Ramsey Monetary Policy and International Relative Prices

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 798 April 2004 RAMSEY MONETARY POLICY AND INTERNATIONAL RELATIVE PRICES Ester Faia and Tommaso Monacelli International Finance Discussion Papers numbers 797-807 were presented on November 14-15, 2003 at the second conference sponsored by the International Research Forum on Monetary Policy sponsored by the European Central Bank, the Federal Reserve Board, the Center for German and European Studies at Georgetown University, and the Center for Financial Studies at the Goethe University in Frankfurt. NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. The views in this paper are solely the responsibility of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or any other person associated with the Federal Reserve System. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or author. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp.

Ramsey Monetary Policy and International Relative Prices ∗ Ester Faia Tommaso Monacelli † ‡ Universitat Pompeu Fabra IGIER, Universita` Bocconi and CEPR January 2004 (first draft October 2003) Abstract Weanalyzewelfaremaximizingmonetarypolicyinadynamictwo-countrymodelwithprice stickinessandimperfectcompetition. Inthiscontext,atypicaltermsoftradeexternalityaffects policyinteraction between independent monetary authorities. Unlike theexisting literature, we remainconsistenttoapublicfinanceapproachbyanexplicitconsiderationofallthedistortions that are relevant to the Ramsey planner. This strategy entails two main advantages. First, it allows an accurate characterization of optimal policy in an economy that evolves around a steady-state which is not necessarily efficient. Second, it allows to describe a full range of alternative dynamic equilibria when price setters in both countries are completely forwardlooking and households’ preferences are not restricted. In this context, we study optimal policy both in the long-run and along a dynamic path, and we compare optimal commitment policy underNashcompetitionandundercooperation. Byderivingasecondorderaccuratesolutionto thepolicyfunctions,wealsocharacterizethewelfaregainsfrominternationalpolicycooperation. Keywords. Optimal Monetary Policy, Ramsey planner, Nash equilibrium, Cooperation, sticky prices, imperfect competition. JEL Classification Number: E52, F41. We thank Gianluca Benigno, Albert Marcet and Pedro Teles for useful discussions. We also thank Fiorella de ∗ Fiore, Jordi Gali, Stephanie Schmitt-Grohe, Alan Sutherland, Jaume Ventura and seminar participants at AUEB, EuropeanCentralBank,Riksbank,UniversitatPompeuFabra,andInternationalResearchForuminMonetaryPolicy forcomments. EsterFaiagratefullyacknowledgessupportfromtheDSGEgrant. Allerrorsareourownresponsibility. Correspondence to Universitat Pompeu Fabra, Ramon Trias Fargas 25, Barcelona, Spain. E-mail: es- † ter.faia@econ.upf.es, homepage: http://www.econ.upf.es/˜faia/. Correspondence to IGIER Universita’ Bocconi, Via Salasco 3/5, 20136 Milan, Italy. Email: ‡ tommaso.monacelli@uni-bocconi.it, Tel: +39-02-58363330, Fax: +39-02-58363332, Web page: www.igier.unibocconi.it/monacelli. 1

Non-Technical Summary In the classic approach to the study of optimal policy in dynamic economies (Ramsey (1927), Atkinson and Stiglitz (1980), Lucas and Stokey (1983), Chari, Christiano and Kehoe (1992)), and in a typical public finance spirit, a social planner maximizes household's welfare subject to a resource constraint, to the constraints describing the equilibrium in the private sector economy, and via an explicit consideration of all the distortions that characterize both the long-run and the cyclical behavior of the economy. In this paper we study welfare maximizing monetary policy in a dynamic two-country model with price stickiness and imperfect competition. In this context, a typical terms of trade externality affects policy competition between independent monetary authorities. Unlike the existing literature, we remain consistent to a public finance spirit by an explicit consideration of all the distortions that are relevant to the Ramsey planner. This strategy entails two main advantages. First, it allows an accurate characterization of optimal policy in an economy that evolves around a steady-state which is not necessarily efficient. Second, it allows to describe a full range of alternative dynamic equilibria when price setters in both countries are completely forward-looking and household's preferences are not restricted. In particular, we study optimal policy both in the long-run and along a dynamic path, and we compare optimal commitment policy under Nash competition and under cooperation. Our analysis can be summarized in terms of three main contributions. First, we show that policy competition in an international setting leads welfare maximizing but independent policymakers to generally deviate from the prescription of price stability. Intuitively, in an open economy, the wedge between the marginal rate of substitution (between consumption and leisure) and the marginal rate of transformation depends not only on the fact that markups are time-varying (due to monopolistic competition coupled with sticky prices), but also on the dynamic behavior of the terms of trade. Hence each country tries to engineer price level movements to try to tilt relative prices in its own favor. It is only when policy is set in a centralized fashion by a world Ramsey planner that the two countries manage to coordinate their policies and thus replicate very closely the equilibrium under flexible prices. Second, and more generally, our approach allows to study optimal policy in dynamic economies that evolve around a steady-state which is not necessarily efficient. In that, it differs crucially from a recurrent strategy in the recent optimal monetary policy New-Keynesian literature that forces another (complementary) policy instrument (e.g., fiscal subsidies) to offset selected sources of distortion (e.g., market power). Third, we argue that, in this framework, welfare gains from cooperation, although positive, are small. To reach this conclusion, once the efficiency conditions of the corresponding optimal policy problem have been characterized, we resort to second order approximation methods (in the neighborhood of the specified Ramsey steadystate). This is required to account for the fact that, when business cycle fluctuations are centered around a distorted steady state, stochastic volatility affects the first moments of those variables that are critical for household's welfare evaluation.

1 Introduction In the classic approach to the study of optimal policy in dynamic economies (Ramsey (1927), Atkinson and Stiglitz (1980), Lucas and Stokey (1983), Chari, Christiano and Kehoe (1992)), and in a typical public finance spirit, a Ramsey planner maximizes household’s welfare subject to a resource constraint, to the constraints describing the equilibrium in the private sector economy, and via an explicit consideration of all the distortions that characterize both the long-run and the cyclical behavior of the economy. Recently there has been a resurgence of interest for a Ramsey-type approach in dynamic generalequilibriummodelswithnominalrigidities. Khan,KingandWolman(2003)analyzeoptimal monetary policy in a closed economy where the relevant distortions are imperfect competition, staggered price setting and monetary transaction frictions. Schmitt-Grohe and Uribe (2003), and Siu (2003) focus on the joint optimal determination of monetary and fiscal policy. The robust conclusion of these studies - that optimal policy is associated to the prescription of stable inflation - is indeed rooted in the principle that the planner tries to eliminate the distortions induced by fluctuations in the aggregate price level, whether stemming from relative price misalignments or from resource costs of resetting prices. In this paper we characterize welfare maximizing monetary policy in a two-country world where financial markets are complete, policymakers act under commitment and compete in a Nash equilibrium. Botheconomiesarecharacterizedbytwodistortions: outputisinefficientlylow(dueto thepresenceofmonopolisticcompetitivegoodsmarkets)andfirmsfacequadraticcostsofadjusting prices. However,andrelativetoacooperativesettingenforcedbyaworldRamseyplanner,openness perseaddsafurtherinefficiencytypicaloftheoutcomeunderaNashequilibrium. Thisinefficiency stemsfromthemonopolypowerthateachcountrycanexertonitsowntermsoftrade,andtherefore from an externality that the policy competition motive necessarily entails.1 Relativetothecorrespondingclosedeconomyliterature, aRamsey-typeapproachhasreceived much less attention in the analysis of optimal monetary and exchange rate arrangements for open economies. Cooley and Quadrini (2003) analyze monetary policy interaction in a model with perfectly competitive goods markets, flexible prices and limited financial markets participation. Their model is essentially static in nature and highlights the presence of a systematic inflation bias induced by international policy competition. Our framework differs from theirs in the fact that prices are sticky (so that nominal exchange rate movements exert an effect on international 1The idea that terms of trade spillovers generate an externality and therefore room for international (monetary and/orfiscal)policycoordinationisalreadydiscussed(althoughwithinad-hocmodels)inCanzoneriandHenderson (1991), Persson and Tabellini (1995) and dates back in the trade literature at least to Johnson (1954). Chari and Kehoe (1990) discuss the specific role of terms of trade distortions for optimal fiscal policy in a two-country generalequilibriummodel. Morerecently,seeCorsettiandPesenti(2000),Tille(2001),BenignoandBenigno(2003), Sutherland(2002). 2

relative prices), goods markets are imperfectly competitive and agents operate in a fully dynamic environment. A Ramsey-type approach has also been employed in a certain stream of the so-called New Open Economy Macroeconomics literature (which instead typically features nominal rigidities and imperfect competition). This is the case - for instance - in the work of Benigno and Benigno (2003b), Corsetti and Pesenti (2002), and Devereux and Engel (2003). However, although elegant, thesearestylizedframeworksinwhichtheanalysisofoptimalpolicyissimplifiedbytheassumption that prices (or wages) are predetermined one-period. Such an assumption is restrictive, for it typically generates a Lucas-type aggregate supply curve in which the forward-looking nature of inflation is neglected, and along with it the channel through which the anticipation of future policy conduct comes to play a role.2 Our work differs from the aforementioned contributions in that it employs optimizing producers’ price setting decisions that are forward-looking, thereby rendering the corresponding optimal policy problem inherently dynamic. Our analysis can be summarized in terms of three main contributions. First, we show that policycompetitioninaninternationalsettingleadswelfaremaximizingbutindependentpolicymakers to generally deviate from the prescription of price stability. Intuitively, in an open economy, the wedge between the marginal rate of substitution (between consumption and leisure) and the marginal rate of transformation depends not only on the fact that markups are time-varying (due to monopolistic competition coupled with sticky prices), but also on the dynamic behavior of the terms of trade. Hence each country tries to engineer price level movements to try to tilt relative prices in its own favour. On the other hand, when policy is set in a centralized fashion by a world Ramsey planner, the two countries manage to coordinate their actions in such a way to replicate very closely the equilibrium dynamics that would prevail under purely flexible prices (thereby mimicking closely the outcome of a corresponding closed economy). Second,andmoregenerally,ourapproachallowstostudyoptimalpolicyindynamiceconomies that evolve around a steady-state which is not necessarily efficient. In that, it differs crucially from a recurrent approach in the recent New-Keynesian literature that forces another (complementary) policy instrument (e.g., fiscal subsidies) to offset second order effects of stochastic uncertainty on the mean levels of variables.3 The same approach resorts to a two-step strategy that involves, at first, taking a log-linear approximation of the competitive equilibrium conditions, and then a quadratic approximation of the correct households’ utility function. In particular, resorting to such anapproximationmethodinanopeneconomyrequiresspecificassumptionsonpreferences, suchas log-utility and unitary elasticity of substitution between goods produced in different countries. Yet 2It is by now well understood that this entails a major consequence in that it neglects the sense in which (time consistent)discretionarypoliciesaresuboptimalindynamicenvironmentswithforward-lookingprice(and/orwage) decisions (Woodford, 2003). 3See,forinstance,RotembergandWoodford(1997),Woodford(2003),BenignoandBenigno(2003),Clarida,Gali and Gertler (2002). 3

