Introduction

The Coase theorem states that with well-defined property rights and in the absence of transaction costs, Pareto-efficient allocations will emerge through negotiations among the players to internalize any externality among them, regardless of the initial assignment of property rights (Coase 1960). This result obtains because participants will costlessly recontract around property rights assignments that fail to be Pareto efficient. Coase (1960) also emphasizes the central importance of transaction costs for resource allocation, focusing on efficient property right structures when transaction costs are significant.

As argued by a number of scholars, the bargaining mechanism over property rights in the Coase theorem can be fruitfully framed in terms of cooperative game theory (Arrow 1979; Davis and Whinston 1965). Focusing on core theory, a branch of cooperative game theory, the Coase theorem can be interpreted as: with zero transaction costs, the grand coalition will always emerge regardless of the initial allocation of property rights among the players, and irrespective of whether or not the core of a superadditive characteristic function exists. (Telser (1994) provides a compelling discussion of the power of core theory). Aivazian and Callen (1981) and a number of their subsequent papers employ cooperative game theory and core theory to show that while the Coasean efficiency result is robust for the case of a two-person game, it may fail when there are more than two players. We review these results drawing on Aivazian and Callen (1981, 2003), Aivazian et al. (1987), and Aivazian et al. (2009) as well as on some of the papers that commented on the original 1981 Aivazian-Callen study.

Aivazian and Callen (1981) show that the Coasean efficiency result may fail in a zero transaction cost environment with at least three players and two externalities, in which there are gains from cooperating and forming coalitions to internalize the externalities. Specifically, in their example, the core is empty under one set of property rights, but nonempty under the other. With an empty core, cycling among coalitions could occur, preventing attainment of the grand coalition and Pareto efficiency. In response to the Aivazian and Callen counterexample to his theorem, Coase (1981) asserts that the zero transaction cost environment underlying his theorem is uninteresting in and of itself. He argues that if transaction costs are imposed on the negotiations in the Aivazian-Callen example, an empty core is less likely to obtain implying that the counterexample is essentially uninformative. However, extending their counterexample to allow for a reasonable transaction cost technology that is convex in the number of coalition partners, Aivazian and Callen (2003) demonstrate that transaction costs tend to aggravate the empty core problem making the breakdown of Coasean efficiency even more likely.

The Empty Core Argument without and with Transactions Costs

The original Aivazian and Callen (1981) analysis involves two polluting firms (A and B) and a laundry (C) and shows that when the polluting firms are liable (when the laundry has the property rights), the Pareto efficient outcome emerges; but when they are not liable, the core is empty and negotiations cycle without necessarily converging to the grand coalition or any other specific outcome. This can be demonstrated by representing the Aivazian and Callen (1981) example in the form of the following normalized characteristic function where V denotes joint coalitional profits:

$$ \mathrm{V}\left(\mathrm{i}\right)=0\kern0.36em \mathrm{all}\kern0.5em \mathrm{i}=\mathrm{A},\mathrm{B},\mathrm{C} $$
(1a)
$$ \mathrm{V}\left(\mathrm{A},\mathrm{B}\right)=\mathrm{a},\mathrm{V}\left(\mathrm{A},\mathrm{C}\right)=\mathrm{b},\mathrm{V}\left(\mathrm{B},\mathrm{C}\right)=\mathrm{c} $$
(1b)
$$ \mathrm{V}\left(\mathrm{A},\mathrm{B},\mathrm{C}\right)=\mathrm{d} $$
(1c)

where a, b, c, d are positive constants, and d > a, b, c, for superadditivity. The Pareto optimal outcome corresponds to the grand coalition outcome V(A,B,C). Note that the characteristic function will be different under different property rights since what each coalition can guarantee itself depends on the prevailing property rights arrangements (Shubik 1984, Chap. 19). Necessary and sufficient condition for the core to be empty (when A and B are not liable) is

$$ \mathrm{d}< 1/2\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right). $$
(2)

If the latter inequality obtains, the grand coalition outcome is not guaranteed so that specific property rights matter for efficiency.

Aivazian and Callen (2003) extend their 1981 paper to allow for transaction costs in the negotiation process by making the reasonable assumption that the costs of forming a coalition are convex in the number of players in the coalition, that is, coalition formation costs increase at an increasing rate with the number of coalition partners. This is reasonable since the number of communication channels required to obtain agreement among coalition members is also convex in the number of members. Two important conclusions emerge. First, if the core is empty in the absence of transaction (coalition formation) costs, then it is necessarily empty with such costs. Second, even if the core is not empty in the absence of transaction costs, such costs could generate an empty core.

What Have We Learned from the Empty Core Argument?

