Dynamic Controllability with Overlapping Targets: Or Why Target Independence May Not be Good for You

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1 Dynamc Conrollably wh Ovrlappng Targs: Or Why Targ Indpndnc May No b Good for You Ncola Acoclla Unvrsy of Rom La Sapnza Govann D Barolomo Unvrsy of Rom La Sapnza and Unvrsy of Tramo Andrw Hughs Hall Vandrbl Unvrsy and CEPR 3 March, 2006 Absrac. W gnralz som rcn rsuls dvlopd n sac polcy gams wh mulpl playrs, o a dynamc conx. W fnd ha h classcal hory of conomc polcy, sac or dynamc, can b usfully appld o a sragc conx of dffrnc gams: f on playr sasfs h Goldn Rul, hn hr all ohr playrs polcs ar nffcv wh rspc o h dynamc arg varabls shard wh ha playr. Or no Nash Fdback Equlbrum can xs, unlss hy all shar arg valus for hos varabls. W xnd hos rsuls o h cas whr hr ar also non-dynamc args, o show ha polcy ffcvnss (a Nash qulbrum) can connu o xs f som playrs sasfy h Goldn Rul bu arg valus dffr bwn playrs n hr non-dynamc args. W dmonsra h praccal mporanc of hs rsuls by showng how polcy ffcvnss (a polcy qulbrum) can appar or dsappar wh small varaons n h xpcaons procss or polcy rul n a wdly usd modl of monary polcy wh h possbly of arg ndpndnc. JEL Classfcaon: C72, E52, E6. Kywords: Polcy gams, polcy nffcvnss, sac conrollably, xsnc of qulbra, Nash Fdback Equlbrum.

2 . Inroducon Th ssu of h ffcvnss of publc polcy s cnral o conomc analyss. Th nal conrbuons by Tnbrgn, Thl and ohrs sad h condons for polcy ffcvnss, boh sac and dynamc, n a paramrc conx. In h las wo dcads a nw approach o conomc polcy problms has dvlopd, mmun from h Lucas (976) crqu, n whch h sragc nracons bwn h govrnmn, cnral bank and ohr agns ar modlld xplcly. 2 Howvr, absrac condons for polcy ffcvnss hav no bn sudd n ha conx unl rcnly. Acoclla and D Barolomo (2004, 2006) provd gnral condons for polcy nffcvnss and qulbrum xsnc n sac LQ-gams of h knd sad by h classcal hory of conomc polcy, and show how hs can b profably usd o dfn som gnral proprs of polcy gams. Ths papr xnds h sam da of conrollably o dynamc dffrnc gams, and n ha conx w consdr h mporanc of arg ndpndnc (as opposd o nsrumn ndpndnc) whch has bn a pon of parcular conrovrsy n monary polcy dsgn. Our approach s o consdr h Nash Fdback Equlbrum for LQ-dffrnc gams, 3 and drv condons for polcy nffcvnss and h qulbrum xsnc for ha cas. W hn dmonsra h usfulnss of our rsuls by showng how asly polcy ffcvnss, or a polcy qulbrum, can appar or dsappar wh small varaons n h xpcaons procss or h polcy rul n a sandard modl of monary hory llusrang, as w do so, how cran varaons n h problm can prm or ak away h opporuny for polcy makrs o opra wh dffrng arg valus for hr polcy objcvs. To do hs, w mak us of som proprs of spars marcs snc narly all conomc modls dsplay sparsnss. Th rs of h papr s organzd as follows. Scon 2 dfns basc concps and nroducs a formal framwork o dscrb LQ-dffrnc gams. Scon 3 drvs wo horms sang a suffcn condon for polcy nffcvnss and a ncssary condon for h qulbrum xsnc n h radonal Tnbrgn framwork. Scon 4 provds a formal rlaxaon of Tnbrgn (952,956); Thl (964). 2 Hughs Hall (984,986), Lvn and Brocnr (994), Aarl al (997), Engwrda al (2002), Pappa (2004). 3 No ha w ar concrnd hr wh dynamc conrollably n h sns of achvng cran arg valus n succssv prods of m, bu no n h alrnav (classcal) sns of rachng hos arg valus only afr a cran numbr of m prods has lapsd. 2

