Merge branch 'ramsey_tests' into 'master'

Expand Ramsey tests and fix consistency of utility and welfare

Closes #1821

See merge request Dynare/dynare!1945

doc/manual/source/the-model-file.rst

@@ -4220,7 +4220,8 @@ Computing the stochastic solution
     variance-covariance of the endogenous variables. Contains
     theoretical variance if the ``periods`` option is not present and simulated variance
     otherwise. Only available for ``order<4``. At ``order=2`` it will be be
-    a second-order accurate approximation. At ``order=3``, theoretical moments
+    a second-order accurate approximation (i.e. ignoring terms of order 3 and 4 that would
+    arise when using the full second-order policy function). At ``order=3``, theoretical moments
     are only available with ``pruning``. The variables are arranged in declaration order.
 
 .. matvar:: oo_.var_list
@@ -10229,7 +10230,8 @@ with ``discretionary_policy`` or for optimal simple rules with ``osr``
 
     At the current stage, the stochastic context does not support the ``pruning`` option.
     At ``order>3``, only the computation of conditional welfare with steady state Lagrange 
-    multipliers is supported.
+    multipliers is supported. Note that at `order=2`, the output is based on the second-order
+    accurate approximation of the variance stored in `oo_.var`.
     
     *Example (stochastic context)*
 

matlab/evaluate_planner_objective.m

@@ -43,7 +43,8 @@ function planner_objective_value = evaluate_planner_objective(M_,options_,oo_)
 
 % 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
+% 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 )
@@ -173,8 +174,7 @@ if options_.ramsey_policy
             Ey = oo_.mean;
             Eyhat = Ey - ys(dr.order_var(nstatic+(1:nspred)));
             
-            var_corr = Eyhat*Eyhat';
-            Eyhatyhat = oo_.var(:) + var_corr(:);
+            Eyhatyhat = oo_.var(:)
             Euu = M_.Sigma_e(:);
             
             EU = U + Uy*gy*Eyhat + 0.5*((Uyygygy + Uy*gyy)*Eyhatyhat + (Uyygugu + Uy*guu)*Euu + Uy*gss);
@@ -262,8 +262,7 @@ elseif options_.discretionary_policy
     Ey = oo_.mean;
     Eyhat = Ey - ys(dr.order_var(nstatic+(1:nspred)));
     
-    var_corr = Eyhat*Eyhat';
-    Eyhatyhat = oo_.var(:) + var_corr(:);
+    Eyhatyhat = oo_.var(:);
     Euu = M_.Sigma_e(:);
     
     EU = U + Uy*gy*Eyhat + 0.5*(Uyygygy*Eyhatyhat + Uyygugu*Euu);

tests/optimal_policy/neo_growth_common.inc

@@ -9,7 +9,7 @@ gamma = 1;
 delta = 0.012;
 alpha = 0.4;
 rho = 0.95;
-s = 0.007;
+s = 0.07; % increased by a factor of 10 from 0.007 to increase risk correction
 
 model;
 c^(-gamma)=beta*c(+1)^(-gamma)*(alpha*exp(z(+1))*k^(alpha-1)+1-delta);

tests/optimal_policy/neo_growth_ramsey.mod

@@ -54,6 +54,15 @@ if abs(cond_W_hand_L_SS - planner_objective_value.conditional.steady_initial_mul
    error('Inaccurate conditional welfare with Lagrange multiplier set to its steady-state value');
 end;
 
+if abs(oo_.mean(strmatch('U',M_.endo_names,'exact'))-oo1.oo_.mean(strmatch('U',M1.M_.endo_names,'exact')))>1e-6
+    error('Utility inconsistent');
+end
+
+if abs(planner_objective_value.unconditional-oo_.mean(strmatch('U',M_.endo_names,'exact'))/(1-beta)) > 1e-6;
+   error('Unconditional welfare assessment does not match utility');
+end;
+
+
 initial_condition_states = zeros(M1.M_.endo_nbr,M1.M_.maximum_lag);
 initial_condition_states(1:M1.M_.orig_endo_nbr,:) = repmat(oo1.oo_.dr.ys(1:M1.M_.orig_endo_nbr),1,M1.M_.maximum_lag);
 shock_matrix = zeros(1,M1.M_.exo_nbr);

tests/optimal_policy/neo_growth_ramsey_common.inc

@@ -1,4 +1,4 @@
-var k z c;
+var k z c U;
 
 varexo e;
 
@@ -9,15 +9,17 @@ gamma = 1;
 delta = 0.012;
 alpha = 0.4;
 rho = 0.95;
-s = 0.007;
+s = 0.07; % increased by a factor of 10 from 0.007 to increase risk correction
 
 model;
 k=exp(z)*k(-1)^(alpha)-c+(1-delta)*k(-1);
 z=rho*z(-1)+s*e;
+U=ln(c);
 end;
 
 steady_state_model;
 z = 0;
+U=ln(c);
 end;
 
 planner_objective ln(c);