Size, spillovers and soft budget constraints

Abstract

There is much evidence against the so-called “too big to fail” hypothesis in the case of bailouts to subnational governments. We look at a model where districts of different size provide local public goods with positive spillovers. Matching grants of a central government can induce socially-efficient provision, but districts can still exploit the intervening central government by inducing direct financing. We show that the ability and willingness of a district to induce a bailout and district size are negatively correlated. Furthermore, we argue that these policies can be equilibrium strategies.

Appendices

Appendix A: Spillovers

In this appendix, we show that the main results of the paper are robust to a different specification of the spillover effect, namely that the spillover effect is increasing in the size of the district where the public good is provided.

In the main text, the spillover effect is related to the per capita amount of local public goods. When the central government considers a bailout, the costs of a bailout are increasing in district size, but the spillover effect does not. In this appendix, the spillover effect is, however, increasing in the total amount of local public goods. The spillover effect is thus increasing in district size. We argue that even in this case the main conclusions still hold.

We assume that an individual in district i gets a benefit κn _j v(g _j ) of the public goods g _j provided in district j (i≠j). In a noncooperative equilibrium the utility of an individual in district i thus is

$$v(g_{i})+\sum_{j\neq i}\kappa n_{j}v(g_{j})+y-t_{i}, $$

where t _i is given by (2). It is straightforward to show that the noncooperative equilibrium is again given by (3).

The socially optimal or efficient outcome is now determined by the following maximization problem:

$$\max_{g_{i}} n_{i}v(g_{i})+\sum _{j\neq i}n_{j}\kappa n_{i}v(g_{i})+n_{i}y-n_{i}p_{i}g_{i}. $$

Let $\hat{g}_{i}$ again denote the socially optimal or efficient outcome, where $\hat{g}_{i}$ satisfies the following first-order condition of this maximization problem

$$ \left \{ \begin{array}{l@{\quad}l} v^{\prime}(\hat{g}_{i}) = \frac{p_{i}}{1+(N-n_{i})\kappa} & \mbox{if} \ v(\hat{g}_{i})+\sum_{j\neq i}n_{j}\kappa v(\hat{g}_{i}) > p_{i}\hat{g}_{i}, \\[6pt] \hat{g}_{i}=0 & \mbox{otherwise.} \end{array} \right . $$

(16)

A comparison of the first-order conditions (16) with (3) yields that there is again underprovision of public goods. As in Sect. 2, it is possible, however, to find a system of matching grants that induces the optimal outcome. Individuals choose to provide the socially optimal amount of public goods when the central government chooses the following matching transfers $\hat{m}_{i}$:

$$\hat{m}_{i} = \frac{N\kappa}{1+(N-n_{i})\kappa}. $$

We now focus on soft budget constraints. As in Sect. 3.1, we first analyze the central government bailout policy. The central government maximizes the payoff of an individual located outside the district that might get a bailout, and this optimization problem can be written as

$$\max_{\underline{m_{i}}} \kappa n_{i}v(g_{i}+ \underline{m_{i}})-T_{\mathit{BO}}, $$

where T _BO is given by (11). The first-order condition of this maximization problem is given by

$$ \left \{ \begin{array}{l@{\quad}l} \kappa v^{\prime}(g_{i}+\underline{m_{i}})=\frac{p_{i}}{N} & \mbox{if}\ \kappa v^{\prime}(g_{i})>\frac{p_{i}}{N}, \\[6pt] \underline{m_{i}}=0 & \mbox{otherwise.} \end{array} \right . $$

(17)

Condition (17) makes it possible to characterize the central government’s bailout policy.

Lemma 3

There exists critical values $\underline{\kappa_{C}(p_{i})}$ and $\overline{g_{C}(p_{i})}$ such that:

• 1. If $\kappa<\overline{\kappa_{C}(p_{i})}$ the central government does not provide district i a bailout, even when district i chooses a zero level of own-contribution to local public good provision;

• 2. If $\kappa>\overline{\kappa_{C}(p_{i})}$ the central government provides a bailout to district i if and only if $g_{i}<\overline {g_{C}(p_{i})}$.

