Omal onear Polc elegaon uner Incomlee Echange Rae Pass-Through aln Aolfson ockholm chool of Economcs, earmen of Economcs, P.O. Bo 65, -3 83 ockholm E/EFI Workng Paer eres n Economcs an Fnance No. 477 Ocober Absrac The cenral bank s omal objecve funcon s analze n a small oen econom moel allowng for ncomlee echange rae ass-hrough. The resuls ncae ha socal welfare can onl be margnall mrove b nclung an elc echange-rae erm n he elegae objecve funcon, rresecve of he egree of ass-hrough. An mlc resonse o he echange rae, hrough Consumer Prce Ine (CPI nflaon argeng s, however, benefcal. Welfare can, moreover, be enhance b aonng a cenral banker wh a greaer reference for neres rae smoohng han ha of he soce, as a resul of surassng some of he sablzaon bas arsng uner a screonar olc. Consequenl, here are welfare gans from monear olc nera. The omal egree of neres rae smoohng s ncreasng n he egree of ass-hrough. Kewors: Echange rae ass-hrough, nflaon argeng, neres rae nera, monear olc, small oen econom JE: E5, E58, F4 Tel.: 46 8 736 96 47, fa: 46 8 3 3 7, e-mal: maln.aolfson@hhs.se I woul lke o hank Aners Vren, Paul öerln (aonall for rovng GAU rounes, Ulf öersröm, aranne Nessén, an Carl Walsh for ver valuable suggesons an scussons. I have also benefe from commens from semnar arcans a Unvers of Calforna, ana Cruz. Fnancal suor from Jan Wallaner s Founaon s graefull acknowlege. Par of hs work was one whle vsng he Research earmen a verges Rksbank, an Unvers of Calforna, ana Cruz.
. Inroucon In an oen econom, echange rae movemens affec nflaon hrough rec changes n mor rces as well as va aggregae eman, whch s nfluence b aleraons n he relave rce beween foregn an omesc goos. In he resence of echange rae surbances, he olc maker can no sablze eman whou creang flucuaons n nflaon, because he echange rae has hs wofol effec on boh he eman an sul relaons. The olc maker hus faces a rae-off beween nflaon an ouu varabl. In conras o he close econom seng, hs rae-off occurs for all es of shocks ha ener he econom, snce all ajusmens of he olc conrolle neres rae also generae movemens n he echange rae. The echange rae works as an aonal ransmsson mechansm of monear olc, beses he raonal neres rae channel. Conser, for eamle, a osve eman shock. In a close econom, hs eman shock can be comleel counere, whou affecng nflaon, b sml rasng he neres rae. In an oen econom framework, he ncrease n he neres rae ncrease also affecs he echange rae, whch arecaes an, n urn, fees no boh nflaon an ouu. Ths mles ha he cenral bank s force o rae off reuce ouu varabl for nflaon varabl (see e.g. Walsh (999. ovemens n he erms of rae can consequenl affec he rae-off beween monear olc objecves. Accorngl, because of he eosure o echange rae flucuaons, here have been some suggesons ha he esgn of he omal olc ffers beween close an oen economes (see e.g. uherlan (.,3 Is ossble o ease he goal conflc, an mnsh he rae-off, b assgnng a fferen objecve funcon o he oen econom-olc maker? houl he olc maker use a fferen nflaon measure, or erhas elcl reson o echange rae flucuaons? B srcl argeng consumer rce (CPI nflaon, he olc maker ma nuce large neres rae varabl, snce such a olc ams a neuralzng he rce effec of ever echange rae flucuaon. On he oher han, b onl focusng on omesc Noe ha hs s conngen uon a cenral bank loss funcon ha oes no enalze neres rae changes, as well as on a forwar-lookng moel whou ransmsson lags of monear olc. On he oher han, Clara e al. ( avocae ha he close an oen econom olc objecves are somorhc as long as he erms of rae are rooronal o he ouu ga. 3 For a relae scusson abou relave rce changes an he arorae goal varables for he cenral bank n a close econom, see Aok (. Aok suggess ha hs resuls can be ale o a small oen econom, where accorngl omesc nflaon shoul be comleel sablze. However, Aok assumes mor rces o be full fleble, an ha he law of one rce hols. On he conrar, when he egree of rce sckness ffers across secors, Bengno ( fns ha he omal monear olc n a currenc area mles ha a larger olc wegh shoul be gven o he nflaon rae wh a hgher egree of nomnal rg.
conons, he olc maker can creae ecessve echange rae volal, whch ma be a subomal oucome (vensson (. 4 However, because of elberae rce scrmnaon or nomnal rges, an echange rae movemen nee no necessarl generae a one-o-one change n mor rces (.e. here ma be an ncomlee echange rae ass-hrough, whch s wh he mac on he econom mgh be que lme. If he egree of ass-hrough s small, he effec of echange rae changes s mnor an here s less nee for he olc maker o ajus he neres rae o such surbances. In he face of, for eamle, eman shocks an echange rae surbances, hs mles ha he conflc beween nflaon an ouu objecves s reuce, whch shfs he olc froner owars he close econom oucome (see Aolfson (. Consequenl, also he (omal oen-econom olc objecve ma be eenen on he egree of ass-hrough. The urose of hs aer s o analze he omal objecve, n erms of socal welfare, ha shoul be ursue b a screonar olc maker of an oen econom wh ncomlee echange rae ass-hrough. Wha are he effecs of echange rae flucuaons on he omze loss funcon, an s he omal olc objecve conngen uon he egree of ass-hrough? In arcular, s relevan o augmen he elegae objecve funcon wh an echange-rae sablzaon erm ha ma mrove he olc maker s conrol over he nflaon-ouu varabl rae-off? Ths aer also eals wh oher was of correcl secfng he olc objecve o mgae he sablzaon bas, or less neral olc resonses, ha arses uner a screonar olc (see e.g. Woofor (999. Are here an gans from assgnng a fferen objecve o he olc maker as comare o soce, for eamle n erms of he egree of neres rae smoohng? oreover, how oes ncomlee ass-hrough affec hs sablzaon bas? The analss s erforme whn an aggregae sul aggregae eman moel ajuse for ncomlee echange rae ass-hrough. The resuls ncae ha he socal welfare mrovemens of nclung an echange-rae erm among he olc objecves are small. Consequenl, a rec, an elc, sablzaon of nomnal or real echange raes aears o be reunan, boh when ass-hrough s lme an when s comlee. However, an nrec resonse o he echange rae, hrough argeng CPI nflaon raher han omesc nflaon, s 4 Neverheless, raher han changng he nflaon measure o arge, hgh neres rae an echange rae volal mgh be avoe f he olc maker s more concerne abou he real effecs of he olc (.e. fleble nflaon argeng wh some concern for ouu sablzaon, whch makes he olc more graual. 3
