CUBO A Mathematical Journal Vol.10, N o ¯ 03, (65–82). October 2008 Equilibrium Cycles in a Two-Sector Economy with Sector Specific Externality Miki Matsuo, Kazuo Nishimura Institute of Economic Research, Kyoto University email: nishimura@kier.kyoto-u.ac.jp Tomoya Sakagami Department of Economics, Kumamoto Gakuen University and Alain Venditti† CNRS-GREQAM ABSTRACT In this paper, we study the two-sector CES economy with sector-specific externality (feedback effects). We characterize the equilibrium paths in the case that allows neg- ative externality, and show how the degree of externality may generate equilibrium cycles around the steady state. RESUMEN En este artculo estudiamos economia de dos-sector CES con externalidad de sector- especifico (efecto de retroalimentacin). Nosotros caracterizamos la trajectoria de equi- †This paper has been written while Alain Venditti was visiting the Institute of Economic Research of Kyoto University. He thanks Professor Kazuo Nishimura and all the staff of the Institute for their kind invitation. 66 Miki Matsuo et al. CUBO 10, 3 (2008) librio en el caso que permite externalidad negativa, e demonstramos como el grado de externalidad puede generar ciclos de equilibrio alrededor del estado regular. Key words and phrases: Difference equations, nonlinear dynamics, bifurcation, two-periodic cycle, multiple equilibria. Math. Subj. Class.: 37G10, 39A11, 91B50, 91B62, 91B64, 91B66. 1 Introduction The aim of this paper is to show the existence of equilibrium cycles around the steady state in the two-sector model with CES production function and sector specific externality.1 A representative agent has concrete expectations on the level of externality and make a decision assuming that the externality is not affected by his own choice of decision variables. However, externalities come from the average values of capital and labor on the market. Therefore, if a representative agent chooses values of decision variables, externalities also vary as everybody also takes the same decision. Over the last decade, an important literature has focused on the existence of locally inde- terminate equilibria in dynamic general equilibrium economies with technological external effects. Local indeterminacy means that there exists a continuum of equilibria starting from the same initial condition, all of which converging to the same steady state. It is now well-known that local indeterminacy is a sufficient condition for the existence of endogenous fluctuations generated by purely extrinsic belief shocks which do not affect the fundamentals, i.e. the preferences and tech- nologies.2 Indeed, in presence of local indeterminacy, by randomizing beliefs over the continuum of equilibrium paths, one may construct equilibria defined with respect to shocks on expectations, and thus provide an alternative to technology or taste shocks to get propagation mechanisms and to explain macroeconomic volatility. Benhabib and Nishimura [3, 4] proved that indeterminacy may arise in a continuous time economy in which the production functions from the social perspective have constant return to scale. Benhabib, Nishimura and Venditti [5] studied the two-sector model with sector specific external effects in discrete time framework. They provided conditions in which indeterminacy may occur even if the production function