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Trade Prelim Notes
Selffulfilling Debt Crises
This problem is from Tim Kehoe. It showed up on the trade prelims in 2016 (spring) and 2018.
Setup:
Dump of LaTeX solution
\newpage Problem 4
Consider a small open economy with a governemen tthat borrows. In
each period, value of output is
\[
y(z)=Z^{1z}\bar{y}
\]
$Z\in(0,1)$ and $z=0$ iff the government defaults or has ever defaulted.
Tax revenue is
\[
\theta y(z)
\]
with constant $\theta\in(0,1)$ . The consumers consume
\[
c=(1\theta)y(z)
\]
The fovernemnt seeks to maximize the expected discounted value of
\[
u(c,g)=\log c+\gamma\log g
\]
$B$ is level of governemnt debt, $\zeta\in[0,1]$ is the realization
of a sunspot variable.
The governemtn offers $B'$ to international bankers, who have same
discount factor, risk neutral.
Bond is bought at price $q(B',B,z,\zeta).$Then the governemnt choose
whether to default.
governemnt spending determined by
\[
g+zB=\theta y(z)+q(B',s)B'
\]
If the goverenment defaults, setting $z=0$, then assume that $z_{1}=0$
forever after. And $q(B',s)=0.$
\subsection*{a) Define recursive equilibrium.}
A recursive equiliibrum in this economy is a value function for the
government $V(s)$ and policy functions $B'(s)$, $z(B',s,q),$ $g(B',s,q)$
such that
At the beginning of the period, given the policy and price functions,
the governement chooses $B'$ to solve
\[
V(s)=\max_{B'}\left[\log c+\gamma\log g+\beta EV(s')\right]
\]
\[
\text{s.t. }c=(1\theta)y(z(B',s,q(B',s)))
\]
\[
g(B',s,q(B',s))+z(B',s,q(B',s))B=\theta y(z)+q(B',s)B'
\]
Bond prices are equal to riskfree price times probability of repayment
\[
q(B'(s),s)=\beta Ez(B'(s'),s',q(B'(s'),s'))
\]
At the end of the period, given the value fucntion, $B'$ and $q$,
the government chooses z and g to solve
\[
\max_{z,g}\left[\log c+\gamma\log g+\beta EV(B',z,\zeta')\right]
\]
\[
\text{s.t. }c=(1\theta)y(z)
\]
\[
g+zB=\theta y(z)+qB'
\]
\[
z\in\{0,1\}\ \ \text{and}\ \ z=0\text{ if }z_{1}=0
\]
\subsection*{b)Assume bankers expect gov to default if $\zeta>1\pi$ and if such
expectation is self fulfilling. Find a level of debt $\bar{b}$ such
that no default occurs if $B\protect\leq\bar{b}$ but does in equilibirum
if $B>\bar{b}.$}
Let's first consider what the value function looks like if the government
is garunteed not to default and has never defaulted before. And where
the level of debt is kept at a steady state $B$ (That is, $B'(s)=B$).
\[
q(B,s)=\beta
\]
\[
V(B,1,\zeta)=\log\left((1\theta)\bar{y}\right)+\gamma\log\left(\theta\bar{y}B+\beta B\right)+\beta EV(B,1,\zeta')
\]
If there is no chance of default, then $V(B,1,\zeta)=V(B,1,\zeta')$
for all $\zeta'$. And so:
\[
(1\beta)V(B,1,\zeta)=u\left((1\theta)\bar{y},\theta\bar{y}B+\beta B\right)
\]
\[
V(B,1,\zeta)=\frac{\log(1\theta)\bar{y}+\gamma\log\left(\theta\bar{y}B+\beta B\right)}{1\beta}
\]
If the gov defaults and bankers don't lend this period, then the goverenment
will collect $\theta Z\bar{y}$ forever, spending all of it on $g.$
$\sum_{t=0}^{\infty}\beta^{t}=\frac{1}{1\beta}$. Thus timediscounted
expected utility of exiting from the financial market if bankers do
not lend is
\[
\frac{1}{1\beta}\left(\log((1\theta)Z\bar{y})+\gamma\log(\theta Z\bar{y})\right)
\]
Now suppose that the bankers all decide to blacklist the government
and never lend, but the gov decides not to default. Then they pay
off the current debt and then starting collect and spend $\theta\bar{y}$
starting from next period unti lthe end of of time. $\sum_{t=1}^{\infty}\beta^{t}=\frac{\beta}{1\beta}$
so the expected discounted utility of repaying the current debt if
the bankers decide never to lend again is
\[
\log(1\theta)\bar{y}+\gamma\log\left(\theta\bar{y}B\right)+\frac{\beta}{1\beta}\left(\log\left((1\theta)\bar{y}\right)+\gamma\log\left(\theta\bar{y}\right)\right)
\]
