Céline Teulière

Eric Marchand

Laurent Eck

3D model-based tracking for UAV indoor localisation

3D model-based tracking for UAV indoor localisation

Céline Teulière, Eric Marchand, Laurent Eck

Abstract-This paper proposes a novel model-based tracking approach for 3D localisation. One main difficulty of standard model-based approach lies in the presence of low-level ambiguities between different edges. In this work, given a 3D model of the edges of the environment, we derive a multiple hypotheses tracker which retrieves the potential poses of the camera from the observations in the image. We also show how these candidate poses can be integrated into a particle filtering framework to guide the particle set toward the peaks of the distribution. Motivated by the UAV indoor localisation problem where GPS signal is not available, we validate the algorithm on real image sequences from UAV flights.

Index Terms-model-based tracking, particle filtering, UAV application

I. INTRODUCTION A. Overview

The past fifteen years have seen a major growth of interest in unmanned aerial vehicles (UAV) [START_REF] Valavanis | Advances in Unmanned Aerial Vehicles. State of the Art and the Road to Autonomy[END_REF]. UAV have strong potential applications such as surveillance, search and rescue, or inspection for maintenance. For navigation, existing systems mainly rely on the fusion of GPS and inertial measurements. However, such approaches are usually unsuitable for urban or indoor environments, where GPS is not available or imprecise. For this reason, recent works propose to make use of the visual information provided by one or two cameras [START_REF] Hérissé | A terrainfollowing control approach for a VTOL unmanned aerial vehicle using average optical flow[END_REF] [6] [START_REF] Klose | Markerless, vision-assisted flight control of a quadrocopter[END_REF] for UAV indoor localisation and navigation. Besides being passive, light and cheap sensors, cameras are usually necessary to provide a visual feedback in a surveillance or inspection task, and they provide a rich source of information on the environment. However, using vision for UAV control is also particularly challenging since strong constraints have to be taken into account: during a UAV flight, image quality can be poor (see Figure 2) with substantial noise and motion blur, and large motion disturbances occur, especially with small platforms like those usually considered for indoor missions.

To ensure safe vision-based navigation and control tasks, the robustness of the extraction of visual information is crucial. In particular, the choice of the visual features and their ability to be robustly matched and tracked over time will determine the good achievement of the task. Feature points, which can be C. Teulière was with CEA, LIST, 18 route du Panorama, Fontenayaux-Roses, F-92265, France. She is now with the Blaise Pascal University, Pascal Institute, Clermont-Ferrand, France. e-mail: celine.teuliere@univbpclermont.fr E. Marchand is with Université de Rennes 1, IRISA, Inria, Lagadic Project, Rennes, France. email: marchand@irisa.fr L. Eck is with CEA, LIST, Sensorial and Ambient Interfaces Laboratory, 91191 -GIF-sur-Yvette CEDEX, France. email: laurent.eck@cea.fr This work was realized in the context of the French ANR national project SCUAV (ANR Psirob 06 174032 SCUAV project ref ANR-06-ROBO-0007-02) extracted and tracked in any textured environment, have been used in various aerial applications, from structure from motion or SLAM [START_REF] Meingast | Vision based terrain recovery for landing unmanned aerial vehicles[END_REF] [START_REF] Angeli | 2D simultaneous localization and mapping for micro aerial vehicles[END_REF] to optical flow computation [START_REF] Hérissé | A terrainfollowing control approach for a VTOL unmanned aerial vehicle using average optical flow[END_REF]. However, in indoor structured environments with plain floors and walls, feature points are less frequent, leading to robustness issues. They are also sensitive to the typical noise produced by transmission interferences (see Figure 2).

Among the different vision-based localisation approaches, SLAM techniques are particularly interesting in unknown cluttered environments, but they are prone to suffer from those issues related to the use of feature points, and they do not provide an absolute localisation. In this paper, we propose a robust model-based tracking approach to estimate the 3D pose of the UAV through its motion. The vision system thus requires a 3D model of the edges of the environment. Although this condition is not suitable for unstructured outdoor missions, it seems reasonable in the context of indoor inspection tasks, for the maintenance of modelled industrial sites for example. Moreover, edges are very frequent in such structured environments, they offer a good degree of invariance to pose and illumination changes and are easy to detect even in presence of noise or blur. Edges have also been already used in the context of outdoor UAV applications, in particular to track linear structures such as pipes, or landing runway [START_REF] Rathinam | Vision based following of locally linear structures using an unmanned aerial vehicle[END_REF] [20] [3] [START_REF] Petit | Tracking complex targets for space rendezvous and debris removal applications[END_REF]. In such scenes the linear structures are generally distinct in the environment. In a structured environment however, one difficulty of using edges is that they suffer from having very similar appearances. Therefore, some ambiguities can occur typically when different edges get close to each other, which can lead to wrong matches and tracking failures.

To address this issue and ensure the robustness of the pose estimation, our 3D visual tracking approach takes into account multiple hypotheses. Our first contribution is to propose a novel method to retrieve multiple pose hypotheses from image measurements, using a clustering algorithm (Section II). These hypotheses, along with an associated confidence measure, provide a simple approximation of the probability density function. From this, we show that choosing the "best" hypothesis provides us with a tracking system with better performances than single hypothesis registration methods as described in [START_REF] Drummond | Real-time visual tracking of complex structures[END_REF], [START_REF] Comport | Real-time markerless tracking for augmented reality: The virtual visual servoing framework[END_REF]. Second, we derive from this an enhanced approach by combining top-down and bottom-up methods, in a particle filter framework. The hypotheses that result from the image are used to guide the particle set towards regions of interest in the space of 3D poses, allowing to consider fewer particles (Section III). We provide simulation results as well as validation on real sequences from UAV flights. To demonstrate the suitability of such a vision system in a UAV navigation task, we finally provide experimental results on a quad-rotor aerial vehicle (Section IV). This paper extends our shortest presentations of our tracker in [START_REF] Teulière | Using multiple hypothesis in model-based tracking[END_REF] with additional details and experiments.

B. Related work

Our work is mainly related to the literature 3D pose estimation in computer vision.

The problem of estimating the pose of a moving camera with respect to its modelled environment in real-time has been widely investigated in the past years (see [START_REF] Lepetit | Monocular model-based 3D tracking of rigid objects: A survey[END_REF] for a survey) and different approaches have been proposed to address it. Most of these approaches can be divided into two categories:

• Registration methods use non linear optimisation techniques (Newton minimisation, virtual-visual servoing,...) to find the pose which minimises a given reprojection error between the model and the image edges [START_REF] Lowe | Fitting parameterized three-dimensional models to images[END_REF], [START_REF] Drummond | Real-time visual tracking of complex structures[END_REF], [START_REF] Comport | Real-time markerless tracking for augmented reality: The virtual visual servoing framework[END_REF]. The robustness of these methods has been improved by using robust estimation tools [START_REF] Armstrong | Robust object tracking[END_REF], [START_REF] Drummond | Real-time visual tracking of complex structures[END_REF], [START_REF] Comport | Real-time markerless tracking for augmented reality: The virtual visual servoing framework[END_REF]. In the UAV context [START_REF] Kemp | Visual control of a miniature quad-rotor helicopter[END_REF] applied modelbased tracking to UAV navigation. On main difficulty of such approaches alone is that they can fail in case of large displacements or wrong edge matching, especially in cluttered environment. • Bayesian methods, on the other hand, have been used

to perform the same task by estimating the probability density associated to the pose. This can be achieved by Kalman filtering when the probability density function (p.d.f.) can be represented by an uni-modal Gaussian distribution. More recently, the improvement of computational performances has allowed to consider particle filtering approaches [START_REF] Pupilli | Real-time camera tracking using known 3D models and a particle filter[END_REF], [START_REF] Klein | Full-3d edge tracking with a particle filter[END_REF], [START_REF] Nuske | Visual localisation in outdoor industrial building environments[END_REF]. Instead of going from the low level edges to retrieve the camera pose, particle filtering uses a set of hypotheses on the possible camera poses (the particles). The likelihood of each particle is then measured in the image. Since the space of all possible poses is large, the main issue is to keep a fair representation of the different modes of the state probability distribution while using few particles.

To overcome the edge matching issue of registration-based methods, [START_REF] Vacchetti | Combining edge and texture information for real-time accurate 3d camera tracking[END_REF] proposed to include multiple low level hypotheses in the robust registration method, showing improved performances. However it still maintain a unique hypothesis on the camera pose. Also motivated by the robustness required for UAV application, [START_REF] Kemp | Multi-modal tracking using texture changes[END_REF] proposed to use RANSAC to maintain a multi-modal representation of the posterior density in a particle-filter inspired way. Our tracking approach differs from [START_REF] Kemp | Multi-modal tracking using texture changes[END_REF] on two main points:

• First, we propose a novel method for generating multiple edge hypotheses, based on k-mean clustering principle (see Section II). • Second, while in [START_REF] Kemp | Multi-modal tracking using texture changes[END_REF] all the pose hypotheses are derived from the image, in our approach we propose to combine hypotheses generated at the pose level, as propagated through a classic condensation scheme [START_REF] Isard | ICONDENSATION: Unifying low-level and high-level tracking in stochastic framework[END_REF], with the multiple hypotheses generated thanks to our k-mean based registration process (see Section III).

The next two sections describe our vision-based approach in details.

II. MULTIPLE HYPOTHESES REGISTRATION

Our multiple hypotheses registration algorithm relies on a similar basis as the ones used in [START_REF] Comport | Real-time markerless tracking for augmented reality: The virtual visual servoing framework[END_REF], [START_REF] Drummond | Real-time visual tracking of complex structures[END_REF] and [START_REF] Vacchetti | Combining edge and texture information for real-time accurate 3d camera tracking[END_REF]. Assuming the camera parameters and an estimate of the pose are known, the 3D model is first projected into the image according to that pose, which can be the previous one or a prediction obtained from a filter.

Formally, the projection of an edge L i of the 3D model according to the pose c M w 1 will be denoted by E i = L i ( c M w ). Each projected edge E i is sampled, giving a set of points {e i,j } (see Figure 1). From each sample point e i,j a search is performed along the edge normal to find strong gradients. As illustrated in Figure 1, two close edges can lead to matching ambiguities. In [START_REF] Comport | Real-time markerless tracking for augmented reality: The virtual visual servoing framework[END_REF] the point of maximum likelihood with regard to the initial point e i,j is selected from the exploration step. It is denoted by e i,j in the following. A non linear optimisation approach is then used to estimate the camera pose which minimises the errors between the selected points and the projected edges [START_REF] Lowe | Fitting parameterized three-dimensional models to images[END_REF], [START_REF] Comport | Real-time markerless tracking for augmented reality: The virtual visual servoing framework[END_REF], [START_REF] Drummond | Real-time visual tracking of complex structures[END_REF], that is:

c M w = arg min c Mw i,j d ⊥ (E i , e i,j ) (1)
where

d ⊥ (E i , e i,j ) = d ⊥ (L i ( c M w ), e i,j
) is the squared distance between the point e i,j and the projection E i of the linear segment L i of the model. The quantity to minimise is then expressed by:

S = 1 N e i j ρ d ⊥ (E i , e i,j ) (2)
1c Mw ∈ SE( 3) is an homogeneous matrix that gives the position of the camera in the scene frame. It is given by

c Mw = c Rw c tw 0 1
where c Rw ∈ SO( 3) is a rotation matrix and c tw ∈ R 3 is a translation vector. where N e is the total number of sampled points, and ρ is a robust estimator.

