diff --git a/matlab/evaluate_planner_objective.m b/matlab/evaluate_planner_objective.m
index 26e23b4a15e1533faa7ae35e8a7aee99f0e9517b..878b06ec30bc6eccf1a6006180fb324995b177ba 100644
--- a/matlab/evaluate_planner_objective.m
+++ b/matlab/evaluate_planner_objective.m
@@ -42,6 +42,13 @@ function planner_objective_value = evaluate_planner_objective(M_,options_,oo_)
% The approximated conditional expectation of the planner's objective function starting from the non-stochastic steady-state and allowing for future shocks thus verifies
% W(y,0,1) = Wbar + 0.5*Wss
+% 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:
+% 1) the second-order derivatives of the policy and transition functions h and g are zero.
+% 2) the unconditional expectation of states coincides with its steady-state, which entails E(yhat) = 0
+% Therefore, the unconditional welfare can now be approximated as
+% E(W) = (1 - beta)^{-1} ( Ubar + 0.5 ( U_xx h_y^2 E(yhat^2) + U_xx h_u^2 E(u^2) )
+% 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.
+
% INPUTS
% M_: (structure) model description
% options_: (structure) options
@@ -76,26 +83,91 @@ beta = get_optimal_policy_discount_factor(M_.params, M_.param_names);
ys = oo_.dr.ys;
planner_objective_value = zeros(2,1);
-if options_.order == 1
- [U] = feval([M_.fname '.objective.static'],ys,zeros(1,exo_nbr), M_.params);
- planner_objective_value(1) = U/(1-beta);
- planner_objective_value(2) = U/(1-beta);
-elseif options_.order == 2
+if options_.ramsey_policy
+ if options_.order == 1
+ [U] = feval([M_.fname '.objective.static'],ys,zeros(1,exo_nbr), M_.params);
+ planner_objective_value(1) = U/(1-beta);
+ planner_objective_value(2) = U/(1-beta);
+ elseif options_.order == 2
+ [U,Uy,Uyy] = feval([M_.fname '.objective.static'],ys,zeros(1,exo_nbr), M_.params);
+
+ Gy = dr.ghx(nstatic+(1:nspred),:);
+ Gu = dr.ghu(nstatic+(1:nspred),:);
+ Gyy = dr.ghxx(nstatic+(1:nspred),:);
+ Gyu = dr.ghxu(nstatic+(1:nspred),:);
+ Guu = dr.ghuu(nstatic+(1:nspred),:);
+ Gss = dr.ghs2(nstatic+(1:nspred),:);
+
+ gy(dr.order_var,:) = dr.ghx;
+ gu(dr.order_var,:) = dr.ghu;
+ gyy(dr.order_var,:) = dr.ghxx;
+ gyu(dr.order_var,:) = dr.ghxu;
+ guu(dr.order_var,:) = dr.ghuu;
+ gss(dr.order_var,:) = dr.ghs2;
+
+ Uyy = full(Uyy);
+
+ Uyygygy = A_times_B_kronecker_C(Uyy,gy,gy);
+ Uyygugu = A_times_B_kronecker_C(Uyy,gu,gu);
+
+ %% Unconditional welfare
+
+ old_noprint = options_.noprint;
+
+ if ~old_noprint
+ options_.noprint = 1;
+ end
+ var_list = M_.endo_names(dr.order_var(nstatic+(1:nspred)));
+ [info, oo_, options_] = stoch_simul(M_, options_, oo_, var_list); %get decision rules and moments
+ if ~old_noprint
+ options_.noprint = 0;
+ end
+
+ oo_.mean(isnan(oo_.mean)) = options_.huge_number;
+ oo_.var(isnan(oo_.var)) = options_.huge_number;
+
+ Ey = oo_.mean;
+ Eyhat = Ey - ys(dr.order_var(nstatic+(1:nspred)));
+
+ var_corr = Eyhat*Eyhat';
+ Eyhatyhat = oo_.var(:) + var_corr(:);
+ Euu = M_.Sigma_e(:);
+
+ EU = U + Uy*gy*Eyhat + 0.5*((Uyygygy + Uy*gyy)*Eyhatyhat + (Uyygugu + Uy*guu)*Euu + Uy*gss);
+ EW = EU/(1-beta);
+
+ %% Conditional welfare starting from the non-stochastic steady-state
+
+ Wbar = U/(1-beta);
+ Wy = Uy*gy/(eye(nspred)-beta*Gy);
+
+ if isempty(options_.qz_criterium)
+ options_.qz_criterium = 1+1e-6;
+ end
+ %solve Lyapunuv equation Wyy=gy'*Uyy*gy+Uy*gyy+beta*Wy*Gyy+beta*Gy'Wyy*Gy
+ 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);
+ Wyygugu = A_times_B_kronecker_C(Wyy,Gu,Gu);
+ Wuu = Uyygugu + Uy*guu + beta*(Wyygugu + Wy*Guu);
+ Wss = (Uy*gss + beta*(Wy*Gss + Wuu*M_.Sigma_e(:)))/(1-beta);
+ W = Wbar + 0.5*Wss;
+
+ planner_objective_value(1) = EW;
+ planner_objective_value(2) = W;
+ else
+ %Order k code will go here!
