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Federal Reserve Bank of Dallas Globalization and Monetary Policy Institute Working Paper No. 321 https://doi.org/10.24149/gwp321 Good Policies or Good Luck? New Insights on Globalization and the International Monetary Policy Transmission Mechanism* Enrique Martínez-García Federal Reserve Bank of Dallas July 2017 Abstract The open-economy dimension is central to the discussion of the trade-offs that monetary policy faces in an increasingly integrated world. I investigate the monetary policy transmission mechanism in a two-country workhorse New Keynesian model where policy is set according to Taylor (1993) rules. I find that a common monetary policy isolates the effects of trade openness on the cross-country dispersion, and that the establishment of a currency union as a means of deepening economic integration may lead to indeterminacy. I argue that the common (coordinated) monetary policy equilibrium is the relevant benchmark for policy analysis showing that in that case open economies tend to experience lower macro volatility, a flatter Phillips curve, and more accentuated trade-offs between inflation and slack. Moreover, I show that the trade elasticity often magnifies the effects of trade integration (globalization) beyond what conventional measures of trade openness would imply. I also discuss how other features such as the impact of a stronger anti-inflation bias, technological diffusion across countries, and the sensitivity of labor supply to real wages influence the quantitative effects of policy and openness in this context. Finally, I conclude that the theoretical predictions of the workhorse open-economy New Keynesian model are largely consistent with the stylized facts of the globalization era started in the 1960s and the Great Moderation period that followed. JEL codes: C11, C13, F41 * Enrique Martinez-Garcia, Research Department, Federal Reserve Bank of Dallas, 2200 N. Pearl Street, Dallas, TX 75201. 214-922-5262. [email protected]. I would like to thank Charles Engel, Serguei Maliar, María Teresa Martínez-García, Alexander Ueberfeldt, Wesley Wilson, Mark A. Wynne, Carlos Yepez, and many participants at the Fourth International Symposium in Computational Economics and Finance (2016) for helpful suggestions and comments. I also acknowledge the excellent research assistance provided by Valerie Grossman. The views in this paper are those of the author and do not necessarily reflect the views of the Federal Reserve Bank of Dallas or the Federal Reserve System. 1 Introduction An ongoing topic of discussion among policymakers is how best to think about the role of openness for the conduct of monetary policy (Fisher (2005), Fisher (2006), Bernanke (2007), Trichet (2008), and more recently Draghi (2015) and Kaplan (2017)). Policymakers increasingly recognize that exchange rate movements and trade linkages cannot be ignored in guiding policy, yet in many cases the closed-economy model— these days a variant of the workhorse closed-economy New Keynesian model (Woodford (2003))— largely remains the starting point for policy analysis. Much research has been devoted to explore how openness in‡uences policy analysis and to what extent the closed-economy setting o¤ers a useful approximation for policy-making, whenever major economies are in fact increasingly interconnected. The international macro literature has raised questions like: How do natural rates and potential output depend on foreign developments? Is the Phillips curve relationship between domestic in‡ation and domestic slack ‡atter or steeper for open economies and what does that entail? Can greater openness contribute to lower volatility and alter the persistence and cross-country comovement of macro aggregates characteristic of the Great Moderation period? And, perhaps most importantly, how does openness in‡uence the policy trade-o¤s confronted by policymakers under a Taylor (1993)-type monetary policy regime? The main purpose of this paper is to investigate the conduct of monetary policy under alternative monetary policy regimes within the workhorse two-country New Keynesian model that explicitly incorporates a role for openness. I build on the model of Martínez-García and Wynne (2010) and Martínez-García (2015b) characterized by ‡exible nominal exchange rates, trade openness, and asymmetric shocks across countries— which provides a straightforward extension of the standard three-equation (closed-economy) New Keynesian model to an open-economy setting. The main postulate of the model is that trade and economic integration have altered the environment in which monetary policy must be conducted. In fact, it shows that even modest trade linkages expose the domestic economy to signi…cant impacts from foreign shocks as well as foreign policies. With the aid of this workhorse model, I analytically investigate the implications of monetary policy independence for open economies (re‡ected in asymmetries in the policy rule across countries). I explore alternative monetary policy regimes involving di¤erent degrees of international monetary policy coordination as well— particularly those arising from coordinated common monetary policies under ‡exible exchange rates or from a monetary union. The role of the monetary policy framework has received increased attention in the current monetary policy debate in light of notable policy regime changes among some major economies— such as the gradual coordination of national policies and eventual adoption of the common European currency (euro) among European Union (EU) countries. Figure 1 illustrates the aggregate and cross-country dispersion patterns on in‡ation and growth for the 11 member states of the EU that gave up their own independent monetary policies to give birth to the euro in 1999 (Haan (2010)).1 The experience with European monetary union has further raised awareness about many international monetary policy coordination issues more broadly such as the signi…cance of stabilizing aggregate rather than domestic measures of in‡ation and slack or whether (and how) the monetary policy framework should take into account the dispersion across countries (even across regions within a country). 1A number of other countries have since then adopted the euro as well. 1 FigureFigure 1. Common European E¤ects Growth In‡ation. 1. Common EuropeanCurrency: Currency: Effects onon Growth and and Inflation Euro-Area 11 Aggregate Real GDP Growth and Inflation Percent, Q/Q, SAAR 15 Euro = Common monetary policy 10 Std. Deviation on Cross-Country Differentials Percent, Q/Q, SAAR 25 Euro = Common monetary policy 20 5 15 0 10 -5 -10 Real GDP Growth 5 Headline CPI Inflation -15 0 1980 1986 1992 1998 2004 2010 2016 1980 1986 1992 1998 2004 2010 2016 Sources: National Statistical Offices and Central Banks; OECD; IMF; author's calculations. Notes: The data includes the 11 countries that adopted the euro initially (Euro-Area 11). All series are at quarterly frequency and aggregated using time-varying PPP-weights from the IMF. Quarterly growth rates are calculated in log-differences times 400. Cross-country differentials are computed relative to the Euro-Area 11 aggregates. 2 From a theoretical perspective, this paper contributes to the existing debate in two ways: First, it ‡eshes out an analytical framework that helps us understand how the monetary policy transmission mechanism under di¤erent policy regimes is altered by the degree of openness and, second, it provides closed-form solutions that are tractable and facilitate a positive analysis of international monetary policy transmission and coordination. An important conclusion of this paper is that Taylor (1993)-type policy rules involving some form of international monetary policy coordination contribute to decoupling the dynamics of the aggregates from those that characterize the dispersion across countries. This has an impact on the propagation of shocks across countries but also on macro volatility as well that varies with the openness of the economy. I …nd that generally the impact of globalization is underpredicted by standard measures of trade openness. In fact, I show that the e¤ects of the trade channel in the model do not depend solely on the extent of trade openness— but critically depend on the trade elasticity of substitution between locally-produced and imported goods too. The trade channel gives greater signi…cance to foreign developments on domestic macro aggregates than what standard trade openness measures would suggest given that demand shifts across countries— which depend on the trade elasticity of substitution— propagate the e¤ects of foreign shocks indirectly also through movements in international relative prices (real exchange rates, terms of trade). Furthermore, I illustrate in this paper some of the pros and cons of explicit agreements among central banks for the coordination of monetary policy and for the formation of a monetary union (Benigno (2004), Woodford (2010)). I show that a common monetary policy is an important benchmark— for international coordination, but also to understand how policy asymmetries across countries propagate and modify the equilibrium dynamics under a common monetary policy. In this paper I also show that deeper economic integration in the form of a monetary union has no bearing on the aggregate dynamics of the countries that adopt the common currency and common monetary policy, but may result in an indeterminate solution at the country-level unlike under international monetary policy coordination. The approach I pursue here is to inspect the mechanism of the workhorse open-economy New Keynesian model more closely by focusing on its three main building blocks in log-linear form: the open-economy Phillips curve, the open-economy dynamic IS curve, and the monetary policy rule (speci…ed in the form of conventional Taylor (1993)-type rules) for each country. The orthogonalization method I use to solve the model helps me analytically characterize the main macro variables of each country in terms of aggregate measures and the corresponding di¤erences between the two countries.2 This paper makes an important methodological contribution illustrating how to decompose the model solution whenever monetary policy rules di¤er across countries. My main contribution here is to make the theoretical case for why trade openness matters more than what we thought in the international transmission of shocks: 1. The model shows that the trade channel provides a plausible avenue to explain a number of stilldebated stylized facts in the international macro literature (such as the …ndings of Roberts (2006) and IMF WEO (2013) on the ‡attening of the Phillips curve or Ciccarelli and Mojon (2010), Duncan and 2 This technique is related to the work of Aoki (1981), Fukuda (1993), and Obstfeld and Rogo¤ (1996) aimed at solving multicountry models. More recent (and related) applications can be found in Benigno (2004), Kabukcuoglu and Martínez-García (2014), and Martínez-García (2015b), among others. 3 Martínez-García (2015), and Bianchi and Civelli (2015) on the dynamics of in‡ation). The international macro literature has intensely debated whether the Great Moderation was the result of good luck, good monetary policy internationally, or changes in the structure of the economy (Benati and Surico (2009), Woodford (2010)).3 This paper provides a novel analytical assessment of the trade channel and its signi…cance showing that changes resulting from greater trade integration can reduce volatility and in‡uence the trade-o¤s faced by policy-makers. It also alters the propagation of shocks and even the contribution of di¤erent shock types (productivity shocks, cost-push shocks, monetary policy shocks) to the business cycle. 2. The paper expands on the existing literature on the monetary policy transmission mechanism in an open economy setting (Benigno (2004), Woodford (2010)). I conclude that a common monetary policy (a form of monetary policy coordination) has the e¤ect of isolating the e¤ects of the trade channel to operate solely through the cross-country dispersion but not on the macro aggregates. Most notably, I show that generally the persistence of the macro variables of interest is largely una¤ected by either the features of the monetary policy rule or the strength of the trade channel. I also show that the establishment of a currency union as a means of deepening economic integration may in turn lead to indeterminacy. This is a novel insight that, to my knowledge, has not been documented before in the optimal currency area literature. Shocks are not measured directly, only their consequences. As Martínez-García and Wynne (2014) pointed out, the patient’s temperature might rise only slightly and brie‡y when sick if medication is used quickly and e¤ectively. The environment in which the patient is treated also in‡uences his condition and the resulting temperature spike. But in either case, the "shock" that caused the temperature to raise may be the same. Analogously, I argue in this paper that structural change— greater trade integration— as well as good monetary policy have e¤ectively altered the vulnerabilities to shocks of the economy. Hence, monetary policy and trade openness do in‡uence the adverse e¤ects of even large shocks and there implications appear largely consistent with the stylized facts of the Great Moderation. The rest of the paper is organized as follows. In Section 2, the log-linear approximation to the equilibrium conditions is discussed. Section 3 analyzes the monetary policy framework investigated in the paper and de…nes alternative monetary policy regimes resulting in the adoption of a common monetary policy— international monetary policy coordination— and in the formation of a monetary union. Section 4 characterizes the analytical solution of the linear rational expectations model using Taylor (1993)’s monetary policy set-up as a benchmark. It then o¤ers a detailed assessment of the policy trade-o¤s between slack and in‡ation and the implications for volatility amongst open economies— under independent (asymmetric) monetary policies, under common (coordinated) monetary policies, and within a currency area. Finally, Section 5 outlines some concluding remarks and possible extensions of this research agenda. The Appendix provides proof of some of the key results presented in the paper and a detailed derivation of the analytical solution of the model. It also includes a description of the building blocks of the two-country workhorse New Keynesian model.4 3 Good luck refers to the hypothesis that shocks hitting the economy during the Great Moderation were smaller than the large, adverse shocks of the preceding decades. 4 A companion (on-line) Technical Appendix is also available upon request with a detailed derivation of the approximated linear rational expectations model. 4 2 Log-Linear Equilibrium Dynamics I postulate a two-country dynamic stochastic general equilibrium (DSGE) model with complete asset markets and optimizing agents. I abstract from capital accumulation— considering linear-in-labor technologies. I also adopt a cashless economy speci…cation where money plays the sole role of unit of account (Woodford (2003), Chapter 2). There is a mass one of varieties produced in each country and all those varieties are traded between the two countries. Business cycle ‡uctuations are driven by country-speci…c productivity shocks, cost-push shocks, and monetary policy shocks. I assume the law of one price (LOOP) holds at the variety level as …rms price all their sales (domestic and foreign) in units of their local currency and quote them in the other country’s currency at the prevailing bilateral nominal exchange rate. The model features two standard distortions in the goods market that are characteristic in the open-economy New Keynesian literature (Martínez-García and Wynne (2010)), monopolistic competition in production and staggered price-setting behavior à la Calvo (1983) (nominal rigidities). Nominal rigidities preserve monetary policy neutrality in the long run while allowing a break from it in the short run— monetary policy has no real e¤ects in either the long run or the short run under perfect competition and ‡exible prices. The two-country New Keynesian model provides a tractable environment under monetary non-neutrality for the purpose of studying the role of monetary policy in the international propagation of shocks.5 I characterize a deterministic, zero-in‡ation steady state for the model, and log-linearize the equilibrium conditions around that steady state.6 I solve for the approximated linear rational expectations model assuming small ‡uctuations around the steady state driven by country-speci…c productivity shocks, cost-push shocks, and monetary policy shocks. The shocks are invariant to the speci…cation of the model. I denote gbt ln Gt ln G as the deviation of a variable in logs from its steady state. I use the superscript to distinguish variables (and parameters) that are speci…c to the Foreign country from those that correspond to the Home country. I identify the frictionless allocation by marking the corresponding variables with an upper bar. As shown in Table 1, the log-linearized equilibrium conditions can be summarized with an open-economy Phillips curve, an open-economy dynamic investment-savings (IS) equation, and a Taylor (1993) rule for monetary policy in each country. In this sense, the open-economy New Keynesian model is a straightforward extension of the standard three-equation (closed-economy) New Keynesian model. The system of equations in Table 1 pins down Home and Foreign CPI in‡ation (quarter-over-quarter changes), bt and bt , Home and Foreign slack (deviations of output from potential absent all frictions), x bt b b and x bt , and Home and Foreign short-term nominal interest rates, it and it . Table 1 also includes a standard de…nition relating output in each country, ybt and ybt , to the country’s corresponding output potential plus b b slack— Home and Foreign output can be expressed as ybt yt + x bt and ybt yt + x bt , respectively. The description of the model in Table 1 is completed with a pair of Fisherian equations for the real interest rates in the Home and Foreign countries de…ned as rbt bit Et [bt+1 ] and rbt bit Et bt+1 , respectively. The natural (real) rates of interest that prevail absent all frictions in the model for the Home and Foreign countries are denoted b rt and b rt . The natural rates are a function of Home and Foreign potential output 5 This framework can be generalized to include backward-looking terms as well. For a method to solve linear rational expectations models with backward-looking and forward-looking terms, see Martínez-García (2016). 