Global COE Hi-Stat Discussion Paper Series 017

Research Unit for Statisticaland Empirical Analysis in Social Sciences (Hi-Stat)

Political Mergers as Coalition Formation:
Evidence from Japanese Municipal Amalgamations

Hi-Stat Discussion Paper

Eric Weese

December 2008

Hi-Stat
Institute of Economic Research
Hitotsubashi University
2-1 Naka, Kunitatchi Tokyo, 186-8601 Japan
http://gcoe.ier.hit-u.ac.jp

Political Mergers as Coalition Formation:
Evidence from Japanese Municipal Amalgamations∗
Eric Weese†November 2008

Abstract
Political coalition formation games can describe the formation and dissolution ofnations, as well as the creation of coalition governments, the establishment of politicalparties, and other similar phenomena. These games have been studied from a theoretical perspective, but the models have not been used extensively in empirical work.
This paper presents a method of estimating political coalition formation models with
many-player coalitions, and then applies this method to the recent heisei municipal
amalgamations in Japan to estimate structural coefficients that describe the behaviourof municipalities. The method enables counterfactual analysis, which in the Japanesecase shows that the national government could increase welfare via a counter-intuitive
policy involving transfers to richer municipalities conditional on their participation ina merger.
Keywords: Coalition Formation, Municipal Mergers, JapanJEL codes: C71, H77

1

Introduction

In recent years, issues surrounding political coalition formation have attracted considerable
interest from both theorists and policy makers. For example, Alesina and Spolaore [1997]
examine from a theoretical perspective what size of countries will form under different conditions. If a coalition is defined as the residents of a certain geographic area, then theformation and dissolution of countries is a political coalition formation game, one that has∗

I would like to thank my advisors: Daron Acemoglu and Abhijit Banerjee. Esther Duflo provided
extremely valuable comments on several drafts of this paper. I would also like to thank Victor Chernozhukov,
Tal Gross, Konrad Menzel, Parag Pathak, Stephen Ryan and the members of the MIT Political Economy
working group for helpful suggestions. The paper has benefited from comments made by attendees atpresentations at MIT and Hitotsubashi University. The seminar at Hitotsubashi was made possible by
Masayoshi Hayashi and Motohiro Sato, who also explained to me some of the finer points of Japanese local
public finance. I am grateful for their support and encouragement.
†
Department of Economics, MIT, Cambridge, MA, USA. Email: weese@mit.edu.

1obvious practical relevance: the dissolution of Yugoslavia, current conflicts in Georgia, andpossible de facto reunification of the island of Cyprus all involve decisions about how many
countries ought to exist and where borders should be drawn. If a coalition is defined as a setof political parties, then the formation of a government can be seen as a political coalition
formation game, and similarly if a coalition is defined as a set of individual members ofa legislature, then the existence of parties in the first place is the outcome of some political coalition formation game. Once again the practical importance of understanding thesetypes of political coalition formation games is borne out by recent news: in the 2007 Belgian national elections it was not obvious even long after the election which parties would
form a coalition government, and in Canada in 2005 the defection of members of parliament
temporarily saved the government from collapse.
In some of these cases it is possible to change the rules governing the coalition formation game, with new rules leading to a different and more efficient coalition structure. For
example, a recent proposal in Canada was that members of parliament should be required
to stand for a by-election if changing their party affiliation between general elections. Had
this rule been in force during recent parliaments, different coalition structures might haveresulted, leading to different governments and different policy outcomes. Similarly, different
laws regarding how municipalities can cooperate to provide public goods, or how farmers
can establish agricultural cooperatives, could lead to very different coalition structures withvery different welfare implications. If so, then it is important to make sure that the “right”
law is in place. In order to predict the results of different laws, however, it is first necessary
to develop a model of the behaviour of the players participating in the coalition formation
game, and then use this model to predict the changes in behaviour that would result fromthe imposition of a different set of laws. Although models of coalition formation date back at
least to von Neumann and Morgenstern [1944], relatively few empirical papers have made useof such models, and in general these papers have not examined the effect of possible changes
in the rules of the coalition formation game being studied.1 There is no immediately obvious estimation strategy for these coalition formation models, since neither existence nor
uniqueness of a stable coalition structure is guaranteed.
This paper presents a method of estimating the structural parameters of a politicalcoalition formation model. The method is then applied to a recent set of Japanese municipal
mergers (the heisei daigappei), where the national government fixed a set of transfer policies
and individual municipalities chose, given these policies, what merger if any they wished1

Desirable properties of some specific forms of coalition formation games, such as two-sided matchinggames, have led to extensive empirical study of those game forms [Roth, 2008]. Empirical research on
general coalition formation models has also been limited by computational feasibility: the number of coalition
structures increases exponentially with the number of players.

2to participate in. The parameters that determine municipal preferences over mergers are
estimated, and these estimates are then used to predict the effect of alternative national
government transfer policies. The heisei mergers are particularly attractive from a modelling
perspective, as government policy allowed mergers to occur only during 1996-2006, andthus the resulting coalition structure can plausibly be treated as the outcome of a single
period coalition formation game.2 Furthermore, the mergers are of interest from a policy
perspective, since due to efficiencies of scale the smaller municipalities spend over $10,000
per capita providing the same services that larger municipalities provide for slightly over
$1,000, and almost all of this difference was being subsidized by the national government.
Overall, then, the paper makes two contributions: first, the method of analysing politicalcoalition formation games, and second, the specific results of this analysis in the Japanesecase.
The methodological contribution consists of the use of simulated maximum likelihood
estimation to obtain structural parameters describing players’ preferences over coalitions
when the observed coalition structure can be treated as the outcome of a cooperative form
hedonic coalition formation game with non-transferable utility. Two ways of overcoming
problems related to non-existence or mulitiplicity of stable coalition structures are presented.
First, all players are assumed to have the same preferences over coalitions, resulting in theexistence of a unique stable coalition structure [Farrell and Scotchmer, 1988]. A secondand distinct strategy is to allow players’ preferences over coalitions to differ, but restrict thetypes of blocking coalitions that can form. This guarantees existence but not uniqueness of astable coalition structure [Ray and Vohra, 1997], and thus estimation requires an additional
assumption regarding which one of the set of stable coalition structures is actually selected.
The advantages of this approach, however, are that the distributional assumption required on
idiosyncratic preferences is less restrictive and a wider variety of covariates can be includedin the specification.
This method is then applied to the case of Japanese municipal mergers. Following Alesina
and Spolaore [1997], there are economies of scale in the production of public goods, butalso benefits to having local policies specifically tailored to match local preferences. This2

In general, a problem with applying political coalition formation models to observed data is that politicalcoalitions once formed tend to persist, and changes that do occur are often separated by large time periods.
The extremely high cost of any realignment means that the stability of existing borders does not provide
much information, and it is not clear what it means for there to be a “stable” coalition structure, if changes
to this structure occur over time at a slow but constant rate. The Japanese data used in this paper mostly
avoids this problem. The fiscal crisis of the 1990s precipitated such significant changes in intergovernmental
transfers that in many cases the old municipal borders were effectively untenable, thus leading to a very large
number of mergers during the window when mergers were allowed. Furthermore, during the 1970-1995 period,
national policy had made municipal mergers extremely unattractive, and thus boundaries remained effectively
unchanged even though demographic changes were rendering these boundaries increasingly inefficient.

