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%	PACKAGES AND THEMES
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%	TITLE PAGE
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\title[]{Macro II} % The short title appears at the bottom of every slide, the full
% title is only on the title page

\author{Professor Griffy} % Your name
\institute[University at Albany, SUNY] % Your institution as it will appear on the bottom of
% every slide, may be shorthand to save space
{
UAlbany  \ % Your institution for the title page
}
\date{Spring 2024} % Date, can be changed to a custom date

\begin{document}

\begin{frame}
  \titlepage % Print the title page as the first slide
\end{frame}




% ----------------------------------------------------------------------------------------
%	PRESENTATION SLIDES
% ----------------------------------------------------------------------------------------

% ------------------------------------------------
\section{Introduction} % Sections can be created in order to organize your presentation into discrete blocks, all sections and subsections are automatically printed in the table of contents as an overview of the talk
% ------------------------------------------------

\begin{frame}
  \frametitle{Introduction}
  \begin{itemize}
  \item Today: Asset pricing
  \item The ``Lucas Tree Model.''
  \item Homework due Thursday.
  \item Test in two weeks-ish
  \item New homework sometime soon...
  \end{itemize}
\end{frame}

% ------------------------------------------------
\section{Rational Expectations (Competitive) Equilibrium}
% ------------------------------------------------


\begin{frame}\frametitle{Rational Expectations Competitive Equilibrium}
Definition: Given the set of exogenous stochastic processes $\left\{
x_{t}\right\} $, and initial conditions $a_{0}$, a rational expectations
equilibrium is a set of stochastic processes for prices $\left\{
p_{t}\right\} $ and quantities $\left\{ q_{t}\right\} $ such that:

\begin{itemize}
\item Given $\left\{ p_{t}\right\} $, $\left\{ q_{t}\right\} $ is consistent
with optimal behavior on the part of consumers, producers and (if relevant)
the government.

\item Given $\left\{ p_{t}\right\} $, $\left\{ q_{t}\right\} $ satisfies the
government's budget constraints and borrowing restrictions.

\item $\left\{ p_{t}\right\} $ satisfies any market-clearing
conditions.
\end{itemize}
\end{frame}

\section{Lucas \textquotedblleft Tree\textquotedblright\ Model}

\begin{frame}\frametitle{Lucas Tree Overview}
\begin{itemize}
\item We are given equilibrium quantities and equilibrium demand
functions---back out equilibrium prices.
\item We know the function D(p) and the quantity $q_{0}$: now find $p_{0}$.
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Compare with Literature on Consumption}
\begin{itemize}

\item Consumption: Take rates of return as given, solve for consumption.

\item Asset Pricing: Take consumption as given, solve for rates of return.

\end{itemize}
\end{frame}

\begin{frame}\frametitle{Model Structure}

\begin{itemize}
\item Preferences: $n$ identical consumers, maximizing
\begin{align*}
& E_{0}\left( \sum\nolimits_{t=0}^{\infty }\beta ^{t}u\left( c_{t}\right)
\right) , \\
& \beta \in \left( 0,1\right) \text{,\quad }u^{\prime }\left( \cdot \right)
>0\text{,\quad }u^{\prime \prime }\left( \cdot \right) \leq 0.
\end{align*}

\item Endowment: one durable \textquotedblleft tree\textquotedblright\ per
individual. Each period, the tree yields some \textquotedblleft
fruit\textquotedblright\ ($d_{t}\equiv $ dividends).

\item Technology: Fruit cannot be stored. Dividends are exogenous and follow
a time-invariant Markov process
\begin{equation*}
\Pr \left( \left. d_{t+1}\leq y\right\vert d_{t}=x\right) =F\left(
y,x\right) ,\forall t,
\end{equation*}%
with density $f\left( y,x\right) $.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Solution strategy}

\begin{itemize}
\item Find the competitive equilibrium allocation via the social planner's
problem. Assume equal weights on each person's utility (parallel to equal
endowments).
\item i.e., use welfare theorems.

\item Calculate the FOC for individuals with the opportunity to buy/sell
specific assets.

\item Evaluate FOC at the competitive equilibrium allocation.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Step 1: Social planner's problem}

\begin{itemize}
\item Use a representative agent

\item Social planner solves
\begin{align*}
\underset{\left\{ c_{t}\right\} _{t=0}^{\infty }}{\max }& E_{0}\left(
\sum\nolimits_{t=0}^{\infty }\beta ^{t}u\left( c_{t}\right) \right) \\
& s.t.\quad c_{t}\leq d_{t}.
\end{align*}

