Production technologies with ratio inputs and outputs

Applications of data envelopment analysis often incorporate inputs and outputs stated as proportions or percentages, which are typically used to represent socio-economic and quality characteristics of the production process. As is well known, the use of such ratio measures is inconsistent with the assumption of convexity required by the conventional variable and constant returns-to-scale (VRS and CRS) technologies, and with the additional assumption of scalability in the case of CRS. Several existing approaches to modelling technologies with ratio data assume that either we know the exact volume numerators and denominators of all ratio measures or, alternatively, that we do not have such information. The former approaches are not always realistic and the latter are equivalent to benchmarking each decision making unit against a significantly reduced subset of observed units, which has a negative impact on the discriminating power of the model. In this paper, we develop new technologies under the assumptions of VRS and CRS that bridge the gap between the two known approaches. They are applicable in a general scenario in which we can specify some lower and upper bounds for the numerators or denominators of the ratio measures, which should be unproblematic in most practical settings. We demonstrate the usefulness and advantages of the developed approach by an application in the context of school education.