Kazuo Nishimura

Thomas Seegmuller

Alain Venditti

Fiscal Policy, Debt Constraint and Expectation-Driven Volatility

Keywords: JEL classification: E32, H20, H68 Indeterminacy, endogenous cycles, public debt, income taxation

Imposing some constraints on public debt is often justified regarding sustainability and stability issues. This is especially the case when the ratio of public debt over GDP is restricted to be constant. Using a Ramsey model, we show that such a constraint can however be a fundamental source of indeterminacy, and therefore, of expectation-driven fluctuations. Indeed, through the intertemporal budget constraint of the government, income taxation negatively depends on future debt, i.e. on the expected level of production. This mechanism ensures that expectations on the future tax rate may be self-fulfilling. We show that this is promoted by a larger ratio of debt over GDP.

Introduction

To dampen the effects of the last financial crisis, many countries have engaged in expansionist fiscal policies. Such policies have been carried out even in countries that already experienced large levels of public debt. This partly explains the sovereign debt crisis that followed. To decrease the associated insolvency risk of public debt, there is now an increasing consensus to promote fiscal policies that fulfill some constraints on public debt. In accordance for instance with the Maastricht treaty, the rule stipulating that the ratio of public debt over GDP has to be less than a maximal value is often advocated to promote sustainability and stability.

Assuming that this constraint is binding, as now in the US and most European countries, we show that imposing such a rule can however be a source of macroeconomic instability related to self-fulfilling expectations on the future income tax rate. Indeed, if agents expect an increase of the future tax rate, they will invest less, implying a lower income in the future. According to the debt constraint, debt emission should be lower, and therefore the income tax rate has to increase today to satisfy the government intertemporal budget constraint.

To address this issue, we consider a Ramsey [START_REF] Ramsey | A mathematical theory of saving[END_REF] model extended by the introduction of a public sector. A constant level of public spending is financed through debt and distortionary taxation on income and debt earnings. To avoid insolvency of public debt, we introduce a debt constraint defined as a constant ratio of debt over GDP. 1 This ratio is considered as a policy parameter fixed by the government. The tax rate adjusts at each period to fulfill the intertemporal budget constraint of the government. Of course, when the ratio of debt over GDP is set at zero, there is no debt and the tax rate is counter-cyclical: a larger current income implies that a lower tax rate is needed to finance the public spending. When a positive level of debt is issued, the endogenous tax rate also depends on future income and capital level: if capital and, therefore, income raise at the next period, debt emission can be larger. Hence, current public spending is financed by further debt and a lower tax rate is needed to fulfill the government budget constraint. This means that the tax rate decreases with the next period level of income and capital.

Since labor is exogenous, the main effect of this fiscal policy goes through a distortion that affects the return of assets, especially capital. At a steady state, the tax rate is counter-cyclical, i.e. negatively depends on the level of capital: a larger amount of capital means a lower cost of debt reimbursement, more debt emission and a larger tax base. Hence, capital has two opposite effects on the after-tax interest rate: a positive one since the tax rate is counter-cyclical and a negative one as marginal productivity of capital is decreasing. These two opposite effects explain the multiplicity of steady states, generically an even number. They also explain that a larger level of debt-output ratio or public spending may raise capital accumulation and welfare. Obviously, both of these policy parameters have a positive effect on the tax rate, but the government intertemporal budget constraint may be balanced by an increase of capital, which implies a lower cost of debt reimbursement, more debt emission and a larger tax base.

Above all, the debt-output ratio affects dynamics, through the occurrence of expectation-driven fluctuations. To highlight this question, we first show that without debt, locall indeterminacy is always ruled out. However, the steady state can loose its saddle-path stability if the elasticity of capital-labor substitution is large enough. Moreover, if inputs are sufficiently substitutable and the intertemporal substitution in consumption is high enough, cycles of period two can also occur. This loss of stability comes from the above mentioned counter-cyclicality of the tax rate that implies an after-tax interest rate which is increasing with respect to current capital. Through the Euler equation, it promotes unstable dynamic paths and deterministic oscillations.

In contrast, when the debt-output ratio is positive, indeterminacy may occur. The loss of saddle-path stability also requires a large enough input substitution. However, if the elasticity of intertemporal substitution in consumption is large enough, the steady state is not unstable anymore, but becomes indeterminate through the occurrence of a Hopf bifurcation. Deterministic cycles occur, but also expectation-driven fluctuations. As explained earlier, when the debt-output ratio is positive, the tax rate negatively depends on next period capital stock, as it decreases with respect to new debt emission. This channel mainly explains the indeterminacy of the tax rate. Any expectation of a larger tax rate tomorrow implies less investment, i.e. a lower future stock of capital, which raises the current tax rate. We also note that the larger the debt-output ratio, the larger the range of elasticities of intertemporal substitution in consumption for indeterminacy. In this sense, a larger debt-output ratio promotes the volatility due to expectation-driven fluctuations. Hence, fixing the level of debt over output, a government faces a trade-off between volatility and welfare evaluated at the steady state. Reducing this ratio, fluctuations due to the volatility of expectations may be ruled out, but at the cost of a decreased stationary welfare.

The key ingredient driving our results is a distortion, explained by the endogenous tax rate, that affects the real interest rate. In fact, a huge literature has studied the (de)stabilizing role of non-linear tax rates on local indeterminacy. One can refer to Schmitt-Grohé and Uribe [START_REF] Schmitt-Grohé | Balanced-budget rules, distortionary taxes, and aggregate instability[END_REF] or Guo and Lansing [START_REF] Guo | Capital-labor substitution and equilibrium indeterminacy[END_REF] for income taxation, Giannitsarou [START_REF] Giannitsarou | Balanced budget rules and aggregate instability: the role of consumption taxes[END_REF] or Nourry et al. [START_REF] Nourry | Aggregate instability under balanced-budget consumption taxes: A re-examination[END_REF] for consumption taxation, Lloyd-Braga et al. [START_REF] Lloyd-Braga | Market distortions and local indeterminacy: A general approach[END_REF] for a critical approach. In contrast to us, these papers do not consider variable public debt and indeterminacy requires endogenous labor. The closest paper to ours is surely Schmitt-Grohé and Uribe [START_REF] Schmitt-Grohé | Balanced-budget rules, distortionary taxes, and aggregate instability[END_REF], where a constant level of public spending is financed through income taxation. Since they consider a balanced budget, the tax rate depends on current income only. 2 Therefore, in contrast to our result, indeterminacy requires a sufficiently elastic labor supply, which is at odds with empirical evidence (Blundell and MaCurdy [START_REF] Blundell | Labour supply: A review of alternative approaches[END_REF], Rogerson and Wallenius [START_REF] Rogerson | Micro and macro elasticities in a life cycle model with taxes[END_REF]).

In fact, the interplay between debt, capital and dynamics has been studied in a few papers only. Most of them consider endogenous growth frameworks. For instance, Greiner [START_REF] Greiner | An endogenous growth model with public capital and sustainable government debt[END_REF] exhibits the existence of a Hopf cycle, but there is no indeterminacy. The main ingredient for his result relies on his constraint on public debt sustainability which is very different from our, stipulating that the primary surplus is a linear function of income and debt. In Futagami et al. [START_REF] Futagami | Debt policy rule, productive government spending, and multiple growth paths[END_REF], the debt constraint concerns the ratio of debt over capital. There exist two steady states and local indeterminacy may emerge. In contrast to us, it is explained by productive public spending, while the tax rate is constant. However, as shown by Minea and Villieu [START_REF] Minea | Debt policy rule, productive government spending, and multiple growth paths: A note[END_REF], the mutiplicities are ruled out when the constraint is rather on the ratio of debt over output. Minea and Villieu [9] also consider productive public spending and a constant tax rate. Their sustainability constraint imposes that the deficit over output is constant. A type of global indeterminacy is exhibited that allows to explain a trap. We show here that we can get simultaneously global indeterminacy associated to the multiplicity of steady states, and local indeterminacy with persistent fluctuations around a steady state.

This paper is organized as follows. In the next section, we present the model. In Section 3, we analyze steady states and welfare. In Section 4, we show that indeterminacy occurs under a positive debt-output ratio only and we provide economic intuitions for our main results. A conclusion is given in Section 5, whereas most technical details are provided in the Appendix.

The model

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

Households

There is a representative infinitely lived household who supplies inelastically one unit of labor, holds k 0 > 0 as capital endowment and b 0 0 as initial public debt. Denoting c t his consumption at period t, the consumer's intertemporal utility function is given by:

+∞ t=0 β t u (c t ) (1)
where β ∈ (0, 1) is the discount factor and the utility function satisfies the following assumption:

Assumption 1. u (c) is C 0 over [0, +∞), C 2
over (0, +∞) and satisfies u ′ (c) > 0, u ′′ (c) < 0. In addition, the Inada condition lim c→0 u ′ (c) = +∞ holds. For further reference, we introduce the following elasticity: ǫ cc (c) ≡ -cu ′′ (c)/u ′ (c) > 0.

Each household derives income from wage, capital and government bonds that allow to finance public debt. Denote r t the real interest rate on physical capital, rt the return on government bonds, w t the real wage, δ ∈ (0, 1) the rate of capital depreciation and τ t ∈ [0, 1) the tax rate on income. The household maximizes (1) facing the budget constraint:

c t + k t+1 + b t+1 = (1 -τ t )[r t k t + w t ] + [1 + (1 -τ t )r t ]b t + (1 -δ)k t (2)
Utility maximization gives:

u ′ (c t ) βu ′ (c t+1 ) = R t+1 (3) u ′ (c t ) βu ′ (c t+1 ) = 1 + (1 -τ t+1 )r t+1 (4)
with R t+1 ≡ (1 -τ t+1 )r t+1 + 1 -δ and the transversality conditions:

lim t→+∞ β t u ′ (c t ) k t+1 = 0 and lim t→+∞ β t u ′ (c t ) b t+1 = 0 (5)
must hold. We get at the equilibrium the equality R t+1 = 1 + (1 -τ )r t+1 since physical capital k t+1 and government bonds b t+1 are perfectly substitutable saving assets.

Firms

A representative firm produces the final good y t , using a technology with constant returns y t = AF (K t , L t ). We denote

F (k t , 1) ≡ f (k t ), with k t ≡ K t /L t .
The production function f (k) satisfies:

Assumption 2. f (k) is C 0 over [0, +∞), C 2 over (0, +∞) and satisfies f ′ (k) > 0, f ′′ (k) < 0. In addition, the conditions lim k→0 f ′ (k) = +∞, lim k→+∞ f ′ (k) = 0 and lim k→+∞ f (k) = +∞ hold.
Profit maximization gives:

r t = Af ′ (k t ) ≡ r (k t ) and w t = Af (k t ) -k t Af ′ (k t ) ≡ w (k t ) (6)
In the following, we denote by s (k) ≡ kf ′ (k) /f (k) ∈ (0, 1) the capital share in total income and σ (k

) ≡ [s (k) -1] f ′ (k) / [kf ′′ (k)]
≥ 0 the elasticity of capital-labor substitution. We derive the following useful relationships:

r ′ (k)k/r(k) ≡ -(1 -s(k))/σ(k) and w ′ (k)k/w(k) ≡ s(k)/σ(k) (7)

Government

At each time t, a constant level of public spending G > 03 and debt issued at period t -1 are financed by taxation of income and debt yields, at the rate τ t ∈ [0, 1), and by issuing new debt b t+1 . The government faces the following budget constraint:

G + (1 + rt )b t = τ t (y t + rt b t ) + b t+1 (8)
We will assume that debt is a fixed proportion of GDP, namely b t = αy t . In accordance for instance with the Maastricht criteria, this restriction can be seen as a stabilizing constraint on debt that prevents any explosive behavior. It has previously been introduced by de la Croix and Michel [START_REF] De La Croix | A Theory of Economic Growth, Dynamics and Policy in Overlapping Generations[END_REF], and Minea and Villieu [START_REF] Minea | Debt policy rule, productive government spending, and multiple growth paths: A note[END_REF]. In the following, α will be seen as a policy parameter under the control of the government. The case without debt is of course obtained when α = 0.