\item Solution: $c_{t}=d_{t}$, $\forall t$ (\textit{non-storable good!}%
).
\item What does this mean?
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Definitions}
% \begin{itemize}

\begin{eqnarray*}
c_{t} &=&\text{consumption}, \\
p_{t} &=&\text{price of a tree}=\text{price of stock,} \\
x_{t} &=&\text{total resources} \\
s_{t+1} &=&\text{number of trees/shares of stock}, \\
R_{t} &=&\text{gross return on one-period risk-free bond,} \\
R_{t}^{-1} &=&\text{price of a one-period, risk-free \underline{\text{%
discount bond}},} \\
b_{t+1} &=&\text{risk-free discount bonds.}
\end{eqnarray*}%

% \end{itemize}
\end{frame}

\begin{frame}
\frametitle{Step 2: Representative consumer's problem}

\begin{align*}
\underset{\left\{ c_{t},b_{t+1},s_{t+1}\right\} _{t=0}^{\infty }}{\max }&
E\left( \left. \sum\nolimits_{t=0}^{\infty }\beta ^{t}u\left( c_{t}\right)
\right\vert I_{0}\right) \\
s.t.\quad & c_{t}+p_{t}s_{t+1}+R_{t}^{-1}b_{t+1}=x_{t}, \\
& x_{t}=\left( p_{t}+d_{t}\right) s_{t}+b_{t}, \\
& \underset{J\rightarrow \infty }{\lim }\beta ^{J}E_{t}\left( u^{\prime
}\left( c_{t+J}\right) p_{t+J}s_{t+J+1}\right) =0 \\
& \underset{J\rightarrow \infty }{\lim }\beta ^{J}E_{t}\left( u^{\prime
}\left( c_{t+J}\right) b_{t+J+1}\right) =0, \\
& s_{0},\ b_{0}\text{ given}
\end{align*}%
\begin{itemize}
\item where $I_{0}$ is the information set at time 0.
\end{itemize}
\end{frame}


\begin{frame}\frametitle{Consumer's problem}
\begin{itemize}

\item Consumer $i$ picks $c_{t}^{i}$, $b_{t+1}^{i}$ and $s_{t+1}^{i}$ on the
basis of
\begin{equation*}
I_{t}^{i}=\left\{
\begin{array}{l}
\left\{ d_{t-m},p_{t-m},R_{t-m}\right\} _{m=0}^{t}, \\
\left\{ s_{t+1-m}^{j},b_{t+1-m}^{j}\right\} _{m=0}^{t+1},\;\forall j\neq i,
\\
\left\{ c_{t-m}^{j},x_{t-m}^{j}\right\} _{m=0}^{t},\;\forall j\neq i, \\
\left\{ s_{t-m}^{i},b_{t-m}^{i},x_{t-m}^{i}\right\} _{m=0}^{t},\left\{
c_{t-m}^{i}\right\} _{m=1}^{t},%
\end{array}%
\right\} ,
\end{equation*}

\item It turns out that $d_{t}$ summarizes the state of the aggregate
economy, with $p_{t}=p\left( d_{t}\right) $ and $R_{t}=R\left( d_{t}\right) $%
.
\item It is the only stochastic variable, and aggregate resources equal $d_{t}$.

\item Because $d_{t}$ is a time-invariant Markov process, the consumer's
problem is time-invariant
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Recursive formulation}
% \begin{itemize}

Bellman's functional equation: \vspace{-0.1in}
\begin{align*}
\hspace{-0.75in}V(x_{t}& ,d_{t})= \\
& \underset{\lambda _{t}\geq 0}{\min }\underset{c_{t}\geq 0,\ s_{t+1},\
b_{t+1}}{\max }u\left( c_{t}\right) +\lambda _{t}\left(
x_{t}-c_{t}-p_{t}s_{t+1}-R_{t}^{-1}b_{t+1}\right) \\
& +\beta \int V\big(\left( p\left( d_{t+1}\right) +d_{t+1}\right)
s_{t+1}+b_{t+1},d_{t+1}\big)dF\left( d_{t+1},d_{t}\right) .
\end{align*}

\begin{itemize}
\item The FOC for an interior solution are: \vspace{-0.05in}
\begin{eqnarray*}
u^{\prime }\left( c_{t}\right) &=&\lambda _{t}, \\
\lambda _{t}p_{t} &=&\beta \int \frac{\partial V\left[ t+1\right] }{\partial
x_{t+1}}\left( p\left( d_{t+1}\right) +d_{t+1}\right) dF\left(
d_{t+1},d_{t}\right) , \\
\lambda _{t}R_{t}^{-1} &=&\beta \int \frac{\partial V\left[ t+1\right] }{%
\partial x_{t+1}}dF\left( d_{t+1},d_{t}\right) .
\end{eqnarray*}%