precisely these assumptions already constrain the form of the optimal policy to coincide, somewhat artificially, with the one that implements the flexible price allocation. Furthermore, if not satisfied, the same conditions do not allow to study each country’s policymaker’s problem independently, forcing to ignore those equilibria that emerge under policy competition and to restrict the analysis only to the world planner’s policy design problem.4 Third, we argue that, in this framework, welfare gains from cooperation, although positive, are small. To reach this conclusion, once the efficiency conditions of the corresponding optimal policy problem have been characterized, we resort to second order approximation methods (in the neighborhood of the specified Ramsey steady-state).5 This is required to account for the fact that when business cycle fluctuations are centered around a distorted steady state stochastic volatilityaffectsthefirstmomentsofthosevariablesthatmaybecriticalforthehousehold’swelfare evaluation.6 The remainder of the paper is organized as follows. Section 2 and 3 describe respectively the economic environment and the features of the equilibrium. Section 4 derives the form of the constraints that are relevant to the planner’s policy problem. Section 5 analyzes optimal policy under commitment. Section 6 explores the welfare gains from cooperation. Section 7 concludes. 2 The Model Theworldeconomyconsistsoftwocountries, thatwelabelthemHomeandForeign. Eacheconomy is populated by infinite-lived agents, whose total measure is normalized to unity. 2.1 Domestic Households Let’s denote by C t ≡ [(1 − α)η 1 C H η −η ,t 1 +αη 1 C F η , −η t 1 ] η −η 1 a composite consumption index of domestic and imported bundles of goods, where α is the balanced-trade steady state share of imported goods (i.e., an inverse measure of home bias in consumption preferences), and η > 0 is the elasticity of substitution between domestic and foreign goods. Each bundle is composed of imperfectly substitutable varieties (with elasticity of substitution ε > 1). Optimal allocation of expenditure 4Morerecently,BenignoandWoodford(2003)show(withinaclosedeconomymodel)howtopreserveaquadratic approximationofthehousehold’swelfareobjectiveinthecaseinwhichtheeconomyfluctuatesaroundanon-efficient steady-state. This per se requires taking a second order approximation also of (some of) the underlying equilibrium conditions. Benigno and Benigno (2003a) and Pappa (2003) apply this approximation method to a two-country optimal policy dynamic model. 5Incidentally, one may want to notice that this entails a strategy which exactly reverses the logic of the approximationmethoddescribedabove, and largely employed in therecent literature. 6ForthedevelopmentandtheapplicationofsecondorderapproximationmethodsforwelfareevaluationseeBergin and Tchakarov (2003),Schmitt-Grohe andUribe(2003), Kolmann(2002), Kimand Kim(2002),Sims (2001). 4

within each variety of goods yields: P (i) ε P (i) ε H,t − F,t − C (i)= C ; C (i) = C (1) H,t H,t F,t F,t P P H,t F,t µ ¶ µ ¶ 1 (cid:18) 1 (cid:18) 1 (cid:18) 1 (cid:18) where C H,t ≡ 0 [C H,t (i) −(cid:18) di](cid:18) − 1 and C F,t ≡ 0 [C F,t (i) −(cid:18) di](cid:18) − 1. Optimal allocation of expenditure between domestic and foreign bundles yields: R R P η P η H,t − F,t − C =(1 α) C ; C =α C (2) H,t t F,t t − P P t t µ ¶ µ ¶ where P t ≡ [(1 − α)P H 1 −,t η +αP F 1 ,−t η ]1 − 1 η is the CPI index. We assume the existence of complete markets for state-contingent money claims expressed in units of domestic currency.7 Let st = s ,....s denote the history of events up to date t, where s 0 t t { } is the event realization at date t. The date 0 probability of observing history st is given by ρ(st). The initial state s0 is given so that ρ(s0)=1. Agents maximize the following expected discounted sum of utilities over possible paths of consumption and labor: ∞ E βtU(C ,N ) (3) 0 t t ( ) t=0 X where where E denotes the mathematical expectations operator conditioned on s , and N 0 0 t {} denoteslaborhours.8 Weassumethatperiodutilityisseparableinitsarguments. Atthebeginning of time t the households receive a nominal labor income of W N . To insure their consumption t t patternagainstrandomshocksattimettheydecidetospendν B innominalstatecontingent t+1,t t+1 securitieswhereν ν(st+1 st)isthepricingkernelofthestatecontingentportfolio. Eachstate t,t+1 ≡ | contingent asset B pays one unit of domestic currency at time t+1 and in state st+1. Hence t+1 the sequence of budget constraints, after considering the optimal expenditure conditions (1) and (2), assumes the following form: 1 P C + ν B W N +τ +B + Γ (i) (4) t t t+1,t t+1 t t t t t ≤ 0 st+1 Z X 7Given that, in our setting, the law of one price holds continually, the unit of denomination of the payoffs of state-contingentassetsisnotstrictlyrelevant. Alternatively,e.g.,inthecaseinwhichdeviationsfromthelawofone price are due to consumer currency pricing, as in Devereux and Engel (2003), the distinction between nominal and real payoffs would be relevant for thespecification of theequilibrium. 8Hencethe expression for lifetimeutility is equivalent to writing ∞ βtU C(st) ρ(st) X t=0 Xst ¡ ¢ whereρ(st)=ρ(s s ). t 0 | 5

whereτ aregovernmentnettransfersofdomesticcurrencyandΓ (i)aretheprofitsofmonopolistic t t firm i, whose shares are owned by the domestic residents.9 Households choose the set of processes C ,N and bonds B , taking as given the set of processes P , W , v and the t t ∞t=0 t+1 ∞t=0 t t t+1,t ∞t=0 { } { } { } initial wealth B so as to maximize (3) subject to (4). 0 For any given state of the world, the following set of efficiency conditions must hold: W t U = U (5) c,t n,t P − t P U t c,t+1 β =ν (6) t+1,t P U t+1 c,t lim E ν B =0 (7) t t+j,t t+j j { } →∞ where U ,t defines the first order derivative of utility with respect to its argument κ = C,N. Our separability assumption implies U = U = 0. Equation (5) equates the CPI-based real wage cn,t nc,t to themarginal rate of substitution between consumption and leisure. Optimality requires that the first order conditions (5), (6) and the no-Ponzi game condition (7) are simultaneously satisfied.10 The conditional expected return on the state contingent asset is given by R , so that, by n,t arbitrage, it holds R 1 E ν n−,t t t+1,t ≡ { } 2.2 Law of One Price and Foreign Demand We assume throughout that the law of one price holds, implying that P (i) = P (i) for all F E F∗ i [0,1], where is the nominal exchange rate, i.e., the price of foreign currency in terms of home ∈ E currency, and P (i) is the price of foreign good i denominated in foreign currency. Let’s denote by F∗ BF foreign households’ holdings of the state contingent bond denominated in domestic currency. The budget constraint of the foreign representative household will read: BF BF 1 P C + ν t+1 W N +τ + t + Γ (i) (8) t∗ t∗ t+1,t t∗ t∗ ∗t ∗t ≤ t t 0 st+1 E E Z X The efficiency condition for bonds’ holdings is P U β t∗ E t c∗ ∗ ,t+1 =ν (9) t+1,t P U t∗+1E t+1 c∗ ∗ ,t 9Eachdomestic household ownsan equal shareof the domestic monopolistic firms. 10Notice that we do not introduce money explicitly, but rather think of it as playing the role of nominal unit of account. For the sake of simplicity, this allows us to abstract from an additional distortion stemming from the presenceoftransactionsfrictions. SeeKhan,KingandWolman(2003)forananalysisinwhichtransactionsfrictions interact with monopolistic competition andpricestaggering in a welfare maximizing monetary policy problem. 6

Foreign demand for domestic variety i must satisfy: ε P H∗,t (i) − C (i) = C (10) H∗,t P H∗,t à H∗,t ! P H∗,t (i) − ε P H∗,t − η = α C P ∗ P t∗ à H∗,t ! µ t∗ ¶ The remaining efficiency conditions characterizing the foreign economy are then exactly symmetric to the ones of the domestic economy described above. 2.3 Domestic Producers Each monopolistic firm i produces a homogenous good according to: Y (i) =A N (i) (11) t t t The cost minimizing choice of labor input implies: W t =mc A (12) t t P H,t where mc denotes the real marginal cost. Changing output prices is subject to some costs. We follow Rotemberg (1982) and model the cost of adjusting prices for each firm i equal to: θ P (i) 2 H,t ψ (i) 1 (13) t ≡ 2 P − H,t µ ¶ where the parameter θ measures the degree of price stickiness. The higher the θ the more sluggish is the adjustment of nominal prices. If θ = 0 prices are flexible. The cost of price adjustment renders the domestic producer’s pricing problem dynamic. Each producer chooses the price P (i) H,t of variety i to maximize its total market value: E ∞ βtλ D t (i) (14) t t P ( H,t ) t=0 X subject to the constraint P (i) ε H,t − Y (i) (C +C ) (15) t ≤ P H,t H∗,t H,t µ ¶ where βtλ measures the marginal utility value to the representative producer of additional profits t expressed in domestic currency, and where 7

D (i) P (i)Y (i) W θ P (i) 2 t H,t t t H,t N 1 t P ≡ P − P − 2 P (i) − H,t H,t H,t H,t 1 µ − ¶ The first order condition of the above problem reads 0 = λ P H,t (i) − (cid:18) C t W (1 ε)+ε W t P H,t (i) − 1 (16) t P P − A P P H,t H,t à t H,t H,t ! µ ¶ µ ¶ P (i) 1 P (i) P (i) H,t H,t+1 H,t+1 λ θ 1 + βλ θ 1 − t P (i) − P (i) t+1 P (i) − P (i)2 H,t 1 H,t 1 H,t H,t µ − ¶ − µ ¶ Let’s define p PH,t(i) as the relative price of domestic variety i and π PH,t as the gross H,t ≡ PH,t H,t ≡ PH,t 1 domestic producer inflation rate. It is useful to see that the above condition can be−rewritten as e W 0 = λ CWp ε (1 ε)+ε t (17) t t H,t− − A P − t H,t µ ¶ p π ,t H,t H λ θ πe 1 t Ht,1 p − p H,t 1 H,t 1 µ − ¶ − e p p H,t+1 H,t+1 + βλ θ π 1 π t+1 e H,t+1 p e− H,t+1 p 2 H.t H,t µ ¶ e e 3 Equilibrium in the Home Econeomy e We focus our attention on a symmetric equilibrium where all domestic producers charge the same price, adopt the same technology and therefore choose the same demand for labor. This implies that p =1, N (i) =N , Γ (i)=Γ for all i, t. H,t t t t t In such an equilibrium equation (17) will simplify to e λ εA N ε 1 t t t λ π (π 1)=βE λ π (π 1) + mc − (18) t H,t H,t t t+1 H,t+1 H,t+1 t − { − } θ − ε µ ¶ The total net supply of bonds must satisfy B +BF =0 t t Market clearing for domestic variety i must satisfy: Y (i) = C (i)+C (i)+ψ (i) (19) t H,t H∗,t t P H,t (i) − ε P H,t − η P H∗,t − η = (1 α)C + α C +ψ (i) P P − t P ∗ t∗ t µ H,t ¶ " µ t ¶ µ t∗ ¶ # 8

