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Productive Capacity of Biodiversity: Crop Diversity and Permanent Grasslands in Northwestern France


Previous studies on the productive capacity of biodiversity emphasized that greater crop diversity increases crop yields. We examined the influence of two components of agricultural biodiversity—farm-level crop diversity and permanent grasslands—on the production of cereals and milk. We focused on productive interactions between these two biodiversity components, and between them and conventional inputs. Using a variety of estimators (seemingly unrelated regressions and general method of moments, with or without restrictions) and functional forms, we estimated systems of production functions using a sample of 3960 mixed crop-livestock farms from 2002 to 2013 in France. The estimates highlight that increasing permanent grassland proportion increased cereal yields under certain conditions and confirm that increasing crop diversity increases cereal and milk yields. Crop diversity and permanent grasslands can substitute each other and be a substitute for fertilizers and pesticides.

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Fig. 1


  1. This method is often used in ecosystem services valuation studies (Perrings 2010). Another method consists of stochastic frontier analysis, such as by Omer et al. (2007), Amsler et al. (2017) and Ang et al. (2018).

  2. Interspecific diversity refers to diversity among crop species, while intraspecific diversity refers to diversity among genetic varieties of the same crop.

  3. Mountain regions in France have more permanent grasslands but less crop production.

  4. We excluded southwestern France from our analysis since it has a notably smaller area of permanent grasslands than northwestern France does (especially due to its warmer climate).

  5. Organic fertilizer (manure) is a crucial control variable since it is correlated with permanent grassland area. Excluding it from the estimation would have overestimated the productivity of permanent grasslands.

  6. A random individual effect could have been specified, but the Durbin-Wu-Hausman test indicated that an individual fixed effect was preferable.

  7. Moreover, the milk yield equation had two more regressors than that for cereals (cow feed and health expenses).

  8. In France, the first stage (land-use decisions) usually occurs in autumn, while the second stage (variable input applications) usually occurs in spring.

  9. Most studies on the productivity of biodiversity have used log–log production functions (e.g., Di Falco and Zoupanidou, 2017). However, because approximately one-third of our observations had no permanent grassland, we could not estimate this function without transforming the data.

  10. The nine annual climatic variables are total rainfall, days of rain, total snowfall, days of snowfall, wind speed, humidity, and minimum, maximum and mean temperatures measured.

  11. We attempted to add similar interactions for milk production, but performances of the models decreased considerably (e.g. several variable inputs had negative productivities for milk).

  12. LAUs (Local Administrative Units) are building blocks of the NUTS (Nomenclature of Territorial Units for Statistics) used by the European Union statistical system.

  13. This result was not surprising: milk-producing farms with a larger proportion of permanent grasslands are usually considered the most extensive (Ryschawy et al. 2012).

  14. Equations of the variable input applications instrumented with prices and subsidies showed R2 = 0.16–0.34 (results available upon request). Price ratios had significant effects and expected signs. In addition, we tested the assumption of short-term optimization by estimating the influence of the other exogenous variables on crop diversity. Ordinary-least-square estimation showed R2 = 0.03 in the within form (results available upon request), which suggested little endogenous bias in crop diversity and tended to support the assumption of very short-term optimization.

  15. Addition of an interaction variable between pesticide application and a trend highlighted that pesticide productivities were positive at the beginning of the period but negative at the end (Appendix 4). This result may have been due to a change in pesticide quality: farmers applied different types of pesticides during the period, and the pesticides that remained by the end may have been less effective. Since milk yields increased over the period, this may have been a temporal conjuncture confound.

  16. Recall that systems (4) and (5) can be estimated only using Models 1 and 2 due to the interaction terms between the biodiversity indicators and variable inputs.

  17. The relation between variable inputs and biodiversity productive capacity may differ in developing regions, where variable inputs are limiting inputs.

  18. In subsequent model development, area-based subsidies of the European Union’s Common Agricultural Policy were not considered in the empirical estimation since they were decoupled from yields before the beginning of our panel.

