The social cost problem, rights and the (non)empty core
Stéphane Gonzaleza,∗, Alain Marcianob , Philippe Solala
a
Université de Saint-Etienne, CNRS, GATE L-SE UMR 5824, F-42023 Saint-Etienne, France
b
Université de Montpellier, CNRS UMR 5774 LAMETA, France
Abstract
The “Coase theorem” is made of the efficiency and the neutrality theses. Using cooperative
game theory, we show that these two theses are not compatible: there exist only two types
of rights assignments that guarantee a nonempty core. Thus, the efficiency thesis holds –
there exist (two types of) rights assignments under which the core of the game is nonempty
– but the neutrality thesis does not hold – for all other rights assignments the core is empty.
This complements the results found in the literature on Coase and the core. Our paper
also adds two more results. Indeed, we add two principles that are not discussed in the
literature about the Coase theorem: a democratic principle for rights assignments and
a fairness principle for the monetary payoffs distributed to the agents, that we translate
into two natural properties for a solution. We show, and this is our second main result,
that the nonempty core requirement is not compatible with our democratic and fairness
properties. Thus, under democratic and fairness criteria, the efficiency thesis does not
hold.
Keywords: Coase theorem – Core – Neutrality and efficiency theses – Rights –
(Im)possibility results.
JEL: C7 – D2 – K1
1. Introduction
The objective of this article is to demonstrate new results about the so-called “Coase
theorem” George Stigler “invented” in 1966. More precisely, to understand our approach,
∗
Corresponding author: Stéphane Gonzalez, [email protected].
For helpful comments received, the authors want to thank the participants of the 11th SING conference
in Odense, july 2016. Financial support by the National Agency for Research (ANR) – research program
“DynaMITE: Dynamic Matching and Interactions: Theory and Experiments”, contract ANR-13-BSHS10010 – and the “Mathématiques de la décision pour l’ingénierie physique et sociale” (MODMAD) project
is gratefully acknowledged. Stéphane Gonzalez acknowledges the support of the Programme Avenir Lyon
Saint-Etienne of Université de Lyon, within the program “Investissement d’Avenir”(ANR-11-IDEX-0007).
Alain Marciano also acknowledges the support of the National Agency for Research (ANR) – research
program “Investissements d’avenir”, contract ANR-10-LABX-l l-0l.
Email addresses: [email protected] (Stéphane Gonzalez),
[email protected] (Alain Marciano), [email protected] (Philippe
Solal)
Preprint submitted to Elsevier
November 10, 2016
one must recall that the “Coase theorem” is made of two theses. First, the efficiency thesis
– according to which in the absence of transaction costs and if property rights are well
defined, individuals will always bargain to reach an optimal allocation of resources and that
corresponds to Stigler’s claim that “[t]he Coase theorem thus asserts that under perfect
competition private and social costs will be equal” (1966, 113). Second, the invariance
or neutrality thesis – according to which the outcome of bargaining is independent of the
distribution of rights and that can also be traced back to Stigler who wrote that “[t]he
manner in which the law assigns liability will not affect the relative private marginal costs
of cattle and grain” (1966, 113), or “no matter where the law places the liability for
damages” (1966, 113).1 Thus, stricto sensu, the “Coase theorem” holds if both theses are
verified simultaneously. This is what we study in this paper.
Let us start by describing the cooperative model we use in this paper. This will allow
us to re-phrase the two theses and then present our results.
In our model, we consider a large class of social cost problems in which are involved
a fixed and finite set of at least two victims and one polluter. As always, the activity
of the latter creates damages that affect the former. More precisely, we assume that
those damages can be described by nondecreasing functions on the set of activity levels
of the polluter. In addition, the benefit function of the polluter is also assumed to be
nondecreasing on his or her set of activity levels. Therefore, victims and polluter have
conflicting interests. In order to iron those conflicts out and to solve the problem of social
cost, a negotiation will take place with the objective to sign a binding agreement about
how much activity the polluter will be able to undertake and how much compensation
victims should receive. Now, we interpret the possibility (permission) granted to each
group of agents who want to form a coalition to sign such binding agreements about the
level of activity of the polluter as a right; these rights form a mapping of rights. We
then assume that a coalition that signs a binding agreement always achieves efficiency:
the agreement allows the coalition to select the best contract for itself. Thus, to put
it in other words, a right for a coalition is here interpreted as the permission to sign
efficient agreements for itself without consideration for the interest of nonmembers. We
further assume that mappings of rights should satisfy natural conditions. The first one
is a condition of sovereignty saying that the grand coalition is entitled to sign binding
agreements with the polluter. The second one is a condition of monotonicity saying that if
a coalition of agents is entitled to negotiate binding agreements, then every larger coalition
inherits this right. The third property is a condition of effectivity of the law saying that if
a coalition is entitled to sign efficient binding agreements for itself, then the nonmembers
cannot form a coalition to prevent this coalition from exercising its rights. The last
property is a condition of independence of law from the economic characteristics, saying
that the right to control an issue is independent on the benefit or damages functions of
its members, though the set of feasible agreements depend on these functions. From each
1
Over the years, the “Coase theorem” has formulated and re-formulated many times. But the version
that remains particularly clear was the immediately post-Stigler one put forward by Calabresi who wrote
that “[i]f people are rational, bargains are costless, and there are no legal impediments to bargains, transactions will ex hypothesis occur to the point where bargains can no longer improve the situation; to the
point, in short, of optimal resource allocation” (1968, 68).
2
of these mappings and the benefit and damage functions of the agents, we construct a
cooperative game with transferable utility. Therefore, a solution for the class of social
cost problems is a mapping formed by a mapping of rights and a monetary payoff vector
for each social cost problem.
In this framework, the two theses that constitute the “Coase theorem” can be expressed
as follows. The efficient thesis means that, given a social cost problem and a rights
assignment, agents are able to sign an agreement that maximizes the social welfare and
that no coalition of agents can rationally block this agreement. Or, to put it in the terms of
cooperative game theory, the core of the cooperative game associated with the social cost
problem and the rights assignment is nonempty2 – a property that we call Core stability.
And the neutrality thesis thus means that, given a social cost problem, the core of the
cooperative game is nonempty whatever the rights assignment.
Then, the question of the validity of the “Coase theorem” relates to the validity of
both theses simultaneously. We then show that both theses are not compatible. This is
the first main result we establish. We demonstrate that the neutrality thesis is not valid
in a general setting but that Core stability holds only under specific rights assignments
and also characterize those rights assignments under which the core is nonempty. More
precisely (see Proposition 6), we demonstrate that, for each social cost problem, Core
stability is satisfied if and only if either the polluter is not liable for the damages his or her
activity generates or, to the contrary, if the polluter is liable for the damages he or she can
generate and the entire set victims is the only coalition that has the right to sign binding
agreements with the polluter. This means that a (core) stable negotiated outcome to a
problem of social cost exists but this outcome is not independent from the distribution of
rights. Even if bargaining clearly does not fail – since the core is nonempty –, the “Coase
theorem”, stricto sensu, does not hold – the neutrality thesis is not satisfied.