6 The steady state of the model with nominal rigidities and monopolistic competition is the same as that of the frictionless model (under perfect competition and ‡exible prices). Asymmetries in the policy rule across countries do not a¤ect the steady state of the model that remains una¤ected by the policy parameters and otherwise symmetric. 5 growth— where Home and Foreign potential output, b y t and b y t , depend exclusively on the Home and Foreign productivity shocks, b at and b at , respectively. Table 1 - Open-Economy New Keynesian Model Home Country bt NKPC x bt ) bit Monetary policy rbt Fisher equation Output decomposition Natural interest rate Potential output bt NKPC Dynamic IS Monetary policy Fisher equation Potential output (1 ) h )u bt i+ u bt rbt b rt + (1 [ b at + (1 )b at ] bt + (1 h vbt = u )u bt i ) rbt b rt + b yt h rbt bt bt + m bt + x x bit Et bt+1 b bt h ybt =i y t + x i h b ) Et b y t+1 yt + y t+1 Et b rbt 1+' +' )b at + b at ] [(1 i b yt i )x bt + x bt + vbt ] (' + ) [(1 (1 b rt h i ) Et b y t+1 b y t + (1 Foreign Country x bt bit h ) rbt bt + x x bt + m bt bit Et [bt+1 ] ybt = b yt + x bt 1+' +' Et bt+1 + h (1 b rt (1 b yt Et x bt+1 Output decomposition Natural interest rate i h Et b y t+1 h b rt )x bt + vbt ] vbt = (1h (Et [b xt+1 ] Dynamic IS (' + ) [ x bt + (1 Et (bt+1 ) + b rt i b yt i Composite Parameters (1 h ) 1 ( ( ( 1) 1)(1 2 ) 1)(1 2 )2 (1 1+ 1 2 )(1 i'+ ) ) = (1 1+( h ) 1 2 (1 1 2 ( '+ )( 1+(1 '+ )( ; (2 )(1 2 ) 1)(2 )(2(1 )) 1+( 1)(2 ) 1+( 1)(2 )(2(1 ) 1)(2 )(2(1 ; )) 1)(2 )(2(1 )) i ; )) i ; : Apart from productivity shocks, the model includes two other country-speci…c exogenous shocks: costpush shocks, u bt and u bt , and monetary policy shocks, m b t and m b t . As indicated in Table 2, all shocks follow bivariate VAR(1) stochastic processes but only productivity shocks incorporate spillovers explicitly.7 Shock 7 Productivity shock spillovers capture technological di¤usion across countries. In turn, di¤usion does not appear so signi…cant for either monetary policy shocks or cost-push shocks. 6 innovations can be correlated across countries, but not across the three di¤erent shock types. 2.1 Table 2 - Country-Speci…c, Exogenous Shocks ! ! ! ! b at b at 1 b "at a a;a + b at ! b at 1 b "at!! a;a !a Productivity shock a 2 2 b "t 0 a;a a a N ; a 2 2 b "t ! 0 ! a;a a! a ! b "ut 0 u bt 1 u bt u + b "ut !! 0 u bt 1 u bt ! u ! Cost-push shock u 2 2 0 b "t u;u u u N ; u 2 2 0 b "t ! u ! u;u u ! ! m bt 0 m bt 1 b "m m t + 0 !m m b t! m bt 1 b "m t !! Monetary shock 2 2 b "m 0 m;m m m t N ; 2 2 b "m 0 m;m m m t The Frictionless Dynamics The dynamics of the frictionless environment with perfect competition and ‡exible prices and wages are rt and b rt , and potential output, b y t and b y t , as shown in Table 1. summarized by each country’s natural rate, b The labor market imperfections that motivate the cost-push shocks, the exogenous marginal cost shifters u bt and u bt , are also absent in the frictionless environment. With ‡exible prices and wages, monetary neutrality holds in the short run as well as in the long run and therefore neither monetary policy shocks nor the monetary policy rule a¤ect the frictionless allocation either. Hence, the natural rate and output potential of each country are solely driven by Home and Foreign productivity shocks, b at and b at , respectively.8 The Natural Rates. I …nd that the composite coe¢ cient domestic potential growth to the domestic natural rate is that determines the contribution of expected = 1 assuming households only include locally- produced goods in their consumption basket ( = 0). Not surprisingly, when imports are not valued by households and there is no role for trade, only domestic potential output growth determines the domestic natural rate. In turn, when there is room for trade but no local-production bias ( = expected domestic potential growth in the domestic natural rate is = 1 2 1 2 ), the weight on re‡ecting the share in production of both countries (production of varieties is equally distributed across countries). More generally, the weight on expected domestic potential growth satis…es that 8 > < > : 1 2 1 2 < < (1 ) < 1 if = (1 ) if = 1; < (1 )< if 0 < (1 ) 1+( 1+( 1)(2 ) 1)(2 )(2(1 )) > 1; (1) < 1; h i ) for any degree of openness 0 < < 21 . To prove the result in (1), I …rst note that 1+( 1+( 1)(2 1)(2 )(2(1 )) > 0 h i ) for all > 0 and 1+( 1+( 1)(2 1)(2 T 1. Then, the fact that > 21 if > 1 follows naturally )(2(1 )) S 1 if 8 Productivity shocks enter into the dynamics of the model only through their impact on the Home and Foreign natural rates, b rt and b rt , and the Home and Foreign output potential, b y t and b y t , as indicated in Table 1. 7 from the de…nition of the composite coe¢ cient whenever > 1 2 1+(1 2 ) and 0 < 1 2 1+(1 2 ) . In turn, if 0 < < 1, I …nd that < 1 holds only < 1. The interpretation of (1) is that the weight on expected domestic potential growth can be lower than what the domestic consumption share (1 ) alone would imply and concurrently the weight on expected foreign potential growth can be larger than what the import share would entail only whenever > 1. This re‡ects that the domestic natural rate captures not just the domestic growth potential but also the aggregate demand shifts across countries prompted by concurrent changes in terms of trade. Whenever the trade elasticity 0< < satis…es that 0 < 1 2 1+(1 2 ) < 1, the e¤ects are attenuated instead and in some instances (if < 1) they can even lead to a reversal whereby > 1. The Potential Output. The composite coe¢ cient weighting domestic productivity on domestic poten1) ( '+ )( tial output satis…es that = 1 if = 0 and = 1 + 21 if = 12 . Hence, only domestic 1) ( '+ )( productivity shocks enter into the speci…cation of domestic potential whenever the trade channel is shut down ( = 0). The weight on foreign productivity (1 ) and its sign when = 1 2 depend on: the preference ratio 0 < '+ < 1 which is a function of the inverse of the Frisch elasticity of labor supply ' > 0 and of the inverse of the intertemporal elasticity of substitution > 0; and the product > 0 which is related to the trade elasticity of substitution between Home and Foreign goods > 0. More generally, the weight on domestic productivity shocks satis…es that 8 > < for any given degree of openness 0 < > : > 1 if > 1; = 1 if = 1; 1 2 < < 1 2. domestic potential output must satisfy that 1 1+ 1 2 ( '+ )( 1+(1 '+ 1)(2 )(2(1 )( )) 1)(2 )(2(1 )) (2) < 1 if 0 < < 1; Analogously, the weight on foreign productivity shocks in S 0 if T 1. This implies that an open economy displays a higher positive impact of domestic productivity on domestic potential than under the closed-economy speci…cation (and a negative e¤ect of foreign productivity on domestic potential) only if the trade elasticity satis…es that > 1. The contribution of Home and Foreign productivity shocks is seen to depend on the preference ratio '+ Whenever as well, which describes the characteristics of the labor market response. > 1, the income e¤ect on domestic production from a foreign productivity shock is out- weighed by the substitution e¤ect that shifts aggregate demand towards the relatively cheaper foreign goods and drags domestic labor and production down. I adopt the case where > 1 as the economically-relevant benchmark to match the expected signs in the transmission across countries of productivity shocks. In the special case where = 1, I …nd that domestic potential output is fully insulated from foreign productivity shocks through trade ( of openness = 1) and identical to its closed-economy counterpart irrespective of the degree (Cole and Obstfeld (1991)).9 Whenever 0 < < 1, a positive foreign productivity shock drives domestic potential output up while positive domestic productivity shocks have an attenuated e¤ect on domestic potential which is lower than in the closed-economy case. 9 In the special case where = 1, full insulation from foreign shocks can be achieved through ‡uctuations in international relative prices (terms of trade) alone irrespective of the assumptions made on the international asset market structure. 8 2.2 The Open-Economy Phillips Curve (NKPC) An important takeaway from the open-economy Phillips curve in Table 1 is that foreign slack— not just domestic slack— plays a central role in modelling domestic in‡ation. Unlike in the closed-economy model, the domestic economy moving above its potential does not necessarily lead to higher domestic marginal costs and in‡ation for open economies whenever there is growing slack elsewhere— as this weighs down on imported goods in‡ation and shifts aggregate demand away from domestic goods through movements in the terms of trade. Therefore, the key insight from the open-economy model is that both Home and Foreign slack (not just domestic slack) help gauge domestic in‡ation. The slope of the Phillips curve in the closed-economy case is given by (1 (' + ) where the closed-economy slope is a function of the Calvo (1983) price stickiness parameter ral discount factor )(1 ) — , the intertempo- , the inverse of the intertemporal elasticity of substitution , and the inverse of the Frisch elasticity of labor supply '. The slope of the domestic open-economy Phillips curve on domestic slack can be writtenh as (' + ) — which is a function of the i closed-economy slope and the composite (2 )(1 2 ) weight (1 ) 1 ( 1) '+ . Analogously, the slope of the domestic 1+( 1)(2 )(2(1 )) open-economy Phillips curve on foreign slack can be expressed as (' + ) (1 ). The sum of the slopes on domestic and foreign slack in the open-economy case equals the closed-economy slope. In the case where there is no endogenous trade ( = 0), I recover the standard closed-economy Phillips curve speci…cation with bias ( = 1 2 ), = 1 which depends solely on domestic slack. Abstracting from local-production I obtain equal weights on the slack of both countries where = 1 2 consistent with the equal shares both countries have in production. More generally, it follows that the weight coe¢ cient that 8 > < for any degree of openness 0 < It can be seen that 1+( > : 1 2 1 2 < < (1 ) < 1 if = (1 ) if = 1; < (1 )< if 0 < satis…es > 1; (3) < 1; < 21 . (2 )(1 2 ) 1)(2 )(2(1 )) Hence, it follows from the de…nition of > 0 for any degree of openness 0 < that S (1 < 1 2 and for all > 0. T 1. In (3), there is a lower bound on ) if how low the slope of the open-economy Phillips curve on domestic slack can go whenever > 1 given by the equal shares in production of each country. To show > h this, notice that in the case where i 1 ( 1)(2 )(1 2 ) 1 1+( > the weight on domestic slack satis…es that = (1 ) 1 + 1 '+ 1)(2 )(2(1 )) i h ( 1)(2 )(1 2 ) (1 ) 1 for any degree of trade openness 0 < < 21 given that 0 < '+ < 1. 1+( 1)(2 )(2(1 )) h i ( 1)(2 )(1 2 ) ) Then, straightforward algebra shows that (1 ) 1 = (1 ) 1+( 1+( 1)(2 1)(2 1+( 1)(2 )(2(1 )) )(2(1 )) > 1 2. (1 By construction, therefore, it holds from here that the weight on domestic slack satis…es that ) and the weight on foreign slack satis…es that <1 < 1 2 whenever 1 0 Here, 1 2 + '+ I …nd that ! (1 2 ) 1 2(1 > 0) whenever < 1 requires that ) < 1 only if '+ > 1 > 2 + '+ 1 (2 )(2(1 (1 2 )(2(1 9 )) )) (1 2 ) where 0 < ! 1 2(1 2 +( '+ ) 1 (2 )(2(1 (1 2 )(2(1 < < 1) while the slope on 1 > max 0; 1 1 < > 1. The slope on domestic slack can be ‡atter than the closed-economy slope (i.e., foreign slack can have a positive sign (i.e., 1 1 2 . )(1 2 ) 1 2(1 ) .10 Given this, I see that 0 < )) )) < 1 . 2 As a result, whenever These …ndings also imply that the domestic slope of the open-economy Phillips curve must be ‡atter than the closed-economy slope and strictly more so than what the domestic share in consumption (1 entail only in the benchmark case where ) would > 1. Naturally, the slope on foreign slack is signi…cantly higher than what the import share ( ) alone would warrant in that case. Whenever is determined by the domestic share alone ( = (1 )). In turn, if 0 < = 1, the domestic slope < 1, the ‡attening of the open-economy Phillips curve is instead attenuated and can even be reversed in some cases. Therefore, the two-country New Keynesian model o¤ers additional economic insight connecting trade openness ( ) to the so-called ‡attening of the Phillips documented in the empirical literature (Roberts (2006), IMF WEO (2013)). I want to highlight here that the ‡attening of the domestic slope of the openeconomy Phillips curve depends not only on the strength of the trade channel, but also on the preference ratio 0 < '+ < 1 and therefore on features of the labor market. The higher the ratio composite coe¢ cient 2.3 will fall below (1 '+ is, the lower the ). The Open-Economy IS Equation The open-economy dynamic IS equations in Table 1 show that slack in each country is tied to developments in both Home and Foreign aggregate demand. More speci…cally, to the wedge between the actual real interest rate— the opportunity cost of consumption today versus consumption tomorrow (b rt and rbt )— and the natural b b rate that would prevail in the frictionless equilibrium (rt and rt ). Abstracting from local-production bias in consumption ( = 12 ), the open-economy dynamic IS equation in both countries can be rewritten in terms of the local interest rate gap alone as real interest rate deviations from the natural rate must equalize across b countries (i.e., rbt b rt rbt rt ). The natural rates of both countries equalize across countries in this rt case (i.e., b b rt )— hence, real interest rate equalization across countries (b rt local-production bias. counterparts with 11 rbt ) occurs if there is no The open-economy dynamic IS equations naturally reduce to their closed-economy = 1 whenever the import share is set to zero ( = 0). From the de…nition of the slope of the domestic IS equation on domestic interest rate deviations (1 ) 1 2 (1 1 2 ) , it follows > (1 1 2 (1 1 2 ) given that ) > 1 for all > 0 and 0 < < 1 2. More generally, I …nd that 8 < > 1 > (1 : 1> for any degree of openness 0 < > (1 ) ; if > ) ; if 0 < 1 2 1+(1 2 ) > 1 2 < 1+(1 2 ) 0; (4) ; < 21 . In other words, an open economy has a slope on domestic interest rate deviations that is larger than in the closed-economy case— and concurrently a negative slope on foreign interest rate deviations— whenever > 1 2 1+(1 2 ) the trade elasticity is su¢ ciently low (0 < < where 0 < 1 2 1+(1 2 ) 1 2 1+(1 2 ) < 12 . In turn, for cases in which ), a reversal is possible whereby the slope on domestic interest rate deviations is lower than in the closed-economy case and the slope on foreign interest rate deviations becomes positive. the preference ratio 0 < '+ < 1 is large enough such that the trade elasticity given by 0 < 1 1 Di¤erences <1 1 2 + '+ (1 2 ) '+ ! 1 2(1 > ) 1 (2 )(2(1 (1 2 )(2(1 )) , )) there is a non-empty range of values for for which the model produces > 1. in the consumption baskets imply that each country’s consumption demand responds di¤erently to countryspeci…c shocks, which is re‡ected in the cross-country di¤erences in natural rates of interest. 10 Aggregate demand responds to deviations of each country’s real interest rate from its own natural rate as those deviations shift aggregate consumption across time. Whenever the real interest rate is above its natural rate, more consumption today is being postponed for consumption tomorrow than would otherwise occur in the frictionless environment. Ceteris paribus, this implies a demand shortfall today (a fall in output relative to potential) and the expectation of slack unwinding in the future. Analogously, when the real rate is below the natural rate, the resulting boost in consumption today (at the expense of future consumption) leads to a temporary increase in the output gap that is nonetheless expected to dissipate over time. Hence, an open economy has to grapple with a steeper dynamic IS curve on domestic real interest rate deviations from the domestic natural rate— not just with a ‡atter open-economy Phillips curve. The slope of the open-economy IS curve depends on how open the economy is to trade ( ) and on the trade elasticity ( ). The intuition for this result is straightforward: A given rise in the real interest rate impacts aggregate demand through two channels, domestic consumption and trade. A shock that is met by a rise in the ex-ante real interest rate drags domestic consumption but also induces a terms of trade deterioration that erodes the foreign demand of domestic goods, lowering domestic output relative to its potential further. However, features of the labor market such as the Frisch elasticity of labor supply do not enter into the slope of the IS curve unlike what happens with the Phillips curve slope. 2.4 The Stochastic Processes I characterize the dynamics of potential output and of the natural rate of interest based on the frictionless allocation described in Table 1 and the shock processes in Table 2. The potential output of the Home and Foreign countries, b y t and b y t , are de…ned as a convex combination of the Home and Foreign productivity shocks, b at and b at . As shown in the Appendix, given the VAR(1) structure of the productivity shocks, the following bivariate VAR(1) stochastic process characterizes the dynamics of potential output ! b yt b yt ! b "yt b "yt where 2 y a a;a a;a a 0 N 0 2 a = a;a ; ! 1 1 + 1 y;y y;y 1 2 y 2 ( ) +2 2 y;y b yt b yt b "yt b "yt !! ! ; (5) ; (6) 2 1+' +' = ! ! ( ) + 2 (1 2 ( ) +2 )+ (1 a;a a;a (1 a;a ) + (1 (1 ) 2 ) + (1 ) ) 2 ; (7) 2 ; (8) de…ne the volatility and the correlation of the potential output innovations. In the closed-economy case ( = 0), output potential would be of = 2 a 2 in (2), it follows then that ( ) + 2 2 and 2 y;closed = 1 so the volatility and correlations for the innovations on 1+' +' and (1 y;y ;closed ) + (1 = ) 2 a;a , respectively. Given the de…nition 2 > ( ) + 2 (1 ) + (1 ) 2 =1 2 ( ) +2 (1 )+ a;a (1 ) ( )2 +2 a;a (1 )+(1 )2 a;a a;a 2 < a;a whenever 11 > 1 and 0 < a;a < 1. More generally, I can conclude that open economies (0 < across countries (0 < 1 2) < where the productivity innovations are positively correlated T 2 y < 1) satisfy that a;a 2 y;closed and S y;y T1( if y;y ;closed > 0). In other words, open economies display higher volatility of potential output innovations and lower correlations > 1.12 than their closed-economy counterparts in the benchmark case where Analogously, a simple characterization of the natural rates in the Home and Foreign countries, b rt and b rt , can be derived from the bivariate stochastic VAR(1) process for the productivity shocks, b at and b at . As seen in the Appendix, the following VAR(1) stochastic process characterizes the dynamics of the natural rates b rt b rt b "rt b "rt where 2 r r;r ! ! a a;a a;a a N = 2 2 a = a;a ( 0 1+' +' ( 1) 2 1) ! 