Optimal Devaluations

Transcription

1 USC FBE MACROECONOMICS AND INTERNATIONAL FINANCE WORKSHOP preened by Juan Pablo Nicolini FRIDAY, Feb. 24, :30 pm 5:00 pm, Room: HOH-601K Opimal Devaluaion Conanino Hevia Univeriy of Chicago Juan Pablo Nicolini Univeridad Di Tella February 2005 Abrac We analyze opimal fical, moneary and exchange rae policy in a imple mall open econonomy model wih price eing fricion. We perform our analyi in he radiion of opimal dynamic Ramey problem. We characerize opimal allocaion and he governmen policie ha implemen he opimal allocaion. INTRODUCTION The purpoe of hi paper i o provide a heoreical framework o characerize opimal fical, moneary and exchange rae policy in a mall open economy model wih varying degree of price eing rericion. The conribuion of hi paper i o carry on he analyi following he dynamic Ramey lieraure. Thu, he mapping from policie o allocaion i derived from a fully ariculaed dynamic general equilibrium moneary model wih axe. An imporan conequence of hi approach i ha we can joinly udy opimal fical and moneary policy. In addiion, he explici inroducion of preference provide a naural welfare crieria o evaluae policie. PRELIMINARY AND INCOMPLETE 1

2 We conider a model in which a fracion of firm i rericed o e price one period in advance and characerize he opimal cyclical properie of he Ramey oluion. For hi economy, we fir exend reul derived in Correia, Nicolini and Tele (2001a) and how ha he e of implemenable allocaion i independen of he price eing rericion. Thu, he opimal allocaion under icky price i he ame a he opimal allocaion under flexible price. We hen how ha he cyclical properie of opimal hor run moneary policy depend on he naure of he hock driving he cycle. In ummary, if he boom i caued by a hock o he echnology of final good (non-radable), opimal moneary policy mu be procyclical and a devaluaion mu follow; if i i driven by a echnology hock o inermediae good (radeable), opimal moneary policy i counercyclical and he exchange rae mu decreae. Finally, if he boom i induced by an inernaional erm of rade hock, opimal moneary policy i procyclical and he exchange rae depend on he ource of he erm of rade hock: if i i driven by a decreae in he price of imporable, heexchangeraemuberevalued,whileifiidrivenbyanincreaeinhepriceof exporable, he exchange rae mu be devalued. Anoher remarkable reul i ha neiher he opimal allocaion nor he policy ha implemen i depend on he degree of price ickine. There i an exenive lieraure ha udie opimal moneary and exchange rae policie and characerize i in erm of i cyclical properie. Obviouly, hee properie do depend on he mapping from policie o allocaion ha i derived form he paricular model ued and on he welfare crierion ued. Mo of he lieraure ha ued reduced form model no explicily derived from preference and echnologie. Our reul differ form mo of he lieraure, omeime becaue of he paricular model we ue, omeime becaue of he welfare crieria ued. The model we analyze i very imple. A uch, i ha a lea wo weaknee we wan o dicu. Fir, a mo of he modern lieraure, we impoe ad-hoc reric- 2

3 ion on he price eing proce, inead of modelling he price eing deciion and deriving he opimal price eing rule. Thu, we ake a a fundamenal parameer he fracion of firm ha can adju price wihin he period. Therefore, he model i ubjec o he Luca criique, and hi raie doub of i uefulne for policy analyi. We do no view hi a a ignifican problem, ince we how ha boh he opimal allocaion and he opimal policy are independen of ha aumed fundamenal parameer. Thu, poenial change in he parameer due o change in policy will no aler our concluion regarding opimal policy. Second, a model a imple a hi one i no able o replicae he evidence of open economie, paricularly a he buine cycle frequency we will be focuing on. Why performing opimal policy exercie in model ha are no able o mach he daa?. Thi i indeed a eriou horcoming, bu here doe no eem o be obviouly beer choice available. We wen ahead wih he analyi, depie hi iue, for wo reaon: fir, we hope ha he inuiion we unravel here will prove ueful o underand he working of moneary and fical policy in model ha can replicae oberved paern for aggregae variable a buine cycle frequencie, if hee do exhibi price ickine. Second, we wan o explore he implicaion of price eing rericion for he conduc of opimal fical and moneary policy in open economie, above and beyond he empirical relevance of hi rericion in explaining real ime dynamic. Since many ime policy advice i offered baed on he alleged working of model wih icky price, clarifying he way hee model work wa, for u, a naural queion o raie. The characerizaion of opimal moneary and exchange rae policie i an old ime queion. There alo eem o be a cerain conenu wih repec o he way he nominal exchange rae ough o be managed given hock uch a governmen pending, real exchange rae or produciviy hock. On he oher hand, hee queion have only very recenly ared being addreed in general equilibrium dynamic mod- 3