3

tradeoff creates an optimal size for municipalities, but pre-existing borders may not create
municipalities of this size. These preferences over municipal characteristics imply preferences
over coalitions, and the parameters determining these preferences are estimated by applying
the method just described to data on the mergers that actually occurred, with the functional
form and some parameters for the cost of providing services derived from existing national
government estimates. Geographical features of the data allow the set of possible coalitions
to be reduced to the point where the model is computationally tractable.
The estimated parameters show that, as expected, municipalities prefer to be in coalitions that offer higher levels of public goods and have lower population. There are somedifferences between the two methods of estimation: the estimated magnitude of the aversion
to amalgamation is higher when all municipalities are assumed to have the same preferences
over coalitions, and the preference for high income fellow residents lower. The methods of
estimation yield estimates for the optimal population of municipalities of 75,000 or 150,000.
Recent Japanese estimates place the lower bound on the efficient population of a municipality at 100,000 [Ministry of Internal Affairs 2003] or 120,000 [Hayashi 2002] and the ability ofthe model to predict an optimal size close to this, despite using a very different technique,suggests that the magnitudes of the estimated coefficients are reasonable. If the average levelof spending on municipal services is assumed to be optimal, then the implied willingness topay for small jurisdiction size is about 0.5% of income to cut municipal population in half.
The estimated parameters are then used to examine an alternative national policy wherea financial incentive is provided for relatively rich municipalities to participate in mergers.
A budget-balanced version of this policy results in higher utility, equivalent on average toan increase in income of 0.3%, with positive effects at the 5th through 95th percentiles. The
result is somewhat counter-intuitive as the problem the national government was attempting
to solve was the high cost of supporting small, poor municipalities. The result is consistent
with theory, however, since a regressive conditional transfer – taxing everyone and transferring money to the residents of richer municipalities that participate in mergers – provides
an incentive for richer municipalities to merge with their neighbours, who then benefit fromhigher levels of public goods. The very poorest municipalities, however, consisting of approximately 5% of the population, are made slightly worse off by this scheme, since they
are never considered as potential merger partners by the richer municipalities. The richest
municipalities, consisting of a similar fraction of the population, are similarly made worse
off because they never participate in mergers but pay the additional tax.
This transfer scheme mimics the transfers that the municipalities themselves would offer,
if they were able to make such commitments. A second counterfactual policy is thus examined, where the national government allows decentralized negotiations over transfers to take4

place between municipalities. In this case, where the game is effectively converted into atransferable-utility game, the outcome depends heavily on the bargaining power of different
types of municipalities. While this policy increases the number mergers that occur, it also
leads to potentially very large transfers from poor municipalities to richer ones, and theexact amount of the transfers cannot be known in advance without knowing the bargaining
method by which municipalities divide the benefits of a merger.3
The major contribution of this paper is the development of an empirical framework for theestimation of political coalition formation models, one that takes into account the theoreticalcharacteristics of solutions to the coalition formation game, and also allows the analysis ofcounterfactual policies. This is an advance over previous techniques: the closest related
work is Brasington [2003], which considers each potential pairwise merger in isolation fromother potential mergers.4 More specifically, in models of the type used by Brasington andothers, the probability that players 1 and 2 will form a coalition is unaffected by the otheroptions that 1 or 2 might have. The method presented below is thus the first to take into
account that the presence of a player 3 and an attractive {1, 3} coalition may disrupt a {1,2} coalition that would otherwise form. In addition, other recent empirical political coalition
formation papers, such as Alesina et al. [2004], focus on describing patterns that are observedin political boundaries, while this paper predicts how counterfactual policies would change
the set of boundaries forming.5 With suitable modifications, the method used in this paper
could be applied to other types of coalition formation games, possibly in other fields as wellas in political economy.
The rest of the paper has the following structure. The general estimation strategy ispresented in Section 2, including both the version imposing a restriction on the form of
players’ preferences and the version using instead a restriction on the types of blocking
coalitions. The use of this strategy in the Japanese case is then described in Section 3,
and potential alternative national government policies are analysed using counterfactual
simulations in Section 4.
3

This result is somewhat similar to the theory presented by Armstrong and Vickers [2007] regarding
control of corporate mergers.
4
The restriction to pairwise mergers follows from the use of the Poirier [1980] bivariate probit model.
5
Brasington [2003], Alesina et al. [2004], and most of the other existing empirical studies of politicalmergers focus on American school districts. Miceli [1993], the earliest example yet found, examines thetrade-off that Connecticut school districts faced between efficiencies of scale and locally optimal education
quality. Alesina et al. [2004] use a much larger dataset, and examine the relationship between county-level
heterogeneity and the number of school districts and other local jurisdictions. While the estimates in eachof these papers imply a type of coalition formation game, they do not present an explicit coalition formation
model.

5

2

TheoryNotation follows that of Banerjee et al. [2001] and Bogomolnaia and Jackson [2002]. Specifically, let N be the set of players, and S ⊂ N a coalition of these players. Π is the setof all possible coalition structures, where a coalition structure π ∈ Π is a set of coalitions{S1 , . . . , SK } such that every player is in exactly one of these coalitions. Suppose that playeri ∈ N has preferences i defined over the set {S ⊂ N |i ∈ S}, with ≺i indicating a strict
preference. The extension of these preferences to partitions is easy: if π(i) is the coalition
that municipality i belongs to in partition π, then π i π 0 if π(i) i π 0 (i). Let π ≺S π 0
for some coalition S if ∀i ∈ S, π i π 0 and at least one of these preferences is strict. The
observed coalition structure is treated as the result of a “hedonic coalition formation game”,
where the payoff to each player depends only on the coalition to which it belongs, and not on
what other coalitions occur. This is the “hedonic aspect” introduced by Dreze and Greenberg [1980], except without the possibility of transfers. The inability to negotiate transfers
prevents some coalitions from forming:
Example 1. Let N = {1, 2}, and ui a utility function describing the preferences of player iover coalitions, withu1 ({1}) = u1 ({1, 2}) + ,

>0u2 ({1, 2}) > u2 ({2})
For small , the stable coalition structure is {{1,2}} if transfers are possible, but {{1}, {2}}if they are prohibited
Ideally, given a set of preferences, there would exist a unique stable partition: First, thesolution set is defined using the von Neumann and Morgenstern [1944] “stable set”:
Definition 1. ΠVNM is a stable set with respect to (Π, <) for some binary operator < if
1. @π, π 0 ∈ ΠVNM where π < π 0 (Internal stability)
2. ∀π ∈
/ ΠVNM , ∃π 0 ∈ ΠVNM where π < π 0 (External stability)
The goal is to define < in a way that is intuitively plausible yet at the same time guarantees that the stable set exists, but this turns out not to be trivial. Consider, for example, thefollowing definition of <: π < π 0 if ∃S ∈ π 0 such that π ≺S π 0 and ∀S 0 ∈ (π \ π 0 ), (S 0 \ S) ∈ π 0
or is empty. Unfortunately, with this definition not only is a stable set not guaranteed toexist, but in general it is not possible to devise another plausible method of selecting a single
partition as the solution of this type of coalition formation game [Barberà and Gerber, 2007].
The following “roommates problem” illustrates this point:
6

Example 2 (Gale and Shapley 1962). Suppose N = {1, 2, 3} and preferences are{1, 2, 3} ≺1 {1} ≺1 {1, 3} ≺1 {1, 2}{1, 2, 3} ≺2 {2} ≺2 {1, 2} ≺2 {2, 3}{1, 2, 3} ≺3 {3} ≺3 {2, 3} ≺3 {1, 3}
With these preferences, no stable partition exists.
Nevertheless, when the Japanese municipalities actually played a coalition formation
game, an outcome did occur. The problem is then how to treat observed outcomes suchas this one when attempting to estimate parameters. There are at least three ways toproceed: to move to a non-cooperative game structure, to restrict preferences, or to relax
the requirements for stability.
A non-cooperative game is guaranteed to provide a set of equilibrium outcomes, but
it is difficult to use in this case as no information is available about the way in which
the municipalities actually negotiated, or who made what offers, and so forth. Thus, thespecification of the rules of the game would be essentially arbitrary. If the equilibria did notdepend on the rules, then the lack of information about the negotiation process would not be
important, but it is fairly easy to see that in this sort of coalition formation game, different
rules produce different outcomes. For example, if there are a finite number of periods in whicha proposer can propose a coalition or coalition structure, then the probability with which
various municipalities are selected to be the proposer will change the types of proposals madeand accepted. Radically different parameter estimates could be obtained by using different
probabilities of having a municipality selected as proposer, and there is no information
available on what reasonable proposer weights would be, or even whether the proposer typeframework is appropriate. Thus, non-cooperative form games will not be used as part of theestimation strategy. The other two potential solutions given above will be used, however.
First, preferences will be restricted so as to ensure that a unique stable partition exists.
Second, a more general utility function will be used, but certain types of deviations will not
be allowed. This ensures the existence of a stable partition, but not its uniqueness, and soestimation becomes somewhat more complicated.

