The Coase Theorem, the Nonempty Core, and the Legal Neutrality Principle

Bertrand Crettez

doi:10.1515/rle-2018-0027

Abstract

The Coase theorem states that where there are externalities and no transaction costs resource allocation is Pareto-optimal and independent of the stakeholders’ legal position. This result has been challenged many times. In the cooperative game approach to resource allocation, the refutation is made by constructing a three-person game which has an empty core under one set of liability rules—which implies that optimal allocations are coalitionally unstable–and a nonempty core under another set. In this example, however, the probability that the core is non-empty is rather high (5/6). Yet, even if coalitionally stable Pareto-optimal arrangements are likely, to establish the plain validity of the Coase theorem it must be shown that the legal neutrality statement also holds. We show that for the three-person cooperative game example mentioned above, the probability that the two assertions of the Coase theorem hold can be as low as 3/8.

Keywords: Coase theorem; cooperative games; core; legal neutrality

JEL Classification: C71; C78; D23; D62; K0

Acknowledgements

I am grateful to Régis Deloche and Gabrielle Smart for very helpful comments on a previous version of this paper. I also thank a referee for stimulating comments and suggestions.

Appendix

A Proof of Proposition 2

Proof

^[13]

To establish the result we shall first assume that θ₁≡v(AB)/v(ABC) is first drawn from a uniform distribution. Let θ₄ denote v^l(C)/v(ABC). For each realization θ₁ the probability that v^l(C)/v(ABC) be lower than 1−v(AB)/v(ABC), namely θ4≤1−θ1 is 1−θ1. Now we must check that inequation (12) holds. But we must take care of the fact that vl(C)v(ABC)≤v(AC)v(ABC)≡θ2 and vl(C)v(ABC)≤v(BC)v(ABC)≡θ3 (the superadditivity property holds by assumption).

Therefore, the probability that the legal neutrality property holds when θ₁ is given and θ4≤1−θ1 is the probability that: θ4≤θ2, θ4≤θ3andθ2+θ3≤2−θ1. This probability is equal to the shaded area in Figure 1. We have

(28)Pθ4≤θ2,θ4≤θ3,θ2+θ3≤2−θ1∣θ4≤1−θ1=1−∫1−θ11∫2−θ1−θ21dθ3dθ2−∫0θ4∫0θ4dθ2dθ3.

We can check that:

(29)Pθ4≤θ2,θ4≤θ3,θ2+θ3≤1−θ1∣θ4≤1−θ1=1−θ122−θ42.

Now the probability that the legal neutrality property holds, θ₁ being given, is

(30)∫01−θ1Pθ4≤θ2,θ4≤θ3,θ2+θ3≤2−θ1∣θ4≤1−θ1dθ4,=∫01−θ11−θ122−θ42dθ4,=1−θ1−θ122+θ132−(1−θ1)33.

Finally, the probability that the legal neutrality property holds is equal to

(31)P[θ4≤θ2θ4≤θ2,θ4≤2−θ1,θ2+θ3≤1−θ1]=∫011−θ1−θ122+θ132−(1−θ1)33dθ1=18.

□

Figure 1:

Probability that the legal neutrality property holds (θ₁ and θ₄ being given).

B Proof of Proposition 3

To compute the probability that the legal neutrality principle holds let us first assume that θ₁, θ₂ and θ₃ are defined as in the preceding part of the appendix. Let us also define θ2′=vl(AC)v(ABC) and θ3′=vl(BC)v(ABC). The probability that the legal neutrality principle holds is equal to the probability that θ2+θ3≤2−θ1, θ2≤θ2′≤1, θ3≤θ3′≤1 and θ2′+θ3′≤2−θ1. To compute this probability we shall first compute the probability that θ2′+θ3′≤2−θ1 given that θ2+θ3≤2−θ1, θ₁, θ₂ and θ₃ being fixed. To proceed there are four zones (see Figure 2). We shall compute the aforementioned probability in each of these zones.

$Figure 2: Probability that θ2+θ3≤2−θ1\theta_2 + \theta_3 \leq 2- \theta_1.$

Figure 2:

Probability that θ2+θ3≤2−θ1.

