Tax Evason and Audtng n a Fdral Economy * Svn Stöwhas Chrstan Traxlr Unvrsty of Munch Unvrsty of Munch Frst Draft Ths Vrson: Aprl 004 Plas do not quot or ct wthout prmsson of th Authors. Abstract Ths papr analyzs th rlaton btwn tax audtng and fscal qualzaton n th contxt of fscal comptton. W ncorporat a modl of tax vason by frms nto a standard tax comptton framwork whr rgonal govrnmnts us thr audt rats as a stratgc nstrumnt to ngag n fscal comptton. It s wll known that n such a stuaton fscal qualzaton can mtgat th nffcncs from tax comptton. W compar th rgons choc of audt polcs for thr dffrnt cass: A scnaro of unconfnd comptton wthout ntrrgonal transfrs, a scnaro wth a gross rvnu qualzaton schm and fnally, a scnaro wth nt rvnu qualzaton, whr not only th rvnus from taxaton but also th rgons audtng costs ar shard. W show that fscal comptton lads to audt rats whch ar nffcntly low from th prspctv of total rvnu maxmzaton. Whl n gnral gross rvnu qualzaton aggravats th nffcncy, nt rvnu qualzaton maks th dcntralzd choc of audtng polcs mor ffcnt. JEL-classfcaton cod: H6; H71; H77 Kywords: Tax Evason; Fscal Comptton; Fscal Equalzaton; Audtng * W would lk to thank Andras Hauflr for hlpful commnts and dscussons. Dpartmnt of Economcs, Akadmstr. 1/II, D-80799 Munch, Grmany. Emal: Svn.Stowhas@lrz.un-munchn.d Munch Graduat School of Economcs, Kaulbachstrass 45, D-80539 Munch, Grmany. Emal: Chrstan.Traxlr@lrz.un-munchn.d, Phon/Fax: +49 89 180-3303/-696; Corrspondng Author. 0
1. Introducton Much of th work n th thory and mprcs of taxaton taks tax collcton as gvn or costlssly xcutd, ffctvly assumng that tax authorts hav full nformaton about ndvduals or frms tax lablts. Of cours, ths s not vry ralstc. In fact, w obsrv sgnfcant lvls of tax vason n almost all dvlopd countrs. 1 In ordr to fght vason, tax authorts hav to spnd som rsourcs on audtng. If rvnu maxmzaton would b th prmary objctv, a tax plannr would choos xpndturs on audtng such that th margnal gans (lss tax vason, hghr rvnus from taxaton and dtctd vason) qual th margnal rsourc costs from an ncras n tax collcton fforts. Howvr, thr s som vdnc that tax authorts spnd lss on tax nforcmnt than what would b optmal. Lnk t al. (1998) as wll as a rport by th Arbtnhmrkammr Brmn (001) argu that som rgons n Grmany put too lttl ffort nto tax nvstgaton. In Blgum, thr rpatdly wr publc dscussons whch clamd that th Flmsh rgon ar too lax n thr tax nforcmnt polcy. Ancdotal vdnc coms from ncom tax vason n Grmany: In 1998, Grman tax authorts nspctd hundrds of banks whch wr undr suspcon to support tax vason by transfrrng non dclard ncom of thr customrs to bank accounts abroad. Although ths nspctons wr hghly succssful n dtctng cass of tax vason, thy ndd n a dsastr. In fact, du to th low numbr of tax nvstgators, authorts could only xamn a vry small fracton of all cass n dtal. As a rsult, most vadrs scapd punshmnt and a consdrabl amount of rvnu was lost for th govrnmnt. Clarly, on could hav prvntd such a stuaton by spndng mor on tax collcton,.. by ncrasng th numbr of tax nvstgators. If w tak ths vdnc as ndcatng an nffcntly low lvl of spndng on audtng, w hav to ask for th rasons that lad to ths nffcncy. 1 Accordng to stmatons of Schndr and Ernst (000) th nformal sctor accounts for approxmatly 10 to 30 prcnt of total GDP n OECD countrs and for mor than 50 prcnt n som lss dvlopd countrs. If w tak ths as a proxy for th dgr of tax vason, ths fgurs mphasz how mportant t s to allow for an mprfct tax collcton systm. Ths s th pont mad n th arly ltratur on tax vason, s for xampl Kolm (1973). For th cas of a wlfar maxmzng govrnmnt, Slmrod and Ytzhak (1987) argu that th optmal lvl of audtng ls blow ths pont. 1
For th Grman cas, thr ar at last two altrnatv xplanatons whch both orgnat from th fdral structur of th country. 3 On s tax comptton. At frst glanc, tax comptton btwn Grman rgons (Ländr) sms to b nglgbl. Th statutory tax rat s dtrmnd on th natonal lvl by th fdral govrnmnt and, unlk as n othr fdral countrs such as Swtzrland, rgons hav no drct nstrumnt for taxaton (.g. rgon spcfc surtaxs). On th othr hand, whl taxng powr ls n th hands of th fdral govrnmnt, tax collcton and thus th choc of th audtng polcy s n commsson of th rgonal govrnmnts. In th prsnc of tax vason, howvr, tax rvnus ar not dtrmnd xclusvly by th statutory tax rat st at th fdral lvl, but also by th nforcmnt polcy st at th rgonal lvl. By choosng th audtng probablty, ach rgon can dtrmn ts ffctv tax rat. Accordngly, th audt polcy bcoms an altrnatv stratgc tool for tax comptton and rgons mght compt va thr ffctv rathr than thr statutory tax rat. 4 As has bn shown by Crmr and Gahvar (000), countrs wll thn nd up wth lss than optmal audt rats, vn f (statutory) tax rats ar harmonzd. An altrnatv xplanaton for th rathr low audt rats n Grmany can b found n th fscal qualzaton schm. In th currnt Grman ntrstat transfr systm (Ländrfnanzausglch), rdstrbuton s basd on pr-capta gross rvnus rathr than on pr-capta nt rvnu (tax rvnus nt of tax nforcmnt costs). Ths mpls an asymmtrc tratmnt of audtng costs on th on hand and tax rvnus on th othr. Whl addtonal tax rvnu that mrgs from a hghr audt rat n on rgon has to b shard wth all othr rgons va th fscal qualzaton schm, th costs for an ncras n th audt rat hav to b fully born by th rgon tslf and cannot b dductd from th tax bas. Consdrng ths fact, t s straghtforward to argu that th currnt Grman fscal qualzaton schm dstorts th rgons ncntvs, causng nffcntly low xpndturs on tax collcton and hnc an audt frquncy that s too low. In th prsnc of tax comptton, howvr, a fscal qualzaton schm can mtgat harmful comptton. As has bn shown by Köthnbürgr (00), fscal qualzaton may nduc th rgons to ntrnalz thr fscal xtrnalts. If ths s th cas, fscal qualzaton could lad to 3 To som xtnt, our argumnts can also b appld on othr fdral countrs such as Canada, Span, Blgum and som Latn Amrcan Countrs. 4 Kn and Marchand (1997) and Snn (003, Chaptr ) dscrb a smlar problm whr xpndturs on publc nfrastructur can b usd as a stratgc substtut for taxs.
