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1 COPYRIGHT NOTICE: Jord Galí: Monetary Polcy, Inflaton, and the Busness Cycle s publshed by Prnceton Unversty Press and copyrghted, 28, by Prnceton Unversty Press. All rghts reserved. No part of ths book may be reproduced n any form by any electronc or mechancal means (ncludng photocopyng, recordng, or nformaton storage and retreval) wthout permsson n wrtng from the publsher, except for readng and browsng va the World Wde Web. Users are not permtted to mount ths fle on any network servers. Follow lnks for Class Use and other Permssons. For more nformaton send emal to: permssons@pupress.prnceton.edu

2 7 Monetary Polcy and the Open Economy All the models analyzed n earler chapters assumed a closed economy: households and frms were not able to trade n goods or fnancal assets wth agents located n other economes. Ths chapter relaxes that assumpton by developng an open economy extenson of the basc New Keynesan model analyzed n chapter 3. The framework ntroduces explctly the exchange rate, the terms of trade, exports, and mports, as well as nternatonal fnancal markets. It also mples a dstncton between the consumer prce ndex that ncludes the prce of mported goods and the prce ndex for domestcally produced goods. Such a framework can n prncple be used to assess the mplcatons of alternatve monetary polcy rules for an open economy. Because the framework nests as a lmtng case the closed economy model of chapter 3, t allows the exploraton of the extent to whch the openng of the economy affects some of the conclusons regardng monetary polcy obtaned for the closed economy model: n partcular, the desrablty of a polcy that seeks to stablze nflaton (see chapter 4). It s also worth analyzng what role, f any, the exchange rate plays n the optmal desgn of monetary polcy and/or what s the measure of nflaton that the central bank should seek to stablze. Fnally, the framework can be used to determne the mplcatons of alternatve smple rules, as was done n chapter 4 for the closed economy. The analyss of a monetary open economy rases a number of ssues that a modeler needs to confront, and whch are absent from ts closed economy counterpart. Frst, a choce needs to be made between the modellng of a large or small economy,.e., between allowng or not, respectvely, for repercussons n the rest of the world of developments (ncludng polcy decsons) n the economy beng modelled. Second, the exstence of two or more economes subject to mperfectly correlated shocks generates an ncentve to trade n assets between resdents of dfferent countres n order to smooth ther consumpton over tme. Hence, a decson must be made regardng the nature of nternatonal asset markets and, more specfcally, the set of securtes that can be traded n those markets, wth Ths chapter s based on Galí and Monacell (25), wth the notaton modfed for consstency wth earler chapters. Secton 7.3 on the transmsson of monetary polcy shocks contans orgnal materal.

3 5 7. Monetary Polcy and the Open Economy possble assumptons rangng from fnancal autarky to complete markets. Thrd, one needs to make some assumpton about frms abltes to dscrmnate across countres n the prce they charge for the goods they produce ( prcng to market versus law of one prce ). Furthermore, whenever dscrmnaton s possble and prces are not readjusted contnuously, an assumpton must be made regardng the currency n whch the prces of exported goods are set ( local currency prcng,.e., prces are set n the currency of the mportng economy versus producer currency prcng,.e., prces are set n the currency of the producer s country). Other dmensons of open economy modellng that requre some choces nclude the allowance or not of nontradeable goods, the exstence of tradng costs, the possblty of nternatonal polcy coordnaton, and so on. A comprehensve analyss of those dfferent modellng dmensons and how they may affect the desgn of monetary polcy would requre a book of ts own, thus t s clearly beyond the scope of ths chapter. The modest objectve here s to present an example of a monetary open economy model to llustrate some of the ssues that emerge n the analyss of such economes and whch are absent from ther closed economy counterparts. In partcular, a small open economy model s developed, wth complete nternatonal fnancal markets, where the law of one prce holds. Then, n the dscusson of the model s polcy mplcatons and n the notes on the lterature n secton 7.6, there s reference made to a number of papers that adopt dfferent assumptons and brefly dscuss the extent to whch ths leads ther fndngs to dffer from those obtaned here. The framework below, orgnally developed n Galí and Monacell (25), models a small open economy as one among a contnuum of (nfntesmally small) economes makng up the world economy. For smplcty, and n order to focus on the ssues brought about by the openness of the economy, the possble presence of ether cost-push shocks or nomnal wage rgdtes s gnored. The assumptons on preferences and technology, combned wth the Calvo prcesettng structure and the assumpton of complete fnancal markets, gve rse to a hghly tractable model and to smple and ntutve log-lnearzed equlbrum condtons. The latter can be reduced to a two-equaton dynamcal system consstng of a New Keynesan Phllps curve and a dynamc IS-type equaton, whose structure s dentcal to the one derved n chapter 3 for the closed economy, though ts coeffcents depend on parameters that are specfc to the open economy whle the drvng forces are a functon of world varables (that are taken as exogenous to the small open economy). As n ts closed economy counterpart, the two equatons must be complemented wth a descrpton of how monetary polcy s conducted. After descrbng the model and dervng a smple representaton of ts equlbrum dynamcs, secton 7.3 analyzes the transmsson of monetary polcy shocks, emphaszng the role played by openness n that transmsson. Secton 7.4 turns to the ssue of optmal monetary polcy desgn, focusng on a partcular case for

4 7.. A Small Open Economy Model 5 whch the flexble prce allocaton s effcent. Under the same assumptons t s straghtforward to derve a second-order approxmaton to the consumer s utlty, whch can be used to evaluate alternatve polcy rules. Secton 7.5 assesses the merts of two dfferent Taylor-type rules, a polcy that fully stablzes the CPI, and an exchange rate peg. Secton 7.6 concludes wth a bref note on the related lterature. 7. A Small Open Economy Model The world economy s modelled as a contnuum of small open economes represented by the unt nterval. Snce each economy s of measure zero, ts performance does not have any mpact on the rest of the world. Dfferent economes are subject to mperfectly correlated productvty shocks, but t s assumed that they share dentcal preferences, technology, and market structure. Next, the problem facng households and frms located n one such economy wll be descrbed n detal. Before dong so, a bref remark on notaton s n order. Because the focus s on the behavor of a sngle economy and ts nteracton wth the world economy, and n order to lghten the notaton,varables are used wthout an -ndex to refer to the small open economy beng modelled. Varables wth an [, ] subscrpt refer to economy, one among the contnuum of economes makng up the world economy. Fnally, varables wth an astersk superscrpt (*) correspond to the world economy as a whole. 7.. Households A typcal small open economy s nhabted by a representatve household who seeks to maxmze E β t U(C t,n t ) () t= where N t denotes hours of labor, and C t s a composte consumpton ndex defned by [ ] η η η η C η η η η t ( α) (C H,t ) + α (C F,t ) (2) where C H,t s an ndex of consumpton of domestc goods gven by the constant elastcty of substtuton (CES) functon ( ) ε ε ε C H,t C H,t (j ) ε dj