3 h wo horms for h cas of spars conomc sysms. Scon 5 llusras h applcaon of our rsuls o on of h mos wdly usd modls n monary hory. Th papr nds wh som conclusons and som das for furhr rsarch. 2. Th Basc Sup W consdr h problm whr n playrs ry o mnmz hr ndvdual quadrac crron. Each playr conrols a dffrn s of npus o a sngl sysm, whch s dscrbd by h followng dffrnc quaon: () x( + ) = Ax( ) + Bu ( ) N M whr N s h s of h playrs, x, s h vcor of h sas of h sysm; u m () s h (conrol varabl) vcor ha playr can manpula; and A M M and B M m () ar full-rank marcs dscrbng h sysm paramrs whch (for smplcy) ar consan. Th crron playr N ams o mnmz s + (2) J ( u, u,..., u ) = ( x ( ) x ) Q ( x ( ) x ) whr 2 n = 0 ( ) M x s a vcor of arg valus. For playr, h rlvan sub-sysm of () s: (3) x( + ) = Ax ( ) + Bjuj( ) j N whr ( ) M( ) M A and ( ) m() M B j ar appropra sub-marcs of A and B. W assum ha all marcs ar of full rank, and ha ( ) () hs assumpons s sraghforward. M m. Th conomc nrpraon of Th Nash Fdback Equlbrum can now b dfnd as follows. ( 2 n ) ( * ) ( * ( ) * ( ) ( ) *, 2,...,..., ( )) u Dfnon (Nash Fdback Equlbrum): A vcor u * ( ) u * ( ), u * ( ),..., u * ( )..., u * ( ) a Nash Fdback Equlbrum f ( ) J u J u u u u n = s for any ( ) and for any playr, whr u () s a fdback sragy gvn h nformaon a prod. 3

4 Opraonally, a fdback sragy mans ha a conngn rul (dpndn on h sysm s sa vcor) s provdd for ach playr, and ha h ruls hmslvs can b oband from h backward rcursons of dynamc programmng (Holly and Hughs Hall, 989: 76-79). 3. Th Goldn Rul and h Equlbrum Proprs In ordr o apply h radonal hory of conomc polcy o sudy h proprs of Nash Fdback Equlbrum, w frs rcall h radonal Tnbrgn da of sac conrollably: Dfnon (Goldn Rul): A polcymakr sasfs h Goldn Rul of conomc polcy f h numbr of s ndpndn nsrumns (a las) quals h numbr of s ndpndn args. Scond, w nd o rdfn polcy nffcvnss, snc s classcal dfnon 4 canno b manand n h ralm of mul-playr polcy gams whr polcs bcom ndognous varabls. Insad, h followng dfnon of nffcvnss can b appld. 5 Dfnon (nffcvnss): A polcy s nffcv f h qulbrum valus of h args ar nvr affcd by changs n h paramrs of s crron funcon. Conrollably, n h rms of h Goldn Rul of conomc polcy, nffcvnss and h Nash Fdback Equlbrum xsnc ar rlad hrough h followng wo horms. Thorm (nffcvnss): Provdd ha an qulbrum xss, f on playr sasfs h Goldn Rul, all h ohr playrs polcs ar nffcv wh rspc o h arg varabls shard wh ha playr. Proof. W sar by assumng ha h polcymakrs valu funcons ar quadrac, 6 V( x) = ( x() x) P( x() x), whr P ar ngav dfn symmrc marcs so ha hr ar no rdundan args (and for h sak of smplcy, m ndxs ar omd). By usng h ranson law o lmna h nx prod sa, h n Bllman quaons bcom: 4 Th classcal dfnon of polcy nffcvnss mpls ha auonomous changs n h polcymakr s nsrumns can hav no nflunc on h args (Hughs Hall, 989). Howvr ha dos no allow for h possbl blockng movs by ohr polcy playrs n h gam. W hrfor adop a mor gnral dfnon hr. 5 S Gylfason and Lndbck (994). 6 Indd, w know ha h valu funcon mus b convx for a soluon o xs (s.g. Başar and Olsdr, 995; Sargn, 987: 42-48; Docknr al, 2000). S also Engwrda (2000a, 2000b) for a mor advancd xposon. 4