hghr tax rats whch ar an quvalnt to hghr audt frquncs n our cas. As a rsult, t s not clar whthr th currnt Grman fscal qualzaton schm tnds to dcras or ncras audt rats as compard to th cas wthout any fscal qualzaton. As has bn outlnd abov, th choc of th audt rat may b affctd by tax comptton and by th dsgn of th fscal qualzaton schm. Whl ths aspcts hav bn dscussd sparatly n th ltratur, th am of th prsnt papr s to combn both aspcts n a sngl framwork and to analyz how thr ntracton affcts th choc of th audt polcy. W us a modfd vrson of th standard modl of tax vason by th frm (.g., Crmr and Gahvar, 1993) and ncorporat t nto a typcal tax comptton sttng. In ach rgon of a fdral conomy a rprsntatv frm uss a fxd factor and mobl captal to produc a consumpton good. Frms hav to pay a tax on captal and dcd on how much of th taxs to vad. In makng ths dcsons, ach frm taks th vason costs as wll as th tax rat and th audtng probablty nto account. Local tax authorts choos thr audt rats (consdrng ts audtng costs and th mpact on th vason lvl) n ordr to maxmz nt rvnus. To kp th analyss tractabl, w tak th cntral govrnmnt polcy, spcally th statutory tax rat, as xognously gvn. For ths sttng, w compar th choc of rgon spcfc audt rats for thr dffrnt cass. Th frst scnaro dscrbs a stuaton wthout any fscal qualzaton schm whr th choc of th audt rat s only affctd by fscal comptton. W show that ths wll rsult n nffcnt low lvls of audtng, whch rsmbls th classcal tax comptton rsult (Zodrow and Mszkowsk, 1986) as wll as th fndngs of Crmr and Gahvar (000). In scnaro two w ntroduc a systm of gross rvnu qualzaton, whch capturs th cntral faturs of th currnt Grman transfr mchansm (th Ländrfnanzausglch). Fnally, scnaro thr dscrbs an altrnatv fscal qualzaton schm undr whch rdstrbuton s basd on a nt rvnu concpt, whr not only th rvnus from taxaton but also th rgons audtng costs gt shard. W show that, n gnral, a systm of gross rvnu qualzaton lads to vn lowr spndng on tax nforcmnt than th cas of unconfnd comptton. Howvr, w can dfn som condtons undr whch th postv ffct from th gross rvnu qualzaton th ntrnalzaton of fscal xtrnalts domnat th dstorton du to th asymmtrc tratmnt of rvnus and audtng costs. Undr ths condtons, gross rvnu qualzaton wll lad to hghr audt rats than undr scnaro on. W furthr show, that mantanng th 3
dgr of ntrrgonal rdstrbuton a systm of nt rvnu qualzaton would unambguously ncras audt rats n comparson to th cas of gross rvnu qualzaton. Th papr s organsd as follows: Scton prsnts th basc modl. Scton 3 provds us wth th bnchmark scnaros whr tax authorts maxmz thr rvnu n th absnc of a fscal qualzaton schm. Nxt, w analyz th rsults undr th two dffrnt dsgns of a fscal qualzaton schm and compar th choc of th audt rat for svral cass. In Scton 5 w sum up and dscuss our rsults. Scton 6 concluds wth som polcy mplcatons.. Basc Modl In ths scton w charactrz th prvat sctor of th modl. Aftr th cntral and rgonal govrnmnt st th tax rat and th local nforcmnt polcy (as dscussd n th followng sctons), frms and consumrs mak thr dcsons, takng th polcy varabls as gvn. 5.1 Frms Consdr an conomy wth n rgons, ach nhabtd by a sngl rprsntatv houshold. In a prfctly compttv ndustry frms produc on homognous prvat good (numrar). Th producton procss n ach rgon uss prfctly mobl captal k and a fxd, mmobl factor. To smplfy th analyss, w assum that all rgons ar ndowd wth th sam amount of th fxd factor, normalzng t to unty. Ths can also b ntrprtd as only consdrng rgons of th sam sz. Th tchnology s allowd to dffr btwn rgons and s rprsntd by a standard noclasscal producton functon f( k ), whr th fxd factor s supprssd. Frms hav to pay a unt tax on captal at a rat t. Ths statutory tax rat s qual for th whol conomy. Howvr, ach frm can try to vad taxs by concalng a shar of th captal mployd. To concal nputs rqurs th us of rsourcs by th frm. Followng th ltratur, w assum that th costs of vason ar convx n and lnar n th tax bas: 6 g ( ) k wth g > 0 and g > 0. Wth a 5 Hnc w omt commtmnt problms. For a gam thortcal analyss wth no commtmnt s Rnganum and Wld (1986). 6 Assumng that tax vason costs ncras lnarly n th tax bas smplfs th analyss a lot. Wth ths spcfcaton th frms vason dcson wll b ndpndnt of th amount of captal mployd 4
probablty p th vason gts dtctd, and th frm has to pay th statutory taxs plus a fn that s proportonal to th taxs vadd (compar Ytzhak, 1974). Th pnalty rat, dnotd by s 1, s qual for all rgons. Wth probablty 1 p th frm gts away wth th vason and pays only taxs on th dclard amount of captal. Expctd profts π ar dfnd as π = f( k) rk g( ) k p( tk + ( s 1) tk) ( 1 p )( 1 ) tk, whr r s th factor prc for captal. W can smplfy ths xprsson to π = f ( k ) ( r + g( ) + t ) k wth t t( 1 + ps), (1) whr t s th xpctd or ffctv tax rat 7 n rgon. A rsk nutral frm now chooss k and to maxmz ts xpctd proft. Th frst ordr condtons ar gvn by f ( k ) = r + g( ) + t, () g ( ) = ( 1 ps) t. (3) Equaton (3) dfns a frm s optmal vason lvl. For th rst of th papr, w wll assum that thr s an ntror soluton wth vason n qulbrum (.., ps < 1 has to hold). From ths quatons on can asly drv two basc rsults. Frst, frms wll concal mor f th statutory tax rat ncrass or th dtcton probablty dcrass a standard rsult for modls of frm tax vason. Scond, an ncras n th audt rat wll ras (pr unt) captal costs and hnc dcras captal dmand. Ths s analogous to th ffct of a tax ncras on mobl captal n tax comptton modls. 8. Captal Markt In ach rgon a rprsntatv houshold s ndowd wth captal k (and th mmobl factor). Th total captal supply to th conomy k s fxd and markt clarng rqurs n n k = k = k. (4) = 1 = 1 (compar quaton 3). Our qualtatv rsults drvd n th followng hold also undr lss rstrctv assumptons. 7 Not that rvnus from dtctd vason (ncludng pnalts) ar also ncludd n th dfnton of th ffctv tax rat. 8 Ths rsults ar drvd n Lmma 1 n th Appndx. 5