7

2.1

Restricted Preferences Approach

Consider the following restriction on the form of ui , the utility that player i derives from a
coalition:ui (S) = u(S) + αiu(S) = v(XS ; θ) + S
Here v is a function of characteristics XS of S, taking parameters θ. The econometrician
observes XS and knows the functional form of v, and the objective is to estimate the parameters θ. The error term S is of a known distribution. The important restriction here is that
if S ≺i S 0 then ∀j, S ≺j S 0 . That is, all agents have identical preferences over coalitions.
Theorem 1 (Farrell and Scotchmer 1988). If all agents have identical preferences overcoalitions, a generically unique stable partition exists.
Proof. The unique stable partition can be constructed as follows:
0. Let V 0 be the set of all potential coalitions, and start with π 0 = ∅ and k = 0kk
1. Find Smax
such that ∀S ∈ V k , u(S) < u(Smax)
kk
2. Set π k+1 = π k ∪ {Smax} and V k+1 = {S|S ∈ V k , S ∩ Smax
= ∅}3. If V k+1 6= ∅, repeat from 1.

This restriction on the idiosyncratic error term is strong: it implies that all the unobserved
characteristics of a coalition are enjoyed equally by all its members. For example, in the caseof municipal mergers, it rules out the possibility that a large municipality merging with asmaller neighbour might take advantage of its dominance on the new amalgamated municipal
council in order to geographically skew public spending. The major benefit of placing thisrestriction on the error term is that it guarantees uniqueness, and thus estimation does not
require any assumption about an equilibrium selection rule.6
6

It may be possible to weaken the restrictions on the error term somewhat in the future by instead
assuming the monotonic median voter property Acemoglu et al. [2008b].

8

Suppose that partition π0 is actually observed. The parameters θ can be estimated via
simulated maximum likelihood. The likelihood of π0 occurring is
Z
L(π0 stable |θ) =I(π0 stable |θ, )ZP (π0 stable |θ, 0 )=
0where 0 denotes the vector {S |S ∈ π0 }. This integral can be numerically approximated bytaking a set E0 of random draws of 0 and calculating
1 X
P (π0 stable |θ, 0 )|E0 |  ∈E
00Because of the “convenient error partitioning” [Train, 1995] of the above, the probability
can be expanded into a product of independent events.7 Let V be the set of all potential
coalitions. ThenP (π0 stable |θ, 0 ) =YP (u(S 0 ) < max u(S)|0 , θ)S∈perpS 0

S 0 ∈V

where perpS 0 = {S|S ∈ π0 , S ∩ S 0 6= ∅} is the set of “perpetrators” necessary to deviate to
S 0 . The likelihood function used for optimization is thusL(π0 stable |θ) =1 X YP (u(S 0 ) < max u(S)|0 , θ)S∈perpS 0|E0 |  ∈E S 0 ∈V
0

2.2

0

Relaxed stability requirements

Now suppose that a less restrictive form was imposed on preferences:ui (S) = v(Xi , XS ; θ) + iSHere, the utility a player derives from a coalition can depend on interactions between theplayer’s characteristics and those of the coalition, and similar the  for a given coalition canvary across players. In this case, the existence, but not uniqueness of a stable partition canbe guaranteed so long as some restrictions are placed on the types of blocking coalitions
That is, once the 0 have been drawn, and thus the u(S) are known for S ∈ π0 , the events u(S 0 ) > u(S)and u(S 00 ) > u(S) are independent. This conditional independence allows conditional probabilities to be
expressed as products of the relevant independent events.
7

9

that can form. In particular, only two types of potential deviations will be considered whenevaluating whether a given partition is stable: refinements, where subcoalition of a single
existing coalition breaks off to form a coalition, and coarsenings, where two or more existingcoalitions merger in order to form a new coalition.
To solve this problem, Ray and Vohra [1997] only allow deviating coalitions to force
refinements of a partition, and Diamantoudi and Xue [2007] show that this creates a stableset. Because hedonic games are simpler than the “equilibrium coalition structures” that Ray
and Vohra examine, refinements and coarsenings will be treated identically. Otherwise, thetheory follows that presented in Ray and Vohra.
Definition 2. π → π 0 if ∃S such that either π %S π 0 or π &S π 0 , where
1. π %S π 0 if π 0 \ π = S such that π ≺S π 0 and S =S

Q for some Q ⊂ π

2. π &S π 0 if ∃S ∈ π 0 such that π ≺S π 0 anda) π \ π 0 = S 0 with S 0 =S

Q0 for some Q0 ⊂ π 0b) @Q̃ such that Q0 → Q̃The recursion is well defined since Q0 is a proper subset of π 0 . Using the terminologyof Ray and Vohra, π is blocked by π 0 if either there is a set of coalitions in π that are
unanimously in favour of merging to create π 0 , or there is a subset of “perpetrators” in πthat are unanimously in favour of deviating from their current coalition. In the former case
π 0 is the coarsening that results from the merger, while in the latter it is a refinement that
includes a coalition for these perpetrators and some arrangement of the “residual” left behindwhen the perpetrators deviated, such that the configuration of perpetrators and residual isstable.
Theorem 2. Let  be the transitive closure of →. Then
1. Π∗ = {π|@π 0 such that π → π 0 } is a stable set with respect to (Π, ).
2. Π∗ is unique.
3. Π∗ contains a Pareto optimal partition.
Proof (existence). By construction, Π∗ is internally stable. Now take some π ∈/ Π∗ . Then∃{π1 , . . . , πm } ⊂ Π such that π → π1 → · · · → πm and either πm ∈ Π∗ or there is a cyclewith πm = πl for some l < m. If there is such a cycle, then it must contain both mergers anddissolutions. Suppose that πk %S πk+1 , and let S1+ ⊂ S be the set of agents that strictly10prefer πk+1 to πk . If πk+1 % πk+2 then S2+ ⊃ S1+ since no agent can be made worse off by amerger. If πk+1 &S 0 πk+2 then S2+ = (S1+ \ R) ∪ P where R is some subset of the residual,+
and P 6= ∅ is some subset of the perpetrators, and (R ∪ P ) ⊂ S 0 . Since Sm−l+1= ∅, at somepoint the agents in S2+ must be made worse off. This can only happen via refinements, andonly if there is a residual smaller than S2+ . The latter, though, implies that some subset of
S2+ cannot be made worse off, and thus S + can never be empty. Thus a cycle cannot exist,
and πm ∈ Π∗ .
(uniqueness). Suppose that Π∗∗ is also a stable set with respect to (Π, ). Consider the
bipartite directed graph defined by  with Π∗∗ \ Π∗ and Π∗ \ Π∗∗ as the two sets of nodes.
Every node must have in-degree of at least one, but there can be no cycles. The only suchgraph is empty, and thus Π∗∗ = Π∗ .
(PO element). Let ΠPO ⊂ Π be the set of Pareto optimal partitions, and
the ParetoPO∗dominance operator. Suppose that Π ∩ Π = ∅ and consider the directed graph definedby  ∪
with ΠPO and Π∗ as two sets of nodes.
There is no particular reason to believe that this particular partition is more attractive
as solution than the other partitions in Π∗ . There may be, however, some “solutions” thatseem particularly unattractive: {π ∈ Π∗ |∃π 0 ∈ Π∗ , π
π 0 }. Unfortunately, computational
limitations make placing further restrictions on the solution set infeasible.

