Adds mod-file of a well-documented non-linear New Keynesian model with
recursive formulation of price setting equations where the steady state file is used to update parameters and invoke a non-linear solver.time-shift
Johannes Pfeifer
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/*
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* This file implements the Baseline New Keynesian DSGE model described in
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* much detail in Jesús Fernández-Villaverde (2006): "A Baseline DSGE
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* Model", available at http://economics.sas.upenn.edu/~jesusfv/benchmark_DSGE.pdf
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*
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* The parametrization is based on the estimated version of this model in
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* Jesús Fernández-Villaverde (2010): "The econometrics of DSGE models",
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* SERIEs, Vol. 1, pp. 3-49, DOI 10.1007/s13209-009-0014-7
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*
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* This implementation was written by Benjamin Born and Johannes Pfeifer. In
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* case you spot mistakes, email us at jpfeifer@gmx.de
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*
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* This mod-file implements a non-linearized version of the New Keynesian
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* model based on a recursive formulation of the price and wage setting
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* equations. Moreover, it makes use of a steady state file to i) set
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* parameters that depend on other parameters that are potentially estimated
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* and ii) solve a nonlinear equation using a numerical solver to find the steady
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* state of labor.
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* The model is written in the beginning of period stock notation. To make the model
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* conform with Dynare's end of period stock notation, we use the
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* predetermined_variables-command.
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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 (C) 2013 Dynare Team
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*
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* This file is part of Dynare.
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*
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* Dynare 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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* Dynare 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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* You should have received a copy of the GNU General Public License
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* along with Dynare. If not, see <http://www.gnu.org/licenses/>.
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*/
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var d //preference shock
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c //consumption
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mu_z //trend growth rate of the economy (from neutral and investment specific technology)
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mu_I //growth rate of investment-specific technology growth
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mu_A //growth rate of neutral technology
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lambda //Lagrange multiplier
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R //Nominal Interest rate
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PI //Inflation
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r //rental rate of capital
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x //investment
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u //capacity utilization
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q //Tobin's marginal q
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f //variable for recursive formulation of wage setting
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ld //aggregate labor demand
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w //real wage
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wstar //optimal real wage
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PIstarw //optimal wage inflation
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PIstar //optimal price inflation
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g1 //variable 1 for recursive formulation of price setting
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g2 //variable 2 for recursive formulation of price setting
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yd //aggregate output
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mc //marginal costs
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k //capital
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vp //price dispersion term
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vw //wage dispersion term
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l //aggregate labor bundle
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phi //labor disutility shock
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F; //firm profits
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varexo epsd epsphi epsmu_I epsA epsm;
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predetermined_variables k;
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parameters h //consumption habits
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betta //discount factor
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gammma1 //capital utilization, linear term
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gammma2 //capital utilization, quadratic term
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delta //depreciation rate
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kappa //capital adjustment costs parameter
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eta //elasticity of substitution between labor varieties
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epsilon //elasticity of substitution between goods varieties
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varpsi //labor disutility parameter
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gammma //inverse Frisch elasticity
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chiw //wage indexation parameter
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chi //price indexation
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thetap //Calvo parameter prices
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thetaw //Calvo parameter wages
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alppha //capital share
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Rbar //steady state interest rate
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PIbar //steady state inflation
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gammmaR //interest smoothing coefficient Taylor rule
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gammmaPI //feedback coefficient to inflation monetary policy rule
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gammmay //feedback coefficient to output growth deviation in monetary policy rule
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Phi //firms fixed costs
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rhod //autocorrelation preference shock
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rhophi //autocorrelation labor disutility shock
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Lambdamu //steady state growth rate of investmentment-specific technology
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LambdaA //steady state neutral technology growth
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Lambdax //steady state growth rate of investment
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LambdaYd //steady state growth rate of output
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sigma_d //standard deviation preference shock
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sigma_phi //standard deviation labor disutility shock
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sigma_mu //standard deviation investment-specific technology
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sigma_A //standard deviation neutral technology
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sigma_m; //standard deviation preference shock
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//Note that the parameter naming in FV(2010) differs from FV(2006)
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//Fixed parameters, taken from FV(2010), Table 2, p. 37
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delta=0.025;
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epsilon=10;
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eta= 10;
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Phi=0;
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gammma2=0.001;
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//Estimated parameters, taken from FV(2010), Table 3, p. 38, median estimate parameters
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betta =0.998;
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h=0.97;
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varpsi =8.92;
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gammma = 1.17;
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kappa =9.51;
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alppha =0.21;
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thetap =0.82;
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chi = 0.63;
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thetaw =0.68;
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chiw =0.62;
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gammmaR =0.77;
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gammmay =0.19;
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gammmaPI =1.29;
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PIbar = 1.01;
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rhod = 0.12;
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rhophi = 0.93;
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sigma_A = -3.97;
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sigma_d = -1.51;
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sigma_phi =-2.36;
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sigma_mu =-5.43;
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sigma_m =-5.85;
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Lambdamu=3.4e-3;
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LambdaA = 2.8e-3;
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LambdaYd= (LambdaA+alppha*Lambdamu)/(1-alppha);
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/*
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The following parameters are set in the steady state file as they depend on other
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deep parameters that were estimated in the original study. Setting them in the
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steady state file means they are updated for every parameter draw in the MCMC
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algorithm, while the parameters initialized here are only set once for the initial
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values of the parameters they depend on:
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gammma1 as it depends on LambdaA, alppha, Lambdamu, betta, and delta
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Rbar =0 as it depends on PI, LambdaA, alppha, Lambdamu, and betta
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Lambdax
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*/
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/*
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The following model equations are the stationary model equations, taken from
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FV(2006), p. 20, section 3.2.
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*/
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model;
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//1. FOC consumption
