The duality of Shephard’s weakly disposable technology

A part of the recent literature on the treatment of undesirable outputs by weakly disposable models advocates for the use of the Kuosmanen approach to weak disposability. It defines the minimum extrapolation technology satisfying the disposability assumptions specific to weakly disposable models under three different assumptions on convexity: convexity of the technology set, convexity of output sets only and no convexity at all. We contribute to this taxonomy by adding the case of assuming both input and output sets convex, restoring the classical Shephard approach to weak disposability on the output correspondence. After defining the technology through the definition of a directional distance function, we also linearize the corresponding program and show how the application of duality results in a clear, intuitive, and economically relevant interpretation of the assumption of weak disposability on outputs. Finally, we provide an original proof of the existing relationship between weak disposability and returns to scale.