The Coase Theorem and the Empty Core Problem: A Critical Assessment E. Guzzini Polytechnic University of Marche A. Palestrini University of Teramo June 9, 2008
Abstract The purpose of this paper is to give some insights about the debate between Aivazian-Callen (1981, 2003) and Coase (1981) regarding the empty core problem. In particular our analysis concerns the role played by transaction costs in the debate. We show that, under certain assumptions, the presence of positive transaction costs may solve the empty core problem as suggested by Coase in his rejoinder (1981) to the Aivazian-Callen analysis. Our partial different conclusions from the Aivazian-Callen work depend on the introduction in the analysis of the novel category of “transaction costs in changing the state”. In particular we show that both the argumentation of Coase and AivazianCallen about the role of transaction costs are not contradictory. In other terms, we show conditions under which Coase insight (Coase, 1981) is correct in the sense that the presence of transaction costs will eventually destroy the empty core. At the same time we also find conditions that corroborate the Aivazian-Callen results.
Keywords: Coase’s Theorem, Empty Core. JEL classification codes: D6, D7.
1
Introduction
About twenty-five years ago, Aivazian and Callen (1981) found an interesting result on the Coase Theorem: They show that “with more than two participants the Coase theorem cannot always be demonstrated” (Aivazian, Callen, 1981, p. 175). The Coase Theorem, as it is well known, asserts that in the absence of transaction costs, the final outcome of bargaining (the resource allocation among individuals) is efficient and does not depend on the initial distribution
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of rights (or liability rules)1 . In their aforementioned paper Aivazian and Callen found that in the absence of transaction costs if there are more than two parties, and at least two externalities (as pointed out by Mueller, 2003, p. 32), then the process of bargaining may give rise to an empty core. In other terms, to an unstable situation leading to an endless re-negotiation cycle2 . Subsequently, in his rejoinder to the Aivazian and Callen analysis, Coase (1981) stresses, among other things, the important role played by the zero transaction costs assumption in the Aivazian and Callen work in order to get their results. He claims that with positive transaction costs the empty core will eventually disappear3 . Our analysis presents some conditions supporting the Coase argumentation by introducing the transaction costs to change the state (TCCS) category and fixed transaction costs (see par. 3). In the next paragraph we resume the empty core debate. In paragraph 3 it is introduced the new category TCCS. Finally, paragraph 4 concludes.
2
The empty core debate
The example of Aivazian and Callen (1981) can be stated as follows. Suppose there are two factories, A and B, producing as a by product a pollutant (smoke) and that there is a third factory, a laundry C, whose costs are raised by the emission of smoke and which does not produce any kind of pollutant. We use the symbol Π to denote the characteristic function of the cooperative game. In particular Πi stands for the profits for the firms in the absence of bargaining and we, as in Aivazian and Callen (1986), suppose that they amount to: ΠA =3,000, ΠB =8,000, ΠC =24,000. There are no transaction costs and the firms are free to bargain, or may merge among them, with the following particular payoffs: ΠA,B =15,000 (where ΠA,B stands for the joint profit of the coalition formed by A and B after having reached an agreement or after having merged); ΠA,C =31,000; ΠB,C =36,000; ΠA,B,C =40,000. It can be shown that all the firms have an incentive to bargain since Πi,j > 1
See Coase (1960, 1988). It was Stigler (1966, p.113) the first author to introduce the label "Coase Theorem". In Stigler’s worlds the Coase Theorem can be stated as: "...under perfect competition private and social costs will be equal” (Stigler 1966, p. 113, cited by Coase 1988, p. 158). The definition in Mueller (2003) is: “In the absence of transaction and bargaining costs, affected parties to an externality will agree on an allocation of resources that is both Pareto optimal and independent of any prior assignment of property rights” (Mueller, 2003, p. 28). 2 In other terms, the coalition agreements in a super additive cooperative game are not stable (see par. 2). 3 Other solutions, proposed in the literature of the empty core, are: penalty clauses, binding contracts (i.e., the fact that contracts once stipulated are binding for all participants and cannot be breached without the permission of all of them), and bargaining restrictions (Bernholz, 1997, 1999; Mueller 2003?).