Intertemporal equilibrium

As the representative household supplies inelastically one unit of labor, equilibrium on the labor market requires L t = 1. Equilibrium on the asset markets is ensured by 1+(1-τ t )r t = (1-τ t )r (k t )+1-δ where, using (8) and b t = αAf (k t ), the tax rate is endogenously determined by:

τ t = G-αAf (kt+1) Af (kt)(1+αAf ′ (kt)) + α 1-δ+Af ′ (kt) 1+αAf ′ (kt) (9) Let ∆ t ≡ 1 -τ t , with: ∆ t = αAf (kt+1)-G Af (kt)(1+αAf ′ (kt)) + 1-α(1-δ)
1+αAf ′ (kt) [START_REF] Minea | Debt policy rule, productive government spending, and multiple growth paths: A note[END_REF] Taking into account that debt is a fixed proportion of GDP, we define an intertemporal equilibrium as follows: Definition 1. Under Assumptions 1-2, given k 0 > 0,4 an intertemporal equilibrium is a sequence (k t , c t ), for t = 0, 1... + ∞, satisfying:

u ′ (c t ) = βR t+1 u ′ (c t+1 ) (11) c t + k t+1 + αAf (k t+1 ) = R t (k t + αAf (k t )) + ∆ t w(k t ) ( 12
)
where ∆ t is given by [START_REF] Minea | Debt policy rule, productive government spending, and multiple growth paths: A note[END_REF], R t = ∆ t r (k t ) + 1 -δ and the transversality condition

lim t→+∞ β t u ′ (c t ) (k t+1 + αf (k t+1 )) = 0 holds.
3 Steady state analysis

Let ∆(k) ≡ 1+αδ-G/(Af (k)) 1+αAf ′ (k) (13)
A steady state is defined by:

c = (1/β -1)(k + αAf (k)) + ∆(k)w(k) (14)
H(k) ≡ ∆(k)Af ′ (k) = θ/β (15)
with θ ≡ 1 -β(1 -δ) ∈ (0, 1). The stationary value of k is obtained from equation (15) and equation ( 14) determines the stationary level of c. Hence, the existence, uniqueness and multiplicity of steady states is given by the number of solutions k of (15). Any steady state has to satisfy τ ∈ [0, 1), or equivalently ∆(k) ∈ (0, 1]. This restriction leads to the following result:

Lemma 1. Under Assumptions 1-2, there exist k > k > 0 such that any stationary solution k belongs to (k, k].
Proof. See Appendix 6.1.

Existence and multiplicity of steady states

At a steady state, the tax rate is counter-cyclical, i.e. decreasing in k, because we have ∆ ′ (k)k/∆(k) > 0. Since the marginal productivity of capital Af ′ (k) is decreasing, capital k has two opposite effects on the after-tax real interest rate H(k) which explain the multiplicity of steady states.

To analyze the existence and the number of stationary solutions, let us focus on equation (15). We have H(k) = 0, H(k) < θ/β (see Appendix 6.2) and:

ǫ H (k) ≡ H ′ (k)k H(k) = 1 1+αδ-G/(Af (k)) s(k)G Af (k) -1-s(k) σ(k) ∆(k) (16) Therefore, ǫ H (k) > 0 if and only if σ(k) > σ α T (k), with: σ α T (k) ≡ 1-s(k) s(k)G/(Af (k)) ∆(k) (17)
We then derive the following proposition:

Proposition 1. Under Assumptions 1-2, there are bounds δ ∈ (0, 1) and β ∈ (0, 1) such that there generically exist an even number of steady states k ∈ (k, k] if δ ∈ (0, δ) and β ∈ ( β, 1).

Proof. See Appendix 6.2.

Note that Proposition 1 still applies if α = 0 except that k tends to +∞. 5 By direct inspection of equations ( 16) and ( 17), we derive that there are two steady

states if σ(k) -σ α T (k) is decreasing in k. Example: Consider a CES technology f (k) = (sk σ-1 σ + 1 -s) σ σ-1
, with s ∈ (0, 1) and σ ∈ (0, 1]. When σ = 1, we get the Cobb-Douglas formulation We then derive:

f (k) = k s . We have s(k) = f ′ (k)k/f (k) = sk σ-1 σ /(sk σ-1 σ + 1 -s). When σ ≤ 1, s(k) is decreasing in k or constant and thus σ α T (k) is increasing in k. Graphically, we get: ✲ ✻ H(k) k 0 θ/β k k k o k e
Corollary 1. Let Assumptions 1-2 hold, δ ∈ (0, δ), β ∈ ( β, 1) and f (k) = (sk σ-1 σ + 1 -s) σ σ-1 with s ∈ (0, 1) and σ ∈ (0, 1]. Then, there exist two steady states k o , k e ∈ (k, k).
As mentioned earlier, the multiplicity of steady states occur because capital has two opposite effects on the after-tax interest rate ∆(k)Af ′ (k). The marginal productivity of capital decreases with k, but the tax rate is counter-cyclical, implying that ∆(k) raises with k. Indeed, a larger level of capital means, on the one hand, a lower interest rate and therefore a lower cost for debt reimbursement and, on the other hand, a larger income, i.e. a larger debt to finance public spendings and a larger tax base.

Using [START_REF] Rogerson | Micro and macro elasticities in a life cycle model with taxes[END_REF], we further see that ∆(k) does not only depend on the marginal productivity of capital, but also on the level of production. This means that the tax rate is counter-cyclical even if the inputs are arbitrarily large substitutes. On the contrary, the decreasing relationship between f ′ (k) and k is weakened when the elasticity of input substitution is large enough. Therefore, the two opposite effects of capital on the after tax interest rate ∆(k)Af ′ (k) are reinforced when σ is not arbitrarily large, which explains the results of Corollary 1.

In any case, the multiplicity of steady states is a form of global indeterminacy. In endogenous growth models, Futagami et al. [START_REF] Futagami | Debt policy rule, productive government spending, and multiple growth paths[END_REF] and Minea and Villieu [START_REF] Minea | Persistent deficit, growth and indeterminacy[END_REF] similarly obtain a multiplicity of stationary growth paths. However, when a sustainability constraint on the ratio of debt over output is considered, this multiplicity is ruled out (Minea and Villieu [START_REF] Minea | Debt policy rule, productive government spending, and multiple growth paths: A note[END_REF]). This is not the case in our framework since, due to the government budget constraint, the tax rate is endogenous.

As we will see now, the multiplicity of steady states also allows us to identify some contrasted effects of fiscal policy depending on the level of capital.

Comparative statics and welfare

Let k o be a steady state such that ǫ H (k o ) > 0, or equivalently σ(k o ) > σ α T (k o ), and k e a steady state such that ǫ H (k e ) < 0, or equivalently σ(k e ) < σ α T (k e ). Following an increase of one of the two policy parameters α or G, we determine now how the levels of capital, consumption and, therefore, welfare evolve depending on the steady state which is considered.

Proposition 2. Under Assumptions 1-2, we have that:

• a larger α or G increases the levels of capital, consumption and welfare at a steady state k o ;

• a larger α or G decreases the levels of capital, consumption and welfare at a steady state k e .

Proof. See Appendix 6.3.

The tax rate increases with both G and α. Indeed, a larger public spending and a heavier debt burden have to be financed through a higher tax rate. The marginal productivity of capital is of course decreasing with capital, while the tax rate is counter-cyclical, i.e. decreasing with capital. These two effects go in opposite directions. When the first effect dominates (steady state k e ), a larger level of either G or α, that positively affects the tax rate, is mainly compensated by an increase of the marginal productivity of capital, i.e. a decrease of capital. On the contrary, when the second effect dominates (steady state k o ), it is mainly compensated by a raise of capital which negatively affects the tax rate, because of a lower cost of debt and a larger income.

The effects on consumption and welfare are derived taking into account, first, that consumption positively depends on capital and changes in policy parameters mainly modify consumption through their effects on capital and, second, that welfare raises with consumption.

To fix ideas, consider the configuration where there are two steady states (see for instance Corollary 1). Steady states are ranked according to their capital level, i.e. k o < k e (see Figure 1). According to our results, an expansive policy is recommended when the economy is characterized by a low level of capital, whereas at the steady state with the larger level of capital, the government is rather recommended to reduce public spending and/or the debt-output ratio.

Normalized steady state

We now use a standard method to ensure the existence of a steady state, which value is not affected by the structural parameters such that the elasticity of capital-labor substitution, the elasticity of intertemporal substitution in consumption or the share of debt over GDP that will be used as bifurcation parameters in the local stability analysis. A normalized steady state k = 1 is a solution of ∆(1)Af ′ (1) = θ/β, with:

∆(1) = 1+αδ-G/(Af (1)) 1+αAf ′ (1) (18)
Let us introduce the following assumption:

Assumption 3. α < α Max ≡ β/(1 -β)
It is not restrictive as β is usually chosen to be close to one. We derive the following proposition:

Proposition 3. Under Assumptions 1-3, there exists A * > 0 such that k = 1 is a normalized steady state if and only if A = A * > 0. The corresponding value of consumption is c = (1/β -1)(1 + αA * f (1)) + ∆(1)w(1).
Proof. See Appendix 6.4.

In the following, let us denote s ≡ s(1), σ ≡ σ(1) and ǫ cc ≡ ǫ cc (c). As there generically exist an even number of steady states which are characterized by different welfare properties, we need to know whether the normalized steady state is located on an increasing portion of the curve H(k) (as k o ) or on a decreasing portion (as k e ).

Corollary 2. Under Assumptions 1-3, let A = A * and consider σ T ≡ σ α T (1) > 0 as given by ( 17). Then, the normalized steady state k = 1 corresponds to k o when σ > σ T and to k e when σ < σ T .

Proof. See Appendix 6.5.

Note that σ T appears not to depend on α (see Appendix 6.5).

Expectation-driven volatility under a positive debt-output ratio

We start by considering the benchmark case of an economy without debt. We will show that local indeterminacy and therefore sunspot fluctuations cannot arise. In a second step, when the debt-output ratio is positive, we will show that local indeterminacy of the normalized steady state may emerge. We will then prove that a sufficiently large debt-output ratio is destabilizing, promoting expectation-driven fluctuations.

The economy without debt

The economy without debt is obviously obtained when α = 0. The stability properties of the steady state are summarized in the following proposition:

Proposition 4. Under Assumptions 1-2, let A = A * and α = 0. Then, there exist ǫ 0 F > 0, σ 0 F ≥ 0 and σ T > 0 such that the following results generically hold.

• If ǫ cc < ǫ 0 F , the normalized steady state is a saddle for σ < σ T , a source for σ T < σ < σ 0 F , and a saddle for σ > σ 0 F .

• If ǫ cc ǫ 0 F , the normalized steady state is a saddle for σ < σ T and a source for σ > σ T .

Flip and transcritical bifurcations generically occurs when σ crosses respectively σ 0 F and σ T .

Proof. See Appendix 6.6.

This proposition shows that without debt, local indeterminacy never occurs, ruling out expectation-driven fluctuations. However, deterministic endogenous business cycles may emerge through a flip bifurcation if the elasticity of intertemporal substitution in consumption is sufficiently large and inputs are sufficiently substitutes. Otherwise, the steady state is either saddle-path stable or unstable.

The economy with positive debt

Let us now consider the economy with positive debt assuming α > 0. Our aim is to check whether debt has a stabilizing or a destabilizing effect on the economy promoting or not expectation-driven fluctuations. We derive the following results: Proposition 5. Under Assumptions 1-3, let A = A * and α > 0. Then, there exist β ∈ (0, 1), δ ∈ (0, 1), α,

α ǫF ∈ (0, α Max ) with α < α ǫF , ǫ α H ≥ 0, ǫ α F > 0, σ α
F ≥ 0 and σ T > 0 such that when β ∈ ( β, 1) and δ ∈ (0, δ), the following results generically hold. 1 -For any given α ∈ (0, α):

• If ǫ cc ≥ ǫ α
F , the normalized steady state is a saddle for σ < σ T and a source for σ > σ T .

