Allotment and Subcontracting in Procurement Bidding

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1 Allotment and Subcontractng n Procurement Bddng Franços Marechal and Perre-Henr Morand May 2004 Abstract Allotment and subcontractng are the two alternatve mechansms enablng the partcpaton of SMEs n procurement. We compare these two alternatves n the context of a procurement contract awarded by a frst-prce sealed-bd aucton. When the wnnng large frm s constraned wth respect to the degree of subcontractng, we show that only a reducton of the chosen SME s proft can reduce the expected cost of the contract. However, when the large frm s allowed to choose the subcontractng level, subcontractng can be a Pareto domnatng mechansm,.e. smultaneously ncreasng both frms profts and reducng the expected total cost of the contract. JEL classfcaton: D44; L51. Keywords : allotment, subcontractng, procurement, bddng. DEEP-HEC, IEMS, (Unversty of Lausanne, BFSH1, CH-1015 Lausanne, Swtzerland) and CRESE. Tel: ; fax: ; francos.marechal@hec.unl.ch. CRESE, Unversty of Besançon, 45 D av. de l observatore, Besancon, France. Tel: ;perre-henr.morand@unv-fcomte.fr. We are grateful to Tago Rbero for helpful comments. The usual dsclamer apples. 1

2 1 Introducton The World Bank estmates that current spendng and growth nvestment actvty by governments account for 18% of GDP n developed countres, 13% of GDP n developng natons and 19% of GDP n transtonal economes. 1 In most countres, mcroenterprses and smallscale enterprses account for the maorty of frms and a large share of employment and t may be the polcy of governments to provde maxmum practcable opportuntes n ther acqustons from small busness. For example, SMEs account for over 65 % of turnover generated by the prvate sector n the European Unon, but the share of publc contracts won drectly (not takng subcontractng nto account) by SMEs remans low (less than 25 %). For some years already the European Commsson has pad specal attenton to the access and partcpaton of SMEs n the publc procurement market. The Commsson has called for greater partcpaton by SMEs wth a vew to strengthenng ther compettveness and enablng them to contrbute more towards growth, employment and compettveness n the European economy. Smlar concerns can be found n the US Federal Acquston Regulaton (FAR). Indeed, secton of the FAR s devoted to encouragng small busness partcpaton n acqustons Small busness concerns shall be afforded an equtable opportunty to compete for all contracts that they can perform to the extent consstent wth the Government s nterest. When applcable, the contractng offcer shall take the followng actons: (a) Dvde proposed acqustons of supples and servces (except constructon) nto reasonably small lots (not less than economc producton runs) to permt offers on quanttes less than the total requrement. [...] (d) Encourage prme contractors to subcontract wth small busness concerns. 1 Source: Oxford Analytca Academc Database. 2

3 As noted n a report to the US Small Busness Admnstraton, 2 despte clauses n the FAR callng on contract offcers to make specal efforts to sustan small busness partcpaton n procurement, budget cuts and drectves to streamlne the procurement process may be leadng contractng offcers to consoldate small purchases nto larger contracts n the name of a lmted effcency. These knds of procurement effcences mpact small busnesses negatvely because the requrements of larger, mult-faceted contracts can easly outstrp the fnancal or admnstratve capabltes of a small busness, precludng them from competng. Furthermore, the opportunty for small busnesses to subcontract from the larger companes wnnng the bundled contracts may also dmnsh because of a tendency for larger frms to use ther own resources on the contracts they wn. Evdence of the negatve mpact of contract bundlng on small busness was frst presented n the U.S. Small Busness Admnstraton s 1993 report. 3 Smlarly, the sxth European Observatory for SMEs shows that the most mportant reason why SMEs do not try to partcpate n European tenders s that the proects are too large for ther enterprses. It also reveals that there are consderable country dfferences wth regard to the partcpaton of SMEs n the publc procurement market. In Sweden, Italy and Portugal the percentage of SMEs tryng to partcpate n European tenders s lower than 10%, whereas n France t s 45%. In Belgum and Luxembourg about one thrd of the SMEs attempt to partcpate. The prmary way of enablng SMEs partcpaton n publc procurement s to dvde proposed acqustons of supples and servces nto reasonably small lots to permt offers on quanttes less than the total requrement. Ths allotment enables wde small busness partcpaton. Further, some countres have establshed programmes to encourage the use of SMEs n subcontractng wth large busnesses. In such programmes, the government awards 2 The Impact of Contract Bundlng on Small Busness, report by Eagle Eye Publshers, Inc. to the US Small Busness Admnstraton s Offce of Advocacy, (2000). 3 U.S. Small Busness Admnstraton, Study of the Impact of Contract Bundlng on Small Busness Concerns and Practcal Recommendatons (Report to the Commttee on Small Busness of the Unted States Senate and the Commttee on Small Busness of the Unted States House of Representatves, 14 May 1993), 77 pages. 3

