Separate Appendix (Not for publication) to “The World Has More Than Two Countries: Implications of Multi-Country International Real Business Cycle Models” Hirokazu Ishise∗ February 18, 2014
Contents A Data
3
A.1 G7 business cycle moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
A.2 Size of G7 countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
B Models
∗
5
B.1 Computation and calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
B.2 Single-good models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
B.2.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
B.2.2 The baseline production technology and the shock process . . . . . . . . . . .
6
B.2.3 Resource constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
B.3 Modified versions of single-good models . . . . . . . . . . . . . . . . . . . . . . . . .
6
B.3.1 Time-to-build . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
B.3.2 Adjustment friction and variable utilization . . . . . . . . . . . . . . . . . . .
7
B.3.3 Resource constraints in case of a complete market and trade cost . . . . . . .
7
B.3.4 Resource constraints in a bond market . . . . . . . . . . . . . . . . . . . . . .
8
B.4 One-good-per-country models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
B.4.1 Household problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
B.4.2 Complete market case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
B.4.3 Non-state contingent bond case . . . . . . . . . . . . . . . . . . . . . . . . . .
9
B.4.4 A generalization of financial autarky . . . . . . . . . . . . . . . . . . . . . . .
9
B.4.5 Final goods producer problem . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
B.4.6 Intermediate goods producer . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
B.4.7 Market clearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
B.5 Derivation of single-good models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
B.5.1 Equilibrium conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
B.5.2 Transformed equilibrium conditions . . . . . . . . . . . . . . . . . . . . . . .
14
ISER, Osaka U. (Email:
[email protected]) Data and programs are available on request.
1
B.5.3 Non-stochastic steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
B.5.4 Log-linearized equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
B.6 Derivation of one-good-per-country models
. . . . . . . . . . . . . . . . . . . . . . .
20
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
B.6.2 Transformed economy equilibrium conditions . . . . . . . . . . . . . . . . . .
23
B.6.3 Non-stochastic steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
B.6.4 Log-linearized equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
B.6.1 Models
C Robustness Tables
32
D Impulse Response Functions and Intuition
42
2
A
Data
A.1
G7 business cycle moments
Table A1 shows key real business cycle moments. Intra-country moments are obtained from the median (mean) of the G7 countries: Canada, France, (West) Germany, Italy, Japan, the United Kingdom and the United States. The main data sources are quarterly national accounts on “SourceOECD.” If longer data is available through a country source or older SNA system, the main economic indicator data is extrapolated using the growth rates of additional data. All the variables are seasonally adjusted quarterly data. The real values in the data sets are used. Hours and employment are normalized by year 2000 values (average of four quarters in 2000), and then multiplied to obtain labor. Italian labor is calculated only using employment data, since the hours data is missing. Filtering is applied for the longest possible dataset, and then moments are calculated using a limited sample period (1970, 1st quarter to 2006, 4th quarter). The data for Germany and West Germany are combined after filtering. Excluding net exports, variables are taken from natural logs, after which filters are applied. Net exports are filtered after the figures have been divided by real GDP. Further details are explained in the dataset, which is available on request. Cross-country correlations are the medians (means) of the possible 21 (= 7 × 6/2) combinations of cross-country correlations. Table A1 includes U.S. statistics. U.S.-foreign cross-country correlations are the median of the possible six cross-country correlations. The last column shows model moments. The model is a straightforward extension of a textbook closed-economy business cycle model (King et al., 1988) to a two-country, connecting, complete set of state contingent claims, with standard parameterizations.
A.2
Size of G7 countries
In the models, each country has an equivalent steady state per capita output, because of the symmetry of the utility and technologies. The population weight captures both population size and economic size. The economic size is used for considering business cycle properties of developed countries. The data is from Penn World Table 6.2 (Heston et al., 2008). Country GDP is real GDP per capita chain (rgdpch) times total population (POP ). Calculations are the data for 1970, 1985 and 2000. The total world GDP is the sum of the GDPs of all the available countries. The ratios for the United States’s GDP to total world GDP are 26% in 1970, 24% in 1985 and 21% in 2000; for Japan’s are 8% in 1970, 9% in 1985 and 7% in 2000. Similarly, the ratios for Germany’s GDP to the total world GDP for the three years are: 8%, 6% and 5%; for the United Kingdom: 5%, 4% and 3%; for France: 5%, 4% and 3%; for Italy: 4%, 4% and 3%; and for Canada: 2%, 2% and 2%.
3
Table A1: G7 Business Cycle Moments Country
G7 Median Filter BP Standard deviation relative Consumption 0.91 Investment 2.60 Labor 1.04 Net exports 0.61 Correlation to output Consumption 0.77 Investment 0.75 Labor 0.74 Net exports −0.36 Autocorrelation Output 0.93 Consumption 0.94 Investment 0.93 Labor 0.93 Net exports 0.91 Cross-country correlation Output 0.37 Consumption 0.28 Investment 0.23 Labor 0.23 Net exports −0.02
G7 G7 G7 Mean S.D. Median BP BP HP to standard deviation of 0.92 0.12 0.92 2.63 0.35 2.67 1.30 0.53 1.67 0.52 0.21 0.65
USA Mean BP output 0.81 2.82 1.07 0.23
Model
0.76 0.80 0.63 −0.32
0.10 0.11 0.37 0.17
0.74 0.74 0.53 −0.29
0.88 0.96 0.93 −0.46
0.73 0.05 0.97 0.21
0.92 0.94 0.93 0.93 0.91
0.02 0.01 0.01 0.03 0.02
0.82 0.77 0.86 0.59 0.72
0.94 0.94 0.95 0.92 0.95
0.91 0.91 0.79 0.91 0.80
0.41 0.21 0.23 0.23 −0.02
0.19 0.29 0.23 0.30 0.22
0.35 0.28 0.25 0.35 −0.01
0.45 0.38 0.30 0.51 −0.08
−0.72 0.64 −0.99 −0.96 −1.00
0.29 13.21 0.56 3.33
The author’s calculation is based mainly on “SourceOECD.” The sample period is 1970 Q1–2006 Q4. See Appendix A for details. Intra-country moments are medians (means) of the G7 countries. Cross-country correlations are medians (means) of 21 (= 7 × 6/2) statistics. Cross-country correlations of the U.S. column are medians of six pairwise correlations between the U.S. and other G7 countries. The S.D. column shows the standard deviations of seven intra-country moments and 21 cross-country correlations. The BP filter (Baxter and King, 1999) uses parameters BP12 (6, 32). The HP filter (Hodrick and Prescott, 1997) uses parameter λ = 1600. The model is the two-country baseline model (King et al. (1988), with a complete market and no shock diffusion).
4
B
Models
B.1
Computation and calibration
The computation procedure is also designed to compare to the prototypical two-country models. The economy is transformed into a detrended economy, and its non-stochastic steady state is calculated. Model moments are obtained from cyclical behavior around the steady state using the log-linear rational expectation system (Kydland and Prescott, 1982; King et al., 1988).1 The multi-country setting does not greatly inflate the state space of the model; state variables in the system of the baseline model are Ai,t , Ki,t for all i = 1, ..., N . In an N = 10 economy, for example, the total number of state variables is 20. The bond market assumption and the timeto-build technology require additional state variables, but the speed of an increment is a multiple of the number of countries and, hence, standard linear rational expectation solvers can handle the calculations.2 All the moments are calculated for each selection of production technology, shock process and market structure. To compare the implications of the number of countries per se, only the number of countries in the model is changed. The moments are obtained by 100,000-period simulations and filtered using an approximated band-path filter.3
B.2
Single-good models
Time is discrete and indexed by t. The world comprises N countries, indexed by i = 1, ..., N . The countries potentially vary in size, but are symmetric in other respects. The world population is a ∑ unit measure. Each country has a πi fraction of the population, where i πi = 1. The model is described as a social planner’s problem, since the equilibrium allocations are Pareto optimal. The Pareto weight is equivalent to a country’s population size. B.2.1
Households
Households enjoy consumption, Ci,t , and dislike working, Li,t . By normalizing total time endowment to be unity, 1 − Li,t stands for the leisure of a household in i at t. Li,t is used for the production in country i, i.e., labor is immobile across countries. The social planner maximizes the weighted sum of the utility of the stand-in households E0
N ∑ i=1
πi
∞ ∑
βt
ψ (Ci,t (1 − Li,t )1−ψ )1−γ
1−γ
t=0
.
(B1)
1 For example, the output is calculated by yˆi,t = log(Yi,t /Y ), where Y is the steady state output level. Net exports are calculated by:
n cxi,t = yˆi,t −
C C cˆi,t − ˆıi,t , Y Y
where C and I are steady state consumption and investment respectively. 2 The calculation employs AIM implementation of Anderson=Moore algorithm (Anderson, 2008). 3 The parameters are BPK (p, q) = BP12 (6, 32), following the recommendation in Baxter and King (1999). The results obtained using the HP filter (Hodrick and Prescott, 1997) are similar. If the number of countries is relatively small, the moments can be directly calculated from the system. As the number of countries increases, the computational burden prevents the use of a direct calculation method. Simulations give almost identical moments for cases with a small number (e.g., five) of countries.
5
subject to a technology, a shock process and a resource constraint. B.2.2
The baseline production technology and the shock process
The baseline model employs the standard neoclassical production function (King et al., 1988). The output, Yi,t , is produced using capital (Ki,t ) and labor (Li,t ). α 1−α Yi,t = Ai,t Ki,t Li,t
(B2)
Ki,t+1 = (1 − δ)Ki,t + Ii,t
(B3)
where Ai,t is the country-specific total factor productivity (TFP) and Ii,t is investment. TFP has a world common trend component (A¯t ) growing at rate g, and country specific stochastic components (A˜i,t ), such that Ai,t = A¯t A˜i,t = A0 (1 + g)t A˜i,t .
