Production trade-offs in models of data envelopment analysis with ratio inputs and outputs: An application to schools in England

In applications of data envelopment analysis (DEA), the inputs and outputs representing environmental and quality characteristics of the production process are often stated in the form of percentages, ratios and averages, collectively referred to as ratio measures. It is known that the conventional variable and constant returns-to-scale (VRS and CRS) DEA models cannot correctly incorporate such ratio inputs and outputs. This problem has been addressed by the development of Ratio-VRS and Ratio-CRS (R-VRS and R-CRS) models suitable for the incorporation of both volume and ratio inputs and outputs. Such models may, however, depending on the application, lack sufficient discriminatory power. In this paper we address this issue by developing a further extension of the R-VRS and R-CRS models (the latter with the most common fixed type of ratio inputs and outputs) by allowing the specification of production trade-offs between volume inputs and outputs, and, similarly, between ratio measures. As in the case of conventional VRS and CRS models in which the role of production trade-offs is well understood, the specification of such trade-offs in the R-VRS and R-CRS production technologies leads to their controlled expansion and results in improved efficiency discrimination of the resulting DEA models. We illustrate the application of the proposed methodology by the assessment of efficiency of a large sample of secondary schools in England.