Frédéric Dufourt
email: frederic.dufourt@univ-amu.fr

Kazuo Nishimura
email: nishimura@kier.kyoto-u.ac.jp

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

Expectations, beliefs and the business cycle: theoretical analysis *

Keywords: belief-driven business cycles, endogenous fluctuations, expectations, income effect Economic Literature Classification Numbers: C62, E32, O41

When can exogenous changes in beliefs generate endogenous fluctuations in rational expectation models? We analyze this question in the canonical one-sector and two-sector models of the business cycle with increasing returns to scale. A key feature of our analysis is that we express the uniqueness/multiplicity condition of equilibirum paths in terms of restrictions on five critical and economically interpretable parameters: the Frisch elasticities of the labor supply curve with respect to the real wage and to the marginal utility of wealth, the intertemporal elasticity of substitution in consumption, the elasticity of substitution between capital and labor, and the degree of increasing returns to scale. We obtain two clear-cut conclusions: belief-driven fluctuations cannot exist in the one-sector version of the model for empirically consistent values for these five parameters. By contrast, belief-driven fluctuations are a robust property of the twosector version of the model -with differentiated consumption and investment goods -, as they now emerge for a wide range of parameter values consistent with available empirical estimates.

The key ingredients explaining these different outcomes are factor reallocation between sectors and the implied variations in the relative price of investment, affecting the expected return on capital accumulation.

Introduction

Are exogenous changes in agents' expectations a significant cause of observed economic fluctuations? A positive answer to this question requires two necessary ingredients. First, belief-driven fluctuations must be a plausible outcome of realistic business cycle models, occurring for realistic values for all structural parameters. Second, to the extent that they exist, exogenous changes in expectations must generate endogenous dynamics for macroeconomic variables consistent with observed patterns of economic fluctuations, in particular in response to transitory shocks. The present paper addresses the first issue, while a thorough assessment of the second issue is proposed in an accompanying paper (Dufourt et al. [START_REF] Dufourt | Expectations, Beliefs and the Business Cycle: data confrontation[END_REF]).

We analyze this question within two canonical models of the business cycle: the onesector Real Business Cycle (RBC) model of Kydland and Prescott [START_REF] Kydland | Time to Build and Aggregate Fluctuations[END_REF], stripped down to its essential ingredients by King, Plosser and Rebelo [START_REF] King | Production, Growth and Business Cycles[END_REF] and extended to consider positive externalities with increasing returns to scale by Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Increasing Returns[END_REF] ; and the two-sector version of the latter model proposed by Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF]. The main factor distinguishing our analysis from these previous papers is that we do not restrict the specifications of individual preferences and of the production functions to have particular forms, imposing instead minimal sets of assumptions on these functions.

A key novelty of our approach is that we show that the local stability conditions of the steady state -influencing the possibility of emergence of belief-driven fluctuations -can then be characterized in terms of five critical and economically interpretable parameters, which we call critical elasticities. Two elasticities relate to the specification of the production function: the elasticity of substitution between capital and labor and the degree of increasing returns to scale. Three elasticities relate to the specification of individual preferences: the elasticity of intertemporal substitution in consumption (EIS), the Frisch elasticity of the labor supply curve with respect to the real wage and the Frisch elasticity of the labor supply curve with respect to the marginal utility of wealth.

Regarding the latter three elasticities, we show that the standard strict quasi-concavity and normality assumptions typically imposed on instantaneous utility functions naturally translate into restrictions on these elasticities, so that considering these elasticities enables to cover the whole set of admissible utility functions. Moreover, we argue that the Frisch elasticity with respect to the marginal utility of wealth provides a relevant measure of what is often called the degree of "wealth effect" on labor supply decisions, which has recently been shown to play a major role in the local stability properties of dynamic macroeconomic models (see Dufourt et al. [START_REF] Dufourt | Indeterminacy and Sunspots in Two-Sector RBC Models with Generalized No-Income-Effect Preferences[END_REF][START_REF] Dufourt | Sunspot Fluctuations in Two-Sector Models with Variable Income Effects[END_REF], Jaimovich [START_REF] Jaimovich | Income Effects and Indeterminacy in a Calibrated One-Sector Growth Model[END_REF]). We show that our general formulation based on these critical elasticities encompasses as special cases all the standard formulations for individual preferences proposed in the literature, such as the Greenwood et al. (GHH) [START_REF] Greenwood | Investment, Capacity Utilization and the Real Business Cycle[END_REF] formulation with no income effect, Hansen's [START_REF] Hansen | Indivisible Labor and the Business Cycle[END_REF] formulation with separable consumption and labor, and the King et al. (KPR) [START_REF] King | Production, Growth and Business Cycles[END_REF] formulation with constant positive income effect. For each version of the model (onesector and two-sector), we can then derive the range of parameter values consistent with belief-driven fluctuations and compare it with the range of available empirical estimates.

We derive two important conclusions. First, we prove that belief-driven fluctuations in one-sector RBC models can emerge with standard slopes for the labor demand and labor supply curves but are ruled out for any empirically plausible calibrations for the critical elasticities, regardless of the specifications for the individual utility function or the production function. Second, in sharp contrast, we prove that the existence of beliefdriven fluctuations is a robust property of two-sector models, in the sense that they arise for a wide range of empirically credible values for the critical elasticities. For example, we show that sunspot fluctuations are compatible with any value for the wage elasticity of labor supply provided the other critical elasticities are in an appropriate range. Likewise, sunspot fluctuations can occur for an arbitrarily small value for the elasticity of intertemporal substitution (EIS) in consumption provided the degree of income effect is not too small. We provide an in-depth analysis explaining how and why belief-driven fluctuations can emerge in each version of the model and we show that the main sources explaining the drastically different results are, in the two-sector model, factor reallocation between the consumption and the investment sector which, in the presence of externalities, affect the relative price of the investment good. This creates potential capital gains from allocating resources to the investment sector, which leaves the room for an investment boom triggered by optimistic self-fulfilling expectations about the return on capital accumulation.

These main results are derived from the canonical versions of the one-sector and twosector RBC models, in which preferences are defined over current private consumption and leisure, and production functions are based on current capital and labor (with productive externalities). Clearly, if belief-driven fluctuations easily emerge in the stripped down version of the two-sector model, the scope for endogenous fluctuations is conceptually greater in richer models for which our benchmark model can be obtained as a special case. For example, models with variable capital utilization rate as in Wen [START_REF] Wen | Capacity Utilization under Increasing Returns to Scale[END_REF], models with different production functions in the consumption and the investment sectors as in Benhabib and Nishimura [START_REF] Benhabib | Indeterminacy and Sunspots with Constant Returns[END_REF] and Benhabib et al. [START_REF] Benhabib | Indeterminacy and Cycles in Two-Sector Discrete-Time Models[END_REF], or models with static consumption spillovers in the utility function as in Chen and Hsu [START_REF] Chen | Admiration is a Source of Indeterminacy[END_REF] and Alonso-Carrera et al. [START_REF] Alonso-Carrera | Can consumption spillovers be a source of equilibrium indeterminacy?[END_REF] should conceptually allow for an even greater scope for indeterminacy as long as our canonical model can be obtained as a limit case. 1 ' 2 We conclude that far from being an exotic feature or a theoretical curiosity arising under extreme parameter configurations, belief-driven fluctuations should be considered as a central feature of standard dynamic (multisector) macroeconomic models.

The rest of this paper is organized as follows. In Section 2, we present a short review of the related literature. In Section 3, we analyze the aggregate (one-sector) model. We define the general technology and the general utility function considered throughout the paper. We present our new and innovative way of decomposing all the elasticities that characterize preferences, focusing in particular on income effect. We then study the existence and uniqueness of the steady state and we prove that belief-driven fluctuations are not a realistic outcome of standard aggregate models for all income effects. In Section 4, we consider the two-sector model under the same general specification of preferences and technologies as in Section 3. As a result the existence and uniqueness of the steady state is derived under the same basic conditions as in the aggregate case. We show that the existence of expectation-driven fluctuations is a generic property of two-sector models and fully compatible with empirically relevant values for all the structural parameters. Section 5 concludes. All the technical proofs are contained in the Appendix.

Literature review

The endogenous fluctuations and sunspot literature was initiated by the seminal contributions of Azariadis [START_REF] Azariadis | Self-Fulfilling Prophecies[END_REF], Cass and Shell [START_REF] Cass | Do Sunspots Matter?[END_REF], Grandmont [START_REF] Grandmont | On Endogenous Competitive Business Cycles[END_REF], and Woodford [START_REF] Woodford | Stationary Sunspot Equilibria in a Finance Constrained Economy[END_REF]. Yet Benhabib and Farmer (BF) [START_REF] Benhabib | Indeterminacy and Increasing Returns[END_REF] is the first paper to analyze these issues in the standard infinite-horizon one-sector model with endogenous labor supply, the workhorse model of the RBC literature. They show that in this model, indeterminacy occurs under the assumption of a large amount of externalities leading to an upward-sloping labor demand function. While this model subjected to sunspot shocks has been shown to account for the main "stylized facts" of the business cycles at least as well as standard RBC models (see Farmer and Guo [START_REF] Farmer | Real Business Cycles and the Animal Spirits Hypothesis[END_REF]), the assumption of large aggregate IRS in production was found to be inconsistent with the data. 3 Since this seminal contribution, a major challenge in the literature has been to find extensions of this benchmark model capable of generating expectation-driven business-cycles under empirically realistic values for all structural parameters. Two strands of the literature address this challenge. The first seeks to determine how the local indeterminacy properties of the benchmark one-sector model evolve when some assumptions on preferences or the production function are relaxed. The second adds new ingredients to the benchmark model and reconsiders the issue of indeterminacy in the extended models. We briefly review some critical papers in these two strands of the literature.

The BF one-sector model is based on an additively separable utility function and a Cobb-Douglas technology. Many contributions have generalized this formulation to show that the existence of sunspot fluctuations can be compatible with a downwardsloping labor demand function. Pintus [START_REF] Pintus | Indeterminacy with Almost Constant Returns to Scale: Capital-Labor Substitution Matters[END_REF] introduced a general technology into the Benhabib-Farmer framework, while Bennett and Farmer [START_REF] Bennett | Indeterminacy with Non-Separable Utility[END_REF] and Hintermaier [START_REF] Hintermaier | On the Minimum Degree of Returns to Scale in Sunspot Models of Business Cycles[END_REF] considered non-separable preferences as defined by King et al. (KPR) [START_REF] King | Production, Growth and Business Cycles[END_REF], still assuming a Cobb-Douglas technology. Pintus [START_REF] Pintus | Local Determinacy with Non-Separable Utility[END_REF] generalized their formulation to a general production function. Lloyd-Braga et al. [START_REF] Lloyd-Braga | Indeterminacy with Small Externalities: The Role of Non-Separable Preferences[END_REF] considered general homogenous preferences and a general technology. The overall message from this literature is that preference and technology parameters, like the elasticity of intertemporal substitution in consumption, the degree of income effect on labor supply, the degree of IRS in production, and the elasticity of substitution between capital and labor, all interact to influence the local stability properties of the model. Yet in all these models, the existence of expectationdriven business cycles requires at least one structural parameter value which appears to be outside the range of available empirical estimates.

In response to this critique of the sunspot literature, some authors have modified the production structure of the model. Wen [START_REF] Wen | Capacity Utilization under Increasing Returns to Scale[END_REF] proposes a simple extension consisting in introducing a variable capital utilization rate into the Benhabib-Farmer setup, in the spirit of Greenwood et al. [START_REF] Greenwood | Investment, Capacity Utilization and the Real Business Cycle[END_REF], and proves that this extension is sufficient to allow for the existence of sunspot fluctuations under empirically plausible values for the structural parameters. 4 However, unitary elasticities of intertemporal substitution in consumption and of capital-labor substitution are assumed, which restricts the possibilities to successfully match the data in a data confrontation perspective. Considering a more general utility function and a general production function, Dufourt et al. [START_REF] Dufourt | On Sunspot Fluctuations in Variable Capacity Utilization Models[END_REF] show that the variable capital utilization model allows for the emergence of belief-driven fluctuations under a wider set of values for these elasticities if the elasticity of the labor supply curve is large enough. Yet, they also show that the model still faces difficulties in accounting for some crucial empirical facts associated with transitory shocks. Other authors have instead modified the specification for individual preferences. For example, Alonso-Carrera et al. [START_REF] Alonso-Carrera | Can consumption spillovers be a source of equilibrium indeterminacy?[END_REF] introduce consumption spillovers in the utility function and they show that this favors the occurrence of indeterminacy even when the labor supply curve is positively sloped. However, the extent to which such a model submitted to belief shocks can account for the data is left unexplored.

Two-sector models have also been considered, again following the seminal contribution of Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF]. In this paper, Benhabib and Farmer extend their initial formulation to a two-sector economy producing differentiated consumption and investment goods but with the same Cobb-Douglas technology characterized by sector-specific output externalities leading to increasing returns. Building on the fact that capital and labor can be freely allocated between sectors, they prove that the existence of local indeterminacy becomes compatible with a downward-sloping labor demand function. Unlike their one-sector contribution, it clearly appears that when external effects in each sector depend on that sector's aggregate output, factor reallocations across sectors can have strong effects on marginal products and indeterminacy can occur with much smaller externalities. Harrison [START_REF] Harrison | Indeterminacy in a Model with Sector Specific Externalities[END_REF] builds on these results to show that indeterminacy occurs for a minimum value of the externality in the investment sector, even with no externality in the consumption sector. All these conclusions have been recently confirmed by Du-fourt et al. [START_REF] Dufourt | Indeterminacy and Sunspots in Two-Sector RBC Models with Generalized No-Income-Effect Preferences[END_REF] considering GHH preferences instead of additively separable ones. 5 Guo and Harrison [START_REF] Guo | Indeterminacy with Capital Utilization and Sector-Specific Externalities[END_REF] also introduce a variable capital utilization rate into the Benhabib and Farmer model and confirm that lower externalities are required.

Worthy of mention too is all the literature departing from the contribution of Benhabib and Nishimura [START_REF] Benhabib | Indeterminacy and Sunspots with Constant Returns[END_REF] and Benhabib et al. [START_REF] Benhabib | Indeterminacy and Cycles in Two-Sector Discrete-Time Models[END_REF], in which sector-specific externalities are introduced in two-sector models with differentiated private technologies and constant social returns. In such a framework, the existence of local indeterminacy relies on a capital-intensity reversal between the private and social levels and, as shown in Nishimura and Venditti [START_REF] Nishimura | Indeterminacy in Discrete-Time Infinite-Horizon Models with Non-Linear Utility and Endogenous Labor[END_REF], requires extreme values for the elasticity of intertemporal substitution in consumption (EIS) which are not in line with empirical estimates.

All in all, although sunspot fluctuations have been shown to arise more easily in extended model formulations, there is still no general analysis in the literature.

A general aggregate model

We consider a closed economy framework in the spirit of Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Increasing Returns[END_REF] (BF). The economy is composed of a large number of identical infinitely-lived agents and a large number of identical producers. Agents consume, supply labor and accumulate capital. Firms produce the unique final good which can be used either for consumption or investment. All markets are perfectly competitive, but there are externalities in production.

The representative firm: a general technological structure

The production sector is composed of a large number of identical firms which operate under perfect competition. Output Y t is produced by combining labor L t and capital K t . The technology of each firm exhibits constant returns to scale with respect to its own inputs and we consider that each of the many firms benefits from positive externalities due to the contribution of the average levels of labor L and capital K. These external effects are exogenous and not traded in markets. The production function is

Y t = f (K t , L t )e( Kt , Lt ) (1
) with e( Kt , Lt ) the externality variable. We follow BF by assuming that externalities affect the technology in a multiplicative way but we depart from them by not requiring the production function to be Cobb-Douglas. Rather, our production function is general and satisfies:

Assumption 1. f (K, L) is C 2 over R 2
++ , increasing in (K, L), concave over R 2 ++ and homogeneous of degree one. e( K, L) is C 1 over R ++ and increasing in ( K, L). Moreover, for any given L > 0, lim

K→0 f 1 (K, L)e(K, L) = +∞ and lim K→+∞ f 1 (K, L)e(K, L) = 0
and, for any given K > 0, lim

L→0 f 2 (K, L)e(K, L) = +∞ and lim L→+∞ f 2 (K, L)e(K, L) = 0
5 See also Guo and Harrison [27].

