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dynare/matlab/optimal_policy/evaluate_planner_objective.m

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function planner_objective_value = evaluate_planner_objective(M_,options_,oo_)
% function planner_objective_value = evaluate_planner_objective(M_,options_,oo_)
% INPUTS
% M_: (structure) model description
% options_: (structure) options
% oo_: (structure) output results
% OUTPUT
% planner_objective_value (structure)
%
%Returns a structure containing approximations of
% - the unconditional expectation of the planner's objective function in the field unconditional
% - the conditional expectations of the planner's objective function starting from the non-stochastic steady state in the field conditional
% - with Lagrange multipliers initially set to zero in the field zero_initial_multiplier
% - with lagrange multipliers initially set to their initial values in the field steady_initial_multiplier
% Approximations are consistent with the order specified in options_order.
%
% SPECIAL REQUIREMENTS
% none
% ALGORITHM
% Welfare satifies
% W(y_{t-1}, u_t, sigma) = U(h(y_{t-1}, u_t, sigma)) + beta E_t W(g(y_{t-1}, u_t, sigma), u_t, sigma)
% where
% - W is the welfare function
% - U is the utility function
% - y_{t-1} is the vector of state variables
% - u_t is the vector of exogenous shocks scaled with sigma i.e. u_t = sigma e_t where e_t is the vector of exogenous shocks
% - sigma is the perturbation parameter
% - h is the policy function, providing controls x_t in function of information at time t i.e. (y_{t-1}, u_t, sigma)
% - g is the transition function, providing next-period state variables in function of information at time t i.e. (y_{t-1}, u_t, sigma)
% - beta is the planner's discount factor
% - E_t is the expectation operator given information at time t i.e. (y_{t-1}, u_t, sigma)
% The unconditional expectation of the planner's objective function satisfies
% E(W) = E(U)/(1-beta)
% The conditional expectation of the planner's objective function given (y_{t-1}, u_t, sigma) coincides with the welfare function delineated above.
% A first-order approximation of the utility function around the non-stochastic steady state (y_{t-1}, u_t, sigma) = (y, 0, 0) is
% U(h(y_{t-1}, u_t, sigma)) = Ubar + U_x ( h_y yhat_{t-1} + h_u u_t )
% Taking the unconditional expectation yields E(U) = Ubar and E(W) = Ubar/(1-beta)
% As for conditional welfare, a first-order approximation leads to
% W = Wbar + W_y yhat_{t-1} + W_u u_t
% Similarly, taking the unconditional expectation of a second-order approximation of utility around the non-stochastic steady state yields a second-order approximation of unconditional welfare
% E(W) = (1 - beta)^{-1} ( Ubar + U_x h_y E(yhat) + 0.5 ( (U_xx h_y^2 + U_x h_yy) E(yhat^2) + (U_xx h_u^2 + U_x h_uu) E(u^2) + U_x h_ss )
% where E(yhat), E(yhat^2) and E(u^2) can be derived from oo_.mean and oo_.var.
% Importantly, E(yhat) and E(yhat^2) are second-order approximations, which is not the same as approximations computed with all the information provided by decision rules approximated up to the second order. The latter might include terms that are order 3 or 4 for the approximation of E(yhat^2), which we exclude here.
% As for conditional welfare, the second-order approximation of welfare around the non-stochastic steady state leads to
% W(y_{t-1}, u_t, sigma) = Wbar + W_y yhat_{t-1} + W_u u_t + W_yu yhat_{t-1} ⊗ u_t + 0.5 ( W_yy yhat_{t-1}^2 + W_uu u_t^2 + W_ss )
% The derivatives of W taken at the non-stochastic steady state can be computed as in Kamenik and Juillard (2004) "Solving Stochastic Dynamic Equilibrium Models: A k-Order Perturbation Approach".
% 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.
