Sunspot fluctuations in two-sector economies withheterogeneous agents∗

Stefano Bosi†, Francesco Magris‡, Alain Venditti§

Received: Accepted:

Abstract: We study a two-sector model with heterogeneous agents and borrowingconstraint on labor income. We show that the relative capital intensity differenceacross sectors is crucial for the conditions required to get indeterminacy and en-dogenous fluctuations. The main result shows that when the consumption goodis sufficiently capital intensive, local indeterminacy arises while the elasticities ofcapital-labor substitution in both sectors are slightly greater than unity and theelasticity of the offer curve is low enough. Locally indeterminate equilibria are thuscompatible with a low elasticity of intertemporal substitution in consumption anda low elasticity of the labor supply. As recently shown in empirical analysis, theseconditions appear to be in accordance with macroeconomic evidences.

Keywords: Heterogeneous agents, Borrowing constraint, Two-sector model, Inde-terminacy

JEL Classification Numbers: C61, E32, E41

∗We would like to thank R. Becker, J.P. Drugeon and an anonymous referee for usefulcomments and suggestions. The current version also benefited from a presentation at theconference “Public Economic Theory 04”, Beijing, August 2004.

†EPEE, Universite d’Evry, Evry, France‡EPEE, Universite d’Evry, Evry, France§CNRS - GREQAM, 2, rue de la Charite, 13002 Marseille, France. E-mail:

venditti@ehess.univ-mrs.fr

1 Introduction

We consider an infinite horizon model with heterogeneous agents and bor-rowing constraint on labor income in the spirit of Woodford (1986) andGrandmont et al. (1998). Contrary to the initial aggregate formulation,we assume two different technologies producing a consumption good and aninvestment good, respectively. We then appraise the local stability proper-ties of the economy as a function of technologies, i.e. the relative capitalintensity difference across sectors and the elasticity of the interest rate, andpreferences, i.e. the elasticity of the offer curve. Our aim is to give condi-tions for the existence of local indeterminacy when there are heterogeneousagents, some of them being financially constrained, i.e. being unable toborrow against labor income. The main result shows that when the con-sumption good is sufficiently capital intensive, local indeterminacy ariseswhen the elasticities of capital-labor substitution in both sectors are slightlygreater than unity and the elasticity of the offer curve is low enough. Asrecent empirical analyses show,1 these conditions appear to be compatiblewith macroeconomic evidences.

It is now well known that in a wide class of macrodynamic models lo-cally indeterminate equilibria and sunspot fluctuations easily arise. How-ever, most of the conditions require parameter values which differ from stan-dard empirical estimates. In one-sector models with homogeneous agents,either a large amount of increasing returns incompatible with the findingsof Basu and Fernald (1997), jointly with an unconventional slope for thelabor demand, or lower externalities but with extremely large elasticities ofintertemporal substitution in consumption and elasticities of labor supply,have to be considered.2 In one-sector models with heterogeneous agents avery low elasticity of capital-labor substitution is required.3 Finally, in two-sector models, local indeterminacy becomes compatible with constant socialreturns to scale, but requires a very high degree of intertemporal substitutionin consumption.4

In this paper, our strategy consists in starting from a two-sector frame-work but instead of assuming productive externalities, we consider two types

1See Duffy and Papageorgiou (2000) for estimations of the elasticity of capital-laborsubstitution and Takahashi et al. (2004) for some evaluation of the capital intensitydifference in the main developed countries.

2See Benhabib and Farmer (1994), Farmer and Guo (1994), Lloyd-Braga et al. (2006),Pintus (2006).

3See Grandmont et al. (1998).4See, for instance, Benhabib and Nishimura (1998), Benhabib et al. (2002).

1

of representative agents labeled, respectively, “capitalists” and “workers”,and some imperfection on the financial market. The agents are indeed dis-tinguished with respect to their preferences, their degree of impatience andtheir ability to borrow on the credit market: “capitalists” are describedby a logarithmic utility function, i.e. a unitary elasticity of intertempo-ral substitution in consumption. They do not work, consume from the re-turns on their savings and can borrow freely. On the contrary, “workers”are described by a general additively separable utility function defined overconsumption and labor, which is characterized by a higher discount ratethan the “capitalists”. They supply labor elastically and are subject everyperiod to a financial (borrowing) constraint, reflecting their difficulty to fi-nance the consumption good out of their wage income. The economy thenincludes one consumption good, and two assets, outside money, which is inconstant supply, and capital.

Under these hypotheses, in a neighborhood of the monetary steady state,capitalists end up holding the whole capital stock and no money (they fi-nance consumption and investment entirely out of capital income) whereasworkers are forced to convert in money balances the whole amount of theirwage bill.5 It is worth mentioning that our model is similar to the Beckerand Tsyganov (2002) two-sector model with heterogenous agents subject toborrowing constraints. The main differences lie on the facts that we considera monetary economy with a segmented financial market and we allow for anelastic labor supply. The segmentation assumption allows us to easily studythe local stability properties of equilibria which are highly sensitive to theelasticity of the offer curve, i.e. the elasticity of the labor supply. Howeverit has to be considered as a limit configuration based on the following welldocumented facts:- there is a concentration of capital ownership and a lack of access to thefinancial market for agents who do not own capital,6

- a significant proportion of households have limited borrowing opportuni-ties,7

- households with high income generally provide lower labor supply.8

5As initially proved in Becker (1980), the households with the lowest discount rate, i.e.“capitalists”, own all the capital in the long-run.

6As shown in Wolff (1998), in 1995 the richest 1% of US families as ranked by financialwealth own 47% of total household financial wealth, and the top 20% own 93%.

7See Jappelli (1990), Dias-Gimenez et al. (1997) who show that 25% of US householdsare liquidity constrained.

8Holtz-Eakin et al. (1993) find evidences to support the view that large inheritancesdecrease labor participation. See also Cheng and French (2000), Coronado and Perozek

2

It is finally useful to mention that the consideration of “capitalists” with apositive endogenous labor supply would only complexify the analysis with-out drastically altering our main conclusions.9

We show that when the consumption good sector is significantly cap-ital intensive, indeterminacy occurs under mild conditions based on someelasticities of capital-labor substitution slightly larger than unity. WhileCobb-Douglas technologies are widely used in growth theory, recent papershave questioned the empirical relevance of this specification. Duffy and Pa-pageorgiou (2000) for instance consider a panel of 82 countries over a 28-yearperiod to estimate a CES production function specification. They find thatfor the entire sample of countries the assumption of unitary elasticity ofsubstitution may be rejected. Moreover, dividing the sample of countries upinto several subsamples, they find that capital and labor have an elasticity ofsubstitution significantly greater than unity, i.e. contained in [1.01, 1.18](seetable 3, p. 109), for the richest group of countries. Considering an aggregateelasticity of capital-labor substitution defined as a weighted sum of the sec-toral elasticities, the analysis of a CES example shows that our conditionsfall into these estimates. We also prove that local indeterminacy is compati-ble with a large set of values for the elasticity of the offer curve. This impliesthat contrary to the one and two-sector models with productive external-ities, sunspot fluctuations may arise while the elasticity of intertemporalsubstitution in consumption and the elasticity of the labor supply remainlow. In addition we show that in such a configuration, the change of stabilityoccurs only through a flip bifurcation, giving rise to period-two cycles.

