Oguz Umut

Ioan R Ionescu
email: ioan.r.ionescu@gmail.com

Tempering the mechanical response of FCC micro-pillars: an Eulerian plasticity approach

The mechanical response of almost pure single-crystal micro-pillars under compression exhibits a highly localized behavior that can endanger the structural stability of a sample. Recent experiments revealed that the mechanical response of a crystal is very sensible to both the presence of a quenched disorder in the sample, and the orientation of the crystal. In this work, we study the influence of disorder and crystal orientation on the large strain response of a 2D FCC crystal with three active glide planes using a very simple Eulerian plasticity model in the FE framework. Our numerical and theoretical results on clean crystal pillars suggest that a single plane or many gliding planes can be activated depending on the crystal orientation. While in the former case, the deformation is localized, leading to ductile rupture, in the latter, a complex interplay between active planes takes place, resulting in a more uniform deformation. The strain-localization can be avoided when inhomogeneities are engineered inside the crystal, or the crystal orientation is altered because of the activation of multiple slip systems, resulting in a "patchwork" of the distribution of the slip systems.

Introduction

When subjected to an external load, crystalline materials undergo plastic deformation beyond a material dependant elastic limit [START_REF] Wilson | Dislocations and Platic Flow in Crystals by A. H. Cottrell[END_REF][START_REF] Salman | Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity[END_REF]. At the crystal lattice level, the origin of the irreversible plastic deformation is the generation and crystal-symmetry dependent motion of linear crystal lattice defects, dislocations [START_REF] Hull | Introduction to dislocations[END_REF][START_REF] Baggio | Landau-Type theory of planar crystal plasticity[END_REF]. At large scale, the elasto-plastic mechanical response of crystalline materials manifest itself in the form of continuous strain-stress curves that rendered possible the development of continuum phenomenological theories based on some preassigned irreversible plastic flow rules when stresses exceed given thresholds [START_REF] Roters | Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finiteelement modeling: Theory, experiments, applications[END_REF][START_REF] Forest | Micromorphic Crystal Plasticity[END_REF]. Although these theories rely on the homogenisation of spatial heterogeneities such as dislocation cores, grain boundaries and mesoscale dislocations patterns etc., typically present in crystals, they are highly successful in reproducing many of aspects of plastic flow such as workhardening, yield stress and plastic shakedown to mention some [START_REF] Hughes | The microstructural origin of work hardening stages[END_REF][START_REF] Ask | A cosserat crystal plasticity and phase field theory for grain boundary migration[END_REF][START_REF] Mcdowell | Multiscale modeling of interfaces, dislocations, and dislocation field plasticity[END_REF].

Our ability to control plastic flow in applications is of fundamental importance for the reliable mechanical functioning of small devices. In recent years, the single-crystal pillar compression tests became the standard tool to study mechanical response at nano and micro scales [START_REF] Papanikolaou | Quasi-periodic events in crystal plasticity and the self-organized avalanche oscillator[END_REF][START_REF] Maaß | Micro-plasticity and recent insights from intermittent and small-scale plasticity[END_REF][START_REF] Sparks | Nontrivial scaling exponents of dislocation avalanches in microplasticity[END_REF][START_REF] Pan | Rotatable precipitates change the scale-free to scale dependent statistics in compressed ti nano-pillars[END_REF]. These experiments put in evidence unambiguously a non-smooth mechanical response with discernible stressdrops and localized deformation endangering the structural stability of the materials (see Fig. 1). This finding led to the development of new quantitative strategies in order to be able to reach a more smooth and delocalized mechanical response, highly desirable in applications. The general idea in these strategies consists in hindering dislocation motion by introducing solutes or precipitates inside the pillars leading to "dirtier is milder" effect [START_REF] Zhang | Taming intermittent plasticity at small[END_REF][START_REF] Pan | Rotatable precipitates change the scale-free to scale dependent statistics in compressed ti nano-pillars[END_REF][START_REF] Zhang | Plate-like precipitate effects on plasticity of Al-Cu alloys at micrometer to sub-micrometer scales[END_REF]. On the other hand, these strategies are not only limited to "tempering" of mechanical response but have also been used to design nanostructured crystals with ultrahigh strength and large plasticity [START_REF] Wu | Hierarchical nanostructured aluminum alloy with ultrahigh strength and large plasticity[END_REF]. Similarly, crystal orientation has also been shown to have an important effect on the mechanical behavior in experiments such that a low-symmetry orientation results in "milder" behavior [START_REF] Sparks | Nontrivial scaling exponents of dislocation avalanches in microplasticity[END_REF].

From the modeling point of view, these findings constitute a new challenge for the continuum of theories of crystal plasticity. In physical terms, the resulting overall mechanical response of pillars depends on the size, shape, and distribution of the disorder due to complex interactions of dislocations with precipitates at the lattice scale, making a continuum description difficult. However, although atomistic simulations [START_REF] Bulatov | Connecting atomistic and mesoscale simulations of crystal plasticity[END_REF][START_REF] Baruffi | Overdamped langevin dynamics simulations of grain boundary motion[END_REF] or mesoscopic approaches such as discrete dislocation dynamics [START_REF] Ispánovity | Avalanches in 2D dislocation systems: plastic yielding is not depinning[END_REF] and Landau-type theories of crystal plasticity [START_REF] Salman | Modeling of spatio-temporal dynamics and patterning mechanisms of martensites by phase-field and Lagrangian methods[END_REF][START_REF] Salman | Minimal integer automaton behind crystal plasticity[END_REF][START_REF] Salman | On the critical nature of plastic flow: One and two dimensional models[END_REF][START_REF] Baggio | Landau-Type theory of planar crystal plasticity[END_REF][START_REF] Zhang | Variety of scaling behaviors in nanocrystalline plasticity[END_REF] can be used to describe both dislocation precipitate interactions and crys- Taken from [START_REF] Zhang | Taming intermittent plasticity at small[END_REF]. tal orientation, accounting for complex mechanical loading and realistic geometries are not within the reach of even the most developed codes and can only be undertaken in the finite element framework. In this work, we study the mechanical response of a 2D FCC crystal when it is compressed in the framework of a continuum Eulerian viscoplasticity approach [START_REF] Cazacu | Dynamic crystal plasticity: an eulerian approach[END_REF]. Our approach is deliberately minimalist such that the elastic properties of the lattice are neglected as they are assumed to be small in comparison with the large plastic deformations, and we used a rate-dependent (viscoplastic) model of Perzyna type without resorting to physically motivated dislocation-density based models. Since the considered rate of deformation and the viscosity are very low, the numerical results correspond to a quasi-static loading of rate-independent materials. We first consider "clean" crystals (i.e., with a homogeneous yield limit) and investigate the deformation of a crystal with respect to its orientation. Afterward, we incorporate the effects of precipitates on the deformation by assuming a higher stress flow in some pre-selected regions inside the crystal.

The outline of the paper is as follows. We first give a brief description of the continuum Eulerian plasticity approach developed in [START_REF] Cazacu | Dynamic crystal plasticity: an eulerian approach[END_REF] (section 2). In section 3 we deduce from a limit load analysis some theoretical features of slip and kink stationary shear bands. Then we present the numerical implementation [START_REF] Cazacu | Augmented lagrangian method for eulerian modeling of viscoplastic crystals[END_REF] used in the simulations (section 4). Finally, section 5 is dedicated to the presentation of the numerical results.

Model

We begin by recalling from [START_REF] Cazacu | Dynamic crystal plasticity: an eulerian approach[END_REF] the mechanical model used in this paper. The equations governing the motion in a domain  = ( ) of an incompressible rigid-viscoplastic crystal. Let ∶ [0, ] ×  → 3 , the the velocity, ∶ [0, ] ×  → 3×3 , the deviator of the Cauchy stress tensor and ∶ [0, ] ×  → , the pressure (mean stress) ( = -is the Cauchy stress tensor) be the principal un-knowns fields, while the mass density > 0 and the body forces are considered known. The lattice orientation of the crystal is modeled through the slip direction distribution ∶ [0, ] ×  → 3 and the slip plane normal distribution ∶ [0, ] ×  → 3 for each crystallographic system = 1, 2, .., while the slip rate on each system will be denoted by ̇ .

The governing equations of the in-plane deformation of a single crystal in the domain  = Ω × take a simpler form if we make use of only two-dimensional vector and tensor quantities. Rice [START_REF] Rice | Plane strain slip line theory for anisotropic rigid/plastic materials[END_REF] showed that certain pairs of the 3-D systems that are potentially active need to combine in order to achieve plane-strain deformation. Let = 1, ..., be the index of the composite slip system formed by two 3D slip systems and . Let ̄ , ̄ , ̄ , ̄ denote the normalized projections of the 3D slip directions , and normal directions , onto the 1 2 -plane, which are also orthogonal in 2D. We can define the inplane slip directions for each = 1, .. as ̄ = ̄ and the in-plane plane normals as ̄ = ̄ and introduce ̄ =

1 2 ̄ ⊗ ̄ + ̄ ⊗ ̄ and ̄ = 1 2 ̄ ⊗ ̄ -̄ ⊗ ̄ ,
the symmetric and antisymmetric parts of the two-dimensional tensor product of slip directions and normals.

For plane-strain conditions let ̄ ∈ 2×2 be the in-plane Cauchy stress deviator ( ̄ = with , = 1, 2) and let ̄ ∈ 2 be the in-plane velocity ( = ( ̄ , 0), ̄ = ̄ ( , 1 , 2 )). Let also denote by ( ̄ ) ∶= 1 2 ( ̄ + ̄ ) and by ( ̄ ) ∶= 1 2 ( ̄ -̄ ), the symmetric and antisymmetric parts, respectively, of the two-dimensional velocity gradient ̄ .

In applications involving large deformations and high strain rates, the elastic component of the deformation is small with respect to the inelastic one. That is why it can be neglected it here and a rigid-viscoplastic approach adopted (e.g. [START_REF] Hutchinson | Bounds and selfconsistent estimates for creep of polycrystalline materials[END_REF][START_REF] Lebensohn | A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: Application to zirconium alloys[END_REF][START_REF] Kok | A polycrystal plasticity model based on the mechanical threshold[END_REF]). As it is proven in [START_REF] Cazacu | Dynamic crystal plasticity: an eulerian approach[END_REF], the two-dimensional rate of deformation tensor ( ̄ ) can be decomposed into composite plane-strain systems specified by ( ̄ , ̄ ), the normalized projections of the three-dimensional slip directions and normal directions onto the plane. Thus,

( ̄ ) = ∑ =1 ̇̄ ̄ , ( 1
)
where ̇̄ = 2 ̇ , and are the in plane factors specific to each crystal type, we get the in-plane form of the flow rule

̇̄ = 1 ̄ 1 - ̄ | ̄ ∶ ̄ | + ̄ ∶ ̄ , (2)
where ̄ = ∕(2 2 ) and ̄ = ∕ denote the "in-plane" viscosity and yield limit.

