Exact nonradial input, output, and productivity measurement.
The problem under consideration is to create a measure of varying over time or over place of main production characteristics: inputs, outputs, productivity. The Bennet-Bowley measures, originally suggested for cost of living, welfare, and consumer surplus measurements, lie in the base of the author's constructions. The technology of a firm is representing here in terms of abstract ``input-output'' analysis: \(T=\{x \in\mathbb{R}_+^n,y \in \mathbb{R}_+^m:x\text{ can produce }y\}\), i.e. \textit{x} is ``input'' and \textit{y} is ``output''. The basic construction is some generalization of \textit{D. G. Luenberger}'s shortage function [J. Math. Econ. 21, No. 5, 461--481 (1992; Zbl 0766.90011)], namely, \[ \vec {D}_{T} \left( {x,y;g_{x} ,g_{y}} \right) = \max \left\{ {\beta \in \mathbb{R}:\left( {x - \beta g_{x} ,y + \beta g_{y}} \right) \in T} \right\}. \] Here \(\left( {g_{x} ,g_{y}} \right)\) is a reference vector of inputs and outputs. This is the ``directional technology distance function'' (d.t.d.f.). It is a complete function representation of the technology, i.e. \[ (x,y) \in T \Leftrightarrow\vec D_T(x,y;g_x,g_y)\geq 0. \] Any twice-continuously differentiable d.t.d.f. can be approximated with second order by some quadratic function of \(\left( {x,y} \right)\) with the same reference vectors \(\left( {g_{x} ,g_{y}} \right)\). Suitably normalized measures, constructed in terms of d.t.d.f., are shown to be exact and superlative measures (in \textit{W. E. Diewert}'s sense [J. Econom. 4, 115--145 (1976; Zbl 0387.90046)]) of input, output and productivity indicators.
scientific article; zbMATH DE number 1850398