Merge branch 'Ramsey_Example' into 'master'

Add example mod-file for Ramsey and OSR

Closes #1645

See merge request !1694

doc/manual/source/examples.rst

@@ -56,3 +56,9 @@ description, please refer to the comments inside the files themselves.
     Baseline New Keynesian Model estimated in *Fernández-Villaverde
     (2010)*. It demonstrates how to use an explicit steady state file
     to update parameters and call a numerical solver.
+
+``Ramsey_Example.mod``
+
+    File demonstrating how to conduct optimal policy experiments in a 
+    simple New Keynesian model either under commitment (Ramsey) or using
+    optimal simple rules (OSR)
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examples/Ramsey_Example.mod

+/*
+ * 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 <http://www.gnu.org/licenses/>.
+ */
+
+//**********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
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