Asymmetric Capital-Tax Competition, Unemployment and Losses from Capital Market Integration

Transcription

1 Asymmetc Captal-Tax Cmpettn, Unemplyment and Lsses fm Captal Maet Integatn Rüdge Pethg Fede Klleß CESIFO WORKING PAPER NO CATEGORY : PUBLIC FINANCE SEPTEMBER 2009 An electnc vesn f the pape may be dwnladed fm the SSRN webste: fm the RePEc webste: fm the CESf webste: Twww.CESf-gup.g/wpT

2 CESf Wng Pape N Asymmetc Captal-Tax Cmpettn, Unemplyment and Lsses fm Captal Maet Integatn Abstact In a mult-cunty geneal equlbum ecnmy wth mble captal and gd-wage unemplyment, cuntes may dffe n captal endwments, pductn technlges and gd wages. Gvenments tax captal at the suce t maxmze natnal welfae. They accunt f tax base espnses t the tax and tae as gven the wld-maet nteest ate. We specfy cndtns unde whch - n cntast t fee tade wth undstted lab maets - welfae declnes and unemplyment nceases n sme cuntes () when mvng fm autay t tade wthut taxatn and/ () when mvng fm tade wthut taxatn t tax cmpettn. JEL Cde: E24, H25, H87, J64, R3, F2. Keywds: captal taxatn, asymmetc tax cmpettn, gd wages, unemplyment, lsses fm tade. Rüdge Pethg Unvesty f Segen Depatment f Ecnmcs Heldelnstasse Segen Gemany pethg@vwl.ww.un-segen.de Fede Klleß Unvesty f Segen Depatment f Ecnmcs Heldelnstasse Segen Gemany lless@vwl.ww.un-segen.de

3 Asymmetc captal-tax cmpettn, unemplyment and lsses fm captal maet ntegatn The pblem The ntenatnal mblty f captal has massvely nceased ve the last decades, gvenments dstt tade by taxng subsdzng captal, and unemplyment s a pesstent phenmenn n many cuntes. The fee-tade paadgm pmses gans fm ntenatnal tade n a pefectly cmpettve wld n the absence f taxatn but leaves unansweed the questn what the allcatve mpact s f tade when captal s mble, when cuntes suffe fm unemplyment, and when the gvenments engage n captal-tax cmpettn. The pesent pape ams at explng the mpact f captal maet ntegatn n a multcunty ecnmy wth hetegeneus cuntes, pesstent gd-wage unemplyment and captal-tax cmpettn. Each cunty pduces the same cnsumptn gd wth the help f lab and captal, and unemplyment esults fm excessvely hgh and gd wage ates. Cuntes may dffe wth espect t the gd wage ates, captal endwments and pductn technlges. Gvenments levy captal taxes at the suce whse ates ae nt sgn cnstaned and whse evenues ae ecycled t the cnsumes. Gvenments chse the tax t maxmze natnal ncme (= welfae) tang accunt f hw the dmestc fms' demands f captal espnd t the tax. We use that mdel t nvestgate the changes n the cuntes' allcatn and welfae, when mvng fm autay t tade wthut taxatn and fm tade wthut taxatn t tax cmpettn. Thee s a lage and gwng lteatue n captal tax cmpettn. The classc papes f Zdw and Meszws (986) and Wlsn (986) analyze the mpact f captal-tax cmpettn n the pvsn f publc gds when lab maets ae pefectly cmpettve. Sme f the subsequent lteatue t up the ssue f tax cmpettn n the pesence f lab maet dsttns, e.g. Fuest and Hube (999), Lete-Mnte et al. (2003), Egget and Gee (2004), Ogawa et al. (2006) and Anssn and Wehe (2008). Hweve, these studes assume dentcal cuntes and theefe yeld lmted nsght nly n allcatve effects f the tanstn fm autay t tade and tax cmpettn. Symmetc tax cmpettn means that tade des nt tae place and that neffcences f tax cmpettn, f any, ht all cuntes ale. The sze f the cuntes' ppulatns (= lab endwment) may vay as well. Hweve, snce all cuntes ae assumed t suffe fm unemplyment the ppulatn sze s elevant. 2

4 Asymmetc tax cmpettn s studed, e.g., by Wlsn (99), Bucvetsy (99), DePate and Myes (994), Pealta and van Ypesele (2005) and Sat and Thsse (2007). But nne f these cntbutns deals wth lab maet dsttns and unemplyment 2. Pealta and van Ypesele (2005) addess the ssue f 'gans fm tade' and fnd f quadatc pductn functns that " fscal cmpettn edes sme, but nt all, f the gans fm lbealzatn." (bdem, p. 259). As we wll pesent cases f tade lsses n the pesent pape, Pealta and van Ypesele's esult suggests that t s the cmbnatn f asymmetc tax cmpettn and lab maet dsttns that has the ptental f endeng captal maet ntegatn unfavable f sme cuntes. Tang the tme-hned fee-tade paadgm as a efeence, thee s, f cuse, a lage lteatue n tade unde vaus cndtns f secnd best. In the pesent cntext the cntbutns f Kemp and Negsh (970) and Eatn and Panagaya (979) ae wth mentnng wh fcus n gans fm tade when cmmdtes ae taxed and fact maets ae dstted. Hweve, they d nt mdel captal-tax cmpettn. Me ecently, the ssue f gans fm tade has been lned t captal-tax cmpettn, e.g. by Kessle et al. (2003) and Lcwd and Mas (2006), wh analyze captal-tax fnanced edstbutn plces and vtng n ecnmes wthut lab maet dsttns. Al et al. (2009) pesent a mdel f tw cuntes whch ae dentcal except that the lab maet n ne cunty s pefectly cmpettve and unnzed n the the cunty. They detemne cndtns unde whch ethe cunty pefes autay t captal maet ntegatn. In the mdel, n tax cmpettn taes place. Summng up, t u nwledge the cnsequences f captal maet ntegatn n a multcunty ecnmy wth lab maet dsttns and asymmetc captal-tax cmpettn have nt yet been analyzed n the lteatue. The pesent pape ams t fll that gap. It cnsdes 'small cuntes' as n Zdw and Meszws (986) and Wlsn (986) athe than mdelng gvenments playng Nash n tax ates. Wth the cncept f gd-wage unemplyment u mdel elates mst clsely t Ogawa et al. (2006). We devate fm the appach by dppng the ssue f publc-gd pvsn 3, by cnsdeng hetegeneus cuntes and by addessng the cnsequences f mvng fm autay t tade and captal-tax cmpettn. Fuest and Hube (999) and late Ogawa et al. (2006) shw that a gvenment's ptmal captal tax 2 A lmtng case s Sat and Thsse (2007) wh cnsde a lab maet wth hetegeneus slls, cstly tanng and the need f matchng the fms' sll needs. Yet the analyss elates t full emplyment except f a hnt n the cncludng emas that fscal cmpettn mght well tgge unemplyment n a cunty. 3 The ssue f ptmal pvsn f publc gds s dpped n many cntbutns t the tax cmpettn lteatue such as Lete-Mnte et al. (2003), Egget and Gee (2004), Pealta and van Ypesele (2005) and Sat and Thsse (2007). Suppessng the fscal pupse f captal taxatn allws slatng the welfae-maxmzng gvenment's ncentve t stmulate dscuage the use f captal n pductn. 3