5 x x P x x = x x Q x x + Ax + Bjuj P Ax + Bjuj u j N j N (4) ( ) ( ) max ( ) ( ) A Nash Fdback Equlbrum mus sasfy h frs-ordr condons: BPB u =BP A x x + Bju j j N/ whch ylds h followng polcy rul: ' ' ' ' (6) u =( B PB ) B PA( x x ) ( B PB ) B P B u (5) ( ) ( ) j j j N/ Now, o dmonsra Thorm, w focus (whou loss of gnraly) on playr. If playr sasfs h Goldn Rul, hn m() M ( ) Equaon (6) hn bcoms: n = j j= 2 (7) ( ) u B A x x B B u snc P s also nonsngular. Tha mpls: (8) x( ) x + = for all [ 0, + ] = and j B ( ) M( ) M s squar and nonsngular. Thus, f a Nash Fdback Equlbrum xss, h valu of h arg vcor x s m nvaran and only dpnds on h prfrncs of playr, snc n ha cas condon (7) wll hold for all prods [ ] 0, +. Ths compls h proof of Thorm. Thorm 2 (non-xsnc): Th Nash Fdback Equlbrum of h polcy gam dscrbd dos no xs f wo or mor playrs sasfy h Goldn Rul and a las wo of hm shar on or mor arg varabls. Proof. To prov Thorm 2 w only nd o show ha f also anohr playr (.g. playr 2) sasfs hs/hr Goldn Rul, h qulbrum dos no xs. Assum a soluon xss and ha hs soluon mpls h followng opmal polcy vcor u * ( u * * *, u2,..., un ) * * * gvn u () u (), () (9) 3,..., n = a m. Thn, * u and u () mus sasfy h followng sysm (oband from (5)): 2 ( ) * A x x + Bju j B PB B 22PB 2 2 u B P j N/ * BPB 2 B22PB 2 22 u = 2 B22P 2 A2( x2 x2) B2 ju + j j N/2 5

6 Noc ha h frs-parond marx of (9) s always squar; and ha f boh playrs sasfy hr Goldn Rul, hn all h marcs hrn ar also squar. Now assum ha boh playrs shar h sam arg varabls,.. x = x2. In hs cas, w hav A = A2 and Bj = Bj for {, 2} and j N. Th frs-parond marx of (9) hrfor has a zro drmnan ( B B2 = and B2 B22 no xs and, hrfor, = ) and canno b nvrd. Hnc, a coupl ( * *, 2) * u canno b h soluon, as clamd by h horm. u u sasfyng (9) dos Convrsly, consdr now arg spac nsad of nsrumn spac. If h frs wo playrs boh sasfy h Goldn Rul, s asy o show ha by subsung h frs ordr condon for u 2 from (5) no (7) for u, h frs ordr condons for boh playrs canno boh b sasfd unlss hy boh shar h sam arg valus,.. unlss h followng holds: (0) A( x x ) = or x =. 2 0 x 2 7 Nx, consdr h cas whr h frs wo playrs do no shar all hr args. Whn h sysm can b conrolld, hs cas can b solvd by dcomposng h problm of ach playr no wo muually nrdpndn problms: (A) o mnmz h quadrac dvaons of h shard args from hr shard arg valus usng an qual numbr of (arbrary slcd) nsrumns from u, assumng ha non-shard arg valus can b rachd; (B) o mnmz h quadrac dvaons of h non-shard args from hr arg valus wh rspc o h rmanng nsrumns, assumng ha h shard args ar sasfd (and qual o hr arg valus bcaus of h Goldn Rul). Gvn (0), h mpossbly of a soluon now mrgs from h frs-ordr condons for h frs of h wo problms (A). 8 Hnc, as clamd, f a las wo playrs conrol hr sub-sysms and shar a las on arg varabl, h Nash Fdback Equlbrum canno xs. Commn : Thorm gvs a suffcn condon for polcy ffcvnss. Bu hs dos no assur h xsnc of an qulbrum, whch may fal o occur. By conras, Thorm 2 gvs a ncssary condon for an qulbrum o xs snc sas a suffcn condon for 7 x x 2 s no possbl hr bcaus A s of full rank. W consdr h cas whr r(a)<m n h nx scon 8 Noc ha, bcaus h args ar conrollabl, hs rsul s ndpndn of h assgnmn of h nsrumns. 6

