/* * This file implements the baseline New Keynesian model of Jordi Galí (2008): Monetary Policy, Inflation, * and the Business Cycle, Princeton University Press, Second Edition, Chapter 3 * * The model equations and FOC's are obtained without linearization. * * 08.12.2016 */ %---------------------------------------------------------------- % Define Variables %---------------------------------------------------------------- var C ${C}$ (long_name='Consumption') W_real ${\frac{W}{P}}$ (long_name='Real Wage') Pi ${\Pi}$ (long_name='inflation') A ${A}$ (long_name='AR(1) technology process') N ${N}$ (long_name='Hours worked') R ${R^n}$ (long_name='Nominal Interest Rate') realinterest ${R^{r}}$ (long_name='Real Interest Rate') Y ${Y}$ (long_name='Output') Q ${Q}$ (long_name='Bond price') Z ${Z}$ (long_name='AR(1) preference shock process') S ${S}$ (long_name='Price dispersion') Pi_star ${\Pi^*}$ (long_name='Optimal reset price') x_aux_1 ${x_1}$ (long_name='aux. var. 1 recursive price setting') x_aux_2 ${x_2}$ (long_name='aux. var. 2 recursive price setting') MC ${mc}$ (long_name='real marginal costs') i_ann ${i^{ann}}$ (long_name='annualized nominal interest rate') pi_ann ${\pi^{ann}}$ (long_name='annualized inflation rate') r_real_ann ${r^{r,ann}}$ (long_name='annualized real interest rate') P ${P}$ (long_name='price level') log_y ${log(M)}$ (long_name='log output') log_W_real ${log(W/P)}$ (long_name='log real wage') log_N ${log(N)}$ (long_name='log hours') log_P ${log(P)}$ (long_name='log price level') log_A ${log(A)}$ (long_name='log technology level') log_Z ${log(Z)}$ (long_name='log preference shock') nu ${\nu}$ (long_name='AR(1) monetary policy shock process') sigma_a ${\sigma_a}$ sigma_nu ${\sigma_nu}$ sigma_z ${\sigma_z}$ ; %---------------------------------------------------------------- % Define Shock Variables %---------------------------------------------------------------- varexo eps_a ${\varepsilon_a}$ (long_name='technology shock') eps_z ${\varepsilon_z}$ (long_name='preference shock') eps_nu ${\varepsilon_\nu}$ (long_name='monetary policy shock') eps_sigma_a eps_sigma_nu eps_sigma_z ; %---------------------------------------------------------------- % Define Parameters %---------------------------------------------------------------- parameters alppha ${\alpha}$ (long_name='capital share') betta ${\beta}$ (long_name='discount factor') rho_a ${\rho_a}$ (long_name='autocorrelation technology shock') rho_nu ${\rho_{\nu}}$ (long_name='autocorrelation monetary policy 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') epsilon ${\epsilon}$ (long_name='demand elasticity') theta ${\theta}$ (long_name='Calvo parameter') tau ${\tau}$ (long_name='labor subsidy') rho_sigma_a sigma_a_bar eta_a rho_sigma_nu sigma_nu_bar eta_nu rho_sigma_z sigma_z_bar eta_z ; %---------------------------------------------------------------- % Parametrization, p. 67 and p. 113-115 %---------------------------------------------------------------- siggma = 1; varphi=5; phi_pi = 1.5; phi_y = 0.125; theta=3/4; rho_nu =0.5; rho_z = 0.5; rho_a = 0.9; betta = 0.99; alppha=1/4; epsilon=9; tau=0; //1/epsilon; rho_sigma_a = 0.94; sigma_a_bar = -2; eta_a = 0.13; rho_sigma_nu = 0.94; sigma_nu_bar = -2; eta_nu = 0.13; rho_sigma_z = 0.94; sigma_z_bar = -2; eta_z = 0.13; %---------------------------------------------------------------- % First Order Conditions %---------------------------------------------------------------- model; [name='Labor demand'] W_real=C^siggma*N^varphi; [name='Euler