5 ate may be pstve negatve dependng n whethe captal and lab ae substtutes cmplements n pductn 4. These ppetes f the pductn technlgy wll tun ut t have a ma mpact n the allcatve cnsequences f captal maet ntegatn. That s why we wll analyze technlges wth captal and lab beng ethe substtutes cmplements. Ultmately, t s an empcal ssue, f cuse, what the elevant pductn technlgy s le. Hweve, as the petanng empcal evdence s qute cmplex, f nt ambguus, e.g. Glches (969), Begstöm and Panas (992) and Duffy et al. (2004), t appeas t be apppate and necessay clafyng the analytcal cnsequences f altenatve assumptns n pductn technlges. The pape s ganzed as fllws. Afte havng ntduced the mdel n Sectn 2, we dentfy cndtns n Sectn 3 unde whch cuntes gan lse n the tanstn fm autay t tade f captal s ntenatnally mble and gvenments d nt tax captal. In Sectn 4 we fst chaacteze an ndvdual gvenment's ptmal captal tax plcy and shw that t nceases dmestc emplyment f sme gven wld maet ate f nteest. Then we nvestgate the allcatve dsplacement effects that ccu when the ecnmy mves fm tade wthut captal taxatn t captal-tax cmpettn. Unde cetan cndtns that tanstn tuns ut t be welfae deceasng f sme cuntes. Sectn 5 cmbnes the esults fm the tw pevus sectns and dentfes cndtns unde whch sme cuntes suffe a welfae lss and a se n unemplyment n the tanstn fm autay t tax cmpettn. Sectn 7 cncludes. Fmal pfs f all ppstns ae delegated t the Appendx. 2 The mdel: gd wages, mble captal and captal taxatn Cnsde an n-cunty ecnmy n whch each cunty =,, n pduces the amunt y Y (, ) = () f a cnsumptn gd by means f captal nput and lab nput accdng t the stctly cncave 5 pductn functn Y that exhbts pstve fst devatves. The cnsumptn gd and captal ae taded n cmpettve wld maets at pce p and nteest ate 4 If the utput Y(, ) s pduced wth captal nput and lab nput, captal and lab ae sad t be substtutes n pductn, f Y < 0, and cmplements, f Y > 0. 5 Assumng stctly cncave pductn functns s ndspensble because thewse we wuld nt btan welldefned fact demand functns. See equatn (2) belw. Unftunately, n the geneal fm stctly cncave pductn functns gve lmted nsghts nly. Theefe, we wll late cnsde me specfc paametc functnal fms as well. 4 y

6 , espectvely. Immble lab s taded n dmestc maets at the wage ate, =,, n. The gvenments f all cuntes tax captal at the suce such that the 'aggegate' pduce n cunty faces the afte-tax ental ate f captal, w ρ : = + t, whee t s the sgnuncnstaned 6 captal tax ate. Fm maxmzes pfts π : = y w ρ as a pce tae gvng se t the standad fact demand functns = K ( ρ, w ) and = L ( ρ, w ), (2) whee 7 K = Y / D < ρ 0, Kw =Y / D, Lw = Y / D< 0, Lρ = Y / D, and D: = YY ( Y ) 2 0 > ; Y may be pstve negatve. Except f bef efeences t the benchma mdel wth flexble wage ates we fcus exclusvely n scenas f pesstent gd wages that ae suffcently hgh as t mae all cuntes suffe fm unemplyment. Me fmally, dente by the numbe f cnsumes esdng n cunty, let each cnsume ffe ne unt f lab and cnsde stuatns f excess supply f lab, 8 m m L ( ρ, w > ) f all =,, n (3) Accdng t (3), m L ( ρ, w ) > 0 cnsumes ae unemplyed and the L (, w ) ρ bs ffeed by fm ae andmly allcated t cnsumes. As cnsumes spend the ncme n a sngle cnsumptn gd nly we can d wthut utlty functns. The natnal ncme f cunty s x : = + π + t + w, whee s cunty 's aggegate captal endwment and whee pfts π and the tax evenues t ae ecycled t the cnsumes 9. Cmbnng (), (2) and π = y w ρ yelds natnal ncme (= welfae) x as a functn f, t and w: (,, ): (, ), (, ) (, ) X t w = Y K + t w L + t w K + t w. (4) 6 If t < 0, captal s subsdzed. T avd clumsy wdng we wll use the tem tax espectve f the sgn f t. 7 Captal lettes dente functns and subscpts t captal lettes dente fst devatves. T smplfy ntatn we Y Y wte nstead f etc. 8 Me pecsely, (3) s assumed t hld n all equlba t be specfed belw. 9 We need nt specfy the shaes f pfts and tax evenues allcated t ndvdual cnsumes because we efan fm fcusng n utlty dstbutns. 5

7 The gvenment f cunty =,, n s suppsed t maxmze natnal ncme X ( t,, w ) wth espect t the captal tax ate. As the defntn f t (,, ) X t w n (4) shws, t accunts f the mpact f tax vaatns n ts fm's fact demands (2) but taes as gven the wld pce,, f captal. F pedetemned tax ates t,..., tn the cndtn f cleang the wld captal maet s = K ( + t, ) w. (5) If (5) s satsfed, the wld maet f the cnsumptn gd s als cleaed whch fllws fm summng (4) ve all. The cncept f geneal equlbum f the n-cunty ecnmy s staghtfwad: F gven captal endwments,..., n and f pesstent gd wage ates w,..., wn a tax-cmpettn equlbum wth unemplyment s fmally detemned by the set = { n ( ) =,..., n} whee the allcatn ( x, y,, ) =,..., E: t,..., t,, x, y,, n and the nteest ate satsfy the equatns () (5) f ( t,..., t n ), and whee gvenment =,, n chses ts tax ate t as t maxmze (4). { n =,..., n} It wll tun ut t be useful t cnsde als equlba E: t,..., t,, ( x, y,, ) = n whch tax ates ae exgenusly fxed athe than ptmally chsen by the gvenments. We wll call such equlba cnstant-tax tade equlba n cntast t tax-cmpettn equlba as defned n the last paagaph. Nte that the n-tax tade equlbum ( ) t =... = t n 0 s a specal cnstant-tax tade equlbum. When we late nvestgate the ncdence f tax cmpettn, we wll explt an equvalence between tax-cmpettn equlba and cnstant-tax tade equlba whch ases because due t (5) the captal maet equlbum depends n and t,..., tn thugh ρ,..., ρ n nly. In fmal tems, we state that equvalence n Ppstn (Neutalty f unfm vaatns n tax ates). { n,..., n} If E : t,..., t,, ( x, y,, ) = s a tax-cmpettn equlbum a cnstant-tax = { θ n θ θ =,..., n} tade equlbum, E : t,..., t,, ( x, y,, ) equlbum f all θ <. = + + s a cnstant-tax tade θ 6

8 Ppstn s a standad esult n tax ncdence they. Unfm vaatns n captal tax ates ae nn-dsttnay because the ttal supply f captal ( = ) s pefectly pce nelastc. Thus unfm changes n all tax ates can be exactly ffset by changes n the nteest ate f equal sze and ppste sgn. Ppstn wll be used n the pf f u man esult n Sectn 4.2 belw. 3 Fm autay t tade wthut taxatn Ou fst step twad nvestgatng gans lsses fm tax cmpettn s t exple the allcatve changes that ccu when the cuntes mve fm autay t tade n the absence f captal taxatn. F the mdel ntduced n the pevus sectn, the efeence scena f autay s staghtfwad. All captal maets ae natnal and (5) s eplaced by = K ( + t, w ) f =,, n wth dentng the nteest ate n cunty. Nte fst that n autay captal taxatn s nn-dsttnay because the supply f captal s pefectly nelastc n each cunty. Hence we set t 0, f cnvenence. Snce we allw cuntes t dffe n the fundamentals 'captal endwments', 'pductn technlges' and 'wage ates', the equlbum nteest ate n autay wll geneally dffe acss cuntes. Usng the geneal functnal fm () f the pductn functn t s had t specfy ppetes f the mappng fm the fundamentals t the autac equlbum nteest ate. We theefe est t CES pductn functns n Ppstn 2 (Detemnants f the sze f the equlbum nteest ate n autay) Suppse captal s untaxed and cunty 's pductn functn s CES,.e. t satsfes e, (6) e e ( ) = ( + ) Y, a a a whee a > 0, a > 0, a > 0, σ >, 0 b σ, b ] 0,[, and e ( σ ) : = / σ. F any gven elastcty f substtutn, σ > 0, σ, cunty 's autac equlbum nteest ate, a, s deceasng n ts captal endwment, Fllwng an ncease n the gd wage ate, ses / emans unchanged / declnes dependng n whethe. a σ s geate than / equal t / smalle than c ( b ) = / >. 7

