Auctions for Infrastructure Concessions with Demand Uncertainty and Unknown Costs

Transcription

1 MRA Munch sonal REc Achv Auctons fo nfastuctu oncssons wth Dmand Unctanty and Unknown osts Gustavo ombla and Gns d Rus Economcs of nfastuctu and anspot, Unvsty of Las almas 00 Onln at MRA ap o. 03, postd 9. Dcmb :33 U

2 Auctons fo nfastuctu oncssons wth Dmand Unctanty and Unknown osts Gustavo ombla * Gnés d Rus Unvsdad d Las almas d Gan anaa Dpto. Análss Económco Aplcado May 00 Abstact Aucton mchansms commonly usd n pactc fo awadng nfastuctu concsson contacts nduc a bas towads th slcton of concssonas who a optmstc about dmand, but a not ncssaly cost-ffcnt. hs hlps to xplan th fqunt ngotaton of concssons obsvd n pactc. hs pap shows that th fxd-tm natu of contacts s th ky lmnt fo slcton os, and t poposs a btt altnatv mchansm basd on flxbl-tm contacts. hs nw aucton mchansm ducs th pobablty of slcton os and contact ngotaton, and t s smpl nough to consttut a good opton fo concssons n sctos lk tanspot and publc utlts. Kywods: concssons, auctons, ngotaton, nfastuctu JEL cods: D44, L9 Acknowldgmnts: W want to thank Andés Gómz-Lobo, Mª Angls d Futos, odoso éz Amaal, Antono Estach, and smna patcpants at Fundacón Empsa úblca Madd and th unvsts of Zaagoza, Madd- omplutns, Mnéndz layo, and Las almas, fo all th usful commnts and suggstons on a pvous vson of ths pap. Any manng os a of ou xclusv sponsblty. Fnancal suppot fom Y ojct SE s gatfully acknowldgd. * ospondng autho: nombla@mpsaals.ulpgc.s

3 . ntoducton h povson of basc nfastuctu, lk always, bdgs o sapots, has bn tadtonally a sponsblty assumd by govnmnts. Howv, snc th lat 980 s, th s a woldwd tnd towads a mo actv patcpaton of pvat nvstos n th constucton and fnancng of lag nfastuctu pojcts. hs s spcally lvant fo dvlopng counts, but t s ncasngly obsvd also n dvlopd conoms. A popula modl fo th patcpaton of pvat captal n nfastuctu s th concsson contact. hs s an agmnt btwn a govnmnt and a pvat fm gnally a consotum fomd of sval pats fo th constucton o majo habltaton of som nfastuctu. vat nvstos assum all costs latd to th pojct, and n tun thy obtan th ght to opat th assts pow plants, apots, oads, and so foth dung som p-spcfd pod. nvstmnt costs a thus covd fom chags o tolls on nfastuctu s uss. Govnmnts tan ownshp and contol ov th assts and may gulat pcs and qualty. h mchansm fo slctng concssonas to mplmnt nfastuctu pojcts s usually a sald-nvlop typ of aucton. n th aa of govnmnts contacts wth pvat fms, th ltatu on auctons has dvotd a gat dal of attnton to th analyss of compttv bddng fo pocumnt contacts Stak, 97; Holt, 980; ot and Zona, 993; Jof-Bont and sndof, 000, and to study th poblm of th wnn s cus fo a gnal dscusson, s Mlgom, 989; fo an applcaton to th oad scto, s hl, 988; Lvn and Smth, 990. A databas on concssons fo nfastuctu pojcts sgnd dung th 990s n dvlopng counts, compld by th Wold Bank, pots th xstnc of 700 contacts. By scto, 45% of ths contacts a tanspot pojcts, 5% wat, 0% lctcty, and 0% n tlcommuncatons. By gon, 60% a locatd n Latn Amca, 0% n Asa, and 0% n Eastn Euop.

4 Lss attnton has cvd th patcula contxt of concsson contacts, although th a many ntstng qustons about fms slcton and x-post contact ngotaton. Howv, ths s an aa wh sach on auctons s pobably most valuabl fo govnmnts, as stssd by Klmp 999. nfastuctu pojcts hav long conomc lvs gnally abov 5 yas and thfo a bult und hgh unctanty about costs and futu lvls of dmand. hfo, th a two dmnsons to b consdd: asymmtc nfomaton btwn govnmnts and fms gadng costs; and dffnt fms blfs about th futu dmand fo th nfastuctu. n th tmnology of aucton thoy, govnmnts a facd wth a poblm that combns lmnts both fom a pvat-valu modl gadng costs, and a common-valu typ of stuaton latd to th unctanty of dmand. Slcton of th fm wth lowst fasbl cost, and mnmsaton of sks ntoducd by dmand fluctuatons, should b th man tagts that an aucton mchansm should pusu n ths contxt. n patcula, th sk of dmand fluctuatons s on of th man causs of dffcults fo nfastuctu pojcts. h a many ntnatonal xpncs about concssons ntng nto dffcults du to dvatons btwn focastd and actual lvls of dmand 3. On th oth hand, a ducton of th sk of dmand may contbut to dcas concssonas costs, though low sk pma assocatd to th cost of captal. h fqunt obsvd ngotaton of concsson contacts and th nd to bal out concssonas ndcat that slcton mchansms may b falng to achv thos objctvs. Futhmo, th s vdnc that th typ of aucton usd to slct a concssona mght hav an mpact on th futu pfomanc of an nfastuctu pojct. Usng data fom a lag sampl s footnot, Rodan and Sappngton 987, s on of th fw contbutons to th analyss of th poblm of concssonas slcton. ol 986 psnts a modl of ngotaton btwn a govnmnt and a fm contactd fo som pocumnt task, though t dos not xplctly dal wth th pcdng stag of contact tndng. 3 On of th mo damatc xampls was a oad pogam mplmntd n Mxco dung th 990s 5 concssons to buld aound 6,000 km of nw hghways. Actual lvls of dmand w on avag 68 pcnt blow xpctatons, and fo 6 concssons taffc was blow 50 pcnt of focastd lvls. n 997, th Mxcan govnmnt was focd to cov 5 of ths concssons, assumng dbts fo US$ 7.7 bllon. Bsds, losss fo pvat nvstos w stmatd n US$ 3 bllon Fshbn and Babba, 996; Gómz-báñz,

5 Guasch 000 pots that 65 pcnt of all concsson contacts a ngotatd. ntstngly, th pobablty of ngotaton s hgh whn th concssona s slctd though an aucton basd on offs fo mnmum pc 9 p cnt of contacts ngotatd, than whn a mum paymnt s usd as th bddng vaabl 9 p cnt. h objctv of ths pap s to show that aucton mchansms gnally usd n pactc fo awadng nfastuctu concsson contacts a fa fom bng optmal, and to popos a btt altnatv mchansm. t s fomally povd how tadtonal auctons basd on bds fo mnmum pcs o mum paymnts do not gnally slct th most cost-ffcnt canddat. On th contay, thy ntoduc a bas towads th slcton of concssonas who a optmstc about futu dmand lvls, whch s lkly to b on of th man asons bhnd th fqunt ngotaton of contacts. h fxd-tm natu of tadtonal contacts s th basc lmnt that nducs slcton os, so th altnatv poposd s to us a mchansm basd on flxbl-tm contacts. h stuctu of th pap s as follows. Scton psnts th modl usd fo th analyss of auctons. Scton 3 uss a basc scnao wth pvat valus gadng constucton costs, and psnts th concpt of flxbl-tm concsson contacts. Scton 4 ntoducs mantnanc and opaton costs, assumd to b qual acoss fms, to valuat th mpact of ths costs on outcoms. Scton 5 consds vaablty on constucton and mantnanc costs acoss fms, and dscusss n dtal th poposd nw aucton mchansm fo th awad of flxbl-tm concsson contacts. Scton 6 summass th man sults and concluds th pap. 4