4 a contract to a large frm wth the requrement or goal that the large frm purchase x% of the value of ts ntermedate nputs from SMEs. Subcontractng programmes can thus be vewed as an alternatve means of nvolvng SMEs n publc procurement actvtes. Hence, subcontractng and allotment of a procurement contract are the two alternatve ways for SMEs to access publc procurement. Should the publc buyer promote the partcpaton of SMEs by an allotment of procurement contracts or encourage subcontractng practces when contracts are awarded by means of a frst-prce sealed-bd aucton? Ths s the man concern of ths paper. To the best of our knowledge, ths queston has never been addressed n the lterature though subcontractng and allotment have already been consdered separately. The lterature on subcontractng essentally focuses on the agency relaton between a frm and ts subcontractors (e.g. Yun (1999) or Kawasak and McMllan (1987)) and on the mpact of the possblty of subsequent subcontractng among rvals on the competton between two frms (e.g. Kamen et al. (1989) or Gale et al. (2000)). We depart from these analyses, assumng that large frms do not subcontract among rvals but more realstcally wth SMEs. 4 Besdes, no postve studes of the allotment procedure have been done 5. In ths artcle, we consder the procurement of a fxed-prce contract awarded by means of a frst-prce sealed-bd aucton. The am of ths paper s to compare the allotment and subcontractng procedures n order to exhbt some mplcatons of these procedures for the mnmzaton of the total cost of the contract and also for small busness and large frms profts. More precsely, we derve condtons under whch both the wnnng large frm and the wnnng or chosen SME can be better off wth the allotment procedure or wth the subcontractng procedure. We also show, counter-ntutvely, that the publc buyer and the large frms can beneft from the asymmetrc nformaton between SMEs and large frms. Furthermore, when the wnnng large frm s constraned on the subcontractng level, we show that only a reducton of the chosen SME s proft can reduce the expected cost of the contract. 4 Laffont and Trole (1993) have addressed the queston of the value of delegatng subcontractng to a regulated frm n a normatve approach. 5 For a normatve approach of optmal allotment rules, see Morand (2003). 4

5 However, when the large frm s allowed to choose the subcontractng level, subcontractng can be a Pareto domnatng mechansm,.e. smultaneously ncreasng both frms proft and reducng the expected total cost of the contract. It gves strong support that the publc buyer should not constran the level of subcontractng and that contrarly to common vew SMEs are not necessarly better off when a part of a contract has been specfcally allotted to them. The next secton presents the model and ts assumptons. Secton 3 characterzes the allotment procedure whereas the subcontractng procedure s dscussed n secton 4. Secton 5offers a frst comparson of both procedures when the subcontractng level s mposed. Secton 6 deals wth the same comparson n a more general context,.e. when large frms are allowed to choose the subcontractng level. A fnal secton contans concludng remarks. Most of the proofs are relegated to the Appendx. 2 Outlne of the model We consder a procurement contract awarded by a frst-prce sealed-bd aucton. We assume that the supply-sde of the market s composed of n large frms (hereafter LFs) and m small busness frms (hereafter SBFs). All frms are assumed to be rsk-neutral. Let p [0, 1] be the proporton of the contract undertaken by a small busness frm (hereafter SBF), ether by a subcontractng or an allotment procedure. Each large frm (hereafter LF) 1,...,n and each SBF 1,...,m has prvate nformaton about ts own effcency parameter and enterng nto ts respectve cost functon c (,p) 1,...,n and c (,p) 1,...,m. However, t s common knowledge that and are..d. on the nterval [, ], accordng to probablty densty f, wth cumulatve F. 6 Concernng cost functons, we naturally have c (,p) p assume that 2 c (,p) p 2 2 c (,p) p > 0, c (,p) > 0, 2 c (,p) p 2 > 0 and c (,p) < 0 and c (,p) p > 0. We further > 0. We also assume that < 0, whch means that the larger the part of the contract undertaken by the LF, the 6 s assumed to reflect the parameter of the most effcent frm. 5

