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/*
* This model shows how to use the differentiate_forward_vars option to simulate
* perfect foresight models when the steady state is unknown
* or when the model is very persistent. In this file, we consider an RBC model
* with a CES technology and very persistent productivity shock. We set the
* autoregressive parameter of this exogenous productivity to 0.999, so that
* in period 400 the level of productivity, after an initial one percent shock,
* is still 0.67\% above its steady state level.
*
* Written by Stéphane Adjemian. For more information, see
* http://gitlab.ithaca.fr/Dynare/differentiate-forward-variables
*/
var Capital, Output, Labour, Consumption, Efficiency, efficiency, ExpectedTerm;
varexo EfficiencyInnovation;
parameters beta, theta, tau, alpha, psi, delta, rho, effstar, sigma;
/*
** Calibration
*/
beta = 0.990;
theta = 0.357;
tau = 30.000;
alpha = 0.450;
psi = -1.000; // So that the elasticity of substitution between inputs is 1/(1-psi)=1/10
delta = 0.020;
rho = 0.999;
effstar = 1.000;
sigma = 0.010;
model(differentiate_forward_vars);
// Eq. n°1:
efficiency = rho*efficiency(-1) + sigma*EfficiencyInnovation;
// Eq. n°2:
Efficiency = effstar*exp(efficiency);
// Eq. n°3:
Output = Efficiency*(alpha*(Capital(-1)^psi)+(1-alpha)*(Labour^psi))^(1/psi);
// Eq. n°4:
Consumption + Capital - Output - (1-delta)*Capital(-1);
// Eq. n°5:
((1-theta)/theta)*(Consumption/(1-Labour)) - (1-alpha)*(Output/Labour)^(1-psi);
// Eq. n°6:
(((Consumption^theta)*((1-Labour)^(1-theta)))^(1-tau))/Consumption - ExpectedTerm(1);
// Eq. n°7:
ExpectedTerm = beta*((((Consumption^theta)*((1-Labour)^(1-theta)))^(1-tau))/Consumption)*(alpha*((Output/Capital(-1))^(1-psi))+1-delta);
end;
steady_state_model;
efficiency = 0;
Efficiency = effstar;
// Compute some steady state ratios.
Output_per_unit_of_Capital=((1/beta-1+delta)/alpha)^(1/(1-psi));
Consumption_per_unit_of_Capital=Output_per_unit_of_Capital-delta;
Labour_per_unit_of_Capital=(((Output_per_unit_of_Capital/Efficiency)^psi-alpha)/(1-alpha))^(1/psi);
Output_per_unit_of_Labour=Output_per_unit_of_Capital/Labour_per_unit_of_Capital;
Consumption_per_unit_of_Labour=Consumption_per_unit_of_Capital/Labour_per_unit_of_Capital;
// Compute steady state share of capital.
ShareOfCapital=alpha/(alpha+(1-alpha)*Labour_per_unit_of_Capital^psi);
// Compute steady state of the endogenous variables.
Labour=1/(1+Consumption_per_unit_of_Labour/((1-alpha)*theta/(1-theta)*Output_per_unit_of_Labour^(1-psi)));
Consumption = Consumption_per_unit_of_Labour*Labour;
Capital = Labour/Labour_per_unit_of_Capital;
Output = Output_per_unit_of_Capital*Capital;
ExpectedTerm = beta*((((Consumption^theta)*((1-Labour)^(1-theta)))^(1-tau))/Consumption)*(alpha*((Output/Capital)^(1-psi))+1-delta);
end;
shocks;
var EfficiencyInnovation;
periods 1;
values 1;
end;
steady;
check;
model_info;
perfect_foresight_setup(periods=500);
perfect_foresight_solver;
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