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 nonstandard fixed-point theorem by Gale and Mas-Colell's (1975), 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 1 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 different 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 | Differentiated 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 adapting a proof by Florenzano (1999) based on the Gale and Mas-Colell's (1975) fixed-point theorem. 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 infinite-horizon 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 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, ∞) 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.

Assumption 1

The production function F : R 2 + → R + is C 1 , 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) → ∞ either when L > 0 and K → ∞ or when K > 0 and L → ∞. Moreover, (∂F/∂K) (0, m) > δ, where δ ∈ (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) (∞, m) = 0 and (∂F/∂L) (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 ∞ yields 0 = F (1, 0) ≥ (∂F/∂K) (∞, m) ≥ 0. The proof that (∂F/∂L) (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 and leisure time. The leisure demand of agent i at time t is denoted by λ it and that agent's labor supply is l it = 1 -λ 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 .

The aggregate, or total, initial capital endowment is positive.

Assumption 2 k i0 ≥ 0 for i = 1, . . . , m, i k i0 > 0.
Each household maximizes a lifetime utility function which is separable over time: ∞ t=0 β t i u i (c it , λ it ), where β i ∈ (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.

Assumption 4 (Inada conditions) The utility function is differentiable with

∂u i ∂c (0, λ) = ∞ if λ > 0 and ∂u i ∂λ (c, 0) = ∞ if c > 0
Household heterogeneity can arise in terms of endowments (k i0 ), discounting (β i ) and per-period utility (u i ).

In any period, the household faces a budget constraint:

p t [c it + k it+1 -(1 -δ) k it ] ≤ r t k it + w t (1 -λ 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 , λ i ) m i=1 , K, L) , where (p, r, w) ≡ ((p t ) ∞ t=0 , (r t ) ∞ t=0 , (w t ) ∞ t=0 ) ∈ R ∞ × R ∞ + × R ∞ + ; (c i , k i , λ i ) ≡ ((c it ) ∞ t=0 , (k it ) ∞ t=1 , (λ it ) ∞ t=0 ) ∈ R ∞ + × R ∞ × R ∞ + ; (K, L) ≡ ((K t ) ∞ t=0 , (L t ) ∞ t=0 ) ∈ R ∞ + × R ∞ + , with i = 1, . . . , m.
Definition 1 A Walrasian equilibrium p, r, w, ci , ki , λi m i=1 , K, L satisfies the following conditions.

(1) Price positivity: pt , rt , wt > 0 for t = 0, 1, . . . (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: ∞ t=0 β t i u i cit , λit is the value of the pro- gram: max ∞ t=0 β t i u i (c it , λ it ), under the following constraints: budget constraint : pt [c it + k it+1 -(1 -δ) k it ] ≤ rt k it + wt (1 -λ it ) borrowing constraint : k it+1 ≥ 0 leisure endowment : 0 ≤ λ 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. Labor supply is uniformly bounded by m.

Let A denote 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.

Existence of equilibrium

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 sufficiently 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 ≡ {(λ i0 , . . . , λ iT ) : 0 ≤ λ 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 -δ) B k + F (B K , B L ) < B c .
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. 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. The standard argument applies given the existence of an equilibrium for the bounded economy.

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 , λi 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 λit > 0, (2) for any i and any t, cit > 0 and λit > 0, (3) for any i and any t,

∂u i ∂c cit , λit ≥ (1 -δ + rt+1 ) β i ∂u i ∂c cit+1 , λit+1
and, if kit+1 > 0, then

∂u i ∂c cit , λit = (1 -δ + rt+1 ) β i ∂u i ∂c cit+1 , λit+1 , (4)
for any i and any t,

wt ∂u i ∂c cit , λit ≤ ∂u i ∂λ cit , λit (1)
and, if λit < 1, then

wt ∂u i ∂c cit , λit = ∂u i ∂λ cit , λit (5)
Finally, Kt , Lt > 0.

Proof. See Appendix 3. Now define:

qt+1 ≡ max i β i (∂u i /∂c) cit+1 , λit+1 (∂u i /∂c) cit , λit = 1 1 -δ + 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 s=1 qs for t > 0 (3)
Clearly, Qt = t s=1 (1 -δ + 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 -δ + rt+1 ) (4)
and, by induction, 1 = Q0 = QT (1 -δ) T + T t=1 Qt rt (1 -δ) 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 -δ unit of capital and r1 unit of consumption good as its dividend. At period 2, 1 -δ unit of capital will give back (1 -δ)

2 unit of capital and (1 -δ) r2 as its dividend. This leads to the following definition of the Fundamental Value of capital:

FV ≡ ∞ t=1 Qt (1 -δ) 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 efficient 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 →∞ QT (1 -δ) T < 1. More formally: Definition 2 The economy is said to experience a bubble if lim T →∞ QT (1 -δ) T > 0
Otherwise (lim T →∞ QT (1 -δ) 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, ρ > 0 and a strictly increasing sequence (t i ) ∞ i=1 such that rti ≥ ρ for i = 1, 2, . . . For T > t n , we get

QT (1 -δ) T = T s=1 1 -δ 1 -δ + rs ≤ n i=1 1 -δ 1 -δ + rti ≤ 1 -δ 1 -δ+ρ n and 0 ≤ lim sup T →∞ QT (1 -δ) T ≤ lim n→∞ 1 -δ 1 -δ+ρ 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 λit + Qt kit+1 = Qt wt + Qt (1 -δ + 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 t=0 Qt cit + wt λit + QT kiT+1 = T t=0 Qt wt + Q0 (1 -δ + 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 T t=0 Qt wt < ∞, then there is no bubble. Indeed, if this property holds and if ( wt ) ∈ int l ∞ + , then T t=0 Qt < ∞ which implies lim t Qt = 0 and we have no bubble. However, here, we do not know, without additional assumptions, that ( wt ) ∈ int l ∞ + and T t=0 Qt wt < ∞.5 Nevertheless, no bubble holds in our economy.

Conclusion

We have analyzed the existence of the intertemporal equilibrium and the occurrence 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. For this reason, nonstandard fixed-point arguments are needed to prove the existence of a Ramsey equilibrium. Assuming borrowing constraints are the financial market imperfection in our model economy, we adopt a three-pronged 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 by-product of our proof we demonstrate that bubbles cannot exist despite the presence of borrowing constraints because equilibrium interest-rental rates are positive.

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 ≡ △ T +1 with the simplex △ ≡ {(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 -δ) k it ] ≤ r t k it + w t (1 -λ it ) for t = 0, . . . , T with k iT +1 = 0.
Consider the budget set:

C T i (p, r, w) ≡    (c i , k i , λ i ) ∈ X i × Y i × Z i : p t [c it + k it+1 -(1 -δ) k it ] ≤ r t k it + w t (1 -λ it ) t = 0, . . . , T    and its interior B T i (p, r, w) ≡    (c i , k i , λ i ) ∈ X i × Y i × Z i : p t [c it + k it+1 -(1 -δ) k it ] < r t k it + w t (1 -λ it ) t = 0, . . . , T    We denote by BT i (p, r, w) the closure of B T i (p, r, w). It is obvious that, when B T i (p, r, w) = ∅, 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.

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.

