A A Mekonnen

F Lerasle

A Herbulot
email: ariane.herbulot@laas.fr

C Briand
email: cyril.briand@laas.fr

People Detection with Heterogeneous Features and Explicit Optimization on Computation Time

In this paper we present a novel people detector that employs discrete optimization for feature selection. Specifically, we use binary integer programming to mine heterogeneous features taking both detection performance and computation time explicitly into consideration. The final trained detector exhibits low Miss Rates with significant boost in frame rate. For example, it achieves a 2.6% less Miss Rate at 10 -4 FPPW compared to Dalal and Triggs HOG detector with a 9.22x speed improvement.

I. Introduction

In modern era, computer vision is playing a significant role in automated object perception; one such thriving role is automated people detection. Visual people detection, i.e., people detection using visual cameras, is the most prominent mode employed in the literature as cameras are cheap, versatile, and provide rich color and texture information. It is indispensable primarily in surveillance systems, human-machine interaction, robotics, automotive industry, image/video indexing, etc. Evidently, it is also one of the challenging tasks in computer vision due to variations in peoples' appearance, background clutter, illumination, sensor motion, and so forth. In recent years astounding progress have been made by the scientific community [START_REF] Dollar | Pedestrian detection: An evaluation of the state of the art[END_REF], [START_REF] Geronimo | Survey of pedestrian detection for advanced driver assistance systems[END_REF], but there is still room for improvement.

One important discipline where applications of visual people detection is highly proliferating is robotics. In robotic systems that entail people perception, the aforementioned challenges are further exacerbated by real-time requirements, limited computational resources, and sensor motion. A mobile robot needs to be reactive during navigation/interaction in human occupied environments. Thus, its people detection module-which is one component of an entire functioning system-should be fast. The advent of powerful camera systems in the robotic community that provide high resolution omnidirectional images, e.g., the Ladybug series [START_REF] Grey | Spherical vision catalog[END_REF] from Point Grey, stresses this point further urging the need to give extra focus on computation time during detector design.

In this work, we try to give explicit consideration to computation time during detector design. Generally speaking, balancing computation time and detection performance is challenging; best detection results are obtained using complex features and descriptors which are computationally expensive. As an example, Histogram of Oriented Gradients (HOG) [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] is the most discriminant feature thus far, but it is also computationally expensive compared to simple features like Haar variants [START_REF] Viola | Robust real-time face detection[END_REF]. Furthermore, most detectors that improve over HOG either use complex human models, e.g., parts based models [START_REF] Felzenszwalb | A discriminatively trained, multiscale, deformable part model[END_REF], or consider various heterogeneous pool of features, e.g., [START_REF] Walk | New features and insights for pedestrian detection[END_REF], [START_REF] Wojek | A performance evaluation of single and multifeature people detection[END_REF], both of which contribute to added computation time unless explicit computation considerations are made. In line with this, we present a person detector that uses heterogeneous pool of features and makes explicit computation time vs detection trade-off optimization to build a performant detector that leads to a significant gain in computation time while maintaining competitive detection performance.

Related Works:

The entire literature in visual people detection is overwhelming and a discussion on the different techniques is beyond the scope of this paper (please refer to [START_REF] Dollar | Pedestrian detection: An evaluation of the state of the art[END_REF], [START_REF] Geronimo | Survey of pedestrian detection for advanced driver assistance systems[END_REF] for extensive surveys). We will focus on approaches that use heterogeneous pool of features with sliding-window detection paradigm. The best results in visual people detection are obtained using heterogeneous pool of features [START_REF] Dollar | Pedestrian detection: An evaluation of the state of the art[END_REF], [START_REF] Geronimo | Survey of pedestrian detection for advanced driver assistance systems[END_REF]. Heterogeneous features help capture complementary information useful to handle various detection challenges. For example: Wojek et al. [START_REF] Wojek | A performance evaluation of single and multifeature people detection[END_REF] used Haar, HOG, and shape context features. They presented a comparative result obtained using boosting techniques and SVMs as classifiers and demonstrated that the combination of different features successfully outperformed individual variants and even the state-of-the-art at the time. Walk et al. [START_REF] Walk | New features and insights for pedestrian detection[END_REF] also clearly showed they obtained the best detection results when concatenating HOG, Histogram Of Flow [START_REF] Dalal | Human detection using oriented histograms of flow and appearance[END_REF], and Color Self Similarity (CSS) features all together, rather than individual features or a subset of them. Similar conclusions were made by Schwartz et al. [START_REF] Schwartz | Human detection using partial least squares analysis[END_REF] and Hussain and Triggs [START_REF] Hussain | Feature sets and dimensionality reduction for visual object detection[END_REF] using-HOG, color frequency, and co-occurrence features-and-HOG, Local Binary (LBP) and Ternay (LTP) Pattern featuresrespectively.

Given heterogeneous pool of features, different ways can be used to build the final detector. Four main trends can be observed in the literature: (1) Direct concatenation [START_REF] Walk | New features and insights for pedestrian detection[END_REF], [START_REF] Wojek | A performance evaluation of single and multifeature people detection[END_REF] in which the different features are concatenated to make one high dimensional feature vector and an SVM used afterwards for classification. This is computationally costly owing to the complex feature and SVM weights applied in sliding window detection. [START_REF] Hussain | Feature sets and dimensionality reduction for visual object detection[END_REF], [START_REF] Schwartz | Human detection using partial least squares analysis[END_REF] used dimensionality reduction techniques after concatenation which improved detection performance but not detection speed. (2) Direct boosting [START_REF] Dollar | Integral channel features[END_REF], [START_REF] Wojek | A performance evaluation of single and multifeature people detection[END_REF], [START_REF] Gerónimo | Haar wavelets and edge orientation histograms for on-board pedestrian detection[END_REF] where an ensemble classifier is learned using the entire heterogeneous pool of features. The problem here is in boosting, on each iteration, the feature with the least weighted classification error is added to the ensemble irrespective of its computation time. This favors complex features resulting in computationally costly detector. (3) Coarse-to-fine hierarchical arrangement [START_REF] Mogelmose | Two-stage part-based pedestrian detection[END_REF], [START_REF] Pan | Efficient and accurate face detection using heterogeneous feature descriptors and feature selection[END_REF] where a cascade is constructed using cheap features at the initial stages and using complex features

{(x i , y i )} i∈N F F        T PR FPR τ        F F H(•)
Fig. 1: Feature selection and classifier learning framework used at each node of a cascade.

at later stages. This approach is quite advantageous and tries to find a balance between detection performance and speed. The concern is, how to decide which features to use at the different stages systematically? Both [START_REF] Mogelmose | Two-stage part-based pedestrian detection[END_REF], [START_REF] Pan | Efficient and accurate face detection using heterogeneous feature descriptors and feature selection[END_REF] adopt a heuristic based rule and use homogeneous family of features they deemed cheap at the initial stages, and homogeneous complex features at the latter. Finally, (4) via a computation time vs detection trade-off. This notion has been considered by the works of Wu and Nevatia [START_REF] Wu | Optimizing discrimination-efficiency tradeoff in integrating heterogeneous local features for object detection[END_REF], Jourdheuil et al. [START_REF] Jourdheuil | Heterogeneous adaboost with real-time constraints -application to the detection of pedestrians by stereovision[END_REF], and Mekonnen et al. [START_REF] Mekonnen | Person detection with a computation time weighted adaboost[END_REF]. In all cases, they defined a criterion composed of feature detection performance and computation time in a multiplicative manner. But, considering a multiplicative factor masks the contributions from the respective objectives and is not guaranteed to be optimal.

Our proposed framework falls in the 4 th category; but, it can also be considered as a variant of coarse-to-fine hierarchy in which the exact features to use at each cascade node are selected automatically via an optimization step. We use five frequently used heterogeneous features, namely: Haar-like features [START_REF] Viola | Robust real-time face detection[END_REF], Edge Orientation Histogram (EOH) [START_REF] Gerónimo | Haar wavelets and edge orientation histograms for on-board pedestrian detection[END_REF], CSS [START_REF] Walk | New features and insights for pedestrian detection[END_REF], Center Surround Local Binary Patterns (CS-LBP) [START_REF] Heikkilä | Description of interest regions with local binary patterns[END_REF], and HOG [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] in a classical cascaded boosting configuration [START_REF] Viola | Robust real-time face detection[END_REF] with an added explicit optimization step based on Binary Integer Programming (BIP) to select a subset of features that have the least combined computation time and achieve a stipulated detection performance.

Contributions: This paper claims to make two important contributions. First, it presents a BIP formulation to mine heterogeneous features taking both detection performance and computation time into consideration. The authors assert this optimization applied to heterogeneous features is unique in the literature and makes a key contribution. Second, the paper presents a thorough evaluation of the proposed person detector-using both proprietary and public datasets-with detailed analysis of its performance compared to alternative approaches and the state-of-the-art.

II. Framework

The objective in this work is to develop a people detection framework based on heterogeneous features that capture different facets of persons in an image. Our proposed detector training framework takes discriminative power of each feature and its associated computation time into consideration explicitly to select, and subsequently use, a subset of features that fulfill the required detection performance and have the minimum cumulative computation time.

As detection speed is one of our design focus, we adopt the acclaimed Viola and Jones [START_REF] Viola | Robust real-time face detection[END_REF] attentional cascade detector configuration in a sliding window paradigm. To train a strong classifier at each node of the cascade, the framework depicted in figure 1 is employed. For a given set of positive and negative training samples (a total of n samples denoted as {(x i , y i )} i∈{1,...,n} ): First, the features described in § II-A are extracted resulting in the feature set F . For each feature a unique weak learner is trained using the examples provided and is used to characterize the discriminating power of the feature in terms of True Positive Rate (TPR) and False Positive Rate (FPR). Following, pareto-front analysis is used to select a subset of features, F , taking their TPR, FPR, and computation time into account. This step is necessary to reduce the overwhelming total number of features to a tractable size for discrete optimization. Next, binary integer optimization, presented in § III, is used to retain a subset of features, F , that have the required performance-detection plus minimum computation time. Finally, a nodal strong classifier H(•) is trained using the retained feature set F with discrete AdaBoost. Specific design choice motivations and brief descriptions of each block are presented herein below.

