200 lines
6.7 KiB
Modula-2
200 lines
6.7 KiB
Modula-2
/*
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* This file implements the optimal monetary policy under commitment exercise
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* in Jordi Galí (2008): Monetary Policy, Inflation, and the Business Cycle,
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* Princeton University Press, Chapter 5.1.2
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*
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* It demonstrates how to use the ramsey_model command of Dynare.
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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.
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* in contrast to to the chapter, both the nominal interest rate and
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* natural output are not in log-levels, but rather mean 0
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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 © 2015-19 Johannes Pfeifer
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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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var pi ${\pi}$ (long_name='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')
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y ${y}$ (long_name='output')
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r_e ${r^{e}}$ (long_name='efficient interest rate')
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y_e ${y^{nat}}$ (long_name='efficient output')
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x ${x}$ (long_name='welfare-relevant output gap')
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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 interest rate')
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n ${n}$ (long_name='hours worked')
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m_growth_ann ${\Delta m}$ (long_name='money growth')
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u ${u}$ (long_name='AR(1) cost push 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_ann ${\pi^{ann}}$ (long_name='annualized inflation rate')
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p ${p}$ (long_name='price level')
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;
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varexo eps_a ${\varepsilon_a}$ (long_name='technology shock')
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eps_u ${\varepsilon_u}$ (long_name='monetary policy shock');
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parameters alppha ${\alppha}$ (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_u ${\rho_{u}}$ (long_name='autocorrelation cost push shock')
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siggma ${\sigma}$ (long_name='log utility')
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phi ${\phi}$ (long_name='unitary Frisch elasticity')
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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 ${\epsilon}$ (long_name='demand elasticity')
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theta ${\theta}$ (long_name='Calvo parameter')
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;
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%----------------------------------------------------------------
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% Parametrization, p. 52
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%----------------------------------------------------------------
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siggma = 1;
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phi=1;
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phi_y = .5/4;
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theta=2/3;
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rho_u = 0;
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rho_a = 0.9;
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betta = 0.99;
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eta =4;
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alppha=1/3;
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epsilon=6;
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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); //defined on page 47
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#psi_n_ya=(1+phi)/(siggma*(1-alppha)+phi+alppha); //defined on page 48
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#lambda=(1-theta)*(1-betta*theta)/theta*Omega; //defined on page 47
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#kappa=lambda*(siggma+(phi+alppha)/(1-alppha)); //defined on page 49
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#alpha_x=kappa/epsilon; //defined on page 96
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#phi_pi=(1-rho_u)*kappa*siggma/(alpha_x)+rho_u; //defined on page 101
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//1. Definition efficient interest rate, below equation (4)
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r_e=siggma*(y_e(+1)-y_e);
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//2. Definition efficient output
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y_e=psi_n_ya*a;
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//3. Definition linking various output gaps, bottom page 96
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y_gap=x+(y_e-y_nat);
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//4. New Keynesian Phillips Curve eq. (2)
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pi=betta*pi(+1)+kappa*x + u;
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//5. Dynamic IS Curve eq. (4)
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x=-1/siggma*(i-pi(+1)-r_e)+x(+1);
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//6. Definition natural rate of interest eq. (23)
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r_nat=siggma*psi_n_ya*(a(+1)-a);
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//7. Definition real interest rate
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r_real=i-pi(+1);
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//8. Natural output
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y_nat=phi*a;
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//9. Definition output gap
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y_gap=y-y_nat;
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//10. cost push shock, equation (3)
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u=rho_u*u(-1)+eps_u;
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//11. TFP shock
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a=rho_a*a(-1)+eps_a;
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//12. Production function (eq. 13)
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y=a+(1-alppha)*n;
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//13. Money growth (derived from eq. (4))
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m_growth_ann=4*(y-y(-1)-eta*(i-i(-1))+pi);
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//14. Annualized nominal interest rate
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i_ann=4*i;
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//15. Annualized real interest rate
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r_real_ann=4*r_real;
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//16. Annualized natural interest rate
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r_nat_ann=4*r_nat;
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//17. Annualized inflation
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pi_ann=4*pi;
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//18. Definition price level
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pi=p-p(-1);
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//19. Interest Rate Rule that implements optimal solution, eq. (10)
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% i=r_e+phi_pi*pi;
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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_u = 1;
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end;
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//planner objective using alpha_x expressed as function of deep parameters
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planner_objective pi^2 +(((1-theta)*(1-betta*theta)/theta*((1-alppha)/(1-alppha+alppha*epsilon)))*(siggma+(phi+alppha)/(1-alppha)))/epsilon*y_gap^2;
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ramsey_model(instruments=(i),planner_discount=betta);
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resid;
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stoch_simul(order=1,irf=13) x pi p u;
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verbatim;
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%% Check correctness
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Omega=(1-alppha)/(1-alppha+alppha*epsilon); %defined on page 47
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lambda=(1-theta)*(1-betta*theta)/theta*Omega; %defined on page 47
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kappa=lambda*(siggma+(phi+alppha)/(1-alppha)); %defined on page 49
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alpha_x=kappa/epsilon; %defined on page 96
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Psi=1/(kappa^2+alpha_x*(1-betta*rho_u)); %defined on page 99
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Psi_i=Psi*(kappa*siggma*(1-rho_u)+alpha_x*rho_u); %defined on page 101
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phi_pi=(1-rho_u)*kappa*siggma/(alpha_x)+rho_u; %defined on page 101
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a_parameter=alpha_x/(alpha_x*(1+betta)+kappa^2);
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delta = (1-sqrt(1-4*betta*a_parameter^2))/(2*a_parameter*betta);
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p_irf(1)=delta/(1-delta*betta*rho_u);
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x_irf(1)=-kappa*delta/(alpha_x*(1-betta*delta*rho_u));
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u_irf(1)=1;
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for ii=2:options_.irf
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u_irf(ii)=rho_u*u_irf(ii-1);
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p_irf(ii)=delta*p_irf(ii-1)+delta/(1-delta*betta*rho_u)*u_irf(ii);
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x_irf(ii)=delta*x_irf(ii-1)-kappa*delta/(alpha_x*(1-betta*delta*rho_u))*u_irf(ii);
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end
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pi_irf=diff([0 p_irf]);
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if max(abs(u_irf-oo_.irfs.u_eps_u))>1e-10 || max(abs(p_irf-oo_.irfs.p_eps_u))>1e-10 || max(abs(p_irf-oo_.irfs.p_eps_u))>1e-10 || max(abs(x_irf-oo_.irfs.x_eps_u))>1e-10
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error('IRFs do not match theoretical ones')
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end
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