# Is the result a general one? Searching for examples with an empty core

Does the Coase Theorem still hold if the core is empty? And what does it take in our example for the Coase Theorem to break down? We now show how the result stated above breaks down in the presence of an empty core, and isolate the source of the empty core.

Only a slight modification of the previous example is needed to show this. Consider the same example as in the previous section, but now suppose that if F_{1} and F_{2} merge, they can realise joint profits of 15 rather than 12. This seems like a trivial modification. After all, our previous example featured economies of scale, this example simply makes those synergies slightly larger. However, as we will see, modifying the example in this way increases the opportunity cost of F_{1} and F_{2} entering into agreement with the residents. This increase in opportunity cost means that F_{1} and F_{2} must receive greater compensation for entering such an agreement, which in turn means that the residents' net benefit is lower than in the previous case. As we will show, these changes render any agreement - even though it is efficient - unstable.

Under legal regime I, the characteristic function is now:^{9}

Once again this situation can be illustrated diagrammatically (see Figure 3.6.3 below).

Does there exist a stable agreement between all three parties? Again, to be individually rational, the payments must satisfy:

The agreement between F_{1}, F_{2} and R must also be efficient and feasible, which again requires:

In addition, the payments must also be stable against other possible sets of agreements. One possibility is that instead of agreeing to shut down, F_{1} and F_{2} merge and continue to produce. To prevent this from occurring, they must be paid at least 15 jointly. In other words:

*Figure 3.6.3* Diagrammatic representation of the three-party cooperative game with an empty core

Since *x _{1} + x_{2} + x_{R} =* 40, this means that we must also have

*x*25. Similarly, to prevent other agreements from being attractive, we must have:

_{R}<

and:

Again, these two inequalities, together with the feasibility constraint, imply that x_{2} < 9 and x_{1} < 4. But these last two inequalities are clearly inconsistent with the requirement that x_{1} + x_{2} > 15.

Note that in this particular example there are actually *three* externalities at work here. The first two externalities are negative, and are caused by A and B's production reducing C's profit. The second externality is a positive externality - A and B producing together increases the joint profits of both of them. The size of this second externality turns out to be crucial in this example.

To see this even more clearly, adding (3.15), (3.16) and (3.17) gives: which implies:

But this is not possible. In other words, if (3.15), (3.16) and (3.17) hold, then (3.14) cannot hold. Conversely, if (3.14) holds, then at least one (and possibly all) of (3.15), (3.16) and (3.17) cannot hold.

In other words, there is no set of payments which is stable here. To illustrate, suppose R pays F_{2} not to produce. Then R would offer F_{2} a maximum of 36 - 24 = 12. F_{2} would accept a minimum of 8, so suppose R pays F_{2} $8. But F_{2} and F_{1} could come to an agreement in which they both produce, and in which F_{2} receives 8.5 and F_{1} receives 6.5 > 3. Thus F_{2} and F_{1} are better off. F_{2} has no incentive to sign an agreement with R under the original terms.

But now notice that F_{1} and R could form an agreement that makes them both better off than in the situation just examined. In turn, this agreement could be bettered by F_{2} and R - and we are back to where we started. In other words, we are in a cycle of endless negotiating and renegotiating. Thus, beginning from a situation of no liability,

there are no stable two firm coalitions - and the grand coalition is unstable as well. This outcome is very different from the outcome that would obtain if the residents had the property right. Under this legal rule, the residents would again choose production levels of zero for both factories (which is the efficient outcome), and there are no payments that the factories could make which would induce the residents to accept a positive level of production. Thus, *if the core is empty, neither the invariance version nor the efficiency version of the Coase Theorem hold.*

To gain an even deeper understanding of what is going on here, note that the 0-1 normalisation of this situation has *z** _{x}* + z

_{2}+

*z*

*= -3 + -8 + -24 = -35 and*

_{R}*K*

*=*5, and so the normalised version of this game has the following characteristic function:

In other words, *the Aivazian-Callen (1981) example is strategically*

2

*equivalent to a simple majority game in which в**=* 0.8 > 3. This is illustrated in Figure 3.6.4, where the set of points that satisfy the requirements of the core are inside the triangle which is drawn with a broken line, but lie outside the triangle drawn with a solid line. In other words, there is no allocation which satisfies:

*Figure 3.6.4* Diagrammatic representation of the 0-1 normalisation of the three- party cooperative game with an empty core