%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% %%% %%% Stock Market Conditions And Monetary %%% %%% Policy In A DSGE Model For The U.S. %%% %%% %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% %%% %%% by E. Castelnuovo / S. Nistico %%% %%% %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Section 1: PREAMBLE %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % All variables are expressed in log-linearized form (by using a first order % Taylor-approximation around the steady state) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Specification of the endogenous variables %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% var %c_hat, % De-trended consumption %d_hat, % De-trended real dividends Delta_g, % Change of shocks in the public sector Delta_nu, % Change of preference shocks Delta_w_FE, % Growth rate of frictionless real wage Delta_y_FE, % Growth rate of frictionless output level mc, % Marginal costs mw, % (Inverse) wage markup omega, % Real wage gap pi, % Price inflation pi_w, % Wage inflation q_hat, % De-trended equity share q_hat_FE, % De-trended equity share in FE r, % Nominal interest rate rr_FE, % Frictionless real interest rate s, % Real stock price gap %w, % real wage %w_hat, % De-trended real wage w_hat_FE, % Frictionless level of real wage x, % Real output-gap %y_hat, % De-trended output y_hat_FE, % De-trended outpot in FE endo_p, % auxiliary endogenous variable endo_w, % auxiliary endogenous variable b, % AR(1) Shock Delta_a, % AR(1) Shock g, % AR(1) Shock mu_p, % Arma(1,1) Shock mu_w, % Arma(1,1) Shock nu, % AR(1) Shock u_r; % AR(1) Shock % Specification of the exogenous variables %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% varexo epsilon_a, % White noise error term epsilon_b, % White noise error term epsilon_g, % White noise error term epsilon_nu, % White noise error term epsilon_p, % White noise error term epsilon_r, % White noise error term epsilon_w; % White noise error term % Specification of the parameters %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% parameters C_hat_ss, D_hat_ss, Omega_hat_ss, Q_hat_ss, Y_hat_ss, N_ss, alpha, % Output elasticity of labor beta, % "Standard" discount factor beta_tilde, % "Extended" discount factor chi_p, % MA(1) in mu_p chi_w, % MA(1) in mu_w delta, % Leisure weight eta, % Wage indexation g_ss, % Steady-state value of public expenditures over GDP gamma, % Quarterly trend-growth rate Gamma, % Steady-state gross rate of productivity growth h, % Parameter strictly connected to hbar hbar, % Degree to which consumers would like to smooth their consumption with respect to average past level kappa_p, % Auxiliary Parameter kappa_w, % Auxiliary Parameter lambda_p, % Auxiliary Parameter lambda_w, % Auxiliary Parameter mu_p_const, % Constant price markup in steady state mu_w_const, % Constant wage markup in steady state pi_ss % Steady state level of inflation phi_pi, % MP response to inflation phi_r, % MP intertia phi_s, % MP response to stock price gap phi_x, % MP response to output gap Phi_omega, % Auxiliary Parameter Phi_x, % Auxiliary Parameter psi, % Auxiliary Parameter psi_nu, % Auxiliary Parameter r_ss, % Net steady state short-term policy rate rho_a, % Persistance in Delta_a rho_b, % Persistance in b rho_g, % Persistance in g rho_nu, % Persistance in nu rho_p, % Persistance in p rho_r, % Persistance in r rho_w, % Persistance in w rho_tilde, % Steady State of gross interest rate sigma_a, % Standard deviation