Maarten C.W. Janssen 1,2 Vladimir A. Karamychev 1

Transcription

1 TI 7-/ Tbrg Iu Dcuo Papr Do Auco lc Effc Frm? Maar C.W. Ja Vladmr A. Karamychv Eramu Uvr Rordam Tbrg Iu.

2 Tbrg Iu Th Tbrg Iu h u for coomc rarch of h Eramu Uvr Rordam Uvr va Amrdam ad Vr Uvr Amrdam. Tbrg Iu Amrdam Rorraa 3 8 WB Amrdam Th Nhrlad Tl.: 3( Fa: 3( Tbrg Iu Rordam Burg. Oudlaa 5 36 PA Rordam Th Nhrlad Tl.: 3( Fa: 3( Mo TI dcuo papr ca b dowloadd a hp://

3 Do Auco Slc Effc Frm? Maar C. W. Ja Vladmr A. Karamychv Tbrg Iu ad Eramu Uvry Rordam Abrac. Th papr codr a govrm aucog off mulpl lc o frm who comp a mar afr h auco. Frm hav dffr co ad co ffccy prva formao a h auco ag ad h mar compo ag. If oly o lc aucod adard rul ay ha h mo ffc frm w h auco (lc a wll g h hgh prof h afrmar.. ha h hgh valuao for h lc. Th papr argu ha h rul do o gral o h ca of mulpl lc ad afrmar compo. I parcular w drm codo udr whch auco may lc ffc frm ad hrfor lad o a ffc allocao of rourc. Sragc raco h afrmar parcular frm prfrc o comp wh h la co-ffc frm rahr ha wh h mo ffc frm ar rpobl for our rul. Ky Word: Auco co-ffccy afrmar JEL Clafcao: D43 L L3. W ha Larry Auubl By Moldovau ad audc a Europa Uvry Iu Eramu Uvry Rordam ad Tbrg Iu for hlpful comm ad uggo.

4 . Iroduco I may lbralao or prvaao proc govrm vually fac h u how o lc frm ha wll provd h formrly publcly provdd rvc. O of h advaag of ug auco a a lco mcham o of hough ha auco lc h mo co-ffc frm. Mar whr mor co-ffc frm ar acv ypcally yld mor ffc mar oucom ha wh h am mar ar rvd by l co-ffc frm.. ohr hg bg qual co-ffccy good for ovrall coomc wlfar. O forcful am of h vw by Dm (968. Dm argu ha compo for h mar may b a good ubu for compo h mar. Morovr a moopoly co h argu ha h mo co-ffc frm wll w h compo for h mar (rad: wll w h auco. I h papr w wll rfr o h rul a h moopoly rul of Dm. Th rul ha prmad a larg lraur o procurm u ad dd Laffo ad Trol ( pp a ha f o gor h procg capur ad dyamc co of auco ay o ha auco ypcally lc h frm wh h low co. Rcly may govrm hav rld o a combao of compo for ad compo h mar. A mpora ca po h wav of 3G mobl lphoy pcrum auco ha hav b hld aroud h world (.g. Klmprr a b Bmor ad Klmprr ad Jhl ad Moldovau 4 for ovrvw. I all of h 3G auco mulpl lc wr old ad a hr wr mor frm parcpag h auco ha avalabl lc frm had o comp o oba a lc. May govrm formally or formally ad ha ffc agm of frqucy pcrum wa o of h goal o b achvd. Wh co aymmr bw frm ffc agm mpl ha h mo co-ffc frm hould w lc ad dd h Duch govrm amog ohr mod lcg h mo ffc frm a o of h rao for holdg a auco (.g. Ja al.. I h papr w wll argu howvr ha h moopoly rul of Dm do o carry ovr o h ca of mulpl lc.. o h ca whr frm comp a olgopolc faho h afrmar. I parcular hr ar mar codo udr whch h mo co-ffc frm wll o carly oba h lc or v

5 wor ha h la co-ffc frm wll carly cur hm. Th ma rao ha h Dm rul do o gral ha a auco wh mulpl lc a ragc ffc wor aga h Dm rul. Bacally h ragc ffc pr almo all mar g a mply cofrm h fac ha ay frm prfr o comp wh hgh-co frm rahr ha wh low-co frm. Dpdg o h mar codo ad h -a drbuo of frm co h ragc ffc ca b o rog ha h mo co-ffc frm ma l prof h afrmar wh hy comp wh ach ohr ha h la co-ffc frm do. Mor chcally w codr a adard mul-u uform-prc auco whr frm hav prva formao abou hr co ad ovrall coomc ffccy rqur h mo ffc frm o w h auco. A ragy for h frm a fuco pcfyg how a frm bd dpd o ffccy paramr. Th gralao of h Dm moopoly rul o h ca whr mulpl lc ar aucod rqur ha h mor ffc a frm h hghr bd h auco.. hr hould a mooo ymmrc bddg qulbrum whr frm bddg ragy crag hr ffccy paramr. W dfy codo udr whch uch a ymmrc crag bddg qulbrum ad wh do o. A fr mor aly dfabl codo udr whch a ymmrc crag bddg qulbrum fal o ha frm ffccy paramr ar povly corrlad (afflad o ha larg o ow ffccy paramr provd formao abou ohr frm prva formao. I pracc pov afflao of frm ffccy paramr may aurally ar cor whr frm u mlar produco cholog ad prc of pu flucua wh (macrocoomc hoc ha ar commo o all frm. Alravly frm may mplm co-avg cholog ha ar from a ogou ochac proc. I boh ca f a frm mor co-ffc lf fr ha all ohr frm ar mor co-ffc a wll. Thrfor mor ffc frm pc o b compg wh ohr ffc frm (who ar ow o b frc compor ad hu pc o ma l prof h afrmar ha l ffc frm. W wll how ha for ay olgopolc mar o mar how wa h ragc ffc hr ar drbuo of frm yp for whch a ymmrc crag bddg qulbrum do o. I ohr word a crag bddg qulbrum hr a d of wr cur or advr lco ffc pr h rgh of whch dpd o h yp of bddr. Bddr opmally adu hr bd for h ffc ad a h ffc rogr for mor ffc frm hghly ffc frm may adu hr bd o much ha h crag bddg qulbrum ragy droyd. 3

