The role of pesticides and fertilizers in Czech cereal output and TFP growth: A flexible production function with endogenous inputs

Contents lists available at ScienceDirect Food Policy journal homepage: www.elsevier.com/locate/foodpol The role of pesticides and fertilizers in Czech cereal output and TFP growth: A flexible production function with endogenous inputs Lukáš Čechuraa, Subal C. Kumbhakara,b, Zdeňka Žáková Kroupová c,∗ aDepartment of Economics, Faculty of Economics and Management, Czech University of Life Sciences Prague, Czech Republic bDepartment of Economics, Binghamton University, United States of America cDepartment of Economics, Faculty of Economics and Management, Czech University of Life Sciences Prague, Kamycka 129, 165 00 Praha 6-Suchdol, Czech Republic A R T I C L E I N F O JEL classification: D22 L25 Keywords: Productivity Growth components Endogeneity Proxy variable Agriculture Fertilizers Pesticides A B S T R A C T Through a comprehensive analysis of the effects of fertilizers and pesticides on both output and total factor productivity (TFP) growth, this study contributes to the discussion on the impacts of agrochemical use in sustainable cereal production systems. Using FADN data from Czech cereal farmers from 2008 to 2020, the study employs a flexible translog (TL) production function alongside a proxy-variable approach that treats pesticides and fertilizers as endogenous inputs. This approach generates robust results that strengthen the evidence base essential for data-driven agricultural policy formulation. The findings highlight the critical role of fertilizers and pesticides as key inputs driving both output and TFP growth. Simulations suggest that reducing these inputs would lead to a decline in agricultural output; however, this outcome is not inevitable. The study finds that technological advancements, which also serve as important drivers of both output and TFP growth, can mitigate potential declines in agricultural output, ensuring a more sustainable transition. 1. Introduction The effort to make Europe a climate-neutral continent by 2050 under the Green Deal (European Commission, 2020a) places significant pressure on the agricultural sector. At the heart of the Green Deal, the Farm-to-Fork Strategy aims to establish a European food system that ensures food security, nutrition, and public health while reducing its environmental and climate footprint. To achieve a neutral or positive environmental impact within the food system, the strategy calls for reducing the dependence of Europe’s agriculture sector on pesticides and fertilizers. Specifically, it sets ambitious targets: a reduction of at least 20% in fertilizer use, a 50% reduction in chemical pesticide use (compared to 1990 levels), and an increase in the share of agricultural land under organic farming to 25% by 2030 (European Commission, 2020b). However, pesticides and fertilizers have historically been considered essential for increasing agricultural productivity, protecting crops, and maximizing yields (Dobrin et al., 2022). Therefore, ensuring a sufficient supply of affordable food with minimal environmental impacts requires transforming agricultural practices to enhance sustainability. At the farm level, decision-makers face the challenge of compensating for the loss of their systems’ full production potential by integrating new environmentally friendly technologies within a relatively short timeframe. This decision directly impacts both the profitability and ∗Corresponding author. E-mail addresses: [email protected] (L. Čechura), [email protected] (S.C. Kumbhakar), [email protected] (Z. Žáková Kroupová). competitiveness of agricultural producers (Beckman et al., 2020). Agricultural policy should incentivize farmers to adopt these sustainable practices (Cortignani et al., 2022). However, effective policy design requires an understanding of the extent to which productivity growth is driven by inputs such as fertilizers and pesticides, as well as the role of technological change (TC) (O’Donnell, 2010). Several studies have examined the economic implications of reducing agrochemical inputs, particularly under the Farm-to-Fork strategy, employing general or partial equilibrium models (Beckman et al., 2020; Bremmer et al., 2021) or an agroeconomic model (Cortignani et al., 2022). Wesseler (2022) highlighted that a reduction in crop yields leads to a significant decline in European agricultural production, higher food prices, and an overall loss in net welfare. Potential solutions include adopting new technologies to improve resource efficiency, such as precision farming—specifically precision fertilization (Heyl et al., 2023)—and applying modern biotechnologies (Noleppa and Cartsburg, 2021). Additionally, reallocating resources to areas where productivity remains low and has not yet reached its full potential, even in organic agriculture, can be beneficial (Wesseler, 2022). A common theme among these solutions is an emphasis on productivity growth. https://doi.org/10.1016/j.foodpol.2025.102955 Received 9 September 2024; Received in revised form 22 August 2025; Accepted 28 August 2025 Food Policy 136 (2025) 102955 Available online 25 September 2025 0306-9192/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). L. Čechura et al. The economic literature distinguishes between two key concepts of productivity: partial factor productivity (PFP) and total factor productivity (TFP). PFP provides readily available information for understanding the underlying factors driving productivity changes; however, its results can be misleading since it does not account for all production inputs. Moreover, PFP may be higher for some inputs than others, making productivity comparisons between farms or across time difficult. In contrast, TFP offers a comprehensive measure of total productivity change by comparing the aggregate volume of outputs to the aggregate volume of inputs used in production (Wang et al., 2020; Murray, 2016). According to Comin (2010), TFP represents ‘‘the portion of output not explained by the amount of inputs used in production’’, with its growth measured by the Solow (1957) residual. Productivity growth, defined in this way, is considered a non-structural concept (Gong and Sickles, 2020). Empirical analysis of TFP typically employs either a parametric approach, such as the Stochastic Frontier Approach (SFA), or a nonparametric approach, such as the Data Envelopment Approach (DEA). These help approximate frontier technology and isolate the effects of inefficient behavior from other sources of productivity gains, including technological change, economies of scale, and shifts in input–output composition (Balk et al., 2020). Since frontier function estimates may suffer from input endogeneity—where efficiency and productivity levels are known to decision makers when determining input use but remain unobservable to economists—a behavioral framework was incorporated into the estimation of production technology, making it a structural approach to productivity analysis (Gong and Sickles, 2020). This approach enables the measurement of productivity growth within an input-specific framework, distinguishing between variable and quasi-fixed inputs. While the quasi-fixed inputs, such as capital, are predetermined, the use of variable inputs is influenced by short-run productivity changes. This responsiveness of input choice to productivity shocks introduces simultaneity issues in the modeling of the production process, particularly for inputs that can be adjusted in the short term, such as fertilizers and pesticides (Levinsohn and Petrin, 2003). To address the simultaneity problem, which can lead to biased parameter and productivity estimates, Olley and Pakes (1996) and Levinsohn and Petrin (2003) proposed a proxy variable approach, in which a proxy derived from a structural model is used to control the error component associated with productivity. While Olley and Pakes (1996) used investment as a proxy for unobserved productivity shocks, Levinsohn and Petrin (2003) instead employed an intermediate input such as materials and/or energy. The literature suggests that intermediate inputs offer several advantages over investment proxies in estimating production functions. Specifically, they mitigate issues related to lumpy investment data, such as the truncation of firms reporting zero investment. Additionally, intermediate inputs can more accurately capture firms’ responses to productivity shocks due to their lower adjustment costs, thereby providing a more precise proxy and a stronger connection between estimation strategies and economic theory. Ackerberg et al. (2015) provided a detailed comparison of the methodologies developed by Olley and Pakes (1996) and Levinsohn and Petrin (2003), noting that these techniques may suffer from a functional dependence problem. To address this issue, they proposed an alternative approach based on a conditional input demand function, rather than the unconditional function used in the Olley and Pakes and Levinsohn and Petrin methods. More recently, Kumbhakar and Li (2024) applied this proxy variable approach to estimate a production function incorporating both variable and quasi-fixed inputs, thereby addressing the endogeneity of variable inputs in a translog production function. Additionally, Hou et al. (2024) adapted this approach for a stochastic frontier model, introducing a new efficiency estimator based on firm’s profit maximization behavior. This research uses a proxy variable approach1 to examine the consequences of agrochemical input reduction through a retrospective analysis of the growth factors of output and TFP. Specifically, it aims to evaluate the impact of fertilizers and pesticides—considered endogenous inputs—on the production and TFP growth of cereal producers. The empirical analysis focuses on the Czech Republic, where cereal production is a key component of agricultural output, accounting for 32% of total agricultural production (Czech Statistical Office, 2023). Compared to its neighbors, Austria and Germany, the Czech Republic relies on higher levels of fertilizer (nitrogen) and pesticide use to achieve comparable crop yields. This suggests a potential opportunity to reduce pollution while maintaining agricultural productivity (e.g. Wuepper et al., 2023; Wuepper et al., 2020). An in-depth analysis of the Czech farming system can identify constraints that hinder farmers from adopting production practices similar to those in Austria and Germany—where equivalent agricultural productivity is achieved with substantially lower environmental impact—and explore the necessary adjustments and support mechanisms to facilitate sustainable agricultural transitions. The novelty of this study lies in its methodological approach, which evaluates how changes in chemical inputs affect output and TFP growth while addressing the endogeneity of these inputs. Our methodology builds on the existing proxy variable approach by incorporating a flexible representation of the production process, ensuring unbiased estimates of production function parameters and productivity. Additionally, it extends previous approaches by including multiple input variables, offering a more comprehensive analysis. At the empirical level, this study provides new and detailed insights into the role of pesticides and fertilizers in driving output and TFP growth, contributing critical information for agricultural policy formulation. The remainder of this study is structured as follows: Section 2 provides the theoretical background for decomposing output and TFP growth, based on the translog production function that addresses endogeneity. Section 3 outlines the identification and estimation procedures, while Section 4 describes the data used. Section 5 presents the results, followed by Section 6, which discusses the findings and their policy implications. Finally, Section 7 offers concluding remarks. 2. Theoretical background 2.1. Production function and components of output growth We start with the production function, which includes a neutral productivity component, and write it as 𝑌𝑖𝑡 =𝐹(𝑋𝑖𝑡, 𝑄𝑖𝑡, 𝑡) exp(𝜔𝑖𝑡 +𝑣𝑖𝑡),(2.1) where 𝑌 is output, 𝑋 is the vector of variable inputs, 𝑄 is the vector of quasi-fixed (predetermined) inputs, and 𝑡 is the time trend variable. 𝜔 is a persistent productivity shock (which is known and predictable to the farm) or Hicks-neutral productivity, while 𝑣 represents a transitory productivity shock (which is unknown to farms or analysts). The subscript 𝑖 denotes individual farms, while 𝑡 represents time. Since 𝜔𝑖𝑡 is known to the farm, it can be viewed as an unobservable input (from the perspective of the analyst) that is separable from other inputs. Alternatively, it can also be viewed as a shifter of the production 1In general, when an input is endogenous, instrumental variables (IV) are required. However, in this case, we do not rely on external IVs. Instead, we address the issue of endogeneity by incorporating the first-order conditions (FOCs) with respect to each variable input. This method, known as the proxy variable approach, expresses the unobserved productivity term 𝜔 in terms of observable variables derived from the FOCs. In particular, this method does not require additional assumptions beyond profit maximization. The assumptions of the proxy variable approach with respect to our model specification are discussed in Section 3.1.2. Food Policy 136 (2025) 102955 2 L. Čechura et al. function. Incorporating a time-constant farm effect, we can rewrite the production as 𝑌𝑖𝑡 =𝐴(𝑚𝑖𝑡)𝐹(𝑋𝑖𝑡, 𝑄𝑖𝑡, 𝑡) exp(𝑣𝑖𝑡) where ln 𝐴(𝑚𝑖𝑡) = 𝜇𝑖+𝜔𝑖𝑡, with 𝜇𝑖 representing farm-specific effects.2 To analyze the contribution of various factors to output growth, we take the total differential of Eq. (2.1) after applying a logarithmic transformation. After a few algebraic steps, this yields:  𝑌𝑖𝑡 =∑ 𝑗 𝐸𝑋𝑗𝑖𝑡  𝑋𝑗𝑖𝑡 +∑ 𝑚 𝐸𝑄𝑚𝑖𝑡  𝑄𝑚𝑖𝑡 +𝑇 𝐶𝑖𝑡 +𝜔𝑖𝑡 +𝑣𝑖𝑡,(2.2) where 𝐸𝑋𝑗𝑖𝑡 and 𝐸𝑄𝑚𝑖𝑡 represent the output elasticities of variable and quasi-fixed inputs, respectively. Technological change (TC) is defined as 𝑇 𝐶 =𝜕ln 𝐹(.)∕𝜕𝑡, and a dot over a variable indicates its rate of change. The above formulation decomposes output growth into contributions from input use (both variable and quasi-fixed), TC, and productivity improvements (𝜔). For example,  𝑌 can be increased by expanding input use, adopting improved technology, or improving productivity. Rearranging Eq. (2.2) gives  𝑇 𝐹 𝑃𝑖𝑡 =𝑇 𝐶𝑖𝑡 + (𝑆𝑅𝑇 𝑆𝑖𝑡 − 1) ∑ 𝑗 𝑆𝑗𝑖𝑡  𝑋𝑗𝑖𝑡 +∑ 𝑚 𝐸𝑄𝑚  𝑄𝑚𝑖𝑡 +𝜔𝑖𝑡,(2.3) where  𝑇 𝐹 𝑃𝑖𝑡 = 𝑌𝑖𝑡 −∑𝑗𝑆𝑗 𝑋𝑗𝑖𝑡, 𝑆𝑗𝑖𝑡 is the cost share, defined as 𝑆𝑗𝑖𝑡 =𝑊𝑗𝑖𝑡𝑋𝑗𝑖𝑡∕𝐶𝑖𝑡, where 𝑊𝑗 is the price of the variable input 𝑋𝑗 and 𝐶𝑖𝑡 is total cost given by 𝐶𝑖𝑡 =∑𝑗𝑊𝑗𝑖𝑡𝑋𝑗𝑖𝑡. Short-run returns to scale (SRTS) are represented as 𝑆𝑅𝑇 𝑆𝑖𝑡 =∑𝑗𝐸𝑋𝑗𝑖𝑡. If input levels remain unchanged, the change in TFP has two components: TC and productivity growth 𝜔𝑖𝑡. In this case,  𝑇 𝐹 𝑃 is equivalent to  𝑌. These components can be identified, as will be discussed later. However, TFP change is not always intuitive. In a microeconomic context, a positive TFP change for a farm does not necessarily indicate that it is making a profit. To better understand this relationship, we can relate the TFP change to the profitability change (Kumbhakar and Lien, 2009). Another intuitive decomposition involves breaking down output change into its key components such as TC, changes in input use (both quasi-fixed and variable), and productivity change. This approach helps to determine whether output growth is input-driven or TC-driven and quantifies their relative contributions. The decomposition is illustrated in Eq. (2.2). To define profit change as a percentage of total cost, we start with the profit equation (𝜋=∑𝑗𝑃𝑌 −𝑊𝑗𝑋𝑗) treating all inputs as variable for simplicity. 