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Working P a ~ e r9213
RELATIVE PRICE MOVEMENTS IN DYNAMIC GENERAL
EQUILIBRIUM MODELS OF INTERNATIONAL TRADE
by David K. Backus, Patrick J. Kehoe, and Finn E. Kydland
David K. Backus is a professor of economics at
the Stern School of Business, New York
University; Patrick J. Kehoe is a professor of
economics at the University of Pennsylvania and
a consultant at the Federal Reserve Bank of
Minneapolis; and Finn E. Kydland is a professor
of economics at Carnegie-Mellon University,
Pittsburgh, and a research associate of the
Federal Reserve Bank of Cleveland. The authors
thank the National Science Foundation, the
Institute for Empirical Macroeconomics, and the
Center for Japan-U.S. Business and Economic
Studies faculty fellowship program for financial
support. They also thank Ronald Jones, Maury
Obstfeld, and Chris Stefanadis for helpful
comments.
Working papers of the Federal Reserve Bank of
Cleveland are preliminary materials circulated
to stimulate discussion and critical comment.
The views stated herein are those of the authors
and not necessarily those of the Federal Reserve
Bank of Cleveland or of the Board of Governors
of the Federal Reserve System.
November 1992
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ABSTRACT
We examine the behavior of international relative prices from the perspective
of dynamic general equilibrium theory, with particular emphasis on the
variability of the terms of trade and the relation between the terms of t r a d e
and net exports.
We highlight aspects of the theory t h a t a r e critical in
determining these properties, contrast our perspective with those associated
with the Marshall-Lerner condition and the Harberger-Laursen-Metzler effect,
and point out features of the data that have proved difficult t o explain
within existing dynamic general equilibrium models.
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1
Introduction
Relative
prices
are
a
central
feature
of
both
international trade and open-economy macroeconomics.
the
pure
theory
of
Although the emphasis
differs in the two branches of international economics, t o a great extent the
same theory underlies theoretical and empirical work in each.
of
dynamic general equilibrium theory,
or international
Applications
real business
cycle
theory, continue this tradition by extending, a t the aggregate level, several
of the features of
static trade theory t o dynamic and stochastic settings.
What these applications offer, we think, is a deeper understanding of
the
dynamics of trade and relative prices.
The Marshall-Lerner condition is, without question, the most common link
between trade theory and international macroeconomics.
In trade theory, this
elasticity condition on import demand functions determines the direction of
many comparative statics exercises and serves as a stability condition on an
otherwise
process
static
will
theory,
succeed
macroeconomics,
the
telling
in
same
us
whether
establishing
condition
is
a
disequilibrium
equilibrium.
used
to
In
adjustment
international
establish
a
positive
association between the trade balance and the terms of trade or real exchange
rate.
This is the level a t which the theory is presented in most textbooks
and,
indeed,
in the
popular
Mundell-Fleming
and
Dornbusch
macroeconomic
models of open economies.
The macroeconomic branch of international economics has also developed
insights that a r e largely independent of the theory of trade.
approach focuses on the
and the balance of trade.
theory
is
its
suggestion
accounting relation
The absorption
connecting saving,
investment,
What distinguishes this work from static trade
that
the
trade
balance,
or
the
closely
related
current account, reflects the dynamic decisions by agents t o lend or borrow
in international capital markets.
A critical development f o r understanding
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t h e relation between t r a d e and relative prices i s t h e recognition t h a t any
dependence of t h e t r a d e balance on t h e terms of t r a d e implies, as a m a t t e r of
accounting,
a
similar
relation
with
international
borrowing
and
lending.
This connection was noted by Harberger (1950) and Laursen and Metzler (1950)
and l a t e r incorporated by Obstfeld (1982) and Svensson and Razin (1983) into
explicitly dynamic theories
of
t h e balance
of
trade.
These l a t e r papers
emphasize t h e influence of t e r m s of t r a d e movements on permanent income, and
hence on saving.
They argue t h a t persistent changes in t h e t e r m s of t r a d e
have l a r g e r income e f f e c t s t h a n transitory changes, and t h u s give r i s e t o
potentially different relations between t h e t r a d e balance and t h e terms of
trade.
The
effect,
in
their
analysis,
is
a
comparison
between
two
deterministic equilibria.
We approach t r a d e and relative price dynamics f r o m a somewhat different
theoretical tradition, t h a t of dynamic general equilibrium theory.
Like much
of s t a t i c t r a d e theory, we use competitive consumers and producers.
t h a t work, however, our theory i s explicitly dynamic.
Unlike
And unlike t h e modern
approach t o t h e Harberger-Laursen-Metzler e f f e c t , we consider not comparisons
between
different
stochastic
deterministic
theoretical
equilibria,
economies.
We
but
properties
consider
in
the
experiment t h a t applied economists consider in t h e data:
between
trade
Somewhat t o
and
our
relative
surprise,
price
this
variables
approach
along
leads
to
an
of
equilibria
theory
the
in
same
t h e correlations
equilibrium
substantially
path.
different
views of t r a d e and price behavior than suggested by earlier work.
The application of
dynamic general equilibrium theory t o international
t r a d e h a s also led, in a rapidly growing number of papers, t o attempts t o
quantify t h e theory's properties and compare them t o properties of national
economies.
These a t t e m p t s have led t o a clearer understanding of
which
f e a t u r e s of t h e d a t a can be accounted f o r by t h e present s t a t e of theory and
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which
remain
predictions
anomalous.
than
qualitative
This
quantitative
theory
and helps
approach
generates
sharper
t o focus f u r t h e r theoretical
work on clearly defined issues.
We
elaborate
on
the
twin
themes
of
theoretical
development
and
We s t a r t , in section 2, by
quantitative properties in t h e r e s t of t h e paper.
documenting some of t h e salient properties of aggregate t r a d e and relative
prices
a number
for
explicit
of
t h e o b j e c t s of
industrialized
interest
countries.
in theoretical
These
properties
make
economies and serve as a
basis of comparison with t h e theory.
We develop t h e theory in a series of two-country worlds, highlighting as
we go t h e
roles played
by
d i f f e r e n t theoretical
features.
Section 3 i s
devoted t o a n exchange economy in which each country specializes in a single
traded
good.
Here,
the
variability
of
relative
prices
and
between prices and t r a d e are governed by a single parameter:
of substitution between foreign and domestic goods.
the
relation
t h e elasticity
Some of t h e quantitative
properties of t h i s economy change when we consider preferences t h a t are not
additively
separable
between
consumption of home goods.
4.
foreign
and
domestic
goods
and
that
favor
This aspect of t h e theory i s developed in section
We find, among o t h e r things, t h a t agents' risk aversion plays a r o l e in
both t h e dynamics of t r a d e and prices and t h e relation between these t w o
variables.
In section 5, we compare t h i s theory with alternatives based on t h e
Marshall-Lerner condition and t h e Harberger-Laursen-Metzler
that
our
elasticity
condition
is
not
related
to
effect.
the
condition, which i s always satisfied in our symmetric economy.
t o t h e Harberger-Laursen-Metzler
effect,
we find t h a t
We show
Marshall-Lerner
With respect
t h e persistence
of
shocks i s orthogonal t o t h e relation between relative prices and t h e balance
of trade:
f o r given preference parameters, t h e correlation between t r a d e and
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prices is the same whether price changes last one period o r one hundred.
both
comparisons,
dynamic general equilibrium theory
provides
a
In
different
perspective on trade and price fluctuations than does earlier work.
In the
structure
remaining
that
change
theory's predictions.
sections,
we consider
extensions of
the theoretical
some of
i t s quantitative features and
broaden the
In section 6, we explore shocks t o government spending
a s well a s t o aggregate endowments.
A s one might expect, this extension has
the potential t o change equilibrium comovements considerably.
We find, f o r
example, that the sign of the relation between the t r a d e balance and the
terms of trade depends on the relative sizes of shocks t o endowments and
government spending, a s well a s on the elasticity of substitution.
In section 7, we embed the exchange structure of earlier sections into
an economy with endogenous labor supply and capital formation.
element here is capital formation.
dynamics of trade now reflect,
physical
investment.
The
The critical
With this modification, we find that the
t o a large extent, cyclical fluctuations in
most
striking
result
is
an
asymmetric
cross-correlation
function f o r the trade balance and the terms of
which
the
we
label
S-curve.
This feature
does not
arise
in
trade,
exchange
economies, whereby construction investment is zero and the cross-correlation
function is symmetric.
In this sense, the dynamics of capital formation play
an important role in connecting the dynamics of the trade balance and the
relative price of foreign t o domestic goods.
We conclude with a few remarks on the strengths and weaknesses of
existing
dynamic
general
equilibrium theories
of
international
trade,
and
suggestions f o r directions the theory might take in the future.
