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%	PACKAGES AND THEMES
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% ----------------------------------------------------------------------------------------
%	TITLE PAGE
% ----------------------------------------------------------------------------------------

\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: consumption smoothing and permanent income.
  \item ``The income fluctuation problem''
  \item HW2 due Thursday.
  \item HW3 posted on Thursday.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Thinking about Uncertainty in Macroeconomic Models}
  \begin{itemize}
  \item Uncertainty makes macroeconomic models more difficult to solve.
  \item We make assumptions about the environment (preferences, technology, etc.) to decrease complexity of problem.
  \item Euler Equation:
    \begin{align}
      u'(c_{t}) &= \beta E[(1 + \underbrace{r_{t + 1}}_{GE})\underbrace{u'(c_{t+1})}_{Non-linear}]
    \end{align}
  \item Each agent chooses consumption and savings based on a
    \begin{enumerate}
    \item general equilibrium object (given by the decision rules of all other agents)
    \item (potentially highly) non-linear marginal utility.
    \end{enumerate}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Today}
  \begin{itemize}
  \item Today: Think about how workers insure against income risk.
  \item Foundation for consumption smoothing.
  \item Explore using different preferences:
    \begin{enumerate}
    \item Certainty Equivalence - Quadratic Utility.
    \item Constant Absolute Risk Aversion - Exponential Utility.
    \end{enumerate}
  \item These each imply different ways in which agents respond to income shocks and uncertainty.
  \item We will return to this when we study heterogeneous agents.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Risk}
  \begin{itemize}
  \item How do we typically think about risk in economic models?
  \item Absolute Risk Aversion:
    \begin{equation}
      AR = -\frac{u''(c)}{u'(c)}
    \end{equation}
  \item A measure of the agent's risk aversion unconditional upon their level of wealth.
  \item Relative Risk Aversion:
    \begin{equation}
      RRA = -\frac{u''(c)c}{u'(c)}
    \end{equation}
  \item Conditioning upon an agent's wealth, how does his risk aversion change?
  \item Probably most reasonable are ``DARA'' ``CRRA''
  \item These will have different implications for savings and consumption.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{When approximations work}
  \begin{itemize}
  \item For a lot of the distribution, decision rules are linear:
  \end{itemize}
\centering\includegraphics[width=0.7\textwidth]{./ahate1LateCareerHighLearnbyWealth_standard.eps}
\end{frame}

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

\section{Certainty Equivalence} % 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 In the case of quadratic utility, we will see that agents don't change their consumption choices when faced with shocks.
  \item Uncertainty still decreases expected utility, but does not change choices.
  \item Why is this relevant? One solution technique (LQ) assumes that agents have a quadratic utility function (locally risk-neutral).
  \item We will see that this is sometimes not a great assumption.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Quadratic Utility}
  \begin{itemize}
\item Utility is given by the following:
    \begin{equation}
      \max E[\sum_{t = 0}^{\infty}\beta^{t}(aC_{t} - bC_{t}^{2})]
    \end{equation}
    \begin{eqnarray}
     \text{s.t.   } A_{t + 1} &=& (1 + r)A_{t} + Y_{t} - C_{t}\\
     Y_{t + 1} &=& \rho Y_{t} + \epsilon_{t + 1}
    \end{eqnarray}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Euler Equation}
  \begin{itemize}
\item Do the usual steps to find the Euler Equation:
  \begin{equation}
    V(A) = \max_{C, A'} aC_{t} - bC_{t}^{2} + \beta E[V(A')]
  \end{equation}
    \begin{eqnarray}
     \text{s.t.   } A' &=& (1 + r)A + Y - C\\
     Y' = \rho Y + \epsilon'
    \end{eqnarray}
    \begin{equation}
      \der{V}{C} = a - 2bC - \lambda
    \end{equation}
    \begin{equation}
      \der{V}{A'} = -\lambda + \beta E[\der{V}{A'}]
    \end{equation}
    \begin{equation}
      \der{V}{A} = (1 + r)\lambda
    \end{equation}
    \begin{equation}
      \Rightarrow C = \beta(1 + r)E[C']
    \end{equation}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Certainty Equivalence}
  \begin{itemize}
\item Suppose that $\beta = (1 + r)$:
    \begin{equation}
      C = E[C']
    \end{equation}
\item Suppose that there were two states of the world: high and low.
  \begin{equation}
    C = P_{h}C_{h} + P_{l}C_{l}
  \end{equation}
\item This is equivalent to an agent receiving the mean income between both states:
  \begin{equation}
    C = C_{m}
  \end{equation}
\item i.e., workers make savings decisions {\it as though they are receiving the average consumption with certainty}.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Prudence}
  \begin{itemize}
  \item Agents in this economy are not ``prudential.''
  \item That is, they don't change their choices based upon uncertainty about the future.
  \item Another way to express this is in the third derivative of the utility function:
    \begin{equation}
      U''' = 0
    \end{equation}
  \item This captures the response of marginal utility (i.e., decisions) to uncertainty.
  \item Marginal utility changes linearly, so any convex combination is equal to the expected value.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Random Walk}
  \begin{itemize}
\item Can show for the AR(1) case:
  \begin{equation}
    C_{t} - C_{t - 1} = \frac{r}{1 + r - \rho}\epsilon
  \end{equation}
\item Now, consider the case in which income shocks are iid:
  \begin{equation}
    Y_{t + 1} = Y_{t} + \epsilon_{t + 1}
  \end{equation}
\item Then the difference in consumption becomes:
  \begin{equation}
    C_{t} - C_{t - 1} = \epsilon_{t}
  \end{equation}
\item In other words, the agent consumes all of the shock in each period (will also happen with CRRA and autarky).
\item This is a martingale!
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Conclusion}
  \begin{itemize}
\item In the quadratic utility world, uncertainty does not change an agents decision when compared with an identical income stream.
\item In the case of CARA utility, we will see that agents have precautionary savings that result from curvature in the utility function.
\item The choices are the same as they would be under complete markets.
  \end{itemize}
\end{frame}