ε foralli ∈ [0,1]andt. Plugging(19)intothedefinitionofaggregateoutputY t ≡ 0 1 Y(i)1 − 1 ε di ε − 1, and recalling that P H,t = E t P H∗,t , we can express the resource constraint as h R i P η P η H,t − H,t − A N = (1 α) C + α C +ψ (20) t t P − t P ∗ t∗ t µ t ¶ µE t t∗¶ 4 Deriving the Relevant Constraints As mentioned before, the optimal policy is determined by a monetary authority that maximizes the discounted sum of utilities of all agents given the constraints that characterize the competitive economy. Our next task is to select the relations that represent the relevant constraints in the planner’soptimalpolicy problem. Thisamountstodescribingthecompetitive equilibriuminterms of a minimal set of relations involving only real allocations, in the spirit of the primal approach described in Lucas and Stokey (1983). There is a difference, though, between that classic approach and the one followed here, which stems from the impossibility, in the presence of sticky prices, of reducing the planner’s problem to a maximization only subject to a single implementability constraint. Khan, King and Wolman (2003) and Schmitt-Grohe and Uribe (2002) face a similar problem in their analysis of dynamic optimal policy problems in the presence of price stickiness. 4.1 Resource and Budget Constraints Let’s begin by analyzing the domestic goods market equilibrium condition (20). This can be rewritten as A N = (1 α) C Φ η +Qη Φ ηα C +ψ (21) t t t t t t ∗ t∗ t − κU η θ = Φ η (1 α) C + c∗ ∗ ,t α C + (π 1)2 t − t U ∗ t∗ 2 H,t − c,t µ µ ¶ ¶ Symmetrically, the resource constraint in the Foreign country will read: η U θ A N =(Φ )η (1 α ) C + c,t αC + π 1 2 (22) ∗t t∗ ∗t − ∗ t∗ κU t 2 ∗F,t − Ã Ã c∗,t! ! ∗ ¡ ¢ NextweturntothebudgetconstraintoftheHomeconsumers. Bysubstitutingthegovernment budget constraint (which implies τ =0 for all t) into equation (4), imposing (7) and iterating, one t obtains (in units of domestic currency): ∞ ∞ B + z [W N +Γ ] = z P C (23) 0 0,t t t t 0,t t t t=0 st t=0 st XX XX 9

where the price system z is obtained after iteration of equation (6) and can be expressed as (for 0,t each possible state s ) t U P z =βtρ c,t 0 (24) 0,t t P U t c,0 Notice, next, that aggregate real profits can be written as: Γ (1 mc )A N θ(π 1)2 t = − t t t − 2 H,t − (25) P Φ t t where Φ Pt is the CPI to PPI ratio t ≡ PH,t By summing over all possible states s in equation (24), substituting (25), (5) and (44) into t (23), we obtain the present value budget constraint for domestic households (expressed in real terms): B +E ∞ βtU A t N t − θ 2 (π H,t − 1)2 =E ∞ βtU C (26) 0 0 c,t 0 c,t t Φ " t # t=0 t=0 X X where B B0U e . This equation states that the sum of initial financial wealth and expected 0 ≡ P0 c,0 presentdiscountednetincomemustmatchtheexpectedpresenteddiscountedvalueofconsumption. e WeproceedinasimilarfashionfortheForeignhousehold. ThepricesystemzF =ν ν ..ν , 0,t 1,0 2,1 t,t 1 − with ν = 1, expressed in units of domestic currency and obtained from the forward iteration of 0 (9) can be written: U P z 0 F ,t = βtρ t P c∗ ∗ ,t U 0∗ E 0 ≡ z 0∗,t E 0 =z 0,t (27) Ã t∗ c∗ ∗ ,0! E t E t Equating with (24) implies the following condition U P κ c∗ ∗ ,t = E t t∗ Q (28) t U P ≡ c,t t w co h u e n r t e ry Q t di i s s tr t i h b e ut r i e o a n l o e f xc w h e a a n l g t e h.1 r 1 ate B a e n lo d w κ w ≡ e d E P 0 is 0 P c U 0∗ u c∗ U ∗ s c , s 0 ,0 h i o s w a t p h a is ra p m a e r t a e m r e c t a e p r tu si r g in n g als th t e he in u it n ia d l er c l r y o i s n s g risk-sharing arrangement between the two countries. By taking conditional expectations of both sides of (27) and proceeding with similar substitutions to the ones operated in the Home case we obtain12 11See also Chari, Kehoe and McGrattan (2003). 12Inparticular one shouldnote that BF U κ 1 B 0∗ = 0 c,0 P − 0 0 E Since equilibriumrequires B 0 F = − B 0 we obtaien B 0∗ = − B 0 κ − 1. e e 10

A N θ(π 1)2 B κ 1+E ∞ βtU ∗t t∗ − 2 ∗F,t− =E ∞ βtU C (29) − 0 − 0 t=0 c∗ ∗ ,t " Φ ∗t # 0 t=0 c∗ ∗ ,t t∗ X X e 4.2 Risk-sharing and PPP Consider the domestic household maximizing (3) subject to (26). Efficiency requires βtU =Ω z P (30) c,t 0,t t where Ω is the Lagrange multiplier on constraint (26). Notice that this mutiplier is constant across time and states. Symmetrically, for the Foreign household we have: βtU = Ω z P (31) c∗,t ∗ 0∗,t t∗ zF 0,t = Ω P ∗ t t∗ E Ã 0 ! E By combining (30), (31) and (28), and applying the normalization = 1, one can write the 0 E risk-sharing parameter in terms of relative shadow values of net income: Ω κ= (32) Ω ∗ This allows the following definition of risk sharing: Definition 1. Complete international asset markets lead to perfect risk-sharing when the shadow value of the household’s present value budget constraint is equalized across countries. From (32), this in turn requires that κ=1. The risk sharing arrangement has implications on how the marginal utilities of consumption are linked across countries. By combining (26) with (21) and (28), and assuming (for the pure sake of simplicity) that the initial level of wealth is zero (so that B =B =0) one can write 0 0∗ ∞ βtU c,t Φ η 1 (1 α)C + κ U c∗,t η α C C =0 Ω t− − t U ∗ t∗ − t c,t t=0 st ½ µ µ ¶ ¶ ¾ XX and similarly for Foreign βtU U η ∞ c∗,t (Φ )η 1 (1 α )C + κ c∗,t − αC C =0 Ω ∗t − − ∗ t∗ U t − t∗ t=0 st ∗ ( Ã µ c,t ¶ ! ) XX 11

Next we assume cross-country symmetry, so that α = α . It is important to notice that this does ∗ not necessarily imply that PPP holds, unless we make the further restrictive assumption of absence of home bias, which entails α =α = 1.13 ∗ 2 Using (32) one can solve for κ, obtaining: ∞t=0 st βtU c,t Φ η t− 1 Z t − C t κ= (33) P PβtU ( n Φ )η 1Z C o ∞t=0 st c∗,t ∗t − t∗ − t∗ n o U Pη P U η where Z (1 α)C + κ c∗,t αC and Z (1 α)C + κ c∗,t − αC . t ≡ − t Uc,t t∗ t∗ ≡ − t∗ Uc,t t µ ¶ Hence κ³=1 (perfect³risk-sh´aring)´requires ³ ´ ∞ βtU Φ η 1Z C = ∞ βtU (Φ )η 1Z C (34) c,t t− t t c∗,t ∗t − t∗ t∗ − − X t=0 X st n o X t=0 X st n o Noticethatthelastexpressiondoesnotnecessarilyimplyaperfectequalizationofthemarginal utilities of consumption. The latter property follows only in the particular case of absence of home bias,whichinturnimpliesthatPPPholds. Inthiscase,andrecalling(28),conditionκ=1requires U =U , and therefore C =C . It is then easy to verify that (34) also implies Φ =Φ for all t. c,t c∗,t t t∗ t ∗t 4.3 Relative Prices and Price Setting Constraints Below we define a series of relationships linking real quantities to the relevant relative prices in our framework. The terms of trade is the relative price of imported goods: P F,t (35) t T ≡ P H,t It can be related to the CPI-PPI ratio as follows Φ t ≡ P P t =[(1 − α)+α T t 1 − η ]1 − 1 η ≡ g( T t ) (36) H,t with g0 > 0. The terms of trade and the real exchange rate are linked through the following expression: P Φ F,t t = (37) t T P t Φ t = Q t Φ ∗t 13One can easily verify this by manipulating the CPI expression and substituting conditions P = P and H,t E t H∗,t P = P , whichare implied by the fact that the law of one price holds. F,t E t F∗,t 12

where Φ ∗t ≡ P P t∗ =[(1 − α ∗ )+α ∗ T t η − 1]1 − 1 η ≡ g ∗ ( T t ) (38) F∗,t with g ∗0 <0. We now wish to rewrite the relative prices Φ and Φ as a function or real allocations only. t ∗t By combining (36), (37), (38) and (28), and recalling that α one can write ∗ U 1 η (1 α ) α κ c∗,t − Φ = − ∗ − Uc,t h(C ,C ) (39) t  1 (α+³α ) ´  ≡ t t∗ ∗ −     and symmetrically U η 1 (1 α) α κ c∗,t − Φ = − − ∗ Uc,t h (C ,C ) (40) ∗t  1 (α+³α ) ´  ≡ ∗ t t∗ ∗ −     Notice that when η = 1, (39) reduces to Φ = Φ = 1 for all t. In this particular case which t ∗t correspondstoCobb-Douglasconsumptionpreferences, thepolicycompetitionmotiveontheterms of trade vanishes. The CPI level can be linked to the domestic price level and aggregate consumption as follows: P = P Φ . Let’s then define gross CPI inflation as π Pt . This is related to domestic t H,t t t ≡ Pt 1 − producer inflation and aggregate relative consumption as follows: Φ t π =π (41) t H,t Φ t 1 − The condition on optimal bond investment can then be rearranged accordingly. By taking conditional expectations of (6) we obtain U =βE R U (42) c,t t t c,t+1 { } where RnP R =E t t (43) t t P t+1 ½ ¾ is the CPI-based gross real interest rate. Next we need to rearrange the optimality conditions for the production sector. This requires, at first, to express the real marginal cost and the real wage in terms of aggregate real quantities. Hence by combining (5) and (12) we can write 13