2.3

Estimation

Following the notation in the restricted preferences section, let 0 denotes the vector iS , ∀i ∈S, ∀S ∈ π0 . Suppose that partition π0 is actually observed. If only partitions in the solution
set Π∗ are observed, and every partition in Π∗ is assumed to be selected with equal probability,
then the parameters θ can be estimated via maximum likelihood. The likelihood of π0
occurring is
Z
L(π0 |θ) =
Z
=
0

P

I(π0 ∈ Π∗ |θ, )∗π∈Π I(π ∈ Π |θ, )
1∗∗π∈Π P (π ∈ Π |θ, 0 , π0 ∈ Π )P (π0 ∈ Π∗ |θ, 0 ) P

11let pl = P (πl ∈ Π∗ |θ, 0 , π0 ∈ Π∗ ), and rewrite using the central limit theorem:
Z

P (π0 ∈ Π∗ |θ, 0 )

1Z

P (π0 ∈ Π∗ |θ, 0 )1
|Π| · Z̃

0

Z
'
0

→p

Z
0

Z ∼ Generalized Binomial (|Π|, {p0 , p1 , p2 , ...})




|Π|
|Π|
2
X
1 X
1
Z̃ ∼ Normal pl ,(pl )(1 − pl )
|Π| l=0|Π|l=01
P (π0 ∈ Π∗ |θ, 0 ) P|Π|as N → ∞, |Π| → ∞l=0 plprovided that the variance on the normal approximation goes to zero, which will be thecase so long as E[|Π∗ |] goes to infinity as |Π| does. The actual estimation is performed via
numerical approximation of the above integral. Specifically, if E0 is a set of draws from the
distribution of 0 , then the approximation is
1 X
1
P (π0 ∈ Π∗ |θ, 0 ) P|Π||E0 |  ∈E
l=0 pl00Once again because of the convenient error partitioning of the above, the probability can be
expanded into a product of independent events. Let S0↑ be the set of all coalitions that could
be formed by mergers of the coalitions in π0 , and let S0↓ be the set of all coalitions that area subset of a coalition in π0 . ThenP (π0 ∈ Π∗ |θ, 0 ) =Y

P (π0 ⊀S S|θ, 0 )S∈(S0↑ ∪S0↓ )Now approximate the denominator by randomly drawing a set Πl ⊂ Π, and defining l andEl in the same way as 0 and E0 :
|Π|
X
l=0

pl '

|Π| X 1 X
|Πl | π ∈Π |El |  ∈E
l

l

l

Y

P (πl ⊀S S|θ, 0 , l , π0 ∈ Π∗ )l S∈(S ↑ ∪S ↓ )ll

There are two problems with estimating this numerically. First, draws need to be made froml |θ, 0 , π0 ∈ Π∗ , and second, given a draw of l from the correct distribution, the required
probability needs to be calculated efficiently. Fortunately, in both cases an application ofBayes’ Rule is sufficient. To draw from l |θ, 0 , π0 ∈ Π∗ , first define S to be the idiosyncratic
shocks to coalition S ∈ πl . If S is not a potential deviation from pi0 , then S |θ, 0 , π0 ∈ Π∗ =
S since no additional information is provided by the fact that π0 is stable. If S is a potential

12deviation from π0 , then consider the identityf (S |θ, 0 ) = f (S |θ, 0 , π0 ⊀ S)P (π0 ⊀S S|θ, 0 ) + f (S |θ, 0 , π0 ≺S S)P (π0 ≺S S|θ, 0 )f (S |θ, 0 ) is equal to the unconditional density f (S ), which is known by assumption. The
second distribution on the right hand side is a set of truncated distributions because if
π0 ≺S S then it must be true that ui (S) > ui (π0 ), and thus
i > ui (π0 ) − vi (S)and these can be calculated sequentially. Thus, the desired distribution can be drawn by
simulating fromf (S |θ, 0 , π0 ⊀S S) =f (S |θ, 0 ) − f (S |θ, 0 , π0 ≺S S)P (π0 ≺S S|θ, 0 )
P (π0 ⊀S S|θ, 0 )which can be done sequentially for each member of S. From a computational perspective, it
is important to avoid simulation “chatter”, which would occur if a new  were drawn for eachnew proposed θ̂, the simplest way of ensuring that π0 was always in the stable set. Insteadof simulating 0 directly, then, draw quantile indices qj , and create j from qj fresh for each
iteration of θ̂.
The next problem is using these drawn l to calculate the probability
P (πl ⊀S S|θ, 0 , l , π0 ∈ Π∗ )where S 0 is some coalition not in πl . If S 0 is not a potential deviation from π0 , then thecalculation is identical for those done for π0 , described above. However, if S 0 is a potential
deviation from π0 , then the fact that π0 is stable provides additional information that needs
to be taken into account. Consider
P (πl ⊀S 0 S 0 |θ, 0 , l ) =P (πl ⊀S 0 S 0 |θ, 0 , l , π0 ⊀ S 0 )P (π0 ⊀S 0 S 0 |θ, 0 , l ) + P (πl ⊀S 0 S 0 |θ, 0 , l , π0 ≺S 0 S 0 )P (π0 ≺S 0 S 0 |θ, 0 , 1 )and since the left hand side can be calculated, and the second term of the right hand side
has the same set of truncated distributions described just above with respect to the l , then13

rearrangement once again permits calculation:
P (πl ⊀S 0 S 0 |θ, 0 , l , π0 ⊀S 0 S 0 ) =1 − P (πl ≺S 0 S 0 |θ, 0 , l ) − (1 − P (πl ≺S 0 S 0 |θ, 0 , l , π0 ≺S 0 S 0 ))P (π0 ≺ S 0 |θ, 0 , 1 )1 − P (π0 ≺S 0 S 0 |θ, 0 , l )Everything on the right hand side of this equation can be computed quickly, making optimization feasible.

3

Application

Treating municipal mergers as a hedonic coalition formation problem is consistent with anecdotal evidence concerning how mergers are effected. Negotiations regarding compensation
seem to be rare, even though controversy is common and the results of unrest sometimes
significant. Usually, some of the involved municipalities were in favour of a merger whileothers were very much opposed, but those in favour did not promise large transfers to those
opposed in order to secure their cooperation. This suggests that there is some problem with
contractibility in political mergers such that transfers are difficult or impossible, and thus,
as in Acemoglu [2003], the “political Coase theorem” does not hold. It thus seems moreplausible to model mergers as a coalition formation game without transfers. First, a simple
model of public good provisioning is presented, then some of the parameters are estimatedfrom other Japanese data sources. The data is described, and then the remaining parameters
are estimated via the methods presented in the preceding section.

