Using global and local measures to decompose the Malmquist productivity index assuming a variable returns-to-scale technology

Traditional decompositions of the Malmquist productivity index (MPI) often obscure important drivers of total factor productivity change, particularly under variable returns to scale (VRS). Existing VRS-based MPI decompositions rely on averages of local technical changes across individual firms, which may not always be feasible for some production units. This paper proposes a novel five-way decomposition of the MPI based on the Lebesgue measure to address these limitations. We introduce global measures of pure global technical change and global scale efficiency change, together with their local counterparts: pure local technical changes and local scale efficiency change. Our approach employs artificial units randomly generated within the normalized input–output space of a unit hypercube, ensuring robust calculations while avoiding infeasible solutions. The resulting framework provides a more accurate representation of productivity dynamics, illustrated empirically using real-world data. This decomposition opens new avenues for productivity analysis across industries and establishes a foundation for further methodological development.