U n,t mc = Φ (44) t t −U A c,t t This implies that the aggregate condition for optimal pricing (18) can be rewritten as U π (π 1) = βU E π (π 1) + (45) c,t H,t H,t c,t+1 t H,t+1 H,t+1 − { − } U εA N U ε 1 c,t t t n,t Φ − t θ −U A − ε c,t t µ ¶ An analogous condition will hold in Foreign: U π (π 1) = βU E π (π 1) + (46) c∗,t ∗F,t ∗F,t c∗,t t ∗F,t+1 ∗F,t+1 ∗ − ∗ − U c∗,t εA ∗t N©t∗ U n∗,t ªε 1 ∗ θ −U A Φ ∗t − − ε Ã c∗,t ∗t ! ∗ In the following, we formulate a proposition that establishes a mapping between the minimal form expressed above (summarized by conditions (21), (26), (45), for Home, and (22), (29), (46) for Foreign) and the set of allocations describing the (imperfectly) competitive equilibrium in the world economy. Proposition 1. (Part A) For a given initial symmetric wealth level B , any equilibrium 0 allocation C , Y , N , mc , Q , Φ , π , C , C satisfying equations (2), (4), (5)-(7), (12), t t t t t t H,t F,t H,t ∞t=0 { } e (18), (20), the risk-sharing condition (28), along with a symmetric set of conditions holding for Foreign, also satisfies equations (21), (26), (45), (22), (29) and (46). (Part B) By reverse, using allocations C , N , π and C , N , π that satisfy equations (45), (21), (26) and { t t H,t } ∞t=0 { t∗ t∗ ∗F,t} ∞t=0 (46), (22), (29), it is possible to construct all the remaining real allocations, nominal variables and policy instruments for Home and Foreign. Proof. See Appendix A. 5 Optimal Monetary Policy under Commitment We now turn to the specification of the optimal policy problem in a dynamic context. We assume that ex-ante commitment is feasible. In this section we take full advantage of our characterization of the equilibrium conditions (in each country) in terms of a minimal set of relations involving only the choice of allocations for consumption and labor input along with the inflation instrument. A distinctive feature of our Ramsey analysis is that we allow the relevant distortions characterizing the economy to remain explicit, both in the short and in the long-run.14 This implies that 14See King and Wolman(1999) for a closedeconomy analog. 14

the policymaker in each country lacks a set of fiscal instruments necessary to achieve the first best allocation. Each economy is in fact characterized by three distortions. The first two, market power and price stickiness, are common to both the closed and the open economy version of our model. The price stickiness distortion, summarized by the quadratic term in inflation in the resource constraint, is obviously minimized at zero net inflation (i.e., π = 1 for all t). On the other hand, H,t the market power distortion, stemming from the level of activity being inefficiently low, calls for the monetary authority to try to expand output and consumption. What in general distinguishes the analysis of an open economy (and as also emphasized in Corsetti and Pesenti (2000)), is the presence of an additional inefficiency. This stems from the possibility for each country, in the presence of rigid nominal prices, of strategically affecting the termsoftrade, andinturntrytoincreaseitslevelofconsumptionforanygivenleveloflaboreffort. This externality creates per se room for policy competition, and for the possibility of gains from cooperative policies. The interesting aspect concerns the extent to which such policy competition motive may lead each policymaker to try to deviate from the prescription of price stability that would typically characterize optimal policy in the closed economy version of our model. 5.1 Nash Competition We begin by assuming that the policymaker in each country sets policy independently taking as given policy actions in the other country. Definition 2. Let’s define U (C t , N t , π H,t , Ω) ≡ U(C t ,N t )+Ω U c,t C t − AtNt − θ 2 Φ (π t H,t − 1)2 · µ ¶¸ where Ω is the multiplier on constraint (26). Let λ , λ represent sequences of Lagrange { p,t f,t } ∞t=0 multipliersontheconstraints(45)and(21)respectively. Let B begiven. Thenforgivenallocations 0 C and stochastic processes A , A , plans for the control variables C , π , N , t∗ ∞t=0 t ∗t ∞t=0 t H,t t ∞t=0 { } { } e { } and for the costate variables λ ,λ and Ω, represent a first best constrained allocation if p,t f,t ∞t=0 { } they solve the following maximization problem: Choose Λn λ ,λ and Ξn C , π , N to t p,t f,t ∞t=0 t t H,t t ∞t=0 ≡{ } ≡{ } ∞ Min Max E βtE (C ,N ,π ,Ω) (47) { Λn t} ∞t=0 { Ξn t} ∞t=0 0 { t {U t t H,t t=0 X U εA N U Φ ε 1 c,t t t n,t t +λ U π (π 1) βU π (π 1)+ + − p,t c,t H,t H,t c,t+1 H,t+1 H,t+1 − − − θ U A ε c,t t · µ ¶¸ U η θ +λ A N (1 α) C Φ η κη c∗ ∗ ,t Φ ηα C (π 1)2 ΩB f,t t t − − t t − U t ∗ t∗ − 2 H,t − }} − 0 c,t · µ ¶ ¸ e A series of observations on the nature of this policy problem are in order. Notice, first, that the distinctive feature of the commitment problem under Nash competition is that the Home 15

policymaker does not internalize that the relative price Φ =h(C ,C ) depends also on the level of t t t∗ consumption in Foreign. It is key to our analysis that the relative price Φ enters pervasively in t the behavioral relationships characterizing the optimal policy problem. Second, it is of independent interest to notice that the present value budget constraint must be part of the policy maximization problem. In fact, and unlike the closed economy case, it is not implicitly satisfied by a combination of the government budget constraint and of the resource feasibility constraint. This dimension characterizes specifically the policy maximization problem in an open economy as opposed to the corresponding closed-economy case. Third, in the following we assume that, prior to policy implementation, the initial wealth is inelastically supplied (in fact, B = 0 is given) and that policy is chosen taking the initial risk- 0 sharing arrangement as given.15 Thishasthecrucialimplicationthat,alreadyintheNashproblem, e each policymaker is in fact facing the same present value budget constraint. Finally, it is important to notice that, as a consequence of the initial stock of wealth B 0 being exogenously supplied, the multiplier Ω is taken as given in each policymaker’s maximization e problem. In other words, the initial stock of wealth does not depend on the anticipation about the future implementation of policy.16 5.1.1 Non-recursivity and Initial Conditions As a result of the constraint (45) exhibiting future expectations of control variables, the maximization problem as spelled out in (47) is intrinsically non-recursive.17 As first emphasized in Kydland and Prescott (1980), and then developed by Marcet and Marimon (1999), a formal way to rewrite the same problem in a recursive stationary form is to enlarge the planner’s state space with additional (pseudo) costate variables. Such variables, that we denote χ and χ for Home and Foreign t ∗t respectively, bear the crucial meaning of tracking, along the dynamics, the value to the planner of committing to the pre-announced policy plan. Another aspect concerns the specification of the law of motion of these lagrange multipliers. For in our case the forward-looking Phillips curve constraint features a simple one period expectation, the same costate variables have to obey the laws of motion:18 15We believe thisis a realistic assumptiongiventhat the two policymakersare acting under commitment. 16See Schmitt-Grohe and Uribe (2003) for a small openeconomy modelin whichthe probability of future”policy reform” is not negligibleandthereforethe determination of B is endogenous. 0 17See Kydland and Prescott (1977), Calvo (1978). As such the system does not satisfy per se the principle of optimality, according to whichtheoptimaldecisionat time teisatimeinvariantfunctiononly ofasmall setofstate variables. 18Thelawsofmotionoftheadditionalcostatevariableswouldtakeamoregeneralformiftheexpectationshorizon intheforwardlookingconstraint(s)featuredamorecomplicatedstructure,as,forinstance,inthecaseofconstraints inpresent value form. See Marcet andMarimon (1999). 16

χ = λ (48) t+1 p,t χ = λ (49) ∗t+1 ∗p,t A particularly important point concerns the definition of the appropriate initial conditions for χ and χ . Marcet and Marimon (1999) show that for the modified (recursive) Lagrangian in (47) t ∗t to generate a global optimum under time zero commitment it must hold: χ = 0 =χ (50) 0 ∗0 The above condition states that there is no value to the policy planner, in either country and as of time zero, attached to prior commitments. Commitment, in this context, bears exactly the meaning that while it would be technically feasible for the planner (in each country) to satisfy (50) for all t>0, it would also be suboptimal to do so. InAppendix B weshowhowtoreformulatetheoptimalplaninanequivalentrecursivestationary form. First order conditions for time t 1 for the choice of C , N and π imply respectively: t t H,t ≥ λ N p,t t 0 = U +U π (π 1) (λ χ )+ ((cid:18) U Φ +(ε 1)A U ) (51) c,t cc,t H,t H,t − p,t − t θ n,t c,t − t cc,t λ (1 α) Φ η +ηC Φ η 1Φ α C κηU η ηΦ η 1Φ U η ηU η 1U Φ η f,t t t t− c,t ∗ t∗ c∗,t t− c,t c−,t c−,t − cc,t t − − − − h θ³ ´ ³ ´i Ω (A N (π 1)2) U Φ 1 Φ Φ 2U (U C +U ) − t t − 2 H,t − cc,t −t − c,t −t c,t − cc,t t c,t · ¸ ¡ ¢ λ εΦ ε 1 U A p,t t c,t t 0 =U + (U +N U )+λ − U A +λ A Ω (52) n,t n,t t nn,t p,t c,t t f,t t θ θ − Φ t U A c,t t 0=U (2π 1)(λ χ ) θ(π 1) λ Ω (53) c,t H,t − p,t − t − H,t − f,t − Φ t µ ¶ The system of efficiency conditions for Home is completed by the law of motion (48), the initial condition (50) and by the constraints (45) and (21) holding with equality. Notice also that first order conditions evaluated at time t =0 differ for what concerns equation (51), which must feature the additional term ΩB0U .19 − P0 cc,0 Once defined a completely symmetric problem for the policy maker in Foreign, we can state the following definition of a Nash equilibrium: Definition 3 (Nash equilibrium under commitment) The set of processes Λn λ ,λ , t p,t f,t ∞t=0 ≡{ } Λn λ ,λ , Ξn C , π , N , Ξn C , π , N and the multiplier Ω fully t∗ ≡ { ∗p,t ∗f,t } ∞t=0 t ≡ { t H,t t } ∞t=0 t∗ ≡ { t∗ ∗F,t t∗ } ∞t=0 19Which,in particular, disappearsunder the particular assumption of B =0. 0 17