0 2 b rt b rt ! 1 1 + 1 r;r r;r 1 2 r ! b "rt b "rt !! ; (9) ; (10) 2 2 1) +2 2 + a;a 1 2 +( ( +2 +2 ; ! 1 a;a 1 a;a ( 2) 2) 2 2 2 2) +( ; (11) 2 ; (12) and 1 2 a;a ( a (1 1) + (1 0 )@ 1+ 1 1+ 1 0 )@ ( '+ 1) (2 ) 1) (2 ) (2 (1 1+ 1 1+ 1 ( '+ ( '+ ( '+ )) 1) (2 ) 1 A( 1) (2 ) (2 (1 )) +1 a;a 1 A( a) ; +1 a;a (13) a) ; (14) de…ne the volatility and the correlation of the corresponding natural rate innovations. Home and Foreign potential output— as well as their corresponding natural rates— inherit the VAR(1) stochastic structure and some of basic features of the productivity shock process— in particular, the persistence and spillovers of the productivity shocks. In turn, the deep structural parameters of the model— including those tied to the trade channel: degree of openness ( ) and trade elasticity ( )— enter into the variance-covariance matrix. Here the closed-economy case ( = 0) implies that closed-economy natural rate innovations is 2 r;closed ( 1 = while the corresponding cross-country correlation is given by For any degree of openness 0 < 1 2, 1 = (1 ) a;a 1) and a 1+' +' 2 2 a + 2 ( a r;r ;closed ( a 2 a;a , so the volatility of the 2 = 1) +2 a;a ( 1) and ( a a;a ( 1) a 1)2 +2( a 2 1) +2 a 2 = a;a a;a ( 1) a a;a + (1 2 a;a ) 2 ( ) +( a;a 2 a;a )( a ) 1) + a;a a;a a;a 1) +( are linear combinations of the parameters that describe the persistence of the bivariate VAR(1) process 1+(1 '+ )( 1)(2 ) for productivity, a;a and ( a 1), and 21 < (1 ) 1+ 1 < 1. To ensure 1)(2 )(2(1 )) ( '+ )( 1 2 The preference ratio '+ plays a major role in lowering the correlation and increasing the volatility of the output potential for open economies as '+ % 1. Simulations and further details on this point are available from the author upon request. 12 . stationarity of the bivariate productivity shock process, I require a;a + ( a 1) < 0 which in turn implies that 1 + 2 = a;a + ( a 1) < 0. Hence, I derive the following expression13 2 r = 2 r;closed +2 1 a;a " 1+' +' 2 2 (1 ) # 2 a ( ( a;a 2 a 1)) ; (15) which, given that 0 < a;a < 1, implies that 2r < 2r;closed for any open economy. This indicates that the volatility of the natural rate innovations is lower in an open economy than it would be in the closed-economy case for any plausible parameterization of the trade elasticity. Furthermore, it follows from the de…nition of the cross-country correlation of the natural rate innovations that r;r < a;a if and only if the condition 1 2 < 0 holds. The expression for the cross-correlation is as 2 2 2 +2(1 a;a )(1 ) ( a;a ( a 1)) a;a ( a 1) +2( a 1) a;a + a;a ( a;a ) , which implies that follows: r;r = 2 2 2(1 a;a )(1 ( a 1)) ( a 1)2 +2 a;a ( a 1) a;a +( a;a ) ) ( a;a = r;r 1 2 )(1 ) ( a;a ( a 1)) 2 (( a 1)+ a;a ) 2(1 a;a )( a 1) a;a : 2 ) ( a;a ( a 1)) 2(1 a;a )(1 2 (( a 1)+ a;a ) 2(1 a;a )( a 1) a;a r;r ;closed 2(1 + a;a (16) It can be shown that r;r > r;r ;closed so long as the cross-correlation of the productivity innovations 2 ( a 1) a;a < 12 (( a 1) + a;a ) . Given that a;a + ( a 1) < 0, then if a;a satis…es that 1 a;a 1) S 0. As a result, it follows that ( a 1) a;a < 0 and accordingly a;a T 0 it must be the case that ( a the inequality on a;a is satis…ed for any value 0 < a;a < 1. In other words, it holds that r;r > r;r ;closed and this shows that innovations to the natural rate are more highly correlated for open economies than for closed economies for any plausible value of the trade elasticity.14 The trade-weighted de…nition of the cost-push shocks in Table 1, vbt (1 )u bt + u bt and vbt u bt + (1 )u bt , is a convex combination of the country-speci…c cost-push shocks, u bt and u bt , which depends on the degree of openness . Given the VAR(1) structure of u bt and u bt , the following bivariate VAR(1) stochastic process— as seen in the Appendix— characterizes the dynamics of the trade-weighted cost-push shocks ! vbt vbt 1 3 After some algebra, 2 it b "vt b "vt ! is u 0 0 u 0 0 N possible to ! ! ; show 2 v vbt vbt ! 1 1 + ! b "vt ; b "vt !! 1 v;v v;v 1 here that (17) ; ( 1) 2 (18) +2 2 a;a 1 2 +( 2 2) = + 2 a;a 1 ( a 1) a;a + 2 a;a 1 (1 ) ( a 1) . a;a ratio '+ plays a critical role in increasing the cross-correlation while further reducing the volatility of the natural rate for open economies relative to their closed-economy counterparts as '+ & 0. Simulations and further details on the persistence of the natural rate are available from the author upon request. 1) a;a + ( a 1 4 The preference 13 where 2 v v;v = 2 u = u;u 2 (1 ) +2 u;u 2 (1 ) + 2 (1 2 (1 (1 ) +2 ) + (1 u;u 2 ) + ; (19) ; (20) 2 u;u 2 ) + de…ne the volatility and the correlation of the trade-weighted cost-push shock innovations. Here, the closedeconomy counterpart ( = 0) implies that 2 v;closed = 2 u and v;v ;closed = u;u . Finally, the Home and Foreign monetary shock processes, m b t and m b t , in Table 2 do not require any transformation as they enter directly into the linear rational expectations model through the speci…cation of the monetary policy rule of each country. 3 Monetary Policy Framework The Home and Foreign monetary policy rules close the model speci…cation, de…ning a particular monetary policy regime and playing a crucial role in the international transmission of shocks. In Table 1, monetary policy is modelled with a Taylor (1993)-type rule reacting to local conditions as given by each country’s in‡ation and output gap alone when set independently (and asymmetrically). The persistence in policy rates arises from the policy shocks re‡ecting inertia that is extrinsic or exogenous to the policymaking process and out of the policymakers’control. The policy rules can be rewritten in terms of aggregate variables and cross-country di¤erences taking into account that the policy responses can vary across countries. 1 I de…ne aggregate variables in this two-country setting generically as gbtW bt + 12 gbt using production 2g weights and label the di¤erences between the two countries as gbtR gbt and Foreign variables, gbt and gbt , respectively, can be decomposed as 1 gbt = gbtW + gbtR ; gbt = gbtW 2 gbt . It follows that any pair of Home 1 R gb ; 2 t (21) where the superscript identi…es the aggregates (W ) and the di¤erences (R). Di¤erences across countries can gbtW = 12 gbtR and gbt also be expressed in deviations from the aggregates, i.e., gbt dynamics for gbtW and gbtR , gbtW = 1 R bt . 2g Given the the transformation in (21) backs out the corresponding variables for each country, gbt and gbt . With this notation, I can cast the Home and Foreign monetary policy rules in Table 1 in the following canonical form biW t biR t ! W ;W R ;W W x;W R x;W ! bW t x bW t ! + W ;R R ;R W x;R R x;R ! bR t x bR t ! m bW t + m bR t ! ; (22) W bR where biW t is the aggregate short-term nominal interest rate (it di¤erential nominal interest rate ), b t is global in‡ation (bR bW xR bW t di¤erential in‡ation), and x t is the global output gap (b t di¤erential slack). Here, m t is the aggregate monetary policy shock (m bR t is the di¤erential monetary policy shock). I de…ne the aggregate coe¢ cients on monetary policy as di¤erential coe¢ cients as R and R x x x. W + 2 and W x x+ 2 x , and the Independently-set and potentially asymmetric monetary policy rules can then be de…ned in relation to (22). With the de…nitions of the transformed policy 14 coe¢ cients, I can describe any pair of Home and Foreign monetary policy rules as those in Table 1 within the framework given by (22) as follows, De…nition 1 A monetary policy equilibrium with independent monetary policy rules responding to local conditions with varying sensitivities on their policy objectives— i.e., where W ;W be represented in the form of (22) with the following coe¢ cients: R ;W =4 W ;R = R , and R x;W =4 W x;R = = R ;R 6= = and/or W , x W x;W = 6= x— can = W x , R x;R R x. Hence, the Taylor (1993) rules for the aggregate and di¤erence sub-systems whenever monetary policies are set independently across countries can be summarized simply as biW t biR t ! W R W x R x ! bW t x bW t ! + R R x 4 W 4 W x ! bR t x bR t ! + m bW t m bR t ! : (23) I can then consider the case where there is international monetary policy coordination among the two countries which can also be de…ned in relation to the framework in (22) as follows, De…nition 2 A coordinated monetary policy equilibrium is characterized by a common monetary policy rule responding to local conditions only that is followed by both countries. Therefore, international monetary policy coordination requires common Taylor (1993) rule coe¢ cients in both countries— i.e., x = x = as well as that W = R W c ;R = ;W = x . Hence, in the context of (22), this implies that W W R R = 0. In this case the aggregate and = = = x;R ;R x;W ;W R R c W c = x = 0. and x = x while W and = W x;W = = c R x;R = and W x di¤ erence coe¢ cients satisfy Under international monetary policy coordination, the interaction terms drop out from the equation in (22) so that the aggregate policy equation depends only on aggregate variables while the di¤erence equation depends only on di¤erence variables. Hence, the Taylor (1993) rules for the aggregate and di¤erence subsystems for the coordinated monetary policy equilibrium can be summarized in the following terms bic;W t bic;R t ! c 0 c x 0 ! bc;W t x bc;W t ! + 0 c 0 c x ! bc;R t x bc;R t ! + m bW t m bR t ! : (24) Naturally, (24) is a special case of (22) where monetary policy is coordinated and symmetric across countries. The superscript c is used to denote the coordinated monetary policy equilibrium. International monetary policy coordination does not necessarily imply that the short-term nominal interest rates equalize across countries. Coordination is a step in the direction of achieving greater monetary integration but does not establish the same level of policy integration as a monetary (or currency) union. A monetary union has two distinct features: First, a currency union implies that the ‡exible nominal exchange rate is one at each point in time— so the unit of account is the same in both countries. Second, the common monetary policy responds to aggregate economic conditions rather than to the local conditions in each country. Implicit in the two-country model with complete international asset markets is the fact that the uncovered interest rate parity (UIP) condition must hold up to a …rst-order approximation. Hence, given that the UIP condition holds here, setting the nominal exchange rate to be one in every period is equivalent to imposing that the short-term nominal interest rates must be equalized across countries in equilibrium (i.e., bit = bit ). 15 This implies that the common monetary authority for the union is left with one instrument only for the conduct of monetary policy. Moreover, the common monetary policy set by the authorities responds to aggregate rather than local developments. Therefore, a monetary union can be de…ned in relation to (22) as, De…nition 3 A monetary or currency union equilibrium is characterized by a common monetary policy rule applied in both countries where the policy rate responds to the economic conditions of the union (rather than to local developments)— i.e., W ;W = W = c and W x;W W x = = c x. As in the case with monetary policy coordination, there are no interactions between the aggregate and di¤ erential policy equations either— i.e., R ;W = R x;W = W ;R = W x;R = 0. However, interest rate equalization across countries imposes a non-trivial departure from the coordination case shown in (24) as it requires— in the notation of (22)— that and R x;R = W x . R ;R = W Moreover, it also assumes that m bR t = 0. Hence, the Taylor (1993) rules for the aggregate and di¤erence sub-systems for the monetary union equilibrium can be summarized as follows, biu;W t biu;R t ! c 0 c x 0 ! m bR t = 0: bu;W t x bu;W t ! + 0 0 0 0 ! bu;R t x bu;R t ! + m bW t m bR t ! ; (25) The superscript u is used to denote the monetary union equilibrium. In this context, forming a monetary union does not change the perception of aggregate monetary policy relative to the international monetary policy coordination case. However, forming a monetary union has signi…cant implications for the dynamics of the two-country model because— unlike under international monetary coordination— it requires: (a) the responses on the policy di¤erence equation to di¤er from those of the aggregate policy equation; and (b) it 15 also imposes that m bt = m bt = m bW t . This then ensures interest rate equalization across countries. 4 Inspecting the Monetary Policy Mechanism I orthogonalize the linear rational expectations system described in Table 1 and Table 2 to re-express it as two separate and smaller sub-systems for aggregates and for di¤erences between Home and Foreign variables using the corresponding de…nitions introduced in Section 3. This orthogonalization approach focuses our analysis of monetary policy across countries on its impact in the aggregate variables and on the crosscountry dispersion. The Taylor (1993) rules for the aggregate and di¤erence sub-systems can be re-written in canonical form as shown in (22) (25). The NKPC equations for the aggregate and di¤erence sub-systems can be cast into the following form 1 5 The bst = Et bst+1 + (' + ) [ s s x bt + vbts ] ; for s = W; R; (26) requirement that m bt = m bt = m bW t is equivalent to saying that country-speci…c monetary policy shocks are perfectly u correlated (i.e., m;m 1). To ensure that this perfectly correlated monetary shock has the same volatility as the aggregate of the country speci…c shocks presented in Table 2, I need to scale the volatility accordingly to be turn, the persistence of the monetary shock m remains unchanged. 16 2 m;u 2 m 1+ m;m 2 . In where Et (:) are expectations formed conditional on information up to time t, and vbtW = u bW t is the global costpush shock (b vtR = (1 2 )u bR t di¤erential cost-push shock). Furthermore, R slope on the global slack and 0 < (2 1 holds for any degree of openness 0 < the results on the composite coe¢ cient trade elasticity 1 is the composite for the 1) < 1 is the slope on di¤erential slack. I observe that 0 < 1 2 < and for all > max 0; 1 1 2 +( '+ summarized in (3). In turn, whenever is low enough that it lies within the non-empty range 0 < then it holds instead that W R > 1 and accordingly that )(1 2 ) given )) )) , if the 1 <1 2 +( '+ )(1 < ) 1 (2 )(2(1 (1 2 )(2(1 > '+ 1 2(1 R 2 ) 1 2(1 ) > 1. The dynamic IS equations for the aggregate and di¤erence sub-systems are given by Et x bst+1 x bst = s bist s b rt ; for s = W; R; Et bst+1 (27) where the Fisher equations help express the aggregate real interest rate rbtW (di¤erential real rate rbtR ) in terms bW of the aggregate short-term interest rate biR t ) net of expected i inominal interest rate it (di¤erential h nominal h W R W b aggregate in‡ation Et bt+1 (expected di¤erential in‡ation Et bt+1 ). Here, rt is the global natural rate R (b rt di¤erential natural rate). Furthermore, deviations from the natural rate and R I observe that R (2 1 is the slope on the aggregate real interest rate in 1) > 1 is the slope on the di¤erential real interest rate gap. > 1 holds for any degree of openness 0 < the results for the composite coe¢ cient 4.1 W < 1 2 and for all summarized in (4). In turn, 0 < R > 1 2 1+(1 2 ) < 1 if 0 < < > 0 given 1 2 1+(1 2 ) . Independent Monetary Policies The interactions among aggregate and di¤erence variables through the policy rules in (22) imply that the aggregate and di¤erence sub-systems given by (26) (27) cannot be solved separately from each other whenever monetary policy is asymmetric across countries. However, the model can still be solved in deviations from the coordinated (common) monetary policy equilibrium. To establish this, I de…ne generically any variable gbts for s = W; R as gbts = gbts;c + gbts;d where gbts;c corresponds to the solution under the coordinated (common) monetary policy equilibrium and gbts;d gbts gbts;c is the solution of the model with asymmetric monetary policy across countries in deviations from the coordinated monetary policy equilibrium. The superscript d refers to the deviations from the coordinated policy equilibrium under asymmetric monetary policy, while the superscript c denotes the coordinated monetary policy equilibrium as before. Using (24) together with the expectational equations in (26) (27) for s = W; R, I can separately write the sub-systems for aggregates and for the cross-country di¤erences under a coordinated (common) monetary policy as follows x bs;c t bs;c t bis;c t = ! s 1 = (' + ) c x c + s s 0 (' + ) s s x bs;c t bs;c t (' + ) ! s (' + ) s ! ! vbts ; s b rt +m b st ; for s = W; R; 17 ! Et x bs;c t+1 Et bs;c t+1 ! s s (' + ) s ! bis;c t + ::: (28) (29) s;c bs;c where the vector of endogenous variables is x bs;c for s = W; R. t ; b t ; it The coordinated monetary policy coe¢ cients are the same as the weighted aggregates of each country’s c policy coe¢ cients under independent monetary policies— i.e., = W c x and = W x . Hence, taking the di¤erence between (23) and (24), I obtain that bid;W t bid;R t ! W ! W x R x R bd;W t x bd;W t ! R + 4 W R x 4 W x ! bd;R t x bd;R t ! R 4 + bc;R + t R c;W bt + R x x bc;R t 4 R c;W bt xx ! : (30) In turn, the equations that describe the aggregate and di¤erence sub-systems in (28) are exactly the same for the coordinated monetary policy equilibrium and the asymmetric monetary policy equilibrium because: (a) the policy coe¢ cients do not enter into the composite coe¢ cients on the structural relations given by the model— the Phillips curve and IS equations— and (b) all other deep structural parameters of the model are common across countries. Therefore, I can derive the following representation in deviations taking the di¤erence between the corresponding equations with asymmetric policy coe¢ cients and with common policy coe¢ cients, x bs;d t bs;d t ! = s 1 (' + ) s + s (' + ) s which shows that the impact on slack and in‡ation !0 h i 1 Et x bs;d t+1 @ h i A Et bs;d t+1 s;d x bs;d t ; bt only arise through their e¤ect on the policy rate deviations s s (' + ) s ! bis;d t ; for s = W; R; (31) from policy asymmetries across countries can bis;d t for s = W; R given by (30). Hence, policy rate deviations from the coordinated equilibrium are the only channel through which monetary policy gets s;d transmitted and a¤ects slack deviations (b xs;d t ) and in‡ation deviations (b t ) in the model. Deviations from the coordinated equilibrium occur when at least one of the policy coe¢ cients di¤ers 6= 0 and/or R x 6= 0. Otherwise, the only solution possible for the interest d;W d;R rate deviations from (30) is that bit = bit = 0 and, given (31), this implies that x bd;W =x bd;R = bd;W = t t t across countries— i.e., either R bd;R = 0. In short, without asymmetric policies, the solution is fully characterized by the coordinated t s;c s bs bs;c for s = W; R. bs;c monetary policy equilibrium whereby x bst = x t , b t = b t , and it = it Interestingly, for cases in which independent monetary policies lead to asymmetric policy coe¢ cients in the two-country model, I note that the dynamics of the vector of policy rates in deviations given by (30) are purely backward-looking and their random driving processes are given solely by the slack and in‡ation solution in the coordinated monetary policy equilibrium case. Hence, the solution to the coordinated equilibrium ultimately also characterizes the dynamics of the endogenous variables in deviations and the full s;d bs;d solution of the model as well. There is no role for monetary policy shocks in the solution of x bs;d — t ; b t ; it neither for productivity nor for cost-push shocks— except through their impact on the equilibrium solution s;c bs;c x bs;c under a common coordinated monetary policy. t ; b t ; it Furthermore, it also follows that, Proposition 1 The implication from (30) and (31) is that deviations from the coordinated equilibrium under independent and asymmetric monetary policies cannot be solved separately for the aggregate and di¤ erence sub-systems given that (31) implies that policy rate in deviations— whether for the aggregate or for the di¤ erential sub-systems— depend on both aggregate and di¤ erence variables on in‡ation and slack. 