Gambling for redemption and self-fulfilling debt crises

Abstract

We develop a model for analyzing the sovereign debt crises of 2010–2013 in the Eurozone. The government sets its expenditure–debt policy optimally. The need to sell large quantities of bonds every period leaves the government vulnerable to self-fulfilling crises in which investors, anticipating a crisis, are unwilling to buy the bonds, thereby provoking the crisis. In this situation, the optimal policy of the government is to reduce its debt to a level where crises are not possible. If, however, the economy is in a recession where there is a positive probability of recovery in fiscal revenues, the government also has an incentive to smooth consumption and increase debt. Our exercise identifies conditions on fundamentals for which the incentive to smooth consumption dominates, giving rise to a situation where governments optimally “gamble for redemption,” running fiscal deficits and increasing their debt, thereby increasing their vulnerability to crises.

Appendices

Appendix A: Characterization of equilibria of the model without self-fulfilling crisis

Our analysis in Sect. 4 distinguishes between two cases. We consider first case 1, where the government chooses to never violate the constraint $B\le \bar{{B}}(0)$. The optimal government policy is the solution to the dynamic programming problem:

$$\begin{aligned} V(B,a)= & {} \max u\left( (1-\theta )A^{1-a}\bar{{y}},\theta A^{1-a}\bar{{y}}+\beta B^{\prime }-B\right) +\beta EV(B^{\prime },a^{\prime })\nonumber \\&\hbox {s.t. }B\le \bar{{B}}(0). \end{aligned}$$

(A.1)

We write the Bellman’s equation explicitly as

$$\begin{aligned} V(B,0)= & {} \max u\left( (1-\theta )A\bar{{y}},\theta A\bar{{y}}+\beta B^{\prime }-B\right) +\beta (1-p)V(B^{\prime },0)+\beta pV(B^{\prime },1)\nonumber \\ \end{aligned}$$

(A.2)

$$\begin{aligned} V(B,1)= & {} \max u\left( (1-\theta )\bar{{y}},\theta \bar{{y}}+\beta B^{\prime }-B\right) +\beta V(B^{\prime },1). \end{aligned}$$

(A.3)

The first-order condition is

$$\begin{aligned} \beta u_g \left( (1-\theta )A^{1-a}\bar{{y}},\theta A^{1-a}\bar{{y}}+\beta B^{\prime }-B\right) =\beta EV_B (B^{\prime },a^{\prime }), \end{aligned}$$

(A.4)

and the envelope condition is

$$\begin{aligned} V_B (B,a)=-u_g \left( (1-\theta )A^{1-a}\bar{{y}},\theta A^{1-a}\bar{{y}}+\beta B^{\prime }-B\right) . \end{aligned}$$

(A.5)

The envelope condition implies that V(B, a) is decreasing in B. A standard argument—that the operator on the space of functions defined by the Bellman’s equation maps concave value functions into concave value functions—implies that V(B, a) is concave in B.

The first-order condition (A.4) implies that the policy function for debt $B^{\prime }(B,a)$ is increasing in B, while the policy function for government spending g(B, a) is decreasing in B. Our assumption that

$$\begin{aligned} u_g ((1-\theta )A\bar{{y}},\theta A\bar{{y}}-B)>u_g ((1-\theta )\bar{{y}},\theta \bar{{y}}-B) \end{aligned}$$

(A.6)

implies that $B^{\prime }(0,0)>0$ and that it is impossible for $B^{\prime }(B,0)=B$ unless the constraint $B^{\prime }\le \bar{{B}}(0)$ binds, which implies that $B^{\prime }(B,0)>B$.

We now consider case 2, where the government chooses to violate the constraint $B\le \bar{{B}}(0)$ with its sale of debt in period T, defaulting in period $T+1$ unless the private sector recovers. The optimal government policy is the solution to the finite horizon dynamic programming problem:

$$\begin{aligned} V_t (B_t )= & {} \max u\left( (1-\theta )A\bar{{y}},\theta A\bar{{y}}+\beta B_{t+1} -B_t \right) \nonumber \\&+\,\beta (1-p)V_{t+1} (B_{t+1} )+\beta p\frac{u\left( (1-\theta )\bar{{y}},\theta \bar{{y}}+(1-\beta )B_{t+1} )\right) }{1-\beta } \nonumber \\&\hbox {s.t. }B_t \le \bar{{B}}(0). \end{aligned}$$

