Robust monetary policy in the micro-founded two-country model of the. currency union. Kuznetsova Olga, Higher School of Economics, Moscow, 2009

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1 Robus moneary polcy n he mcro-founded wo-counry model of he currency unon Kuznesova Olga, gher School of Economcs, Moscow, 29 For wo-counry model of currency unon wh scy prces he problem of model uncerany s analyzed. By mehods of robus conrol we derve robus moneary polcy ha wors reasonably well even n he wors-case of model perurbaons. We fnd some volaon of Branard prncple and show ha cenral ban s opmal reacon o he economc shocs becomes more aggressve wh an ncrease n s fear of msspecfcaon. Inroducon...2 Chaper. Leraure...5. Types of uncerany Mehods of analyss Choce crera General resuls Robus analyss for euro area...9 Chaper 2. Reference model of moneary unon Consumer problem Fscal polcy Toal demand Frms Deermnsc seady sae Equlbrum wh scy prces Welfare creron Calbraon Cenral ban opmzaon problem Sae space form of he model...24

2 Chaper 3. Robus conrol specfcaon Model uncerany specfcaon Consran robus conrol problem Mulpler or penaly robus conrol problem Relaons beween wo robus programs Robus conrol program soluon Defnon of possble model se θ Some compuaonal resuls...4 Graph. Terms of rade shoc n he wors-case, approxmang model and raonal expecaons whou robusness...43 Graph 2. Welfare benefs from robus polcy applcaon...44 Concluson...45 Appendx A. Dervaon of he reference model...47 ) Soluon of consumer problem ) Governmens ) Frms program...5 4) Deermnsc equlbrum...5 5) Lnearzaon of scy-prces equlbrum ) Dervaon of mcro-founded loss funcon...55 Appendx B. Compuaonal resuls...65 References...7 Inroducon A usual way of elaborang opmal moneary polcy s o desgn n he conex of a parcular model of he economy. owever nobody nows he rue and exremely complex srucure of he economy. In aemp o capure a leas some of he mos mporan economc regulares by means of racable model a ceran degree of smplfcaon s needed. The resulng analycal framewor appears o be more or less sylzed. Consequenly, nobody can be absoluely confden abou he predcng power of any parcular model employed for he moneary polcy 2

3 analyss. Thus, he problem of dealng wh hs model uncerany or uncerany abou he rue srucure of he economy arses. There are wo common approaches o hs problem. The frs one mples polcymaer s learnng abou he rue srucure of he economy. Bu hs process requres a lo of me and possbly he second approach s more approprae o acle uncerany qucly. Ths second mehod s a searchng for he robus moneary polcy ha wors reasonably well across a ceran se of models or model specfcaons. Accordng o he ype of he model se and specfc ype of uncerany he analyss can be made n dfferen ways. The man queson whch s usually n he cenre of aenon n he robus leraure concerns he comparson of robus polces and smple opmal ones, desgned for he parcular model. The semnal resul called Branard conservasm assumes ha robus polcy s less aggressve n he reacon o he economc shocs han a polcy, consruced for a sngle model whou ang no accoun model uncerany. There are many examples of such analyss for he USA and for he euro area and some auhors agree wh hs general fndng whle he ohers do no. All he models used n such analyss n he applcaon o he euro area represen he areawde models whch operae wh aggregaed daa. Bu seems o be no he bes approach, as a euro area represens a se of neracng heerogeneous counres. So n our analyss we adop for hs problem wo-counry mcro-founded model of currency area. We derve a mcro-founded welfare creron on whch a polcy evaluaon s based. I s shown ha erms of rade beween hese regons maer for he populaon welfare and so usng a one-counry model for analyss of such consrucons as moneary unon s unjusfed. 3

4 Then we calbrae hs model for he European Unon ang no accoun he dfferences beween regons. Afer ha we consruc a robus o model uncerany moneary polcy under commmen. Ths consrucon s based on he robus conrol mehodology. We llusrae how he robus conrol echnques naed by ansen, Sargen (2) can be appled for mul-counry models wh raonal expecaons. We show, conrary o Bhan (22), Zacovc, Weland, Rusem (25), ha he man characerscs of robus polcy and economc oucome depends crucally on he parcular parameer a wllngness of he cenral ban o consruc a robus polcy or a fear of model msspecfcaon. To choose a proper exen of model msspecfcaon we adop an error deecon probably approach frsly naed by ansen, Sargen. The res of he paper s organzed as follows: Chaper oulnes exsng leraure on he robus quesons. The wo-counry model s presened n Chaper 2 where he cenral ban s mcro-founded loss funcon s also derved. In Chaper 3 we apply robus conrol echnques for hs model and derve he characerscs of he robus polcy. The las secon concludes and oulnes he possble drecons for he fuure research. 4

5 Chaper. Leraure. Types of uncerany The noon of uncerany can be referred o some emprc varables or o he model srucure as a whole. To clearly undersand he applcaon area of robus heory accurae dsncon of dfferen sources of uncerany s needed. Kngh (92) conrased rs wh uncerany. Rs refers o he processes whch have nown dsrbuons, whle uncerany arses from unnown facors and processes or from he processes whch hardly can be descrbed by sasc erms. Speang abou emprc varables nghan noon of rs corresponds o he aleaory or sascal uncerany, whle he erm uncerany refers o he epsemc or sysemc uncerany. Uncerany abou he model srucure or model uncerany can arse boh from aleaory (possble esmaon errors, par example) and epsemc sources (unnown facors or neracons of economc varables). In boh cases robus echnques can be mplemened, bu he praccal mehod dffers wh he parcular ype of model uncerany. All possble cases of model uncerany can be roughly dvded no wo broad classes. The frs class can be named more or less paramerc uncerany. In hs case he overall srucure of he model s aen as rue bu he values of specfc parameers are unceran. Ths very class can be dvded no 3 ypes accordng o he exen o whch model uncerany s couned as a paramerc (Telow, Muehlen 24): Bayesan uncerany, he mos paramerc case. Under hs ype of model uncerany parameers of he model are assumed o have nown dsrbuons. 5