describe a Nash equilibrium under commitment if they solve the system of equations (51) - (53), equations(45), (21), (26)holdingwith equality, alongwith asimilarsetofconditionsjointlyholding for Foreign. 5.1.2 Nash-Optimal Inflation Rate in the Long-Run To determine the long-run optimal inflation rate associated to the Nash-game described above, one needs to solve the steady state version of the set of efficiency conditions (51)-(53). In the language of the Ramsey-Cass-Koopmans model this amounts to computing the modified golden rule steady state, or, in other words, the unconstrained long-run optimal inflation rate. This contrasts with the golden rule inflation rate, which would correspond to the one that maximizes households’ instantaneous utility under the constraint that the steady state conditions are imposed ex-ante. It is well known that in dynamic economies with discounted utility the two concepts of long-run optimal policy cannot coincide. InAppendix C wecharacterizethesystemofequationsthatdescribesthelong-runsteadystate associated to the optimal policy problem under Nash competition. Under the assumption α =α , ∗ and of zero initial wealth (B =0), the solution to the steady-state of the Nash game is symmetric 0 and features π = π = 1, Φ = Φ = 1, N = N , and C = C , along with a non-zero value H ∗F ∗ ∗ ∗ for the multiplier Ω. Hence the steady state of the solution to the Nash-optimal policy indicates that, if unconstrained, both policymakers would choose to set the economy along a path that would lead to a long-run net inflation rate of zero. The intuition for this result is simple. One can view the modified golden rule as the long-run state of the economy when the discount rate β converges to 1. In this case the steady state version of the Phillips curve relation (45) is vertical, and the policymakerofeithercountrycannotexertanyeffectonmarkupsbysettinginflationratesdifferent from zero. 5.1.3 Optimal Stabilization Policy around the Long-Run Steady-State Wearenowinthepositiontoanalyzethedynamicfeaturesoftheoptimalcommitmentpolicyunder Nash competition. In this section we interpret optimal policy in the sense of optimal stabilization in response to shocks. To this end, we proceed in the following way. After characterizing (for both countries) the stationary allocations associated to the deterministic steady state of the first order conditions (51)-(53) (and symmetric ones for Foreign), we compute a log-linear approximation of the respective policy functions in the neighborhood of the same long-run steady state.20 20From a methodological point view, it may be of independent interest that at this stage,since the Nash-optimal allocationhasbeenalreadycharacterized,wecanlimitourselvestoemploystandardlog-linearapproximationmethods to describe the policy function. On the other hand, when later computing relative conditional welfare of alternative policyequilibria,wewillhavetoresorttoasecondorderapproximationofthesamepolicyfunction. Thisisnecessary to account for the natural effect of stochastic volatility on the first moments of critical variables, as well as for the 18

The spirit of this exercise deserves some further comments. In concentrating on (log-linear) dynamics in the neighborhood of the steady state associated to the efficiency conditions of the Nash-optimal policy, we deviate from the initial requirement (50) in the fact that we set the initial value of the lagged lagrange multipliers equal to their deterministic steady state values, i.e., χ = χ ; χ = χ . It is important to understand that this strategy, as in Khan, King and 0 0 ∗0 ∗0 Wolman (2003), corresponds to focusing on a particular dimension. Namely, optimal stabilization policy in response to bounded shocks that hit in the neighborhood of the long-run steady state. This amounts to implicitly assuming that such a steady state has been already reached after the implementation of the optimal policy plan as of time zero.21 5.1.4 Parameterization In conducting our simulations we employ the following form of the period utility: U(C ,N ) = t t 1 C1 σ 1 N 1+γ . The time unit is meant to be quarters. The discount factor β is equal to 1 σ t− − 1+γ t − 0.99. The degree of risk aversion σ is 1, the inverse elasticity of labor supply γ is equal to 3. As a benchmark value (see below for a discussion) we set η = 2. As in Bergin and Tchakarov (2003), andconsistentwithestimatesbyIreland(2001), wesetthepricestickinessparameterθ equalto50. The elasticity ε between varieties produced by the monopolistic sector is 6. The (inverse) degree of home bias α, identified bytheshare offoreignimported goodsinthedomesticconsumption basket, is set to a default value of 0.4. This implies the existence of a mild home bias, which is assumed to be symmetric across countries (α=α ). ∗ 5.1.5 Response to Productivity Shocks: Nash-Optimal vs. Inflation Targeting Figure1 comparesimpulseresponsesofselectedvariablestoaonepercentriseinHomeproductivity in the case of Nash-commitment with the same responses under (domestic) inflation targeting. The figure is illustrative of the inefficiency associated to policy competition in our context. Since productivity is higher in Home, the adjustment to the equilibrium requires an increase in the demand of domestic relative to foreign goods. This is achieved by means of a terms of trade depreciation, captured by a rise in the CPI to PPI ratio Φ. Recall, in fact, that Φ = Φ( ), i.e., t t T the same ratio is a (positive) function of the terms of trade. The only equilibrium is the one in which the same terms of trade depreciation is achieved via an increase in prices in both countries, Home and Foreign. In fact, and due to risk sharing, both countries face the incentive to increase prices to tilt the terms of trade in their own favor, thereby achieving a relatively higher real income transitional dynamics that characterize the economy in its adjustment towards the long-run steady-state associated to theoptimalpolicy. 21Torephraseit,thiscorrespondstoassumingthattheeconomyhasbeenevolvingaroundsuchasteadystatefor a sufficiently long period of time. 19

and consumption for any given level of labor effort. However, since Home is the country in which productivity is relatively higher, the increase in domestic (producer) prices falls short the increase in foreign (producer) prices. This explains why, for a given nominal adjustment in the exchange rate, the terms of trade depreciate more in a Nash equilibrium relatively to the inflation targeting case. In the resulting dynamics, since aggregate consumption must rise equally in both countries due to risk sharing, the rise in employment exceeds the one that obtains under inflation targeting. ItisalsointerestingtonoticethataNashequilibriumgeneratesadynamicbehavioroftheprice levelthatresemblestheoneinresponsetoacost-pushshock. Thenovelaspectofourresultsisthat thesamedynamicsareobtainedinresponsetoaproductivityshock, whichisnotaimedperse(like in many recent New Keynesian studies of optimal monetary policy) to induce the artificial effect of exogenously drifting the economy away from the efficient allocation. The fact that productivity shocks are a source of price variability under the optimal policy is here an endogenous outcome of the competition on international relative prices.22 Figure 2 illustrates how the incentive to generate price movements vary with a critical parameter, namely the elasticity of substitution between domestic and foreign goods. The figure displays impulse responses (under Nash-commitment) of the same selected variables to a productivity shock for alternative values of η = [1, 2, 3]. The first case nearly corresponds to the benchmark case of Cobb-Douglas preferences typically employed in the linear-quadratic approach to the study of optimal policy for open economies. The literature lacks a consensus on the value of this parameter. Harrigan (1993) and Trefler and Lai (1999) suggest an empirical value as high as 5. Collard and Dellas (2002) derive an estimated value of 2.5. In their quantitative (theoretical) studies, Backus, Kehoe and Kydland (1992) explore a range of η between 0 and 5. Chari et al (2002) set η = 1.5, while Bergin and Tchakarov (2003) set η = 5. Overall, there seems to exist both empirical and theoretical support for the hypothesis that the value of η lies above unity. The figure highlights the coincidence of the Nash-optimal response with a close-to-price stability strategy only in the particular case of η = 1. In this knife-edge case, the income effect of the required terms of trade depreciation (given the relatively higher productivity in Home) balances the incentive to switch expenditure towards Home goods.23 In general, the higher the elasticity of substitution, the larger (at the margin) the incentive for the policymaker to induce a strategic rise in the (producer) price level to try to generate a relative appreciation of its own residents’ real income and purchasing power for any given level of labor effort. 22ForananalysisoftheoptimalpolicysettinginresponsetothistypeofshocksseeWoodford(2003)andClarida et al. (1999). For open economy models withone-periodpredetermined prices seeSutherland (2001). 23See also Benigno and Benigno (2003b). Another way of seeing this is that, from equations (39) and (40), both Φ and Φ ceaseto play any role in the determination of theequilibrium in theparticular caseof η=1. t ∗t 20

5.2 Cooperation Undercooperation, asocial plannerexplicitlyrecognizesthechannelofinterdependencethatworks through the relative prices Φ and Φ . Below we define the world Ramsey planner problem, under t ∗t the assumption that the same planner aims at maximizing the average level of utility of the two countries. We also assume that both countries receive equal weight in the planner’s objective function. Let’s define the world Ramsey period utility objective as: U(C ,N )+U (C ,N ) w(C ,C ,N ,N ,π ,π ,Ωw) t t t∗ t∗ t∗ + U t t t∗ t t∗ H,t ∗F,t ≡ 2 ½ ¾ A N θ(π 1)2 A N θ(π 1)2 Ωw U C t t − 2 H,t − +U C ∗t t∗ − 2 ∗F,t− " c,t à t − Φ t ! c∗ ∗ ,t à t∗ − Φ ∗t !# where Ωw is a constant multiplier on the sum of the constraints (26) and (29). Then the Ramsey maximization problem can be defined as follows. Definition 4. Let λ p,t , λ f,t , λ ∗p,t , λ ∗f,t ∞ t=0 represent sequences of Lagrange multipliers on the constraints (45), (46), (21) and (22) respectively. Let B be given and κ = 1. For any given © ª 0 stochastic processes A , A , plans for the control variables C , π , N , C , π , N , and { t ∗t } ∞t=0 e { t H,t t t∗ ∗F,t t∗ } forthecostate variables λ ,λ ,λ ,λ and Ωw,represent afirst best constrained allocation p,t f,t ∗p,t ∗f,t ∞t=0 { } if they solve the following maximization problem: Choose Λc t ≡ λ p,t ,λ f,t ,λ ∗p,t ,λ ∗f,t ∞ t=0 and Ξc t ≡{ C t , π H,t , N t } ∞t=0 to © ª ∞ Min Max E βtE w(C ,C ,N ,N ,π ,π ,Ωw) (54) { Λc t} ∞t=0 { Ξc t} ∞t=0 0 { t {U t t t∗ t t∗ H,t ∗F,t t=0 X U εA N U Φ ε 1 c,t t t n,t t +λ U π (π 1) βU π (π 1)+ + − p,t c,t H,t H,t c,t+1 H,t+1 H,t+1 − − − θ U A ε c,t t · µ ¶¸ U η θ +λ A N (1 α) C Φ η κη c∗ ∗ ,t Φ η α C (π 1)2 f,t t t − − t t − U t ∗ t∗ − 2 H,t − c,t · µ ¶ ¸ U εA N U Φ ε 1 c∗,t ∗t t∗ n∗,t ∗t +λ ∗p,t " U c∗ ∗ ,t π ∗F,t (π ∗F,t − 1) − βU c∗ ∗ ,t+1 π ∗F,t+1 (π ∗F,t+1 − 1)+ ∗ θ à U c∗,t A ∗t + − ε !# ∗ η U θ +λ A N (1 α ) C (Φ )η κ η c,t (Φ )ηαC π 1 2 2ΩwB ∗f,t ∗t t∗ − − ∗ t∗ ∗t − − U ∗t t − 2 ∗F,t − }} − 0 " à c∗,t! # ∗ ¡ ¢ We defer to Appendix D the description of the first order conditions corresponding to this pelan. The discussion on the non-recursivity structure of the problem follows exactly the logic applied above to the re-definition of the Nash-commitment policy setup. In practice, this will entail specifying an equivalent recursive stationary program in the new world planner’s state space defined by A ,A , χ , χ . t ∗t t ∗t { } 21