Proof. Take k i1 > 0 and

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

Observe that when δ = 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

ε > 0 satisfy m (1 -δ) (B k + ε)+F (B K , B L )+mε < B c and mB k +mε < B K .
We define:

C T ε i (p, r, w) ≡    (c i , k i , λ i ) ∈ X i × Y i × Z i : p t [c it + k it+1 -(1 -δ) k it ] ≤ p t ε + p t (1 -δ) ε + r t (k it + ε) + w t (1 -λ it ) t = 0, . . . , T    B T ε i (p, r, w) ≡    (c i , k i , λ i ) ∈ X i × Y i × Z i : p t [c it + k it+1 -(1 -δ) k it ] < p t ε + p t (1 -δ) ε + r t (k it + ε) + w t (1 -λ it ) t = 0, . . . , T   
Remark 8 ε represents a perturbation of the fundamental economy. In the εeconomy, the firm uses ε as an additional input. ε 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 -δ) (ε + 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). Proof. Take λ it = η < 1, k it+1 = 0, c it = 0. Then, p t ε + p t (1 -δ) ε + r t (ε + k it ) + w t (1 -λ it ) > 0 for any (p t , r t , w t ) ∈ △ and, hence,

α ≡ min (p,r,w)∈△ [pε + p (1 -δ) ε + rε + w (1 -η)] > 0 We have p t [c it + k it+1 -(1 -δ) k it ] = -p t (1 -δ) k it ≤ 0 < α ≤ p t ε + p t (1 -δ) ε + r t (ε + k it ) + w t (1 -λ it )
for any t. So, B T ε i (p, r, w) is non empty. 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 (ε) , r (ε) , w (ε) , ci (ε) , ki (ε) , λi (ε) m i=1 , K (ε) , L (ε)
in the finite-horizon bounded economy E T which satisfies:

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

goods :

m i=1 cit (ε) + kit+1 (ε) -(1 -δ) kit (ε) = F Kt (ε) , Lt (ε) + mε + m (1 -δ) ε capital : Kt (ε) = m i=1 kit (ε) + mε labor : Lt (ε) = m i=1 lit (ε)
for t = 0, . . . , T , where lit (ε) = 1 -λit (ε) denotes the individual labor supply.

(3) Optimal production plans:

pt (ε)F Kt (ε), Lt (ε) -rt (ε) Kt (ε) -wt Lt (ε) is the value of the program: max [p t (ε)F (K t , L t ) -rt (ε)K t -wt (ε)L t ], under the constraints K ∈ Y and L ∈ Z for t = 0, . . . , T . Moreover, pt (ε)F Kt (ε), Lt (ε) -rt (ε) Kt (ε) -wt Lt (ε) = 0 (4) Optimal consumption plans: ∞ t=0 β t i u i cit (ε), λit (ε) is the value of the program: max ∞ t=0 β t i u i (c it , λ it )
, under the following constraints:

pt (ε) (c it + k it+1 ) ≤ pt (ε) ε + [p t (ε) (1 -δ) + rt (ε)] (k it + ε) + wt (ε) (1 -λ it ) ci ∈ X i , ki ∈ Y i , λit ∈ [0, 1], k i0 ≥ 0 given
for t = 0, . . . , T .

Proof. In the spirit of Florenzano (1999), we introduce the reaction correspondences ϕ i (p, r, w, (c h , k h , λ h ) m h=1 , K, L), i = 0, . . . , m + 1 defined on

P × [× m h=1 (X h × Y h × Z h )] × Y × Z
, 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):

ϕ 0 (p, r, w, (c h , k h , λ h ) m h=1 , K, L) ≡        (p, r, w) ∈ P : T t=0 (p t -p t ) ( i [c it + k it+1 -(1 -δ) k it ] -mε -m (1 -δ) ε -F (K t , L t )) + T t=0 (r t -r t ) (K t -mε -m i=1 k it ) + T t=0 ( wt -w t ) (L t -m + m i=1 λ it ) > 0        (5)
Agents i = 1, . . . , m (consumers-workers):

ϕ i (p, r, w, (c h , k h , λ h ) m h=1 , K, L) ≡ B T ε i (p, r, w) if (c i , k i , λ i ) / ∈ C T ε i (p, r, w) B T ε i (p, r, w) ∩ [P i (c i , λ i ) × Y i ] if (c i , k i , λ i ) ∈ C T ε i (p, r, w)
where P i is the ith agent's set of strictly preferred allocations:

P i (c i , λ i ) ≡ ci , λi : T t=0 β t i u i cit , λit > T t=0 β t i u i (c it , λ it ) . Agent i = m + 1 (the firm): ϕ m+1 (p, r, w, (c h , k h , λ h ) m h=1 , K, L) ≡        K, L ∈ Y × Z : T t=0 p t F Kt , Lt -r t Kt -w t Lt > T t=0 [p t F (K t , L t ) -r t K t -w t L t ]        (6)
We observe that ϕ i : Φ → 2 Φi where

Φ ≡ Φ 0 × . . . × Φ m+1 Φ 0 ≡ P Φ i ≡ X i × Y i × Z i , i = 1, . . . , m Φ m+1 ≡ Y × Z and 2 Φ i denotes the set of subsets of Φ i .
ϕ i is a lower semicontinuous convex-valued correspondence for i = 0, . . . , m+ 1.

Let us simplify the notation

v ≡ (p, r, w, (c i , k i , λ i ) m i=1 , K, L) v 0 ≡ (p, r, w) v i ≡ (c i , k i , λ i ) for i = 1, . . . , m v m+1 ≡ (K, L)
We observe the following.

(1) By definition of ϕ 0 (the inequality in (5) is strict):

(p, r, w) / ∈ ϕ 0 (v). (2) (c i , k i , λ i ) / ∈ P i (c i , λ i ) × Y i implies that (c i , k i , λ i ) / ∈ ϕ i (v) for i = 1, . . . , m.
(3) By definition of ϕ m+1 (the inequality in ( 6) is also strict):

(K, L) / ∈ ϕ m+1 (v).
Then, for i = 0, . . . , m + 1, v i / ∈ ϕ i (v). Apply Gale and Mas-Colell (1975) fixed-point theorem. There exists v ∈ Φ such that ϕ i (v) = ∅ for i = 0, . . . , m + 1, that is, there exists v ∈ Φ such that the following holds.

Focus on "agent" i = 0. For every (p, r, w) ∈ P ,

T t=0 (p t -pt ) m i=1 cit + kit+1 -(1 -δ) kit -mε -m (1 -δ) ε -F Kt , Lt + T t=0 (r t -rt ) Kt -mε - m i=1 kit + T t=0 (w t -wt ) Lt -m + m i=1 λit ≤ 0 (7)
Consider i = 1, . . . , m. ci , ki , λi ∈ C T ε i (p, r, w) and B T ε i (p, r, w) ∩ P i ci , λi × Y i = ∅ for i = 1, . . . , m. Then, for i = 1, . . . , m, (c i , k i , λ i ) ∈ C T ε i (p, r, w) = B T ε i (p, r, w) implies T t=0 β t i u i (c it , λ it ) ≤ T t=0 β t i u i cit , λit (8)
Focus on the firm i = m+1. For t = 0, . . . , T and for every (K, L) ∈ Y ×Z, we have

T t=0 [p t F (K t , L t ) -rt K t -wt L t ] ≤ T 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 (9)
for any t. In particular, the equilibrium profit is nonnegative.

pt F Kt , Lt -rt Kt -wt Lt ≥ 0 (10)
Let us show that pt > 0.

