Applying the concept of the core: The Coase Theorem with three or more parties

An example with a non-empty core.

To illustrate the main issues involved in situations in which there are three or more parties, consider the following example. There are two factories, F1 and F2, and a group of residents, R. Both factories emit pollution and this reduces the well-being of the residents. In the absence of production by the firms, the residents are assumed to enjoy utility of 40. If both firms produce, the residents' utility falls to 24. Therefore, production by both factories imposes a negative external cost of 16 (= 40 - 24) on the residents. If only one firm produces, the utility of the residents increases from 24 to 32. So production by only one firm imposes a negative external cost of 8. The marginal external cost to the residents is therefore constant.

If the factories produce alone, they can earn profits of 3 and 8 respectively. If they merge and produce together, then they can jointly earn profits of 11. Therefore, there are no scale effects in production. Clearly, the efficient outcome here is for both factories not to produce.

Suppose first that the factories can produce as much as they wish. Call this legal regime I. Let the characteristic function of this game be v. Denote the value of the grand coalition by v/12R). Then v_{I}(12R) = 40, the total utility available to the parties when the factories and the residents agree that the factories should shut down production. Notation for other possible coalitions is defined in a similar fashion. The characteristic function in this situation is:

Does there exist an efficient agreement between the parties which is stable against all threats? Suppose that the agreement involves payments x_{v} x_{2} and x_{R} to each of the players. To be individually rational, the payments must satisfy:

The agreement between F_{1}, F_{2} and R must also be feasible, and if it is efficient it will not waste any resources. This requires:

In addition, the payments must also be stable against other possible sets of agreements. One possibility is that instead of both agreeing to shut down, the factories F_{1} and F_{2} merge and continue to produce. To prevent this from occurring, they must be paid at least 11 jointly. In other words:

Since x_{1} + x_{2} + x_{R} = 40, this means that we must also have x_{R} < 29. Similarly, to prevent other agreements between groups of only two parties from being attractive, we must have:

and:

Again, these two inequalities, together with the feasibility constraint, imply that x_{2} < 8 and x_{1} < 8. The former condition, together with the restriction that x_{2} > 8, implies that x_{2} = 8. The gains from trade are all shared between factory 1 and the residents. Therefore, any set of

agreements which satisfies the following set of inequalities will be stable against deviations by any deviation by a single party, or any deviation by a subcoalition of 2 parties:

where 0 5. The set of payments in (3.12) is the core of the game when the factories have the legal right to emit pollution.

Now suppose that the factories must first obtain the residents' permission to produce. Call this legal regime II, and let the characteristic function of this game be v_{n}. Denote the value of the grand coalition by v_{n}(12R). Then again we have v_{n}(12R) = 40. The characteristic function in this situation is:

There is no payment that the factories could make to persuade the residents to let them produce. The core is still non-empty, but has a very simple structure: it is simply the point x_{1} = 0, x_{2} = 0, x_{R} = 40. Note, however, that the outcome is the same as that under legal regime I, and is still efficient. Hence both versions of the Coase Theorem hold in this example.

Another way of seeing what is going on in this example is to examine the 0-1 normalisation of the game under each legal rule. Consider legal regime I. The 0-1 normalisation has z_{1} + z_{2} + z_{R} =-3 + -8 + -24 = -35 and K = 5, and so:

Clearly firm 1 and the residents can share the gains from trade between themselves, without giving anything to firm 2. On the other hand, consider legal regime II. Its 0-1 normalisation is v(S) = 0 for any coalition. This is as it should be, since there are no gains from trade if the residents already have the property right and this is the efficient outcome.