Approaches to Solving the Eurozone Sovereign Debt Default Problem

Abstract

The Eurozone sovereign debt crisis stems from a failure in risk management design. In this paper we first present a formal model to clarify the nature of this failure, and then use it to analyse possible solutions. We argue that a long-term solution must involve institutional innovation based on the mutual insurance principle. We also critically discuss existing proposals in the light of our results.

Paper presented at the Conference in Celebration of Gerhard Illing’s 60th Birthday, Munich, 4–5 March 2016.

Appendix

1.1 Portfolio Problem

In country j ∈ {R, S} the consumer’s portfolio choice problem is:

$$\displaystyle{ \max _{b_{j}^{R},b_{j}^{S}}u(c_{0}^{\,j}) +\rho \{ (1-\pi )[u(c_{ H}^{\,j}) +\pi u(c_{ L}^{,j})]\} }$$

(35)

given:

$$\displaystyle\begin{array}{rcl} c_{0}^{\,j} = w_{ 0}^{\,j} + b_{ j}^{0} - q^{S}b_{ j}^{S} - q^{R}b_{ j}^{R}& &{}\end{array}$$

(36)

$$\displaystyle\begin{array}{rcl} c_{H}^{\,j} = w_{ H}^{\,j} + b_{ j}^{S} + b_{ j}^{R}& &{}\end{array}$$

(37)

$$\displaystyle\begin{array}{rcl} c_{L}^{\,j} = w_{ L}^{\,j} + b_{ j}^{S} + b_{ j}^{R}(1 - h)& &{}\end{array}$$

(38)

where h is the proportionate loss of income each investor expects as a result of the haircut imposed by country R in state L. As explained in the text, wage rates w are determined by the government at each date/state. We also impose the no-short sales constraints

$$\displaystyle{ b_{j}^{S},b_{ j}^{R} \geq 0 }$$

(39)

though assume that they do not bind. The Lagrange function for the problem is

$$\displaystyle\begin{array}{rcl} & \mathcal{L} =\text{ }u(c_{0}^{\,j}) +\rho [(1-\pi )u(b_{j}^{S} + b_{j}^{R} + w_{H}^{\,j}) +\pi u(b_{j}^{S} + b_{j}^{R}(1 - h) + w_{L}^{\,j})]& \\ & +\lambda [b_{0}^{\,j} + w_{0}^{\,j} - q^{S}b_{j}^{S} - q^{R}b_{j}^{R} - c_{0}^{\,j}] &{}\end{array}$$

(40)

FOC are²⁷

$$\displaystyle\begin{array}{rcl} \frac{\partial \mathcal{L}} {\partial c_{0}^{\,j}} = u^{{\prime}}(c_{ 0}^{\,j{\ast}}) -\lambda ^{{\ast}} = 0& &{}\end{array}$$

(41)

$$\displaystyle\begin{array}{rcl} \frac{\partial \mathcal{L}} {\partial b_{j}^{S}} =\rho [(1-\pi )u^{{\prime}}(c_{ H}^{\,j{\ast}}) +\pi u^{{\prime}}(c_{ L}^{\,j{\ast}})] -\lambda ^{{\ast}}q^{S} = 0& &{}\end{array}$$

(42)

$$\displaystyle\begin{array}{rcl} & \frac{\partial \mathcal{L}} {\partial b_{j}^{R}} =\rho [(1-\pi )u^{{\prime}}(b_{j}^{S{\ast}} + b_{j}^{R{\ast}} + w_{H}^{\,j}) + (1 - h)\pi u^{{\prime}}(b_{j}^{S{\ast}} + b_{j}^{R{\ast}}(1 - h) + w_{L}^{\,j}]& \\ & -\lambda ^{{\ast}}q^{R} \leq 0; & \\ & \text{ }b_{j}^{R{\ast}}\geq 0;\text{ }b_{j}^{R{\ast}} \frac{\partial \mathcal{L}} {\partial b_{j}^{R}} = 0 &{}\end{array}$$

(43)

Given λ ^∗ > 0, which follows from (36) with a non-satiation assumption, we have:

$$\displaystyle{ b_{0} + w_{0}^{\,j} - (c_{ 0}^{{\ast}} + q^{S}b_{ j}^{S{\ast}} + q^{R}b_{ j}^{R{\ast}}) = 0 }$$

(44)

Recall that h ∈ [0, 1]. Then by straightforward manipulation of the above conditions we have:

Result 1::

h > 0 and b _j ^R∗ > 0 ⇔ q ^R < q ^S

As we would expect, the possibility of a haircut means that if the R-country bonds are to be bought at all, they must be at a discount to the S-country bonds. Indeed:

Result 2::

b _j ^R∗ > 0 ⇔ q ^S − q ^R = γh, where γ ≡ [πρu ^′(b _j ^S∗ + b _j ^R∗(1 − h) + w _L ^j)∕λ ^∗] > 0

Thus the price discount increases with both the size of the haircut, if any, and the probability of the low return state π, as well as with the marginal utility of income in the low state, which also increases with the size of the haircut.