welfare enhancng. The aer also ons ou ha, alhough he ncomlee echange rae asshrough nuces erssen olc resonses o ceran shocks, here are gans from aonng a cenral banker wh a greaer reference for neres rae smoohng han he socal objecve, as roose b Woofor (999. However, he resuls here show ha he omal egree of elc neres rae smoohng ecreases as ass-hrough ecreases. The reason s ha low ass-hrough as such generaes more neral neres rae reacons. In econ, he moel econom, an he cenral bank objecves, are oulne an arameerze. econ 3 conans he omze cenral bank objecves, evaluae from a socal loss raonale, an he resulng olc rae-offs uner fferen es of surbances. Robusness ssues are scusse n econ 4, whle some conclusons are resene n econ 5.. oel The heorecal seng s a forwar-lookng oen econom aggregae sul-aggregae eman moel allowng for ncomlee echange rae ass-hrough. os ror evelomens of oen econom moels have assume he law of one rce o hol, such ha he echange rae asshrough s comlee (see, for eamle, vensson (, an ccallum an Nelson (999. In conras, n he moel use here, he foregn roucer can no full ajus her omesc currenc (mor rce n he face of echange rae changes because of nomnal rce ajusmen coss (à la Roemberg (98. Ths mles a lme echange rae ass-hrough an consequenl, a mofe sul relaon, where he egree of ass-hrough can be alere b sml changng he level of mor rce sckness. 5.. Inflaon, ouu, an neres rae relaons Conser an oen econom wh consumon of boh omescall rouce goos an more foregn goos. The econom consss of an aggregae sul relaon, an aggregae eman relaon, an an neres rae ar conon nnng own eece echange rae changes. The nomnal neres rae s eermne from an elc cenral bank objecve funcon. Ths econom (calle omesc s assume o be small comare o he res of he worl (calle foregn, such ha foregn nflaon, foregn ouu an foregn monear olc are eogenousl gven. 4
The omesc aggregae sul equaon s erve from he roucers omal rce seng relaons assumng nomnal (Roemberg rce sckness, an usng he unerlng consan elasc subsuon (CE funcon for he househols aggregae consumon. Aggregae nflaon n he omesc econom (,.e. consumer rce ne (CPI nflaon, s comose b nflaon of omescall rouce goos ( an mor goos nflaon enoe n he omesc currenc ( (A an (A n he Aen: 6, accorng o he followng (see equaons ( ( κ α E κ α α ( α ( e ε q, where κ enoes he (sea-sae mor share of omesc consumon, α < s a scoun facor, an α, α q, an < α are osve consans. s omesc ouu, s he rce of omescall rouce goos, s he rce of mor goos enoe n he omesc currenc, s he foregn currenc rce of mor goos, an e s he nomnal echange rae (omesc currenc er un of foregn currenc. CPI nflaon s a funcon of eece fuure nflaon, aggregae ouu (or eman, he relave rce of mors (whch can be nerree as a real echange rae 7, an he evaon beween he omal rce of mor goos n he absence of an nomnal rges an he rce acuall charge (.e., e. 8 Ths wege erm s wha makes he ass-hrough ajuse sul relaon fferen from a sanar Phlls curve wh comlee echange rae ass-hrough. The rce sckness mles ha he omesc currenc rce can no be full ajuse o aleraons n, for eamle, he echange rae. Ths creaes a wege beween margnal cos (caure b he rce charge n he foregn marke ajuse for he echange rae; e an he rce acuall charge ( mles ncomlee ass-hrough an shor-run evaons from he law of one rce. 9. Ths The 5 For a more horough scusson of he moel, see Aolfson (. 6 The noaon s as follows; lower case leers enoe logarhmc values (.e. evaons from sea-sae, a suerscr ncaes wheher omesc or mor goos are consere, an foregn varables are reresene b an asersk. A rce enoe n foregn currenc s hus characerze b an asersk. Fnall, E enoes raonal eecaons as of ero. 7 Ths s, n effec, he nverse of he erms of rae, an onl one arcular efnon of he real echange rae. 8 Gven equal eman elasces n he wo esnaons o whch he foregn roucer sells, here are no ncenves o evae from he law of one rce n he absence of nomnal rges, because of he consan elasc subsuon funcon. The omal omesc fle rce s jus he rce charge n he foregn marke correce for he echange rae. 9 In he long run, he roucer eecs o charge he omal fleble rce such ha ( e ~ I(. 5
egree of ass-hrough s conrolle b α, whch s a funcon of he srucural arameer eermnng he mor rce sckness an he mor share of oal consumon. B mosng a larger nomnal rg on he foregn roucer, ncae b a lower α, a smaller echange rae ass-hrough s generae. A hgher cos of ajusng rces mles ha less of an echange rae movemen wll affec he curren rce. Furhermore, hese ajusmen coss lea o graual changes n he rce, mlng ha he roucer alers he rce charge n hs ero n he recon of he eece fuure omal rce. asl, ε s a sul surbance (.e. a omesc cos-ush shock ha s assume o follow he auoregressve rocess, ε τ ε u, where u s an surbance wh zero mean an varance σ. The aggregae eman relaon s obane usng a sanar Euler equaon for he (reresenave househol s neremoral choce of consumon, an he CE funcon: ( E β (E β E ( q ε, β ( E β (E e (E e e E where s he omesc neres rae, s foregn nflaon, an s foregn ouu. ε s a eman shock (e.g. o references ha follows, τ ε u ε, where u s an surbance wh zero mean an varance σ. omesc ouu s a funcon of eece fuure E ouu, he change n he eece fuure relave rce of mors, (, he real neres rae, he change n he eece fuure relave rce of eors, E ( e, an he change n eece fuure foregn ouu. The (change n he relave rce of mors aears hrough s effec on omesc eman for omesc goos, whle he (change n he relave rce of eors an he (change n foregn ouu show u ue o her nfluence on foregn eman for omesc goos. The fference beween he eman relaon n equaon ( an a full ass-hrough eman curve les n he evaon from he law of one rce Robers (995 shows ha he behavour of he aggregae rce s smlar usng he Roemberg (98 aroach for nroucng rce sckness, as when usng he Calvo (983 formulaon (whch, n conras, reners saggereness n he nvual rces. Noe ha he relave rce level affecs he nraemoral allocaon beween consumon of mors an omesc goos, whle he change n he relave rce affecs he neremoral consumon ecson. However, observe aonall ha all fference erms saear when solvng equaon ( forwar; β q ( β E ( s s β e ( e β E ε, s s s usng he arorae ransversal conons. 6
(.e. e, whch makes he relave rce of mors ( an he (nverse of he relave rce of eors ( e verge. The lme ass-hrough s hus mlcl ncororae, also n he aggregae eman relaon, hrough he mor rce (. The echange rae fulflls a mofe uncovere neres rae ar conon, lnkng he eece echange rae change o he fference n omesc an foregn neres raes: (3 e φ ε E e, where s he omesc neres rae, s he foregn neres rae, an ε φ s a rsk remum followng, φ φ φ ε τ φε u, where φ u s an surbance wh zero mean an varance σ φ. Anhng affecng hs neres rae fferenal wll also affec he echange rae (such as foregn, an omesc, nflaon or ouu shocks ha generae some olc resonse whch s wh neenen echange rae shocks can be har o sngush. However, snce rsk remum shocks have he same (shor-run effecs as auonomous surbances o eecaons abou he fuure echange rae, he rsk remum shocks can be nerree as ure echange rae surbances. The foregn econom consss of eogenous AR( rocesses for nflaon an ouu, an a smle Talor rule wh some erssence ae, eermnng he foregn neres rae (see e.g. Clara e al. (: (4 (5 (6 u ρ, ρ u, ( ( b b ρ ρ u, where ρ, ρ, ρ are non-negave coeffcens less han un, an surbances wh zero mean an varance σ, σ, an u, σ, resecvel. u, u are.. ocal references an olc mlemenaon To evaluae he cenral bank s alernave objecves, an erformance, he followng socal loss funcon s assume o reval n he econom: 7
