Interpreting profit-related inefficiencies as Euclidean distances in the context of directional distance functions and data envelopment analysis

In the context of Data Envelopment Analysis (DEA) and Directional Distance Functions (DDF), we introduce an exogenous directional vector based on market prices, normalized via the Euclidean norm, which provides an intuitive geometric interpretation of inefficiency measures. Specifically, we demonstrate that normalized profit inefficiency can be represented as the Euclidean distance between the maximum profit hyperplane and the observed profit hyperplane, allowing for a clear visualization of economic inefficiency gaps. The proposed methodology effectively decomposes overall profit inefficiency into its technical and allocative components: technical inefficiency corresponds to the Euclidean distance from a Decision Making Unit (DMU) to its projection point on the efficient frontier, while allocative inefficiency emerges as the residual gap between the observed and optimal profit conditions. By employing a fixed, market-driven directional vector, our approach ensures direct comparability and industry aggregation of efficiency scores across DMUs. We compare the Euclidean model with other proposals in the literature that measure and decompose profit inefficiency using endogenous directions, and illustrate it empirically using data on financial institutions.