domain: sde qfin sdv shs

text:
Jean Joseph Minviel
email: jean-joseph.minviel@inrae.fr

Marc Benoit

Economies of diversification and stochastic dominance analysis in French mixed sheep farms

Keywords: downside risk exposure, economic resilience, economies of diversification, Massif Central, mixed sheep farms, SERF analysis, stochastic dominance JEL Classifications: D81, L25, Q12

Farm diversification is mainly driven by risk mitigation effects and economic gains related to complementarities between production activities. By combining these two aspects, we investigate diversification economies in a sample of French mixed sheep farming systems and rank these systems using stochastic dominance criteria. Partially diversified systems (Sheep-Grass, Sheep-Crop, Sheep-Landless) and fully diversified systems (Sheep-Grass-Crop-Landless) were evaluated. We find a high degree of diversification diseconomies in the sheep farming systems considered. The results also indicate that the fully diversified system is driven by its risk-reducing effects (including downside risk exposure) and that Sheep-Crop is the dominant system in terms of risk-adjusted returns.

Introduction

The main objective of the present paper is to investigate the existence and the sources of economies of diversification in French mixed sheep farms and to identify mixtures of sheep farming activities that provide greater risk-adjusted returns to farmers. Such an analysis may be useful for producers who think about changing their production mix. In fact, we are interested in two standard economic rationales for diversification strategies. Indeed, in economics it is usually argued that production diversification can have (or can be driven by) two effects: a risk-reducing effect and a scope economies effect [START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF]. Under uncertainty, risk-averse producers have incentives to diversify their production activities [START_REF] Heady | Diversification in resource allocation and minimization of income variability[END_REF]. This could be explained by the fact that production diversification can be seen as a strategy used by farmers to protect themselves against production and market risks. Since farmers are typically risk-averse (they are aware of the variability in their productions and their related prices), they may respond to risk by diversifying their production activities. This could be illustrated by the rule of thumb: "Don't put all your eggs in one basket" (Chavas and Di Falco 2012, p. 26). Diversification can thus help farms to be more resilient (i.e., to withstand external shocks without catastrophic consequences or to absorb downside shocks) in case of a crisis in at least one of their production activities.

Besides risk management, another possible motivation for diversification is the presence of economies of scope [START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF]. Scope economies exist if the cost of joint production of a set of outputs in a diversified firm is lower than the cost of their disjoint production in several specialized firms. The concept of economies of scope focuses on measuring economic gains (in terms of cost reductions) associated with diversified firms in comparison with specialized firms. Analysis of scope economies requires either data on completely specialized firms (or stand-alone production) or partitions of the outputs of the firms in mutually exclusive categories (see, [START_REF] Panzar | Economies of scope[END_REF][START_REF] Grosskopf | Measuring economies of diversification: A frontier approach[END_REF][START_REF] Ferrier | Economies of diversification in the banking industrya frontier approach[END_REF]. Since such data are not always available, the classic definition of economies of scope has been generalized to economies of diversification [START_REF] Grosskopf | Measuring economies of diversification: A frontier approach[END_REF][START_REF] Ferrier | Economies of diversification in the banking industrya frontier approach[END_REF][START_REF] Chavas | Economies of diversification: A generalization and decomposition of economies of scope[END_REF][START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF][START_REF] Malikov | Economies of diversification in the US credit union sector[END_REF]. Economies of diversification measure economic gains (in terms of cost reductions or certainty equivalents) associated with fully diversified firms in comparison with partially diversified firms. Thus, economies of scope are a special case of the more general measure of economies of diversification [START_REF] Eder | Measuring and decomposing economies of diversification: An application to biogas-fuelled cogeneration plants in Austria[END_REF].

Empirically, these two effects are usually investigated separately (e.g., [START_REF] Smale | The contribution of genetic resources and diversity to wheat production in the Punjab of Pakistan[END_REF][START_REF] Di Falco | On crop biodiversity, risk exposure, and food security in the highlands of Ethiopia[END_REF]. Recently, [START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF] introduced a unified framework where both rationales for production diversification are integrated into an applied microeconomics setting using the concept of certainty equivalent (CE). Our paper uses their approach but it departs from [START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF] in three ways. First, in contrast to [START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF], our analysis is based on observed partially diversified farms (see also, [START_REF] Malikov | Economies of diversification in the US credit union sector[END_REF]. Second, while [START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF] consider only production risk, we consider both yield and price risks. Third, we apply the Stochastic Efficiency with respect to a Function (SERF) method [START_REF] Hardaker | Stochastic efficiency analysis with risk aversion bounds: A simplified approach[END_REF], which is widely used to estimate CE values and to compare risky alternatives. As such, our framework has two important features. First, it makes it possible to investigate the existence and the sources of economies of diversification. Second, it allows a ranking of farming activities in terms of stochastic dominance. The SERF method has been used previously to rank crop production systems (e.g., [START_REF] Pendell | Soil carbon sequestration strategies with alternative tillage and nitrogen sources under risk[END_REF][START_REF] Watkins | An economic risk analysis of no-till management and rental arrangements in Arkansas rice production[END_REF][START_REF] Bryant | Valuing Transgenic cotton technologies using a risk/return framework[END_REF][START_REF] Archer | Economic performance of alternative tillage systems in the northern Corn Belt[END_REF][START_REF] Williams | Comparison of runoff, soil erosion, and winter wheat yields from no-till and inversion tillage production systems in northeastern Oregon[END_REF][START_REF] Hignight | An Economic Risk Analysis of Tillage and Cropping Systems on the Arkansas Grand Prairie[END_REF][START_REF] Barham | Mitigating cotton revenue risk through irrigation, insurance, and hedging[END_REF][START_REF] Williams | Risk analysis of tillage and crop rotation alternatives with winter wheat[END_REF][START_REF] Boyer | Investment analysis of cover crop and no-tillage systems on Tennessee cotton[END_REF]Adusumilli et al. 2020). However, to our best knowledge, our paper provides the first application of the SERF method in economics of diversification.

Our framework is applied to a sample of sheep farms in an area in the Center of France, namely the "Massif Central." Besides their main activity (sheep), some farms in the Massif Central have complementary activities (e.g., crop, landless activities). These activities have been developed given, inter alia, market opportunity, difficulties in the sheep sector, and available labor force. Under the European terminology, the Massif Central is classified as a Less Favored Agricultural Area. In these areas, farmers operate under more difficult production conditions, such as steep lands, high altitudes, unfavorable climates (e.g., short growing season and long wintering period), and isolated locations. The Massif Central includes mountain areas (60% of the territory); but also areas immediately adjacent to them, which are known as simple less-favored areas (foothills, plains) (SIDAM 2020) 1 . Eighty-five percent of the territory of the Massif Central is devoted to grazing livestock, including 60% of permanent grassland (SIDAM 2020). In the Massif Central, sheep farming represents the third agricultural production activity in terms of farm numbers (15% of farms), after beef cattle farming (38% of farms) and dairy cattle farming (20% of farms) (SIDAM 2020) (see also appendix A). An advantage of sheep farming is that sheep are well suited to valorize Less Favored Agricultural Areas, such as the Massif Central [START_REF] Benoit | Performances techniques et économiques en élevage ovin viande biologique: Observations en réseaux d'élevage et fermes expérimentales[END_REF]. However, the sheep farms in the Massif Central are exposed to hazards (climate, health, etc.), but also to seasonal and annual price volatility (see appendix B). In this context, our analysis may provide relevant information for designing/selecting mixtures of farming activities that allow farmers to have greater risk-adjusted returns, to reduce their economic risk (including downside risk exposure), or to valorize territorial potentialities (including touristic activities) of the Massif Central.

The remainder of the paper is organized as follows. The second section describes our analytical approach. The third section presents the data used. Results are presented and discussed in the fourth section, and the fifth section provides concluding remarks.

Theoretical, conceptual, and empirical considerations

The Expected Utility Theory and the concept of Certainty Equivalent Our analysis is based on the concept of Certainty Equivalent (CE), which is derived from the Expected Utility Theory (EUT). The EUT postulates that when faced with risky prospects, an individual chooses the alternative that offers the maximum expected utility. Under this theory, the preferences of economic agents are represented by a utility function U(.), and the consequences of their decisions by random variables π associated with probabilities p. The EUT assumes that agents make their decisions as if they were maximizing the expected utility of their random earnings. Thus, they prefer a random gain π 1 to a random gain π 2 , π 1 π 2 , if and only if EU π 1 ≥ EU π 2 . More formally, denote by U π the utility function of a producer with respect to stochastic outcomes (π) (e.g., random net returns). The probability density functions (PDF) that represent the outcomes for n risky alternatives are noted by f 1 π ; f 2 π ; . . . ; f n π , and their corresponding cumulative distribution functions (CDF) are F 1 π ; F 2 π ; . . . ; F n π . The utility of these alternatives can be expressed as follows:

U π EU π Z U π f π dπ Z U π dF π (1)
From the CDF, it is possible to determine which alternative is stochastically dominant. For instance, given two alternatives A and B, with CDFs over outcomes π defined by F a π and

F b π , alternative A will dominate alternative B if F a π ≤ F b π . Graphically, this means that F b π
is closest to the origin of the axes. In other words, the CDF of A must be to the right of the CDF of B. However, for more discriminating power, we will rank our mixed sheep farms by converting their utility into certainty equivalents (CE) as follows [START_REF] Hardaker | Stochastic efficiency analysis with risk aversion bounds: A simplified approach[END_REF][START_REF] Hardaker | Coping with Risk in Agriculture: Applied Decision Analysis[END_REF]:

CE π U 1 EU π (2)
where U 1 is the reciprocal of the utility function U.