% In the deterministic case, resorting to approximations for welfare is no longer required as it is possible to simulate the model given initial conditions for pre-determined variables and terminal conditions for forward-looking variables, whether these initial and terminal conditions are explicitly or implicitly specified. Assuming that the number of simulated periods is high enough for the new steady-state to be reached, the new unconditional welfare is thus the last period's welfare. As for the conditional welfare, it can be derived using backward recursions on the equation W = U + beta*W(+1) starting from the final unconditional steady-state welfare.
% Copyright © 2007-2022 Dynare Team
%
% This file is part of Dynare.
%
% Dynare is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% Dynare is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with Dynare. If not, see <https://www.gnu.org/licenses/>.
dr = oo_.dr;
if isempty(options_.qz_criterium)
options_.qz_criterium = 1+1e-6;
end
exo_nbr = M_.exo_nbr;
nstatic = M_.nstatic;
nspred = M_.nspred;
beta = get_optimal_policy_discount_factor(M_.params, M_.param_names);
if beta>=1
fprintf('evaluate_planner_objective: the planner discount factor is not strictly smaller than 1. Unconditional welfare will not be finite.\n')
end
if options_.ramsey_policy && oo_.gui.ran_perfect_foresight
T = size(oo_.endo_simul,2);
[U_term] = feval([M_.fname '.objective.static'],oo_.endo_simul(:,T-M_.maximum_lead),oo_.exo_simul(T-M_.maximum_lead,:), M_.params);
EW = U_term/(1-beta);
W = EW;
for t=T-M_.maximum_lead:-1:1+M_.maximum_lag
[U] = feval([M_.fname '.objective.static'],oo_.endo_simul(:,t),oo_.exo_simul(t,:), M_.params);
W = U + beta*W;
end
planner_objective_value = struct('conditional', W, 'unconditional', EW);
else
planner_objective_value = struct('conditional', struct('zero_initial_multiplier', 0., 'steady_initial_multiplier', 0.), 'unconditional', 0.);
if isempty(oo_.dr) || ~isfield(oo_.dr,'ys')
error('evaluate_planner_objective requires decision rules to have previously been computed (e.g. by stoch_simul or discretionary_policy)')
else
ys = oo_.dr.ys;
end
if options_.order == 1 && ~options_.discretionary_policy
[U,Uy] = feval([M_.fname '.objective.static'],ys,zeros(1,exo_nbr), M_.params);
Gy = dr.ghx(nstatic+(1:nspred),:);
Gu = dr.ghu(nstatic+(1:nspred),:);
gy(dr.order_var,:) = dr.ghx;
gu(dr.order_var,:) = dr.ghu;
%% Unconditional welfare
EW = U/(1-beta);
planner_objective_value.unconditional = EW;
%% Conditional welfare starting from the non-stochastic steady-state (i) with Lagrange multipliers set to their steady-state value (ii) with Lagrange multipliers set to 0
Wbar = U/(1-beta);
Wy = Uy*gy/(eye(nspred)-beta*Gy);
Wu = Uy*gu + beta*Wy*Gu;
[yhat_L_SS,yhat_L_0, u]=get_initial_state(ys,M_,dr,oo_);
W_L_SS = Wbar+Wy*yhat_L_SS+Wu*u;
W_L_0 = Wbar+Wy*yhat_L_0+Wu*u;
planner_objective_value.conditional.steady_initial_multiplier = W_L_SS;
planner_objective_value.conditional.zero_initial_multiplier = W_L_0;
elseif options_.order == 2 && ~M_.hessian_eq_zero %full second order approximation
[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);
Uyygugy = A_times_B_kronecker_C(Uyy,gu,gy);