(7 mn where E j j β j [ λ ], such ha he socal loss consss of quarac evaons of CPI nflaon an ouu from her (consan an zero arges, an λ s he relave wegh soce us on ouu sablzaon. The ouu arge s assume o be equal o he naural ouu level so ha here s no nflaon bas n he moel. Tha soce cares abou nflaon sems from he fac ha he nomnal rges n he moel cause a relave rce serson beween goos. uch rce serson s ermenal for socal welfare, snce nuces fferences n ouu across oherwse encal roucers. 3 Woofor ( suggess ha he general rce level shoul be sablze n economes wh nomnal rges, n orer o reuce hs rce serson beween fleble an fe rce roucers. ablzaon of (CPI nflaon can reuce he rce serson an hence, unceran abou fuure real consumon, whch s welfare mrovng for he rsk averse consumers. In a close econom, CPI nflaon an omesc nflaon are equvalen. However, hs s no he case n an oen econom, an some argue ha omesc nflaon, raher han CPI nflaon, eermnes he oen econom-welfare creron. 4 On he oher han, CPI nflaon argeng mgh beer mgae he wo sorons ha arse n he moel use here, namel ha omesc an mor rces are boh sck (gven ncomlee echange rae ass-hrough. 5 When he oen econom-olc maker seeks o sablze he econom aroun he fleble rce oucome, he consequences of hgh neres rae varabl mus be consere, snce hs nuces echange rae flucuaons an erms of rae sorons. 6 The heorecal unernnngs of hs objecve funcon are characerze b a secon-orer Talor aromaon of he eece ul of a reresenave househol (see Woofor ( for a close econom ervaon. 3 Noe, however, ha foregn an omesc roucers are no encal n hs moel. For nsance, he use somewha fferen nus n roucon. 4 Bengno an Bengno ( show ha he oen econom welfare creron can be characerze b a loss funcon base on sablzaon of consumon an omesc nflaon, assumng roucer currenc rcng an full asshrough. Gven ncomlee echange rae ass-hrough, uherlan ( erves he welfare funcon n erms of he varances of omesc rces an he nomnal echange rae. However, Corse an Pesen ( show ha he oen econom monear olc objecve can be reresene as (equvalen funcons of eher, eece markus, he consumer rce ne, or he ouu ga an evaons from he law of one rce. As a resul, he conclue ha he use of arorae olc rae-offs s moran (.e. ha omal olces rae off a larger ouu ga for lower mor rces. 5 Bengno ( shows ha a weghe average of wo regonal nflaon raes shoul be argee n an omal currenc area, gven nomnal rges n boh regons. 6 oreover, he consumon bunle, conssng of omesc an foregn goos, s rce n erms of aggregae rces, whch s wh agens nuvel care abou CPI nflaon. 8
As he scoun facor, β, aroaches un, he neremoral loss funcon becomes rooronal o he uncononal mean of he ero loss funcon, mlng ha he followng relaon can be use o quanf he socal references (see e.g. vensson (: λ. (8 E [ ] var( var( The monear olc, n urn, s assume o be mlemene hrough a olc objecve funcon, from whch an elc reacon funcon for he olc nsrumen can be obane. The olc maker s lackng (ceran commmen echnologes so ha she, b assumon, solves her omzaon roblem uner screon an re-omzes ever ero, reang he agens eecaons as gven an neenen of he curren olc choce. The cenral bank ajuss s olc nsrumen,.e. he nomnal neres rae, o mnmze he neremoral loss funcon: (9 E j j j mn { } j j β, where β s a scoun facor, an s he cenral bank ero loss funcon. The queson a han s wheher he cenral bank shoul ursue a fferen objecve han ha of he socal references. Because of he sablzaon bas occurng uner a screonar olc, he objecve ha s elegae o he olc maker ( socal loss funcon (. nee no necessarl be encal o he.3. Polc elegaon Two ssues are sue n hs aer; how are he ror fnngs of (screonar omal olc nera affece b oen econom asecs an ncomlee echange rae ass-hrough? Is here a role for an elc echange rae objecve n he olc maker s loss funcon? The sablzaon bas ma be reuce, an socal welfare mrove, b elegang a olc objecve ncororang, for eamle, some neres rae smoohng, low ouu sablzaon, or he sablzaon of nomnal or real echange raes. Conser, for eamle, ha he olc maker (n conras o he assumon mae above can comm o reeael rasng neres raes. In hs case, a smaller neres rae ajusmen reners he same effec as a screonar olc. The reason s ha eecaons abou he fuure 9
are affece b he curren monear olc choce uner commmen. nce agens are forwar-lookng, he olc maker can, n hs case, elo he rvae agens eecaons abou nflaon an ouu, when mlemenng he monear olc, alhough here s no nflaon bas n hs moel. B jus lookng a equaon (, one sees ha a smaller conracon n aggregae eman els he aequae ro n nflaon f eecaons abou fuure nflaon can be lowere b, for eamle, commng o some olc choce. A commmen olc coul hen mnsh he neres rae varance, reuce he loss an hereb make he rae-off beween nflaon an ouu varabl more effcen. 7 Gven ha he commmen resonse s more neral han he screonar olc, he laer mgh ossbl be mrove uon b makng more erssen. Accorngl, Woofor (999 suggess ha b assgnng a fferen objecve funcon o he olc maker, wh larger wegh on neres rae smoohng han ha of he soce s objecve, he screonar oucome can be brough closer o he commmen soluon. 8 Ths resuls from elong he agens forwar-lookng behavour, an he role of eecaons. Woofor argues ha neres rae erssence mles ha eecaons abou fuure shor neres rae changes el a greaer effec on long raes, an hereb also have a more subsanal mac on aggregae eman. 9 Consequenl, he larger wegh on neres rae smoohng seems o be a wa of smulang a commmen envronmen, whch mroves socal welfare. Walsh (999 shows ha here are gans from aonng a conservave banker n Rogoff s (985 sense (.e. wh a lower egree of ouu sablzaon han soce n an oen econom wh full ass-hrough when usng a non-neral rule (.e. wh no neres rae smoohng. As n Woofor s case of neres rae nera, he reasonng buls on he ossbl of elong he rvae agens eecaons. Conser a cos-ush shock ha rases nflaon. If he olc maker s erceve o emhasze nflaon objecves, eece fuure nflaon wll rse less (comare o f he ouu sablzaon, n conras, s larger. Ths mles ha less of an ouu reucon s neee, whch mroves he nflaon-ouu varabl rae-off. In Rogoff s conservave banker case as well as n Woofor s neres rae nera case, he olc maker s hence erceve o emhasze nflaon objecves, eher recl hrough lower ouu sablzaon or, as n he laer case, nrecl hrough larger neres rae erssence. 7 A smaller nal nomnal neres rae change s requre o aler eman f nflaon eecaons can be lowere, snce he nuce real neres rae wll be larger n hs case (see equaon (. 8 Oher eamles of elegaon schemes ha make monear olc more neral are; nomnal ncome growh argeng (Jensen (, mone growh argeng (öersröm (, an rce level argeng (Vesn (. These schemes are no eal wh n hs aer, however. 9 Tha fuure eece neres raes,.e. he long neres rae, are of morance for ouu can be seen b solvng equaon ( forwar (see Foonoe.