(B4)
˜ t be a vector of log A˜i,t and Moreover, let a ˜ t = Ω˜ a at−1 + εt ,
(B5)
where the diagonal element of Ω(i,i) is autocorrelation parameter ωii , and the off-diagonal element of Ω(i,h) is spillover parameter ωih . The vector of shock, εt , is jointly normal (εt ∼ N ), independently and identically distributed across t, E(εt ) = 0, V (εt )(i,i) = σii2 , and V (εt )(i,h) = σih . B.2.3
Resource constraint
The resource constraint of the baseline model is that the planner collects the entire global output and allocates both consumption and investment, which is identical to the complete market situation in the decentralized model. ∑
πi N Xi,t = 0.
(B6)
i
where N Xi,t is net exports and N Xi,t = Yi,t − Ci,t − Ii,t .
(B7)
Appendix B presents three alternative resource constraints, one including a complete market subject to trade costs, one restricting assets to non-state contingent bonds (risk-free bond market), and the other imposing period-by-period trade balance, which is a multi-country extension of “financial autarky” proposed by Heathcote and Perri (2002).
B.3
Modified versions of single-good models
Regardless of the production technologies or market assumptions, the models can be solved as constrained social planner’s problem.
6
B.3.1
Time-to-build
A simplified version employed in Backus et al. (1992) is expressed as follows: α 1−α Yi,t = Ai,t Ki,t Li,t ,
(B8)
Ki,t+1 = (1 − δ)Ki,t + Vi,t,1 ,
(B9)
Ii,t =
1 J
J ∑
Vi,t,j , and
(B10)
j=1
Vi,t+1,j = Vi,t,j+1
for j = 1, ..., J − 1,
(B11)
where Ii,t is the total investment in country i at t, and the investment is divided into J fractions of Vi,t,j . Each Vi,t,j becomes the working capital stock as j period passes. B.3.2
Adjustment friction and variable utilization
Some of the production functions are special cases of the following model. α 1−α Yi,t = Ai,t ζ(Zi,t )Ki,t Li,t
Ki,t+1 = (1 − δ(Zi,t ))Ki,t + ϕ
(
Ii,t Ki,t
(B12)
) Ki,t ,
(B13)
where Ii,t is the investment, Ki,t is the capital stock, Zi,t is the degree of capital utilization, and ϕ(.) is a function of the capital adjustment friction. The function ζ(.) represents the possibility of variable capital utilization. If ζ(Z) = Z α , the model is equivalent to Baxter and Farr (2005). If ζ(.) = 1 and δ(.) = δ, the capital utilization and depreciation rate are constant, which is reduced to the model of Baxter and Crucini (1993) specifications. Following Baxter and Farr (2005), a set of conditions is imposed on these functions, so that the non-stochastic steady state values are irrelevant with or without friction and/or adjustment assumptions.4 Then, only elasticity parameters are relevant for calculating log-linearized models. B.3.3
Resource constraints in case of a complete market and trade cost
Backus et al. (1992) specify a quadratic trade cost ∑
( ) πi N Xi,t − 0.1(N Xi,t )2 /Y = 0,
i
where Y is the steady state output.
4
The conditions are
) ( ) I I I = , ϕ′ =1 K K K δ(Z) = δ, Z = 1, (
ϕ
where variables without subscripts are non-stochastic steady state values derived in the separate appendix.
7
(B14)
B.3.4
Resource constraints in a bond market
Baxter and Crucini (1995) and Kollmann (1996) consider a model in which only risk-free bonds are available. It is expressed by N Xi,t + Bi,t = Ptb Bi,t+1 ∑ πi Bi,t = 0,
(B15) (B16)
i
where Bi,t is the bond holding of i country and Ptb is the subjective “price” of a bond that the planner faces. This price is the same as the competitive equilibrium price of the non-state-contingent bond.
B.4
One-good-per-country models
Backus et al. (1994), Arvanitis and Mikkola (1996), and Heathcote and Perri (2002) consider a onegood-per-country setting with various financial market arrangements. In this case, a decentralized problem is considered. B.4.1
Household problem
Each representative agent living in i solves
max E0
∞ ∑
βt
( )1−γ ψ Ci,t (1 − Li,t )1−ψ 1−γ
t=0
(B17)
subject to ( Ki,t+1 = (1 − δ)Ki,t + ϕ
Ii,t Ki,t
) Ki,t
(B18)
where ϕ is capital adjustment friction function as explained above. The capital adjustment friction is imposed only in Arvanitis and Mikkola (1996). The budget constraint is one of the following. B.4.2
Complete market case
Backus et al. (1994) consider the complete market model: Pi,t (Ci,t + Ii,t ) +
∑
Ptb (st+1 )Bi,t+1 (st+1 )
st+1
= Bi,t + Pi,t (Wi,t Li,t + Ri,t Ki,t ) .
(B19)
where Pi,t is final good price, Wi,t is real wage, Ri,t is real rental rate, Bi,t+1 (st+1 ) is state contingent claim in i for rewarding a unit of currency, and Ptb (st+1 ) is the price of the claim (for each state at t + 1). Transversality conditions: 0 = E0 lim
t→∞
t ∑ ∏
Pkb (sk+1 )Bi,k+1 (sk+1 ).
k=0 sk+1
8
(B20)
B.4.3
Non-state contingent bond case
Arvanitis and Mikkola (1996) consider the only available asset in the world is non-state contingent period ahead risk free claim, household budget constraint is ηtb b b P (Bi,t )2 − Ti,t 2 t = Bi,t−1 + Pi,t (Wi,t Li,t + Ri,t Ki,t ) . Pi,t (Ci,t + Ii,t ) + Ptb Bi,t +
(B21)
where Ptb is world price of non-state contingent bond, Bi,t is amount of bond holdings, ηtb is bond holdings adjustment tax parameter (which is time-dependent for having stationarity in transformed b is lump-sum transfer financed by bond holding adjustment tax.5 Also, correeconomy) and Ti,t
sponding transversality condition is imposed: 0 = E0 lim
t→∞
B.4.4
t ∏
Pkb Bi,t .
(B22)
k=0
A generalization of financial autarky
Period-by-period trade balance assumption restricts possibility of asset trading that is balanced: Pi,t (Ci,t + Ii,t ) = Pi,t (Wi,t Li,t + Ri,t Ki,t ) .
(B23)
The period-by-period trade balance is a straightforward extension of “financial autarky” assumption proposed by Heathcote and Perri (2002). If there are only two countries, a country-by-country goods side balance automatically implies a period-by-period financial side balance. In the multi-country framework, however, the economies are still possible to trade asset. The asset trading may not hold country-by-country financial side balance, but a financial account of the economy is balanced in the aggregate, ensuring period-by-period financial side balance. B.4.5
Final goods producer problem
Final goods producer in i solves max Pi,t Xi,t −
N ∑
pj,i,t xj,i,t
(B24)
j=1
subject to
Xi,t
1 θ N ∑ θ = υj,i xj,i,t
(B25)
j=1
where pj,i,t is the price of an intermediate product xj,i,t in country j.
This type of tax and transfer is based on Ghironi and Melitz (2005). ηtb is adjusted so that it becomes stationary after removing trend, and detrended version of the parameter is parameterized to be a small number (0.0000125). 5
9
B.4.6
Intermediate goods producer max pi,t xi,t − Pi,t Wi,t Li,t − Pi,t Ri,t Ki,t
(B26)
α 1−α xi,t = Ai,t Ki,t Li,t
(B27)
subject to
B.4.7
Market clearing
Followings are market clearing conditions of the economy: N ∑
πj xi,j,t = πi xi,t
(B28)
j=1
Country’s resource constraint: Ci,t + Ii,t = Xi,t
(B29)
Market clearing of claims if there are state contingent claims: 0=
N ∑
πi Bi,t+1 .
(B30)
i=1
Non-state contingent bond case: 0=
N ∑
πi Bi,t .
(B31)
i=1
and lump-sum transfer is financed by bond holding adjustment tax: ηtb b b Pt (Bi,t )2 = Ti,t . 2
B.5
(B32)
Derivation of single-good models
Objective Function E0
N ∑ ∞ ∑
β t u(Ci,t , Li,t )
i=1 t=0
where u(Ci,t , Li,t ) =
ψ (Ci,t (1 − Li,t )1−ψ )1−γ
1−γ
subject to production technology and resource constraints.
10
.
(CD): Cobb=Douglas Production Function α 1−α Yi,t = Ai,t ζ(Zi,t )Ki,t Li,t
TFP process is defined Ai,t = A0 (1 + g)t A˜i,t = A¯t A˜i,t where ˜ t = Ω˜ a at−1 + εt where Ω(i,i) = ωii and Ω(i,h) = ωih where subscripts (i, h) stands for (i, h), non-diagonal, element of ˜ t is log A˜i,t . ith row of εt is εi,t , εt ∼ N jointly normal and i.i.d. across t, the matrix. ith row of a E(εt ) = 0, V (εt )(i,i) = σii2 and V (εt )(i,h) = σih . Adding to this, combinations of following components generate variations of models:
Choices of Investment Friction (NF): No Friction (Baseline) Ki,t+1 = (1 − δ(Zi,t ))Ki,t + Ii,t (AF): Adjustment Friction ( ) Ii,t Ki,t+1 = (1 − δ(Zi,t ))Ki,t + ϕ Ki,t Ki,t ( ) ( ) I I I ′ ϕ = , ϕ =1 K K K
(TB): Time-to-Build Ki,t+1 = (1 − δ(Zi,t ))Ki,t + Vi,t,1 Ii,t
J 1∑ = Vi,t,j J j=1
Vi,t+1,j = Vi,t,j+1
for j = 1, ..., J − 1.
Choices of Utilization (CU): Constant Utilization δ(Zi,t ) = δ
11
(VU): Variable Utilization δ(Z) = δ,
Z=1
Choice of Resource Constraint (CM): Complete Market ∑
πi [Yi,t − Ci,t − Ii,t ] = 0.
i
(TC): Complete Market with Trade Costs ∑
πi [Yi,t − Ci,t − Ii,t − τ (Yi,t − Ci,t − Ii,t )] = 0.
i ′ τ (0) = 0, τi,t > 0,
(BM): Bond Market Yi,t + Bi,t = Ci,t + Ii,t + Pt Bi,t+1 ∑ πi Bi,t = 0. i
B.5.1
Equilibrium conditions
Although the model is presented as a Social planner’s problem, the resulting system coincides with competitive equilibria. The equilibrium conditions can be expressed as the appropriate combinations of following blocks of equations. All the model uses marginal utility (MU) condition and the production function (CD) condition. In addition, the model is characterized by one of the investment friction conditions ((NF), (AF) or (TB)), one of the utilization conditions ((CU) or (VU)), and one of the resource constraints ((CM), (TC) or (BM)). Denote uc and ul first derivative of utility with respect to C and L. Λs are multipliers associate with various constraints.