The rest of the paper is organized as follows. In Section 2 we describethe behavior of the agents. Section 3 is devoted to the characterization ofthe technology, while in Section 4 we introduce the intertemporal equilib-rium and prove the existence of a unique stationary solution. Section 5presents the characteristic polynomial and the main arguments of the geo-

(2003) who consider data from the stock market boom of the 90’s to exhibit a negativeeffect of wealth on the labor supply.

9Barinci and Cheron (2001) consider a calibrated version of the aggregate Woodford(1986) model augmented to include productive external effects. They show the existence oflocal indeterminacy with an infinite elasticity of labor for the workers, a unitary elasticityof intertemporal substitution in consumption for the capitalists and a Cobb-Douglas tech-nology characterized by a degree of increasing returns between 19.85% and 30%. Barinciet al. (2003) extend this model allowing capitalists (i.e. the unconstrained households) tosupply a variable quantity of labor. The dynamical system then becomes 3-dimensionalbut local indeterminacy is obtained under the same calibration for the fundamentals, ex-cept that the elasticity of the labor supply for unconstrained households is fixed at a lowvalue (1/3) and external effects are lowered to a degree of increasing returns less than 5%.

3

metrical method used to study the local dynamics and bifurcations. Section6 presents the main results. In Section 7 we calibrate the model using a CESspecification. Section 8 discusses the plausibility of our results and provideseconomic intuitions. In Section 9 we conclude the paper. All the proofs aregathered in a final Appendix.

2 Agents and intertemporal optimization

The economy is populated by two types of infinite-lived agents, workers andcapitalists, each of them identical within their own type. Workers consume,supply endogenously labor and are subject to a financial constraint prevent-ing them from borrowing against current as well as future labor income.Capitalists conversely do not work and therefore are subject only to thebudget constraint.

2.1 Capitalists

Capitalists maximize a logarithmic intertemporal utility function∞∑

t=0

βt ln cct

where cc denotes their consumption and β ∈ (0, 1) their discount factor.Since capitalists do not earn any labor income, they are subject to thebudget constraint

cct + pt

[kc

t+1 − (1− δ) kct

]+ qtM

ct+1 ≤ rtk

ct + qtM

ct

where p stands for the price of investment, k for the capital, δ ∈ [0, 1] forthe depreciation rate of capital, M for the money balances, q for the priceof money and r for the interest rate in terms of the consumption good. Inaddition, the usual non-negativity constraints kc

t ≥ 0 and M ct ≥ 0, must be

satisfied. The first order conditions state as follows

cct+1

cct≥ β pt+1(1−δ)+rt+1

pt,

cct+1

cct≥ β qt+1

qt(1)

and hold with equality if kct > 0 and M c

t > 0. We shall focus in the sequelon the case where

pt+1(1−δ)+rt+1

pt> qt+1

qt(2)

holds at all dates. The gross rate of return on capital is then higher thanthe profitability of money holding, and capitalists decide to hold only capital

4

and no money.10 Their optimal policy for all t ≥ 1 takes the explicit form

kct+1 = β (1− δ + rt/pt) kc

t (3)

meanwhile consumption choice is given by cct = (1− β) [pt (1− δ) + rt] kc

t .11

2.2 Workers

Workers maximize their lifetime utility function∞∑

t=0

γt [u (cwt )− γv (lt)] (4)

where cw stands for consumption, u for the per-period utility function, l forlabor supply, v for the per-period disutility of labor and γ ∈ (0, 1) for thediscount factor. The functions u and v satisfy the following properties:

Assumption 2.1. u(c) and v(l) are Cr, with r large enough, for, respec-tively, c > 0 and 0 ≤ l < l∗, where l∗ > 0 is the (possibly infinite) work-ers’ endowment of labor. They satisfy u′ (c) > 0, u′′ (c) < 0, v′ (l) > 0,v′′ (l) > 0 with limc→0 u′ (c) = +∞, limc→+∞ u′ (c) = 0, liml→0 v′ (l) = 0and liml→l∗ v′ (l) = +∞. Consumption and leisure are assumed to be grosssubstitutes, i.e. u′ (c) + cu′′ (c) > 0.

Notice that gross substitutability is nothing but assuming that the elasticityof intertemporal substitution in consumption, εc = −u′(c)/c′′(c)c, is largerthan unity. This restriction implies that the labor supply is an increasingfunction of the real wage. We also assume that workers are more impatientthat capitalists:

Assumption 2.2. γ < β

Workers are subject to the dynamic budget constraint

cwt + pt

[kw

t+1 − (1− δ) kwt

]+ qtM

wt+1 ≤ wtlt + rtk

wt + qtM

wt (5)

where all the variables are those introduced in the capitalists’ budget con-straint with the exception of the wage w in terms of units of consumptiongood. In addition, workers face the borrowing constraint

10According to Becker (1980), the capital income distribution is determined in the long-run steady state by the lowest discount rate.

11A more general utility function for capitalists could be considered as in Barinci (2001).However, with non logarithmic preferences, the optimal policy cannot be explicitly com-puted and we have to deal with the Euler equation which defines an implicit 3-dimensionaldynamical system. In order to reach more tractable conclusions we still consider the sameassumption as in Woodford (1986).

5

cwt + pt

[kw

t+1 − (1− δ) kwt

]≤ rtk

wt + qtM

wt (6)

reflecting their difficulty to borrow against future labor income, and the non-negativity conditions on asset holding kw

t ≥ 0 and Mwt ≥ 0. Straightforward

computations show that at the optimum kwt = 0 if and only if

u′ (cwt ) > γu′

(cwt+1

) pt+1(1−δ)+rt+1

pt(7)

We shall focus in the following on the case where (7) holds at all dates.Workers then choose not to hold capital (kw

t = 0) . Moreover, we assume aconstant money supply, i.e. Mt = M > 0 for every t. Since capitalists donot hold any money we get Mw

t = M and the budget constraint implies

cwt = qtM (8)

We then derive from the liquidity constraint that workers hold the quantityof money balances

qtM = wtlt (9)employers pay to them in exchange to their labor services at the end of eachperiod. From (9) we obtain the following relationship between the growthfactor of labor income and the growth factor of the money price:

wtltwt+1lt+1

=qt

qt+1(10)

Workers maximize (4) subject to (8) and (9). This yields, taking into ac-count (10), the first order condition reflecting the standard trade-off betweenconsumption and labor

U(cwt+1

)= V (lt) (11)

with U (c) ≡ cu′ (c) and V (l) ≡ lv′ (l) .