is decreasing return to scale from the social perspective. Nishimura and Venditti [7] study the interplay between the elasticity of capital-labor substitution and the rate of depreciation of capital, and its influence on the local behavior of equilibrium paths in a neighborhood of the steady state. However, in all these contributions, the existence of local bifurcations as the degree of externalities is modified is not discussed. In this paper, we study the model in Nishimura and Venditti [7], focusing on the external effect of capital-labor ratio in the pure capital good sector and characterize the equilibrium paths 1External effects are feedbacks from the other agents in the economy who also face identical maximizing problems. See Benhabib and Farmer [2] for a survey. 2See Cass and Shell [6]. CUBO 10, 3 (2008) Equilibrium Cycles in a Two-Sector Economy ... 67 in the case that allows negative externality, which was not discussed in their paper. We will focus on the existence of flip bifurcation, i.e. of period-two equilibrium cycles, through the existence of local indeterminacy. In Section 2 we describe the model. We discuss the existence of a steady state and give the local characterization of the equilibrium paths around the steady state in Section 3. Section 4 contains some concluding comments. 2 The model We consider a two-sector model with an infinitely-lived representative agent. We assume that its single period linear utility function is given by u (ct) = ct. We assume that the consumption good, c, and capital good are produced with a Constant Elasticity of Substitution (CES) production functions. ct = [ α1K −ρ c ct + α2L −ρ c ct ]− 1 ρc (1) yt = [ β 1 K −ρ 2 yt + β2L −ρ 2 yt + et ]− 1 ρy (2) where ρc, ρy > −1 and et represents the time-dependent externality (feedback effects) in the capital good sector. Let the elasticity of capital-labor substitution in each sector be σc = 1 1+ρ c ≥ 0 and σy = 1 1+ρ y ≥ 0. We assume that the externalities are as follows: e = bK̄ −ρ y yt − bL̄ −ρ y yt , (3) where b > 0, and K̄y and L̄y represents the economy-wide average values. The representative agent takes these economy-wide average values as given. Definition 1 We call y = [ β 1 K −ρ y y + β2L −ρ y y + e ] − 1 ρy the production function from the private perspective, and y = [ (β 1 + b) K −ρ y y + (β2 − b) L −ρ y y ]− 1 ρy the production function from the social perspective. In the rest of the paper we will assume that α1 + α2 = β1 + β2 = 1 so that the consumption good sector does not have externalities. Notice then that denoting β̂ 1 = β 1 + b and β̂ 2 = β 2 − b, we get also β̂ 1 + β̂ 2 = 1. The investment good sector has externalities but the technology is linearly homogeneous, i.e. has constant returns, from the social perspective. 68 Miki Matsuo et al. CUBO 10, 3 (2008) Remark 1 Notice that the externality (3) may be expressed as follows e = bL̄ −ρ y 2 [ ( K̄y L̄y )−ρ y − 1 ] . (4) Now consider the production function from the social perspective as given in Definition 1. Dividing both sides by Ly, we get denoting ky = Ky/Ly and ỹ = y/Ly ỹ = [ (β 1 + b) k −ρ y y + (β2 − b) ]− 1 ρy . (5) From equations (4) and (5) we derive that the externality is given in terms of the capital-labor ratio in the investment good sector. The aggregate capital is divided between sectors, kt = Kct + Kyt, and the labor endowment