1 𝐶 𝑑𝜋 𝑑𝑡 =𝑃𝑌 𝐶{ 𝑃+ 𝑌}− {∑ 𝑗 𝑆𝑗 𝑊𝑗+∑ 𝑗 𝑆𝑗 𝑋𝑗}.(2.4) 2.2. Proxy variable approach Our goal is to estimate a production function, but this process presents econometric challenges, as first identified by Marschak and Andrews (1944). They recognized that in real-world production, the relationship between inputs and output is often simultaneous, meaning that output influences input choices (e.g., firms adjust labor and capital based on output levels), while inputs also determine output. This simultaneous determination complicates the estimation of production functions, as the standard assumption of a one-directional causal relationship (where input determines output) does not hold. To correctly identify the production function, Marschak and Andrews (1944) argued that simultaneity must be explicitly accounted for. Their central contribution was to address the identification problem in production functions by incorporating simultaneity and random variables into the estimation process. Marschak and Andrews (1944) emphasized that a valid production function must account for the fact that inputs are chosen based on economic behavior (e.g., profit maximization) and that unobservable 2It can capture time-invariant productivity as well because 𝜇𝑖 cannot be separated from 𝜔𝑖 if one writes the productivity component as 𝜔0 𝑖+𝜔𝑖𝑡. shocks affect both inputs and output. To address these challenges, they introduced the framework of simultaneous equations, which helps to identify the true relationship between inputs and output in production models. Their work laid the foundation for later econometric methods, such as estimators developed by Olley and Pakes (1996) and Levinsohn and Petrin (2003), which explicitly account for endogeneity and simultaneity in the estimation of the production function. Another common approach in the applied production literature is the use of instrumental variable (IV) estimators to address these issues. However, identifying valid external instrumental variables is often an ad hoc exercise. For this reason, we adopt the proxy variable approach, assuming expected profit maximization to address endogeneity and simultaneity. Specifically, we classify inputs into endogenous and exogenous (predetermined) categories to ensure a more robust estimation of the production function. We employ a multi-step approach. In step 1, we use share equations (as proposed by Gandhi et al. (2020)), which exclude the persistent productivity shock 𝜔 and therefore are not subject to endogeneity (i.e., correlation between 𝜔 and variable inputs). In this step, we estimate the parameters associated with the variable inputs in the translog production function. We follow the literature that treats state variables as predetermined (Olley and Pakes, 1996; Levinsohn and Petrin, 2003; Gandhi et al., 2020), along with many others who have built on the framework introduced by Olley and Pakes in their wellknown Econometrica paper. Economic reasoning is used to determine whether a variable is quasi-fixed or variable. In Step 2, we treat the previously estimated (identified) parameters as known and incorporate them into the production function. In this step, a proxy for 𝜔 is used to estimate the remaining parameters. Once all parameters are estimated, we recover 𝜔𝑖𝑡 using the first-order conditions (FOCs). The proxy variable approach works, in general, because (i) the FOCs are rewritten (inverted) to express the unobserved 𝜔—which is correlated with the variable inputs, in terms of observables (inputs); and (ii) a Markov assumption is imposed on 𝜔, allowing it to be expressed in terms of lagged values of the input variables and prices (variable inputs and output). 2.3. The production function specification In this section, we parameterize the production function described in Eq. (2.1). We assume that the production process can be well approximated using a translog (TL) production function. Although the proxy variable approach often uses the Cobb–Douglas (CD) function, this functional form can be overly restrictive and may lead to biased results. The advantage of the TL function lies in its flexibility, as its second derivatives are non-zero and the CD function is nested within the TL function, allowing us to test the flexible form against the restrictive form. Drawing on theoretical foundations and empirical evidence on adjustment costs (Hamermesh and Pfann, 1996; Cooper and Haltiwanger, 2006; Mishra et al., 2004; Yang and Shumway, 2016; Hu et al., 2020), our translog model treats capital (𝐾), land (𝐴) and labor (𝐿) as quasifixed inputs (state variables). These factors face adjustment constraints, including time-to-install costs for capital equipment, binding lease agreements for land, as well as employment contracts and training requirements for labor. In contrast, material inputs—fertilizer (𝐹), pesticides (𝐻) (including herbicides and other crop protection products), and other material inputs (𝑂)—are treated as freely varying factors without dynamic implications, allowing for instantaneous adjustment. Adjustment costs formalize the process of a farm’s dynamic production decisions. Seeing as quasi-fixed inputs are subject to adjustment costs, farms determine their levels dynamically at time 𝑡− 1, making them predetermined state variables at time 𝑡. Variable inputs, by contrast, are chosen at time 𝑡, within the production period to maximize profit (short run). Additionally, we include a time variable as a proxy for Food Policy 136 (2025) 102955 3 L. Čechura et al. technological change3 and account for farm-specific effects (𝜇𝑖). Finally, we express the production function (2.1) in logarithmic form as follows, where lowercase letters represent the logarithms of their uppercase counterparts: 𝑦𝑖𝑡 =𝑓(𝑘𝑖𝑡, 𝑎𝑖𝑡, 𝑙𝑖𝑡, 𝑓𝑖𝑡, ℎ𝑖𝑡, 𝑜𝑖𝑡, 𝑡)+𝜇𝑖+𝜔𝑖𝑡+𝑣𝑖𝑡 ≡𝑓(𝑞𝑗𝑖𝑡, 𝑠𝑟𝑖𝑡)+𝜇𝑖+𝜔𝑖𝑡+𝑣𝑖𝑡,(2.5) where the variables in 𝑞𝑗𝑖𝑡 are 𝑘𝑖𝑡, 𝑎𝑖𝑡, 𝑙𝑖𝑡, 𝑡, and the variables in 𝑠𝑟𝑖𝑡 are 𝑓𝑖𝑡, ℎ𝑖𝑡, 𝑜𝑖𝑡. The subscripts 𝑖 and 𝑡 denote the farm and time periods, respectively (𝑖= 1,…, 𝑛, 𝑡= 1,…, 𝑇 ). The full translog form of 𝑓(⋅) can be expressed as 𝑦𝑖𝑡 =𝜇𝑖+∑ 𝑗 𝛽𝑗𝑞𝑗𝑖𝑡 +1 2∑ 𝑗∑ 𝑙 𝛽𝑗𝑙 𝑞𝑗𝑖𝑡 𝑞𝑙𝑖𝑡 (2.6) +∑ 𝑟 𝛼𝑟𝑠𝑟𝑖𝑡 +1 2∑ 𝑟∑ 𝑟′ 𝛼𝑟𝑟′𝑠𝑟𝑖𝑡𝑠𝑟′𝑖𝑡 +1 2∑ 𝑗∑ 𝑟 𝛿𝑗𝑟𝑞𝑗𝑖𝑡 𝑠𝑟𝑖𝑡 +𝜔𝑖𝑡 +𝑣𝑖𝑡 with the symmetry restrictions on 𝛽𝑗𝑙, 𝛼𝑟𝑟′ and 𝛿𝑗𝑟. The above TL function 𝑓(𝑞𝑗𝑖𝑡, 𝑠𝑟𝑖𝑡), where 𝑗=𝑘, 𝑎, 𝑙, 𝑡 and 𝑟=𝑓, ℎ, 𝑜, can be written as 𝑇(𝑠𝑟𝑖𝑡, 𝑞𝑗𝑖𝑡;𝜷1) + 𝑇 𝐿(𝑞𝑗𝑖𝑡;𝜷𝟐), where 𝑇(𝑠𝑟𝑖𝑡, 𝑞𝑗𝑖𝑡;𝛽1) = ∑ 𝑟 𝛼𝑟𝑠𝑟𝑖𝑡 +1 2∑ 𝑟∑ 𝑟′ 𝛼𝑟𝑟′𝑠𝑟𝑖𝑡𝑠𝑟′𝑖𝑡 +1 2∑ 𝑗∑ 𝑟 𝛿𝑗𝑟𝑞𝑗𝑖𝑡 𝑠𝑟𝑖𝑡,(2.7) is a translog function with the variable inputs 𝐹 , 𝐻, 𝑂 and 𝑇 𝐿(𝑞𝑗𝑖𝑡;𝜷2) = 𝜇𝑖+∑ 𝑗 𝛽𝑗𝑞𝑗𝑖𝑡 +1 2∑ 𝑗∑ 𝑙 𝛽𝑗𝑙 𝑞𝑗𝑖𝑡 𝑞𝑙𝑖𝑡 +𝜔𝑖𝑡 +𝑣𝑖𝑡,(2.8) is a translog function with the quasi-fixed inputs 𝐾, 𝐴, 𝐿 and 𝑡. We denote 𝜷1=(𝛼𝑟, 𝛼𝑟𝑟′, 𝛿𝑗𝑟) and 𝜷2= (𝜇𝑖, 𝛽𝑗, 𝛽𝑗𝑙) for 𝑟, 𝑟′=𝑓, ℎ, 𝑜, and 𝑗, 𝑙 =𝑘, 𝑎, 𝑙, 𝑡. The full TL function with 𝐹, 𝐻, 𝑂 and 𝐾, 𝐴, 𝐿, 𝑡 is decomposed into two parts: the first part is a sub-translog function with 𝐹, 𝐻, 𝑂, while the remaining terms are included in the 𝑇 𝐿(⋅) function. 3. Identification and estimation 3.1. Identification This section provides insights into the two-step identification of production function parameters and evaluates the assumptions underlying the proxy variable approach with respect to model specification and estimation strategy. 3.1.1. Two-step identifying procedure of production function parameters The TL function in Eq. (2.6) is identified using a multi-step procedure. We assume that producers maximize their expected short-term profit and choose the variable inputs 𝐹 , 𝐻, 𝑂 in the current period, 𝑡, while the quasi-fixed inputs 𝑄∈ (𝐾, 𝐴, 𝐿) are predetermined (chosen at time 𝑡− 1). The persistent productivity shock 𝜔 is known to the farmer when making the optimal input decision. Identification consists of several steps: the first step identifies the 𝜷1 parameters, while the second step identifies the remaining parameters in 𝜷2. To achieve this, we first express 𝜔𝑖𝑡 in terms of observed data and the estimated parameters from Step 1 (𝜷1) using the FOCs. We then assume a Markov process for 𝜔. These two features allow us to rewrite the production function in terms of 𝜷2 and the dynamics of 𝜔. The details of both steps are provided in Appendix A. 3.1.2. Proxy variable identifying assumptions In the industrial organization (IO) literature, the identification of production functions using the proxy variable approach typically rests on several assumptions (Olley and Pakes, 1996; Levinsohn and Petrin, 2003): (i) monotonicity of the proxy variables, implying a consistent relationship with productivity and no heterogeneous effects; (ii) exogeneity of the state variables with respect to contemporaneous shocks; (iii) timing of input choices, such that variable inputs depend on both 3Since technological change has a long term character, 𝑡 is included among quasi-fixed inputs. observed and unobserved current productivity, while state variables depend only on past productivity; (iv) functional-form restrictions on the production function; (v) validity of the proxies, in that they respond to productivity shocks observed by the firm and are not driven by unobserved factors; (vi) absence of selection bias due to the exit of less productive firms; and (vii) absence of measurement error in input variables. We evaluated these assumptions in relation to our model specification and econometric framework. The first assumption is a necessary condition for identifying parameters when using a non-parametric production function. However, if a parametric production function is employed, monotonicity is not required for at least two reasons: (i) identification does not depend on inverting a production function to obtain unobservable variables, and (ii) estimation relies on functional-form and orthogonality assumptions rather than on monotonicity.45 Thus, the monotonicity of the proxy variables is not required to test in the parametric approach adopted in our study. Second, our model follows the IO literature in which some inputs are considered quasi-fixed (state variables)—determined in the previous period (known as the timing assumption). From an econometric perspective, these variables are treated as predetermined at time 𝑡. A dynamic optimization model based on the sequential decision process in crop production can be used to determine the state variables. In particular, the decision process can be divided into (i) the planting season and (ii) the growing and harvest season. The decision regarding the planted area is based on the information during the planting season in time 𝑡− 1, while yield is determined using information from the growing season in the period of time 𝑡.6 Regarding the third assumption, variable inputs are chosen by maximizing the short-term (variable) profit, where the persistent productivity shock, 𝜔, is observed by the farm (the decision-maker) but not by researchers. This is an important assumption, as the random productivity shock, 𝑣, is always assumed to be unobservable in agricultural production. Thus, while the farm observes the persistent productivity shock, 𝜔, it does not observe the random productivity shock, 𝑣. Consequently, expected profit maximization is applied, which removes the unobserved 𝑣 from the expected profit expression. In this short-term framework, the current value of 𝜔 is used. Lagged values appear in the estimating equation due to the Markov assumption on 𝜔, but the model itself does not incorporate past productivity directly. We assess this assumption using the second specification test described below. With respect to functional-form assumptions, the CD specification remains the most widely used in the literature. However, it has several well-documented limitations.7 In this study, we employ a flexible func4Specifically, the persistent productivity shock 𝜔 can be analytically expressed in terms of observable variables. The general idea is that endogenous variables (demand functions) can be solved in terms of prices and quasi-fixed inputs. For instance, assuming CD production function, 𝑆= 𝑆(𝐾, 𝐴, 𝐿, 𝑃 , 𝑊𝑆, 𝜔), it can be inverted to express 𝜔=𝑆−1(𝐾, 𝐴, 𝐿, 𝑆, 𝑃 , 𝑊𝑆) where 𝐾, 𝐴, 𝐿 are quasi-fixed inputs, 𝑆 is the variable input, 𝑃 is the output price, and 𝑊𝑆 is the price of the variable input 𝑆. The function 𝑆(⋅) is invertible when it is monotonic, as demonstrated by Levinsohn and Petrin (2003). For a CD function, ln 𝑌𝑖𝑡 =𝛽0+𝛽𝐾ln 𝐾𝑖𝑡 +𝛽𝐴ln 𝐴𝑖𝑡 +𝛽𝐿ln 𝐿𝑖𝑡 +𝛽𝑆ln 𝑆𝑖𝑡 +𝜔𝑖𝑡 +𝑣𝑖𝑡, the FOC with respect to 𝑆 is 𝑊𝑆𝑖𝑡𝑆𝑖𝑡∕𝑃𝑖𝑡𝑌𝑖𝑡 =𝛽𝑆𝜃exp(−𝑣𝑖𝑡), where 𝜃=E[exp(𝑣𝑖𝑡)]. This implies that 𝜔𝑖𝑡 = − ln(𝛽𝑆𝜃) + ln(𝑊𝑆∕𝑃)𝑖𝑡 + (1 − 𝛽𝑆) ln 𝑆𝑖𝑡 −𝛽𝐾ln 𝐾𝑖𝑡 −𝛽𝐴ln 𝐴𝑖𝑡 − 𝛽𝐿ln 𝐿𝑖𝑡. Here, the demand function for 𝑆 is always invertible. 5Moreover, farm-level heterogeneity (if relevant) can be incorporated into the production function through farm fixed effects. 6Our distinction is therefore grounded in the intrinsic characteristics of the agricultural production process rather than solely econometric considerations. The sequential decision process is also supported by the composition of our sample: crop production accounts for 74.3% of total agricultural output, indicating its dominance within the production structure. Moreover, we indirectly validate this assumption through our first specification test (Levinsohn and Petrin, 2003). 7Restrictions of CD function include unitary elasticity of substitution, constant production elasticities, and perfect substitutability of inputs, etc. Food Policy 136 (2025) 102955 4 L. Čechura et al. tional form—the translog production function—and test this against the more restrictive CD specification. This generalization, employed in our study, helps address limitations associated with restrictive functionalform assumptions. The proxy validity assumption, tested by the first and the second specification tests, is crucial for empirical applications. In our study, the proxy for the scalar unobservable 𝜔𝑖𝑡 is derived from the FOCs of expected profit maximization. Together with the Markov process assumption on 𝜔𝑖𝑡, this expresses 𝜔𝑖𝑡 in terms of lagged observed variables. The unobserved variable in the proxy is 𝜂𝑖𝑡, where 𝜔𝑖𝑡 = 𝜌𝜔𝑖𝑡−1 +𝜂𝑖𝑡. The term 𝜂𝑖𝑡 is assumed to be independent of all variables at times 𝑡 and 𝑡− 1, and remains unobserved. This assumption is verified using the second specification test. The sixth assumption is introduced for convenience. Empirical evidence shows that it has no impact on results (see footnote 21 of Levinsohn and Petrin (2003)). Therefore, this assumption is routinely adopted in all applications. The final assumption (no measurement error) is standard in all econometric models.8 To summarize, we employ a Wald test to test the functional form and two specification tests to verify the identifying assumptions (Levinsohn and Petrin, 2003). Specifically, the first specification test checks whether consistent estimates are obtained when inputs are split differently into quasi-fixed and variable categories. The second specification test examines whether inputs at times 𝑡 and 𝑡− 1 are correlated with innovations in productivity at time 𝑡.9 Following Levinsohn and Petrin (2003) and Olley and Pakes (1996), we therefore test the proxy validity assumption by examining the correlation between input choices and contemporaneous productivity innovations. An insignificant correlation supports the assumption that the proxies reflect observed productivity shocks rather than unobserved factors. 3.2. Estimation This section briefly discusses the parameter estimation process, which is performed in three steps using nonlinear least squares (NLS) and the nonlinear mixed-effect model estimator (NLME). 3.2.1. Step 1 In this step, we use the FOCs with respect to material inputs to estimate the parameters in 𝑇(⋅), i.e., 𝜷1. The relevant equations are given in (A.4), ln 𝑠𝑆𝑖𝑡 = ln(𝜀𝑆𝑖𝑡𝜃) − 𝑣𝑖𝑡, 𝑆∈𝐹, 𝐻, 𝑂. Given that 𝑣 is i.i.d. and uncorrelated with the variables in 𝜀𝑆, we use NLS to estimate 𝜷1 following Bates and Watts (1988), Bates and Chambers (1992), and Moré (1978). Note that 𝜔 does not appear in the share equations, so no assumptions about it are required to estimate them. The only assumption is that the error term 𝑣 is uncorrelated with the inputs—both quasi-fixed and variable. The procedure is as follows: We perform NLS regressions of ln 𝑠𝐹𝑖𝑡 on ln(𝜀𝐹𝑖𝑡𝜃) and obtain the fitted values  ln(𝜀𝐹𝑖𝑡𝜃) and −𝑣𝑖𝑡. Using 𝑣𝑖𝑡, we estimate 𝜃 from  𝜃= E[exp(𝑣𝑖𝑡)]. We then perform additional NLS regressions ln 𝑠𝐻𝑖𝑡 on ln(𝜀𝐻𝑖𝑡𝜃) and ln 𝑠𝑂𝑖𝑡 on ln(𝜀𝑂𝑖𝑡𝜃), respectively, to obtain the fitted values: (i)  ln(𝜀𝐻𝑖𝑡𝜃) and −𝑣𝑖𝑡, and (ii)  ln(𝜀𝑂𝑖𝑡𝜃) and −𝑣𝑖𝑡. Then, we use 𝑣𝑖𝑡 from these NLS regressions to obtain the second and third estimates of 𝜃 using  𝜃=E[exp(𝑣𝑖𝑡)]. Finally, we compute an average of these three 8To the best of our knowledge, no studies explicitly address measurement error in the covariates. In particular, the IO literature does not provide guidance on handling measurement errors in inputs, especially within structural approaches (proxy variable methods) commonly applied in the estimation of production functions. This remains an important area for future research. 9Levinsohn and Petrin (2003) test for the correlation between inputs in period 𝑡 and innovations in productivity in period 𝑡+ 1. However, in an unbalanced panel datasets, it is methodologically more appropriate and statistically more reliable to examine the correlation between lagged inputs and current innovations in productivity. This approach is also consistent with standard practice in dynamic panel models (Baltagi, 2021; Wooldridge, 2002). estimates of 𝜃. The estimated parameters in these NLS regressions represent 𝜷1×𝜃. Finally, we use the estimated value of 𝜃 to obtain estimates of 𝜷1 by dividing each of the NLS parameters by  𝜃. 3.2.2. Step 2 In this step, we estimate the parameters in 𝜷2. The estimating equation is (A.12): 𝑦𝑖𝑡 =𝑇 𝐿(𝑞𝑗𝑖𝑡;𝜷2) + 𝜌 𝑅𝑖𝑡−1 −𝜌 𝑇 𝐿(𝑞𝑗𝑖𝑡−1;𝜷2) + 𝑒𝑖𝑡 (3.1) where 𝑒𝑖𝑡 =𝑣𝑖𝑡 +𝜂𝑖𝑡 with 𝜂𝑖𝑡 is an i.i.d. productivity innovation uncorrelated with 𝑆𝑖𝑡, where 𝑆∈𝐹 , 𝐻, 𝑂, and any other variables in the model.  𝑅𝑖𝑡−1 = [𝑅𝐹𝑖𝑡−1 +𝑅𝐻𝑖𝑡−1 +𝑅𝑂𝑖𝑡−1]∕3 and 𝑅𝐹𝑖𝑡−1, 𝑅𝐻𝑖𝑡−1, 𝑅𝑂𝑖𝑡−1 are known/estimated, since they consist of data and estimated parameters from Step 1. Remark 1. Since 𝑇 𝐿(𝑞𝑗𝑖𝑡;𝜷2) includes 𝜇𝑖, which is likely to be correlated with 𝑓𝑖𝑡, 𝑓𝑖𝑡−1, ℎ𝑖𝑡, ℎ𝑖𝑡−1, 𝑜𝑖𝑡, 𝑜𝑖𝑡−1 and possibly other variables in the 𝑇 𝐿(.) function, we must first address this issue before estimating (3.1). Rewriting 𝑇 𝐿(𝑞𝑗𝑖𝑡;𝜷2)−𝜌 𝑇 𝐿(𝑞𝑗𝑖𝑡−1;𝜷2) fully to bring 𝜇𝑖 to light using (2.8) and taking a lag, yields 𝑇 𝐿(𝑞𝑗𝑖𝑡;𝜷2) − 𝜌 𝑇 𝐿(𝑞𝑗𝑖𝑡−1;𝜷2) = 𝜇𝑖(1 − 𝜌) + ∑𝑗𝛽𝑗(𝑞𝑗𝑖𝑡 −𝜌 𝑞𝑗𝑖𝑡−1) + 1 2∑𝑗∑𝑙𝛽𝑗𝑙 [𝑞𝑗𝑖𝑡 𝑞𝑙𝑖𝑡 −𝜌 𝑞𝑗𝑖𝑡−1 𝑞𝑙𝑖𝑡−1]. We then substitute this in (3.1) to obtain: 𝑦𝑖𝑡 =𝜇𝑖(1 − 𝜌) + 𝜌 𝑅𝑖𝑡−1 +∑ 𝑗 𝛽𝑗(𝑞𝑗𝑖𝑡 −𝜌 𝑞𝑗𝑖𝑡−1) +1 2∑ 𝑗∑ 𝑙 𝛽𝑗𝑙 [𝑞𝑗𝑖𝑡 𝑞𝑙𝑖𝑡 −𝜌 𝑞𝑗𝑖𝑡−1 𝑞𝑙𝑖𝑡−1] + 𝑒𝑖𝑡.(3.2) Applying first differencing (FD) to Eq. (3.2) eliminates 𝜇𝑖 allowing us to estimate the equation using NLS.10 Note that 𝜌 is identified from the coefficient of  𝑅𝑖𝑡−1 and the 𝜷2 coefficients are identified from the first differences of ∑𝑗𝛽𝑗(𝑞𝑗𝑖𝑡 −𝜌 𝑞𝑗𝑖𝑡−1) + 1 2∑𝑗∑𝑙𝛽𝑗𝑙 [𝑞𝑗𝑖𝑡 𝑞𝑙𝑖𝑡 − 𝜌 𝑞𝑗𝑖𝑡−1 𝑞𝑙𝑖𝑡−1]. However, FD has a drawback: the presence of lagged variables results in a loss of two observations for each 𝑖 (a total of 2𝑁 observations) due to the presence of two lags. An alternative to FD is the correlated random effects model, where 𝜇𝑖=𝑧′ 𝑖𝛾𝑖+𝜉𝑖 where 𝑧𝑖 are time-invariant variables correlated with those in (3.2), i.e., 𝑞𝑗𝑖𝑡 and 𝑞𝑗𝑖𝑡−1 ∀𝑗. In the absence of any 𝑧𝑖 variables, one can use Mundlak’s (1978) formulation which replaces 𝑧𝑖 with the mean farm-level values of 𝑞𝑗, i.e., 𝑞𝑗, 𝑗 = 𝑘, 𝑎, 𝑙, 𝑡. That is, 𝜇𝑖=∑ 𝑗 𝛾𝑗𝑞𝑗𝑖+𝜒𝑖, 𝑗 =𝑘, 𝑎, 𝑙, 𝑡 (3.3) where 𝜒𝑖 is not correlated with the right-hand-side variables in (3.2). Having identified 𝜷1 and 𝜷2, we can estimate 𝜔𝑖𝑡 by averaging 𝜔𝑖𝑡 from Eqs. (A.5), (A.6) and (A.7). Since FD of (3.4) will involve two lags, we use the correlated random effects formulation of Mundlak (1978) to save one observation for each i, i.e. in total 𝑁 observations. That is, we use (3.3) in (3.4) and estimate 𝛾𝑗 together with the other parameters in (3.4). The final model to be estimated empirically is the following: 𝑦𝑖𝑡 = (∑ 𝑗 𝛾𝑗𝑞𝑗𝑖+𝜒𝑖)(1 − 𝜌) + 𝜌 𝑅𝑖𝑡−1 +∑ 𝑗 𝛽𝑗(𝑞𝑗𝑖𝑡 −𝜌𝑞𝑗𝑖𝑡−1) +1 2∑ 𝑗∑ 𝑙 𝛽𝑗𝑙 [𝑞𝑗𝑖𝑡 𝑞𝑙𝑖𝑡 −𝜌 𝑞𝑗𝑖𝑡−1 𝑞𝑙𝑖𝑡−1] + 𝑒𝑖𝑡 (3.4) Given the presence of the 𝜒𝑖 term, we use NLME11 to estimate (3.4). 10 Moreover, the standard errors can be corrected for possible heteroscedasticity. In particular, we can incorporate some exogenous variables 𝑧𝑖𝑡 into Eq. (A.8) to account for factors influencing productivity. This allows identification of key drivers of productivity growth. 11 NLME is a generalization of linear mixed-effects model, where the conditional mean of the outcome, given the random effects, is a non-linear function of both the coefficients and random effects (StataCorp, 2023). Food Policy 136 (2025) 102955 5 L. Čechura et al. 3.2.3. Step 3 Productivity (𝜔𝑖𝑡) is estimated by averaging 𝜔𝑖𝑡 from Eqs. (A.5), (A.6) and (A.7). After estimating 𝜔𝑖𝑡, we can estimate the productivity change component (𝜔𝑖𝑡) as: 𝜔𝑖𝑡 =𝜔𝑖𝑡 −𝜔𝑖𝑡−1 (3.5) Given the significant variation in farm sizes, we employ an aggregate weighted productivity metric and its decomposition to analyze the impact of farm size on overall productivity. In particular, the weighted aggregate productivity (Olley and Pakes, 1996) is given by: 𝜔𝑡=∑ 𝑖 𝑊𝑖𝑡𝜔𝑖𝑡 (3.6) where: 𝑊𝑖𝑡 =𝑌𝑖𝑡∕∑𝑖𝑌𝑖𝑡. The weighted aggregate productivity can be decomposed to unweighted average productivity and the covariance between output and productivity level (Kumbhakar and Li, 2024): 𝜔𝑡=∑ 𝑖 𝑊𝑖𝑡𝜔𝑖𝑡 =∑ 𝑖 [ 𝑊𝑡+ (𝑊𝑖𝑡 − 𝑊𝑡)][ 𝜔𝑡+ (𝜔𝑖𝑡 −𝜔𝑡)] =𝜔𝑡+∑ 𝑖 (𝑊𝑖𝑡 − 𝑊𝑡)(𝜔𝑖𝑡 −𝜔𝑡)≡𝜔𝑡+𝑐𝑜𝑣𝑡(3.7) where 𝜔𝑡= 1∕𝑁𝑡∑𝑖𝜔𝑖𝑡 is the unweighted average productivity,  𝑊𝑡= 1∕𝑁𝑡 represents the unweighted output share and 𝑐𝑜𝑣𝑡≡∑𝑖(𝑊𝑖𝑡 −  𝑊𝑡)(𝜔𝑖𝑡 −𝜔𝑡) is sample covariance between output share and productivity. Here, 𝜔𝑡 is the mean of persistent productivity over farms at time 𝑡, while 𝑐𝑜𝑣𝑡 measures the extent to which additional resources are allocated to more productive farms. The change in mean productivity 𝛥𝜔𝑡 and the change in covariance 𝛥𝑐𝑜𝑣𝑡 represent the strength of the association between output share and productivity change. That is, the larger the covariance, the greater the contribution of farms with highoutput farms to the weighted aggregate productivity change: 𝛥𝜔𝑡= 𝛥 𝜔𝑡+𝛥𝑐𝑜𝑣𝑡. 4. Data This study uses farm-level data derived from the Farm Accountancy Data Network (FADN), a comprehensive database containing harmonized microeconomic data for agricultural holdings. Our initial sample consists of 10,799 observations from the two most prominent farming types contributing to cereal production in the Czech Republic: field crop farms and mixed farms, covering the period 2008 to 2020. Specifically, field crop farms account for 55% observations, while mixed crop and livestock farms 45%. The model differentiates between three variable inputs: fertilizers, pesticides, and other materials; three quasi-fixed inputs: land, capital, and labor; and one output variable representing aggregate production. Fertilizers (𝐹) refer to the cost of purchasing fertilizers and soil improvers. Pesticides (𝐻) are the cost of purchasing plant protection products, traps and baits, bird scarers, anti-hail shells, frost protection, etc. (excluding those used for forests). Other materials (𝑂) are calculated as the sum of the total specific costs and total agricultural overhead, excluding fertilizers and pesticides. The land (𝐴) is measured in hectares of the Utilized Agricultural Area (UAA). Capital (𝐾) is measured as the total current assets. Labor (𝐿) is measured in the Annual Work Unit (AWU). The output variable (𝑌) represents the aggregate production and is calculated as the sum of the value of cereal production, other crop production (measured as the difference between the total value of crop production and the value of cereal production) and other farm output (which is the sum of cow’s milk and milk products, other livestock production and other farm-generated production). Table B.1 in Appendix B provides the FADN codes for these variables. The monetary variables are deflated using the price indices for the Czech Republic from the EUROSTAT database (2010 = 100). Capital input is adjusted for inflation using the index of purchase prices for agricultural production inputs, specifically for materials. The price index for fertilizers and soil adjusts the fertilizer input. The price index for plant protection products and pesticides is used to adjust the pesticide input. The price index of goods and services currently consumed in agriculture is used to deflate the other input materials. The output variable is deflated using the producer price indices of agricultural products (output), namely, the price index for the cereals, the price index for the crops, the price index of milk and milk products, the price index for animal production and the price index for the production of agricultural inputs. In addition, this study incorporates the price index for the production of agricultural goods (excluding fruits and vegetables, 𝑃). Since not all agricultural holdings in the database contain complete information, observations with zero values of cereal production, land, labor, capital, and material are excluded from the dataset. Furthermore, holdings with less than three consecutive years of observations are excluded from the sample. This procedure allows for the use of lags and Mundlak’s device, while also reducing issues related to the entry and exit of agricultural holdings in the database. After this data cleaning procedure, the final dataset consists of 7,269 observations. Table 1 presents the summary statistics for the input and output variables. Furthermore, all log-transformed variables are normalized by dividing them by their respective sample means. This allows the first-order parameters to be interpreted as output elasticities of the respective inputs evaluated at the sample mean. We adopt this approach because it simplifies the interpretation of the estimates. 5. Results 5.1. Production function parameter estimates The estimated parameters are presented in Tables 2and 3. The tables indicate that the parameters conventionally analyzed in the production function estimates are highly significant. This applies to all first-order coefficients as well as the majority of second-order parameters and productivity term. The statistical significance of the secondorder parameters supports the use of a flexible translog form over the more restrictive CD production function. This is formally confirmed by the Wald test, which rejects the null hypothesis of imposing zero restrictions on the second-order parameters, even at 1% significance level (𝜒2∶ 1156.5, 𝑝-value = 0.000). In other words, the application of the CD function, as seen in numerous proxy variable literature studies, could lead to biased results. In addition, the estimation results are evaluated based on whether the estimated parameters satisfy the theoretical consistency conditions of production technology. Theoretically, a production function should be nondecreasing in inputs (monotonicity) and concave in inputs (diminishing marginal returns). The monotonicity conditions for the inputs require 𝛼𝑟>0 for 𝑟=𝑓, ℎ, 𝑜; and 𝛽𝑗>0 for 𝑗=𝑘, 𝑎, 𝑙, respectively. The concavity condition (diminishing marginal returns) requires: 𝛼𝑟𝑟 +𝛼2 𝑟−𝛼𝑟<0 for 𝑟=𝑓, ℎ, 𝑜; and 𝛽𝑟𝑟 +𝛽2 𝑟−𝛽𝑟<0 for 𝑟=𝑘, 𝑎, 𝑙, respectively. Tables 2and 3 show that these conditions are met by the estimates.12 The first specification test suggests the robustness of the model across alternative specifications and validates the correct choice of proxy variables. In particular, Table B.2 in Appendix B presents alternative model estimates: Model 1—baseline model with 3 variable and 3 quasi-fixed inputs; Model 2—with 2 variable inputs (fertilizers and pesticides aggregated into a single agrochemical input) and 3 quasifixed inputs; Model 3—with 4 variable and 2 quasi-fixed inputs; and Model 4—with 3 variable and 2 quasi-fixed inputs. The results indicate consistent robustness across these models. Moreover, Table B.3 shows 12 We restrict our attention to the first principle minors of the Hessian (the matrix of second derivatives of the production function) because our analysis requires only diminishing returns to scale, not convex technologies. While convexity implies diminishing returns to scale, the reverse is not necessarily true. Food Policy 136 (2025) 102955 6 L. Čechura et al. Table 1 Summary statistics of input and output variables. Variable Obs Mean Std.Dev. Min Max Land [hectares] 7269 908.7 983.4 6.5 10160.3 Capital [1,000 CZK] 7269 24200.0 34000.0 30.0 391000.0 Labor [AWU] 7269 23.6 29.6 0.2 232.5 Fertilizers [1,000 CZK] 7269 2581.8 3383.6 1.6 52300.0 Pesticides [1,000 CZK] 7269 2446.3 3075.3 1.3 29100.0 Other materials [1,000 CZK] 7269 20000.0 26800.0 95.1 211000.0 Output [1,000 CZK] 7269 29600.0 38800.0 93.7 310000.0 Table 2 Estimates of input elasticity coefficients (step 1). Variable input - Fertilizers Variable Coef. Std.Err. 𝑝-value Recalculated coef. const. 0.086 0.000 0.000 0.081 𝑓0.017 0.000 0.000 0.016 ℎ0.005 0.000 0.000 0.005 𝑜−0.032 0.001 0.000 −0.030 𝑎0.023 0.000 0.000 0.021 𝑙−0.010 0.001 0.000 −0.009 𝑘−0.002 0.001 0.001 −0.002 t−0.001 0.000 0.000 −0.001 Variable input - Pesticides Variable Coef. Std.Err. 𝑝-value Recalculated coef. const. 0.076 0.000 0.000 0.072 ℎ0.016 0.000 0.000 0.015 𝑓0.005 0.000 0.000 0.005 𝑜−0.017 0.001 0.000 −0.016 𝑎0.013 0.001 0.000 0.012 𝑙−0.008 0.000 0.000 −0.008 𝑘−0.008 0.000 0.000 −0.007 t 0.000 0.000 0.000 0.000 Variable input - Other materials Variable Coef. Std.Err. 𝑝-value Recalculated coef. const. 0.646 0.002 0.000 0.608 𝑜0.236 0.005 0.000 0.222 𝑓−0.062 0.004 0.000 −0.059 ℎ−0.040 0.004 0.000 −0.037 𝑎0.003 0.005 0.635 0.002 𝑙−0.087 0.004 0.000 −0.082 𝑘−0.054 0.003 0.000 −0.050 t−0.004 0.000 0.000 −0.004 Note: The estimated coefficients are 𝜷1×𝜃. Estimates of 𝜷1 (Recalculated coef.) are obtained by dividing each of the NLS parameters by estimated  𝜃 (see Section 3.2.1 for details). no significant differences in the first-order parameters (production elasticities). Figs. B.1 further illustrate that the dynamics of productivity change remain consistent across specifications, while Table B.4 provides statistical evidence of no significant differences in productivity change under alternative model structures. The second specification test evaluates the assumptions of proxy validity. Specifically, it tests whether production inputs in 𝑡 and 𝑡− 1 are uncorrelated with productivity innovations in time 𝑡. Table B.5 in Appendix B confirms that this condition holds across all estimates, even at the 1% significance level. Table 2 presents the estimated parameters for the variable inputs. Specifically, the coefficients represent the first derivatives of the production function with respect to each variable input. The constant term corresponds to the first-order parameter of the production function, i.e., the output elasticity of respective input evaluated at the sample mean. The parameters for variable and quasi-fixed inputs correspond to the squared and cross-product terms, respectively. Finally, the parameter for the time variable provides information on biased technological change.13 13 The time variable is commonly used in applied production analysis as a proxy for technological change. However, it may also capture additional The estimated elasticities of the variable inputs indicate that the overall impact of material inputs is 0.761.14 This value aligns with other studies on Czech agriculture (e.g., Bokusheva and Čechura, 2017) and reflects the material input shares in our dataset. The elasticities of fertilizers and pesticides are similar, at 0.081 and 0.072, respectively. In other words, fertilizers and pesticides account for one-fifth of the impact of material input, suggesting their considerable role in overall production. Furthermore, the role of fertilizers and pesticides increases in material-intensive production systems, as indicated by the secondorder parameters. These inputs also complement each other, unlike other material inputs. Increased fertilizer use can indirectly create favorable conditions for pests, diseases, and weeds, which, in turn, increases the demand for pesticides to protect crops (Ogada et al., 2021). Additionally, the use of fertilizers, pesticides, and other materials increases with land input, but decreases with labor and capital input. This indicates that larger farms tend to have higher material intensity, including greater fertilizer and pesticide use. This aligns with trends in the Czech Republic, where extensive farming systems with lower chemical inputs, particularly organic farming, are primarily small enterprises. In fact, 75% of organic farming enterprises operate on farms of up to 100 hectares (Hlaváčková et al., 2023). Table 3 presents the estimated output elasticities of quasi-fixed inputs. The results reveal that land has the greatest impact on production, with an output elasticity of 0.162. The elasticities of labor and capital are almost the same magnitude, about 0.09 each. The elasticity of land declines in land-intensive production systems. In contrast, the elasticities of labor and capital increase in capital-intensive and labor-intensive production systems. The mutual cross-product parameters indicate a negative association between land and both labor and capital inputs as well as a positive association between labor and capital. These findings are consistent with the first-step results for variable inputs, offering important implications for agricultural production systems and their developmental trajectory. These findings suggest that expanding land use in land-intensive systems may be less effective for increasing output than optimizing other inputs or enhancing land productivity through technological and sustainable practices. Investments in workforce skills, technological advancements, and capital improvements yield proportionally higher returns than land expansion, reinforcing the need for modernization and efficiency-driven strategies in Czech agriculture. The sum of the first-order parameters is 1.114, suggesting only limited economies of scale at the mean of the data. This indicates that farms, on average, operate close to their optimal scale, meaning the benefits of expanding farm size are relatively modest. However, second-order parameters of variable and quasi-fixed inputs suggest that economies of scale are less pronounced in land-intensive farms, whereas they are more significant in material-, labor-, and capitalintensive farms. Estimated scale returns also provide insight into the shadow shares of inputs.15 These shadow shares are: fertilizers (0.073), factors, such as climate change or land degradation, if they follow a similar deterministic trend. 14 The overall impact of material inputs represents the sum of the elasticities fertilizers, pesticides and other material inputs. 15 The input shadow shares are calculated as estimated output elasticities divided by returns to scale. That is, the shadow share of the input represents the elasticities of the output in production with constant returns to scale. Food Policy 136 (2025) 102955 7 L. Čechura et al. Table 3 Production function estimates (step 2). Variable Coef. Std.Error 𝑝-value Variable Coef. Std.Error 𝑝-value R 0.029 0.014 0.034 𝑎×𝑘−0.037 0.010 0.000 𝑎0.162 0.022 0.000 𝑙×𝑘0.039 0.013 0.002 𝑙0.093 0.016 0.000 𝑎×𝑡0.010 0.002 0.000 𝑘0.098 0.009 0.000 𝑙×𝑡0.001 0.002 0.499 𝑡0.001 0.001 0.214 𝑘×𝑡−0.002 0.001 0.049 𝑎2−0.005 0.022 0.811 𝑞𝑎−0.137 0.025 0.000 𝑙20.079 0.025 0.001 𝑞𝑙0.018 0.021 0.390 𝑘20.067 0.009 0.000 𝑞𝑘0.011 0.014 0.412 𝑡20.005 0.000 0.000 𝑞𝑡0.007 0.002 0.004 𝑎×𝑙−0.021 0.020 0.300 const. −0.074 0.029 0.010 Note: 𝑅 represents  𝑅𝑖𝑡−1 = [𝑅𝐹𝑖𝑡−1 +𝑅𝐻𝑖𝑡−1 +𝑅𝑂𝑖𝑡−1]∕3, see estimated model formulation in (3.4). 𝑞𝑗, 𝑗 =𝑘, 𝑎, 𝑙, 𝑡 represent group means of particular variables (see Mundlak’s (1978) formulation). Fig. 1. Sources of output growth. pesticides (0.065); other materials (0.546), with a total shadow share for material inputs of 0.683; land (0.145); labor (0.083); and capital (0.088). Technological change has a positive impact on production, and this impact accelerates over time. However, the first-order parameter is not statistically significant, suggesting that technological change only becomes important later in the study period. Since the cross parameters of fertilizers, pesticides, and other material, as well as those of quasi-fixed inputs (except for labor), are statistically significant, Hicks-neutral technological change can be rejected in favor of biased technological change16 (formally confirmed by the F-test). Specifically, technological change exhibits the following patterns: it is materialand capital-saving, while being land-using. However, given the magnitude of the estimated parameters, the overall impact of biased technological change is limited. 5.2. Sources of output and TFP growth Fig. 1 illustrates the dynamics of output growth and its underlying sources.17 Output growth follows a strong upward trend from 2010 to 2015, then declines in 2016 and 2017, returning to 2013 levels before recovering in the subsequent years. Productivity changes are identified as the main driver of output growth, with their dynamics closely mirroring those of overall output growth (see Table B.6 in Appendix B for reference). Additionally, variable inputs and technological change also contribute significantly to output growth. Variable inputs have a positive 16 Biased technological change offers valuable insights into the dynamics of input shadow shares, reflecting shifts in input productivity driven by technological advancements, including the adoption of new technologies and innovations. 17 Sources of output growth are given by Eq. (2.2). Fig. 2. Sources of TFP growth. effect in the first half of the study period but contribute negatively in the latter part. Technological change exerts a positive impact on output growth toward the end of the period under investigation. Quasi-fixed inputs, however, do not contribute to output growth. The observed patterns indicate a transition from intensive agriculture to more sustainable practices, aligning with the evolution of agricultural policy frameworks and their implementation. The dynamics of TFP closely follow those of output growth (see Fig. 2 and Table B.7 in Appendix B). The drivers18 of TFP growth are also similar, with productivity changes being their primary factor. Variable inputs are the second most significant contributor to TFP dynamics. However, unlike output growth, variable inputs have a considerable negative impact in the first half of the study period. In the second half, the positive and negative impacts of variable inputs closely mirror the fluctuations in TFP growth. Similar to output growth, technological change, similar to its role in output growth, has a positive impact toward the end of the study period. Quasi-fixed inputs do not significantly influence TFP growth. The agricultural productivity of the farm is commonly characterized by considerable heterogeneity. Fig. 3 illustrates the distribution of productivity changes across the study period, highlighting that productivity dynamics vary widely among farms. The pronounced heterogeneity is a key characteristic of Czech cereal production. Moreover, given the substantial dual structure in Czech agriculture, where farm sizes differ significantly,19 we constructed an aggregate productivity measure using output weights (see Eq. (3.6)). The dynamics of weighted aggregate productivity change can be broken down into the unweighted average productivity change (as discussed earlier); and the sample covariance between output share and productivity level. Both are depicted in Appendix B Table B.8.20 The dynamics of productivity appear to be shaped by multiple factors, including management quality, weather conditions, and farming practices. For example, between 2012 and 2015, productivity exhibited an upward trend, driven by favorable weather conditions, stable overwintering, and the optimal distribution of rainfall during the growing season. In contrast, extreme weather events negatively affected cereal 18 Sources of TFP growth are given by Eq. (2.3). 19 Duality in Czech agriculture is discussed by Cechura et al. (2022), Redlichová et al. (2023), Lososová et al. (2023). 20 The table indicates that there are no considerable differences between weighted and unweighted productivity changes. This suggests that farm size, represented by output level, does not significantly affect the productivity dynamics. However, a slight decrease in covariance during the second half of the study period may imply that small farms have contributed increasingly to weighted aggregate productivity changes. Furthermore, this trend may also indicate that more resources are being allocated to more productive farms, specifically to smaller farms. Food Policy 136 (2025) 102955 8 L. Čechura et al. Fig. 3. Productivity change distribution. production in 2017, as cereals experienced temperature stress and water deficits during critical growth stages21. Agricultural policy underwent significant transformations during the study period, reflecting a shift toward greater environmental responsibility, sustainable rural development, and more targeted financial support mechanisms. Since 2015, greening measures and young farmers’ payments have been incorporated into the direct payments scheme. These policy changes may have influenced productivity dynamics, as numerous studies have documented productivity shifts driven by policy instruments (Biagini et al., 2023; Nilsson, 2017; Khafagy and Vigani, 2022; Mary, 2013; Rizov et al., 2013). However, the existing literature does not provide a definitive conclusion regarding the impact of subsidies on productivity. The heterogeneous effects of subsidies are attributed to the varying objectives and implementation, which in turn influence farmers’ behavior and productivity in diverse ways (Biagini et al., 2023; Nilsson, 2017). 5.3. Variable input dynamics and simulation of policy scenarios The drivers of variable input dynamics are illustrated in Fig. 4 (for details, see Table B.9 in Appendix B). The figure indicates that the variable input dynamics are influenced by all variable inputs— namely fertilizers, pesticides, and other materials—in the first half of the study period, while in the second half, they are primarily driven by fertilizers and other materials. Fertilizers appear to have a mostly positive impact on variable input dynamics throughout the study period, with a stronger influence in the first half. Pesticides exhibit a considerable positive impact in the early years, followed by a small negative impact later on. Other materials show fluctuations that mirror the overall changes in variable inputs, with varying contributions over time. Most importantly, since variable inputs are key drivers of both output and TFP growth, Fig. 4 highlights the essential role of fertilizers and pesticides in shaping agricultural productivity. The critical role of fertilizers and pesticides in driving output and TFP growth can have significant implications for the objectives of the Farm-to-Fork strategy. As discussed in the introduction, this strategy aims to: (i) reduce nutrient losses by at least 50%, resulting in a minimum 20% reduction in fertilizer use by 2030; and (ii) reduce the use and risk of chemical pesticides by 50% by 2030. The following sections will explore the potential ramifications of reducing agrochemicals within the context of Czech agriculture. 21 As demonstrated by Lesk et al. (2016), Beillouin et al. (2020), Devot et al. (2023), droughts and extreme heat significantly reduce cereal production, particularly when these climatic extremes occur during critical stages of the growing season. Fig. 4. Decomposition of variable inputs dynamics. Table 4 presents the results of simulations for different scenarios.22 The baseline scenario reflects conditions in 2020, the final year of the study. The simulations illustrate the impact of different levels of fertilizer and pesticide reductions23 on overall agricultural output. Table 4 shows that a 10% overall reduction in fertilizer and pesticide use (Scenario 1) compared to 2020 leads to a 3.5% decrease in agricultural output, with corresponding reductions in the shadow shares of fertilizers and pesticides by 4.2% and 4.9%, respectively. Scenario 2 models a 10% reduction in fertilizers and a 20% reduction in pesticides, leading to a 4.2% decrease in output and reductions in the shadow shares of fertilizers and pesticides by 4.6% and 6.6%, respectively. Finally, Scenario 3 assumes a 20% reduction in both fertilizers and pesticides, resulting in a 5.4% decrease in agricultural output and declines in fertilizer and pesticide shadow shares by 6.1% and 7.1%, respectively. 6. Discussion In line with previous research (Beckman et al., 2020; Bremmer et al., 2021; Jost et al., 2025), the results indicate the considerable impact of fertilizer and pesticide reduction on agricultural output. This suggests that efforts to reduce reliance on agrochemicals could substantially affect farm productivity and profitability. However, estimated output reductions represent an upper bound of expected effects, as they do not account for farmer adaptation strategies. Farmers can mitigate reductions in agrochemical input by adopting alternative management practices, such as cover cropping, mechanical interventions, biological pest control, and various crop rotations (Weisberger et al., 2019; Riemens et al., 2022; Bremmer et al., 2021). Implementing adaptive practices requires adjustments in other material inputs, labor and capital (Beckman et al., 2020). This is consistent with the larger economic patterns observed in the estimation of the production function, which highlight the role of enhancing laborand capital-intensive systems as a pathway for agricultural development that reduces the reliance on agrochemical inputs. However, considering the ongoing outflow of labor from agriculture, these adaptations will likely manifest predominantly through increased capital intensification rather than increased labor 22 The scenarios were selected to assess the sensitivity of production to fertilizer and pesticide reductions, providing insights into the potential impacts of the Farm-to-Fork strategy. While they do not exactly consider a 50% pesticide reduction, they reflect the strategy’s framework. 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