First, we have from the budget constraints:

pt i cit + kit+1 -(1 -δ) kit ≤ mp t ε + mp t (1 -δ) ε + mr t ε + rt i kit + wt i (1 -λit )
Combining with [START_REF] Huang | Asset price bubbles in Arrow-Debreu and sequential equilibrium[END_REF], we get

0 ≥ pt i cit + kit+1 -(1 -δ) kit -F Kt , Lt -m (1 -δ) ε -mε +r t Kt -mε - i kit + wt Lt - i 1 -λit (11)
Combining [START_REF] Bosi | On the Ramsey equilibrium with heterogeneous consumers and endogenous labor supply[END_REF] with [START_REF] Kocherlakota | Bubbles and constraints on debt accumulation[END_REF], we find

0 ≥ T t=0 p t i cit + kit+1 -(1 -δ) kit -F Kt , Lt -m (1 -δ) ε -mε + T t=0 r t Kt -mε - i kit + T t=0 w t Lt - i 1 -λit (12)
and, noticing that (12) holds for any (p, r, w) ∈ P ,

Kt -mε - i kit ≤ 0 (13) Lt - i 1 -λit ≤ 0 (14) i cit + kit+1 -(1 -δ) kit -F Kt , Lt -m (1 -δ) ε -mε ≤ 0 (15)
Observe that (15) implies

i cit ≤ (1 -δ) mB k + F (B K , B L ) + m (1 -δ) ε + mε < B c (16)
Suppose pt = 0. From the consumers' problem, we obtain cit = B c and λit = 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 (17)
and rt > 0, wt > 0.

From ( 13) and ( 14), we have

Kt ≤ mε + mB k < B K and Lt ≤ m < B L . Suppose pt F Kt , Lt -r t Kt -wt Lt = π > 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 = µπ > π = pt F Kt , Lt -rt Kt -wt Lt
which is a contradiction to [START_REF] Gale | An equilibrium existence theorem for a general model without ordered preferences[END_REF].

Assume rt = 0. Then, we have 0 ≥ pt F (K, L)-wt L for any (K, L) ∈ 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) = ∞, we have [p t F (K/L, 1) -wt ] > 0 when L is sufficiently close to 0, leading to a contradiction.

The proof that wt > 0 is similar.

Let us show now that

Kt -mε -

i kit = 0 Lt - i 1 -λit = 0 i cit + kit+1 -(1 -δ) kit -F Kt , Lt -m (1 -δ) ε -mε = 0
Since pt > 0 the budget constraints bind. Combining with ( 13), ( 14), ( 15) and [START_REF] Sorger | Chaotic Ramsey equilibrium[END_REF], we obtain

mp t ε + mp t (1 -δ) ε + mr t ε + rt i kit + wt i 1 -λit = pt i cit + kit+1 -(1 -δ) kit ≤ pt F Kt , Lt + m (1 -δ) ε + mε = rt Kt + wt Lt + pt m (1 -δ) ε + pt mε ≤ mp t ε + mp t (1 -δ) ε + mr t ε + rt i kit + wt i 1 -λit Hence pt i cit + kit+1 -(1 -δ) kit = pt F Kt , Lt + m (1 -δ) ε + mε and rt Kt -mε - i kit + wt Lt - i 1 -λit = 0 (18)
Since pt > 0, we have

i cit + kit+1 -(1 -δ) kit -F Kt , Lt -m (1 -δ) ε -mε = 0
Since rt > 0, wt > 0 and (18) holds, inequalities ( 13) and ( 14) become

Kt -mε - i kit = 0 and Lt - i 1 -λit = 0
The proof of Lemma 4 is now complete. Proof of Theorem 3 Proof. Keeping in mind these results, we now prove Theorem 3. We let ε converge to 0. We denote the allocations and the prices obtained in Lemma 4 by

p (ε) , r (ε) , w (ε) , ci (ε) , ki (ε) , λi (ε) m i=1 , K (ε) , L (ε) We recall that, for any t, pt (ε) + rt (ε) + wt (ε) = 1. Denote p, r, w, ci , ki , λi m i=1 , K, L ≡ lim ε→0 p (ε) , r (ε) , w (ε) , ci (ε) , ki (ε) , λi (ε) m i=1 , K (ε) , L (ε)
For any t and any (K t , L t ) ∈ 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 = i k i0 > 0 and L0 ≤ m < B L .
Let us show that w0 > 0 and rt + wt > 0, for t = 1, . . . , T . If w0 = 0 and p0 > 0 we have

0 = p0 F K0 , L0 -r0 K0 ≥ p0 F (K 0 , L 0 ) -r0 K 0 (19)
for any (K 0 , L 0 ) ∈ Y × Z. Take K 0 = ǫ > 0 and L 0 = L ∈ (0, B L ). We obtain a contradiction:

0 ≥ p0 F (ǫ, L) -r0 ǫ = [p 0 F (1, L/ǫ) -r0 ] ǫ > 0
when ǫ goes to zero since F (1, ∞) = ∞. Hence w0 = 0 implies p0 = 0 and r0 = 1 (because of the unit simplex). However, from [START_REF] Sorger | Differentiated capital and the distribution of wealth[END_REF], 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 ) ∈ Y × Z. Take K t = K ∈ (0, B K ) and L t = ǫ > 0. We obtain a contradiction 0 ≥ pt F (K, ǫ) -wt ǫ = [p t F (K/ǫ, 1) -wt ] ǫ > 0 when ǫ becomes sufficiently close to zero, since F (∞, 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 , λ i ) ∈ B T i (p, r, w), we have

pt cit + kit+1 -(1 -δ) kit < rt kit + wt 1 -λit
for any t. There exists ε > 0 such that for any ε < ε, we have for any t,

pt (ε) cit (ε) + kit+1 (ε) -(1 -δ) kit (ε) < rt (ε) kit (ε) + wt (ε) 1 -λit (ε) < pt (ε) ε + pt (ε) (1 -δ) ε + rt (ε) ε +r t (ε) kit (ε) + wt (ε) 1 -λit (ε) Therefore, T t=0 β t i u i (c it , λ it ) ≤ T t=0 β t i u i cit (ε) , λit (ε) . Let ε go to 0. Then T t=0 β t i u i (c it , λ it ) ≤ T t=0 β t i u i cit , λit . Let (c i , k i , λ i ) ∈ C T i (p, r, w). There exists a sequence (c n i , k n i , λ n i ) ∞ n=1 ⊂ B T i (p, r, w) which converges to (c i , k i , λ i ). For any n, we have T t=0 β t i u i (c n it , λ n it ) ≤ T t=0 β t i u i cit , λit
Let n go to ∞. Then T t=0 β t i u i (c it , λ it ) ≤ T t=0 β t i u i cit , λit . 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 , λi m i=1 , K, L is an equilibrium for the T + 1-horizon economy. for the product topology.

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

0 = p0 F K0 , L0 -ŵ0 L0 -r0 K0 ≥ p0 F (K, L) -ŵ0 L -r0 K for any (K, L) ∈ R 2 + . If ŵ0 = 0 and p0 > 0, then 0 ≥ p0 F (K, L) -r0 K for any (K, L) ∈ R 2
+ . 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 = 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) ∈ 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) ∈ 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 τ ≥ 1, the set

B τ i (p, r, ŵ) is non empty. Fix some τ ≥ 1. Take (c it , k it+1 , λ it ) τ t=0 ∈ B τ i (p, r, ŵ). We have pt [c it + k it+1 -(1 -δ) k it ] < rt k it + ŵt (1 -λ it )
for t = 0, . . . , τ . There exists N > τ such that, for any T ≥ N ,

pt (T ) [c it + k it+1 -(1 -δ) k it ] < rt (T ) k it + wt (T ) (1 -λ it ) for t = 0, . . . , τ . Take T ≥ N . Define cit (T ) , kit+1 (
β t i u i (c it , λ it ) = T t=0 β t i u i (c it , λit ) ≤ T t=0 β t i u i cit (T ), λit (T ) This implies τ t=0 β t i u i (c it , λ it ) ≤ lim T →∞ T t=0 β t i u i cit (T ) , λit (T ) = ∞ t=0 β t i u i ĉit , λit (20)
Now let If kit+1 > 0, then we can take also ε < 0 small enough in absolute value and let it go to zero to obtain the reverse inequality.

