On the Welfare Losses from External Sovereign Borrowing

Abstract

This paper studies the losses to the citizenry when the private agents discount the future at different rates from their government. In the presence of such a disagreement, the private sector may prefer an environment in which the government is in financial autarky. Using a sequence of sovereign debt models, the paper quantifies the potential welfare losses that citizens suffer from the government’s access to international bond markets. While the environment is not necessarily comprehensive, the analysis provides a counterweight to proposals that are designed to ease market access for sovereign borrowers.

Appendices

Appendix 1: Proof of Lemma 1

The functional form describe in Eq. (3) arises from the solutions to the welfare functions obtained in the text. To see that this expression is strictly increasing in $\rho _{\mathrm{{G}}}$, consider the case where $\rho \not =1$.

First note that ${{\overline{b}}} > 0$ and $\rho _{\mathrm{{G}}} > r$ imply that $T > 0$.

Let $\kappa \equiv (\rho _{\mathrm{{G}}} - r)\frac{1-\sigma }{\sigma } \not = 0$. Note that $\rho _{\mathrm{{G}}} > r$ implies that $\kappa$ inherits the sign of $1 - \sigma$, given $\sigma > 0$. Whether $\lambda$ increases with $\rho _{\mathrm{{H}}}$ then depends on whether

$$\begin{aligned} \frac{\rho _{\mathrm{{H}}} e^{ \kappa T} + e^{- T \rho _{\mathrm{{H}}}} \kappa }{\kappa + \rho _{\mathrm{{H}}}} \end{aligned}$$

strictly increases when $0< \sigma < 1$ ($\kappa > 0)$ and decreases when $\sigma > 1$ ($\kappa < 0$).

The derivative with respect to $\rho _{\mathrm{{H}}}$ is

$$\begin{aligned} - \kappa \frac{ e^{-T \rho _{\mathrm{{H}}}} ( 1- e^{T (\kappa + \rho _{\mathrm{{H}}})} + T ( \kappa + \rho _{\mathrm{{H}}}))}{ (\kappa + \rho _{\mathrm{{H}}})^ 2}. \end{aligned}$$

Note that this derivative is continuous for all $\rho _{\mathrm{{H}}} > 0$, equating $\kappa \frac{1}{2} e^{T \kappa } T^2$ at $\rho _{\mathrm{{H}}} = - \kappa$. Hence, the derivative at $\rho _{\mathrm{{H}}} = - \kappa$ is nonzero (as $T >0$) and inherits the sign of $\kappa$.

It suffices to check that

$$\begin{aligned} 1- e^{T (\kappa + \rho _{\mathrm{{H}}})} + T ( \kappa + \rho _{\mathrm{{H}}}) < 0 \end{aligned}$$

for the rest of the domain, $\rho _{\mathrm{{H}}} \not = -\kappa$. But this follows as the above is a strictly concave function of $\rho _{\mathrm{{H}}}$ (given $T > 0$), with a maximum of 0 at $\rho _{\mathrm{{H}}} = -\kappa$.

The proof for the case of $\sigma = 1$ is simpler and left to the reader.

Appendix 2: Proof of Lemma 2

From the value function of government, starting from any s where $b=0$, it follows that it is feasible to set $b' = 0$ (independently of the price). In that case, such a strategy provides a lower bound to the value, and thus:

$$\begin{aligned} V(0, s)&\ge {} u(y(s)) + \beta _{\mathrm{{G}}} \sum _{s'|s} \pi (s'| s) \max \{ V(0, s'), {\underline{V}}(s') \} \\& \ge {} u(y^D(s)) + \beta _{\mathrm{{G}}}\sum _{s'|s} \pi (s'| s) (\theta \{ V(0, s') + (1 - \theta ) {\underline{V}}(s') ) = {\underline{V}}(s), \end{aligned}$$

where the second inequality follows from $y^D(s) \le y(s)$. Thus, $V(0, s) \ge {\underline{V}}(s)$, and

$$\begin{aligned} V(0, s) \ge u(y(s)) + \beta _{\mathrm{{G}}} \sum _{s'|s} \pi (s'|s) V(0, s'). \end{aligned}$$

