Stefano Bosi

Kazuo Nishimura

Alain Venditti

Keywords: Money-in-the-utility-function, two-sector economy, sector-specific externalities, indeterminacy, period-two cycles, sunspot equilibria. Journal of Economic Literature Classification Numbers: C61, E32

We consider a two-sector economy with money-in-the-utility-function and sector-specific externalities. We provide conditions on technologies leading to the existence of local indeterminacy for any value of the interest rate elasticity of money demand, provided the elasticity of intertemporal substitution in consumption is large enough. Moreover, we show that the occurrence of multiple equilibria is intimately linked with the existence of a flip bifurcation and period-two cycles.

Introduction

Endogenous business-cycle fluctuations through the existence of local indeterminacy are known to occur in monetary growth models in which the central bank is assumed to follow an exogenous money growth rule. Particular examples are provided by Fukuda [START_REF] Fukuda | The Emergence of Equilibrium Cycles in a Monetary Economy with a Separable Utility Function[END_REF], Matsuyama [START_REF] Matsuyama | Sunspot Equilibria (Rational Bubbles) in a Model of Money-in-the-Utility-Function[END_REF] and Woodford [START_REF] Woodford | Monetary Policy and Price Level Determinacy in a Cash-in-Advance Economy[END_REF] within optimal growth models with money-in-the-utility-function (MIUF). Local indeterminacy means that there exists a continuum of equilibria starting from the same initial condition, all of which converging to the same attractor such as a steady state. It is now well-known that local indeterminacy is a sufficient condition for the existence of endogenous fluctuations generated by purely extrinsic shocks on beliefs which do not affect the fundamentals, i.e. preferences and technologies. 2 Indeed, in presence of local indeterminacy, randomizing beliefs over a continuum of equilibrium paths generates new stochastic equilibria derived from shocks on expectations. These expectations-driven fluctuations are no longer based on exogenous shocks on fundamentals and represent a convincing rationale to explain macroeconomic volatility through alternative propagation mechanisms.

Formally, local indeterminacy arises when the stable manifold has dimension greater than the number of predetermined variables (i.e., determined the period before and currently taken as given). Within traditional MIUF models, the competitive equilibrium is described by a system of implicit difference equations of order 2 in the capital stock k, namely in k t , k t+1 and k t+2 , and order 1 in real money balances m, namely in m t and m t+1 . Typically, at any time t ≥ 0, we have k t , the current capital stock, as the predetermined variable, and m t and k t+1 , respectively, the current real balances and the future capital stock, as the two forward variables (i.e., depending on future decisions). 3As a result, as soon as the stable manifold has dimension greater or equal to 2, there exists a continuum of converging equilibrium paths, and the steady state is locally indeterminate. In this case, two kinds of local indeterminacy may arise:

-A local indeterminacy of order 1 when the stable manifold is twodimensional. In a given neighborhood of the steady state, the intersection of the plane defined by the forward variables m t and k t+1 with the stable manifold determines a one-dimensional manifold which is the set of multiple starting points for as many equilibrium paths converging to the steady state.

-A local indeterminacy of order 2 when the stable manifold is threedimensional. In this case, in a given neighborhood of the steady state, the intersection of the plane defined by m t and k t+1 with the stable manifold determines a two-dimensional local manifold which is the set of multiple starting points for as many converging equilibrium paths.

Most of the results available in the literature can be derived from MIUF growth models. For instance, Feenstra [START_REF] Feenstra | Functional Equivalence Between Liquidity Costs and the Utility of Money[END_REF] has shown that a cash-in-advance economy can be viewed as a particular case of the MIUF approach, provided that the real balances in the utility function follow a cash-in-advance (CIA) timing. 4 In this case, the preferences are assumed to depend on the cash the household has before entering the goods market, i.e. the stock of money accumulated in the previous period and available before buying new goods. In this context, there is no room for local indeterminacy of order 2, but indeterminacy of order 1 arises provided the interest elasticity of real money demand is low enough, a condition which is hard to justify empirically.

In this paper we consider a two-sector economy with MIUF and CIA timing but we assume also some sector-specific externalities in production. We provide less stringent conditions on the preferences for the occurrence of multiple equilibria. On the one hand, we show that local indeterminacy of order 2 requires a large enough elasticity of intertemporal substitution in consumption, a low enough interest elasticity of real money demand, and a consumption good sector capital intensive at the private level but labor intensive at the social level. On the other hand, when the elasticity of intertemporal substitution in consumption is decreased, or when the interest elasticity of real money demand is increased, we prove the occurrence of local indeterminacy of order 1 through a flip bifurcation.

The paper is organized as follows: The next section sets up the basic model, defines the intertemporal equilibrium and proves the existence of a unique steady state. In section 3, we provide our main results on local indeterminacy. Section 4 contains concluding comments and all the proofs are gathered into a final appendix.

The model 2.1 Technology

There are two goods: a consumption good, c, and a capital good, k. Each good is assumed to be produced with a Cobb-Douglas technology which contains some sector specific externalities. We denote by c and y the outputs of sectors c and k, and by e c and e y the corresponding external effects. The private production functions are thus defined as:

c = e c K α 1 c L α 2 c , y = e y K β 1 y L β 2 y (1)
The externalities e c and e y depend on Ki and Li , which denote the average use of capital and labor in sector i = c, y, and are equal to

e c = Ka 1 c La 2 c , e y = Kb 1 y Lb 2 y ( 2
)
with a i , b i ≥ 0, i = 1, 2. We assume that these economy-wide averages are taken as given by individual firms. At the equilibrium, all firms of sector i = c, y being identical, we have Ki = K i and Li = L i . Denoting αi = α i +a i , βi = β i + b i , the social production functions are defined as

c = K α1 c L α2 c , y = K β1 y L β2 y (3)
We assume in the following that α1 + α2 = 1 and β1 + β2 = 1 so that the returns to scale are constant at the social level, and decreasing at the private level.

Total labor is given by L c +L y = , and the total stock of capital is given by K c + K y = k. We assume complete depreciation of capital in one period so that the capital accumulation equation is y t = k t+1 . 5 Optimal factor allocations across sectors are obtained by solving the following program: max Kc,t,Lc,t,Ky,t,Ly,t

e ct K α 1 ct L α 2 ct s.t. k t+1 = e yt K β 1 yt L β 2 yt k t = K ct + K yt e ct , e yt given
Denote by p t , w t and r t respectively the price of the capital good, the wage rate of labor and the rental rate of the capital good at time t ≥ 0, all in terms of the price of the consumption good. The Lagrangian is:

L t = e ct K α 1 ct L α 2 ct + w t ( t -L ct -L yt ) + r t (k t -K ct -K yt ) + p t e yt K β 1 yt L β 2 yt -k t+1 (4)
For any given (k t , k t+1 ), solving the first order conditions gives input demand functions Kc = K c (k t , k t+1 , e ct , e yt ), Lc = L c (k t , k t+1 , e ct , e yt ), Ky = K y (k t , k t+1 , e ct , e yt ), Ly = L y (k t , k t+1 , e ct , e yt ). We then define the production frontier as

c t = T (k t , k t+1 , e ct , e yt ) = e ct Kα 1 ct Lα 2 ct
Using the envelope theorem we derive:

r t = T 1 (k t , k t+1 , e ct , e yt ), p t = -T 2 (k t , k t+1 , e ct , e yt ) (5)
where T 1 = ∂T /∂k t and T 2 = ∂T /∂k t+1 . The wage rate is then obtained as

w t = T (k t , k t+1 , e ct , e yt ) -r t k t + p t y t .

Preferences

The economy is populated by a large number of identical infinitely-lived agents. We assume without loss of generality that the total population is constant. In each period t ≥ 0, each consumer receives income from capital and labor, and monetary lump-sum transfers from the government. His expenditures concern the consumption good, the investment good and money for the next period t + 1. Let us then denote by M t the stock of money at time t. In each period t ≥ 0, the representative agent is subject to the budget constraint

qt c t + pt k t+1 + M t+1 = rt k t + ŵt + M t + τt ( 6
)
where qt is the price of the consumption good, pt is the price of the investment good, rt is the rental rate of capital, ŵt is the wage rate and τ is the nominal lump-sum transfers issued by the government. Choosing the consumption good as the numéraire, i.e. dividing equation ( 6) by qt , gives a reformulation of the budget constraint

c t + p t k t+1 + q t M t+1 = r t k t + w t + q t M t + τ t (7)
with p t = pt /q t , q t = 1/q t , r t = rt /q t , w t = ŵt /q t and τ t = τt /q t . Money in real terms is thus given by q t M t . The per-period utility function of the representative agents depends on consumption c and real money balances qM. With the CIA timing, the amount of money that facilitates trading at time t corresponds to the cash the household has in advance of goods-market trading, namely

M t = M t (8)
Let us then denote real money balances at time t by m t = q t M t . We assume that the utility function is additively-separable between consumption and real money balances:

U (c, m) = u(c) + v(m)
Assumption 1. u(c) and v(m) are C 2 , increasing, concave and satisfies the boundary conditions

lim c→0 u (c) = +∞, lim m→0 v (m) = +∞, lim c→+∞ u (c) = 0, lim m→+∞ v (m) = 0.
Notice that the production frontier T (k, y, e c , e y ) gives the maximum production level of the consumption good which will be entirely consumed by the representative agent, i.e. c t = T (k t , k t+1 , e ct , e yt ). The intertemporal maximisation program of the representative agent is then as follows

max {ct,k t+1 ,M t+1 } +∞ t=0 +∞ t=0 δ t [u(c t ) + v(q t M t )] s.t. c t + p t k t+1 + q t M t+1 = r t k t + w t + q t M t + τ t k 0 , M 0 given (9)
where δ ∈ (0, 1] denotes the discount factor.