1 Service interdépartemental pour l'animation du Massif Central (i.e., in English, Interdepartmental service for the animation of the Massif Central).

Existence of diversification economies

Following [START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF], economies of diversification (diseconomies of diversification) (ED) exist if:

ED CE π X K k1 CE π k > 0 < 0 (3)
where CE π is the certainty equivalent value of the net return of the fully diversified farms and CE π k is the certainty equivalent value of the net return of the k-th type of partially diversified farms. As aforementioned, in contrast to [START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF], our counterfactual CE π k is defined on observed partially diversified farms (see also, [START_REF] Malikov | Economies of diversification in the US credit union sector[END_REF]. The variable π k is the net income of the k-th type of partially diversified farms, that is, it is measured as the profit generated by all the activities of this kind of farm. For instance, for the Sheep-Crop system, it is measured as the sum of the profit generated by the sheep and the crop production.

Equation (3) indicates that economies of diversification exist (ED > 0) when the certainty equivalent of the fully diversified farms is higher than the one of K partially diversified farms. This may allow us to identify the presence of positive externalities across production activities. Alternatively, diseconomies of diversification exist if ED < 0.

Equation (3) provides a monetary measure of diversification benefits. Assuming that CE π > 0, a relative measure that provides the degree of diversification economies can be defined using the following expression:

DED CE π P K k1 CE π k CE π (4)
The components of diversification economies For a stochastic outcome, π, [START_REF] Pratt | Risk aversion in the small and in the large[END_REF] has shown that the CE can be approximated by:

CE π E π R π (5)
where E π denotes the expected value of π and R π is a risk premium. Thus, the certainty equivalent of a random outcome π, CE π , is the difference between its expected value (E π

) and a risk premium R π . The risk premium (R) is the maximum amount of money that a risk-averse individual is willing to pay to avoid facing a risk. In other words, it is the amount of money that a risk-averse farmer is willing to give up to eliminate risk exposure. That is, the risk premium is the implicit cost of risk bearing. From equation ( 5), the CE can be seen as the risk-adjusted value of the expected net return. Equation (5) has been used by [START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF] to derive the components of diversification economies. In this respect, economies of diversification (diseconomies of diversification) (ED) exist if:

ED ED π ED R > 0 < 0 (6)
Where

ED π E π X K k1 E π k ED R R π X K k1 R π k " #
Equation ( 6) identifies two additive components of the benefits of diversification: the expected income component ED π , and the risk component ED R . The risk premium estimated from the SERF method can be used in the equation of ED R to examine the risk component of diversification economies. However, following [START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF] we use the variance component and the skewness component of the risk premium separately in order to examine the effects of diversification patterns on different aspects of risk exposure. As such, given the utility function U π , the risk premium can be expressed as follows [START_REF] Chavas | Risk Analysis in Theory and Practice[END_REF][START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF]):

R π 1 2 U 00 π U 0 π Var π 1 6 U 000 π U 0 π Skew π (7)
where U h π denotes the derivative of order h of the utility function with respect to π. U 00 =U 0 is known as the Arrow-Pratt risk aversion coefficient and U 000 =U 0 is a measure of downside risk aversion. From equation ( 7), the risk premium can be decomposed into two components:

R π R Var R Skew (8)
where R Var 1 2

U 00 π U 0 π
Var π is the variance component and R Skew 1 6

U 000 π U 0 π
Skew π is the skewness component of the risk premium. As previously mentioned, the variance component and the skewness component can be used separately in the ED R equation to examine the variance and the skewness effects of diversification economies. However, from an empirical standpoint, it is difficult to derive the variance and the skewness components from the SERF estimates 2 . Indeed, in its current form, the SERF method provides the Arrow-Pratt risk aversion coefficients and not the downside risk aversion coefficients. Hence, in this sub-section, we focus mainly on the variance component, while special attention is devoted to the skewness component in the next sub-section. In the estimation of the variance component, the risk aversion coefficients U 00 π U 0 π are obtained from the SERF method and the variance (Var π ) is estimated using econometric techniques as in [START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF] (see the next sub-section for more details).

Diversified production systems and downside risk exposure

In the previous sub-section, we focused mainly on the variance component of the risk component of the diversification economies. However, since the variance does not distinguish between unexpected good and bad events, that is, between upside and downside risk (Di Falco and Chavas 2009), special attention is devoted here to the skewness component. Indeed, we use the third moment (the skewness) of farmers' income distribution (Skew π

) for downside risk exposure3 analysis to provide additional information for diversification strategies [START_REF] Antle | Testing the stochastic structure of production: A flexible-moment based approach[END_REF][START_REF] Kim | Technological change and risk management: An application to the economics of corn production[END_REF][START_REF] Donoso | A decision framework for a farmer who is risk averse in the Arrow-Pratt sense and downside risk averse[END_REF][START_REF] Bozzola | Stability of risk attitude, agricultural policies and production shocks: Evidence from Italy[END_REF]. That is, we assume that higher moments than the variance play an important role in farmers' decision process. This is in line with the fact that agricultural returns are often characterized by extreme loss events and farmers are often downside riskaverse [START_REF] Menezes | Increasing downside risk[END_REF][START_REF] Antle | Testing the stochastic structure of production: A flexible-moment based approach[END_REF][START_REF] Di Falco | Crop genetic diversity, farm productivity and the management of environmental risk in rainfed agriculture[END_REF][START_REF] Koundouri | The effects of EU agricultural policy changes on farmers' risk attitudes[END_REF]. In an agricultural setting, the downside risk is particularly relevant as it identifies the probability of failure. Indeed, an increase in 2

The scope of this study is to use the SERF method to estimate the CE values, to rank the risky alternatives over a range of risk aversion coefficients, and to use the SERF estimates for diversification economies analyses.

the skewness of the income means that downside risk exposure decreases, that is, the probability of failure decreases [START_REF] Di Falco | On crop biodiversity, risk exposure, and food security in the highlands of Ethiopia[END_REF]. This suggests that an increase in the downside risk involves an increase in the skewness of the risk distribution toward low outcomes [START_REF] Menezes | Increasing downside risk[END_REF]. By definition, a downside risk-averse decision maker is impoverished by such a change [START_REF] Di Falco | Crop genetic diversity, farm productivity and the management of environmental risk in rainfed agriculture[END_REF]. In this respect, farmers exhibiting downside risk aversion have incentives to develop management strategies that affect positively the skewness of the distribution of their returns, thus reducing their exposure to downside risk (i.e., by reducing the probability of failure) [START_REF] Di Falco | Crop genetic diversity, farm productivity and the management of environmental risk in rainfed agriculture[END_REF].

We are concerned with the question of how diversification affects the third central moment (the skewness) of the distribution of farm income or whether diversified systems could help farmers to reduce their exposure to downside risk. We use an econometric moment-based approach to this assessment [START_REF] Antle | Testing the stochastic structure of production: A flexible-moment based approach[END_REF][START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF]. This approach has been widely used in the context of risk management in agriculture [START_REF] Kim | Technological change and risk management: An application to the economics of corn production[END_REF][START_REF] Koundouri | Technology adoption under production uncertainty: Theory and application to irrigation technology[END_REF][START_REF] Bozzola | Stability of risk attitude, agricultural policies and production shocks: Evidence from Italy[END_REF]. In order to specify the econometric model, we rewrite the CE as follows [START_REF] Di Falco | Crop genetic diversity, farm productivity and the management of environmental risk in rainfed agriculture[END_REF]:

CE E pg x; z C x; z R x; z (9)
Where p is a vector of output prices and g : is a production function. Equation ( 9) indicates that production factors (x) and contextual drivers (or management decisions) (z) can affect the expected revenue 4 (E pg x; z

), the production cost (C x; z as well as the risk premium (R x; z

). It is worth noting that from equations ( 7) and ( 9), the risk premium is

given by R x; z 1 2 U 00 π U 0 π Var π x; z 1 6 U 000 π U 0 π
Skew π x; z , implying that x and z can also affect the variance and the skewness components.

The econometric estimations follow a two-step procedure. In the first step, the income is regressed on production factors (x) and contextual drivers (z) to provide an estimate of the "mean" (i.e., the first moment) effect:

π f 1 x; z υ ( 10
)
where π is the farm income per hectare, x is a vector of production factors (labor, farm capital, intermediate inputs), and z is a vector of contextual drivers [START_REF] Latruffe | The Role of Environmental and Land Transaction Regulations on Agricultural Land Price: The Example of Brittany[END_REF] including farm-specific characteristics (e.g., diversification patterns, feed autonomy). In equation ( 10), f 1 x; z E π is the mean of π, and υ π f 1 x; z is the usual identically independently distributed error term. In the second step, higher-order moments 5 of the income are given by:

E π f 1 x; z k f k x; z υ; k 2;3 (11)
Thus, f 2 x; z represents the second central moment (the variance), and f 3 x; z is the third central moment (the skewness). Equations ( 10) and ( 11) can be consistently estimated using least-squares methods [START_REF] Kim | Technological change and risk management: An application to the economics of corn production[END_REF][START_REF] Koundouri | Technology adoption under production uncertainty: Theory and application to irrigation technology[END_REF].

