Abstract
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.
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Notes
- 1.
Shephard (1970, p. 223), in his classic book “Theory of Cost and Production Functions”, used a similar argument to prove that the infimum in the cost function is always achieved on the production possibility set. In other words, “inf” can be changed to “min” in the definition of the cost function.
- 2.
Note the similarities between this definition and the minimization of Debreu’s dead loss function.
- 3.
The Hölder norms \( \ell_{q} \) are defined as \( \left\| {(c,p)} \right\|_{q} = \left\{ {\begin{array}{*{20}l} {\left[ {\sum\nolimits_{i = 1}^{m} {\left| {c_{i} } \right|^{q} } + \sum\nolimits_{r = 1}^{s} {\left| {p_{r} } \right|^{q} } } \right]^{{{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-0pt} q}}} ,} \hfill & {{\text{if}}\;\;q \in \left[ {1, + \infty } \right)} \hfill \\ {\hbox{max} \left\{ {\left| {c_{1} } \right|, \ldots ,\left| {c_{m} } \right|,\left| {p_{1} } \right|, \ldots ,\left| {p_{s} } \right|} \right\},} \hfill & {{\text{if}}\;\;q = + \infty } \hfill \\ \end{array} } \right. \).
- 4.
The set of weakly efficient points may be described symmetrically as the set of \( \left( {x,y} \right) \) satisfying the equation \( F\left( {x,y} \right) = 0 \), where \( F \) is the transformation function. Alternatively, one output can be singled out, for example \( y_{1} \), and the weakly efficient set may be described asymmetrically by \( y_{1} = F^{\prime}\left( {x,y_{ - 1} } \right) \) where \( F^{\prime} \) is the transformation function.
- 5.
In that case, \( NS = \left\{ {\left( {c,p} \right) \in R_{ + }^{m + s} :\,\left\| {\left( {c,p} \right)} \right\|_{1} \ge 1} \right\} \) where \( \left\| {\left( {c,p} \right)} \right\|_{1} = c1_{m} + p1_{s} \).
- 6.
In particular, a weighted norm \( \ell_{1} \).
- 7.
If the vector \( g \) has some zero-components then its norm associated does not satisfy the basic property \( \left\| z \right\| = 0\, \Leftrightarrow \,z = 0_{m + s} \).
- 8.
For example, we could assume that Slater’s constraint qualification holds, i.e., there exists at least a point \( \left( {\tilde{c},\tilde{p}} \right) \in R_{ + + }^{m + s} \) such that \( h\left( {\tilde{c},\tilde{p}} \right) > 1 \) (see Mangasarian 1994; p. 78).
- 9.
The same idea was already used by Pastor et al. (2012) in a quite different context, i.e., for modelling the DEA efficiency models in a unified way.
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Aparicio, J., Borras, F., Pastor, J.T., Zofio, J.L. (2016). Loss Distance Functions and Profit Function: General Duality Results. In: Aparicio, J., Lovell, C., Pastor, J. (eds) Advances in Efficiency and Productivity. International Series in Operations Research & Management Science, vol 249. Springer, Cham. https://doi.org/10.1007/978-3-319-48461-7_4
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