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Πi + Πj for all i 6= j in {A, B, C}. Furthermore, it results that ΠA,B,C > Πi,j + Πk for all i 6= j 6= k in {A, B, C}, and therefore all the firms have the incentive to form the grand coalition which indeed represents the Pareto optimal outcome. In the coalition formed by AC (similarly B C ), laundry C by side payments obtains the production of A (B ) to stop. In the grand coalition AB C the laundry obtains, always by side payments, that both the production of B and C are interrupted. Further, let’s indicate by Xi the amount of profit firm i enjoys when participating to the grand coalition: it results, therefore that XA +XB +XC =ΠA,B,C Which will be the final outcome of this situation? Are we sure that the firm will choose the AB C coalition? In order to address the question we have to consider the liability to emit pollutant. If A and B can be considered liable for the pollution, then the final outcome will be ΠA,B,C =40,000 with XA =XB = 0 are XC = 40,000. In this situation C will force the other factories not to produce; at the same time A and B will not be able to pay adequately A in order to continue the production: the maximum sum A and B can pay amount to ΠA,B =15,000 which is less than 16.000=ΠA,B,C − ΠC , the damage they cause to C. We may think that also in the case in which A and B are not liable for the pollution, the final outcome will be ΠA,B,C =40,000, since C will be able to compensate the other factories for their closure. However, it can be shown that in this case the grand coalition situation is unstable, i.e. is not in the core 4 . In order to be stable, the grand coalition should satisfies the following conditions5 XA > ΠA ,
XA + XB > ΠA,B ,
XB > ΠB ,
XC > ΠC
XA + XC > ΠA,C ,
XB + XC > ΠB,C
XA + XB + XC = ΠA,B,C
(1)
(2) (3)
By summing the inequalities in (2) we obtain: 1 XA + XB + XC > [ΠA,B + ΠB,C + ΠA,C ] (4) 2 Then by substituting (3) in (4), we obtain the non-empty core conditions: 1 ΠA,B,C > [ΠA,B + ΠB,C + ΠA,C ] 2 Condition (5) is violated in the numerical example given above: 4 5
Regarding the core theory see, among others, Telser (1994, 1996). Aivazian - Callen (1981, p. 179).
3
(5)
1 ΠA,B,C = 41, 000 < [ΠA,B + ΠB,C + ΠA,C ] = 40, 000. 2 Therefore, if A and B are not liable for the pollution the process of forming coalition is intrinsically unstable6 . Coase (1981, p.187) stressed, among others, the absence of transaction costs in the AC model: «I would not wish to conclude without observing that, while consideration of would happen in a world of zero transaction cost can give us valuable insights, these insights are, in my view, without value axcept as steps on the way to the analysis of the real world of positive transaction costs.» In other terms, he seemed to argue that with positive transaction costs the probability to observe an empty-core may be negligible. In order to solve this part of the Coase rejoinder Aivazian and Callen (2003, pp.290-92) introduce positive transaction costs in their framework: in particular they model transaction costs as an increasing function of the number of agents participating to the bargain showing that the occurrence of an empty-core may be even wider. Following Aivazian and Callen, consider a generic normalized characteristic function Πi = 0 for all i = A, B, C; ΠA,B = a; ΠA,C = b; ΠB,C = c; ΠA,B,C = d where a, b, c, d are positive constants with d > a, b, c. It can be shown that the core of the game is empty if and only if d < (a + b + c)/2 Now introduce a transaction cost C(x) equal to ( xk , if x > 1 C(x) = 0, if x = 1
(6)
where x is the number of agents in the coalition and k > 1 is a parameter. In other terms, C(x) is a convex function in the number of agents. With such transaction cost the characteristic function becomes Πi = 0 for all i = A, B, C; ΠA,B = a − 2k ; ΠA,C = b − 2k ; ΠB,C = c − 2k ; ΠA,B,C = d − 3k . This implies that the core is empty if d < 3k + (a + b + c − 3 · 2k )/2 Note that increasing k will cause the inequality to eventually be true since 3k increase more quickly than 2k . Therefore Aivazian-Callen (2003, p. 6
Consider, for example the situation in which XA = 5, 000, XB = 10, 000 and XC = 25, 000. In this case there are more than one coalitions which could stop the grand coalition. For example A and C would have the incentive to form a new coalition by dividing ΠA,C = 31, 000 in the following way: 5,500 to A and 25,500 to C.
