Key points are not available for this paper at this time.
Douglas W. Caves, Laurits R. Christensen, W. Erwin Diewert, The Economic Theory of Index Numbers and the Measurement of Input, Output, and Productivity, Econometrica, Vol. 50, No. 6 (Nov., 1982), pp. 1393-1414
Caves et al. (Mon,) studied this question.