  19. Since area-based subsidies were decoupled from yields, they influence land allocation among products but not yields.

  20. In France, the first stage usually occurs in autumn, while the second stage usually occurs in spring.

  21. Carpentier and Letort (2012), for example, also made this assumption. We estimated the production functions assuming non-constant return to area, but the estimated parameters were non-significant.

  22. Total forage area equals the sum of the areas of maize silage, temporary grassland and permanent grassland. Note that \(a_{i2t}\) and \(B_{i2t}\) differ:\(B_{i2t}\) provides information only about permanent grasslands. The areas of maize silage and temporary grasslands are ecosystem components captured by \(B_{i1t}\).


  • Amsler C, Prokhorov A, Schmidt P (2017) Endogenous environmental variables in stochastic frontier models. J Econom 199(2):131–140

    Article  Google Scholar 

  • Ang F, Mortimer SM, Areal FJ, Tiffin R (2018) On the opportunity cost of crop diversification. J Agric Econ 69(3):794–814

    Article  Google Scholar 

  • Asunka S, Shumway CR (1996) Allocatable fixed inputs and jointness in agricultural production: more implications. Agric Resour Econ Rev 25:143–148

    Article  Google Scholar 

  • Aviron S, Burel F, Baudry J, Schermann N (2005) Carabid assemblages in agricultural landscapes: impacts of habitat features, landscape context at different spatial scales and farming intensity. Agric Ecosyst Environ 108(3):205–217

    Article  Google Scholar 

  • Baltagi B (2008) Econometric analysis of panel data. Wiley, New York

    Google Scholar 

  • Baudry J, Bunce RGH, Burel F (2000) Hedgerows: an international perspective on their origin, function and management. J Environ Manage 60:7–22

    Article  Google Scholar 

  • Baumgärtner, S. (2006). Measuring the diversity of what? And for what purpose? A conceptual comparison of ecological and economic biodiversity indices. WorkingPaper. Heidelberg: Interdisciplinary Institute for Environmental Economics.

  • Baumol WJ, Panzar JC, Willig RD, Bailey EE, Fischer D, Fischer D (1988) Contestable markets and the theory of industry structure. Harcourt Brace Jovanovich, New York

    Google Scholar 

  • Burel F, Baudry J (2003) Landscape ecology: concepts, methods, and applications. N.H., Science Publishers, Enfield, p 362

    Book  Google Scholar 

  • Carpentier A, Letort E (2012) Accounting for heterogeneity in multicrop micro-econometric models: implications for variable input demand modeling. Am J Agric Econ 94:209–224

    Article  Google Scholar 

  • Chatellier V, Gaigné C (2012) Les logiques économiques de la spécialisation productive du territoire agricole français. Innovations Agronomiques 22:185–203

    Google Scholar 

  • CORPEN (2006) Les émissions d’ammoniac et de gaz azotés à effet de serre en agriculture. MAAP, Paris

    Google Scholar 

  • Desjeux Y, Dupraz P, Kuhlman T, Paracchini ML, Michels R, Maigné E, Reinhard S (2015) Evaluating the impact of rural development measures on nature value indicators at different spatial levels: application to France and The Netherlands. Ecol Ind 59:41–61

    Article  Google Scholar 

  • Díaz S, Settele J, Brondízio E, Ngo H, Guèze M, Agard J, Arneth A, Balvanera P, Brauman K, Butchart S, Chan K, Garibaldi L, Ichii K, Liu J, Subrmanian S, Midgley G, Milo-slavich P, Molnár Z, Obura D, Pfaff A, Polasky S, Purvis A, Razzaque J, Reyers B, Chowdbury R, Shin Y, Visseren-Gamakers I, Bilis K, Zayas C (2020) Summary for policy-makers of the global assessment report on biodiversity and ecosystem services of the Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services. Available from:

  • Di Falco S, Chavas J-P (2006) Crop genetic diversity, farm productivity and the management of environmental risk in rainfed agriculture. Eur Rev Agric Econs 33:289–314

    Article  Google Scholar 

  • Di Falco S, Chavas J-P (2008) Rainfall shocks, resilience, and the effects of crop biodiversity on agroecosystem productivity. Land Econ 84:83–96

    Article  Google Scholar 

  • Di Falco S, Bezabih M, Yesuf M (2010) Seeds for livelihood: crop biodiversity and food production in Ethiopia. Ecol Econ 69:1695–1702

    Article  Google Scholar 

  • Di Falco S, Zoupanidou E (2017) Soil fertility, crop biodiversity, and farmers’ revenues: evidence from Italy. Ambio 46(2):162–172

    Article  Google Scholar 

  • Donfouet HPP, Barczak A, Détang-Dessendre C, Maigné E (2017) Crop production and crop diversity in France: a spatial analysis. Ecol Econ 134:29–39

    Article  Google Scholar 

  • Hooper DU, Chapin Iii FS, Ewel JJ, Hector A, Inchausti P, Lavorel S, Lawton JH, Lodge DM, Loreau M, Naeem S (2005) Effects of biodiversity on ecosystem functioning: a consensus of current knowledge. Ecol Monogr 75:3–35

    Article  Google Scholar 

  • Kim K, Barham BL, Coxhead I (2000) Measuring soil quality dynamics A role for economists, and implications for economic analysis. Agric Econ 25:13–26

    Article  Google Scholar 

  • Kleijn D, Kohler F, Báldi A, Batáry P, Concepción ED, Clough Y, Diaz M, Gabriel D, Holzschuh A, Knop E, Kovács A, Marshall EJP, Tscharntke T, Verhulst J (2009) On the relationship between farmland biodiversity and land-use intensity in Europe. Proc Roy Soc Lond B Biol Sci 276:903–909

    Article  Google Scholar 

  • Klemick H (2011) Shifting cultivation, forest fallow, and externalities in ecosystem services: Evidence from the Eastern Amazon. J Environ Econ Manage 61:95–106

    Article  Google Scholar 

  • Martel G, Aviron S, Joannon A, Lalechère E, Roche B, Boussard H (2019) Impact of farming systems on agricultural landscapes and biodiversity: From plot to farm and landscape scales. Eur J Agron 107:53–62

    Article  Google Scholar 

  • Mainwaring L (2001) Biodiversity, biocomplexity, and the economics of genetic dissimilarity. Land Econ 77:79–83

    Article  Google Scholar 

  • MEA (2005) Ecosystems and human well-being. Island Press, Washington

    Google Scholar 

  • Noack F, Riekhof MC, Di Falco S (2019) Droughts, biodiversity, and rural incomes in the tropics. J Assoc Environ Resour Econ 6(4):823–852

    Google Scholar 

  • Omer A, Pascual U, Russell NP (2007) Biodiversity conservation and productivity in intensive agricultural systems. J Agric Econ 58:308–329

    Article  Google Scholar 

  • Paul C, Hanley N, Meyer ST, Fürst C, Weisser WW, Knoke T (2020) On the functional relationship between biodiversity and economic value. Sci Adv 6(5)

  • Perrings C (2010) The economics of biodiversity: the evolving agenda. Environ Dev Econ 15:721–746

    Article  Google Scholar 

  • Ricketts TH, Regetz J, Steffan-Dewenter I, Cunningham SA, Kremen C, Bogdanski A, Gemmill-Herren B, Greenleaf SS, Klein AM, Mayfield MM (2008) Landscape effects on crop pollination services: are there general patterns? Ecol Lett 11:499–515

    Article  Google Scholar 

  • Ryschawy J, Choisis N, Choisis JP, Joannon A, Gibon A (2012) Mixed crop-livestock systems: an economic and environmental-friendly way of farming? Animal 6:1722–1730

