Pigouvian Taxes: A Strategic Approach

Abstract

This paper analyzes the problem of designing mechanisms to implement efficient solutions in economies with externalities. We provide two simple mechanisms implementing the Pigouvian Social Choice Correspondence in environments in which coalitions can or cannot be formed. Copyright 1999 by Blackwell Publishing Inc.

Full text

Pigouvian Taxes: A Strategic Approach
JOSÊ ALCALDE
University of Alicante
LUIS C. CORCHÓN
University Carlos III, Madrid
BERNARDO MORENO
University of Málaga
Abstract
This paper analyzes the problem of designing mechanisms to imple-
ment efficient solutions in economies with externalities. We pro-
vide two simple mechanisms implementing the Pigouvian Social
Choice Correspondence in environments in which coalitions can
or cannot be formed.
1. Introduction
This paper studies the problem of implementing efficient allocations in
economies with externalities. In these environments, it is well known that
competitive equilibrium fails to be Pareto efficient. Pigou ~1920!pro-
posed solving this problem by establishing a tax system ~called Pigouvian
taxes!in such a way that when agents are price and tax takers a Pareto
efficient allocation is attained.
1
A classical criticism of Pigou’s approach is that the computation of
Pigouvian taxes requires precise knowledge of utility and production
José Alcalde, University of Alicante ~alcalde@merlin fae.ua.es!. Luis C. Corchón, University
Carlos III, Madrid, 126 28903 Getafe, Madrid, Spain ~lcorchon@eco.uc3m.es!. Bernardo
Moreno, University of Málaga.
We are grateful to Salvador Barberà, Carmen Beviá, Martin Peitz, William Thomson, two
anonymous referees, an associated editor, and participants in the Primer Congrès de la
Catalunya Central for very useful comments on a preliminary version of the paper. Alcalde’s
work is partially supported by DGCYT under project PB 94 –1504. Corchón acknowledges
financial support from DGC YT’s project PB 93–0940. The authors also acknowledge finan-
cial support from the Institut Valencià d’Investigacions Económiques.
Received 1 July 1997; Accepted 14 June 1998.
1
Other approaches dealing with the problem of externalities include the classical contribu-
tions by Coase ~1960!and Arrow ~1970!.
1

functions—information that is not in the hands of the regulator. This is
the typical problem dealt with in the Theory of Implementation. In this
framework Varian ~1994!presents a two-stage compensation mechanism imple-
menting in subgame perfect Nash equilibria ~SPE!the Pigouvian Social
Choice Correspondence ~PSCC!, that is, the mapping between the set of
admissible economies and the allocations that are a Pigouvian equilibrium
for the corresponding economy. Varian’s compensation mechanism for
the two-agent case is such that at the first stage each agent announces the
Pigouvian tax for the other agent. The agent generating the externality is
penalized whenever the sum of the Pigouvian taxes is not balanced. At the
second stage, this agent selects the output level.
The compensation mechanism provides a nice solution to the prob-
lem of implementing the PSCC when agents’ preferences are strictly con-
vex. However, this mechanism presents two difficulties. First, in economies
in which one agent has linear preferences, there are SPE of this mecha-
nism yielding allocations that are not Pareto efficient ~see Examples 2.8
and 2.9 below!. Second, and more important, the solution concept used
by Varian, namely SPE, cannot be easily extended to environments in
which coalitions can be formed. We believe that an extension to these
environments is crucial because of their relevance to potential applications.
2
In this paper we present two “simple” mechanisms that implement,
even for linear economies, the PSCC for different equilibrium concepts.
The first mechanism implements the PSCC in strong Nash equilibria.
An important difference between Varian’s mechanism and ours is that
output is not decided by a single individual. As we will see in Examples 2.8
and 2.9, the reason that Pareto inefficient allocations can be supported by
SPE in Varian’s compensation mechanism is that output is decided by a
single individual. In order to avoid this problem, our mechanism deter-
mines the output level through the interaction of all agents. In order to
interpret this procedure, let us think of the level of externality as pollu-
tion. The quantity of pollution must equal the quantity of pollution per-
mits, which are determined by the interaction of all agents. Assuming that
the quantity of pollution can be monitored without cost, and that pollu-
tion and output are in fixed proportions, it follows that output is ~indi-
rectly!determined by the quantity of pollution permits.
2
In the case considered in this paper, Pigouvian taxes can be interpreted as Lindahl prices
of the pollution. Therefore, a Pigouvian equilibrium can be seen as a Lindahl equilibrium.
Nakamura ~1988!presents a feasible and continuous mechanism ~at the cost of the simplicity
of the mechanism!implementing Pigouvian allocations in Nash equilibria but not in strong
equilibria. Walker ~1981!presents a mechanism implementing Lindahl allocations in Nash
equilibria but not in strong equilibria. Peleg ~1996!presents a mechanism implementing the
Lindahl correspondence in Nash and strong equilibria for the case in which all strategically
active agents have strictly increasing preferences in all goods. In our model, agent 0 has
strictly increasing preferences in the public good but preferences for the rest of the agents
are strictly decreasing in this good. Thus, Peleg’s mechanism cannot be directly applied to
our framework.
2