To be more robust to ambiguous cases, our registration method considers multiple low level hypotheses {e i,j,l } corresponding to local extrema of the image gradient along the edge normal in e i,j . Instead of performing one single minimisation from these points as in [START_REF] Vacchetti | Combining edge and texture information for real-time accurate 3d camera tracking[END_REF] resulting in one single pose, we go from these multiple low level hypotheses to multiple hypotheses on the camera pose itself. This process is described in the next section.

A. Determining the underlying edges

In order to retrieve multiple hypotheses for the camera pose from the detected low level hypotheses, we first determine the underlying lines from the set of points {e i,j,l }. The idea is to use the knowledge we have about the linear components of the model, to assign each detected point to a potential edge and to associate a confidence criterion to these candidate edges (see Figure 3). In [START_REF] Kemp | Multi-modal tracking using texture changes[END_REF] this was achieved via a RANSAC approach. Here, we express this as a typical classification problem, and propose to tackle it using a k-mean clustering algorithm [START_REF] Hartigan | Clustering algorithms[END_REF].

In our case the clustering algorithm is slightly modified to group the candidate points into edge hypotheses. For each projected edge E i , the algorithm segments the candidate points {e i,j,l } into k i sets of points or classes (C i 1 , ..., C i ki ). The mean of each of the k i classes corresponds in our case to the line obtained by a least square minimisation on the points of that class. To initialise the algorithm, the number k i of classes for the edge E i is set to the maximum number of candidate points detected, that is: k i = max j {n i,j }. The classes (C i 1 , ..., C i ki ) are initialised using the order in which the hypotheses have been found on the normal. That is, for each class C i m : C i m = {e i,j,m } j . This initialisation is often close to the correct segmentation, allowing the algorithm to converge faster (see Figure 3 (a)). At each iteration of the algorithm, the "mean line" of each class is computed (Figure 3 (b)). Each point is then assigned to the class with the nearest mean line.

Since the potential edges are not supposed to be normal to the initial edge, we add the constraint that two hypotheses e i,j,l1 and e i,j,l2 of a same initial sample point e i,j cannot belong to the same class. The process is summarized in the algorithm 1.

In k-mean algorithms the local convergence is ensured by the finite number of classes and the decrease of the cost function at each iteration. Other ending criteria such as a maximum number of iterations and/or a threshold on the cost function decrease rate are often added to set a maximum computational time. In our case, we do not have an analytical proof of convergence and base the choice of this clustering component on experimental performances we observe. The algorithm is deemed to have converged when the assignments no longer change or the iteration number reaches a given threshold which gives an upper limit in terms of computation time (this threshold was set to 30 in our experiments).

k-mean algorithm for edge clustering. Let {e i,j,l } be the candidate points corresponding to the projected edge E i and k i = max j {n i,j } the maximum number of candidates detected for a sample point. The algorithm is initialised by respecting the order of detection of the candidates: C i l = {e i,j,l } j . Then the process is iteratively run as follows:

1. For each class C i l ∈ {C i 1 , . . . , C i ki } compute the interpolation line of the points in the class. Let r i l denote the residual of the interpolation. 2. For each i, j :

compute the distance from the points e i,j,l to each computed line, the points e i,j,l (for i, j fixed) are then associated to different classes, with higher priority for the points closer to the closest line. The algorithm runs until the assignments no longer change in step 2 or the iteration number exceeds the maximum iteration number.

Algorithm 1: k-mean algorithm for edge clustering.

Finally, the k-mean algorithm corresponding to the initial edge E i provides us with a set of classes C i m = ({e i,j,m } j , r i m ) where r i m is the residual of the least square minimisation. This residual gives a measurement of the confidence we have in the corresponding candidate edge. Note also that only the lines with a sufficient number of points are taken into account. This allows to avoid cases where 2 or 3 well aligned points would then form a candidate edge considered as very likely without this class being meaningful. We thus eliminate classes with less than five points. Figure 3 shows a simple example of the process. Although the contours considered have been restricted to lines in this study, the approach can be easily adapted to other kinds of contours.

In most cases, the number of classes k i does not exceed two or three. Figure 5 gives an example of the lines detected from the teabox sequence.

B. From edge hypotheses to pose hypotheses

Once candidates have been obtained for each edge in the form of sets of points associated to a residual, random weighted draws are performed. We compute weights w i m for each candidate as a function of the residuals. Several choices for computing the weights are possible and we use the following expression that gives satisfying performances:

w i m =    e -λ r i m -r i min r i max -r i min 2 if r i max = r i min 1 otherwise. ( 3
)
where λ is a parameter that can be tuned according to the selectivity that is desired. A value of λ = 1 has been used in our experiments. One weighted draw will denote here the draw of one candidate per edge, that is, for each edge E i a class C i pi is drawn from the k i classes. From each draw, a numerical nonlinear minimisation is performed according to (4), using the set of points corresponding to the picked classes, resulting in a camera pose. We use the virtual visual servoing minimization process [START_REF] Comport | Real-time markerless tracking for augmented reality: The virtual visual servoing framework[END_REF] but any other non-linear minimisation method could be used for this step.

S = 1 N e i e i,j,l ∈C i p i ρ ∆ Ei (e i,j,l ) (4)
Since the optimisation is deterministic, it is only computed when the sets of candidates are different. The weighted draw allows to favour, among all the possible combinations, the ones with the candidates of lowest residual, which are more likely to correspond to a real edge. Several hypotheses on the camera pose are thus obtained from the low level detected hypotheses. The process is summarized in Figure 4. In practice, since the number of candidate lines per edge is small, so will be the number of optimisations to be performed and thus the number of pose candidates obtained. In our experiments we set the number of optimisations (and thus pose hypotheses) to 3.

C 1 1 , w 1 1 C 1 p1 C i pi C n pn C i 1 , w i 1 C i 2 , w i 2 C i 3 , w i 3 C n 1 , w n 1 C n 2 , w n 2

Minimisation

One camera pose Fig. 4. From low level hypotheses, classes of points are extracted. For each projected edge a random weighted draw is performed among the classes to determine the points that will be used for the minimisation process. The minimisation provides an hypothesis on the camera pose. A different draw would lead to another candidate pose.

C. Experiments

To illustrate the benefits of this approach in terms of robustness, Figure 5 shows an example where the multiple hypotheses approach allows to avoid failure. At this particular frame, extracted from a video sequence, a single hypotheses tracker fails. While running the multiple hypotheses algorithm, it appears (see Figure 5) that only one candidate is found for almost each projected edge, except the top back one. For this edge, two candidates have been found, which lead to two different camera poses. Whereas the single hypothesis tracker fails due to a wrong match, the multiple hypotheses tracker finds the correct pose (Figure 5-(2-b)).

To validate the proposed approach, we also used a simulated sequence, for which the ground truth is known. The comparative results between the classic registration method and our multiple hypotheses method are shown in Figure 6.

In the single hypothesis case, only the maximum likelihood point is selected in the search along the normal. The tracking fails when confronted to ambiguities, that is especially when two edges get close to each other, or when a new face appears (Figure 6-(a) and(c)).

In the multiple hypotheses case, the output considered at this stage is the camera pose which gives the lowest residual in the minimisation process. The object is successfully tracked even in cases of ambiguities. However, at some ambiguous frames where two candidates give almost the same residual, the tracker selects the wrong one as the best, which results in some jitter on the camera trajectory (Figure 6 (d)). In the same way, the minimisations which lead to Figure 5 (1-b) and (2b) correspond to minima giving almost the same residuals. By selecting only the "best" one, some information given by other candidates can be lost. Moreover the tracker still needs frame to frame motion to be small to converge and could benefit from a prediction.

The next section presents an enhanced algorithm of our pose estimation tracker that deals with these issues and ensures temporal coherence of our pose estimation.

III. ORIENTED PARTICLE FILTER

To tackle the above issues, we consider particle filtering. Particle filtering offers a very interesting framework in that it provides both temporal coherence through the filtering process and multi-hypotheses handling through the particle representation. More specifically the main interests of the particle filtering framework at this point are:

• the temporal filtering which can incorporate easily some prior knowledge on the UAV motion and prevents the system to fail in case of large occlusion. • the possibility to handle multi-modal representations, which means concretely that, if 2 poses have similar residual, the particle filter will keep particles corresponding to both hypotheses, while the optimisation-based approach presented in the Section II will detect several, choose one, and forget about the others.

The remainder of this section presents how we combined the optimisation process presented above with the particle filtering framework.

A. Overview

As mentioned in the introduction, particle filtering approaches [START_REF] Isard | ICONDENSATION: Unifying low-level and high-level tracking in stochastic framework[END_REF] have been recently introduced in model-based tracking as an alternative to numerical optimisation methods, showing promising performances [START_REF] Pupilli | Real-time camera tracking using known 3D models and a particle filter[END_REF], [START_REF] Klein | Full-3d edge tracking with a particle filter[END_REF], [START_REF] Nuske | Visual localisation in outdoor industrial building environments[END_REF].

Our approach relies on the same basis as these works, that is the classic CONDENSATION algorithm [START_REF] Isard | CONDENSATION: conditional density propagation for visual tracking[END_REF], which represent the probability density function p(x k | z 1:k ) of the state x k which in our case will be the pose at frame k, by a finite set {(s

(i) k , π (i) k )} i=1..N of N samples, or particles, s (i) k associated with the weights π (i) k . Each particle s (i)
k represents a potential camera pose and z 1:k are the observations until frame k, deduced from image measurements. For each new frame, the particles first evolve according to a given dynamic model. Then, the likelihood of every particle is measured in the image and a weight is derived. The output considered is usually the weighted mean of the resulting set of particles. The particle set is updated by performing a random weighted draw among the particles.

It is interesting to note that whereas the tracker presented in the previous section was a bottom-up approach, in which multiple hypotheses on the camera pose were derived from low level hypotheses, particle filtering does the opposite. Multiple hypotheses are made on the camera pose at start, and the likelihood of these hypotheses is measured on the low level, in the image.

The main difficulty with these top-down approaches, as proposed in [START_REF] Klein | Full-3d edge tracking with a particle filter[END_REF] and [START_REF] Nuske | Visual localisation in outdoor industrial building environments[END_REF], results from the great size of the considered state space. For the tracking to be accurate enough, a large number of particles is needed. [START_REF] Klein | Full-3d edge tracking with a particle filter[END_REF] and [START_REF] Nuske | Visual localisation in outdoor industrial building environments[END_REF] use particle annealing method as a hierarchical approach to reduce the particle number. However, the likelihood functions proposed still need to be very fast to compute, since they have to be called for each particle. These functions may not distinguish enough between low level ambiguities.

In this paper, we propose to use particles resulting from our multiple hypotheses tracker to guide the particle set towards the local maxima of the distribution. The coherence is ensured using the so-called importance sampling method [START_REF] Isard | ICONDENSATION: Unifying low-level and high-level tracking in stochastic framework[END_REF].