4

The expected net income is given by E π E pg x; z C x; z :

5 Recall that high-order moments (e.g., moments of orders 2 and 3) of a random variable π is given by E π E π k , with k 2; 3. The SERF method

The SERF method has been developed to estimate the certainty equivalent (CE) of the return of a set of risky alternatives in order to rank them over a range of risk aversion coefficients. In the present study, we also used the CE values estimated with the SERF method to investigate the existence and the components of economies of diversification for our sample of mixed sheep farms.

The SERF method estimates the CE values using equations ( 1) and ( 2). It therefore requires the specification of a utility function. [START_REF] Schumann | Stochastic Efficiency Analysis Using Multiple Utility Functions[END_REF] showed that the use of different types of utility functions, such as the power or negative exponential utility functions, is unlikely to affect the results of the SERF method. In this study, we use the power utility function to estimate the CE values. This utility function exhibits a decreasing absolute risk aversion, namely individuals are willing to take more risks when their wealth increases. In the implementation of the SERF method, we do not need to know the decision-maker's risk aversion coefficient φ [START_REF] Richardson | Ranking risky alternatives: Innovations in subjective utility analysis[END_REF]. It requires only to set a lower and an upper value for φ (e.g., φ 0, for risk-neutral producers and φ 4, for extremely risk-averse producers) (see [START_REF] Anderson | Risk Analysis in Dryland Farming Systems[END_REF]. This enables the SERF method to rank risky alternatives upon a plausible range of risk aversion coefficients [START_REF] Schumann | Stochastic Efficiency Analysis Using Multiple Utility Functions[END_REF]). This is an important point because producers with different degrees of risk aversion are likely to have different preferences among risky alternatives [START_REF] Hardaker | Coping with Risk in Agriculture[END_REF].

The SERF method has many advantages over other methods usually used for comparing risky alternatives. For instance, the direct implementation of the expected utility approach requires consistent specification of the utility function of the decision maker [START_REF] Anderson | Agricultural Decision Analysis[END_REF]. However, it has been shown that accurate elicitation of the utility function of the decision makers is not a clear-cut issue [START_REF] King | Risk efficiency models[END_REF]. First-order and second-order stochastic dominance overcome the need to define a utility function, but they often lead to meaningless results [START_REF] Schumann | Stochastic Efficiency Analysis Using Multiple Utility Functions[END_REF]). Another approach, called stochastic dominance with respect to a function (SDRF) could be used to compare risky alternatives [START_REF] Meyer | Choice among distributions[END_REF]. The SDRF method does not require an explicit definition of a utility function, but only a lower and an upper absolute risk aversion coefficient. However, the SDRF method performs only pairwise comparisons [START_REF] Hardaker | Stochastic efficiency analysis with risk aversion bounds: A simplified approach[END_REF]). The mean-variance (MV) method could also be used to compare risky alternatives. However, the MV approach often leads to inconclusive results [START_REF] Hardaker | Stochastic efficiency analysis with risk aversion bounds: A simplified approach[END_REF]). The SERF method overcomes the limitations of the previous methods. In addition, unlike the SDRF, the SERF method compares simultaneously many risky alternatives [START_REF] Hardaker | Stochastic efficiency analysis with risk aversion bounds: A simplified approach[END_REF][START_REF] Meyer | Stochastic efficiency analysis with risk aversion bounds: A correction[END_REF]. Another advantage of the SERF method is that it considers the entire distribution of the net return, but not only one point of measurement, as does the MV method.

In SERF analysis, the term stochastic efficiency is usually used to interpret the results. However, in the present study, to avoid any confusion with the term stochastic frontier analysis (SFA) commonly used in efficiency analysis [START_REF] Minviel | Dynamic stochastic analysis of the farm subsidy efficiency link: Evidence from France[END_REF][START_REF] Minviel | A dynamic stochastic frontier approach with persistent and transient inefficiency and unobserved heterogeneity[END_REF], we use the term stochastic dominance. Indeed, we acknowledge that SERF is a variant (an improved method or a generalized form) of stochastic dominance analysis (see also, [START_REF] Schenk | How do farmers react to varying water allocations? An assessment of how the attitude to risk affects farm incomes[END_REF]. Note also that the SERF procedure is suggested to be a method of stochastic dominance analysis [START_REF] Meyer | Stochastic efficiency analysis with risk aversion bounds: A correction[END_REF][START_REF] Hardaker | Stochastic efficiency analysis with risk aversion bounds: A comment[END_REF]. This is related to the fact that stochastic dominance analysis is often used to order risky prospects [START_REF] Lien | Assisting whole-farm decision-making through stochastic budgeting[END_REF].

Data description

The data used in this study come from surveys carried out by the Joint Research Unit Herbivores (UMRH), of the French National Research Institute for Agriculture, Food and the Environment (INRAE). These data were collected in a network of suckler sheep farms monitored annually over the period 1987 to 2017. During these successive surveys, many variables were collected on the structure and management of the flock, as well as on the technical and economic results of the farms. More precisely, our dataset includes information on: (i) the workforce (family, salaried, temporary); (ii) the flock (lambing, animal movements, and weights of animals sold); (iii) areas (crops, grassland); (iv) equipment (exhaustive list and costs); (v) buildings and installations; (vi) intermediate consumption (quantities and prices); (vii) sales (types of animals and crops, quantities, prices); (viii) subsidies (coupled, decoupled, unit amounts); and (ix) investments, loans, social contributions, taxes, salaries paid, and insurance. The farms are mainly located in the north of the Massif Central and its periphery, and they are distributed between plain areas with grassland breeds and mountain and foothill areas with rustic breeds.

Our empirical analysis is undertaken on an unbalanced panel of 1,139 farm-year observations from 134 mixed sheep farms over the period from 1987 to 2017. Over this period, we keep farms that were surveyed for at least 3 years in order to have a certain level of variability in the data. Based on the strategies used by farmers to mitigate economic risks, the following sheep farming systems are examined in the present study: Sheep-Grass: This system consists of farms that have neither annual crops nor other livestock. These are farms with 100% of their land dedicated to grass. Sheep-Crop: This system concerns farms that have sheep, grass, and more than 15% of the usable agricultural area in annual crops such as cereals and oilseeds. Sheep-Landless: This system refers to sheep-grass farms with off-land activities, such as poultry and agro-tourism. Sheep-Grass-Crop-Landless: This system consists of sheep farms that have all the previous categories (i.e., grass, crop, and off-land activities).

This partition of the sheep farms has been done in line with [START_REF] Grosskopf | Measuring economies of diversification: A frontier approach[END_REF] and [START_REF] Ferrier | Economies of diversification in the banking industrya frontier approach[END_REF], in order to investigate the existence of economies of diversification. That is, the partially diversified farms are defined in the sense they produce at least one output in common. In other words, if we consider a case of three outputs and two farms: one farm must be specialized in the production of output 1, the other farm specialized in the production of output 2, while both farms must produce output 3. The main characteristics of the farms operating in these four systems are summarized in Table 1.

The SERF method does not allow explicit treatments of panel data. Hence, in the present study, pooled data are used. It is worth noting that the SERF method accounts for individual effects given the fact that it does not use a fixed risk aversion coefficient for all the farms in the sample and that it estimates the CE for each farm according to the production systems. The SERF analysis can also be used to rank risky alternatives by type of decision makers [START_REF] Richardson | Ranking risky alternatives: Innovations in subjective utility analysis[END_REF]. One way to account for time effects is to split the data into sub-periods and apply the SERF method per period. To avoid any ad hoc splitting, we need information on events that may cause structural breaks in the data. Since such information is not available in our dataset, we apply the SERF method to the pooled data over our study period. However, as indicated previously, to go a step further in our analysis, we use an econometric approach for examining time effects and the effects of other variables (including diversification patterns) on downside risk exposure. The econometric approach used can be seen as complementary to the SERF method.

Results and discussion

Diversification economies

We use the certainty equivalent (CE) values estimated using the SERF method to compute economies of diversification (ED) in the French sheep farms located in the Massif Central using equation (3). Such economies would exist if the CE value of the fully diversified farms is greater than the sum of the CE values of the partially diversified farms. The results concerning the existence of ED are shown in Figure 1.

Figure 1 indicates that all the values of the ED indicator are negative. This suggests that there are no economies of diversification in the sheep farming systems considered. That is, our results highlight the existence of diseconomies of diversification in our sheep farming systems. In order words, our results indicate that the fully diversified system considered does not generate more risk-adjusted returns (CE) than the partially diversified systems. To go a step further in our analysis, we also investigate the degree of diversification economies (DED) using equation ( 4), and the results are plotted in Figure 2. Notes SD: Std. dev.

Figure 2 indicates that there is a relatively high degree of diseconomies of diversification in the sheep farming system considered. Indeed, all the values of the indicator of the degree of economies of diversification are negative and amount to -0.68 on average. This suggests that the fully diversified sheep system reaches only 32% of the risk-adjusted return (CE) generated by the partially diversified systems altogether. Using a quite different approach from ours, [START_REF] Fleming | Synergies, scope economies and scale diseconomies on farms in Norway[END_REF] showed that the existence of synergies (i.e., output complementarities) does not necessarily result in economic gains from diversification. Indeed, despite the existence of synergies, they found significant diseconomies for small-scale farming operations. Using the same approach as [START_REF] Fleming | Synergies, scope economies and scale diseconomies on farms in Norway[END_REF], [START_REF] Coelli | Diversification economies and specialisation efficiencies in a mixed food and coffee smallholder farming system in Papua New Guinea[END_REF] found weak evidence for diversification economies between subsistence food production, coffee, and cash food production, and diseconomies of diversification between coffee and cash food production. However, following [START_REF] Hajargasht | A dual measure of economies of scope[END_REF], [START_REF] Wimmer | Diversification economies in dairy farming: Empirical evidence from Germany[END_REF] measure diversification economies as cost complementarities between individual outputs. They found that small dairy farms are more likely to benefit from diversification between milk and livestock production, while larger farms tend to benefit from diversification between milk and crop production.