7 h oppos. Howvr, may no b suffcn for xsnc. 9 sasfd, Thorm 2 s no (and vc vrsa). No also ha f Thorm s Commn 2: Th mporanc of hs rsuls for conomc polcy s xmplfd by Thorm 2. I says ha f wo ndpndn polcy auhors, say fscal polcy makrs and h cnral bank, dcd o pursu dffrn nflaon args, hn h Nash qulbrum may no xs and h conomy may no b abl o rach an qulbrum whn boh polcy makrs ry o opmz hr polcs. Th condons for hs o happn ar no parcularly srngn. In ohr words, xcp for cran spars conoms dscussd blow, arg ndpndnc may b unhlpful no bcaus fscal and monary polcs canno b coordnad proprly, bu bcaus h undrlyng qulbrum canno b rachd f boh polcy makrs ry o opmz hr polcy chocs ndpndnly. 4. A Gnralzaon: Spars Economc Sysms W now rlax Thorms and 2 n a way whch may prov mporan n conomc modls, bu whch s lss ofn obsrvd n physcal sysms. Mos conomc modls dsplay sparsnss. Tha s o say, whn wrn n srucural form, hy ypcally rla ach ndognous varabl o jus on or wo ohr ndognous varabls; and a small numbr of laggd ndognous varabls, conrol varabls, or prdrmnd varabls. In ha cas, h srucural modl from whch () s drvd can b wrn as () x( + ) = Cx( + ) + Dx() + Fu () N whr C, D and F ar spars marcs (prdomnanly zro marcs, wh jus a fw nonzro lmns pr row). For h sak of smplcy w assum ha all h playrs shar all h arg varabls (as dscussd n h prvous scon, hs assumpon can b asly rlaxd). In ha cas, h ndx on marcs A can b rmovd, oghr wh h scond ndx on h B marcs. In hs suaon, () has (2) = ( ) and = ( ) A I C D B I C F 9 Exsnc s a rahr complx mar n hs conx. For xampl, bng n a dynamc sysm, sably s also rqurd. S Engwrda (2000a, 2000b). 7

8 whr ( I C) xss by vru of h normalzaon n (), rrspcv of h dfnons of C, D and F. Bu A and B ar now no longr of full rank. Howvr, w can pr-mulply () by a prmuaon marx T; and nsr T T (whr T = T, a propry of prmuaon marcs) no h frs wo rms on h rgh of (). Ths allows us o spara hos arg varabls whch ar affcd drcly by dynamc adjusmns ovr m, from hos whch ar no. W g h rordrd sysm: (3) x ( + ) = Ax ( ) + Bu ( ) whr x() = Tx() N, ( ) and ( ) A= I TCT TDT B = I TCT TFT. Bu hs formulaon A 0 hn mpls A = A2 0 whr A s a squar full rank marx of ordr l, A2 (Ml) l, and whr l s h numbr of arg varabls n h sysm ha ar drcly subjc o dynamc adjusmns (.. h rank of C). Hnc Ml arg varabls ar no drcly subjc o dynamc adjusmns. Thy appar n h scond sub-vcor of x (). Now w can rwork Thorm 2. W g: Thorm 3 (nffcvnss and non-nuraly n spars conoms). If h args of on (and only on) playr whch ar drcly subjc o dynamc adjusmns also sasfy h Goldn Rul among hmslvs, hn h polcs of all ohr playrs wll b nffcv wh rspc o hr dynamc args. Convrsly, no Nash Fdback Equlbrum xss n hs polcy gam f wo or mor playrs sasfy h Goldn Rul for hr dynamc args unlss hy happn o shar arg valus for hos varabls. Bu h Nash qulbrum may sll xs f h Goldn Rul s sasfd and h arg valus for h non-dynamc args dffr across playrs; and h polcs of h ohr playrs wll sll b ffcv for hos args vn f on (or som) playr sasfs h Goldn Rul. Proof. Rcall ha, unl now, f playrs and 2 sasfy h Goldn Rul, hr racon funcons mply A( x x ) ( x ) A x (no ha 2 = 0 =. In a spars conomc sysm, h quvaln condon s 2 0 sll xss f s squar, and h Goldn Rul appls o playr B ). W now wr x as h frs l lmns of x (corrspondng o h frs l lmns, or dynamc args, n x ) and x 2 as h rmanng Ml lmns of x. Smlarly, w dfn x 2 8

