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IDE DISCUSSION PAPER No. 269
International Real Business Cycles:
A Re-Visit
Quoc Hung NGUYEN*
December 2010
Abstract
It is well known that several quantitative properties of international real business cycle models
with are at odds with the data. First, the cross-country correlations are much higher for
consumption than for output, while in the data the opposite is true (the BKK puzzle). Second,
cross-country correlations of employment and investment are negative, while in the data they are
positive. This paper quantitatively shows that preferences with a zero income effect on labor
supply help generate a correct cross-country correlation in employment even without any
restrictions on financial markets. In a bond economy, a zero income effect in labor supply,
combined with time-to-build investment, can generate a positive cross-country correlation in
investment, and the BKK puzzle is also resolved when the inter-temporal elasticity of
substitution in labor supply is low.
Keywords: International Real Business Cycles, Income Effects, GHH Preferences
JEL classification: E21, E32, F41
* Research Fellow, Macroeconomic Analysis Group, Development Studies Center,
IDE ([email protected])
The Institute of Developing Economies (IDE) is a semigovernmental,
nonpartisan, nonprofit research institute, founded in 1958. The Institute
merged with the Japan External Trade Organization (JETRO) on July 1, 1998.
The Institute conducts basic and comprehensive studies on economic and
related affairs in all developing countries and regions, including Asia, the
Middle East, Africa, Latin America, Oceania, and Eastern Europe.
The views expressed in this publication are those of the author(s).
Publication does
not imply endorsement by the Institute of Developing Economies of any of the views
expressed within.
INSTITUTE OF DEVELOPING ECONOMIES (IDE), JETRO
3-2-2, WAKABA, MIHAMA-KU, CHIBA-SHI
CHIBA 261-8545, JAPAN
©2010 by Institute of Developing Economies, JETRO
No part of this publication may be reproduced without the prior permission of the
IDE-JETRO.
International Real Business Cycles: A Re-Visit∗
Quoc Hung Nguyen
†
This version: December, 2010
Abstract
It is well known that several quantitative properties of international real business cycle models with are at odds with the data. First, the cross-country correlations are much higher for consumption than for output, while in the data the
opposite is true (the BKK puzzle). Second, cross-country correlations of employment and investment are negative, while in the data they are positive. This paper
quantitatively shows that preferences with a zero income effect on labor supply
help generate a correct cross-country correlation in employment even without any
restrictions on financial markets. In a bond economy, a zero income effect in labor
supply, combined with time-to-build investment, can generate a positive crosscountry correlation in investment, and the BKK puzzle is also resolved when the
inter-temporal elasticity of substitution in labor supply is low.
Keywords: International Real Business Cycles, Income Effects,
GHH Preferences
∗
I am especially grateful to Michael B.Devereux for continual academic support and encouragement.
I would like to thank Paul Beaudry, Viktoria Hnatkovska, Amartya Lahiri, and Henry Siu for their
useful comments and feedback. All errors are solely mine.
†
Institute of Developing Economies. Address: 3-2-2 Wakba, Mihama-ku, Chiba-shi, Chiba, 261-8545
Japan. Tel.: +81-43-299-9693; Fax: +81-43-299-8545. Email: [email protected]
1
Introduction
It is well known that several quantitative properties of international business cycle models are at odds with empirical data. First, the cross-country correlations are much higher
for consumption than for output, while in the data the opposite is true (the BKK puzzle).
Second, cross-country correlations of employment and investment are negative, while in
the data they are positive. Baxter (1995) wrote in her paper that: “It has proved
particularly difficult to write down plausibly-parameterized models which can generate
positive comovement of labor and investment across countries...Thus a major challenge
to the theory is to develop a model which can explain international comovement in labor
input and investment.”
The main reason for the negative comovements is that in the model of one-good
with the internationally mobile capital, there is a strong tendency to move capital to
the most productive location in response to persistent productivity shocks. The movement of capital to the more productive country leads to a rise in labor returns there
accompanied by a fall in labor returns in the other country, hence inducing the negative comovement in labor and output in the model via the substitution effect. As a
result, labor input in different country is negatively correlated unless the cross-country
correlation of the innovations to the country-specific shocks is very high. Baxter (1995)
argues that although the innovations to Solow residuals are positively correlated across
countries, this correlations is not strong enough to overcome the natural mechanisms
leading to negative comovement.