    Article  Google Scholar 

  • Steffan-Dewenter I, Münzenberg U, Bürger C, Thies C, Tscharntke T (2002) Scale-dependent effects of landscape context on three pollinator guilds. Ecology 83(5):1421–1432

    Article  Google Scholar 

  • Tilman D, Reich PB, Knops J, Wedin D, Mielke T, Lehman C (2001) Diversity and productivity in a long-term grassland experiment. Science 294(5543):843–845

    Article  Google Scholar 

  • Thenail C (2002) Relationships between farm characteristics and the variation of the density of hedgerows at the level of a micro-region of bocage landscape. Study case in Brittany. France Agric Syst 71:207–230

    Article  Google Scholar 

  • van Rensburg TM, Mulugeta E (2016) Profit efficiency and habitat biodiversity: the case of upland livestock farmers in Ireland. Land Use Policy 54:200–211

    Article  Google Scholar 

  • Wooldridge JM (2015) Introductory econometrics: a modern approach. Cengage Learning, Boston

    Google Scholar 

  • Zellner A (1962) An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. J Am Stat Assoc 57(298):348–368

    Article  Google Scholar 

  • Zhang W, Ricketts TH, Kremen C, Carney K, Swinton SM (2007) Ecosystem services and dis-services to agriculture. Ecol Econ 64:253–260

    Article  Google Scholar 

Download references


The authors thank the three anonymous reviewers for their suggestions. The authors thank Sylvain Cariou for his help with data management. This research was funded by the Horizon 2020 program of the European Union (EU) under Grant Agreement No. 633838 (PROVIDE project, It is completed with the support of the French National Research Agency (ANR-16-CE32-0005, Soilserv, 2016–2020) and the Convergence Institute CLAND. This article does not necessarily reflect the view of the French Agence Nationale de la Recherche or the EU and in no way anticipates the European Commission’s future policy.

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Appendix 1: Very Short-Term Optimization

We considered a risk-neutral farmer who maximizes her annual profit \({\pi }_{it}\) by adjusting her applications of variable inputs (\({{\varvec{X}}}_{{\varvec{i}}{\varvec{t}}}\)) according to her quasi-fixed input levels (\({{\varvec{Z}}}_{{\varvec{i}}{\varvec{t}}}\)) and levels of biodiversity productive capacity (\({\varvec{B}}_{{{\varvec{it}}}}\)). We wrote the general farmer’s program as follows:

$$\pi_{it} = \mathop {\max }\limits_{{{\varvec{X}}_{it} }} \left\{ {E({\varvec{p}}_{it} )^{\prime } {\varvec{Y}}_{it} - E\left( {{\varvec{w}}_{it} } \right)^{\prime } {\varvec{X}}_{it} + S_{it} ;\left( {{\varvec{Y}}_{it} ,{\varvec{X}}_{it} ,{\varvec{Z}}_{it} ,{\varvec{B}}_{it} ,{\varvec{A}}_{it} } \right) \in T} \right\}$$

where \(E\left( {{\varvec{p}}_{it} } \right)\) and \(E\left( {{\varvec{w}}_{it} } \right)\) are the farmer’s expected prices, \(S\) sums the area-based subsidies received by the farm,Footnote 18 and \({\text{T}}\) is the production feasible plan of the multi-output farm. Program (6) defined the multi-output multi-input profit function that represents \({\text{T}}\) if \({\text{T}}\) is bounded compact and quasi-convex in (\({\varvec{X}}_{it}\),\({\varvec{Y}}_{it}\)) for each \({\varvec{Z}}_{it}\), \({\varvec{B}}_{it}\) and \(A_{it}\) (McFadden, 1978).