Proof of Lemma 3

• (1) From condition (17), it follows that when g _i =0 a necessary condition for $\underline{m_{i}}>0$ is $\kappa v^{\prime}(0)>\frac{p_{i}}{N}$. Hence, for $\kappa<\underline{\kappa_{C}(p_{i})}=\frac{p_{i}}{Nv^{\prime }(0)}$ the central government never provides a bailout.

• (2) Let $\overline{g_{C}(p_{i})}$ be such that $\kappa v^{\prime}(\overline {g_{C}(p_{i})})=p_{i}/N$. Then for $g_{i}<\overline{g_{C}(p_{i})}$ it holds that κv′(g _i )>p _i /N. From condition (17), it then follows that the central government will provide a bailout. □

As in Sect. 3.2, we now focus on the local government’s bailout policy. In the remaining of this Appendix, we assume that the spillover effect is important enough, that is, κ is bigger than the inverse of the minimum district size. Individuals within district i prefer to induce a bailout with t _i =0 over an optimal level of public good provision $g_{i}=\hat {g_{i}}$ when

which, using expressions (5), (9), and (12), can be rewritten as

$$ v(\hat{g}_{i})-v(\underline{m_{i}}\,) < v^{\prime}( \hat{g}_{i})\hat{g}_{i}-\kappa n_{i}v^{\prime}( \underline{m_{i}}\,)\underline{m_{i}}-\frac{c_{\mathit{BO}}}{N}. $$

(18)

Condition (18) makes it possible to show how district size and the local government’s bailout policy are related.

Lemma 4

There exists a critical value $\overline{n_{L}(p_{i})}$ such that if $n_{i}<\overline{n_{L}(p_{i})}$ and if the central government is willing to give a bailout to district i, then the local government of district i will induce a bailout.

Proof of Lemma 4

First note that when the central government is not willing to give a bailout, the local government will not induce a bailout since the per-capita costs of inducing would be c _BO /N.

Secondly, look at the case in which the central government is willing to give a bailout. The left-hand side of (18) increases more when n _i increases than the right-hand side if

$$0 > v^{\prime\prime}(\hat{g}_{i})\frac{\partial\hat{g}_{i}}{\partial n_{i}} \hat{g}_{i}-\kappa v^{\prime}(\underline{m_{i}}\,) \underline{m_{i}} . $$

When v(g)=ln(g+1), then this inequality can be rewritten as

$$\bigl(1+(N-n_{i})\kappa\bigr) (N-n_{i}) \kappa^{2}N<p_{i}\bigl(\bigl(1+(N-n_{i})\kappa \bigr)^{2}-N\kappa\bigr) $$

and since (from (16)) p _i ≤(1+(N−n _i )κ), a sufficient condition for this inequality to hold is that $\kappa>\frac {1}{n_{i}}$.

For v(g)=g ^1−α/(1−α), expression (15) can be rewritten as

$$\bigl(1+(N-n_{i})\kappa\bigr)^{1/\alpha-2}<(\kappa N)^{1/\alpha-1} $$

and this inequality holds for all values of $\kappa>\frac{1}{N}$ when $\frac{1}{2}<\alpha<1$.

This leads to three possibilities. Firstly, when (18) holds for all possible n _i then bailouts always take place, and this is the case when $\overline {n_{L}(p_{i})}=N/2$. Secondly, when (18) does not hold for any n _i then bailouts never take place and this is the case when $\overline{n_{L}(p_{i})}=0$. Finally, when neither of these two does hold, then by the intermediate value theorem there exists an $\overline{n_{L}(p_{i})}$ such that condition (18) holds if and only if $n_{i}<\overline {n_{L}(p_{i})}$. □

Appendix B: Proof of Proposition 1

From Lemma 1, it follows that a condition for the central government to provide a bailout is $n_{1}<\overline{n_{C}(p_{1})}$ and from Lemma 2 it follows that a condition for the local government to induce a bailout is $n_{1}<\overline{n_{L}(p_{1})}$. It should therefore hold in an equilibrium in which bailouts can take place (which, in addition, requires that the central government has no incentive to change the system of matching grants), that bailouts will be induced and provided when $n_{1}<\min\{\overline{n_{C}(p_{1})},\overline{n_{L}(p_{1})}\}$.