3.1

Municipal Public Goods Model

Suppose that at time t, each municipality m provides a public good of level gmt to its
residents at a total cost of cm (gmt ), with g including public goods such as local roads,
elementary education, waste disposal, and some health care. cm is assumed to be subadditivein population: cm0 < 2cm if the population of m0 is twice as large as that of municipality m.
To pay for this service provision, the municipality levies taxes at rate τmt (possibly restrictedby the national government to τ̄ ), and receives transfers Tmt from the national government
such that the intertemporal budget constraint is satisfied:
Z ∞
e
0

−rt

Z ∞
cm (gmt )dt =0

14e−rt (τmt Ym + Tmt )dt

where Ym is the tax base of the municipality, and r is the interest rate. If characteristics ofthe municipality are constant across time, the discount rate is the same as the interest rate,
and there is no uncertainty, then the optimal tax rate and level of government services does
not change across time, and everything is perfectly smoothed. Thus, the t subscripts are
dropped, and any short term transfers received (as in the next section) are assumed to be
perfectly smoothed out. Thus, the budget constraint is treated asZ ∞cm (gm ) = τm Ym + r

e−rt Tmt dt

0

In addition to providing general services of level gm , the municipal council also has control
over some (costless) local policies, as in Alesina and Spolaore [1997]. Different individuals
have different ideal points regarding these policies, and thus as the population of a municipality grows, so does the utility loss due to having to impose a constant policies over theentire municipality. Thus, there is a tradeoff between providing the public good cheaply, andhaving government policies that are carefully tailored to residents’ desires. A formal modelof this tradeoff is not presented here, but instead, controlling for the economies of scale inproduction of public goods, higher population will enter the utility function negatively.8

3.2

Japanese Local Public FinanceMochida [2006] provides an excellent summary of the development and current state of theJapanese local finance system. Post-war Japanese fiscal policy placed great emphasis on
the provision of equal quality public goods across the country, and established a national
standards for the general services that were provided by local governments. To ensure thatevery municipality had sufficient funds to offer the specified services at or above the standard
quality level, the national government developed a complicated system of transfers, called
8

The Alesina and Spolaore [1997] model can be extended to two dimensions in order to handle actual
geographic data, but the reduced form approach is used here because it is difficult to justify any particularfunctional form assumption on the utility loss due to government policies being set away from a resident’s ideal
point. Although multi-dimensional policy spaces in general suffer from cycles, if preferences are Euclidean
and the number of voters is large, then the plausible set of policies, following a reasonable definition such as
the yolk, lies within a small and shrinking region. With an assumption on how the policy is selected withinthat region, the remaining difficulty would be how preferences of voters determine preferences over mergers.
One possibility here would be to assume that voters will move randomly within whatever merger they join,
and thus do not apply their geography-based policy preferences to decisions regarding which coalition to
join.

15

the “Local Allocation Tax”.9 The transfer to municipality m is determined by the equation
Tm = max(c̃m (ḡ) − .75τ̄ Ym , 0)(1)where ḡ is the minimum quality of services the municipality is expected to provide, c̃m (ḡ)
the estimated cost to the municipality of providing those services (referred to as “Standard
Financial Need” in official documents), τ̄ the fixed tax rate that the municipality is required
to charge. c̃m varies significantly from municipality to municipality, based on a formuladeveloped by central ministries. The exposition of this formula consists of approximately
280 pages of Japanese legal text; however, as Figures 2 and 3 show, c̃m can be approximatedquite well by the linear regressionc̃m (ḡ) = ḡ(β0 + β1 P OPm ) + vm

(2)

These figures and equations effectively duplicate those found in an official Ministry of Financepublications, as the linear approximation is well known and commonly used.10 The positiveand significant intercept shown in Table 1 reflects the fact that the central ministries believedthat there were economies of scale in the production of public goods, and thus per capita
costs would be higher in municipalities with lower population. At current exchange rates,ḡ β̂0 is a little more than $10M. Thus, if the national government estimated costs correctly,with c̃(ḡ) ' c(ḡ), the per capita cost of providing ḡ quality public goods in Ashiyasu village(population 567) is roughly $22,000, compared with roughly $1,400 in Sakai city (population
790,000).
With almost half of Japanese municipalities having a population less than 10,000, thedecision to provide additional subsidies to smaller municipalities due to their size was an
expensive one. Although there were provisions for municipalities to merge, there was little
incentive for them to do so, because if a coalition S decided to form a new (amalgamated)
municipality, TS would be calculated identically to Equation 1, above:
TS = max(c̃S (ḡ) − .75τ̄ YS , 0)(3)

Thus almost all savings would be passed to the national government, and even a slight
preference for smaller population jurisdictions ensured that residents would be opposed to
9

The slightly-confusing name is due to the fact that it is an allocation to local governments from taxes
collected by the national government.
10
A new formula introduced after the period of interest is explicitly based on a linear function of population
and area.

16

mergers.11
During the financial difficulties of the early 1990s, the national government implemented
a series of reforms designed to reduce the total transfers provided to municipalities while
minimizing the negative effects of this decrease. First, the government substantially reduced
transfers to the smallest municipalities by revising the Local Allocation Tax. This can beapproximated as
Tm = max(c̃newm (ḡ) − .75τ̄ Ym , 0)
new
c̃new
+ β1 P OPm ) + vm
m (ḡ) = ḡ(β0as shown in Table 2, with β0new being a little more than half the size of β0 .12 Second,municipalities that merged between 1995 and 2005 would not have their transfers lowered
due to the merger for at least ten years starting from the date of the merger. That is,
TSnew =

X

Tmnew

m∈S

would be provided for the decade following the merger. This resulted in a strong financial
incentive for municipalities to merge, as shown by the utility functions derived in the appendix. By 2006 there were only 1821 municipalities remaining, down from 3232 at the startof the merger period in 1995. Figure 4 shows the mergers in Shizuoka Prefecture. Mergers
were voluntary, and needed to be approved by the municipal council of every participating
municipality.13 The parameters to the utility function can thus be estimated by examining
the mergers that actually occurred.

3.3

Data

The initial laws implementing the new incentive scheme were passed in 1995, and thus it
would be optimal to use data from before this period. However, some later data is currently
used due to data availability issues. Almost all of the merger negotiations and approvals
occurred near the end of the 1996-2006 window, with most occurring after 2002, and thelatest data used is from 2000. Thus, it seems like this should not be a huge concern. Due to
11

In general, the division of a municipality was prohibited. In one case, such a split did occur, but bothof the resulting municipalities were immediately merged with different neighbours.
12
Previously, mayors were responsible for delivering hundreds of “agency delegated functions” from higher
levels of government, making them bureaucrats as well as elected officials, and making it possible (at leastin theory) for central ministries to fire a mayor for not performing a delegated function according to specifications. “Agency delegated functions” were also abolished, and responsibility was devolved in many cases
to mayors and municipal councils.
13
In about a third of cases, referenda were held. Nominally, these were consultative, but in general themunicipal council would not vote opposite to a referendum result

17

disorganization, coordination failure, or for political reasons, a few mergers occurred after
the end of the merger window. These “late” merging municipalities did not benefit from the
incentive scheme described above, and are not considered in the estimation described below.
There were 3382 municipalities at the start of the merger period, divided into 47 prefectures (similar to US states). Since mergers do not cross prefectural boundaries, eachprefecture is treated as a separate game in the following sections. Surface area data for eachmunicipality is obtained from a 1996 survey conducted by Geographical Survey Institute, an
arm of the Japanese national government. Municipal population data comes from the 2000
national census as reported by the Home Affairs Ministry, but should shortly be replacedwith 1995 census data. Taxable income from 1996 is used as a proxy for the municipal tax
base, with income data, as well as the list of mergers that actually occurred come from the
Asahi Shimbun minryoku. To construct the set of possible coalitions, as described in moredetail later, information on which municipalities share a border is taken from Global Map
files for Japan.
The financial data to generate Table 1 is from the shichouson betsu kessan joukyou shirabe,
which is an official national government report of municipal finances. The 1996-1997 fiscal
year data is used as this is the first year available electronically. To determine the newtransfer policy, as shown in Table 2, the 2006-2007 fiscal year data is also used. Becauseof the large transfers from the national treasury to local governments, this data is handledquite carefully by officials in the central ministries and is generally regarded as accurate,particularly the sections produced by the central government itself. The isolated incidents
of fraud reported generally relate to variables reported by the municipalities, which are notused in this paper.
There are no missing values in any of the financial data or surface area data. In the
population data, approximately 6 values are missing because one merger took place before
the data was issued, and thus the old municipalities were not reported. This will be dealt
with by finding an older version of the data. The 23 special wards covering the area of
pre-war Tokyo city, although having powers similar to municipalities, are excluded fromthe analysis because any enlargement of this sui generis area would likely involve adding
more wards, rather than changing the borders of existing ones. The 12 “designated cities”,
which have some powers normally reserved for prefectures, are omitted from the financial
calculations because their additional responsibilities increase their required spending, butthey are included in the rest of the analysis as regular cities. The categories of “core city”
and “special city” were created after the merger period, and thus do not directly affectthe data. However, the policy-setting powers of these municipalities are slightly different,
and this may have provided an incentive for coalitions to form that met the threshold for18

classification into one of these categories. This is currently ignored, but could be consideredlater. Similarly, the policy distinctions between city (shi), town (chou), and village (son)are ignored, although they could be added as dummy variables at a later date. Counties(gun) are statistical divisions, and have no government structure; however, because they
are historical units, by some accounts county lines are reflected in the patern of mergers
observed. This data is also available, and could be taken into account at a later time.