Program (6) represented the farmer’s annual production decisions, which we divided into a two-stage optimization process that isolated the estimated yield functions. The first stage occurs at the beginning of the agricultural year, when the farmer sows her land based on decoupled area subsidies \({s}_{ijt}\) (with\({S}_{it}={\sum }_{j}{{s}_{ijt}a}_{ijt}\)) and expected margins per ha\({E(\omega }_{ijt}^{ })\), with her land-use decisions being composed of J components\({a}_{ijt}\). \({E(\omega }_{ijt}^{ })\) depends on the farmer’s price expectations during this stage (usually in October in France). Unlike prices, \({s}_{ijt}\) is known and depends only on the type of land use (arable or grasslands).Footnote 19 The second stage (i.e. very short-term optimization) occurs during the agricultural year when the farmer optimizes gross margins of each area based on variable input application given her land use, which is assumed to be fixed (Asunka and Shumway 1996). Following Carpentier and Letort (2012) and Bareille and Letort (2018), we assumed that farmers know input prices (\(E\left({{\varvec{w}}}_{{\varvec{i}}{\varvec{t}}}\right)={{\varvec{w}}}_{{\varvec{i}}{\varvec{t}}}\)) but have naïve expectations of output prices (\(E\left({p}_{ijt}\right)={p}_{ijt-1}\)). However, because the first stage (land-use decisions) occurs ca. 3–6 months before the second stage (variable input applications),Footnote 20 expectations of variable input prices may differ between the two stages (due to new information), which may lead to differences between expected and realized margins. This difference justified the very short-term optimization. Specifically, we broke down (6) into a first-stage optimization (7) followed by a second-stage optimization (8):

$$\pi_{it} = \mathop {\max }\limits_{{a_{i1t} , \ldots ,a_{iJt} }} \left\{ {\mathop \sum \limits_{j = 1}^{J} a_{ijt} \left[ {E\left( {\omega_{ijt} \left( {p_{ijt - 1} ,E\left( {{\varvec{w}}_{it} } \right),{\varvec{Z}}_{it} } \right)} \right) + s_{ijt} } \right];\mathop \sum \limits_{j = 1}^{J} a_{ijt} = A_{it} } \right\}$$
$$E\left( {\omega_{ijt} \left( {p_{ijt - 1} ,{\varvec{w}}_{it} ,{\varvec{Z}}_{it} } \right)} \right) = \mathop {\max }\limits_{{x_{ijt} }} \left\{ {p_{ijt - 1} \cdot y_{ijt} - {\varvec{w}}_{it}^{\prime } {\varvec{x}}_{ijt} ; y_{ijt} \le f_{j} \left( {{\varvec{x}}_{ijt} ;{\varvec{B}}_{it} ,{\varvec{Z}}_{it} ,{\varvec{Y}}_{ - ijt} } \right)} \right\}$$

where the vector \({\varvec{x}}_{{{\varvec{ijt}}}}\) contains the variable input applied per ha of product j such that \(\mathop \sum \limits_{j} a_{ijt} {\varvec{x}}_{{{\varvec{ijt}}}}\) are the components of \({\varvec{X}}\). We assumed that \({\text{T}}\) is defined completely by the J output-specific frontiers \(f_{j} ( \cdot )\) such that \(Y_{ijt} \le a_{ijt} f_{j} ( \cdot )\) where \(Y_{ijt}\) is output production at the farm level and \({\varvec{Y}}_{{ - {\varvec{ijt}}}}\) represents the vector of the outputs besides j. The output-specific frontiers thus consider technological jointness at the farm level (e.g. organic fertilization, on-farm cereal consumption). Function \(f_{j} ( \cdot )\) is nonnegative, nondecreasing, linearly homogenous and concave in \({\varvec{x}}_{ijt}\). Note that \(f_{j}\)(⋅) does not depend on \(a_{ijt}\) explicitly (i.e. we assumed that marginal short-run returns to area are constant in output area).Footnote 21 In the econometric strategy, we focused only on the second stage (8), in which variable inputs are determined based on the exogenous land-use decisions and related biodiversity indicators.