3 Technology

We assume that the consumption good Y 0 and the investment good Y 1

in each period t are produced by two different constant returns to scaletechnologies employing capital and labor as productive inputs:

Y i = F i(Ki, Li

)

i = 0, 1, where(K0, L0

)and

(K1, L1

)denote the amount of inputs used

respectively in consumption and investment sectors. At equilibrium we have:

K0 + K1 = K = N ckc, L0 + L1 = L = Nwl

K and L stand for aggregate capital and labor, N c and Nw denote respec-tively the number of capitalists and workers, kc is the stock of capital held

6

by each capitalist and l is the labor supply of each worker. All the previousvariables can be normalized with respect to the size Nw of the labor force:

yi ≡ Y i/Nw

ki ≡ Ki/Nw, li ≡ Li/Nw

k ≡ K/Nw = N ckc/Nw, l ≡ L/Nw

i = 0, 1. Observe that at the equilibrium k0 + k1 = k and l0 + l1 = l.Without loss of generality assume a constant ratio n between capitalists

and workers: k = kcN c/Nw ≡ nkc. Homogeneity of production functionsimplies:

yi = f i(ki, li

)

where f i ≡ F i/Nw is the per-worker production in sector i = 0, 1.

Assumption 3.1. Each production function f i : R2+ → R+, i = 0, 1, is Cr,

with r large enough, increasing in each argument, concave, homogeneous ofdegree one and such that for any x > 0, limy→0 f i

1 (y, x) = limy→0 f i2 (x, y) =

+∞, limy→+∞ f i1 (y, x) = limy→+∞ f i

2 (x, y) = 0.

For any given(k, y1, l

), we define a temporary equilibrium by solving the

following problem of optimal allocation of factors between the two sectors:

max{k0,k1,l0,l1}

f0(k0, l0

)

s.t. y1 ≤ f1(k1, l1

)

k0 + k1 ≤ k

l0 + l1 ≤ l

k0, k1, l0, l1 ≥ 0

(12)

The associated Lagrangian is

L = f0(k0, l0

)+ p

[f1

(k1, l1

)− y1

]+ r

[k − k0 − k1

]+ w

[l − l0 − l1

]

Solving the corresponding first order conditions give optimal demand func-tions for capital and labor, namely k0

(k, y1, l

), l0

(k, y1, l

), k1

(k, y1, l

)and

l1(k, y1, l

). The resulting value function

T(k, y1, l

)= f0

(k0

(k, y1, l

), l0

(k, y1, l

))

is called the social production function and describes the frontier of theproduction possibility set. The constant returns to scale of technologiesimply that T

(k, y1, l

)is homogeneous of degree one and thus concave non-

strictly. We will assume in the following that T(k, y1, l

)is at least C2.

It is easy to show from the first order conditions that the rental rate ofcapital, the price of investment good and the wage rate satisfy

7

T1(k, y1, l

)= r, T2

(k, y1, l

)= −p, T3

(k, y1, l

)= w

Concavity of T(k, y1, l

)implies

T11(k, y1, l

)≤ 0, T22

(k, y1, l

)≤ 0, T33

(k, y1, l

)≤ 0

However the signs of the cross derivatives are not obvious. Consider thusthe relative capital intensity difference across sectors defined as follows

b ≡ a01

(a11a01− a10

a00

)(13)

witha00 ≡ l0/y0, a10 ≡ k0/y0, a01 ≡ l1/y1, a11 ≡ k1/y1

the input coefficients in each sector. Then b > 0 if and only if the investmentgood is capital intensive. It is shown in Bosi et al. (2005) that

T12 = −T11b, T31 = −T11a ≥ 0, T32 = T11ab

with a ≡ k0/l0 > 0 the capital-labor ratio in the consumption good sector.It follows therefore that the sign of T12 and T32 crucially depends on thesign of the capital intensity difference across sectors b.12 It is also easy toshow that T22

(k, y1, l

)and T33

(k, y1, l

)may be written as

T22 = T11b2, T33 = T11a

2

These expressions will be useful to study the dynamical properties of theequilibrium paths.

4 Intertemporal equilibrium and steady state

Since within each type all agents are identical we can focus on symmetricequilibrium. Coupling the capital accumulation equation (3) and the work-ers’ first order condition (11) with equilibrium conditions in factor markets,and recalling that cw

t+1 = wtlt, we can introduce the intertemporal equilib-rium with perfect foresight in terms of k and l. In each period t, kt is apredetermined variable (in order to simplify notation, we will set c = cw

and k = kc) and k0 > 0 is the initial stock of physical equipment.

Definition 4.1. For any given initial capital stock k0 > 0, an intertemporalequilibrium with perfect foresight is a sequence {kt+1, lt}∞t=0 > 0 satisfying

kt+1 − β

[1− δ − T1(kt,kt+1−(1−δ)kt,lt)

T2(kt,kt+1−(1−δ)kt,lt)

]kt = 0

U (T3 (kt, kt+1 − (1− δ) kt, lt) lt)− V (lt−1) = 0(14)

12As initially proved in a two-sector optimal growth model with inelastic labor by Ben-habib and Nishimura (1985), the cross derivative T12 is positive if and only if the invest-ment good is capital intensive, i.e. b > 0.

8

together with the transversality condition13

limt→+∞

βt(pt/ct)kt+1 = 0 (15)

Before going through the stability analysis of system (14), our first con-cern is to prove the existence of a stationary solution.

Definition 4.2. An interior steady state equilibrium is a stationary se-quence {kt+1, lt}∞t=0 = {k∗, l∗}∞t=0 > 0 that satisfies the two-dimensionalsystem

−T1(k,δk,l)

T2(k,δk,l) = β−1 − (1− δ)

U (T3 (k, δk, l) l) = V (l)(16)

We can show that Assumptions 2.1, 2.2 and 3.1 guarantee the existence andthe uniqueness of the steady state.

Proposition 4.1. Under Assumptions 2.1, 2.2 and 3.1, there exists aunique steady state (k∗, l∗) > 0 solution of (16).

It is easy to verify that at the steady state 1 − δ + r/p = 1/β > 1.Moreover, under Assumption 2.2 we get 1−δ+r/p < 1/γ. It follows thereforethat conditions (2) and (7) are satisfied in a neighborhood of (k∗, l∗).

5 Characteristic polynomial and geometric method

Our aim consists now in analyzing the dynamics of system (14) aroundits stationary solution as well as along bifurcations. Let us introduce theexpressions of the following elasticities all evaluated at the steady state: theelasticity of the interest rate

εr ≡ −T11kT1∈ (0,+∞)

the elasticity of the real wageεw ≡ −T33l

T3∈ (0,+∞)

and the elasticity of the offer curve λ (l) ≡ U−1 (V (l))

ε ≡ V ′lU ′c ∈ (1,+∞)

It is straightforward to show that the elasticity of the labor supply with re-spect to the real wage is equal to εlw = 1/(ε−1). Notice that considering the

13Michel (1990) shows that equations (14) together with the transversality condition(15) provide necessary and sufficient conditions for an equilibrium path.