• If ǫ α H < ǫ cc < ǫ α F , the normalized steady state is a saddle for σ < σ T , a source for σ T < σ < σ α F and a saddle for σ > σ α F .

• If ǫ cc < ǫ α H , the normalized steady state is a saddle for σ < σ T , a sink for σ T < σ < σ α F and a saddle for σ > σ α F . 2 -For any given α ∈ (α, α ǫF ):

• If ǫ cc > ǫ α
H , the normalized steady state is a saddle for σ < σ T and a source for σ > σ T .

• If ǫ α F ≤ ǫ cc < ǫ α H , the normalized steady state is a saddle for σ < σ T and a sink for σ > σ T .

• If ǫ cc < ǫ α F , the normalized steady state is a saddle for σ < σ T , a sink for σ T < σ < σ α F and a saddle for σ > σ α F . 3 -For any given α ∈ (α ǫF , α Max ):

• If ǫ cc > ǫ α
H , the normalized steady state is a saddle for σ < σ T and a source for σ > σ T .

• If ǫ cc < ǫ α H , the normalized steady state is a saddle for σ < σ T and a sink for σ > σ T .

In each case, flip and transcritical bifurcations generically occur when σ crosses respectively σ α F and σ T .

Proof. See Appendix 6.7.

When the elasticity of intertemporal substitution in consumption 1/ǫ cc is too low, i.e. smaller than 1/ǫ α H , the steady state is either saddle-path stable or unstable. Such a result also arises in the absence of public debt (see Proposition 4). On the contrary, when the elasticity of intertemporal substitution in consumption 1/ǫ cc is large enough, i.e. larger than 1/ǫ α H , the steady state becomes locally indeterminate when σ crosses σ T from below. It is important to note from the expression (27) (as given in Appendix 6.5) that the critical value σ T does not depend on the debt-output ratio α, and is quite low when β ∈ ( β, 1) and δ ∈ (0, δ) (as θ is close to zero). This means that as soon as the debt-output ratio is positive, expectation-driven fluctuations may occur. Contrary to most of existing contributions considering one-sector growth models, indeterminacy is obtained here in an economy with exogenous labor. By continuity, we conjecture that expectation-driven fluctuations would still occur under a weakly elastic labor supply, which is a realistic assumption (Blundell and MaCurdy [START_REF] Blundell | Labour supply: A review of alternative approaches[END_REF], Rogerson and Wallenius [START_REF] Rogerson | Micro and macro elasticities in a life cycle model with taxes[END_REF]). This is explained by the fact that the fiscal policy we consider introduces a distortion ∆ t that does not only depend on current capital k t , but also on future capital k t+1 , and is in contrast to many contributions that analyzed the role of non-linear tax on expectation-driven fluctuations. The closest paper to ours is surely Schmitt-Grohé and Uribe [START_REF] Schmitt-Grohé | Balanced-budget rules, distortionary taxes, and aggregate instability[END_REF], where a constant level of public spending is financed through income taxation. Since they do not consider debt, the tax rate depends on current income only and indeterminacy requires a sufficiently elastic labor supply, which implies some unconventional labor supply function that is decreasing with respect to the wage rate. Such a restrictive condition is obviously not necessary in our framework.

It is also worthwhile to note that, for a given level of σ larger than σ T , when the elasticity of intertemporal substitution in consumption 1/ǫ cc increases and crosses the value 1/ǫ α H from below, the steady state becomes locally indeterminate through a Hopf bifurcation generating quasi-periodic endogenous fluctuations: 6Corollary 3. Under Assumptions 1-3, let A = A * , α > 0, β ∈ (β, 1) and δ ∈ (0, δ). Then the following results generically hold.

1 -For any given α ∈ (0, α), if σ T < σ < σ α F , the normalized steady state is a sink for ǫ cc < ǫ α H and becomes a source for ǫ α H < ǫ cc < ǫ α F . 2 -For any given α ∈ (α, α ǫF ), if σ > σ T , the normalized steady state is a sink for ǫ α F ǫ cc < ǫ α H and becomes a source for ǫ cc > ǫ α H . 3 -For any given α ∈ (α ǫF , α Max ), if σ > σ T , the normalized steady state is a sink for ǫ cc < ǫ α H and becomes a source for ǫ cc > ǫ α H . In each case, a Hopf bifurcation generically occurs when ǫ cc crosses ǫ α H , Note that in contrast to σ T , the critical values ǫ α H , ǫ α F and σ α F depend on the debt-output ratio α (see Appendix 6.7). In particular, ǫ α H and ǫ α H are respectively increasing and decreasing functions of α while σ α F is an increasing function of α, so that the range of values of ǫ cc and σ enlarges when α increases. Therefore, for given values of ǫ cc and σ, when α raises from 0, we can derive from the above analysis some precise effect of debt on the stability properties of the normalized steady state.

Corollary 4. Under Assumptions 1-3, let A = A * and consider the critical values α, ǫ α

H , ǫ α F , σ α F and σ T > 0 as given in Proposition 5. Assume that ǫ cc ∈ (ǫ α H , ǫ 0 F ) and σ > σ 0 F (> σ T ). Then, there exist β ∈ (0, 1), δ ∈ (0, 1), α F ∈ (0, α) and α H ∈ (α, α Max ) such that when β ∈ (β, 1) and δ ∈ (0, δ), the normalized steady state is a saddle for α ∈ [0, α F ), a source for α ∈ (α F , α H ) and a sink for α ∈ (α H , α Max ). Moreover, flip and Hopf bifurcations generically occur when α crosses respectively α F and α H .

Proof. See Appendix 6.8.

Corollary 4 clearly shows that debt has a destabilizing effect, generating endogenous cycles and self-fulfilling expectations. It is worth noting however that this destabilizing effect has to be mitigated by the welfare analysis. Indeed, as shown by Corollary 2, along a steady state k o , around which indeterminacy may occur, a raise of α improves the stationary welfare.

Economic intuition

We now focus on the economic mechanisms that generate the loss of saddlepath stability and the emergence of indeterminacy. Recall that the dynamics are governed by the Euler equation describing the intertemporal trade-off faced by households and their budget constraint. Using ( 2) and ( 8), this budget constraint can also be written:

c t + k t+1 -(1 -δ)k t + G = y t = Af (k t ) (19)
and using for simplification the utility function u(c) = c 1-ǫcc /(1-ǫ cc ), the Euler equation becomes:

ct+1 ct ǫcc = β(1 -δ + ∆ t+1 r t+1 ) (20)
where r t+1 is given by ( 6) and ∆ t+1 by [START_REF] Minea | Debt policy rule, productive government spending, and multiple growth paths: A note[END_REF].

When compared with the standard Ramsey model, these two equations are modified first by the constant level of public expenditures G > 0, and second by the fact that the usual intertemporal households' trade-off is now affected by a distortion ∆ t+1 = 1 -τ t+1 lowering the return of capital. A direct inspection of [START_REF] Minea | Debt policy rule, productive government spending, and multiple growth paths: A note[END_REF] shows that ∆ t+1 is a function of k t+1 and k t+2 , where the derivatives ∂∆ t+1 /∂k t+2 and ∂∆ t+1 /∂k t+1 , evaluated at the normalized steady state, are respectively given by the following expressions

B 4 (α) ≡ αs 1 + αA * f ′ 0 (21) B 5 (α, σ) ≡ G A * f -α s(1 + αA * f ′ ) + αA * f ′ 1-s σ 1 + αδ -G A * f (1 + αA * f ′ ) 2 (22)
When the debt-output ratio α is equal to 0, we have ∂∆ t+1 /∂k t+2 = 0 and ∂∆ t+1 /∂k t+1 > 0. The tax rate is then counter-cyclical, decreasing with production and capital of the same period. This distortion explains the instability of the steady state. Of course, in the limit case where G = 0, the tax rate is zero, i.e. there is no distortion, and σ T tends to +∞. The steady state is unique and saddle-path stable in this last case.

To give a more intuitive explanation of instability, using ( 7), ( 18) and ( 22), we compute the elasticity of ∆ t+1 r t+1 with respect to k t+1 which is given by

sG/(A * f ) 1-G/(A * f ) -1-s σ
This expression is strictly positive for σ > σ T . As shown in Proposition 4, this is a necessary condition to have instability. This condition means that the after-tax interest rate is increasing in capital of the same period and its role on instability can be understood as follows. Suppose that one deviates from the steady state by an increase of capital k t . Using (19), this generates a raise of income and therefore, of c t and k t+1 . The above discussion allows us to conclude that the after-tax interest rate raises. Using (20), this implies a raise of consumption growth c t+1 /c t that explains unstable dynamic paths.

Proposition 4 also establishes that a cycle of period 2 occurs through a flip bifurcation if the elasticity of capital-labor substitution crosses the value σ 0 F , which is larger than σ T , ensuring that the after-tax interest rate is still increasing in capital. It also requires that ǫ cc is sufficiently small (ǫ cc < ǫ 0 F ). To explain the emergence of such a cycle, consider than one deviates from the steady state through a decrease of c t exactly compensated by an increase of k t+1 . This implies an increase of the after-tax rate ∆ t+1 r t+1 . Because ǫ cc is small, we derive from the Euler equation ( 20) an arbitrarily large raise of consumption growth c t+1 /c t . This means that c t+1 strongly raises. Therefore, even if k t+1 increases, and income at period t + 1 too, k t+2 reduces, which explains oscillations and the existence of cycles of period 2.

When the debt-output ratio α is strictly positive, we have ∂∆ t /∂k t+1 > 0, i.e. the tax rate τ t negatively depends on future capital k t+1 . When the production at the next period reduces, future debt needs to reduce as well because of the debt constraint. Therefore, public spending is further financed by taxation. As shown in Proposition 5, expectation-driven fluctuations may occur if ǫ cc is low enough. As we will explain now, the negative link between τ t and k t+1 is a key mechanism to get local indeterminacy.

Indeed, if households expect a larger tax rate at the next period, they expect a lower return for their investment. Therefore, k t+1 reduces. Through the effect explained above, the current tax rate raises. Hence, depending on expectations on future tax rates, alternative dynamic paths can be constructed. However, to be self-fulfilling, these dynamic paths should satisfy the inter-temporal household's choice (20). On the one hand, a lower return for k t+1 means that ∆ t+1 r t+1 decreases. On the other hand, because capital k t is predetermined, the decrease of k t+1 implies a raise of consumption c t (see equation ( 19)). The decrease of k t+1 can also generate a decrease of future consumption c t+1 , because it lowers the income Af (k t+1 ). This last effect is fully rationalized by the inter-temporal household's choice (20). Indeed, if ǫ cc is low enough, the decrease of ∆ t+1 r t+1 implies a strong fall of consumption growth c t+1 /c t , which is explained not only by a larger consumption at period t, but also a smaller consumption at period t + 1. In this case, expectations are self-fulfilling and the associated dynamic path experiences oscillations through the reversal of consumption through time. We finally note that a larger α reinforces the intertemporal channel on the tax rate ∂∆ t+1 /∂k t+2 which is the source of indeterminacy, and which shows that a larger ratio of debt over GDP promotes expectation-driven fluctuations (Corollaries 3 and 4).

Conclusion

In this paper we consider a Ramsey model extended to include a public sector with public spending financed by income taxation and public debt. In order to guarantee sustainability, debt is assumed to be a fixed proportion of GDP. The tax rate, which adjusts to satisfy the government budget constraint, then depends on current capital, through income and debt reimbursement, and on next period capital through expected debt emission.

