On two definitions of productivity

Caves et al. in 1982 introduced a productivity measure ''that does not proceed from a continuous timer presentation''. They named these indexes after Sten Malmquist who in 1953 used ratios of distance functions to formulate quantity indexes in the consumer context. In a later paper, Diewert proposed an alternative definition of productivity also based on distance functions, which he attributed to Hicks and Moorsteen. In this letter we provide necessary and sufficient conditions for the Malmquist productivity index to equal the Hicks-Moorsteen index.