A. Features

Five different feature families are considered, namely: Haar like, CS-LBP, CSS, EOH, and HOG. This choice is motivated by two aspects: (1) their frequent use in the literature for person detection, and (2) their complementary nature. EOH and HOG capture edge distributions, CSS focuses on color symmetry, Haar-like and CS-LBP on intensity and texture variations. The feature pool of each family is extracted from a 128 × 64 pixels human template window. Haar like: Here, the extended set proposed by Lienhart and Maydt [START_REF] Lienhart | An extended set of haar-like features for rapid object detection[END_REF] which includes tilted variants, is used. The pool is generated by extracting feature values at all positions and scales in the template window with the extended Haar set. CS-LBP: Computes per pixel CS-LBP [START_REF] Heikkilä | Description of interest regions with local binary patterns[END_REF] value by taking and modulating the intensity difference of center symmetric pixels for all the neighboring pixels. For each pixel, we privilege a 3 × 3 pixel region which results in a scalar integer between 0 and 16. Then, considering a rectangular region within the human template, a histogram with 16 bins is computed to signify one feature of this family. For all possible positions and scales of the rectangular region a distinct feature (which is a histogram) is computed and added in to the set of CS-LBP feature pool. CSS: Color self similarity, proposed by Walk et al. [START_REF] Walk | New features and insights for pedestrian detection[END_REF], captures pairwise similarities of spatially localized color distributions and can be used to capture the left and right symmetry of persons' clothing (upper body and lower body). The computation first starts by subdividing the given template into non-overlapping regions called blocks. For each block a 3 × 3 × 3 HSV color histogram is constructed. Then, the similarity of block with the rest of the blocks is determined via histogram intersection. Instead of concatenating all computed similarities like Walk et al. [START_REF] Walk | New features and insights for pedestrian detection[END_REF], we define a single CSS feature to be a vector of scalar values that are obtained by intersecting the histogram of one block with the rest of the blocks. The CSS feature pool set is then determined by computing this vector for all blocks. By dividing the template into blocks of 8 × 8 pixels, a total of 128 feature vectors, each with 127 dimensions, are obtained. EOH: This feature pool is generated exactly as described by Geronimo et al. [START_REF] Gerónimo | Haar wavelets and edge orientation histograms for on-board pedestrian detection[END_REF]: edge orientation histogram followed by ratios of magnitude of two bins to get a single scalar feature value and doing this for all positions and scales of rectangular subregions for histogram computation within the template window. HOG: The HOG feature pool set is constructed as follows: Given the template window, it is divided into overlapping blocks and a 36 dimensional histogram of oriented gradients is computed just like [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]. But, rather than concatenating all block histograms to make one high dimensional feature, we consider concatenating a subset spanning a rectangular region. The HOG feature pool is generated by considering all possible positions, width, and hight of the rectangular region. The features range from a 36 dimensional vector, a single block, to 3780 dimensional one, all blocks in the template.

Table I summarizes the total number of features, the scaled maximum and minimum feature computation time (τ max and τ min ), and the exact weak learner used in each feature family. For CS-LBP families Linear Discriminant Analysis combined with a decision tree (which is trained after re-projection) is privileged as SVM leads to overwhelming training period (due to the high number of CS-LBP features).

B. Pareto-front extraction

Given all set of features, F , along with their trained associated weak learners, and characterized by three parameters: TPR, FPR, and computation time (τ), pareto-front analysis is used to find the optimal solutions that make up the pareto optimal set-the solutions that cannot be improved in one objective function without deteriorating their performance in at least one of the rest. The subset of features that are pareto optimal with respect to TPR, FPR, and computation time, denoted F , are extracted and passed on to be used for the discrete optimization step.

C. Feature selection and cascade classifier learning

The final and decisive feature selection step is performed by the BIP optimizer and is discussed in § III. This module provides the set F . Finally, the nodal strong classifier, H(•), is built with discrete AdaBoost by using the F feature set.

The complete classifier used for detection, however, contains multiple nodes forming a cascade. The cascade construction starts with all positive training samples and a subset of the negative training samples (equivalent to the positive ones) to learn the set of relevant features and classifiers for the initial cascade node. Once this is done, all negative training samples in the dataset are tested with it. All those that get classified correctly are rejected while all those labeled as positive samples (false positives) are retained along with the positive samples for training the following nodes. This step is repeated until all negative training samples are exhausted. This data mining technique makes it possible to use vast number of negative training samples.

III. Discrete optimization feature selection

The BIP based feature selection applied to heterogeneous features makes the core of this work's contribution. The detailed optimization formulation to select a subset of features that fulfill a stipulated nodal FPR k , TPR k , with the minimum combined computation time possible is provided as follows (k denotes the node index):

Definition of parameters:

The following are list of parameters used in the optimization specification (applies to a cascade node k). B = {0, 1} denotes a binary set.

• N = {1, ..., n}: set of training sample indexes with n ∈ Z; a total of n training samples indexed by i;

• M = {1, ..., m}: set of weak learners indexes with m ∈ Z; a total of m weak learners indexed by j;

• y + ∈ B n , y + = y + i i∈N ; y -∈ B n , y -= y - i i∈N y + i = 1 if i is positive 0 otherwise y - i = 1 if i is negative 0 otherwise • H ∈ B nxm where H = h i, j i∈N j∈M with h i, j ∈ {0, 1}
h i, j = 1 if weak learner detects sample i as positive 0 otherwise

• TPR k ∈ [0, 1]: minimum true positive rate set at the considered node (k) of the cascade;

• FPR k ∈ [0, 1]: maximum false positive rate at the node;

• T ∈ R m : with T = τ j j∈M computation time of weak learner j.

Decision Variables:

In BIP, the decision variables are restricted to binary values, values from the set B = {0, 1}. The BIP decision variables are the following.

• v ∈ B m , v = v j j∈M v j ∈ {0, 1}: v j = 1 if weak learner j is selected, else v j = 0; • t ∈ B n , t i ∈ {0, 1}: t i = 1 if
a positive sample i has been detected as positive (true positive) by at least one selected weak learner, else t i = 0;

• f ∈ B n , f i ∈ {0, 1}: f i = 1 if a negative sample i has been detected as positive (false positive) by at least one selected classifier, else f i = 0. Let vector p, p = {p i } i∈N = Hv denote the total number of weak learners that have labeled each training sample i as positive.

Objective Function and Constraints:

min T T v (1) s.t t i ≤ y + i • p i ∀i (2) f i ≥ y - i • h i, j • v j ∀(i, j) (3) t 1 ≥ y + 1 • TPR k (4) f 1 ≤ y - 1 • FPR k (5) v ∈ B n ; t = {t i } i∈N , f = { f i } i∈N ; t, f ∈ B n (6) • 1 is l 1 norm.
The objective function (1) aims at minimizing the computation time. Constraints (2)-( 5) express that a given rate of detection quality has to be reached (depending on the number of true and false positives). Constraints (2) links v j and t i variables (via p i ) so that t i = 0 if positive image i is not correctly detected by at least one selected classifier. Constraints (3) links v j and f i variables so that f i = 1 if a negative image i has been recognized as positive by at least one selected classifier. Constraint (4) expresses that the stipulated TPR k of true positives, obtained with the selected classifiers, has to be reached. Similarly, constraint [START_REF] Viola | Robust real-time face detection[END_REF] expresses that the stipulated FPR k of false positives, obtained with the selected classifiers, must not be exceeded. In this formulation, there are a total of (n • (m + 1) + 2) binary variables in the BIP, which could be huge for large n and m values. The final subset of features F corresponds to only the selected features, i.e., non zero v entry; since each feature indexed by j is associated with a unique weak learner h j , F also represents the subset of weak learners retained

IV. Experiments and Results

In this section the different experiments carried out to investigate the performance of the proposed framework and obtained results along with commentaries are presented. The evaluation is focused on the following two aspects:

(1) Feature selection strategy evaluation: Here, the aim is to analyze the pros and cons of using BIP over other simpler alternatives. The proposed BIP based feature selection and classifier learning strategy, labeled as BIP+AdaBoost, is compared with two other modes. First, Pareto+AdaBoost which discards the BIP block in the framework and directly trains a nodal strong classifier with discrete adaboost using the features retained by the pareto-front extraction block. And second, Random+AdaBoost which directly builds a nodal classifier using randomly sampled features from the total feature pool (proportional to each feature pool family size) and AdaBoost.

(2) General comparative evaluation with the state-of-the-art: In this part, the performance of the trained BIP+AdaBoost is compared with the prominent approaches in the literature.

A. Evaluation Criteria

For detector performance evaluation, we use two approaches: (1) The per window approach, whereby a Detection Error Trade-off (DET) curve with Miss Rate versus False Positives Per Window (FPPW) is generated by using cropped positive and negative windows; and (2) the per image approach which shows Miss Rate versus False Positives per Image (FPPI). The first curve is used to compare experimental variants of the proposed framework with respect to Dalal and Triggs HOG [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] (aspect 1), and the second is used to determine how our best approach plays out compared to the different techniques in the literature (aspect 2). To summarize the performance, the Miss Rate at 10 -4 FPPW and the logaverage miss rate are used in the first and second approaches respectively.

Another criterion that is taken into account is the average computation time. For a cascade detector the average computation time for a given candidate window is affected by the FPR of each node. Let K be the total number of nodes in the cascade, FPR k be the false positive rate and τ k be the total computation time of the k th node during detection. Assuming the nodal FPR characteristics hold on a generic input image, the average time spent on a test candidate window, T av , can be estimated as T av = τ 0 + K-1 k=1 ( k-1 z=0 FPR z )τ k . Using Dalal and Triggs [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] detector, which takes ζ HOG per window, as a reference, the Average Speed Up (ASU) over it is determined as ASU = ζ HOG T av . Consequently, the ASU values reported henceforth are with respect to Dalal and Triggs detector.

B. Dataset

For evaluation, two different datasets are considered: The Ladybug dataset 1 , which is a proprietary dataset compiled from indoor laboratory environment using the Ladybug2 spherical camera; and the INRIA public dataset [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF], a publicly available dataset most predominantly used for benchmarking people detectors in the literature. A detailed description is not provided here due to space considerations, but table II summarizes the actual data used for training and testing purposes. The Ladybug dataset is used for training and testing the framework using cropped windows. On the INRIA dataset cropped windows are used for training. For testing, both cropped windows and full images are used for a per window and full image evaluation respectively. In both datasets, the cropped negative windows are uniformly sampled from provided person free full images.

C. Training

Each cascade node training (learning) is governed by two provided parameters: the nodal TPR k and FPR k for node k (TPR k is always 1.0). The training is done so the final trained nodal classifier conforms to these stipulated performance requirements. Each cascade node is built using a subset of the total negative training samples and all positive samples. This set is initially divided into a 60% training and a 40% validation set. The weak learners are trained using the 60% training set. Then, TPR and FPR values corresponding to each weak learner are determined based on the validation set. All subsequent computations, i.e., pareto-front analysis and feature selection via BIP are performed using the weak learners performance conferred on the validation set. Once the pertinent features are selected, the corresponding weak learners are re-trained using the combined training and validation set within the discrete AdaBoost to build the per node final strong classifier, i.e., H(•). The complete cascaded classifier is then learned as explained in § II-C. For the associated weak learners, decision trees of depth 2, 3, and 3 are used for Haar like, EOH, and LBP features, respectively, after detection performance and over-fitting trade off analysis on a validation set.

