Bargained haircuts and debt policy implications

Abstract

We extend the Cole and Kehoe model (J Int Econ 41:309–330, 1996) by adding a Rubinstein bargaining game between creditors and debtor country to determine the share of debt repayment in a sovereign debt crisis. Ex-post, the possibility of partial repayment avoids the costly case of total default, as seen recently in Greece. Ex-ante, the effects are to increase the sovereign debt cap and delay the fiscal adjustment. In other words, expectations of a haircut in times of crisis relax leverage restrictions implied by financial markets and make government more lenient, suggesting caution with haircut adoption, especially when risk-free interest rates are low.

Appendix

First, we highlight that, as already extensively discussed in the original Cole and Kehoe model (1996), a government that cares sufficiently more about private than public consumption or is sufficiently farsighted is guaranteed to have a non-empty crisis zone (i.e., $\overline{B}>\underline{B}$). Second, we assume the existence of a non-empty crisis zone for $\phi =0$, and we show, in the end of this appendix, that the condition $\overline{B}>\underline{B}$ remains as long as $\phi$ has an upper limit.Next, we are going to detail the three conditions—(i), (ii) and (iii)—that ensure the result of our proposition, taking as given the existence of the crisis zone for $\phi =0$. To derive such conditions, note that the floor of the crisis zone, $\underline{B},$ is defined as the highest stationary debt level under which not defaulting $(z=1)$ is better than defaulting $(z=\phi <1),$ even when there is a speculative attack, i.e., $q=0$ and new debt is not available. Then, to compute the floor of the crisis zone, we need to compare $V(s,B^\prime ,0,1)$ with $V(s,B^\prime ,0,\phi )$. Formally, $V(s,B^\prime ,0,1)$ is given by : 

$$\begin{aligned} V(s,B^\prime ,0,1)&=\underline{C1}+\underline{G1}\\ \underline{C1}&\equiv \frac{\left( 1-\theta \right) \left( f\left( K^{n}\right) -\delta K^{n}\right) }{1-\beta }\\ \underline{G1}&\equiv \frac{\left( 1-\beta ^{T}\right) v\left( g_{n} -\frac{B\left( 1-\beta \right) }{1-\beta ^{T}}\right) +\beta ^{T}v\left( g_{n}\right) }{1-\beta }\\ g_{n}&\equiv \theta \left[ f\left( K^{n}\right) -\delta K^{n}\right] \\ K^{n}\text { solves }\frac{1}{\beta }&=(1-\theta )\left[ f^\prime (k^{\prime })-\delta \right] +1\quad \text {and}\\ \frac{\partial V(s,B^\prime ,0,1)}{\partial B}&=\frac{\partial \underline{G1} }{\partial B}=-v^\prime \left( g_{n}-\frac{B\left( 1-\beta \right) }{1-\beta ^{T}}\right) <0 \end{aligned}$$

where the payment of $\frac{B\left( 1-\beta \right) }{1-\beta ^{T}}$ must be made at maturities $1,2,\ldots ,T$, and the value function is decreasing in the current debt level as expected. $V(s,B^\prime ,0,\phi )$ is given by:

$$\begin{aligned}&V(s,B^\prime ,0,\phi )=\underline{C0}+\underline{G0}\\&\underline{C0}\equiv \left( 1-\theta \right) \left[ \alpha f\left( K^{n}\right) -\delta K^{n}\right] +K^{n}-K^{d}+\frac{\beta \left( 1-\theta \right) \left[ \alpha f\left( K^{d}\right) -\delta K^{d}\right] }{1-\beta }\\&\underline{G0}\equiv v\left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -\frac{\phi B\left( 1-\beta \right) }{1-\beta ^{T} }\right) \\&\quad \qquad +\,\beta \frac{\left( 1-\beta ^{T-1}\right) v\left( g_{d}-\frac{\phi B\left( 1-\beta \right) }{1-\beta ^{T}}\right) +\beta ^{T-1}v\left( g_{d}\right) }{1-\beta }\\&g_{d}\equiv \theta \left[ \alpha f\left( K^{d}\right) -\delta K^{d}\right] \\&K^{d}\text { solves }\frac{1}{\beta }=(1-\theta )\left[ f^\prime (k^{\prime })\alpha -\delta \right] +1\\&\frac{\partial V(s,B^\prime ,0,\phi )}{\partial B}=\frac{\partial \underline{G0}}{\partial B}\text { which is given by:}\\&\left( \frac{-\phi }{1-\beta ^{T}}\right) \left\{ \left( 1-\beta \right) v^\prime \left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -\phi B_{T}\right) +\left( \beta -\beta ^{T}\right) v^\prime \left[ g_{d}-\phi B_{T}\right] \right\} \\&\text {where, } B_{T}\equiv \frac{B\left( 1-\beta \right) }{1-\beta ^{T}} \end{aligned}$$

