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\begin{center}
{\Large Demand-Deposit Contracts and the\vspace{0.4in} Probability of Bank
Runs}
\bigskip
\bigskip
{\large Itay Goldstein}
{\large Ady Pauzner}
\bigskip
\bigskip
\bigskip
Journal of Finance (2005)\pagebreak
{\large The Model}
\end{center}
\begin{itemize}
\item Three periods: $t=0,1,2$. Single homogeneous good, no aggregate
uncertainty
\item Continuum $\left[ 0,1\right] $\ of \textit{ex ante} identical,
risk-averse consumers with an endowment of $1$\ unit of the good in period $%
0 $
\item Beginning of period $1$: idiosyncratic, \textbf{unobservable}
preference shock:
\begin{itemize}
\item with probability $\lambda >0$: early type, $u=u\left( c_{1}\right) $
\item with probability $\left( 1-\lambda \right) $: late type, $u=u\left(
c_{1}+c_{2}\right) $
\end{itemize}
\item $u\left( c\right) $ well-behaved, $u\left( 0\right) =0$, $CRRA>1$ for $%
c\geq 1$\pagebreak
\item One investment technology:%
\begin{equation*}
\begin{tabular}{|l|l|l|l|}
\hline
Date & $t=0$ & $\;\ t=1$ & $\;\ \ t=2$ \\ \hline
& $1\;\ \ \ \ \rightarrow $ & $\;\ 1\;\ \ \ \ \ \ \rightarrow $ & $\;\
\left\{
\begin{tabular}{l}
$R$ with prob. $p\left( \theta \right) $ \\
\\
$0$ with prob. $1-p\left( \theta \right) $%
\end{tabular}%
\right. $ \\ \hline
\end{tabular}%
\end{equation*}%
\newline
where $\theta \sim \left[ 0,1\right] $\ is the state of the economy
(realized in period $1$, revealed publicly in period $2$).
\item Assume: $p%
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\left( \theta \right) >0$\ and $E_{\theta }\left[ p\left( \theta \right) %
\right] u\left( R\right) >u\left( 1\right) $\pagebreak
\item Autarky: early types consume $1$ and late types consume $R$ with prob.
$p\left( \theta \right) $
\item Social planner (can observe \textit{ex post} types):
\end{itemize}
\begin{center}
$\max\limits_{c_{1}}\lambda u\left( c_{1}\right) +\left( 1-\lambda \right)
u\left( \frac{1-\lambda c_{1}}{1-\lambda }R\right) E_{\theta }\left[ p\left(
\theta \right) \right] $
\end{center}
\qquad FOC: $u%
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\left( c_{1}^{FB}\right) =Ru%
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{\acute{}}%
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\left( \frac{1-\lambda c_{1}^{FB}}{1-\lambda }R\right) E_{\theta }\left[
p\left( \theta \right) \right] $
\qquad In the optimum, $c_{1}^{FB}>1$\ since at $c_{1}=1$: $1\cdot u%
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\left( 1\right) >Ru%
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\left( R\right) E_{\theta }\left[ p\left( \theta \right) \right] $
\qquad since $CRRA>1$ and $E_{\theta }\left[ p\left( \theta \right) \right]
<1$\pagebreak
\begin{itemize}
\item Types are unobservable. How to implement the first-best allocation?
Set-up a bank offering a demand-deposit contract. Bank max expected utility
of consumers
\item Period-one return $r_{1}$\ fixed and promised unless the bank runs out
(sequential service), period-two return $\widetilde{r_{2}}$\ stochastic
\item Consumers give their entire endowment to a bank in exchange for an
incentive compatible, $u\left( r_{1}\right) \leq u\left( \frac{1-\lambda
c_{1}}{1-\lambda }R\right) E_{\theta }\left[ p\left( \theta \right) \right] $%
, demand deposit contract that sets%
\begin{equation*}
r_{1}=c_{1}^{FB}.
\end{equation*}
\item BUT: the solution above is implemented as \textbf{an} equilibrium.