Could the R-country’s bonds ever actually be worthless? Here we have:

Result 3::

π < 1 and $u^{{\prime}}(.) > 0 \Leftrightarrow q^{R} > 0\forall h \in [0,1]$.

Note that the worst the haircut can be is h = 1. Then if q ^R = 0 and b _j ^R∗ ≥ 0 we have from (43):

$$\displaystyle{ (1-\pi )u^{{\prime}}(b_{ j}^{S{\ast}} + b_{ j}^{R{\ast}} + w_{ H}^{\,j}) \leq 0 }$$

(45)

which is a contradiction if there is some chance that the better state will occur and consumers are non-satiated. Thus there is always some price at which the R-country bonds will be bought because it yields a positive return in at least one state.

1.2 First Order Conditions for Country R’s Problem

Here we simply present the first order conditions. Interpretation and discussion are given in the text of the paper.

Attaching multipliers λ ₀, λ _L , λ _H to the respective budget constraints and μ to the haircut constraint, and denoting the Lagrange function of the problem by $\mathcal{L}$, we have the first order conditions²⁸:

$$\displaystyle\begin{array}{rcl} \frac{\partial \mathcal{L}} {\partial w_{0}^{R}} = n^{R}(u_{ 0}^{{\prime}}-\lambda _{ 0}) = 0& &{}\end{array}$$

(46)

$$\displaystyle\begin{array}{rcl} \frac{\partial \mathcal{L}} {\partial w_{L}^{R}} = n^{R}(\rho \pi u_{ L}^{{\prime}}-\lambda _{ L}) = 0& &{}\end{array}$$

(47)

$$\displaystyle\begin{array}{rcl} \frac{\partial \mathcal{L}} {\partial w_{H}^{R}} = n^{R}(\rho (1-\pi )u_{ H}^{{\prime}}-\lambda _{ H}) = 0& &{}\end{array}$$

(48)

$$\displaystyle\begin{array}{rcl} \frac{\partial \mathcal{L}} {\partial I^{R}} = -\lambda _{0} +\lambda _{L}F_{L}^{{\prime}}(I^{R}) +\lambda _{ H}F_{H}^{{\prime}}(I^{R}) = 0& &{}\end{array}$$

(49)

$$\displaystyle\begin{array}{rcl} & \frac{\partial \mathcal{L}} {\partial B^{R}} = n^{R}\rho \pi u_{L}^{{\prime}} \frac{b_{R}^{R}} {(B^{R})^{2}} \{ \frac{\partial T} {\partial B^{R}}B^{R} - [P^{R} + T(B^{R},P^{R})]\}& \\ & +\lambda _{0}q^{S} -\lambda _{L}C_{R}^{{\prime}}-\lambda _{H} +\mu (1 - \frac{\partial T} {\partial B^{R}}) = 0 &{}\end{array}$$

(50)

$$\displaystyle\begin{array}{rcl} & \frac{\partial \mathcal{L}} {\partial P^{R}} = n^{R}\rho \pi u_{L}^{{\prime}}\frac{b_{R}^{R}} {B^{R}} (1 + \frac{\partial T} {\partial P^{R}}) +\lambda _{L}(C_{R}^{{\prime}}- 1) -\mu (1 + \frac{\partial T} {\partial P^{R}}) \leq 0,& \\ & \text{ }P^{R} \geq 0,\text{ } \frac{\partial \mathcal{L}} {\partial P^{R}}P^{R} = 0 &{}\end{array}$$

(51)

$$\displaystyle\begin{array}{rcl} & \frac{\partial \mathcal{L}} {\partial \mu } = B^{R} - T(B^{R},P^{R}) - P^{R} \geq 0,\text{ }\mu \geq 0,\text{ }\frac{\partial \mathcal{L}} {\partial \mu } \mu = 0\text{ }&{}\end{array}$$

(52)

Recall that ∂T∕∂P ^R = −1 when T > 0. Here we focus on the interpretation of conditions (50)–(52). There are a number of solution possibilities, depending inter al. on what kind of equilibrium country S will be in.

