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\begin{document}

\title{Zero lower bound in a New Keynesian model}
\author{Sébastien Villemot}
\date{June 12, 2018}
\maketitle

\begin{enumerate}
\item Open the file \texttt{nk3.mod}. Observe that it describes a minimal New
  Keynesian model with 3 equations (Euler equation, Phillips curve, Taylor rule)
  and three shocks.
\item Run the file. Observe the IRFs. In particular, notice that the zero-lower
  bound is crossed for the IRF related to the productivity shock (recall that
  IRFs are plotted by Dynare in deviation from the steady state for a positive
  one-standard-deviation shock).
\item Modify the file by enforcing the zero-lower bound in the Taylor rule. You
  will need the \texttt{max} operator. What do you observe in the IRF? Why?
\item Transform the file into a perfect foresight simulation. You will need to
  completely rewrite the \texttt{shocks} block, by hard coding a positive
  productivity shock (in period 1) of the size of one standard deviation of the
  stochastic version. Display the interest rate with \texttt{rplot}. What do
  you observe?
\item Now, transform the previous example by adding a \emph{negative}
  (\texttt{i.e.} inflationary) one-standard-deviation productivity shock in
  period 5, in two different ways. In the first version, the shock will be
  anticipated; in the second one, the shock will be unexpected (for the latter,
  you will need to run \texttt{perfect\_foresight\_solver} twice, and
  manipulate matrices by hand). Verify that in the first version the ZLB is
  never hit, and in the second version the ZLB is left at the time of the
  unexpected shock (and is therefore shorter than in question 4).
\item Finally, transform the file into an extended path simulation. You will need
  to revert the \texttt{shocks} block into the form used in the stochastic case.
  Again, use \texttt{rplot} to display the interest rate. Observe that the ZLB is
  enforced on the simulation path.
\end{enumerate}

\end{document}