There is another, bank run equilibrium.\pagebreak
\item Panic-based runs: if everyone withdraws in period $1$: $r_{1}\cdot
\frac{1}{r_{1}}>0$ (Nash equilibrium)
\item Game played by late consumers has two Nash equilibria:
\begin{itemize}
\item a "good", no run, equilibrium in which all late consumers withdraw in
period $2$
\item a "bad", run, equilibrium in which late consumers panic and try to
withdraw in period $1$
\end{itemize}
\item When setting optimal $r_{1}$, need to know \textbf{how likely} each
equilibrium is. Do banks increase welfare? Solve backwards: period one
first, then period-zero decisions\pagebreak
\end{itemize}
\begin{center}
{\large Unique Equilibrium in Period 1}
\end{center}
\begin{itemize}
\item $\theta $ is realized but not revealed at the beginning of period $1$
\item Private signals: each agent receives a private signal about $\theta
_{i}=\theta +\varepsilon _{i}$, i.i.d. (uniform on $\left[ -\varepsilon
,\varepsilon \right] $). Note: no one has advantage in terms of the quality
of the signal
\item The signal $\theta _{i}$\ has two effects:
\begin{itemize}
\item provides info about $R$ (the higher $\theta _{i}$, the lower the
incentive to run)
\item provides info about signals of others (the higher $\theta _{i}$, the
more probable others got a high signal, too, the lower incentive to
run)\pagebreak
\end{itemize}
\item Early consumers always withdraw in period $1$
\item Late consumers compare the expected payoffs from withdrawing in period
$1$\ and $2$. This payoff depends on $\theta $\ and proportion $n$\ of
consumers demand early withdrawal.
\item Signal $\theta _{i}$\ provides info on both $\rightarrow $\ actions
depend on signals
\item Assume there are two extreme ranges of fundamentals at which agents'
behavior is known: the lower range $\left[ 0,\theta _{LB}\left( r_{1}\right) %
\right] $ and the upper range $\left[ \theta ^{UB}\left( r_{1}\right) ,1%
\right] $\pagebreak
\item For $\theta <\theta _{LB}-2\varepsilon $, all late consumers receive
signals below $\theta _{LB}-\varepsilon $\ and everybody runs: $n=1$
\item For $\theta >\theta ^{UB}+2\varepsilon $, all late consumers receive
signals above $\theta ^{UB}+\varepsilon $\ and only early consumers
withdraw: $n=\lambda $
\item When choosing the equilibrium action, a consumer must take into
account the equilibrium actions at nearby signals etc.
\item Theorem 1: There is a unique equilibrium in which late consumers
withdraw of they observe a signal below threshold $\theta ^{\ast }\left(
r_{1}\right) $\ and do not run above\pagebreak
\item Strategic complementarity property: an agent's incentive to take an
action is influenced by how many other agents take that action
\item A late consumer's utility differential is:%
\begin{equation*}
\nu \left( \theta ,n\right) =\left\{
\begin{tabular}{l}
$p\left( \theta \right) u\left( \frac{1-nr_{1}}{1-n}R\right) -u\left(
r_{1}\right) \;$if $\frac{1}{r_{1}}\geq n\geq \lambda $ \\
\\
$0-\frac{1}{n}\frac{u\left( r_{1}\right) }{r_{1}}\;$if$1\geq n\geq \frac{1}{%
r_{1}}$%
\end{tabular}%
\right.
\end{equation*}
\item Proof: Show that there exists a unique threshold equilibrium, i.e.
equilibrium in which all late consumers run if their signal is below some
common threshold and do not run above. Need to show that the utility
differential is equal to zero when $\theta $\ equals the threshold\pagebreak
\item What proportion of consumers runs at every realization of $\theta $?