Our second main result relates two additional dimensions – democracy and fairness –
that we introduce to supplement the discussion of Core stability, which is the main issue
discussed in the literature. First, the democratic nature of the set of feasible multilateral
negotiations is a consequence of one of the characteristics of our model, namely the possibly large number of victims affected by pollution. Indeed, in this type of situations, a
victim would have the power to unilaterally veto negotiations between the polluter and
the other victims if, when leaving the grand coalition, this victim would prevent or block
any possibility for the others to negotiate a contract with the polluter. We claim that
avoiding this kind of veto power for a victim is a minimum democratic requirement to
guarantee. To this end, we introduce a condition of no veto power for a victim, that can
be characterized as follows: a distribution of rights satisfies the property of No veto power
for a victim if any coalition formed by the polluter and the entire set of victims except one
retains its right to negotiate binding agreements. Then, second, we also take into account
fairness. Clearly, victims of accidents, pollution or, more broadly, negative externalities
suffer from losses while, in the meantime, the polluter earns benefits. This clearly, as it
is well known, comes from the fact that the polluter does not bear the social costs that
2
This formulation is standard in the cooperative approach of the “Coase theorem”. For more details
on this strand of literature, we refer the reader to section 4 of the article.
3
result from his or her actions. We argue that, to guarantee fairness, victims should be
fully compensated for their losses. We thus introduce a property of full compensation: a
solution satisfies the property of Full compensation if the payoff component of the solution is a nonnegative vector; this ensures that the victims are fully compensated for the
damages and that the polluter as an interest to produce. Then, and this constitutes our
second main result (Proposition 5), there is no solution on the class of social cost problems
which satisfies simultaneously Full compensation, No veto power for a victim and Core
stability. Complementarily, we show that the efficiency thesis is consistent with either
democracy or fairness. Proposition 3 establishes that there is only one mapping of rights
which allows to reconcile No veto power for a victim and Core stability; it consists in
distributing rights to all coalitions in which the polluter can belong to, which amount to
saying that the polluter is not considered as liable for the damages he or she can generate. And, Proposition 7 shows that there are only two mappings of rights which allow to
reconcile Full compensation and Core stability. They consist in distributing rights to the
full set of victims and/or the grand coalition only, meaning that unanimity is required to
negotiate with the polluter.
The article is organized as follows. Section 2 sets up our formal apparatus and introduce the three properties for a solution to the social cost problems. Section 3 contains our
main results. Section 4 describes our contribution’s relation to the existing literature on
the “Coase theorem”, and also clarifies a few points about our methodology.
2. Preliminaries
2.1. Cooperative games
Let N be a fixed and finite set of n agents. Subsets of N are called coalitions, while N
is called the grand coalition. A cooperative game with transferable utility or simply a
TU-game is a pair (N, v), where v is a function v : 2N −→ R such that v(∅) = 0. For each
coalition S ⊆ N , v(S) describes the worth of the coalition S when its members cooperate.
The worth v(S) is thus what the members of S can accomplish together without the
assistance of members of N \ S. Let V be the class of all TU-games. A payoff vector
x ∈ Rn of a TU-game (N, v) ∈ V is an n-dimensional vector giving a payoff xi ∈ R to each
agent i ∈ N . For each S ⊆ N and each x ∈ Rn , denote by x(S) the sum of the payoffs xi ,
i ∈ S. A payoff vector is efficient with respect to (N, v) if x(N ) = v(N ); it is coalitionally
rational if x(S) ≥ v(S) for every possible coalition S.
The core of (N, v) ∈ V, denoted by C(N, v), is the set, possibly empty, of efficient and
coalitionally rational payoff vectors:
n
C(N, v) = x ∈ R : ∀S ⊆ N, x(S) ≥ v(S) and x(N ) = v(N ) .
(1)
The interpretation of the core is that no group of agents has an incentive to split from the
grand coalition N and form a smaller coalition S since they collectively receive at least as
much as what they can obtain for themselves as coalition.
The so-called Bondareva-Shapley theorem (Bondareva, 1962; Shapley, 1967) provides a
sufficient and necessary condition under which the core of a TU-game is nonempty. First,
4
we introduce the concept of balanced map. A balanced map λ : 2N −→ [0, 1] is such that:
X
λ(∅) = 0, and ∀i ∈ N,
λ(S) = 1.
S3i
Denote by B(N ) the set of balanced maps over N .
Proposition 1 (Bondareva-Shapley Theorem)
For each (N, v) ∈ V, C(N, v) 6= ∅ if and only if for each balanced map λ ∈ B(N ), the
following inequality holds:
X
λ(S)v(S) ≤ v(N ).
(2)
S⊆N
2.2. Social cost problems
As mentioned in the introduction, we analyze situations in which there are one polluter
and more than one potential victim affected by the activity of the polluter. This means
that we consider a finite and fixed set of agents M ∪ {p} of size m + 1, where m ≥ 2.
Agent p undertakes an activity at level z ∈ Z that generates a private benefit Bp (z) ≥ 0.
The private benefit function Bp : Z −→ R is nonnegative and nondecreasing on a subset
Z ⊆ R containing 0. The production of z ∈ Z by p generates also a negative externality.
This external damage potentially affects each agent i ∈ M through the function Di (z).
Precisely, for each i ∈ M , the external damage function Di : Z −→ R is nonnegative and
nondecreasing on Z. We further assume that if p is not active, i.e. z = 0, then, for each
i ∈ M , Di (0) = Bp (0) = 0.
A social cost problem P on M ∪ {p} is described by a tuple (Z, Bp , (Di )i∈M ). Let P
be the class of all social cost problems P on the finite set of agents M ∪ {p}.
∗ ,
Let P = (Z, Bp , (Di )i∈M ) ∈ P. For each nonempty coalition S ⊆ N , denote by ZP,S
the set of socially optimal levels of production for coalition S. We have:
X
∗
ZP,S = arg max 1p (S)Bp (z) −
Di (z) ,
z∈Z
i∈S\{p}
where 1p (S) = 1 if S 3 p, and 1P (S) = 0 otherwise. For each P and each S, we assume
∗
is nonempty. This requirement is satisfied, for instance, if for each P , Z is a
that ZP,S
finite set, or Bp and Di , i ∈ M are continuous on a compact subset Z of R. In the sequel,
∗ , to
for P ∈ P, and each non empty coalition S, we will use the notation zS∗ , instead of zP,S
∗
∗
∗
represent an element of ZP,S . Denote by ZP the union of sets ZP,S , S ⊆ M ∪ {p}, S 6= ∅.
∗
Note that 0 ∈ ZP,S
whenever S 63 p.
Example 1 In many applications the benefit function is concave and the external damage
functions are convex. A specific instance of such social problems is the social cost problem
√
P̃ where Z = [0, 1], Bp (z) = z, and for i ∈ M , Di (z) = ci z for some ci > 0.