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

\section{Constant Absolute Risk Aversion} % 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 to CARA World}
  \begin{itemize}
  \item Now, use CARA preferences to think about world in which certainty equivalence does not hold.
  \item Now, we will allow agents to be prudential in their savings response to future uncertainty.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Constant Absolute Risk Aversion Utility}
  \begin{itemize}
  \item The maximization problem is given by
    \begin{equation}
      \max E[\sum_{t = 0}^{\infty}-\frac{1}{\alpha}\exp(-\alpha C_{t})]
    \end{equation}
    \begin{eqnarray}
      \text{s.t.   } A_{t + 1} &=& A_{t} + Y_{t} - C_{t}\\
Y_{t} &=& \rho Y_{t-1} + \epsilon_{t}, \epsilon_{t}\sim N(0, \sigma^{2})
    \end{eqnarray}
\item Key difference: first derivative (i.e., policy functions), no longer linear.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Euler Equation}
  \begin{itemize}
  \item Bellman Equation (implicitly assume $\beta = \frac{1}{1 + r}$):
    \begin{equation}
      V(A) = \max_{C, A'} -(\frac{1}{\alpha})\exp(-\alpha C) + E[V(A')]
    \end{equation}
    \begin{eqnarray}
      \text{s.t.   } A' &=& A + Y - C\\
Y' &=& \rho Y + \epsilon'
    \end{eqnarray}
    \begin{equation}
      \der{V}{C} = \exp(-\alpha C) - \lambda
    \end{equation}
    \begin{equation}
      \der{V}{A'} = -\lambda + E[\der{V}{A'}]
    \end{equation}
    \begin{equation}
      \der{V}{A} = \lambda
    \end{equation}
    \begin{equation}
      \Rightarrow \exp(-\alpha C) = E[\exp(-\alpha C')]
    \end{equation}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Euler Equation}
  \begin{itemize}
  \item Bellman Equation (implicitly assume $\beta = (1 + r)$):
    \begin{equation}
      \exp(-\alpha C) = E[\exp(-\alpha C')]
    \end{equation}
\item For normally distributed random variables, the following holds:
  \begin{equation}
    E[exp(x)] = exp(E[x] + \sigma_{x}^{2}/2)
  \end{equation}
\item Thus, we can rewrite the Euler Equation as
  \begin{equation}
      \exp(-\alpha C) = E(exp(-\alpha C' + \alpha^{2}\sigma^{2}/2))
  \end{equation}
  \begin{equation}
      \Rightarrow C' = C + \frac{\alpha\sigma^{2}}{2} + \nu
  \end{equation}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Policy Function}
  \begin{itemize}
  \item Policy function:
  \begin{equation}
      \Rightarrow C' = C + \frac{\alpha\sigma^{2}}{2} + \nu
  \end{equation}
  \item This says that consumption is {\it increasing} ex-ante in response to uncertainty, measured by $\sigma^{2}$.
  \item What does this mean about life-cycle consumption?
  \item We would expect it to be upward-sloping, at least initially.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Consumption in time t}
  \begin{itemize}
  \item Can show:
    \begin{equation}
      C_{t} = (\frac{1}{T - t})A_{t} + Y_{t} - \frac{\alpha(T - t - 1)\sigma^{2}}{4}
    \end{equation}
\item Certainty equivalence: last term is equal to zero. i.e., cake-eating problem.
\item Agents consume less than they would if their income stream was certain!
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Prudence}
  \begin{itemize}
  \item What is different in this case?
  \item Agents are prudential: $U''' > 0$.
  \item The Euler Equation is given by:
    \begin{equation}
      \exp(-\alpha C) = E[\exp(-\alpha C')]
    \end{equation}
  \item Suppose $C = C'$, then consider Jensen's Inequality:
    \begin{equation}
      exp(-\alpha E(C)) < E[\exp(-\alpha C)]
    \end{equation}
  \item This needs to hold in equilibrium, thus agents must decrease current consumption.
  \item Agents save in excess of what they would under certainty!
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{CARA Utility}
  \begin{itemize}
  \item When CARA agents cannot perfectly insure, they change their choices from the certainty equivalence (quadratic utility) case.
  \item Unfortunately, CARA has some problems: Marginal utility is finite when consumption is equal to zero.
  \item CRRA utility will solve this problem, but is more challenging to solve.
  \end{itemize}
\end{frame}

