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%_____________ Opening slide _______________________

\title[Buffer Stock Theory]{Theoretical Foundations of Buffer Stock Saving}
\author[Carroll]{Chris Carroll}
\institute[JHU]{Johns Hopkins University}
\date[\today]{\today}

\begin{document}\bibliographystyle{\econtexBibStyle}

\begin{frame}[plain]
  \titlepage
\end{frame}


%_____________ 1st section  ____________
\section{Introduction}
\subsection{Motivation}

\begin{frame}
\frametitle{Drawbacks of Numerical Solutions}


\pause A Black Box \pause
\begin{itemize}
\item Can Construct Solution to Model Without Really Understanding It
\item Hard Even To Be Sure Your Numerical Solution Is {\it Right}
\item Little Intuition for How Results Might Change With
\begin{itemize}
\item Calibration
\item Structure
\end{itemize}
\item {\it Very} Hard To Teach!
\end{itemize}

\medskip\medskip
\pause I Am A {\it Big} Fan Of Numerical Methods
\begin{itemize}
\item Have Done A Good Deal Of Work With Them Myself
\item But As A Result, Have Felt All These Drawbacks Keenly
\end{itemize}



\end{frame}

\begin{frame}
\frametitle{The Gap This Paper Fills}

\pause Foundations For Microeconomic Household's Problem With
\begin{itemize}
\item Uncertain Labor Income
\item No Liquidity Constraints
\item CRRA Utility
\item (Problem with Liquidity Constraints Is A Limiting Case)
\end{itemize}

\end{frame}
\begin{frame}{Key Result}
\pause
Restrictions On Parameter Values Such That \pause
\begin{itemize}
\item Problem Defines A Contraction Mapping
\begin{itemize}
\item $\Rightarrow~~\exists $ A Unique Consumption Function $\cFunc(m)$
\end{itemize}
\item There Is A `Target' Ratio Of Assets to Permanent Income
\begin{itemize}
\item Requires A Key `Impatience' Condition To Hold
\item Good News
\begin{itemize} \item Condition Is Weaker (Easier To Satisfy) Than Previous Papers Imposed \end{itemize}
\end{itemize}
\end{itemize}

\end{frame}

\section{The Problem}

\begin{frame}

Limit as horizon $T$ goes to infinity of 
\input \EqDir/supfn

\input \EqDir/DBCparts

\input \EqDir/tShkDef

\begin{itemize}
\item $\util(\bullet)=\bullet^{1-\CRRA}/(1-\CRRA)$; $\Ex_{t}[\pShk_{t+n}]=\Ex_{t}[\tShk_{t+n}]=1~\forall~n>0$; $\beta < 1, \CRRA > 1$
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{Surely This Problem Has Been Solved?}

\pause No.
\begin{itemize}
\item Can't Use Stokey et.\ al.\ theorems because CRRA utility
\item Lit thru \cite{mnUnique} Can't Handle Permanent Shocks
\item Must Use Boyd's `Weighted' Contraction Mapping Theorem 
\item Surprisingly Subtle
\end{itemize}

\pause Fortunately, the Conclusions Are Simple!

\end{frame}

\subsection{The Perfect Foresight Problem}

\begin{frame}
\frametitle{Benchmark: Perfect Foresight Model}

Definitions: \smallskip

\begin{tabular}{llcl}
   Absolute Patience Factor & $\Pat$ & = & $(\Rfree \Discount)^{1/\CRRA}$
\\ Return Patience Factor & $\PatR$ & = & \Pat/\Rfree
\\ Perfect Foresight Growth Patience Factor & $\PatPGro$ & = & \Pat/\PGro
\end{tabular}

\medskip

\begin{tabular}{l|lcl|l} \hline
   Name                                 & \multicolumn{3}{c|}{Condition}    & Implication 
\\ \hline (AIC) Absolute Impatience Condition  & $\Pat$  & $<$ & 1 & $\cLev$ $\downarrow$ over time
\\ (RIC) Return Impatience Condition    & $\PatR$ & $<$ & 1 & $\cLev/\aLev$ $\downarrow$ over time
\\ (PF-GIC) Growth Impatience Condition & $\PatPGro$ & $<$ & 1 & $\cLev/\pLev$ $\downarrow$ over time
\end{tabular}

\medskip

\end{frame}

\begin{frame}
\frametitle{When Does A Useful Limiting Solution Exist?}

Finite Human Wealth (FHWC) condition:
\begin{eqnarray}
\PGro & < & \Rfree
\end{eqnarray}

\pause\medskip
Return Impatience Condition:
\begin{eqnarray}
\PatR & < & \Rfree
\end{eqnarray}

\end{frame}

\begin{frame}
\frametitle{What If There Are Liquidity Constraints?}

\pause 

\begin{itemize}
\item FHWC is {\it not} necessary for solution to exist
\item Other Key Condition For Useful Solution is

`Perfect Foresight Finite Value of Autarky Condition (PF-FVAC)':
\begin{eqnarray}
\beta \PGro^{1-\CRRA} & < & 1  
\end{eqnarray}

\item Without RIC, Constraints Are Irrelevant
\begin{itemize}
\item Because Wealth Always Wants To Rise, So Constraint Never Binds
\end{itemize}
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{Liquidity Constraints and Uncertainty}

\begin{itemize}
\item Introduce permanent shocks to income
\item Finite Value of Autarky Condition Becomes
\input \EqDir/FVAC
\end{itemize}