5 52 7. Monetary Polcy and the Open Economy where j [, ] denotes the good varety. C F,t s an ndex of mported goods gven by ( γ ) γ γ C γ F,t (C,t ) d where C,t s, n turn, an ndex of the quantty of goods mported from country and consumed by domestc households. It s gven by an analogous CES functon ( ) ε ε ε C,t C,t (j ) ε dj. Note that parameter ε > denotes the elastcty of substtuton between varetes produced wthn any gven country. 2 Parameter α [, ] can be nterpreted as a measure of openness. 3 Parameter η> measures the substtutablty between domestc and foregn goods from the vewpont of the domestc consumer, whle γ measures the substtutablty between goods produced n dfferent foregn countres. Maxmzaton of () s subject to a sequence of budget constrants of the form P H,t (j ) C H,t (j ) dj + P,t (j ) C,t (j ) dj d + E t {Q t,t+ D t+ } D t + W t N t + T t (3) for t =,, 2,... where P H,t (j ) s the prce of domestc varety j. P,t (j ) s the prce of varety j mported from country. D t+ s the nomnal payoff n perod t + of the portfolo held at the end of perod t (and whch ncludes shares n frms), W t s the nomnal wage, and T t denotes lump-sum transfers/taxes. The prevous varables are all expressed n unts of domestc currency. Q t,t+ s the stochastc dscount factor for one-perod-ahead nomnal payoffs relevant to the domestc household. Assume that households have access to a complete set of contngent clams, traded nternatonally. As dscussed below, each country produces a contnuum of dfferentated goods, represented by the unt nterval. 2 Notce that t s rrelevant to thnk of ntegrals lke the one n (2) as ncludng or not the correspondng varable for the small economy beng modelled, because ts presence would have a neglgble nfluence on the ntegral tself (n fact, each ndvdual economy has a zero measure). The prevous remark also apples to many other expressons nvolvng ntegrals over the contnuum of economes (.e., over ) that the reader wll encounter below. 3 Equvalently, α s a measure of the degree of home bas. Note that n the absence of some home bas the households n the small open economy would attach an nfntesmally small weght to local goods, and consumpton expendtures would be allocated to mported goods (except for an nfntesmally small share allocated to domestc goods).

6 7.. A Small Open Economy Model 53 The optmal allocaton of any gven expendture wthn each category of goods yelds the demand functons ( PH,t (j ) ) ε C H,t (j ) = P H,t ( P,t (j ) ) ε C H,t ; C,t (j ) = P,t C,t (4) ε for all, j [, ], where P H,t ( P H,t (j ) ε dj ) s the domestc prce ndex (.e., an ndex of prces of domestcally produced goods) and P,t ( P,t (j ) ε dj ) ε s a prce ndex for goods mported from country (expressed n domestc currency) for all [, ]. Combnng the optmalty condtons n (4) wth the defntons of prce and quantty ndexes P H,t, C H,t, P,t, and C,t yelds P H,t (j ) C H,t (j ) dj = P H,t C H,t and P,t (j ) C,t (j ) dj = P,t C,t. Furthermore, the optmal allocaton of expendtures on mported goods by country of orgn mples ( ) γ C,t = P,t P F,t C F,t (5) γ γ for all [, ] where P F,t ( P,t d ) s the prce ndex for mported goods, also expressed n domestc currency. Note that (5), together wth the defntons of P F,t and C F,t, mples that total expendtures on mported goods can be wrtten as P,t C,t d = P F,t C F,t. Fnally, the optmal allocaton of expendtures between domestc and mported goods s gven by ( ) η ( ) η PH,t PF,t C H,t = ( α) C t ; C F,t = α C t (6) P t P t where P t [( α) (P H,t ) η + α(p F,t ) η ] s the CPI. 4 Note that under the assumpton of η = or, alternatvely, when the prce ndexes for domestc and foregn goods are equal (as n the steady state descrbed below), parameter α corresponds to the share of domestc consumpton allocated to mported goods. It s also n ths sense that α represents a natural ndex of openness. Accordngly, total consumpton expendtures by domestc households are gven by P H,t C H,t + P F,t C F,t = P t C t. Thus, the perod budget constrant can be rewrtten as η P t C t + E t {Q t,t+ D t+ } D t + W t N t + T t. (7) 4 It s useful to notce, for future reference, that n the partcular case of η =, the CPI takes the form P t = (P H,t ) α (P F,t ) α, whle the consumpton ndex s gven by C t = ( α) ( α) α α C α H,t C α F,t.