(c it , k it+1 , λ it ) ∞ t=0 ∈ R ∞ + × R ∞ + × [0, 1] ∞ satisfy: pt [c it + k it+1 -(1 -δ) k it ] ≤ rt k it + ŵt (1 -λ it ) for t = 0, . . . , ∞. In this case, (c it , k it+1 , λ it ) τ t=0 ∈ C τ i (p,
(4) Since λit > 0, we can choose 0 < ε < λit . Define c it = cit + wt ε and λ it = λit -ε. The budget constraint is satisfied. In addition, we have for ε ∈ 0, λit Now, if λit < 1, then we can take ε > 0 such that λit + ε < 1 and let ε go to zero to get the reverse inequality.

0 ≥ u i (c it , λ it ) -u i cit ,
(5) We have Ct + Kt+1 = F Kt , Lt + (1 -δ) 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].

If Lt = 0, then we have

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 . Since F is differentiable and concave, we have for any t rt ≥ lim a→∞ (∂F/∂K) (a, 1)

Suppose the economy has a bubble in prices. Then, from Lemma 1, rt converges to zero. But from [START_REF] Tirole | Asset bubbles and overlapping generations[END_REF], 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 -δ) Kt = Kt F 1, Lt / Kt + 1 -δ and choose ε > 0 such that F (1, ε) + 1 -δ < 1. There exists T such that for any t > T , K t+1 ≤ Kt F 1, Lt / Kt + 1 -δ < [F (1, ε) + 1 -δ] Kt . This implies Kt → 0 when t tends to infinity, and Ct → 0 too.

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

wt ∂u i ∂c cit , λit ≤ ∂u i ∂λ cit , λit (23)
It is easy to see that wt = (∂F/∂L) Kt , Lt . Since Lt / Kt converges to 0, according to [START_REF] Tirole | On the possibility of speculation under rational expectations[END_REF], we have lim t→∞ wt = lim t→∞ (∂F/∂L) 1, Lt / Kt > 0. We claim that, for any i, lim t→∞ λit = 0. Assume the contrary: lim t→∞ λjt = λ > 0 for some j. Since lim t→∞ Ct = 0, we have lim t→∞ cit = 0 for any i. Thus, Hence, for any i, lim t→∞ λit = 0 and lim t→∞ Lt = m, that is a contradiction.

( 2 )

 2 Market clearing: goods : m i=1 cit + kit+1 -(1 -δ) kit = F Kt , Lt ; t = 0, 1, . . ., where l it = 1 -λ it denotes the individual labor supply.

Theorem 3

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

Lemma 3

 3 Under Assumptions 1, 2 and 3, the set B T ε i (p, r, w) is nonempty and C T ε i (p, r, w) = BTε i (p, r, w). Moreover the correspondence B T ε i is lower semicontinuous (lsc).

8,

  Appendix 2: existence of equilibrium in an infinite-horizon economy Proof of Theorem 4 Proof. We will denote by p (T ) , r (T ) , w (T ) , ci (T ) , ki (T ) , λi (T ) m i=1 , K (T ) , L (T ) an equilibrium for the T + 1-horizon economy and p, r, ŵ, ĉi , ki , λi m i=1 ) , r (T ) , w (T ) , ci (T ) , ki (T ) , λi (T ) m i=1 , K (T ) , L (T )

  T ) , λit (T ) T t=0 by cit (T ) = c it , kit+1 (T ) = k it+1 and λit (T ) = λ it for t = 0, . . . , τ, and cit (T ) = kit+1 (T ) = λit (T ) = 0 for t = τ + 1, . . . , T . Obviously, (c it (T ), kit+1 (T ), λit (T )) T t=0 ∈ C T i (p(T), r(T), w(T)). Hence τ t=0

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 Proof. First, observe that the production function F satisfies Assumption 1 and lim b→0 + (∂F/∂L) (1, b) > 0

  cjt , λjt λjt ≤ u j cjt , λjt -u j (c jt , 0) ≤ u j cjt , λjt ≤ u j (A, 1) (23) implies a contradiction: ∞ = lim t→∞ wt ∂u j ∂c cjt , λjt ≤ lim sup t ∂u j ∂λ cjt , λjt ≤ u j (A, 1) λ < ∞

  converging to (c it , k it+1 , λ it ) τ t=0 .We then have, from (20):Then 0 ≥ u i cit -ε, λit -u i cit , λit +β i u i cit+1 + ε (1 -δ + rt+1 ) , λit+1 -u i cit+1 , λit+1

	≥	∂u i ∂c		cit -ε, λit (-ε)
		+β i	∂u i ∂c	cit+1 + ε (1 -δ + rt+1 ) , λit+1 ε (1 -δ + rt+1 )
	0 ≥ -	∂u i ∂c	cit -ε, λit
		+β i	∂u i ∂c	cit+1 + ε (1 -δ + rt+1 ) , λit+1 (1 -δ + rt+1 )
	if ε > 0 and small enough.
	Let ε go to zero. Then,
		0 ≥ -	∂u i ∂c	cit , λit + β i	∂u i ∂c	cit+1 , λit+1 (1 -δ + rt+1 )
	sequence c n it , k n it+1 , λ n it i (p, r, ŵ) τ τ t=0 n ⊂ B τ ∞	r, ŵ). There exists a
						β t i u i (c n it , λ n it ) ≤	β t i u i ĉit , λit
					t=0	t=0

  λit ≥ ∂u i ∂c (c it , λ it ) wt ε + ∂u i ∂λ (c it , λ it ) (-ε)

	0 ≥	∂u i ∂c	(c it , λ it ) wt -	∂u i ∂λ	(c it , λ it )
	Let ε go to zero. Then
			0 ≥ wt	∂u i ∂c	cit , λit -	∂u i ∂λ	cit , λit

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 (β). 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 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.

Our proof is quite general and holds even if some initial individual capital endowments are zero and the capital depreciation rate equals one.

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 ∞ + when l ∞ is endowed with the supnorm topology.

Let n go to ∞:

We have proved that ĉ, λ 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 p, r, ŵ, ĉi , ki , λi

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

9 Appendix 3: non-existence of bubbles Proof of Claim 5 Proof. (1) cit + wt λit + kit+1 = (1 -δ + rt ) kit + wt . Suppose cit > 0 and λit = 0. By Assumption 4, we can decrease cit and increase λit to have a higher utility for period t. Hence λit > 0. The converse is proved by the same argument.

(2) We first prove that cit = λit = 0 for any t is excluded. Suppose it is not true. Then

that is a contradiction. 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 λi1 > 0 (because of point ( 1)). Hence, ci0 = λi0 = 0. Define

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

β t i u i cit , λit for ε sufficiently close to 0. This leads to a contradiction. Hence ci0 > 0 and λi0 > 0. By induction, we obtain also cit > 0 and λit > 0 for any i and any t.