2
F i r s t Look a t t h e Data
Since the ultimate objective of our theory is t o account f o r empirical
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regularities, we start by looking at some of t h e properties of international
relative prices.
denoted p,
We focus here on t h r e e variables.
i s t h e r a t i o of
The t e r m s of
t h e import price deflator t o t h e export price
deflator, both taken f r o m national income and product accounts.
denoted nx,
i s t h e r a t i o of
output in current prices.
on t h e country,
trade,
exports minus imports,
Net exports,
in c u r r e n t prices,
to
Real output, denoted y, i s GDP o r GNP, depending
in base-year
(generally 1985).
prices
All t h r e e variables
a r e constructed with d a t a taken from t h e OECD's Quarterly National Accounts.
In table 4.1 we r e p o r t various properties of
t h r e e countries:
t h e t e r m s of
trade f o r
Japan, t h e United Kingdom, and t h e United States.
s t a t i s t i c s r e f e r t o Hodrick-Prescott
filtered variables,
of t r a d e and r e a l output a r e logarithms.
These
and both t h e t e r m s
Many of t h e same properties a r e
reported in Backus, Kehoe, and Kydland (1992b) and Blackburn and Ravn (1991)
for
additional
developed
countries
and
in
Mendoza
(1992) f o r
developing
The sample period in table 4.1 r u n s f r o m 1955:2 t o 1989:4, which
countries.
enables us t o look separately at t h e periods before and a f t e r t h e collapse of
Bretton Woods.
We see, f o r a s t a r t , t h a t movements in t h e t e r m s of t r a d e have been both
variable
and
persistent.
The
standard
deviation
of
terms
of
trade
f1uc;uations
ranges f r o m 2.71 percent in t h e United Kingdom t o 5.97 percent
in Japan.
Both here and in our earlier paper, variability of t h e t e r m s of
t r a d e i s considerably
larger
in
Japan
than
in
other
countries.
We
are
unsure, at t h i s point, how much of t h i s additional variability reflects t r u e
relative price movement and how much has t o do with differences in t h e manner
in which t r a d e prices a r e constructed.
Both Alterman (1991) and Graboyes
(1991) r a i s e questions concerning t h e quality of c u r r e n t t r a d e prices in t h e
United States, and these problems may be g r e a t e r in earlier periods and other
countries.
Alterman (1991) estimates t h a t improved price d a t a exhibit about
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30 percent less variability t h a n those reported here.
Nevertheless, we a r e
probably on s a f e ground in claiming substantial variability of t h e t e r m s of
trade
in
all
countries.
Persistence
is
evident
in
the
autocorrelations,
Table 4.1 also verifies t h a t
which a r e generally in t h e neighborhood of 0.8.
t h e r e h a s been much more variability of t h e t e r m s of t r a d e since t h e advent
of floating exchange r a t e s than before, a f e a t u r e s t r e s s e d by Mussa (1986)
f o r r e a l exchange r a t e s ( r a t i o s of consumer price indexes converted at spot
exchange r a t e s ) .
Standard deviations of t h e t e r m s of t r a d e a r e typically t w o
t o t h r e e times l a r g e r in t h e l a t t e r period.
We also include, in table 4.1, correlations of t h e t e r m s of t r a d e with
net exports and r e a l output.
coherence across periods,
and
net
exports
have
With respect t o net exports, we find g r e a t e r
but little across countries.
generally
been
positively
The t e r m s of t r a d e
correlated
in
S t a t e s and negatively correlated in Japan and t h e United Kingdom.
table 4.2, which covers t h e post-Bretton
the
United
We s e e in
Woods period f o r 10 countries, t h a t
t h e United S t a t e s i s an outlier in t h i s regard:
The correlation between t h e
t r a d e balance and t h e t e r m s of t r a d e i s negative f o r every other country.
With respect t o output, t h e r e h a s been no regularity in t h e correlation with
t h e t e r m s of t r a d e , either over time o r across countries.
The contemporaneous correlation between net e x p o r t s and t h e t e r m s of
t r a d e f a i l s t o capture a n important regularity t h a t appears when we examine
t h e complete cross-correlation
function:
t h e correlations, t h a t is, between
pt and n ~ ~ f +
o r ~various
,
leads and lags k.
r e f e r s t o k=O.
The contemporaneous correlation
For positive k t h e correlations pertain t o net exports and
p a s t prices, and f o r negative k t h e reverse.
We find f o r Japan and t h e
United Kingdom t h a t t h i s function h a s an asymmetric S shape, which we call
t h e S-curve.
For Japan and t h e United Kingdom, t h i s f e a t u r e appears not only
in t h e postwar period as a whole, but in t h e pre-
and post-Bretton Woods
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subperiods as well.
For t h e United States, the same p a t t e r n is evident only
in the earlier period.
Our earlier paper (Backus, Kehoe, and Kydland, 1992b,
figure 1) documents t h i s pattern in eight of eleven countries.
In t h e remainder of t h e paper, we examine these properties from the
perspective
of
a
series
of
successively
more
complex
dynamic
general
equilibrium models, bringing additional d a t a t o bear when t h e theory suggests
it.
For now, we note t h a t t h e t e r m s of t r a d e has been highly variable and
persistent in all t h r e e countries, t h a t i t s correlation with net exports i s
generally negative,
and t h a t t h e cross-correlation
function f o r net exports
and t h e t e r m s of t r a d e is often asymmetric (the S-curve property).
3
A Dynamic Exchange Economy
One of
the
simplest dynamic general equilibrium models of
economy has two countries t h a t t r a d e specialized endowments.
from
an
initial
date
0
to
a
terminal
date
T,
possibly
a world
Let time t r u n
infinite.
The
evolution of this endowment i s stochastic, given by a "Debreu tree" t h a t we
describe in notation adapted from Lucas (1984).
The s t a t e zL, a n element of
t
t h e s e t Z , denotes t h e history of the economy from d a t e 0 through t.
t
of these possible s t a t e s occurs with a probability n ( z 1.
Each
Country 1, which
we call the home country, is endowed with a stream of positive quantities of
the
home
Likewise,
good,
country
denoted
2,
the
t
{y (z 1)
1
foreign
or,
in
country,
shorthand
is
notation,
endowed
with
simply
the
(yl).
positive
sequence (y 1 of quantities of t h e foreign good. We denote the prices of t h e
2
t
t
t
domestic and foreign goods in s t a t e z by ql(z
and q2(z 1.
As in section
t
t
t
2, we define t h e t e r m s of trade, p ( z 1 = q ( z 1/q (z 1, as t h e relative price
2
1
of imports t o exports.
Each country i s represented by a single consumer, who stands in f o r a
large number of like agents.
The preferences of t h e consumer of country i
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a r e characterized by the expected utility function
with O<p<l, U(c) = cl-'/(l-'),
parameter.
We r e f e r t o
and p O .
a s the risk aversion
Both agents consume composites of the foreign and domestic goods,
described by the Armington aggregator functions,
c (a b )
1 1' 1
=
G(al,bl),
c (b , a )
2 2 2
G(b2,a2),
=
with
1-a
1-a l/(l-a)
[wa
+ b
1
,
G(a,b) =
where a. and b. a r e the quantities of the domestic and foreign good consumed
1
1
by the agent
of
country
i.
Thus,
the
agents
of
each country consume
combinations of foreign and domestic goods, a s expressed in the function G.
This
theoretical
device,
due
to
Armington
(19691,
computable s t a t i c general equilibrium trade models.
is
widely
used
in
In this economy i t is
equivalent t o giving consumers preferences over foreign and domestic goods
directly.
The
two
parameters,
substitution and shares of
cr=l/a.
a>O
and
w>O,
govern
the
foreign and domestic goods.
elasticity
of
The elasticity is
With w=l, the two consumers have identical preferences, and with w>l
they exhibit a preference f o r home goods:
If home and foreign goods sell f o r
the same price, agents consume more of the home good than the foreign good.
The budget constraint of the domestic agent is
The foreign agent faces an analogous constraint.
A competitive equilibrium
quantities
( a b. 1 and prices
i' 1
in this economy consists of
(qi) such that
state-contingent
( i ) consumers maximize utility
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given prices and budget constraints and (ii) quantities s a t i s f y t h e resource
constraints,
f o r each s t a t e z
Negishi-Mantel
t
.
We find i t convenient t o compute a n equilibrium using t h e
algorithm in which, f o r any initial distribution of resources,
a competitive allocation i s associated with a P a r e t o optimum.
i s t h e solution t o a
welfare
( A ,A ),
1 2
weights
maximize C.A.u
1 1
problem of t h e form:
choose
quantities
For
some choice of
(a a b b )
1' 2' 1' 2
subject t o t h e resource constraints.
i'
Each optimum
in
each
positive
state
to
The supporting prices
can then be identified with t h e Lagrange multipliers on t h e constraints o r
derived
from
describes
consumers'
this
first-order
procedure
in
a
conditions.
similar
context
Backus
and
(1992,
section 2 )
discusses
alternative
decentralization schemes.