6 A cod mor urprg codo udr whch a ymmrc crag bddg qulbrum fal o whr frm ffccy paramr ar -a dpd. Dp frm yp bg -a dpd h yp of frm ha w lc wll b corrlad. Th o bcau all wg frm oubd h frm wh h hgh loog bd. A h po form of pov corrlao (afflao acually h oly corrlao ha rlva for drmg h opmal bddg ragy h uv rao for h oc of a ymmrc crag bddg qulbrum h h am a abov. W wll how ha h ca of acally dpd yp a crag qulbrum fal o oly f h ragc ffc uffcly rog. Wh o of h wo codo hold a ymmrc crag bddg qulbrum do o. Th mpl ha oly ( aymmrc qulbra whch dffr frm hav dffr bddg fuco or ( h qulbrum bddg fuco ar o mooo or ( frm u radom bddg rag or (v a dcrag qulbrum. I all of h four ca hr a la a pov probably ha l ffc frm wll bd mor ha mor ffc frm ad hrfor oba h lc. Thu f frm yp ar hghly corrlad or h ragc ffc rog ough h ovrall oucom ffc wh pov probably. 3 W alo how by ma of a ampl ha f frm yp ar corrlad ad h ragc ffc rog ough a uqu mooo ymmrc bddg qulbrum dcrag. I h ca auco alway lc h la ffc frm mplyg lowr ovrall wlfar ha ay ohr lco mcham. Th r of h papr orgad a follow. Sco dcrb h wo-ag modl wh a auco ag ad a mar compo ag. I Sco 3 w h provd cary ad uffc codo for a crag qulbrum o ad llura wha h gral codo mply ca of Brrad ad Couro compo h afrmar. Th co 4 w provd a ampl whch a uqu mooo ymmrc bddg qulbrum ad ha dcrag. Sco 5 coclud ad provd a dcuo of rlad lraur ad rmag u. Th appd coa all proof. 3 Thr ar ow may papr udyg ffcc crad by auco du o afrmar compo. For a rc ovrvw Jhl ad Moldovau (6. Hopp Jhl ad Moldovau (6 focu o h raco bw cumb ad ra Ja (6 codr how auco may lad o coordao h mar ag ad Ja ad Karamychv (6 udy lco ffc mar whr dmad ucray mpora. Jhl ad Moldovau (996 ad Jhl Moldovau ad Sacch (996 how ha v wh o lc aucod ffcc may ar du o h c of ral bw h bddr. 4

7 . Th Modl Acc o h afrmar lmd o h frm ha hav obad lc o opra h mar. Th govrm alloca hgh bddg frm ad w aum ha N lc a mul-u auco o h ( frm parcpa h auco. I h olgopolc afrmar frm comp by mulaouly choog a valu of h ragc varabl. Dpdg o h mar w ca rpr a hr a prc p or a quay q or ay ohr rlva ragc varabl. Th prof of frm drmd by h lvl of ha frm ad h ohr ( - frm choo ad by h frm ffccy paramr. A w aum ha ymmrc all for ca b wr a ( whr - a vcor of ragc varabl cho by all ohr frm. To hor oao w do h paral drvav of a follow: / / for / / c. Th ffccy paramr povly fluc h prof of frm by rducg oal a wll a margal co. Thrfor for a ypcal co fuco f( q of a frm w wll hav: ( q f f q q > ad ( q. f q Wh frm comp qua (Couro compo q ad frm prof fuco gv by ( q q q p( q f ( q o ha f > ad f. Wh frm comp prc (dffrad q > Brrad compo p ad frm prof fuco gv by ( p p q( p p p f ( q( p p whr ( q p p frm mar dmad o ha f > ad f q. q I ordr o ur h c uqu ad ably of h Nah qulbrum h afrmar frm margal prof fuco mu afy a ably rqurm. W follow Bulow al. (985 ad aum h ca of ragc complm whr ha ( > ha (. ad ca of ragc ubu whr 5