Intertemporal equilibrium

The generalised Lagrangian corresponding to the intertemporal optimization problem ( 9) is

L = +∞ t=0 δ t [u(c t ) + v(q t M t )] + +∞ t=0 δ t λ t [r t k t + w t + q t M t + τ t -c t -p t k t+1 -q t M t+1 ] (10)
with λ the non-negative Lagrange multiplier associated with constraint [START_REF] Matsuyama | Sunspot Equilibria (Rational Bubbles) in a Model of Money-in-the-Utility-Function[END_REF].

The first-order conditions with respect to c t , k t+1 and M t+1 are given by

λ t = u (c t ) (11)
λ t p t = δλ t+1 r t+1 (12)
λ t q t = δq t+1 [λ t+1 + v (q t+1 M t+1 )] (13)
with r t and p t as defined by [START_REF] Feenstra | Functional Equivalence Between Liquidity Costs and the Utility of Money[END_REF]. By manipulating ( 11)-(13), we get the standard Euler equation u (c t )p t = δu (c t+1 )r t+1 (14) and the arbitrage condition between consumption and real balances

u (c t )q t = δq t+1 [u (c t+1 ) + v (q t+1 M t+1 )] (15)
From the input demand functions defined in Section 2.1 together with the external effects (2) considered at the equilibrium, we define the equilibrium factors demand functions Kc = Kc (k t , k t+1 ), Ky = Ky (k t , k t+1 ), Lc = Lc (k t , k t+1 ), Ly = Ly (k t , k t+1 ) so that êc = êc (k t , k t+1 ) = Ka 1 c La 2 c and êy = êy (k t , k t+1 ) = Kb 1 y Lb 2 y . From [START_REF] Feenstra | Functional Equivalence Between Liquidity Costs and the Utility of Money[END_REF], prices now satisfy

r(k t , k t+1 ) = T 1 (k t , k t+1 , êc (k t , k t+1 ), êy (k t , k t+1 )) p(k t , k t+1 ) = -T 2 (k t , k t+1 , êc (k t , k t+1 ), êy (k t , k t+1 )) (16)
and the consumption level at time t is given by

c(k t , k t+1 ) = T (k t , k t+1 , êc (k t , k t+1 ), êy (k t , k t+1 )) (17)
In order to complete the definition of the intertemporal equilibrium, we need to specify the supply of money. Government follows a simple monetary rule: in each period it issues lump-sum transfers of money balances at the constant rate θ -1 > 0, so that in period t the supply of money M s t satisfies M s t = θ t M s 0 , with M s 0 = M 0 the initial amount of money balances. Thus nominal transfers are given by τ t = (θ -1)q t M s t . We then get the first order conditions ( 14), (15) evaluated at êc and êy , namely the Euler equation

u (c(k t , k t+1 ))p(k t , k t+1 ) = δu (c(k t+1 , k t+2 ))r(k t+1 , k t+2 , ) (18)
and the arbitrage condition between consumption and real balances

u (c(k t , k t+1 ))m t = δ θ m t+1 [u (c(k t+1 , k t+2 )) + v (m t+1 )] (19)
Any solution {(k t , m t )} +∞ t=0 which also satisfies the transversality conditions

lim t→+∞ δ t u (c(k t , k t+1 ))p(k t , k t+1 )k t+1 = 0 (20) lim t→+∞ δ t u (c(k t , k t+1 ))m t = 0 (21)
is called an equilibrium path.

Steady state

A steady state is defined as k t = k * and m t = m * for all t. Along a stationary equilibrium, we have indeed a decreasing path for the price of money balances q t+1 /q t = 1/θ which is associated with the expansion of money supply M t+1 /M t = θ so that real money balances m t remain constant. Notice also that the steady state value of consumption c * is directly obtained from k * through the production frontier, namely c * = T (k * , k * , êc (k * , k * , êy (k * , k * )) ≡ T * . The stationary levels of capital k * and real money balances m * are obtained as solutions of

r(k,k) p(k,k) = δ -1 and v (m) u (c) = θ-δ δ > 0 (22)
Notice that the stationary capital stock k * only depends on the characteristics of the technologies and is independent from the per-period utility function. As a result, once the existence of k * is ensured, we get c * = T * and the stationary real money balances m * are directly obtained from the per-period utility function associated with money holding. Then we get the following conclusion:

Proposition 1. Under Assumption 1, there exists a unique steady state (k * , m * ).

Proof : See Appendix 5.1

In the following, (k * , m * ) will be called the Modified Golden Rule.

Main results

In order to provide a clear-cut local stability analysis of the equilibrium paths in a neighborhood of the Modified Golden Rule, we specify the per-period utility function with respect to consumption as follows:

u(c) = c 1-σ 1-σ
with σ ≥ 0. Hence, the elasticity of intertemporal substitution in consumption is given by c = 1/σ. With such a formulation, we derive the characteristic polynomial building on some results provided in Nishimura and Venditti [START_REF] Nishimura | Indeterminacy in Discrete-Time Infinite-Horizon Models with Non-Linear Utility and Endogenous Labor[END_REF]. Let us introduce the interest elasticity of real money demand evaluated at the steady state:

ε m = -v (m * ) v (m * )m * > 0 (23)
Linearizing equations ( 18) and ( 19) around (k * , m * ) gives:

Proposition 2. The characteristic polynomial is

P(x) = x - θεm θεm-(θ-β) x - α 2 δ(α 2 β 1 -α 1 β 2 ) x -β1 -α1 α2 -σ (1-δβ 1 )( α2 α 1 β 2 + α1 α 2 β 1 )+δβ 1 α 1 β 2 α2 (1-δβ 1 )(α 2 β 1 -α 1 β 2 ) x -(1-δβ 1 ) α1 α 2 +δ β1 α 1 β 2 δ α2 (1-δβ 1 )(α 2 β 1 -α 1 β 2 ) (x -1) ≡ (x -Λ) P(x) Proof : See Appendix 5.2.
Proposition 2 shows that the first characteristic root x 1 = Λ depends on the properties of the utility derived from money holding, while the two others, x 2 and x 3 , depend on the parameters of the technologies and σ.

Let us start with the first characteristic root x 1 . The following Lemma provides a precise characterization:

Lemma 1. Under Assumption 1, x 1 ∈ (-1, 0) if and only if ε m ∈ (0, (θ - β)/2θ).
Proof : Obvious from Proposition 2.

Let us consider now the two other roots x 2 and x 3 . Note that if the utility function is linear in consumption, i.e. σ = 0, these two roots are

x 2 = α 2 δ(α 2 β 1 -α 1 β 2 ) , x 3 = β1 -α1 α2 < 1
We then have

x 2 ∈ (-1, 0) ⇔ α 1 β 2 -α 2 β 1 > α 2 /δ x 3 ∈ (-1, 1) ⇔ β1 > α1 -α2 (24)
These conditions have clear economic interpretations. Based on the private and social production functions as defined respectively by ( 1) and ( 3), factor intensities are determined by the parameters of the Cobb-Douglas technologies. If α 1 β 2 > (<)α 2 β 1 , the consumption (investment) good sector is capital intensive at the private level, and if α1 > β1 the consumption (investment) good sector is capital intensive at the social level. Then, the first condition requires that the consumption good is suffiently capital intensive at the private level. The second condition holds if the consumption good is labor intensive at the social level but can be also satisfied even if the consumption good is also capital intensive at the social level.