Appendix 2: Allocation of Variable Inputs Between Outputs

We considered the case in which variable inputs are allocable inputs [which corresponds to \({{\varvec{x}}}_{{\varvec{i}}{\varvec{j}}{\varvec{t}}}\) in relations (8)]. Without loss of generality, we considered two outputs (j = 1 for cereals and j = 2 for milk) and solved the second stage (8) for \({x}_{ijkt}\) (\({x}_{ijkt}\) being the kth element of \({{\varvec{x}}}_{{\varvec{i}}{\varvec{j}}{\varvec{t}}}\)). With \({Y}_{2}={a}_{2}{y}_{2}\) and the area devoted to milk production \({a}_{2}>\) 0 (which corresponds to the total forage areaFootnote 22 and is exogenous in the second stage), we obtained the following first-order conditions:

$$\frac{{\partial f_{2} \left( {x_{i2t} ;\;{\varvec{B}}_{it} ,{\varvec{Z}}_{it} ,Y_{i1t} } \right)}}{{\partial x_{i2kt} }} = \frac{{w_{kt} }}{{p_{2t - 1} + \frac{{a_{i1t} }}{{a_{i2t} }}p_{1t - 1} \frac{{\partial y_{i1t} }}{{\partial y_{i2t} }}}}$$

where \(\partial {y}_{i1t}/\partial {y}_{i2t}\) represents additional cereal yields due to the increase of one unit of milk yield (which is null when there is no jointness). Farmers apply \({x}_{i2kt}\) on \({a}_{i2t}\) until the sum of the expected marginal productivity of \({x}_{i2kt}\) on \({y}_{i2t}\) and its indirect marginal productivities on \({y}_{i1t}\) equals \({w}_{kt}\). Like the common short-term maximization conditions, the previous relation highlights that an increase in the expected price of one output leads to increased input use (because \({f}_{j}()\) is concave in \({{\varvec{x}}}_{ijt}\)). Because the above relation is valid for each input and output, we obtained:

$${{\frac{{\partial f_{1} ( \cdot )}}{{\partial x_{i11t} }}} \mathord{\left/ {\vphantom {{\frac{{\partial f_{1} ( \cdot )}}{{\partial x_{i11t} }}} {\frac{{\partial f_{2} ( \cdot )}}{{\partial x_{i21t} }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial f_{2} ( \cdot )}}{{\partial x_{i21t} }}}} = \ldots = {{\frac{{\partial f_{1} ( \cdot )}}{{\partial x_{i1Jt} }}} \mathord{\left/ {\vphantom {{\frac{{\partial f_{1} ( \cdot )}}{{\partial x_{i1Jt} }}} {\frac{{\partial f_{2} ( \cdot )}}{{\partial x_{i2Jt} }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial f_{2} ( \cdot )}}{{\partial x_{i2Jt} }}}} = \frac{{p_{2t - 1} + p_{1t - 1} \frac{{a_{i1t} }}{{a_{i2t} }}\frac{{\partial y_{i1t} }}{{\partial y_{i2t} }}}}{{p_{1t - 1} + p_{2t - 1} \frac{{a_{i2t} }}{{a_{i1t} }}\frac{{\partial y_{i2t} }}{{\partial y_{i1t} }}}}$$

The ratios of marginal input productivities of cereals for milk are equal if variable inputs are actually allocable inputs. We used relation (9) for the shared variable inputs (fertilizers, pesticides, seeds and fuel) as parameter restrictions in Model 3 (SUR) and Model 4 (GMM).

