Adds the discretionary case to the evaluate_planner_objective function
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@ -42,6 +42,13 @@ function planner_objective_value = evaluate_planner_objective(M_,options_,oo_)
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% The approximated conditional expectation of the planner's objective function starting from the non-stochastic steady-state and allowing for future shocks thus verifies
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% The approximated conditional expectation of the planner's objective function starting from the non-stochastic steady-state and allowing for future shocks thus verifies
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% W(y,0,1) = Wbar + 0.5*Wss
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% W(y,0,1) = Wbar + 0.5*Wss
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% In the discretionary case, the model is assumed to be linear and the utility is assumed to be linear-quadratic. This changes 2 aspects of the results delinated above:
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% 1) the second-order derivatives of the policy and transition functions h and g are zero.
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% 2) the unconditional expectation of states coincides with its steady-state, which entails E(yhat) = 0
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% Therefore, the unconditional welfare can now be approximated as
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% E(W) = (1 - beta)^{-1} ( Ubar + 0.5 ( U_xx h_y^2 E(yhat^2) + U_xx h_u^2 E(u^2) )
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% As for the conditional welfare, the second-order formula above is still valid, but the derivatives of W no longer contain any second-order derivatives of the policy and transition functions h and g.
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% INPUTS
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% INPUTS
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% M_: (structure) model description
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% M_: (structure) model description
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% options_: (structure) options
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% options_: (structure) options
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@ -76,26 +83,91 @@ beta = get_optimal_policy_discount_factor(M_.params, M_.param_names);
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ys = oo_.dr.ys;
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ys = oo_.dr.ys;
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planner_objective_value = zeros(2,1);
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planner_objective_value = zeros(2,1);
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if options_.order == 1
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if options_.ramsey_policy
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[U] = feval([M_.fname '.objective.static'],ys,zeros(1,exo_nbr), M_.params);
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if options_.order == 1
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planner_objective_value(1) = U/(1-beta);
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[U] = feval([M_.fname '.objective.static'],ys,zeros(1,exo_nbr), M_.params);
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planner_objective_value(2) = U/(1-beta);
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planner_objective_value(1) = U/(1-beta);
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elseif options_.order == 2
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planner_objective_value(2) = U/(1-beta);
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elseif options_.order == 2
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[U,Uy,Uyy] = feval([M_.fname '.objective.static'],ys,zeros(1,exo_nbr), M_.params);
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Gy = dr.ghx(nstatic+(1:nspred),:);
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Gu = dr.ghu(nstatic+(1:nspred),:);
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Gyy = dr.ghxx(nstatic+(1:nspred),:);
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Gyu = dr.ghxu(nstatic+(1:nspred),:);
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Guu = dr.ghuu(nstatic+(1:nspred),:);
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Gss = dr.ghs2(nstatic+(1:nspred),:);
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gy(dr.order_var,:) = dr.ghx;
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gu(dr.order_var,:) = dr.ghu;
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gyy(dr.order_var,:) = dr.ghxx;
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gyu(dr.order_var,:) = dr.ghxu;
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guu(dr.order_var,:) = dr.ghuu;
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gss(dr.order_var,:) = dr.ghs2;
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Uyy = full(Uyy);
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Uyygygy = A_times_B_kronecker_C(Uyy,gy,gy);
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Uyygugu = A_times_B_kronecker_C(Uyy,gu,gu);
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%% Unconditional welfare
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old_noprint = options_.noprint;
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if ~old_noprint
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options_.noprint = 1;
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end
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var_list = M_.endo_names(dr.order_var(nstatic+(1:nspred)));
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[info, oo_, options_] = stoch_simul(M_, options_, oo_, var_list); %get decision rules and moments
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if ~old_noprint
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options_.noprint = 0;
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end
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oo_.mean(isnan(oo_.mean)) = options_.huge_number;
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oo_.var(isnan(oo_.var)) = options_.huge_number;
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Ey = oo_.mean;
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Eyhat = Ey - ys(dr.order_var(nstatic+(1:nspred)));
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var_corr = Eyhat*Eyhat';
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Eyhatyhat = oo_.var(:) + var_corr(:);
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Euu = M_.Sigma_e(:);
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EU = U + Uy*gy*Eyhat + 0.5*((Uyygygy + Uy*gyy)*Eyhatyhat + (Uyygugu + Uy*guu)*Euu + Uy*gss);
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EW = EU/(1-beta);
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%% Conditional welfare starting from the non-stochastic steady-state
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Wbar = U/(1-beta);
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Wy = Uy*gy/(eye(nspred)-beta*Gy);
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if isempty(options_.qz_criterium)
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options_.qz_criterium = 1+1e-6;
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end
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%solve Lyapunuv equation Wyy=gy'*Uyy*gy+Uy*gyy+beta*Wy*Gyy+beta*Gy'Wyy*Gy
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Wyy = reshape(lyapunov_symm(sqrt(beta)*Gy',reshape(Uyygygy + Uy*gyy + beta*Wy*Gyy,nspred,nspred),options_.lyapunov_fixed_point_tol,options_.qz_criterium,options_.lyapunov_complex_threshold, 3, options_.debug),1,nspred*nspred);
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Wyygugu = A_times_B_kronecker_C(Wyy,Gu,Gu);
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Wuu = Uyygugu + Uy*guu + beta*(Wyygugu + Wy*Guu);
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Wss = (Uy*gss + beta*(Wy*Gss + Wuu*M_.Sigma_e(:)))/(1-beta);
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W = Wbar + 0.5*Wss;
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planner_objective_value(1) = EW;
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planner_objective_value(2) = W;
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else
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%Order k code will go here!
