A superlative indicator for the Luenberger-Hicks-Moorsteen productivity indicator: Theory and application

Abstract

Consisting of the difference between an output indicator and an input indicator, the Luenberger-Hicks-Moorsteen (LHM) productivity indicator allows straightforward interpretation. However, its computation requires estimating distance functions that are inherently unknown. This paper shows that a computationally simple Bennet indicator is a superlative indicator for the LHM indicator when one can assume profit-maximizing behavior and the input and output directional distance functions can be represented up to the second order by a quadratic functional form. We also show that the Luenberger-and LHM-approximating Bennet indicators coincide for an appropriate choice of directional vectors. Focusing on a large sample of Italian food and beverages companies for the years 1995 − 2007, we empirically investigate the extent to which this theoretical equivalence translates into similar estimates. We find that the Bennet indicator is a close empirical alternative to the LHM indicator for the sample.

Analysing determinate components of an approximated Luenberger–Hicks–Moorsteen productivity indicator: An application to German dairy‐processing firms

“…To do so, we rely on a productivity framework based on the Luenberger–Hicks–Moorsteen (LHM) indicator introduced by Briec and Kerstens (2004). As mentioned by Ang and Kerstens (2017, 2020), the LHM indicator has several interesting theoretical properties. Traditionally, productivity change has been expressed as a ratio between productivity levels.…”

“…Let us, without loss of generality, normalise the vector of input and output prices in period $$t$$ as $$({\check{{\bf{w}}}}_{t},{\hat{{\bf{p}}}}_{t})\equiv (\frac{{{\bf{w}}}_{t}}{{{\bf{w}}}_{t}\cdot {{\bf{g}}}^{x}},\frac{{{\bf{p}}}_{t}}{{{\bf{p}}}_{t}\cdot {{\bf{g}}}^{y}})$$. Ang and Kerstens (2020) show that the following Bennet indicator approximates the LHM indicator developed by Briec and Kerstens (2004): $$$BLHM\equiv B({{\bf{x}}}_{t},{{\bf{y}}}_{t},{{\bf{x}}}_{t+1},{{\bf{y}}}_{t+1};{\check{{\bf{w}}}}_{t},{\check{{\bf{w}}}}_{t+1},{\hat{{\bf{p}}}}_{t},{\hat{{\bf{p}}}}_{t+1}).$$$…”

“…Our approach focuses on a decomposition of a price‐normalised Bennet indicator that approximates the LHM indicator, instead of decomposing the LHM indicator itself. Ang and Kerstens (2020) show that the Bennet indicator, by which input (output) prices are separately normalised, is a superlative indicator for the LHM indicator in Diewert (1976)'s sense: that is, if there is profit‐maximising behaviour and the directional distance function can be represented by a quadratic functional form up to the second order with time‐invariant second‐order coefficients. This price‐normalised Bennet indicator coincides with the LHM indicator.…”

Economic and Environmental Decomposition of Luenberger-Hicks-Moorsteen Total Factor Productivity Indicator: Empirical Analysis of Chinese Textile Firms With a Focus on Reporting Infeasibilities and Questioning Convexity

“…But, it is clear that the LHM TFP indicator is nowhere as popular as the Luenberger productivity indicator: a Google Scholar search on 12 April 2022 obtained 187 results for the expression "Luenberger-Hicks-Moorsteen productivity", while a search for "Luenberger productivity" yields 2510 hits. [18] seem to be the first to systematically compare the Luenberger and the LHM TFP indicators. [20] show that the Bennet indicator is a superlative indicator for the LHM indicator under certain conditions.…”

How limiting is finance for Dutch dairy farms? A dynamic profit analysis

“…Using a linear programming procedure in line with Aigner and Chu (1968), we approximate a dynamic directional distance function by a quadratic functional form. Following Ang and Kerstens (2023), we adapt the approximation of the static directional distance function (see, for example, Färe et al, 2005 and Ang & Kerstens, 2020) to the dynamic context. The shadow prices are determined by exploiting the dual relationship between the dynamic directional distance function and the dynamic profit function.…”