/* * This file implements the baseline New Keynesian model of Jordi Galí (2015): Monetary Policy, Inflation, * and the Business Cycle, Princeton University Press, Second Edition, Chapter 3 * * THIS MOD-FILE REQUIRES DYNARE 4.5 OR HIGHER * * Notes: * - all model variables are expressed in deviations from steady state, i.e. in contrast to * to the chapter, both the nominal interest rate and natural output are not in log-levels, but rather mean 0 * * 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 © 2016-20 Johannes Pfeifer * 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 . */ var pi ${\pi}$ (long_name='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') 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_ann ${\pi^{ann}}$ (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 ${\frac{w}{p}}$ (long_name='real wage') mu ${\mu}$ (long_name='markup') mu_hat ${\hat \mu}$ (long_name='markup gap') ; varexo eps_a ${\varepsilon_a}$ (long_name='technology 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_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 ${\epsilon}$ (long_name='demand elasticity') theta ${\theta}$ (long_name='Calvo parameter') ; %---------------------------------------------------------------- % Parametrization, p. 67 and p. 113-115 %---------------------------------------------------------------- siggma = 1; varphi=5; phi_pi = 1.5; phi_y = 0.125; theta=3/4; rho_z = 0.5; rho_a = 0.9; betta = 0.99; eta =3.77; %footnote 11, p. 115 alppha=1/4; epsilon=9; %---------------------------------------------------------------- % First Order Conditions %---------------------------------------------------------------- model(linear); //Composite parameters #Omega=(1-alppha)/(1-alppha+alppha*epsilon); %defined on page 60 #psi_n_ya=(1+varphi)/(siggma*(1-alppha)+varphi+alppha); %defined on page 62 #lambda=(1-theta)*(1-betta*theta)/theta*Omega; %defined on page 61 #kappa=lambda*(siggma+(varphi+alppha)/(1-alppha)); %defined on page 63 [name='New Keynesian Phillips Curve eq. (22)'] pi=betta*pi(+1)+kappa*y_gap; [name='Dynamic IS Curve eq. (23)'] y_gap=-1/siggma*(i-pi(+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; [name='Definition real interest rate'] r_real=i-pi(+1); [name='Definition natural output, eq. (20)'] y_nat=psi_n_ya*a; [name='Definition output gap'] y_gap=y-y_nat; [name='TFP shock'] a=rho_a*a(-1)+eps_a; [name='Production function (eq. 14)'] 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); [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_ann=4*pi; [name='Output deviation from steady state'] yhat=y-steady_state(y); [name='Definition price level'] pi=p-p(-1); [name='resource constraint, eq. (12)'] y=c; [name='FOC labor, eq. (2)'] w-p=siggma*c+varphi*n; [name='definition real wage'] w_real=w-p; [name='definition nominal money stock'] m_nominal=m_real+p; [name='average price markup, eq. (18)'] mu=-(siggma+(varphi+alppha)/(1-alppha))*y+(1+varphi)/(1-alppha)*a; [name='average price markup, eq. (20)'] mu_hat=-(siggma+(varphi+alppha)/(1-alppha))*y_gap; end; %---------------------------------------------------------------- % define shock variances %--------------------------------------------------------------- shocks; var eps_a = 0.5^2; //unit shock to preferences end; planner_objective 0.5*((siggma+(varphi+alppha)/(1-alppha))*yhat^2+epsilon/0.0215*pi^2)/100; discretionary_policy(instruments=(i),irf=20,planner_discount=betta, periods=0) y_gap pi_ann y n w_real p yhat;