'Rescuing' the Coase Theorem, part I: The existence of multiple externalities requires multiple legal rules

Can the Coase Theorem be rescued in the three-party example? One possibility, following Mueller (2003), is to note the fact that there are really multiple externalities in this example: since the factories can merge and obtain a higher payoff that exceeds the sum of the payoffs they could obtain separately, there is also a positive externality between the factories. However, it is also important to note that it was only when this additional externality became sufficiently large that problems emerged. In any case, recognising this, let us introduce an additional legal rule to address this additional externality. After all, the Coase Theorem states that the legal rule does not matter for efficiency as long as some legal rule is in place.

Suppose, then, that we introduce an 'auxiliary' legal rule that addresses the positive externality and governs the actions that F: and F2 are permitted to take on their own. For example, suppose that there is a rule of no liability for pollution, but that this is combined with a legal rule that states that F: and F2 must effectively act as a merged entity and cannot produce separately. This rules out the coalitions {F1}, {F2}, {F1R} and {F2R}. Then, if a no liability rule applies to the factories, agreements between the merged entity {F1F2} and the residents will be stable and so the efficient outcome will occur.

On the other hand, suppose that the auxiliary legal rule states that F1 and F2 cannot act as a merged entity and must therefore produce separately, but that they are permitted to enter into contracts with the residents, both individually and separately. This would rule out the coalition {F1F2}. Then, again, if the grand coalition was to form there is an allocation which is stable. For example, the allocation x1 = 3 + a, x2 = 9 - a, xR = 28 where 0 < a < 1 is stable against deviations from sub-coalition when the coalition {F1F2} is not permitted to form.

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