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/*
* This file implements the optimal monetary policy under commitment exercise
* in Jordi Galí (2008): Monetary Policy, Inflation, and the Business Cycle,
* Princeton University Press, Chapter 5.1.2
*
* It demonstrates how to use the ramsey_model command of Dynare.
*
* 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 2015-2024 Johannes Pfeifer
*
* 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 <https://www.gnu.org/licenses/>.
*/
var pi ${\pi}$ (long_name='inflation')
y_gap ${tilde y}$ (long_name='output gap')
y_nat ${y^{nat}}$ (long_name='natural output')
y ${y}$ (long_name='output')
r_e ${r^{e}}$ (long_name='efficient interest rate')
y_e ${y^{nat}}$ (long_name='efficient output')
x ${x}$ (long_name='welfare-relevant output gap')
r_nat ${r^{nat}}$ (long_name='natural interest rate')
r_real ${r^r}$ (long_name='real interest rate')
i ${i}$ (long_name='nominal interest rate')
n ${n}$ (long_name='hours worked')
m_growth_ann ${\Delta m}$ (long_name='money growth')
u ${u}$ (long_name='AR(1) cost push 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_ann ${\pi^{ann}}$ (long_name='annualized inflation rate')
p ${p}$ (long_name='price level')
;
varexo eps_a ${\varepsilon_a}$ (long_name='technology shock')
eps_u ${\varepsilon_u}$ (long_name='monetary policy shock');
parameters alppha ${\alppha}$ (long_name='capital share')
betta ${\beta}$ (long_name='discount factor')
rho_a ${\rho_a}$ (long_name='autocorrelation technology shock')
rho_u ${\rho_{u}}$ (long_name='autocorrelation cost push shock')
siggma ${\sigma}$ (long_name='log utility')
phi ${\phi}$ (long_name='unitary Frisch elasticity')
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. 52
%----------------------------------------------------------------
siggma = 1;
phi=1;
phi_y = .5/4;
theta=2/3;
rho_u = 0;
rho_a = 0.9;
betta = 0.99;
eta =4;
alppha=1/3;
epsilon=6;
%----------------------------------------------------------------
% First Order Conditions
%----------------------------------------------------------------
model(linear);
//Composite parameters
#Omega=(1-alppha)/(1-alppha+alppha*epsilon); //defined on page 47
#psi_n_ya=(1+phi)/(siggma*(1-alppha)+phi+alppha); //defined on page 48
#lambda=(1-theta)*(1-betta*theta)/theta*Omega; //defined on page 47
#kappa=lambda*(siggma+(phi+alppha)/(1-alppha)); //defined on page 49
#alpha_x=kappa/epsilon; //defined on page 96
#phi_pi=(1-rho_u)*kappa*siggma/(alpha_x)+rho_u; //defined on page 101
//1. Definition efficient interest rate, below equation (4)
r_e=siggma*(y_e(+1)-y_e);
//2. Definition efficient output
y_e=psi_n_ya*a;
//3. Definition linking various output gaps, bottom page 96
y_gap=x+(y_e-y_nat);
//4. New Keynesian Phillips Curve eq. (2)
pi=betta*pi(+1)+kappa*x + u;
//5. Dynamic IS Curve eq. (4)
x=-1/siggma*(i-pi(+1)-r_e)+x(+1);
//6. Definition natural rate of interest eq. (23)
r_nat=siggma*psi_n_ya*(a(+1)-a);
//7. Definition real interest rate
r_real=i-pi(+1);
//8. Natural output
y_nat=phi*a;
//9. Definition output gap
y_gap=y-y_nat;
//10. cost push shock, equation (3)
u=rho_u*u(-1)+eps_u;
//11. TFP shock
a=rho_a*a(-1)+eps_a;
//12. Production function (eq. 13)
y=a+(1-alppha)*n;
//13. Money growth (derived from eq. (4))
m_growth_ann=4*(y-y(-1)-eta*(i-i(-1))+pi);
//14. Annualized nominal interest rate
i_ann=4*i;
//15. Annualized real interest rate
r_real_ann=4*r_real;
//16. Annualized natural interest rate
r_nat_ann=4*r_nat;
//17. Annualized inflation
pi_ann=4*pi;
//18. Definition price level
pi=p-p(-1);
//19. Interest Rate Rule that implements optimal solution, eq. (10)
% i=r_e+phi_pi*pi;
end;
%----------------------------------------------------------------
% define shock variances
%---------------------------------------------------------------
shocks;
var eps_u = 1;
var eps_a = 1;
end;
//planner objective using alpha_x expressed as function of deep parameters
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;
ramsey_model(instruments=(i),planner_discount=betta);
varobs pi x y u a;
stoch_simul(order=1,irf=13,partial_information) x pi u;
check;
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