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\hypersetup{pdftitle={Problem Set on Mathematics for Macroeconomics}}
\begin{document}

\title{Problem Set on Mathematics for Macroeconomics}
\author{Pascal Michaillat}
\date{}
\paperurl{https://pascalmichaillat.org/x/}
\begin{titlepage}
\maketitle
\end{titlepage}

\section*{Problem 1}

Let $\a \in (0,1)$, $\d \in (0,1)$, $\r \in (0,1)$, and $\s>0$. Impose that $\r+\d<1$. Given $k(0)$, we want to find the function $c(t) $ to maximize 
\begin{equation*}
\int_{0}^{\infty}e^{-\r t} \frac{c(t)^{1-\s}-1}{1-\s} dt,
\end{equation*}
subject to the law of motion
\begin{equation*}
\dot{k}(t) =k(t)^{\a}-c(t)-\d k(t).
\end{equation*}

\begin{enumerate}
\item Which variable do you choose as a state variable? Which variable do you choose as a control variable? Write down the current-value Hamiltonian and derive the optimality conditions.
\item The Euler equation is the first-order differential equation that characterizes the optimal function $c(t)$. Determine the Euler equation.
\item Suppose $\a =1$ and $\s =1$. Show that the system describing the optimal functions $\{k(t),c(t)\}$ reduces to a linear, homogeneous system of first-order differential equations. Show that the system is unstable by computing the eigenvalues. 
\item Suppose $\a <1$ and $\s >0$. Show that the system describing the optimal functions $\{k(t),c(t)\}$ reduces to a nonlinear system of first-order differential equations. Use a phase-diagram to show that the steady state of the system is a saddle point. Explain how you draw the phase diagram.
\end{enumerate}

\section*{Problem 2}

Let $\b \in (0,1)$ and $r>0$. Given $k_{0}>0$, we want to find sequences $\bc{c_{t},k_{t+1}}_{t=0}^{\infty}$ to maximize 
\begin{equation*}
\sum_{t=0}^{\infty}\b^{t} \ln{c_{t}}, 
\end{equation*}
subject to the constraints
\begin{equation*}
k_{t+1}=(1+r) k_{t}-c_{t}
\end{equation*}
for all $t\geq 0$.

\paragraph{Lagrangian} We first solve the maximization problem using the Lagrangian method.
\begin{enumerate}
\item Write down the Lagrangian of the problem.
\item Derive the first-order condition(s) of the maximization problem.
\item Derive the Euler equation.
\end{enumerate}

\paragraph{Dynamic Programming} Next we solve the maximization problem using the dynamic programming method.
\begin{enumerate}\setcounter{enumi}{3}
\item Which variable do you choose as a state variable? Which variable do you choose as a control variable? Write down the Bellman equation.
\item Derive the first-order condition associated with the Bellman equation.
\item Derive the Benveniste-Scheinkman equation.
\item Derive the Euler equation. Compare it with the Euler equation obtained with the Lagrangian method and discuss.
\item Suppose that the policy function takes the form $h(k)=A (1+r) k$ where $A\in (0,1) $. Derive $A$.
\item Suppose that the value function takes the form $V(k)=B+D \ln{k},$ where $B$ and $D$ are constants. Using the expression for the policy function that you derived in the previous question, derive $B$ and $D$.
\end{enumerate}
\end{document}