Joseph Mirabel

Florent Lamiraux

Manipulation planning: addressing the crossed foliation issue

This paper deals with manipulation planning. First, we propose a new tool called the constraint graph to describe the various motion constraints relative to a manipulation planning problem. Then, we describe a problem arising for some manipulation planning problems called the crossed foliation issue. We propose a extension of RRT algorithm that explores the leaves of the foliations generated by motion constraints and that solves the crossed foliation problem. Finally, we show a wide variety of problem that our approach can solve.

I. INTRODUCTION

Manipulation planning as a computational geometry problem has raised a lot of interest for the past fourty years. Pioneering works by [START_REF] Wilfong | Motion planning in the presence of movable obstacles[END_REF] and [START_REF] Alami | A geometrical approach to planning manipulation tasks. the case of discrete placements and grasps[END_REF] first considered low dimensional problems where robots and objects move in translation. [START_REF] Simon | Manipulation planning with probabilistic roadmaps[END_REF] is the first work that applies random motion planning methods developed a few years earlier [START_REF] Kavraki | Probabilistic roadmaps for path planning in high-dimensional configuration spaces[END_REF] to the manipulation planning problem. Recently, the domain has regained in interest with various variants where papers propose approaches that tackle the inherent complexity of manipulation planning. The domain is traditionally divided into several categories. Navigation among movable obstacles (NAMO) consists in planning a path for a robot that needs to move obstacles in order to reach the goal configuration [START_REF] Stilman | Planning among movable obstacles with artificial constraints[END_REF], [START_REF] Nieuwenhuisen | An effective framework for path planning amidst movable obstacles[END_REF], [START_REF] Dalibard | Manipulation of documented objects by a walking humanoid robot[END_REF]. Rearrangement planning consists in automatically finding a sequence of manipulation paths that moves several objects from initial configurations to a specified goal configuration [START_REF] Krontiris | Dealing with difficult instances of object rearrangement[END_REF], [START_REF] Ota | Rearrangement of multiple movable objects-integration of global and local planning methodology[END_REF], [START_REF] Lertkultanon | A single-query manipulation planner[END_REF]. Multi-arm motion planning has also given rise to a lot of papers [START_REF] Gharbi | Roadmap composition for multi-arm systems path planning[END_REF], [START_REF] Harada | A manipulation motion planner for dual-arm industrial manipulators[END_REF], [START_REF] Dobson | Planning representations and algorithms for prehensile multi-arm manipulation[END_REF]. From a geometric point of view, manipulation planning is a hybrid problem where discrete states (gripper A holds object B) are defined by continuous constraints on the positions of objects and robots. States are connected by manipulation trajectories that give rise to the underlying structure of a graph the nodes of which are the discrete states [START_REF] Mirabel | Hpp: a new software for constrained motion planning[END_REF]. This structure although not expressed as such is present in various papers [START_REF] Berenson | Task space regions: A framework for pose-constrained manipulation planning[END_REF], [START_REF] Jentzsch | MOPL: A multimodal path planner for generic manipulation tasks[END_REF], [START_REF] Hauser | Randomized multi-modal motion planning for a humanoid robot manipulation task[END_REF]. The partially discrete nature of the problem has also given rise to integration of task and motion planning techniques [START_REF] Cambon | A hybrid approach to intricate motion, manipulation and task planning[END_REF], [START_REF] Barry | A hierarchical approach to manipulation with diverse actions[END_REF], [START_REF] Garrett | Ffrob: An efficient heuristic for task and motion planning[END_REF], [START_REF] Srivastava | Combined task and motion planning through an extensible plannerindependent interface layer[END_REF], [START_REF] Havur | Hybrid reasoning for geometric rearrangement of multiple movable objects on cluttered surfaces[END_REF].

A. The crossed foliation issue

Surprisingly, the geometric structure of the problem, apparently well understood [START_REF] Vendittelli | Decidability of Robot Manipulation Planning: Three Disks in the Plane[END_REF] has not been investigated in depth and some simple problems are out of reach for most *This work has been partially supported by the national PSPC-Romeo 2 project, has received funding from the European Communitys Seventh Framework Programme (FP7/2007-2013) under grant agreement n 609206 and n 608849.

1 CNRS, LAAS, 7 avenue du colonel Roche, F-31400 Toulouse, France 2 Univ de Toulouse, LAAS, F-31400 Toulouse, France placement robot o b j e c t object grasp q init q goal possible connection Fig. 1: If the grasp is unique, any pair of grasp configurations are connectable by a transfer path. of the previously cited papers that explore the configuration space using RRT-like methods.

Let us consider the problem of manipulation defined by a cylindrical vertical object like a bottle, a table and a simple 6-dof manipulator robot with a gripper. The configuration space of the system is C [-π, π] 6 × SE(3). This problem is defined by two states 1) placement: the subspace of configurations of the system where the object is standing on the table: height of object is constant, roll and pitch of the object are equal to 0. 2) grasp: the subspace of configurations where the object is grasped by the robot. Admissible motions in placement are called transit paths, while admissible motions in grasp are called transfer paths. Along transit paths, the position of the object on the table remains constant. This constraint defines a foliation of placement. Leaves of the foliation are parameterized by the 3 remaining degrees of freedom of the object on placement: horizontal translation and yaw angle of the object.

In many cases grasp is defined by a fixed relative position between the gripper and the object. In this case, grasp is not foliated (or composed of only one leaf). In this case, extending a RRT-tree from the initial configuration and from the goal configuration will solve the problem of moving the object at another configuration. The trees will connect in grasp space.

In the general case, objects can be grasped in a continuous way. In our example, the relative orientation of the object with respect to the gripper is free around the vertical axis of the object. In this case, grasp is also foliated. The foliation is parameterized by the relative angle of the object with respect to the gripper along the object axis. In this case, RRT-trees rooted in placement will never meet, since the probability placement robot o b j e c t object grasp q init q goal Fig. 2: The crossed foliation issue: if grasp and placement are foliated, trees rooted at initial and goal configurations will never meet. that both trees start exploring grasp with the same grasp is equal to 0. We call this issue the crossed foliation issue. Figures 1 and2 illustrate the reasoning.

To our knowledge, this issue has never been stated and solved in a general way. Note that [START_REF] Simon | Manipulation planning with probabilistic roadmaps[END_REF], [START_REF] Harada | A manipulation motion planner for dual-arm industrial manipulators[END_REF] and [START_REF] Lertkultanon | A single-query manipulation planner[END_REF] overcome the issue by searching grasp ∩ placement and by approximating the non-feasible solution path by a sequence of admissible manipulation paths.

B. Contribution

The contributions of this paper are 1) to propose a new concept (briefly introduced in [START_REF] Dalibard | Manipulation of documented objects by a walking humanoid robot[END_REF]) called the constraint graph that encodes the constraints relative to any manipulation problem, 2) to propose an algorithm that automatically builds the constraint graph of a given problem, 3) to propose a simple algorithm called Manipulation RRT that solves manipulation problems encoded in a constraint graph, including problems featuring the crossed foliation issue.

Experimental results are proposed at the end of the paper. They provide an example of manipulation problem where Romeo robot holds a placard with two hands and needs to perform a sequence of one hand and two hand grasps to rotate the object. This example would be very difficult to solve with other existing approaches.