8 W aaly h ca whr h govrm orga a mul-u uform-prc auco o alloca h lc whr all h wg frm pay h am lc f w whch qual o h hgh o-wg bd. Th uform-prc auco allow u o mplfy h poo of rul whl pg h formulao of h afrmar compo ag qu gral. I h ma body of h papr w aum ha ral of lc o allowd. I h fal co w dcu how allowg for dffr auco forma ad ral of lc afr h auco wll affc our rul. A frm ffccy paramr.. h yp of frm prva formao h auco ag. Th pror o drbuo of yp dod by F( -. Th drbuo ha a f uppor [ ] ad aumd o b waly afflad (hu allowg for acal dpdc. A frm ubm a bd b bad o. W do a mooo ymmrc qulbrum bddg fuco by b( o ha frm bd b( qulbrum. Dpdg o wha formao rvald mmdaly afr h auco hld hr dffr caro ca b codrd: a A prva formao caro whr hr frm yp or h wg bd bcom publc. b A mprfc formao caro whr oly h bd of h wg frm bu o hr yp bcom publc. c A full formao caro whr yp of all wg frm bcom publc. I wha follow w maly focu o h prva formao caro. Th mprfc formao caro oo complcad o aaly a galg u gfcaly complca frm bddg bhavor. A ach frm ha a cv o prd o b mor ffc ha acually h auco ag would hav o b aalyd a a N frm galg gam wh ach playr bg a dr ad a poal rcvr of gal. 4 Th full formao caro o h ohr had do o m o b ralc. Morovr h caro ca b aalyd a mlar way o h prva formao caro a w wll brfly dca fooo 5 blow. I h prva formao caro a yp payg a lc f w ad choog h afrmar ha a pcd mar prof codoal o wg h auco of 4 Gor (3 coa a aaly of how gl-ag galg affc bddg bhavor auco. 6

9 ( ( b( w b( E l whr w dca all wg frm ohr ha frm by h d ad all loog frm by d l. Mamg pcd prof wh rpc o ad aumg ha all ohr wg frm choo opmally yld h fr-ordr codo dfg frm afrmar Nah qulbrum ragy. I ca h bddg fuco b( mooocally cra w do a b ( ( ad wr h fr-ordr codo a follow: * * ( ( E K l whr do h yp of frm ha ubm h h hgh bd amog (N frm ohr ha frm drmg h lc f w.. w b ( (. Udr h aumpo ha w mad abou h prof fuco ( h fr-ordr codo uquly df a mar ag Nah qulbrum ragy *. 3. Wh a crag bddg qulbrum do o W ar ow rady o aaly h cary ad uffc codo for a crag ymmrc bddg qulbrum o. W fr drv h codo for h gral ca dcrbd h prvou co. Th w aaly wo of crcumac (frm yp bg dpdly drbud ad afflad yp udr whch h cary codo cao b afd o ha h auco ag do o hav a crag ymmrc bddg qulbrum. Wh dpd yp w alo dca wha h codo mply ca of Brrad ad Couro compo. L b ( ( b a crag ymmrc qulbrum bddg fuco ad * ( b h corrpodg frm afrmar Nah qulbrum ragy. Dog a frm rducd-form prof by * * ( ( (.. allow u o wr h pcd prof of yp codoal o gg a lc a ( ( ( E ( v. l Th fuco v ( ( a frm valuao fuco whch ud h auco ag o drm h opmal bddg ragy. Th followg propoo drv a qulbrum bddg fuco ad cary ad uffc codo for a crag ymmrc bddg qulbrum o. 7

10 Propoo. If > ad v v > for all h hr v ( [ ] a uqu ymmrc crag bddg qulbrum gv by b ( ( v (. If h ( qulbrum h v ad v ( v ( for all [ ]. Th am mad Propoo ca b udrood a follow. Suppo ha. I ohr word uppo ha frm ad aohr frm l u ay frm m hav h am yp.. o ha hy oghr drm h auco prc w ad > m for all ohr wg frm. I h ca frm ad m comp for oly o rmag lc. Thy wll bd hr r pcd mar prof v ( ( o ha h qulbrum bddg fuco b ( ( mu afy b ( ( v ( (. Suppo ow ha margally largr ha. Th ordr o g a lc frm mu g a margally hghr pcd prof v ( ( ha frm m o ha frm ca bd margally hghr ha frm m. Thu v ( ( mu b a crag fuco of a.. v ( ( v ( ( ( v ( h fr cary codo. Th ohr cary codo h guara ha h acual bd v ( ( of frm dd hghr ha h bd v ( ( of frm m. O h ohr had f b ( ( v ( a rcly crag bddg fuco h uffc codo ( v > whch bacally h cod-ordr codo for prof mamao guara ha a frm ha o profabl dvao from b ( (. From h po of vw of a wg frm h yp of all ohr ( - wg frm ar afflad v f h yp ar -a dpd. Th dgr of h afflao gral drmd by frm ow yp ad by h -a drbuo of yp. I h lm ca wh yp of all compor of frm ar prfcly corrlad.. wh for all h paral drvav of h rducd-form prof fuco ca b aalycally calculad. Th h co of h followg lmma. Lmma. I ca for all h paral drvav of h rducd-form prof fuco of frm a ar: ( > ( ad 8

11 ( ( (. ( Lmma ur ou o b uful h r of h co. Ug a couy argum ay o ha h qual > ad alo hold f compor yp ar hghly (bu o prfcly corrlad ad f ad ar clo o ach ohr (bu o cocd. Hc accordac wh Lmma f h po corrlao of all wg frm yp larg h drc ffc alway pov.. ach frm wa o b mor ffc ad h drc ragc ffc gav.. ach frm wa o comp wh l ffc frm. I follow ha hr a o h auco ag of h modl. O o had h pov drc ffc forc a frm o bd hghr f h frm mor ffc. O h ohr had h gav ragc ffc fluc a frm valuao hrough h po afflao ad forc a frm o bd lowr f h frm mor ffc. A rg coquc of Lmma ha h auco prc w b ( ( cao b codrd a u co for a frm h afrmar ag... Th bcau h auco prc ffc h drbuo of frm compor ad hrough hr afrmar ragy drcly fluc a frm prof. Sacally dpd yp Ug Propoo ad Lmma a a gral ool w ow fr aaly h ca whr frm yp ar dpd. L h frm ffccy paramr b dcally ad dpdly drbud ovr a compac ad boudd uppor accordac wh a arbrary wc dffrabl drbuo fuco dy dod by f ( a crag qulbrum fal o. Propoo. If h codo ( ( ( F ( whch. Th followg propoo a a codo udr whch hold a h h auco ag do o hav a ymmrc crag bddg qulbrum. 9

