%slides_overborrowing.tex
%Slides for Chapter 12 of ``Open Economy Macroeconomics'' by Martín Uribe and Stephanie Schmitt-Grohé, Princeton University Press 2017. 
%© Martín Uribe and Stephanie Schmitt-Grohé.

\input{slides_preamble}

\begin{slide}
\begin{minipage}{5in}
\rightline{\includegraphics[width=4.5in]{USG_cover.eps}}

\centerline{\tiny  Princeton University Press,  2017}
\end{minipage}
\begin{minipage}{4.5in}
\centering 
{\small 
slides 

chapter 12   

financial frictions 

and aggregate instability

}

\vspace{1in}

\end{minipage}
\end{slide}

\topleft{Chapter 12: Financial Frictions and Aggregate Instability
}
\topright{Uribe \& Schmitt-Groh\'e}

\begin{slide}
{\bf Motivation}

$\bullet$ Emerging economies suffer from excess aggregate volatility, as documented in chpater 1. 

$\bullet$ They also suffer from sudden stops, defined as rapid and large reversals in the current account with depressed levels of aggregate activity. 


$\bullet$  Emerging countries are often said to overborrow, that is, to  hold excessive levels of external debt. 

$\bullet$ A central question for policymakers of individual  countries and multilateral institutions  is the desirability and optimal design of macroprudential  policy. 

$\bullet$ This chapter sheds light on these issues from the vantage point of models with  financial frictions taking the form of collateral constraints. 
\end{slide}

\begin{slide}
Questions Analyzed in this chapter: 

\begin{enumerate}
\item  Do collateral constraints amplify regular business cycles? 
\item  Do collateral constraints deepen large recessions? 

\item  Do collateral constraints 
open the door to  nonfundamental uncertainty,  exacerbating aggregate instability? 

\item  Do collateral constraints  lead to overborrowing?

\item  Can the presence of collateral constraints  justify the use of  counter macroprudential policy?
\end{enumerate}
\end{slide}



\begin{slide}

{\bf Narrative of how 
collateral constraints 
 amplify  the cycle}\\
During booms the value of collateral $\uparrow$, borrowing $\uparrow$, more borrowing further  raises the value of collateral, leading to even more borrowing and hence even larger expansions of aggregate demand. 
Busts drive down value of collateral, the collateral constraint  binds {\bf (financial crisis)},  agents deleverage {\bf (Fire Sale of Assets)}, $\rightarrow$  price of collateral $\downarrow$ {\bf (Fisherian Deflation)}   $\rightarrow$  further
 deleveraging $\rightarrow$ rapid
contraction is aggregate demand for goods and services and forced current-account surpluses {\bf (sudden stop)}. 

Formulations of this idea in open economy macroeconomics

$\bullet$ Auernheimer and  Garc\'{\i}a-Saltos (2000)
\\
$\bullet$
Mendoza (2002, 2010) 
\\
$\bullet$
Uribe (2006, 2007) 
\\
$\bullet$
Lorenzoni (2008)
\\
$\bullet$
Jeanne and Korinek  (2010)
\\
$\bullet$
Bianchi (2011)  
\\
$\bullet$
Korinek (2011)
\\
$\bullet$
Benigno, Chen, Otrok, Rebucci, and Young  (2013, 2014)\\
$\bullet$
Schmitt-Groh\'e and Uribe (2016)
\end{slide}




\begin{slide}
\leftline{\bf  Section 12.1 Stock Collateral Constraints}
Preferences: 
$$
\sum_{t=0}^{\infty} \beta^t \ln c_t
$$
Sequential Budget Constraint: 
\begin{equation*}
c_t + d_{t} + q_t k_{t+1}  = y_t + \frac{d_{t+1}}{1+r} + q_t k_t
\end{equation*}
Technology: 
\begin{equation*}
y_t = A_t k_t^{\alpha}
\end{equation*}
Stock Collateral Constraint: 
\begin{equation*}
d_{t+1} \le \kappa q_t k_{t+1}; \quad \mbox{ with $0\le \kappa<1$}
\end{equation*}
{\small Note: Price  $q_t$ is taken as  exogenous by individual agent but is endogenous for the economy as a whole $\rightarrow$ pecuniary externality.}

Capital is in fixed supply, so in equilibrium: $k_t = k>0$ for all $t$.


\end{slide}

\begin{slide}
A (bubble-free) equilibrium is a set of sequences $c_t>0$,   $d_{t+1}$,
$\mu_t$, and $q_t\ge0$ satisfying 
\begin{eqnarray}
\label{eq:ovb_q_bc_eqm}
c_t + d_t  &=& y_t + \frac{d_{t+1}}{1+r}
\\
\label{eq:ovb_q_d_eqm}
\frac{1}{c_t} \left[\frac1{1+r}- \mu_t\right] &=&  \beta\frac{1}{c_{t+1}}
\\
\label{eq:ovb_q_k_eqm}
\frac{q_t}{c_t} 
\left[1 - \kappa \mu_t \right] 
&=& 
 \frac{\beta}{c_{t+1}}\left[ q_{t+1} +  \alpha \frac{y_{t+1}}{k} \right] 
\end{eqnarray}
\begin{equation}
\label{eq:ovb_q_slack_eqm}
\mu_t \left(\kappa q_t k-d_{t+1}\right)=0;\quad \mu_t\ge0;\quad
d_{t+1} \le  \kappa q_t k
\end{equation}
\begin{equation}
\label{eq:ovb_q_limq_eqm}
\lim_{t\rightarrow \infty} (1+r)^{-t}q_t = 0
\end{equation}
\begin{equation}
\label{eq:ovb_q_limd_eqm}
 d_0 = \sum_{t=0}^{\infty}\frac{y_t -c_t}{(1+r)^t}
\end{equation}
given $d_0<\frac{1+r}ry\equiv${\it natural debt limit,} $A_t$ and $y_t \equiv A_tk^{\alpha}$. Instead of \eqref{eq:ovb_q_limd_eqm}, we could have written  $
\lim_{t\rightarrow \infty} (1+r)^{-t}d_t = 0 $.   Assume that 
\[
\beta(1+r) = 1
\]
\end{slide}


\begin{slide}
{\bf  The Steady-State Equilibrium}

Assume that  $A_t = A>0 \quad \forall t.$ Then $  y_t = y\equiv Ak^{\alpha} >0 \quad \forall t.$

A steady-state equilibrium  is  a set of constant sequences  
$c_t  c^*>0$, 
$d_{t+1}=d^*$,
$\mu_t=\mu^*$, 
and $q_t=q^*\ge0$
for $t\ge0$,
satisfying  \eqref{eq:ovb_q_bc_eqm}-\eqref{eq:ovb_q_limd_eqm}. Note, `steady-state equilibrium' and  `steady state,' are not equivalent concepts, for the steady-state equilibrium must respect 
the given iniital condition $d_0$, whereas in the steady state $d_t=d^*$ for all $t\ge 0$.   

By \eqref{eq:ovb_q_d_eqm} $ \fbox{$\mu^* = 0$}$, and  so first two expression in \eqref{eq:ovb_q_slack_eqm} are also satisfied; 
by \eqref{eq:ovb_q_k_eqm} $\fbox{$q^* = \frac{\alpha y }{r k}>0$}$;  
and by \eqref{eq:ovb_q_limd_eqm} 
$\fbox{$c^* = y - \frac r{1+r} d_0>0$}$. 
This expressiion for $c^*$
together with \eqref{eq:ovb_q_bc_eqm} yields
$
\fbox{$d^*=d_0$}.
$
Finally, the collateral constraint
(last expression in \eqref{eq:ovb_q_slack_eqm})
 is satisfied if $d_0\le \kappa q^* k$, or
\[\fbox{$d_0\le \frac{ \kappa \alpha y } r$}\]
This expression yields the highest level of debt sustainable in  a steady-state equilibrium. 
A steady-state equilibrium exists for any level of initial debt $d_0$ that satisfies this condition. 
\end{slide}


\begin{slide}
{\bf No Amplification of Regular Shocks}\\
To illustrate lack of amplification of shocks of regular size, consider a negative unexpected output shock in $t=0$

Initial condition: $t<0$ the economy was in a steady state and $d_0=d^*$. 
Then in period $0$ it is learned that 
\[
A_t = \left\{\begin{array}{ll}A^L & t=0\\A>A^L&t\neq 0\end{array} \right.
\]

$\Rightarrow y_t = \left\{\begin{array}{ll}y^L\equiv A^L k^{\alpha} & t=0\\y\equiv Ak^{\alpha} >y^L&t\neq 0\end{array} \right.$ 


To discuss amplification we must indicate amplification relative to what. Here we mean relative to the economy without the collateral constraint. 

Next we   derive 2 intermediate results. One is the characterization of the steady state and the other is the characterization of the unconstrained economy. Let's begin with the steady state.  
\end{slide}





\begin{slide}
{\bf Response of the Economy Without  A Collateral Constraint (NC).  }
\[
c^{NC} = c^* - \frac{r}{1+r} \left(y-y^L\right)<c^*
\]
\[
d^{NC} = d_0+(y-y^L)>d_0 
\]
\[
ca_0^{NC} = \frac{y^L-y}{1+r}<0 = ca^*
\]
\[
tb_0^{NC} = tb^* + \frac{y^L-y}{1+r} < tb^*
\]
\[
q^{NC} = q^* = \frac{\alpha y }{rk}
\]

 reduce consumption by less than output and use 
 the current account to do so, that is, let trade balance deteriorate and borrow more


\end{slide}
\begin{slide}
Now go back to the collateral-constraint economy (CC). 

If the NC solution satisfies the equilibrium conditions of the CC economy, then we say that  there is lack of amplification. Need to check if   
\[
d_{t+1}\le \kappa q_t k \quad ?
\]
\[\mbox{Recall  }
d_1^{NC} = d_0 + (y-y_L); \quad q_0^{NC} = q^*; \mbox{ and } d_0 >
 \kappa q^* k
\]
It does if $d_0 + (y-y_L) \le \kappa q^* k$, that is, if 
\[
\fbox{$y-y^L\le \kappa q^* k - d_0$}
\]
then there is no amplification. 

$\bullet$ if shock is small, ie $y-y^L$ small. 
\\
$\bullet$ if not too indebted, ie $d_0$ small. 
\\
$\bullet$ if weaker constraint, ie $\kappa$ large. 

