Abstract
Path-based cardinal characterizations of closed and nonempty sets are defined, and their basic properties are detailed. Differential properties and applications to performance measurement are considered.
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Notes
A translation of an article published in Russian in 1924.
Part a is a special case of a result reported by Aczél (1966, p.18). Therefore, I only prove part b by adapting Aczél’s proof of a.
The hypograph of \(f:{{\mathbb{R}}}^{N}\to {\mathbb{R}}\) is \(\left\{\left(x,y\right):f\left(x\right)\ge y\right\}\).
My thanks to Bert Balk and Rolf Färe for comments that suggested this example.
The remainder of this paragraph is drawn almost word for word from Johnson (1913, pp. 506-507). The changes I have made are notational and omit some verbiage.
Carlson (1939, see pp. 16-17), who lists Johnson (1913) as his earliest source on the elasticity of production defines the function coefficient (elasticity of production), as a directional derivative without using that specific terminology. Courant (1936), which represents a translation of an earlier German work, discusses the concept as a derivative in a direction.
A more ordinary convention in welfare economics is to define the equivalent benefit measure as the negative of \(RS\left({x}^{1},{x}^{0},{C}^{1}\right)\) on the presumption that C0 is the original indifference set.
In the same setting, the compensating and equivalent variations differ.
Bennet (1920), Fisher (1921) established a convention followed by, among others, Caves et al. (1982), and Chambers (1996, 2002). Aczél (1990) shows that the geometric mean is the only merged relative score that satisfies multiplicativity, positive homogeneity, and symmetry. I follow the convention, while noting that further investigation is required to determine whether Aczél’s axioms are reasonable in a path-based setting.
A favorite saying my father often quoted on our walks in the woods that I later learned is best attributed to the the 18th century English landscape architect, William Kent.
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Acknowledgements
This paper was prepared for the Plenary Session of the European Workshop on Efficiency and Productivity Analysis, Porto, 2022 honoring Shawna Grosskopf and Rolf Färe as the winners of the 2022 Lifetime Achievement Award. My thanks to Chris Chambers, Bert Balk, Rolf Färe, and Walter Briec for comments on earlier versions.
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Chambers, R.G. Path-based set representations. J Prod Anal 60, 249–256 (2023). https://doi.org/10.1007/s11123-023-00691-2
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DOI: https://doi.org/10.1007/s11123-023-00691-2