12 W hav argud abov ha frm yp (codoal o wg ar -po afflad v f h yp ar -a dpd. Propoo u h fac ha f clo o h uppr d of h drbuo h afflao raoably rog bcau h yp of all wg frm wll b bw ad. Toghr wh a uffcly rog ragc ffc gv by h qualy Propoo a crag bddg qulbrum fal o. 5 W wll llura Propoo by codrg om pcfc fucoal form udr Couro ad dffrad Brrad compo. Th ampl rgh h da covyd h co amly ha f h ragc ffc rog ough a ymmrc crag bddg qulbrum fal o. Th ampl dfy codo rm of dmad lac ad cro-prc ffc guarag a rog ough ragc ffc. Eampl : Couro compo. L frm hav coa margal co o ha h co fuco f ( q q( c wh c ad l mar dmad b characrd by a coa prc lacy r.. Q p r. Udr Couro compo a frm prof fuco bcom / r ( q q q ( q ( c o ha qulbrum oupu lvl a ar gv by q q q * r ( r ( c r. Propoo ll u ha a uffc codo for a ymmrc crag bddg qulbrum o o ca b obad by vgag h paral drvav of h prof fuco a h po. O ca aly calcula hm o b: ( c r ( r ( r ( ( r rq c ( r ( r ( ( r c * rq * 5 O ca ha h rul of Propoo ad alo hold ru h full formao caro dcud abov. Th rao ha udr h codo of Lmma all wg frm hav h am yp ad hrfor Lmma hold ru boh (full ad prva formaoal caro. Coquly Propoo hold boh g.

13 q * ad. Subug h valu o h codo of Propoo w g ha f r a ymmrc crag bddg qulbrum do o. Thu f /( dmad rlavly lac h ragc ffc rog ough o doma h drc ffc. Th qu uv a h ca a mall rduco oal oupu (du o hghr co ha larg prc (ad hu prof ffc. O h ohr had w do o wa h o-c rul o b drv b by h fac ha h mar ag qulbrum o wll-dfd. I ay o ha h ably codo ( ha w mpod rduc h ca o r > /.. h prc lacy of dmad hould o b oo mall. Coquly f r h codo of Propoo a wll a h ably aumpo mod Sco ar afd for ay -a dpd drbuo of yp. Thu for ay umbr of lc ha ar aucod hr valu of h prc lacy of dmad uch ha a crag bddg qulbrum do o. Eampl : Dffrad Brrad compo. If o h corary frm comp prc wh h am co fuco a Eampl ad frm mar dmad ar lar ad gv by ( p p p a q p h h prof fuco bcom ( p p ( c ( ( p a p p. Equlbrum prc a ca b aly calculad o b gv by p ( ( c /( a( * p p. Aga o vo Propoo w hav o vga h paral drvav of h prof fuco. O ca aly ha a h po hy ar: ( c ( a( ( a( a 6 ( /( /( ad ar 6 I h ampl qua ar ragc ubu f r a ragc complm f r ( / / ( a. >

14 ( c ( a( ( a( a ad. A Eampl w d a wo cora o accou. O o had ubug h valu o h codo of Propoo w ca chc ha f a > 4 /( 3( a ymmrc crag bddg qulbrum do o. Th paramr a maur h ragc ffc h ca ad h ffc mu b uffcly rog. O h ohr had h ably codo rduc h ca o Thu f /( ( a.. h ragc ffc cao b oo larg. a 4 3( a ymmrc crag bddg qulbrum do o for ay -a dpd drbuo of frm yp. Aga ay o ha for ay hr ar paramr valu uch ha h h ca. Afflad yp I h ampl abov w hav how ha f h ragc ffc rogr ha h drc ffc h for ay dpd pror drbuo of frm yp a crag qulbrum do o. I ur ou ha ha v f h ragc ffc wa (bu ll a crag qulbrum may ll fal o provdd frm yp ar a afflad. 7 I h qul w codr a mar ag wh codo ha commoly hold olgopoly mar amly >.. frm prfr bg mor co-ffc hmlv ad.. frm prfr compg wh l co-ffc compor. For mplcy w aum ha frm co fuco gv by f ( q q( c c ad ha all ffccy paramr wh probably a half ar uformly ad dpdly drbud ovr h rag [ ] ad wh probably a 7 I ay o ha f h ragc ffc complly ab.. f o ha frm hav local moopol h Dm moopoly rul cou o hold.