How often are those conditions encountered in a more realistic stochastic economy? 

\end{slide}
\begin{slide}
{\bf Quantitative Result: No amplification of regular business cycles}

Mendoza (AER, 2010) makes this point in the context of a quantitative stock collateral constraint model. The model is  more empirically realistic and hence much more complex than our model here. It has capital accumulation, imported inputs, labor, working capital constraints and is driven by 3 shocks, TFP, interest rate, and import price. Each shock is discretized with 2 values, so that the total number of exogenous grid points is 8. 
\end{slide}
\begin{slide}
Lack of amplification as documented in  Mendoza (AER, 2010)

Stock collateral constraint: $d_{t+1} + \mbox{working capital} \le \kappa q_t k_{t+1}$

{\small $\kappa = 0.20$; picked to match observed frequency of a Sudden Stop (3.3\%), which is defined as a binding CC constraint {\bf and} trade-balance-to-output ratio 2 percentage points above mean } 

\centerline{\begin{tabular}{lrr}
\hline
\hline 
Variable&\multicolumn{2}{c}{Std. Dev. in \%}\\
&No CC& CC \\
\hline
output&3.90&3.85\\
consumption&4.21&3.69\\
investment&13.85&13.45\\
%debt to output&17.57&12.57\\
$q_t$&3.33&3.23\\
%\\
%leverage&8.32&5.86\\
\hline \hline
\end{tabular}
}
\centerline{\tiny Source: Table 3 of Mendoza (2010).}

mean debt to output is 32.6\% in no CC and 10.4\% in CC economy. 

Prob of binding CC is 9.54\%

%Mean leverage is 0.266 in No CC model and 0.232 in CC model with $\kappa = 0.3$

%What surprises me is that nevertheless the probability of a binding CC is 9.54 percent. 
\end{slide}



\begin{slide}
We have thus provided a partial answer to Q1, namely, we have shown analytically and numerically that  a stock collateral constraint  may not  amplify the cycle.



\end{slide}



\begin{slide}
\underline{What if the contraction is large?}

Large$:= y-y^L > \kappa q^* k - d_0$ 

We will next show that in response to a large negative output shock: 

\begin{tabular}{ll}
$\bullet$ Fisherian deflation& $q_0<q_0^{NC}$\\
$\bullet$ Amplification of contraction of demand& $c_0<c_0^{NC}$\\
$\bullet$ Fire sale/deleveraging& $d_1<d_1^{NC}$\\
$\bullet$ Less TB deterioration& $tb_0>tb_0^{NC}$\\
$\bullet$ Less CA deterioration& $ca_0>ca_0^{NC}$\\
\end{tabular}

... and welfare is lower than in the NC economy.

Hence, yes, financial frictions amplify busts when they trigger a binding collateral constraint. 

\end{slide}

\begin{slide}

Intuition: $A^L\downarrow$; HH want to borrow more to smooth consumption. hits CC. HH thinks if he  sells 1 unit of $k_{t+1}$ he gets $1\times q_t$. and use  proceeds to consume and to reduce debt 
\[
\Delta(\kappa q_t k_{t+1}) = \kappa q_t \Delta k_{t+1}
\]
Must use $\kappa q_t$ to reduce debt. 
But can still use $(1-\kappa)q_t >0$ to consume more (recall $\kappa<1$). 

However, in equilibrium selling capital cannot  increase $c$.  Capital is in fixed supply, and the fall in prices ends up being so large that the decline in the value of collateral makes  debt go down and $c$ in fact fall more than output, or the trade balance to improve. 
\end{slide}

\begin{slide}
To show this formally, combine \eqref{eq:ovb_q_d_eqm} with \eqref{eq:ovb_q_k_eqm} to obtain
\begin{equation}
\label{eq:ovb_q_q_betatilde}
 q_{t+1} =  \tilde{\beta}^{-1}_t  q_t -r q^*
\end{equation}
with
\begin{equation}
\label{eq:ovb_q_betatilde}\tilde{\beta}_t \equiv \beta \frac{1 - (1+r) \mu_t }{1 - \kappa \mu_t } 
\end{equation}
Note that $\tilde{\beta}_t>0$
\[
\begin{array}{ll}
\tilde{\beta}_t 
= \beta& \mbox{if } \mu_t=0 
\\
\tilde{\beta}_t 
< \beta & \mbox{if } \mu_t>0 
\end{array}
\]

When constraint is binding $(\mu_t>0)$, it is as if agents are more impatient. 
\end{slide}

\begin{slide}
{\bf  Figure 12.1 Phase Diagram of the Price of Capital}
\\\begin{minipage}{6in}\rightline{\includegraphics[width=6.5in]{ovb_q_phaseq.eps}}
%produced with plot_q_phase.m  
%in directory \uribe\book\overborrowing\fire_sales
\end{minipage}\begin{minipage}{4in}in any eqm: 

$q_t\le q^*$

with large shock CC binds in at least 1 period

let $T$ be the first period it binds, $\tilde{\beta}^{-1}_T>\beta^{-1}>1$ 

$\Rightarrow q_T<q^*$

$\Rightarrow q_0<q^*$


\end{minipage}

{\bf We have therefore shown that in this economy collateral constraints exacerbate the effects of large negative shocks.} (that is we have addressed Q2)


\end{slide}


\begin{slide}
{\bf Section 12.2 Stock Collateral Constraints and Self-fulfilling Financial Crises}

Equilibrium multiplicity can  also be a contributor excess volatility. 

The existing related literature has either ignored this issue (exceptions are Menodoza 2005 and  Jeanne and Korinek 2010) or added assumptions that are meant to guarantee a unique equilibrium.

We now  show that  self-fulfilling financial crises arise for  plausible specifications in the current model. The self-fulfilling crisis coexists with an unconstrained equilibrium and features: a contraction in demand $c\downarrow$, a Fisherian deflation $q\downarrow$, and a fire sale $d\downarrow$.

Intuition: Agents become pessimistic and believe the value of collateral will be low, based on this belief they deleverage (firesale). The fire sale results in lower prices of capital, confirming the pessimistic beliefs.  
\end{slide}


\begin{slide}

Assume that 

$A_t = A$ for all $t\ge0$ 

$d_0<\kappa q^*k$ ($\Rightarrow$ the unconstrained equilibrium exists)

We wish to show that there exists a second equilibrium in which the collateral constraint binds in period $0$ and the economy is in a steady state beginning in period 1.
\end{slide}

\begin{slide}
\begin{eqnarray}
\label{eq:ovb_q_pdv_uniqueness}
d_0 &=& \frac{1+r}ry -\frac{c_1}r - c_0,
\\
\label{eq:ovb_q_bc_uniqueness}
c_1 &=& y-\frac{r}{1+r}d_1,
\\
\label{eq:ovb_q_d_uniqueness}
\frac1{c_0}[1-(1+r)\mu_0] 
&= &
\frac1{c_1}
\\
\label{eq:ovb_q_k_uniqueness}
\frac{q_0}{c_0}(1-\kappa\mu_0) 
&=& \frac{\beta}{c_1}(q^*+\alpha y/k)
\\
\label{eq:ovb_q_slack_uniqueness}
\mu_0(\kappa q_0k-d_1)&=&0
\\
\label{eq:ovb_q_cc_uniqueness}
d_1 &\le& \kappa q_0k
\end{eqnarray}
Now solve~\eqref{eq:ovb_q_pdv_uniqueness}-\eqref{eq:ovb_q_k_uniqueness} for obtain $q_0$ as an increasing function of $d_1$ 
\begin{equation}
\label{eq:ovb_q_q0_d1}
\kappa q_0(d_1) k = \kappa q^* k
\left[
\frac{(1+r)c^*+d_1-d_0}
{(1+r)c^*+(\kappa-r)(d_1-d_0)}
\right].
\end{equation}

\end{slide}

\begin{slide}


\centerline{\bf \small Figure 12.2 Stock Collateral Constraints and Self-fulfilling Financial Crisis}

\begin{minipage}{6in}\rightline{\includegraphics[width=6.5in]{ovb_q_plot_example_internal.eps}}
%produced with  plot_example_internal.m in directory 
%\uribe\book\overborrowing\fire_sales
\end{minipage}\begin{minipage}{4in}\rightline{
$\kappa q_0(d_1) k = \kappa q^* k
\left[
\frac{(1+r)c^*+d_1-d_0}
{(1+r)c^*+(\kappa-r)(d_1-d_0)}
\right]$
}

\bigskip 

Sufficient condition \\for $\underline{d}>0$: $d_0>y$ 
\end{minipage}

\end{slide}

\begin{slide}
Observations on the figure: \\
-- at least 2 equilibria: Unconstrained equilibrium at point $A$ and constrained eqm at point $B$
\\
-- at $B$: CC is binding $\rightarrow$ financial crisis; $q$ is low, ie crisis has Fisherian deflation; $d_1<d_0$, ie fire-sale or deleveraging. 
\\
-- sufficient condition for existence of eqm $B$: $d_0/y>1$. If time unit is a quarter, then debt to annual output of 25\% is sufficient.
\\
--bad economic fundamentals make the economy more vulnerable to a self-fulfilling financial crisis.


-- Point $B$ is welfare inferior to point $A$ and  at $B$ debt is lower, therefore at $B$ there is  `underborrowing'.
\\
-- we have either borrowing the optimal amount or less, that is, we have no overborrowing. 
This finding is at odds with the overborrowing result stressed in the literature.