5
2.3. Coalitions and rights assignment
A consequence of having more than two (potential or actual) victims is that the latter
can perfectly well form a coalition with the objective to sign an agreement regarding the
level of activity and, accordingly of pollution, of p. This is very different from what Coase
analyzed, and it is closer to the kind of analysis led by Tullock (1958) or Buchanan (1961;
see also Buchanan and Tullock, 1962, chapters 10 and 11). Our perspective also differs
from the one adopted in most of the literature on Coase and the core – where there is only
one victim. One of the rare exceptions is Peter Bernholz (1997), who took into account
what he names an organization and that is very close to our coalitions. In effect, in our
model a coalition corresponds to what Bernholz named an organization in his analysis,
that is a “subset of society . . . with more than one member . . . [and] . . . having the right
to control an issue” (1997, 422)
There are nonetheless differences with Bernholz’s approach. The first and main one is
that we do not analyze the internal functioning of coalitions and the contracts that should
be signed to avoid opportunistic behaviors. We are interested in the impact of rights on
pollution and on the activity of the polluter. This means that, in our analysis, “the right
to control an issue” that characterizes a coalition is not the consequence of the internal
structure of the coalition but is the consequence of a certain distribution of rights that is
given ex ante to the agents. To put it in other words, the set of agreements that a coalition
can sign is determined by the distribution of property rights. Second, we assume that the
polluter p can be part of the coalition, whereas in Bernholz’s model there is no polluter
distinct from the victims. In our model, the population is partitioned into a polluter
and a set of victims which suffer from the activity of the polluter. This assumption is of
particular importance for the rest of our analysis but it is perfectly legitimate: polluters
and pollutees can bargain together to determine how much pollution is acceptable and
will be authorized. We do not mean to say, at this point, that the reduction of pollution
will necessarily result from a bargaining involving polluter and pollutees but we leave this
possibility open.
The assumption that property rights determine economic possibilities is consistent
with the law and economics literature and the idea that one must take the existing legal
structure – the existing distribution of rights – as a starting point for trying to envisage how
to solve this very problem. As James Buchanan put it, “some structure, any structure, of
well-defined rights is a necessary starting point for the potential “trades” that are required
to remove the newly-emergent interdependence.” (1972, 442) This is not only one of –
if not the – most fundamental and well known insights of law and economics analyses,
brought up by Coase (1959, 1960, among others).
Formally, this means that the set of feasible levels of production that a coalition can
∗ }
choose is summarized by a mapping of rights φ which assigns a subset φP (S) ∈ {∅, ZP,S
∗
to each P = (Z, Bp , (Di )i∈M ) ∈ P and each S ⊆ M ∪ {p}. The situation φP (S) = ZP,S
means that S is entitled to control the activity of p either to negotiate a socially optimal
level of production if S 3 p or to prevent p from being active if S 63 p. In case φP (S) = ∅,
coalition S is neither entitled to produce if S 3 p nor entitled to prevent the activity of
p if S 63 p. Thus, φ assigns rights to some coalitions, which allow them to implement an
optimal contract without consideration of the nonmembers of this coalition. This means
6
that, in our framework, a right is viewed as the permission to choose some actions that
reduce the set of possible outcomes in a specific way.
We assume that φ satisfies five independent and natural conditions: for each P =
(Z, Bp , (Di )i∈M ) ∈ P, we have:
(C.1) φP (∅) = ∅;
∗
(C.2) φP (M ∪ {p}) = ZP,M
∪{p} ;
∗ , then, for each T ⊇ S, φ (T ) = Z ∗ ;
(C.3) If φP (S) = ZP,S
P
P,T
∗ , then φ (N \ S) = ∅.
(C.4) If φP (S) = ZP,S
P
(C.5) For each P, P 0 ∈ P and each S ⊆ M ∪ {p}, if φP (S) 6= ∅, then φP 0 (S) 6= ∅.
Condition (C.1) is clear: there is no assignment of rights associated with the empty
coalition.
Condition (C.2) is a condition of sovereignty. It indicates that the grand coalition
M ∪ {p} has always the right to implement a socially optimal level of production for
the whole society. The consequences of this condition differ, depending on the initial
distribution of rights. In the case where p has the right to pollute – is not liable for
the damages its activity can create –, then (C.2) means that all the victims can form a
coalition with p and bargain to reach an agreement; conversely, if the victims have the
right not to be polluted – that is if p is liable for the damages its activity can generate
– then (C.2) means that p can try to form a coalition with all the victims and try to
bargain with them. Condition (C.3) is a monotonicity condition. It is straightforward in
the case of two agents, a tortfeasor and a victim. In such a situation (C.3) means that if
the polluter has the right to pollute, then the coalition made of the polluter and the victim
– with the objective to bargain – has also the right to control pollution; conversely, if the
victim have the right to prevent pollution – by controlling the activity of p –, then the
polluter must form a coalition with the victim to continue its activity. Here, we generalize
it. Thus, if a coalition S has the right to control the activity of the polluter p, then every
coalition T containing S is also entitled to control the activity of p. For instance, assume
that p is liable for the damages generated by his or her activity. If S does not contain
p and S is entitled to control the activity of p, then the members of S have the right to
prevent p to pollute since p does not belong to S and is liable for the damages he or she
can induce. If coalition T ⊇ S contains p, then, by virtue of monotonicity, p is now a
member of a coalition that has the right to control the level of pollution. It follows that
the victims in T and p will bargain to choose an optimal contract for T . Reciprocally, if
p has the right to pollute, any coalition containing p has the right to bargain with p in
order to control the level of pollution.
Condition (C.4) is a condition of effectivity of the law, meaning that if S has the right
to implement an optimal contract without consideration of the members of (M ∪ {p}) \ S,
then coalition (M ∪ {p}) \ S cannot prevent S to exercise its right. Once again, this is
7
obvious when, as it is the case in “The Problem of Social Cost”, there are only two agents:
the tortfeasor and the victim cannot have the same rights simultaneously. Our condition
(C.4) generalizes the case to many agents.
∗ , then φ (T ) = ∅
Then, combining (C.3) and (C.4), we obtain that if φP (S) = ZP,S
P
for each T ⊆ (M ∪ {p}) \ S. That is: if victims have the right not to be polluted, and if
a coalition S of victims is formed to which p does not belong – p ∈ (M ∪ {p}) \ S –, then
p cannot have the right to pollute and the victims have the right to prevent p to pollute.
In that case, the only way for p to go on with its activity is to form the coalition S ∪ {p}
with the victims. The consequence is then that now p will no longer be able to be liable
for the damages its activity generates. Monotonicity allows to transfer the right of the
victims to the polluter. The other victims in M \ S have no right to oppose to p. They
are also obliged to negotiate with the members of S ∪ {p}. Reciprocally, if p has the right
to pollute, then no group of victims has the right not to be polluted; the only way for the
victims to control pollution is to form a coalition with p.
Finally, (C.5) is a condition of independence of law from the economic characteristics.
It indicates that the shape of Bp and Di , i ∈ M , do not influence the right to control the
level of pollution, even if the shape of these functions affects the set of optimal contracts.
∗ , then (C.5), φ 0 (S) = Z ∗
∗
Indeed, if φP (S) = ZP,S
P
P 0 ,S , but ZP,S is not necessarily equal to
ZP∗ 0 ,S .
Let ΦP be the set of all correspondences of rights satisfying (C.1), (C.2), (C.3), (C.4)
and (C.5) that we can construct from P ∈ P.