15 half hy ar uformly ad dpdly drbud ovr h rag [ ]. If h yp of frm margally blow.. ε whr ε > mall pc o comp wh frm whch yp ar drbud ovr [ ε ] whch ca: ( ( ( ε v ( ε ε E ( ε ε [ ε] b. If o h ohr had h yp of frm margally abov.. ε h frm pc o comp wh frm whch yp ar drbud ovr [ ε ] whch ca: ( ( ( ε v ( ε ε E ( ε ε [ ε] b. I ay o ha h bddg fuco dcouou ad acually dcra a : lm ε ( ( ( b ( ε b ( ε E ( ( [ ] du o. Hc a ymmrc crag bddg qulbrum fal o o mar how wa h ragc ffc. 4. O dcrag bddg qulbra Th aaly h prvou co lad u o a aural quo amly whhr hr mar rucur ad yp drbuo for whch o oly a crag qulbrum fal o bu ad a ymmrc dcrag qulbrum do. I uch a qulbrum h la-ffc frm alway ubm h hgh bd ad oba h lc o comp h afrmar. I h co w fr argu ha h ragc ffc ad -a afflao of frm yp ar boh cary for a dcrag qulbrum o. N w provd a ampl of pcfc mar codo udr whch a dcrag qulbrum. Wh a mooo ymmrc qulbrum bddg fuco a dcrag fuco frm valuao fuco v (- ( mu b dfd a follow v ( ( ( E (. Th followg propoo drv a qulbrum bddg fuco ad cary ad uffc codo for a ymmrc dcrag bddg qulbrum o. l 3

16 Propoo 3. If ad v v for all h hr v ( [ ] a uqu ymmrc dcrag bddg qulbrum gv by b (- ( v (-. If h ( qulbrum h v ad v ( v ( for all. [ ] Th proof of Propoo 3 mlar o h proof of Propoo ad hrfor omd. Codr fr h codo ( ( E ( ( v ( v l.. If frm yp ar dpd h Udr h aumpo ha frm prfr bg mor co-ffc hmlv w hav ad hrfor v >. Hc a dcrag qulbrum vr ha ( > f yp ar acally dpd. If o h ohr had hr o ragc ffc.. ad frm hav local moopol w hav v ( ( >. Hc alo h ca a dcrag qulbrum do o. I h rmag of h co w provd a ampl of mar codo whr a uqu ymmrc bddg qulbrum ha dcrag. To h d w a h followg drbuo F * of frm ffccy paramr. L a macrocoomc fudamal (.g. r ra ol prc h growh ra of h coomy c. b drbud ovr h rval [ ] accordac wh a arbrary (wc dffrabl drbuo fuco F ( ( Pr. Th for ay gv l all b dpdly [ ] ad uformly drbud ovr h rval ( ( whr ( ε ( ( ε ( ad ε ( a paramr.. l h codoal drbuo F ( ( Pr b F ( ( for ( ( ( ( [ ]. Th rao why w codr h pcfc drbuo ha for mall valu of ε f a frm ha a yp h drbuo of yp of all ohr frm codoal o cocrad o a mall ghborhood of. Thrfor all frm ha ar compg h afrmar hav appromaly h am yp h Nah qulbrum almo ymmrc ad a dcrag qulbrum bddg fuco ca b aalycally calculad 4

17 ( ( Fgur. Suppor of h codoal drbuo F ( ( Pr. [ ] h lm wh ε covrg o ro. Fgur how h uppor ( ( codoal drbuo F (. of h Propoo 4. L N frm wh coa margal co c > comp a auco for lc l h wg frm comp qua a mar wh coaly lac dmad Q r p ad l frm ffccy paramr b drbud accordac wh h drbuo F *. If h prc lacy r af r r ( 3 3 ~ r > h hr a ε o ha for all ( ε ymmrc bddg qulbrum ha dcrag. ε ~ h auco ag ha a uqu I h ampl codrd Propoo 4 h dmad lacy r mu o b oo mall (r > / ordr o ur ha h afrmar Nah qulbrum ad abl. O h ohr had r mu b mall ough ( r r( 5 o ha h ragc ffc

18 uffcly rog. Th mmum umbr of lc for h dcrag qulbrum o 3. Th ma rao ha h ragc ffc hould b rog ough ad h ffc g rogr h largr h umbr of frm compg h mar plac Dcuo ad Cocluo I h arcl w hav how ha wh mulpl lc ar aucod o frm whch comp a afrmar h lc do o hav o d up h had of h mo ffc frm. Th mpl ha auco may cra afrmar ffcc. Th ma rao for h rul ar h prc of a formaoal raly ad h fac ha raoal bddr a a d of advr lco or wr cur o accou. Jhl ad Moldovau (6 argu ha ca frm afrmar prof dpd o prva formao h had of ohr wg frm ( our ca co hr a formaoal raly. Th d of advr lco or wr cur ha pr our co ha frm prfr o comp wh l ffc frm ad ha wh h auco lc h mo ffc frm bddg frm hav o a h lco ffc o accou. W hav dfd codo udr whch ffc frm dow hr bd o much mor ha l ffc frm ha a crag bddg qulbrum do o. Th modl dvlopd h papr do o f o h ow adard aumpo of h afflad valuao modl (Mlgrom ad Wbr 98. A mpora aumpo h afflad valuao modl ha a playr valuao a crag fuco of h ow gal a wll a of h prva gal rcvd by all ohr playr. I our ca whr frm rcv a gal of hr co paramr frm valuao a crag fuco of ow gal bu a dcrag fuco of h gal of ohr frm. Morovr bu l mpora a frm oly car abou h gal rcvd by ohr wg frm. I h papr w hav focud our ao oly o a adard mul-u uformprc auco. I ca b how howvr ha ohr mulaou-bd mul-u 8 Formally h propoo ad for h ca whr h umbr of rd frm u o mor ha h umbr of avalabl lc. Th a chcal rqurm a ohrw w cao gra ou h pro h prof fuco. 6

