Download Working Paper No. 321 - Good Policies or Good Luck? New Insights

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)). The bar with black margins
indicates the range of the import share that is considered plausible for the U.S.
60
References
Aoki, M. (1981). Dynamic Analysis of Open Economies. New York, NY: Academic Press.
Benati, L. and P. Surico (2008). Evolving U.S. Monetary Policy and the Decline of In‡ation Predictability.
Journal of the European Economic Association 6 (2-3), 634–646.
Benati, L. and P. Surico (2009). VAR Analysis and the Great Moderation. American Economic Review 99 (4), 1636–1652.
Benigno, P. (2004). Optimal Monetary Policy in a Currency Area. Journal of International Economics 63 (2), 293–320.
Bernanke, B. S. (2007). Globalization and Monetary Policy. Speech given at the Fourth Economic Summit,
Stanford Institute for Economic Policy Research, Stanford, March 2.
Bianchi, F. and A. Civelli (2015). Globalization and In‡ation: Evidence from a Time Vaying VAR. Review
of Economic Dynamics 18 (2), 406–433.
Calvo, G. A. (1983). Staggered Prices in a Utility-Maximizing Framework. Journal of Monetary Economics 12 (3), 383–398.
Carlstrom, C. T., T. S. Fuerst, and M. Paustian (2009). In‡ation Persistence, Monetary Policy, and the
Great Moderation. Journal of Money, Credit and Banking 41 (4), 767–786.
Ciccarelli, M. and B. Mojon (2010). Global In‡ation. The Review of Economics and Statistics 92 (3),
524–535.
Clark, T. E. (2009). Is the Great Moderation Over? An Empirical Analysis. Federal Reserve Bank of
Kansas City Economic Review (Fourth Quarter), 5–42.
Cole, H. L. and M. Obstfeld (1991). Commodity Trade and International Risk Sharing. How Much Do
Financial Markets Matter? Journal of Monetary Economics 28 (1), 3–24.
Draghi, M. (2015). Global and Domestic In‡ation. Speech to the Economic Club of New York, New York,
December 4.
Duncan, R. and E. Martínez-García (2015). Forecasting Local In‡ation with Global In‡ation: When
Economic Theory Meets the Facts. Federal Reserve Bank of Dallas Globalization and Monetary Policy
Institute Working Paper No. 235 .
Engel, C. (2009). Currency Misalignments and Optimal Monetary Policy: A Reexamination. NBER
Working Paper no. 14829 (Cambridge, Mass., National Bureau of Economic Research, April).
Fisher, R. W. (2005). Globalization and Monetary Policy. Warren and Anita Marshall Lecture in American
Foreign Policy, Federal Reserve Bank of Dallas, November 3.
Fisher, R. W. (2006). Coping with Globalization’s Impact on Monetary Policy. Remarks for the National
Association for Business Economics Panel Discussion at the 2006 Allied Social Science Associations
Meeting, Boston, January 6.
Fukuda, S.-i. (1993). International Transmission of Monetary and Fiscal Policy. A Symmetric N-Country
Analysis with Union. Journal of Economic Dynamics and Control 17 (4), 589–620.
Galí, J., M. Gertler, and J. D. López-Salido (2001). European In‡ation Dynamics. European Economic
Review 45, 1237–1270.
Haan, J. (2010). The European Central Bank at Ten, Chapter In‡ation Di¤erentials in the Euro Area: A
Survey, pp. 11–32. Springer.
Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press.
IMF WEO (2013). The Dog that Didn’t Bark: Has In‡ation Been Muzzled or Was it Just Sleeping?
Chapter 3. World Economic Outlook (IMF), April 2013.
Kabukcuoglu, A. and E. Martínez-García (2014). What Helps Forecast U.S. In‡ation? - Mind the Gap!
Mimeo.
61
Kaplan, R. S. (2017). Assessment of Current Economic Conditions and Implications for Monetary Policy.
Essay by President Robert S. Kaplan, Federal Reserve Bank of Dallas, February 13.
Martínez-García, E. (2015a). Globalization: The Elephant in the Room That Is No More. Globalization
and Monetary Policy Institute 2014 Annual Report (February), 2–9.
Martínez-García, E. (2015b). The Global Component of Local In‡ation: Revisiting the Empirical Content
of the Global Slack Hypothesis. In W. Barnett and F. Jawadi (Eds.), Monetary Policy in the Context
of Financial Crisis: New Challenges and Lessons, pp. 51–112. Emerald Group Publishing Limited.
Martínez-García, E. (2016). System Reduction and Finite-Order VAR Solution Methods for Linear Rational Expectations Models. Globalization and Monetary Policy Institute Working Paper no. 285.
September.
Martínez-García, E. and M. A. Wynne (2010). The Global Slack Hypothesis. Federal Reserve Bank of
Dallas Sta¤ Papers, 10. September.
Martínez-García, E. and M. A. Wynne (2014). Assessing Bayesian Model Comparison in Small Samples.
Advances in Econometrics 34, 71–115. Bayesian Model Comparison.
Obstfeld, M. and K. S. Rogo¤ (1996). Foundations of International Macroeconomics. MIT Press.
Roberts, J. M. (2006). Monetary Policy and In‡ation Dynamics. International Journal of Central Banking 2 (3), 193–230.
Stock, J. H. and M. W. Watson (2003). Has the Business Cycle Changed and Why? In M. Gertler and
K. Rogo¤ (Eds.), NBER Macroeconomics Annual, Volume 17, pp. 159–230. MIT.
Taylor, J. B. (1993). Discretion versus Policy Rules in Practice. Carnegie-Rochester Conference Series on
Public Policy 39, 195–214.
Trichet, J.-C. (2008). Globalisation, In‡ation and the ECB Monetary Policy. Lecture at the Barcelona
Graduate School of Economics, Barcelona, February 14.
Woodford, M. (2003). Interest and Prices. Foundations of a Theory of Monetary Policy. Princeton, New
Jersey: Princeton University Press.
Woodford, M. (2010). Globalization and Monetary Control. In J. Galí and M. J. Gertler. (Eds.), International Dimensions of Monetary Policy, NBER Conference Report, Chapter 1, pp. 13–77. Chicago:
University of Chicago Press.
62