\end{itemize}
\end{frame}

\begin{frame}\frametitle{Euler Equations}
\begin{itemize}

\item Note that (by Benveniste-Scheinkman)
\begin{equation*}
\frac{\partial V\left[ t\right] }{\partial x_{t}}=\lambda _{t},
\end{equation*}%
\ so that
\begin{eqnarray*}
p_{t} &=&\beta E_{t}\left( \frac{u^{\prime }\left( c_{t+1}\right) }{%
u^{\prime }\left( c_{t}\right) }\left( p_{t+1}+d_{t+1}\right) \right) , \\
R_{t}^{-1} &=&\beta E_{t}\left( \frac{u^{\prime }\left( c_{t+1}\right) }{%
u^{\prime }\left( c_{t}\right) }\right) .
\end{eqnarray*}%

\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Step 3: Equilibrium}

\begin{itemize}
\item Intuition

% \begin{itemize}
\item Agents allocate resources based on beliefs about future prices and
consumption

\item These decision rules determine processes for market clearing prices
and quantities.

\item In a rational expectations equilibrium, the actual processes must be
consistent with the beliefs.
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}

\item \textbf{Sequential} definition: Given the stochastic process $\left\{
d_{t}\right\} _{t=0}^{\infty }$ and the initial endowments $s_{0}=1$ and $%
b_{0}=0$, a rational expectations equilibrium consists of the stochastic
processes $\left\{ c_{t},s_{t+1},b_{t+1},p_{t},R_{t}\right\} _{t=0}^{\infty
} $ such that:

\begin{itemize}
\item Given the process for prices $\left\{ p_{t},R_{t}\right\} $, $\left\{
c_{t},s_{t+1},b_{t+1}\right\} $ solves the consumer's problem.

\item All markets clear: $c_{t}=d_{t}$, $s_{t+1}=1$, and $b_{t+1}=0$, $%
\forall t$.
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}

\item \textbf{Recursive definition}: \ given the random variable $d_{0}$,
the conditional distribution $F\left( d_{t+1},d_{t}\right) $, and the
initial endowments $s_{0}=1$ and $b_{0}=0$, a recursive rational
expectations equilibrium consists of pricing functions $p\left( d\right) $
and $R\left( d\right) $, a value function $V\left( x,d\right) $, and
decision functions $c\left( x,d\right) $, $s\left( x,d\right) $, and $%
b\left( x,d\right) $ such that:

\begin{itemize}
\item Given the pricing functions $p\left( d\right) $ and $R\left( d\right) $%
, the value and policy functions $V\left( x,d\right) $, $c\left( x,d\right) $%
, $s\left( x,d\right) $, and $b\left( x,d\right) $ solve the consumer's
problem.

\item Markets clear: for $x=p\left( d\right) +d$, $c\left( x,d\right) =d$, $%
s\left( x,d\right) =1$, and $b\left( x,d\right) =0$.
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Backing out prices}
\begin{itemize}

\item Find $R\left( d_{t}\right) $: impose the equilibrium allocation, $%
c_{t}=d_{t}$, to get
\begin{align}
R_{t}^{-1}& =\beta E_{t}\left( \frac{u^{\prime }\left( d_{t+1}\right) }{%
u^{\prime }\left( d_{t}\right) }\right)  \notag \\
& =\beta \frac{1}{u^{\prime }\left( d_{t}\right) }E_{t}\left( u^{\prime
}\left( d_{t+1}\right) \right) .  \tag{EE}
\end{align}

\item Find $p\left( d_{t}\right) $: impose the equilibrium allocation, $%
c_{t}=d_{t}$, to get
\begin{equation}
p_{t}=\beta E_{t}\left( \frac{u^{\prime }\left( d_{t+1}\right) }{u^{\prime
}\left( d_{t}\right) }\left( p_{t+1}+d_{t+1}\right) \right) .
\tag{EE$^{\prime }$}
\end{equation}%

\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Bond Price}

\begin{itemize}
\item Recall equation (EE):
\begin{align}
R_{t}^{-1}& =\beta \frac{1}{u^{\prime }\left( d_{t}\right) }E_{t}\left(
u^{\prime }\left( d_{t+1}\right) \right) ,  \tag{EE} \\
R_{t}& =\frac{u^{\prime }\left( d_{t}\right) }{\beta E_{t}\left( u^{\prime
}\left( d_{t+1}\right) \right) }.  \notag
\end{align}

\item The price of a discount bond increases (return falls) in $\beta $.