19 auco.g. a pay-your-bd auco alo uffr from afrmar ffccy provdd h ragc ffc or h -a yp corrlao uffcly rog. Frm dr o comp wh l ffc compor rpobl for uch ffccy. Th aaly much mor complcad ca of qual auco whr lc ar old o-by-o. I ay o ha h la lc d up had of h mo ffc rmag frm. Nvrhl h ragc ffc mgh cra ffc allocao llg prcdg lc. W hav o allowd for ral h papr. Ral op up h pobly ha ca of a ffc allocao of lc a ffc frm buy a lc from a l ffc frm. Such a gl raaco would alo b muually bfcal bcau for gv compor ffccy lvl a ffc frm ma mor prof ha a l ffc frm do. Howvr much l clar whhr uch a raaco fabl ca ohr frm ar alo allowd o raac o ha a qual ral mar would mrg. Aalyg h modl whl allowg for rllg ur ou o b qu complcad. Apar from h fac ha rllg omm o allowd or prohbvly coly hr aohr good rao o o codr h pobly of rllg. For ampl ca a dcrag qulbrum mo ffc frm oghr ma l prof ha la ffc frm do. If oghr wh rllg w allow frm o ma d paym o ohr frm for o llg hr lc o ca how ha h lc holdr (oghr ca oubd a offr of a mor ffc frm a h prof hy would loo wh h w frm comp h mar ar largr ha h prof h wcomr could ma. Hc a dcrag bddg qulbrum yld a -po ffc allocao from h prpcv of h coalo of wg frm. Th papr do o codr h quo whhr a opmal mcham h pr uao. Th a rg bu o-rval quo. Wha clar howvr ha h rul obad by Auubl (4 do o drcly apply a h pr modl do o afy h adard aumpo of h afflad valuao modl (a dcad abov. 7

20 Rfrc Auubl L. 4. A Effc Acdg Bd Auco for Mulpl Obc. Amrca Ecoomc Rvw 94 (5 p Bmor K. ad P. Klmprr.. Th Bgg Auco Evr: Th Sal of h Brh 3G Tlcom Lc. Ecoomc Joural C Bulow J. J. Gaooplo ad P. Klmprr Mul-mar Olgopoly: ragc ubu ad complm. Joural of Polcal Ecoomy 93 p Dm H Why Rgula Ul? Joural of Law ad Ecoomc p Gor J. 3. Bddg for h Fuur. Joural of Ecoomc Thory 8 ( p Hopp H. Ph. Jhl ad B. Moldovau. 6. Lc Auco ad Mar Srucur Joural of Ecoomc ad Maagm Sragy (forhcomg Ja M.C.W.. D Draad Kw? (A Idpd Evaluao of h Duch UMTS auco commod by h Duch Parlam. Ja M.C.W. 6. Auco a Coordao Dvc. Europa Ecoomc Rvw 5 p Ja M.C.W. ad V. Karamychv 6. Slco Effc Auco for Moopoly Rgh. Joural of Ecoomc Thory (forhcomg. Jhl P. ad B. Moldovau Sragc No-Parcpao. RAND Joural of Ecoomc 7( pp Jhl P. ad B. Moldovau.. Auco wh Dowram Iraco amog Buyr. Rad Joural of Ecoomc Jhl P. ad B. Moldovau. 3. A Ecoomc Prpcv o Auco. Ecoomc Polcy Jhl P. ad B. Moldovau. 6. Allocav ad Iformaoal Eral Auco ad Rlad Mcham R. Bludll W. Nwy ad T. Pro (d. Th Procdg of h 9h World Cogr of h Ecoomrc Socy Cambrdg Uvry Pr. Jhl P. B. Moldovau ad E. Sacch How (o o Sll Nuclar Wapo Amrca Ecoomc Rvw 86(4 pp

21 Klmprr P. a. Wha rally mar auco dg? Joural of Ecoomc Prpcv 6 p Klmprr P. b. How o o ru auco: h Europa 3G mobl lcom auco. Europa Ecoomc Rvw 46 p Laffo J.J. ad J. Trol.. A Thory of Icv Procurm ad Rgulao. MIT Pr. Mlgrom P. ad R. Wbr. 98. A Thory of Auco ad Compv Bddg. Ecoomrca 5 p

22 Appd Proof of Propoo. L u df Z o b h yp of h frm ha ubm h h hgh bd amog all (N - frm ohr ha frm.. Z h h hgh ordr ac amog. W do h drbuo of Z codoal o by ( ( Pr Z ad h corrpodg dy fuco by ( G g. Suppo ha all frm ohr ha follow h bddg fuco b ( ( ad Z a a valu. W codr a frm whch ha a co paramr ad whch bd b ( (y. If y frm loo h auco ad rcv o prof. If o h ohr had y > frm g a lc a h auco prc w b prof v ( ( ( b ( paramr ad a bd b ( (y ( ( ( y v ( b ( ( ( whch yld h codoal pcd o frm. Th ucodoal pcd prof of a frm wh co ( g ( V d. y ( Mamg V ( y wh rpc o y yld h fr-ordr codo arg ma( y y ( ( (.. b v. Th cary cod-ordr codo h ( ca ca b wr a v. Fally ( b ( v ( a crag fuco oly f v v. Suppo ow ha > ad v v >. I ordr o chc ha ( ( ( v ( dd a opmal bd w valua ( V ( V ( y b y : V v ( ( ( ( V y v ( v ( ( g ( y y v ( ( ( d g ( d Th how ha frm ha o profabl dvao. d for ay Proof of Lmma. If a frm g a lc ha a yp ad choo whra all compor hav h am yp h mar prof of frm ca b wr a follow: ( ( E.