However, there is another effect, which is not often emphasized, that causes the cross
country negative comovement of employment in these models: the income/wealth effect.
Intuitively, when a positive productivity shock hits the foreign country, consumers in
the home country expect “spillover” to the home productivity and an increase in wealth
because of risk sharing via financial markets, hence increasing consumption goods and
leisure. In other words, the positive wealth effect, combined with the substitution effect that already raise leisure with declining home wage rates, magnifies the decrease
of the home country’s labor supply in response to a foreign country’s positive productivity shock. As a result, labor inputs are negatively correlated across country despite
positive correlations in productivity innovations. In this paper, I quantitatively shows
1
that without the income effect on labor supply, relatively small positive correlations in
cross-country productivity innovations, as suggested by empirical studies, are sufficient
to generate significant positive comovement in employment even without any restrictions
on financial markets.1
The paper then shows that when there are delays in investment (time-to-build),
restricted financial markets of state non-contingent bonds, combined with a zero income
effect on labor supply, can account for the positive comovement in investment. The
intuition goes as follows: it takes time for firms to implement many planned projects
each time so by the time firm can implement new projects, productivity differences
between countries have become relatively less significant, which limits the cross-country
capital movement, especially under a restricted financial market.
Finally, the paper shows that under an economy of restricted financial markets and
time-to-build investment, the BKK puzzle is resolved when the intertemporal elasticity
of substitution in labor supply is low.2 Intuitively, restricted financial market structures limit risk-sharing capacity of consumers, hence lower correlation in consumption
across countries. Moreover, when the elasticity is low, labor supply is less responsive
to a country-specific productivity shocks and because of non-separability, the countryspecific effect on consumption is lower. Consequently, the cross-country correlation in
consumption becomes significantly weaker.
Devereux et al (1992) is the first to introduce non-separable preferences without
income effects into two-country one good models. They quantitatively show that there
is a realistic cross-country correlation of consumption in a bond economy model with
independently random productivity shocks. However, they did not compare this value
of correlation in consumption to that of output and did not examine the possibility of
the positive comovement in employment, output and investment. My work also differs
from their paper in another critical aspect. I consider a different productivity shock
structure: there is persistence in productivity shocks and positive correlations in crosscountry innovations, as suggested by empirical studies. Moreover, I show that under
this productivity shock structure, positive comovement in employment and output and a
1
It means that the positive comovement result holds both in the complete markets where where
people in both countries can trade a full-set of contingent claims.
2
The value is still in the range suggested by empirical studies.
2
realistic cross-country correlation in consumption can be generated even under complete
financial markets.
There has been also a growing literature focusing on the role of credit market imperfections in explaining the positive comovements. Backus et al. (1992), Baxter and
Crucini (1995) and Heathcote and Perri (2002) assume financial autarky, i.e. countries
cannot trade financial claims. These papers find that extreme restrictions in the trade
of financial assets, by largely reducing international capital mobility, can generate the
positive cross-country correlation in output. Kehoe and Perri (2002) analyze a model
where the borrowing constraints of a country are endogenous and change over the cycle. In particular, they consider the case in which the borrowing capacity of a country
depends on the value that the country attributes to future access to international financial markets. The borrowing constraint requires that, in each period, allocations
have a higher discounted utility than would prevail if the country were excluded from
all further intertemporal and international trade. When the foreign country is hit by
a positive shock, its output cannot increase too much otherwise the value of defaulting
would become higher than the penalty of being excluded from international financial
markets in the future. Therefore, the flow of capital from the home country to the foreign country is limited, which help to account for the positive cross-country correlations
under relative small correlations in productivity innovations.
My work takes a different approach from the above literature in examining the
anomalies between the data and theoretical models. It shows that many properties
of international business cycle models can be reconciled with the data by simply assuming a zero wealth effect in labor supply via non-separable GHH preferences, hence
attempting to provide insights from different point of view.
The structure of this paper is as follows. Section 2 introduces models’ setting and
calibration. Section 3 discusses quantitative results. Conclusions follow in section 4.
2
Model
The world consists of two countries: the home country and the foreign country. Consumers in each country value leisure and consumption of the single tradable produced
good while labor is internationally immobile. Firms in each country produce the single
3
good by identical Cobb-Douglass production functions and are subject to exogenous
shocks to total factor productivity.