II. MANIPULATION GRAPH

Let C denote the Cartesian product of the configuration spaces of all robots and objects considered. We make use of the following concepts:

a) Numerical constraint f : C 1 mapping from C to R m
where m is an integer. q ∈ C is said to satisfy the constraint iff b) Parameterized numerical constraint g: same as equality constraint, except that the constraint is satisfied iff g(q) = g 0 where g 0 ∈ R m is called the parameter of the numerical constraint. c) Projection proj on a numerical constraint f : mapping from a subspace of C to C such that for any q in the input subspace, proj(q) satisfies the numerical constraint. In our implementation, projection is obtained by using Newton-Raphson algorithm. f) Transition: a transition contains a parameterizable numerical constraint g. The constraint graph contains a transition T between two states S 0 and S 1 if there exists 1)

f (q) = 0 (resp. f (q) ≤ 0 ) if f is an equality (resp. inequality) constraint.
q 0 ∈ S 0 , q 1 ∈ S 1 2) a path γ from [0, 1] to C such that γ(0) = q 0 and γ(1) = q 1 ∀t ∈ [0, 1], g(γ(t)) = g(q 0 )
Note that: 1) the existence of a transition does not imply the existence of a path between the corresponding states, (for instance placement on a table unreachable by a robot may be connected to grasp by a transition), 2) the constraint graph is not unique and depends on the partition of the admissible configuration space into numerical constraints, 3) for most manipulation problems, it is easy to produce an effective constraint graph. Figure 3 displays the constraint graph of the simple example described in Section I-A

A. Construction of the constraint graph

In this section, we explain how a constraint graph is automatically built from a set of objects and robot grippers.

1) Placement: they are defined by contact between convex polygons attached to objects or to the environment. the contact constraint between two polygons P 0 and P 1 is satisfied iff the center of P 1 (projection of the center of mass of object 1 onto P 1 plane) lies inside P 0 and if P 0 and P 1 normals are colinear. We denote by CONTACT(L 1 , L 2 ) the numerical constraint that enforces contact between either polygon of two sets L 1 (attached to the object) and L 2 (attached to the environment). This single constraint is defined by evaluating the constraint defined by the closest pair of polygons in L 1 and L 2 .

2) Grasp: We attach to each gripper a frame centered at the center of the gripper (between the jaws), and to each object one or several frames called handles. A gripper can grasp an object when both frames coincide (up to possible degrees of freedom for symmetric objects, like free rotation around axis for cylindrical objects, free translation for long and thin objects,...). We denote by

GRASP(g, h)

the numerical constraint defined by the grasp of handle h by gripper g. The dimension of the constraint can be less than 6 if the grasp does not constrain all the degrees of freedom (5 for cylindrical objects for instance).

Grippers, handles and contact surfaces are part of the description of the robots, objects, environment and need to be provided.

As explained above grasp and contact constraints may partially constrain the relative position of an object with respect to a gripper or to the environment. Once an object is grasped, however or once it is released in contact, the relative position becomes fully constrained. We thus define FIXED(obj, g) where obj is an object and g is either a robot gripper or the environment as the parameterizable numerical constraint that keeps the relative position of the object with g constant.

B. Algorithm

Algorithm 1 describes the construction of the constraint graph relative to a problem of manipulation where some objects are grasped by robot grippers. One or several handles are attached to each object. obj.handles denotes the set of handles of object obj (line 19). The algorithm loops over all possible combinations of "some grippers hold some handles" (Lines 7, 8) and creates a state for each combination. Method GETSTATE returns the state built with the combination of grasps f 1 given as input (Line 14). Then transitions are created from the new state to each state defined by the same set of grasps minus one (lines [START_REF] Gharbi | Roadmap composition for multi-arm systems path planning[END_REF][START_REF] Harada | A manipulation motion planner for dual-arm industrial manipulators[END_REF][START_REF] Dobson | Planning representations and algorithms for prehensile multi-arm manipulation[END_REF][START_REF] Mirabel | Hpp: a new software for constrained motion planning[END_REF][START_REF] Berenson | Task space regions: A framework for pose-constrained manipulation planning[END_REF].

Function MAKESTATE loops over all ungrasped objects and creates constraints so that those objects stay in stable contact pose (Line 19). Then it loops over each grasp defined by f 1 (Line 23) and creates the corresponding constraint. Lines 22 and 27 store foreach state a set of parameterized for all injective mapping f 1 from G to H do 9: C grasp ←GRASP(g 1 , f 1 (g 1 ))

S 1 ← MAKESTATE(f 1 , G ) 10: states ← states ∪ {S 1 } 11: for all g 1 ∈ G do 12: G ← G \ {g 1 } 13: f 0 ← f 1 restricted to G
25: T 1 .constraints.ADD(S 1 .transConstr) numerical constraints that will be inserted in edges by function MAKETRANSITIONS.

S.constraints.ADD(C grasp

III. MANIPULATION PLANNING

In this section, we explain how we use the previously built contraint graph to plan manipulation paths in order to solve the crossed foliation issue.

The number of nodes of the constraint graph may quickly grow when the number of objects and grippers increases. The user may provide as input only a subset of relevant grasps to algorithm 1 so that the result remains tractable. This subset can be provided as a grasp-placement table as in [START_REF] Tournassoud | Regrasping[END_REF], [START_REF] Lertkultanon | A single-query manipulation planner[END_REF].

A. Exploration using the constraint graph

For each state of the constraint graph, we define a probability distribution over all transitions starting from this state. By default the uniform distribution is a reasonable option.

Then our manipulation planning algorithm consists in exploring the transitions of the constraint graph as follows.

1) shoot a random configuration q rand , 2) find the closest node q near in the current roadmap, 3) find the state of this node in the constraint graph, 4) sample a transition getting out of this state, 5) extend q near along the transition up to q new following Algorithm 2, 6) try to connect q new to other connected components of the roadmap (Algorithm 3).

Algorithm 2 Constrained extension 1: function CONSTRAINEDEXTEND(q near ,q rand ,T ) 2:

S ← T .TARGETNODE 3:

g t ← T .CONSTRAINT 4: f s ← S.CONSTRAINT 5:
proj ← PROJECTOR(f s = 0, g t = g t (q near ))

6:

q proj ←proj(q rand ) 7:

p 1 ← INTERPOLATE(q near , q proj ) 8:
pathProj ← PROJECTOR(g t = g t (q near ))

9:
p 2 ← PROJECTPATH(p1, pathProj) 10:

p 3 ← TESTCOLLISION(p 2 )
11:

q new ← FINALCONFIGURATION(p 3 ) 12:
return q new , p 3 Algorithm 2 describes Step 5 above. g t denotes the parameterized numerical constraint of transition T (Line 3). f s denotes the numerical constraint of target state S (Line 4). g right hand side is initialized with q near (Line 5). q proj is obtained by projecting q rand (Line 6). Line 9, the path is projected, by discretizing and projecting on the transition constraint. The resulting path is tested for collision. In case of collision or projection failure, p 3 is the part of the path that was successfully projected and collision-free. After calling the constrained extension algorithm, q new is added as a new roadmap node, and p 3 is added as a new roadmap edge starting from q near . Algorithm 3 Connect configurations along a transition 1: function CONNECT(q 1 , q 2 ) 2:

S 1 ←STATE(q 1 ) 3: S 2 ←STATE(q 2 ) 4: T ←TRANSITION(S 1 , S 2 ) 5:
if T is None then return failure 6:

g ← T .CONSTRAINT 7:
if g(q 2 ) = g(q 1 ) then return failure 8: proj ← PROJECTOR(g = g(q 1 )) if no collision and projection success then return p 3

13:

else return failure Algorithm 3 tries to connects new nodes created by Algorithm 2 to other connected components of the roadmap, as in any classical implementation of RRT algorithm. First the function looks for a transtion between the configurations. If a transition exists, the function checks that q 1 and q 2 are on the same leaf of the transition foliation. If not the function returns failure.

As explained in the introduction, if states S 1 and S 2 are foliated, Line 7 will always return failure.

B. Crossed foliation transition

The crossed foliation issue arises when two transitions connecting two nodes back and forth are both foliated. In Algorithm 1, lines 32 and 33, it corresponds to the parameterized constraints S 1 .transConstr and S 2 .transConstr to be both non-empty.