Firms rent capital units at the real rental rate r t and hire labor at the unit real wage w t . The profit maximization program of the representative firm, max

{Yt,Lt,Kt} Y t -w t L t -r t K t ,
leads to the standard demand function for capital K t and labor L t :

r t = f 1 (K t , L t )e( Kt , Lt ) (2)
w t = f 2 (K t , L t )e( Kt , Lt ) (3
) As will become clear, the production function and the optimal decisions of firms influence the local dynamics of the model through four crucial elasticities: the elasticity of output with respect to capital stock s(K, L) (which, at equilibrium, is also the share of capital in total income), the elasticity of capital-labor substitution σ(K, L), and the elasticities of the externality variable with respect to labor, ε eL ( K, L), and capital, ε eK ( K, L):

s(K, L) = Kf 1 (K,L) f (K,L) ∈ (0, 1), σ(K, L) = -(1-s(K,L))f 1 (K,L) Kf 11 (K,L)
> 0 (4)

ε eK ( K, L) = e 1 ( K, L) K e( K, L) , ε eL ( K, L) = e 2 ( K, L) L e( K, L)
(5) Obviously, the choice of a Cobb-Douglas production function, as in BF, implies σ(K, L) = 1, whereas the use of a general production function entails σ(K, L) ∈ (0, +∞). To simplify notation, we will for now denote by s, σ, ε eK and ε eL the corresponding elasticities evaluated at the steady state. In order to allow for a direct comparison with BF, the externalities are also expressed as follows:

ε eK = sΘ k ε eL = (1 -s)Θ l (6)
where Θ k , Θ l ≥ 0 are the degrees of increasing returns to scale in capital and labor. BF assume output externalities, implying Θ k = Θ l = Θ. We allow for a more general formulation in which external effects can be factor-specific and independent of each-other.

Finally, we make a standard assumption requiring that the aggregate (i.e., taking external effects into account) labor demand and capital demand functions are decreasing in the real wage and in the rental rate of capital, respectively:

Assumption 2. Θ k < (1 -s)/sσ ≡ Θk and Θ l < s/(1 -s)σ ≡ Θl
It is well known from BF analysis that when the slope of the aggregate labor demand curve is positive and greater than the slope of the aggregate labor supply curve, indeterminacy and sunspot fluctuations can occur in the one-sector infinite-horizon model. We rule out this possibility here because it entails extremely high degrees of increasing returns to scale that are at odds with the data. We will be more specific about realistic degrees of IRS later in the paper.

The representative household: a general utility function

The economy is composed of a continuum of mass 1 of identical households. In each period, the representative household is endowed with units of time. Given the real wage w t and the rental rate of capital r t , the household decides how much of its available time to allocate to leisure time L t and hours worked l t , and how much to consume c t . It also rents its capital stock k t to the representative firms, and accumulates capital according to the following capital accumulation constraint:

k t+1 = (1 -δ + r t )k t + w t l t + d t -c t (7)
where δ ∈ (0, 1) is the capital depreciation rate, and d t are potential dividends redistributed ex-post by firms.

In each period, the utility that the household derives from consumption and leisure is described by a general instantaneous utility function u(c, L). As in the case of the productive side of the economy, we want our analysis to be as general as possible. We thus make the following minimum standard assumptions on the utility function:

Assumption 3. u(c, L) is C 2 over R 2
++ increasing in each argument, strictly quasiconcave in (c, L), and satisfies the Inada conditions

lim c→0 u 1 (c, L) = +∞, lim c→+∞ u 1 (c, L) = 0, lim L→0 u 2 (c, L) = +∞ and lim L→+∞ u 2 (c, L) = 0.
The Inada conditions are introduced to ensure an interior optimum. Furthermore, to avoid basing our analysis of the local stability conditions and of the occurrence of sunspot fluctuations on exotic features regarding individual preferences, we introduce the following standard assumption on consumption and leisure: Assuming that the intertemporal utility function is additively separable over time, the representative consumer solves the following lifetime utility maximization program (where β ∈ (0, 1) is the subjective discount factor):

max {ct, lt, k t+1 } t=0...∞ +∞ t=0 β t u(c t , -l t ) s.t. k t+1 = (1 -δ + r t )k t + w t l t + d t -c t , t = 0...∞ k 0 given (8)
Denoting by λ t the Lagrange multiplier on constraint [START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF] and R t = 1 -δ + r t the net return factor on capital, the first-order conditions can be written as

u 1 (c t , -l t ) = λ t , (9)
u 2 (c t , -l t ) u 1 (c t , -l t ) = w t ( 10
)
λ t = βR t+1 λ t+1 (11
) Equation ( 10) describes the optimal consumption-leisure trade-off, while equations ( 9) and [START_REF] Bennett | Indeterminacy with Non-Separable Utility[END_REF] jointly describe the optimal arbitrage between consumption and saving (i.e., the Euler equation). An optimal path must also satisfy the transversality condition: lim

t→+∞ β t λ t k t+1 = 0 (12)
Following Rotemberg and Woodford [START_REF] Rotemberg | Oligopolistic Pricing and the Effects of Aggregate Demand on Economic Activity[END_REF], we can rewrite the optimality conditions (9-10) in terms of time-invariant Frisch consumption-demand and labor-supply curves involving the real wage w t and the marginal utility of wealth λ t : c t = c(w t , λ t ), l t = l(w t , λ t ) (13) As was the case for the productive size of the economy, we show later that the local dynamics of the model around the steady state is determined by a limited number of critical elasticities. Denote by

cw = c 1 (w,λ)w c , cλ = c 2 (w,λ)λ c , lw = l 1 (w,λ)w l , lλ = l 2 (w,λ)λ l , (14)
the Frisch elasticities of the demand and supply functions [START_REF] Campbell | Asset Prices, Consumption and the Business Cycle[END_REF], and by cc = -u 1 (c,L) u 11 (c,L)c , (15) the elasticity of intertemporal substitution in consumption. We can easily prove the following Lemma: Lemma 1. The three critical elasticities cc , lw and lλ are related to the individual utility function by

cc = -u 1 u 11 c , lw = 1 l -u 11 u 2 u 11 u 22 -u 12 u 21 , lλ = 1 l u 21 u 1 -u 11 u 2 u 11 u 22 -u 12 u 21 (16)
Moreover, the remaining two elasticities cw and cλ are related to cc , lw , and lλ through the following equations

cw = C ( lw -lλ ) , cλ = -cc + C 1 -lλ lw lλ ( 17
)
where C = θ(1-s)/(θ-sβδ) < 1 is the steady state ratio of wage income over consumption (wl/c), which is independent of the specification of the individual utility function, and

θ = 1 -β(1 -δ).
Proof. See Appendix 6.1.

An implication of this Lemma is that, as far as the representative consumer's decision is concerned, the dynamic properties of the model are completely determined by the three elasticities lw , lλ , and cc , in addition to the parameter C, which is independent of the specification of individual preferences.

Using this Lemma, we can immediately derive the following Proposition:

Proposition 1. Under Assumption 3, cc > 0 and lw > 0. Moreover, under Assumption 4, lλ ≥ 0, cλ ≤ 0, and thus

cc ≥ C lλ ( lw -lλ ) lw ≡ N cc . (18)
Proof. See Appendix 6.2.

The importance of this Proposition is that it shows how assumptions on preferences (namely Assumptions 3 and 4) naturally translate into restrictions on the critical elasticities cc , lw , and lλ . Note that these restrictions ( cc > 0, lw > 0, lλ ≥ 0, and cc ≥ N cc ) are actually much simpler than working directly with the standard concavity and normality assumptions based on the utility function.

Another nice feature of considering these elasticities, instead of considering the firstorder and second-order derivatives of u(•), is that the former have a clear economic interpretation. The EIS in consumption cc and the Frisch elasticity of labor supply lw have a well-known interpretation that requires no further discussion. On the other hand, since λ t is the marginal utility of wealth, the elasticity lλ captures the extent to which a change in the household's expected wealth over its entire lifetime affects the current labor supply decision. Indeed, when, for any reason, lifetime income decreases, the intertemporal budget constraint obtained from aggregating [START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF] over time becomes more restrictive, and λ t increases as the household's consumption choices are more constrained.

As implied by the normality assumption, it follows that consumption and leisure decrease while hours worked increase. The elasticity lλ captures the extent to which such a change in lifetime income affects the current labor supply decision.

In short, lλ is a properly defined measure of the wealth effect on labor supply. This elasticity is particularly important because the recent literature has shown that the intensity of this wealth effect plays a significant role in the local stability properties of many dynamic macroeconomic models (see Dufourt et al. [START_REF] Dufourt | Indeterminacy and Sunspots in Two-Sector RBC Models with Generalized No-Income-Effect Preferences[END_REF][START_REF] Dufourt | Sunspot Fluctuations in Two-Sector Models with Variable Income Effects[END_REF], Jaimovich [START_REF] Jaimovich | Income Effects and Indeterminacy in a Calibrated One-Sector Growth Model[END_REF]). However, this literature only addressed particular utility functions in which the wealth effect is known to be either "positive" (but not precisely defined) or zero. Moreover, these specific utility functions also require the introduction of other cross-restrictions on the three critical elasticities defined above. With our analysis, however, the intensity of the wealth effect on labor supply can be chosen independently of the values given to the other two elasticities ( lw and cc ).

To better illustrate these points, the following Proposition clarifies the restrictions implied by some of the most widely used classes of utility functions:

Proposition 2. Under KPR preferences, u(c, L) = c 1-γ 1-γ v(L),
with γ > 0, γ = 1 log(c) + log(v(L)), with γ = 1 with L = -l and v(L) increasing and concave (if γ ≤ 1) or decreasing and convex (if γ > 1), the critical elasticities satisfy:

cc = 1 γ , lw = L l 1 (1-cc)v (L)L/v(L)-v (L)L/v (L) > 0, lλ = cc lw . Under generalized GHH preferences, u(c, l) = 1 1-γ c -l 1+χ 1+χ 1-γ
, with γ > 0 and χ ≥ 0, the critical elasticities satisfy:

cc = 1 γ 1 -C 1+χ , lw = 1 χ , lλ = 0. Under Generalized Hansen [28] preferences, u(c, l) = c 1-γ 1-γ -l 1+χ
1+χ , with γ > 0 and χ ≥ 0, the critical elasticities satisfy:

cc = 1 γ , lw = lλ = 1 χ . (19)
According to Proposition 2, in the case of KPR preferences, only two of the three critical elasticities cc , lw , and lλ are independent since they are related through the equation lλ = cc lw . In the case of generalized GHH preferences, the restriction lλ = 0 is well-known, having been introduced on purpose to eliminate the wealth effect on labor supply. However, there is often far less awareness that, with this class of preferences, changing the calibration of the preference parameter χ to change the value of the wage elasticity of the labor supply, lw = 1/χ, meanwhile generates a change in the EIS in consumption, cc . In fact, Proposition 2 implies that changing the calibration of χ to χ = χ also requires adjusting the calibration of γ to γ = (1 -C/(1 + χ )) cc if one wants to keep the initial value of the EIS unchanged. Finally, under generalized Hansen preferences, a strong restriction relating the two Frisch elasticities of the labor supply curve is introduced, since we have in this case: lw = lλ .

The particular case of the Jaimovich-Rebelo formulation Jaimovich and Rebelo (JR) [START_REF] Jaimovich | Can News About the Future Drive the Business Cycles?[END_REF] were the first to discuss the importance of the income effect on the occurrence of indeterminacy and sunspot fluctuations. Their discussion is based on the following specification of the instantaneous utility function:

u(c t , l t , X t ) = ct- l 1+χ t 1+χ Xt 1-γ -1 1-γ (20) with X t = c φ t X 1-φ t-1 and φ ∈ [0, 1].
As they state, this specification nests as polar cases the GHH utility function (when φ = 0) and the KPR utility function (when φ = 1) formulations. The magnitude of the income effect is therefore controlled by varying the value of γ between these two extremes.

Assuming γ = 1 and Θ k = Θ l = Θ, Jaimovich [START_REF] Jaimovich | Income Effects and Indeterminacy in a Calibrated One-Sector Growth Model[END_REF] shows that, for some values of Θ compatible with a negatively-slopped labor demand function, there exist two bounds 0 < φ < φ < 1 such that when φ ∈ (φ, φ) local indeterminacy and sunspot fluctuations occur under realistic values for all the structural parameters. The conclusion is that the income effect has a non-linear effect on the range of values consistent with indeterminacy, which appears to arise under intermediate amounts of income effect, and to be ruled out with either low or high amounts.

A difference between the JR specification and our specification is that, except for the two polar cases φ = 0 and φ = 1, the JR utility function assumes that an additional state variable X t enters the utility function:

X t = t-1 s=0 c φ(1-φ) s t-s X (1-φ) s 0
It follows that the utility function at time t depends on the whole history of past consumption decisions. The result is that the consumption demand and labor supply decisions no longer write c(w t , λ t ) and l(w t , λ t ) but c(w t , λ t , X t ) and l(w t , λ t , X t ). In short, compared to our specification, such a formulation introduces two additional elasticities cX and lX associated with consumption habits which generate a dynamic link between current and future consumption and labor supply decisions -what Jaimovich refers to as a form of "dynamic" income effects. 6To avoid the complexities of introducing additional state variables, Nourry et al. [START_REF] Nourry | Aggregate Instability under Balanced-Budget Consumption Taxes: A Re-examination[END_REF] and Dufourt et al. [START_REF] Dufourt | Sunspot Fluctuations in Two-Sector Models with Variable Income Effects[END_REF] consider a modified JR utility function which only involves current-period variables, namely

u(c t , l t ) = ct- l 1+χ t 1+χ c φ t 1-γ -1 1-γ (21)
We recover the two polar cases of a GHH and a KPR utility function associated with φ = 0 and φ = 1, respectively. Moreover, it is now possible to vary the values of the three critical elasticities cc , lw , and lλ by considering alternative calibrations for the three parameters γ, χ, and φ. However, the critical elasticities are very cumbersome combinations of these parameters, and when one parameter is adjusted so as to change the value of one elasticity, the other parameters also have to be adjusted to maintain the value of the other elasticities constant. It is therefore much better to work with a general utility function and calibrate the three critical elasticities directly, as we do in this paper.

Intertemporal equilibrium and steady state

At a symmetric general equilibrium of the economy, prices {w t , r t , λ t } adjust so that all markets clear at any date t, with the externality variables satisfying (k t , l t ) = (k t , l t ) for any t. Imposing the latter equalities in the set of physical constraints and optimality conditions (1)-( 3), [START_REF] Bennett | Indeterminacy with Non-Separable Utility[END_REF], and ( 13), we obtain that a symmetric general equilibrium satisfies in any t,

λ t = β(1 -δ + r t+1 )λ t+1 k t+1 = (1 -δ)k t + y t -c t r t = f 1 (k t , l t )e(k t , l t ) w t = f 2 (k t , l t )e(k t , l t ) ( 22
)
c t = c(w t , λ t ) l t = l(w t , λ t ) y t = f (k t , l t
)e(k t , l t ) together with the initial condition k 0 given and the transversality condition [START_REF] Boldrin | Habit Persistence, Asset Returns, and the Business Cycle[END_REF].

From these dynamic equations, we immediately derive that if a steady state exists, the rental rate of capital at the steady state is

r * = 1-β(1-δ) β ≡ θ β
In order to study the existence and uniqueness of a steady state, we analyze the existence of a 6-uple (k * , y * , l * , c * , w * , λ * ) solution to the set of equations

f 1 (k * , l * )e(k * , l * ) = θ β , f 2 (k * , l * )e(k * , l * ) = u 2 (c * , -l * ) u 1 (c * , -l * ) c * = f (k * , l * )e(k * , l * ) -δk * , w * = f 2 (k * , l * )e(k * , l * ) y * = f (k * , l * )e(k * , l * ), c * = c(w * , λ * ) (23)
Note that, for analytical convenience, instead of considering the Frisch labor supply equation l * = l(w * , λ * ), we reintroduce the initial optimality condition involving the marginal rate of substitution between consumption and labor. We first prove the following Lemma:

Lemma 2. At the steady state, the ratios y * /k * , c * /k * and w * l * /c * satisfy

y * k * = θ sβ , c * k * = θ-sβδ sβ , and w * l * c * = θ(1-s) θ-sβδ ≡ C Proof. See Appendix 6.3.
Using this Lemma, we derive the following Proposition:

Proposition 3. Under Assumptions 1-4, a unique steady state generically exists. Moreover, for any given calibration of structural parameters, there always exists a value * > 0 such that when = * , the steady state is constant across calibrations with l * = l * < * .