Along a stationary solution, capital has two opposite effects on the after-tax interest rate: a positive one since the tax rate is counter-cyclical and a negative one through the marginal productivity. These two opposite effects explain the multiplicity of steady states and also imply that a larger level of debt-output ratio or public spending can foster capital accumulation and welfare. Indeed, at a steady state, they both imply a stronger fiscal pressure, which may be dampened by a raise of income.

Since income taxation decreases following a larger level of future debt emission, the tax rate becomes counter-cyclical with respect to the expected income. This relationship is a key mechanism for indeterminacy. Indeed, if agents expect a larger future tax rate, they decrease investment. This means lower debt emission and therefore a larger tax rate today to balance the budget. We show that the occurrence of expectation-driven volatility associated to this indeterminacy is promoted by a larger debt-output ratio.

Appendix

Proof of Lemma 1

Any steady state has to satisfy τ ∈ [0, 1), or equivalently ∆(k) ∈ (0, 1]. On the one hand, ∆(k) > 0 requires:

Af (k) > G 1+αδ This is satisfied for k > k, where k is defined by Af (k) = G/(1 + αδ).
On the other hand, ∆(k) 1 is equivalent to:

α[δ -Af ′ (k)] G Af (k) (23)
When k increases from 0 to +∞, the left-hand side increases from -∞ to αδ > 0, while the right-hand side decreases from G/(Af (0)) > 0 to 0. Therefore, there is a unique k, defined by α[δ -Af ′ (k)] = G/(Af (k)), such that inequality (23) is satisfied for all k k. Substituting k in (23), one can easily check that k < k.

Note that when α = 0, k keeps a finite and strictly positive value, whereas k tends to +∞. 7

Proof of Proposition 1

Let us consider the expression

H(k) = ∆(k)Af ′ (k). When k = k, we have obviously H(k) = 0 < θ/β. Moreover, when k = k, we get H(k) = Af ′ (k) = δ -G/[αAf (k)] < θ/β since G/[αAf (k)] > δ -θ β = -1-β β
Let us now consider the expressions of ǫ H (k) and σ α T (k) respectively given by ( 16) and (17). Since σ α T (k) = 0, we have ǫ H (k) = +∞. Let us then consider the values of k such that ǫ H (k) = 0. By continuity of ǫ H (k), there must exist at least one solution of this equation in the interior of (k, k) if θ is sufficiently close to 0, i.e. δ close to 0 and β close to 1. However multiple solutions may arise. More precisely, we get ǫ H (k) = 0 if and only if

σ(k) = 1-s(k) s(k)G/(Af (k)) ∆(k) = σ α
T (k) Assume first that δ = 0. In this case, k = +∞. Over the set of values of k that are solutions of this equation, let us denote k maximal value which satisfies

∆( k) = 1-G/(Af ( k) 1+αAf ′ ( k) = σ( k) s( k)G/(Af ( k)) 1-s( k)
We can then compute

H( k) = σ( k) s( k)Gf ′ ( k)/(f ( k)) 1-s( k)
It follows that there generically exists an even number of steady states solutions of H(k) = 0 if and only if H( k) > (1 -β)/β. We conclude therefore that there exists β 0 ∈ (0, 1) as given by

β 0 = 1 + σ( k) s( k)Gf ′ ( k)/(f ( k)) 1-s( k) -1
such that H( k) > (1 -β)/β if and only if β ∈ (β 0 , 1). By continuity, there exist δ ∈ (0, 1) and thus β δ ∈ (0, 1) such that the same results holds if δ ∈ (0, δ) and β ∈ (β δ , 1). The result follows considering β = max δ∈(0, δ) β δ .

Proof of Proposition 2

Using ( 13) and (15), we get the following elasticity:

ǫ H,G (k) ≡ ∂H(k) ∂G G H(k) = -G/(Af (k)) 1+αδ-G/(Af (k)) < 0 We derive that (dk/k)/(dG/G) = -ǫ H,G (k)/ǫ H (k) has the sign of ǫ H (k).
Using ( 13) and (15) again, we get:

ǫ H,α (k) ≡ ∂H(k) ∂α α H(k) = - α(1/β-1)
1+αδ-G/(Af (k)) < 0 Therefore, (dk/k)/(dα/α) = -ǫ H,α (k)/ǫ H (k) has the sign of ǫ H (k).

Using [START_REF] Rogerson | Micro and macro elasticities in a life cycle model with taxes[END_REF] and ( 14), we note that the stationary level of consumption can also be written c = Af (k) -δk -G. We derive that:

dc dα = ∂c ∂k ∂k ∂α = -(Af ′ (k) -δ) k α ǫH,α(k) ǫH (k)
which has the sign of ǫ H (k). We also find that:

dc dG = 1 ǫH (k) s(k) 1+αδ-G/(Af (k)) Af ′ (k)(1-G/(Af (k))-δ Af ′ (k) + 1-s(k) σ(k) 1 1+αAf ′ (k)
Using ( 13) and (15), we can show that [1 -G/(Af (k))]Af ′ (k) > δ, which allows us to conclude dc/dG has the sign of ǫ H (k). We finally observe that the welfare evaluated at a steady state is increasing in c. Therefore, the results follow from the fact that ǫ H (k o ) > 0 and ǫ H (k e ) < 0.

Proof of Proposition 3

The existence of a normalized steady state k = 1 is ensured if there is a unique

A > 0 solving Γ(A) ≡ f ′ (1) A(1+αδ)-G/f (1) 1+αAf ′ (1) = θ β (24)
Obviously, Γ ′ (A) > 0, Γ(0) = -f ′ (1)G/f (1) < θ/β and Γ(+∞) = (1 + αδ)/α. Under Assumption 3, we have Γ(+∞) > θ/β, which means that there is a unique A * as given by

A * = θ β +Gs(1) f ′ (1)(1-α(1-β) β ) (25)
that solves Γ(A * ) = θ/β. We need also to verify that 1 ∈ (k, k]. We get 1 > k if and only if 1 + αδ > G/(A * f (1)) which obviously follows from (24). Let us focus now on 1 < k which holds if and only if G/(A * f (1)) > α(δ -A * f ′ (1)). Using the expression of A * , we easily show that A * f ′ (1) > δ. The stationary consumption level is finally derived from (14).

Proof of Corollary 2

The normalized steady state satisfies (c, k) = (1, (1/β -1)(1 + αA * f (1)) + ∆(1)w( 1)). Let us denote f = f (1), f ′ = f ′ (1) and ∆ = ∆(1). As shown by equations ( 16) and (17), k = 1 is located on an increasing portion of H(k) if and only if σ > σ α T (1) ≡ 1-s sG/(A * f ) ∆(1) Using ( 24) and (25), we get:

G A * f = sG(β-α(1-β)) θ+sβG and ∆(1) = θ β β-α(1-β) θ+sβG (26)
We then derive that

σ α T (1) ≡ σ T = (1-s)θ s 2 βG (27)

Proof of Proposition 4

We characterize the stability properties and the occurrence of local bifurcations by linearizing the dynamic system ( 11)-( 12) around the normalized steady state (c, k) = (1, (1/β -1)(1 + αA * f (1)) + ∆(1)w( 1)).

Lemma 6.1. Under Assumptions 1-3, the characteristic polynomial is given by P (λ) ≡ λ 2 -T (α, σ)λ + D(α) = 0, where D(α) and T (α, σ) are the determinant and the trace of the associated Jacobian matrix:

D(α) = ǫccB3(α) ǫcc 1- θα β(1+αδ-G/(A * f )) +θ B 4 (α) ∆ B 1 (α)-δ B 2 (α) T (α, σ) = 1 + D(α) + B 1 (α)-δ B 2 (α) θ 1-s σ - B 4 (α) ∆ - B 5 (α,σ) ∆ ǫcc 1- θα β(1+αδ-G/(A * f )) +θ B 4 (α) ∆ B 1 (α)-δ B 2 (α)
with

B 1 (α) ≡ θ β β -α(1 -β) + α(θ + sβG) s[β -α(1 -β)] + α(θ + sβG) > δ (28) B 2 (α) ≡ s β -α(1 -β) + α(θ + sβG) s[β -α(1 -β)] + α(θ + sβG) > 0 (29) B 3 (α) ≡ 1 -δ + θ β 1 -α(1 -δ) 1 + αδ -G/(A * f ) (30) B 4 (α) ≡ αs 1 + αA * f ′ 0 (31) B 5 (α, σ) ≡ G A * f -α s(1 + αA * f ′ ) + αA * f ′ 1-s σ 1 + αδ -G A * f (1 + αA * f ′ ) 2 (32)
Proof. Linearizing the dynamical system (11)-( 12) around k = 1, we obtain:

ǫ cc + θ B4(α) ∆ B 1 (α)-δ B 2 (α) 1- θα β(1+αδ-G/(A * f )) dct+1 c = ǫ cc + B 1 (α)-δ B 2 (α) θ 1- θα β(1+αδ-G/(A * f )) 1-s σ -B5(α,σ) ∆ -B4(α) ∆ B3(α) 1- θα β(1+αδ-G/(A * f )) dct c - B3(α)θ 1- θα β(1+αδ-G/(A * f )) 1-s σ -B5(α,σ) ∆ -B4(α) ∆ B3(α) 1- θα β(1+αδ-G/(A * f )) dkt k 1 - θα β(1+αδ-G/(A * f )) dkt+1 k = -B1(α)-δ B2(α) dct c + B 3 (α) dkt k
where B 1 (α), B 2 (α), B 3 (α), B 4 (α) and B 5 (α, σ) are given by ( 28)-(32). Since T (α, σ) and D(α) are respectively the trace and the determinant of the associated Jacobian matrix, the lemma follows after simplifications.

We may now prove Proposition 4. Let α = 0. Using Lemma 6.1, we get

D(0) = 1 -δ + θ β 1 1 -G/(A * f ) = 1 β + sG ≡ D 0 > 1 (33) T (0, σ) = 1 + D 0 + θ ǫ cc θ sβ -δ 1 -s σ -s G/(A * f ) 1 -G/(A * f ) ≡ T 0 (σ) (34)
Equation (33) implies that the steady state is never a sink. Using (34), we conclude that T 0 (σ) is decreasing in σ with T 0 (0) = +∞ and

T 0 (+∞) = 1 + D 0 -θ ǫcc θ sβ -δ s G/(A * f ) 1-G/(A * f ) (35)
We then have T 0 (+∞) < 1 + D 0 and T 0 (+∞) < -1 -D 0 if and only if

ǫ cc < ǫ 0 F , with ǫ 0 F ≡ ( θ sβ -δ)sβG/(A * f ) 2 1+ 2-δ A * f ′ (36)
Graphically we derive from these results:

✻ ✲ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ T D 1 -1 -2 2 1 + T (0, σ) + D 0 = 0 1 -T (0, σ) + D 0 = 0 σ → 0 σ = +∞ σ 0 T Figure 2: The case ǫ cc ǫ 0 F . ✻ ✲ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ T D 1 -1 -2 2 1 + T (0, σ) + D 0 = 0 1 -T (0, σ) + D 0 = 0 σ → 0 σ = +∞ σ 0 T σ 0 F Figure 3: The case ǫ cc < ǫ 0 F .
Let σ 0 T be defined by 1 -T 0 (σ 0 T ) + D 0 = 0 and σ 0 F by 1 + T 0 (σ 0 F ) + D 0 = 0. Straightforward computations using (33)-(34) give

σ 0 F ≡ (1-s)θ( θ sβ -δ) 2s(ǫ 0 F -ǫcc)A * f ′ 1+ 2-δ A * f ′ (37)
and σ 0 T = σ T as given by (27). We get 1 -T 0 (σ) + D 0 < 0 for σ < σ T , 1-T 0 (σ)+D 0 > 0 and 1+T 0 (σ)+D 0 > 0 for σ ∈ (σ T , σ 0 F ), and 1+T 0 (σ)+D 0 < 0 for σ > σ 0 F . The results follow by setting σ 0 F = +∞ when ǫ cc ≥ ǫ 0 F .