When the utility function is non-linear, i.e. if σ > 0, it can be shown that under the conditions given in (24), we still get x 2 ∈ (-1, 0) and x 3 ∈ (-1, 1) provided the value of σ is not too large. More precisely, we have:

Lemma 2. Under Assumption 1, let α 1 β 2 -α 2 β 1 > α 2 /δ and β1 > α1 -α2 . Then x 2 ∈ (-1, 0) and x 3 ∈ (-1, 1) if and only if σ ∈ [0, σ * ) with σ * = [δ(α 1 β 2 -α 2 β 1 )-α 2 ][ β1 -( α1 -α2 )](1-δβ 1 ) 2[(1-δβ 1 )[ α1 α 2 (1+δβ 1 )+δ α2 α 1 β 2 ]+δα 1 β 2 ( β1 +δβ 1 )] (25) Proof : See Appendix 5.3.
Building on these results, we may now discuss the multiplicity of equilibrium paths. As seen in the Introduction, local indeterminacy arises when the stable manifold has dimension greater than the number of predetermined variables. Equations ( 18) and (19) define a system of implicit difference equations of order 2 in the capital stock k and order 1 in the real money balances m. Since k t has been decided at period t -1, it plays the role of predetermined variable at time t while, on the contrary, m t and k t+1 can be chosen at time t and play the role of forward variables. More precisely, even if the stock of money M t is a predetermined variable, the real balances m t = q t M t inherit the status of forward variable from the relative price q t . As a result, the equilibrium is unique if one characteristic root is stable, i.e. belongs to the unit circle, and two characteristic roots are unstable, i.e. lie outside the unit circle. In other words, uniqueness rests on the existence of one-dimensional stable manifold. Conversely, as soon as the stable manifold has dimension greater or equal to 2, there exists a continuum of converging equilibrium paths and the steady state is locally indeterminate. In this case, two kinds of local indeterminacy may arise:

-A local indeterminacy of order 1 when the stable manifold is twodimensional. In a given neighborhood of the steady state, the intersection of the plane defined by the forward variables m t and k t+1 with the stable manifold is a one-dimensional manifold. From each point of this manifold starts an equilibrium trajectory converging to the steady state.

-A local indeterminacy of order 2 when the stable manifold is threedimensional. Indeed, in a given neighborhood of the steady state, the intersection of the plane defined by m t and k t+1 with the stable manifold gives a two-dimensional local manifold which contains multiple starting points for as many converging equilibrium paths.

The following Proposition provides clear-cut conditions for the existence of local indeterminacy of both the orders. Proposition 3. Let Assumption 1 holds and consider the critical bound σ * as defined by (25). Then the steady state is locally indeterminate of order 2 if and only if α 1 β 2 -α 2 β 1 > α 2 /δ, β1 > α1 -α2 , σ ∈ [0, σ * ) and ε m ∈ (0, (θ -β)/2θ). Moreover: i) For any given ε m ∈ (0, (θ -β)/2θ), the steady state is locally indeterminate of order 1 when σ > σ * , where σ * is a flip bifurcation value giving rise to period-two cycles. If the cycles of period 2 arise in the right (respectively, left) neighbourhood of σ * , they are locally indeterminate of order 2 (respectively, locally indeterminate of order 1).

ii) For any given σ ∈ [0, σ * ), the steady state is locally indeterminate of order 1 when ε m > ε * m = (θ -β)/2θ, where ε * m is a flip bifurcation value giving rise to period-two cycles. If the cycles of period 2 arise in the right (respectively, left) neighbourhood of ε * m , they are locally indeterminate of order 2 (respectively, locally indeterminate of order 1).

Proof : See Appendix 5.4.

Proposition 3 shows that under some appropriate restrictions on the capital intensity differences at the private and social levels, a large enough elastic-ity of intertemporal substitution in consumption and a low enough interest elasticity of real money demand are necessary and sufficient conditions for the occurrence of local indeterminacy on the full dimensional phase space. Moreover, when the elasticity of intertemporal substitution in consumption is decreased, or when the interest elasticity of real money demand is increased, we prove the occurrence of local indeterminacy of order 1 through a flip bifurcation. Case ii) is particularly worthwhile as it is based on a large enough interest elasticity of real money demand, a condition that matches empirical evidences. We also show that period-two cycles are intimately connected with the existence of multiple equilibria.

Concluding comments

We have shown that local indeterminacy and business-cycle fluctuations based on self-fulfilling prophecies occur in a two-sector monetary economy with productive externalities and money in an additively separable utility function. More precisely, we have provided conditions on technologies leading to the existence of local indeterminacy for any value of the interest rate elasticity of money demand, provided the elasticity of intertemporal substitution in consumption is large enough. Indeed, while local indeterminacy of order 2 requires a low enough interest elasticity of real money demand, local indeterminacy of order 1 can be obtained even with a large interest elasticity of real money demand. However, as in this last case the restriction on the elasticity of intertemporal substitution in consumption appears to be disputed with respect to empirical evidences, it would be interesting to generalize the formulation of preferences by considering a non-separable utility function and to check whether this condition could be relaxed. This is left for future research.

Appendix

Proof of Proposition 1

Consider first equation ( 22). We need to compute the partial derivatives of T . From the Lagrangian [START_REF] Shell | Monnaie et Allocation Intertemporelle[END_REF] we derive the first order conditions:

r = e c α 1 K α 1 -1 c L α 2 c = pe y β 1 K β 1 -1 y L β 2 y (26) w = e c α 2 K α 1 c L α 2 -1 c = pe y β 2 K β 1 y L β 2 -1 y (27)
Using K c = k 0 -K y , L y = 1 -L c , and merging the above equations gives:

A = 0 dk t+2 T 1 δB -dk t+1 T 1 (A + δB) + dk t θ δ T 1 A + dm t+1 c * (θεm-(θ-β)) δσm * εm -dm t θc * δσm * = 0 with T * 11 T * 12 = - α 2 δ(α 2 β 1 -α 1 β 2 ) , T * 22 δT * 12 = -β1 -α1 α2 , T * 2 1 c * T * 12 = -α 1 α2 T * 11 T * 12 1+ β1 T * 12 T * 11 1-δβ 1 A = α1 α 1 1-δβ 1 +δβ 1 α2 β1 α1 β2 α 1 β 2 α 2 β 1 " 1+ T * 22 δT * 12 « 1+ β1 T * 12 T * 11 , B = α2 α 2 β 2 β2 (1-δβ 1 ) 1-β1 " 1-α1 β2 α2 β1 α 2 β 1 α 1 β 2 « +δβ 1 " 1+ T * 22 δT * 12 « 1+ β1 T * 12 T * 11
Then we derive the associated Jacobian matrix: The result follows after straightforward simplifications.

J =         0

Proof of Lemma 2

Consider the expression between brackets in the characteristic polynomial given in Proposition 2, namely:

P(x) = x - α 2 δ(α 2 β 1 -α 1 β 2 )
x -β1 -α1 α2 -σ (1-δβ 1 )( α2 α 1 β 2 + α1 α 2 β 1 )+δβ 1 α 1 β 2 α2 (1-δβ 1 )(α 2 β 1 -α 1 β 2 )

x -(1-δβ 1 ) α1 α 2 +δ β1 α 1 β 2 δ α2 (1-δβ 1 )(α 2 β 1 -α 1 β 2 ) (x -1)

Under the conditions given in (24), we get x 2 ∈ (-1, 0) and x 3 ∈ (-1, 1) when σ = 0. When σ increases the two roots decrease and the loss of stability is obtained when one root goes outside the unit circle through -1. The bifurcation value of σ, denoted σ * , is then obtained by solving P(-1) = 0.

Proof of Proposition 3

The first part is immediately derived from Lemmas 1 and The existence of a flip bifurcation is generically obtained when one characteristic root crosses the value -1, which happens, for any given ε m ∈ (0, (θ -β)/2θ), when σ crosses σ * from below. In addition, period-two cycles appear either in a right or left neighbourhood of σ * according to the properties of the nonlinear part of the system of difference equations, while their stability properties depend on the stability properties of the steady state (see Ruelle [START_REF] Ruelle | Elements of Differentiable Dynamics and Bifurcation Theory[END_REF]). Therefore, it follows that, generically, σ * is a flip bifurcation value giving rise to periodtwo cycles for any σ in its right (resp. left) neighborhood, which are locally indeterminate of order 2 (resp. locally indeterminate of order 1). This proves case i). Case ii) is proved with a similar argument for any given σ ∈ [0, σ * ) using ε m as bifurcation parameter. In this case, ε * m = (θ -β)/2θ is also a flip bifurcation value.

See Azariadis[START_REF] Azariadis | Self Fulfilling Prophecies[END_REF], Cass and Shell[START_REF] Cass | Do Sunspots Matter ?[END_REF], Shell[START_REF] Shell | Monnaie et Allocation Intertemporelle[END_REF], Woodford[START_REF] Woodford | Stationary Sunspot Equilibria: the Case of Small Fluctuations Around a Deterministic Steady State[END_REF].

Indeed, while the stock of money Mt is a predetermined variable, the real money balances mt = qtMt depend on a relative price qt which is a forward variable.

See for instance Carlstrom and Fuerst[START_REF] Carlstrom | Timing and Real Indeterminacy in Monetary Models[END_REF] and Woodford[START_REF] Woodford | Monetary Policy and Price Level Determinacy in a Cash-in-Advance Economy[END_REF].