D. Results and Discussions 2

Ladybug Dataset: The main results obtained with the Ladybug dataset are depicted in figure 2 and summarized in table III. Clearly Pareto+AdaBoost results in the best detection performance, 2.9% MR, followed by Dalal and Triggs detector trained on this dataset, 3.0%, at 10 -4 FPPW. In terms of detection, BIP+AdaBoost trails behind Random+AdaBoost with marginal loss. But, the most important result to notice is that BIP+AdaBoost results in a drastic 42.7x speed up over Dalal and Triggs with only a 7% loss in MR at 10 -4 FPPW. The main reason for this speed up is that BIP+AdaBoost systematically uses cheap features in the intial stages of the cascade and only starts using computationally expensive features at later stages. The trained classifier has 10 cascade nodes with CSS features initially appearing at the 6 th node and HOG at the final stage. Apparently, Pareto+AdaBoost and Random+AdaBoost result in worsened speeds. This is because AdaBoost always privileges the most discriminant feature, irrespective of computation cost, from the pool of features passed to it, and both pareto front extraction and random sampling are likely to pass such kind of complex features. Hence, the set of features selected in the first node result in a set that is effectively computationally demanding than Dalal and Triggs detector. These results are obtained using a fixed nodal FPR of 0.5 for all constructed nodes and the obtained results are very precise that altering the FPR is not necessary. 3 and summarized in table IV. As this dataset is challenging, two variants of the BIP+AdaBoost classifier are trained. In the first case, a fixed nodal FPR of 0.5 is used for all nodes, called BIP+AdaBoost(Fix). In the second case, an adaptive FPR is employed which starts at 0.3 in the initial stage and continues training nodes, whenever a solution for the BIP optimization does not exist, this constraint is relaxed/incremented by 0.1 and the procedure continues from that node likewise until all negative samples are depleted. This is called BIP+AdaBoost(Ad). Again, the best detection results at 10 -4 FPPW are obtained by the Random+AdaBoost and Pareto+AdaBoost variants. But, this time both variants of BIP+AdaBoost beat Dalal and Triggs detector at 10 -4 by more than 2%. On top of this, the BIP+AdaBoost(Fix) achieves a 15.6x speed up while that of BIP+AdaBoost(Ad) trails with a 9.22x speed up. As the initial FPR constraints are stringent on the BIP+AdaBoost(Ad) variant, it will favor relatively discriminative features with increased computation time. But, this contributes to its superior detection performance, over BIP+AdaBoost(Fix), throughout the FPPW range shown in figure 3. Observe in table IV, there are more proportions of Haar like features (5.4% more) and less proportions of HOG features (2.0% less) in the fixed variant compared to the adaptive variant resulting in the increase speed up. Figure 5 shows histogram of the selected features, with relative proportions, for the first 9 nodes of both the fixed and adaptive variants. Clearly, the fixed variant initially uses cheaper features and increases along the cascade both in number and complexity. On the contrary, for the variable variant, complex features appear in the initial nodes and increase in number along the cascade. Figure 4 illustrates a few of the selected features overlaid on an average human gradient image for BIP+AdaBoost(Ad). Observe that all selected features capture discriminant facets of people. Finally, figure 6 shows the comparative evaluation of BIP+AdaBoost (Ad) detector on the INRIA dataset using the full image evaluation criteria. Comparative evaluations are taken from [START_REF] Dollar | Pedestrian detection: An evaluation of the state of the art[END_REF]; the reader is referred to this survey for explanation of each detector (as space does not permit here).

To generate these results, a Pairwise Max non-maximal suppression [START_REF] Dollar | Pedestrian detection: An evaluation of the state of the art[END_REF] with an overlap threshold of 0.65 is used. Again, here, BIP+AdaBoost(Ad) does well achieving a log-average miss rate of 47%. At lower FPPI values, less than 0.1 FPPW, the BIP variant consistently supersedes Dalal and Triggs HOG.

Using the computation speed reported in [START_REF] Dollar | Pedestrian detection: An evaluation of the state of the art[END_REF] for people more than 100 pixels in a 640×480 image, our detectors achieves 2.

V. Conclusions

In conclusion, a novel framework based on heterogeneous pool of features and discrete optimization for developing a computation time and detection performance optimized person detector has been presented. The proposed framework has been validated thoroughly using proprietary and public datasets. The results obtained conform to our aims and result in a faster detector with competitive detection performance amongst the state-of-the-art.

In the near future, we plan to investigate ways to achieve more faster versions of the detector by focusing on implementation optimization and specialized accelerator hardwares like Graphical Processing Units (GPUs).

Fig. 2 :

 2 Fig. 2: DET of different detectors trained and tested on the Ladybug dataset.

Fig. 3 :

 3 Fig. 3: DET of different detectors trained and tested on the INRIA dataset.

Fig. 5 :

 5 Fig. 5: Histogram of selected features in the first 9 nodes of the model trained on the INRIA dataset using both fixed FPR of 0.5 and adaptive FPR.

Fig. 4 :

 4 Fig. 4: Sample depictions (overlaid on an average human gradient image) of the heterogeneous features selected at different nodes of the cascade trained on the INRIA dataset using an adaptive FPR. Black rectangular regions show Haar features, blue is for CS-LBP, green boxes represent CSS features and their position indicates the reference block, and finally, violet shows the spatial region spanned by the concatenated HOG blocks.

Fig. 6 :

 6 Fig. 6: Comparative full image evaluation on the INRIA test set.

  3 frames per second (fps) for the adaptive variant, and 3.9 fps for the fixed FPR variant trained on the INRIA dataset. These values are amongst the top best only exceeded by FPDW which achieves approximately 6.0 fps. But, actually FPDW uses the underlying principles of ChnFeats and optimizes the detection process by approximating the features over scale space. Similar techniques can be used to further improve the fps of our detector. On the other hand, the model trained on the Ladybug dataset, achieves 10.6 fps on the simpler dataset. This is an added advantage as a majority of the methods in the state-of-the-art do not have the ability to automatically change the complexity of the trained detector based on the dataset; examples include Dalal and Triggs HOG and HogLbp which have fixed size feature vector irrespective of dataset.

TABLE I :

 I Feature pool summary. Time is reported relatively as a multiple of the smallest feature computation time, u = 0.0535µs.

	Feature Type	No of features	τ min	τ max	Weak Learner
	Haar like	672,406	1.0u	3.48u	Decision Tree
	EOH	712,960	4.83u	317.75u	Decision Tree
	CS-LBP	59,520	15.45u	393.64u	LDA + Decision Tree
	CSS	128	1017.94u 1017.94u SVM
	HOG	3,360	489.72u	51420.56u SVM

TABLE II :

 II Summary of the different dataset used for training and testing.

	Dataset	Training		Test
		pos win.	neg win.	pos win.	neg win.	full images
	Ladybug 1	1, 990	488, 992	1, 000	319, 653	-
	INRIA [4]	2, 416	2.55 × 10 6	1, 132	2.00 × 10 6	288

TABLE III :

 III Summary of the cascade detector trained on the Ladybug dataset. Miss Rate is reported at 10 -4 FPPW. Similar results obtained for the INRIA dataset are shown in figure

	Detector		Feature Proportion	MR	ASU
		Haar	CSLBP CSS	EOH HOG
	Dalal and Triggs [4]	-	-	-	-	100% 3.0% 1.0x
	Pareto + AdaBoost	10.7% 0.0%	0.0% 0.0% 83.7% 2.9% 0.7x
	Random + AdaBoost	51.6% 6.2%	1.5% 36.0% 4.7% 8.0% 0.6x
	BIP + AdaBoost	54.3% 8.6%	8.5% 25.7% 2.8% 10.0% 42.7x
	INRIA Dataset:

TABLE IV :

 IV Summary of the cascade detector trained on the INRIA datasets. Miss Rate is reported at 10 -4 FPPW.

	Detector		Feature Proportion	MR	ASU
		Haar	CSLBP CSS	EOH HOG
	Dalal and Triggs [4]	-	-	-	-	100% 11.0% 1.0x
	Pareto + AdaBoost	42.8% 14.5%	7.8% 25.6% 9.3% 7.0% 0.4x
	Random + AdaBoost	26.3% 10.8%	3.7% 53.5% 5.6% 6.0% 0.4x
	BIP + AdaBoost (Fix)	60.4% 10.8%	8.0% 9.7% 11.0% 8.0% 15.6x
	BIP + AdaBoost (Ad)	55.0% 14.6%	8.1% 9.3% 13.0% 7.4% 9.22x

Please see http://homepages.laas.fr/aamekonn/ladybug dataset/

All figures in this section are best viewed in color.

Acknowledgment

This work was supported by a grant from the French National Research Agency (ANR) under grant number ANR-12-CORD-0003.


Kazuo Nishimura

Carine Nourry

Thomas Seegmuller

Alain Venditti
email: alain.venditti@univ-amu.fr

Emmanuelle Augeraud-Véron

Larry Blume

Pedro Bom

Stefano Bosi

Raouf Boucekkine

Jean-Pierre Drugeon

Karine Gente

Andreas Irmen

Mouez Fodha

Fabien Prieur

Xavier Raurich

Michael Reiter

Gerhard Sorger

Emmanuel Thibault

On the (de)stabilizing effect of public debt in a Ramsey model with heterogeneous agents *

Keywords: Endogenous cycles, heterogeneous agents, public spending, public debt, borrowing constraint Journal of Economic Literature Classification Numbers: C62, E32, H23

We introduce public debt in a Ramsey model with heterogenous agents and a public spending externality affecting utility which is financed by income tax and public debt.

We show that public debt considered as a fixed portion of GDP can have a stabilizing or destabilizing effect depending on some fundamental elasticities. When the public spending externality is weak and the elasticity of capital labor substitution is low enough, public debt can only be destabilizing, generating damped or persistent macroeconomic fluctuations.

Whereas when the public spending externality and the elasticity of capital labor substitution are strong enough, public debt can be stabilizing, driving to monotone convergence an economy experiencing damped or persistent fluctuations without debt.