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291) may conclude that the introduction of positive transaction costs make the occurrence of the empty core even more likely. Before introducing the TCCS category, we note that Aivazian and Callen restrict their attention to the case in which k > 1. We conclude this section by considering the other cases which are: 1) k = 1 In this case there C(x) becomes: ( x, if x > 1 C(x) = (7) 0, if x = 1 This means that there are fixed transaction costs normalized to 1 for each agent participating to the bargain. These costs includes, for example, notary and legal expenses for the bargain and so on. Since 3k = 3 · 2k /2, with k = 1 we come back to the "original" Aivazian-Callen (1981) result as described in equation (5). Therefore we may conclude that the introduction of fixed transaction costs leaves the Aivazian and Callen conclusion unaffected. 2) k < 1 In this case C(x) is a concave function in the number of agents participating to the coalition. Since 3k < 3 · 2k /2 with k < 1, the introduction of transaction costs reduces the occurrence of the empty core giving some support to the aforementioned Coase’s argumentation. In the following, we will continue to maintain the Aivazian-Callen assumption of convex transaction costs w.r. to the cardinality of the coalition (k > 1) and we will introduce a novel transaction cost category.
3
“Costs to change the state” and transaction costs
In this section we introduce a simple notation in order to analyze the emptycore debate. In the following N = {A, B, C} is the set of economic agents and S is the set of all the possible coalitions. We define Ci (s, ti ) the non productive costs for reaching the coalition s afforded by agent i, where s ∈ S, i ∈ N . The variable ti represents the past history of individual i. Now define Γi (s) the "information-independent" transaction cost component; i.e., the cost which depends only on the specific type of contract agent-i has to stipulate in order to reach s 7 and ∇i (s, ti ) = Ci (s, ti ) − Γi (s) so that we can obtain the following transaction cost decomposition: Ci (s, ti ) = Γi (s) + ∇i (s, ti ) where ∇i (s, ti ) is defined to be the additional transaction cost depending both on s and also on the individual history ti . In other words, we assume 7
As described above, in the Aivazian-Callen (2003) framework it depends on the number of agents participating to the bargain and hence on s.
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that if agent-i reaches s by starting with an history ti he/she would have to afford a different cost than by starting with a different one, say t′i . We call this component transaction cost to change the state (TCCS as defined in the introduction). Now, transaction costs considered by Aivazian and Callen refer only to Γ. The main difference between Γ and ∇ is that Γ is the "standard" transaction cost, which is necessary for the negotiation or the bargaining. ∇ is the cost depending also on the actual characteristics of the agent and hence on his/her history. In the Aivazian and Callen analysis the role of Γ is played by xk (see eq. 6). In our analysis total transaction cost is equal to: Ci (s, ti ) = Γi (s) + ∇i (s, ti ) = θi xk + ∇i (s, ti ) where xk has the same meaning of the Aivazian-Callen framework, θi is a fraction8 in [0, 1] whereas, ∇i (s, ti ) 9 is the additional cost depending also on agent-i history. In the following ti will indicate, with a little abuse of terminology, the number of transactions faced by agent-i before to reach state s. To understand the role of ∇i (ti ), let’s assume that ∇i (ti ) is differentiable and its first derivative is monotone; we have to consider, therefore, two possible cases: Condition 1)
∂∇i (ti ) ∂ti
≡ ∇′i (ti ) > 0.
