Public debt, bailouts, and common bonds

Abstract

We look at a model where countries of different sizes provide local public goods with positive spillovers. Matching grants can give rise to optimal expenditure levels, but countries can induce bailouts. We study the characteristics of these bailouts in a subgame-perfect Nash equilibrium and how these characteristics are affected by the introduction of common bonds. Partial substitution of common for sovereign bonds has two implications. First, it lowers the average and marginal borrowing costs of countries which may be eligible for bailouts. This effect leads to higher borrowing in these countries irrespective of their bailout expectations. Second, the lower borrowing costs mitigate the incentives of countries to induce a bailout and, therefore, constrain the parameter set for which a soft budget constraint equilibrium exists. As a result, the introduction of common bonds can also be in the interest of those countries that provide the bailouts.

Appendix

Proof of Lemma 2

The proof consists of two parts. First, we show that the utility difference can be both positive and negative. Since $g_{j\tau }^*=\hat{g}_{j\tau }$, no difference in utility can come through the externalities of foreign public good provision. Second, (25) and (26) show that $g_{i1}^*=\hat{g}_{i1}$ and $g_{i2}^*<\hat{g}_{i2}$. Therefore, utility from public good provision is lower in a soft budget constraint equilibrium than in the optimal outcome. Third, $g_{i1}^*=\hat{g}_{i1}$ and $b_i^*>\hat{b}_i$ from (27) imply $t_{i1}^*<\hat{t}_{i1}$ and $c_{i1}^*>\hat{c}_{i1}$. Thus, higher borrowing increases utility from private consumption in period 1. As a result, $U_i^*>\hat{U}_i$ if $t_{i2}^*$ is sufficiently smaller or not too much larger than $\hat{t}_{i2}$, which would augment second-period consumption $c_{i2}^*$ relative to $\hat{c}_{i2}$. However, $t_{i2}^*\gtreqless \hat{t}_{i2}$ depends on the reaction of public debt to the bailout. If public debt increases only marginally relative to its optimal level, then $t_{i2}^*<\hat{t}_{i2}$ and the possibility exists that country i chooses to induce a bailout. The stronger $b_i$ reacts to the bailout, the larger $t_{i2}^*$ is, and the lower the utility difference $U_i^*-\hat{U}_i$. In the extreme, a large difference $b_i^*-\hat{b}_i$ leads to a sufficiently large difference $t_{i2}^*-\hat{t}_{i2}>0$ and $U_i^*<\hat{U}_i$. However, whether $t_{i2}^*-\hat{t}_{i2}>0$ actually leads to $U_i^*<\hat{U}_i$ depends also on the time preference parameter. We show this explicitly by plugging the first-order conditions (13), (14), (25), (26), and (27) in (23.1) and showing that $U_i^*<\hat{U}_i$ holds if and only if the following condition is satisfied:

$$\begin{aligned} \begin{aligned} \delta _i > \frac{b_i^*-\hat{b}_i}{(1+r_i(b_i^*))b_i^*-(1+r_i(\hat{b}_i))\hat{b}_i+g_{i2}^*-\underline{m_i}-\hat{g}_{i2}- \ln ( {g_{i2}^*})+({\hat{g}_{i2}})}. \end{aligned} \end{aligned}$$

(33)

Therefore, the government in country i may decide not to induce a bailout when it is has a high time preference parameter $\delta _i$.

Subsequently, we differentiate the utility difference with respect to $n_i$. Using the Envelope Theorem, any effect of $n_i$ on the equilibrium $\hat{t}_{i\tau }, \hat{b}_i, t_{i\tau }^*, b_i^*$ has zero effect on $\hat{U}_i$ and $U_i^*$, respectively. Moreover, the effects through $\hat{t}_{j\tau }, \hat{b}_j,t_{j\tau }^*$, and $b_j^*$ cancel out, which leaves

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}n_i}\left[ U_i^*-\hat{U}_i\right] =\frac{\delta _i}{g_{i2}^*}\frac{\mathrm{d} \underline{m_i}}{d n_i}<0. \end{aligned}$$

(34)

Therefore, there are three cases. Either (23) is always negative, which means $\overline{n_L}=0$, or it is always positive and thus $\overline{n_L}=N/2$, or it changes sign as $n_i$ increases. The last case means that by the intermediate value theorem, there exists $\overline{n_L}\in [0,N/2]$, such that (23) holds if and only if $n_i<\overline{n_L}$. $\square$

Proof of Proposition 1

To show that $\mathcal{{S}}$ can constitute a subgame-perfect Nash equilibrium, we have to show that no actor has an incentive to change its strategy. We use backward induction.