(MU): Marginal Utility ψ(1−γ)−1
uc (i, t) = ψCi,t
(1 − Li,t )(1−ψ)(1−γ)
ψ(1−γ)
ul (i, t) = −(1 − ψ)Ci,t ΛYi,t = πi uc (i, t)
12
(1 − Li,t )(1−ψ)(1−γ)−1
(CD): Cobb-Douglas Production Function α α 1−α Yi,t = Ai,t Zi,t Ki,t Li,t Yi,t 1 − ψ Ci,t (1 − α) = Li,t ψ 1 − Li,t
(NF): No Friction Ki,t+1 = (1 − δ(Zi,t ))Ki,t + Ii,t [ ] Yi,t+1 uc (i, t) = Et βuc (i, t + 1) α + 1 − δ(Zi,t+1 ) Ki,t+1 ΛK i,t = uc (i, t) ϕ′ (i, t) = 1 (AF): Adjustment Friction Ki,t+1 = (1 − δ(Zi,t ))Ki,t + ϕ(i, t)Ki,t [ ( )] Yi,t+1 Ii,t+1 uc (i, t + 1) uc (i, t) ′ ′ = E β ϕ (i, t + 1)α + 1 − δ(Z ) + ϕ(i, t + 1) − ϕ (i, t + 1) t i,t+1 ϕ′ (i, t) ϕ′ (i, t + 1) Ki,t+1 Ki,t+1 ΛK i,t = uc (i, t) ϕ(i, t) = ϕ(Ii,t /Ki,t ) ϕ′ (i, t) = dϕ(i, t)/d(Ii,t /Ki,t ) (TB): Time-to-Build Ki,t+1 = (1 − δ(Zi,t ))Ki,t + Vi,t,1 Ii,t =
J 1∑ Vi,t,j J j=1
Vi,t+1,j = Vi,t,j+1 [ ] −1 Y K ΛK = E β Λ αY K + Λ (1 − δ(Z )) t i,t+1 i,t+1 i,t, i,t+1 i,t+1 i,t+1 1 Y 1 V Λi,t = ΛK Λ i,t − J G4 i,t−1,1 1 V 1 Y Λi,t = ΛVi,t,1 − Λ J G4 i,t−1,2 ... 1 Y 1 V Λi,t = ΛVi,t,J−2 − Λ J G4 i,t−1,J−1 1 Y Λ = ΛVi,t,J−1 J i,t ϕ′ (i, t) = 1
13
(CU): Constant Utilization Zi,t = 1 δ(Zi,t ) = δ (VU): Variable Utilization ϕ′ (i, t)ΛYi,t α
Yi,t ′ = ΛK i,t δ (Zi,t )Ki,t Zi,t
(CM): Complete Market ∑
πi (Yi,t − Ii,t − Ci,t ) = 0
i
uc (i, t) = uc (h, t) (TC): Complete Market with Trade Costs ∑
πi (Yi,t − Ii,t − Ci,t − τ (Yi,t − Ii,t − Ci,t )) = 0
i
uc (i, t) uc (h, t) ′ = 1 − τ′ 1 − τi,t h,t (BM): Bond Market ∑
πi Bi,t = 0
i
Yi,t + Bi,t = Ci,t + Ii,t + Pt Bi,t+1 uc (i, t + 1) Pt = βEt uc (i, t) B.5.2
Transformed equilibrium conditions
Denote variables with tilde as detrended values. The table summarizes various factors to detrend variables. Detrending factors detrend factor 1 1−α
A¯t
ψ(1−γ) 1−α
A¯t
ψ(1−γ)−1 1−α
variables Yi,t , Ci,t , Ii,t , Ki,t , Bi,t , Vi,t,j M ul (i, t), ΛL i,t , Λi,t
A¯t
V uc (i, t), ΛYi,t , ΛK i,t , Λi,t,j
none
Li,t , Zi,t
14
For notational convenience, denote G’s as growing factors, and define shortcut notation of adjustment friction functions Growing factors notation
value
G1
(1 + g)
G2
G11−α
G3
βG1 1−α
1
ψ(1−γ)
G4
ψ(1−γ)−1 1−α
βG1
(MU) ( )(1−ψ)(1−γ) e ψ(1−γ)−1 1 − L e i,t uec (i, t) = ψ C i,t )(1−ψ)(1−γ)−1 ( ψ(1−γ) e e 1 − Li,t uel (i, t) = −(1 − ψ)Ci,t e Yi,t = πi uec (i, t) Λ (CD) ei,t Z eα K e α e 1−α Yei,t = A i,t i,t Li,t (1 − α)
ei,t Yei,t 1−ψ C = e i,t e i,t ψ 1−L L
(NF) e i,t+1 = (1 − δ(Z ei,t ))K e i,t + Iei,t G2 K [ ( )] Yei,t+1 uec (i, t) = G4 Et uec (i, t + 1) α + 1 − δ(Zei,t+1 ) e i,t+1 K ϕ′ (i, t) = 1 (AF) e i,t+1 = (1 − δ(Z ei,t ))K e i,t + ϕ(i, t)K e i,t G2 K [ ( )] ei,t+1 ei,t+1 Y I uec (i, t) uec (i, t + 1) = G4 Et ϕ′ (i, t + 1)α + 1 − δ(Zei,t+1 ) + ϕ(i, t + 1) − ϕ′ (i, t + 1) e i,t+1 e i,t+1 ϕ′ (i, t) ϕ′ (i, t + 1) K K
15
(TB) e i,t+1 = (1 − δ(Z ei,t ))K e i,t + Vei,t,1 G2 K J 1∑e e Vi,t,j Ii,t = J j=1
G2 Vei,t+1,j = Vei,t,j+1 [ ] −1 Y K eK e e e e e Λ = E G Λ α Y K + Λ (1 − δ( Z )) t 4 i,t+1 i,t+1 i,t+1 i,t, i,t+1 i,t+1 1 eV 1 eY eK Λ =Λ Λ i,t − J i,t G4 i,t−1,1 1 eY eV eV − 1 Λ Λ =Λ i,t,1 J i,t G4 i,t−1,2 ... 1 eY e Vi,t,J−2 − 1 Λ eV Λi,t = Λ J G4 i,t−1,J−1 1 eY eV Λ =Λ i,t,J−1 J i,t ′ ϕ (i, t) = 1 (CU) ei,t = 1 Z ei,t ) = δ δ(Z (VU) e Yi,t α ϕ′ (i, t)Λ
Yei,t ′ e eK e =Λ i,t δ (Zi,t )Ki,t e Zi,t
(CM) ∑
[ ] ei,t = 0 πi Yei,t − Iei,t − C
i
uec (i, t) = uec (h, t) (TC) ∑
[ ] ei,t − τ (Yei,t − Iei,t − C ei,t ) = 0 πi Yei,t − Iei,t − C
i
uec (h, t) uec (i, t) = ′ ′ 1 − τi,t 1 − τh,t
16
(BM) ∑
ei,t = 0 πi B
i
ei,t = C ei,t + Iei,t + Pt G2 B ei,t+1 Yei,t + B uec (i, t + 1) Pt = βG4 Et uec (i, t) B.5.3
Non-stochastic steady state
At the non-stochastic steady state, market structure does not matter. Due to symmetric assumption, steady state values are the same across countries so that we drop country subscripts. Also, ad hoc capital adjustment friction and variable utilization do not affect by construction. Denote Ξ as the e i,t . The following system include cases with time-to-build (TB), adjustment steady state value of Ξ friction (AF), and variable utilization (VU). If time-to-build is not used, set J = 1. ) ( 1 1 r= −1+δ α G4 ( ( ))−1 1−ψ 1 1 1 − GJ2 −1 L= 1+ 1− (G2 − 1 + δ)r ψ 1−α J 1 − G2 1
K = r α−1 L V1 = (G2 − 1 + δ)K Vj+1 = G2 Vj I=
J 1∑ Vj J j=1
Y = rK C =Y −I Z=1 δ ′ = αr uc = ψC ψ(1−γ)−1 (1 − L)(1−ψ)(1−γ) ΛY = πuc 1 ΛVJ−1 = ΛY J ΛVJ−2 = (1 +
1 1 Y ) Λ G4 J
... ΛV1 = (1 + ... + ΛK = (1 +
1 GJ−2 4
1 GJ−1 4
1 ) ΛY J
1 ) ΛY J
B=0 P = βG4
17
B.5.4
Log-linearized equations
˜ i,t − log Ξ, deviation from the steady state, except for Let ξˆi,t = log Ξ nxi,t − nx ˆbi,t = bi,t − B , n cxi,t = . Y Y (MU) ubc (i, t) = (ψ(1 − γ) − 1)ˆ ci,t − (1 − ψ)(1 − γ)
L ˆ li,t 1−L
(CD) yˆi,t = a ˆi,t + αˆ zi,t + αkˆi,t + (1 − α)ˆli,t 1 ˆ yˆi,t = cˆi,t + li,t 1−L
(NF) ) 1 ′ 1 I ( kˆi,t+1 = kˆi,t − δ (Z)Z zˆi,t + ˆıi,t − kˆi,t G2 G2 K ( ) ubc (i, t) = E ubc (i, t + 1) + (1 − G4 (1 − δ)) E yˆi,t+1 − E kˆi,t+1 − G4 δ ′ (Z)ZE zˆi,t+1 ˆ K = ubc (i, t) λ i,t ϕb′ (i, t) = 0 (AF) ) 1 ′ 1 I ( kˆi,t+1 = kˆi,t − δ (Z)Z zˆi,t + ˆıi,t − kˆi,t G2 G2 K ( ) ubc (i, t) − ϕb′ (i, t) = E ubc (i, t + 1) + (1 − G4 (1 − δ)) E yˆi,t+1 − E kˆi,t+1 ′′ I
ϕ ϕb′ (i, t) = − ′K ϕ K ˆ λ = ubc (i, t)
−G4 δ ′ (Z)ZE zˆi,t+1 − G4 G2 E ϕb′ (i, t + 1) ( ) kˆi,t − ˆıi,t
i,t
18
(TB) ) 1 ′ 1 V1 ( vˆi,t,1 − kˆi,t kˆi,t+1 = kˆi,t − δ (Z)Z zˆi,t + G2 G2 K J 1∑ Iˆıi,t = Vj vˆi,t,j J j=1
vˆi,t+1,j = vˆi,t,j+1
( ) ˆ K = (1 − G4 (1 − δ)) E ubc (i, t + 1) + E yˆi,t+1 − E kˆi,t+1 λ i,t ′ ˆk +G4 (1 − δ)E λ ˆi,t+1 i,t+1 − G4 δ (Z)ZE z ˆV ˆ K − 1 ΛY ubc (i, t) = 1 ΛV λ ΛK λ i,t J G4 1 i,t−1,1 ˆ V − 1 ΛY ubc (i, t) = 1 ΛV λ ˆV ΛV1 λ i,t,1 J G4 2 i,t−1,2 ... 1 Y 1 V ˆV ˆV ΛVJ−2 λ bc (i, t) = Λ λ i,t,J−2 − Λ u J G4 J−1 i,t−1,J−1 ˆV ubc (i, t) = λ i,t,J−1
ϕb′ (i, t) = 0 (CU) zˆi,t = 0 (VU) ubc (i, t) + ϕb′ (i, t) + yˆi,t =