is normalized to one and divided between sectors, Lct + Lyt = 1. The capital accumulation equation is kt+1 = yt, as the capital depreciates completely in one period. To simplify we assume that both technologies are characterized by the same properties of substitution, i.e. ρc = ρy = ρ. The consumer optimization problem will be given by max ∞ ∑ t=0 δ t [ α1K −ρ ct + α2L −ρ ct ]− 1 ρ s.t. yt = [ β 1 K −ρ yt + β2L −ρ yt + et ]− 1 ρ 1 = Lct + Lyt kt = Kct + Kyt yt = kt+1 k0, {et} ∞ t=0 given (6) where δ ∈ (0, 1) is the discount factor. pt, rt, and wt respectively denote the price of capital goods, the rental rate of the capital goods and the wage rate of labor at time t ≥ 03. For any sequences {et} ∞ t=0 of external effects that the representative agent considers given, the Lagrangian at time 3We normalize the price of consumption goods to one. CUBO 10, 3 (2008) Equilibrium Cycles in a Two-Sector Economy ... 69 t ≥ 0 is defined as follows: Lt = [ α1K −ρ ct + α2L −ρ ct ]− 1 ρ + pt [ [ β 1 K −ρ yt + β2L −ρ yt + et ]− 1 ρ − kt+1 ] + rt (kt − Kct − Kyt) + wt (1 − Lct − Lyt) . (7) Then the first order conditions derived from the Lagrangian are as follows: ∂Lt ∂Kct = α1 ( ct Kct ) − rt = 0, (8) ∂Lt ∂Lct = α2 ( ct Lct ) − wt = 0, (9) ∂Lt ∂Kyt = ptβ1 ( yt Kyt ) − rt = 0, (10) ∂Lt ∂Lyt = ptβ2 ( yt Lyt ) − wt = 0. (11) From the above first order conditions, we derive the following equation, ( α1�α2 β 1 �β 2 ) = ( Kct�Lct Kyt�Lyt )1+ρ . (12) If α1/α2 > (<) β1/β2, the consumption (capital) good sector is capital intensive from the private perspective. For any value of (kt, yt) , solving the first order conditions with respect to Kct, Kyt, Lct, Lyt gives these inputs as functions of capital stock at time t and t + 1, and external effect, namely: Kct = Kc (kt, yt, et) , Lct = Lc (kt, yt, et) , Kyt = Ky (kt, yt, et) , Lyt = Ly (kt, yt, et) . For any given sequence {et} ∞ t=0 , we define the efficient production frontier as follows: T ∗ (kt, kt+1, et) = [ α1Kc (kt, yt, et) −ρ + α2Lc (kt, yt, et) −ρ ]− 1 ρ . Using the envelope theorem we derive the equilibrium prices,4 T2 (kt, kt+1, et) = −pt, (13) T1 (kt, kt+1, et) = rt. (14) 4See Takayama for the envelope theorem, pp160-165. Using the envelope theorem, we get ∂Lt ∂kt = ∂T ∂kt and ∂Lt ∂kt+1 = ∂T ∂kt+1 . This is equivalent to (13) and (14). 70 Miki Matsuo et al. CUBO 10, 3 (2008) Next we solve the intertemporal problem (6). In this model, lifetime utility function becomes U = ∞ ∑ t=0 δ t T ∗ (kt, kt+1, et) . From the first order conditions with respect to kt+1, we obtain the Euler equation T2 (kt, kt+1, et) + δT1 (kt+1, kt+2, et+1) = 0. (15) The solution of equation (15) also has to satisfy the following transversality condition lim t→+∞ δ t ktT1 (kt, kt+1, et) = 0. (16) We denote the solution of this problem {kt} ∞ t=0. This path depends on the choice of sequence {et} ∞ t=0 . If the sequence {et} ∞ t=0 satisfies et = bKy (kt, yt, et) −ρ − bLy (kt, yt, et) −ρ , (17) then {k̂t} ∞ t=0 is called an equilibrium path. Along an equilibrium path, the expectations of the representative agent on the externalities {et} ∞ t=0 are realized. Definition 2 {kt} ∞ t=0 is an equilibrium path if {kt} ∞ t=0 satisfies (15), (16) and (17). Solving the equation (17) for et, we derive et that is given as a function of (kt, kt+1), namely et = ê (kt, kt+1). Let us substitute ê (kt, kt+1) into