6 Srucured Knghan uncerany. In hs case uncerany s assumed o be locaed n some parcular parameers of he model. The rue values of hese parameers are supposed o le beween he nown mnmum and maxmum possble levels. Unsrucured Knghan uncerany, he less paramerc varan. Under such uncerany neher s locaon nor s naure are specfed. Ths case can be analyzed as a game played by he cenral baner agans a malevolen naure (Onas, Soc 2). The second class aes no accoun much more severe exen of uncerany han he frs one so called specfcaon uncerany. No only he precse parameers values bu even core srucural model characerscs are reaed as unnown. In he macroeconomc models such core characerscs can represen a characer of expecaons (forward- or bacward-loong), adherence o mcrofoundaons or, par example, lag srucure. In all cases hs class of uncerany adms he rue economy o be consderably dfferen from he gven model, and no only by he value of some parameers..2 Mehods of analyss Le possble nds of model uncerany, all analyzng robus echnques can also be dvded n wo classes. The frs one refers o he fs class of uncerany he paramerc one and mples ha he polcymaer or economs has only one reference model. The real srucure of he economy may dffer from hs model and he polcymaer beleves ha he rue economy les n he specfed neghborhood of a baselne model (Branard, 967). Ths neghborhood ncludes 6

7 all possble devaons from he reference framewor and one can nerpre hs approach as analyss of a se of smlar bu no dencal models (Gannon, 22). The second approach o he analyss mples a much more serous exen of uncerany specfcaon uncerany. In hs case such mporan characerscs as a nd of expecaons (bacward- or forward-loong or even a hybrd case) or lag srucure of varables are aen as unceran. Thus he rs of specfcaon error s so hgh ha he use of a sngle reference model s no approprae. Moreover, moneary polcy rules based only on a sngle model may cause very bad performance f he rue economy dffers from he baselne model by characer of expecaons mplemened (Levn, Wllams 23). Tha s why he followers of hs approach propose he analyss of a se of dsnc or compeng models wh very dfferen core characerscs..3 Choce crera Despe of general approach chosen we need o have crera accordng o whch one can evaluae he robusness of moneary polcy rule (a rule whch wors reasonably well n all models from he model se (eher smlar or compeng models)). The fs possble way s model or Bayesan averagng (Broc, Durlauf, Wes (24)) when he parameers of fnal rule mnmze a weghed sum of polcymaer s losses under dsnc models from he usng se. The weghs used are poseror probables of each model o be he rue one. There are several problems concernng hs mehod. The frs one s he dffculy wh compung hese probables. The second concerns he fac ha large dffcul models f daa beer han he smaller ones only due o he ncluson of large se of varables, 7

8 no because of her beer qualy. Bu n hs case he weghs of hese large models wll be unreasonably hgh n comparson wh smpler consrucons. Boh problems concerned here are no a queson f equal weghs are chosen as n Levn, Wllams (23). Ths mehod dffers from he Bayesan averagng because model weghs do no correspond o poseror probables. Mnmax creron mples mnmzaon of polcymaer s losses n he wors varan from he se of possble oucomes. Ths choce allows expecng a reasonably well performance of hs robus rule n all possble cases. Some auhors ry o combne advanages of dfferen choce varans and some hybrd crera arse. Kueser, Weland (28) ulzes an ambguy averson prncple when model weghs are aached le n he Bayesan mehod bu o he wors possble oucomes exra weghs are gven. Performance maxmzaon creron s very smlar o he mn-max one. In hs case he problem s represened as a zero-sum game beween a polcymaer and a malevolen naure, whch s amed o maxmze he losses (Gannon, 22). Under sably maxmzaon creron (for example n Onas and Soc (22)) polcy s chosen o maxmze a se of models, whch are sable under such a rule. Fnally, faul olerance approach naed by Levn, Wllams (23) mples frsly consrucng he opmal polcy rule for each model from he usng se. Then a researcher assesses a model s faul olerance by analyzng devaons of one of he polcy parameers from s opmal value, holdng he oher parameers fxed. If he loss funcon s relavely nsensve o changes n all parameers of he polcy rule, he model s consdered as a faul oleran one. Then zones of muual olerance are compued. For hese zones a compromse polcy can be 8

9 consruced. And hs polcy s consdered as a robus moneary polcy under model uncerany..4 General resuls The core characerscs of obaned robus rules crucally depend on he choce crera appled. Many auhors ry o compare hese dfferen rules n order o recognze some regulary n her performance. The frs resul was so called Branard s prncple accordng o whch a polcy under uncerany mus be less aggressve han a ceran one. Ths prncple s confrmed by some auhors (Bhan (22), Zacovc, Weland, Rusem (25), ec) and rejeced by ohers (Onas, Soc (22), Leemo and Södersröm (28)), who have found a more aggressve han a ceran polcy robus rules. Generally speang model averagng echnques lead o comparavely moderae polcy reacon whle mn-max approach mples more aggressve moneary rules (Zacovc, Weland, Rusem, 25). Bu even hs very cauous saemen s no always he ruh. Leemo and Södersröm (25) sae ha he exen of polcy reacon depends on he shoc ype, he source of uncerany and even on he wllngness of polcymaer o mplemen robus polcy (hs robus preferences)..5 Robus analyss for euro area The exen of model uncerany for euro area and consequenly he pernence of robus echnques are very hgh. There are several reasons for. Frs of all n comparson wh he USA models, consrucng models of euro area s relavely new and unelaboraed doman. Secondly he rue neracons beween European auhores are really complex. Whle moneary polcy s decded unquely by he 9

10 European Cenral Ban he fscal measures are underaen n each counry ndependenly whch n urn provoe addonal quesons when consrucng moneary polcy. Moneary auhory canno monor enrely he fscal moves and measures, so ha s an addonal source of uncerany. Thrdly he queson of daa aggregaon arses. There s also a more normave problem of necessary cenral ban s reacons on he counry specfc shocs or wha s he rue welfare creron for hese models. As he frs ECB s presden Wllem Dusenberg once sad, hs s no a rval as gven he large unceranes ha we are facng due o he esablshmen of a mul-counry moneary unon. No only can we expec some of he hsorcal relaonshps o change due o hs shf n regme, bu also, n many cases, here s a lac of comparable and cross-counry daa seres ha can be used o esmae such relaonshps. So, many auhors deal wh he problem of model uncerany relave o he euro area. Alavlla, Cccarell (28) shows on he models of euro area ha model uncerany really represens a danger for he moneary polcy and model averagng can be a useful ool for he polcy conduc. The whole doman of analyss concerns moneary polcy under nflaon uncerany. Unguelon, Coenen, Smes (23), Jaasela (25), Adald, Coenen, McAdam, Svero (25) and Coenen (27) found ha under hgh degree of nflaon he opmal moneary polcy rules are more aggressve and moreover, such rules are more robus. Some of hese arcles are based on he sole baselne model whle ohers examne dfferen ses of compeng framewors. The man general recommendaon for he moneary auhory s o assume hgher exen of nflaon perssence when consrucng moneary polcy. So, opmal robus polcy should mply relavely hgh degree of nera.

