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{\noindent Instructor: \textit{Professor Griffy}\\ Due: \textit{Mar., 2nd 2023}\\ AECO 701}

\begin{center} \Large Problem Set 3 \end{center} \vspace{1em}

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\noindent \textbf{Problem 1: Linear difference equation economy (Sargent and Ljundqvist exercise 2.13 slightly modified)}
A household has preferences over consumption processes $\{c_{t}\}_{t = 0}^{\infty}$ that are ordered by

\begin{align}
  -0.5\sum_{t=0}^{\infty}\beta^{t}[(c_{t} - 30)^{2} + 0.000001b_{t}^{2}]
\end{align}

where $\beta = 0.95$. The household chooses a consumption, borrowing plan to maximize (1) subject to the sequence of budget constraints

\begin{align}
  c_{t} + b_{t} = \beta b_{t + 1} + y_{t}
\end{align}

for $t\geq 0$, where $b_{0}$ is an initial condition, $\beta^{-1}$ is the one-period gross risk-free interest rate, $b_{t}$ is the household's one-period debt that is due in period $t$, and $y_{t}$ is its labor income, which obeys the second-order autorefressive process

\begin{align}
  (1 - \rho_{1}L - \rho_{2}L^{2}) y_{t + 1} = (1 - \rho_{1} - \rho_{2})5 + 0.05 w_{t + 1}
\end{align}

where $\rho_{1} = 1.3$ and $\rho_{2} = -0.4$.

\begin{enumerate}[label=\textbf{\alph*})]
\item Define the state of the household at $t$ as $x_{t} = [1\; b_{t}\; y_{t}\; y_{t - 1}]'$ and the control as $u_{t} = (c_{t} - 30)$. Then express the transition law facing the household in the form (2.4.22 in your book).


\item Use a computer to compute the eigenvalues of A, the transition matrix.




\end{enumerate}

\noindent \textbf{Problem 2: Contraction Mapping Theorem.} Blackwell's Sufficiency Conditions for a contraction are given by the following:

T is monotone if for $f(x) \leq g(x)$ $\forall x\in X$, then
  \begin{equation}
    \label{eq:monotonicity}
    Tf(x)\leq Tg(x)\hspace{3mm} \forall x\in X
  \end{equation}

T discounts if for some $\beta\in(0, 1)$ and any $a\in\mathcal{R}_{+}$
  \begin{equation}
    \label{eq:discounting}
    T(f + a)(x)\leq Tf(x) + \beta a\hspace{3 mm}\forall x\in X
  \end{equation}

Now, consider a neoclassical growth model where a social planner maximizes

    \begin{align}
      V(k,z) &= \max_{k'} u(c) + \beta E[V(k',z)]\\
      \text{s.t. } c + k' &= z k^{\alpha}\\
      z &\sim iid
    \end{align}


\begin{enumerate}[label=\textbf{\alph*})]

  \item Prove that the Bellman equation specified above satisfies the monotone requirement.

  \item Prove that the Bellman equation specified above satisfies the discounting requirement.

  \end{enumerate}



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