6 . A modl to analys auctons fo nfastuctu concssons A smpl famwok s poposd to analys auctons fo th awad of concsson contacts. h numb of uss Q who wll dmand svcs fom a nw nfastuctu to b bult 4 s unknown at th momnt of daftng th contact. All uss a qual, and ach of thm dmands on unt of svc fom th nfastuctu. h only avalabl nfomaton about dmand s a ang of fasbl valus [Q mn, Q.]. n od to psnt a stuaton of mum unctanty about futu dmand, t s consdd that Q s unfomly dstbutd ov that ang.. Onc that unctanty about th stat of natu s solvd th nfastuctu s bult and opn to th publc, dmand s found to tak a patcula valu Q *. hs lvl s consdd to b constant ov tm,.. th wll b Q * uss ach ya dung th whol lf of th concsson. h a fms bddng fo th contact, whch a dffnt n tms of cost ffcncy. wo typs of costs a consdd: constucton costs, =,..,, whch psnt all nvstmnts qud bfo th nfastuctu s opatv; and mantnanc and opaton costs M, =,..,, whch a annual fxd xpnss n whch th concssona ncus fo th povson of svcs qupmnt and psonnl and to mantan th nfastuctu pas, podc vsons, and so foth. M s assumd to b constant ov tm, and dos not vay wth th lvl of dmand. h ason s that only mantnanc and opaton costs of fxd natu a lvant fo th analyss. All oth vaabl costs that dpnd on Q a fomally quvalnt to a ducton n th pc pad by ach us. As ndcatd by notaton, t s assumd that th a dffnt fms typs gadng both constucton costs and mantnanc and opaton costs M. hs assumpton s justfd by th 5

7 fact that fms manags hav som dg of contol ov fms costs, bcaus thy mght hav dffnt sklls, o may xt mo o lss ffot n ducng costs. n patcula, t s assumd that s dawn fom a unfom dstbuton wth suppot [ mn, ], and M fom anoth ndpndnt unfom dstbuton ov [M mn, M ]. Fo fxd-tm concsson contacts, th total numb of yas dung whch th concssona opats th nfastuctu s pdtmnd by th govnmnt, and dnotd as. h concssona knows that, onc th lvl of dmand Q * s vald, t wll cv a total vnu qual to Q * dung th lf of th concsson. h pc may b st by th govnmnt o, n th cas of an aucton basd on pc offs, t wll b th pc bd fo by th wnn. t s consdd that, apat fom nvstmnt and mantnanc costs, th concssona must mak a lump-sum paymnt Z to th govnmnt. hs paymnt s mad at th bgnnng of th concsson, and t could tak ngatv valus a subsdy s dmandd to opat th concsson, o zo no paymnt s qud. n som cass, Z mght b th bddng vaabl at th aucton, whl n oths t s pspcfd by th govnmnt. h pojct s assumd to b bult n ya t=0, whn th concssona maks all nvstmnts, and t stats cvng uss n ya t=, whn th fm ntats th collcton of vnus though tolls o taffs, and ncung nto mantnanc and opaton costs. Wthout loss of gnalty, th dscount at s assumd to b qual to on, snc consdng a dscount at δ < dos not alt th basc sults of th pap. All fms a consdd to b sk-nutal and to ms xpctd pofts fo th whol lf of th concsson. Fo a gvn blf on futu dmand Q, xpctd pofts fo fm a qual to Π = Q - M - - Z. 4 houghout th pap, all dscussons focus on a cas wh th nfastuctu to b bult s nw, but th sam agumnts apply to a pojct to nhanc o habltat xstng assts oad nlagmnts, scond unways at apots, mpovmnt of pow tansmsson lns, and so foth. Dmand unctanty, howv, wll b gnally low fo 6

8 Sval altnatvs xst fo th dsgn of auctons to tnd concsson contacts. h most common a to nvt offs fo pcs, awadng th contact to th lowst pc; o offs fo f paymnts to th govnmnt, wh th hghst poposd f wns. hs a th two mchansms analysd n ths pap. Auctons fo concssons a gnally opn only to thos fms who pass a plmnay fltng pocss. Oth auctons lss fquntly obsvd n pactc a basd on bds fo nvstmnts, volums of svc, o duaton of th contact. n som cass, sval bddng vaabls can b usd smultanously, wth som wghtd cton to slct th aucton s wnn. 3. Scnao wth no mantnanc costs: fxd-tm and flxbl-tm contacts Usng th poposd famwok, t s possbl to study th optmal statgs pusud by fms comptng fo a concsson contact at an aucton, and th outcoms that a obtand. W wll consd ntally a smpl scnao wth no mantnanc costs M = 0. h two most common typs of auctons to awad contacts mnmum pc o mum paymnt a xamnd and compad, and thn t s shown how an aucton to awad a flxbl-tm concsson contact outpfoms tadtonal auctons. 3. Mnmum pc aucton onsd an aucton at whch canddats must psnt sald-nvlops wth th bds fo th pc that thy popos to chag uss fo th us of nfastuctu. Envlops a opnd smultanously and th wnn s th fm wth th lowst pc. o possbl modfcaton o ngotaton ov th poposd pc s pmttd aft th aucton s solvd. h contact tm s st by th govnmnt and known n advanc by bdds. habltaton than fo gnfld pojcts. 7

9 Assum that all canddats sha a common blf Q about th xpctd lvl of dmand ths assumpton wll b lat laxd. hs could b th cas fo xampl f th govnmnt povds sults fom dmand studs n ppaaton fo th aucton, and patcpants do not nvst to hav oth altnatv stmats. onsd th bst statgy to b pusud by a fm wth constucton cost. n gnal, bds wll b basd on fms typs.. on th latv cost ffcncy, so t s assumd that th pc offd s a functon of. W thn sach fo symmtc qulba, wh all fms us th sam ul to calculat th bds. hus, th pc offd by fm wll b =, wth > 0. Usng th known pobablty dstbuton of constucton costs, t s possbl fo any fm to comput th pobablty of wnnng th aucton wth a patcula off. ompang s bd wth that of any oth fm j, fm wns f < j, o, applyng th nvs functon -, f - < - j = j. hfo, by choosng cafully ts bd, fm knows that t would wn th contact aganst any fm j wth th sam pobablty of th vnt [ j > - ]. Snc th a - vals at th aucton, th pobablty fo fm to wn th contact wth a bd s qual to: pob = pob [ > ] = j mn All fms a sk nutal, so ach of thm wll calculat ts optmal bd by msng xpctd pofts, gvn th pobablty of bng slctd as concssona as ndcatd by. Whn all fms sha a common blf Q about th futu lvl of dmand, th poblm that fm solvs to comput ts optmal bd can b xpssd as: Max Π = Q Z wh = - mn s a constant that can b gnod. h fst od condton of poblm s a dffntal quaton n tms of th optmal functon : 8