18 In light of this, the e¤ects of monetary policy asymmetries can be interpreted as a modi…cation (or transformation) of the common monetary policy equilibrium which introduces non-separabilities between the aggregate solution and the solution that explains the cross-country dispersion. An important takeaway from all of this is that a common monetary policy undoes the modi…cation of the dynamic propagation of shocks that arises from policy asymmetries. It also leads to a coordinated solution whereby aggregate and di¤erential dynamics are perfectly separable (with no spillovers between them). 4.2 International Coordination vs. Monetary Union Using the aggregate and di¤erential monetary policy rules under monetary policy coordination in (24) to replace bis;c in (28) for s = W; R, the sub-system of equations that determines in‡ation and slack for the t aggregates and for the cross-country di¤erentials can be written in the following form s 1+ s s c x c x s (' + ) s 1+ c s 1 s (' + ) + s s (' + ) c s (' + ) ! or more compactly as ! x bs;c t bs;c t s s;c where s;c 1+ openness 0 < s ( c+ x Et x bs;c t+1 s 1 ('+ ) s c (' + ) ) s s s c x > 0, < 21 , I have noted that W c (1 c W = 1, R ! ) + 1+ ! (' + ) > 0, = (' + ) s b rt s (' + ) s 0 + Et bs;c t+1 c (' + ) (' + ) 1 + s s = ! bs;c t s (' + ) ! x bs;c t 1 s;c = ! s c x vbts m b st > 0, and > 1 for all ! ; c x s s (' + ) ! Et x bs;c t+1 Et bs;c t+1 ! s b rt vbts m b st (32) + ::: (33) 0. Moreover, for any degree of > 1 2 1+(1 2 ) > 0, and 0 < R <1 1 . )(1 2 ) 2(1 ) I can write the aggregate and di¤erence sub-systems of expectational equations in (33) in canonical form for all > max 0; 1 1 2 +( '+ as s;c zbts;c = As;c Et zbt+1 + B s;cb "st ; for s = W; R; where the vector zbts;c s;c (b xs;c t ; bt ) T (34) includes in‡ation (bs;c xs;c t ) and slack (b t ) for s = W; R under a com- mon monetary policy and the driving processes can be represented by the vector b "st = The matrices of structural parameters As;c s B s;c s;c (' + ) c s 1 s;c (' + ) ! s s s (' + ) c (1 s + ) 1+ s vbts ; b rt m b st ! s c x T . and characterize the dynamics of each sub-system. s s c (' + ) 1 + (' + ) s x Hence, in‡ation and slack depend on both cost-push shocks (b vts ) and on deviations between the natural s s rate (b rt ) and the monetary policy shocks (m b t ). It is worth noticing here that the propagation of monetary shock innovations in the model is largely the same as that of innovations to the natural rate but of the 19 ! ; opposite sign. In turn, cost-push shocks have distinct e¤ects on in‡ation and slack and propagate di¤erently. The degree of openness does not enter into the aggregate sub-system (s = W ) described here and neither does the intratemporal trade elasticity of substitution between Home and Foreign goods . Therefore, neither the composition of the consumption basket nor the degree of substitutability between local and imported goods a¤ects the aggregate allocation. In other words, the strength of the trade channel does not in‡uence the aggregate dynamics in the coordinated monetary policy equilibrium. The only deep structural parameters that a¤ect the aggregate dynamics are the Calvo (1983) parameter , the intertemporal discount factor , the inverse of the intertemporal elasticity of substitution , the inverse of the Frisch elasticity of labor supply ', and the policy parameters c c x. and In turn, the deep structural parameters that determine the strength of the trade channel ( and ) a¤ect the cross-country dispersion only under a common monetary policy rule through the di¤erential sub-system R s = R— appearing in the composite coe¢ cients The Stochastic Forcing Processes. and R . Given the characterization of the natural rates for each country based on the frictionless allocation and the productivity shocks (equations (9) (14)) and the maintained assumptions on the cost-push and monetary shocks (shown in Table 2), I derive the stochastic forcing s processes for b b st , and vbts for s = W; R (see the Appendix for further details). The forcing processes can rt , m be described as follows s b rt m b st vbts s bs r rt 1 = "rs +b "rs t t ; b "ms +b "ms t t ; b b st 1 mm = +b "vs t ; b "vs t + ; bts 1 vv = N 0; 2 rs N 0; ; 2 ms 2 vs N 0; (35) ; (36) : (37) ; (38) The persistence of the natural rate and cost-push shocks is given by, while m W r = a v = u; a;a R r = a a;a (39) is the known persistence of the monetary policy shock process. The volatility term for the aggregate natural rate can be tied to parameters of the productivity shock and other structural parameters of the model as 2 rW 2 r = 2 a given the derivations of 1+ r;r = 2 1+ and 1+ r;r a;a 2 2 1+' +' a;a 2 2 r 2 a in (11) 1+' +' 2 ( 1) 2 +2 1 2 +( 2 2) 2 ( a;a +( a 1)) ; (14) and the fact that (40) 1 + 2 = a;a +( a 1) < 0. Analogously, the volatility term for the di¤erence natural rate can be tied to parameters of the productivity 20 shock and other structural parameters of the model as 2 rR 2 2 r 1 r;r = 2 2 a 1 a;a given the derivations of 2 r and =2 2 a a;a 1+' +' (2 ( 2 2) 1 2 1) (2 1) ( ( a;a ; 1)) a (41) (14).16 The volatility terms for the aggregate and di¤erence in (11) r;r 2 1+' +' 2 1 monetary policy shocks are given as 2 mW 1+ 2 m m;m 2 mR ; 2 2 m 2 1 ; m;m (42) which depend solely on the variance-covariance of the monetary shocks. Finally, the volatility terms for the aggregate and di¤erence cost-push shocks are given as 2 vW 1+ 2 u u;u 2 vR ; 2 2 2 (1 2 u 2 ) 1 ; u;u (43) which depend on the variance-covariance matrix of the country-speci…c cost-push shocks (but also on the degree of openness for the di¤erence cost-push shocks). c The coordinated policy parameters, c x, and a¤ect neither the aggregate driving processes nor the di¤erence driving processes. The only structural parameters that a¤ect the dynamics of the aggregate forcing processes are the inverse of the intertemporal elasticity of substitution and the inverse of the Frisch elasticity of labor supply ' and they in‡uence only the volatility of the aggregate natural rate shock innovations. The parameters and ' also a¤ect the volatility of the di¤erence natural rate shock innovations. However, in the case of the driving processes in di¤erences, I observe that the degree of openness and the trade elasticity of substitution between the Home and Foreign goods 2 rR di¤erence natural rate process di¤erence cost-push process 2 vR (through the composite coe¢ cients through the term (1 a¤ect the volatility of the and ) and the volatility of the 2 ) as well. The importance of the cost-push shocks to explain cross-country di¤erences declines with the openness of the economy ( ) because the volatility 2 vR declines with it— which implies that both monetary as well as natural rate di¤erence shocks may acquire a larger role the more open the economy becomes. 1 6 Notice that 2 ( follows that ( ( 1 1 1) = 2) 2) 1+( 1+ 1 '+ implies that 2 = = = = 41 2 2 (1 4(2 1) 4(2 1) 2 (2 1)(2 )(2(1 ( 1)(2 )(2(1 0 )@ 2 (1 2 1) (2 )) )) ! 1+ 1 1+ 1 0 0 @ )@ = (1 ( '+ ( '+ 1+ 1 a;a ( ( a 21 13 A5 1) (2 ) (2 (1 )) 1) (2 ) (2 (1 1) : )) 1)(2 ) 1)(2 )(2(1 a;a 1) (2 ) 1) (2 ) (2 (1 ( '+ 1+( 1+( 1) (2 ) (2 (1 '+ 1+( h ) 1) (2 ) 1+( 1+ 1 1) and )) )) 13 A5 13 A5 a;a )) ( i as in Table 1. Hence, it a 1) a;a ( ( a a 1) 1) c Finally, I want to point out that while the coordinated policy coe¢ cients, c x, and do not in‡uence the driving processes either for the aggregate case or for the di¤erential case, deeper economic integration through the formation of a monetary union as de…ned in this paper will surely alter the monetary shock 2 mW process. In that policy regime, the volatility of the aggregate does not necessarily change relative to what would be implied by the maintained assumptions on the Home and Foreign monetary policy shocks in Table 2. However, as expected from perfectly correlated monetary shocks across countries, the volatility of 2 mR the di¤erence will then have to be set to 4.2.1 = 0. Determinacy Properties Under the assumption that b "st is stationary, then (34) has a unique nonexplosive solution in which the vector T zbts;c s;c s;c (b xs;c are inside the unit circle for t ; b t ) is stationary whenever both eigenvalues of the matrix A each sub-system (s = W; R). The eigenvalues corresponding to the matrix As;c can be written as 1 2 s;c 1 where s;c s;c s 1+ ( c+ x r s;c 1 ('+ ) 2 s c s;c )2 ( ) 4 s;c ! ; s;c > 0 and 1+ and 0 < R < 1 2, < 1 for all 1 unit circle if and only if s;c 2 2 q 2 4 s;c < 2 whenever s;c 2 ( s;c ) ( ) s;c as: ( s;c 2 ( s;c ) 4 s;c <( W = s;c s s;c c x ( + R s;c ( + s;c s;c ) 4 s;c )2 ( s (' + ) 4 s;c ! ; (44) ) > 0 hold given that 0. Moreover, for any degree > 1 for all 1 1 2(1 ) . 2 +( '+ )(1 2 ) < s1 < s2 . Therefore, both q 2 s;c s;c 2 s;c 2 c x > 0 and = 1, r + > max 0; 1 of the model, it naturally follows that 0 s;c 2 W it also holds that + c (' + ) > 0 and the policy coe¢ cients satisfy that of openness 0 < 1 2 s;c 2 > 1 2 1+(1 2 ) > 0, For standard parameterizations eigenvalues of As;c lie inside the < 1. This inequality holds, in turn, s;c . Taking the square on both sides of the inequality— i.e., 2 s;c 2) — and then, re-arranging terms, the inequality can be rewritten ) < 1. From here it follows that further algebraic manipulations, if and only if c s 2 < 1 if and only if 1 ('+ ) + s c x 1+ s s 1+ ( ( c s ) x + ('+ ) c + ('+ ) s c ) x < 1 or, after > 1. Proposition 2 An open-economy variant of the Taylor principle which requires that c + 1 ('+ ) s c x > 1 for each s = W; R is needed to ensure the uniqueness and existence of the nonexplosive solution for the aggregate and di¤ erential sub-systems under a coordinated monetary policy equilibrium. The standard Taylor principle ( c > 1) is su¢ cient, but not necessary, to prove existence and uniqueness of the solution. Moreover, the open-economy Taylor principle reduces to the closed-economy variant which simply requires c + c x 1 ('+ ) > 1 whenever > max 0; 1 1 2 +( '+ )(1 2 ) 1 2(1 ) . Existence and uniqueness of a coordinated monetary policy equilibrium depends on the policy parameters c > 0 and c x c x 0. In the case where irrespective of the degree of openness = 0, the standard Taylor principle requiring with determinacy given that 1 ('+ ) s > 1 holds as it does in the closed-economy case. Whenever the common monetary policy involves a positive response to the output gap in each country ( Taylor principle can be relaxed whereby c c c x > 0), then the standard > 0 can fall to some extent below one and still be consistent > 0. 22 Here, I observe that all R W 2 +( '+ c = 1 and + R = 1 and 0 < 1 > max 0; 1 < W )(1 1 ('+ ) 1 2(1 2 ) c x R ) c > (2 + 1) < 1 for any degree of openness 0 < < 1 2 and for (as implied by the results in (3)). Then, it must follow that 1 ('+ ) c x. c As a result, the inequality c x 1 ('+ ) + >1 su¢ ces to ensure existence and uniqueness of a solution for both the aggregate and di¤erential sub-systems— irrespective of the openness to trade or the trade elasticity . In turn, whenever are plausible values of the trade elasticity and this implies that R W > = 1. In those cases, it follows that and, therefore, the inequality solution— where R low enough such that 0 < c 1 ('+ ) + R c x c + 1 (2 )(2(1 (1 2 )(2(1 )) )) , 1 <1 1 ('+ ) > '+ 2 +( '+ R c x c < )(1 + 2 ) there 1 2(1 ) c x 1 ('+ ) > 1 is needed to ensure existence and uniqueness of a is a function of trade openness and of the trade elasticity . Hence, the determinacy of the solution to the open-economy model under a coordinated monetary policy does not depend on the degree of trade openness ( ) and the trade elasticity ( ) for most plausible parameterizations unless the preference ratio '+ is su¢ ciently high and the trade elasticity is su¢ ciently low. Accordingly, the criterion to ensure existence and uniqueness of the open-economy solution is exactly the same as in the closed-economy case in most instances and, in particular, in the benchmark case where > 1. The conventional Taylor principle ( c > 1) is su¢ cient but not necessary in all cases where 1 , the range of values for )(1 2 ) 2(1 ) consistent with determinacy depends on the intertemporal discount factor through (1 Whenever > max 0; 1 1 2 +( '+ also on the closed-economy slope of the Phillips curve and 0 < <1 c 1 )(1 2 ) through the composite coe¢ cient 1 2(1 2 +( '+ R ), (' + ) > 0. In the case where > 0. below one that are ) > 0 and depends '+ > 1 (2 )(2(1 (1 2 )(2(1 )) )) determinacy also depends on the strength of the trade channel (which is a function of the import share In short, the determinacy condition when c x c x and the trade elasticity ). > 0 depends on how patient households are and how steep the closed-economy Phillips curve is— it depends on trade parameters only in rather special cases. Assessing the Determinacy Properties on a Monetary Union. From the de…nition of the monetary policy equilibrium under both international coordination and monetary union in Section 3, I note that both have the same aggregate policy equation but di¤er on the di¤erence policy equation instead. Hence, it is straightforward that the counterpart of (34) for the aggregate case (s = W ) is identical in both monetary policy regimes with the matrices of structural parameters in the monetary union case satisfying that AW;u = AW;c and B W;u = B W;c . Therefore, an analogous set of derivations to the ones used for the coordinated monetary policy equilibrium case implies that determinacy of the aggregate in the monetary union case requires c + 1 ('+ ) c x > 1. The counterpart of (34) for the di¤erential sub-system under a monetary union can be represented with R u (1 ) 1 R;u R;u the following matrices of structural parameters: A R R R R (' + ) (' + ) + 1+ ! R R (' + ) u 1 R;c R;c and B where R;c > 0. R R R u R 1+ ( u + ('+ ) R u ) (' + ) 1 + (' + ) x x Here, the policy parameters are written generically as u and ux so that an analogous set of calculations 23 u x ! to those discussed under a coordinated monetary policy equilibrium implies that determinacy of the solution to the di¤erence sub-system under this generic representation of a monetary union equilibrium would u 1 require u + x > 1. Given the de…nition of a monetary union proposed in Section 3, the ('+ ) R only set of policy coe¢ cients that are consistent with nominal interest rate equalization across countries are u = ux = 0. As a result, I conclude that a monetary union is inconsistent with the determinacy of the di¤erence sub-system. c 1 Proposition 3 The closed-economy variant of the Taylor principle which simply requires c + ('+ x > ) 1 su¢ ces to ensure determinacy of the aggregate dynamics within a monetary union that has a common monetary policy and results in short-term nominal interest rate equalization in equilibrium. In turn, the cross-country dispersion does not satisfy the conditions for determinacy. This holds true irrespective of the degree of openness ( ) and the trade elasticity ( ). This is an aspect of the formation of a monetary union that has not received much attention in the existing literature (Benigno (2004), Woodford (2010))— more focused on the positive and normative implications for the union as a whole rather than on the impact on each one of the local economies (or regions) that become part of the currency union. Intuitively, however, this is not a completely unexpected result either. The di¤erence sub-system is insulated from the aggregate sub-system as is the case under international monetary policy coordination. As such, the di¤erence sub-system is isomorphic to a closed-economy New Keynesian model where the interest rate (the interest rate di¤erential in the di¤erence sub-system case) is kept constant at zero in every period. The indeterminacy of constant interest rates within the closed-economy New Keynesian model is a well-known result already in the literature (Woodford (2003)). 