6 greater s the mpact of an ncrease n effcency. Smlarly, concernng SBFs cost functon, we assume that 2 c (,p) p > 0. 3 The allotment procedure In ths secton, we consder that by law, the contract has to be sub-dvded nto two lots, so that SBFs can have drect access to one of the lots. Then, n LFs compete for the award of apart(1 p a ) of the contract and m SBFs compete for the award of the remnder p a.wth ths allotment (a) procedure, we assume that subcontractng s not allowed. The expected proft oflf n a frst-prce sealed-bd aucton s Eπ a (b c (,p a )) (1 F (b 1 (b ))) n 1. (1) Then, each LF chooses a bd b to maxmze Eπ a. The dervatve of the expected proft wth respect to s deπ a (c (,p a )) (1 F (b 1 (b ))) n 1. (2) d Followng standard dervatons, 7 we obtan the optmal bddng strategy of LF b a ( )c (,p a ) R c (,p a ) (1 F ()) n 1 d (1 F ( )) n 1 1,...,n. The optmal bddng strategy s equal to the true cost plus a strategc mark-up. Contrary to conventonal developments n aucton theory, the strategc mark-up s not drectly related to the effcency parameter but to the cost functon and so s affected by the margnal varaton of ths cost nduced by the varaton of the effcency parameter. Ths s the margnal competton effect, whch turns out to be crucal n the followng. 8 Note that by assumpton, c (,p a) > 0. Actually, the more the cost s senstve to the effcency parameter, the greater s the dfference between the wnner and the second bdder and so the more the wnner can ncrease ts strategc mark-up. 7 See the appendx. 8 Ths effect was frst depcted by Marechal and Morand (2003), but n a less general framework. 6

7 Smlarly, the optmal bddng strategy of SBF s b a ( )c (,p a ) R c (,p a ) (1 F ()) m 1 d (1 F ( )) m 1 1,...,m. Let us denote g ( )n(1 F ( )) n 1 f ( ) d as the densty functon of the lowest cost of the n LFs, and h ( )m(1 F ( )) m 1 f ( ) d as the densty functon of the lowest cost of the m SBFs. Gven b a ( ) and b a ( ), the expected total cost of the proect s ETC a (b a ( )) g( )d b a ( ) h( )d. (3) From the pont of vew of the publc buyer, the expected proft of the wnnng LF s Z Ã Z! Eπ a(1) c (, p a ) (1 F ()) n 1 d g( )d. (4) The expected proft of the wnnng SBF s Z Ã Z! Eπ a(1) c (, p a ) (1 F ()) m 1 d h( )d. (5) Remark 1 For both the wnnng LF and the wnnng SBF, the expected proft onlydepends on the margnal competton effect. Indeed, n an aucton, the proft derved by the wnner s equal to the expected strategc mark-up. 4 The subcontractng procedure In ths secton, we consder that the contract s not sub-dvded so that SBFs only have access to publc procurement by means of subcontractng. We consder that the proporton of the contract to be subcontracted s not constraned by regulaton. 9 So, only LFs compete for the award of the whole contract but may subcontract a part of ths contract. Accordng to e.g. the US subcontractng regulaton for contracts that s expected to exceed $500,000 (1,000,000 for constructon), 10 we assume that each LF s 9 We further relax ths assumpton, consderng that the subcontractng level s mposed by procurement rules. Ths wll enable us to hghlght some specfc results. 10 Exceptons to ths rule can be found n Secton of the US Federal Acquston Regulaton. 7

8 requred to submt the potental subcontractng plan before the award of the contract. Then, each LF has to choose both the subcontractng level p( ) and a bd b. The barganng process between the wnnng LF and the chosen SBF s modelled n a very smple way. The payment receved by the SBF depends on the barganng power of both frms. Furthermore, although the nformaton s asymmetrc, we assume that the barganng process results n the most effcent SBF beng chosen. (1) Therefore, we can consder that the LF pays the SBF at a cost c ( b, p( )), where b > the expected true effcency parameter of the chosen (and most effcent) SBF. Roughly speakng, the closer b s to (1), the greater s the barganng power of the LF. However, under asymmetrc nformaton, no barganng process can perform better than an optmal aucton. Hence, c ( b, p( )) s bounded below by c ( (2),p( )), the second lowest cost of the m SBFs, whch would correspond to the result of an aucton mechansm. Note that we wll also consder a context of complete nformaton between LFs and SBFs as a benchmark stuaton. Ths case corresponds to a context n whch LFs know, contrary to the publc buyer, the true costs of the SBFs. The wnnng LF can then choose the most effcent SBF, characterzed by (1), and ths latter receves no rents. When a LF s selected, t has to support a total cost whch s equal to the sum of ts own cost (the non-subcontracted part of the contract) plus the negotated cost of the chosen subcontractor (as long as a part of the contract s subcontracted). The expected proft oflf when t chooses to subcontract (s) a part p( ) s Eπ s ( ) ³ b c (,p( )) c ( b, p( )) (1 F (b 1 (b ))) n 1. (6) The optmal subcontractng level p( ) maxmzes (6). The dervatve of Eπ s ( ) wth respect 8