First, consider ($\mathcal{{S}}4$). Note that when the governments of countries j are asked to provide a bailout, these governments conclude that $\delta _i=\delta _\mathrm{low}$, as otherwise the government of country i would not make this request. This thus automatically takes away the information asymmetry that existed in period 1. Due to the subgame perfectness, the governments of countries j cannot stick to a non-bailout policy, even though from a welfare perspective, they may prefer to do so. Given this restriction, (21) describes their bailout decision.

Then, consider ($\mathcal{{S}}3$). When countries j give a bailout as (21) describes, then the previous section shows that (13) and (14) describe the utility maximizing choices, these choices are also maximizing the utility of the citizens in countries j if country i does not induce a bailout.

Considering ($\mathcal{{S}}2$), note that when $\delta _\mathrm{high}$ is large enough, i.e., close to one, it follows from (14) that the government of country i does not borrow money and that it has no incentive to consume more in period 1 at the cost of lower public good provision in country i in period 2. It will thus not induce a bailout in period 2. On the other hand, when $\delta _i=\delta _\mathrm{low}$ is small enough, i.e., close to zero, it gets more attractive for individuals in country i to consume more in period 1 at the expense of consumption and public good provision in period 2, and choose a $b_i$ big enough to change all period 2 income into period 1 consumption. In addition, for a lower $\delta$, the period 2 costs associated with a bailout for the individuals in country i matter less when they take their decision in period 1 on a debt level that implies inducing a bailout in period 2.

Finally, consider ($\mathcal{{S}}1$). Changing the system of matching grants in a way to avoid bailouts leads to public good levels that are too high when country i does not induce a bailout, that is, when $\delta _i=\delta _\mathrm{high}$. Since $\delta _i$ is not known when the system of matching grants is designed, the only way to give the government of country i an incentive to increase its public good provision is by increasing the size of the matching grant. Let $g_i^a$ denote the amount of public goods in country i when the system of matching grants is adjusted in such a way that the government of country i will never induce a bailout, and let $T^a$ denote the additional tax that has to be raised to finance the adjustment in the system of matching grants. The benefits of doing so are the additional spillovers that can be expected when a bailout is avoided minus the loss due to suboptimal decision making when, even without adjustments, no bailout would have been induced: $p^L\left[ \ln (g_i^a)-\ln (g_i^*) -T^a\right] +(1-p^L)\left[ \ln (g_i^a)-\ln (\hat{g}_i)-T^a\right]$. While the first term might be positive, the second term is surely negative, so for $p^L$ small enough, there is no incentive to change the system of matching grants. $\square$

Proof of Proposition 2

The first result exists both in the case of hard and soft budget constraints. Totally differentiating equation (31) with respect to $b_i$ and $\bar{b}$, we get

$$\begin{aligned} \frac{\mathrm{d}b_i}{\mathrm{d}\bar{b}} = \frac{r_{i}^{\prime }(b_i)}{2r_{i}^{\prime }(b_i)+b_ir_{i}^{\prime \prime }(b_i)-\overline{b}r_{i}^{\prime \prime }(b_i)}. \end{aligned}$$

(35)

Moreover, taking into consideration that $r^{\prime }(b_i)$ is homogenous of degree $h-1$ and, therefore, $r^{\prime \prime }(b_i)b_i=(h-1)r^{\prime }(b_i)$, we get

$$\begin{aligned} \frac{\mathrm{d} b_i^*}{\mathrm{d}\bar{b}}= & {} \frac{1}{2+(h-1)\left( 1-\bar{b}/b_i^*\right) }>0,\\ \frac{\mathrm{d} \hat{b}_i}{\mathrm{d}\bar{b}}= & {} \frac{1}{2+(h-1)\left( 1-\bar{b}/\hat{b}_i\right) }>0. \end{aligned}$$

The second result follows directly from Eq. (28), which determines the size of the bailout and is independent of $\bar{b}$.

The third result is related to the previous one. Since both the bailout and $g_{i2}^*$ are determined only by $\kappa , n_i$ and N, common bonds and the borrowing of country i do not influence the incentive of the remaining countries to increase the bailout above zero.

The fourth result states that the critical value $\overline{n_L}$ goes down. It is the case if the introduction of common bonds lowers the value of $U_i^*-\hat{U}_i$ for any $n_i$. In this case, either $\overline{n_L}$ declines (if an interior value for it exists), or it remains at 0 or N / 2. Therefore, showing that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\bar{b}}(U_i^*-\hat{U}_i)<0 \end{aligned}$$

suffices to prove this result. We can show that it always holds. First, due to the independence of $g_{i\tau }^*,\hat{g}_{i\tau }, g_{j\tau }^*$, and $\hat{g}_{j\tau }$ of common bonds, the utility difference is not affected by $\bar{b}$ through public good consumption. Nevertheless, since common bonds lead to an increase in borrowing in each equilibrium, taxes decline in period 1 and may rise in period 2. Using Eqs. (13), (14), (16), (25), (26), and (35), we can show that

$$\begin{aligned} \frac{\mathrm{d} t_{i1}^*}{\mathrm{d}\bar{b}}&=-\frac{(1-\kappa )n_i}{n_i+\kappa (N-n_i)}\frac{\mathrm{d}b_i^*}{\mathrm{d}\bar{b}}<0, \end{aligned}$$

