Schulz, Norbert

English

Research Papers in Economics

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Schulz, Norbert
Working Paper
Thoughts on the nature of vetoes when
bargaining on public projects
Würzburg economic papers, No. 17
Provided in cooperation with:
Julius-Maximilians-Universität Würzburg
Suggested citation: Schulz, Norbert (2000) : Thoughts on the nature of vetoes when bargaining
on public projects, Würzburg economic papers, No. 17, http://hdl.handle.net/10419/484691
Nr. 17
Thoughts on the Nature of Vetoes When
Bargaining on Public Projects
Norbert Schulz
May 2000
Universität Würzburg
Lehrstuhl für Volkswirtschaftslehre,
insbesondere Industrieökonomik
Sanderring 2, D-97070 Würzburg
norbert.schulz@mail.uni-wuerzburg.de
Tel.: +49-931-312961
W. E. P.
Würzburg Economic Papers2
Thoughts on the Nature of Vetoes When Bargaining on Public Projects
Norbert Schulz
What is a veto? Usually a veto is connected to a proposal. A veto can prevent the proposal
from being carried out. If the proposal has some positive value to an agent it is not in his
interest to veto. This poses the question what positive value means. This in turn calls for a
comparison with some alternative. Often this is the status quo. If the proposal yields higher
value according to the preferences of an agent when compared to the status quo there will
be no veto. But this refers to a take it or leave it situation. Even if one agent is better off
with the proposal when compared with the status quo he might oppose to the proposal if
there is a better alternative available in his view. If after rejecting the proposal he and
others have a right to suggest another version of a proposal, he will do so.
In principle the situation is then one of bargaining. Consider therefore for a moment the
bargaining about the division of a cake of fixed size among two agents. If the status quo
gives the lowest possible utility for both agents any take it or leave it proposal will be
accepted by both agents at least as long as the proposal specifies a positive share for both
of them. If one agent can reject the proposal and suggest another he does not necessarily
accept the offer. In the context of a potentially infinite number of bargaining rounds a
specific equilibrium occurs (Rubenstein (1982)). This is due to the fact that any agent can
veto the proposal made by the other agent and make a counter proposal in the next round.
This leads to a more lenient choice of the share claimed by the first proposer. If this share
is too high the other agent will not accept. Note that the result of the bargaining process is a
Nash equilibrium of the underlying game. If we ask the question which proposal will be
accepted , i.e. which one is not confronted by a veto of at least one agent, in a bargaining
process, it is thus natural to look at a Nash equilibrium of the “proposal game”.
In the following we will direct our attention to cost sharing of a public project. Note that as
long as the project size is fixed we could apply the noncooperative bargaining approaches
just outlined. For simplicity let V be the value of the public project to each of the two
agents. Let the cost be 1. Denote by xi the share which i covers. Then the payoffs can be
written as V – 1 + 1 – x1 for agent 1 and V – 1 + 1 – x2 for agent 2. Agent 1 tries to
minimize x1 which is the same as to say he tries to maximize x2. Rearranging the names of
the agents gives the same structure as the original bargaining game for the division of a
fixed sized cake.3
The contributions to the literature concerning this problem without any deficits in
information have assumed an infinite number of bargaining rounds. Indeed looking at the
one shot variant of the game yields a continuum of Nash equilibria. If one agent offers a
share this proposal perfectly determines the other agents best response: the remaining
share. In a one period context the share implicitely proposed by the opponent is the
maximizing choice given the share of the opponent. This fact drives the multiplicity of
equilibria. In our context the cake is not fixed. It is the size of the public project which is
variable. Therefore an agent being confronted with a proposal for financing the project has
the option to propose a different sized project. For example if the current proposal appears
to impose too much cost on one agent he will not accept and propose a smaller sized
project. This provides an idea to model the result of a bargaining process: It is a Nash
equlibrium of the proposal game. As long as a proposition is made as to who has to finance
how much an agent will not accept it, if given the proposed contributions of the remaining
agents his own proposed contribution does not maximize his payoff. In this case, he will
make a counteroffer. Or stated inversely: an agent will accept a proposal of contributions
iff his own proposed contribution is his best response to the proposed contributions of the
remaining agents. A Nash equlibrium is often described as a situation where no agent has
an incentive to deviate unilaterally. An incentive to deviate is an incentive to impose a
veto. Hence, it is indeed quite natural to model the "veto-free" result of a bargaining
process as a Nash equilibrium. In such an equilibrium the agents do not coordinate their
incentives. Therefore this may lead to Pareto inferior solutions.
Comparing with the Chen/Ordeshook paper this view of a proposal game is quite different.
In their context each proposal survives potential vetoes if it is Pareto superior. This not so
in our approach. A Pareto superior proposal may be blocked by some agent because given
the contributions of the remaining agents he can improve his situation by a different
proposal.
Consider a situation where some committee consisting of representatives of n regions have
to decide on the level of a public good. Suppose that the public good influences the utility
of the residents of all regions alike and that the chances of being reelected are the higher
the higher this utility is. The committee has a budget R which it can spend on the public
good or for favors to special interest groups. Favors to the latter increase the chances of
being reelected as well. Suppose that without the public good regions would distribute the
budget equally among themselves. Hence, each region is "entitled" to a share R/n.4
A proposal for the public good consists of levels of contributions xi of the representatives
for this good. These contributions are no longer available for favors to special interest
groups. The utility of a representative can than be written as
) ( ) (
1
i
n
i
i i x
n
R
v x u − + ∑ θ
=
,
where u denotes the aggregate utility of residents derived from the public good. This can be
taken as measure of the potential of voters for the representatives. As each representative is
only elected by the local residents only fraction of the total potential is relevant in his
decision making. This is denoted by θ i. v denotes the utility derived from serving special
interests, e.g. in order to solicit support for an election campaign. Both u and v are assumed
to be concave.
Suppose that a certain proposal is made with respect to the size of the public good, X. To
make things simple suppose that the contributions of all regions are always equal, although
this can be derived as a result, if θ i does not depend on i. This can be interpreted as a
situation where X is subtracted from the total budget R and the rest of the budget can be
spend on favors.
The central assumption is: a representative will veto against such a proposal, if the own
contribution is not maximizing his utility given the contributions of the remaining regions.
If his maximizing contribution is smaller he will argue for a reduced size of the public
good and vice versa. The maximizing size is characterized by the following first order
conditions:
0 ) ( )
1
( = − ′ −
−
+ ′ θ i i x
n
R
v X
n
n
x u ,
where we have assumed that all representatives are identical in terms of θ . Therefore no
proposal has a chance of being passed, which does not satisfy
(1) 0 ) ( ) ( = − ′ − ′ θ
n
X
n
R
v X u
By concavity of u and v the left hand side is a decreasing function of X. If X = 0, we assume
that the left hand side is positive:
) ( ) 0 (
n
R
v u ′ > ′ θ5
Hence, it is assumed that it is more helpful winning an election if some means are spend on
the public good rather than spending all money on interest groups. Likewise we assume
that spending all money on the public good is worse than spending at least some on favors:
) 0 ( ) ( v
n
R
u ′ < ′ θ .
This implies that the first order condition has exactly one solution, X
nc. Incidentally, it is
the Nash equilibrium of the contribution game, where each representative simultaneously
proposes a contribution.
Using the implicit function theorem, the solution of (1) is decreasing in n:
0 ) ( ) (
1
) (
2 =
− − ′′ +
∂
∂





 − ′′ + ′′ θ
n
X R
n
X R
v
n
X
n
X R
v
n
X u
Therefore, an increasing number of representatives leads to a lower level of public goods.
The utility level of a representative is therefore
) ( ) (
n
X R
v X u
−
+ θ ,
where X denotes the solution of (1). Now suppose, that the representatives collude. This
means that they would try to maximize this expression by their choice of X. Then the first
order condition is
(2) 0 ) (
1
) ( =
− ′ − ′ θ
n
X R
v
n
X u
which by comparing with (1) yields a higher solution, X
c. Hence, the egoistic concerns of
the individual representatives harm the provision of the public good. Given that the other
representatives contribute according to (2) a specific representative will veto such a level as
it is higher than the one that maximizes his utility.
If the proposal comes from outside the committee any proposal above X
c will be blocked
by the committee, if the representatives recognize their joint power . Note that X
c is
decreasing in n (apply the implicit function theorem to (2)). If the egoistic concerns of the
representatives cannot be coordinated, any proposal above X
nc will be vetoed by some
member of the committee.6
Given the complete symmetry of representatives, abolishing the unanimity rule and opting
for a majority rule would not change anything as long as symmetric contributions are used.
The interests of all representatives are symmetric. If one has a reason to veto, each
representative has one too. However, if representatives are different, e.g. in terms of θ i, this
will turn out differently. The intuition is quite clear: A low θ i means that the public good is
not that important to the representative. Given a right to veto he will only accept a quite
low size of the public good. In a majority vote this can be overturned. Note however, that
in the case of asymmetric representatives contributions have to be asymmetric as well (just
look at (1)). This implies that a proposal would not only specify the size of the public good
but also a financing scheme. This could be the case, if the project is proposed by a central
government together with a proposal to finance it out of federal state finances.
Rather than elaborating on this idea at the present stage I would like to extend the analysis
on a project of a different type. Here the contributions of the regions are in kind rather than
purely financial. What I have in mind is e.g. a system of highways. If one region makes a
highway available, this also increases the value of the highway of another region. This
implies that the contributions are complements to each other rather than substitutes as in
the situation above. I want to test whether the conclusions derived above are robust in this
sense.
Suppose that the proposal consists in contributions for each region xi and that it generates
utility
) , , , ( 2 1 n x x x u ! ,
where u increases in each argument, is concave, and has positive mixed second partial
derivatives. Assume also that u is symmetric in the sense that the first partial derivatives
are equal for all i if evaluated at a common level of xi = x, i = 1, 2, ..., n.
As above suppose the interests of representatives can be modelled as
) ( ) , , , ( 2 1 i n i x
n
R
v x x x u − + θ ! .
Again suppose that a representative will vote against any proposal, if given the
contributions of the remaining representatives the proposal does not correspond to the
maximizing choice of xi . This choice is characterized by the first order condition7
(3) 0 ) ( ) , , , ( 2 1 = − ′ −
∂
∂
θ i n
i
x
n
R
v x x x
x
u
! ,
where we concentrate on the symmetric case: θ  does not depend on i. It is therefore
sensible to concentrate on symmetric solutions:
(4) 0 ) ( ) , , , ( = − ′ −
∂
∂
θ x
n
R
v x x x
x
u
i
!
Differentiating with respect to x gives
) (
1
2
x
n
R
v
x x
u n
j j i
− ′′ + ∑
∂ ∂
∂
θ
=
.
Assuming that the sum of second partial derivatives is negative, which is consistent with
the concavity assumption, implies that the left hand side of (4) is decreasing in x. Imposing
similar assumptions as above yields a unique solution of (4)
Suppose now that the regions are not represented by one representative each but by a
smaller number. Suppose that k regions are represented by one representative and that there
are m of them. So km = n. Suppose that the utility of such a representative is
) ( ) , , , ( 2 1 ∑ − + θ
∈ k i
i n i x
n
kR
v x x x u ! .
This implies the first order conditions
0 ) ( ) ,..., , ( = 




 − ′ −
∂
∂
θ x
n
R
k v x x x
x
u
i
,
if we concentrate again on the symmetric case. The left hand side decreases in x and
increases in k. Hence a higher number of k yield a higher value of x. In this sense, the result
obtained above is robust. A smaller number of potential vetoes yields a higher supply of the
public good.
What happens, if all n representatives coordinate? Then the first order conditions are
0 ) (
1
= − ′ − ∑
∂
∂
θ
=
x
n
R
v
x
u n
j j
.
As by assumption the derivatives are all equal this yields:8
0 ) (
1
= − ′ −
∂
∂
θ x
n
R
v
n x
u
i
This parallels (2) exactly. Hence coordination leads to a higher provision of the public
good. The same holds true if only m representatives coordinate. Their coordinating
equilibrium would be characterized by
0 ) (
1
= 




 − ′ −
∂
∂
θ x
n
R
k v
n x
u
i
.
From this it follows that x increases in k.
Finally note that the results do not depend on the status of strategic subsitutes versus
complements. Note also that the above results generalize the results on the underprovision
of public goods. As has been suggested at least since Olson (1965) an increasing number of
agents enhances the problem of underprovision of public goods. The above analysis
supports this view. It should be noted, however, that such a result is dependent on the
alternative use of funds relative to contributions to a public good and on the structure of
preferences. For differing results with respect to the impact of group size on the problem of
underprovision see e.g. Cornes and Sandler (1984).
References:
Chen, Yan and Peter C. Ordeshook (1998): Veto Games: Spatial Committees under
Unanimity Rule, Public Choice, vol. 97, pp. 617-643
Cornes, Richard and Todd Sandler (1984): Easy Riders, Joint Production, and Public
Goods, Economic Journal, vol 94, pp. 580-598
Olson; Mancur (1965): The Logic of Collective Action, Harvard University Press
Rubinstein, A. (1982): Perfect Equlibrium in a Barganining Model, Econometrica, vol 50,
pp.97-1099
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Steffen Meyer
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Länderfinanzausgleich
04 Horst Entorf Der deutsche Aktienmarkt, der Dollar und der Aussenhandel
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16 Timo Wollmershäuser ESZB-Devisenbestand - quo vadis?
Further information:
http://www.wifak.uni-wuerzburg.de/vwl1/wephome.htm

A comment on Yin, Xiangkan and Yew-kwang Ng: quantity precomment and Bertrand competition

and Timo Wollmershäuser Exchange rate policies for the EU accession countries (and other emerging market economies)

and Timo Wollmershäuser Monetary policy and exchange rate targeting in open economies

Capacity constrained price competition and entry deterrence in heterogeneous product markets

Currency and financial crises - lessons from the Asian crises for China?

Ebert und Steffen Meyer Die Berücksichtigung der Gemeindefinanzen im Länderfinanzausgleich

Entorf Der deutsche Aktienmarkt, der Dollar und der Aussenhandel

Loef und Hans G. Monissen Monetary Policy and monetary reform: Irving Fisher's contributions to monetary macroeconomics

Monissen Explorations of the Laffer curve

Monissen Knut Wicksell und die moderne Makroökonomik

Monissen Reflections on the optimal size of government

Pension reform during the demographic transition

Promotional banks as an instrument for improving the financing situation of small and medium sized enterprises in the transition economies of Central and

The monetary policy of the ECB: pretence and reality

Third-degree price discrimination in an oligopolistic market

Timo Wollmershäuser ESZB-Devisenbestand - quo vadis? Further information: http://www.wifak.uni-wuerzburg.de/vwl1/wephome.htm

Veto Games: Spatial Committees under Unanimity Rule,

Thoughts on the nature of vetoes when bargaining on public projects

http://econstor.eu/bitstream/10419/48469/1/570611733.pdf

Thoughts on the nature of vetoes when bargaining on public projects

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