For t h e optimum problem, l e t u s denote t h e Lagrange multipliers on t h e
resource constraints in s t a t e z
foreign goods, respectively.
t
t
t
by ql(z 1 and q2(z ) f o r t h e domestic and
The Lagrange multipliers correspond t o prices
in t h e associated competitive equilibrium.
price
functions
t
t
qi(z 1 = / 3 t n ( z t ) ~ i ( z1,
by
Qi
If, f o r each i, we define spot
then
the
optimum
problem
t
s e p a r a t e s into a number of identical problems, one f o r each s t a t e z , of t h e
form
max
{a ,bi}
subject t o t h e
state are
t
t
t
resource
constraints.
t
/3 n ( z 1 {AIU(G[al(z ),bl(z 11)
+
t
t
A2U(G[b2(z 1,a2(z 11))
The f i r s t - o r d e r
conditions f o r each
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t
t
t
t
t
The t e r m s of t r a d e i s p(z 1 = q ( z )/q ( z 1 = Q ( z )/Ql(z ).
2
1
2
In
this
preferences,
section,
w=l,
for
we
which
restrict
the
ourselves
analysis
can
to
the
be
done
case
of
identical
analytically.
The
equilibrium allocation i s then
f o r countries i=1,2, with consumption s h a r e s s = hil/'/,Y
i
one.
.A!/'
J J
t h a t sum t o
Backus (1992) describes how t h e welfare weights, and hence t h e shares,
a r e related t o t h e endowments.
The properties of interest, however, do not
depend on t h e choice of weights, s o we can skip t h i s additional step.
The
supporting prices a r e , up t o a f a c t o r of proportionality,
and the equilibrium t e r m s of t r a d e i s
The t r a d e balance in country 1 is, in units of t h e domestic good,
In equilibrium, t h e r a t i o of t h e t r a d e balance t o output ( t h e form of t h e
t r a d e variable used in table 4.1) i s
Along any equilibrium path f o r this economy t h e welfare weights,
hi,
are
2
constant.
The consumption shares, s a r e functions of t h e welfare weights
i'
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alone, so they a r e constant as well.
Thus, fluctuations in both t h e t r a d e
balance and t h e t e r m s of t r a d e a r e driven, along any equilibrium path, only
by movements
prices
y2/y1,
and
in t h e endowment
quantities
y2/y1.
ratio,
a r e functions of
a
In t h i s
sense,
equilibrium
single s t a t e variable,
the
ratio
and a r e t h u s indirectly related t o each other.
Although t h e theory i s f a i r l y simple, we can s t a r t t o compare some of
i t s properties with those of t h e d a t a
--
in particular, t h e variability and
persistence of t h e t e r m s of t r a d e , p, and t h e correlation of t h e t e r m s of
t r a d e with n e t exports.
variability
cr=l/a.
of
Clearly, t h e variability of
t h e endowment ratio,
y2/y1,
p i s governed by t h e
and t h e substitution
parameter,
A given amount of variability of t h e endowment r a t i o can produce as
much price variability as we like if cr i s small enough.
As a n example,
consider t h e standard deviation of t h e US t e r m s of t r a d e reported in table
3 percent, f o r t h e period as a whole.
4.1:
This r e f e r s t o t h e s t a n d a r d
deviation of t h e Hodrick-Prescott f i l t e r e d logarithm.
The standard deviation
of t h e f i l t e r e d logarithm of t h e r a t i o of Japanese t o US output i s about 2.2.
To generate t h e amount of price variability we see in t h e United S t a t e s ,
then, we need an elasticity of about cr=0.73 (since 2.2/0.73=3.01.
This i s
only a rough calculation, since t h e d a t a r e f e r t o a world with more t h a n one
country and in which a large p a r t of t h e variability of output appears in
investment, which i s obviously absent here.
played
by
variability.
the
As
elasticity
a
rule,
of
the
But i t i s suggestive of t h e role
substitution
theory
can
in
generating
produce
any
relative
amount
of
price
price
variability we like if t h e elasticity of substitution i s a f r e e parameter.
With
regard
to
persistence,
the
terms
autocorrelation properties of t h e endowment ratio.
the
autocorrelation
of
the
Japan/US
output
ratio
of
trade
inherits
the
To continue our example,
(the filtered
logarithm,
t h a t is) i s 0.7, s o the autocorrelation of t h e t e r m s of t r a d e , in our theory,
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is also 0.7.
table
4.1,
This is slightly less than we see f o r t h e US t e r m s of t r a d e in
but
the
discrepancy
is
not
large,
either
economically
or
statistically.
The final issue concerns t h e relation between t h e t r a d e balance and t h e
t e r m s of trade.
The contemporaneous relation is summarized by
Proposition I.
Let o=l in t h e Armington aggregator, G.
Then t h e relation
between t h e t r a d e balance, n x /y
and t h e t e r m s of t r a d e , p, is governed by
1 1'
c=l/a,
t h e elasticity of substitution between foreign and domestic goods.
c > l , t h e t w o variables
a r e positively related;
if
If
6 1 , they a r e negatively
related.
More
precisely,
consider
x1=y'/y'
with x > x l .
2 1'
ratio, p < p l .
two
states
with
endowment
ratios
Since p i s a decreasing function of t h e endowment
If c > l , then nx/y < nx1/y'.
Now consider the t r a d e balance.
In t h i s case, t h e s t a t e with higher p also has higher nx/y
variables
are,
in
this
x=y /y
and
2 1
sense,
positively
related.
If
and t h e t w o
c<l, the
reverse
is
true.
A similar result is implicit in Stockman and Svensson (1987, section
5.3).
If c = l , t h e t r a d e balance is constant, as noted recently by Cole and
(1991).
Obstfeld
correlation
is
Except
either
+1 o r
for
-1,
nonlinearities
unless
c=l,
and
when
nonstationarities,
the
trade
the
balance
is
constant and t h e correlation i s not defined.
The dynamics of t h e relation between t r a d e and prices, like t h e dynamics
of prices, a r e determined completely by t h e dynamics of t h e endowment ratio.
Except
for
nonlinearities
and
nonstationarities,
the
cross-correlation
function f o r t h e t r a d e balance and t h e t e r m s of t r a d e is t h e same as t h e
autocorrelation function of t h e endowment ratio.
t h e logarithm of
Then if
the endowment
c > l , t h e cross-correlation
By way of example, suppose
r a t i o is AR(l), with autocorrelation
p>O.
function f o r the t r a d e balance and t h e
clevelandfed.org/research/workpaper/index.cfm
t e r m s of t r a d e i s tent-shaped:
correlation between p
t
and nx
The contemporaneous correlation i s +1 and t h e
t+k
is p l k ' .
If c < l , t h e function i s V-shaped.
Even with o t h e r autocorrelation patterns,
t h e cross-correlation function will
be symmetric since t h e autocorrelation function is.
therefore,
i s incapable of
The exchange economy,
reproducing t h e asymmetric correlation functions
pictured in f i g u r e 4.1.
4
P r e f e r e n c e f o r Home Goods
When consumers in t h e t w o countries have preferences t h a t f a v o r t h e i r
respective home goods (w>l in t h e Armington aggregator GI, t h e behavior of
t r a d e and prices changes somewhat.
When preferences f o r domestic and foreign
goods a r e n o t additively separable, we find t h a t t h e r i s k aversion p a r a m e t e r ,
7 , a s well
a s t h e elasticity of
goods, c=l/a,
substitution between foreign and domestic
plays a role in t h e relation between t h e t r a d e balance and t h e
t e r m s of t r a d e .
The simplest case i s y=a:
risk aversion
( 7 ) i s t h e inverse of
substitution
elasticity f o r foreign and domestic goods (c=l/a).
restriction,
preferences,
between
the
considerably.
foreign
including
and
the
domestic
aggregator,
goods,
are
additively
simplifying
the
With t h i s
separable
the
analysis
We compute an equilibrium, as before, f r o m an optimum.
The
r e a d e r may verify t h a t t h e equilibrium allocations, f o r countries i=1,2, a r e
a
a
where t h e consumption shares, s 1 = (ohl)1/~[(whl)1/'+h~'7~, s2 = 1-sa sb =
1' 1
b
b
1/'/[~11/'+(w~2)1/'l,
and s = 1-5
a r e constant along any equilibrium
A1
2
1'
path.
The only difference f r o m t h e symmetric case i s t h a t t h e consumption
s h a r e s now d i f f e r across goods, with l a r g e r values of w leading t o l a r g e r
a
s h a r e s of home good consumption, s and sb
1
2'
The supporting prices are
clevelandfed.org/research/workpaper/index.cfm
so the equilibrium terms of trade is
A s before, the relative price p is driven by the endowment ratio.
The only
change is the factor of proportionality, which does not affect the properties
of the logarithm of p.
by
the
variability
of
The variability of the terms of trade is determined
the
endowment
ratio,
y2/y1,
and
the
elasticity
of
substitution, cr=l/a.