1) State space: The state space considered is the special Euclidean group SE(3) of all possible pose matrices which transform points from homogeneous world coordinates to the camera coordinate frame.

c M w = c R w c t w 0 1
where c R w ∈ SO( 3) is a rotation matrix and c t w ∈ R 3 is a translation vector. SE( 3) is the group of rigid body transformations.

2) Particles propagation on SE(3): As in [START_REF] Klein | Full-3d edge tracking with a particle filter[END_REF], the propagation model that has been considered in the experiments is a simple Gaussian noise centered on the previous pose. Since SE( 3) is a Lie group, there exists an exponential map between SE(3) and its Lie algebra se(3). Gaussian noise is first added on the canonical exponential coordinates and the resulting pose matrix is computed using the exponential map. Formally,

x pred = M σ .x (5)
where M σ = exp(v), v ∼ N 0,σ 2 I6 , v ∈ se(3) and σ is the vector of the covariances associated to the components of v.

3) Mean and averaging on SE(3): Since the addition is not a binary operation on SO(3) (and thus SE(3)), the arithmetic mean R = 1 N N i=1 R i of a set of rotation matrices R i is usually not a rotation. It is however possible to define a meaningful average as the point in SO(3) which minimises the sum of squared distances to the considered points. This distance can be extrinsic when we use the vector space embedding SO(3) or intrinsic when a Riemannian distance is considered [START_REF] Fletcher | IEEE Int. Conf. on Computer Vision and Pattern Recognition[END_REF]. In this work an extrinsic distance was used, following [START_REF] Moakher | Means and averaging in the group of rotations[END_REF]. The average rotation is thus computed as the arithmetic mean R, followed by the unique projection onto SO(3) given by the unique polar factor in the polar decomposition of R. Let R = UΣV be the singular value decomposition of R, then the mean rotation R m is given by:

R m = VU if det(R) > 0 VHU otherwise, (6)
where

H = diag(1, 1, -1).
Finally, the weighted mean of the particle set is computed using this average rotation and the arithmetic mean of the translations.

4) Likelihood evaluation: Each particle represents a potential camera pose which has to be evaluated according to image measurements. In [START_REF] Klein | Full-3d edge tracking with a particle filter[END_REF] the contours are projected according to the particle s to evaluate, and the ratio between the number n of pixels of the projected contours which do correspond to an edge in the image, and the total number v of pixels on the visible contours is computed. The likelihood of the particle s is then derived from this ratio by:

p(z | x = s) = e λ n v (7)
where λ is a parameter to be tuned. To decide whether a pixel do correspond to an edge in the image, a distance map [START_REF] Fabbri | 2D euclidean distance transform algorithms: A comparative survey[END_REF] is first computed, providing for each pixel of the image the distance to the closest edge and its direction (see Figure 7). Then a threshold on the distance has to be set to determine the inlier/outlier count. The distance map has to be computed only once per frame, which make the likelihood value very fast to compute. [START_REF] Klein | Full-3d edge tracking with a particle filter[END_REF] showed the computation can be performed in real time on a graphics processing unit (GPU).

In this paper, the distance map is directly used to compute a mean distance:

d(s) = 1 N i d i (8)
where d i is the distance given by the distance map for the pixel i, that is the distance between the pixel i and the closest contour in the image. The pixels i are the pixels of the projected edges. The use of the direction to the nearest edge could improve the discriminative power of the distance function. However, we found that this measure was accurate enough in our experiments. Figure 8 shows the shape of the distance d with respect to in-plane translations for the frame of Figure 7. The likelihood is derived from this distance by:

p(z | x = s) ∝ e -λ d(s)-d min dmax -d min 2 if d max = d min 1 otherwise. (9)

B. Importance sampling

In our work we propose to use the registration method presented in Section II to guide the particle set towards the regions of interests. The approach is inspired from hybrid particle filters like [START_REF] Sullivan | Guiding random particles by deterministic search[END_REF] [4] [START_REF] Shan | Real-time hand tracking using a mean shift embedded particle filter[END_REF] where some particles are moved to local maxima of the likelihood by a local optimisation. In our case, the optimisation corresponds to the multiple hypotheses tracker described in Section II. To reduce computational complexity, the optimisation is only applied to a subset of particles whom likelihood is above a given percentage of the maximum likelihood. The resulting set of particles still provides a good representation of the main modes of the density, but it is not directly sampled from the prior distribution

f k (x k ) = p(x k | z 1:k-1
) as required by Bayesian filtering theory. However, the new particles {(s * k (i) )} i=1..N * can be regarded as sampled from an importance function g k (x k ). A corrective term f /g can then be applied to the weights of the particles according to the importance sampling theory [START_REF] Isard | ICONDENSATION: Unifying low-level and high-level tracking in stochastic framework[END_REF] to maintain a fair distribution. As in [START_REF] Bray | Smart particle filtering for 3D hand tracking[END_REF], f k (x k ) and g k (x k ) are approximated by Gaussian mixtures to evaluate the corrective term:

f k (x k ) = 1 N N i N (s k (i) , Σ)(x k ) (10)
g k (x k ) = N N + N * 1 N N i=1 N (s k (i) , Σ)(x k )+ 1 N * N * i=1 N (s * k (i) , Σ)(x k ) (11)
where N (s k (i) , Σ) denotes the 6-dimensional normal distribution of covariance Σ centered on the exponential coordinates of the pose s. The whole algorithm is summarized in the algorithm 2.

Note that to avoid a loss of diversity in the particles, known as the sample impoverishment issue, the initial particles are not removed but combined with the optimised ones.

Besides, the addition of optimised particles generated from a bottom-up optimisation process as well as the resampling step also prevent the approach from degeneracy issues.

C. Experiments 1) Comparative results:

To underline the improvement in robustness brought by particle filtering framework, the tracker was tested on different sequences taken from a UAV. Comparative results are presented in the Figures 9 and10.

The window sequence presents important frame to frame motion and occlusions, as much as great illumination changes. Results are shown in Figure 9. The registration process alone fails when the occlusion is too important (Figure 9 1-c).

Algorithm summary.

Given the set (s k-1 (i) , 1 N ) i=1..N of N particles of equal weights 1 N at frame k -1, the algorithm goes as follow: • Propagation of the particles according to [START_REF] Comport | Real-time markerless tracking for augmented reality: The virtual visual servoing framework[END_REF] to get a set (s

(i) k , 1 N +N * ) i=1.
.N +N * . • Weight computation for each particle using a corrective term:

π (i) k ∝ f k (s (i) k ) g k (s (i) k ) p(z k | x k = s (i) k ), with N +N * i=1 π (i) k = 1.
See III-A4. It gives the set (s

(i) k , π (i) k ) i=1..N +N *
• Estimation of the tracking result as the weighted mean of the particle set (see III-A3). • Resampling by performing a weighted draw of N particles among the N + N * particles Algorithm 2: Algorithm summary.

Embedded in a particle filtering framework, the window is tracked all along the sequence. This experiment shows the interest of introducing temporal filtering, with respect to the optimisation-based approach of Section II.

Thanks to the optimisation of some of the particles, a small number of particles is needed. For the window sequence of Figure 9, only 100 particles were used.

Figure 10 shows comparative results for a complex structured sequence with ambiguities (see rows 3 and 5), noise (2nd row for instance) and sometimes few information when few edges are visible in the image (4th row). The 1st column presents the results from the single optimisation approach. On the 2nd column a particle filter with 25 particles has been applied, which is not enough to track the camera pose. With 200 particles the filter does not diverge anymore but the accuracy is still poor (3rd column). Our hybrid approach (4th column) allows good estimation with very few particles (25 particles only were used in this example).

2) Computational complexity and execution time: In the different tracking approaches considered in this paper, for a given image and model, the computational complexity depends on different parameters:

• the number K of optimisations,

• the average number M of iterations per optimisation,

• the number N of particles when particle filtering is used. If M denotes the average number of iterations necessary for the classic registration method, then the complexity of this method is C = O(M ).

The multiple hypotheses optimisation process of Section II requires the computation of several optimisations when ambiguities are met. In practice, the number K of optimisations needed for a real robustness improvement does not exceed 4 or 5, and this approach is still suitable for real time: C = O(M ).

In the case of particle filtering integration III, the complexity mainly depends on the number of particles and the optimisations applied: C = O(N ×M ×K). However, the number N of particles required is significantly lower than for classic particle filtering. Moreover, the mean number of iterations required by the optimisation process is reduced thanks to the prediction of the particle filter, and the fact that the optimisation is only applied to the best particles.

In the example of Figure 10 the execution time of the hybrid approach was 10 frames per second, without specific optimisation, on a standard PC computer. Since particle filtering is well adapted to parallelisation, the approach is suitable for real-time applications.

IV. SUITABILITY OF THE APPROACH FOR UAV POSITIONING TASKS.

In Sections II and III we presented a multiple hypotheses framework to estimate the pose of a camera and we showed how it allows to improve the tracking robustness with regard to the constraints associated with UAV flight. This section presents an experimental validation of the suitability of the presented approach in the UAV indoor navigation context. Since our current implementation of the algorithm of Section III is still slow for UAV control, we consider here a simplified version to validate the feasibility of the global approach. We thus estimate the pose from our multiple hypotheses tracking (Section II) with 3 optimisations, and fuse it with inertial data in a Kalman filter instead of particle filtering. With such simplification, the pose estimation is still better than the classical approach, as shown in Section II, since it still retrieves several poses from the images. However, compared to the particle filter, Kalman filtering is mono-modal and does not keep a memory of all the candidate poses when one is selected. The main objective is to test our vision-based approach in a real UAV experiment, faced with the challenges of flight conditions, to validate the feasibility of the overall system.

A. Experimental setup

The experiments have been conducted on the quad-rotor (X4-flyer) developed by the CEA LIST (Figure 11). The UAV sends the images from its embedded camera to the ground station (PC) via a wireless analogical link of 2.4GHz. Images are processed on the ground station with a framerate of 20Hz. A scene was built, combining planar and 3D objects (see Figure 2). The tracking initialisation has not been considered in this paper. During the experiment, the tracking is automatically initialised by detecting a black dot in its first position (Figure 2). Then the vehicle can locate itself thanks to the model-based tracking, without using the dot anymore.

B. Accuracy of the estimation

To evaluate the accuracy of the estimation of the position of the UAV, we compared it to a ground truth obtained using a metrologic laser tracker Leica 2 of micrometric precision.

Figures 12 and13 show the comparative results obtained for the pose estimation for a flight in the modelled scene.

The standard deviation for the pose estimation is about 16cm in translation which is much better than the localisation achieved with standard GPS. The main limiting factor here is the accuracy of the 3D model, which was obtained by coarsely measuring the different elements.

experiment also shows the robustness of the approach to typical interference noise that was present in the flight (see Figure 2 (b)).

C. Navigation

In [START_REF] Teulière | 3D model-based tracking for UAV position control[END_REF] we show that such a localisation method can be used for the UAV control. We do not focus here on the control strategy which is not the aim of this paper. We show however that the proposed approach was successfully validated on a way-point navigation trajectory.

The task considered was to autonomously reach several set points successively: first the vehicle is stabilised 2 meters above the dot target (p = (0, 0, -2)). Then the set points are successively set to (0, -2, -2), (-2, -2, -2), (0, -2, -2), (0, 0, -2), (-2, 0, -2) (see Figure 14b).

