Graph Efficiency and Parametric Distance Function with an Application to Spanish Savings Banks

Distance functions were initially defined on the input or output production possibility sets by Shephard (1953, 1970) and extended to a graph representation of the technology by Färe, Grosskopf and Lovell (1985) through their graph hyperbolic distance function. Since then, different techniques, non-parametric-DEA and parametric-SFA, have been used in order to calculate these distance functions, but in the latest case no study is known to have relaxed the restrictive input or output orientation. What we propose is to relax such partial dimensionality by defining and estimating a parametric hyperbolic distance function which simultaneously allows for the maximum equiproportionate expansion of outputs and reduction of inputs. Particularly, we introduce a translog hyperbolic distance function showing how this specification complies with the conventional properties that the hyperbolic distance function satisfies. Finally, to illustrate its applicability in efficiency analysis, we implement it using a data set of Spanish savings banks.