In sum, the existence of economies of diversification in mixed farms remains an open empirical question.

In Figure 3, we shed light on the sources of the observed diseconomies of diversification. Note that the sources of the observed diseconomies of diversification are not reported here as presented in equation ( 7), that is, we do not present directly ED π and ED R . Indeed, for presentation convenience, we report in Figure 3 ED π =E π and ED R =R π . That is, we have scaled these measures in a similar way to the equation (4). Figure 3 shows that the mean return effect is negative while the risk effect is positive. The negative value for the mean return effect highlights that the average net return per family labor unit generated by the fully diversified system is lower than the one generated by the partially diversified systems altogether. The positive value of the risk effect indicates that the variance of the fully diversified system is lower than the one of the partially diversified systems altogether. Overall, the results from Figure 3 indicate that although the fully diversified system does not generate more returns than the partially diversified systems, it ensures more stable net returns. This suggests that farmers could consider the diversified system as a safer option for their farms [START_REF] Sarwosri | Are risk attitudes and time preferences crucial factors for crop diversification by smallholder farmers?[END_REF] in the sense that it allows farmers to spread production and market risks [START_REF] Villano | Farming systems and productivity gaps: Opportunities for improving smallholder performance in the Forest-Savannah transition zone of Ghana[END_REF]. These results are also in line with the portfolio theory, which suggests that diversified production systems should face lower risk [START_REF] Markowitz | Portfolio selection[END_REF][START_REF] Markowitz | Foundations of portfolio theory[END_REF][START_REF] Paut | Reducing risk through crop diversification: An application of portfolio theory to diversified horticultural systems[END_REF][START_REF] Mosnier | Does mixing livestock farming enterprises improve farm and product sustainability?[END_REF]. They are also line with the idea that classical concepts (such as income maximization) that are often used to guide farm management are increasingly replaced by other concepts such as stability [START_REF] Darnhofer | Adaptiveness to enhance the sustainability of farming systems[END_REF][START_REF] Astigarraga | Production flexibility in extensive beef farming systems[END_REF].

Stochastic dominance analysis

This section presents the ranking of our four sheep farming systems using the SERF method, which is known as a more discriminating form of stochastic dominance analysis [START_REF] Schenk | How do farmers react to varying water allocations? An assessment of how the attitude to risk affects farm incomes[END_REF]. However, for comparison purposes, CDF curves and the mean-standard deviation ranking for our sheep farming systems are depicted in Figures 4 and5, respectively. Indeed, some general conclusions can be drawn from each of these methods.

The CDF curves show that the Sheep-Landless system is closest to the origin of the axes and does not intersect with the other sheep farming systems. This suggests that the Sheep-Landless system is dominated by the other systems. However, it is not feasible to rank the other systems using their CDFs since they intersect each other.

A general conclusion that can be drawn from Figure 5 is that the Sheep-Landless system has the lowest standard deviation for the net return per family labor unit. This suggests that this system could be a good strategy to mitigate economic risks. This could be explained by the off-land activities, which may be less sensitive to climatic context. In addition, in the Sheep-Landless system, the breeders work along integrated lines, mainly for poultry, that is, as subcontractors for agribusiness companies. In this system, the production is paid at a remunerative price even if the market conditions are unfavorable. Therefore, the market risk is shared between the breeders and the integrator. However, the Sheep-Landless system has also a disadvantage compared to the other systems: it has the lowest average net return per family labor unit. Among the farms studied, these farms have developed off-ground activities generally because they have a lack of land in relation to the available labor force; and sometimes because sheep profitability is too low. As our results show, the off-ground activities did not upset their economic results. This result is in line with [START_REF] Ripoll-Bosch | Role of self-sufficiency, productivity and diversification on the economic sustainability of farming systems with autochthonous sheep breeds in less favoured areas in Southern Europe[END_REF] who found that production diversification might enhance farm flexibility but it is not necessarily related to economic performance. Similarly, [START_REF] Villagra | Productive and income contributions of sheep, goat, and cattle, and different diversification schemes in smallholder production systems of Northern Patagonia, Argentina[END_REF] found that sheep represented the main source of income across different diversification schemes in Argentina, but it was not the most profitable livestock activity. Note that Figure 5 also indicates that the Sheep-Grass (i.e., completely specialized) system exhibits the highest variability in the net return per family labor unit. This result highlights an interest in farms' diversification [START_REF] Chavas | On the role of risk versus economies of scope in farm diversification with an application to Ethiopian farms[END_REF]. By considering relatively similar systems to ours, [START_REF] Prache | Diversité dans la filière Petits Ruminants : Une source de résilience ?[END_REF] found that the system that exhibits the highest variability in terms of net returns over the past 10 years is the Sheep-Crop system. They argued that the Sheep-Crop system was very impacted by the climatic context and the fall of cereal prices. Our result is slightly different from the one of [START_REF] Prache | Diversité dans la filière Petits Ruminants : Une source de résilience ?[END_REF] maybe because we consider the net return per family labor unit instead of the absolute value of the net return.

Nevertheless, from Figure 5, it appears that it is not feasible to rank our farming systems using the mean-standard deviation criterion. Indeed, while the Sheep-Landless system has the lowest mean and the lowest standard deviation, we cannot classify the other systems using both their means and their standard deviations. For instance, the Sheep-Crop system has the largest mean net return while the Sheep-Grass system has the largest standard deviation. Thus, with the mean-standard deviation criterion, none of the sheep farming systems has the largest mean net return and smallest standard deviation. In addition, none of them has the lowest average net return and the highest standard deviation. That is why their ranking under the mean-standard deviation criterion is difficult. However, it is worth noting that Figure 5 indicates that both the Sheep-Crop system and the fully diversified system would be preferred to the Sheep-Grass system by any risk-averse producer, since these systems generate higher net returns at lower risk (variance).

Figure 6 reports the results of the SERF method. These results are obtained by estimating the CE values by including the total subsidies received by farmers. Figure 7 reports the same results but without including the subsidies in the estimations. This may be useful for investigating the role of the subsidies in the ranking of the different systems. The SERF analysis (Figure 6) confirms the fact that the Sheep-Landless system is dominated by the other ones in terms of risk-adjusted net return per family labor unit, as shown in the CDF curves (see Figure 4). Interestingly, the SERF analysis clarifies the rank of the other systems. Indeed, the SERF analysis (Figure 6) indicates that the Sheep-Crop system is the dominant alternative across all levels of risk aversion coefficients (RAC). Following [START_REF] Anderson | Risk Analysis in Dryland Farming Systems[END_REF], the relative RAC is equal to 0 for risk-neutral decision makers, 0.5 for slightly risk-averse decision makers, 1 for normally risk-averse decision makers, 2 for moderately risk-averse decision makers, and 4 for extremely risk-averse decision makers. By definition and as observed from Figures 6 and7, the CEs decrease with an increase in the RAC. This means that the most risk-averse farmers have higher risk premiums than less risk-averse farmers. In this line, note that that the Sheep-Crop system exhibits the highest risk-adjusted return per family labor unit, regardless of the values of the RAC. This could be explained by the fact that this system benefits from larger structures (see Table 1) and that it is located in areas with relatively good agricultural potential [START_REF] Benoit | Performances techniques et économiques en élevage ovin viande biologique: Observations en réseaux d'élevage et fermes expérimentales[END_REF][START_REF] Venineaux-Delvalle | Facteurs de résilience des systèmes ovins viande[END_REF]. Indeed, as industrial crops are more frequent in the rotations in the Sheep-Crop system, they provide access to various co-products to feed the sheep and the highest sheep feed autonomy (see Table 1). Another possible explanation is that the Sheep-Crop system facilitates the production of off-season lambs because it offers the possibility of feeding animals from co-products and crops produced on the farm (see [START_REF] Prache | Diversité dans la filière Petits Ruminants : Une source de résilience ?[END_REF], which may be valued by the market. Indeed, it is well known that from the economic law (supply and demand), off-season products can be better valued by the market. That is, they can be sold at more attractive prices, offsetting higher production costs.

The SERF analysis conducted without considering the subsidies received by farmers (Figure 7) provides a similar pattern. However, note that when the subsidies are taking into account (Figure 6), the Sheep-Grass and the fully diversified systems perform similarly; while without accounting for the subsidies (Figure 7), the Sheep-Landless and the completely diversified systems perform similarly. This may be understood in the sense that the performance of our fully diversified sheep systems relies mainly on the subsidies received by the farmers.

As previously mentioned, the SERF approach can be considered as a more discriminating form of the stochastic dominance analytical framework [START_REF] Schenk | How do farmers react to varying water allocations? An assessment of how the attitude to risk affects farm incomes[END_REF]. In this respect, it is worth noting that the stochastic dominance analysis based on the SERF approach (Figures 6 and7) provides more concrete evidence to explain the ranking of risky alternatives than the stochastic dominance analysis based on the CDF (Figure 4) (see also, [START_REF] Richardson | Ranking risky alternatives: Innovations in subjective utility analysis[END_REF].

