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

Indeterminacy and sunspots in two-sector RBC models with generalized no-income-effect preferences *

Keywords: Indeterminacy, sunspots, two-sector model, sector-specific externalities, real business cycles Classification Numbers: C62, E32, O41

We analyze sunspot-driven fluctuations in the standard two-sector RBC model with moderate increasing returns to scale and generalized no-income-effect preferences à la Greenwood, Hercovitz and Huffman [13]. We provide a detailed theoretical analysis enabling us to derive relevant bifurcation loci and to characterize the steady-state local stability properties as a function of various structural parameters. We show that local indeterminacy occurs through flip and Hopf bifurcations for a large set of values for the elasticity of intertemporal substitution in consumption, provided that the labor supply is sufficiently inelastic. Finally, we provide a detailed quantitative analysis of the model. Computing, on a quarterly basis, a new set of empirical moments related to two broadly defined consumption and investment sectors, we are able to identify, among the set of admissible calibrations consistent with sunspot equilibria, the ones that provide the best fit of the data. The model properly calibrated solves several empirical puzzles traditionally associated with two-sector RBC models.

Introduction

The aim of this paper is to provide a detailed theoretical and empirical assessment of the sunspot-driven two-sector Real Business Cycle model with productive externalities and increasing returns to scale, considering the Greenwood, Hercovitz and Huffman [13] (GHH) specification for individual preferences, characterized by the lack of income effect on labor choices. Compared to the previous literature in which the formal theoretical analysis of such models and their data confrontation step are largely divorced, we argue that providing a complete characterization of the local stability properties of the model as a function of various structural parameters is a crucial ingredient for a successful data confrontation strategy. Following this approach, we are able not only to derive new theoretical configurations for which the two-sector RBC model is locally indeterminate, but also to improve several well-known counterfactual predictions of this model when submitted to sunspot-driven shocks.

The recent literature suggests that by comparison to their one-sector equivalents, two-sector RBC models are able to generate local indeterminacy with much lower degrees of increasing returns to scale. [START_REF] Basu | Returns to Scale in US Production: Estimates and Implications[END_REF] Yet, this result has often been obtained under relatively narrow specifications for technology and/or preferences, without much attention to robustness and domain of validity issues. Besides, in many cases, this result has been obtained through numerical simulations, without explicit consideration of the types of local bifurcations identified (and their implications for the local dynamics around the steady-state).

The first contribution of this paper is thus theoretical. It aims to provide a general theoretical analysis of local indeterminacy and local bifurcations in the canonical twosector RBC model. Starting from the Benhabib and Farmer [4]'s formulation with increasing social returns, we consider the generalized specification of GHH preferences, enabling us to thoroughly analyze the interplays between increasing returns to scale, intertemporal substitution effects and labor supply elasticity in the emergence of local indeterminacy.

It is known that with GHH preferences and constant social returns, local indeterminacy occurs for sufficiently inelastic labor supply (Nishimura and Venditti [23]). Yet, for increasing social returns, this result has been extended only for the specific case of a logarithmic specification, and in fact essentially through numerical simulations (Guo 1 and Harrison [15]). [START_REF] Baxter | Are Consumer Durables Important for Business Cycles ?[END_REF] We prove here that this result holds quite generally, in particular for a large set of values for the elasticity of intertemporal substitution (EIS) in consumption. We also prove the existence of an upper bound on the labor supply elasticity above which local indeterminacy never arises, and we show that this upper bound is decreasing with the degree of increasing social returns. We finally exhibit the existence of flip and Hopf bifurcations in the parameter space and we provide the analytical expressions for these bifurcations. [START_REF] Benhabib | Indeterminacy and Increasing Returns[END_REF] This allows us to show how a change in the EIS in consumption drastically affects the range of values for the other structural parameters for which the steady-state is locally indeterminate.

The second contribution of the paper is empirical. While two-sector RBC models are able to generate local indeterminacy and endogenous sunspot fluctuations with much lower degrees of increasing returns to scale than one-sector models, they also tend to make several inaccurate empirical predictions. In particular, the literature has identified several empirical puzzles associated with such models:4 the consumption-investment cyclicality puzzle (the inability to generate simultaneous procyclical comovements of consumption and investment with output), the consumption volatility puzzle (the tendency to generate a volatility of consumption that exceeds that of output), the labor comovement puzzle (the inability to generate procyclical movements in sectoral hours worked), and the hours worked volatility puzzle (the inability to generate sufficiently volatile hours worked relatively to output).

We first start by computing on a quarterly basis a new set of empirical moments related to two broadly defined consumption and investment sectors, adapting a methodology initially proposed in Baxter [2] with annual data. Then, we show that, by considering the general GHH specification for individual preferences together with appropriate calibrations, all four empirical puzzles mentioned above can be resolved. Improving the model's predictions requires to find better compromises between the various economic mechanisms -EIS, income effects, wage elasticity of labor supply -identified as crucial for the local stability properties of the model. We show that the best performing calibrations are typically close to the boundary of the set of admissible calibrations consistent with indeterminacy, near the Hopf bifurcation locus identified in the theoretical analysis. This implies that appropriate calibrations must depart from the traditional [START_REF] Baxter | Are Consumer Durables Important for Business Cycles ?[END_REF] In one-sector models with GHH preferences, the results are drastically different: Meng and Yip [19] and Nishimura et al. [21] have shown that local indeterminacy cannot arise. Jaimovich [18], using a specification that nests the GHH formulation as a special case, has proved that a minimum amount of income effect is necessary for local indeterminacy. logarithmic specification for the utility function extensively considered in the literature.

As a whole, we significantly generalize and improve the conclusions obtained by Guo and Harrison [15] in the particular case of GHH preferences with a logarithmicconsumption utility function. On the theoretical ground, we exhibit the existence of a Hopf bifurcation, in addition to the flip bifurcation they identify, and we show how a change in the EIS in consumption drastically affects the range of values for the other parameters for which the steady-state is locally indeterminate. In particular, for any degree of increasing returns to scale, we can get indeterminacy with much larger labor supply elasticities. On the empirical ground, we provide a detailed dataset enabling to derive relevant business cycle statistics to which two-sector DSGE models can be compared. We use these data to show that considering generalized GHH preferences and a calibration of structural parameters consistent with a much larger labor supply elasticity than in Guo and Harrison [15] enables to solve the four empirical puzzles traditionally associated with two-sector RBC models with indeterminacy.

The rest of this paper is organized as follows. We present the model and we characterize the intertemporal equilibrium and the steady state in Section 2. In Section 3, the complete set of conditions for indeterminacy are derived. In Section 4, we provide detailed simulations in order to discuss the ability of the model to account for the main features of observed business cycles. Some concluding remarks are provided in Section 5, whereas all the technical details are contained in an Appendix.

The model

We consider a standard infinite-horizon two-sector Real Business-Cycle (RBC) model with productive externalities and GHH preferences. Households are infinitely-lived, accumulate capital, and derive utility from consumption and leisure. Firms produce differentiated consumption and investment goods using capital and labor, and sell them to consumers. All markets are perfectly competitive.

Production

Firms in the consumption sector produce output Y ct according to a Cobb-Douglas production function:

Y ct = z t K α ct L 1-α ct ( 1
)
where K ct and L ct are capital and labor allocated to the consumption sector, and z t is an exogenously evolving total factor productivity (TFP) level.

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

Y It = z t A t K α It L 1-α It (2)
where K It and L It are the numbers of capital and labor units used in the production of the investment good, and A t is the externality parameter. Following Benhabib and Farmer [4], we assume that the externality depends on the average levels KIt and LIt of capital and labor used in the investment sector, such that:

A t = KαΘ It L(1-α)Θ It (3)
with Θ ≥ 0.5 These economy-wide averages are taken as given by individual firms.

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 = αY ct K ct = p t αY It K It , (4)
ω t = (1 -α)Y ct L ct = p t (1 -α)Y It L It (5)
where r t , p t and ω 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.

