Sunspot fluctuations in two-sector economies with heterogeneous agents

Transcription

1 Sunspot fluctuations in two-sector economies with heterogeneous agents Stefano Bosi, Francesco Magris, Alain Venditti Received: Accepted: Abstract: We study a two-sector model with heterogeneous agents and borrowing constraint on labor income. We show that the relative capital intensity difference across sectors is crucial for the conditions required to get indeterminacy and endogenous fluctuations. The main result shows that when the consumption good is sufficiently capital intensive, local indeterminacy arises while the elasticities of capital-labor substitution in both sectors are slightly greater than unity and the elasticity of the offer curve is low enough. Locally indeterminate equilibria are thus compatible with a low elasticity of intertemporal substitution in consumption and a low elasticity of the labor supply. As recently shown in empirical analysis, these conditions appear to be in accordance with macroeconomic evidences. Keywords: Heterogeneous agents, Borrowing constraint, Two-sector model, Indeterminacy JEL Classification Numbers: C61, E32, E41 We would like to thank R. Becker, J.P. Drugeon and an anonymous referee for useful comments and suggestions. The current version also benefited from a presentation at the conference Public Economic Theory 04, Beijing, August EPEE, Université d Evry, Evry, France EPEE, Université d Evry, Evry, France CNRS - GREQAM, 2, rue de la Charité, Marseille, France. venditti@ehess.univ-mrs.fr

2 1 Introduction We consider an infinite horizon model with heterogeneous agents and borrowing constraint on labor income in the spirit of Woodford (1986) and Grandmont et al. (1998). Contrary to the initial aggregate formulation, we assume two different technologies producing a consumption good and an investment good, respectively. We then appraise the local stability properties of the economy as a function of technologies, i.e. the relative capital intensity difference across sectors and the elasticity of the interest rate, and preferences, i.e. the elasticity of the offer curve. Our aim is to give conditions for the existence of local indeterminacy when there are heterogeneous agents, some of them being financially constrained, i.e. being unable to borrow against labor income. The main result shows that when the consumption good is sufficiently capital intensive, local indeterminacy arises when the elasticities of capital-labor substitution in both sectors are slightly greater than unity and the elasticity of the offer curve is low enough. As recent empirical analyses show, 1 these conditions appear to be compatible with macroeconomic evidences. It is now well known that in a wide class of macrodynamic models locally indeterminate equilibria and sunspot fluctuations easily arise. However, most of the conditions require parameter values which differ from standard empirical estimates. In one-sector models with homogeneous agents, either a large amount of increasing returns incompatible with the findings of Basu and Fernald (1997), jointly with an unconventional slope for the labor demand, or lower externalities but with extremely large elasticities of intertemporal substitution in consumption and elasticities of labor supply, have to be considered. 2 In one-sector models with heterogeneous agents a very low elasticity of capital-labor substitution is required. 3 Finally, in twosector models, local indeterminacy becomes compatible with constant social returns to scale, but requires a very high degree of intertemporal substitution in consumption. 4 In this paper, our strategy consists in starting from a two-sector framework but instead of assuming productive externalities, we consider two types 1 See Duffy and Papageorgiou (2000) for estimations of the elasticity of capital-labor substitution and Takahashi et al. (2004) for some evaluation of the capital intensity difference in the main developed countries. 2 See Benhabib and Farmer (1994), Farmer and Guo (1994), Lloyd-Braga et al. (2006), Pintus (2006). 3 See Grandmont et al. (1998). 4 See, for instance, Benhabib and Nishimura (1998), Benhabib et al. (2002). 1

3 of representative agents labeled, respectively, capitalists and workers, and some imperfection on the financial market. The agents are indeed distinguished with respect to their preferences, their degree of impatience and their ability to borrow on the credit market: capitalists are described by a logarithmic utility function, i.e. a unitary elasticity of intertemporal substitution in consumption. They do not work, consume from the returns on their savings and can borrow freely. On the contrary, workers are described by a general additively separable utility function defined over consumption and labor, which is characterized by a higher discount rate than the capitalists. They supply labor elastically and are subject every period to a financial (borrowing) constraint, reflecting their difficulty to finance the consumption good out of their wage income. The economy then includes one consumption good, and two assets, outside money, which is in constant 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 finance consumption and investment entirely out of capital income) whereas workers are forced to convert in money balances the whole amount of their wage bill. 5 It is worth mentioning that our model is similar to the Becker and Tsyganov (2002) two-sector model with heterogenous agents subject to borrowing constraints. The main differences lie on the facts that we consider a monetary economy with a segmented financial market and we allow for an elastic labor supply. The segmentation assumption allows us to easily study the local stability properties of equilibria which are highly sensitive to the elasticity of the offer curve, i.e. the elasticity of the labor supply. However it has to be considered as a limit configuration based on the following well documented facts: - there is a concentration of capital ownership and a lack of access to the financial market for agents who do not own capital, 6 - a significant proportion of households have limited borrowing opportunities, 7 - households with high income generally provide lower labor supply. 8 5 As initially proved in Becker (1980), the households with the lowest discount rate, i.e. capitalists, own all the capital in the long-run. 6 As shown in Wolff (1998), in 1995 the richest 1% of US families as ranked by financial wealth own 47% of total household financial wealth, and the top 20% own 93%. 7 See Jappelli (1990), Dias-Gimenez et al. (1997) who show that 25% of US households are liquidity constrained. 8 Holtz-Eakin et al. (1993) find evidences to support the view that large inheritances decrease labor participation. See also Cheng and French (2000), Coronado and Perozek 2