Econometric results on downside risk exposure Prior to the econometric analysis, we test whether the farm income exhibited skewness. Using the test proposed by D 'Agostino, Balanger, and D'Agostino (1990), we find that the skewness is statistically different from zero, with a p-value of 0.000. We, therefore, reject the null hypothesis of normality due to skewness at the 1% significance level. Table 2 reports the econometric results on the effects of the diversified systems on downside risk exposure measured through the third moment (the skewness) of the farm income.

Recall that an increase in the skewness means that downside risk exposure decreases, that is, the probability of failure decreases [START_REF] Di Falco | On crop biodiversity, risk exposure, and food security in the highlands of Ethiopia[END_REF]. This suggests that variables that are positively associated with the skewness involve a reduction in the downside risk exposure. Most of the estimated coefficients shown in Table 2 are statistically different from zero at the 5% significance level. This indicates that most of the explanatory variables used have significant effects on the downside risk exposure. Except for farm capital, all the significant variables have a positive effect on skewness, meaning that they can reduce farm exposure to downside risk. More specifically, we find that farm capital has a statistically significant and negative effect on skewness. This result suggests that the existing capital stock tends to increase downside risk by causing the collapse of farmers' net income under unfavorable production or market conditions. The coefficients of the intermediate inputs and the number of livestock units are positively related to the skewness, suggesting that these variables appear to be downside risk decreasing. This may be due to the fact that these variables are proportional to the production levels. The findings related to capital and intermediate input effects are consistent with Di Falco and Veronesi (2014), while the findings on livestock units are consistent with [START_REF] Finger | Determinants of downside risk exposure of dairy farms[END_REF]. We find that labor has a significant positive effect on skewness. This implies that the availability of labor can help farmers to manage downside risk by reducing the likelihood of farm activities disruption. Similar results can be found in [START_REF] Bozzola | Stability of risk attitude, agricultural policies and production shocks: Evidence from Italy[END_REF]. The variable feed autonomy is positively related to the skewness, suggesting that feed autonomy contributes to the reduction of downside risk exposure by lowering farm production costs. This could also be understood in the sense that feed autonomy allows farmers to avoid production failure under unfavorable input market conditions. The positive and significant coefficient of the time trend suggests that downside risk exposure decreases with time. Note that the coefficient of the Sheep-Landless system is not statistically significant. Thus, our estimates do not provide enough evidence on the potential of this system for managing downside risk exposure.

The coefficients associated with the Sheep-Crop system and the fully diversified system (Sheep-Grass-Crop-Landless) are positive and significant. This suggests that compared to the Sheep-Grass system (the reference category), the Sheep-Crop system and the fully diversified system can help farmers to reduce exposure to downside risk. Such results are consistent with many studies showing that farm diversification is a production strategy that allows farmers to manage exposure to downside risk [START_REF] Kim | Technological change and risk management: An application to the economics of corn production[END_REF][START_REF] Di Falco | On crop biodiversity, risk exposure, and food security in the highlands of Ethiopia[END_REF]. The coefficient of the Sheep-Crop system is significant at the 5% level while the coefficient of the fully diversified system is significant at the 1% level. These results indicate that the fully diversified system can be very effective in reducing exposure to downside risk [START_REF] Di Falco | Crop genetic diversity, farm productivity and the management of environmental risk in rainfed agriculture[END_REF]. These results could also be interpreted in terms of economic resilience. Indeed, resilience can be defined as the capacity of a system to withstand external shocks without catastrophic consequences [START_REF] Holling | Resilience and stability of ecological systems[END_REF]. In this respect, our results indicate that diversification increases the ability of the production systems to absorb downside shocks [START_REF] Di Falco | Crop genetic diversity, farm productivity and the management of environmental risk in rainfed agriculture[END_REF]. In this line, our results suggest that diversification strategies are successful risk management strategies that make farms more resilient to unfavorable production or market conditions. Our results also suggest that the fully diversified system would be the most resilient [START_REF] Di Falco | Crop genetic diversity, farm productivity and the management of environmental risk in rainfed agriculture[END_REF].

Concluding remarks

This study investigates the existence and the components of economies of diversification in sheep farming systems in France. It also identifies stochastic dominant mixtures of sheep farming activities in terms of risk-adjusted net returns. To examine the economies of diversification, we compare partially diversified systems (Sheep-Grass, Sheep-Crop, and Sheep-Landless) with fully diversified systems (Sheep-Grass-Crop-Landless) in terms of certainty equivalents of their net returns per family labor unit. The certainty equivalent of the net return is a risk-adjusted measure that makes it possible to integrate two economic rationales for production diversification: risk-reducing effect and scope economies effect. To identify the stochastic dominant mixtures of sheep farming activities, we use the stochastic efficiency with respect to a function (SERF) analysis, which is also known as a generalized method of stochastic dominance analysis.

Our empirical analysis is undertaken on an unbalanced panel of 1,139 observations from 134 mixed sheep farms located in an area of the Center of France, namely the Massif Central, over the period 1987 to 2017. The Massif Central is classified in the so-called Less Favored Agricultural Area (LFA). An advantage of sheep farming is that sheep are well suited to valorize LFA. In spite of that, sheep farmers in the Massif Central have to face risk and uncertainty due to unforeseen climate conditions, sanitary issues, and price volatility. In this context, our analysis may provide relevant information for designing/selecting mixtures of farming activities that allow farmers to have greater risk-adjusted returns, or to reduce their economic risk. This may be insightful for producers who think about changing their production mix in order to continue to valorize the potentialities of their territory.

The results indicate that there are no economies of diversification in the sheep farming systems considered. That is, we find that the fully diversified system considered does not generate more risk-adjusted returns (CE) per family labor unit than the partially diversified systems. In addition, we find that all the values of the indicator of the degree of economies of diversification are negative and amount to -0.685 on average. This suggests a relatively high degree of diseconomies of diversification in the sense that the completely diversified sheep system reaches only 31.5% of the risk-adjusted returns (CE) generated by the partially diversified systems altogether. Regarding the sources of the observed diseconomies of diversification, we find that the average net return per family labor unit generated by the fully diversified system is lower than the one generated by the partially diversified systems altogether, while the variance component of the fully diversified system is lower than the one of the partially diversified systems. This suggests that the observed diseconomies of diversification are mainly due to the mean return effect. That is, the diversified system does not generate more returns than the partially diversified systems, but it ensures more stable returns. However, since the variance does not distinguish between upside and downside shocks (i.e., good and bad events), special attention has been devoted to the effects of diversification on downside risk exposure. In this respect, our results indicate that the Sheep-Crop and the fully diversified systems (Sheep-Grass-Crop-Landless) considered can be effective in reducing exposure to downside risk.

The dominance stochastic analysis conducted using the SERF method reveals that the dominant system in terms of risk-adjusted returns is the Sheep-Crop one. This may be due to the fact that these systems: (i) benefit from more attractive prices (for off-season products); and (ii) concern large farms located in areas favorable to cash crops. We also find that the returns of the Sheep-Landless system exhibit the lowest variability, while the ones of the fully specialized (Sheep-Grass) system exhibit the largest variability. Note also that the variability of the returns of the fully specialized (Sheep-Grass) system is rather close to that of the Sheep-Crop system (Figure 5). We recognize these differences may reflect the heterogeneity of the diversification strategies of farmers. For instance, we admit that each farmer trades off risk and returns at a personal (or individual) rate. In addition, diversification decisions could also be related, among other things, to soil quality, microclimate, historical context, territorial norms and constraints, and soil conservation strategies. It could also depend on: (a) individual circumstances; (b) resource availability (e.g., workforce, land, fixed capital); (c) farmers' skills; (d) farmers' desire to contribute to social and environmental objectives; (e) abilities; (f) incentives and existence of marketing channels (or market opportunity); and (g) extension service for a given production (see also, [START_REF] Mcnally | Farm diversification in England and Wales d what can we learn from the farm business survey?[END_REF][START_REF] Meert | Farm household survival strategies and diversification on marginal farms[END_REF][START_REF] Leck | Agriculturewho cares? An investigation of 'care farming' in the UK[END_REF][START_REF] Mcfadden | Exploring the concept of farm household innovation capacity in relation to farm diversification in policy context[END_REF][START_REF] Suess-Reyes | The future of family farming: A literature review on innovative, sustainable and succession-oriented strategies[END_REF][START_REF] Morris | Farm diversification, entrepreneurship and technology adoption: Analysis of upland farmers in Wales[END_REF]. For instance, it is recognized that larger farms and farms with greater assets are more likely to diversify their production activities [START_REF] Mcinerney | Patterns, Performance and Prospects in Farm Diversification[END_REF][START_REF] Ilbery | Farm diversification as an adjustment strategy on the urban fringe of the West Midlands[END_REF][START_REF] Mcnally | Farm diversification in England and Wales d what can we learn from the farm business survey?[END_REF].