If we specify the two-dimensional vectors ̄ and ̄ by their polar coordinates ̄ = (cos , sin ), ̄ = (-sin , cos ), then the angles between two systems, say -, do not change in time, and it is sufficient to compute the change in orientation of only one of the composite in-plane slip systems, for instance = 1 , by using the equation

+ ̄ ⋅ ∇ = 1 2 ⎛ ⎜ ⎜ ⎝ ∑ =1 ̇̄ -( 1 2 -2 1 ) ⎞ ⎟ ⎟ ⎠ . ( 3
)
Then the orientation of f all other -1 composite plane strain systems can be obtained from the relation

( ) = ( ) + (0) -1 (0).
The yield limits ̄ of each slipping system can be considered as constants, but they can vary in time if hardening effects are taken into consideration. Since the slip on each system produces hardening on all slip systems, the slip resistances ̄ have an Eulerian evolution law of the form:

̄ + ̄ ⋅ ∇ ̄ = ∑ =1 ℎ | ̇̄ |. (4)
The matrix ℎ , called hardening matrix, describes the slip resistance on system which is caused by slip on system . As shown by Franciosi [START_REF] Franciosi | The concepts of latent hardening and strain hardening in metallic single crystals[END_REF], the matrix ℎ is not constant. Expressions for the hardening matrix, ℎ = ℎ ( ), on the cumulated shears on all systems are widely used in FE simulations of polycrystals (see [START_REF] Peirce | An analysis of nonuniform and localized deformation in ductile single crystals[END_REF][START_REF] Anand | A computational procedure for rateindependent crystal plasticity[END_REF]). Hardening laws that use dislocation densities on all slip planes as internal variables have also been proposed (e.g. [START_REF] Teodosiu | A theory of finite elastoviscoplasticity of single crystals[END_REF][START_REF] Teodosiu | A theory of finite elastoviscoplasticity of single crystals[END_REF]).

Let us see now how the in-plane model is applied at an FCC crystal, which corresponds to the case used in the next sections. For a FCC crystal, which has 12 potentially active 3-D slip systems, let 3 axis be parallel to [110] in the crystal basis, which means that the plane-strain plane

Stationary shear bands in a FCC crystal

We shall try to understand how the in-plane model, presented in the previous section, can describe the mechanism of stationary shear bands in "clean" FCC crystals. For simplicity, we will suppose that the slip resistance for all systems is equal to , and no hardening is taken into consideration. This means that the in-plane yield limit for each systems, ̄ 1 = √ 3 and ̄ 2 = ̄ 3 = 2 ∕ √ 3 are constant. Moreover we consider a vanishing viscosity ( ̄ = 0), i.e., we deal with a rigid-perfectly plastic model.

x3 [001] q b1 b2 b3 _ _ f _ x1 x1 x2 [110] [110] _ V h V a q = q b q= q 0 q= q 0 l l/ c o s (a ) n t
The approach used here to describe a shear band is not following Assaro and Rice (see [START_REF] Asaro | Strain localization in ductile single crystals[END_REF] but also [START_REF] Asaro | Mechanics of Solids and Materials[END_REF][START_REF] Marano | Intragranular localization induced by softening crystal plasticity: Analysis of slip and kink bands localization modes from high resolution FFT-simulations results[END_REF][START_REF] Borja | Bifurcation of elastoplastic solids to shear band mode at finite strain[END_REF]) for elastic-plastic models with hardening. For our simple rigidplastic model, another method, used in modeling ductile rupture, called limit load analysis, seems to be more appropriate. To model shear bands, where strains are localized on some surfaces, the associate (collapse) velocity has to exhibit discontinuities. The block decomposition method, which was intensively used in the analytical developments (see for instance [START_REF] Hill | The Mathematical Theory of Plasticity[END_REF][START_REF] Kachanov | Foundations of the theory of plasticity amsterdam[END_REF]), consists in considering only velocity fields generated by a decomposition of the structure in rigid blocks separated by discontinuity surfaces. The plastic dissipation power is then minimized on this particular class of functions (see also [START_REF] Ionescu | Discontinuous velocity domain splitting in limit analysis[END_REF]). The main difference between the present model and a classical plasticity approach of the block decomposition method is the role played by the evolution of the crystal orientation in the shear band through (3).

The "clean" crystal, occupying the domain (0, )×(0, ) and having the homogeneous orientation = 0 , is splinted into two rigid regions (see Fig. 2) separated by a shear band in the direction = (cos( ), sin( )). The upper region is animated by a velocity -∕ ( ) , where is the vertical velocity of the upper plateau, while the bottom region is at rest. In the shear band region, denoted by  and having the width ℎ, the crystal has the orientation = . The associated velocity field ̄ has the strain rate ( ̄ ) = -∕(ℎ sin( )) ⊗  , where  is the indicator function of the shear band  (  ( ) = 1 if ∈  and

 ( ) = 0 if not).
The slip rates on each system in the shear band region ̇̄ , = 1, 2, 3 can be determined by minimizing the inter-

nal plastic dissipation power ( ̇̄ 1 , ̇̄ 2 , ̇̄ 3 ) = ∑ 3 =1 ̄ | ̇̄
|, under the kinematic constraints [START_REF] Anand | A computational procedure for rateindependent crystal plasticity[END_REF]. Moreover, we can use the closed form from [START_REF] Cazacu | Dynamic crystal plasticity: an eulerian approach[END_REF] to find the analytic expressions of ̇̄ for any crystal orientation . The evolution of the crystal orientation in the shear band can be obtained from the differential equation ( 3), which reads ̇ = [ ̇̄ 1 + ̇̄ 2 + ̇̄ 3 + ∕(ℎ sin( ))]∕2. For stationary (permanent) shear bands the crystal orientation in the shear band is constant in time, hence we have a supplementary equation for the slip rates

̇̄ 1 + ̇̄ 2 + ̇̄ 3 = - ℎ sin( ) .
Using this last equation we can prove that there exists only three possible orientations of the crystal in a stationary shear band = , -, + and for each orientation there exists only one active slipping system ̇̄ ≠ 0.

Let us analyse now the link between the crystal orientation in the shear band and on the clean crystal 0 . We have to check that the requirement of continuing equilibrium ̄ ⋅ = ̄ 0 ⋅ is compatible with the rigid-perfect plastic law | ̄ 0 ̄ ⋅ ̄ | ≤ ̄ for all . We found that if = then there exists a restriction on 0 but there are no restrictions for = ± . The case 0 = is always possible and following [START_REF] Marano | Intragranular localization induced by softening crystal plasticity: Analysis of slip and kink bands localization modes from high resolution FFT-simulations results[END_REF], we called it slip band while for 0 ≠ we called it kink band. Having in mind the link between the shear band angle and the orientation of the crystal in the shear band we obtain that the slip bands are possible only for a special orientation of the cystal 0 = , -, + .

The total plastic dissipation power associated to the ve-

locity field ̄ , ( ̄ ) = ∑ 3 =1 ∫  ̄ | ̇̄ |, is then given by  = 2 ̄ 1 ∕| sin(2 )| if = and  = 2 ̄ 2 ∕| sin(2 )| if = ±
. Moreover the plastic dissipation power is independent of the shear band thickness ℎ. That is why the shear band can be considered with a vanishing width, which corresponds to a discontinuous velocity field. We expect that a stationary band is stable if the total plastic dissipation power is minimum. Since ̄ 1 > ̄ 2 = ̄ 3 we deduce that the orientation in the shear band is = ± . The angle of the shear band which corresponds to the minimum of the plastic dissipation power , is = ± ∕4 and  = 2 ̄ 2 . In conclusion we expect that a stable stationary shear band has an angle of = ± ∕4 while the crystal orientation in the shear band is = ± ∕4 + or = ± ∕4 -.

Finally, let us mention that the above analysis tells us when the shear bands can be expected, but it does not guarantee the existence of them. Moreover, the Eulerian configuration considered here has an idealized geometry corresponding to the beginning of the shear band formation.

As we can see from the numerical simulations after large strains, these configurations are no more realistic, and the analysis became much more complicated.

Numerical approach

We recall from [START_REF] Cazacu | Augmented lagrangian method for eulerian modeling of viscoplastic crystals[END_REF] the principal features of the numerical scheme used in this paper. The time implicit (backward) Euler scheme for time discretization gives a set of nonlinear equations for the velocities and lattice orientation ( , ). At each time iteration, an iterative algorithm is developed to solve these nonlinear equations. Specifically, a mixed finite-element and Galerkin discontinuous strategy are proposed. The variational formulation for the velocity field is discretized using the finite element method, while a Galerkin discontinuous method with an upwind choice of the flux is adopted for solving the hyperbolic equations that describe the evolution of the lattice orientation. It is to be noted that in the case of the rigid-viscoplastic model, additional difficulties arise from the non-differentiability of the plastic terms. To overcome these difficulties, we have used a modified version of the iterative decomposition-coordination formulation coupled with the augmented Lagrangian method (introduced in [START_REF] Glowinski | Augmented Lagrangian and Operator Splitting method in Non-Linear Mechanics[END_REF]). The adopted visco-plastic model contains as a limit case the inviscid Schmid law. For low viscosities and moderate strain rates, the iterative decomposition coordination formulation coupled with the augmented Lagrangian method works very well, and no instabilities are present. If the computational Eulerian evolves in time (see the last section), an ALE (Arbitrary Eulerian-Lagrangian) description of the crystal evolution has to be implemented.

Numerical simulations

In this section, we analyze the in-plane compression of a micro-pillar. The micro-pillar is modeled as an FCC crystal with [110] axis of the crystal is along 3 (see Fig. 3(a) for a schematic representation and the end of section 2 for more details). The material coefficients considered correspond to tantalum (Ta) of density = 16, 650Kg/m 3 and of slip resistance for all systems equal to 0 = 66MPa (i.e. ̄ 1 0 = 114.31MPa, ̄ 2 0 = ̄ 3 0 = 76.23MPa) and no hardening is considered. In all the computations, the initial configuration of the micro-pillar has a rectangular section of × , with =1 m, =2 m. The rate of deformation is very low (10 -4 s -1 ) such that the loading could be considered as quasi-static and the time interval [0, ] was chosen such that the final height corresponds to a quarter of the initial height of the sample, i.e., a 25% reduction.

To capture the shear bands, we have used an adaptive mesh technique with respect to the strain rate norm | ( ̄ )|. That means that the regions where the slip rate is larger will have a fine mesh while outside the mesh is coarse. The ratio between the sizes of the fine and coarse mesh was 1∕8. Since we deal with an implicit numerical scheme, the chosen time step is large and corresponds to a deformation of 0.1% between two steps in time. That is why the computational cost is low, and simulations can be performed on a personal computer.

Firstly we investigate the compression of a "clean" homogeneous crystal, i.e., with a uniform yield limit and initial orientation = 0 . In Fig. 3, we show the final configurations of the pillars for three different initial orientations. We remark that the numerical scheme was able to reproduce the shear bands and the ductile rupture associated with it [START_REF] Zhang | Plate-like precipitate effects on plasticity of Al-Cu alloys at micrometer to sub-micrometer scales[END_REF]. Note that capturing a discontinuous phenomenon, such as rupture, is a challenging task for a continuous FE technique. We distinguish two types of deformation: (i) a thin shear band separating two rigid regions (as in (b) and (c)) and (ii) a more diffuse deformation configuration as in (d).

