Measuring technical efficiency under variable returns to scale using Debreu's loss function

This paper proposes a model that makes two contributions to the measurement of technical efficiency under a technology with variable returns to scale. First, the criteria for identifying an optimal benchmark are not limited to technical dominance and Pareto efficiency, but also include maximum average productivity, defined as the ratio between a weighted linear aggregate of outputs and inputs.

Second, the paper contributes a conceptual basis for correcting the shadow prices of inputs and outputs to reflect the influence of returns to scale. Debreu's loss function is used to value inefficiency as the difference between the virtual input and output using the shadow prices of the supporting hyperplane at the optimal reference. The efficiency score is a virtual profitability index with endogenous shadow prices that reflect the valuation of inputs and outputs with a microeconomic rationale, i.e., it is not a distance measure based on aggregation with exogenous weights of the difference between observed and optimal quantities.

Two further results follow from these contributions. First, the radial input-output orientation to maximise productivity is endogenous. It is conditioned by the nature of the returns to scale. Second, the efficiency measure based on the loss function exhibits the desirable properties in a radial context, including the indication property, because the efficiency score incorporates non-radial slack.