In this model, the foreign country is distinguished from the home country by means
of a star attached to all foreign-country variables. When there are no stars, the variable,
parameter, or function is assumed to be identical across countries. All variables are in
per capita terms.
Preferences. The representative household in each country maximizes its expected
lifetime utility defined over random sequences of consumption goods (ct ) and labor disutility (lt ):
U = E0
∞
β t U ct , lt ,
Home country;
(2.1)
β t U c∗t , lt∗ ,
Foreign country
(2.2)
t=0
∗
U = E0
∞
t=0
I consider two types of preferences in this paper. The first one is “standard” separable
Cobb-Douglas preferences, which have been used in the international business cycle
literature of one good two country models.3
1−σ
cγt (1 − lt )1−γ
−1
U(ct , lt ) =
1−σ
(2.3)
Parameter γ will determine the value of hours at the steady state.
The second one is non-separable GHH preferences:
lω
(ct − κ ωt )1−σ − 1
U(ct , lt ) =
1−σ
(2.4)
Similar to γ, parameter κ will determine the value of hours at the steady state and parameter ω determine the elasticity of labor supply. It is well known that GHH preferences
imply zero elasticity of leisure to income.
Technology. Production functions are in Cobb-Douglass forms, hence exhibiting constant returns to scale; production of the single final good requires the input of both labor
and capital. Capital used in production in a specific country is not necessarily owned
3
The type of preferences was used by King, Plosser, and Rebelo, 1988
4
by residents of that country; thus, kt represents capital in place in the home country,
not necessarily capital owned by residents of the home country. Labor is internationally
immobile.
yt = At (kt )α (lt )1−α
yt∗ = A∗t (kt∗ )α (lt∗ )1−α
Home country;
(2.5)
Foreign country.
(2.6)
where At represents the stochastic level of productivity home country and kt is the
capital stock installed home country at time t.
Productivity evolves according to the bivariate autoregressive process:
a1 a2 log(At )
t+1
log(At+1 )
=
+ ∗
a2 a1 log(A∗t )
t+1
log(A∗t+1 )
(2.7)
where a1 measures the persistence in productivity shocks and a2 measures the degree
of international spillovers. The variance in the innovations is denoted by σ2 and the
correlation between t and ∗t is σ12 .
I follow Kydland and Prescotts (1982) to assume (four-period) time-to-build investment technology. That is, in the time-to-build models:
it = ω1 s1t + ω2 s2t + ω3 s3t + ω4 s4t
(2.8)
where it is the investment at time t and sjt is the volume of projects j periods away
from completion at the beginning of period t and ωj is the resource cost associated
with work on a project j periods away from completion, for j = 1, 2, 3, 4. Investment
projects progress according to sj,t+1 = sj+1,t for j = 1, 2, 3; and starts during period t
are represented by s4t . The capital stock thus evolves according to:
2
φ s1t
−δ
kt+1 = (1 − δ)kt + s1t − kt
2
kt
(2.9)
This law of motion includes investment adjustment costs governed by φ and is such that
there are no adjustment costs in steady state. Notice that when ω1 = 1, ω2 = ω3 = ω4 =
0, we have the regular one-time-to-build investment.
Market Structure. I assume that there is frictionless international trade in output,
so that there is an unified world resource constraint for the single produced good:
(yt − ct − it ) + (yt∗ − c∗t − i∗t ) = 0
5
(2.10)
Regarding the financial structure, I consider both complete-markets and bond economies.
The difference of the two structures lies in the number of assets available to the agents.
When markets are complete, the representative agents in both countries can trade a
full set of contingent claims. Accordingly, the budget constraint of the home country’s
representative household can be expressed as:
ct + it +
p(st+1 , st )b(st+1 ) = yt + b(st )
(2.11)
st+1
where st indicates the state in period t and b(st+1 ) denotes the quantity of contingent
claims purchased in period t and paying off one unit of consumption the following period,
conditional on the state of the world being st+1 next period. p(st+1 , st ) denotes the price
of these contingent assets.
By contrast, in a bond economy, there is only one-period real discount bonds. Let
bt+1 denote the per capita quantity of these discount bonds purchased by the home
economy, which mature in period t + 1, and pbt is its price at time t.