10 ' = Q + Z 3 Du to th unfom dstbuton assumd fo constucton costs, th soluton to quaton 3 tuns out to b a lna functon n tms of and oth paamts: + Z = + 4 Q Q hs optmal bds xhbt som ntstng popts. Fst, whn all fms sha th sam xpctd dmand Q, t s cla that th mchansm slcts th most ffcnt fm, bcaus th canddat wth th lowst constucton cost wns th contact. Scond, th pc offd allows th concssona to obtan som postv pofts, gvn by th scond tm of th ght-hand sd of xpsson 4. Whth fm to off a pc = + Z/ Q, ts xpctd poft would b zo, thfo th tm - / ndcats th nfomaton nt that th wnn xtacts. hs nt dcass wth th numb of fms patcpatng at th aucton, t s zo fo th last ffcnt fm wth =, and t achs a mum fo th most ffcnt fm. 3. Maxmum paymnt aucton A scond tadtonal modl to awad concsson contacts s an aucton at whch fms mak offs fo a paymnt Z to th govnmnt, and th contact s won by th hghst bd. hs paymnts Z could vn b ngatv, n whch cas canddats dmand subsds fo th constucton of th pojct, and th contact s awadd to th fm dmandng th lowst subsdy hghst Z n that contxt. n ths typ of aucton, th pc s st by th govnmnt, as wll as th duaton of th contact. As n th pvous cas, ach fm wll calculat ts bd Z as a functon of ts typ, and agan w sach fo a symmtc qulbum n whch all canddats us th sam functon Z, wth Z <0, 9

11 to comput th bds Z. h pobablty of fm wnnng th aucton s now th pobablty of th vnt Z > Z j, fo any j oth than. n tms of constucton costs, ths can b xpssd as: pob = pob [ Z > Z ] = pob [ Z Z < ] = Z Z j 5 j Optmal statgs a thn dvd fom th msaton of xpctd pofts: Max Z Π = Q Z Z Z 6 h fst od condton of poblm 6 s a dffntal quaton smla to 3, whch ylds: Z = Q 7 Optmal statgs on paymnts Z sha th sam popts of. amly, f fms had th sam blfs about futu dmand, th mchansm would slct th most ffcnt fm n tms of constucton costs. h concssona s abl to xtact som nfomatonal nt, bcaus th poposd paymnt to th govmnt ylds a postv xpctd poft qual to - /. t s woth notcng that ths nt s xactly th sam that th wnn of th pc aucton obtans, whch s a vnu-quvalnc sult n th contxt of ths modl. 5 hs sult has som pactcal lssons. t tlls a govnmnt that th fom usd to aucton a concsson contact s lvant fom th pont of vw of th fm, whch can xpct th sam lvl of pofts. Howv, th conomc mpact of ach patcula typ of aucton s dffnt. Whn a pc aucton s usd, uss of nfastuctu pay fo th nt that th concssona obtans, though pcs somwhat hgh than th fasbl mnmum. Manwhl, whn usng a paymnt aucton, th 5 A wll known sult fom aucton thoy s th Rvnu Equvalnc hom Vcky, 96; Myson, 98; Rly and Samulson, 98, whch stats that, f bdds a sk-nutal, th fom usd by a sll to aucton a good fstand scond-pc sald bds, Englsh and Dutch auctons ylds th sam xpctd vnus. n ou modl, all auctons a fst-pc sald bds, but sults ndcat that, n tms of xpctd vnus, t dos not matt whch bddng vaabl s usd. 0

12 nt - / taks th fom of a low than optmal f fo th govnmnt, so taxpays a n fact payng fo that nt. Anoth lsson s th mpotanc of havng as many fms as possbl comptng fo a concsson contact. h numb of bdds ducs th nfomaton nt that a concsssona may xtact. n th lmt, f tnds to b suffcntly lag, ths nt would b zo, obtanng th known sult of convg to al valus Wlson, 977; Mlgom, oblms suffd by tadtonal systms: slcton os and sk of ngotaton As ndcatd whn analysng optmal bddng uls gvn by 4 and 7, th tadtonal pc and paymnt auctons wll slct th most ffcnt canddat lowst n ths contxt, but only whn all canddats sha th sam xpctd dmand lvls Q. hs assumpton wll hadly hold n pactc, gvn th dffcults assocatd wth accuat dmand focastng. hfo, t s mo alstc to consd that Q wll b a andom vaabl dawn fom som suppot [Q mn, Q ], and that ach canddat may hav a dffnt blf Q. o psnt a stuaton of mum dmand unctanty, w wll consd a unfom dstbuton Q U [Q mn, Q ]. h mpact of blfs about dmand on th bds submttd by canddats can b assssd by studyng xpssons 4 and 7. t can b obsvd that d / dq < 0 and dz / dq > 0, whch mans that thos canddats wth hgh xpctd dmand lvls wll tnd to submt btt bds low pcs and hgh paymnts, spctvly, than oth canddats wth smla costs but low dmand xpctatons. hs cats two typs of poblms. Fst, th mchansms do not long guaant th slcton of th most ffcnt canddat, and th possblty of slcton os appas. Scond, and latd to th fom, th slcton of ovoptmstc concssonas wth costs hgh than optmal

13 ncass th sk that contact ngotaton 6 mght b qud to bal out bankuptd concssonas, n stuatons of low dmand. a Slcton o: onsd an aucton at whch canddats a ankd accodng to th constucton costs, so that < <...<. A slcton o occus f th aucton mchansm slcts a fm j oth than. h condton qud fo that vnt s smply that fm j s bd tuns out to b btt than fm s a low pc o a hgh paymnt, spctvly, at ach typ of aucton. hs can only occu whn th blf of fm j about dmand s suffcntly hgh than fm s,.. Q j /Q = +λ, wth λ > 0 psntng th dg of latv optmsm of fm j. Usng xpssons 4 and 7 fo optmal bds, t s possbl to dtmn th qud sz of λ fo a fm oth than to wn ach typ of aucton and thus to b slctd as concssona : c aucton: j λ > 8 + Z + aymnt aucton: j λ > 9 Q h basc factos that affct th possblty of makng slcton os a flctd n xpssons 8 and 9. h cost gap btwn fms j - and λ a dctly latd, thfo a wd ang of fasbl constucton costs maks os to b lss lkly bcaus, on avag, th dffnc btwn and j wll tnd to b lag. A lag numb of bdds ducs th pobablty of slcton os fo both typs of auctons. t s notcabl that th contact tm dos not hav an ffct on th pobablty of o fo th aucton pc, whl t dos on th paymnt aucton, ncasng th possblty of slcton os fo long contacts. 6 houghout th pap, w assum that ngotaton s only honstly sought by fms,.. th s no statgc bddng antcpatng th possblty of contact ngotaton. Fms comput th bds basd on al costs and dmand

14 t s possbl to comput xplct xpssons fo th pobablty of slcton os, basd on th assumd unfom dstbutons fo dmand blfs Q U[Q mn,q ] and constucton costs U[ mn, ]. h dmand blf of any fm j, latv to that of th most ffcnt fm n tms of costs, s by dfnton a andom vaabl + λ = Q j / Q, wth pobablty dstbuton functon: F+ λ = [ Q + λ Q ] + λ Q mn Q mn [ Q Q + λ ] ; f Q Q mn mn ; f < + λ + λ Q Qmn + λ Q Q mn 0 At an aucton wth bdds, th xpctd valus fo costs of ach of thm can b calculatd fo any gvn valu of th most ffcnt fm. Bcaus th cost of th manng - bdds must b unfomly dstbutd btwn and, any fm at poston j n th ankng of costs wll hav an xpctd constucton cost qual to j = + j - /. hus, t s fasbl to comput, fo any fm j, th qud latv dmand blf to ovcom th gap j -, ncssay fo fm j to submt an off btt than th on fom th most ffcnt fm. Fo th pc aucton, and dsgadng stuatons n whch a fm wth cost { 3,..., } could outbd th most ffcnt fm 7, th pobablty of slcton os, pob s, s qual to: [ Q θ Q ] mn pob s = d mn θ Q Q mn + + Z wh θ =. + + Z xpctatons. hfo, ngotaton only taks plac whn dmand tuns out to b low and th concssona facs a stuaton wth Π * < 0. Fo a modl of ngotaton and statgc bddng, s Wang