7 54 7. Monetary Polcy and the Open Economy As n prevous chapters, the perod utlty functon s specalzed to be of the form U(C,N) C σ N+ϕ. Thus, the remanng optmalty condtons for the σ +ϕ household s problem can be rewrtten as C t σ N t ϕ = W t whch s the standard ntratemporal optmalty condton. In order to derve the relevant ntertemporal optmalty condton note that the followng relaton must hold for the optmzng household n the small open economy P t V t,t+ σ σ C t = ξ t,t+ βc t + P t P t+ where V t,t+ s the perod t prce (n domestc currency) of an Arrow securty,.e., a one-perod securty that yelds one unt of domestc currency f a specfc state of nature s realzed n perod t +, and nothng otherwse, and where ξ t,t+ s the probablty of that state of nature beng realzed n t + (condtonal on the state of nature at t). Varables C t+ and P t+ on the rght sde should be nterpreted as representng the values taken by the consumpton ndex and the CPI at t + condtonal on the state of nature to whch the Arrow securty refers to beng realzed. Thus, the left sde captures the utlty loss resultng from the purchase of thearrow securty consdered (wth the correspondng reducton n consumpton), whereas the rght sde measures the expected one-perod-ahead utlty gan from the addtonal consumpton made possble by the (eventual) securty payoff. If the consumer s optmzng the expected utlty gan, t must exactly offset the current utlty loss. Gven that the prce of Arrow securtes and the one-perod stochastc dscount factor are related by the equaton Q t,t+ V t,t+, (9) can be rewrtten as 5 ξ t,t+ ( ) σ ( ) C t+ P t β = Q t,t+ () C t P t+ whch s assumed to be satsfed for all possble states of nature at t and t +. Takng condtonal expectatons on both sdes of () and rearrangng terms, a conventonal stochastc Euler equaton can be derved { ( ) σ ( ) } C t+ P t (8) (9) Q t = β E t () C t P t+ where Q t E t {Q t,t+ } denotes the prce of a one-perod dscount bond payng off one unt of domestc currency n t +. 5 Note that under complete markets a smple no room for arbtrage argument mples that the prce of a one-perod asset (or portfolo) yeldng a random payoff D t+ must be gven by V t,t+ D t+ where { Vt,t+ } the sum s over all possble t + states. Equvalently, that prce can be wrtten as E t ξ t,t+ D t+. Thus, the one-perod stochastc dscount factor can be defned as Q t,t+ Vt,t+ ξ t,t+.

8 7.. A Small Open Economy Model 55 For future reference, recall that (8) and () can be respectvely wrtten n log-lnearzed form as w t p t = σc t + ϕn t c t = E t {c t+ } ( t E t {π t+ } ρ) (2) σ where lowercase letters denote the logs of the respectve varables, t log Q t s the short term nomnal rate, ρ log β s the tme dscount rate, and π t p t p t s CPI nflaton (wth p t log P t ) Domestc Inflaton, CPI Inflaton, the Real Exchange Rate, and the Terms of Trade: Some Identtes Next, several assumptons and defntons are ntroduced, and a number of denttes are derved that are extensvely used below. Blateral terms of trade between the domestc economy and country s defned as S,t = P,t,.e., the prce of P H,t country s goods n terms of home goods. The effectve terms of trade are thus gven by S t P F,t P H,t ( ) γ γ = d S,t whch can be approxmated (up to frst order) around a symmetrc steady state satsfyng S,t = for all [, ] by s t = s,t d (3) where s t log S t = p F,t p H,t. Smlarly, log-lnearzaton of the CPI formula around the same symmetrc steady state yelds p t ( α) p H,t + αp F,t = p H,t + αs t. (4) It s useful to note, for future reference, that (3) and (4) hold exactly when γ = and η =, respectvely. It follows that domestc nflaton, defned as the rate of change n the ndex of domestc goods prces,.e., π H,t p H,t+ p H,t, and CPI nflaton are lnked accordng to the relaton π t = π H,t + α s t (5)

9 56 7. Monetary Polcy and the Open Economy whch makes the gap between the two measures of nflaton proportonal to the percent change n the terms of trade, wth the coeffcent of proportonalty gven by the openness ndex α. Assume that the law of one prce holds for ndvdual goods at all tmes (both for mport and export prces), mplyng that P,t (j ) = E,t P,t (j ) for all, j [, ], where E,t s the blateral nomnal exchange rate (the prce of country s currency n terms of the domestc currency), and P,t (j ) s the prce of country s good j expressed n terms of ts own currency. Pluggng the prevous assumpton nto ε the defnton of P,t yelds P,t = E,t P, where P ( P (j ) ε dj ),t,t,t s country s domestc prce ndex. In turn, by substtutng nto the defnton of P F,t and log-lnearzng around the symmetrc steady state, p F,t = = e t + p t (e,t + p,t )d where p,t p,t (j ) dj s the (log) domestc prce ndex for country (expressed n terms of ts own currency), e t e,t d s the (log) effectve nomnal exchange rate, and p t p,t d s the (log) world prce ndex. Notce that for the world as a whole, there s no dstncton between CPI and domestc prce level, nor between ther correspondng nflaton rates. Combnng the prevous result wth the defnton of the terms of trade yelds the expresson s t = e t + p t p H,t. (6) Next, a relatonshp s derved between the terms of trade and the real exchange t rate. Frst, the blateral real exchange rate s defned wth country as Q,t E,t P, P t.e., the rato of the two countres CPIs, both expressed n terms of domestc currency. Let q t q,t d be the (log) effectve real exchange rate, where q,t log Q,t. It follows that q t = (e,t + p t p t )d = e t + p t p t = s t + p H,t p t = ( α) s t where the last equalty holds only up to a frst-order approxmaton when η =. 6 6 The last equalty can be derved by log-lnearzng symmetrc steady state, whch yelds p t p H,t = αs t. P t η =[( α) + α S ] η P H,t t around a

10 7.. A Small Open Economy Model Internatonal Rsk-Sharng Under the assumpton of complete markets for securtes traded nternatonally, a condton analogous to (9) must also hold for the representatve household n any other country, say country V t,t+ (C ) σ = ξ t,t+ β(c t + ) σ E t P t t E t + P t + where the presence of the exchange rate terms reflects the fact that the securty purchased by the country s household has a prce V t,t+ and a unt payoff expressed n the currency of the small open economy of reference, and hence, needs to be converted to country s currency. The prevous relaton can be wrtten n terms of our small open economy s stochastc dscount factor as ( C ) σ ( t+ Pt )( t ) E β = Q t,t+. (7) E Ct Pt+ t+ Combnng () and (7), together wth the defnton for the real exchange rate defnton gves C t = ϑ C Q,t σ (8) t for all t, and where ϑ s a constant that wll generally depend on ntal condtons regardng relatve net asset postons. Henceforth, and wthout loss of generalty, symmetrc ntal condtons are assumed (.e., zero net foregn asset holdngs and an ex-ante dentcal envronment), n whch case ϑ = ϑ = for all. Takng logs on both sdes of (8) and ntegratng over yelds c t = c t + q t (9) ( σ ) α = c t + s t σ where c t c t d s the ndex for world consumpton (n log terms), and where the second equalty holds only up to a frst-order approxmaton when η =. Thus, the assumpton of complete markets at the nternatonal level leads to a smple relatonshp lnkng domestc consumpton wth world consumpton and the terms of trade A Bref Detour: Uncovered Interest Party and the Terms of Trade Under the assumpton of complete nternatonal fnancal markets, the equlbrum prce (n terms of the small open economy s domestc currency) of a rskless bond denomnated n country s currency s gven by E,t Q t = E t {Q t,t+ E,t+ },