9

elasticity of intertemporal substitution in consumption εc = −u′(c)/c′′(c)cand the inverse of the elasticity of the marginal disutility of labor εl =v′(l)/v′′(l)l, the elasticity ε may also be expressed as

ε =(1 + 1

εl

)/

(1− 1

εc

)(17)

It follows that ε = 1, i.e. εlw = +∞, if and only if εl = +∞, while ε = +∞,i.e. εlw = 0, if and only if either εl = 0 or εc = 1. Therefore, the elasticityof the labor supply εlw may be equivalently appraised through εl.

Denoting θ ≡ β−1−(1− δ), let us also define the share of capital in totalincome s ≡ rk/

(T + py1

)∈ (0, 1) and the relative capital intensity across

sectors b ∈ (−∞, 1/θ),14 again evaluated at the steady state. Linearizingsystem (14) around (k∗, l∗) yields to the following Proposition:

Proposition 5.1. Under Assumptions 2.1, 2.2 and 3.1, the characteristicpolynomial is P (λ) = λ2 − Tλ + D with

T = 1 + D − (1− ε) εrβθ(1−θb)(1−δb)

1−εr[(1−δb)2s/(1−s)+βθ(1−θb)b] (18)

andD = ε 1−εrβθ(1−θb)[1+(1−δ)b]

1−εr[(1−δb)2s/(1−s)+βθ(1−θb)b] (19)

As in Grandmont et al. (1998), we study the variations of the trace T andthe determinant D in the (T,D) plane as the elasticity of the offer curve ismade to vary continuously within (1,+∞). Notice indeed that both T andD are linear with respect to ε. When the latter covers the interval (1,+∞),the locus of points (T (ε) , D (ε)) consists in a half-line ∆ (T ) with slope

ψ = 1− εrβθ(1−θb)(1−δb)1−εrβθ(1−θb)b (20)

Notice also that the origin (T1, D1) of ∆ (T ) lies on the line T = 1+D. Themethod simply consists in locating ∆ (T ) in the plane (T,D) , which meansto study its origin and its slope. Specifically, if T and D lie in the interiorof the triangle ABC depicted in Fig. 1, the stationary solution is a sink,hence locally indeterminate. In the opposite case, it is locally determinate:it is either a saddle when |T | > |1 + D| , or a source otherwise.

This geometrical method may also be exploited to characterize bifurca-tions. Indeed, as it is shown in Fig. 1, when ∆ (T ) goes through the lineD = −T−1 (at ε = εF ) one eigenvalue is equal to −1 and we get a flip bifur-cation. When ∆ (T ) intersects the interior of the segment BC (at ε = εH)

14The fact that b must be lower than 1/θ comes from the positivity constraint on theprice of investment p (see Bosi et al. (2005)).

10

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$$

$$

$$

$$

$$

$$

AT

D

B

%%%%%%%%%%

&&

&&

&&

&&

&& ∆′

∆′′

εH

εF

Figure 1: Geometrical analysis.

the modulus of the complex conjugate eigenvalues is one and the system un-dergoes a Hopf bifurcation.15 As shown in Proposition 5.1 , for given valuesof β, δ and thus θ, the half-line ∆ (T ) depends on the technological param-eters: the share of capital in total income s, the capital intensity differenceacross sectors b, the elasticity of the interest rate εr and the elasticity of thereal wage εw. It is easy to show that all these parameters are linked throughthe following relationship:

εw = εr (1− δb)2 s1−s

Thus, for some fixed value of s, we can vary independently both εr andb. Our aim is now to characterize the origin (T1, D1) ≡ (T (1) , D (1)), theslope ψ and the endpoint (T (∞) , D (∞)) of the half-line ∆ (T ) when b andεr are made to vary within their domain of definition. By observing thatexpressions (18) and (19) are first-order polynomials in εr, it is easier toproceed by first fixing the value of b and then by considering variations ofεr. By repeating this procedure with different values of b, we will be ableto appraise the whole evolution of the local dynamics and bifurcations. Ofcourse, b must fall within the range compatible with positive prices. As it isshown in details in Bosi et al. (2005), this requires b < 1/θ. The followingLemma shows that two types of geometrical configurations, associated withdifferent properties of the slope ψ, may be exhibited:

Lemma 5.1. Under Assumptions 2.1, 2.2 and 3.1, the following properties15The uniqueness of the steady state rules out the occurrence of transcritical bifurca-

tions.

11

hold:i) The slope satisfies ψ (εr) ∈ (1− δ + 1/b, 1) for any εr > 0;ii) limε→+∞D (ε) = + (−)∞ if and only if D1 > (<) 0.

Lemma 5.1-i) implies that if b < −1/ (1− δ), the slope ψ (εr) is includedin the interval (0, 1) for any εr ≥ 0, while if b ∈ (−1/ (1− δ) , 1/θ), the slopemay be positive or negative depending on the value of εr. More precisely,it can be shown that for large values of εr the slope ψ is negative, while itis positive when εr admits lower values. Lemma 5.1-ii)emphasizes the factthat the localization of the endpoint (T (∞) , D (∞)) of the half-line ∆ (T )depends upon the value of its initial point (T1, D1). As shown in Fig. 1,when D1 ∈ (0, 1), then D (∞) = +∞ and as illustrated by ∆′, the steadystate is locally indeterminate when ε ∈ (1, εH) while a Hopf bifurcationoccurs when ε crosses εH . Similarly, when D1 ∈ (−1, 0), then D (∞) = −∞and as illustrated by ∆′′, the steady state is locally indeterminate whenε ∈ (1, εF ) while a flip bifurcation occurs when ε crosses εF .