Proof. See Appendix 6.4.

An implication of Proposition 3 is that, when analyzing how alternative calibrations for the structural parameters affect the stability properties of the model, it is possible to maintain the steady state (k * , y * , l * , c * , w * , λ * ) unchanged by adjusting the value for * accordingly. In other words, we can follow the usual practice of "calibrating" the level of hours worked at the steady state without difficulty.

Local stability analysis

We now carry out a thorough analysis of the local stability properties of the steady state when the dynamics is defined by [START_REF] Grandmont | On Endogenous Competitive Business Cycles[END_REF]. In order to do so, we log-linearize the set of equations in [START_REF] Grandmont | On Endogenous Competitive Business Cycles[END_REF] around the unique steady state. Using Lemmata 1 and 2, we obtain (where hatted variables denote percentage deviations from the steady state):

λ t = λ t+1 + θ r t+1 k t+1 = (1 -δ) k t + θ sβ y t -θ-sβδ sβ c t r t = -1-s σ + sΘ k k t + 1-s σ + (1 -s)Θ l l t w t = s σ + sΘ k k t + -s σ + (1 -s)Θ l l t c t = C ( lw -lλ ) w t + C 1 -lλ lw lλ -cc λ t l t = lw w t + lλ λ t y t = s(1 + Θ k ) k t + (1 -s) (1 + Θ l ) l t
This is a system of seven equations in seven variables, only two of these equations being dynamic. To analyze the local stability properties of the model, we first reduce the system by using the five static equations to eliminate five variables, y t , c t , l t , w t , and r t , from the dynamic equations. The obtained system of minimal dimension -two dynamic equations in two variables, k t and λ t -can be expressed as:

k t+1 λ t+1 = J 11 J 12 J 21 J 22 k t λ t ≡ J k t λ t
where J is the Jacobian matrix of the underlying non-linear 2-dimensional system evaluated at the steady state, which is given in the following Proposition:

Proposition 4. The elements of the Jacobian matrix J are:

J 11 = 1 β 1 + θΘ k + θ(1-s)( 1 σ +Θ k )( lw Θ l + lλ ) 1+ lw [ s σ -Θ l (1-s)] , J 12 = 1 sβ θ(1-s) lλ lw ( lw Θ l + lλ ) 1+ lw [ s σ -Θ l (1-s)] + (θ -sβδ) cc J 21 = θ 1-s σ -sΘ k -lw σ [Θ l (1-s)+sΘ k ] 1+ lw [ s σ -Θ l (1-s)]+ lλ θ(1-s)( 1 σ +Θ l) J 11 , J 22 = 1+ lw [ s σ -Θ l (1-s)]+θ{ 1-s σ -sΘ k -lw σ [Θ l (1-s)+sΘ k ]}J 12 1+ lw [ s σ -Θ l (1-s)]+ lλ θ(1-s)( 1 σ +Θ l)
Proof. See Appendix 6.5.

The local dynamics of the model is thus determined by the nine structural parameters constituting the matrix J: four of them concern the productive side of the economy, namely s, σ, Θ k , and Θ l , four of them concern individual preferences, namely β, cc , lw , lλ , and finally there is the depreciation rate of capital δ.

Using the geometrical methodology of Grandmont et al. [START_REF] Grandmont | Capital-Labor Substitution and Competitive Nonlinear Endogenous Business Cycles[END_REF] as presented in Appendix 6.6, we prove that there exist two bifurcation loci in the parameter space such that, when cc is increased from 0 to +∞, a change in the stability properties of the steady state occurs when cc crosses any of the two loci. These results are formally summarized in the following Lemma: Lemma 3. Under Assumptions 1-4, let Ω = (β, δ, s, σ, Θ k , Θ l ) be the set of structural parameters. For any ω ∈ Ω such that σ ≤ σ ≡ θ/(1 -β), there exist two bifurcation curves crossing the 3-dimensional plane ( lw , lλ , cc ) and generating a change in the local stability properties of the steady state:

-a flip bifurcation curve F cc ( lw , lλ ) associated with one real eigenvalue of J crossing -1, -a (degenerate) transcritical bifurcation curve T cc ( lw , lλ ) associated with one real eigenvalue of J crossing 1.

These bifurcation curves appear for any lw > lw , with

lw ≡ 1-s-σsΘ k (1-s)Θ l +sΘ k .
There also exists one critical bound lλ ( lw ) such that D = 1 when lλ = lλ ( lw ). This critical bound exists for any

(Θ k , Θ l ) such that Θ k ∈ [0, Θ k ) and Θ l ∈ (Θ l , Θl ), with Θ k ≡ sβ (1-s)σ-(1-β) , Θ l ≡ 1-β+Θ k β .
The formal expressions of the bifurcation curves and the critical bound are given in Appendix 6.6.

Proof. See Appendix 6.6.

The following Theorem now provides a complete picture of the local stability properties of the aggregate model. Theorem 1. Under Assumptions 1-4, let σ ≤ σ ≡ θ/(1 -β) and consider the bifurcation curves, critical bound, and thresholds defined in Lemma 3. Then the following results hold:

Case 1 -Low wage elasticity of labor supply: lw ∈ (0, lw ).

The steady state is a saddle-point.

Case 2 -High wage elasticity of labor supply: lw > lw .

-Under low capital externalities Θ k ∈ (0, Θ k ), the steady state is

i) for Θ l ∈ [0, Θ l ), -a saddle-point if cc ∈ [0, T cc ) ∪ ( F cc , +∞), -a source if cc ∈ ( T cc , F cc ). ii) for Θ l ∈ (Θ l , Θl ), -a saddle-point if cc ∈ [0, T cc ) ∪ ( F cc , +∞), -a source if cc ∈ ( T cc , F cc ) and lλ < lλ ( lw ), -a sink if cc ∈ ( T cc , F cc ) and lλ > lλ ( lw ).
-Under large capital externalities Θ k ∈ (Θ k , Θk ), the steady state is

-a saddle-point if cc ∈ [0, T cc ) ∪ ( F cc , +∞), -a source if cc ∈ ( T cc , F cc ). Proof. See Appendix 6.7.

Discussion

Theorem 1 -characterizing the local stability properties of the one-sector model for any specification of individual preferences, any specification for the production function, and any degrees of IRS in capital and labor consistent with downward-sloping labor and capital demand curves -considerably generalizes existing results surveyed in the literature review. 7 Three main results can be drawn from this Theorem. First, from a theoretical standpoint, it is possible to identify an area in the parameter space such that indeterminacy and sunspot fluctuations exist in the one-sector model, even though externalities are mild enough to ensure downward-sloping capital demand and labor demand Yet, this area of indeterminacy occurs for empirically implausible configurations of parameter values (Result 1). Second, under various minor additional assumptions on parameters frequently encountered in the literature, local indeterminacy is excluded (Result 2). Third, for parameter values in the range of available empirical estimates, the steady-state is a saddle path (Result 3). Before elaborating further on these conclusions, we first provide an explanation as for why indeterminacy can occur in this standard model.

Interpretation for indeterminacy:

Case 2(ii) of Theorem 1 shows that indeterminacy and sunspot fluctuations can occur in the one-sector model (with standard slopes on the labor supply and labor demand curves) when the following necessary conditions are met: Θ l ∈ (Θ l , Θl ), lw > lw , lλ > lλ ( lw ) and cc > T cc ( lw , lλ ) . To understand why these conditions are needed, it is simpler to start with a situation of fixed labor supply ( lw = lλ = 0) and analyze why indeterminacy and sunspot equilibria cannot occur in this case. Consider the Euler equation λ t = β(1 -δ + r t+1 )λ t+1 and assume that agents expect an increase in the future return on capital accumulation, r t+1 . Can such an expectation be self-fulfilling? According to the Euler equation, an increase in r t+1 is associated with an increase in the marginal utility of wealth λ t and an incentive to substitute investment for consumption (since cλ < 0), leading to a lower consumption c t , a higher investment i t and a higher capital stock k t+1 at t + 1. With a fixed labor supply, the larger capital stock k t+1 generates a decrease in the marginal productivity of capital at t + 1 and thus a decrease in r t+1 = f 1 (k t+1 , l)e(k t+1 , l). The initial expectation cannot be self-fulfilling.

To make this expectation self-fulfilling, the negative influence of the larger capital stock on the real return on capital must be more than compensated by a proportionately larger positive influence associated with an increase in labor at t + 1. This is only possible if both the labor demand curve and the labor supply curve are sufficiently elastic. Positives externalities in labor Θ l ∈ (Θ l , Θl ) make a step in that direction by rendering the aggregate labor demand curve more elastic with respect to the real wage. Consider now the log-linearized versions of the Frisch consumption-demand and labor-supply curves:

c t = C ( lw -lλ ) cw w t + C 1 - lλ lw lλ -cc cλ <0 λ t l t = lw w t + lλ λ t
At period t, an increase in λ t generates an increase in the labor supply curve due to the positive income effect, lλ > 0. Since the capital stock is predetermined at t, labor demand is unchanged and the increase in labor supply generates a decrease in the equilibrium real wage w t . At t + 1, the change in labor supply must be sufficiently large to compensate the negative effect of capital accumulation on the real return on capital, which requires lλ to be very large, i.e. lλ > lλ ( lw ) > lw , as shown in Theorem 1. However, for the story to remain fully consistent, one must also ensure that consumption still decreases at t and investment still increases at t when λ t increases. In the log-linearized version of the Frisch consumption demand above, one easily observe that cw = C ( lwlλ ) < 0 when lλ > lw , so that the first component is positive ( w t < 0 since w t decreases at t). A decrease in consumption c t then requires cλ to be sufficiently negative. This in turn requires that the intertemporal substitution effect be sufficiently large, i.e. that cc > T cc ( lw , lλ ) as shown in Theorem 1.

Result 1: Implausibility of indeterminacy in the one-sector model Indeterminacy requires at the same time a sufficiently large degree of IRS in labor, a sufficiently large wage elasticity of the labor supply curve, a sufficiently large degree of income effect on labor supply, and a sufficiently large EIS in consumption: Θ l ∈ (Θ l , Θl ), lw > lw , , lλ > lλ ( lw ) and cc > T cc ( lw , lλ ). However, it is easy to verify that these four conditions can only be simultaneously satisfied for extremely high (thus highly unrealistic) values for several critical elasticities. To see this, consider that the wage elasticity of the labor supply curve is close to the lower bound lw required for indeterminacy. From Appendix 6.6, we know that T cc tends to +∞ when lw tends to lw , making the condition cc > T cc ( lw , lλ ) impossible to satisfy for plausible EIS values. Conversely, consider now that the aggregate labor supply curve is very elastic (due, for example, to labor indivisibility at the individual level combined with perfect unemployment insurance, as in Hansen [START_REF] Kydland | Time to Build and Aggregate Fluctuations[END_REF] and Rogerson [88]). We know that lλ ( lw ) is an increasing function of lw so the condition lλ > lλ ( lw ) required for indeterminacy now imposes very large degrees of income effect on the labor supply. Moreover, T cc ( lw , lλ ) is increasing in lλ , so the large degree of income effect has a retroactively large effect on the value for the EIS in consumption required for indeterminacy ( cc > T cc ( lw , lλ )). No empirically realistic calibration ensures this outcome.

To fix ideas, consider a simple calibration with σ = 1, Θ k = 0.25, Θ l set to its upper bound Θ l (the most favorable case for indeterminacy), and lw = 3 (the value advocated by Rogerson Result 2: Impossibility of indeterminacy under additional assumptions frequently made in the literature Consider some various additional restrictions on structural parameters frequently imposed in the literature. We derive from Theorem 1 the following Proposition: Proposition 5. Under Assumptions 1-4, for any σ > 0, local indeterminacy is ruled out in the following cases: i)

Θ k = Θ l , ii) lw = 0, iii) lλ = 0, iv) Θ l = 0.
Proof. See Appendix 6.8.

Case i) of Proposition 5 corresponds to the case of output externalities considered in, e.g., Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Increasing Returns[END_REF], and was initially proved by Hintermaier [START_REF] Hintermaier | On the Minimum Degree of Returns to Scale in Sunspot Models of Business Cycles[END_REF] in the case of a Cobb-Douglas technology. Here, we extend this result to any production function. Cases ii), iii), and iv) correspond, respectively, to the case of an inelastic labor supply, to the case of GHH preferences with no-income effect on labor supply, and to the case in which externalities occur solely through the aggregate capital stock. In all these cases, indeterminacy is ruled out under otherwise standard assumptions regarding the utility and production functions defined in Assumptions 1-4.

Result 3: Saddle-path stability for realistic calibrations

Finally, we can introduce some restrictions on the nine structural parameters (s, σ, Θ k , Θ l , β, cc , lw , lλ and δ) influencing the local stability properties of the steady state based on empirical considerations. We take advantage of the fact that a narrow range of empirical estimates exist for several of these parameters. In particular, it is widely accepted in the literature that, at a quarterly frequency, the subjective discount factor is close to β = 0.99, consistent with a long-run annual return on capital of around 4%. Likewise, empirical estimates for the annual depreciation rate of capital are typically around 10%, implying δ = 0.025. In the US, the share of capital income in total income is typically estimated around 30%, implying a capital elasticity in the production function close to s = 0.3. Estimates for other critical elasticities are often more variables across empirical studies, but a range including most available empirical estimates can nonetheless be defined. For example, based on the recent empirical literature (see e.g. León-Ledesma et al. [START_REF] León-Ledesma | Identifying the Elasticity of Substitution with Biased Technical Change[END_REF], Klump et al. [START_REF] Klump | The Normalized CES Production Function -Theory and Empirics[END_REF], Duffy and Papageorgiou [START_REF] Duffy | A Cross-Country Empirical Investigation of the Aggregate Production Function Specification[END_REF] and Karagiannis et al. [START_REF] Karagiannis | Variable Elasticity of Substitution and Economic Growth: Theory and Evidence[END_REF]), a plausible range for the capital-labor elasticity of substitution is σ ∈ (0, 2). Likewise, using the various empirical estimates provided by Campbell [START_REF] Campbell | Asset Prices, Consumption and the Business Cycle[END_REF], Vissing-Jorgensen [START_REF] Vissing-Jorgensen | Limited Asset Market Participation and the Elasticity of Intertemporal Substitution[END_REF],

and Gruber [START_REF] Gruber | A Tax-Based Estimate of the Elasticity of Intertemporal Substitution[END_REF], a plausible range for the EIS in consumption is cc ∈ (0, 2). Finally, estimates of increasing returns to scale by Basu and Fernald [START_REF] Basu | Returns to Scale in US Production: Estimates and Implications[END_REF] for US manufacturing industry provide a value of around 10% with standard deviation 0.33, which enables us to define a range of empirically credible values for the aggregate degree of IRS in the model, Θ = (1 -s)Θ l + sΘ k , of Θ ∈ (0, 0.43). Regarding the Frisch wage-elasticity of the labor supply curve, it is well known from the literature that for various reasons this elasticity can be large at the aggregate level even though it is small at the individual level (see e.g. Rogerson and Wallenius [START_REF] Rogerson | Micro and Macro Elasticities in a Life Cycle Model with Taxes[END_REF], and Prescott and Wallenius [START_REF] Prescott | Aggregate Labor Supply[END_REF] for a discussion). Our choice here is to not restrict this elasticity a priori in order to include Hansen's [START_REF] Hansen | Indivisible Labor and the Business Cycle[END_REF] specification of individual preferences -associated with an infinitely elastic aggregate labor supply curve -into the analysis, since these preferences are widely used in the DSGE literature. This leads us to introduce the following Assumption: Assumption 5. Realistic structural parameters: β = 0.99, δ = 0.025, s = 0.3, σ ∈ (0, 2), cc ∈ (0, 2) and Θ ∈ (0, 0.43) with Θ = (1 -s)Θ l + sΘ k .

We then obtain the following Proposition which follows directly from Theorem 1: Proposition 6. Under Assumptions 1-5, the steady state is a saddle-point.