Indvduals nvst thr captal n a larg numbr of frms dstrbutd ovr th whol conomy. By holdng a fully dvrsfd portfolo, thy avod th potntal rsks assocatd wth tax vason by frms. In th captal markt qulbrum thr wll b a captal allocaton whch quats th nt rturn, so th arbtrag condton f ( k ) g( ) t = r (5) has to b fulflld for all rgons. 3. Rgonal Govrnmnts Polcy wthout Fscal Equalzaton In ths and th subsqunt sctons w analys th publc sctor s dcson makng. As a bnchmark scnaro w consdr a fdral conomy wthout any ntrrgonal rdstrbuton. Our analyss concntrats on th dcson problm of a typcal rgonal govrnmnt whch taks th polcy varabls st by th cntral govrnmnt,.. th tax and pnalty rat as wll as th rdstrbuton paramtr (ntroducd n th nxt scton), as xognously gvn. W assum rvnu maxmzng rgonal govrnmnts (or tax authorts), whch s standard n th ltratur on tax vason. 9 Ths can b justfd thr by a Lvathan govrnmnt or by a wlfar maxmzng govrnmnt n th cas of consumrs whch rcv sgnfcantly hghr margnal utlts from publc than from prvat consumpton (s g. Kanbur and Kn, 1993). Th only polcy varabl controlld by th rgonal govrnmnt s th audt rat, whch dtrmns th captal allocaton and hnc th rgons rvnus from taxs and pnalts. To provd a crtan audt frquncy th rgon has to ncur som costs. Ths costs ar convx n th dtcton probablty and do also dpnd on th sz of th tax bas,.. th lvl of frms captal nputs. 10 Analogous to th vason costs of th frm w assum th govrnmnt s audtng costs to b lnar n th captal bas 11. Th total dtcton costs of a rgon ar thn dfnd as cp ( ) k, whr 9 In Scton 5 w dscuss th cas of a wlfar maxmzng govrnmnts. 10 In th prsnc of a plannr who prfctly knows th sz of th tax bas, t s not clar at frst sght, why tax vason should b possbl. Howvr, f thr ar many producrs and th tax authorty dos not know th xact dstrbuton of th captal among th frms, tax vason s a possbl outcom. 11 On could asly thnk of a dtcton tchnology charactrzd by margnal audtng costs whch ar ncrasng (dcrasng) n th tax bas. Ths would tnd to lowr (ras) th dtcton probablty n rgons wth a hgh lvl of captal mployd. Howvr, non-lnarty would not affct our rsults n a qualtatv way. 6
c(p ) dnots th audtng costs pr unt of captal (for a gvn p ), wth c > 0, c > 0 and c (0) = 0. In th absnc of an ntrrgonal dstrbuton mchansm, th nt rvnu of rgon s gvn by k ( t c( p )), (7) wth th ffctv tax rat t as dfnd abov. 3.1 Closd Rgons Lt us brfly consdr th cas of mmobl captal. Th rvnu would b gvn by (7) wth k = k and a rvnu maxmzng plannr would st a dtcton probablty accordng to th condton t c = ( p ), (8) whr t p s drvd n Lmma n th Appndx. 1 Th LHS of (8) dpcts th margnal bnfts (pr unt of captal) of a hghr audt rat: Spndng mor ffort on audtng wll ncras ffctv taxaton and rvnus snc a smallr shar of th tax bas wll b concald. W can stat that, n th absnc of captal moblty, ach rgon sts ts audt rat to quat margnal bnfts pr unt of captal and margnal audtng costs pr unt of captal. 13 3. Opn Rgons Lt us now allow for captal moblty btwn rgons. Whl th statutory tax rat n th conomy s harmonzd and hnc thr s no scop for standard tax comptton btwn rgons, th dtcton polcy acts as a stratgc substtut for th rgonal polcymakr: 14 By rducng ts audtng probablty, a rgon can lowr th ffctv tax rat. Ths wll rduc captal costs and attract mobl captal from othr rgons. Takng th captal markt rsponss nto account and consdrng th polcy of th othr rgons as fxd, th frst ordr condton thn bcoms 1 Th scond ordr condton s fulflld snc c > 0 and t s concav n p (shown n Lmma n th Appndx). For th rst of th papr, scond ordr condtons ar dscussd n th Appndx A.1. 13 Condton (8) s ndpndnt of th amount of captal, snc captal supply s fxd n th closd conomy. Clarly ths ndpndnc s also drvn by th assumpton that frms vason costs as wll as total audtng costs ar lnar n k. 14 Th argumnt s comparabl to th fndng that comptton va publc xpndturs (.g. on nfrastructur) can srv as a substtut for comptton n tax rats. S Kn and Marchand (1997), and Snn (003, Chaptr ). 7
t k k + ( t c( p )) = c ( p ) k. (9) In ordr to as notaton and nsur comparablty wth th rsults drvd n th nxt sctons, w rarrang condton (9) and gt t k + = k t MC k wth MC c ( p) k + c( p) p, (10) whr MC dnots th xtndd margnal costs. 15 Not that k s a functon of th dtcton polcs of all rgons, k( p, p ), whr th vctor p s dfnd as p = ( p1,..., p 1, p+ 1,..., pn ). Th frst ordr condtons drvd for th uncoordnatd choc thn dtrmn a systm of racton functons p = R( p ) and th qulbrum of th Cournot-Nash gam s gvn by th ntrscton of ths racton functons. 16 Compard to th cas of a closd conomy, w now fnd th total margnal costs on th RHS of condton (9). Th margnal bnft on th LHS conssts of two ffcts: Th margnal ncras n th ffctv tax rat (wghtd wth th amount of captal mployd) and th margnal captal outflow whch follows from an ncras n p (wghtd wth th nt rvnu pr unt of captal). Whl ths captal outflows clarly lowr th margnal bnft of th rgon, thy nlarg th forgn tax bas and, ctrs parbus, ras nt rvnus n th rst of th conomy. Snc rgonal dcson makrs gnor ths wll known fscal xtrnalts, thy st dtcton probablts at an nffcntly low lvl from th prspctv of th whol conomy. Ths wll, n turn, lad to nffcntly hgh lvls of tax vason by th frms. Bfor w summarz ths rsults, w want to clarfy th concpt of nffcncy n our framwork wth rvnu maxmzng govrnmnts: Dfnton 1. () Dtcton probablts ar st nffcntly f a cntral plannr could ncras total nt rvnus n th conomy through a chang n th audt rat(s) (n on or mor rgons). () Othrws, dtcton probablts ar st ffcntly. () Th frms vason rats ar calld (n)ffcnt f thy ar assocatd wth an (n)ffcnt dtcton probablty. 15 Not that w ncorporat a partal ffct from th captal outflow th rducton n total audtng costs du to a dcras n th tax bas nto th dfnton of 8 MC. 16 In scton A. n th Appndx w show that for all scnaros consdrd n th papr, th rgons audt rats ar stratgc complmnts.