%% Unconditional welfare
old_noprint = options_.noprint;
if ~old_noprint
options_.noprint = 1;
end
var_list = M_.endo_names(dr.order_var(nstatic+(1:nspred)));
if options_.pruning
fprintf('evaluate_planner_objective: pruning option is not supported and will be ignored\n')
end
oo_=disp_th_moments(dr,var_list,M_,options_,oo_);
if ~old_noprint
options_.noprint = 0;
end
if any(isnan(oo_.mean)) || any(any(isnan(oo_.var)))
fprintf('evaluate_planner_objective: encountered NaN moments in the endogenous variables often associated\n')
fprintf('evaluate_planner_objective: with either non-stationary variables or singularity due e.g. including\n')
fprintf('evaluate_planner_objective: the planner objective function (or additive parts of it) in the model.\n')
fprintf('evaluate_planner_objective: I will replace the NaN with a large number, but tread carefully,\n')
fprintf('evaluate_planner_objective: check your model, and watch out for strange results.\n')
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)));
Eyhatyhat = oo_.var(:);
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);
planner_objective_value.unconditional = EW;
%% Conditional welfare starting from the non-stochastic steady-state (i) with Lagrange multipliers set to their steady-state value (ii) with Lagrange multipliers set to 0
Wbar = U/(1-beta);
Wy = Uy*gy/(eye(nspred)-beta*Gy);
Wu = Uy*gu + beta*Wy*Gu;
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);
Wyygugy = A_times_B_kronecker_C(Wyy,Gu,Gy);
Wuu = Uyygugu + Uy*guu + beta*(Wyygugu + Wy*Guu);
Wss = (Uy*gss + beta*(Wy*Gss + Wuu*M_.Sigma_e(:)))/(1-beta);
Wyu = Uyygugy + Uy*gyu + beta*(Wyygugy + Wy*Gyu);
[yhat_L_SS,yhat_L_0, u]=get_initial_state(ys,M_,dr,oo_);
Wyu_yu_L_SS = A_times_B_kronecker_C(Wyu,yhat_L_SS,u);
Wyy_yy_L_SS = A_times_B_kronecker_C(Wyy,yhat_L_SS,yhat_L_SS);
Wuu_uu_L_SS = A_times_B_kronecker_C(Wuu,u,u);
W_L_SS = Wbar+Wy*yhat_L_SS+Wu*u+Wyu_yu_L_SS+0.5*(Wss+Wyy_yy_L_SS+Wuu_uu_L_SS);
Wyu_yu_L_0 = A_times_B_kronecker_C(Wyu,yhat_L_0,u);
Wyy_yy_L_0 = A_times_B_kronecker_C(Wyy,yhat_L_0,yhat_L_0);
Wuu_uu_L_0 = A_times_B_kronecker_C(Wuu,u,u);
W_L_0 = Wbar+Wy*yhat_L_0+Wu*u+Wyu_yu_L_0+0.5*(Wss+Wyy_yy_L_0+Wuu_uu_L_0);
planner_objective_value.conditional.steady_initial_multiplier = W_L_SS;
planner_objective_value.conditional.zero_initial_multiplier = W_L_0;
elseif (options_.order == 2 && M_.hessian_eq_zero) || options_.discretionary_policy %linear quadratic problem
[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),:);
gy(dr.order_var,:) = dr.ghx;
gu(dr.order_var,:) = dr.ghu;
Uyy = full(Uyy);
Uyygygy = A_times_B_kronecker_C(Uyy,gy,gy);
Uyygugu = A_times_B_kronecker_C(Uyy,gu,gu);
Uyygugy = A_times_B_kronecker_C(Uyy,gu,gy);
%% Unconditional welfare
old_noprint = options_.noprint;
if ~old_noprint
options_.noprint = 1;
end
var_list = M_.endo_names(dr.order_var(nstatic+(1:nspred)));
oo_=disp_th_moments(dr,var_list,M_,options_,oo_);
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)));
Eyhatyhat = oo_.var(:);
Euu = M_.Sigma_e(:);
EU = U + Uy*gy*Eyhat + 0.5*(Uyygygy*Eyhatyhat + Uyygugu*Euu);
EW = EU/(1-beta);
planner_objective_value.unconditional = EW;
%% Conditional welfare starting from the non-stochastic steady-state
