Daniele A Di

Jean-Marc Gratien
email: j-marc.gratien@ifpenergiesnouvelles.fr

Lowest order methods for diffusive problems on general meshes: A unified approach to definition and implementation

In this work we propose an original point of view on lowest order methods for diffusive problems which aims at laying the pillars of a C++ multi-physics, FreeFEM-like platform. The key idea is to regard lowest order methods as (Petrov)-Galerkin methods based on possibly incomplete, broken polynomial spaces defined from a gradient reconstruction. After presenting some examples of methods entering the framework, we show how implementation strategies common in the finite element context can be extended relying on the above definition. Several examples are provided throughout the presentation, and programming details are often concealed to help the reader unfamiliar with advanced C++ programming techniques.

Introduction

An increasing amount of attention has recently been given to the discretization of diffusive problems on general meshes. Lowest order methods possibly featuring conservation of physical quantities are traditionally employed in industrial applications where computational cost is a crucial issue. In this context, the main interest of handling general meshes is to reduce the number of elements required to represent complicate domains. In sedimentary basin modeling, non-standard elements may also appear due to the erosion of geological layers. Different ways to adapt finite volume and finite element methods to general, possibly non-conforming polyhedral meshes have been proposed. In the context of cell centered finite volume methods, we recall, in particular, the classical works of Aavatsmark, Barkve, Bøe and Mannseth [START_REF] Aavatsmark | Discretization on unstructured grids for inhomogeneous, anisotropic media, Part I: Derivation of the methods[END_REF] and Edwards and Rogers [START_REF] Edwards | Finite volume discretization with imposed flux continuity for the general tensor pressure equation[END_REF] on multipoint fluxes. More recently, two ways of extending the mixed finite element philosophy to general meshes have been proposed independently by Brezzi, Lipnikov, Shashkov and Simoncini [START_REF] Brezzi | A family of mimetic finite difference methods on polygonal and polyhedral meshes[END_REF][START_REF] Brezzi | Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes[END_REF] (mimetic finite difference methods) and by Droniou and Eymard [START_REF] Droniou | A mixed finite volume scheme for anisotropic diffusion problems on any grid[END_REF] (mixed/hybrid finite volume methods). Yet another perspective is considered by Eymard, Gallouët and Herbin [START_REF] Eymard | Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces[END_REF], who show, in particular, that face unknowns can be selectively used as Lagrange multipliers for the flux continuity constraint or be eliminated using a consistent interpolator (SUSHI scheme). The strong link between the strategies above has been highlighted by Droniou, Eymard, Gallouët and Herbin [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF]. A slightly different approach based on the analogy between lowest order methods in variational formulation and discontinuous Galerkin methods has been proposed by the author in [START_REF] Di | Cell centered Galerkin methods[END_REF][START_REF] Di | A compact cell-centered Galerkin method with subgrid stabilization[END_REF][START_REF] Di | Cell centered Galerkin methods for diffusive problems[END_REF] (cell centered Galerkin methods). The key advantage of this approach is that it largely benefits from the well-established theory for discontinuous Galerkin methods applied to diffusive problems [START_REF] Di Pietro | Mathematical aspects of discontinuous Galerkin methods[END_REF]. All of the methods above have been (or can be) extended to several classical problems for which the discretization of second order diffusive terms is central.

In this work we present a unified implementation covering a wide range of lowest order methods and applications based on similar experiences in the context of finite element methods. Finite element libraries have nowadays reached a good level of maturity, and user-friendly front-ends are provided in several cases. Just to mention a few, we recall Feel++ [START_REF] Prud'homme | A domain specific embedded language in C++ for automatic differentiation, projection, integration and variational formulations[END_REF] (formerly known as Life), FEniCS [START_REF] Logg | DOLFIN: Automated finite element computing[END_REF], FreeFEM++ [START_REF] Danaila | Simulation numérique en C++[END_REF]. Our goal is to show that similar tools can be conceived and implemented for lowest order methods. The starting point is to reformulate the method at hand as a (Petrov)-Galerkin scheme based on possibly incomplete broken affine spaces. This new unified perspective, drawing on the lines of [START_REF] Di | Cell centered Galerkin methods for diffusive problems[END_REF], allows, in particular, to recycle many ideas originally developed for finite elements. A major difference, however, is that the lowest order methods considered herein are often based on reconstructions of first order differential operators which may depend on problem data such as the diffusion coefficient or the boundary condition. As a consequence, the classical approach based on a table of degrees of freedom computed from a mesh and a finite element (see, e.g., [START_REF] Ern | Theory and Practice of Finite Elements[END_REF]) is no longer adequate. This issue is solved by introducing the programming counterpart of tensor-valued linear combinations of (globally numbered) degrees of freedom. This concept allows, in particular, to reproduce a finite element-like matrix assembly with local contributions stemming from integrals over mesh elements and faces. A further layer of abstraction is added by defining a domain-specific language (DSL) for variational formulations. The DSL is closely inspired by that of Feel++, the most noticeable differences being the type-based identification of test and trial functions and the possibility to store linear and bilinear forms as expressions independently of their algebraic representation. Another novelty is the introduction of tensor-like notation for systems of PDEs. Domain-specific languages and generative programming are an established tool to break down the complexity of industrial applications by distinguishing the actors that tackle different aspects of the problem, and providing each of them with means of expression as close as possible to his/her technical jargon. An important advantage of the DSL is that it potentially allows to combine lowest order methods with more standard discretizations techniques in a seamless way.

In the presentation we try to avoid all technicalities and to pinpoint the main difficulties as well as the proposed solutions. Although the language of choice is C++, the listings are rather to be intended as pseudo-code since simplifications are often made to improve readability. The actual implementation is based on the Arcane framework [START_REF] Grospellier | The arcane development framework[END_REF], a proprietary platform conjointly developed at CEA-DAM and IFP Energies nouvelles which takes care of technical aspects such as memory management, parallelism and post-processing.

The material is organized as follows. In §2 we propose a unified perspective and show how several lowest order methods can fit in there for a simple diffusion problem. In §3 we discuss the implementation. More specifically, we first discuss the solutions to the issues that arise when trying to mimic the finite element approach and then present a DSL which allows to conceal the related technicalities.

Definition

Discrete setting

Let Ω ⊂ | d , d ≥ 2, denote a bounded connected polyhedral domain. The first ingredient in the definition of lowest order methods is a suitable discretization of Ω . We denote by T h a finite collection of nonempty, disjoint open polyhedra T h = {T } forming a partition of Ω such that h = max T ∈T h h T and h T denotes the diameter of the element T ∈ T h . Admissible meshes include general polyhedral discretizations with possibly nonconforming interfaces; see Figure 1. Mesh nodes are collected in the set N h and, for all T ∈ T h , N T contains the nodes that lie on the boundary of T . We say that a hyperplanar closed subset F of Ω is a mesh face if it has positive (d-1)-dimensional measure and if either there exist T 1 , T 2 ∈ T h such that F ⊂ ∂ T 1 ∩ ∂ T 2 (and F is called an interface) or there exist T ∈ T h such that F ⊂ ∂ T ∩ ∂ Ω (and F is called a boundary face). Interfaces are collected in the set F i h , boundary faces in F b h and we let F h := F i h ∪ F b h . For all T ∈ T h we set

F T := {F ∈ F h | F ⊂ ∂ T }. (1)
Symmetrically, for all F ∈ F h , we define

T F := {T ∈ T h | F ⊂ ∂ T }.
The set T F consists of exactly two mesh elements if F ∈ F i h and of one if F ∈ F b h . For all mesh nodes P ∈ N h , F P denotes the set of mesh faces sharing P, i.e.

F P := {F ∈ F h | P ∈ F}. (2)
For every interface F ∈ F i h we introduce an arbitrary but fixed ordering of the elements in T F and let n F = n T 1 ,F = -n T 2 ,F , where n T i ,F , i ∈ {1, 2}, denotes the unit Fig. 1 Left. Mesh T h Right. Pyramidal submesh P h normal to F pointing out of T i ∈ T F . On a boundary face F ∈ F b h , n F denotes the unit normal pointing out of Ω . The barycenter of a face F ∈ F h is denoted by x F := F x/|F| d-1 . For each T ∈ T h we identify a point x T ∈ T (the cell center) such that T is star-shaped with respect to x T . For all F ∈ F T we let

d T,F := dist(x T , F).
It is assumed that, for all T ∈ T h and all F ∈ F T , d T,F > 0 is comparable to h T . Starting from cell centers we can define a pyramidal submesh of T h as follows:

P h := {P T,F } T ∈T h , F∈F T ,
where, for all T ∈ T h and all F ∈ F T , P T,F denotes the open pyramid of apex x T and base F, i.e.,

P T,F := {x ∈ T | ∃y ∈ F \ ∂ F, ∃θ ∈ (0, 1) | x = θ y + (1 -θ )x T }.
The pyramids {P T,F } T ∈T h , F∈F T are nondegenerate by assumption. Let S h be such that S h = T h or S h = P h .

For all k ≥ 0, we define the broken polynomial spaces of total degree ≤ k on S h ,

P k d (S h ) := {v ∈ L 2 (Ω ) | ∀S ∈ S h , v |S ∈ P k d (S)}, with P k d (S)
given by the restriction to S ∈ S h of the functions in P k d . Remark 1 (Admissible mesh sequence). In the context of a priori convergence analysis for vanishing mesh size h it is necessary to bound some quantities uniformly with respect to h. This leads to the concept of admissible mesh sequence. This topic is not addressed in detail herein since our focus is mainly on implementation. For a comprehensive discussion we refer to [START_REF] Brezzi | A family of mimetic finite difference methods on polygonal and polyhedral meshes[END_REF][START_REF] Brezzi | Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes[END_REF][START_REF] Droniou | A mixed finite volume scheme for anisotropic diffusion problems on any grid[END_REF][START_REF] Eymard | Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces[END_REF][START_REF] Di | Cell centered Galerkin methods for diffusive problems[END_REF]; see also [START_REF] Di Pietro | Mathematical aspects of discontinuous Galerkin methods[END_REF]Chapter 1].

We close this section by introducing trace operators which are of common use in the context of nonconforming finite element methods. Let v be a scalar-valued function defined on Ω smooth enough to admit on all F ∈ F h a possibly two-valued trace. To any interface

F ⊂ ∂ T 1 ∩ ∂ T 2 we assign two nonnegtive real numbers ω T 1 ,F and ω T 2 ,F such that ω T 1 ,F + ω T 2 ,F = 1,
and define the jump and weighted average of v at F for a.e. x ∈ F as

v F (x) := v |T 1 -v |T 2 , { {v} } ω,F (x) := ω T 1 ,F v |T 1 (x) + ω T 2 ,F v |T 2 (x). (4) If F ∈ F b h with F = ∂ T ∩∂ Ω , we conventionally set { {v} } ω,F (x) = v F (x) = v |T (x)
. The subscript ω is omitted from the average operator when ω T 1 ,F = ω T 2 ,F = 1 2 . The dependence on x and on the face F is also omitted if no ambiguity arises.

An abstract perspective

The key idea to gain a unifying perspective is to regard lowest order methods as nonconforming methods based on incomplete broken affine spaces that are defined starting from the space of degrees of freedom (DOFs) ❱ h . More precisely, we let

❚ h := | T h , ❋ h := | F h ,
and consider the following choices:

❱ h = ❚ h or ❱ h = ❚ h × ❋ h .
In every case the elements of ❱ h are indexed with respect to the mesh entity they belong to. Other choices for ❱ h are possible but are not considered herein for the sake of conciseness. To fix the ideas, one can assume that the choice ❱ h = ❚ h corresponds to cell centered finite volume (CCFV) and cell centered Galerkin (CCG) methods, while the choice ❱ h = ❚ h × ❋ h leads to mimetic finite difference (MFD)

and mixed/hybrid finite volume (MHFV) methods.

The key ingredient in the definition of the broken affine space is a piecewise constant linear gradient reconstruction G h : ❱ h → [P 0 d (S h )] d (the linearity of G h is a founding assumption for the implementation discussed in §3). Starting from G h , we can define the linear operator R h :

❱ h → P 1 d (S h ) such that, for all v h ∈ ❱ h , ∀S ∈ S h , S ⊂ T S ∈ T h , ∀x ∈ S, R h (v h ) |S = v T S + G h (v h ) |S •(x -x T S ) ∈ P 1 d (S h ). (5)
The operator R h maps every vector of DOFs onto a piecewise affine function belonging to P 1 d (S h ). Hence, we can define a broken affine space as follows:

V h = R h (❱ h ) ⊂ P 1 d (S h ). (6)
The operator R h is additionally assumed to be injective, so that a bijective operator can be obtained by restricting its codomain. The next section presents some examples covering the methods listed above.

F F F F (a) G F = L-groups containing the face F x T x T 2 x T 1 F 1 F 2 (b) L-construction Fig. 2 L-construction

Examples

In this section we focus on the model problem

-∇•(κ∇u) = f , u = 0, (7)
where f ∈ L 2 (Ω ) and κ ∈ [P 0 d (T h )] d is a piecewise constant, uniformly elliptic tensor field (possibly resulting from a homogeneization process). Problem [START_REF] Danaila | Simulation numérique en C++[END_REF] provides the paradigm to illustrate how selected lowest order methods can be recast in the framework of §2.2.

The G-method

As a first example we consider the special instance of CCFV methods analyzed in [START_REF] Agélas | The G method for heterogeneous anisotropic diffusion on general meshes[END_REF]. A preliminary step consists in presenting the so-called L-construction introduced in [START_REF] Aavatsmark | A compact multipoint flux approximation method with improved robustness[END_REF]. The key idea of the L-construction is to use d cell and boundary face values (provided, in this case, by the homogeneous boundary condition) to express a continuous piecewise affine function with continuous diffusive fluxes. The values are selected using d neighboring faces belonging to a cell and sharing a common vertex. More precisely, we define the set of L-groups as follows:

G := {g ⊂ F T ∩ F P , T ∈ T h , P ∈ N T | card(g) = d} ,
with F T and F P given by ( 1) and ( 2) respectively. It is useful to introduce a symbol for the set of cells concurring in the L-construction: For all g ∈ G , we let

T g := {T ∈ T h | T ∈ T F , F ∈ g}.
Let now g ∈ G and denote by T g an element T g such that g ⊂ F T g (this element may not be unique). For all v h ∈ ❱ h we construct the function ξ g v h piecewise affine on the family of pyramids {P T,F } F∈g, T ∈T g such that: (i) ξ g v h (x T ) = v T for all T ∈ T g and

ξ g v h (x F ) = 0 for all F ∈ g ∩ F b h ; (ii) ξ g v h is affine inside T g and is continuous across every interface in the group: For all F ∈ g ∩ F i h such that F ⊂ ∂ T 1 ∩ ∂ T 2 , ∀x ∈ F, ξ g v h |T 1 (x) = ξ g v h |T 2 (x);
(iii) ξ g v h has continuous diffusive flux across every interface in the group: For all

F ∈ g ∩ F i h such that F ⊂ ∂ T 1 ∩ ∂ T 2 , (κ∇ξ g v h ) |T 1 •n F = (κ∇ξ g v h ) |T 2 •n F .
For further details on the L-construction including an explicit formula for ξ g v h we refer to [START_REF] Agélas | The G method for heterogeneous anisotropic diffusion on general meshes[END_REF]. For every face F ∈ F h we define the set G F of L-groups containing F,

G F := {g ∈ G | F ∈ g}, (8)
and introduce the set of nonnegative weights {ς g,F } g∈G F such that ∑ g∈G F ς g,F = 1.

The trial space for the G-method is obtained as follows: (i) let

S h = P h and ❱ h = ❚ h ; (ii) let G h = G g h with G g h such that ∀v h ∈ ❚ h , ∀T ∈ T h , ∀F ∈ F T , G g h (v h ) |P T,F = ∑ g∈G F ς g,F ∇ξ g v h |P T,F .
We denote by R g h the reconstruction operator defined as in [START_REF] Brezzi | Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes[END_REF] with [START_REF] Agélas | The G method for heterogeneous anisotropic diffusion on general meshes[END_REF] is then equivalent to the following Petrov-Galerkin method:

G h = G g h and let V g h := R g h (❱ h ). The G-method of
Find u h ∈ V g h s.t. a g h (u h , v h ) = Ω f v h for all v h ∈ P 0 d (T h ), where a g h (u h , v h ) := -∑ F∈F h F { {κ∇ h u h } }•n F v h with ∇ h broken gradient on S h .
Remark 2 (An unconditionally stable method). The main drawback of the G-method is that stability can only be proven under quite stringent conditions; see, e.g., [START_REF] Agélas | The G method for heterogeneous anisotropic diffusion on general meshes[END_REF]Lemma 3.4]. A possible way to circumvent this difficulty has recently been proposed by one of the authors [START_REF] Di | A compact cell-centered Galerkin method with subgrid stabilization[END_REF] in the context of CCG methods. The key idea is to use V g h both as a trial and test space, and modify the discrete bilinear form to recover both consistency and stability. Since the discrete functions in V g h are discontinuous across the lateral faces of the pyramids in P h , least-square penalization of the jumps is required to assert stability in terms of coercivity. The resulting method also enters the present framework, but is not detailed here for the sake of conciseness.

A cell centered Galerkin method

The L-construction is used to define a trace reconstruction in the CCG method of [START_REF] Di | Cell centered Galerkin methods[END_REF][START_REF] Di | A compact cell-centered Galerkin method with subgrid stabilization[END_REF]. More specifically, for all F ∈ F i h , we select one group g F ∈ G F with G F defined by [START_REF] Di | Cell centered Galerkin methods[END_REF] and introduce the linear trace operator T g h : ❚ h → ❋ h which maps every vector of cell centered DOFs v h ∈ ❚ h onto a vector

(v F ) F∈F h ∈ ❋ h such that v F = ξ g F v h (x F ) if F ∈ F i h , 0 if F ∈ F b h . (9)
The trace operator T g h is then employed in a gradient reconstruction based on Green's formula and inspired from [START_REF] Eymard | Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces[END_REF]. More precisely, we introduce the linear gradient operator G green h

: ❚ h × ❋ h → [P 0 d (T h )] d such that, for all (v T , v F ) ∈ ❚ h × ❋ h and all T ∈ T h , G green h (v T , v F ) |T = 1 |T | d ∑ F∈F T |F| d-1 (v F -v T )n T,F . (10)
The discrete space for the CCG method under examination can then be obtained as

follows: (i) let S h = T h and ❱ h = ❚ h ; (ii) let G h = G ccg h with G ccg h such that ∀v h ∈ ❱ h , G ccg h (v h ) = G green h (v h , T g h (v h )). (11)
The reconstruction operator defined taking 5) is denoted by R ccg h , and the corresponding discrete space by

G h = G ccg h in (
V ccg h := R ccg h (❚ h ).
The last ingredient to formulate the discrete problem is a suitable definition of the weights in the average operator. To this end, we let, for all

F ∈ F i h such that F ⊂ ∂ T 1 ∩ ∂ T 2 , ω T 1 ,F = λ T 2 ,F λ T 1 ,F +λ T 2 ,F , ω T 2 ,F = λ T 1 ,F λ T 1 ,F +λ T 2 ,F , where λ T i ,F := κ |T i n F •n F for i ∈ {1, 2}. Set, for all (u h , v h ) ∈ V ccg h ×V ccg h , a ccg h (u h , v h ) := Ω κ∇ h u h •∇ h v h -∑ F∈F h F [{ {κ∇ h u h } } ω •n F v h + u h { {κ∇v h } } ω •n F ] + ∑ F∈F h η γ F h F F u h v h , ( 12
) with ∇ h broken gradient on T h , γ F = 2λ T 1 ,F λ T 2 ,F λ T 1 ,F +λ T 2 ,F on internal faces F ⊂ ∂ T 1 ∩ ∂ T 2 and γ F = κ |T n F •n F on boundary faces F ⊂ ∂ T ∩ ∂ Ω .
The user-dependent parameter η should be chosen large enough to ensure stability. The CCG method reads

Find u h ∈ V ccg h s.t. a ccg h (u h , v h ) = Ω f v h for all v h ∈ V ccg h . (13)
The bilinear form a ccg h has been originally introduced by Di Pietro, Ern and Guermond [START_REF] Di Pietro | Discontinuous Galerkin methods for anisotropic semi-definite diffusion with advection[END_REF] in the context of dG methods for degenerate advection-diffusion-reaction problems. In particular, for κ = 1 d , the method (13) coincides with the Symmetric Interior Penalty (SIP) method of Arnold [START_REF] Arnold | An interior penalty finite element method with discontinuous elements[END_REF] associated to the bilinear form

a sip h (u h , v h ) = Ω ∇ h u h •∇ h v h -∑ F∈F h F [{ {∇ h u h } }•n F v h + u h { {∇ h v h } }•n F ] + ∑ F∈F h η h F F u h v h . (14)
For further details on the link between CCG and discontinuous Galerkin methods we refer to [START_REF] Di | Cell centered Galerkin methods[END_REF][START_REF] Di | A compact cell-centered Galerkin method with subgrid stabilization[END_REF][START_REF] Di | Cell centered Galerkin methods for diffusive problems[END_REF].

A hybrid finite volume method

As a last example we consider a variant of the SUSHI scheme of [START_REF] Eymard | Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces[END_REF]; see also [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF] for a discussion on the link with the MFD methods of [START_REF] Brezzi | A family of mimetic finite difference methods on polygonal and polyhedral meshes[END_REF][START_REF] Brezzi | Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes[END_REF]. This method is based on the gradient reconstruction ( 10), but stabilization is achieved in a rather different manner with respect to [START_REF] Di Pietro | Discontinuous Galerkin methods for anisotropic semi-definite diffusion with advection[END_REF]. More precisely, we define the linear residual operator

r h : ❚ h × ❋ h → P 0 d (P h ) as follows: For all T ∈ T h and all F ∈ F T , r h (v T h , v F h ) |P T,F = d 1 2 d T,F v F -v T -G green h (v T h , v F h ) |T •(x F -x T ) .
The discrete space for SUSHI method with hybrid unknowns is then obtained as follows: (i) let

S h = P h and ❱ h = ❚ h × ❋ h ; (ii) let G h = G hyb h with G hyb h such that, for all (v T h , v F h ) ∈ ❚ h × ❋ h , all T ∈ T h and all F ∈ F T , G hyb h (v T h , v F h ) |P T,F = G green h (v T h , v F h ) |T + r h (v T h , v F h ) |P T,F n T,F . (15)
Denote by R hyb h the reconstruction operator defined by [START_REF] Brezzi | Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes[END_REF] with G h = G hyb h . The SUSHI method with hybrid unknowns reads

Find u h ∈ V hyb h s.t. a sushi h (u h , v h ) = Ω f v h for all v h ∈ V hyb h , with a sushi h (u h , v h ) := Ω κ∇ h u h •∇ h v h
and ∇ h broken gradient on P h . Alternatively, one can obtain a cell centered version by setting ❱ h = ❚ h and replacing G hyb h defined by [START_REF] Edwards | Finite volume discretization with imposed flux continuity for the general tensor pressure equation[END_REF] by

G h = G cc h with G cc h such that ∀v h ∈ ❚ h , G cc h (v h ) = G hyb h (v h , T g h (v h )), (16)
and trace operator T g h defined by ( 9). This variant coincides with the version proposed in [START_REF] Eymard | Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces[END_REF] for homogeneous κ, but it has the advantage to reproduce piecewise affine solutions of (7) on T h when κ is heterogeneous. The discrete space obtained taking

G h = G cc h in (6) is labeled V cc h .
Listing 1 Implementation of the bilinear form ( 14) using the DSL of §3

Implementation

The goal of this section is to lay the foundations for a DSL which transposes the mathematical concepts of §2 into practical implementations. To illustrate the capabilities of the DSL in a nutshell, compare Listing 1 with the expression of the bilinear form a sip h [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF]. The DSL is here embedded in C++, which allows to exploit the optimization features of the compiler as well as several constructs from the host language. The material is organized as follows: §3.1 introduces the algebraic backend aiming at replacing the table of DOFs in the context of a element-like assembly procedure; §3.2 deals with more abstract concepts that mimic function spaces, linear and bilinear forms to offer a functional front-end.

Algebraic back-end

In this section we focus on the elementary ingredients to build the terms appearing in the linear and bilinear forms of §2, which constitute the back-end of the DSL presented in §3.2.

Linear combination

The point of view presented in §2 naturally leads to finite element-like assembly of local contributions stemming from integrals over elements or faces. However, a few major differences have to be taken into account: (i) the stencil of the local Listing 2 Implementation of the gradient reconstruction G green h [START_REF] Di | A compact cell-centered Galerkin method with subgrid stabilization[END_REF] for an element T ∈ T h . Bufferization is a means to improve efficiency 1 LinearCombination<0> vT; 2 vT += LinearCombination<1>::Term(I T ,1.); 3 LinearCombination<1, Buffer> buffer;

4 for(F ∈ F T ) { 5 const LinearCombination<0> & vF = T h .eval(F); 6 buffer += |F| d-1 |T | d (vF-vT)n T,F ;
7 } 8 LinearCombination<1, Vector> GT; 9 buffer.compact(GT);

contributions may vary from term to term; (ii) the stencil may be data-dependent, as is the case for the methods of §2 based on the L-construction; (iii) the stencil may be non-local, as DOFs from neighboring elements may enter in local reconstructions. All of the above facts invalidate the classical approach based on a global table of DOFs inferred from a mesh and a finite element in the sense of Ciarlet. Our approach to meet the above requirements is to (i) drop the concept of local element, and to refer to DOFs by a unique global index; (ii) introduce the concept of LinearCombination (with template parameters to be specified in what follows), which realizes a linear application from ❱ h onto the space ❚ r of real tensors of order r ≤ 2.

In practice, a LinearCombination is an efficient mapping of the DOFs in ❱ h onto the corresponding coefficients in ❚ r . A LinearCombination l r can indeed be thought of as a list of couples (I, τ l,I ) I∈■ l where ■ l ⊂ ❱ h is the stencil (i.e., a vector of global DOFs) and τ l,I ∈ ❚ r , I ∈ ■ l , are the corresponding coefficients. To account for strongly enforced boundary conditions, LinearCombination also contains a constant coefficient τ l,0 , so that the evaluation at v h ∈ ❱ h (obtained by calling the function LinearCombination.eval(v h )) actually returns

l r (v h ) = ∑ I∈■ l τ l,I v I + τ l,0 ∈ ❚ r .
It is useful to define efficient operations such as the sum and subtraction of linear combinations, as well as different kinds of products by constants. This allows, e.g., to implement the gradient G green h defined by (10) as described in line 6 of Listing 2. When needed, each DOF I can be represented as a LinearCombination containing only the couple (I, 1). As a result, both the hybrid version with face unknowns [START_REF] Edwards | Finite volume discretization with imposed flux continuity for the general tensor pressure equation[END_REF] and the cell centered version [START_REF] Di Pietro | Mathematical aspects of discontinuous Galerkin methods[END_REF] of the gradient reconstruction can be obtained from Listing 2 by simply changing the value returned by the trace interpolator T h .eval(F) in line 5. We also pinpoint that the tensor order is a template parameter of LinearCombination to reduce the need for dynamic allocation.

In the implementation, particular care must be devoted to expressions containing the sum or subtraction of two linear combinations l r 1 and l r 2 , since this involves computing the intersection of the corresponding set of DOFs, say ■ l 1 and ■ l 2 respectively. To overcome this difficulty, complicate expressions are computed in two steps: a first step in which duplicate DOF indices are allowed, followed by a compaction stage where the algebraic sums of the corresponding coefficients are performed. This is obtained by changing the value of the second template parameter of LinearCombination. Specifically, in Buffer-mode a LinearCombination efficiently supports adding terms and can appear in the left-hand side of an assignment operator, while Vector-mode (default) only allows to traverse its elements in a fixed order; c.f. lines 3 and 8 of Listing 2.

Linear and bilinear contributions

Exploiting the concept of LinearCombination, it is possible to devise a unified treatment for local contributions stemming from integrals over elements or faces. We illustrate the main ideas using the an example: For a given T ∈ T h and for u h , v h ∈ V ccg h , we consider the local contribution A loc associated to the term

T κ∇ h u h •∇ h v h .
For the sake of simplicity we focus on the case when the constant coefficient τ l,0 is zero (in the example, this corresponds to the homogeneous Dirichlet boundary condition in problem [START_REF] Danaila | Simulation numérique en C++[END_REF]). The key remark is that both (κ∇ h u h ) |T = κ |T ∇(u h|T ) and (∇ h v h ) |T = ∇(v h|T ) can be represented as objects of type LinearCombination<1>, say l 1 u = (J, τ l u ,J ) J∈❏ and l 1 v = (I, τ l v ,I ) I∈■ . The associated local contribution reads

A loc = [|T | d τ l v ,I •τ l u ,J ] I∈■, J∈❏ . (17)
Generalizing the above remark, one can implement local terms in matrix assembly as BilinearContributions which can be represented as triplets (■, ❏, A loc ) containing two vectors of DOF indices ■ and ❏ and the local matrix A loc . Observe, in particular, that ■ and ❏ play the same role as the lines of the table of DOFs corresponding to test and trial functions supported in T in standard finite element implementations. As such, they are related to the lines and columns of the global matrix A to which A loc contributes,

A(■, ❏) ← A(■, ❏) + A loc . (18)
When the LinearCombinations concurring to a local term take values in ❚ r , the vector inner product in [START_REF] Eymard | Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces[END_REF] should be replaced by the appropriate tensor contraction. An additional argument γ in BilinearContribution serves as a multiplicative factor for the whole expression (in the above example, γ = |T | d ). More generally, γ can be a function of space and time, and may depend on discrete variables. Similarly, LinearContributions serve to represent right-hand side contributions. LinearContributions are not detailed here for the sake of brevity. A Listing 1. The gradient reconstruction implicitly fixes both the choice (6) for the algebraic space of DOFs and the choice (3) for S h . The programming counterparts of the function spaces appearing in §2 are listed in Table 1.

P 0 d (T h ) T h Polynomial<d, 0> Null V g h P h Polynomial<d, 1> GFormula V ccg h T h Polynomial<d, 1> GreenFormula<LInterpolator> V hyb h P h Polynomial<d, 1> SUSHIFormula<HybridUnknowns> V cc h P h Polynomial<d, 1> SUSHIFormula<LInterpolator>
The key role of a FunctionSpace is to bridge the gap between the algebraic representation of DOFs and the functional representation used in the methods of §2. This is achieved by the functions Gh and Rh, which are the C++ counterpart of the linear operators G h and R h respectively; see §2.1. More specifically, (i) for all S ∈ S h , Gh(S) returns a vector-valued linear combination corresponding to the (constant) restriction G h|S ; (ii) for all S ∈ S h and all x ∈ S, Rh(S, x) returns a scalar-valued linear combination corresponding to R h|S (x) defined according to [START_REF] Brezzi | Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes[END_REF]. The linear combinations returned by Gh and Rh can be used to generate LinearContributions and BilinearContributions to build linear and bilinear terms as described above. A FunctionSpace also defines the types TestFunction and TrialFunction that correspond to the mathematical notions of test and trial functions in variational formulations. The main difference between a TestFunction and a TrialFunction is that the latter is associated to a vector of DOFs which is stored in memory. In addition, when used to define bilinear contributions, test and trial functions are associated to the lines and columns of the local matrix respectively. We conclude by observing that the choice of identifying test and trial functions by their type is in contrast with the approach of [20, §3.4], where special keywords accomplish this task. plied to a test or trial function, a unary modifier is an object capable of returning a LinearCombination at evaluation. By default, linear and bilinear forms are represented by vectors and (sparse) matrices, but other representations are possible resulting, e.g., in matrix-free implementations. In contrast with [START_REF] Prud'homme | A domain specific embedded language in C++ for automatic differentiation, projection, integration and variational formulations[END_REF], the expression corresponding to a linear or bilinear form is stored instead of being evaluated on-the-fly. This allows, in particular, to change the operations actually performed at evaluation according to a context. Changing the representation of linear and bilinear forms thus amounts to changing the context of evaluation. An example of evaluation is provided in lines 16-17 of Listing 1, where the global matrix A is assembled according to the expression of ah and to the procedure defined in MatrixContext. During the evaluation, each term in the expression of ah generates a corresponding BilinearContribution, which is in turn assembled as described in §3.1.

To conclude, we present a more complicate example involving a system of PDEs. More specifically, we consider the Stokes problem:

-△u + ∇p = f in Ω , ∇•u = 0 in Ω , u = 0 on ∂ Ω , with p Ω = 0 to ensure well-posedness. Let X h := [V ccg h ] d × P 0 d (T h )/|. In Listing 5 we present the implementation of the CCG method of [9, §3]: Find (u h , p h ) ∈ X h such that

a h (u h , v h ) + b h (v h , p h ) -b h (u h , q h ) + s h (p h , q h ) = Ω f •v h , ∀(v h , q h ) ∈ X h
where a h (u h , v h ) := ∑ d i=1 a sip h (u h,i , v h,i ), b h (p h , v h ) := -∑ F∈F i h F p h { {v h } }•n F and s h (p h , q h ) := ∑ F∈F i h h F F p h q h . Notice, in particular, the use of the sum keyword.

toire Jean Kuntzmann, University of Grenoble) are gratefully acknowledged. We also wish to thank all the contributors to the Arcane platform.

Listing 5

 5 Linear and bilinear formsLinear and bilinear forms are obtained as sums of linear and bilinear terms resulting from the composition of TestFunctions and TrialFunctions (or unary modifications thereof) by suitable tensor contractions. Examples of tensor contractions in Listing 1 are the dot and * operators. Products by functions of space, time and possibly discrete variables are also allowed. In Listing 1 we also display examples of geometric operators such as N() and H(), which allow to access face normals and element diameters respectively. Unary modifiers encountered in Listing 1 are grad, avg and jump, corresponding, respectively, to the broken gradient on S h and to the average and jump operators defined by[START_REF] Arnold | An interior penalty finite element method with discontinuous elements[END_REF].When ap-Stokes problem 1 CCGSpace::VectorTrialFunction uh(d); 2 CCGSpace::VectorTestFunction vh(d); 3 P0Space::TrialFunction ph; 4 P0Space::TestFunction qh; 5 Range::Index i(Range(0,dim-1)); 6 Form2 ah, bh, sh; 7 ah = integrate(All<Cell>::items(T h ), 8 sum(i)(dot(grad(uh(i)),grad(vh(i))) )) 9 +integrate(Internal<Face>::items(T h ), 10 sum(i)(-dot(fn,avg(grad(uh(i))))) * jump(vh(i)) 11 -jump(uh(i)) * dot(N(),avg(grad(vh(i)))) 12 +η/H() * jump(uh(i)) * jump(vh(i)))); 13 bh =-integrate(Internal<Face>::items(T h ),jump(ph) * dot(N(),avg(vh))); 14 sh = integrate(Internal<Face>::items(T h ),H() * jump(ph) * jump(qh));

Table 1

 1 span and gradient template parameters for the discrete spaces of §2

	Space S h	span	gradient

Acknowledgements. Fruitful discussions with Christophe Prud'homme (Labora-

Listing 3 Assembly of a bilinear and linear contribution (A represents here the global matrix b the global right-hand side vector) 1 LinearCombination<r> l r u , l r v ; 2 / / Assemble a bilinear contribution into the left-hand side 3 BilinearContribution<r> blc(γ, l r u , l r v ); typical assembly pattern is described in Listing 3. In particular, line 4 is the programming counterpart of (18).

Functional front-end

A further level of abstraction can be reached defining a DSL that allows to conceal all technical details and provide only the relevant components in a form as close as possible to the mathematical formulations of §2. We focus here, in particular, on the programming equivalent of discrete spaces and bilinear forms.

Function spaces

Incomplete broken polynomial spaces defined by ( 6) are mapped onto C++ types conforming to the FunctionSpace concept detailed in Listing 4. The actual types are generated by a helper template class parametrized by a containing polynomial space, labeled span, and a piecewise constant gradient reconstruction, labeled gradient (labels for template arguments are here defined using the boost::parameter library). An example of usage is provided in lines 2-4 in


Daniel Laskar
email: laskar@pse.ens.fr

Imprecision of central bank announcements and credibility

Keywords: Central bank transparency, central bank announcements, imprecise announcements, credibility JEL classification: E58, E52 central bank transparency, central bank announcements, imprecise announcements, credibility

published or not. The documents may come

Introduction

The basic economic reason in favor of central bank (CB) transparency is that more transparency allows the private sector (PS) to have more information. A more informed PS should make better decisions, which should be beneficial. Of course, things may be more complicated. The theoretical literature on CB transparency has exhibited different situations where such an argument may not hold and where, as a consequence, transparency may be harmful the CB usually cares about its credibility and therefore is trying to make announcements which will appear to be true ex post [START_REF] Crawford | Strategic Information Transmission[END_REF] . Second, the CB does not always make precise quantitative announcements. A part of the CB's announcements consists in some rather qualitative imprecise statements [START_REF] Van Der Cruijsen | The Economic Impact of Central Bank Transparency: A Survey[END_REF] .

The aim of the present paper is to develop an analysis which simualtenously and explicitly considers these two aspects, and examine how they interact. If there are signalling costs due to the credibility concern of the CB, it may be possible to make credible precise announcements. Then, does this mean that the CB will choose to make these precise announcements? The CB may actually prefer to make imprecise announcements. But to examine this point, we must consider how imprecise announcements can be equilibria when there are some signalling costs. This mean that the analysis, which exhibits these imprecise announcements as the only possible equilibria in the cheap talk case, has to be reconsidered in the presence of such costs. These are the issues that we will consider.

In order to simplify the analysis, to be able to solve the model analytically, and to easily relate this analysis to the previous studies found in the literature, we consider a simple static model of credibility of monetary policy. The CB has some useful private information on some underlying variable affecting monetary policy, and makes some announcement about this variable. As the CB is concerned by its credibility, it is assumed that the CB bears some cost when it does not transmit its information in a truthful way. The model is presented in Section 2.

As indicated above, we consider equilibria with precise announcements, and also equilibria with imprecise announcements. These are considered in Section 3. In Section 4, we examine the issue of the choice of the CB between these equilibria. In particular, we want to know whether, and under what conditions, the CB prefers imprecise rather than precise announcements. This is important because, as we will see, from the point of view of society, imprecise announcements are less efficient than precise announcements. Finally, Section 5 concludes.

Model

We consider a very standard and simple static model of monetary policy, where a credibility issue arise because the CB has a desired level of employment which is greater than the level of employment desired by the private sector [START_REF] Gersbach | Forward Guidance for Monetary Policy: Is It Desirable?[END_REF] .

The private sector (PS) is made of a large number of identical sectors and, in each sector, the wage setters want to stabilize employment according to the utility function

U P S = -n 2 (1)
As all sectors are identical, the variable n can be taken to represent aggregate employment. In (1) , the desired level of employment of the PS has been normalized to zero. Once the nominal wage has been fixed, employment is determined by labor demand according to

n = -α (w -π) + η (2)
where w represents the (log of the) nominal wage (which, as all sectors are alike, is the same in all sectors), π is the inflation rate, and wπ is the (log of the) real wage (because the last period (log of the) price level p -1 has been normalized to zero). Equation (2) indicates that employment is a decreasing function of the real wage. The variable η is a zero-mean random variable which represents a shock which affects employment (a productivity shock). We assume that the CB has some private information about η and, for simplicity, the CB will be assumed to know η 8 . The timing of the decisions of the CB and of the PS is as follows. At the beginning of the period, the CB makes some announcement about the shock η. After having heard the CB's announcement, the PS determines the nominal wage variable w, and, subsequently, the CB determines the inflation rate π. The CB has the utility function

U CB = -(n -µ) 2 -ψ(π -π) 2 -ξ (η an -η) 2
(3) [START_REF] Gersbach | Forward Guidance for Monetary Policy: Is It Desirable?[END_REF] The model is taken from Rogoff (1985) and is in the line of the models of credibility of monetary policy of Barro and Gordon (1983) and Kydland and Prescott (1977). [START_REF] Kydland | Rules Rather Than Discretion: The Time Inconsistency of Optimal Plans[END_REF] We have chosen to present the analysis by considering that the variable on which the CB has private information is the shock η (Therefore, in terms of signalling theory, the "type" of the CB is given by the value of η). However, we could as well have taken some other variable which affect monetary policy. For example, we could have assumed that the variable on which the CB has private information is the CB's inflation target π (see (3) just below), as in the signalling model of Palmqvist (1999). Then, the CB's "type" would be given by the value of π. It could easily be seen that, in this case, we would end up with exactly the same formal analysis and results as those we obtain here with the variable η.

The CB wants to stabilize the level of employment n around a desired level µ > 0, which is therefore greater than the level of employment desired by the PS [START_REF] Palmqvist | Why Central Banks Announce their Objectives: Monetary Policy with Discretionary Signalling[END_REF] . The CB also wants to stabilize inflation around some desired level π.

The variable η an is the "announced value" of η. It is assumed that the PS can observe η ex post, and the term ξ (η anη) 2 in (3) represents the cost incurred by the CB, due to a loss of credibility, when the CB makes its announcement. This cost is equal to zero when the CB transmits its information in a truthful way, by simply announcing the true value η. But, in general, when the CB makes some other announcements, the CB will bear some credibility cost, because the lack of truthfulness in the announcement will impair the credibility of the CB [START_REF] Rogoff | The Optimal Degree of Commitment to an Intermediate Monetary Target[END_REF] .

In the case of precise announcements, where the CB announces a value of η, then η an is simply this announced value. In the case of imprecise announcements, where the CB announces that η belongs to some interval, it will be assumed, for simplicity, that η an is the mean of this interval. This announced value η an would correspond to the value of η expected by the PS if the PS did not take into account any other information than what the CB says. Note that even if η did in fact belong to the announced interval, the CB, in general, would still be penalized, because, by being imprecise, the CB would not be telling the whole truth. Thus, if we have η an 6 = η, the PS would be mislead by such an announcement, because the imprecise announcement would tend to give wrong expectations to the PS, which is penalized.

Let z e the expectation that the PS has on any variable z after the CB's announcement. Being small, each wage setter takes expected inflation π e as given. Maximizing the expected utility EU P S under the constraint (2) gives the nominal wage w = π e + η e α (4) [START_REF] Palmqvist | Why Central Banks Announce their Objectives: Monetary Policy with Discretionary Signalling[END_REF] A usual arguments involve the presence of distortions due to taxes. [START_REF] Rogoff | The Optimal Degree of Commitment to an Intermediate Monetary Target[END_REF] Our model takes as exogenous this cost due to a loss of credibility, and does not try to explain why the CB would care to loose credibility by making untruthful announcements. This cost takes into account the fact that, in the real world, CBs do care about their credibility and about making truthful announcements. Some quadratic cost of deviating from an announced policy because the CB cares about its credibility, is introduced in Gerbasch and Hahn (2008), who use this cost function to study the trade-off between gains from commitment and losses of flexibility implied by forward guidance.

From (2) and (4) , we get11

n = α (π -π e ) + η -η e (5)
Then, the CB chooses π which maximize U CB . From (3) and (5) , the first order condition is

α (n -µ) + ψ (π -π) = 0 (6)
which, using (5) , can be written

π = 1 α 2 + ψ £ α 2 π e + ψπ + αµ -α (η -η e ) ¤ (7)
Taking the expected value of each member of (7) and substracting to (7)

gives π -π e = - α α 2 + ψ (η -η e ) (8)
From ( 5) and (8) we get

n = ψ α 2 + ψ (η -η e ) (9)
From (1), (3) , (6) and (9) we get

U CB = - α 2 + ψ ψ • ψ α 2 + ψ (η -η e ) -µ ¸2 -ξ (η an -η) 2 (10)
U P S = - µ ψ α 2 + ψ ¶ 2 (η -η e ) 2 (11)
The rest of the paper will be devoted to the issue of the announcement strategy of the CB.

Equilibria

We will first examine whether there exists an equilibrium where the CB announces a precise value of η. Then, we will also look for equilibria where the CB makes some imprecise announcements.

Precise announcements

We consider the case where the CB announces some precise value η an . This announced value is a function of the true value η. From (10) , we see that the CB has an incentive to manipulate the expectation η e of the PS. The CB would like η e to be equal to η-α 2 +ψ ψ µ, and therefore to be smaller than η by an amount proportional to the gap µ between the desired levels of employment of the CB and of the PS. The reason is that, according to (4) , a lower value of η e decreases the nominal wage w. This tends to increase employment and therefore to make employment closer to the CB's employment target. This means that the CB would like the PS to believe that the employment shock η is smaller than what it is in reality.

Thus, it seems natural to look for an equilibrium where the CB's strategy is η an = ηθ, where θ > 0 is some given "bias" in the CB's announcement [START_REF] Stein | Cheap Talk and the Fed: A Theory of Imprecise Policy Announcements[END_REF] . At the equilibrium, the PS takes the strategy of the CB as given and forms its expectations accordingly. This implies

η e = η an + θ (12)
At the equilibrium, the CB knows that the expectation η e of the PS is given by (12) . From (10) and (12) , we get

U CB = - α 2 + ψ ψ • ψ α 2 + ψ (η -η an -θ) -µ ¸2 -ξ (η an -η) 2 (13)
The equilibrium condition is that the strategy η an = ηθ is an optimal strategy when the CB maximizes U CB given by (13) . From (13) , the first order condition dU CB dη an = 0 for maximizing U CB yields η an = η -ψθ+(α 2 +ψ)µ ψ+(α 2 +ψ)ξ . Consequently, the strategy η an = ηθ is an equilibrium if and only if we have ψθ+(α 2 +ψ)µ ψ+(α 2 +ψ)ξ = θ, which is equivalent to θξ = µ. Therefore, we get: Proposition 1 In the cheap talk case ξ = 0, there is no equilibrium with precise announcements.

In the case ξ > 0, there is an equilibrium with precise announcements. It is given by η an = η -e θ, where the equilibrium announcement bias e θ is given by

e θ = µ ξ (14)
The result can be interpreted in the following way. If the CB announced a value of η an marginally smaller than η -θ, the CB would have the marginal benefit of more favorable expectations of the PS, but it would also have to bear the marginal cost due to the corresponding greater loss of credibility. The marginal benefit is obtained by differentiating the first term of the right hand side of (13) . Using the equality η an = ηθ, this gives a marginal benefit equal to 2µ. The marginal cost is obtained by differentiating the second term of (13) , which (using η an = ηθ) gives a marginal cost equal to 2ξθ. An equilibrium is obtained when the marginal cost is equal to the marginal benefit.

Imprecise announcements

Precise announcements exist in the case ξ > 0 but may not be the unique equilibrium. As the analysis of Crawford and Sobel (1982) suggests, there may also exist equilibria with imprecise announcements. Crawford and Sobel (1982) only considered the cheap talk case ξ = 0, but some similar analysis and results may also hold in the case ξ > 0. Therefore, we will also look for equilibria with imprecise announcements in the context of our model. As in the illustrative example of Crawford and Sobel (1982), we will make a uniform distribution assumption in order to simplify the analysis. We will assume that η is uniformly distributed on [η m , η M ] [START_REF] Vickers | Signalling in a Model of Monetary Policy with Incomplete Information[END_REF] . Then, consider a partition (a 0 , a 1 , ..., a N-1 , a N ) of [η m , η M ], where we have a 0 = η m < a 1 < ... < a N-1 < a N = η M . In the same way as for the equilibrium with perfect precision, the CB's announcement strategy may have some bias. Consider a given bias θ > 0. Then, the strategy of the CB consists in announcing that η belongs to the interval [a i-1θ, a iθ] whenever η belongs to the interval ]a i-1 , a i ] , for i = 1, ..., N 14 . [START_REF] Vickers | Signalling in a Model of Monetary Policy with Incomplete Information[END_REF] As a uniform distribution has a finite support, such an assumption, however, leads to the undesirable property that the CB may announce values outside the support. For, as the equilibrium with perfect precision has underlined, the CB's announcements may exhibit some bias θ which leads the CB to announce a value which is lower than the true value. This implies that the announced value would be ouside the support when η is close enough (i.e. by an amount smaller than θ) to the lower limit η m of this support. Therefore, we may rather view the uniform distribution assumption as a limit case. We could assume that the support of η is infinite but that the distribution is very close to a uniform distribution. Formally, we could assume that, inside the interval [η m , η M ] , the density is constant, and that the probability that η does not belong to this interval is equal to ε. Then, we could consider the uniform distribution as a limit case case when ε goes to zero.

14 It would remain to specify the strategy when η is equal to η m = a 0 . We could assume, for example, that, in this case, the CB announces that η belongs to [a 0 , a 1 ] . (However, As the PS knows the CB's strategy, it takes into acount this bias θ when it determines its expectations, and therefore the PS knows the true interval to which η belongs. Therefore, because of the uniform distribution assumption on [η m , η M ], when η belongs to ]a i-1 , a i ] we have

η e = a i-1 + a i 2 (15)
On the other hand, we have previously assumed that the "announced value" η an , which enters the loss function (10), is equal to the mean of the announced interval. As this announced interval is [a i-1θ, a iθ] , this gives

η an = a i-1 + a i 2 -θ (16)
From (15) and (16) , we see that (12) and therefore also (13) are still valid in the case of imprecise announcements.

For each given value of the bias parameter θ > 0, we will look for partition equilibria (a 0 , a 1 , ..., a N-1 , a N ). Note that we can limit the analysis to the case θ < e θ because any partition equilibrium associated to a bias θ ≥ e θ, if it existed, would be more imprecise than the equilibrium with precise announcements without having a smaller bias. It would therefore never be chosen by the CB 15 .

We obtain the following result (see Appendix 1 for the proof):

Proposition 2 To each value of the announcement bias θ such that 0 ≤ θ < e θ, we can associate an equilibrium partition 16 (a * 0 , a * 1 , ..., a * N * -1 , a * N * ). The number N * of intervals of the equilibrium partition is

N * = * - 1 2 + 1 2 µ 1 + 2 B ¶1 2 + ( 17
)
where hxi denotes the smallest integer greater than or equal to x, and where B is given by

B = 1 η M -η m α 2 + ψ (α 2 + ψ) ξ + ψ (µ -ξθ) > 0 (18)
other assumptions, like announcing the value η m , would lead to the same results, because the probability that η is equal to η m is equal to zero.) 15 From (20) below, it will be clear that both the imprecision and the bias of the announcement decrease the expected utility of the CB. 16 As in Crawford and Sobel (1982), there is actually a finite number of equilibria obtained for all values of N such that N ≤ N * . However, as in Crawford and Sobel (1982), all equilibria corresponding to N < N * are dominated by the equilibrium obtained for N = N * (i.e. they are worse for both players). They will therefore be disgarded for the static comparative analysis that we will do.

Let σ 2 η be the residual variance of η that the PS expects to have after having heard the CB's announcement. We have

σ 2 η = 1 3 • 1 4N * 2 + B 2 (N * 2 -1) ¸(η M -η m ) 2 (19)
When we have ξ > 0, the residual variance σ 2 η is a decreasing function of the bias θ : the greater the bias, the smaller the imprecision of the announcement. Thus, there is a trade-off between the imprecision and the bias of the announcement. The imprecision goes to zero when θ goes to e θ.

The residual variance σ 2 η is a decreasing function of the weight ξ17 .

Proposition 2 exhibit similarities with the results obtained in the cheap talk analysis of Crawford and Sobel (1982). In fact, it can be shown that, in the cheap talk case ξ = 0, through some adequate change of variables, the present model can be put under the form of the illustrative model of Crawford and Sobel (1982) (see Appendix 1). Furthermore, in the case ξ > 0, the analysis leads to the same kind of solutions as in the case ξ = 0, provided that some parameter of the model is modified. The change concerns parameter B, which, according to (18) , depends on θ and ξ.

Thus, this result generalizes the result of Crawford and Sobel (1982) to the case ξ > 0. As (18) indicates, in the cheap talk case ξ = 0, the equilibrium is the same whatever the bias θ is18 , but, in the case ξ > 0, parameter B, and therefore the equilibrium, depend on the bias θ.

As indicated in Proposition 2, the imprecision, given by σ 2 η , is smaller when the bias θ increases. In order to interpret this result, first note that, in the cheap talk case ξ = 0, imprecise announcements can be an equilibrium because it may not be advantageous for the CB to announce a lower interval and have the private sector have the implied lower expected value for η. For, from (10) , it is beneficial for the CB to have η e lower than η by some amount, but this can be overdone. If η e is too low, this could be worse. Such a case will occur when, broadly speaking, intervals are large enough. Second, when we have ξ > 0, there is an additional cost to announce the lower interval. Consequently, a partition with intervals of smaller lengths could become an equilibrium because of this additional cost when we have ξ > 0. The greater θ is, the greater this cost is (because the corresponding marginal cost is equal to 2ξθ, as we have previously seeen), and, therefore, the smaller the lengths of intervals can be. This implies a smaller imprecision of the announcements when θ increases. This means that there is a trade-off between the imprecision and the bias of the announcement.

As the imprecision goes to zero when θ goes to e θ, the equilibrium with precise announcements is the limit case of the equilibria with imprecise announcements when θ goes to e θ.

Finally, for a given bias θ, as Proposition 2 indicates, we have the rather intuitive result that, when the weight ξ increases, the imprecision of the announcement is reduced. 4 Optimal choice of the central bank

Results

In this section we assume ξ > 0 (which means that CB announcements are not cheap talk) 19 . We consider the choice of the CB between the equilibria we have considered. We consider the ex ante choice of the CB, when the CB does not yet know the value of the shock η.

These equilibria are obtained by taking a value of θ belonging to the closed interval h 0, e θ i , the value θ = e θ corresponding to the equilibrium with precise announcements. For all the equilibria considered, we have η an = η eθ. Replacing η an by η eθ in (10) , and noting that we have E(η eη) = 0 and E(η eη) 2 = σ 2 η , we get

EU CB = - µ ψ α 2 + ψ + ξ ¶ σ 2 η -ξθ 2 - α 2 + ψ ψ µ 2 (20)
The expected utility of the CB is a decreasing function of σ 2 η and of θ. Therefore, the CB dislikes both imprecision and bias. The CB dislikes imprecision for two reasons. First, imprecise announcements lead the PS to make less efficient decisions. Second, when there is some imprecision in the announcement, the CB does not tell the whole truth and is therefore penalized.

The CB dislikes a bias in the announcement because of its credibility concern (the CB does not tell the truth). And there is no gain for the CB in having an announcement bias because, at the equilibrium, the PS corrects for the bias: the CB cannot manipulate the expectations of the private sector by introducing an announcement bias.

In the last section, we have seen that there is a trade-off between imprecision and bias. This implies that the CB has to make a choice between the imprecision and the bias of its announcements. In this section we will consider this choice and determine the optimal amount of bias and imprecision that the CB would like. The CB chooses θ ∈ h 0, e θ i which maximizes EU CB , where σ 2 η is the function of θ given by (17) , (18) and (19) . In order to be able to solve this problem analytically, we will limit our analysis to the case where the credibility problem of the CB is, in some sense, small. It will be assumed to be small relatively to the amount of uncertainty that the PS has on the variable on which the CB has some private information. More precisely, from (10) , we can see that the CB would like to have the PS believe that η e is equal to η -α 2 +ψ ψ µ instead of η. This gap α 2 +ψ ψ µ is at the origin of the credibility problem of the CB. We will assume that the magnitude of this gap is small when compared to the amount of uncertainty that the PS has on η. Under our assumption of a uniform distribution for η, this amount of uncertainty can be represented by the length η Mη m of the support of this distribution. Therefore, we will assume that

1 η M -η m α 2 +ψ ψ µ is small. As, from (18) , we have 0 < B ≤ 1 η M -η m α 2 +ψ
ψ µ, this implies that B is small. Then, we find (see Appendix 2 for a proof): (and therefore we have 0 < θ < e θ), and the imprecision c σ 2 η is a decreasing function of ξ, satisfying lim ξ→ξ c σ 2 η = 0. -In the case ξ ≥ ξ, the CB prefers to have precise announcements (we have c σ 2 η = 0). In this case, we have b θ = e θ = µ ξ . Therefore the bias is a decreasing function of ξ, and goes to zero when ξ goes to infinity.

In the case 0 < ξ < ξ, the bias e θ = µ ξ , which would be required to have the equilibrium with precise announcements, is too high. The CB prefers the partition equilibrium with the bias θ = µ ξ , which is lower than e θ. In this case, an increase in ξ reduces the imprecision without changing the bias. When ξ reaches the threshold value ξ , the bias e θ becomes equal to θ, and the CB prefers the equilibrium with precise announcements. Further increases in ξ has only the effect of reducing the bias, which is equal to e θ = µ ξ , because the announcement with perfect precision is always preferred by the CB in this case.

Implications

The cheap talk case appears as an extreme case where precise announcements do not exist. But, as soon as the CB cares about its credibility, precise announcements can be an equilibrium. However, we have shown that the CB may nonetheless choose to make imprecise announcements. Thus, our model is consistent with the fact that, in the real world, we may observe both imprecise announcements and a credibility concern by the CB. However, as Proposition 3 also indicates, precise announcements can also be an equilibrium. It is not necessary to have an infinite weight ξ, and therefore to have truthful announcements, in order to have precise announcements. It is enough that this weight becomes larger than or equal to some threshold value ξ.

This last result has welfare implications. For, as it is usually the case in the literature, let us assume that society cares about employment and inflation, and, for simplicity, consider the case where it weights these two objectives in the same way as the CB. This means that society's utility function U s can be obtained by replacing ξ by zero in the expressions giving the CB's utility function. From (20) , we therefore have

EU s = - ψ α 2 + ψ σ 2 η - α 2 + ψ ψ µ 2 (21)
Equality (21) underlines that society only cares about the imprecision of the announcement. Society does not care about the bias of the announcement because it does not have the credibility concern that the CB has and, as we have seen, the presence of an announcement bias does not change the behavior of the PS because the PS corrects for the bias (we have η e = η an + θ). Consequently, society always prefers precise anouncements to any equilibrium with imprecise announcements. This implies that, as long as the CB chooses to make imprecise announcements, CB announcements are suboptimal. This suboptimality gets smaller when the weight ξ gets larger. Such a suboptimality completely disappears when the weight ξ becomes larger than or equal to ξ.

As a consequence, if we are in a situation where we have ξ < ξ, any factor which increases the weight given by the CB to transmitting its information in a truthful way, may be beneficial. In fact, it has been argued that the observed shift of CBs toward more transparency could be explained by the previous move of CBs toward more independence. The reason is that this made it more necessary for the CBs to explain their policies and therefore to be more transparent. This kind of argument is usually given as the other main reason for having CB transparency, beside the economic reason. From this point of view, the present analysis allows us to relate in some way these two kinds of arguments for transparency: the economic reason and the shift toward more CB independence. The increased pressure put on an independent CB to give its information in a transparent way may be interpreted as an increase in the weight ξ in our model. Our analysis can therefore be used to examine the implications of this stronger requirement for transparency, due to greater CB independence, in a model which takes into account some economic consequences of transparency. From our analysis, as we could expect, this greater concern about being transparent makes the CB more transparent, in the sense that, from proposition 3, either the bias or the imprecision of CB announcements decreases when the weight ξ increases. When we start from a situation where we have ξ < ξ, this increase in the CB's pressure for transparency due to the CB's increased independence is beneficial, because it takes the form of reduced imprecision. If its effect on ξ is large enough, it could even completely eliminate the suboptimality of announcements by leading the CB to make precise announcemnts 20 .

Conclusion

We have considered a model where the CB has some useful private information but where a credibility problem may prevent the CB to transmit this information to the PS. We have assumed that the CB, which is concerned by its credibility, has to bear some cost when it does not truthfully reveal its private information. We have shown that this makes precise announcements of the CB credible, and that this would allow the CB to transmit its private information to the PS. The required marginal cost of deviating from the announcement is produced by some announcement bias (a greater bias giving a greater marginal cost). This implies that precise announcements exhibit some announcement bias.

However, beside this equilibrium with precise announcements, there are also equilibria with imprecise announcements. To each level of some an-20 Independent CBs may also exhibit a less acute credibility problem. This effect will reinforce the effect, going through a larger weight ξ, that we have just considered. For example, if more independence also implies a smaller value of µ (µ represents the gap between the CB's and the PS's employment targets), then the value of ξ, given in Proposition 3, decreases, which also makes the condition ξ ≥ ξ more easily fulfilled. nouncement bias, there corresponds some imprecision of announcements. The greater the bias is, the more precise the announcements can be. The reason is that a greater bias implies a greater marginal cost of deviating and, consequently, more precise announcements become credible. At the limit, when the bias becomes large enough, precise announcements become possible.

As a consequence, there is a trade-off between the imprecision and the bias of the CB's announcements. As both the bias and the imprecision are costly to the CB, this implies that the CB has to make some optimal choice between the imprecision and the bias of its announcements.

We have studied this optimal choice of the CB. In order to be able to solve this problem analytically, we have limted the analysis to the case where the credibility problem of the CB is small, in proportion to the amount of uncertainty concerning the variable on which the CB has some private information. We have found that when the weight given to the credibility concern (which makes untruthful announcements costly) is smaller than some threshold value, then the CB prefers to make imprecise announcements. And, the lower this weight is, the more imprecise the annoucements chosen by the CB is. Although precise announcements can be an equilibrium, the CB prefers to make imprecise announcements as long as this weight is not too large. Thus, in this case, we have both imprecise announcements and some credibility concern of the CB, which are two features which can be observed in the real world.

On the other hand, when the weight given to the credibility concern is greater than or equal to this threshold value, precise announcements are preferred by the CB. This means that it is not necessary to have an infinite weight, and consequently to constrain the CB to truthfully reveal its information, if we want the CB to make precise announcements. What is required is only that the CB prefers to eliminate all imprecision in its announcements.

From the point of view of society, only the imprecision of the CB's announcements matter, because the PS can correct for the bias but cannot correct for the imprecision of annoucements. Therefore, when the CB prefers to make imprecise announcemnts, this is suboptimal from the point of view of society. However this suboptimality disappears when the CB prefers precise announcements, which occurs when the CB gives a large enough weight to truthfully revealing its information.

Our analysis also permits us to relate two factors which, in the literature, were given as separate explanations to the observed shift of CBs toward more transparency. The first is that the previous shift toward CB independence required CBs to explain their policies to the public, and, for that, required them to be more transparent. In our model, this can be interpreted as an increase in the weight given to truthfully transmitting the information that CBs have. The second factor concerns the beneficial economic effects of transparency. This beneficial effect is taken into account in our model because, in this model, more information allows the PS to make better decisions. Our analysis underlines that the first factor leads the CB to transmit its useful private information to the PS with more precision, which is beneficial from the point of view of society. Furthemore, if CB independence increases the weight given to more truthful announcements by a sufficient amount, this could lead to full transmission of the CB's information through precise announcements.

The model we used was a simple static model of monetary policy, in which a credibility issue arises. This has allowed us to simplify the analysis and to compare it to other existing studies. However, it would be useful to try to deal with the same kind of issues in a dynamic model. As we know21 , a dynamic model of monetary policy may lead to credibility issues even in the absence of a gap between the employment targets of the CB and of the wage setters (while introducing such a gap was necessary to get a credbility issue in our static model).

APPENDICES

Proof of Proposition 2

For a given θ, the equilibrium condition is that the CB does not (strictly) prefer to announce that η belongs to an other interval than [a i-1θ, a iθ] when η belongs to ]a i-1 , a i ] . And, by continuity, this has to hold also when η belongs to the closed interval [a i-1 , a i ]. For a partition (a 0 , ..., a N ), there are only N discrete possible values for η an which are given by η an = a j-1 +a j 2

, j = 1, ..., N. Therefore, a partition is an equilibrium if and only if, when η belongs to [a i-1 , a i ] , the value η an = a i-1 +a i 2 is at least as good as any other value η an = a j-1 +a j 2

, j = 1, ..., N, j 6 = i. But, as the function U CB given by ( 13) is a concave quadratic function of η an , this is actually equivalent to having η an = a i-1 +a i 2

be at least as good as η an = a i +a i +1 2 and η an = a i-2 +a i-1

2

, which are the two possible adjacent values for η an . These two conditions can be written

U CB ¡ a i-1 +a i 2 ¢ -U CB ¡ a i +a i+1 2 ¢ ≥ 0, i = 1, ...N -1, and U CB ¡ a i-1 +a i 2 ¢ - U CB ¡ a i-2 +a i-1 2 ¢ ≥ 0, i = 2, ..., N, for all η belonging to [a i-1 , a i ]. Using (13) ,
the first of these two conditions is equivalent to f (i, η) ≥ 0 for i = 1, ..., N -1, and for all η belonging to [a i-1 , a i ], where we define

f (i, η) ≡ µ ξ + ψ α 2 + ψ ¶ (a i+1 + 2a i + a i-1 -4η) + 4(µ -ξθ) (22)
As f is a decreasing function of η, the inequality f (i, η) ≥ 0 holds for all η ∈ [a i-1 , a i ] if and only if we have f (i, a i ) ≥ 0. Therefore the first condition can be written f (i, a i ) ≥ 0 for i = 1, ..., N -1.

In the same way, it can easily be seen that the second condition is equivalent to f (i, a i ) ≤ 0 for i = 1, ..., N -1.

Consequently, the equilibrium condition can be written f (i, a i ) = 0 for i = 1, ..., N -1. Using (22) , this leads to the difference equation

a i+1 -2a i + a i = -4 (η M -η m ) B ( 23
)
where B is given by (18) of proposition 2. If we introduce the "normalized" variables η

0 ≡ η M -η η M -η m and a 0 i ≡ η M -a N -i η M -ηm , then η 0 belongs to [0, 1] and (a 0 0 , a 0 1 , ...a 0 N ) is a partition of [0, 1]
where we have a 0 0 = 0, a 0 N = 1 and a 0 0 < a 0 1 < ... < a 0 N , as in Crawford and Sobel (1982). From (23), the a 0 i satisfiy the difference equation a 0 i+1 -2a 0 i + a 0 i = 4B, i = 1, ..., N -1. This is a difference equation formally identical to the one of the illustrative example of Crawford and Sobel (1982) (equation 21 p. 1441). However, note that, from (18) ,in our model, the constant term 4B on the right hand side of this difference equation depends on the weight ξ and on the bias θ 22 .

According to the solutions found in the illustrative example of Crawford and Sobel (1982), this leads to a finite number of solutions. However, the equilibrium with the "finest" partition can be selected as the only relevant partition equilibrium, because this equilibrium partition dominates all the other partition equilibria (it always gives higher utility levels for both the CB and the PS). This is the partition which has the greater number of interval N * . The value of N * is given by (17) (see Crawford and Sobel (1982) p.1441) Then, using the equality σ2 η = (η Mη m ) 2 σ 2 η 0 and the result for σ 2 η 0 found in Crawford and Sobel (1982) (equation 25 p.1442), we get equation (19) of Proposition 2. 22 In the case ξ = 0, our model is actually formally identical to the one of the illustrative example of Crawford and Sobel (1982). If we take b ≡ -

α 2 +ψ ψ 1 η M -η m µ, V CB ≡ α 2 +ψ ψ 1 (η M -η m ) 2 U CB and V P S ≡ ³ α 2 +ψ ψ ´2 1 (η M -η m ) 2 U P S , we can write our model under the form V CB = -(η 0e -η 0 -b)
Futhermore, Crawford and Sobel (1982) have shown that σ 2 η 0 is an increasing function of B, which goes to zero when B goes to zero. This result, together with (18) , implies that σ 2 η is a decreasing function of both θ and ξ, and that σ 2 η goes to zero when θ goes to e θ.

Proof of

From (18) and (20), we get

dEU CB dθ = 2ξg (θ) (25)
where the function g (.) of θ is given by

g (θ) = 1 2 1 η M -η m dσ 2 η dB -θ (26)
Therefore, from (25) , as we have ξ > 0, the sign (and the nullity) of dEU CB dθ is the same as g (θ) . From (19) , when we replace N * by x, we have:

dσ 2 η dB = 1 3 (η M -η m ) 2 • - 1 2
x 0 (B)

x(B) 3 + 2B 2 x(B)x 0 (B) + 2B ¡ x(B) 2 -1 ¢ ¸(27)
where x(B) is the function given by (24) . Therefore, from (24) , the derivative x 0 (B) is

x 0 (B) = - 1 2 B -2 µ 1 + 2 B ¶ -1 2 (28)
From (17) and (28) , when B is small, we have the equivalences:

x(B) v (2B) - 3 (η Mη m )2 . Consequently, from (26) , g(θ) and therefore also dEU CB dθ have the sign of 1 6 (η Mη m )θ when B is small.

Consequently, in the case 1 6 (η Mη m ) < e θ, which is eqivalent to having ξ < ξ ≡ 1 η M -η m 6µ, there is an interior solution b θ = 1 6 (η Mη m ) = µ ξ . This is the equilibrium with imprecise announcements which corresponds to the partition equilibrium obtained for the optimal bias b θ = 1 6 (η Mη m ). In this case, as long as we have ξ < ξ, when ξ increases, the bias remains equal to [START_REF] Barro | A Positive Theory of Monetary Policy in a Natural-Rate Model[END_REF] 6 (η Mη m ) . Therefore, from Proposition 2, when ξ increases, the imprecision σ 2 η decreases, and goes to zero when ξ goes to ξ. In the case ξ ≥ ξ , or equivalently In the above proof, the function hxi has been replaced by x, where x, given by (24) , is the term inside h.i in (17) . We will now argue that, when B is small, this can be justified. As, from (18) , there is a one to one relationship between θ and B, the optimization problem of the CB can be seen as finding the value of B which maximizes EU CB . Then, consider the problem of maximizing EU CB where B, instead of being a continuous variable, is a discrete variable obtained by only taking the values of B which, according to (24) , give integer values of x. Then, for this optimization problem, where B takes these discrete values, having hxi instead of x would make no difference. Therefore, any error due to the substitution of x to hxi would be negligible if we can show that the difference between taking this discrete B variable instead of the continuous B variable would itself be negligible. This would be the case if the lengths of the intervals between two adjacent values of the corresponding discrete B variable are negligible.

Consider such an interval [hxi , hx + 1i] . By definition, the length of such an interval is equal to R hx+1i hxi ¯dB dx ¯dx. This would be negligible if ¯dB dx ¯is negligible. But dB dx is equal to 1 x 0 (B) , and, according to (29) , we have x 0 (B) v -(2B) -3 2 . This implies ¯dB dx ¯∼ (2B)

Proposition 3

 3 Let b θ and c σ 2 η be the values of θ and σ 2 η chosen by the CB. Define ξ ≡ 6µ η M -η m > 0. Then, we have: -In the case 0 < ξ < ξ, the CB prefers to have imprecise announcements (we have c σ 2 η > 0). In this case, the bias b θ is equal to θ = 1 6 (η Mη m ) = µ ξ

1 2x 3 2 2 η

 132 0 (B) v -(2B) -B→0 (2B 2 x(B)x 0 (B)) = -1 2 ; and lim B→0 (2B (x(B) 2 -1)) = 1. Using (27) , these equalities imply lim B→0 dσ dB = 1

  1 6 (η Mη m ) ≥ e θ, we have dEU CB dθ > 0 for all θ ∈ h 0, e θ h , and we have dEU CB dθ ≥ 0 for θ = e θ. We therefore get the corner solution b θ = e θ, which corresponds to precise announcements. This gives Proposition 3.

2. 2

 2 Justification to replacing hxi by x in (17)

  Proposition 3 2.1 Proof when we replace hxi by x in (17) . In (17) we will replace hxi , and therefore the integer N * , by the real variable x. As Appendix 2.2 below shows, this can be justified when B is small. From (17) , we have

	σ 2	Consider a partition equilibrium of bias θ ∈	h 0, e θ h . The residual variance
		x ≡ -	1 2	+	1 2	µ 1 +	B 2	¶1 2

η is given by (17) , (18) and (19) .

For some recent survey of the theoretical and empirical literature, see, for example,[START_REF] Blinder | Central Bank Communication and Monetary Policy: A Survey of Theory and Evidence[END_REF] and van der Cruijsen and Eijffinger (2007).

Standard references on these issues are[START_REF] Barro | A Positive Theory of Monetary Policy in a Natural-Rate Model[END_REF] and[START_REF] Kydland | Rules Rather Than Discretion: The Time Inconsistency of Optimal Plans[END_REF].

Models of signalling have been applied to monetary policy. For models where, through the signalling of the CB, the PS learns the CB's type, see, for example,[START_REF] Palmqvist | Why Central Banks Announce their Objectives: Monetary Policy with Discretionary Signalling[END_REF] and[START_REF] Vickers | Signalling in a Model of Monetary Policy with Incomplete Information[END_REF].

See the cheap talk model of[START_REF] Crawford | Strategic Information Transmission[END_REF].[START_REF] Stein | Cheap Talk and the Fed: A Theory of Imprecise Policy Announcements[END_REF] has applied this kind of analysis to monetary policy when the CB has some private information on the exchange rate target.

Thus,[START_REF] Blinder | Central Bank Credibility: Why Do We Care? How Do We Build It?[END_REF] has sent a questionnaire on issues related to CB credibility to the heads of all CBs which are menbers of the BIS. The answers (84 central bankers responded) indicate that "living up to its word" is ranked first in importance.

For example, the Fed has made rather imprecise qualitative statements on its future policy (see[START_REF] Rudebusch | Revealing the Secrets of the Temple: The Value of Publishing Central Bank Interest Rate Projections[END_REF]).

In the literature on CB transparency, the analysis has often relied on an expectationsaugmented Phillips curve model, given by the equation n = α (ππ e ) + ε, where the shock ε is exogenously given. However, as we see here, the shock ε should depend on transparency, because, according to (5) , ε is equal to ηη e , and therefore depends on the expectations of the private sector (more generally, on this point, see Laskar (2010)).

In fact, this particular form does not restrict the generality of the analysis. For, because of the linear quadratic structure of the model, we would only have to consider the linear CB's announcement strategy η an = θ 0 + θ 1 η, where θ 0 and θ 1 are coefficients. But, then, it can easily be shown that such a strategy can be an equilibrium only if we have θ 1 = 1. Therefore, no other equilibrium strategies would be found by using this extended set of strategies.

Note that, as the inequality θ < e θ = µ ξ has to be satisfied, then , for a given θ, the weight ξ has to satisfy the inequality ξ < µ θ .

Therefore, in the cheap talk case, it is not useful to introduce such a bias in the analysis.

In the cheap talk case ξ = 0, the CB's choice is trivial because only one equilibrium remains. For the equilibrium with perfect precision does not exist, and there is only one partition equilibrium because, from (18) , the bias θ has no effect. This is the equilibrium found in[START_REF] Crawford | Strategic Information Transmission[END_REF].

See Clarida et al. (1999) in the context of the new Keynesian model which has been widely used in the literature.

and V P S = -(η 0eη 0 ) 2 , which is similar to the model of the illustrative example of[START_REF] Crawford | Strategic Information Transmission[END_REF].

. When B is small, this is negligible relatively to B. Therefore, by having replaced hxi by x in the proof, any error we could have made on the optimal announcement chosen by the CB, would be negligible.