Denote by Piθ2≤θ2′≤1,θ3≤θ3′≤1,θ1+θ2′+θ3′≤2∣θ1+θ2+θ3≤2 the probability referring to zone i (i = I,II,III,IV).

Zone I
We can check that
(32)PI=∫θ22−θ1−θ3∫θ32−θ1−θ2′dθ3′dθ2′,
(33)=2−θ1−θ322−2−θ1−θ3θ2+θ222.
Zone II
(34)PII=∫1−θ12−θ1−θ3∫θ32−θ1−θ2′dθ3′dθ2′+∫θ21−θ1∫θ31dθ3′dθ2′,
(35)=2−θ1−θ322−1−θ122−(1−θ3)θ2.
Zone III
(36)PIII=∫θ21∫1−θ12−θ1−θ2′dθ3′dθ2′+∫θ21∫θ31−θ1dθ3′dθ2′,
(37)=12−θ2+θ222+1−θ1−θ3(1−θ2).
Zone IV
(38)PIV=∫θ21−θ1∫θ31dθ3′dθ2′+∫1−θ11∫θ31−θ1dθ3′dθ2′+∫1−θ11∫1−θ12−θ1−θ2′dθ3′dθ2′,
(39)=(1−θ3)1−θ1−θ2+1−θ1−θ3θ1−12+θ1+1−θ122.

Now we have

(40)P[θ2≤θ2′,θ3≤θ3′,θ1+θ2′+θ3′∣θ1+θ2+θ3≤2]=∫01P(θ1)dθ1,

where

(41)∫01P(θ1)dθ1=∫1−θ11∫1−θ12−θ1−θ2PI(θ1,θ2,θ3)dθ3dθ2+∫01−θ1∫1−θ11PII(θ1,θ2,θ3)dθ3dθ2+∫1−θ11∫01−θ1PIII(θ1,θ2,θ3)dθ3dθ2+∫01−θ1∫01−θ1PIV(θ1,θ2,θ3)dθ3dθ2,

where we have stressed that the P_i are all functions of θ₁, θ₂ and θ₃. We next compute each integral in the above sum.

To compute the first integral we first get after tedious computations

(42)∫1−θ12−θ1−θ2PI(θ1,θ2,θ3)dθ3=1−θ236−θ2(2−θ1)(1−θ2)+θ22(2−θ1−θ2)2−(1−θ1)2+θ222(1−θ2)

Then we obtain

(43)∫1−θ11∫1−θ12−θ1−θ2PI(θ1,θ2,θ3)dθ3dθ2=124−(1−θ1)6+(1−θ1)24−1−θ136+(1−θ1)424.

We also have

(44)∫01−θ1∫1−θ11PII(θ1,θ2,θ3)dθ3dθ2=1−θ16+(1−θ1)412−(1−θ1)24,

(45)∫1−θ11∫01PIII(θ1,θ2,θ3)dθ3dθ2=1−θ16−(1−θ1)24+(1−θ1)412,

and

(46)∫1−θ11∫01PIV(θ1,θ2,θ3)dθ3dθ2=θ1(1−θ1)22+θ1(1−θ1)32+(1−θ1)44.

Using these expressions, we finally get

(47)∫01P(θ1)dθ1=19120

Finally, the probability that the legal neutrality principle holds (and the Coase theorem) is:

(48)P[θ1+θ2+θ3≤2]×∫01P(θ1)dθ1=56×19120=19144.

References

Aivazian, Varouj A., and Jeffrey L. Callen. 1981. “The Coase Theorem and the Empty Core,” 24 (1)Journal of Law and Economics 175–181.10.1086/466979Suche in Google Scholar

Aivazian, Varouj A., and Jeffrey L. Callen. 2003. “The Core, Transaction Costs, and the Coase Theorem,” 14 Constitutional Political Economy 287–299.10.1023/B:COPE.0000003859.10184.f3Suche in Google Scholar

Allen, Douglas W. 2015. “The Coase Theorem: Coherent, Logical, and not Disproved,” 11 (2)Journal of Institutional Economics 379–390.10.1017/S1744137414000083Suche in Google Scholar

Benoit, Jean-Pierre, and Lewis Kornhauser. 2002, “Game Theoretic Analysis of Legal Rules and Institutions, Chapter 60,” in Aumann Robert and Sergiu Hart, eds. Handbook of Game Theory with Economic Applications, vol. 3. London: North-Holland.10.1016/S1574-0005(02)03023-0Suche in Google Scholar