Case A: Assume T(B ^R, P ^R) ≡ 0 and B ^R − P ^R > 0 so that μ = 0. There is no transfer and a haircut. The conditions then imply:

$$\displaystyle\begin{array}{rcl} & q^{S} =\delta _{L}[C_{R}^{{\prime}} +\beta _{ R}^{R}\frac{P^{R}} {B^{R}}] +\delta _{H}&{}\end{array}$$

(53)

$$\displaystyle\begin{array}{rcl} & \delta _{L}[\beta _{R}^{R} + C_{R}^{{\prime}}(B^{R}) - 1] < 0 \Rightarrow \text{}P^{R} = 0;\text{ }& \\ & P^{R} > 0 \Rightarrow \beta _{R}^{R} + C_{R}^{{\prime}}(B^{R},P^{R}) = 1 &{}\end{array}$$

(54)

where δ _H ≡ ρ(1 −π)u _H ^′∕u ₀ ^′ and δ _L ≡ ρπu _L ^′∕u ₀ ^′ can be thought of as the planner’s discount factors for time and risk, and β _R ^R ≡ n ^R b _R ^R∕B ^R. Note that we rule out the case in which T(B ^R, P ^R) ≡ 0 and μ > 0 since that implies no default even in state L, which is uninteresting.

Case B: Here there is a positive transfer and, from the earlier analysis, the equilibrium condition determining the relationships between T on the one hand and B ^R, P ^R on the other is

$$\displaystyle{ 1 -\frac{n^{S}b_{R}^{S}} {B^{R}} = C_{S}^{{\prime}}(B^{R} - P^{R} - T^{{\ast}}) }$$

(55)

As already shown, the comparative statics on this condition yield ∂T ^∗∕∂P ^R = −1, but the sign of ∂T ^∗∕∂B ^R is ambiguous²⁹:

$$\displaystyle{ \frac{\partial T^{{\ast}}} {\partial B^{R}} = -(\frac{n^{S}b_{R}^{S}} {(B^{R})^{2}} - C_{S}^{{\prime\prime}})/C_{ S}^{{\prime\prime}}\lesseqgtr 0 }$$

(56)

and so we have that ∂T ^∗∕∂B ^R > 0 ⇔ (n ^S b _R ^S∕(B ^R)² − C _S ^′′) < 0. This has to be taken into account in interpreting the first order conditions in this case, in particular (50) above.

In (51), the assumption that B ^R − T(B ^R, P ^R) − P ^R > 0, so there is still a positive haircut, implies the condition:

$$\displaystyle\begin{array}{rcl} & \delta _{L}(C_{R}^{{\prime}}(B^{R}) - 1) < 0 \Rightarrow \text{}P^{R} = 0;\text{ }& \\ & P^{R} > 0 \Rightarrow C_{R}^{{\prime}}(B^{R},P^{R}) = 1 &{}\end{array}$$

(57)

The difference to case A results from the fact that ∂T ^∗∕∂P ^R = −1, the crowding out result.

Case C: Here, there is a complete bailout, and therefore no haircut in state L. The haircut constraint is binding with B ^R − P ^R = T ^∗(B ^R, P ^R), and so we can substitute from the haircut constraint into the state L resource constraint to obtain

$$\displaystyle{ F_{L}(I^{R}) - n^{R}w_{ L}^{R} - [P^{R} + C_{ R}(T(B^{R},P^{R}))] \geq 0 }$$

(58)

while dropping the haircut term from the consumer’s state L utility function. As a result, we have the first order conditions:

$$\displaystyle\begin{array}{rcl} & \frac{\partial \mathcal{L}} {\partial B^{R}} =\lambda _{0}q^{S} -\lambda _{L}C_{R}^{{\prime}} \frac{\partial T} {\partial B^{R}} -\lambda _{H} = 0&{}\end{array}$$

(59)

$$\displaystyle\begin{array}{rcl} & \frac{\partial \mathcal{L}} {\partial P^{R}} = -\lambda _{L}(1 + C_{R}^{{\prime}} \frac{\partial T} {\partial P^{R}}) \leq 0,& \\ & \text{ }P^{R} \geq 0,\text{ } \frac{\partial \mathcal{L}} {\partial P^{R}}P^{R} = 0 &{}\end{array}$$

(60)

with the interpretation given in the text.