W nglct standard wlfar aspcts n ths dfnton. Hnc, a mor ffcnt audt rat just mans that ths polcy wll nduc hghr total nt rvnus. Morovr, w call a frm s vason lvl nffcnt f t s assocatd wth an nffcnt dtcton rat (albt frms always choos an vason rat whch s optmal for thm,.. proft maxmzng; Compar condton (3)). Usng Dfnton 1 w can stat Proposton 1. In an conomy wth opn rgons and wthout ntrrgonal rdstrbuton, th uncoordnatd choc of dtcton probablts by rvnu maxmzng govrnmnts wll lad to an nffcntly low lvl of dtcton fforts and, thrfor, to an nffcntly hgh lvl of tax vason. Proposton 1 s smlar to th classcal undrprovson rsult n tax comptton (s Zodrow and Mszkowsky, 1986) whr th uncoordnatd choc of tax rats lads to nffcntly low lvls of taxaton. Snc n our modl th statutory tax rat s harmonzd, t s th dtcton rat whch srvs th rgons as stratgc substtut. In fact, w obsrv tax comptton, but wth an altrnatv nstrumnt. Ths rsult rsmbls th fndngs by Crmr and Gahvar (000) who study tax comptton wth audt probablts as an altrnatv nstrumnt. Thy show that comptton n tax and audt rats lads to an undrprovson of publc goods. In addton, thy dmonstrat that harmonzng tax rats alon wll not solv th undrprovson problm. Barrd from comptng n th statutory tax rat, rgons lowr thr ffctv tax rats through cuttng audt rats. As a rsult, tax vason ncrass. Proposton 1 dscrbs our bnchmark rsult whch w wll compar to th rgons uncoordnatd choc n th prsnc of two dffrnt fscal qualzaton mchansms. Bfor w com to ths cass n scton 4, lt us tak a look at th cntralzd soluton to th problm. 3.3 Cntralzd Audtng Polcy Suppos that a cntral plannr would choos a dtcton polcy for ach rgon n ordr to maxmz th sum of all rvnus, max k( t c( p) ). p1,..., p n n = 1 Th n frst ordr condtons ar gvn by t k kj k c ( p) + ( t c( p) ) + ( tj c( pj) ) = 0. (11) j 9
Ths condtons thn charactrz a st of ffcnt audt rats for th conomy. Whl th frst and scond trm also show up n quaton (9), th last trm rprsnts th rvnu spllovrs cratd by a rgon s dtcton polcy. 17 For th cas of dntcal rgons, th scond and thrd trm n condton (11) cancl out and w gt t c = ( p ), (1) whch s quvalnt to th frst ordr condton n th cas of a closd conomy. Morovr, on can show that vn wth rgonal dffrncs n producton tchnologs, ths s th only soluton whch satsfs (11). 18 Th ntuton for ths rsult s clar: Snc th problm of maxmzng th nt rvnu can b solvd n pr unt of captal -trms, htrognty wth rspct to producton tchnologs whch bascally lads to an asymmtrc captal allocaton dos not altr th rsult drvd undr prfct symmtry. In addton, ths shows that thr s a unqu ffcnt audt rat for th whol conomy, vn n th cas of dffrnt producton tchnologs. Howvr, on cannot ntrprt ths as a strong pont n favour of a cntral plannr mplmntng a unqu ffcnt audtng polcy, snc th rsult s drvn by two smplfyng assumptons: Frst, w assum th frms vason costs as wll as th audtng costs of th rgons to b lnar n th tax bas. 19 Scond, w also mak th assumpton that audtng costs pr unt of captal ar dntcal across rgons. (On can asly s from (1) that htrognous costs would rsult n dffrnt audt rats.) 4. Rgonal Govrnmnts Polcy wth Fscal Equalzaton In ths scton, w ntroduc ntrrgonal rdstrbuton systms. W analys two dffrnt fscal qualzaton schms: Gross rvnu qualzaton and nt rvnu qualzaton. 0 17 Not that condton (11) corrsponds to th cas whr on would provd ach rgon wth a corrctv subsdy n ordr to ntrnalz ts fscal xtrnalts. Ths rplcats th rsult of Wldasn (1989) for our framwork of fscal comptton n audt rats. 18 A prov s gvn n scton A.3 n th Appndx. 19 Thrfor, nthr th vason dcson by frms nor th rgons choc of a dtcton polcy s affctd by th amount of captal mployd. Compar Lmma 1 n th Appndx and condton (1). 0 In a comparabl framwork, Köthnbürgr (00) studs tax comptton undr tax rvnu as wll as tax bas qualzaton. 10
4.1 Gross Rvnu Equalzaton Consdr th followng smpl rdstrbuton mchansm: Each of th n rgons contrbuts a shar α (0 < α 1) of ts gross rvnus tk to th rdstrbuton systm and rcvs a shar 1 n of th total rvnus dstrbutd, α tk j j. 1 Ths fscal qualzaton schm capturs a cntral fatur of th currnt Grman ntrstat transfr systm (Ländrfnanzausglch): Whl rvnus obtand undr a gvn dtcton polcy hav to b shard wth all th othr rgons, th costs for mantanng a crtan audt rat hav to b fully born by th rgon tslf. Th nt rvnu of a rgon s thn gvn by n α ( 1 α) tk cp ( ) k + kt j j n. (13) j= 1 Th frst ordr condton for th uncoordnatd choc of a rvnu maxmzng rgonal govrnmnt bcoms n t k α kj t ( 1 α) k + t + tj + k = MC n j= 1 p p (14) wth MC as dfnd abov. Th frst trm on th LHS dpcts th dstorton ntroducd by th fscal qualzaton schm, whras th scond trm shows that a part of th fscal spllovrs wll b ntrnalzd va th rdstrbuton mchansm. For a propr analyss, w hav to dstngush btwn an conomy wth many small or fw larg opn rgons. A) Small opn rgons In th cas of many small rgons (n ), th mpact of a sngl rgon s dtcton polcy on th tax bas n th rst of th conomy bcoms nglgbl. Hnc, th scond trm on th LHS of (14) vanshs and th frst ordr condton bcoms t k ( 1 α) k + t = MC. (14a) If w compar (14a) wth condton (10), th rsult n th scnaro wthout any ntrrgonal rdstrbuton, on can asly s that th gross rvnu qualzaton mchansm dstorts th rgon s choc of dtcton fforts. Whl th margnal costs MC ar unaffctd, th rdstrbuton systm clarly rducs th margnal gans from an ncras n th audt rat. Ths s th cas, snc fscal qualzaton lads to 1 Apparntly th mchansm s budgt balancng for any α. 11