of Delta_a sigma_b, % Standard deviation of b sigma_ceta, % Standard deviation of ceta sigma_g, % Standard deviation of g sigma_nu, % Standard deviation of nu sigma_p, % Standard deviation of p sigma_r, % Standard deviation of r sigma_w, % Standard deviation of w theta_p, % Price rigidity theta_w, % Wage rigidity Theta, % Auxiliary Parameter varphi, % Inverse steady-state Frisch elasticity of labor supply varpi, % Price indexation xi; % Turnover rate %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Section 2: Calibration of parameters %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% alpha = 0.3600; chi_p = 0.6598; chi_w = 0.8572; delta = 1.4456; eta = 0.3543; g_ss = 0.1800; gamma = 0.0047; r_ss = 1.4300; hbar = 0.8269; mu_p_const = 0.2010; mu_w_const = 0.2551; phi_pi = 1.6745; phi_r = 0.7533; phi_s = 0.1181; phi_x = 0.0232; pi_ss = 0.8100; rho_a = 0.3111; rho_b = 0.9071; rho_g = 0.9692; rho_nu = 0.0772; rho_p = 0.8445; rho_w = 0.9600; sigma_a = 0.0102; sigma_b = 0.0059; sigma_ceta = 0.0673; sigma_g = 0.0106; sigma_nu = 0.0314; sigma_p = 0.1120; sigma_r = 0.0031; sigma_w = 0.4704; theta_p = 0.7404; theta_w = 0.6325; varpi = 0.1516; xi = 0.1292; beta = 1/(1+r_ss); Gamma = log(gamma); h = (hbar)/(Gamma); D_hat_ss = ((alpha+mu_p_const)/(1+mu_p_const))*(((1+g)*(1-alpha))/((1+g)*(1-alpha)+delta*(1-h)*(1+mu_p_const)*(1+mu_w_const)))^(1-alpha); Q_hat_ss = (D_hat_ss)*(((1+pi_ss)*(1+gamma))/((1+r)-(1+pi_ss)*(1+gamma))); Omega_hat_ss = Q_hat_ss+D_hat_ss; N_ss = ((1-alpha)*(1+g_ss))/((1-alpha)*(1+g_ss)+delta*(1-h)*(1+mu_p_const)*(1+mu_w_const)); varphi = (N_ss)/(1-N_ss); Y_hat_ss = (N_ss)^(1-alpha); C_hat_ss = (Y_hat_ss)/(1+g_ss); psi = xi*(1-beta*(1-xi))/(1-xi)*(Omega_hat_ss/C_hat_ss); beta_tilde = (beta*(1-h))/(1-h+psi); lambda_p = mu_p_const*((1-theta_p)*(1-beta_tilde*theta_p)*(1-alpha))/(theta_p*(mu_p_const+alpha)); lambda_w = mu_w_const*((1-theta_w)*(1-beta_tilde*theta_w))/(theta_w*(mu_p_const+varphi*(1+mu_w_const))); kappa_p = lambda_p*alpha/(1-alpha); kappa_w = lambda_w*(1/(1-h)+varphi/(1-alpha)); Phi_omega = (1-beta_tilde)*(1-alpha)/(alpha+mu_p_const); Phi_x = (1-beta_tilde)*mu_p_const/(alpha+mu_p_const); psi_nu = (psi*(beta*(1-xi)*rho_nu))/((1+psi-h)*(1-beta*(1-xi)*rho_nu)); rho_tilde = log(1+r_ss); Theta = (1-h)/(1+psi-h); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Section 3: Model Equations %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Specification of the model relations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% model (linear); % Central equations of the economy %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x-h*x(-1) = Theta*(x(+1)-h*x)+psi*Theta*s-(1-h)*Theta*(r-pi(+1)-rr_FE); % Equation outputgap s = beta_tilde*s(+1)+Phi_x*x(+1)-Phi_omega*omega(+1)-(r-pi(+1)-rr_FE)+b; % Stockprice-gap pi-varpi*pi(-1) = beta_tilde*(pi(+1)-varpi*pi)+lambda_p*omega+kappa_p*x+lambda_p*(mu_p-mu_p_const); % New Keynesian Philips Curve for price inflation dynamics pi_w-eta*pi(-1)-eta*Delta_a(-1) = beta_tilde*(pi_w(+1)-eta*pi-eta*Delta_a)+kappa_w*x-h/(1-h)*lambda_w*x(-1)+lambda_w*omega+lambda_w*(mu_w-mu_w_const); % New Keynesian Philips Curve for wage inflation dynamics omega = omega(-1)+pi_w-pi-Delta_w_FE; % Equation for real wage gap r = (1-phi_r)*(phi_pi*pi+phi_x*x+phi_s*s)+phi_r*r(-1)+u_r; % Taylor Rule for short-term nominal interest rates % Frictionless equilibrium %%%%%%%%%%%%%%%%%%%%%%%%%% y_hat_FE = h*(1-alpha)/(1-h*alpha+(1-alpha)*varphi)*y_hat_FE(-1)+(1-alpha)/(1-h*alpha+(1-h)*varphi)*(g-h*g(-1)-h*Delta_a); % Frictionless