4 It is finally useful to mention that the consideration of capitalists with a positive endogenous labor supply would only complexify the analysis without drastically altering our main conclusions. 9 We show that when the consumption good sector is significantly capital intensive, indeterminacy occurs under mild conditions based on some elasticities of capital-labor substitution slightly larger than unity. While Cobb-Douglas technologies are widely used in growth theory, recent papers have questioned the empirical relevance of this specification. Duffy and Papageorgiou (2000) for instance consider a panel of 82 countries over a 28-year period to estimate a CES production function specification. They find that for the entire sample of countries the assumption of unitary elasticity of substitution may be rejected. Moreover, dividing the sample of countries up into several subsamples, they find that capital and labor have an elasticity of substitution significantly greater than unity, i.e. contained in [1.01, 1.18](see table 3, p. 109), for the richest group of countries. Considering an aggregate elasticity of capital-labor substitution defined as a weighted sum of the sectoral elasticities, the analysis of a CES example shows that our conditions fall into these estimates. We also prove that local indeterminacy is compatible with a large set of values for the elasticity of the offer curve. This implies that contrary to the one and two-sector models with productive externalities, sunspot fluctuations may arise while the elasticity of intertemporal substitution in consumption and the elasticity of the labor supply remain low. In addition we show that in such a configuration, the change of stability occurs only through a flip bifurcation, giving rise to period-two cycles. The rest of the paper is organized as follows. In Section 2 we describe the behavior of the agents. Section 3 is devoted to the characterization of the technology, while in Section 4 we introduce the intertemporal equilibrium and prove the existence of a unique stationary solution. Section 5 presents 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 negative effect of wealth on the labor supply. 9 Barinci and Chéron (2001) consider a calibrated version of the aggregate Woodford (1986) model augmented to include productive external effects. They show the existence of local indeterminacy with an infinite elasticity of labor for the workers, a unitary elasticity of intertemporal substitution in consumption for the capitalists and a Cobb-Douglas technology characterized by a degree of increasing returns between 19.85% and 30%. Barinci et al. (2003) extend this model allowing capitalists (i.e. the unconstrained households) to supply a variable quantity of labor. The dynamical system then becomes 3-dimensional but local indeterminacy is obtained under the same calibration for the fundamentals, except that the elasticity of the labor supply for unconstrained households is fixed at a low value (1/3) and external effects are lowered to a degree of increasing returns less than 5%. 3

5 metrical method used to study the local dynamics and bifurcations. Section 6 presents the main results. In Section 7 we calibrate the model using a CES specification. Section 8 discusses the plausibility of our results and provides economic intuitions. In Section 9 we conclude the paper. All the proofs are gathered in a final Appendix. 2 Agents and intertemporal optimization The economy is populated by two types of infinite-lived agents, workers and capitalists, each of them identical within their own type. Workers consume, supply endogenously labor and are subject to a financial constraint preventing them from borrowing against current as well as future labor income. Capitalists conversely do not work and therefore are subject only to the budget constraint. 2.1 Capitalists Capitalists maximize a logarithmic intertemporal utility function β t ln c c t t=0 where c c denotes their consumption and β (0, 1) their discount factor. Since capitalists do not earn any labor income, they are subject to the budget constraint c c t + p t [ k c t+1 (1 δ) k c t ] + qt M c t+1 r t k c t + q t M c t where p stands for the price of investment, k for the capital, δ [0, 1] for the depreciation rate of capital, M for the money balances, q for the price of money and r for the interest rate in terms of the consumption good. In addition, the usual non-negativity constraints k c t 0 and M c t 0, must be satisfied. The first order conditions state as follows c c t+1 c c t β p t+1(1 δ)+r t+1 p t, c c t+1 c c t β q t+1 q t (1) and hold with equality if kt c > 0 and Mt c > 0. We shall focus in the sequel on the case where p t+1 (1 δ)+r t+1 p t > q t+1 q t (2) holds at all dates. The gross rate of return on capital is then higher than the profitability of money holding, and capitalists decide to hold only capital 4

6 and no money. 10 Their optimal policy for all t 1 takes the explicit form k c t+1 = β (1 δ + r t /p t ) k c t (3) meanwhile consumption choice is given by c c t = (1 β) [p t (1 δ) + r t ] k c t Workers Workers maximize their lifetime utility function γ t [u (c w t ) γv (l t )] (4) t=0 where c w stands for consumption, u for the per-period utility function, l for labor supply, v for the per-period disutility of labor and γ (0, 1) for the discount factor. The functions u and v satisfy the following properties: Assumption 2.1. u(c) and v(l) are C r, with r large enough, for, respectively, c > 0 and 0 l < l, where l > 0 is the (possibly infinite) workers endowment of labor. They satisfy u (c) > 0, u (c) < 0, v (l) > 0, v (l) > 0 with lim c 0 u (c) = +, lim c + u (c) = 0, lim l 0 v (l) = 0 and lim l l v (l) = +. Consumption and leisure are assumed to be gross substitutes, i.e. u (c) + cu (c) > 0. Notice that gross substitutability is nothing but assuming that the elasticity of intertemporal substitution in consumption, ɛ c = u (c)/c (c)c, is larger than unity. This restriction implies that the labor supply is an increasing function of the real wage. We also assume that workers are more impatient that capitalists: Assumption 2.2. γ < β Workers are subject to the dynamic budget constraint c w [ t + p t k w t+1 (1 δ) kt w ] + qt Mt+1 w w t l t + r t kt w + q t Mt w (5) where all the variables are those introduced in the capitalists budget constraint with the exception of the wage w in terms of units of consumption good. In addition, workers face the borrowing constraint 10 According to Becker (1980), the capital income distribution is determined in the longrun steady state by the lowest discount rate. 11 A more general utility function for capitalists could be considered as in Barinci (2001). However, with non logarithmic preferences, the optimal policy cannot be explicitly computed and we have to deal with the Euler equation which defines an implicit 3-dimensional dynamical system. In order to reach more tractable conclusions we still consider the same assumption as in Woodford (1986). 5