The reason wh nomnal or real echange rae sablzaon mgh be benefcal for socal welfare, or o be more recse, mgh reuce he screonar sablzaon bas, s he echange rae s role of ransmng monear olc. B allevang flucuaons n, for eamle, he nomnal echange rae, he olc maker ges a beer chance of conrollng he nflaon-ouu varabl rae-off n he face of ceran shocks, such as omesc eman shocks, as menone n he Inroucon. nce ever neres rae ajusmen also mles a change n he echange rae, such an aonal echange rae sablzaon-objecve coul nernalze he acual mac cause b an neres rae resonse, whch as well fees no he econom hrough he echange rae. Hence, hs suggess ha he oal effec of monear olc s aken no accoun, an ha conserable varaon n he echange rae mgh be avoe. However, recall ha f he olc maker us some wegh on neres smoohng, he volal n he nomnal echange rae s robabl alrea ke small. Alernavel, b recl sablzng he real echange rae (.e. eher he relave rce of mors or he relave consumer rce he cenral bank mgh, sragh off, acheve a more sable nflaon rae. The welfare mrovemen wll, consequenl arse from a more effcen sablzaon of consumon, whch s affece b relave rce sorons ha mgh ossbl be mgae b an echange rae objecve. However, noe ha he echange-rae augmene olces mgh, nrecl, reuce he sablzaon bas, snce he ml a more neral reacon funcon. The cenral bank ero objecve funcon s quarac n evaons of CPI nflaon an ouu from her consan arges (normalze o zero, an quarac n varaons of he neres rae. Ths objecve funcon s augmene wh fferen echange-rae erms accorng o he followng: (a λ ν (, (b e λ ν ( e ( e e µ, (c ( TOT PPP λ ν ( µ ( (, λ ν ( µ ( e ( e, where λ s he relave wegh on ouu sablzaon, ν s he arameer eermnng he rae of neres rae smoohng, an µ an µ are he relave weghs on µ e, ( ( e For a somewha relae scusson of ncororang he echange rae no smle olc rules, see, e.g., Ball (999, an Talor (.
echange rae sablzaon. The cenral bank s benchmark objecve s o recl mlemen he socal references, whch s accomlshe b assgnng equaon (a wh λ λ an ν. Aonng a olc maker wh references for neres rae smoohng an low ouu sablzaon mles ν > an λ < λ, resecvel. The echange rae s ncororae no he olc objecve hrough quarac evaons (from zero arges of eher; b he nomnal echange rae fference, c he relave mor rce (.e. he nverse of he erms of rae, or he relave rce beween foregn an omesc CPIs (hereafer calle evaons from Purchasng Power Par (PPP. Gven ncomlee echange rae ass-hrough, he wo real echange rae efnons n equaons (c an ( are no equvalen. The frs echange rae characerzaon s, n fac, a sub-se of he laer real echange rae efnon. Tha s, evaons from PPP n equaon ( conss of wo ars; he relave rce of mors, as n equaon (c, as well as an elc erm caurng evaons from he aw of One Prce. Polc ajusmens of he nomnal neres rae wll fee no he econom va he real neres rae an he echange rae. The real neres rae an he echange rae boh affec aggregae eman whch, n urn, affecs nflaon, bu he echange rae also has a rec effec on nflaon hrough changes n mor rces. The wo comonens n CPI nflaon, ha s, nflaon of omesc goos ( an nflaon of mor goos (, are lnke fferenl o he ransmsson channels of monear olc. The nflaon of mor goos onl resons o echange rae aleraons, whle he nflaon of omesc goos s affece b real neres rae changes (.e. va aggregae eman changes as well as b echange rae changes. nce he egree of ass-hrough affecs he een o whch echange rae movemens have an mac on he econom, wll nfluence he monear olc ransmsson as well as he egree of eosure o foregn shocks, such as echange rae surbances. A ossble eenson s o nclue sablzaon of he nomnal echange rae level n he elegae olc. However, hs requres a fferen sae-sace reresenaon where he (ossbl non-saonar level of he echange rae s nrouce. When he sae vecor conans non-saonar varables, s hough unclear wheher he numercal algorhm caures he soluon o he olc maker s roblem. However, a feasble alernave (keeng he sae vecor saonar s o ean he sae-sace reresenaon wh several lagge changes n he nomnal echange rae. In he lm, argeng he sum of hese erms wll aromae echange rae level argeng. Noe, however, ha such an aroach oes no seem o change he man resuls obane n he aer. Tha s, ( e ( κ ( ( e.
