Indeterminacy in discrete-time infinite-horizon models with non linear utility and endoenous labor Kazuo NISHIMURA Institute of Economic Research, Kyoto University, Japan and Alain VENDII CNRS - GREQAM, Marseille, France First version: June 2004 ; Revised: March 2006 Abstract: We consider a discrete-time two-sector model with sector specific externalities in which the technoloies are iven by CES functions with asymmetric elasticities of capital-labor substitution, and the preferences of the representative aent are iven by a CES additively separable utility function defined over consumption and leisure. We first show that when the labor supply is infinitely elastic, the steady state is always saddle-point stable, no matter what the elasticity of intertemporal substitution in consumption and the size of externalities are. We then prove that when the elasticity of intertemporal substitution in consumption is sufficiently lare, local indeterminacy requires a low enouh elasticity of the labor supply. Keywords: Sector specific externalities, elasticity of intertemporal substitution in consumption, elasticity of labor sypply, elasticity of capital-labor substitution, local indeterminacy. Journal of Economic Literature Classification Numbers: C62, E32, O4. his paper benefited from comments received durin a presentation at the International Conference on Instability and Fluctuations in Intertemporal Equilibrium Models, Marseille, June 2005. We thank all the participants and in particular J. Gali and K. Shimomura. We are also rateful to an anonymous referee for useful comments and suestions.
Introduction Over the last decade, a lare number of papers have established the fact that locally indeterminate equilibria and sunspots fluctuations may arise in infinite-horizon rowth models with external effects in production. hese contributions also show that there exist sinificant differences between onesector and two-sector models. As initially established by Benhabib and Farmer [, in one-sector models local indeterminacy requires some increasin returns based on externalities comin from capital and more importantly labor, a stronly elastic labor supply, and a lare enouh elasticity of intertemporal substitution in consumption which may however remains within plausible intervals. 2 On the contrary, as proved in Benhabib and Nishimura [3, in two-sector models with sector-specific externalities, local indeterminacy is compatible with constant returns at the social level, does not require some elastic labor supply and is mainly based on a technoloical mechanism comin from the broken duality between the Rybczynski and Stolper-Samuelson effects. 3 However, most of the contributions on twosector models are based on the assumption of a linear utility function, i.e. an infinite elasticity of intertemporal substitution in consumption. Of course, indeterminacy remains possible by continuity when the utility function is sufficiently close to linear. When we compare with the results characterizin one-sector models mentioned above, at least two open questions on two-sector models remain: i) findin a lower bound on the elasticity of intertemporal substitution in consumption above which local indeterminacy occurs; ii) characterizin the role of an elastic labor supply in the existence of local indeterminacy. In order to provide answers to these questions, we study in this paper the in- See Benhabib and Farmer [2 for a survey. 2 See also Lloyd-Braa et al. [7 and Pintus [. 3 Benhabib and Nishimura [3 consider a continuous-time model with Cobb-Doulas technoloies. he extension to the discrete-time case is iven by Benhabib et al. [4. wo-sector models with CES technoloies and non-unitary elasticities of capital-labor substitution are considered in Nishimura and Venditti [8, 9.
teractions between the technoloical mechanism from which local indeterminacy oriinates, and the preference mechanisms based on the consumptionleisure trade-off and the intertemporal substitution of consumption. We shall then assume that the two sectors are characterized by CES technoloies with asymmetric elasticities of capital-labor substitution and that the preferences of the representative aent are iven by a CES additively separable utility function defined over consumption and leisure. he cases with Cobb-Doulas technoloies, a linear utility function with respect to consumption, or inelastic labor will be considered as particular specifications. Considerin in a first step the limit case of an infinitely elastic labor supply, we show that the steady state is always a saddle-point for any values of the elasticity of intertemporal substitution in consumption and any amount of sector-specific external effects. his result is drastically different from the one obtained in one-sector models in which local indeterminacy is mainly obtained under the assumption of an infinitely elastic labor supply. 4 While this confiuration makes the occurrence of local indeterminacy easier in one-sector models, it necessarily implies saddle-point stability in two-sector models. his conclusion also echoes one of the results provided by Bosi et al. [5. hey show that in a two-sector optimal rowth model with leisure and additive separable preferences, the steady-state is saddle-point stable when the elasticity of labor supply is infinite. In a second step, we consider an infinite elasticity of intertemporal substitution in consumption. We then show that the local stability properties of the steady state do not depend on the elasticity of the labor supply. Under the assumption of a capital intensive consumption ood at the private level, we also ive precise conditions on the elasticities of capital-labor substitution for the existence of local indeterminacy which are compatible with Cobb-Doulas technoloies in both sectors. Buildin on these results, we introduce in a third step a non-linear utility in consumption but under the assumption of an inelastic labor supply. his restriction allows to precisely isolate the effect of the intertemporal substitutability. We use a eometrical method provided by Grandmont et al. [6. 4 See for instance Benhabib and Farmer [, Lloyd-Braa et al. [7 or Pintus [. 2
It is based on a particular property characterizin the product and sum of characteristic roots, which allows to provide a complete characterization of the local stability properties of the steady state and of the occurrence of bifurcations. We then show under some simple conditions on the CES coefficients of technoloies that the steady state is locally indeterminate if and only if the elasticity of intertemporal substitution in consumption is lare enouh. Moreover, we prove that period-two cycles always occur as the elasticity of intertemporal substitution in consumption is decreased and the steady state becomes saddle-point stable. Finally, we consider the eneral model with a finite elasticity of intertemporal substitution in consumption and a finite elasticity of the labor supply. Usin aain the eometrical method of Grandmont et al. [6, we show that contrary to one-sector models, when the elasticity of intertemporal substitution in consumption is sufficiently lare, the steady state is locally indeterminate if and only if the elasticity of the labor supply is low enouh. Moreover, period-two cycles always occur as the elasticity of the labor supply is increased and the steady state becomes saddle-point stable. he paper is oranized as follows: Section 2 presents the basic model, the intertemporal equilibrium and the steady state. Preliminary results are provided in Section 3 while Section 4 contains the main contributions of the paper. Some concludin comments are provided in Section 5 and all the proofs are athered in a final Appendix. 2 he model 2. he basic structure We consider a discrete-time two-sector economy havin an infinitely-lived representative aent with sinle period CES utility function defined over consumption c and leisure L = l: uc) = c σ σ l+γ +γ with σ 0 and γ 0. he elasticity of intertemporal substitution in consumption is thus iven by ɛ c = /σ while the elasticity of the labor supply is iven by ɛ l = /γ. 3