11 The smlar resul s obaned by Bhan, Sahuc (22) who examne parameer uncerany and fnd ha for he model of euro area Branard prncple holds. Smlar resul s obaned by Zcovc, Weland, Rusem (25) n spe of mn-max creron appled. Kueser, Weland (28) search for he robus polcy usng a hybrd approach whch combnes Bayesan and mn-max feaures. The weghs aached o he compeve euro models are compued accordng o he Bayesan prncple bu he exra weghs are added o he wors possble varans. All he papers concernng robus polcy characerscs n he euro zone deal wh area-wde models. Bu Bengno (24) showed ha erms of rade beween dfferen counres also maer and hus mus be n he opmal polcy consrucon. In our paper we llusrae a robus polcy consrucon for he wo-counry model of moneary unon.

12 Chaper 2. Reference model of moneary unon In hs paper we assume a unque cenral ban ha decdes for he moneary polcy n a wo-counry currency unon. Ths ban has n s possesson a sngle mcro-founded model wh scy prces ha s aen as reference bu here are some doubs concernng s qualy, hus he moneary auhory acles wh a model uncerany problem. So, he cenral ban needs o consruc no smply an opmal polcy for hs model bu a robus polcy for he possble msspecfcaon. Possble model dsorons are nroduced as choces of some malevolen agen whch ams a he cenral ban s losses maxmzaon. So he man as s a resoluon of mn-max problem by means of robus conrol echnques naed by ansen, Sargen (2). In hs paper we apply he wo-counry opmzng model wh scy prces descrbed n Bengno (24). The dfference of hs paper from he base model of Bengno s he choce of α - parameer of prce nera. Whle Bengno doesn specfy s value for he fuure calculaons, n our analyss we need o have some specfc numbers. So on a base of European daa n dvde counres of EMU n wo regons accordng o her prce rgdy and compue he parameer α for each of regons. Then hese values are used o derve he robus moneary polcy for he unon. The whole dervaon of he fnal equaons s presened n he Appendx A and here we nroduce only he man feaures of he model. The currency unon consss of wo counres or regons ( and F). The populaon of hs unon represens a un-connuum where agens from [, n) nerval belong o he -counry and he res [n, ] are F-counry s nhabans. 2

13 Each counry has an ndependen local governmen, whch deermnes fscal polcy n a correspondng regon. Moneary measures are defned by a sngle cenral ban. Each agen s smulaneously producer of a sngle dfferenaed good and consumer of all good ypes manufacured n he unon. So here s an nerregonal rade whle mgraon of labor force s absen. Number of goods, produced n he regon, equals o n, so hs parameer represens also an economc sze of hs regon or he share of unon GDP produced n he regon. 2. Consumer problem j j s j M s j max U = E β U ( Cs ) + L ; ε s V ( ys ; z s ) () s= P s.. E, { q B } B M + + j j j P ( + R ) P ( + ) M + B p j y j Q B + + C + j j, j j, j ( τ ) P P R P (2) where j [,] s agen s ndex, - counry s ndex ( or F ). β - neremporal dscoun rae. Uly funcon of agen j a perod s posvely depends on hs consumpon j C s, hs soc of real money balances M P j s s (hs can be nerpreed as some lqudy preferences of agens) and negavely on he j ' s supply of a dfferenaed good ( j y s ), as represens labor dsuly. j y s s a funcon of labor hours and a erm V ( ) ε s s a counry-specfc lqudy preference shoc, whle z s represens a producvy shoc n counry. 3

14 E X + sands for he expeced n he perod value of varable X n he perod + ; E s an operaor of raonal expecaons. Every agen consumes home and foregn good bundles, whch are subsues. So ndex j C s a composon of consumpon ndces for home and foregn goods: С j = j n j ( C ) ( CF ) n n ( n) n n (3) Whn each bundle dsnc producs are subsues wh an elascy of subsuon σ. Indces of j agen s consumpon of home and foregn goods are represened by he followng relaons: σ n σ σ σ j σ j С = c ( h) dh n (4) σ σ σ σ j σ j СF = c ( f ) df n (5) n j where c h s a quany of a good h [, n) j he j agen and F regon. Correspondng prce ndces are: from he counry consumed by c f s he same for he good f [ n,] produced n he Regon s prce ndex: n P P P P T = We also nroduce erms of rade n he followng way: P F n F. 4

15 Prce ndces of and F bundles sold on he regon s n σ P p ( h) dh n mare: σ σ PF p ( f ) df n n σ (6) where p ( h) s a prce of a good h sold n he regon, he same s assumed for he good f. Under assumpon of zero ransacon coss every good j mus be F sold a equal prces n boh regons, or p ( j) = p ( j) j. Consumer s budge consran (2) ncludes: B he real value a me of he agen j s porfolo of conngen j, secures ssued n regon and denomnaed n uns of he consumpon-based prce ndex wh one-perod maury q he vecor of he secury prces j B he agen j s holdng of he nomnal one-perod non-conngen bond denomnaed n he unon currency R he nomnal neres rae, moneary auhory s nsrumen τ a regonal proporonal ax on nomnal ncome Q nomnal lump-sum ransfers from he fscal auhory of regon j, o he agen j 2.2 Fscal polcy Fscal polces are deermned by he local governmens n each counry separaely. Each governmen collecs axesτ, deermnes ransfers 5 Q and j,

16 purchases only producs produced n s own counry. We don deal wh he problem of fscal polcy deermnaon, so we don solve any programs for he ransfers or axes. We only ae hese values as gven under assumpon ha he neremporal budge consran s held: E τ Y Q G = s= ( + R ) s = So, he whole sums of governmen expendures are gven by G and F G respecvely. I should be noed ha we assume a cenral ban whch aes he fscal measures as gven and auonomous and so here s no space for he fscal-moneary neracons and even for he opmal consrucon of fscal polcy. Bu one can ncorporae an addonal assumpons concernng he fscal polcy and even he relaons beween fscal and moneary auhores neracons o expand an analyss o he opmal consrucon of fscal measures or ang no accoun game aspecs of polcy deermnaon. For example, he smlar analyss can be made for he model of Beesma, Jensen (25), where polcy neracons can be aen no accoun. 2.3 Toal demand The oal demand for each good s a sum of all prvae agens demands and he demand of he correspondng governmen: σ d p h n j = + P y h T C dj G σ d p f n j F = + P F y f T C dj G (7) 6