2.4. Social cost TU-games
The results a coalition can achieve also depend on the behavior of the nonmembers. More
precisely, there are situations in which the behavior of the nonmembers of S affect the
gains of the members of S, and situations in which the behavior of the nonmembers of
S do not affect the gains of the members of S. Then, the worth of a coalition S will
depend on whether or not they believe the nonmembers will behave in a way that will
affect them. This represents a large number of cases. We propose to restrict our attention
to the so-called pessimistic (expectation) rule. This rule, inspired by Aumann (1967) and
discussed by Hart and Kurz (1983), assume that the members of any coalition S believe
or expect that nonmembers S will from the partition which minimizes the worth of S. As
noted by Bloch and van Nouweland (2014), the pessimistic rule is most in line with the
idea that a coalition should consider the worth that it can guarantee itself independent
of the behavior of nonmembers of S - an idea that underlines the very definition of a
TU-game. From this assumption, the TU-game (M ∪ {p}, vφP ) is defined as:
X

∗) −
B
(z
Di (zS∗ )

p
S



i∈S\{p}

X
vφP (S) =
−
max
Di (zT∗ )
T
⊆(M
∪{p})\S



i∈S\{p}


0
if S 3 p and φP (S) 6= ∅,
if S 63 p and φP ((M ∪ {p}) \ S) 6= ∅,
otherwise.
(3)
8
Several comments are in order.
If S 3 p and φP (S) 6= ∅, then the members of S has the right to negotiate an optimal
∗ . By (C.3) and (C.4), for each T ⊆ M \S, it holds that φ (T ) = ∅.
production level in ZS,P
P
So, whatever the partition of M \ S, coalition S can guarantee
X
vφP (S) = Bp (zS∗ ) −
Di (zS∗ ).
i∈S\{p}
If S 63 p and φP ((M ∪ {p}) \ S) 6= ∅, then the polluter p does not cooperate with
the members of coalition S, and there is at least one coalition in (M ∪ {p}) \ S including
p which can cooperate with the polluter to negotiate an optimal level of production for
itself. Under the pessimistic rule, agents in S expect that nonmembers of S will form the
partition which minimizes the worth of S. Because every coalition of the partition not
containing p has no impact on the worth S, the maximal impact of pollution caused by
the nonmembers of S is determined by the coalition T 3 p, T ⊆ (M ∪ {p}) \ S, such that
φP (T ) 6= ∅ and which minimizes the worth of S, i.e.
X
vφP (S) = −
max
Di (zT∗ ).
T ⊆(M ∪{p})\S
i∈S\{p}
Note that this coalition T is necessarily minimal (with respect to set inclusion) among the
coalitions in (M ∪{p})\S which have the right to negotiate an optimal level of production.
In all other cases, either p does not belong to S and φP ((M ∪{p})\S) = ∅, or p belongs
to S and φP (S) = ∅. In both situations p has not the right to pollute or to contract an
optimal level of production with some victims, and so no activity is undertaken. Therefore,
the worth of S is equal to zero.
2.5. Solutions and properties for a solution
A solution to a class of problems of social cost is a pair formed by a mapping of rights
and payoff vectors.
Formally, a (singled-valued) solution on P is a function F defined on P which assigns
a mapping of rights and a payoff vector F (P ) = (φP , xP ) ∈ ΦP × RM ∪{p} to each social
cost problem P ∈ P. The rights assignment part of the solution has been discussed above.
Regarding the “monetary” part of the solution, i.e. the payoff vector xP , the interpretation
is as follows. First, xP,i , i ∈ M , represents the quantity of money received by the victim i
after having suffered a damage represented by Di and having received a transfer from p.
Second, xP,p represents what p obtains after having produced a certain quantity of goods,
received a certain benefit Bp , and transferred a certain amount to the victim. If xP,i ≥ 0
for i ∈ M , then this means that i has been integrally compensated for the activity of p,
and xP,p ≥ 0 means that after having produced at a certain level for a benefit represented
by Bp , and proceeded to the transfers, p obtains a surplus (otherwise, p will not produce).
We now introduce three properties for a solution F on P. The first one is a propriety
of efficiency and stability captured by the core. It indicates that, for each social cost
problem P , acceptable payoff vectors are those who belong to the core of the TU-game
(M ∪ {p}, vφP ). Thus, this property implies that, for each social cost problem, the core of
9
the corresponding TU-game must be nonempty.
Core stability A solution F satisfies Core stability on P if, for each P ∈ P, F (P ) = (φP , xP )
is such that xP ∈ C(M ∪ {p}, vφP ).
The second property contains a principle of democracy, which is captured by the idea
that no victim can veto the negotiations between the rest of the victims and the polluter.
It indicates that if a victim leaves the grand coalition, which is sovereign by hypothesis,
the rest of the victims and the polluter retain their right to negotiate binding agreements.
This property echoes the property of No veto power introduced by Maskin (1999) in the
context of implementation of social choice functions. Note that No veto power for a victim is satisfied by any solution assigning rights to coalitions formed by a majority of the
population, i.e. F is such that, for each P , φP (S) 6= ∅ if |S| > (m + 1)/2.
No veto power for a victim A solution F satisfies No veto power for a victim on P, if, for each
∗
P ∈ P, F (P ) = (φP , xP ) is such that, for each i ∈ M , φP ((M ∪{p})\{i}) = ZP,(M
∪{p})\{i} .
Let us note that, by condition (C.4), No veto power for a victim implies that each
victim, individually, has no right to oppose to other coalitions: for i ∈ M , φP ({i}) = ∅.
The last property reflects a principle of fairness that bears on the amount of compensation that victims receive. Here, fairness is captured by the idea that each victim should
be fully compensated for the damages the polluter generates by his or her activities, and
that the latter earns, having received a certain benefit and transferred a certain amount
of money to each victim, a nonnegative payoff.
Full compensation A solution F satisfies Full compensation on P if, for each P ∈ P,
F (P ) = (φP , xP ) is such that xP ≥ (0, . . . , 0).
3. Results
We begin by proving that when the polluter is not liable for the externalities he or she
generates, the core of the TU-game constructed from this mapping of rights and defined
as in (3) is nonempty whatever the social cost problem. So, let us define, for each social
cost problem P = (Z, Bp , (Di )i∈M ) ∈ P, the mapping of rights φdP ∈ ΦP as follows:
∗
φdP ({p}) = ZP,{p}
and φdP (S) = ∅ if S 63 p.
Each φdP indicates that the polluter p is not liable for the pollution damages and so has
∗
the right to select an optimal level of production in ZP,{p}
. By the monotonicity condition
(C.3), every coalition of victims S containing the polluter p has also the right to select an
∗ , i.e. φ (S) = Z ∗
optimal level of production in ZP,S
P
P,S for each S 3 p. By condition (C.4),
if S 63 p, then S is not entitled to prevent the polluter p to produce, which means that p
can exercise his or her right to pollute. All in all, φdP says that p can veto any coalition
of victims in the sense that if p leaves such a coalition, the latter will suffer the maximal
10
damages. From (3), for each social cost problem P = (Z, Bp , (Di )i∈M ) ∈ P, the TU-game
(M ∪ {p}, vφp ) rewrites as:
P
X

∗) −
B
(z
Di (zS∗ ) if S 3 p.

p
S


X i∈S\{p}
vφd (S) =
(4)
P

Di (zp∗ )
if S 63 p.

 −
i∈S\{p}
Proposition 2 For each P = (Z, Bp , (Di )i∈M ) ∈ P, it holds that C(M ∪ {p}, vφd ) 6= ∅.
To prove Proposition 2 we need an intermediary result.
Lemma 1 Let λ ∈ B(N ) be a balanced map on the agent set N containing at least two
elements. Consider any two distinct agents i and j in N . It holds that:
X
X
λ(S) =
λ(S).