The moel (.e. equaons (-(6 can be reresene n sae-sace form, mlng ha he cenral bank s omzaon roblem can be eresse as a lnear-quarac roblem (see he Aen. The cenral bank s objecve funcon, equaons (9 an (, closes he moel. In he screonar case, he cenral bank s reacon funcon wll relae he neres rae o he reeermne varables of he moel, an hese reacon coeffcens are unravele b erang on he value funcon. 3 The moel s solve b numercal mehos, escrbe n e.g. öerln (999, an herefore requres some arameerzaon..4. Parameerzaon To llusrae he monear olc rae-off uner fferen olc objecves, an varng egrees of echange rae ass-hrough, he socal loss s calculae usng he choce of moel arameers an shock varances shown n Table. These are chosen along he lnes of e.g. vensson (. 4 Table : Parameer values ocal references ul relaon eman relaon Foregn econom hock erssence hock varance β.99 κ.3 β q.6 ρ.8 τ.8 σ.4 λ.5 α.99 β.35 ρ.8 τ.8 σ.6 α.56 β e.8 ρ.8 τ φ.8 σ.8 φ α q.7 β.7 b.5 σ.5 α {3,.6,.5,.3} b.5 σ. σ nce he egree of echange rae ass-hrough n hs moel s generae b he nomnal rg mose on he foregn roucer, ass-hrough s hghl eenen on he eogenousl gven egree of mor rce sckness. The level of ajusmen coss (.e. he level of nomnal rg caure n arameer α s chosen such ha he egree of aral echange rae ass- 3 In he commmen case, he curren behavour of monear olc aonall affecs he rvae agens eecaons, whch s wh he omal commmen olc also eens on he shaow rces of he forwar-lookng varables. 4 The arameers are base on unerlng ee moel arameers ha ml he followng; a scoun facor elng an annual neres rae of 4% (assumng quarerl eros, a rce elasc of eman generang a % marku over margnal cos, an mor share conssng of 3 % of oal consumon, an eor share of 3 % of aggregae eman, an neremoral elasc of subsuon of.5, an a arameer lnkng ouu o margnal coss such ha he sea-sae ouu elasc of margnal coss s.8. surbance varances are more or less aken from eemo an Røslan (. 3
hrough s.99,.66,.33, an.9, resecvel. In he frs case, an echange rae movemen, consequenl, mmeael alers he mor rce b 99 % of he echange rae movemen. Hence, hs se of values caures he sanar oen econom case of almos full ass-hrough, an hree nermeae cases of ncomlee ass-hrough. The emrcal evence seems o sugges ha also small oen economes le n one of he nermeae caegores. Aolfson (997 reors % mmeae, aral, echange rae ass-hrough, an anoher % whn a monh, o aggregae wesh mor rces, whereas Naug an Nmoen (996 oban somehng lke a % ass-hrough er quarer for aa on aggregae Norwegan mors. 3. Omal olc objecves - resuls The moel s numercall solve, resulng n an elc reacon funcon for he cenral bank, as well as he ranson mar for he sae varables (see he Aen. The ranson mar s subsequenl use o calculae he asmoc varances of, for eamle, nflaon an ouu whch, n urn, eermne he olc rae-off an he socal loss (see equaon (8, uner he varous olc objecves. 3.. Omal neres rae nera an omal ouu sablzaon Fgures an llusrae ha socal loss can be reuce b aonng a screonar olc maker wh a larger egree of neres rae smoohng or wh a lower egree of ouu sablzaon, comare o he references of he soce. 5 Ths mles ha he oen-econom olc maker can elo eecaons abou nflaon an ouu as n he commmen case, whch consequenl rouces a beer rae-off beween nflaon an ouu sablzaon. I seems ha such a reucon n he loss can be acheve rresecve of wheher ass-hrough s comlee or ncomlee. Fgure shows ha he nflaon-ouu rae-off froner s suae closer o he orgn as he olc maker us more wegh on neres rae sablzaon (.e. larger ν. In some sense, he olc maker comms o connuousl fghng nflaon, hereb mlng ha curren nflaon can be lowere wh a smaller ouu reucon. Furher, he omal rae of neres 5 The econom s h b a combnaon of all surbances, wh varances secfe n Table (see he Aen for he varance-covarance mar. Recall ha onl he socal welfare an he nflaon-ouu rae-off are affece b he sze of he shocks. The olc maker s reacon funcon s ceran equvalen an hus neenen of he surbances covarance mar. 4
rae smoohng aears o be smaller n he cases wh more lme ass-hrough (cf. Fgures a an. Fgure a shows ha aonng a Rogoff-conservave olc maker reuces he socal loss. Noe, however, ha he averse effecs of rvng ouu sablzaon enrel o zero are farl large. oreover, he omal egree of ouu sablzaon oes no seem o be eenen on he egree of ass-hrough. Hence, alhough he olc maker s reacon funcon s alrea base on lagge varables, such as he ror relave rce, some aonal olc nera seems o be reucng he socal loss. 6 The reason for hs s ha aonal olc nera, for nsance generae b an neres rae smoohng objecve, mles ha he agens eecaons abou fuure olc are affece. Ths s esecall aaren n he moel use here, where he erssence comes from eogenous surbances. The nomnal rce rg er se onl reners forwar-lookng behavour of he agens, an oes no ml an backwar-lookng comonens n he equlbrum relaons. Boh he sul an he eman relaons hus lack elc backwar-lookng erms, so ha her eenence on as values of he enogenous varables onl comes from he olc rule (see equaons ( an (. 7 Table slas he omal olc weghs, an he resulng socal loss, when elegang equaon (a o he olc maker. Neher he reucon n socal loss nor he omal wegh on ouu sablzaon are eenen on he egree of ass-hrough, bu he omal egree of neres rae smoohng s ncreasng n he egree of ass-hrough (see Table. 8,9 The omal wegh on neres rae sablzaon s hus larger n he full ass-hrough case, comare o f ass-hrough s low. The reason for he nerrelaon beween he neres rae smoohng an he echange rae asshrough s ha ncomlee ass-hrough nuces some nheren erssence no he olc reacon funcon. Ths mgh be elane b conserng an echange rae surbance. A low 6 eals on he reacon funcon are foun n Table A n he Aen. 7 Noe ha even f he eman an sul relaons were more backwar-lookng, changes n he olc rule woul aler he agens eecaons. However, some forwar-lookng comonens are necessar for neral olc-makng o be omal. 8 Comarng he absolue loss level across fferen ass-hrough cases s of lme neres, snce hese cases reresen fferen srucural economes. 9 The omal neres rae nera s also ncreasng n he egree of ass-hrough for gr (no shown. λ.5, when narrowng he 5