9 Clealy, expandng the captal endwment nceases captal abundance and hence educes the pce f captal ( ). Inceasng the wage ate maes captal scace ( ) less scace a ( ) dependng n whethe captal and lab ae substtutes ( σ > c ) cmplements a ( σ < c ). Ppstn 2 can be cnvenently used t cmpae the autac equlbum nteest ate f dffeent cuntes whse pductn functns ae CES. T see that suppse sme cuntes and ae chaactezed by the paametes (, w ) and (, ) mplctly Ppstn 2 defnes a functn, say a w and bseve that h h R, such that ha = R ( h, wh ) f h =,. If bth cuntes use the same pductn functn ( R = R = R), they have the same autac nteest ate ( a = a ), f and nly f R(, w) = R(, w). Meve, the nequalty a > a hlds, f, cetes pabus, ethe { < w w } { > and σ > c } { w < w and σ < c }. Suppse nw that all cuntes have attaned the autac equlbum and the bdes ae subsequently pened f tade n captal and the cnsumptn gd whle all gvenments efan fm taxatn. It s staghtfwad fm (5) (wth t 0 f all ) that the n-tax tade mn max equlbum nteest ate, dented, satsfes, mn, when s the smallest and a a a max a mn a max a s the lagest autac equlbum nteest ate f all cuntes and <. F any cunty wth autac nteest ate a the allcatve cnsequences f the tanstn fm autay t the n-tax tade equlbum clealy depend n the sgn f the dffeence a. We tae ths dffeence as u pnt f depatue f analyzng the mpact f mvng fm autay t the n-tax tade equlbum and explctly allw f dffeent pductn technlges. The esults ae summazed n Ppstn 3 (Tanstn fm autay t n-tax tade) Suppse all gvenments efan fm taxatn ( t 0 f all ) and cnsde the tanstn f the n-cunty ecnmy fm ts autay equlbum (subscpt a) t ts ze-tax tade equlbum (subscpt ). () The allcatve mpacts f that tanstn ae summazed n Table. () Suppse the cases 7 n Table apply and the pductn functns ae Cbb- Duglas, defned by 8

10 ( ) = Y, α β f all wth α > 0, β > 0and α + β = : b <. (7) If the pductn functns ae the same acss cuntes ( α = α and β = β, all ) then the fllwng equvalences hld: > > < < = ω = a = a x = xa, (8) < < > > whee : nw = and ω : =. β c n w β c () If Case 6 n Table apples and the pductn functn s CES, (6), cunty lses fm tade f the elastcty f substtutn n pductn, σ, satsfes ( )( w + q ) a w σ > c + > c, whee q : = w a σ σ and c : = >. b > a = a < a L < 0 L > 0 L = 0 L > < 0 L = 0 L > 0 L < 0 Case N y x a a y -? - 0 +? + a x ) a? (-) ? (-) ) ) F detals see the Ppstns 2 and 2 + Table : Allcatve mpacts f the tanstn fm autay t n-tax tade A few emas n Table ae n de. The tp w dstngushes the cases n whch cunty 's autac equlbum nteest ate, a, s lwe than, equal t hghe than the wld maet nteest ate n the n-tax tade equlbum,. Nte that ne can cmbne that nfmatn wth the esults establshed n Ppstn 2 t tace the dffeence a t dffeences n captal endwments and gd wages 0. The secnd w n Table elates t ppetes f 0 Recall that Ppstn 2 allws dentfyng the captal endwments and the wage ates as detemnants f the dffeences a f the case f dentcal CES pductn functns. 9

11 the lab demand f cunty 's fm, whch n tun ae detemned by ppetes f the pductn functn as shwn n equatn (2). T be me specfc, f pductn functns ae Cbb-Duglas, (7), we have Y > 0. Hence f Cbb-Duglas the Cases, 4 and 7 apply. CES pductn functns (6) exhbt Y < > 0, f and nly f σ > < c. Theefe, such functns ae examples f the Cases 2, 4 and 6, f σ > c, f the Cases, 4 and 7, f σ < c, and f the nfe-edge Cases 3 and 5, f σ = c. Case 4 s tval but nt entely unnteestng f sme cnclusns n the next sectn. Fgue : Illustatn f the cases and 6 n Table The esults lsted n Table can be easly llustated. In Fgue we tae up the Cases and 6 f Table that ae unclea n the sgn f x x a and leave the llustatn f the the cases t the eade. In bth panels f Fgue, ya s gven by the aea 0BC, and y s gven by 0AE. In Case captal s expted, and the value f these expts s equal t the aea DE n the left panel f Fgue. In Case 6 captal s mpted, and the value f these mpts s equal t the aea DE n the ght panel f Fgue. It fllws that n Case [Case 6] we have x = 0AED such that x s smalle than / equal t / geate than x a, f and nly f the aea ABFE [ABCF] s geate than / equal t / smalle than DFC [DFE]. Accdng t Table cunty unambguusly gans fm tade n the Cases 2, 3 and 7, whle the welfae change n the Cases and 6 emans unclea f geneal pductn functns. T gan addtnal nsghts n changes f natnal ncme n the unclea cases, we have specfed 0

12 the pductn technlgy n Ppstn 3 by Cbb-Duglas functns thus estctng the fcus f Ppstn 3 t the Cases 7 f Table, as agued abve. The stng esult f Ppstn 3 s that wth Cbb-Duglas functns cuntes lse fm tade and suffe fm hghe unemplyment - n Case f Table and they gan fm tade and eny hghe emplyment - n Case 7 f Table. Ppstn 3 demnstates that cuntes wth CES pductn functns may als lse fm tade f the elastcty f substtutn s lage enugh. As mentned abve, f CES pductn functns we can use Ppstn 2 t specfy cndtns n fundamentals unde whch a cunty wll expt mpt captal afte the bdes ae pened. F Cbb-Duglas functns Ppstn 3 establshes an even me nfmatve clea elatnshp between the dffeence n nteest ates, a, n the ne hand and captal endwments and wages f all cuntes, n the the hand. It s theefe wthwhle analyzng and ntepetng (8) n sme me detal. Suppse fst the cuntes dffe n the captal endwments nly. If the level f gd wages s the same n all cuntes, we have ω = f all such that (8) s tuned nt > > < = = x = x < < > a a Unde these cndtns cunty lses [gans] fm tade, f and nly f ts captal endwment exceeds [falls sht f] the cuntes' aveage captal endwment,. If captal s elatvely. abundant ( > ), cunty 's equlbum nteest ate n autay s lw ( < ) and s a bund t se when captal s ntenatnally taded. Recall that K < 0 hlds f all stctly cncave pductn functns, and that L < 0 hlds because f (7). Theefe cunty wll use less captal and lab and wll cnsequently pduce less utput, s much less, that the (new) evenues fm exptng captal d nt cmpensate f the eductn n utput. 2 Cnvesely, cuntes wth a elatvely small captal endwment ( < ) wll face a lwe nteest ate n tade equlbum ( such that the exta value f utput s geate than the expendtue n captal mpts. Ths ea > ) whch, n tun, bsts the nput f bth captal and lab Case 7 apples f me geneal pductn functns as well. 2 Nte that f the wage ate wee flexble, wuld shn t este the full emplyment equlbum n the lab maet whch wuld then tend t bst dmestc pductn. w