The second mechanism doubly implements the PSCC in Nash and
strong Nash equilibria. This mechanism is more complicated than the
first one but it has the additional advantage that implementation occurs
in both Nash and strong Nash equilibria. Therefore, the implementation
of the desired result does not depend on the possibilities of coalition
formation.
The two mechanisms are simple. Agents announce allocations and
prices, and the mechanisms mimic the role of competitive markets. The
first mechanism is continuous but there are Nash equilibria yielding allo-
cations that are not Pareto efficient. The second mechanism is discontin-
uous but it can be made continuous at the cost of complicating the
mechanism. Thus, the trade-off between the two mechanisms is that of
simplicity versus robustness of the equilibrium concept.
The rest of the paper is organized as follows. Section 2 presents the
model. Implementation of Pigouvian SCC in strong Nash equilibria using
a continuous mechanism is reported in Section 3. Section 4 presents a
mechanism that doubly implements the Pigouvian SCC in Nash and strong
Nash equilibria.
2. The Model
Let us consider the following externality problem involving n11 agents.
Let I5$0,1,...,n%be the set of agents. Agent 0 consumes qunits of a
good generating an external effect on the other agents. Agents’ prefer-
ences are representable by a utility function, u
i
, which depends on two
variables. The first one, T
i
[R, plays the role of ~a transfer of!numér-
aire,
3
whereas the second, q[R
1
, measures the consumption of the good
that generates the externality. For simplicity, we assume that each u
i
is
quasilinear, but all of our results are still valid in the case of nonquasilin-
ear utility functions. Thus, agent i’s utility function can be expressed as
u
i
~T
i
,q!5T
i
1v
i
~q!,
where v
i
is concave, strictly increasing for agent 0, and strictly decreasing
for the other agents.
4
Alternatively, for interpretative convenience, we
3
In order to simplify the presentation, we assume that agents can incur unbounded debts.
An alternative assumption is that T
i
is nonnegative and agents prefer any feasible allocation
to any unfeasible allocation. In any case, the mechanisms presented in this paper can be
made feasible at the cost of some extra complications.
4
As was pointed out by Starrett ~1972!, externalities introduce nonconvexities that may
preclude the existence of a Pigouvian equilibrium. In our case, for qlarge enough, agents
1,...,nwould suffer a damage larger than any possible payoff they could obtain. In such a
case these agents would like to shut down business ~or move to a different location!and buy
an infinite amount of pollution permits. The existence of a Pigouvian equilibrium can be
restored if pollution cannot exceed a suitably chosen upper bound for which agents 1,...,n
wish to remain active. See Boyd and Conley ~1997!for an alternative interpretation of this
issue.
3