In this case, as explained in Figure 2, the exploration algorithm described in Section III-A will never succeed in connecting configurations from different trees.

To solve this issue, we add between S 1 and S 2 a new type of transition called crossed foliation transition. This type of transition stores the parameterized numerical constraints of transition S 1 → S 2 as g 1 , and the parameterized numerical constraints of transition S 2 → S 1 as g 2 .

When extending a node belonging to S 1 through this transition, the extension algorithm picks a random configuration q 1 in another connected component of the current roadmap and lying in state S 1 . Constraint g 2 = q 2 (q 1 ) is then added to the projector (Line 5 of Algorithm 2). As a consequence, q proj is on the same leaf as q 1 for the transition linking S 2 to S 1 . As a consequence, Algorithm 3 may connect these configurations.

When adding crossed foliation transition to the graph, we need to redefine the probability distribution of edges getting out of each node. Thus, when extending a node of a state, the algorithm alternates between regular extension and crossed foliation extension.

IV. EXPERIMENTAL RESULTS

In this section, we show some experimental results demonstrating the effectivity of our algorithm. Algorithms are implement in an open source software called HPP [START_REF] Mirabel | Hpp: a new software for constrained motion planning[END_REF]: https://humanoid-path-planner.github.io/hpp-doc. Experiments are run on a standard 2.4GHz, RAM 4Go, 4 cores, desktop computer.

A. Romeo holding a placard

In this example, Romeo holds a placard. The manipulation planning problem consists in rotating the placard around the vertical axis. To do so, the robot needs to go through a sequence of states where the robot holds the placard by the right hand, the left hand and both hands. Applying a task planning based approach for this problem seems difficult since the minimal sequence of tasks highly depends on the workspace of the robot arms and is very difficult to precompute.

The results obtained after 20 runs of our algorithms are displayed in Table I. The variance of the computation time is surprisingly high. We chose not to interrupt path planning after a threshold time as usually done for difficult problems. Note that the model of the robot is very accurate (each hand has four fingers with 3 segments) and a lot of time

B. Rearrangement planning

On these 4 test cases, Baxter robot moves boxes on a table. In cases 1 and 2, only the left arm is used. In cases 3 and 4, both arms are used. The constraint graph is given in Figure 6. In the first case, the box positions are only shifted and the problem is monotone. In the other 3 cases, the boxes are to be permuted so the solutions to these 3 problems are not monotone.

Table II, the accompanying video and Figure 4 summarizes the results. The solver is able to find solutions in all the four cases. For cases 1 and 2, the problem is not difficult and the solution comes quickly. Cases 3 and 4 corresponds to artificially-hard toy problems, yet the planner is able to discover a solution in a reasonable amount of time. Case 1 shows that our approach is not as efficient as task planning based approaches on monotone cases. In contrast, we can solve non-monotone instances, as shown in cases 2, 3, 4. These cases shows the ability to discover new common valid placement. Case 3 and 4 also show the ability of the planner to consider simultaneous manipulation.

C. Grasping behind a door

We demonstrate here the implementation of a more complex manipulation planning problem. A mobile (humanoid) robot has to grasp an object and place it inside a fridge while opening the fridge door. Once more, the sequence of high level actions is not given. We also demonstrate that this approach, valid for mobile robots, can simply be extended to humanoid robots. First, we generate a path for the sliding robot. Then the path can be post processed [START_REF] Dalibard | Dynamic walking and whole-body motion planning for humanoid robots: an integrated approach[END_REF], [START_REF] Carpentier | A versatile and efficient pattern generator for generalized legged locomotion[END_REF] generate a walking trajectory. We successfully planned a path where Romeo robot takes an object and puts it in a fridge. The solution found is included in the video and summarized in Figure 7.

V. CONCLUSION

The main contribution of this paper is the introduction of the constraint graph as a tool to model and solve manipulation planning problems. We propose an algorithm using this tool to explore the configuration space of the system, but we truly think that many existing manipulation planning approaches (task planning based, multi-arm manipulation, navigation among movable obstacles,...) could be implemented using this tool.

The purely geometric algorithm that we propose in this paper is certainly not the most efficient on any benchmark, but it is the most general that we know about.

Future improvements include solving explicit constraints in an explicit way: when an object is rigidly fixed to a gripper, the position of the object depends on the position of the gripper. Expressing this as a numerical constraint is unefficient. Better handling collision-checking by not testing collision for objects that are fixed together will also improve performance.

Fig. 3 :

 3 Fig. 3: Constraint graph of a robot with one gripper manipulating one object.

  d) Constraint graph: graph the nodes of which are called States and the edges of which are called Transitions. The space of admissible configurations of a manipulation problem is the union of the states. e) State: a state contains a numerical constraint. A configuration belongs to the state iff it satisfies the numerical constraint.

Algorithm 1 7 :

 17 Build the constraint graph 1: G ← set of grippers, 2: H ← set of handles, 3: O ← set of objects (with contact polygons), 4: P ← set of contact polygons of the environment 5:for all subset G of G by increasing cardinal do 8:

  for all obj ∈ O|f 1 (G ) ∩ obj.handles = ∅ do 20: C contact ←CONTACT(obj.polygons, P ) 21: S.constraints.ADD(C contact ) 22: S.transConstr.ADD(FIXED(obj, env)) 23:for all g 1 ∈ G do 24:

) 26 :T 1

 261 if C grasp .dimension < 6 then27:S.transConstr.ADD(FIXED(f 1 (g 1 ).obj, g 1 ))28:return S 29: function MAKETRANSITIONS(S 0 , S 1 ) 30:T 0 ←EMPTYTRANSITIONBETWEEN(S 0 , S 1 ) 31: ←EMPTYTRANSITIONBETWEEN(S 1 , S 0 ) 32:T 0 .constraints.ADD(S 0 .transConstr) 33:

Fig. 4 :

 4 Fig. 4: Baxter permutes the position of 3 boxes.

Fig. 5 :

 5 Fig. 5: Humanoid robot Romeo holding a placard. In final configuration, the placard is rotated by 180 degrees. The manipulation planning algorithm needs to explore manifolds defined by right, left and both hand grasps. Min Median Max Number of nodes 42 1370 7002 Time of computation (s) 19 880 6500

Fig. 6 :

 6 Fig. 6: Constraint graph for Case 1 with Baxter robot: 2 boxes and considering only the right arm. The constraint graph is produced by an extension of Algorithm 1 that inserts waypoint corresponding to approaching positions of the gripper in front of the object.

Fig. 7 :

 7 Fig. 7: Romeo puts a box in a fridge.

TABLE I :

 I Romeo holding a placard is spent in collision checking. Moreover, the solution path goes through several "narrow passages" since grasping the pole requires to avoid a lot of collisions between the object and the fingers. Half of the time, the problem is solved in less than 15 minutes. The accompanying video shows a path solving the manipulation problem.

  to

	Case Nb 1 2 2 3 4 3	Arm Right Both	Min 0.15 1.1 3.7 76	Time (s) Med 1.4 4.6 18 397	Max 3.5 10.6 60 1028	Min 11 42 103 664	Nb nodes Med 55 199 273 3659 8830 Max 141 482 832

TABLE II :

 II Minimum / median / maximum solving time and number of nodes, over 20 runs, for various cases with Baxter robot. "Nb" represents the number of boxes.