Solving this recursion yields that

$$\begin{aligned} V(0, s) \ge V^A(s). \end{aligned}$$

Appendix 3: Disagreement About the Default Value

The fact that $\lambda _{\mathrm{{D}}}$ is large in the CE model raises the question of to what extent does the government value default differently from households. We can compute $\lambda _{\mathrm{{D}}}^G$ in a manner similar to that of $\lambda _{\mathrm{{D}}}$. Specifically, let $V_0$ denote the government’s expected value conditional on zero debt. Let $V^{ND}(0)$ be constructed in the same was as $W^{ND}(0)$ but replacing $\beta _{\mathrm{{H}}}$ with $\beta _{\mathrm{{G}}}$. We find $\lambda _{\mathrm{{D}}}^G=-0.38\%$. This is the value in Figure 9 at which $\lambda _{\mathrm{{D}}}$ intersects the $\rho _{\mathrm{{H}}}=\rho _{\mathrm{{G}}}$ vertical line. For $\rho _{\mathrm{{H}}}=r$, for example, $\lambda _{\mathrm{{D}}}$ is approximately $-1\%$, or nearly three times $\lambda _{\mathrm{{D}}}^G$.

This difference between $\lambda _{\mathrm{{D}}}$ and $\lambda _{\mathrm{{D}}}^G$ can be further decomposed. At the time of a default, the government and households disagree about the expected present value of post-default consumption due to the potential differences in discount factor. Suppose, instead, that households and the government valued the default state the same. That is, in the period of default, suppose households preferences become identical to the government’s for all future periods (including post-re-entry). Using this alternative preference specification, we can compute ${\hat{W}}_0$ and ${\hat{W}}^{ND}(0)$. Specifically, let $T(h_t)$ denote the first time of default in history $h_t$, where we set $T(h_t)=t$ if no default has occurred as of t. Then,

$$\begin{aligned} {\hat{W}}_0=\sum _{s_0}\pi ^\infty (s_0)\sum _{t=0}^\infty \sum _{h_t}\pi (h_t|h_0=(s_0,0))\beta _{\mathrm{{H}}}^{T(h_t)}\times \beta _{\mathrm{{G}}}^{t-T(h_t)}u\left( C(h_t)\right) , \end{aligned}$$

with ${\hat{W}}^{ND}(0)$ defined accordingly by replacing $C(h_t)$ with $c^{ND}(h_t)$ in the above. We can then define

$$\begin{aligned} 1+{{\hat{\lambda }}}_{\mathrm{{D}}}&\equiv \left( \frac{{\hat{W}}_0}{{\hat{W}}^{ND}(0)}\right) ^{\frac{1}{1-\sigma }}. \end{aligned}$$

This is the loss due to default costs, but evaluated such that the government and household preferences agree post-default. However, prior to default, the household discounts with a different discount factor. This leads to the following decomposition:

$$\begin{aligned} \frac{1+\lambda _{\mathrm{{D}}}}{1+\lambda _{\mathrm{{D}}}^G}&= \frac{1+\lambda _{\mathrm{{D}}}}{1+{{\hat{\lambda }}}_{\mathrm{{D}}}}\times \frac{1+{{\hat{\lambda }}}_{\mathrm{{D}}}}{1+\lambda _{\mathrm{{D}}}^G}. \end{aligned}$$

In Fig. 11, we plot the ratio on the left-hand side as well as the two components from the right-hand side as we vary $\rho _{\mathrm{{H}}}$. In the figure, we see that the majority of the disagreement when the household is relatively patient is due to the first ratio on the right. This ratio captures the households disagreement post-default.

Fig. 11

The two vertical lines represent the market interest rate and the discount factor of the government in the CE calibration