23 Mamg h pro wh rpc o yld h followg fr-ordr codo: * * * ( ( ( ( E. Dffrag wh rpc o ad ag o accou ha frm of yp ad all ohr frm ar of yp ad choo * ( ad valuag h rulg pro a yld: ad (. Solvg hm oghr provd u wh h followg paral of h Nah qulbrum ragy ( a : ad ( ( (. Th ubug h pro o ( / ad ( ( / fally yld ( > ( ad ( ( (. ( I quay compo g ad whra prc compo for ubu > ad ubu ( > ad ( >. Thrfor ( > wh ar ragc wh ar ragc complm (.

24 Proof of Propoo. W wll how ha udr h codo of h propoo h cary c codo for a crag qulbrum fal a v v. To h d w fr calcula frm afrmar Nah qulbrum ragy ad paral drvav. Th w calcula frm valuao fuco v ad paral drvav. I h lm wh ( h Nah qulbrum ragy ca b wr a ( ( whr ( ( ad. Droppg argum all fuco valuad a ( h fr-ordr codo ( l E * * for dpd yp h fr-ordr appromao bcom ( ( ( ( E E lm A ( h fr-ordr codo mpl. Th dog E E lm ~ h fr-ordr codo mpl h followg wo quao drmg ad : ( ( ~ E. Solvg h ym yld ( E ~ Hc frm valuao fuco h lm ( v ( ca b wr a

25 ( ( ( ( E( ( ~ ( ( ( ( E ( ( v ( ( ~ E Fally h cary codo v ( ( ( v for a crag qulbrum o fal a > v f ( E v ( ( I accordac wh Lmma. Hc h abov qualy ca b wr a ~ E ( ( ( >. Calculag E ~ yld: ~. ~ E E lm ( lm ( f ( ( f ( d d lm f f (. ( ( f '( Th la qualy hold a a qualy f f ( > f ( (o ha '( f f ( (. ( Th d h proof. ad a a rc qualy f. Thrfor h cary c codo fal Proof of Propoo 4. Fr w drv frm afrmar Nah qulbrum ragy whch w do by ( ;ε o mpha dpdc o ε udr h aumpo ha h auco ag all hy follow a crag bddg fuco b ( (. Udr h aumpo ha h prof fuco wc cououly dffrabl ( ; ε cououly dffrabl wh rpc o all argum ad w rpr a a fr-ordr Talor pao. W how ha h bddg fuco b ( ( do o afy h cod ordr codo hc a crag ymmrc bddg qulbrum do o. Scod w rpa h prvou rc for a dcrag bddg fuco 3

26 b (- ( ad aaly codo udr whch b (- ( dd a qulbrum bddg fuco. From ow o w do by a yp of a frm. Typ of all ohr (wg ad loog frm ar ( ad l rpcvly ad w df ( ε λ / ( ( ε ( ε 4ε /( ε ad ( ( ε 4ε ( ε /( ε [ ] o ha for ay : ( ( ad ( (. Thu for gv codoal drbuo F ( ( Pr ha h uppor [ ( ( ] ad codoal drbuo F ( ( λ λ [ λ ( λ( ] whr boh λ ( ad ( ( ( / ε (. 5 λ ad ( ( / ε (. 5 λ. Dog ;ε λ ar boudd: 9 λ Pr ha h uppor lf ad paral drvav valuad a ; a ( ( ( ; ( ( ( ; ( ( ( ; ε ( 3 ( ( ; allow ;ε ad ad ( ;ε o b wr h fr-ordr appromao a ( ( ( 3 ( ; ε ( λ ε ad ( ( ( ( 3 ( ; ε ( λ λ ε. Droppg argum l all fuco valuad a w wr h fr-ordr codo 9 Th qual ca b obad by mmg λ ( ad mamg ( 4 λ wh rpc o.

27 l E ; ; K ε ε a follow: ( ( ε λ λ λ λ λ ε ε λ λ ε λ ε l l l E E E lm K L fr ε. I h ca h fr-ordr codo mpl whch for a gv prof fuco r c / yld h ymmrc afrmar Nah qulbrum ragy ( co 3: r c r r ;. I h ca r c r c ( r r c r r ( r r c r r ad ( c r. Ug w rwr h fr-ordr codo for ε ad λ a follow: ( ε ε E l lm 3 o ha ( ε ε H lm 3 whr w dfd fuco l E H ± ± ±. 5