\end{frame}



\subsection{The Real Problem}
\begin{frame}
\frametitle{Contraction Mapping Requirements}

Finite Value of Autarky Condition: Same As In Liq Constr Problem!
\input \EqDir/FVAC

`Weak Return Impatience Condition' (WRIC)

\begin{eqnarray}
 0 \leq & \pZero^{1/\CRRA} \PatR & < 1 \label{eq:WRIC}
\end{eqnarray}

\end{frame}

\begin{frame}
\frametitle{Requirement For Existence Of A Target}

Definitions: `Uncertainty-Adjusted' Growth:
\input \EqDir/PGroAdj

Adjusted Growth Patience Factor:
\input \EqDir/PatPGroAdj

Growth Impatience Condition:
\input \EqDir/GIC

Why?  Because it can be shown that
\begin{eqnarray}
 \lim_{m_{t} \rightarrow \infty} \Ex_{t}\left[\frac{\mRat_{t+1}}{\mRat_{t}}\right] & = & \PatPGroAdj  \label{eq:mGrowth}
\end{eqnarray}

\end{frame}

\section{Features Of the Solution}
\subsection{Five Propositions}
\begin{frame}
\frametitle{Five Propositions}

\begin{enumerate}
\item $\lim_{m_{t} \rightarrow \infty} \Ex_{t}[\cLev_{t+1}/\cLev_{t}] = \Pat$
\item $\lim_{m_{t} \rightarrow 0} \Ex_{t}[\cLev_{t+1}/\cLev_{t}] = \infty$
\item $\exists$ a unique target value of $m$, called $\check{m}$
\item $\Ex_{t}[\cLev_{t+1}/\cLev_{t} | m_{t} = \check{m}] = \PGro - \epsilon$
\item $\left(\frac{d \Ex_{t}[\cLev_{t+1}/\cLev_{t}]}{d m_{t}}\right) < 0$
\end{enumerate}

\end{frame}

\subsection{The Target Saving Figure}
\begin{frame}
\frametitle{The Target Saving Figure}
\centerline{\includegraphics[width=4in]{\FigDir/cGroTargetFig.pdf}}
\end{frame}

\subsection{Bounds On The Consumption Function}
\begin{frame}
\frametitle{Bounds On the Consumption Function}
\centerline{\includegraphics[width=4in]{\FigDir/cFuncBounds.pdf}}
\end{frame}

\begin{frame}
\frametitle{The Marginal Propensity to Consume}
\centerline{\includegraphics[width=4in]{\FigDir/MPCLimits.pdf}}
\end{frame}

\subsection{The Consumption Function and Target Wealth}
\begin{frame}
\frametitle{The Consumption Function and Target Wealth}
\centerline{\includegraphics[width=4in]{\FigDir/cRatTargetFig.pdf}}
\end{frame}



\section{A Small Open Buffer Stock Economy}

\subsection{The Invariant Distribution}
\begin{frame}
\frametitle{Convergence To The Invariant Distribution}

\cite{szeidlInvariant} Proves Existence of an Invariant Distribution of 
$\mRat, \cRat, \aRat,$ etc.

\centerline{\includegraphics[height=2.5in]{\FigDir/SimCDFsConverge.pdf}}



\end{frame}


\subsection{Balanced Growth Equilbrium}
\begin{frame}
\frametitle{Balanced Growth Equilibrium}

Achieved When Cross Section Distribution Reaches Invariance
\begin{eqnarray}
  \YLev_{t+1}/\YLev_{t} = \CLev_{t+1}/\CLev_{t} & = & \PGro
\end{eqnarray}

Fisherian Separation Fails, Even Without Liquidity Constraints!
\medskip\medskip

\pause Insight:
\begin{itemize}
\item  Precautionary Saving $\approx$ Liquidity Constraints
\item If $\grave{\cFunc}(m)$ is solution for constrained consumer, 
\end{itemize}
\pause
\begin{equation}
\lim_{ \wp \downarrow 0} \cFunc(m;\wp) = \grave{\cFunc}(m)
\end{equation}


\end{frame}

\begin{frame}
\frametitle{The MPC Out Of Permanent Shocks}

\url{http://econ.jhu.edu/people/ccarroll/papers/MPCPerm.pdf}
\medskip

Lots of Recent Papers Trying to Measure the MPCP

\medskip
Paper Proves:

\begin{itemize}
\item MPCP $< 1$
\item But not a lot less:
\begin{itemize}
\item 0.75 to 0.95 (annual rate) for wide range of parameter values
\end{itemize}
\end{itemize}

\end{frame}

\section{Conclusions}
\begin{frame}

\begin{itemize}
\item Defined Conditions Under Which Widely Used Problem Has Solution
\begin{itemize}
\item Finite Value of Autarky Condition Guarantees Contraction (with WRIC)
\item Growth Impatience Condition Prevents $m \rightarrow \infty$
\end{itemize}
\item Economy Of Buffer Stock Consumers Exhibits Balanced Growth
\begin{itemize}
\item Even In Absence of General Equilibrium Adj of Interest Rate
\end{itemize}
\end{itemize}

\end{frame}

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