17 Or, namng a unon-wde ndex of consumpon C W = j C dj d p h n W y ( h) = T C + G P σ σ d p f n W F y ( f ) = T C + G P F (8) Or on he regon level: Y Y F n W T C + G = P n W F T C + G = F P (9) 2.4 Frms Producers n he model are monopolss on her producs mares. They se prces accordng o he Calvo (983). Each seller faces a probably ( α ) of adjusng hs prce. Ths ype of nomnal rgdy mples ha dynamcs of prce level n every counry can de represened by he followng ln: J ( α ) P = P + p% () σ J σ J, α J, σ So, he erm α can be nerpreed as an exen of nomnal rgdy n he correspondng regon. Wh a help of hs parameer heerogeney beween regons can be nroduced no analyss (le n Brssms, Soda, 28). 7

18 2.5 Deermnsc seady sae Unforunaely, hs model canno be solved n he closed-form. Consequenly, he analyss s made n he erms of devaons from he deermnsc equlbrum. Deermnsc seady sae whou any shocs and zero nflaon rae s aen as a base for he analyss. In he followng analyss a noaon X corresponds o he value of varable X n he deermnsc seady sae. So we assume ha C = cons ε = G = z = π = and T = cons Y = cons We smplfy he fuure analyss resrcng our aenon only o he case of equal ax raes τ F = τ and T = and F Y = Y = C 2.6 Equlbrum wh scy prces In he followng analyss only equlbra whch are close o he deermnsc case descrbed earler are dscussed. So he soluon can be made n erms of small flucuaons from he deermnsc case and an approprae log-lnearzaon of he model s usually conduced. In he log-lnearzed equlbrum condons by X % we denoe he devaon of logarhmc of varable X from he seady sae when prces are flexble, whle X ˆ s he same devaon under scy prces. In oher words X% Xˆ X ( fl. prces) ln X X ( s. prces) ln X Dfferences beween hese wo devaons are named by small leers: 8

19 x = Xˆ X % In some cases s useful o deermne a unon-wde varable X W, whch represens a weghed average of specfc counres values: = + ( ) X nx n X W F, or a relave varable, whch s smply he dfference beween some values for dfferen counres: X X X R F. So, he law of moon of he economy s presened by he followng equaons: ( π ) E Cˆ = Cˆ + ρ Rˆ E () W W W + + ( ) ˆ ˆ ˆ W Y = n T + C + g (2) ˆ F ˆ ˆ W F Y = nt + C + g (3) ˆ ˆ F T = T + π π (4) ( n) ( Tˆ T% ) y E π W = T + C + β π + ( ˆ % ) F F F W F π = nt T T + C y + β Eπ + (5) (6) where C T UCCC ρ ( α β )( α ) ρ + η UC = α + ση VyyC and η + η Vy = C ρ + η VY εε Y V C yy The frs hree equaons deermne he relaons beween consumpon, governmen spendng, oupu gap, expeced fuure nflaon and he value of nomnal neres rae. To smplfy fuure calculaons we assume ha a fscal polcy s auonomous from any moneary decsons and s reaed as nown wh cerany, so g + = E g +. In hs case we can rearrange equaons (-3) no (7): 9

20 E y = y + Rˆ E π ρ W W W + + (7) Ths equaon s a usual form of IS-curve for he hole currency area, whch deermnes an oupu gap, whch depends posvely on s fuure expeced value, expeced fuure nflaon and negavely on he nomnal neres rae. Equaons (5-6) descrbe supply sde of he unon economy and sand for he New Keynesan Phllps curves. From hese equaons nflaon raes n he unon regons are deermned by he unon-wde oupu gap, expecaons of fuure nflaon and he unon erms of rade. Usually nsde erms of rade are omed from such ype of analyss, based on he unon-wde models, so he opmal polcy s consruced for he aggregae levels of nflaon and oupu. Bu from he equaons (5-6) s clear ha ang no accoun rade flows beween regons can be mporan for polcy consrucng. For hs very purpose we offer a more-hanone-counry model for analyzng robusness characerscs. Equaon (4) goes explcly from he defnon of erms of rade and represens a dynamc of hs varable deermned by s pas value and he curren nflaon raes n boh counres. So he cenral ban s as has o se a nomnal rae mnmzng s welfare funcon subjec o he equaons (4-7). Thus, n he model here are 4 forwardloong varables ( T π,, π y F W ) and one polcy conrol varable ( R ). 2.7 Welfare creron We assume ha cenral ban s benevolen and res o maxmze socal welfare W + gven by W = E β w - an expeced weghed sum of all fuure values of average uly n he unon w U ( C ) V y ( j), z dj. 2

21 The second-order approxmaon of he welfare funcon s based on he Beesma (25) and gves he followng form of welfare creron (see Appendx for deals): + W = E β L, where one perod loss s gven by ( ) ˆ γ ( π ) γ ( π ).. ( ε ) W 2 F 3 L = Λ y + n n Γ T T% + + F + p + o (8) Where..p. sands for he erms ndependen from polcy and he las par of hs relaon 3 ε ncludes all parameers of more-han-second order of approxmaon / σ and Λ = n / + n / C F C ( + η ) / σ F ( ρ η ) Γ = ( n / + n / ) + C n / C γ = n / + n / F C C F ( n) / C + ( ) γ = n / n / F F C C C (9) (2) (2) (22) We can see ha n hs model he loss funcon L has a usual quadrac form. Bu conrary o he semnal represenaon where he loss value s defned by he oupu gap and nflaon raes, erms of rade are also presen n hs funcon. I can be explaned by he reference model srucure, as he Phllps curves srucure ncludes rade beween regons and aes no accoun no only a currency area as a whole bu also relaonshps whn a unon. 2.8 Calbraon In our calbraon we parly follow Bengno (24). Thus we chose a value of elascy of producng dfferenaed goods η equal o.67. Parameer of 2

22 neremporal subsuon β equals o.99. The degree of monopolsc compeon σ s aen o be equal o Rs-averson coeffcen ρ s assumed o be /6. Moreover, s assumed ha he shoc T % follows he auo-regressve process of he nd T% =.95T% + ε, where he erm ε sands for he whe-nose process wh varance.86. The man dffculy concerns he choce of prce nera parameerα. In hs secon we don follow Begnno (24) who allows hese parameers o vary across a wde range of possble values. In conras, our choce of hese values s based on he esmaons of by Vermeulen a al (27). Table Frequency of prce changes and counry weghs n Euro GDP (%) (2) (3) Belgum France Germany Ialy Porugal.23.5 Span Euro area.22 Noe: (2): Frequency of prce changes ( α ) Source: Vermeulen a al (27) (3): Counry wegh n Euro GDP (%) 22