) ( δ ′′ Z ˆ K (i, t) zˆi,t + kˆi,t + λ 1+ ′ δ
(CM) ∑
πi (Y yˆi,t − Iˆıi,t − Cˆ ci,t ) = 0
i
ubc (i, t) = ubc (h, t) (TC) ∑
πi (Y yˆi,t − Iˆıi,t − Cˆ ci,t ) = 0
i
ubc (i, t) + τi′′ (0) (Y yˆi,t − Iˆıi,t − Cˆ ci,t ) = ubc (h, t) + τh′′ (0) (Y yˆh,t − Iˆıh,t − Cˆ ch,t )
19
(BM) ∑
πiˆbi,t = 0
i
ubc (i, t) − E ubc (i, t + 1) = ubc (h, t) − E ubc (h, t + 1) C I yˆi,t + ˆbi,t = cˆi,t + ˆıi,t + G3ˆbi,t+1 Y Y TFP process b t = Ωb a at−1 + εbt
B.6 B.6.1
Derivation of one-good-per-country models Models
Household problem Each representative agent living in i solves
max E0
∞ ∑
βt
( )1−γ ψ Ci,t (1 − Li,t )1−ψ 1−γ
t=0
(B33)
subject to ( Ki,t+1 = (1 − δ)Ki,t + ϕ
Ii,t Ki,t
) Ki,t
(B34)
where ϕ is capital adjustment friction function as explained above. The budget constraint is one of the following. Complete market case Pi,t (Ci,t + Ii,t ) +
∑
Ptb (st+1 )Bi,t+1 (st+1 )
st+1
= Bi,t + Pi,t (Wi,t Li,t + Ri,t Ki,t ) .
(B35)
where Pi,t is final good price, Wi,t is real wage, Ri,t is real rental rate, Bi,t+1 (st+1 ) is state contingent claim in i for rewarding a unit of currency, and Ptb (st+1 ) is the price of the claim (for each state at t + 1). Transversality conditions: 0 = E0 lim
t→∞
t ∑ ∏
Pkb (sk+1 )Bi,k+1 (sk+1 ).
k=0 sk+1
20
(B36)
Non-state contengent bond case Household budget constraint is ηtb b b P (Bi,t )2 − Ti,t 2 t = Bi,t−1 + Pi,t (Wi,t Li,t + Ri,t Ki,t ) . Pi,t (Ci,t + Ii,t ) + Ptb Bi,t +
(B37)
where Ptb is world price of non-state contingent bond, Bi,t is amount of bond holdings, ηtb is bond holdings adjustment tax parameter (which is time-dependent for having stationarity in transformed b is lump-sum transfer financed by bond holding adjustment tax. Also, correspondeconomy) and Ti,t
ing transversality condition is imposed: 0 = E0 lim
t→∞
t ∏
Pkb Bi,t .
(B38)
k=0
A generalization of financial autarky Period-by-period trade balance assumption restricts possibility of asset trading that is balanced: Pi,t (Ci,t + Ii,t ) = Pi,t (Wi,t Li,t + Ri,t Ki,t ) .
(B39)
Final goods producer problem Final goods producer in i solves max Pi,t Xi,t −
N ∑
pj,i,t xj,i,t
(B40)
j=1
subject to
Xi,t
1 θ N ∑ θ υj,i xj,i,t =
(B41)
j=1
where pj,i,t is the price of an intermediate product xj,i,t in country j. Intermediate goods producer max pi,t xi,t − Pi,t Wi,t Li,t − Pi,t Ri,t Ki,t
(B42)
α 1−α xi,t (st ) = Ai,t Ki,t Li,t
(B43)
subject to
21
Market clearing Followings are market clearing conditions of the economy: N ∑
πj xi,j,t = πi xi,t
(B44)
j=1
Country’s resource constraint: Ci,t + Ii,t = Xi,t
(B45)
Market clearing of claims if there are state contingent claims: 0=
N ∑
πi Bi,t+1 .
(B46)
i=1
Non-state contingent bond case: 0=
N ∑
πi Bi,t .
(B47)
i=1
and lump-sum transfer is financed by bond holding adjustment tax: ηtb b b Pt (Bi,t )2 = Ti,t . 2
(B48)
Terms of trade
E Pi,t
θ−1 θ θ ∑ 1 θ−1 θ−1 υi,j pi,j,t = /Pi,t .
(B49)
j̸=i
Similarly, the price index of the importing goods is
I Pi,t
θ−1 θ 1 θ ∑ θ−1 θ−1 = υj,i pj,i,t /Pi,t .
(B50)
j̸=i
The terms of trade, defined as the relative price of exports to imports, is E I T oTi,t (st ) = Pi,t /Pi,t .
(B51)
Detrending Let variables with tilde ˜ as detrended variables. It is defined, for a case of Xi,t , ˜ i,t = Xi,t /g1t X
22
(B52)
Other cases are summarized in the table: Detrending factor
Value
Variables
ga
ga
Ai,t
1 1−α
g1
ga
Xi,t , Ci,t , Ii,t , Ki,m,t , xi,j,t , Bi,t , Wi,t , exj,i,t , Yi,t , xi,t
g3
uL,i,t
g4
ψ(1−γ) g1 ψ(1−γ)−1 g1
1
–
Li,t , Ri,t , Pi,t , pi,j,t , ej,i,t , pi,t
Λi,t , uC,i,t
Also, let ηtb = η b g1−t . B.6.2
Transformed economy equilibrium conditions α ˜ 1−α ˜ i,t x ˜i,t = A˜i,t K Li,t
( ϕ˜i,t = ϕ
I˜i,t ˜ i,t K
( ϕ˜′i,t = ϕ′
I˜i,t ˜ i,t K
) (B54)
)
˜ i,t+1 = (1 − δ)K ˜ i,t + ϕ˜i,t K ˜ i,t g1 K
˜ i,t X
(B53)
1 θ N ∑ θ υj,i x ˜j,i,t =
(B55)
(B56)
(B57)
j=1
πi x ˜i,t =
∑
πj τ˜i,j,t x ˜i,j,t
(B58)
j
p˜i,t (1 − α)
uL,i,t x ˜i,t ˜ i,t = P˜i,t −˜ = P˜i,t W li,t u ˜C,i,t
(B59)
[ ( )] ˜ii,t+1 u ˜C,i,t 1 ′ ˜ i,t+1 + = βg4 Et u ˜C,i,t+1 R 1 − δ + ϕ˜i,t+1 − ϕ˜i,t+1 ϕ˜′i,t ϕ˜′i,t+1 k˜i,t+1
(B60)
˜ i,t − C˜i,t − I˜i,t , 0=X
(B61)
23
If complete market u ˜C,i,t P˜i,t /µi = u ˜C,j,t P˜j,t /µj
(B62)
If incomplete market, 0=
N ∑
˜i,t B
(B63)
i=1
(
[
)
˜i,t = βg4 Et P˜b t 1 + η b B
] u ˜C,i,t+1 /P˜i,t+1 . u ˜C,i,t /P˜i,t
[ ] [ ] ˜i,t = B ˜i,t−1 + P˜i,t W ˜ i,t L ˜ i,t + R ˜ i,t K ˜ i,t . P˜i,t C˜i,t + I˜i,t + P˜b t B
(B64)
(B65)
If financial autarky, ˜ i,t = W ˜ i,t L ˜ i,t + R ˜ i,t K ˜ i,t X B.6.3
(B66)
Non-stochastic steady state
Normalization of the price Setting P¯N = 1 as normalization. Rental rate and capital labor ratio ¯ K) ¯ = I/ ¯K ¯ and ϕ′ (I/ ¯ K) ¯ = 1. From Euler equation, By assumption, ϕ(I/ ¯i = R ¯ = 1 − 1 + δ. R βg4
(B67)
Capital-labor ratio (denoting κi ) is expressed by ¯i ¯i K α W κ ¯i = ¯ = ¯ . 1−α R Li
(B68)
¯ i = I¯i , (g1 − 1 + δ)K
(B69)
¯i α W ¯ I¯i = (g1 − 1 + δ) ¯ Li , 1−α R
(B70)
Investment Capital accumulation implies:
or
24
Absorption From household budget constraint and resource constraint, ¯i = W ¯ iL ¯i + R ¯K ¯i X ¯ ¯ iL ¯i + R ¯ α Wi ¯li = W ¯ 1−α R 1 ¯ ¯ = Wi Li 1−α
(B71)
With resource constraint and investment condition, α 1 − (g1 − 1 + δ) R ¯ ¯ ¯ Wi Li = C¯i . 1−α
(B72)
¯ ¯ i = 1 − ψ Ci , W ¯i ψ 1−L
(B73)
α 1 − (g1 − 1 + δ) R ¯ ¯ ¯ Wi Li = C¯i . 1−α
(B74)
Labor Combining two equations
to obtain ψ ¯= L
1−ψ ¯ 1 − (g1 − 1 + δ)α/R 1−α
. +
(B75)
ψ 1−ψ
¯i = W ¯ , etc. Consider the symmetric situation so that P¯i = P¯ = 1, W Wage and price level 1 From θ−1 θ N −1 θ ∑ θ−1 θ−1 P¯i = υj,i p¯j,i ,
(B76)
j=1
p¯j,i = p¯j τ¯j,i and ¯αW ¯ 1−α , p¯j αα (1 − α)1−α A¯j = P¯j R j
25
(B77)
Let ¯ α α−α (1 − α)−[1−α] . Ψ≡R
(B78)
θ−1 θ N ) θ −1 ( ∑ θ−1 θ−1 −1 ¯ ¯ 1−α ¯ ¯ Pi = υj,i ΨAj Pj Wj τ¯j,i ,
(B79)
To obtain
j=1
or
∑ ¯ ¯ 1−α τ¯j,i P¯i = Ψ Pj Wj 1 θ ¯ Aj υj,i j
θ θ−1
θ−1 θ
.