Proof. See Appendix 6.9.

Proposition 6 generalizes a well-known property of the RBC model with constant returns to scale to the case of increasing returns to scale associated with positive externalities: for any empirically plausible calibration of structural parameters, the steadystate is a saddle path, regardless of the specification of individual preferences and the specification of the production function.

A general two-sector model

As emphasized by Jaimovich and Rebelo [START_REF] Jaimovich | Can News About the Future Drive the Business Cycles?[END_REF], aggregate and sectoral comovement are central features of business-cycles. We now assess whether indeterminacy and sunspot fluctuations are a more likely outcome of multisector infinite horizon models. There are of course many possible ways of constructing multisector economies. To facilitate comparison with the existing literature, we choose to focus our analysis on a two-sector model similar to the one analyzed by Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF], except that we do not restrict the specifications of the utility and the production functions. 9Thus, we consider a two-sector economy in which firms produce differentiated consumption and investment goods using capital and labor. As in Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF], we assume that capital and labor are perfectly mobile across sectors, and that both sectors produce their goods with the same technology at the private level. However, we assume, as in Dufourt et al. [START_REF] Dufourt | Indeterminacy and Sunspots in Two-Sector RBC Models with Generalized No-Income-Effect Preferences[END_REF], that only the firms in the investment good sector are affected by productive externalities. This choice is based on the fact that empirical estimates for the degree of IRS in the consumption sector are close to zero, while they are positive and significant in the investment sector (see e.g. Harrison [START_REF] Harrison | Returns to scale and Externalities in the Consumption and Investment Sectors[END_REF]).

Given these assumptions, firms in the consumption sector produce output Y ct according to the production function:

Y ct = f (K ct , L ct ) ( 24
)
where K ct and L ct are capital and labor allocated to the consumption sector.

In the investment sector, output Y It is also produced according to the same production function, but is affected by a productive externality

Y It = f (K It , L It )e( KIt , LIt ) (25)
where K It and L It are the numbers of capital and labor units used in the production of the investment good, and e( KIt , LIt ) is the externality variable. The functions f (., .) and e(., .) of course satisfy Assumption 1. Following Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF], we also restrict the specifications of externalities to consider output externalities, satisfying Θ k = Θ l = Θ ≥ 0.10 Recall from Proposition 5 that under such a restriction, local indeterminacy is completely ruled out in the aggregate model.

Assuming that factor markets are perfectly competitive and that capital and labor inputs are perfectly mobile across the two-sectors, the first-order conditions for profit maximization of the representative firm in each sector are:

r t = f 1 (K ct , L ct ) = p t f 1 (K It , L It )e( KIt , LIt ), (26)
w t = f 2 (K ct , L ct ) = p t f 2 (K It , L It )e( KIt , LIt ) (27)
where r t , p t , and w t are respectively the rental rate of capital, the price of the investment good, and the real wage rate at time t, all in terms of the price of the consumption good, which is chosen here as the numeraire.

As in the previous section, we restrict the degree of IRS to ensure that the capital and labor demand functions are negatively sloped. Under output externalities, this is ensured by the following Assumption, replacing Assumption 2 above:

Assumption 6. Θ < s/(1 -s)σ
Denoting by i t the investment, the budget constraint faced by the representative household is

c t + p t i t = r t k t + w t l t + d t, (28)
where again dividends d t are zero at equilibrium. The law of motion of the capital stock is:

k t+1 = (1 -δ)k t + i t (29)
The household then maximizes its present discounted lifetime utility max

{k t+1 ,ct,lt,it} ∞ t=0 +∞ t=0 β t u(c t , -l t ) (30)
subject to [START_REF] Hansen | Indivisible Labor and the Business Cycle[END_REF], [START_REF] Harrison | Indeterminacy in a Model with Sector Specific Externalities[END_REF], and k 0 given. The first-order conditions and the transversality condition are the same as ( 9)-( 12), with the return factor now defined as:

R t+1 = (1-δ)p t+1 +r t+1 pt (31)

Intertemporal equilibrium and steady state

We consider symmetric rational expectation equilibria which consist of prices {r t , p t , w t } t≥0 and quantities {c t , l t , i t , k t , Y ct , Y It , K ct , K It , L ct , L It } t≥0 , with the externality variables satisfying (K It , L It ) = (K It , L It ) for any t, thereby satisfying the household's and the firms' first-order conditions as given by ( 9)-( 11) and ( 26)-( 27), the technological and budget constraints ( 24)-( 25) and ( 28)-( 29), the market equilibrium conditions for the consumption and investment goods

c t = Y ct , i t = Y It , (32)
with GDP defined as y t = c t + p t i t , the market equilibrium conditions for capital and labor

K ct + K It = k t , L ct + L It = l t , (33)
and the transversality condition [START_REF] Boldrin | Habit Persistence, Asset Returns, and the Business Cycle[END_REF]. Combining ( 24)-( 25) and firms' first-order conditions ( 26)-( 27), we derive p t e(K It , L It ) = 1 and that the equilibrium capital-labor ratios in the consumption and the investment sectors are identical and equal to

k t /l t = K ct /L ct = K It /L It = sw t / ((1 -s)r t ).
Combining this with Assumption 1, aggregate output y t can be rewritten as y t = f (k t , l t ), and the first-order conditions with respect to capital and labor in the consumption and investment sectors can be expressed as r t = f 1 (k t , l t ) and w t = f 2 (k t , l t ). It follows that a symmetric general equilibrium satisfies in any t,

λ t = βλ t+1 (1 -δ)p t+1 + r t+1 p t (34)
k t+1 = (1 -δ)k t + y t -c t p t (35)
r t = f 1 (k t , l t ) ( 36
)
w t = f 2 (k t , l t ) ( 37
)
p t = 1 e(K It , L It ) ( 38
)
c t = c(w t , λ t ) (39)
l t = l(w t , λ t ) (40)
y t = f (k t , l t ) (41)
K ct = sc t r t (42)
L ct = (1 -s)r t K ct sw t ( 43
)
k t = K It + K ct ( 44
)
l t = L It + L ct (45
) together with the initial condition k 0 given and the transversality condition [START_REF] Boldrin | Habit Persistence, Asset Returns, and the Business Cycle[END_REF].

It is easy to show that the same conclusion as in Proposition 3 applies here: under Assumptions 1, 3, 4, and 6, there exists a unique steady state. Moreover, this steady state can be maintained constant across calibrations by adjusting the value of accordingly. 11

Local stability analysis

As in the former Section, we log-linearize the system of equations ( 34)-( 45) around the steady state. Once again, this system contains only two dynamic equations, so that it can be reduced to a system of minimal dimension, i.e. a system involving two dynamic equations in two variables k t and λ t . This reduced system can be expressed as:

k t+1 λ t+1 = J 11 J 12 J 21 J 22 k t λ t ≡ J k t λ t
where the Jacobian matrix J is given in the following Proposition:

Proposition 7. The elements of the Jacobian matrix J are:

J 11 = A 22 B 11 +B 21 A 21 , J 12 = A 22 B 12 -B 22
A 21

, J 21 = B 11 , J 22 = -B 22 with

A 21 = 1 + θ(1-s) σ lλ 1+ s lw σ -(1-δ)Θ sδ θ(1-s) 2 lλ lw 1+ s lw σ + (θ -sβδ) cc , A 22 = θ(1-s) σ 1+ s lw σ + θ(1-δ)Θ δ 1+ s lw σ + (1-s) lλ σ 1+ s lw σ B 11 = 1+Θ sβ θ(1-s) 2 lλ lw 1+ s lw σ + (θ -sβδ) cc , B 12 = 1 β 1 + θ(1-s) σ lλ 1+ s lw σ + θΘ 1+ s lw σ + (1-s) lλ σ 1+ s lw σ B 21 = 1 -Θ sβδ θ(1-s) 2 lλ lw 1+ s lw σ + (θ -sβδ) cc , B 22 = θΘ βδ 1+ s lw σ + (1-s) lλ σ 1+ s lw σ
Proof. See Appendix 6.10.

We can thus carry out the same kind of analysis as in Section 2 and provide a detailed local stability analysis of the steady state, considering a family of economies parameterized by the three elasticities ( cc , lw , and lλ ) that govern the EIS in consumption, the wage elasticity, and the income effect, and by the technological parameters σ and Θ governing the elasticity of capital-labor substitution and the degree of increasing returns to scale (IRS) in the investment sector.

Similar to the previous Section, we prove in Appendix 6.11 that there exist three bifurcation loci in the parameter space such that, when cc is increased from 0 to +∞, a change in the stability properties of the steady state occurs when cc crosses any of the three loci. We establish the following Lemma: Lemma 4. Under Assumptions 1, 3, 4 and 6, let Ω = (β, δ, s, σ, Θ) be the set of structural parameters. For any ω ∈ Ω, there exist three bifurcation curves crossing the 3dimensional plane ( lw , lλ , cc ) and generating a change in the local stability properties of the steady state:

-a flip bifurcation curve F cc ( lw , lλ ) combined with one real eigenvalue of J crossing -1, -a Hopf bifurcation curve H cc ( lw , lλ ) combined with two complex conjugate eigenvalues of J crossing the unit circle, -a (degenerate) transcritical bifurcation curve T cc ( lw , lλ ) combined with one real eigenvalue of J crossing 1.

There also exist four critical bounds ¯ lλ ( lw ) , ˜ lλ ( lw ) , ˆ lλ ( lw ) and lλ ( lw ) such that:

-H cc ( lw , lλ ) = 0 when lλ = ¯ lλ ( lw ) ,

-F cc ( lw , lλ ) = 0 when lλ = ˜ lλ ( lw ) , -H cc ( lw , lλ ) = T cc ( lw , lλ ) when lλ = ˆ lλ ( lw ) , -F cc ( lw , lλ ) = T cc ( lw , lλ ) when lλ = lλ ( lw )
. The formal expressions of these bifurcation curves and critical bounds are given in Appendix 6.11.

Proof. See Appendix 6.11.

The critical bounds help us to define areas in the 3-dimensional plane where the bifurcation curves exist (or not) when cc is gradually increased from 0 to +∞. For example, the flip bifurcation exists whenever lλ ∈ (0, ˜ lλ ). 12 The transcritical bifurcation has a vertical asymptote at lw = (σ sup -σ)/s, so that the transcritical bifurcation exists whenever lw > lw . Finally, if lw ≤ lw , the Hopf bifurcation exists whenever lλ ∈ (0, ¯ lλ ), whereas if lw > lw , the Hopf bifurcation exists whenever lλ ∈ (ˆ lλ , ¯ lλ ).

It is also easy to show that whenever both curves exist, the flip and Hopf bifurcations satisfy 0 < F cc < H cc < ∞. Likewise, whenever both curves exist, the flip and transcritical bifurcations satisfy 0 < F cc < T cc < ∞ if lλ < lλ , and 0 < T cc < F cc < ∞ if lλ > lλ . Following a similar discussion as in Section 3.4, we introduce the following Assumption on the empirically relevant values of the structural parameters, excluding here the elasticity of intertemporal substitution in consumption which is used as a bifurcation parameter: Assumption 7. β = 0.99, δ = 0.025, s = 0.3, σ ∈ (0, 2) and Θ ∈ (0, 0.43).

We can now establish the following Theorem, providing a complete picture of the stability properties of the 2-sector model.

( lw ) = ˆ lλ ( lw ). Then, if σ > σ inf with σ inf ≡ δθ(1-s)(1+Θ) Θ[4β(1-δ)+δθ]
(46) we have:

Case 1 -Low wage elasticity of labor supply: lw ∈ [0, lw ).

i) when lλ ∈ [0, ˜ lλ ), the steady state is -a saddle-point for any cc ∈ [0, F cc ), -a sink for any cc ∈ ( F cc , H cc ), -a source for any cc > H cc . ii) when lλ ∈ (˜ lλ , ¯ lλ ), the steady state is -a sink for any cc ∈ [0, H cc ), -a source for any cc > H cc . iii) when lλ > ¯ lλ , the steady state is a source for any cc ≥ 0.

Case 2 -Intermediate wage elasticity of labor supply: lw ∈ ( lw , lw ). i) when lλ ∈ [0, lλ ), the steady state is -a saddle-point for any cc ∈ [0, F cc ), -a source for any cc ∈ ( F cc , T cc ), -a saddle-point for any cc > T cc . ii) when lλ ∈ ( lλ , ˆ lλ ), the steady state is -a saddle-point for any cc ∈ [0, F cc ), -a sink for any cc ∈ ( F cc , T cc ), -a saddle-point for any cc > T cc . iii) when lλ ∈ (ˆ lλ , ˜ lλ ), the steady state is -a saddle-point for any cc ∈ [0, F cc ), -a sink for any cc ∈ ( F cc , H cc ), -a source for any cc ∈ ( H cc , T cc ), -a saddle-point for any cc > T cc . iv) when lλ ∈ (˜ lλ , ¯ lλ ), the steady state is -a sink for any cc ∈ [0, H cc ), -a source for any cc ∈ ( H cc , T cc ), -a saddle-point for any cc > T cc . v) when lλ > ¯ lλ , the steady state is -a source for any cc ∈ [0, T cc ), -a saddle-point for any cc > T cc . Case 3 -High wage elasticity of labor supply: lw > lw . i) when lλ ∈ [0, lλ ), the steady state is -a saddle for any cc ∈ [0, F cc ), -a source for any cc ∈ ( F cc , T cc ), -a saddle-point for any cc > T cc . ii) when lλ ∈ ( lλ , ˜ lλ ), the steady state is -a saddle for any cc ∈ [0, F cc ), -a sink for any cc ∈ ( F cc , T cc ), -a saddle-point for any cc > T cc . iii) when lλ ∈ (˜ lλ , ˆ lλ ), the steady state is -a sink for any cc ∈ [0, T cc ), -a saddle-point for any cc > T cc .

iv) when lλ ∈ (ˆ lλ , ¯ lλ ), the steady state is -a sink for any cc ∈ [0, H cc ), -a source for any cc ∈ ( H cc , T cc ), -a saddle-point for any cc > T cc . v) when lλ > ¯ lλ , the steady state is -a source for any cc ∈ [0, T cc ), -a saddle-point for any cc > T cc .

Proof. See Appendix 6.12.

It is worth noticing that the lower bound σ inf on the capital-labor elasticity of substitution as given by [START_REF] Rogerson | Micro and Macro Elasticities in a Life Cycle Model with Taxes[END_REF] implies that local indeterminacy requires a strictly positive amount of externalities. However, the lower bounds is extremely close to zero even for very low values of Θ, i.e. for instance σ inf = 0.016 when Θ = 0.01.

Discussion

Theorem 2 enables us to derive the fourth and most central result of our paper:

Result 4: plausibility of indeterminacy in the two-sector model Indeterminacy and belief-driven fluctuations are robust properties of the standard twosector model of the business cycle, emerging for a large range of empirically-consistent values for all the critical elasticities considered: lλ , lw , cc , Θ and σ.

Before providing numerical examples supporting this claim, we first provide a conceptual explanation as for why indeterminacy easily emerges in the two-sector model.