This case implies that TCCS is increasing with the number of bargains occurred. A possible interpretation of this condition may be found in reputational concerns. [MATTERE GREENWALD-STIGLITZ] Reputation, as it is well known, depends on the past behavior of the individual and this information is contained in ti , that is the number of coalitions made by the agent in the past. If an agent pass from s to s′ (i.e., from one coalition to another one) may give a signal of unreliability to other agents and this negative signal may involve an additional transaction cost for him/her regarding future transactions. For example he would experience more difficulty in persuading a potential partner about his/her reliability and this difficulty may lead to a TCCS increase. As noted by Macaulay (1963, p. 58) quoted by Halonen (2002, p. 539): "Businessmen often prefer to rely on a ’a man’ word’s in a brief letter, a handshake, or ’common honesty and decency’ - even when the transaction involves exposure to serious risks". The reason of this behavior may be found in reputational concerns. 8 9
Obviously, θA + θB + θC = 1, and θi > 0. In the following for notational convenience we will write ∇i (ti ) instead of ∇i (s, ti ).
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Condition 2)
∇′i (ti ) ≤ 0.
In this case, on the contrary, are involved phenomena such as learning by doing, learning by interacting with others, positive externalities depending on the number of transactions agents made. Therefore, the more are the transactions the individual made, the less are the costs of each one of them. The Aivazian and Callen analysis changes as follows: the normalized characteristic function becomes Πi = 0 10 for all i = A, B, C; ΠA,B = a − 2k − ∇A (t′A ) − ∇B (t′B ) ΠA,C = b − 2k − ∇A (t′′A ) − ∇C (t′′C ) ′′′ ΠB,C = c − 2k − ∇B (t′′′ B ) − ∇C (tC )
ΠA,B,C = d − 3k − ∇A (tA ) − ∇B (tB ) − ∇C (tC ) where ti < t′i , t′′i , t′′′ i
(8)
In other terms, since our fundamental concern is to analyze the stability of the grand-coalition, we assume that ti precedes the others, that is the grand coalition is the first one to be formed, while the other coalitions are subsequently formed. Since it is not important, in our analysis, to specify the formation order of two-agents coalitions, we assumed, without loss of generality, that the coalition A, B is formed after the grand coalition, the coalition A, C is formed after the A, B one, and so on. The empty core condition now changes in d < 3k +
a + b + c − 3 · 2k − (FA + FB + FC ) − G 2
(9)
where G = ∇A (tA ) + ∇B (tB ) + ∇C (tC ) FA =
∇A (t′A ) + ∇A (t′′A ) 2
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The single coalition is not affected by transaction costs since it is a non-recurrent state in the bargaining loop.
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FB =
∇B (t′B ) + ∇B (t′′′ B) 2
∇C (t′′C ) + ∇C (t′′′ C) 2 To analyze equation (9) we return to Condition 1) and Condition 2). FC =
Condition 1) ∇′i (ti ) > 0. By inequality (8), under Condition 1) it is straightforward to note that (10)
FA + FB + FC − G > 0. and there exists FA + FB + FC
sufficiently high such that the empty core condition (9) does not hold. Condition 2) that
∇′i (ti ) ≤ 0. By inequality (8) it is straightforward to note (11)
FA + FB + FC − G < 0. k
Now, since Aivazian and Callen showed that d < 3k + a+b+c−3·2 , by 2 adding FA + FB + FC − G < 0 the empty core condition is a fortiori verified. An obvious consequence is that there not exist FA + FB + FC such that the empty core condition (9) does not hold. Therefore under condition 2) the Aivazian and Callen result is robust to the introduction of TCCS. To be complete, condition 1) does not imply, per se, the absence of an empty core. In other terms, such condition shows only the stability of the grand coalition, but does not assure that the grand coalition itself may be chosen at the beginning of the game. In order to have a convenience to participate to the bargain, for each agent it must result11 : di > θi 3k + ∇i (ti ) ∀i ∈ N
(12)
11 Note that di stands for the share of payoff that goes to agent i. Note also that equation (12) implies: d > 3k + G. To be precise, equation (12) is not sufficient to assure that the Coase theorem holds as shown by Anderlini and Felli (2006) if transaction costs are payable ex ante. We do not tackle this issue in the present paper.