(B80)
Under the symmetry,
∑ ¯ 1−α Ψ τ¯1 j,i 1=W θ ¯ υj,i Aj j
θ θ−1
θ−1 θ
.
(B81)
¯. This gives W Consumption and absorption ¯ and L, ¯ Given W ψ ¯ ¯ W (1 − L), 1−ψ ¯L ¯ Z¯ = (1 − α)W
C¯ =
(B82) (B83)
Capital-labor ratio, marginal utility, absorption
κ ¯=
¯ α W ¯. 1−α R
(B84)
¯ I¯ = (g1 − 1 + δ)¯ κL,
(B85)
¯ (1−ψ)(1−γ) , u ¯C = ψ C¯ ψ(1−γ)−1 [1 − L]
(B86)
¯, u ¯L = −¯ uC W
(B87)
26
Check ¯ Z¯ = C¯ + I.
(B88)
Goods production, exporting, and relative price The price of goods in j is p¯j =
1 ¯−1 −α ¯ ¯ A κ ¯ WP. 1−α
(B89)
Then, the price at country i is p¯j,i = p¯j τ¯j,i .
(B90)
[ ] 1 1−θ z¯j,i = P¯ Z¯ 1−θ υj,i p¯−1 . j,i
(B91)
implying
Output and investment z¯i =
∑
πj τ¯i,j z¯i,j /πi
(B92)
j
Y¯ = Z¯
(B93)
ex ¯ j,i = πi p¯j,i z¯j,i
(B94)
∑ ¯ i= 1 IM ex ¯ j,i /P¯i πi
(B95)
∑ ¯ i= 1 EX ex ¯ i,j /P¯i πi
(B96)
Trade values
j̸=i
j̸=i
P¯iI =
∑
θ−1 −1 θ−1
θ θ−1
υj,i p¯j,i
j̸=i
27
θ
/P¯i ,
(B97)
P¯iE =
∑
θ−1 −1 θ−1
θ θ−1
υi,j p¯i,j
θ
/P¯i
(B98)
j̸=i
B.6.4
Log-linearized equations
The economy’s dynamics is analyzed by first order log-linearized economy. • # number of equations • ♡ combining to reduce number of variables • ♢ redundant conditions (not used) ˆi,t is log-deviations from the stead state, e.g., All the variables excluding B Cˆi,t = log(Ci,t /C¯i )
(B99)
Bond holding is linear approximation (not log-linear approximation): ¯ ˆi,t = Bi,t − Bi = Bi,t . B Y¯i Y¯i
(B100)
¯i L ˆ (#N ) : 0 = −ˆ uC,i,t + (ψ(1 − γ) − 1)Cˆi,t − (1 − ψ)(1 − γ) ¯ i Li,t 1−L
(B101)
HH problem
ˆ i,t + Cˆi,t + (#N ) : 0 = −W
¯i L ˆ ¯ i Li,t 1−L
¯ tR ˆ i,t+1 − βg4 g1 Et ϕˆi,t+1 (#N ) : 0 = −ˆ uC,i,t + ϕˆi,t + Et u ˆC,i,t+1 + βg4 RE
(B102)
(B103)
( ) ˆ i,t (#N ) : 0 = −ϕˆi,t − ηϕ Iˆi,t − K
(B104)
ˆ i,t+1 + (1 − δ)K ˆ i,t + (g1 − 1 + δ)Iˆi,t (#N ) : 0 = −g1 K
(B105)
Final goods producer ¯ iθ X ˆ i,t + (#N ) : 0 = −X
∑ j
28
υj,i x ¯θj,i x ˆj,i,t
(B106)
¯X ˆ i,t + (#N ♢) : 0 = −P¯ Pˆi,t − X
∑
p¯j,i x ¯j,i (ˆ pj,i,t + x ˆj,i,t )
(B107)
j
ˆ i,t + (θ − 1)ˆ (#N 2 ♡) : 0 = −ˆ pj,i,t + Pˆi,t + (1 − θ)X xj,i,t
(B108)
Intermediate goods producer and trade ˆ i,t + (1 − α)L ˆ i,t (#N ) : 0 = −ˆ xi,t + Aˆi,t + αK
(B109)
ˆ i,t − Pˆi,t + pˆi,t + x ˆ i,t (#N ) : 0 = −W ˆi,t − L
(B110)
ˆ i,t − Pˆi,t + pˆi,t + x ˆ i,t (#N ) : 0 = −R ˆi,t − K
(B111)
(#N ) : 0 = −πi x ¯i x ˆi,t +
∑
πj τ¯i,j x ¯i,j x ˆi,j,t
(B112)
j
(#N 2 ♡) : 0 = −ˆ pi,j,t + pˆi,t
(#N ♢) : 0 = −πi p¯i x ¯i (ˆ pi,t + x ˆi,t ) +
∑
πj p¯i,j x ¯i,j (ˆ pi,j,t + x ˆi,j,t )
(B113)
(B114)
j
Resource constraints ¯X ˆ i,t − C¯ Cˆi,t − I¯Iˆi,t (#N ) : 0 = X
(B115)
(#1) : 0 = PˆN,t
(B116)
(#N − 1) : 0 = −ˆ uC,i,t + Pˆi,t + u ˆC,j,t − Pˆj,t
(B117)
Price normalization
Complete market case
Incomplete market case ( ) ¯B ˆi,t Et u ˆC,i,t+1 − u ˆC,i,t − Et Pˆi,t+1 + Pˆi,t − η b P¯ b X ( ) ¯B ˆj,t + Et u ˆC,j,t+1 − u ˆC,j,t − Et Pˆj,t+1 + Pˆj,t − η b P¯ b X
(#N − 1) : 0 = −
29
(B118)
¯X ˆ i,t − P¯ b X ¯B ˆi,t + X ¯iB ˆi,t−1 (#N ) : 0 = − P¯ X ( ) ( ) ¯ iL ¯ W ˆ i,t + L ˆ i,t + P¯ R ¯K ¯ R ˆ i,t + K ˆ i,t + P¯ W
(# : 1♢) : 0 =
∑
ˆi,t πi B
(B119)
(B120)
i
Financial autarky case (One of N equation is not used.) ) ( ) ( ∑ ˆ i,m,t + kˆi,m,t . (B121) ¯ ˆ i,t + L ˆ i,t + P¯ R ¯X ˆ i,t + P¯ W ¯L ¯ W k¯i,m R (#N − 1(+1♢)) : 0 = −P¯ X m
Shock process (#N ) : Aˆi,t = ρa Aˆi,t−1 + σε εi,t
(B122)
(#N ) : εi,t ∼ N (0, 1)
(B123)
# of state variables, equations, and total variables ˆ i,t , Aˆi,t • State variables (#2N ): K ˆ i,t is predetermined • K • # of equations: 11N + N 2 • # of variables: 11N + N 2 ˆ i,t , Iˆi,t , X ˆ i,t , W ˆ i,t , R ˆ i,t , Pˆi,t , x • (#N × 7) : u ˆC,i,t , Cˆi,t , L ˆi,t , pˆi,t , ϕˆi,t • (#N 2 × (2 − 1)) : zˆi,j,t , pˆi,j,t (by ♡, pˆi,j,t is dropped.) • Exogenous shocks: εˆi,t ˆi,t , N additional equations. • Incomplete market case: N additional state variables, B Remarks • One of the #N equations is redundant. • pˆj,i,t can be dropped from the system.
30
Additional variables of interests ( ) ( ) ¯ iL ¯i W ˆ i,t + L ˆ i,t + R ¯K ¯i R ˆ i,t + K ˆ i,t Y¯i Yˆi,t = W
(B124)
ec xi,j,t = pˆi,j,t + x ˆi,j,t
(B125)
∑ 1 ex ¯ i,j ec xi,j,t − Pˆi,t ¯ i πi P¯i EX
(B126)
∑ 1 ex ¯ j,i ec xj,i,t − Pˆi,t ¯ i πi P¯i IM
(B127)
d i,t = EX
j̸=i
d i,t = IM
j̸=i
¯ i + EX ¯ i EX d i,t − IM ¯ i − IM ¯ i IM d i,t EX d N X i,t = Y¯i + Y¯i Yˆi,t θ ∑ −1 θ θ − 1 ( ¯I ¯ ) 1−θ θ−1 θ−1 I Pˆi,t = υj,i p¯j,i pˆj,i,t − Pˆi,t Pi Pi θ
(B128)
(B129)
j̸=i
θ ∑ −1 θ θ − 1 ( ¯E ¯ ) 1−θ θ−1 θ−1 E ˆ υi,j Pi,t = Pi Pi p¯i,j pˆi,j,t − Pˆi,t θ
(B130)
j̸=i
E I Td oT i,t = Pˆi,t − Pˆi,t
(B131)
ˆ i,t − (1 − α)L ˆ i,t T[ F P i,t = Yˆi,t − αK
(B132)
31
C
Robustness Tables
Following tables and figures are full versions of Table 1 and figures in the main text.