Introduction

The last financial crisis has shed the light on the problem of large public debt in developed countries, in particular in Europe. In many advanced countries, debt levels have increased dramatically during the two last decades, now reaching extremely high amounts. The control of the growth rate of public spendings has became a major concern for economists and policy-makers while public deficits digging. A heavily indebted country may appear as fragile, for many reasons, among which solvency, or simply because it is unlikely to raise sufficient funds to deal with a large negative shock on its economy. The Maastricht treaty introduced a rule on the maximal amount a country may contract, limiting the debt to 60% of GDP, but this limit has been exceeded by almost all european countries. Indeed, most advanced countries are characterized both by large amounts of public debt and large fluctuations of GDP. Given this fact, two types of questions become central. The first concerns the relationship between the level of debt and growth, the second focuses on the relationship between macroeconomic stability and debt level. But whereas the literature has focused recently on the first question, little attention has been paid to the second.

While subject to a recent controversy, 1 the paper of [START_REF] Reinhart | Growth in a time of debt[END_REF] indeed shows that a gross public debt exceeding 90% of nominal GDP on a sustained basis may have a significant negative impact on the growth rate. 2 On the basis of this type of result, the IMF has strongly advised European countries over the last years to decrease their debt. The main objective was to boost growth but also to stabilize the economies. Indeed, since 2008, most advanced economies have been characterized by large fluctuations of GDP.

In a OECD Economics Department Policy note, Sutherland, Hoeller, 1 See Herndon, Ash and Pollin (2013). 2 See also [START_REF] Reinhart | Public debt overhangs: advanced-economy episodes since 1800[END_REF].

1 Merola and Ziemann (2012) argue that the level of government debt has a significative impact on business cycle characteristics. They identify the characteristics of a "low debt" business cycle and a "high debt" business cycle aggregating the countries according to their level of debt. In countries with high debt, the cycle is more pronounced, with phases of expansions longer and larger and recessions also more pronounced. The arguments for such differences usually rely on the "vulnerability" of high public debt economies. Government then have less latitude to run the appropriate fiscal policy in case of negative shocks. Moreover, [START_REF] Ilzetzki | How big (small?) are fiscal multipliers?[END_REF] show that the impact of governments expenditure shocks depends crucially on public indebtness, as the fiscal multipliers seems to be lower in high debt countries.

This paper proposes to study the question of the relationship between debt and fluctuations in the simple framework of the neoclassical Ramsey model (1928). Our aim is to precisely discuss the effect of public debt on the macroeconomic stability of one country in the optimal growth model. As we focus on business cycle properties, we will consider a model with heterogeneous agents, which allows the emergence of endogenous fluctuations. Actually, in the standard [START_REF] Ramsey | A mathematical theory of saving[END_REF] model with one representative agent, one sector and usual assumptions, the economy monotonically converges to the steady state. With many agents, conclusions may differ: Introducing borrowing constraints, the Ramsey conjecture holds at the steady state, i.e. the most patient holds all the capital [START_REF] Becker | On the long-run steady state in a simple dynamic model of equilibrium with heterogeneous households[END_REF]). Dynamics can be non-monotone and endogenous cycles can occur around the steady state for a weak elasticity of capital-labor substitution, i.e. when capital income monotonicity fails [START_REF] Becker | On the long-run steady state in a simple dynamic model of equilibrium with heterogeneous households[END_REF]Foias (1987, 1994)). 3We focus on government intervention as a source of macroeconomic fluc-tuations when government spending is financed through taxes on income and public debt. Public spending is useful because it improves households' utility of consumption as an externality. Even with linear income taxation, endogenous fluctuations (flip bifurcation) become compatible with plausible values of the elasticity of capital-labor substitution in the economy without debt. 4When public debt is introduced as a fixed proportion of GDP, we show that it can have a stabilizing or destabilizing effect depending on the value of the elasticity of capital-labor substitution.

If the elasticity of capital labor substitution is low, and the marginal utility of consumption with respect to public spendings is weak, i.e. the variation of individuals' welfare is not very sensitive to variations of the available public good, we prove that public debt can have a destabilizing effect. Indeed, economies characterized by a high level of the debt-output ratio are more subject to fluctuations. They can actually experience endogenous oscillations whereas the same economies are characterized by saddle-point stability if their level of debt is low.

Conversely, when the elasticity of capital labor substitution is high enough, and the elasticity of the marginal utility with respect to public spendings is large, i.e. the variation of individuals' welfare is very sensitive to variations of the available public good, we prove that public debt has a strong stabilizing effect. Indeed a large enough amount of public debt can stabilize the economy by guaranteeing monotone convergence toward the steady state.

The destabilizing effect of public debt associated to cases where public spending does not matter a lot for agents decisions can be explained follow-ing an intuition which is closely related to the intuition given by [START_REF] Becker | On the long-run steady state in a simple dynamic model of equilibrium with heterogeneous households[END_REF]Foias (1987, 1994). Without debt, endogenous fluctuations indeed occur only when the elasticity of capital-labor substitution is low enough, meaning that the "capital income monotonicity" fails. Capital income is then a decreasing function of capital and a high capital at one period is compatible with a lower investment at the next period. For indebted countries, as now the most patient agents own both assets, capital and debt, endogenous fluctuations occur when the "asset income monotonicity" fails, which is less restrictive and thus true for larger sets of values for the elasticity of capital labor substitution. The set of economies subject to these fluctuations is then larger if the economies are sufficiently indebted.

The stabilizing effect of public debt associated to cases where public spending matters a lot for agents decisions can be explained as follows. Assume that the elasticity of capital labor substitution is large enough and that at a given period, capital is high, meaning income is high. Fluctuations occur if the consumer's intertemporal trade-off is compatible with a decrease of capital at the following period, associated to a higher return to capital.

Without debt, public spending is procyclical, following mechanically the capital labor ratio through taxes on income, and endogenous fluctuations are compatible with the intertemporal trade-off. But when public spending is also related to the repayment of a previous debt and the contraction of a new one, it can be countercyclical, and the fluctuations cannot occur anymore as they become incompatible with the intertemporal trade-off. Thus when the welfare associated to consumption depends strongly of the level of public spendings, a high enough amount of debt can prevent the existence of endogenous fluctuations. 5We finally show that, depending on the value of the elasticity of capitallabor substitution, public debt can be used at the same time to stabilize the economy and to affect the degree of inequalities. Indeed, when capital and labor are weakly substitutable, a low debt can stabilize and decrease the inequalities as capital is growing. On the contrary, when the elasticity of capitallabor substitution is larger, a large debt can stabilize and decrease or increase the inequalities as capital is growing depending on whether the elasticity has intermediary or large values. This paper is organized as follows. The model is presented in the next section. Section 3 is devoted to the steady state analysis. In section 4, we study the occurrence of endogenous business cycles and economic interpretations.

Section 5 provides some concluding remarks. Proof and technical details are provided in the Appendix.

The model

We consider a discrete time economy (t = 0, 1, ..., ∞), with three types of agents, heterogeneous households, firms and a government.

Households

There are H heterogeneous infinitely lived households, indexed by i = 1, . . . , H with H 2, who supply inelastically labor and face borrowing constraints.

They have heterogeneous capital and debt endowments (k i0 , b i0 0), and heterogeneous preferences, i.e. different discount factors and different instantaneous utilities in consumption. Households are ranked according to their

discount factors: 0 ≤ β H ≤ . . . ≤ β 2 < β 1 < 1.
overlapping generations.

Household i derives utility for consumption c it at period t. Moreover, we assume that public spending G t affects welfare, as an externality on utility for consumption. Utility is non separable between consumption and public spending at each period, but separable over time:

+∞ t=0 β t i u i (c it , G t ) (1)
Public spending thus corresponds to a pure public good providing services to consumers, e.g. schools, infrastructures, .... Each household derives income from wage, capital and government bonds that allow to finance public debt. Denote r t the real interest rate on physical capital, rt the return of government bonds, w t the real wage and δ ∈ (0, 1) the rate of depreciation of capital. In addition, each household pays taxes on labor income, capital income and on the remuneration of bonds' holding, at a constant rate τ ∈ (0, 1). 6 Any household i maximizes (1) facing the budget constraint:

c it + k it+1 + b it+1 = (1 -τ )[r t k it + w t + rt b it ] + (1 -δ)k it , (2)
and the borrowing constraint on individual capital and government bonds holding k it , b it ≥ 0. The utility function satisfies the following assumption:

Assumption 1. u i (c i , G) is a continuous function defined on [0, +∞) × [0, +∞), and C 2 on (0, +∞) × (0, +∞). u i (c i , G) is strictly increasing (u ic (c i , G) > 0), strictly concave (u icc (c i , G) < 0)
with respect to its first argument, and the marginal utility of consumption increases with respect to public spending (u icG (c i , G) 0). 7 For further reference, we introduce the following elasticities:

8 icc ≡ -u icc c i /u ic > 0, icG ≡ u icG G/u ic 0 (3)
In addition, the Inada condition lim c i →0 u ic (c i , G) = +∞ is satisfied.

We consider here that private consumption and government services are Edgeworth complements, icG ≥ 0, following [START_REF] Ni | An empirical analysis on the substitutability between private consumption and government purchases[END_REF] who provides empirical support for this assumption.

Utility maximization gives:

u ic (c it , G t ) u ic (c it+1 , G t+1 ) ≥ β i R t+1 , with equality when k it+1 > 0 (4) u ic (c it , G t ) u ic (c it+1 , G t+1 ) ≥ β i (1 -τ )r t+1 , with equality when b it+1 > 0 ( 5
)
with R t+1 = (1 -τ )r t+1 + 1 -δ and the transversality conditions lim t→+∞

β t i u ic (c it , G t ) k it+1 = 0 and lim t→+∞ β t i u ic (c it , G t ) b it+1 = 0 (6)
Note that for all i = 1, . . . , H, c it are forward variables while k it and b it are predetermined variables.

Firms

A representative firm produces the final good y t , using capital k t and labor l t under a constant returns to scale technology y t = F (k t , l t ). As labor is inelastic and there are H ≥ 1 households who supply one unit of labor, aggregate labor is constant, l t = H. Therefore, following [START_REF] Becker | On the long-run steady state in a simple dynamic model of equilibrium with heterogeneous households[END_REF] and [START_REF] Becker | A characterization of Ramsey equilibrium[END_REF], we denote F (k t , H) ≡ f (k t ). The production function f (k) satisfies:

Assumption 2. f (k) is a continuous function defined on [0, +∞) and C 2 on (0, +∞), strictly increasing (f (k) > 0) and strictly concave (f (k) < 0). In addition, the conditions lim k→0 f (k) = +∞ and

lim k→+∞ f (k) < θ/[β 1 (1 - τ )], with θ ≡ 1 -β 1 (1 -δ), are satisfied.
Profit maximization gives:

r t = f (k t ) ≡ r (k t ) (7)
and due to constant returns to scale and perfect competition, profits are zero, so that

Hw t = f (k t ) -r t k t ≡ Ω (k t ) (8)
In the following, we denote by

s (k) ≡ kf (k) /f (k) ∈ (0, 1) the capital share in total income and σ (k) ≡ [s (k) -1] f (k) / [kf (k)]
≥ 0 the elasticity of capital-labor substitution. We derive the following useful relationships:

r (k)k/r(k) ≡ -(1 -s(k))/σ(k) and Ω (k)k/Ω(k) ≡ s(k)/σ(k) (9)

Government

Public spending G t is financed by total income taxation and debt. Since

y t = H i=1 r t k it + Hw t ,
the budget constraint faced by the government at period t writes:

G t + rt b t = τ (y t + rt b t ) + b t+1 (10)
where τ ∈ (0, 1) is the constant proportional tax rate on households' total income. Here, b t corresponds to the stock of real debt used to finance public investment. As a consequence, b t is a predetermined variable as capital.