11 58 7. Monetary Polcy and the Open Economy where Q t s the prce of the bond n terms of country s currency. The prevous prcng equaton can be combned wth the domestc bond prcng equaton Q t = E t {Q t,t+ } to obtan a verson of the uncovered nterest-party condton E t {Q t,t+ [exp{ t } exp{ t } (E,t+ /E,t )]} =. Log-lnearzng around a perfect foresght steady state, and aggregatng over, yelds the famlar expresson t = t + E t { e t+ }. (2) Combnng the defnton of the (log) terms of trade wth (2) yelds the stochastc dfference equaton s t = ( t E t {π t + }) ( t E t {π H,t+ }) + E t {s t+ }. (2) As shown n appendx 7., the terms of trade are pnned down unquely n the perfect foresght steady state. That fact, combned wth the assumpton of statonarty n the model s drvng forces and unt relatve prces n the steady state, mples that lm T E t {s T }=. 7 Hence, (2) can be solved forward to obtan s t = { [( E t t+k k= π t +k+ ) ( t+k π H,t+k+ )] } (22).e., the terms of trade are a functon of current and antcpated real nterest rate dfferentals. It must be ponted out that whle equatons (2) and (22) provde a convenent (and ntutve) way of representng the connecton between terms of trade and nterest rate dfferentals, they do not consttute an addtonal ndependent equlbrum condton. In partcular, t s easy to check that (2) can be derved by combnng the consumpton Euler equatons for both the domestc and world economes wth the rsk sharng condton (9) and equaton (5). Next, attenton s turned to the supply sde of the economy. 7 The assumpton regardng the steady state mples that the real nterest rate dfferental wll revert to a zero mean. More generally, the real nterest rate dfferental wll revert to a constant mean, as long as the terms of trade are statonary n frst dfferences. That would be the case f, say, the technology parameter had a unt root or a dfferent average rate of growth relatve to the rest of the world. Those cases would have persstent real nterest rate dfferentals.

12 7.. A Small Open Economy Model Frms Technology A typcal frm n the home economy produces a dfferentated good wth a lnear technology represented by the producton functon Y t (j ) = A t N t (j ) where a t log A t follows the AR() process a t = ρ a a t + ε t, and where j [, ] s a frm-specfc ndex. 8 Hence, the real margnal cost (expressed n terms of domestc prces) wll be common across domestc frms and gven by mc t = ν + w t p H,t a t where ν log( τ), wth τ beng an employment subsdy whose role s dscussed later n more detal Prce Settng As n the basc model of chapter 3, t s assumed that frms set prces n a staggered fashon. In partcular, a measure θ of (randomly selected) frms sets new prces each perod, wth an ndvdual frm s probablty of reoptmzng n any gven perod beng ndependent of the tme elapsed snce t last reset ts prce. As shown n chapter 3, the optmal prce-settng strategy for the typcal frm resettng ts prce n perod t can be approxmated by the (log-lnear) rule p H,t = µ + ( βθ) (βθ) k E t {mc t+k + p H,t+k } (23) k= ε where p H,t denotes the log of newly set domestc prces, and µ log ε s the log of the (gross) markup n the steady state (or, equvalently, the equlbrum markup n the flexble prce economy). 9 8 An extenson of the analyss to the case of decreasng returns consdered n chapter 3 s straghtforward. In order to keep the notaton as smple as possble the analyss here s restrcted to the case of constant returns. 9 p H,t s used to denote newly set prces nstead of p t (used n chapter 3), because n ths chapter letters wth an astersk refer to world economy varables.

13 6 7. Monetary Polcy and the Open Economy 7.2 Equlbrum 7.2. Aggregate Demand and Output Determnaton Consumpton and Output n the Small Open Economy Goods market clearng n the home economy requres Y t (j ) C (j ) d = ( PH,t (j ) ) ε = C H,t (j ) + H,t P H,t [ ( ) P η ) γ ( P H,t P H,t F,t ( α) C t + α P t ( E,tPF,t P t ) η C t d] (24) for all j [, ] and all t, where CH,t (j ) denotes country s demand for good j produced n the home economy. Notce that the second equalty has made use of (5) and (6) together wth the assumpton of symmetrc preferences across countres, ( ) ε ( ) γ ( P ) η whch mples C (j ) = α P H,t(j ) P H,t F,t C H,t. P H,t E,t P P t F,t t Pluggng (24) nto the defnton of aggregate domestc output ε Y t [ Y t (j ) ε dj ] ε yelds ( ) P η ) γ ( ) P η H,t P H,t F,t Y t = ( α) C t + α C t d P t ( E,tPF,t ( ) [ η ( ) γ η ] PH,t E,t PF,t η = ( α) C t + α Q C,t t d P t P H,t [ ] ( ) η PH,t ( = C t ( α) + α ) γ η η S S σ t,t Q,t d P t P t (25) where the last equalty follows from (8), and where S denotes the effectve terms t of trade for country, whle S,t denotes the blateral terms of trade between the home economy and country. Notce that n the partcular case of σ = η = γ = the prevous condton can be wrtten exactly as Y t = C t S α. (26) t Here one must use the fact that under the assumpton η =, the CPI takes the form P t = (P H,t ) α (P F,t ) α, thus mplyng P t = ( PF,t ) α α. P H,t P H,t = S t