may write T
i
5t
i
q. In such a case, t
i
is interpreted as a per-unit tax. An
economy is a list e5~u
0
,u
1
,..,u
n
!. Let Edenote the set of all admissible
economies. Given an economy e, we say that an allocation z 5~T
0
,T
1
,...,T
n
,q!
is feasible for eif q[R
1
and (
i[I
T
i
50. Let Zdenote the set of all
feasible allocations. In order to simplify notation, we extend preferences
~and utility functions!over allocations in the following way. We say that
agent iweakly prefers allocation z5~T
0
,...,T
n
,q!to z
'
5~T
0
'
,...,T
n
'
,q
'
!
iff u
i
~T
i
,q!≥u
i
~T
i
'
,q
'
!. In this case, we will also write u
i
~z!≥u
i
~z
'
!.
The properties we introduce next are minimal conditions that should
be satisfied by any solution to be considered satisfactory. The minimal
requirements for an allocation are
1. Pareto efficiency. Given an economy e, we say that a feasible alloca-
tion zis efficient if there is no feasible allocation z
'
that is weakly
preferred to zby all the individuals, and strictly preferred by at
least one of the agents. That is, z[Zis Pareto efficient if there is
no z
'
[Zsuch that, u
i
~z
'
!≥u
i
~z!for all i[I, and u
j
~z
'
!.u
j
~z!
for some j[I.
2. Individual rationality. Given an economy e, we say that a feasible
allocation zis individually rational if it yields no fewer payoffs to all
agents than the allocation where neither exchange nor production
occurs. An allocation z[Zis individually rational whenever u
i
~z!≥
u
i
~0,0!for all i[I.
We are interested in social choice correspondences satisfying these two
minimal conditions. It is well known that, in a competitive equilibrium,
the level of output chosen by agent 0 fails to be Pareto efficient and
individually rational. Pigou ~1920!pointed out a solution to this problem—
intervention by a regulator who imposes a tax system. The idea of estab-
lishing a tax system to solve the problem of externalities is credited to
Pigou; however, he did not address the problem of distributing the income
generated by these taxes.
DEFINITION 2.1: The Pigouvian Social Choice Correspondence (PSCC),f
p
:Er
R
n
3R
1
,associates to each economy, e 5~u
0
,u
1
,..,u
n
![E, a tax0transfer
system, ~t
i
*
!
i[I
[R
n
, and an output q
*
[R
1
such that both of the following
hold:
1. q
*
maximizes u
i
~T
i
,q!5u
i
~t
i
*
q,q!5t
i
*
q1v
i
~q!for all i [I
2. (
i[I
t
i
*
50.
Note that the quantity qmay be interpreted as a public good produced by
a firm ~agent 0!and paid by all agents. The only difference with the
standard public good case is that utility functions of agents 1 to nare
decreasing on q. In such a case, given an economy e5~u
0
,u
1
,..,u
n
![E,t
i
is agent i’s individualized price of the public good, and condition 2 in
Definition 2.1 represents the usual way of defining the price at which the
firm sells the public good.
4

The PSCC has good properties. For instance, it is Pareto efficient and
individually rational. However, in many cases the regulator does not have
access to the information needed to compute the Pigouvian taxes. This is
where implementation theory comes into the picture. The following con-
cepts are standard in the literature.
DEFINITION 2.2: A mechanism G, is a list of strategy spaces, M 5~M
i
!
i[I
, and
an outcome function, f :3
i[I
M
i
rR
n
3R
1
.
Given e5~u
0
,u
1
,..,u
n
![E, let ~G,e!denote a normal form game. Let xbe
an equilibrium concept ~i.e., Nash equilibrium, etc.!and x~G,e!be the set
of x-equilibria of the game ~G,e!. In this paper we focus our attention on
Nash and strong Nash equilibria.
DEFINITION 2.3: A profile of messages m 5~m
0
,m
1
,...,m
n
![M is a Nash
equilibrium of the game ~G,e!if for all i [I and for all m
i
'
[M
i
, we have
that u
i
~f~m
i
,m
2i
!! ≥u
i
~f~m
i
'
,m
2i
!!.Let N~G,e!be the set of Nash equi-
libria of the game ~G,e!.
DEFINITION 2.4: A profile of messages m 5~m
0
,m
1
,...,m
n
![M is a strong
Nash equilibrium of the game ~G,e!if there is no S #I and m
S
'
[3
i[S
M
i
,
satisfying u
i
~f~m
S
'
,m
2S
!!.u
i
~f~m
S
,m
2S
!! for all i [S.Let S ~G,e!be the
set of strong Nash equilibria of the game ~G,e!.
DEFINITION 2.5: The mechanism Gimplements the PSCC f
P
:ErR
n
3R
1
in
x-equilibrium if for all e [Ewith x~G,e!ÞB,f~x~G,e!! 5f
P
~e!.
Let xand x
'
be two concepts of equilibrium, and x~G,e!~respectively,
x
'
~G,e!! be the set of x-equilibria ~respectively, x
'
-equilibria!of the game
~G,e!.
DEFINITION 2.6: The mechanism Gdoubly implements the PSCC f
P
:ErR
n
3
R
1
in x-equilibrium and x
'
-equilibrium if for all e [Ewith x~G,e!ÞB,
and x
'
~G,e!ÞB,f~x~G,e!! 5f~x
'
~G,e!! 5f
P
~e!.
Let us introduce Varian’s ~1994!compensation mechanism for the two
agents case.
DEFINITION 2.7 ~The Compensation Mechanism; Varian 1994!:
•Announcement stage: Agents 0 and 1 simultaneously announce the mag-
nitude of the appropriate Pigouvian per-unit tax; denote the announcement
offirm0bym
0
and the announcement of agent 1 by m
1
.
•Choice stage: The regulator makes side payments to the agents so that they
face the following payoff functions:
u
0
~q,m
0
,m
1
!5m
1
q1v
0
~q!2a~m
0
1m
1
!
2
u
1
~q,m
0
,m
1
!5m
0
q1v
1
~q!.
The parameter a.0is of arbitrary magnitude. Agent 0 selects the level of
output q.
5