32
33
0.57 5.20 0.32 1.19
0.18 15.82 0.64 4.01 0.47 −0.01 0.99 0.25 0.91 0.91 0.79 0.91 0.80 −0.54 0.67 −0.97 −0.83 −1.00
0.81 0.04 0.99 0.22 0.91 0.91 0.80 0.91 0.80 −0.01 0.51 −0.05 −0.04 −0.05
−0.54 0.67 −0.97 −0.83 −1.00
0.91 0.91 0.81 0.91 0.80
0.97 0.38 0.96 0.08
B 0.9
A 0.1
20 0.05 of output 0.19 13.94 0.59 3.51
0.18 0.58 0.31 0.26 0.32
0.91 0.91 0.79 0.91 0.80
0.53 −0.00 0.99 0.24
0.18 15.47 0.63 3.92
C 0.1
−0.50 0.65 −0.79 −0.73 −0.81
0.91 0.91 0.80 0.91 0.80
0.95 0.25 0.96 0.11
0.48 7.39 0.39 1.78
D 0.8
−0.06 0.53 −0.12 −0.11 −0.12
0.91 0.91 0.80 0.91 0.80
0.80 0.04 0.99 0.21
0.20 13.84 0.58 3.49
E 0.1
0.37 0.28 0.23 0.23 −0.02
0.93 0.94 0.93 0.93 0.91
0.77 0.75 0.74 −0.36
0.91 2.60 1.04 0.61
Data
Table A1.
N = 19, {{0.1} × 9, {0.01} × 10}. Cross-country correlations are of first two countries. The data (median of G7 countries) is taken from
large (D) countries. Cross-country correlations are of the two small countries (C), and the large and one of the small countries (D). E:
moments are for the first (A) and second (B) countries. C, D: N = 3, {0.1, 0.1, 0.8}. Intra-country moments are for the small (C) and
N and size patterns: The first five columns represent models of symmetric size. Sizes are 1/N . A, B: N = 2, {0.1, 0.9}. Intra-country
N 2 3 5 10 Size 0.5 0.33 0.2 0.1 Standard deviation relative to standard deviation Consumption 0.29 0.24 0.22 0.20 Investment 13.21 13.56 13.75 13.87 Labor 0.56 0.57 0.58 0.58 Net exports 3.33 3.42 3.47 3.50 Correlation to output Consumption 0.73 0.75 0.77 0.80 Investment 0.05 0.04 0.04 0.04 Labor 0.97 0.98 0.99 0.99 Net exports 0.21 0.21 0.21 0.22 Autocorrelation Output 0.91 0.91 0.91 0.91 Consumption 0.91 0.91 0.91 0.91 Investment 0.79 0.79 0.80 0.80 Labor 0.91 0.91 0.91 0.91 Net exports 0.80 0.80 0.80 0.80 Cross-country correlation Output −0.72 −0.36 −0.17 −0.06 Consumption 0.64 0.60 0.56 0.53 Investment −0.99 −0.50 −0.25 −0.11 Labor −0.96 −0.48 −0.23 −0.10 Net exports −1.00 −0.50 −0.25 −0.11
Table C2: Business Cycle Moments of Baseline Model
34
B 0.9 1.84 6.60 1.25 0.46 0.28 0.34 0.26 0.11 0.86 0.93 0.91 0.92 0.92 −0.49 0.94 −0.20 0.57 −1.00
A 0.1 0.66 5.18 0.85 1.65 −0.06 −0.09 0.84 0.69 0.91 0.93 0.88 0.92 0.92 −0.49 0.94 −0.20 0.57 −1.00
0.20 0.91 0.24 0.59 0.32
0.92 0.93 0.87 0.92 0.92
0.00 0.03 0.87 0.62
0.58 5.23 0.79 1.55
C 0.1
−0.47 0.92 −0.27 0.44 −0.81
0.88 0.93 0.90 0.92 0.92
0.39 0.30 0.30 0.17
1.47 6.03 0.97 0.72
D 0.8
−0.08 0.84 −0.03 0.16 −0.12
0.92 0.93 0.87 0.92 0.92
0.37 0.17 0.87 0.47
0.50 5.37 0.64 1.37
E 0.1
0.37 0.28 0.23 0.23 −0.02
0.93 0.94 0.93 0.93 0.91
0.77 0.75 0.74 −0.36
0.91 2.60 1.04 0.61
Data
from Table 1.
E: N = 19, {{0.1} × 9, {0.01} × 10}. Cross-country correlations are of first two countries. The data (median of G7 countries) is taken
large (D) countries. Cross-country correlations are of the two small countries (C), and the large and one of the small countries (D).
moments are for the first (A) and second (B) countries. C, D: N = 3, {0.1, 0.1, 0.8}. Intra-country moments are for the small (C) and
N and size patterns: The first five columns represent models of symmetric size. Sizes are 1/N . A, B: N = 2, {0.1, 0.9}. Intra-country
N 2 3 5 10 20 Size 0.5 0.33 0.2 0.1 0.05 Standard deviation relative to standard deviation of output Consumption 1.01 0.75 0.60 0.51 0.48 Investment 5.68 5.46 5.39 5.37 5.37 Labor 0.87 0.75 0.69 0.65 0.64 Net exports 1.38 1.38 1.38 1.38 1.38 Correlation to output Consumption 0.20 0.25 0.30 0.35 0.37 Investment 0.06 0.10 0.13 0.16 0.17 Labor 0.63 0.75 0.82 0.86 0.88 Net exports 0.56 0.52 0.50 0.48 0.47 Autocorrelation Output 0.90 0.91 0.91 0.92 0.92 Consumption 0.93 0.93 0.93 0.93 0.93 Investment 0.89 0.88 0.87 0.87 0.87 Labor 0.92 0.92 0.92 0.92 0.92 Net exports 0.92 0.92 0.92 0.92 0.92 Cross-country correlation Output −0.76 −0.40 −0.19 −0.07 −0.02 Consumption 0.93 0.90 0.88 0.85 0.84 Investment −0.38 −0.22 −0.10 −0.02 0.02 Labor 0.27 0.20 0.19 0.19 0.19 Net exports −1.00 −0.50 −0.24 −0.11 −0.05
Table C3: Business Cycle Moments of Backus et al. (1992) Model
35
0.83 1.77 0.12 0.11
1.18 3.12 0.44 1.05 0.84 0.73 0.01 −0.30 0.91 0.91 0.90 0.90 0.90 0.48 −0.05 −0.24 0.55 −1.00
0.94 0.89 −0.34 −0.64 0.91 0.91 0.90 0.90 0.90 0.37 0.13 0.07 0.06 −0.05
0.48 −0.05 −0.24 0.55 −1.00
0.91 0.91 0.90 0.90 0.90
1.00 0.99 0.94 −0.54
B 0.9
A 0.1
20 0.05 of output 1.21 3.09 0.29 0.84
0.41 0.15 0.13 0.56 0.32
0.91 0.91 0.90 0.90 0.90
0.86 0.76 −0.04 −0.35
1.19 3.11 0.41 1.01
C 0.1
0.46 −0.03 −0.21 0.36 −0.81
0.91 0.91 0.90 0.90 0.90
0.99 0.98 0.83 −0.61
0.87 1.91 0.11 0.19
D 0.8
0.38 0.11 0.04 0.00 −0.12
0.91 0.91 0.90 0.90 0.90
0.94 0.89 −0.30 −0.62
1.19 3.01 0.28 0.81
E 0.1
0.37 0.28 0.23 0.23 −0.02
0.93 0.94 0.93 0.93 0.91
0.77 0.75 0.74 −0.36
0.91 2.60 1.04 0.61
Data
Table 1.
N = 19, {{0.1} × 9, {0.01} × 10}. Cross-country correlations are of first two countries. The data (median of G7 countries) is taken from
large (D) countries. Cross-country correlations are of the two small countries (C), and the large and one of the small countries (D). E:
moments are for the first (A) and second (B) countries. C, D: N = 3, {0.1, 0.1, 0.8}. Intra-country moments are for the small (C) and
N and size patterns: The first five columns represent models of symmetric size. Sizes are 1/N . A, B: N = 2, {0.1, 0.9}. Intra-country
N 2 3 5 10 Size 0.5 0.33 0.2 0.1 Standard deviation relative to standard deviation Consumption 1.00 1.07 1.14 1.19 Investment 2.40 2.65 2.86 3.02 Labor 0.22 0.25 0.27 0.28 Net exports 0.56 0.67 0.75 0.81 Correlation to output Consumption 0.95 0.94 0.94 0.94 Investment 0.89 0.88 0.88 0.89 Labor 0.16 −0.04 −0.18 −0.29 Net exports −0.43 −0.51 −0.57 −0.62 Autocorrelation Output 0.91 0.91 0.91 0.91 Consumption 0.91 0.91 0.91 0.91 Investment 0.90 0.90 0.90 0.90 Labor 0.90 0.90 0.90 0.90 Net exports 0.90 0.90 0.90 0.90 Cross-country correlation Output 0.46 0.43 0.41 0.38 Consumption −0.08 0.01 0.07 0.11 Investment −0.32 −0.14 −0.03 0.04 Labor −0.37 −0.17 −0.05 0.02 Net exports −1.00 −0.50 −0.24 −0.11
Table C4: Business Cycle Moments of Baxter and Crucini (1995) Model
36
B 0.9 0.81 1.69 0.13 0.04 1.00 1.00 0.99 −0.57 0.90 0.90 0.90 0.90 0.90 0.46 0.19 0.07 0.93 −1.00
A 0.1 0.94 2.16 0.18 0.43 0.96 0.92 0.35 −0.41 0.91 0.90 0.90 0.90 0.90 0.45 0.19 0.07 0.93 −1.00
0.39 0.22 0.18 0.91 0.32
0.91 0.90 0.90 0.91 0.90
0.97 0.94 0.34 −0.46
0.95 2.16 0.17 0.41
C 0.1
0.44 0.19 0.08 0.92 −0.81
0.90 0.90 0.90 0.90 0.90
1.00 1.00 0.98 −0.65
0.83 1.74 0.12 0.08
D 0.8
0.37 0.22 0.17 0.76 −0.12
0.91 0.90 0.90 0.91 0.90
0.99 0.97 0.36 −0.72
0.96 2.17 0.10 0.32
E 0.1
0.37 0.28 0.23 0.23 −0.02
0.93 0.94 0.93 0.93 0.91
0.77 0.75 0.74 −0.36
0.91 2.60 1.04 0.61
Data
Table 1.
N = 19, {{0.1} × 9, {0.01} × 10}. Cross-country correlations are of first two countries. The data (median of G7 countries) is taken from
large (D) countries. Cross-country correlations are of the two small countries (C), and the large and one of the small countries (D). E:
moments are for the first (A) and second (B) countries. C, D: N = 3, {0.1, 0.1, 0.8}. Intra-country moments are for the small (C) and
N and size patterns: The first five columns represent models of symmetric size. Sizes are 1/N . A, B: N = 2, {0.1, 0.9}. Intra-country
N 2 3 5 10 20 Size 0.5 0.33 0.2 0.1 0.05 Standard deviation relative to standard deviation of output Consumption 0.88 0.91 0.93 0.96 0.97 Investment 1.91 2.02 2.10 2.17 2.20 Labor 0.13 0.12 0.11 0.11 0.10 Net exports 0.23 0.27 0.30 0.33 0.34 Correlation to output Consumption 0.99 0.99 0.99 0.99 0.99 Investment 0.98 0.97 0.97 0.97 0.97 Labor 0.73 0.60 0.48 0.36 0.30 Net exports −0.50 −0.59 −0.66 −0.71 −0.74 Autocorrelation Output 0.90 0.91 0.91 0.91 0.91 Consumption 0.90 0.90 0.90 0.90 0.90 Investment 0.90 0.90 0.90 0.90 0.90 Labor 0.91 0.91 0.91 0.91 0.91 Net exports 0.90 0.90 0.90 0.90 0.90 Cross-country correlation Output 0.45 0.42 0.39 0.37 0.36 Consumption 0.19 0.20 0.21 0.22 0.23 Investment 0.06 0.11 0.14 0.17 0.18 Labor 0.86 0.83 0.80 0.77 0.74 Net exports −1.00 −0.50 −0.24 −0.11 −0.05
Table C5: Business Cycle Moments of Baxter and Farr (2005) Model
37
B 0.9 0.55 2.63 0.38 0.01 0.91 0.96 0.91 0.06 0.90 0.91 0.89 0.88 0.95 0.13 0.74 −0.58 −0.59 −0.34
A 0.1 0.56 4.12 0.36 0.23 0.93 0.86 0.92 −0.58 0.90 0.91 0.88 0.89 0.88 0.14 0.74 −0.57 −0.59 −0.34
0.18 0.68 −0.07 −0.27 0.30
0.90 0.91 0.89 0.89 0.89
0.92 0.92 0.93 −0.61
0.51 3.64 0.39 0.14
C 0.1
0.17 0.68 −0.31 −0.25 −0.23
0.90 0.91 0.89 0.89 0.96
0.90 0.97 0.93 0.15
0.50 2.75 0.40 0.01
D 0.8
0.55 0.72 0.46 0.43 0.16
0.90 0.91 0.89 0.89 0.93
0.93 0.98 0.97 −0.33
0.43 2.92 0.43 0.01
E 0.1
0.37 0.28 0.23 0.23 −0.02
0.93 0.94 0.93 0.93 0.91
0.77 0.75 0.74 −0.36
0.91 2.60 1.04 0.61
Data
Table 1.
N = 19, {{0.1} × 9, {0.01} × 10}. Cross-country correlations are of first two countries. The data (median of G7 countries) is taken from
large (D) countries. Cross-country correlations are of the two small countries (C), and the large and one of the small countries (D). E:
moments are for the first (A) and second (B) countries. C, D: N = 3, {0.1, 0.1, 0.8}. Intra-country moments are for the small (C) and
N and size patterns: The first five columns represent models of symmetric size. Sizes are 1/N . A, B: N = 2, {0.1, 0.9}. Intra-country
N 2 3 5 10 20 Size 0.5 0.33 0.2 0.1 0.05 Standard deviation relative to standard deviation of output Consumption 0.55 0.50 0.47 0.44 0.41 Investment 3.24 3.22 3.20 3.19 3.22 Labor 0.38 0.40 0.42 0.43 0.43 Net exports 0.10 0.07 0.06 0.05 0.05 Correlation to output Consumption 0.91 0.90 0.91 0.92 0.93 Investment 0.91 0.94 0.96 0.97 0.98 Labor 0.91 0.93 0.95 0.96 0.97 Net exports −0.55 −0.62 −0.65 −0.67 −0.69 Autocorrelation Output 0.90 0.90 0.90 0.90 0.90 Consumption 0.91 0.91 0.91 0.91 0.92 Investment 0.88 0.89 0.89 0.89 0.89 Labor 0.88 0.89 0.89 0.89 0.89 Net exports 0.88 0.89 0.90 0.90 0.90 Cross-country correlation Output 0.17 0.19 0.27 0.41 0.57 Consumption 0.81 0.73 0.69 0.69 0.74 Investment −0.57 −0.22 0.01 0.24 0.47 Labor −0.60 −0.27 −0.03 0.21 0.45 Net exports −1.00 −0.50 −0.21 0.05 0.33
Table C6: Business Cycle Moments of Backus et al. (1994) Model
38
B 0.9 0.80 1.67 0.14 0.07 1.00 1.00 0.98 −0.44 0.91 0.91 0.90 0.90 0.90 0.51 −0.05 0.30 0.84 −0.57
A 0.1 0.86 1.90 0.12 0.10 0.99 0.98 0.84 −0.53 0.91 0.91 0.90 0.90 0.90 0.32 0.26 0.13 0.69 0.73
0.33 0.30 0.24 0.63 0.25
0.91 0.91 0.90 0.90 0.90
1.00 0.99 0.91 −0.61
0.85 1.87 0.12 0.08
C 0.1
0.36 0.27 0.19 0.72 −0.53
0.91 0.91 0.90 0.90 0.90
1.00 1.00 0.98 −0.53
0.80 1.68 0.14 0.06
D 0.8
0.60 0.60 0.58 0.62 0.20
0.91 0.91 0.90 0.90 0.90
1.00 1.00 0.98 −0.69
0.80 1.71 0.14 0.03
E 0.1
0.37 0.28 0.23 0.23 −0.02
0.93 0.94 0.93 0.93 0.91
0.77 0.75 0.74 −0.36
0.91 2.60 1.04 0.61
Data
Table 1.
N = 19, {{0.1} × 9, {0.01} × 10}. Cross-country correlations are of first two countries. The data (median of G7 countries) is taken from
large (D) countries. Cross-country correlations are of the two small countries (C), and the large and one of the small countries (D). E:
moments are for the first (A) and second (B) countries. C, D: N = 3, {0.1, 0.1, 0.8}. Intra-country moments are for the small (C) and
N and size patterns: The first five columns represent models of symmetric size. Sizes are 1/N . A, B: N = 2, {0.1, 0.9}. Intra-country
N 2 3 5 10 20 Size 0.5 0.33 0.2 0.1 0.05 Standard deviation relative to standard deviation of output Consumption 0.82 0.82 0.82 0.82 0.82 Investment 1.81 1.83 1.83 1.84 1.85 Labor 0.13 0.13 0.13 0.12 0.12 Net exports 0.05 0.05 0.04 0.04 0.04 Correlation to output Consumption 1.00 1.00 1.00 1.00 1.00 Investment 0.99 0.99 0.99 0.99 0.99 Labor 0.97 0.97 0.98 0.98 0.98 Net exports −0.52 −0.65 −0.74 −0.82 −0.85 Autocorrelation Output 0.91 0.91 0.91 0.91 0.91 Consumption 0.91 0.91 0.91 0.91 0.91 Investment 0.90 0.90 0.90 0.90 0.90 Labor 0.90 0.90 0.90 0.90 0.90 Net exports 0.90 0.90 0.90 0.90 0.90 Cross-country correlation Output 0.38 0.35 0.39 0.49 0.62 Consumption 0.31 0.31 0.36 0.47 0.61 Investment 0.13 0.20 0.29 0.43 0.59 Labor 0.66 0.56 0.53 0.58 0.67 Net exports −0.96 −0.49 −0.21 0.05 0.32
Table C7: Business Cycle Moments of Arvanitis and Mikkola (1996) Model
39
0.90 0.91 0.89 0.89 – 0.62 0.78 0.53 0.49 –
0.90 0.91 0.89 0.89 – 0.50 0.75 0.35 0.27 –
0.29 0.81 −0.22 −0.50 –
0.90 0.91 0.89 0.88 – 0.48 1.00 0.97 −0.49 –
0.90 0.91 0.89 0.89 –
0.96 0.97 0.91 –
0.94 0.98 0.96 –
0.93 0.98 0.95 –
0.30 0.75 −0.03 −0.20 –
0.90 0.91 0.89 0.89 –
0.92 0.97 0.92 –
0.54 2.63 0.38 –
0.63 2.27 0.30 –
0.93 0.96 0.90 –
C 0.1
B 0.9
10 20 A 0.1 0.05 0.1 deviation of output 0.49 0.46 0.58 2.72 2.78 2.51 0.39 0.41 0.35 – – –
0.31 0.76 −0.06 −0.26 –
0.90 0.91 0.89 0.89 –
0.95 0.97 0.91 –
0.60 2.40 0.33 –
D 0.8
0.52 0.71 0.41 0.35 –
0.90 0.91 0.89 0.89 –
0.94 0.98 0.96 –
0.46 2.78 0.40 –
E 0.1
0.37 0.28 0.23 0.23 −0.02
0.93 0.94 0.93 0.93 0.91
0.77 0.75 0.74 −0.36
0.91 2.60 1.04 0.61
Data
of G7 countries) is taken from Table 1.