7 c w t [ + p t k w t+1 (1 δ) kt w ] rt kt w + q t Mt w (6) reflecting their difficulty to borrow against future labor income, and the nonnegativity conditions on asset holding kt w 0 and Mt w 0. Straightforward computations show that at the optimum kt w = 0 if and only if u (c w t ) > γu ( c w t+1) pt+1 (1 δ)+r t+1 p t (7) We shall focus in the following on the case where (7) holds at all dates. Workers then choose not to hold capital (kt w = 0). Moreover, we assume a constant money supply, i.e. M t = M > 0 for every t. Since capitalists do not hold any money we get Mt w = M and the budget constraint implies c w t = q t M (8) We then derive from the liquidity constraint that workers hold the quantity of money balances q t M = wt l t (9) employers pay to them in exchange to their labor services at the end of each period. From (9) we obtain the following relationship between the growth factor of labor income and the growth factor of the money price: w t l t w t+1 l t+1 = q t q t+1 (10) Workers maximize (4) subject to (8) and (9). This yields, taking into account (10), the first order condition reflecting the standard trade-off between consumption and labor U ( c w t+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 scale technologies employing capital and labor as productive inputs: Y i = F i ( K i, L i) i = 0, 1, where ( K 0, L 0) and ( K 1, L 1) denote the amount of inputs used respectively in consumption and investment sectors. At equilibrium we have: K 0 + K 1 = K = N c k c, L 0 + L 1 = L = N w l K and L stand for aggregate capital and labor, N c and N w denote respectively the number of capitalists and workers, k c is the stock of capital held 6

8 by each capitalist and l is the labor supply of each worker. All the previous variables can be normalized with respect to the size N w of the labor force: y i Y i /N w k i K i /N w, l i L i /N w k K/N w = N c k c /N w, l L/N w i = 0, 1. Observe that at the equilibrium k 0 + k 1 = k and l 0 + l 1 = l. Without loss of generality assume a constant ratio n between capitalists and workers: k = k c N c /N w nk c. Homogeneity of production functions implies: y i = f i ( k i, l i) where f i F i /N w is the per-worker production in sector i = 0, 1. Assumption 3.1. Each production function f i : R+ 2 R +, i = 0, 1, is C r, with r large enough, increasing in each argument, concave, homogeneous of degree one and such that for any x > 0, lim y 0 f1 i (y, x) = lim y 0 f2 i (x, y) = +, lim y + f1 i (y, x) = lim y + f2 i (x, y) = 0. For any given ( k, y 1, l ), we define a temporary equilibrium by solving the following problem of optimal allocation of factors between the two sectors: f 0 ( k 0, l 0) max {k 0,k 1,l 0,l 1 } The associated Lagrangian is s.t. y 1 f 1 ( k 1, l 1) k 0 + k 1 k l 0 + l 1 l k 0, k 1, l 0, l 1 0 L = f 0 ( k 0, l 0) + p [ f 1 ( k 1, l 1) y 1] + r [ k k 0 k 1] + w [ l l 0 l 1] (12) Solving the corresponding first order conditions give optimal demand functions for capital and labor, namely k 0 ( k, y 1, l ), l 0 ( k, y 1, l ), k 1 ( k, y 1, l ) and l 1 ( k, y 1, l ). The resulting value function T ( k, y 1, l ) = f 0 ( k 0 ( k, y 1, l ), l 0 ( k, y 1, l )) is called the social production function and describes the frontier of the production possibility set. The constant returns to scale of technologies imply that T ( k, y 1, l ) is homogeneous of degree one and thus concave nonstrictly. We will assume in the following that T ( k, y 1, l ) is at least C 2. It is easy to show from the first order conditions that the rental rate of capital, the price of investment good and the wage rate satisfy 7

9 T 1 ( k, y 1, l ) = r, T 2 ( k, y 1, l ) = p, T 3 ( k, y 1, l ) = w Concavity of T ( k, y 1, l ) implies T 11 ( k, y 1, l ) 0, T 22 ( k, y 1, l ) 0, T 33 ( k, y 1, l ) 0 However the signs of the cross derivatives are not obvious. Consider thus the relative capital intensity difference across sectors defined as follows ( ) b a a11 01 a 01 a 10 a 00 (13) with a 00 l 0 /y 0, a 10 k 0 /y 0, a 01 l 1 /y 1, a 11 k 1 /y 1 the input coefficients in each sector. Then b > 0 if and only if the investment good is capital intensive. It is shown in Bosi et al. (2005) that T 12 = T 11 b, T 31 = T 11 a 0, T 32 = T 11 ab with a k 0 /l 0 > 0 the capital-labor ratio in the consumption good sector. It follows therefore that the sign of T 12 and T 32 crucially depends on the sign of the capital ( intensity difference across sectors b. 12 It is also easy to show that T 22 k, y 1, l ) ( and T 33 k, y 1, l ) may be written as T 22 = T 11 b 2, T 33 = T 11 a 2 These expressions will be useful to study the dynamical properties of the equilibrium paths. 4 Intertemporal equilibrium and steady state Since within each type all agents are identical we can focus on symmetric equilibrium. Coupling the capital accumulation equation (3) and the workers first order condition (11) with equilibrium conditions in factor markets, and recalling that c w t+1 = w tl t, we can introduce the intertemporal equilibrium with perfect foresight in terms of k and l. In each period t, k t is a predetermined variable (in order to simplify notation, we will set c = c w and k = k c ) and k 0 > 0 is the initial stock of physical equipment. Definition 4.1. For any given initial capital stock k 0 > 0, an intertemporal equilibrium with perfect foresight is a sequence {k t+1, l t } t=0 [ ] > 0 satisfying k t+1 β 1 δ T 1(k t,k t+1 (1 δ)k t,l t) T 2 (k t,k t+1 (1 δ)k t,l t) k t = 0 (14) U (T 3 (k t, k t+1 (1 δ) k t, l t ) l t ) V (l t 1 ) = 0 12 As initially proved in a two-sector optimal growth model with inelastic labor by Benhabib and Nishimura (1985), the cross derivative T 12 is positive if and only if the investment good is capital intensive, i.e. b > 0. 8