Full depreciation is introduced in order to simplify the analysis.

where

To simplify notation let:

From the envelope theorem we get: 26), ( 28) and (30) we obtain:

and from ( 26), ( 29), ( 31) and (34):

By the derivation of g, we have, for any equilibrium path, the identity ∆ = e 1/β 2 y α 1 β 2 (g/y) 1/β 2 . Substituting this into (34) and (35) gives after simplifications:

Substituting (36) into the definition of g, we find

Considering that:

and substituting e * y into the above equation, finally gives

Consider now the monetary part of equation ( 22) evaluated at k * with c * = T * . We get: v (m) = θ-δ δ u (T * ) The boundary conditions in Assumption 1 guarantee the existence and uniqueness of a solution m * .


Camille Daujeard

Jean-Philip Brugal

Marie-Hélène Moncel

Paul Fernandes

Vincent Delvigne

Audrey Lafarge

Jeanne-Marie Le Pape

Jean-Paul Raynal

Lionel Gourichon

Hommes

Jeanne-Marie Le Pape

Neanderthals, Carnivores and Caprines in two Upper Pleistocene sites of

Keywords: Néanderthal, caprinés, archéozoologie, taphonomie, bivouac Neanderthal, Caprines, Zooarchaeology, Taphonomy, Bivouac

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Pleistocene faunal assemblages dominated by caprine and carnivore remains are common from Western and Central Europe to the Near East, in particular in rocky and highland areas [START_REF] Brugal | Stratégie d'exploitation et mode de vie des populations du Paléolithique moyen : exemples des sites du Sud de la France[END_REF][START_REF] Moncel | Mixed occupation during the Middle Palaeolithic : Case study of a small pit-cave-site of Les Pêcheurs (Ardèche, south-eastern France)[END_REF]Daujeard dir., 2018). These sites generally testify of brief human occupations and belong to diversified ecosystems [START_REF] Gamble | Large mammals, climate and resource richness in Upper Pleistocene Europe[END_REF]. Caprine-dominant accumulations are found in specific contexts related to karstic environments, in narrow and deep caves, rockshelters or pitfalls. Among ungulates, caprine species (Capra, Rupicapra, Hemitragus, Ovis) are peculiar as most of them usually live on uneven ground and rocky landscapes in opposition to species living in forests or plains. Due to these differences in biotope, they are subject to specific modes of acquisition (hunting or scavenging) by both non-human and human predators.

Caprines are known to take refuge in shelters or caves during special climatic events (great heat, storms or heavy rain or snowfalls) and may also sometimes naturally die inside (cf. infra). They are often associated with other cave dwellers, such as ursids and other carnivores, whose presence is due to denning, natural death or predation. They may also be preys and be associated with remains of other ungulates (especially cervids) living in the surrounding plateaus and valleys. The scarcity or the absence of hyenids in these sites is noteworthy while canids and some medium-and small-sized felids (panther, lynx) are more frequent. The Pleistocene sites rich in caprine remains also raise specific issues such as the relationships between Neanderthal groups and these rupicolous herbivores. Indeed, since the Lower Palaeolithic, the relationship between small bovids and hominids represents an interesting case study about specific ecosystems and cynegetic behaviors [START_REF] Moncel | Mixed occupation during the Middle Palaeolithic : Case study of a small pit-cave-site of Les Pêcheurs (Ardèche, south-eastern France)[END_REF][START_REF] Yravedra | Neanderthal exploitation of ibex and chamois in southwestern Europe[END_REF]. The reasons for these caprine accumulations are therefore a great challenge which can be addressed integrating ecology, ethology, taphonomy and contextual conditions. Many scenarios are possible which would be discussed : i) natural deaths in shelters or pitfalls ; ii) prey or carrion accumulated and/or modified by predators (humans, carnivores) and iii) mixed scenarios.

This study is focused on two small cave sites in south-eastern France which yielded Middle Palaeolithic archaeological layers with Neanderthal and animal occupations. At Les Barasses II and Abri des Pêcheurs, ibex is the dominant taxon, associated with few other ungulates (cervids, equids and bovines) and various carnivores such as ursids (cave bears) and canids (wolves and foxes). The Middle Palaeolithic layers of these two sites have been recently revised by fieldworks and collection studies [START_REF] Moncel | Mixed occupation during the Middle Palaeolithic : Case study of a small pit-cave-site of Les Pêcheurs (Ardèche, south-eastern France)[END_REF][START_REF] Moncel | Nouvelles données sur les occupations humaines du début du Pléistocène Supérieur de la moyenne vallée du Rhône (France). Les sites de l'Abri des Pêcheurs, de la Baume Flandin, de l'Abri du Maras et de la Grotte du Figuier (Ardèche)[END_REF][START_REF] Daujeard | Exploitation du milieu animal par les Néanderthaliens dans le Sud-Est de la France[END_REF]Daujeard dir., 2018). In order to understand the site function both for human and non-human predators and to infer the origin of the bone accumulation, especially the ibex ones, we will question the role of all potential factors or agents involved in such composite assemblages. The topography of the sites, the global environment and climatic condition with seasonality factors, will also constitute basic information to understand the presence of ibex populations in these caves and allow a brief comparison with similar western European sites, to propose a general interpretative framework.

Site presentation

The sites of Les Barasses II and Abri des Pêcheurs are located in the Ardèche region, along tributary rivers on the right bank of the Rhône Valley, within limestone cliffs and plateau environments at 200-250 m a.s.l. (fig. 1). Close to each other (about 20 km) and both with layers dated to the Marine Isotopic Stages (MIS) 5b-d to 3, they yield a rich faunal corpus composed of various carnivores and herbivores. These two caves present a similar morphology with one single narrow gallery (fissure-like) and steep cliff entrance.

The site of Abri des Pêcheurs is a small and narrow southeast facing cave (total surface of 20 m²) first excavated in the 1970s [START_REF] Lhomme | Un nouveau gisement paléolithique dans la moyenne vallée du Rhône : l'Abri des Pêcheurs à Casteljau (Ardèche)[END_REF]. The opening is situated at 26 meters above the present-day Chassezac River. Recent excavations have provided new chronostratigraphic data for the entire Middle Palaeolithic sequence [START_REF] Moncel | Nouvelles données sur les occupations humaines du début du Pléistocène Supérieur de la moyenne vallée du Rhône (France). Les sites de l'Abri des Pêcheurs, de la Baume Flandin, de l'Abri du Maras et de la Grotte du Figuier (Ardèche)[END_REF]. The cave looks like a pit with a stratigraphic sequence of almost four meters thick. The Middle Palaeolithic faunal assemblage comes from different sectors (from the bottom of the cave S4 to S2) found in layers of limestone rocks in a clay matrix. The deposit is related to a relatively cold sub-phase of the end of the MIS 5 (especially 5b) for the S4, to a wetter period contemporary of the MIS 4-3 for the S3-2 [START_REF] Debard | Le Quaternaire du Bas-Vivarais d'après l'étude des remplissages d'avens, de grottes et d'abris sous roche. Dynamique sédimentaire, paléoclimatique et chronologie[END_REF][START_REF] Masaoudi | Datation d'ossements et de planchers stalagmitiques provenant de l'abri des Pêcheurs (Ardèche) par la méthode des déséquilibres des familles de l'uranium[END_REF][START_REF] Hazzazi | Paléoenvironnement et chronologie des sites du Pleistocène moyen et supérieur, Orgnac 3, Payre et Abri des Pêcheurs (Ardèche, France) d'après l'étude des rongeurs[END_REF][START_REF] Valladas | Contribution des méthodes nucléaires à la chronologie des cultures paléolithiques entre 300 000 et 35 000 BP[END_REF][START_REF] Moncel | Mixed occupation during the Middle Palaeolithic : Case study of a small pit-cave-site of Les Pêcheurs (Ardèche, south-eastern France)[END_REF].

Les Barasses II is also a small and narrow cavity of almost 35 m² opening at 50 meters above the present-day Ardèche River. The cave opens in a steep and rocky south-facing cliff and was first excavated by J. Combier [START_REF] Combier | Le Paléolithique de l'Ardèche dans son cadre paléoclimatique[END_REF][START_REF] Combier | Rapport de fouilles de Balazuc, Ardèche, circonscription archéologique Rhône-Alpes[END_REF]. Since 2011, new investigations were undertaken to better understand the sequence, especially its poorly explored lowest part (Daujeard dir., 2018). The sedimentary sequence is composed of a three meter silty deposit including weathering clasts from the limestone bedrock. Two main lithostratigraphic units are subdivided in eight stratigraphic layers. The Upper unit (layers 2-4) is composed of brownish colored silts with some altered limestone fragments due to phosphatation. The Lower unit (layers 6-8) is a brown-yellowish silty sediment richer in rock fragments. Past and recent datings on bone and calcite suggest that the Middle Palaeolithic layers were deposited from MIS 5d to 3. The Upper unit has been dated by ESR/U-series, providing a chronology from 48 ± 5 to 61 ± 5 ka (MIS 3). For the lowest part of the sequence, datings of 111 ± 13 ka, 89 ± 6 ka, 107 ± 13 and 79 ± 8 were obtained using the same method [START_REF] Richard | Contribution of ESR/U-series dating to the chronology of late Middle Palaeolithic sites in the middle Rhône valley, southeastern France[END_REF]Daujeard dir., 2018). According to the palaeoecological information, a colder episode coincides with the deposit of the lower part of the sequence characterized by steppe vegetation and a rigorous environment, however with some forested patches. During the deposit of the upper part of the sequence, the climate was less severe with temperate and wooded biotopes (Foury et al., 2016 ;Rufa et al., 2018 ;Daujeard dir., 2018).