Thus, both individuals simultaneously select Pigouvian taxes at the
first stage and agent 0 selects the output level at the second stage. Varian
~1994!shows that this mechanism implements in SPE the Pigouvian Social
Choice Correspondence when agents’ preferences are strictly convex. How-
ever, we provide two examples which show that Varian’s results are only
valid when preferences are convex but not strictly convex.
Example 2.8: Agent 0 has linear preferences ~constant returns to scale!.
Consider the following profit functions
v
0
~q!5pq 2cq,p.c.0
v
1
~q!52
d
2q
2
d.0.
The Pigouvian allocation is q
*
5~p2c!0d,t
0
*
5c2p,t
1
*
5p2c. Note
that q
*
5~p2c!0d,m
0
5t
1
*
,m
1
5t
0
*
is an SPE of Varian’s compen-
sation mechanism yielding the Pigouvian allocation. However, setting
[q50, [m
0
5p2c,[m
1
5c2pis a subgame perfect equilibrium because
agent 0’s payoff is identically zero in the second stage.
Example 2.9 ~Linear externality!:
Consider the following functions
v
0
~q!5pq 2Aq
c
,c.1, p,A.0
v
1
~q!52dq,d.0
The Pigouvian allocation is q
*
5@~ p1d!0cA#
10~c21!
,t
0
*
52t
1
*
52d.
Then agent 1’s payoff is zero. Note that q
*
,m
0
5t
1
*
,m
1
5t
0
*
is an SPE
of Varian’s compensation mechanism yielding the Pigouvian alloca-
tion. However, there are SPE where agent 1 chooses any arbitrary m
1
~which yields an undesirable outcome in the second stage!because
agent 1 makes zero payoff in the second stage.
We have seen that the compensation mechanism, in the case of econ-
omies in which exactly one agent has linear preferences, does not solve
the problem of finding a mechanism that implements the Pigouvian SCC.
More seriously, coalition formation cannot be considered because an exten-
sion of SPE to deal with this possibility does not exist. Thus, we look for
new mechanisms in which the problem of coalition formation can be
dealt with.
3. A Continuous Mechanism Implementing the Pigouvian
Correspondence in Strong Nash Equilibria
This section presents a simple and continuous mechanism that imple-
ments, in strong Nash equilibria, the Pigouvian Social Choice Correspon-
dence. For the sake of concreteness, think of the classical example of the
6