Effect of the microbiota of children with autism spectrum disorder(ASD) on behavior and various biological parameters in germ-free mice

Léa Roussin

Autism spectrum disorder

•Studies show a rate of heritability lower than 50%, highlighting an important impact of environnement on the disorder •It is important to characterize and better understand these environmental factors

•Notably: Prevalence of GI symptoms 4X higher in ASD patients→implication of the Gut-microbiota-brain axis embracingasd.com tacanow.org

Microbiota-gut-brain axis in ASD

•Many studies in the past decade show a difference in microbiota composition in ASD patients compared to neurotypical individuals •However, there is great variability in the results that could be in part caused by the impact of environment and alimentation on microbiota, or differences in the choice of patients and control groups: •Th1/Th2 ratio is lower in the AG group, highlighting a less inflammatory profile in those mice compared to S-AG and AG groups.

Difference in microbiota composition

One way anova

From

  Dr. Ruth Ann Luna's presentation at the 74th Annual Meeting Society of Biological Psychiatry,Chicago (May 2019) →Controls groups used: difference in microbiota composition in unrelated neurotypical controls compared to neurotypical siblings of ASD individuals. → GI symptoms: difference in microbiota composition depending on presence or absence of GI symptoms behavior in the ASD groups compared to their respective sibling groups. •Potential differences between A and AG groups DO/(DO+NDO)*100 DO=mean time interacting with displaced objets NDO=mean time interacting with non displaced objets Cognition: spatial memory •Decreased recognition index in the AG group compared to S-AG and A.→ Microbiota from children with ASD and GI symptoms can impact spatial memory in germ-

First results

Principal coordinate Analysis (PcoA):

→Microbiota from each group forms a distinct cluster in PcoA analysis, highlighting that our 4 groups form distinct microbiota populations

Unifrac plot ordination

Conclusions and perspectives

Behavior:

•Altered spacial memory in germ-free C57BL/6 mice that received microbiota from the AG group No differences between groups for the other behavioral tests.

•Another study is planned to study behavior of Germ-free mice (2022)

Other parameters: •Some differences in SFCAs profiles and T cell populations (analysis still ongoing)

• A lot left to analyse (gut inflammation/permeability, neuroinflammation, 5-HT etc).

month stay at Utrecht University for some of the experiments!

Thanks for your attention!

PhD supervisors:

Laurent Naudon Sylvie Rabot

The Amipem team:


Benjamin Pacaud

Loïc Leclercq
email: loic.leclercq@univ-lille.fr

Véronique Nardello-Rataj

Self-assembly of 1-Decanol/-Cyclodextrin/Polyoxometalate in Colloidal Structures: a new lightning to understand POM catalysis in Pickering Emulsions

).

 Face with this finding, a deeper investigation of the PAC system was undertaken by resorting to 1 H NMR, DLS, conductimetry, zetametry, TEM, and molecular dynamic experiments. It was thus shown this system can be described as core-shell structure consisting of a CD/1-decanol inclusion complexes core coated with a POM shell. These systems, based on colloidal tectonics concepts, 5 which constitute a new class of nanostructured materials, can play important roles in various catalytic processes and offer sustainable ecofriendly solutions.


Arthur Desbois
email: arthur.desbois@gmail.com

Tristan Venot

Fabrizio De

Vico Fallani

Marie-Constance Corsi
email: marie-constance.corsi@inria.fr

HappyFeat -An interactive and efficient BCI framework for clinical applications

Keywords: BCI, Signal Processing, Classification, Motor Imagery, Functional Connectivity, Brain Networks

Brain-Computer Interface (BCI) systems allow to perform actions by translating brain activity into commands. Such systems require training a classification algorithm to discriminate between mental states, using specific features from the brain signals. This step is crucial and presents specific constraints in clinical contexts.

HappyFeat is an open-source software making BCI experiments easier in such contexts: effortlessly extracting and selecting adequate features for training, in a single GUI. Novel features based on Functional Connectivity can be used, allowing graph-oriented approaches. We describe HappyFeat's mechanisms, showing its performances in typical use cases, and showcasing how to compare different types of features.

Introduction

Brain-Computer Interface (BCI) systems allow to transcribe brain signals into commands. For this purpose, classification algorithms are used to discriminate between different mental states [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF]. The field of possible applications is vast, ranging from communication to prosthesis control and poststroke rehabilitation [START_REF] Pfurtscheller | Future prospects of ERD/ERS in the context of brain-computer interface (BCI) developments[END_REF]. Multiple BCI paradigms exist, such as P300 and Steady-State Visual Evoked Potentials (SSVEP) [START_REF] Lotte | Electroencephalography (EEG)-Based Brain-Computer Interfaces[END_REF]. We choose to focus on Motor Imagery (MI), as it is prominent in prospective therapeutic usages (e.g. post-stroke rehabilitation [START_REF] Pichiorri | Braincomputer interface based motor and cognitive rehabilitation after stroke -state of the art, opportunity, and barriers: summary of the BCI Meeting[END_REF]), which aligns with our current research efforts aiming to improve the use BCI in clinical settings. In MI, the subject actively imagines a movement without actually performing it, in order to command a virtual or real device (e.g. moving an object on a screen, controlling a robotic arm). Consequently, MI offers a high level of interactivity and requires a strong active implication from the subject.

MI protocols consists of multiple phases [START_REF] Pfurtscheller | Motor imagery and direct brain-computer communication[END_REF][6] (illustrated in Figure 1): an acquisition phase of training data; an offline analysis phase usually including pre-processing, extraction of features of interest (based on e.g. power spectra, functional connectivity), feature selection and classification algorithm training; a closed-loop online BCI usage using the trained classification algorithm.

The performance of a BCI system, while dependent on internal (e.g. concentration, fatigue [START_REF] Myrden | Effects of user mental state on EEG-BCI performance[END_REF] and ease with BCI) and external factors (e.g. montage of electrodes), is strongly linked to the correct training of the classification algorithm. Therefore, the choice of adequate features that capture the user's intent is crucial.

The offline analysis phase, leading to the choice of those features, should be made as short and efficient as possible for two reasons:

• A complete MI experiment is a long and strenuous process lasting multiple hours, along which the subject's concentration and motivation can drop. Patients' conditions increase the need for a fast feature selection process.

• Too long a time between the different phases may result in low classification accuracy, and therefore poor BCI performance. Indeed, as time passes, many physical parameters may change: the impedance or localization of Electroencephalographic (EEG) sensors (leading to change in EEG signals characteristics and quality), the subject's motivation, but also their mental activity itself. These changes mean that at the time of the final online classification phase, the subject's EEG signals and brain activity might not match with the ones used by the BCI experimenter to select features and train the classifier anymore.

Therefore, there is a crucial need for providing assistance to the experimenter during the training phase, to identify the most relevant and robust features.

Software solutions already exist today to manipulate BCI systems, or to analyze acquired EEG data. For creating and manipulating complete BCI systems, one can cite OpenViBE [START_REF] Renard | OpenViBE: An Open-Source Software Platform to Design, Test and Use Brain-Computer Interfaces in Real and Virtual Environments[END_REF], which offers compatibility with a wide range of EEG hardware, practical modular tools to edit BCI systems as "scenarios", and powerful signal processing capabilities. It also allows interfacing with Virtual or Augmented Reality headsets and other virtual environments such as video games. Timeflux [START_REF] Clisson | Timeflux: an open-source framework for the acquisition and near real-time processing of signal streams[END_REF] is another software solution, allowing experimenters to acquire and process EEG signals in real-time, with a high degree of scenario and interface flexibility. Another existing solution is BCI2000 [START_REF] Schalk | BCI2000: a generalpurpose brain-computer interface (BCI) system[END_REF], which allows a lower level of customization and modularity. However, none of the aforementioned solutions provide tools to interactively identify the best features to use for training and fine-tuning a BCI system.