4.2.2 Characterization of the Solution I examine the equilibrium for the aggregate and di¤erence variables in response to natural rate shocks, monetary policy shocks, and cost-push shocks. I focus on the solution to the coordinated monetary policy equilibrium only, but it follows from my preceding discussion that the solution for the aggregates is exactly the same under a monetary union (while the cross-country dispersion is indeterminate). The description of each sub-system (s = W; R) given by (34) is completed with the corresponding driving processes characterized in equations (35) (37). A detailed derivation of the solution conditional on each type of shock is found in the Appendix. The law of motion describing analytically the endogenous propagation of each individual shock j = r; m; v— i.e., for the natural rate shock (j = r), the monetary policy shock (j = m), and the cost-push shock (j = v)— on the endogenous variables x bs;c;j ; bs;c;j for s = W; R de…nes the unique equilibrium under a t t common (coordinated) monetary policy as bs;c;j t bs;c;j t = = s;c s;c;j bt ; 0;j x s;c s;c;j 1;j b t 1 + (45) s;c;j ; t s;c;j t N 0; 2 s;c;j ; (46) where the nominal short-term interest rate is given by bis;c;j = c bs;c;j + cx x bs;c;j for j = r; v and is t t t c s;c;j c s;c;j s bis;c;j = bt + xx bt +m b t if j = m. Equation (45) identi…es the policy trade-o¤ between in‡ation t and slack in each sub-system s = W; R conditional on each individual shock j = r; m; v, while equation (46) 24 states that in‡ation in each sub-system inherits the autoregressive structure of the corresponding driving process of each shock (equations (35) (37)). Natural Rate Shocks and Monetary Policy Shocks. I consider …rst the case where the sub-system in (34) is solely driven by either natural rate shocks (equation (35)) or monetary policy shocks (equation (36)), as both these shocks pose very di¤erent policy trade-o¤s than the cost-push shocks. Taking the conjectured solution in (45) (46) as given and assuming that the common monetary policy is consistent with determinacy of the solution for both sub-systems, I can verify by the method of undetermined coe¢ cients that, Proposition 4 The solution for j = r; m given in (45) s;c 1;r s;c 0;j 2 s;c;j = s;c 1;m s r; = (46) satis…es that m; (47) s = (' + ) s;c ; j = r; m; 1 1;j 0 = @ s;c 1;j 1 1 s;c 1;j where the innovations in (46) are de…ned as (48) (' + ) + s s;c;r t c x = arising from natural rate shock innovations and as + s s (' + ) s s s;c 1;j c ('+ ) s;c 1;r (1 s;c;m t = )(1 s;c 1;r + s c x )+ s 12 A 2 js ; j = r; m; (49) s ('+ ) s ('+ ) s s ( s c s s;c 1;r ) b "rs t when s;c s;c c + ('+ ) s s (1 ( c s;c x) 1;m )(1 1;m + 1;m ) when they are the result of monetary policy shock innovations instead. The volatility terms for the shocks, 2 (42). js for j = r; m and for s = W; R, are de…ned in (40) The key takeaways from this result are as follows: s;c 1. The endogenous persistence implied by natural rate shocks ( s;c 1;r ) and monetary policy shocks ( 1;m ) is entirely determined by the properties of the productivity and monetary shock processes, respectively. In other words, no deep structural parameters appear to a¤ect the persistence of the endogenous variables— let alone those related to the strength of the trade channel ( and ). 2. The positive comovement between in‡ation and slack given by s;c 0;j depends critically on the slope of the Phillips curve. For the aggregate solution (s = W ), only the slope of the closed-economy Phillips curve matters ( W = 1) and therefore is completely invariant to the openness ( ) and the trade elasticity ( ) that characterize the trade channel in the model. In turn, the ‡attening of the Phillips curve discussed in Sub-section 2.2 shows up in the trade-o¤ implicit in the di¤erential solution > (s = R). As noted before, 0 < R < 1 holds for any degree of openness 0 < < 21 and for all 1 1 given the results on the composite coe¢ cient summarized in )(1 2 ) 2(1 ) (3). Figure 2.A and Figure 3.A illustrate that the trade-o¤ R;c 0;j for j = r; m declines with the degree of openness ( ) while the trade-o¤ is lower the larger the trade elasticity ( ) is.17 These …ndings show that the trade channel matters for the cross-country dispersion trade-o¤— implying that more max 0; 1 2 +( '+ 17 I use the following standard parameterization to produce Figure 2.A, Figure 2.B, Figure 3.A, and Figure 3.B: = 0:99, = 2, = 0:75, a = 0:95, m = 0:9, a;a = m;m = 0:25, and a;a = 0:025. The policy parameters are set as in the standard Taylor (1993) rule: c = 1:5 and cx = 0:5. The volatility of the shocks is normalized to one in this case: 2a = 2m = 1. 25 b "ms t open economies (with lower di¤erential x bR;c;j t R;c 0;j ; j = r; m) should experience larger movements in the output gap for any given change in the cross-country in‡ation di¤erential bR;c;j . Aggregate t trade-o¤s should be una¤ected by the trade channel. Moreover, the policy parameters c and c x have no e¤ect on either the aggregate or the di¤erential trade-o¤s. Interestingly, the preference ratio '+ plays a very signi…cant role, showing how important the features of the labor market (the inverse of the Frisch elasticity) can be. 3. Figure 2.A and Figure 3.A illustrate how R;c 0;r < W;c 0;r while R;c 0;m = R;c 0;m for the closed-economy case ( = 0). This is entirely due to the fact that the model permits cross-country spillovers for the productivity shocks ( a;a 6= 0) and not for the monetary shocks. In fact, given the maintained assumptions on the parameters of the model, it holds that R;c 0;r S W;c 0;r whenever a;a T 0. This shows that cross-country shock spillovers (di¤usion) are another important feature of the economy with a sizeable quantitative impact on the equilibrium trade-o¤s that can arise between in‡ation and slack. In other words, even though more open economies generally face lower R;c 0;r and R;c 0;m , the quantitative size of these e¤ects will depend on key features of the economy such as the extent of technological di¤usion across countries ( 4. The volatility of in‡ation 2 s;c;j a;a ) or the sensitivity of the labor supply to wages ('). for j = r; m reveals the importance of the trade channel on accounting for the declines in macro volatility during the Great Moderation period. The more open economies generally experience lower di¤erential in‡ation volatility from natural rate shocks and monetary policy shocks, and the volatility declines tend to be accentuated the higher the trade elasticity turn, the trade parameters ( and is. In ) have no e¤ect on the volatility of the aggregates. Figure 2.B and Figure 3.B illustrate the volatility implications of the model for a given one-standard-deviation shock on productivity impacting the natural rate and on monetary policy, respectively. On the one hand, I observe that the volatility on the cross-country in‡ation dispersion tends to be higher than the volatility of aggregate in‡ation for both shocks whenever the degree of openness ( ) is su¢ ciently low. This relationship changes as gets larger (closer to = 12 ) and the volatility di¤erential converges to zero. On the other hand, the …ndings extend a well-known prediction of the closed-economy New Keynesian model to the open-economy case— it shows that the New Keynesian mechanism accentuates markedly the role of monetary policy shocks while it dilutes somewhat the role of productivity shocks (embedded in the natural rate shocks). Not surprisingly, this is a model in which monetary policy shocks end up playing a dominant role in explaining the variations in in‡ation and slack as well.18 5. I should also point out that the persistence of the productivity shock process is another crucial factor explaining the very low volatility pass-through shown in Figure 2.B— in particular, lower values of a signi…cantly increase the implied volatility (a reduction of a from 0:95 to 0:85 will double and even more than triple the cross-country dispersion volatility depending on '+ and can raise aggregate volatility more than sevenfold). Moreover, the aggregate and di¤erential volatility generally decline the higher the preference ratio '+ is— once again showcasing the importance of the inverse of the Frisch s;c;j output is equal to slack plus potential output (b yt +x bs;c;j for s = W; R and for all shocks j = r; m; v). Potential t output depends solely on productivity shocks. Hence, actual output and slack are the same for the monetary shocks (j = m) and cost-push shocks (j = v). While the contribution of productivity shocks to driving slack might be mitigated here, their e¤ect on output potential has to be factored in when assessing the volatility of actual output. In this regard, it should be noted that open economies tend to experience more volatile potential. 1 8 Actual 26 elasticity of labor supply (') in this model. In general, I …nd that the stronger the anti-in‡ationary bias of the common (coordinated) monetary policy (the higher c x activity (the higher c is) or the higher the bias on economic is), the lower the in‡ation volatility arising from natural rate shocks or monetary policy shocks (the lower 2 s;c;j is for j = r; m). Hence, I argue that in terms of explaining the decline in macro volatility over the past several decades (in particular during the Great Moderation period), both greater economic integration as well as monetary policies around the world more focused on …ghting in‡ation may have contributed to this. The autoregressive processes for in‡ation in (46) conditional on j = r; m are independent of each other and can therefore be aggregated to describe the dynamics of in‡ation in response to the gap between the natural s rate and the monetary policy shocks b rt m b st . I de…ne the in‡ation process in this case as bts;c;r m = bs;c;r + bs;c;m . In the knife-edge case where there are no productivity shock spillovers across countries t t ( = 0) and the monetary policy shocks are assumed to have the same persistence as productivity shocks a;a do ( m = a ), s;c 1;r then I can easily see that = s;c 1;m , s;c 0;r = s;c 0;m , and 2 s;c;r 2 rs = 2 s;c;r 2 rs .19 In other words, whenever the bivariate productivity and monetary shock processes have common persistence and no crosscountry spillovers, a one-standard-deviation increase in the natural rate innovation b "rs t has the exact same pattern of propagation on in‡ation and slack as a one-standard-deviation decline on the monetary policy for s = W; R. This also implies that bs;c;r shock innovation b "ms t t bs;c;r t m = s;c s;c;r m 1;r b t 1 + s;c;r m , t s;c;r m t where N 0; 2 s;c;r m m follows an AR(1) process given by: and 2 s;c;r m = 2 s;c;r + 2 s;c;m . In general, however, natural rate shock innovations and monetary policy shock innovations are not the mirror image of each other as a result of di¤erences in the key parameters that determine the persistence and cross-country spillovers of the underlying productivity and monetary shocks in Table 2. In the economicallys;c 1;r relevant case where obtain that m bs;c;r t bs;c;r t m s;c;r m t where s;c;r m 1 s;c 2 1;m s;c;r s;c 1;m , 6= based on well-known aggregation results (Hamilton (1994), Chapter 4), I follows an ARMA(2; 1) process of the following form: = s;c 1;m N 0; + s;c 1;r 2 s;c;r m bts;c;r 1 m s;c s;c s;c;r m m m s;c;r m + s;c;r + s;c;r ; t 1;r 1;m b t 2 1 t 1 ! s;c 2 s;c 2 1;m s;c;r + 1;r s;c;m 2 ; s;c;r m = s;c;r m 1 ; (50) (51) is the solution to the quadratic equation + s;c 2 1;r s;c;m s;c;r m 2 1 h s;c 2 2 s;c;r 1;m s;c 2 s;c 2 + 1;m s;c;r 1;r s;c;m + 1+ + 1+ =0 s;c 2 1;r 2 s;c;m i s;c;r m 1 + ::: (52) that ensures the invertibility of the MA part of the ARMA(2; 1) process. 1 9 This could easily be generalized as well to a case where international monetary policy shock spillovers are permitted and they are equal to the productivity spillovers given by a;a . 27 Figure 2.A The Trade-o¤ Between In‡ation and Slack Arising from Natural Rate Shocks Figure 2.A The Trade-Off Between Inflation and Slack Arising from ( s;c 0;r for s = W; R). Natural Rate Shocks (χ0,r,s for s=W,R) (γ/(ϕ+γ)) = 0.1 (γ/(ϕ+γ)) = 0.9 (γ/(ϕ+γ)) = 0.5 50 50 50 45 45 45 40 40 40 35 35 35 30 30 30 25 25 25 20 20 20 15 15 15 10 10 10 5 5 5 0 0 0 σγ = 0.5 σγ = 1 σγ = 1.5 χ0,r,s for s=W,R σγ = 2 σγ = 2.5 σγ = 3 σγ = 3.5 σγ = 4 σγ = 4.5 χ0,r,W 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 σγ = 5 ξ Note: The bar with black margins indicates the range of the import share that is referenced in the text. 28 Figure 2.B Inflation Volatility fromNatural Natural Rate (for(for s=W,R) Figure 2.B In‡ ation Volatility from RateShocks Shocks s = W; R). (γ/(ϕ+γ)) = 0.9 0.09 0.09 0.09 0.08 0.08 0.08 0.07 0.07 0.07 0.06 0.06 0.06 0.05 0.05 0.05 0.04 0.04 0.04 0.03 0.03 0.03 0.02 0.02 0.02 0.01 0.01 0.01 0 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 σγ = 0.5 σγ = 1 σγ = 1.5 σγ = 2 σγ = 2.5 σγ = 3 σγ = 3.5 σγ = 4 σγ = 4.5 σγ = 5 s=W 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 (γ/(ϕ+γ)) = 0.5 0.1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Volatility for s=W,R (γ/(ϕ+γ)) = 0.1 0.1 ξ Note: The bar with black margins indicates the range of the import share that is referenced in the text. 29 Figure 3.A The Trade-o¤ Between In‡ation and Slack Arising from Monetary Policy Shocks Figure 3.A The Trade-Off Between Inflation and Slack Arising from ( s;c 0;m for s = W; R). Monetary Policy Shocks (χ0,m,s for s=W,R) (γ/(ϕ+γ)) = 0.1 (γ/(ϕ+γ)) = 0.9 (γ/(ϕ+γ)) = 0.5 16 16 16 14 14 14 σγ = 0.5 σγ = 1 12 12 12 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 σγ = 2 σγ = 2.5 σγ = 3 σγ = 4 σγ = 4.5 σγ = 5 χ0,m,W 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 σγ = 3.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 χ0,m,s for s=W,R σγ = 1.5 ξ Note: The bar with black margins indicates the range of the import share that is referenced in the text. 30 Figure 3.B Inflation Volatility fromMonetary Monetary Policy (for(for s=W,R) Figure 3.B In‡ ation Volatility from PolicyShocks Shocks s = W; R). (γ/(ϕ+γ)) = 0.1 (γ/(ϕ+γ)) = 0.5 (γ/(ϕ+γ)) = 0.9 4 4 4 3.5 3.5 3.5 3 3 3 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 σγ = 0.5 Volatility for s=W,R σγ = 1 σγ = 1.5 σγ = 2 σγ = 2.5 σγ = 3 σγ = 3.5 σγ = 4 σγ = 4.5 σγ = 5 0.5 0 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 s=W ξ Note: The bar with black margins indicates the range of the import share that is referenced in the text. 31 The solution for j = r; m given in (45) s;c 1;r s;c 0;j 2 s;c;j = s;c 1;m s r; = (46) satis…es that m; (53) s = (' + ) s;c ; j = r; m; 1 1;j 0 = @ s;c 1;j 1 1 s;c 1;j (54) + where the innovations in (46) are de…ned as s s;c;r t c x + arising from natural rate shock innovations and as (' + ) s s s;c 1;r (1 s;c;m t s;c 1;r + )(1 s A s;c 1;j c c x )+ 2 js ; j = r; m; (55) s s ('+ ) = 12 s s (' + ) s s ('+ ) ('+ ) = ( s;c 1;r c ) s s s b "rs t when s;c s;c c + ('+ ) s s (1 ( c s;c x) 1;m )(1 1;m + 1;m ) when they are the result of monetary policy shock innovations instead. The volatility terms for the shocks 2 js for j = r; m and for s = W; R are de…ned in (40) (42). Cost-Push Shocks. Taking the conjectured solution in (45) (46) as given and assuming that the common (coordinated) monetary policy is consistent with determinacy of the solution for both sub-systems, I can verify by the method of undetermined coe¢ cients (as shown in the Appendix) that, Proposition 5 The solution for j = v given in (45) s;c 1;v s;c 0;v 2 s;c;v = v; 1+ = = s 0 @ s c x c s;c 1;v s;c 1;v ! s;c 1 s;c s;c 1;v c (46) satis…es that (56) ; (57) 1+ s;c 1;v s;c;v t s s;c 1;v c x s;c + c (' + ) 1+ s s;c 1;v c x s;c (1+ s c (1 s;c c x 1;v s;c 1+ ) ) s;c 1;v ('+ ) 12 A 2 vs ; (58) c b "vs c ) s;c t (1 s;c )( )+ ( )(1 1;v when arising from cost-push shock innovations. The volatility term for the cost-push shock 2vs for s = W; R where the innovations in (46) are de…ned as = s;c 1;v c s;c 1;v s c x s;c 1;v is de…ned in (43). The key takeaways from this result are as follows: 1. The endogenous persistence arising from cost-push shocks s;c 1;v is entirely determined by the properties of the shock process itself (as it was the case with the endogenous persistence arising from natural rate shocks and monetary policy shocks as well). These results on endogenous persistence suggest that variations in the persistence of in‡ation are generally not symptomatic of greater economic integration (openness) or changes in monetary policy so long as those policy changes are coordinated. 2. The comovement between in‡ation and slack given by the trade-o¤ s;c 0;v is negative (unlike the positive comovement arising from natural rate shocks and monetary policy shocks). The equilibrium trade-o¤ from cost-push shocks does not depend on the slope of the Phillips curve (unlike what happens with 32 b "ms t natural rate shocks and monetary policy shocks)— instead, the equilibrium trade-o¤ depends critically on the slope of the IS curve and the common (coordinated) monetary policy rule coe¢ cients ( c x ). For the aggregate solution (s = W ), the aggregate IS curve slope is W c and = 1 and therefore the trade-o¤ is invariant to the openness ( ) and the trade elasticity ( ) that characterize the trade channel in this model. As noted before in regards to the natural rate shocks and the monetary policy shocks, the strength of trade linkages does not matter for the aggregate trade-o¤— it matters only for the cross-country trade-o¤. The steepening of the IS curve discussed in Sub-section 2.3, in fact, shows up in the trade-o¤ implicit in the di¤erential solution (s = R). In that case, degree of openness 0 < coe¢ cient < 1 2 and for all > 1 2 1+(1 2 ) R > 1 holds for any > 0 given the results on the composite summarized in (4). Figure 4.A illustrates that the trade-o¤ s;c 0;v for j = r; m declines in absolute value with the degree of trade openness ( ) and also shows that the absolute value of the trade-o¤ is lower the larger the trade elasticity ( ) is.20 This implies that more open economies (with lower R;c 0;v ) should experience larger movements in the output gap di¤erential x bR;c;v for any given t while aggregate trade-o¤s remain una¤ected by change in the cross-country in‡ation di¤erential bR;c;v t the trade channel. Interestingly, the preference ratio '+ , and the inverse of the Frisch elasticity of labor supply (') in particular, play no role in the trade-o¤ arising from the cost-push shocks (unlike what happens with natural rate shocks and monetary policy shocks). 3. The policy parameters c and cx have a major e¤ect on the aggregate and the di¤erential trade-o¤s. In general, the stronger the anti-in‡ationary bias of the common (coordinated) monetary policy (the higher c is) the lower the trade-o¤ for cost-push shocks in absolute value will be (the lower s;c 0;v is). Similarly, the weaker the bias on economic activity of the common (coordinated) monetary policy (the lower c x is), the lower the trade-o¤ for cost-push shocks in absolute value will be (the lower 4. The volatility of in‡ation 2 s;c;v s;c 0;v is). con…rms similar patterns to those described for the other shocks— particularly on the role of the trade channel explaining the declines in macro volatility during the Great Moderation period. The results indicate that the New Keynesian mechanism tends to amplify the volatility of the underlying cost-push shocks (more in line with what happens for monetary policy shocks than for the natural rate shocks), and volatility tends to be lower the larger the preference ratio '+ is in connection with the features of the labor market. While the volatility of the aggregates is una¤ected by the degree of openness ( ) and the trade elasticity ( ), the strength of the trade channel a¤ects the volatility of the cross-country in‡ation dispersion in a rather non-linear way depending again on the preference ratio '+ . Figure 4.B illustrates the volatility implications of the model for a given one-standard-deviation cost-push shock. From that …gure, I observe that for low enough levels of openness ( ), an increase in the parameter may lead to an increase in the volatility of the cost-push shocks. For large enough levels of openness, a higher towards zero as gets arbitrarily close to 1 2. will then lead to lower volatility which converges This nonlinearity in the model adds an additional layer of complexity to explain shifts in the contribution of the di¤erent shocks resulting from greater trade integration. For large enough values of the preference ratio '+ , a higher trade elasticity tends to widen the impact on volatility. 