9 and x 22 o b h assocad sub-vcors of x 2. Ths parons conform o ha n A. Our horm now follows from h fac ha boh A ( x x ) = and A ( x x ) 2 0 =, and hnc x = x2 (snc A and A 2 dffr n dmnson and A s of full rank), wll b ndd o sasfy h rplacmn for (0) n hs cas: namly, A( x x ) s conssn wh A ( x x ) 2 = 0. Tha compls h proof.. Howvr x x = 0 5. An Exampl W urn now o som smpl xampls o llusra h usfulnss of hs rsuls n pracc. Consdr an conomy ha can b dscrbd by h followng wll-known modl: (4) y = ρ y + απ ( π ) β( π ) + ε (5) = c0 + c( π π ) + c2y (6) π π = d( π π ) wh 0< d <. Equaon (4) s an laboraon of h sandard workhors modl whch has bn par of h hory of monary polcy snc h Barro-Gordon modl was nroducd n 983. I consss of a shor run Phllps curv wh prssnc ( ρ 0 ), s whn a sandard Lucas supply funcon (long run Phllps curv) and laborad o nclud h ffcs of nrs ra changs on oupu. I could hrfor b nrprd as hr a dynamc opn conomy Phllps curv; or a nw Kynsan IS curv wh dynamcs. In ha conx, y s h dvaon of oupu from s naural ra (h oupu gap); π s h ra of nflaon, and π h xpcd ra of nflaon n h prva scor; s h nomnal ra of nrs ( ra of nrs); and ε a supply shock wh man zro and consan varanc. π, h corrspondng ral Th chf polcy nsrumn (conrol varabl) n hs xampl wll b. Equaon (5) s hrfor a Taylor rul: c 0 s a consan rm, rflcng conrol rrors or h qulbrum ra of nrs; π s h arg nflaon ra, and drmnacy (h Taylor prncpl) suggss c >. Fnally, (6) says ha xpcaons ar formd by h adapv prncpl (w mprov on ha blow); and all paramrs, n all hr quaons, ar dfnd o b posv. Ths modl has lags n all hr ndognous varabls: y, π and π. 9

10 To oban h rducd form of (4)-(6), corrspondng o (), w rnormalz (5) on π. Ths hn ylds, corrspondng o (), (7) α α β y ρ 0 0 y β ε cc 2 0 π = π + c + π c. 0 0 π 0 d d π 0 0 From hr w can drmn h valu of A for hs modl, usng (2). I s ρ d( α β) ( d)( α β) (8) A=Δ ρc2c d( α β) c2c ( d)( α β) c2c 0 d( + c2cα) ( + c2cα)( d) whr Δ= + αcc, h drmnan of h Jacoban marx n (7), s nonzro as long as 2 c 0 2 α c +, a condon whch always holds. Bu (8) canno b r-organzd o dlvr zros n h rgh hand column (h condon ha allows on arg o b dcoupld). Hnc, f hr ar mulpl polcy makrs n hs modl, hy would hav o s dncal arg valus for h oupu gap, h nflaon ra, and h nflaon xpcaons ha hy wan h marks o hav, f hr s o b an qulbrum for h polcy gam; and f hos args ar o b conrollabl. Morovr hr could b compng polcy makrs, wh h cnral bank usng nomnal nrs ras o conrol nflaon bu anohr auhory (h govrnmn) sng h long rm nflaon arg π ; or whr fscal polcy makrs ry o modra h ffcs of monary polcy by mans of ax braks or suabl budgary polcs; or whr polcy makrs ry o nflunc nflaon xpcaons by sng nrmda args, or by alkng h xchang ra up or down (hs would rqur an xra consan rm n (6) and hnc h hrd quaon of (7)). Ths ar all suaons ha ar common n pracc. Th Bank of England s an xampl of h frs cas; h US, or Ialy and Franc n h Euro s an xampl of h scond; and Turky or many hgh nflaon counrs of h hrd. Nx w consdr a varan on hs xampl. Suppos, bcaus of daa rvsons, polcy makrs rcognz ha s dffcul o masur h currn oupu gap accuraly, and us a mor rlabl pas masur y n quaon (5) nsad. Suppos also ha h prva scor, prhaps for smlar rasons, fnd ha mprfc xpcaons nroduc oo much volaly no 0

