The structure of production technologies with ratio inputs and outputs

Applications of efficiency and productivity analysis in which some inputs and outputs are given in the form of percentages, averages and other types of ratio measures are sufficiently common in the literature. In two recent papers, the authors developed the variable and constant returns-to-scale technologies with both volume and ratio types of inputs and outputs, referred to as the R-VRS and R-CRS technologies. These technologies are generally nonconvex and have a complex structure. In this paper we explore this in detail. We show that the R-VRS technology can be stated as the union of a finite number of specially constructed standard VRS technologies. Similarly, the R-CRS technology in which all ratio inputs and outputs are of the fixed type, which are typically used to represent environmental and quality factors, can be stated as the union of a finite number of partial polyhedral cones. We show that these results have important theoretical, including conceptual, implications.