ass-hrough mles ha he echange rae movemen s onl ncororae no he mor rce o a small een. Because of hs low ass-hrough, he move owars he long-run sea-sae (wh comlee ass-hrough s graual, whch mles ha he echange rae surbance has a rolonge effec on nflaon. Hence, when ass-hrough s low, he olc maker wll requre a smaller, bu more erssen, resonse o an echange rae surbance (.e. a rsk remum shock. Table : ocal loss ( an omze olc arameers ( λˆ, νˆ, equaon (a Pass-hrough Benchmark Omzng he olc weghs concernng: ν neres rae ouu ouu an neres rae jonl λ.5 νˆ λ.5 ν Rel. λˆ Rel. λˆ νˆ Rel..99.368.3.996..857...78.66.4.3.995..847..9.773.33.648.3.994..84..7.77.9 9.56..995..849..4.787 Noe: The omze olc weghs are esablshe b a gr search, wh se., over he values -. uch a resonse can also be seen from he olc reacon funcon. Comare o he full ass- φ hrough case, he olc maker resons less o a rsk remum shock ( ε when ass-hrough s small, whle concurrenl ajusng s neres rae more o he lagge neres rae ( (see Table Ab n he Aen. The reacon coeffcen on s hus ecreasng n he egree of ass-hrough. Consequenl, also he omal neres rae nera (ν wll be eenen on he egree of ass-hrough. nce he egree of acual neres rae erssence (measure as he reacon coeffcen on he lagge neres rae s larger when ass-hrough s small, s no necessar o nuce as large neres rae smoohng (.e. ncreasng ν as n he full asshrough case. The gans from aonng a Rogoff banker aears o be larger han he gans from omzng he egree of neres rae smoohng,.e. aonng a Woofor banker (see Table. Alhough he rvng mechansm of boh welfare mrovemens works hrough he eecaon channel, he rec emhass on nflaon objecves (.e. lower λ aears o have a greaer nfluence on agens eecaons han he neres rae smoohng objecve. The amoun of omal nera, or erssence n he olc resonses, s hough eenen on he 6
egree of ouu sablzaon. arger weghs on ouu sablzaon an on neres rae smoohng boh ml ha nflaon s more grauall brough back o he argee level of nflaon. Ths smlar role hus mles ha less aonal neres rae erssence, n he form of an neres rae smoohng objecve, s requre when he wegh on ouu sablzaon s large. Consequenl, he ave value of omzng he egree of neres rae smoohng becomes larger, he lower s he egree of ouu sablzaon ( λ. 3.. Omal echange rae sablzaon Now, conser he queson of wheher he echange rae shoul be ncororae n he olc objecve n orer o enhance socal welfare. Coul some form of echange rae sablzaon caure he same e of favourable olc nera, as nuce b an neres rae smoohng objecve, an reuce he screonar sablzaon bas? Fgures 3a-3c llusrae he socal loss uner he hree echange-rae augmene loss funcons (.e. equaons (b (. The welfare gans from elegang an echange rae objecve, f an, aear o be mnor. Table 3 confrms hs oucome, slang he socal loss an he omze echange rae arameers for a gven level of (searael omze ouu an neres rae sablzaon. 3 The olc maker oes no seem o be able o mrove socal welfare hrough equaon (b b lmng flucuaons n he nomnal echange rae. The omze wegh on he echange rae fference, µˆ e, s zero n all ass-hrough cases (see Table 3. 3 Hence, rng o smooh ou changes n he nomnal echange rae oes no creae he same kn of olc nera ha was foun o be benefcal for he socal loss above. 3 In conras, a osve wegh on nomnal echange rae sablzaon aears o creae ecessve varabl n ouu an relave rces (no shown. In fac, unresrane echange rae ajusmens mgh be helful n allevang surbances requrng relave rce ajusmens, whch hen make hs kn of olc erssence ermenal. 3 Recall ha he coeffcens lace on each objecve are relave weghs, mlng ha s ffcul o comare he sze of secfc olc weghs beween he benchmark case an cases ncororang aonal varables. 3 Noe ha uherlan ( fns ha CPI nflaon argeng creaes oo lle volal n he nomnal echange rae, an ha socal welfare mroves f he echange rae level s nclue n he cenral bank objecve funcon. Consequenl, he omal wegh on he nomnal echange level rae urns ou o be negave n uherlan s full ass-hrough moel (whle he omal ouu sablzaon s zero. Ths effecvel relcaes omesc nflaon argeng, whch s he frs-bes oucome n ha arcular moel, mlng ha he echange rae elemen of CPI nflaon shoul omall be offse. On he oher han, n he moel use here, argeng CPI nflaon s welfare enhancng (see he scusson below. 3 Noe, however, ha here seems o be some role for echange rae sablzaon (.e. µ e > f he sablzaon bas s worse or, n oher wors, f he olc maker uses a more graual olc (see econ 4.. below. 7
Neher argeng he relave rce of mors n equaon (c, (.e. sablzng he (nverse of he erms of rae, nor allevang evaons from PPP hrough equaon (, seems o conserabl mrove socal welfare. The omze olc weghs on he fferen real echange rae efnons are zero, or close o zero, n mos ass-hrough cases (see Table 3. 33 Neher he full ass-hrough case, nor he wo nermeae ass-hrough cases, ncaes ha an of he echange-rae erms shoul be ncororae no he olc maker s objecve funcon. I s onl for he case wh he smalles ass-hrough ha some real echange rae sablzaon s welfare mrovng o an noceable een. Ths s somewha surrsng, gven ha he eenure swchng effecs (or he relave rce sorons cause b echange rae movemens ough o be small n hs case, a leas n he face of rsk remum shocks. However, noe ha he volal of he enogenousl eermne echange rae ncreases as ass-hrough ecreases (see Table A n he Aen. When ass-hrough s low, a counr-secfc shock can no be absorbe b he (eogenousl sck mor rce, an he requre relave rce ajusmens are herefore generae hrough lager movemens n he enogenousl eermne echange rae. Ths mles ha he uncononal varance of he relave rce of mors s, n fac, larger when ass-hrough s low han when ass-hrough s comlee, whch nuces a olc resonse o he real echange rae n he former case. 34 Table 3: ocal loss ( an omze echange rae olc arameers Pass- equaon (a equaon (b equaon (c equaon ( hrough λˆ νˆ µˆ e Rel. ˆ µ Rel. ( ˆ µ Rel. ( e.99.. 7.46....66..9 7.76....33..7 6.683. -..998..9..4 5.8..4.979..983 Noe: The ouu an neres rae sablzaon s searael omze o reflec he margnal avanage of ncororang an echange-rae erm. The omze echange rae weghs are esablshe b a gr search, wh se., over he values - o. However, some negave arameers o no el a soluon o he ssem. 33 The resuls o no seem o be affece b he egree of olc nera ( ν. Even whou an neres rae smoohng, he omal echange rae weghs are low n mos cases (see Table A3 n he Aen. 34 Noe ha evereu an Engel ( show ha wh zero ass-hrough, he omal olc objecve n her moel s neenen of he echange rae volal. When ass-hrough s zero, he omal olc s herefore conssen wh fe echange raes n resonse o real shocks. In he full ass-hrough case, he echange rae s, n conras, emloe for sablzng consumon, alhough here s now a welfare cos of echange rae volal. However, hs volal s rae off for he benefs of echange rae ajusmens n reucng consumon varance. 8