3.4

Estimation

Suppose that for any coalition S, the utility of municipality i in the coalition is given byui (S) = θ1 log ((1 − τS )yi ) + θ2 log gS + θ3 log POPS + θ4 log AREAS + φXS + iS
where XS are other fixed characteristics of the coalition.14 The third term reflects the valuethat residents place on smaller jurisdictions that can tailor government policies more carefully
to reflect local concerns, and the fourth term is included because heterogeneity of policy ideal
points may be greater when the same population is spread out over a larger area. Income
per capita and a dummy for whether the coalition is actually a merger (i.e. non-singleton)
being the only column in XS . In the future, however, additional interaction terms could be
included. For estimation via the restricted preferences method, the restriction iS = jS = S
is required.
While de jure municipalities were given the power to set taxes as part of the reforms,
there is a belief that de facto they do not have much authority to change tax rates, and there
appear to have been very few significant shifts in tax rates. Thus, estimation is performed
assuming that local governments cannot change the tax rate, which is fixed at τ̄ . This tax
revenue, combined with transfers, then determines the amount of general services provided.15
The determination of V , the set of potential mergers that need to be checked during
estimation, is slightly more problematic. There are a number of large mergers observed,
with the largest involving 15 municipalities. Almost all observed mergers are geographicallycontiguous; however, even after restricting V to contiguous coalitions of size 15 or less, there
are still over 1016 possibilities, which is computationally infeasible. Many of these coalitions
look very different than the actually observed coalitions, however. In particular, they tend
14
The existence of a representative agent is not obvious, since if many different potential merger partners
exist, and individuals are allowed to have arbitrary preferences, then Arrow’s Impossibility Theorem applies.
If the only choice that needs to be made is which merger to pick, then “intermediate preferences” would
guarantee that the median voter is the representative agent. To have such a voter exist, though, heterogeneity
must be one dimensional and have a specific form [Grandmont 1978].
15
Since the transfers are phased out after ten years, the present value of this amount is used to determine
the effective transfer after smoothing.

19

to be a thin line of municipalities, stretching almost all the way across a prefecture. The
actually observed coalitions, on the other hand, look like ellipses. Figure 5 shows a merger
that actually occurred (Hanamatsu city, in Shizuoka prefecture, involving 12 municipalities),while Figure 6 shows a typical randomly generated contiguous coalition of the same size. The
randomly generated coalition in this case suffers from a defect that is not considered in theutility function given above: because of its elongated shape, travel time to a single centrallylocated city hall other such facility would be extremely high for residents starting in certain
parts of the amalgamated municipality. Similarly, the cost of visiting the more remote partsof the new municipality would likely be excessive for centrally located municipal bureaucrats.
These sorts of coalitions are thus ruled out of consideration by the use of a restriction related
to the surface area to perimeter ratio of the coalition. Because the RAND-ESU [Wernicke
2006] algorithm used to generate the coalitions is limited in the types of restrictions it
can accomodate, this restriction is formulated in terms of graph theory characteristics, anapproximation which makes implementation computationally feasible.
The randomly generated potential coalition shown in Figure 6 differs from actually observed coalitions in that it borders over 30 other municipalities, whereas the actually observed
coalitions of this size never border more than 9. A restriction is placed on the number ofmunicipalities that the coalition can border is introduced, with an upper bound based on
the maximum in the actually observed mergers. This restriction dramatically reduces thenumber of large coalitions that need to be considered: with 15-municipality coalitions, only 1
coalition in 10 billion has a small enough number of neighbours. This reduces the total number of alternatives that need to be considered to about 5 million, which is computationallyfeasible.
Another problem, only relevant to the estimation via relaxed stability requirements, is theestimation of the size of the stable set. Since the number of partitions grows exponentiallywith the number of municipalities, it is not possible to examine all partitions. The total
number of partitions is unknown but bounded above by the Bell numbers (Sloane #A000110),
which are greater than 10100 for larger prefectures such as Hokkaidou. Thus, instead, arandom sample is drawn; however, it is not trivial to randomly sample from a set which is
too large to be enumerated. Thus, random draws are obtained using Markov chains. Let
the state space X be the set of all partitions, and the transition matrix P be
Pxy = k if y can be created by either breaking apart one coalition in x into subcoalitions,or merging coalitions in x together to create one new coalition
Pxy = 0 ∀y 6= x that do not meet the above conditionP
Pxx = 1 − y6=x Pxy
Since the state space is finite, this creates a valid transition matrix for sufficiently small
20

k. P describes a reversible Markov chain, since a transition from x to y via merging implies
a possible transition from y to x via a breakup, and vice versa. The chain is connected,
since any partition can be obtained by first breaking all coalitions down to singletons, andthen constructing the desired partition. There is thus a unique stationary distribution,
since the chain satisfies the detailed balance condition [Robert and Casella 2004]. Moreover,the stationary distribution gives equal probability to each state, and thus draws from thestationary distribution are equivalent to random draws from the set of all partitions. These
draws can be performed using the Metropolis-Hastings algorithm.
Unfortunately, the number of transitions that need to be considered is too large to becomputationally feasible. The number of ways a size 15 coalition could be broken down
into sub coalitions could be as high as a billion, and thus it is not practical to enumerate
all the possible transitions. Instead, the approach introduced by Wernicke [2006] for the
RAND-ESU algorithm will be used. Let p be a length 15 vector of “cut probabilities” andlet Q be a set of subcoalitions of S that form a partition of S. Then enumerate a member ofQ
Q with probability i∈{1,...,|S|} pi . This gives each member of Q an equal probability of being
enumerated, but allows for a much smaller, randomly selected set to be considered. So longas this randomly selected Q̃ is re-randomized each time a given state is reached, then theproperties of the above Markov chain are unchanged.

3.5

Results

The results are shown in Table 3. Results are consistent with the theory, with high government services and low population being preferred, as predicted. Having richer fellow
residents also appears to be preferred, but the magnitude and statistical significance of thiseffect depends on the method used. The dummy for mergers is negative under both estimation methods, but statistically significant only in the restricted preferences estimates,making it unclear whether there was indeed a strong preference for the status quo. The
ratio of the coefficients on government services and population imply that a municipality
would be willing to accept an increase in population of one log point in exchange for an
increase in the level of services of either .06 or .12 log points. This implies that, the optimal
population size for a municipality is 75,000 or 150,000, respectively. This accords well withJapanese estimates of the efficient size for a municipality. The Ministry of Internal Affairs’“Standard Municipality” has a size of 100,000, and the Ministry [2003] has estimated thatthat is the minimum efficient size for a municipality. Furthermore, Hayashi [2002] finds thatthe smallest city of efficient scale has a population of 120,000. This estimate is particularlyinteresting, since Hayashi uses third-party ratings of municipal service quality, which is not21

used in this paper, but recovers roughly the same population target. Total spending on municipal government services accounts for approximately 10% of income, so, if the assumptionof Cobb-Douglas utility is correct, and if the tax rate specified by the national government
is approximately the efficient level, then the implied willingness to pay for small jurisdiction
size is about 0.5% of income to cut municipal population in half.
Although the sign on surface area is not as expected in the relaxed stability requirements
column, the magnitude is tiny. The smaller standard deviation of the error term in therelaxed stability requirements estimates is an indication that the variables included in theestimation are relatively more important in determining choices. Additional data that could
be used in the future includes commuting patterns, age and education distribution, andpossibly industrial sectors. In some cases it might be possible to also replace means withmedians, since the variance (but not the precise distribution) of income is known for eachmunicipality. Finally, prefecture fixed effects could be considered, although this presents
some computational issues.