+ fprintf('\nevaluate_planner_objective: order>2 not yet supported\n')
+ planner_objective_value(1) = NaN;
+ planner_objective_value(2) = NaN;
+ return
+ end
+elseif options_.discretionary_policy
+
[U,Uy,Uyy] = feval([M_.fname '.objective.static'],ys,zeros(1,exo_nbr), M_.params);
Gy = dr.ghx(nstatic+(1:nspred),:);
Gu = dr.ghu(nstatic+(1:nspred),:);
- Gyy = dr.ghxx(nstatic+(1:nspred),:);
- Gyu = dr.ghxu(nstatic+(1:nspred),:);
- Guu = dr.ghuu(nstatic+(1:nspred),:);
- Gss = dr.ghs2(nstatic+(1:nspred),:);
-
gy(dr.order_var,:) = dr.ghx;
gu(dr.order_var,:) = dr.ghu;
- gyy(dr.order_var,:) = dr.ghxx;
- gyu(dr.order_var,:) = dr.ghxu;
- guu(dr.order_var,:) = dr.ghuu;
- gss(dr.order_var,:) = dr.ghs2;
Uyy = full(Uyy);
@@ -125,9 +197,10 @@ elseif options_.order == 2
Eyhatyhat = oo_.var(:) + var_corr(:);
Euu = M_.Sigma_e(:);
- EU = U + Uy*gy*Eyhat + 0.5*((Uyygygy + Uy*gyy)*Eyhatyhat + (Uyygugu + Uy*guu)*Euu + Uy*gss);
+ EU = U + Uy*gy*Eyhat + 0.5*(Uyygygy*Eyhatyhat + Uyygugu*Euu);
EW = EU/(1-beta);
+
%% Conditional welfare starting from the non-stochastic steady-state
Wbar = U/(1-beta);
@@ -136,27 +209,23 @@ elseif options_.order == 2
if isempty(options_.qz_criterium)
options_.qz_criterium = 1+1e-6;
end
- %solve Lyapunuv equation Wyy=gy'*Uyy*gy+Uy*gyy+beta*Wy*Gyy+beta*Gy'Wyy*Gy
- 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);
+ %solve Lyapunuv equation Wyy=gy'*Uyy*gy+beta*Gy'Wyy*Gy
+ 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);
Wyygugu = A_times_B_kronecker_C(Wyy,Gu,Gu);
- Wuu = Uyygugu + Uy*guu + beta*(Wyygugu + Wy*Guu);
- Wss = (Uy*gss + beta*(Wy*Gss + Wuu*M_.Sigma_e(:)))/(1-beta);
+ Wuu = Uyygugu + beta*Wyygugu;
+ Wss = beta*Wuu*M_.Sigma_e(:)/(1-beta);
W = Wbar + 0.5*Wss;
planner_objective_value(1) = EW;
planner_objective_value(2) = W;
-else
- %Order k code will go here!
- fprintf('\nevaluate_planner_objective: order>2 not yet supported\n')
- planner_objective_value(1) = NaN;
- planner_objective_value(2) = NaN;
- return
end
if ~options_.noprint
if options_.ramsey_policy
fprintf('\nApproximated value of unconditional welfare: %10.8f\n', planner_objective_value(1))
fprintf('\nApproximated value of conditional welfare: %10.8f\n', planner_objective_value(2))
elseif options_.discretionary_policy
- fprintf('\nApproximated value of unconditional welfare with discretionary policy: %10.8f\n\n', EW)
+ fprintf('\nApproximated value of unconditional welfare with discretionary policy: %10.8f\n', planner_objective_value(1))
+ fprintf('\nApproximated value of conditional welfare with discretionary policy: %10.8f\n', planner_objective_value(2))
+
end
end
diff --git a/tests/discretionary_policy/Gali_discretion.mod b/tests/discretionary_policy/Gali_discretion.mod
index fc596311eb0ef3a8fcef5b4193af8d20860f9232..af32cfa54916ae68cd1e4cd66781ae825cc5e09a 100644
--- a/tests/discretionary_policy/Gali_discretion.mod
+++ b/tests/discretionary_policy/Gali_discretion.mod
@@ -1,5 +1,5 @@
/*
- * This file implements the baseline New Keynesian model of Jordi Gal� (2008): Monetary Policy, Inflation,
+ * This file implements the baseline New Keynesian model of Jordi Gal� (2008): Monetary Policy, Inflation,
* and the Business Cycle, Princeton University Press, Chapter 5
*
* This implementation was written by Johannes Pfeifer.
@@ -9,7 +9,7 @@
*/
/*
- * Copyright (C) 2013-15 Johannes Pfeifer
+ * Copyright (C) 2013-21 Johannes Pfeifer
*
* This is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
@@ -130,7 +130,7 @@ end
%Compute theoretical objective function
V=betta/(1-betta)*(var_pi_theoretical+alpha_x*var_y_gap_theoretical); %evaluate at steady state in first period
-if isnan(oo_.planner_objective_value) || abs(V-oo_.planner_objective_value)>1e-10
+if any( [ isnan(oo_.planner_objective_value(2)), abs(V-oo_.planner_objective_value(2))>1e-10 ] )
error('Computed welfare deviates from theoretical welfare')
end
end;
@@ -144,6 +144,6 @@ end;
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
discretionary_policy(instruments=(i),irf=20,discretionary_tol=1e-12,planner_discount=betta) y_gap pi p u;
-if isnan(oo_.planner_objective_value) || abs(V-oo_.planner_objective_value)>1e-10
+if any( [ isnan(oo_.planner_objective_value(1)), abs(V-oo_.planner_objective_value(1))>1e-10 ] )
error('Computed welfare deviates from theoretical welfare')
end