/* * This file implements the New Keynesian model with price and wage rigidities under optimal policy * with commitment (Ramsey) of Jordi Galí (2015): Monetary Policy, Inflation, and the Business Cycle, Princeton * University Press, Second Edition, Chapter 6.4 * * THIS MOD-FILE REQUIRES DYNARE 4.6 OR HIGHER * * Notes: * - all model variables are expressed in deviations from steady state, i.e. in contrast to * to the chapter, the nominal interest rate, natural output, and the natural real wage are not in log-levels, but rather mean 0 * - in the LOM for the discount rate shock z the shock enters with a minus sign in this mod-file to generate the * IRF to a -0.5% shock * * This implementation was written by Johannes Pfeifer. In case 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 © 2020 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 whether to use interest rate or money growth rate rule @#define money_growth_rule=1 var pi_p ${\pi^p}$ (long_name='price inflation') y_gap ${\tilde y}$ (long_name='output gap') y_nat ${y^{nat}}$ (long_name='natural output') //(in contrast to the textbook defined in deviation from steady state) y ${y}$ (long_name='output') yhat ${\hat y}$ (long_name='output deviation from steady state') r_nat ${r^{nat}}$ (long_name='natural interest rate') r_real ${r^r}$ (long_name='real interest rate') i ${i}$ (long_name='nominal interrst rate') n ${n}$ (long_name='hours worked') m_real ${(m-p)}$ (long_name='real money stock') m_growth_ann ${\Delta m}$ (long_name='money growth annualized') m_nominal ${m}$ (long_name='nominal money stock') nu ${\nu}$ (long_name='AR(1) monetary policy shock process') a ${a}$ (long_name='AR(1) technology shock process') r_real_ann ${r^{r,ann}}$ (long_name='annualized real interest rate') i_ann ${i^{ann}}$ (long_name='annualized nominal interest rate') r_nat_ann ${r^{nat,ann}}$ (long_name='annualized natural interest rate') pi_p_ann ${\pi^{p,ann}}$ (long_name='annualized inflation rate') z ${z}$ (long_name='AR(1) preference shock process') p ${p}$ (long_name='price level') w ${w}$ (long_name='nominal wage') c ${c}$ (long_name='consumption') w_real $\omega$ (long_name='real wage') w_gap ${\tilde \omega}$ (long_name='real wage gap') pi_w ${\pi^w}$ (long_name='wage inflation') w_nat ${w^{nat}}$ (long_name='natural real wage') mu_p ${\mu^p}$ (long_name='markup') pi_w_ann ${\pi^{w,ann}}$ (long_name='annualized wage inflation rate') ; varexo eps_a ${\varepsilon_a}$ (long_name='technology shock') eps_nu ${\varepsilon_\nu}$ (long_name='monetary policy shock') eps_z ${\varepsilon_z}$ (long_name='preference shock innovation') ; parameters alppha ${\alpha}$ (long_name='capital share') betta ${\beta}$ (long_name='discount factor') rho_a ${\rho_a}$ (long_name='autocorrelation technology shock') rho_nu ${\rho_{\nu}}$ (long_name='autocorrelation monetary policy shock') rho_z ${\rho_{z}}$ (long_name='autocorrelation monetary demand shock') siggma ${\sigma}$ (long_name='inverse EIS') varphi ${\varphi}$ (long_name='inverse Frisch elasticity') phi_pi ${\phi_{\pi}}$ (long_name='inflation feedback Taylor Rule') phi_y ${\phi_{y}}$ (long_name='output feedback Taylor Rule') eta ${\eta}$ (long_name='semi-elasticity of money demand') epsilon_p ${\epsilon_p}$ (long_name='demand elasticity goods') theta_p ${\theta_p}$ (long_name='Calvo parameter prices') epsilon_w ${\epsilon_w}$ (long_name='demand elasticity labor services') theta_w ${\theta_w}$ (long_name='Calvo