Proof of Proposition 5

From Lemma 6.1, using (28)-(31), we observe that D(α) is increasing in ǫ cc . When ǫ cc tends to 0, D(α) tends to 0, while, using (24) and (25), when ǫ cc tends to +∞, we get under Assumption 3:

lim ǫcc→+∞ D(α) ≡ D +∞ (α) = B3(α) 1- θα β(1+αδ-G/(A * f )) = 1+sβG-α(1-δ)(1-β) β-α(1-β) > 1
Therefore, there exists a value ǫ α H , as given by

ǫ α H ≡ (θ+βGs)βαs B 1 (α)-δ B 2 (α) βGs+(1-β)(1+αδ) (38)
such that D +∞ (α) ∈ (0, 1) for ǫ cc < ǫ α H , D +∞ (α) = 1 for ǫ cc = ǫ α H and D +∞ (α) > 1 for ǫ cc > ǫ α H . Consider now the trace T (α, σ) as given in Lemma 6.1. We immediately get T (α, 0) = +∞. When σ tends to +∞, we have:

T (α, +∞) = 1 + D(α) - B 1 (α)-δ B 2 (α) θs G/(A * f ) 1+αδ-G/(A * f ) ǫcc 1- θα β(1+αδ-G/(A * f )) +θ B 4 (α) ∆ B 1 (α)-δ B 2 (α)
Therefore, T (α, +∞) < 1 + D(α). We can also show that:

∂T (α,σ) ∂σ = -1 σ 2 B 1 (α)-δ B 2 (α) θ(1-s) 1+αA * f ′ ǫcc 1- θα β(1+αδ-G/(A * f )) +θ B 4 (α) ∆ B 1 (α)-δ B 2 (α) < 0 Finally we get T (α, +∞) -1 -D(α) if ǫ cc ǫ α F , whereas T (α, +∞) < -1 - D(α) if ǫ cc < ǫ α F , with ǫ α F ≡ βs B 1 (α)-δ B 2 (α) (G/(A * f )-2α) 2 1+ 2-δ A * f ′ (39)
Let us now provide a useful technical result: Lemma 6.2. Under Assumptions 1-3, there exist β ∈ (0, 1) and δ ∈ (0, 1) such that when β ∈ ( β, 1) and δ ∈ (0, δ), then ǫ α H is an increasing function of α while ǫ α F is a decreasing function of α. Moreover, there exist α, α ǫF ∈ (0, α Max ) with α < α ǫF such that:

i) ǫ α H < ǫ α F if and only if α ∈ [0, α), ii) ǫ α F > 0 if and only if α ∈ [0, α ǫF ).
Proof. Using (28) and (29), we get:

B1(α)-δ B2(α) = (θ/β-sδ)(β-α(1-β))+α(θ+sβG)(θ/β-δ) s[β-α(1-β)+α(θ+sβG)] (40
ǫ α H = (θ + sβG) α(a1α+a2) a3α 2 +a4α+a5 with a 1 ≡ s(1 -β)(G + δ), a 2 ≡ θ -sβδ, a 3 ≡ δ(1 -β)(δ + sG) a 4 ≡ δ(1 -β) + (δ + sG)(1 -β + sβG), a 5 ≡ 1 -β + sβG (41)
The sign of ∂ǫ α H /∂α is given by the sign of

A(α) = (a 1 a 4 -a 2 a 3 )α 2 + 2a 1 a 5 α + a 2 a 5
Using (41), we get a 1 a 5 > 0, a 2 a 5 > 0 and

a 1 a 4 -a 2 a 3 = s(1 -β)(G + δ)[δ(1 -β) + (δ + sG)(1 -β + sβG)] -(θ -sβδ)δ(1 -β)(δ + sG) ≡ ϕ(δ) Since ϕ(0) = s 2 G 2 (1 -β)(1 -β + sβG) > 0
, there exists δ > 0 such that when δ ∈ (0, δ), ϕ(δ) > 0 and ǫ α H is increasing in α.

Similarly, under Assumption 3, (26), ( 40) and (39) yield lim α→0 ǫ α F = ǫ 0 F as given by (36) and lim α→αMax ǫ α F < 0. Moreover, substituting (26) and ( 40) into (39), we obtain: 3 -If α ∈ (α ǫF , α Max ), we get the same configuration as in Figure 4 when ǫ cc > ǫ α H and the same configuration as in Figure 7 when ǫ cc < ǫ α H . In all these different cases, there exists a unique value σ α T , defined by 1 -T (σ α T ) + D = 0 and such that σ α T = σ T as given by (27), which does not depend on α.

ǫ α F = b3-αb2-
If ǫ cc < ǫ α F , there is also a unique value σ α F , defined by 1+T (α, σ α F )+D(α) = 0 and given by σ

α F ≡ (1-s)θ B 1 -δ B 2 2(ǫ α F -ǫcc)A * f ′ 1+ 2-δ A * f ′ (43)
The proposition follows from these different figures.

Proof of Corollary 4

As shown in Lemma 6.2, when β ∈ ( β, 1) and δ ∈ (0, δ), ǫ α H is an increasing function of α while ǫ α F is a decreasing function of α. We get graphically: When α = α, the corresponding value of ǫ α H is denoted ǫ α H . Let us assume that ǫ cc ∈ (ǫ α H , ǫ 0 F ). Then, there exist α ∈ (0, α) and α H ∈ (α, α Max ) such that ǫ cc < ǫ α F if and only if α ∈ [0, α) and ǫ cc > ǫ α H if and only if α ∈ [0, α H ). Let us consider now the critical value σ α F as given by (43). Using (25), (28) and (29), we get We easily derive that lim α→ α-σ α F = +∞ Moreover, for any given ǫ cc < ǫ α F , the sign of the derivative ∂σ α F /∂α is given by the sign of the following expression

ǫ α F ǫ α H 0 β α Max ǫ αMax F α α ǫF ǫ α H ǫ 0 F ǫ cc α H α
Φ(α) = -[θ -sβδ + α(1 -β)s(δ + G)] [1 + α(δ + sG)] × (1-β)(θ+sβG) [β-α(1-β)] 2 (ǫ α F -ǫ cc ) + ∂ǫ α F ∂α 2 -δ + θ+sβG) β-α(1-β) - 2 -δ + θ+sβG) β-α(1-β) (ǫ α F -ǫ cc )δ(1 -s)(θ + sβG)
We know that if β ∈ ( β, 1) then ∂ǫ α F /∂α < 0. Therefore, there exist β σF ∈ (0, 1) and δ σF ∈ (0, δ) such that σ α F is a monotone increasing function of α when β ∈ (β σF , 1) and δ ∈ (0, δ σF ). Let us denote β = max{ β, β σF } and δ = min{ δ, δ σF }.

Let us finally assume that β ∈ (β, 1), δ ∈ (0, δ) and σ > σ 0 F (> σ T ) with σ 0 F as given by (37). Then, there exists α F ∈ (0, α) such that σ > σ α F if and only if α ∈ [0, α F ). The results follow from Proposition 5.

Figure 1 :

 1 Figure 1: Existence and multiplicity of steady states

  Moreover, substituting (40) into (38), we obtain:

α 2 b1 2 (b4+αb5+α 2 b6) with b 1 ≡ 1 +Figure 4 : 1 +FFigure 5 : 1 +FFigure 6 : 2 - 1 +

 2114151621 Figure 4: The case α ∈ (0, α) with ǫ cc ǫ α F .✻

Figure 7 :

 7 Figure 7: The case α ∈ (α, α ǫF ) with ǫ α F ≤ ǫ cc < ǫ α H .

Figure 8 :

 8 Figure 8: Effects of α when ǫ cc ∈ (ǫ α H , ǫ 0 F ).

  θ+sβG β-α(1-β) )(ǫ α F -ǫcc)[1+α(δ+sG)]

Note that a similar constraint has been considered by de la Croix and Michel[START_REF] De La Croix | A Theory of Economic Growth, Dynamics and Policy in Overlapping Generations[END_REF], or more recently by Minea and Villieu[START_REF] Minea | Debt policy rule, productive government spending, and multiple growth paths: A note[END_REF].

For their simulations, they consider an extension with debt, but taken as constant.

Public spending neither enters the utility function, nor the technology. Since public spending is constant, this is not an issue.

To be consistent with the debt constraint, we should have b 0 = αAf (k 0 ).

Note that if G = 0, which is excluded in this paper, our analysis of multiplicity does not apply. Indeed, in this case, the tax rate is zero, there is no distortion, and there is a unique steady state satisfying f ′ (k) = θ/β, as usually in the optimal growth model.

As shown by Grandmont et al.[START_REF] Grandmont | Capital-labour substitution and competitive nonlinear endogenous business cycles[END_REF], the existence of a Hopf bifurcation also allows to generate sunspot fluctuations far from the steady state, i.e. in the neighborhood of the quasiperiodic cycle.

Of course, if G further tends to 0, we also have that k tends to 0.

* This work has been carried out thanks to the support of the A*MIDEX project (n°ANR-11-IDEX-0001-02) funded by the "Investissements d'Avenir" French Government program, managed by the French National Research Agency (ANR), and of the Japan Society for the Promotion of Science, Grant-in-Aid for Specially Promoted Research #23000001 and #(B)23330063.


Introduction

A wide-spread perception among economists is that macroeconomic business-cycle fluctuations are not driven only by shocks on technologies or preferences, but also by changes in expectations about the fundamentals. A major strand of the literature focussing on fluctuations derived from agents' beliefs is based on the concept of sunspot equilibria that dates back to the early works of Shell [START_REF] Shell | Monnaie et Allocation Intertemporelle[END_REF], Azariadis [START_REF] Azariadis | Self Fulfilling Prophecies[END_REF] and Cass and Shell [START_REF] Cass | Do Sunspots Matter ?[END_REF]. As shown by Woodford [START_REF] Woodford | Stationary Sunspot Equilibria. The Case of Small Fluctuations Around a Deterministic Steady State[END_REF], the existence of sunspot equilibria is closely related to the equilibrium indeterminacy under perfect foresight, i.e. the existence of a continuum of equilibrium paths converging toward one steady state from the same initial value of the state variable.

Since the contribution of Reichlin [START_REF] Reichlin | Equilibrium Cycles in an Overlapping Generations Economy with Production[END_REF], the possible co-existence of dynamic efficiency, i.e., Pareto-optimal equilibrium paths, and local indeterminacy in OLG models has been widely discussed in the literature. This is an important question in terms of stabilization policies. Indeed, when local indeterminacy occurs under dynamic efficiency, the introduction of a public policy based on taxes and transfers could at the same time stabilize the economy and reach the Pareto optimal steady state along which all generations get an equal level of welfare. On the contrary, when local indeterminacy occurs under dynamic inefficiency, stabilization policies targeting the steady state leave room for welfare losses. Although Reichlin [START_REF] Reichlin | Equilibrium Cycles in an Overlapping Generations Economy with Production[END_REF] has shown that locally indeterminate dynamically efficient equilibria can occur in an aggregate model with a Leontief technology, Cazzavillan and Pintus [START_REF] Cazzavillan | Dynamic Inefficiency in an Overlapping Generations Economy with Production[END_REF] have recently proved that this result is not robust to the introduction of any positive capital-labor substitution.