\item The price increases (return falls) in expected future marginal
utility, and decreases in current marginal utility: reflects consumption
smoothing motive.
\item Recall from last time: implies that more uncertainty raises bond price if convex preferences.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Stock prices}

\begin{itemize}
\item Recall equation (EE$^{\prime }$):
\begin{equation}
p_{t}=\beta E_{t}\left( \frac{u^{\prime }\left( d_{t+1}\right) }{u^{\prime
}\left( d_{t}\right) }\left( p_{t+1}+d_{t+1}\right) \right) .
\tag{EE$^{\prime }$}
\end{equation}
\item Define the expected rate of return on stocks as
\begin{equation*}
E_{t}\left( R_{t}^{s}\right) =E_{t}\left( \frac{p_{t+1}+d_{t+1}}{p_{t}}%
\right) .
\end{equation*}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Equity premium}

\begin{itemize}
\item The expected rate of return on stocks is
\begin{equation*}
E_{t}\left( R_{t}^{s}\right) =E_{t}\left( \frac{p_{t+1}+d_{t+1}}{p_{t}}%
\right) .
\end{equation*}

\item Recall that \vspace{-0.1in}
\begin{equation*}
E_{t}\left( XY\right) =E_{t}\left( X\right) E_{t}\left( Y\right)
+C_{t}\left( X,Y\right) .
\end{equation*}

\item Rewrite (EE$^{\prime }$): \vspace{-0.1in}
\begin{eqnarray*}
1 &=&\beta E_{t}\left( \frac{u^{\prime }\left( d_{t+1}\right) }{u^{\prime
}\left( d_{t}\right) }\left( \frac{p_{t+1}+d_{t+1}}{p_{t}}\right) \right) \\
&=&\beta E_{t}\left( \frac{u^{\prime }\left( d_{t+1}\right) }{u^{\prime
}\left( d_{t}\right) }R_{t}^{s}\right) \\
&=&\beta E_{t}\left( \frac{u^{\prime }\left( d_{t+1}\right) }{u^{\prime
}\left( d_{t}\right) }\right) E_{t}\left( R_{t}^{s}\right) +C_{t}\left(
\beta \frac{u^{\prime }\left( d_{t+1}\right) }{u^{\prime }\left(
d_{t}\right) },R_{t}^{s}\right)
\end{eqnarray*}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Risk premium}
\begin{itemize}
\item Insert (EE) and rearrange:

\begin{equation*}
\hspace{-0.5in}1=R_{t}^{-1}E_{t}\left( R_{t}^{s}\right) +C_{t}\left( \beta
\frac{u^{\prime }\left( d_{t+1}\right) }{u^{\prime }\left( d_{t}\right) }%
,R_{t}^{s}\right) ,\vspace{-0.15in}
\end{equation*}%
\begin{eqnarray*}
\hspace{-0.5in}E_{t}\left( R_{t}^{s}\right) &=&R_{t}-R_{t}C_{t}\left( \beta
\frac{u^{\prime }\left( d_{t+1}\right) }{u^{\prime }\left( d_{t}\right) }%
,R_{t}^{s}\right) , \\
&=&R_{t}-\frac{u^{\prime }\left( d_{t}\right) }{\beta E_{t}\left( u^{\prime
}\left( d_{t+1}\right) \right) }C_{t}\left( \beta \frac{u^{\prime }\left(
d_{t+1}\right) }{u^{\prime }\left( d_{t}\right) },R_{t}^{s}\right) , \\
&=&R_{t}-\frac{C_{t}\left( u^{\prime }\left( d_{t+1}\right)
,R_{t}^{s}\right) }{E_{t}\left( u^{\prime }\left( d_{t+1}\right) \right) }.
\end{eqnarray*}

% \begin{itemize}
\item The expected return on stocks equals the return on the risk-free bond
plus the \underline{risk-premium}, which is $-\frac{C_{t}\left( \cdot ,\cdot
\right) }{E_{t}\left( \cdot \right) }$.
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Equity premium: a puzzle?}
\begin{itemize}

\item If the covariance $C_{t}\left( \cdot ,\cdot \right) $ is negative,
which we normally expect, there is an equity premium.

\item Interpretation

\begin{itemize}
\item The most desirable assets yield well when marginal utility is high ($%
C_{t}\left( \cdot ,\cdot \right) >0$). Risk-aversion means that agents
prefer assets that act like insurance.

\item Investors are willing to sacrifice return if $C_{t}\left( \cdot ,\cdot
\right) >0$, and they will demand higher returns if $C_{t}\left( \cdot
,\cdot \right) <0$.
\end{itemize}

\item Mehra and Prescott (1985): the observed equity premium is difficult to
reconcile with this formula.
\end{itemize}
\end{frame}

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

\begin{frame}\frametitle{Equity premium: a puzzle?}
\begin{itemize}
\item Mehra and Prescott (1985): the observed equity premium is difficult to
reconcile with this formula.
\centering\includegraphics[width=0.5\textwidth]{./EquityPremium.png}
\item But most other assets have $> 10\%$ excess returns.
\end{itemize}
\end{frame}

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

% \section{Conclusion}
% % ------------------------------------------------


\begin{frame}
  \frametitle{Conclusion}
  \begin{itemize}
  \item Homework due tonight.
  \item Midterm in two weeks.
  \item Check for new homework over the weekend.
  \end{itemize}
\end{frame}


\end{document}