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d*(c-h*c(-1)*mu_z^(-1))^(-1)-h*betta*d(+1)*(c(+1)*mu_z(+1)-h*c)^(-1)=lambda;
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//2. Euler equation
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lambda=betta*lambda(+1)*mu_z(+1)^(-1)/PI(+1)*R;
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//3. FOC capital utilization
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r=gammma1+gammma2*(u-1);
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//4. FOC capital
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q=betta*lambda(+1)/lambda*mu_z(+1)^(-1)*mu_I(+1)^(-1)*((1-delta)*q(+1)+r(+1)*u(+1)-(gammma1*(u(+1)-1)+gammma2/2*(u(+1)-1)^2));
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//5. FOC investment
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1=q*(1-(kappa/2*(x/x(-1)*mu_z-Lambdax)^2)-(kappa*(x/x(-1)*mu_z-Lambdax)*x/x(-1)*mu_z))
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+betta*q(+1)*lambda(+1)/lambda*mu_z(+1)^(-1)*kappa*(x(+1)/x*mu_z(+1)-Lambdax)*(x(+1)/x*mu_z(+1))^2;
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//6-7. Wage setting
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f=(eta-1)/eta*wstar^(1-eta)*lambda*w^eta*ld+betta*thetaw*(PI^chiw/PI(+1))^(1-eta)*(wstar(+1)/wstar*mu_z(+1))^(eta-1)*f(+1);
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f=varpsi*d*phi*PIstarw^(-eta*(1+gammma))*ld^(1+gammma)+betta*thetaw*(PI^chiw/PI(+1))^(-eta*(1+gammma))*(wstar(+1)/wstar*mu_z(+1))^(eta*(1+gammma))*f(+1);
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//8-10. firm's price setting
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g1=lambda*mc*yd+betta*thetap*(PI^chi/PI(+1))^(-epsilon)*g1(+1);
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g2=lambda*PIstar*yd+betta*thetap*(PI^chi/PI(+1))^(1-epsilon)*PIstar/PIstar(+1)*g2(+1);
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epsilon*g1=(epsilon-1)*g2;
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//11-12. optimal inputs
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u*k/ld=alppha/(1-alppha)*w/r*mu_z*mu_I;
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mc=(1/(1-alppha))^(1-alppha)*(1/alppha)^alppha*w^(1-alppha)*r^alppha;
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//13. law of motion wages
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1=thetaw*(PI(-1)^chiw/PI)^(1-eta)*(w(-1)/w*mu_z^(-1))^(1-eta)+(1-thetaw)*PIstarw^(1-eta);
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//14. law of motion prices
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1=thetap*(PI(-1)^chi/PI)^(1-epsilon)+(1-thetap)*PIstar^(1-epsilon);
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//15. Taylor Rule
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R/Rbar=(R(-1)/Rbar)^gammmaR*((PI/PIbar)^gammmaPI*((yd/yd(-1)*mu_z)/exp(LambdaYd))^gammmay)^(1-gammmaR)*exp(epsm);
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//16-17. Market clearing
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yd=c+x+mu_z^(-1)*mu_I^(-1)*(gammma1*(u-1)+gammma2/2*(u-1)^2)*k;
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yd=(mu_A*mu_z^(-1)*(u*k)^alppha*ld^(1-alppha)-Phi)/vp;
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//18-20. Price and wage dispersion terms
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l=vw*ld;
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vp=thetap*(PI(-1)^chi/PI)^(-epsilon)*vp(-1)+(1-thetap)*PIstar^(-epsilon);
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vw=thetaw*(w(-1)/w*mu_z^(-1)*PI(-1)^chiw/PI)^(-eta)*vw(-1)+(1-thetaw)*(PIstarw)^(-eta);
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//21. Law of motion for capital
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k(+1)*mu_z*mu_I-(1-delta)*k-mu_z*mu_I*(1-kappa/2*(x/x(-1)*mu_z-Lambdax)^2)*x=0;
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//22. Profits
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F=yd-1/(1-alppha)*w*ld;
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//23. definition optimal wage inflation
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PIstarw=wstar/w;
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//exogenous processes
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//24. Preference Shock
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log(d)=rhod*log(d(-1))+epsd;
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//25. Labor disutility Shock
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log(phi)=rhophi*log(phi(-1))+epsphi;
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//26. Investment specific technology
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log(mu_I)=Lambdamu+epsmu_I;
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//27. Neutral technology
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log(mu_A)=LambdaA+epsA;
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//28. Defininition composite technology
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mu_z=mu_A^(1/(1-alppha))*mu_I^(alppha/(1-alppha));
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end;
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shocks;
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var epsd; stderr exp(sigma_d);
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var epsphi; stderr exp(sigma_phi);
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var epsmu_I; stderr exp(sigma_mu);
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var epsA; stderr exp(sigma_A);
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var epsm; stderr exp(sigma_m);
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end;
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steady;
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check;
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stoch_simul(order=1,irf=20) yd c R PI;
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@ -0,0 +1,94 @@
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function [ys,check] = NK_baseline_steadystate(ys,exe)
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global M_ lgy_
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if isfield(M_,'param_nbr') == 1
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NumberOfParameters = M_.param_nbr;
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for i = 1:NumberOfParameters
