Commit f532de0f by Normann Rion Committed by NormannR

### Adds the discretionary case to the evaluate_planner_objective function

parent e880d1bc
 ... ... @@ -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
 /* * 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
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