Tools for offline analysis of EEG signals also exist today, such as Brainstorm [START_REF] Tadel | Brainstorm: A User-Firendly Application for MEG/EEG Analysis[END_REF], EegLab [START_REF] Delorme | EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis[END_REF] or FieldTrip [START_REF] Oostenveld | FieldTrip: Open Source Software for Advanced Analysis of MEG, EEG, and Invasive Electrophysiological Data[END_REF]. While these software embark wide selections of statistical and signal processing methods and are useful for research on neurophysiological phenomena, they lack capabilities in the domain of machine-learning, and the possibility to interface with BCI in a direct workflow.

Finally, the open-source software package MNE-Python [START_REF] Gramfort | MNE software for processing MEG and EEG data[END_REF] provides extensive functionalities allowing experimenters to manipulate EEG data and analyze them with signal-processing and machinelearning methods using scikit-learn [START_REF] Pedregosa | Scikit-learn: Machine Learning in Python[END_REF], some of which HappyFeat makes use of.

Description and features

HappyFeat is a Python framework, consisting in a main application and various scripts and automation routines, allowing to facilitate the use of MI-based BCI pipelines. This is achieved by grouping all necessary manipulations and parameters in an unified graphical user interface (GUI), and making the steps of feature extraction, feature selection and classifier training as effortless and seamless as possible, so that experimenters may focus on building an efficient BCI.

The [16][17]. This approach is illustrated in Figure 2)

• HappyFeat uses OpenViBE in the background for the extraction and training parts, as a fast and efficient processing engine, taking advantage of its optimized C++ implementation of signal processing methods (notably using the Eigen1 library). The generation and manipulation of OpenViBE scenarios is entirely automated via scripts and templates, removing the inherent risk of mistakes in a time constrained environment. Feature visualization and selection use tools from MNE-Python [START_REF] Gramfort | MNE software for processing MEG and EEG data[END_REF] and scipy [START_REF] Virtanen | SciPy 1.0: fundamental algorithms for scientific computing in Python[END_REF]. The GUIs are built with PyQt2 .

• HappyFeat puts the emphasis on reproducibility, by keeping track of all manipulations (EEG sessions file lists, signal processing steps and parameters, classification attempts) and allowing to save, load and export previous work.

Two main use-cases are targeted:

• Using MI in a clinical setting (e.g. stroke rehabilitation), by greatly reducing the risks of mistakes during the offline analysis and the time needed to perform this step, quickly bridging the gap between EEG data acquisition and online BCI usage.

• Exploring new, alternative metrics for discriminating between mental states. To this aim, prototypes for prospective methods need to be validated on signal databases, before moving on to experimental conditions. HappyFeat helps bridging this gap, and provides a framework in which such methods can be tested, after implementation.

Mechanisms

HappyFeat's main GUI and mechanisms are shown in Figure 3 • HappyFeat allows the experimenter to choose between different metrics for discriminating between mental states, such as Power Spectral Density (PSD) [START_REF] Diez | A comparative study of the performance of different spectral estimation methods for classification of mental tasks[END_REF] or network estimators based on Functional Connectivity (e.g. Node Strength) [START_REF] Nolte | Identifying true brain interaction from EEG data using the imaginary part of coherency[END_REF][START_REF] Cattai | Characterization of Mental States through Node Connectivity between Brain Signals[END_REF][START_REF] Cattai | Phase/Amplitude Synchronization of Brain Signals During Motor Imagery BCI Tasks[END_REF], both of which are described in the Annex section. Experimenters can also use a mix of two different estimators, allowing to classify using both PSD and Node-strength features for example.

• Data loading: Experimenters can select different EEG recordings (either directly after acquisition during a BCI experiment, or using pre-recorded signals) from which to compute metrics and extract features. Parameters relevant to the chosen classification metric can be edited. OpenViBE scenarios are automatically created, updated and ran in the background using relevant information provided by the experimenter, without needing any additional manipulation.

• Visualization tools allow to analyze and select features of interest, accumulating statistics across selected EEG runs. Such figures allow for a comfortable and easy selection of adequate features for training the classification algorithm. Experimenters may open any number of visualization windows, allowing to compare R 2 values between MI conditions (e.g. MI vs REST ) as a channel-frequency map, as power densities for a given channel, as a topography map for a given frequency range, etc. Examples of such visualizations are given in Figures 4 and5.

• Classifier training can be done iteratively in a trial-and-error way. Using a set of EEG runs (from which features have been extracted previously in HappyFeat) selected by the experimenter, and FOIs selected in the previous step, a classification algorithm (such as Linear Discriminant Analysis (LDA) [START_REF] Sanei | Classification and Clustering of Brain Signals[END_REF]) is trained in a few seconds, and the application provides a training accuracy score.

At this point, if the training accuracy score is satisfactory, experimenters can proceed with the last step of the experiment (i.e. the closed-loop online BCI), using an OpenViBE scenario automatically updated with the trained classifier and the selected features. In the case of an insufficient score, experimenters may simply go back to the previous steps, either directly trying other features to try training the classifier again; or modify their visualizations to select other features; or extract features from other EEG signals. Going back and forth between these steps only takes a few seconds, and manipulations are limited to the strict minimum.

• A Session & Settings Management System allows to export and import "workspaces", allowing experimenters to keep track of manipulations previously done on EEG recording sessions: extraction results and corresponding parameters, training attempts and accuracy scores with the corresponding feature set, etc.

Example of metric comparison

The common metric used in MI protocols to discriminate between mental tasks is the Power Spectral Density (PSD), which is available in HappyFeat. Alternative metrics based on functional connectivity [START_REF] Cattai | Characterization of Mental States through Node Connectivity between Brain Signals[END_REF][START_REF] Cattai | Phase/Amplitude Synchronization of Brain Signals During Motor Imagery BCI Tasks[END_REF] are proposed. As the raw connectivity matrices are difficult to interpret to physically store, having a dimension of (nb.channels × nb.channels × nb.frequencies), the metrics proposed in HappyFeat are network metrics, such as the node-strength, obtained by summing the weights of all connections for each channel. The resulting matrix of node-strengths has a dimension of (nb.channels×nb.f requencies), which can be analyzed using the same tools as the PSD. Similarly to the PSD, sets of (channel, frequency) can be used as features for training the classification algorithm. HappyFeat also proposes to use a mix of different metrics (i.e. PSD and Node-Strength) to train the classifier.

To illustrate this, two subjects from Venot et al.'s BRACCIO protocol [START_REF] Venot | Mental imagery timing affects hybrid BCI control of robotic arms[END_REF] were selected (both male, aged 25, right-handed). In this protocol, subjects were asked to perform either MI of the right hand, or no MI ("Rest"), with visual feedback in the form of a moving robotic arm. 120 trials (60 per class) were used.

Table 1 gives the accuracies obtained for both subjects when training a classifier with particular features, using PSD, coherence-based node strength (NS-COH), imaginary part of coherence -based node strength (NS-iCOH) and mixing metrics.

Figures 4 and5 show the channel-frequency R 2 maps and metric comparison for Subject 1, obtained with HappyFeat's visualization tools.