Interpretation for indeterminacy

What makes indeterminacy much easier to obtain in the two-sector model compared to it's one-sector counterpart? The key answer lies on the Euler equation, which is now expressed as λ t = βλ t+1 R t+1 , with R t+1 = ((1 -δ)p t+1 + r t+1 )/p t the real return on capital accumulation. The latter no longer exclusively depends on the capital rental rate at t + 1, r t+1 , as in the one-sector model, but also on the evolution of the relative price of the investment good between t and t + 1. For example, an increase in p t+1 /p t generates a capital gain which must be taken into account in the decision of how much to consume and invest. Consider now as before that agents expect an increase in this real return on capital accumulation, with R t+1 replacing r t+1 in the 2-sector model. As in the one-sector model, this leads to an increase in the marginal utility of wealth λ t which, since cλ < 0, creates an incentive to substitute investment for consumption, leading to a higher investment i t and a higher capital stock k t+1 at t + 1. When lw ∈ [0, lw ) and the labor supply is almost fixed with respect to the wage rate, the larger capital stock generates a decrease in the capital rental rate r t+1 . However, at t, the increase in the demand for investment also creates an incentive to reallocate capital and labor from the consumption to the investment sector: Both K It and L It increase, while K ct and L ct decrease. With positive externalities e(K It , L It ) > 0 in the investment sector, this reallocation of inputs increases the supply of the investment good which more than compensates the increase in demand. As a result, the equilibrium price p t = 1/e(K It , L It ) decreases, which generates an expected capital gain from investing in the investment good (with p t+1 > p t ). When this expected capital gain is large enough, the expected rate of return on capital accumulation R t+1 = ((1-δ)p t+1 +r t+1 )/p t can increase even though the expected rental rate of capital r t+1 decreases. This makes the initial expectation that R t+1 will increase self-fulfilling. This situation is depicted in case 1(i) of Theorem 2, covering the almost fixed labor supply case with lw ∈ [0, lw ), where indeterminacy emerges under positive externalities Θ > 0 in the investment sector for any positive cc ∈ ( F cc , H cc ) which guarantees a large enough substitution between investment and consumption. In this case indeed, lλ ∈ [0, ˜ lλ ) is also small, the labor supply is also almost fixed with respect to the marginal utility of wealth, so that cλ can be sufficiently negative if cc is large enough. In case 1(ii) however, since lλ ∈ (˜ lλ , ¯ lλ ) is larger and the labor supply is more affected by the marginal utility of wealth, cλ can be sufficiently negative even for arbitrarily small EIS cc ∈ [0, H cc ). This complex mechanism associated with factor reallocation and variations in the relative price of the investment good in the presence of positive externalities is what makes the two-sector model much more prone to endogenous fluctuations triggered by exogenous changes in beliefs. Obviously, if indeterminacy is possible in the version with fixed labor supply, the scope for indeterminacy is even larger when the labor supply curve is allowed to vary. Indeed, in this case, the increase in equilibrium labor reduces the extent to which the capital rental rate r t+1 decreases when capital accumulates, which in turn reduces the extent to which the relative price of the investment good p t has to decrease to generate the requested increase in the expected return factor R t+1 . Clearly, it is no longer required to have a sufficiently large variations in equilibrium labor for indeterminacy to emerge (i.e.,. lw , lλ and cc no longer have to be very large, as in the one sector model).

Graphical illustrations

To illustrate our claim in Result 4, we represent in the 3-dimensional plane of Figure 1 the local stability properties of the two-sector model for economies displaying different degrees of Frisch labor supply elasticities and different EIS in consumption ( lλ , lw and cc ), while the other parameters are fixed and calibrated according to the benchmark calibration β = 0.99, δ = 0.025, s = 0.3, σ = 1 and Θ = 0.3. 13 The value σ = 1 corresponds to a Cobb-Douglas production function while the value Θ = 0.3 for the 13 As implied by Proposition 1, there always exist admissible utility functions u(c, -l) such that lλ , lw and cc reach their targeted value at the steady-state. It should then be emphasized that Figure 1 (and related figures below) do not represent what happens when one structural parameter of a specific utility function changes, but rather represents how economies differing in terms of their critical elasticities lλ , lw and cc have similar or different local stability properties, independently of the implicit utility function considered. This is, in our view, the relevant approach to take since deep preference parameters of specific utility functions are never observed while estimates for the critical elasticities can be obtained from the data. degree of IRS in the investment sector is close to the point estimate obtained by Harrison [START_REF] Harrison | Returns to scale and Externalities in the Consumption and Investment Sectors[END_REF] for the US economy (4-digit data). We see that, unlike with the one-sector model, there now exists a wide range of values for ( lw , lλ , cc ) such that the steady state is locally indeterminate and beliefdriven fluctuations emerge. Indeterminacy occurs for a large set of values of the EIS in consumption in the range cc ∈ (0, 2) and for various configurations for the wage and wealth elasticities of the labor supply curve ( lw , lλ ). This includes very large values for the latter two elasticities (as in Hansen's type of preferences with infinitely elastic labor supply, lw = lλ = +∞) or very small values for the wage-elasticity lw consistent with micro-level estimates (also covering the fixed labor supply case). 14 Clearly, the range of values associated with indeterminacy is entirely consistent with the range of credible empirical estimates defined in Assumption 5.

For better clarity, Figure 2 displays the stability property areas when the range of values considered for lw and lλ is restricted to (0,5), close to the value of lw = 3 advocated by Rogerson and Wallenius [START_REF] Rogerson | Micro and Macro Elasticities in a Life Cycle Model with Taxes[END_REF] and Prescott and Wallenius [START_REF] Prescott | Aggregate Labor Supply[END_REF] to calibrate dynamic macroeconomic models. As can be seen, indeterminacy arises in this configuration for a very wide range of values for cc in the realistic interval (0,2), including values that are arbitrarily close to 0.

14 As is well kown, a lengthy discussion exists in the literature about how to calibrate the Frisch wageelasticity of the aggregate labor supply curve. Both theoretical considerations and empirical evidence point toward small values at the individual level but greater values at the aggregate level (see for example Rogerson and Wallenius [START_REF] Rogerson | Micro and Macro Elasticities in a Life Cycle Model with Taxes[END_REF] for a discussion). Meanwhile, it is well known that standard RBC-DSGE models do not perform well when the wage elasticity of the aggregate labor supply is too low, which explains the popularity of the class of preferences suggested by Hansen [START_REF] Hansen | Indivisible Labor and the Business Cycle[END_REF].

On the other hand, there is very little empirical evidence on the wealth-elasticity lλ . The main difficulty is that this elasticity captures the effect of a marginal increase in intertemporal wealth on labor supply, and that exogenous variations enabling this elasticity to be identified are very difficult to find in the data (see the lengthy discussion and relevant references to the literature in Kimball and Shapiro, [START_REF] Kimball | Labor Supply: Are Income and Substitution Effects Both Large or Both Small?[END_REF]). Analyzing data from a thought-experiment survey conducted by the Health and Retirement Study (HRS), Kimball and Shapiro [START_REF] Kimball | Labor Supply: Are Income and Substitution Effects Both Large or Both Small?[END_REF] tend toward the conclusion that the elasticities lw and lλ are rather small. Yet, their estimations assume an equality between both elasticities (as imposed by Hansen's type of preferences), while from a theoretical standpoint there is no reason to assume that they are equal.

Robustness

Since we emphasized that five critical elasticities have a strong influence on the local stability properties of the 2-sector model , it is worthwhile to analyze how the elasticities related to the productive side of the economy influence the local stability properties of the two-sector model displayed in Figures 1 and2 in the case of a Cobb-Douglas production function (σ = 1) and Θ = 0.3. In Figure 3, we display how these stability properties vary when we consider four different values for Θ, namely Θ ∈ (0.1, 0.2, 0.3, 0.4). We see that increasing the degree of increasing returns to scale has the effect of shifting down the flip of Hopf bifurcation curves, so that indeterminacy is easier to obtain for larger degrees of IRS. Note however that in all cases there exist a range of values cc ∈ (0, 2) such that indeterminacy prevails, including values arbitrarily close to 0 We see that the area for which the steady-state is a sink tends to shrink when larger values for σ are considered, especially for larger intensities of the wealth effect lλ . Moreover, the transcritical bifurcation curve shifts to the right when σ increases, implying that indeterminacy is also harder to obtain for large wage elasticities of the labor supply curve lw . However, in all cases, a significant range of parameter configurations in the set of empirically plausible values (as defined in Assumption 7) are consistent with an indeterminate steady-state, in particular for the EIS in consumption cc ∈ (0, 2). This reinforces our claim that indeterminacy and the emergence of expectation-driven fluctuations are robust properties of the standard two-sector model.

Specific utility functions

Since Theorem 2 applies to any form of instantaneous utility functions satisfying the usual assumptions, results regarding any particular utility function can be derived as a direct application of this Theorem. We consider here the three most popular utility functions used in the macroeconomic literature, namely the KPR, generalized Hansen and GHH utility functions, with cross restrictions on the three critical elasticities cc , lw , and lλ summarized in Proposition 2. Using these cross restrictions enables us to obtain a slightly modified lower bound σinf compared to the one, σ inf , introduced in the general case (see Theorem 2), which is slightly larger:

Assumption 8. σ > σinf ≡ (1-β+Θθ)δ(1-s)(1+Θ) Θ[4(1-δ)Θβ+δ(1-β+Θθ)]
We then obtain for the KPR case: Corollary 1. Under Assumptions 1, 3, 4, 6, 7 and 8, consider a KPR utility function such that lλ = cc lw . Then there exist bifurcation curves 0 < F cc < H cc such that the steady state is:

-a saddle-point for any cc ∈ [0, F cc ), -a sink for any cc ∈ ( F cc , H cc ), -a source for any cc > H cc , The formal expressions for the flip and Hopf bifurcation curves ( F cc and H cc ) are given in Appendix 6.13.

Proof. See Appendix 6.13. Notice that with a KPR utility function there is no transcritical bifurcation. Moreover, the existence of local indeterminacy is compatible with an arbitrarily large wage-elasticity of the labor supply curve. As an illustration, Figure 5 displays the local stability properties of the steady state in the KPR case as a function of lw , cc and Θ when the production function is Cobb-Douglas (σ = 1). We see that with a KPR utility function, indeterminacy can easily be obtained for moderate values for the EIS in consumption cc ∈ (0, 2) and an arbitrarily large elasticity of the labor supply curve, provided that the degree of IRS is large enough.

With GHH preferences, we obtain: Corollary 2. Under Assumptions 1, 3, 4, 6, 7 and 8, consider a GHH utility function such that lλ = 0. Then there exist a critical bound lw ≡ (σ sup -σ)/s, with σ sup = (1 -s)(1 + Θ)/Θ, and bifurcation curves 0 < F cc < H cc such that the following cases hold: i) when lw ∈ (0, lw ), then the steady state is:

-a saddle-point for any cc ∈ [0, F cc ), -a sink for any cc ∈ ( F cc , H cc ), -a source for any cc > H cc . ii) when lw > lw , the steady state is unstable for any cc ∈ [0, F cc ) and saddle-point stable for any cc > F cc . The formal expressions for the flip and Hopf bifurcation curves ( F cc and H cc ) are given in Appendix 6.14.

Proof. See Appendix 6.14. A graphical illustration of this case in the 3-dimensional plane defined by ( lw , Θ, cc ) is displayed in Figure 6, again in the case σ = 1. Notice that with a GHH utility function, the critical bound lw actually corresponds to a transcritical bifurcation curve when lw is taken as bifurcation parameter. However, this critical bound is independent of the elasticity of intertemporal substitution in consumption cc , implying that the locus is vertical with respect to cc . As can be seen, compared to the KPR case, the existence of local indeterminacy for realistic cc ∈ (0, 2) is somewhat more difficult to obtain, as the range of parameter values consistent with a sink equilibrium is shrunk by this transcritical bifurcation curve, implying that indeterminacy is ruled out for large elasticities of the labor supply curve.

With generalized Hansen preferences, we obtain: Corollary 3. Under Assumptions 1, 3, 4, 6, 7 and 8, consider a generalized Hansen utility function such that lλ = lw . Then there exist critical bounds F lw < lw = (σ supσ)/s < H lw , σinf < σ F < σ H < σ sup and bifurcation curves 0 ≤ F cc ≤ H cc ≤ T cc ≤ +∞ such that the steady state is:

-a saddle-point for any cc ∈ [0, F cc ), -a sink for any cc ∈ ( F cc , H cc ), -a source for any cc ∈ ( H cc , T cc ), -a saddle-point for any cc > T cc . . Moreover:

-

F cc = 0 if σ > σ F and lw > F lw , -H cc = 0 if σ > σ H and lw > H lw , -T cc = +∞ if lw < lw .
The formal expressions for the flip, Hopf and transcritical bifurcation curves ( F cc , H cc and T cc ), as well as those of all critical bounds ( F lw , H lw , σ F and σ H ), are given in Appendix 6.15.

Proof. See Appendix 6.15.

Figure 7 displays again the corresponding stability properties associated with a Cobb-Douglas technology (σ = 1). We see that with generalized Hansen preferences, indeterminacy is very easy to obtain since a large range of values cc ∈ (0, 2) exists such that the steady-state is a sink, including values arbitrarily close to 0. This is true for a large set of values for the wage labor supply elasticity, including values that are either small or arbitrarily large.

Concluding comments

This paper established two main results: First, the existence of belief-driven fluctuations in the standard one-sector RBC model with increasing returns to scale is ruled out for any empirically plausible calibration regarding the critical elasticities defining the utility function and the production function. Second, on the contrary, the existence of such fluctuations is a very likely outcome of the standard two-sector version of the model, in the sense that they now arise for a large set of empirically plausible values for these elasticities. While this latter result is true in general, it also holds when some of the most frequently used utility function in the macroeconomic literature are considered (KPR, generalized Hansen and GHH utility functions) even though the model is restricted in this case by cross-restrictions on the critical elasticities. Hence, far from being an exotic feature emerging under extreme assumptions, the possibility of expectation-driven fluctuations is a likely outcome of workhorse models in the macroeconomic literature, provided they are at least bisectoral (arguably a reasonable assumption).

At this stage, a critical issue remains. Are business cycles triggered by exogenous changes in expectations important quantitatively? So far, the literature has been largely negative. In a well-known paper, Schmitt-Grohé [START_REF] Schmitt-Grohé | Endogenous Business Cycles and the Dynamics of Output, Hours, and Consumption[END_REF] showed that a standard two-sector model submitted to beliefs shocks is unable to account for several defining features of observed business cycle, in particular in response to transitory shocks. However, the model and its extensions are evaluated under the maintained assumptions of KPR preferences. Whether the same negative outcome prevails for a larger set of preferences is an open question. In Dufourt et al. [START_REF] Dufourt | Expectations, Beliefs and the Business Cycle: data confrontation[END_REF], we address this issue and we show that the belief-driven two-sector model can actually account for all the dimensions of observed business cycles emphasized by Schmitt-Grohé [START_REF] Schmitt-Grohé | Endogenous Business Cycles and the Dynamics of Output, Hours, and Consumption[END_REF], as well as other dimensions of the data underlined in the recent literature (see e.g. Beaudry et al., [START_REF] Beaudry | Putting the Cycle Back into Business Cycle Analysis[END_REF]). We refer the reader to this paper for a thorough discussion.

Appendix

Proof of Lemma 1

From the definition of lw and lλ as given by ( 15)-( 16), a total differentiation of the optimality conditions ( 9)- [START_REF] Benhabib | Indeterminacy, Aggregate Demand, and the Real Business Cycle[END_REF] gives u 11 dc t -u 12 dl t = dλ t u 21 dc t -u 22 dl t = dλ t w t + λ t dw t Solving this system with respect to dl t yields to the expressions [START_REF] Dufourt | Sunspot Fluctuations in Two-Sector Models with Variable Income Effects[END_REF] and [START_REF] Dufourt | On Sunspot Fluctuations in Variable Capacity Utilization Models[END_REF]. We also derive cw = -

u 12 u 1 u 11 u 22 -u 12 u 21 w c = wl c ( lw -lλ ) cλ = u 1 u 11 c -u 12 u 11 c u 11 u 2 -u 21 u 1 u 11 u 22 -u 12 u 21 = -cc + wl c 1 -lλ lw lλ
We show in Appendix 6.3 below that at the steady state

wl c = θ(1-s) θ-sβδ ≡ C < 1 (47)
The result follows.

Proof of Proposition 1

By Assumption 3, u is an increasing function over R 2 ++ , implying (c, l, u 1 , u 2 ) > 0. Moreover, the strict quasi-concavity of u implies u 11 < 0 and u 11 u 22 -u 12 u 21 > 0. Using Lemma 1, we straightforwardly obtain cc > 0 and lw > 0. By Assumption 4, c and L are normal goods. The normality of L requires u 21 u 1 -u 11 u 2 ≥ 0. Combined with the strict quasi-concavity of u, we straightforwardly obtain lλ ≥ 0. Using a similar reasoning, we obtain that the normality of c requires cλ ≤ 0 and therefore, using Lemma 1, cc ≥ C lλ ( lwlλ )/ lw ≡ N cc .