(A.7)

We solve this problem by backward induction with the terminal value function:

$$\begin{aligned} V_T (B_T )= & {} \max u\left( (1-\theta )A\bar{{y}},\theta A\bar{{y}}+\beta pB_{T+1} -B_T \right) \nonumber \\&+\,\beta (1-p)\frac{u((1-\theta )Z\bar{{y}},\theta Z\bar{{y}}))}{1-\beta }+\beta p\frac{u((1-\theta )\bar{{y}},\theta \bar{{y}}+(1-\beta )B_{T+1} ))}{1-\beta } \nonumber \\&\hbox {s.t. }\bar{{B}}(1)\ge B_{T+1} \ge \bar{{B}}(0). \end{aligned}$$

(A.8)

We then choose the value of T for which $V_0 (B_0 )$ is maximal. As long as the constraint $B_{T+1} \ge \bar{{B}}(0)$ binds, we can increase the value of $V_0 (B_0 )$ by increasing T.

The algorithm for calculating the optimal policy function is a straightforward application of backward induction. We work backward from the period in which the government borrows at price $\beta p$ and defaults in the next period unless a recovery of the private sector occurs. Define

$$\begin{aligned} V_T (B)= & {} \max u((1-\theta )A\bar{{y}},\theta A\bar{{y}}+\beta pB^{\prime }-B) \nonumber \\&+\,\beta (1-p)\frac{u((1-\theta )Z\bar{{y}},\theta Z\bar{{y}}))}{1-\beta }+\beta p\frac{u((1-\theta )\bar{{y}},\theta \bar{{y}}+(1-\beta )B^{\prime }))}{1-\beta } \nonumber \\&\hbox {s.t. }\bar{{B}}(0)\le B^{\prime }\le \bar{{B}}(1). \end{aligned}$$

(A.9)

The steps of the algorithm are as follows:

• 1. Solve for the value function $V_0 (B)$ and the policy function $B_0 ^{\prime }(B)$ on a grid of bonds B on the interval $[{\underline{B}},\bar{{B}}(0)]$. We can set the lower limit $\underline{B}$ equal to any value, including a negative value. In an application with a given initial stock of debt, we could set $\underline{B}=B_0$. We have already solved this problem analytically. The solution is $B^{\prime }(B)=\min [\hat{{B}}^{\prime }(B),\bar{{B}}(1)]$ unless $B^{\prime }(B)<\bar{{B}}(0)$, in which case ${B}_0^{'} (B)=\bar{{B}}(0)$. Consequently,

  $$\begin{aligned} B_0 ^{\prime }(B)=\max \left[ {\bar{{B}}(0),\min [\hat{{B}}^{\prime }(B),\bar{{B}}(1)]} \right] . \end{aligned}$$

  (A.10)

  The values of B for which $B^{\prime }(B)<\bar{{B}}(0)$ are those for which it is not optimal to set $T=0$.

• 2. Let $t=0$, and set $\tilde{B}_0 =\bar{{B}}(0)$.

• 3. Solve for the value function $V_{t+1} (B,0)$ and the policy function $B_{t+1} ^{\prime }(B)$ in the Bellman’s equation

  $$\begin{aligned} V_{t+1} (B,0)= & {} \max u((1-\theta )A\bar{{y}},\theta A\bar{{y}}+\beta B^{\prime }-B) \nonumber \\&+\,\beta (1-p)V_t (B^{\prime },0)+\beta p\frac{u((1-\theta )\bar{{y}},\theta \bar{{y}}+(1-\beta )B))}{1-\beta }.\nonumber \\ \end{aligned}$$

  (A.11)

  Let $\tilde{B}_t$ be the largest value of B for which $V_{t+1} (B,0)\ge V_t (B,0)$.

• 4. Repeat step 3 until $\tilde{B}_t =\underline{B}$.