In the second case, we modeled the variable inputs as non-allocable inputs (Baumol et al. 1988). We broke down program (6) into programs (10) (land-use decisions) and (11). (variable input application). Unlike in program (12), the farmer cannot optimize each margin separately in the second stage. We obtained:

$$\pi_{it} = \mathop {\max }\limits_{{a_{i1t} ; \ldots ;a_{iJt} }} \left\{ {\mathop \sum \limits_{j = 1}^{J} a_{ijt} \left[ {E\left( {\omega_{ijt} \left( {p_{ijt - 1} ,E\left( {{\varvec{w}}_{it} } \right),{\varvec{Z}}_{it} } \right)} \right) + s_{ijt} } \right];\mathop \sum \limits_{j = 1}^{J} a_{ijt} = A_{it} } \right\}$$
$$E\left( {\omega_{ijt} \left( {p_{ijt - 1} ,{\varvec{w}}_{{{\varvec{it}}}} ,{\varvec{Z}}_{{{\varvec{it}}}} } \right)} \right) = \mathop {\max }\limits_{{x_{it} }} \left\{ {p_{ijt - 1} \cdot y_{ijt} - {\varvec{w}}_{{{\varvec{it}}}}^{\prime } {\varvec{x}}_{{{\varvec{it}}}} ; y_{ijt} \le g_{j} \left( {{\varvec{x}}_{{{\varvec{it}}}} ;{\varvec{B}}_{{{\varvec{it}}}} ,{\varvec{Z}}_{{{\varvec{it}}}} ,{\varvec{Y}}_{{\user2{ - ijt}}} } \right)} \right\}$$

where \({\varvec{x}}_{{{\varvec{it}}}}\) is the vector of variable input applied per ha at the farm level such that \({\varvec{X}}_{{{\varvec{it}}}} = A{\varvec{x}}_{{{\varvec{it}}}}\). \(E\left( {y_{k} } \right)\) and \(E\left( {\varvec{x}} \right)\) defined in (10) are the solutions of (11) in which \({\varvec{w}}\) is imperfectly known. The vector of yields \({\varvec{y}}_{{{\varvec{it}}}}\) is composed of \(J\) yields \(y_{ijt}\). The function \(g_{j} \left( {{\varvec{x}}_{it} ;{\varvec{B}}_{{{\varvec{it}}}} ,{\varvec{Z}}_{{{\varvec{it}}}} ,{\varvec{Y}}_{{ - {\varvec{ijt}}}} } \right)\) is the yield function of \(y_{ijt}\), which differs from function \(f_{j} ( \cdot )\) by the form of the modelling of the variable inputs. We assumed that \({\text{T}}\) is defined completely by the K output-specific frontiers \(g_{j} ( \cdot )\) such that \(Y_{ijt} \le a_{j} g_{j} ( \cdot )\). Like function \(f_{j} ( \cdot )\), \(g_{j} ( \cdot )\) is nonnegative, nondecreasing, linearly homogenous and concave in \({\varvec{x}}_{{{\varvec{it}}}}\).

The variable input in program (11) was optimized in the very short term for all products at the same time (here, only milk and cereals), which led to the following:

$$a_{i1t} p_{1t - 1} \left( {\frac{{\partial g_{1} \left( {{\varvec{x}}_{{{\varvec{it}}}} ;{\varvec{B}}_{{{\varvec{it}}}} ,{\varvec{Z}}_{{{\varvec{it}}}} ,Y_{i2t} } \right)}}{{\partial x_{i2kt} }} + \frac{{\partial y_{1it} }}{{\partial y_{12t} }}\frac{{\partial g_{2} \left( {{\varvec{x}}_{{{\varvec{it}}}} ;{\varvec{B}}_{{{\varvec{it}}}} ,{\varvec{Z}}_{{{\varvec{it}}}} ,Y_{i21} } \right)}}{{\partial x_{i2kt} }}} \right) + a_{i2t} p_{2t - 1} \frac{{\partial g_{2} \left( {{\varvec{x}}_{{{\varvec{it}}}} ;{\varvec{B}}_{{{\varvec{it}}}} ,{\varvec{Z}}_{{{\varvec{it}}}} ,Y_{i21} } \right)}}{{\partial x_{i2kt} }} = w_{kt}$$

The sum of the direct and indirect marginal productivities of \({\varvec{x}}_{{{\varvec{it}}}}\) equals \({\varvec{w}}\), which prevented deriving parameter restrictions between outputs and inputs as was done in Models 3 and 4. Modeling variable inputs as non-allocable inputs led to direct estimation of the within transformation of system (2), with instrumentation (Model 2) or without instrumentation (Model 1) of the variable input applications.