The t r a d e balance f o r country 1 in this economy, expressed a s a ratio t o
domestic output, is
We can see that preference f o r home goods (w>l) has the effect of damping
fluctuations in the balance of trade, since larger values of w imply smaller
Comovements, however, do not change:
values of sb
1'
The sign of the effect
of the endowment ratio on the balance of t r a d e once more hinges on whether
cr=l/a
is
greater
parameters.
or
Thus,
less
the
than
one
properties
and
does
described
not
in
the
depend
on
previous
any
other
section
for
identical preferences (w=l) apply t o this economy a s well.
It should be clear, then, that any influence of
parameter,
w,
on
price
behavior
between domestic and foreign goods
allocation
sign of
solution,
of
goods between
a-y.
but
straightforward.
intuition
--
operate
nonseparabilities
influenced, in this
conditions do not
behind
through
different values of a and y.
countries is
The first-order
the
must
the home preference
the
admit a
equilibrium
The
case, by
the
simple analytic
allocation
is
fairly
A s before, the variables of interest a r e functions of the
clevelandfed.org/research/workpaper/index.cfm
endowment r a t i o , x=y /y
2 1'
In s t a t e s with high values of x, t h e r a t i o s of
foreign t o domestic good consumption, b./ai,
1
r a t i o , c /c
2
of
1'
a r e also high:
t h e endowment r a t i o x.
and t h e aggregate consumption
They a r e increasing functions, in other words,
Neither
property is surprising:
If
t h e r e is
relatively more of t h e foreign good, then in equilibrium both agents consume
relatively
more,
and
aggregate
consumption
favors
the
country
preferences weight t h e foreign good more (country 2, since w>l).
whose
In simple
terms, let t h e foreign and domestic goods be bananas and apples, and suppose
the
foreign
apples.
agent
prefers
bananas
(w>l) and
the
domestic
agent
prefers
Then t h e two statements a r e , f i r s t , t h a t in s t a t e s with relatively
more bananas, both agents consume relatively more bananas than apples and,
second, t h a t aggregate consumption by t h e foreign agent, who h a s a s t r o n g e r
preference f o r bananas, r i s e s proportionately more t h a n consumption by t h e
domestic agent.
Proofs of
both of
these statements a r e included in t h e
appendix.
From t h i s s t a r t i n g point, w e can deduce t h e e f f e c t s on t h e relative
price of t h e foreign good and t h e balance of trade.
From t h e first-order
conditions, reported
Consider t h e price.
in t h e previous section, w e see
t h a t t h e t e r m s of t r a d e satisfies the relations
where f o r convenience w e have dropped explicit dependence of variables on t h e
state, z
t
.
Since b /a and b2/a2
1 1
a r e increasing functions of t h e endowment
r a t i o x , t h e t e r m s of t r a d e is a decreasing function.
t h e relative
supply of
t h e foreign good
Thus, a n increase in
lowers i t s relative price.
With
somewhat g r e a t e r e f f o r t we can characterize t h e magnitude of t h e decline.
The f i r s t s t e p i s t o r e l a t e t h e endowment r a t i o t o bl/al.
constraints imply
The resource
clevelandfed.org/research/workpaper/index.cfm
b
s1=bl/y2
where
is
the
(possibly
state-dependent)
share
of
foreign
good
Thus, with 0 1 , the proportional
consumption by the agent of country 1.
change in b /a can be either greater o r less than the proportional change in
1 1
x, depending on whether sb increases o r decreases with x.
1
The second step
follows from the first-order conditions f o r the foreign good:
Since
c /c
2 1
is
an
increasing
function
of
the
endowment
ratio,
sb
1
is
increasing in x if p a , decreasing if ;r<a, and constant if ;r=a (this being
the additively separable case described earlier).
Thus, the (absolute value
of the) slope of the relation between the logarithms of the terms of trade,
p, and the endowment ratio, x, is greater than a if y>a, less than a if y<a,
and equal t o a ( a s we've already seen) if y=a.
risk
aversion
tends
to
increase
the
variability
Other things equal, greater
of
the
terms
of
trade
relative t o t h a t of the endowment ratio.
The contemporaneous relation between the trade balance and the terms of
t r a d e is characterized by
Proposition 2.
positive
number
Let w>l (preference f o r home goods).
cr*
such
that
if
cr>cr*,
the
trade
Then there exists a
balance,
nx /y
1 1'
is
positively related t o the terms of trade, p, and if crccr*, the two variables
a r e negatively related.
Furthermore, if ;r>l, then cr*<l.
Thus, the risk aversion parameter, ;r, has an influence on comovements between
t h e trade balance and the terms of trade.
The proof
consists largely of
expressing the ratio of the trade balance t o output in a convenient form.
As
in Proposition 1, both the trade balance and the terms of trade a r e monotonic
clevelandfed.org/research/workpaper/index.cfm
functions of t h e endowment r a t i o x=y /y
2 1'
parameter values, decreasing in x.
a
With w=l, sl i s constant.
The t e r m s of t r a d e is, f o r a l l
As f o r t h e t r a d e balance, we let
Since bl/al
is, as we have shown, increasing in
t h e endowment r a t i o x=y /y
t h e proposition follows immediately with cr*=l.
2 1'
a
With w>l, t h e dependence of s on x introduces an additional element of
1
dependence on x.
If ;r>l and a<;r, sa i s increasing in x.
1
t r a d e balance i s decreasing in x.
parameters,
Thus, f o r a s 1 t h e
Since t h e solution i s continuous in t h e
t h i s i s t r u e f o r values of a slightly g r e a t e r than 1 a s well.
For l a r g e values of a , however, t h e t r a d e balance is increasing in x , as we
show
in
cr*=l/a*<l
t h e appendix
(result [A41).
Thus,
there
a r e numbers a*>l and
t h a t divide regions of positive and negative comovement between t h e
t r a d e balance and t h e t e r m s of trade.
In s h o r t , t h i s economy i s much like t h a t of t h e previous section.
Its
similarities and differences a r e illustrated by t h e following example.
Example.
1 2 1 2
Let
there
be
two
s t a t e s , with
unconditional
probabilities
(Nonlinearities a r e irrelevant here, since t w o points c a n always
be connected by a s t r a i g h t line.)
Let t h e aggregate endowments (y y
1' 2
of
t h e domestic and foreign goods be (l+e,l-E) in s t a t e 1, (I-e,l+e) in s t a t e 2,
with e>O.
Thus, t h e standard deviations of y
1
and y
2
a r e e and t h e standard
is, f o r small e ,
deviation of t h e logarithm of t h e endowment r a t i o y /y
2 1
approximately 2e.
The transition probabilities between s t a t e s i and j a r e
The persistence parameter p governs t h e autocorrelation function f o r
any
random variable adapted t o the s t a t e , with t h e k-order autocorrelation equal
to p
k
.
clevelandfed.org/research/workpaper/index.cfm
Case 1:
identical preferences (w=l).
The t e r m s of t r a d e i s
- ~ state 1
which equals [ ( 1 + ~ ) / ( 1 - ~ ) 1 in
The
is, f o r small c ,
approximately
With c=0.05 and a=5, t h e standard deviation i s 0.500.
The r a t i o of
s t a n d a r d deviation of
a(2c).
and [ ( l - c ) / ( l + c l ~ - ~in s t a t e 2.
t h e logarithm
of
p
t h e t r a d e balance t o output is
Thus, f o r cr=l/a<l, t h e correlation between p and t b is -1,
correlation i s +l.
and f o r cr>l t h e
For a = l , t h e t r a d e balance is zero in both s t a t e s and t h e
correlation i s not defined.
Note t h a t t h e persistence parameter, p, h a s no
bearing on t h i s correlation.
The cross-correlation
function f o r t h e t r a d e
balance and t h e t e r m s of t r a d e has t h e f o r m
where cr=l/a.
Case 2:
As noted earlier, t h i s function i s symmetric in k.
preference f o r home goods (w>l).
c=0.05, w=2, and 7=2.
To make t h i s concrete, l e t
With a=5, t h e standard deviation of t h e logarithm of
t h e t e r m s of t r a d e i s 0.497, slightly smaller than in t h e case of identical
preferences.
There
is
a
critical
value
cr*=O.885<1 of
the
elasticity
of
substitution such t h a t f o r cr>cr*, t h e t r a d e balance and t h e t e r m s of t r a d e a r e
positively
correlated,
and
cross-correlation function i s
which remains symmetric.
f o r crccr* they
a r e negatively
correlated.
The
clevelandfed.org/research/workpaper/index.cfm
5
T h e Marshall-Lerner Condition a n d t h e Harberger-Laursen-Metzler E f f e c t
The theory of
data
t h e l a s t two sections
a r e relatively
straightforward.
and i t s relation t o time series
We have
found,
however,
that
this
theory gives u s a much different understanding of t h e relation between t r a d e
and
relative
prices
Marshall-Lerner
than
that
condition
Harberger-Laursen-Metzler
underlying
or
the
effect.
either
1980s
the
widely
cited
on
the
literature
In this section, we contrast our work with
these other approaches.