S63i
S3j
S3i
S63j
Proof. Take λ, i and j as hypothesized. We have:
X
1 =
λ(S)
S3i
=
X
λ(S) +
X
λ(S) −
S3j
= 1−
λ(S)
S3i
S63j
S3i
S3j
=
X
X
λ(S) +
S3j
S63i
X
λ(S) +
S3j
S63i
X
λ(S)
S3i
S63j
X
λ(S),
S3i
S63j
and so
X
λ(S) =
X
λ(S),
S63i
S3j
S3i
S63j
as desired.
We are now in position to prove Proposition 2.
Proof.
Pick any P ∈ P. By the Bondareva-Shapley theorem (see Proposition 1), it
suffices to prove that, for each balanced map λ ∈ B(M ∪ {p}), it holds that:
X
λ(S)vφd (S) ≤ vφd (M ∪ {p}),
P
P
S⊆M ∪{p}
11
where (M ∪ {p}, vφd ) is given by (4). So, pick any balanced map λ ∈ B(M ∪ {p}). By
P
definition of (M ∪ {p}, vφd ), we have:
P
X
X
X
λ(S)vφd (S) =
λ(S)vφd (S) +
λ(S)vφd (S)
P
P
P
S63p
S3p
S⊆M ∪{p}
=
X
λ(S)Bp (zS∗ ) −
S3p
=
X
λ(S)Bp (zS∗ )
−
S3p
X
λ(S)
X
S3p
i∈S
i6=p
X
X
λ(S)
S3p
Di (zS∗ ) −
X
λ(S)
S63p
Di (zS∗ )
−
i∈S
i6=p
X
Di (zp∗ )
i∈S
X
Di (zp∗ )
X
λ(S) .
S63p
S3i
i∈M
(5)
Consider the polluter p and any victim i ∈ M . By Lemma 1, we have:
X
X
λ(S) =
λ(S).
S63p
S3i
S3p
S63i
Therefore, equality (5) can be rewritten as:
X
λ(S)vφd (S) =
X
P
X
λ(S)Bp (zS∗ )−
S3p
S⊆M ∪{p}
λ(S)
X
S3p
X
Di (zS∗ )−
i∈S
i6=p
λ(S)
S3p
X
Di (zp∗ )
.
i∈(M ∪{p})\S
(6)
By construction, for each zS∗ ∈ ZP,S ∗ and each zp∗ ∈ ZP,{p} , zp∗ ≥ zS∗ . Because, for each
i ∈ M , Di are nondecreasing functions, zp∗ ≥ zS∗ implies Di (zp∗ ) ≥ Di (zS∗ ). Therefore, from
equality (6), we obtain:
X
X
X
X
X
X
∗
∗
∗
λ(S)vφd (S) =
λ(S)Bp (zS ) −
λ(S)
Di (zS ) −
λ(S)
Di (zp )
P
S3p
S⊆M ∪{p}
≤
X
λ(S)Bp (zS∗ )
−
S3p
=
X
≤
X
S3p
i∈S
i6=p
X
X
λ(S)
S3p
S3p
Di (zS∗ )
i∈S
i6=p
−
X
S3p
i∈(M ∪{p})\S
λ(S)
X
i∈(M ∪{p})\S
X
λ(S) Bp (zS∗ ) −
Di (zS∗ )
S3p
i∈M
X
λ(S) max Bp (z) −
Di (z)
S3p
z∈Z
i∈M
X
= max Bp (z) −
Di (z)
z∈Z
i∈M
= vφd (M ∪ {p}),
P
where
the third equality comes from the fact that, by definition of a balanced map,
P
S3p λ(S) = 1. This completes the proof of Proposition 2.
12
Di (zS∗ )
Define the set of solutions Fd as follows:
F ∈ Fd
if
∀P ∈ P,
F (P ) = (φdP , xP ) for some xP ∈ C(M ∪ {p}, vφd ).
P
Proposition 2 helps to show that the only way to obtain a solution satisfying Core stability
and No veto power for a victim is to pick a solution in Fd .
Proposition 3 The solutions in Fd are the only solutions on P satisfying Core stability
and No veto power for a victim.
Proof. By Proposition 2, Fd 6= ∅. By definition of Fd , any F ∈ Fd satisfies Core stability.
By definition of φd , for each P = (Z, Bp , (Di )i∈M ) ∈ P, φdP ((M ∪{p})\{i}) 6= ∅. Therefore,
any F ∈ Fd satisfies No veto power for a victim.
It remains to show that F 6∈ Fd violates Core stability or No veto power for a victim. It
suffices to show that if F 6∈ Fd satisfies No veto power for a victim, then it violates Core
stability. To this end, pick any P = (M ∪{p}, Z, Bp , (Di )i∈M ) ∈ P, and any φP ∈ ΦP \{φdP }.
Among the elements of ΦP \ {φd } only two types of mappings of rights can be used to
construct a solution F satisfying No veto power for a victim. To show this point, consider
the following two remaining and exclusive cases.
∗
(a) Assume that φP ({i}) = ZP,{i}
for some victim i ∈ M . By (C.4), φP ((M ∪ {p}) \
{i})) = ∅. Therefore, any solution F constructed from such a correspondence of
rights will violate No veto power for a victim.
∗
(b) Assume φP ({i}) = ∅ for each victim i ∈ M . Note that φP ({p}) = ZP,{p}
is not
d
possible since otherwise φP = φP . So, φP ({p}) = ∅ as well. If, for some i ∈ M ,
φP (M ∪ {p} \ {i}) = ∅, then the corresponding solution F will violate No veto power
∗
for a victim. Thus, we must have φP ((M ∪ {p}) \ {i}) = ZP,(M
∪{p})\{i} for each
∗
i ∈ M . At this step, we conclude that either φP (M ) = ZP,M or φP (M ) = ∅ are
possible.
From (a) and (b), conclude that φ is either of the form φP ({i}) = ∅ for each i ∈ M ∪{p} and
∗
. Next,
φP (M ) = ∅ or of the form φP ({i}) = ∅ for each i ∈ M ∪ {p}, and φP (M ) = ZP,M
consider the social cost problem P̃ given in Example 1, and any such above mentioned
mappings of rights φP̃ . The worth of the grand coalition and the worth of each coalition
of size m in (M ∪ {p}, vφP̃ ) are given by:
X −1
- vφP̃ (M ∪ {p}) = 4
ci
;
i∈M
- For each i ∈ M , vφP̃ ((M ∪ {p}) \ {i}) =
4
X
−1
cj
j∈M \{i}
∗
- vφP̃ (M ) = 0 whatever the value of φP̃ (M ) ∈ {∅, ZP,M
}.
13
;
As a final step, choose the balanced map λ ∈ B(M ∪ {p}) given by:
λ(S) = m−1 if |S| = m and λ(S) = 0 otherwise.
Straightforward computations give:
X
X
λ(S)vφP̃ (S) =
4m
i∈M
S⊆M ∪{p}
−1
X
cj
j∈M \{i}
X −1
> 4
ci
= vφP̃ (M ∪ {p}).
i∈M
By the Bondareva-Shapley theorem (see Proposition 1), (M ∪ {p}, vφP̃ ) has an empty core.
