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\begin{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%	Homework title, name, and date
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\noindent Instructor: \textit{Professor Griffy}\\ Due: \textit{May 8th, 2025}\\ AECO 701}

\begin{center} \Large Problem Set 6: The Heterogeneous Agent Model \end{center} \vspace{1em}

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

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

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

\begin{problem}{1} \textbf{Huggett Model.} On the campus cluster, you will find code to solve the Aiyagari model with a labor-leisure choice. Please start from that code (email me if you cannot access the cluster). Please note that this code is highly sensitive to initial guesses!

The Huggett (1993) Model is given by

  \begin{equation}
    V(a,\epsilon;\psi) = u(c) + \beta E[V(a',\epsilon';\psi')]
  \end{equation}

subject to

\begin{align}
  c + a' &\leq (1 + r(\psi))a + \epsilon\\
  \epsilon&\sim \text{\it Markov}, \Pi(\epsilon'|\epsilon)\\
  \psi' &= \Psi(\psi)
\end{align}

Assume the following calibration:

\begin{center}
 \begin{tabular}{||c c||}
 \hline
 Parameter & Value   \\ [0.5ex]
 \hline\hline
 $u(c)$ & $\frac{c^{1 - \sigma}}{1 - \sigma}$  \\
 \hline
 $\beta$ & 0.993  \\
 \hline
 $\sigma$ & 1.5  \\
 \hline
   a' &$\geq$ -2 \\
 \hline
\text{a grid} & [-2, 12]\\
 \hline
\text{a nodes} & 100\\
 \hline\hline
\end{tabular}
\end{center}

\begin{equation} \pi_{t} =
\begin{bmatrix}
  0.925 & 0.075 \\ 0.5 & 0.5
\end{bmatrix}
\end{equation}

\begin{equation} \epsilon =
\begin{bmatrix}
  1.0 \\ 0.1
\end{bmatrix}
\end{equation}

Note that market clearing is given by

\begin{equation}
  \int_{a\times\epsilon}a'd\psi = 0
\end{equation}

\end{problem}

  \begin{enumerate}

  \item Solve the model. Plot the decision rules for savings across the a grid for an agent in employment state 1 and employment state 2.

  \item Plot the stationary distribution of wealth.

  \end{enumerate}
% \newpage
% \begin{problem}{2} \textbf{Aiyagari Model.} Now we will extend the problem to include a firm, as in Aiyagari (1994). In this economy, the household's problem is given by

%   \begin{equation}
%     V(k,l;\psi) = u(c) + \beta E[V(k',l';\psi')]
%   \end{equation}

% subject to

% \begin{align}
%   c + k' &\leq (1 + r(K,L) - \delta)k + w(K,L)l\\
%   k' &\geq 0 \\
%   ln(l') &= \rho ln(l) + \sigma(1 - \rho^{2})^{\frac{1}{2}}\epsilon',\;\epsilon\sim N(0,1)\\
%   \psi' &= \Psi(\psi)
%   % K &= \int_{k\times l}k'd\psi\\
%   % L &= \int_{k\times l}l d\psi\\
% \end{align}

% and the firm's problem is given by

% \begin{align}
%   \Pi = \max_{K,L} F(K,L) - r(K,L)K - w(K,L)L
% \end{align}

% % \end{problem}

% Assume the following calibration:

% \begin{center}
%  \begin{tabular}{||c c||}
%  \hline
%  Parameter & Value   \\ [0.5ex]
%  \hline\hline
%  $u(c)$ & $\frac{c^{1 - \mu}}{1 - \mu}$  \\
%  \hline
%  $F(K, L)$ & $K^{\alpha}L^{1 - \alpha}$  \\
%  \hline
%  $\beta$ & 0.96  \\
%  \hline
%  $\delta$ & 0.08 \\
%  \hline
%  $\alpha$ & 0.36 \\
%  \hline
% \text{k Grid} & [0, 18]\\
%  \hline
% \text{k nodes} & 100\\
%  \hline\hline
% \end{tabular}
% \end{center}

% % \begin{equation} \pi_{t} =
% % \begin{bmatrix}
% %   0.925 & 0.075 \\ 0.5 & 0.5
% % \end{bmatrix}
% % \end{equation}

% % \begin{equation} \epsilon =
% % \begin{bmatrix}
% %   1.0 \\ 0.1
% % \end{bmatrix}
% % \end{equation}

% Note that market clearing is given by

% \begin{align}
%   \int_{k\times l}ld\psi = L\\
%   \int_{k\times l}k'd\psi = K
% \end{align}

% and that $\mu$, $\sigma$, and $\rho$ will be given in the following parts.

%   \end{problem}

%   \begin{enumerate}

%   \item Use Tauchen's method to approximate the AR(1) process for $\rho = 0.6$ and $\sigma = 0.2$. Write out the resulting grid and transition matrix.

%   \item Find the ergodic distribution of employment using this grid and transition matrix. Call this $L$. Note that you will need to calculate this for each $\rho$, $\sigma$ pair.

%   \item Solve the model. Plot the decision rules for savings across the k grid for $\rho = 0.6$, $\sigma = 0.2$, and $\mu = 3$.

%   \item Plot the stationary distribution of wealth for $\rho = 0.6$, $\sigma = 0.2$, and $\mu = 3$.

%   \item Now pick $\sigma = 0.4$ and plot the net return to capital for $\rho = \{0,0.3,0.6,0.9\}$ and $\mu = \{1,3,5\}$.

%   \end{enumerate}

\end{document}