Vincent Duwicquet
email: vincent.duwicquet@univ-lille1.fr

Jacques Mazier
email: mazier@univ-paris13.fr

Jamel Saadaoui
email: saadaoui@unistra.fr

Interest rates, Eurobonds and intra-European exchange rate misalignments: The challenge of sustainable adjustments in the eurozone

Keywords: JEL Classification: F31, F32, F37, F41, E12 Euro crisis, Exchange rate misalignments, Eurobonds, Interest rates

The euro crisis sheds light on the nature of alternative adjustment mechanisms in a heterogeneous monetary union. Exchange rate adjustments being impossible, it remains very few efficient alternative mechanisms. At the level of the whole eurozone the euro is close to its equilibrium parity. But the euro remains overvalued for Southern European countries, France included, and largely undervalued for Northern European countries, especially for Germany. This paper gives a new evaluation of these exchange rate misalignments inside the eurozone thanks to a FEER approach. In a second step, we use a two-country SFC model of a monetary union with endogenous interest rates and Eurobonds. Overvaluations amount to negative competitiveness shocks in Southern countries. In this respect, three main results are found. Firstly, an increase of intra-European financing by banks of northern countries or other institutions could contribute to reduce the debt burden and induce a partial recovery but public debt would increase. Secondly, the implementation of Eurobonds as a tool to partially mutualize European sovereign debt would have a rather similar positive impact with a public debt limited to 70 percent of GDP. Thirdly, Eurobonds could also be used to finance large European projects which could impulse a stronger recovery in the entire zone with stabilized current account balances.

Introduction

"The second implication of the absence of fiscal transfers is that countries need to invest more in other mechanisms to share the cost of shocks. Even with more flexible economies, internal adjustment will always be slower than it would be if countries had their own exchange rate. Risk-sharing is thus essential to prevent recessions from leaving permanent scars and reinforcing economic divergence."

Mario [START_REF] Draghi | Stability and prosperity in monetary union[END_REF].

The euro crisis sheds light on the nature of alternative adjustment mechanisms in a heterogeneous monetary union. Adjustment mechanisms are defined in a broad sense as mechanisms that ensure a return to the initial situation or, possibly, to recover towards full employment after a slowdown. It remains very few efficient alternative mechanisms in the absence of exchange rate flexibility as underlined by the quotation of Mario Draghi. Budgetary policy could play a major role. In the United States, budgetary policy stabilizes 20 percent of shocks on the GDP [START_REF] Italianer | Systèmes budgétaires et amortissement des chocs régionaux : implications pour l'Union économique et monétaire[END_REF]. But there is no equivalent in the European case.

Well integrated capital markets, with portfolio diversification and intra-zone credit, have been proposed as a powerful adjustment mechanism by the "international risk sharing" approach.

Intra-zone credit and capital income from international portfolio could absorb near from 40 to 60 percent of the shocks [START_REF] Asdrubali | Dynamic risk sharing in the United States and Europe[END_REF]). These results have been used during the 2000 by proponents of liberal economic policies in the EU to promote a deeper financial integration instead of having to develop a federal budget (European Commission, 2007;[START_REF] Trichet | Le processus d'intégration européenne[END_REF] 1 . 1 Mario [START_REF] Draghi | Stability and prosperity in monetary union[END_REF] acknowledges the crucial role of budgetary policies and that this approach "the less public risk-sharing we want, the more private risk-sharing we need" could be insufficient in case of financial storms in the future. However, he concludes that Members States should achieve structural reforms to have sound public finances in order to be able to deal with periods of financial and economic turmoil. This approach is still present in the last Action Plan of the European Commission (2015) on the Capital Market Union with the goal of creating one single market for shares, bonds and securitized bank loans. However, the theoretical basis, the empirical methodology and the results seem to be highly questionable [START_REF] Clévenot | Partage du risque interrégional[END_REF].

Consequently, relative wage and price flexibility are proposed in order to take place, at least partially, of exchange rate adjustments. Actually, these mechanisms allow only a very slow and partial return to equilibrium with an important cost in terms of growth and employment and with large differences between countries, due to huge structural specificities. They are more inefficient when they are implemented simultaneously in interdependent countries, as it is the case in the eurozone, especially in the Southern European countries. They are more efficient in a largely opened economy like Ireland than in rather closed ones like Greece or even Portugal [START_REF] Mazier | Interdependency and adjustments in the European Union[END_REF].

At the level of the whole eurozone, the current account is close to equilibrium and the fiscal deficit is smaller than in many other OECD countries. The euro is close to its equilibrium parity but intra-European imbalances are huge.

The euro is strongly overvalued for Southern European countries, France included, and largely undervalued for Northern European countries, especially for Germany [START_REF] Jeong | Exchange rate misalignments at world and European levels: A FEER approach[END_REF]. These overvaluations slow growth and induce fiscal and current deficits in the South while undervaluations boost growth in the North via exports, especially towards the rest of the eurozone, and deficits are reduced. This situation is equivalent to implicit positive transfers in favor of the North and negative transfers at the detriment of the South, which are largely ignored in the public debate.

In order to investigate these issues, [START_REF] Duwicquet | Désajustements de change, fédéralisme budgétaire et redistribution[END_REF] have used a two-country Stock-Flow Consistent (hereafter, SFC) model of a monetary union along the lines of [START_REF] Godley | Comprehensive accounting in simple open economy macroeconomics with endogenous sterilization or flexible exchange rates[END_REF], 2007a[START_REF] Godley | A simple model of three economies with two currencies: The Eurozone and the USA[END_REF], [START_REF] Lavoie | A fully coherent post-Keynesian model of the Eurozone[END_REF] and Duwicquet andMazier (2010, 2011). The model described the real sector and assets and liabilities of economic agents in order to analyze financial integration in a consistent manner. A federal budget has been introduced with federal expenditures and social transfers financed by federal taxes and Eurobonds issuance. The stabilizing role of such a federal budget has been confirmed facing asymmetric shocks or, equivalently, exchange rate misalignments within the monetary union. Similarly, the stabilizing role of Eurobonds, used to finance European investment projects, has been illustrated. But the model was limited to exogenous interest rates, which can only be regarded as a preliminary step, as we have witnessed large movements of interest rates in Southern European countries since the onset of the euro crisis. This paper is organized as follow. In a first part, we give a new evaluation of these exchange rate misalignments inside the eurozone, using a FEER approach, and we discuss the structural heterogeneity of the eurozone. In a second part, we introduce an SFC model of a monetary union with endogenous interest rates. With this model, we examine to what extent these asymmetric evolutions due to intra-European misalignments can be adjusted. Interest rates on public bonds are now endogenous. Fiscal policy is partially endogenous and reacts to financial markets evolution with the implementation of budget cuts. An increase of intra-zone financing allows to contain upward pressure on interest rates. Eurobonds are introduced and used in two ways, on the one hand, in order to mutualize a part of the European public debts and, in the other hand, to finance European investments in growth sectors. A combination of tax rebate and budget cuts is also investigated.

Intra-European exchange rates misalignments and structural heterogeneity

Since the beginning of the 2000s, a surge of current account imbalances within the eurozone has been observed in spite of a rather balanced current account for the whole area. On the one side, Northern European countries have accumulated huge current account surpluses and on the other side, Southern European countries have run important current account deficits. After 2009, current account deficits of Southern European countries have been reduced mainly because of restrictive policies and internal devaluations. These evolutions reflect, at least partially, the increasing exchange rate misalignments inside the eurozone. Thanks to a FEER approach, introduced by Williamson (1983), Jeong et alii (2010) and Duwicquet et alii (2013) have shown a divergence within the eurozone between some countries increasingly undervalued (like Germany, Austria, Netherlands and Finland) and others increasingly overvalued (like Greece, Portugal, Spain and France).

