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Adds the discretionary case to the evaluate_planner_objective function

time-shift
Normann Rion 2021-02-18 11:55:47 +01:00 committed by NormannR
parent e880d1bcc3
commit f532de0f29
2 changed files with 99 additions and 30 deletions
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121
matlab/evaluate_planner_objective.m
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@ -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

8
tests/discretionary_policy/Gali_discretion.mod
View File

@ -1,5 +1,5 @@
/*
* This file implements the baseline New Keynesian model of Jordi Galí (2008): Monetary Policy, Inflation,
* This file implements the baseline New Keynesian model of Jordi Gal<EFBFBD> (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