Generalised commensurability properties of efficiency measures: Implications for productivity indicators

We analyse the role of new weak and strong commensurability conditions on efficiency measures and especially on productivity measurement. If strong commensurability fails, then a productivity index (indicator) may exhibit a homogeneity bias yielding inconsistent and contradictory results. In particular, we show that the Luenberger productivity indicator is sensitive to proportional changes in the input-output quantities, while the Malmquist productivity index is not affected by such changes. This is due to the homogeneity degree of the directional distance function under constant returns to scale. In particular, the directional distance function only satisfies the weak commensurability axiom in general. However, if the directional distance function is a diagonally homogeneous function of the technology, then the directional distance function satisfies strong commensurability. This explains why the direction of an arithmetic mean of the observed data works well. Numerical examples and an empirical illustration are proposed. Under a translation homothetic technology, the Luenberger productivity indicator is not affected by any additive directional transformation of the observations.