%------------------------------------------------------------------------------------------------------------------------------------- % One country RBC model with flexible labour (hours worked) and government spending. % % It is a version of % Eric M. Leeper, Michael Plante, Nora Traum (2009), Dynamics of Fiscal financing in the United States, Journal of Econometrics % % U(c,n) = BETA * u_b * [ (c(t) - hc(t-1)) ** (1-gamma)/(1-gamma) - u_l* n**(1-KAPPA)/(1-KAPPA) ] % % The paper is modified to allow for either IAC or CAC % investment adjustment costs as in Christiano Eichenbaum and Evans (2005) % capital adjustment costs as in Backus, Routledge and Zin (2007) % % h subscript stands for home country % UPPERCASE is a parameter or steady state value % lowercase variables are in logarithms % % In this version I alter the original leeper file "leeperorig.mod" model equations: % (A.4)and multiply the errors on the shocks by the variances %--------------------------------------------------------------------------------------------------------------------------------------- %-------------------------------------------------------------------------- % 40 Endogenous variables %-------------------------------------------------------------------------- var uc_h ul_h y_h c_h i_h k_h q_i_h r_k_h l_h w_h r_h b_h v_h delv_h g_h tk_h tl_h tc_h tau_k_h tau_l_h tau_c_h z_h u_b_h u_l_h u_i_h u_a_h u_g_h u_tk_h u_tl_h u_tc_h u_z_h ; %-------------------------------------------------------------------------- % 9 Exogenous shocks (error terms) %-------------------------------------------------------------------------- varexo e_u_b_h e_u_l_h e_u_i_h e_u_a_h e_u_g_h e_u_tk_h e_u_tl_h e_u_tc_h e_u_z_h; parameters BETA GAMMA H KAPPA DELTA0 DELTA1 DELTA2 ALPHA CHI ZETA UPS G_Y_bar B_Y_bar TAU_K_h_bar TAU_L_h_bar TAU_C_h_bar Y_h_bar B_h_bar Z_h_bar G_h_bar RHO_B RHO_L RHO_I RHO_G RHO_A RHO_TK RHO_TL RHO_TC RHO_Z SIGMA_B_H SIGMA_L_H SIGMA_I_H SIGMA_A_H SIGMA_G_H SIGMA_TK_H SIGMA_TL_H SIGMA_TC_H SIGMA_Z_H EPSI_G EPSI_K EPSI_L EPSI_Z GAMMA_G GAMMA_K GAMMA_L GAMMA_Z PHI_KL PHI_KC PHI_LC ; %-------------------------------------------------------------------------- % Calibration: %-------------------------------------------------------------------------- %Discount factor BETA = 0.99; DELTA0 = 0.025; %Capital share ALPHA = 0.30; %Steady state ratio of governemnt spending to GDP G_Y_bar = .0922; %Steady state ratio of governemnt debt to GDP B_Y_bar = .3396; %Steady state capital tax rate TAU_K_h_bar = 0.38; %Steady state labour tax rate TAU_L_h_bar = 0.297; // Steady state consumption tax rate TAU_C_h_bar = 0.099; %Parameter values based on Table 2 in Leeper et al (2009). I use the mean of the posterior distribution under column titled "All Adjust". %Risk aversion GAMMA = 2.7; %Iinverse Frish elasticity KAPPA = 1.9; %Habit formation H = 0.50; %Investment adjustment cost parameters and is 0 <= CHI CHI = 5.5; %Capital adjustment cost parameters and 0 < UPS <= 1 UPS = 