(i) As it is predicted by the simple model given in section 3, the shear band deformation mechanism is related to a sin- gle active slipping system while the other two are inactive.

To see that we have plotted in Fig. 4 the slip rate ̇̄ 2 and ̇̄ 3 for the pillar with the initial orientation 0 = ∕2 (corresponding to Fig. 3(b)) at four different levels of deformation. The slip rate ̇̄ 1 was not plotted because this system is completely inactive. We remark that during all the compression process, the slipping system = 2 is very active in the shear band, while the slipping system = 3, which was active at the beginning, is almost inactive at the end. From the orientation distribution, plotted in a color scale in Fig. 3, we see that at the end of the deformation process the crystal has two "rigid regions" where the pillar has the same orientation as before the compression, separated by a "shear band region" where the orientation is different. In both cases, we deal with a kink band. Let analyze now the orientation, of the shear band. In the case (b), the band is not exactly a straight line, and its overall angle varies in time between 45 and 55 , not far from = ∕4, predicted by the theoretical considerations of section 3. In the case (c), the shear band is a straight line perfectly orientated at ∕4.

(ii) In the third case, corresponding to an initial orientation of 0 = 0 , the deformation mechanism is quite different. In this case two slipping systems = 2 and = 3 are active while the system = 1 is inactive (i.e., ̇̄ 1 = 0). We have plotted in Fig. 5 two snapshots of the slip rate ̇ distribution, corresponding to directions = 2 and = 3 at four levels of compression. We remark that there exists a large (band) region where the deformation is not vanishing, between two small rigid regions. This deformation region is split into several regions where one of the slipping system = 2 or = 3 is active, but the two systems are not active in the same place, forming a "patchwork" of the distribution of the slipping systems. Each active slipping system is related to some specific shear bands, oriented at an angle around ∕4, but we do not deal with one single shear band, which accumulates all the deformation. These multiple shear bands give a diffuse overall deformation. Con- cerning the crystal orientation, as we can see in 3 (d) that the pillar seems to have three regions of orientations: two with the initial orientation 0 in the rigid regions and another one in the deformation region, which could be assimilated to a kink band. However, the orientation in this last region is not uniform, with multiple bands of initial orientation.

We now turn our focus into "non-clean" crystals with some pre-existent inhomogeneities. This case can be considered as a size-related effect since, generally speaking, with increasing size, it becomes harder to manufacture "clean" crystals [START_REF] Weiss | Ice: the paradigm of wild plasticity[END_REF]. However, it is also possible to engineer the disorder into the small crystals [START_REF] Zhang | Taming intermittent plasticity at small[END_REF]. In this work, we use some circular regions with a lower yield limit into the computational domain as a proxy for "non-clean" crystals, see Fig 6(a). For simplicity, the orientation of these seven defects was taken to be the same as in the crystal 0 . A more realistic approach would be, of course, to use dislocationdensity based models that we will consider in future work.

For the study of the effect of inhomogeneities on macroscopic deformation, we choose the crystal orientation 0 = ∕2 as in this case, we observed a strong localization of deformation, see Fig. 3(b). The final deformation state for the inhomogeneous case is shown in Fig. 6(b), where we observe that the kink band formation is suppressed in fa- vor of a more uniform deformation state. To elucidate this behavior, we show in Fig. 7, the evolution of the spatial distribution of deformation rates ̇ 2 (upper row) and ̇ 3 (lower row), where we observe that both are steadily active during all the deformation history. Note that we did not include ̇ 3 since it vanished everywhere almost all the time.

Conclusions

We studied the in-plane compression of micro-pillars using a "minimal" two-dimensional Eulerian plasticity approach. The model shows a very strong localization of deformation, and we found a qualitative agreement with experiments. From theoretical considerations, based on the rigid block decomposition method of the associated Eulerian limit load problem, we expect that the stationary kink bands are oriented at ∕4 form the compression axis. Numerical simulations on clean single crystal pillar, which partially confirm the theory, show that the compression process is not stable with respect to the initial orientation, and it is very difficult to predict the final shape of the pillar. However, the principal mechanism of deformation seems to be the kink shear band which separate two rigid blocks, associated with one single active slip system. If two possible slip systems are active, then the deformation is diffuse but localized in a large width shear band. Besides, our results show that strain-localization can be avoided when inhomogeneities are engineered inside the crystal, or the crystal orientation is altered. This is because of the activation of multiple slip systems, resulting in a "patchwork" of the distribution of the slip systems. The natural extension of our work will be to consider three-dimensional systems and incorporate dislocation-density based constitutive rules.

Figure 1 :

 1 Figure 1: Scanning electron microscopy images showing the deformed states pillars: (a) the localisation of deformation of an almost disorder-free Al crystal. Adding quenched disorder leads to a more uniform deformation as shown in (b) and (c).Taken from[START_REF] Zhang | Taming intermittent plasticity at small[END_REF].

1 2

 2 is the plane [ 110] -[001]. To describe the orientation of the crystal we denote by the angle counterclockwise between the 1 axis and [ 110] direction. The three active composite in-plane slip systems ̄ 1 , ̄ 2 , ̄ 3 will be specified by the angles 1 = , 2 = + , and 3 = -, with = arctan( √ 2), while the corresponding in-plane factors are 1 = 1∕ √ 3, 2 = 3 = √ 3∕2.

Figure 2 :

 2 Figure 2: A schematic representation of a shear band in a rigid-perfectly plastic model.

Figure 3 :

 3 Figure 3: (a): The initial geometry of a "clean" single crystal with a homogeneous initial crystal orientation 0 in the plane strain compression configuration. The final states of "clean" crystals with different initial orientation 0 : (b) 0 = ∕2, (c) 0 = 20 , (d) 0 = 0. The distribution of the orientation misfit sin(2( -0 )) is plotted in a color scale.

Figure 4 :

 4 Figure 4: The slip rates ̇̄ 2 (up) and ̇̄ 3 (bottom) distribution for a "clean" crystal with initial orientation 0 = ∕2 at four different levels of deformation.

Figure 5 :

 5 Figure 5: The slip rates ̇̄ 2 (up) and ̇̄ 3 (bottom) distribution for a "clean" crystal with initial orientation 0 = 0 at four different levels of deformation.

Figure 6 :

 6 Figure 6: Initial (a) and final (b) states of the crystal with inhomogeneities (shown in blue in (a)) in which plastic yield threshold is smaller. The distribution of the orientation misfit sin(2( -0 )) is plotted in a color scale.

Figure 7 :

 7 Figure 7: The slip rates ̇ 2 (up) and ̇ 3 (bottom) spatial distribution in a "non-clean" crystal at four different levels of deformation.

Acknowledgements. This work was partially supported by the grants of the French Research Agency ANR-17-CE24-0027 and ANR-19-CE08-0010-01.

scales. Acta Mater. 128, 351-364.


Emmanuelle Augeraud-Véron

Giorgio Fabbri

Katheline Schubert

VOLATILITY-REDUCING BIODIVERSITY CONSERVATION UNDER STRATEGIC INTERACTIONS

Keywords: Biodiversity, insurance value, land conversion, recursive preferences, stochastic dierential games, Q15, O13, O20, C73, D62

How can decentralized individual decisions ineciently reduce the ability of biodiversity to mitigate ecological and environmental variability and then its natural insurance role? In this article we present a simple theoretical set-up to address this question and to evaluate some policy options.

We study a model of strategic competition among farmers for the conversion of a natural forest to agricultural land. Unconverted forest land allows to conserve biodiversity, which contributes to reducing the volatility of agricultural production. Agents' utility is given in terms of a Kreps Porteus stochastic dierential utility capable of disentangling risk aversion and aversion to uctuations.

We characterize the land used by each farmer and her welfare at the Nash equilibrium, we evaluate the over-exploitation of the land and the agents' welfare loss compared to the socially optimal solution and we study the drivers of the ineciencies of the decentralized equilibrium.

After characterizing the value of biodiversity in the model, we use it to obtain a decomposition which helps to study the policy implications of the model by identifying in which cases the allocation of property rights is preferable to the introduction of a tax on land conversion. Our results suggest that enforcing property rights is more relevant in case of stagnant economies while taxing land conversion may be more suited for rapidly developing economies.

Introduction

According to FAO (2018) projections, agricultural production is expected to increase by 70% between 2005 and 2050 to feed a population that is expected to reach 9.1 billion people by 2050. Whereas part of this increase will come from higher agricultural yields, another part will inevitably require the extension of agricultural land (by 5% per year, FAO, 2018), at the expense of natural ecosystems. This land conversion phenomenon is, and will be in the future, one of the main drivers of biodiversity destruction (IPBES, 2019).

The evaluation of the phenomenon demands to appraise the trade-o between agricultural production and the damages caused by biodiversity loss. It is part of a broader reexion on the development of theoretical tools for identifying the economic value of biodiversity. Traditional approaches evaluate direct and indirect use values (see for instance Simpson et al., 1996, for the case of the pharmaceutical industry or Maille and Mendelsohn, 1993, for ecotourism) or, in a context of uncertainty, consider the option and quasi-option values of biodiversity (see e.g. [START_REF] Lasserre | Species preservation and biodiversity value: A real option approach[END_REF]Kassar, 2004, or Di Falco and[START_REF] Di Falco | Crop biodiversity, risk management and the implications of agricultural assistance[END_REF].

A more recent stream of literature introduced and exploited the notion of insurance value of biodiversity. The idea comes from the recognition that a series of services oered by biodiversity have an impact on the variability of ecological and then economic conditions. It is the case, for example, with regulating services (control of the local climate, ood control, regulation of soil fertility, pollination, etc.) or biocontrol services (pest control, resistance to plant invasion, disease control, etc.). In a series of contributions [START_REF] Baumgärtner | The insurance value of biodiversity in the provision of ecosystem services[END_REF], [START_REF] Quaas | Uncertainty and sustainability in the management of rangelands[END_REF], [START_REF] Quaas | Natural vs. nancial insurance in the management of public-good ecosystems[END_REF], [START_REF] Baumgärtner | Managing increasing environmental risks through agrobiodiversity and agrienvironmental policies[END_REF] and [START_REF] Baumgärtner | The economic insurance value of ecosystem resilience[END_REF] measured the insurance value as the reduction of the risk premium that can be achieved by relying on conservative ecosystem management. Augeraud-Véron et al. ( 2019) 1 analyzed the total value of biodiversity as the welfare gain that can be attributed to biodiversity conservation in a dynamic framework, distinguishing between the notion of risk and the notion of intertemporal substitutability. That study also highlighted the insurance value of biodiversity in a dynamic set-up.

1 See also [START_REF] Quaas | Insurance value of natural capital[END_REF] for a similar argument applied to a model with natural capital.