10 together with the transversality condition 13 lim t + βt (p t /c t )k t+1 = 0 (15) Before going through the stability analysis of system (14), our first concern is to prove the existence of a stationary solution. Definition 4.2. An interior steady state equilibrium is a stationary sequence {k t+1, l t } t=0 = {k, l } t=0 > 0 that satisfies the two-dimensional system T 1(k,δk,l) T 2 (k,δk,l) = β 1 (1 δ) (16) U (T 3 (k, δk, l) l) = V (l) We can show that Assumptions 2.1, 2.2 and 3.1 guarantee the existence and the uniqueness of the steady state. Proposition 4.1. Under Assumptions 2.1, 2.2 and 3.1, there exists a unique 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 therefore that 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) around its stationary solution as well as along bifurcations. Let us introduce the expressions of the following elasticities all evaluated at the steady state: the elasticity of the interest rate ε r T 11k T 1 the elasticity of the real wage (0, + ) ε w T 33l T 3 (0, + ) and the elasticity of the offer curve λ (l) U 1 (V (l)) ε V l U c (1, + ) It is straightforward to show that the elasticity of the labor supply with respect to the real wage is equal to ɛ lw = 1/(ε 1). Notice that considering the 13 Michel (1990) shows that equations (14) together with the transversality condition (15) provide necessary and sufficient conditions for an equilibrium path. 9

11 elasticity of intertemporal substitution in consumption ɛ c = u (c)/c (c)c and the inverse of the elasticity of the marginal disutility of labor ɛ l = v (l)/v (l)l, the elasticity ε may also be expressed as ) ) ε = (1 + 1 / (1 1 (17) ɛl ɛc It follows that ε = 1, i.e. ɛ lw = +, if and only if ɛ l = +, while ε = +, i.e. ɛ lw = 0, if and only if either ɛ l = 0 or ɛ c = 1. Therefore, the elasticity of the labor supply ɛ lw may be equivalently appraised through ɛ l. Denoting θ β 1 (1 δ), let us also define the share of capital in total income s rk/ ( T + py 1) (0, 1) and the relative capital intensity across sectors b (, 1/θ), 14 again evaluated at the steady state. Linearizing system (14) around (k, l ) yields to the following Proposition: Proposition 5.1. Under Assumptions 2.1, 2.2 and 3.1, the characteristic polynomial is P (λ) = λ 2 T λ + D with and T = 1 + D (1 ε) D = ε ε rβθ(1 θb)(1 δb) 1 ε r[(1 δb) 2 s/(1 s)+βθ(1 θb)b] 1 ε rβθ(1 θb)[1+(1 δ)b] 1 ε r[(1 δb) 2 s/(1 s)+βθ(1 θb)b] (18) (19) As in Grandmont et al. (1998), we study the variations of the trace T and the determinant D in the (T, D) plane as the elasticity of the offer curve is made to vary continuously within (1, + ). Notice indeed that both T and D 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 ε rβθ(1 θb)b (20) Notice also that the origin (T 1, D 1 ) of (T ) lies on the line T = 1 + D. The method simply consists in locating (T ) in the plane (T, D), which means to study its origin and its slope. Specifically, if T and D lie in the interior of 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 bifurcations. Indeed, as it is shown in Fig. 1, when (T ) goes through the line D = T 1 (at ε = ε F ) one eigenvalue is equal to 1 and we get a flip bifurcation. When (T ) intersects the interior of the segment BC (at ε = ε H ) 14 The fact that b must be lower than 1/θ comes from the positivity constraint on the price of investment p (see Bosi et al. (2005)). 10

12 D ε B H C A ε F Figure 1: Geometrical analysis. the modulus of the complex conjugate eigenvalues is one and the system undergoes a Hopf bifurcation. 15 As shown in Proposition 5.1, for given values of β, δ and thus θ, the half-line (T ) depends on the technological parameters: the share of capital in total income s, the capital intensity difference across sectors b, the elasticity of the interest rate ε r and the elasticity of the real wage ε w. It is easy to show that all these parameters are linked through the following relationship: ε w = ε r (1 δb) 2 s 1 s Thus, for some fixed value of s, we can vary independently both ε r and b. Our aim is now to characterize the origin (T 1, D 1 ) (T (1), D (1)), the slope ψ and the endpoint (T ( ), D ( )) of the half-line (T ) when b and ε r are made to vary within their domain of definition. By observing that expressions (18) and (19) are first-order polynomials in ε r, it is easier to proceed by first fixing the value of b and then by considering variations of ε r. By repeating this procedure with different values of b, we will be able to appraise the whole evolution of the local dynamics and bifurcations. Of course, b must fall within the range compatible with positive prices. As it is shown in details in Bosi et al. (2005), this requires b < 1/θ. The following Lemma shows that two types of geometrical configurations, associated with different properties of the slope ψ, may be exhibited: Lemma 5.1. Under Assumptions 2.1, 2.2 and 3.1, the following properties 15 The uniqueness of the steady state rules out the occurrence of transcritical bifurcations. T 11