Proof of Lemma 2

We know from constant-returns-scale of the technology at the private level that We conclude from this

w * l * c * = θ(1-s) θ-sβδ ≡ C < 1 (48)

Proof of Proposition 3

Considering again that R * = r * + 1 -δ = 1/β, we get

f 1 (k * , l * )e(k * , l * ) ≡ g(k * , l * ) = θ β (49)
It is then easy to compute under Assumption 2

g 1 (k * ,l * )k * g(k * ,l * ) = sΘ k -1-s
σ < 0 Therefore, applying the implicit function theorem, we conclude that there exists a unique function k(.) such that k * = k(l * ). Considering that

g 2 (k * ,l * )l * g(k * ,l * ) = (1 -s)Θ l + 1-s σ we conclude that k (l * )l * k(l * ) = - (1-s)Θ l + 1-s σ sΘ k -1-s σ > 0 Recalling now that y * k * = θ sβ and c * = θ-sβδ sβ k * (50)
we derive

c * = c(l * ) = θ-s(k(l * ),l * )βδ s(k(l * ),l * )β k(l * ) ≡ h(l * )k(l * ) Straightforward computations give h (l * )l * h(l * ) = θ(1-s) θ-sβδ 1 -1 σ (1-s)Θ l +sΘ k sΘ k -1-s σ
and we easily conclude under Assumption 2

c (l * )l * c(l * ) = - (1-s) Θ l[ s(1-β)+ θ(1-s) σ ]+ θsΘ k σ + θ-sβδ σ -θsΘ k (θ-sβδ)(sΘ k -1-s σ )
> 0

Moreover we also get from ( 25)

w * = w(l * ) = f 2 (k(l * ), l * )e(k(l * ), l * )
and thus

w (l * )l * w(l * ) = - 1-s σ Θ l + s σ Θ k sΘ k -1-s σ > 0
Consider then the third equation of (26) which becomes

u 2 (c(l * ), -l * ) u 1 (c(l * ), -l * ) ≡ ψ(l * ) = w(l * ) (51)
Under Assumptions 2, 3 and 4, we get

ψ * )l * ψ(l * ) = c (l * )l * c(l * ) lλ cc lw + 1 cc lw cc -C lλ 1 -lλ lw ≥ 0 (52)
It follows that the existence of a unique steady state value l * is obtained if g (l * ) = w (l * ). Straightforward computations show that this condition is satisfied if

c (l * )l * c(l * ) lλ cc lw + 1 cc lw cc -C lλ 1 -lλ lw -w (l * )l * w(l * ) = 0 (53)
Such a condition is generically satisfied so that the existence and uniqueness of a steady state is generically ensured. Now let us normalize the steady state considering the value l * corresponding to the average amount of working hours relative to the total amount of time . Substituting l * = l * into equation ( 51), we get

u 2 (c( l * ), -l * ) u 1 (c( l * ), -l * ) ≡ φ( ) = w( l * ) (54) Straightforward computations give φ ( ) φ( ) = -l * 1 cc lw cc -C lλ 1 -lλ lw
It follows under Assumptions 3 and 4 that φ ( ) /φ( ) ≤ 0 and there exists a unique value * > l * solution of equation ( 54). We conclude finally that if = * , then the unique steady state (k * , l * , c * ) is such that l * = l * .

Proof of Proposition 4

From the optimality conditions ( 9)- [START_REF] Benhabib | Indeterminacy, Aggregate Demand, and the Real Business Cycle[END_REF] and Lemma 1, we derive

l t = lw w t + lλ λ t ( 55
)
c t = C 1 -lλ lw l t -cc λ t (56)
and ( 25) implies

w t = sΘ k + s σ k t + (1 -s)Θ l -s σ l t Using this expression in (55) yields l t = lw s( 1 σ +Θ k ) 1+ lw [ s σ -Θ l (1-s)] k t + lλ 1+ lw [ s σ -Θ l (1-s)] λ t (57)
Using (57) into (56) gives

c t = Cs( 1 σ +Θ k )( lw -lλ ) 1+ lw [ s σ -Θ l (1-s)] k t -cc - lλ C 1-lλ lw 1+ lw [ s σ -Θ l (1-s)] λ t (58)
Using [START_REF] Gruber | A Tax-Based Estimate of the Elasticity of Intertemporal Substitution[END_REF], the system of difference equations describing the intertemporal equilibrium can be stated as follows

f (k t , l(k t , λ t ))e(k t , l(k t , λ t )) + (1 -δ t )k t -c(k t , λ t ) -k t+1 = 0 β [1 -δ + f 1 (k t+1 , l(k t+1 , λ t+1 ))e(k t+1 , l(k t+1 , λ t+1 ))] λ t+1 -λ t = 0 (59)
Linearizing the first equation around the steady state using ( 50), ( 57) and (58) gives after simplifications

k t+1 = k t 1 β 1 + θΘ k + θ(1-s)( 1 σ +Θ k )( lw Θ l + lλ ) 1+ lw [ s σ -Θ l (1-s)] + λ t 1 sβ lλ θ(1-s) Θ l + lλ lw 1+ lw [ s σ -Θ l (1-s)] + (θ -sβδ) cc (60)
Linearizing the second equation of (59) around the steady state gives

λ t+1 = λ t + k t+1 sΘ k -1-s σ θ -l t+1 Θ l + 1 σ θ(1 -s) (61)
Using (57) finally gives

λ t+1 1 + lλ θ(1-s)( 1 σ +Θ l) 1+ lw [ s σ -Θ l (1-s)] -k t+1 θ 1-s σ -sΘ k -lw σ [s(Θ k -Θ l )+Θ l ] 1+ lw [ s σ -Θ l (1-s)] = λ t (62)
Equations ( 60) and (62) can be expressed as follows 1 0

-A 21 A 22 k t+1 λ t+1 = B 11 B 12 0 1 k t λ t with A 21 = θ 1-s σ -sΘ k -lw σ [s(Θ k -Θ l )+Θ l ] 1+ lw [ s σ -Θ l (1-s)] A 22 = 1 + lλ θ(1-s)( 1 σ +Θ l) 1+ lw [ s σ -Θ l (1-s)] B 11 = 1 β 1 + θΘ k + θ(1-s)( 1 σ +Θ k )( lw Θ l + lλ ) 1+ lw [ s σ -Θ l (1-s)] B 12 = 1 βs lλ lw θ(1-s)( lw Θ l + lλ ) 1+ lw [ s σ -Θ l (1-s)] + (θ -sβδ) cc
The Jacobian matrix J follows after straightforward computations and simplifications.

Proof of Lemma 3

We easily derive from Proposition 4 the following characteristic polynomial

P(λ) = λ 2 -λT ( cc ) + D (63)
with

D = 1 β 1 + θ Θ k [1+s lw σ +(1-s) lλ] +(1-s)Θ l( lw σ -lλ) 1+ lw [ s σ -Θ l (1-s)]+ lλ θ(1-s)( 1 σ +Θ l) T ( cc ) = 1 + D + θ(θ-sβδ)(1-s) βs cc 1 σ - sΘ k 1-s -lw σ Θ l + sΘ k 1-s + lλ 1 σ s(1-β) θ-sβδ + CsΘ k 1-s +Θ l( s(1-β) θ-sβδ + C σ )+C lλ lw 1 σ - sΘ k 1-s 1+ lw [ s σ -Θ l (1-s)]+ lλ θ(1-s)( 1 σ +Θ l)
The analysis of the local stability properties of the model is based on the geometrical methodology of Grandmont et al. [START_REF] Harrison | Returns to scale and Externalities in the Consumption and Investment Sectors[END_REF]. In Figure 8, we draw a graph in the tracedeterminant (T , D) space where three relevant lines are considered: line AC (D = T -1) along which one eigenvalue of D is equal to 1, line AB (D = -T -1) along which one eigenvalue of D is equal to -1 and segment BC (D = 1, |T | < 2) along which the two eigenvalues of D are complex conjugates with modulus equal to 1. These three lines divide the space (T , D) into three different types of regions according to the number of eigenvalues with modulus smaller than, equal to, and greater than 1. This determines whether the steady state is a sink (locally indeterminate), a source (locally unstable) or a saddle-point (see the corresponding areas in Figure 8).

Then, for any particular calibration of structural parameters, we can compute the trace and determinant using the expression for the Jacobian matrix obtained in Proposition 4 and assess in which area the model is located. We can also assess how these local stability properties change when the calibration of any particular parameter is varied over its admissible range. In the one-sector stochastic growth model considered so far, the analysis is greatly simplified by observing that D does not depend on cc , implying that the pair (T ( cc ), D) describes an horizontal line in the (T , D) space when cc increases from 0 to +∞. As a result, any Hopf bifurcation related to a Determinant equal to 1 is generically ruled out.

To prove the possible existence of local indeterminacy we need to show that there exist some parameters' configurations such that D < 1 and 1 -T ( cc ) + D > 0. It is easy to show from the expression of T ( cc ) that a necessary condition to get 1 -T

( cc ) + D > 0 is lw > 1- σsΘ k 1-s Θ l + sΘ k 1-s ≡ lw
Let us now write the determinant as

D = 1 β 1+ lw [ s σ -Θ l (1-s)(1-θ σ )]+ lλ θ(1-s) σ +θΘ k [1+s lw σ +(1-s) lλ] 1+ lw [ s σ -Θ l (1-s)]+ lλ θ(1-s)( 1 σ +Θ l)
Since under Assumption 2 the expression s σ -Θ l (1 -s) 1 -θ σ is necessarily positive for any σ > 0, we get D > 0 for any σ > 0. Moreover, D ≤ 1 if and only if Θ l > Θ l and lλ ≥ lλ ( lw ) with

Θ l ≡ 1-β β + Θ k β , lλ ( lw ) ≡ 1-β+θΘ k + lw s(1-β+θΘ k ) σ -Θ l (1-s)(1-β-θ σ ) θ(1-s)(1-β+Θ k )(Θ l -Θ l ) (64)
Under σ ≤ σ ≡ θ/(1 -β) we get 1 -β -θ σ ≤ 0 so that lλ ( lw ) > 0 for any Θ l > Θ l and lw ≥ 0. We need therefore to show that Θ l < Θl which is obtained if and only if

Θ k < Θ k ≡ sβ (1-s)σ -(1 -β) with Θ k ∈ (0, Θk ) under Assumption 5 and σ ≤ σ.
Obviously, we conclude that D > 1 when Θ l < Θ l for any lλ ≥ 0, or when Θ k ∈ [0, Θ k ), Θ l ∈ (Θ l , Θl ) and lλ < lλ ( lw ).

Let us compute the critical values F cc ( lw , lλ ) and T cc ( lw , lλ ) respectively associated with flip and transcritical bifurcations. The first one is obtained as the solution of 1 -T ( cc ) + D = 0, namely

T cc ≡ lλ 1 σ s(1-β) θ-sβδ + CsΘ k 1-s +Θ l( s(1-β) θ-sβδ + C σ )+C lλ lw 1 σ - sΘ k 1-s lw σ Θ l + sΘ k 1-s -1 σ - sΘ k 1-s (65)
while the second one is obtained as the solution of 1 + T ( cc ) + D = 0, namely

F cc ≡ 2 1+β+θΘ k + lw (1+β)( s σ -Θ l (1-s))+ θΘ l (1-s) σ + lλ θ(1-s)[ 1+β σ +βΘ l +Θ k ] θ(θ-sβδ)(1-s) sσ Θ l + sΘ k 1-s ( lw -lw ) + lλ θ(1-s)(θ-sβδ) s 1 σ s(1-β) θ-sβδ + CsΘ k 1-s +Θ l( s(1-β) θ-sβδ + C σ )+C lλ lw 1 σ - sΘ k 1-s θ(θ-sβδ)(1-s) sσ Θ l + sΘ k 1-s ( lw -lw )
(66)

Proof of Theorem 1

We immediately derive for any Θ l :

1 -T (0) + D < 0 and lim cc→+∞ T ( cc ) = ±∞ when lw ≶ lw Case 1 -Let us consider first the case with a low wage elasticity for the labor supply, i.e. lw < lw . We get the following two graphical configurations depending on the values of Θ l , Θ k and lλ : As D does not depend on cc and, when lw < lw , the determinant D satisfies D > 1 or D ∈ (0, 1) depending on the values of Θ l and lλ , and we get an horizontal line characterizing the variation of T ( cc ) when cc is varied over [0, +∞). Obviously, this line cannot cross the line BC. Moreover, as 1 -T (0) + D < 0, the starting point when cc = 0 is located below the line AC and we have lim cc→+∞ T ( cc ) = +∞. The steady state is then a saddle-point for any cc ≥ 0.

Case 2 -Let us consider now the case with a high wage elasticity for the labor supply, i.e. lw > lw , and low capital externalities, i.e. Θ k ∈ [0, Θ k ). We get the following graphical configuration: When Θ l < Θ l or Θ l ∈ (Θ l , Θl ) and lλ < lλ ( lw ), we have D > 1 but now lim cc→+∞ T ( cc ) = -∞. Local indeterminacy cannot arise but the steady state is not always a saddle-point and can be a source. Indeed, the steady state is saddle-point stable for any cc ∈ [0, T cc ) ∪ ( F cc , +∞) and locally unstable when cc ∈ ( T cc , F cc ). When Θ l ∈ (Θ l , Θl ) and lλ > lλ ( lw ), we get D ∈ (0, 1) with lim cc→+∞ T ( cc ) = -∞. It follows that the line now crosses the triangle ABC and we get indeterminacy: As 1 -T (0) + D < 0, the starting point when cc = 0 is located below the line AC and the steady state is saddle-point stable. As cc increases, T ( cc ) will cross the line AC when cc = T cc , implying the existence of a degenerate transcritical bifurcation since the steady state is unique. When cc ∈ ( T cc , F cc ), the steady state is locally indeterminate. When cc = F cc , a flip bifurcation generically occurs leading to the existence of period-two cycles in a right or left neighborhood of F cc . Finally, when cc > F cc , the steady state is again saddle-point stable.

Case 3 -Let us finally consider the case with a high wage elasticity for the labor supply, i.e. lw > lw , and large capital externalities, i.e. Θ k ∈ (Θ k , Θk ). Since in this case Θ l > Θl , we have necessarily Θ l ∈ (0, Θ l ) and thus D > 1. We then get the same configuration as Figure 10. The steady state is saddle-point stable for any cc ∈ [0, T cc ) ∪ ( F cc , +∞) and locally unstable when cc ∈ ( T cc , F cc ).

Proof of Proposition 5

We easily conclude that under Assumptions 1-4, local indeterminacy is ruled out for any σ > 0 in the following cases: i) when Θ k = Θ l = Θ, we get

D = 1 β 1 + θ Θ(1+ lw σ ) 1+ lw [ s σ -Θ l (1-s)]+ lλ θ(1-s)( 1 σ +Θ l) > 1 β ii) when lw = 0 we get lim lw →0 1 -T ( cc ) + D = -∞
iii) when lλ = 0 we get

D = 1 β 1 + θ Θ k + lw σ [sΘ k +(1-s)Θ l ] 1+ lw [ s σ -Θ l (1-s)] > 1 β iv) when Θ l = 0 we get D = 1 β 1 + θ Θ k [1+ s lw σ +(1-s) lλ] 1+ s lw σ + lλ θ(1-s) σ > 1 β 6.9 Proof of Proposition 6
The critical value T cc given in Lemma 3 provides a lower bound on cc to get local indeterminacy. Since lλ > lλ ( lw ), we can derive the following lower bound for T cc :

T cc > lλ ( lw ) 1 σ s(1-β) θ-sβδ + CsΘ k 1-s +Θ l( s(1-β) θ-sβδ + C σ )+C lλ ( lw ) lw 1 σ - sΘ k 1-s lw σ Θ l + sΘ k 1-s -1 σ - sΘ k 1-s ≡ T cc ( lw )
T cc ( lw ) is a decreasing function of lw over ( lw , +∞) with lim lw → lw = +∞. Straightforward computations then show that under Assumption 5, T cc > 2 when lλ > lλ ( lw ). As a result local indeterminacy is ruled out and the steady state is always a saddle-point.