In Galor [START_REF] Galor | A Two-Sector Overlapping-Generations Model: a Global Characterization of the Dynamical System[END_REF] type two-sector OLG models, with one pure consumption good and one pure investment good, the conclusion is not so clear-cut. Indeed, Drugeon et al. [START_REF] Drugeon | On Efficiency and Local Uniqueness in Two-Sector OLG Economies[END_REF] and Nourry and Venditti [START_REF] Nourry | Local Indeterminacy Under Dynamic Efficiency in a Two-Sector Overlapping Generations Economy[END_REF] have proved that local indeterminacy is ruled out when the steady state is dynamically efficient provided the sectoral technologies are not too close to Leontief functions. The intuition of this result is quite simple. Starting from an arbitrary equilibrium, consider another one in which the agents expect a higher rate of investment at time t leading to some higher capital stock at time t + 1. This expectation will be self-fulfilling provided the amount of saving at date t is large enough to support the increase of the investment good output which directly provides the capital stock of the next period. When the equilibrium is dynamically efficient, the share of first period consumption is large enough to generate a stationary capital stock lower than the Golden Rule and thus prevents the agent from saving enough. At the same time, when the sectoral technologies have elasticities of capital-labor substitution far enough from zero, any transfer of capital in the investment good sector is followed by a decrease of labor so that the final output cannot generate a large enough increase of capital. 1 As a result, the initial expectation cannot be realized as an equilibrium, and, under dynamic efficiency, local indeterminacy together with fluctuations based on local sunspots are ruled out.

However, in a multisector framework, the assumption of a unique consumption good is highly peculiar and is likely to generate singular properties. When heterogeneous sectors are introduced, it is quite common to consider instead that multiple consumption goods co-exist. 2 In such a case, the existence of additional substitution mechanisms between the different consumption goods suggests that new conclusions could be obtained. Unfortunately, the literature on OLG models has almost exclusively focused on Galor-type formulations. Among the few contributions dealing with a different framework, the most noteworthy contribution is the paper of Kalra [START_REF] Kalra | Cyclical Equilibria in Multi-Sector Productive Economies: the Role of Substitution and Factor Intensity[END_REF] which studies the existence of cyclical equilibria in a generalization of the standard two-sector model by assuming that both goods are consumed. 3Beside the pure consumption good, the second sector produces a mixed good which can be either consumed or used as capital. Such a formulation, which is much in line with standard national accounting data than the usual Galor-type formulation,4 allows for the existence of intratemporal substitution in consumption. Kalra then shows that local indeterminacy through the occurrence of a Hopf bifurcation is more likely to occur than in the standard Galor-type model. 5 However, as he does not discuss the dynamic efficiency property of the equilibrium, nothing is said about the question initially raised by Reichlin [START_REF] Reichlin | Equilibrium Cycles in an Overlapping Generations Economy with Production[END_REF].

The purpose of this paper is then to complement the analysis of Kalra and to explore the existence of dynamically efficient endogenous fluctuations. A simple intuition suggests indeed that new conclusions can be obtained. With two consumption goods, everything else equal, the stock of capital in the next period supported by the amount of saving at date t is lower than in the standard case since it is based on a mixed good output decreased by the part which is used for consumption. In such a case, a given amount of saving may be compatible with the expectation of a higher capital stock at time t + 1 if a larger part of the mixed good output is consumed. Based on this new channel, local indeterminacy and the existence of expectations-driven fluctuations can become compatible with dynamic efficiency under standard sectoral technologies with larger elasticities of capital-labor substitution.

We consider a simplified version of the Kalra's model assuming a unitary elasticity of intratemporal substitution between the two consumption goods. We provide a simple condition for dynamic efficiency. Then we show that, according to the previous intuition, when the pure consumption good is capital intensive, local indeterminacy and expectations-driven fluctuations occur for dynamically efficient competitive equilibria with larger sectoral elasticities of capital-labor substitution than in the standard model, provided the share of the pure consumption good in utility is low enough. We thus show that Reichlin's [START_REF] Reichlin | Equilibrium Cycles in an Overlapping Generations Economy with Production[END_REF] result is more robust in a two-sector OLG economy with multiple consumption goods than in the standard Galor-type formulation.

The paper is organized as follows: Section 2 presents the model. Section 3 proves the existence of a steady state and provides a condition for dynamic efficiency. Section 4 contains our main results on the co-existence of local indeterminacy and dynamic efficiency. Some results under dynamic inefficiency are also provided in order to establish a better understanding of the relationship between the Pareto optimality and the dynamic properties of competitive equilibria. Concluding comments are in Section 5 and the proofs are gathered in a final Appendix.

The model 2.1 The production side

We consider an economy with two produced goods, one pure consumption good y 0 which cannot be used as capital, and one mixed good y which can be either consumed or invested, i.e. used as capital. There are two inputs, capital and labor. We assume complete depreciation of capital within one period and that labor is inelastically supplied. We then get

y t -z t = k t+1 (1)
with z t the consumption part of the mixed good in period t, and k t+1 the total amount of capital in period t + 1. Each good is produced with a constant returns to scale technology such that y 0 = f 0 (k 0 , l 0 ) and y = f 1 (k 1 , l 1 ), with k 0 + k 1 ≤ k, k being the total stock of capital, and l 0 + l 1 ≤ , being the total amount of labor.

Assumption 1. Each production function f i : R 2 + → R + , i = 0, 1, is C 2 , increasing in each argument, concave, homogeneous of degree one and such that for any x > 0,

f i 1 (0, x) = f i 2 (x, 0) = +∞, f i 1 (+∞, x) = f i 2 (x, +∞) = 0.
For any given (k, y, ), profit maximization in the representative firm of each sector is equivalent to solving the following problem of optimal allocation of productive factors between the two sectors:

τ (k, y, ) = max k 0 ,k 1 ,l 0 ,l 1 ≥0 f 0 (k 0 , l 0 ) s.t. y ≤ f 1 (k 1 , l 1 ), k 0 + k 1 ≤ k and l 0 + l 1 ≤ (2)
The social production function τ (k, y, ) gives the maximal output of the consumption good. Under Assumption 1, τ (k, y, ) is homogeneous of degree one, concave and twice continuously differentiable. 6 Denoting w the wage rate, r the gross rental rate of capital and p the price of the mixed good, all in terms of the price of the pure consumption good, we derive

r = τ 1 (k, y, ), p = -τ 2 (k, y, ), w = τ 3 (k, y, ) (3)

The consumption side

In each period t, N t agents are born, and they live for two periods. In their first period of life (when young), the agents are endowed with one unit of labor that they supply inelastically to firms. Their income is equal to the real wage. They allocate this income between current consumption and savings which are invested in the firms. In their second period of life (when old ), they are retired and their income resulting from the return on the savings is entirely consumed. Each agent is assumed to have one child so that population is constant, i.e., N t = N . The preferences of a representative agent born at time t are defined over his consumption bundle for each of the two produced goods, c 0 t , c 1 t , when young, and d 0 t+1 , d 1 t+1 , when old. They are summarized by the utility function

U (c 0 t , c 1 t , d 0 t+1 , d 1 t+1 ) = u(C t , D t+1 /B) with C t = (c 0 t ) θ (c 1 t ) 1-θ , D t+1 = (d 0 t+1 ) θ (d 1 t+1 ) 1-θ (4) and 7 u(C t , D t+1 /B) = C 1-1/γ t + δ(D t+1 /B) 1-1/γ γ/(γ-1) (5)
where 0 < θ ≤ 1, δ > 0 is the discount factor, γ > 0 is the elasticity of intertemporal substitution in consumption and B > 0 is a scaling parameter. C t and D t+1 can be interpreted as composite goods derived from the two consumption goods y 0 t and z t with

y 0 t = T (k t , y t , t ) = N c 0 t + d 0 t and z t = N c 1 t + d 1 t (6)
It follows that θ is the share of the pure consumption good and 1 -θ the share of the mixed good in the composite goods C and D.

A young agent born in period t has first to solve two static problems of optimal composition of his two composite goods. Denoting by π t the consumer price index in terms of the pure consumption good, we get the following optimization program for C t max

c 0 t ,c 1 t ≥0 c 0 t θ c 1 t 1-θ s.t. c 0 t + p t c 1 t = π t C t (7)
The corresponding program for D t+1 is similar with d 0 t+1 + p t+1 d 1 t+1 = π t+1 D t+1 . Solving the associated first order conditions gives:

c 0 t = θπ t C t , c 1 t = (1-θ)πtCt pt , π t = pt 1-θ 1-θ θ -θ (8)
with similar expressions for d 0 t+1 and d 1 t+1 , namely

d 0 t+1 = θπ t+1 D t+1 , d 1 t+1 = (1-θ)π t+1 D t+1 p t+1 (9)
Under perfect foresight, and considering w t and R t+1 as given, a young agent also has to solve an intertemporal allocation problem in order to maximize his utility function over his life-cycle: max

Ct,D t+1 ,φt≥0 u(C t , D t+1 /B) s.t. w t = π t C t + φ t and R t+1 φ t = π t+1 D t+1
Solving the first order conditions gives:

C t = wt/πt 1+δ γ [R t+1 πt/(Bπ t+1 )] γ-1 ≡ α(v t )w t /π t (10)
with v t = R t+1 π t /(Bπ t+1 ) and α(v t ) ∈ (0, 1) the propensity to consume of the young, or equivalently the share of first period consumption spending over the wage income. We also get the saving function

φ t = φ (w t , v t ) ≡ (1 -α (v t )) w t (11)
In the rest of the paper we introduce the following standard Assumption:

Assumption 2. γ > 1
Such a restriction implies that the saving function ( 11) is increasing with respect to the gross rate of return R.

3 Dynamic efficiency of competitive equilibrium

Perfect-foresight competitive equilibrium

Total labor is given by the number of young households and is normalized to one, i.e., = N = 1. From now on, let τ (k, y, 1) = T (k, y) and τ i (k, y, 1) = T i (k, y), i = 1, 2, 3. Using (1), ( 3) and ( 11), a perfect-foresight competitive equilibrium is defined as a sequence {k t , y t } t≥0 that satisfies (1 -α (R t+1 π t /(Bπ t+1 ))) w t = k t+1 and y t -z t = k t+1 with R t+1 = r t+1 /p t . Using ( 6), ( 8) and ( 9), we conclude that a perfect-foresight competitive equilibrium satisfies the following system of two difference equations:

k t+1 + T 3 (kt,yt) 1-α - T 1 (k t+1 ,y t+1 ) T 2 (k t ,y t )B T 2 (k t ,y t ) T 2 (k t+1 ,y t+1 ) 1-θ T 2 (kt,yt) = 0 k t+1 -yt θ -(1-θ)[T 3 (kt,yt)+T 1 (kt,yt)kt] θT 2 (kt,yt) = 0 ( 12
)
with k 0 given. It is worth noting at this point that if θ = 1 the second difference equations reduces to k t+1 = y t and we get the standard two-sector OLG model with one pure consumption good and one pure investment good studied in Drugeon et al. [START_REF] Drugeon | On Efficiency and Local Uniqueness in Two-Sector OLG Economies[END_REF] and Nourry and Venditti [START_REF] Nourry | Local Indeterminacy Under Dynamic Efficiency in a Two-Sector Overlapping Generations Economy[END_REF]. 8

A normalized steady state

A steady state (k t , y t ) = (k * , y * ) for all t satisfies

k * + T 3 (k * ,y * ) 1-α - T 1 (k * ,y * ) T 2 (k * ,y * )B T 2 (k * ,y * ) = 0 k * -y * θ -(1-θ)[T 3 (k * ,y * )+T 1 (k * ,y * )k * ] θT 2 (k * ,y * ) = 0 (13)
We consider a family of economies parameterized by the elasticity of intertemporal substitution in consumption γ. We follow the same procedure as in Drugeon et al. [START_REF] Drugeon | On Efficiency and Local Uniqueness in Two-Sector OLG Economies[END_REF]: we use the scaling parameter B and the share θ to ensure the existence of a normalized steady state (NSS) (k * , y * ) which remains invariant as γ is varied.

Let us define the maximum admissible value of capital k solution of: k -f 1 ( k, 1) = 0

Under Assumption 1 we have indeed

f 1 (k, 1) > k if k < k, while f 1 (k, 1) < k if k > k.
Obviously, the NSS must be such that (k * , y * ) ∈ (0, k) × (0, k). We then get: Proof : See Appendix 6.1.

In the rest of the paper we assume θ = θ(k * , y * ) and B = B(k * , y * , γ) so that the share of capital in total income, as given by

s = s(k * , y * ) = r * k * T (k * ,y * )+p * y * (14)
and α = α(R * /B(k * , y * , γ)) remain constant as γ is made to vary.

