where SUB S ðZÞ is the degree of cost subadditivity at Z when m ¼ l. Generally, m 6¼ l. When m ¼ l,
the output vector of any duopolistic firm is proportionate to the output vector of the original monopolist. Therefore, it is reasonable to select such a particular case (that is, the case when
m ¼ l) as a benchmark for our analysis. It follows that a value of SPECðZ; kÞ greater than one
implies that specialized production is a source of efficiency for the provision of rail services.
In order to preserve the validity of the subadditivity test, some restrictions have to be imposed
on the output levels that can be produced by any duopolistic firm. These define an admissible
region that considers only cost configurations which are compatible with the range of output
levels observed in the data. Thus, no hypothetical firm in the two-firm industry should be allowed
to produce less of either output than the firm for which data are available. Moreover, any hy-
pothetical firm in the two-firm industry should be precluded from specializing in either output
to a greater extent than the firm for which data are available (see Evans and Heckman, 1984).
Finally, Roller (1990) introduced the additional constraint that marginal costs must be positive.