Measuring productivity when technology is heterogeneous using a latent class stochastic frontier model

Abstract

We examine an extension of the latent class stochastic frontier model (LCSFM) to productivity estimation and the decomposition of productivity change into technical change, output-oriented technical efficiency change, and scale change. We base our productivity estimation on a Multi-class Grifell-Tatjé, Lovell & Orea Malmquist (GLOM) index. An advantage of this new productivity index is to account for classes' posterior probabilities to derive individual farm parameters. In addition, we extend our analysis to estimate a metafrontier GLOM productivity index to explore potentialities when all firms use the best available technologies. An empirical application to a sample of French sheep and goat farms observed between 2002 and 2021 confirms the necessity to account for technological heterogeneity when measuring productivity change. Among the two classes of farms identified by the LCSFM, the intensive class experiences TFP gains, while the extensive class sees its TFP worsening. However, the gap between intensive and extensive technologies seems to reduce over time. Finally, the multi-class GLOM reveals technical change as the primary driver of productivity for French goat and sheep farms.

Appendices

Appendix

Appendix 1: Diewert (1976) 's Quadratic Lemma

$$ \begin{aligned} \ln D_{O} \left( {t + 1} \right) - \ln D_{O} \left( t \right) = & \\ & \frac{1}{2}\mathop \sum \limits_{q = 1}^{Q} \left( {\frac{{\partial \ln D_{O} \left( {t + 1} \right)}}{{\partial \ln y_{q}^{t + 1} }} + \frac{{\partial \ln D_{O} \left( t \right)}}{{\partial \ln y_{q}^{t} }}} \right)\ln \left( {\frac{{y_{q}^{t + 1} }}{{y_{q}^{t} }}} \right) \\ & + \frac{1}{2}\mathop \sum \limits_{k = 1}^{K} \left( {\frac{{ - \partial \ln D_{O} \left( {t + 1} \right)}}{{\partial \ln x_{k}^{t + 1} }} + \frac{{ - \partial \ln D_{O} \left( t \right)}}{{\partial \ln x_{k}^{t} }}} \right)\ln \left( {\frac{{x_{k}^{t + 1} }}{{x_{k}^{t} }}} \right) \\ & + \frac{1}{2}\left( {\frac{{\partial \ln D_{O} \left( {t + 1} \right)}}{{\partial \ln T^{t + 1} }} + \frac{{\partial \ln D_{O} \left( t \right)}}{{\partial \ln T^{t} }}} \right)\ln \left( {\frac{{T^{t + 1} }}{{T^{t} }}} \right) \\ \end{aligned} $$

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As $\frac{\partial {\text{ln}}{D}_{O}}{\partial {\text{ln}}T}=1$, in our case where $T$ is one external shift factor, this becomes

$$ \begin{aligned} \ln D_{O} \left( {t + 1} \right) - \ln D_{O} \left( t \right) = & \\ & \frac{1}{2}\mathop \sum \limits_{q = 1}^{Q} \left( {\frac{{\partial \ln D_{O} \left( {t + 1} \right)}}{{\partial \ln y_{q}^{t + 1} }} + \frac{{\partial \ln D_{O} \left( t \right)}}{{\partial \ln y_{q}^{t} }}} \right)\ln \left( {\frac{{y_{q}^{t + 1} }}{{y_{q}^{t} }}} \right) \\ & + \frac{1}{2}\mathop \sum \limits_{k = 1}^{K} \left( {\frac{{ - \partial \ln D_{O} \left( {t + 1} \right)}}{{\partial \ln x_{k}^{t + 1} }} + \frac{{ - \partial \ln D_{O} \left( t \right)}}{{\partial \ln x_{k}^{t} }}} \right)\ln \left( {\frac{{x_{k}^{t + 1} }}{{x_{k}^{t} }}} \right) \\ & + \ln \left( {\frac{{T^{t + 1} }}{{T^{t} }}} \right) \\ \end{aligned} $$

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Appendix 2

See Table 

Table 6 Translog estimation for the pooled and the LCSFM models

6.