## 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.

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## Notes

A well-established political economy literature has provided several models that generate impatient policy makers. See, for example, Alesina and Tabellini (1990) and Azzimonti (2011) for closed economy environments and Amador (2003), Aguiar and Amador (2011) and Cuadra and Sapriza (2008) for environments with external sovereign debt.

See Lledó et al. (2017) for a survey of existing rules.

Malaysia is an example where the fiscal rule includes explicit limits on external public debt, in addition to domestic.

These models are based on the original contribution of Eaton and Gersovitz (1981).

A recent contribution here is Gordon and Guerron-Quintana (2018), which introduces investment to the Eaton–Gersovitz framework.

For example, the government may decide to spend resources on goods that are not as valued by the households (or just channel the external funds to the private accounts of politically connected entities).

See also Amador and Phelan (2018) for a model as well as other references.

See also the work of Azzimonti et al. (2016) for a closed economy counterpart of the benefits of balanced budget rules starting from any particular level of debt.

We derived this calculation in continuous time, but given the deterministic nature of the environment, the same quantitative results hold (approximately) in a discrete time version.

To compute the implicit annual discount factor, we compound the annual discount rate for one unit of time: \(\beta _{\mathrm{{G}}} = e^{-\rho _{\mathrm{{G}}}}\).

The model has also been extended to incorporate bargaining among creditors and the sovereign after a default, production, and risk-averse lenders.

An alternative reference would be to allow the government to save but not borrow. Doing this in principle will raise the value of financial constraints from the perspective of the households, biasing our current results against finding welfare gains from financial market access.

See Aguiar et al. (2019) for a discussion of maturity choice under lack of commitment.

In principle, we can take the expectation over any initial distribution. We choose the ergodic for simplicity of exposition.

Note that we start the history from no exclusion in the initial period. The reason is that a government never defaults with zero debt.

Our specification of the income process is the same as the specification in Chatterjee and Eyigungor (2012), including both a persistent component and a transitory component. In all the simulations we perform, we similarly set the transitory endowment component to its lowest value in a period where a default occurs. The transitory component is small and needed to facilitate the convergence of the numerical computations in the long-term bond case. Its presence is not necessary for the one-period bond calculations (and its effects there are not significant). We refer the reader to Chatterjee and Eyigungor (2012) for more details.

We compare our results to Aguiar and Gopinath (2006)’s transitory shocks model (Model I). They obtained a 0.25 (quarterly) debt-to-output ratio. The only difference between our numerical exercise and their transitory shocks model is the specification of output process.

This last is related to the cost of business cycles. See Lucas (1987).

In Chatterjee and Eyigungor (2012), the output process is the sum of two components: \(y(s) = {{\tilde{y}}}(s) + m(s)\), where \({{\tilde{y}}}(s)\) follows a persistent Markov chain and

*m*(*s*) is an i.i.d. process. In the period where default is triggered, the transitory component is set to its lowest level in the support, while it reverts back to its normal stochastic process in subsequent periods. See the related discussion in footnote 21.Chatterjee and Eyigungor (2012) use a different specification for the long-term bond. They assume that a given bond matures with an idiosyncratic probability \(\lambda\), in which case it pays a unit to its holder. If the bond does not mature, it pays a coupon

*z*. The budget constraint (using hats to represent these alternative bonds) is now$$\begin{aligned} c = y(s) - (\lambda + (1 - \lambda )z) {\hat{b}} - \hat{q}(s, b') ({\hat{b}}' - (1-\lambda ) {\hat{b}}) \end{aligned}$$and the pricing equation

$$\begin{aligned} \hat{q} = {\mathbb {E}}\left[ 1({{\text {No default}}}) \frac{\lambda + (1-\lambda ) (z + \hat{q}') }{R} \right] \end{aligned}$$This is equivalent to our formulation with the following change in variables: \(q = {\hat{q}} / (\lambda + (1 - \lambda ) z)\), \(b = (\lambda + (1 - \lambda ) z) {\hat{b}}\), \(\delta = 1 - \lambda\).

This excess variability of consumption with respect to income is one of the quantitative successes of the sovereign debt model. The other is the counter-cyclicality of the trade balance, which is related: the country borrows more in good times generating a more volatile consumption process.

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This paper was prepared for the International Monetary Fund 19th Jacques Polak Annual Research Conference. We thank Cristina Arellano, Javier Bianchi, Doireann Fitzgerald, and Christopher Phelan for helpful discussions. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.

## 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

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

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

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

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:

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

Solving this recursion yields that

### 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,

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

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:

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.

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Aguiar, M., Amador, M. & Fourakis, S. On the Welfare Losses from External Sovereign Borrowing.
*IMF Econ Rev* **68**, 163–194 (2020). https://doi.org/10.1057/s41308-019-00103-2

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DOI: https://doi.org/10.1057/s41308-019-00103-2

### JEL Classification

- F34
- F41