4 el. The policy implicaion derived form he model we analyze are a odd wih he convenional widom many ime. In addiion, ome of hoe policy implicaion do no appear robu o mall change in he environmen. Our fir, very imple approximaion ugge ha general equilibrium economic doe no eem o uppor he convenional widom in all dimenion. THE MODEL We conider a mall-open moneary economy model wih icky price populaed by a repreenaive houehold which derive uiliy from a non-radeable final good and leiure. There i a coninuum of producer of non-radeable inermediae good indexed by i [0, 1], which are ued in he producion of he final good, and firm ha produce a radeable inpu. Each firm i in he non-radeable inermediae ecor i monopoliic compeiive and we aume ha a fracion α [0, 1] of hem i exogenouly conrained o e price in period condiional on he informaion available a mo up o period 1 (i.e. hey e price one period in advance). The re of he firmbehavecompeiively. Time i dicree and denoe he ae of he economy in period, whichbelong o a finie e of even S. =( 0, 1,..., ) i he hiory of even up o and including period andwedenoebyπ ( ) he probabiliy a of period 0 of he hiory condiional on a given iniial even 0. Noe ha hi formulaion allow for arbirary ochaic procee. The governmen iue a complee e of one period nominal bond coningen on he hiorie +1. Specifically, a bond indexed by he hiory +1 pay one uni of domeic currency condiional on +1 and zero oherwie, which co Q ( +1 ) uni of domeic currency. We denoe by Q ( τ ) hepriceofoneuniofdomeiccurrency in τ in uni of currency a,whereq( τ ) Q ( τ τ 1 ) Q ( τ 1 τ 2 )...Q ( +1 ). We alo aume ha here i an inernaional credi marke where one period ae 4

5 coningen ecuriie denominaed and priced in foreign currency are raded a a price Q ( +1 ). An analogou definiion for Q ( τ ) hold. Throughou he paper a upercrip i ued o denoe any foreign variable. The commodiie in hi economy are a non-radeable finalgood,labor,aconinuum of non-radeable inermediae good and wo radeable inermediae inpu. The domeic economy i able o produce only one of he radeable inpu and he oher mu be necearily impored. Even hough boh inpu can be impored from abroad, o avoid confuion le u agree o call he former he home inpu and he laer he foreign inpu. Each commodiy a period i indexed by he parial hiory,ohecommodiy pace i he e of coningen equence whoe h componen i a funcion of. A i uually aumed in he opimal axaion lieraure, he governmen ha o finance an exogenou equence of expendiure g ( ) which doe no generae uiliy o he houehold. The e of policy inrumen alo belong o he pace of pace of coningen equence. Moneary policy coni of rule for he ock of money M ( ) and he nominal exchange rae beween domeic and foreign currency ε ( ). Fical policy coni of linear ax rae on labor τ n ( ), axe on dividend τ d ( ) and linear axe on he receip from foreign ecuriie τ ( ). The inroducion of axe on foreign ecuriie i ueful for wo reaon: fir, i roughly capure he idea of axing inernaional capial flow and econd, hey urn ou o be imporan for he implemenaion of he Ramey allocaion (laer on we dicu he conequence of eliminaing hoe axe). Moreover, ince axaion of dividend i equivalen o lump-um axe, any opimal policy will have τ d ( 1 )=1. To make he problem inereing, we aume ha he proceed from he dividend axe are no enough o 1 Taxing dividend more han 100% i no feaible ince we aume ha firm can produce zero in he period where τ d ( ) > 1 (or e an infinie price) and we focu in equilibria where all good are produced. 5

6 finance he equence of governmen conumpion g ( ) and hence, he governmen mu ue diorionary inrumen o finance he remaining par of i expendiure. A governmen policy i defined a Ω {M,ε,τ n,τ }, where we ue he compac noaion x o denoe he coningen equence {x ( )} for any variable x. Preference are decribed by he expeced uiliy funcion: X X β u c,n π (1) =0 where 0 <β<1, c ( ) and n ( ) are conumpion and labor a repecively. We inroduce a cah-in-advance conrain for purchae of he final good c ( ): of he oal value of final good raded, we aume, a in Ireland (1996), ha a ochaic fracion 1/v ( ) of i ha o be paid wih currency and he remaining fracion can be acquired wih credi o be cleared a he beginning of he nex period. We follow he iming ued in Luca (1984) where each ime period i divided ino wo nonoverlapping rading eion. Specifically, a he beginning of period every agen oberve he curren ae and he governmen announce he amoun of money ha will be inroduced ino he economy or wihdrawn from i during he curren period. The fir eion coni of rading in ecuriie. A he beginning of he eion all previou deb and rade credi are honored and firm diribue he previou period dividend, if any. The ecuriie ha are raded aferward include: domeic and inernaional one period nominal ae coningen bond, domeic currency and hare in he non-radeable inermediae good firm dividend. In he econd rading eion, exchange of good, inpu and producion aciviie are carried ou. The houehold pli ino a worker-hopper pair. The worker ell labor o he firm and he hopper acquire he houehold planned conumpion for he curren period, paying a fracion 1/v ( ) wih he currency acquired in he previou rading eion and he remaining par i purchaed in exchange for claim o be honored in he ecuriie marke eion of he following period. Worker and oher 6