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fprintf('\nevaluate_planner_objective: order>2 not yet supported\n')
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planner_objective_value(1) = NaN;
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planner_objective_value(2) = NaN;
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return
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end
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elseif options_.discretionary_policy
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[U,Uy,Uyy] = feval([M_.fname '.objective.static'],ys,zeros(1,exo_nbr), M_.params);
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[U,Uy,Uyy] = feval([M_.fname '.objective.static'],ys,zeros(1,exo_nbr), M_.params);
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Gy = dr.ghx(nstatic+(1:nspred),:);
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Gy = dr.ghx(nstatic+(1:nspred),:);
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Gu = dr.ghu(nstatic+(1:nspred),:);
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Gu = dr.ghu(nstatic+(1:nspred),:);
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Gyy = dr.ghxx(nstatic+(1:nspred),:);
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Gyu = dr.ghxu(nstatic+(1:nspred),:);
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Guu = dr.ghuu(nstatic+(1:nspred),:);
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Gss = dr.ghs2(nstatic+(1:nspred),:);
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gy(dr.order_var,:) = dr.ghx;
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gy(dr.order_var,:) = dr.ghx;
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gu(dr.order_var,:) = dr.ghu;
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gu(dr.order_var,:) = dr.ghu;
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gyy(dr.order_var,:) = dr.ghxx;
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gyu(dr.order_var,:) = dr.ghxu;
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guu(dr.order_var,:) = dr.ghuu;
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gss(dr.order_var,:) = dr.ghs2;
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Uyy = full(Uyy);
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Uyy = full(Uyy);
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@ -125,9 +197,10 @@ elseif options_.order == 2
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Eyhatyhat = oo_.var(:) + var_corr(:);
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Eyhatyhat = oo_.var(:) + var_corr(:);
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Euu = M_.Sigma_e(:);
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Euu = M_.Sigma_e(:);
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EU = U + Uy*gy*Eyhat + 0.5*((Uyygygy + Uy*gyy)*Eyhatyhat + (Uyygugu + Uy*guu)*Euu + Uy*gss);
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EU = U + Uy*gy*Eyhat + 0.5*(Uyygygy*Eyhatyhat + Uyygugu*Euu);
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EW = EU/(1-beta);
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EW = EU/(1-beta);
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%% Conditional welfare starting from the non-stochastic steady-state
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%% Conditional welfare starting from the non-stochastic steady-state
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Wbar = U/(1-beta);
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Wbar = U/(1-beta);
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@ -136,27 +209,23 @@ elseif options_.order == 2
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if isempty(options_.qz_criterium)
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if isempty(options_.qz_criterium)
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options_.qz_criterium = 1+1e-6;
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options_.qz_criterium = 1+1e-6;
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end
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end
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%solve Lyapunuv equation Wyy=gy'*Uyy*gy+Uy*gyy+beta*Wy*Gyy+beta*Gy'Wyy*Gy
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%solve Lyapunuv equation Wyy=gy'*Uyy*gy+beta*Gy'Wyy*Gy
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Wyy = reshape(lyapunov_symm(sqrt(beta)*Gy',reshape(Uyygygy + Uy*gyy + beta*Wy*Gyy,nspred,nspred),options_.lyapunov_fixed_point_tol,options_.qz_criterium,options_.lyapunov_complex_threshold, 3, options_.debug),1,nspred*nspred);
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Wyy = reshape(lyapunov_symm(sqrt(beta)*Gy',reshape(Uyygygy,nspred,nspred),options_.lyapunov_fixed_point_tol,options_.qz_criterium,options_.lyapunov_complex_threshold, 3, options_.debug),1,nspred*nspred);
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Wyygugu = A_times_B_kronecker_C(Wyy,Gu,Gu);
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Wyygugu = A_times_B_kronecker_C(Wyy,Gu,Gu);
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Wuu = Uyygugu + Uy*guu + beta*(Wyygugu + Wy*Guu);
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Wuu = Uyygugu + beta*Wyygugu;
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Wss = (Uy*gss + beta*(Wy*Gss + Wuu*M_.Sigma_e(:)))/(1-beta);
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Wss = beta*Wuu*M_.Sigma_e(:)/(1-beta);
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W = Wbar + 0.5*Wss;
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W = Wbar + 0.5*Wss;
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planner_objective_value(1) = EW;
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planner_objective_value(1) = EW;
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planner_objective_value(2) = W;
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planner_objective_value(2) = W;
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else
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%Order k code will go here!