Bernholz, Peter. 1997. “Property Rights, Contracts, Cyclical Social Preferences, and The Coase Theorem: A Synthesis,” 13 European Journal of Political Economy 419–442.10.1016/S0176-2680(97)00027-XSuche in Google Scholar

Bergstrom, Ted. 2017, “When was Coase Right?” Departemental Working paper, UC Santa Barbara.Suche in Google Scholar

Chipman, John S., and Guoqiang Tian. 2012. “Detrimental Externalities, Pollution Rights, and the “Coase Theorem”,” 49 Economic Theory 309–327.10.1007/s00199-011-0602-1Suche in Google Scholar

Coase, Ronald. 1960. “The Problem of Social Cost,” 3 The Journal of Law & Economics 1–44.10.1057/9780230523210_6Suche in Google Scholar

Coase, Ronald. 1981. “The Coase Theorem and the Empty Core: Comment,” 24 (1)Journal of Law and Economics 183–187.10.1086/466980Suche in Google Scholar

Chipman, John S. 1998, “A Close Look at the Coase Theorem,” in James M. Buchanan and Bettina Monissen, eds. The Economists’ Vision : Essays in Modern Economic Perspectives; for Hans G. Monissen on the Occasion of His 60th Birthday, 131–162. Frankfurt/Main: Campus Verlag.Suche in Google Scholar

Ferey, Samuel, and Pierre Dehez. 2016a, “Multiple Causation, Apportionment and the Shapley Value,” 45 (1)Journal of Legal Studies 143–171.10.1086/685940Suche in Google Scholar

Gonzales, Stéphane, and Alain Marciano. 2017, “De Nouveaux Éclairages sur le Théorème de Coase et la Vacuité du Coeur,” Revue D’économie Politique 579–600.10.3917/redp.274.0579Suche in Google Scholar

Gonzales, Stéphane, Marciano Alain, and Philippe Solal. 2016, “The Social Cost Problem, Rights and the (Non)Empty Core” Working paper, Université de Saint-Etienne.10.2139/ssrn.2863952Suche in Google Scholar

Hurwicz, Leonid. 1995, “What is the Coase Theorem?,” 7 Japan and the World Economy 49–74.10.1016/0922-1425(94)00038-USuche in Google Scholar

Medema, Steven. 2017, “The Coase Theorem at Sixty” Working paper, University of Colorado Denver.Suche in Google Scholar

Maschler, M., Solan Eilon, and Shmuel Zamir. 2013, Game Theory. Cambridge, UK: Cambridge University Press.10.1017/CBO9780511794216Suche in Google Scholar

Parisi, Francesco. 2008. “Coase Theorem,” in Steven Durlauf and Lawrence Blume, eds. The New Plagrave Dictionary of Economics. London: Plagrave Macmillan.10.1057/978-1-349-95121-5_517-2Suche in Google Scholar

Robson, Alex. 2014, “Transaction Costs Can Encourage Coasean Bargaining,” 160 Public Choice 539–549.10.1007/s11127-013-0117-3Suche in Google Scholar

Zaho, Jingang. 2017. “A Reexamination of the Coase Theorem,” Journal of Mechanism and Institution Design, forthcoming.Suche in Google Scholar

Supplementary Material

The online version of this article offers supplementary material (DOI:https://doi.org/10.1515/rle-2018-0027).

Published Online: 2019-10-17

Sie haben derzeit keinen Zugang zu diesem Inhalt.

Supplementary Material Details

The Coase Theorem, the Nonempty Core, and the Legal Neutrality Principle

Abstract

Acknowledgements

Appendix

A Proof of Proposition 2

Proof

B Proof of Proposition 3

References

Supplementary Material

Artikel in diesem Heft

Artikel in diesem Heft

The Coase Theorem, the Nonempty Core, and the Legal Neutrality Principle

Artikel

Abstract

Acknowledgements

Appendix

A Proof of Proposition 2

Proof

B Proof of Proposition 3

References

Supplementary Material

Zusatzmaterial

Artikel in diesem Heft

Artikel in diesem Heft

Artikel in diesem Heft