Malmquist productivity index in DEA for special cases

The Malmquist Productivity Index (MPI) is a major development in data envelopment analysis (DEA), aiming to assess productivity changes over time. The MPI estimates the total factor productivity growth of a decision-making unit (DMU) with multiple inputs and outputs. However, the special case of a single input alongside multiple outputs or multiple inputs with a single output has not been thoroughly studied. These configurations have some unique features, particularly regarding computational costs and the selection of lower bounds—commonly known as non-Archimedean epsilon—for the dual input and output weights in the multiplier DEA model. Moreover, DEA-based MPI often overlooks the role of epsilon in determining efficiency measures. This paper proposes a new approach for measuring DEA-based MPI with epsilon in single-input or single-output data sets. Four scenarios are designed, each based on varying epsilon values, to address settings with a single input and multiple outputs (SIMO) as well as settings with multiple inputs and a single output (MISO), with the aim of determining the optimal epsilon values. The proposed method ensures that no input or output is deemed non-instrumental to the production technology when estimating the MPI. The methodology’s effectiveness is demonstrated through an analysis of productivity growth in 18 OECD countries from 2005 to 2021.