A high inter-subject variability can be observed. Subject 1 shows similar performances between metrics, with a slight advantage for PSD. Subject 2 illustrates how using NS-COH, NS-iCOH and mixing them with PSD can lead to performance improvement.

Impacts

HappyFeat offers a novel complete integrated workflow, allowing to perform all steps of the offline analysis leading to feature selection in BCI setup. Two main use-cases are targeted, whose impacts are methodological and scientific on one side, and clinical and therapeutic on the other.

• On the one hand, HappyFeat's potential impact on research using BCI is noteworthy. Novel and innovative algorithms (e.g. based on graph-theory and functional connectivity [START_REF] Cattai | Characterization of Mental States through Node Connectivity between Brain Signals[END_REF][START_REF] Cattai | Phase/Amplitude Synchronization of Brain Signals During Motor Imagery BCI Tasks[END_REF]) can be validated on pre-recorded data, and compared to one another or to state of the art techniques (PSD). HappyFeat provides an efficient framework in which new methods for discriminating between mental states can be tested and benchmarked.

• On the other hand, HappyFeat helps using MI in a clinical setting (e.g. stroke rehabilitation), by greatly reducing the risks of mistakes during the offline analysis and the time needed to perform this step, quickly bridging the gap between EEG data acquisition and online BCI usage. HappyFeat's mechanisms and graphical interface have been designed to be easily used by experimenters and clinicians without strong programming skills, in order to facilitate the introduction of BCI methods in healthcare. By making therapeutic protocols using BCI feasible and realistically applicable in real-life, HappyFeat aims to facilitate the design of innovative training programs to improve neuro-rehabilitation, in order to improve patients' quality of life in the long run.

HappyFeat has been instrumental in a real-life experimental study using BCI to control a robotic arm [START_REF] Venot | Exploring strategies for multimodal BCIs in an enriched environment[END_REF][23], making the acquisition, analysis and BCI steps feasible in the same session, and allowing in a further study on recorded signals to compare training performance of PSD and Spectral Coherence. Such a comparison is illustrated in figure 4, figure 5 and table 1 with two subjects from this study.

As another example of real-life impact, in the context of the BCINET research project [START_REF]bcinet: Non-invasive decoding of brain communication patterns to ease motor restoration after stroke[END_REF], Hap-pyFeat will serve as the cornerstone software for feature analysis with Functional Connectivity. This protocol aims to evaluate training effects and recovery in stroke patients, using BCI and non-invasive stimulation techniques.

HappyFeat's workspace management system allows for reproducible research, by enabling to import or share work sessions between different users. The flexibility offered by metric comparison and the open-source nature of the project will help to develop the use of BCI in multiple research domains and applications not limited to healthcare, and as a pedagogic tools for users new to BCI.

Limitations & Perspectives

Flexibility

In order to offer a safe and risk-free workflow, we chose to build HappyFeat around the concept of fixed pipelines, trading OpenViBE's high level of flexibility in designing BCI systems for fixed, efficient pipelines with reproducible results. Nevertheless, even though HappyFeat is designed as a turnkey software solution, more experienced experimenters are free to modify scenarios run along the pipeline and templates used to generate them if the need arises, for example to fine-tune the signal processing chain in the feature extraction step, or to edit the type of feedback provided in the online BCI scenario. An in-depth guide helping to edit HappyFeat's pipelines is available in the software documentation.

Proposed algorithms and methods

At the time of writing, the only machine-learning algorithm proposed for the classification step is Linear Discriminant Analysis (LDA). Other methods shall be made available in the future, such as Support Vector Machine (SVM) or Riemannian Geometry based methods [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF].

Regarding discriminant metrics based on network and graph theory, the only method available is the node strength, calculated from the coherence-based connectivity matrix. Other network-based metrics will soon be available, such as network laterality or betweenness centrality [START_REF] Gonzalez-Astudillo | Network-based brain-computer interfaces: principles and applications[END_REF]. Moreover, HappyFeat proposes to mix different metrics, but at the time of writing only PSD + NS-COH or PSD + NS-iCOH may be chosen. A mechanism allowing to mix NS-COH + NS-iCOH or any combination of future implemented metrics will be proposed in a later version of the software.

In practice, feature selection may be realized manually or using automatic method such as Common Spatial Pattern (CSP) [START_REF] Sanei | Classification and Clustering of Brain Signals[END_REF]. However, while such methods offer simplicity and speed, they imply reducing control over the selection process, and a reduced level of interpretability, both of which are crucial when testing prospective feature such as connectivity-based metrics. The workflow proposed in HappyFeat is therefore a trade-off between speed, ease-of-use and human interpretability. It should be noted that such automatic methods could still be proposed in the interface in the future.

A time-frequency ERD/ERS analysis tool is provided in the visualization part of the GUI, but only allows to compare the averaged spectra of trial against a "baseline" defined as the EEG signal acquired right before the stimulation cue (i.e., MI trial vs. MI baseline, and REST trial vs. REST baseline). This tool will be improved in a future version to allow comparing conditions between themselves (i.e. REST vs. MI), and to be based on Morlet wavelets [START_REF] Brodu | Comparative study of band-power extraction techniques for Motor Imagery classification[END_REF].

Processing engine, BCI software dependency At the time of writing, HappyFeat is built upon the manipulation of OpenViBE scenarios, taking advantage of this software's high level of modularity and its signal processing capabilities. However, work is ongoing to show that other modular BCI software can be used as processing engines (such as Timeflux [START_REF] Clisson | Timeflux: an open-source framework for the acquisition and near real-time processing of signal streams[END_REF]), without modifying the mechanisms of HappyFeat.

Conclusions

We propose a software to facilitate usage of MI-based BCI for multiple types of experimenters: clinicians often without technical or programming background, and researchers whose focus is more targeted on exploring new features. By helping experimenters manage reproducible pipelines, and by reducing the time and effort necessary to select adequate features for classification, HappyFeat fills a gap in the BCI world between offline analysis of neurophysiological phenomena using pre-recorded signals, and live BCI applications.

HappyFeat was designed as an open-source project and its usage can be acknowledged by citing this article. HappyFeat is continuously updated with new features and regularly maintained. Suggestions of improvements, as well as further developments, can be addressed to the corresponding authors of this article. (node n i )

Figure 2: From brain activity to Functional Connectivity and Networks. Spectral coherence (or imaginary coherence) between EEG signals from electrodes n i and n j is computed from their Power Spectral Densities and Cross Spectral Density. Applying this operation to every electrode pairs yields a connectivity matrix, visualized here at frequency f . From this matrix, network estimators can be extracted. The "Node strength" of a given node is obtained by summing all connections to this node. Computing the node strength for all frequencies in the considered range yields a "Node Strength Density" which can be analyzed and manipulated similarly to the usual Power Spectral Density. Courtesy of J. Gonzalez-Astudillo [START_REF] Astudillo | Development of Network Features for Brain-Computer Interfaces[END_REF] For each EEG run:

Metric (class1) Metric (class2) Metadata

Extraction scenario

Accumulating runs:

Training scenario

Online BCI scenario Trials from selected EEG runs + Selected features Accuracy score

Annex

Installation & Requirements

HappyFeat is available as a PyPI Python package: python -m pip install happyfeat. Alternatively, the source code can be downloaded at https://github.com/Inria-NERV/happyFeat. It requires Python 3.9 and OpenViBE 3.5.0, which can be downloaded at https://openvibe.inria.fr/. Further information can be found in the official documentation (https://happyfeat.readthedocs.io/) and in the README.md file at the top level of the repository.