The elasticity condition of
Proposition 1 i s an example of
a common
result in price theory, t h a t t h e sign of something depends on whether utility
i s more o r less concave than t h e log (cr=l).
surprising about it.
difference between
In t h a t sense, t h e r e i s nothing
What we find interesting in t h e present setting i s t h e
this
condition
and t h e Marshall-Lerner
condition,
often
cited as t h e determining f a c t o r in t h e sign of t h e relation between t h e t r a d e
balance
and
the
terms
of
trade.
The Marshall-Lerner
condition
is
the
centerpiece of textbook treatments of t h i s issue; among many other examples,
see Dornbusch
(1980, p.
591,
Ethier
(1988, pp.
402-71,
and Krugman
and
Obstfeld (1991, p. 423).
The Marshall-Lerner condition applies t o a s t a t i c economy, but we can
review i t s logic in a one-period
treatment follows Kemp (1987).
version of
our two-country
world.
Our
We start by deriving conditions on s t a t i c
import demand functions t h a t allow us t o express t h e balance of t r a d e as a n
increasing
function
of
the
terms
of
trade.
Among
the
conditions
for
equilibrium in t h e s t a t i c economy a r e t h e resource constraint f o r t h e home
good,
a l + a 2 = yl,
and t h e f i r s t agent's budget constraint,
clevelandfed.org/research/workpaper/index.cfm
These conditions together imply balanced trade:
Utility
maximization
by
both
agents,
subject
to
their
respective
budget
constraints, r e s u l t s in import demand functions, say a2(p) and bl(p).
These
functions define net exports as a function of t h e t e r m s of trade:
This function i s increasing in p if
+
&
&*
> 1,
where
=
c
-(abl/ap)[p/bl)
and
=
e*
a r e the
domestic and
foreign
(aa2/ap)(p/a2)
import
elasticities.
The
inequality
is the
well-known Marshall-Lerner condition.
We now apply t h i s analysis t o our economy.
The relevant import demand
functions a r e
bl(p)
where, as before, cr=l/a
domestic goods.
=
yl/(p+pD)
i s t h e elasticity of substitution between foreign and
The import elasticities a r e therefore
E:
=
cr-1
(l+crp )/(p+pC),
clevelandfed.org/research/workpaper/index.cfm
Adding these two expressions, we find t h a t f o r all positive p and cr,
E:
In other words,
+
t h e Marshall-Lerner
economy, regardless of
condition i s always satisfied
t h e value of
foreign and domestic goods.
when
consumers
t h e elasticity of
in this
substitution between
This example illustrates a more general result,
The Marshall-Lerner
cited by Ethier (1988, section A.3):
satisfied
> 1.
E*
in
the
two
countries
condition is always
have
identical
no
connection
homothetic
preferences.
Clearly,
the
Marshall-Lerner
elasticity condition of
condition
has
Proposition 1 and therefore
with
the
has no bearing on the
correlation between t h e t r a d e balance and the terms of t r a d e f o r time series
d a t a generated by economies like ours.
The difference in results stems, we
think, from t h e difference between dynamic modeling and t h e s t a t i c analysis
t h a t underlies the Marshall-Lerner
condition.
Despite t h e intuitive appeal
of t h e l a t t e r , we find t h a t when the dynamics a r e worked out explicitly, we
get a different interpretation of this property of the data.
The theoretical
s t a t e of
t h e art regarding
t h e relation
between
the
t r a d e balance and t h e t e r m s of trade, however, i s not t h e Marshall-Lerner
condition
but
the
1980s
revival
of
the
Harberger-Laursen-Metzler
initiated by Obstfeld (1982) and Svensson and Razin (1983).
effect
These papers,
and others t h a t followed, start with the central insight of the absorption
approach:
that
trade
imbalances
reflect
differences
between
investment.
The theoretical economies of these two papers a r e deterministic
but share with ours t h e f e a t u r e t h a t dynamics a r e explicit.
saving and
They come,
however, t o much different conclusions regarding t h e f a c t o r s t h a t lead t o a
positive association between t h e t r a d e balance and t h e t e r m s of trade.
These
papers suggest t h a t two f a c t o r s a r e critical in determining t h e pattern of
clevelandfed.org/research/workpaper/index.cfm
comovements: the persistence of the shock and the form of dependence of the
discount factor, o r r a t e of time preference,
on future utility.
Transitory
shocks typically lead, in their analysis, t o movements in the terms of trade
and the trade balance of
opposite sign.
We find,
in contrast,
that
the
relation between the trade balance and the terms of trade is independent of
the dynamics of prices.
The aforementioned papers also find that the effect of permanent shocks
depends on the behavior of the discount factor.
p.
100) put
it:
"A permanent
A s Svensson and Razin (1983,
terms-of-trade
deterioration
...
causes
a
deterioration or improvement in the real trade balance, depending on whether
...
the
the r a t e
level
of
of
time preference decreases o r
welfare."
Obstfeld
increases,
(1982) assumes
that
respectively,
the
rate
of
with
time
preference is increasing in utility and therefore predicts a decline in the
trade balance.
production
must
In his words (1982, p.
experience
a
fall
in
2511, "...an
aggregate
economy specialized in
spending
and
a
current
[account] surplus a s a result of an unanticipated, permanent worsening in i t s
terms of trade."
In both papers there is no effect of a permanent change in
price on the t r a d e balance if the r a t e of time preference is constant.
In
our economies, the r a t e of time preference is constant, fixed by the discount
factor
p.
The conclusion should then be t h a t permanent movements in the
terms of trade have no effect on the trade balance.
We find, instead, t h a t
the relation between the trade balance and the terms of trade is determined
by the elasticity of substitution, regardless of the persistence of shocks.
As
in
our
analysis
of
the
between our approach and that of
stems, in part,
Marshall-Lerner
the
difference
the Harberger-Laursen-Metzler
literature
from our definition of
the
condition,
issue.
In our
analysis,
the
relation between the trade balance and the terms of trade pertains t o the
correlation between these two variables f o r a single time series realization,
clevelandfed.org/research/workpaper/index.cfm
like t h e q u a r t e r l y series f o r
States
described
in
table
Japan,
4.1.
the
United
This
Kingdom,
corresponds,
in
and t h e
our
United
theoretical
economies, t o t h e correlation between t h e t w o variables along a n equilibrium
path.
In t h e analysis of
however,
the
Obstfeld (1982) and Svensson and Razin
Harberger-Laursen-Metzler
effect
pertains
between t w o different deterministic equilibria:
to
a
(19831,
comparison
one in which t h e t e r m s of
t r a d e i s high, and one in which i t i s low.
Apparently these t w o thought
experiments emphasize much different f e a t u r e s of t h e theory.
We would a r g u e
t h a t our thought experiment i s closer t o what we have in mind when we compare
theory and data.
A closer look also suggests, as brought out in Backus
(1992) and Stockman and Svensson (19871, t h a t t h e theory requires explicit
treatment
of
the
stochastic
structure
of
the
economy,
something
that
deterministic analysis obviously cannot provide.
One way in which these t w o points of view might be reconciled is t o
consider economies in which agents have more limited ability t o hedge risk
than they do in t h e complete market economies of sections 3 and 4.
e f f e c t , income e f f e c t s play a
1980s analysis of t h e Harberger-Laursen-Metzler
central role.
In t h e
In our economies, however, t h e r e a r e no income e f f e c t s along
a n equilibrium path.
budget constraint.
With complete markets, each agent has a single, date-0
As a result, each has a state-invariant
marginal utility
of income, reflected in t h e constant welfare weights of our optimum problem.
Backus (19921, Kehoe and Richardson (19851, and Mendoza (1992) suggest t h a t
some of t h e flavor of t h e Harberger-Laursen-Metzler
literature may c a r r y over
t o dynamic stochastic settings with some types of market incompleteness.
In short, we find t h a t explicit dynamic stochastic analysis of t r a d e and
relative prices
their
leads t o much
comovements.
Even
different views
the
role
of
of
that
Marshall-Lerner condition, may need t o be reconsidered.
t h e f a c t o r s determining
textbook
standard,
the
clevelandfed.org/research/workpaper/index.cfm
Government Spending
6
The theory t h u s f a r h a s focused on fluctuations in t r a d e and prices
arising f r o m movements in endowments.
Here we consider a n extension t o t h e
exchange economy of section 3 in which we have, in addition, exogenous shocks
t o government
(1992),
Buiter
spending.
(19891,
Related
Hodrick
Reynolds (1991), and Yi (1991).
analyses have been
(19881,
Macklem
provided
(19901,
by
Baxter
Obstfeld
(1989),
We f i n d t h i s extension both interesting in
i t s own right and a useful s t e p toward introducing a wide range of impulses
into dynamic general equilibrium models of trade:
shocks, f o r example, t o
t a x e s , t a r i f f s , and possibly even monetary policies.
To keep t h e analysis
simple, we r e s t r i c t ourselves t o t h e case of symmetric preferences (w=l
in
t h e Armington aggregator GI.
In t h i s new economy, t h e government i s a n additional consumer of goods.