Preferences

We consider an economy populated by a continuum of unit mass of identical infinitelylived agents. The representative agent enters each period t with a capital stock k t inherited from the past. He then supplies elastically an amount l t ∈ [0, ) of labor (with > 0 his exogenous time endowment), rents its capital stock k t to the representative firms in the consumption and investment sectors, consumes c t , and invests i t in order to accumulate capital. Following Greenwood-Hercovitz-Huffman [13] (GHH), the instantaneous utility function is

U (c t , l t ) = ct- Bl 1+χ t 1+χ 1-σ 1-σ (6)
with B > 0 a normalization constant, σ > 0 and χ ≥ 0. The essential feature of this specification is that the marginal rate of substitution between consumption and leisure is independent of consumption, as

U 2 (ct,lt) U 1 (ct,lt) = -Bl χ t ( 7
)
This property illustrates the lack of income effect associated with the agent's labor supply.

Denoting by y t the GDP, the budget constraint faced by the representative household is

c t + p t i t ≡ y t = r t k t + ω t l t (8)
Assuming that capital depreciates at rate δ ∈ (0, 1) in each period, the law of motion of the capital stock is:

k t+1 = (1 -δ)k t + i t (9)
The household then maximizes its expected present discounted lifetime utility max

{k t+1 ,ct,lt,it} ∞ t=0 E 0 +∞ t=0 β t ct-Bl 1+χ t 1+χ 1-σ 1-σ , (10)
with β ∈ (0, 1) the discount factor, subject to (8), (9) and k 0 given. The first-order conditions for an interior solution to this optimization problem (with l t < ) are

c t - Bl 1+χ t 1+χ -σ = βE t c t+1 - Bl 1+χ t+1 1+χ -σ r t+1 +(1-δ)p t+1 pt (11)
ω t = Bl χ t (12)
Equation ( 11) is the standard stochastic Euler equation, and ( 12) corresponds to the trade-off between consumption and leisure. With GHH preferences, as suggested by (7), the income elasticity of intertemporal substitution in labor is zero, and the Frisch wage elasticity of the labor supply is 1/χ.

Intertemporal equilibrium and steady state

We consider symmetric rational expectation equilibria which consist of prices {r t , p t , ω 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 that satisfy the household's and the firms' first-order conditions as given by ( 4)-( 5) and ( 11)- (12), the technological and budget constraints (1) to ( 3) and ( 8)-( 9), and the market equilibrium conditions. All firms in the investment sector being identical, we have KIt = K It and LIt = L It for any t. At the equilibrium, the production function in the investment good sector is then given by

Y It = z t K α(1+Θ) It L (1-α)(1+Θ) It (13)
We thus have increasing social returns which size is measured by Θ. The market clearing conditions for the consumption and investment goods give

c t = Y ct , (14)
i t = Y It , (15)
while the market clearing conditions for capital and labor yield

K ct + K It = k t (16)
L ct + L It = l t (17)
For a given initial capital stock k 0 , any solution that also satisfies the transversality condition lim

t→+∞ β t E 0 c t - Bl 1+χ t (1+χ) -σ k t+1 = 0
is called an equilibrium path. An interior steady state is defined by constant equilibrium quantities and prices in the non-stochastic environment (z t = z) such that l < . We provide the following Proposition:

Proposition 1. Assume that Θ = (1 -α)/α and consider χ and B such that

χ ≡ αΘ 1-α(1+Θ) , B ≡ (1 -α)δ χ βα 1-θ χ(1+Θ) Θ χ-χ (18)
with θ ≡ β(1 -δ). Then there exists a unique interior steady state if and only if (χ -χ)(B -B) > 0.

Proof : See Appendix 6.1.

We are now able to provide a detailed local stability analysis of the steady-state, considering a family of economies parameterized by the three parameters that govern the wage elasticity of labor supply, the EIS in consumption and the degree of increasing returns to scale (IRS) in the investment sector, σ, χ and Θ. Indeed, in appendix 6.2 we show that, at the steady state and for given parameters (β, α, δ), the EIS in consumption cc ≡ -U 1 (c, l)/(U 11 (c, l)c) is a function of (σ, χ), namely

cc (σ, χ) = 1 σ 1 - 1-α (1+χ)(1-βδα 1-θ ) (19)
Thus, cc (σ, χ) is decreasing with respect to σ but increasing with respect to χ.

Theory

It is easy to show that the set of equations describing an equilibrium path can be reduced to a 3-dimensional dynamic system involving c t , k t and z t (the equation specifying the exogenous process for the TFP level simplifying to z t = z in a deterministic environment).

Combining (1)-( 2) and firms' first-order conditions (4)-( 5), we derive p t A t = 1 and that the equilibrium capital-labor ratios in the consumption and the investment sectors are identical and equal to

a t ≡ k t /l t = K ct /L ct = K It /L It = αω t / ((1 -α)r t ) , with ω t = (1 -α)z t a α
t and r t = αz t a α-1 t . Combining the labor supply equation (12) with aggregate labor demand ω t = (1α)z t a α t yields

l t = (1-α)zt B 1 α+χ k α α+χ t = l(k t , z t ) (20)
and thus

a t = k t /l(k t , z t ) = a(k t , z t ) ω t = (1 -α)z t (a(k t , z t )) α = ω(k t , z t ), r t = αz t (a(k t , z t )) α-1 = r(k t , z t ) (21)
Using the equilibrium conditions (4), ( 8) and ( 16), we also derive :

y t = z t k α t (l(k t , z t )) 1-α = y(k t , z t ) K ct = r(kt,zt)ct α = K c (k t , c t , z t ) K It = k t -K c (k t , c t , z t ) = K I (k t , c t , z t )
Finally, combining (3) with p t A t = 1 leads to

p t = (a(k t , z t )) (1-α)Θ K I (k t , c t , z t ) -Θ = p(k t , c t , z t ) (22)
We conclude from this analysis that, although aggregate labor l t , the capital labor ratio a t , and output y t , are control variables, their equilibrium values at t are all pinned down by the initial aggregate capital stock k t and the TFP level z t , independently of households' expectations of future economic conditions. [START_REF] Cass | Do Sunspots Matter?[END_REF] This is also the case at equilibrium for the real wage ω t and the real interest rate r t. Yet, the relative price of capital p t also depends on the consumption level c t , which adjusts at t according to households' expectations (see (11)). It follows that the economic adjustments at work in period t rest exclusively on how much consumption c t is substituted to current investment i t in the current period, or, equivalently, how many units of the initial capital stock k t = K ct + K It and of total hours worked l t = L ct + L It are allocated to the consumption and the investment sectors, respectively.

A complete description of the dynamics can now be obtained by referring to the households' first-order condition (11) and the budget constraint (8) rewritten as follows

E t   r(k t+1 ,z t+1 )+(1-δ)p(k t+1 ,c t+1 ,z t+1 ) c t+1 - B(l(k t+1 ,z t+1 )) 1+χ 1+χ σ   = p(kt,ct,zt) β c t -B(l(kt,zt)) 1+χ 1+χ -σ (23)
k t+1 = (1 -δ)k t + y(kt,zt)-ct p(kt,ct,zt) (24)
together with an exogenous process for the TFP level z t .

When the TFP level is constant (z t = z), equations ( 23)-( 24) implicitly define a twodimensional dynamic system in (c t , k t ). Log-linearizing this system in a neighborhood of the interior steady-state (c, k) yields a Jacobian matrix for which the characteristic polynomial is given in Appendix 6.2. In this Appendix, we also show that for a given value of χ, the Trace and Determinant of the Jacobian matrix are linear functions of σ, so that we can use the geometrical methodology described in Grandmont et al. [12] to study the local stability properties of the steady state. Indeed, for a given Θ, as σ is varied over (0, +∞), the Trace and Determinant move along a line denoted ∆ χ in Figures 1 and2, and whose location depends on the value of χ.