6 Main results

As shown by Lemma 5.1, two different configurations for the capital intensitydifference b have a priori to be considered, namely b < −1/ (1− δ) and b ∈(−1/ (1− δ) , 1/θ). However, we will only focus on the case b < −1/ (1− δ)for the following two main reasons:

i) First it has been shown recently by Takahashi et al. (2004) that overthe last three decades, the main developed countries are characterized by acapital-intensive aggregate consumption good sector, i.e. b < 0.

ii) Second, in the case b ∈ (−1/ (1− δ) , 1/θ), the consideration of pa-rameter values compatible with quarterly data imply that b has to belong toa small interval around 0. In other words, this configuration is equivalent toa small perturbation of the aggregate formulation considered by Grandmontet al. (1998). A simple intuition therefore suggests that the occurrence oflocal indeterminacy will require low elasticities of capital-labor substitution(i.e. less than the share of capital in total income), at least in one of thetwo sectors.16 But such a restriction cannot be supported by the recentempirical estimates of Duffy and Papageorgiou (2000). We then introducethe following assumption:

Assumption 6.1. b < −1/ (1− δ)16The detailed results on this case are provided in the GREQAM Working paper avail-

able at: http://www.vcharite.univ-mrs.fr/pp/venditti/publications.htm

12

Under Assumption 6.1, Lemma 5.1 implies that the slope ψ is positiveand less than one. We know from equation (19) in Proposition 5.1 thatD (ε) is a linear function of ε. To get local indeterminacy, we then needto find conditions for D1 ∈ (−1, 1). To this end, exploiting Lemma 5.1allows to show that there exist some critical values for, respectively, theshare of capital in total income s∗ and the elasticity of interest rate ε∗r ,such that if s ≤ s∗ or εr ∈ (0, ε∗r), the slope of ∆ (T ) is positive and lowerthan one and either D1 > 0 (and limε→+∞D (ε) = +∞) or D1 < −1 (andlimε→+∞D (ε) = −∞). As a consequence ∆ (T ) remains in the saddle pointregion and the steady state is always locally determinate. Conversely, whens > s∗ and εr > ε∗r , as it is shown in Fig. 2, one has D1 ∈ (−1, 0) andlimε→+∞D (ε) = −∞. It follows that for low elasticities of the offer curveε, the half-line ∆ (T ) crosses the interior of the triangle ABC and thereforethe steady state is locally indeterminate. Then ∆ (T ) intersects the lineD = −T − 1 at ε = εF and a flip bifurcation generically occurs. Eventually,for ε > εF the steady state becomes a saddle, thus locally determinate.

!

"

##

##

##

##

##

##

#

C

$$

$$

$$

$$

$$

$$

AT

D

B

%%%%%%%%%%

∆εF

Figure 2: b < −1/ (1− δ) with s > s∗ and εr > ε∗r .

All these results are summarized in the following Proposition:

Proposition 6.1. Let Assumptions 2.1, 2.2, 3.1 and 6.1 hold. Then thereexist s∗ ∈ (0, 1) and ε∗r > 0, such that:

(i) If s ≤ s∗ or εr ∈ (0, ε∗r), then the steady state is a saddle (locallydeterminate) for all ε > 0.

(ii) If s > s∗ and εr > ε∗r, then there exists εF > 1 such that the steadystate is a sink (locally indeterminate) when ε ∈ (1, εF ) and a saddle whenε > εF . A flip bifurcation generically occurs at ε = εF .

13

Proposition 6.1 shows that local indeterminacy requires a large enoughshare of capital in total income, a large enough elasticity of interest rate anda low enough elasticity of the offer curve. We know that in the one-sectormodel studied by Grandmont et al. (1998) a necessary condition to getindeterminacy is an elasticity of capital-labor substitution lower than theshare of capital. In order to understand the implications of our results interms of the elasticity of capital-labor substitution, we need to know how tointerpret the elasticity of the interest rate in terms of the elasticity of capital-labor substitution. The main difference with the one-sector formulation thenappears: in a two-sector model, each technology is characterized by someparticular substitutability properties and we need to take into account twodistinct elasticities of capital-labor substitution. Denoting σi the elasticityof sector i = 0, 1 and using the recent contribution of Drugeon (2004), wemay however define an aggregate elasticity of substitution between capitaland labor, denoted Σ, which is obtained as a weighted sum of the sectoralelasticities σi,17 and then derive an expression for the elasticity of the interestrate εr, namely:

Σ = y0+py1

py1ky0

(py1k0l0σ0 + y0k1l1σ1

), εr =

(l0

y0

)2 w(y0+py1)Σ

(21)

In the particular case with identical elasticities across sectors σ0 = σ1 = σ,we find as in the one-sector model that the elasticity of the interest ratewill be high enough provided σ is low enough.18 However, as soon as thereare some asymmetries between sectors, larger elasticities of capital-laborsubstitution may become compatible with a large elasticity of the interestrate. Indeed εr also depends on the factors and output prices, the amountof factors used in each sectors, and the production levels. More preciseresults remain difficult to obtain at this point of the analysis since the capitalintensity difference b, the outputs and the amounts of capital and labor usedin each sector are functions of the elasticities of capital-labor substitutionσ0 and σ1. Numerical simulations in a CES economy will give additionalconclusions in Section 7. In particular, it will be shown that contrary to theone-sector formulation, indeterminacy is possible under sectoral elasticities

17As shown in Drugeon (2004), the weighted sum is defined with the aggregate sharesof capital and labor income in national income, the aggregate shares of the consump-sion good and the investment good in the national product, and the sectoral shares ofcapital and labor income in total production cost. The expression (21) is obtained afterstraightforward simplifications.

18Notice that in a one-sector model we have p = 1, y0 = y1 = y/2, l0 = l1 = 1/2,k0 = k1 = k/2, σ0 = σ1 = σ and py1k0l0σ0 + y0k1l1σ1 = ykσ, so that equations (21)become Σ = σ and εr = (1− s)/σ.

14

of capital-labor substitution leading to an aggregate elasticity Σ which isconsistent with recent empirical estimates.

Consider now the condition on the elasticity of the offer curve. As shownby (17), ε is defined from the elasticity of intertemporal substitution inconsumption εc and the inverse of the elasticity of the marginal disutility oflabor εl. The restriction ε ∈ (1, εF ) in Proposition 6.1 can thus be stated as

εc [1 + εl(1− εF )] < −εlεF (22)and to be satisfied it requires

1 + εl(1− εF ) < 0 ⇔ εl > 1εF−1 ≡ εl (23)

i.e. a large enough elasticity of the labor supply. Condition (22) is then

εc > εlεFεl(εF−1)−1 ≡ εc (24)

i.e. a large enough elasticity of intertemporal substitution in consumption.However, depending on the value of the critical bound εF , local indetermi-nacy may be compatible with low values for these elasticities. In particular,straightforward computations show that for some finite value of εl satisfying(23), the lower bound on εc as given by (24) may remain close to 1.

7 A CES economy

To provide some quantitative insights of the plausibility of indeterminacy inour model, we consider the classical example of a CES economy. We retainthe following functional forms

u(cw) = (cw)1−η1/(1− η1), v(l) = l1+η2/(1 + η2) (25)

with η1 ∈ (0, 1), η2 ≥ 0 for preferences and

f0(k0, l0) =[α10(k0)−ρ0 + α00(l0)−ρ0

]−1/ρ0

f1(k1, l1) =[α11(k1)−ρ1 + α01(l1)−ρ1

]−1/ρ1

with α00 +α10 = α01 +α11 = 1, ρ0, ρ1 > −1 for technologies. It follows obvi-ously that the elasticity of the offer curve is ε = (1 + η2)/(1− η1) ≥ 1 whilethe elasticities of capital-labor substitution are σi = 1/(1 + ρi), i = 0, 1.We calibrate the structural parameters in a standard way compatible withquarterly data by choosing β = 0.99, δ = 0.025 and thus θ ≈ 0.0351. Itfollows that the bound for the capital intensity difference b given in As-sumption 6.1 is −1/(1−δ) ≈ −1.025. In the RBC literature, Cobb-Douglastechnologies are usually considered and standard calibrations are based oncapital shares in the consumption and investment good sectors which are

15

such that α10,α11 ∈ (0.2, 0.6). Notice that with CES technologies, the cap-ital shares now also depends on the parameters ρi as this clearly appears inthe formulation of the capital intensity difference b given in Appendix 10.5.