In the same vein, the different systems could co-exist given specific farm characteristics and many contextual drivers. In this line, [START_REF] Villano | Evidence of farm-level synergies in mixed-farming systems in the Australian Wheat-Sheep Zone[END_REF] argued that the choice of diversification may be a function of a number of factors outside the farmer's control. In addition, previous studies (e.g., [START_REF] Mcnally | Farm diversification in England and Wales d what can we learn from the farm business survey?[END_REF][START_REF] Pardos | The diversity of sheep production systems in Aragón (Spain): Characterisation and typification of meat sheep farms[END_REF][START_REF] Maye | Farm diversification, tenancy and CAP reform: Results from a survey of tenant farmers in England[END_REF][START_REF] Hansson | Farmers' motives for diversifying their farm business: The influence of family[END_REF][START_REF] Morris | Farm diversification, entrepreneurship and technology adoption: Analysis of upland farmers in Wales[END_REF] found significant heterogeneity among farmers in their motives for diversifying their production activities. Using econometrics techniques, the effects of the diversified system considered on downside risk exposure were examined by controlling for some of the above factors (available in our data set). However, further research on the production capacities of the farms, their characteristics, and their contextual drivers could provide additional information on their diversification strategies. As such, it would be difficult to prescribe a specific sheep farming system to farmers from our results. However, from an economic standpoint, an important goal of farm diversification is to find combinations of farming activities that provide the best compromise between the farm return and its variability. In addition, as our results show, diversified production systems could also be driven by their ability to reduce downside risk exposure. In this respect, we believe that, in addition to producers who think about changing their production mix, our results could provide insights to policy makers and extension professionals. They can inform policy-makers' decisions on the components of the observed diseconomies of diversification in mixed sheep farms in the Massif Central. Extension professionals can distribute the information from our results to help farmers to make better diversification decisions.

Figure 1 .

 1 Figure 1. Economies (diseconomies) of diversification in French sheep farms.

Figure 2 .

 2 Figure 2. Degree of economies (diseconomies) of diversification (DED) in French sheep farms.

Figure 3 .

 3 Figure3. Sources of diversification economies.

Figure 4 .

 4 Figure 4. Cumulative distribution functions (CDFs) of the different sheep farming system (€/AWU).

Figure 5 .

 5 Figure 5. Mean-standard deviation plot of the four sheep farming systems.

Figure 6 .

 6 Figure 6. Certainty equivalent (CE) curves of net return (€/AWU) for the different sheep farming systems for a range of risk aversion coefficients (RAC).

Figure 7 .

 7 Figure 7. Certainty equivalent (CE) curves of net return (€/AWU) for the different sheep farming system (without subsidies) for a range of risk aversion coefficients (RAC).

  www.cambridge.org/core/terms.

Table 1 .

 1 Main characteristics of the farms over the entire study period for the four farming systems

					Sheep-grass-
		Sheep-grass	Sheep-crop	Sheep-landless	crop-landless
		Mean (SD)	Mean (SD)	Mean (SD)	Mean (SD)
	Usable agricultural	89.55 (50.01)	111.71 (57.63)	82.69 (32.07)	90.54 (36.81)
	area (ha)
	Main fodder area (ha)	89.54 (50.03)	89.38 (45.38)	74.97 (29.84)	78.13 (32.05)
	Permanent grassland	16.05 (23.19)	7.74 (17.57)	3.56 (5.47)	5.62 (10.30)
	(ha)
	Crop area	0 (0)	20.61 (24.93)	7.52 (5.07)	12.08 (10.02)
	Forage autonomy for	79.10 (8.90)	73.68 (9.27)	69.86 (10.58)	71.66 (10.97)
	sheep
	Feed autonomy for	79.14 (8.91)	84.08 (8.18)	79.66 (9.26)	81.35 (10.02)
	sheep
	Ovine Livestock units	78.79 (30.63)	82.99 (34.73)	70.09 (25.24)	75.17 (37.75)
	Annual work unit	1.52 (0.53)	1.39 (0.47)	1.48 (0.41)	1.66 (0.56)
	(AWU)
	Gross product	85,169 (33,817) 100,513 (52,949) 103,945 (42,368)	127,182 (71,471)
	Operational costs	25,958 (11,085) 33,301 (18,680) 38,063 (15,982) 48,584.39 (36,770.13)
	Fixed costs	33,925 (15,293) 41,460 (24,766) 45,111 (22,483)	51,099 (24,553)
	Subsidies	16,528 (14,276) 14,129 (15,387) 19,249 (12,243)	25,230 (18,099)
	Net return	25,286 (14,559) 25,752 (18,724) 20,772 (15,407)	27,498 (21,478)
	Number of	215	736	82	106
	observations

Table 2 .

 2 Econometric estimates for the skewness of farm income

	Estimates	Std. Err.

Agricultural and Resource Economics Review

We thank an anonymous referee for raising this point.

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Acknowledgements. The authors would like to thank the editor and the anonymous reviewers for their helpful and relevant suggestions that helped us to improve significantly the quality of this article.

Funding statement. This research received no specific grant from any funding agency, commercial or notfor-profit sectors.


A Etiemble

N Besnard

J Adrien

P Tran-Van

L Gautier

B Lestriez

E Maire
email: eric.maire@insa-lyon.fr

Quality control tool of electrode coating for lithium-ion batteries based on X-ray radiography

Keywords: lithium ion battery, electrode, processing, X-ray radiography

A simple and efficient method, based on X-ray radiography, is developed to check the quality (homogeneity of the thickness, presence of defects) of NMC-, LFP-and NMC/LFPbased electrode coating for Li-ion batteries at the scale of several cm 2 with a resolution of 20µm. As a first step, the attenuation coefficient of NMC-and LFP-based coating is experimentally determined according to the Beer-Lambert law. Then, the attenuation coefficient of each active material is estimated from these experimental results and X-ray attenuation databases, which allows establishing an attenuation law for any coating composition. Finally, thanks to this relationship, the thickness can be evaluated in each spot of the film and the defects, such as pinholes or broad edges with gradual decrease of the thickness coating, can be detected. The analysis of NMC-, LFP-and NMC/LFP-based electrodes shows that the coating quality decreases as coating thickness increases and as the nanometric vs. micrometric material content increases in the coating composition. This reveals detrimental aspects of nanomaterials with respect to their use in composite electrode manufactured through conventional slot-die or casting process.

Introduction

Lithium-ion batteries are manufactured by assembling three foils: the negative electrode, the positive electrode and the separators [1-4]. After blending of active materials, additive, binder and electrolyte, the electrodes are made by a coating process of the active layer on a copper (anode) or aluminum (cathode) substrate. After drying, the coating is then compressed (calendered) to optimize the porosity. Electrodes and separators are cut and, finally, the cells are stacked. In this process, the coating and the compression plays a major role on the battery performances [START_REF] Lanza | Automated optical detection of particles and defects on a Li-ion-cell surface using a single-point analysis[END_REF]. The initial capacity of the battery but also the cycle life is strongly influenced by the processing and electrode fabrication [START_REF] Lanza | Automated optical detection of particles and defects on a Li-ion-cell surface using a single-point analysis[END_REF][START_REF] Kenney | Modelling the impact of variations in electrode manufacturing in lithium-ion battery modules[END_REF]. In addition, simulations of the impact of variations in electrode manufacturing on Li-ion batteries show that the electrode thickness and density are the most significant parameters to optimize the useful capacity of battery module [START_REF] Kenney | Modelling the impact of variations in electrode manufacturing in lithium-ion battery modules[END_REF].

The electrodes have to be continuous, homogeneous and smooth (i.e. a low thickness deviation). The coating method and the slurry properties are the main factors playing a role on process efficiency [START_REF] Reinhart | Research and demonstration center for the production of large-area lithium-ion cells[END_REF]. Also, appropriate slurry rheological characteristics, density, porosity, viscosity, adhesion have to be optimized [START_REF] Bauer | Rheological properties and stability of NMP based cathode slurries for lithium ion batteries[END_REF][START_REF] Cho | Evaluation of slurry characteristics for rechargeable lithium-ion batteries[END_REF][START_REF] Kim | Effect of mixing sequences on the electrode characteristics of lithium-ion rechargeable batteries[END_REF][START_REF] Porcher | Design of aqueous processed thick LiFePO4 composite electrodes for high-energy lithium battery[END_REF]. The coating process must result in layers with very homogeneous thickness and density distribution as many heterogeneities in the layer might result in undesired local aging of the electrode [START_REF] Takamura | Charge/discharge efficiency improvement by the incorporation of conductive carbons in the carbon anode of Li-ion batteries[END_REF] or poorer rate performance [START_REF] Smith | The effect of microstructure on the galvanostatic discharge of graphite anode electrodes in LiCoO2-based rocking-chair rechargeable batteries[END_REF].

In this framework, an efficient monitoring of the electrode parameters is required in order to evaluate the process and to detect possible defects in the early stage of the production. In addition, a detection of the electrode coating edges on both sides is required in order to optimally adjust the laser cut during the electrode tailoring Radioscopic inspection is one of the most non-destructive testing used for inline monitoring of industrial process, such as lightweight material production, electronic component soldering or food production [START_REF] Hanke | X-ray based methods for non-destructive testing and material characterization[END_REF]. The use of X-ray radiography also requires ionizing radiation protection but much less restrictive than beta transmission and doesn't induce expensive cost associated with storage and transportation of radioactive sources. X-ray radiography allows visualizing a three-dimensional body, the attenuation of which is projected in a two-dimensional plane. As such, it provides a precise information and localization of structural heterogeneities and defects.

Li1Ni1/3Co1/3Mn1/3O2 (NMC), compound is a high-capacity electrode material for lithium ion battery which exhibits great structural stability, a higher capacity a lower cost and a better thermal stability than the traditional LiCoO2 cathode. Its high bulk conductivity allows using it as micrometric particles or even clusters of micrometric particles in composite electrodes [START_REF] Seid | Multiscale electronic transport in Li1+xNi1/3-uCo1/3-vMn1/3-wO2: a broadband dielectric study from 40 Hz to 10 GHz[END_REF]. Such large particles allow higher solid content in the composite electrode layer facilitating a higher energy density, which makes this compound an active material of choice for electric and hybrid electric vehicles [START_REF] Zheng | Cathode performance as a function of inactive material and void fractions[END_REF]. LiFePO4 (LFP) compound possesses a better thermal stability than NMC. As a consequence of low bulk conductivity, it can only be used as nanometric particles or as clusters of nanometric particles if these ones are coated by a thin layer of conducting carbon [START_REF] Seid | Multiscale electronic transport mechanism and true conductivities in amorphous carbon-LiFePO4 nanocomposites[END_REF]. Additionally, LFP operates at a low voltage, which makes it less attractive for electric and hybrid electric vehicles. Blending different active materials is an approach followed by several automotive battery suppliers intended to optimize the performance of the battery with respect to the automotive operating requirements. By blending two cathode materials the shortcomings of the parent materials can be minimized and the resulting blend can be tailored to have a higher energy or power density coupled with enhanced stability and lower cost [START_REF] Chikkannanavar | A review of blended cathode materials for use in Li-ion batteries[END_REF].

In this paper, a simple and efficient method, based on X-ray radiography, is developed to evaluate the thickness and to detect the defects of positive electrodes based on NMC, LFP, and blends of NMC and LFP for Li-ion batteries.

Experimental

Materials

Several samples with different composition and thickness have been produced for the present study. Table 1 summarizes composition, thickness (estimated on SEM images) and calculated porosity of the all the studied electrodes. The porosity derives from the electrode weight, thickness and material density. These were prepared with a NMC compound (Cellcore ® MX10) and/or a LFP one (PhoLiCat® FE100), both supplied by UMICORE. The NMC is micrometric material in the form of dense spherical clusters whose size ranges from 5 to 17µm (D50 = 9.7µm). The LFP is a nanometric material that is more or less agglomerated and the particles/agglomerates size ranges from 0.5 to 10µm. The BET measured specific surfaces are 0.28 and 18-20 m² g -1 for NMC and LFP, respectively. Carbon black (Cb) was used as conductive additive, polyvinylidene difluoride (PVdF) as binder and aluminum foil as substrate.

X-ray Radiography

The X-ray radiography analysis of the electrodes was performed at a pixel size of 20µm using a laboratory tomograph (Phoenix V tome X) described in [START_REF] Maire | In Situ Experiments with X ray Tomography: an Attractive Tool for Experimental Mechanics[END_REF]. The X-ray source was operated with a tungsten cathode at a voltage of 80kV and a current of 240µA. Each acquired radiograph consisted in an average of 3 radiographs for each electrode. The electrodes are put orthogonally to the incident beam between the X-ray source and the detector. In this condition, it can be considered that the beam cross the electrode thickness in a straight line.

Results and Discussion

Beer-Lambert Law

X-ray radiography physics is based on the Beer-Lamber law (Eq. 1), which depicts the relationship between the number N of transmitted photons of energy E and the number N0 of the incident photons of energy E [START_REF] Maire | On the application of X-ray microtomography in the field of materials science[END_REF].

𝑁 𝑁 0 = 𝑒𝑥𝑝 [-∫ 𝜇(𝑆)𝑑𝑆 𝑆∈𝑟𝑎𝑦 ] (1)
where µ is the attenuation coefficient. In the photoelectric domain, the attenuation coefficient, µ, typically follows the relationship:

𝜇 = 𝐾𝜌 𝑍 4 𝐸 3 (2)
where K is a constant, ρ and Z respectively are the density and the atomic number of the investigated material and E is the energy of the incident photon [START_REF] Maire | On the application of X-ray microtomography in the field of materials science[END_REF]. For a polychromatic incident beam, like the one emitted by a X Ray tube in a laboratory tomograph, the Beer-Lambert law has to be integrated over the whole photon energy spectrum.

The contrast observed in the X-ray radiograph of a bulky material is explained by this attenuation law because each point of a detector placed behind the sample is situated in front of a different path, each path exhibiting a different value of the integral of µ.

The integral of µ can be approximated by assuming that the X-rays behave like a monochromatic beam exhibiting the average equivalent photon energy of the X-ray tube spectrum. Strictly speaking, we are aware of the fact that this is slightly wrong especially after crossing the samples as the X-Ray beam is modified (is filtered) differently by every different material. This can however be estimated to be a good first approximation, especially because our samples are rather thin. It will be shown later that this approximation is validated by our different measurements.

Experimentally, all this leads to the relationship (Eq. 3) between the incident intensity I0 and the outgoing intensity I.

𝐼 = 𝐼 0 𝑒 -𝜇𝐿 (3)
where L is the length through the material (the attenuation of the X rays through air before and after the sample is very small and can be neglected) and µ is the attenuation coefficient corresponding to the average equivalent energy.

Figure 1 shows the relative outgoing X-ray intensity histograms in electrode and air areas for various thicknesses of NMC (Fig. 1A) and LFP electrodes (Fig. 1B). The outgoing intensities are distributed according to a Gaussian function, whose parameters are characteristic of the various paths through the material. This Gaussian distribution has roughly the same spread in the air area for all the samples. The peak center and shape of the Gaussian distribution vary depending on the composition (attenuation coefficient) and the thickness (length through the material) of the electrodes. In order to investigate the thickness heterogeneity, the equivalent average photon energy and the attenuation of the electrode materials must be defined.

Equivalent photon energy and attenuation coefficients

At this point, the average equivalent energy of the photon X-ray beam emitted by the X-Ray tube is not known. We determine an approximation of this energy in this section using standards with known attenuation but also using our electrodes of known thickness and porosity composed of unblended materials. We will also use the fact that the carbon component of the electrodes (Cb and PVdF) exhibit negligible attenuation.

The attenuation through titanium and aluminum foils of known measured thickness (14 µm for aluminium and 20 µm for titanium) has firstly been measured using equation [START_REF] Reinhart | Research and demonstration center for the production of large-area lithium-ion cells[END_REF] and measurements of I and I0 from radiographs such as those shown in Fig 1 . µ was found to be equal to 0.0023µm -1 for Ti and 0.00043µm -1 for Al.

In the case of electrodes perpendicular to the incident beam, Equation (3) can be expressed as a relationship between the outgoing intensity, I, electrode thickness, T, and porosity, ε:

𝑙𝑛 ( 𝐼 𝐼 𝑎𝑖𝑟 ) = 𝐶 -𝜇𝑇(1 -𝜀) (4)
where Iair, the outgoing intensity in air area and C, a constant associated especially with the attenuation due to the aluminium current collector and the diffraction. The porosity is considered homogeneous inside the electrodes and the outgoing intensities are evaluated as the peak center of the Gaussian distribution (Fig. 1).

Figure 2 shows ln(I/Iair)/(1-ε) as a function of the thickness T, estimated on crosssection SEM images, for NMC and LFP electrodes (the blended electrodes were excluded from this analysis). The Beer Lambert law is clearly nicely respected in this figure and a linear evolution is observed for all materials in agreement with equation (4).

The linear fitting parameters are summarized in Table 2. The most important information to extract from this table is the value of the slope, µ, that corresponds to the attenuation coefficient of each material for the average equivalent energy of the beam. previously can be superimposed on these curves as horizontal lines as shown in Figure 3 and the energy corresponding to the crossing of the µ curves with these particular values of µ can be extracted from this construction. The figure shows that these energies for all the investigated materials are all very close to 27 ± 2 keV. The equivalent energy of the photon X-Rays can then be assumed to be close to 27 keV. This is in relatively good agreement with what can be expected for the X-Ray tube operated at 80 kV in this study and this validates the concept of average equivalent energy that we used from simplification in the present study.

For any blended NMC/LFP/PVdF/Cb electrodes, the relationship between the outgoing intensity, I, electrode thickness (µm), T, and porosity, ε, can be established, by taking into consideration these attenuation coefficients, as:

𝑙𝑛( 𝐼 𝐼 𝑎𝑖𝑟 ) (1-𝜖) = 𝐶 -(3.35𝑓 𝑁𝑀𝐶 + 1.02𝑓 𝐿𝐹𝑃 ) * 𝑇 * 10 -3 (5)
with C sets at -0.02, -0.04 and -0.03, respectively for FLP, NMC and blended LFP/NMC cathodes and, f, the volume density of each highly absorbent material in the composite electrode composition.

The values of T, calculated with the equation ( 5) and averaged over the electrode surface, are shown in Figure 4 and are in fair agreement with the SEM estimation for all the electrodes.

Thickness heterogeneity

Thanks to Equation 5 and considering that the phase distribution is homogeneous in the thickness of the electrode, the fluctuation of the intensity in the radiographs can be attributed to thickness fluctuations only and this fluctuation of T can thus be mapped for every pixel of the detector, each pixel corresponding to a path through the electrode.