0.500; %Capital and/or Investment adjustment cost parameters and 0 <= ZETA <= 1 %ZETA =1 Pure investment adjustment costs as in Christiano and Eichenbaum (2005) %ZETA =0 Pure capital adjustment costs as in Backus, Routledge and Zin (2007) ZETA = 1.0; %Capital utilization DELTA2 = 0.29; %Government spending debt coefficient GAMMA_G = 0.23; %Capital tax rate debt coefficient GAMMA_K = 0.39; %Labour tax rate debt coefficient GAMMA_L = 0.049; %Transfers debt coefficient GAMMA_Z = 0.5; %Capital tax rate output coefficient EPSI_K = 1.7; %Labour tax rate output coefficient EPSI_L = 0.36; %Government spending output coefficient EPSI_G = 0.34; %Transfers output coefficient EPSI_Z = 0.13; %Capital-Labour co-term PHI_KL = 0.19; %Capital-Consumption co-term PHI_KC = 0.024; %Labour-Consumption co-term PHI_LC = -0.028; %Parameters for the exogenous shocks RHO_A = 0.96; SIGMA_A_H = 0.62; RHO_B = 0.66; SIGMA_B_H = 7.0; RHO_L = 0.99; SIGMA_L_H = 2.8; RHO_I = 0.55; SIGMA_I_H = 6.4; RHO_G = 0.97; SIGMA_G_H = 3.1; RHO_TK = 0.93; SIGMA_TK_H = 4.4; RHO_TL = 0.97; SIGMA_TL_H = 3.0; RHO_TC = 0.93; SIGMA_TC_H = 4.0; RHO_Z = 0.94; SIGMA_Z_H = 3.4; %-------------------------------------------------------------------------- % Steady states %-------------------------------------------------------------------------- R_h_bar = 1/BETA; R_K_h_bar = (1/BETA + DELTA0-1) / (1-TAU_K_h_bar); DELTA1 = R_K_h_bar * (1-TAU_K_h_bar); K_h_bar = ( (((1-H)*((R_K_h_bar/ALPHA)*(1-G_Y_bar)-DELTA0))^(-GAMMA)*(1-TAU_L_h_bar)*(1-ALPHA)/(1+TAU_C_h_bar))/ (R_K_h_bar/ALPHA)^((KAPPA+ALPHA)/(1-ALPHA)) ) ^(1/(KAPPA+GAMMA)); A1 = (((1-H)*((R_K_h_bar/ALPHA)*(1-G_Y_bar)-DELTA0))^(-GAMMA)*(1-TAU_L_h_bar)*(1-ALPHA)/(1+TAU_C_h_bar)); A2 = (R_K_h_bar/ALPHA)^((KAPPA+ALPHA)/(1-ALPHA)); A3 = (A1/A2)^(1/(KAPPA+GAMMA)); //disp('Steady state A1'); disp(A1); //disp('Steady state A2'); disp(A2); //disp('Steady state A3'); disp(A3); Y_h_bar = (R_K_h_bar/ALPHA) *K_h_bar; C_h_bar = ((R_K_h_bar/ALPHA)*(1-G_Y_bar)-DELTA0) *K_h_bar; I_h_bar = DELTA0 *K_h_bar; L_h_bar = (R_K_h_bar/ALPHA)^(1/(1-ALPHA)) *K_h_bar; G_h_bar = G_Y_bar * Y_h_bar; B_h_bar = B_Y_bar * Y_h_bar; Z_h_bar = TAU_K_h_bar*ALPHA*Y_h_bar + TAU_L_h_bar*(1-ALPHA)*Y_h_bar + TAU_C_h_bar*C_h_bar - Y_h_bar * (G_Y_bar + ((1-BETA)/BETA)*B_Y_bar); TK_h_bar = TAU_K_h_bar * ALPHA * Y_h_bar; TL_h_bar = TAU_L_h_bar * (1-ALPHA) * Y_h_bar; TC_h_bar = TAU_C_h_bar * C_h_bar; Uc_h_bar = (C_h_bar-H*C_h_bar) ^(-GAMMA) ; Ul_h_bar = -(L_h_bar ^ KAPPA) ; W_h_bar = (1-ALPHA)* Y_h_bar / L_h_bar ; DELV_h_bar = DELTA0 ; disp('Steady state Uc_h_bar'); disp(Uc_h_bar); disp('Steady state Ul_h_bar'); disp(Ul_h_bar); disp('Steady state y_h'); disp(Y_h_bar); disp('Steady state c_h'); disp(C_h_bar); disp('Steady state i_h'); disp(I_h_bar); disp('Steady state l_h'); disp(L_h_bar); disp('Steady state W_h'); disp(W_h_bar); disp('Steady state k_h'); disp(K_h_bar); disp('Steady state g_h'); disp(G_h_bar); disp('Steady state b_h'); disp(B_h_bar); disp('Steady state z_h'); disp(Z_h_bar); disp('Steady