13 sult cnfms the gans fm tade stated aleady unde me geneal cndtns n the Case 7 f Table. T fcus n the le played by gd wages, suppse all cuntes ae endwed wth the same amunt f captal, =... = n =, but dffe wth espect t the wage ates. Invng (8), we establsh the fllwng equvalences: > > > < = ω w = w = a x = xa, whee < < < > βc βc w w : =. n Hence n that case, thee s a pstve numbe w, the same f all cuntes, such that cunty lses / s equally well ff / gans fm tade, f and nly f ts wage s abve / equal t / belw w. The magntude f the theshld value w s unclea. We wuld le t nw, n patcula, hw w elates t w : = w /n, the aveage wage. One can shw 3 that ( ) w> w, f and nly f βc β c + βc. n > w / w > 0 In the wds, n an ecnmy wth a suffcently lage numbe f cuntes nly thse cuntes lse fm tade whse wage ate s well abve the aveage wage ate. Nte that the theshld value whch needs t be exceeded f the numbe f cuntes t be lage enugh depends n the level f wage ates and n paametes f the Cbb-Duglas technlgy. T hghlght the cnsequences f gd wages n the tanstn fm autay t tade fm anthe pespectve, we establsh w Ppstn 4 (Tanstn fm autay t n-tax tade wth flexble wages) Suppse that all gvenments efan fm taxatn ( t 0 f all ), that wages ae flexble, and that pductn functns ae Cbb-Duglas, (7), the same acss cuntes. The tanstn f the n-cunty ecnmy fm autay (subscpt a) t ts n-tax tade equlbum (subscpt ) s chaactezed by > > < > βγ nm = m = a w = wa x = x < < > > a, whee m = (9) βγ m 3 The pf s pvded at the end f the Appendx. 2

14 As expected, Ppstn 4 cnfms f the mdel at hand the geneal gans-fm-tade esult f ecnmes wth a full set f pefectly cmpettve maets. In the pesent cntext the pupse f Ppstn 4 s t use (9) as a efeence f futhe ntepetatn f the equvalences (8) fm Ppstn 3. T eep the expstn smple, we estct that cmpasn t wage ates and lab endwments satsfyng the cndtn ω = m =, whch hlds, e.g., f we set w = w2 =... = wn n Ppstn 3 and m = m2 =... = mn n Ppstns 4. Obvusly, n that case the fst equvalences n (8) and (9) ae the same. Hweve, (9) shws that mantanng full emplyment f lab afte penng the bdes esults n a lwe wage ate n captalch cuntes and a hghe wage ate n captal-p cuntes. In the captal-ch cunty we fnd that w < wa < w, whee the wage ates w and wa, espectvely, ae equlbum ates n autay and tade n the flexble-wage scena f Ppstn 4, whle w s the gd wage ate f Ppstn 3. Hence when the wage ate s gd, allwng f tade wdens the dffeence between the gd wage ate and the espectve equlbum wage ates n case f flexble wages ( ) ( w w > w w a ). As a cnsequence, the lab maet dsequlbum s aggavated and unemplyment ses, whch n tun educes natnal ncme (last equvalence n (8)). In cntast, f the captal-p cunty we fnd the nequaltes whch mply that penng the bdes f tade educes the dffeence between the gd wage ate and the espectve equlbum wage ates n case f flexble wages wa < w < w ( w w ) < ( w w ) a. Theefe tade dmnshes the lab maet dsequlbum and thus ases emplyment as well as natnal ncme. 4 Fm tade wthut taxatn t captal tax cmpettn Havng clafed the allcatve cnsequences f the tanstn fm autay t tade wthut taxatn n the pevus sectn we nw tae as u pnt f depatue the tade equlbum wthut taxatn and analyze the allcatve mpact f tax cmpettn. As a fst step twad that end t s necessay and useful t tae a clse l at the gvenment's ptmzatn calculus. 4. Ppetes and mplcatns f an ndvdual gvenment's ptmal captal tax By assumptn, gvenments chse the captal tax as t secue the maxmum natnal ncme (= welfae), (4), f any gven nteest ate. The esultant ptmal tax ates ae chaacte- 3

15 zed n Ppstn 5 (Ppetes f the ptmal captal tax). 4, 5 Gvenment 's ptmal tax ate satsfes the cndtn wl t = = K ρ wy ρ Y. (0) If the pductn functns ae CES, (6), Cbb-Duglas, (7), espectvely, (0) s tuned nt t = σ σ ρ σ σρ + ( / ) a a cw σ c and t = β < 0. () whee a > 0, a > 0, σ, c : / ( b ) = and ρ : = + t. Snce L >0 Y < < > 0, the ptmal tax ate s negatve f the pductn functn s Cbb- Duglas. In case f CES pductn functns we have Y < > 0, f and nly f σ > < c, and theefe t < > 0 f and nly f σ < > c. When pductn functns ae Cbb-Duglas, the ptmal tax ates ae unfm acss cuntes, f and nly f the technlgy s the same n all cuntes. Inteestngly, dffeences wth espect t the captal endwments and wage ates d nt tanslate nt dffeences n ptmal tax ates. When pductn functns ae CES, the ptmal tax ates ae the same acss cuntes, f and nly f all cuntes have dentcal pductn functns and dentcal wage ates. Dffeences n captal endwments d nt matte. Cases f unfm dentcal tax ates wll be f sme nteest n Sectn 4.2 belw. By pesuppstn, f the gvenment sets ts tax ate accdng t (0), the cunty's ncme s maxmzed. But t s nt clea hw ncme maxmzatn changes the level f emplyment, n patcula, f we allw f dffeent sgns f Y. We pvde the answe n Ppstn 6 (Incme maxmzatn always pmtes emplyment) 4 Nte that n (0) the tems L and K ρ ρ ae functns f t. 5 The equatn (0) has been deved by Fuest and Hube (999) and late by Ogawa et al. (2006). Fuest and Hube (999) emply a ght-t-manage mdel f wage baganng f the lab maet. When they mdel gvenments maxmzng utlty f a gven wage ate; they deve an ptmal tax ate n the equatn (26) equal t (0) f e ' = 0. Ogawa et al. (2006) deve (0) n a setup whee gvenments levy a head tax and a captal tax and fnance a publc gd. Iespectve f these dffeences, the atnale f levyng the captal tax n the pesent mdel s the same as n Fuest and Hube (999) and Ogawa et al. (2006). 4

16 t Let be the ptmal tax ate f gvenment (satsfyng (0)), f sme nteest ate pevals. Iespectve f the sgn f (.e. the sgn f ) t s tue that the gvenment's ptmal tax ate nceases emplyment alng wth natnal ncme as cmpaed t the n-tax stategy ( +, ) > (, ) L t w L w. t T see the atnale f that synegsm bseve that the change n natnal ncme esultng fm a small vaatn n the tax ate s gven by Y X YK YL K = + = ( ) K wl t ρ l ρ ρ ρ + = ρ ρ tk ρ + wl, ρ f the wage ate s gd. If the lab maet wee pefectly cmpettve, the tem YL ρ wth Y = w wuld be absent. As an mmedate mplcatn ptmalty wuld eque t abstan fm taxatn ( X t = 0 t =0). Hweve, snce the wage ate s gd, the devatve f natnal ncme wth espect t the tax ate s equal t the sum f the cmpnents ( ρ ) K ρ and wl ρ. T ntepet these tems suppse captal s taxed ( t ρ 0 = > ) n the ntal stuatn and the tax ate s nceased, dt = dρ > 0. We then bseve the patal magnal beneft effect K ρ > 0 f nceased captal expt evenues educed expendtues n captal mpts, and ρ < 0 s the magnal cst f educed utput. Hence ρ <. The K ρ ( ) 0 K ρ sgn f wl ρ depends n whethe Y > 0 0 Y <. If Y > 0, the dp n captal nput (fllwng dt = dρ > 0 ) dmnshes the magnal pductvty f lab,, cetes pabus. Y Hweve, the fst-de cndtn f pft maxmzatn, Y = w, and the gd wage ate w nduce the fm t 'este' the fme level f Y. It des s by educng ts lab nput, L ρ < 0 such that wl ρ < 0 s anthe magnal cst tem yeldng ( ρ ) 0 X = K + wl < t ρ ρ, f t > 0. T attan the fst-de cndtn f a maxmum f t s theefe necessay t chse ρ < and hence a captal subsdy, t < 0, n ths scena. x Analgus aguments apply t the case Y < 0. Recall the qualfcatn f Ppstn 6 that gvenment 's pmtn f emplyment s subect t the cndtn that t taes the nteest ate as gven. T fx u deas tae as the baselne the n-tax tade equlbum wth ts equlbum nteest ate and suppse that all gven 5