Let us consider a capital intensive consumption good with α00 = 0.4,α11 = 0.6. Assuming that ρ0 ∈ [−0.64,−0.46] and ρ1 ∈ [−0.152,−0.13],the associated relative capital intensity difference satisfies Assumption 6.1.The corresponding elasticities of substitution in each sector are thus σ0 ∈[1.85, 2.77] and σ1 ∈ [1.146, 1.179]. In order to provide comparisons with theestimates for the richest group of countries given in Duffy and Papageorgiou(2000), namely [1.01, 1.18] (see table 3, p. 109),19 we have to consider theaggregate elasticity of capital-labor substitution Σ defined by (21). Weeasily get from the previous values of σ0 and σ1 that Σ has to belong to theinterval [1.116, 1.178] which perfectly fits with the estimates of Duffy andPapageorgiou. Numerical computations then show that the steady stateis locally indeterminate for all values of the elasticity of the offer curvesuch that ε ∈ (1, εF ), with εF ∈ (4.02, 278.4) depending on the values of ρi

considered, and when ε crosses εF from below a flip bifurcation occurs, givingrise to period-two cycles. More precisely if we set ρ0 = −0.57 (i.e. σ0 = 2.32)and ρ1 = −0.15 (i.e. σ1 = 1.176) we get Σ ≈ 1.173 and εF ≈ 73.1.20

It is worthwhile remarking that within such an example indeterminacyarises in correspondence to an aggregate elasticity of factor substitutionvery close to the unitary Cobb-Douglas case, condition known for eliminat-ing such a phenomenon in the one-sector framework. Notice also that thelower bound on εl as defined by (23) is εl ≈ 1.38%. It follows that if, in ac-cordance with stylized facts,21 we choose low values for the elasticity of thelabor supply, i.e. for instance εl ∈ (0.1, 4), the lower bound on εc given by(24) is εc ∈ (1.017, 1.177). Therefore, contrary to one and two-sector modelswith productive externalities, local indeterminacy remains compatible witha small elasticity of intertemporal substitution in consumption and a lowelasticity of the labor supply.22

19The larger bound of the interval is obtained with raw labor while the lower bound isobtained with human capital adjuted labor. Since we do not include human capital in ourmodel, we consider the estimations based on raw labor.

20All these numerical simulations are obtained from expressions given in Appendix 10.5.21See Blundell and McCurdy (1999).22In one-sector models with aggregate externalities, local indeterminacy is obtained

with an infinitely elastic labor supply and a large elasticity of intertemporal substitution(see Lloyd-Braga et al. (2006), Pintus (2006)). In two-sector models with sector-specificexternalities, local indeterminacy is obtained with a close to infinite elasticity of intertem-poral substitution in consumption (see Benhabib and Nishimura (1998), Benhabib et al.(2002)). All these restrictions are not consistent with recent macroeconomic evidences.

16

Remark : As shown in Barinci (2001), if a non logarithmic utility func-tion for capitalists is considered, the Euler equations define an implicit 3-dimensional dynamical system parameterized by the elasticity of intertempo-ral substitution in consumption of capitalists, denoted η. Actually, Barinciand Drugeon (2005) prove that when η = 1, the dynamical system can bereduced to dimension 2 since β−1 > 1 happens to be a trivial characteristicroot. It follows that the characteristic roots of the general 3-dimensionaldynamical system are continuous with respect to η. As a result, all ourprevious conclusions obtained with η = 1 are robust to small perturbationsof η around 1.

8 Plausibility of sunspots and economic intuitions

The numerical simulations clearly show that local indeterminacy arises withplausible values for the elasticities of capital-labor substitution, the elastic-ity of intertemporal substitution in consumption and the elasticity of thelabor supply when the consumption good is sufficiently capital-intensive.This configuration appears to be consistent with national accounting dataof the main developed countries over the last three decades. Indeed in a re-cent contribution, Takahashi et al. (2004) aggregate sectoral data in order toget a two-sector representation of the Japanese, U.S. and German economiesfrom 1955 to 2000. They find that, while the U.S. and German economies arecharacterized by a capital-intensive aggregate consumption good sector overthe whole period, the Japanese economy experiences a capital-intensity re-versal in 1975, the aggregate consumption good sector being labor-intensivebefore and capital-intensive since then. These findings may be explainedby the fact that, within developed countries, consumption goods with anincreasing amount of technological content have become a growth engine.Our results then provide a theoretical background to explain why developedcountries are characterized by a strong macroeconomic volatility based onexpectations-driven fluctuations.

It remains now to understand the economic mechanisms at the core ofthese results. Actually, if the consumption good is sufficiently capital in-tensive, there exist in our model two main forces from which endogenousfluctuations originate: a pure technological mechanism based on factor al-locations across sectors, and a monetary mechanism based on the cash-in-advance constraint. As initially shown in Benhabib and Nishimura (1985),the technological mechanism refers to the Rybczinsky effect. Starting fromone equilibrium paths, consider an instantaneous increase in the capital

17

stock kt. This results in two opposing forces:- Since the consumption good is more capital intensive than the in-

vestment good, the trade-off in production becomes more favorable to theconsumption good. Moreover, the Rybczinsky effect implies a decrease ofthe output of the capital good yt. This tends to lower the investment andthe capital stock in the next period kt+1.

- In the next period the decrease of kt+1 implies again through the Ry-bczinsky effect an increase of the output of the capital good yt+1. Thismechanism is explained by the fact that the decrease of kt+1 improves thetrade-off in production in favor of the investment good which is relativelyless intensive in capital. Therefore this tends to increase the investment andthe capital stock in period t + 2, kt+2.

The monetary mechanism, as shown Bosi et al. (2005), is based on theagents’ expectations. Let us consider again the CES instantaneous utilityfunction as given in (25). Equations (1) and (7) then become respectively

cct+1/cc

t = βit+1,(cwt+1/cw

t

)η1 > γit+1 (26)

withit+1 ≡ [rt+1 + (1− δ) pt+1]/pt

the nominal interest factor. Starting from one equilibrium path, let us try toconstruct an alternative equilibrium. For this purpose, assume that agentscollectively revise their expectations in reaction to a given sunspot signal andcome to believe that the nominal interest factor will undergo an appreciation.It follows that to re-establish (26), future consumptions cc

t+1 and cwt+1 must

be driven up. When mixed with the technological effect, this expectationbecomes self-fulfilling and a new equilibrium path can be defined.