Let us s a y t h a t in s t a t e zt t h e government of country i consumes t h e quantity
t
g.(z ) of i t s home good.
This spending i s financed with lump-sum t a x e s , say
1
t
ri(z
).
An equilibrium then consists of quantities (a and b i ) , prices (qi),
i
a n d government policies ( g . , ~ . ) such t h a t
1
prices
and
budget
1
constraints,
( i ) agents maximize utility given
(ii) quantities
satisfy
the
resource
constraints,
a n d (iii) policies satisfy governments' budget constraints.
With
this
structure,
the
endowments" yi-gi, r a t h e r than y
section 3 with little change.
economy
i'
is
equivalent
to
one
with
"net
and we can apply most of t h e r e s u l t s of
The equilibrium allocation includes
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for
i=1,2,
with consumption
w e l f a r e weights A..
1
shares s
i
x!/'1 /z.x!/'J
=
J
f o r some choice of
The equilibrium t e r m s of t r a d e i s
The variability of t h e t e r m s of t r a d e is governed, then, by t h e variability
of
t h e net endowment r a t i o and t h e elasticity of
practice,
t h e addition of
substitution, cr=l/a.
In
government purchases has l i t t l e influence on t h e
variance of p, since g is only a f r a c t i o n of output and i s generally less
variable.
The same reasoning applies t o persistence:
purchases of
Introducing government
goods and services does l i t t l e t o change our prediction t h a t
relative prices retain t h e persistence of output ratios.
The most interesting consequences of government purchases concern trade.
If w.=y.-g.
1
1
1
i s t h e endowment net of government purchases, net exports in
country 1 a r e
t
nxl(z 1 =
t
(1-sl)wl(z 1
-
t
t -a
t
[w2(z )/wl(z 11 slw2(z
The r a t i o of net exports t o output i s
t
t
t
t 1-a
t
t
nxl(z l/yl(z 1 = ((1-sll - sl[w2(z )/wl(z 11 } wl(z )/yl(z 1.
In our earlier analysis, g
1
w a s z e r o and t h e last t e r m was, therefore, one.
As a result, t h e e f f e c t of t h e endowment r a t i o on t h e t r a d e balance, and t h e
and t h e t e r m s of
association between movements in t h e t r a d e balance, nx /y
1 1'
t r a d e , p, depended only on t h e value of cr=l/a.
positive; f o r cr<l t h e reverse.
of t h e net endowment, w
1'
For cr<l t h e association was
Here we find a n additional f a c t o r , t h e r a t i o
t o t o t a l output, y
1'
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We can g e t some idea of t h e contributions of output and government
spending shocks on t r a d e and price fluctuations by considering special cases.
Consider, f i r s t , t h e case in which g
1
i s proportional t o y
1'
Then wl/yl
is
constant, and t h e relation between t h e t r a d e balance and t h e t e r m s of t r a d e
i s determined by cr, as i t w a s in Proposition 1.
With a>l ( o r u<l), t h e t r a d e
balance and t h e t e r m s of t r a d e a r e positively related along a n equilibrium
path; with cr<l, they a r e negatively related.
y. a r e constant and g
1
1
i s t h e only shock.
Alternatively, suppose outputs
Then t h e t r a d e balance and the
t e r m s of t r a d e a r e positively associated, regardless of t h e value of cr.
This
example i s like many others in economics in which t h e comovement between t w o
endogenous variables depends on t h e source of t h e i r fluctuations.
This analysis suggests a
second
look at t r a d e and price
special attention paid t o government purchases.
t h e r e h a s been l i t t l e regularity
of
government
purchases
a c r o s s countries
relative
to
that
correlation between these two variables.
of
data,
with
As we see in table 4.2,
in
either t h e
real
output,
variability
or
in
the
The same statement applies t o t h e
correlations of government purchases with t h e t r a d e balance and t h e t e r m s of
trade.
That i s not t o say t h a t government purchases have not played a role
in t r a d e and price behavior, but t h a t t h i s role i s not simple enough t o show
u p in summary s t a t i s t i c s of t h i s form.
Froot and Rogoff
(1991) document
somewhat stronger indications of a relation between government spending and
r e a l exchange r a t e s using different methods.
7
T r a d e a n d C a p i t a l Formation
In t h e exchange economies of sections 3 and 4, we compared properties of
t h e d a t a with analogous properties of
theoretical economies.
closer look.
t r a d e and relative prices in simple
This analysis brought up two questions t h a t deserve a
We found, f o r one thing, t h a t t h e variability of t h e t e r m s of
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t r a d e i s governed by the variability of the output ratio and the elasticity
of
substitution between foreign and domestic goods.
By choosing a small
enough elasticity, the theory can generate literally any amount of
price variability.
in
the
theory
elasticity.
not,
for
relative
The question, in this case, i s whether price variability
and
the
data
are
close
for
reasonable
values
of
this
In another respect, we found t h a t the exchange economy could
any
choice
of
parameter
values,
mimic
the
data:
cross-
the
correlation function f o r the t r a d e balance and the terms of trade.
In the
d a t a this function i s generally asymmetric, a feature we document in figure
4.1 and label the S-curve.
In the exchange economy, however, the function i s
symmetric by construction,
since both the t r a d e balance and the terms of
t r a d e a r e functions of the same s t a t e variable.
The question here i s whether
this property changes when we introduce physical capital formation.
The introduction of capital formation brings us closer t o the theme of
the absorption approach t o the balance of payments:
t h a t fluctuations in
t r a d e reflect differences between saving and investment.
A s a matter of pure
accounting, this connection i s undeniable, but i t also shifts one's
away from within-period
save and invest.
relative prices t o the intertemporal
attention
decisions t o
Thus, Sachs (1981) argues that trade deficits often reflect
investment booms and Stockman and Svensson (19871 tie both t r a d e and relative
prices t o fluctuations in, among other things, fixed capital formation.
We
continue this tradition by introducing capital formation t o an economy t h a t
i s otherwise like our earlier ones.
Kehoe, and Kydland (1992b).
The structure i s adapted from Backus,
The emphasis, a s in earlier sections, i s on the
dynamics of the t r a d e balance and the terms of trade.
The theoretical
economy has the following elements.
before, two countries that specialize in different goods.
There are,
as
Preferences of the
representative consumer in each country i a r e characterized by a n expected
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utility function of t h e f o r m
where c
it
and n
it
a r e consumption and hours worked in country i, U(c,l-n) =
[cp(l-n)'-'1'-~/(1-7),
and 720.
The primary difference in preferences f r o m
section 3 i s t h e appearance of
t h e economy of
leisure
in agents'
utility
functions.
Goods in t h e t w o countries, labeled
country
2,
are
produced
using
a
f o r country 1 and
k,
and
capital,
labor,
n,
b
with
8 1-8
homogeneous production functions of t h e same form, F(k,n) = k n
for
linear
.
This
gives r i s e t o t h e date-t resource constraints,
in countries 1 and 2, respectively.
The quantity y
i, measured in units of t h e local good, and a
t w o goods in country i.
exchange
setting
of
it
it
denotes GDP in country
and bit
denote uses of t h e
If k and n a r e constant, t h i s reduces t o t h e pure
section
3,
with
proportionate output fluctuations.
productivity
shocks
giving
The vector zt = (zlt,z2t)
rise
to
i s a stochastic
shock t o productivity whose properties will be described shortly.
Consumption,
investment, and government spending in each country a r e
composites of t h e foreign and domestic goods, with
1-a 1-all/(l-a)
where, as before, G(a,b) = [wa +b
both
positive,
and
the
domestic goods i s cr=l/a.
elasticity
of
.
The parameters a and w a r e
substitution
between
foreign
and
As noted earlier, t h i s s t r u c t u r e i s widely used in
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s t a t i c general equilibrium models of trade.
Capital stocks evolve according
to
where 6 i s t h e depreciation rate.
the
exchange
economy
of
Two differences between t h i s economy and
sections
3-5
are
the
introduction
of
capital
formation and t h e assumption here t h a t government spending may have some
foreign content.
Finally, t h e underlying shocks t o our economy a r e independent bivariate
autoregressions.
where
E
Z
The technology shocks follow
i s distributed normally and independently over time with variance
The correlation between t h e technology shocks, z
vz'
by t h e off-diagonal elements of A and VZ.
1
and z
i s determined
2'
Similarly, shocks t o government
spending a r e governed by
where
g
t
=
( g , g 1 and
It 2t
cg i s distributed
normally
with
variance
V
g'
Technology shocks, z, and government spending shocks, g, a r e independent.
With these elements, and t h e parameter values listed in table 4.3, we
can approach t h e behavior of t h e t e r m s of trade.
The critical parameters,
f o r our purposes, a r e t h e elasticity of substitution, cr, which we s e t equal
t o 1.5, and t h e steady-state r a t i o of imports t o GDP, which w e s e t equal t o
0.15 by choosing w appropriately.
In t h i s benchmark version of t h e economy,
foreign and domestic goods a r e b e t t e r substitutes t h a n they would be with
Cobb-Douglas preferences (cr=l) and imports a r e , on average, 85 percent of
GDP.
