% --------------------------------------------------------------
%                         Preamble
% --------------------------------------------------------------

\documentclass[11pt]{article}

\usepackage{amsmath,amsthm,amssymb}
\usepackage{cancel}
\usepackage[margin=1in]{geometry}
\usepackage{graphicx}
\usepackage{hyperref}
% \usepackage{listings}
\usepackage{xcolor}
\usepackage{enumitem}
\usepackage[procnames]{listings}

\lstset{language=Python,
        basicstyle=\ttfamily\small,
        keywordstyle=\color{keywords},
        commentstyle=\color{green},
        stringstyle=\color{red},
        showstringspaces=false,
        identifierstyle=\color{black},
        procnamekeys={def,class}}

\newenvironment{theorem}[2][Theorem]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{lemma}[2][Lemma]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{exercise}[2][Exercise]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{problem}[2][Problem]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{question}[2][Question]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{corollary}[2][Corollary]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}

\newcommand{\E}{\mathbb{E}}
\newcommand{\argmin}{\operatornamewithlimits{argmin}}
\newcommand{\argmax}{\operatornamewithlimits{argmax}}

\begin{document}

% --------------------------------------------------------------
%                         Start here
% --------------------------------------------------------------

{\noindent Instructor: \textit{Professor Griffy}\\ Due: \textit{Mar. 11th, 2026}\\ AECO 701}

\begin{center} \Large Problem Set 4 \end{center} \vspace{1em}

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

\vspace{.5em} \noindent \textbf{Income Fluctuations with CARA Utility}

\begin{problem}{1} \textbf{Solving for Consumption.} You're asked to study an optimal savings plan when households face fluctuating income. The exponential (or CARA) utility function is tractable and it allows for closed-form solutions using a guess-and-verify method. Consider an agent with the following utility maximization problem:
\begin{equation} \label{eq: 1}
\E \sum\limits_{t=1}^{\infty} \left(\frac{1}{1+\delta} \right)^t u(c_t)
\end{equation}
\begin{center} subject to \end{center}
\begin{equation} \label{eq: 2}
y_t = \phi_0 + \phi_1 y_{t-1} + \varepsilon_t, \: \: \: \varepsilon_t \sim N(0,\sigma)
\end{equation}
\begin{equation} \label{eq: 3}
\delta > 0, \qquad 0 < \phi < 1,
\end{equation}
where utility takes the CARA form $u(c) = -\frac{1}{\theta}e^{-\theta c}$.
\end{problem}
\begin{enumerate}
\item The recursive formulation of this problem is given by
\begin{equation} \label{eq: 4}
V(A,y) = \max\limits_c \left\{ u(c) + \beta \E[V(A', y')]\right\}
\end{equation}
\begin{equation} \label{eq: 5}
\text{s.t.} \qquad A' = (1+r)A + y - c. \\
\end{equation}
Take the first-order condition in consumption and solve for the within period relationship between assets and consumption. \\\\
\noindent

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

\item Guess that the value function takes the form
\begin{equation} \label{eq: 6}
V(A,y) = - \frac{1}{\theta r}e^{-\theta r (A + ay + \overline{b})}.
\end{equation}
Using the relationship you derived in part (a), show that the candidate optimal consumption rule takes the form
\begin{equation} \label{eq: 7}
c^* = r(A + ay + a_0),
\end{equation}
where we define
\begin{equation} \label{eq: 8}
a_0 = \overline{b} + \frac{1}{\theta r}ln(1+r).
\end{equation}
Note that $a = \frac{1}{1+r- \phi_1}$, which means that $ay$ is the present value of human wealth given by
\begin{equation} \label{eq: 9}
h_t = \sum\limits_{t=0}^{\infty} \left(\frac{1}{1+r} \right)^t y_t = \frac{y_t}{1+r-\phi_1}.
\end{equation} \\


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

\item Using our guess of the value function, we can rewrite the Bellman Equation as
\begin{equation} \label{eq: 10}
V(A,y) = \frac{r}{1+r}V(A,y) - \left(\frac{1}{1+\delta} \right)\frac{1}{\theta r} \E\left[exp\left(-\theta r \left(A' + ay' + \overline{b}\right)\right)\right].
\end{equation}
Plug in the equation for the evolution of assets for $A'$ and the AR(1) process that determines income for $y'$, as well as your guess for $V$, and show that consumption is equal to
\begin{equation} \label{eq: 11}
c = r \left\{A + \frac{1-a+ a\phi_1}{r}y + \frac{a\phi_0}{r}+ \frac{1}{\theta r^2} \left[ln\left(\frac{1+\delta}{1+r} \right) - \ln\left(\E\left[exp\left(-\theta r a \varepsilon '\right)\right]\right)\right] \right\}.
\end{equation}
(Two hints: 1. Derivatives are not required!; 2. Remember that $exp(a+b) = exp(a) \times exp(b)$) \\\\
% --------------------------------------------------------------

\item Using the method of undetermined coefficients (aka guess and verify - set your two solutions for consumption equal), solve for $\overline{b}$ using your solution obtained in part (b). \\\\
 %--------------------------------------------------------------

\item Show that this solution for consumption can be written as
\begin{equation} \label{eq: 12}
c^* = r(A + h - \Gamma (r)),
\end{equation}
where $h = a(y+ \frac{\phi_0}{r})$ is human wealth and $\Gamma (r) = \frac{1}{\theta r^2}[ln(\E[exp(-\theta r a \varepsilon')]) - ln(\frac{1+\delta}{1+r})]$ is the difference between precautionary savings and impatience caused by a distaste for lower consumption. \\\\
\end{enumerate}
% --------------------------------------------------------------
% --------------------------------------------------------------
%                         End here
% --------------------------------------------------------------
\end{document}