However, studying the optimal trade-o between agricultural production and biodiversity conservation is not enough. As various external eects are involved, the study of the decentralized solution is also necessary. Indeed, even if in some countries the number of farmers is decreasing, agriculture remains a highly distributed activity, for which farmers' productive choices act as an externality on the outcomes of others.

We study in this paper the case of a forest-rich developing economy needing to decide how much forest to be converted to agriculture and how much to be kept undeveloped, for biodiversity conservation. Among the many motives a country could have for conserving biodiversity we choose to investigate the natural insurance against uctuations of agricultural productivity motive. The stabilizing role of biodiversity has been largely documented and the ecological mechanisms behind this phenomenon are well understood (see for instance [START_REF] Loreau | Biodiversity and ecosystem stability: A synthesis of underlying mechanisms[END_REF]. The role of common pool resources such as natural forests as an insurance device for rural poor is also well documented [START_REF] Baland | Commons as insurance and the welfare impact of privatization[END_REF]Francois, 2005, Delacote, 2009), in a context where they do not have access to formal insurance and to credit markets [START_REF] Banerjee | Giving credit where it is due[END_REF].

In this spirit we develop and study a decentralized model where two farmers 2 can obtain property rights on an undeveloped land, for instance a natural forest. Each farmer chooses the share of her land she converts to agriculture, then produces and consumes the harvest from her plot. The evolution of agricultural productivity over time is inuenced by the endogenous choices about the size of the plots. Indeed, the part of land which is not used for farming is devoted to biodiversity conservation and the services provided by biodiversity reduce the volatility of agricultural productivity. The two farmers/players coordinate on a Nash equilibrium of the economy. [START_REF] Noack | Droughts, Biodiversity, and Rural Incomes in the Tropics[END_REF] provides a nice illustration of the relevance of our model.

The authors use panel data covering 7,556 households in 23 developing countries, combined with data on droughts, the major environmental reason for crop losses, data on natural biodiversity and data on the timing of the agricultural cycle.

The intrinsic volatility of agricultural yields comes from weather events, here droughts, whereas natural biodiversity has a stabilizing role. One of Noack et al.

(2019)'s results is that the negative impact of droughts on crop income declines 2

The results for the n-farmer case are also provided, see Section 6.

when natural biodiversity increases. They are able to estimate the magnitude of the eect and obtain that reducing biodiversity by one standard deviation below the regional mean almost doubles the negative impact of droughts on income.

In this context the attitudes of farmer towards income intertemporal uctuations and towards risk are essential. In order to disentangle aversion to uctuations and aversion to risk we use a recursive utility set-up, initially introduced in the seminal works of [START_REF] Epstein | Substitution, risk aversion, and the temporal behavior of consumption and asset returns: an empirical analysis[END_REF] and of Due and Epstein (1992). The relevance of recursive utilities in natural resources and environmental economic models is well documented (see for instance [START_REF] Knapp | Dynamic resource management: Intertemporal substitution and risk aversion[END_REF][START_REF] Epaulard | Optimally eating a stochastic cake: a recursive utility approach[END_REF][START_REF] Howitt | Estimating intertemporal preferences for natural resource allocation[END_REF] but this is, as far as we know, the rst work where a dynamic environmental economic model with strategic interaction and EpsteinZin preferences is studied and solved.

The model is completely solved by explicitly nding the Nash equilibrium of the dierential game and characterizing for each farmer the share of land used for agricultural activities. These shares are constant over time. For any farmer, impatience drives land conversion, whereas risk and its perception play in the opposite direction. What the other farmer does also plays a role: a farmer will convert more land to farming when the other farmer does so if her aversion to uctuations is smaller than unity, and she will convert less land when the other farmer converts more if her aversion to uctuations is high.

We then compare, in the symmetric situation with uniform preference and technology parameters, the equilibrium in the strategic competition set-up and the socially optimal choice. As expected, since the agents only partly internalize the negative eects of their actions, an over-exploitation of the forest systematically arises, so that the land available for biodiversity conservation is always lower in the decentralized case than at the social optimum.

We proceed with welfare comparisons and some policy implications of our results. We show how to decompose the per-capita welfare gain from biodiversity conservation in two parts: the welfare gain of moving from the open access equilibrium to the Nash equilibrium, and the welfare gain of moving from the Nash equilibrium to the optimum. Depending on which of the two is prevalent we can identify in which cases the allocation of property rights is preferable to the introduction of a land conversion tax. In particular we show that enforcing property rights is more relevant in case of stagnant technological progress in agricultural activities while policies consisting in pricing the volatility externality (for instance through a tax on land conversion) may be more suited for rapidly developing economies averse to uctuations.

Finally we extend our result to the more general case of more than 2 farmers. We show that the more farmers there are the larger land conversion and the bigger the volatility externality. Moreover, if the aversion to uctuations is less than unity, when the number of farmers increases every individual share of land devoted to farming increases as well. In this sense, a small aversion to uctuations is detrimental to biodiversity conservation. Conversely, if farmers have a large aversion to uctuations, when the number of farmers increases each of them converts less land. The pressure of an increasing population on a xed land area may then either cause a race to land conversion detrimental to biodiversity conservation, or a reduction of each farmer's plot mitigating biodiversity destruction, depending on their aversion to uctuations.

The paper proceeds as follows. In Section 2 we present the main elements of the model which is completely solved in Section 3. Section 4 is devoted to the comparison between the two-players setting and the social optimum benchmark while in Section 5 we discuss the policy hints that we get from the model. In Section 6 we look at the results for the n-player extension of the model and in

Section 7 we conclude. The appendices contain technical details and the proofs of all the results.

The model

We develop and study a decentralized version of the model proposed by [START_REF] Augeraud-Véron | The value of biodiversity as an insurance device[END_REF].

We consider an agricultural economy with a stock L = 1 of land, initially undeveloped for instance a natural forest. This forest shelters biodiversity, which is a common pool resource. Several farmers contemplate converting the forest to agricultural land. Each one can obtain from the state property rights on a plot, and then has to decide how much of his plot to convert to farming

3

. The forest left intact supports biodiversity and provides ecosystem services.

As explained in Augeraud-Véron et al. ( 2019), there is a close link between biodiversity and space, synthesized in the classical species-area curve, initially introduced by [START_REF] Arrhenius | Species and Area[END_REF] and [START_REF] Gleason | On the relation between species and area[END_REF] and well anchored in the ecological literature. According to the species-area curve, the long term species richness S(t) is constrained by the size of the natural habitat 1 -f (t). The relationship is non-linear and specic to any ecological and geographical context. A widespread empirically-based specication is the power function used fo instance by [START_REF] Brook | Catastrophic extinctions follow deforestation in Singapore[END_REF]

S(t) = 1 (1 -f (t)) 2 with 1 > 0 and 0 < 2 < 1
Then, the evolution of biodiversity B(t), dened as the variety of ecological elements present in the habitat, including genes, species, functional traits, communities etc., can be modeled as:

η Ḃ(t) = -(B(t) -S(t)) with η ≥ 0,
where η the relaxation time of biodiversity (see [START_REF] Lafuite | Time-delayed biodiversity feedbacks and the sustainability of social-ecological systems[END_REF]. When this relaxation time is negligible, biodiversity takes instantaneously its long term value, that is B(t) = S(t), meaning that it responds instantaneously to habitat conversion.

We describe here a 2-player setting: 4 we suppose that 2 players/farmers can appropriate the forest for farming purposes. They are indexed by i ∈ {1, 2}.

3 We will contrast in Subsection 5.2 this institutional setting where farmers can obtain prop- erty rights on the forest to the case where the forest remains in open access, meaning that there is no incentive for the farmers to refrain from converting all of it to farming. 4 The n-player extension is discussed in Section 6.

Farmer i can appropriate some share f i (t) of total land, although the following constraint needs to be veried:

(1)

f 1 (t) + f 2 (t) ≤ 1, t ≥ 0.
We denote by f (t) := f 1 (t) + f 2 (t) ∈ [0, 1] and 1 -f (t) ∈ [0, 1] the shares of land respectively used for farming and left undeveloped, that is devoted to biodiversity conservation. We assume that the agricultural production output of farmer i at time t is given by:

(2)

Y i (t) = f i (t)A i (t)
where A i (t) is the productivity 5 of a unit of land farmed by agent i at time t, whose dynamics is described by the following stochastic dierential equation:

(3)

dA i (t) = α i A i (t) dt + σ i f (t) 1/2 A i (t) dW (t) A i (0) = A 0 i
where W is a real standard Brownian motion W : [0, +∞) × Ω → R, adapted to some ltration F t , dened on a complete probability space (Ω, F, P). In such an expression, α i ∈ R represents some xed and exogenous rate of technological progress in farming activities (it can be equal to 0), and the term σ i f (t) 1/2 measures the volatility of agricultural productivity. Total volatility has an exogenous component σ i , stemming for instance from weather events, and an endogenous component f (t) 1/2 , common to both farmers, increasing in the total share of land converted to agriculture, that is decreasing in the biodiversity level.

6 It is in this sense that biodiversity appears in the model as insurance against adverse outcomes.

The two farmers play a non-cooperative game with Markovian strategies so that farmer i chooses the size of land to cultivate in the form

f i = Ψ i (A 1 , A 2 )
5 An intense debate has developed, following the [START_REF] Green | Farming and the fate of wild nature[END_REF] contribution, on the im- portance of preserving biodiversity also in the context of land used for agriculture, in particular in relation to dierent farming practices (extensive vs intensive land use). The discussion about land sparing vs land sharing remains alive (see for instance Fisher et al., 2014, Kremen, 2015 or Renwick and Schellhorn, 2016). In the simplied context of the model presented here we abstract from this issue and we consider only an average productivity whose growth is exogenous and independent of the level of biodiversity.

6 [START_REF] Augeraud-Véron | The value of biodiversity as an insurance device[END_REF] write more generally this endogenous component as: f (t) χ/2 , χ ≥ 1. We restrict ourselves to the case χ = 1 for simplicity, without loss of generality.

where

Ψ i : R 2 + → [0, 1] is a measurable function. A couple of Markov strategies Ψ := (Ψ 1 , Ψ 2 ) is admissible if (1) is veried for all positive couple of real numbers (A 1 , A 2 ), i.e. if Ψ 1 (A 1 , A 2 ) + Ψ 2 (A 1 , A 2 ) ≤ 1, for all (A 1 , A 2 ) ∈ R 2 +
and if the two equations (3) have a unique solution. 7 We denote by C the set of admissible strategy couples:

C := Ψ := (Ψ 1 , Ψ 2 ) : R 2 + → [0, 1] 2 measurable : Ψ is an admissible strategy couple
Given a strategy Ψ 1 of Player 1 (respectively Ψ 2 of Player 2), we denote by

C 2 (Ψ 1 ) (respectively C 1 (Ψ 2 )) the set C 2 (Ψ 1 ) := {Φ : R + → [0, 1] measurable : (Ψ 1 , Φ) ∈ C} (and similarly for C 1 (Ψ 2 )).
We suppose that at each time t ≥ 0 all the production is consumed:

(4) as the expected value of utility on each trajectory. We denote by θ i > 0 (and θ i = 1) the (constant) relative risk aversion of Player i and by φ i > 0 her aversion to uctuations i.e. the inverse of her intertemporal elasticity of substitution. The case θ i = φ i corresponds to the usual time additive case. The discount rate of Player i is denoted by ρ i > 0.