here are two oods: the pure consumption ood, c, and the pure capital ood, k. Each ood is assumed to be produced with a CES technoloy which contains some sector specific externalities. We denote by c and y the outputs of sectors c and k, and by e c and e y the correspondin external effects. he private production functions are thus defined as: ) ) c = α Kc ρc + α 2 L ρc c + e ρc c, y = β Ky + β 2 L y + e y with ρ c, ρ y > and γ c = / + ρ c ) 0, γ y = / + ρ y ) 0 the elasticities of capital-labor substitution in each sector. he externalities e c and e y depend on K i and L i, which denote the averae use of capital and labor in sector i = c, y, and will be equal to e c = a K ρ c c + a L ρ c 2 c, e y = b K ρ y y + b L ρ y 2 y ) with a i, b i 0, i =, 2. We assume that these economy-wide averaes are taken as iven by individual firms. At the equilibrium, all firms of sector i = c, y bein identical, we have K i = K i and L i = L i. Denotin ˆα i = α i +a i, ˆβ i = β i + b i, the social production functions are defined as c = ˆα ) Kc ρc + ˆα 2 Lc ρc /ρc, y = ˆβ Ky + ˆβ ) / 2 L y 2) he returns to scale are therefore constant at the social level, and decreasin at the private level. We will assume in the followin that ˆα + ˆα 2 = and ˆβ + = so that the production functions collapse to Cobb-Doulas in the particular case ρ c = ρ y = 0. otal labor is iven by L c +L y = l, and the total stock of capital is iven by K c + K y = k. We assume complete depreciation of capital in one period so that the capital accumulation equation is y t = k t+. 5 Optimal factor allocations across sectors are obtained by solvin the followin proram: ) /ρc max α K ρc ct + α 2 L ρc ct + e ct K c,t,l c,t,k y,t,l y,t ) / s.t. k t+ = β K yt + β 2 L yt + e yt l t = L ct + L yt k t = K ct + K yt e ct, e yt iven 5 Full depreciation is introduced in order to simplify the analysis and to focus on the role of preferences. Of course, local indeterminacy may also occur under partial depreciation see Nishimura and Venditti [0 for some results with a linear utility function). 4
Denote by p t, w t and r t respectively the price of the capital ood, the wae rate of labor and the rental rate of the capital ood at time t 0, all in terms of the price of the consumption ood. he Laranian is: L t = α K ρc ct +α 2 L ρc ct σ +e ct) σ ρc l+γ t +γ + w tl t L ct L yt ) [ ) + r t k t K ct K yt ) + p t β K yt + β 2 L yt + e yt k t+ 3) For any iven k t, k t+, l t ), solvin the first order conditions ives input demand functions K c = K c k t, k t+, l t, e ct, e yt ), L c = L c k t, k t+, l t, e ct, e yt ), K y = K y k t, k t+, l t, e ct, e yt ), L y = L y k t, k t+, l t, e ct, e yt ). We then define the production frontier as c t = k t, k t+, l t, e ct, e yt ) = Usin the envelope theorem we derive: ) /ρc α K ρ c c + α L ρ c 2 c + e ct r t = k t, k t+, l t, e ct, e yt ) p t = 2 k t, k t+, l t, e ct, e yt ) 4) w t = 3 k t, k t+, l t, e ct, e yt ) where = k t, 2 = k t+ and 3 = l t. he representative consumer s optimization proram is then iven by [ max δ t σ k t, k t+, l t, e ct, e yt ) l+γ t {k t,l t} t=0 σ + γ t=0 s.t. k 0, e ct ) + t=0, e yt) + t=0 iven with δ 0, ) the discount factor. he first order conditions ive the Euler equation and the equation describin the consumption/leisure trade-off p t c σ t = δr t+ c σ t+ w t c σ t = l γ t From the input demand functions toether with the external effects ) considered at the equilibrium we may define the equilibrium factors demand fonctions ˆK i = ˆK i k t, k t+, l t ), ˆLi = ˆL i k t, k t+, l t ) so that ê c = ê c k t, k t+, l t ) = a ˆK ρ c c b ˆL ρ y 2 c. From 4) prices now satisfy 5) + a 2 ˆL ρ c c and ê y = ê y k t, k t+, l t ) = b ˆK ρ y y + 5
rk t, k t+, l t ) = k t, k t+, l t, ê c k t, k t+, l t ), ê y k t, k t+, l t )) pk t, k t+, l t ) = 2 k t, k t+, l t, ê c k t, k t+, l t ), ê y k t, k t+, l t )) 6) wk t, k t+, l t ) = 3 k t, k t+, l t, ê c k t, k t+, l t ), ê y k t, k t+, l t )) and the consumption level at time t is iven by ck t, k t+, l t ) = k t, k t+, l t, ê c k t, k t+, l t ), ê y k t, k t+, l t )) 7) We then et equations 5) evaluated at ê c and ê y : pk t, k t+, l t )ck t, k t+, l t ) σ = δrk t+, k t+2, l t+ )ck t+, k t+2, l t+ ) σ wk t, k t+, l t )ck t, k t+, l t ) σ = l γ t 8) Any solution {k t, l t )} + t=0 is called an equilibrium path. which also satisfies the transversality condition lim t + δt ck t, k t+, l t ) σ pk t, k t+, l t )k t+ = 0 2.2 Steady state and characteristic polynomial A steady state is defined by k t = k, y t = y = k, l t = l and is determined as a solution of δrk, k, l) pk, k, l) = 0 wk, k, l)ck, k, l) σ l γ 9) = 0 he methodoloy used in this paper consists in approximatin equations 8) in order to compute the steady state and the characteristic polynomial. We will assume the followin restriction on parameters values: Assumption. ρ y, ˆρ y ) with ˆρ y ln ˆβ /[lnδβ ) ln ˆβ > 0 Such a restriction is quite standard with CES technoloies. Indeed when the elasticity of capital-labor substitution is less than, the Inada conditions are not satisfied and corner solutions cannot be a priori ruled out. Assumption precisely uarantees positiveness and interiority of all the steady state values for the input demand functions K c, K y, L c and L y. We start by provin existence and uniqueness of the steady state k, l ). As mentioned in Remark in Appendix 6.2, the production frontier, i.e. 6
equation 7), when evaluated alon a stationary solution, is homoeneous of deree one. It follows that the price functions 6) are homoeneous of deree zero. Denotin κ = k /l, a steady state may thus be characterized as a pair κ, l ) solution of: δrκ, κ, ) pκ, κ, ) = 0 wκ, κ, )cκ, κ, ) σ l γ+σ = 0 0) Notice from the first equation that κ is only determined by technoloical characteristics and does not depend on preferences. Once κ is obtained, the second equation ives the steady state value of the labor supply l. Proposition. κ, l ) > 0, such that: κ = l = δβ ) + Under Assumption, there exists a unique steady state α β 2 +ρc α 2 β [ [» σ δβ ) + α β 2 +ρc α 2 β Proof : See Appendix 6.. ) + +ρc) δβ ) + ˆβ δβ ) + ˆβ α β 2 δβ ) β ρc α β ˆα +ˆα 2 +ρc 2 α 2 β Consider the followin notations + ˆβ ) ρc +ρc) ) + δβ ) + ˆβ ) ρc+) +ρc σ ρc +ρc) i k t, k t+, l t ) = i k t, k t+, l t, ê c k t, k t+, l t ), ê y k t, k t+, l t ))/ k t i2 k t, k t+, l t ) = i k t, k t+, l t ), ê c k t, k t+, l t ), ê y k t, k t+, l t ))/ k t+ i3 k t, k t+, l t ) = i k t, k t+, l t, ê c k t, k t+, l t ), ê y k t, k t+, l t ))/ l t for i =, 2, 3. he linearization of the Euler equation around κ, l ) ives the followin characteristic polynomial: heorem. Under Assumption, the characteristic polynomial is { ) ) } Px) = γ 3 32 x + l x + δ + σ 2 c δbx A) x ) [ σ 2 c + C) 2 D x ˆβ [ δβ ) + x δβ ) + 7 κ σ γ+σ
with δ = { δβ ) + = ˆβ δβ ) A = ˆα α +ρ y)δβ )» B = ˆα +ρ y) 2 β 2 α 2 ) ρc C = ˆα 2 α β 2 +ρc ˆα α 2 β D 2 c 3 [ ) α β 2 +ρc α 2 β [ ) ˆα ˆβ ρc 2 α2 β +ρc ˆα 2 ˆβ α β 2 δβ ) + ˆβ ) } ρc +ρc) ) ρc +ρc) δβ ) + + ˆβ [ + δβ ) + +δβ +ρ c) ˆα 2 ˆβ α β 2 + «ˆα ˆβ2 α 2 β δ»» +ρ y)δβ ) + δβ ) + ++ρ c) δβ + ˆβ [ «δβ ) + δβ ) + ˆβ ˆα α 2 β ++ρ ˆα 2 ˆβ α β 2 c)δβ + «δ»» +ρ y)δβ ) + δβ ) + ++ρ c) δβ + ˆβ δβ ) + ˆβ ) ρc+) +ρc) > 0 δβ ) + ˆβ ) + +ρc)»» +ρ y)δβ ) + δβ ) + ++ρ c) δβ + ˆβ ˆα» +ρ c)+ρ y)δβ ) + δβ ) + C ) = ˆα α β 2 +ρc α α 2 β = α = 2 32 l c Proof : See Appendix 6.2. [ δβ ) + κ ˆα α C + C) We have now to study the stability properties of the steady state dependin on the value of preference and technoloical parameters. As in Nishimura and Venditti [8, if the elasticities of capital-labor substitution are identical across sectors, the consumption ood is capital intensive at the private level if and only if α β 2 α 2 β > 0 while it is capital intensive at the social level if and only if ˆα ˆβ2 ˆα 2 ˆβ > 0. However, with asymmetric elasticities of capital-labor substitution, the capital intensity differences between sectors also depend on the prices and the parameters ρ c and ρ y. As shown in Nishimura and Venditti [9, we have the followin characterization at the steady state: Proposition 2. Under Assumption, at the steady state: i) the consumption investment) ood sector is capital intensive from the 8