V. CONCLUSIONS

In this paper, we proposed a novel model-based tracking system. Our multiple hypotheses registration process provides us with multiple pose hypotheses by performing several minimizations, corresponding to different sets of points. We show how these hypotheses can be integrated into a particle filter. The multiple hypothesis tracker is applied to the best particles to move them to the local maxima of the likelihood function. The particle set is therefore guided toward the candidate poses emerging from the multiple hypothesis tracker. Although the state space is large, a small number of particles are needed. In that case, the tracking approach benefits from both the temporal coherence and multi-modal representation of the Bayesian framework as well as the accuracy of a registration process. The resulting approach has been validated on different sequence from UAV flights. A simplified approach has been tested on positioning tasks on a quad-rotor UAV. The accuracy of the estimates has been evaluated and the experiment shows the feasibility of the proposed approach in indoor structured environments.

Fig. 1 .

 1 Fig. 1. In classic edge based tracking, the model is projected into the image plane and points are sampled on the projected edges. A search is performed along the normal (top). When multiple strong edges are close in the image, the exploration of the normal can lead to ambiguities (bottom).

Fig. 2 .

 2 Fig. 2. (a) Example of 3D model, (b) corresponding scene (undistorted) and (c) tracking result.

Fig. 3 .

 3 Fig. 3. An example of the k-mean computation, with k = 3. Each class is represented by a different color. (a) Initialisation of the classes of points. (b) Mean lines computation. (c) The final segmentation is obtained in one step.

(Fig. 5 .

 5 Fig. 5. Example of multiple hypotheses ambiguity. On the top frame, all the candidate lines and their corresponding points have been represented. The gray level of the line corresponds to its likelihood. When tracking the topback edge, two hypotheses have been found, one of them corresponding to the top-front edge. (1-a) and (2-a) show two different draws from the initial set of candidates, resulting in two different camera poses (1-b) and (2-b). In the first draw, the back edge has been mixed up with the front one, leading to a tracking failure. Using multiple hypotheses allows to be more robust to such situations.

Fig. 6 .

 6 Fig. 6. Comparative results on a simulated sequence with auto-occlusion. The basic algorithm with single hypothesis (1-a), (1-b) fails when a new face appears. While considering multiple hypotheses (2-a), (2-b), the object is successfully tracked. The ground truth for the camera pose is shown in (c).

Fig. 7 .

 7 Fig. 7. Window frame (left) and its distance map (right). The darkest values correspond to the smallest distances.

Fig. 8 .

 8 Fig. 8. Distance function with respect to x and y translations for the window frame. The zero position corresponds to the true camera pose.

Fig. 9 .

 9 Fig. 9. Window sequence. The registration method alone (first row) fails when large occlusions occur (1-c). The multiple hypotheses tracker embedded in particle filtering framework (second row) tracks the object successfully.

Fig. 11 .

 11 Fig. 11. Quad-rotor UAV.

Fig. 10 .

 10 Fig. 10. Comparative results for the apartment sequence.

Fig. 12 .

 12 Fig. 12. Position estimated with the model-based tracking (blue) and ground truth (red).

Fig. 13 .

 13 Fig. 13. Position estimated with the model-based tracking (blue) and ground truth (red).

Fig. 14 .

 14 Fig. 14. Example of camera view with model reprojection (a) and required trajectory (b).

Figure 15 ,

 15 Figure15, shows the estimated and ground truth trajectories.The system was able to localise itself despite noise and large interframe motion.

Fig. 15 .

 15 Fig.[START_REF] Kemp | Visual control of a miniature quad-rotor helicopter[END_REF]. UAV trajectory as estimated by our vision algorithm (blue) and ground truth (red).

  Distance measurement for the optimised particles.• Combination of the particles s k

					, giving the
	new set: (s k • Distance measurement for each particle, computed from (i) , 1 . N ) i=1..N
	equation (8), to determine which particle to optimise
	• Optimisation of the best particles: the multiple
	hypotheses tracker is applied from the camera poses
	corresponding to the best particles (with a distance
	below a given threshold). One optimisation can lead
	to several hypotheses. A set of optimised particles
	(s * k	(i)	, 1 N * )	i=1..N * is obtained.
				(i)	k and s *	(i)

•


Robert Becker
email: becker@indiana.edu.

Stefano Bosi
email: stefano.bosi@univ-evry.fr.

Cuong Le Van
email: cuong.le-van@univ-paris1.fr.

Thomas Seegmuller
email: thomas.seegmuller@univ-amu.fr.

On existence and bubbles of Ramsey equilibrium with borrowing constraints ⇤

Keywords: Existence of equilibrium, bubbles, Ramsey model, heterogeneous agents, borrowing constraint, endogenous labor. JEL classification: C62, D31, D91, G10

We study the existence of equilibrium and rational bubbles in a Ramsey model with heterogeneous agents, borrowing constraints and endogenous labor.

Applying a Kakutani's fixed-point theorem, we prove the existence of equilibrium in a time-truncated bounded economy. A common argument shows this solution to be an equilibrium for any unbounded economy with the same fundamentals. Taking the limit of a sequence of truncated economies, we eventually obtain the existence of equilibrium in the Ramsey model.

In the second part of the paper, we address the issue of rational bubbles and we prove that they never occur in a productive economy à la Ramsey.

Introduction

Frank [START_REF] Ramsey | A mathematical theory of saving[END_REF] seminal article on optimal capital accumulation ends with a famous conjecture: "... equilibrium would be attained by a division into two classes, the thrifty enjoying bliss and the improvident at the subsistence level ". In the long run, the most patient agent(s) would hold all the capital, while the others would consume at the minimum level necessary to sustain their lives. [START_REF] Becker | On the long-run steady state in a simple dynamic model of equilibrium with heterogeneous households[END_REF] demonstrated the Ramsey conjecture in the case of a stationary equilibrium when households face borrowing constraints. Without such a constraint, markets are complete and the impatient households would borrow against the future stream of their labor incomes, consume more in the present and accept their consumption converges to zero as time tends to infinity (Le [START_REF] Van | Existence of a competitive equilibrium in a one sector growth model with heterogeneous agents and irreversible investment[END_REF], and [START_REF] Becker | Optimal growth with heterogeneous agents and the twisted turnpike: an example[END_REF]). 1 In contrast, borrowing constraints result in impatient agents' positive consumption (equal to wage) at a steady state.

In the last three decades, the framework introduced by Becker (1980) has been used for di↵erent purposes. For instance, Becker andFoias (1987, 1994) and [START_REF] Sorger | On the structure of Ramsey equilibrium: cycles, indeterminacy, and sunspots[END_REF][START_REF] Sorger | Chaotic Ramsey equilibrium[END_REF] prove that persistent cycles of period two as well as chaotic solutions arise when the capital income monotonicity fails. Under additional market imperfections (strategic behavior on capital markets and progressive capital taxation), [START_REF] Becker | Strategic Ramsey equilibrium dynamics[END_REF], [START_REF] Sarte | Progressive taxation and income inequality in dynamic competitive equilibrium[END_REF] or [START_REF] Sorger | On the long-run distribution of capital in the Ramsey model[END_REF][START_REF] Sorger | Di↵erentiated capital and the distribution of wealth[END_REF][START_REF] Sorger | Strategic saving decisions in the infinite-horizon model[END_REF] prove that impatient households may hold capital in the long run. [START_REF] Bosi | On the Ramsey equilibrium with heterogeneous consumers and endogenous labor supply[END_REF] extend the Ramsey model with heterogeneous households to endogenous labor.

Our Ramsey model with heterogeneous households, endogenous labor and borrowing constraints addresses two important issues: the existence of equilibrium on the one hand and rational bubbles on the other hand.

To the best of our knowledge, the existence of bubbles has never been considered in such a context, while the existence of the equilibrium was tackled only by [START_REF] Becker | The existence of Ramsey equilibrium[END_REF] and [START_REF] Bosi | On the Ramsey equilibrium with heterogeneous consumers and endogenous labor supply[END_REF]. In the latter, labor supply is endogenous, but the existence of the intertemporal equilibrium is shown only in a neighborhood of the steady state. [START_REF] Becker | The existence of Ramsey equilibrium[END_REF] focus on the model with inelastic labor supply, but provide a global existence argument. Their proof rests on the introduction of a tâtonnement continuous map in which each fixed point yields an equilibrium. However, their argument no longer works when labor is elastically supplied. Our proof of existence holds in this more general and challenging case and, by the way, a proof of nonexistence for rational bubbles is also provided. 2We show the existence of Ramsey equilibrium in three steps. [START_REF] Becker | On the long-run steady state in a simple dynamic model of equilibrium with heterogeneous households[END_REF] We start by considering a bounded time-truncated economy and adapt a Kakutani's fixed-point argument to account for the sequential budget constraints in the model's household sector. 3 (2) This solution remains an equilibrium as the uniform bounds are relaxed. (3) Taking the limit of a sequence of time-truncated economies, we eventually prove the existence of equilibrium in the infinitehorizon economy. To the best of our knowledge, there are no papers that prove the existence of equilibrium under imperfections in the financial markets (borrowing constraint) in the case of capital accumulation and endogenous labor supply.

Our setup is also suitable to address the important issue of existence of rational bubbles in a general equilibrium model. The seminal models on the existence of rational bubbles in a general equilibrium context are overlapping generations (OG) economies [START_REF] Tirole | Asset bubbles and overlapping generations[END_REF]). Financially constrained economies seem to share some of the same properties found in OG models. In connection with the [START_REF] Tirole | On the possibility of speculation under rational expectations[END_REF] idea that new traders should enter the market each period, more recent contributions have shown that bubbles may exist in exchange economies with heterogeneous infinite-lived households facing some borrowing constraints [START_REF] Kocherlakota | Bubbles and constraints on debt accumulation[END_REF], [START_REF] Huang | Asset price bubbles in Arrow-Debreu and sequential equilibrium[END_REF]). Conversely, we prove that rational bubbles fail to exist when production is taken into account because of a positive interest rate. Our condition for ruling out any bubble is less demanding than the one one finds in [START_REF] Kocherlakota | Bubbles and constraints on debt accumulation[END_REF] or in [START_REF] Huang | Asset price bubbles in Arrow-Debreu and sequential equilibrium[END_REF], that is an endowment growth rate lower than the interest rate.

Section 2 introduces the model. Sections 3 and 4 focus on the definition and the existence of a Ramsey equilibrium. Sections 5 proves that rational bubbles never emerge. Technical details are gathered in Appendices 1 to 3.

The Ramsey model

The Ramsey equilibrium model specifies the behavior of a finite collection of households and the profit motive governing production. Each household is infinitely-lived and enjoys a felicity, or reward, at each time based on its consumption and leisure time. Lifetime utility is the discounted sum of felicities and each household's discount factor is a given constant. The production technology is defined by a one-sector model with a single all purpose consumption-capital good. Households supply capital goods and labor services to the production sector at each time. Time is discrete and there is an infinite horizon. Markets are perfectly competitive and households act with perfect foresight when composing their consumption and investment decisions. Each household's budget and the productivity, the proof of [START_REF] Becker | The existence of Ramsey equilibrium[END_REF] can not be carried over our model under endogenous labor supply. Indeed, their assumptions allow to have the capital per head bounded away from zero and, since the labor supply is exogenous in their paper, the paths of capital stock are bounded away from zero. In our paper, since labor supply is endogenous, labor and capitals are no longer ensured to be bounded away from zero.