7 eller of inermediae good iue rade credi which are alo eled a omorrow ecuriie rading eion. Finally, we aume ha he marke for foreign currency i open during he whole period. The cah in advance conrain faced by he houehold i P ( ) c ( ) v ( ) M (2) where P ( ) i he price of he final good in period. The ubindex in he price level i no redundan, ince ome firm e price condiional on he informaion available a period 1. We inerpre v ( ) a a velociy hock. Since boh he e of domeic bond and he e of foreign bond pan he e of ae S, we aume, wihou lo of generaliy, ha houehold only hold domeic bond. A he beginning of he period, afer all deb and rade credi are honored and houehold oberve he curren ae, he ecuriie rading eion open and houehold chooe currency M ( ) and one period domeic bond B ( +1 ),ubjec o he wealh conrain M + X B +1 Q +1, W, (3) +1 where W ( ) i he amoun of nominal wealh afer deb and rade credi are honored. 2 Nominal wealh evolve according o W +1 = M + B +1 + W n 1 τ n P c. (4) The houehold problem i o maximize (1), by choice of c ( ), n ( ), M ( ) and B ( +1 ) ubjec o (2), (3), (4) and an arbirarily large negaive lower bound for B ( +1 ) /P ( ). The fir order condiion for he houehold can be ummarized in 2 Since we have aumed ha here i full axaion of dividend, we can diregard rade in he inermediae good firm hare. 7

8 he following equaion u n ( ) u c ( ) = W ( ) (1 τ n ( )) v ( ) P ( ) (R ( ) 1+v ( )) (5) βu c ( +1 ) π ( +1 ) u c ( ) = P +1 ( +1 ) P ( ) Q ( +1 ) R ( ) v ( ) R ( +1 ) 1+v ( +1 ) R ( +1 ) v ( +1 ) R ( ) 1+v ( ) (6) and he cah-in-advance conrain (2), where u c ( ) and u n ( ) denoe he marginal uiliy of conumpion and labor in period, hiory,andwherer( ) h P i 1 Q +1 (+1 ) i he nominal inere rae beween period and +1. Final good a period are produced from he coninuum of non-radeable inermediae good wih he conan reurn o cale echnology y Z 1 = y θ θ 1 θ 1 i, θ ; θ>1 (7) 0 where y ( ) denoe he final good, y (i, ) he inermediae inpu of variey i and θ i he elaiciy of ubiuion beween inermediae good. Since inermediae good producer are monopoliic compeiive firm, exience of an equilibrium require θ>1. Final good producer chooe inpu y (i, ) and oupu y ( ) o maximize P y Z 1 0 p i, y i, di ubjec o he producion funcion (7), where p (i, ) denoe he price of he inermediae good of ype i in period. Thi problem give he demand funcion y d i, P ( θ ) = y. (8) p (i, ) The zero profi condiion deermine he equilibrium price P ( ) a he following funcion of p (i, ): P Z 1 = p i, 1 θ di 0 8 1/(1 θ)

9 Theechnologyoproduceheinermediaegoodi [0, 1] i he conan reurn o cale producion funcion y i, = F n y i,,x i,,h i,,z y (9) where n y (i, ) i labor, x (i, ) denoe he home inpu, h (i, ) he foreign inpu and z y ( ) i an aggregae produciviy hock common acro varieie i [0, 1] aifying F/ z y > 0. The inroducion of he wo ype of radeable inermediae good i done in order o analyze how he opimal policy repond o erm of rade hock defined a d ( ) p x ( ) /p h ( ) where p x ( ) and p h ( ) denoe he price in foreign currency of he home-produced radeable inermediae good and foreign inermediae good repecively. To deermine a nominal exchange rae, we aume ha all inernaional exchange of good are carried ou in foreign currency. The domeic price in local currency of each radeable good are denoed by p x ( ) and p h ( ) repecively. We find i convenien o idealize he exience of inermediarie carrying ou all he inernaional exchange of radeable inermediae good. The aumpion ha hoe inermediarie demand foreign currency o impor good, or receive foreign currency when hey ell expor abroad doe no have oher effec han deermining a nominal exchange rae. Since he inermediary aciviy coni mainly of ranporing he inermediae inpu from he dock o he firm or vicevera, abence of arbirage opporuniie implie he purchaing power pariy (PPP) condiion p x = ε p x (10) p h = ε p h. Oupu of he home-produced radeable inermediae good i given by he linear producion funcion X = z x n x 9