private perspective if and only if δβ ) + ˆβ ) ρc +ρc) ) < >) α β 2 +ρc ) α 2 β ii) the consumption investment) ood sector is capital intensive from the social perspective if and only if δβ ) + ˆβ ) ρc +ρc) ) < >) ˆα ˆβ2 +ρc ˆβ2 β ˆα 2 ˆβ ) ρc +)+ρc) ) ˆβ β 2 Remark: If both techonoloies are Cobb-Doulas with ρ c = ρ y = 0, we et the followin expressions: = {δβ [ α β 2 α 2 β }, δ = ˆβ [ ˆα ˆα 2 ˆβ Condition ) becomes < >)α β 2 /α 2 β and condition ) becomes < >)ˆα ˆβ2 /ˆα 2 ˆβ. It follows that / is positive if and only if the consumption ood is capital intensive at the private level while / is positive if and only if the consumption ood is capital intensive at the social level. Moreover, notice from ) and the expression / in heorem that as in the Cobb-Doulas formulation, / is positive if and only if the consumption ood is capital intensive at the private level. On the contrary, when ρ c ρ y 0, the sin of / does not directly depend on the sin of the capital intensity difference across sectors at the social level. In order to simplify the exposition in the rest of the paper, we will discuss the local stability properties of the steady state dependin on the sin of the differences α β 2 α 2 β and ˆα ˆβ, 6 and the values of the elasticities of substitution in both sectors. We will only refer to capital intensities when the results are economically interpreted. We now introduce the followin standard definition. Definition. A steady state k is called locally indeterminate if there exists ɛ > 0 such that from any k 0 belonin to k ɛ, k + ɛ) there are infinitely many equilibrium paths converin to the steady state. 6 We have indeed ˆα ˆβ2 ˆα 2 ˆβ = ˆα ˆβ. 9
If both roots of the characteristic equation have modulus less than one then the steady state is locally indeterminate. If a steady state is not locally indeterminate, then we call it locally determinate. 3 Preliminary results Before analyzin the eneral confiuration with σ, γ > 0, we start by considerin three polar cases: in the first one we assume a linear utility function with recpect to labor, i.e. an infinite elasticity of the labor supply. he second extreme confiuration will be based on the assumption of linear utility with respect to consumption, i.e. an infinite elasticity of intertemporal substitution in consumption. A third particular case concerns the model with inelastic labor and non-linear utility function with respect to consumption. From these polar cases, we will be able to study the eneral formulation. 3. Infinite elasticity of the labor supply: σ > 0, γ = 0 Consider the case of a linear utility function with respect to labor, i.e. γ = 0. he characteristic polynomial then reduces to [ Px) = x ˆβ δβ ) + [x δβ ) + = 0 and the characteristic roots can be explicitely computed, namely x = ˆβ δβ ) +, x 2 = δβ ) + > Assumption also implies x 0, ) so that we et: heorem 2. Under Assumption, if γ = 0 the steady state is saddle-point stable. his result implies that for any intertemporal elasticity of substitution in consumption and any amount of external effects, the steady state is locally determinate. his conclusion may be compared to one of the main results provided by Bosi et al.[5. hey show that in a two-sector optimal rowth model with leisure and additive separable preferences, the steady-state is saddle-point stable when the elasticity of the labor supply is infinite utility is linear in labor). 0
his conclusion points out a role of the labor supply which is the complete opposite of the one obtained in one-sector models. Indeed, as shown in Lloyd-Braa et al. [7 and Pintus [, the occurrence of local indeterminacy in areate models requires the consideration of lare elasticities of labor supply. Such a drastic difference is explained by the fact that there exists a discontinuity between one-sector and two-sector models. Indeed, if the capital intensity difference at the private level is equal to zero, then / = and the characteristic polynomial is no loner well-defined. 3.2 Infinite elasticity of intertemporal substitution in consumption: σ = 0, γ > 0 Consider the case of a linear utility function with respect to consumption, i.e. σ = 0, in which the elasticity of intertemporal substitution in consumption is infinite. he characteristic polynomial then reduces to ) ) Px) = x + δx + Aain the characteristic roots can be explicitely computed, namely x =, x 2 = δ his case has been analyzed by Nishimura and Venditti [9. However, a new conclusion is exhibited here: Proposition 3. Under Assumption, let σ = 0. hen the local stability properties of the steady state only depend on the CES coefficients α i, β i, a i, b i, ρ i ), i = c, y, and do not depend on the elasticity of the labor supply ɛ l = /γ. When the utility function is additively separable and the elasticity of intertemporal substitution in consumption is infinite, the local stability properties are completely characterized by the production side of the model. A similar conclusion is obtained by Bosi et al. [5 in a discrete-time two-sector optimal rowth model with leisure and additively separable preferences. As shown in Nishimura and Venditti [9, local indeterminacy cannot hold when the investment ood is capital intensive at the private level. We then need to introduce the followin assumption:
Assumption 2. he consumption ood is capital intensive at the private level. We will focus in the rest of the paper on confiurations with α β 2 > α 2 β. With Cobb-Doulas technoloies this condition is equivalent to Assumption 2. Indeed, as shown in Benhabib, Nishimura and Venditti [4, local indeterminacy in a Cobb-Doulas economy with linear preferences is obtained if and only if α β 2 α 2 β > α 2 /δ and ˆα 2 > ˆα ˆβ. he first inequality is then considered as a new Assumption: Assumption 3. α β 2 α 2 β > α 2 /δ A slihtly more restrictive condition than the second inequality will be introduced to et clear-cut conclusions. We will however exclusively focus on parameter confiurations of CES technoloies compatible with a Cobb- Doulas production function in both sectors. 7 heorem 3. Under Assumptions -3, let ˆα 2 > ˆα. hen there exist ρ c, 0) and ρ c 0, + ) with the followin property: For any iven ρ c ρ c, ρ c ), there exists ρ y 0, ˆρ y ) such that the steady state is locally indeterminate for all ρ y, ρ y ). Proof : See Appendix 6.3. In order to ive more precise economic interpretations, assume first that the technoloies in both sectors are Cobb-Doulas, i.e. ρ c = ρ y = 0. Condition ˆα 2 > ˆα implies that the areate share of labor in the consumption ood sector is hiher than the areate share of capital. his inequality, which also implies ˆα 2 > ˆα ˆβ, may be satisfied when, at the social level, the investment ood is capital intensive, i.e. ˆα ˆβ < 0, and is also compatible with a consumption ood which is capital intensive at the social level, i.e. ˆα ˆβ > 0. 7 We provide here a more precise formulation for some results exhibited in Nishimura and Venditti [9. We ive indeed conditions for local indeterminacy which are valid for some intervals of values for both ρ c and ρ y. In [9 on the contrary, for a fixed value of ρ c, we provide some interval for ρ y in which local indeterminacy holds.