Let T be such that $\tilde{B}_T =\underline{B}$. We can prove that $\underline{B}<\tilde{B}_{T-1}<\tilde{B}_{T-2}<\cdots<\tilde{B}_1 <\bar{{B}}(0)$. Our algorithm divides the interval $[\underline{B},\bar{{B}}(0)]$ into subintervals $[\underline{B},\tilde{B}_{T-1} )$, $[\tilde{B}_{T-1} ,\tilde{B}_{T-2} ),{\ldots },\,[\tilde{B}_1 ,\bar{{B}}(0;p,0)]$. If the initial capital stock $B_0$ is in the subinterval $[\tilde{B}_t ,\tilde{B}_{t-1} ]$, then the optimal government policy is to increase B, selling debt B, $\bar{{B}}(0)<B\le \bar{{B}}(1)$, in period $t-1$, and defaulting in period t unless the private sector recovers. The optimal sequence of debt is $B_0$, $B_{t-1} ^{\prime }(B_0 )$, $B_{t-2} ^{\prime }(B_{t-1} ^{\prime }(B_0 )){\ldots }, B_0 (B_1 (\cdots (B_{t-1} ^{\prime }(B_0 ))))$.

Appendix B: The algorithm for computing an equilibrium in the general model

The algorithm computes the four debt thresholds, the value functions, and the policy functions.

• 1. Compute the value function $V\left( {B,a,z_{-1} ,\zeta } \right)$ of being in the default state, where $B=0$ and $z_{-1} =0$. To simplify notation, we denote it $V_d (a)$. Notice that these values are independent of the sunspot $\zeta$, which becomes irrelevant after a default has occurred. The value function of defaulting in normal times, where $a=1$, is

  $$\begin{aligned} V_d (1)=u(\left( {1-\theta } \right) Zy,\theta Zy)+\beta V_d (1) \end{aligned}$$

  (B.1)

  which is just a constant:

  $$\begin{aligned} V_d (1)=\frac{1}{1-\beta }u((1-\theta )Zy,\theta Zy). \end{aligned}$$

  (B.2)

  Similarly, in a recession, where $a=0$,

  $$\begin{aligned} V_d (0)=u((1-\theta )AZy,\theta AZy)+\beta pV_d (1)+\beta (1-p)V_d (0) \end{aligned}$$

  (B.3)

  which is also a constant:

  $$\begin{aligned} V_d (0)= & {} \frac{1}{1-\beta \left( {1-p} \right) }u((1-\theta )AZy,\theta AZy) \nonumber \\&+\,\frac{\beta p}{\left( {1-\beta \left( {1-p} \right) } \right) \left( {1-\beta } \right) }u((1-\theta )Zy,\theta Zy) \end{aligned}$$

  (B.4)

  Notice that the value functions become those obtained above whenever the state of the economy determines that a self-fulfilling debt crisis happens or has happened in the past. To simplify notation, from this point on, we describe how to compute the value functions in the case of no default and drop the variable $z_{-1}$ that determines whether a government has defaulted in the past and the sunspot $\zeta$ as arguments of the value functions. That is, from this point on, V(B, a) is the value function if default has not happened today or anytime in the past.

• 2. Guess initial values for the thresholds $\bar{{b}}(0)$, $\bar{{b}}(1)$, $\bar{{B}}(0)$, $\bar{{B}}(1)$, where $\bar{{b}}(0)<\bar{{b}}(1)<\bar{{B}}(0)<\bar{{B}}(1)$, and the associated prices. (We could also modify the algorithm to calculate an equilibrium in the case where $\bar{{b}}(0)<\bar{{B}}(0)<\bar{{b}}(1)<\bar{{B}}(1)$.)

• 3. Perform value function iteration on a finite grid of values of debt to compute the value function in normal times, $a=1$. Guess an initial value function in normal times if default has not happened in the past, $\tilde{V}(B,1)$, and an optimal debt policy, which is needed to recursively compute the prices, $\tilde{q}({B}^{'},1)$, defined as in equation (57), in the case with multiperiod debt. Then:

  • 3.1. For values of initial debt $B\le \bar{{B}}(1)$, the value function is

    $$\begin{aligned} V(B,1)=\max [V_1 (B,1),V_2 (B,1)], \end{aligned}$$

    (B.5)

    where $V_1 ,V_2$ are the value functions if next period bonds are in the regions $B^{\prime }\le \bar{{b}}(1)$ or $\bar{{b}}(1)<B^{\prime }\le \bar{{B}}(1)$, respectively:

    $$\begin{aligned} V_1 ({B,1})= & {} \max u((1-\theta )y,\theta y+\tilde{q}({B}^{'},1)(B^{\prime }\nonumber \\&-(1-\delta )B)-\delta B)+\beta \tilde{V}(B^{\prime },1) \nonumber \\&\hbox { s.t. }B^{\prime }\le \bar{{b}}(1) \end{aligned}$$