(36a)

$$\begin{aligned} \frac{\mathrm{d} \hat{t}_{i1}}{\mathrm{d}\bar{b}}&=-\frac{(1-\kappa )n_i}{n_i+\kappa (N-n_i)}\frac{\mathrm{d}\hat{b}_i}{\mathrm{d}\bar{b}}<0, \end{aligned}$$

(36b)

$$\begin{aligned} \frac{\mathrm{d}t_{i2}^*}{\mathrm{d}\bar{b}}&=\frac{(1-\kappa )n_i}{n_i+\kappa (N-n_i)}\left[ r_j(b_j^*)-r_i(b_i^*)\right. \nonumber \\&\quad \left. + \left( 1+r_i(b_i^*)\left( 1+\varepsilon _i^*\left( 1-\frac{\bar{b}}{b_i^*}\right) \right) \right) \frac{\mathrm{d}b_i^*}{\mathrm{d}\bar{b}}\right] \gtreqless 0, \end{aligned}$$

(36c)

$$\begin{aligned} \frac{\mathrm{d} \hat{t}_{i2}}{\mathrm{d}\bar{b}}&=\frac{(1-\kappa )n_i}{n_i+\kappa (N-n_i)}\left[ r_j(\hat{b}_j)-r_i(\hat{b}_i) \right. \nonumber \\&\quad \left. + \left( 1+r_i(\hat{b}_i)\left( 1+\hat{\varepsilon }_i\left( 1-\frac{\bar{b}}{\hat{b}_i}\right) \right) \right) \frac{\mathrm{d}\hat{b}_i}{\mathrm{d}\bar{b}}\right] \gtreqless 0. \end{aligned}$$

(36d)

Using the above equations, we can derive

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\bar{b}}(U_i^*-\hat{U}_i)= & {} -\frac{1}{1-\kappa }\left[ \frac{\mathrm{d} t_{i1}^*}{\mathrm{d}\bar{b}}-\frac{\mathrm{d} \hat{t}_{i1}}{\mathrm{d}\bar{b}}+\delta _i\left( \frac{\mathrm{d} t_{i2}^*}{\mathrm{d}\bar{b}}-\frac{\mathrm{d} \hat{t}_{i2}}{\mathrm{d}\bar{b}}\right) \right] \\= & {} \frac{-\delta _i n_i}{n_i+\kappa (N-n_i)}\left[ \rho +r(\hat{b}_i)-r(b_i^*)\right] \end{aligned}$$

Taking into consideration the first-order conditions of governments, given by Eqs. (14) and (31), we can present the above utility difference change as

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\bar{b}}(U_i^*-\hat{U}_i)=\frac{-\delta _i n_i h}{n_i+\kappa (N-n_i)}\left[ (r(b_i^*)-r(\hat{b}_i))+\left( \frac{r(\hat{b}_i)\bar{b}}{\hat{b}_i}-\frac{r(b_i^*)\bar{b}}{b_i^*}\right) \right] \le 0. \end{aligned}$$

(37)

The first term in the square brackets is positive $(r(b_i^*)-r(\hat{b}_i))>0$, while the last term is negative and increasing in the amount of common bonds. The whole term in the square brackets is multiplied by a negative number, and we can conclude that the change in the utility difference attains its largest value when $\bar{b}$ approaches its upper limit $b_i$. In this case, full substitution of sovereign for common debt is achieved, and both $b_i^*$ and $\hat{b}_i$ equal $\bar{b}$. In this case, we can show that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\bar{b}}(U_i^*-\hat{U}_i)\Bigg |_{{\text {full substitution}}} =0. \end{aligned}$$

(38)

Therefore, the marginal effect of common bonds on the incentives to induce a bailout is always negative, except for full substitution of national for common bonds. Thus, the impact of a discrete introduction of common bonds from $\bar{b}=0$ to $\bar{b}>0$ is also negative. Note that common bonds have a larger positive effect on second-period taxes in an equilibrium with a bailout than in one without. Therefore, the possible negative tax differential $t_{i2}^*-\hat{t}_{i2}$, which is responsible for the inducement of a bailout, declines in absolute value. Lastly, $\mathrm{d}(U_i^*-\hat{U}_i)/\mathrm{d}\bar{b}$ is most negative when $\bar{b}=0$, because the positive effect vanishes. Therefore, the marginal effect on $\overline{n_L}$ is strongest at the lowest possible common bond ceiling. It vanishes at full substitution of national for common bonds. $\square$