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paramname = deblank(M_.param_names(i,:));
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eval([ paramname ' = M_.params(' int2str(i) ');']);
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end
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check = 0;
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end
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%% Enter model equations here
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options=optimset(); % set options for numerical solver
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% the steady state computation follows FV (2006), section 4.1
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PI=PIbar;
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u=1;
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q=1;
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d=1;
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phi=1;
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m=0;
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zeta=1;
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mu_z=exp(LambdaYd);
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mu_I=exp(Lambdamu);
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mu_A=exp(LambdaA);
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%set the parameter Lambdax
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Lambdax=mu_z;
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%set the parameter gammma1
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gammma1=mu_z*mu_I/betta-(1-delta);
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r=1*gammma1;
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R=1+(PI*mu_z/betta-1);
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%set Rbar
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Rbar=R;
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PIstar=((1-thetap*PI^(-(1-epsilon)*(1-chi)))/(1-thetap))^(1/(1-epsilon));
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PIstarw=((1-thetaw*PI^(-(1-chiw)*(1-eta))*mu_z^(-(1-eta)))/(1-thetaw))^(1/(1-eta));
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mc=(epsilon-1)/epsilon*(1-betta*thetap*PI^((1-chi)*epsilon))/(1-betta*thetap*PI^(-(1-epsilon)*(1-chi)))*PIstar;
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w=(1-alppha)*(mc*(alppha/r)^alppha)^(1/(1-alppha));
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wstar=w*PIstarw;
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vp=(1-thetap)/(1-thetap*PI^((1-chi)*epsilon))*PIstar^(-epsilon);
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vw=(1-thetaw)/(1-thetaw*PI^((1-chiw)*eta)*mu_z^eta)*PIstarw^(-eta);
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tempvaromega=alppha/(1-alppha)*w/r*mu_z*mu_I;
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ld=fsolve(@(ld)(1-betta*thetaw*mu_z^(eta-1)*PI^(-(1-chiw)*(1-eta)))/(1-betta*thetaw*mu_z^(eta*(1+gammma))*PI^(eta*(1-chiw)*(1+gammma)))...
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-(eta-1)/eta*wstar/(varpsi*PIstarw^(-eta*gammma)*ld^gammma)*((1-h*mu_z^(-1))^(-1)-betta*h*(mu_z-h)^(-1))*...
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((mu_A*mu_z^(-1)*vp^(-1)*tempvaromega^alppha-tempvaromega*(1-(1-delta)*(mu_z*mu_I)^(-1)))*ld-vp^(-1)*Phi)^(-1),0.25,options);
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l=vw*ld;
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k=tempvaromega*ld;
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x=(1-(1-delta)*(mu_z*mu_I)^(-1))*k;
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yd=(mu_A/mu_z*k^alppha*ld^(1-alppha)-Phi)/vp;
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c=(mu_A*mu_z^(-1)*vp^(-1)*tempvaromega^alppha-tempvaromega*(1-(1-delta)*(mu_z*mu_I)^(-1)))*ld-vp^(-1)*Phi;
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lambda=(1-h*betta*mu_z^(-1))*(1-h/mu_z)^(-1)/c;
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F=yd-1/(1-alppha)*w*ld;
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f=(eta-1)/eta*wstar*PIstarw^(-eta)*lambda*ld/(1-betta*thetaw*mu_z^(eta-1)*PI^(-(1-chiw)*(1-eta)));
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f2=varpsi*d*phi*PIstarw^(-eta*(1+gammma))*ld^(1+gammma)/(1-betta*thetaw*(PI^chiw/PI)^(-eta*(1+gammma))*(wstar/wstar*mu_z)^(eta*(1+gammma)));
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g1=lambda*mc*yd/(1-betta*thetap*PI^((1-chi)*epsilon));
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g2=epsilon/(epsilon-1)*g1;
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%% end own model equations
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for iter = 1:length(M_.params)
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eval([ 'M_.params(' num2str(iter) ') = ' M_.param_names(iter,:) ';' ])
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end
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if isfield(M_,'param_nbr') == 1
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if isfield(M_,'orig_endo_nbr') == 1
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NumberOfEndogenousVariables = M_.orig_endo_nbr;
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else
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NumberOfEndogenousVariables = M_.endo_nbr;
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end
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ys = zeros(NumberOfEndogenousVariables,1);
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for i = 1:NumberOfEndogenousVariables
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varname = deblank(M_.endo_names(i,:));
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eval(['ys(' int2str(i) ') = ' varname ';']);
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end
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else
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ys=zeros(length(lgy_),1);
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for i = 1:length(lgy_)
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ys(i) = eval(lgy_(i,:));
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end
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check = 0;
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end
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