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:
$$\pi_{it} = \mathop {\max }\limits_{{{\varvec{X}}_{it} }} \left\{ {E({\varvec{p}}_{it} )^{\prime } {\varvec{Y}}_{it} - E\left( {{\varvec{w}}_{it} } \right)^{\prime } {\varvec{X}}_{it} + S_{it} ;\left( {{\varvec{Y}}_{it} ,{\varvec{X}}_{it} ,{\varvec{Z}}_{it} ,{\varvec{B}}_{it} ,{\varvec{A}}_{it} } \right) \in T} \right\}$$
(6)
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):
$$\pi_{it} = \mathop {\max }\limits_{{a_{i1t} , \ldots ,a_{iJt} }} \left\{ {\mathop \sum \limits_{j = 1}^{J} a_{ijt} \left[ {E\left( {\omega_{ijt} \left( {p_{ijt - 1} ,E\left( {{\varvec{w}}_{it} } \right),{\varvec{Z}}_{it} } \right)} \right) + s_{ijt} } \right];\mathop \sum \limits_{j = 1}^{J} a_{ijt} = A_{it} } \right\}$$
(7)
$$E\left( {\omega_{ijt} \left( {p_{ijt - 1} ,{\varvec{w}}_{it} ,{\varvec{Z}}_{it} } \right)} \right) = \mathop {\max }\limits_{{x_{ijt} }} \left\{ {p_{ijt - 1} \cdot y_{ijt} - {\varvec{w}}_{it}^{\prime } {\varvec{x}}_{ijt} ; y_{ijt} \le f_{j} \left( {{\varvec{x}}_{ijt} ;{\varvec{B}}_{it} ,{\varvec{Z}}_{it} ,{\varvec{Y}}_{ - ijt} } \right)} \right\}$$
(8)
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:
$${{\frac{{\partial f_{1} ( \cdot )}}{{\partial x_{i11t} }}} \mathord{\left/ {\vphantom {{\frac{{\partial f_{1} ( \cdot )}}{{\partial x_{i11t} }}} {\frac{{\partial f_{2} ( \cdot )}}{{\partial x_{i21t} }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial f_{2} ( \cdot )}}{{\partial x_{i21t} }}}} = \ldots = {{\frac{{\partial f_{1} ( \cdot )}}{{\partial x_{i1Jt} }}} \mathord{\left/ {\vphantom {{\frac{{\partial f_{1} ( \cdot )}}{{\partial x_{i1Jt} }}} {\frac{{\partial f_{2} ( \cdot )}}{{\partial x_{i2Jt} }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial f_{2} ( \cdot )}}{{\partial x_{i2Jt} }}}} = \frac{{p_{2t - 1} + p_{1t - 1} \frac{{a_{i1t} }}{{a_{i2t} }}\frac{{\partial y_{i1t} }}{{\partial y_{i2t} }}}}{{p_{1t - 1} + p_{2t - 1} \frac{{a_{i2t} }}{{a_{i1t} }}\frac{{\partial y_{i2t} }}{{\partial y_{i1t} }}}}$$
(9)
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:
$$\pi_{it} = \mathop {\max }\limits_{{a_{i1t} ; \ldots ;a_{iJt} }} \left\{ {\mathop \sum \limits_{j = 1}^{J} a_{ijt} \left[ {E\left( {\omega_{ijt} \left( {p_{ijt - 1} ,E\left( {{\varvec{w}}_{it} } \right),{\varvec{Z}}_{it} } \right)} \right) + s_{ijt} } \right];\mathop \sum \limits_{j = 1}^{J} a_{ijt} = A_{it} } \right\}$$
(10)
$$E\left( {\omega_{ijt} \left( {p_{ijt - 1} ,{\varvec{w}}_{{{\varvec{it}}}} ,{\varvec{Z}}_{{{\varvec{it}}}} } \right)} \right) = \mathop {\max }\limits_{{x_{it} }} \left\{ {p_{ijt - 1} \cdot y_{ijt} - {\varvec{w}}_{{{\varvec{it}}}}^{\prime } {\varvec{x}}_{{{\varvec{it}}}} ; y_{ijt} \le g_{j} \left( {{\varvec{x}}_{{{\varvec{it}}}} ;{\varvec{B}}_{{{\varvec{it}}}} ,{\varvec{Z}}_{{{\varvec{it}}}} ,{\varvec{Y}}_{{\user2{ - ijt}}} } \right)} \right\}$$
(11)
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)