17 ments ntduce the ptmal captal tax tang the nteest ate as gven. The clea mplcatn f Ppstn 6 s that all cuntes wll ase the ncme as well as the level f emplyment. The bad news s, hweve, that the wld captal maet wll nt be n equlbum anyme. Hence t este equlbum the wld nteest ate wll have t change, n geneal, gvng se t the pssblty that tax cmpettn may patly ffset pehaps even defeat the gvenment's efft t pmte bth welfae and emplyment. We wll study that ssue n the fllwng subsectn. 4.2 The mpact f tax cmpettn as cmpaed t tade wthut taxatn Statng fm the scena f tade wthut taxatn we wsh t detemne hw tax cmpettn changes the allcatn attaned n the n-tax tade equlbum. In the wds, we nw tae as a benchma the n-tax tade equlbum and cmpae the petanng allcatn wth the allcatn attaned n tax-cmpettn equlbum. Analgus t the le f the dffeence a n case f the tanstn fm autay t tade (Ppstn 3), the change n the nteest ate fm (n-tax tade equlbum) t (tax-cmpettn equlbum) wll nw tun ut t play an mptant le. It s theefe mptant t nw what detemnes the sgn f the dffeence all, then. A clea-cut answe s pssble, f sgn t s the same f all. If t < > because < K ( + t, w ). In that case must se t este equ- lbum n the captal maet, (5). Lewse, f t > 0 f all, then because > K ( t, + w ). Nw must declne t este equlbum n the captal maet. If technlges ae mxed such that t < 0 f sme cuntes and t > 0 f thes, the dffeence may tae n ethe sgn. Keepng these pelmnaes n mnd we ae nw eady t cmpae the allcatns f the ntax tade equlbum and the tax-cmpettn equlbum n Ppstn 7 (Tanstn fm tade wthut taxatn t tax cmpettn) Cnsde the n-tax tade equlbum E : t 0,..., t 0,, ( x, y,, ) < { =,..., } n n 0 f = = = and { n,..., n} the asscated tax-cmpettn equlbum E : t,..., t,, ( x, y,, ) =, whee = x, y,, t satsfes (0) f all. The allcatn ( ) =,..., n x, y,, f the tax-cmpettn equlbum devates fm the n-tax tade allcatn ( ) =,..., n as shwn n Table 2. 6

18 sgn sgn L ρ sgn ( t + ) ( ) y y x x N. > t + < ) L ρ < 0 t + > L ρ > 0 t + > ? - 0 L ρ t + = = 0 L ρ = t = < L 0 ρ > ) : = L ρ < 0 t + < t + > ? + t + < ? - Table 2: Tanstn fm tade wthut taxatn t tax cmpettn Table 2 calls f sme cmments. The vey fst clumn lsts the pssble cnstellatns f the espectve equlbum nteest ates analgus t the tp w n Table. As n Table, each f the Cases - 8 f Table 2 s chaactezed by the sgn f L ρ (secnd clumn) but a new attbute s the sgn f t + (thd clumn). The meanng and le f the tem t + s made pecse n the pf f Ppstn 7 n the Appendx. Thee we shw that f the nteest ate f the n-tax tade equlbum pevals (athe than the nteest ate f the taxcmpettn equlbum) and f all gvenments chse tax-cmpettn equlbum s attaned. 6 t +, then the allcatn f the It s nfmatve t cmbne the esults f Table 2 wth the nfmatn pvded n Ppstn 5. Recall fm () that L ρ < 0 hlds, n patcula but nt nly, f Cbb-Duglas functns and CES functns satsfyng σ < c. Yet f the technlgy s Cbb-Duglas, the Cases and 2 apply nly f the Cbb-Duglas functns dffe acss cuntes. Othewse Case 4 apples. Case 5 epesents a nfe-edge case cespndng t the Cases 3 and 5 f Table whch happens t ccu when all pductn functns ae when all pductn functns ae 6 Cnsde f example the cnstellatn > ( = 0 > ) and L ρ < 0 n Table 2. Owng t L ρ < 0 we have t 0 such that may tae n ethe sgn geneatng ne f the Cases, 2 4. < t + 7

19 CES wth σ = c. The ppetes f pductn functns leadng t the Cases 6, 7 and 8 n Table 2 can be ndentfed n analgy t the ppetes n the Cases, 2 and 3. Nte fnally that f > and L 0 ρ > (Case 3), t + < 0 s nt a feasble utcme. If < and L 0 ρ < (Case 6), t + > 0 s nt a feasble utcme ethe. A necessay cndtn f the Cases 3 and 6 t ccu s that thee ae technlges exhbtng Y > 0 f sme cuntes and Y < 0 f thes. In thse specal tax-cmpettn equlba n whch the tax ates t,..., tn ae unfm acss cuntes (Case 4 and tvally Case 5 f Table 2) 7 the allcatns f the tax-cmpettn equlbum E and the n-tax tade equlbum. Tax cmpettn then has n allcatve mpact at all. E cncde as a cnsequence f Ppstn Cnsde next the tax ncdence n the plausble case f tax cmpettn whee tax ates dffe acss cuntes. Fcusng n emplyment we bseve, athe unexpectedly, that vaatns n emplyment ae clea n sgn unde all cndtns: Unless tax ates ae unfm (see abve) emplyment ethe mpves shns. F ethe sgn f the dffeence emplyment declnes [nceases] f ( ) 0 [ t ( t + ) > 0 t t + < ]. The cnstellatn t < 0 and ( t ) 0 + > vce vesa ccus unde tw cndtns: () t and must exhbt ppste sgns and () t must be suffcently clse t ze. Whle all sgns f changes n fact nputs and ncme ae clea, the changes n the level f utput ae ambguus n the Cases 3, 7 and 8 whee sgn ( ) ( ) s negatve. The stng esult f Ppstn 7 s that cuntes may lse fm tax cmpettn f the scena f tade wthut taxatn s taen as the baselne. Unde the cndtns specfed n Ppstn 7 cunty may suffe a welfae lss and such a lss can ccu unde vaus assumptns. Cunty 's technlgy may satsfy Y > 0 (Cases 2 and 6) Y < (Cases 3 and 8) the cnstellatn > (Cases 2 and 3) 0 < (Cases 6 and 8) may be gven. T see the dvng fce f the welfae lss, cnsde Case 2 n Table 2 as an example. t + wll be pstve, f cunty 's tax ate t s negatve but elatvely small n abslute value (see 7 Case 4 apples, e.g., when all cuntes pduce wth dentcal Cbb-Duglas functns when all cuntes pduce wth dentcal CES functns and the gd wage ates ae dentcal. See u cmments n Ppstn 5. 8