and again, the debt payment of $\frac{\phi B\left( 1-\beta \right) }{1-\beta ^{T}}$ must be made during T periods, and the value function is decreasing in the current debt level as expected. For B sufficiently close to zero, it is trivial to conclude that $V(s\left( B\rightarrow 0\right) , B^\prime ,0,\phi )<V(s\left( B\rightarrow 0\right) , B^\prime ,0,1),$ since after the default, the productivity becomes lower without significantly improving the public expenditure. Moreover, if B is high enough, higher than $g_{n}\frac{1-\beta ^{T}}{\left( 1-\beta \right) },$ then $z_{t}=1$ is not an option as g cannot be negative. To ensure $z_{t}=\phi$ as the unique feasible option for this high debt level, $\phi$ must be lower than $\frac{g_{d}}{g_{n}}$.¹⁸ Then, to assure the existence of the floor of the crisis zone, $\underline{B},$ it is sufficient to have $\phi <\frac{g_{d}}{g_{n}}$ and $\frac{\partial V(s,B^\prime ,0,1)}{\partial B}<\frac{\partial V(s,B^\prime ,0,\phi )}{\partial B},$ or, more than sufficient to have : 

$$\begin{aligned} \frac{v^\prime \left( \theta \left( f\left( K^{n}\right) -\delta K^{n}\right) -\frac{B\left( 1-\beta \right) }{1-\beta ^{T}}\right) }{v^\prime \left( g_{d}-\phi \frac{B\left( 1-\beta \right) }{1-\beta ^{T} }\right) }>1>\phi \end{aligned}$$

assured by

$$\begin{aligned}&\theta \left[ f\left( K^{n}\right) -\delta K^{n}\right] -\theta \left[ \alpha f\left( K^{d}\right) -\delta K^{d}\right]<\frac{B\left( 1-\beta \right) }{1-\beta ^{T}}\left( 1-\phi \right) \\&\phi <1-\frac{\theta \left( f\left( K^{n}\right) -\delta K^{n}\right) -\theta \left( \alpha f\left( K^{d}\right) -\delta K^{d}\right) }{\frac{B\left( 1-\beta \right) }{1-\beta ^{T}}} \end{aligned}$$

i.e., funds raised by the government due to the partial default must be higher than the reduction of the government revenue from tax collection.

The cap of the crisis zone, $\overline{B},$ is defined as the highest stationary debt level under which not defaulting $(z=1)$ is better than defaulting $(z=\phi <1)$ when there is no speculative attack, i.e., $q=\beta$ and new lending is available. Then, to compute the cap of the crisis zone, we need to compare $V(s,B^\prime ,\beta ,1)$ with $V(s,B^\prime , \beta ,\phi ).$ Formally, $V(s,B^\prime ,\beta ,1)$ is given by : 

$$\begin{aligned}&V(s,B^\prime ,\beta ,1)=\overline{C1}+\overline{G1}\\&\overline{C1}\equiv \frac{\left( 1-\theta \right) \left( f\left( K^{n}\right) -\delta K^{n}\right) }{1-\beta }\\&\overline{G1}\equiv \frac{v\left( g_{n}-B\left( 1-\beta \right) \right) }{1-\beta }\\&\frac{\partial V(s,B^\prime ,\beta ,1)}{\partial B}=\frac{\partial \overline{G1}}{\partial B}=-v^\prime \left( g_{n}-B\left( 1-\beta \right) \right) \end{aligned}$$