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


\section{Permanent Income} % 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{Permanent Income Hypothesis}
  \begin{itemize}
  \item Theory developed by Milton Friedman that describes how agents allocate resources over their lifetime.
  \item Consumption is based on not just current income, but expectations over future income as well.
  \item Implies that agents want to consumption smooth, rather than consume out of transitory income shocks.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Lifetime Budget Constraint}
\begin{itemize}
\item Solve the flow budget constraint forward%
\begin{eqnarray*}
A_{0} &=&\frac{1}{1+r}A_{1}-\left( y_{0}-c_{0}\right) \\
&=&\frac{1}{1+r}\left( \frac{1}{1+r}A_{2}-\left( y_{1}-c_{1}\right) \right)
-\left( y_{0}-c_{0}\right) \\
&=&-\sum\nolimits_{t=0}^{T}\left( \frac{1}{1+r}\right) ^{t}\left(
y_{t}-c_{t}\right) \\
&&+\left( \frac{1}{1+r}\right) ^{T+1}A_{T+1},
\end{eqnarray*}

\item Impose No-Ponzi condition requiring, $A_{T+1}=0,$ to yield
\begin{equation*}
A_{0}=-\sum\nolimits_{t=0}^{T}\left( \frac{1}{1+r}\right) ^{t}\left(
y_{t}-c_{t}\right)
\end{equation*}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Lifetime Budget Constraint}
\begin{itemize}
\item Rearrange to derive the present value budget constraint%
\begin{equation}
\sum\nolimits_{t=0}^{T}\left( \frac{1}{1+r}\right) ^{t}c_{t}\left(
I_{t}\right) =A_{0}+\sum\nolimits_{t=0}^{T}\left( \frac{1}{1+r}\right)
^{t}y_{t}\left( I_{t}\right) ,  \tag{PVBC}
\end{equation}

% \begin{itemize}
\item Holds for all realized $\left\{ y_{t}\right\} $

\item Not an expectation

\item Right-hand side of (PVBC) is lifetime wealth

\item (PVBC) does not imply that the time path of consumption is known in
advance
\end{itemize}

\end{frame}

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

\begin{frame}
  \frametitle{Derivation}
  \begin{itemize}

\item In finite-horizon case with $J\equiv T,$ expected present value budget
constraint (EPVBC) becomes%
\begin{align}
& E_{t}\left\{ \sum\nolimits_{j=0}^{J}\left( \frac{1}{1+r}\right)
^{j}c_{t+j}\right\} =W_{t}.  \tag{EPVBC} \\
& W_{t}\equiv A_{t}+E_{t}\left\{ \sum\nolimits_{j=0}^{J}\left( \frac{1}{1+r}%
\right) ^{j}y_{t+j}\right\} .  \notag
\end{align}

\item As $J\rightarrow \infty ,$ (EPVBC) follows from (FBC) and
(ENPG)
\end{itemize}
\end{frame}