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The validity of equation (12) assures that each agent has an incentive to shift from the individual initial state to the grand coalition. It implies that G is not too high. Under condition 1), the reputation example may be an important case in which it is reasonable to assume that equation (12) holds. Indeed if an agent decides to join the grand coalition starting from the initial situation in which, as it is assumed in the coasian literature, he/she belongs to the one-member coalition, probably he/she has only in a very limited way a reputational concern. On the contrary, once an individual has joined a coalition with at least two agents, the decision to pass to another one implies the breach of the previous one, and therefore individual’s reputation is likely to be affected. This analysis implies the following results Result 1 Under condition 1, if di > θi 3k + ∇i (ti ) ∀i ∈ N , then there exist FA + FB + FC sufficiently high and G sufficiently small such that the game has a core not empty, i.e. equation (9) does not hold whereas equation (12) does hold. It could be easily shown that, in this case, G < d − 3k and that FA + k FB + FC < d − 3k − G − a+b+c−3·2 . The meaning of the first inequality 2 is straightforward; the second inequality implies that the advantage to leave the grand coalition is more than counterbalanced by the transaction costs necessary to join the two members-coalitions. Result 2 Under condition 2, there not exist any FA + FB + FC sufficiently high such that the empty core can be eliminated, i.e. equation (9) does hold. [RIGA DI SPIEGAZIONE]
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Conclusions
In this paper we analyze the role of transaction costs in the Aivazian-Callen and Coase debate about the empty core. In particular, we first investigate the role of the various kind transaction cost about the number of coalition members and we find that if such transaction costs are concave, then Coase’s insight (1981) is correct. Second, by focusing on the Aivazian-Callen assumption of convex transaction costs w.r. to the cardinality of the coalition (k > 1) and we introduce the novel TCCS transaction cost category. In this case, we show that the presence of positive transaction costs may solve, under some assumptions, the empty core problem as suggested by Coase in his rejoinder to the Aivazian9
Callen analysis. Indeed, the first result of our work is that with convex transaction costs (in the number of transactions/bargains), there exists a threshold in the sum of two-coalitions above which the empty core disappears. However, if the TCCS are concave on the number of transactions, then there do not exist such threshold, and therefore the Aivazian-Callen analysis fully holds. To conclude, our analysis shows that both the argumentation of Coase and Aivazian-Callen about the role of transaction costs are not contradictory. In other terms, we show conditions under which Coase insight (Coase, 1981) is correct in the sense that the presence of transaction costs will eventually destroy the empty core. At the same time we also find conditions that corroborate the Aivazian-Callen results. Da mettere Stiglitz,....e aggiungere Kahn (1995).
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[11] Halonen M. (2002), ’Reputation and the allocation of ownership’, Economic Journal, 112, pp. 539-558. [12] Macaulay S. (1963), ’Non-contractual relations in business: a preliminary study’, American Sociological Review, 12:4, pp. 55-67. [13] Mueller D.C. (2003) ’Public Choice III’, Cambridge University Press, Cambridge. [14] Telser L.G. (1994) ’The usefulness of Core Theory in Economics’. Journal of Economic Perspectives, 8, pp. 151-164. [15] Telser L.G. (1996) ’Competition and the Core’. Journal of Political Economy, 104:1, pp. 85-107.
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