Imposing concavity and the null-jointness property on the production possibilities frontier in case of polluting technologies

Economic theory requires the directional distance functions used to study the properties of production possibility sets of polluting technologies to be concave in both outputs, while the implied production possibilities frontier (PPF) is required to be concave with respect to the bad output. However, existing estimation frameworks do not preclude the estimation of convex PPFs. We analyze geometrical properties of the quadratic approximation to the directional output distance functions to derive a constraint that guarantees PPF concavity and consider the issue of imposing the property of null-jointness on the production possibilities set, which is also required by theory. We simulate a dataset corresponding to a concave PPF and show that in case concavity and null-jointness constraints are not imposed, it is possible that the conventional estimation framework may lead to erroneous conclusions with respect to the type of curvature of both the directional output distance function, and the PPF.