/*
* This file replicates the model studied in:
* Lawrence J. Christiano, Roberto Motto and Massimo Rostagno (2007):
* "Notes on Ramsey-Optimal Monetary Policy", Section 2
* The paper is available at http://faculty.wcas.northwestern.edu/~lchrist/d16/d1606/ramsey.pdf
*
* Notes:
* - This mod-files allows to simulate a simple New Keynesian Model with Rotemberg price
* adjustment costs under three different monetary policy arrangements:
* 1. a Taylor rule with a fixed inflation feedback coefficient alpha
* -> set the Optimal_policy switch to 0
* 2. a Taylor rule where the inflation feedback coefficient alpha is chosen
* optimally to minimize a quadratic loss function (optimal simple rule (OSR))
* -> set the Optimal_policy switch to 1 and the Ramsey switch to 0
* 3. fully optimal monetary under commitment (Ramsey)
* -> set the Optimal_policy switch to 1 and the Ramsey switch to 1
*
* - The Efficent_steady_state switch can be used to switch from an distorted steady state
* due to a monopolistic distortion to one where a labor subsidy counteracts this
* distortion. Note that the purely quadratic loss function in the OSR case does not capture
* the full welfare losses with a distorted steady state as there would be a linear term
* appearing.
*
* - This files shows how to use a conditional steady state file in the Ramsey case. It takes
* the value of the defined instrument R as given and then computes the rest of the steady
* state, including the steady state inflation rate, based on this value. The initial value
* of the instrument for steady state search must then be defined in an initval-block.
*
* - The optim_weights in the OSR case are based on a second order approximation to the welfare function
* as in Gali (2015). The relative weight between inflation and output gap volatility is essentially
* given by the slope of the New Keynesian Phillips Curve. Note that the linear terms that would be
* present in case of a distorted steady state need to be dropped for OSR.
*
* - Due to divine coincidence, the first best policy involves fully stabilizing inflation
* and thereby the output gap. As a consequence, the optimal inflation feedback coefficient
* in a Taylor rule would be infinity. The OSR command therefore estimates it to be at the
* upper bound defined via osr_params_bounds.
*
* - The mod-file also allows to conduct estimation under Ramsey policy by setting the
* Estimation_under_Ramsey switch to 1.
*
* This implementation was written by Johannes Pfeifer.
*
* If you spot mistakes, email me at jpfeifer@gmx.de
*
* Please note that the following copyright notice only applies to this Dynare
* implementation of the model.
*/
/*
* Copyright (C) 2019 Dynare Team
*
* This 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.
*
* It 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.
*
* For a copy of the GNU General Public License,
* see .
*/
//**********Define which monetary policy setup to use ***********
@#ifndef Optimal_policy
@#define Optimal_policy=1
@#ifndef Ramsey
@#define Ramsey=1
@#endif
@#endif
//**********Define whether to use distorted steady state***********
@#ifndef Efficent_steady_state
@#define Efficent_steady_state=0
@#endif
@#ifndef Estimation_under_Ramsey
@#define Estimation_under_Ramsey=0
@#endif
var C $C$ (long_name='Consumption')
pi $\pi$ (long_name='Gross inflation')
h $h$ (long_name='hours worked')
Z $Z$ (long_name='TFP')
R $R$ (long_name='Net nominal interest rate')
log_C ${\ln C}$ (long_name='Log Consumption')
log_h ${\ln h}$ (long_name='Log hours worked')
pi_ann ${\pi^{ann}}$ (long_name='Annualized net inflation')
R_ann ${R^{ann}}$ (long_name='Annualized net nominal interest rate')