13 hold: i) The slope satisfies ψ (ε r ) (1 δ + 1/b, 1) for any ε r > 0; ii) lim ε + D (ε) = + ( ) if and only if D 1 > (<) 0. Lemma 5.1-i) implies that if b < 1/ (1 δ), the slope ψ (ε r ) is included in the interval (0, 1) for any ε r 0, while if b ( 1/ (1 δ), 1/θ), the slope may be positive or negative depending on the value of ε r. More precisely, it can be shown that for large values of ε r the slope ψ is negative, while it is positive when ε r admits lower values. Lemma 5.1-ii)emphasizes the fact that the localization of the endpoint (T ( ), D ( )) of the half-line (T ) depends upon the value of its initial point (T 1, D 1 ). As shown in Fig. 1, when D 1 (0, 1), then D ( ) = + and as illustrated by, the steady state is locally indeterminate when ε (1, ε H ) while a Hopf bifurcation occurs when ε crosses ε H. Similarly, when D 1 ( 1, 0), then D ( ) = and as illustrated by, the steady state is locally indeterminate when ε (1, ε F ) while a flip bifurcation occurs when ε crosses ε F. 6 Main results As shown by Lemma 5.1, two different configurations for the capital intensity difference 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 over the last three decades, the main developed countries are characterized by a capital-intensive aggregate consumption good sector, i.e. b < 0. ii) Second, in the case b ( 1/ (1 δ), 1/θ), the consideration of parameter values compatible with quarterly data imply that b has to belong to a small interval around 0. In other words, this configuration is equivalent to a small perturbation of the aggregate formulation considered by Grandmont et al. (1998). A simple intuition therefore suggests that the occurrence of local 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 the two sectors. 16 But such a restriction cannot be supported by the recent empirical estimates of Duffy and Papageorgiou (2000). We then introduce the following assumption: Assumption 6.1. b < 1/ (1 δ) 16 The detailed results on this case are provided in the GREQAM Working paper available at: 12

14 Under Assumption 6.1, Lemma 5.1 implies that the slope ψ is positive and less than one. We know from equation (19) in Proposition 5.1 that D (ε) is a linear function of ε. To get local indeterminacy, we then need to find conditions for D 1 ( 1, 1). To this end, exploiting Lemma 5.1 allows to show that there exist some critical values for, respectively, the share of capital in total income s and the elasticity of interest rate ε r, such that if s s or ε r (0, ε r), the slope of (T ) is positive and lower than one and either D 1 > 0 (and lim ε + D (ε) = + ) or D 1 < 1 (and lim ε + D (ε) = ). As a consequence (T ) remains in the saddle point region and the steady state is always locally determinate. Conversely, when s > s and ε r > ε r, as it is shown in Fig. 2, one has D 1 ( 1, 0) and lim ε + D (ε) =. It follows that for low elasticities of the offer curve ε, the half-line (T ) crosses the interior of the triangle ABC and therefore the steady state is locally indeterminate. Then (T ) intersects the line D = T 1 at ε = ε F and a flip bifurcation generically occurs. Eventually, for ε > ε F the steady state becomes a saddle, thus locally determinate. B D A ε F C T Figure 2: b < 1/ (1 δ) with s > s and ε r > 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 there exist s (0, 1) and ε r > 0, such that: (i) If s s or ε r (0, ε r), then the steady state is a saddle (locally determinate) for all ε > 0. (ii) If s > s and ε r > ε r, then there exists ε F > 1 such that the steady state is a sink (locally indeterminate) when ε (1, ε F ) and a saddle when ε > ε F. A flip bifurcation generically occurs at ε = ε F. 13

15 Proposition 6.1 shows that local indeterminacy requires a large enough share of capital in total income, a large enough elasticity of interest rate and a low enough elasticity of the offer curve. We know that in the one-sector model studied by Grandmont et al. (1998) a necessary condition to get indeterminacy is an elasticity of capital-labor substitution lower than the share of capital. In order to understand the implications of our results in terms of the elasticity of capital-labor substitution, we need to know how to interpret the elasticity of the interest rate in terms of the elasticity of capitallabor substitution. The main difference with the one-sector formulation then appears: in a two-sector model, each technology is characterized by some particular substitutability properties and we need to take into account two distinct elasticities of capital-labor substitution. Denoting σ i the elasticity of sector i = 0, 1 and using the recent contribution of Drugeon (2004), we may however define an aggregate elasticity of substitution between capital and labor, denoted Σ, which is obtained as a weighted sum of the sectoral elasticities σ i, 17 and then derive an expression for the elasticity of the interest rate ε r, namely: Σ = y0 +py 1 py 1 ky 0 ( py 1 k 0 l 0 σ 0 + y 0 k 1 l 1 σ 1 ), εr = ( ) l 0 2 w(y 0 +py 1 ) y 0 Σ (21) In the particular case with identical elasticities across sectors σ 0 = σ 1 = σ, we find as in the one-sector model that the elasticity of the interest rate will be high enough provided σ is low enough. 18 However, as soon as there are some asymmetries between sectors, larger elasticities of capital-labor substitution may become compatible with a large elasticity of the interest rate. Indeed ε r also depends on the factors and output prices, the amount of factors used in each sectors, and the production levels. More precise results remain difficult to obtain at this point of the analysis since the capital intensity difference b, the outputs and the amounts of capital and labor used in each sector are functions of the elasticities of capital-labor substitution σ 0 and σ 1. Numerical simulations in a CES economy will give additional conclusions in Section 7. In particular, it will be shown that contrary to the one-sector formulation, indeterminacy is possible under sectoral elasticities 17 As shown in Drugeon (2004), the weighted sum is defined with the aggregate shares of capital and labor income in national income, the aggregate shares of the consumpsion good and the investment good in the national product, and the sectoral shares of capital and labor income in total production cost. The expression (21) is obtained after straightforward simplifications. 18 Notice that in a one-sector model we have p = 1, y 0 = y 1 = y/2, l 0 = l 1 = 1/2, k 0 = k 1 = k/2, σ 0 = σ 1 = σ and py 1 k 0 l 0 σ 0 + y 0 k 1 l 1 σ 1 = ykσ, so that equations (21) become Σ = σ and ε r = (1 s)/σ. 14