11 h sysm, and fnd chapr o us smpl laggd xpcaons nsad: π = d. Th modl now has no lags n π. Solvng hrough () and (2), w now g (9) ρ d( α β) 0 A =Δ c2c ρ d( α β) c2c 0. 0 d( + c2cα ) 0 Ths allows our ponal polcy makrs o dsagr on h (nrmda) nflaon args hy announc o h marks ( π ), bu sll hav conrollabl arg varabls and a rachabl Nash qulbrum. Ths happns bcaus hr s now a dlay bfor som of h arg varabls ar affcd by h polcy nsrumns. So hy can s polcs o rach som agrd args frs, allowng dffrncs o prss lswhr, and hn us hm agan o rach h ohr arg valus lar. π A srongr vrson of hs rsul s oband f h conmporanous oupu gap s rsord o h Taylor rul (5), bu xpcaons ar raonal. Tha mans (6) s rplacd by (20) π = π v whr v s a random xpcaons rror wh man zro. Ths s h form of h modl ha mos horss would favor. I mpls ha w now hav no lags n hrπ or π, and ha (2) 0 0 A =Δ Γ c2c 0 0 cc ( 2 ) Γ= Δ+. Evdnly, n hs modl, h polcy makrs could hav cc α β ρ whr ( ) dffrn arg valus for boh π and π and sll rach a Nash qulbrum oucom for hr arg varabls. Onc agan, dffrn polcy makrs (n govrnmn and h cnral bank) could hav arg ndpndnc (and hnc dffrn nflaon args) and sll xpc o rach an qulbrum poson. Bu could nvrhlss prov o b a dram snc f xpcaons ar no raonal (bcaus s oo xpnsv o gahr h ncssary nformaon accuraly), or f s dffcul o masur h currn oupu gap rlably, hn hy wll no b abl o rach hs dalzd qulbrum or ndd any ohr soluon whch allows boh o opmz hr polcs.