r_real ${r^{ann,real}}$ (long_name='Annualized net real interest rate')
y_nat ${y^{nat}}$ (long_name='Natural (flex price) output')
y_gap ${r^{gap}}$ (long_name='Output gap')
;
varexo epsilon ${\varepsilon}$ (long_name='TFP shock')
;
parameters beta ${\beta}$ (long_name='discount factor')
theta ${\theta}$ (long_name='substitution elasticity')
tau ${\tau}$ (long_name='labor subsidy')
chi ${\chi}$ (long_name='labor disutility')
phi ${\phi}$ (long_name='price adjustment costs')
rho ${\rho}$ (long_name='TFP autocorrelation')
@# if !defined(Ramsey) || Ramsey==0
pi_star ${\pi^*}$ (long_name='steady state inflation')
alpha ${\alpha}$ (long_name='inflation feedback Taylor rule')
@# endif
;
beta=0.99;
theta=5;
phi=100;
rho=0.9;
@# if !defined(Ramsey) || Ramsey==0
alpha=1.5;
pi_star=1;
@# endif
@# if Efficent_steady_state
tau=1/(theta-1);
@# else
tau=0;
@# endif
chi=1;
model;
[name='Euler equation']
1/(1+R)=beta*C/(C(+1)*pi(+1));
[name='Firm FOC']
(tau-1/(theta-1))*(1-theta)+theta*(chi*h*C/(exp(Z))-1)=phi*(pi-1)*pi-beta*phi*(pi(+1)-1)*pi(+1);
[name='Resource constraint']
C*(1+phi/2*(pi-1)^2)=exp(Z)*h;
[name='TFP process']
Z=rho*Z(-1)+epsilon;
@#if !defined(Ramsey) || Ramsey==0
[name='Taylor rule']
R=pi_star/beta-1+alpha*(pi-pi_star);
@#endif
[name='Definition log consumption']
log_C=log(C);
[name='Definition log hours worked']
log_h=log(h);
[name='Definition annualized inflation rate']
pi_ann=4*log(pi);
[name='Definition annualized nominal interest rate']
R_ann=4*R;
[name='Definition annualized real interest rate']
r_real=4*log((1+R)/pi(+1));
[name='Definition natural output']
y_nat=exp(Z)*sqrt((theta-1)/theta*(1+tau)/chi);
[name='output gap']
y_gap=log_C-log(y_nat);
end;
steady_state_model;
Z=0;
@# if !defined(Ramsey) || Ramsey==0
R=pi_star/beta-1; %only set this if not conditional steady state file for Ramsey
@# endif
pi=(R+1)*beta;
C=sqrt((1+1/theta*((1-beta)*(pi-1)*pi-(tau-1/(theta-1))*(1-theta)))/(chi*(1+phi/2*(pi-1)^2)));
h=C*(1+phi/2*(pi-1)^2);
log_C=log(C);
log_h=log(h);
pi_ann=4*log(pi);
R_ann=4*R;
r_real=4*log((1+R)/pi);
y_nat=sqrt((theta-1)/theta*(1+tau)/chi);
y_gap=log_C-log(y_nat);
end;
@# if defined(Ramsey) && Ramsey==1
//define initial value of instrument for Ramsey
initval;
R=1/beta-1;
end;
@# endif
shocks;
var epsilon = 0.01^2;
end;
@#if Optimal_policy==0
//use Taylor rule
stoch_simul(order=2) pi_ann log_h R_ann log_C Z r_real y_nat;
@#else
@# if !defined(Ramsey) || Ramsey==0
//use OSR Taylor rule
//set weights on (co-)variances for OSR
optim_weights;
pi theta/((theta-1)/phi);
y_gap 1;
end;
//define OSR parameters to be optimized
osr_params alpha;
//starting value for OSR parameter
alpha = 1.5;
//define bounds for OSR during optimization
osr_params_bounds;
alpha, 0, 100;
end;
//compute OSR and provide output
osr(opt_algo=9) pi_ann log_h R_ann log_C Z r_real;
@# else
//use Ramsey optimal policy
//define planner objective, which corresponds to utility function of agents
planner_objective log(C)-chi/2*h^2;
//set up Ramsey optimal policy problem with interest rate R as the instrument,...
// defining the discount factor in the planner objective to be the one of private agents
ramsey_model(instruments=(R),planner_discount=beta,planner_discount_latex_name=$\beta$);
//conduct stochastic simulations of the Ramsey problem
stoch_simul(order=1,irf=20,periods=500) pi_ann log_h R_ann log_C Z r_real;
evaluate_planner_objective;
@# if Estimation_under_Ramsey==1
datatomfile('ramsey_simulation',{'log_C'})
estimated_params;
rho,0.5,uniform_pdf, , ,0,1;
end;
varobs log_C;
estimation(datafile=ramsey_simulation,mode_compute=5);
@# endif
@# endif
@# endif