23 We ae a frequency of prce changes as a proxy for he probably o change a prce ( α ). We dvde counres n wo groups accordng o he followng scheme: f a frequency of prce changes s lower or equal o.22 (average frequency for he unon), he counry belongs o he regon. If hs frequency s hgher han.22, he counry s a par of F regon. Thus for he counres wh avalable daa regon consss of Germany, Span and Ialy, whle regon F consss of France, Belgum and Porugal. Accordng o he Table regon produces around 7% of unon oupu, so we calbrae he regon sze o he.7. Accordng o he correspondng weghs we assume ha an average frequency of prce change n he regon equals o.7, whle he same rao for he regon F comes o.23. F These values correspond o he model parameers α =.83 and α =.77 Now we compue he correspondng weghs n he welfare funcon: Λ =.58 Γ =.24 γ F =.82 γ = γ =.8 So, we can see ha he wegh aached o he scer regon ( ) n he loss funcon s hgher han a wegh of a more flexble one whch corresponds o he man fndngs of papers sudyng quesons of nomnal nera n he models of euro-area (see, for example, Adald, Coenen, McAdam, Svero (25)). 2.9 Cenral ban opmzaon problem Accordng o he made assumpons and he calbraon from he prevous secon we can compue values of he man parameers of he model. So, he cenral ban s program can be rewren as: 23

24 max E + β L ( π ) ( π ) 2 W ˆ F = + % L y T T (23) W W W E y+ = y + 6 R Eπ + π =.3( Tˆ T% ) +.5y +.99E π s.. π =.4( Tˆ T% ) +.y +.99E π ˆ ˆ F T T% = T T% + π π + e e =.95e + ε W + F W F + Ths specfcaon represens he problem of search for he opmal moneary polcy on he base of one reference model. From here s seen ha he man concern of he cenral ban s nflaon n he regon wh he hghes prce scness. In he followng secons we dscuss a polcy of cenral ban whch aes hs model as reference one bu fears ha realy can be raher dfferen from hs base framewor. In he nex secon we sae he program of robus polcy consrucon and hen we fnd parameers of he moneary polcy under model uncerany. 2. Sae space form of he model For he convenence of furher analyss he program (23) can be rewren n he usual sae space form. For hs we frsly expand he low of moon of erms of rade o he perod + and ae he raonal expecaon: Tˆ T% = Tˆ T% + π π +.95e F E Tˆ T% = Tˆ T% + E π E π +.95e F (24) From hs, usng (4), we oban: 24

25 ( ˆ % ) ˆ % F π ( ˆ % + + ) ( ˆ W π % ) ( ˆ % F W ) π π E T T = T T +. +.*.4 T T.*.y W. +.*.3 T T +.*.5y +.95e =.44 T T y +.95e (25) From oher equaons of (23) we oban expressons for he all expecaons of forward-loong varables: ( ˆ % ) ( ˆ % ) ( ˆ % ) E y =.2 T T +.4y 4.24π.82π + 6R W W F + E π =.3 T T.5y +.π W + E π =.4 T T.98y +.π F W F + (26) So he sae space form of our model s he followng: e =.95e + ε + + ( ˆ % + + ) ( ˆ %) ( ˆ + %) ( ˆ + % ) ( ˆ + %) W F E T T =.44 T T.55 y.π +.π +.95e W W F E y =.2 T T +.4y 4.24π.82π + 6R E π =.3 T T.5 y +.π W F W F E π =.4 T T.98y +.π (27) Or n he bref form: e e + A BR C E z = z ε + (28) Where T% ˆ T W y z = π F π - a vecor of forward-loong varables, E z + s an expeced n he perod fuure value of vecor z. A s marx of sze 5 5 of correspondng coeffcens from (27). B = and C =, showng ha only a predeermned varable e s allowed o be affeced be shocs. 25

26 Mnmzng loss funcon can be also rewren n he followng form: E β ( x' Qx ) (29) = where x e = z and Qs marx of sze 5 5 : n( n) Γ Q = Λ γ γ (3) So, he problem (53) can be rewren as: mn E β ( x' Qx ) R = e e + s.. A BR C E z = z ε + (3) 26

27 Chaper 3. Robus conrol specfcaon 3. Model uncerany specfcaon In modelng he robus program we mplemen a ansen-sargen s approach whch s also called robus conrol. We assume ha he cenral ban has (57) as a reference model descrbng he economy. Bu a he same me hs moneary auhory fears ha hs reference consrucon doesn correspond o he real sae of naure properly here s a rs of msspecfcaon. In oher words some perurbaons of modeled economy from he real one are allowed. The possble sources of hese perurbaons are some unnown varables or processes. To accoun hs possble msspecfcaon moneary auhory analyses only a class of alernave models, whch canno be dsngushed from he reference one wh help of sascal mehods. In oher words a se of possble perurbaons s lmed - ncludes only such perurbaons, whch wh some fxed probably wll no be dscovered. The reason o mpose hs resrcon on he possble msspecfcaon s que clear for grea perurbaons, when he real economy dffers consderably from he reference one, here s no any reason o ae any decson on he base of hs concree model and adapaon of he model o realy s needed. So he as for he cenral ban s o consruc no an opmal for hs model polcy bu a polcy, whch performs reasonably well even f here s any perurbaon. For hs purpose a mn-max creron s appled a robus polcy s such one ha produces he smalles losses n he case of he wors model perurbaon. For hs wors-case creron we can suppose ha perurbaons from he reference model ae he form of some addonal shocs 27 υ + s whch are added o he

28 sandard ε + s n he model (57) and are nduced by so called malevolen naure or evl agen, whch res o maxmze losses of he cenral ban. So he robus program can be represened by smulaneous wo-agens game, where he evl agens chooses a perurbaon from he reference model υ + s, he cenral ban decdes he values of s nsrumen neres rae. The se of possble perurbaons s modeled by he resrcon on he evl agen s measuresυ + s. In general here are hree possbles for he cenral ban o nerac wh prvae agens who form expecaons n he economy: ) Commmen o he opmal polcy 2) Commmen o some smple polcy rule (for example, semnal Taylor rule) 3) Dscreon ere we assume commmen case. In hs suaon several assumpons concernng he prvae percepons of he possble msspecfcaon can be made: ) Populaon has he same fear of msspecfcaon as he cenral ban 2) Populaon s more cauous han moneary auhory 3) Populaon s more careless In our analyss we consder he frs smples case, when populaon and cenral ban have he same reference model and he same deas concernng possble msspecfcaons. In he res wo cases he populaon s reference model and s se of model devaons should be specfed. Moreover, we need decde f he evl agen comms or no. Followng ansen, Sargen we analyze a case when boh opmzng agen comm. The man reason for hs assumpon s he fac ha our malevolen agen s only a meaphor used o consruc and solve he mn-max problem. The cenral ban under uncerany 28

