ps2.tex

\documentclass[letterpaper, 11pt, leqno]{article}
\usepackage{paper}
\usepackage{math}
\usepackage{exam}
\hypersetup{pdftitle={Problem Set on Optimal Control}}
\begin{document}

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

\section*{Problem 1}

Consider the following optimal growth problem: Given initial capital $k_{0}>0$, choose a consumption path $\bc{c_{t}}_{t\ge 0}$ to maximize utility
\begin{equation*}
\int_{0}^{\infty}e^{-\r t} \ln{c_{t}}\,dt 
\end{equation*}
subject to the law of motion of capital
\begin{equation*}
\dot{k}_{t} =f(k_{t}) -c_{t}-\d k_{t}.
\end{equation*}
The discount rate $\r>0$, and the production function $f$ satisfies
\begin{equation*}
f(k) =a k^{\a},
\end{equation*}
 where $\a \in (0,1)$ and $a>0$.
\begin{enumerate}
\item Write down the present-value Hamiltonian.
\item Show that the Euler equation is
\begin{equation*}
\frac{\dot{c}_{t}}{c_{t}} =\a a k_{t}^{\a -1} - (\d +\r).
\end{equation*}
\item Solve for the steady state of the system.
\end{enumerate}

\section*{Problem 2}

Consider the following investment problem: Given initial capital $k_{0}$, choose the investment path $\bc{i_{t}} _{t \ge 0}$ to maximize profits
\begin{equation*}
\int_{0}^{\infty}e^{-r t}\bs{f(k_{t})-i_{t}-\frac{\c}{2}\cdot\frac{i_{t}^{2}}{k_{t}}}\,dt 
\end{equation*}
subject to the law of motion of capital (we assume no capital depreciation)
\begin{equation*}
\dot{k}_{t} =i_{t}.
\end{equation*}
The interest rate $r>0$, the capital adjustment cost $\c>0$, and the production function $f$ satisfies $f'>0$ and $f''<0$.
\begin{enumerate}
\item Write down the current-value Hamiltonian. 
\item Use the optimality conditions for the current-value Hamiltonian to derive the following differential equations:
\begin{align*}
\dot{k}_{t} &=\frac{q_{t}-1}{\c} \cdot k_{t} \\
\dot{q}_{t} &=r q_{t}-f'(k_{t}) -\frac{1}{2 \c} \cdot (q_{t}-1)^{2}
\end{align*}
\item Solve for the steady state.
\end{enumerate}

\end{document}