equations (13) and (14) and define the equilib- rium prices as pt = pt (kt, kt+1) , rt = rt (kt, kt+1) . Then the Euler equation (15) evaluated at {kt} ∞ t=0 is −p (kt, kt+1) + δr (kt+1, kt+2) = 0. (18) We have the following lemma. Lemma 1 The partial derivatives of T (kt, kt+1, et) with respect to kt and kt+1 are given by T1 (kt, kt+1, ê (kt, kt+1)) = α1 [ α1 + α2 ( α1β2 α2β1 ) 1+ρ ρ ( ( g k t+1 ) ρ β 2 −b − (β 1 +b) β 2 −b )ρ]− 1+ρ ρ T2 (kt, kt+1, ê (kt, kt+1)) = T1(kt, kt+1,ê(kt,kt+1)) β 1 ( g kt+1 ) 1+ρ where g = g (kt, kt+1) = { Kyt ∈ [0, kt] | α1β2 α2β1 = ( kt−Kyt 1−Lyt(Kyt,kt+1) )1+ρ ( Lyt(Kyt ,kt+1) Kyt )1+ρ } CUBO 10, 3 (2008) Equilibrium Cycles in a Two-Sector Economy ... 71 and Lyt (Kyt, kt+1) = ( k −ρ t+1 −(β 1 +b)K −ρ yt β 2 −b ) − 1 ρ . 3 Steady state Definition 3 A steady state is defined by kt = kt+1 = yt = k ∗ and is given by the solution of T2 (k ∗, k∗, e∗) + δT1 (k ∗, k∗, e∗) = 0 with e∗ = ê (k∗, k∗) . In the rest of the paper we assume the following restriction on parameters’ values that guar- antees all the steady state values are positive. Assumption 1 The parameters δ, β 1 , b and ρ satisfy (δβ 1 ) −ρ 1+ρ < β 1 + b. We obtain the steady state value. Proposition 1 In this model, there exists a unique stationary capital stock k∗ such that: k∗ = { 1 + ( α1β2 α2β1 ) −1 1+ρ (δβ 1 ) −1 1+ρ [ 1 − (δβ 1 ) 1 1+ρ ] }−1 [ 1−β̂ 1 (δβ 1 ) −ρ 1+ρ β̂ 2 ] 1 ρ . (19) To study local behavior of the equilibrium path around the steady state k∗, we linearize the Euler equation (15) at the steady state k∗ and obtain the following characteristic equation δT12λ 2 + [δT11 + T22] λ + T21 = 0, or δλ 2 + [ δ T11 T12 + T22 T12 ] λ + T21 T12 = 0. (20) As shown in Nishimura and Venditti [7], the expressions of the characteristic roots are as follows: Proposition 2 The characteristic roots of Equation (20) are λ1 = 1 (δβ 2 ) 1 1+ρ [ ( β 1 β 2 ) 1 1+ρ − ( α1 α2 ) 1 1+ρ ], (21) λ2(b) = (δβ 2 ) 1 1+ρ [ β 1 +b β 1 ( β 1 β 2 ) 1 1+ρ − β 2 −b β 2 ( α1 α2 ) 1 1+ρ ] δ . 72 Miki Matsuo et al. CUBO 10, 3 (2008) The roots of the characteristic equation determine the local behavior of the equilibrium paths. The sign of λ1 is determined by factor intensity differences from the private perspective. 5 We now characterize the equilibrium paths in this model. In particular we can show that the local behavior of equilibrium path around the steady state changes according to the degree of external effect in the capital good sector. Definition 4 A steady state k∗ is called locally indeterminate if there exists ε such that for any k0 ∈ (k ∗ − ε, k∗ + ε) , there are infinitely many equilibrium paths converging to the steady state. As there is one pre-determined variable, the capital stock, local indeterminacy occurs if the stable manifold has two dimension, i.e. if the two characteristic roots are within the unit circle. We will also present conditions for local determinacy (for saddle-point stability) in which there exists a unique equilibrium path. Such a configuration occurs if the stable manifold has one dimension, i.e. if one root is outside the unit circle while the other is inside. When the investment good is capital intensive, local indeterminacy and flip bifurcation cannot occur. Proposition 3 Suppose that the capital good sector is capital intensive from the private perspec- tive, i.e. α2β1 > α1β2. Then the characteristic roots