As we can see in table 1, on average between 2005 and 2010, Germany, Austria, Netherlands and Finland have been undervalued by 13 percent while Greece, Portugal, Spain and France have been overvalued by 23 percent. These intra-European exchange rate misalignments reflect a strong structural heterogeneity between European countries at several levels (nature of the international specialization, size and productivity of the firms, R&D effort and qualification of the labor force). They are at the heart of the current woes of the eurozone. However, since the onset of the euro crisis in 2010, a reduction of misalignments has been observed for most of the Southern European countries. Irish, Spanish, Italian and even Portuguese euros seemed no more overvalued in 2013. But Greek and French euros remained overvalued by around 15 percent and German euro undervalued by around 20 percent in 2014. These movements have been mainly driven by large real effective devaluations in Ireland, Spain, Portugal and Greece, as shown in figure 1 with the evolutions of the relative unit labor cost (RULC). These politics of internal devaluation have been very painful and has led to a deep recession in Greece, as in other Southern European countries. We have observed a reduction of current account deficits mainly due to the reduction of imports. However, ex-post improvement of public finances had been limited.

The financial crisis of 2008 was, in any case, due to a rise of public deficits and debts.

Nevertheless, it has led to a huge increase in the government deficits and debts due to measures implemented to rescue banks and support the economic activity and, also, to the fall of tax incomes induced by the recession. The European Central Bank (hereafter, ECB) switched to a more active monetary policy. But the weakest European countries faced difficulties to finance their deficits as the financial markets feared sovereign debt default. As they could not attack national currencies thanks to the monetary union, the financial markets concentrated their attacks on the public bonds. Interest rates soared, first, in Ireland and in Greece, then in other Southern European countries while they remained very low in Germany, but also in France. The true existence of the monetary union was at stake and a reversal only took place after the announcement of the ECB in September 2012 to intervene without restriction on the public bonds secondary markets in case of necessity (see figure 2).

But this ECB intervention was accompanied by very restrictive fiscal policies in the framework of the Fiscal Pact and by structural reforms in favor of more liberalization.

On the whole, results have been uneven amongst member states. Real devaluation has been inefficient in Greece and, to a large extent, in Portugal due to their limited degree of trade openness and the social costs have been important. Combined with restrictive fiscal policies, it has led to a deep recession which has limited the improvement of public finances. The public debt ratio has increased massively.

In Ireland and, to a less extent, in Spain the real devaluation has been more operational thanks to the role played by the export sector but the initial negative shock has not been offset and the rate of unemployment remained important (in January 2016, the unemployment rate was 8.9 percent in Ireland and 20.5 percent in Spain).

In Germany, between 2000 and 2008, sharp wage and productivity adjustments have led to a large reduction of the German relative unit labor cost. These adjustments have been preserved

during the crisis.

On the opposite, in France, the successive governments have been reluctant to implement cost adjustments and the euro remained overvalued. The strategy adopted has been, since 2013, to reduce costs through tax rebates (around 1.5 percent of GDP). The target of these measures is not clear and they raise, at least, two questions. If the government wants to improve the competitiveness, this measure is inaccurate as all the firms, including those of the non-tradable sector like banks and retailers, can benefit from the tax rebates. Consequently, the transfer in favor of the tradable sector is too limited, compared with the cost disadvantage which currently prevail. A larger transfer would be necessary but could not be easily supported by public finances. If the target is to improve employment, as it seems to be more the case with the "Pacte de responsabilité et de solidarité", the past experiences show that efficiency is not warranted and the problem of cost-competitiveness, which cannot be ignored, is not solved. The government is aware of these limits and has completed his array of measures by re-launching industrial policy measures (major industrial projects, innovation policy, etc.) to improve nonprice competitiveness. Nevertheless, this kind of measures takes a long time (around 10 years)

to be fully operational.

The risk is, therefore, that any target can be reached. The competitiveness problem will remain and financing the current account deficit might become more important. Tax rebates could have a limited impact on employment, since profit rates have been squeezed during the crisis, especially for the export sector. The financing of the tax rebates will imply substantial expenditures cuts. Thus, the more likely outcome would be a long lasting period of stagnation.

In this context, it is worthwhile, using an SFC model of a monetary union, to assess various alternative economic policies scenarios which try to tackle this problem of intra-European misalignments.

SFC modeling of adjustment mechanisms in a monetary union

The structure of the model

An SFC model allows to obtain a consistent description of assets and liabilities of all associated real and financial flows in a monetary union. This monetary union is composed of two countries ( n and s ) with an asymmetry of size. The country n is five times larger than the country s .

This configuration eases the exploration of the adjustment mechanisms of the country s facing the rest of the monetary union.

We introduce in this model the possibility of public federal expenditures and Eurobonds. This In table 2, we describe the balance sheet in terms of assets (written with a positive sign) and liabilities (written with a negative sign) of each sector: households, firms, government, commercial banks, the single central bank and a federal budget.

F G B FB ECB HH F G B  C n k  s k  0 D n bd  n bd  s bd  s bd  0 Cs n hh  n h  h  s hh  s h  0 Cr n l  n n l  0 n s l  0 s n l  s s l  R n rf  ns rf rf  s rf  0 Bd n nn pb b  nn pb b  n ns pb b  0 s sn pb b  s ss pb b  E n bte  0 Bi n bt  n n bt  0 n s bt  0 s n bt  s s bt  Eq n nn pe eh  n nn pe ee  n ns pe eh  n ns pe ee  0 nn pe e  s sn pe eh  s sn pe ee  s ss pe eh  s ss pe ee  0 ss pe e  W n vh  n v  n d  n vb  n de  s vh  s v  s d  s vb  0  0 0 0 0 0 0 0 0 0 0
Notes: For the agents in the economy, HH stands for households, F for firms, G for national government, B for private banks and FB stands for federal budget or a federal institution in charge of the emission of Eurobonds.

For the type of financial assets held and issued in the economy, C stands for physical capital, D for deposits, Cs for cash, Cr for credit, R for advances of the central bank, Bd for bonds, E for Eurobonds, Bi for bills, Eq for equities and W stands for wealth.  Budgetary policy is partially endogenous and is linked to financial markets. When interest rates on sovereign debt increase, the national government can reduce public expenditures in reaction.

 The possibility to increase intra-zone financing is introduced in order to reduce the pressure on interest rates. This can be achieved through foreign banks purchases of public bonds or Treasury bills, through the European Stability Mechanism or even through direct intervention of the central bank on the public bond markets. 2 When there are two symbols ( n and s ), the subscript denotes the country where the asset is held, the superscript the country where the asset is issued. For example, s n bt represents the amount of bills held by country n and issued by the country s .

 The role of Eurobonds is examined in two ways. On the one hand, Eurobonds are aimed at mutualizing a part of sovereign debt in the eurozone. On the other hand, Eurobonds could be used to finance European investment projects in several sectors namely education, health and innovation.

 Last the rather traditional policy mix combining tax rebates and expenditures cuts (roughly the French government strategy) is also discussed.

The main equations are presented below. The model has been calibrated to represent the structure of the European Monetary Union. The value of coefficients and the entire model are available in the working paper version of this text 3 .

The demand side

The model dynamics relies essentially on the investment function. As we can see below, investment reacts positively to the rate of profit and to variation of aggregate demand. It responds negatively to the debt structure and to credit costs.

0 1 2 3 4 4 ( 1) ( 1) ( 2) ( 1) ( 1)
n n n n n n n n nn n sn s n n n up y l g k k k k k rl k rl k y k             [1]
where g stands for the rate of accumulation of physical capital; up represents the amount of undistributed profits; l is the firms' indebtedness; rl is the credit cost and y is the gross domestic product.

At the macroeconomic level, an increase in investment spending will generate more profits.

These profits will be, on the one hand, distributed in part to shareholders (here, households and other firms) and, on the other hand, undistributed.

The household consumption function includes a positive wealth effect. This wealth effect describes the behavior of households which target a constant ratio between wealth and disposable income. ( 1)

n n n n n n c a a yh a vh     [2]
where vh stands for the households' wealth and yh for the disposable income with capital gains.