\end{slide}

\begin{slide}
{\bf  Section 12.3 Flow Collateral Constraints}

{\tiny Based on  Schmitt-Groh\'e and Uribe (NBER WP 22264, May 2016) }




{Households maximize}
\begin{equation*}
\label{eq:ovb_flowcc_utility}
\mathbb{E}_0 \sum_{t=0}^{\infty}\beta^t \, \frac{c_t^{1-\sigma}-1}
{1-\sigma}
\end{equation*}
subject to
\begin{equation*}
\label{eq:ovb_flowcc_aggregator}
c_t =  \left[a {c^T_t}^{1-1/\xi} + (1-a){ c^N_t}^{1-1/\xi}
\right]^{1/(1-1/\xi)}
\end{equation*}
\begin{equation*}
\label{eq:ovb_flowcc_budget_constraint}
c^T_t + p_t c^N_t + d_t =  y_t^T + p_t y^N
+\frac{d_{t+1}}{1+r_t}
\end{equation*}
\begin{equation*}
\label{eq:ovb_flowcc_collateral_constraint}
d_{t+1} \le \kappa( y_t^T + p_t y^N)
\end{equation*}

where $c_t=$consumption; $c^T_t,c^N_t=$consumption of tradables, nontradables; $d_{t+1}=$ debt assumed in $t$ and maturing in $t+1$; $y^T_t,y^N=$ endowments of tradables, nontradables; $p_t=$relative price of nontradables; 
 $r_t=$ interest rate. 
\end{slide}

\begin{slide}
{\bf \large Observations}

$(1)$ The last constraint is the flow collateral constraint (CC). It says that the amount of debt issued in period $t$, $d_{t+1}$, cannot exceed a fraction $\kappa$ of income, $y^T_t+p_ty^N$. 

$(2)$ From the individual agent's point of view, the CC is well behaved:
the larger is $d_{t+1}$, the closer he gets to hitting the collateral constraint. This is because he takes as exogenous  all of the objects on the RHS of the collateral constraint (in particular $p_t$).

$(3)$ Also, from the perspective of the individual agent, the collateral constraint defines a convex set of feasible debt levels: if $d'$ and $d''$ satisfy the collateral constraint, then so does the debt level  $\alpha\,d'+(1-\alpha)d''$, for any $\alpha\in[0,1]$.

$(4)$ As we will see shortly,  (2) and (3) do not hold  from an aggregate perspective. 
\end{slide}

\begin{slide}
\centerline{\bf \large Three Equilibrium Conditions of Interest}
\[
d_{t+1} \le \kappa( y_t^T + p_t y^N)
\]
\[
p_t = \frac{1-a}{a}
\left(
\frac{c_t^T}{y^N}
\right)^{1/\xi}
\]
\[
c^T_t + d_t =  y_t^T 
+\frac{d_{t+1}}{1+r_t}
\]

These three conditions give rise to the following  {\bf equilibrium collateral constraint}
\[
\fbox{$
d_{t+1} \le \kappa
\left[
 y^T_t
+ 
\left(
\frac{1-a}a
\right)
\left(
y^T_t +\frac{d_{t+1}}{1+r_t}-d_t
\right)^{1/\xi}{y^N}^{1-1/\xi}
\right]
$}
\]
\end{slide}

\begin{slide}
{\bf \large Observations}

(1)  $d_{t+1}$, appears on both the RHS and the LHS of the equilibrium CC. 

(2)  Because $\xi>0$,  the equilibrium value of collateral  increases with the level of debt, giving rise to the possibility that the higher is $d_{t+1}$ the less tight is the collateral constraint. 

(3)  Moreover, collateral 
(i.e., the RHS of the collateral constraint) 
 is in general nonlinear in $d_{t+1}$, giving rise to the possibility that the set of debt levels that satisfy the equilibrium collateral constraint is nonconvex, that is, if $d'$ and $d''$ satisfy the equilibrium collateral constraint, then $\alpha\,\,d'+(1-\alpha)d''$ may not for some $\alpha \in (0,1)$.
\end{slide}
 
\begin{slide}
{\bf Section 12.4 Flow Collateral Constraints and Self-Fulfilling Financial Crises}

{ Some Simplifying Assumptions}

(1)  $r_t=r$, $y^T_t= y^T$. 

(2) $\sigma=1/\xi = 2$. 

(3) $\beta(1+r) = 1$. 

(4) $a=0.5$.  

(5) $y^N=1$. A normalization.  

The equilibrium collateral constraint then becomes
\[
\fbox{$
d_{t+1} \le \kappa
\left[
 y^T
+ 
\left(
y^T +\frac{d_{t+1}}{1+r}-d_t
\right)^2
\right]
$}
\]
\end{slide}

\begin{slide}
{\bf \large The Unconstrained Equilibrium}\\
 Euler equation says\\
\centerline{$\frac{c^T_{t+1}}{c^T_t} = \beta(1+r)=1$}\\
A constant consumption path implies a constant debt path
\[
d_t = d_0
\]
for all $t$. 

{\bf \large Constrained Equilibria}\\
In the event of a binding collateral constraint, the Euler equation becomes
\[
\frac{c^T_{t+1}}{c^T_t} =
\frac1{\sqrt{1-(1+r)\mu_t}}
\]
where $\mu_t$ is the multiplier associated with the collateral constraint. A binding collateral constraint introduces deviations from perfect consumption smoothing. 
\end{slide}

\begin{slide}
\begin{center}
{\bf The Long-Run Equilibrium Collateral Constraint}
\[
\fbox{$
d \le \kappa
\left[
 y^T
+ 
\left(
y^T -rd/(1+r)\right)^2
\right]
$}
\]
\centerline{\includegraphics[width=6in]{ovb_flowcc_ss_slides.eps}}
%produced with plot_flowcc_ss.m
%in directory z:\uribe\book\overborrowing\flowcc_analytical
%for slides set slides_on=1; 
\end{center}
\end{slide}

\begin{slide}
{\bf \large Observations}

(1) The RHS of the long-run (LR) collateral constraint is quadratic. 

(2) The expression under the power of 2 is steady state consumption, $y^T-rd/(1+r)$. 

(3) The LR collateral constraint achieves a minimum when long-run consumption is 0, that is, at the natural debt limit. 

(4) This means that for all relevant values of debt (i.e., all values below the natural debt limit), the LR collateral constraint is well behaved, that is, the larger is debt the tighter it gets. 

(5) For any initial debt $d_0<\tilde{d}$, 
an equilibrium is $d_{t+1}=d_0$ and $c^T_t = y-rd_0/(1+r)$ for all $t$. In these equilibria, the collateral constraint never binds. 

(6) The LR collateral constraint binds at $\tilde{d}$. No steady state equilibrium is possible to the right of $\tilde{d}$. 


\end{slide}

\begin{slide}
{\bf  The Short-Run Equilibrium Collateral Constraint}
{\bf  and  Multiple Equilibria}
\[
\fbox{$
d \le \kappa
\left[
 y^T
+ 
\left(
y^T +\frac{d}{1+r}-d_0
\right)^2
\right]
$}
\]
\centerline{\includegraphics[width=6in]{ovb_flowcc_multiple_slides.eps}}
%%in directory z:\uribe\book\overborrowing\flowcc_analytical
%You must change the value of d0 in line 140 to d0=6.7; 
%You must also adjust line 185 for the text. search for ssg
%produced with plot_flowcc_ss.m
%in directory z:\uribe\book\overborrowing\flowcc_analytical

\end{slide}


\begin{slide}
{\bf \large Observations}

{\small (1) The slope of the short-run (SR)  CC is proportional to  $c^T_0$. So an equilibrium must be on an upward sloping range of the SR CC. 

(2) Suppose the initial debt level is $d_0$.  The solution $d_{t+1}=d_0$ for all $t\ge 0$ does not violate the SR CC, because point $A$ lies above the 45-degree line. Since this solution also satisfies all other equilibrium conditions, it is indeed an equilibrium. Are there more? 

(3) Point $C$ also satisfies the SR CC, since it is on the 45-degree line. But, because the slope of the  SR CC is negative, $c^T_0$ is negative. So we rule out $C$. 

(4) Another candidate is 
point $B$. 
 This solution satisfies the SR CC since it is on the 45-degree line. It also satisfies the LR CC. And because the slope of the SR CC is positive, 
$c^T_0$ is positive. 
 But we must check that the Euler equation is satisfied at point $B$ for $\mu_0\ge0$ (next slide). 

(5) At point $B$, the economy experiences a self-fulfilling financial crisis, caused by an arbitrary desire to deleverage. 
\par }
\end{slide}

\begin{slide}
{\bf \large The Euler Equation in Period 0}
\[
\frac{c^T_1}{c^T_0} = \frac1{\sqrt{1-\mu_0(1+r)}}
\]
where $\mu_0$ is the multiplier associated with the collateral constraint. Is $\mu_0\ge 0$ at point $B$? Yes, because at that point $c^T_1/c^T_0>1$.  This corroborates that point $B$ on the previous graph is indeed an equilibrium. 


The equilibrium at point $B$ is costly in terms of welfare, because the initial deleveraging requires a drop in consumption, which implies  a deviation from the perfect consumption smoothing induced by equilibrium $A$. 
\end{slide}


\begin{slide}
\begin{center}
{\bf Three Equilibria}

Figure 12.5
\centerline{\includegraphics[width=7.5in]{ovb_flowcc_multiple3_slides.eps}}

%produced with plot_flowcc_multiple3.m
%in directory z:\uribe\book\overborrowing\flowcc_analytical
\end{center}
\end{slide}
\begin{slide}
{\bf \large Observations}

{\small
(1) Suppose the initial debt level is $d_0$.  As before the  solution $d_{t+1}=d_0$ for all $t\ge 0$ does not violate the SR CC, because point $A$ lies above the 45-degree line. Since this solution also satisfies all other equilibrium conditions, it is indeed an equilibrium. 

(2) Again another candidate is point $B$. 
 This solution satisfies the SR CC since it is on the 45-degree line. It also satisfies the LR CC. And because the slope of the SR CC is positive, 
$c^T_0$ is positive. 
And by the same argument as before, we can show that $\mu_0>0$. So $B$ is an equilibrium. 

(3) Point $C$ also satisfies the SR CC, since it is on the 45-degree line. And this time  the slope of the  SR CC at point $C$ is positive, so $c^T_0$ is positive. We need to check that $\mu_0>0$, which is indeed the case here.  