λ1 and λ2(b) are positive with λ1 > 1. Next we present our results assuming that the capital good is labor intensive from the private perspective, i.e. α2β1 − α1β2 < 0. Equilibrium period-two cycles may occur in this case through a flip bifurcation. We will also get local indeterminacy of equilibria. By rewriting equation (21), the characteristic roots are λ1 = − 1 (δβ 2 ) 1 1+ρ [ ( α1 α2 ) 1 1+ρ − ( β 1 β 2 ) 1 1+ρ ], (22) λ2(b) = − (δβ 2 ) 1 1+ρ [ β 2 −b β 2 ( α1 α2 ) 1 1+ρ − β 1 +b β 1 ( β 1 β 2 ) 1 1+ρ ] δ . To get λ1 ∈ (−1, 0), we need however to suppose a slightly stronger condition than simply ensuring the capital good sector to be labor intensive from the private perspective. The capital intensity difference α1β2 − α2β1 needs to be large enough and the discount factor has to be close enough to 1. Proposition 4 Assume that (α1β2) 1 1+ρ − (α2β1) 1 1+ρ > α 1 1+ρ 2 and δ ∈ (δ3, 1) with δ3 = β −1 2 [ (β 1 /β 2 ) 1 1+ρ − (α1/α2) 1 1+ρ ]−1−ρ < 1. 5If α2β1 − α1β2 > 0, the capital good sector is capital intensive from the private perspective. CUBO 10, 3 (2008) Equilibrium Cycles in a Two-Sector Economy ... 73 Then there exist b(δ) > 0 and b(δ) > b(δ) such that the steady state is saddle point for b ∈ (0, b (δ)), undergoes a flip bifurcation when b = b (δ), becomes locally indeterminate for b ∈ ( b (δ) , b (δ) ) and is again saddle-point stable for (b (δ) , +∞). Generically, there exist period-two cycles in a left (right) neighborhood of b (δ) that are locally indeterminate (saddle-point stable). Next we still assume that the capital good is labor intensive from the private perspective with α2β1 − α1β2 < 0, but make λ1 an unstable root, i.e. λ1 < −1. As a result local indeterminacy cannot occur but period-two cycles may still exist through a flip bifurcation. Two cases need to be considered: (α1β2) 1 1+ρ − (α2β1) 1 1+ρ > α 1 1+ρ 2 and δ ∈ (0, δ3), as well as (α1β2) 1 1+ρ − (α2β1) 1 1+ρ < α 1 1+ρ 2 . The following result is proved along the same lines as Proposition 4. Proposition 5 Suppose that the capital goods sector is labor intensive from the private perspective and let δ4 = β 1 ρ 2 [ (β 1 /β 2 ) 1 1+ρ − (α1/α2) 1 1+ρ ] 1+ρ ρ . Assume also that one of the following sets of conditions hold: i) (α1β2) 1 1+ρ − (α2β1) 1 1+ρ > α 1 1+ρ 2 and δ ∈ (0, δ∗) with δ∗ = min{δ3, δ4}, ii) (α1β2) 1 1+ρ − (α2β1) 1 1+ρ < α 1 1+ρ 2 , ρ > 0 and δ ∈ (0, δ4), Then there exist b(δ) > 0 and b(δ) > b(δ) such that the steady state is totally unstable for b ∈ (0, b (δ)), undergoes a flip bifurcation when b = b (δ), becomes saddle-point stable for b ∈ ( b (δ) , b (δ) ) and is again totally unstable for (b (δ) , +∞). Generically, there exist period-two cy- cles in a left (right) neighborhood of b (δ) that are locally saddle-point stable (unstable). Remark 2 Consider the production function from the social perspective as given in Definition 1 and recall from (5) that we can write it as follows ỹ = [ (β 1 + b) k −ρ y y + (β2 − b) ]− 1 ρy . (23) According to b ≷ β 2 , the following inequality holds: for any η > 1, [ (β 1 + b) (ηky) −ρ + (β 2 − b) ]− 1 ρ ≷ [ (β 1 + b) (ηky) −ρ + η−ρ (β 2 − b) ]− 1 ρ = η [ (β 1 + b) k−ρy + (β2 − b) ]− 1 ρ . If b is larger than β 2 , the function ỹ exhibits increasing returns while if b is smaller than β 2 the function ỹ exhibits decreasing returns. As we consider in Proposition 5 values of δ close to zero, the role of b on the local stability properties of the steady state is multiple. Indeed, starting from a low amount of externalities, an increase of b contributes to saddle-point stability and the existence of cycles through a flip 74 Miki Matsuo et al. CUBO 10, 3 (2008) bifurcation. But then if b is increased too much, total instability occurs since the returns to scale becomes increasing as shown in the previous Remark. 