10 where z x ( ) i a echnology hock and n x ( ) i labor. Compeiive pricing enure ha he value of he marginal produc equal he nominal wage rae, W ( ), p x z x = W. (11) Inwhafollow,iiconvenienocharacerizeheunicoofproducionofany non-radeable inermediae good firm i. Thi co minimizaion problem i given by min Wn y + p x x + p h h n y,x,h ubjec o F (n y,x,h,z y )=1. Since he echnology i conan reurn o cale, he uni co funcion i increaing, homogeneou of degree one and concave in he inpu price, and decreaing in z y. The homogeneiy of degree one of he uni co funcion implie ha i can be wrien a Wϕ µ px W, p x p h,z y W p x However, uing he equilibrium condiion (11) and ha he PPP condiion (10) imply p x ( ) /p h ( )=p x ( ) /p h ( )=d ( ), he uni co funcion become W φ z x,z y,d where φ ( ), defined a µ φ (z x,z y,d ) ϕ 1, 1 1, z x z x d,z y, depend only on he exogenou hock and aifie φ/ z x < 0, φ/ z y < 0, φ/ d < 0 and (z x φ) / z x > 0. We will find i convenien o ue he horer noaion φ ( ) inead of φ (z x ( ),z y ( ),d ( )). Moreover, he conan reurn o cale aumpion implie ha he inpu raio are he ame acro all non-radeable inermediae good firm, hu for all i [0, 1] x (i, ) n y (i, ) = κ x and h (i, ) n y (i, ) = κ h. (12) 10

11 I follow, hen, ha where y i, = n y i, F, (13) F F 1,κ x,κ h,z y iheameforallypei [0, 1]. The indury rucure in he non-radeable inermediae good ecor i monopoliic compeiion a in Dixi and Sigliz (1977). A menioned above, we inroduce price ickine by auming ha he firm in he inerval i [0,α] are conrained o e price a period condiional on he informaion available up o period 1. Equivalenly, we can hink of hoe firm, called icky firm, a eing price one period in advance. The remaining fracion of firm, called flexible firm, i allowed o e price a condiional on all he informaion available a ha period. We now conider he problem of he inermediae good firm. Sricly peaking, full axaion of dividend imply ha he pricing and producion deciion of he firm are indeerminae. We find i convenien, inead, o hink of each firm maximizing aferax dividend for τ d j < 1, and hen conidering he limiing economy a lim j τ d j =1 for all. Flexible firm face he aic opimizaion problem of maximizing nominal dividend max p i, W φ y d i, p (i, ) ubjec o he demand funcion (8). The opimal pricing rule deermine he price a a conan mark-up over he marginal co, p fl p i, = θ θ 1 W φ for i (α, 1]. (14) Sicky firm e price a period condiional on informaion available up o period 1. Sincep (i, 1 ) doe no depend on he new informaion available a period, i could be he cae ha along he equence τ d j < 1 for ome j, hefirm run negaive 11

12 dividend if forced o aify all he demand, o i will be opimal o produce zero. On he oher hand, if he price i higher han he marginal co, demand deermine producion. For any finie price p (i, 1 ), here will be poiive demand of ha variey, hence exience of equilibrium wih finie price require poiive producion of each ype. Throughou we aume ha hock are ufficienly mall o ha icky firm produce poiive quaniie. A icky price firm chooe he period price condiional on informaion a 1. I problem i max p (i, 1 ) X Q 1 p i, 1 W φ y d i, where y d (i, )=p (i, 1 ) θ P ( ) θ y ( ). The pricing rule i p 1 p i, 1 = θ X ψ W φ (15) θ 1 where ψ Q ( 1 ) P θ ( ) y ( ) P Q ( 1 ) P θ ( ) y ( ) Inead of eing a conan mark-up over he marginal co a he flexible firm do, icky firm e a conan mark-up over a weighed average of he marginal co acro all ae. We now decribe he aggregae conrain of he economy wih he re of he world. The rade balance meaured in uni of he home-produced inermediae good i defined a TB = X Z 1 0 x i, di p h ( ) p x ( ) Z 1 0 h i, di, (16) where X ( ) R 1 x (i, 0 ) di areheneexporofhehomeproducedinermediae good and p h( ) R 1 h (i, p x ( ) 0 ) di denoe he impor of foreign inermediae inpu meaured in uni of he home-produced inermediae good. 12

13 The equaion ha deermine he evoluion of he counry ne foreign ae a period, in foreign currency, i given by X B +1 Q +1 = B + p x TB +1 where B ( ) denoe he ne holding of foreign ecuriie of he economy a a whole. Solving he previou equaion aring from period 0 forward and focuing on bounded real allocaion, we obain he economy foreign ecor feaibiliy conrain, 0=B 0 + X X =0 p x TB Q 0 where B 0 i he iniial holding of foreign ae. Auming ha foreign ecuriie are priced by rik neural inveor we obain Q +1 = β p x ( ) p x ( +1 ) π +1 where i wa aumed ha foreign inveor have he ame dicoun facor β a domeic houehold. Thu, he foreign ecor conrain become X X β TB =0 π = B 0 p 0 (17) Abence of arbirage opporuniie beween domeic and foreign bond imply Q +1 = Q ( +1 ) ε ( ) (1 τ ( +1 )) ε ( +1 ). (18) Final good marke-clearing i given by c + g = y (19) and marke clearing in he labor marke require n = n x + 13 Z 1 0 n y i, di. (20)