5.2.1 Ramsey Steady-State A deterministic Ramsey steady state is a set of allocations C,C ,N,N ,π ,π ,Ωw that solves { ∗ ∗ H ∗F } the steady-state version of the efficiency conditions associated to the program under Definition 3. In Appendix E we characterize such system of equations. Under the assumption of zero initial net wealth (B = 0), this steady state has a symmetric solution in which π = π = 1. Hence, and 0 H ∗F exactly like in the steady state version of the efficiency conditions of the Nash problem analyzed above, the unconstrained long run optimal inflation policy is associated with price stability in both countries. 5.2.2 Optimal Response to Shocks around the Ramsey Steady-State: Nash vs. Cooperation Figure 3 displays impulse responses to a normalized one percent increase in home productivity and compareselectedvariablesunderNash-commitment versusCooperation-commitment. Underpolicy cooperation, the planner coordinate the responses of both policy makers to achieve the required termsoftradedepreciationonlybymeansofanominal exchangeratedepreciation. Inotherwords, it is optimal for the Ramsey planner to have both countries targeting very closely the flexible price allocation. This results in a dampened dynamic of the CPI-PPI ratio Φ under cooperation. The crucial aspect is that this is now compatible with a smooth path of the price level (the response of the price level, measured in percent deviation from steady state, barely deviates from zero) and withasmootherresponseofemployment, foranygivenvariationin consumption(inturnequalized across countries). It is also interesting to notice, under the Ramsey cooperative regime, that while the response of the price level resembles the one that would obtain in a closed economy under the optimal policy (see for instance King and Wolman, 1997), so does not the response of employment. For an intuition, consider the (equilibrium) real marginal cost equation (44). As already emphasized, the closed economy version of that equation obtains in the case Φ =1. Hence, in a closed economy, it t is optimal, in response to a rise in productivity, to fully absorb the rise in productivity by means of an equal increase in consumption (and output), while keeping employment constant. In an open economy, the equilibrium requires a rise in Φ ( i.e., a real depreciation). Hence, coordinating on t stableprices(i.e.,constantrealmarginalcosts),requiresariseinconsumptionwhichissmallerthan the one in productivity and, under the benchmark parameterization, also a rise in employment.24 24The size of the rise in employment will be, in turn, a function of the elasticity η (with a smaller η implying a smaller rise) and of the labor supply elasiticity 1 (with a higher elasticity requirying a smaller risein employment). γ 22

6 Welfare Analysis and Dynamic Features of the Ramsey Policy We now turn to a characterization of the dynamic properties of the alternative policy regimes analyzed so far.25 We illustrate our numerical analysis in terms of cyclical properties of selected variables and welfare levels associated to each policy arrangements. We report results under three alternative parameter scenarios: 1) High home bias, in which the value of α = α is set to 0.1; 2) ∗ Small home bias in which α = α =0.45; 3) Low elasticity of substitution, in which η =0.1. ∗ Some observations on the computation of welfare are in order. First, one cannot safely rely on standard first order approximation methods to compare the relative welfare implied by each policy arrangement. In fact, in an economy like ours, in which distortions exert an effect both in the short and in the long-run, stochastic volatility affects both first and second moments of those variables thatarecriticalforhousehold’swelfare. Sinceinafirstorderapproximationofthemodel’ssolution the expected value of a variable is always equal to its non-stochastic steady state, the effects of volatility on mean values of variables is by construction neglected. Hence policy arrangements can be correctly ranked only by means of a higher order approximation of the policy functions.26 This last observation suggests also that our welfare metric needs to be correctly chosen. In particular one needs to focus on the conditional expected discounted utility of the representative agent. Thisisnecessaryexactlytotakeintoaccountoftransitionaleffectsfromthedeterministicto the different stochastic steady states respectively uimplied by each alternative policy arrangement. We proceed in the following steps. First, we compute a second order approximation of the policy function(s) around the long-run deterministic steady-state implied by each policy regime under scrutiny.27 Then we assume that both economies are subject to a stationary distribution of 25More in line with the present analysis is the one of Kollmann (2003) and Bergin and Tchakarov (2003), who studyoptimizinglinearinterestraterulesandperformwelfarecalculationsbasedonasecondorderapproximationof themodel’s equilibriumconditions. Our paper differs crucially inthat itcharacterizes equilibriumallocations under theRamseypolicy,withoutrestrictingtheformofthepolicyfunctiontotheargumentsofapre-specified(log-linear) interest rate rule. 26See Kim and Kim (2002) for an analysis of the inaccuracy of welfare calculations based on log-linear approximations indynamic openeconomies. SeeKimetal. (2003)andSchmitt-GroheandUribe(2004)for amoregeneral discussion. 27Theset of optimality conditions of theRamsey plan canbedescribedas follows: E f(Y ,Y ,X ,X )=0 (55) t t+1 t t+1 t where E denotes the mathematical expectations operator, conditional on information available at time t, Y is the t t vector of endogenous non-predetermined variables and X [x ,x ] is the state vector. Here x denotes the t 1,t 2,t 1,t ≡ vector of (pseudo) co-state variables [χ , χ ], while x is the vector of exogenous variables [A ,A ] which follows a t ∗t 2,t t ∗t stochastic process. x 2,t+1 =zx 2,t +ησε t+1 ; ε t i.i.d.N(0,Σ) (56) ∼ The scalar σ andη are knownparameters. The solution to themodel is of the form: Y =g(X ,σ) (57) t t 23

(productivity shocks) and generate, for given initial conditions, artificial time series of length T = p 500 periods. We compute mean, standard deviation and implied presented discounted utility for any given random draw. We then iterate the computation T = 1000 times and average across n experiments.28 Along the lines of Lucas (1987), the measure of welfare cost (of business cycles) that we associate to each policy is the proportional upward shift in the consumption process that would be required to make the representative household indifferent between its random consumption allocation and a nonrandom consumption allocation with the same mean. Hence such measure is defined as the fraction ∆ that satisfies the following equality between conditional utilities: ∞ ∞ E βtU((1+∆)C ,N )= βtU(E (C ,N )) (59) 0 t t 0 t t t=0 t=0 X X In Table 1 we report second moments of selected variables under Nash competition and Cooperation. In particular, we report the cyclical properties of the CPI to PPI ratio (our key relative price) meant as a proxy of the terms trade. Hence we see that, across parameter scenarios, and relative to cooperation, Nash competition is indeed a source of enhancement of inflation volatility. In particular this happens in a low home bias scenario. Intuitively, under this scenario, economies are more open to trade, and therefore, at the margin, more open to a policy competition motive over their terms of trade. However, deviations from price stability, once measured in terms of second moments of inflation, remain per se rather small under Nash competition and barely different from the regime of Cooperation. In the same Table we also report the measure ∆ of the welfare costs associated to the alternative policy scenarios. Hence it is clear that cooperation delivers welfare gains relative to a Nash competition arrangement. However, such welfare numbers remain quite small. In absolute terms, andacrossallpolicyscenarios,theupwardshiftinconsumptionneededtomakethehouseholdindifferent between a random and a nonrandom allocation range between a minimum of 0.0113 percent (achieved under Cooperation with high home bias) to a maximum of 0.0157 percent (achieved under Nash competition with low home bias). In relative terms, welfare gains from Cooperation are also rather small. This result, however, is hardly surprising. That both policies were delivering X t+1 =h(X t ,σ)+−ησε t+1 (58) where Y is the vector of control variables, equation (57) is the policy function and equation (58) is the transition t function. We compute a second order expansion of the functions g(x ,σ) and h(x ,σ). Schmitt-Grohe and Uribe t t (2002) show, crucially, that, up to a second order, the coefficients of the policy functions attached to terms that are linearinthestatevectorx areindependentofthesizeofthevolatilityoftheshock(s). Toevaluatenumericallythe t first and second order derivatives of the policy functions we employ the Matlab codes compiled by Schmitt-Grohe and Uribe, available at thewebsite http://www.econ.duke.edu/˜grohe/. 28Weemployastandardparameterizationfortheinnovationstotheproductivityprocesses,andassumeVar(εa)= t Var(εa∗)=(0.01)2, with persistence ρa =ρa∗ =0.9. t 24

very similar dynamics in response to the same distribution of shocks was already apparent from our previous impulse response analysis. It is of some interest, however, to see that, in our exercise, such gains may depend on the comparative statics on two critical parameters that identify ”openness” and that affect the relationship between the terms of trade and the CPI to PPI ratio, namely α and η . Hence our exercise indicate that welfare gains from cooperation are minimized when the elasticity of substitution between domestic and foreign goods η is small, and maximized when the home bias is low (i.e., α = 0.45). Intuitively, these two scenarios correspond, respectively, to a case in which the policy competition motive is either reduced in scope or, alternatively, magnified. 7 Golden Rule So far we have concentrated only on the optimal short-run stabilization policy around a deterministic Ramsey steady-state. This is the steady state associated to the efficiency conditions that describe the optimal policy under commitment (modified golden rule). In this long-run, the policymaker faces no incentive to use the inflation instrument to affect the markup via an exploitation of the Phillips curve. However, if the policymaker was forced to maximize households’ utility under the constraint that the steady state conditions are imposed ex-ante, this may lead to the presence of an inflationary or deflationary bias. In analogy with the terminology of the neoclassical growth model, and as in King and Wolman (1997), we can define this as the policy maker’s golden rule. To understand whether openness coupled with policy competition is responsible for either an inflationaryordeflationarybias,letusfocusonhowtherelevantdistortionsinteractinagoldenrule steady state. It is first instructive to derive the markup function from the competitive equilibrium of the domestic open economy. By combining the steady-state version of (5) and (12) we can write U 1P 1 − n = H = [Φ( )] − 1 U µ P µ(π ,N) T c H where the function µ(π ,N) derives from the steady-state version of (18) as H εN µ(π ,N)= (60) H θπ (π 1)(1 β)+(ε 1)N H H − − − Hence efficiency in any given steady state of the economy requires µ(π ,N)Φ( ) =1 (61) H T To gain intuition on how the international relative price distortion (summarized by the wedge Φ( ) which is increasing in ) interacts with the markup distortion from the view point of a given T T country, let’s assume, for the sake of argument, that producer prices are fully flexible, so that the 25