4 Concluding remarks In this paper we have characterized the local dynamics of equilibrium paths depending on the size of external effects b. We have shown that when the consumption good is capital intensive, the effect of b on the local dynamics of equilibrium path depends on the value of the discount factor. If the discount factor is close enough to one and the capital intensity difference is large enough, local indeterminacy occurs for intermediary values of b while saddle-point stability is obtained when b is low enough or large enough. On the contrary, if the discount factor is low enough, local indeterminacy cannot occur. But the existence of equilibrium cycles and saddle-point stability require intermediary values of b while total instability is obtained when b is low enough or large enough. 5 Appendix 5.1 Proof of Lemma 1 We shall derive the first partial derivatives of T (kt, kt+1, et) along an equilibrium path. The first order conditions derived from the Lagrangian are as below: α1 ( ct Kct ) − rt = 0, (A1.1) α2 ( ct Lct ) − wt = 0, (A1.2) ptβ1 ( yt Kyt ) − rt = 0, (A1.3) ptβ2 ( yt Lyt ) − wt = 0. (A1.4) In the equilibrium the equation (2) is rewritten as Lyt = ( y −ρ t − (β1 + b) K −ρ yt β 2 − b )− 1 ρ . (A1.5) From the first order conditions (A1.1)-(A1.4), α1β2 α2β1 = ( Kct Lct ) 1+ρ ( Lyt Kyt ) 1+ρ . CUBO 10, 3 (2008) Equilibrium Cycles in a Two-Sector Economy ... 75 Substituting Kct = kt − Kyt, Lyt = 1 − Lct into the equation, α1β2 α2β1 = ( kt − Kyt 1 − Lyt )1+ρ ( Lyt Kyt )1+ρ . (A1.6) By solving equations (A1.5) and (A1.6) with respect to Kyt and substituting yt = kt+1, we have Kyt = g (kt, kt+1) . From the equation (A1.1), rt = α1 [ α1 + α2 ( Kct Lct )ρ]− 1+ρ ρ . Using the equation (A1.6) we have rt = α1 [ α1 + α2 ( α1β2 α2β1 ) 1+ρ ρ ( g (kt, kt+1) Lct )ρ ]− 1+ρ ρ . And then from (A1.5) rt can be rewritten as the following equation by substituting ( g(kt,kt+1) Lyt )ρ = ( g(kt ,kt+1) yt ) ρ β 2 −b − (β 1 +b) β 2 −b 6 , rt = α1    α1 + α2 ( α1β2 α2β1 ) 1+ρ ρ   ( g(kt,kt+1) yt )ρ β 2 − b − (β 1 + b) β 2 − b   ρ    − 1+ρ ρ . (A1.7) Moreover from the equation (A1.3), we have pt = rt β 1 ( g (kt, kt+1) yt )1+ρ . (A1.8) Therefore we get T1 and T2 from the envelope theorem which gives T1 = rt, T2 = −pt. � 5.2 Proof of Proposition 1 By definition k∗ satisfies T2 (k ∗, k∗, e∗) + δT1 (k ∗, k∗, e∗) = 0 with e∗ = ê (k∗, k∗) . In the steady state, g∗ = g (k∗, k∗) and y∗ = k∗. Using Lemma 1, the Euler equation is − r β 1 ( g∗ y∗ )1+ρ + δr = 0. 6Substitute the equation (A1.5) into ( g Lyt )ρ . 