15 h pobablty of slcton o fo th paymnt aucton, pob s Z, can b smlaly calculatd. ts xpsson s slghtly mo complx, bcaus n ths cas th condton to b satsfd fo fm to b slctd nstad of fm vas wth Q. Q [ Q θ, Q Qmn ] mn Q mn θ, Q Q Q pob sz = dq d mn wth θ, Q = +. Q b ontact ngotaton: h condtons fo a concsson contact to nt nto fnancal poblms and qu to b ngotatd a smla n natu to thos of slcton os. h vnt Π * < 0 occus whn th actual dmand fo th nfastuctu Q * s low, and falls blow xpctatons. Rvnus obtand fom uss mght thn sult nsuffcnt to cov fo nfastuctu costs, so th concssona s focd to go bankupt unlss th govnmnt accpts to ntoduc changs n th contact tms, lk sng th pc, povdng subsds o xtndng th lf of th concsson. As dscbd abov, ths s a pactc too fquntly obsvd fo al concssons. onsd th andom vaabl µ = Q * / Q, psntng th dvaton of actual dmand Q * wth spct to th focastd valu usd by th wnn whn computng ts bd. h condton fo a concsson contact awadd though a pc aucton to b ngotatd s: + Z µ < 3 + Z + 7 otal pobablty assocatd to an vnt n whch a fm j {3...} outbds fms and s of scond od compad to th pobablty of fm batng fm at th aucton. An vnt wth a fm j > wnnng th aucton s only lkly to tak plac whn fm has a cost clos to, and all fms hav costs hgh than, whch s a stuaton wth xtmly low pobablty. 4

16 Usng th fact that µ s unfomly dstbutd on th suppot [Q mn /Q, Q /Q ], th pobablty of contact ngotaton, pob wll b qual to: pob = + Z + Z Q Q Q mn Q mn Q mn + + Z Q mn Q mn 4 wh = /+ mn + /+, and = -/+ mn + /+ a th xpctd valus fo th constucton costs of th fst and scond bst fms n th cost ankng of bdds. A smla condton to 3 s dvd fo th cas of th paymnt aucton, fom whch th cospondng pobablty of contact ngotaton pob Z s calculatd: pob Z Q Qmn mn = 5 Q Q + Q Q mn mn An xamnaton of xpssons and 4 and of and 5, spctvly, vals that factos that affct th pobablts of slcton os and contact ngotaton a vy smla. hus, fo xampl, th contact tm dos not affct th pobablty of ngotaton fo a pc aucton pob, whl t dos fo th paymnt aucton. Long pods ncas th pobablty of ngotaton, bcaus an optmstc fm ls on obtanng substantal vnus fom th concsson whn submttng ts bd, and ths assumpton s nfocd by th duaton of th contact. t s not supsng to obsv that pob s and pob a hghly colatd bcaus both ffcts a lnkd th sam appls to pob s Z and pob Z. An aucton mchansm s mo lkly to mak slcton os whn th wnn has a blf fo hgh xpctd dmand, whch n tun ss th pobablty of contact ngotaton whn dmand falls blow xpctatons. hs s th fom that th wnns cus taks n th contxt of ths modl. Evn though th pobablts of slcton o and ngotaton gnally vay n th sam dcton, howv, th pobablts of 5

17 ngotaton a mo affctd by dmand unctanty. t can b povd that th ang Q Q mn of fasbl dmand has a lag ffct ov pob {,Z} than ov pob s {,Z}. Expssons 4 and 5 fo th pobablts of ngotaton of contacts can b usd to ty to xplan th obsvd fact about pob bng much hgh n pactc than pob Z Guasch, 000. h condton fo pob > pob Z s: < Q mn + + Q + Z Q mn mn 6 ondton 6 allows to compa pc and paymnt auctons. h conclusons dvd fom ths condton a qut valng. Low pcs.., chags to uss, shot contact tms, and hgh f paymnts to govnmnts a all factos that mak contact ngotaton to b a mo lkly vnt fo th mnmum pc aucton than fo th mum paymnt aucton. hs th tagts appa fquntly n many nfastuctu concsson pogams announcd by govnmnts. hfo, t s lkly that many of th fquntly obsvd concsson contacts ngotatons could b patly causd by th typ of aucton mchansm usd fo th slcton of concssonas and th valus chosn fo, Z and. 3.4 Flxbl-tm concsson contacts oblms suffd by tadtonal aucton mchansms stm fom a basc common fatu: unctanty about futu dmand lvls ntoducs unctanty about total vnu xpctd by fms. hs nducs bds submttd by fms to b contamnatd by th blfs on dmand, and thfo not to psnt accuatly th undlyng constucton costs. Fomally, ths can b obsvd by th psnc of dmand xpctatons Q on xpssons 4 and 7, whch ndcat that fms nclud th blfs n th bds fo pcs and paymnts. 6

18 h da of usng concsson contacts wth flxbl tms, poposd fo th oad scto by Engl t al 997 fo a dtald dscusson, s ol, 997, 999, ams at bakng ths lnk btwn dmand unctanty and vnu unctanty. A flxbl-tm contact woks as follows: fms submt bds fo th total vnu that thy want to obtan fom th concsson R. An aucton basd on ths typ of contact awads th concsson to th lowst bd R < R <...<R. h wnn thn s nsud to obtan a vnu qual to R by xtndng th lf of th concsson as ncssay. f actual dmand s Q *, th contact lasts untl Q * Q * = R, at whch pont th concsson nds. Bfo unctanty about dmand s solvd, fms hav total xpctd vnus qual to R = Q. adtonal auctons basd on a fxd-tm thus foc xpctd vnus to b a functon of th unctan dmand, R = RQ, and bds a computd on th bass of ths unctan vnus. Manwhl, f th tm w flxbl and ndognously dtmnd by dmand, total vnu could b gadd as constant by ach fm. o matt what ts blf about dmand mght b, submttng a bd fo vnu R would yld that amount f th fm s slctd as concssona. f a fm xpcts a dmand lvl Q, and submts a bd R, t can xpct th concsson to last fo a pod Q = R / Q. hfo, ts xpctd vnu would b R = Q Q = R. hs nducs bds that a not affctd by blfs about futu dmand. hs typ of flxbl-tm concsson contacts has alady bn appld n pactc, although th numb of xampls aound th wold s vy ducd 8. n a contxt wth dscount factos δ small than on, fms patcpatng at auctons fo flxbl-tm contacts must submt bds fo dscountd t t vnus, R = = δ Q. hs s th ason why ths typ of aucton has bn namd as lastpsnt-valu of vnu LVR, bcaus th concsson s awadd to th canddat wth th lowst bd fo dscountd vnu R. 8 o ou knowldg, th fst xpnc wth flxbl-tm concssons was a UK oad pojct Datfod bdg. A wll 7