The choice of elasticity i s in t h e range of estimates f r o m a large
number of studies, a s documented by Whalley (1985, ch. 4).
Estimates of t h e
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elasticity a r e generally close t o one, and often slightly larger.
The import
s h a r e i s slightly l a r g e r than we s e e in t h e United S t a t e s , Japan, o r a n
aggregate of European countries (with intra-European t r a d e netted out), but
smaller than we s e e f o r most countries individually.
A number
of
properties
of
t h e theoretical
parameter settings a r e reported in table 4.4.
in t h e t e r m s of trade.
economy
with
alternative
Consider, f i r s t , fluctuations
The standard deviation of t h e t e r m s of t r a d e with our
benchmark parameter values i s 0.48 percent, which i s a f a c t o r of s i x less
than we see f o r t h e United S t a t e s in table 4.1.
With smaller values of cr,
t h e theoretical economy generates g r e a t e r variability of t h e t e r m s of trade.
A s illustrated in f i g u r e 4.2, t h e standard deviation of p g e t s larger as we
decrease c , and f o r cr less than 0.03 t h e standard deviation exceeds 2.
Thus,
i t appears t h a t while t h e theory can produce as much variability in t h e t e r m s
of t r a d e as we s e e in t h e d a t a , i t requires a n elasticity of
substitution
much smaller t h a n most existing estimates.
The value of cr required t o match t h e variability of t h e t e r m s of t r a d e
in US d a t a i s substantially smaller in t h i s model (less than 0.03) than in
our calculation in section 3 f o r t h e United S t a t e s and Japan ( f o r which we
estimated t h a t w=0.73 would be sufficient).
of t h i s difference.
Three f a c t o r s account f o r most
The f i r s t i s t h a t t h e theoretical economy has, in t h e
benchmark case, about 25 percent less variability in t h e output r a t i o t h a n we
calculated f o r Japan and t h e United States.
Modifications of t h e theory t h a t
bring t h e magnitude of business cycles closer t o t h e d a t a will also bring t h e
theory
and
data
closer
together
second f a c t o r i s capital formation.
by
setting
8=0
in
the
with
respect
price
variability.
The
If we eliminate capital (which w e can do
production
function),
considerably g r e a t e r price variability, despite
of outputs.
to
the
economy
less variability
The final f a c t o r i s t h e import share.
generates
in t h e r a t i o
If t h e import s h a r e i s
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raised
from 0.15 t o
value of cr.
0.25,
the variability
increases substantially a t
every
For cr=1.5, the benchmark value, the standard deviation of the
relative price rises from 0.48 t o 0.58.
second
A
property
of
the
model
is
the
between net exports and the terms of trade.
contemporaneous
correlation
In the data, this correlation
has been positive f o r the United States and negative f o r Japan and the United
Kingdom
(see
benchmark
expect
table
4.1).
In
the
theoretical
economy
we
find,
Propositions
elasticity
of
increases
with
1 and
substitution.
cr,
and
is
We
2,
this
see
in
positive
for
correlation
figure
4.3
elasticities
is
the
A s we might
parameter values, that the correlation is -0.41.
from
for
sensitive
that
the
greater
the
correlation
than
This feature, too, is strongly influenced by capital formation.
to
cr*=2.76.
In the model
without capital (8=0), the economy is much like that described in Proposition
2, with a critical elasticity cr*=0.94.
For cr>cr*, the trade balance and the
terms of trade a r e positively correlated; f o r crccr*, the reverse.
A third property of interest is the impact of government spending on the
correlation between the trade balance and the terms of trade.
We see in
table
correlation
4.4
that
with
only
shocks t o
government
spending,
the
between net exports and the terms of trade shifts from negative t o positive.
This
mirrors
productivity,
a
z,
similar
and
result
government
in
section
spending,
6.
g,
With
we
find
shocks
that
to
the
both
former
dominate, in the sense that the economy's properties a r e similar t o those
with shocks t o productivity alone.
Finally,
we
look
at
the
exports and the terms of trade.
complete
cross-correlation
function
for
net
A s pictured in figure 4.4, this correlation
has the same asymmetric shape we documented f o r the data in figure 4.1.
Some
intuition f o r this behavior is provided by figure 4.5, in which we graph the
dynamic responses t o a one-standard-deviation
shock t o domestic productivity.
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Following t h i s shock,
increase.
output and t h e relative price of
foreign goods both
Consumption also rises, but by less than output.
Investment grows
initially by much more than consumption, as resources a r e t r a n s f e r r e d t o t h e
home
country
to
exploit
its expected f u t u r e productivity
capital accumulates, t h i s resource t r a n s f e r diminishes.
advantage.
As
The t r a d e balance,
which i s t h e difference, at market prices, between output and t h e sum of
consumption and investment,
by surplus.
exhibits an initial period of
deficit, followed
These dynamic responses give r i s e t o t h e asymmetric cross-
correlation function of figure 4.4.
In short,
much of
t h e intuition f o r t h i s dynamic general equilibrium
t r a d e model i s available f r o m t h e exchange economy of sections 3-5.
What t h e
exchange economy misses completely a r e the dynamics of t h e relation between
t h e t r a d e balance and t h e t e r m s of trade:
t h e asymmetric cross-correlation
function t h a t we documented in t h e d a t a and
cross-correlation
labeled t h e S-curve.
function between these two variables
exchange economy f o r all parameter values.
t h a t t h e dynamics of
The
is symmetric in the
In t h i s section, we have seen
capital formation provide
a plausible basis f o r a n
asymmetric pattern.
8
F i n a l Thoughts
We have argued t h a t a dynamic general equilibrium approach t o aggregate
t r a d e theory provides both a new level of understanding of t h e interrelations
between t r a d e and price movements and a framework in which these relations
can be quantified.
With regard t o t h e former, we have seen t h a t t h e relation
between t r a d e and price variables is much different f r o m t h a t suggested by
t h e Marshall-Lerner
condition cited in most textbooks.
With r e g a r d t o t h e
l a t t e r , we suggest t h a t t h e dynamic relation between t r a d e and t h e relative
price of foreign goods can be understood as a consequence of t h e dynamics of
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capital formation.
Thus, the dynamics of
trade variables a r e inseparable
from the dynamics of the r e s t of the economy.
Future work will undoubtedly focus, however, not on these contributions,
but on dimensions in which the theory, in i t s current incarnation, provides a
relatively poor approximation t o the dynamics of actual economies.
obvious example i s the variability of the terms of trade.
The most
In the economy of
section 7, and in Stockman and Tesar (1991) a s well, the standard deviation
of the terms of t r a d e is substantially smaller than we estimate in the data.
This discrepancy between theory and data helps t o motivate theories in which
monetary
policy
influences
relative
prices
(Grilli
and
Roubini
1992,
Schlagenhauf and Wrase 19921 and in which international market segmentation,
possibly in conjunction with imperfect competition, also plays a part (Dumas
1992, Giovannini 1988, Lapham 1990).
Ongoing research will likely tell us
how important each of these features is, and how they modify the lessons of
the theory outlined above.
clevelandfed.org/research/workpaper/index.cfm
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clevelandfed.org/research/workpaper/index.cfm
APPENDIX:
Proof o f Proposition 2
The algebra behind Proposition 2 i s s t r a i g h t f o r w a r d but f a i r l y tedious.
We start by reducing t h e economy t o t w o equations in t w o unknowns.
the
consumption
shares
of
the
first
agent,
sa=a /y
1 1 1
unknowns
are
b
s =b /y
1 1 2'
The f i r s t - o r d e r conditions and resource constraints then imply
The
and
and
b a
where m=b /a -xs /s >O and v=2/a>0.
With these substitutions, t h e t w o
1 1 - 1 1
a
equations determine s and sb as functions of t h e endowment r a t i o x=y /y
1
1
2 1'
Note t h a t if either w = l
o r a=;r, t h e r i g h t side of
(A21 i s one and sa i s
1
constant.
Preliminaries:
a
Thus, s and s:
1
(a1 If we differentiate t h e f i r s t equation we g e t
a r e positively related
and we can use t h i s relation
b
s u b s t i t u t e out any d s 's we get.
1
b a
( b ) Differentiate t h e r a t i o (s /s 1:
1 1
b a
d ( s /s 1
1 1
=
b
1
(S /S
a
b a v
a a
1 [(s1/S 1 10 -11 dsl/sl.
1
(c) Differentiate m:
dm/m
(dl Inequalities.
b a v
a a
= dx/x + I ( s l / s l l w -11 dsl/sl.
From (A11 and w>l:
b a v
( s l / s l ) w -1
=
a b
(sl-sll/(l-s:)
> 0.
to
clevelandfed.org/research/workpaper/index.cfm
We now prove t h e claims in t h e t e x t .
a
We show t h a t sl is increasing in x if 7>a, and decreasing if 7<a.
1.
