The political economy of municipal consortia and municipal mergers

Abstract

This paper analyses both the policy and welfare consequences of municipal mergers vs municipal consortia in the presence of spillovers and inequalities in the districts size. Citizens of different municipalities have a common interest in internalising spillovers, but conflicting interests on how to reach that goal. We address the question of when, from a welfare point of view, the central government should legislate in favour of municipal mergers and when, instead, it should prefer municipal consortia. Results provide some useful insights for the design of local governments reforms.

Appendices

Appendix 1: Nash Bargaining Solution and Kalai and Smorodinsky Solution to bargaining problems

The first formal treatment of the bargaining agreement is due to John Nash, who defined a bargaining situation as one in which individuals have the opportunity to collaborate for mutual benefits in more than one way (Nash 1950).

There are two important ideas here. First, individuals must be in a position to generate a cooperative surplus. Second, because there are different ways in which they can cooperate, they cannot realize any of the surplus until they have decided exactly how they intend to cooperate. They must therefore come to some agreement about how their interaction will be organized.

Nash begins by defining the bargaining space. This space is constructed by first plotting the set of possible cooperative outcomes in terms of their pay-offs as in Fig. 5. Each point in this space represents a particular allocation of utility to the two players, and so can be represented $(u_1,u_2)$.

In this context, it is important to know the expected utility that each player can achieve by refusing to cooperate. This is referred to as the disagreement point, and the associated outcome as the non-cooperative outcome. Together, the bargaining space and the disagreement point specify a bargaining problem.

One common solution to bargaining problems is provided by Ehud Kalai and Meir Smorodinsky. According to Kalai and Smorodinsky (1975), important elements in a bargaining problem are the disagreement point, the bargaining space, and the “utopia point” or “claim point”. The utopia point is the outcome that would satisfy each player’s maximum claim, or “best-case” scenario.

When the bargaining problem is graphed, the claim point is easy to locate. Its coordinates are given by the maximum utility level each player can obtain. This is illustrated, for a typical bargaining space, in Fig. 5.

Fig. 5

A bargaining problem with utopia point. The area S is the bargaining space

The Kalai and Smorodinsky (1975) is based on the following axioms.

• 1. Efficiency: if a is a point in S such that there is another point b which is Pareto-superior to it, then a is not the bargaining solution.

• 2. Symmetry: if S is symmetric, then the bargaining solution is a point of the form (a, a), that is, a point on the line $u_1 = u_2$.

• 3. Monotonicity: if a second bargaining set T is given, which has the same claim point as S, but where player x’s maximum feasible utility level is higher at every utility level that player y may demand, then the bargaining solution of T should give player x at least as high a utility level as in S.

Informally, the last axiom states that if the bargaining space is expanded in such a way that the welfare of one player can be improved without worsening that of the other, then that player should get at least as much in the expanded problem as in the initial one.

Kalai and Smorodinsky (1975) demonstrate that there is a unique point that satisfies all three axioms. If one constructs a line joining the disagreement point and the claim point, the bargaining solution is located at the point where this line intersects the Pareto frontier. This is illustrated in Fig. 6.

Fig. 6

A bargaining problem with utopia points

Alternatively, one can use the Nash Bargaining Solution (Nash 1950). However, we prefer the approach proposed by Kalai and Smorodinsky (1975) since the independence of irrelevant alternatives (IIA) axiom, at the bases of the NBS, has been deeply criticized (Luce and Raiffa 1957) for a set of bargaining situations. Kalai and Smorodinsky (1975) replace the IIA axiom with the property of individual monotonicity. This property implies that players must not suffer from an enlargement of the bargaining set that leaves the maximum utilities attainable by both players unchanged. Based on this axiom, Kalai and Smorodinsky (1975) propose a solution where both parties make equal proportional concessions from their respective favoured points. In our case, where cooperation is imposed by the central government, we recur to the Kalai and Smorodinsky approach that, being based on a different set of axioms, ensures a more general solution.

Appendix 2

We maximize Eq. (24) for delegate B, under constraint (25) taking into account condition (23).

