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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%	Homework title, name, and date
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{\noindent Instructor: \textit{Professor Griffy}\\ Due: \textit{Apr. 13th, 2023}\\ AECO 701}

\begin{center} \Large Problem Set 5: The RBC Model \end{center} \vspace{1em}

% --------------------------------------------------------------
% --------------------------------------------------------------

% \vspace{.5em} \noindent \textbf{The RBC Model}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{problem}{1} \textbf{RBC Model with Labor-Leisure Choice.}  The RBC model with an intensive margin is given by the following:

  \begin{equation}
    \max_{C_{t}, K_{t + 1}}\sum_{t = 0}^{\infty}\beta^{t}U(C_{t}, 1 - L_{t})
  \end{equation}

subject to

\begin{eqnarray}
  C_{t} + K_{t + 1} &=& z_{t}F(K_{t}, L_{t}) + (1 - \delta)K_{t}\\
  z_{t}&\sim& \text{\it Markov}
\end{eqnarray}

\end{problem}

  \begin{enumerate}

  \item Write the recursive formulation of this problem.

  \item Assume the following calibration:

\begin{center}
 \begin{tabular}{||c c||}
 \hline
 Parameter & Value   \\ [0.5ex]
 \hline\hline
 $U(C, 1 - L)$ & $ln(C) + \theta ln(1 - L)$  \\
 \hline
 $F(K, L)$ & $K^{\alpha}L^{1 - \alpha}$  \\
 \hline
 $\beta$ & 0.99  \\
 \hline
 $\rho$ & 0.9  \\
 \hline
 $\delta$ & 0.04 \\
 \hline
 $\alpha$ & 0.3 \\
 \hline
 $\theta$ & 2  \\
 \hline
\text{K Grid} & [4, 9]\\
 \hline\hline
\end{tabular}
\end{center}

\begin{equation} \pi_{t} =
\begin{bmatrix}
  0.9 & 0.1 \\ 0.7 & 0.3
\end{bmatrix}
\end{equation}

\begin{equation} z_{t} =
\begin{bmatrix}
  12 \\ 8
\end{bmatrix}
\end{equation}


Solve the model using grid search. Note that you will need to solve the model each time for each possible state.

\item[c] Start the model in the ``high state,'' and run the model forward for 100 periods. In period 101, a highly persistent recession starts (10 periods); simulate the response of labor to this recession. Discuss the response of hours as a result of this shock.

  \end{enumerate}

\begin{problem}{2} \textbf{Solving for Consumption.}  Now, assume the following about the economy:


  \begin{equation}
    U(C, 1 - L) = ln(C) + \theta(1 - L)
  \end{equation}

  \end{problem}

\begin{enumerate}

  \item Write the recursive formulation of this problem.

  \item Using the calibration above, solve this version of the model. Use a K Grid of [7, 13].

  \item Simulate the model for 100 periods and then use the same shock as above. Discuss the response of hours and compare with the results from the model with an intensive margin.

  \end{enumerate}

\end{document}