Therefore, we conclude from (a) and (b) that if F 6∈ Fd satisfies No veto power for a victim,
then it violates Core stability. This completes the proof of Proposition 3.
Proposition 3 states that the efficiency thesis is valid even if one imposes the requirement that no victim has the right to veto the rest of the society. But, it also indicates
that under this democratic requirement, there is only one possible rights assignment compatible with the efficiency thesis. Under this assignment, the polluter is not liable for the
damages his or her activity generates. This implies that, even if one restricts the rights
assignments to those which satisfy No veto power for a victim, the neutrality thesis fails
to be true. It should be noted that Proposition 3 continues to be true if one restricts
the domain of social cost problems to the subset of instances where the benefit functions
are concave and the damage functions are convex. To see this, it suffices to note that
Proposition 2 remains true on this subdomain and the Example 1 used in the Proposition
3 to prove the emptiness of the core belongs to this subdomain. So, we have the following
corollary.
Corollary 1 Assume that the domain of social cost problems under consideration is restricted to instances where the benefit functions are concave and the damage functions
are convex. Then, the solutions in Fd are the only solutions on this subdomain satisfying
Core stability and No veto power for a victim.
Furthermore, it is easy to show that under No veto power for a victim there is no
solution satisfiying Full compensation, from which we conclude that there is no stable,
democratic and fair solutions in our sense for the social cost problems. These two facts
are collected in the following propositions.
Proposition 4 Each solution F ∈ Fd violates Full compensation.
Proof. Pick any F ∈ Fd and any P ∈ P. By (4) the corresponding TU-game (M ∪{p}, vφd )
is such that:
X
∗
∗
vφd ({p}) = Bp (zp∗ ) and vφd (M ∪ {p}) = Bp (zM
Di (zM
∪{p} ) −
∪{p} ).
i∈M
Of course, vφd ({p}) ≥ vφd (M ∪ {p}) ≥ 0. Combining the core constraints on coalitions
P
P
{p} and M ∪ {p} with the latter inequality, yields:
X
xP,p ≥ vφd ({p}) ≥ vφd (M ∪ {p}) = xP,p +
xP,i ≥ 0.
P
P
i∈M
14
In P there exist instances P such that vφd ({p}) > vφd (M ∪ {p}). For such a P , we have:
P
xP,p > xP,p +
X
P
xP,i ≥ 0 =⇒
i∈M
X
xP,i < 0,
i∈M
which is not compatible with Full compensation.
Combining Proposition 3 with Proposition 4, we obtain the following impossibility
result.
Proposition 5 There is no solution on P satisfying Core stability, No veto power for a
victim and Full compensation.
From Proposition 5, the efficiency thesis is no longer true in presence of Full compensation and No veto power for a victim. From Proposition 3, we know that the neutrality
thesis does not hold. Therefore, it remains to characterize the set of solutions satisfying
Core stability only, or equivalently, the set of rights assignments under which the efficiency
thesis holds. Note that this constitutes the main question studied in the cooperative game
theory literature related to the “Coase theorem”. So far, and to the best of our knowledge, this question has not received a clear answer. We fill up this gap in Proposition 6
below by showing that there are only two types of rights assignment compatible with Core
stability: either the polluter is not liable for his or her activity, or he or she is liable and
negotiations over a pollution level require unanimity. To show this impossibility result,
we need a definition.
For each P = (Z, Bp , (Di )i∈M ) ∈ P, define the (two) mappings of rights φuP ∈ ΦP as
follows:
∗
u
∗
u
φuP (M ∪ {p}) = ZP,M
∪{p} , φP (M ) ∈ {∅, ZP,M }, and φP (S) = ∅ otherwise.
The mappings of rights φuP indicate that the polluter p needs the agreement of the set
of victims M to produce at some activity level, meaning that an agreement requires
unanimity. Denote by ΦuP these two mappings of rights. Next, let us introduce the set of
solutions Fu as follows:
F ∈ Fu
if
∀P ∈ P,
F (P ) = (φuP , xP ) for some xP ∈ C(M ∪ {p}, vφuP ).
For each P ∈ P, and whatever the value of φuP (M ), (M ∪ {p}, vφuP ) is such that:
X
∗
∗
vφuP (M ∪ {p}) = Bp (zM
Di (zM
otherwise.
∪{p} ) −
∪{p} ) ≥ 0, and vφu (S) = 0
i∈M
It follows that each F ∈ Fu satisfies Core stability, Full compensation, but violates
No veto power for a victim. Consequently, Fu ∩ Fd = ∅, and each F in Fu ∪ Fd satisfies
Core stability. The following result establishes that Fu ∪ Fd is the unique set of solutions
satisfying Core stability.
Proposition 6 The solutions in Fu ∪ Fd are the only solutions on P satisfying Core stability.
15
Proof. As noted above, Fu ∪ Fd satisfies Core stability. To prove that Fu ∪ Fd is the
largest set solutions satisfying Core stability, take any F on P which satisfies Core stability
but violates No veto power for a victim. To show: F ∈ Fu . By Proposition 3, F 6∈ Fd .
So, there is P = (M ∪ {p}, Z, Bp , (Di )i∈M ) ∈ P for which F (P ) = (φP , xP ) is such that
φP ((M ∪ {p}) \ {i}) = ∅ for at least one i ∈ M . There are two cases to consider.
(a) φP ((M ∪ {p}) \ {i}) = ∅ for each i ∈ M . By condition (C.3), φP (S) = ∅ for each
∗ .
S 3 p, S 6= M ∪ {p}. Next, assume that there is S ( M such that φP (S) = ZP,S
This means that there is i ∈ M such that S ⊆ M \ {i} ⊆ (M ∪ {p}) \ {i}. Applying
∗
∗
(C.3), φP (S) = ZP,S
implies φP ((M ∪ {p}) \ {i}) = ZP,(M
∪{p})\{i} , a contradiction.
Therefore, we necessarily have, for each S ( M , φP (S) = ∅. It follows that φP ∈ ΦuP .
By definition of a solution F , for each P 0 ∈ P \ {P }, we have φP 0 (S) = ∅ if and only
if φP (S) = ∅. Conclude that F ∈ Fu .
∗
(b) φP ((M ∪ {p}) \ {j}) = ZP,(M
∪{p})\{j} for some j ∈ M . Define by I and J the
following nonempty subsets of agents:
I = i ∈ M : φP ((M ∪ {p}) \ {i}) = ∅ ,
and
J = j ∈ M : φP ((M ∪ {p}) \ {j}) 6= ∅ .
By condition (C.5), the sets I and J do not depend on P . For the rest of the proof
assume, without loss of generality, that 2 ∈ I and 1 ∈ J. Consider the social problem
P = (M ∪ {p}, Z, Bp , (Di )i∈M ) ∈ P defined as:
∀z ∈ Z = [0, 1], Bp (z) = z 3/4 , D1 (z) = 4mz, D2 (z) = z, and Di (z) = 0, ∀i ∈ M \{1, 2}.
For the grand coalition, coalition M , and coalitions of size m containing p, we have:
- vφP (M ∪ {p}) =
27
;
256(4m + 1)3
- vφP (M ) = 0;
27 ;
- vφP ((M ∪ {p}) \ {1}) = 256
- For each i ∈ I, vφP (M ∪ {p} \ {i}) = 0;
- For each j ∈ J \ {1}, vφP (M ∪ {p} \ {j}) = vφP (M ∪ {p}).