9 to s deπ s ( ) d p( ) Ã c (,p( ) c ( b!, p( )) p( ) p( ) {z } 0 (1 F (b 1 (b ))) n 1. c (,p( ) {z } >0 c ( b, p( )) {z } S0 ³ c ( The term,p( )) can ether be postve, negatve or equal to zero, snce a more effcent ³ LF may be a better or a poorer barganer. Obvously, the term (c (,p( )) p( ) c (,p( )) p( ) vanshes snce p( ) maxmzes Eπ s ( ). of LF Followng the dervatons of the prevous secton, we obtan the optmal bddng strategy 11 b s ( ) c (,p( )) c ( b, p( )) R ³ (c (,p()) c (,p()) (1 F ()) n 1 d (1 F ( )) n 1 1,...,n. Note that, compared to the prevous bddng strategy b a ( ), the margnal competton effect stll appears but n the context of a new specfc effect. Indeed, c (,p( )) of the LF s effcency on the barganng result. reflects the mpact If c (,p( )) < 0, amoreneffcent LF s supposed to be a better barganer (barganng specalzed). > 0, amoreneffcent LF s then a poorer barganer (producton specal- If c (,p( )) zed). Fnally,thecasewhere c (,p( )) 0reflects the assumpton that the LF s effcency does not affect the barganng result. 11 In order to satsfy dbs ( ) d > 0, we assume n the followng that (c (,p( )) c (,p( )) > 0. 9

10 The expected total cost of the contract s ETC s (b s ( )) g( )d. (7) The expected proft of the wnnng LF s Z " Z Ã c (, p() c ( b! #, p()) (1 F ()) n 1 d g( )d. (8) The expected proft of the chosen SBF s Z Ã Z! ³ c ( b, p( )) c (,p( )) g( )d h( )d h c ( b, p( )) c ( (1),p( )) g( )d. (9) Before turnng to consder the man concern of our paper, an nterestng queston to nvestgate s whether the publc buyer and/or the LFs and the SBFs are better off under asymmetrc nformaton. Note that the chosen SBF gets no rents when, contrary to the publc buyer, the LF knows the SBFs costs. Then obvously, from (9), SBFs are better off under asymmetrc nformaton, snce > 0 when b > (1). Somewhat counter-ntutvely we have the followng lemma Lemma 1 The wnnng LF can be better off when nformaton s asymmetrc (.e. when SBFs have prvate nformaton about ther cost) f t s producton specalzed. In ths case, both the chosen SBF and the wnnng LF can be better off when nformaton s asymmetrc between them. Proof. See the Appendx. Ths lemma can be easly nterpreted. Snce each LF ncorporates the negotated cost of the subcontractor nto ts bddng strategy, the global negotated cost s neutral for the LF; only the margnal competton effect of effcency on negotated cost matters. Under complete nformaton between LFs and SBFs, ths latter vanshes. So the wnnng LF can be better off f asymmetrc nformaton unables t to ncrease ts mark-up. Furthermore, the goal of mnmzng total costs can be better acheved under ncomplete nformaton, accordng to the followng 10

11 Lemma 2 The expected cost of the contract may be lower under ncomplete nformaton than under complete nformaton f the wnnng LF s hghly barganng specalzed. Proof. See the Appendx. Indeed, when the LF s barganng specalzed, t gets more rents under ncomplete nformaton than under complete nformaton, whch ncreases expected total costs. Let us now compare n the followng the two procurement procedures. 5 A frst comparson In order to generate some nterestng frst results, we now consder that the subcontractng level s mposed by the publc buyer. Roughly speakng, ths frst comparson wll hghlght the problem of delegaton. Indeed, the publc buyer has to choose between two procedures: ether the wnnng LF s delegated to subcontract a gven part of the proect or the contract s dvded so that SBFs can drectly compete for the award of thesamegvenpartof the contract. Note that ths frst comparson does not ncorporate the ndrect mpact of the modfcaton of the subcontractng level whch s analyzed n the next secton. When subcontractng s mposed, the analyss can be derved n a straghtforward fashon from the prevous secton, substtutng p( )p a. The optmal bddng strategy of LF becomes b s ( ) c (,p a )c ( b, p a ) R ³ c (,p a) c (,p a ) (1 F ()) n 1 d (1 F ( )) n 1, 1,...,n wth the assumpton that c (,p a) c (,p a ) > 0 to ensure that b s ( ) s ncreasng. 5.1 Comparson of profts wth mposed subcontractng From (8) and (4), when p( )p a, the dfference n the wnnng LF s expected profts derved under the two procedures s Eπ a(1) Ã c ( b!, p a ) (1 F ()) n 1 d g( )d. (10) 11

