Has anyone formalised Coase's argument?

Question

Coase's 'theorem' says that under certain conditions (low 'transaction costs', 'full information'?), bargaining can be expected to produce Pareto efficient outcomes. Of course, Coase did not prove a theorem as such. My question, then, is whether Coase's argument has been formalised into a theorem.

Before I pose the question, let me briefly comment on what do not seem to be adequate formalisations:

• I remember hearing that Coase's argument is 'just' an application of the First Welfare Theorem (FTW). The idea is that, if there is a market for all goods (e.g. for pollution rights), then we can just apply the FTW to show that the outcome must be efficient. This strikes me as completely wrong. The FWT assumes that outcomes are generated through competitive markets (e.g. there must be a market for 'pollution rights' whose price everyone takes as given). If you read Coase, however, you'll see that this is absolutely not what he had in mind. Instead, Coase imagined a process of bargaining (not price taking!); e.g. the polluter might be bribed to cut back on their pollution.

• One might alternately try to formalise Coase within the context of a bargaining model, e.g. Rubinstein's alternating offers model. Such models can yield efficient outcomes (e.g. this happens under alternating offers). However, I believe that Coase's insight is not supposed to depend on the exact details of the bargaining process; so one might worry that this approach fails to correctly capture the key to his argument.

Answer 1

My question, then, is whether Coase's argument has been formalised into a theorem.

Short answer is yes. See:

Aivazian V.A., Callen J.L. (2017) The Coase Theorem and the Theory of the Core. In: Marciano A., Ramello G. (eds) Encyclopedia of Law and Economics. Springer, New York, NY.

Aivazian and Callen (1981) and a number of their subsequent papers use cooperative game theory and core theory to show that the Coasean efficiency result is not robust when there are more than two players. Drawing primarily on their results, this chapter systematically explains the main argument and its extensions as follows. First, the Coase theorem could 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. Second, transaction costs will tend to aggravate the empty core problem and make it more likely that the Coasean efficiency result will fail. Third, Pareto optimality can be achieved when the core is empty by the imposition of constraints on the bargaining process and the use of penalty clauses and binding contracts. Overall, the results indicate that it is important to distinguish between transaction costs (when the core exists) and costs due to the empty core because each has different implications for rationalizing institutions. This chapter also summarizes experimental results indicating that the existence of the core is an important determinant of negotiations generally and the Coase theorem in particular. It also points out that some of the problems raised for Coasean efficiency by the empty core also arise under alternative (non-core) notions of coalitional stability.

and

Zhao J. (2018) A Reexamination of the Coase Theorem. Journal of Mechanism and Institution Design 3: 111-132.

This paper makes three advances: 1) It fixes the empty-core problem of the Coase theorem; 2) it provides the smallest upper bound of transaction costs below which the optimal or efficient outcomes can be achieved; and 3) it establishes two mathematical theorems that capture the main insights and major aspects of the Coase theorem. A simpler version of the theorems says that in a coalitional production economy without transaction costs, the maximal payoff will be produced by the optimal firms and be allocated in the always non-empty core.