(4) Point $C$ entails a larger drop in the value of collateral and more deleveraging than the crisis associated with point $B$.   Thus, the contraction in aggregate demand is also larger making point $C$ a more severe self-fulfilling financial crisis than point $B$. 
\par }
\end{slide}
\begin{slide}
\begin{center}
{\bf A Unique Equilibrium}

\bigskip

\centerline{\includegraphics[width=5.5in]{ovb_flowcc_unique_slides.eps}}

%produced with plot_flowcc_unique.m
%in directory z:\uribe\book\overborrowing\flowcc_analytical
\end{center}
\end{slide}

\begin{slide}
{\bf Section 12.5: Debt Dynamics in a Stochastic Economy \\with A Flow Collateral Constraint}

Questions:\\
$\bullet$ How to solve this model numerically\\
$\bullet$ How to handle the possibility of multiplicity\\
$\bullet$ How to calibrate the economy\\
$\bullet$ What is the effect of the  constraint on eqm debt dynamics\\
$\bullet$ Will the economy hit the collateral constraint\\


\end{slide}



\begin{slide}
{\bf The Flow Collateral Constraint}

\[
d_{t+1} \le \kappa  (y^T_t + p_t y^N_t)
\]





\end{slide}



\begin{slide}
{\bf \large Equilibrium:}   
  $\{c_t$, $c^T_t$, $d_{t+1}$,  $\lambda_t$, $\mu_t$, $p_t\}$     satisfying  
\begin{eqnarray}
%\label{eq:ovb_flowcc_stochastic_bc}
c^T_t + d_t &=& y^T_t + \frac{d_{t+1}}{1+r_t}
\\
c_t 
&
= 
&
\left[
a {c^T_t}^{1-\frac1{\xi}}
+(1-a) {c^N_t}^{1-\frac1{\xi}}
\right]^{\frac1{1-\frac1{\xi}}}
\\
\lambda_t
& = &
a  c_t^{-\sigma}
\left(
\frac{c^T_t}{c_t}
\right)^{-1/\xi}
\\
\lambda_t
\left[
\frac1{1+r_t} -\mu_t
\right]
&=&
\beta \mathbb{E}_t \lambda_{t+1}
\\
p_t
& = &
\frac{1-a}a \left(
\frac{c^T_t}{y^N}
\right)^{1/\xi}
\\
c^N_t 
&=&
y^N_t
\end{eqnarray}
\begin{equation}
d_{t+1} \le \kappa\left[ y^T_t
+ p_ty^N_t\right], \quad \mu_t \left[
\kappa
\left(
y^T_t + p_ty^N_t
\right)
 -d_{t+1}
\right]
=0,
\quad
\mu_t \ge 0
\end{equation}
given exogenous $\{y^T_t, y^N_t, r_t\}$ and $d_0$. 
\end{slide}


\begin{slide}
{\bf \large Exogenous Driving  Processes:}

{\bf Empirical Measure of $y^T_t$:}  sum of Argentine GDP in agriculture, manufacturing, fishing, forestry, and mining. Quadratically detrended. 

{\bf Empirical Measure of $r_t$:} Sum of Argentine EMBI+ plus 90-day Treasury-Bill rate minus a measure of U.S.\ expected inflation. 

{\bf Constant Endowment of $y^N_t$:} $y^N_t = y^N = 1$.

\begin{center}
\begin{tabular}{cc}
{\tiny \bf  (a) Traded Output, $y^T_t$}&
{\tiny \bf (b) Interest Rate, $r_t$}\\
\includegraphics[width=4in]{dnwr_lgdp_traded_data.eps}&
\includegraphics[width=4in]{dnwr_arg_int_rate_data.eps}
\end{tabular}
\end{center}
\end{slide}


\begin{slide}
Estimate the following AR(1) system using Argentine data over the period 1983:Q1---2001:Q3:


\begin{equation*}
\label{eq:AR(1)_yT_and_r}
\left[
\begin{array}{c}
\ln  y^T_t\\
\ln \frac{1+r_t}{1+r}
\end{array}
\right]
=
A 
\left[
\begin{array}{c}
\ln  y^T_{t-1}\\
\ln \frac{1+r_{t-1}}{1+r}\end{array}
\right]
+ 
\epsilon_t, \quad \epsilon_t \sim N\left(\emptyset, \Sigma_{\epsilon}\right)
\end{equation*}

{\bf  OLS Estimate}

\[
A = \left[
\begin{array}{cc}
0.79&-1.36\\
-0.01&0.86
\end{array}
\right] ;\quad 
\Sigma_{\epsilon} = \left[
\begin{array}{cc}
0.00123&-0.00008\\
-0.00008&0.00004
\end{array}
\right]; \quad 
  r=0.0316.\]
\end{slide}


\begin{slide}
\centerline{\bf Some Unconditional Summary Statistics of the Driving Process}

\bigskip

\centerline{\begin{tabular}{lcc}
\hline\hline
Statistic&$y^T_t$&$r_t$\\
\hline
Std. Dev.&12\%&6\%yr\\
Serial Corr.&0.95&0.93\\
Corr($y^T_t,r_t$)&\multicolumn{2}{c}{-0.86}\\
Mean&1&12\%yr\\
\hline\hline
\end{tabular}}

Comments:\\ (1) High volatility of tradable $y^T_t$ and $r_t$; \\(2) negative  correlation between $y^T_t$ and $r_t$, when it rains it pours; \\(3) High mean country interest rate.  
\end{slide}

\begin{slide}



\centerline{{\bf How to solve the model numerically?} --- 3 complications}


{\bf 1.} There is an occasionally binding constraint. $\rightarrow$ no perturbation. 

\leftline{{\bf 2.} There is an externality. $\rightarrow$ cannot be cast as value function problem. }

$\Rightarrow$ These two considerations suggest using an  Euler equation iteration procedure over a discretized state space. 

\vspace{0.4in}

\leftline{{\bf 3.} There may exist multiple eqa. $\rightarrow$ impose eqm selection criterion.}
 

\end{slide}



\begin{slide}
{\bf Equilibrium Selection} 



{\bf (b)   }If for a given
current state $(y^T_t,r_t,d_t)$ there are one or more  values of $d_{t+1}$ for which
all equilibrium conditions are satisfied  pick the largest one for which the collateral constraint is binding. 

{\bf (c)   }
If for a given
current state $(y^T_t,r_t,d_t)$ there are one or more  values of $d_{t+1}$ for which
all equilibrium conditions are satisfied 
pick the smallest one for which the collateral constraint is binding. 


Criterion (b) favors  equilibria like point $B$ and criterion (c) favors equilibria like point $C$ in figure 12.5.
[   One could in principle adopt other equilibrium selection criteria, including ones in which nonfundamental uncertainty (sunspot realizations) affects the real allocation. ]




\end{slide}



\begin{slide}
\centerline{\bf \large Discretization of the Driving Process}

$\bullet$ How to pick the grid?

The first and last values  of the grids 
for $\ln y^T_t$ and $\ln (1+r_t)/(1+r)$
are set to $\pm \sqrt{10}$ times the 
respective standard deviations ($\pm 0.3858$ and $\pm 0.0539$, respectively). 


Why $\sqrt{10} $ std(x)? Somewhat arbitrary, try to get low probability of visiting the endpoints of the grid. 


$\ln y^T_t$ grid has  21 equally spaced points\\

$\ln \frac{1+r_t}{1+r}$  grid has 11 equally spaced points\\

Why 21 and 11. Again somewhat arbitrary. Tradeoff between getting the variance and serial correlation right and not having too many gridpoints. 

\end{slide}


\begin{slide}

$\bullet$ How to construct the  transition probability matrix?

Construct the transition probability matrix of the state $(\ln y^T_t , \ln((1+r_t)/(1+r)))$ using the  simulation approach, in particular the Matlab code \verb7tpm.m7 proposed in Schmitt-Groh\'e and Uribe (2009), which consists in simulating  a time series of length 1,000,000 drawn from the AR(1) system above and  associating each observation in the time series  with one of the 231 possible discrete states by distance minimization.
 
The resulting discrete-valued time series is used to compute the probability of transitioning from a particular discrete state in one period to a particular discrete state in the next period. 

The resulting transition probability matrix, stored in \verb7tpm.mat7,  captures well the covariance matrices of order 0 and 1.
\end{slide}


\begin{slide}

Note. Some combinations of $(y^T_i, r_i)$ are never visited. We remove those states, resulting in 145 possible pairs $(y^T_i, r_i)$ instead of 231. 
Thus we have $ny=145$ grid points for the exogenous state. 


An alternative method for computing the transition probability matrix of the exogenous state is the quadrature based method proposed by Tauchen and Hussey (1991).


\end{slide}


\begin{slide}
\centerline{ \bf Functional Forms and Parameter Values}


$\bullet$ Following the business-cycle literature we assume that the time unit is one quarter. 

$\bullet$ $\kappa = 1.2$ ($\Rightarrow$upper limit on debt 
$= 30$ percent of annual output). 


$\bullet$ Assume a CRRA form for preferences and a CES form for the aggregator  of tradables and nontradable 
\[
U(c) = \frac{c^{1-\sigma}-1}{1-\sigma}
\]
\[
A(c^T,c^N) = 
\left[
a (c^T)^{1-\frac{1}{\xi}}
+ (1-a) (c^N)^{1-\frac{1}{\xi}}
\right]^{\frac1{1-\frac1{\xi}}}
\]
with $\sigma=2, \xi=1/2, a=0.26$. 





\end{slide}




\begin{slide}
{\bf The case of Equal Intra- and Intertemporal Elasticities of Substitution}

We consider the case
\[
\xi = \frac1{\sigma}
\]

Why this case is of interest:

$\bullet$ It is  empirically plausible, see Akinci (2011). 

$\bullet$ Preferences become separable in $c^T_t$ and $c^N_t$, which in some applications facilitates the characterization of equilibrium (although not in the application we are interested in here.) 
\[
U(A(c^T_t,c^N_t)) = 
\frac{a{c^T_t}^{1-\sigma}
+
(1-a) {c^N_t}^{1-\sigma} -1}
{1-\sigma}
\]

\end{slide}

\begin{slide}
{\bf We need to make the model  stationary
 --- how?}

Make households impatient
\[
\beta (1+r) < 1.
\]

Specifically, pick $\beta=0.9635$   to match the average external-debt-to-output ratio (in the model without the collateral constraint)
of 23 percent per year observed in Argentina over the period 1983-2001 (Lane and Milesi-Ferretti, 2007). This value implies that  $\beta (1+r) =0.9939<1$ which means that the subjective discount rate,$\beta^{-1}-1$,  is 63 basis points per quarter below the pecuniary discount rate, $r$. 