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

\begin{frame}
\begin{itemize}
\item Equation (EE') and the law of iterated expectations imply that
\begin{eqnarray*}
E_{t}\left( c_{t+2}\right) &=&E_{t}\left( E_{t+1}\left( c_{t+2}\right)
\right) \\
&=&E_{t}\left( c_{t+1}\right) \\
&=&c_{t},
\end{eqnarray*}%
so that
\begin{eqnarray*}
\hspace{-0.5in}E_{t}\left\{ \sum\nolimits_{j=0}^{J}\left( \frac{1}{1+r}%
\right) ^{j}c_{t+j}\right\} &=&c_{t}\sum\nolimits_{j=0}^{J}\left( \frac{1}{%
1+r}\right) ^{j} \\
&\equiv &R_{J}c_{t}.
\end{eqnarray*}%
\end{itemize}
\end{frame}

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

\begin{frame}\frametitle{Pemanent Income}
\begin{itemize}
\item Define permanent income as the constant value of future income such
that its present value equals the present value of actual income and assets%
\begin{equation*}
y_{t}^{P}\equiv \frac{1}{R_{J}}W_{t}
\end{equation*}

\item (EPVBC) becomes%
\begin{equation*}
c_{t}=\frac{1}{R_{J}}W_{t}\equiv y_{t}^{P}
\end{equation*}%
where
\begin{equation*}
\sum\nolimits_{j=0}^{J}\left( \frac{1}{1+r}\right) ^{j}y_{t}^{P}=W_{t}.
\end{equation*}

\item Therefore, consumption equals permanent income
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Overview}
\begin{itemize}
\item A temporary change in income leads to a permanent change in expected
consumption: consumption smoothing extends the effects of income changes
over time

\item The effect of a change in current income on current consumption
depends on its effect on permanent income

% \begin{itemize}
\item Permanent changes in income have larger consumption effects than
temporary changes
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Empirical Implications (Friedman (1957))}
\begin{itemize}
\item Consider the linear projection of consumption on total income%
\begin{equation*}
\widehat{c}_{t}=\alpha _{1}+\alpha _{2}y_{t}
\end{equation*}

\begin{itemize}
\item For a cross-section of households at a point in time, $\alpha _{1}>0,$
and $\alpha _{2}$ is much less than 1

\item For a country over time, $\alpha _{1}\approx 0,$ and $\alpha _{2}$ is
closer to 1
\end{itemize}

\item Define transitory income%
\begin{equation*}
y_{t}^{T}=y_{t}-y_{t}^{P}.
\end{equation*}

\item Suppose $C\left( y_{t}^{T},y_{t}^{P}\right) =0$\end{itemize}
\end{frame}


\begin{frame}\frametitle{Friedman (1957)}
\begin{itemize}
\item The coefficient $\alpha _{2}$ is given by%
\begin{eqnarray*}
\alpha _{2} &=&\frac{C\left( y_{t},c_{t}\right) }{V\left( y_{t}\right) }=%
\frac{C\left( y_{t}^{T}+y_{t}^{P},y_{t}^{P}\right) }{V\left(
y_{t}^{T}+y_{t}^{P}\right) } \\
&=&\frac{V\left( y_{t}^{P}\right) }{V\left( y_{t}^{P}\right) +V\left(
y_{t}^{T}\right) }.
\end{eqnarray*}

\item Cross-section data: $V\left( y_{t}^{T}\right) $ is large because of
wide variance of household transitory income implying small $\alpha _{2}$

\item Time-series data: $V\left( y_{t}^{T}\right) $ is small because
transitory income averages out across households in the aggregate implying
large $\alpha _{2}$ close to one\end{itemize}
\end{frame}

\end{itemize}
\end{itemize}

\begin{frame}\frametitle{Hall (1978)}
\begin{itemize}
\item Consider an alternative equation
\begin{equation*}
\widehat{c}_{t}=\alpha _{1}+\alpha _{2}c_{t-1}+\gamma x_{t-1},
\end{equation*}%
where $x_{t-1}$ is some other variable

\item Recall that under linear quadratic preferences
\begin{equation*}
E_{t}\left( c_{t+1}\right) =\frac{a}{b}\left[ 1-\frac{1}{\beta \left(
1+r\right) }\right] +\frac{1}{\beta \left( 1+r\right) }c_{t},
\end{equation*}%
so that $\gamma =0$. Nothing should predict consumption except lagged
consumption

\item There is some evidence that $\gamma \neq 0$

\item Perhaps permanent income changes over time and the change takes time
for agents to realize so that $c_{t-1}$ is not affected, but $c_{t}$
is\end{itemize}
\end{frame}

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

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


\begin{frame}
  \frametitle{Conclusion}
  \begin{itemize}
  \item Next: Cover Asset Pricing and Lucas Tree
  \item Please let me know if you can't access the cluster!
  \end{itemize}
\end{frame}


\end{document}