an mplct taxaton of a rgon s gross rvnu. Hnc, rgons wll unambguously choos a lowr audt frquncy than n th absnc of fscal qualzaton. Furthrmor, t s straghtforward to show that th audt rat wll dcras as α ncrass. Proposton. () Undr a systm of gross rvnu qualzaton, a small opn rgon maxmzng ts nt rvnu wll choos a dtcton probablty whch s vn lowr than n th cas wthout fscal qualzaton. () Th dtcton probablty s dcrasng n α. Ths rsult s quvalnt to Proposton 3 n Köthnbürgr (00). As n hs analyss, th nffcncy du to th dstortd ncntvs adds to th nffcncy du to fscal comptton: Dtcton fforts wll b rducd furthr blow th ffcnt lvl and th amount of vason wll b hghr than n our rfrnc cas wthout any ntrrgonal rdstrbuton systm. B) Larg opn rgons Snc th dcsonmakr of a larg rgon taks nto account th mpact of th local dtcton polcy on th total rvnus dstrbutd, w can rarrang condton (14) and gt 1 ( 1 α( 1 )) k k t t MC n + + = p p n. (14b) p t k j α j j As n th cas of small opn rgons, gross rvnu qualzaton ntroducs a dstorton. Howvr, snc larg rgons consdr th stratgc ffct of thr polcs th mpact of thr audt rat on total rvnus rdstrbutd th dstorton bcoms 1 1 1 > 1 α. Statd dffrntly, for larg rgons th mplct smallr: α( n ) taxaton of gross rvnus drops blow th rat α obsrvd for small rgons. Furthrmor, as rflctd n th scond trm on th LHS of condton (14b), a part of th fscal xtrnalts 3 gts ntrnalzd va th rdstrbuton mchansm. Agan, ths s du to th fact that larg rgons n contrast to small ons tak nto account th stratgc ffct of thr polcs. If w compar ths cas wth th bnchmark scnaro, th ntroducton of th gross rvnu qualzaton schm shows an ambguous ffct on th choc of audt Ths rsult, whch prvals n a smlar way n Köthnbürgr (00), s also confrmd n an mprcal study for Grmany by Bartt t al. (00). k 3 j Not that th total spllovr would b ( t j c( p j )). j 1 1
rats. Whl th mplct taxaton of gross rvnus tnds to lowr audtng fforts, th ntrnalzaton of th spllovrs wll act n th othr drcton. Comparng condton (14b) and (10), w can fnd a thrshold lvl such that, for n kj t j j < nˆ 1 + t k k + t, (15) th scond ffct domnats. W sum up ths rsults n Proposton 3. Suppos a rvnu maxmzng rgonal govrnmnt undr gross rvnu qualzaton. () For vry larg opn rgon thr xsts a thrshold n ˆ such that for gvn audtng polcs p n th othr rgons and n < nˆ ( n > nˆ ), th dtcton probablty chosn undr a systm of gross rvnu qualzaton wll b hghr (lowr) than n th cas wthout fscal qualzaton. () For n < nˆ ( n > nˆ ) th dtcton probablty s ncrasng (dcrasng) n α. () Th dtcton probablty s dcrasng n n. (v) If n nˆ mn, wth nˆmn mn { nˆ1,..., nˆn }, th qulbrum dtcton probablts undr a systm of gross rvnu qualzaton wll b unambguously hghr than n th cas wthout fscal qualzaton. Whl gross rvnu qualzaton has an unambguously ngatv ffct n th cas of small rgons (s Proposton ), t could ncras audt rats for larg opn rgons. 4 Ths rsult s to som xtnt surprsng. At frst glanc, a gross rvnu qualzaton schm sms to lad to a clar nffcncy: Th asymmtrc tratmnt of audtng costs on th on hand, and tax rvnus on th othr, unambguously dstorts a rgon s choc of th audtng ffort. Howvr, n th prsnc of fscal comptton, th rdstrbuton systm has a furthr ffct: It works as a corrctv subsdy and nducs (larg) rgons to ntrnalz part of th fscal xtrnalts. Whl th dstorton from th mplct taxaton tnds to lowr audt rats, th corrctv subsdy works n th othr drcton. If thr ar n < nˆ jursdctons, th scond ffct domnats th frst on and gross rvnu qualzaton maks th dcntralzd choc mor ffcnt, compard to th bnchmark scnaro. Th ntuton for ths rsult s clar-cut: If th numbr of rgons dmnshs, th mplct taxaton of gross rvnus bcoms smallr and th dgr of ntrnalzaton gts hghr - rrspctv of α (s condton (14b)). 4 Anothr ssu w do not tak up hr, s th fact that larg rgons fac a lowr captal lastcty than small rgons bcaus of a dffrnt mpact of thr dtcton polcs on th ntrst rat. Compar Wldasn (1988), Bucovtsky (1991). 13
Barng ths n mnd, also th scond part of Proposton 3 bcoms obvous: If th gross qualzaton systm nducs a rgon to ras (lowr) ts audtng ffort for th cas of n < nˆ ( n nˆ > ), an ncras n α furthr amplfs th ffcncy nhancng (rducng) ffct of th mchansm. Th thrd part of th proposton holds, snc mor rgons wll always ncras th dstorton of th rdstrbuton and dcras th dgr of ntrnalzaton. In part (v) of th proposton w just us th rsult from () to mak a statmnt about th qulbrum audt rats. It s straghtforward that audt rats must ncras f all racton functons shft upward. Howvr, snc n n s only a suffcnt but not a ncssary condton, thr s scop for an ˆmn ncras n audt rats, vn f th condton s not fulflld. 4. Nt Rvnu Equalzaton Lt us now ntroduc an altrnatv systm of ntrrgonal rdstrbuton. Instad of gross rvnu qualzaton, w consdr a mchansm whch s basd on nt rvnu qualzaton. Each rgon contrbuts a shar 0 < α 1 of ts nt rvnus tax rvnus nt of audtng costs and rcvs a shar 1 n of th total rvnus dstrbutd, α kj ( tj c( pj) ). Wth ths mchansm, th rvnu of a rgon bcoms n α ( 1 α) k( t cp ( ) ) + kj ( tj cp ( j) ) n. (16) j= 1 Th frst ordr condton s gvn by n t k α kj t ( 1 α) k t MC ( tj c( pj) ) k c ( p) + + + = 0 n p p. (17) j = 1 As bfor, w dscuss ths condton sparatly for th cass of small and larg opn rgons. A) Small opn rgons As undr gross rvnu qualzaton th scond trm on th LHS of condton (17) vanshs for th cas of many small rgons. W can rwrt (17) as t k + = k t MC, (17a) whch s dntcal to condton (10) n th bnchmark scnaro. Whl gross rvnu qualzaton dstorts th audtng polcy of small rgons (Proposton ), th dstorton dsappars undr nt rvnu qualzaton. W can stat 14
