\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{geometry}
\geometry{top=1in,bottom=1in,left=1in,right=1in}
\setlength{\parindent}{0in}
\setlength{\parskip}{2ex}
\begin{document}
\begin{center}
\begin{tabular*}{6.5in}{l@{\extracolsep{\fill}}r}
\multicolumn{2}{c} {\bfseries Problem Set 7 } \\
\multicolumn{2}{c} {Due Thursday, October 29}
\end{tabular*}
\end{center}
This problem asks you to derive some of the implications of a model of stock prices
and consumption due originally to Robert Lucas, ``Asset Prices in an Exchange
Economy,'' \emph{Econometrica}, November 1978, pp.1429-1445.
Consider an economy populated by a large number of identical consumers, each of
whom has preferences described by the additively-time separable utility function
\begin{equation}
\sum_{t=0}^{\infty} \beta^t u(c_{t}), \tag{1}
\end{equation}
where the discount factor satisfies $0<\beta<1$ and the single-period utility
function $u$ is strictly increasing, strictly concave, and satisfies $\lim_{c
\rightarrow 0} u'(c)=\infty$, this last assumption allowing us to ignore
nonnegativity constraints on consumption in all of the analysis that follows.
Each consumer finances his or her consumption by trading equity shares in the
economy's productive assets: let's call them ``fruit trees.'' Each share in each
tree provides a dividend in the form of $d_{t}$ pieces of ``fruit'' during each
period $t=0,1,2,.
..$, where fruit is the economy's only consumption good. Let
$s_{t}$ denote the number of shares carried by a representative consumer into each
period $t=0,1,2,.
..$, and let $p_{t}$ denote the price of each share in each tree
during each period $t=0,1,2,.
..$.
Then, as sources of funds during each period $t=0,1,2,.
..$, the representative
consumer has his or her dividend payments $d_{t}s_{t}$ and the total value
$p_{t}s_{t}$ of the shares carried into the period.
And as uses of funds during
each period $t=0,1,2,.
..$, the consumer has his or her consumption $c_{t}$ and the