Wbar = U/(1-beta);
Wy = Uy*gy/(eye(nspred)-beta*Gy);
Wu = Uy*gu + beta*Wy*Gu;
%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);
Wyygugy = A_times_B_kronecker_C(Wyy,Gu,Gy);
Wuu = Uyygugu + beta*Wyygugu;
Wss = beta*Wuu*M_.Sigma_e(:)/(1-beta);
Wyu = Uyygugy + beta*Wyygugy;
[yhat_L_SS,yhat_L_0, u]=get_initial_state(ys,M_,dr,oo_);
Wyu_yu_L_SS = A_times_B_kronecker_C(Wyu,yhat_L_SS,u);
Wyy_yy_L_SS = A_times_B_kronecker_C(Wyy,yhat_L_SS,yhat_L_SS);
Wuu_uu_L_SS = A_times_B_kronecker_C(Wuu,u,u);
W_L_SS = Wbar+Wy*yhat_L_SS+Wu*u+Wyu_yu_L_SS+0.5*(Wss+Wyy_yy_L_SS+Wuu_uu_L_SS);
Wyu_yu_L_0 = A_times_B_kronecker_C(Wyu,yhat_L_0,u);
Wyy_yy_L_0 = A_times_B_kronecker_C(Wyy,yhat_L_0,yhat_L_0);
Wuu_uu_L_0 = A_times_B_kronecker_C(Wuu,u,u);
W_L_0 = Wbar+Wy*yhat_L_0+Wu*u+Wyu_yu_L_0+0.5*(Wss+Wyy_yy_L_0+Wuu_uu_L_0);
planner_objective_value.conditional.steady_initial_multiplier = W_L_SS;
planner_objective_value.conditional.zero_initial_multiplier = W_L_0;
elseif options_.order > 2 || ~options_.discretionary_policy
% Computes the welfare decision rule
[W] = k_order_welfare(dr,M_,options_);
% Appends the welfare decision rule to the endogenous variables decision
% rule
for i=0:options_.order
dr.(['g_' num2str(i)]) = [dr.(['g_' num2str(i)]); W.(['W_' num2str(i)])];
end
% Amends the steady-state vector accordingly
[U] = feval([M_.fname '.objective.static'],ys,zeros(1,exo_nbr), M_.params);
ysteady = [ys(oo_.dr.order_var); U/(1-beta)];
% Generates the sequence of shocks to compute unconditional welfare
i_exo_var = setdiff([1:M_.exo_nbr],find(diag(M_.Sigma_e) == 0));
nxs = length(i_exo_var);
chol_S = chol(M_.Sigma_e(i_exo_var,i_exo_var));
exo_simul = zeros(M_.exo_nbr,options_.ramsey.periods);
if ~isempty(M_.Sigma_e)
exo_simul(i_exo_var,:) = chol_S*randn(nxs,options_.ramsey.periods);
end
yhat_start = zeros(M_.endo_nbr+1,1);
[moment] = k_order_mean(options_.order, M_.nstatic, M_.npred, M_.nboth, M_.nfwrd+1, M_.exo_nbr, 1, options_.ramsey.drop, yhat_start, exo_simul, ysteady, dr);
% Stores the result for unconditional welfare
planner_objective_value.unconditional = moment(end);
% Conditional welfare
% Gets initial values
[yhat_L_SS,yhat_L_0, u] = get_initial_state(ys,M_,dr,oo_);
% Conditional welfare (i) with Lagrange multipliers set to their
% steady-state values
yhat_start(M_.nstatic+1:M_.nstatic+M_.npred+M_.nboth) = yhat_L_SS;
[~,sim] = k_order_mean(options_.order, M_.nstatic, M_.npred, M_.nboth, M_.nfwrd+1, M_.exo_nbr, 1, 0, yhat_start, u, ysteady, dr);
planner_objective_value.conditional.steady_initial_multiplier = sim(end,1);
% Conditional welfare (ii) with Lagrange multipliers set to 0
yhat_start(M_.nstatic+1:M_.nstatic+M_.npred+M_.nboth) = yhat_L_0;
[~,sim] = k_order_mean(options_.order, M_.nstatic, M_.npred, M_.nboth, M_.nfwrd+1, M_.exo_nbr, 1, 0, yhat_start, u, ysteady, dr);
planner_objective_value.conditional.zero_initial_multiplier = sim(end,1);
end
end
if ~options_.noprint
if options_.ramsey_policy
if oo_.gui.ran_perfect_foresight
fprintf('\nSimulated value of unconditional welfare: %10.8f\n', planner_objective_value.unconditional)
fprintf('\nSimulated value of conditional welfare: %10.8f\n', planner_objective_value.conditional)