The disposable income of households is defined as the sum of after-tax labor incomes (wages)

and after-tax capital incomes (interest rates and dividends). A part of disposable income augmented with capital gains is consumed whereas the residual saving corresponds to bank deposits, money holdings and to financial assets (bonds supplied by the State and equities supplied by private firms). The financial wealth covers a large array of financial assets (bank deposits, cash money, equities and bonds).

The government receives taxes from households and banks, spends and pays interests. The public deficit is financed by issuance of bonds and Treasury bills. Supply of Treasury bills balances the gap between public deficit and bonds issuance thus:

( 1) ( 1)

n n n n n n n n n n n n n bt gn r bt b t tb teb pb b ps cl tf               [3]
where bt is the amount of T-bills; gn stands for the national public expenditures; r is the interest rate on T-bills; b represents the amount of bond issued; t are the taxes paid by the households; tb are the taxes paid by commercial banks; teb are the taxes paid by the ECB; tf are the taxes paid by firms; ps stands for social benefits and cl for firms' social contributions.

The banking sector

The central bank supplies money and provides an unlimited amount of refinancing to private banks at the key interest rate ( ib ) thus acting as the lender of last resort. The interest rate on bank deposit ( id ) is simply determined with a margin on the key interest rate of the central bank.

n s n s n n n n n n n n rf h l l bt bt bp bd               [4] 2b id ib m  [5]
The central bank does not make any profit as in [START_REF] Godley | Monetary Economics: An Integrated Approach to Credit, Money, Income, Production and Wealth[END_REF]. Thus interests paid to the central bank are equal to taxes paid to the State. This is in line with the practice of most modern central banks in the world economy.

Besides, commercial banks supply the entire amount of demanded credit:

ns n n n n n n n s n l inv up pe e pe ee pe ee          [6]
The credit market is open to foreign banks. We suppose that banks of the smaller country (country s ) do not lend to firms of the larger country n ( 0 n s l  ). Bank loans are allocated between domestic and foreign firms relatively to their respective trade openness. The interest rate on bank loans is endogenous and depends on the lagged value of Treasury bills' rate of each country and on their own lagged value.

(1 ) ( 1) . ( 1)

n n n rl a rl a r      [ 7]
(1 ) ( 1) . ( 1)

s s s rl a rl a r      [8]
Treasury bills play a key role in the model resolution. Regarding the rest of the union (the northern country), we assume that the southern country does not hold bills issued by the northern country which finances its public deficit only domestically: After an increase of public deficit, the public deficit remains financed by commercial banks.

However, the level of interest rates is higher. This tightening of financial conditions is partially transmitted to rates on bank loans granted to firms and to interest rates on public bonds which are supposed to be equal to interest rates on Treasury bills.

Baseline scenario

Our model represents a monetary union characterized by a sluggish growth in the baseline scenario (around 1 percent per year).

                     [12]
This shock deteriorates the current account of the southern country and improves external trade of the northern country. Consequently, we observe a decline of national income in the South and an increase of national income in the North. In order to investigate the current developments of the eurozone crisis, we compare the effect of this shock in different versions of the model.

In addition to the baseline scenario, five versions of the model will be examined.

Alternative scenarios of economic policies

Scenario 1: Budget cuts

In this first scenario, public expenditures become endogenous and react to rising interest rates on Treasury bills:

12 ( 1) ( 1)
n gg n gg n n gn a gn a r bt     [START_REF] Godley | A simple model of three economies with two currencies: The Eurozone and the USA[END_REF] In line with the objectives of the revised Stability and Growth Pact as well as aims of the Fiscal Compact, we assume that the government targets to reach a debt-GDP ratio of 70 percent in period 45. To achieve this challenge, the government progressively reduces its public expenditures. The speed of public expenditures reduction is governed by the evolution of interest rates. The year of the shock, public expenditures decrease by 0.2 percent of GDP relatively to the baseline scenario. In the baseline scenario, public expenditures amount to 19.5 percent of GDP in period 45. In this first scenario, they drop to 12 percent of GDP in period 45.

Scenario 2: Intra zone financing

We investigate, here, implications of financial support granted by the northern country to the southern country. In the wake of a loss of competitiveness in the southern country, the issuance of public securities will rise to finance an increasing deficit. We assume that private banks of the northern country will sustain a supplementary demand to bring down interest rates. This scenario can also be seen as an illustration of the European Stability Mechanism where northern countries grant loans with low rates of interest to southern countries. Similar effects are also expected if the ECB purchases directly Treasury bills of southern countries. In each case, the southern country receives financial aid to substantially reduce the debt burden.

Scenario 3: Issuance of Eurobonds

In this scenario, Eurobonds are issued in order to partially mutualize sovereign debt of southern countries. We assume that there is threshold (a debt-GDP ratio of 60 percent) from which Eurobonds are issued to finance public debt in the eurozone as a substitute to national debt.

Nevertheless, national governments have to pay interest on issued Eurobonds. Southern countries must be committed to stabilize their public debt.

  60% ( 1) ( 1)
                 [15]
Each government may appeal the issuance of Eurobonds ( n bte for the northern government and s bte for the southern government).

  60%

                [16]   60% ( 1) ( 1
                [17]
The aggregate supply of Eurobonds is obtained by the sum of the two countries.

ns bte bte bte  [START_REF] Mazier | A multi-speed Europe: is it viable? An SFC approach[END_REF] Demand for Eurobonds simply depends on the interest rate ( reuro ) and the level of GDP of the entire eurozone (

Scenario 4: Issuance of Eurobonds and European projects

To complete the previous scenario, Eurobonds are used as a tool to finance European projects in innovative sectors. Southern countries as well as northern countries can use Eurobonds in order to stimulate their economic growth.   in period 10 in scenario 6). However, these measures are not sufficiently devoted to the tradable sector and the total effect on employment, which is the other main target, is very uncertain. That is why the government toolkit includes also industrial policy measures such as innovation and technology policy or relocation policy. These measures are complex to design and to manage and their effects are only felt in the long run. As an illustration and in an optimistic way, it is assumed that after period 30 the non-price competitiveness of country s is improved (the import income elasticity of country s , 1s  declines from 1 to 0.98 while the import income elasticity of country n , 1n  increases from 1 to 1.02 in scenario 7).

Adjustments in the monetary union and economic policies

In figure 3, we can observe the evolution of interest rates and public debt in the southern country in the baseline scenario (competitiveness loss in the southern country) and in the first four versions of the model.

In the baseline scenario, we assume that any adjustment mechanism is implemented to face the competitiveness loss. Thus, this competitiveness loss widens the external deficit and in the same time increases the need of external financing. In addition, the negative impact of trade deficit on the GDP implies a diminution of taxes collected by the government and thus an increase of the public deficit. On the Treasury bills market, interest rates increase alongside the debt increase and the slowdown of GDP. This "snowball" effect implies a tremendous increase in debt levels (140 percent of GDP in period 45) and of interest rates (4.5 percent in period 45). In order to eschew another "Greek drama", European authorities can react by implementing various alternative economic policies in order to achieve more sustainable adjustments.

In the first scenario, the government tries to reduce its public expenditures in order to prevent an increase of interest rates. The long run purpose of this policy is to reach a debt-to-GDP ratio limited to 70 percent. However, due to the Keynesian multiplier effect, public expenditures reduction puts a huge strain on economic activity as we can see in figure 4. Interest rates are reduced compared with the baseline scenario but still rise in the medium run and reach 2.8 percent in period 45 due to a smaller demand of Treasury bills induced by the decline of the activity. Current account in percent of GDP - In the second scenario, we assume that intra-zone financing is important thanks to an eased demand from private banks of the Northern countries or to the implementation of a European

Stability Mechanism. This allows to keep interest rates at low level (2.4 percent in period 45) in spite of a huge increase of public debt-to-GDP ratio (130 percent in period 45). The negative impact on economic growth is largely offset in the long run but the competitiveness problem is not solved (see figure 4). We can recall that the Treaty ratified in March 2012 which gives an institutional background to the European Stability Mechanism stipulates that members States must reach a debt-to-GDP ratio of 60 percent in the medium run. The results of the second scenario will be greatly affected if the goal targeted by the European Stability Mechanism was respected. In such a case the result in terms of relative growth rates would be largely similar to those of the first scenario.