\item Function $n\left( \theta ,\theta ^{%
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}\right) $: specifies the proportion of agents who run when fundamentals are
$\theta $\ and all consumers run at signals below $\theta ^{%
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%BeginExpansion
{\acute{}}%
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}$ and do not run at signals above $\theta ^{%
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%BeginExpansion
{\acute{}}%
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}$
\item $n\left( \theta ,\theta ^{\ast }\left( r_{1}\right) \right) =\left\{
\begin{tabular}{l}
$1\;$if $\theta \leq \theta ^{\ast }\left( r_{1}\right) -\varepsilon $ \\
$\lambda +\left( 1-\lambda \right) \left( \frac{1}{2}+\frac{\theta ^{\ast
}\left( r_{1}\right) -\theta }{2\varepsilon }\right) \;$if $\theta \in
\theta ^{\ast }\left( r_{1}\right) \pm \varepsilon $ \\
$\lambda \;$if $\theta \geq \theta ^{\ast }\left( r_{1}\right) +\varepsilon $%
\end{tabular}%
\right. $\ \pagebreak
\end{itemize}
\begin{center}
{\large Do Banks Increase Welfare?}
\end{center}
\begin{itemize}
\item Threshold signal $\theta ^{\ast }\left( r_{1}\right) $: a late type
must be indifferent between withdrawing in period $1$\ and $2$\
\item His posterior distribution of $\theta $: uniform over $\left[ \theta
^{\ast }\left( r_{1}\right) -\varepsilon ,\theta ^{\ast }\left( r_{1}\right)
+\varepsilon \right] $
\item Beliefs: the proportion of people who run is $n\left( \theta ,\theta
^{\ast }\left( r_{1}\right) \right) $;
\item Posterior distribution of $n$\ is uniform over $\left[ \lambda ,1%
\right] $\pagebreak
\item Indifference condition$:$%
\begin{equation*}
\dint\limits_{n=\lambda }^{1/r_{1}}u\left( r_{1}\right)
+\dint\limits_{n=1/r_{1}}^{1}\frac{1}{nr_{1}}u\left( r_{1}\right)
=\dint\limits_{n=\lambda }^{1/r_{1}}p\left( \theta ^{\ast }\right) u\left(
\frac{1-nr_{1}}{1-n}R\right)
\end{equation*}
\end{itemize}
\qquad Solving for $\theta ^{\ast }$: $\lim\limits_{\varepsilon \rightarrow
0}\theta ^{\ast }\left( r_{1}\right) =p^{-1}\left( \frac{u\left(
r_{1}\right) }{r_{1}}\frac{\left( 1-\lambda r_{1}+\ln \left( r_{1}\right)
\right) }{\tint\limits_{n=\lambda }^{1/r_{1}}u\left( \frac{1-r_{1}n}{1-n}%
R\right) }\right) $
\begin{itemize}
\item Theorem 2: $\theta ^{\ast }\left( r_{1}\right) $ is increasing in $%
r_{1}$\pagebreak
\end{itemize}
\begin{center}
{\large Decision in Period 0}
\end{center}
\begin{itemize}
\item Choose $r_{1}$\ to max expected utility:%
\begin{equation*}
\begin{tabular}{ll}
$\lim\limits_{\varepsilon \rightarrow 0}EU\left( r_{1}\right) =$ & $%
\dint\limits_{0}^{\theta ^{\ast }\left( r_{1}\right) }\frac{1}{r_{1}}u\left(
r_{1}\right) d\theta +\dint\limits_{\theta ^{\ast }\left( r_{1}\right)
}^{1}\lambda u\left( r_{1}\right) +\left( 1-\lambda \right) \cdot $ \\
& $p\left( \theta \right) u\left( \frac{1-\lambda r_{1}}{1-\lambda }R\right)
d\theta $%
\end{tabular}%
\end{equation*}
\item Theorem 3: If $\theta _{LB}$\ is not too large, the optimal $r_{1}$\
must be larger than $1$
\item Liquidity provision is optimal and banks increase welfare even though
panic-based bank runs occur in the optimum: $\theta ^{\ast }\left(
r_{1}\right) >\theta _{LB}\left( r_{1}\right) $\ \pagebreak
\item FOC for $r_{1}$:
\end{itemize}
\qquad $\lambda \dint\limits_{\theta ^{\ast }\left( r_{1}\right) }^{1}\left[
u%
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\left( r_{1}\right) -p\left( \theta \right) Ru%
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\left( \frac{1-\lambda r_{1}}{1-\lambda }R\right) \right] d\theta =\frac{%
\partial \theta ^{\ast }\left( r_{1}\right) }{\partial r_{1}}\left[ \lambda
u\left( r_{1}\right) +\left( 1-\lambda \right) \cdot \right. $
\qquad $\left. p\left( \theta ^{\ast }\left( r_{1}\right) \right) u\left(
\frac{1-\lambda r_{1}}{1-\lambda }R\right) -\frac{u\left( r_{1}\right) }{%
r_{1}}\right] +\dint\limits_{0}^{\theta ^{\ast }\left( r_{1}\right) }\left[
\frac{u\left( r_{1}\right) -r_{1}u%
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\left( r_{1}\right) }{r_{1}^{2}}\right] d\theta $
\begin{itemize}
\item Theorem 4: The optimal $r_{1}$\ is lower than $c_{1}^{FB}$
\item Need to cut back on liquidity provision because of the possibility of
runs\pagebreak
\end{itemize}
\end{document}