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\hypersetup{pdftitle={Problem Set on Dynamic Programming}}
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\title{Problem Set on Dynamic Programming}
\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 consumption $\bc{c_{t}} _{t =0}^{\infty}$ to maximize utility
\begin{equation*}
\sum_{t=0}^{\infty}\b^{t} \ln{c_{t}}
\end{equation*}
subject to the resource constraint
\begin{equation*}
k_{t+1}=a k_{t}^{\a}-c_{t}.
\end{equation*}
The parameters satisfy $0<\b<1$, $a>0$, $0<\a <1$.

\begin{enumerate}
\item Derive the optimal law of motion of consumption $c_{t}$ using a Lagrangian.
\item Identify the state variable and the control variable. 
\item Write down the Bellman equation.
\item Derive the following Euler equation: 
\begin{equation*}
c_{t+1}=\b  \a  a k_{t+1}^{\a -1} c_{t}.
\end{equation*}

\item Derive the first two value functions, $V_{1}(k)$ and  $V_{2}(k)$, obtained by iteration on the Bellman equation starting with the value function $V_{0}(k) \equiv 0$. 
\item The process of determining the value function by iterations using the Bellman equation is commonly used to solve dynamic programs numerically. The algorithm is called \textit{value function iteration}. For this optimal growth problem, one can show using value function iteration that the value function is
\[V(k) =\k +\frac{\ln{k^{\a}}}{1-\a \b},\]
where $\k$ is a constant. Using the Bellman equation, determine the policy function $k'(k)$ associated with this value function.
\item In light of these results, for which reasons would you prefer to use the dynamic-programming approach instead of the Lagrangian approach to solve the optimal growth problem? And for which reasons would you prefer to use the Lagrangian approach instead of the dynamic-programming approach?
\end{enumerate}

\section*{Problem 2}

Consider the problem of choosing consumption $\bc{c_{t}}_{t=0}^{\infty}$ to maximize expected utility
\begin{equation*}
\E[0]{\sum_{t=0}^{\infty}\b^{t} u(c_{t})}
\end{equation*}
subject to the budget constraint
\begin{equation*}
c_{t}+p_{t} s_{t+1}=(d_{t}+p_{t}) s_{t}.
\end{equation*}
$d_{t}$ is the dividend paid out for one share of the asset, $p_{t}$ is the price of one share of the asset, and $s_{t}$ is the number of shares of the asset held at the beginning of period $t$. In equilibrium, the price $p_{t}$ of one share is solely a function of dividends $d_{t}$. Dividends can only take two values $d_{l}$ and $d_{h}$, with $0<d_{l}<d_{h}$. Dividends follow a Markov process with transition probabilities 
\begin{equation*}
\P{d_{t+1}=d_{l}\mid d_{t}=d_{l}} =\P{d_{t+1}=d_{h}\mid d_{t}=d_{h}} =\r
\end{equation*}
with $0.5<\r 1$.

\begin{enumerate}
\item Identify state and control variables. 
\item Write down the Bellman equation.
\item Derive the following Euler equation: 
\begin{equation*}
p_{t} u'(c_{t}) =\b  \E{(d_{t+1}+p_{t+1})  u'(c_{t+1}) \mid d_{t}} .
\end{equation*}
\item Suppose that $u(c) =c$. Show that the asset price is higher when the current dividend is high.
\end{enumerate}

\section*{Problem 3}

Consider the following optimal growth problem: Given initial capital $k_{0}>0$, choose consumption and labor $\bc{c_{t},l_{t}}_{t=0}^{\infty}$ to maximize utility
\begin{equation*}
\sum_{t=0}^{\infty}\b^{t} u(c_{t},l_{t})
\end{equation*}
subject to the law of motion of capital
\begin{align*}
k_{t+1}&=a_{t} f(k_{t},l_{t}) -c_{t}.
\end{align*}
In addition, we impose $0\le l_{t}\le 1$. The discount factor $\b \in (0,1)$. The function $f$ is increasing and concave in both arguments. The function $u$ is increasing and concave in $c$, decreasing and convex in $l$. 

\paragraph{Deterministic case} First, suppose $a_{t}=1$ for all $t$.

\begin{enumerate}
\item What are the state and control variables?
\item  Write down the Bellman equation.
\item Derive the following optimality conditions: 
\begin{align*}
\pd{u(c_{t},l_{t})}{c_{t}} &=\b  \pd{u(c_{t+1},l_{t+1})}{c_{t+1}} \cdot \pd{f(k_{t+1},l_{t+1})}{k_{t+1}}\\
\pd{u(c_{t},l_{t})}{c_{t}} \cdot \pd{f(k_{t},l_{t})}{l_{t}} &=-\pd{u(c_{t},l_{t})}{l_{t}}.
\end{align*}
\item Suppose that the production function $f\bp{k,l} =k^{\a}\cdot l^{1-\a}$. Determine the ratios $c/k$ and $l/k$ in steady state.
\end{enumerate}

\paragraph{Stochastic case} Now, suppose $a_{t}$ is a stochastic process that takes values $a_{1}$ and $a_{2}$ with the following transition probabilities: $\P{a_{t+1}=a_{1}\mid a_{t}=a_{1}} =\P{a_{t+1}=a_{2}\mid a_{t}=a_{2}} =\r.$
\begin{enumerate}\setcounter{enumi}{4}
\item Write down the Bellman equation.
\item Derive the optimality conditions.
\end{enumerate}


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