    (B.6)

    and

    $$\begin{aligned} V_2 \left( {B,1} \right)= & {} \max u((1-\theta )y,\theta y+\tilde{q}({B}^{'},1)(B^{\prime }-(1-\delta )B)-\delta B)\nonumber \\&\quad +\,\beta (1-\pi )\tilde{V}(B^{\prime },1)+\beta \pi V_d (1) \nonumber \\&\hbox { s.t. }\bar{{b}}(1)<B^{\prime }\le \bar{{B}}(1). \end{aligned}$$

    (B.7)

  • 3.2. For high values of initial debt, $B>\bar{{B}}(1)$, set $V(B,1)=V_d (1)$.

  • 3.3. If there is multiperiod debt, compute the pricing function, q(B, 1), recursively, as in Eq. (57), using the optimal policy function.

  • 3.4. If $\max \limits _B \left| {V(B,1)-\tilde{V}(B,1)} \right| >\varepsilon$ and $\max \limits _B \left| {q(B,1)-\tilde{q}(B,1)} \right| >\varepsilon$, where $\varepsilon$ is a preset convergence criterion, then $\tilde{V}\left( {B,1} \right) =V\left( {B,1} \right)$ and $\tilde{q}\left( {B,1} \right) =q\left( {B,1} \right)$ and go to 3.1. Else, go to 4.

• 4. Perform value function iteration on a finite grid of values of debt to compute the value function in a recession, $a=0$. Guess an initial value function if we are in a recession and the government has not defaulted in the past: $\tilde{V}({B,0})$, and an optimal debt policy, which is needed to recursively compute the prices, $\tilde{q}({B}^{'},0)$, defined as in Eq. (58). Remember the value function in normal times, V(B, 1), is already known from step 3. Then:

  • 4.1. For values of initial debt where $B\le \bar{{B}}(0)$, the value function is

    $$\begin{aligned} V(B,0)=\max [V_1 (B,0),V_2 (B,0),V_3 (B,0),V_4 (B,0)], \end{aligned}$$

    (B.8)

    where $V_1$, $V_2$, $V_3$, $V_4$ are the associated value functions if next period bonds are in the regions $B^{\prime }\le \bar{{b}}(0)$, $\bar{{b}}(0)<B^{\prime }\le \bar{{b}}(1)$, $\bar{{b}}(1)<B^{\prime }\le \bar{{B}}(0)$, $\bar{{B}}(0)<B^{\prime }\le \bar{{B}}(1)$, respectively:

    $$\begin{aligned} V_1 (B,0)= & {} \max u((1-\theta )Ay,\theta Ay+\tilde{q}(B^{\prime },0)(B^{\prime }-(1-\delta )B)-\delta B) \nonumber \\&\quad +\,\beta pV(B^{\prime },1)+\beta (1-p)\tilde{V}(B^{\prime },0) \nonumber \\&\hbox { s.t. }B^{\prime }\le \bar{{b}}(0) \end{aligned}$$

    (B.9)
    $$\begin{aligned} V_2 (B,0)= & {} \max u((1-\theta )Ay,\theta Ay+\tilde{q}(B^{\prime },0)(B^{\prime }-(1-\delta )B)-\delta B) \nonumber \\&\quad +\,\beta pV(B^{\prime },1)+\beta (1-p)\pi V_d (0)+\beta (1-p)(1-\pi )\tilde{V}(B^{\prime },0) \nonumber \\&\hbox { s.t. }\bar{{b}}(0)<B^{\prime }\le \bar{{b}}(1) \end{aligned}$$

    (B.10)
    $$\begin{aligned} V_3 (B,0)= & {} \max u\left( (1-\theta )Ay,\theta Ay+\tilde{q}(B^{\prime },0)(B^{\prime }-(1-\delta )B)-\delta B\right) \nonumber \\&\quad +\,\beta p\pi V_d (1)+\beta p(1-\pi )V(B^{\prime },1) \nonumber \\&\quad +\,\beta (1-p)\pi V_d (0)+\beta (1-p)(1-\pi )\tilde{V}(B^{\prime },0) \nonumber \\&\hbox { s.t. }\bar{{b}}(1)<B^{\prime }\le \bar{{B}}(0) \end{aligned}$$