12 Lemma 3 If the barganng power of the LF does not depend on ts own effcency, then allottng a part of the contract or mposng upon the LF to subcontract the same part yelds the same proft for the wnnng LF. Proof. If c (,p a ) 0, then Eπ a(1). Q.E.D. Corollary 1 If c (,p a ) < 0,.e. a more neffcent LF s a better barganer (barganng specalzed), then the wnnng LF gets a hgher proft wthallotment. If c (,p a ) > 0,.e. a more neffcent LF s a poorer barganer (producton specalzed), then the wnnng LF gets a hgher proft wth subcontractng. From the pont of vew of the wnnng LF, only the margnal competton effect matters. Even f a LF s a poorer barganer, the global competton effect does not matter snce ts bd covers the cost of the SBF. The key element s the gap between the LF s cost and the second expected net cost,.e. ts cost plus the cost pad to the subcontractor. In a nutshell the second bdder s less effcent than the frst one and therefore f a less effcent frm s a better barganer the gap between both net costs s reduced. Ths leads to a reducton n markup for the wnner. Conversely, the same argument apples when a less effcent frm s a poorer barganer. We now turn to the comparson of expected profts of the wnnng SBF. We have Eπ a(1) (11) Ã Z! c ( b c (, p a ), p a ) c (,p a ) (1 F ()) m 1 d h( )d. Equaton (11) shows that a SBF prefers to be a subcontractor than to partcpate n the aucton f the gan from the barganng process s hgher than the expected strategc mark-up obtaned n the aucton. Recall that standard developments n aucton theory show that the wnner s expected proft, n a frst-prce sealed-bd aucton, corresponds to the dfference between ts own cost 12

13 and the (condtonal) expected second cost. 12 Equaton (11) can, therefore, be rewrtten as Eπ a(1) c ( b, p a ) c ( (1),p a ) c ( b, p a ) c ( (2),p a ), ³ c ( (2),p a ) c ( (1),p a ) where c ( (2),p a ) reflects the expected second lowest cost among m SBFs. Under our framework (wth symmetrc, rsk-neutral bdders and ndependent prvate values), t s well known that the optmal barganng nsttuton s an aucton, and that a frst-prce aucton s revenue-equvalent to other aucton mechansms. Hence, n a context of asymmetrc nformaton, we have c ( b, p a ) c ( (2),p a ), snce an aucton s the most powerful tool to extract rents. Consequently, a SBF always prefers to be a subcontractor than to partcpate n an aucton (obvously, SBFs are better off wth the allotment procedure f LFs know the SBFs costs). 5.2 Comparson of total costs wth mposed subcontractng From (3) and (7), the comparson between expected total costs, when the publc buyer uses ether the subcontractng or the allotment rule, yelds ETC s ETC a c ( b, p a ) c (,p a ) R c (,p a ) (1 F ()) n 1 d (1 F ( )) n 1 g( )d (1 F ()) m 1 d (1 F ( ) m 1 h( )d. R c (,p a ) Snce the subcontracted part of the contract corresponds to the allotted part, the cost of the wnnng LF s the same under both procedures. Nevertheless, the cost of the wnnng SBF dffers, because t ether represents the cost the LF negotates and ncorporates nto ts own bd, or the expected second cost among the m SBFs when the contract s allotted. Therefore, only rents conceded to the wnnng LF and the chosen SBF modfy the expected total cost of the contract. We then have the followng lemma 12 See e.g. Krshna (2002) and Klemperer (1999) for surveys of aucton theory. 13