19 n th contxt of ou modl, wth no dscount at, a fm patcpatng at an LVR aucton wll comput ts bd accodng to ts typ, followng som ul R, wth R > 0. As n th cass of pc and paymnt auctons, w sach fo a symmtc qulbum wth all fms usng th sam ul R to comput th bds. h pobablty of wnnng th aucton wth an off R has an xpsson smla to and 6. h xpctd contact-tm s = R /Q, so th poblm solvd by ach fm s now: Max R Π = Q R Q Z R R 7 As n pvous cass, th fst-od condton of ths msaton poblm s a dffntal quaton on R, whch ylds th lna soluton: R = + Z + 8 Expsson 8 vals th nomous advantags of awadng a flxbl-tm concsson contact ov th tadtonal fxd-tm contacts. Fst, no dmand blf Q nts xpsson 8, whch ndcats that bds a f fom th ffct of dmand unctanty. hus, any possblty of havng slcton os s lmnatd. o matt th dstbuton of blfs acoss canddats, ths mchansm wll always slct th most ffcnt on n tms of costs, whch wll b th fm submttng th off fo lowst vnu. Scond, ngotaton nv taks plac, bcaus th fm cvs xactly th amount qustd as total vnu. n th tmnology of aucton thoy, ths mchansm allows th slcton of concssona to b tansfomd fom a pvat- and common-valu poblm, to a much smpl pvat-valu stuaton, n whch fms only dff accodng to th constucton costs. documntd xpnc s th concsson of a hlan hghway Santago-Valpaaíso-Vña dl Ma, awadd though an aucton basd on bds fo total dscountd vnus Gómz-Lobo and Hnojosa,

20 Futh xamnaton of optmal offs fom 8 ndcats that xpctd pofts a postv fo th wnn. Bds fo vnu cov fo all costs pad by th fm + Z, and allow fo som xta nt, whch s qual to th nfomaton nts that fms xpctd to xtact fom th tadtonal fxd-tm aucton systms. hs vnu quvalnc sult can b xpssd as: Π = Π Z = Π LVR = 9 Although th th analysd aucton mchansms a x-ant quvalnt fo fms, n tms of xpctd vnus, thy pfom vy dffntly x-post. As t has bn alady xamnd, pc and paymnt auctons hav postv pobablts of makng slcton os, and t s also lkly that th concsson contact could b qud to b ngotatd n th vnt of low dmand fo th nfastuctu. Manwhl, th LVR typ of aucton ylds optmal sults. o slcton o s v mad, bcaus fms bds a compltly f fom blfs about dmand. Mo ntstngly, a flxbl-tm concsson s nv ngotatd. n stuatons of low dmand, th only ffct s that th duaton of th contact s automatcally xtndd to allow th fm to collct vnus fom chags dung a long pod, and thus obtan th qustd total amount R. Flxbl-tm concsson contacts, howv, suff at last fom fou dawbacks. Fst, bcaus th fm s nsud to cv ts clamd vnu, t has no ncntvs at all to pomot th us of nfastuctu, as pontd out by ol 997. hs mans that uss may nd up cvng a poo svc fom th nfastuctu opato. Scond, and latd to th fom, th concssona may not mantan adquat qualty and safty standads, an aspct whch should b stctly supvsd by govnmnts. hd, fo lag pojcts th amount of qud nvstmnts s usually hgh, and th pod ndd to cov thm wll b typcally long. Although at th nd of th concsson th fm wll hav obtand ts total qustd vnu, t mght xpnc som cash-flow shotags dung ntmdat pods, whch mght affct ts ablty to pay dbts. A ncssay condton fo 9

21 a smooth pfomanc of flxbl-tm concssons s th xstnc of wll-dvlopd captal makts, whch allow concssonas to obtan bdg-loans fo such pods. h s a fouth lmtaton of flxbl-tm contacts, whch tuns out to b qut sous, and t ass du to th xstnc of mantnanc and opaton costs. n th scnao consdd n ths scton, n whch fms only ncu nto constucton costs, bdds a ndffnt to th duaton of th concsson. Howv, whn th concssona facs annual xpnss to opat an nfastuctu, long pods mply hgh costs that mght affct ts poftablty and, n xtm cass, may vn qu contact ngotaton. hs latt poblm s xtnsvly analysd n th nxt sctons, wh a modfd LVR mchansm s poposd to solv th poblms posd by mantnanc costs M. Bfo tunng to th study of th cas wth mantnanc and opaton costs, som fgus a povdd to assss th magntud of th poblms dscussd. 3.5 Smulatons of pc, paymnt and LVR auctons n od to valuat pobablts of slcton os and ngotaton of concsson contacts, som smulatons hav bn pfomd. aamts a chosn to psnt a typcal cas fo nfastuctu concssons, n tms of th latv sz of vaabls nvolvd. Usng a patcula bnchmak of fnc, t s possbl to assss th mpact of ach facto by ntoducng changs n paamts and studyng th mpact on outcoms. h bnchmak cas cosponds to an aucton wth =5 potntal canddats. Fms constucton costs a andomly dawn fom a unfom dstbuton U [00;,000], and dmand blfs fom an ndpndnt dstbuton Q U [0; 30]. h contact-tm s st at =30 yas fo th fxdtm auctons. c s fxd at =0.75 fo th cas of mum paymnt aucton; and paymnt at Z=00, fo th pc aucton. Wth ths valus, th xpctd ang of total vnu fo fms s [5, 0

22 675]. Fo th avag typ of fm, wth cost = 550, ths mpls that pofts may vay btwn [- 35, 5], so th pojct s qut sky and only attactv to fms wth low costs. abl shows th outcoms of ach typ of aucton aft pfomng 5,000 smulatons, at ach of whch fv bdds w andomly chosn gadng both th cost typs and blfs on futu dmand lvls. Bds fom all fms a computd and valuatd to dtmn th aucton s wnn. h aucton s outcom s thn compad wth th actual ankng of fms costs to calculat th pobablty of slcton o. [ SER ABLE ] Fo th bnchmak cas consdd, th pobablts of slcton os and contact ngotaton a consdably hgh. Fgus ndcat that pc and paymnt auctons a slctng on avag concssonas wth costs abov th optmal lvl = 5 mn + /6. h basc ason xplanng slcton poblms s th bas towads optmstc fms. hs bas can b assssd by compang th avag xpctd dmand lvls by aucton wnns wth th unbasd xpctd dmand Q =0. Manwhl, an aucton basd on th LVR mchansm ylds optmal outcoms: th s no bas n th slcton of concssona, whch mpls optmal slcton, and th pobablty of contact ngotaton s qual to zo. n contast to th stuaton dscbd n th al wold, wth a hgh pobablty of ngotaton fo th pc aucton than fo th paymnt aucton, sults show smla valus fo both typs n ths cas. hs s smply du to th sz of paamts chosn fo th bnchmak cas: t s possbl to obtan asly stuatons n whch pob s lag o low than pob Z by changng th valus of, Z o.