markup is always equal to a constant value of ε . We therefore temporarily abstract from the ε 1 − price-stickinessdistortion. Bymakinguseof(36), wecanrewritearelationshipbetweenthedesired terms of trade and the markup which reads 1 α η 1 = − (62) T µη 1 (1 α) µ − − − ¶ Notice that, independently of the values of η and α, is always decreasing in µ. One should T view equation (62) as an iso-efficiency condition. Hence, and in order to keep the economy at the maximum welfare in steady-state, a higher markup calls for more appreciated terms of trade. The intuition is simple. The presence of imperfectly competitive output markets makes desirable to expand output towards the efficient level. While in an closed economy with flexible prices this is always welfare improving, in an open economy the same rise in output requires (in equilibrium) also a depreciation of the terms of trade, which hurts the purchasing power of domestic consumers. Equation(62)showsthat,atthemarginandforany given level of foreign consumption,itisoptimal to have the terms of trade depreciate less, or equivalently let output expand less relatively to the case of an imperfectly competitive closed economy. However, such a strategic incentive characterizes also the optimal reaction function in Foreign. To formalize the policy game let’s define, for any given C , the golden rule Nash steady state for ∗ Home as the triplet π ,C,N gr argmax U(C,N) (63) H { } ≡ { } subject to a (steady state) pricing-implementability condition εN U Φ(C,C ) ε 1 n ∗ π (π 1)(1 β) − − (64) H H − − ≤ θ U − ε c µ ¶ and to a (steady state) feasibility constraint κU η θ N (1 α) C [Φ(C,C ∗ )]η + c∗ ∗ [Φ(C,C ∗ )]ηα ∗ C ∗ + (π H 1)2 (65) ≤ − U 2 − c µ ¶ whereitisunderstoodthatC istakenasgivenfromtheviewpointofthepolicymakerinHomebut ∗ is instead chosen optimally by the policymaker in Foreign. First order conditions of this problem defineHomepolicymaker’sreactionfunctionforanygivenlevelofForeignconsumption. Anexactly symmetric problem characterizes the reaction function of the policymaker in Foreign. The solution to the joint system of equations pins down the Nash equilibrium, and is reported inFigure 4. Thedashed line showsthesolutionoftheNashgameforaselectednumberofvariables asafunctionofthe(inverse)homebiasparameterα. Thisparameterisanaturalindexofopenness in our context. In the simulations, we set the vector [σ, γ, η] = [1, 3, 2], while we maintain that 26

α = α . Hence we see that for α 0 the rate of (producer) inflation that maximizes steady-state ∗ → welfare is positive and coincides exactly with the one of the corresponding closed economy with sticky prices and monopolistic competition. As α turns positive, i.e., both economies become open to trade, the desired steady-state inflation rate decreases below the one of the corresponding closed economy. The intuition is simple. As explained above, policy competition calls for a strategic reduction in consumption (relative to the closed economy case) in order to obtain an appreciation of the terms of trade and a reduction in work effort. For any given level of foreign consumption, and in the absence of any asymmetric shock, this is welfare improving. However, in a Nash-game where policy objectives collide, terms of trade and work effort remain constant, while consumption ends up being slightly lower. This entails, for any level of openness (as measured by α), a lower steady-state rate of inflation relative to a closed economy. On the other hand, if two policymakers coordinate their actions in order to avoid a strategic manipulation of the terms of trade, they can enjoy higher consumption for any given level of work effort. Under Cooperation, we assume that the planner chooses simultaneously the two triplets U(C,N)+U(C ,N ) π , C, N and π , C , N in order to maximize average utility ∗ ∗ , subject to { H } { ∗F ∗ ∗ } 2 the constraints (64), (65) and to the corresponding equations for Foreign. Figure 5 (in solid line) shows the outcome of the optimal cooperative policy game. Notice that once again the optimal inflation rate lies below the one of the corresponding closed economy. Yet the inflation rate under policy cooperation lies above the one prevailing under Nash competition. Intuitively, under cooperation, the planner induces both policymakers to avoid any strategic reduction in consumption aimed at appreciating the terms of trade. In equilibrium, the terms of trade and work effort are unchanged, and consumption is higher relative to the Nash outcome. This, ceteris paribus, entails also a relatively higher inflation rate. Is the result of a Nash deflationary bias (relative to Cooperation) robust to parameter values? In fact it is not. Intuitively, and as emphasized, although in a different framework, also by Sutherland (2002) and Benigno and Benigno (2003b), the gains from strategically manipulating the terms of trade should be decreasing in the elasticity of substitution between domestic and foreign goods, for this is a measure of the strength of the expenditure switching effect. We therefore employ an alternative parameterization, with high labor supply elasticity and low international elasticity of substitution. This implies setting [σ, γ, η] = [1, 1, 0.7]. We see that in this case there exist values for the degree of openness such that the inflation rate under Nash lies above the one under policy cooperation. It is only when the economy becomes extremely open (i.e., the degree of home bias is extremely small) that the Nash deflationary bias result re-emerges. The above discussion on the golden-rule incentives for an optimal inflation policy can be summarized as follows: Result 1 (Open economy bias). In an open economy with price adjustment costs and monopo- 27

listiccompetition, the(producer)inflationratethatmaximizessteady-stateutilityliesmonotonically below the one of the corresponding closed economy. This holds under both policy competition and cooperation. The interesting aspect of Result 1 is that, while in a closed economy with sticky prices and monopolisticcompetitionpricestabilitycannotimplementthesteady-statemaximizationofwelfare, it can indeed do so when the economy is open, due to the additional effect of the international relative price distortion that pushes the efficient inflation rate downwards. Result 2 (Nash bias ). There is no monotonic ranking between the golden rule inflation rate under Nash and the one under Cooperation. Hence policy competition may lead either to a deflationary or inflationary bias depending on the degree of trade openness and on the elasticity of substitution between domestic and foreign goods. ThisresultdiffersfromtheoneofCooleyandQuadrini(2003),whofindthatpolicycompetition is necessarily associated with an inflationary bias. The reason lies in the structure of their model, whichfeatures flexible pricesand an unambiguos positive outputeffect ofreal appreciations. Inour context,andcrucially,pricesaresticky(sothatmonetaryauthoritiescanexertadirecteffectonthe terms of trade) and the effect of international relative price movements on output strictly depends on the value of the elasticity of substitution between domestic and foreign goods. In particular, under our preferred parameterization of η >1 (with log utility), real appreciations exert a negative effect on output. 8 Conclusions We have laid out a typical public finance framework for the analysis of welfare maximizing monetary policy within an economy characterized by three distortions: market power, rigidity in the adjustment of producer prices and international terms of trade externality. The main advantage of our approach, relative to the existing literature, is that it allows to characterize optimal policy in a fully dynamic open economy setting while maintaining all the distortions completely spelled out both in the short and in the long run. Despite the generality of the approach, our modelling framework remains restrictive in three main dimensions. First, in assuming that the law of one price for traded goods holds continually. Second,inallowinghouseholdstoobtainfullrisksharingviainternationalfinancialmarkets. Third, in not allowing households to invest in physical capital. Amending on all these features should aim at generating less trivial dynamics of the current account than the ones generated here via the only movements in the trade balance. Such dynamics may be of first order importance for the welfare evaluation along two dimensions. First, they would more critically affect the transition from deterministic to stochastic steady state. Second, they would impinge on the transition from one policy regime to another. For instance from Nash-competition to cooperation, or from the 28

optimal commitment policy to a fixed exchange rate arrangement. Given the flexibility and the rigor of a Ramsey-based approach, all these issues will certainly be the source of new research efforts in international macroeconomics in the near future. 29

A Proof of Proposition 1 (Part A). The proof follows from the substitutions and the rearrangements of Section 4 that lead to the minimal form. (Part B). For given productivity processes A , A and using the allocations t ∗t C ,C ,N ,N ,Φ , Φ satisfying the optimal plan, one can obtain the optimal allocation for { t t∗ t t∗ t ∗t } ∞t=0 thereal wage, outputandreal marginal costfrom(5), theaggregateversionof(11)and (12), which are symmetric across countries. Using the producer inflation rates π , π obtained by the H,t ∗F,t optimal plan and the relative prices { Φ t , Φ ∗t } ∞t=0 one can obtain thenCPI inflatioon rates from (41) and the analog for Foreign. For given optimal paths C ,C the path for the real exchange { t t∗ } ∞t=0 rate follows from (28). Given Φ , Φ , this allows to solve for the terms of trade from (37). By { t ∗t } ∞t=0 rewriting C = (1 α)Φ η C and C = α(Φ )ηC from (2) one obtains consumption demand H t t F,t t t t − T for domestic and foreign goods. For given consumption, the path for the real interest rate is given by (42), which implies, given CPI inflation, a path for the nominal rate via equation (43). B The Stationary Dynamic Policy Problem Below we derive the stationary form of the policy problem under Nash commitment. We illustrate the argument only for the Home policymaker’s problem, since the problem in Foreign is exactly symmetric. Let’s consider the optimal plan as formulated in equation (47) in the text. By applying the law of iterated expectations and by grouping expectations and multipliers that share the same date one obtains: Min Max E (C , N , π , Ω) { Λt } ∞t=0 { Ξt } ∞t=0 0 {U 0 0 H,0 U εA N U Φ ε 1 c,0 0 0 n,0 0 +λ U π (π 1)+ + − p,0 c,0 H,0 H,0 − θ U A ε c,0 0 · µ ¶¸ +λ A N (1 α)C Φ η κη U c∗ ∗ ,0 η Φ η α C θ (π 1)2 f,0 0 0 − − 0 0 − U 0 ∗ 0∗ − 2 H,0 − c,0 · µ ¶ ¸ U εA N U Φ ε 1 c,1 1 1 n,1 1 +β U(C ,N )+(λ βλ )(U π (π 1))+λ + − 1 1 p,1 p,0 c,1 H,1 H,1 p,1 { − − θ U A ε c,1 1 µ µ ¶¶ +λ A N (1 α)C Φ η κη U c∗ ∗ ,1 η Φ η α C θ (π 1)2 +... ΩB f,1 1 1 − − 1 1 − U 1 ∗ 1∗ − 2 H,1 − }} − 0 c,1 · µ ¶ ¸ Notice that this problem is not time-invariant due to the fact that the constraints atetime zero lack the term βλ (U π (π 1)). For this reason we amplify the state space to introduce p, 1 c,0 H,0 H,0 − − − a new (pseudo) costate variable χ and define a new policy functional (C ,N π ,χ ,Ω) t t t, H,t t W ≡ (C ,N ,π ,Ω) χ (U π (π 1)). We then write the optimal policy plan in the following t t H,t t c,t H,t H,t U − − form: 30