The third and the fourth scenario analyze the impact of an issuance of Eurobonds in the eurozone. We can observe that interest rates increase less rapidly in the fourth scenario than the third scenario. In the fourth scenario, Eurobonds finance investments in innovative sectors therefore economic growth is stronger and upward pressures on interest rates are weaker. These growth gaps can be observed in the figure 4. We compute adjustments on GDP thanks to the following formula:

competitiveness loss with adjustment competitiveness loss without adjustment competitiveness loss without adjustment

GDP GDP RelativeGDP GDP  
Initially, the GDP drops after the negative competitiveness shock. The implementation of European projects financed by Eurobonds (scenario 4) absorbs completely the competitiveness loss in the long run as GDP returns to its value before the shock in period 45. Eurobonds issuance to partially mutualize sovereign debt (scenario 3) induces a partial adjustment. We can observe that intra-zone financing (scenario 2) appears to be more efficient than Eurobonds issuance alone (scenario 3). The implementation of a European Stability Mechanism, aimed at providing low interest rates to governments and firms, stimulates investment. In terms of relative growth, the worst case is the first scenario where governments implement drastic budget cuts in order to achieve a debt-to-GDP ratio of 70 percent in the long run. The GDP drops by 30 percent in relative terms in period 45. The slowdown of economic activity induces a decrease of imports and then a massive adjustment of the current account balance. Without any policy reactions (baseline scenario) after the competitiveness loss, external deficits of the southern country steadily increase and reach 3.5 percent of GDP in period 45. In other scenarios, we observe a stabilization of the external deficit around 2 percent in the long run. In figure 5, we can analyze the consequences of the various scenarios in the northern country in terms of growth and public debt. Again, drastic budget cuts in the southern country have negative impact on economic activity (even) in the northern country. In the long run, the fall of GDP will bring public debt to 65 percent of GDP. In other scenarios, public debt increases less thanks to a stronger growth, particularly in the fourth scenario.

According to our numerical simulations, the emission of Eurobonds constitutes a useful tool to ensure a strong recovery in the entire eurozone. Figure 6 shows levels of public debt and evolution of interest rates on Treasury bills and Eurobonds in the third and the fourth scenario.

As growth is stronger in the fourth scenario, interest rates on national T-bills are lower when Eurobonds play a role in financing the real economy. Conversely, the interest rate on Eurobonds is slightly higher in the fourth scenario (1.9 percent) than in the third scenario (1.6 percent).

Regarding levels of public debt, again, European debt in Eurobonds is higher in the fourth (20 percent of GDP) relatively to the third scenario (10 percent of GDP). Nevertheless, European indebtedness remains sustainable as well as national indebtedness in spite of the fact that national governments have to pay interests on these issued Eurobonds.

We now move towards the last scenarios with tax rebate and public expenditures cuts. In scenario 5, the reduction of the social contributions paid by firms partly offsets the effect of overvaluation of the euro for southern countries and their loss of competitiveness. The GDP fall is less noticeable but the current account balance remains deteriorated while the public deficit and debt increase a lot, inducing a substantial increase of interest rates (see figure 7). To avoid this unsustainable worsening of the public finance, public expenditures are cut of an amount equivalent to the tax rebate (scenario 6). These cuts partially limit the rise of public debt and of the interest rate but at the expense of the GDP growth which returns to the depressed baseline scenario.

This strategy of the French "Pacte de responsabilité et de solidarité" uses simultaneously the accelerator and the brake and can have only a limited effect. The initial tax rebate represents a high cost for the public finances without being targeted on the tradable sector, mainly due to European competition rules. The only way of escape would be the success of industrial and innovation policies able to improve at medium term the non-price competitiveness, as it is illustrated in the scenario 7. Source: authors' calculations.

Conclusion

If European authorities do not react by implementing new economic policies to achieve sustainable adjustments, the intra-European exchange rates misalignments and the subsequent competitiveness loss in southern countries induce economic stagnation in southern countries.

Current account imbalances will continue to diverge and public debt dynamics will not be stabilized. Restrictive fiscal policies, as they have been implemented in southern countries, can contain interest and public indebtedness but at the cost of a deeper recession. This policy-mix based on tax rebate to improve competitiveness and public expenditures cuts, illustrated by the French "Pacte de responsabilité et de solidarité", has only a limited effect.

Increasing intra-European financing by banks of northern countries or by the European Stability

Mechanism or even by the intervention of the ECB itself could contribute to reduce the debt burden and induce a partial recovery. But the problem of competitiveness of the southern countries would not be solved and public debt would increase (scenario 2).

Implementation of euro-bonds as a tool to partly mutualize European sovereign debt would have a rather similar positive impact, but with a public debt limited to 70 percent of GDP, which could be considered as an important advantage (scenario 3). Furthermore, Eurobonds could also be used to finance large European projects which could impulse a stronger recovery in the entire eurozone with stabilized current account imbalances (scenario 4). To improve non-price competitiveness, it could (and should) be completed by more structural policies (industrial and innovation policies) which are complex to implement and effective only in the long run. These various monetary regimes are a more straightforward solution to the problem of competitiveness of southern countries and allow a more efficient adjustment at short term, with a more balanced growth regime at medium term [START_REF] Mazier | A multi-speed Europe: is it viable? An SFC approach[END_REF]. They could also be completed by structural policies to improve non-price competitiveness. However, the main difficulty, raised by this alternative strategy, is the transition period which would be difficult to manage with the risk of capital flights and multiple bank crises. In case of real appreciation (a decrease of ), import in volume increases while exports decreases with lagged effects of the exchange rate variations but current account is improved thanks to cheaper imports. Last rising domestic output gap has a negative impact on current account while foreign output gap has an opposite effect. The underlying current account is the current account corrected by the effects of past and present exchange rate variations and by the effects of the domestic and foreign output gaps:

      UND m x CA Y M Y X Y R M Y R         
By substitution, we obtain: Source: authors' calculations.

         

Figure 1 :

 1 Figure 1: Real effective exchange rates based on unit labor cost

Figure 2 :

 2 Figure 2: 10-years government real interest rates in percent

  will open the road to investigate stabilizing effects of Eurobonds. Firms can accumulate both real and financial capital. They can finance their investments by non-distributed profits, bank loans or equities. Commercial banks are able to supply credit and to ration credit. The single central bank (i.e. the ECB) refinances the commercial banks. Households hold banking deposits, bonds and equities. The two national governments issue bonds and Treasury bills.

  Beyond physical capital ( k ), eight kinds of monetary or financial assets are distinguished 2 : bank deposits ( bd ) held by households, bonds issued by governments ( . pb b ) and held by households of both countries, loans ( l ) supplied by each commercial bank to firms of the two countries, equities issued by firms ( . pe e ) and held by households and firms of both countries, Treasury bills issued by each State ( bt ) and held by commercial banks of both countries, high powered money ( h ) held by households ( hh ) as well as by commercial banks (mandatory reserves), advances supplied by the central bank to commercial banks ( rf ) and finally Eurobonds ( bte ) issued by a federal authority and held by banks. Our model relies on the main features of the contributions of Duwicquet and Mazier (2010, 2011) and Duwicquet et al. (2013). Nevertheless, several crucial changes are included to examine current developments in the eurozone crisis:  Interest rates on Treasury bills supplied by the State are endogenous. The demand of Treasury bills by private banks is an increasing function of interest rate. Thus, in case of an insufficient demand, this mechanism induces upward pressures on interest rates.