Total public expenditures, that are the sum of public spendings G t and the reimbursement of debt contracted the previous period rt b t , are financed by the new issue of debt b t+1 and taxation of (capital and labor) income and of remuneration of bonds. In order to match the constraint imposed by the Maastricht treaty, we will assume that public debt cannot exceed a fixed proportion α 0 of GDP, namely b t ≤ αy t

We will focus in the following on equilibria where this constraint is binding, i.e. b t = αy t . The case without debt is of course obtained when α = 0 and α is defined as the debt-output ratio. Note that when the constraint is binding, α can also be interpreted as a policy parameter that allows to manage public debt excluding its explosive path (see de la Croix and Michel (2002), p.230-233). The parameter α will allow us to discuss the stability properties of the economy according to the level of the debt-output ratio.

Intertemporal equilibrium

An intertemporal equilibrium can be defined as follows:

Definition 1. Under Assumptions 1 and 2, an equilibrium is a sequence

(r t , rt , w t , k t , b t , G t , (k it , b it , c it ) H i=1 ) +∞ t=0
satisfying the optimal behavior of households ( 2), ( 4), ( 5) and ( 6), profit maximization ( 7) and ( 8), the government constraint [START_REF] De La Croix | A Theory of Economic Growth, Dynamics and Policy in Overlapping Generations[END_REF], the equilibrium conditions on the asset markets

k t = H i=1 k it and b t = H i=1 b it = αf (k t ),

Steady state analysis

Since the tax rate on income is constant, we can derive the existence of a steady state along which the equality R = (1 -τ )r holds as physical capital k and governments bonds b are perfectly substitutable saving assets.

Proposition 1. Under Assumptions 1 and 2, let α ∈ [0, α) with α = τ β 1 /(1- β 1 )
. Then there exists a steady state defined by the following properties:

(S1) r = f (k) = θ/[β 1 (1 -τ )], r = R/(1 -τ ) = f (k) + (1 -δ)/(1 -τ ) and w = [f (k) -kf (k)]/H are constant; (S2) R = 1/β 1 < 1/β 2 ≤ . . . ≤ 1/β H ; (S3) k = k 1 > 0, b = b 1 = αf (k 1 ) > 0 and k i = b i = 0 for i ≥ 2; (S4) c 1 = (k 1 + b 1 )(R -1) + (1 -τ )w and c i = (1 -τ )w for i ≥ 2; (S5) G = τ (rk 1 + Hw) + (1 -R)b 1 ≡ ∆f (k 1 ) with ∆ = τ -α(1-β 1 ) β 1 . Proof : See Appendix 6.1.
This Proposition shows that because of the borrowing constraints, there exists a steady state. In accordance with the so-called Ramsey (1928) conjecture and the seminal contribution of [START_REF] Becker | On the long-run steady state in a simple dynamic model of equilibrium with heterogeneous households[END_REF], the most patient household holds the whole capital stock and government debt. Note that the debt-output ratio has to be lower than α, because otherwise, the public debt burden is too heavy and is no more compatible with a positive government spending.

Endogenous business cycles under public spending externalities

In the neighborhood of the steady state exhibited in Proposition 1, the intertemporal equilibrium can be summarized by a two-dimensional dynamical system given by the patient household's trade-off between present and future consumption and his budget constraint. Indeed, using Definition 1, an intertemporal equilibrium can be redefined as a sequence (c 1t , k t ) ∞ t=0 , satisfying:

u 1c (c 1t , G t ) u 1c (c 1t+1 , G t+1 ) = β 1 R(k t+1 ) (11)
k t+1 + αf (k t+1 ) = R(k t )[k t + αf (k t )] + (1 -τ ) Ω(k t ) H -c 1t (12) with R(k t+1 ) = (1 -τ )f (k t+1 ) + 1 -δ, G t = f (k t ) [τ -αR(k t )] + αf (k t+1 ),
and where c 1t is a forward variable and k t is the only predetermined variable.

We characterize the stability properties of the steady state and the occurrence of local bifurcations by linearizing the dynamic system ( 11)-( 12) around the steady state (c 1 , k) and computing the Jacobian matrix J, evaluated at this steady state.

Lemma 1. Under Assumptions 1 and 2, the characteristic polynomial is given by P (λ) ≡ λ 2 -T λ + D = 0, where:

D = B 2 (α)-1cG ∆ 1cc B 3 (α)B 1 (α) 1+ 1cG ∆ 1cc αsB 1 (α) ≡ D (α) , T = 1 + D(α) + (1-s)θB 1 (α) σ 1cc 1+ 1cG ∆ 1cc αsB 1 (α) ≡ T (α) (13)
with

B 1 (α) = 1-τ β 1 (1-τ )+αθ (1 -β 1 ) 1 + αθ β 1 (1-τ )s + (1-s)θ sH > 0 B 2 (α) = 1 β 1 1 -β 1 θ(1-s)(1-τ ) σ[β 1 (1-τ )+αθ] 1 -1 H + αθ β 1 (1-τ )s B 3 (α) = s(τ β 1 -α) β 1 + α(1-s)θ σβ 1 (14)
Proof : See Appendix 6.2.

As shown by [START_REF] Becker | On the long-run steady state in a simple dynamic model of equilibrium with heterogeneous households[END_REF]Foias (1987, 1994)

The Ramsey economy without debt

The equilibrium without debt is obviously obtained when α = 0. We then get the following Proposition:

Proposition 2. Under Assumptions 1-2, let α = 0. Then there exist ¯ 1cG > 1cG ≥ 0, σ < σ < σ CIM and ¯ 1cc > 0 such that the following results hold: i) if σ > σ, the steady state is saddle-point stable with monotone convergence when 1cG ∈ (0, 1cG ), saddle-point stable with damped oscillations when 1cG ∈ ( 1cG , ¯ 1cG ), undergoes a flip bifurcation for 1cG = ¯ 1cG , and becomes locally unstable when 1cG > ¯ 1cG ;

ii) if σ ∈ (σ, σ), the steady state is saddle-point stable with damped oscillations when 1cG ∈ (0, ¯ 1cG ), undergoes a flip bifurcation for 1cG = ¯ 1cG , and becomes locally unstable when 1cG > ¯ 1cG ;

iii) if σ ∈ (0, σ) and 1cc < ¯ 1cc , the steady state is saddle-point stable with damped oscillations when 1cG ∈ (0, ¯ 1cG ), undergoes a flip bifurcation for 1cG = ¯ 1cG , and becomes locally unstable when 1cG > ¯ 1cG . In all cases, saddle-point stable (locally unstable) period-two cycles occur in a right (left) neighborhood of ¯ 1cG .

Proof : See Appendix 6.3.

This Proposition shows that endogenous business cycles emerge through a flip bifurcation even when the capital income monotonicity assumption is satisfied, provided the government spending externality is large enough. Indeed, if σ > σ CIM , case i) of Proposition 2 applies. This is at odds with the non monotonicity of capital income required to get period-two cycles in the model without public spending externalities [START_REF] Becker | On the long-run steady state in a simple dynamic model of equilibrium with heterogeneous households[END_REF]Foias (1987, 1994)).

Of course, when there is no externality, i.e. 1cG = 0, we get as a Corollary the same conclusions as in [START_REF] Becker | On the long-run steady state in a simple dynamic model of equilibrium with heterogeneous households[END_REF]Foias (1987, 1994):

Corollary 1. Under Assumptions 1-2, let α = 0 and 1cG = 0. Then there exist σ < σ < σ CIM and ¯ 1cc > 0 such that the following results hold: i) if σ > σ, the steady state is saddle-point stable with monotone convergence for any 1cc > 0;

ii) if σ ∈ (σ, σ), the steady state is saddle-point stable with damped oscillations for any 1cc > 0;

iii) if σ ∈ (0, σ), the steady state is saddle-point stable with damped oscillations when 1cc ∈ (0, ¯ 1cc ), undergoes a flip bifurcation for 1cc = ¯ 1cc , and becomes locally unstable when 1cc > ¯ 1cc . Moreover, saddle-point stable (locally unstable) period-two cycles occur in a right (left) neighborhood of ¯ 1cc .

The Ramsey economy with debt

Let us now consider the case with debt assuming α ∈ (0, α). Our aim is to check whether debt has a stabilizing or a destabilizing effect on the economy.

Put differently, we are looking for some conditions on the share α of debt over GDP that allow to generate or rule out endogenous fluctuations. Building on the results derived in the no-debt case, we show that the conclusions strongly depend on the value of the elasticity of capital-labor substitution.

We start by analyzing the standard formulation of [START_REF] Becker | On the long-run steady state in a simple dynamic model of equilibrium with heterogeneous households[END_REF]Foias (1987, 1994) without public spending externality in the utility function. Indeed, it is worth noting from Lemma 1 that when 1cG = 0, the Determinant and Trace of the characteristic polynomial are linear functions of the share α. This monotonicity property allows to derive the following clear-cut conclusions: Proposition 3. Under Assumptions 1-2, let 1cG = 0. Then there exist β 1 ∈ (0, 1), σ > σ > σ > σ, ¯ 1cc > ˜ 1cc and α ≥ ᾱ > α ≥ 0 such that when β 1 ∈ (β 1 , 1) the following results hold: i) Public debt does not have any impact on the local stability properties of the steady state when σ ≥ σ, or σ ∈ (σ, σ) and 1cc ≤ ˜ 1cc , or σ ∈ (0, σ) and

1cc ∈ (0, ˜ 1cc ] ∪ (¯ 1cc , +∞).
ii) Public debt has a destabilizing effect in the following cases:

-when σ ∈ (σ, σ), or σ ∈ (σ, σ) and 1cc ≤ ˜ 1cc . Indeed, the steady state is saddle-point stable with monotone convergence when α ∈ [0, α) and saddle-point stable with damped fluctuations when α ∈ (α, α).