C i (t) = Y i (t).
The standard additive case is sometime called separable case since the utility of agents can be written as the integral over time of the discounted instantaneous expected utility (so the time and the probability are somehow separated in two dierent integrals). The same thing cannot be done in context of recursive utility 7 This second requirement is not very strong: the term f is bounded (by 0 and 1) so the system of equations for A has always linear growth, so that, by standard existence and uniqueness results for the solution of a system of SDE (see Chapter V of [START_REF] Protter | Stochastic Integration and Dierential Equations, Second Edition[END_REF], we only need to have weak Lipschitz-type continuity in the diusion term (w.r.t. of A i ) and some mild regularity of Ψ to guarantee the existence and the uniqueness of the solution.

(as, for instance in the case of Epstein-Zin-Weil preferences that we use here).

In this case the utility is expressed through the evolution of a suitable stochastic dierential equation as more formally explained at the beginning of Appendix A.

The interested reader can look at the foundational works by Due andEpstein (1992a, 1992b) and [START_REF] Due | PDE solutions of stochastic dierential utility[END_REF] and at the more recent papers by [START_REF] Lazrak | A generalized stochastic dierential utility[END_REF] or [START_REF] Kraft | Stochastic dierential utility as the continuous-time limit of recursive utility[END_REF].

In this framework the problem focuses on the study of a pair of Hamilton-Jacobi-Bellman (HJB) equations of the following form:

ρ i 1 -θ i 1 -φ i V i (A 1 , A 2 ) = max f i ∈[0,1-f j ] (f i A i ) 1-φ i 1 -φ i 1 ((1 -θ i )V i (A 1 , A 2 )) 1-φ i 1-θ i -1 + ∂V i ∂A 1 A 1 α 1 + ∂V i ∂A 2 A 2 α 2 + 1 2 (f 1 + f 2 ) σ 1 σ 2 , HV i σ 1 σ 2 where V i (A 1 , A 2 )
is the value function of Player i, HV i the Hessian matrix of V i and •, • the scalar product on R 2 (we did not write explicitely the dependence

of ∂V i ∂A 1
and HV i on (A 1 , A 2 ) to lighten the notation).

In this setting we implicitly make the assumption that the choice of developing land made by a farmer at any date is completely reversible. However, given the particular structure of the model, introducing an irreversibility constraint (i.e., in formal term, admitting only weakly increasing paths of the shares of land used for farming) would not signicantly change the situation. Indeed the best strategies we identify are constant over time, so they remain so if we restrict the set of possible choices.

More technical details on the formulation we use can be found in Appendix A.

The study of the HJB equations is carried out in Appendix B.

The characterization of the strategic interaction equilibrium

In this section we solve the model and get an explicit description of the Nash equilibrium. Let's start by dening the quantity (5)

x i = ρ i -α i (1 -φ i ) θ i σ 2 i 2 , for i = 1, 2.
that will be frequently used below

8

. The strategies of the two players to balance are described in the following theorem.

Theorem 3.1. Suppose that ( 6)

ρ i > (1 -φ i )α i , for i = 1, 2
(it implies in particular x 1 and x 2 positive), ( 7)

φ 1 + φ 2 > 1
and that the following transversality condition is veried:

(8)

ρ i > (1 -θ i ) α i -θ i σ 2 i 2 x 1 + x 2 φ 1 + φ 2 -1 , for i = 1, 2.
Then, if

f1 := φ 2 x 1 + (1 -φ 1 )x 2 φ 1 + φ 2 -1 ∈ (0, 1) (9)
and ( 10)

f2 := φ 1 x 2 + (1 -φ 2 )x 1 φ 1 + φ 2 -1 ∈ (0, 1 -f1 ),
the couple

Ψ 2 (A 1 , A 2 ) ≡ f2 , Ψ 1 (A 1 , A 2 ) ≡ f1
is a Nash equilibrium of Markov strategies. Moreover it is the unique Nash equilibrium of Markov strategies where strategies are constant and deterministic.

Proof. See Appendix B.

The equilibrium described in Theorem 3.1 is a true Markov perfect Nash equilibrium in the sense that, given the described (constant) strategy Φ 1 of Player 1, the described strategy Ψ 2 is the unique best response of Player 2 among the whole set of (constant and non-constant) admissible Markov strategies (and viceversa exchanging the role of Player 1 and Player 2).

This equilibrium is unique among equilibria where at least one the the players plays a constant deterministic strategy but it can fail to be unique in the set of all admissible couples strategies. In other words there could exist couples of Markov strategy (non-constant and deterministic for both players) which are equilibria.

The problem of course is that agents could coordinate on equilibria which are 8 Quantity x i is only a notation to ease the reading and has no intrinsic interpretation. dierent from the one described. A sort of meta-analytical justication can be used to defend the selection of the described equilibrium precisely because of its simplicity: an Occam's razor argument suggests indeed that the players could more easily coordinate on the simplest and more easily computable equilibrium.

Using the expressions of the land used by each player at the equilibrium we can easily determine the total amount of land converted for agricultural purposes and therefore, in a complementary way, the land available to preserve biodiversity.

The result is contained in the corollary below.

Corollary 3.2. Suppose that hypotheses of Theorem 3.1 are satised. Then the total land area devoted to farming is ( 11)

f := f1 + f2 = x 1 + x 2 φ 1 + φ 2 -1 .
To complete the analysis of the system at the equilibrium we can nally calculate the utility of each of the players:

Proposition 3.3. Suppose that the hypotheses of Theorem 3.1 are veried. Then, along the described equilibrium, the welfare of the two players is

(12) V i (A 1 , A 2 ) = β i 1 -θ i A 1-θ i i with ( 13
)
β i = θ i σ 2 i 2 f φ i i - 1-θ i 1-φ i
where the expression of x i and fi are given respectively in (5) and in ( 9)-(10).

Since V 1 is only a function of A 1 , we will simply write V 1 (A 1 ) instead of V 1 (A 1 , A 2 ). Similarly for V 2 (A 2 ).
Proof. It is a corollary of the proof of Theorem 3.1. See Appendix B.

A rst study of the results of the model is possible using the comparative analysis contained in the following proposition. Proposition 3.4. Suppose that the hypotheses of Theorem 3.1 are satised. Then we have the following dependencies of fi on the parameters characterizing Player

i: ∂ fi ∂ρ i > 0, ∂ fi ∂σ i < 0, ∂ fi ∂θ i < 0,
and on the parameters characterizing Player j:

       ∂ fi ∂ρ j > 0, ∂ fi ∂σ j < 0, ∂ fi ∂θ j < 0, if φ i < 1 ∂ fi ∂ρ j = 0, ∂ fi ∂σ j = 0, ∂ fi ∂θ j = 0, if φ i = 1 ∂ fi ∂ρ j < 0, ∂ fi ∂σ j > 0, ∂ fi ∂θ j > 0, otherwise Proof. See Appendix B.
The trade-o agent i faces when she chooses the amount of land f i she converts to agriculture is indeed between consuming more today (higher f i ) and having a smaller uncertainty tomorrow (higher 1 -f i ). Therefore the fact that ∂ fi ∂ρ i > 0 is rather intuitive: the higher is the subjective discount rate, the lower the importance of the future and then the higher the relative weight the player assigns to present consumption and the higher the f i she will choose. The eects of θ i and σ i are pure risk-aversion eects: the higher σ i (respectively θ i ) the higher the future volatility (respectively its perception) and then the higher the eort of the agent to reduce it.

To understand the eects of preference parameters of Player j on the value of fi it is instructive to have a look at the best response of Player i to a generic choice fj of Player j. It reads (see equations ( 46) and (48) in Appendix B):

fi = x i + (1 -φ i ) fj φ i Then: d fi d fj      > 0 if φ i < 1 = 0 if φ i = 1 < 0 otherwise
If Player i's aversion to intertemporal uctuations φ i is smaller than 1, her best response is to increase her share of land devoted to farming when Player j increases her own. In this case, there is strategic complementarity. It is the reverse if φ i is larger than 1: Player i, who is now strongly averse to intertemporal uctuations, will decrease her share of land devoted to farming when Player j increases her own. Now the decisions of the two players are strategic substitutes.

The total amount of land, that has to be divided between three uses: farming by Player 1, farming by Player 2, and biodiversity conservation, is xed. When Player j chooses to convert more forest to farming, Player i, expecting more volatility of agricultural productivity in the future, may either increase f i to increase its average consumption at each date, or, on the contrary, decrease f i to counteract the volatility increase due to the behavior of Player j. The rst eect is stronger than the second one when Player i is not very averse to intertemporal uctuations, and vice-versa. In case of strategic complementarity, the discount rate and the risk parameters of Player j aect the behavior of Player i in the same way as her own do. It is the reverse in case of strategic substitutability.

When φ i = 1 the two eects compensate exactly, and the decision of Player i is independent of the choice made by Player j.

The symmetric case and the comparison with the optimal solution

To compare now the results with the optimal solution presented by Augeraud-Véron et al. ( 2019) we consider the case where parameters (α, σ, ρ, θ and φ) do not depend on i. We also suppose that the initial value of A is the same for both farmers. More precisely we compare the decentralized solution at the symmetric equilibrium with the optimal solution chosen by a planner which gives in the social welfare function the same weight to each farmer. We start by re-stating the result of Section 3 in this circumstances. In the next two subsections we will compare the results with the social planner benchmark in two perspectives: the biodiversity conservation and the welfare of the agents.

Dene

x = ρ -α(1 -φ) θ σ 2 2 .
The results of Theorem 3.1 and Proposition 3.3 read as follows.

Corollary 4.1. Suppose that

(14) ρ > (1 -φ)α
(it implies in particular that x is positive),

φ > 1/2

and that the following transversality condition is veried:

(16)

ρ (1 + 2(φ -θ)) > (1 -θ)α.
Suppose that ( 17)

fh := x 2φ -1 ∈ 0, 1 2 .
Then the couple

Ψ 2 (A 1 , A 2 ) ≡ fh , Ψ 1 (A 1 , A 2 ) ≡ fh
is a Nash equilibrium of Markov strategies. Moreover it is the unique Nash equilibrium of Markov strategies where strategies are constant and deterministic. The total land used for farming is then

(18) f := 2 fh = 2x 2φ -1 ≤ 1.
The corresponding welfare of each farmer is

(19) V h (A) = β h 1 -θ A 1-θ with (20) β h = θ σ 2 2 2 -φ f φ -1-θ 1-φ
.