Consider now eneral asymmetric elasticities of capital-labor substitution. heorem 3 shows that local indeterminacy still occurs with elasticities of capital-labor substitution which are arbitrarily lare in the investment ood sector. his cannot be the case with symmetric elasticities. 3.3 Inelastic labor supply: σ > 0, γ = + Assumin inelastic labor is equivalent to considerin a zero elasticity of the labor supply, i.e. γ = +. In such a case the utility function becomes uc) = c σ σ and the roots of the characteristic polynomial may be approximated by the roots of the followin equation ) ) x + x + δ + σ 2 c δbx A) x ) = 0 3) A similar formulation has been studied by Nishimura et al. [ but under the assumption that both sectors have the same elasticity of capital-labor substitution, i.e. ρ c = ρ y = ρ. We provide here a much more detailed analysis usin the eometrical method provided by Grandmont et al. [6. It is based on a particular property characterizin the product Det) and sum r) of characteristic roots, which allow to provide a complete characterization of the local stability properties of the steady state and of the occurrence of bifurcations. he characteristic equation 3) is written as with Detσ) = δ x 2 rσ)x + Detσ) = 0 4) +σ 2 c +σδ 2 c B A, rσ) = δ +σa+δb) 2 c +σδ 2 c B 5) We analyze the local stability of the steady state by studyin the variations of rσ) and Detσ) in the r, Det) plane when the inverse of the elasticity of intertemporal substitution in consumption σ varies continuously. Solvin the two equations in 5) with respect to σ shows that when σ covers the interval [0, + ), Detσ) and rσ) vary alon a line, called in what follows, which is defined by the followin equation Det = S r + A + δ δb 3 «+A+δB) + + δ + A δb δ «6)
with the slope of. S = A δb δ + + δ + A δb 7) Since γ = +, labor is inelastic and l =. Moreover, κ does not depend on σ or γ, as shown in Proposition. he steady state then remains the same alon the line. Fiure provides a possible illustration of. Det B C σ σ H σ - 0 - A r Fiure : Stability trianle and line. We also introduce three other relevant lines: line AC Det = r ) alon which one characteristic root is equal to in 4), line AB Det = r ) alon which one characteristic root is equal to in 4), and sement BC Det =, r < 2) alon which the characterisitc roots in 4) are complex conjuate with modulus equal to. hese lines divide the space r, Det) into three different types of reions accordin to the number of characteristic roots with modulus less than. When r, Det) belons to the interior of trianle ABC, the steady state is locally indeterminate. Let σ, σ H and σ in 0, + ) be the values of σ at which the line respectively crosses sements AB, BC and AC. hen as σ respectively oes throuh σ, σ H or σ, a flip, Hopf or transcritical bifurcation is enerically expected to occur. We have shown that a unique steady state always exists so that if the critical value σ exists, it will be only associated with a loss of stability of the steady state. We compute the startin and end points of the pair rσ), Detσ)) on the line as σ moves from 0 to +. Consider first the startin point 4
when σ = 0. Straihtforward computations from 5) ive Det0) = δ, r0) = δ Det0) and r0) correspond to the values of the characteristic roots obtained in Section 3.2 under an infinite elasticity of intertemporal substitution in consumption. heorem 3 ives sufficient conditions for r0), Det0)) to belon to the trianle ABC. If ˆα 2 > ˆα, there exist ρ c, 0) and ρ c 0, + ) with the followin property: for any iven ρ c ρ c, ρ c ), we can define a function ρ y ρ c ) 0, ˆρ y ) such that local indeterminacy occurs for all ρ y, ρ y ρ c )). We state these conditions as an Assumption in order to focus on the case with local indeterminacy when utility is linear in consumption with such values ρ c, ρ c and a function ρ y ρ c ). Assumption 4. ˆα 2 > ˆα, ρ c ρ c, ρ c ) and ρ y, ρ y ρ c )) with ρ c, 0), ρ c 0, + and ρ y ρ c ) 0, ˆρ y ). Under this Assumption, the startin point of with σ = 0 is within the trianle ABC as indicated in Fiure 2. Consider now the end point of the locus rσ), Detσ)) when σ = +. From 5) we et Notice that Det+ ) = A δb, r+ ) = + A δb 8) r+ ) + Det+ ) = 0 9) so that the point r+ ), Det+ )) lies on the line AC. It follows that the critical value σ mentioned on Fiure is equal to +. Based on these results, in order to locate the line we finally need to study the slope S and how Detσ) and rσ) vary with σ. Lemma. Under Assumptions -4, there exists σ 0, + ) such that lim σ σ Detσ) =, lim σ σ + Detσ) = +, and Detσ) and rσ) are monotonically decreasin in σ over each of the intervals [0, σ ) and σ, + ). Moreover, S 0, ). Proof : See Appendix 6.4. he critical value σ in Lemma is defined from 5) and is equal to 5