14 7.2. Equlbrum 6 More generally, and recallng that s t d =, the followng frst-order loglnear approxmaton to (25) s derved around the symmetrc steady state ( ) y t = c t + αγ s t + α η q t σ αω = c t + s t (27) σ where ω σγ + ( α) (ση ). Notce that σ = η = γ = mples ω =. A condton analogous to the one above wll hold for all countres. Thus, for a generc country t can be rewrtten as yt = ct + αω s. By aggregatng over all σ t countres, a world market clearng condton can be derved as y t y t d = c t d c t (28) where y t and c t are ndexes for world output and consumpton (n log terms), and where the man equalty follows, once agan, from the fact that s t d =. Combnng (27) wth (9) and (28) yelds y t = y t + s t (29) σ α σ where σ α +α(ω ) >. Fnally, combnng (27) wth Euler equaton (2) gves αω y t = E t {y t+ } ( t E t {π t+ } ρ) E t { s t+ } σ σ α = E t {y t+ } ( t E t {π H,t+ } ρ) E t { s t+ } σ σ = E t {y t+ } ( t E t {π H,t+ } ρ)+ α E t { y t+ } (3) σ α where (σ γ ) + ( α)(σ η ) = ω. Note that, n general, the degree of openness nfluences the senstvty of output to any gven change n the domestc real rate t E t {π H,t+ }, gven world output. In partcular, f > (.e., for relatvely hgh values of η and γ ), an ncrease n openness rases that senstvty (.e., σ α s smaller). The reason s the drect negatve effect of an ncrease n the real rate on aggregate demand and output s amplfed by the nduced real apprecaton (and the consequent swtch of expendture toward foregn goods). Ths wll be partly offset by any ncrease n CPI nflaton relatve to domestc nflaton nduced by the expected real deprecaton, whch would dampen the change n the consumpton-based real rate t E t {π t+ } whch s the one ultmately relevant for aggregate demand relatve to t E t {π H,t+ }.

15 62 7. Monetary Polcy and the Open Economy The Trade Balance Let nx t ( )( ) Y t P t C Y P t H,t denote net exports n terms of domestc output, expressed as a fracton of steady state output Y. In the partcular case of σ = η = γ =, t follows from (25) that P H,t Y t = P t C t for all t, thus mplyng a balanced trade at all tmes. More generally, a frst-order approxmaton yelds nx t = y t c t αs t, whch combned wth (27) mples a smple relaton between net exports and the terms of trade ( ω nx t = α σ ) s t. (3) Agan, n the specal case of σ = η = γ =, nx t = for all t, though the latter property wll also hold for any confguraton of those parameters satsfyng σ(γ ) + ( α) (ση ) =. More generally, the sgn of the relatonshp between the terms of trade and net exports s ambguous, dependng on the relatve sze of σ, γ, and η The Supply Sde: Margnal Cost and Inflaton Dynamcs Aggregate Output and Employment ε Let Y t [ Y t (j ) ε dj ] ε represent an ndex for aggregate domestc output, analogous to the one ntroduced for consumpton. As n chapter 3, one can derve an approxmate aggregate producton functon relatng the prevous ndex to aggregate employment. Hence, notce that ( ) ε Y t P t (j ) N t N t (j ) dj = dj. A t P t As shown n chapter 3, however, varatons n d t ( P t (j ) ) ε dj around P t the perfect foresght steady state are of second order. Thus, and up to a frstorder approxmaton, the followng relatonshp between aggregate output and employment holds as Margnal Cost and Inflaton Dynamcs n the Small Open Economy y t = a t + n t. (32) As was shown n chapter 3, the (log-lnearzed) optmal prce-settng condton (23) can be combned wth the (log lnearzed) dfference equaton descrbng the evoluton of domestc prces (as a functon of newly set prces) to yeld an equaton

16 7.2. Equlbrum 63 determnng domestc nflaton as a functon of devatons of margnal cost from ts steady state value π H,t = βe t {π H,t+ }+λ mc t (33) where λ ( βθ)( θ). Thus, relatonshp (33) does not depend on any of the θ parameters that characterze the open economy. On the other hand, the determnaton of real margnal cost as a functon of domestc output n the open economy dffers somewhat from that n the closed economy, due to the exstence of a wedge between output and consumpton, and between domestc and consumer prces. Thus, n the present model, mc t = ν + (w t p H,t ) a t = ν + (w t p t ) + (p t p H,t ) a t = ν + σc t + ϕn t + αs t a t = ν + σy t + ϕy t + s t ( + ϕ) a t (34) where the last equalty makes use of (9) and (32). Thus, t can be seen that the margnal cost s ncreasng n the terms of trade and world output. Both varables end up nfluencng the real wage through the wealth effect on labor supply resultng from ther mpact on domestc consumpton. In addton, changes n the terms of trade have a drect effect on the product wage for any gven consumpton wage. The nfluence of technology (through ts drect effect on labor productvty) and of domestc output (through ts effect on employment and, hence, the real wage for gven output) s analogous to that observed n the closed economy. Fnally, usng (29) to substtute for s t, the prevous expresson for the real margnal cost n terms of domestc output and productvty, as well as world output, can be rewrtten as mc t = ν + (σ α + ϕ) y t + (σ σ α ) y t ( + ϕ) a t. (35) Generally, n the open economy, a change n domestc output has an effect on margnal cost through ts mpact on employment (captured by ϕ) and the terms of trade (captured by σ α, whch s a functon of the degree of openness and the substtutablty between domestc and foregn goods). World output, on the other hand, affects margnal cost through ts effect on consumpton (and, hence, the real wage as captured by σ ) and the terms of trade (captured by σ α ). Note that the sgn of ts mpact on margnal cost s ambguous. Under the assumpton of > (.e., hgh substtutablty among goods produced n dfferent countres), σ>σ α, mplyng that an ncrease n world output rases the margnal cost. Ths s so because n that case the sze of the real apprecaton needed to absorb the change n relatve supples s small wth ts negatve effects on margnal cost more than offset by the postve effect from a hgher real wage. Notce that n