else
fprintf('\nApproximated value of unconditional welfare: %10.8f\n', planner_objective_value.unconditional)
fprintf('\nApproximated value of conditional welfare:\n')
fprintf(' - with initial Lagrange multipliers set to 0: %10.8f\n', planner_objective_value.conditional.zero_initial_multiplier)
fprintf(' - with initial Lagrange multipliers set to steady state: %10.8f\n\n', planner_objective_value.conditional.steady_initial_multiplier)
end
elseif options_.discretionary_policy
fprintf('\nApproximated value of unconditional welfare with discretionary policy: %10.8f\n', planner_objective_value.unconditional)
fprintf('\nApproximated value of conditional welfare with discretionary policy:\n')
fprintf(' - with initial Lagrange multipliers set to 0: %10.8f\n', planner_objective_value.conditional.zero_initial_multiplier)
fprintf(' - with initial Lagrange multipliers set to steady state: %10.8f\n\n', planner_objective_value.conditional.steady_initial_multiplier)
end
end
function [yhat_L_SS,yhat_L_0, u]=get_initial_state(ys,M_,dr,oo_)
% initialize Lagrange multipliers to their steady-state values in yhat_L_SS
yhat_L_SS = ys;
% initialize Lagrange multipliers to 0 in yhat_L_0
yhat_L_0 = zeros(M_.endo_nbr,1);
if ~isempty(M_.aux_vars)
mult_indicator=([M_.aux_vars(:).type]==6);
mult_indices=[M_.aux_vars(mult_indicator).endo_index];
else
mult_indices=[];
end
non_mult_indices=~ismember(1:M_.endo_nbr,mult_indices);
if ~isempty(M_.endo_histval)
% initialize endogenous state variable to histval if necessary
yhat_L_SS(non_mult_indices) = M_.endo_histval(non_mult_indices);
yhat_L_0(non_mult_indices) = M_.endo_histval(non_mult_indices);
else
yhat_L_0(non_mult_indices) = ys(non_mult_indices);
end
yhat_L_0 = yhat_L_0(dr.order_var(M_.nstatic+(1:M_.nspred)),1)-ys(dr.order_var(M_.nstatic+(1:M_.nspred)));
yhat_L_SS = yhat_L_SS(dr.order_var(M_.nstatic+(1:M_.nspred)),1)-ys(dr.order_var(M_.nstatic+(1:M_.nspred)));
if ~isempty(M_.det_shocks)
if ~all(oo_.exo_simul(1,:)==0)
fprintf(['\nevaluate_planner_objective: oo_.exo_simul contains simulated values for the initial period.\n'...
'evaluate_planner_objective: Dynare will ignore them and use the provided initial condition.\n'])
end
u =oo_.exo_steady_state;
periods=[M_.det_shocks(:).periods];
if any(periods==0)
fprintf(['\nevaluate_planner_objective: M_.det_shocks contains values for the predetermined t=0 period.\n'...
'evaluate_planner_objective: Dynare will ignore them. Use histval to set the value of lagged innovations.\n'])
end
if any(periods>1)
fprintf(['\nevaluate_planner_objective: Shock values for periods not contained in the initial information set (t=1) have been provided.\n' ...
'evaluate_planner_objective: Note that they will be ignored.\n'])
end
shock_indices=find(periods==1);
if any(cellfun(@(x) ~strcmp(x, 'level'), { M_.det_shocks(shock_indices).type }))
fprintf(['\nevaluate_planner_objective: Shock values need to be specified in level.\n'])
end
u([M_.det_shocks(shock_indices).exo_id])=[M_.det_shocks(shock_indices).value];
else
u = oo_.exo_simul(1,:)'; %first value of simulation series (set by simult.m if periods>0), 1 otherwise
end
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