Classification metrics

Motor Imagery BCI is based on Event-Related Desynchronization/Synchronization (ERD/ERS) [START_REF] Pfurtscheller | Event-related EEG/MEG synchronization and desynchronization: basic principles[END_REF], which corresponds to a decrease/increase of signal power between different motor-related tasks in specific frequency bands, for EEG signals in the sensorimotor cortex. This phenomenon is observable in both motor execution and motor imagery.

The discrimination between mental states, and therefore between MI tasks, is done by using a classification algorithm, with adequately selected classification features. Here, we will describe two types of features: the first is the signal's Power Spectral Density (PSD), which is the most commonly used feature [START_REF] Hwang | EEG-Based Brain-Computer Interfaces: A Thorough Literature Survey[END_REF]; the second is Coherence-based Functional Connectivity [START_REF] Cattai | Characterization of Mental States through Node Connectivity between Brain Signals[END_REF].

Power Spectral Density

The PSD of the recorded EEG signals is widely used as a discriminant feature in MI BCIs. One of the most common ways of estimating the PSD is by using Welch's method [START_REF] Diez | A comparative study of the performance of different spectral estimation methods for classification of mental tasks[END_REF], which consists in splitting the signal into overlapping segments, applying a window function on the segments, computing the periodograms of the windowed segments (via a Discrete Fourier Transform), and finally averaging the squared magnitude of the individual periodograms.

As described in Diez et al. [START_REF] Diez | A comparative study of the performance of different spectral estimation methods for classification of mental tasks[END_REF], Burg's method is more relevant in the context of EEG signals. Notably, it allows high frequency resolution for short data inputs, which makes more sense in the context of mental tasks lasting a few seconds. It implies fitting an autoregressive (AR) model to the signal, by minimizing the forward and backward prediction errors, then computing the DFT of the AR coefficients.

Functional Connectivity

As an alternative to PSD, metrics based on Functional Connectivity (FC) have been studied in the last years. FC allows the observation of brain activity as a network of time-varying connections between areas. Algorithms and analysis from network and graph theory can be applied on the connectivity matrix.

In order to estimate the degree of interaction between electrodes, the simplest measure is the coherence, which can be seen as a generalization of correlation in the frequency domain.

With S xy (f ) is the cross-spectrum of complex signals x and y at frequency f , and S xx (f ) and S yy (f ) the spectra of x and y at frequency f , we define coherence and its variants [START_REF] Nolte | Identifying true brain interaction from EEG data using the imaginary part of coherency[END_REF][START_REF] Cattai | Characterization of Mental States through Node Connectivity between Brain Signals[END_REF][START_REF] Cattai | Phase/Amplitude Synchronization of Brain Signals During Motor Imagery BCI Tasks[END_REF] Computing one of those metrics yields a connectivity (or adjacency) matrix, with (nb.channels × nb.channels × nb.frequencies) coefficients, which can also be seen as a weighted network. The information contained in this matrix can be exploited or re-formatted in many ways: for example summing all the weights associated to one edge (or channel) yields the node strength. Other metrics can be explored, such as for example laterality, centrality, or betweenness [START_REF] Vico Fallani | Graph analysis of functional brain networks: practical issues in translational neuroscience[END_REF] [START_REF] Gonzalez-Astudillo | Network-based brain-computer interfaces: principles and applications[END_REF].

Processing speed One way of evaluating how HappyFeat can help in a BCI experiment is by measuring the processing times of each segment.

The feature extraction step is the most time consuming, and its processing time depends on the metric chosen, as computing Connectivity matrices using Autoregressive (AR) models [START_REF] Schlögl | Analyzing event-related EEG data with multivariate autoregressive parameters[END_REF] is more costly than PSDs, but also on the used parameters: the higher the AR model order, or the lower the time between two connectivity measurements, the costlier it gets. Of course, the number of channels has an important impact on processing times, linear for PSD, and quadratic for Connectivity.

Table 2 shows a summary of the measured times, using a computer with 3 GHz 4 core CPU (8 threads) and 32 GB of RAM. Signal and extraction parameters can be found in the same table. PSD estimation is performed with Burg's method [START_REF] Diez | A comparative study of the performance of different spectral estimation methods for classification of mental tasks[END_REF], Node strength is computed from Connectivity matrices using Coherence (NS-COH) [START_REF] Cattai | Characterization of Mental States through Node Connectivity between Brain Signals[END_REF].

Param

Figure 1 :

 1 Figures

Figure 3 :Figure 4 :

 34 Figure 3: HappyFeat main GUI. The interface is split in 3 parts: in the leftmost part, experimenters can perform Feature Extraction from recorded EEG signals available in the workspace. The central part is dedicated to Feature Visualization for signals that have undergone extraction, allowing to select adequate features for training. The last and rightmost part allows the experimenter to train the classifier using selected features and signal files. Along the way, certain parameters can be set, while other parts of the interface help to remind acquisition settings or last training attempts.

Figure 5 :

 5 Figure 5: Comparison of MI conditions "Rest" and "MI" in Subject 1, using NS-COH (top row) and NS-iCOH (bottom row). R 2 values represented as a frequency-sensor map (A, D), as a "brain topography" mapped on a scalp (B, E) at frequency 12Hz, and direct comparison of NS averaged over trials for sensor C3 (C) and C1 (F). Figures A and B share the same color scale for R 2 values, as do figures D and E. The black curves in C and F show the R 2 values, scaled from 0 to 1.

  as:• Spectral Coherence:COH xy (f ) = |Sxy(f )| √ Sxx(f ).Syy(f )• Imaginary part of Coherence:iCOH xy (f ) = |Im(Sxy(f ))| √ Sxx(f ).Syy(f )

software assistant for feature extraction and selection. It

  following contains a description of HappyFeat's main features and mechanisms.

	Key features
	• HappyFeat is a proposes an
	exploration-oriented workflow, where experimenters can extract, visualize and select Features
	of Interest (FOI) for training as many times as needed, in a short time, until a satisfying
	classification training accuracy is reached.
	• Every operation from signal loading and feature extraction to classifier training is handled
	from a unified, dashboard-like GUI, removing the need to use different softwares for data
	acquisition, feature analysis, classifier training and online classification, and to manage data
	formatting across the different environments.
	• Along with the commonly used Power Spectral Density (PSD), HappyFeat enables to work
	with Functional

Connectivity, allowing to use novel network-based approaches based on recent research

Table 1 : Comparison of training accuracies when using connectivity-based features.

 1

	Metric	Feature (channel;frequency) Training Accuracy (%)
	Subject 1
	PSD	C3;13	91.67
	NS-COH	C3;10	81.67
	PSD + NS-COH	C3;13 + C3;10	86.67
	NS-iCOH	C1;11	80.00
	PSD + NS-iCOH	C3;13 + C1;11	91.65
	Subject 2
	PSD	C3;11	83.35
	NS-COH	C1;12	96.65
	PSD + NS-COH	C3;11 + C1;12	93.35
	NS-iCOH	P5;11	80.00
	PSD + NS-iCOH	C3;11 + P5;11	88.35

Table 2 : HappyFeat's main operations processing times.

 2

	1	Param 2

https://eigen.tuxfamily.org/index.php?title=Main Page

https://doc.qt.io/qtforpython-5/

Acknowledgements & Funding

The research leading to these results has received funding from the program "Investissements d'avenir" ANR-10-IAIHU-06. The authors acknowledge support from European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 864729). The content is solely the responsibility of the authors and does not necessarily represent the official views of any of the funding agencies.