Dynamic efficiency

From the homogeneity of τ , considering that k * T 2 /T 3 = (T 2 /T 1 )(k * T 1 /T 3 ) = -s/(R(1 -s)), we derive the stationary gross rate of return along the NSS:

R * = s (1-α)(1-s) (15)
It is well-known that in OLG models if the capital-labor ratio exceeds the Golden-Rule level, the economy is dynamically inefficient. In our two-sector model, the Golden-Rule level, denoted k, is characterized from the total stationary consumption which is given by the sum of the social production function with the consumption part of the mixed good, namely

C + D = T (k, y) + p[y -k]. Denoting R(k, y) = -T 1 (k, y)/T 2 (k, y), k satisfies as usual R( k, ŷ) ≡ R = 1.
Under-accumulation of capital is obtained if and only if R * > 1. As in Drugeon et al. [START_REF] Drugeon | On Efficiency and Local Uniqueness in Two-Sector OLG Economies[END_REF] we get: Proposition 2. Under Assumption 1, let α = 1 -s/(1 -s). Then an intertemporal competitive equilibrium converging towards the NSS is dynamically efficient if α ∈ (α, 1) and dynamically inefficient if α ∈ (0, α).

Dynamic inefficiency can be avoided if the amount of savings is not too large.

Endogenous business cycles

Our aim is to show that a two-sector OLG model with two consumption goods provides new results on the existence of endogenous business cycles derived from agents' beliefs. This type of fluctuations is based on the concept of sunspot equilibria. As shown by Woodford [START_REF] Woodford | Stationary Sunspot Equilibria. The Case of Small Fluctuations Around a Deterministic Steady State[END_REF], the existence of sunspot equilibria is closely related to the equilibrium local indeterminacy, i.e. the existence of a continuum of equilibrium paths converging toward the NSS from the same initial capital stock. In our framework, local indeterminacy occurs when the characteristic roots associated with the linearization of equations ( 12) around the NSS are less than 1 in absolute value.

We focus on two particular aspects of the model: the share θ of the pure consumption good in the composite goods C and D, and the share α of first period consumption spending in total income. Let us also introduce the relative capital intensity difference across sectors and the elasticity of the rental rate of capital, respectively

b ≡ l 1 y k 1 l 1 -k 0 l 0 and ε rk = -T 11 (k * , y * )k * /T 1 (k * , y * ) ( 16
)
Note that ε rk is negatively linked to the sectoral elasticities of capital-labor substitution (see Drugeon [START_REF] Drugeon | On Consumptions, Inputs and Outputs Substitutabilities and the Evanescence of Optimal Cycles[END_REF]).

A labor intensive pure consumption good

Let us consider first the case b > 0. We derive from the homogeneity of τ and equations (3

) that at the NSS b < b M ax with b M ax = s R * [1-θ[(1-s)α+s]] > 0 (17)
We get the following results:

Proposition 3. Under Assumptions 1-2, any equilibrium path is unique and monotone when the pure consumption good is labor intensive (b ∈ (0, b M ax )).

Proof : See Appendix 6.2.

This Proposition implies that endogenous fluctuations and local indeterminacy are ruled out when b ∈ (0, b M ax ). It complements the analysis of Kalra [START_REF] Kalra | Cyclical Equilibria in Multi-Sector Productive Economies: the Role of Substitution and Factor Intensity[END_REF] in the case of a labor intensive pure consumption good. First it confirms that endogenous period-2 cycles through the existence of negative characteristic roots and a flip bifurcation are ruled out. Second, it proves that endogenous quasi-periodic cycles through the existence of a Hopf bifurcation are also ruled out in the case of a unitary elasticity of intratemporal substitution between the two consumption goods. Therefore, Proposition 2 in Kalra [START_REF] Kalra | Cyclical Equilibria in Multi-Sector Productive Economies: the Role of Substitution and Factor Intensity[END_REF] requires an elasticity sufficiently lower than one.

A capital intensive pure consumption good

Let us consider from now on the case b < 0 and focus in a first step on dynamically efficient equilibria with α > α. We assume θ < 1/2 to provide conditions for the existence of local indeterminacy through the occurrence of a Hopf bifurcation. The motivation to consider such a configuration comes from the fact that when θ is close to 1, a Hopf bifurcation cannot occur under dynamic efficiency (see Nourry and Venditti [START_REF] Nourry | Local Indeterminacy Under Dynamic Efficiency in a Two-Sector Overlapping Generations Economy[END_REF]). We also restrict the share of capital to get a positive value for the bound α = 1 -s/(1 -s) and to be compatible with standard empirical values:

Assumption 3. s ∈ [1/3, 1/2) Proposition 4. Under Assumptions 1-3, let α > s/(1 -s)(> α) and b ∈ (-(1 -α)s/ [α(1 -θ + θs)] , 0). Then there exist θ ∈ (s/[(1 -s)α + s], 1/2], ˜ rk < rk < ¯ rk and γ H , γ F > γ T > 1 such that when θ ∈ (s/[(1 -s)α + s], θ),
the following results hold:

1 -If rk ∈ ( rk , ¯ rk ), then γ F > γ H and the NSS is locally indeterminate when γ ∈ (γ T , γ H ) and undergoes a Hopf bifurcation when γ = γ H . Moreover, there generically exist locally indeterminate (resp. locally unstable) quasi-periodic cycles when γ ∈ (γ H , γ H + ) (resp. γ ∈ (γ H -, γ H )) with > 0, i.e. when the bifurcation is super (resp. sub-) critical.

2 -If rk ∈ (˜ rk , rk ), then γ H > γ F and the NSS is locally indeterminate when γ ∈ (γ T , γ F ) and undergoes a flip bifurcation when γ = γ F . Moreover there generically exist locally indeterminate (resp. saddle-point stable) period-two cycles when γ ∈ (γ F , γ F + ε) (resp. γ ∈ (γ F -ε, γ F )) with ε > 0, i.e. when the bifurcation is super (resp. sub-) critical.

Proof : See Appendix 6.3.

Remark 1. Whether the bifurcation is super or sub-critical is driven by the sign of some coefficient computed from the second and third order approximations of the dynamical system [START_REF] Reichlin | Equilibrium Cycles in an Overlapping Generations Economy with Production[END_REF]. This property determines whether the bifurcation leads to the occurrence of locally indeterminate or unstable (resp. saddle-point stable) quasi-periodic (resp. period-two) cycles near the bifurcation value.

Proposition 4 shows that when the share θ of the pure consumption good in the composite goods is low enough, local indeterminacy and expectationsdriven fluctuations arise when rk ∈ (˜ rk , ¯ rk ), i.e. for strictly positive but intermediary values of the sectoral elasticities of capital-labor substitution. We then get more general conclusions for a wider range of elasticities of capital-labor substitution than in the case θ = 1 in which the occurrence of local indeterminacy is only obtained through a flip bifurcation and requires to consider sectoral technologies very close to Leontief functions (see Nourry and Venditti [START_REF] Nourry | Local Indeterminacy Under Dynamic Efficiency in a Two-Sector Overlapping Generations Economy[END_REF]). The intuition for the existence of dynamically efficient expectations-driven fluctuations with multiple consumption goods given in the introduction is thus confirmed.

We thus show that Reichlin's [START_REF] Reichlin | Equilibrium Cycles in an Overlapping Generations Economy with Production[END_REF] result is more robust in a two-sector OLG economy with multiple consumption goods than in the Galor-type formulation. The introduction of a public policy based on taxes and transfers can at the same time stabilize the economy and reach the Pareto optimal steady state along which all generations get an equal level of welfare. 9In order to establish a better understanding of the relationship between the Pareto optimality and the dynamic properties of competitive equilibria, let us finally focus on dynamically inefficient equilibria with α < α. We successively consider the existence of local indeterminacy through flip and Hopf bifurcations. As mentioned in Remark 1, the two following Propositions will encompass the super and sub-critical bifurcations. Let us denote b = min

1 R * (1-θ)-θ , -(1-α)s α[1-θ+θs] Proposition 5. Under Assumptions 1-3, let α < α. If one of the following conditions is satisfied: i) θ ∈ (s/[1 -α(1 -s)], 1/2) and b < b, ii) θ ∈ (1/2, [s 2 + α(1 -s)]/[2s 2 + α(1 -s) 2 ]) and b ∈ (1/[R * (1 -2θ)], b
), then there exists γ F > 1 such that the NSS is locally indeterminate when γ > γ F and undergoes a flip bifurcation when γ = γ F . Moreover, there generically exist locally indeterminate (resp. saddle-point stable) period-two cycles when γ ∈ (γ F -ε, γ F ) (resp. γ ∈ (γ F , γ F + ε)) with ε > 0, i.e. when the bifurcation is super (resp. sub-) critical.

Proof : See Appendix 6.4.

When compared with Proposition 4, Proposition 5 shows that considering dynamically efficient equilibria implies restrictions for the existence of local indeterminacy that are not necessary under dynamic inefficiency. Indeed, expectations-driven fluctuations now occur without any condition on the elasticity of the rental rate of capital ε rk and are thus compatible with any values of the sectoral elasticities of capital-labor substitution.

This conclusion is also confirmed when local indeterminacy is appraised through the existence of a Hopf bifurcation. Let us indeed denote

θ = s+(1-α)(1-s)+(1-s) √ (α-α)(1-α) s+3(1-α)(1-s) and b = min - s R * [θ[(1-s)α+s]-s] , (1-θ)R * -θ+θ √ 1-R * R * [(1-θ) 2 R * -θ(2-3θ)]
In order to simplify the formulation, we focus on the standard value s = 1/3 for the share of capital in total income. We then get: Proposition 6. Under Assumptions 1-2, let s = 1/3. There exist α ∈ (0, α) and γ F > γ H > 1 such that if α ∈ (α, α), θ ∈ (s/[(1 -s)α + s], θ) and b < b, the NSS is locally indeterminate when γ ∈ (γ H , γ F ), undergoes a flip bifurcation when γ = γ F and undergoes a Hopf bifurcation when γ = γ H . Moreover there generically exist locally indeterminate (resp. saddle-point stable) period-two cycles when γ ∈ (γ F , γ F + ε) (resp. γ ∈ (γ F -ε, γ F )) with ε > 0, i.e. when the bifurcation is super (resp. sub-) critical, and locally indeterminate (resp. locally unstable) quasi-periodic cycles when γ ∈ (γ H -, γ H ) (resp. γ ∈ (γ H , γ H + )) with > 0, i.e. when the bifurcation is super (resp. sub-) critical.

Proof : See Appendix 6.5.

Propositions 5 and 6 prove that the existence of expectations-driven fluctuations are more likely to occur under dynamic inefficiency and thus confirm the crucial role of the saving behavior. Moreover, in this case, any stabilization policy targeting the steady state leaves room for welfare losses associated with productive inefficiency.

Concluding comments

We have considered a two-sector OLG economy with two consumption goods which enter the utility function in both periods of life through a composite good. To simplify we have assumed that the share of each consumption good into the composite one is constant. We have proved that although dynamically efficient competitive equilibria are less likely to be locally indeterminate than dynamically inefficient ones, the existence of Pareto optimal expectations-driven fluctuations becomes compatible with standard sectoral technologies if the share of the pure consumption good is low enough.

