Productivity Interpretations of the Farrell Efficiency Measures and the Malmquist Index and Its Decomposition

The ratio definition of efficiency has the form of a productivity measure. But the weights are endogenous variables and they do not function well as weights in a productivity index proper. It is shown that extended Farrell measures of efficiency can all be given an interpretation as productivity measures as observed productivity relative to productivity at the various projection points on the frontier. The Malmquist productivity index is the efficiency score for a unit in a period relative to the efficiency score in a previous period, thus based on a maximal common expansion factor for outputs or common contraction factor for inputs not involving any individual weighting of outputs or inputs, as is the case if a Törnqvist or ideal Fisher index is used. The multiplicative decomposition of the Malmquist productivity index into an efficiency part and a frontier shift part should not be taken to imply causality. The role of cone benchmark envelopments both for calculating Malmquist indices of productivity change and for decomposing the indices into an efficiency change term and a frontier shift term is underlined, and connected to the index property of proportionality and circularity, adding the use of a fixed benchmark envelopment. The extended decomposition of the efficiency component by making use of scale efficiency is criticised.