Proposton 4. Undr a systm of nt rvnu qualzaton wth α < 1, a small opn rgon maxmzng ts nt rvnu wll choos th sam audt rat as n th bnchmark scnaro wthout fscal qualzaton. Th ntuton for ths rsult s clar: In contrast to th scnaro of gross rvnu qualzaton, thr s no asymmtrc tratmnt of rvnus and audtng costs undr nt rvnu sharng. Thrfor, rvnu maxmzng rgons would unambguously choos a hghr audt rat aftr th chang from gross to nt rvnu qualzaton. Th only nffcncy rmanng arss from th comptton for th mobl tax bas. As n th bnchmark scnaro, audtng fforts wll b nffcntly low and th vason lvl of th frms wll b nffcntly hgh. An ntrstng pont to not s that Proposton 4 holds tru for any α < 1. Snc ntrrgonal rdstrbuton dos not ntroduc any dstorton, thr s no qutyffcncy trad off n th choc of α. B) Larg opn Rgons As bfor, th govrnmnt of a larg opn rgon consdrs th mpact of ts audtng polcy on th tax bas n th rst of th conomy. W can rarrang condton (17) and gt t k kj k + t + β ( tj c( p) ) = MC wth j β α n ( 1 α) + α. (17b) As n th cas of small opn rgons dscussd abov, rdstrbuton va nt rvnu qualzaton schm dos not dstort th dcntralzd choc of dtcton probablts. An nffcncy du to th fscal comptton btwn rgons stll rmans. Howvr, snc th dcsonmakr of a larg opn rgon ncorporats th stratgc ffct of ts audtng polcy, a fracton β of th fscal xtrnalty gts ntrnalzd. If w contrast th mpact of gross rvnu qualzaton wth that of nt rvnu qualzaton for larg rgons, th lattr systm ntroducs no dstorton but a hghr corrctv subsdy. 5 Hnc, w can stat 5 Although β > α n th comparson wth th cas of gross rvnu qualzaton (quaton (14b)) s not that clar, snc th rdstrbuton volum s smallr n th cas of nt rvnu qualzaton. Nvrthlss w wll argu, that (for a gvn α ) th dgr of th xtrnalty whch gts ntrnalzd s hghr wth nt than wth gross rvnu sharng. Ths wll hold f audtng costs c(p ) pr unt ar rathr small rlatv to th pr unt ffctv tax rat. 15
Proposton 5. () Undr a systm of nt rvnu qualzaton, a larg opn rgon maxmzng ts rvnu wll st an audt rat whch s hghr than th rat chosn undr a gross rvnu qualzaton mchansm, and () also hghr than th rat chosn n th bnchmark scnaro wthout any fscal qualzaton. () Furthrmor, th dtcton probablty s ncrasng n α, and (v) for α = 1, th audtng polcy s st ffcntly. (v) Th dtcton probablty s dcrasng n n. Proposton 5 s th man rsult of our analyss: For larg opn rgons, a nt rvnu qualzaton schm (partly) ntrnalzs fscal xtrnalts wthout ntroducng any dstorton. Hnc, undr ths mchansm, th dcntralzd choc of dtcton polcs wll unambguously rsult n mor ffcnt (.. hghr) audtng fforts as compard to th choc undr both, th cas of a gross rvnu qualzaton schm and th cas wthout any ntrrgonal rdstrbuton. Incorporatng audtng costs nto th rdstrbuton mchansm wll, f at all, only slghtly rduc th amount rdstrbutd 6 (for a gvn α ) but ncras dtcton probablts. Frms would vad lss and tax rvnus would ncras. Snc a hghr dgr of rdstrbuton dos not nduc any dstorton but ncrass th dgr of ntrnalzaton, audtng fforts wll ncras as α ncrass. Hnc, as n Proposton 3 (), thr s no qutyffcncy trad-off. For th lmt cas of α = 1, th nt rvnu sharng rul provds a mchansm to mplmnt an ffcnt soluton (accordng to our noton of ffcncy from Dfnton 1). Part (v) of th proposton s straghtforward and follows th ntuton of Proposton 3 (), dscussd abov. 5. Dscusson Lt us brfly sum up th rsults prsntd so far. In our bnchmark scnaro, w showd that rgons choos nffcntly low audtng fforts, n ordr to attract mobl captal. Whl n such a sttng fscal qualzaton can mtgat ths nffcncy, ths s not qut tru n th cas of a gross rvnu qualzaton schm. Hr, th asymmtrc tratmnt of tax rvnus and costs crats an addtonal nffcncy. Howvr, f w swtch to nt rvnu qualzaton,.., f w ncorporat th audtng costs nto th sharng rul, ths wll rmov th dstorton nhrnt to th gross sharng systm. Snc nt rvnu qualzaton also provds for a 6 W assum, ralstcally, that t c( p ). 16
mchansm to ncorporat (part of) th fscal xtrnalty, t domnats both, th cas wthout fscal qualzaton as wll as th cas of gross rvnu sharng. By now, thr ar two mportant qustons w hav not addrssd. Frst, w want to ask f our rsults xtnds for th cas of wlfar maxmzng govrnmnts. Ths s quvalnt to th quston whthr or not our dfnton of ffcncy (Dfnton 1) s msladng. Scond, on mght ask f th cntral govrnmnt can us ts polcy tools, spcally th statutory tax rat, to countrbalanc th rgons nffcntly low dtcton probablts (.., th nffcntly low ffctv tax rats). 