29 derves s polcy as f hs evl naure exsed. Tha s why we suppose ha evl agen arrves and solves s program only when he cenral ban opmzes Consran robus conrol problem As we ve already dscussed n he prevous secon, here s some resrcon on he evl agen s measures. We assume he followng neremporal consran of he malevolen naure: + ' + = E β υ υ η (32) where υ s a vecor of dsurbances naed by he malevolen naure n he economy. And η s a possble oal exen of model msspecfcaon. In oher words, (32) represens he allowed se of perurbaons dscussed earler. If we magne all possble perurbed models as a cloud around he reference one and η s s radus. Ths represenaon of possble model se corresponds o he consran robus conrol problem of he cenral ban n he followng form: mn max E β ( x' Qx ) R = e e = s.. A BR C E z + z E υ + ' + = β υ υ η ( ε υ ) + + (33) Sze of possble perurbaons, η s defned by he cenral ban s fear of msspecfcaon: hgher fear allows hgher possbles of he malevolen agen and bgger devaons of he real economy from he reference one and s refleced by hgher η n (32). gher fear of msspecfcaon a he same me sands for hgher preferences for he robusness. 29

30 For example, f we assume ha he cenral ban has no any fear of msspecfcaon or has no preferences for robusness, η =, and we oban he program equvalen o (3). Noce ha we analyze polcy decsons for he momen and evl acons for he momen +. Ths means ha he value of hs sraegc shoc s nown a he me. (Alernave assumpon supposes ha he evl agens acons are no nown a he prevous momen or are hdden by he sochasc errors of hs perod). For a gven law of moon of economy he oal welfare depends only on he polcymaer s and malevolen agen s seps and he economc shocs. So we rewre problem (33) n he followng form: mn max β L ( R, υ) R = υ + = β D ( υ) η (34) where R s a sequence of cenral ban s decsons and υ s a sequence of sraegc shocs naed by malevolen agen: D υ = υ ' υ Mulpler or penaly robus conrol problem Anoher possble represenaon of robus conrol program s so called mulpler or penaly problem. The program (33) aes he followng form: β x Qx θυ+ υ+ = mn max E ( ' ' ) R υ e e = s.. A BR C E z + z ( ε υ ) + + (35) where θ represens a se of possble devaons of reference model from he real economy. When hs erm θ s low hs se s very large, when θ s hgh, here 3

31 are only very lmed dsorons. The case of θ corresponds o he usual opmal conrol program when no accoun of possble msspecfcaons s aen. Rewrng n he same form as (34), we oban: β L R υ θ β D + υ (36) = = mn max (, ) R υ 3.4. Relaons beween wo robus programs In hs secon we derve relaons beween wo robus programs presened earler. Frs of all we rename he overall losses of he consran robus program as K ( η ) = mn max β L ( R, υ) R υ Η ( η ) = Η ( η ) = υ : β D + ( υ) η = (37) The resulng level of opmzed funcon for he penaly problem s named: (38) = = = + V θ mn max β L ( R, υ) θ β D ( υ) R υ Now we reformulae consran robus problem (34) n he Lagrangan form: mn max mn (, ) R υ θ β L R υ θ β D + υ η (39), = = where θ s a Lagrange mulpler for he malevolen agen s consran. Changng he order of opmzaon (we can do f we consder only cases when equlbrum exss) we oban mn mn max β L ( R, υ) θ β D ( υ) η θ R υ + = = = = mnv + θη θ ( θ ) (4) From (4) s seen ha for a gven θ he las erm of hs expresson (θη ) doesn nfluence he choce of R andυ. So he followng saemen s rue: 3

32 Saemen. For he gven η * R and * υ are soluons of consran robus program. Then exss a * θ (equal o he opmal Lagrange mulpler from he consran robus program) such ha he correspondng penaly robus program has he same soluon and K mnv η * = θ + θη * (4) θ Now we consder ha for some follows ha for any η and θ K V V K θ * η θ * η * θ he penaly program s resolved. From (4) η θ + θη, so for any η (42) From here we conclude ha V max K (43) θ * η θ * η η Or, usng he defnon of K ( η ), V * θ maxmn max β L (, ) * R υ θ η η R υ Η ( η ) = (44) We maxmze he rgh sde of he expresson f we pu β D + ( υ) nsead of η, as he consran n (64) holds. = * max mn max β L ( R, υ) θ β D + ( υ) η R υ Η ( η ) = = * β L R υ θ η η R υ Η ( η ) = max mn max (, ) = (45) We rewre he penaly robus program 32

33 * * R υ = = V θ = mn max β L ( R, υ) θ β D ( υ) = * max mn β L ( R, υ) θ β D + ( υ) υ R = = = = + * max max mn β L ( R, υ) θ β D + ( υ) η υ Η( η ) R = = = (46) * Tang no accoun (45) and (46) we rewre he defnon of V ( θ ) : V η υ Η ( η ) * * max max mn L ( R, ) R = θ = β υ θ η = ( η ) = max K η * θ η (47) For gven η he las par of expresson, * θ η doesn nfluence he choce of R and υ, so he soluon of penaly robus corresponds o he soluon of he consran robus program. Saemen 2. For some * θ * R and * υ are soluons of penaly robus program. Then * * he consran robus program wh η = β D + υ has he same soluon. = So, we ve showed ha wo robus programs are ghly relaed. The sze of possble uncerany s gven by η n he consran robus program and by θ n he penaly program. These wo measures are nerconneced n he way presened by Saemens and 2. From (4) and (47) we can see ha hgher θ means lower correspondng value of η and vce versa. So when we spea abou hgher uncerany we mean hgher η and lowerθ Robus conrol program soluon 33

34 In hs and he nex secons we acle wh he penaly robus program rememberng ha all hese resuls can be easly represened as he soluon of correspondng consran problem. Ths problem s solved numercally and here we presen some echncal deals of soluon proposed n Gordan, Soderlnd (24). Gordan, Soderlnd (24) assume he followng robus polcy program: { u} { υ} ' β + + θυ+ υ+ = mn max E ( x ' Qx u ' Ru 2 xuu ' ) x x ( ε υ ) + s.. A Bu C E x = x 2 C C = Οn 2 n, where x s a n vecor of predeermned or bacward-loong varables, x2 s n 2 vecor of forward-loong varables, vecor ' ' neres equal o (, 2 ) x x. ' x s a vecor of all varables of u s a vecor of cenral ban nsrumens of sze. ε + s an d n vecor of shocs and υ + s an n vecor of sraegc shocs, naed by evl agen. Q and R are symmerc marces. Our program (35) s a prvae case of such of Gordan, Soderlnd (24) wh zero-marx and he sole polcy nsrumen neres rae, so =. n =, as we have only one predeermned varable e and here are 4 forward-loong varables n he model, so our vecor of forward-loong varables s of sze n 2 = 4. Because he core srucure of our model and such proposed n Gordan, Soderlnd (24) are dencal we now pass o he soluon algorhm of our specfc program. Frs of all we defne he error of expecaons for he perod + : = z E z (48) z ξ Then we noe he oal error of he model as: 34