3.3. omesc nflaon argeng I mgh be argue ha he low omze weghs on he fferen echange rae objecves above orgnae from he fac ha he echange rae s alrea nrecl ncororae n he olc maker s objecve funcon hrough argeng CPI nflaon an ouu, whch s wh a rec am s reunan. nce nflaon an ouu are boh affece b echange rae flucuaons, an agens are forwar-lookng, fuure echange rae eecaons are alrea ncororae no he realze oucomes of nflaon an ouu (see e.g. equaon (. If hs s he case, suggess ha he echange rae shoul be gven a fferen emhass f he olc maker, n conras, arges omesc nflaon. nce he rec echange rae comonen ha eners CPI nflaon hrough mor rces, s flere ou from omesc nflaon, ceran effecs of an echange rae movemen are remove. 35,36 Consequenl, f sablzng he echange rae s rul moran, he omze wegh mus be larger f he olc maker arges omesc nflaon. Comarng he socal loss uner CPI nflaon argeng an omesc nflaon argeng ncaes ha he former scheme roves larger welfare (cf. Tables an 4. The fference beween he wo argeng schemes s abou 3% for he full ass-hrough case, bu becomes smaller as he egree of ass-hrough ecreases. Ths suggess ha he nrec echange rae comonen of CPI nflaon s, n fac, benefcal for he olc oucome. Noe also ha he nomnal echange rae volal s slghl lower when argeng CPI nflaon (see Table A n he Aen. However, alhough aears o be beer for he olc maker o arge CPI nflaon, he welfare mrovemen of nclung an echange-rae erm s no larger uner omesc nflaon argeng (see Table 4b. The rec echange rae resonses o no seem o be gven hgher emhass an overall, he omze echange rae coeffcens aear o be que smlar o he ones uner CPI argeng. Consequenl, elc echange rae sablzaon oes no la a welfare enhancng role, rresecve of wheher he olc maker arges CPI nflaon or omesc nflaon, whle an mlc echange rae resonse hrough CPI nflaon argeng s benefcal. 35 ese hs, he echange rae sll nfluences omesc nflaon boh va eenure swchng effecs n aggregae eman, as well as va more nu nermeaes. 36 Recall ha here are no ransmsson lags of he monear olc n hs moel, whch mles ha he olc maker wll reac o all surbances, ermanen as well as emorar. If, n conras, he olc maker faces ransmsson lags n he effec of monear olc, neres rae ajusmens o emorar flucuaons n he real echange rae (.e. evaons from urchasng ower ar mgh rouce ecessve varaons n boh nflaon an ouu, whch s clearl unesrable for he olc maker (see e.g. Talor (. In ha case, he olc maker woul asre o eacl fler ou emorar nflaonar mulses (semmng from shor-run earures from he law of one rce, from he nflaon measure she s argeng. 9
Table 4a: ocal loss (, an omze ouu an neres rae sablzaon ( λˆ, νˆ, uner omesc nflaon argeng equaon (a, - argeng Pass-hrough Omzng he olc weghs concernng: ν neres rae ouu ouu an neres rae jonl λ.5 λ.5 ν νˆ Rel. λˆ Rel. λˆ νˆ Rel..99.95..997..839..6.777.66.64..996..84..7.77.33.856..994..849..7.77.9 9.75..994..878..6.8 Table 4b: ocal loss (, an omze echange rae sablzaon ( µˆ e, ˆ µ, ˆ µ ( e, uner omesc nflaon argeng ( - argeng Pass- equaon (a equaon (b equaon (c equaon ( hrough νˆ µˆ e Rel. ˆ µ Rel. ˆ µ e Rel. λˆ ( (.99..6 7.87..988...66..7 7.4..993...33..7 6.879. -..997..9..7 5.346..3.985..98 As scusse n econ.., seems as f soce shoul be concerne abou CPI nflaon when ass-hrough s ncomlee. Noe, however, ha he resuls are qualavel robus o changng he socal references, such ha soce values sable omesc nflaon (.e. evaluang he elegae olces from a socal loss funcon of he form; var( λ var(. In arcular, he resuls sugges ha he olc maker shoul arge CPI nflaon, even f omesc nflaon s an argumen n he socal loss funcon (cf. Tables 5a an 5b. The reason s ha echange rae flucuaons nrecl also nfluence omesc nflaon. ovemens n he echange rae affec he omesc roucer s margnal cos hrough nermeae mors an hrough relave rce changes n aggregae eman. B argeng CPI nflaon, he olc maker acheves an mlc resonse o he echange rae, whch also reuces he omesc nflaon varabl. nce CPI nflaon argeng generaes lower
echange rae volal, whch mroves he varance rae-off beween omesc nflaon an ouu, hs s helful for sablzng omesc nflaon. Table 5a: ocal loss ( var( λ var(, an omze olc arameers, elegang CPI nflaon argeng - argeng Pass- equaon (a equaon (b equaon (c equaon ( hrough νˆ µˆ e Rel. ˆ µ Rel. ˆ µ e Rel. λˆ ( (.99.. 7.643....66.. 7.5....33..7 7.77. -..997..9..4 6.345..5.97..98 Table 5b: ocal loss ( var( λ var(, an omze olc arameers, elegang omesc nflaon argeng - argeng Pass- equaon (a equaon (b equaon (c equaon ( hrough νˆ µˆ e Rel. ˆ µ Rel. ˆ µ e Rel. λˆ ( (.99..7 7.94..99...66..7 7.735..997...33..7 7.468....9..5 6.58..4.98..98 4. Robusness 4.. Increase sablzaon bas an echange-rae augmene olces How o he echange-rae augmene olc funcons erform when he screonar sablzaon bas s worse? Tha s, alhough he elc echange rae sablzaon s no welfare mrovng comare o elegang a olc omze wh resec o ouu an neres rae sablzaon, he echange rae ma la a role f, for some reason, a conservave Rogoffbanker can no be aone.
A larger wegh on ouu sablzaon mles ha he elegae olc s more graual, comare o he case when a Rogoff-conservave banker, wh lower ouu wegh, s aone. Ths, n urn, has consequences for he echange-rae augmene olc funcons. Graual neres rae resonses ml ha rsk remum shocks el rolonge earures of he echange rae from s long-run level, for eamle. Noe also ha echange rae volal s hgher when ouu sablzaon s larger (no shown. Consequenl, here seems o be more reason o sablze he echange rae n hs case. As a resul, when he olc wegh on ouu sablzaon s ke a he same level as he socal references (.e. λ λ.5, he omze wegh on he nomnal echange rae change, n equaon (b, urns ou osvel n ever ass-hrough case. oreover, hs reuces socal loss b u o 5% (see Table A4 n he Aen. On he oher han, he resuls regarng he real echange-rae augmene olc funcons are somewha more ambguous. As s he case when ouu sablzaon s omze, sablzaon of he erms of rae, or of he earures from PPP, can no reuce socal loss n he full ass-hrough case, bu mroves welfare when ass-hrough s low. 4.. Oher arameerzaons Can an echange rae objecve la a welfare mrovng role f he econom becomes more oen, where oenness s measure n erms of mor an eor shares? A more oen econom mles ha he eosure o foregn surbances ncreases, an ha he mac on boh eman an sul relaons s greaer, hereb requrng larger neres rae resonses. However, alhough he real echange rae affecs nflaon an ouu more sgnfcanl n hs case, he echange rae er se s less nfluence. nce he omesc secor s alrea affece, here s less nee for echange rae nuce relave rce ajusmens, whch lower he echange rae volal (no shown. Therefore, neher nomnal, nor real, echange rae sablzaon els an subsanal welfare enhancemen, comare o elegang a olc funcon wh omal ouu an neres rae sablzaon (see Table A5a n he Aen. Consequenl, he resuls aear o be qualavel robus o changng he egree of oenness. The morance of he echange rae channel also ncreases f he rsk remum becomes more erssen, or f he relave varances of rsk remum surbances an oher foregn shocks ncrease. Perssen shocks warran more rolonge neres rae resonses, whch s wh he omal neres rae nera s slghl larger n hs case (see Table A5b n he Aen.