{\tiny Note that here, contrary to the analysis in section 4.10.8 of Chapter 4, we are able to match the desired debt to output ratio, with a relatively small amount of impatience. [Suggestion for replication: use a different stationarity inducing device, say EDEIR,  and analyze the sensitivity of the results to this modification.] }
\end{slide}





\begin{slide}
{\bf How to pick the grid for debt, $d_t$}

Use $nd = 501$ equally spaced points for $d_t$ in the interval $[\underline{d}, \overline{d}]$

How to pick the first and last points of the grid?

First, try $\underline{d} = 0$ and $\overline{d} = d^n$, where
$d^n$ is equal to the  natural debt limit, 
\[
d^n \equiv  \underline{y}^T\frac{1+\overline{r}}{\overline{r}}=8.3416.
\] 

It turns out that $d^n$ is never visited in eqm.
To have a more efficient grid, we thus set  $[\underline{d}, \overline{d}]  = [0,3.5]$. 


Overall grid size: $n = ny\times nd = 145 \times 501 = 72,645$ points.
\end{slide}


\begin{slide}
Other issues complicating the numerical algorithm: 


$\bullet$  for some current states $(y^T_t,r_t,d_t)$
 there exists no  value of $d_{t+1}$ that ensures both the satisfaction of the collateral constraint and positive consumption of tradables. 

$\bullet$  some debt choices   lead with positive probability to areas of the state  space for which either consumption is non-positive or 
 the aggregate collateral constraint is violated in the next period. 

$\Rightarrow$ to address those issues we introduce a  path-finder refinement of the solution algorithm that avoids such debt choices. 

 
The Matlab program \verb7constrained.m7 computes the equilibrium policy function. It uses the refinement \verb7pathfinder.m7.    And \verb7simu.m7 produces simulated time series of variables of interest.  


\end{slide}


\begin{slide}
\centerline{\bf Summary of the Calibration of the Flow-Collateral-Constraint Economy}

\bigskip

\centerline{\tiny
\begin{tabular}{ccl}
\hline\hline
Parameter&Value&Description\\
\hline
$\kappa$&$0.3\times 4$&Parameter of collateral constraint \\
$\sigma$&2&Inverse of intertemporal elasticity of consumption\\
$\beta$&    0.9635&Quarterly subjective discount factor\\
$r$&    0.0316&Steady state quarterly country  interest rate\\
$\xi$&0.5&Elasticity of substitution between tradables and nontradables\\
$a$&0.26&Share of tradables in CES aggregator \\
$y^N$&1&Nontradable output\\
$y^T$&1&Steady-state tradable output\\
\hline
\multicolumn{3}{c}{Discretization of State Space}\\
\hline
$n_{y^T}$ &21& Number of  grid points for $\ln y^T_t,$ equally spaced\\
$n_{r}$ &11& Number of  grid points for $\ln [(1+r_t)/(1+r)],$ equally spaced\\
$n_d$&501&Number of grid points for $d_t,$ equally spaced\\
$
\left[
\ln \underline{y}^T,
\ln \overline{y}^T
\right]$&[-0.3858,0.3858]&Range for  tradable output\\
$\left[
\ln 
\left(
\frac{1+\underline{r}}
{1+r}
\right),
\ln 
\left(
\frac{1+\overline{r}}
{1+r}
\right)
\right]$
&[-0.0539,0.0539]&Range for interest rate\\
$[\underline{d},\overline{d}]$&[0,3.5]&Range for debt\\
\hline\hline
\end{tabular}
}
\centerline{\tiny Note. The time unit is one quarter. }
\end{slide}

\begin{slide}
\centerline{\bf \large  Multiple Binding Debt Levels   In the Stochastic Economy\label{fig:ovb_flowcc_multiple3_stochastic}}


\centerline{\includegraphics[width=5.5in]{ovb_flowcc_multiple3_stochastic.eps}}

{\tiny Note. The value of collateral is evaluated at the state $(y^T_t,r_t,d_t)=(0.7633,0.0541,1.5960)$. The state space has  26,024  states (or 36 percent of all states)  with multiple binding debt levels.  \par}
%produced with 
%study_slope.m
%in directory 
%z:\joint\underborrowing\mfiles\quarterly
\end{slide}

\begin{slide}

\centerline{\bf Figure 12.6: External  Debt Densities\label{fig:ovb_flowcc_plotd}}

\centerline{\includegraphics[width=7.5in]{ovb_flowcc_plotd.eps}}

{\small Note. Replication Matlab program \verb7plotdu.m7.}
%produced with plotdu.m  
%set if 1>2
%in directory %z:\joint\underborrowing\mfiles\quarterly
\end{slide}


\begin{slide}
Observations on the figure

--Equilibrium selection criterion (b) gives rise to a different debt density than equilibrium selection criterion (c).  Difference in densitities suggests  that there is also equilibrium multiplicity in the stochastic economy. 

--more pessimistic criterion (c) yields mean debt, 12\% of output. 

--criterion (b) yields slightly higher mean debt of 12.4\% of output. 
 
--ignoring eqm selection might cause non-convergence  of the Euler-equation iteration procedure. 
\end{slide}


\begin{slide}
\centerline{\bf The Unconstrained Economy}

(no externality, eqm can be expressed as solution to a value function problem)

\[
v(y^T,r,d) = \max_{c^T,d'}
\left\{
U(A(c^T,y^N)) + \beta \mathbb{E}
\left[
v({y^T}',r',d')\left|y^T,r
\right.\right]
\right\}
\]
subject to 
\[
c^T+d = y^T + \frac{d'}{1+r},
\] 
where a prime superscript denotes next-period values. 


\end{slide}

\begin{slide}

\begin{minipage}[h]{5in}
\leftline{\tiny \bf  Figure 12.7: External Debt Densities }
\leftline{\tiny \bf  With And Without 
a Collateral Constraint\label{fig:ovb_flowcc_plotdu}}

\bigskip

\rightline{\includegraphics[width=5in]{ovb_flowcc_plotdu.eps}}
 
{\tiny Replication program \verb7plotdu.m.7}
%produced with plotdu.m  
%set if 1<2
%in directory %z:\joint\underborrowing\mfiles\quarterly
\end{minipage}
\begin{minipage}{4in}

\leftline{\bf Observations} 

\vspace{1cm}

{\small 

\leftline{$\bullet$ mean debt is larger without constraint}

{$\bullet$ debt to output is 23\% versus 12\% with constraint \par}


\leftline{$\bullet$ constraint compresses debt distribution}

\vspace{1cm}

\begin{tabular}{llll}
&(b)&(c)&NoCC\\
\hline 
$E\left(d_t\right)$&1.74&1.69&2.89\\
$E\left(\frac{d_{t+1}}{4y_t}\right)$&12.4&12.0& 23.3\\
std($d_t$)&0.18&0.17&0.67\\
std$\left(\frac{d_{t+1}}{4y_t}\right)$&0.022&0.023&0.121\\
corr$(d_t, d_{t-1})$&0.983&0.984&0.999 
\end{tabular}}
\end{minipage}
\end{slide}

\begin{slide}
\centerline{Yet, the constraint almost never binds ...}

\begin{minipage}[h]{5in}
\centerline{\tiny \bf  Figure 12.8}
\centerline{\tiny \bf The Equilibrium Distribution of Leverage \label{fig:ovb_flowcc_plotdu}}

\bigskip

\rightline{\includegraphics[width=5in]{ovb_flowcc_plot_leverage_slides.eps}}
 
{\tiny Replication program \verb7plot_leverage.m7.}
%produced with plot_leverage_slides.m  
%in directory %z:\joint\underborrowing\mfiles\quarterly

\end{minipage}
\begin{minipage}{4in}


{\small 
\[\mbox{leverage } = \frac{d_{t+1}}{y^T_t + p_t y^N}
\]

$\bullet$ households stay well clear of upper bound on leverage, $\kappa$.

$\bullet$ this precautionary savings arises because, as we will see shortly, hitting the constraint is very painful.


$\bullet$ In 1 million quarters the constraint binds
\\
-- 287 times in economy (c) and 
\\-- 1,113 times in economy (b) 

$\bullet$ The prob that leverage exceeds $\kappa$ in the unconstrained economy is  18.6\%


 [Extension: Show that the frequency of a binding constrained rises when agents are made more impatient.] 

}


\end{minipage}

\end{slide}

\begin{slide}
{\bf Section 12.6 Financial Amplification}


Even if the constraint binds so rarely, are regular business cycle fluctuations different in the collateral constrained economy than in the economy without the collateral constraint?


\end{slide}

\begin{slide}
\centerline{\bf \small Table 12.2: No Amplification of Regular Business Cycles}

\bigskip

\centerline{\tiny \begin{tabular}{lccccccccc}
\hline 
\hline 
&\multicolumn{3}{c}{Std.~Dev.}&\multicolumn{3}{c}{Serial Corr.}
&\multicolumn{3}{c}{Corr.~w.~Output}\\
\cline{2-10}
Indicator & $CC$&No CC
&R&CC& No CC&R
&CC& No CC&R\\
\hline
Tradable Output, $y^T_t$&  0.12&  0.12&  0.12&  0.94&  0.94&  0.94&  0.97&  0.95&  0.96\\
Interest Rate, $r_t$&  0.02&  0.02&  0.02&  0.90&  0.90&  0.90&  -0.91&  -0.92&  -0.93\\
Output, $y^T_t+p_t y^n$&  0.96&  0.99&  0.95&  0.93&  0.95&  0.94&  1.00&  1.00&  1.00\\
Consumption of Tradables, $c^T_t$&  0.15&  0.16&  0.15&  0.92&  0.95&  0.94&  0.99&  0.99&  0.99\\
Relative Price of Nontradables, $p_t$&  0.85&  0.87&  0.83&  0.92&  0.95&  0.94&  1.00&  1.00&  1.00\\
Trade Balance, $tb_t$&  0.05&  0.06&  0.05&  0.61&  0.79&  0.67&  -0.65&  -0.71&  -0.63\\
Current Account, $ca_t$&  0.04&  0.04&  0.04&  0.30&  0.30&  0.32&  -0.27&  0.13&  -0.13\\
Capital Control Tax, $\tau_t$ in percent&  & &  0.96& &  &  0.10&  & &  -0.11\\
\hline
\hline
\end{tabular}
}

{\tiny Note. 
All moments are unconditional. 
CC stands for the collateral-constraint economy under equilibrium selection criterion (c), No CC for the economy without a collateral constraint, and R for the economy with Ramsey optimal capital control policy. \par}
%produced by running table_r.m in
%z:\joint\underborrowing\mfiles\quarterly

\end{slide}


\begin{slide}
{\bf Observations on the table}

$\bullet$ Amplitude of business cycle (std.) not increased in CC economy. (Mendoza AER 2010 first pointed this out in context of a stock collateral contraint economy). Why? Because of precautionary savings main effect of CC is to shift the debt density to the left. 