The higher the debt, the more tempting it is for the government to
default when the bankers stop lending. We can use the two above equations
to find the $B$ that makes the government indifferent between default
and repayment:
\[
\log(1\theta)\bar{y}+\gamma\log\left(\theta\bar{y}B\right)+\frac{\beta}{1\beta}\left(\log\left((1\theta)\bar{y}\right)+\gamma\log\left(\theta\bar{y}\right)\right)=\frac{1}{1\beta}\left(\log((1\theta)Z\bar{y})+\gamma\log(\theta Z\bar{y})\right)
\]
\[
\gamma\log\left(\theta\bar{y}B\right)=\frac{1}{1\beta}\log\left((1\theta)\theta^{\gamma}Z^{\gamma+1}\bar{y}^{\gamma+1}\right)\frac{\beta}{1\beta}\log\left((1\theta)\theta^{\gamma}\bar{y}^{\gamma+1}\right)\log(1\theta)\bar{y}
\]
\[
\gamma\log\left(\theta\bar{y}B\right)=\log\left((1\theta)\theta^{\gamma}Z^{\gamma+1}\bar{y}^{\gamma+1}\right)^{\frac{1}{1\beta}}\log\left((1\theta)\theta^{\gamma}\bar{y}^{\gamma+1}\right)^{\frac{\beta}{1\beta}}\log(1\theta)\bar{y}
\]
\begin{align*}
\log\left(\theta\bar{y}B\right) & =\log\left(\frac{\left((1\theta)\theta^{\gamma}Z^{\gamma+1}\bar{y}^{\gamma+1}\right)^{\frac{1}{1\beta}}}{\left((1\theta)\theta^{\gamma}\bar{y}^{\gamma+1}\right)^{\frac{\beta}{1\beta}}(1\theta)\bar{y}}\right)^{\frac{1}{\gamma}}\\
& =\log\left(\frac{\left(\theta^{\gamma}Z^{\gamma+1}\bar{y}^{\gamma+1}\right)^{\frac{1}{1\beta}}}{\left(\theta^{\gamma}\bar{y}^{\gamma+1}\right)^{\frac{\beta}{1\beta}}\bar{y}}\right)^{\frac{1}{\gamma}}\\
& =\log\left(\frac{\left(Z^{\gamma+1}\right)^{\frac{1}{1\beta}}}{\bar{y}}\theta^{\gamma}\bar{y}^{\gamma+1}\right)^{\frac{1}{\gamma}}\\
& =\log\left(Z^{\frac{\gamma+1}{1\beta}}\theta^{\gamma}\bar{y}^{\gamma}\right)^{\frac{1}{\gamma}}\\
& =\log Z^{\frac{\gamma+1}{\gamma(1\beta)}}\theta\bar{y}
\end{align*}
\[
\bar{b}=\left(1Z^{\frac{\gamma+1}{\gamma(1\beta)}}\right)\theta\bar{y}
\]
If $B\leq\bar{b}$, then the penalty to production is enough to make
the governemnet pay its debts, and so bankers can know with certainty
that there will be no defaulting this period.
\subsection*{c) Suppose that $B_{0}>\bar{b}$ and the government chooses to run
down its debt to $B_{T}\protect\leq\bar{b}$\textmd{\normalsize{}
in} $T$ periods. }
Prove that it is not optimal to set $B_{T}<\bar{b}$. Prove that it
is optimal for gov to set $g_{t}$ constant as long as $B_{t}>\bar{b}$
and no crisis occurs. Find expresions for $g_{t}$ and $B_{t}$that
depend on $B_{0}$ and $T.$ Find expresion for expected discounted
value of utiltiy of running down the debt that starts at $B_{0}$
to $\bar{b}$ in $T$ periods. Find the limit of these expresions
when $T=\infty$

Suppose that the government is trying to decide whether to reduce
its debt to $B_{T}<\bar{b}$ or $\bar{b}$. The time to reduce is
the same in either case, and so the probability of debt crisis and
default is the same in either plan. If the gov defaults, they ignore
debt levels for the rest of time. Thus when choosing between the two
plans, the government only needs to consider $c$ and $g$ during
these rundown period.
If the optimal plan involves going to the lower debt level, then the
government could follow that plan until they reach the point where
debt is $\bar{b}$ and then take the money which the plan says is
for further debt repayment and instead use that money on additional
expenditure in the current period. This is a strictly better alternate
plan which runs down debt to $\bar{b}$. Therefore, by contradiction,
the optimal plan cannot feature $B_{T}<\bar{b}$ as its target debt.

In running down the debt, the government is essentially redirecting
a portion of tax revenue from spending to repaying debts. Because
of the logarithmic form of the objective fucntion, it is valuable
to smooth government spending over time. And so taking a constant
chunk from production each period is better than taking one large
chunk from production in one period.