parameter wages') lambda_p ${\lambda_p}$ (long_name='composite parameter Phillips Curve') lambda_w ${\lambda_w}$ (long_name='composite parameter wage Phillips Curve') ; %---------------------------------------------------------------- % Parametrization, p. 67 and p. 113-115 %---------------------------------------------------------------- siggma = 1; varphi=5; phi_pi = 1.5; phi_y = 0.125; theta_p=3/4; rho_nu =0.5; rho_z = 0.5; rho_a = 0.9; betta = 0.99; eta =3.77; %footnote 11, p. 115 alppha=1/4; epsilon_p=9; epsilon_w=4.5; theta_w=3/4; %---------------------------------------------------------------- % First Order Conditions %---------------------------------------------------------------- model(linear); //Composite parameters #Omega=(1-alppha)/(1-alppha+alppha*epsilon_p); %defined on page 166 #psi_n_ya=(1+varphi)/(siggma*(1-alppha)+varphi+alppha); %defined on page 171 #psi_n_wa=(1-alppha*psi_n_ya)/(1-alppha); %defined on page 171 #aleph_p=alppha*lambda_p/(1-alppha); %defined on page 172 #aleph_w=lambda_w*(siggma+varphi/(1-alppha)); %defined on page 172 [name='New Keynesian Phillips Curve eq. (18)'] pi_p=betta*pi_p(+1)+aleph_p*y_gap+lambda_p*w_gap; [name='New Keynesian Wage Phillips Curve eq. (22)'] pi_w=betta*pi_w(+1)+aleph_w*y_gap-lambda_w*w_gap; [name='Dynamic IS Curve eq. (22)'] y_gap=-1/siggma*(i-pi_p(+1)-r_nat)+y_gap(+1); [name='Definition natural rate of interest eq. (24)'] r_nat=-siggma*psi_n_ya*(1-rho_a)*a+(1-rho_z)*z; w_gap=w_gap(-1)+pi_w-pi_p-(w_nat-w_nat(-1)); [name='Definition natural wage, eq (16)'] w_nat=psi_n_wa*a; [name='Definition markup'] mu_p=-alppha/(1-alppha)*y_gap-w_gap; [name='Definition real wage gap, p. 171'] w_gap=w_real-w_nat; [name='Definition real interest rate'] r_real=i-pi_p(+1); [name='Definition natural output, eq. (20)'] y_nat=psi_n_ya*a; [name='Definition output gap'] y_gap=y-y_nat; [name='Monetary policy shock'] nu=rho_nu*nu(-1)+eps_nu; [name='TFP shock'] a=rho_a*a(-1)+eps_a; [name='Production function, p. 171'] y=a+(1-alppha)*n; [name='Preference shock, p. 54'] z = rho_z*z(-1) - eps_z; [name='Money growth (derived from eq. (4))'] m_growth_ann=4*(y-y(-1)-eta*(i-i(-1))+pi_p); [name='Real money demand (eq. 4)'] m_real=y-eta*i; [name='Annualized nominal interest rate'] i_ann=4*i; [name='Annualized real interest rate'] r_real_ann=4*r_real; [name='Annualized natural interest rate'] r_nat_ann=4*r_nat; [name='Annualized inflation'] pi_p_ann=4*pi_p; [name='Annualized wage inflation'] pi_w_ann=4*pi_w; [name='Output deviation from steady state'] yhat=y-steady_state(y); [name='Definition price level'] pi_p=p-p(-1); [name='resource constraint, eq. (12)'] y=c; [name='definition real wage'] w_real=w-p; [name='definition real wage'] m_nominal=m_real+p; end; steady_state_model; lambda_p=(1-theta_p)*(1-betta*theta_p)/theta_p*((1-alppha)/(1-alppha+alppha*epsilon_p)); %defined on page 166 lambda_w=(1-theta_w)*(1-betta*theta_w)/(theta_w*(1+epsilon_w*varphi)); %defined on page 170 end; %---------------------------------------------------------------- % define shock variances %--------------------------------------------------------------- shocks; var eps_a = 1; var eps_z = 1; end; %---------------------------------------------------------------- % generate IRFs for technology shock under optimal policy, replicates Figures 6.3, p. 182 %---------------------------------------------------------------- //planner objective, uses lambda_w and lambda_p updated