σ = c δb 2 20) Notice also that, as shown in Section 3.2, when σ = 0 the characteristic roots are real. Under Assumption 4, since S 0, ), we conclude that complex characteristic roots cannot occur and any real root cannot be equal to for finite values of σ. herefore, startin from r0), Det0)) into the trianle ABC, when σ increases, the point σ), Dσ)) decreases alon a line as σ 0, σ ), oes throuh when σ = σ and finally decreases from + as σ > σ until it reaches the end point r+ ), Det+ )) which is located on the line AC, as shown in the followin Fiure: Det σ = + σ = 0 r σ Fiure 2: Local indeterminacy with low σ and γ = +. We derive from heorem 3 and Fiure 2: heorem 4. Let Assumptions -4 hold. hen there exists σ 0, + ) such that the steady state is locally indeterminate when σ [0, σ) and saddle-point stable when σ > σ. Moreover, the steady state underoes a Flip bifurcation when σ = σ so that locally indeterminate resp. saddle-point stable) periodtwo cycles enerically exist in left resp. riht) neihbourhood of σ. he bifurcation value σ in Lemma is defined as the solution of + r + Det = 0 and is equal to σ = ) ) δ c 2A+δB) 2 Moreover, as shown in Fiure 2, we necessarily have σ < σ. 2) heorem 4 provides conditions for the occurrence of local indeterminacy which are compatible with the standard followin intuition formulated 6
for instance in Benhabib and Nishimura [3: expectations-driven fluctuations are more likely under a hih elasticity of intertemporal substitution in consumption ɛ c = /σ. When the elasticity of intertemporal substitution is decreased and crosses ɛ c = / σ, the steady state becomes saddle-point stable throuh a Flip bifurcation and endoenous equilibrium cycles are enerated in a neihbourhood of ɛ c. 4 Main results: σ > 0 and γ > 0 We finally consider the eneral confiuration with σ, γ > 0. From the three polar cases previously analyzed, we will be able to provide a clear picture of the local stability properties of equilibria. Indeed, as in Section 3.3, a simple eometrical method based on the variations of r and Det in the r, Det) plane can be applied. We easily derive from the characteristic polynomial in heorem the expressions of Det and r: Detσ, γ) = γ 3 rσ, γ) = γ 3 32 l δ +σ 2 «c A σ 2 c +C) 2 D ˆβ δβ ) + γ 3 32 l +σ 2 «c δb σ 2 c +C) 2 Dδβ ) + 32 l + δ σa+δb) 2 «c +σ 2 c +C) 2 D + ˆβ δβ ) + γ 3 32 l +σ 2 «c δb σ 2 c +C) 2 Dδβ ) + ) ) Solvin the two equations in ) with respect to γ shows that when the inverse of the elasticity of labor supply γ covers the interval [0, + ), Detσ, γ) and rσ, γ) also vary alon a line, denoted γ, which is defined as Det = S γ σ) r+! + ˆβ δβ ) + with δβ ) S γ σ) = + δβ ) + δ δ +σ 2 «c A σa+δb) 2 c + ˆβ δβ ) + ˆβ δβ ) + +σ 2 «c δb δβ ) + the slope of γ. + δ σa+δb) 2 c + δ «+ + ˆβ δβ ) + + σa+δb) 2 «c «! +σ 2 c δb δ «+ + ˆβ δβ ) + +σ 2 c «A! +σ 2 c δb 23) «24) We compute the startin and end points of the pair rσ, γ), Detσ, γ)) on γ as γ moves from 0 to +. From ) we et 7
Detσ, 0) = ˆβ δβ, Detσ, + ) = δ +σ 2 c +σδ 2 c B A rσ, 0) = δβ ) + + ˆβ δβ ), rσ, + ) = σa+δb) 2 c +σδ 2 c B + δ 25) Of course, Detσ, 0) and rσ, 0) correspond to the values of the characteristic roots obtained in Section 3. under an infinite elasticity of the labor supply while Detσ, + ) and rσ, + ) correspond to the values of the characteristic roots obtained in Section 3.3 under inelastic labor. Notice that under Assumption the end point rσ, 0), Detσ, 0)) satisfies ) rσ, 0) + Detσ, 0) = δβ ) + δβ ) + ˆβ ) δβ ) < 0 so that it lies below the line AC with Detσ, 0) > and rσ, 0) > 2. Geometrically, for some iven value of σ, the end point rσ, + ), Detσ, + )) on the line γ belons to the line which is obtained by settin γ to be + and which is analyzed in Section 3.3. Based on these results, in order to locate the line γ we finally need to study the slope S γ σ) and how Detσ, γ) and rσ, γ) vary with γ. introduce an additional Assumption to et clear-cut results: Assumption 5. α β 2 α 2 β > ˆα ˆα 2 ˆβ Assumption 3 implies α β 2 /α 2 β >. We Assumption 5 is then compatible with ˆα ˆβ2 /ˆα 2 ˆβ > and ˆα ˆβ2 /ˆα 2 ˆβ <, i.e. with both capital intensity differences at the social level when the technoloies are Cobb-Doulas. Lemma 2. Under Assumptions -5, consider the critical value σ as defined by 20). hen there exist σ σ, + ) and σ 2 σ, + ) such that the followin results hold: 8 i) Det 2 σ, γ) > 0 when σ 0, σ ), Det 2 σ, γ) = 0 when σ = σ and Det 2 σ, γ) < 0 when σ > σ ; ii) r 2 σ, γ) > 0 when σ 0, σ 2 ), r 2 σ, γ) = 0 when σ = σ 2 and r 2 σ, γ) < 0 when σ > σ 2 ; 8 he partial derivatives of Detσ, γ) and rσ, γ) with respect to γ are respectively denoted Det 2σ, γ) and r 2σ, γ). 8
iii) S γ σ) > 0 when σ 0, min{σ, σ 2 }) max{σ, σ 2 }, + ) and S γ σ) < 0 when σ min{σ, σ 2 }, max{σ, σ 2 }). Proof : See Appendix 6.5. Buildin on our previous results, we are lookin for conditions for the occurrence of local indeterminacy which are compatible with the case of inelastic labor. Accordin to heorem 4, we thus introduce a restriction on the elasticity of intertemporal substitution in consumption which ensures that rσ, + ), Detσ, + )) belons to the trianle ABC. Assumption 6. σ [0, σ) with σ > 0 as defined by 2). Since σ < σ, Assumption 6 implies σ < min{σ, σ 2 } so that Detσ, γ) and rσ, γ) are increasin functions of γ with S γ σ) > 0. Moreover, we et the followin result: Lemma 3. Under Assumptions -6, for any iven σ [0, σ), there exists γ 0, + ) such that lim γ γ + Detσ, γ) =, lim γ γ Detσ, γ) = +, and S γ σ) 0, ) for all γ > 0. Proof : See Appendix 6.6. All these results may be summarized with the followin eometrical representation: Det γ γ = 0 σ = + σ = 0 r γ = + γ γ Fiure 3: Local indeterminacy with low σ and lare γ. When γ is lare enouh, i.e. when the elasticity of labor supply is low enouh, the steady state is locally indeterminate. We finally derive from heorem 4, Lemmas 2-3 and Fiure 3: 9