17 64 7. Monetary Polcy and the Open Economy the specal cases α = and/or σ = η = γ =, whch mply σ = σ α, the domestc real margnal cost s completely nsulated from movements n foregn output. How does the degree of openness affect the senstvty of margnal cost and nflaton to changes n domestc and world output? Note also that, under the same assumpton of hgh substtutablty ( > ) consdered above, an ncrease n openness reduces the mpact of a change n domestc output on margnal cost (and, hence, on nflaton), for t lowers the sze of the requred adjustment n the terms of trade. By the same token, t rases the postve mpact of a change n world output on margnal cost by lmtng the sze of the assocated varaton n the terms of trade and, hence, ts countervalng effect. Fnally, and for future reference, note that under flexble prces, mc t = µ for all t. Thus, the natural level of output n the open economy s gven by y t n = + a a t + y t (36) where v µ, a +ϕ >, and α σ α. Note that the sgn of the σ α +ϕ σ α +ϕ σ α +ϕ effect of world output on the domestc natural output s ambguous, dependng on the sgn of the effect of the former on domestc margnal cost, whch n turn depends on the relatve mportance of the terms of trade effect dscussed above Equlbrum Dynamcs: A Canoncal Representaton In ths secton the lnearzed equlbrum dynamcs for the small open economy s shown to have a representaton n terms of output gap and domestc nflaton analogous to ts closed economy counterpart. Let ỹ t y t yt n denote the domestc output gap. Gven (35) and the fact that y t s nvarant to domestc developments, t follows that the domestc real margnal cost and the output gap are related accordng to mc t = (σ α + ϕ) y t. Combnng the prevous expresson wth (33) the followng verson of the New Keynesan Phllps curve for the open economy can be derved π H,t = βe t {π H,t+ }+κ α ỹ t (37) where κ α λ (σ α + ϕ). Notce that for α = (or σ = η = γ = ) the slope coeffcent s gven by λ (σ + ϕ) as n the standard, closed economy New Keynesan Phllps curve. More generally, note that the form of the nflaton equaton for the open economy corresponds to that of the closed economy, at least as far as domestc nflaton s concerned. The degree of openness α affects the dynamcs of nflaton only through ts nfluence on the slope of the NKPC,.e., the sze of the nflaton response to any gven varaton n the output gap. If > (whch

18 7.3. Equlbrum Dynamcs under an Interest Rate Rule 65 obtans for hgh values of η and γ,.e., under hgh substtutablty of goods produced n dfferent countres), an ncrease n openness lowers σ α, dampenng the real deprecaton nduced by an ncrease n domestc output and, as a result, the effect of the latter on margnal cost and nflaton. Usng (3) t s straghtforward to derve a verson of the so-called dynamc IS equaton for the open economy n terms of the output gap where ỹ t = E t {ỹ t+ } ( t E t {π H,t+ } rt n ) (38) σ α r n ρ σ α a ( ρ a )a t + α σ αϕ Et { y t + } (39) t σ α + ϕ s the small open economy s natural rate of nterest. Thus, t s seen that the small open economy s equlbrum s characterzed by a forward lookng IS-type equaton smlar to that found n the closed economy. Two dfferences can be ponted out, however. Frst, as dscussed above, the degree of openness nfluences the senstvty of the output gap to nterest rate changes. Second, openness generally makes the natural nterest rate depend on expected world output growth, n addton to domestc productvty. 7.3 Equlbrum Dynamcs under an Interest Rate Rule Next, the equlbrum response of our small open economy to a varety of shocks s analyzed. In so dong, t s assumed that the monetary authorty follows an nterest rate rule of the form already assumed n chapter 3, namely t = ρ + φ π π H,t + φ y ỹ t + v t (4) where v t s an exogenous component, and where φ π and φ y are non-negatve coeffcents chosen by the monetary authorty. Combnng (37), (38), and (4), the equlbrum dynamcs for the output gap and domestc nflaton can be represented by means of the system of dfference equatons [ ] [ ] y t Et{ y n = A t + } π α + B E { π } α ( r t v t ) (4) H,t n where r t rt n ρ, and t t+ [ ] [ σ α βφ π A α α ; B σ α κ α κ α + β(σ α + φ y ) T α wth α σα +φ y +κ α φ π. Note that the prevous system takes the same form as the one analyzed n chapter 3 for the closed economy, wth the only dfference lyng κ α ]

19 66 7. Monetary Polcy and the Open Economy n the fact that some of the coeffcents are a functon of the open economy n parameters α, η, and γ, and that r t s now gven by (39). In partcular, the con- dton for a locally unque statonary equlbrum under rule (4) takes the same form as shown n chapter 3, namely κ α (φ π ) + ( β) φ y >, (42) whch s assumed to hold for the remander of ths secton. Secton 7.3. uses the prevous framework to examne the economy s response to an exogenous monetary polcy shock,.e., an exogenous change n v t.gven the somorphsm wth the closed economy model of chapter 4, many of the results derved there can be exploted. The analyss of the effects of a technology shock (or a change n world output), whch s not pursued below, goes along the same lnes as n chapter 3. Frst, one should determne the mplcatons of the shock consdered for the natural nterest n rate r t and then proceed to solve for the equlbrum response of the output gap and domestc nflaton exactly as done below for the case of a monetary polcy shock, n gven the symmetry wth whch v t and r enter the equlbrum condtons. t 7.3. The Effects of a Monetary Polcy Shock Assume that the exogenous component of the nterest rate v t follows an AR() process v t = ρ v v t + ε t v where ρ v [, ). n The natural rate of nterest s not affected by a monetary polcy shock so r t = for all t for the purposes of ths exercse. As n chapter 3, let us guess that the soluton takes the form ỹ t = ψ yv v t and π t = ψ πv v t, where ψ yv and ψ πv are coeffcents to be determned. Imposng the guessed soluton on (37) and (38) and usng the method of undetermned coeffcents, and y t = ỹ t = ( βρ v ) v v t π H,t = κ α v v t where v ( βρv )[σ α ( ρ v )+φ y ]+κ α (φ π ρ v. It can be easly shown that as long as ) (42) s satsfed, v >. Hence, as n the closed economy, an exogenous ncrease n the nterest rate leads to a persstent declne n output and nflaton. The sze Of course, as n chapter 3, t must be taken nto account that a technology shock or a shock to world output also leads to a varaton n the natural output level, thus breakng the dentty between output and the output gap.