16 of capital-labor substitution leading to an aggregate elasticity Σ which is consistent with recent empirical estimates. Consider now the condition on the elasticity of the offer curve. As shown by (17), ε is defined from the elasticity of intertemporal substitution in consumption ɛ c and the inverse of the elasticity of the marginal disutility of labor ɛ l. The restriction ε (1, ε F ) in Proposition 6.1 can thus be stated as ɛ c [1 + ɛ l (1 ε F )] < ɛ l ε F (22) and to be satisfied it requires 1 + ɛ l (1 ε F ) < 0 ɛ l > 1 ε F 1 ɛ l (23) i.e. a large enough elasticity of the labor supply. Condition (22) is then ɛ c > ɛ l ε F ɛ l (ε F 1) 1 ɛ c (24) i.e. a large enough elasticity of intertemporal substitution in consumption. However, depending on the value of the critical bound ε F, local indeterminacy may be compatible with low values for these elasticities. In particular, straightforward computations show that for some finite value of ɛ l satisfying (23), the lower bound on ɛ c as given by (24) may remain close to 1. 7 A CES economy To provide some quantitative insights of the plausibility of indeterminacy in our model, we consider the classical example of a CES economy. We retain the following functional forms u(c w ) = (c w ) 1 η 1 /(1 η 1 ), v(l) = l 1+η 2 /(1 + η 2 ) (25) with η 1 (0, 1), η 2 0 for preferences and f 0 (k 0, l 0 ) = [ α 10 (k 0 ) ρ 0 + α 00 (l 0 ) ρ 0 f 1 (k 1, l 1 ) = [ α 11 (k 1 ) ρ 1 + α 01 (l 1 ) ρ 1 ] 1/ρ0 ] 1/ρ1 with α 00 +α 10 = α 01 +α 11 = 1, ρ 0, ρ 1 > 1 for technologies. It follows obviously that the elasticity of the offer curve is ε = (1 + η 2 )/(1 η 1 ) 1 while the elasticities of capital-labor substitution are σ i = 1/(1 + ρ i ), i = 0, 1. We calibrate the structural parameters in a standard way compatible with quarterly data by choosing β = 0.99, δ = and thus θ It follows that the bound for the capital intensity difference b given in Assumption 6.1 is 1/(1 δ) In the RBC literature, Cobb-Douglas technologies are usually considered and standard calibrations are based on capital shares in the consumption and investment good sectors which are 15

17 such that α 10, α 11 (0.2, 0.6). Notice that with CES technologies, the capital shares now also depends on the parameters ρ i as this clearly appears in the formulation of the capital intensity difference b given in Appendix Let us consider a capital intensive consumption good with α 00 = 0.4, α 11 = 0.6. Assuming that ρ 0 [ 0.64, 0.46] and ρ 1 [ 0.152, 0.13], the associated relative capital intensity difference satisfies Assumption 6.1. The corresponding elasticities of substitution in each sector are thus σ 0 [1.85, 2.77] and σ 1 [1.146, 1.179]. In order to provide comparisons with the estimates 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 the aggregate elasticity of capital-labor substitution Σ defined by (21). We easily get from the previous values of σ 0 and σ 1 that Σ has to belong to the interval [1.116, 1.178] which perfectly fits with the estimates of Duffy and Papageorgiou. Numerical computations then show that the steady state is locally indeterminate for all values of the elasticity of the offer curve such that ε (1, ε F ), with ε F (4.02, 278.4) depending on the values of ρ i considered, and when ε crosses ε F from below a flip bifurcation occurs, giving rise to period-two cycles. More precisely if we set ρ 0 = 0.57 (i.e. σ 0 = 2.32) and ρ 1 = 0.15 (i.e. σ 1 = 1.176) we get Σ and ε F It is worthwhile remarking that within such an example indeterminacy arises in correspondence to an aggregate elasticity of factor substitution very close to the unitary Cobb-Douglas case, condition known for eliminating such a phenomenon in the one-sector framework. Notice also that the lower bound on ɛ l as defined by (23) is ɛ l 1.38%. It follows that if, in accordance with stylized facts, 21 we choose low values for the elasticity of the labor supply, i.e. for instance ɛ l (0.1, 4), the lower bound on ɛ c given by (24) is ɛ c (1.017, 1.177). Therefore, contrary to one and two-sector models with productive externalities, local indeterminacy remains compatible with a small elasticity of intertemporal substitution in consumption and a low elasticity of the labor supply The larger bound of the interval is obtained with raw labor while the lower bound is obtained with human capital adjuted labor. Since we do not include human capital in our model, we consider the estimations based on raw labor. 20 All these numerical simulations are obtained from expressions given in Appendix See Blundell and McCurdy (1999). 22 In 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-specific externalities, local indeterminacy is obtained with a close to infinite elasticity of intertemporal substitution in consumption (see Benhabib and Nishimura (1998), Benhabib et al. (2002)). All these restrictions are not consistent with recent macroeconomic evidences. 16