-when σ ∈ (σ, σ) and 1cc > ˜ 1cc . Indeed, the steady state is saddlepoint stable with monotone convergence when α ∈ [0, α), saddle-point stable with damped fluctuations when α ∈ (α, ᾱ), undergoes a flip bifurcation when α = ᾱ and becomes locally unstable with oscillations when α ∈ (ᾱ, α).

-when σ ∈ (σ, σ) and 1cc > ˜ 1cc . Indeed, the steady state is saddle-point stable with damped fluctuations when α ∈ [0, ᾱ), undergoes a flip bifurcation when α = ᾱ and becomes locally unstable with oscillations when α ∈ (ᾱ, α).

-when σ ∈ (0, σ) and 1cc ∈ (˜ 1cc , ¯ 1cc ). Indeed, the steady state is saddle-point stable with damped fluctuations when α ∈ [0, ᾱ), undergoes a flip bifurcation when α = ᾱ and becomes locally unstable with oscillations when α ∈ (ᾱ, α).

Moreover, in all cases where ᾱ exists, there are saddle-point stable (locally unstable) period-two cycles in a right (left) neighborhood of ᾱ.

Proof : See Appendix 6.4

Proposition 3 provides a complete picture of the impact of public debt on the local stability properties of the steady state when there is no public spending externality in preferences, i.e. 1cG = 0. We have then clearly shown that when σ is low enough and 1cc is sufficiently large, public debt has a destabilizing effect as it may create damped and/or persistent macroeconomic fluctuations.

Let us finally consider the formulation with public spending externality in utility in order to check whether the destabilizing effect of debt is robust.

When 1cG > 0 the analysis becomes much more complex as the Determinant and Trace of the characteristic polynomial are no longer linear functions of α. We may however provide some results which show that depending on the values of σ and 1cG , the previous conclusion does not necessarily hold as a sufficiently large public debt may have a stabilizing or destabilizing effect.

Proposition 4. Under Assumptions 1-2, let δ < s < 1/2 and β 1 > 1/(1 + s -δ). Then there exist σ, σ > 0, satisfying σ CIM > σ > σ > σ > σ, ¯ 1cG > 1cG > 0, ¯ 1cc > 0 and ᾱ ∈ [0, α) such that the following results hold:

1-Public debt has a stabilizing effect in the following cases: i) when σ > σ. Indeed, the steady state is saddle-point stable with monotone convergence when α ∈ (ᾱ, α) no matter what are the stability properties of the steady state in the economy without debt (i.e. for any 1cG > 0);

ii) when σ ∈ (σ, σ) and 1cG > ¯ 1cG . Indeed, the steady state is saddlepoint stable with damped oscillations when α ∈ (ᾱ, α) while in the economy without debt it is unstable with possible persistent fluctuations (period-two cycles) in the left neighborhood of ᾱ.

2-Public debt has a destabilizing effect in the following cases: i) when σ ∈ (σ, σ) and 1cG ∈ (0, 1cG ). Indeed, the steady state is saddle-point stable with damped oscillations when α ∈ (ᾱ, α) while in the economy without debt it is saddle-point stable with monotone convergence;

ii) when σ ∈ (σ, σ) and 1cG ∈ (0, ¯ 1cG ). Indeed, the steady state is totally unstable with oscillations when α ∈ (ᾱ, α) and there exist saddle-point stable (locally unstable) period-two cycles in a right (left) neighborhood of ᾱ, while in the economy without debt it is saddle-point stable with monotone convergence (when 1cG ∈ (0, 1cG )) or damped oscillations (when 1cG ∈ ( 1cG , ¯ 1cG ));

iii) when 1cG ∈ (0, 1cG ) and either σ ∈ (σ, σ), or σ ∈ (0, σ) and 1cc ∈ (0, ¯ 1cc ). Indeed, the steady state is totally unstable with oscillations when α ∈ (ᾱ, α) and there exist saddle-point stable (locally unstable) periodtwo cycles in a right (left) neighborhood of ᾱ, while in the economy without debt it is saddle-point stable with damped oscillations.

Proof : See Appendix 6.5

Case 1-i) in Proposition 4 covers the configuration in which the capital income monotonicity (i.e. inequality [START_REF] Klump | Factor substitution and factor-augmenting technical progress in the United States: a normalized supply-side system approach[END_REF]) holds. This result is particularly interesting in the case where 1cG > 1cG in which there exist damped or persistent fluctuations in the economy without debt. There is indeed a level of public debt ᾱ > 0 above which the economy does not fluctuate anymore and converges monotonically toward its steady state. Public debt has here a strong stabilizing effect. In case 1-ii) with a large public spending externality, i.e. 1cG > ¯ 1cG , public debt also has a stabilizing effect by ruling out persistent fluctuations.

Cases 2-i) and ii) on the contrary imply that when the capital income monotonicity is not satisfied and the public spending externality is not too large, a large enough level of public debt with respect to GDP may destabilize the economy by generating endogenous fluctuations while the economy without debt is characterized by monotone convergence towards the steady state.

Public debt has now a strong destabilizing effect.

In case 2-iii), public debt still has a destabilizing effect but which is less radical as it amplifies fluctuations by generating persistent cycles while the economy without debt is characterized by damped fluctuations.

To summarize, Propositions 3 and 4 show that public debt has a stabilizing effect for large values of σ and 1cG , while it has a destabilizing effect for low values of σ and 1cG . We then need to provide some economic intuitions for these results. But before that we can derive some conclusions on the consequences of the stabilization effect of public debt on inequalities.

The impact on inequalities

As initially shown by [START_REF] Becker | On the long-run steady state in a simple dynamic model of equilibrium with heterogeneous households[END_REF], inequalities occur in Ramsey models as the most patient holds all the capital and thus receives capital and labor incomes, while all the other agents only receive labor income. A possible measure of inequalities can then be provided by the ratio of patient over impatient agents' incomes which is proportional to the following expression

I(k) = R(k) [k + αf (k)] Ω(k)
Straightforward computations show that

I (k) ≷ 0 ⇔ σ ≷ 1+ αθ β 1 (1-τ ) 1+ αθ β 1 s(1-τ ) [θ(1-s)+s] ≡ σ I (16)
and we derive the following result:

Lemma 2. There exist β 1 ∈ (0, 1) and δ ∈ (0, 1) such that when δ ∈ (0, δ)

and β 1 ∈ (β 1 , 1), then σ I > 1 for any α ∈ [0, α).

Proof : See Appendix 6.6

Depending on whether σ is larger or lower than σ I , inequalities increase or decrease along an optimal growth path where capital is monotonically growing. Considering this result together with Propositions 3 and 4, we can then derive some conclusions on the impact of public debt on inequalities when the proportion α is used as a policy instrument to stabilize the economy, i.e. that leads to a saddle-point steady state with monotone convergence.

Corollary 2. Under Assumptions 1-2, let δ ∈ (0, δ) and β 1 ∈ (β 1 , 1) with β 1 and δ as given by Lemma 2. Consider also the bounds σ > σ > σ and 1cG as given by Proposition 4. Then there exists α I ∈ (0, α) such that the following results hold:

1-When σ > σ I , the steady state is saddle-point stable with monotone convergence and inequalities are pro-cyclical if α ∈ (α I , α). A large enough public debt stabilizes but increases inequalities.

2-When σ ∈ (σ, σ I ), the steady state is saddle-point stable with monotone convergence and inequalities are counter-cyclical if α ∈ (α I , α). A large enough public debt stabilizes and decreases inequalities.

3-When σ ∈ (σ, σ) and 1cG ∈ (0, 1cG ), the steady state is saddle-point stable with monotone convergence and inequalities are counter-cyclical if α ∈ [0, α I ). A low enough public debt stabilizes and decreases inequalities.

Note that when σ is lower than σ, a low enough public debt leads to a saddle-point steady state but with damped fluctuations. In such a case, the impact on inequalities is less clear as inequalities, being counter-cyclical, will successively increase and decrease along the fluctuations.

Economic interpretation

To give an economic intuition of our previous results, we recall that using the

utility function u 1 (c 1t , G t ) = c 1-1cc 1t G 1cG t /(1 -1cc
), the two dynamic equations that govern the dynamics can be written:

c 1t + k t+1 + b t+1 = R(k t )(k t + b t ) + (1 -τ )w(k t )/H ≡ I t (17) c 1t+1 c 1t 1cc Gt G t+1 1cG = β 1 R(k t+1 ) ( 18
)
with b t = αf (k t ), G t = f (k t )(τ -αR(k t )) + αf (k t+1 ) and R(k t ) = 1 -δ + (1 - τ )f (k t ).
As shown by Propositions 3 and 4, when 1cG is weak enough, public debt can only have a destabilizing effect. To understand this property, assume for simplicity that there is no public spending externality ( 1cG = 0) and consider first that there is no debt (α = 0). As stressed by [START_REF] Becker | The local bifurcation of Ramsey equilibrium[END_REF], a

necessary condition to have oscillations and cycles of period 2 is that capital income monotonicity fails, i.e. capital income is decreasing in capital. More precisely, as shown in Corollary 1, we need that

σ < θ(1 -s)(1 -1/H) ≡ σ.
In this case, following an increase of k t , the income I t decreases, implying a decrease of k t+1 and explaining non-monotone dynamic paths and endogenous cycles. Note that 1cc needs to be large enough to prevent any intertemporal arbitrage.

Consider now the case with debt α > 0. Since the patient household holds two assets, capital and public debt, we argue that oscillations and instability are explained by the same mechanism than before, except that it requires now the lack of asset income monotonicity. Asset income

R(k t )(k t + b t ) = R(k t )(k t + αf (k t )) is decreasing in k t if σ < θ(1 -s)(1 + b/k)/(1 + sb/k) with b/k = αθ/[sβ 1 (1 -τ )].
It follows therefore that an increase of k t will be followed by a decrease of k t+1 if the income I t is again a decreasing function

of k t , namely if σ < θ(1-s)(1-1/H+b/k) 1+sb/k ≡ σα
with σα an increasing function of α. Assume then that when α = 0, the economy is not subject to fluctuations with σ ∈ (σ, σα ). 9 Then there necessarily exists a level of debt-output ratio above which σ < σα and endogenous fluctuations occur. This explains that public debt has a destabilizing effect since when α raises, the range of input substitutions for saddle-point stability with monotone convergence reduces.

Consider finally the case where 1cG is large enough. Proposition 4 shows that when σ is large enough (in particular if σ > σ(> σ)), public debt has on the contrary a stabilizing effect. To understand this property, assume for simplicity that 1cc = 0 and consider first that there is no debt (α = 0).