14 An allocaion ℵ and a price yem P are coningen equence ℵ = {c, n, M, B, B, n x, n y (i), x (i), h (i), X, y d (i)} and P = {Q, Q, p x, p m, p (i), W } for i [0, 1]. Given a governmen policy Ω, an allocaion ℵ and a price yem P are an equilibrium if (i) houehold olve heir uiliy maximizaion problem; (ii) hepriceofhehome inpu aifie he compeiive pricing equaion (11); (iii) final good producer olve heir maximizaion problem; (iv) inermediae good producer ac opimally, ha i, (12) hold for all i [0, 1] and hey follow he pricing rule (14) if i (α, 1] and (15) if i [0,α]; (v) he final good and inpu marke clearing condiion (19), (20) are aified; (vi) he economy-wide feaibiliy conrain (17) hold; and (vii) he no-arbirage condiion (18) and (10) hold. 3 EQUILIBRIUM ALLOCATIONS: NECESSARY CONDITIONS An allocaion rule ℵ (Ω) i he equilibrium mapping from he e of policie Ω ino allocaion (which of coure, can be empy). The Ramey problem coni chooing a policy Ω uch ha he reuling allocaion rule ℵ (Ω) maximize he houehold uiliy among he e of equilibrium allocaion induced by he policy Ω. Theraegy we follow below coni in deriving a e of neceary condiion ha an equilibrium allocaion ℵ (Ω) ha o aify. We hen proceed o maximize he uiliy of he houehold ubjec o he e of allocaion derived from he neceary condiion. Since hoe condiion need no be ufficien for an equilibrium, he obained allocaion olve a relaxed problem ha may no be implemenable a an equilibrium. We hen how ha he allocaion i an equilibrium in our environmen and hence, i i indeed he opimal Ramey allocaion. For he re of he paper, we aume ha B 0 =0 and B0 =0, o ha here i no iniial wealh o ax. 3 By Walra Law, he budge conrain of he governmen, implici in he previou equaion, i alo aified. 14

15 The nex propoiion ake derive he neceary condiion ha an equilibrium allocaion ha o aify: Propoiion 1: If an equilibrium exi for a given policy Ω, hen he equilibrium allocaion ℵ (Ω) aifie he following hree condiion: i) Feaibiliy, c + g = F h αn y θ 1 θ +(1 α) n fl y i θ 1 θ θ 1 θ (21) where F ( ) wa defined in (13). ii) Curren accoun uainabiliy, where and iii) X X β TB π =0 (22) =0 TB = z x h n αn y =0 Proof: In he appendix. +(1 α) n fl y F φ i X X β uc c + u n n π =0 (23) Condiion iii) follow from he houehold ineremporal budge conrain and i include he opimizaion problem of all he agen in he economy. Inereingly, noe ha neiher i), ii) nor iii) include he velociy hock v ( ). Thi implie ha he (relaxed) opimal allocaion will be independen of i, in paricular, moneary policy will repond o accommodae any change in v ( ). Correia, Nicolini and Tele (2004) go one ep furher and prove, in a cloed economy verion of hi model, ha in fac, he previou condiion are ufficien o characerize all allocaion ha any planner wih preference increaing in c and decreaing in n would chooe. 15

16 Remark: Uing andard argumen (ee for example, Chari, Chriiano and Kehoe 199X) i can be hown ha wih he available policy inrumen, i), ii) and iii) above are neceary and ufficien condiion for an equilibrium allocaion in a flexible price economy. In oher word, an equilibrium wih flexible price aify hoe condiion and converely, if an allocaion aifie i), ii) and iii) above, hen here are price and governmen policie which implemen ha allocaion a an equilibrium wih flexible price. THE RAMSEY PROBLEM We aume ha here i a commimen echnology hrough which he governmen bind ielf forever o a paricular policy Ω choen a period 0. Alo, he governmen i able o make raional foreca abou fuure acion and price, in paricular, he governmen know he allocaion rule ℵ (Ω) mapping policie o allocaion. The relaxed Ramey problem i o maximize (1), by choice of ª c, n, n x,n y,n fl y, ubjec o he neceary condiion (21), (22) and (23). The fir order condiion of hi problem imply, afer ome algebra, u c 1+λ + λ u cc ( ) c ( ) + λu u c ( nc n = ηz x φ (24) ) u n 1+λ + λ u nn ( ) n ( ) + λu cn c = ηz x (25) u n ( ) n y = n fl y where λ i he Lagrange muliplier on (23) and η i he muliplier on (22). Since n y ( )=n fl y ( )=n y ( ), feaibiliy can be wrien a c + g = n y F. (26) Equaion (24), (25) and (26) ogeher wih conrain (22) and (23) compleely 16