The flow budget constraints for the bond economy are:
ct + it + pbt bt+1 +
πb
(bt+1 )2 = yt + bt ;
2
πb ∗ 2
(bt+1 ) = yt∗ + b∗t ;
2
The world market clearing condition for bonds is:
c∗t + i∗t + pbt b∗t+1 +
b(st+1 ) + b∗ (st+1 ) = 0;
4
home country
(2.12)
foreign country
(2.13)
complete markets
bt+1 + b∗t+1 = 0 bond economy
(2.14)
(2.15)
Calibration. This paper follows closely calibration from Baxter and Crucini (1995),
Kollmann (1996), and Kehoe and Perri (2002). See Table 1 for details.
In particular, parameter ω, which determines the inter-temporal elasticity of substitution in labor supply,
5
is set to 2 as a benchmark. The unit benchmark elasticity
is equal to the value implied by standard preferences as in form (2.3). For sensitivity
4
Following Boileau et al (2008) and others, I impose quadratic portfolio adjustment costs to induce
stationarity in incomplete markets. See Boileau et al (2008) for more details about other methods.
1
5
The inter-temporal elasticity of substitution in labor supply is approximately ω−1
.
6
analysis, ω is set from 1.58
6
to 6, which then implies the intertemporal elasticity of
substitution varies from 1.7 to 0.2 accordingly. This is a range suggested by empirical
studies.
Parameters, κ, γ in GHH preferences and Cobb-Douglass preferences, respectively
are chosen such that the hours of working in the steady state are 0.25.
Portfolio adjustment costs parameter, πb is set to 0.0005 such that the implied volatility of the ratio of net exports to output in bond economy models is the same in corresponding financial complete market models.
In regular one-time-to-build investment environment, investment adjustment cost
parameter, φ is set such that the ratio of investment volatility to that of output match
the data, which is equal to 3.24. In the four-time-to-build investment environment, φ is
set to zero.
Table 1: Parameter values
Parameters
Preferences
β = 0.99, σ = 2
hours at s.s l = 0.25
ω = 2 as benchmark
Technology
α = 0.3, δ = 0.03
ω1 = ω2 = ω3 = ω4 = 0.25
Productivity shocks
a1 = 0.95, a2 = 0
var(1 )=var(2 )=0.072 , corr(1 ,2 )=0.25
Adjustment cost
3
πb = 0.0005
Quantitative Results
There are four types of models with different financial market structures and different
investment technologies: models with complete financial markets and models with restricted one period state non-contingent bonds, models with regular one-time-to-build
6
1.58 is the value used by Devereux et all (1992) in their two-country model; the value was first used
by Greenwood et al (1988) in a closed-economy model.
7
investment and models with four-time-to-build investment. I solve and simulate these
models by the perturbation method 7 and Figures 1-4 present impulse responses of these
models with respect to one unit of positive shock in the foreign countries. For comparison, I combine impulse responses of models with Cobb-Douglas separable preferences
and those with non-separable GHH preferences in one figure.
8
Table 2 and Table 3
provide business cycle statistics from data and those implied by these models.
Figures 1-4 show differences between the impulse responses of models with GHH
preferences and those with Cobb-Douglas preferences, particularly in consumption and
labor input. In response to a positive shock in the foreign country, consumptions in
both countries exhibit a relatively smooth and similar dynamic pattern for models with
Cobb-Douglas preferences while for GHH preferences, the impulses are more responsive
and move in opposite directions. This explains why the cross-country correlation in
consumption is higher in Cobb-Douglas preferences. The impulse responses also present
evidence of more risk-sharing in the complete markets, where consumption in the home
country increases relatively more than it does in a bond economy.
By contrast, employment (hours) in home country decreases significantly less in
models with GHH preferences. The reason is straightforward. With a zero wealth effect
in labor supply, consumers in the home country consume less leisure in response to a
positive productivity shock in the foreign country, hence reducing labor supply by a
relatively smaller amount compared to a positive wealth effect case. Tables 2 and 3
show that the cross-country correlations in labor and output are positive in the models
with GHH preferences while it is negative in the models with Cobb-Douglas preferences.
Figures 3- 4 show impulse responses of models with time-to-build investment. As
pointed out in Kydland et al (1982), models with time-to-build display a more persistence
in all time series variables. The above arguments for consumption and labor input
dynamics hold for models with time-to-build investment. However, there is a significant
difference between models with complete markets and models with restricted financial
markets. Restricted financial markets, combined with time-to-build investment, can
largely reduce the international capital mobility. The last column of Table 2 shows that
7
8
For more details, see Schmitt-Grohe and Uribe (2004)
In each figure, ω is chosen such that the implied standard deviations of output for the two types of
preferences are the same.
8
there is a positive cross-country correlation in investment in a bond economy model with
time-to-build investment.
Table 4 presents the implied business cycle statistics when I vary the intertemporal elasticity of substitution in labor supply from 1.7 to 0.2, which are in the range
of empirical studies. The cross-country correlation in consumption decreases when the
elasticity is lower or labor supply becomes less responsive to shocks. As a result, when
the elasticity is about
1
3
the PKK puzzle is resolved, i.e., the cross-country correlation in
consumption is smaller than that of output. Intuitively, restricted financial market structures limit risk-sharing capacity of consumers, hence lower correlation in consumption
across countries. Moreover, when labor supply is less responsive to a country-specific
productivity shocks and because of non-separability, the country-specific effect on consumption is lower. Therefore, the cross-country correlation in consumption becomes
relatively weaker.
4
Conclusions
This paper quantitatively shows that the anomaly between models and empirical data in
cross-country correlation in labor can be resolved by the introduction of non-separable
GHH preferences. It also proposes solutions to anomalies in cross-country correlation in
investment and the BKK puzzle.
9
Figure 1: IRs: Complete Markets, One-time-to-build
Productivity shock
Consumption to Af
1
1
c
cf
0.8
0.5
0.6
0.4
0
0.2
a
af
−0.5
0
5
10
15
0
20
0
5
Hours to Af
10
15
20
Investment to Af
1.5
20
1
i
if
10
0.5
0
0
−0.5
−1
−10
h
hf
0
5
10
15
−20
20
0
5
Output to Af
15
20
TB to Af
2
4
y
yf
1.5
2
0.5
1
0
0
−0.5
−1
0
5
10
15
tby
3
1
−1
10
−2
20
0
5
10
15
Solid line: GHH preference; Dashed line:Cobb−Douglas preference
10
20
Figure 2: IRs: Bond Economy, One-time-to-build
Productivity shock
Consumption to Af
1
1
c
cf
0.8
0.5
0.6
0.4
0
0.2
0
a
af
0
5
10
15
−0.5
20
0
5
Hours to Af
10
15
20
Investment to Af
1
20
i
if
10
0.5
0
0
−10
h
hf
−0.5
0
5
10
15
−20
20
0
5
Output to Af
10
15
20
TB to Af
2
4
y
yf
1.5
tby
3
1
2
0.5
1
0
0
−0.5
−1
0
5
10
15
−1
20
0
5
10
15
Solid line: GHH preference; Dashed line:Cobb−Douglas preference
11
20
Figure 3: IRs: Complete Markets, Four-time-to-build
Productivity shock
Consumption to Af
1
1.5
c
cf
1
0.5
0.5
0
0
a
af
−0.5
0
5
10
15
−0.5
20
0
5
Hours to Af
10
15
20
Investment to Af
1.5
20
1
i
if
10
0.5
0
0
−0.5
−1
−10
h
hf
0
5
10
15
−20
20
0
5
Output to Af
15
20
TB to Af
3
3
y
yf
2
tby
2
1
1
0
0
−1
−1
−2
10
0
5
10
15
−2
20
0
5
10
Solid line: GHH preference; Dashed line: Cobb−Douglas preference
12
15
20
Figure 4: IRs: Bond Economy, Four-time-to-build
Productivity shock
Consumption to Af
1
1
c
cf
0.8
0.5
0.6
0.4
0
0.2
a
af
−0.5
0
5
10
15
0
20
0
5
Hours to Af
15
20
Investment to Af
1
10
0.5
5
0
0
−0.5
−1
10
−5
h
hf
0
5
10
15
i
if
−10
20
0
5
Output to Af
10
15
20
TB to Af
2
2
y
yf
1.5
tby
1
1
0.5
0
0
−1
−0.5
−1
0
5
10
15
−2
20
0
5
10
Solid line: GHH preference; Dashed line:Cobb−Douglas preference
13
15
20
Table 2: Business Cycles Statistics: GHH Preferences
Economy with
One-time-to-build
Four-time-to-build
Data
CM
Bond
CM
Bond
Consumption
0.79
0.71
0.81
0.72
0.88
Investment
3.24
3.24
3.24
3.24
1.83
Employment
0.63
0.5
0.5
0.5
0.5
Net Exports/GDP
0.09
0.65
0.64
0.69
0.43
Consumption
0.87
0.96
0.96
0.96
0.91
Investment
0.93
0.5
0.5
0.43
0.7
Employment
0.86
1
1
1
1
Net Exports/GDP
-0.36
0.17
0.06
0.22
0.21
GDP
0.51
0.2
0.19
0.21
0.51
Consumption
0.32
0.68
0.28
0.69
0.72
Investment
0.29
-0.7
-0.7
-0.7
0.18
Employment
0.43
0.2
0.19
0.21
0.51
Statistics
Std.dev rel. to GDP
Domestic Comovement
Corr. with GDP
International Correlation
Home and Foreign
Notes: The statistics in the Data column are taken from Kehoe and Perri (2002), which are calculated
from U.S. quarterly time series, 1970:1-1998:4 and an aggregate of 15 European countries. All relevant
time series, except ratio of net exports to output, have been logged and HP-filtered. CM indicates a
complete financial market economy while bond indicates an economy where people can only trade
one-period non-contingent bond. The inter-temporal elasticity of substitution in labor supply is set to
1
14
Table 3: Business Cycles Statistics: Cobb-Douglas Preferences
Economy with
One-time-to-build
Four-time-to-build
Data
CM
Bond
CM
Bond
Consumption
0.79
0.41
0.51
0.4
0.63
Investment
3.24
3.24
3.24
4.33
3.47
Employment
0.63
0.58
0.61
0.58
0.49
Net Exports/GDP
0.09
0.78
0.81
1.10
0.83
Consumption
0.87
0.7
0.58
0.7
0.7
Investment
0.93
0.51
0.51
0.27
0.41
Employment
0.86
0.93
0.86
0.93
0.79
Net Exports/GDP
-0.36
0.52
0.5
0.48
0.38
GDP
0.51
-0.45
-0.35
-0.5
0.04
Consumption
0.32
0.86
0.42
0.85
0.57
Investment
0.29
-0.76
-0.71
-0.88
-0.61
Employment
0.43
-0.87
-0.86
-0.9
-0.64
Statistics
Std.dev rel. to GDP
Domestic Comovement
Corr. with GDP
International Correlation
Home and Foreign
Note: The statistics in the Data column are taken from Kehoe and Perri (2002), which are calculated
from U.S. quarterly time series, 1970:1-1998:4 and an aggregate of 15 European countries. All relevant
time series, except ratio of net exports to output, have been logged and HP-filtered. CM indicates a
complete financial market economy while bond indicates an economy where people can only trade
one-period non-contingent bond.
15
Table 4: Business Cycles Statistics: Sensitivity to Elasticity
ω
Statistics
Data
1.58
2
3
4
5
6
Consumption
0.79
0.91
0.88
0.85
0.84
0.83
0.83
Investment
3.24
1.74
1.83
1.94
1.99
2.02
2.03
Employment
0.63
0.63
0.5
0.33
0.25
0.2
0.17
Net Exports/GDP
0.09
0.42
0.43
0.43
0.45
0.47
0.45
Consumption
0.87
0.93
0.91
0.88
0.87
0.85
0.85
Investment
0.93
0.66
0.7
0.73
0.75
0.75
0.76
Employment
0.86
1
1
1
1
1
1
Net Exports/GDP
-0.36
0.19
0.21
0.23
0.24
0.24
0.25
GDP
0.51
0.46
0.51
0.52
0.52
0.52
0.52
Consumption
0.32
0.76
0.72
0.59
0.52
0.48
0.46
Investment
0.29
0.1
0.18
0.27
0.31
0.33
0.35
Employment
0.43
0.46
0.51
0.52
0.52
0.52
0.52
Std.dev rel. to GDP
Domestic Comovement
Corr. with GDP
International Correlation
Home and Foreign
Note: The statistics in the Data column are taken from Kehoe and Perri (2002), which are calculated
from U.S. quarterly time series, 1970:1-1998:4 and an aggregate of 15 European countries. All relevant
time series, except ratio of net exports to output, have been logged and HP-filtered. The model
statistics are computed from an model economy with GHH preferences, non-contingent bond, and
time-to-build investment. Omega is related to the intertemporal elasticity of substitution in labor
supply: the higher omega is the lower the elasticity
16
References
[1] Backus, D., P. Kehoe, and F. Kydland 1992, “International Real Business Cycles”,
Journal of Political Economy 100, 745-775.
[2] Baxter, M., 1995, “International Trade and Business Cycles,”Working Paper 5025,
NBER.
[3] Baxter, M., M. Crucini 1995, “Business cycles and the asset structure of foreign
trade”, International Economic Review 36, 821-854.
[4] Boileau M., M. Normandin, 2008, “Closing international real business cycle models
with restricted financial markets,”Journal of International Money and Finance 27,
733-756.
[5] Devereux M., A. Gregory, G. Smith (1992), “Realistic cross-country consumption
correlations in a two-country, equilibrium, business cycle model”, Journal of Intenational Money and Finance, 11, 3-16.
[6] Correia, I., J. Neves, and S. Rebelo, 1995, “Business Cycles in a Small Open Economy”, European Economic Review 39 (6), 1089-1113.
[7] Greenwood, J., Z. Hercowitz and G. Huffman, 1988, “Investment, Capacity Utilization, and the Real Business Cycle”, American Economic Review 78, 402-417.
[8] Heathcote, J., Perri, F., 2002, “Financial autarky and international business cycles,”
Journal of Monetary Economics 49 (3), 601627.
[9] King R., C. Plosser, and S. Rebelo 1988 “Production, growth and business cycles:
The basic neoclassical model,” Journal of Monetary Economics, 21, 195-232.
[10] Kydland F., E. Prescott, 1982 “Time to build and aggregate fluctuations,” Econometrica, 50(6), 1345-1370.
[11] Kehoe P., F. Perri 2002, “International business cycles with endogenous incomplete
markets”, Econometrica 70(3), 907-928.
17
[12] Kollmann R., 1996, “Incomplete asset markets and the cross-country consumption
correlation puzzle”, Journal of Economic Dynamic and Control, 20, 945-961.
[13] Mendoza, Enrique G., 1991, “Real Business Cycle in a Small Open Economy”,
American Economic Review, 81(4), 797-818.
[14] Monacelli, Tommaso, 2008, “New Keynesian Models, Durable Goods, and Collateral
Constraints”, Journal of Monetary Economics, forthcoming.
[15] Schmitt-Grohe, Stephanie and Uribe, Martin. (2003), “Closing Small Open Economy Models”, Journal of International Economics, 61, 163-185.
[16] Schmitt-Grohe, Stephanie and Martin Uribe. (2004), “Solving Dynamic General
Equilibrium Models Using a Second-Order Approximation to the Policy Function”,
Journal of Economic Dynamics and Control 28, 645-858.
18
Appendix A: Theoretical Model
1
Solution Method
I solve the models by the perturbation method.9 Particularly the set of optimality
conditions of the economy can be expressed as follows:
Et {F (Yt+1 , Yt , Xt+1 , Xt )} = 0
(1.1)
Et is the mathematical expectation operator conditional on information available at
time t, Yt is the vector of non-predetermined variables, and Xt = [x1t , x2t ] is the state
variable vector, x1t is endogenous predetermined state variables while x2t is exogenous
state variables. Particularly, x2t follows exogenous process given as:
x2t+1 = Λx2t + η̃σ̄t+1
(1.2)
where η̃, σ̄ are given parameter. The solution of the optimal plan is of the form:
Yt = g(Xt , σ̄)
(1.3)
Xt+1 = h(Xt , σ̄) + η̄σ̄t+1
(1.4)
where η̄ = [∅, η̃] , these equations describe the policy and transition functions respectively. I compute a first order expansion of the two functions around the deterministic
steady state.
2
Solving the bond economy model
The representative agent in Home country chooses sequences {ct , lt , kt+1 , bt+1 , sit } to
solve the problem
U = E0
∞
β t U ct , lt
(2.5)
t=0
subject to
ct + it + pbt bt+1 +
9
πb
(bt+1 )2 = At (kt )α (lt )1−α + bt
2
For more details, see Schmitt-Grohe and Uribe (2004)
i
(2.6)
2
φ s1t
kt+1 = (1 − δ)kt + s1t − kt
−δ
2
kt
it =
4
(2.7)
ωi sit
(2.8)
i=1
sj,t+1 = sj+1,t
for j = 1, 2, 3
(2.9)
The Lagrangian associated with the Home agent’s optimization problem can be written as:
L = E0
∞
βt
U(.) + λt At ktα lt1−α + bt − ct −
t=0
4
i=1
πb
ωi sit − pbt bt+1 − b2t+1
2
+λt ν2t (s2t − s1t+1 ) + λt ν3t (s2t − s1t+1 ) + λt ν4t (s2t − s1t+1 )
2
φ s1t
+λt qt (1 − δ)kt + s1t − kt
− δ − kt+1
2
kt
(2.10)
The optimality conditions associated with the Home representative agent’s problem
are:
(2.11)
Uc ct , lt = λt
(2.12)
λt pbt + πb bt+1 = βλt+1
yt
(2.13)
Ul ct , lt = −λt (1 − α)
lt
2 φ s1t+1
αyt
φs1t+1
s1t+1
+ qt+1 1 − δk −
− δk +
− δk
λt qt = βEt λt+1
kt
2
kt
kt
kt
(2.14)
s1t+1
− ω1 λt+1 = 0
(2.15)
− δk
−λt ν2t + βEt λt+1 qt+1 1 − φ
kt
−λt ν3t + βEt {λt+1 ν2t+1 − ω2 λt+1 } = 0
(2.16)
−λt ν4t + βEt {λt+1 ν3t+1 − ω3 λt+1 } = 0
(2.17)
−ω4 λt + λt ν4t = 0
πb
ct + it + pbt bt+1 + (bt+1 )2 = At (kt )α (lt )1−α + bt
2
2
φ s1t
kt+1 = (1 − δ)kt + s1t − kt
−δ
2
kt
(2.18)
ii
(2.19)
(2.20)
yt = At ktα lt1−α
sj,t+1 = sj+1,t
(2.21)
for j = 1, 2, 3
(2.22)
Similar conditions for the Foreign representative agent.
The world market clearing condition for bonds is:
bt+1 + b∗t+1 = 0 bond economy
(2.23)
In solving the system, I replace the budget constraint for Foreign representative agent:
c∗t + i∗t + pbt b∗t+1 +
πb ∗ 2
(bt+1 ) = yt∗ + b∗t ;
2
foreign country
(2.24)
by the unified world resource constraint for the single produced good:
(yt − ct − it ) + (yt∗ − c∗t − i∗t ) = 0
3
(2.25)
Solving the complete market model
When financial markets are complete, the competitive equilibrium is Pareto optimal.
Hence, we can derive the equilibrium system using an equal weight planner problem.
The planner maximizes the sum of expected lifetime utilities
∞
t β U ct , lt + β t U c∗t , lt∗
U = E0
(3.26)
t=0
subject to:
ct + it + c∗t + i∗t = At (kt )α (lt )1−α + A∗t (kt∗ )α (lt∗ )1−α
2
φ s1t
kt+1 = (1 − δ)kt + s1t − kt
−δ
2
kt
2
φ s∗
∗
= (1 − δ)kt∗ + s∗1t − kt∗ 1t∗ − δ
kt+1
2
kt
it =
4
(3.27)
(3.28)
(3.29)
ωi sit
(3.30)
ωi s∗it
(3.31)
i=1
i∗t
=
4
i=1
sj,t+1 = sj+1,t
for j = 1, 2, 3
iii
(3.32)
s∗j,t+1 = s∗j+1,t
for j = 1, 2, 3
(3.33)
The Lagrangian associated with the social planner’s optimization problem can be set
up similarly as the one in the bond economy above. The optimality conditions associated
with this problem are straightforward.
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2008
2008
2008
2008
2008
2008
The Changing Nature of Employment and the Reform of Labor
2008
and Social Security Legislation in Post-Apartheid South Africa
Technology Choice, Change of Trade Structure, and A Case of
2008
Hungarian Economy
The Policy Making Process in FTA Negotiations: A Case Study
2008
of Japanese Bilateral EPAs
Rural to Urban Migration: A District Level Analysis for India
2008
No.
Author(s)
Title
136
Nicolaus Herman SHOMBE
Causality relationship between Total Export and Agricultural
GDP and Manufacturing GDP case of Tanzania
2008
135
Ikuko OKAMOTO
The Shrimp Export Boom and Small-Scale Fishermen in
Myanmar
2008