Take the balanced map λ ∈ B(M ∪ {p}) given by:
λ(S) = m−1 if |S| = m and λ(S) = 0 otherwise.
16
We have:
X
λ(S)vφP (S) =
S⊆M ∪{p}
X
`∈M ∪{p}
=
X
j∈J\{1}
=
vφP ((M ∪ {p}) \ {`})
m
vφP ((M ∪ {p}) \ {j}) vφP ((M ∪ {p}) \ {1})
+
m
m
vφ ((M ∪ {p}) \ {1})
|J| − 1
vφP (M ∪ {p}) + P
.
m
m
By the Bondareva-Shapley theorem (see Proposition 1), if
vφ ((M ∪ {p}) \ {1})
|J| − 1
vφP (M ∪ {p}) + P
> vφP (M ∪ {p}),
m
m
that is, if
vφP ((M ∪ {p}) \ {1}) > vφP (M ∪ {p})(m − |J| + 1),
then the social cost TU-game (M ∪{p}, vφP ) has an empty core. Because |J| ∈ {1, . . . , m−
1},
mvφP (M ∪ {p}) ≥ vφP (M ∪ {p})(m − |J| + 1) ≥ 2vφP (M ∪ {p}).
Therefore, to show that the core of the above social cost TU-game is empty whatever
the value taken by |J|, that is whatever the number of victims without veto power, it is
enough to verify that:
vφP (M ∪ {p} \ {1}) > mvφP (M ∪ {p}),
which is a routine exercice left to the reader.
From (a) and (b), we conclude that if F on P satisfies Core stability but violates No
veto power for a victim, then F ∈ Fu . Therefore, Fd ∪Fu is the largest subclass of solutions
satisfying Core stability, as desired.
As for Proposition 3, it is easy to notice that Proposition 6 continues to hold if one
restricts the domain to instances where the benefit functions are concave and the damage
functions are convex.
Corollary 2 Assume that the domain of social cost problems under consideration is restricted to instances where the benefit functions are concave and the damage functions are
convex. Then, the solutions in Fd ∪ Fu are the only solutions on this subdomain satisfying
Core stability.
By Proposition 4, every solution in Fd violates Full compensation. Combining Proposition 4 with Proposition 6, we obtain the following result.
Proposition 7 The solutions in Fu are the only solutions on P satisfying Core stability
and Full compensation.
17
Proposition 7 establishes that the efficiency thesis is valid even if one imposes the
requirement that each victim is compensated for the damages they suffer from the activity
of the polluter. But, by Proposition 6 and the fact that the solutions in Fu satisfy Full
compensation, there is only one possible right assignment compatible with the efficiency
thesis.
4. Related literature and complementary explanations
4.1. Methodology: the axiomatic approach
From the cooperative game theory viewpoint, the question of the validity of the “Coase
theorem” relates mainly to the nonemptiness of the core of a game in which there are
negative externalities and a problem of social cost. To be more precise, the question
amounts to asking: is there one or many distribution of rights under which the core is
nonempty?
One of the older works on the core and externalities is maybe the article by Shapley
and Shubik (1969) who studied the situation where a set of polluters decide on where
to dump their garbage. The worth a coalition is defined by the total damages that the
members suffer from the nonmembers if the latter decide to dump their garbage in the
members’ backyards. Therefore, and contrary to our model, here agents are all potential
polluters and pollute each other. The efficient coalition is the grand coalition in which
each member dump is garbage in his or her backyard. Except in the case of two agents,
Shapley and Shubik showed that this game has an empty core. Ambec and Kervinio (2016)
reconsidered this problem with a spatial model externalities and proposed a general index
for testing whether the core of the game is empty or not.
Still in the cooperative approach, Aivazian and Callen (1981) constructed a cooperative
game in which there are two polluting firms and one victim and demonstrate that the core
of this game is nonempty when the two polluters are liable for the damages generated by
their activity. Thus, in this case, they claimed that the “Coase theorem” holds and that
a stable and efficient allocation of resources can be reached though a bargaining process.
But, the core of the game is empty when the polluters are not liable, which means that
each efficient allocation can be blocked by a coalition, and so firms can endlessly negotiate.
None of the two theses that form the “Coase theorem” are verified. The theorem “breaks
down did they conclude (see also Aivazen, Callen et Lipnowsky, 1987; Aivazen and Callen,
2003; Aivazian, Callen and McCracken, 2009)”3 . In other words, Aivazian and Callen
showed – through an example and without acknowledging it – that one cannot have the
efficiency and the neutrality theses satisfied at the same time. This is also what we
demonstrate by relying on an axiomatic study.
Indeed, beyond the fact that we consider one polluter and several victims, which, by the
way, allows to avoid the difficulty put forward by Magnan de Bornier (see footnote 3), our
approach differs from the above mentioned works in that we provide an axiomatic study
3
Magnan de Bornier (1996) raised a different type of critique against Aivazian and Callen: their conclusions hold only if the two polluters can merge; without this condition, the core is nonempty and the
theorem holds.
18
of the social cost problems. As explained by Thomson (2001), the axiomatic study begins
with the specification of the domain of the social cost problems, and the formulation of a
list of desirable properties, called also axioms, of solutions for the domain. It ends with
a description of the classes of solutions satisfying various combinations of the properties.
It should also offer, among other things, a discussion of whether plausible alternative
specifications of the domain would affect the conclusions. This is precisely that we do in
this article. We specify a large class of social cost problems with monotonic benefit and
damage functions. We design desirable properties for a solution. Each of these properties
incorporates a stability (Core stability), a democratic (No veto power for a victim) or a
fairness (Full compensation) principle. Then, Proposition 3, Proposition 5, Proposition
6, and Proposition 7 characterize the solutions satisfying various combinations of these
properties. The proof of these propositions reveals that certain (im)possibility results
continue to hold when we restrict the domain to the social cost problems where the benefit
functions are concave and the damages functions are convex.
By analyzing a large class of social cost problems from the axiomatic point of view, we
also depart from Coase’s original approach in “The Problem of Social Cost”, who mainly
studies stylized examples, and are closer to what Calabresi did (1961, 1965). On these
differences, we refer the reader to Medema (2014). At last, we think that the axiomatic
method is relevant for law and economics by exposing the possible logical incompatibility
between economic and legal principles.
4.2. Transaction costs
Our article is not related to the cooperative game theoretical literature on the positive
or zero transaction costs and the “Coase theorem”, which represents another particularly
important aspect in any discussion of the “Coase theorem”4 . In our model, we assume,
as most of the literature does, that there are no transaction costs. To understand the
legitimacy of this assumption, one must go back to the difference between the “Coase
theorem” and “The Problem of Social Cost”.