Lithic assemblages

The lithic material of Abri des Pêcheurs (ancient and recent excavations) is composed of 315 pieces for S2, 649 for S3 and 222 for S4 ; mainly made in quartz (up to 95 %) and flint artefacts. Local quartz pebbles were flaked in situ by unifacial or bifacial core technologies to provide irregular end-products rarely retouched. Small flint flakes (unretouched or retouched) and some small cores on flake were introduced into the site already knapped. Some flint flakes are Levallois ones [START_REF] Moncel | Mixed occupation during the Middle Palaeolithic : Case study of a small pit-cave-site of Les Pêcheurs (Ardèche, south-eastern France)[END_REF]. Twenty-seven different types of flint were identified, collected from alluvial deposits and colluviums from semi-local outcrops in an area extending in several directions up to 30 km from the site [START_REF] Fernandes | Ressources minérales et comportements au Paléolithique moyen : Payre et l'Abri des Pêcheurs (Ardèche, France)[END_REF].

At Les Barasses II, the lithic corpus totals 813 pieces for the two main units (respectively 195 and 618 from Upper and Lower units). The local raw materials are mainly basalt, limestone, quartzite and quartz pebbles probably collected in the river bed under the cave and partially worked in situ (mostly flakes and one pebble tool). Fifteen types of flint were identified, collected from local outcrops up to 30 km mainly eastwards [START_REF] Fernandes | ) : entre Néandertaliens, bouquetins et carnivores[END_REF]. The flint artefacts are diversified and represent circa 45 % (a bit less in the lower part of the sequence), mostly composed of products of reduction processes (small flakes and larger elongated flakes) and some small modified cores on flakes. The core technologies are various (Discoid, Levallois, Kombewa, Quina). The presence of very small flakes indicates an in situ activity of knapping or alteration. Scrapers and some points (including broken ones) compose the tool kits.

In short, the Middle Palaeolithic occupation layers of the two sites yield a relatively small quantity of artefacts compared to other sites in the region [START_REF] Moncel | The variability of the Middle Palaeolithic on the right bank of the Middle Rhône Valley (southeast France) : Technical traditions or functional choices ?[END_REF]. The reduction sequences are incomplete and indicate regular introduction of ready-made tool kits made of flint coming from a restricted area (< 20-30 km), and also quartz and volcanic local stones. Those lithic assemblages may be related to short-term human occupations.

Materiel and methods

We will compare the faunal sets on the base of abundance of taxa, skeletal representation, degree of bone breakage, anatomical connection, age mortality profile and presence of anthropogenic and carnivore marks, focusing on ibex remains. The methodological approach and raw data have been presented in different previous publications for the two sites [START_REF] Moncel | Mixed occupation during the Middle Palaeolithic : Case study of a small pit-cave-site of Les Pêcheurs (Ardèche, south-eastern France)[END_REF][START_REF] Daujeard | Neanderthal and animal karstic occupations from southern Belgium and south-eastern France : Regional or common features ?[END_REF]Daujeard dir., 2018) and resumed here for comparative purposes.

The analysis concerns the faunal remains of medium-and large-sized mammals coming from recent (R) excavations (C. Daujeard for Les Barasses II and M.H. Moncel for the S2 to S4 of Abri des Pêcheurs) and ancient (A) excavations (G. Lhomme for S4 of Abri des Pêcheurs). Bone breakage and surface modifications were distinguished between those made by rodents, carnivores (gnaw-marks) or hominins (cut and percussion marks), as well as climatic and edaphic modifications (including cracking, desquamation, polish, concretion, root etchings, natural abrasion, chemical corrosion and oxide colorations) [START_REF] Binford | Bones : Ancient men and modern myths[END_REF][START_REF] Lyman | Vertebrate Taphonomy[END_REF]. Following [START_REF] Haynes | A guide for differentiating mammalian Carnivore taxa responsible for gnaw damage to herbivore limb bones[END_REF] and [START_REF] Blumenschine | An experimental model of the timing of hominid and Carnivore influence on archaeological bone assemblages[END_REF], carnivore marks were classified as follows : pits, punctures, scores, notches or corrosion by gastric acids, and their location on long bones were recorded. Ontogenic age-atdeath for ibex is based on dental eruption and replacement patterns and then on tooth wear [START_REF] Grant | The use of tooth wear as a guide to the age of domestic ungulates[END_REF][START_REF] Couturier | Détermination de l'âge du bouquetin des Alpes (Capra aegagrus ibex ibex) à l'aide des dents et des cornes[END_REF][START_REF] Couturier | Le bouquetin des Alpes[END_REF]. Three age groups were distinguished : juvenile, adult and old. However not all these taphonomical criterions will be fully developed in this contribution, and most were exposed in previous publications [START_REF] Moncel | Mixed occupation during the Middle Palaeolithic : Case study of a small pit-cave-site of Les Pêcheurs (Ardèche, south-eastern France)[END_REF][START_REF] Daujeard | Neanderthal and animal karstic occupations from southern Belgium and south-eastern France : Regional or common features ?[END_REF]Daujeard dir., 2018). The Lower unit of Les Barasses II contains a total of 7,040 faunal remains (sieving and coordinated) of which only 348 were studied for taphonomy (number of recorded bone remains with readable surfaces) and with a number of identified specimens (NISP) of 181. The Upper unit has a total of 3,487 remains, including 349 that are taphonomically studied and a NISP of 222 (Daujeard dir., 2018) (tab. 1). The total NISP for the sector 4 of Abri des Pêcheurs is of 1,900 for the earliest excavations and of 55 for the recent ones, while the NISP from the sectors 2 to 3 is of 118 [START_REF] Moncel | Mixed occupation during the Middle Palaeolithic : Case study of a small pit-cave-site of Les Pêcheurs (Ardèche, south-eastern France)[END_REF][START_REF] Daujeard | Neanderthal and animal karstic occupations from southern Belgium and south-eastern France : Regional or common features ?[END_REF] (tab. 2). For the sector 4 of Abri des Pêcheurs, given that the ibex series coming from the first excavations is the most abundant, we preferentially used those data for the graphs representing the carnivore/herbivore distribution and age categories (fig. 23).

Zooarchaeological and taphonomic data

Ibex remains are largely abundant at Les Barasses (80.5 % of the NISP herbivore and 57.7 % of the NISP total for the Upper unit and respectively 57.7 % and 43.6 % for the Lower unit ; tab. 1 and fig. 2b), followed by cervids such as red deer and reindeer, and by other secondary herbivores like bovines, horses, wooly rhinoceroses and chamois. Among the carnivores, which represent respectively 27.9 % and 17.7 % of the total identified specimens in the Upper and Lower units (tab. 1 and fig. 2a), we find in order of frequency ; red foxes, cave bears, and wolves, and a few panthers, wild cats and mustelids. Some lagomorphs and beavers are present.

Red fox is present in the Upper unit with one juvenile, two mature adults and one old individual, and in the Lower unit by one juvenile and one adult. For the cave bear, the Upper unit counts two juveniles and one mature adult, and the Lower unit yields only juveniles. The wolf is the third most frequent carnivore in the cave, distributed all along the sequence, with the presence a minima of one young and one mature adult in the Upper unit and one old individual in the Lower unit. The panther is rare and present in both units while small carnivores, felids and mustelids, are only present in the Lower unit. All these communities of species are in correspondence with the local biotopes present around the two caves.

The faunal association of Abri des Pêcheurs is also dominated by ibex (89.9 % of the NISP herbivore and 52.5 % of the NISP total for S2-3 and respectively 97.5 % and 81.1 % for S4 ; tab. 2 and fig. 2b), completed by cervids (red deer, reindeer and very few roe deer). Carnivores are abundant with 40.7 % of the total identified specimens for S2-3 and up to 16.9 % for S4 (tab. 2 and fig. 2a), mainly wolf and cave bears, few foxes, and some scarce cave hyena remains. Carnivores are mostly adults with some old individuals. Beavers are also present (S2-3). This varied spectrum is closely related to the local environment, with many rocky cliffs and some forests and open grassland species along the rivers and on the plateau.

Skeletal profiles for ibex are similar between the two sites (in Daujeard dir., 2018 : 137, fig. 69 andin Moncel et al., 2008 : 389, fig. 6) with all the skeletal parts represented (tab. 3 et 4), indicating the presence/introduction of some complete carcasses into the site. Limb bones, head (mainly isolated teeth) and axial portions are dominant, whereas small foot bones, such as phalanges, carpals and tarsals are fewer.