state r_h'); disp(R_h_bar); disp('Steady state r_k_h'); disp(R_K_h_bar); disp('Steady state tk_h'); disp(TK_h_bar); disp('Steady state tl_h'); disp(TL_h_bar); disp('Steady state tc_h'); disp(TC_h_bar); %-------------------------------------------------------------------------- % Model %-------------------------------------------------------------------------- model; #R_K_h_ss = (1/BETA + DELTA0-1) / (1-(TAU_K_h_bar)); #K_h_ss = ( (((1-H)*(((R_K_h_ss)/ALPHA)*(1-G_Y_bar)-DELTA0))^(-GAMMA)*(1-(TAU_L_h_bar))*(1-ALPHA)/(1+(TAU_C_h_bar)))/ ((R_K_h_ss)/ALPHA)^((KAPPA+ALPHA)/(1-ALPHA)) ) ^(1/(KAPPA+GAMMA)); #Y_h_ss = ((R_K_h_ss)/ALPHA)*(K_h_ss); #G_h_ss = G_Y_bar * (Y_h_ss); #B_h_ss = B_Y_bar * (Y_h_ss); #C_h_ss = (((R_K_h_ss)/ALPHA)*(1-G_Y_bar)-DELTA0) *(K_h_ss); #Z_h_ss = (TAU_K_h_bar)*ALPHA*(Y_h_ss) + (TAU_L_h_bar)*(1-ALPHA)*(Y_h_ss) + (TAU_C_h_bar)*(C_h_ss) - (Y_h_ss) * (G_Y_bar + ((1-BETA)/BETA)*B_Y_bar); %MU of consumption uc_h = exp(u_b_h)*(exp(c_h) - H*exp(c_h(-1))) ^(-GAMMA) ; %MU of hours worked ul_h = -exp(u_b_h)*exp(u_l_h)*exp(l_h) ^ KAPPA ; %Equation 6: Endogenous depreciation rate of the capital stock exp(delv_h) = DELTA0 + DELTA1*(exp(v_h) - 1) + (DELTA2/2)*(exp(v_h) - 1)^2; %marginal product of labour exp(w_h) = (1-ALPHA) * exp(y_h)/exp(l_h) ; %marginal product of capital exp(r_k_h) = ALPHA * exp(y_h)/(exp(v_h)*exp(k_h(-1))) ; %Equation (A.1): Euler equation - intertemporal condition uc_h/(1+exp(tau_c_h)) = BETA*exp(r_h)*uc_h(+1)/(1+exp(tau_c_h(+1))); %Equation (A.2): Intratemporal condition ul_h + uc_h * exp(w_h) * (1-exp(tau_l_h))/(1+exp(tau_c_h)) = 0; %Equation (A.3): marginal conditions for capital exp(q_i_h) = ( exp(q_i_h(+1))*(1-exp(delv_h)) + exp(q_i_h(+1))*( ( (1-ZETA)*(1-UPS)*DELTA0/UPS)*( ( exp(u_i_h(+1))*exp(i_h(+1))/exp(k_h))^UPS * DELTA0^(-UPS) - 1 ) ) + (1-exp(tau_k_h(+1)))*ALPHA * exp(y_h(+1))/exp(k_h) ) / exp(r_h) ; %Equation (A.4): marginal conditions for intensity of capital exp(r_k_h)*(1-exp(tau_k_h)) = exp(q_i_h)*( DELTA1 + DELTA2*(exp(v_h) - 1) ) ; %Equation (A.5): marginal conditions for investment 1.0 = exp(q_i_h)* ( ZETA*( 1.0 - 0.5*CHI - 1.5*CHI*(exp(u_i_h)*exp(i_h)/exp(i_h(-1)))^2 + 2.0*CHI*(exp(u_i_h)*exp(i_h)/exp(i_h(-1))) ) + (1-ZETA)* ( (exp(u_i_h)*exp(i_h)/exp(k_h(-1)))^(UPS-1) * DELTA0^(1-UPS) ) ) + exp(q_i_h(+1)) *ZETA *CHI *( (exp(u_i_h(+1))*exp(i_h(+1))/exp(i_h))^3 -(exp(u_i_h(+1))*exp(i_h(+1))/exp(i_h))^2 ) / exp(r_h) ; %Equation (A.6): final goods market equilibrium exp(c_h) + exp(i_h) + exp(g_h) = exp(y_h) ; %Equation (A.7): Capital stock accumulation (adjusted to allow for either IAC and/or CAC) %Note if ZETA=1 it is the same as Leeper et al (2009). exp(k_h) = (1-exp(delv_h))*exp(k_h(-1)) + ZETA *( (1 - 0.5 * CHI * (exp(u_i_h)*exp(i_h)/exp(i_h(-1))-1.0)^2) * exp(i_h) ) + (1-ZETA)*( ( (exp(u_i_h)*exp(i_h)/exp(k_h(-1)))^UPS * DELTA0^(1-UPS) - (1-UPS)* DELTA0 )*(exp(k_h(-1))/UPS) ); %Equation (A.8): Government budget exp(b_h) + exp(tk_h) + exp(tl_h) + exp(tc_h) = exp(r_h(-1))*exp(b_h(-1)) + exp(g_h) + exp(z_h) ; %Equation (A.9): Total capital tax revenue exp(tk_h) = exp(tau_k_h) * ALPHA * exp(y_h) ; %Equation (A.10): Total labour tax revenue exp(tl_h) = exp(tau_l_h) * (1-ALPHA) * exp(y_h) ; %Equation (A.11): Total consumption tax revenue exp(tc_h) = exp(tau_c_h) * exp(c_h) ; %Equation (A.12): Production function exp(y_h) = exp(u_a_h) * (exp(v_h)*exp(k_h(-1)))^ALPHA * exp(l_h)^(1-ALPHA); %Fiscal rules %Equation 11: Government spending policy rule g_h = - EPSI_G*y_h - GAMMA_G*b_h(-1) + u_g_h + ( EPSI_G*log(Y_h_ss) + GAMMA_G*log(B_h_ss) + log(G_h_ss) ) ; %Equation 12: Capital tax policy rule tau_k_h = EPSI_K*y_h + GAMMA_K*b_h(-1) + PHI_KL*u_tl_h + PHI_KC*u_tc_h + u_tk_h + (-EPSI_K*log(Y_h_ss) - GAMMA_K*log(B_h_ss) + log(TAU_K_h_bar) ) ; %Equation 13: Labour tax policy rule tau_l_h = EPSI_L*y_h + GAMMA_L*b_h(-1) + PHI_KL*u_tk_h + PHI_LC*u_tc_h + u_tl_h + (-EPSI_L*log(Y_h_ss) - GAMMA_L*log(B_h_ss) + log(TAU_L_h_bar) ) ; %Equation 14: Consumption tax policy rule tau_c_h = PHI_KC*u_tk_h + PHI_LC*u_tl_h + u_tc_h + log(TAU_C_h_bar) ; %Equation 15: Transfers tax policy rule z_h = - EPSI_Z*y_h - GAMMA_Z*b_h(-1) + u_z_h + ( EPSI_Z*log(Y_h_ss) + GAMMA_Z*log(B_h_ss) + log(Z_h_ss) ) ; % Exogenous shocks %Equation 1: Taste shock u_b_h = RHO_B * u_b_h(-1) + e_u_b_h; %Equation 2: Leisure shock u_l_h = RHO_L * u_l_h(-1) + e_u_l_h; %Equation 5: Utilization shock u_i_h = RHO_I * u_i_h(-1) + e_u_i_h; %Equation 8: Technology shock u_a_h = RHO_A * u_a_h(-1) + e_u_a_h; u_g_h = RHO_G * u_g_h(-1) + e_u_g_h; u_tk_h = RHO_TK * u_tk_h(-1) + e_u_tk_h; u_tl_h = RHO_TL * u_tl_h(-1) + e_u_tl_h; u_tc_h = RHO_TC * u_tc_h(-1) + e_u_tc_h; u_z_h = RHO_Z * u_z_h(-1) + e_u_z_h; end; %-------------------------------------------------------------------------- % Computation %-------------------------------------------------------------------------- initval; uc_h = (Uc_h_bar); ul_h = (Ul_h_bar); y_h = log(Y_h_bar); c_h = log(C_h_bar); i_h = log(I_h_bar); l_h = log(L_h_bar); k_h = log(K_h_bar); w_h = log(W_h_bar); g_h = log(G_h_bar); b_h = log(B_h_bar); z_h = log(Z_h_bar); r_h = log(R_h_bar); r_k_h = log(R_K_h_bar); tk_h = log(TK_h_bar); tl_h = log(TL_h_bar); tc_h = log(TC_h_bar); tau_k_h = log(TAU_K_h_bar); tau_l_h = log(TAU_L_h_bar); tau_c_h = log(TAU_C_h_bar); v_h = 0; delv_h = log(DELV_h_bar); q_i_h = 0; u_b_h = 0; u_l_h = 0; u_i_h = 0; u_a_h = 0; u_g_h = 0; u_tk_h = 0; u_tl_h = 0; u_tc_h = 0; u_z_h = 0; end; %-------------------------------------------------------------------------- % shocks block %-------------------------------------------------------------------------- shocks; var e_u_b_h = SIGMA_B_H ^2; var e_u_l_h = SIGMA_L_H ^2; var e_u_i_h = SIGMA_I_H ^2; var e_u_a_h = SIGMA_A_H ^2; var e_u_g_h = SIGMA_G_H ^2; var e_u_z_h = SIGMA_Z_H ^2; var e_u_tk_h = SIGMA_TK_H ^2; var e_u_tl_h = SIGMA_TL_H ^2; var e_u_tc_h = SIGMA_TC_H ^2; end; steady; check; stoch_simul(order=1,irf=0); %-------------------------------------------------------------------------- % Observables: %-------------------------------------------------------------------------- varobs c_h i_h l_h g_h tl_h tk_h tc_h b_h z_h; %-------------------------------------------------------------------------- % Priors: 43 parameters to estimate %-------------------------------------------------------------------------- estimated_params; GAMMA, gamma_pdf, 1.75, 0.5; KAPPA, gamma_pdf, 2, 0.5; DELTA2, gamma_pdf, 0.7, 0.5; CHI, gamma_pdf, 5, 0.25; H, beta_pdf, 0.5, 0.2; RHO_B, beta_pdf, 0.7, 0.2; RHO_L, beta_pdf, 0.7, 0.2; RHO_I, beta_pdf, 0.7, 0.2; RHO_G, beta_pdf, 0.7, 0.2; RHO_A, beta_pdf, 0.7, 0.2; RHO_TK, beta_pdf, 0.7, 0.2; RHO_TL, beta_pdf, 0.7, 0.2; RHO_TC, beta_pdf, 0.7, 0.2; RHO_Z, beta_pdf, 0.7, 0.2; stderr e_u_b_h, inv_gamma_pdf, 0.01, 1; stderr e_u_l_h, inv_gamma_pdf, 0.01, 1; stderr e_u_i_h, inv_gamma_pdf, 0.01, 1; stderr e_u_a_h, inv_gamma_pdf, 0.01, 1; stderr e_u_g_h, inv_gamma_pdf, 0.01, 1; stderr e_u_tk_h, inv_gamma_pdf, 0.01, 1; stderr e_u_tl_h, inv_gamma_pdf, 0.01, 1; stderr e_u_tc_h, inv_gamma_pdf, 0.01, 1; stderr e_u_z_h, inv_gamma_pdf, 0.01, 1; GAMMA_G, gamma_pdf, 0.4, 0.20; GAMMA_K, gamma_pdf, 0.4, 0.20; GAMMA_L, gamma_pdf, 0.4, 0.20; GAMMA_Z, gamma_pdf, 0.4, 0.20; EPSI_K, gamma_pdf, 1, 0.30; EPSI_L, gamma_pdf, 0.5, 0.25; EPSI_G, gamma_pdf, 0.07,0.05; EPSI_Z, gamma_pdf, 0.20,0.10; PHI_KL, normal_pdf, 0.25, 0.10; PHI_KC, normal_pdf, 0.05, 0.10; PHI_LC, normal_pdf, 0.05, 0.10; end; %-------------------------------------------------------------------------- % Estimation %-------------------------------------------------------------------------- %estimation(datafile = pc_data,mode_compute=6,mh_replic=0); estimation(datafile = pc_data,mode_compute=0, mode_file = leeper_est_mode,nobs=193,first_obs=1, mh_replic=50000, mh_nblocks=2, mh_drop=0.005, mh_jscale=0.80, mode_check); %-------------------------------------------------------------------------- % Options %-------------------------------------------------------------------------- % mh_replic : the number of draws for MH algorithm, default = 2000 % mh_nblocks: the number of parallel chains for MH algorithm, default = 2 % mh_drop: fraction of intially generated parameter vectors to be dropped as % a burn in before doing posterior simulations, default = 0.5 % mh_jscale: scale parameter of the jumping distributions covariance matrix % Default (0.2) is rarely satisfactory, tune for an acceptance ratio of 25%-33%. % Increase if acceptance ratio is too high,decrease if too low. % Consider parameter specific values for this scale parameter, this can be done in the [estimated params], page 49 block. %-------------------------------------------------------------------------- % Algorithms %-------------------------------------------------------------------------- % If mh_nblocks = 1; convergence diagnostics of Geweke (1992, 1999) is computed % If mh_nblocks = 2; convergence diagnostics of Brooks and Gelman (1998) is used instead % If mh_replic > 2000; MCMC diagnostics are generated