18 Remark: As shown in Barinci (2001), if a non logarithmic utility function for capitalists is considered, the Euler equations define an implicit 3- dimensional dynamical system parameterized by the elasticity of intertemporal substitution in consumption of capitalists, denoted η. Actually, Barinci and Drugeon (2005) prove that when η = 1, the dynamical system can be reduced to dimension 2 since β 1 > 1 happens to be a trivial characteristic root. It follows that the characteristic roots of the general 3-dimensional dynamical system are continuous with respect to η. As a result, all our previous conclusions obtained with η = 1 are robust to small perturbations of η around 1. 8 Plausibility of sunspots and economic intuitions The numerical simulations clearly show that local indeterminacy arises with plausible values for the elasticities of capital-labor substitution, the elasticity of intertemporal substitution in consumption and the elasticity of the labor supply when the consumption good is sufficiently capital-intensive. This configuration appears to be consistent with national accounting data of the main developed countries over the last three decades. Indeed in a recent contribution, Takahashi et al. (2004) aggregate sectoral data in order to get a two-sector representation of the Japanese, U.S. and German economies from 1955 to They find that, while the U.S. and German economies are characterized by a capital-intensive aggregate consumption good sector over the whole period, the Japanese economy experiences a capital-intensity reversal in 1975, the aggregate consumption good sector being labor-intensive before and capital-intensive since then. These findings may be explained by the fact that, within developed countries, consumption goods with an increasing amount of technological content have become a growth engine. Our results then provide a theoretical background to explain why developed countries are characterized by a strong macroeconomic volatility based on expectations-driven fluctuations. It remains now to understand the economic mechanisms at the core of these results. Actually, if the consumption good is sufficiently capital intensive, there exist in our model two main forces from which endogenous fluctuations originate: a pure technological mechanism based on factor allocations across sectors, and a monetary mechanism based on the cash-inadvance constraint. As initially shown in Benhabib and Nishimura (1985), the technological mechanism refers to the Rybczinsky effect. Starting from one equilibrium paths, consider an instantaneous increase in the capital 17

19 stock k t. This results in two opposing forces: - Since the consumption good is more capital intensive than the investment good, the trade-off in production becomes more favorable to the consumption good. Moreover, the Rybczinsky effect implies a decrease of the output of the capital good y t. This tends to lower the investment and the capital stock in the next period k t+1. - In the next period the decrease of k t+1 implies again through the Rybczinsky effect an increase of the output of the capital good y t+1. This mechanism is explained by the fact that the decrease of k t+1 improves the trade-off in production in favor of the investment good which is relatively less intensive in capital. Therefore this tends to increase the investment and the capital stock in period t + 2, k t+2. The monetary mechanism, as shown Bosi et al. (2005), is based on the agents expectations. Let us consider again the CES instantaneous utility function as given in (25). Equations (1) and (7) then become respectively c c ( t+1 /cc t = βi t+1, c w t+1 /c w ) η1 t > γi t+1 (26) with i t+1 [r t+1 + (1 δ) p t+1 ]/p t the nominal interest factor. Starting from one equilibrium path, let us try to construct an alternative equilibrium. For this purpose, assume that agents collectively revise their expectations in reaction to a given sunspot signal and come to believe that the nominal interest factor will undergo an appreciation. It follows that to re-establish (26), future consumptions c c t+1 and cw t+1 must be driven up. When mixed with the technological effect, this expectation becomes self-fulfilling and a new equilibrium path can be defined. The simultaneous consideration of technological and monetary effects therefore generates sunspot fluctuations while the elasticities of capital-labor substitution, the elasticity of intertemporal substitution in consumption and the elasticity of the labor supply may be fixed at some standard values. However, all this story crucially depends on the assumption of a consumption good which is sufficiently capital intensive. If on the contrary, either the consumption good is weakly capital intensive or the investment good is capital intensive, the technological effect does not occur as in one-sector models. In order to generate sunspot fluctuations, the monetary mechanism therefore requires extreme parameterizations for the structural elasticities, i.e. low elasticities of capital-labor substitution (lower than the share of capital), or large elasticities of intertemporal substitution in consumption. 18

