A remark on the definition of the Färe-Lovell measure for infeasible production vectors: implication for the Luenberger productivity indicator

In a recent paper Mahlberg and Sahoo (2011) introduced a non-radial Luenberger indicator in order to avoid the problem of slacks. They suggest to evaluate the Luenberger indicator by using a directional slacks-based measure. This paper demonstrates, by numerical examples, that directional slacks-based measures do not allow to characterize the technology. Along this line, it follows that a Luenberger indicator based upon a directional slacks-based measure may have some drawback. Mahlberg and Sahoo (2011) mention these problems and suggest to replace non-negative signs of the input-specific inefficiency factors by non-positive signs. In this paper an alternative formulation is proposed. It characterizes both the input (output) sets and the input (output) efficient subset. Its application to the construction of a suitable Färe-Lovell type productivity indicator is suggested.