The simultaneous consideration of technological and monetary effectstherefore generates sunspot fluctuations while the elasticities of capital-laborsubstitution, the elasticity of intertemporal substitution in consumption andthe elasticity of the labor supply may be fixed at some standard values. How-ever, all this story crucially depends on the assumption of a consumptiongood which is sufficiently capital intensive. If on the contrary, either the con-sumption good is weakly capital intensive or the investment good is capitalintensive, the technological effect does not occur as in one-sector models. Inorder to generate sunspot fluctuations, the monetary mechanism thereforerequires extreme parameterizations for the structural elasticities, i.e. lowelasticities of capital-labor substitution (lower than the share of capital), orlarge elasticities of intertemporal substitution in consumption.

18

9 Concluding remarks

In this paper we consider a two-sector infinite horizon model with hetero-geneous agents and borrowing constraint on labor income. We show thatthe relative capital intensity across sectors plays a relevant role with respectto the emergence of indeterminacy and deterministic as well as sunspotfluctuations. The main result is obtained when the consumption sector issignificantly more capital intensive than the investment sector. Indeed lo-cal indeterminacy comes about for elasticities of capital-labor substitutionslightly greater than unity and for a broad range of values for the elasticityof the offer curve which are compatible with plausible values for the elas-ticity of intertemporal substitution in consumption and the elasticity of thelabor supply.

These conditions appear to be consistent with recent macroeconomicempirical evidences. On the one side, concerning the capital intensity differ-ence, Takahashi et al. (2004) find that over the last three decades the mainOECD countries are characterized by a capital-intensive consumption goodsector. On the other side, concerning the substitutability properties, Duffyand Papageorgiou (2000) show that capital and labor have an elasticity ofsubstitution greater than unity in the richest group of countries.

The main criticisms that can be addressed to this model concern thefacts that the financial market is segmented and that “capitalists” do notsupply labor. Even though such assumptions can be considered as a limitconfiguration based on the evidences suggesting that a significant proportionof households have limited borrowing constraints, and that households withhigh income generally provide lower labor supply, it would be more realisticto consider that both types of households work and hold simultaneously thetwo types of assets. The former point has been considered by Barinci etal. (2003) within an aggregate model and, as suggested in the introduction,could be also considered within our two-sector framework without drasticallyaltering the main conclusions. The latter extension is more difficult to carryout since it requires to introduce two important modifications within our ba-sic framework. The first consists in considering as in Bosi and Magris (2002)a partial borrowing constraint. If we also assume that both types of house-holds have the same discount factor, such a partial constraint forces workersto finance one part of their total expenditures from money and the otherfrom savings. The second modification consists in introducing as in Bosiet al. (2005) a fractional cash-in-advance constraint that forces capitaliststo hold money in order to finance part of their consumption expenditures.This work is left for future research.

19

10 Appendix

10.1 Proof of Proposition 4.1

Since T (k, δk, l) is homogeneous of degree one, (16) can be rewritten as

−T1(κ,δκ,1)T2(κ,δκ,1) = β−1 − (1− δ) , U (T3 (κ, δκ, 1) l) = V (l) (27)

with κ ≡ k/l. Consider the first equation in (27): this is equivalent to theequation defining the stationary capital stock of a two-sector optimal growthmodel with inelastic labor supply. Then, the proof of Theorem 3.1 in Beckerand Tsyganov (2002) applies and there exists a unique solution κ∗ of thefirst equation of (27). Consider now the second equation in (27) evaluated atκ∗ with the fact that c∗ = wl∗. In view of the definition of U and V , rewriteit as T3 (κ∗, δκ∗, 1) u′ (T3 (κ∗, δκ∗, 1) l) = v′ (l). Then, under Assumption 1,it is easy to verify that such an equation possesses a unique solution l∗.

10.2 Proof of Proposition 5.1

By linearizing (14), we obtain the following expression for the Jacobian J∗:

J∗ =[

−bεw a (1− εw)1− bϑεr −aϑεr

]−1 [− [1 + (1− δ) b] εw aε

1− [1 + (1− δ) b]ϑεr 0

](28)

with ϑ ≡ βθ (1− θb), from which we get the expressions for the trace T andthe determinant D.

10.3 Proof of Lemma 5.1

To prove point i) of the Lemma, it is sufficient to look at expression (20).Point ii) follows directly from a check of (19).

10.4 Proof of Proposition 6.1

Assume that b < −1/(1− δ). When εr moves from zero to +∞, ψ decreasescontinuously from one to 1− δ + 1/b ∈ (0, 1− δ). This in particular meansthat ψ ∈ (0, 1) for all εr ≥ 0. We also notice that from (19) with ε = 1,T1 = 1 + D1 so that the origin of ∆ belongs to D = T − 1.

Now, let us define the following useful formulas

z ≡ 1−δb1−θb

1βθ

s1−s , z1 ≡ 1−εrβθ(1−θb)b

εrβθ(1−θb)(1−δb) > 1, z2 ≡ 2−εrβθ(1−θb)[1+(2−δ)b]εrβθ(1−θb)(1−δb) > z1

Then one easily verifies that ∂D1/∂εr < 0 if and only if z < 1. Still, in theinterval under study for b, straightforward computations show that:

- when z < 1 then D1 ∈ (0, 1) for every εr and D (+∞) = +∞,

20

- when 1 < z < z1 then D1 > 1 and D (+∞) = +∞,- when z1 < z < z2 then D1 < −1 and D (+∞) = −∞,- when z > z2, then D1 ∈ (−1, 0) and D (+∞) = −∞.Recall that, since b < −1/ (1− δ), the slope ψ is positive and less than

one. Therefore, we face two possible subcases:(i) if z < z2, then the ∆-line does not cross the triangle ABC and the

steady state is a saddle;(ii) if z > z2, then ∆-line crosses the triangle ABC and, as shown in Fig.

2, there exists εF > 1 such that the steady state is a sink when ε ∈ (1, εF )and a saddle when ε > εF .

