Duality of Shephard’s weakly disposable technology under a directional output distance function

In a recent paper Kuosmanen and Matin (2011) have developed a dual formulation of the weakly disposable Kuosmanen’s technology (Kuosmanen, 2005). Their work sheds new light on the economic interpretation of weak disposability and allows estimation of shadow prices for undesirable/bad outputs. More precisely, they have derived the multiplier formulation for the Kuosmanen weakly disposable technology and have found a ‘limited liability condition’ as the economic interpretation of weak disposability. Considering further the traditional Shephard technology, they argued that: “Obviously, the duality theory of linear programming does not apply to the classic Shephard technology because it is generally non-convex. This explains why the dual formulations of the weakly disposable technology are currently unavailable” (p.505). We prove here that by considering a directional output distance function there exists a duality for the Shephard weakly disposable technology. Moreover it turns out with a clear, intuitive and relevant economic interpretation of the weak disposability axiom. Thanks to our dual framework, we finally provide an original proof of the existing relationship between weak disposability and returns to scale.