$\bullet$ Serial Correlations also little affected by the presence of the CC. 

$\bullet$ These findings are important because they suggest that it is unlikely that emerging economies are more volatile than rich countries because they face more severe borrowing limits of the type studied here. 


\end{slide}


\begin{slide}
What about amplification of more dramatic events, like a boom bust episode. 


Definition of a boom bust episode: 

In quarter $t=-20,$ $y^T_t$ is below the mean, in quarter $t=-10,$ $y^T_t$ is at least 1 std above its mean (the boom). Then in the course of 2.5 years, $y^T$ falls to 1 standard deviations below the mean, that is, in $t=0$ $y^T_t$ is one standard deviation below the mean (the bust). 


The assumed driving process implies that this type of boom bust episode occurs once every 130 years. In this sense it is a rare event. 
\end{slide}

\begin{slide}
\centerline{\bf  Figure 12.9: No Amplification of Boom-Bust Episodes}
\centerline{\includegraphics[width=11in]{ovb_flowcc_boom_bust_slides.eps}}
\centerline{\tiny 
\rule{1.4cm}{1mm} CC 
\hspace{2in}
\rule{2mm}{1mm} \rule{2mm}{1mm} \rule{2mm}{1mm} \rule{2mm}{1mm} No CC
}



{\tiny Note. CC indicates the economy with the collateral constraint under equilibrium selection criterion (c). Replication program \verb7boom_bust.m7.} 

%produced with boom_bust.m  using simu_constrained_c.mat 
%in directory 
%z:\joint\underborrowing\mfiles\quarterly
\end{slide}

\begin{slide}
Observations on the figure: 

- dynamics in CC and no-CC economies are not very different. 

- we can see that the value of collateral displays a clear boom-bust pattern. Yet, these variations in the value of collateral do not lead to a boom bust pattern in external debt. Debt is flat over the entire boom-bust episode. Thus, this model does not have the feature that expansions in the value of collateral during booms lead to more debt. Similarly, the collapse of the value of collateral during the bust, leaves debt unaffected. 

\end{slide}

\begin{slide}
Finally, let's look at the dynamics when the collateral constraint actually binds. 

As explained earlier this happens rarely, for equilibrium selection criterion (c), this happens once every 870 years. 

Again, we simulated the economy for 1e6 periods, and then averaged over all windows in which the collateral constraint binds.
The window starts 20 quarters before the collateral constraint binds ($t=0$) and end 20 quarters after the constraint binds. 

For comparison, we also show the behavior of the unconstrained economy during the same periods, i.e., having experienced the identical paths of $y^T_t$ and $r_t$  

\end{slide}

\begin{slide}
\centerline{\bf Figure 12.10: Amplification During   Financial Crises}
\centerline{\includegraphics[width=11in]{ovb_flowcc_typical_crisis_slides.eps}}
\centerline{\tiny 
\rule{1.4cm}{1mm} CC 
\hspace{2in}
\rule{2mm}{1mm} \rule{2mm}{1mm} \rule{2mm}{1mm} \rule{2mm}{1mm} No CC
}

{\tiny Note. Replication program \verb7typical_crisis.m7.}
%produced by running 
%typical_crisis.m  in 
%z:\joint\underborrowing\mfiles\quarterly\
\end{slide}

\begin{slide}
Observations: 

- when does the constraint bind? after the economy got one big negative surprise after another and each negative shock larger than the previous one. 
After 5 years of such bad luck, finally, the constraint binds. 

- the financial crisis is not preceded by a build-up in debt. 

- the collateral constraint becomes binding because the value of collateral falls, not because debt rises. 

- once that happens the entire decline in the value of collateral (between periods $t=-1$ to period $t=0$) must be accommodated by a reduction in debt, $\rightarrow$ deleveraging. 

- at that point $c^T$ falls, and hence $p$ falls, setting of a Fisherian debt deflation and a firesale. 

- the crisis is short lived. Once the stock of debt was reduced, the economy returns immediately to `normal'. 

\end{slide}

\begin{slide}
{\bf Section 12.7 Optimal Capital Control Policy} 

Models with endogenous collateral constraints (stock or flow) display a pecuniary externality. 

The existing related literature has stressed that the pecuniary externality induces {\bf overborrowing} in the sense that a planner who internalizes the pecuniary externality would borrow less. 

This is the topic of this section. 
\end{slide}

\begin{slide}
Assumptions: 

-- the government has commitment
\\
-- the government is benevolent
\\
-- the government has access to state contingent capital control taxes, $\tau_t$, and lump sum taxes. 

We will show: \\
-- capital control tax results in full internalization of pecuniary externality. 

-- one calibration will yield underborrowing and another overborrowing. 

\end{slide}
\begin{slide}
$\tau_t =$ proportional tax on debt assumed in period $t$; $\tau_t>0$ capital control tax , $\tau_t<0$ borrowing subsidy


Financed with lump sum taxes: $\ell_t=$ lump-sum taxes in period $t$

Tax revenue: $\tau_t \frac{d_{t+1}}{1+r_t}$


Government budget constraint in period $t$: $\tau_t \frac{d_{t+1}}{1+r_t}= \ell_t$

The Household budget constraint:\\ $c^T_t + p_t c^N_t + d_t = y^T_t + p_t y^N_t + (1-\tau_t) \frac{d_{t+1}}{1+r_t} + \ell_t$



Interest rate on foreign borrowing was 
\[(1+r_t)
\]
and now is 
\[
\frac{1+r_t}{1-\tau_t}> 1+r_t , \quad \mbox{if } \tau_t >0
\]
\end{slide}
\begin{slide}
Competitive equilibrium in the economy with capital control taxes are processes~$c^T_t$, $d_{t+1}$, $\lambda_t$, $\mu_t$, and  $p_t$ satisfying 
\begin{eqnarray}
\label{eq:ovb_ramsey_bc}
c^T_t + d_t 
&=& y^T_t + \frac{d_{t+1}}{1+r_t}
\\
\label{eq:ovb_ramsey_lambda}
\lambda_t &=& 
U'(A(c^T_t,y^N_t)) A_1(c^T_t,y^N_t)
\\
\label{eq:ovb_ramsey_Euler}
\left(
\frac{1-\tau_t}{1+r_t}-\mu_t
\right)
\lambda_t 
&=& \beta \mathbb{E}_t \lambda_{t+1}
\\
\label{eq:ovb_ramsey_p}
p_t& =& \frac{A_2(c^T_t,y^N_t)}
{A_1(c^T_t,y^N_t)}
\\
\label{eq:ovb_ramsey_cc}
d_{t+1} &\le&
\kappa 
\left[
y^T_t +p_ty^N_t
\right]
\\
\label{eq:ovb_ramsey_slackness}
\mu_t [\kappa (y^T_t+p_ty^N_t)-d_{t+1}]&=&0
\\
\label{eq:ovb_ramsey_mu}
\mu_t &\ge& 0
\end{eqnarray}
given $\{\tau_t\}$, $\{y^T_t\}$ and $\{r_t\}$, and  $d_0$. 
\end{slide}
\begin{slide}
How to pick $\tau_t$?
To  maximize 
\begin{equation*}
%\label{eq:ovb_ramsey_utility}
\mathbb{E}_0
\sum_{t=0}^{\infty}
\beta^t U(A(c^T_t,y^N_t))
\end{equation*}
subject to~\eqref{eq:ovb_ramsey_bc}-\eqref{eq:ovb_ramsey_mu}. 
\end{slide}
\begin{slide}

{\underline{Claim}:} $\{c^T_t\}$ and $\{d_{t+1}\}$ satisfy~\eqref{eq:ovb_ramsey_bc}-\eqref{eq:ovb_ramsey_mu}
 if and only if they satisfy~\eqref{eq:ovb_ramsey_bc} and 
\begin{equation}
\label{eq:ovb_ramsey_collateral_lessrestricted}
d_{t+1} \le 
\kappa 
\left[
y^T_t + 
\frac{A_2(c^T_t,y^N_t)}
{A_1(c^T_t,y^N_t)}y^N_t
\right].
\end{equation}


\underline{Proof:}
Suppose $\{c^T_t\}$ and $\{d_{t+1}\}$ satisfy~\eqref{eq:ovb_ramsey_bc} and~\eqref{eq:ovb_ramsey_collateral_lessrestricted}. Show that they also satisfy~\eqref{eq:ovb_ramsey_bc}-\eqref{eq:ovb_ramsey_mu}. (To show the reverse is also needed, but as it is trivial not shown here.)

Pick $\{p_t\}$ to satisfy~\eqref{eq:ovb_ramsey_p}. Then by 
~\eqref{eq:ovb_ramsey_collateral_lessrestricted}, \eqref{eq:ovb_ramsey_cc} holds. 
\\
Pick $\{\mu_t\}=0$ $\forall t$, then  \eqref{eq:ovb_ramsey_slackness}
and \eqref{eq:ovb_ramsey_mu} hold 
\\
Pick $\lambda_t$ to satisfy\eqref{eq:ovb_ramsey_lambda}.
\\
Pick $\tau_t$ to satisfy~\eqref{eq:ovb_ramsey_Euler}
\rule{3mm}{3mm}

[Can you show that $\tau_t$ is not unique? Ie, $\exists$ other picks for $\mu_t$ and $\tau_t$ that are consistent with the same allocation?]
\end{slide}
\begin{slide}
The Ramsey Optimal Capital Control Tax Problem 

\begin{equation}
\label{eq:ovb_ramsey_utility}
\max_{\{c^T_t, d_{t+1}\} } \mathbb{E}_0
\sum_{t=0}^{\infty}
\beta^t U(A(c^T_t,y^N_t))
\end{equation}
subject to
\[
\tag{\ref{eq:ovb_ramsey_bc}}
c^T_t + d_t 
= y^T_t + \frac{d_{t+1}}{1+r_t}
\]
\[
\tag{\ref{eq:ovb_ramsey_collateral_lessrestricted}}
d_{t+1} \le 
\kappa 
\left[
y^T_t + 
\frac{A_2(c^T_t,y^N_t)}
{A_1(c^T_t,y^N_t)}y^N_t
\right]
\]

We have thus shown that with a capital control tax the Ramsey planner fully internalizes the pecuniary externality. 