Therefore, we proceed by maximizing the following Lagrangian function

$$\begin{aligned} {{\mathcal {L}}}(g_B, g_S, \lambda )&= 1+\sqrt{g_{{B}}}+k\sqrt{g_{{S}}}-p \left( g_{{B}}+g_{{S}} \right) \nonumber \\&\quad +\, \lambda \, \left( \left( 1-\theta \right) \left( \sqrt{g_{{B}}}+k \sqrt{g_{{S}}}-p \left( g_{{B}}+g_{{S}} \right) \right) \right. \nonumber \\&\quad \left. -\,\theta \, \left( \sqrt{g_{{S}}}+k\sqrt{g_{{B}}}-p \left( g_{{B}}+g_{{S}} \right) \right) \right) . \end{aligned}$$

(64)

For simplicity, we can define:

$$\begin{aligned} \gamma _B=\sqrt{g_B}; \quad \gamma _S=\sqrt{g_S}. \end{aligned}$$

(65)

It follows that Eq. (64) can be written as:

$$\begin{aligned} {{\mathcal {L}}}(\gamma _B, \gamma _S, \lambda )&= 1+ \gamma _{{B}}+k\gamma _{{S}}-p \left( {\gamma _{{B}}}^{2}+{\gamma _{{S}} }^{2} \right) \nonumber \\&\quad +\,\lambda \, \left( \left( 1-\theta \right) \left( \gamma _{{B}}+k\gamma _{{S}}-p \left( {\gamma _{{B}}}^{2}+{\gamma _{{S}}}^ {2} \right) \right) \right. \nonumber \\&\quad \left. -\,\theta \, \left( \gamma _{{S}}+k\gamma _{{B}}-p \left( {\gamma _{{B}}}^{2}+{\gamma _{{S}}}^{2} \right) \right) \right) , \end{aligned}$$

(66)

under the constraints $\gamma _B>0$ and $\gamma _S>0$.

The first order conditions are:

$$\begin{aligned}&\left( 1-\theta \right) \left( \gamma _{{B}}+k\gamma _{{S}}-p \left( { \gamma _{{B}}}^{2}+{\gamma _{{S}}}^{2} \right) \right) -\theta \, \left( \gamma _{{S}}+k\gamma _{{B}}-p \left( {\gamma _{{B}}}^{2}+{\gamma _{{S}}}^{2} \right) \right) =0, \end{aligned}$$

(67)

$$\begin{aligned}&\lambda ={\frac{k-2\,p\gamma _{{S}}}{k\theta +2\,p\gamma _{{S}}+\theta -k -4\,p\theta \,\gamma _{{S}}}}, \end{aligned}$$

(68)

$$\begin{aligned}&\lambda =-{\frac{1-2\,p\gamma _{{B}}}{k\theta +2\,p\gamma _{{B}}+\theta -1-4\,p\theta \,\gamma _{{B}}}}. \end{aligned}$$

(69)

The solutions of the system of Eqs. (67), (68) and (69) are:

$$\begin{aligned} \left( \gamma _{{B}}, \gamma _{{S}}\right) = \left( \bar{\gamma _B}(\theta ), \bar{\gamma _S}(\theta ) \right) \end{aligned}$$

(70)

where,

$$\begin{aligned} \bar{\gamma _B}(\theta )=\sqrt{2}\left( \frac{1-\theta -k\theta - \sqrt{(1+k^2)(1-2\theta +2\theta ^2)-4k\theta (1-\theta )} }{(1-2\theta )4\,p}\right) \end{aligned}$$

(71)

and

$$\begin{aligned} \bar{\gamma _S}(\theta )=\sqrt{2} \left( \frac{1-\theta -k\theta +\sqrt{(1+k^2)(1-2\theta +2\theta ^2)-4k\theta (1-\theta )} }{(1-2\theta )4\,p}\right) . \end{aligned}$$

(72)

Now, we need to make sure that the above solutions are on the Pareto frontier. In order to do this, we note that, from Eqs. (14) and (15), the allocation that maximizes the utility of delegate B is $\gamma _{{B}}^M = \dfrac{1}{2p}$ and $\gamma _{{S}}^M = \dfrac{k}{2p}$. Furthermore, note that $\dfrac{\partial \bar{\gamma _B} }{\partial \theta }>0 \; \forall \; p>0$ and that $\dfrac{\partial \bar{\gamma _S} }{\partial \theta }<0 \; \forall \; p>0$. Note also that:

$$\begin{aligned}&\bar{\gamma _B}(\theta ) = \gamma _{{B}}^M \Leftrightarrow \theta = \frac{1+k^2}{4k}, \end{aligned}$$

(73)

$$\begin{aligned}&\bar{\gamma _S}(\theta ) = \gamma _{{S}}^M \Leftrightarrow \theta = \frac{1+k^2}{4k}. \end{aligned}$$

(74)