14 Lemma 4 If there s asymmetrc nformaton between LFs and SBFs (about SBFs costs) and f LFs effcency does not modfy ther barganng power, then allottng a part of the contract reduces the total expected cost compared to mposng upon the LF the requrement to subcontractthesamepart. 13 Proof. See the Appendx. Snce p( )p a, the expected total cost s only affected by rents conceded to both LFs and SBFs. When the barganng power of the LF does not depend on ts own effcency, from lemma 3, the proft of the wnnng LF s the same. So the expected total cost dffers only by the SBF s rents and under asymmetrc nformaton, the allotment procedure performs better n reducng expected total costs. Obvoulsy, under complete nformaton between LFs and SBFs, the SBF s rents vansh. So, the cost mnmzng goal s better acheved when the publc buyer delegates the selecton of the SBF. The comparson between expected profts of the chosen SBF and expected costs of the contract hghlghts the trade-off between cost effcency concerns and a publc polcy whch favors SBFs. Indeed, we have the followng proposton Proposton 1 When the LF s constraned on the subcontractng level, only a reducton of the chosen SBF s proft can reduce the expected cost of the contract. Hence, there does not exst any Pareto domnatng mechansm. Proof. See the Appendx. 6 Subcontractng vs allotment The prevous results clearly rely on the assumpton that the subcontractng level s mposed by law, and corresponds to the allotted part. In order to hghlght the mpact of the choce of 13 Assumng complete nformaton between LFs and SBFs on SBFs costs may also requre the assumpton of complete nformaton among SBFs. In ths case, the result of the bddng process among the SBFs would actually reman unchanged n terms of expected cost whch wll n each case correspond to the second lowest cost. 14

15 the subcontractng level by the LF, we now return to the general case where the law enables LFs to choose the subcontractng level p(). 6.1 Comparson of profts Let us frst analyze the expected profts of the wnnng LF. Comparng expected profts n each stuaton, we get Eπ a(1) (12) " Z Ã c ( b! #, p()) (1 F ()) n 1 d g( )d µ [c (, p()) c (, p a )] (1 F ()) n 1 d] g( )d. [ The second lne of (12) refers to the mpact of the margnal competton effect of the barganng power. Ths effect already exsted n the case where subcontractng was mposed, and may ether be postve or negatve dependng on specfc assumptons about the barganng technology. The thrd lne of (12) reflects the mpact of the choce of the subcontractng level. Ths mpact may renforce or reverse the prevous effect dependng on the condton p( ) Q p a. We then have the followng lemma Lemma 5 If the barganng power of the LF does not depend on ts own effcency or f amoreneffcent LF s a better barganer (barganng specalzed), then the wnnng LF s better off wththeallotmentprocedurethanwththesubcontractngprocedurefthelfs nduced to subcontract a larger part of the contract relatve to the allotted part. Proof. See the Appendx. If these condtons are not all satsfed, the results are less clear-cut: the wnnng LF may be better off wth subcontractng or wth allotment dependng on both the sensblty of the optmal choce of p and the margnal competton effect of the barganng power. Indeed, 15

16 from (12), we have > (<) Eπ a(1) f c ( b, p( )) [c (,p( ))] {z } >0 by assumpton > (<) [c (,p a )]. (13) Consder now the case of the wnnng (or chosen n the case of subcontractng) SBF. The dfference n the expected profts derved from each procedure s Eπ a(1) " h( )d. ³ c ( b, p( )) c (,p( )) c (, p a ) g( )d (1 F ()) m 1 d (14) # As depcted by the second lne of (14), the dfference between expected profts depends on both the global comparson of costs and the margnal competton effect. Intutvely, the wnnng SBF s better off wth subcontractng f the gan from the barganng on costs s hgher than the expected strategc mark-up n the auctonng of the allotted part. 6.2 Pareto domnatng mechansm The last secton enables us to show that both the wnnng LF and the chosen SBF can smultaneously prefer ether the subcontractng procedure or the allotment procedure. In contrast to proposton 1, snce the LF can choose the subcontractng level, ths result s not necessarly at odds wth an ncrease n expected total costs. Thus, we have the followng proposton Proposton 2 When the LF s allowed to choose the subcontractng level, subcontractng can be a Pareto domnatng mechansm,.e. smultaneously ncreasng both frms proft and reducng the expected total cost. We now gve an ntutve explanaton of ths man result (a formal proof s gven n the appendx). The expected total cost for the publc buyer s equal to the expected true 16