23 4. Scnao wth qual mantnanc and opaton costs As a fst stp towads th analyss of th mo gnal cas n whch fms dff both on constucton and mantnanc costs, consd ntally a scnao n whch w ntoduc som fxd annual xpnss on mantnanc and opaton of nfastuctu assts, but ths costs a known fo all pats and a qual acoss fms, M = M. Fo fxd-tm auctons, th xstnc of ths costs s lvant. t only changs slghtly th functons and Z usd by fms to comput th bds s xpssons 4 and 7. Fms smply accommodat fo th hgh costs and obtan th sam xpctd pofts - /. n th cas of th pc aucton, bds a ncasd by a facto of M/Q, whch mans that ach us s qud to pay a sha of th annual mantnanc costs. Manwhl, fo th paymnt aucton, bds Z a ducd by an amount M, whch s qual to total mantnanc and opaton costs dung th complt lf of th concsson. n ths latt cas, th wnn s smply subtactng thos costs fom th paymnt poposd to b mad to th govnmnt, so mantnanc costs a n fact tansfd to taxpays. h only modfcaton woth to b mntond fo fxd-tm auctons, wth spct to th analyss of th pvous scton, s that th concsson tm now has an mpact on th pobablts of slcton o and ngotaton fo th cas of th pc aucton. n patcula, condton 8 fo th slcton of a fm j oth than th bst fm s tansfomd so that paamt now affcts th condton. h ffct of on pobablts, howv, mans bng mo mpotant fo th cas of th paymnt aucton. Anoth lvant fatu s that pobablts of slcton os and ngotaton fo th paymnt aucton a not affctd by th sz of mantnanc cost M, whl pobablts fo th pc aucton do chang accodng to M.

24 4. Effct of mantnanc costs ov LVR auctons h ntoducton of mantnanc costs M has a damatc ffct on th flxbl-tm contact. o als ths, consd th poblm that a fm wth mantnanc cost M patcpatng at an LVR aucton must solv now. As bfo, whn submttng a bd fo vnu R, th canddat s dtmnng ts xpctd tm fo th contact, whch wll b qual to = R /Q. akng nto account th mpact of R on th pobablty of wnnng, fm calculats ts optmal bd by solvng: Max R Π R Q M Z = R R Q 0 Although n pncpl ths msaton poblm sms to b almost dntcal to 7, th psnc of M n th xpctd pofts has a stong mpact on th soluton: Q R = + Z + Q M As t can b obsvd fom xpsson, n th psnc of mantnanc and opaton costs, bds submttd und LVR auctons suff xactly fom th sam poblms dtctd fo th tadtonal fxd-tm auctons. Blfs about futu dmand Q a usd whn computng bds fo vnu R, mplyng th possblty of dstotd nfomaton fo th slcton of concssona. Agan, t s possbl that an ovoptmstc fm may outbd th most ffcnt fm n tms of costs, nducng th mchansm to mak slcton os. Evn mo sous s th ffct that M has on th pobablty of contact ngotaton, whch now s dffnt fom zo. Although und a flxbl tm contact a concssona s guaantd to obtand ts clamd R vnu, t s no long mmdat that ths amount of vnu wll always b suffcntly lag to cov fo all costs. n th vnt of a low dmand stuaton Q * Q mn, th contact tm must b xtndd, whch causs total 3

25 mantnanc costs fo th lf of th concsson to s. Rngotaton mght thn b qud n cass wh + Z + M * Q * > R. 4. A nw poposd mchansm: last-psnt-valu of nt vnu LVR h mpact of mantnanc costs and th assocatd poblms can b avodd by dsgnng a nw typ of aucton, also basd on flxbl-tm concsson contacts. As th am of th aucton mchansm s to xtact nfomaton fom fms gadng th tu constucton costs, th objctv should b to nsu that th bddng vaabl s as clos as possbl to. hs can b don by nsung fms that th annual mantnanc cost M wll b covd n any cas. h mchansm s smla n natu to LVR, but ylds btt sults. Fms submt bds B fo th total amount of mony thy dmand to cov fo nvstmnt costs. Fom vnus collctd fom uss, ach ya a pat M s consdd to b dstnd to cov fo mantnanc costs, whl th st s accountd fo as nt ncom pcvd. Whn total nt ncom pcvd s qual to th bd B, th concsson nds. Fomally xpssd, fm wth a bd B can thfo xpct to hold th concsson untl Q = B + M, and th xpctd tm s thn qual to = B / Q -M. hs nw aucton mchansm wll nd xactly th sam optmal sults n a contxt wth dscount ats δ <, thfo t s namd as last-psnt-valu of nt vnu LVR, bcaus th contact s awadd to th canddat wth th lowst bd B Bdds patcpatng at an LVR aucton wll comput th bds B accodng to som functon B whch s assumd to b shad by all patcpants. Each fm thn solvs: Max B Π B Q M Z = B B Q M 4

26 Solvng fo B shows that ths msaton poblm s xactly th sam that th fm was confontd wth at an LVR aucton whn th a no mantnanc and opaton costs s xpsson 7. h soluton to, thfo, stos th good popts about slcton and no ngotaton that w dvd n that cas. Optmal bds a qual to: B = B = + Z + 3 Expsson 3 ndcats why a LVR aucton outpfoms any oth aucton mchansm n ths contxt of qual mantnanc and opaton costs. Submttd bds B a compltly ndpndnt fom blfs about futu dmand, and B > 0, thfo th mchansm guaants th slcton of th most ffcnt canddat. h aucton s wnn xtacts som nfomaton nt fom th concsson, whch taks xactly th sam xpctd valu as n th oth auctons. Π = Π Z = Π LVR = Π LVR = 4 Rsults fom smulatons pfomd fo ths scnao wth constant mantnanc costs acoss fms a psntd n tabl. t can b obsvd that th LVR aucton mchansm suffs xactly fom th sam poblms as auctons basd on fxd-tm concssons, although ts pobablts fo slcton os and contact ngotaton a small. Manwhl, th LVR aucton always slcts th canddat wth lowst cost, and t has zo pobablty of contact ngotaton. [ SER ABLE ] 5. Scnao wth dffnt mantnanc and opaton costs h most lvant cas fo th analyss of auctons fo nfastuctu concssons s a contxt n whch fms may dff both on th constucton costs and also on th mantnanc and opaton costs M. h logc bhnd th assumpton of dffnt M s acoss fms s th sam that 5

27 justfs th xstnc of dffncs n constucton costs. Fms can vay accodng to th ffcncy lvls, fo xampl bcaus manags xt mo o lss ffot n contollng costs. h xstnc of dffnt mantnanc and opaton costs poss som dffcults fo th fomal analyss of auctons xpctd outcoms. n patcula, fm s typs a no long unfomly dstbutd ov som ang of constucton costs, bcaus th vaablty of mantnanc costs ntoducs som changs n th pobablts of ach typ of fm. A scond ssu s th dfnton of cost ffcncy. otal costs xpctd by ach fm a now qual to = + M, thfo th dfnton of th lowst-cost fm now dpnds on th contact-tm. Fo flxbl-tm concssons s dffcult to dtmn x-ant whch fm has th lowst cost, du to th fact that actual duaton of th contact * may chang accodng to dmand. Som ul s qud to calculat pobablts of slcton os, and to valuat bds f t s dcdd to us som fom of b-dmnsonal aucton. 5. obablty dstbuton of fms typs Usng th assumpton of unfom ndpndnt dstbutons fo constucton costs, U [ mn ; ], and fo mantnanc costs, M U [M mn ; M ], t s possbl to dv th pobablty dstbuton of fms typs. A typ now s dfnd as th valu of total costs, = + M. As dscussd abov, ths typs may vay accodng to th contact-tm pdfnd by th govnmnt. Fo a gvn valu of, th dstbuton functon of s qual to: M mn ; f mn a F + ; f < 5 F = a a a b [ + ] ; f < F b + b b b M 6