4

Counterfactual simulations

A major advantage of having coefficient estimates for a structural model is the ability to
conduct counterfactual analysis. Here, two alternative national government policies will be
examined: first, an incentive scheme for richer municipalities that participate in mergers; second, national government enforcement of transfers negotiated between municipalities during
the coalition formation process.

4.1

Incentives to merge

Suppose that the national government offered an additional, budget balanced incentive forcertain municipalities to merge. The targeted municipalities should be those that are most
likely to be opposed to mergers that would benefit other municipalities, and the most likely
municipalities to fall into that category are richer municipalities. Consider the policy thatoffered a transfer equivalent to 0.3% of income, to residents of richer municipalities that participated in a merger, where a “richer” municipality is defined as one that had above average
income per capita in more than half of the potential mergers they could have participatedin. This transfer would be paid for by an increase in the income tax on everyone. This
type of subsidy preserves the existence of a unique stable partition, since it is equivalent
to increasing um for rich municipalities while at the same time decreasing u(S) for those
singleton coalitions that consist of a single rich municipality.

22

There are two different types of questions that could be asked regarding the effect of thispolicy. One is whether, conditional on the observed outcome occurring, the counterfactual
policy would have yielded a better outcome. The other is whether the counterfactual policy
would have yielded a better outcome without any information about what outcome occurs
under the actually implemented policy. The difference concerns the way the  are drawn.
In this section, the first case will be considered, while in the next section the latter will be
estimated.
Given that the observed outcome occurred under the actually implemented policy, andassuming that the true θ are exactly the estimated θ̂, the distribution of  is no longer
i.i.d.; however, draws can be made from {|π0 stable, θ} via Gibbs sampling. Then, for eachof these draws of , the stable partition under the counterfactual policy can be computed
via the algorithm given in the proof of Theorem 1. The changes in utility, averaged over
several simulations, are shown in Figures 7 and 8. In one representative simulation, outof 3822 municipalities, 784 received the new incentive transfer but also merged under theoriginal policy, and there were 176 municipalities that participated in mergers that did notoccur under the original policy. The mean change in utility is equivalent to an increase
in income of 0.3%.16 However, even though almost all quantiles of the utility distribution
are shifted upwards, the new policy is not rank preserving, and a significant number ofmunicipalities would be better off under the actual policy. In each of the simulations, there
were a few dozen municipalites that were in mergers under the actual policy, but whose
merger partners abandoned them for another coalition under the alternative policy. Thus,
not all municipalities would be in favour of switching to the alternative policy.

4.2

Transferable utility

Suppose that the central government offered to enforce whatever transfers resulted fromdecentralized negotiations amongst municipalities. That is, if municipality m promised tomake a certain transfer from its current residents to the current residents of municipality n,
the central government would ensure that this was actually carried out. Assume that thetransfers negotiated are “small”, in the sense that a linear approximation of utility aroundtransfers of zero is reasonable:Pj∈S zij) + ... + iSui (S, z) = θ1 log((1 − τS )yi +POPiPθ1 j∈S zij' θ1 log((1 − τS )yi ) ++ ... + iS(1 − τS )yi · POPi
16

The fact that this is the same as the size of the additional transfer offered is purely coincidental.

23

WherePz is total transfers received from other municipalities. Now define(1 − τS )yi POPi
· ui (S)θ1X'
zij + θ1 log((1 − τS )yi ) + ... + iSũi (S, z) =j∈Sand thus ũ approximates a standard transferable utility cooperative game. This does nothave a unique solution, but is covered by the Ray and Vohra [1997] approach detailed above,and so will result in some sort of stable set. If random  are drawn, then some idea ofthe efficiency gain of enforcing the TU game, versus simply administering a fixed incentive
structure, can be obtained. The exact transfers, however, depend on exactly how the surplus
from the coalition is divided. It seems extremely unlikely that poor municipalities would havemore bargaining power than rich municipalities, and thus an “equitable” distribution of thesurplus seems to be the most optimistic scenario. A more pessimistic scenario would give
most of the bargaining power to richer municipalities, with a resulting increase in post-merger
inequality.
The nucleolus is used as a “best case” equitable division. It is the allocation which maximizes over all potential deviating coalitions S 0 the smallest difference between the amountallocated to the members of S 0 and that which they could obtain if they deviated. 100
separate draws of  were performed, and the resulting nucleoli were averaged. This wascompared to the no-transfer case, using the same 100 draws of . The results are showngraphically in Figure 9. Poorer municipalities are worse off, relative to the no-transfers case,while richer municipalities are much better off. Thus, making transfers feasible may or may
not be optimal for the national government, depending on its social welfare function.

5

Conclusion

This paper estimated the parameters determining preferences in a cooperative form political coalition formation game, using two different sets of assumptions and definitions of thesolution set.17 The results are consistent with intuition, and are used to examine potential
17

One potential incentive for mergers that is not considered in this paper is the gappei tokurei sai, specialbond issues allowed for municipalities planning amalgamation. Currently, the estimation strategy assumes
that these bonds, subsidized by the national government, exactly eliminate any direct financial cost of
merging, such as the construction of a new city hall. It is widely believed, though, that these merger bonds
allowed significant capital expenditures beyond the actual costs of amalgamation. This additional financial
incentive to merge could bias estimates, and an attempt will be made to take it into account in futureversions of this paper. This point is due to Prof. Miyazaki.

24alternative national government policies.18 Counterfactual simulations suggest that an alternative incentive scheme that rewarded relatively rich municipalities for merging would haveresulted in welfare improvements under most reasonable social welfare functions. Allowing
transfers to be negotiated between municipalities may or may not be superior, dependingon the national government’s aversion to inequality and the bargaining power of the various
municipalities. The latter is likely unknown to the national government, and thus even if
transfers between municipalities could be enforced, it may not be beneficial for the national
government to do so. Of course, further work could consider other alternative national
government policies.19
At least as important as the implications to government policy, however, is the methodology developed. A coalition formation game without transfers accurately describes many
real-world phenomena, but it is rarely estimated in the empirical literature. As the priceof computing power decreases, however, the number of uses of this sort of model that arefeasible should increase. Although the game presented in this paper could be estimated only
because the geographical nature of the data permitted a large number of possible coalitions
to be discarded, in the future such restrictions should be less necessary. The results given
above, then, are hopefully only the first of many applications of coalition formation modelsof this type to actual data.

18

Technically income yi should be income of the median voter, but right now mean income is used. It
would be possible to estimate the median income, given the data that is available, but this has not yet beendone.
19
For example, one possibility that was not considered in this paper is that of a tax on negotiated transfers.
In the simplest price control model, a tax redistributed to the consumer should be able to mimic a pricecontrol, but with the assurance that the consumers with the highest willingness to pay obtain the good. To
the extent that the inability to make transfers is like a price control at zero, then, it could be that the optimal
policy for the government – rather than specifying a fixed incentive scheme to encourage rich municipalities
to merge with their neighbours – would be to allow transfers, but tax them heavily and redistribute therevenue obtained to the poorest municipalities. Overall, the problem bears some resemblance to the classic
rent control problem.

25

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Figure 1:

Prefectures of Japan

Shizuokaµ0 0.5 1

29

2

3

Decimal Degrees
4Table 1: Dependent variable is c(gS ), cost of providing general services (’96-’97 fiscal year)(Intercept)
POPULATION

I
1294.6(23.0)
136.4
(0.3)

II
808.4
(24.4)
136.0
(0.3)
4.3
(0.1)
0.4
(4.8)
−1070.4(69.0)
324.1(54.9)

III
834.3(25.2)
136.6
(0.3)
3.6
(0.1)
0.3
(4.9)
−779.8(104.3)369.8(54.2)

3220

3216

X
3216

AREA
INCOME.INEQ
INCOME
IS.CITY
POP*INCOME.INEQPOP*INCOME
POP*IS.CITY
PREFECTURE
N

IV
792.2
(27.1)
142.3
(1.7)
3.8
(0.1)
− 20.9(4.3)−164.9(69.1)− 16.2(59.2)
1.1(0.1)− 30.5
(1.0)
5.4
(2.2)
3216

V
902.7
(21.2)
142.5
(1.3)
2.9
(0.1)
− 12.4(3.3)−483.4(79.8)
295.4(48.1)
0.2
(0.1)
− 8.6
(1.5)
− 1.7
(1.7)
X
3216

Units: U1,000,000 (roughly $10,000) per year. POPULATION is in thousands of residents, AREA is in square kilometers, INCOME is in U1,000,000 per capita per year, INCOME.INEQ is the coefficient of variation of income, IS.CITY is a dummy variable coded as
1 if the municipality in question is a city, and zero if it is a village or town. PREFECTURE
is a set of dummy variables for each of the 47 prefectures, with the restriction that the sumof the coefficients on these variables must equal zero. Designated cities and special wards
are excluded from the regression because they have additional responsibilities devolved fromthe prefectural governments, and thus have higher (and non-comprable) expenditures per
capita.

30

Table 2: Dependent variable is c(gS ), cost of providing general services
fiscal year
’96-’97 ’06-’07
899.9
582.2
(43.9) (59.5)
129.4
131.5
(0.5)
(0.6)
4.6
4.6
(0.2)
(0.2)
1194
1194

(Intercept)
POPULATION
AREA
N

Units: U1,000,000 (roughly $10,000) per year. POPULATION is in thousands of residents,
AREA is in square kilometers, designated cities and special wards are excluded as in Table 1.
The sample is further restricted to those municipalities that did not participate in a merger
in order to have the same sample in both periods. Thus, the change in coefficients representsa change in national government transfer policy on the same group of municipalities during
the period in question. Inflation during this period was negligible.
Table 3: Dependent variable is ui (S), utility to municipality i from merger S

log(gs )log(POPULATIONS )log(AREAS )log(INCOMES )IS.MERGERS
σrestricted
preferences
1.00(0.05)−0.06(0.01)−0.13(0.01)
0.05(0.03)− 0.18(0.01)
0.35

relaxed stability
requirements
1.00(0.001)− 0.12(0.0004)
0.01(0.0004)
1.58(0.001)− 0.001(0.001)
0.05

The coefficient on government services (gS ) is normalized to 1, with the standard deviation
σ of the error term (S and iS , respectively) determined by this normalization. INCOME
is income per capita. IS.MERGER is a dummy variable equal to 0 if the coalition S isa singleton (and thus would not imply a merger) and 1 otherwise. The standard errors
presented in the second column assume that the error introduced by numerically estimating
the size of the solution set is not important.
31

Figure 2:"Standard Financial Need" of Japanese Municipalities('96−'97 fiscal year)~c i(g) (¥100,000,000s ≈ $1,000,000s)

1000

800

600

400

200

200

400

Population (1000s of residents)

32

600

Figure 3:

10

5

Pi

~
c i(g)(¥100,000s ≈ $1000s per capita)"Standard Financial Need"(Per capita, log scale)

1
1

10

100

Population (1000s of residents)

33

1000

Figure 4:

Decimal Degrees
0.2

1995 borders

4

2006 borders

Shizuoka Prefecture

0 0.05 0.1

34Figure 5: Hanamatsu City in Red

344
●

207
●

215
●

201
●

361
488

220

210

●

●

●

428

●

●

342

381
382
383

●

●

206
325
341
203
205
●

●

●

●

●

204

●

●

487
●

427

326
321
327

●

●

●

462

●

●

486
●

●

426
●

323 322
●

401
214
217
522
209
461
212
425
218 485 213
523 521402
424
202
216 446 423
211
221
484
505
481442 445422
503
502
483482 447
444

●

●

●

●

●

●

329208
●

●

324
●

●

328
●

●

●

●

●

●

●

●

●

307
306
●

●

●

●

●

●

●

305
●

●

●

●

●

●

●

●

302

●

●

301

●

219

●

●

●

●

●

●

●

304
●

421
●

This connected graph represents Shizuoka Prefecture (also shown in Figure 4).
Hanamatsu City is 22202.

35

“Old”

Figure 6: Random Contiguous Coalition in Red

344
●

207
●

215
●

201
●

361
488

220

210

●

●

●

428

●

●

342

381
382
383

●

●

206
325
341
203
205
●

●

●

●

●

204

●

●

487
●

427

326
321
327

●

●

●

462

●

●

486
●

●

426
●

323 322
●

401
214
217
522
209
461
212
425
485
218
213
523 521402
424
202
216 446 423
211
221
484
505
481442 445422
503
502
483482 447
444

●

●

●

●

●

●

329208
●

●

324
●

●

328
●

●

●

●

●

●

●

●

●

307
306
●

●

●

●

●

●

●

305
●

●

●

●

●

●

●

●

302

●

●

301

●

219

●

●

●

●

●

●

●

304
●

421
●

This connected graph represents Shizuoka Prefecture (also shown in Figure 4).

36

Figure 7:

Effect of Alternative Policy by Quantile●

Difference in utility at quantile (alternative − actual)

0.3

●

0.2

●

●

●
●

●●

●

●

●

●
●
●

●
●
●●
●
●
●
●
●
●●

●

0.1

●

●

●
●
● ●
● ●●● ● ●●
●
●● ●
●
●
●
●●
● ●
●

40

60

●
●●● ● ● ●●
●
●

0.0

●
●●
●
●
●

●

●

●

●●
●
●
● ● ●
●● ●

●

●

●●
●

●
●
●● ●●

●

●
●

●

●

●
●
●

−0.1

−0.2
●

0

20

Utility quantile
Change in distribution of utility, weighting all municipalities equally.

37

80

100

Figure 8:

Effect of Alternative Policy by Quantile (Population Weighted)●

Difference in utility at quantile (alternative − actual)

0.8
●

0.6
●

●

●

0.4
●
●
●
●

●
●

0.2

●
●
●

●
●●
●
●

●
●●
●
●
●
● ●●
●
●
●●
●●
●
●
●
●
●
●
●
● ●● ●●●●● ●●● ● ●
●● ●

●

●

●

●

●

●

●
●

●
●
●

●
●

●

●
●

●

●
●
●

●

●

●

●

0.0

●

●
●

●●

●

●

● ●●●●●●

●

●
●

●

●

0

20

40

60

80

Utility quantile
Change in distribution of utility, weighting municipalities by population.

38

100

Figure 9:

Utility change from allowing transfers

4

U(with transfers) − U(no transfers)

2

0

−2

−4

−6

−0.5

0.0

0.5

log(Income per capita)
The slope is statistically significant (t=4), although this is not taking into account thatthe data is generated from a simulation and the simulation itself is based on estimatedcoefficients.

39