3 Our proof is quite general and holds even if some initial individual capital endowments are zero and the capital depreciation rate equals one. constraint also reflects a borrowing constraint at each time.

Assumptions governing household behavior and relationships with the production sector are detailed below.

The production sector

Consumption goods and new capital goods are produced at each time. The technology is represented by a constant returns to scale production function: F (K t , L t ), where K t and L t denote the input demands for capital and labor at time t. Profit maximization occurs at each time. All intertemporal decisions reside with households as there are no adjustment costs in the production sector. Under competition zero profits are earned at each time in this sector. Standard assumptions are imposed on F as well as a boundary condition when the labor supply is maximal. This additional condition simplifies the existence argument.

Let R + = [0, 1) and R 2 + = R + ⇥ R + . Let m denote the maximum possible labor supply within any period. This can occur if each household provides one unit of labor at each time and there are m households. Note that all labor services provided by the households are alike.

Notation: (@F/@K ) (0, m) ⌘ lim ✏!0+ F (✏,m) ✏ , (@F/@K ) (1, m) ⌘ lim ✓!+1 (@F/@K ) (✓, m) , (@F/@L) (1, 1) ⌘ lim ✓!+1 (@F/@L) (1, ✓) It can be shown that lim ✏!0+ F (✏,m) ✏ = lim ✏!0+ (@F/@K ) (✏, m), lim ✓!+1 F (✓,m) ✓ = lim ✓!+1 (@F/@K ) (✓, m) and lim ✓!+1 F (1,✓) ✓ = lim ✓!+1 (@F/@L) (1, ✓).
Assumption 1 The production function

F : R 2 + ! R + is C 1 in the interior of R 2
+ , homogeneous of degree one, strictly increasing, concave and strictly concave separately in K > 0 and L > 0. Inputs are essential: F (0, L) = F (K, 0) = 0. Limit conditions for production hold: F (K, L) ! 1 either when L > 0 and K ! 1 or when K > 0 and L ! 1. Moreover, (@F/@K ) (0, m) > , where 2 (0, 1] denotes the rate of capital depreciation.

Remark 1 Since F is homogeneous of degree one and F (1, 0) = 0, we obtain (@F/@K ) (1, m) = 0 and (@F/@L) (1, 1) = 0. Indeed,

F (K, mL) (@F/@K ) (K, mL) K = (@F/@K ) (K/L, m) K.
This implies F (1, mL/K) (@F/@K ) (K/L, m). Letting K/L go to 1 yields 0 = F (1, 0) (@F/@K ) (1, m) 0. The proof that (@F/@L) (1, 1) = 0 is left to the reader.

Households

We consider an economy without population growth where m households work and consume. Each household i is endowed with k i0 units of capital at period 0 and one unit of time per period which may be divided between labor supply The aggregate, or total, initial capital endowment is positive.

Assumption 2 k i0 0 for i = 1, . . . , m,

P i k i0 > 0.
Each household maximizes a lifetime utility function which is separable over time:

P 1 t=0 t i u i (c it , it )
, where i 2 (0, 1) is agent i's discount factor.

Assumption 3 u i : R 2 + ! R is C 0 , strictly increasing and concave. Without loss of generality, we assume u i (0, 0) = 0 for any i.

For the proof of non-existence of bubbles, the following additional assumption is required.

Notation: @ui @c (0, ) ⌘ lim ✏!0+ ui(✏, ) ✏ , @ui @ (c, 0) ⌘ lim ✏!0+ ui(c,✏) ✏ It can be shown that lim ✏!0+ ui(✏, ) ✏ = lim ✏!0+ (@u i /@c) (✏, ) and lim ✏!0+ ui(c,✏) ✏ = lim ✏!0+ (@u i /@ ) (c,✏ ) Assumption 4 (Inada conditions) The utility function is C 1 in the interior of R 2 + , (@u i /@c) (0, ) = 1 if > 0 and (@u i /@ ) (c, 0) = 1 if c > 0.
Household heterogeneity can arise in terms of endowments (k i0 ), discounting ( i ) and per-period utility (u i ).

In any period, the household faces a budget constraint:

p t [c it + k it+1 (1 ) k it ]  r t k it + w t (1 it )
It is known that, in economies with heterogenous discounting factors and no borrowing constraints, the more impatient agents borrow, consume more and work less in the short run than the most patient agents. Over the longer run those impatient agents consume less and work more in order to repay their debts to patient agents (see Le [START_REF] Van | Equilibrium dynamics in an aggregative model of capital accumulation with heterogeneous agents and elastic labor[END_REF]). In our model, as in [START_REF] Becker | On the long-run steady state in a simple dynamic model of equilibrium with heterogeneous households[END_REF], agents are prevented from borrowing: k it 0 for i = 1, . . . , m and t = 1, 2, . . . This constraint implies the model has an incomplete market structure as no markets exist where agents can borrow against their future wage income in order to consume more today.

Definition of equilibrium

Let an infinite-horizon sequences of prices and quantities be denoted by:

(p, r, w, (c i , k i , i ) m i=1 , K, L) where (p, r, w) ⌘ ((p t ) 1 t=0 , (r t ) 1 t=0 , (w t ) 1 t=0 ) 2 R 1 ⇥ R 1 + ⇥ R 1 + ; (c i , k i , i ) ⌘ ((c it ) 1 t=0 , (k it ) 1 t=1 , ( it ) 1 t=0 ) 2 R 1 + ⇥ R 1 ⇥ R 1 + ; (K, L) ⌘ ((K t ) 1 t=0 , (L t ) 1 t=0 ) 2 R 1 + ⇥ R 1 + , with i = 1, . . . , m. Definition 1 A Walrasian equilibrium p, r, w, ci , ki , ¯ i m i=1
, K, L satisfies the following conditions.

(1) Price positivity: pt , rt , wt > 0 for t = 0, 1, . . .

(2) Market clearing:

goods : m X i=1 ⇥ cit + kit+1 (1 ) kit ⇤ = F Kt , Lt ; capital : Kt = m X i=1 kit ; labor : Lt = m X i=1 lit ,
for t = 0, 1, . . ., where l it = 1 it denotes the individual labor supply. (3) Optimal production plans: pt F Kt , Lt rt Kt wt Lt is the value of the program: max [p t F (K t , L t ) rt K t wt L t ], under the constraints K t 0 and L t 0 for t = 0, 1, . . ..

(4) Optimal consumption plans:

P 1 t=0 t i u i cit , ¯ it is the value of the pro- gram: max P 1 t=0 t i u i (c it , it ), under the following constraints: budget constraint : pt [c it + k it+1 (1 ) k it ]  rt k it + wt (1 it ) borrowing constraint : k it+1 0 leisure endowment : 0  it  1 capital endowment : k i0 0 given for t = 0, 1, . . .
The following observation is a critical feature of our economic model.

Remark 2 Under Assumption 1, individual and aggregate capital supplies, individual and aggregate consumption demands are uniformly bounded by max {K 0 , K ⇤ }, where K ⇤ is the unique solution to the equation ( 1) x + F (x, m) = x and K 0 ⌘ P m i=1 k i0 is the initial aggregate capital. This solution namely exists under the assumption (@F/@K ) (0, m) > . Labor supply is uniformly bounded by m. We denote by A the common bound of the feasible consumption and capital stocks sequences.

Let us prove now the existence of a Walrasian equilibrium. The proof will be articulated in two parts. First, we consider an equilibrium with a finite horizon. Then, we let the horizon go to infinity to obtain a Walrasian equilibrium as limit of a sequence of finite-horizon economies.

Our main result is the existence of an equilibrium for this heterogeneous agent capital accumulation economy. The proof is given in two major steps. First, we prove the existence of equilibrium in finite-horizon economies. This demonstration has two parts: an existence theorem with an artificial bound on agents choice sets, and an extension of this theorem when those bounds are relaxed. The last step proves an equilibrium exists in the infinite-horizon economy. That economy is viewed as a limit of a sequence of truncated economies; the infinite horizon equilibrium prices and quantities are naturally shown to be limits of their finite horizon counterparts.

Consider a finite-horizon bounded economy which goes on for T + 1 periods: t = 0, . . . , T . Choose su ciently large quantity bounds B c , B k , and so on, with:

X i ⌘ {(c i0 , . . . , c iT ) : 0  c it  B c } = [0, B c ] T +1 ; Y i ⌘ {(k i1 , . . . , k iT ) : 0  k it  B k } = [0, B k ] T ; Z i ⌘ {( i0 , . . . , iT ) : 0  it  1} = [0, 1] T +1 ; Y ⌘ {(K 0 , . . . , K T ) : 0  K t  B K } = [0, B K ] T +1 ; Z ⌘ {(L 0 , . . . , L T ) : 0  L t  B L } = [0, B L ] T +1 , where mB k < B K , m < B L , m (1 ) B k +F (B K , B L ) < B c , min {B K , B c , B L } > A (A

is the common bound of the feasible allocations, see Remark 2).

Recall that k i0 is given and that the borrowing constraint inequalities k it 0 model the imperfection in the credit market. 4Let E T denote this bounded economy with technology and preferences as in Assumptions 1 to 3. Let X i , Y i and Z i be the ith consumer-worker's bounded sets for consumption demand, capital supply and leisure demand respectively (i = 1, . . . , m). The sets Y and Z are the bounded sets constraining the production sector's capital and labor demands at each time.

Theorem 3 Under the Assumptions 1, 2 and 3, there exists an equilibrium p, r, w, ch , kh , ¯ h m h=1 , K, L for the finite-horizon bounded economy E T .

Proof. See Appendix 1.

The unbounded economy, with relaxed artificial bounds, is shown to possess an equilibrium price system and allocation.

Corollary 1 Any equilibrium of E T is an equilibrium for the finite-horizon unbounded economy.

Proof. See Appendix 1.

Our main result is the following existence theorem for the infinite-horizon economy.

Theorem 4 Under the Assumptions 1, 2 and 3, there exists an equilibrium in the infinite-horizon economy with endogenous labor supply and borrowing constraints.

Proof. We consider a sequence of time-truncated economies and the associated equilibria. We prove that there exists a sequence of equilibria which converges, when the horizon T goes to infinity, to an equilibrium of the infinite-horizon economy. Appendix 2 contains the formal proof.

Non-existence of bubbles

There is considerable interest in whether or not a perfect foresight equilibrium capital asset price sequence is consistent with the notion of a rational pricing bubble. Some researchers points out that bubbles may occur with heterogeneous infinite-lived households facing borrowing constraints in an exchange economy [START_REF] Kocherlakota | Bubbles and constraints on debt accumulation[END_REF], [START_REF] Huang | Asset price bubbles in Arrow-Debreu and sequential equilibrium[END_REF]). We show this does not carry over to a model with productive capital accumulation.

Let p, r, w, ci , ki , ¯ i m i=1 , K, L denote an equilibrium in the infinite horizon economy. We will take pt = 1 for any t. We know that, for any t, rt > 0 and wt > 0.

