Loss Distance Functions and Profit Function: General Duality Results

  • Juan AparicioEmail author
  • Fernando Borras
  • Jesus T. Pastor
  • Jose L. Zofio
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 249)


The concept of loss distance functions is introduced and compared with other functional representations of the technology including the Hölder metric distance functions (Briec 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’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.


Loss distance functions Directional distance functions Hölder distance functions Duality Profit function 


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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Juan Aparicio
    • 1
    Email author
  • Fernando Borras
    • 2
  • Jesus T. Pastor
    • 1
  • Jose L. Zofio
    • 3
  1. 1.Center of Operations Research (CIO)Universidad Miguel Hernandez de ElcheElche, AlicanteSpain
  2. 2.Department of Statistics, Mathematics and Computer ScienceUniversity Miguel Hernandez of ElcheAlicanteSpain
  3. 3.Departamento de Analisis Economico: Teoria Economica e Historia EconomicaUniversidad Autonoma de MadridMadridSpain

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