Among the ibex populations all age groups (juvenile, adult and old) are represented in the two sites, although adults (mature and old) are the most abundant (fig. 3). We can note the existence of both sexes, based on cranium (horn cores) and pelvis fragments, as well as metrical differences for some bone elements. Seasonal indices for ibex (periods of dental eruption for juveniles) at Les Barasses II give individual deaths from summer to winter. No seasonal evidence could be found for the other ungulate species.

Regarding bone preservation, the three main climatic and edaphic modifications (weathering, corrosion and concretions) have slightly affected the bone assemblages. Most of the bone surfaces are easily readable and cracking and flaking are moderate. Trampling is the post-depositional process which had the greatest impact, with spatial disruptions and abrasion marks (striations and polishes). The bone assemblages are highly fragmented. The few complete elements are mostly small articular bones or vertebrae of carnivores and ungulates, with some very well-preserved ibex vertebrae found in anatomical connections. two assemblages. Most of the bone surface modifications are pits, punctures and scores (fig. 4). Digested remains are scarce and count about twenty small carnivorous articular bones and unidentified fragments. Human cut marks on bones are globally uncommon with 0.5 % of cut-marked remains in the Upper unit and 0.8 % in the Lower unit (Daujeard dir., 2018). However, the percentage is higher when considering only the herbivore stock, respectively 12 % of cut marks in the Upper unit and 23 % in the Lower unit, mainly found in stylopodial and zeugopodial skeletal parts. This main hominid exploitation occurred on secondary species such as red deer, reindeer or bovines ; the ibex seems relatively less affected with 6.7 % of butchery marks from the Upper unit and nearly 19.6 % from the Lower unit. We notice that percussion marks and green bone fractures are both more numerous in the Lower units, which can be related to the greatest impact of humans in these layers. Skinning, defleshing, dismembering, scraping and marrow extraction are all recognized butchery actions in the two units.

A relatively different picture is observed for other ungulate remains at Abri des Pêcheurs [START_REF] Moncel | Mixed occupation during the Middle Palaeolithic : Case study of a small pit-cave-site of Les Pêcheurs (Ardèche, south-eastern France)[END_REF]. Globally speaking, cut-marked elements are less frequent, only visible on few bones whatever the sectors : five from ibex, four from red deer and only one on reindeer (this count excludes the unidentified bone splinters). Percussion marks are also rare on ibex and reindeer long bones. Butchery marks are often located on metapodials, suggesting dismembering and skinning processes (figs. 5-6). Carnivore activity (gnawing marks) is largely present (10.9 % for ibex remains in S2-3 and around a third for identified species in S4, including carnivores themselves), although with a low degree of bone destruction, lower that that observed in hyena dens [START_REF] Moncel | Mixed occupation during the Middle Palaeolithic : Case study of a small pit-cave-site of Les Pêcheurs (Ardèche, south-eastern France)[END_REF].

Thus, in both sites, taking into account the relative frequency of each species, the humans seem to have preferentially processed secondary ungulates, especially cervids. Ibex were sometimes exploited, as maybe it was more the case at the Lower unit of Les Barasses, although in a less systematic way. Carnivore actions are also apparent and could be related to moderate modifications left by canids such as wolves and foxes. Finally, no clear evidence of fire-use exists at Les Barasses II. On the other hand, a large fireplace is present in the S4 of Abri des Pêcheurs, which induced the accidental burning of many bones.

Discussion

What types of accumulations for the various animals in these two small caves ?

The ibex is the main herbivore present in both sites of Les Barasses II and Abri des Pêcheurs. Several points stressed the presence of complete skeletons of many individuals of different age classes, mainly adults of both sexes, sometimes found in articulated connection (S4 -Abri des Pêcheurs). The study of bone surfaces showed a clear impact of carnivores for all the assemblages. The low degree of modifications and dimensions of the tooth marks shows that small-to-mediumsized carnivores (mainly canids) consumed the carcasses. Cut-marked elements are fewer, mostly present on some long bones of non-caprine elements. These bone surface modifications, along with skeletal and age representations, suggest diverse origins to explain the presence of this 'rocky species' within these two caves : carnivore, human and natural.

Indeed, according to the moderate carcass exploitation and the relative bone completeness, the ibex accumulations seem to be partly driven by several nonexclusive natural reasons such as trapping, wintering, calving or sheltering. The caprines are known to use karstic cavities as a shelter during harsh climatic conditions and, in our cases, they were also probably trapped due to difficulty of access (caves have a steep entrance) complicated by potential development of snowdrifts in winter time (deposit under cold glacial conditions) (see Moncel et al., 2008 and reference herein). Nevertheless, seasonal analyses based on mandibles highlight deaths all year long, excepting during spring. Their absence during this season could be the mobility of herds looking for better pastures and/or birth places in different regions, like the mountainous landscapes of the Vivarais about ten kilometers away. Humans and carnivores could be other minor agents of the ibex accumulation and modifications, either by hunting or scavenging.

The secondary ungulate species (cervids and bovines) are only represented by a few long bone elements showing some more anthropic and/or carnivore modifications. In our opinion it corresponds to direct predator activities or scavenging especially from human groups which transport carcass parts, and not complete skeleton, for in situ consumption during transient or crossing occupations.

Inversely, other species such as chamois, horse or small mammals, are represented by a very few limb elements and isolated teeth, bearing only carnivore marks. They could account for carnivore prey from medium-to small-sized predators.

Fox remains are numerous at Les Barasses II. Age and skeletal profiles, including juveniles and all parts of the skeleton, indicate denning activities, notably in the Lower unit where the presence of mustelids, lagomorphs and bird remains bearing small-sized carnivore tooth marks, also point towards a great impact of the fox on this small vertebrate assemblage (accumulation and modification).

The large predators, wolves, cave hyenas and panthers are mainly represented by scattered skeletal elements belonging to adults and old individuals. Their presence in the cave mostly results from natural deaths in the cave, and possibly interspecific competition and encounters.

The case of cave bears is peculiar and they are well known for their hibernating behavior (winter lethargy). They are among the dominant carnivores and include foetal and juvenile individuals. Some of them died during winter, indicating hibernation. Animal or human trampling has caused significant damage (displacement, striations and fragmentation), particularly in the upper part of the sequence of Les Barasses II and in the S4 of Abri des Pêcheurs, where more cave bears used the cave.

What type of accumulations for small bovids in European Pleistocene rocky landscapes ?

In order to decipher carnivore, human or natural origin within such small bovids assemblage, a description of the topographic (aspect of the cave and surroundings) and taphonomic features (age and skeletal profiles, completeness of bones, rates and distribution of the cut and tooth marks, seasonality and so on) can be used. Data will be also discussed regarding eco-ethological data of modern caprines and carnivores. Here, we aim to categorize some of the famous and already published (without being exhaustive) Holocene or Pleistocene caprinedominant sites into four categories.

1. Natural accumulations : Holocene and Pleistocene fossil records yield natural caprine accumulations in caves, shelters or pitfalls in rocky landscapes, sometimes in highland mountains. Those caprine-dominant sites present almost complete carcasses and bone elements, some being in anatomical association, and moderate bone surface modifications, which may be related to natural deaths from animals finding themselves trapped, or during sheltering. These sites reflect the particular behaviour of these ungulates ; visiting caves, sometimes deep karstic networks and natural traps. As stated above, the presence of living caprines in cavities can be explained by the need to find a refuge against heat or cold weather [START_REF] Couturier | Le bouquetin des Alpes[END_REF][START_REF] Schaller | Mountain Monarchs -Wild sheep and goat of the Himalaya[END_REF]. This type of actualistic observation broadens the scope of interpretations for the fossil record. Natural accumulations of ibex have been observed among Pleistocene and Holocene fossil faunal accumulations in rocky landscapes : e.g. Grotte de l'Eglise in southwestern France [START_REF] Griggo | -Étude paléontologique du Locus III de la Grotte de l'Eglise (Dordogne). Gisement ante Würmien à Bouquetin[END_REF], Les Barasses II and Les Pêcheurs in southeastern France (this study), Observatoire cave in Monaco [START_REF] Brugal | -Premières observations sur les guildes de carnivores de la grotte de l'Observatoire (Principauté de Monaco)[END_REF] and probably in some layers of the Hortus cave (Lumley, De, 1972) and finally in some higher altitude caves in the Alps (for example the Tempiette cave) [START_REF] Blant | Analyse chronologique des données d'occupation de bouquetins (Capra ibex) dans les grottes des Alpes suisses[END_REF]Luret, Castel, 2012 ;[START_REF] Griggo | Un piège à bouquetins et chamois du Mésolithique. La Grotte Tempiette (Entremont-le-Vieux, Savoie)[END_REF] among others). This behavior seems to be common to many caprine species, and we could also mention the case of the tahr (genus Hemitragus) in the early Pleistocene site of l'Escale (Bouches-du-Rhône, Bonifay 1974Bonifay -1975) ) and in the late Middle Pleistocene of Les Cèdres cave (Var, Defleur, Crégut-Bonnoure, 1995). However, these accumulations are also partly due to predators and thus may belong to the fourth category (cf. infra).