2 0 I use the following standard parameterization to produce Figure 4.A and Figure 4.B: = 0:99, = 2, = 0:75, u = 0:9, and u;u = 0:25. The policy parameters are set as in the standard Taylor (1993) rule: c = 1:5 and cx = 0:5. The volatility of the shock is normalized to one in this case: 2u = 1. 33 5. In general, I …nd that the stronger the anti-in‡ationary bias of the common (coordinated) monetary c policy (the higher 2 s;c;v ). in‡ation volatility arising from cost-push shocks (the lower given by c c x is) or the smaller the bias on economic activity (the lower is), the lower the Unlike for the anti-in‡ationary bias , the sensitivity to slack in the common (coordinated) monetary policy rule given by c x has very di¤erent implications for in‡ation volatility depending on the nature of the shock. An increase c x in tends to reduce the volatility arising from natural rate shocks and monetary policy shocks, but it increases the volatility from cost-push shocks. My interpretation of this is that pursuing in‡ation stabilization more strongly may have added to the decline in volatility during the Great Moderation, but the e¤ects on macro volatility (and on the contribution of the di¤erent shocks over the business cycle) from changes in the policymakers’ response to slack appear to play a mixed role in the Great Moderation instead. Backing Out the Country-Level Solution. Given the dynamics for s = W; R in (45) (46) under the terms of Proposition 4, the transformation in (21) backs out the corresponding variables for each individual country. Using the persistence terms for the natural rate derived in (38), it follows that in‡ation and the output gap in response to natural rate shocks, bc;r and x bc;r in the Home country, must satisfy that t t bc;r t where = = bW;c;r + 21 bR;c;r t t ('+ ) 1 a bc;r t 1 1 ( 1 ( a a + a;a ) ('+ ) a + a;a + ) 1 1 ( stands for domestic in‡ation, x bW;c;r + t a a a;a x bc;r t 1 21 R ) R ('+ ) ( a 1 c;r bt 2x a;a + ) x bR;c;r t 1 1 ( 1 a ) a + a;a 1 ( R a a a;a ) 1 bt ;c;r 2x ; (59) is the domestic slack, and x bt ;c;r is foreign slack. The relationship between domestic in‡ation and domestic and foreign slack depends on the cross-country spillovers for the productivity shocks ( a;a ). Domestic in‡ation is related to a convex combination of domestic and foreign slack by a common scaling factor term R ('+ ) . 1 a When there are no cross-country spillovers ( a;a = 0), only the matters implying that domestic slack outweighs its contribution to aggregated global slack while foreign slack tends to be below its contribution given that 0 < (2 < and for all > max 0; 1 1 1) < 1 for any degree of openness 1 (as implied by the results in (3)). This )(1 2 ) 2(1 ) shows that the equilibrium trade-o¤ between domestic in‡ation and domestic and foreign slack is in fact 0< 1 2 R 2 +( '+ largely dominated by domestic slack even in the open-economy case. 34 Figure 4.A The Trade-o¤ Between In‡ation and Slack Arising from Cost-Push Shocks ( Figure 4.A The Trade-Off Between for s = W; R). Inflation and Slack Arising from Cost-Push Shocks (χ0,v,s for s=W,R) 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -0.2 σγ = 0.5 χ0,v.s for s=W,R -0.4 σγ = 1 σγ = 1.5 -0.6 σγ = 2 σγ = 2.5 σγ = 3 -0.8 σγ = 3.5 σγ = 4 -1 σγ = 4.5 σγ = 5 -1.2 -1.4 χ0,v,W ξ Note: This part of the solution does not depend on the preference ratio (γ/(ϕ+γ)). The bar with black margins indicates the range of the import share that is referenced in the text. 35 s;c 0;v 4.Bation Inflation Volatilityfrom fromCost-Push Cost-Push Shocks (for(for s=W,R) Figure Figure 4.B In‡ Volatility Shocks s = W; R). (γ/(ϕ+γ)) = 0.1 (γ/(ϕ+γ)) = 0.5 (γ/(ϕ+γ)) = 0.9 4 4 4 3.5 3.5 3.5 3 3 3 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 σγ = 0.5 Volatility for s=W,R σγ = 1 σγ = 1.5 σγ = 2 σγ = 2.5 σγ = 3 σγ = 3.5 σγ = 4 σγ = 4.5 σγ = 5 0.5 0 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 s=W ξ Note: The bar with black margins indicates the range of the import share that is referenced in the text. 36 Furthermore, the domestic in‡ation process in response to natural rate shocks, bc;r t , follows a sim0: bc;r = t ple AR(1) process whenever 1 2 2 2 W;c;r + a;a = c;r c;r W;R and W;R 2 R;c;r + c;r a bt 1 W;c;r R;c;r . t t E c;r t , + c;r t where N 0; 2 c;r 2 c;r given by = It also follows from well-known aggregation results (Hamilton (1994)) that domestic in‡ation in response to natural rate shocks, bc;r t , follows an ARMA(2; 1) process of the following form whenever cross-country productivity spillovers matter ( bc;r t = c;r t where h c;r 1 1+( 2 a bc;r t 1 N 0; ( 2 c;r a a;a 2 c;r ; )( ( = + a a;a a ) bc;r t 2+ 2 W;c;r ) a;a c;r t + + c;r c;r t 1; 1 1 2 2 ( a c;r 1 + a;a ) 2 R;c;r a;a 6= 0): c;r a W;R + ! (60) ; (61) is the solution to the quadratic equation 2 a a;a ) h ( a 2 W;c;r h ( 2 W;c;r ) a;a 1 2 2 + a a;a 1+( ) 1 2 2 + a 2 W;c;r + + ( a + a;a 2 a;a 1 2 2 2 R;c;r ) ( 2 R;c;r ) a + a;a + c;r a W;R + (1 + ( ) 2 R;c;r a + + a i ( c;r 2 1 ) + ::: )( a i c;r W;R = 0 a;a a;a )) c;r W;R i c;r 1 + ::: (62) that ensures the invertibility of the MA part of the ARMA(2; 1) process. Aggregate and di¤erence shocks E(b "rW b "rR t t ) = 0 given that E b "rW "rR = are uncorrelated in the model— i.e., c;r W;c;r R;c;r t b t W;R rW rR 1 "rt 2 E ((b b "rt )) = 0— which further simpli…es the characterization of the stochastic process for "rt +b "rt ) (b domestic in‡ation. This comes to show that even modest international cross-country spillovers can have signi…cant e¤ects on the dynamics of domestic in‡ation and on the patterns of propagation of natural rate shocks. Similar derivations would help characterize the dynamics of foreign in‡ation in response to real interest rate shocks and the corresponding trade-o¤ of foreign in‡ation with domestic and foreign slack. Given the dynamics for s = W; R in (45) (46) under the terms of Proposition 4, the transformation in (21) can be used again to back out the endogenous variables of each individual country. It follows that in‡ation and the output gap in response to monetary policy shocks, bc;m and x bc;m in the Home country, t t must satisfy that bc;m t = ('+ ) W;c;m ) R R;c;m + 12 ('+ x bt x bt 1 1 m m ;c;m 1 c;m R 1 bt + 1 bt ; 2x 2x bW;c;m + 21 bR;c;m = t t ('+ ) 1 m 1+ R (63) where bc;m stands for domestic in‡ation, x bc;m is the domestic slack, and x bt ;c;m is foreign slack. Domestic t t in‡ation is related to a convex combination of domestic and foreign slack by a common scaling factor R Then, only the term of openness 0 < < m . matters implying that domestic slack outweighs its contribution to aggregated global slack while foreign slack tends to have a lower contribution given that 0 < 1 2 ('+ ) 1 and for all > max 0; 1 1 R (2 1) < 1 for any degree 1 2(1 (as implied by the results in ) )(1 2 ) (3)). This shows that the equilibrium trade-o¤ between domestic in‡ation and domestic and foreign slack 2 +( '+ is in fact largely dominated by domestic slack whenever we are dealing with monetary policy shocks (as it happened with natural rate shocks as well). 37 The domestic in‡ation process in response to monetary policy shocks, bc;m , follows a simple AR(1) t process: where 2 c;m 2 W;c;m = + 1 2 2 bc;m = t 2 R;c;m + c;m m bt 1 c;m W;R c;m ; t + c;m t c;m W;R and 2 c;m N 0; E ; (64) W;c;m R;c;m t t . As before, I should note here that aggregate and di¤erence monetary policy shocks are uncorrelated in the model— i.e., c;m W;R E(b "mW b "mR ) mW mR m m m m t t 1 = 0 given that E b "t b "t + b "t "t ) (b = 2 E ((b "t b "t )) = 0— which furW;c;m R;c;m mW mR ther simpli…es the characterization of the stochastic process for domestic in‡ation. This shows that the volatility of in‡ation at the country-level is largely dominated by the volatility of the world aggregate (as it was the case in response to natural rate shocks too). Finally, given the dynamics for s = W; R in (45) (46) under the terms of Proposition 5, the trans- formation in (21) helps me back out the solution for the endogenous variables of each individual country in response to a cost-push shock. It follows that in‡ation and the output gap, bc;v and x bc;v in the Home t t country, must satisfy that bc;v t = 1+ 1 cx 1 ( c = bW;c;v + 12 bR;c;v t t 1+ 1 cx 1 ( c v 1+ v) (1 v v) R v )+ R (1 v )+ x bW;c;v t c x R c x 1 c;v bt 2x R 1+ 1 2 R ( c x v c v) (1 + 1 R (1 x bR;c;v t v )+ v )+ R R (65) c x c x 1 bt ;c;v 2x ; where bc;v stands for domestic in‡ation, x bc;v is the domestic slack, and x bt ;c;v is foreign slack. Domestic t t in‡ation is once again related to a convex combination of domestic and foreign slack by a common scaling factor 1+ 1 cx 1 ( c v v) . However, unlike for the case of natural rate shocks and monetary shocks, the slope R arising from the Phillips curve relationship does not a¤ect the weight of domestic and foreign slack in the resulting equilibrium trade-o¤. In turn, the slope R arising from the IS curve matters showing that domestic slack tends to outweigh its contribution to aggregated global slack while foreign slack tends to have a lower contribution given that R > 1 for any degree of openness 0 < 1 2 < and for all > 1 2 1+(1 2 ) >0 (as implied by (4)). This shows that the nature of the shock is important when determining the equilibrium trade-o¤ between domestic in‡ation and domestic and foreign slack. However, the …ndings based on the cost-push shocks con…rm the fact that the trade-o¤ for domestic in‡ation tends to be largely dominated in equilibrium by domestic slack (as it happened with natural rate shocks and monetary policy shocks as well). The domestic in‡ation process in response to cost-push shocks, bc;v t , also follows a simple AR(1) process: where 2 c;v = 2 W;c;v + 1 2 2 bc;v t = 2 R;c;v + c;v v bt 1 c;v W;R and + c;v t ; c;v W;R c;v t E N 0; W;c;v R;c;v t t di¤erence cost-push shocks are uncorrelated in the model— i.e., that E b "vW "vR t b t = 1 "vt 2 E ((b +b "vt ) (b "vt 2 c;v c;v W;R ; (66) . Not surprisingly, aggregate and W;c;v R;c;v E(b "vW b "vR t t ) vW vR = 0 given b "vt )) = 0— which simpli…es the characterization of the stochastic process for domestic in‡ation (as it did for natural rate shocks and monetary policy shocks). This result shows that the nature of the shocks driving the business cycle can be very important, but also validates the claim that the volatility of in‡ation at the country-level is largely dominated by the volatility of the world aggregate (across all types of shocks). Hence, global in‡ation shares signi…cant features with domestic 38 in‡ation (see, e.g., Ciccarelli and Mojon (2010), Duncan and Martínez-García (2015), and Bianchi and Civelli (2015) on the dynamics of in‡ation). 4.3 Discussion The solution of the model shows that local macro variables display strong common movements even when all shocks are country-speci…c— that is, even without common shocks driving the international business cycle. I explore the distinct patterns of shock propagation— looking at natural rate shocks, cost-push shocks, and monetary policy shocks— and the di¤erent roles monetary policy plays on the aggregates of the two countries and the cross-country dispersion. The results of this paper illustrate how di¤erent monetary policy regimes impact the dynamics of open economies establishing that: 1. In general there are negative foreign productivity shock spillovers to domestic potential output. Full insulation from foreign productivity shocks appears as a knife-edge case (Cole and Obstfeld (1991)). In the economically-relevant cases, the more open the economy ( ) is and the higher the trade elasticity of substitution ( ), the larger the negative spillover into domestic potential output. The reason being that the income e¤ect of foreign productivity shocks is then dominated by the strength of the substitution e¤ect leading to an aggregate demand shift across countries that drags domestic potential output down. I also …nd that the negative spillovers tend to be larger whenever the inverse of the Frisch elasticity of labor supply (') is smaller relative to the intertemporal elasticity of substitution ( ). Hence, the sensitivity of labor supply to wage rate changes can be important in determining the magnitude of the spillovers for any given strength of the trade channel mechanism. Furthermore, open economies in general display higher volatility and lower cross-country correlations of potential output innovations than their closed-economy counterparts. 2. The weight of expected foreign output potential growth on the domestic natural rate exceeds the import share (a standard measure of trade openness) in the economically-relevant cases. Positive spillovers from faster foreign potential growth— whether due to foreign or domestic productivity growth— increase the domestic natural rate. The more open the economy ( ) is and the higher the trade elasticity is ( ), the larger the e¤ect of foreign potential growth beyond what can be attributed to the import share alone— and the smaller the e¤ect of domestic potential growth is— on the domestic natural rate. When potential output growth is related to the Home and Foreign productivity shocks, then the volatility of the natural rate innovations is lower and their cross-country correlations are higher for open economies than for closed economies. This shows that greater openness contributes to some extent to explaining the decline in macro volatility— albeit not for output potential— characteristic of the Great Moderation (Roberts (2006), Clark (2009), Martínez-García (2015a)). It’s worth noting that the inverse of the Frisch elasticity of labor supply ' (a key feature of the labor market) a¤ects the natural rate process as well magnifying these di¤erences between open and closed economies. 3. In general an open economy has a ‡atter Phillips curve so that there is a smaller reduction in domestic in‡ation for any given decline in domestic slack. A more open economy ( ) and one with a larger trade elasticity ( ) would have a ‡atter Phillips curve on domestic slack and simultaneously a positive and larger slope on foreign slack. I also …nd that the slope on domestic slack becomes ‡atter and conversely the slope on foreign slack becomes steeper for smaller values of the inverse of the Frisch elasticity of labor 39 supply '— indicating that the responsiveness of labor supply is a key factor magnifying the impact of the trade channel. This theoretical insight implies that the trade channel provides a consistent explanation for the empirical evidence on the ‡attening of the Phillips curve documented, among others, by Roberts (2006) and IMF WEO (2013). Moreover, I show that a more open economy ( ) or one with a higher trade elasticity ( ) generally has a steeper investment-savings (IS) curve with respect to deviations of the domestic real interest rate from its corresponding domestic natural rate. In this case, changes in the policy rate in an open economy have larger e¤ects on domestic slack than in an otherwise identical closed economy. Labor market features like the inverse of the Frisch elasticity of labor supply ' have no bearing on the slope of the open-economy IS curve, though. 4. I show that monetary policy asymmetries across countries can be interpreted as alterations of the monetary policy equilibrium that emerges under international monetary policy coordination. Interestingly, some form of monetary integration— either a common coordinated monetary policy or deeper monetary integration in the form of a currency union— leads to a policy equilibrium whereby aggregate dynamics and cross-country dispersion are characterized by perfectly separable structural systems of equations which can, therefore, be investigated on their own. Moreover, I …nd that the strength of the trade channel implied by the degree of openness of the economy ( ) but also by the trade elasticity of substitution between locally-produced and imported goods ( ) a¤ects the cross-country dispersion only and not the aggregate dynamics under a common monetary policy. The key message here is that greater economic integration through trade has no e¤ect on the aggregate dynamics of the two countries so long as some sort of international monetary policy cooperation is achieved. 5. The criterion to ensure uniqueness and determinacy of the open-economy solution is exactly the same as in the closed-economy case in most instances under the case of international monetary policy coordination. This criterion also ensures the determinacy of the aggregate solution under a monetary union— but, in turn, the short-term nominal interest rate equalization across countries resulting from monetary union leads to indeterminacy in the solution that characterizes the dispersion across countries. Hence, an important distinction between international policy coordination and a monetary union— where this paper adds novel insight on the existing theoretical literature on currency unions (Benigno (2004), among others)— is that the latter may induce indeterminacy (and hence multiple equilibria) unlike the former. 6. The macroeconomic persistence implied by the model is entirely determined by the properties of the underlying processes for the productivity shocks, the monetary policy shocks, and the cost-push shocks. The results of the paper generally suggest that variations in the macroeconomic persistence— particularly on in‡ation which has been the main focus of a strand of the New Keynesian literature (Benati and Surico (2008), Carlstrom et al. (2009))— are not necessarily symptomatic of greater trade integration (openness). Moreover, changes in monetary policy, so long as those changes are internationally coordinated, do not appear to play a role either. The role of asymmetric monetary policies across countries on persistence is left for future research. 