20 the pf f Ppstn 7) 8. Hence n tax-cmpettn equlbum we fnd that ( +, ) = L ( + t +, w ) < L ( w ) L t w, because L ρ < 0 s pesuppsed n Case 2, and ( +, ) = ( + +, ) < ( ) K t w K t w K, w because K ρ < 0. The staghtfwad mplcatn s that the utput shns ( y ). On the the hand, snce < y <, cunty 's value f expts ses the cst f ts mpts declnes. Hweve that patal ncme ncease s smalle than the lss fm educed pductn because the pe unt cst f captal as an nput, + t +, s hghe than, the pe unt evenue fm nceased captal expts fm educed captal mpts. 5 Fm autay t tax cmpettn Ppstn 3 scutnzed the shft fm autay t the n-tax tade equlbum and Ppstn 7 analyzed the allcatve mpact f mvng fm the n-tax tade equlbum t the taxcmpettn equlbum. It s theefe necessay cmbnng bth steps n an efft t answe the questn what the allcatve cnsequences ae f ndvdual cuntes f mvng fm autay t tax cmpettn. As f ncme changes, clse nspectn f the Tables and 2 eveals that the sgn f thse changes s unclea n seveal cases. Obvusly, snce the sgn f the dffeence x xa s unclea n the Cases and 6 f Table, the net welfae change fm autay t tax cmpettn s bund t be ambguus. Hweve, even f the patal welfae effects n the Tables and 2 ae clea n sgn, the net effect s als ambguus, wheneve the patal effects exhbt ppste sgns. Althugh t s nt pssble t fully explt the cmplex nfmatn pesented n the Ppstns 3 and 7, we estct u attentn t changes n unemplyment and welfae and select sme specfc cases n Ppstn 8 (Lsses fm autay t tax cmpettn). Cnsde the tanstn f the n-cunty ecnmy fm autay t tax cmpettn. () Cunty suffes a welfae lss, f all cuntes use dentcal Cbb-Duglas pductn functns, (7), and f > ω hlds. 8 F example, f all pductn functns ae Cbb-Duglas and dffe acss cuntes wth espect t the paamete β, we cnclude fm > 0 (whch s pesuppsed n Case 2) and t = β < 0 fm equatn () that t = t + = β + > 0, f β, β,..., β. 2 n β s a suffcently small cmpnent n ( ) 9

21 () Cunty suffes a welfae lss, f the pductn functns satsfy Y > 0 f all (as e.g. n case f Cbb-Duglas CES wth σ < c ), f a = and f t s small enugh n abslute value elatve t the the cuntes' ptmal tax ates. () Cunty suffes a welfae lss, f a (mplyng > > ), f ts pductn functn σ > c + w + q w, and s CES satsfyng ( )( )/ - ethe f Y < 0 f all and t s small enugh elatve t the the cuntes' ptmal tax ates, - f all cuntes use the same CES pductn functn as cunty and wage ates ae the same acss cuntes. (v) In all scenas f the Ppstns 8, 8 and 8 cunty suffes fm nceasng unemplyment n the tanstn fm autay t tax cmpettn. The pncpal message f Ppstn 8 s that captal maet lbealzatn wth tax cmpettn can lead t sng unemplyment and welfae lsses unde vaus cndtns. Welfae lsses can be deved by cmbnng n vaus ways the cuntes' fundamentals,.e. the wage ates, captal endwments and pductn technlges. It s cnceded that all cases pesented n Ppstn 8 mae use f cndtns that ae me less estctve. Yet all these cndtns ae suffcent but nt necessay. In u vew, t s theefe safe t cnectue that lsses fm tax cmpettn ae nt an elusve phenmenn. Tae f example Ppstn 8. Its ange and elevance - s cetanly lmted because the cndtn a = s vey specal, f t s fulflled at all f any cunty. Hweve, a welfae lss wll als ccu f the cndtn f Ppstn 8 s eplaced by a = a enugh. When the cndtn as lng as f the dffeence a s small (Case 4 f Table ) s weaened n ths nn-gus way, welfae lsses can als be dentfed n the Cases 3, 6 and 8 f Table 2. Meve, n all cases f unfm ptmal tax ates whch we dentfed n u emas n equatn () sme cuntes lse when mvng fm autay t tax cmpettn, f and nly f they lse n the tanstn fm autay t tade wthut taxatn. a = 7 Cncludng emas We have shwn that unemplyment maedly changes the mpact f captal maet lbealzatn and captal-tax cmpettn amng hetegeneus cuntes as cmpaed t the case f 20

22 pefectly cmpettve lab maets. Wth autay as the efeence scena, the ntductn f ntenatnal captal mblty and tax cmpettn tuned ut t have the ptental f educng the welfae and/ f exacebatng unemplyment n sme cuntes. Hence such cuntes wll nt be n fav f captal maet lbealzatn unless they succeed n emvng wage gdty. Snce we allwed cuntes t dffe wth espect t captal endwments, gd wage ates and pductn technlges, thee s a geat vaety f utcmes and welfae changes whch can hadly be chaactezed cmpletely. Nnetheless, we dentfed a numbe f specfc cases whee cuntes suffe hghe unemplyment and a welfae lss dung the tanstn fm autay t tax cmpettn and taced the easns f that utcme. As culd be expected, less geneal assumptns n pductn functns yelded me nfmatve esults. F example, n case f Cbb-Duglas technlgy we wee able t fully chaacteze the allcatve dsplacement effects. We shwed that n the tanstn fm autay t tax cmpettn cuntes fae the bette, cetes pabus, the geate s the captal endwment the lwe s the gd wage ate. The gd-wage assumptn s a vey smple and case way t mdel unemplyment gven the geat vaety f sphstcated and cmplex thees f nn-cmpettve wage fmatn develped n lab ecnmcs (Ncel 990). One small step n elaxng that assumptn n futue eseach w wuld be t etan dwnwad gdty but allw f upwad flexble wages. Me cmplex and aguably me ealstc lab maet thees have aleady been emplyed n sme studes f captal-tax cmpettn wth lab-maets mpefectns sme f whch we have efeenced n the Intductn. Hweve, as we pnted ut, nne f these studes tacles unemplyment and hetegeneus cuntes. The tade-ff between ealstc cmplexty n mdelng, tactablty and nfmatve nsghts appeas t necesstate and waant cmpmses. Refeences Al, M., Lete-Mnte, M., and Llyd-Baga, T. (2009), "Unnzed lab maets and glbalzed captal maets", Junal f Intenatnal Ecnmcs 78, Anssn, T., and Wehe, S. (2008), "Publc gds, unemplyment and plcy cdnatn", Regnal Scence and Uban Ecnmcs 38, Begstöm, V., and Panas, E. E. (992), "Hw bust s the captal-sll cmplementaty hypthess?", The Revew f Ecnmcs and Statstcs 74,

23 Bucvetsy, S. (99), "Asymmetc tax cmpettn", Junal f Uban Ecnmcs 30, 7-8 DePate, J. A., and Myes, G. M. (994), "Stategc captal tax cmpettn: A pecunay extenalty and a cectve devce", Junal f Uban ecnmcs 36, Duffy, J., Papagegu, C. and Peez-Sebastan, F. (2004), "Captal-sll cmplementaty? Evdence fm a panel f cuntes", The Revew f Ecnmcs and Statstcs 86, Eatn, J., and Panagaya, A. (979), "Gans fm tade unde vaable etuns t scale, cmmdty taxatn, taffs and fact maet dsttns", Junal f Intenatnal Ecnmcs 9, Egget, W., and Gee, L. (2004), "Fscal plcy, ecnmc ntegatn and unemplyment", Junal f Ecnmcs 82, Fuest, C., and Hube, B. (999), "Tax cdnatn and unemplyment", Intenatnal Tax and Publc Fnance 6, 7-26 Glches, Z. (969), "Captal-sll cmplementaty", The Revew f Ecnmcs and Statstcs 5, Kemp, M., and Negsh, T. (970), "Vaable etuns t scale, cmmdty taxes, fact maet dsttns and the mplcatns f tade gans", The Swedsh Junal f Ecnmcs 72, - Kessle, A.S., Lülfesmann, C., and Myes, G. M. (2003), "Ecnmc vesus pltcal symmety and the welfae cncen wth maet ntegatn and tax cmpettn", Junal f Publc Ecnmcs 87, Lete-Mnte, M., Machand, M., and Pesteau, P. (2003), "Emplyment subsdy wth captal mblty", Junal f Publc Ecnmc They 5, Lcwd, B., and Mas, M. (2006), "Tax ncdence, maty vtng and captal maet ntegatn", Junal f Publc Ecnmcs 90, Ncell, S. (990), "Unemplyment: A suvey", Ecnmc Junal 00, Ogawa, H., Sat, Y., and Tama, T. (2006), "A nte n unemplyment and captal tax cmpettn", Junal f Uban Ecnmcs 60,