20 7.3. Equlbrum Dynamcs under an Interest Rate Rule 67 of the effect of the shock relatve to the closed economy benchmark depends on the values taken by a number of parameters. More specfcally, f the degree of substtutablty among goods produced n dfferent countres s hgh (.e., f η and γ are hgh, then ω> ) then v can be shown to be ncreasng n the degree of openness, thus mplyng that a gven monetary polcy shock wll have a larger mpact n the small open economy than n ts closed economy counterpart. Usng nterest rate rule (4) can determne the response of the nomnal rate, takng nto account the central bank s endogenous reacton to changes n nflaton and the output gap t = [ v (φ π κ α + φ y ( βρ v )) ] v t. Note that as n the closed economy model, the full response of the nomnal rate may be postve or negatve, dependng on parameter values. The response of the real nterest rate (expressed n terms of domestc goods) s gven by r t = t E t {π H,t+ } = [ v ((φ π ρ v )κ α + φ y ( βρ v )) ] v t whch can be shown to ncrease when v t rses (because the term n square brackets s unambguously postve). Usng (29) can uncover the response of the terms of trade to the monetary polcy shock s t = σ α y t = σ α ( βρ v ) v v t. The change n the nomnal exchange rate s gven n turn by e t = s t + π H,t = σ α ( βρ v ) v v t κ α v v t. Thus, a monetary polcy contracton leads to an mprovement n the terms of trade (.e., a decrease n the relatve prce of foregn goods) and a nomnal exchange rate apprecaton. Note that, n the long run, the terms of trade revert back to ther orgnal level n response to the monetary polcy shock, whle the (log) levels of both domestc prces and the nomnal exchange rate experence a permanent change of sze κ α v (gven an ntal shock of sze normalzed to unty). ρ v Hence, the exchange rate wll overshoot ts long-run level n response to the monetary polcy shock, f and only f, σ α ( βρ v )( ρ v )>κ α ρ v whch requres that the shock s not too persstent. It can be easly shown that the prevous condton corresponds to that for an ncrease n the nomnal nterest rate

21 68 7. Monetary Polcy and the Open Economy n response to a postve v t shock. Note that, n that case, the subsequent exchange rate deprecaton requred by the nterest party condton (2) leads to an ntal overshootng. 7.4 Optmal Monetary Polcy: A Specal Case Ths secton derves and characterzes the optmal monetary polcy for the small open economy descrbed above, as well as the mplcatons of that polcy for a number of macroeconomc varables. The analyss, whch follows closely that of Galí and Monacell (25), s restrcted to a specal case for whch a second-order approxmaton to the welfare of the representatve consumer can be easly derved analytcally. Its conclusons should thus not be taken as applyng to a more general envronment. Instead, ths exercse s presented as an llustraton of the approach to optmal monetary desgn to an open economy. Let us take as a benchmark the basc New Keynesan model developed n chapter 3. As dscussed n that chapter, under the assumpton of a constant employment subsdy τ that neutralzes the dstorton assocated wth frms market power, the optmal monetary polcy s the one that replcates the flexble prce equlbrum allocaton. The ntuton for that result s straghtforward: Wth the subsdy n place, there s only one effectve dstorton left n the economy, namely, stcky prces. By stablzng markups at ther frctonless level, nomnal rgdtes cease to be bndng, snce frms do not feel any desre to adjust prces. By constructon, the resultng equlbrum allocaton s effcent, and the prce level remans constant. In an open economy and as noted, among others, by Corsett and Pesent (2) there s an addtonal factor that dstorts the ncentves of the monetary authorty beyond the presence of market power: the possblty of nfluencng the terms of trade n a way benefcal to domestc consumers. Ths possblty s a consequence of the mperfect substtutablty between domestc and foregn goods, combned wth stcky prces (that render monetary polcy non-neutral). As shown below, and as dscussed by Bengno and Bengno (23) n the context of a two-country model, the ntroducton of an employment subsdy that exactly offsets the market power dstorton s not suffcent to render the flexble prce equlbrum allocaton optmal, for, at the margn, the monetary authorty would have an ncentve to devate from t to mprove the terms of trade. For the specal parameter confguraton σ = η = γ = the employment subsdy that exactly offsets the combned effects of market power and the terms of trade dstortons can be derved analytcally, thus renderng the flexble prce equlbrum allocaton optmal. That result, n turn, rules out the exstence of an average nflaton (or deflaton) bas and allows the focus on polces consstent wth zero average nflaton n a way analogous to the analyss for the closed economy found

22 7.4. Optmal Monetary Polcy: A Specal Case 69 n chapter 4. Perhaps not surprsngly, and as shown below, the polcy that maxmzes welfare n that case requres that domestc nflaton be fully stablzed, whle allowng the nomnal exchange rate (and, as a result, CPI nflaton) to adjust as needed n order to replcate the response of the terms of trade that would be obtaned under flexble prces. One may wonder to what extent the optmalty of strct domestc nflaton targetng s specfc to the specal case consdered here or whether t carres over to a more general case. The optmal polcy analyss undertaken n Faa and Monacell (27), usng a model nearly dentcal to the one consdered here, suggests that whle the optmal polcy nvolves some varaton n the domestc prce level, the latter s almost neglgble from a quanttatve pont of vew, thus makng strct domestc nflaton targetng a good approxmaton to the optmal polcy (or at least condtonal on the productvty shocks consdered here). Usng a dfferent approach, de Paol (26) reaches a smlar concluson, except when an (mplausbly) hgh elastcty of substtuton s assumed. 2 But even n the latter case, the losses that arse from followng a domestc nflaton targetng polcy are neglgble. 3 More generally, t s clear that there are several channels n the open economy that may potentally render a strct domestc nflaton polcy suboptmal, ncludng a nonuntary elastcty of substtuton, local currency prcng, ncomplete fnancal markets, and so on, all of whch are unrelated to the sources of polcy tradeoffs that may potentally arse n the closed economy. The quanttatve sgnfcance of the effects of those channels (ndvdually or jontly) stll needs to be explored n the lterature, and ts analyss s clearly beyond the scope of ths chapter. Wth that consderaton n mnd, let us next turn to the analyss of the optmal polcy n the specal case mentoned above The Effcent Allocaton and Its Decentralzaton Let us frst characterze the optmal allocaton from the vewpont of a socal planner facng the same resource constrants to whch the small open economy s subject n equlbrum (n relaton to the rest of the world), gven the assumpton of complete markets. In that case, the optmal allocaton must maxmze U(C t,n t ) subject to () the technologcal constrant Y t = A t N t, () a consumpton/output possbltes set mplct n the nternatonal rsk-sharng condtons (8), and () the market clearng condton (25). 2 Those results are condtonal on productvty shocks beng the drvng force. Not surprsngly, n the presence of cost-push shocks of the knd consdered n chapter 5, stablzng domestc nflaton s not optmal (as n the closed economy). 3 In solvng the optmal polcy problem for the general case, de Paol (26) adopts the lnear quadratc approach orgnally developed n Bengno and Woodford (25), whch replaces the lnear terms n the approxmaton to the households welfare losses usng a second-order approxmaton to the equlbrum condtons. Faa and Monacell (27) solve for the Ramsey polcy usng the orgnal nonlnear equlbrum condtons as constrants of the polcy problem.

