Indicators and indexes of directional output loss and input allocative inefficiency

We extend Grosskopf et al.'s method and create an output loss indicator using directional output distance functions that allows non‐radial efficiency gains in output. We compare our new output loss indicator with an indicator of input allocative inefficiency and derive the necessary and sufficient condition for equivalence between the two indicators. We also present an output loss index and corresponding input allocative inefficiency index and consider how indexes are related. Then we extend our analysis to productivity change and derive the necessary and sufficient condition for an output loss change indicator to be equivalent to an indicator of input allocative inefficiency change.