small countries (D). E: N = 19, {{0.1} × 9, {0.01} × 10}. Cross-country correlations are of first two countries. The data (median
small (C) and large (D) countries. Cross-country correlations are of the two small countries (C), and the large and one of the
country moments are for the first (A) and second (B) countries. C, D: N = 3, {0.1, 0.1, 0.8}. Intra-country moments are for the
N and size patterns: The first five columns represent models of symmetric size. Sizes are 1/N . A, B: N = 2, {0.1, 0.9}. Intra-
N 2 3 5 Size 0.5 0.33 0.2 Standard deviation relative to standard Consumption 0.63 0.57 0.53 Investment 2.29 2.51 2.63 Labor 0.30 0.35 0.38 Net exports – – – Correlation to output Consumption 0.95 0.93 0.93 Investment 0.97 0.97 0.97 Labor 0.90 0.91 0.93 Net exports – – – Autocorrelation Output 0.90 0.90 0.90 Consumption 0.91 0.91 0.91 Investment 0.89 0.89 0.89 Labor 0.89 0.89 0.89 Net exports – – – Cross-country correlation Output 0.61 0.47 0.43 Consumption 0.92 0.84 0.77 Investment 0.20 0.12 0.20 Labor −0.14 −0.08 0.07 Net exports – – –
Table C8: Business Cycle Moments of Heathcote and Perri (2002) Model
Figure C1: Cross-country Correlations and Number of Countries in the Baseline Model
Cross−country correlations
1
0.5
←Y ←C ←I, L ←NX
0
Output Consumption Investment Labor Net exports
−0.5
−1 2
5
8 11 14 Number of countries
17
20
The solid lines show the model’s cross-country correlations, given the number of countries in the model. The filled symbols represent positive values whereas the open symbols negative values. The interrupted lines represent data values taken from Table A1. The multi-country versions of King et al. (1988), with no shock diffusion and complete market.
40
Figure C2: Cross-country Correlations and Number of Countries in Baxter and Farr (2005) Model
Cross−country correlations
1
0.5
←Y ←C ←I, L ←NX
0
Output Consumption Investment Labor Net exports
−0.5
−1 2
5
8 11 14 Number of countries
17
20
The solid lines show the model’s cross-country correlations, given the number of countries in the model. The filled symbols represent positive values whereas the open symbols negative values. The interrupted lines represent data values. The multi-country versions of Baxter and Farr (2005), with no shock diffusion and a bond only market.
Figure C3: Cross-country Correlations and Number of Countries in Arvanitis and Mikkola (1996) Model
Cross−country correlations
1
0.5
←Y ←C ←I, L ←NX
0
Output Consumption Investment Labor Net exports
−0.5
−1 2
5
8 11 14 Number of countries
17
20
The solid lines show the model’s cross-country correlations, given the number of countries in the model. The filled symbols represent positive values whereas the open symbols negative values. The interrupted lines represent data values. The multi-country versions of Arvanitis and Mikkola (1996), no shock diffusion and a bond only market.
41
D
Impulse Response Functions and Intuition
Impulse response functions (IRFs) of the baseline single-good model illustrate the mechanisms behind the moment results. Figures D4 and D5 are IRFs of output, consumption, investment, labor, net exports and TFP. The X axis represents the model time period, with the shock hitting the economies at period 0. The vertical axis shows the percentage deviations of the variables from the steady state values. Both models use baseline single-good technology, a complete market assumption and a shock that excludes the cross-country diffusion component (ωih = 0). The figures are drawn from correlated shocks; at period 0, a 1% positive TFP shock hits Country 1 and a 0.12% positive TFP shock hits Country 2. Other countries in the ten-country model do not experience shocks. Figure D4 is derived from a two-country model, and Figure D5 from a ten-country model. The upper and lower panels in Figure D4 show IRFs for countries 1 and 2, respectively. Similarly, The upper and middle panels in Figure D5 are the IRFs of Country 1 and Country 2, respectively. Since the remaining countries are completely symmetric, the lower panel of Figure D5 shows IRFs of the remaining countries. Figure D4 indicates that, after the positive productivity shock in the two-country model, Country 1 experiences a boom and Country 2 a recession; a large positive TFP shock stimulates investment in Country 1; and the investment spike leads to a higher labor supply in Country 1. Output rises since capital stock, labor and TFP increase. Country 2 also experiences a positive productivity shock (TFP line of Country 2). Nonetheless, in the two-country model, investment in Country 2 decreases, since the higher marginal return to investment in Country 1 attracts investment away from Country 2 to Country 1, as indicated by large changes in net exports. The reduction in investment leads to a decrease in the labor supply in Country 2. The output then decreases, despite the positive TFP shock, because of a large decline in input. The complete market assumption ensures consumption increments in both countries and the consumption correlation is strongly positive. The implications are drastically different in a ten-country model. As in the two-country model, Country 1 experiences a boom (the upper panel of Figure D5), by inducing investments from other countries. In addition, the correlated shocks in the ten-country model lead to a boom in the secondhighest TFP country. The middle panel of Figure D5 shows that, in Country 2, there is a slight rise in investment in the ten-country model. Other variables also increase, although the magnitude of the increments is smaller than in the case of Country 1. In the ten-country model, the diminishing marginal returns for capital in the highest TFP country (Country 1) lead to an investment inflow to the second-highest TFP country (Country 2). As a result, cross-country correlations of investment, labor and output are positive between Countries 1 and 2. At the same time, internal investments decrease in Countries 3 to 10 (the lower panel of Figure D5). Cross-country correlations among these 28 (= 8 × 7/2) pairs are positive because all these countries experience reductions in output, investment and labor. This particular correlated shock delivers two types of positive cross-country correlations of investment: one increasing pair and 28 decreasing pairs. The other 16 (= 10 × 9/2 − 1 − 28) combinations are negatively correlated.
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Figure D4: Impulse Response Functions (Two-country Model) (+1% TFP shock to Country 1, +.12% TFP shock to Country 2.) Country 1
Deviations from the steady state
4 3 2 1 Output Consumption Investment Labor Net expopts TFP
0 −1 −2 −5
0
5 Period
10
15
Country 2
Deviations from the steady state
4 Output Consumption Investment Labor Net expopts TFP
3 2 1 0 −1 −2 −5
0
5 Period
43
10
15
Figure D5: Impulse Response Functions (Ten-country Model) (+1% TFP Shock to Country 1, +.12% TFP Shock to Country 2, No Shock to Others) Country 1
Deviations from the steady state
4 3 2 1 Output Consumption Investment Labor Net expopts TFP
0 −1 −2 −5
0
5 Period
10
15
Country 2
Deviations from the steady state
4 Output Consumption Investment Labor Net expopts TFP
3 2 1 0 −1 −2 −5
0
5 Period
10
15
Country 3, ..., Country 10
Deviations from the steady state
4 Output Consumption Investment Labor Net expopts TFP
3 2 1 0 −1 −2 −5
0
5 Period
44
10
15
References Anderson, Gary S. (2008) “Solving Linear Rational Expectation Models: A Horse Race,” Computational Economics, Vol. 31, No. 2, pp. 95–113, March. Arvanitis, Athanasios V. and Anne Mikkola (1996) “Asset-Market Structure and International Trade Dynamics,” American Economic Review, Vol. 86, No. 2, pp. 67–70, May. Backus, David K., Patrick J. Kehoe, and Finn E. Kydland (1992) “International Real Business Cycles,” Journal of Political Economy, Vol. 100, No. 4, pp. 745–775, August. (1994) “Dynamics of the Trade Balance and the Terms of Trade: The J-Curve?” American Economic Review, Vol. 84, No. 1, pp. 84–103, March. Baxter, Marianne and Mario J. Crucini (1993) “Explaining Saving-Investment Correlations,” American Economic Review, Vol. 83, No. 3, pp. 416–436, June. (1995) “Business Cycles and the Asset Structure of Foreign Trade,” International Economic Review, Vol. 36, No. 4, pp. 821–854, November. Baxter, Marianne and Dorsey D. Farr (2005) “Variable Capital Utilization and International Business Cycles,” Journal of International Economics, Vol. 65, No. 2, pp. 335–347, March. Baxter, Marianne and Robert G. King (1999) “Measuring Business Cycles: Approximate Band-Pass Filters for Economic Time Series,” Review of Economics and Statistics, Vol. 81, No. 4, pp. 575–593, November. Ghironi, Fabio and Mark J. Melitz (2005) “International Trade and Macroeconomic Dynamics with Heterogeneous Firms,” Quarterly Journal of Economics, Vol. 120, No. 3, pp. 865–915, August. Heathcote, Jonathan and Fabrizio Perri (2002) “Financial autarky and international business cycles,” Journal of Monetary Economics, Vol. 49, No. 3, pp. 601–627, April. Heston, Alan, Robert Summers, and Bettina Aten (2008) “Penn World Table version 6.2,” Center for International Comparisons at the University of Pennsylvania (CICUP). Hodrick, Robert J. and Edward C. Prescott (1997) “Postwar US Business Cycles: An Empirical Investigation,” Journal of Money, Credit and Banking, Vol. 29, No. 1, pp. 1–16, February. King, Robert G., Charles I. Plosser, and Sergio T. Rebelo (1988) “Production, Growth and Business Cycles I: The Basic Neoclassical Model,” Journal of Monetary Economics, Vol. 21, No. 2/3, pp. 195–232, March-May. Kollmann, Robert (1996) “Incomplete Asset Markets and the Cross-country Consumption Correlation Puzzle,” Journal of Economic Dynamics and Control, Vol. 20, No. 5, pp. 946–961, May. Kydland, Finn and Edwar Prescott (1982) “Time to Build and Aggregate Fluctuations,” Econometrica, Vol. 50, No. 6, pp. 1345–1370, November.
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