23 7 7. Monetary Polcy and the Open Economy Consder the specal case of σ = η = γ =. In that case, (9) and (26) mply the exact expresson C t = Y t α (Y t ) α. The optmal allocaton (from the vewpont of the small open economy, whch takes world output as gven) must satsfy U n (C t,n t ) = ( α) U c (C t,n t ) whch, under the assumed preferences and gven σ =, can be wrtten as C t N t C t N t ϕ = ( α) C t N t thus mplyng a constant employment N = ( α) +ϕ. Notce, on the other hand, that the flexble prce equlbrum n the small open economy (wth correspondng varables denoted wth an n superscrpt) satsfes = MCt n ε ( τ) U n = (S n ) α n,t A t t U n c,t = ( τ) Y t n (N n t ) ϕ n n C A t t C t = ( τ) (N t n ) +ϕ where the term on the rght sde of the second equalty corresponds to the real wage (net of the subsdy) normalzed by productvty, and where the thrd equalty follows from (26). Hence, by settng τ such that ( τ)( α) = ε s satsfed or, equvalently, ν = µ + log( α), the optmalty of the flexble prce equlbrum allocaton s guaranteed. As n the closed economy case, the optmal monetary polcy requres stablzng the output gap (.e., ỹ t = for all t). Equaton (37) then mples that domestc prces are also stablzed under that optmal polcy (.e., π H,t = for all t). Thus, n the specal case under consderaton, (strct) domestc nflaton targetng (DIT) s ndeed the optmal polcy Implementaton and Macroeconomc Implcatons Ths secton dscusses the mplementaton of a domestc nflaton targetng polcy and characterzes some of ts equlbrum mplcatons. Whle that polcy has been shown to be optmal only for the specal case consdered above, the mplcatons of that polcy for the general case wll also be consdered.

24 7.4. Optmal Monetary Polcy: A Specal Case Implementaton As dscussed above, full stablzaton of domestc prces mples ỹ t = π H,t = n n for all t. Ths n turn mples that y t = y t and t = r t wll hold n equlbrum for all t, wth all the remanng varables matchng ther natural levels at all tmes. n For the reasons dscussed n chapter 4, an nterest rate rule of the form t = r t s assocated wth an ndetermnate equlbrum, and hence, does not guarantee that the outcome of full prce stablty be attaned. That result follows from the equvalence between the dynamcal system descrbng the equlbrum of the small open economy and that of the closed economy of chapter 4. As shown there, the ndetermnacy problem can be avoded, and the unqueness of the prce stablty outcome restored by havng the central bank follow a rule that makes the nterest rate respond wth suffcent strength to devatons of domestc nflaton and/or the output gap from target. More precsely, the central bank can guarantee that the desred outcome s attaned f t commts to a rule of the form t = r n + φ π π H,t + φ y ỹ t (43) t where κ α (φ π ) + ( β)φ y >. Note that, n equlbrum, the term φ π π H,t + n φ y ỹ t wll vansh (because ỹ t = π H,t = ), mplyng that t = r t for all t Macroeconomc Implcatons Under strct domestc nflaton targetng, the behavor of real varables n the small open economy corresponds to the one that would be observed n the absence of nomnal rgdtes. Hence, t s seen from the nspecton of equaton (36) that domestc output always ncreases n response to a postve technology shock at home. As dscussed earler, the sgn of the response to a rse n world output s ambguous, however, and t depends on the sgn of, whch n turn depends on the sze of the substtutablty parameters γ and η and the rsk averson parameter σ. The natural level of the terms of trade s gven by s t n = σ α (y t n y t ) = σ α ( + a a t y t ) σ + ϕ where σ >. Thus, gven world output, an mprovement n domestc α+ ϕ technology always leads to a real deprecaton through ts expansonary effect on domestc output. On the other hand, an ncrease n world output always generates an mprovement n the domestc terms of trade (.e., a real apprecaton), gven domestc technology. Gven that domestc prces are fully stablzed under DIT, t follows that DIT n e t = s t p t,.e., the nomnal exchange rate moves one for one wth the

25 72 7. Monetary Polcy and the Open Economy (natural) terms of trade and (nversely) wth the world prce level. Assumng constant world prces, the nomnal exchange rate wll nhert all the statstcal propertes of the natural terms of trade. Accordngly, the volatlty of the nomnal exchange rate under DIT wll be proportonal to the volatlty of the gap between the natural level of domestc output (n turn related to productvty) and world output. In partcular, that volatlty wll tend to be low when domestc natural output dsplays a strong postve comovement wth world output. When that comovement s low (or negatve), possbly because of a large dosyncratc component n domestc productvty, the volatlty of the terms of trade and the nomnal exchange rate under DIT wll be enhanced. The mpled equlbrum process for the CPI can also be derved. Gven the constancy of domestc prces t s gven by p t DIT = α(e t DIT + p t ) = αs n. t Thus, t s seen that under the DIT regme, the CPI level wll also vary wth the (natural) terms of trade and wll nhert ts statstcal propertes. If the economy s very open, and f domestc productvty (and hence, the natural level of domestc output) s not much synchronzed wth world output, CPI prces could potentally be hghly volatle, even f the domestc prce level s constant. An mportant lesson emerges from the prevous analyss: Potentally large and persstent fluctuatons n the nomnal exchange rate, as well as n some nflaton measures (lke the CPI), are not necessarly undesrable, nor do they requre a polcy response amed at dampenng such fluctuatons. Instead, and especally for an economy that s very open and subject to large dosyncratc shocks, those fluctuatons may be an equlbrum consequence of the adopton of an optmal polcy, as llustrated by the model above The Welfare Costs of Devatons from the Optmal Polcy Under the partcular assumptons for whch strct domestc nflaton targetng has been shown to be optmal (.e., log utlty and unt elastcty of substtuton between goods of dfferent orgn), t s relatvely straghtforward to derve a second-order approxmaton to the utlty losses of the domestc representatve consumer resultng from the optmal polcy devatons. Those losses, expressed as a fracton of steady state consumpton, can be wrtten as [ ] W = ( α) β t ε π 2 + ( + ϕ) ỹ 2. (44) 2 λ H,t t t=