17 costs (ETrC ) plus the profts conceded to the frms. Let us wrte the expected total costs respectvely for the subcontractng procedure and the allotment procedure ETC s ETrC s, ETC a ETrC a Eπ a(1) Eπ a(1). Snce the LF chooses the subcontractng level whch mnmzes ETrC, we necessarly have ETrC s <ETrC a. Then, subcontractng can be Pareto mprovng f >Eπ a(1) and >Eπ a(1), and smultaneously ETrC s or ETrC a ETrC s {z } >0 < ETrC a Eπ a(1) Eπ a(1) > Eπ a(1) Eπ a(1), {z } {z } >0 >0 that s f the ncrease n both frms profts does not offset the dfference between expected true costs. Obvously, ths result could not be obtaned wth mposed subcontractng snce we would always have ETrC a ETrC s. Further, the allotment procedure can never be a Pareto mprovng mechansm snce t would requre the followng nequalty to be satsfed ETrC a ETrC s {z } >0 whch clearly cannot be true. < Eπ a(1) {z } <0 Eπ a(1), {z } <0 17

18 7 Concluson Ths paper has focused on the comparson of two aternatve mechansms enablng the partcpaton of SMEs n procurement. We hghlght the mpact of the choce of the subcontractng level and of the barganng process on the bddng strategy of LFs. Compared to tradtonal developments n aucton theory, these margnal competton effects ether ncrease or decrease the famlar strategc mark-up. It enables us to derve some new nsghts nto the desgn of procurement rules. We have derved condtons under whch both the wnnng LF and the wnnng or chosen SME can be better off wth the allotment procedure or wth the subcontractng procedure. Specfcaly, f the barganng power of the LF does not depend on ts own effcency or f a more neffcent LF s a better barganer, then the wnnng LF s better off wth the allotment procedure than wth the subcontractng procedure f, compare to the allotted part, the LF s nduced to subcontract a larger part. We have also explaned why the publc buyer and LFs can beneft fromtheasymmetrc nformaton between SMEs and LFs. Furthermore, when the wnnng LF s constraned on the subcontractng level, we have shown that only a reducton of the chosen SME s proft can reduce the expected cost of the contract. Nevertheless, when the LF s allowed to choose the subcontractng level, subcontractng can be a Pareto domnatng mechansm,.e. smultaneously ncreasng both frms proft and reducng the expected total cost of the contract. An extenson of ths model could be to consder an allotted contract but to allow the wnnng LF to subcontract a part of the resdual proect.inthscontext,smeswouldhave two ways to access publc procurement. 18

19 8 Appendx Dervaton of the optmal bddng strategy of LF Snce we search for a symmetrc Nash equlbrum of the bddng functon, we have b b ( ). Then,from(2),wehave deπ a (c (,p a )) (1 F ( )) n 1. d By ntegraton, and wth Eπ a 0, we obtan µ Eπ a (c (, p a )) ( ) (1 F ()) n 1 d, From ths equaton and (1), we obtan the optmal bddng strategy of LF. Q.E.D. Proofoflemma1.From (8), we can consder the dfference between the expected profts of the wnnng LF under asymmetrc nformaton (a) and under complete nformaton (c). Indeed we have Eπ s(a)(1) > Eπ s(c)(1) Z " Ã c ( b, p()) Obvously, under complete nformaton, c ( (1) c ( (1),p( )) (A1)! #,p()) (1 F ()) n 1 d g( )d > 0. 0snce whatever the effcency of the wnnng LF s. So, ths latter can always pay the subcontractor at a cost c ( b, p()) c ( (1),p()). Then, a condton for (A1) to be satsfed s.e. f the LF s producton specalzed. Q.E.D. c ( b, p( )) > 0, (A2) Proofoflemma2.From (7), the dfference between the expected total costs under ncomplete and under complete nformaton s ETC s(a) ETC s(c) c ( b, p( )) c ( (1),p( )) R ³ c (,p( )) (1 F ()) n 1 d (1 F ( )) n 1 g( )d. 19

20 Snce c ( b, p( )) c ( (1),p( )) > 0, a condton for ETC s(a) <ETC s(c) to hold s c (,p( )) (A2). So, we cannot have both ETC s(a) <ETC s(c) and Eπ s(a) Further, we can have ETC s(a) <ETC s(c) f.e. f c (,p( )) > R ³ c (,p( )) (1 F ()) n 1 d (1 F ( )) n 1 h c ( b, p( )) c ( (1),p( )) g( )d s not too hgh. Q.E.D. ProofofLemma4.Assumng (c (,p a )) ETC s ETC a Z Ã c ( b, p a ) c (, p a ) c ( b, p a ) c ( (2),p a ). 0, we get R c (,p a ) < 0, whch s n contradcton wth >Eπ s(c). g( )d! (1 F ()) m 1 d (1 F () m 1 g()d As wth the comparson of the wnnng SBF s proft, the comparson of expected total costs s based on the dfference between negotated and auctoned lots. Clearly, we obtan ETC s < ETC a f the wnnng LF knows the SBFs costs. If these costs are prvate nformaton, then obvously ETC s >ETC a. Q.E.D. Proofofproposton1.From (3) and (7), we have Recall that ETC s ETC a c ( b, p a ) c ( (2),p a ) Z R c (,p a ) (1 F ()) n 1 d (1 F ( )) n 1 g( )d. Eπ a(1) c ( b, p a ) c ( (2),p a ), and that Z Ã Eπ a(1) c ( b!, p a ) (1 F ()) n 1 d g( )d. 20