Since σ > σ, following an increase of k t , the income I t increases. There are oscillations and cycles of period 2 if k t+1 decreases. This implies an increase of R(k t+1 ), but requires also a strong increase of c 1t . When 1cG is equal to 0 or is sufficiently low, this is not compatible with the Euler equation ( 18), since the right-hand side is increasing and the left-hand side is almost constant. On the contrary, when 1cG is sufficiently large, since G t = τ f (k t ), the dominant effect on the left-hand side of ( 18) comes from

G t /G t+1 = f (k t )/f (k t+1 ). When k t
increases and k t+1 decreases, this ratio increases, meaning that the left-hand side of the Euler equation becomes compatible with the raise of R(k t+1 ).

Consider now the case with debt α > 0. We have:

dGt G = f (k) β 1 G s(τ β 1 -α) + θ(1-s)α σ dkt k + s b G dk t+1 k (19)
This means that for any α > 0, G t is procyclical (dG t /dk t+1 > 0) with respect to k +1 . On the contrary, depending on whether α is low or sufficiently large (in any case larger than τ β 1 ), G t is either procyclical or countercyclical (dG t /dk t < 0) with respect to k t . Consider a sequence of capital stock with oscillations, i.e. k t larger, k t+1 lower and k t+2 larger again. When α is sufficiently large, G t is lower, G t+1 larger, meaning that the ratio G t /G t+1 is lower (i.e. procyclical with respect to k t+1 ). By direct inspection of equation ( 18), this is not compatible with a larger R(k t+1 ), explaining that when inputs are high substitutes, public debt has a stabilizing effect since a sufficiently large α promotes saddle-point stability with monotone convergence.

Our results also show that when α is low or equal to zero, government spending is procyclical and the externalities destabilize as in the standard literature (see for instance [START_REF] Cazzavillan | Public spending, endogenous growth, and endogenous fluctuations[END_REF] and [START_REF] Zhang | Public services, increasing returns, and equilibrium dynamics[END_REF]). On the contrary, we prove that when α is large enough, government spending becomes countercyclical along the cycle and the externalities stabilize. These opposite conclusions are easily explained by the fact that in these two cases, as 1cG > 0, government spending has opposite effects on marginal utility.

Conclusion

Public debt is introduced in a Ramsey model with heterogenous agents and a public spending externality affecting utility. Public spending is financed by income tax and debt, which is assumed to be a fixed proportion of GDP.

We show that depending on the size of the externality and the value of the elasticity of capital-labor substitution, public debt can have a stabilizing or destabilizing effect by ruling out, or promoting the occurrence of endogenous fluctuations. When the public spending externality is weak, government debt can only be destabilizing, by creating damped or persistent macroeconomic fluctuations when the elasticity of capital labor substitution is low enough.

But when the public spending externality is strong enough, public debt can also be stabilizing for large values of the elasticity of capital labor substitution, driving to saddle-point stability, and thus monotone convergence, an economy experiencing damped or persistent fluctuations without debt. We also show that when the ratio of public debt over GDP is used as a policy instrument to stabilize the economy, it can also decrease or increase the degree of inequalities depending on whether the elasticity of capital-labor substitution is large or low.

6 Appendix

Proof of Proposition 1

The proof of this Proposition consists in three steps.

Step 1. For i = 1, (S2)-(S4) satisfy the optimality conditions ( 2), ( 4) and [START_REF] Becker | The local bifurcation of Ramsey equilibrium[END_REF]. Moreover, since c 1 , k 1 , b 1 and G are constant and 0 < β 1 < 1, the transver-

sality conditions lim t→+∞ β t 1 u 1c (c 1 , G) k 1 = 0 and lim t→+∞ β t 1 u 1c (c 1 , G) b 1 = 0 hold.
Step 2. For i ≥ 2, given G, consider the feasible sequence kit , bit , cit , starting from ki0 = bi0 = 0. We now compare this path with the stationary solution c i , such that k i = b i = 0 and c i = (1 -τ )w, and show that the stationary solution is optimal. +∞ t=0

β t i [u i (c i , G) -u i (c it , G)] ≥ +∞ t=0 β t i u ic ((1 -τ )w, G) [(1 -τ )w -cit ] = u ic ((1 -τ )w, G) lim T →+∞ β T i ( kiT+1 + biT+1 ) + (1/β i -1/β 1 ) T t=1 β t i ( kit + bit ) -( ki0 + bi0 )/β 1 ≥ -u ic ((1 -τ )w, G) ( ki0 + bi0 )/β 1 = 0
Step 3. Under Assumption 2, there is a unique finite and strictly positive value of k such that f (k) = θ/β 1 (1 -τ ). We further note that:

1. If R > 1 β 1 , i.e. f (k) > θ/β 1 (1 -τ )
, then it is optimal for the most patient household to increase capital. This cannot be a stationary solution because of decreasing returns in capital.

If

R < 1/β 1 < 1/β 2 ≤ . . . ≤ 1/β H , i.e. f (k) < θ/(1 -τ )β 1 , each
household decumulates to zero. Then k t tends to 0 and f (k t ) to +∞, violating stationarity.

It follows that, if α ∈ [0, α) with α = τ β 1 /(1 -β 1 ), then k 1 = k, b 1 = αf (k 1 ), G = τ (rk 1 + Hw) + (1 -R)b 1 ≡ ∆f (k 1 ) with ∆ = τ -α(1-β 1 ) β 1 > 0.

Proof of Lemma 1

Linearizing the dynamic system ( 11)-( 12) around the steady state, we obtain:

dk t+1 k = B 2 (α) dkt k -B 1 (α) dc 1t c 1 1 + 1cG ∆ 1cc αsB 1 (α) dc 1t+1 c 1 -1cG ∆ 1cc B 3 (α) -αs -(1-s)θ∆ σ 1cG + αsB 2 (α) dk t+1 k = dc 1t c 1 -1cG ∆ 1cc B 3 (α) dkt k
with B 1 (α), B 2 (α) and B 3 (α) given by [START_REF] Ilzetzki | How big (small?) are fiscal multipliers?[END_REF]. We then derive the following

linear system   dc 1t+1 c 1 dk t+1 k   = J   dc 1t c 1 dkt k   with J =    1-1cG ∆ 1cc B 3 (α)-αs- (1-s)θ∆ σ 1cG +αsB 2 (α) B 1 (α) 1+ 1cG ∆ 1cc αsB 1 (α) 1cG ∆ 1cc B 2 (α) B 3 (α)-αs- (1-s)θ∆ σ 1cG +αsB 2 (α) -B 3 (α) 1+ 1cG ∆ 1cc αsB 1 (α) -B 1 (α) B 2 (α)   
Since T and D represent respectively the trace and the determinant of J, the result follows after straightforward simplifications.

Proof of Proposition 2

We start by stating a property that applies for any α ∈ [0, α).

Lemma 6.1 For any α ∈ [0, α), there exists one root λ 1 > 1 solution of the characteristic polynomial P (λ) ≡ λ 2 -T λ + D = 0.

Proof. Straightforward computations from Proposition 1 give

P (1) = 1 -T + D = - (1-s)θB 1 (α) σ 1cc 1 + 1cG ∆ 1cc αsB 1 (α) < 0
Since lim λ→±∞ P (λ) = +∞, the result follows.

Let α = 0. We get from Proposition 1:

D(0) = 1 β 1 1 -(1-s)θ σ 1 -1 H -1cG 1cc s 1 -β 1 + (1-s)θ sH and P (-1) = 2 β 1 1 + β 1 -(1-s)θ σ 1 -1 H -1 2 1cc 1 -β 1 + (1-s)θ sH -1cG 1cc s 1 -β 1 + (1-s)θ sH Let us introduce the following bound σ ≡ (1 -s)θ 1 -1 H such that 1 -(1-s)θ σ 1 -1 H 0 ⇔ σ σ It follows that when σ > σ, D(0) 0 if and only if 1cG [1-(1-s)θ σ (1-1 H )] 1cc s(1-β 1 + (1-s)θ sH ) ≡ 1cG
On the contrary, when σ < σ, D(0) < 0 for any 1cG > 0. Let us now introduce a second bound σ ≡

(1-s)θ(1-1 H ) 1+β 1 < σ such that 1 + β 1 -(1-s)θ σ 1 -1 H 0 ⇔ σ σ
It follows that when σ > σ, P (-1) 0 if and only if

1cG 1+β 1 - (1-s)θ σ 1-1 H -1 2 1cc (1-β1+ (1-s)θ sH ) s 1cc (1-β1+ (1-s)θ sH ) ≡ ¯ 1cG
with ¯ 1cG > 1cG . Finally, when σ ∈ (0, σ), we can define the following bound

¯ 1cc ≡ (1-s)θ(1-β 1 + (1-s)θ sH ) 2(1+β 1 )(σ-σ)
> 0 and we conclude that when 1cc < ¯ 1cc , P (-1) 0 if and only if 1cG ¯ 1cG .

Recall then that P (1) > 0 and lim λ→±∞ P (λ) = +∞ for any α ∈ [0, α). On this basis, we conclude from Lemma 6.1 that the following results hold: i) if σ > σ, then -the second root of the characteristic polynomial satisfies λ 2 ∈ (0, 1) when 1cG ∈ [0, 1cG ) and the steady state is saddle-point stable with monotone convergence;

-the second root of the characteristic polynomial satisfies λ 2 ∈ (-1, 0) when 1cG ∈ ( 1cG , ¯ 1cG ), and the steady state is saddle-point stable with damped oscillations;

-the second root of the characteristic polynomial satisfies λ 2 < -1 when 1cG > ¯ 1cG and the steady state is totally unstable. But when 1cG = ¯ 1cG , P (-1) = 0 and a flip bifurcation occurs so that saddle-point stable (locally unstable) period-two cycles occur in a right (left) neighborhood of ¯ 1cG .

ii) if σ ∈ (σ, σ), the same results as in i) are satisfied with 1cG = 0.

iii) if σ ∈ (0, σ), the same results as in ii) are satisfied provided 1cc < ¯ 1cc .

On the contrary, when 1cc > ¯ 1cc , both characteristic roots are outside the unit circle and the steady state is totally unstable for any 1cG ≥ 0. Note also that if 1cG = 0, ¯ 1cc becomes a flip bifurcation value.

