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

Abstract

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.

Appendices

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,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).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),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).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 k^th 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 area²² 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)