where $B\left( 1-\beta \right)$ is the interest payment on total debt. The value function is decreasing in the current debt level as expected. $V(s,B^\prime ,\beta ,\phi )$ is given by : 

$$\begin{aligned}&V(s,B^\prime ,\beta ,\phi )=\overline{C0}+\overline{G0}\\&\overline{C0}\equiv \left( 1-\theta \right) \left[ \alpha f\left( K^{n}\right) -\delta K^{n}\right] +K^{n}-K^{d}+\frac{\beta }{1-\beta }\left[ \left( 1-\theta \right) \left( \alpha f\left( K^{d}\right) -\delta K^{d}\right) \right] \\&\overline{G0}\equiv v\left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -B_{T}\left( \phi -\beta ^{T}\right) \right) \\&\quad \qquad +\,\frac{\beta \left[ \left( 1-\beta ^{T-1}\right) v\left( g_{d}-\phi B_{T}\right) +\beta ^{T-1}v\left( g_{d}\right) \right] }{1-\beta }\\&B_{T}\equiv \frac{B\left( 1-\beta \right) }{1-\beta ^{T}}\quad \text {and}\quad q_{_{T} }=\beta ^{T}\\&\qquad \,\, \left( -\frac{\partial V(s,B^\prime ,\beta ,\phi )}{\partial B}\right) \text { is given by:}\\&\frac{\left( 1-\beta \right) \left( \phi -\beta ^{T}\right) }{1-\beta ^{T} }v^\prime \left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -B_{T}\left( \phi -\beta ^{T}\right) \right) \\&\qquad +\,\phi \frac{\beta \left( 1-\beta ^{T-1}\right) }{1-\beta ^{T}}\left[ v^\prime \left( g_{d}-\phi B_{T}\right) \right] \end{aligned}$$

For B sufficiently close to zero, it is trivial to conclude that $V(s\left( B\rightarrow 0\right) , B^\prime ,\beta ,\phi )<V(s\left( B\rightarrow 0\right) , B^\prime ,\beta ,1),$ since after the default, the productivity becomes lower without significantly improving the public expenditure. Moreover, if B is high enough, higher than $\frac{g_{n} }{\left( 1-\beta \right) },$ then $z_{t}=1$ is not an option as g cannot be negative. To ensure $z_{t}=\phi$ as the unique feasible option for this high debt level, $\phi$ must be lower than $\left( 1-\beta ^{T}\right) \frac{g_{d}}{g_{n}}$. Given this condition, to ensure the existence of the cap of the crisis zone, $\overline{B},$ it is sufficient to have $\frac{dV(s,B^\prime ,\beta ,1)}{dB}<\frac{dV(s,B^\prime ,\beta ,\phi )}{dB},$ or, by considering $\left( \phi \le \beta ^{T}\right)$, it is sufficient to have:

$$\begin{aligned}&\frac{v^\prime \left( g_{n}-B\left( 1-\beta \right) \right) }{v^\prime \left( g_{d}-\phi B_{T}\right) }>\phi \beta \\&\text {assured by}\\&g_{n}-B\left( 1-\beta \right)<g_{d}-\phi B_{T}\\&\phi B_{T}+\left( g_{n}-g_{d}\right)<B\left( 1-\beta \right) \\ \quad&\text {or}\\&\phi <\left( 1-\beta ^{T}\right) \frac{B\left( 1-\beta \right) -\left( g_{n}-g_{d}\right) }{B\left( 1-\beta \right) } \end{aligned}$$

Then, we have two conditions for the existence of $\overline{B}$. First, partial default must occur on the principal amount of the debt, i.e., partial default must imply negative return on bonds $\left( \phi <\beta ^{T}\right)$. Note that, after the payment of $\phi$, the rate of return is equal to $\left( \frac{\phi }{\beta ^{T}}-1\right)$. Second, the flow reduction of the government revenue from both tax collection $\left( g_{n}-g_{d}\right)$ and partial repayment of maturing debt $\left( \phi B_{T}\right)$ must be lower than the total interest payment on the total debt $B\left( 1-\beta \right)$. Therefore, the conditions for $\underline{B}$ and $\overline{B}$ to be well defined are the following:

$$\begin{aligned}&\phi B_{T}+\left( g_{n}-g_{d}\right)<\frac{B\left( 1-\beta \right) }{1-\beta ^{T}}\\&\phi B_{T}+\left( g_{n}-g_{d}\right)<B\left( 1-\beta \right) \\&\phi B_{T}+\left( g_{n}-g_{d}\right)<\beta ^{T}B_{T}+\left( g_{n} -g_{d}\right) \\&\phi <\left( 1-\beta ^{T}\right) \frac{g_{d}}{g_{n}} \end{aligned}$$

and noting that the first condition is redundant, we can focus only in the following three conditions:

$$\begin{aligned}&(\hbox {i})\quad \phi B_{T}+\left( g_{n}-g_{d}\right)<B\left( 1-\beta \right) \\&(\hbox {ii})\quad \phi<\beta ^{T}\\&(\hbox {iii})\quad \phi <\left( 1-\beta ^{T}\right) \frac{g_{d}}{g_{n}} \end{aligned}$$

Again, as already discussed in the original CK96 paper, a government that cares sufficiently more about private than government consumption or is sufficiently farsighted is guaranteed to have a crisis zone $(\overline{B}>\underline{B})$. For a non-empty crisis zone, conditions (i), (ii) and (iii) are sufficient to ensure the result of proposition, i.e., the higher the $\phi ,$ the higher are the limits $\overline{B}$ and $\underline{B}$. Note that since $\underline{B}$ is increasing in $\phi$ and $V(s,B^\prime ,0,1)$ does not depend on $\phi$, it is sufficient to show that $\frac{\partial V(s,B^\prime ,0,\phi )}{\partial \phi }<0,$ which is true, since : 

$$\begin{aligned} \left[ \left( 1-\beta \right) v^\prime \left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -\frac{\phi B\left( 1-\beta \right) }{1-\beta ^{T}}\right) \right. \\ \left. +\,\beta \left( 1-\beta ^{T-1}\right) v^\prime \left( g_{d}-\frac{\phi B\left( 1-\beta \right) }{1-\beta ^{T}}\right) \right] >0 \end{aligned}$$

and to have $\overline{B}$ increasing in $\phi ,$ it is sufficient to show that $\frac{\partial V(s,B^\prime ,\beta ,\phi )}{\partial \phi }<0,$ which is true:

$$\begin{aligned}&-\,B_{T}\ \left\{ v^\prime \left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -B_{T}\left( \phi -\beta ^{T}\right) \right) \right. \\&\qquad \left. +\,\frac{\beta \left[ \left( 1-\beta ^{T-1}\right) v^\prime \left( g_{d}-\phi B_{T}\right) \right] }{1-\beta }\right\} <0 \end{aligned}$$

Finally, in order to assure that the crisis zone characterized under total default; $\left[ \underline{B}_{\phi =0},\overline{B}_{\phi =0}\right]$ remains non-empty under partial default ($\phi >0$), i.e., to assure that $\underline{B}_{\phi >0}$ remains smaller than $\overline{B}_{\phi >0}$ when partial repayment is allowed, it is sufficient to show that the resulted reduction in the crisis zone is limited:

or it is sufficient to have

where,

$$\begin{aligned}&\hbox {LHS}=-v^\prime \left( g_{n}-\frac{B\left( 1-\beta \right) }{1-\beta ^{T}}\right) \\&\qquad \times \left( \begin{array}[c]{c} \frac{\partial V(s,B^\prime ,\beta ,\phi )}{\partial \phi }-\frac{\left( 1-\beta \right) \left( \phi -\beta ^{T}\right) v^\prime \left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -B_{T}\left( \phi -\beta ^{T}\right) \right) }{1-\beta ^{T}}\\ -\frac{\phi \beta \left( 1-\beta ^{T-1}\right) v^\prime \left( g_{d}-\phi B_{T}\right) }{1-\beta ^{T}} \end{array}\right) \\&\hbox {RHS}=-v^\prime \left( g_{n}-B\left( 1-\beta \right) \right) \\&\qquad \times \left( \frac{\partial V(s,B^\prime ,0,\phi )}{\partial \phi }-\left( \frac{\phi }{1-\beta ^{T}}\right) \left\{ \begin{array}[c]{l} \left( 1-\beta \right) v^\prime \left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -\phi B_{T}\right) \\ +\left( \beta -\beta ^{T}\right) v^\prime \left[ g_{d}-\phi B_{T}\right] \end{array}\right\} \right) \end{aligned}$$