Proof of Proposition 3

If 1cG = 0, we easily derive from (13) that when α is varied over the interval [0, α), D and T evolve along the following line

D = 1-s(1-1 H ) 1-s(1-1 H )-1-s 1cc (1-β1-θ H ) T - σβ 1[ 1-s(1-1 H )]+ σ(1-s) 1cc (1-β1-θ H )+ θ(1-s) H 1+ θ(1-s) 1cc s 1-s(1-1 H )-1-s 1cc (1-β1-θ H ) ≡ ST -C (20)
We then need to compute the starting and end points of the line. We get from Lemma 1:

D(0) = σ-θ(1-s)(1-1 H ) σβ 1 T (0) = σ(1+β 1 )-θ(1-s)(1-1 H ) σβ 1 + θ(1-s)[1-β 1 + θ(1-s) sH ] σβ 1 1cc D(α) = σ 1-τ + τ θ 1-β 1 -θ(1-s)(1-τ ) 1-1 H + τ θ (1-β 1 )s(1-τ ) σβ 1 1-τ + τ θ 1-β 1 T (α) = σ(1+β 1 ) 1-τ + τ θ 1-β 1 -θ(1-s)(1-τ ) 1-1 H + τ θ (1-β 1 )s(1-τ ) σβ 1 1-τ + τ θ 1-β 1 + θ(1-s)(1-τ ) (1-β 1 ) 1+ τ θ (1-β 1 s(1-τ + θ(1-s) sH σβ 1 1cc 1-τ + τ θ 1-β 1
As shown previously, we have D(0) 0 if and only if σ σ and P (-1)| α=0 > 0 if σ ≥ σ or σ < σ and 1cc < ¯ 1cc , while P (-1)| α=0 < 0 if σ < σ and 1cc > ¯ 1cc .

We also derive that D(α) 0 if and only if ≡ ˜ 1cc while P (-1)| α= α < 0 if σ < σ and 1cc > ˜ 1cc . We also easily derive that there exists β 1 ∈ (0, 1) such that when β 1 ∈ (β 1 , 1), σ > σ, so that we have the following ranking: σ > σ > σ > σ. Moreover, when σ < σ we have ¯ 1cc > ˜ 1cc .

σ θ(1-s)(1-τ ) 1-1 H + τ θ (1-β 1 )s(1-τ ) 1-τ + τ θ 1-β 1 ≡ σ with σ > σ, and P (-1)| α= α > 0 if σ ≥ θ(1-s)(1-τ ) 1-1 H + τ θ (1-β 1 )s(1-τ ) ( 1+β
Recall finally from Lemma 6.1 that P (1) < 0 and lim λ→±∞ P (λ) = +∞ for any α ∈ [0, α). We then conclude from all this:

-If σ > σ then D(0) > 0, D(α) > 0, P (-1)| α=0 > 0 and P (-1)| α= α > 0. It follows that the steady state is saddle-point stable with monotone convergence for any α ∈ [0, α).

-If σ ∈ (σ, σ) then D(0) > 0, D(α) < 0, P (-1)| α=0 > 0 and P (-1)| α= α > 0. It follows that there exists α ∈ (0, 1) such that the steady state is saddlepoint stable with monotone convergence when α ∈ [0, α) and saddle-point stable with damped fluctuations when α ∈ (α, α).

-If σ ∈ (σ, σ) then D(0) > 0, D(α) < 0, P (-1)| α=0 > 0 and P (-1)| α= α 0 if and only if 1cc ˜ 1cc . Therefore, when 1cc ≤ ˜ 1cc we get the same conclusion as in the previous case, but when 1cc > ˜ 1cc , there exist α > ᾱ > α > 0 such that the steady state is saddle-point stable with monotone convergence when α ∈ [0, α), saddle-point stable with damped fluctuations when α ∈ (α, ᾱ), undergoes a flip bifurcation when α = ᾱ and becomes locally unstable with oscillations when α ∈ (ᾱ, α). Moreover, there exist saddle-point stable (locally unstable) period-two cycles in a right (left) neighborhood of ᾱ.

-If σ ∈ (σ, σ) then D(0) < 0, D(α) < 0, P (-1)| α=0 > 0 and P (-1)| α= α 0 if and only if 1cc ˜ 1cc . Therefore, when 1cc ≤ ˜ 1cc the steady state is saddlepoint stable with damped fluctuations for any α ∈ [0, α), but when 1cc > ˜ 1cc , there exists ᾱ ∈ (0, α) such that the steady state is saddle-point stable with damped fluctuations when α ∈ [0, ᾱ), undergoes a flip bifurcation when α = ᾱ and becomes locally unstable with oscillations when α ∈ (ᾱ, α). Moreover, there exist saddle-point stable (locally unstable) period-two cycles in a right (left) neighborhood of ᾱ.

-If σ ∈ (0, σ) then D(0) < 0, D(α) < 0, P (-1)| α=0 0 if and only if 1cc ¯ 1cc and P (-1)| α= α 0 if and only if 1cc ˜ 1cc , with ¯ 1cc > ˜ 1cc .

Therefore, when 1cc ≤ ˜ 1cc the steady state is saddle-point stable with damped fluctuations for any α ∈ [0, α) and when 1cc > ¯ 1cc the steady state is locally unstable with fluctuations for any α ∈ [0, α). On the contrary, when 1cc ∈ (˜ 1cc , ¯ 1cc ), there exists ᾱ ∈ (0, α) such that the steady state is saddle-point stable with damped fluctuations when α ∈ [0, ᾱ), undergoes a flip bifurcation when α = ᾱ and becomes locally unstable with oscillations when α ∈ (ᾱ, α).

Moreover, there exist saddle-point stable (locally unstable) period-two cycles in a right (left) neighborhood of ᾱ.

The results follow.

Proof of Proposition 4

Let α ∈ (0, α). In the limit case α = α, we get ∆ = 0 and thus from Lemma 1: Assume that δ < s < 1/2 and β 1 > 1/(1 + s -δ). It follows that σ CIM > σ > σ. Obvious computations also show that σ, σ > σ. We then conclude from Lemma 6.1 and Proposition 2 that there exists ᾱ ∈ [0, α) such that the following results hold: i) if σ > σ, for any 1cG > 0 the steady state is saddle-point stable with monotone convergence when α ∈ (ᾱ, α).

D(α) = 1 -(1-s)θ
ii) if σ ∈ (σ, σ), for any 1cG > 0 the steady state is saddle-point stable with damped oscillations when α ∈ (ᾱ, α).

iii) if σ ∈ (0, σ), for any 1cG > 0 the steady state is totally unstable with oscillations when α ∈ (ᾱ, α).

The results follow considering Proposition 2 and the fact that 0 < σ < σ < σ < σ < σ CIM .

Proof of Lemma 2

Let us consider the expression of the bound σ I as given by [START_REF] León-Ledesma | Identifying the elasticity of substitution with biased technical change[END_REF]. We get σ I > 1 if and only if g(α) ≡ 1 -θ -αθ 2 β 1 (1-τ )s > 0

  and the no-arbitrage condition (1 -τ )r t = 1 -δ + (1 -τ )r t , the equilibrium on the labor market being satisfied sincel t = H.The existence of the intertemporal equilibrium is an issue that we do not address in this paper. The interested reader can refer to Becker et al. (1991), Bosi and Seegmuller (2010) or Becker et al. (2013). In the next section, we show the existence of a steady state. Since we focus on local dynamics around such an equilibrium, we consider that, by continuity, an intertemporal equilibrium exists in a neighborhood of the steady state.

σβ 1 s 1 s

 11 and P (-1) = 2 2 -(1-s)θ σβ It follows that D(α) < 0 if and only if σ < (1 -s)θ/(β 1 s) ≡ σ, and P (-1) < 0 if and only if σ < (1 -s)θ/(2β 1 s) ≡ σ.

  Recent papers have explored the empirical value of the elasticity of capitallabor substitution and questioned the empirical relevance of the Cobb-Douglas specification which is widely used in growth theory. They find that capital and labor have an elasticity of substitution significantly different than unity.

	et al. (2013) have recently exhibited the existence of endogenous fluctuations
	through the occurrence of local indeterminacy in a similar Ramsey model with
	government spending financed by linear income taxes but without public debt
	and augmented to include endogenous labor.

, the existence of endogenous fluctuations in a standard Ramsey model with heterogeneous agents and borrowing constraints can be obtained only if the capital income monotonicity assumption is not satisfied. Actually, capital income monotonicity holds if f (k)k is an increasing function of k. At the steady state, it can be easily shown that the capital income monotonicity holds if

σ > 1 -s ≡ σ CIM

(15)

However, empirical evidences for both gross substitutability (elasticity above one) and gross complementarity (elasticity below one) of capital and labor are obtained in the literature. For instance, Duffy and Papageorgiou (2000) report robust estimates that are contained in [1.24, 3.24] and Krusell et al. (2007) find an elasticity of substitution between unskilled labor and equipment of 1.67. On the contrary, Chirinko (2008), Klump et al. (2007) and León-Ledesma et al. (2010) provide robust estimates in the range [0.4, 0.6]. When s = 0.3, we get

σ CIM = 0.7 a value which is precisely in between all the available empirical estimates. We then need to study both cases σ < σ CIM and σ > σ CIM .

We start by considering an economy without debt. Our aim is then to show that, since public spending externalities affect utility of consumption, endogenous business cycles can occur in the Ramsey model with heterogeneous agents for any value of the elasticity of capital-labor substitution, and in particular even in the case where capital income monotonicity holds. Nishimura

See also[START_REF] Sorger | On the structure of Ramsey equilibrium: cycles, indeterminacy, and sunspots[END_REF] for the existence of more complex dynamics.

In Nishimura et al. (2013), the existence of endogenous fluctuations through the occurrence of local indeterminacy is analyzed in a Ramsey model with government spending, financed from a constant income tax only, and endogenous labor.

See alsoBom and Lighthart (2014) for related results in a small open economy with

We could assume different tax rates on labor income, capital income and remuneration of bonds, but this does not alter our results.

We denote u ixj (x 1 , x 2 ) = ∂u i (x 1 , x 2 )/∂x j and u ixj x h (x 1 , x 2 ) = ∂ 2 u i (x 1 , x 2 )/∂x j ∂x h .

For simplicity, we omit the arguments of the functions.

The upper bound σα is given by σα= θ(1-s)[1-1/H)s(1-τ )(1-β1)+τ θ] (1-τ )(1-β1)+τ θand occurs as the destabilizing effect of debt requires a low enough elasticity of capital-labor substitution.

* 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), and of the Japan Society for the Promotion of Science, Grant-in-Aid for Specially Promoted Research #23000001 and #(B)23330063.

We have g(0) > 0 and g (α) < 0. We then need to evaluate this function at the upper bound α = τ β 1 /(1 -β 1 ). Using the expression of θ we derive

Assuming δ = 0 we get

and this expression is strictly positive when β 1 = 1. Then there exist β 1 ∈ (0, 1) and δ ∈ (0, 1) such that when δ ∈ (0, δ) and β 1 ∈ (β 1 , 1), then σ I > 1 for any α ∈ [0, α).