papermill ~agent 0!and the fishermen ~agents 1,...,n!. In this context, it
is reasonable to assume that agents know the damages and profits caused
by the papermill output and that they can meet together and discuss the
quantity of pollution permits. Let us introduce the mechanism G
S
that
formalizes the rules under which discussions are organized. Let M
i
5R
2
be the strategy space for each i[I. A strategy for agent i[Iis a pair
m
i
5~m
i
1
,m
i
2
![R
2
. The first component is interpreted as her proposed
Pigouvian tax, m
i
1
. The second component, m
i
2
, is interpreted as the
incremental level of pollution permits. A profile of strategies is a list m5
~m
0
,m
1
,...,m
n
![R
2~n11!
. The outcome function is
f~m
0
,m
1
,...,m
n
!5~t
0
~m!,...,t
n
~m!,q~m!!,~3.1!
where q~m!5max$0, (
i50
n
m
i
2
%and, for each iin I,t
i
~m!5
min$m
i
1
,2(
jÞi
m
j
1
%.
In this mechanism each agent i[Ireceives a compensation ~or pays
a tax!of T
i
~m!5t
i
~m!q~m!. Her payoff is t
i
~m!q~m!1v
i
~q~m!!. Thus, in
mechanism G
S
the quantity of pollution permits is unanimously decided
by all agents and each agent’s tax is given by the taxes announced by the
others, except when she states a “high” tax ~or a “low” subsidy!for herself.
A strong Nash equilibrium of the game formed by this mechanism and
the payoff functions of each agent yields a self-enforcing agreement that
no group of agents would like to renegotiate.
5
We first state a result connecting strong Nash and Nash equilibria for
this mechanism.
LEMMA 3.1: [m5~[m
0
,[m
1
,..., [m
n
!is a strong Nash equilibrium of the above
mechanism if and only if ~i![m is a Nash equilibrium and ~ii!(
i50
n
[m
i
1
50.
Proof: To prove the necessary condition, let [m5~[m
0
,[m
1
,..., [m
n
!be a
strong Nash equilibrium of G
S
which does not satisfy condition ~ii!.
First, consider the case where q~[m!.0. Then consider the case where
each agent plays the strategy Km
i
5~[m
i
1
2(
i[I
@t
i
~[m!0~n11!# ,[m
i
2
!. Since
the transfer received by each agent is larger than before and the level
of qis unchanged, all agents will benefit from this deviation. Notice
that, when condition ~ii!is not satisfied, (
i[I
t
i
~[m!is negative. Second,
suppose that q~[m!50. Consider the following agents’ strategies. For
agent 0, Km
0
5~2(
i51
n
t
i
,q!and for each i51,...,n,Km
i
5~(
jÞi
t
j
,0!,
where qand t
i
,i51,...,n, are positive real numbers satisfying
q(
i51
n
t
i
,v
0
~q!2v
0
~0!. Notice that, by construction, all agents prefer
the outcome associated with these strategies rather that the equilib-
rium outcome. A contradiction.
In order to prove the sufficient condition, let [mbe a Nash equi-
librium satisfying condition ~ii!in Lemma 3.1. Suppose this is not a
strong Nash equilibrium, so there is a coalition S#Nand strategies
5
Notice that we do not assume the outcome of the meeting can be written in the form of a
contract that can be enforced by courts, as in the famous Coase Theorem.
7