We introduce the following parameter restrictions:

Assumption 1. α < 1/2, Θ ∈ (Θ, Θ) and δ ∈ (0, δ), with Θ = δ/(1-δ), Θ = (1-α)/α, and δ = √ [(1-β)(1+Θ)(1-α(1+Θ))+4βΘ 2 ] 2 +4β 2 Θ 3 (1-α(1+Θ))-[(1-β)(1+Θ)(1-α(1+Θ))+4βΘ 2 ] 2βΘ(1-α(1+Θ))
These restrictions cover all empirically plausible configurations since capital shares are typically less than 50% of GDP in industrialized economies. Moreover, using a standard calibration consistent with quarterly US data, (α, β) = (0.3, 0.99), Assumption 1 implies Θ ≈ 0.0256, Θ ≈ 2.33 and δ > 0.06. These bounds define intervals for Θ and δ which largely cover the range of available empirical estimates for these parameters. [START_REF] Dufourt | Sunspot Equilibria and the Expectation-Driven Phillips Curve[END_REF] Note also that the bound χ on the inverse of the labor supply elasticity, as given by (18), cannot be arbitrarily close to zero since the amount of externalities satisfies Θ > Θ.

When Θ ∈ (Θ, Θ), depending on the value of χ, the line ∆ χ has two possible locations. We have indeed the following geometrical configurations that provide a full picture of the local stability properties of the steady state: When χ = χ, the line ∆ χ merges with the line generated by AC along which D = T -1 and one characteristic root is equal to 1. If χ > χ, ∆ χ crosses the triangle ABC in which both characteristic roots have a modulus less than 1 and local indeterminacy [START_REF] Dufourt | Sunspot Equilibria and the Expectation-Driven Phillips Curve[END_REF] For example, Basu and Fernald [4] obtain a point estimates for the degree of IRS in the durable manufacturing sector in the US economy of 0.33, with standard deviation 0.11 (see also Harrison [17]) for other estimates in a similar range). A typical estimate for the annual depreciation rate in the US is 10%, implying δ = 0.025. arises (see Figure 1). Indeed, when σ increases from 0, the value of the pair (T , D) varies along ∆ χ . The steady state is first unstable for σ ∈ [0, σH ), then becomes locally indeterminate when σ ∈ (σ H , σF ) and is finally saddle-point stable for σ > σF . When σ crosses σH , one pair of complex characteristic roots crosses the unit circle and a Hopf bifurcation occurs generating quasi-periodic endogenous fluctuations. When σ crosses σF one negative characteristic root crosses the value -1 and a flip bifurcation occurs generating period-two cycles.

T E d d d d d d d d d d T D σ = +∞ σ = 0 σH σF 4 4 4 4 4 4 4 4 4 4 4 4 4 4 A B C ∆ χ ∆ χ -1 1 -1 1
On the contrary, if χ < χ, the line ∆ χ is located on the right of the triangle ABC and local indeterminacy is ruled out (see Figure 2). Depending on the value of σ the steady state is either saddle-point stable or unstable with the possible existence of a flip bifurcation and period-two cycles.

T E d d d d d d d d d T D σ = +∞ σ = 0 σF A B C ∆ χ ∆ χ -1 1 -1 1 Figure 2: Local determinacy with GHH preferences when χ ∈ [0, χ).
We then get the following Proposition:

Proposition 2. Under Assumption 1, let χ ≡ αΘ/[1 -α(1 + Θ)].
Then the following results hold: i) If χ > χ, the steady state is saddle-point stable when σ > σF , undergoes a flip bifurcation at σ = σF , becomes locally indeterminate when σ ∈ (σ H , σF ), undergoes a Hopf bifurcation when σ = σH and becomes locally unstable when σ ∈ [0, σH ).

ii) If χ ∈ [0, χ), the steady state is locally unstable when σ > σF , undergoes a flip bifurcation at σ = σF and becomes saddle-point stable when σ ∈ [0, σF ).

The Hopf and flip bifurcation values are respectively defined as:

σH = Θ(1-δ)(1-β)(1-θ) (1-βδα 1-θ )-1-α 1+χ δα[1-β+(1-θ)Θ] and σF = (1-θ)[(χ+α)Θ[2(1-δ)(1+β)+(1-θ)δ]-(1-θ)δχ(1-α)(1+Θ)] (1-βδα 1-θ )-1-α 1+χ 2(χ+α)δα[1+β+(1-θ)Θ]
with σF > σH .

Proof : See Appendix 6.2.

Interpretation. Proposition 2 shows that local indeterminacy in a two-sector model with GHH preferences can only occur if the aggregate labor supply curve is sufficiently inelastic (i.e., if χ > χ) and the EIS in consumption cc (σ, χ) is in an intermediate range (i.e., such that σ ∈ (σ H , σF )). While a similar conclusion has been obtained in models with constant social returns (see e.g. Benhabib and Nishimura [5]), [START_REF] Farmer | Real Business Cycles and the Animal Spirits Hypothesis[END_REF] this is a new conclusion in models with increasing social returns. Moreover, as ∂χ/∂Θ > 0, larger externalities requires less elastic labor supply curves for the existence of indeterminacy. Finally, since with α < 1/2, we have Θ < α/(1 -α) < Θ, indeterminacy is compatible with standard negatively sloped capital and labor equilibrium demand functions.

Proposition 2 also strongly generalizes the conclusions obtained on a numerical basis by Guo and Harrison [15] in the particular case of GHH preferences with a logarithmicconsumption utility function (σ = 1). First, we provide an explicit analytical expression for the threshold 1/χ on the labor supply elasticity above which local indeterminacy is ruled out. Second, we exhibit the existence of a Hopf bifurcation, in addition to the flip bifurcation identified by Guo and Harrison [15]. Third, we show how a change in the EIS in consumption drastically affects the range of values for the other parameters for which the steady-state is locally indeterminate. In particular, since ∂ σH /∂χ > 0 and ∂ σF /∂χ > 0, for any degree of increasing returns to scale Θ, considering smaller values for σ enables to obtain indeterminacy with much larger labor supply elasticities (i.e. smaller χ s). This property will be fundamental in our data confrontation analysis undertaken in the next section. As an illustration to Proposition 2, Figure 3 displays the determinacy-indeterminacy areas in the (Θ, χ, σ) space, once a fixed calibration (α, δ, β) = (0.3, 0.025, 0.99) for the other three structural parameters is chosen. The indeterminacy zone is the interior area delimited by the flip (upper curve) and the Hopf (lower curve) bifurcation loci. The figure confirms, numerically, that a large set of values for (Θ, χ, σ) are consistent with indeterminacy -including low values for the degree of increasing returns to scale -, and that indeterminacy with a sufficiently elastic labor supply (χ sufficiently close to 0) requires low values for σ (typically less than 0.5). The configuration considered by Guo and Harrison [15], obtained as the cross section of the three-dimensional plane at σ = 1, clearly strongly restricts the range of admissible configurations regarding the various economic mechanisms -EIS, income effects, wage elasticity of labor supply -important in the business cycle properties of the model, in particular as concerns the labor supply elasticity. [START_REF] Garnier | Intertemporal Substitution, Labor Supply Elasticity and Sunspot Fluctuations in Continuous-Time Models[END_REF] Our aim is now to explore if considering generalized GHH preferences enables to provide better empirical performance when the model is submitted to sunspot shocks, in particular as regards the four empirical puzzles mentioned in the introduction.

Quantitative analysis

We now turn to the quantitative analysis of the model. Our aim is to investigate if a two-sector model with GHH preferences is able to reproduce the main features of observed business cycles when subject to indeterminacy and sunspot disturbances. Previous papers in the literature have already performed this data confrontation step, using different types of preferences, [START_REF] Grandmont | Nonlinear Difference Equations, Bifurcations and Chaos : An Introduction[END_REF] and have identified several empirical regularities that are hardly accounted for by two-sector models: the consumption/investment cyclicality puzzle (the inability to generate procyclical comovements of consumption and investment with output), the consumption volatility puzzle (the propensity to generate a volatility of consumption larger than that of output), the labor comovement puzzle (the inability to generate procyclical comovements in sectoral hours worked), and the hours worked volatility puzzle (the inability to generate sufficiently volatile hours worked relatively to output).

The main contribution of this section is to show that many of these difficulties can actually be overcome by adopting a sufficiently general specification for GHH preferences and an appropriate calibration for structural parameters. It is at this stage, we argue, that benefiting from the theoretical characterization of the local stability properties of the model is important as it helps to find calibrations of structural parameters that, at the same time, (i) are consistent with an indeterminate steady-state, thus enabling sunspot shocks to play a role in the business cycle; (ii) leave additional degrees of freedom with respect to the relative intensities of the various underlying economic mechanisms (intertemporal substitution in consumption, labor supply elasticity, etc.) important in the dynamic properties of the business cycle.

