# Trade Prelim Notes

## Self-fulfilling Debt Crises

This problem is from Tim Kehoe. It showed up on the trade prelims in 2016 (spring) and 2018.

# Dump of LaTeX solution

\newpage Problem 4 Consider a small open economy with a governemen tthat borrows. In each period, value of output is $y(z)=Z^{1-z}\bar{y}$ $Z\in(0,1)$ 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 risk-free 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 time-discounted 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*} $\bar{b}=\left(1-Z^{\frac{\gamma+1}{\gamma(1-\beta)}}\right)\theta\bar{y}$ 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 $\tilde{g}(T,B_{0})=\theta\bar{y}-\frac{1-\beta(1-\pi)}{1-\left(\beta(1-\pi)\right)^{T}}\left(B_{0}-(\beta(1-\pi))^{T-1}\beta\bar{b}\right)$ $\tilde{g}(\infty,B_{0})=\lim_{T\to\infty}\tilde{g}(T,B_{0})=\theta\bar{y}-\left(1-\beta(1-\pi)\right)\left(B_{0}\right)$ 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 t-1 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)^{T-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}\\ & +\left(\beta(1-\pi)\right)^{T-2}\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 a-d} 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 risk-free 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.