Appendix

Proof of Proposition 1

Let (k * , y * ) ∈ (0, k)×(0, k). Solving the second equation in [START_REF] Reichlin | Endogenous Fluctuations in a Two-Sector Overlapping Generations Economy[END_REF] with respect to θ and using the homogeneity of τ gives θ (k * , y * ) =

T (k * ,y * )
T (k * ,y * )-T 2 (k * ,y * )(y * -k * ) ∈ (0, 1) Using [START_REF] Kalra | Cyclical Equilibria in Multi-Sector Productive Economies: the Role of Substitution and Factor Intensity[END_REF] and solving the first equation in [START_REF] Reichlin | Endogenous Fluctuations in a Two-Sector Overlapping Generations Economy[END_REF] with respect to B gives

B(k * , y * , γ) = -T 1 (k * ,y * ) T 2 (k * ,y * ) -k * T 2 (k * ,y * ) δ γ [T 3 (k * ,y * )+k * T 2 (k * ,y * )]

Proof of Proposition 3

From (10), we derive 16) and T 11 < 0. Consider rk as given by ( 16) together with

α (v) = (1 -γ)α(v)(1 -α(v))/v (
T 1 k * /T 3 = s/(1 -s), -T 1 /T 2 = R * = s/(1 -α)(1 -s) and the fact that the homogeneity of τ (k, y, ) implies a = [(1 -α)(1 -s)(1 -θb) -(1 -θ)b] k * /[(1 -α)(1 -s)].
Total differenciation of ( 12) using ( 3), ( 10), ( 14) and (18) evaluated at the NSS gives the characteristic polynomial P γ (λ) = λ 2 -λT θ (γ) + D θ (γ) with

D θ (γ) = s b (1-s)α(γ-1)θ+θ(1-s)α-s(1-θ) +(1-α)(1-s) α(γ-1)(1-α)(1-s) 2 θb T θ (γ) = 1+α(γ-1) rk [1-(1-θ)R * b] 2 +θ 2 b 2 R * +θR * b rk (1-θ) bα s +θbα+1-α α(γ-1)[1-(1-θ)R * b]θb rk
When θ = θ(k * , y * ) and B = B(k * , y * , γ), the NSS and α remain constant as γ is made to vary, and as in Grandmont et al. [START_REF] Grandmont | Capital-Labor Substitution and Competitive Nonlinear Endogenous Business Cycles[END_REF], we can study the variations of T θ (γ) and D θ (γ) in the (T , D) plane. Solving T and D with respect to α(γ -1) yields the following linear relationship ∆(T ):

D = ∆(T ) = S θ T + M (19)
where M is a constant term and the slope S θ of ∆(T ) is

S θ = rk sR * [1-(1-θ)R * b][(1-s)α+s](θ-θ 1 )(b-b 1 ) (1-s){s+θR * b rk α[1-θ+θs](b-b 2 )} (20) with θ 1 = s (1-s)α+s , b 1 = - s R * [(1-s)α+s](θ-θ 1 ) and b 2 = -(1-α)s α(1-θ+θs) (21)
For a given θ = θ(k * , y * ), as γ spans the interval (1, +∞), T θ (γ) and D θ (γ) vary linearly along the line ∆(T ). As γ ∈ (1, +∞), the fundamental properties of ∆(T ) are characterized from the consideration of its extremities. The starting point of the pair (T θ (γ), D θ (γ)) is indeed obtained when γ = +∞: lim

γ→+∞ rk ≥ - 1 s[(1-s)α+s](b-b 1 ) b(1-s) 2 θ(α-α) P(b) + θR * bα[1-θ+θs] s (b-b 2 ) ≡ rk (> ˆ rk ) (27)
Moreover, there exists ¯ rk > rk such that T θ (γ) = 2 when rk = ¯ rk and the half-line is given by ∆. Therefore T θ (γ) ∈ (-2, 2) as long as rk ∈ ( rk , ¯ rk ).

Let us denote ε the value of ε such that the denominator of the ratio in ( 27) is equal to zero. The maximal admissible value of ε is such that ε = min{ε, 1/2 -θ 1 }. It follows that when θ ∈ (θ 1 , θ) with θ = θ 1 + ε and rk ∈ ( rk , ¯ rk ), we get a half-line above ∆ 1 as shown on Figure 1. This proves the first part of Proposition 4.

2 -Note from (20) that lim rk →ˆ rk S θ = ∞. When rk = ˆ rk , the half-line is thus given by ∆ and there exists ˜ rk ∈ (ˆ rk , rk ) such that when θ ∈ (θ 1 , θ) and rk = ˜ rk , we get a half-line such that ∆. As a result, if rk ∈ (˜ rk , rk ) the half-line is given by ∆ 2 as shown on Figure 1. This proves the second part of Proposition 4.

The bifurcation values γ H , γ F and γ T are respectively defined as the solutions of

D θ (γ) = 1, P γ (-1) = 1 + T θ (γ) + D θ (γ) = 0, P γ (1) = 1 -T θ (γ) + D θ (γ) = 0
with the corresponding values of θ and rk .

D ∞ (T ∞ θ , D ∞ ) γ F γ H γ T γ T ∆ 1 ∆ 2 ∆ ∆ ∆ ∆ 0 T E T D d d d d d d d d d d d d d d l l
Figure 1: Hopf and flip bifurcations with α > s/(1 -s)(> α).

Proof of Proposition 5

Consider the bounds defined in (21) and (24). α < α implies D ∞ < 1. We get θ 2 < 1/2 < θ 1 and P ∞ (-1) > 0 if and only In both cases the flip bifurcation value γ F is defined as the solution of P γ (-1) = 1 + T θ (γ) + D θ (γ) = 0. It is easy to show that (1 -θ) 2 R * -θ(2 -3θ) < 0 if and only if θ ∈ (θ -, θ + ) with θ 1 > θ -. Moreover, θ 1 < θ + if and only if g(α) = [(1 -s)α) + s] (α -α)(1 -α) -α 2 (1 -s) + α(1 + 2s) -2s > 0, with g(0) < 0 and g(α) ≤ 0 under Assumption 3. When s = 1/3, we get lim α→α g (α) < 0. It follows that there exists α 1 ∈ (0, α) such that when α ∈ (α 1 , α), g(α) > 0. As a result we conclude that when α ∈ (α 1 , α), θ ∈ (θ 1 , θ +

if i) θ ∈ (θ 2 , 1 /

Proposition 1 .

 1 Under Assumptions 1-2, let (k * , y * ) ∈ (0, k) × (0, k). Then there exist unique values θ(k * , y * ) ∈ (0, 1) and B(k * , y * , γ) > 0 such that (k * , y * ) is a steady state if and only if θ = θ(k * , y * ) and B = B(k * , y * , γ).

  Then (k * , y * ) is a NSS if and only if θ = θ(k * , y * ) and B = B(k * , y * , γ).

  18) Under Assumption 1, we get from the first order conditions of program (2): T 12 = -T 11 b < 0, T 22 = T 11 b 2 < 0, T 31 = -T 11 a > 0 and T 32 = T 11 ab, with a ≡ k 0 /l 0 > 0, b as defined by (

2 )

 2 and b < b 4 , or, ii) θ > 1/2 and b ∈ (b 3 , b 4 ). Moreover, D 1 θ = +∞ if and only if θ > θ 1 and b < b 1 with b 1 < b 3 < b 4 . This implies that when P ∞ (-1) > 0 then D 1 θ = -∞ and any Hopf bifurcation is ruled out. Also, when D θ(γ) = -1, T θ (γ) < 0 if b < b 2 with b 2 > b 3 if θ < θ 3 = s 2 +α(1-s) 2s 2 +α(1-s) 2 ∈ (1/2, 1)Based on all this, we now prove Proposition 5. We focus here on the case in which P ∞ (1) > 0 and P ∞ (-1) > 0. As b < 0 we already know that P ∞ (1) > 0.i) Assume first that θ ∈ (θ 2 , 1/2) and b < b 4 . This implies P ∞ (-1) > 0 and thusD 1 θ = -∞. A flip bifurcation occurs if T θ (γ) < 0 when D θ (γ) = -1. This is obtained provided b < b 2 . Part i) isthen proved by assuming b < b with b = min{b 2 , b 4 }. We get indeed the half-line ∆ as shown on Figure 2. ii) Assume now that θ ∈ (1/2, θ 3 ) and b ∈ (b 3 , b). We get as previously P ∞ (-1) > 0, D 1 θ = -∞ and T θ (γ) < 0 when D θ (γ) = -1, which proves part ii).

Figure 2 :

 2 Figure 2: Flip bifurcation with α < α.

Figure 3 :

 3 Figure 3: Flip and Hopf bifurcations with α ∈ (α, α).

  ) and b < b + , we get P(b) < 0. Therefore assuming θ ∈ (θ 1 , θ + ) and b < min{b 1 , b + }, we derive from (25) that there exists α∈ [α 1 , α) such that if α ∈ (α, α), T θ (γ) > -2 when D θ (γ) = 1.Proposition 6 is then proved under the following conditions: α ∈ (α, α), θ ∈ (s/[(1 -s)α + s], θ) and b < b with θ = θ + and b = min{b 1 , b + }. We get indeed the half-line ∆ as shown on Figure 3. The bifurcation values γ H and γ F are finally defined as the solutions of D θ (γ) = 1 and P γ (-1) = 1 + T θ (γ) + D θ (γ) = 0

On the contrary, when the technologies are close to Leontief, capital and labor are complement and both can be transfered in the investment good sector so that the increase of the final output may compensate the lack of savings and lead to an increase of capital compatible with the expectations. As shown in Nourry and Venditti[START_REF] Nourry | Local Indeterminacy Under Dynamic Efficiency in a Two-Sector Overlapping Generations Economy[END_REF], local indeterminacy then becomes compatible with dynamic efficiency.

[START_REF] Benhabib | Stability of Equilibrium in Dynamic Models of Capital Theory[END_REF] See for instance Benhabib and Nishimura[START_REF] Benhabib | Stability of Equilibrium in Dynamic Models of Capital Theory[END_REF] for an optimal growth infinite-horizon model with many consumption

goods.[START_REF] Cass | Do Sunspots Matter ?[END_REF] See also Drugeon[START_REF] Drugeon | On Sectoral Supply Functions and Heterogeneous Consumptions in a Simple Life-Cycle Economy[END_REF] for the analysis of saddle-point stability in a similar model with

endogenous labor.[START_REF] Cazzavillan | Dynamic Inefficiency in an Overlapping Generations Economy with Production[END_REF] National accounting data are based on subdivisions of the productive sectors in which many goods are both

final and intermediary.[START_REF] Drugeon | On Consumptions, Inputs and Outputs Substitutabilities and the Evanescence of Optimal Cycles[END_REF] For instance Reichlin[START_REF] Reichlin | Endogenous Fluctuations in a Two-Sector Overlapping Generations Economy[END_REF] proves the existence of a Hopf bifurcation in a standard OLG model with Leontief technologies and Kalra[START_REF] Kalra | Cyclical Equilibria in Multi-Sector Productive Economies: the Role of Substitution and Factor Intensity[END_REF] extends this conclusion to the case of technologies with substitutable factors.

See Benhabib and Nishimura[START_REF] Benhabib | Stability of Equilibrium in Dynamic Models of Capital Theory[END_REF].

All the results in this paper can be obtained with a general concave and homothetic utility function u(Ct, Dt+1/B).

See alsoGalor [8] and Venditti[START_REF] Venditti | The Two Sector Overlapping Generations Model: a Simple Formulation[END_REF].

See Nourry and Venditti[START_REF] Nourry | Local Indeterminacy Under Dynamic Efficiency in a Two-Sector Overlapping Generations Economy[END_REF] for an illustration of such a policy in the Galor-type OLG model.

* This paper is dedicated to Professor Kazuo Nishimura, a very close friend, who has made outstanding contributions to economic theory. This work was supported by French National Research Agency Grant (ANR-08-BLAN-0245-01). We thank H. d'Albis, R. Boucekkine, J.M. Grandmont, T. Seegmuller, M.

l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l

Using the expressions of D θ (γ) and T θ (γ) allows to show that when

with Based on all this, we now prove Proposition 4. Let α > s/(1 -s)(> α) so that θ 2 > 1/2 > θ 1 .

1 -Assume that θ = θ 1 + ε ≤ 1/2 with ε > 0 small and b > b 2 (> b 1 ). It follows that P ∞ (-1) < 0 and the term between brackets in (25) is negative. Moreover T θ (γ) ≥ -2 if and only if