We will develop these points more thoroughly in sections 4.2 and 4.3. Before this, we describe shortly the dataset we constructed and the empirical moments we derived in order to evaluate the model.

Data

One difficulty when evaluating multisector models is to compute empirical series whose definitions are broadly consistent with the corresponding variables in the model. In the case of two-sector models, Baxter [2] made a crucial step in this direction. Using standard input-output tables for the US economy, she shows that most of the SIC onedigit industries are clearly recognizable as producing predominantly either consumption goods or investment goods. She uses this observation to identify two broadly-defined consumption and investment sectors.

Yet, one limitation of the Baxter dataset is that it is based on annual series, while DSGE models are typically evaluated according to their ability to make accurate predictions on the covariations of macroeconomic variables at business cycle frequency (i.e., with quarterly data). For this reason, we constructed our own dataset formed with series taken from various sources but available at quarterly frequency. [START_REF] Grandmont | Nonlinear Difference Equations, Bifurcations and Chaos : An Introduction[END_REF] Our implicit definition for aggregate consumption, aggregate investment and aggregate output is standard, while we followed the Baxter methodology to define series on hours worked in the consumption and investment sectors. More precisely, we use quarterly data from the Bureau of Economic Analysis and define consumption as the sum of personal consumption expenditure in non-durable goods and services, and investment as the sum of private fixed investment and personal consumption expenditures in durable goods. [START_REF] Grandmont | Capital-Labor Substitution and Competitive Nonlinear Endogenous Business Cycles[END_REF] To obtain per capita variables, both series are divided by the population aged 16 and over. Finally, output is the sum of consumption and investment thus defined.

In order to construct series for hours worked in the consumption and investment sectors, we used data collected by the Current Employment Statistics program and available on the FRED database at the Federal Reserve Bank of Saint-Louis. The CES program provides series on the number of production and non-supervisory employees (together with their average weekly hours of work) at the sectoral level for the period 1964-Q1 to 2014-Q2. [START_REF] Greenwood | Investment, Capacity Utilization and the Real Business Cycle[END_REF] Following Baxter's approach, we allocate hours worked in a specific industry either to the consumption or the investment sector according to the predominant final use of the output of this industry as consumption or investment goods (see the NIPA 2012 Input-Output table available in the supplementary material). We thus define hours worked in the investment sector as the total number of (production and non-supervisory) employees in the Mining and logging, Construction, Manufacturing durables and Professional and business services sectors, multiplied by the average weekly hours of work in each sector. Similarly, we define hours worked in the consumption sector as the total number of employees, multiplied by the corresponding series on average weekly hours worked, in the Manufacturing nondurables, Trade, transportation and utilities, Information, Leisure and hospitality and Other services sectors. To obtain per capita variables, all these series were divided by the population aged 16 and over.

Table 1 provides the main summary statistics (second-order moments), after all series were detrended using the HP filter. The results display all the well-known stylized facts concerning aggregate consumption, investment, output and hours worked, so that we do not comment on them further. The main new features concern the comovements of hours worked in the consumption and investment sectors. As Table 1 indicates, hours worked in the consumption sector appear to be significantly less volatile than output and total hours worked. By contrast, hours worked in the investment sector are twice as volatile as output. Fluctuations in each variable are very persistent (the firstorder autocorrelation coefficients are above 0.9) and are strongly positively correlated with fluctuations in output. Finally, hours worked in the consumption and in the investment sectors are very strongly positively correlated (0.96). Note that most of these features are consistent with those obtained by Baxter [2], with the main difference that we obtain substantially larger autocorrelation and contemporaneous correlation coefficients. [START_REF] Gruber | A Tax-Based Estimate of the Elasticity of Intertemporal Substitution[END_REF] This is what we expected, taking account of the fact that our dataset is [START_REF] Greenwood | Investment, Capacity Utilization and the Real Business Cycle[END_REF] Industries are defined according to the following classification, which slightly differs from the standard NAICS classification: (1) Mining and Logging, (2) Construction , (3) Manufacturing (durable and nondurable) Goods, (4) Trade, Transportation and Utilities, (5) Information, (6) Professional and Business Services, (7)

Explaining the puzzles

We now provide a formal analysis enabling us to explain how an exogenous change in expectations (sunspot shock) is transmitted to the economy through various economic mechanisms (intertemporal substitution in consumption, wage elasticity of labor supply, capital accumulation, etc.) important in the business cycle properties of the model. This enables us to understand why most parameter calibrations lead to the four empirical puzzles mentioned above, in particular when the utility function is restricted to be logarithmic in consumption (σ = 1). In the next section, we make use of this analysis to identify the best strategy to solve these puzzles, and we provide simulation results corroborating our main analysis.

As usual in the RBC literature, in order to understand the effects of a sunspot shock, it is useful to analyze how the labor market works in this model. Indeed, consider the Euler equation for consumption with generalized GHH preferences, (11), and denote by R t+1 = (r t+1 + (1 -δ)p t+1 )/p t the real rate of return of investing into an 3.37, respectively. The correlation between these two series is 0.87, and the first-order autoregressive coefficients are 0.32 and 0.46, respectively. additional unit of capital between t and t + 1. Assume that there is a positive change in expectations concerning this rate of return. Clearly, these optimistic expectations must lead consumers to substitute investment for consumption. The extent to which such substitution occurs depends, at the same time, on the degree of consumption smoothing consumers wish to have over the business cycle (which in turn depends on the preference parameters σ and χ, which govern the value of the elasticity of intertemporal substitution in consumption cc , see (19)), but also on how these consumers adjust their labor supply in response to the shock: a larger initial increase in hours worked implies a smaller decrease in consumption for any given change in the expected real rate of return, and vice versa. [START_REF] Guo | Indeterminacy with No-Income Effect Preferences and Sector-Specific Externalities[END_REF] To understand how hours worked l t adjust, one must therefore refer to the functioning of the labor market. We have shown with equation ( 20) that l t fluctuates proportionally with capital, k t , and the TFP level, z t . We can also show that output y t evolves in strict proportion with hours worked at equilibrium. Indeed, using (5), we obtain Bl 1+χ t = ω t l t = (1 -α)y t , so that:

y t = B 1-α l 1+χ t . (25)
We can now provide an explanation for the main empirical puzzles associated with the two-sector RBC model submitted to sunspot disturbances.

Autonomous sunspot disturbances. When changes in expectations of the rate of return R t+1 occur independently of any change in fundamentals, we can treat the TFP level as constant, z t = z. In this case, (20) and (25) show that, with GHH preferences, the lack of income effect in labor supply implies that total hours worked and output remain constant in the immediate aftermath of a sunspot shock. Over time, as consumers substitute investment for consumption, the capital stock accumulates and there is a corresponding gradual increase in hours worked and output, occurring through [START_REF] Guo | Indeterminacy with No-Income Effect Preferences and Sector-Specific Externalities[END_REF] To verify this, observe that the log-linearized version of equation ( 11) is:

-

1 cc ( ct -ξ lt) = - 1 cc (Et ct+1 -ξEt lt+1) + Et Rt+1.
where ξ ≡ (1 -α) / (1 -βαδ/(1 -θ)), see Appendix 1. Along a monotone convergent path to the steady state, the marginal utility of consumption λt ≡ ct -ξ lt gradually reverts back to steady-state. Let us define ϑt ≡ Et λt+1/ λt, with ϑt < 1. We then obtain:

ct = - cc 1 -ϑt EtRt+1 + ξ lt.
Clearly, if the initial reaction of lt is small (or even zero), an increase in the expected rate of return of investment must generate a significant decrease in consumption, with multiplicative coefficient cc/(1ϑt). As capital accumulates over time, lt gradually increases away from 0 since there is an increase in labor demand: the reaction of consumption becomes less and less negative, and may even become positive if the increase in hours worked is sufficiently large.

an increase in labor demand. [START_REF] Harrison | Indeterminacy in a Model with Sector Specific Externalities[END_REF] The extent to which these two variables fluctuate along the business cycle depends on χ, the (inverse of the) wage-elasticity of labor supply: larger χ s imply smaller fluctuations in hours worked and output, and vice versa. Since indeterminacy requires a sufficiently inelastic labor supply (see Proposition 2), following an expected increase in the rate of return of investment, consumption is substituted for investment along a constant (in period t) and then weakly volatile (in periods onwards) aggregate output. This implies that consumption and investment must go in opposite direction following a sunspot shock, and that the volatilities of consumption and investment are larger than the volatility of output. We thus have an explanation for both the consumption-investment cyclicality puzzle and the consumption-volatility puzzle. Obviously, the consumption-volatility puzzle is magnified when the wage-elasticity of labor supply is low (i.e., when χ is large)

Explaining the sectoral labor comovement puzzle also follows directly from this analysis. Since K It /K ct = L It /L ct , an increase in the production of the investment good is obtained through both an increase in capital, K It , and hours worked, L It , allocated to the investment sector. Since total hours worked l t remain constant in the immediate aftermath of the shock, hours worked in the consumption and the investment sectors must also move in the opposite direction.