Km
s
[M
S
such that u
i
~T
i
~Km
s
,[m
2s
!,q~Km
s
,[m
2s
!! .u
i
~T
i
~[m!,q~[m!! for
each i[S. Let us consider the following cases:
~a!t
i
~Km
s
,[m
2s
!5t
i
~[m!for some i[S. Then, u
i
~t
i
~Km
s
,[m
2s
!q,q!5
u
i
~t
i
~[m!q,q!for all q[R
1
,soq~[m![arg max
q
u
i
~t
i
~Km
s
,[m
2s
!q,q!
if, and only if [q[arg max
q
u
i
~t
i
~[m!q,q!. A contradiction.
~b!t
i
~Km
s
,[m
2s
!Þt
i
~[m!for all i[S. Since (
j[I
t
j
~Km
s
,[m
2s
!≤05
(
j[I
t
j
~[m!, there should be an agent i[Ssuch that t
i
~Km
s
,[m
2s
!,
t
i
~[m!. Therefore, u
i
~t
i
~[m!q~Km
s
,[m
2s
!,q~Km
s
,[m
2s
!! ≥u
i
~t
i
~Km
s
,[m
2s
!3
q~Km
s
,[m
2s
!,q~Km
s
,[m
2s
!! .u
i
~t
i
~[m!q~[m!,q~[m!!. Note that for agent i
there is a strategy Um
i
such that q~Um
i
,[m
2i
!5q~Km
s
,[m
2s
!. Since [mis
a Nash equilibrium, we get a contradiction. n
The main result in this section is given in the following theorem.
THEOREM 3.2: The mechanism G
S
implements in strong Nash equilibria the
Pigouvian Social Choice Correspondence.
Proof: Let e5~u
0
,u
1
,..,u
n
![Ebe an economy. We first prove that every
Pigouvian equilibrium for ecan be supported by a strong Nash equi-
librium of the game ~G,e!.
Let ~T
0
*
,...,T
n
*
,q
*
![R
n
3R
1
be a Pigouvian allocation for e.We
show that the strategy profile Um, where Um
i
5~t
i
*
,~q
*
0n11!! and t
i
*
5
T
i
*
0q
*
for each i[I, is a strong Nash equilibrium for ~G
S
,e!yielding
this allocation. Note that, given other agents’ strategies, no unilateral
deviation by an agent can force a lower tax ~higher subsidy!t
i
for
her. Therefore, each agent i[Iselects her incremental output level
m
i
2
so to maximize her payoff t
i
*
~m
i
2
1@nq
*
0~n11!# ! 1v
i
~m
i
2
1@nq
*
0
~n11!# !. Since ~T
0
*
,...,T
n
*
,q
*
!is a Pigouvian allocation for e,Um
i
2
maximizes the function above. Thus Umis a Nash equilibrium of ~G
S
,e!.
Since (
i[I
Um
i
1
50, by Lemma 3.1, Umis a strong Nash equilibrium of
~G
S
,e!.
On the other hand, let Um[R
2~n11!
be a strong Nash equilibrium of
~G
S
,e!. By Lemma 3.1, Umis a Nash equilibrium of ~G
S
,e!and (
i[I
Um
i
1
50.
Since Umis a Nash equilibrium of ~G
S
,e!,q~Um!maximizes t
i
~Um!q1v
i
~q!
for each i[I. Given that (
i[I
Um
i
1
50, each agent i[Iwill behave as
a price-taker because she can not decrease her own tax. Therefore,
~t
1
~Um!q~Um!,...,t
n
~Um!q~Um!,q~Um!! is a Pigouvian allocation for e.n
4. Double Implementation of the Pigouvian Correspondence
This section introduces a mechanism that doubly implements the Pigou-
vian Social Choice Correspondence in Nash and strong Nash equilibria. It
is related to the one presented by Corchón and Wilkie ~1996!. The moti-
vation for this mechanism is that in some cases the planner lacks infor-
mation not only about agents’ preferences but also about the coalition
8