Proof of Proposition 7

From (45) we derive

w t = s σ k t -l t (67)
Using this into (55) and (56) then gives

l t = lλ 1+ s lw σ λ t + s lw 1+ s lw σ k t c t = C 1 -lλ lw lλ 1+ s lw σ -cc λ t + C s σ ( lw -lλ ) 1+ s lw σ k t (68)
Equation (67) then becomes

w t = s σ 1 1+ s lw σ k t -lλ 1+ s lw σ λ t (69)
From the prices r t and p t as given by ( 44) and ( 46) we finally derive:

r t = (1-s) σ lλ 1+ s lw σ λ t -1 1+ s lw σ k t p t = -Θ sβδ sθ 1+ s lw σ + (1-s) lλ σ 1+ s lw σ k t + θ(1-s) 2 lλ 1+ s lw σ + (θ -sβδ) cc ) λ t
Tedious computations based on these results allow to get from the system of difference equations ( 42)-( 43):

0 1 A 21 -A 22 k t+1 λ t+1 = B 11 B 12 B 21 -B 22 k t λ t with A 21 = 1 + θ(1-s) σ lλ 1+ s lw σ -(1-δ)Θ sδ θ(1-s) 2 lλ lw 1+ s lw σ + (θ -sβδ) cc A 22 = θ(1-s) σ 1+ s lw σ + θ(1-δ)Θ δ 1+ s lw σ + (1-s) lλ σ 1+ s lw σ B 11 = 1+Θ sβ θ(1-s) 2 lλ lw 1+ s lw σ + (θ -sβδ) cc B 12 = 1 β 1 + θ(1-s) σ lλ 1+ s lw σ + θΘ 1+ s lw σ + (1-s) lλ σ 1+ s lw σ B 21 = 1 -Θ sβδ θ(1-s) 2 lλ lw 1+ s lw σ + (θ -sβδ) cc B 22 = θΘ βδ 1+ s lw σ + (1-s) lλ σ 1+ s lw σ
The Proposition follows.

Proof of Lemma 4

We easily derive from Proposition 5 the Determinant and Trace:

D = B 11 B 22 +B 12 B 21 A 21 T = 1 + D + (B 21 -A 21 )(1-B 12 )+B 11 (A 22 -B 22 ) A 21
The characteristic polynomial is then

P(λ) = λ 2 -λT ( cc , lw , lλ , σ, Θ) + D( cc , lw , lλ , σ, Θ) (70)
with

D( cc , lw , lλ , σ, Θ) = 1 β    1 + Θθ 1+ s lw σ + (1-s) lλ σ 1+ s lw σ + θ(1-s) lλ σ - Θ(1-δ)(θ-sβδ) sδ C 2 lλ lw + cc(1+ s lw σ )    T ( cc , lw , lλ , σ, Θ) = 1 + D( cc , lw , lλ , σ, Θ) + θ(θ-sβδ) sβ lλ[ (1-s)(1-C)+ΘsC σ ]+ C 2 lλ lw [ 1-s σ -Θ(1-1-s σ )]+ cc[ 1-s σ -Θ(1-1-s σ )-Θ s σ lw ] 1+ s lw σ + θ(1-s) lλ σ - Θ(1-δ)(θ-sβδ) sδ C 2 lλ lw + cc(1+ s lw σ ) ≡ 1 + D( cc , lw , lλ , σ, Θ) + X ( cc , lw , lλ , σ, Θ)
It is easy to derive that as the parameter cc is varied over the interval (0, +∞), D( cc , lw , lλ , σ, Θ) and T ( cc , lw , lλ , σ, Θ) are linked through a linear relationship ∆(T ) such that D = ∆(T ) = S( lw , lλ , σ, Θ)T + M with S( lw , lλ , σ, Θ) = ∂D( cc, lw , lλ ,Θ)/∂ cc ∂T ( cc, lw , lλ ,Θ)/∂ cc which does not depend on cc . Straightforward computations show that ∂D( cc , lw , lλ , σ, Θ)/∂ cc > 0 and, un-der Assumptions 1, 3, 4, 6 and 7, ∂T ( cc , lw , lλ , σ, Θ)/∂ cc > 0. It follows that S( lw , lλ , σ, Θ) > 0 and thus ∆(T ) is a line in the space (T , D) with a positive slope. In order to locate this line, we need to compute the starting and end points (T (0, lw , lλ , σ, Θ), D(0, lw , lλ , σ, Θ)) and (T (+∞, lw , lλ , σ, Θ), D(+∞, lw , lλ , σ, Θ)). We easily get

D(0, lw , lλ , σ, Θ) = 1 β 1 + Θθ 1+ s σ lw + (1-s) σ lλ 1+ s σ lw + θ(1-s) σ lλ - Θ(1-δ)θ(1-s) sδ 2 lλ lw ≡ D 0 X (0, lw , lλ , σ, Θ) = θ(θ-sβδ) lλ [(1-s)(1-C)+ΘsC]( lw -ˆ lw ) sβσ lw 1+ s σ lw + θ(1-s) σ lλ - Θ(1-δ)θ(1-s) sδ 2 lλ lw with ˆ lw ≡ ΘC lλ (σ-σsup) (1-s)(1-C)+ΘsC and σ sup ≡ (1-s)(1+Θ)
Θ Under Assumption 7, we obviously have σ < 2 < σ sup so that the bound ˆ lw < 0 is no longer relevant. It follows that D 0 satisfies:

-D 0 > 1/β if and only if lλ < 0 lλ ( lw )( lw ) with 0 lλ ( lw ) ≡ θ(1-s)sδ σ lw + [ θ(1-s)sδ σ lw ] 2 +4Θ(1-δ)θ(1-s)(1+ s lw σ )sδ lw 2Θ(1-δ)θ(1-s) -D 0 ∈ (-∞, 1) if and only if lλ ∈ ( 0 lλ ( lw ), ¯ lλ ( lw )) with ¯ lλ ( lw ) ≡ θ(1-s)sδ lw (1-β+Θ) σ + θ(1-s)sδ lw (1-β+Θ) σ 2 +4Θ(1-β)(1-δ)θ(1-s)(1+ s lw σ )sδ lw (1-β+Θθ) 2Θ(1-β)(1-δ)θ(1-s) > 0 lλ ( lw ) -D 0 ∈ (1, 1/β) if and only if lλ > ¯ lλ ( lw ).
We also immediately conclude that 1 -T (0, lw , lλ , σ, Θ) + D(0, lw , lλ , σ, Θ) = -X (0, lw , lλ , σ, Θ) < 0 if and only if lλ < 0 lλ ( lw ). Moreover, we easily derive that 1 + T (0, lw , lλ , σ, Θ) + D(0, lw , lλ , σ, Θ) > 0 if and only if lλ ∈ (0, 0 lλ ( lw )) ∪ (˜ lλ ( lw ), +∞) with

˜ lλ ( lw ) ≡ θδ lw σ 2(1-s)s(1+β+Θ)+(θ-sβδ)[(1-s)(1-C)+ΘsC] + √ ∆ 2Θθ(1-s) 2(1-δ)(1+β)+ θδ(σ-σsup) σ ∈ ( 0 lλ ( lw ), ¯ lλ ( lw )) and ∆ = θδ lw 2(1-s)s(1+β+Θ)+(θ-sβδ)[(1-s)(1-C)+ΘsC] σ 2 + 8 1 + s lw σ sδ lw (1 + β + Θθ)Θθ(1 -s) 2(1 -δ)(1 + β) + θδ(σ-σsup) σ (71) while 1 + T (0, lw , lλ , σ, Θ) + D(0, lw , lλ , σ, Θ) < 0 if and only if lλ ∈ ( 0 lλ ( lw ), ˜ lλ ( lw )). Finally, we have ˆ lλ ( lw ) = -σ(1-s) (1-β+Θθ)(1-β)(1-δ)(1+ s σ lw )-θ σ (1-β+Θ)sδ( lw -lw ) +σ √ ∆ 2Θθ(1-β)(1-δ)(1-s)σsup and ∆ = (1 -s) 2 (1 -β + Θθ)(1 -β)(1 -δ) 1 + s σ lw -θ σ (1 -β + Θ)sδ( lw -lw ) 2 + 4Θθ(1-β)(1-δ)(1-s)σsup σ 1 + s σ lw (1 -β + Θθ
)sδ( lwlw ) Note that ˆ lλ ( lw ) > 0 if and only if lw > lw . By convention, we will consider that ˆ lλ ( lw ) = 0 when lw ∈ (0, lw ).

When lλ < ˆ lλ ( lw ), the Hopf bifurcation is always ruled out. In order to locate the ∆(T ) line we need to check whether D = -1 can occur, and if yes, we need to know the sign of T when D = -1. If the sign is positive then the ∆(T ) line is located below the triangle ABC and local indeterminacy is ruled out. On the contrary, if the sign is negative then the ∆(T ) line may cross the triangle ABC leading to the possible existence of local indeterminacy. Solving D( cc , lw , lλ , σ, Θ) = -1 with respect to cc gives

¯ cc ( lw , lλ ) = (1+ s σ lw )(1+β+Θθ)+ θ(1-s) σ lλ (1+β+Θ)-Θθ(1+β)(1-δ)(1-s) 2 lλ sδ lw Θ(1+β) (1-δ) sδ (θ-sβδ)(1+ s σ lw ) Straightforward computations show that ¯ cc ( lw , lλ ) > 0 if and only if lλ > ¯ lλ ( lw ) with ¯ lλ ( lw ) ≡ θ(1-s)sδ lw (1+β+Θ) σ + θ(1-s)sδ lw (1+β+Θ) σ 2 +4Θ(1+β)(1-δ)θ(1-s)(1+ s lw σ )sδ lw (1+β+Θθ) 2Θ(1+β)(1-δ)θ(1-s) ∈ ( 0 lλ ( lw ), ¯ lλ ( lw ))
If lλ > ¯ lλ ( lw ) we derive that

T (¯ cc , lw , lλ , σ, Θ) = χ(¯ cc , lw , lλ , σ, Θ)
and straightforward computations yield T (¯ cc , lw , lλ , σ, Θ) ≥ 0 if and only if lλ ≥ lλ ( lw ) with

lλ ( lw ) ≡ -(1-s) (1-β+Θθ)(1+β)(1-δ) δ + θ(1+β+Θ)(σsup-σ) σ + s lw δσ (1+β)[θ(1-δ)-δ]+Θθ[(1+β)(1-δ)-δ] + √ ∆ 2Θ(1+β)(1-δ)θ(1-s)σsup δσ < 0 lλ ( lw ) and ∆ = (1 -s) 2 (1-β+Θθ)(1+β)(1-δ) δ + θ(1+β+Θ)(σsup-σ) σ + s lw δσ (1 + β)[θ(1 -δ) -δ] + Θθ[(1 + β)(1 -δ) -δ] 2 + 4 1 + s σ lw (1 + β + Θθ)s( lw -lw ) Θ(1+β)(1-δ)θ(
X ( H cc ( lw , lλ ), lw , lλ , σ, Θ) = - (1-β)(θ-sβδ) lλ sβ [(1-s)(1-C)+ΘsC]( lw -ˆ lw ) lw 1+ Θs H cc ( lw , lλ ) lw ( lw -lw ) lλ [(1-s)(1-C)+ΘsC]( lw -ˆ lw ) Θθ[1+ s σ lw + (1-s) σ lλ]
We easily derive that X ( H cc ( lw , lλ ), lw , lλ , σ, Θ) < 0 and thus T ( H cc ( lw , lλ ), lw , lλ , Θ) < 2 if and only if lλ ∈ (ˆ lλ ( lw ), ¯ lλ ( lw )).

We need now to provide a condition to get and thus X ( H cc (0, lλ ), 0, lλ , σ, Θ) = -θδ(σsup-σ) βσ(1-δ)

We then derive that if σ > σ inf with

σ inf = δθ(1-s)(1+Θ) Θ[4β(1-δ)+δθ]
then X ( H cc (0, lλ ), 0, lλ , σ, Θ) > -4 and X ( H cc ( lw , lλ ), lw , lλ , σ, Θ) > -4 for any lw ≥ 0. Case 1 -Let us consider in a first step the case lw ∈ (0, lw ) where ˆ lλ ( lw ) = 0. We get the following geometrical characterizations of the ∆(T ) line depending on the value of lλ . When lλ < 0 lλ ( lw ) or lλ ∈ ( 0 lλ ( lw ), ˜ lλ ( lw )), we have: As shown by these Figures, increasing cc from 0, the steady state is first saddle-point stable. Still increasing cc leads to the existence of a flip bifurcation giving rise to the existence of period-two cycles when cc crosses F cc ( lw , lλ ). Above F cc ( lw , lλ ) the steady state then becomes locally indeterminate and when cc crosses the bound H cc ( lw , lλ ), a Hopf bifurcation occurs giving rise to the existence of periodic cycles. Above H cc ( lw , lλ ), the steady state is totally unstable for any cc ∈ ( H cc ( lw , lλ ), +∞). When lλ ∈ (˜ lλ ( lw ), ¯ lλ ( lw )), we obviously get the following case where the flip Finally, when lλ > ¯ lλ ( lw ), we get the following case where the Hopf bifurcation no longer exists, i.e. Case 2 -Let us consider now the case lw ∈ ( lw , ¯ lw ) where 0 < ˆ lλ ( lw ) < ˜ lλ ( lw ). We get lλ ( lw ) < 0 lλ ( lw ) < ¯ lλ ( lw ) < ˜ lλ ( lw ) < ¯ lλ ( lw ) and lλ ( lw ) < ˆ lλ ( lw ) < ˜ lλ ( lw ). However, the bounds 0 lλ ( lw ) and ¯ lλ ( lw ) may be lower or larger than ˆ lλ ( lw ) but this does not really impact the local stability results. It is worth noticing that when lw = lw , lλ ( lw ) = ˆ lλ ( lw ) = 0. We then obtain the following geometrical characterizations of the ∆(T ) line depending on the value of lλ and lw . Recall that lw ∈ ( lw , ¯ lw ) and consider for now that 0 lλ ( lw ) < ¯ lλ ( lw ) < ˆ lλ ( lw ). When lλ < lλ ( lw ) we have When lλ ∈ ( lλ ( lw ), ˜ lλ ( lw )) we get the following three cases: In these three cases, local indeterminacy arises if cc ∈ ( F cc ( lw , lλ ), T cc ( lw , lλ )) and saddle-point stability holds outside this interval. The same conclusion would be obtained is 0 lλ ( lw ) and/or ¯ lλ ( lw ) were larger than ˆ lλ ( lw ). When lλ > ˆ lλ ( lw ), the Hopf bifurcation may again occur as long as lλ < ¯ lλ ( lw ). We get indeed: As shown by these Figures, when lλ ∈ (ˆ lλ ( lw ), ˜ lλ ( lw )), increasing cc from 0, the steady state is first saddle-point stable. Still increasing cc leads to the existence of a flip bifurcation giving rise to the existence of period-two cycles when cc crosses F cc ( lw , lλ ). Above F cc ( lw , lλ ) the steady state then becomes locally indeterminate and when cc crosses the bound H cc ( lw , lλ ), a Hopf bifurcation occurs giving rise to the existence of periodic cycles. Above H cc ( lw , lλ ), the steady state is locally unstable when cc ∈ ( H cc ( lw , lλ ), T cc ( lw , lλ )) and saddle-point stable when cc ∈ ( T cc ( lw , lλ ), +∞). Since the steady state is generically unique, T cc ( lw , lλ ) corresponds to a degenerate transcritical bifurcation. When lλ ∈ (˜ lλ ( lw ), ¯ lλ ( lw )), we get T cc ( lw , lλ ) = 0 and when lλ > ¯ lλ ( lw ) we get H cc ( lw , lλ ) = 0. Case 3 -Let us consider finally the case lw > ¯ lw where ˜ lλ ( lw ) < ˆ lλ ( lw ) < ¯ lλ ( lw ). Therefore, using all our previous results, we get the following geometrical characterizations of the ∆(T ) line depending on the value of lλ . When lλ < lλ ( lw ), local indeterminacy is ruled out as the ∆(T ) does not cross the triangle ABC as in Figure 16. On the contrary, when lλ ∈ ( lλ ( lw ), ˜ lλ ( lw )), local indeterminacy can arise as we get the following three cases where the ∆(T ) crosses the triangle ABC as in Figures 17,18 and 19. When lλ ∈ (˜ lλ ( lw ), ˆ lλ ( lw )) the flip bifurcation no longer exists. When lλ ∈ (ˆ lλ ( lw ), ¯ lλ ( lw )), the Hopf bifurcation exists as the same configuration as Figure 21 occurs. When lλ > ¯ lλ ( lw ), the Hopf bifurcation no longer exists and local indeterminacy is again ruled out as the same configuration as Figure 22 occurs.