The expression of f given in ( 18) is particularly transparent. One can rst easily verify that f is always a decreasing function of θ. This is the same risk hedging reaction we have seen in Proposition 3.4 for the dependence of the values of f1 and f2 on θ 1 and θ 2 . On the other hand we can observe that f is a decreasing function of φ if and only if α < 2ρ. This condition relates to the comparison of the magnitudes of the growth and discount rates. When the trend of agricultural productivity is smaller than twice the discount rate, farmers' future prospects are not good. Then the more averse to uctuations they are the more willing they are to insure against bad outcomes in the future by conserving more biodiversity.

The opposite occurs when α > 2ρ.

Notice that when α > 2ρ, the two parameters characterizing preferences play in opposite directions. This is a strong argument in favor of disentangling the two parameters. See the discussion in Subsection 4.2.

4.1. The eect of strategic competition on land use and biodiversity.

We recall that the part f * of land used for farming in the social planner model of Augeraud-Véron et al. ( 2019) is equal to (21)

f * = x φ .
The restrictions on the parameters which are needed to obtain this result are (14), a transversality condition which reads:

(22) ρ (1 + (φ -θ)) > (1 -θ)α,
and the condition x φ ∈ [0, 1]. So, all in all, the conditions that allow to work in the intersection of the hypotheses needed for the current results and for the social planner case are the following: ( 14), ( 15), and ( 23)

ρ (1 + min [φ -θ, 2(φ -θ)]) > (1 -θ)α.
When they are satised, ( 24)

f * = x φ if x < φ, 1 otherwise f = x φ-1 2 if x < φ -1 2 , 1 otherwise.
The optimal share of land used for farming increases with the discount rate ρ, decreases when intrinsic volatility σ increases, decreases when risk aversion θ increases, decreases when aversion to uctuations φ increases if α < ρ, and vice versa. Notice that, for ρ < α < 2ρ, f * increases with φ whereas f decreases with φ.

Dene Λ = f f * the ratio of total land converted to farming at the Nash equi- librium and at the optimum. We have:

(25) Λ =      φ φ-1 2 , if x < φ -1 2 φ x , if φ -1 2 ≤ x < φ 1, if φ ≥ x.
Proposition 4.2. Suppose that ( 14), ( 15) and ( 23) are satised. Then Λ ≥ 1 so that the land used for biodiversity conservation is weakly smaller in the strategic competition situation than that chosen by the social planner. Λ is strictly greater than 1 as soon as the optimal land devoted to farming is strictly smaller than 1.

Proof. Direct inspection of formulas in (25). Proposition 4.2 shows that a tragedy of the commons mechanism is at work in the model: the exploitation of the natural resource is larger in the strategic competition set-up than at the social optimum and then the land used for biodiversity conservation is smaller. This is what we expected since the two players do not completely internalize the volatility externality: the conversion decisions of each farmer aect the volatility of agricultural productivity and then the utility of both. In the following proposition we try to dig a little deeper into this result. Proposition 4.3. Suppose that ( 14), ( 15) and ( 23) are satised and that x ≤ φ -1 2 , so that the solution is interior at the optimum and at the Nash equilibrium. Then Λ does not depend on the risk aversion θ and is decreasing in the aversion to uctuations φ.

Proof. Direct inspection of the rst formula in (25).

The previous proposition identies the parameters that play a role in the gap between the optimal choice of land conversion and the outcome of a strategic competition for land appropriation between two identical farmers. The result is sharp: even if the risk aversion has a role in establishing the share of land devoted to production it has no impact on the gap between the optimal and the decentralized solutions. Conversely the elasticity of intertemporal substitution parameter does play a role: the smaller it is the higher is Λ, that is the further is the Nash solution from the optimal one. 4.2. A technical excursus: comments on the role of the Epstein-Zin-Weil utility. In this paper we have chosen to use the Epstein-Zin-Weil specication of preferences. Of course as a general statement there are several good reasons to disentangle the eect of risk aversion and intertemporal elasticity of substitution: they are related to dierent characteristics of the individuals, one concerning intra-temporal preferences, the other (as the name suggests) intertemporal behavior.

At the level of our model the relevance of this distinction is immediately clear once we look at the realistic possible values of the parameters (see for instance [START_REF] Barro | Environmental protection, rare disasters and discount rates[END_REF]. Plausible calibrations of the inverse of the intertemporal elasticity of substitution φ are smaller than 1 while the relative risk aversion parameter θ is typically largely higher than 1 (3 and more). This fact, besides pointing out once again the unlikelihood of the expected utility case (ϕ = θ = φ), has qualitative implications in the model.

Disentangling the eect of risk aversion and intertemporal elasticity of substitution often gives the possibility of more deeply understanding some behaviors of the model. To clearly make the point consider for instance the value of the variable x which appears several times in the paper. Its expression is

x = ρ -α(1 -φ) θ σ 2 2
while its expected utility counterpart is given by

x EU = ρ -α(1 -ϕ) ϕ σ 2 2
where ϕ is the unique parameter appearing in the expected utility case. One can readily see that the derivative ∂x EU ∂ϕ does not have a univocal sign while the two derivatives ∂x ∂θ and ∂x ∂φ are respectively always negative and positive (under assumption ( 6)) so that introducing the two-parameters Epstein-Zin-Weil specication allows to resolve the ambiguity that we observe in the expected utility case. This explanatory eect of the Epstein-Zin-Weil specication on x brings similar results to more interesting variables. If we look for instance at the expected utility counterpart of (18) (we could comment in a similar way ( 9) and ( 10)), it reads as

fEU = 2x EU 2ϕ -1 = 2 ρ-α(1-ϕ) ϕ σ 2 2 2ϕ -1 .
The dependence of this expression on ϕ relies on various relative magnitudes of the parameters and they are not easy to justify, while the signs of the two derivatives ∂x ∂θ and ∂x ∂φ is clear as we stressed in the discussion below Corollary 4.1: f is always a decreasing function of θ and it is an increasing function of φ if and only if α > 2ρ. As discussed above these results have a neat economic interpretation.

Observe that in particular, as the two variables indicate dierent notions and characteristics, their eects on the agent's behavior can be very dierent. We have already mentioned above that, as far as α > 2ρ the role of θ and φ in establishing the size of the land devoted to farming (and then the size of the land devoted to biodiversity conservation) at the equilibrium f is opposite. This is also true for the size of the land devoted to farming at the social optimum f * .

5. The effect of strategic competition on welfare and policy implications of the model 5.1. The per-capita welfare gain of moving from the Nash equilibrium to the optimum. We restrict ourselves to the most interesting case where parameters are such that the solution at the Nash equilibrium is interior ( f < 1),

which guarantees that the optimal solution is interior as well.

The welfare of each farmer at the Nash equilibrium is given in Proposition 3.3.

At the optimum, the planner sets the level of f at f * and then she gives half of total production to each farmer 9

. Since the production is linear in productivity, the welfare of each farmer is equal to the total welfare V p computed in Augeraud-Véron et al. ( 2019), calculated for a productivity level equal to A/2, where (26)

V p (A) = 1 1 -θ β p A 1-θ with (27) β p = θ σ 2 2 (f * ) φ -1-θ 1-φ
.

We dene the welfare gain of moving from the Nash equilibrium to the optimum as the permanent percentage increase in consumption that has to be given to each farmer in order to increase her welfare up to the optimal level. This welfare gain is denoted ω and dened by:

(28)

V h ((1 + ω)A) = V p A 2 .
Using equations ( 26), ( 27), ( 19) and ( 20) we obtain:

(29) ω = 1 2 φ φ -1 2 φ 1 1-φ -1
and can prove the following proposition: Proposition 5.1. Suppose that hypotheses of Corollary 4.1 are veried. ω is always positive, does not depend on risk aversion and is a decreasing function of aversion to uctuations.

Proof. See Appendix B. 5.2. Decomposition of the total value of biodiversity and policy implications of the model. In reference to Lucas' welfare cost of uctuations [START_REF] Lucas | Models of business cycles[END_REF], the total value of biodiversity can be dened as the welfare gain from conserving biodiversity at its optimal value compared to no conservation at all. More precisely, is the percentage increase in consumption society is willing to accept at all dates to give up the optimal level of biodiversity in favor of no biodiversity at all (see [START_REF] Augeraud-Véron | The value of biodiversity as an insurance device[END_REF].

The case where the whole forest is converted to farming and no biodiversity is conserved can be interpreted as the consequence of a forest in open access, where property rights are absent or poorly dened or enforced. Then each farmer appropriates 1/2 of total land and all land is converted to farming. Following Augeraud-Véron et al. ( 2019), the utility in this case is 10 :

(30) V oa (A) = 1 1 -θ β oa A 1-θ
with, when the restriction on parameters

(31) ρ > (1 -φ) α -θ σ 2 2 is satised, ( 32
)
β oa = ρ -(1 -φ) α -θ σ 2 2 -1-θ 1-φ
.

The per capita welfare gain from biodiversity conservation is then λ dened by: (33)

V oa (1 + λ) A 2 = V p A 2 .
Using equations ( 26), ( 27), ( 30) and ( 32) we obtain:

(34)

λ = φf * + (1 -φ) f * φ 1 1-φ -1.
10 Indeed the utility is computed in Augeraud-Véron et al. ( 2019) using a sort of trick that can be used only because we consider a constant control. They observe that the welfare in the situation where we impose that all the land is used for farming is indeed the value function of a dierent (auxiliary) optimization problem where the agents can only choose to take f 1 = f 2 = 1/2 so we can compute the utility in the situation where we impose f = 1/2 just computing the value function of the problem where the agents, instead of maximizing on the set [0, 1] maximize on the set (singleton) {1/2} and so they are obliged to choose f 1 = f 2 = 1/2. λ decreases when the discount rate ρ increases, increases (resp. decreases) when the trend of agricultural productivity increases if φ < 1 (resp. > 1), increases with the intrinsic volatility σ, increases with the risk aversion θ and is an ambiguous function of the aversion to uctuations φ. Augeraud-Véron et al. (2019) show that when α < ρ, λ unambiguously increases with φ. In this case, a higher aversion to uctuations translates into more biodiversity conservation and a higher value of biodiversity. In the opposite case where α > ρ, the eect of φ on λ is ambiguous. Then, increasing aversion to uctuations decreases the share of land devoted to biodiversity conservation and may at the same time increase the total value of biodiversity.

The total insurance value of biodiversity λ can be decomposed in two parts: the welfare gain of moving from the open access equilibrium to the Nash equilibrium, denoted λ, and the welfare gain of moving from the Nash equilibrium to the optimum, ω. Assessing the relative magnitudes of these two components is important for policy. Indeed, if the insurance value of biodiversity is mostly due to the departure from the open access situation, then the appropriate policy consists in dividing total land equally among farmers and giving them secure property rights on their plot; if on the contrary it is mostly due to the lack of internalization of the volatility externality, then the appropriate policy consists in pricing the volatility externality, for instance through a tax on land conversion.