12 6. Concludng Rmarks Ths papr rprsns an amp o gnralz som rcn rsuls dvlopd n sac polcy gams o a dynamc modl. W fnd ha h classcal hory of conomc polcy can b usfully appld o a sragc conx of dffrnc gams: namly, f on playr sasfs h Goldn Rul, hr all h ohr playrs polcs ar nffcv wh rspc o hr dynamc arg varabls shard wh ha playr or no Nash Fdback Equlbrum xss whou xac agrmn on all h (dynamc) arg valus. W llusra h usfulnss of our rsuls wh rfrnc o a modl ncorporang a Taylor rul, a dscrpon of xpcaons formaon and a rlaon ha can b nrprd as hr a dynamc opn conomy Phllps curv or a Nw- Kynsan IS curv wh dynamcs. Small varaons n h modl spcfcaon can brng, or ak away, polcy ffcvnss allowng h polcy makrs h laud o dsagr on non, on or svral of h xac arg valus n hr (common) objcvs. Lkws, our gnral rsuls show how asly arg ndpndnc, n a world whr nsuonal and polcy ndpndnc ar consdrd mporan, can prov o b counrproducv f polcy makrs ry o opmz hr chocs. Ths rsuls lad o hr obvous opcs for furhr rsarch. Frs. our horms ar basd on a spcfc concp of srong conrollably, usually known as sac conrollably: ha s, h arg valus ar nndd o b rachd n succssv m prods. I s wll known, n fac, ha n gnral fwr nsrumns han args ar ndd o conrol a dynamc sysm whn h args ar o b rachd only afr a gvn numbr of m prods hav lapsd. Onc h horms ar rformulad n rms of ha form of dynamc conrollably, may b possbl o dfn mor gnral and lss srngn condons han hos dscussd hr. Scond, w hav nroducd conmporanous and adapv xpcaons basd on h argumn ha backward lookng modls f h daa br han forward lookng modls (Gal and Grlr, 999). Bu ha may conflc wh acual pracc. For xampl, Cnral Banks rac o nflaon forcass, and h prva scor may b forward lookng n hr wag bargans or ass holdngs. I would b usful o chck f our xampls connu o apply n such cass. Thrd, our rsuls ar dsgnd o dal wh cass of dvolvd dcson makng whn a sngl conomy, whr h govrnmn, cnral bank, mployrs and unons ar concrnd wh oupu, mploymn, nflaon for ha conomy and hav a vary of fscal, monary and labour mark nsrumns o rach hr own args. I would b nrsng, 2

13 hrfor, o xnd our analyss o a mul-counry sng whr som args (for xampl, xchang ras, blaral rad balancs, and nflaon f n a currncy unon) ar hld n common, bu h ohr args ar no. Rfrncs Aarl, B. van, A. L. Bovnbrg and M. G. Rah (997), Is hr a Tragdy for a Common Cnral Bank? A Dynamc Analyss, Journal of Economc Dynamcs and Conrol, 2: Acoclla, N. and G. D Barolomo (2004), Non-Nuraly of Monary Polcy n Polcy Gams, Europan Journal of Polcal Economy, 20: Acoclla, N. and G. D Barolomo (2006), Tnbrgn and Thl M Nash: Conrollably n Polcy Gams, Economcs Lrs, 90: Başar, T. and G.J. Olsdr (995), Dynamc Noncooprav Gam Thory, scond don, Acadmc Prss Lmd, London. Docknr, E., S. Jorgnsn, N. Van Long, and G. Sorgr (2000), Dffrnal Gams n Economcs and Managmn Scncs, Cambrdg Unvrsy Prss, Cambrdg. Engwrda, J.C. (2000a), Fdback Nash Equlbra n h Scalar Infn Horzon LQ-gam, Auomaca, 36: Engwrda, J.C. (2000b), Th Soluon S of h N-Playr Scalar Fdback Nash Algbrac Rcca Equaons, IEEE Transacons on Auomac Conrol, 45: Engwrda, J.C., B. van Aarl and J. Plasmans (2002), Cooprav and Noncooprav Fscal Sablzaon Polcs n EMU, Journal of Economc Dynamcs and Conrol, 26: Gal, J. and M. Grlr (999), Inflaon Dynamcs: A Srucural Economrc Analyss, Journal of Monary Economcs, 44: Gylfason, T. and A. Lndbck (994), Th Inracon of Monary Polcy and Wags, Publc Choc, 79: Holly, S. and A.J. Hughs Hall (989), Opmal Conrol, Expcaons and Uncrany, Cambrdg Unvrsy Prss, Cambrdg. Hughs Hall, A.J. (984), Noncooprav Srags n Dynamc Polcy Gams and h Problm of Tm Inconssncy, Oxford Economc Paprs, 36: Hughs Hall, A.J. (986), Auonomy and h Choc of Polcy n Asymmrcally Dpndn Economs, Oxford Economc Paprs, 38: Hughs Hall, A.J. (989), Economrcs and h Thory of Economc Polcy: Th Tnbrgn-Thl Conrbuons 40 Yars On, Oxford Economc Paprs, 4: Lvn, P. and A. Brocnr (994), Fscal Polcy Coordnaon and EMU, Journal of Economc Dynamcs and Conrol, 8:

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