21 Thus, subcontractng can be a Pareto domnatng mechansm f the followng condtons are smultaneously satsfed Obvously, ETC s <ETC a >Eπ a(1) >Eπ a(1). (A3) ETC s < ETC a (A4) Z R c (,p a ) c ( b, p a ) c ( (2),p a ) < (1 F ()) n 1 d (1 F ( )) n 1 g( )d, and >Eπ a(1) c ( b, p a ) c ( (2),p a ) > 0. (A5) (15) and (A5) mply R c (,p a ) (1 F ()) n 1 d (1 F ( )) n 1 g( )d < 0 c ( b, p a ) < 0. (A6) However, > Eπ a(1) c ( b, p a ) Ã c ( b, p a ) (1 F ()) n 1 d! g( )d > 0 > 0. (A7) Snce (A6) and (15) cannot both be satsfed, subcontractng cannot be a Pareto mprovng mechansm. The same reasonng apples to the allotment procedure. Q.E.D. Proofoflemma5 Recall that comparng expected profts n each stuaton, we get Eπ a(1) " Z Ã c ( b! #, p()) (1 F () n 1 d g( )d µ [c (, p()) c (, p a )] [ (1 F () n 1 d] g( )d. 21

22 ³ The sgn of [c (,p( )) c (,p a)] s related to the sgn of the cross dervatve 2 c (,p) p. More precsely, snce we assume that 2 c (,p) p < 0, we have [c (,p( )) c (,p a )] < (>)0 f p( ) > (<) p a. If c (,p()) 0 and p( ) >p a, then Eπ a(1) >. Q.E.D. Proofofproposton2.The dfference n expected total costs of the contract, when the LF s allowed to choose the subcontractng level, s ETC s ETC a c (,p( )) c ( b, p( )) c (,p a ) c (,p a ) [c (,p()) c (,pa)] c (,p()) (1 F ( )) n 1 R c (,p a ) (1 F ()) n 1 d g( )d (1 F ()) m 1 d (1 F ( ) m 1 h( )d. (A8) Consder the specal case of n m and, 14 then h( )g( ) and (15) becomes ETC s ETC a c (,p( )) c ( b, p( )) c (,p a ) c (,p a ) c ( b, p()) c (, p()) c (,pa) c (,p a) {z } >0 by assumpton (1 F ( )) n 1 (1 F ())n 1 d (A9) g( )d. From (15), suffcent condtons for ETC s <ETC a to be satsfed are c (,p( )) c (,p( )) < c (,p a ) c (,p a) c (, p()) c ( b, p()) c (, p a ) c (, p a ) < 0. (A10) Gven (13) and (A10), we have both ETC s <ETC a and >Eπ a(1) c (,p a) < c (,p( )) c (,p( )) < c (,p a) c (,p a ) c (, p()) c ( b, p()) c (, p a ) c (, p a ) < 0 f. (A11) 14 Smlar (but tedous) dervatons can be made for more general stuatons. Our am here s to provde some suffcent condtons. 22

23 Then, subcontractng can be a Pareto mprovng procedure f condtons (A11) and > Eπ a(1) are smultaneously satsfed. Gven (14), h c ( b, p()) c (, p()) g()d > >Eπ a(1) µ c (, p a ) f (1 F ()) m 1 d h()d,.e. f the gan from the barganng on costs s hgher than the expected strategc mark-up n the auctonng of the allotted part. Then, suffcent condtons for a Pareto mprovement are c (,p a ) < c (,p( )) c (,p( )) < c (,p a ) c (,p a) c (,p( )) c ( b, p( )) c (,p a ) c (,p a ) < 0. R hc ( b, p()) c (, p()) g()d > R ³ c (,p a) (1 F ()) m 1 d h()d Q.E.D. 23

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