28 Propr of ( H ( ± ar drvd h followg lmma whch prov afr h proof of h propoo. Lmma. L b drbud accordac wh h drbuo fuco F *. Th ( ± ( H for mall ε ca b wr a follow: ( ( ( ( ( ± H ( ± ε o( ε Ug Lmma w rwr ( 3 ( 3 ( ( ( a follow: Pluggg h pro o h fr-ordr codo yld: ( ( E l lm ε ( ( ( ( ( ( ( ( ε ( ( ( ( Solvg r /( c oghr wh h abov pro for ad ( 3 fally yld: ( ( r( ( c ( ( r( ( ( c ( ( ad ( r( 3 ( c ( (. ( Hc h afrmar Nah qulbrum ragy ( ;ε for mall ε ad ( ( ; ( (( ( ( ε ( c ( ( r ( ε. ( Th d h aaly of h afrmar ag of h gam. I ordr how ha a crag ymmrc bddg qulbrum do o for mall ε w calcula frm valuao fuco paramr rrco ( ( ; v ( v ( ; ε ad vrfy ha for gv. Ug h fr-ordr appromao for ( ( ; ε frm valuao fuco v ( ; ε h fr-ordr appromao ca b wr a follow: 6

29 ( l ( ( ( 3 ( ( ( ( 3 ( ( λ ε ( λ λ ε K ( ( ( 3 ( ( λ λ ε l ( ( ( H ( 3 ( lm ε ( ( E ( ( ; ε ( ; ε v E E ( r ε ( (( ( ( ε ( c ( ( ε ( l Subug pro for ad fuco v ;ε : ( fally yld h followg valuao v Bu h v ( r( c r ( r ( ( ; provdd r( r ( c ( r r r r( ( r ( r ( ( ( ( ε ( c ( ( ( r ( r ( ( r ( r ( r ( ( ( 3 3 r ( c ( r( 3 ( r ( r( r r. Th mpl ha ; ε for uffcly mall (bu rcly v pov ε o ha a crag ymmrc bddg qulbrum do o. Suppo ow ha hr a ymmrc dcrag qulbrum bddg fuco b (- (. I a mlar way a abov frm valuao fuco a follow: v Bu h v ( r( c r ( r ( ( ; for r r( v r ( c ( r r r ad ( ( ; ( c ( r r r ( v ( ; ε ca b wr r( ( r ( r ( ( ( ( ε ( c ( ( ( r ( r ( ( r ( r. ( r ( ( ( 3 3 r ( c ( r( 3 ( r ( r( r ( ( ( r ( r ( c ( ( ( r ( r ( ( r ( r By couy argum hr a ~ ε > o ha ( ( ; ε ad. ( v ( ; ε for all ε ~ ε ad fabl. Thrfor h propod fuco ( ( b ( v ( dd dcrag ad a uqu ymmrc qulbrum bddg fuco. v 7

30 Proof of Lmma. W do drbuo fuco a follow: ( ( F Pr F F ( Pr( ad > ( Pr( > ad h corrpodg d b f f f ( df ( / d ( df ( d ad / ( df ( d. / ( [ ] A ~ U ( follow ha for ( ( F ( Ad hrfor f Hc f ( ( ( ( ( ( ( ( ( ( ( f ( ( ( ( ( ( ( f ( f f f( ( ( f ( ( ( f f ( ( ( f ( ( ( f f : ( ( ( ( ( ( ε( ( ( ε( f f whr ad ar a dfd Proof of Propoo 4. W df ~ H a Wh wr a H ( ma( ( ( m( ( ( ( m( ( ( ( ( ( E( ad codr wo ca. ( ( ( l ( ~ H ( ( ( E H ( ( ( for (( H ( ~ ( P ( / Q ( l. Hc ca b whr 8

31 ad P ( f ( ( Q b Wh ( f( f ( ( ( ( ( ( ( ( ( ( f ( f ( d d ( ( ( ( ( ( ( ( f ( ( ( ( d d. ( (( ( ~ H ( ( (. Hc H ( ca b wr a ad H ( ( ( E H ( ( for ( ( ad ( H ( ~ ( P ( / Q ( whr l ( ( ( ( ( ( f ( ( ( f ( P d d ( ( ( ( f ( ( ( ( ( ( f ( ( ( ( ( ( Q d d. ~ for I ordr o valua appromao Hc ( H ( ad paral for mall valu of ε w u h 3 rd -ordr 3 3 ε ε ε ε o ε 8 6 o ha 3 3 ( ε 4ε ( ε ( ε ( ( ε o( ε 3 3 ( ε 4ε ( ε ( ε ( ( ε o( ε ( ( ε ( ( ε o( ε ( ( ε ( ( ε o( ε '( o( ad '( o( Th uform covrgc wh rpc o ( ( ( ( lm lm ( ε ad mpl h followg pro for ad : ( ( ε ( ( ε o( ε ad ( ( ε ( ( ε o( ε. W codr ca ε ε ad paraly. ( ( of h lm lm lm ( ε. ad 9

32 a L. Ug h abov Talor pao yld Q ( f ( f f ( ( ε Q Q P ( ' ( ε ( f ( f ' ( ( f ( ( ( ( f ε ( ε ( f ( ( ε ( Q ( f ( ( ε ( ε ( f ( f ( f ( ( P ' ( ad ( ε P Hc H H ( Q ( ( f ε ( ( ( f ( ( f ( ε. ( ( ( ε ( ( ( ( ( ( ( H. ad ( ( ( ( b L. A P P ad Q Q follow ha H ( ( ( H (. Smlarly a P P / P P / Q / Q / ad Q / Q / a ( follow ha H ( ( ( H ( ad H H. ( ( /( ( o( ( ad H ( / o( H / / ( Thu a H ( cououly dffrabl wh h paral b wr a ( ( ( ( ( H ( ε o( ε. Th pro for ( ad hrfor ca H mmdaly follow from H H. 3