heorem 5. Let Assumptions -5 hold. hen, for any iven σ satisfyin Assumption 6, there exists γ 0, + ) such that the steady state is locally indeterminate when γ > γ and becomes saddle-point stable when γ < γ. Moreover, the steady state underoes a Flip bifurcation when γ = γ so that locally indeterminate resp. saddle-point stable) period-two cycles enerically exist in a riht resp. left) neihbourhood of γ. heorem 5 shows that the steady state is locally indeterminate provided that the elasticity of intertemporal substitution in consumption is lare enouh while the elasticity of labor supply is low. We also immediately derive from Lemma 2 that the consideration of endoenous labor does not provide any additional room for the occurrence of local indeterminacy. Indeed, the existence of multiple equilibria aain requires σ to be lower than the bound σ exhibited in the case of inelastic labor. 9 5 Concludin comments he main objective of this paper has been to discuss jointly the roles of the elasticity of intertemporal substitution in consumption and the elasticity of the labor supply on the local indeterminacy properties of the lon-run equilibrium. We have considered a discrete-time two-sector model with sector specific externalities in which the technoloies are iven by CES functions with asymmetric elasticities of capital-labor substitution, and the preferences of the representative aent are iven by a CES additively separable utility function defined over consumption and leisure. Different specific confiurations have been studied in order to identify the influence of each precise parameter. In a first step we have clearly showed that when inelastic labor is considered, the existence of local indeterminacy is obtained if and only if the elasticity of intertemporal substitution in consumption is lare enouh. his result confirms basic intuitions obtained within one-sector models. In a second step, we have proved that the consid- 9 Indeed, when σ σ, σ ) and Det 2σ, γ) > 0, or when σ > σ and Det 2σ, γ) < 0, the pair r, Det) is necessarily outside of the trianle ABC for any γ > 0. 20
eration of endoenous labor within a two-sector model does not introduce any additional dimensions for the occurrence of local indeterminacy. Indeed we have shown on one hand that when the labor supply is infinitely elastic, local indeterminacy is ruled out and the steady state is always saddle-point stable, no matter what the elasticity of intertemporal substitution in consumption and the size of externalities are. On the other hand, we have proved that when the elasticity of intertemporal substitution in consumption is lare, local indeterminacy requires a low enouh elasticity of the labor supply. hese results show that the effects of the elasticity of the labor supply appear to be in complete opposition with the main conclusions obtained within one-sector models. 6 Appendix 6. Proof of Proposition From the Laranian 3) we derive the first order conditions: α c/k c ) +ρc r = 0 26) α 2 c/l c ) +ρc w = 0 27) pβ y/k y ) + r = 0 28) pβ 2 y/l y ) + w = 0 29) Usin K c = k 0 K y, L y = L c, and manipulatin 26)-29) ive L y = y ˆβ Ky )/ ˆβ ) / 2 30) ) ) α β 2 k0 K +ρc ) + y Ly = 3) α 2 β L y K y By solvin 30)-3) with respect to K y and substitutin y = k, we et K y = k 0, k, l). From 26) and 30) we et ) ρc r = α {ˆα + ˆα α β 2 +ρc 2 α 2 β Moreover we have from 28) and 27) /y) ˆβ p = w β y ) +, w = r β 2 β /y) ˆβ 2 } ) ρc+) +ρc ρc +ρc) ) + 32)
From the envelope theorem we finally conclude that = r, 2 = p and 3 = w. A steady state k, l ) is a solution of equations 9) with y = k. Usin the derivatives of in the definition of k ives k, k, l ) = δβ ) + k 33) Constant returns to scale at the social level imply that k, k, l) is homoeneous of deree see Remark in Appendix 6.2). Denotin κ = k/l and ḡ = κ, κ, ), we may then derive from 33) ḡ = δβ ) + κ 34) When k 0 = k, equations 30)-3) with the fact that /y = ḡ/κ ive ) Kc = lκ δβ ) +, L y = lκ δβ ) δβ ) + + ˆβ ) 35) and L c = l L y. Under Assumption, k > K c > 0 and l > L y > 0. Substitutin these input demand functions into equation 3) we et the expression of κ. From 2), 34) and 35) we compute c = [ «κ l δβ ) + ρc α β ˆα +ˆα 2 +ρc δβ ) 2 α 2 β + ˆβ ) ρc+) ρc 36) +ρc) Substitutin this expression into wc σ l γ = 0 finally ives l. 6.2 Proof of heorem Various preliminary lemmas are necessary to prove heorem. Denote in what follows, 2, 3 the partial derivatives of with respect to k 0, k, l. Lemma 6.. Under Assumption, at the steady state the followin hold = +ρc K c, 2 = [+)++ρc)ly/lc)ly/y), y 3 = +ρc L c with, K c, L y, L c respectively iven by equations 33)-35) and ) = + + +ρc K c + + ρ y ) + + ρ c ) Ly ˆβ Ly L c Proof : From equation 3) we et α β 2 α 2 β = k 0 ) +ρc y ˆβ ) + { l otally differenciatin this expression ives after simplification ) ) y ˆβ } ρc