Similarly to the denition of λ, λ is dened by: (35)

V oa (1 + λ) A 2 = V h A 2 .
It is easy to see that the decomposition reads:

(36)

1 + λ = (1 + ω)(1 + λ).
5.3. Illustrative simulations. To get more insights on the relative magnitude of the three welfare gains identied above we perform illustrative simulations using the following values of the parameters: θ = 3, ρ = 0.04 and σ = 0.25, in three congurations: α = 0.035 < ρ, ρ < α = 0.05 < 2ρ, 2ρ < α = 0.085.

The value chosen for θ is empirically relevant (see for instance [START_REF] Barro | Environmental protection, rare disasters and discount rates[END_REF]. The value of ρ is in the standard range considered in dynamic models and it is for instance consistent with the data from [START_REF] Lopez | The social discount rate: estimates for nine Latin American countries[END_REF]. The range of values of α is consistent with the growth rates observed in developing countries (see e.g.

World Bank Group, 2018). We have chosen on purpose a very high volatility of agricultural productivity, implying a variance σ 2 of 6.25 %, in the case where all land is converted to farming, to reect rstly the fact that agricultural output is more volatile than global output, and second that without the dampening eect of biodiversity this volatility will become very high. Finally we suppose that aversion to uctuation φ is smaller than risk aversion θ. All in all, our calibration is related to the case of biodiversity-rich developing countries where agricultural productivity is very volatile, deforestation and land conversion are an issue, and farmers have a preference for the early resolution of uncertainty [START_REF] Gollier | The Economics of Risk and Time[END_REF], in the sense that they are more risk averse than they are concerned about smoothing consumption.

11 (lower panel) as functions of aversion to uctuations φ, in the three congurations identied as relevant for the relative magnitudes of α and ρ. The range of φ is such that the restrictions necessary to obtain interior solutions and the transversality conditions are satised, and there is preference for early resolution of uncertainty. In the lower panel the restriction on parameters (31) is added to ensure the existence of the welfare gains.

α < ρ ρ < α < 2ρ 2ρ < α 1.8 2.0 2.2 2.4 2.6 2.8 3.0 ϕ 0.1 0.2 0.3 0.4 0.5 0.6 f ,f * 1.5 2.0 2.5 3.0 ϕ 0.2 0.4 0.6 f ,f * 1.0 1.5 2.0 2.5 3
In the rst and second cases depicted on Figure 1, the trend of agricultural productivity is smaller than twice the discount rate. Then the welfare gain of conserving the optimal level of biodiversity is increasing in the aversion to uctuations, whereas the welfare gain from moving from the Nash equilibrium to the optimum is decreasing in the aversion to uctuations. Then as φ increases the welfare gain of conserving the optimal level of biodiversity becomes mostly explained by the welfare gain of moving from the open access situation to the Nash equilibrium. These results suggest that the most relevant policy option in poorly productive agricultural economies is to put in place land property rights to move away from open access.

In the third case, where the agricultural productivity trend is very high compared to the discount rate, the eects may be very dierent since the welfare gain of conserving the optimal level of biodiversity is decreasing in the aversion to uctuations. We exhibit in the simulation a case where this welfare gain is mostly explained by the welfare gain of moving from the open access situation to the Nash equilibrium when φ is small, then more and more explained by the welfare gain of moving from the open access situation to the Nash equilibrium as φ increases.It suggests that in highly productive economies averse to uctuations the best policy option may be to tax land conversion to internalize the volatility externality.

Extension to the n-player case

The results obtained for the two-player case can be extended to the n-player case. In this section we briey sketch the results in the general case.

As proved in Appendix C the best response function of Player i to the decisions of the other players is:

(37) fi = 1 φ i x i + (1 -φ i ) j =i fj .
As in the two-player case, there is strategic complementarity if φ i < 1, strategic substitutability if φ i > 1, and the decision of Player i is independent of the choices of the other players when φ i = 1. Moreover, the interior Nash equilibrium is characterized by the following set of strategies:

(38) fi = min           x i    n j=1 j =i φ j -(n -2)    + (1 -φ i ) n j=1 j =i x j n j=1 φ j -(n -1) , 1 - n j=1 j =i f j          
.

When preferences parameters are the same for all players and φ > 1 -1 n , according to (38), the interior optimal quantity of land devoted to farming for each farmer in the n-player game is given by fh

= x nφ + 1 -n ∈ 0, 1 n
and then the total land devoted to farming is

f = x φ -n-1 n ≤ 1.
If can be noticed that More interestingly, how individual choices depend on the number of farmers is radically dierent whether the aversion to intertemporal uctuations is less or more than 1. We have seen in Section 3 that farmers' decisions are strategic complements when φ < 1; then, when the number of farmers increases every individual share of land devoted to farming increases as well. In this sense, a small aversion to intertemporal uctuations is detrimental to biodiversity conservation.

d fh dn      > 0 if φ < 1 = 0 if φ = 1 < 0 otherwise
Conversely, if farmers have a large aversion to intertemporal uctuation (φ > 1), farmers' decisions are strategic substitutes and when the number of farmers increases each one reduces its own farming.

In the n-player situation, the relationship 1 + λ = (1 + ω)(1 + λ) still holds, with λ unchanged and (see Appendix C)

ω = 1 n φ φ + 1 n -1 φ 1 1-φ -1
We can see that

∂ω ∂n = 1 -1 n   φ φ+ 1 n -1 φ n   1 1-φ 1 + n(φ -1) thus as φ > 1 -1
n , ω is an increasing function of n. The more farmers there are the bigger the gain of putting in place land property rights.

Conclusion

Biodiversity is a multiform concept, as are the ways to calculate its economic value. Biodiversity is in particular a ow of services that allows to reduce the variability of ecological and economic conditions and can therefore be thought of, among other things, as a natural form of insurance. What is the loss of biodiversity insurance value that results from the decentralized decision of individuals and volatility externalities? Under which conditions is this loss most pronounced?

What policies are best suited to counter this loss?

In this article we propose a theoretical framework to approach these questions.

We consider an agricultural economy with free access and use of a natural forest.

A trade-o arises between the forest conversion to agricultural activity and the loss of biodiversity and consequently of its role of natural insurance this conversion entails.

We study the solution of the model in a context of strategic competition among farmers, quantifying the over-exploitation of the forest and the welfare loss with respect to the social optimum. Agents' utility is given in terms of a Kreps Porteus stochastic dierential utility capable of disentangling risk aversion and aversion to uctuations.

For empirically plausible values of the parameters, the model suggests that enforcing property rights is more relevant in case of stagnant technological progress in agricultural activities while policies consisting in pricing the volatility externality (for instance through a tax on land conversion) may be more suited for rapidly developing economies averse to uctuations.

The structure of the proposed benchmark model is kept simple to allow the model to be solved explicitly, but a number of possible extensions would be of interest. We mention here some of them.

A a rst extension of the model could include production storage technologies.

In this way the consumption at any given time would not need to be equal to production and the smoothing of the consumption trajectory could go through a new channel compared to the model presented in this paper.

A second possible generalization could take into account catastrophic events that may have an impact on production/productivity and whose frequency or magnitude could be endogenously inuenced by conservation choices as suggested by a great deal of evidence (for example, see [START_REF] Isbell | Biodiversity increases the resistance of ecosystem productivity to climate extremes[END_REF].

A third way to extend the current results could consist in considering the dierent impacts of dierent agricultural models on biodiversity following the well-developed discussion about land sparing vs land sharing, since a part of the biodiversity could be preserved also in agricultural environments depending on the farming practices.

where the aggregator F can be chosen 12 in the following form [START_REF] Epstein | Substitution, risk aversion, and the temporal behavior of consumption and asset returns: an empirical analysis[END_REF], Svensson, 1989and and Due and Epstein, 1992a) :

(40) F 1 (C, V ) = ρ 1 -φ i (1 -θ i )V      C ρ (1 -θ i )V 1 1-θ i   1-φ i -1    .
One of the most characteristic properties of Epstein-Zin-Weil utilities, apart from the partular case where φ i = θ i , is that they are not separable in the sense that they cannot be written as a double integral (one with respect to time and one, hope, with respect to the probability measure). For this reason, in the general case utility can only be expressed by an expression of the form (39).

In the innite horizon case the utility is dened as

U 1 (Ψ 1 , Ψ 2 ) = lim T →∞ U 0,T 1 (Ψ 1 , Ψ 2 ).
Proving that this utility is well dened is not in general an easy problem (one has to be sure that the backward equation has a solution and that the limit exists). Results for existence are given for instance by Due and Epstein (1992a) in the nite horizon case, and by [START_REF] Due | PDE solutions of stochastic dierential utility[END_REF] in the innite horizon Kreps Porteus case (i.e. the case we are interested in). Given a strategy Ψ 2 of farmer 2 we say that

Ψ 1 ∈ C 1 (Ψ 2 ) is a best response to Ψ 2 if (41) Ψ 1 ∈ arg max Φ∈C 1 (Ψ 2 ) U 1 (Φ, Ψ 2 ).
Similarly, given a strategy Ψ 1 of farmer 1,

Ψ 2 ∈ C 2 (Ψ 1 ) is a best response to Ψ 1 if (42) Ψ 2 ∈ arg max Φ∈C 2 (Ψ 2 ) U 1 (Ψ 1 , Φ).
We say that the couple (Ψ 1 , Ψ 2 ) is a Nash equilibrium of Markov strategies if both ( 41) and (42) are veried.

ρ i 1 -θ i 1 -φ i V i (A 1 , A 2 ) = max f i ∈[0,1-f j ] (f i A i ) 1-φ i 1 -φ i 1 ((1 -θ i )V i (A 1 , A 2 )) 1-φ i 1-θ i -1 + ∂V i ∂A 1 A 1 α 1 + ∂V i ∂A 2 A 2 α 2 + 1 2 (f 1 + f 2 ) σ 1 σ 2 , HV i σ 1 σ 2
We guess and check the following solution:

V 1 (A 1 , A 2 ) = β 1 A 1-θ 1 1 1-θ 1 for some positive constant β 1 . Then ρ 1 1 1 -φ 1 β 1 A 1-θ 1 1 = max f 1     (f 1 A 1 ) 1-φ 1 1 -φ 1 1 β 1 A 1-θ 1 1 1-φ 1 1-θ 1 -1 + β 1 A 1-θ 1 1 α 1 -θ 1 σ 2 1 2 (f 1 + f 2 )     i.e. ρ 1 1 -φ 1 = max f 1    f 1-φ 1 1 (1 -φ 1 )β 1-φ 1 1-θ 1 1 + α 1 -θ 1 σ 2 1 2 (f 1 + f 2 )    i.e. ( 43
) ρ 1 1 -φ 1 -α 1 = max f 1    f 1-φ 1 1 (1 -φ 1 )β 1-φ 1 1-θ 1 1 -θ 1 σ 2 1 2 (f 1 + f 2 )   
The maximizer f 1 on (0, +∞) (we will check later that point of maximum will be in (0, 1 -f 2 ]) is given by the solution of (44)

f -φ 1 1 β 1-φ 1 1-θ 1 1 = θ 1 σ 2 1 2 If we replace β 1-φ 1 1-θ 1 1
taken from ( 44) into (43) we get:

(45) ρ 1 -α 1 (1 -φ 1 ) = θ 1 σ 2 1 2 [f 1 φ 1 -(1 -φ 1 )f 2 ] i.e. ( 46
) f 1 = 1 φ 1 (x 1 + (1 -φ 1 )f 2 )
where x i is dened as in (5). If this expression belongs in [0, 1 -f 2 ], using classical results for optimization of problems with stochastic dierential utilities and innite horizon (see for instance Appendix C of Due et Epsten, 1992a), the feedback optimal control of Player 1 when she knows that Player 2 plays Ψ 2 (A 1 , A 2 ) ≡ f 2 (i.e. her optimal response) is Ψ 1 (A 1 , A 2 ) ≡ f 1 if a transversality condition in the form (see Smith, 1996):

(47)

lim t→∞ E 0 e -ρ 1 t V 1 (A 1 (t)) = 0
is veried. In this expression, and in the following, to lighten the notation, since V 1 is only a function of A 1 , we simply write V 1 (A 1 ).

