Loss Distance Functions and Profit Function: General Duality Results

The concept of loss distance functions is introducedPastor, J.T. and compared with other functional Borras, F. representations of the technology including the Hölder metric distance functions (BriecBriec, W. and Lesourd in J Optim Theory Appl 101(1):15–33, 1999), the directional distance functions due to Chambers et al. (J Econ Theory 70(2):407–419 1996; J Optim Theory Appl 98(2):351–364 1998), and the Shephard, R.W. Shephard’s input and output distance functions as particular cases of the directional distance functions. Specifically, it is shown that, under appropriate normalization conditions defined over the (intrinsic) input and output prices, the loss distance functions encompass a wide class of both well-known and much less known distance functions. Additionally, a dual correspondence is developed between the loss distance functions and the profit function, and it is shown that all previous dual connections appearing in the literature are special cases of this general correspondence. Finally, we obtain several interesting results assuming differentiability.