A multistep estimation approach for primal production functions accounting for technology choice: a comparative evaluation 

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Recovering latent technology

Unfortunately, no data exists on the adoption of LI-CMPs by French farmers. More-over, farm accountancy data, even with cost accounting, does not contain any indi-cator enabling us to identify farmers using LI-CMPs. For instance, seed cultivars are not reported. Similarly, if purchased seed expenditures vary with sowing densities, these expenditures may also vary with seed prices and the share of seeds produced farmers themselves. This information lacking implies that we can only consider in-ferring farmers’ CMPs from their yield and chemical input use levels, that is to say indirectly. Different approaches can be considered to reveal latent structure in a data set.
Among those approaches is clustering analysis. Clustering is an unsupervised learning method whose aims at finding structure in a data set or, to say it differently it is “the art of finding groups in data” (Kaufman and Rousseeuw, 2009). Clustering analysis results in defining several groups among the data. To be meaningful, the ob-tained partition must separate observations that are very dissimilar while gathering similar observations. Clustering algorithms are thus trying to minimize the within group distance while maximizing the between-group distance. There exists different metrics to measure distance as well as different partitioning algorithms. Hence the need to generally compare the results obtained with different algorithms and metric to get robust results.
Clustering analysis is used in a lot of research areas and among them is agriculture. It has been used mainly to build farm classifications. For instance, it has been used by Gebauer (1987) to build a typology of farm households in Germany. Such type of typology was also used by Perrot (1990) in France and Maseda, Diaz, and Alvarez (2004) in Spain. Those typologies were foremost used to describe the state of the agricultural sector at a specific time. Clustering analysis was also used to identify different cultural systems by Bellon, Lescourret, and Calmet (2001) and Renaud-Gentié, Burgos, and Benoît (2014). Clustering is used here to summarize the detailed agronomic information available for each farm to build homogeneous groups.
Another method to recover latent variable is to use a latent class model (see Bartholomew, Knott, and Moustaki, 2011). Whereas clustering analysis assigns each observation to a class, latent class models rather estimate a probability to belong to each class. Given the number of classes we expect to have, we use a mixture model to estimate (i) the a posteriori probability to belong to each group and (ii) the parame-ters of the dependent variable distribution. Generally, the estimation of the mixture likelihood is performed with the Expectation-Maximization algorithm. In fact, latent class models can be seen as a probabilistic clustering where the distance metric is substituted by the likelihood that evaluates the model consistency. Plus, contrary to clustering methods that require a second step to estimate the separated production functions, latent class models have the merit to compute both the production func-tions and the CMP affiliation in one step. In that case, this one step procedure has advantages over the two step procedure because it avoids (i) biasing the estimates of the second stage equation if some errors are affecting the first stage and (ii) losing information. The latest point is emphasized by Orea and Kumbhakar (2004): the information that is contained within a class is not used to estimate the technology of other classes. This is the very reason why, in their work, Orea and Kumbhakar (2004) prefer to use a latent class framework to estimate different production frontiers.
The work of Orea and Kumbhakar (2004) was conducted on Spanish banks and proved that latent class models allow to control adequately for unobserved hetero-geneity. Latent class frontier models were also used in the agricultural literature by Alvarez and Corral (2010), Martinez Cillero et al. (2018) and Renner, Sauer, and El Benni (2021) for instance. Alvarez and Corral (2010) distinguish intensive from extensive dairy farm. As for Martinez Cillero et al. (2018), they estimated three dif-ferent categories among Irish beef farms. Renner, Sauer, and El Benni (2021) consider stochastic frontier with latent class so to evaluate the impact of technology choice and change on productivity and efficiency of Swiss dairy farms. If production frontiers are found to be significantly different across technology, no doubt that this techno-logical heterogeneity also needs to be accounted for when considering production functions.
The pitfall of the clustering and latent class approaches as described before is that it requires restrictive assumptions on the technology choice across time. Indeed, when having panel data at your disposal, the previous methods entail supposing either (i) that farmers have the same technology for each year (e.g., Alvarez and Corral, 2010; Martinez Cillero et al., 2018) or (ii) that farmers can change technology each year (e.g., Dakpo et al., 2021; Orea, Perez, and Roibas, 2015). Allowing technology change each year might be problematic. Even if CMP choices share more similarities with crop variety choices – usually considered as short run choices and modelled as such (e.g., Michler et al., 2018; Suri, 2006) – than with irrigation technologies – modelled as investment choices (e.g., Genius et al., 2014) – existence of transition costs makes frequent changes not appealing. Yet, technology change needs to be allowed, especially when considering short run decisions. Hence, the model needs to allow the possibility of change while making it costly so that individuals cannot switch every year from technology to technology. Such modelling effort was made in Dakpo et al. (2021) by considering explicitly a time trend among the variables that are used to separate the data in different groups. However, the dynamics of technology adoption is not explicitly modelled. If one is interested in the determinants of technological change to evaluate the impact of a public policy, this approach is not appropriate. A solution is to model the dynamics of technological adoption as a Markov process. Markov models are particularly adapted to model situation where there can be regime switching every year. They were introduced in econometrics by Goldfeld and Quandt (1973) and then largely diffused with the Markov-switching regression model used by Hamilton (1989). The Markov-switching models defined by these authors are based on a latent variable of which we want to model the dynamics. They were used in economics for example to explore the dramatic breaks that can occur in economic time series (see Chauvet and Hamilton, 2006; Hamilton, 1989). It is of special interest in financial economics as abrupt changes are common in financial data (see, e.g., Bonomo and Garcia, 1996; Cecchetti, Lam, and Mark, 1988)). Markov models are also used in the agricultural literature. For instance, Benoît, Le Ber, and Mari (2001) use hidden Markov model (HMM) to study the dynamics of crop rotation. Markov chain were also implemented by Miller et al. (2017) to investigate the adoption path of precision agriculture technologies in farms. The advantage of Markov models is their flexibility: transition matrices can either be (i) time invariant or change across time, (ii) be arbitrary fixed or can be defined by a logistic model.

Thesis outline

As stated previously, the main objective of the thesis is to estimate and to compare technology specific production functions. We attempted, in this general introduction, to stress the need for separate functions to account for heterogeneous technology. Yet, because technology adoption might be subject to selection biases, estimating technol-ogy specific production functions is not enough. A standard approach to deal with this technology selection issue in agricultural production is the endogenous regime switching (ERS) framework. In Chapter 1, we present the standard ERS framework as well as en extension to the case with endogenous covariates. We also present an associated estimation procedure to this ERS extension. Such extension and its estimation procedure are of particular interest in Chapter 2. As a matter of fact, in this chapter we intend to study how yields respond – and to what extent the re-sponse differs based on the adopted technology – to changes in variable input uses, especially pesticide uses. The perspective adopted in Chapter 3 is slightly different. We still want to compare CMP specific production functions, but in an unobserved technology choice context. Thus, we need to develop a framework that permits to uncover the CMP choice.
The second chapter of this thesis is mainly a theoretical chapter. We start by pre-senting the standard approaches to account for technology in agricultural production functions. In particular, we present the endogenous regime switching framework that permits to account for technology selection issues. Chapter 1 contribution lies in our considering an extension of the ERS framework for the endogenous covariates case. We consider an easily tractable control function approach with two sets of control functions. The first one controls for endogenous sample selection issue asso-ciated to technology choice whereas the second controls for input use endogeneity. In particular, we show in this chapter that the expression of the so-called inverse Mills ratio used in the widely used Heckman’s two step approach needs to be adapted in ERS models when regressors are endogenous.
The extended ERS model we present in Chapter 1 as well as its estimation ap-proach directly derives from the research question we try to answer in Chapter 2. In this chapter, we intend to estimate and compare the production functions of low-input and high-input farmers. In particular, we want to study how yields respond to input uses with a focus on pesticides. This objective encourages us to consider (i) a primal production function with (ii) a damage abatement function so we can consider the protective role of pesticides. The primal function comes with the well-known input endogeneity estimation issue whereas the damage abatement function entails non-linearity estimation issues. Both issues are tackled with the approach proposed in Chapter 1, i.e. a multi-step estimation approach relying on control func-tions. Thus, Chapter 2 is an empirical application of the extended ERS model and estimation technique we developed in Chapter 1.
The empirical analysis presented in Chapter 2 uses rich, high resolution panel data on Swiss wheat production (617 observations, from 2009 to 2015), containing detailed information on output and input uses (i.e. on pesticide uses, mechanical weed control, fertilizer uses, work and machinery), obtained from field journals. We make use of the fact that parallel to conventional wheat production, a low-input (“Ex-tenso”) wheat production system exists in Switzerland. Thus, we can compare the conventional and low-input production functions, in particular how yield responds to input uses in both functions. Unfortunately, our empirical analysis suffers from weak instruments in the input use equations. It means that the estimation results of the production functions have to be interpreted cautiously as we do not prop-erly account for the input use endogeneity. Despite that pitfall, this article seems to confirm the presence of selection biases affecting both the input demand and yield level models. In particular, unobserved characteristics of high-input farmers seem to boost up their pesticide use levels whereas unobserved characteristics of low-input farmers seem to allow them to reach greater yield levels. Even if estimates from the production functions need to be considered cautiously, this chapter argues in favor of controlling for selection biases when estimating input use and yield level functions and when evaluating the treatment effect of low-input adoption on both input use and yield levels.
Finally, Chapter 3 aims at proposing statistical and micro-econometric approaches for uncovering CMPs used by farmers when these are not available in the data set. Given that intensity in the use of chemical inputs of a CMP is directly related to the yield level targeted by this CMP, our methods aim to identify CMPs used by farmers based on their yield and chemical input use levels, information that are generally available in most cost accounting data set. We first considered more exploratory ap-proaches such as clustering and latent class models to (i) identify the CMP classes and (ii) estimate CMP specific production functions. Yet, when considering panel data, those approaches suffer from their static perspective. We need either to suppose that technology is stable on the whole period, an assumption that is problematic when the size of the panel is increasing. Or, we need to suppose that the technology choice is independent from the technology observed at the previous period. This second assumption is as unsatisfactory as the first one. We need to consider the adoption process as a dynamic one. We will then assume that CMP choices can be modelled as a Markovian processes and use a hidden Markov chain model to estimate CMP specific production functions. Another aspect we integrate in our modelling frame-work is farm unobserved heterogeneity, which can affect both crop production and farmer choices. We account for the unobserved farm heterogeneity by considering a random parameter model. Finally, our model combines different elements that are generally considered separately: latent technologies, unobserved heterogeneity and the dynamics of CMP choices. Our model can thus be considered as an endogenous Markov switching model. Plus, by disclosing the effect of CMP returns, our model allows to investigate the effects of economic incentives on CMP choices, which is critical from a public policy perspective.
We illustrate our approaches by investigating French farmers’ CMPs for win-ter wheat based on a panel dataset from 1998 to 2014 covering la Marne, a (highly productive) arable crop production area located in eastern France. We manage to distinguish three different CMPs: a high-yielding, an intermediate and a low-input CMPs. Even if low-input practices are associated to lower profit levels, they are still attracting an irreducible share of farmers. This shows that technology return is not sufficient to explain technology adoption and confirms the need to account for farmer preferences when studying adoption decision. In addition, our results show that, to encourage farmers with more intensive practices, price instruments on pesticide inputs are quite ineffective when wheat prices are high. This is due to the fact that profits depend more from the selling price of wheat than the input expenses that are small compared to the yield value. This confirms the previous findings on the inelastic demand of pesticides when wheat prices are high. A public policy implication is that, when crop prices are high, economic incentives targeting input prices might not be the best instrument to encourage a change in practices.
Overall, the results of this PhD thesis argue for the need to consider CMP in economists’ production functions. Farmers using different CMPs tend to have differ-ent input uses and yield levels. Moreover, this CMP choice induces selection biases on both input use and yield levels. The unobserved factors implying selection biases might also impact the farmers’ response to public policy. In particular, the response to public policies aiming at reducing pesticide uses might be heterogeneous across CMPs. This potential response heterogeneity should be considered by public author-ities in order to adequately evaluate the impact of a specific policy, or more generally agri-environmental programs.
An other valuable input from this PhD thesis comes from our investigation into the CMP choice mechanisms. In particular, we show that the standard taxation instru-ment is not a sufficient incentive for low-input adoption. Differences in pesticide uses between low- and high-input CMPs do not compensate for the yield loss, especially in a context of high crop prices. On the other hand, price premiums for low-input wheat might help into encouraging farmers to adopt this more agri-environmental practice. This is the strategy that have been adopted by Switzerland, where low-input practices are much more established among wheat producers. Low-input produc-ers benefit from both a price premium and a direct payment (400 CHF/ha). This might seem a rather expensive policy in comparison to the pesticide uses saving. The viability of low-input practices in wheat production as a high-input versus or-ganic in-between might be questioned. In particular, a cost-benefit analysis of both production practices might be considered to answer that question.

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A multistep estimation approach for primal production functions accounting for technology choice: a comparative evaluation1

Historically, agricultural economics was interested in quantifying “the contribution of inputs to output variations” and evaluate the production elasticities (Mundlak, 2001). Yet, as pointed out by Marschak and Andrews (1944), direct estimation of the production function suffers from biased estimates due to inputs endogeneity. Such endogeneity issue comes from what analysts assume that farmer’s inputs choice is ra-tional while most drivers of this choice remains unobserved to them. In that context, dual approach gain popularity in the early 1970s as a way to get around input endo-geneity issue (e.g., Griliches and Mairesse, 1995). Duality relies on the assumption that farmer’s input demand answers profit maximization objective. This approach was first developed by Klein (1953) and applied by Wolfson (1958). Chambers et al. (1988) even wrote an entire book on “dual” approaches applied to production analysis. Yet, Mundlak (2001) deplores that, because it relies on a direct behavioral function for input demand, dual approach rarely questions the relationship between inputs and outputs.

Table of contents :

General Introduction 
0.1 Agronomic principles and brief history of “Low Input” CMPs
0.1.1 “Low-Input” CMPs as induced technological innovations
0.1.2 Recent trends in input and wheat prices, and their potential impacts on “Low-Input” CMPs
0.2 Production function, technology and endogeneity
0.3 Recovering latent technology
0.4 Thesis outline
1 A multistep estimation approach for primal production functions accounting for technology choice: a comparative evaluation 
1.1 Introduction
1.2 A review of the techniques to assess the effects of production technologies on farmer choices and production outcomes
1.3 Standard approaches to account for agricultural technology choices
1.3.1 Considering a technology shifting effect in the yield supply and/or input demand models
1.3.2 Considering technology as a regime associated to specific production and input demand models
1.3.3 Technology specific production function in a panel data ERS model
1.4 A new modelling framework to account for technology choice in agricultural production using the primal
1.4.1 Technology specific yield and input demand functions in a ERS framework
1.4.2 Insights for the estimation procedure
1.4.3 Identifying assumptions, model nonlinearity and other estimation methods
1.5 Discussion
1.6 Appendices
1.6.1 Calculation details
2 Estimation of production functions in low- and high-input production practices
2.1 Introduction
2.2 Background
2.2.1 The economic and agronomic principles of low-input production system
2.2.2 How to integrate technology in farmers’ production function?
2.3 Methodological framework: from a standard ERS to an « extended » ERS model
2.4 Econometric implementation
2.4.1 A Cobb-Douglas crop production function with a damage abatement part
2.4.2 An original, multistep, estimation procedure
2.4.3 Robustness checks
2.5 Data
2.6 Insights from the technology choice and the input uses equations
2.7 Production function estimation results
2.8 Discussion and Conclusion
2.8.1 The input endogeneity issue
2.8.2 General considerations
2.8.3 Conclusion
2.9 Appendices
2.9.1 Results from the separability and asymmetry tests
2.9.2 Additional descriptive statistics
2.9.3 Estimation results from yield offer functions
2.9.4 Estimation results when considering the TFI pesticide indicator 77
2.9.5 Results when using global pesticide use variable in the production function
2.9.6 Results when using revenue instead of yield in the production function
2.9.7 Results when using a Translog specification for the productive part
3 Crop production models accounting for latent CMP choices 
3.1 Introduction
3.2 Agronomic principles and brief history of “Low Input” CMPs
3.3 A Random Parameter Hidden Markov Model for modelling production choices accounting for CMPs
3.3.1 Crop production models accounting for CMP choices
3.3.2 Latent CMPs models
3.3.3 A model with dynamic CMP choice
3.4 Sketch of the estimation procedure
3.5 Data
3.6 Insights from “exploratory” analyses
3.6.1 Presentation of the exploratory approaches
3.6.2 Insights from results and comparison with the HMM approach
3.7 Results
3.7.1 Random parameters ex-post distribution
3.7.2 Characteristics of the three CMP categories
3.7.3 Simulations results
3.8 Discussion and Conclusion
3.9 Appendices
3.9.1 Detailed estimation procedure
3.9.2 Insights about k-means and AHC algorithms
3.9.3 Additional results for the « exploratory analyses »
3.9.4 Distribution of random parameters from RPHMM
3.9.5 Simulation results
3.9.6 Determinants of input uses and yield
General Conclusion


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