Figure 5 and Figure 6 show the thickness mapping for NMC, LFP, and blended NMC/LFP electrodes. Figure 7 shows the width at half maximum of the thickness histograms, which reflects its heterogeneity, as a function of the average thickness. Note that the resulting thickness heterogeneity values are slightly overestimated by this method due to the inherent dispersion of the intensity of the incident beam, as illustrated in Figure 1 for the air area (used as reference of the incident beam) and the variability of porosity/density. Also, X-ray tomography investigations show that the surface standard deviation of NMC density is about 3% at this resolution (not shown), namely less than the thickness heterogeneity estimated by X-ray radiography. In addition, the porosity/PVdF/Cb distribution variability should decrease as the LFP content increases due to its nanometric morphology.

For NMC electrodes, the thickness heterogeneity is low and slightly increases as the thickness increases. Also, the thickness appears spatially homogenous in this material (Fig. 5A and5B) with a granular aspect, which could be associated with the clusters/porosity repartition. The heterogeneity of LFP electrodes thickness is much higher than for NMC electrodes and increases more significantly as the thickness increases (Fig. 5C and5D).

Large thinner areas are observed on the thickness mapping and could be associated with the coating step during the electrode preparation. The thickness heterogeneity of blended NMC/LFP electrodes also appears high, similar to LFP ones, with exception of the electrode D that has the largest roughness variations of all electrodes (Fig. 7). One should note that electrode D has the higher Cb content of all electrodes (Table 1).

In manufacturing lithium-ion battery electrodes, the slurry is coated using a slot-die process or cast onto a transfer roll by a doctor blade, which then transfers the slurry to the current collector. The coating quality depends on many parameters, among which the colloidal stability, the rheological properties and the surface tension of the electrode slurry, the roll speed, the gap width between the slot-die or the doctor blade and the current collector Putting all these results together, it is suggested that marked shear thinning behavior and/or strong elasticity are detrimental properties of electrode slurries with respect to the quality of the resulting coating. An increase of surface roughness with an increase of coating thickness was also measured by Schmitt et al. on Graphite/PVdF-based slurries for negative electrodes

[22].

Defects detection

During the slurry coating, pinholes can be left over inside the electrodes due to gas release. This kind of defect, which is detrimental to the calendering step, is not desirable. Some pinholes, characterized by local low attenuation coefficient (i.e. thickness) as shown in Figure 8A, can be observed in the thickness mapping of NMC, LFP, and blended NMC/LFP electrodes (Fig. 5 and6). For both NMC and LFP electrodes, they are mostly observed for the highest coating thickness. This is particularly obvious for the LFP electrodes (Fig. 5D). The 25/75 blended NMC/LFP electrode, which is the thickest of this group, appears also impacted by this phenomenon. From these results, the thickness seems to be the major parameter, acting negatively on pinhole formation. The difference of maximum thickness between NMC and LFP electrode specimens does not allow investigating precisely the composition influence on this defect appearance. Pinholes could find their origin from air bubbles trapped in the viscous electrode slurries that are more easily removed when the slurry is forced to flow through a narrow gap to form a thin coated layer than through large ones. Shear is also likely to destroy these bubbles and shear is more important for thinner electrodes.

The slurry coating on the copper or aluminum substrate induces broad edges with gradual decrease of the thickness coating. The laser cutting has to be as precise as possible in order to maximize the consistent output electrode surface area. Figure 8B shows thickness mapping, from the X-ray radiography, of the coating edge for the thickest (P) NMC electrode.

A thickness gradient and a large defect with a trench aspect are observed at the edge of the active layer. This demonstrates the efficiency of the X-ray radiography to evaluate precisely the electrode coating and that this technique allows detecting easily the defects and the edges of the films. In the same way, layer scratch, which can occur during the electrode manufacturing, could be characterized by X-ray radiography as drastic local thickness variation.

Conclusion

This study has demonstrated that X-ray radiography, which is a simple and nondestructive analytical technique, is a powerful tool for checking the quality of electrode coating at the scale of several cm 2 with a resolution of 20µm. Radiography is fast so it could also be implemented on much larger surfaces and/or with a different resolution by multiplying the number of fields of view. Methods to evaluate from the acquired radiographs through careful image processing the homogeneity of the electrode thickness and of components distribution, as well as the presence of defects, were described. Care must be taken because of the polychromaticity of the incident beam in a standard laboratory radiographic system. We have tackled this difficulty by assuming and evaluating the average equivalent energy of the beam.

By this X-ray radiography analysis of NMC-, LFP-and NMC/LFP-based electrodes, we quantify the decrease in coating quality with increase in coating thickness and with increase in nanometric vs. micrometric material content in the coating composition. This reveals detrimental aspects of nanomaterials with respect to their use in composite electrode manufactured through conventional slot-die or casting process.

Manufacturing, well-performing thick electrodes is a critical issue for lithium ion battery application to raise the energy and power densities. Finally, this new analytical technique could be used by industry engineering or researchers to quantify the quality of their coatings at large scale. The setup used here is dedicated to research and does not allow us to achieve roll to roll process control. There would be no difficulties however, to adapt such a device to an in-line control of the fabrication process of such electrodes directly on a fabrication line in the future.

Table captions

Table 1

Composition, thickness (estimated on SEM images) and porosity of NMC, LFP and blended NMC/LFP cathodes.

Table 2

Linear fitting parameters of equation ( 4) for Al foil, Ti foil, NMC and LFP cathodes. Figure 4 NMC (m-p), LFP (q-t) and blended NMC/LFP (h, d, l) cathodes thickness evaluated on SEM images and by radiography.

Figure captions

Figure 7

Width at half maximum of the thickness histogram as a function of the average thickness for NMC, LFP and blended NMC/LFP cathodes.

Figure 8

Thickness mapping of (A) pinholes in the thickest LFP cathode (t) and (B) coating edge for the thickest NMC cathode (p)

[ 3 ]

 3 . An efficient method needs a resolution of a few micrometers to assure complete defect detection and a large visual field to match up the industrial scale. Optical microscopy, beta transmission, infrared radiation and laser caliper are currently applied during industrial electrode manufacturing [1, 2, 11, 12]. Thickness heterogeneity, pinholes, divots and agglomerates in the electrodes however are not easily identified during the coating process [12]. In addition, beta transmission measurement is an expensive and environmentally hazardous (due to ionizing radiation) technique [12]. It requires radioactive sources, which necessitates specialized shielding, services and high levels of security. Raman microscopy has also been previously evaluated as a possible inline characterization technique of battery electrodes [13], but there is however still a need for conventional, efficient and low cost inspection tools in Li-ion batteries quality control.

Figure 3

 3 Figure3shows the attenuation coefficient of NMC, LFP, PVdF, Cb, Al and Ti as a

[ 22 ]

 22 . The rheological properties of NMP-based electrode slurries containing NMC or LFP active materials, carbon black (Cb) conductive additive, and polyvinylidene difluoride (PVdF) binder have been studied by Bauer et al. [5]. As a consequence of smaller and nanometric particle size, LFP-based slurries show marked shear thinning behavior (large viscosity variations with shear/flow rate) and stronger elasticity than NMC-based ones. However, increasing CB and PVdF contents in NMC-based slurries increases their shear thinning behavior and elasticity. These two characters are due in both types of slurries to the establishment of PVdF bridges between LFP and/or CB nanoparticles. Here, the larger roughness variations are observed for slurries that contain these nanoparticles. In particular, high PVdF and Cb contents (electrode D) seem to significantly impact roughness variations.

Figure 1

 1 Figure 1Outgoing X-ray intensity histograms in electrode and air areas for (A) NMC and (B) LFP cathodes.

Figure 2

 2 Figure 2ln(I/Iair)/(1-ε) as a function of the cathode or foil thickness, estimated on SEM images.

Figure 3 X

 3 Figure 3 X-ray attenuation coefficients of NMC, LFP, PVdF, Cb, Al and Ti as a function of the photons energy [21].

Figure 5

 5 Figure 5Thickness mapping of (A, B) NMC and (C, D) LFP cathodes.

Figure 6

 6 Figure 6Thickness mapping of (A) 75/25, (B) 50/50 and (C) 25/75 blended NMC/LFP cathodes.

  Fig.1

  Fig. 5

Table 1

 1

	Ref.	NMC (vol. %)	LFP (vol. %)	PVdF (vol. %)	Cb (vol. %)	Estimated thickness (µm)	Porosity (%)
	M	82.1	0	11.5	6.3	34	33
	N	82.1	0	11.5	6.3	46	32
	O	82.1	0	11.5	6.3	61	34
	P	82.1	0	11.5	6.3	69	34
	H	58.3	24.8	10.9	6.0	73	34
	D	33.8	43.2	14.9	8.2	77	29
	L	17.5	67.2	9.9	5.4	83	35
	Q	0	85.5	9.4	5.1	44	34
	R	0	85.5	9.4	5.1	61	37
	S	0	85.5	9.4	5.1	76	35
	T	0	85.5	9.4	5.1	88	37
	Table 2
			µ (µm -1 )		C/(1-ε)	R 2
	NMC/Cb/PVdF	0.0027 ±0.0002	-0.041 ±0.010	0.986
	LFP/Cb/PVdF	0.0016 ±0.0001	-0.021 ±0.004	0.997
	Al		0.00043 ± 0.00001	-0.0005±0.0004	0.998
	Ti		0.0023 ±0.0001	-0.032 ±0.007	0.993

Acknowledgements

The authors would like to acknowledge IMD (Institut de la Mobilité Durable -Sustainable Mobility Institute) for financial support.