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