5.1 Wlfar Maxmzaton 7 On can asly xtnd th modl prsntd n ths papr to a framwork for a standard wlfar analyss. W add consumrs, who rcv ncom from thr ndowmnts of captal and th fxd factor and who drv utlty from prvat consumpton as wll as from a rgonal publc good. A wlfar maxmzng rgonal plannr would thn choos an audt rat (or quvalntly, a lvl of th publc good) that maxmzs th utlty of th rprsntatv consumr. Th most mportant dffrnc to th cas of rvnu maxmzaton s th apparanc of a furthr xtrnalty. A wlfar maxmzng plannr wll now also consdr th pcunary ffcts of ts dtcton polcy: a hghr audt rat wll rduc th captal ncom as wll as th fxd factor ncom of th rgons consumr. Th lattr ffct arss, snc a captal outflow wll lowr th margnal productvty of th fxd factor. Howvr, th loss of captal n on rgon wll lad to an nflow of captal and thrfor to hghr fxd factor ncoms n th rst of th conomy. On th othr hand, all captal ownrs, domstc as wll as forgn, wll rcv lowr ncom from captal snc an ncras n audt rats wll dcras th ntrst rat. Snc a rgonal plannr dos not tak nto account th mpact on factor ncoms n th rst of th conomy, thr s a pcunary xtrnalty (whch can b thr postv or ngatv). Hnc, n th cas of unconfnd fscal comptton, thr ar now two xtrnalts a fscal and a pcunary spllovr whch rndr th dcntralzd choc of audt rats nffcnt (n trms of wlfar maxmzaton). How dos th ntroducton of gross or nt rvnu qualzaton affct ffcncy n ths cas? As dscussd abov, rvnu qualzaton provds a mchansm whch maks th dcntralzd plannr ntrnalz (a part of) th fscal xtrnalts. Ths 7 In an xtnson to ths papr (avalabl upon rqust) w provd for a formal modl wth rgonal govrnmnts maxmzng wlfar nstad of rvnus. 17
also holds for th wlfar maxmzng plannr. Th pcunary xtrnalty, howvr, wll not gt ntrnalzd va an ntrrgonal rdstrbuton mchansm. It s not possbl to rach th frst bst soluton. Nvrthlss, our man rsult of scton 4 holds n th wlfar maxmzaton framwork: th dcntralzd choc undr th systm of nt rvnu qualzaton wll lad to mor ffcnt audtng polcs than undr gross rvnu qualzaton. 8 Agan, ths s du to th asymmtrc tratmnt of tax rvnus and audtng costs nhrnt n th gross rvnu sharng schm. 5. Cntral Govrnmnt Polcy Tools Snc w do not modl th dcson of a cntral govrnmnt, w can not quantfy whthr or not an nffcntly low ffctv tax rat wll prval. Th cntral govrnmnt mght countrbalanc nffcntly low dtcton probablts by rasng th statutory tax rat. Howvr, a comparatv statc analyss shows that ths polcy s not ncssarly fasbl: applyng th mplct functon thorm on th frst ordr condtons of th dffrnt scnaros, w gt p 0 t for all th cass consdrd. 9 Hnc, an ncras n th statutory tax rat mght furthr rduc audtng fforts. Th othr polcy nstrumnt avalabl to th cntral govrnmnt s th pnalty rat. Snc punshmnt s costlss and nforcs a lowr lvl of vason, a cntral plannr could st s and thr would b no vason at all (s Kolm, 1973). Howvr, strong pnalts ar probably not fasbl: If a frm would go bankrupt bcaus of a vry svr punshmnt, th pnalty may not b crdbl. Kpng ths rstrcton n mnd and assumng that s s fxd at som crdbl fnt rat (whr ps < 1 holds) appars plausbl. 6. Concluson In ths papr, w hav analyzd th dcntralzd choc of audt rats, assumng that rgonal plannrs maxmz thr tax rvnus. Whl th statutory tax rat, modld as a sourc tax on captal, s xognously fxd by th cntral govrnmnt, rgons 8 In th xtnson w show undr whch condtons th nt rvnu qualzaton wll lad to mor ffcnt audt rats compard to th cas wthout any ntrrgonal rdstrbuton. 9 Th comparatv statc rsults ar drvd n scton A.4 n th Appndx. 18
dcd on how much to spnd on tax audtng. By sttng a crtan dtcton probablty, rgons dtrmn th ffctv tax rat frms hav to pay. Thus, rgonal audtng polcs dtrmn th allocaton of captal as wll as th xtnt of tax vason by frms. Nt rvnus ar gvn by th rvnus from taxaton plus pnalts from dtctd vason mnus th costs to provd th chosn audt rat. As a bnchmark rsult w show that fscal comptton wll lad to dtcton probablts whch ar nffcntly low from th prspctv of total rvnu maxmzaton. Ths nffcncy s cratd by th fact that rgonal audtng polcs gnrat (postv) fscal xtrnalts. In such a framwork, a fscal qualzaton schm has an ffcncy nhancng potntal, snc ntrrgonal rdstrbuton provds a mchansm to ntrnalz th fscal xtrnalty (s Köthnbürgr, 00). Howvr, a systm of gross rvnu qualzaton, whch maks rgons bar th full audtng costs whl tax rvnus wll b shard, also ntroducs a dstorton. Th asymmtrc tratmnt of nforcmnt costs and gross rvnus drvs a wdg btwn margnal bnfts and margnal costs of an ncras n th dtcton rats. Ths tnds to lowr audt rats. For th cas of many small rgons, only th dstortonary ffct rmans. Hnc, th gross rvnu qualzaton mchansm wll lad to vn lowr lvls of tax nforcmnt than n th bnchmark scnaro wthout fscal qualzaton. In th cas of larg rgons, th ffct from th ntrnalzaton and th dstorton work n oppost drctons. If th numbr of rgons n th conomy s blow a crtan thrshold, th ffcncy nhancng ffct wll domnat, and rgons wll choos a hghr audt rat than n th bnchmark cas. Nxt, w ntroduc an altrnatv ntrrgonal rdstrbuton systm: nt rvnu qualzaton. Compard to th gross rvnu sharng, thr s no dstorton snc undr ths nw mchansm tax rvnus as wll as audtng costs gt shard. For small rgons ths lads to th sam rsult as n th rfrnc cas wthout any fscal qualzaton. Howvr, du to th (partal) ntrnalzaton of th fscal xtrnalty, nt rvnu sharng would unambguously nduc hghr dtcton probablts for larg rgons. Thrfor, a swtch from a gross to a nt rvnu qualzaton systm would mak th dcntralzd choc of audt rats mor ffcnt. Th polcy mplcatons of ths rsults ar straghtforward. A fdral govrnmnt whch, on th on hand, qualzs tax rvnus btwn ts rgons but, on th othr, mposs th costs of tax collcton upon ths rgons, wll fac a hghr dgr of tax vason. Ths may wll b th cas for Grmany, snc th currnt Grman ntrstat transfr systm (Ländrfnanzausglch) corrsponds to gross rvnu qualzaton. 19
