α-Returns to scale and multi-output production technologies

This contribution proposes a specification of strictly increasing and decreasing returns to scale in multi-output technologies. Along this line a notion of α-returns to scale is derived from that of homogeneous multi-output technology. For a large class of technologies we establish necessary and sufficient conditions characterizing strictly increasing and strictly decreasing returns to scale to scale. Furthermore, a relationship between input, output and graph distance functions is established. These connections lead naturally to a link between the various Malmquist indexes and the Chavas–Cox productivity index. Finally, we show that these concepts can be implemented in a DEA context using a piecewise homogeneous constant elasticity substitution–transformation model due to [Färe, R., Grosskopf, S., Njinkeu, D., 1988b. On piecewise reference technologies. Management Science 34, 1507–1511].