28 wh = - mn ; a = mn + M ; M = M - M mn ; b = + M mn ; mn = mn + M mn ; = + M. Bcaus constucton costs a gnally much lag than mantnanc and opaton costs fo most typs of nfastuctu, t s assumd that a < b th oth altnatv cas dos not mply any oth sgnfcant chang than som modfcatons n th xpssons fo F. Expsson 5 shows that th pobablty dstbuton of s not unfom. t xhbts a lna pat n th ang [ a, b ], but th functon s convx on [ mn, a ], and concav on [ b, ]. hs flcts th fact that xtm typs wth vy low o vy hgh total costs hav small pobablty masss. Gvn ths th-pat pobablty dstbuton, th calculaton of optmal bds fo th dffnt typs of auctons bcoms n gnal mo complx. Statgs must now b a functon of full costs, and thfo th pobablty of wnnng wth ach possbl bd of th fom f has to b calculatd fom F, wh f wll b th cospondng functon, Z o R, fo ach typ of aucton. 5. mpact of mantnanc and opaton costs M on fxd-tm concssons h msaton poblms fo computng optmal bds to patcpat at a mnmum pc o mum paymnt aucton a not bascally altd fom thos cass analysd n pvous sctons. h only dffnc s th chang n th dfnton of fms typs. n ths scnao, fms bds a functons of al full costs = +M. h pobablty of a fm wnnng an aucton wth a bd =, o Z =Z, now vas accodng to th ang wh th bd falls. n gnal, fo any bd o Z th wll b th possbl cass, accodng to th possblts of havng th - < a ; a - b ; o b < -. h two xtm cass yld non-lna solutons, whl th ntmdat cas s lna, du to th fact that typs a unfomly dstbutd btwn a and b. A complt dvaton of th optmal 7

29 statgy, takng nto account th whol dstbuton of, fo th cas of a pc aucton s psntd n dtal n th appndx Fo nfastuctu pojcts wth lag constucton costs, th xtm cass can b dsgadd, bcaus whn M s small latv to, a mn and b. onsdng that all bds submttd by canddats l wthn th ang [ a, b ], optmal bds fo pc auctons and paymnt auctons wll hav lna foms, smla to 4 and 7, spctvly. M = = + Z + a Q 6 Z M = Z = Q a 7 A smpl xamnaton of xpssons 6 and 7 ndcats that ths solutons sha th sam fatus of thos analysd on pvous sctons. Both and Z and functons that dpnd on blfs Q, thfo outcoms wll agan b plagud fom th sam poblms of slcton os and possbl contact ngotaton dscussd abov. 5.3 mpact of mantnanc and opaton costs M on LVR and LVR concssons As t s th cas fo fxd-tm concssons, optmal bddng statgs to follow at an LVR aucton a not fundamntally altd. h only majo chang s that bds fo vnu R a calculatd as a functon of total costs. Assumng that all bds l wthn th ang [ a, b ], ylds th soluton: R Q M = R = + Z + a Q M 8 A compason of xpssons 8 and shows that sults obtand fo th cas wth qual mantnanc costs M =M a not altd fo LVR auctons whn vaablty on M s ntoducd. 8

30 h bd fo vnu R s slghtly mo complx, to tak nto account th tu valu of mantnanc and opaton costs fo ach fm, but ts man fatus man unaltd, ncludd th ffct of blfs on futu dmand on bds R. Fo th LVR aucton, th xstnc of dffnt valus fo M acoss fms ntoducs changs. Bcaus th govnmnt now dos not hav accuat nfomaton about th sz of ths costs, a fst altnatv could b to us th xpctd avag valu M av = M mn +M /, to dtmn th pat of vnus collctd fom chags that wll b assgnd ach ya to cov fo mantnanc costs. Fms a agan nvtd to submt bds on th vnu, nt of mantnanc and opaton costs, taht thy dmand to cov th nvstmnt costs, and th contact s awadd to th lowst bd. haft, whn th actual lvl of dmand Q * s known, th duaton of th contact wl b qual to * = B / Q * - M av onsdng that fms wll comput th bds B basd on th al full costs 9, B =B, th poblm that thy solv s: Max B Π = Q Q M M av B M Z B B a 9 Optmal statgs a thn: B = Q M av M B = + Z + a Q M 30 Expsson 30 shows how, n th psnc of dffnt mantnanc costs acoss fms, th mchansm LVR dos not hav th popty of lmnatng compltly th psnc of dmand 9 Anoth possbl altnatv could b to consd that fms only tak nto account constucton costs to calculat bds,.. B = B. n that cas, t s found that th optmal statgy s B = Q -M av [ +Z+ - /] / Q -M av, whch s smla to 30 and ylds on avag th sam pofts. 9

31 blfs n bds B. hfo, t s possbl that ths aucton mchansm could also mak slcton os. A caful xamnaton of xpssons 8 and 30 ndcats that, although thy appa to b smla, th ffct of Q on R and B s dffnt. hs ffct can b chckd by takng th cospondng dvatvs: R Q = Q M M Z M + + < a 0 3 B Q = M Q av M M + Z + M a > < 0 3 h ngatv sgn of dvatv R / Q ndcats th bas towads slctng optmstc canddats fo th LVR aucton. Hgh xpctd valus Q sult n low bds, thus ncasng th pobablty of slcton os. hs s th sam ffct that had bn alady dtctd n th cas of qual mantnanc costs. Manwhl, dstotons ntoducd by dmand blfs Q on bds fo nt vnus at a LVR aucton a small obsv that R / Q > B / Q. Moov, th ffct of Q on th bd may vay accodng to th typ of fms. Fo thos canddats wth mantnanc costs M > M av th sgn of B / Q s ngatv, and th ffct s th sam as obsvd fo R / Q. But, fo canddats wth costs M small than M av, th ffct s xactly th oppost. n ths latt cas, mo optmstc fms tnd to s th bds, thus dcasng th pobablty of wnnng th aucton. h combnaton of both ths ffcts s lkly to sult n btt slcton outcoms fo LVR auctons than fo th oth mchansms. A compason of all outcoms dvd n th contxt of ths scnao wth dffnt M costs fo th fou auctons xamnd agan povds a vnu quvalnc sult: x-ant pofts xpctd by th wnn of any of th auctons a agan qual. h xstnc of dffncs acoss fms allow thos 30

32 fms wth low mantnanc costs to obtan hgh pofts than thos obtand n th cas of constant costs, M =M compa th sult wth xpsson 4 abov. Π = Π Z = Π LVR = Π LVR = + M av M 33 Howv, ths xpctd vnu-quvalnc dos not mply that th x-post pfomanc of all typs of mchansms s th sam. hs can b povd by numcal smulatons fo ths scnao, whch val that flxbl-tm concssons pfom gnally btt than th tadtonal mchansms. abl 3 psnts th sults obtand fom ths smulatons, basd agan on th bnchmak cas, and assumng that M U[3,7]. [SER ABLE 3] As xpctd, th LVR aucton now psnts postv pobablts of slcton os and ngotaton. Howv, t s ntstng to obsv how ths mchansm ylds btt sults than th altnatv LVR, and also outpfoms th tadtonal auctons basd on fxd-tm contacts. 5.4 LVR aucton basd on bds wth two dmnsons hs fnal scton ams to complt th analyss of fasbl auctons to awad concsson contacts fo nfastuctu pojcts by analysng a possbl fnmnt of th LVR aucton. hs nw poposd mchansm has bn povd to yld optmal outcoms n smpl scnaos wth constant mantnanc costs, claly outpfomng both tadtonal aucton modls and th LVR typ of aucton whch dos not tak nto account xplctly th xstnc of mantnanc costs n th aucton dsgn. Howv, n a al contxt of stong asymmts of nfomaton, t s possbl that a govnmnt could b compltly unnfomd about th sz of mantnanc costs M, so t wll b unabl to us an LVR aucton as dscbd n th pvous scton whch mpls th us of som known avag mantnanc cost M av. h fnmnt poposd s thn to nvt fms to submt 3