\end{slide}

\begin{slide}
Overborrowing or Underborrowing in the Analytical Example of Section 12.4
\end{slide}


\begin{slide}
\begin{center}
{\bf \large Underborrowing!!}

\medskip

\centerline{\includegraphics[width=5in]{ovb_flowcc_multiple3_slides.eps}}

%produced with plot_flowcc_multiple.m
%in directory z:\uribe\book\overborrowing\flowcc_analytical
\end{center}
{\small
The competitive equilibrium at point $A$
can be supported with $\tau_t=0$ for all $t$. 
It is also the first-best allocation, so it must be  Ramsey optimal.  

Thus, if  agents coordinate on equilibrium $A$, there is neither overborrowing nor underborrowing.  

But if agents coordinate on equilibrium $B$ or $C$, the economy suffers underborrowing. }
\end{slide}



\begin{slide}
{\bf \large Implementation of Ramsey Optimal Equilibrium}


The Ramsey optimal tax  rate in this economy is $\tau_t=0$ at all times. 

However, announcing the policy $\tau_t=0$ for all $t$ does not guarantee that the Ramsey  optimal equilibrium will emerge. Indeed, this tax policy also supports the deleveraging equilibria $B$ or $C$. 

What capital control policy can induce the Ramsey-optimal equilibrium?  Consider a debt-dependent feedback rule for $\tau_t$:
\[\tau_t = \tau(d_{t+1},d_t)\]
satisfying $\tau(d,d)=0$ and $\tau>0$ if $d_{t+1}<d_t$.

\end{slide}

\begin{slide}
{\bf{ \large Implementation}  (continued)}

Under this tax-policy rule, the Euler equation in period 0 becomes:
\[
\frac{c^T_1}{c^T_0} = \frac1{\sqrt{1-\tau(d_1,d_0)-(1+r)\mu_0}}
\]
(1) In the intended (Ramsey) equilibrium, $c^T_1/c^T_0=1$, $d_1=d_0$, and $\mu_0=0$, so the Euler equation holds and  $\tau(d_1,d_0) = 0$. 

(2) In the unintended equilibrium (points $B$ or $C$), $c^T_1/c^T_0>1$, and $d_1<d_0$.  Make $\tau(d_1,d_0)>0$ and so large that $\mu_0$ has to be negative for the Euler equation to hold. Since $\mu_0$ must be nonnegative, this capital-control policy rules out the unintended equilibrium.

(3) $\Rightarrow$  The thread of imposing capital controls in response to outflows eliminates self-fulfilling crises. 

\end{slide}


\begin{slide}
{\bf Section 12.8  Overborrowing and Underborrowing in the Stochastic Economy}

Computation: The Ramsey optimal allocation is relatively easy to compute because the Ramsey problem can be cast in the form of a Bellman equation problem. 

The recursive version of the Ramsey problem of maximizing~\eqref{eq:ovb_ramsey_utility}
subject to \eqref{eq:ovb_ramsey_bc} and \eqref{eq:ovb_ramsey_collateral_lessrestricted} is given by
\[
v(y^T,r,d) = \max_{c^T,d'}
\left\{
U(A(c^T,y^N)) + \beta \mathbb{E}
\left[
v({y^T}',r',d')\left|y^T,r
\right.\right]
\right\}
\]
subject to 
\begin{eqnarray*}
c^T+d& =& y^T + \frac{d'}{1+r}
\\d' &\le& \kappa 
\left[
y^T + 
\frac{1-a}a
\left(
\frac{c^T}{y^N}
\right)^{\frac1{\xi}}
y^N
\right]\end{eqnarray*}
where a prime superscript denotes next-period values.  

Observation:  Although the constraints of this control problem may not represent a convex set in tradable consumption and debt,  the fact that the Ramsey allocation is the result of a utility maximization problem, implies that its solution is generically unique.

\end{slide}


\begin{slide}
\centerline{Debt Densities With Optimal Capital Controls: A Case of  Underborrowing}
\centerline{\includegraphics[width=6in]{ovb_flowcc_plotd_ramsey_slides.eps}}
%produced with plotd_ramsey.m  
%in directory %c:\data\uribe\book\overborrowing\flowcc

\end{slide}

\begin{slide}
Observations on the figure: 

Equilibrium selection criterion (c) $E(D/Y^{annual}) = 12.0$ percent

Equilibrium selection criterion (b) $E(D/Y^{annual}) = 12.4$ percent

Ramsey optimal capital control policy: $E(D/Y^{annual}) = 13.1$ percent


Therefore there is {\bf underborrowing} in the unregulated economy. 
(between 0.7 and 1.1 percent of output)

Agents engage in excessive precautionary savings. 

How often is there a crisis under the Ramsey policy? 179 times in  one million periods. [and 287 and 1,133 times, respectively, under selection criteria (c) and (b).] 




\end{slide}



\begin{slide}
For comparison let's look now at a different calibration of the model, namely, that studied in Bianchi (AER 2011 )---the central reference for the overborrowing result in the quantitative flow collateral constraint literature. 

Bianchi has a different driving force, $y^T_t$ and $y^N_t$ are stochastic but  $r_t$ is not. 

\end{slide}
\begin{slide}
{\bf The Driving Process of  Bianchi 2011}\\
The natural logarithms of the traded and nontraded endowments follow a bivariate AR(1), which is estimated on annual HP-filtered Argentine data spanning the period 1965 to 2007. 
Traded GDP: Manufacturing and primary products. Nontraded GDP: remaining components.
%(Q: is this $p_t y^N_t$ or really $y^N_t$? Javier told us that it is really y^N_t and not p_t y^N_t)

\begin{equation}
\label{eq:bianchi_AR(1)_yT_and_yN}
\left[
\begin{array}{c}
\ln  y^T_t\\
\ln y^N_t
\end{array}
\right]
=
 \left[
\begin{array}{cc}
0.901&-0.453\\
0.495&0.225
\end{array}
\right] 
\left[
\begin{array}{c}
\ln  y^T_{t-1}\\
\ln  y^N_{t-1}
\end{array}
\right]
+ 
\epsilon_t,
\end{equation}

where $\epsilon_t\sim N(\emptyset,\Omega_{\epsilon})$, with 
$
\Omega_{\epsilon} = \left[
\begin{array}{cc}
0.00219& 0.00162 \\
0.00162& 0.00167
\end{array}
\right].
$

\end{slide}

\begin{slide}
{\bf Some Unconditional Summary Statistics of the Bianchi (2011) Driving Process }

\vspace{1in} 

\centerline{\begin{tabular}{lcc}
\hline\hline
Statistic&$\ln y^T$&$\ln y^N$\\
\hline
Std. Dev.&6\%&6\%\\
Serial Corr.&0.53&0.62\\
Corr($\ln y^T_t,\ln y^N_t$)&\multicolumn{2}{c}{0.83}\\
\hline\hline
\end{tabular}
}

{\bf Observations:}\\
(1) High volatility of tradable and nontradable endowments. 
\\
(2) Strong positive  correlation between $y^T_t$ and $y^N_t$. 
\end{slide}


\begin{slide}
{\bf Discretization of the State Space}

There are 4 distinct grid points for $ln(y^T)$,  $$\left[\begin{array}{r} -0.1093\\
   -0.0347\\
    0.0347\\
    0.1093\end{array}\right]$$


and 16 distinct pairs  $(y^T,y^N)$. 

There are 800 grid points for $d_t$. 

The  total grid has $16\times 800 = 12,800$ points. 

\end{slide} 
\begin{slide}

\centerline{\bf Summary of the Calibration}


Time unit is one year. 

\vspace{1in}

\centerline{\tiny 
\begin{tabular}{ccl}
\hline\hline
Parameter&Value&Description\\
\hline
$\kappa$&$0.33$&Parameter of collateral constraint \\
$\sigma$&2&Inverse of intertemporal elasticity of consumption\\
$\beta$&0.91&Subjective discount factor\\
$r$& 0.04& Interest rate (annual)\\
$\xi$&0.83&Elasticity of substitution between tradables and nontradables\\
$a$&0.31&Weight on  tradables in CES aggregator \\
$y^N$&1&Steady-state nontradable output\\
$y^T$&1&Steady-state tradable output\\
$n_{y}$&16& Number of  grid points for $(\ln y^T_t,\ln y^N_t)$\\
$n_d$&800&Number of grid points for $d_t,$ equally spaced\\
$\left[
\ln \underline{y}^T,
\ln \overline{y}^T
\right]$&[-0.1093,0.1093]&Range for  tradable output\\
$\left[\ln \underline{y}^N,
\ln \overline{y}^N
\right]$&[-0.1328,0.1328]&Range for  nontradable output\\
$[\underline{d}/(1+r),\overline{d}/(1+r)]$&[0.4 1.02]&Range for debt\\
\hline\hline
\end{tabular}
}
\end{slide} 

\begin{slide}
{\bf Comment on Impatience}

The calibration implies that consumers  are quite impatient:

$r = 0.04$; 

$\beta = 0.91$; 

$\beta (1+r) =    0.95 $

The high degree of impatience influences 
the role of optimal capital controls. The Ramsey planner is constantly negotiating a trade off between, on the one hand, allowing impatient consumers to frontload consumption and, on the other hand, preventing the economy from  hitting  the collateral constraint. 

The high degree of impatience determines  how far apart the debt densities are between the CC and the no CC economy. See the next figure

\end{slide}


\begin{slide}
\centerline{\bf Debt Densities under the Bianchi Calibration} 

\centerline{\includegraphics[width=4in]{ovb_flowcc_plot_bianchi_unconstrained_d.eps}}
%produced with  plot_bianchi_d.m in 
%z:\uribe\book\overborrowing\flowcc\bianchi

\end{slide}


\begin{slide}
Let's next turn ask how much overborrowing there is in this calibrated economy. Recall overborrowing is defined as the amound of borrowing in the economy in which agents fail to internalize the pecuniary externality compared to the economy in which they do, or in which they do not but fact optimally set capital control taxes that makes households behave as if they internalized the pecuniary externality. 