Proof of Corollary 1

With KPR preferences such that lλ = cc lw , we get D( cc , lw , σ, Θ) = 1 We obviously need σ > σ F to get F cc > 0. We easily show that T ( H cc , lw , σ, Θ) < 2 and that solving H cc = F cc allows to prove the existence of σinf > σ F as given by σinf ≡ (1-β+Θθ)δ(1-s)(1+Θ) Θ[4(1-δ)Θβ+δ(1-β+Θθ)]

such that T ( H cc , lw , σ, Θ) > -2 if σ > σinf . Notice that the bound obtained here is different from the one derived in the Proof of Theorem 2 as we consider a specific value lλ = cc lw which modifies the computations.

Proof of Corollary 2

With GHH preferences such that lλ = 0, we get D( cc , lw , σ, Θ) = 1

Assumption 4 .

 4 Consumption c and leisure L are normal goods.

Theorem 2 .

 2 Let Assumptions 1, 3, 4, 6 and 7 hold. Consider the bifurcation curves and critical curves defined by Lemma 4, and define by lw = (σ sup -σ)/s, with σ sup = (1 -s)(1 + Θ)/Θ, and by lw the unique solution of ˜ lλ

Figure 1 :

 1 Figure 1: Local stability properties of the two sector model. Benchmark calibration with σ = 1 and Θ = 0.3.

Figure 2 :

 2 Figure 2: Local stability properties for low labor suply elasticities (benchmark calibration with σ = 1 and Θ = 0.3).

ΘFigure 3 :

 3 Effects of varying Θ on the indeterminacy areaIn Figure4, we display the local stability properties for four different values for the elasticity of substitution between capital and labor σ, namely σ ∈ (0.5, 1, 1.5, 2).

Figure 4 :

 4 Effects of varying σ on the indeterminacy area

Figure 5 :

 5 Figure 5: Local stability properties with KPR utility function (σ = 1).

Figure 6 :

 6 Figure 6: Local stability properties with GHH utility function (σ = 1).

Figure 7 :

 7 Figure 7: Local stability properties with generalized Hansen utility function (σ = 1).

rk y = s and wl y = 1

 1 -s Considering that at the steady state we have R * = 1/β with R * = r * +1-δ = sy * /k * +1-δ we get y * k * = θ sβ with θ = 1 -β(1 -δ). It follows from the capital accumulation equation evaluated at the steady state that c = y -k and thus c * k * = θ-sβδ sβ

Figure 8 :

 8 Figure 8: Area of local indeterminacy.

Figure 9 :

 9 Figure 9: lw < lw

Figure 10 :

 10 Figure 10: lw > lw and Θ k ∈ [0, Θ k ), with Θ l < Θ l or Θ l ∈ (Θ l , Θl ) and lλ < lλ ( lw )

Figure 11 :

 11 Figure 11: lw > lw and Θ k ∈ [0, Θ k ), with Θ l ∈ (Θ l , Θl ) and lλ > lλ ( lw ).

Figure 12 :

 12 Figure 12: ∆(T ) line when lw ∈ (0, lw ) and lλ < 0 lλ ( lw ).

Figure 13 :Figure 14 :

 1314 Figure 13: ∆(T ) line when lw ∈ (0, lw ) and lλ ∈ ( 0 lλ ( lw ), ˜ lλ ( lw )).

Figure 15 :

 15 Figure 15: ∆(T ) line when lw ∈ (0, lw ) and lλ > ¯ lλ ( lw ).

Figure 16 :

 16 Figure 16: ∆(T ) line when lw ∈ ( lw , ¯ lw ) and lλ < lλ ( lw ).

Figure 17 :Figure 18 :

 1718 Figure 17: ∆(T ) line when lw ∈ ( lw , ¯ lw ) and lλ ∈ ( lλ ( lw ), 0 lλ ( lw )).

Figure 19 :

 19 Figure 19: ∆(T ) line when lw ∈ ( lw , ¯ lw ) and lλ ∈ ( ¯ lλ ( lw ), ˆ lλ ( lw )).

Figure 20 :Figure 21 :

 2021 Figure 20: ∆(T ) line when lw ∈ ( lw , ¯ lw ) and lλ ∈ (ˆ lλ ( lw ), ˜ lλ ( lw )).

Figure 22 :

 22 Figure 22: ∆(T ) line when lw ∈ ( lw , ¯ lw ) and lλ > ¯ lλ ( lw ).

Figure 23 :

 23 Figure 23: ∆(T ) line when lw > ¯ lw and lλ ∈ (˜ lλ ( lw ), ˆ lλ ( lw )).

  (1-s) cc]- Θ(1-δ)(θ-sβδ) cc sδ [C cc lw +1+ s lw σ ] T ( cc , lw , σ, Θ) = 1 + D( cc , lw , σ, Θ) + θ(θ-sβδ) cc sβ lw [ (1-s)(1-C)+ΘsC σ ]+ C cc lw Θ(σsup-σ) σ + sΘ σ ( lwlw ) 1+ s lw σ [s+θ(1-s) cc]- Θ(1-δ)(θ-sβδ) cc sδ [C cc lw +1+ s lw σ ] ≡ 1 + D( cc , lw , σ, Θ) + X ( cc , lw , σ, Θ) with σ sup ≡ (1-s)(1+Θ)Θ and lw ≡ σsup-σ s Under Assumption 7 we obviously have σ < 2 < σ sup . Applying the same technique as in the Proof of Theorem 2 we can compute the Hopf bifurcation value

2 + 8 1

 28 )Θ(1-δ)(θ-sβδ) sδ 1 + s lw σ -2(1-s)θ lw σ (1 + β + Θ) + θ(θ-sβδ) s lw (1-C)s σ 1-s s -Θ + Θ σ (σ sup -σ) + s lw σ (1 + β + Θθ) Θ(θ-sβδ)C lw s 2(1+β)(1-δ) δ -θ σ (σ sup -σ) and σ F ≡ δθ(1-s)(1+Θ) Θ[2(1+β)(1-δ)+θδ]

1 -≡ 1 +

 11 ccΘ(1-δ)(θ-sβδ) sδ T ( cc , lw , σ, Θ) = 1 + D( cc , lw , σ, Θ) + θ(θ-sβδ) β cc Θ σ ( lwlw ) (1+ lw σ )[1-ccΘ(1-δ)(θ-sβδ) sδ ] D( cc , lw , σ, Θ) + X ( cc , lw , σ, Θ) with σ sup ≡ (1-s)(1+Θ)Θ and lw ≡ σsup-σ s

  and Wallenius[70] andPrescott and Wallenius [68] to calibrate the wage elasticity of the aggregate labor supply curve in standard RBC/DSGE models). Indeterminacy requires in this case lλ > lλ 55 and cc > T cc 1169. If, at the other extreme, the wage elasticity of the labor supply curve is increased to 1000 for the same other parameter values (approximating Hansen's[START_REF] Kydland | Time to Build and Aggregate Fluctuations[END_REF] type of preferences with an infinitely elastic labor supply curve), indeterminacy now requires lλ > lλ 1259 and

	cc > T cc	254. Clearly, no configuration is empirically realistic. 8

  1-s)σsup δσ Note that lλ ( lw ) is obviously such that F cc ( lw , lλ ) = T cc ( lw , lλ ) when lλ = lλ ( lw ). lw > lw such that ˆ lλ ( lw ) < ˜ lλ ( lw ) if and only if lw ∈ ( lw , ¯ lw ) and ˆ lλ ( lw ) = ˜ lλ ( lw ) if and only if lw = ¯ lw . Let us consider now the value of the Trace when cc = H cc ( lw , lλ ). We get T ( H cc ( lw , lλ ), lw , lλ , Θ) = 2 + X ( H cc ( lw , lλ ), lw , lλ , Θ) with

	6.12 Proof of Theorem 2

Under Assumptions 1, 3, 4, 6 and 7, straightforward computations show that if lw > lw , ˆ lλ ( lw ) < ¯ lλ ( lw ). We need now to check whether ˆ lλ ( lw ) ≷ ˜ lλ ( lw ). Since ˆ lλ ( lw ) = 0 < ˜ lλ ( lw ) when lw = lw , obvious computations then show that there exists a unique value ¯

  T ( H cc ( lw , lλ ), lw , lλ , σ, Θ) > -2 or equivalently X ( H cc ( lw , lλ ), lw , lλ , σ, Θ) > -4 for any lw ≥ 0. Straightforward computations give X ( H cc ( lw , lλ ), lw , lλ , σ, Θ) =

	θ(θ-sβδ) sβ	lw lλ[	(1-s)(1-C)+ΘsC σ	]+	ΘC 2 lλ σ	(σsup-σ)+	Θs lw σ	H cc	( lw -lw )
	lw 1+	s lw σ +	θ(1-s) lλ σ	-	Θ(1-δ)(θ-sβδ) sδ	[C 2 lλ +( lw -lw ) H cc (1+	s lw σ )]

For example, in models with variable capital utilization, the limit case would be obtained as the one for which the elasticity of the depreciation rate to capital utilization variations is zero. In models with different sectoral production functions (e.g., in terms of their capital-labor intensity or their capital-labor elasticity of substitution), the limit case would be the one for which these elasticities are the same in both sectors. In models with static consumption spillovers, the limit case would the one for which the effect of aggregate consumption on individual preferences is nullified, etc.

On the other hand, the case of models involving additional state variables is trickier, because the limit case enabling to recover our benchmark model may be associated with a change in the dimensionality of the dynamic system. When this happens, a discontinuity in the local stability properties of the model may (or may not) occur, so that a specific analysis for the larger model is required. This potentially concerns, for example, models with habit formation in the utility function (see e.g. Boldrin et al.[START_REF] Boldrin | Habit Persistence, Asset Returns, and the Business Cycle[END_REF] and Jaimovich and Rebelo[START_REF] Jaimovich | Can News About the Future Drive the Business Cycles?[END_REF]) for which past consumption is an additional state variable disappearing from the dynamic system when the influence of past consumption is nullified, or New-Keynesian models with sticky prices in which the past inflation rate is a state variable disappearing in the limit case of flexible prices.

See e.g.Basu and Fernald [4] for empirical estimates of aggregate IRS in the US economy and Aiyagari[START_REF] Aiyagari | The Econometrics of Indeterminacy: An Applied Study -A Comment[END_REF] for a critique of macroeconomic sunspot models relying on a upward sloping aggregate labor demand curve.

See also Benhabib and Wen[START_REF] Benhabib | Indeterminacy, Aggregate Demand, and the Real Business Cycle[END_REF] for a discrete-time version of the same model.

As we will show later on, absent these additional ingredients, sunspot fluctuations cannot occur under empirically relevant calibrations for any size of income effect.

The literature shows that sunspot fluctuations require a sufficiently large (possibly larger than one) elasticity of capital-labor substitution and/or a sufficiently large elasticity of intertemporal substitution in consumption. Moreover, local indeterminacy is ruled out under GHH preferences with no-income effect and KPR preferences (seeBennett and Farmer [11], Hintermaier[START_REF] Hintermaier | On the Minimum Degree of Returns to Scale in Sunspot Models of Business Cycles[END_REF], Lloyd-Braga et al.[START_REF] Lloyd-Braga | Indeterminacy with Small Externalities: The Role of Non-Separable Preferences[END_REF], and Pintus[START_REF] Pintus | Indeterminacy with Almost Constant Returns to Scale: Capital-Labor Substitution Matters[END_REF][START_REF] Pintus | Local Determinacy with Non-Separable Utility[END_REF]).

Similar unrealistic values appear for the whole range of potential calibrations regarding σ, Θ k and Θ l .

In Benhabib and Farmer [7], consumers have Hansen's type of individual preferences and the production functions are Cobb-Douglas.

This assumption allows us to avoid having to consider a much larger number of cases.

A proof of this statement can be provided upon request.

To simplify notations, from now on we no longer explicitly mention the dependence of the critical bounds on lw and the dependence of the bifurcation curves on lw and lλ.

NRA (reference: ANR-17-EURE-0020), and from the Excellence Initiative of Aix-Marseille University -A*MIDEX. We thank an anonymous referee and the

We conclude here that since σ < 2 < σ sup , X (+∞, lw , lλ , σ, Θ) > 0 and thus 1 -T (+∞, lw , lλ , σ, Θ) + D(+∞, lw , lλ , σ, Θ) = -X (+∞, lw , lλ , Θ) < 0 if and only if lw > lw . We then get the following conclusions:

-if lλ < 0 lλ ( lw ) then 1 -T (0, lw , lλ , σ, Θ) + D(0, lw , lλ , σ, Θ) < 0 for any lw ≥ 0 while 1 -T (+∞, lw , lλ , σ, Θ) + D(+∞, lw , lλ , σ, Θ) > 0 if and only if lw < lw ;

-if lλ > 0 lλ ( lw ) then 1 -T (0, lw , lλ , σ, Θ) + D(0, lw , lλ , σ, Θ) > 0 for any lw ≥ 0 while 1 -T (+∞, lw , lλ , σ, Θ) + D(+∞, lw , lλ , σ, Θ) > 0 if and only if lw < lw .

Let us compute the critical values H cc ( lw , lλ ), T cc ( lw , lλ ) and F cc ( lw , lλ ) respectively associated with Hopf, transcritical and flip bifurcations. The first one H cc ( lw , lλ ) is obtained solving the equality D( cc , lw , lλ , σ, Θ) = 1 with respect to cc . Straightforward computations yield the value

It follows obviously that for any given lw , H cc ( lw , lλ ) ≥ 0 if and only if lλ ≤ ¯ lλ ( lw ). The critical value T cc ( lw , lλ ) is obtained as the solution of the equality X ( cc , lw , lλ , σ, Θ) = 0, namely

We have ˆ lw < 0 and thus lw -ˆ lw > 0. So, obviously, T cc ( lw , lλ ) > 0 if and only if lw > lw . By convention, we will consider that T cc ( lw , lλ ) = +∞ when lw ∈ (0, lw ). Note that since the steady state is generically unique, T cc ( lw , lλ ) corresponds to a degenerate transcritical bifurcation.

The critical value F cc ( lw , lλ ) is finally obtained as the solution of the equality 1 +

and ∆ as given by (71). Obviously, - lλ ( lw ) < 0 and we conclude that F cc ( lw , lλ ) ≥ 0 if and only if lλ ≤ ˜ lλ ( lw ). This critical value corresponds to a flip bifurcation giving rise to the existence of period-two cycles.

As this will become obvious later on, we need now to check that H cc ( lw , lλ ) ≤ T cc ( lw , lλ ). This inequality is satisfied if and only if lλ ≥ ˆ lλ ( lw ) with

Applying the same technique as in the Proof of Theorem 2 we can compute the Hopf bifurcation value

and the flip bifurcation value

Assuming σ > σ F implies that F cc > 0 for any lw > 0. However, we easily get

We then derive that if σ > σinf with

for any lw ≥ 0. Notice that the bound obtained here is different from the one derived in the Proof of Theorem 2 as we consider a specific value lλ = 0 which modifies the computations.

Proof of Corollary 3

Under generalized Hansen preferences such that lλ = lw , we get

Applying the same technique as in the Proof of Theorem 2 we can compute the Hopf bifurcation value

and the transcritical bifurcation value

Assuming σ > σ implies ˆ lw < 0 so that the existence of F cc relies only on the bounds σ F and F lw . Straightforward computations show that as long as σ < σ H , X ( H cc , lw , σ, Θ) < 0 and thus T ( H cc , lw , σ, Θ) < 2 for any lw ≥ 0. We need finally to provide a condition to get T ( H cc , lw , σ, Θ) > -2 or equivalently X ( H cc , lw , σ, Θ) > -4 for any lw ≥ 0. Straightforward computations give X ( H cc , 0, lλ , σ, Θ) = -δ(1-β+Θθ)(σsup-σ) βσ(1-δ)Θ

We then derive that if σ > σinf with σinf = (1-β+Θθ)δ(1-s)(1+Θ) Θ[4(1-δ)Θβ+δ(1-β+Θθ)] ∈ (σ F , 2) then X ( H cc , 0, σ, Θ) > -4 and X ( H cc , lw , σ, Θ) > -4 for any lw ≥ 0. Notice that the bound obtained here is different from the one derived in the Proof of Theorem 2 as we consider a specific value lλ = lw which modifies the computations.