formation possibilities. Thus we need a mechanism that copes with cases
in which coalitions can or cannot be formed.
To simplify the presentation we introduce a mechanism that is discon-
tinuous. Nevertheless, the mechanism can be made continuous by making
the proper modifications ~see Corchón and Wilkie 1996!. Thus the trade-
off between the mechanism presented in Section 3 and the one in this
section is that of simplicity versus a more robust concept of implementation.
We introduce the mechanism G
H
. Let M
i
5R
2
be the strategy space
for each i[I. A strategy for agent i[Iis a pair m
i
5~m
i
1
,m
i
2
![R
2
. For
each i[I,m
i
1
is the tax ~or subsidy!she proposes for herself, and m
i
2
is an
incremental quantity on the production level. A profile of strategies is
written as Km5~Km
0
,Km
1
,..., Km
n
![R
2~n11!
. The outcome function is given
by
f~Km!5~T
0
~Km!,...,T
n
~Km!,q~Km!! ~4.1!
with q~Km!5max$0, (
i50
n
m
i
2
%if (
j[I
m
j
1
50 and q~Km!50 in any other case,
and T
i
~Km!5m
i
1
q~Km!if (
j[I
m
j
1
50 and 2g6(
i50
n
m
i
1
6otherwise, where the
parameter g.0 can be chosen to be arbitrarily small.
A natural interpretation can be provided for this mechanism. Each
agent announces her proposed Pigouvian tax, m
i
1
. The second component
for each agent’s strategy, m
i
2
, is the incremental level of pollution permits.
The output level is q~Km!and she receives a compensation ~or pays a tax!
of T
i
~Km!. So her payoff will be T
i
~Km!1v
i
~q~Km!!. This compensation depends
on two factors. First, if the announced taxes balance the sum of transfers
among agents, each agent’s compensation is q~Km!times the tax she
announced. Second, if such a balance does not hold, no output is pro-
duced and each agent is penalized.
We next introduce our main result of this section.
THEOREM 4.1: The mechanism G
H
doubly implements in Nash and strong Nash
equilibria the Pigouvian Social Choice Correspondence.
Proof: Let e5~u
0
,u
1
,..,u
n
![Ebe an economy. We will first prove that
every Pigouvian equilibrium can be supported by a strong Nash equi-
librium of the game ~G
H
,e!.
Let ~T
0
*
,...,T
n
*
,q
*
![R
n
3R
1
be a Pigouvian allocation for e.We
show that the strategy profile Um, where Um
i
5~t
i
*
,@q
*
0~n11!# ! and t
i
*
5
~T
i
*
0q
*
!for each i[I, is a strong Nash equilibrium of ~G
H
,e!yielding
such an allocation.
First, it is clear that such strategies yield the desired outcome.
Now we show that Um[S~G
H
,e!. Consider a coalition S#Iand a
deviation [m
S
of agents in this coalition. If Um
i
1
5[m
i
1
for all i[S, any
agent in Scan get any q
'
[R
1
at a price t
i
*
. By definition of a
Pigouvian allocation the agents cannot improve payoffs with any q
'
Þ
q
*
.If Um
i
1
Þ[m
i
1
for some i[S, we should consider two cases.
9

~i!If (
i[S
[m
i
1
1(
j[I\S
Um
j
1
Þ0, then q~[m
S
,Um
2S
!50. Suppose that
agents in Sobtain a higher payoff. But then for all i[S
v
i
~0!.2g
*
(
i[S
[m
i
1
1(
j[I\S
Um
j
1
*
1v
i
~0!.t
i
*
q
*
1v
i
~q
*
!,
contradicting that ~T
0
*
,...,T
n
*
,q
*
!is a Pigouvian allocation for e.
~ii!If (
i[S
[m
i
1
1(
j[I\S
Um
j
1
50, then (
i[S
[m
i
1
5(
i[S
Um
i
1
. In this case,
there should be an agent i[Ssuch that [m
i
1
,Um
i
1
. Suppose that
igets a higher payoff. Since [m
i
1
,Um
i
1
, we have that
u
i
~Um
i
1
q~[m
S
,Um
2S
!,q~[m
S
,Um
2S
!! ≥u
i
~[m
i
1
q~[m
S
,Um
2S
!,q~[m
S
,Um
2S
!!
.u
i
~Um
i
1
q~Um!,q~Um!!,
which contradicts that ~T
0
*
,...,T
n
*
,q
*
!is a Pigouvian allocation for
e. Thus, Umis a strong Nash equilibrium and hence a Nash equi-
librium.
On the other hand, let [m5~[m
0
,[m
1
,..., [m
n
![N~G
H
,e!.We
show that this yields a Pigouvian allocation for e. Consider two
cases.
Case A: (
i[I
[m
i
1
Þ0. In such a case, for each agent there is a strategy,
namely Um
i
5~2(
jÞi
[m
j
1
,2(
jÞi
[m
j
2
!yielding q~Um
i
,[m
2i
!5
T
i
~Um
i
,[m
2i
!50, a contradiction.
Case B: (
i[I
[m
i
1
50. But then, given [m
2i
, each agent is a price-taker
so the outcome associated with the Nash equilibrium is a Pigou-
vian allocation for e.
To conclude the proof, let [m5~[m
0
,[m
1
,..., [m
n
![S~G
H
,e!. Since
S~G
H
,e!#N~G
H
,e!,[mis a Nash equilibrium of this game. Therefore,
its outcome is a Pigouvian allocation for e.n
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Pigouvian Taxes 281
11