Several lessons emerge from the original Aivazian and Callen (1981) paper and the literature it has spawned (see, e.g., Bernholz (1997), De Bornier (1986), Hurwicz (1995), and Mueller (2003)). First, the Coase theorem may break down when there are more than two participants because the core of the negotiations may be empty under one set of property rights and nonempty under another. As a consequence, even in the absence of other transactions costs, the empty core is likely to impose particular costs of its own. Specifically, cycling induced by the empty core will tend to increase bargaining costs, diminish the value of the exchange opportunity as the negotiation process is prolonged, and the exchange is postponed, potentially keep at least one coalition unsatisfied and yield a non-Pareto allocation of resources if eventually the grand coalition does not form (Aivazian and Callen (1981, 2003); Shubik (1983), p. 150–151). In fact, Bernholz (1997) argues that the empty core in the Aivazian and Callen example is equivalent to cyclical social preferences. Second, the empty core problem arises as the number of participants increases only when additional participants bring in additional externalities (see Mueller (2003), De Bornier (1986), and the discussion in Aivazian and Callen (2003) on this point). Third, transaction costs may well aggravate the empty core problem, especially if they are incurred prior to the bargaining process (Aivazian and Callen 2003; Anderlini and Felli 2006). Fourth, Pareto optimality may be achieved when the core is empty if there are reputational (transaction) costs with the breaking of agreements (Guzzini and Palestrini (2009), or, from a normative perspective, by the imposition of constraints on the bargaining process (e.g., limiting negotiations to only certain sub-coalitions) and the use of penalty clauses and binding contracts (Bernholz 1997; Telser 1994; Aivazian and Callen 2003). As a consequence, it is important to distinguish between transaction costs (when the core exists) and costs due to the empty core because each has a different implication for rationalizing institutions. As Aivazian and Callen (2003, p. 291–292) emphasize:

It is wrong to conclude, therefore, that once transaction costs are introduced, then the problem of the empty core disappears and a Pareto optimal solution obtains. Rather, in such circumstances negotiations may break down more quickly and which specific coalition structure (the grand coalition or a proper sub-coalition) obtains cannot be specified a priori. Even if transaction costs were to force an equilibrium, nothing insures that the equilibrium is Pareto optimal … It may seem difficult to distinguish empirically between institutional arrangements that arise because of the nonexistence of the core from those that arise from the transaction costs of bargaining when there is a core. After all, the nonexistence of the core will also manifest itself in transaction costs, through the opportunity cost of (negotiation) time. However, the fact that an empty core can arise in the absence of bargaining costs, although these costs exacerbate the empty core problem, means that the costs generated by an empty core are fundamentally different from the transactions costs of bargaining. Indeed, what is unique about the empty core is that, in addition to direct bargaining costs, it gives rise to costs such as the erosion of the value of the exchange opportunity as it is postponed or the possibility of settling down to a non-Pareto optimal coalition (a proper sub-coalition).

Non-core Coalitional Stability

Many of the issues raised by the empty core also arise under alternative (non-core) notions of coalitional stability. As Aivazian et al. (1987) argue, for the Coase theorem to obtain, the grand coalition must be stable and, moreover, no other coalition can be similarly stable because otherwise a Pareto optimal allocation cannot be guaranteed. Aivazian et al. (1987) extend the Aivazian-Callen (1981) example to Aumman and Maschler (1964) bargaining set notions of coalitional stability by showing that while a specific Pareto optimal allocation of resources obtains for one set of property rights, a non-Pareto optimal allocation may well obtain for another set of property rights that involves bargaining. Indeed, under one type of bargaining set stability, they find that every coalition but the grand coalition is stable, completely vitiating the Coase theorem.

Testing the Implications of the Empty Core

It is unlikely that archival data are available that would allow one to test the implications of the empty core problem for the Coase theorem. Instead, Aivazian et al. (2009) investigate the Coase theorem experimentally in a bargaining game in which the final allocation of payoffs differ in terms of whether the core exists and in the initial allocation of property rights among the players. The experimental results indicate that the existence of the core is an important determinant of negotiations generally and the Coase theorem in particular. They find that when the core is empty and property rights are ill defined, Coasean efficiency breaks down. In particular, the number of non-Pareto optimal agreements and negotiation rounds with cycling are significantly larger when the core is empty than when it exists, particularly when property rights are ill defined.

Conclusion

The upshot of the empty core issue for the Coase Theorem can be summarized as follows (Aivazian and Callen 2003, p. 296):

“In the real world opportunities for exchange are sometimes manifold and the bargaining strategies potentially complex. The Coase Theorem masks this reality by presupposing that exchange occurs between two parties … with more than two parties, and at least two externalities, coalitional behavior may predominate. In which case, under some property rights arrangements the core may not exist; as a result, the Coase Theorem may fail to hold. Transactions costs may well exacerbate the empty core problem. In such circumstances, specific property rights arrangements, and contractual schemes such as penalty clauses, binding contracts, and restrictions on the sequence of bargaining, may emerge to attenuate the problems engendered by the nonexistence of the core”

Cross-References