33 bds wth two dmnsons: a total vnus to b obtand fom th concsson, nt of mantnanc costs B ; and b annual avag xpnss on mantnanc and opaton costs E. h analyss of ths poposd vaaton of LVR s ntstng both fom a pactcal pont of vw bcaus t could b an altnatv fo govnmnts to aucton concsson pojcts, and also fom a thotcal pspctv. h a not many xampls n th aucton ltatu dalng wth stuatons n whch bdds submt bds n mo than on dmnson h, 993; Banco, 997. Howv, as pontd out by McAf and McMllan 987, t s fqunt that whn awadng contacts, govnmnts a ntstd n mo than on dmnson of nfomaton.g. qualty, pcs, nvstmnt lvls, and so foth. t s thn ntstng to ty to comput qulbum statgs whn fms a qustd to submt ths typ of b-dmnsonal bds. h fst stp s to calculat th pobablty fo a fm to wn th aucton wth a bd B,E. n od to do that, w nd fst to stablsh how th govnmnt wll valuat offs. Du to th fact that, fo a flxbl-tm concsson, th contact tm s x-ant undtmnd, th s not a unqu opton to valuat whch off s th bst n tms of total xpctd costs. On altnatv s to st som abtay pod of fnc v, and calculat xpctd total costs as = B + E v, awadng th contact to th lowst off. Oth optons a to announc a ang of pods [ mn, ], and comput avag xpctd costs ov that ang, o to choos th fm wth lowst costs dung mo pods wthn that ang. t can b povd that whn fms know th ang of pods usd fo valuaton, all ths cta a quvalnt to us th ntmdat pont of th ang as a pod of fnc,.. v = mn + /. hfo, fo a smpl xposton, w wll wok wth a sngl pod v fo th valuaton of bds. t s no long fasbl n ths contxt to us gnal statgs of th fom B = B, E = E, bcaus valuaton of bds would thn mply th us of a jont-nvs functon [B + E v ] -, fom whch t s mpossbl to tv ndvdual functons fo B, E. nstad, a smpl way to 3

34 calculat pobablts of wnnng th aucton s to assum som patcula foms fo th functons B and E that can b latd to som andom vaabl wth known pobablty dstbuton. atual canddats a lna foms fo B and E, n tms of constucton and mantnanc costs, spctvly. t s thn assumd that fms choos th bds fo nt vnu as a functon of constucton costs, B = α + β, and bds fo mbusmnt of mantnanc costs as a functon of th tu costs, E = γ M. 0 Usng ths uls, th pobablty of fm wnnng th aucton s qual to th pobablty of th vnt B + E v < B j + E j v, fo any possbl j, whch can b xpssd as: pob = v v pob [ B + E < + β + γ M ] α 34 j j h pobablty dstbuton of α + β j + γ M j v can b calculatd fom th unfom dstbutons of j and M j t has a fom vy smla to that of, s xpsson 5. hfo, t s possbl to dv an xplct xpsson fo pob, whch can b usd to ms xpctd pofts. Assumng, as n pvous sctons, that only th lna pat of th pobablty dstbuton s lvant whch s quvalnt to say that all bds = B + E v fall wthn a gvn ang, th pobablty of fm wnnng th aucton wth a bd s qual to: pob γ M a = + β β β 35 wh a = α + β mn + γ M v. 0 Anoth possbl opton s to mak both B and E dpndnt on full cost..g. B = α+β, E = α +β. Howv, ths s quvalnt to th foms adoptd, mplyng only a ntptaton of coffcnts α, β and γ. h sam agumnt appls to th choc of ntoducng a constant tm α only n th functon of B and not on E. 33

35 onsdng that th concsson wll last fo an xpctd tm = B /Q -E, dscadng som constants tms, and dnotng by θ = α + β + γ v M mn +M /, th poblm that fm solvs to dtmn ts optmal bd B, E can b xpssd as: v B E B Max, B E Π = Q M Z θ 36 Q E h two fst-od condtons of ths poblm consttut a non-lna systm of quatons n tms of B and E : v θ B E + Z Q M 0 Q M = Q M B Q E v v Q M θ B E B Z 0 = Q E h xst two solutons, B, E and B, E, fo ths systm of quatons, and fo ach of thm t s possbl to obtan xplct xpssons fo paamts α, β and γ, usng th assumd foms fo th optmal statgs. Both of thm hav som conomc ntptaton. h fst soluton B, E s a bd that ylds xpctd pofts qual to zo, so any fm wll only submt ths bd n absnc of any oth btt opton. h scond soluton B, E ylds an xpctd poft qual to Π = Q v +M v + Z,.., th fm ts to xtact fom th concsson all th xstnt suplus xpctd fom th pojct, computd on th bass of a contact-tm qual to th valuaton pod v. hs scond soluton s n fact a whol famly of fasbl bds, wth E = E, M, Q, and B = v Q E, all of whch yld th sam xpctd lvl of poft Π. Fms that hav ngatv xpctd valus fo ths poft lvl a cas that mght occu fo hgh-cost fms wth low xpctd dmands wll opt fo th oth altnatv bd B, E, o wll dop out fom th aucton. 34

36 Optmal statgs fo fms dvd fo th contxt of ths b-dmnsonal aucton can b asly ntptd. Du to th fact that bds B, E must b vntually ducd to a sngl-dmnson sco fo valuaton puposs, canddats know that thy can ty to xtact nts fom th concsson, th by sng bds B ov al constucton costs, o sng th oth dmnson E ov al mantnanc costs M. along th statgy n such a way that th total bd = B + E v s kpt constant dos not affct th pobablty of wnnng, but th ffcts of movng B o E can b vy dffnt on th xpctd poft lvls. Whn a fm consds that th concsson wll last fo a long pod than th on usd fo valuaton, t wll b ntstd n ncasng th gap E -M, bcaus ach addtonal ya as opato of th nfastuctu povds that dffnc as xta pofts. On th contay, f a canddat xpcts th concsson to b shot, t wll b mo wadng to ty to xtact nts fom th dffnc btwn al and dclad constucton costs, B -. h man concluson fo ths poposd fnmnt of th LVR mchansm s that blfs Q stll nt th calculaton of optmal statgs. hfo, onc agan t s concludd that th mchansm s suscptbl to mak slcton os, as t s th cas wth th oth aucton systms studd n ths pap. vthlss, mpcal smulatons pfomd wthn th famwok of th bnchmak cas analysd abov ndcat that ths vaant of LVR ylds asonabl good outcoms. hs maks t a good canddat to b usd n pactc by govnmnts that want to mplmnt auctons basd on th da of flxbl-tm contacts, but a compltly unnfomd about th possbl ang of valus fo mantnanc and opaton costs of an nfastuctu. [ SER ABLE 4] On ntstng fndng fom th mpcal analyss s that ths typ of aucton sms to favou low cost-fms, as vald fom th avag sz of wnns, whch s much low than fo oth auctons. hs sults n hgh xpctd pofts fo concssonas as a sult of lag 35