35 ξ + ε = ξ + z + 5 (49) So he program (36) can be rewren as follows: β x Qx θυ + υ+ = mn max E ( ' ' ) (5) s.. x = Ax + BR + ξ + Cυ The Lagrangan of hs represenaon s: L ( x' Qx θυ ' υ ) = E β = + 2 ρ ' + ( Ax + BR + ξ+ + Cυ + x + ) (5) where ρ + s a vecor of sze 5 of correspondng langrangan mulplers. The frs-order condons wh respec o ρ +, x, R and υ + are: I5 Ο5 Ο5 Ο5 5 x + A5 5 B5 C5 Ο5 5 x ξ + Ο5 5 Ο5 Ο5 β A' 5 5 R + βq5 Ο5 Ο5 I 5 R = Ο5 + (52) Ο 5 Ο Ο B' 5 υ + 2 Ο 5 Ο Ο Ο 5 υ + Ο Ο 5 Ο Ο C ' 5 E ρ+ Ο 5 Ο θi Ο 5 ρ Ο where Οn sands for zero-marx of n sze. Or f we rename he marces of correspondng coeffcens n (52) we oban x + x ξ+ R R 5 G + Ο = D + υ + 2 υ + Ο Eρ+ ρ Ο (53) e Then we ae condonal expecaon of (53) and nong ha ρ s a shadow prce correspondng o he predeermned varable shadow prces for he z we oban: e and ρ s a vecor ( 4 ) z of 35

36 e + e z z ρ ρ + z z z + e R GE = D, or posng = z R + R ρ and λ = we oban: υ+ e υ + 2 υ + ρ e e ρ + ρ GE = D λ + λ + (54) Vecor e = z ρ has nal condons: he frs erm s he predeermned varable wh some nal condon e and he second represens shadow prces for he forward-loong varables and hus can be chosen freely a he nal perod o be z equaled o zero, so ha ρ =. Then we use decomposon proposed n he Gordan, Soderlnd (24). Square marces G and D can be represened by he followng decomposon: G = VSZ D = VTZ, when Z s a ranspose of conjugae of Z. Then, V V = Z Z = I and S and T are upper rangular. ence, (54) can be rewren as: + VSZ E VTZ λ = + λ (55) Premulply (55) by V + SZ E TZ λ = + λ (56) As S and T are upper rangular, (56) can be rewren so ha a he frs places sable soluons go. Sably of he soluon s checed by he correspondng egenvalue s modulus should be less han. If we denoe now 36

37 θ = Z δ λ, (57) where θ corresponds o he sable roos and δ - o he unsable ones, we oban: SE θ θ = T δ, or, as S and T are upper rangular, + δ + S S θ T T θ E =, (58) θθ θδ + θθ θδ S δδ δ + T δδ δ Ο Ο where J j comes for he correspondng par of J marx. For every sable soluon δ =, so Sθθ Eθ = + Tθθ θ or Eθ + = Sθθ Tθθθ (59) z z Then, usng e + Ee + = Ceε + and from Gordan (24) ρ E ρ =, we have + + e+ e + Ceε + E z z + E+ ρ = = + ρ + Ο (6) θ Z θ Z δ θ From (42) Z λ = δ = Z λθ Z λδ δ (6) And as δ =, Z θ = θ λ Z δ (62) ence, E Z E C ε Ο e = θ ( θ+ θ + ) = (63) In assumpon ha Z θ s nverble and usng (44), we oban: θ C ε e + + = Sθθ Tθθθ + Z θ Ο 4 (64) Then e + C eε + + = Z z θθ + Z θ Sθθ Tθθ θ ρ = = + + Ο 4 (65) + e + e + Or Z z θ Sθθ Tθθ Z θ M z z ρ = ρ + = + + Ο 4 ρ Ο 4 37 e e C ε e C ε (66)

38 z R e e And λ = = Zλθ Z θ = N z z υ + ρ ρ e ρ (67) From he las equaon (67) one can derve an opmal robus polcy, whch s consruced as some reacon on he values of predeermned varables and shadow prces of forward-loong varables: e R = ( Zλθ Z θ ) R z ρ (68) Equlbrum dynamcs of he model depends crucally on he acons aen by he evl agen n realy. And here we dsngush wo varans: wors-case model and approxmang one. The wors case aes place when he evl agen s acon realzes n realy. So such a model s descrbed by (66-67). The oppose suaon, when he cenral ban consrucs he robus polcy bu he evl agen doesn ac refers as an approxmang model. We oban from approxmang model by pung all sraegc shocs n (67) equal o zero. Surely, he resulng dynamcs of he economy depends crucally on he value of msspecfcaon fears, θ. The nex secon s devoed o he defnon of reasonable value for hs parameer Defnon of possble model se θ Followng Denns (29), we choose he value of hs parameer on he bass of error deecon probably mehod. The man dea under hs approach s ha model from he possble model se canno be easly dsngushed usng he avalable daa. In oher words, he cenral ban canno decde f real daa are 38

39 generaed wh he acons of malevolen agen or n he realy here s no any an-auhory measures. The frs suaon, when he malevolen naure aes all possble resources o nfluence he cenral ban s losses, s named wors case model ( W ). The second case, when he cenral ban nsure agans hs agen by robus polcy bu he evl naure doesn aes any measures s reaed as approxmang model ( A ). So, accordng o he error deecon mehod, he cenral ban should no be able o dsngush beween hs o models, ( W ) and ( A ), usng all avalable nformaon. In he oppose case, when he can ruly decde f here s any model deecon, a polcymaer has no any need of robus polcy, he smply adops he model o he realy. Probably of error π θ : ( LA LW W ) ( LW LA A) π θ = Pr > / 2 + Pr > / 2 (69) Where L s a lelhood funcon. The frs par of rgh-hand expresson sands for he probably o rea he model as an approxmang case whle n realy malevolen naure nerrups he daa generang process and he second par s probably o ae he model as a wors-case one whle here are no any naure acons. Ths probably s compued wh a help of smulaons and depends crucally on he value ofθ. We need o choose allowed error probably and fnd correspondng sze of msspecfcaon. 39