It follows that, for $\theta \ge \frac{1+k^2}{4k}$ delegate B has no incentives to set $\bar{\gamma _B}(\theta ) > \gamma _{{B}}^M$ and $\bar{\gamma _S}(\theta ) < \gamma _{{S}}^M$, since it will obtain a smaller utility. In fact, the allocation $(\gamma _{{B}}^M,\gamma _{{S}}^M )$ corresponds to the maximum feasible utility for delegate B (see Sect. 3). It follows that, in order to satisfy condition (23), the set of solutions becomes (Dittrich and Städter 2015):

$$\begin{aligned}&\gamma _{{B}} = {\left\{ \begin{array}{ll} \dfrac{1}{2p}, &{}\quad \text{ if } \theta \ge \frac{1+k^2}{4k} \\ \bar{\gamma _B}(\theta ), &{}\quad \text{ if } \frac{1}{2}<\theta < \frac{1+k^2}{4k}; \end{array}\right. } \end{aligned}$$

(75)

$$\begin{aligned}&\gamma _{{S}} = {\left\{ \begin{array}{ll} \dfrac{k}{2p}, &{}\quad \text{ if } \theta \ge \frac{1+k^2}{4k} \\ \bar{\gamma _S}(\theta ), &{}\quad \text{ if } \frac{1}{2}<\theta < \frac{1+k^2}{4k}. \end{array}\right. } \end{aligned}$$

(76)

After substituting (65), we obtain (31) and (32).

Note that when $\theta$ tends to $\frac{1}{2}$ the AKSS converges to the KSS. In fact it is:

$$\begin{aligned}&\lim _{\theta \rightarrow \frac{1}{2}^+} g_B^{C\theta } = g^{C}; \end{aligned}$$

(77)

$$\begin{aligned}&\lim _{\theta \rightarrow \frac{1}{2}^+} g_S^{C\theta } = g^{C}. \end{aligned}$$

(78)

Appendix 3

Note that:

$$\begin{aligned} W^C_B-W^D_B>0 \iff {k}^{2}+4\,k\alpha _{{B}}+2\,{\alpha _{ {B}}}^{2}-2\,k-4\,\alpha _{{B}}+1 >0. \end{aligned}$$

(79)

For $\alpha _B>\frac{1}{2}$, condition (79) is satisfied when:

$$\begin{aligned} k \ne {\hat{k}}, \end{aligned}$$

(80)

where ${\hat{k}}= 1-\sqrt{2}\sqrt{{\alpha _{{B}}}^{2}}-2\,\alpha _{{B}}$.

Note that ${\hat{k}}<0$ for each $\alpha _B \in \left( \frac{1}{2},1\right)$ and that $k \in \left( 0,1\right]$. It follows that condition (79) is always satisfied and that $W^C_B-W^D_B >0$ for each value of $\alpha _B$ and k in the domain.

Appendix 4

Note that $W^M_S-W^D_S> 0\,$ if $\, 2\,{\alpha _{{B}}}^{2}- {k}^{2}- ( 2\,\alpha _{{B}}-4 ) k-2 >0$.

Let

$$\begin{aligned} g(\alpha _B,k)= 2\,{\alpha _{{B}}}^{2}- {k}^{2}-\left( 2\,\alpha _{{B}}-4 \right) k-2. \end{aligned}$$

Let

$$\begin{aligned} 1/2< \alpha _B< 1;\quad 0 < k\le 1. \end{aligned}$$

Then,

$$\begin{aligned} g(\alpha _B,k)>0\quad \text{ for } \text{ any } \; k \in \left( \check{k}({\alpha _{{B}}}),1\right] \end{aligned}$$

where

$$\begin{aligned} \check{k}({\alpha _{{B}}})=2-\alpha _{{B}}-\sqrt{3\,{\alpha _{{B}}}^{2}-4\,\alpha _{{B}}+2}. \end{aligned}$$

Note that:

$$\begin{aligned}&g(\alpha _B,k)>0 \Leftrightarrow 2 -\alpha _{{B}}-\sqrt{3\,{\alpha _{{B}}}^{2}-4\,\alpha _{{B}}+2}<k< 2 -\alpha _{{B}}+\sqrt{3\,{\alpha _{{B}}}^{2}-4\,\alpha _{{B}}+2},\\&0 \le 2 -\alpha _{{B}}-\sqrt{3\,{\alpha _{{B}}}^{2}-4\,\alpha _{{B}}+2} < 1 \quad \forall \quad \alpha _{{B}} \in \left( \frac{1}{2}, 1 \right] ,\\&2 -\alpha _{{B}}+\sqrt{3\,{\alpha _{{B}}}^{2}-4\,\alpha _{{B}}+2} > 1 \quad \forall \quad \alpha _{{B}} \in \left( \frac{1}{2}, 1 \right] . \end{aligned}$$

It follows that the unique admissible solutions occurs at $k \in ( \check{k}({\alpha _{{B}}}),1]$.