24 Pealta, S., and van Ypesele, T. (2005), "Fact endwments and welfae levels n an asymmetc tax cmpettn game", Junal f Uban Ecnmcs 57, Sat, Y., and Thsse, J.-F. (2007), "Cmpetng f captal when lab s hetegeneus", Eupean Ecnmc Revew 5, Wlsn, J. D. (986), "A they f nteegnal tax cmpettn", Junal f Uban Ecnmcs 9, Wlsn, J. D. (99), "Tax cmpettn wth nteegnal dffeences n fact endwments", Regnal Scence and Uban Ecnmcs 2, Zdw, G. R., and Meszws, P. (986), "Pgu, Tebut, ppety taxatn, and the undepvsn f lcal publc gds", Junal f Uban Ecnmcs 9, Appendx Pf f Ppstn. In the tax-cmpettn equlbum { n ( ),..., n} = ) E : = t,..., t,, x, y,, the captal maet s n equlbum, by pesuppstn: = K ( + t, : = θ w. Cnsde ], [ θ and defne t : = t + f all and θ. Then we bvusly have = K ( t, w) K ( t, w) + = + Pf f Ppstn 2. Unde cndtns f pefect cmpettn the elastcty f substtutn eads σ = dq w / q d w /, whee : q =. Reaange ths equatn t btan σ wˆ σ ˆ = ˆq. ( ) Futheme, cnsde σ wˆ ˆ ˆ ( ) qˆ = as well as ˆ cw + q c q = σ wˆ σ ˆ w + q w + q σ ˆ = ˆq t btan afte sme eaangements f tems. n ˆ ( σ c ) w + q ˆ w = + σ w + cq σ w + cq w ˆ, whee ˆ = ˆa ( ) c : = / b >. If we set = and hence fllws. That pves Ppstn 2. ˆ = ˆ n that equatn, and = a 23

25 Pf f Ppstn 3. Ppstn 3: Dente by x and x a the natnal ncmes when the nteest ate s and, espectvely. Obvusly, that mples a x xa = f = a, and f dx / d s mntne n we ae able t detemne the sgn f the dffeence x xa. Dffeentatn f x wth espect t yelds dx ( wl d = ) (A) and dx d wl = =. Suppse that > a and 0. Then a L > ( ) ( a ) = K,w < K,w = dx and theefe 0 d >, pvng x > xa f the case 2 n Table. Alng the same lnes we shw that x > xa n the cases 3 and 7 and x < xa n case 5 n Table. In the cases and 6 the sgn f dx / d s unclea. Ppstn 3: (a) We fst bseve that the tem y tuns nt y = q = b α ( ) q when the pductn functn s Cbb-Duglas. Inteest ates may dffe acss cuntes n autay but they ae unfm n the asscated ze-tax equlbum. We als nw that b α b α q = q b α b = q =, because α q α = s the fst- α de cndtn f pft maxmzatn. Hence Fm ths nfmatn the equatns (, ) Y αγ b α y = =, whee γ : =. α αγ α = and a Y (, a ) a = (A2) fllw, whee the ndexes a and efe t the autay equlbum and the ze-tax equlbum, espectvely. We wte the dffeence n ncme f cunty autay t fee-tade as and cnsde (A2) t btan a ( αγ ) + αγ fllwng a swtch fm (, ) (, ) ( ) x = x x = Y Y a a a a αγ ( αγ ) αγ ( ) ( αγ ) ( ) x x a = a + + αγ ( ) = ( αγ ) = + +. (A3) 24

26 a (b) Next we detemne the vaables, and f the Cbb-Duglas pductn functn. The fst-de cndtns f pft maxmzatn β = yeldng αw α β = α l and w α β β l = mply β β β β c α β w = 0, whee c : =. (A4) b In autay, we have a = such that (A4) s tuned nt β β β β ( β ) c = w. (A5) αβ a In case f fee tade we cnvet (A4) nt ( β) c βc ( β) c βc = w. Inve the equlbum α β cndtn n the wld captal maet, = ( ) c βc ( ) α, equvalently, β β c βc β w, t btan = = c ( ) c ( ) β c c w β β β : = = α β and n n β β β α β βc c w c = n β β β c w ( ) c β ( β ) c = αβ (A6) n T detemne tems, we cmbne (A4) and (A6) whch yelds, afte sme eaangement f β c nw = ω, whee ω : =. (A7) β c w (c) Inve fm (A5) and fm (A6) cmbned wth (A7) and bseve that a β β β β > β β < a ( ) ( ) ( β c ) c β > β β αβ w ω αβ w < ω <. (A8) > (d) We nw nset (A5), (A6) and (A7) n (A3): αγ = [ ] x c c ω κ ω κ β β ( + αγ ) αβ ( β) ( β) + β ( β ) c β w 25

27 = ( β) c ( β) c δ ω + αγ ( + αγ ) a, (A9) whee β β αβ δ : = and ω β ( β ) c β w κ : =. Fm (A9) t s staghtfwad that x 0, f = ω = κ ω =. T specfy hw αγ x espnds t changes n cnsde the devatves f (A9) ( αγ ) d x d = αδ ( β ) c ω β b α (A0) ( ) ( β ) 2 αγ 2 αδ c β = ω d ( β ) c α α. (A) d x b These devatves mply ( αγ ) d x > > b 0 ω < < d α and 2 ( αγ ) d x > β b 0 2 > ω < + <. d α α ( αγ ) d x d b = 0 s attaned f = α ω b and nsetng = α ω n (A) yelds d ( x ) ( β ) 2 2 c 2 2 αγ αδ β = > 0 d ( β ) c α. Hence αγ x attans ts unque mnmum at b = <. We have shwn abve that x = 0, f =. F ths value f α ω ω ω (A0) tuns nt ( αγ ) d x d = αβ γ δ ( β ) c 0 > β. Theefe, at ts mnmum αγ x s negatve. Meve, snce < 0 f = 0 accdng t (A9), we cnclude that > 0 f x x 0, ω, = 0 f = ω and < 0 f ω, nω. x x d = > f all [, ] a Ppstn 3: In Case 6 wl ( ) 0 x < x. Snce CES mples L a dx = ( σ c ) w + q dx we fnd that 0 s suffcent f d > hlds, f f all [, ] a 26

28 σ w q > c + whee φ : = φ w + σ a w and q : = a σ. Snce s deceasng and φ s nceasng n, t fllws that [, ] a f all φ φ. That pves Ppstn 3. Pf f Ppstn 4. The stategy f pf s smla t that f Ppstn 3. Cunty 's a ncme s xa = y = α m β n autay and a α β ( ) ( ) x = y = m (A2) unde fee tade. In the latte case pft maxmzatn mples α β α m =. (A3) Cmbnng (A2) and (A3) yelds α β α β α β α α = + ( α ) x m m m α m β = + α α m β. Hence ( α) α α β α β α β a. x = x x = m + m m (A4) Next we shw that = m. Fm (A3) we have α = β αm. Invng = fm (5) yelds β α m α =, α α β α = m and n α α β ( ) ( α ) = α m α n. Inset fm the last equatn n α = β αm t btan = m. Use ths nfmatn t tun (A4) nt ( α) α α α β α α β α β x = m m + m m m. (A5) It s staghtfwad fm (A5) that x = 0 f = m. T specfy hw x espnds t changes n when s ept cnstant, cnsde the devatves f (A5), d x d ( ) 2 β α α α d x = αm m and ( ) 2 2 = β α α α m. d 27