Let $\tilde{g}(T,B_{0})$ be the optimal government spending when
running down an initial debt of $B_{0}$ over $T$ periods. From HW5,
we know that
\[
\tilde{g}(T,B_{0})=\theta\bar{y}\frac{1\beta(1\pi)}{1\left(\beta(1\pi)\right)^{T}}\left(B_{0}(\beta(1\pi))^{T1}\beta\bar{b}\right)
\]
\[
\tilde{g}(\infty,B_{0})=\lim_{T\to\infty}\tilde{g}(T,B_{0})=\theta\bar{y}\left(1\beta(1\pi)\right)\left(B_{0}\right)
\]
And let $\tilde{V}(T,B_{0})$ be the time discounted expected value
of the same plan above. Again, from the hw,
\begin{align*}
\tilde{V}(T,B_{0}) & =P\left(\text{No crisis occurs}\right)\left(\text{Value of following plan for T periods+}\beta^{t}V(\bar{b},1,\zeta)\right)\\
& +\sum_{t=0}^{T}\left[\left(P(\text{crisis at time exactly t})\right)\left(\text{Value of following plan for t1 periods+}\beta^{t}V(0,0,\zeta)\right)\right]
\end{align*}
\begin{align*}
\tilde{V}(T,B_{0})= & \frac{1\left(\beta(1\pi)\right)^{T}}{1\beta(1\pi)}\left(\log\left((1\theta)\bar{y}\right)+\gamma\log\left(\tilde{g}(T,B_{0})\right)\right)\\
& +\frac{1\left(\beta(1\pi)\right)^{T1}}{1\beta(1\pi)}\frac{\beta\pi\left(\log\left((1\theta)Z\bar{y}\right)+\gamma\log\left(\theta Z\bar{y}\right)\right)}{1\beta}\\
& +\left(\beta(1\pi)\right)^{T2}\frac{\beta\left(\log\left((1\theta)\bar{y}\right)+\gamma\log\left(\theta\bar{y}\right)\right)}{1\beta}
\end{align*}
\begin{align*}
\tilde{V}(\infty,B_{0})= & \frac{1}{1\beta(1\pi)}\left(\log\left((1\theta)\bar{y}\right)+\gamma\log\left(\tilde{g}(\infty,B_{0})\right)\right)\\
& +\frac{1}{1\beta(1\pi)}\frac{\beta\pi\left(\log\left((1\theta)Z\bar{y}\right)+\gamma\log\left(\theta Z\bar{y}\right)\right)}{1\beta}
\end{align*}
\subsection*{d) Using the answers to part c, write down a formula that determines
a value of debt $\bar{B}(\pi)$ such that the government would choose
to default if $B>\bar{B}(\pi)$ even if international bankers do not
expect a default. }
If the government is going to run down their debt, they will choose
the time period of the rundown to maximize expected discounted value
of the objective function. To find the threshold at which the government
will always choose to default, we find the point at which the government
is indifferent between defaulting and the optimal rundown policy.
that is,
\begin{align*}
\frac{1}{1\beta}\left(\log((1\theta)Z\bar{y})\right)+\frac{\beta}{1\beta}\left(\gamma\log(\theta Z\bar{y})\right)+\gamma\log(\theta Z\bar{y}+\beta(1\pi)\bar{B}(\pi))=\max_{T}\tilde{V}(T,\bar{B}(\pi))
\end{align*}
Not sure how to solve this algebraicaly. But the above can be solved
for $\bar{B}(\pi)$. At any debt levels higher than this, the government
has higher utility from just taking the loaned money $\beta(1\pi)\bar{B}(\pi)$
than they lose from default penalties.
\subsection*{e) construct recursive equilibirum based on ad}
A recursive equiliibrum in this economy is a value function for the
government $V(s)$ and policy functions $B'(s)$, $z(B',s,q),$ $g(B',s,q)$
such that
At the beginning of the period, given the policy and price functions,
the governement chooses $B'$ to solve
\[
V(s)=\max_{B'}\left[\log c+\gamma\log g+\beta EV(s')\right]
\]
\[
\text{s.t. }c=(1\theta)y(z(B',s,q(B',s)))
\]
\[
g(B',s,q(B',s))+z(B',s,q(B',s))\delta B=\theta y(z)+q(B',s)\left(B'(1\delta)B\right)
\]
Bond prices are equal to riskfree price times probability of repayment
\[
q(B'(s),s)=\beta Ez(B'(s'),s',q(B'(s'),s'))
\]
At the end of the period, given the value fucntion, $B'$ and $q$,
the government chooses z and g to solve
\[
\max_{z,g}\left[\log c+\gamma\log g+\beta EV(B',z,\zeta')\right]
\]
\[
\text{s.t. }c=(1\theta)y(z)
\]
\[
g+z\delta B=\theta y(z)+q\left(B'(1\delta)B\right)
\]
\[
z\in\{0,1\}\ \ \text{and}\ \ z=0\text{ if }z_{1}=0
\]
Here, $\delta$ is the fraction of the debt rundown each period, which
can be derived from the $g$ policy function in part c
\subsection*{f) Use this model to interpret Mexican financial crisis of Dec 1994
through 1995. Discuss the strengths and weaknesses of this model.}
Mexico issued short term debt in Pesos but garunteed repayment in
American Dollars. This meant that devaluation of the Peso could lead
to the real value of the debt to increase. Thus when capital began
flowing out of Mexico, the Mexican government's debt increased without
them issuing additional bonds.
As such, what they thought was a safe level of debt (with $B<\bar{b}$
) entered into the crisis zone. Then a sunspot occured in the form
of the central bank running out of cash reserves, and so Mexico wasn't
able to raise enough new funds to pay off the old bonds and so was
forced to default.