  The global demand for Treasury bills issued by the northern country depends on the level of interest rate ( Consequently, we have the following interest rate determination for the northern country:

  In the model, we use the following determination of interest rates:

Scenario 5 :

 5 Tax rebate and public expenditures cutsThis scenario describes roughly the French "Crédit d'impôt pour la compétitivité et l'emploi (CICE)" and the "Pacte de responsabilité et de solidarité" the government reduces the social contributions paid by the firms to partly compensate the competitiveness loss due to the overvaluation ( 1.5 trs  for 10 ti  in period 10 in scenario 5). To avoid an increase of the public debt, public expenditures are cut in the same proportion (

Figure 3 : 1 :

 31 Figure 3: Evolutions of public debt and interest rate in the southern country

Figure 4 :

 4 Figure 4: Relative GDP and current account in the southern country

Figure 5 : 1 :

 51 Figure 5: Relative GDP and public debt in the northern country

Figure 6 :

 6 Figure 6: Interest rate and public debt in scenario 3 and 4

4 :

 4 European investment) Public debt (Scenario 3: Eurobonds issuance) Source: authors' calculations.

Figure 7 : 5 :

 75 Figure 7: Tax rebate and public expenditures cuts (scenarios 5 to 7)

where

  YGAPF is the average output gap of the main partners; R , the logarithm of the real exchange rate (an increase of R indicates a depreciation); , long run export and import volume elasticities, respectively.

Figure B1 .

 B1 Figure B1. Impact of a competitiveness loss on country S GDP (in percent relative to the baseline)

  demand for bonds: v0 = 0.02 Household demand for bonds: v0 = 0.05 Household demand for bonds: v0 = 0.08 Household demand for bonds: v0 = 0.1 Household demand for bonds: v0 = 0.15 -

  demand for treasury bills: a1 = 0 Bank demand for treasury bills: a1 = 0.1 Bank demand for treasury bills: a1 = 0.3 Bank demand for treasury bills: a1 = 0.5 Bank demand for treasury bills: a1 = 0demand for treasury bills: a2 = 0 Bank demand for treasury bills: a2 = 1.3 Bank demand for treasury bills: a2 = 2 Bank demand for treasury bills: a2 = 3 Bank demand for treasury bills: a2 = effect (Country s): k1 = 0.2 Profit effect (Country s): k1 = 0.4 Profit effect (Country s): k1 = 0.525 Profit effect (Country s): k1 = 0.7 Profit effect (Country s): k1 = 0price elasticity: µ2 = 0.1 Import price elasticity: µ2 = 0.3 Import price elasticity: µ2 = 0.5 Import price elasticity: µ2 = 0.7 Import price elasticity: µ2 = 0.9

Table 2 : Balance sheet of a monetary union

 2

	Country n

  Banks purchase a limited amount of Treasury bills with a demand which depends positively on the rate of interest. Thus interest rates become endogenous, as they adjust the supply of Treasury bills determined by the public deficit (which has to be financed) and the private demand of Treasury bills in each country.Bills issued by the southern country and domestically held in the private sector ( s

																				s bt ) as well as
	bills held in the rest of the union ( s bt ) depends on the interest rates differential between the two
						n
	countries:
										s
								s s bt	ss s 12 a r a r ss n 
									y
										s
								n n bt	ns s 12 a r a r ns n 
									y
	By summing demands of these two countries, we obtain the global demand for Treasury bills
	issued by the southern country:
	bt	s			1 ss a	s r		2 a	ss	n r	  s y 	1 ns a	s r		2 a	ns	n r		n y
	The interest rate on Treasury bills issued by the southern country becomes endogenous and we
	can write:
				s r		bt	s	  a 11     22 ss s ns ss n s a r y a y a y n ns  r n  	y	n		[9]

  Several sensitivity tests have been conducted on the most relevant parameters (see appendix D). They indicate a rather good robustness of the results.

	equations:
	log( ) n im			0	n		1 n 	log( ) n y			2	log	  	2 ns log ns w ti w yy         	ti	  	[11]
	s log( ) im			0	s		1 s	s log( ) y		2	log		2 sn w ti w log	ti
															yy
															sn

From this baseline scenario, we simulate an asymmetric loss of competitiveness in the southern country due to an exchange rate misalignment. As underlined by Blanchard and Milesi-Ferretti (2012), overvaluations induce distortions in prices of internationally traded goods. Thus, cumulative effects of overvaluations amount to negative competitiveness shock. To illustrate this loss of competitiveness, the term ti is equal to 10 between periods 10 and 45 in the import

  )

	d
	s
	If			then
	y
	s
		s
	bte	g	r	bt	b	t	tb	teb	pb	b	ps	cl	tf	ge
	s	s	s	s	s	s	s	s	s	s	s	s	s	s

  However, the creation of a European institution in charge of the emission of Eurobonds would face strong political obstacles. The northern countries fear that Eurobonds would give to the southern countries the opportunity to continue irrelevant policies. They would ask that the launching of Eurobonds would be accompanied by more restrictive fiscal policy in the respect of the Stability pact and by a stricter monitoring of national fiscal policies. This could generate numerous political tensions between Member States. Actually, the European Stability Mechanism organizes the rescue of countries facing difficulties only under the condition of a strict control of the public finance. In such a configuration, Eurobonds would not be more efficient than the present institutional framework. On the opposite the southern countries could argue that a part of the debt induced by the overvaluation of their euro could be financed by Eurobonds without being subjected to tougher constraints.Ultimately, as pointed out by[START_REF] Stiglitz | The Euro and its threat to the future of Europe[END_REF], the efficiency of these institutional innovations inside the monetary union could be compared with an alternative framework where the possibility of intra-European exchange rate adjustments would be reintroduced thanks to a new type of monetary regime (cohabitation of a global euro with national euros, new European Monetary System with a euro reduced to a simple ECU, exit of the Germany or of southern countries).

Notes: Forecasts for 2016 based on IMF WEO April 2016; See[START_REF] Jeong | Exchange rate misalignments at world and European levels: A FEER approach[END_REF] for a complete description of the model of world trade and the methodology used to compute misalignments. Source: authors' calculations. A positive (negative) number indicates an undervaluation (overvaluation) expressed in percent of the observed value. See appendix A for details about the methodology used to correct current account balances from the effect of differences between output gaps in the eurozone. This correction, especially important after the crisis (due to a desynchronization of business cycles in the euro area), is based on Bayoumi and Faruqee (1998).

The EViews codes used for the numeric simulations are, also, available from the authors upon request.

This research has been supported by a grant from the University of Strasbourg (Initiative d'Excellence (IdEx) -attractivité 2014). We would like to express our gratitude to the scientific committee of the University of Strasbourg. Table 1: Misalignments in real effective terms (in percent) EU FRA GER ITA SPA AUT FIN IRL NLD PRT GRC 1994 -

Appendix A: Methodological note on the underlying current account

A simple foreign trade model is used for all the countries with export and import equations for goods and services related to real exchange rates, domestic output gap for import and foreign output gap for export. The interested reader could consult the IMF occasional paper 167 in which this correction is completely described [START_REF] Bayoumi | A calibrated model of the underlying current account[END_REF]. Lagged effects of exchange rate variations are spread on three years ( t : 0.6; 1 t  : 0.25 and 2 t  : 0.15). Export price in domestic currency is independent from the real exchange rate while on the contrary import price in domestic currency depends immediately and completely of the exchange rate variation. The current account in percentage of GDP can be written as: