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J. Dreze
CORE, Université Catholique
de Louvain, Belgium,
M. Le Breton
Universite de Toulouse I –
GREMAQ and IDEI, France,
A. Savvateev
New Economic School,
S. Weber
Southern Methodist University, USA
A PROBLEM
OF FOOTBALL
BARS: VERTICALLY
AND HORIZONTALLY
DIFFERENTIATED
PUBLIC GOODS
Recent decades, there was a continuing interest in problems and challenges
arising from the necessity to provide club goods (i.e. excludable local public goods)
for the population's need. Literature tackled several distinct approaches, such as
«voting by your feet» tradition starting with seminal paper [Tiebout, 1956], vertical
conflict as explained in [Jehiel, Scotchmer, 1997], and horizontal differentiation
tradition initiated by [Mas-Colell, 1980]. Let me overview these in turn.
[Tiebout, 1956] suggested decentralization of the different tastes problem by
creating multiple communities in which public good was supplied in various differing
qualities, quantities and characteristics; along this, tax schemes and the extension of
the tax burden were different. Of these communities, every citizen were to choose
the best suitable for his/her tastes. Therefore, social optimality was conjectured to
be reached in a decentralized way by people' decisions to migrate and free mobility
assumption.
This approach was rather tentative, without a proper mathematical explanation
(Tiebout himself was a geographer). Quite a number of alternative models were
designed in order to prove or reject the Tiebout's hypothesis. Up to now, however,
there is no unified mathematical approach convincing scientists in the correctness of
the «voting by your feet» approach.
One of the major critique is devoted to the free mobility assumption. Looking
at the Europe and especially Russia, one can notice that population is extremely
immobile. At the same time, people usually move quite far to visit public places of
their interest, such as music clubs, football stadiums and bars, swimming pools etc.
needless to mention hospitals, local administration and the like.
This creates alternative view to think about local public goods. Namely, we
can assume population to habit in the fixed manner, according to a certain distribution either on the line or, more probable, on the plane. Then, given the distribution
of the demand towards a certain public activity such as watching football, swim-
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ming or whatever, the society as a whole should decide upon how many centers to
create in which this demand is to be met, where to place these centers, how to allocate the population between the centers, and the not least, how to finance the cost of
this economic activity through the contributions of the customers-users of the activity.
Of these, the first three questions are rather straightforward and belong to the
theory of optimization. In each distinct situation, the optimal number, addresses and
assignment scheme could be calculated, as it is done in various approaches starting
with [Christaller, 1933]. The most questionable is the last point. Namely, how to
allocate cost in such a way that nobody is dissatisfied with his/her share of monetary
contribution?
And here, two alternative traditions exist. First looks for the solutions which
are individually rational, hence, Nash-stable, and this appears to be not too demanding [Konishi, 1998; Haimanko, 2004]. The second view of stability admits «coalitional threats», i.e. cooperative rejection of citizens when they form groups of dissatisfied users to start up a new jurisdiction with the newly-located center, once
such an opportunistic behavior honors all the members of the jurisdiction-to-be.
On this way, literature experienced the two big waves. Until [Bogomolnaia et
al., 2005a, 2005b], scientists believed that there exist a possibility to allocate costs
such as to preclude any incentives to secede and form new jurisdictions [Alesina,
Spolaore, 1997; Le Breton, Weber, 2003]. The specifics of all these models was that
the geographical dimension was assumed to be 1, namely, that people lived on the line.
Moreover, either the distribution on the line was uniform [Alesina, Spolaore, 1997],
or the space of feasible allocation rules was large enough [Le Breton, Weber, 2003].
However, once we get rid of uniformity and transferable utility paradigm simultaneously [Weber, Zamir, 1985; Bogomolnaia et al., 2005a, 2005b], or just of unidimentionality [Dreze et al., 2007], the existence of coalition-proof allocation
schemes was illustrated to fail. One of the possible solutions to this negative information is to finance a (small) part of costs from outside, say by the government, but
only in case people follow the prescribed optimal agreement on the utilization of
public goods; once some group wishes to secede, their members pay the whole burden.
In [Dreze et al., 2007], it is shown that, at least in the uniform distribution
case, the share needed to be covered is remarkably small, namely it is less than 0,002.
This observation gives us a right to tell that the problem is not serious, after all.
Things however stop to look that optimistic once vertical differentiation of
tastes enters this story. Indeed, consider football: some of us need to watch it every
week, every match of our favorite team, whereas some others ignore football altogether or, which is the most problematic case, watch say only the national team's
performances. What will change in this vertically-horisontally differentiated story?
[Jehiel, Scotchmer, 1997] convinced us that in the case of purely vertical differentiation, the problem of stability is not that severe. More surprisingly then is the
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fact that, as we demonstrate below, given horizontal differentiation, even a slight
difference in tastes precludes the existence of the core allocation rules (i.e. rules that
do not allow for secession). And the degree of the gap between optimality and stability is times higher then 0,002 obtained in the two-dimentional case.
Indeed, even in the case of the uniform distribution of the total demand on the
line and the just two different types of demand, the degree of instability is higher
than the 0,4 number! In this case, it is difficult to find sources to finance such a gap.
This calls for alternative approaches to solve the efficiency-stability problem.
Next section specifies economic environment in which we expect the problem
of nonexistence of coalition-proof cost allocation schemes to be pronounced.
The model and its main outcomes.
Now, let me briefly present the mathematical model. Consider a society of
football fans that are populating a very long linear city, say the whole line [Dreze et
al., 2005]. The distribution of citizens is uniform, and so is the total demand for
watching football games. People watch games in bars (or pubs), the latter being
located in a vertices of a net to be constructed.
Besides horizontal differentiation inherently presented in the model introduced, there is another dimension of heterogeneity, namely, different citizens need
different frequencies to watch football, as explained above. Denote the (inelastic,
per unit of time) demand expressed by a given ciziten by , and we identify any
citizen with the two characteristics, (t, ), where t is his address in the city (i.e. location, or coordinate, on the real line). If a citizen (t, ) visits a bar located at the point
m, his transport cost is equal to ∙|t – m|.
Now, we define optimal net of bars as the one which minimizes per capita total
cost, including aggregated transportation costs and the costs for maintaining the net
of bars. The latter one is assumed to be equal to the fixed cost g of one bar times the
total number of bars. The average cost is defined as the limit of the average cost over
the very big segment of that real line. We prove the following «folk theorem» result.
Result
 The optimal organization of the society into «football jurisdictions» using
one and the same bar, consists of equal-sized connected intervals on the line.
 The size is determined by the per location total demand , and fixed cost
g of maintaining any given bar. (Note that the optimal size of the jurisdiction does
not depend in any way on the distribution F(∙) of this total demand  among classes
of football fans with equal intensity of preferences.)
 Within any jurisdiction, citizens are assigned to one and the same bar
placed at the center of the corresponding interval.
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Now, about allocation schemes that support this optimal organization.
Theorem 1.
 Except when all the citizens have the same intensity  of preferences towards football, there does not exist any coalition-proof allocation scheme.
 For every distribution F(∙), the «discrepancy index», δ could be calculated
which points to the minimal possible per unit of the tax burden, per capita subsidy
which secure stability.
 Given this δ, there exists the UNIQUE allocation rule in the δ-core, i.e. the
δ-secession-proof allocation rule.
 This allocation rule is Rawlsian, which means that it is purely Egalitarian –
every citizen ends up with the same total cost, transportation plus tax burden.
Example.
Consider the case when there are two demand groups within the population:
real fans, which watch football with intensity  = 2, and modest fans, with intensity
equal to 1. We calculated the index δ for this case, and it is equal to
Quite a number!
2  1  0,4 .
To conclude, let me give an apparent prospect to our findings. Namely, here
we assumed that the demand for football is perfectly inelastic. The obvious new step is
to admit for varying demand, and see what happens in this case. Is it true that the
problem of instability will appear to be even more severe, or the opposite is true –
the most problematic case is the one with exogenously given, invariant fixed demand?
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