20 9 Concluding remarks In this paper we consider a two-sector infinite horizon model with heterogeneous agents and borrowing constraint on labor income. We show that the relative capital intensity across sectors plays a relevant role with respect to the emergence of indeterminacy and deterministic as well as sunspot fluctuations. The main result is obtained when the consumption sector is significantly more capital intensive than the investment sector. Indeed local indeterminacy comes about for elasticities of capital-labor substitution slightly greater than unity and for a broad range of values for the elasticity of the offer curve which are compatible with plausible values for the elasticity of intertemporal substitution in consumption and the elasticity of the labor supply. These conditions appear to be consistent with recent macroeconomic empirical evidences. On the one side, concerning the capital intensity difference, Takahashi et al. (2004) find that over the last three decades the main OECD countries are characterized by a capital-intensive consumption good sector. On the other side, concerning the substitutability properties, Duffy and Papageorgiou (2000) show that capital and labor have an elasticity of substitution greater than unity in the richest group of countries. The main criticisms that can be addressed to this model concern the facts that the financial market is segmented and that capitalists do not supply labor. Even though such assumptions can be considered as a limit configuration based on the evidences suggesting that a significant proportion of households have limited borrowing constraints, and that households with high income generally provide lower labor supply, it would be more realistic to consider that both types of households work and hold simultaneously the two types of assets. The former point has been considered by Barinci et al. (2003) within an aggregate model and, as suggested in the introduction, could be also considered within our two-sector framework without drastically altering the main conclusions. The latter extension is more difficult to carry out since it requires to introduce two important modifications within our basic framework. The first consists in considering as in Bosi and Magris (2002) a partial borrowing constraint. If we also assume that both types of households have the same discount factor, such a partial constraint forces workers to finance one part of their total expenditures from money and the other from savings. The second modification consists in introducing as in Bosi et al. (2005) a fractional cash-in-advance constraint that forces capitalists to hold money in order to finance part of their consumption expenditures. This work is left for future research. 19

21 10 Appendix 10.1 Proof of Proposition 4.1 Since T (k, δk, l) is homogeneous of degree one, (16) can be rewritten as T 1(κ,δκ,1) T 2 (κ,δκ,1) = β 1 (1 δ), U (T 3 (κ, δκ, 1) l) = V (l) (27) with κ k/l. Consider the first equation in (27): this is equivalent to the equation defining the stationary capital stock of a two-sector optimal growth model with inelastic labor supply. Then, the proof of Theorem 3.1 in Becker and Tsyganov (2002) applies and there exists a unique solution κ of the first equation of (27). Consider now the second equation in (27) evaluated at κ with the fact that c = wl. In view of the definition of U and V, rewrite it as T 3 (κ, δκ, 1) u (T 3 (κ, δκ, 1) l) = v (l). Then, under Assumption 1, it is easy to verify that such an equation possesses a unique solution l Proof of Proposition 5.1 By linearizing (14), we obtain the following expression for the Jacobian J : [ ] 1 [ ] J bε = w a (1 ε w ) [1 + (1 δ) b] εw aε (28) 1 bϑε r aϑε r 1 [1 + (1 δ) b] ϑε r 0 with ϑ βθ (1 θb), from which we get the expressions for the trace T and the determinant D 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) Proof of Proposition 6.1 Assume that b < 1/(1 δ). When ε r moves from zero to +, ψ decreases continuously from one to 1 δ + 1/b (0, 1 δ). This in particular means that ψ (0, 1) for all ε r 0. We also notice that from (19) with ε = 1, T 1 = 1 + D 1 so that the origin of belongs to D = T 1. Now, let us define the following useful formulas 1 1 θb βθ z 1 δb s 1 s, z 1 1 εrβθ(1 θb)b ε > 1, rβθ(1 θb)(1 δb) z 2 2 εrβθ(1 θb)[1+(2 δ)b] ε rβθ(1 θb)(1 δb) > z 1 Then one easily verifies that D 1 / ε r < 0 if and only if z < 1. Still, in the interval under study for b, straightforward computations show that: - when z < 1 then D 1 (0, 1) for every ε r and D (+ ) = +, 20

22 - when 1 < z < z 1 then D 1 > 1 and D (+ ) = +, - when z 1 < z < z 2 then D 1 < 1 and D (+ ) =, - when z > z 2, then D 1 ( 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 < z 2, then the -line does not cross the triangle ABC and the steady state is a saddle; (ii) if z > z 2, then -line crosses the triangle ABC and, as shown in Fig. 2, there exists ε F > 1 such that the steady state is a sink when ε (1, ε F ) and a saddle when ε > ε F. One could easily prove that z z 2 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 < z 2 if and only if ε r (0, ε r), with ε r 2(1 s) s(1 δb) 2 +(1 s)βθ(1 θb)[1+(2 δ)b] > Computations for the CES economy Using the recent contribution of Nishimura and Venditti (2004), we get k /l = ( 1 α10 α 01 1+ρ 0 ( α 11 α 00 α 11 k 1 /l = κ [1 δ ( α 11 θ θ ) ρ 1 1+ρ 1 ) 1+ρ1 α11 ρ 1 (1+ρ 0 ) α 01 1 δb ) 1 1+ρ 1 ] l 1 /l = κ δ ( ) 1 ( α 11 1+ρ 1 ( α 11 θ θ ) ρ 1 1+ρ 1 α11 ) 1 ρ 1 α 01 b = ( ) 1 ( ) 1 ( α 11 1+ρ 1 θ [1 α10 α 01 1+ρ 0 ( α 11 α 00 α 11 ( ) r = α 10 [α ρ0 ( 10 + α α10 α 01 1+ρ 0 ( α α 00 α 11 w = r α 01 α 11 ( ( α11 θ ) ρ 1 1+ρ 1 α11 ) 1+ρ1 ρ 1 α 01 κ θ ) ρ 1 1+ρ 1 α11 α 01 θ ) ρ 1 1+ρ 1 α11 α 01 From this we derive p = r/θ, k 0 = k k 1, l 0 = l l 1 and ] ) ρ1 ρ0 ρ 1 (1+ρ 0 ) ] 1+ρ ) 0 ρ0(1+ρ1) ρ 0 ρ 1 (1+ρ 0 ) y 0 = [ α 10 (k 0 ) ρ 0 + α 00 (l 0 ) ρ 0] 1/ρ0, y 1 = [ α 11 (k 1 ) ρ 1 + α 01 (l 1 ) ρ 1 All the numerical simulations are finally based on these expressions. 21 ] 1/ρ1

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