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Sébastien Villemot
perfect-foresight-slides
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22ff11c9
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22ff11c9
authored
1 year ago
by
Sébastien Villemot
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Incorporate some improvements by Johannes Pfeifer
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873b1be9
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deterministic.tex
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...
...
@@ -206,6 +206,9 @@ rplot c;
\item
Symmetric process for variables with more than one lag
\item
With future uncertainty, the transformation is more elaborate
(but still possible) on variables with leads
\item
\texttt
{
write
\_
latex
\_
original
\_
model
}
and
\texttt
{
write
\_
latex
\_
dynamic
\_
model
}
show model before and after
substitution
\end{itemize}
\end{frame}
...
...
@@ -226,12 +229,37 @@ f(\bar y, \bar y, \bar y, \bar u) = 0
\item
Even for a given set of exogenous and parameter values, some (nonlinear) models have
several steady states
\item
The steady state is computed by Dynare with the
\texttt
{
steady
}
command
\item
That command internally uses a nonlinear solver
\item
Two possibilities for the steady state in Dynare:
\begin{itemize}
\item
If analytical steady state is known, can be given in
\texttt
{
steady
\_
state
\_
model
}
block
\item
Otherwise, numerical procedure based on a nonlinear solver, with the
\texttt
{
steady
}
command
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle
{
Towards a solution
}
\begin{itemize}
\item
In each period, agent needs to choose control variables (
\textit
{
e.g.
}
consumption) today, taking into account:
\begin{itemize}
\item
the known time path of the exogenous variables
\item
the current endogenous states that cannot be altered anymore
(
\textit
{
e.g.
}
capital)
\item
the consequences of today’s actions for future decisions
\end{itemize}
\item
Assume finite horizon problem that terminates in period
$
T
$
with
\begin{itemize}
\item
initial states in
$
y
_
0
$
given
\item
terminal choice
$
y
_{
T
+
1
}$
known
\end{itemize}
$
\Rightarrow
$
problem boils down to simultaneous equations system with
$
n
_
y
\times
T
$
equations in
$
n
_
y
\times
T
$
unknowns
(where
$
n
_
y
$
is the number of endogenous)
\end{itemize}
\end{frame}
\begin{frame}
\frametitle
{
A two-boundary value problem
}
...
...
@@ -440,7 +468,7 @@ where
\begin{frame}
\frametitle
{
Laffargue-Boucekkine-Juillard algorithm (3/5)
}
Normal iteration:
Normal iteration
of the triangularization
:
\begin{small}
\begin{equation*}
\begin{pmatrix}
...
...
@@ -664,7 +692,7 @@ $$d_T = (B_T-A_TD_{T-1})^{-1}(f(y_{T+1},y_T,y_{T-1},u_T)+A_Td_{T-1})$$
A
_{
t
}
=
{
A
^{
\star
}}
e
^{
a
_{
t
}}
\\
a
_{
t
}
=
\rho\,
a
_{
t-1
}
+
\varepsilon
_
t
\end{gather*}
where
$
\varepsilon
_
t
$
is an exogenous shock
.
where
$
\varepsilon
_
t
$
is an exogenous shock
and
$
k
_
0
$
given.
\end{frame}
\begin{frame}
...
...
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