Swtchng to a nt rvnu qualzaton schm would hardly affct rdstrbuton, th prmary objctv of fscal qualzaton. At th sam tm, audt rats would ncras, tax vason dcras and total nt rvnus would b hghr. Clarly, th frst bst soluton could b rachd through a cntralzd choc of tax nforcmnt polcs, as practsd n most othr fdral countrs and as rcntly proposd by th Grman Mnstry of Fnanc. Howvr, buldng up a nw nsttutonal framwork wll b costly and wll tak som tm. In th lght of currnt budgtary problms, a swtch to a nt rvnu qualzaton systm could b a mor ffctv polcy n th short run. Appndx [Prlmnary] Lmma 1 Usng th mplct functon thorm on quatons () and (3) and smplfy, w gt k t r st + = < 0. (A.1) g st = < 0. (A.) g 1 ps = > 0. (A.3) g Not that r p = 0 for small rgons. Lmma Th ffctv tax rat s gvn by t t( 1 + ps). W can asly drv t = t ( 1 ps ) + s. (A.4) Usng (A.) from Lmma 1 and ps < 1 w gt t st = t ( 1 ps ) s + > 0 g. (A.5) From ths w can drv ( ) t st stg t ( ) ( 1 = + ps ) + s p g p g, (A.6) 0
whch s ngatv f th frst ordr ffct domnats. A.1 Scond Ordr Condtons () SOC for th cas of an opn conomy wthout any rdstrbuton; FOC s gvn by (9). k k t t ( t c( p) ) + c ( p) k c ( p) 0 p p p + <, (A.7) I II - From th FOC w s that th trm n brackts n II must b postv. Snc k < 0 (Lmma 1), II s ngatv. - From Lmma and c > 0 w know that trm III s ngatv. r r k st + f st + f k p p p - Snc th sgn of = s not clar p ( f ) (ngatv, f f = 0 ), trm I could b thr postv or ngatv. Snc I s an ffct of scond ordr, SOC < 0 should always hold. For small rgons th partal ffcts on th ntrst rat would vansh, but th analyss would not chang. () III A. Stratgc Complmnts Usng th mplct functon thorm on FOC (9), w obtan j A B k k t ( t c( p) ) + c ( p) j j =, (A.8) SOC whr SOC stands for th th LHS of condton (A.7). k Snc and th trm n th brackts n B must b postv, th whol trm s j r r k f st + f p pj p k j postv. Th sgn of = p s ambguous pj ( f ) (ngatv, f f = 0 ), hnc trm A could b thr postv or ngatv. Snc SOC 1
s ngatv, th RHS n (A.8) s postv, f A s postv or f A s ngatv and gts domnatd by B. Snc A s an ffct of scond ordr, > 0 should always hold. j A.4 Comparatv Statcs () Impact of th statutory tax rat t on p t Not frst, that n an asymmtrc qulbrum t j. Thrfor any chang n t t th tax rat statutory wll hav an mpact on th captal allocaton. * Th Cas wthout ntrrgonal rdstrbuton. Usng th mplct functon thorm on condton (9) w can drv I IV II III 1 k k t k t t = ( t c( p) ) + + c ( p) + k t SOC t t t. (A.9) t Th sgn of trm I s ambguous. (Howvr th trm bcoms unambguously ngatv k f th frst ordr ffct n,.. th frst trm on th RHS of t k s 1 r r k = + st f st f p t f + ( f ) t p t + p domnats.) t Th frst ordr ffct, dpctd n II, s ngatv. Th sgn of th trm III s dtrmnd by th ambguous sgn of k t. From Lmma on can asly show that trm IV s postv. () Impact of th th rdstrbuton paramtr α on p * Th Cas wth gross rvnu sharng. Usng th mplct functon thorm on condton (16), w gt + 1 t k 1 kj t = k t tj k + + + α SOC n j. (A.10) - For small rgons, n and th postv trm n th brackts vanshs: < 0. α - For larg rgons t s straghtforward to show, that th trm n th brackts bcoms postv (ngatv), f n < nˆ ( n nˆ > ). Hnc n nˆ 0. α
() Impact of th th numbr of rgons n on p * Th Cas wth gross rvnu sharng. Usng th mplct functon thorm on condton (16), w gt for α > 0 1 α kj t = t 0 SOC j k + n < n j p p. (A.11) * Th Cas wth nt rvnu sharng. Usng th mplct functon thorm on condton (19), w gt 1 β kj t = ( t ( )) ( ) 0 SOC j c p j k c p + n n < j, (A.1) β snc < 0. n Rfrncs Arbtnhmrkammr Brmn (001), Sturlch Btrbsprüfung 1999 m Ländrvrglch. Bartt, Chrstan; Hubr, Brnd and Lchtblau, Karl (00), A tax on tax rvnu: th ncntv ffcts of qualzng transfrs: vdnc from Grmany, Intrnatonal Tax and Publc Fnanc, 9, 631-649. Bucovstsky, Sam (1991), Asymmtrc tax comptton, Journal of Urban Economc, 30, 167-181. Crmr, Hlmut and Gahvar, Frouz (1993), Tax vason and optmal commodty taxaton, Journal of Publc Economcs, 50, 61-75. Crmr, Hlmut and Gahvar, Frouz (000), Tax vason, fscal comptton and conomc ntgraton, Europan Economc Rvw, 44, 1633-1657. Kanbur, Rav and Kn, Mchal (1993), Jux sans Frontèrs: Tax comptton and tax coordnaton whn countrs dffr n sz, Amrcan Economc Rvw, 83(4), 877-89. Kn, Mchal and Marchand, Maurc (1997), Fscal Comptton and th Pattrn of Publc Spndng, Journal of Publc Economcs, 66, 33-53. 3
Köthnbürgr, Marko (00), Tax comptton and Fscal Equalzaton, Intrnatonal Tax and Publc Fnanc, 9, 391-408. Kolm, Srg-Chrstoph (1973), A not on optmal tax vason, Journal of Publc Economcs,, 65-70. Lnk, Thomas; Fug, Hd and Schndr, Frdrch (1998), Zurück zu mhr Födralsmus: En Vorschlag zur Nugstaltung ds Fnanzausglchs n dr BRD untr bsondrr Brückschtgung dr ökonomschn Thor dr Poltk. Workng Papr 9805, Dpartmnt of Economcs, Johanns Kplr Unvrsty Lnz. Rnganum, Jnnfr and Wld, Lous (1986), Equlbrum Vrfcaton and Rportng Polcs n a Modl of Tax Complanc, Intrnatonal Economc Rvw, 7(3), 739-760. Snn, Hans-Wrnr (003), Th Nw Systms Comptton, Oxford. Schndr, Frdrch and Ernst, Domnk (000), Shadow Economs: Sz, Causs and Consquncs, Journal of Economc Ltratur, 38(1), 77-114. Slmrod, Jol and Ytzhak, Shlomo (1987), Th optmal sz of a Tax Collcton Agncy, Scandnavan Journal of Economcs, 89(), 183-19. Wldasn, Davd (1988), Nash Equlbra n Modls of Fscal Comptton, Journal of Publc Economcs, 35, 9-40. Wldasn, Davd (1989), Intrjursdctonal Captal Moblty: Fscal Extrnalty and a Corrctv Subsdy, Journal of Urban Economcs, 5, 193-1. Ytzhak, Shlomo (1974), A not on Incom tax vason: A thortcal Analyss, Journal of Publc Economcs, 3, 01-0. Zodrow, Gorg and Mszkowsk (1986), Pgou, Tbout, Proprty Taxaton, and th Undrprovson of Local Publc Goods, Journal of Urban Economcs, 19, 356-370. 4