equation'] Q=betta*(C(+1)/C)^(-siggma)*(Z(+1)/Z)/Pi(+1); [name='Definition nominal interest rate)'] R=1/Q; [name='Aggregate output'] Y=A*(N/S)^(1-alppha); [name='Definition Real interest rate'] R=realinterest*Pi(+1); [name='Monetary Policy Rule'] R=1/betta*Pi^phi_pi*(Y/steady_state(Y))^phi_y*exp(nu); [name='Market Clearing'] C=Y; [name='Technology shock'] log(A)=rho_a*log(A(-1))+eps_a*exp(sigma_a); sigma_a=(1-rho_sigma_a)*sigma_a_bar + rho_sigma_a*sigma_a(-1)+eta_a*eps_sigma_a; [name='Preference shock'] log(Z)=rho_z*log(Z(-1))-eps_z*exp(sigma_z); sigma_z=(1-rho_sigma_z)*sigma_z_bar + rho_sigma_z*sigma_z(-1)+eta_z*eps_sigma_z; [name='Monetary policy shock'] nu=rho_nu*nu(-1)+eps_nu*exp(sigma_nu); sigma_nu=(1-rho_sigma_nu)*sigma_nu_bar + rho_sigma_nu*sigma_nu(-1)+eta_nu*eps_sigma_nu; [name='Definition marginal cost'] MC=W_real/((1-alppha)*Y/N*S); [name='Aggregate prices'] 1=theta*Pi^(epsilon-1)+(1-theta)*(Pi_star)^(1-epsilon); [name='Price dispersion'] S=(1-theta)*Pi_star^(-epsilon/(1-alppha))+theta*Pi^(epsilon/(1-alppha))*S(-1); [name='FOC price setting'] Pi_star^(1+epsilon*(alppha/(1-alppha)))=x_aux_1/x_aux_2*(1-tau)*epsilon/(epsilon-1); [name='Auxiliary price setting recursion 1'] x_aux_1=C^(-siggma)*Y*MC+betta*theta*Pi(+1)^(epsilon+alppha*epsilon/(1-alppha))*x_aux_1(+1); [name='Auxiliary price setting recursion 2'] x_aux_2=C^(-siggma)*Y+betta*theta*Pi(+1)^(epsilon-1)*x_aux_2(+1); [name='Definition price level'] Pi=P/P(-1); [name='Definition log output'] log_y = log(Y); [name='Definition log real wage'] log_W_real=log(W_real); [name='Definition log hours'] log_N=log(N); [name='Annualized inflation'] pi_ann=4*log(Pi); [name='Annualized nominal interest rate'] i_ann=4*log(R); [name='Annualized real interest rate'] r_real_ann=4*log(realinterest); [name='Definition log price level'] log_P=log(P); [name='Definition log TFP'] log_A=log(A); [name='Definition log preference'] log_Z=log(Z); end; %---------------------------------------------------------------- % Steady state values %--------------------------------------------------------------- steady_state_model; sigma_a = sigma_a_bar; sigma_nu = sigma_nu_bar; sigma_z = sigma_z_bar; A=1; Z=1; S=1; Pi_star=1; P=1; MC=(epsilon-1)/epsilon/(1-tau); R=1/betta; Pi=1; Q=1/R; realinterest=R; N=((1-alppha)*MC)^(1/((1-siggma)*alppha+varphi+siggma)); C=A*N^(1-alppha); W_real=C^siggma*N^varphi; Y=C; nu=0; x_aux_1=C^(-siggma)*Y*MC/(1-betta*theta*Pi^(epsilon/(1-alppha))); x_aux_2=C^(-siggma)*Y/(1-betta*theta*Pi^(epsilon-1)); log_y = log(Y); log_W_real=log(W_real); log_N=log(N); pi_ann=4*log(Pi); i_ann=4*log(R); r_real_ann=4*log(realinterest); log_P=log(P); log_A=0; log_Z=0; end; %---------------------------------------------------------------- % Write Latex File for the Model %---------------------------------------------------------------- write_latex_dynamic_model; %---------------------------------------------------------------- % Define the Monetary Policy Shock %---------------------------------------------------------------- resid(1); steady; check; shocks; var eps_nu = 0.25^2; //1 standard deviation shock of 25 basis points, i.e. 1 percentage point annualized var eps_z = 0.5^2; //unit shock to preference var eps_a = 1^2; //unit shock to technology var eps_sigma_a = 0.25^2; var eps_sigma_nu = 0.5^2; var eps_sigma_z = 1; end; %---------------------------------------------------------------- % Solve the Model %---------------------------------------------------------------- stoch_simul(order = 3, pruning, irf=20) Y Pi nu log_A log_Z sigma_a sigma_z sigma_nu;