# The core and transaction costs

The example in the previous section showed that even in the absence of transaction costs, the legal rule will, in general, matter for both production and efficiency, unless we add auxiliary legal rules. As we show in this section, introducing general transaction costs into the analysis does not alter this conclusion. The analysis in this section shows that if general rather than specific transactions costs are introduced, a situation in which the core is empty does not transform the outcome into one where we have a non-empty core, just because general transaction costs are present. Moreover, in situations in which the core *is* non-empty, introducing general transaction costs can transform the situation into one in which the core suddenly becomes empty. These results are due to Aivazian and Callen (2003).

Following Aivazian and Callen (2003), let us illustrate these results using the following simple specification of transaction costs:

where *k >* 1. These transaction costs are illustrated in Figure 3.6.5 for various coalition sizes and various values of k.

Following Aivazian and Callen (2003), consider the following characteristic function:

Suppose that the core is non-empty. A sufficient condition for this to be the case is as follows. Let *x _{1} >* 0,

*x*0,

_{2}>*x*0 be in the core. Then we require:

_{R}>Which, summing across all three inequalities, implies:

*Figure 3.6.5* Transaction costs which increase with the size of the coalition

Thus, suppose that *d > *^{a} + ^ + ^{C}. To see if the legal rule matters for

efficiency here, suppose again that there is a rule of no liability in place and that F_{1} and F_{2} produce separately. This means that the initial utility levels are v(*F _{1}) =* 0, v(

*F*0, v(

_{2}) =*R) =*0. To be individually rational, the payments must satisfy:

The agreement between F_{1}, F_{2} and R must also be efficient and feasible. When transaction costs are present this 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 *a* jointly, less the transaction costs incurred when they negotiate. In other words:

Similarly, to prevent other agreements from being attractive, we must have:

and:

Again, adding these inequalities together yields the following condition for the core to be non-empty:

Therefore, for the core to be non-empty, we require: or:

The right-hand side is positive as long as *k **>* 1, which we have assumed to be the case. Thus, if *d* is positive and less than 2.3^{к} - 3.2^{к}, *even if the core is not empty in the absence of transaction costs, it will become empty once transaction costs are introduced.* In such situations, neither the invariance version nor the efficiency version of the Coase Theorem will hold.