratio m (w 2,t ) are arbitrarily lose to 0 onverges (almost surely) to one as measures the severity of the finanial onstraint as it dereases with the ash flow m and inreases with the working apital needed to push the bid down to. Proposition 5 illustrates the onventional wisdom that autions [still] work well if raising ash for bids is easy as in our model, for a fixed m, it is easy to raise ash for bids from internal or external resoures if is lose to one or if is small, respetively. Next, Theorem 1 and Corollary 4(i), whih are the basis for our first main result, show that the ease to raise ash from internal resoures is not su ient for autions to work well. Theorem 1. If m > 4 and (2m)+ (m) > (0), then lim "1 µ ({m}) = 1. Taking as given, and sine we assume that is ontinuous, the seond hypothesis in Theorem 1 says that m is su iently small. For instane, the linear funtion (w) = w satisfies this su the finanial onstraint is su ient ondition when m < 3,i.e. m iently severe in the sense that m > 3. This theorem says that when > 4 and the exogenous ashflow is not too large in the sense that (2m)+ (m) > (0), then the laggard s ash is equal to m most of the time as inreases to one. 43 In the more di ult ase in whih the transition probabilities depend on the state, the invariant distribution assoiated to the limit transition probabilities as inreases to one has an easy haraterization. This is beause the transition probabilities beome degenerate and onentrate its probability in one point only, either m or + m, and thus any distribution with support in {m, + m} is an invariant distribution. Sine there are multiple invariant distributions, we annot apply a ontinuity argument to haraterize what happens when the ost of working apital is small. 21