2. Human accumulations : the predation and significant consumption of caprines by Palaeolithic human groups has been identified in many Pleistocene sites in France such as (among others) Caune de l'Arago for the mountain sheep [START_REF] Monchot | Les petits Bovidés (genres Ovis, Hemitragus, Capra et Rupicapra) de la Caune de l'Arago (Tautavel, Pyrénées-Orientales[END_REF][START_REF] Monchot | La chasse au mouflon au Pléistocène moyen : l'exemple de la Caune de l'Arago (Tautavel, France)[END_REF][START_REF] Rivals | Les petits bovidés pléistocènes dans le bassin méditerranéen et le Caucase. Etude paléontologique, biostratigraphique, archéozoologique et paléoécologique[END_REF], West cave of Le Portel [START_REF] Gardeisen | Middle Palaeolithic Subsistence in the West Cave of « Le Portel » (Pyrénées, France)[END_REF], Tournal cave [START_REF] Magniez | Taphonomic study of the Middle and Upper Palaeolithic large mammal assemblage from Tournal Cave (Bize-Minervois, France)[END_REF], Baume des Peyrards [START_REF] Daujeard | Exploitation du milieu animal par les Néanderthaliens dans le Sud-Est de la France[END_REF] or in the Magdalenian layer of les Eglises [START_REF] Delpech | Activités de chasse et boucherie dans la grotte des Eglise[END_REF] for the ibex, Scladina cave in Belgium [START_REF] Moncel | Halte de chasse au chamois au Paléolithique moyen : la couche 5 de la grotte Scladina (Sclayn, Namur, Belgique)[END_REF] for the chamois and Bau de l'Aubesier in France [START_REF] Fernandez | -Stratégies de chasse des Néanderthaliens du Bau de l'Aubesier (Monieux, Vaucluse) : choix et opportunisme[END_REF] for the tahr. In the Iberian Peninsula, ibex and chamois were frequently hunted by Neanderthals in similar contexts [START_REF] Yravedra | Neanderthal exploitation of ibex and chamois in southwestern Europe[END_REF]. Cut marks, percussion marks, green bone breakage and the presence of all age classes and high-meat portions for ibex carcasses are the main indicators of a human predation.

3. Carnivores accumulations : carnivores as potential predators and accumulators of caprines in caves or shelters are a priori uncommon. The most common species associated with remains of caprines are canids (wolf, cuon and fox) and medium-sized felids such as panther and lynx. Some of these taxa, considered as non-accumulators of carcasses on the basis of published studies, may have accumulated remains indirectly through their faeces [START_REF] Mallye | -Dhole (Cuon alpinus) as a bone accumulator and new taphonomic agent ? The case of the Noisetier Cave (French Pyrenees)[END_REF]. They may also have been direct accumulators as well, behaving differently during the Pleistocene under harsher climatic and competitive conditions. Some fossil series would add weight to this argument. For example, those discovered in the Noisetier Cave in France suggest indirect accumulations (coprocoenosis) of large canids and those from the sites of Racó del Duc and Los Rincones in Spain and Soulabé cave in France indicate that large felids (i.e., panthers) may have been predators and bone accumulators [START_REF] Sauqué | Bone accumulation by leopards in the Late Pleistocene in the Moncayo massif (Zaragoza, NE Spain)[END_REF]2016 ;[START_REF] Sauqué | -Leopards as taphonomic agents in the Iberian Pleistocene, the case of Racó del Duc (Valencia, Spain)[END_REF]Fosse et al., this volume). The Middle Palaeolithic site of Esquilleu in Spain may highlight carnivores (hyenids and canids) as the main agents in the introduction of caprines in the site, with some human activity evidence [START_REF] Yravedra | Taphonomic implications for the Late Mousterian of South-West Europe at Esquilleu Cave (Spain)[END_REF]. The type of carnivores, the presence and distribution of tooth marks on the caprine carcasses, the degree of preservation of the bone elements (complete, cylinders or fragments) and the age (most vulnerable individuals) and skeletal profiles are criteria which can be used to identify the impact of carnivores on a caprine assemblage. According to [START_REF] Couturier | Le bouquetin des Alpes[END_REF], ibex and especially young individuals have been and still are prey for the wolf, although the chamois is currently favored. Moreover, as Couturier also points out, the important presence of fox in certain caprine-assemblages should not be underestimated.

It is an active carcass scavenger and can also attack very young individuals.

4. Mixed accumulations : usually, carnivores and even human accumulations of caprine remains go along with some natural and recurrent presence of those rupicolous animals. This is actually the majority of the caprine-dominant Pleistocene accumulations, which share criteria from the three types of accumulations described above. Some examples among others are the mixed accumulations at L'Hortus (Lumley De, 1972 ;[START_REF] Boulbes | Les grands herbivores de la grotte moustérienne de l'Hortus (Valflaunès, Hérault) : Etude paléontologique, archéozoologique, paléoécologique[END_REF], Les Cèdres caves [START_REF] Defleur | Le gisement paléolithique moyen de la grotte des Cèdres (Le Plan-d'Aups, Var)[END_REF] and the two cave sites presented here, Les Barasses II and Abri des Pêcheurs, for the French Middle Palaeolithic [START_REF] Daujeard | Exploitation du milieu animal par les Néanderthaliens dans le Sud-Est de la France[END_REF][START_REF] Moncel | Mixed occupation during the Middle Palaeolithic : Case study of a small pit-cave-site of Les Pêcheurs (Ardèche, south-eastern France)[END_REF]Daujeard dir., 2018), Zafarraya in Spain [START_REF] Caparrós | Did Neanderthals and Carnivores Compete for Animal Nutritional Resources in the Surroundings of the Cave of Zafarraya ?[END_REF] and Buraca Escura in Portugal for the Middle and Upper Palaeolithic [START_REF] Brugal | Modalités d'occupations au Paléolithique supérieur dans la grotte de Buraca Escura (Redinha, Pombal, Portugal)[END_REF][START_REF] Brugal | Carnivores pléistocènes (Hyénidés, Canidés, Félidés) dans les grottes du Portugal[END_REF][START_REF] Arceredillo | Paléolithique moyen et supérieur à Buraca Escura (Redinha, Pombal, Portugal) : comparaisons taphonomique et archéozoologique[END_REF]. All document a scarce human exploitation.

Conclusion

At Les Barasses II and Abri des Pêcheurs, ibex remains appear to have been mostly naturally accumulated. These two sites thus provided convenient shelters not only for humans but also for various animals, including caprines. Whether sheltered or trapped, the animals died in these small and narrow caves of peculiar topography and difficult accesses, and may have been consumed by secondary predators. We cannot rule out a few cases of direct predation by humans or medium -and large -carnivores either and we must note the predominance of canids, and especially wolves in these caves. We have highlighted punctual human activities in these sites due to some butchery marks mainly observed on secondary herbivore species. Besides Neanderthal occupations, there is evidence of the presence of small, medium, and large-sized carnivores (birth or death places). The caves have been frequently used by many animals and humans for sheltering, hibernating, denning or procuring food supplies. The scant human archaeological evidence illustrates recurrent and short-term visits. The caves were used for bivouacs (transient/resting sites) involved in larger scale mobility strategies. Human subsistence was mostly oriented towards cervids and bovines, rather than caprines inhabiting the cliffs surrounding the caves. The sites were not real hunting places but rather a waypoint briefly used (on a day-base) with ready-made lithic tools and some parts of herbivore carcasses brought in (snack).

The biological and chronological similarities between these faunal assemblages could indicate recurrent visits of the cave by Neanderthal groups sharing the same behavioral pattern. It could also suggest a specific type of occupation of these rocky environments by humans and animals, underlining at some point a determinant influence of topography on subsistence behaviours.

HOMMES ET CAPRINÉS HUMANS AND CAPRINES de la montagne à la steppe, de la chasse à l'élevage From mountain to steppe, from hunting to husbandry

Lionel Gourichon, Camille Daujeard, Jean-Philip Brugal L es 39 e Rencontres Internationales d'Archéologie et d'Histoire d'Antibes ont porté sur la thématique de la relation Homme-Caprinés au cours des temps préhistoriques et historiques. Dans une démarche diachronique et pluridisciplinaire, archéozoologues, paléontologues, écologues, taphonomistes, ethnologues, anthropologues, archéologues et historiens se sont regroupés pour esquisser et illustrer l'Histoire de cette longue et étroite relation autour de ce titre ambitieux : « Hommes et Caprinés : de la montagne à la steppe, de la chasse à l'élevage ». Les interactions entre les Hommes et les Caprinés débute bien avant la domestication, dès le Paléolithique inférieur, au coeur des milieux rocheux d'Eurasie. À partir du Néolithique, la domestication de certaines espèces (mouton, chèvre) a joué un rôle majeur dans les changements socio-économiques des sociétés humaines au Proche-Orient, en Europe et en Afrique, et transformé les paysages par le pastoralisme, jusqu'au temps actuels et sur tous les continents. À travers les pratiques de subsistance, de la prédation à la domestication, et des pratiques pastorales les plus diverses, complétée par les dimensions biologiques, écologiques et sociales, l'iconographie et la symbolique, ce recueil met en lumière ce groupe particulier d'ongulés, les Caprinés, et leurs exploitations sensu lato par les groupes humains.

