//************************************************ // Master Thesis // // --------------- // Donat Bünger // // stochastic //************************************************ // 1. Declare variables and parameters // ----------------------------------- var mc rk w K L Y X_pnr X_pnu X_pdr X_pdu E_u E_r pi u z I K_bar q C_u C_r r zeta X_wnu X_wnr X_wdu X_wdr rl B B_L T e_B yg s1 s2 s3 s4 e_m;// varexo eps_B eps_m; parameters alpha beta_u beta_r zeta_p lam_f omega_r omega_u delta gamma A a_2 S_2 h sigma_u sigma_r zeta_w v lam_w chi_w kappa pi_s N M sigma_B rho_B phi_T rho_r phi_pi phi_y sigma_m zeta_2 rho_z sigma_z rho_zeta chi_p R_L q_u rk_s zeta_s I_s K_bar_s T_s C_u_s C_r_s BL_s B_s; // 2. Calibrate parameters values // ----------------------------------- alpha =0.33; beta_u =0.997558; zeta_s =0.001806; beta_r =beta_u/(1+zeta_s); zeta_p =0.5; lam_f =1.45; omega_r =0.2666; omega_u =0.7334; delta =0.025; A =0.918; a_2 =0.1836; S_2 =3.917; sigma_u =1.8360; sigma_r =1.8360; zeta_w =0.5; v =1.9585; lam_w =1.05; h =0.6029; chi_w =0.6143; gamma = 0.00616675; kappa = 0.00483; pi_s = 0.00489625; N =0.9867; M =0.918; sigma_B =-0.3433; rho_B =0.8135; phi_T =1.4448; rho_r =0.7068; phi_pi =1.7026; phi_y =0.3672; sigma_m =0.17; zeta_2 =0.012846; rho_z =0.3857; sigma_z =0.3433; rho_zeta =0.8135; chi_p = omega_u / (omega_u + omega_r*(((1-beta_u*zeta_p)/(1-beta_r*zeta_p))*(1/A))); rk_s = (1/((omega_u*A*beta_u+omega_r*beta_r)/(omega_u*A+omega_r)))*exp(gamma)-(1-delta); q_u = 1/(1+((omega_r/omega_u)*(1/A))); R_L = (exp(gamma)*pi_s+1)/beta_r; I_s = (exp(gamma)-(1-delta))*(alpha/(1+lam_f))*(1/rk_s); BL_s = (zeta_s)^(1/rho_zeta)*(R_L-kappa); B_s = BL_s/N; T_s = (-1)*(1-(1/beta_u))*B_s-((1/(R_L-kappa))-(R_L/(R_L-kappa))*(1/(pi_s*exp(gamma))))*BL_s; C_r_s =(1-I_s)/(omega_u*M+omega_r); C_u_s = M*C_r_s; K_bar_s = exp(gamma)*(alpha/(1+lam_f))*(1/rk_s); // 4. Declare the model's dynamics // ----------------------------------- model(linear); //1. real marginal cost mc = alpha*rk + (1-alpha)*w; //equation 1 //2. Capital demand K = w - rk + L; //equation 2 //3. Technology Y = alpha*K + (1-alpha)*L; //3 //4. Price setting: p=price n=nominator, d= denominator, u=unrestricted, r= restricted p=price X_pnu = (1-beta_u*zeta_p)*(E_u+Y+mc) + beta_u*zeta_p*(((1+lam_f)/lam_f)*pi(1)+X_pnu(1)); //4 X_pnr = (1-beta_r*zeta_p)*(E_r+Y+mc) + beta_r*zeta_p*(((1+lam_f)/lam_f)*pi(1)+X_pnr(1)); //5 X_pdu = (1-beta_u*zeta_p)*(E_u+Y) + beta_u*zeta_p*((1/lam_f)*pi(1)+X_pdu(1)); //6 X_pdr = (1-beta_r*zeta_p)*(E_r+Y) + beta_r*zeta_p*((1/lam_f)*pi(1)+X_pdr(1)); //7 //5. LOM prices (eq.8) pi = ((1-zeta_p)/zeta_p)*(chi_p*X_pnu + (1-chi_p)*X_pnr - chi_p*X_pdu - (1-chi_p)*X_pdr); //8 //6. Effective capital (9) K = -z + u + K_bar(-1); //9 //7. LOM capital K_bar = (1-delta)*exp((-1)*gamma)*(K_bar(-1)-z) + (1-(1-delta)*exp((-1)*gamma))*I; //10 //8. Capital utilization rk*rk_s = u*a_2; //11 //9. LOM of Q //12 q = ((omega_u*A*beta_u+omega_r*beta_r)/(omega_u*A+omega_r))*exp((-1)*gamma)*(rk_s*rk(1)+(1-delta)*q(1)) - z(1) + q_u*(((1+zeta_s)/(1+q_u*zeta_s))*E_u(1)-E_u) + (1-q_u)*(((1/(1+q_u*zeta_s))*E_r(1)-E_r)); //10. Investment