One could easily prove that z ≤ z2 if s ≤ s∗ with

s∗ ≡ βθ(1−θb)[1+(2−δ)b]

βθ(1−θb)[1+(2−δ)b]−(1−δb)2∈ (0, 1)

and that when s > s∗, then z < z2 if and only if εr ∈ (0, ε∗r), with

ε∗r ≡2(1−s)

s(1−δb)2+(1−s)βθ(1−θb)[1+(2−δ)b]> 0

10.5 Computations for the CES economy

Using the recent contribution of Nishimura and Venditti (2004), we get

k∗/0∗ =

“α10α01α00α11

” 11+ρ0

((α11θ )

ρ11+ρ1 −α11α01

) 1+ρ1ρ1(1+ρ0)

1−δb ≡ κ∗

k1/0∗ = κ∗[1− δ

(α11θ

) 11+ρ1

]

l1/0∗ = κ∗δ(

α11θ

) 11+ρ1

((α11θ )

ρ11+ρ1 −α11

α01

)− 1ρ1

b =(

α11θ

) 11+ρ1

[1−

(α10α01α00α11

) 11+ρ0

((α11θ )

ρ11+ρ1 −α11

α01

) ρ1−ρ0ρ1(1+ρ0)

]

r = α10

[α10 + α00

(α10α01α00α11

) ρ01+ρ0

((α11θ )

ρ11+ρ1 −α11

α01

) ρ0(1+ρ1)ρ1(1+ρ0)

]− 1+ρ0ρ0

w = rα01α11

((α11θ )

ρ11+ρ1 −α11

α01

) 1+ρ1ρ1

From this we derive p = r/θ, k0 = k∗ − k1, l0 = 0∗ − l1 and

y0 =[α10(k0)−ρ0 + α00(l0)−ρ0

]−1/ρ0 , y1 =[α11(k1)−ρ1 + α01(l1)−ρ1

]−1/ρ1

All the numerical simulations are finally based on these expressions.

21

References

[1] Barinci, J.P.: Factors substitutability, heterogeneity and endogenousfluctuations in a finance constrained economy. Economic Theory 17,181-195 (2001).

[2] Barinci, J.P., Drugeon, J.P.: On outward-looking comparison utility,heterogeneous preferences and the third dimension: a geometric per-spective. WP 05-07, University of Evry - EPEE (2005).

[3] Barinci, J.P., Cheron, A.: Sunspots and the business cycle in a financeCcnstrained economy. Journal of Economic Theory 97, 30-49 (2001).

[4] Barinci, J.P., Cheron, A., Langot, F.: Liquidity constraints, heteroge-neous households and sunspots fluctuations. WP 02-17, University ofEvry - EPEE (2003).

[5] Basu, S., Fernald, J.: Returns to scale in US production: estimates andimplications. Journal of Political Economy 105, 249-283 (1997).

[6] Becker, R.: On the long-run steady state in a simple dynamic model ofequilibrium with heterogeneous households. Quarterly Journal of Eco-nomics 95, 375-382 (1980).

[7] Becker, R., Tsyganov, E.: Ramsey equilibrium in a two-sector modelwith heterogeneous households. Journal of Economic Theory 105, 188-225 (2002).

[8] Benhabib, J., Farmer, R.: Indeterminacy and increasing returns. Jour-nal of Economic Theory 3, 19-41 (1994).

[9] Benhabib, J., Nishimura, K.: Competitive equilibrium eycles. Journalof Economic Theory 35, 284-306 (1985).

[10] Benhabib, J., Nishimura, K.: Indeterminacy and sunspots with con-stant returns. Journal of Economic Theory 81, 58-96 (1998).

[11] Benhabib, J., Nishimura, K., Venditti, A.: Indeterminacy and cycles intwo-sector discrete-time model. Economic Theory 20, 217-235 (2002).

[12] Blundell, R., McCurdy, T.: Labour supply: a review of alternativeapproaches. In Ashenfelter, O., Card, D. (eds.) Handbook of LaborEconomics. Amsterdam: North-Holland 1999.

22

[13] Bosi, S., Magris, F.: Endogenous business cycles: capital-labor sub-stitution and liquidity constraint. Journal of Economic Dynamics andControl 26, 1901-1926 (2002).

[14] Bosi, S., Magris, F., Venditti, A.: Competitive equilibrium cycles withendogenous labor. Journal of Mathematical Economics 41, 325-349(2005).

[15] Bosi, S., Magris, M., Venditti, A.: Multiple equilibria in a cash-in-advance two-sector economy. International Journal of Economic Theory1, 131-149 (2005).

[16] Cheng, I., French, E.: The effect of the run-up in the stock market onlabor supply. Federal Reserve Bank of Chicago Economic Perspectives24, 48-65 (2000).

[17] Coronado, J., Perozek, M.: Wealth effects and the consumption ofleisure: retirement decisions during the stock market boom of the 1990s.Board of Governors of the Federal Reserve System, Finance and Eco-nomics Discussion Series, 2003-20 (2003).

[18] Dias-Gimenez, J., Quadrini, V., Rios-Rull, J-V.: Dimension of inequal-ity: facts on the U.S. distributions of earnings, income and wealth. Fed-eral Reserve Bank of Minneapolis Quarterly Review 21, 3-21 (1997).

[19] Drugeon, J.P.: On consumptions, inputs and outputs substitutabilitiesand the evanescence of optimal cycles. Journal of Difference Equationsand Applications 10, 473-487 (2004).

[20] Duffy, J., Papageorgiou, C.: A cross-country empirical investigation ofthe aggregate production function specification. Journal of EconomicGrowth 5, 87-120 (2000).

[21] Farmer, R., Guo, J.T.: Real business cycles and the animal spirit hy-pothesis. Journal of Economic Theory 63, 42-73 (1994).

[22] Grandmont, J.M., Pintus, P. de Vilder, R.: Capital-labor substitutionand competitive nonlinear endogenous business cycles. Journal of Eco-nomic Theory 80, 14-59 (1998).

[23] Holtz-Eakin, D., Joulfaian, D., Rosen, H.: The Carnegie conjecture:some empirical evidence. Quarterly Journal of Economics 108, 413-435(1993).

23

[24] Jappelli, T.: Who is Credit Constrained in the US Economy ? Quar-terly Journal of Economics 105, 219-234 (1990).

[25] Lloyd-Braga, T., Nourry, C., Venditti, A.: Indeterminacy with smallexternalities: the role of non-separable preferences. Forthcoming in In-ternational Journal of Economic Theory (2006).

[26] Michel, P.: Some clarifications on the tranversality condition. Econo-metrica 58, 705-723 (1990).

[27] Nishimura, K., Venditti, A.: Capital depreciation, factors substitutabil-ity and indeterminacy. Journal of Difference Equations and Applica-tions 10, 1153-1169 (2004).

[28] Pintus, P.: Indeterminacy with almost constant returns to scale:capital-labor substitution matters. Economic Theory 28, 633-649(2006).

[29] Takahashi, H., Mashiyama, K., Sakagami, T.: Measuring capital inten-sity in the postwar Japanese economy. Meiji Gakuin University WorkingPaper, Tokyo (2004).

[30] Wolff, E.: Recent trends in the size distribution of household wealth.Journal of Economic Perspectives 12, 131-150 (1998).

[31] Woodford, M.: Stationary sunspot equilibria in a finance constrainedeconomy. Journal of Economic Theory 40, 128-137 (1986).

24