40 For example, 5% error probably corresponds o zero robusness (he case of sandard opmal polcy n he model wh raonal expecaons). Lower π ( θ ) means hgher msspecfcaon fear and hgher robus preferences (or lower θ ) Some compuaonal resuls We ve analyzed several varans of polcy robusness. Correspondng resuls of compuaons are summarzed n Appendx B, where noons M and N sand for he wors-case, M a and Na descrbe law of moon of approxmang model. Fv represens a malevolen agen reacon on he predeermned varables. Wha we are neresed n are he coeffcens of he robus polces, whch are e = represened by R ( Zλθ Z θ ) R z ρ The moneary polcy reacons under dfferen preferences for robusness are summarzed n Table2. Table 2. Parameers of robus moneary polcy e R = ( Zλθ Z θ ) R z ρ Error deecon probably θ r r2 r3 r4 r5 2% % % %

41 The mos mporan s he frs coeffcen r, whch represens reacon on he erms of rade. As hs parameer s subjec o he shoc nfluence, hs coeffcen also reflecs polcy reacon o he shocs. If we rac he module of he correspondng coeffcen as a degree of polcy aggressveness, we can see ha aggressveness of polcy reacon on he predeermned varable e rses wh ncrease n he preferences for robusness. Ths relaon can be explaned by he fac ha wh hgher fear of msspecfcaon he cenral ban supposes ha here s hgher possbly ha he shoc n he economy s no smply..d process bu s naed by he malevolen agen, and he reacs more aggressvely. ere we see some volaon of Branard prncple for he moneary unon and resul smlar o he sandard mn-max sraegy of more aggressve reacon o any shocs. Anoher neresng pon s he sgn of polcy reacon on he shoc: f here s a posve shoc, he cenral ban rases he neres rae. There s a clear nuon under hs fac: accordng o (23) hs shoc ncreases nflaon n he regon and decreases nflaon n he second par of he unon. Bu accordng o he coeffcens n he polcy loss funcon, he hng of s mos concern s nflaon n he regon wh hgher nflaon perssence, as s wegh n losses s much more mporan han oher varables. So n response o such shoc he reasonable response of cenral ban s an ncrease n neres rae wha decreases oal oupu n he unon and hus sablzes nflaon n he regon of he hghes neres. 4

42 Now we smulae conduc of economy under a shoc of unon s erms of rade, Impulse-response funcons are summarzed n he Graph, where he followng noaons are used: ε TT Terms of rade V Value of sraegc evl shoc Y Oupu gap L Value of loss funcon ph Inflaon n he regon WC Wors-case model pf Inflaon n he F regon A Approxmang model R Moneary polcy nsrumen RE Raonal expecaons whou robusness 42

43 Graph. Terms of rade shoc n he wors-case, approxmang model and raonal expecaons whou robusness TTWC YWC phwc pfwc RWC VWC LWC x 2 TTA x 22 YWA x 2 pha x 2 pfa x 2 RA VA x 42 LWA 5 5 TTRE YRE phre pfre RRE x -5 VRE 5 5 LRE

44 We can see from hs pcure ha malevolen naure reacs on hs shoc by edng some sraegc shoc (wors-case model). Knowng hs, he cenral ban ncreases neres rae. The resul of hs measure s a moderaon of he s nflaon rse. Resul of polcy-naure game n he wors case s a rse of oupu gap and nflaon n he regon rse, whle regon F s nflaon decreases. The las graph n he frs column represens losses assocaed wh he wors-case model. We can see, ha malevolen acons cause suffcen welfare losses. When here s no malevolen acon, he losses of naon are praccally nl (see column 2), even f robus polcy s appled. So, n hs case he robus polcy can properly couneraac any exernal shoc. If we consder RE case, when cenral ban aes he possbly of msspecfcaon as zero, he losses are hgher han n he case of a smple robus polcy. We compare hese wo polces defnng he dfference beween oal welfare under robus polcy and under non-robus one. Resul s presened n he followng graph: Graph 2. Welfare benefs from robus polcy applcaon

45 We can see ha robus polcy enals some welfare benefs n comparson wh he alernave. So here s an evden bass for robus polcy consrucon. Concluson For he mcro-founded wo-counry model of currency unon we have consruced a robus polcy under commmen. We ve found ha he cenral ban reacs on he shocs more aggressvely when hgher exen of possble msspecfcaon s admed, hus Branard prncple s volaed. We ve shown echncal deals of robus conrol opmzaon n he adapaon for wo-counry models. There are many possble applcaons of hs mehod for he consrucon of moneary polcy n he currency area. Frs of all we ve consdered only he case of erms of rade shoc. The analyss can be easly exended o oher economc shocs. For example, echnologcal changes can be aen no accoun. Secondly, we ve consruced a full commmen polcy. Bu he conrol of such polcy s raher dffcul, so he problem of polcy nconssency can arse. Conrol of smple moneary rules (for example, of a Taylor ype) s easer and so such rules can conrbue o he populaon confdence o he cenral ban measures. So he fs possble exenson s consrucon of robus moneary polcy rule. Thrdly, n our model here s no sance for moneary and fscal polcy neracons. The nfluence of cenral ban preferences for robusness on he economy under assumpon of sraegc neracons beween moneary and fscal 45

46 auhores can be analyzed, for example, on he base of wo-counry model of Beesma, Jensen (25). Then, our analyss can be easly exended o he case of ree or more counres. For example, represenaon of EMU as a unon of Germany, France and Ialy, he larges European economes, mgh provde he research wh some mporan exensons. Fnally, we ve dscussed only a case of wo-counry world. Neverheless, neracons wh he res of he world va nernaonal rade and capal flows can be analyzed. 46

47 Appendx A. Dervaon of he reference model ) Soluon of consumer problem Soluon of he consumer problem consss of 3 sages: ) Solvng a problem () subjec o (2) one deermnes an opmal value of consumpon ndex j C, an opmal real balances soc M P,an opmal oupu suppled. Ths problem can be easly rewren: j s j M + s j max E β U ( C+ s ) + L, ε+ s V ( y+ s, z+ s ) + s= P + s + E M + B p j y j Q j j, j, j + s + s + s + s j + s B + s + + ( τ ) C + P s P s P s λ s j j s=, j B + s M + s E + s { q + sb } P + s ( + R + s ) P + s From where we oban he relevan frs-order condons: M R LM, ε = U C ( C ) P P + R (A.) P U C ( C ) = ( + R ) β E U C ( C + ) P + j ( + ) λ EU C = E + P C s s + s (A.2) (A.3) (A.) ells us ha n he opmal consumer choce margnal rae of subsuon beween consumpon and real money balances holdngs mus concde wh he prce of real money balances n erms of consumpon ndex. (A.2) represens a usual Euler equaon for he forward-loong model. Accordng o (A.2) he margnal uly of consumpon n he perod mus equals o he expeced 47