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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 2021} % 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{Announcements}
  \begin{itemize}
  \item Today: Start heterogeneous agent models.
  \item First: income fluctuation problem.
  \item New homework uploaded on Wednesday.
  \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{Thinking about Uncertainty in Macroeconomic Models}
  \begin{itemize}
  \item Market clearing:
    \begin{align}
      \sum_{i = 1}^{N}((1 + r_{t + 1})a_{i,t + 1} + w_{i,t + 1} - c_{i,t + 1} - a_{i,t + 2}) &= 0
    \end{align}
  \item Wealth + Income - (Consumption + Savings) = 0
  \item Now we have to find decision rules that satisfy
    \begin{align}
      u'(c_{i,t}) &= \beta E[(1 + r_{t + 1})u'(c_{i,t+1})]
    \end{align}
  \item Imposing decision rules as a function of worker state $(\hat{S}_{i,t})$:
    \begin{align}
      \sum_{i = 1}^{N}((1 + r_{t + 1})a_{i,t + 1}(\hat{S}_{i,t + 1}) + w_{i,t + 1}(\hat{S}_{i,t + 1})) \\- \sum_{i=1}^{N}(c_{i,t + 1}(\hat{S}_{i,t + 1}) - a_{i,t + 2}(\hat{S}_{i,t + 2})) &= 0
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Thinking about Uncertainty in Macroeconomic Models}
  \begin{itemize}
  \item Typical assumptions in macroeconomics are a convex combination of
    \begin{enumerate}
    \item certainty equivalence:
    \begin{align}
      u'(\bar{c}_{i,t}) &= \beta E[(1 + \underbrace{r_{t + 1}}_{GE})\underbrace{u'(\bar{c}_{i,t+1})}_{Closer\;to\;Linear}]
    \end{align}
  \item linearized decision rules:
    \begin{align}
      \sum_{i = 1}^{N}&((1 + r_{t + 1})a_{i,t + 1} + w_{i,t + 1} - c_{i,t + 1} - a_{i,t + 2}) = 0\\
      \sum_{i = 1}^{N}&((1 + r_{t + 1})\beta_{a}\hat{S}_{i,t + 1} + \beta_{w}(\hat{S}_{i,t + 1}) - \beta_{c}\hat{S}_{i,t + 1} - \beta_{a}\hat{S}_{i,t + 2}) = 0
    \end{align}
  \end{enumerate}
\item Can be expressed as matrix \& solved quickly on computer.
\end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{So far}
  \begin{itemize}
  \item We've thought about worlds in which markets are complete:
    \begin{enumerate}
    \item agents can share risk perfectly;
    \item contract on any feasible consumption stream ex-ante (Arrow-Debreu) or ex-post (sequential).
    \item implies representative agent.
    \end{enumerate}
  \item Today: a different route. Workers cannot insure against income uncertainty.
  \item Explore using different preferences:
    \begin{enumerate}
    \item Certainty Equivalence - Quadratic Utility.
    \item Constant Absolute Risk Aversion - Exponential Utility.
    \item Constant Relative Risk Aversion.
    \end{enumerate}
  \item These each imply different ways in which agents respond to income shocks and uncertainty.
  \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{align}
     \text{s.t.   } A' &= (1 + r)A + Y - C\\
     Y' &= \rho Y + \epsilon'
    \end{align}
    \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).
  \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}
  \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} &=& 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 $\beta = (1 + r)$):
    \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' &=& 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{Constant Relative 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{CRRA Preferences}
  \begin{itemize}
  \item Now, we will start to think about an economy in which agents have {\bf C}onstant {\bf R}elative {\bf R}isk {\bf A}verse preferences.
  \item i.e., power utility.
  \item What else does this mean? Key difference:
  \item Agents are very unhappy when they starve:
    \begin{align}
      u'(0) = \infty
    \end{align}
  \item Seems like a reasonable assumption.
  \item Cover this in heterogeneous agent models next time.
  \end{itemize}
\end{frame}

