This paper develops index number procedures for making comparisons under very general circumstances. Malmquist input, output, and productivity comparisons are defined for structures of production with arbitrary returns to scale, substitution possibilities and biases in productivity change. For translog production structures, Tornqvist output and input indexes are shown to equal the mean of two Malmquist indexes. The Tornqvist productivity index, corrected by a scale factor, is shown to equal the mean of two Malmquist productivity indexes. Similar results are given for making cost of living comparisons under general structures of consumer preferences.