Verified Commit 89b6ecf9 authored by Johannes Pfeifer 's avatar Johannes Pfeifer Committed by Stéphane Adjemian
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Manual: add warnings regarding the use of auxiliary variables

(@stepan removed trailing spaces)
parent cc04ef6e
......@@ -3327,6 +3327,34 @@ blocks in the model structure and use this information to aid the
solution process. These solution algorithms can provide a significant
speed-up on large models.
.. warning:: Be careful when employing auxiliary variables in the context
of perfect foresight computations. The same model may work for stochastic
simulations, but fail for perfect foresight simulations. The issue arises
when an equation suddenly only contains variables dated ``t+1`` (or ``t-1``
for that matter). In this case, the derivative in the last (first) period
with respect to all variables will be 0, rendering the stacked Jacobian singular.
*Example*
Consider the following specification of an Euler equation with log utility:
::
Lambda = beta*C(-1)/C;
Lambda(+1)*R(+1)= 1;
Clearly, the derivative of the second equation with respect to all endogenous
variables at time ``t`` is zero, causing ``perfect_foresight_solver`` to generally
fail. This is due to the use of the Lagrange multiplier ``Lambda`` as an auxiliary
variable. Instead, employing the identical
::
beta*C/C(+1)*R(+1)= 1;
will work.
.. command:: perfect_foresight_setup ;
perfect_foresight_setup (OPTIONS...);
......@@ -10113,6 +10141,51 @@ with ``discretionary_policy`` or for optimal simple rules with ``osr``
Optimal policy under commitment (Ramsey)
----------------------------------------
Dynare allows to automatically compute optimal policy choices of a Ramsey planner
who takes the specified private sector equilibrium conditions into account and commits
to future policy choices. Doing so requires specifying the private sector equilibrium
conditions in the ``model``-block and a ``planner_objective`` as well as potentially some
``instruments`` to facilitate computations.
.. warning:: Be careful when employing forward-looking auxiliary variables in the context
of timeless perspective Ramsey computations. They may alter the problem the Ramsey
planner will solve for the first period, although they seemingly leave the private
sector equilibrium unaffected. The reason is the planner optimizes with respect to variables
dated ``t`` and takes the value of time 0 variables as given, because they are predetermined.
This set of initially predetermined variables will change with forward-looking definitions.
Thus, users are strongly advised to use model-local variables instead.
*Example*
Consider a perfect foresight example where the Euler equation for the
return to capital is given by
::
1/C=beta*1/C(+1)*(R(+1)+(1-delta))
The job of the Ramsey planner in period ``1`` is to choose :math:`C_1` and :math:`R_1`, taking as given
:math:`C_0`. The above equation may seemingly equivalently be written as
::
1/C=beta*1/C(+1)*(R_cap);
R_cap=R(+1)+(1-delta);
due to perfect foresight. However, this changes the problem of the Ramsey planner in the first period
to choosing :math:`C_1` and :math:`R_1`, taking as given both :math:`C_0` and :math:`R^{cap}_0`. Thus,
the relevant return to capital in the Euler equation of the first period is not a
choice of the planner anymore due to the forward-looking nature of the definition in the second line!
A correct specification would be to instead define ``R_cap`` as a model-local variable:
::
1/C=beta*1/C(+1)*(R_cap);
#R_cap=R(+1)+(1-delta);
.. command:: ramsey_model (OPTIONS...);
|br| This command computes the First Order Conditions for maximizing
......
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