    (B.11)
    $$\begin{aligned} V_4 (B,0)= & {} \max u((1-\theta )Ay,\theta Ay+\tilde{q}(B^{\prime },0)(B^{\prime }-(1-\delta )B)-\delta B) \nonumber \\&\quad +\,\beta p\pi V_d (1)+\beta p(1-\pi )V(B^{\prime },1)+\beta (1-p)V_d (0) \nonumber \\&\hbox { s.t. }\bar{{B}}(0)<B^{\prime }\le \bar{{B}}(1). \end{aligned}$$

    (B.12)

  • 4.2. For high values of initial debt, $B>\bar{{B}}(0)$, set $V(B,0)=V_d (0)$.

  • 4.3. If there is multiperiod debt, compute the pricing function, q(B, 0), recursively, as in Eq. (58), using the optimal policy function.

  • 4.4. If $\max \limits _B \left| {V\left( {B,0} \right) -\tilde{V}\left( {B,0} \right) } \right| >\varepsilon$ and $\mathop {\max }\limits _B \left| {q\left( {B,0} \right) -\tilde{q}\left( {B,0} \right) } \right| >\varepsilon$, then $\tilde{V}\left( {B,0} \right) =V\left( {B,0} \right)$ and $\tilde{q}({B,0})=q({B,0})$ and go to 4.1. Else, go to 5.

• 5. Update the threshold values:

  • 5.1. Choose $\bar{{b}}_{new} (0)$ to be the highest point in the grid for B for which

    $$\begin{aligned}&u((1-\theta )Ay,\theta Ay-\delta B)+\beta pV((1-\delta )B,1)\nonumber \\&\quad +\,\beta (1-p)V((1-\delta )B,0)\ge V_d (0). \end{aligned}$$

    (B.13)

  • 5.2. Choose $\bar{{b}}_{new} (1)$ to be the highest point in the grid for which

    $$\begin{aligned} u((1-\theta )y,\theta y-\delta B)+\beta V((1-\delta )B,1)\ge V_d (1). \end{aligned}$$

    (B.14)

  • 5.3. Choose $\bar{{B}}_{new} (0)$ to be the highest point in the grid for which

    $$\begin{aligned}&V\left( {B,0} \right) \ge u((1-\theta )ZAy,\theta ZAy+q(B^{\prime },0)(B^{\prime }(B,0)-(1-\delta )B)) \nonumber \\&\quad +\,\beta pV_d (1)+\beta (1,p)V_d (0), \end{aligned}$$

    (B.15)

    where $q(B^{\prime },0)$ are the prices computed in step 5.

  • 5.4. Choose $\bar{{B}}_{new} (1)$ to be the highest point in the grid for which

    $$\begin{aligned}&V(B,1)\ge u((1-\theta )Zy,\theta Zy+q(B^{\prime },1)(B^{\prime }(B,1)-(1-\delta )B))\nonumber \\&\quad +\,\beta V_d (1). \end{aligned}$$

    (B.16)

  • 5.5. If $\left| {\bar{{b}}_{new} (0)-\bar{{b}}(0)} \right| >\varepsilon$ or $\left| {\bar{{b}}_{new} (1)-\bar{{b}}(1)} \right| >\varepsilon$ or $\left| {\bar{{B}}_{new} (0)-\bar{{B}}(0)} \right| >\varepsilon$ or $\left| {\bar{{B}}_{new} (1)-\bar{{B}}(1)} \right| >\varepsilon$, then $\bar{{b}}(0)=\bar{{b}}_{new} (0),\bar{{b}}(1)=\bar{{b}}_{new} (1),\bar{{B}}(0)=\bar{{B}}_{new} (0),\bar{{B}}(1)=\bar{{B}}_{new} (1)$ and go to 3. Else, exit.

    Notice that the lower threshold in normal times, $\bar{{b}}(1)$, can be computed directly since no information about policy functions is required. Hence, the iterative procedure would not be necessary, but we choose to do it this way to be consistent with the computation of the upper thresholds and the lower threshold in recession, which do depend on the policy function and hence require an iterative procedure.