Claim 5 Under Assumptions 1, 2, 3 and 4 (Inada), we have:

(1) cit > 0 if and only if ¯ it > 0, (2) for any i and any t, cit > 0 and ¯ it > 0, (3) for any i and any t,

@u i @c cit , ¯ it (1 + rt+1 ) i @u i @c cit+1 , ¯ it+1
and, if kit+1 > 0, then

@u i @c cit , ¯ it = (1 + rt+1 ) i @u i @c cit+1 , ¯ it+1 , (4)
for any i and any t,

wt @u i @c cit , ¯ it  @u i @ cit , ¯ it (1)
and, if Now define:

¯ it < 1, then wt @u i @c cit , ¯ it = @u i @ cit , ¯ it (
qt+1 ⌘ max i i (@u i /@c) cit+1 , ¯ it+1 (@u i /@c) cit , ¯ it = 1 1 + rt+1
The ratio qt has a natural interpretation as a market discount factor. The maximum condition on the right-hand side (above) shows that this ratio reflects the marginal rate of substitution between t and t + 1 for the highest marginal valuation consumer. Let

Q0 ⌘ 1 (2) Qt ⌘ t Y s=1 qs for t > 0 (3)
Clearly, Qt = Q t s=1 (1 + rs ) 1 for t > 0. Qt is the present value of a unit of capital of period t with focal date t = 0. This present value is implicitly defined via the current value prices system that arises as an equilibrium configuration in our main theorem. For any t, we obtain:

Qt = Qt+1 (1 + rt+1 ) (4)
and, by induction, 1 = Q0 = QT (1 ) T + P T t=1 Qt rt (1 ) t 1 . We define below the fundamental value of capital considered as a long-lived asset with focal date t = 0. At date 1, one unit of this asset will give back 1 unit of capital and r1 unit of consumption good as its dividend. At period 2, 1 unit of capital will give back (1 ) 2 unit of capital and (1 ) r2 as its dividend. This leads to the following definition of the Fundamental Value of capital:

FV ⌘ 1 X t=1 Qt (1 ) t 1 rt
Given that the price of capital at t = 0 is expected, with our normalizations, to be 1 if capital is priced in an e cient market (i.e. the present value of a unit of capital is its present value of future rental rates), we say there is a bubble if FV = 1 lim T !1 QT ( 1) T < 1. More formally:

Definition 2 The economy is said to experience a bubble if lim T !1 QT (1 ) T > 0
Otherwise (lim T !1 QT (1 ) T = 0), there is no bubble.

The crucial question concerns the existence of bubbles in a productive economy. We show that a productive economy experiences no bubbles. The proof rests on the following lemma.

Lemma 1 If the economy experiences a bubble, then rt converges to zero.

Proof. It is equivalent to prove that, if rt does not converge to zero, there are no bubbles. If rt does not converge to zero, there are, equivalently, ⇢ > 0 and a strictly increasing sequence (t i ) 1 i=1 such that rti ⇢ for i = 1, 2, . . . For T > t n , we get QT ( 1)

T = T Y s=1 1 1 + rs  n Y i=1 1 1 + rti  ✓ 1 1 +⇢ ◆ n and 0  lim sup T !1 QT (1 ) T  lim n!1 ✓ 1 1 +⇢ ◆ n = 0
Now, we are able to prove the No-Bubble Theorem.

Theorem 6 Under the Assumptions 1, 2, 3 and 4, our productive economy experiences no bubble.

Proof. See Appendix 3.

Remark 7

We have for any t and any i Qt cit + Qt wt ¯ it + Qt kit+1 = Qt wt + Qt (1 + rt ) kit

If we consider the capital as a long-lived asset, we can consider the sequence of perfectly foreseen equilibrium wages at each time, wt , as if it is agent i's endogenously determined labor income present value or endowment. Summing from t = 0 to t = T , we get:

T X t=0 Qt cit + wt ¯ it + QT kiT +1 = T X t=0 Qt wt + Q0 (1 + r0 ) ki0
Following [START_REF] Huang | Asset price bubbles in Arrow-Debreu and sequential equilibrium[END_REF] among others, we may say that, when the interest rate is high, in the sense that P T t=0 Qt wt < 1, then there is no bubble. Indeed, if this property holds and if ( wt ) 2 int l 1 + , then P T t=0 Qt < 1 which implies lim t Qt = 0 and we have no bubble. However, here, we do not know, without additional assumptions, that ( wt ) 2 int l 1 + and P T t=0 Qt wt < 1.5 Nevertheless, no bubble holds in our economy.

Conclusion

We have analyzed the existence of the intertemporal equilibrium and the occurrence of rational bubbles in a Ramsey model with heterogeneous agents, borrowing constraints and endogenous labor.

The assumed market incompleteness from the borrowing constraint nullifies the equivalence between the planner's and the market solution characteristic of the complete markets modeling framework. Assuming borrowing constraints are the financial market imperfection in our model economy, we adopt a threepronged proof strategy: existence in a (1) bounded and, then, (2) an unbounded truncated economy, and, last, (3) existence in an infinite-horizon economy as limit of a sequence of truncated unbounded economies. Moreover, as a byproduct of our proof we demonstrate that bubbles cannot exist despite the presence of borrowing constraints because equilibrium interest-rental rates do not converge to zero. One can observe for the existence of an equilibrium in a truncated economy, we still use the demand approach and the Kakutani's fixed-point theorem.

Therefore, this paper adds to the existing Ramsey equilibrium literature on two fronts. We provide a simple general proof of existence of an equilibrium (because of the endogenous labor supply and the weaker assumptions on the fundamentals); on the other hand, our arguments also furnish a proof that bubbles are nonexistent in a productive economy.

Appendix 1: existence of equilibrium in a finitehorizon economy

The proof of Theorem 3 requires some ingredients which are given below. Define a bounded price set P ⌘ 4 T +1 with the simplex 4 ⌘ {(p, r, w) : p, r, w 0, p + r + w = 1}

Focus now on the budget constraints:

p t [c it + k it+1 (1 ) k it ]  r t k it + w t (1
it ) for t = 0, . . . , T with k iT +1 = 0. Consider the budget set:

C T i (p, r , w) ⌘ 8 <
: Lemma 2 Under Assumptions 1, 2 and 3, if w 0 > 0 and r t + w t > 0, for t = 1, . . . , T , then the set B T i (p, r, w) is nonempty.

(c i , k i , i ) 2 X i ⇥ Y i ⇥ Z i : p t [c it + k it+1 (1 ) k it ]  r t k it + w t (1 it ) t = 0, . . . , T 9
(c i , k i , i ) 2 X i ⇥ Y i ⇥ Z i : p t [c it + k it+1 (1 ) k it ] < r t k it + w t (1 it ) t = 0, . . . , T 9
Proof. Take k i1 > 0 and i0 < 1 such that

p 0 k i1 < w 0 (1 i0 )  r 0 k i0 + w 0 (1 i0 ). Take k i2 > 0, i1 < 1 such that p 1 k i2  r 1 k i1 + w 1 (1 i1
) and so on.

Observe that when = 1, the set B T i (p, r, w) is empty if r t = w t = 0 and p t = 1 for some t and, when k i0 = 0, this set is empty if p 0 = w 0 = 0 and r 0 = 1. For that reason, at the beginning, we introduce the following sets.

Let " > 0 satisfy m (1 ) (B k + ")+F (B K , B L )+m" < B c and mB k +m"< B K . We define:

C T " i (p, r, w) ⌘ 8 < : (c i , k i , i ) 2 X i ⇥ Y i ⇥ Z i : p t [c it + k it+1 (1 ) k it ]  p t " + p t (1 ) " + r t (k it + ") + w t (1 it ) t = 0, . . . , T 9 = ; B T " i (p, r, w) ⌘ 8 < : (c i , k i , i ) 2 X i ⇥ Y i ⇥ Z i : p t [c it + k it+1 (1 ) k it ] < p t " + p t (1 ) " + r t (k it + ") + w t (1 it ) t = 0, . . . , T 9 =
; Remark 8 " represents a perturbation of the fundamental economy. In the "economy, the firm uses " as an additional input. " and k it are the same capital good and experiences the same depreciation during the production process. When the process ends, they are resold at the same price p t to earn p t (1 ) (" + k it ).

The next lemma plays a critical role. The perturbation of the fundamental economy yields that each agent has a positive income at each time. As in standard competitive equilibrium proofs for finite exchange and/or production economies, this is required to show all agents are, in fact, finding their utility maximizing bundles subject to a budget constraint (the cheaper point property in standard equilibrium theories).

Lemma 3 Under Assumptions 1, 2 and 3, the set B T " i (p, r, w) is nonempty and C T " i (p, r, w) = BT" i (p, r, w). Moreover the correspondence B T " i is lower semicontinuous (lsc). Hence, the correspondence C T " i is continuous.

Proof. Take it = ⌘ < 1, k it+1 = 0, c it = 0. Then, p t " + p t (1 ) " + r t (" + k it ) + w t (1 it ) > 0 for any (p t ,
[p" + p (1 ) " + r" + w (1 ⌘)] > 0
We have

p t [c it + k it+1 (1 ) k it ] = p t (1 ) k it  0 < ↵  p t " + p t (1 ) " + r t (" + k it ) + w t (1 it ) 12
Documents de travail du Centre d'Economie de la Sorbonne -2014.40 for any t. So, B T " i (p, r, w) is nonempty. The proof of the remaining assertions is easy.

The following lemma is crucial for the proof of Theorem 3.

Lemma 4 Under the Assumptions 1, 2 and 3, there exists p (") , r (") , w (") , ci (") , ki (") , ¯ i (")

m i=1 , K (") , L (")
in the finite-horizon bounded economy E T which satisfies:

(1) price positivity: pt (") , rt (") , wt (") > 0 for t = 0, . . . , T , (2) market clearing:

goods : m X i=1 ⇥ cit (") + kit+1 (") (1 ) kit (") ⇤ = F Kt (") , Lt (") + m" + m (1 ) " capital : Kt (") = m X i=1 kit (") + m" labor : Lt (") = m X i=1 lit (")
for t = 0, . . . , T , where lit (") = 1 ¯ it (") denotes the individual labor supply.

(3) Optimal production plans: pt (")F Kt ("), Lt (") rt (") Kt (") wt Lt (") is the value of the program: max [p t (")F (K t , L t ) rt (")K t wt (")L t ], under the constraints K 2 Y and L 2 Z for t = 0, . . . , T . Moreover, pt (")F Kt ("), Lt (") rt (") Kt (") wt Lt (") = 0 (4) Optimal consumption plans:

P 1 t=0 t i u i cit ("), ¯ it (") is the value of the program: max P 1 t=0 t i u i (c it , it )
, under the following constraints:

pt (") (c it + k it+1 )  pt (") " + [p t (") (1 ) + rt (")] (k it + ") + wt (") (1 it ) ci 2 X i , ki 2 Y i , ¯ it 2 [0, 1], k i0 0 given for t = 0, . . . , T .
Proof. We introduce the reaction correspondences ' i i = 0, . . . , m + 1 where i = 0 denotes an "additional" agent, i = 1, . . . , m the consumers, and i = m + 1 the firm. These correspondences are defined as follows.