T he 39 th Rencontres Internationales d'Archéologie et d'Histoire d'Antibes focused on the topic of the Human-Caprine relationship in prehistoric and historical times. Within a diachronic and multidisciplinary approach, zooarchaeologists, paleontologists, ecologists, taphonomists, ethnologists, anthropologists, archaeologists and historians gathered to sketch and illustrate the history of this long and close relationship through this ambitious topic : "Humans and Caprines: from mountain to steppe, from hunting to husbandry". The interactions between Humans and Caprines starts well before the domestication, since the lower Palaeolithic, in the rocky Eurasian regions. From the Neolithic onwards, the domestication of some species (sheep, goat) played a crucial role in the socio-economic changes in the Near East, Europe and Africa, and transformed the landscapes through pastoralism till the present time and on all continents. Through subsistence practices, from predation to domestication, as well as through a large variety of dimensions: social, economic, biological, ecological, iconographic and symbolic, this volume deals with this particular group of ungulates, the Caprines, and their exploitation sensu lato by human groups.

Fig. 1 .

 1 Fig. 1. a) Map of France ; b) Situation of Les Barasses II and Abri des Pêcheurs on the right bank of the Rhône Valley with those of some other Middle Palaeolithic sites (modified from Rufà et al., 2018) ; c) and e) The entrance of Les Barasses II cave and its location in the cliff ; d) and f) The cave of Abri des Pêcheurs and its location within the surrounding cliffs.

  Fig. 2. a) Large Taxonomical Distribution (Meso = mesofauna, Herb = herbivore, Carn = Carnivore) (% NISP) ; b) Detail of Herbivore distribution by taxa (% NISP). Bar = Les Barasses II, Up = Upper units (total NISP = 222) and Low = Lower (total NISP = 181) units ; Pêch = Abri des Pêcheurs, S2-3 (R) = sectors 2 and 3 for recent excavations (total NISP = 118) and S4 (total) = sector 4 for ancient and recent excavations (total NISP = 1,955).

Fig. 3 .

 3 Fig. 3. Age profile of ibex (MNI = Minimum number of individuals). Bar = Les Barasses II, Up = Upper (total MNI = 22) and Low = Lower (total MNI = 5) units. Pêch = Abri des Pêcheurs, S4 (A) = sector 4 for ancient excavations (total MNI = 39).

Fig. 4 .

 4 Fig. 4. Different stages of carnivore bone surface modifications (pits, punctures and scores) for ibex remains at Les Barasses II (Upper units -Combier excavations ; DaujearD, 2008). a) calcanei ; b) proximal ulnas ; c) scapulas ; d-e) proximal and distal metatarsals and f) distal metapodial.

Fig. 5 .

 5 Fig. 5. a) Dismembering cut marks on the medial face of a red deer talus (Pech86-C6) and b) on the distal part of a reindeer metatarsal (Pech86-C9) from the sector 4 of Abri des Pêcheurs (scale = 1 cm ; the red line drawings were made in the past).

Fig. 6 .

 6 Fig. 6. a) Percussion notches on shaft fragments of a large ungulate humerus (Pech78-C9-D9) and b) of a red deer humerus (Pech89-D10) (scale = 1 cm).

  Environnement et Archéologie' in which the first two authors participated. The fieldworks were financially and scientifically supported by the French Ministry of Culture : the Service régional de l'Archéologie Auvergne-Rhône-Alpes (excavations at Abri des Pêcheurs by M.-H. Moncel and at Barasses II by C. Daujeard and collective research program (PCR) 2005-2016 by J.-P. Raynal and M.-H. Moncel).Finally, we thank Philippe Fosse for his precious comments.

  Complete long bones are scarce, with some long bone cylinders. For ibex we found only one complete radius and one metacarpal from the Upper unit of Les Barasses II. At Les Pêcheurs, at least 11 % of the ibex remains are complete, most of them are metapodials or short and compact bones (talus, phalanxes). Dry bone breakage (weathering, trampling or sedimentary pressure) as well as green bone breakage (carnivores and humans) are both responsible of that intense . At Les Barasses II for example, green bone fractures are mostly associated to carnivore tooth marks throughout the sequence, with 32.5 % in the Lower unit and 40.8 % in the Upper unit. On the other hand, 6 % of the green bone fractures are associated with percussion marks in the Upper unit and 22 % in the Lower unit, suggesting some human activity of marrow extraction.At Les Barasses II, carnivore activity is revealed by a high proportion of gnawing marks among the identified specimen, between one third and half of the NISP according to layers. On the whole, we count 9.2 % of carnivore tooth-marked elements in the Upper unit and 12.1 % in the Lower unit, with almost half of the ibex remains bearing carnivore traces, respectively 50.1 % and 44.6 % for the

			NISP	NISP % NISP	%NISP % MAU	%MAU Survival rates	Survival rates
	cranial remains cranial remains (skull)	45	2.9 %	4	3.1 % 70.3 %	16.7 % 57.7 %	10.0 %
	maxillary mandible	28	1.8 %	2	1.6 % 43.8 %	16.7 % 35.9 %	10.0 %
	mandible isolated teeth	255	10 16.6 %	7.8 %	100.0 %	60.0 %
	humerus humerus	66	4.3 %	2	1.6 % 62.5 %	33.3 % 51.3 %	20.0 %
	radius-ulna radius-ulna	92	6.0 %	7	5.5 % 82.8 %	66.7 % 67.9 %	40.0 %
	carpals carpals	27	1.8 %	2	1.6 % 7.0 %	5.6 % 5.8 %	3.3 %
	metacarpal metacarpal	41	2.7 %	6	4.7 % 50.0 %	66.7 % 41.0 %	40.0 %
	femur	femur	90	5.9 %	5	3.9 % 68.8 %	50.0 % 56.4 %	30.0 %
	patella	tarsals	6	0.4 %	7	5.5 % 9.4 %	19.4 % 7.7 %	11.7 %
	tibia	metatarsal	72	4.7 %	1	0.8 % 48.4 %	16.7 % 39.7 %	10.0 %
	tarsals	metapodials	76	4.9 %	3	2.3 % 19.8 %	25.0 % 16.2 %	15.0 %
	phalanges metatarsal	42	2.7 %	9	7.0 % 43.8 %	12.5 % 35.9 %	7.5 %
	ribs metapodial	35	21 2.3 %	16.4 % 23.4 %	21.8 % 19.2 %	13.1 %
	cervical vertebrae phalanx I	64	4.2 %	2	1.6 % 22.3 %	9.5 % 18.3 %	5.7 %
	thoracic vertebrae phalanx II	28	1.8 %	3	2.3 % 10.5 %	7.7 % 8.7 %	4.6 %
	lumbar vertebrae phalanx III	23	1.5 %	5	3.9 % 9.0 %	27.8 % 7.4 %	16.7 %
	sacrum sesamoids	10	0.7 %	2	1.6 % 1.3 %	66.7 % 1.1 %	40.0 %
	ribs	unidentified vertebrae 99	6.4 %	2	1.6 % 7.0 %	0.0 % 5.7 %	0.0 %
	coxal costal cartilage	5	0.3 %	1	0.8 %	16.7 %	10.0 %
	scapula vertebrae	296	2 19.3 %	1.6 % 35.6 %	33.3 % 29.2 %	20.0 %
	upper isolated teeth sternebrae	14	10 0.9 %	7.8 %
	coxal	lower isolated teeth	71	14 4.6 %	10.9 % 100.0 %	82.1 %
	unidentified isolated teeth scapula 51	3.3 %	8	6.3 % 50.0 %	41.0 %
	NISP	Total NISP	1536	128 100.0 %	100.0 %

Tab. 3. Anatomical distribution of the ibex remains from the Upper units of Les Barasses II (Number of Identified Specimens : NISP ; Minimal Animal Unit : MAU and Survival rates : MAU*100/NMImax). fragmentationTab. 4. Anatomical distribution of the ibex remains from the S4 (A) of Abri des Pêcheurs (Number of Identified Specimens : NISP ; Minimal Animal Unit : MAU and Survival rates : MAU*100/ NMImax).

Acknowledgments

The theme of this contribution comes partly from the Caprines Project initiated by the GDR 3591 (Research Group, 2013-2016) of CNRS-INEE 'Taphonomie,