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/*
* This file implements the multi-country RBC model with time to build,
* described in Backus, Kehoe and Kydland (1992): "International Real Business
* Cycles", Journal of Political Economy, 100(4), 745-775.
*
* The notation for the variable names are the same in this file than in the paper.
* However the timing convention is different: we had to taken into account the
* fact that in Dynare, if a variable is denoted at the current period, then
* this variable must be also decided at the current period.
* Concretely, here are the differences between the paper and the model file:
* - z_t in the model file is equal to z_{t+1} in the paper
* - k_t in the model file is equal to k_{t+J} in the paper
* - s_t in the model file is equal to s_{J,t}=s_{J-1,t+1}=...=s_{1,t+J-1} in the paper
*
* The macroprocessor is used in this file to create a loop over countries.
* Only two countries are used here (as in the paper), but it is easy to add
* new countries in the corresponding macro-variable and completing the
* calibration.
*
* The calibration is the same than in the paper. The results in terms of
* moments of variables are very close to that of the paper (but not equal
* since the authors a different solution method).
*
* This implementation was written by Sebastien Villemot. Please note that the
* following copyright notice only applies to this Dynare implementation of the
* model.
*/
/*
* Copyright © 2010 Dynare Team
*
* This file is part of Dynare.
*
* Dynare is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* Dynare is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with Dynare. If not, see <https://www.gnu.org/licenses/>.
*/
@#define countries = [ "H", "F" ]
@#define J = 4
@#for co in countries
var C_@{co} L_@{co} N_@{co} A_@{co} K_@{co} Z_@{co} X_@{co} LAMBDA_@{co} S_@{co} NX_@{co} Y_@{co};
varexo E_@{co};
parameters beta_@{co} alpha_@{co} eta_@{co} mu_@{co} gamma_@{co} theta_@{co} nu_@{co} sigma_@{co} delta_@{co} phi_@{co} psi_@{co} rho_@{co}_@{co};
@#endfor
// Lagrange multiplier of aggregate constraint
var LGM;
parameters rho_@{countries[1]}_@{countries[2]} rho_@{countries[2]}_@{countries[1]};
model;
@#for co in countries
Y_@{co} = ((LAMBDA_@{co}*K_@{co}(-@{J})^theta_@{co}*N_@{co}^(1-theta_@{co}))^(-nu_@{co}) + sigma_@{co}*Z_@{co}(-1)^(-nu_@{co}))^(-1/nu_@{co});
K_@{co} = (1-delta_@{co})*K_@{co}(-1) + S_@{co};
X_@{co} =
@# for lag in (-J+1):0
+ phi_@{co}*S_@{co}(@{lag})
@# endfor
;
A_@{co} = (1-eta_@{co})*A_@{co}(-1) + N_@{co};
L_@{co} = 1 - alpha_@{co}*N_@{co} - (1-alpha_@{co})*eta_@{co}*A_@{co}(-1);
// Utility multiplied by gamma
# U_@{co} = (C_@{co}^mu_@{co}*L_@{co}^(1-mu_@{co}))^gamma_@{co};
// FOC with respect to consumption
psi_@{co}*mu_@{co}/C_@{co}*U_@{co} = LGM;
// FOC with respect to labor
// NOTE: this condition is only valid for alpha = 1
psi_@{co}*(1-mu_@{co})/L_@{co}*U_@{co}*(-alpha_@{co}) = - LGM * (1-theta_@{co})/N_@{co}*(LAMBDA_@{co}*K_@{co}(-@{J})^theta_@{co}*N_@{co}^(1-theta_@{co}))^(-nu_@{co})*Y_@{co}^(1+nu_@{co});
// FOC with respect to capital
@# for lag in 0:(J-1)
+beta_@{co}^@{lag}*LGM(+@{lag})*phi_@{co}
@# endfor
@# for lag in 1:J
-beta_@{co}^@{lag}*LGM(+@{lag})*phi_@{co}*(1-delta_@{co})
@# endfor
= beta_@{co}^@{J}*LGM(+@{J})*theta_@{co}/K_@{co}*(LAMBDA_@{co}(+@{J})*K_@{co}^theta_@{co}*N_@{co}(+@{J})^(1-theta_@{co}))^(-nu_@{co})*Y_@{co}(+@{J})^(1+nu_@{co});
// FOC with respect to stock of inventories
LGM=beta_@{co}*LGM(+1)*(1+sigma_@{co}*Z_@{co}^(-nu_@{co}-1)*Y_@{co}(+1)^(1+nu_@{co}));
// Shock process
@# if co == countries[1]
@# define alt_co = countries[2]
@# else
@# define alt_co = countries[1]
@# endif
(LAMBDA_@{co}-1) = rho_@{co}_@{co}*(LAMBDA_@{co}(-1)-1) + rho_@{co}_@{alt_co}*(LAMBDA_@{alt_co}(-1)-1) + E_@{co};
NX_@{co} = (Y_@{co} - (C_@{co} + X_@{co} + Z_@{co} - Z_@{co}(-1)))/Y_@{co};
@#endfor
// World ressource constraint
@#for co in countries
+C_@{co} + X_@{co} + Z_@{co} - Z_@{co}(-1)
@#endfor
=
@#for co in countries
+Y_@{co}
@#endfor
;
end;
@#for co in countries
beta_@{co} = 0.99;
mu_@{co} = 0.34;
gamma_@{co} = -1.0;
alpha_@{co} = 1;
eta_@{co} = 0.5; // Irrelevant when alpha=1
theta_@{co} = 0.36;
nu_@{co} = 3;
sigma_@{co} = 0.01;
delta_@{co} = 0.025;
phi_@{co} = 1/@{J};
psi_@{co} = 0.5;
@#endfor
rho_H_H = 0.906;
rho_F_F = 0.906;
rho_H_F = 0.088;
rho_F_H = 0.088;
initval;
@#for co in countries
LAMBDA_@{co} = 1;
NX_@{co} = 0;
Z_@{co} = 1;
A_@{co} = 1;
L_@{co} = 0.5;
N_@{co} = 0.5;
Y_@{co} = 1;
K_@{co} = 1;
C_@{co} = 1;
S_@{co} = 1;
X_@{co} = 1;
E_@{co} = 0;
@#endfor
LGM = 1;
end;
shocks;
var E_H; stderr 0.00852;
var E_F; stderr 0.00852;
corr E_H, E_F = 0.258;
end;
steady;
check;
stoch_simul(order=1, hp_filter=1600, nograph);
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