Agent i = 0 (the "additional" agent):

' 0 : ⇥ m h=1 (X h ⇥ Y h ⇥ Z h ) ⇥ Y ⇥ Z ! P with ' 0 ((c h , k h , h ) m h=1 , K, L) ⌘ arg max {H (p, r, w, (c h , k h , h ) m h=1 , K, L) : (p,
H (p, r, w, (c h , k h , h ) m h=1 , K, L) ⌘ T X t=0 p t X i [c it + k it+1 (1 ) k it ] m" m (1 ) " F (K t , L t ) ! + T X t=0 r t K t m" m X i=1 k it ! + T X t=0 w t L t m + m X i=1 it !
The correspondence ' 0 is upper semicontinuous (from the Maximum Theorem) and nonempty, convex, compact-valued.

Agents i = 1, . . . , m (consumers-workers):

' i : P ! X i ⇥ Y i ⇥ Z i with ' i (p, r, w, K, L) ⌘ arg max ( T X t=0 t i u i (c it , it ) : (c i , k i , i ) 2 C T " i (p, r, w
)

)
The correspondences (' i ) m i=1 are upper semicontinuous (from the Maximum Theorem) and nonempty, convex, compact-valued.

Agent i = m + 1 (the firm):

' m+1 : P ! Y ⇥ Z with ' m+1 (p, r, w) ⌘ arg max ( T X t=0 [p t F (K t , L t ) r t K t w t L t ] : (K, L) 2 Y ⇥ Z )
Again, ' m+1 is upper semicontinuous (from the Maximum Theorem). Clearly, it is nonempty, convex and compact-valued.

Apply the Kakutani Theorem. There exists a sequence p, r, w, ch , kh ,

¯ h m h=1 , K, L satisfying (p, r, w) 2 ' 0 ch , kh , ¯ h m h=1 , K, L ci , ki , ¯ i m i=1 2 ' i (p, r, w)
for any i and K, L 2 ' m+1 (p, r, w) Explicitly we have the following.

(1) For every (p, r, w) 2 P ,

T X t=0 (p t pt ) m X i=1 ⇥ cit + kit+1 (1 ) kit ⇤ m" m (1 ) " F Kt , Lt ! + T X t=0 (r t rt ) Kt m" m X i=1 kit ! + T X t=0 (w t wt ) Lt m + m X i=1 ¯ it !  0 (5)
(2

) For i = 1, . . . , m, (c i , k i , i ) 2 C T " i (p, r, w) implies T X t=0 t i u i (c it , it )  T X t=0 t i u i cit , ¯ it (6)
(3) Finally, for t = 0, . . . , T and for every (K, L) 2 Y ⇥ Z, we have

T X t=0 [p t F (K t , L t ) rt K t wt L t ]  T X t=0 ⇥ pt F Kt , Lt rt Kt wt Lt ⇤ .

This is possible if and only if

pt

F (K t , L t ) rt K t wt L t  pt F Kt , Lt rt Kt wt Lt (7)
for any t. In particular, the equilibrium profit is nonnegative.

pt F Kt , Lt rt Kt wt Lt 0 [START_REF] Huang | Asset price bubbles in Arrow-Debreu and sequential equilibrium[END_REF] Let us show that pt > 0. First, we have from the budget constraints:

pt X i ⇥ cit + kit+1 (1 ) kit ⇤  mp t " + mp t (1 ) " + mr t " + rt X i kit + wt X i (1 ¯ it )
Combining with (8), we get 0 pt

X i ⇥ cit + kit+1 (1 ) kit ⇤ F Kt , Lt m (1 ) " m" ! +r t Kt m" X i kit ! + wt Lt X i 1 ¯ it ! (9)
Combining ( 5) with ( 9), we find

0 T X t=0 p t X i ⇥ cit + kit+1 (1 ) kit ⇤ F Kt , Lt m (1 ) " m" ! + T X t=0 r t Kt m" X i kit ! + T X t=0 w t Lt X i 1 ¯ it ! (
Lt X i 1 ¯ it  0 (12) X i ⇥ cit + kit+1 (1 ) kit ⇤ F Kt , Lt m (1
) " m"  0 [START_REF] Sarte | Progressive taxation and income inequality in dynamic competitive equilibrium[END_REF] Observe that [START_REF] Sarte | Progressive taxation and income inequality in dynamic competitive equilibrium[END_REF] implies

X i cit  (1 ) mB k + F (B K , B L ) + m (1 ) " + m" < B c (14)
Suppose pt = 0. From the consumers' problem, we obtain cit = B c and ¯ it = 1 for any i. That is a contradiction with [START_REF] Sorger | On the structure of Ramsey equilibrium: cycles, indeterminacy, and sunspots[END_REF]. Hence, pt > 0.

We want to prove now that, for any t, pt F Kt , Lt rt Kt wt Lt = 0 [START_REF] Sorger | Chaotic Ramsey equilibrium[END_REF] and rt > 0, wt > 0. From ( 11) and ( 12 which is a contradiction to [START_REF] Bosi | On the Ramsey equilibrium with heterogeneous consumers and endogenous labor supply[END_REF]. Assume rt = 0. Then, we have 0 pt F (K, L) wt L for any (K, L) 2 Y ⇥ Z. Take 0 < K < B K and 0 < L < B L . We obtain 0 L [p t F (K/L, 1) wt ]. Since pt > 0 and lim L!0 F (K/L, 1) = 1, we have [p t F (K/L, 1) wt ] > 0 when L is su ciently close to 0, leading to a contradiction.

The proof that wt > 0 is similar. Let us show now that Kt m"

X i kit = 0 Lt X i 1 ¯ it = 0 X i ⇥ cit + kit+1 (1 ) kit ⇤ F Kt , Lt m (1
) " m" = 0

Since pt > 0 the budget constraints bind. Combining with [START_REF] Van | Existence of a competitive equilibrium in a one sector growth model with heterogeneous agents and irreversible investment[END_REF], [START_REF] Ramsey | A mathematical theory of saving[END_REF]

i kit + wt X i 1 ¯ it = pt X i ⇥ cit + kit+1 (1 ) kit ⇤  pt ⇥ F Kt , Lt + m (1 ) " + m" ⇤ = rt Kt + wt Lt + pt m (1 ) " + pt m"  mp t " + mp t (1 ) " + mr t " + rt X i kit + wt X i 1 ¯ it Hence pt X i ⇥ cit + kit+1 (1 ) kit ⇤ = pt ⇥ F Kt , Lt + m (1 ) " + m" ⇤ and rt Kt m" X i kit ! + wt " Lt X i 1 ¯ it # = 0 (16)
Since pt > 0, we have

X i ⇥ cit + kit+1 (1 ) kit ⇤ F Kt , Lt m (1
) " m" = 0 Since rt > 0, wt > 0 and ( 16) holds, inequalities [START_REF] Van | Existence of a competitive equilibrium in a one sector growth model with heterogeneous agents and irreversible investment[END_REF] and ( 12) become Kt m" X i kit = 0 and Lt

X i 1 ¯ it = 0
The proof of Lemma 4 is now complete.

Proof of Theorem 3

Keeping in mind these results, we now prove Theorem 3. We let " converge to 0. We denote the allocations and the prices obtained in Lemma 4 by p (") , r (") , w (") , ci (") , ki (") , ¯ i (") m i=1 , K (") , L (")

We recall that, for any t, pt (") + rt (") + wt (") = 1. Denote

p, r, w, ci , ki , ¯ i m i=1 , K, L ⌘ lim "!0 p (") , r (") , w (") , ci (") , ki (") , ¯ i (") m i=1 , K (") , L (")
For any t and any (K

t , L t ) 2 Y ⇥ Z, we have 0 = pt F Kt , Lt rt Kt wt Lt pt F (K t , L t ) rt K t wt L t K0 = P i k i0 > 0 and L0  m < B L .
Let us show that w0 > 0 and rt + wt > 0, for t = 1, . . . , T . 17

= p0 F K0 , L0 r0 K0 p0 F (K 0 , L 0 ) r0 K 0 (17)
for any (K 0 , L 0 ) 2 Y ⇥ Z. Take K 0 = ✏ > 0 and L 0 = L 2 (0, B L ). We obtain a contradiction:

0 p0 F (✏, L) r0 ✏ = [p 0 F (1, L/✏) r0 ] ✏ > 0
when ✏ goes to zero since F (1, 1) = 1. Hence w0 = 0 implies p0 = 0 and r0 = 1 (because of the unit simplex). However, from (17), K0 = 0 which is impossible.

Assume that wt = 0 for some t 1. The same argument previously used implies pt = 0 and rt = 1.

Assume rt = 0 for some t 1 and pt > 0. Then,

0 pt F (K t , L t ) wt L t for any (K t , L t ) 2 Y ⇥ Z. Take K t = K 2 (0, B K ) and L t = ✏ > 0. We obtain a contradiction 0 pt F (K,✏ ) wt ✏ = [p t F (K/✏, 1) wt ] ✏ > 0
when ✏ becomes su ciently close to zero, since F (1, 1) = 1. Hence rt = 0 implies pt = 0 and wt = 1. From Lemma 2, the set

B T i (p, r, w) is nonempty. Taking (c i , k i , i ) 2 B T i (p, r, w), we have pt [c it + k it+1 (1 ) k it ] < rt k it + wt (1 it )
for any t. There exists " > 0 such that for any " < ", we have for any t,

pt (") [c it + k it+1 (1 ) k it ] < rt (") k it + wt (") (1 it ) < pt (") " + pt (") (1 ) " + rt (") " +r t (") k it + wt (") (1 it ) Therefore, P T t=0 t i u i (c it , it )  P T t=0 t i u i cit (") , ¯ it (") . Let " go to 0. Then P T t=0 t i u i (c it , it )  P T t=0 t i u i cit , ¯ it . Let (c i , k i , i ) 2 C T i (p, r, w). There exists a sequence (c n i , k n i , n i ) 1 n=1 ⇢ B T i (p, r, w) which converges to (c i , k i , i ). For any n, we have T X t=0 t i u i (c n it , n it )  T X t=0 t i u i cit , ¯ it Let n go to 1. Then P T t=0 t i u i (c it , it )  P T t=0 t i u i cit , ¯ it .
The prices (p t , wt ) are strictly positive since the utility functions u i are strictly increasing. The price rt is strictly positive since we have proved above that rt = 0 implies pt = 0. It is easy to check that the list p, r, w, ci , ki , ¯ i m i=1 , K, L is an equilibrium for the T + 1-horizon economy.

Proof of Corollary 1

Let p, r, w, ch , kh , ¯ h m h=1 , K, L with pt , rt , wt > 0, t = 0, . . . , T , be an equilibrium of E T .

Let (c i , k i , i ) verify

P T t=0 t i u i (c it , it ) > P T t=0 t i u i cit , ¯ it .
We want to prove that this allocation violates at least one budget constraint, that is that there exists t such that pt

[c it + k it+1 (1 ) k it ] > rt k it + wt (1 it ) (18)
Focus on a strictly convex combination of (c i , k i , i ) and ci , ki , ¯ i : 1) kit [START_REF] Tirole | On the possibility of speculation under rational expectations[END_REF] it ( ) ⌘ it + (1 ) ¯ it with 0 < < 1. Notice that we assume that the bounds satisfy B c , B k , B K > A and B L > m in order ensure that we enter the bounded economy when the parameter is su ciently close to 0. Entering the bounded economy means (c i ( ) , k i ( ) , i ( )) 2 X i ⇥ Y i ⇥ Z i . In this case, because of the concavity of the utility function, we find [START_REF] Sorger | Strategic saving decisions in the infinite-horizon model[END_REF]. Thus p, r, w, ch , kh , ¯ h m h=1 , K, L is also an equilibrium for the unbounded economy.

c it ( ) ⌘ c it + (1 ) cit k it ( ) ⌘ k it + (
If Lt = 0, then we have 0 = F ( Kt , Lt ) = rt Kt + wt Lt = rt Kt Hence Kt = 0, since rt > 0. As above, this contradicts the point (2) of the claim.