17 characerize he relaxed Ramey allocaion, where he rade balance become TB = z x h n n y F φ i. (27) Specifically, equaion (24) and (25) can be ued o olve for conumpion and labor a a funcion of he muliplier λ and η. Thenwecanue(26)andhelabormarke clearing condiion o olve for n y ( ) and n x ( ) a a funcion of λ and η. A hi poin he allocaion i expreed a a funcion of he muliplier. The preen value conrain (22) and (23) can be ued o find he value for η and λ. Since he e of equilibrium allocaion i included in he e of allocaion ha aify condiion i), ii) and iii) of Propoiion 1, i follow ha if he oucome of he relaxed Ramey problem can be implemened a an equilibrium allocaion, hen i i he opimal allocaion. We hen have, Propoiion 2: The relaxed Ramey allocaion can be implemened a an equilibrium wih icky price and hence i i, indeed, he Ramey allocaion. Proof: Inheappendix. p fl In he opimal allocaion y ( )=y fl ( ), hen i follow from (8) ha p ( 1 )= ( )=P ( ) for all. In oher word, he governmen undoe he price ickine by manipulaing policy inrumen o make he price level fully predicable one period in advance. Thi reul i reminicen o he Diamond and Mirrle (1971) precripion according o which i i no opimal o ax inermediae good. In fac, he differen price of he varieie i [0, 1] due o price ickine i iomorphic o an economy wih flexible price bu in which a ax on he firm i [0,α] i included. Undoing he price ickine i equivalen o eliminaing ha ax rae (ee Correia, Nicolini and Tele (2004) for a furher developmen of hi idea). Noe ha velociy hock v ( ) and he fracion of icky firm α in he economy are irrelevan for he Ramey allocaion. I alo follow from (24), (25), (26) and (27) 17

18 ha he Ramey allocaion c ( ), n x ( ), n y ( ), n ( ) and TB( ) a any only depend on he realizaion of he ochaic procee daed a period,a governmen purchae g ( ), produciviy hock z x ( ) and z y ( ), and he erm of rade hock d ( ) and no on he hiory of realizaion. Furher, governmen expendiure hock do no affec c ( ) nor n ( ). The proof follow direcly from he fac ha c ( ) and n ( ) canbeolvedaafuncionofhe hock z x ( ), z y ( ) and d ( ), and of he wo muliplier λ and η, which do no depend on he paricular realizaion of g ( ). Aume, for example, ha governmen expendiure increae: equaion (26) how ha in order o keep conumpion conan, he amoun of labor allocaed o he producion of final good ha o increae o mee he higher governmen expendiure. Equaion (27) how ha he rade balance decreae. In oher word, he availabiliy of inernaional credi allow he planner o inure all governmen expendiure hock hrough borrowing and lending. Of coure, more finalgoodhaveobeproduced,omorelaboriallocaedohe inermediae good ecor. Hence he rade balance decreae for wo reaon: fir, inermediae good firm demand more radeable inpu of boh ype and econd, a he ame ime he counry reduce he producion of he home inpu o allocae more labor o he producion of inermediae good and hence, o final good.. In wha follow we find i convenien o aume ha uiliy i eparable beween conumpion and labor, u (c, n) =U (c) V (n) where U 0 (c) > 0, U 00 (c) < 0, V 0 (n) > 0 and V 00 (n) > 0. The eparabiliy of conumpion and labor urn ou o be a convenien aumpion. In hi cae he fir order condiion wih repec o c ( ) and n ( ) become U 0 c µ 1+λ 1+ U 00 (c ( )) c ( ) = ηz U 0 (c ( x φ (28) )) V 0 n µ 1+λ 1+ V 00 (n ( )) n ( ) = ηz V 0 (n ( x (29) )) 18

19 We furher aume ha he lef ide of equaion (28) i decreaing in c ( ) and he lef ide of (29) i increaing in n ( ). 4 We now udy how he opimal allocaion change wih he differen hock. Sar wih a poiive hock o he inermediae good echnology z y ( ). I follow from equaion (28) and (29), and he fac ha φ ( ) i decreaing in z y ( ),hac( ) increae and n ( ) remain conan. The exac reallocaion of labor beween ecor and he change in he rade balance i indeerminae. A poiive hock o he erm of rade d ( )=p x ( ) /p h ( ), ha qualiaively he ame effec a a produciviy hock o he final good echnology. Indeed, boh work hrough a reducion in he marginal co of producion in he final good ecor. Apoiivehockohehomeinpuechnologyz x ( ) ha a differen effec. Equaion (28) implie ha n ( ) increae, and given ha z x ( ) φ ( ) i increaing in z x ( ), c ( ) decreae. No only labor i reallocaed from he inermediae good ecor o he radeable inpu ecor bu i i alo opimal o increae aggregae labor a a whole. The rade balance increae for wo reaon: inermediae good firm reduce he demand of radeable inpu and oal producion of he home inpu increae. Inuiively, he hock o he radeable ecor deermine when i i good ime o expor and when i i no. When he counry produciviy in he radeable ecor increae, i i opimal o allocae more labor o i, o reduce labor allocaed o he inermediae good ecor and hence, o reduce conumpion. Furher, ince conumpion and leiure are boh normal, leiure alo decreae. The nex able ummarize he reul. 4 If U 00 (C) C/U 0 (C) and V 00 (N) N/V 0 (N) are roughly conan, hi requiremen i aified. 19

20 c ( ) n ( ) n x ( ) n y ( ) z x ( ) + + z y ( ),d ( ) + =?? g ( ) = = + v ( ) = = = = DECENTRALIZATION We now udy he policy implicaion of he opimal allocaion. Queion uch a under wha circumance flexible exchange rae are uperior o fixed exchange rae or vice-vera follow from our analyi. A we howed in he proof of propoiion 2, here i an indeerminacy in he implemenaion of he Ramey allocaion. In paricular, here i an equilibrium which implemen he Ramey allocaion for any equence of inere rae R ( ). (I i raighforward o eliminae hi indeerminacy: by exending he model o incorporae a cah-credi good eing, a in Luca and Sokey (1983), he indeerminacy diappear.) InourcaeiiconvenienofocuonheequilibriumwhereR ( )=1 (i.e. he Friedman rule), ince i i he only equilibrium where he labor ax rae and he price level of he final good P ( 1 ) doe no depend on he velociy hock v ( ). A he Friedman rule houehold are indifferen abou wha level of money o chooe and he cah-in-advance ceae o bind. We focu on he equilibrium where, in fac, he cah in advance conrain hold a equaliy. In he proof of propoiion 2 we provide he decenralizaion for any equence of inere rae R ( ). Herewewriedownherelevanequaionwhenhegovernmen follow he rule R ( )=1for all. The equaion ha deermine he decenralizaion 20

21 of he Ramey allocaion if α>0 are given by, P 0 ( 1 )=p 0, P +1 = P X 1 β U 0 (c (+1 )) π (+1 ) U 0 (c ( )) +1 (30) M = P ( 1 ) c ( ) (31) v ( ) µ V 0 (n ( )) θ 1 1 τ n U 0 (c ( )) = ( ) (32) θ φ ( ) W = θ 1 θφ ( ) P 1 (33) The decenralizaion of he equilibrium i obained recurively a follow: Given he iniial price P 0 ( 1 )=p 0,hefir equaion i he condiion ha juifie he inere rae R ( )=1and deermine recurively he price of he final good. The cah in advance condiion (31) deermine he money upply a period ha juifie he price P ( 1 ) given he allocaion and he velociy hock. Then (32) pin down he labor ax rae, and (33) deermine he equilibrium wage rae. The price of he home inpu follow from he pricing equaion (11). Finally, he exchange rae and he domeic price of he foreign inpu follow from he PPP condiion (10). Noe ha even hough he final good price i icky, he re of he price (i.e. wage rae, exchange rae and price of inpu) do depend on he curren realizaion of he ae. Given he previou equilibrium price and policie, Q ( +1 ) follow from (6) and he ax on foreign bond i deermined from (18). SincevelociyhockareirrelevaninheRameyallocaion,hegovernmenaccommodae he money upply o eliminae any flucuaion in v ( ), a can be een in equaion (31). Noe, furher, ha he re of he policie and nominal price are alo independen of he velociy hock; in paricular, he exchange rae ε ( ) doe no vary wih v ( ). If we inerpre velociy hock a moneary hock (or LM hock), a in Lahiri and Vegh (2003), he previou reul i conien wih he view ha he exchange rae hould no repond o moneary hock. 21

22 We conider now a poiive hock o he inermediae good echnology z y ( ). The Ramey allocaion precribe increaing conumpion and leing he aggregae amoun of labor conan. Since he final good price P ( 1 ) i given, i follow from (31) ha he ock of money M ( ) increae. Furher, ince φ ( ) decreae wih z y ( ), i follow from (33) ha he nominal wage increae and hence, from (11) and (10)heinpupricep x ( ) and p h ( ) increae and he nominal exchange rae ε ( ) alo increae. Now aume ha here i an increae in he erm of rade d ( )=p x ( ) /p h ( ). The qualiaive effec on he opimal allocaion i, a hown in he previou ecion, idenical o a poiive hock o he final good echnology z y ( ),namely,c ( ) increae and n ( ) remainconan. Ainheprevioucae,heincreaeinc ( ) iaainedhroughanincreaeinheockofmoneym ( ), he nominal wage increae and he domeic price of he home inpu alo increae. The behavior of he exchange rae, however, depend on how he increae in he erm of rade i made: hrough an increae in p x ( ), a decline in p h ( ), or ome oher change uch ha he d ( ) increae. According o he PPP equaion (10) for good x, if he increae in he erm of rade i driven by an increae in p x ( ), he movemen in he exchange rae and he domeic price of he home inpu i indeerminae. If, however, he increae in he erm of rade i driven by a decline in he p h ( ), equaion (10) for he good h and he fac ha p x ( ) increae imply ha ε ( ) mu go up. The domeic price of he foreign inpu could eiher increae or decreae. Finally conider a poiive hock o he home inpu echnology z x ( ).Theoluion o he Ramey problem implie ha aggregae labor increae, and conumpion decreae. Since he price level remain conan, (31) implie ha he governmen wihdraw money from he economy. Since φ ( ) decreae wih z x ( ), (33) implie ha he nominal wage rae increae. Alo, he domeic price of he home inpu 22