How much overborrowing is there?

\end{slide}




\begin{slide}

\centerline{\bf Modest Amount of Overborrowing under the Bianchi Calibration}
\begin{minipage}{6.5in}
\rightline{\includegraphics[width=7in]{ovb_bianchi_plotd_ramsey_slides.eps}}
%produced with  plotd_ramsey.m in 
%z:\joint\macroprudential\mfiles_bianchi

\end{minipage}\begin{minipage}{3.5in}  

{\tiny 

Under Ramsey Optimal Capital Controls: 

\bigskip 

\leftline{\tiny $E\left(
\frac{d_{t+1}}{(1+r)y_t}\right) = 28.5$\%}



}

\bigskip 


\leftline{\tiny With collateral constraint:}
\leftline{\tiny  $E\left(
\frac{d_{t+1}}{(1+r)y_t}\right) = 29.2$\%}

\bigskip 

{\tiny $\Rightarrow$ Pecuniary externality leads to overborrowing of 0.7 percentage points of output


(These results are our replication of those reported in Bianchi. He reports, 28.6\% and 29.2\%, respectively.)

}
\end{minipage}
\end{slide}





\begin{slide}
This suggests that the main role of optimal capital control taxes is not a sizable reduction in the amount of debt. 

What do the taxes affect?

Frequencey of crisis (defined as a binding collateral constraint) falls from once every {\bf 12 years} in the unregulated economy to once every {\bf 26 years} in the Ramsey economy. 


This suggests that the main role of optimal capital controls is to avoid a binding collateral constraint. 
\end{slide}





\begin{slide}
We conclude, depending on the calibration the pecuniary externality can lead  either to  overborrowing or to underborrowing vis-\`a-vis the allocation under Ramsey optimal capital control taxes. 


\end{slide}

\begin{slide}
Note on the literature: 


An exception to the standard overborrowing result  is Benigno, Chen, Otrok, Rebucci, and Young (2013) who obtain underborrowing by replacing the assumption of an endowment economy maintained in Bianchi (2011) with the assumption that output is produced with labor. In their production economy,  the social planner  sustains more debt than in the unregulated economy  by engineering sectoral employment allocations conducive to elevated values of the collateral in terms of tradable goods. The underborrowing result obtained in this section   is complementary but different from  that of Benigno et al. Here, underborrowing arises even in the context of an endowment economy  and is due to an inefficiently high level of precautionary savings in an environment prone to self-fulfilling crises. 


\end{slide}

\begin{slide}
{\bf Section 12.9 Is Optimal Capital Control Policy Macroprudential?}

Macroprudential used to mean two slightly different things. 

Meaning 1: On average there is a tax on capital inflows, ie, $\mathbb{E} \tau_t >0$

Meaning 2: Taxes on capital flows are cyclical, in particular, high during good times and low during bad times, ie, capital controls are countercylical, $corr(\tau_t, y_t)>0$.


Calibration 1: USG, Chapter 12. mean($\tau_t)=0$ 

Calibration 2: Bianchi, AER 2011. mean($\tau_t$) = 4.2\% ; median($\tau_t$) = 2.5\%

According to meaning 1 optimal capital controls fail to be macroprudential in the USG calibration but are indeed macroprudential in the Bianchi calibration. 


However, neither calibration implies that the optimal capital control tax is raised during booms and then lowered during the bust, and hence the optimal capital control tax fails to be macroprudential in this precise sense. 



\end{slide}


\begin{slide}
\centerline{\bf Are Optimal Capital Controls Macroprudential? }\centerline{USG calibration, boom bust}

\centerline{\includegraphics[width=11in]{ovb_flowcc_boom_bust_ramsey_slides.eps}}
\centerline{\tiny 
\rule{1.4cm}{1mm} CC 
\hspace{0.5in}
\rule{.3cm}{1mm} \rule{1mm}{1mm} \rule{.3cm}{1mm} \rule{1mm}{1mm} \rule{.3cm}{1mm} \rule{1mm}{1mm}  Ramsey
}
%produced with boom_bust_ramsey.m  
%in directory 
%c:\data\uribe\book\overborrowing\flowcc
\end{slide}

\begin{slide}
\centerline{\bf Macroprudential Capital Controls in a Financial Crisis. USG calibration}
\centerline{\includegraphics[width=11in]{ovb_flowcc_typical_crisis_ramsey_slides.eps}}
\centerline{\tiny 
\rule{1.4cm}{1mm} CC 
\hspace{0.5in}
\rule{.3cm}{1mm} \rule{1mm}{1mm} \rule{.3cm}{1mm} \rule{1mm}{1mm} \rule{.3cm}{1mm} \rule{1mm}{1mm}  Ramsey
}
%produced with boom_bust_ramsey.m  
%in directory 
%c:\data\uribe\book\overborrowing\flowcc
\end{slide}



\begin{slide}
\centerline{\bf \small Are Optimal Capital Controls Cyclical? Bianchi calibration}



\centerline{\includegraphics[width=10in]{z:/joint/macroprudential/bianchi_typical_boom_bust_slides.eps}}
%produced with  typical_boom_bust.m in 
%z:\joint\macroprudential\mfiles_bianchi
%make sure to pick rows and cols for paper and not for slides
\centerline{\rule{3cm}{1mm}: unregulated economy \hspace{1in} \rule{2mm}{1mm} \rule{2mm}{1mm} \rule{2mm}{1mm} \rule{2mm}{1mm} \rule{2mm}{1mm} \rule{2mm}{1mm} : Ramsey economy}

{\tiny Source: Schmitt-Groh\'e and Uribe (IMF ER, 2017). 
Definition of boom-bust episode: $y^T_{-3}>1$ and $y^T_0<1$; given grid this implies that during a typical boom bust episode output falls from  5\% above mean    to  5\% below  mean over 3 years. Frequency, 12.3\%.
Each line is the mean across all windows containing 
a boom-bust cycle
in a time series of 1 million years. 
For the capital-control tax rate, the figure displays the median instead of the mean across   windows because this variable is  skewed, with an unconditional mean of 4.2 percent and an unconditional median of 2.5 percent. Because the capital control tax rate is indeterminate when the collateral constraint binds under the Ramsey policy,  this variable   is given a number only if the collateral constraint is slack under the Ramsey policy.    Replication file typical\_boom\_bust.m in sgu\_endowment\_shocks.zip. \par}
\end{slide}


\begin{slide}
Observation: 

Over the typical boom bust cycle the optimal capital control tax is not countercyclical. It is lowered during booms and raised during recessions. 

\end{slide}


\begin{slide}
What is the role of Ramsey optimal capital control taxes? --- To avoid a binding constraint. 


$\bullet$ Capital Control Taxes are positive on average
$
\mbox{ Median($ \tau_t$)} = 0.0258
$

this should lower debt. Mean debt in the Ramsey economy is 0.926 (or 28.5\% of output ) as opposed to  0.9483 (or 29.2\% of output) in the unregulated economy. 

Positive taxes help the economy stay clear of a binding constraint. 

$\bullet$ Capital control taxes are quite volatile, std($\tau_t$) is 4.2\%. They are moved around to  avoid a  binding collateral constraint.
 
$\bullet$ Frequency of binding constraint in Ramsey 3.9\% (or once every 26 years) and in unregulated economy 8.5\% (or once every 12 years). 

Why is it so important to avoid a binding constraint? Because it leads to a deep (albeit short) contraction: 

\end{slide}


\begin{slide}
\centerline{\bf The Typical Financial Crisis in the Bianchi Economy}

\centerline{\includegraphics[width=10in]{z:/joint/macroprudential/bianchi_typical_crisis_slides.eps}}
%produced with  typical_crisis.m in 
%z:\joint\macroprudential\mfiles_bianchi
%make sure to pick rows and cols for paper and not for slides
\centerline{\rule{3cm}{1mm}: unregulated economy \hspace{1in} \rule{2mm}{1mm} \rule{2mm}{1mm} \rule{2mm}{1mm} \rule{2mm}{1mm} \rule{2mm}{1mm} \rule{2mm}{1mm} : Ramsey economy}

{\tiny Source: Schmitt-Groh\'e and Uribe (IMF ER 2017). Note. Each line is the mean across all  11-year windows 
containing a binding collateral constraint 
in the center in a one-million-year time series from the 
  unregulated economy. For the capital-control tax rate, the figure displays the median instead of the mean across   windows because this variable is  skewed, with an unconditional mean of 4.2 percent and an unconditional median of 2.5 percent. Because the capital control tax rate is indeterminate when the collateral constraint binds under the Ramsey policy,  this variable   is given a number only if the collateral constraint is slack under the Ramsey policy.    Replication file typical\_crisis.m in sgu\_endowment\_shocks.zip.\par }


\end{slide}


\begin{slide}

Observation: 

Ramsey planner {\bf raises} capital control taxes in run up to crisis and {\bf lowers} them once crisis is over. $\Rightarrow$ optimal capital controls are {\bf not} countercyclical. And not macroprudential in that sense.  


\end{slide}
\begin{slide}
{\bf \large What about unconditionally?}

corr($\tau_t, \ln y_t$) = {\bf -0.84}, 


%for it is raised during bad times and lowered during good times. 

%What about capital flows?
corr($\tau_t, \ln c^T_t$) = {\bf -0.88} 

%That is, under optimal policy taxes are lowered when debt increases. 

$\Rightarrow$ optimal capital controls are {\bf not} countercyclical in that sense either. 

\end{slide}

\begin{slide}
{\bf Summary of Findings  on the Cyclicality of Optimal Capital Control Taxes}



\begin{itemize}
  \item Ramsey optimal capital control taxes make households fully internalize the collateral constraint induced pecuniary externality. 
    \item Ramsey optimal capital control taxes are found to be {\bf procyclical}, they  are raised during recessions and lowered during booms. Therefore, the pecuniary externality does not support adoption of cyclical macroprudential policy.  
  \item What drives the result?--- Ramsey planner navigates a tradeoff between allowing agents to frontload consumption as much as possible and avoiding a binding collateral constraint. 
    \end{itemize}
\end{slide}


\end{document}