[ + ρ y ) + + ρ c )k 0 ) + + ρ y ) { ) + + ρ c ) y ˆβ } y ˆβ = + ρ c )k 0 ) dk 0 + + ρ y ) { ) + + ρ c ) y ˆβ } y ˆβ y ˆβ ) + ) y ˆβ y ) + ) ˆβ ˆβ d dy y dy Notice from 30) and 3) that ) /y) ˆβ ) ) + = L y and α β 2 +ρc Ky +ρc α 2 β L y = Kc 37) L c Substitutin 37) into the previous total differenciation and considerin dy = dk toether with = + + +ρc we derive [ +ρ d = dk c 0 K c + dk his completes the proof. K c + + ρ y ) + + ρ c ) Ly L c ) ˆβ + ρ y ) + + ρ c ) Ly L c y ) Ly ) Ly y dl +ρ c L c Lemma 6.2. Under Assumption, at the steady state the followin holds ) = 2y Kc L y L c, K 3 = c L c with, K c, L y, L c respectively iven by equations 33)-35). Proof : From Lemma 6. and equation 30) we have ) 2 y = +ρc) K c Kc L y L c he result follows. Remark : From Lemma 6.2 we derive that at the steady state the function k, y, l) is homoeneous of deree since k, k, l) = k+ 2 k+ 3 l. It follows from 36) that the production frontier, i.e. equation 7), when evaluated alon a stationary solution, is also homoeneous of deree. Consider the followin notations i k 0, k, l) = i k 0, k, l, ê c k 0, k, l), ê y k 0, k, l))/ k 0 i2 k 0, k, l) = i k 0, k, l, ê c k 0, k, l), ê y k 0, k, l))/ k i3 k 0, k, l) = i k 0, k, l, ê c k 0, k, l), ê y k 0, k, l))/ l and ij k, k, l ) ij for i, j =, 2, 3. 23
Lemma 6.3. Under Assumption, at the steady state, k 0 = k = y = k and the followin hold: k,k,l ) k,k,l ) = y k,k,l ) k,k,l ) = β y ) Kc L y L c = 3 k,k,l ) 32 k,k,l ) [ α2 β ˆα ˆβ2 K c L y α β 2 ˆα 2 L c + ˆβ ) 2 L y y 2 k,k,l ) k,k,l ) = k,k,l ) k,k,l k,k,l ) ) k,k,l ) 3 k,k,l ) k,k,l ) = Kc L c Kc L y L c 23 k,k,l ) k,k,l ) = k,k,l ) 3 k,k,l ) k,k,l ) k,k,l ) ) = 33 k,k,l ) 32 k,k,l ) with, K c, L y, L c respectively iven by equations 33)-35). Proof : By definition we have = r k 0, = r k, 3 = r l, 2 = p k 0, = p k, 23 = p l, Simple computations ive ) ρc ) ρc r k 0 = + ρ y ) r +ρc α r ˆα 2 α β 2 +ρc α 2 β ) p k 0 = + [ r β y k 0 + + ρ y )r [ w k 0 = β 2 β δβ ) r k = + ρ y ) + ˆβ ) r δβ ) k 0 + ˆβ ) ρc ) ρc r +ρc α r ˆα 2 α β 2 +ρc α 2 β ) p k = + [ r β y k + + ρ y )r 2y y w k = β 2 β δβ ) r + ˆβ ) [ r δβ ) k ) ρc r +ρc α r ˆα 2 α β 2 l = + ρ y ) ) p l = + [ r β y l + + ρ y)r 3 w l = β 2 δβ ) + ˆβ ) [ r δβ ) β l 3 = w k 0 32 = w k 33 = w l ) δβ ) + ˆβ ) ρc +ρc) y ) + + ρ y )r y ) + ˆβ ) ρc +ρc α 2 β δβ ) + ˆβ ) ρc +ρc) y ) + + ρ y )r y ) δβ ) + ˆβ ) ρc +ρc) y + ˆβ ) + + ρ y )r y ) ) ) 2 y y 2 y y ) 3 3 24
Substitutin r/α ) from 32), and usin 37) we et ) [ ) = + ρ y )r ˆα ˆβ2 α2 β Kc L y y ˆα 2 α β 2 L c + ˆβ ) 2 38) L y Let us denote ) [ ) E = ˆα ˆβ2 α2 β Kc L y y ˆα 2 α β 2 L c + ˆβ ) 2 L y Followin the same procedure we obtain 2 = + ρ y) r = + ρ y) r 23 = + ρ y) r β y Let us now denote We also et ) + β y ) + 2 y β y y F = E), ) + 3 E) = + ρ y)r 2y E y E), 3 = + ρ y)r 3 E ) + [ ) Ly L y ˆβ2 y E 3 = + ρ y)r β 2 β F, 32 = + ρ y)r β 2 hen we conclude = y 2 y = 3 32, 2 y β y F, 33 = + ρ y)r β 2 β 3 F [ ) = ˆα ˆβ2 α2 β Kc L y β y ˆα 2 α β 2 L c + ˆβ ) ρ ) ρ 2 L y y 2 =, 3 = 3y 2 y = 33 32, 23 = 3 Considerin Lemmas 6. and 6.2 completes the proof. We may finally prove heorem. We first evaluate / and /. Substitutin 33) and 37) into the formulas iven in Lemma 6.3 ives the expressions provided in heorem. We need now to compute the characteristic polynomial. From 7), consider the followin notations c = ck, k, l )/ k t, c 2 = ck, k, l )/ k t+, c 3 = ck, k, l )/ l t From Lemma 6., we et c = It follows that [ˆα k ) ρc + ˆα 2 y ˆβ ) / ) ρ c /ρ c 39) 25
) +ρc ) c = ˆα c K c ) + ˆα 2 ˆβ +ρc ) + c Ly L c ) +ρc ) [ c 2 = ˆα c K 2 c ˆα +ρc 2 c Ly L c y ) + ˆβ Ly ) +ρc ) [ +ρc ) c 3 = ˆα c K 3 c + ˆα c 2 L c + ˆβ + Ly 3 Notice now that the first order conditions 26)-29) imply ) +ρ ) +ρ ) +ρ c K c = c α L c = w α 2 = Kc α L c Finally, usin 3) we et after simplifications [ ) c = ˆα α ˆα 2 ˆβ α β 2 ˆα ˆβ2 α 2 β A [ ) c 2 = δ ˆα 2 β 2 α 2 2 ˆβ δ β ˆα α 2 β ˆα 2 ˆβ α β 2 δ B [ ) c 3 = ˆα +ρc ) ˆα 2 Kc α ˆα L c 3 ˆα 2 ˆβ α 2 β G ˆα ˆβ2 α β 2 ) + 2 Considerin Lemma 6.3, total differenciation of equations 8) ives [ dk t+2 δ [ + σδ 2 c B + dk t+ + δ σδ 2 c δb + A) [ [ [ + dk t + σδ 2 c A + dl t+ δ 3 σ 2 c G + dl t 3 + σδ 2 c G = 0 = 0 dk t+ [ + σδ 3 c 32 B + dk t [ σ 3 c 32 A + dl t [ 3 σ 3 c 32 G γ 3 32 l Solvin the first equation with respect to dk t+2 and substitutin the result into the second equation considered one period forward ives [ [ dk t+ {[ + σδ 3 c 32 B σδ2 2 c B + σδ 2 3 c 32 [ + dk t [ + σδ 3 c 32 B + σδ 2 c A {[ [ } + dl t+ + σδ 2 c B δγ 3 32 + σδ 2 3 l c 32 [G + δ 3 B + dl t [ + σδ 3 c 32 B [ 3 + σδ 2 c G = 0 [A + δ } B 26
3 edious computations finally ive the characteristic polynomial { ) ) } Px) = γ 3 32 x + l δx + + σδ 2 c δbx A) x ) [ ) [ ) + σ 2 c δx 3 32 + 3 32 + δ x G + δ 3 B + G A 3 Considerin Lemmas 6.-6.3 and the definitions of A, B, G, we can show that 3 32 32 G + δ 3 B = ) ρc = ˆα 2 α β 2 +ρc ˆα α 2 β + δ = C [A 3 G δβ ) + ˆβ ) ρc+) +ρc) + δ = δ ˆβ δβ ) + + C) δβ ) + ) = ˆα α β 2 +ρc α α 2 β D + C) δβ ) + δβ ) + ˆβ C ) + +ρc) + C) δβ ) + 40) A final substitution of all these expressions into Px) ives the formulation of the characteristic polynomial provided in heorem. Consider now Lemma 6.. Usin 33)-35) ives the followin expressions = 2 =» +ρ c)δβ ) + δβ ) + ˆβ»» +ρ y)δβ ) + δβ ) + ++ρ c) δβ + ˆβ»» ) δβ +ρ y) δβ ) + ++ρ c) δβ + ˆβ»» +ρ y)δβ ) + δβ ) + ++ρ c) δβ + ˆβ Substitutin and 2 into A and B, and considerin the values of / and / then ives the expressions provided in heorem. We finally have to compute 2/c and 3 / 32 l. From 39) and 35) we et after simplifications ) c = [ δβ ) + lκ +ρc 4) α Usin Lemma 6. it follows therefore that [ k δβ ) + + C) c = Moreover from 38) we obtain after simplifications k =» +C) +ρ c)+ρ y)δβ ) + C» +ρ y)δβ ) + δβ ) + 27 ++ρ c)» δβ + ˆβ