Finally, we can infer from ( 25) that there is a close relationship between the preference parameter χ and the relative volatility of hours worked with respect to output, σ l /σ y . Indeed, using (25), we get: σ l /σ y = 1/(1 + χ). Thus, any calibration for χ significantly greater than 0 must generate a volatility of hours worked that is substantially lower than the volatility of output. Since, in the data, this ratio is close to 1 (see Table 1), we have in this case an explanation for the hours worked-volatility puzzle.

Correlated sunspot and technological disturbances. When sunspot shocks are correlated with technological shocks, the analysis differs significantly. Indeed, equation (20) shows that a positive technological shock z t shifts the aggregate labor demand curve up and thus leads to an increase in total hours worked. Then, when this shock is associated with a change in the expected rate of return R t+1 (sunspot shock), the substitution of investment for consumption now occurs along an increasing output level. Clearly, if the increase in output is sufficiently large, both consumption and investment may now rise in response to the shock, and the rise in consumption may be smaller than the rise in output. This would solve the consumption-investment cyclicality puzzle and the consumption volatility puzzle. As for the sectoral labor comovement puzzle, the same logic applies: if l t increases sufficiently in response to the shock, L It and L ct may now rise simultaneously.

This line of reasoning of course requires that the variations in hours worked and in output be "sufficiently large". Otherwise, the same logic as above would apply. This shows again that successful calibrations must combine an indeterminate steady-state with a sufficiently elastic labor supply. Moreover, a low calibrated value for χ remains a condition for solving the hours worked volatility puzzle since the relative volatility of hours worked, σ l /σ y = 1/(1 + χ), is independent from the assumptions made about the correlation between shocks.

Note finally that the fact that technological shocks generate an increase in hours worked does not imply that sunspots are unimportant in the business cycle. As explained above, sunspots determine the extent to which expectations about the real rate of return of investment adjust after the observed increase in the TFP level z t . For a given volatility in hours worked and in output (determined by χ), "optimistic" expectations on R t+1 translate into larger increases in investment and smaller increases in consumption, and vice-versa. Thus, sunspot shocks critically affect the relative volatilities of consumption and investment with respect to output. As such, they are a key ingredient for solving the consumption volatility puzzle.

Simulation results

Armed with the considerations above, we can now turn to data confrontation. Our qualitative discussion suggests that the ability of the model to account for observed stylized facts depends both on the values ascribed to structural parameters (in particular, the wage-elasticity of labor supply) and on the properties of the sunspot shocks (are they an autonomous source of disturbances or do they contribute to amplify the effects of real shocks on the economy?). In order to make more quantitative assessments, we now assume that the TFP level evolves according to:

ln z t+1 = ρ ln z t + ε t+1 (26)
with autoregressive coefficient ρ ∈ (0, 1) and ε t+1 a normally distributed technological shock with standard deviation σ ε . The resulting 3-dimensional dynamic system obtained from log-linearizing ( 23)-( 24) and ( 26) around the deterministic steady state with z = 1 can be written as:

     c t+1 k t+1 z t+1      = J      c t k t z t      +    η t+1 0 ε t+1    ( 27
)
where η t+1 ≡ c t+1 -E t c t+1 , the forecast error of consumption, satisfies E t η t+1 = 0. As shown in Appendix 6.3, introducing such an exogenous stochastic TFP level does not change the stability properties of the model. Thus, when the steady-state is a sink, the three eigenvalues of the matrix J have modulus lower than 1 and the log-linear system (27) is stable. [START_REF] Harrison | Returns to scale and Externalities in the Consumption and Investment Sectors[END_REF] In this case, the forecast error η t is not pinned down by economic fundamentals and can be a source of business cycles (the sunspot shock). Still, theory does not impose restrictions on the correlation between this sunspot shock and the fundamental disturbance ε t . We assume in this respect that the forecast error satisfies

η t = ιε t + υ t
where ι is a scalar and υ t is a white noise shock with variance σ υ, uncorrelated with ε t . The "pure" sunspot shock υ t reflects extrinsic uncertainty in the sense of Cass and Shell (1983), while the scalar ι governs the extent to which households' expectations are influenced by technological shocks and attenuate or amplify the effects of these shocks on the economy. Following Benhabib and Farmer (1996), we can now evaluate the model considering two alternative assumptions regarding the role of sunspots in the economy: (i) sunspot shocks are purely exogenous and are the only source of disturbances: (σ υ > 0 and σ ε = 0, see Table 2), and (ii) sunspot shocks are perfectly correlated with technological shocks, and determine the extent to which agents change their expectations in response to these shocks (σ υ = 0 and σ ε > 0, see Table 3).

Calibration. In order to make quantitative assessments, a proper calibration of structural parameters is required. The preference parameter B is just a normalization variable which does not influence the second-order properties of the model. Thus, one only needs to ensure that the condition required for an interior steady state is satisfied (see Proposition 1). Following the standard practice in the Real Business Cycle literature, we set β = 0.99, implying a net annual return on capital of around 4%, δ = 0.025, implying a 10% annual depreciation rate of physical capital, α = 0.3, consistent with a labor share in income of 70%, and ρ = 0.95, consistent with the large persistence in estimated Solow residuals in the US economy. We calibrate the degree of increasing return to scale to Θ = 0.33, the point estimate obtained by Basu and Fernald [4] for the degree of IRS in the durable manufacturing industry. [START_REF] Jaimovich | Income Effects and Indeterminacy in a Calibrated One-Sector Growth Model[END_REF] Based on this calibration, we obtain from ( 18) that χ = 0.17, and we know from Proposition 2 that χ > χ is required for indeterminacy. We set χ = 1/3, implying a wage elasticity of aggregate labor supply of 3. This value is the one recommended by Rogerson and Wallenius (2009) and Prescott and Wallenius (2011) to calibrate business cycle models, based on both theoretical considerations and cross-country tax analysis. [START_REF] Meng | On Indeterminacy in One-Sector Models of the Business Cycle with Factor-Generated Externalities[END_REF] We also obtain from Proposition 2 that indeterminacy occurs when σ ∈ (σ H , σ F ) = (0.17, 0.35) or, using (19), when the elasticity of intertemporal substitution in consumption satisfies cc ∈ ( ccF , ccH ) = (0.86, 1.83). There is no agreement in the empirical literature about the precise value of this parameter, since most estimates typically vary between 0 and 2. [START_REF] Mulligan | Capital Interest and Aggregate Intertemporal Substitution[END_REF] Yet, indeterminacy clearly occurs for a large set of values in this interval. Based on the most recent estimates by Gruber [14], who concluded for values around 2, our choice is to calibrate cc close to its maximum value consistent with indeterminacy, namely cc = 1.80. It is worthwhile to emphasize that with this calibration, the model is located near the Hopf bifurcation locus in the parameter space. As emphasized by Dufourt et al. [7], models placed in this configuration -having two complex conjugate eigenvalues with modulus close to 1 -are more likely to account for the persistent and non-monotone dynamics of convergence to the steady-state following transitory shocks observed in the data.