23 decreae. To ee hi, noe ha from (33) and (11) we obain P 1 = θ θ 1 p x z x φ, hence, ince z x ( ) φ ( ) increae wih z x ( ), p x ( ) ha o decreae. Furher, from he PPP condiion (10) imply ha boh, he nominal exchange rae ε ( ) and he domeic price of he foreign inpu decline. A can be een from he previou analyi, he exchange rae implied by he opimal allocaion doe change in he preence of he real hock z y, z x, p x and p h. Thi concluion i alo conien wih he radiional view ha he exchange rae hould vary when real hock hi he economy. The following able ummarize he finding of hi ecion. M ( ) ε ( ) τ n ( ) z x ( )? z y ( ),d ( ) + +? g ( ) = = = v ( ) = = 23

24 Appendix Proofofpropoiion1: We now prove ha i), ii) and iii) have o be aified in any equilibrium. Condiion i) follow by uing (7) and (13) ino he neceary condiion (19). Condiion ii) follow from he definiion of equilibrium, and he equaliy for he rade balance follow from (12), (20) and by noing ha he co minimizaion problem of he inermediae good firm implie z x F φ = z x + κ x + κ h ( ) d ( ) We now how ha iii) i neceary. Ue (3) and (4) o obain he houehold ineremporal budge conrain X X Q ( 0 ) P c + M R 1 W 1 τ n n =0 R ( ) =0 Uing he cah in advance conrain (2) a equaliy and rearranging we find X X ½ Q 0= ( 0 ) P c R ( ) 1+v ( ) W 1 τ n n ¾ R ( ) v ( ) =0 Inroducing (5) and (6) ino he previou equaion we obain (23) QED. Proofofpropoiion2: We now prove ha here i an equilibrium which i conien wih he allocaion of he relaxed Ramey problem. Below we find a price yem P and a governmen policy Ω uch ha he propoed allocaion aifie condiion i) o vii) in he definiion of equilibrium. Aume fir ha α>0and pick an arbirary equence R ( ). The relaxed Ramey allocaion implie ha he price level i fully predicable and P ( 1 )=p fl ( 1 )= p ( 1 ).GivenP ( 1 ), equaion (14) deermine he nominal wage W = θ 1 θφ ( ) P 1 (A1) 24

25 hen (11) deermine p x ( ) and he PPP condiion (10) pin down he nominal exchange rae ε ( ) and he price p h ( ). Inroducing (A1) ino (5) we obain he labor ax rae τ n ( ) conien wih he houehold inraemporal condiion. Given any equence of price P ( 1 ) and he arbirary equence R ( ), he houehold condiion (6) can be ued o find he price Q ( +1 ),whicharegivenby Q +1 = βπ (+1 ) u c ( +1 ) R ( +1 ) v ( +1 ) P ( )[R( ) 1+v ( )] P +1 ( +1 )[R( +1 ) 1+v ( +1 )] u c ( ) R ( ) v ( ) (A2) We have he addiional rericion P +1 Q ( +1 )=1/R ( ) hahaobeaified. Summing he previou equaion over all +1 and rearranging we obain u c ( ) v ( ) P ( )[R( ) 1+v ( )] = X βπ ( +1 ) u c ( +1 ) R ( +1 ) v ( +1 ) P +1 ( +1 )[R( +1 ) 1+v ( +1 )] +1 (A3) When α>0, he price a depend on 1, hence he previou equaion can be wrien a P +1 = P 1 R ( ) 1+v ( ) u c ( ) v ( ) X βπ (+1 ) u c (+1 ) R (+1 ) v (+1 ) [R ( +1 ) 1+v ( +1 )] +1 (A4) Given P 0 ( 1 )=p 0, he allocaion and he arbirary equence R ( ), (A4) deermine he whole equence of price P ( 1 ) which juifie he equence R ( ).Furhermore, he price level P ( 1 ) can be juified by eing M ( ) according o M = c ( ) v ( ) P 1 (A5) Given he nominal exchange rae ε ( ) and he price Q ( +1 ) obained above, he arbirage condiion (18) pin down he ax on foreign bond τ ( ). Noe ha by conrucion, flexible and icky price firm will find i opimal o e p ( 1 )=p fl ( 1 )=P ( 1 ). Given c and g, ue (26) o obain n y ( ),he demand for inpu x (i, ) and h (i, ) follow from (12) and n x ( ) follow from he labor marke clearing condiion (20). The holding of domeic bond follow from 25

26 olving he houehold budge conrain a period forward: B = X X Q (j ) Pj j 1 c j + M j R j 1 W j n j 1 τ n j R ( j ) j j= and he holding of foreign bond follow from olving he economy-wide conrain aring from period forward: B = X X j z h x j n j n y j F j φ j i Q j p x j= j Since all he condiion of an equilibrium are aified, he relaxed Ramey allocaion i implemenable and herefore i i, indeed, he opimal allocaion. If α =0(i.e. in a flexible price economy), price need no be fully predicable. Beide he indeerminacy in he nominal inere rae R ( ), here i indeerminacy in he price level, ince any price equence P ( ) aifying (A3) i implemenable a an equilibrium, in paricular he one derived for he cae α>0. An eay way o eliminae he indeerminacy of he nominal inere rae R ( ),i by adding a credi good. The marginal rae of ubiuion beween cah and credi goodpindownheinererae,ahowninlucaandsokey(1983).qed. 26

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