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

% \begin{frame}
%   \frametitle{Heterogeneous Agent Models}
%   \begin{itemize}
%   \item Workers change their behavior in response to uncertainty.
%   \item First wave of heterogeneous agent models: how do aggregates change when {\it individual idiosyncratic} uncertainty is uninsurable.
%   \item In other words: when agents must accumulate {\it precautionary savings} to insure against income shocks.
%   \item Key ``first wave'' papers (no particular order):
%     \begin{itemize}
%     \item Huggett (1993): Incomplete markets exchange economy with GE interest rate.
%     \item Imrohoroglu (1989): Individual and aggregate uncertainty with fixed interest rate.
%     \item Aiyagari (1994): Incomplete markets production economy with GE interest rate.
%     \item Bewley (1986): Individual uncertainty with fixed interest rate.
%     \end{itemize}
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Heterogeneous Agent Models}
%   \begin{itemize}
%   \item We can generically write the worker's problem as
%     \begin{align}
%       \max_{\{c_{t},i_{t},l_{t}\}_{t=0}^{\infty}}& E \sum_{t = 0}^{\infty}\beta^{t}u(c_{t})\\
%       \text{s.t.  } c_{t} + i_{t} &\leq r_{t}a_{t} + w_{t}l_{t}\\
%       a_{t + 1} &= (1 - \delta)a_{t} + i_{t}\\
%       a_{t + 1} &\geq \underline{a}_{t} \\
%       w_{t} &\sim F\\
%       c_{t}\geq 0&, l_{t}\geq 0, a_{0}\text{ given}
%     \end{align}
%   \item How we deal with prices $r_{t}, w_{t}$ and choices $c_{t}, i_{t}, l_{t}$ is central to equilibrium.
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Recursive Formulation}
%   \begin{itemize}
%   \item Can be written as
%     \begin{align}
%       V(a)&= u(c) + \beta E[V(a')]
%       \text{s.t.  } c + i &\leq ra + wl\\
%       a' &= (1 - \delta)a + i\\
%       a' &\geq \underline{a} \\
%       w &\sim F\\
%       c\geq 0&, l\geq 0, a_{0}\text{ given}
%     \end{align}
%   \item Under fairly general conditions, this inherits same properties as non-stochastic version.
%   \item Key: optimizing a concave return function subject to a convex budget.
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Idiosyncratic Markov Income Uncertainty}
%   \begin{itemize}
%   \item Suppose $wl = e$, $F[e'] = \pi(e'|e)$
%   \item Two states: $e_{l}, e_{h}$
%   \item Can be written as
%     \begin{align}
%       V(a,e)&= u(c) + \beta\sum_{e'}\pi(e'|e)V(a',e')
%       \text{s.t.  } c + a' &\leq (1 + r)a + e\\
%       a' &\geq \underline{a} \\
%       c\geq 0&, a_{0}\text{ given}
%     \end{align}
%   \item (Huggett, 1993)
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Equilibrium}
%   \begin{itemize}
%   \item Define a distribution of agents over assets $as$ and endowments $e$, $\psi$.
%   \item Stationary equilibrium: aggregate state ($\psi$) is unchanging.
%   \item Agents move around distribution, but LLN $\rightarrow$ $\psi' = \psi$
%   \item Define $\psi(B)$ such that given transition function $P$:
%     \begin{align}
%       \psi(B) &= \int_{S}P(x, B)d\psi
%     \end{align}
%   \item $P(x, B)$ the probability that an agent with state $x$ will have state $B\in\beta_{S}$ next period.
%   \item $B$ is a subset of the state space.
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Stationary Equilibrium}
%   \begin{itemize}
%   \item Roughly summarizing Huggett, 1993:
%     A stationary equilibrium for this economy is a tuple ($c, a', r, \psi$) that satisfy
%     \begin{enumerate}
%     \item $c$ and $a'$ solve the workers problem taking prices as given.
%     \item Markets clear:
%       \begin{enumerate}
%       \item consumption = production: $\int c(x)d\psi = \int ed\psi$
%       \item no net savings: $\int a(x)d\psi = 0$
%       \end{enumerate}
%     \item $\psi$ is stationary:
%       \begin{align}
%       \psi(B) &= \int_{S}P(x, B)d\psi
%       \end{align}
%       for all $B\in\beta_{S}$
%     \end{enumerate}
%   \end{itemize}
% \end{frame}

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

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

\begin{frame}
  \frametitle{Next time}
  \begin{itemize}
  \item First wave of heterogeneous agent models: how do aggregates change when {\it individual idiosyncratic} uncertainty is uninsurable.
  \item In other words: when agents must accumulate {\it precautionary savings} to insure against income shocks.
  \item Key ``first wave'' papers (no particular order):
    \begin{itemize}
    \item Huggett (1993): Incomplete markets exchange economy with GE interest rate.
    \item Imrohoroglu (1989): Individual and aggregate uncertainty with fixed interest rate.
    \item Aiyagari (1994): Incomplete markets production economy with GE interest rate.
    \item Bewley (1986): Individual uncertainty with fixed interest rate.
    \end{itemize}
  \item Start this on Wednesday, talk about how to solve them next Monday.
  \end{itemize}
\end{frame}

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


% \begin{frame}
%   \frametitle{Conclusion}
%   \begin{itemize}
%   \item Started talking about uncertainty.
%   \item Next time: first wave heterogeneous agent models (Aiyagari (1993), Huggett, Imrohoroglu,)
%   \item Project 1 due Friday.
%   \end{itemize}
% \end{frame}


\end{document}