29 Snce the fst devatve mples d x d 0 m and > > < < 2 d x > 0, x fm (A5) d attans ts unque mnmum at = m. Pf f Ppstn 5. Equatn (0) s staghtfwad fm the fst-de cndtn f maxmzng (4) wth espect t t : X YK YL K t = ρ + l ρ = ( ) ρ ρ ρ K + wl ρ = tk ρ ρ + wl = 0. () fllws fm cmbnng the equatn (0) wth (6) and (7) and the petanng fact demand functns. Pf f Ppstn 6. Suppse that L ρ < 0 and hence t < 0. Statng fm 0, successve eductns f that L ρ > 0 t and hence decease t ρ = + and theefe ncease L ( t, w) t = +. Suppse next t > 0. Statng fm t = 0 successve nceases n ncease ρ = + t and theefe als ncease L ( t, w) +. Pf f Ppstn 7. The pf pceeds n seveal steps. Fst we pve the Clam: Asscated wth the equlbum E s an 'auxlay' cnstant-tax tade equlbum 9 { n ( ) =,..., n} E : = t,..., t,, x, y,, wth the fllwng ppetes: x, y,, (a) ( ) = ( ), =,..., n x, y,, ( t t ) = ( t + t + ) =,..., n,..., n t,..., n, = and hence = =.,..., n n E cntans pstve and negatve tax ates, f the tax ates t,..., t n (b) ( t t ) n E dffe acss cuntes; thewse t =... = t n = 0. The exstence f the cnstant-tax tade equlbum E : = t,..., t n,, ( x, y,, ) { =,..., n} as defned n pat (a) f the Clam fllws mmedately fm Ppstn. Nte that (5) mples = K (, w ) = K ( t, w ) + = K ( t, w ) + + = K ( + t, w ). Pat (b) f the Clam pstulates that f tax ates t,..., tn dffe acss 9 E pvdes the nfmatn abut the tax ates needed f shftng fm the n-tax tade equlbum allcatn t the tax-cmpettn equlbum allcatn whle eepng unchanged the nteest ate pevalng n the ntax tade equlbum. 28

30 cuntes, the tax ates t,..., t need t cntan negatve and pstve cmpnents. Suppse n nt. Then sgn t mn = sgn t max wth t mn 0 and t max ae the mnmum and maxmum cmpnents f ( t t ) K (, w ) (, t max 0, whee t mn,..., n, espectvely. That bvusly mples = K + t w ) cntadctng the fact that E s an equlbum. If the tax ates t,..., tn ae unfm acss cuntes, the (asscated) tax ates t,..., t n ae als unfm because by defntn f t t s tue that t : = + t f all and satsfy t =... = t n = 0 by cnstuctn f E. Ths cmpletes the pf f the Clam. Wth ths nfmatn we pceed t establsh Ppstn 7. We fst fcus n the change n emplyment nduced by mvng fm E t E by pvng the equvalence ( t t ) > < 0 L ( + t, w) = L ( + t, w) > < (, ) L w. Cnsde the case that ( ) ( t 0 and t 0. We nw that t 0 L ρ 0 and theefe L + t, w L, w ). Smlaly, f t 0 and t 0, we have t 0 L 0 and theefe ( ) ( L + t, w L, w shw that ( t t ) 0 L ( + t, w ) L (, w ) The sgn f the dffeence equal t the sgn f ( t Kρ ) ) as well. Based n ths nfmatn t s staghtfwad t., snce K ρ < 0. (secnd clumn f Table 2) s easly calculated as beng It emans t pve the sgns n the last clumn f Table 2. Set ρ = + and dffeentate x = y + wth espect t t : t ρ dx dt = YK + YL K = t K + wl = t + K wl ρ ρ ρ ρ ρ ρ ρ K ρ. (A6) dx Fm t 0 dt = = wl dx fllws sgn t 0 sgn L = dt = and f dx / dt s mntne n we ae t able t detemne the sgn f the dffeence x x. Cmbne / wl ρ + t = + t ( ) t btan ( ) ( ) ( ) wl ρ wl ρ + t, w wl ρ + t, w = = K K + t, w K + t, w ρ ρ ρ K ρ fm (A6) wth = t and hence 29

31 dx dt ( ) ( ) ρ ρ = t t K = K We cnclude fm (A7) that x > x <. (A7) > and t < 0 (Case ) < and t > 0 (Case 7) > and t > 0 (Cases 2 and 3) < and t < 0 (Cases 6 and 8). The Case 4 n Table 2 s bvus. Pf f Ppstn 8. Ppstn 8 fllws fm cmbnng Case f Table wth Case 4 f Table 2 and Ppstn 3. Ppstn 8 fllws fm cmbnng Case 4 f Table wth Case 2 f Table 2. The fst pat f Ppstn 8 fllws fm cmbnng Case 6 f Table wth Case 8 f Table 2 and Ppstn 3. The secnd pat f Ppstn 8 fllws fm cmbnng Case 6 f Table wth Case 4 f Table 2 and Ppstn 3. ( w ) Pf f the clam: + βc βc n > 0 βc > w > w. w We eaange the nequalty w w > < 0 and btan w w > < 0 βc w βc w > n < n βc w < βc w n > n βc ( w ) w βc < > n + βc ( ) βc + w βc < βc w > n. (A8) Wth w > 0 f =,, n the nequalty (A8) pfs the clam. 30

32 CESf Wng Pape Sees f full lst see Twww.cesf-gup.g/wpT (addess: Pschngest. 5, 8679 Munch, Gemany, ffce@cesf.de) 2733 Fancesc Cnnella and Jachm Wnte, Sze Mattes! Bdy Heght and Lab Maet Dscmnatn: A Css-Eupean Analyss, July Samuel Bwles and Sanda Planía Reyes, Ecnmc Incentves and Scal Pefeences: A Pefeence-based Lucas Ctque f Publc Plcy, July Gay Butless, Lessns f the Fnancal Css f the Desgn f Natnal Pensn Systems, July Helmuth Ceme, Fuz Gahva and Pee Pesteau, Fetlty, Human Captal Accumulatn, and the Pensn System, July Hans Jale Knd and Fan Stähle, Maet Shaes n Tw-Sded Meda Industes, July Pamela Campa, Alessanda Casac and Pala Pfeta, Gende Cultue and Gende Gap n Emplyment, August Sebastan Gechet, Supplementay Pvate Health Insuance n Selected Cuntes: Lessns f EU Gvenments?, August Lef Danzge, Endgenus Mnpsny and the Pevese Effect f the Mnmum Wage n Small Fms, August Yan Dng and Jhn Whalley, A Thd Beneft f Jnt Nn-OPEC Cabn Taxes: Tansfeng OPEC Mnply Rent, August Valentna Bsett, Cal Caa and Massm Tavn, Clmate Change Mtgatn Stateges n Fast-Gwng Cuntes: The Benefts f Ealy Actn, August Chstna Felfe, The Wllngness t Pay f Jb Amentes: Evdence fm Mthes Retun t W, August Jög Fane, Chstan Kanzw, Wlfgang Lennge and Alexanda Väth, Efft Maxmzatn n Asymmetc N-Pesn Cntest Games, August Bun S. Fey and Pal Pamn, Mang Wld Hetage Tuly Glbal: The Cultue Cetfcate Scheme, August Fan N. Calend, Is Scal Secuty behnd the Cllapse f Pesnal Savng?, August Catena Lesegang and Mac Runel, Cpate Incme Taxatn f Multnatnals and Fscal Equalzatn, August 2009