in steady_state_model-block planner_objective 0.5*((siggma+(varphi+alppha)/(1-alppha))*y_gap^2+ epsilon_p/lambda_p*pi_p^2+epsilon_w*(1-alppha)/lambda_w*pi_w^2); ramsey_model(instruments=(i),planner_discount=betta, planner_discount_latex_name = $\delta$); stoch_simul(order=1,irf=16,noprint) y_gap pi_p_ann pi_w_ann w_real; evaluate_planner_objective; oo_baseline=oo_; %flexible wage case set_param_value('theta_w',0.0000000001); set_param_value('theta_p',3/4); stoch_simul(order=1,irf=16,noprint) y_gap pi_p_ann pi_w_ann w_real; evaluate_planner_objective; oo_flexible_wages=oo_; %flexible price case set_param_value('theta_w',3/4) set_param_value('theta_p',0.000000001) stoch_simul(order=1,irf=16,noprint) y_gap pi_p_ann pi_w_ann w_real; evaluate_planner_objective; oo_flexible_prices=oo_; figure('Name','Dynamic Responses to a technology shock under optimal policy') subplot(2,2,1) plot(1:options_.irf,oo_baseline.irfs.y_gap_eps_a,'-o',1:options_.irf,oo_flexible_wages.irfs.y_gap_eps_a,'-d',1:options_.irf,oo_flexible_prices.irfs.y_gap_eps_a,'-s') ylim([-0.1 0.1]) title('Output gap') ll=legend('baseline','flexible wages','flexible prices'); set(ll,'Location','SouthEast'); subplot(2,2,2) plot(1:options_.irf,oo_baseline.irfs.pi_p_ann_eps_a,'-o',1:options_.irf,oo_flexible_wages.irfs.pi_p_ann_eps_a,'-d',1:options_.irf,oo_flexible_prices.irfs.pi_p_ann_eps_a,'-s') title('Price Inflation') subplot(2,2,3) plot(1:options_.irf,oo_baseline.irfs.pi_w_ann_eps_a,'-o',1:options_.irf,oo_flexible_wages.irfs.pi_w_ann_eps_a,'-d',1:options_.irf,oo_flexible_prices.irfs.pi_w_ann_eps_a,'-s') title('Wage inflation') subplot(2,2,4) plot(1:options_.irf,oo_baseline.irfs.w_real_eps_a,'-o',1:options_.irf,oo_flexible_wages.irfs.w_real_eps_a,'-d',1:options_.irf,oo_flexible_prices.irfs.w_real_eps_a,'-s') title('Real wage') %---------------------------------------------------------------- % generate first row of Table 6.1, p. 186 %---------------------------------------------------------------- shocks; var eps_a = 1; end; set_param_value('theta_w',3/4); set_param_value('theta_p',3/4); stoch_simul(order=1,irf=16,noprint) y_gap y_gap pi_p pi_w; oo_baseline=oo_; y_gap_pos=strmatch('y_gap',var_list_ ,'exact'); pi_p_pos=strmatch('pi_p',var_list_ ,'exact'); pi_w_pos=strmatch('pi_w',var_list_ ,'exact'); %read out current parameter values par.alppha=M_.params(strmatch('alppha',M_.param_names,'exact')); par.epsilon_p=M_.params(strmatch('epsilon_p',M_.param_names,'exact')); par.epsilon_w=M_.params(strmatch('epsilon_w',M_.param_names,'exact')); par.siggma=M_.params(strmatch('siggma',M_.param_names,'exact')); par.varphi=M_.params(strmatch('varphi',M_.param_names,'exact')); par.lambda_w=M_.params(strmatch('lambda_w',M_.param_names,'exact')); par.lambda_p=M_.params(strmatch('lambda_p',M_.param_names,'exact')); variance.y_gap=oo_.var(y_gap_pos,y_gap_pos); variance.pi_p=oo_.var(pi_p_pos,pi_p_pos); variance.pi_w=oo_.var(pi_w_pos,pi_w_pos); L=0.5*((par.siggma+(par.varphi+par.alppha)/(1-par.alppha))*variance.y_gap+ par.epsilon_p/par.lambda_p*variance.pi_p+par.epsilon_w*(1-par.alppha)/par.lambda_w*variance.pi_w) labels={'sigma(pi_p)';'sigma(pi_w)';'sigma(tilde y)';'L'}; headers={' ';'Optimal'}; values=[sqrt([variance.pi_p;variance.pi_w;variance.y_gap]);L]; options_.noprint=0; dyntable(options_,'Evaluation of Simple Rules',headers,labels,values,size(labels,2)+2,4,3) if any(isnan(values)) error('Parameter updating in steady state went wrong') end