We will verify it at the equilibrium.

A similar argument can be used for Farmer 2 responding to a constant strategy of Farmer 1. The equation which corresponds to (46) is then

(48) f 2 = 1 φ 2 (x 2 + (1 -φ 2 )f 1 )
The solution of the system ( 46)-( 48) gives ( 9)-( 10). Thanks to hypotheses of the Theorem the couple Ψ 2 (A 1 , A 2 ) ≡ f2 , Ψ 1 (A 1 , A 2 ) ≡ f1 is an admissible couple. When the players use this couple the total area devoted to farming is, as described in (11), f := f1 + f2 = x 1 +x 2 φ 1 +φ 2 -1 . To conclude the proof we only need to see that, along the equilibrium path the condition (47) and its counterpart for Player 2 are veried. We can see that

E 0 e -ρ i t V i (A(t)) = β i 1 -θ i lim t→∞ e -ρ i t E 0 A i (t) 1-θ i = β i 1 -θ i A 0 lim t→∞ e -ρ i t e (1-θ i )(α i -θ i σ 2 i 2 f )t ,
and one can easily see that both (for i = 1, 2) these expressions goes to 0 under the conditions (8).

Finally we can observe that, since the solution of the two equations ( 46) and ( 48) is unique, so it is the equilibrium of constant strategies.

Remark B.1. Observe that, s long as the drift and the diusion terms in equation (3) remains linear in A there are several degrees of freedom with respect to our specication in order to obtain existence of a Markov perfect equilibrium of constant strategy. More precisely, if we consider the generic specication

dA i (t) = α i (f 1 , f 2 )A i (t) dt + σ i (f 1 , f 2 )A i (t) dW (t)
there exists a constant and deterministic equilibrium (Ψ 1 , Ψ 2 ) = ( f1 , f2 ) if the following conditions are veried for some β 1 , β 2 > 0:

0 < f1 ∈ arg max f 1    f 1-φ 1 1 (1 -φ 1 )β 1-φ 1 1-θ 1 1 + α 1 (f 1 , f2 ) -θ 1 σ 1 (f 1 , f2 ) 2 2    0 < f2 ∈ arg max f 2    f 1-φ 2 2 (1 -φ 2 )β 1-φ 2 1-θ 2 2 + α 2 ( f1 , f 2 ) -θ 2 σ 2 ( f1 , f 2 ) 2 2    f1 + f2 < 1.
This system has a solution for several specication of α i and θ i . The problem (and the reason why we concentrated on the version that we study in the paper) is that very rarely the solution of the system can be written explicitly and then studied in detail.

Proof of Proposition 3.4. By direct computation we have

∂ f1 ∂ρ 1 = ∂x 1 ∂ρ 1 φ 2 φ 1 + φ 2 -1 > 0 ∂ f1 ∂θ 1 = ∂x 1 ∂θ 1 φ 2 φ 1 + φ 2 -1 < 0 ∂ f1 ∂σ 1 = ∂x 1 ∂σ 1 φ 2 φ 1 + φ 2 -1 < 0
and similarly when we compute ∂ f2 ∂ρ 2 , ∂ f2 ∂θ 2 and ∂ ln f2 ∂σ 2 . Conversely, when we compute the derivatives of f1 w.r.t. to parameters of player 2 we get

∂ f1 ∂ρ 2 = ∂x 2 ∂ρ 2 1 -φ 1 φ 1 + φ 2 -1 ∂ f1 ∂θ 2 = ∂x 2 ∂θ 2 1 -φ 1 φ 1 + φ 2 -1 ∂ f1 ∂σ 2 = ∂x 2 ∂σ 2 1 -φ 1 φ 1 + φ 2 -1
whose sign depend on the value of φ 1 compared to 1. However the sign of f i w.r.t. φ i remains unspecied as clear looking at the following expression:

∂ f1 ∂φ 1 = φ 2 φ 1 + φ 2 -1 ∂x 1 ∂φ 1 - x 1 + x 2 φ 1 + φ 2 -1
Proof of Proposition 4.2. Since, by assumption φ > 1/2, the result is straightforward.

Proof of Proposition 4.3. By direct computation we have

∂ ln Λ ∂φ = 1 φ - 2 2φ -1 = - 1 φ (2φ -1) < 0.
Proof of Proposition 5.1. The positivity of ω for all φ > 1 2 follows by observing that the limit of ω when φ tends to innity is 0.

We have ln [1 + ω) = -1 1 -φ ln 2 + φ ln 1 -1 2φ By direct computation of the derivative we obtain:

d ln (1 + ω) dφ = 1 1 -φ 1 1 -φ ln φ 2φ -1 - 1 2φ -1 = 1 1 -φ 1 1 -φ ln 1 + 1 -φ 2φ -1 - 1 2φ -1 < 1 1 -φ 1 1 -φ 1 -φ 2φ -1 - 1 2φ -1 = 0.
Appendix C. Proofs in the n-player case Optimal response and Nash equilibrium. We assume that the parameters are the same for all players. Suppose rst that all players j = i play a constant strategy Ψj(A 1 , .., A n ) ≡ f j . Then the optimal response of Player i can be found as the solution (in feedback form) of an optimal control whose Bellman equation may be written as:

ρ i 1 -θ i 1 -φ i V i (A 1 , ..., A n ) = max f 1 ∈[0,1-j =i f j ] (f i A i ) 1-φ i 1 -φ i 1 ((1 -θ i )V i (A 1 , ..., A n )) 1-φ i 1-θ i -1 +   n j=1 ∂V i ∂A j α j   + 1 2 (f 1 + f 2 )    σ 1 ... σ n    , HV i    σ 1 ... σ n   
where •, • denotes the scalar product on R n .

We guess and check the following solution: V i (A 1 , ..., A n ) = β i A 1-θ i 1-θ for some positive constant β i . Then

ρ 1 -φ β i A 1-θ i = max f i     (f i A i ) 1-φ 1 -φ 1 β i A 1-θ i 1-φ 1-θ -1 + β i A 1-θ 1 i   α -θ σ 2 2 n j=1 f j       i.e. ρ 1 -φ β i A 1-θ i = max f i     (f i A i ) 1-φ 1 -φ 1 β i A 1-θ i 1-φ 1-θ -1 + β i A 1-θ 1 i   α -θ σ 2 2 n j=1 f j      
i.e.

(49) ρ 1 -φ = max

f i   f 1-φ i 1 -φ 1 β 1-φ 1-θ i +   α -θ σ 2 2 n j=1 f j    
The maximizer f i on (0, +∞) (we will check later that point of maximum will be in (0, 1j =i f j ]) is given by the solution of (50)

f -φ i β 1-φ 1-θ i = θ σ 2 2
If we replace β 1-φ 1-θ i taken from ( 50) into (49) we get:

ρ -α (1 -φ) = θ σ 2 2   φf i -(1 -φ) j =i f j   i.e.
(51)

f i = x + (1 -φ) j =i f j φ
When all the strategies are symmetric, then

f i = x nφ -n + 1
Thus using (50),

x nφ-n+1 -φ θ σ 2 2 = β 1-φ 1-θ i
Per-capita welfare gain and gain from biodiversity conservation. Let us now compute the per-capita welfare gain of moving from Nash equilibrium to optimum, dened as ω such that

V p A n = V h ((1 + ω) A) As V p (A) = 1 1-θ β p A 1-θ with β p = θ σ 2 2 (f * ) φ -1-θ 1-φ ω = 1 n nφ nφ + 1 -n φ 1 1-φ -1.
Finally, the per capita welfare gain from biodiversity conservation λ is

V p A n = V oa (1 + λ) A n
The value of λ is the same as in the 2-player case

λ = φf * + (1 -φ) f * φ 1 1-φ -1.

Farmer

  i's target is to maximize an innite horizon, continuous time, Epstein-Zin-Weil utility function with Kreps Porteus utility. This specication of utilities makes it possible to disentangle the notions of aversion to risk and aversion to uctuations which are identied (and conated) in the usual time additive expected utility function i.e. in the specications where the target can be written

Figure 1 .

 1 Figure 1. Land conversion (upper panel) and welfare gains (lower panel) as functions of aversion to uctuations

  Not surprisingly, as the number of farmers increases, the total share of land devoted to biodiversity conservation decreases. The more farmers there are the bigger the volatility externality. The open access situation corresponds to the case of free entry, where new farmers appropriate land until all land is converted to farming.

  and Storch et al. (2012):

This result relies on the fact that all farmers have the same preferences and the same initial productivity. We also suppose that the planner attaches the same weight to each farmer in the social welfare function.

Acknowledgements

The work of Giorgio Fabbri is supported by the French National Research Agency in the framework of the Investissements d'avenir program (ANR-15-

 Renwick, A., and Schellhorn, N. (2016). A perspective on land sparing versus land sharing. in Learning from agri-environment schemes in Australia, 117. Simpson, R.D., Sedjo, R.A. and Reid, J.W., 1996. Valuing We spend some words to recall how this utility is dened. We start by dening the utility for the nite horizon problem with horizon T > 0. Chosen a couple of strategies (Ψ 1 , Ψ 2 ) ∈ C we have, through (3), the dynamics of A 1 and A 2 and then, thanks to (2) the processes Y i , and (with (4)) the consumptions paths. The Epstein-Zin-Weil utility (for player 1, the situation for player 2 is completely analogous) is dened (see Due andEpstein, 1992a, 1992b) as the solution at time t = 0 of the following integral equation (it is a backward SDE with nal condition equal to 0):

Proof of Theorem 3.1. Suppose rst that Player 2 plays a constant strategy

Then the optimal response of Player 1 can be found as the solution (in feedback form) of an optimal control whose Bellman equation may be written as:

12 Several ordinally equivalent choices of the aggregator are possible, the one described in (40) has the advantage of eliminating the variance term that in general appears in (39), it is indeed the unique with this property, see Due and Epstein (1992b) and [START_REF] Due | PDE solutions of stochastic dierential utility[END_REF].