We differentiate (A2) and find, a f t e r rearrangement:
If we substitute t h e expression f o r dm/m
[ ( c ) above1 we g e t a n equation of
the form
so that
a
sl i s increasing in x
if
B/A>O,
and decreasing otherwise.
The
coefficient of dx/x i s
Since w>l, B has t h e same sign as 7-a.
a a
The coefficient of d s /s i s
1 1
a
Clearly if 7<a, A i s positive and d s /dx i s negative.
1
A/7
=
1-a
l+v 1-a
2+v) 1-a
(w+m ) (l+w m
) + 8 (1-w
m
,
a b
where 8=(s -s )(7-a)/;r<l.
1 1
in t h i s case, too.
2.
If p a , then
Combining terms makes i t clear t h a t A i s positive
Thus, the sign of dsa/dx i s t h e same as t h e sign of 7-a.
1
An immediate corollary i s t h a t c /c i s increasing in x:
2 1
The f i r s t -
order conditions imply
The behavior of
c
2
/C
1'
sy with respect t o x implies t h a t t h e consumption ratio,
i s increasing in x , as claimed in t h e text.
3.
We now show t h a t b /a =m i s increasing in x.
1 1
From (A31 we have
clevelandfed.org/research/workpaper/index.cfm
From the definition of B,
Since A>O, m is increasing in x.
4.
We turn t o the dependence of the trade balance on x.
A s in the
text,
Differentiating, we find
which is positive if
From (A4) and (AS) we can show that this inequality holds f o r large enough a.
clevelandfed.org/research/workpaper/index.cfm
Table 4.1
Properties of the Trade Balance and the Terms of Trade
Standard
Dev. (%)
Country
Japan
United Kingdom
United States
Autocorrelation
CrossCorrelation
Period
P
nx
P
1955-89
1955-70
1971-89
5.97
2.17
7.76
.97
.98
.94
.87
.73
.87
nx
.77
.66
.83
1955-89
1955-70
1971-89
2.71
1.51
3.38
1.08
.78
1.21
.76
.38
.79
.66
.54
.65
- .54
1955-89
1955-70
1971-89
2.99
1.31
3.84
.45
.30
.55
.82
.65
.84
.80
.79
.80
.30
.28
.30
(P,nx>
(P,Y)
- .46
- .09
- .55
- .51
-. 15
-.60
.41
- .27
.20
.56
.10
- .09
.47
- .23
Variables are p, the terms of trade, logarithm; nx, the ratio of net exports to output; y, real output,
logarithm. Data are quarterly from the OECD's Quarterly National Accounts. Statistics refer to
Hodrick-Prescott filtered variables.
SOURCE: Authors' calculations.
clevelandfed.org/research/workpaper/index.cfm
Table 4.2
The Trade Balance, the Terms of Trade, and Government Purchases
Standard
Deviation
Country
Y
g
Australia
1.47
Austria
Cross-Correlations
@,nx>
(g,nx>
1.90
-.I1
-.I5
1.27
.45
- .25
Canada
1.49
1.16
-.06
France
.91
.66
- .50
Germany
1.47
1.22
-.09
Italy
1.70
.69
Japan
1.48
Switzerland
.I1
-.I5
(g,~) (g,~)
.15
.28
-.02
.17
-.23
-.22
-.45
.24
-.I1
-.I6
.23
-.66
.ll
-.42
1.54
-.51
.19
-.35
.02
1.94
1.01
-.61
-.I5
-.29
.28
United Kingdom
1.60
1.07
-.60
-.06
-.01
.06
United States
1.93
1.47
.31
.ll
-.28
.13
-.01
.12
The sample period is 1971:l to 1989:4. Variables are p, the terms of trade, logarithm; nx, the ratio of net exports to output; y, real output, logarithm; g, real
government purchases of goods and services, logarithm. Data are quarterly from the
OECDYsQuarterly National Accounts. Statistics refer to Hodrick-Prescott filtered
variables.
SOURCE: Authors' calculations.
clevelandfed.org/research/workpaper/index.cfm
Table 4.3
Benchmark Parameter Values
Preferences
Technology
8 = 0.36, 6 = 0.025, a = l l a = 1.5,
import share = 0.15
Forcing Processes
var ~f = var
E:
= 0.008522,
C O ~ T ( E=~0.258
,E~
gt = 0
SOURCE: Authors' calculations.
clevelandfed.org/research/workpaper/index.cfm
Table 4.4
Properties of Theoretical Economies with Capital Formation
Standard
Deviation (%)
Autocorrelation
Correlation
nx
Y
P
nx
Y
P
.30
(.02)
1.38
(. 18)
.48
.63
(. 10)
.83
(.05)
-.64
(.M)
.61
(.07)
Large Elasticity
.33
(.03)
1.41
(.18)
.35
(.05)
.63
(.07)
.64
(. 18)
Small Elasticity
.37
(.03)
1.33
(. 18)
.76
(.07)
.61
(.07)
Big Share
.63
(.04)
1.37
(. 18)
.58
(.08)
.59
(.07)
Small Share
.08
(.01)
1.38
(.18)
.43
(.06)
Two Shocks
.33
(.03)
1.33
(. 15)
Govt. Shocks
.16
(.03)
.17
(.02)
Economy
Benchmark
( W Y ) (WP)
@,P)
(.07)
-.41
(.08)
.49
(.14)
.88
(.03)
-.57
(.08)
-.05
(.09)
.43
(.14)
.63
(. 10)
.77
(.05)
-.66
(.07)
-.80
(.09)
.51
(.16)
.64
(. 10)
.83
(.04)
-.61
(.07)
-.41
(.07)
(. 13)
.62
(.07
.63
(. 10)
.81
(.05)
-.65
(.07)
-.41
(.08)
.48
(.14)
.57
(.07)
.62
(.08)
.65
(.08)
.78
(.06)
-.57
(.15)
-.05
(.17)
.39
(.17)
.30
(.05)
.67
(.ll)
.67
(.08)
.67
(. 11)
-.55
(.13)
1.00
(.00)
-.55
(.13)
.52
Variables are defined in Table 4.1. Statistics refer to Hodrick-Prescott filtered variables. Entries are averages
over 20 simulations of 100 quarters each; numbers in parentheses are standard deviations. Parameters as in
Table 4.3, except large elasticity, a = 2.5; small elasticity, a = 0.5; big share, import share = 0.25; small
share, import share = 0.05; two shocks, mean of g = diag(0.2,0.2), B = diag(0.95,0.95), and V, =
diag(0.0042,0.0042);government shocks, as in two shocks plus z, = 1, all t.
SOURCE: Authors' calc~.llations.
clevelandfed.org/research/workpaper/index.cfm
Fig. 4.1 S-CURVES IN THE DATA
JAPAN
0.75
ALL
U
-0.50 --
\
/ I ' \ \
-0.75
I
I
I
I
I
I
I
1
d-/ /
\
>
I
I
I
I
I
I
I
PRE-72
POST-72
/
I
I
I
I
I
1
1
1
--
,
,
,
I
I
I
I
I
1
UNITED KINGDOM
0.75 7
UNITED STATES
0.75 7
--
\
/
/
/
\
--0.75
!
\
/
ALL
/
PRE-72
POST-72
- - I _
I
I
-8
I
I
I
I
I
I
I
I
-6
-4
I
I
I
I
I
I
I
I
I
I
I
I
-2
0
2
Lag k of p behind nx
SOURCE: Authors' calculations.
I
I
I
I
4
I
I
I
I
6
---I
I
8
clevelandfed.org/research/workpaper/index.cfm
Fig. 4.2 VARIABILmY OF THE TERMS OF TRADE
Elasticity of Substitution, o
SOURCE: Authors' calculations.
clevelandfed.org/research/workpaper/index.cfm
Fig. 4.3 CORRELATION OF THE TRADE BALANCE
AND TERMS OF TRADE
0.8 -
0.6 -0.4 --
3
2
4
5
Elasticity of Substitution, o
SOURCE: Authors' calculations.
6
clevelandfed.org/research/workpaper/index.cfm
Fig. 4.4 S-CURVE FOR THE BENCHMARK ECONOMY
Lag k of p over nx
SOURCE: Authors' calculations.
clevelandfed.org/research/workpaper/index.cfm
Fig. 4.5 DYNAMIC RESPONSES TO DOMESTIC
PRODUCTIVITY SHOCK
...
..
1.00 --
120
8
*,
s
C)
8y 0.80 --
OUTPUT
..
-..
3
X
3m 0.60--
---.- - - - - -
- - - _- - - - - - _- - - - - - - - - _ _ PRODUCI'M~
b
o
u
SHOCK
0.40 --
8
- - 2 -
020--
/
I
I
-
-
TERMS OF TRADE
4
4
-
4
-
4
-
4
-
4
4
4
- 2 .
- 0 2 0 l ~ I 1 1 I I I 1 I I I I I I I I 1 I I I I I I I
0
3
6
9
12
15
18
21
Time in Quarters
SOURCE: Authors' calculatio~is.
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