decision //13 0 = q - exp(2*gamma)*S_2*(z+I-I(-1)) + ((omega_u*A*beta_u+omega_r*beta_r)/(omega_u*A+omega_r))*exp(2*gamma)*S_2*(z(1)+I(1)-I); //11. marginal utilities E_r = (1/(1-beta_r*h))*(-1)*(sigma_r/(1-h))*((1+beta_r*h*h)*C_r-beta_r*h*C_r(1)-h*C_r(-1)); //14 E_u = (1/(1-beta_u*h))*(-1)*(sigma_u/(1-h))*((1+beta_u*h*h)*C_u-beta_u*h*C_u(1)-h*C_u(-1)); //15 //12. Euler equation, unconstrained, short-term bonds E_u = r + E_u(1)-z(1)-pi(1); //16 //13. Euler equation, unconstrained, long-term bonds zeta+E_u = (R_L/(R_L-kappa))*rl + E_u(1)-z(1)-pi(1)-(kappa/(R_L-kappa))*rl(1); //17 //14. Euler equation, constrained, long-term bonds (eq 21) E_r = (R_L/(R_L-kappa))*rl + E_r(1)-z(1)-pi(1)-(kappa/(R_L-kappa))*rl(1); //18 //15. Wage setting X_wnu = (1-zeta_w*beta_u)*((1+v)*L + ((1+lam_w)/lam_w)*(1+v)*w) + zeta_w*beta_u*(((1+lam_w)/lam_w)*(1+v)*(pi(1)+z(1))+X_wnu(1)); //19 X_wnr = (1-zeta_w*beta_r)*((1+v)*L + ((1+lam_w)/lam_w)*(1+v)*w) + zeta_w*beta_r*(((1+lam_w)/lam_w)*(1+v)*(pi(1)+z(1))+X_wnr(1)); //20 X_wdu = (1-zeta_w*beta_u)*(E_u + L + ((1+lam_w)/lam_w)*w) + zeta_w*beta_u*((1/lam_w)*(pi(1)+z(1))+X_wdu(1)); //21 X_wdr = (1-zeta_w*beta_r)*(E_r + L + ((1+lam_w)/lam_w)*w) + zeta_w*beta_r*((1/lam_w)*(pi(1)+z(1))+X_wdr(1)); //22 //16. LOM wages w = (1-zeta_w)*(1/(1+((1+lam_w)/lam_w)*v))*(chi_w*(X_wnu-X_wdu)+(1-chi_w)*(X_wnr-X_wdr))+zeta_w*(w(-1)-pi-z); //23 //17. Budget constraint B_s*B+(N/(R_L-kappa))*B_L*B_s = (1/beta_u)*B_s*(B(-1)+r(-1))+B_s*(N/(R_L-kappa))*(1/beta_r)*B_L(-1) //24 + B_s*(((1-exp((-1)*gamma)*(1/pi_s)*kappa)*R_L)/(R_L-kappa))*(N*rl/(R_L-kappa)) - T - B_s*((1/beta_u)+(N/(R_L-kappa))*(1/beta_r))*(z+pi); //18. Long-term bond policy B_L = (R_L/(R_L-kappa))*rl + rho_B*(((-1)*(R_L/(R_L-kappa))*rl(-1))+B_L(-1))+e_B; //25 e_B = sigma_B*eps_B; //26 //19. Transfers feedback rule T = T_s* (phi_T*((B(-1)+(1/(R_L-kappa))*N*B_L(-1)-(R_L/((R_L-kappa)*(R_L-kappa)))*N*rl(-1))/(1+(1/(R_L-kappa))*N))); //27 //20. Monetary policy r = s1(-1); //zero-lower bound trick for 4 periods //28 s1 = s2(-1); //29 s2 = s3(-1); //30 s3 = s4(-1); //31 s4 = rho_r*r(3)+(1-rho_r)*(phi_pi*pi(4) + phi_y*(Y(4)-Y+z(1)+z(2)+z(3)+z(4)))+ e_m; //32 //r = rho_r*r(-1)+(1-rho_r)*(phi_pi*pi + phi_y*(Y-Y(-4)+z(-3)+z(-2)+z(-1)+z))+ e_m; e_m = sigma_m*eps_m; //33 //21. Term premium zeta = zeta_2*B_L; //34 //32 //22. Resource constraint Y = omega_u*C_u_s*C_u + omega_r*C_r_s*C_r + I_s*I //35 + exp((-1)*gamma)*rk_s*K_bar_s*u; //23. remaining shocks z = rho_z*z(-1); //36 yg =Y-Y(-1); //log-linearized output growth //37 end; resid(1); steady; // 6. Simulate the stochastic model // ---------------------------------------- shocks; var eps_B = 1; end; check; stoch_simul(order=1,irf=30, relative_irf, periods=500, nograph); P = 4/(rl_eps_B-kappa); for ii = 1:8 figure(1) subplot(4,2,ii) if ii == 1; jm = plot([yg_eps_B]); set(jm(1),'linestyle','-','LineWidth',1,'color',[0.4 0 0]); end if ii == 2; jm = plot([Y_eps_B]); set(jm(1),'linestyle','-','LineWidth',1,'color',[0.4 0 0]); end if ii == 3; jm = plot([pi_eps_B]); set(jm(1),'linestyle','-','LineWidth',1,'color',[0.4 0 0]); end if ii == 4; jm = plot([r_eps_B]); set(jm(1),'linestyle','-','LineWidth',1,'color',[0.4 0 0]); end if ii == 5; jm = plot([rl_eps_B]); set(jm(1),'linestyle','-','LineWidth',1,'color',[0.4 0 0]); end if ii == 6; jm = plot([zeta_eps_B]); set(jm(1),'linestyle','-','LineWidth',1,'color',[0.4 0 0]); end if ii == 7; jm = plot([B_L_eps_B]); set(jm(1),'linestyle','-','LineWidth',1,'color',[0.4 0 0]); end if ii == 8; jm = plot([P]); set(jm(1),'linestyle','-','LineWidth',1,'color',[0.4 0 0]); end if ii == 1; title('Output Growth','color','k','fontname','times','fontsize',14, 'fontweight','b'); end if ii == 2; title('Output Level','color','k','fontname','times','fontsize',14, 'fontweight','b'); end if ii == 3; title('Inflation','color','k','fontname','times','fontsize',14, 'fontweight','b'); end if ii == 4; title('r (FFR)','color','k','fontname','times','fontsize',14, 'fontweight','b'); end if ii == 5; title('rl (10 Y Yield)','color','k','fontname','times','fontsize',14, 'fontweight','b'); end if ii == 6; title('zeta (Risk Premium)','color','k','fontname','times','fontsize',14, 'fontweight','b'); end if ii == 7; title('BL (Long term bond holdings)','color','k','fontname','times','fontsize',14, 'fontweight','b'); end if ii == 8; title('Long-term bond market value (PL)','color','k','fontname','times','fontsize',14, 'fontweight','b'); end if ii == 1; xlabel('Quarters after shock','color','k','fontname','times','fontsize',11); ylabel('Deviation from SS','color','k','fontname','times','fontsize',11);end if ii == 2; xlabel('Quarters after shock','color','k','fontname','times','fontsize',11); ylabel('Deviation from SS','color','k','fontname','times','fontsize',11);end if ii == 3; xlabel('Quarters after shock','color','k','fontname','times','fontsize',11); ylabel('Deviation from SS','color','k','fontname','times','fontsize',11);end if ii == 4; xlabel('Quarters after shock','color','k','fontname','times','fontsize',11); ylabel('Deviation from SS','color','k','fontname','times','fontsize',11);end if ii == 5; xlabel('Quarters after shock','color','k','fontname','times','fontsize',11); ylabel('Deviation from SS','color','k','fontname','times','fontsize',11);end if ii == 6; xlabel('Quarters after shock','color','k','fontname','times','fontsize',11); ylabel('Deviation from SS','color','k','fontname','times','fontsize',11);end if ii == 7; xlabel('Quarters after shock','color','k','fontname','times','fontsize',11); ylabel('Deviation from SS','color','k','fontname','times','fontsize',11);end if ii == 8; xlabel('Quarters after shock','color','k','fontname','times','fontsize',11); ylabel('Deviation from SS','color','k','fontname','times','fontsize',11);end if ii == 1; line([0 30],[0 0], 'color','k'); end if ii == 2; line([0 30],[0 0], 'color','k'); end if ii == 3; line([0 30],[0 0], 'color','k'); end if ii == 4; line([0 30],[0 0], 'color','k'); end if ii == 5; line([0 30],[0 0], 'color','k'); end if ii == 6; line([0 30],[0 0], 'color','k'); end if ii == 7; line([0 30],[0 0], 'color','k'); end if ii == 8; line([0 30],[0 0], 'color','k'); end end //legend('without zlb','with zlb',1);