However, here are sll no welfare mrovemens from elcl sablzng he echange rae, as long as he elegae olc s omze wh resec o ouu an neres rae sablzaon. arger varances of he foregn surbances, n urn, ml ha he omal olc mus be somewha more aggressve o offse hese shocks. nce hs s reflece b a lower egree of neres rae smoohng, welfare s no mrove b ang an elc echange-rae erm o he olc maker s oher (omze objecves (see Table A5c n he Aen. 5. Conclusons The omal screonar olc objecve s analze whn a forwar-lookng aggregae sul-aggregae eman moel, ajuse for ncomlee echange rae ass-hrough. The monear olc rae-off beween nflaon an ouu varabl s ease as he egree of asshrough ecreases, snce he echange rae channel hen ransms monear olc, an foregn surbances, o a smaller een. Ths mles ha also he elegae olc objecve funcon s eenen on he egree of ass-hrough n he econom. However, here are onl small welfare mrovemens, f an, of ncororang an elc nomnal, or real, echange-rae erm n he olc maker s omze objecve funcon. Neher he sablzaon of nomnal echange rae changes, nor of real echange raes (.e. eher he relave rce of mors, or evaons from Purchasng Power Par, oes mrove he olc maker s erformance n erms of socal welfare. However, an nrec echange rae resonse, acheve b argeng CPI nflaon, aears o be suffcen for reucng he socal loss somewha, comare o f he olc maker arges omesc nflaon. The reason s ha eecaons abou fuure echange raes affec boh nflaon an ouu, an ha CPI nflaon argeng generaes lower echange rae volal. oreover, CPI nflaon argeng seems o o beer han omesc nflaon argeng, rresecve of wheher he echange rae ass-hrough s lme or comlee. Ths resul s neher eenen on he soce s references for CPI nflaon sablzaon or omesc nflaon sablzaon. ll, here are oher asecs han he relave rce sorons above ha can be alere, such ha a fferen objecve han ha of he soce s elegae o he olc maker. The resuls ncae ha some of he sablzaon bas ha occurs uner a screonar olc can be mgae hrough aonng a Rogoff-conservave banker wh lower wegh on ouu sablzaon or hrough aonng an neres rae smoohng olc maker ( Woofor banker. B elong he agens eecaons abou fuure olc, he nflaon-ouu rae- 3
off s mae more favourable whch, n urn, reuces socal loss. The welfare mrovemen of ure neres rae smoohng aears o be low, whle combnng a low wegh on ouu sablzaon wh neres rae smoohng gves a szeable effec. The omze wegh on ouu sablzaon aears, n rncle, o be neenen of he egree of ass-hrough. In conras, he olc wegh on he omze neres rae nera s ncreasng n he egree of ass-hrough. The reason s ha ncomlee, an graual, asshough requres a rolonge neres rae resonse when, for eamle, a rsk remum surbance hs he econom. Ths nheren neres rae erssence consequenl mles ha less aonal neres rae nera, n he form of an neres rae smoohng objecve, s neee when ass-hrough s ncomlee. 4
5 Aen A.. The cenral bank s omzaon roblem The wo comonens n aggregae (CPI nflaon, ha s nflaon of mor goos an nflaon of omesc goos, follow: (A ( e E κ α β, (A ( ε α α κ β q ( ( E. To formulae he sae-sace reresenaon of he moel, he followng enes are aonall use: (A3 ( (, (A4 e e e ( (. Ths mles ha he comlee moel,.e. he ssem of equaons (-(6, he hree shock rocesses 37, lus (A-(A4, can be rewren n sae-sace form: (A5,,,, ~ ~ ~ E ~ B A A υ, [ ] ( (, e ε ε ε φ, [ ] e,, [ ] ~ u u u u u u φ υ, where, s a 9 vecor of reeermne sae varables,, s a 4 vecor of forwar-lookng varables, an ~ υ s a 3 vecor of surbances, 37 Tha s, ε τ ε u, u ε τ ε, an φ φ φ φ ε τ ε u.
6, ( ( ( ~ ( κ κ κ β β β β κ β β β κ ρ ρ a q a q e q b b A, ( ~ ( a e e e A κ φ β β ρ β ρ β τ τ τ ρ ρ ρ [ ] ( ~ e B β β. The olc maker s conrol roblem s a sanar sochasc lnear-quarac roblem: (A6 { } [ ] { }, ( E mn ( E ' mn ( E ~ mn ( ' ' ' ' ' J R U Q J R U U Q J Q J β β β where Q, U an R are marces mang he goal varables (.e.,, ( o he sae varables. In he screonar case, he value funcon wll be quarac n he reeermne sae varables, V J ω,, (, an he forwar-lookng varables wll be a lnear funcon of he reeermne varables, H,, (see e.g. öerln (999. V s a mar
7 an ω s a scalar, boh o be eermne b erang on he value funcon. The frs orer conon of hs roblem relaes he neres rae o he reeermne varables, F,. Ths also mles ha he reeermne varables can be wren as,,, υ, where s a mar eenng on F, H an he srucural arameers n he sae-sace reresenaon. The namcs of he ssem uner screon s hus he followng: (A7a,, υ, (A7b H,,, (A7c F,. Ths ssem has a sable soluon f he number of sable egenvalues n equals he number of reeermne varables. The numercal algorhm hen caures he soluon, ha s, unravels he coeffcens n he reacon funcon, F, an n H (see Aolfson ( for more eals. A.. Varance-covarance marces The uncononal varance-covarance mar of he surbance vecor, υ, s gven b [ ] 4 9 Σ Σ υ υ, where υ Σ s efne as:, ( ( ( ( ( ( b b b b b b σ σ σ σ σ ρ σ ρ σ σ ρ σ σ ρ σ ρ σ φ υ Gven ha he namcs of he ssem can be wren as (A7, he asmoc uncononal varance-covarance mar of he reeermne varables s gven b: (A8 ' υ, (A9 ( [ ] ( vec ( vec υ n I,
8 usng ( ( ( B A B A vec vec vec, an ( ( ( B A C ABC vec vec ' (see Ruebusch an vensson (999. The varables of neres can be wren as a funcon of he reeermne varables, [ ] [ ], T F T H T T T T T T T z mlng ha he varance-covarance mar of he neres varables s (A ' z T T.
Table Aa: Reacon funcon, coeffcens n F ( F,, benchmark case Equaon (a, λ. 5, ν ε φ ε ε ( ( e.99.33.93 -.679 3.596.93.95 -.48.66.3.868 -.686 3.67.868.377.7.33.8.83 -.688 3.697.83.487.8.9..84 -.678 3.859.84.53.3 Table Ab: Reacon funcon, coeffcens n F ( F,, wh neres rae smoohng Equaon (a, λ. 5, ν. ε φ ε ε ( ( e.99.5.34.93 -.66 3.53.93.9 -.4.66.5.5.84 -.655 3.53.84.36.67.33.33..797 -.647 3.59.797.46..9.37.4.786 -.634 3.66.786.477.5 Table Ac: Reacon funcon, coeffcens n F ( F,, wh omze olc weghs Passhrough Passhrough Passhrough Equaon (a, omze olc weghs ε φ ε ε ( ( e.99.8.5.734 -.448.5.734.76 -.54.66.37.48.63 -.389.89.63.7.5.33.64.44.573 -.367.898.573.49.5.9.39.39.579 -.399.75.579.37.85 Noe: The omze olc weghs are λ {.,.,.,.}, ν {.,.9,.7,.4} for ass-hrough {.99,.66,.33,.9}, resecvel. 9