hen we et from the two previous expressions [ 2 c = α ˆα +ρ y)δβ ) he result is derived from the fact that» + δβ ) + ++ρ c) δβ + ˆβ» +ρ c)+ρ y)δβ ) + δβ ) + C 2 c = 2 c 42) Finally, to compute 3 / 32 l, consider 32) and 36) with 40). We et 3 32 = l C = 2 c l c C = 2 l c [ δβ ) κ ˆα + C+C) α 43) Notice that 42) and 43) do not depend on the preference parameters σ and γ. 6.3 Proof of heorem 3 Our stratey consists in considerin a fixed value for ρ c and varyin ρ y in order to find intervals of values in which local indeterminacy occurs. o simplify the analysis, we have to impose some restrictions on the parameters ρ c and ρ y such that the followin function is monotone decreasin. ρ y ) = δβ ) + ˆβ ) ρc +ρc) 44) Lemma 6.4. Under Assumption, there is a ρ c ˆρ y, + ) such that for any iven ρ c, ρ c ) there exists ρ y 0, ˆρ y such that ρ y ) is a monotone decreasin function for all ρ y, ρ y ). Proof : Available upon request. We may now start the proof of heorem 3. Consider first the root x. o simplify the exposition we will study its inverse: [ ) x = δβ ) + α β 2 +ρc δβ ) α 2 β + ˆβ ) ρc +ρc) We have shown in Nishimura and Venditti [9 that if x > 0 then necessarily x 0, ) so that local indeterminacy cannot occur. We have therefore to find conditions to et x <. We know that ρ y, ˆρ y ) with ˆρ y > 0. Notice that L Hôpital rule ives: 28
It follows that lim ρ y) = δβ ) ρ y 0 lim ρ y 0 x = δβ [ Under Assumption 3, we then et lim ρ x y 0 ρc lim ρ x y 0 ρc=0 ) α β 2 α 2 β ρc +ρc) 45) +ρc δβ ) = δβ < ρc +ρc) herefore, we derive from Lemma 6.4 that there exist ρ c, 0) and ρ 2 c 0, ρ c such that lim 0 x < for any ρ c ρ c, ρ 2 c). Consider now ) lim ρ x = δβ ) α β +ρc 2 +ρc α y 2 β ˆβ2 It follows that lim x +ρc < if and only if < δα β 2 /α 2 ). Notice then that Assumption 3 can be equivalently written as α β 2 α 2 β > + δβ > δβ and thus δ α β 2 α 2 > It follows that under Assumption 3 the inequality for any ρ c >. Finally we have ˆβ +ρc 2 < δα β 2 /α 2 ) holds lim x = δβ ) +ˆ if ρ c < ˆρ y ρ y ˆρ y [ ) = δβ ) +ˆ α β 2 +ˆ α 2 β if ρ c = ˆρ y = if ρ c > ˆρ y We derive from all this and Lemma 6.4 the followin results: under Assumption 3, a) for a iven ρ c, ρ c), there exists ρ y, 0) such that x < for any ρ y, ρ y); b) for a iven ρ c ρ c, ˆρ y ), there exists ρ 2 y 0, ˆρ y ) such that x < for any ρ y, ρ 2 y). Consider now the root x 2 : [ x 2 = ˆβ ) δβ ) + ˆα ˆβ ρc 2 α2 β +ρc ˆα 2 ˆβ α β 2 δβ ) + ˆβ ) ρc +ρc) 29
We have shown in Nishimura and Venditti [9 see pae 46) that if x 2 > 0 then necessarily x 2 0, ). Since we are lookin for the occurrence of local indeterminacy, we have therefore to find conditions to et x 2 >. Usin aain 45), we et lim ρ y 0 x 2 = ˆβ [ ) ˆα ˆβ ρc 2 α2 β +ρc ˆα 2 ˆβ α β 2 δβ ) Under Assumption 3, we derive lim = if and only if α β 2 ρc α 2 β δβ ) / > ρ y 0 x 2 lim ρ y 0 x 2 +) = ˆβ if and only if α β 2 α 2 β δβ ) / < [ = ˆβ ˆα > if and only if ˆα ρc=0 ˆα 2 > ˆα ˆβ 46) herefore, assumin that ˆα 2 > ˆα ˆβ, there exist ρ 3 c [, 0) and ρ 4 c 0, ρ c such that for any ρ c ρ 3 c, ρ 4 c), lim 0 x 2 >. Consider now ) lim x 2 = δβ ) ρc ρc +ρc ˆα α β 2 +ρc ρ ˆα y 2 α 2 β > α β 2 α 2 δ ) ρc +ρc > ˆα ˆα 2 If we assume that ˆα 2 > ˆα, Assumption 3 implies that there exists ρ 5 c, 0) such that the above inequality will be satisfied for any ρ c ρ 5 c, ρ c ). Finally under Assumption we have lim x 2 ρ y ˆρ y 0, ) if ρ c < ˆρ y = [ ) ˆα α2 β +ˆ ˆα 2 ˆβ α β 2 = if ρ c ˆρ y, ρ c ) if ρ c = ˆρ y herefore there exists ρ 6 c 0, ρ c ) such that lim ˆρ y x 2 > for any ρ c < ρ 6 c. We derive from all this and Lemma 6.4 the followin results: under Assumption 3 and ˆα 2 > ˆα, c) for any iven ρ c max{ρ 3 c, ρ 5 c}, ρ 4 c), there exists ρ 4 y 0, ˆρ y ) such that x 2 > for all ρ y, ρ 4 y). d) for any iven ρ c max{ρ 3 c, ρ 5 c}, min{ρ 4 c, ρ 6 c}), x 2 > for all ρ y, ˆρ y ). he final result is derived from a)-c) considerin ρ c = max{ρ c, ρ 3 c, ρ 5 c}, ρ c = min{ρ 2 c, ρ 4 c} and ρ y = min{ρ 2 y, ρ 3 y, ρ 4 y}. 30
6.4 Proof of Lemma It is easy to derive from heorem that [ + δ = δβ ) + δβ ) + ˆβ δβ + ˆβ ) = δβ ) + α β 2 +ρc α 2 β +C C δβ ) + ˆβ ) ρc [ +ρc) ˆβ + ˆα α 2 β ˆα 2 α β 2 C It follows that A > 0 and B > 0. Moreover, direct computations ive with A δb = ˆβ δβ» +ρ y)δβ ) + δβ ) +» +ρ y)δβ ) + δβ ) + X = ˆα 2 ˆβ α β 2 ˆα ˆβ2 α 2 β [ ˆβ δβ ) +δβ+ρ c) ˆα 2 ˆβ α β 2 + ˆα ˆβ2 α 2 β δ X +δβ+ρ c) ˆα 2 ˆβ α β 2 + ˆα ˆβ2 α 2 β δ + + ˆβ δβ ) + ««47) Since ˆβ /δβ >, a sufficient condition for A/δB to be reater than is X <. On the contrary, X > is a necessary condition for A/δB to be less than. Notice that X > if and only if Considerin aain heorem, let us now denote ˆα ˆβ2 ˆα 2 ˆβ < α β 2 α 2 β 48) 2 c H We derive from the above results that H > 0. Straihtforward computations from 5) also ive Det σ) = A HδB δb «δ» +σδ 2 c B, r σ) = 2 Recall that when σ = 0, the characteristic roots are x =, x 2 = δ HA+δB)» + +σδ 2 c B + «δ 2 49) Under Assumption 4, we derive from the proof of heorem 3 that x, 0) and x 2, ). It follows that / > 0 while /δ may be positive or neative. If /δ > 0 then we et + / + /δ > 0. If /δ < 0, then we necessarily have /δ > and we find aain + / + /δ > 0. herefore we et r σ) < 0. he sin of Det σ) is iven by the sin of the followin expression 3