Results. Results obtained using this benchmark calibration are reported in Tables 2 and3 (Table 2 refers to the case of autonomous sunspots, and Table 3 to the case of correlated sunspots). For comparison purposes, we also provide results obtained when the specification of GHH preferences is restricted to be logarithmic in consumption, as in Guo and Harrison [15]. Note that for this latter specification to be consistent with [START_REF] Jaimovich | Income Effects and Indeterminacy in a Calibrated One-Sector Growth Model[END_REF] This point estimate is roughly consistent with those obtained by Harrison [17] on the degree of externality in the investment sector. Using 2-digit data, Harrison [17] obtains a point estimate of 0.41. With 4-digit data, the point estimate is 0.29. Note that the aggregate degree of IRS in the model remains very small (equal to 7%), since the share of investment in GDP is 21%. [START_REF] Meng | On Indeterminacy in One-Sector Models of the Business Cycle with Factor-Generated Externalities[END_REF] See Prescott and Wallenius (2011) for a discussion of the factors that make the wage elasticity of aggregate labor supply significantly differ from the corresponding elasticity at the micro level. [START_REF] Mulligan | Capital Interest and Aggregate Intertemporal Substitution[END_REF] While early studies suggest quite low values for this elasticity, more recent estimates provide a much more contrasted view. For example, Mulligan [20] repeatedly obtained estimates above unitytypically in the range 1.1-2.1 -, and Vissing-Jorgensen and Attanasio [26] found estimates significantly above 1 using the Epstein-Zin preference specification, which enables to separately estimate the IES coefficient from the degree of risk aversion. Gruber [14] also provides robust estimates of this elasticity around 2. In light of all these studies, we assume that a plausible range for cc is (0, 2). indeterminacy, the interval (σ H , σ F ) must now contain 1, which only occurs if the wage elasticity of labor supply is sufficiently low, i.e. if the preference parameter χ is larger than χ F ≈ 5.5. [START_REF] Nishimura | Indeterminacy in Aggregate Models with Small Externalities: an Interplay Between Preferences and Technology[END_REF] To maximize the model's chances evaluated in this configuration, we calibrate χ to a value close to this minimum value consistent with indeterminacy, namely: χ = 6.

The statistics reported in Tables 2 and3 are the averaged second-order moments obtained by simulating 200 series of 202 observations each (the length of our dataset), and after detrending all series using the HP filter. [START_REF] Nishimura | Indeterminacy in Discrete-Time Infinite-Horizon Models with Non-Linear Utility and Endogenous Labor[END_REF] As clearly appears, and for the reasons explained in subsection 4.2., the model performs poorly when sunspot shocks are assumed to be an autonomous source of the business cycle (see Table 2). In particular, it exhibits many of the counterfactual predictions associated with two-sector indeterminate RBC models: consumption and investment are much more volatile than output (the consumption volatility puzzle), consumption and investment move in opposite direction, leading to a negative correlation coefficient between investment and output (the consumption/investment cyclicality puzzle), sectoral hours worked also move in opposite direction, leading to a negative correlation between them (the sectoral labor volatility puzzle). Furthermore, in the logarithmic consumption case, the relative volatility of hours worked with output, 1/(1 + χ) = 1/7 ≈ 0.14 , is extensively underestimated (in the case of generalized GHH preferences, this elasticity, albeit too low, is much closer to its empirical counterpart: 1/(1 + χ) = 1/(1 + 1/3) = 0.75). More generally, although these inaccurate predictions hold for both specifications of individual preferences, they are clearly worse when the utility function is restricted to be logarithmic in consumption. As explained above, the reason is that, when the elasticity of labor supply is low, the reallocation of resources between the investment and the consumption sector occurs along a constant (in the shock period) and then weakly volatile output level. As a result, all the above mentioned puzzles are significantly magnified in this configuration.

When sunspots are assumed to be perfectly correlated with technological shocks, things are substantially different. Now, the outward shift in labor demand generated by positive technological shocks implies that consumption is substituted for investment along an increasing output level. The extent to which output increases in response to these shocks depends again on the wage elasticity of labor supply, while the extent to which consumers allocate this extra production between consumption and investment Data preferences preferences . The unobserved parameter ι is the key parameter in this dimension: a small value for ι implies a small initial response of consumption and a large initial response of investment, and vice versa. Thus, when ι is calibrated in a proper range, the model can generate a correct initial ordering in the relative amplitudes of the responses of consumption, investment and output, which is a crucial step toward solving the consumption-volatility puzzle. [START_REF] Nishimura | Indeterminacy and Expectation-Driven Fluctuations with Non-Separable Preferences[END_REF] In the model with generalized GHH preferences, the range of values for which the initial response of consumption is procyclical but smaller than the initial response of output is ι ∈ (0, 2.1). In the model with logarithmic-consumption GHH preferences, the corresponding interval is ι ∈ (0, 1.1). To make things comparable, we calibrate ι so as to imply, in both versions of the model, that a 1% positive technological shock leads to a 1% drop in the relative price of investment, p t . Results are reported in RBC models under indeterminacy. The relative volatilities of consumption, investment and output have the correct ordering and are relatively close to their empirical counterparts. The contemporaneous cross-correlations of consumption and investment with output have the correct sign, and so does have the correlation between sectoral hours worked (even though this correlation remains too low compared to the data). The relative volatility of total hours worked remains unchanged at 0.75. By contrast, the model with σ = 1 remains clearly unsatisfactory, for the same reasons as those described above: although technological shocks imply an outward shift in labor demand, the low elasticity of labor supply implies that variations in total hours worked and output are negligible. The model is then unable to solve the empirical puzzles. This clearly shows that the generalized specification of GHH preferences is the only one that can improve the model's predictions in these dimensions.

Concluding comments

Although multi-sector Real Business Cycle models are based on a pervasive feature of the data and require smaller degrees of increasing returns to scale for indeterminacy than aggregate models, they are usually associated with several empirical shortcomings:

the labor comovement and consumption-investment cyclicality puzzles, and the consumption and hours worked volatility puzzles. Many contributions have tried to solve these shortcomings and the traditional strategy is to increase the degree of increasing returns. The main contribution of this paper is to show that a strong interaction between the theoretical and the empirical analysis of the model allows to significantly improve its empirical predictions. This is obtained by finding better compromises between the various economic mechanisms important in both the local stability and business cycle properties of the model. Considering generalized GHH preferences, we have provided a detailed theoretical analysis of the local stability properties of the steady state in order to get a full picture of the configurations giving rise to local indeterminacy and sunspot fluctuations. We have shown that local indeterminacy occurs through flip and Hopf bifurcations for a large set of values of the EIS in consumption provided that the labor supply is sufficiently inelastic. Moreover, the existence of expectations-driven fluctuations is consistent with a mild amount of increasing returns.

Building on this detailed theoretical analysis, we have been able to find, among the set of admissible parameter configurations consistent with sunspot equilibria, the ones that provide the best fit of the data. We have shown that a properly calibrated model is able to solve most empirical puzzles traditionally associated with two-sector RBC models.

Appendix

Proof of Proposition 1

Without loss of generality, let z t = 1. Equation (11) evaluated at the steady state gives r/p = (1 -θ)/β, with θ = β(1 -δ). We also derive from (4) that r/p = αδ k/ KI and thus KI / k = βδα/(1 -θ). From (4)-( 5) and ( 16)-( 17) , we derive that K I /L I = K c /L c = k/l and thus that L I /l = KI / k = βδα/(1 -θ). Observing from ( 9) and ( 13) that δ = ī/ k = a α(1+Θ)-1 L I /l L Θ I , we can express the capital-labor ratio as

ā ≡ k l = δ Θ 1-α(1+Θ) βα 1-θ 1+Θ 1-α(1+Θ) l Θ 1-α(1+Θ) (28) assuming of course that Θ = Θ ≡ (1 -α)/α.
Combining now the labor demand equation ω = (1 -α)ā α (with ā given by (28)) and the labor supply equation (12), we obtain the steady-state level of hours worked:

l = 1-α B 1-α(1+Θ) χ[1-α(1+Θ)]-αΘ δ αΘ χ[1-α(1+Θ)]-αΘ βα 1-θ α(1+Θ) χ[1-α(1+Θ)]-αΘ (29) Under Θ = Θ ≡ (1 -α)/α, let χ ≡ αΘ/(1 -α(1 + Θ)
). An interior steady-state in which (12) holds also requires l < , or equivalently:

B > (1 -α) δ χ βα 1-θ χ(1+Θ) Θ
χ-χ ≡ B when χ > χ or B < B when χ < χ Once l is obtained, it is straightforward to derive the steady-state values of all other endogenous variables. For example, using k = āl , we get

k = δ Θ 1-α(1+Θ) βα 1-θ 1+Θ 1-α(1+Θ) l (1-α)(1+Θ) 1-α(1+Θ) (30) Likewise, we derive from (1) that c = āα L c = āα (L -L I ) = āα (1 -βδα/(1 -θ))l, and thus: c = 1 -βδα 1-θ δ αΘ 1-α(1+Θ) βα 1-θ α(1+Θ) 1-α(1+Θ) l 1-α 1-α(1+Θ) (31)

Proof of Proposition 2

Let us introduce the following elasticities:

cc = -U 1 (c,l) U 11 (c,l)c , lc = -U 2 (c,l) U 21 (c,l)c , cl = -U 1 (c,l) U 12 (c,l)l , ll = -U 2 (c,l) U 22 (c,l)l (32)
We get at the steady state

cc = lc , 1 ll -1 cl = -χ < 0 with cc = 1 σ 1 - 1-α (1+χ)(1-βδα 1-θ ) (33) and cl = -cc (1-βαδ 1-θ ) 1-α = 1 σ(1-α) 1-α 1+χ -1 -βδα 1-θ (34)
Considering the first-order condition (11), the households' budget constraint (8), the capital accumulation equation ( 9) and the equilibrium value of labor as given by (20) with z t = 1, we get the following system of three equations

c t - Bl 1+χ t 1+χ -σ = β c t+1 - Bl 1+χ t+1 1+χ -σ r t+1 +(1-δ)p t+1 pt l t = 1-α B 1 α+χ k α α+χ t c t + p t [k t+1 -(1 -δ)k t ] = r t k t + ω t l t
Using the price equations at the equilibrium ( 21)-( 22), total differentiation of this system in a neighborhood of the steady state gives after simplifications:

   1 cc - (1-δ)(1-θ)Θ(1-βαδ 1-θ ) δα α cl +(1-α)(1-θ)χ+ (1-θ)(1-δ)Θ(1+χ) δ χ+α 0 βα (1-θ)(1+Θ)      c t+1 k t+1   =    1 cc - (1-θ)Θ(1-βαδ 1-θ ) βδα α cl + (1-θ)Θ(1+χ) βδ χ+α -1 -βδα 1-θ α 1+χ χ+α + (1-δ)β (1-θ)(1+Θ)      c t k t   ⇔ M   c t+1 k t+1   = N   c t k t   (35)
Considering the expression of cc as given by (33), the matrix M is invertible if and only if

σ = Θ(1-δ)(1-θ) (1-βδα 1-θ )-1-α 1+χ δα ≡ σ ∞ (36)
Under this condition, let us denote J = M -1 N . We then get

  c t+1 k t+1   = J   c t k t   =   J 11 J 12 J 21 J 22     c t k t   (37)
with When cc increases the ∆ χ (T ) half-line crosses the triangle ABC depending on the slope and on the location of the end-point (T χ (0), D χ (0)) obtained when σ = 0. Assume from now on that α < 1/2 and Θ ∈ (Θ, Θ) with Θ ≡ δ/(1 -δ) and Θ = (1 -α)/α. It follows that the slope satisfies S χ > 0 for any χ. Moreover, we have S χ ≤ 1 if and only if χ ≥ αΘ/[1 -α(1 + Θ)] ≡ χ. Notice also that ∂S χ /∂χ < 0 with lim χ→+∞ S χ = (1 -δ)Θ 2 / (1 -δ)Θ 2 + (1 -α(1 + Θ))δ < 1 and ∂D χ (σ)/∂σ > 0. We have indeed: We then derive that when σ = 0 P 0 (1) = (1-θ)δ[χ(1-α(1+Θ))-αΘ] (χ+α)θΘ P 0 (-1) = χ{Θ[2(1-θ(1-δ))+δ(1-θ)α]-δ(1-θ)(1-α)}+α(1+θ)(2-δ)Θ (χ+α)θΘ

J 11 = 1 cc - (1-θ)Θ ( 1-βαδ 1-θ ) βδα 1 cc - (1-θ)Θ(1-δ) ( 1-βαδ 1-θ ) δα + (1-θ)(1+Θ) βα(χ+α) (1-βδα 1-θ ) α cl +(1-α)(1-θ)χ+
It follows that P 0 (1) > 0 if and only if χ > χ. Moreover, it can be easily shown that when Θ > Θ, P 0 (-1) > 0 for any χ ≥ 0. The Hopf bifurcation value σH > 0 such that D χ (σ) = 1 is given by

σH = Θ(1-δ)(1-β)(1-θ) (1-βδα 1-θ )-1-α 1+χ δα[1-β+(1-θ)Θ]
Similarly, the flip bifurcation value σF > 0 such that P σ (-1) = 1 + T χ (σ) + D χ (σ) = 0 is given by σF =

(1-θ)[(χ+α)Θ[2(1-δ)(1+β)+(1-θ)δ]-(1-θ)δχ(1-α)(1+Θ)] (1-βδα 1-θ )- We easily derive that

lim χ→+∞ T χ (σ H ) = 2 -δ[1-β+(1-θ)Θ](1-α(1+Θ))
such that when δ ∈ (0, δ), lim χ→+∞ T χ (σ H ) ∈ (-2, 2). Moreover, we have ∂T χ (σ H )/∂χ < 0 and T χ (σ H ) = 2, so that T χ (σ H ) ∈ (-2, 2) for any χ > χ. Therefore, the ∆ χ line, when χ > χ, is located as in Figure 1. On the contrary, when χ ∈ [0, χ), the ∆ χ line is located as in Figure 2 and the steady state is either saddle-point stable or unstable. It is worth noting that when σ crosses the singular value σ ∞ for which the matrix M is non-invertible, D changes its sign as it crosses infinity but in the neighborhood of σ ∞ the steady state remains locally unstable or saddle-point stable depending on whether χ > χ or χ < χ.

The three-dimensional stochastic dynamic system

Consider the stochastic dynamic system as given by ( 23)-( 24) and (26). Log-linearizing this system around the deterministic steady state with z = 1, and denoting η t+1 = c t+1 -E t c t+1 , ε t+1 = z t+1 -E t z t+1 , with E t η t+1 = E t ε t+1 = 0, we derive: and J as given by (37). Obvious computations then give D( J) = ρD, T ( J) = ρ + T and S( J) = D + ρT with D( J), T ( J), S( J) respectively the determinant, trace and sum of principal minors of J, and D, T as given by (39). It follows therefore that the degree-three characteristic polynomial associated with (41) is P (λ) = (λ -ρ)P cc (λ), so that two eigenvalues of matrix J are the same as those of the 2-dimensional dynamic system described by the matrix J, while the third eigenvalue is equal to ρ.

M      c t+1 k t+1 z t+1      = Ñ      c t k t z t      + M   

Figure 1 :

 1 Figure 1: Local indeterminacy with GHH preferences when χ > χ.

Figure 3 :

 3 Figure 3: Local stability properties in the (Θ, χ, σ) space Note: The indeterminacy zone is the interior area delimited by the flip (upper curve) and the Hopf (lower curve) bifurcation loci.

