Modifying the example.

Now consider a slightly modified version of our previous example. Suppose now that if the factories merge, they enjoy joint profits of 12 instead of 11. This could be because there are economies of scale in the production of the good produced by the factories, which makes their joint profits higher than the sum of their profits separately.

Also, suppose that the externalities are not symmetric. Specifically, suppose factory 1 imposes a negative external cost of 9 on the residents, whilst factory 2 imposes a negative external cost of 4. All other values remain the same as in our previous example. The efficient outcome is again for both factories to shut down.

Let us now consider the characteristic functions of the games that are generated by our two legal regimes for this new example. Under legal regime I, we have:

Does there exist an efficient agreement between the parties which is stable against all threats? We proceed as we did before. Suppose that the agreement involves payments x1, x2 and xR to each of the players. Individual rationality and efficiency require:

This situation is illustrated in Figure 3.6.1. The diagram is an equilateral triangle with a height of 40. The distance from any point in the triangle to the side opposite the vertex labelled with a party represents the payment that the party on that vertex receives. So, for example, the vertex labelled xR = 40 represents set of payments where the residents receive 40 and F1 and F2 receive nothing.

In addition to individual rationality and efficiency, the payments must also be stable against other possible sets of agreements. One

Diagrammatic representation of the three-party cooperative game

Figure 3.6.1 Diagrammatic representation of the three-party cooperative game

possibility is that instead of both agreeing to shut down, the factories F and F2 merge and continue to produce. To prevent this from occurring, they must be paid at least 12 jointly. In other words:

Since x1 + x2 + xR = 40, this means that we must also have xR < 28. Similarly, to prevent other agreements between groups of only two parties from being attractive, we must have:


Again, these two inequalities, together with the feasibility constraint, imply that x2 < 9 and x1 < 4.

The shaded area in Figure 3.6.2 shows the range of allocations which satisfy all of these inequalities. Clearly, this set is non-empty. For example, the allocation:

where 0 < a < 1, satisfies all of the inequalities discussed above.

Thus, if the residents pay factory 1 an amount equal to 4 - a, and pay factory 2 an amount equal to 8 + a, they will obtain utility of 40,

Diagrammatic representation of the three-party cooperative game with non-empty core

Figure 3.6.2 Diagrammatic representation of the three-party cooperative game with non-empty core

less total payments of 12, giving them final utility of 28. Both factory 1 and factory 2 would accept these payments, since they are both strictly better off than they were producing on their own. In addition, 1 and 2 do not have an incentive to merge and keep producing, since they would not gain from doing so. Similarly, neither the factories nor the residents would have an incentive to break this agreement and negotiate a separate agreement in which only one of the factories agrees to shut down. The 0-1 normalisation of this situation has z1 + z2 + zR = -3 + -8 + -24 = -35 and K = 5, and so:

Since Vj (12) + Vj (2 R) + vR (1R) = 0.2 + 0.8 + 0.8 = 1.8, any allocation that is in the core will have to have 2(x1 + x2 + xR) > 1.8, which is permissible given that we also have x1 + x2 + xR = 1 for allocations in the core.

Now suppose that the residents have the property right to prevent the factories from producing. The residents would choose production levels of zero for both (which is efficient), and there are no payments that the factories could make which would induce the residents to accept a positive level of production. The core again consists of a single point, (0,0,40). Thus, we have illustrated what is actually a more general result: if the core is non-empty, then the invariance and efficiency versions of the Coase Theorem both hold.

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