7. The comovement between in‡ation and slack arising from natural rate shocks and monetary policy shocks is positive while it is negative when the driving force is the cost-push shock. The strength of trade linkages does not matter for the aggregate trade-o¤ irrespective of the shock one considers— it matters 40 only for the cross-country dispersion trade-o¤. It is important to note then that in general the more open an economy is ( ) and the higher the trade elasticity ( ), the larger the movements in the output gap di¤erential seen in equilibrium for any given change in the cross-country in‡ation di¤erential. This result shows that greater economic integration plays an important role in the equilibrium tradeo¤s that emerge in the two-country model across all types of shocks (natural rate shocks derived from productivity, monetary policy shocks, and cost-push shocks). Quantitatively, there are other important economic features that will also have a large e¤ect on the size of the trade-o¤s between in‡ation and slack: On the one hand, cross-country spillovers in productivity (technological di¤usion, a;a ) and features of the labor market such as the inverse of the Frisch elasticity ' matter a great deal for the trade-o¤s arising from productivity shocks and monetary shocks while a common (coordinated) policy rule does not. On the other hand, the inverse of the Frisch elasticity of labor supply ' does not a¤ect the trade-o¤s arising from cost-push shocks but the coe¢ cients on the policy rule do ( c argue based on these results that a monetary policy rule with a stronger anti-in‡ation bias less weight on slack c x) and c c x ). I (or with tends to exacerbate the movements in slack for any given change in in‡ation, albeit only when those movements are driven by cost-push shocks. These …ndings bring new light to the ongoing debate on the role of good luck and better monetary policies (Benati and Surico (2008), Woodford (2010)), suggesting that greater economic integration through trade (structural change) may be more important than often thought. 8. The paper makes the case for greater economic integration as one of the leading causes of the decline in macro volatility characteristic of the Great Moderation period (Stock and Watson (2003), Clark (2009), Martínez-García (2015a)). The results derived from the model show that the New Keynesian mechanism downplays the contribution of productivity shocks and magni…es that of monetary policy shocks and cost-push shocks in the open economy (as well as in the closed economy). Volatility tends to be lower the smaller the inverse of the Frisch elasticity of labor supply ' is. In general, I …nd that the volatility of the aggregates is una¤ected by the degree of openness of the economy ( ) and the trade elasticity ( ) but that open economies tend to experience lower volatility of the cross-country in‡ation dispersion— a feature that becomes more accentuated the larger the trade elasticity is. Interestingly, the relationship is non-linear and in the case of cost-push shocks, greater trade integration when the initial degree of openness is rather low can result in higher macro volatility before macro volatility starts to decline. In turn, pursuing a common policy of in‡ation stabilization with a stricter antiin‡ation bias c adds to the decline in volatility seen during the Great Moderation. The e¤ects on macro volatility (and on the contribution of the di¤erent shocks over the business cycle) from changes in the policymakers’response to slack 5 c x appear less obvious as a factor explaining the Great Moderation. Concluding Remarks Theory suggests that the impact of international monetary policy coordination isolates the e¤ect of the trade channel on the dynamics of cross-country dispersion with no impact on the aggregates. In turn, the …ndings in this paper suggest that there are signi…cant challenges to assess the role a currency union plays because it can lead to indeterminacy in the cross-country dispersion. Inspecting the mechanism more closely allows me to show that open economies tend to experience lower volatility in the cross-country dispersion of in‡ation 41 and varying ‡uctuations in slack di¤erentials for a given change in the in‡ation di¤erential. The model with a common (coordinated) monetary policy appears to provide a consistent explanation for some of the stylized facts that have characterized international business cycles over the past several decades (particularly during the Great Moderation period). In particular, it shows that greater openness tends to lead to lower macro volatility and even to a ‡atter Phillips curve as documented in the existing literature (Stock and Watson (2003), Roberts (2006), Clark (2009), IMF WEO (2013), and Martínez-García (2015a)). Generally, greater trade elasticity tends to magnify the e¤ects of trade integration beyond what conventional measures of trade openness (like simple trade shares) would imply. I also note that the coe¢ cients in the common Taylor (1993)-type monetary policy rule can play an important role as well. In particular, the anti-in‡ation bias of policymakers appears to play a very signi…cant role driving macro volatility. Other structural features of the economy unrelated to policy or the trade channel play a major role in how country-speci…c shocks are propagated internationally as well. For instance, I …nd that cross-country spillovers in the productivity shocks capturing technological di¤usion or the sensitivity of the labor supply to real wages (through the inverse of the Frisch elasticity of labor supply) can have very large e¤ects on the quantitative magnitudes implied by the model. In contrast, persistence generally appears inherited from the properties of the underlying stochastic processes driving the economy and largely una¤ected by the trade channel, by monetary policy, or by any other structural features of the model. Finally, the paper also makes a novel technical contribution by proposing a decomposition method that permits isolating the behavior of a country with respect to the aggregate in a multi-country setting and furthermore to solve the model allowing monetary policies to be asymmetric (independently-set) across countries. 42 Appendix A Proofs Derivation of the bivariate stochastic process for potential output. The potential output of both countries can be expressed as a linear transformation of the productivity shocks as, b yt b yt 1+' +' b at b at 1 1 : Assuming invertibility, the vector of potential output inherits the VAR(1) stochastic structure of the productivity shocks. Accordingly, the potential output process takes the following stochastic form, b yt b yt 1 1 where, a;a a;a a b "at b "at 2 a;a a 2 a 1 1+' +' b "at b "at a 1 0 0 N 2 a ; 2 a a;a 1 1 = 1 1 1 1 ; b yt b yt 1 + ::: 1 ; 1 1 2 : 1 1 Hence, it follows from here that, 1 1 a a;a a;a a 1 1 a a;a a;a a = 1 ; which implies that the structure of the VAR(1) for the potential output inherits the persistence structure of the underlying productivity shocks. Moreover, I can simplify the notation by expressing the innovations to the output potential process in the following terms, b "yt b "yt 1+' +' b "yt b "yt 1 0 0 N b "at b "at 1 ; 1+' +' 2 ; 2 a 1 1 a;a a;a 2 a 2 a 2 a 1 1 T ! ; where, 1+' +' 2 1 a;a = 2 a 1+' +' 2 2 a 1+' +' 2 = 0 @ 2 a 1 2 a a;a 2 a 2 a 1 2 2 ( ) + 2 a;a (1 ) + (1 ) 2 2 ) + a;a (1 ) a;a ( ) + 2 (1 2 ( ) +2 a;a 1 ( )2 +2 (1 )+ a;a (1 )2 ( )2 +2 a;a (1 )+(1 )2 (1 T 1 ) + (1 ) 2 2 ::: ( )2 +2 (1 )+ a;a (1 )2 2 2 ( ) +2 a;a (1 )+(1 ) a;a a;a 1 43 ( ) + 2 (1 ) + a;a (1 ) 2 2 ( ) + 2 a;a (1 ) + (1 ) a;a 1 A: 2 ! Hence, I can de…ne the volatility and the correlation of the output potential innovations in the following fashion, 2 y 2 1+' +' = 2 a = a;a 2 ( ) +2 2 y;y ( ) + 2 (1 2 ( ) +2 )+ (1 a;a (1 a;a ) + (1 (1 a;a ) 2 ; 2 : 2 ) + (1 ) ) Derivation of the bivariate stochastic process for the natural rate of interest. The natural rates of both countries can be expressed as a linear transformation of the productivity shocks as, b rt b rt 1+' +' ( (1 1+' +' 1+' +' + (1 + (1 ) (1 ) (1 )) ( + (1 + (1 ) (1 ) (1 )) ( ( (1 1 2 2 1 b at b at where, + (1 ) (1 ( 1 = (1 a;a ( 2 =( ) (1 ) + (1 1) + (1 a ( ( ) (1 )) '( '( ) (1 ( ))) ( + (1 + (1 ) (1 ) (1 )) + (1 + (1 ) (1 ) (1 )) Et [ b at+1 ] Et b at+1 ))) ))) 1 a a;a 0 ( 1) (2 (1 ) 1)) + ( 1) (2 (1 ) 1) a 1) + (1 '( )@ ' )) ( '( '( ( ( b at b at a;a a 1 ; ) = (1 + (1 (1 ))) 1)(2(1 1)(2(1 ( + (1 ) 1))+ ) 1)2 )+ (1 + + (1 a;a ( ( ( 1)(2(1 1)(2(1 + (1 ) 1))+ ) 1)2 )+ + 1 ))) a;a 2 ) (1 a;a a) ; ) (1 ))) ( (1 + A; 1) a a) : a;a Assuming invertibility, the vector of natural rates inherits the VAR(1) stochastic structure of the productivity shocks. Accordingly, the natural rates take the following stochastic form, 1 b rt b rt b "at b "at 1 2 a a;a 1 2 2 1 a;a a 2 1 0 0 N 2 a ; a;a a;a 2 a 2 a 2 a b rt b rt : 1 1+' +' + 1 1 2 2 1 where, 1 1 2 2 ( = 1 2 1) ( 2 2) 1 2 ( 1) 0 = @ ' 1 ( 2 2) 2 ( + ') (2 (1 ( 1) (2 (1 Hence, it follows that, 44 2 ; 1 ) 1) ) 2 1) + 1 A ( a 1) 2 ( 2 a;a ) : b "at b "at ; 1 1 2 a 2 1 a;a = a;a 1 ( 1) 2 2 2) ( 1 2 a 2 1 1 2 a a;a 2 1 a;a a 1 2 2 a a;a a;a a = 1 ; which implies that the structure of the VAR(1) for the natural rates inherits the persistence structure of the underlying productivity shocks. Moreover, I can simplify the notation as follows, b "rt b "rt 1+' +' b "rt b "rt where, N 1+' +' 2 = = = 0 0 2 2 a 2 2 a 2 2 1 ; 1 2 2 1 2 a a;a 2 a a;a 2 1+' +' 2 1+' +' 2 1 2 2 1 2 a;a 2 2 1 2 a +2 1 a;a 2 a a;a 1 2 2 1 2 a 2 a 1 2 +( 2 + a;a 1 2 2 1 2 2) 2 2) ( 2) 0 @ ( 2 a T 2 a 1) 2 2 +2 1 ( 1 ) + 2 a;a a;a 2 +( 2 a;a 1 2 2 1 ! ; T a;a + 2 a;a 2 ( 1) + 2 1 2 a 2 a 2 a 1) a;a 1) 2 a ; 1 2 ( ( b "at b "at 2 1+' +' 2 2 1+' +' 2 1 2 + ( 2) 2 2 + ( 2) 1 a;a 1 ( 1) 2 2 +2 1 ( 1 ) +2 a;a a;a ( 2 + a;a 2 +( 1 ( 2) 2 2 a;a 2 ! 2 1 ) +2 1 2 ) +2 1 a;a ( 2 + a;a 1 2 +( 1 2) Hence, I de…ne the volatility and the correlation of the natural rate innovations in the following fashion, 2 r r;r = 2 2 a = a;a ( 2 1+' +' ( 1) 2 1) 2 +2 +2 2 1) +2 2 + a;a 1 2 +( ( 1 a;a 1 a;a ( 2) 2) 2 2 +( 2 2) ; 2 : Derivation of the stochastic process for world and cross-country di¤erence variables. The Home and Foreign shocks are summarized in (9) (14) and Table 2. Accordingly, the aggregate shocks are, 1 W b rt = 2 1 b rt b rt 1 ; m bW t = 1 2 1 m bt m bt 1 ; vbtW = u bW t = 1 2 1 u bt u bt 1 Hence, the structure of the shocks can be summarized as follows, W b rt b "rW t = 1 2 = ( = 1 2 1 a + 1 1 a;a 1 ) 1 2 a a;a a;a a 1 b "rt b "rt 1 N b rt b rt 0; 1 b rt b rt 1 1 + 1 21 r 4 + 1 2 1 1 1 1 2 1 1 1 r;r 45 b "rt b "rt 1 b "rt b "rt =( r;r 1 a + 1 1 W a;a )b rt ; 1 : +b "rW t ; ( 2) 2 2) 2 1 A: 1 1 1 1 Analogously, it must follow that, where 1 4 1 1 m bW t where r;r 1 2 = r;r 1 2 = 1 1 1 1 1 m 2 1 1 1 2 = b "mW t 1 4 1 0 0 m m bt m bt 1 m;m m;m 1 1 m b "m t b "m t 1+ m bt m bt 1 1 N 1 1 1 1 2 1 1 4 1+ m;m where 1 4 1 1 1 b "vW t u bW t = = N 21 u 0; 4 1 1 u;u 1 u;u btW 1 uv = +b "vW t ; 1 1 2 1+ Then, I derive the di¤erence forcing processes R b rt 1 = vbtR = b rt b rt 1 u;u 1 1 1 1 1 2 ) where b "rW t 1 1 = = ( a a;a 1 = a;a a;a a ) 1 r;r b "rt b "rt 1 1 1 a 1 1 N 1 Analogously, it must follow that, r;r m bR t where 1 b "mR t 1 = = 1 m 1 1 1 1 1 m;m m;m 1 1 1 0 0 m b "m t b "m t m bt m bt 0; 1 1 u bt u bt 1 1 + N 1 1 1 2 r 1 r;r 1 =( + 1 1 1 2 m =2 1 46 1 m;m b "m t b "m t 1 1+ 2 m m;m 2 . 1+ 2 u u;u . 2 ; W a a;a 1 1 1 1 1 N 0; r;r implies that b "rW t r;r m bt m bt 0; 1 1 b "rt b "rt ; : b "rt b "rt 1 1 + 1 1 + 1 =2 1 m 1 b rt b rt b rt b rt 1 1 ; m bt m bt 1 1 . 2 )u bR t similarly as, btR = (1 m bR t and v Hence, the structure of the shocks can be summarized as follows, R b rt 1 m;m u;u ; m bR t = 2 )u bR t = (1 (1 N 0; m;m implies that b "vW t u;u R b rt , implies that b "mW t 1 r;r 2 +b "mW ; t 1 1 b "m t b "m t 1+ 2 r N 0; bW mm t 1 = 1 Finally, I can say that, vbtW 1 b "m t b "m t 1 2 1 m 0; = 1 2 + 1 1 + 2 1 implies that b "rW t r;r = 1 m;m N 0; 2 b "m t b "m t bR mm t 1 implies that b "mW t 1 ; 2 r +b "mR ; t m;m 1 rt )b +b "rW t ; 1 r;r 1 1 ; N 0; 2 2 m 1 . m;m . Finally, I can say that, vbtR b "vR t = where B 2 )u bR t = (1 (1 (1 2 ) (1 1 1 u 2 )b "uR t 1 1 2 ) u bt u bt 1 (1 u;u u;u 1 1 u 0 0 u + (1 u bt u bt 1 2 ) b "ut b "ut 1 1 1 N =2 1 1 + (1 1 2 ) b "ut b "ut 0; (1 2 ) = 2 2 u btR 1 uv 1 implies that b "vW t u;u The Model Solution x bs;c;j t s;c s;c;j 1;j b t 1 = 1 = s;c 0;j + s;c;j ; t s;c;j t c bs;c;j = t s;c s s;c s b rt = h i Et x bs;c;j t+1 + s bs r rt 1 (' + ) s (' + ) "rs +b "rs t t ; b s s;c s u;u u;u 1 2 2 ) 2 u 1 1 1 (1 i h Et x bs;c;j t+1 + s b rt N 0; c s s;c m b st ; 2 rs h i ) Et bs;c;j t+1 + ; m b st = (' + ) b st 1 mm s s;c s u;u corresponding to the model (67) bs;c;j + t I can express the model whenever vbts = 0 for all s = W; R as follows, s;c 1 ; Natural Rate Shocks and Monetary Shocks = x bs;c;j t 1 b "ut b "ut bs;c;j ; for all s = W; R and for all shocks j = r; m; v; t where the nominal short-term interest rate is given by bis;c;j = t B.1 2 s;c;j N 0; +b "vR t ; N 0; 2 (1 I conjecture that the solution for the endogenous variables x bs;c;j ; bs;c;j ; bis;c;j t t t under a common (coordinated) monetary policy can be expressed as, bs;c;j t 1 1 1 1 1 2 ) 1 1 + c s;c;j bt xx s b rt 1+ "ms +b "ms t t ; b +m b st . m b st ; s N 0; (68) c x 2 ms h i Et bs;c;j t+1 + ::: ; where the forcing processes for s = W; R are characterized elsewhere in the paper. 1 Step 0. I replace the conjecture for x bs;c;j = s;c bs;c;j in the corresponding expectational equations of the t t 0;j model so that I can express both of them in terms of in‡ation alone as, " # i h s s 1 s;c;j 1 s;c;j s s;c;j c s;c s;c s;c b + b = E b (1 ) E b rt m b st ; t t s;c t s;c t+1 t+1 + 0;j bs;c;j t or, simply, 0;j = s;c s;c s (' + ) (' + ) s s Et s b rt h 1 s;c 0;j i bs;c;j t+1 + s;c m b st ; 47 s (' + ) s + 1+ s c x h i Et bs;c;j t+1 + :: ; . bs;c;j = t bs;c;j t = s;c s;c 1+ " s s;c 0;j c (1 (' + ) s;c 0;j + + Step 1. Replace the conjecture for bs;c;j = t expectations in terms of current in‡ation, bs;c;j t bs;c;j t = s;c = s;c 1+ " s s;c 0;j c (1 (' + ) s;c 0;j + ! s s;c s;c;j 1;j b t 1 s;c;j ; t + s s;c s;c 0;j + s + # c x 1+ s;c s;c;j 1;j b t s 1 s ) s s;c s;c 0;j s 1 s i h ) Et bs;c;j t+1 + ! # c x 1+ bs;c;j t s b rt m b st s;c 1 1+ s;c 0;j s c (1 s s;c = s;c 1 s (' + ) = s;c 1;j ) s (' + ) 1 2 rs 0; s + s;c 0;j s bs "rs "rs N r rt 1 + b t ; b t ms ms s "t bt 1 +b "t ; b mm = s;c s;c;j t N 0; 2 s;c;j s b rt m b st ; s;c s;c;j 1;j b t + s (' + ) m b st : to express the in‡ation s s;c s b rt s (' + ) s s b rt m b st ; s s;c s;c 0;j = m b st ; i h Et bs;c;j t+1 + and re-write the system of equations as follows, bs;c;j t s b rt s b rt + m b st ; 1+ s s b rt s;c 1;j c x ; 2 ms N 0; m b st ; : Step 2. Replace the solution for in‡ation in the natural rate shock process as follows, 0 1 s bs;c;r t = bs;c;r t = s;c s;c 0;j s s;c;r r bt 1 +@ 1 s s;c;r r bt 1 +@ 1 0 s;c 1+ s;c 0;j s c (1 s s;c s;c ) s;c 0;j s + Ab "rs "rs t t ; b + 1+ s c x and in the monetary shock process as, bs;c;m t bs;c;m t = = s;c;m m bt 1 s;c;m m bt 1 0 @ 0 @ s s;c s;c 0;j 1 s;c 1+ s;c 0;j s (1 s;c 1 s;c (' + ) c s s 1 s;c 1;j ) (' + ) s;c 0;j + s 2 rs N 0; s (' + ) 1 s (' + ) s;c 1;j 1 Ab "ms "ms t ; b t + s;c 1;j 1 Ab "rs "rs t t ; b N 0; s 1+ s c x ; s;c 1;j 2 ms 1 N 0; 2 rs ; ; Ab "ms "ms t ; b t N 0; 2 ms Step 3. I then apply the method of matching coe¢ cients to equate this formula with the conjecture above, imposing enough restrictions to ensure that both solutions are identical, 48 : 0 @ s;c 1;r s s;c s;c 0;r 1 s;c s;c 0;r 1+ s c (1 1 0 A = @ s;c;r t = @ s;c 1;r ) s r; = 0 s s;c s;c 1 1 s (' + ) s;c 1 s;c 0;r 1+ s s + s;c 0;r c (1 s;c 1;r ) and 0 @ s;c 1;m s s;c s;c 0;m 1 s;c s;c 0;m 1+ s c (1 s;c 1;m ) 1 A s;c;m t s;c 0;j The implicit formula for = 0 @ s;c 1+ s s;c s;c 0;j 1 s;c s;c 0;j 1+ s c+ x ( s c ) s c (1 s;c 1;j ) A; Ab "rs t ; = @ 1 0 @ = s s;c s;c 1 s (' + ) s;c 0;m s;c s;c 0;m 1+ s s + + s s;c s;c 0;m 1 s (' + ) c (1 s;c 1;m ) for j = r; m comes down to, 1 ('+ ) s;c 1;r c x 1 m; 0 s;c 0;j given that s + 1+ 1 s s;c s;c 0;r s (' + ) 1+ 1 s s;c 1;m c x 1 A; Ab "ms t : (' + ) s s;c ; 1 1;j = > 0. Then, it holds that, 1 0 A=@ s;c 1;j 1 s;c 1;j 1 s + c x s s (' + ) + (' + ) s s c s;c 1;j 1 A; and I can summarize the results as follows, s;c 1;r s;c;r t 0 @ = s;c;m t = h i Notice that E s r; = s;c;j t s;c 1;m = = (' + ) s s;c ; j = r; m; 1 1;j (' + ) 1 0 @ s;c 1;r s;c 1;r 1 s + c x 1 s;c 1;m 1 s;c 1;m + 2 s;c;j V h + s s;c;j t i 49 c x = 1 s s (' + ) s (' + ) = 0 and also that for all shocks j = r; m. s;c 0;j m; + s s s (' + ) s s s;c 1;m c ('+ ) (1 Ab "rs t ; s;c 1;r c s;c 1;j )(1 s;c 1;j + s c x )+ s 1 Ab "ms t : 2 s ('+ ) s s ( c s;c 1;j ) 2 js B.2 Cost-Push Shocks s rt = m b st = 0 for all s = W; R as follows, I can express the model whenever b x bs;c;v = t s;c bs;c;v = t s;c vbts = Et x bs;c;v t+1 + s;c (' + ) bts 1 vv s +b "vs "vs t ; b t c (1 Et x bs;c;v t+1 + s (' + ) 1 + s s;c c x N 0; vbts ; 2 vs ) Et bs;c;v t+1 s s;c s s;c s (' + ) c (' + ) + s 1+ c x vbts ; Et bs;c;v t+1 + ::: ; where the forcing processes for s = W; R are characterized elsewhere in the paper. 1 bs;c;v in the corresponding expectational equations of Step 0. I replace the conjecture for x bs;c;v = s;c t t 0;v the model so that I can express both of them in terms of in‡ation alone as, # " s s 1 s;c;v 1 s;c;v c s;c s;c + s;c = Et (1 ) Et bs;c;v (' + ) c vbts ; s;c b t s;c b t+1 t+1 0;v bs;c;v t 0;v = s;c s;c (' + ) 1 + s (' + ) Et s h 1 s;c 0;v c x = s;c bs;c;v t = bs;c;v t = s;c = s;c s;c 1+ " s;c 0;v s c (1 ) Et bs;c;v t+1 ! s;c 0;v + + Step 1. Replace the conjecture for bs;c;v = t expectations in terms of current in‡ation, bs;c;v t 1+ " s;c 0;v (' + ) s c (1 = + s;c 0;v ! + bs;c;v t = vbts s;c s;c 0;v s;c 1 1+ s s;c 0;v s;c 1 = s;c bts 1 vv 1+ s # c x 1+ 1+ c x (' + ) c s c (' + ) +b "vs t ; b "vs t (1 1 s;c 0;v # ) N 0; s + 2 vs : s;c;v t vbts ; s;c s c x s;c s;c;v 1;v b t c + vbts ; s;c s s (' + ) 1 + vbs ; s;c t 1;v s + c x 1+ s c x c x vbts : to express the in‡ation vbs ; s;c t 1;v Step 2. Replace the solution for in‡ation in the natural rate shock process as follows, 50 Et bs;c;v t+1 + ::: (' + ) 1 + 2 s;c;v N 0; (' + ) (' + ) 1 + s Et bs;c;v t+1 + s s;c s;c 0;v s c (' + ) and re-write the system of equations as follows, bs;c;v t + s s;c s;c;v s;c;v ; 1;v b t 1 + t s;c s;c;v 1;v b t ) s 1 s s (' + ) s;c s;c 0;v s 1 s (' + ) s s;c vbts ; or, simply, bs;c;v t i bs;c;v + t+1 c x vbts ; bs;c;v t bs;c;v t 0 @ s;c;v v bt 1 = s;c;v v bt 1 = 0 +@ s s;c s;c 0;v s;c 1 s;c 0;v 1+ (' + ) c s c (1 s;c s;c 1 s;c 1;v ) 1 (' + ) s + s;c 0;v Ab "vs "vs t ; b t s (' + ) 1 + s 1 2 vs N 0; c x + s 1+ s;c 1;v c x ; 1 Ab "vs "vs t ; b t 2 vs N 0; : Step 3. I then apply the method of matching coe¢ cients to equate this formula with the conjecture above imposing enough restrictions to ensure that both solutions are identical, 0 @ s;c 1;v s s;c s;c 0;v 1 s;c s;c 0;v 1+ (' + ) c s c (1 s;c 1;v ) = 1 A = @ s;c;v t s;c 0;v The implicit formula for s;c 0 @ s;c 1+ ( s s;c s;c 0;v 1 s 1+ s;c 0;v c+ x s;c 1 0 @ = s;c s;c 1 s s;c 0;v 1+ 1 s (' + ) s;c s;c 0;v s (' + ) 1 + s;c 0;v s + (' + ) c s c (1 c x + s;c 1;v ) comes down to, s;c 0;v given that v; 0 1 ('+ ) (' + ) c s c (1 s c = s s;c 1;v s;c 1;v c x c ! s;c 1;v c x A; Ab "vs t : ; > 0. Then, it holds that, ) s;c 1;v ) s 1+ s 1+ 1 1 1 0 s;c A=@ s;c s;c 1;v 1 s 1+ s;c 1;v c s;c 1;v c x + s;c (' + ) s 1+ c s;c 1;v c x c (1 ) s;c 1;v and I can summarize the results as follows, s;c 1;v s;c;v t Notice that E [ = v; 0 = @ s;c;v ] t s;c 0;v = s 1+ s c s;c 1 s;c s;c 1;v = 0 and also that s;c 1;v s;c 1;v c x 1+ c s;c 1;v 2 s;c;v V[ s ! ; s;c 1;v c x s;c + s;c;v ] t for the cost-push shock. 