Choose Λ λ ,λ and Ξ C , π , N to t p,t f,t ∞t=0 t t H,t t ∞t=0 ≡{ } ≡{ } ∞ Min Max E βtE (C ,N π ,χ ,Ω) (66) { Λt } ∞t=0 { Ξt } ∞t=0 0 { t {W t t, H,t t t=0 X U εA N U Φ ε 1 c,t t t n,t t +λ U π (π 1)+ + − p,t c,t H,t H,t − θ U A ε c,t t · µ ¶¸ U η θ +λ A N (1 α) C Φ η κη c∗ ∗ ,t Φ η α C (π 1)2 ΩB f,t t t − − t t − U t ∗ t∗ − 2 H,t − }}− 0 c,t · µ ¶ ¸ with law of motion for the new costate e χ =λ t+1 p,t and initial condition χ =0 0 Following Marcet and Marimon (1999), one can show that this new maximization program is now saddlepointstationary in the amplifiedstate space A , χ . Firstorder conditionsofthisproblem t t { } exactly replicate conditions (51)-(53) in the text. An exactly symmetric argument is applied to the design of the policy problem in Foreign, which will involve specifying an amplified state space A , χ , with law of motion χ =λ and initial condition χ =0. ∗t ∗t ∗t+1 ∗p,t ∗0 { } C Steady State of the Nash-Optimal Policy Problem The steady state version of the efficiency conditions (51)-(53) of the Nash problem is derived by imposing λ =λ =λ =χ. This implies: p,t p,t 1 p − λ N 0 = U + p ((cid:18) U Φ +(ε 1)U ) λ (1 α) Φη +ηCΦη 1Φ c n c cc f − c θ − − − +λ α C κηU η ηΦη 1Φ U η ηU η 1U Φ¡η ¢ f ∗ ∗ c∗ − c c− c− − cc − θ Ω (N (π ¡ 1)2) U Φ 1 Φ Φ 2U ¢(U C +U ) H cc − c − c cc c − − 2 − − − · ¸ ¡ ¢ λ εΦ ε 1 U p c 0=U + (U +NU )+λ − U +λ Ω n n nn p c f θ θ − Φ U c 0= θ(π 1) λ Ω H f − − − Φ µ ¶ Thesystemiscompletedbythesteadystateversionof(4),(45),(21)alongwithsymmetricequation forForeign. Onecaneasilyverifythatthesolutionisavector Ω, C, C , N, N , π π λ λ λ λ ∗ ∗ H, ∗F, F, ∗F, p, ∗p, with π H, =π ∗F, =1, C =C ∗ , N = N ∗ , Φ= Φ ∗ =1. n o 31

D First Order Conditions of the Policy Cooperation Problem First order conditions for the Ramsey problem under Cooperation at time t 1 read: ≥ 1 λ N ε ε 1 p,t t 0 = U +U π (π 1) (λ χ )+ U Φ +λ − A N U (67) 2 c,t cc,t H,t H,t − p,t − t θ n,t c,t p,t θ t t cc,t µ ¶ λ f,t (1 α) Φ η t +ηC t Φ η t− 1 Φ c,t λ f,t α ∗ C t∗ κη U c∗,t η ηΦ η t− 1 Φ c,t U c−,t η ηU c−,t η − 1 U cc,t Φ η t − − − ∗ − +λ εN t∗ U n∗ ³ ,t Φ ∗c,t λ η(Φ ´ )η 1Φ (1 α ¡ )C ¢ h i ∗p,t θ − ∗f,t ∗t − ∗c,t − ∗ t∗ µ ¶ ³ ´ λ ∗f,t ακ − η U c∗,t − η η(Φ ∗t )η − 1Φ ∗c,t C t U c η ,t +(Φ ∗t )η U c η ,t +ηU c η ,−t 1U cc,t C t − ∗ +Ωw(A N ¡ θ (π ¢ h 1)2) U Φ 1 Φ Φ 2U ³ Ωw(U C +U ´i )+ t t − 2 H,t − cc,t −t − c,t −t c,t − cc,t t c,t +Ωw(A N θ (π 1)2) ¡ U Φ Φ 2 ¢ ∗t t∗ − 2 ∗F,t − c∗ ∗ ,t − ∗c,t ∗t− ¡ ¢ λ εΦ ε 1 U A U + p,t t (U +N U )+λ − U A +λ A +Ωw c,t t =0 (68) n,t n,t t nn,t p,t c,t t f,t t θ θ Φ t µ ¶ U A U (2π 1)(λ χ ) θ(π 1) λ +Ωw c,t t =0 (69) c,t H,t − p,t − t − H,t − f,t Φ t µ ¶ ∂Φ The expression for Φ ∗t reads: ∗c,t ≡ ∂Ct α Φ ∗c,t = − 1 ∗ α (Φ ∗t )2 − η U c∗,t η − 1 κη − 1 Φ t η − 2Φ c,t U c−,t (η − 1) − U c−,t ηU cc,t Φ t η − 1 µ − ∗¶ h i ¡ ¢ The set of analogous conditions for Foreign variables at time t 0 will involve an expression for ≥ Φ ∂Φt : c ∗ ,t ≡ ∂C t∗ α Φ c ∗ ,t = − 1 α Φ2 t− η (U c,t )η − 1κ1 − η (Φ ∗t )η − 2Φ ∗c ∗ ,t U c∗,t − (η − 1) − U c∗,t − η U c∗c,t (Φ ∗t )η − 1 µ − ¶ h i ¡ ¢ ¡ ¢ In addition all constraints must hold with equality. Also, when evaluated at time t = 0, condition (67) must feature the additional term ΩwB0. − 0 P0 E Steady State of the Policy Cooperation Problem Under Cooperation the steady state version of the efficiency conditions (67)-(69) reads: 32

1 λ Nε ε 1 p 0 = U + U Φ +λ − NU c n c p cc 2 θ θ µ ¶ λ (1 α) Φη +ηCΦη 1Φ λ α C κη(U )η ηΦη 1Φ U η ηU η 1U Φη f − c f ∗ ∗ c∗ − c c− c− − cc − − − ∗ − εN U Φ +λ ∗ n∗¡ ∗c λ η(Φ¢)η 1Φ (1 α )C £ ¤ ∗p θ − ∗f ∗ − ∗c − ∗ ∗ µ ¶ ³ ´ λ ακ η(U ) η η(Φ )η 1Φ CUη +(Φ )η Uη +ηUη 1U C ∗f − c∗ − ∗ − ∗c c ∗ c c− cc − ∗ +Ωw(N θ (π h 1)2) U Φ 1 Φ Φ 2U ¡ Ωw(U C +U ) ¢ + i H cc − c − c cc c − 2 − − − +Ωw(N θ (π 1)2) ¡ U Φ Φ 2 ¢ ∗ − 2 ∗F − c∗ ∗ − ∗c ∗− ¡ ¢ λ εΦ ε 1 U 0 =U + p (U +NU )+λ − U +λ +Ωw c n n nn p c f θ θ Φ µ ¶ U 0= θ(π 1) λ +Ωw c H f − − Φ µ ¶ The system is completed by the steady state version of (4), (45), (21) along with symmetric equationforForeign. Onecaneasilyverifythatthesolutionisagainavector Ω, C, C , N, N ,π π λ λ λ ∗ ∗ H, ∗F, F, ∗F, p with π H =π ∗F =1, C =C ∗ , N = N ∗ , Φ= Φ ∗ =1. n 33

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Table 1 Volatility and Welfare under Alternative Policy Regimes High Home Bias Low Home Bias Low Elasticity Nash Cooperation Nash Cooperation Nash Cooperation Consumption 1.553 1.443 1.834 1.695 1.561 1.491 Labor 0.227 0.196 0.184 0.100 0.155 0.142 PPI Inflation 0.053 0.005 0.072 0.006 0.041 0.009 CPI/PPI Ratio 0.698 0.699 0.240 0.237 1.552 1.541 Welfare cost ∆ 0.0122 0.0113 0.0157 0.0146 0.0118 0.0118 Note: Standard deviations are in %. The welfare cost (in %) is the proportional upward shift in the consumption process that would make the representative household indifferent between its random consumption allocation and a nonrandom consumption allocation with the same mean

Domestic Producer Price Level Home Employment 0.025 0.14 0.12 0.02 Nash 0.1 0.015 0.08 0.06 0.01 0.04 0.005 IT 0.02 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Home Consumption Home CPI/PPI Ratio 0.7 0.35 0.6 0.3 0.5 0.25 0.4 0.2 0.3 0.15 0.2 0.1 0.1 0.05 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Figure 1. Impulse Responses to a Positive Home Productivity Shock: Nash-Optimal vs. Inflation Targeting

Home Producer Price Level 0.05 0.045 0.04 eta = 3 0.035 0.03 0.025 eta = 2 0.02 0.015 eta = 1.01 0.01 0.005 0 0 5 10 15 20 25 30 Figure 2. Response to a Productivity Shock under Nash-Optimal: Effect of Varying the Elasticity of Substitution

Home Producer Price Level Home Employment 0.02 0.1 0.09 0.015 0.08 Nash 0.07 0.01 0.06 0.005 0.05 0.04 0 0.03 0.02 -0.005 Cooperation 0.01 -0.01 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Home Consumption Home CPI/PPI Ratio 0.7 0.25 0.6 0.2 0.5 0.15 0.4 0.3 0.1 0.2 0.05 0.1 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Figure 3. Impulse Responses to a Home Productivity Shock: Nash vs. Cooperation

Home Gross Producer Inflation CPI/PPI Ratio (Terms of Trade) 1.0005 1 closed 1 1 0.9995 1 0.999 0.9985 1 0.998 1 0.9975 1 0.997 0.9965 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 openness openness Home Employment Home Consumption 0.9554 1 0.995 0.9554 0.99 0.9553 0.985 0.98 0.9553 0.975 0.9552 0.97 0.965 0.9552 0.96 0.9551 0.955 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 openness openness Figure 4. Effect of Varying Openness on the Golden Rule Steady-State under Nash Competition

Golden Rule Steady-State Inflation: high elasticity of substitution 1.0005 1 0.9995 0.999 0.9985 0.998 0.9975 0.997 0.9965 0.996 0.9955 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 openness noitalfni ssorg s.s Golden Rule Steady-State Inflation: low elasticity of substitution 1.0001 1 0.9999 0.9998 0.9997 0.9996 0.9995 0.9994 0.9993 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 openness noitalfni ssorg s.s Cooperation closed economy Nash Nash Cooperation Figure 5. Effect of Varying Openness on the Golden Rule Steady-State Inflation Rate