T

  θ)(1+Θ) α cl +(1-α)(1-θ)χ+ (1-θ)(1-δ)Θ(1+χ) δ J 21 = -(1-θ)(1+Θ) βα 1 -βδα 1-θ , J 22 = (1-θ)(1+Θ) β 1+χ χ+α + (1-δ)β (1-θ)(1+Θ)(38)Using (34), we then derive after simplifications the characteristic polynomialP cc (λ) = λ 2 -T λ + D with D = D χ (σ) = = T χ (σ) = 1 + D χ (σ) + (1-θ) 2 δ[χ(1-α(1+Θ))-αΘ] (1-βδα 1-θ )-1-α 1+χ β(χ+α) σδα-(1-δ)(1-θ)Θ (1-βδα 1-θ )-1-α 1+χ(39)We conclude that when σ is varied over the interval [0, +∞), D and T are linked through a linear relationship D = ∆ χ (T ) = T S χ + C with a slopeS χ = (χ+α)(1-δ)Θ 2 χδ(1-α)(1+Θ)-(χ+α)Θ[1-(1-δ)(1+Θ)]In other words, ∆ χ (T ) corresponds to a half-line in the (T , D) plane with a starting point (T χ (+∞), D χ (+∞)) obtained when σ = +∞ such that:D χ (+∞) = (1-θ)(1+Θ)+θ β = D(+∞) T χ (+∞) = (1-θ)(1+Θ)+(θ+β) β = T (+∞) = 1 + D(+∞)It follows that when σ = +∞, P +∞ (1) = 0 and P +∞ (-1) = 2(1 + D(+∞)). Note also that D(+∞) > 1.

D

  χ (0) = lim σ→0 D χ (σ) = 1 β = D(0) T χ (0) = lim σ→0 T χ (σ) = 1 + 1 β -δ(1 -θ) χ(1-α(1+Θ))-αΘ (χ+α)θΘ

θΘ 2 ≤ 2 2 -

 222 It follows that limχ→+∞ T χ (σ H ) > -2 if and only if h(δ) ≡ δ 2 βΘ(1 -α(1 + Θ)) + δ (1 -β)(1 + Θ)(1 -α(1 + Θ)) + 4βΘ 4βΘ 2 < 0 Let Ω = (1 -β)(1 + Θ)(1 -α(1 + Θ)) + 4βΘ 2 2 + 4β 2 Θ 3 (1 -α(1 + Θ)). Therefore, there exists δ ∈ (0, 1) as given by δ= √ Ω-[(1-β)(1+Θ)(1-α(1+Θ))+4βΘ 2 ]2βΘ(1-α(1+Θ))

  Assuming that (36) holds, the matrix M is invertible with | M | = |M | and we get

  Leisure and Hospitality, and (8) Other Services.

			I. Volatility
	(absolute and relative standard deviations)
	(x)	Y	C	pI	L	L C	L I
	σ x	1.92 0.89 4.51 2.35 1.65 4.57
	σ x,Y =	1	0.47 2.35 1.22 0.86 2.38
			II. Persistence
		(first-order autocorrelation)
	(x)	Y	C	pI	L	L C	L I
	ρ x	0.91 0.86 0.91 0.93 0.93 0.92
			III. Covariations
	(contemporaneous correlations with output)
	(x)	Y	C	pI	L	L C	L I
	ρ x,y	1	0.87 0.98 0.89 0.87 0.88
		Sectoral labor comovements:
			corr(L

C , L I ) = 0.

96 Table 1: US Data -Cyclical properties quarterly.

Table 2 :

 2 Model properties, autonomous sunspots depends on how their expectations about the real return of investment are affected in this new economic environment (i.e., on the sunspot shock)

	Rel. std. dev.:
	Cons. (C)	0.47	3.57	12.41
	Inv. (pI)	2.35	12.17	44.50
	Hours (L C + L I )	1.22	0.75	0.14
	Hours cons (L C )	0.86	3.48	12.18
	Hours inv. (L I )	2.38	12.18	44.68
	Corr. with output
	Cons. (C)	0.87	0.38	0.30
	Inv. (pI)	0.98	-0.02	-0.20
	Hours (L C + L I )	0.89	1	1
	Hours cons. (L C )	0.87	0.31	0.24
	Hours inv. (L I )	0.88	-0.04	-0.22
	Corr. (L C , L I )	0.96	-0.96	-0.99

Table 3 .

 3 As can be observed, the model with generalized GHH preferences is now relatively successful, as it solves all the empirical puzzles traditionally associated with 2 sector Data preferences preferences

	Rel. std. dev.:
	Cons. (C)	0.47	0.80	3.42
	Inv. (pI)	2.35	2.30	14.94
	Hours (L C + L I )	1.22	0.75	0.14
	Hours cons (L C )	0.86	0.59	3.81
	Hours inv (L I )	2.38	2.09	14.43
	Corr. with output
	Cons. (C)	0.87	0.92	-0.36
	Inv. (pI)	0.98	0.87	0.61
	Hours: (L C + L I ) 0.89	1	1
	Hours cons (L C )	0.87	0.83	-0.55
	Hours inv (L I )	0.88	0.82	0.58
	Corr. (L C , L I )	0.96	0.39	-0.99

Table 3 :

 3 Model properties, correlated sunspots.

See Grandmont[START_REF] Grandmont | Nonlinear Difference Equations, Bifurcations and Chaos : An Introduction[END_REF][START_REF] Grandmont | Nonlinear Difference Equations, Bifurcations and Chaos : An Introduction[END_REF] for a simple presentation of bifurcation theory.

See e.g. Benhabib and Farmer[START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF],Harrison [16] and Guo and Harrison[START_REF] Guo | Indeterminacy with No-Income Effect Preferences and Sector-Specific Externalities[END_REF].

We do not consider externalities in the consumption good sector as they do not play any crucial role in the existence of multiple equilibria.

As discussed in section 3.2 below, this property results from the lack of income effect in labor supply.

In two-sector models with constant social returns à la Benhabib and Nishimura[START_REF] Benhabib | Indeterminacy and Sunspots with Constant Returns[END_REF] and GHH preferences, local indeterminacy remains compatible with an infinitely elastic labor supply provided σ is close enough to 0 (see Nishimura and Venditti[START_REF] Nishimura | Indeterminacy and Expectation-Driven Fluctuations with Non-Separable Preferences[END_REF]).

Observe also that in this cross-section, there is no Hopf bifurcation in the (χ, Θ) plane. The Hopf bifurcation emerges only when σ is sufficiently smaller than 1.

See e.g. Benhabib and Farmer[START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF],Harrison [16], Guo and Harisson[START_REF] Guo | Indeterminacy with No-Income Effect Preferences and Sector-Specific Externalities[END_REF].

We provide a new Appendix (available as supplementary material downloadable from the web page of the Journal of Economic Theory) in which we give precise details on how we built our dataset. All the series can also be downloaded as supplementary material.

Those data can easily be found in NIPA tables provided by the BEA. See in particular NIPA tables 1.1.4 and 1.1.5.

The Baxter analysis, based on HP-filtered annual data with smoothing coefficient λ = 400, reports relative volatility coefficient of hours worked in the consumption and investment sector of 0.87 and

To illustrate these mechanisms, impulse response functions to autonomous and correlated sunspot shocks, obtained under our benchmark calibration described below, are provided in the Appendix available as supplementary material.

Yet, as concerns the nonlinear system from which (27) is an approximation, one must ensure that the model remains in the basin of attraction of the stable steady state. This is particularly true for calibrations that imply that the model is near a subcritical Hopf bifurcation in the parameter space, since in this case the basin of attraction is delimited by an invariant closed curve surrounding the steady-state. Thus, strictly speaking, the distribution of shocks must be truncated to avoid that an extreme realization makes endogenous variables leave the stable neighborhood of the steady state.

As shown in Guo and Harrison[START_REF] Guo | Indeterminacy with No-Income Effect Preferences and Sector-Specific Externalities[END_REF], if the degree of increasing returns to scale in the investment sector is calibrated to Θ = 0.3 (instead of Θ = 0.33 retained here), the minimum value for χ consistent with indeterminacy jumps to χF ≈ 15.

Note that the results in the logarithmic consumption case differ from those reported by Guo and Harrison[START_REF] Guo | Indeterminacy with No-Income Effect Preferences and Sector-Specific Externalities[END_REF] because their calibration is slightly different and they did not filter their simulated data.

Yet, the ability of the model to actually solve this puzzle depends on the various internal transmission mechanisms (elasticity of labor supply, elasticity of intertemporal substitution in consumption, etc.) at work in the economy.

Grant (ANR-08-BLAN-0245-01) and by Japan Society for the Promotion of Science, Grant-in-Aid for Specially Promoted Research #23000001 and #(B)23330063. We are indebted to Jean-Michel Grandmont for many insightful comments and suggestions that helped us to greatly improve the successive versions of this paper.