51 1+ c (' + ) s s;c 1;v c x s;c = (1 s;c s;c 1;v )( c c (1 (1+ s;c 1;v s )+ s;c 1;v ) s;c c x 1;v s;c 1+ ) ( 1 Ab "vs t : ('+ ) s c x 2 c s;c 1;v )(1 c ) s;c 1;v 2 vs 1 A; C The Building Blocks of the Model Here I describe the main features of the open-economy New Keynesian framework maintaining the symmetry in the structure of both countries but allowing for di¤erences in the monetary policy. I illustrate the model with the …rst principles from the Home country unless otherwise noted, and use the superscript to denote Foreign country variables (or parameters). I focus my attention on the principal elements of departure from previous treatments about the role that the monetary policy regime plays in the international transmission of country-speci…c shocks as that is where the main interest of my analysis lies. The Representative Household. The lifetime utility of the representative household in the Home country is additively separable in consumption, Ct , and labor, Lt , i.e., X+1 =0 Et 1 1 (Ct+ ) 1 1+' 1+' (Lt+ ) ; (69) where 0 < < 1 is the subjective intertemporal discount factor, > 0 is the inverse of the intertemporal elasticity of substitution, and ' > 0 is the inverse of the Frisch elasticity of labor supply. The scaling factor > 0 pins down labor in steady state. The representative household maximizes its lifetime utility in (69) subject to the following sequence of budget constraints which holds across all states of nature ! t 2 , i.e., Z Z Qt (! t+1 ) BtF (! t+1 ) Pt Ct + Qt (! t+1 ) BtH (! t+1 ) + St (70) ! t+1 2 ! t+1 2 BtH 1 (! t ) + St BtF 1 (! t ) + Wt Lt + P rt Tt ; where Wt is the nominal wage in the Home country, Pt is its consumer price index (CPI), Tt is a nominal lump-sum tax (or transfer) from the Home government, and P rt are (per-period) nominal pro…ts from all …rms producing the Home varieties. I denote the bilateral nominal exchange rate as St indicating the units of the currency of the Home country that can be obtained per unit of the Foreign country currency at time t. Similarly, I de…ne the problem of the representative household in the Foreign country. The household’s budget includes a portfolio of one-period Arrow-Debreu securities (contingent bonds) internationally traded, issued in the currencies of both countries and in zero net supply. That is, the pair BtH (! t+1 ) ; BtF (! t+1 ) indicates the portfolio of contingent bonds issued by both countries and held by the representative household of the Home country. Access to a full set of internationally-traded, one-period Arrow-Debreu securities completes the local and international asset markets recursively. The prices of the Home and Foreign contingent bonds expressed in their currencies of denomination are denoted Qt (! t+1 ) and Qt (! t+1 ), respectively.21 Under complete asset markets, standard no-arbitrage results imply that Qt (! t+1 ) = St+1S(!t t+1 ) Qt (! t+1 ) for every state of nature ! t 2 . Hence, Home and Foreign households can e¢ ciently share risks domestically as well as internationally— this implies that the intertemporal marginal rate of substitution is equalized across countries at each possible state of nature, and accordingly it follows that Ct Ct 1 Pt 1 = Pt I de…ne the bilateral real exchange rate as RSt risk-sharing condition in (71) implies that, RSt = Pt 1 St Pt St Ct Ct 1 St Pt Pt 1 : (71) , so by backward recursion the perfect international Ct Ct ; (72) 2 1 The price of each bond in the currency of the country who did not issue it is converted at the prevailing bilateral exchange rate with full exchange rate pass-through under the LOOP. 52 S P C 0 0 0 where is a constant that depends on initial conditions. If the initial conditions correspond P0 C0 to those of the symmetric steady state, then the constant is simply equal to one. Yields on redundant one-period, uncontingent nominal bonds in the Home country are derived from the price of the contingent Arrow-Debreu securities, which results in the following standard stochastic Euler equation for the Home country, # " Pt 1 Ct+1 ; (73) = Et 1 + it Ct Pt+1 where it is the riskless Home nominal interest rate. The representative household’s optimization problem also produces a labor supply equation of the following form, Wt = Pt ' (Ct ) (Lt ) ; (74) plus the budget constraint of the Home country household given by (70), the initial conditions, and the appropriate (no-Ponzi games) transversality conditions. An analogous Euler equation, labor supply equation, and household budget constraint (with the corresponding initial conditions and transversality conditions) can be derived for the Foreign country. Ct is the CES aggregator of both countries’goods for the Home country household and is de…ned as, Ct = (1 ) 1 1 CtH +( ) 1 1 1 CtF ; (75) where > 0 is the elasticity of substitution between the consumption bundle of locally-produced goods CtH and the consumption bundle of the foreign-produced goods CtF . The share of imported goods in the 1 consumption basket of the Home country satis…es that 0 < 2 , so these preferences allow for localconsumption bias. Similarly, the CES aggregator for the Foreign country is de…ned as Ct = ( ) 1 1 CtH + (1 ) 1 1 1 CtF ; (76) where CtF and CtH are respectively the consumption bundle of foreign-produced goods and of the homeproduced goods for the Foreign country household, and identi…es the share of imported goods in the Foreign consumption basket. The consumption sub-indexes aggregate the consumption of the representative household of the bundle of di¤erentiated varieties produced by each country and are de…ned as follows, CtH = Z 1 1 Ct (h) Z 1 ; CtF = dh 0 CtH = Z 1 Ct (f ) 1 1 df ; (77) 0 1 Ct (h) 1 1 dh ; CtF = 0 Z 1 Ct (f ) 1 1 df ; (78) 0 where > 1 is the elasticity of substitution across the di¤erentiated varieties within a country. The CPIs that correspond to this speci…cation of consumption preferences are, h Pt = (1 ) PtH 1 + PtF 1 i11 ; Pt = 53 h PtH 1 + (1 ) PtF 1 i11 ; (79) and, PtH = Z 1 1 Pt (h) 1 1 dh ; PtF = = Z 1 1 Pt (f ) 1 1 df ; (80) 0 0 PtH Z 1 1 Pt (h) 1 1 dh ; PtF = 0 Z 1 1 Pt (f ) 1 1 df ; (81) 0 where PtH and PtF are the price sub-indexes corresponding to the bundle of varieties produced locally in the Home and Foreign countries, respectively. The price sub-index PtF represents the Home country price of the bundle of Foreign varieties while PtH is the Foreign country price for the bundle of Home varieties. The price of the variety h produced in the Home country is expressed as Pt (h) and Pt (h) in units of the Home and Foreign currency, respectively. Similarly, the price of the variety f produced in the Foreign country is quoted in both countries as Pt (f ) and Pt (f ), respectively. Each household decides how much to allocate to the di¤erent varieties of goods produced in each country. Given the structure of preferences, the utility maximization problem implies that the representative household’s demand for each variety is given by, Ct (h) = Pt (h) PtH CtH ; Ct (f ) = Ct (h) = Pt (h) PtH CtH ; Ct (f ) = Pt (f ) PtF Pt (f ) PtF CtF ; CtF ; (82) (83) while the demand for the bundle of varieties produced by each country is simply equal to, CtH CtH = = (1 ) PtH Pt PtH Pt PtF Pt Ct ; CtF = Ct ; CtF = (1 ) PtF Pt Ct ; (84) Ct : (85) These equations relate the demand for each variety— whether produced domestically or imported— to the aggregate consumption of the country. The Firms’Price-Setting Behavior. Each …rm located in either the Home or Foreign country supplies its local market and exports with its own di¤erentiated variety operating under monopolistic competition. I assume producer currency pricing (PCP), so these …rms set prices by invoicing all sales in their local currency. The PCP assumption implies that the LOOP holds at the variety level— i.e., for each variety h produced in the Home country, it must hold that Pt (h) = St Pt (h) (similarly, for each variety f produced in the Foreign country, Pt (f ) = St Pt (f )). Hence, it follows naturally that for the same bundle of varieties, the conforming price sub-indexes in both countries must satisfy that PtH = St PtH (and, similarly, that PtF = St PtF ). PF The bilateral terms of trade T oTt = St PtH de…ne the Home country value of the imported bundle of t goods from the Foreign country in its own currency relative to the Foreign value of the bundle of the Home country’s exports (quoted in the currency of the Home country at the prevailing bilateral nominal exchange rate). Under the LOOP, terms of trade can simply be expressed as, T oTt = PtF PtF = : St PtH PtH (86) Even though the LOOP holds, the assumption of local-product bias in consumption introduces deviations from purchasing power parity (PPP) at the level of the consumption basket. For this reason, Pt 6= St Pt St Pt and therefore the bilateral real exchange rate between both countries deviates from one— i.e., RSt Pt = 54 h +(1 )(T oTt )1 (1 )+ (T oTt )1 i11 6= 1 if 6= 12 .22 Given households’ preferences in each country, the demand for any variety h produced in the Home country is given as, Yt (h) Ct (h) + Ct (h) = (1 = PtH Pt (h) PtH (1 Pt PtH Pt Pt (h) PtH ) Ct + 1 RSt ) Ct + PtH Pt Pt (h) PtH Ct (87) Ct : Similarly, I can derive the demand for each variety f produced by the Foreign country …rms. Firms maximize pro…ts subject to a partial adjustment rule à la Calvo (1983) at the variety level. In each period, every …rm receives either a signal to maintain their prices with probability 0 < < 1 or a signal to re-optimize them with probability 1 . At time t, the re-optimizing …rm producing variety h in the Home country chooses a price Pet (h) optimally to maximize the expected discounted value of its pro…ts, i.e., ) ( X+1 Pt U Ct+ t+ ; (88) Yet;t+ (h) Pet (h) (1 ) M Ct+ Et ( ) =0 Ct Pt+ U subject to the constraint that the aggregate demand given in (87) is always satis…ed at the set price Pet (h) for as long as that price remains unchanged. Yet;t+ (h) indicates the demand for consumption of the variety h produced in the Home country at time t + ( > 0) whenever the prevailing prices have remained unchanged since time t— i.e., whenever Pt+ (h) = Pet (h). An analogous problem describes the optimal price-setting behavior of re-optimizing …rms in the Foreign country. Firms produce their own varieties subject to a linear-in-labor technology. I assume homogeneity of the labor input and within-country labor mobility— although labor remains immobile across countries— ensuring that wages equalize across …rms in a given country but not necessarily across countries. Hence, the (beforesubsidy) nominal marginal cost in the Home country M Ct can be expressed as, Wt At M Ct ; (89) where the nominal wage rate is denoted by Wt and Home productivity shocks are given by At . A similar expression holds for the Foreign country’s (before-subsidy) nominal marginal cost. Productivity shocks are described with the following bivariate stochastic process, 1 At = (A) "at "at a N (At 1) 0 0 a At a 2 a ; a;a 1 e"t ; At = (A) a;a 1 2 a a;a 2 a 2 a ; a (At 1) a;a At a 1 a e "t ; (90) (91) where A is the unconditional mean of the process, a and a;a capture the persistence and cross-country T spillovers, and ("ut ; "ut ) is a vector of Gaussian innovations with a common variance 2a and possibly correlated across both countries a;a . Various rationalizations of the "cost-push" shock UUt in (88) have been advanced in the literature, but following Galí et al. (2001) I simply assume that labor market imperfections give rise to a "wage mark-up," shifting the e¤ective (before-subsidy) nominal marginal costs M Ct exogenously. I say that this "cost-push" 2 2 For more in-depth analysis on the role of international price-setting on PPP and the design of optimal monetary policy, see Engel (2009). 55 shock is described by a bivariate stochastic process of the following form, 1 u Ut = (U ) "ut "ut (Ut N 1) u 0 0 u 1 e"t ; Ut = (U ) 2 u ; u;u 2 u u u;u 2 u Ut u 1 2 u u e "t ; ; (92) (93) where U is its unconditional mean (normalized to one), u captures the persistence of the process, and T ("ut ; "ut ) is a vector of Gaussian innovations with a common variance 2u and also possibly correlated across both countries u;u . The optimal pricing rule of the re-optimizing …rm h of the Home country at time t is given by, i h X+1 Yet;t+ (h) Ut+ M C ( ) Et (CPt+t+) t+ U =0 Pet (h) = (1 ) ; (94) i h X+1 (C ) 1 t+ Yet;t+ (h) ( ) Et Pt+ =0 where is a time-invariant labor subsidy which is proportional to the nominal marginal cost. An analogous expression can be derived for the optimal pricing rule of the re-optimizing …rm f in the Foreign country to pin down Pet (f ). Given the inherent symmetry of the Calvo-type pricing scheme, the price sub-indexes in both countries for the bundles of varieties produced locally, PtH and PtF , respectively, evolve according to the following law of motion, PtH PtF 1 1 = PtH 1 1 = PtF 1 1 + (1 + (1 ) Pet (h) ) Pet (f ) 1 ; (95) ; (96) 1 linking the current-period price sub-index to the previous-period price sub-index and to the symmetric pricing decision made by all the re-optimizing …rms. In turn, the LOOP relates these price sub-indexes to PtH and PtF with full pass-through of the nominal exchange rate St . Fiscal and Monetary Policy. Monopolistic competition in production introduces a distortive steadystate mark-up between prices and marginal costs, 1 , which is a function of the elasticity of substitution across varieties within a country > 1. Home and Foreign governments raise lump-sum taxes from households in order to subsidize labor employment and eliminate the steady-state mark-up distortion. An optimal (timeinvariant) labor subsidy proportional to the marginal cost set to be = 1 in every country neutralizes the steady-state monopolistic competition mark-ups in the pricing rule (equation (94) in steady state). I model monetary policy in the Home country according to a standard Taylor (1993)-type rule on the short-term nominal interest rate, it , i.e., " # x Mt Yt t ; (97) 1 + it = 1 + i M Yt where i denotes the nominal interest rate in steady state, and > 0 and x 0 represent the sensitivity Pt of the monetary policy rule to changes in in‡ation and the output gap, respectively. t Pt 1 is the (gross) CPI in‡ation rate and is the corresponding steady-state in‡ation rate. Yt de…nes the actual output of the Home country, YYt is the output gap in levels, and Y t the potential output level of the Home country. t Alternatively, I also consider a"speci…cation whereby # the monetary policy index responds to aggregates instead— i.e., 1 + it = 1 + i MtW MW W t W YtW Y x where the superscript W denotes the corresponding W t aggregate— in order to evaluate the role of international monetary policy coordination. Moreover, I write an index of monetary policy analogous to (97) for the Foreign country assuming distinct policy parameters 56 > 0 and x 0 in order to allow for independent and asymmetric monetary policy rules across countries. The Home and Foreign monetary policy shocks, Mt and Mt , are described by a bivariate stochastic process: Mt = (M ) "m t "m t 1 m (Mt N 1) 0 0 m m 1 e"t ; Mt = (M ) 2 m ; m;m 2 m m m;m 2 m Mt 2 m m 1 m e "t ; (98) ; (99) where M is its unconditional mean (normalized to one), m captures the persistence of the process, and m T 2 ("m t ; "t ) is a vector of Gaussian innovations with a common variance m and possibly correlated across both countries m;m . D Supplementary Figures These additional …gures are not explicitly referred to in the main body of the paper, but serve to illustrate the main composite coe¢ cients of the two-country model in Table 1. Figure A1. Weight of the Weight Domesticon Natural Rate on Expected Potential Domestic Growth ( ). Figure A1. Domestic Natural Rate Expected Domestic Potential Growth (Θ) 1 0.9 σγ = 0.5 σγ = 1 0.8 σγ = 1.5 Θ σγ = 2 σγ = 2.5 0.7 σγ = 3 σγ = 3.5 σγ = 4 0.6 σγ = 4.5 σγ = 5 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ξ Note: This aspect of the model does not depend on the preference ratio (γ/(ϕ+γ)). The bar with black margins indicates the range of the import share that is referenced in the text. 57 Figure A2. The Productivity Weight on Domestic Productivity in Domestic Potential Figure A2. Domestic Weight on Domestic Potential Output ( ). Output (Λ) (γ/(ϕ+γ)) = 0.1 (γ/(ϕ+γ)) = 0.5 (γ/(ϕ+γ)) = 0.9 2.6 2.6 2.6 2.2 2.2 2.2 σγ = 0.5 σγ = 1 1.8 1.8 σγ = 1.5 1.8 Λ σγ = 2 σγ = 2.5 1.4 1.4 σγ = 3 1.4 σγ = 3.5 σγ = 4 0.6 0.6 0.6 σγ = 4.5 σγ = 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 1 ξ Note: The bar with black margins indicates the range of the import share that is referenced in the text. 58 Figure A3. The Open-Economy Phillips Curve Slope on Domestic Slack Relative to the Figure A3.Closed-Economy The Open-EconomyPhillips Phillips Curve Slope on Domestic Slack Curve Slope ( ). Relative to the Closed-Economy Phillips Curve Slope (κ) (γ/(ϕ+γ)) = 0.1 (γ/(ϕ+γ)) = 0.5 (γ/(ϕ+γ)) = 0.9 1 1 1 0.9 0.9 0.9 σγ = 0.5 σγ = 1 0.8 0.8 σγ = 1.5 0.8 κ σγ = 2 σγ = 2.5 0.7 0.7 σγ = 3 0.7 σγ = 3.5 σγ = 4 0.5 0.5 0.5 σγ = 4.5 σγ = 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.6 ξ Note: The bar with black margins indicates the range of the import share that is referenced in the text. 59 Figure A4. The Slope of the Open-Economy Dynamic IS Curve on the Domestic Interest Figure A4. The Slope of the Open-Economy Rate Gap ( ). Dynamic IS Curve on the Domestic Interest Rate Gap (Ω) 10 9 8 σγ = 0.5 7 σγ = 1 Ω 6 σγ = 1.5 σγ = 2 5 σγ = 2.5 4 σγ = 3 σγ = 3.5 3 σγ = 4 2 σγ = 4.5 1 σγ = 5 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ξ Note: This aspect of the model does not depend on the preference ratio (γ/(ϕ+γ)). 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