level of real output w_hat_FE = h*(1-alpha)/(1-h*alpha+(1-alpha)*varphi)*w_hat_FE(-1)-alpha/(1-h*alpha+(1-h)*varphi)*(g-h*g(-1)-h*Delta_a); % Frictionless level of real wage rr_FE = rho_tilde+1/(1-h)*(Delta_y_FE(+1)-Delta_g(+1)+Delta_a(+1)-Delta_nu(+1))+h/(1-h)*(Delta_nu(+1)-Delta_y_FE+Delta_g-Delta_a)+psi/(1-h)*q_hat_FE-psi/(1-h)^2*(y_hat_FE-g-h*y_hat_FE(-1)+h*g(-1)+h*Delta_a)-(psi_nu*(1+psi-h)+psi)/(1-h)*Delta_nu(+1); % Equation for frictionless real interest rate q_hat_FE = beta_tilde*q_hat_FE(+1)+(1-beta_tilde)*y_hat_FE(+1)-rr_FE+Delta_a(+1); % Equation for stock price level Delta_w_FE = w_hat_FE-w_hat_FE(-1); % Identity for frictionless wage growth % Demand side of the economy %%%%%%%%%%%%%%%%%%%%%%%%%%%% %y_hat = c_hat+g; % A.43 good market clearing %(c_hat-h*c_hat(-1)+h*Delta_a) = (1-h)/(1+psi-h)*(c_hat(+1)-h*c_hat+Delta_a(+1))+psi*(1-h)/(1+psi-h)*q_hat-(1-h)^2/(1+psi-h)*(r-pi(+1)-rho_tilde-Delta_a(+1))-(1-h)*(1+psi_nu)*Delta_nu(+1); % A.44 dynamic consumption path %q_hat-b = beta_tilde*(q_hat(+1)-b(+1))+(1-beta_tilde)*d_hat(+1)-(r-pi(+1)-rho_tilde)+Delta_a(+1); % A.45 share pricing %d_hat = y_hat-(1-alpha)/(alpha+mu_p_const)*mc; % A.46 real dividends % Supply side of the economy %%%%%%%%%%%%%%%%%%%%%%%%%%%% %pi-varpi*pi(-1) = beta_tilde*(pi(+1)-varpi*pi)+lambda_p*mc+lambda_p*(mu_p-mu_p_const); % A.47 New Keynesian Philips Curve for price inflation %mc = w_hat+alpha/(1-alpha)*y_hat; % A.48 marginal costs %pi_w-eta*pi(-1)-eta*Delta_a(-1) = beta_tilde*(pi_w(+1)-eta*pi-eta*Delta_a)+lambda_w*mw+lambda_w*(mu_w-mu_w_const); % A.49 New Keynesian Philips Curve for wage inflation %mw = (h/(1-h)+varpi/(1-alpha))*y_hat-h/(1-h)*(y_hat(-1)-Delta_a-g(-1))-1/(1-h)*g-w_hat; % A.50 (inverse) wage markup % Stochastic structure of the model %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Delta_a = rho_a*Delta_a(-1)+epsilon_a; g = rho_g*g(-1)+epsilon_g; nu = rho_nu*nu(-1)+epsilon_nu; u_r = rho_r*u_r(-1)+epsilon_r; b = rho_b*b(-1)+epsilon_b; mu_p = (1-rho_p)*mu_p_const+rho_p*mu_p(-1)+epsilon_p-chi_p*endo_p(-1); % ARMA (1,1) price markup mu_w = (1-rho_w)*mu_w_const+rho_w*mu_w(-1)+epsilon_w-chi_w*endo_w(-1); % ARMA (1,1) wage markup % Additional auxiliary equations and identities %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% endo_p = epsilon_p; endo_w = epsilon_w; Delta_g = g-g(-1); Delta_nu = nu-nu(-1); Delta_y_FE = y_hat_FE-y_hat_FE(-1); s = q_hat-q_hat_FE; mc = omega+(alpha)/(1-alpha)*x; mw = (1/(1-h)+(varphi)/(1-alpha))*x-(h)/(1-h)*x(-1)-omega; end; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Section 4: Computation %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % compute and display Eigenvalues and check the Blanchard-Kahn conditions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% resid; %steady; check; %resid; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Section 5: Shocks %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % All shocks are white noise and crossequation uncorrelated %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% shocks; var epsilon_a = sigma_a^2; var epsilon_g = sigma_g^2; var epsilon_nu = sigma_nu^2; var epsilon_r = sigma_r^2; var epsilon_b = sigma_b^2; var epsilon_p = sigma_p^2; var epsilon_w = sigma_w^2; end; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Section 6: Simulation %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% stoch_simul(irf=40, order=1) r rr_FE s pi pi_w x omega;