where

$$\begin{aligned}&\frac{\partial V(s,B^\prime ,\beta ,\phi )}{\partial \phi }=-B_{T}v^\prime \left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -B_{T}\left( \phi -\beta ^{T}\right) \right) \\&\quad -B_{T}\frac{\beta \left( 1-\beta ^{T-1}\right) }{1-\beta }v^\prime \left( g_{d}-\phi B_{T}\right) \\&\frac{\partial V(s,B^\prime ,0,\phi )}{\partial \phi }=-B_{T}v^\prime \left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -\phi B_{T}\right) \\&\quad -B_{T}\frac{\beta \left( 1-\beta ^{T-1}\right) }{1-\beta }v^\prime \left( g_{d}-\phi B_{T}\right) \\&\hbox {so}\\&\hbox {LHS}=v^\prime \left( g_{n}-\frac{B\left( 1-\beta \right) }{1-\beta ^{T} }\right) \\&\quad \times \left( \begin{array} [c]{c} \left( \phi +B-\beta ^{T}\right) \left( \frac{1-\beta }{1-\beta ^{T}}\right) v^\prime \left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -B_{T}\left( \phi -\beta ^{T}\right) \right) \\ +\left( \phi +B\right) \left( \frac{\beta -\beta ^{T}}{1-\beta ^{T}}\right) v^\prime \left( g_{d}-\phi B_{T}\right) \end{array} \right) \\&\hbox {RHS}=v^\prime \left( g_{n}-B\left( 1-\beta \right) \right) \\&\quad \times \left( \begin{array} [c]{c} \left( \phi +B\right) \left( \frac{1-\beta }{1-\beta ^{T}}\right) v^\prime \left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -\phi B_{T}\right) \\ +\left( \phi +B\right) \left( \frac{\beta -\beta ^{T}}{1-\beta ^{T}}\right) v^\prime \left( g_{d}-\phi B_{T}\right) \end{array} \right) \end{aligned}$$

and to have non-empty crisis zone at least for $T=1$ and a positive $\phi$, after replacing the debt level where derivative is evaluated, it is sufficient to have:

$$\begin{aligned}&\left( \phi +\underline{B}\right) v^\prime \left( g_{n}-\overline{B}\left( 1-\beta \right) \right) \left( v^\prime \left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -\phi \underline{B}\right) \right) \\&\qquad -\left( \phi +\overline{B}-\beta \right) v^\prime \left( g_{n}-\underline{B} \right) \left( v^\prime \left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -\phi \overline{B}+\beta \overline{B}\right) \right) \le 0 \end{aligned}$$

And it is sufficient

$$\begin{aligned} \frac{\left( v^\prime \left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -\phi \underline{B}\right) \right) }{v^\prime \left( g_{n}-\underline{B}\right) }\le \frac{\left( \phi +\overline{B}-\beta \right) \left( v^\prime \left( \theta \left( \alpha f\left( K^{n}\right) -\delta K^{n}\right) -\phi \overline{B}+\beta \overline{B}\right) \right) }{\left( \phi +\underline{B}\right) v^\prime \left( g_{n}-\overline{B}\left( 1-\beta \right) \right) } \end{aligned}$$

As in the original Cole and Kehoe model, depending on the set of parameters, the crisis zone is non-empty. For a particular example, and remembering that a government that cares sufficiently more about private than public consumption or that is sufficiently farsighted assures a non-empty crisis zone (i.e., $\Delta >0$), suppose $v^\prime (.)$ tends to a constant $\kappa$. Then, the previous condition becomes

$$\begin{aligned} \beta \le \overline{B}_{\phi =0}-\underline{B}_{\phi =0}\equiv \Delta \end{aligned}$$

and $\beta <\Delta$ makes the incentive to smooth the private consumption important enough to assure a non-empty crisis zone for, at least, $T=1$, and small positive $\phi$.