Proof of Theorem 6

First, observe that the production function F satisfies Assumption 1 and

lim b!0 + (@F/@L) (1, b) > 0 ( 21
)
Since F is homogeneous of degree one, we have, for K > 0 and L > 0, (@F/@K ) (K, L) = (@F/@K ) (K/L, 1) and (@F/@L) (K, L) = (@F/@L) (1, L/K). Let Kt , Lt be an equilibrium sequence of aggregate capital stocks and labors. Observe that rt = (@F/@K ) Kt / Lt , 1 .

Suppose the economy has a bubble in prices. Then, from Lemma 1, rt converges to zero. But, in this case, Kt / Lt tends to infinity, or equivalently, Lt / Kt goes to 0. Since Kt is positive and bounded above, we obtain Lt ! 0. Recall that Ct + Kt+1 = F Kt , Lt + ( 1) Kt = Kt ⇥ F 1, Lt / Kt + 1 ⇤ and choose " > 0 such that F (1, ") + 1 < 1. There exists T such that for any t > T , Kt+1  Kt ⇥ F 1, Lt / Kt + 1 ⇤ < [F (1, ") + 1 ] Kt . This implies Kt ! 0 when t tends to infinity, and Ct ! 0 too. Thus, lim t!1 cit = 0 for any i.

Reconsider the first-order conditions of point (4) of Claim 5:

wt @u i @c cit , ¯ it  @u i @ cit , ¯ it (22)
It is easy to see that wt = (@F/@L) Kt , Lt . Since Lt / Kt converges to 0, according to (21), we have lim t!1 wt = lim t!1 (@F/@L) 1, Lt / Kt > 0. We will show that lim t!1 ¯ it = 1 for any i. Indeed, at equilibrium, we have for any i cit + kit+1  (1 + rt ) kit + wt 1 ¯ it Since lim t!1 cit , kit , rt = 0, lim t!1 wt > 0, we have lim t!1 ¯ it = 1 for any i.

Thus, lim t!1  wt @u j @c cjt , ¯ jt = 1 Since @u j @ cjt , ¯ jt ¯ jt  u j cjt , ¯ jt u j (c jt , 0)  u j cjt , ¯ jt  u j (A, 1)

where A is defined in Remark 2, (22) implies a contradiction:

1 = lim t!1 wt @u j @c cjt , ¯ jt  lim sup t @u j @ cjt , ¯ jt  u j (A, 1) < 1

5 ) 8

 58 Finally, Kt , Lt > 0. Proof. See Appendix 3. Documents de travail du Centre d'Economie de la Sorbonne -2014.40

= ;

 = We denote by BT i (p, r, w) the closure of B T i (p, r, w). It is obvious that, whenB T i (p, r, w) 6 = ?, then C T i (p, r, w) = BT i (p,r, w). Nonemptiness of B T i is crucial for the existence of demands. The following result is very useful for our proof of existence of a Walrasian equilibrium. 11 Documents de travail du Centre d'Economie de la Sorbonne -2014.40

10 ) 15

 1015 Documents de travail du Centre d'Economie de la Sorbonne -2014.40 and, noticing that (10) holds for any (p, r, w) 2 P , Kt m" X i kit  0 (11)

  ), we have Kt  m" + mB k < B K and Lt  m < B L . Suppose pt F Kt , Lt rt Kt wt Lt = ⇡ > 0. Choose µ > 1 such that µ Kt < B K and µ Lt < B L . We have pt F µ Kt , µ Lt rt µ Kt wt µ Lt = µ⇡ >⇡ = pt F Kt , Lt rt Kt wt Lt

  Documents de travail du Centre d'Economie de la Sorbonne -2014.40 If w0 = 0 and p0 > 0 we have 0

18

  Documents de travail du Centre d'Economie de la Sorbonne -2014.40

4

  Documents de travail du Centre d'Economie de la Sorbonne -2014.40 and leisure time. The leisure demand of agent i at time t is denoted by it and that agent's labor supply is l it = 1 it . This individual's capital supply and consumption demands at time t are denoted by k it and c it , respectively. Agents allocate their income at time t to consumption, c it , and capital accumulation, denoted k it+1 .

  r t , w t ) 2 4 and, hence,

	↵ ⌘ min (p,r,w)24

  (c it ( ) , it ( )) is an equilibrium for this economy, there exists t 2 {0, . . . , T } such that pt [c it ( ) + k it+1 ( ) (1) k it ( )] > rt k it ( ) + wt (1

	T X	t i u i T X	t i u i (c it , it ) + (1	)
	t=0						t=0
										it ( ))
	Replacing (19), we get
	pt c it + (1 > rt ⇥ k it + (1	) cit + k it+1 + (1 ) kit ⇤ + wt 1 ⇥	) kit+1 (1 it + (1 ) ¯ it	⇤	)	⇥	k it + (1	) kit	⇤
	that is
	>	pt [c it + k it+1 (1 [r t k it + wt (1 it )] + (1 ) k it ] + (1 ) ⇥	) pt rt kit + wt 1 ¯ it ⇥ cit + kit+1 (1 ⇤	) kit	⇤
	Since pt	⇥	cit + kit+1 (1	) kit	⇤	= rt kit + wt 1 ¯ it , we obtain

T X t=0 t i u i cit , ¯ it > T X t=0 t i u i cit , ¯ it Since (c i ( ) , k i ( ) , i ( )) 2 X i ⇥Y i ⇥Z i and p, r, w, ch , kh , ¯ h m h=1 , K, L

The introduction of a labor-leisure arbitrage implies in addition that impatient agents work less today to enjoy the leisure time but more tomorrow to repay their debt (Le[START_REF] Van | Equilibrium dynamics in an aggregative model of capital accumulation with heterogeneous agents and elastic labor[END_REF]).

To construct this continuous map, the authors require the intertemporal utility to be continuous for the product topology on the whole space of sequences and the productivity at the origin larger than the inverse of the time preference ( ). In our paper, we only require the utility to be continuous for the product topology on the feasible set and the productivity at the origin to be larger than the capital depreciation rate.However, even if we take the assumptions of[START_REF] Becker | The existence of Ramsey equilibrium[END_REF] on the utility function 2 Documents de travail du Centre d'Economie de la Sorbonne -2014.40

Documents de travail du Centre d'Economie de la Sorbonne -2014.40

The credit constraint might be generalized by requiring: h i  k it with h i < 0 given. This specification is left for another paper.

Here, we consider the interior intl 1 + when l 1 is endowed with the supnorm topology.

suggestions have helped us to simplify and improve the paper. This work also benefits from comments by Monique Florenzano and the participants to the international conference New Challenges for Macroeconomic Regulation held on June 2011 in Marseille. This work has been carried out thanks to the support of the A*MIDEX project (n ANR-11-IDEX-0001-02) funded by the "Investissements d'Avenir" French Government program, managed by the French National Research Agency (ANR) , Indiana University.

for the product topology. We claim that ŵ0 > 0, ŵt + rt > 0 for any t 1. Indeed, we always have

+ . Take K > 0 and let L go to infinity to get a contradiction. Hence ŵ0 = 0 implies p0 = 0 and r0 = 1. In this case we will have K0 = 0 which is impossible since K0 = P i k i0 > 0. We conclude that ŵ0 > 0. Assume ŵt = 0 and pt > 0 for some t 1. Then 0 pt F (K, L) rt K for any (K, L) 2 R 2 + . Take K > 0 and let L go to infinity to have a contradiction. Now assume rt = 0 and pt > 0 for some t 1. Then 0 pt F (K, L) ŵt L for any (K, L) 2 R 2 + . Take L > 0 and let K go to infinity: a contradiction arises. Then, rt + ŵt > 0 for any t. From Lemma 2, for any ⌧ 1, the set

for t = 0, . . . , ⌧. There exists N > ⌧ such that, for any T N ,

Documents de travail du Centre d'Economie de la Sorbonne -2014.40

This implies

Now let (c it , k it+1 , it )

We then have, from (20):

⌘

Let n go to 1:

⌘ solves the consumer's problem in the infinite-horizon economy. The prices (p t , ŵt ) are strictly positive thanks to the strict increasingness of the utility functions. The price rt > 0 since we have proved rt = 0 implies pt = 0.

It is now easy to check that the list

, K, L⌘ is an equilibrium for the infinite-horizon economy.

Appendix 3: non-existence of bubbles

Proof of Claim 5

(1) cit + wt ¯ it + kit+1 = (1 + rt ) kit + wt . Suppose cit > 0 and ¯ it = 0. By Assumption 4, we can decrease cit and increase ¯ it to have a higher utility for period t. Hence ¯ it > 0. The converse is proved by the same argument.

(2) We first prove that cit = ¯ it = 0 for any t is excluded. Suppose it is not true. Then P 1 t=0 t i u i cit , ¯ it = 0. Define k it = c it = it = 0 for any t 1 and c i0 + w0 i0 = (1 + r0 ) k i0 + w0 with c i0 > 0 and i0 2 (0, 1). Then

Without loss of generality, we can assume that t = 1 is the first period where the consumption and leisure are positive, i.e. ci1 > 0 and ¯ i1 > 0 (because of point ( 1)). Hence, ci0 = ¯ i0 = 0. Define

The sequence (c i , k i , i ) belongs to the budget set of agent i. And we have, by Assumption 4 (Inada),

for " suciently close to 0. This leads to a contradiction. Hence ci0 > 0 and ¯ i0 > 0. By 21 Documents de travail du Centre d'Economie de la Sorbonne -2014.40 induction, we obtain also cit > 0 and ¯ it > 0 for any i and any t.

(3) We have cit + kit+1 = kit (1

cit ", ¯ it ( ")

if " > 0 and small enough.

Let " go to zero. Then,

If kit+1 > 0, then we can take also " < 0 small enough in absolute value and let it go to zero to obtain the reverse inequality.

(4) Since ¯ it > 0, we can choose 0 < " < ¯ it . Define c it = cit + wt " and it = ¯ it ". The budget constraint is satisfied. In addition, we have for " 2 0, ¯ it 0 u i (c it , it ) u i cit , ¯ it @u i @c (c it , it ) wt " + @u i @ (c it , it ) ( ") 0 @u i @c (c it , it ) wt @u i @ (c it , it )

Let " go to zero. Then 0 wt @u i @c cit , ¯ it @u i @ cit , ¯ it Now, if ¯ it < 1, then we can take " > 0 such that ¯ it + " < 1 and let " go to zero to get the reverse inequality.

(5) We have Ct + Kt+1 = F Kt , Lt + (1 ) Kt . If Kt = 0, then Ct = 0 and cit = 0 for any i contradicting point [START_REF] Becker | Optimal growth with heterogeneous agents and the twisted turnpike: an example[END_REF].