Appendix 3: Verification of Parameter Restrictions for a Log-Linear Production Function and Unobserved Variable Input Application

We considered system (2) when the variable inputs were assumed to be private (Appendix 2). We verified the parameter restriction (9) when the production functions had a log-linear form (and assuming \(\partial y_{i1t} /\partial y_{i2t} = \partial y_{i2t} /\partial y_{i1t} = 0\), as in system (2)). We calculated marginal productivities of \(x_{ikt} = X_{ikt} /A_{it}\) (\(k \in \left[ {1;4} \right]\)) for cereals and milk. Noting that \(X_{ikt} = a_{i1t} x_{i1kt} + a_{i2t} x_{i2kt}\), we obtained respectively:

$$\left\{ {\begin{array}{*{20}c} {\frac{{\partial \log \left( {y_{i1t} } \right)}}{{\partial x_{ikt} }} = \gamma_{k1} \frac{{a_{i1t} }}{{A_{it} }}} \\ {\frac{{\partial \log \left( {y_{i2t} } \right)}}{{\partial x_{ikt} }} = \gamma_{k2} \frac{{a_{i2t} }}{{A_{it} }}} \\ \end{array} } \right.$$

Which is equivalent to:

$$\left\{ {\begin{array}{*{20}c} {\frac{{\partial y_{i1t} }}{{\partial x_{ikt} }} = \gamma_{k1} \frac{{a_{i1t} }}{{A_{it} }}y_{i1t} } \\ {\frac{{\partial y_{i2t} }}{{\partial x_{ikt} }} = \gamma_{k2} \frac{{a_{i2t} }}{{A_{it} }}y_{i2t} } \\ \end{array} } \right.$$

Thus, we obtained \(\forall k \in \left[ {1;4} \right]:\)

$$\frac{{\frac{{\partial y_{i1t} }}{{\partial x_{ikt} }}}}{{\frac{{\partial y_{i2t} }}{{\partial x_{ikt} }}}} = \frac{{\gamma_{k1} a_{i1t} y_{i1t} }}{{\gamma_{k2} a_{i2t} y_{i2t} }}$$

Because \(a_{i1t} y_{i1t}\) and \(a_{i2t} y_{i2t}\) do not depend on \(x_{ikt}\), we had the three valid restrictions, which held if we added \({Y}_{\mathrm{i}2t}\) to the cereal yield function explicitly or vice-versa (see program (9), Appendix 2).

Appendix 4: Alternative Estimates of Model 2

See Table 6.

Table 6 Estimates of Model 2 without or with an additional interaction term for pesticides (N = 3960)

Appendix 5: Additional Estimates of Log-Quadratic Production Functions

See Table 7.

Table 7 Estimates of log-quadratic production functions (system (3)) with Models 1–3 (N = 3960)

Appendix 6: Seemingly Unrelated Regression Estimates of Systems (4) and (5)

See Table 8.

Table 8 SUR estimates of log-quadratic production functions with Model 1 (N = 3960)

Appendix 7: Estimates of System (2) with all Alternative Measures of Permanent Grassland Proportion (Farm, Municipality, District and Province Scales)

See Table 9.

Table 9 Estimates of system (2) with all indicators for permanent grasslands and Model 4 (N = 2344)

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Bareille, F., Dupraz, P. Productive Capacity of Biodiversity: Crop Diversity and Permanent Grasslands in Northwestern France. Environ Resource Econ 77, 365–399 (2020).

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  • Agriculture
  • Biodiversity
  • Ecosystem services
  • Pesticides
  • Productivity