As in Sect. 3.2, we look at the utility function v(g)=g ^1−α/(1−α) and for tractability, we focus on α=1/2. The possible values for the cost parameter p ₁ of district 1 are p ^L and p ^H, with p ^L<p ^H, where $\mathit{Prob}[p_{1}=p^{L}]=\mathit{Prob}[p_{1}=p^{H}] =\frac{1}{2}$. Note that these probabilities (but not the actual value of p ₁) are know by the central government when the central government chooses the matching grants.

First note that when v(g)=2g ^1/2, and when it is efficient to provide a positive amount of local public goods in district 1, then this amount is given by

$$ \hat{g}_{1} = \biggl(\frac{n_{1}+(N-n_{1})\kappa}{n_{1}p_{1}} \biggr)^{2} $$

(19)

and if a bailout is given to district 1 than the amount of public goods is given by

$$\underline{m_{1}} = \biggl(\frac{N\kappa}{n_{1}p_{1}} \biggr)^{2}. $$

We impose the condition p ^H>2. Note that it is socially optimal to provide no public good in district 1 when p ₁=p ^H if

$$ n_{1}v(\hat{g}_{1})+ (N-n_{1} )\kappa v( \hat{g}_{1})-n_{1}p^{H}\hat{g}_{1} < 0 $$

(20)

using (19) it follows that (20) holds if p ^H>2/n ₁. Since a district should consist of at least one individual, a sufficient condition is p ^H>2. With p ^H>2, it is thus too costly to provide a local public good in region 1.

No change in the matching grant

The central government does not have an incentive, from the aggregate social welfare point of view, to change the matching grant to district 1 if for any matching grant m the expected payoff is lower than for $\hat{m}_{1}$. There are two possibilities after changing the matching grant. In the first, individuals in district 1 start providing public goods when p ₁=p ^H, and in the second individuals do not. In the following, let g(L,m) denote the amount of public goods individuals in district i provide when the matching grants are m and p ₁=p ^L, while g(H,m) denotes the amount of public goods with m and p ₁=p ^H.

In the first case, the government does not have an incentive to change the matching grant if

(21)

In the second case, the government does not have an incentive to change the matching grant if

(22)

When the matching grant differs from $\hat{m}_{1}$, an amount of public goods is provided in district 1 that differs from the efficient one, so the net aggregate payoff from providing public goods in district 1 decreases. Recall that n ₁ and p ^H are such that it is efficient to provide no public good in district 1 when p ₁=p ^H. Using (20), this implies that for all g(H,m) the following inequalities hold

This implies for (21)

so it is sufficient to look at condition (22).

With a change in matching grants the central government tries to avoid a bailout. A bailout is less attractive for individuals in district 1 when they get a higher matching grant. On the other hand, however, the more the matching grant exceeds the optimal grant $\hat{m}_{1}$, the lower the net aggregate social welfare, since the left-hand side of (22) is decreasing in m. The social-welfare maximizing central government therefore tries to find the smallest matching grant m ^∗ such that individuals in district 1 are indifferent between providing public goods and inducing a bailout. This m ^∗ is implicitly given by

It follows from Sect. 3.2 that bailouts are more attractive for individuals in smaller districts, so the smaller the district the bigger the m ^∗ that makes the individuals indifferent between providing public goods and inducing a bailout. The left-hand side of inequality (22) is, however, decreasing in m while the right-hand side does not depend on m, so an increase in m=m ^∗ makes it more likely that inequality (22) is satisfied and that it is optimal not to change the system of matching grants.

Note that when p ₁=p ^L and $g(L,\hat{m}_{1})=\hat{g}_{1}$, inequality (22) is not satisfied for any c _BO >0. When m=1, however, inequality (22) can be written as

$$c_{\mathit{BO}} < (1-\kappa)^{2} \biggl(\frac{1}{2}N-n_{1} \biggr) $$

and is thus satisfied for some c _BO >0 since, by assumption, n _i <N/2. As argued above, a decrease in n ₁ increases m ^∗, so it follows from the intermediate value theorem that there exists an n ^∗ such that when n ₁<n ^∗ then the central government does not have an incentive to change the matching grant to district 1. □