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fprintf('\nevaluate_planner_objective: order>2 not yet supported\n')
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planner_objective_value(1) = NaN;
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planner_objective_value(2) = NaN;
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return
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end
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end
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if ~options_.noprint
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if ~options_.noprint
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if options_.ramsey_policy
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if options_.ramsey_policy
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fprintf('\nApproximated value of unconditional welfare: %10.8f\n', planner_objective_value(1))
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fprintf('\nApproximated value of unconditional welfare: %10.8f\n', planner_objective_value(1))
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fprintf('\nApproximated value of conditional welfare: %10.8f\n', planner_objective_value(2))
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fprintf('\nApproximated value of conditional welfare: %10.8f\n', planner_objective_value(2))
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elseif options_.discretionary_policy
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elseif options_.discretionary_policy
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fprintf('\nApproximated value of unconditional welfare with discretionary policy: %10.8f\n\n', EW)
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fprintf('\nApproximated value of unconditional welfare with discretionary policy: %10.8f\n', planner_objective_value(1))
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fprintf('\nApproximated value of conditional welfare with discretionary policy: %10.8f\n', planner_objective_value(2))
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end
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end
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end
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end
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@ -1,5 +1,5 @@
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/*
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/*
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* This file implements the baseline New Keynesian model of Jordi Galí (2008): Monetary Policy, Inflation,
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* This file implements the baseline New Keynesian model of Jordi Gal<EFBFBD> (2008): Monetary Policy, Inflation,
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* and the Business Cycle, Princeton University Press, Chapter 5
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* and the Business Cycle, Princeton University Press, Chapter 5
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*
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*
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* This implementation was written by Johannes Pfeifer.
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* This implementation was written by Johannes Pfeifer.
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@ -9,7 +9,7 @@
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*/
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*/
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/*
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/*
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* Copyright (C) 2013-15 Johannes Pfeifer
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* Copyright (C) 2013-21 Johannes Pfeifer
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*
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*
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* This is free software: you can redistribute it and/or modify
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* This 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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* it under the terms of the GNU General Public License as published by
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@ -130,7 +130,7 @@ end
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%Compute theoretical objective function
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%Compute theoretical objective function
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V=betta/(1-betta)*(var_pi_theoretical+alpha_x*var_y_gap_theoretical); %evaluate at steady state in first period
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V=betta/(1-betta)*(var_pi_theoretical+alpha_x*var_y_gap_theoretical); %evaluate at steady state in first period
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if isnan(oo_.planner_objective_value) || abs(V-oo_.planner_objective_value)>1e-10
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if any( [ isnan(oo_.planner_objective_value(2)), abs(V-oo_.planner_objective_value(2))>1e-10 ] )
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error('Computed welfare deviates from theoretical welfare')
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error('Computed welfare deviates from theoretical welfare')
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end
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end
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end;
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end;
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@ -144,6 +144,6 @@ end;
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V=var_pi_theoretical+alpha_x*var_y_gap_theoretical+ betta/(1-betta)*(var_pi_theoretical+alpha_x*var_y_gap_theoretical); %evaluate at steady state in first period
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V=var_pi_theoretical+alpha_x*var_y_gap_theoretical+ betta/(1-betta)*(var_pi_theoretical+alpha_x*var_y_gap_theoretical); %evaluate at steady state in first period
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discretionary_policy(instruments=(i),irf=20,discretionary_tol=1e-12,planner_discount=betta) y_gap pi p u;
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discretionary_policy(instruments=(i),irf=20,discretionary_tol=1e-12,planner_discount=betta) y_gap pi p u;
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if isnan(oo_.planner_objective_value) || abs(V-oo_.planner_objective_value)>1e-10
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if any( [ isnan(oo_.planner_objective_value(1)), abs(V-oo_.planner_objective_value(1))>1e-10 ] )
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error('Computed welfare deviates from theoretical welfare')
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error('Computed welfare deviates from theoretical welfare')
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end
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end
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