76 Miki Matsuo et al. CUBO 10, 3 (2008) Thus, g∗ = (δβ 1 ) 1 1+ρ k∗. (A2.1) As y∗ = k∗ at the steady state, the equation (A1.5) becomes, L ∗ y = k ∗ ( 1 − (β 1 + b) (δβ 1 ) −ρ 1+ρ β 2 − b )− 1 ρ . (A2.2) Using K∗c = k ∗ − K∗y and L ∗ c = 1 − L ∗ y, L∗c = 1 − k ∗ ( 1 − (β 1 + b) (δβ 1 ) −ρ 1+ρ β 2 − b )− 1 ρ , (A2.3) K∗c = k ∗ ( 1 − (δβ 1 ) 1 1+ρ ) . (A2.4) Then equation (A1.6) can be rewritten as follows; ( K∗c L∗c )1+ρ ( L∗y g∗ )1+ρ = α1β2 α2β1 . (A2.5) Substituting these input demand functions into the above equation and solving with respect to k∗, we can get k∗ = [ 1 + ( α1β2 α2β1 ) −1 1+ρ (δβ 1 ) −1 1+ρ ( 1 − (δβ 1 ) 1 1+ρ ) ]−1 [ 1 − (β 1 + b) (δβ 1 ) −ρ 1+ρ β 2 − b ] 1 ρ . Then k∗ is well defined if and only if (δβ 1 ) −ρ 1+ρ < 1 β 1 + b . � 5.3 Proof of Proposition 2 We give some lemmas in order to derive the characteristic roots. Lemma 2 At the steady state the following holds g1 = 1 + ρ ∆Kc , g2 =   (1 + ρ) Lρy + (1 + ρ) L 1+ρ y Lc ∆   y−1−ρ β 2 − b , CUBO 10, 3 (2008) Equilibrium Cycles in a Two-Sector Economy ... 77 where ∆ = 1 + ρ g + 1 + ρ Kc + ( (1 + ρ) Lρy + (1 + ρ) L1+ρy Lc ) β 1 + b β 2 − b g−1−ρ. Proof. From equation (A2.5) we get α1β2 α2β1 = g−1−ρ (k − g) 1+ρ ( y−ρ − (β 1 + b) g−ρ β 2 − b )− 1+ρ ρ { 1 − ( y−ρ − (β 1 + b) g−ρ β 2 − b )− 1 ρ }−1−ρ . Totally differentiating this equation, we have the following relationship, [(1 + ρ) g−1 + (1 + ρ) (k − g) −1 + (1 + ρ) ( y−ρ − (β 1 + b) g−ρ β 2 − b )−1 β 1 + b β 2 − b g−1−ρ + (1 + ρ) { 1 − ( y−ρ − (β 1 + b) g−ρ β 2 − b )− 1 ρ }−1 ( y−ρ − (β 1 + b) g−ρ β 2 − b )− 1+ρ ρ β 1 + b β 2 − b g−1−ρ]dg = (1 + ρ) (k − g) −1 dk + (1 + ρ) ( y−ρ − (β 1 + b) g−ρ β 2 − b )−1−1 y−1−ρ β 2 − b dy + (1 + ρ) { 1 − ( y−ρ − (β 1 + b) g−ρ β 2 − b )− 1 ρ }−1 ( y−ρ − (β 1 + b) g−ρ β 2 − b )− 1+ρ ρ y−1−ρ β 2 − b dy. (A3.1.1) Notice from equation (A1.5) L −ρ y = y−ρ − (β 1 + b) g−ρ β 2 − b (A3.1.2) and (A2.5) ( α1β2 α2β1 ) 1 1+ρ = K∗c L∗c ( g∗ L∗y )−1 . (A3.1.3) Then substituting these equations and dyt = dkt+1 into (A3.1.1) gives RHS = 1 + ρ Kc dkt + ( (1 + ρ) Lρy + (1 + ρ) L1+ρy Lc ) y1−ρ β 2 − b dkt+1, LHS = [ 1 + ρ g + 1 + ρ Kc + ( (1 + ρ) Lρy + (1 + ρ) L1+ρy Lc ) β 1 + b β 2 − b g−1−ρ ] dg, where we denote ∆ ≡ 1 + ρ g + 1 + ρ Kc + ( (1 + ρ) Lρy + (1 + ρ) L1+ρy Lc ) β 1 + b β 2 − b g−1−ρ, 78 Miki Matsuo et al. CUBO 10, 3 (2008) and we derive ∆dg = 1 + ρ Kc dkt + [ (1 + ρ) Lρy + (1 + ρ) L1+ρy Lc ] y1−ρ β 2 − b dkt+1. Therefore dg = 1 + ρ ∆Kc dkt + [(1 + ρ) Lρy + (1 + ρ) L 1+ρ y Lc ] ∆ y1−ρ β 2 − b dkt+1. Lemma 3 At the steady state the following holds g1y = (g − g2y) y g ( 1 − Kc Lc Ly Ky ) −1 with g, Kc, Ly, Lc respectively given by equations (A2.1) − (A2.4). Proof. From equation (A1.5) we get ( L−ρy + β 1 + b β 2 − b K−ρy ) y−1 = y−ρ β 2 − b y−1. Substituting this equation into g2, g2 = [(1 + ρ) Lρy + (1 + ρ) L 1+ρ y Lc ](L−ρy + β 1 +b β 2 −b K−ρy )y −1 ∆ . Using the expression of ∆ we derive g2y = g + (1 + ρ) ∆ Ly Lc − (1 + ρ) ∆ g Kc . Then, g − g2y = (1 + ρ) ∆Kc g ( 1 − Ly Lc Kc g ) , (A3.2.1) g1y = 1 + ρ ∆Kc y. (A3.2.2) From equations (A3.2.1) and (A3.2.2), we finally get g1y = (g − g2y) y g ( 1 − Kc Lc Ly Ky )−1 . Lemma 4 Under Assumption 1, at the steady state, kt = kt+1 = yt = k ∗ and the following holds V11 (k ∗, k∗) V12 (k∗, k∗) = − y g ( 1 − Kc Lc Ly Ky ) −1 , CUBO 10, 3 (2008) Equilibrium Cycles in a Two-Sector Economy ... 