As it is clear from the quotations recalled in the introduction, Stigler did not formally
and explicitly use the assumption of zero transaction costs in his formulation of the “Coase
theorem” but his reference to “perfect competition” indicates that he was assuming a world
without frictions. Coase, for his part, explicitly introduced this ‘heroic” (McKean, 1970,
31) assumption in “The Problem of Social Cost”. However, Coase also and very rapidly –
after one third of the paper – dismissed this assumption as unrealistic: “the assumption
. . . that there were no costs involved in carrying out market transactions is, of course, a
very unrealistic assumption” (Coase, 1960, 15). Indeed, despite what the literature has
assumed afterwards, a large part of Coase’s paper was devoted to an analysis of situations
in which there are transaction costs. The reason was straightforward: to Coase, “[w]e do
not do well to devote ourselves to a detailed study of the world of zero transaction costs”
(Coase, 1981, 187) or “[i]t would not seem worthwhile to spend much time investigating
the properties of such a world” (Coase, 1988, 15). It could be interesting but as “a
4
The non-cooperative game theoretical literature has also investigated this aspect. See, among others,
Anderlini and Felli (2001, 2006).
19
preliminary” (1988, 15) that would precede the “development of an analytical system
capable of tackling the problems posed by the real world of positive transaction costs”
(1988, 15), which was the real objective pursued by Coase and that has been missed or
forgotten by commentators. It may be viewed as “surprising how little systematic followup has been to Coase’s own call for examining positive transaction costs” (Robson and
Skaperdas, 2008, 110). However, this is perfectly consistent with Stigler’s version of the
“Coase theorem”. This is the first reason that leads us to make this assumption.
The second justification is that the cooperative game literature that exists on positive
vs. zero transaction costs has reached unambiguous but contradictory conclusions. For
their part, Aivazian and Callen, in their original paper, assumed that transaction costs
are nil. This was precisely what Coase replied to them: in the real world, transaction
costs are positive and so firms cannot negotiate endlessly; they are bound by contracts,
that they cannot break at will. Later, Aivazian and Callen (2003) took transaction costs
in their analysis, and concluding that this does not modify their result – the core remains
empty – but “exacerbate” it (2003, 296). For their part, Guzzini and Palestrini (2009)
were less affirmative. They introduced another type of friction and showed that in the
Aivazian- Callen’s example the core is non empty. But they added that their result “does
not invalidate the Aivazian-Callen’s analysis: it just shows that by introducing this type
of friction in the bargain process, the Coase argumentation (1981) may be supported [...]
we conclude by claiming again that if, in one hand [transaction costs] seem to mitigate the
empty core, on the other hand, we cannot say that the efficient outcome will be reached
for sure” (7). More recently, Robson (2014) also concluded that transaction costs can
sometimes mitigate the instability problem posed by an empty core. In a three-agent
situations, it is always possible, as Robson demonstrated, to find a set of transaction costs
which, when introduced into a frictionless bargaining situation, will cause an empty core
to become nonempty.
In view of these inconclusive results, we have chosen to propose one more of one of
those analyses that Coase viewed as preliminary steps towards more realistic analyses and
to assume away transaction costs.
4.3. Neutrality of rights
This reasoning about transaction costs has particularly important consequences in terms
of rights.
Indeed, as it has been emphasized earlier, the “Coase theorem” also implies that the
law has no impact on economic activities. More precisely, the complementarity of the
two theses implies that the neutrality thesis holds only when there are no transaction
costs. At least, this is what Stigler – and the literature on the “Coase theorem” after
him – argued. This is what Chipman and Tian (2012) demonstrated, in a recent article,
with their “Coase neutrality theorem”. Following in the footsteps of Hurwicz (1995), they
provided a necessary and sufficient condition under which the neutrality thesis holds in
a competitive setting with two commodities (money and pollution) and two price-taker
agents.
And, of course, when there are transaction costs, there is no neutrality of the law.
Legal rules – and how are assigned rights – do matter.
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This is exactly what Coase argued. He was interested in analyzing situations in which
there are transaction costs because he was interested in showing that the law matters and
in understanding how various institutions – including the market – perform when there are
positive transaction costs, which happens all the time; he was interested to know whether
or not one can rely on the pricing system in cases in which there are transaction costs and,
accordingly, if other institutional arrangements are required to complement or replace the
market. In other words, for Coase, the law is certainly not neutral for the efficiency of the
allocation of resources. It affects transactions and exchanges and the object of economics
should indeed be to understand the nature of such impact.
In this paper, we demonstrate that the situation is more complex: even when there are
no transaction costs, law matters and the distribution of rights affects the outcome of a
negotiation between agents to remove an externality; even when there are no transaction
costs, the neutrality thesis does not hold. This negative result continues to be true when
one imposes a democratic principle to rights assignments, i.e. when one reduces the feasible
set of rights assignments.
4.4. Enforcement
“Enforcement” is another aspect that is discussed – even if not so frequently – in the
literature about the “Coase theorem”, to which we must say a few words. Actually,
enforcement is a twofold problem. First, there is no guarantee that a polluter is going to
respect the agreement that bargaining has allowed to reach. This issue could very well
have been but was not discussed in the frame of Coase’s model, with two players only. The
reason was probably that, as Robson and Skaperdas (2008, 2) noted, it is presupposed that
there exists a legal system that guarantee a costless enforcement of agreements reached
by the parties. But, as Robson and Skaperdas showed, when enforcement is costly and
because the probability to win a trial is always lower than 1, the “Coase theorem” does not
hold. In this article, we do not take into account the problems that can be caused by this
kind of enforcement: the choice the activity level of the polluter is made by bargaining,
and it is assumed that the polluter will respect what has been agreed on. The reason is
that our primary focus is not on how agents arrive at an efficient contract but what are
the efficient contracts which satisfy desirable properties.
Although we agree that these behaviors are important and represent a possible threat
to the existence of a negotiated solution to a problem of social cost, we do not take them
explicitly into account since, in a cooperative setting, contracts are binding and agent
cannot exploit their strategic power. Our primary purpose is to demonstrate that, when
there are many victims and one polluters, under certain distributions of rights, the core
of a cooperative game associated with a social cost problem is nonempty, but that this
result is incompatible with fairness and democratic requirements. The analysis of free
riding behaviors in a noncooperative framework is well studied in the recent article by
Ellingsen and Paltseva (2016). We also refer the reader to Parisi (2003), and Luppi and
Parisi (2012) who completed the analysis of the role of enforcement institutions.
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5. Conclusion
The “Coase theorem” has received a lot of attention among economists, but there are
still no completely definitive conclusions about its validity. In this paper, we propose new
results about whether or not the theorem holds. Certainly, our results are partial – limited
to the case of one polluter and many victims and we do not take transaction costs into
account. These are clearly the next steps that should be taken. But we nonetheless put
forward important results.
We show that the two theses, efficiency and neutrality, upon which the theorem rests
cannot be satisfied at the same time: the efficiency thesis can be satisfied but under only
two specific assignment of rights. This result confirms and complements the results reached
in the literature about the core and the Coase theorem – since we analyze a different
situation and we use a more general approach than the one usually used. However, and
this is also important, if there were only two agents – a polluter and a victim – as in
Coase’s original article, the set of mappings of rights reduces to the mappings φu and φd .
It could then easily be shown that, in that case, the core is nonempty whatever the rights
assignment. Therefore, the efficiency and the neutrality theses hold in the two agents case.
In addition, we also enrich this literature by adding an impossibility result. We show
that Core stability cannot be satisfied under democratic (no veto power for a victim) and
fairness (full compensation of victims) constraints; even if Core stability is compatible with
fairness (full compensation of victims). This means that efficiency could be satisfied – an
agreement could be signed by the polluter and the victims to reach an optimal allocation
of resources – but at the expense of, either democracy or fairness.
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