79 V22 (k ∗, k∗) V12 (k∗, k∗) = − g β 1 y [ β 1 (β 2 − b) β 2 Kc Lc Ly g + (β 2 − b) ( g Ly )ρ − ( g y )ρ] , V21 (k ∗, k∗) V12 (k∗, k∗) = V22 (k ∗, k∗) V12 (k∗, k∗) V11 (k ∗, k∗) V12 (k∗, k∗) , where g, Kc, Ly, Lc are given by equations (A2.1) − (A2.4), respectively. Proof. Let V (kt, kt+1) denote Ti (kt, kt+1, ê (kt, kt+1)) for i = 1, 2. By definition, V ∗ 11 = ∂T1 ∂kt = ∂r ∂kt , V ∗ 12 = ∂T1 ∂kt+1 = ∂r ∂kt+1 , V ∗ 21 = ∂T2 ∂kt = − ∂p ∂kt , V ∗ 22 = ∂T2 ∂kt+1 = − ∂p ∂kt+1 . Computing the these equations, we have V ∗ 11 = ∂r ∂kt = − (1 + ρ) α − ρ 1+ρ 1 r 1+2ρ 1+ρ α2 β 2 − b ( α1β2 α2β1 ) ρ 1+ρ ( g y )ρ g1 g , V ∗ 12 = ∂r ∂kt+1 = − (1 + ρ) α − ρ 1+ρ 1 r 1+2ρ 1+ρ α2 β 2 − b ( α1β2 α2β1 ) ρ 1+ρ ( g y )ρ ( g2y − g yg ) , ∂p ∂kt = 1 β 1 ∂r ∂kt ( g y )1+ρ + (1 + ρ) r β 1 ( g y )1+ρ g1 g , ∂p ∂kt+1 = 1 β 1 ∂r ∂kt+1 ( g y ) 1+ρ + (1 + ρ) r β 1 ( g y ) 1+ρ ( g2y − g yg ) . From equation (A1.7), ( r α1 ) ρ 1+ρ = [ α1 + α2 ( α1β2 α2β1 ) ρ 1+ρ ( g Lc )ρ ] . Substituting the above equation into V ∗ 11 , and using (A3.1.2) and (A3.1.3) we obtain V ∗ 11 = − (1 + ρ) r ( g y )ρ g1 g [ α1β̂2 α2 ( α2β1 α1β2 ) K∗c L∗c L∗y g∗ + β̂ 2 ( y Ly )ρ ]−1 , where A ≡ α1β̂2 α2 ( α2β1 α1β2 ) K∗c L∗c L∗y g∗ + β̂ 2 ( y Ly )ρ . 80 Miki Matsuo et al. CUBO 10, 3 (2008) We can calculate V ∗ 21 , V ∗ 12 , and V ∗ 22 as we did previously, V ∗ 21 = − (1 + ρ) r β 1 ( g y ) 1+ρ g1 g [ 1 − ( g y )ρ A−1 ] , V ∗ 12 = − (1 + ρ) r ( g y )ρ ( g2y − g yg ) A−1, V ∗ 22 = − (1 + ρ) r β 1 ( g y )1+ρ ( g2y − g yg )[ 1 − ( g y )ρ A−1 ] . Then we get V ∗ 11 V ∗ 12 = g1y g2y − g , V ∗ 22 V ∗ 12 = g β 1 y [ A− ( g y )ρ] , We shall now prove Proposition 2. From Lemma 4 the characteristic polynomial may be rewritten as G (λ) = ( λ + V ∗ 11 V ∗ 12 )( δλ + V ∗ 22 V ∗ 12 ) . Then the characteristic roots are λ1 = − V ∗ 11 V ∗ 12 , λ2 = − V ∗ 22 δV ∗ 12 . (A3.3.1) We can calculate V ∗ 11 V ∗ 12 and V ∗ 22 V ∗ 12 by substituting the following relationship Kc Lc Ly g = ( α1β2 α2β1 ) 1 1+ρ , g y = (δβ 1 ) 1 1+ρ , ( g Ly )ρ = (δβ 1 ) ρ 1+ρ − β̂ 1 β̂ 2 , g − g2y = (1 + ρ) ∆Kc g ( 1 − Ly Lc Kc g ) , and we obtain the first root by substituting all the above equations into the expressions given in Lemma 4 λ1 = − 1 (δβ 2 ) 1 1+ρ [ ( α1 α2 ) 1 1+ρ − ( β 1 β 2 ) 1 1+ρ ]. CUBO 10, 3 (2008) Equilibrium Cycles in a Two-Sector Economy ... 81 Moreover we can rewrite A by using these equations, A =β̂ 2 α1 α2 ( α2β1 α1β2 )− ρ 1+ρ + (δβ 1 ) ρ 1+ρ − β̂ 1 . From Lemma 4, we finally have the second characteristic root, λ2 = − (δβ 2 ) 1 1+ρ [ β 2 −b β 2 ( α1 α2 ) 1 1+ρ − β 1 +b β 1 ( β 1 β 2 ) 1 1+ρ ] δ . � 5.4 Proof of Proposition 3 Notice from (21) that λ1 > 0. Denoting 7 δ1 ≡ β −1 2 [ (β 1 /β 2 ) 1 1+ρ − (α1/α2) 1 1+ρ ]−1−ρ > 1 then we obtain λ1 = (δ1/δ) 1 1+ρ > 1 for 0 < δ < 1. Since (β 1 + b)/β 1 > 1 and (β 2 − b)/β 2 < 1, λ2(b) is always positive. � 5.5 Proof of Proposition 4 If (α1β2) 1 1+ρ −(α2β1) 1 1+ρ > α 1 1+ρ 2 and δ ∈ (δ3, 1), then −1 < λ1 < 0. The size of λ2(b) is determined in the following way. Notice that λ2(b) is increasing in b. For b = 0, λ2 (0) = 1/δλ1 < −1 by the above hypothesis and for b = β 2 , λ2 (β2) = (δβ1) −ρ 1+ρ . (i) If −1 < ρ < 0, λ2 (β2) < 1. Therefore there exist b (δ) ∈ (0, β2) and b (δ) > β2 such that λ2 < −1 for b ∈ (0, b (δ)) , −1 < λ2 < 1 for any b ∈ ( b (δ) , b (δ) ) and λ2 > 1 for any b > b (δ) . (ii) If ρ = 0, λ2 (β2) = 1. Therefore λ2 (b) < −1 for b ∈ (0, β2 − 2α2) , −1 < λ2 (b) < 1 for b ∈ (β 2 − 2α2, β2) and λ2 (b) > 1 for b > β2. (iii) If ρ > 0, λ2 (β2) > 1. Therefore there exist b (δ) and b (δ) in (0, β2) such that λ2 (b) < −1 for b ∈ (0, b (δ)) , −1 < λ2 (b) < 1 for b ∈ ( b (δ) , b (δ) ) , and λ2 (b) > 1 for b > b (δ). In each of these three cases, when b = b(δ), λ2(b) = −1 and λ ′ 2 (b)|b=b(δ) > 0. 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