A NOTE ON TARIFF POLICY, INCREASING RETURNS, AND ENDOGENOUS FLUCTUATIONS

Transcription

1 Macroeconomic Dynamics, 2009, Page 1 of 13. Prined in he Unied Saes of America. doi: /s A NOTE ON TARIFF POLICY, INCREASING RETURNS, AND ENDOGENOUS FLUCTUATIONS YAN CHEN Shandong Universiy YAN ZHANG Shanghai Jiao-Tong Universiy We show ha he inroducion of a consan ariff or subsidy levied on foreign energy can lead o a rich se of endogenous flucuaions around he unique seady sae, including sable 2-, 4-, 8-, and 15-cycles, quasiperiodic orbis, and chaos. This is demonsraed in a sandard neoclassical growh model wih social increasing reurns o scale. Numerical exercises could be viewed from a mehodological perspecive as illusraing ha capial income axes and ariffs are equivalen in generaing endogenous flucuaions because Guo and Lansing [Guo, J.T. and K.J. Lansing (2002) Fiscal policy, increasing reurns and endogenous flucuaions. Macroeconomic Dynamics 6, ] show ha a consan capial ax or subsidy has he same effec on he model dynamics in a one-secor closed economy. Keywords: Tariff Policy, Global Indeerminacy, Chaos 1. INTRODUCTION The lieraure on rade axes suggess ha energy axes (or ariffs) have a high efficiency cos when governmen needs o raise revenue by using ax insrumens. 1 Dynamic sochasic general equilibrium (DSGE) models wih energy in producion have been used widely o analyze represenaive agen problems [Roemberg and Woodford (1994); de Miguel and Manzano (2006)]. However, mos of hese papers are concerned wih local seady-sae analyses and deerminacy cases. Neverheless, ariffs on inermediae goods (say, impored energy), which ac like a ax on he reurns o facors of producion, may generae indeerminacy in a way similar o facor income axes. We wish o hank an associae edior, wo anonymous referees, Wen Yi, Kim Jinill, and he audience a he 2009 MEA meeing for helpful commens and Kevin Lansing for his consrucive criicisms on he previous version of his paper. We especially hank he edior, William A. Barne, and Jess Benhabib for encouraging us o wrie his noe. Megha Jain, Bei Zhang and He Qinying s proofreading is also acknowledged. Shanghai Jiao-Tong Universiy provides financial suppor o his projec. All remaining errors are of course our own. The wo auhors made equal conribuions o his paper. Address correspondence o: Yan Zhang, Economics Deparmen, School of Economics, Anai College of Economics & Managemen, Shanghai Jiao-Tong Universiy, 535 Fa Hua Zhen Road, Shanghai , People s Republic of China; laurencezhang@yahoo.com. c 2009 Cambridge Universiy Press /09 1

2 2 YAN CHEN AND YAN ZHANG Recenly, Zhang (2009) showed ha local indeerminacy can emerge when ariff raes levied on impored energy are endogenously deermined by a balancedbudge rule wih a consan level of governmen expendiures (or lump-sum ransfers). In ha paper, we show ha, in he presence of fiscal increasing reurns caused by endogenous ariffs, ariffs and labor income axes [Schmi-Grohe and Uribe (1997)] are equivalen in generaing (local) indeerminacy. However, we did no sysemaically invesigae he equivalen relaionship beween (exogenous) ariffs and (exogenous) capial income axes from he perspecive of global indeerminacy. In his noe, we show ha in a sandard neoclassical growh model wih social increasing reurns o scale, as in Aguiar-Conraria and Wen (2005, 2007, 2008; henceforh ACW), he inroducion of a consan ariff or subsidy levied on foreign energy can lead o a rich se of endogenous flucuaions around he unique seady sae, including sable 2-, 4-, 8-, and 15-cycles, quasiperiodic orbis, and opological chaos. Similar resuls can be found in Guo and Lansing (2002). Their paper shows ha a consan capial ax or subsidy has he same effec on he model s dynamics in a one-secor closed economy wih social increasing reurns o scale, hence proving ha (exogenous) ariffs and (exogenous) capial income axes are equivalen in generaing global indeerminacy. Sudying global dynamics in a numerical exercise, we show ha a supercriical flip bifurcaion can occur as he ariff rae passes a posiive flip-bifurcaion value. As he ariff rae is furher increased beyond he flip bifurcaion value, he model exhibis a series of period-doubling bifurcaions and chaos. Moreover, a supercriical Hopf bifurcaion can occur as he energy subsidy rae passes a posiive Hopf bifurcaion value. Furher increases in he subsidy rae beyond he Hopf bifurcaion value may break he orbi up ino a regular 15-cycle. We also noice ha when we solve for a benchmark ariff policy ha closes he gap beween he social and privae marginal producs of impored energy in he ACW srucure, we need o se consan subsidy raes o be linked direcly o he increasing-reurns parameer. All of hese resuls are in line wih previous works of Cazzavillan (1996), Chrisiano and Harrison (1999), Pinus, Sands, and de Vilder (2000), Guo and Lansing (2002), and Coury and Wen (2009). We confirm ha purely local analysis can be misleading because in some region of he parameer space, models can exhibi global indeerminacy when hey are locally deerminae and ha global dynamics can be quie differen from local dynamics. The paper is organized as follows. In he nex secion, we presen he model. In Secion 3, we sudy he model s dynamics wih consan subsidy/ariff raes and provide inerpreaions of our findings. Finally, Secion 4 concludes. 2. THE BASIC MODEL The exended model of ACW consiss of hree ypes of agens: firms, households, and he governmen. To remain comparable wih he analysis of Guo and Lansing

3 TARIFF POLICY AND ENDOGENOUS FLUCTUATIONS 3 (2002), we use he version of he ACW model wih social increasing reurns o scale due o he presence of producive exernaliies. 2 A represenaive household maximizes a discouned sream of uiliies over her lifeime by choosing sequences of consumpion {c } =0, hours {n } =0, and he sock of capial {k +1 } =0 : subjec o max ( ) β log c B n1+γ, B>0, (1) 1 + γ =0 c + [k +1 (1 δ) k ] = r k + w n + T, k 0 given, (2) where δ (0, 1) is he capial depreciaion rae, β (0, 1) is he discoun facor, γ 0 is he inverse of he ineremporal elasiciy of subsiuion in labor supply, r is he renal rae of capial, w is he real wage, and T is he lump-sum (ariff revenue) ransfer/ax o he agen, i.e., T can be negaive. The household receives income by supplying capial and labor services o firms and views r, w, and T as being exogenously given. In equilibrium, he firs order opimaliy condiions are given by Bn γ = w, (3) c 1 = β (r δ), (4) c c +1 plus a ransversaliy condiion. Equaion (3) requires ha he household s marginal rae of subsiuion beween consumpion and leisure be equal. Equaion (4) is he consumpion Euler equaion. The producion echnology is given by y = z k a k n a n o a o, (5) where a k + a n + a o = 1, o is he impored energy, and z is a measure of producion exernaliies and is defined as a funcion of average aggregae oupu, which individual firms ake as given: z = ( k a k n a n o a ) o η, η 0. (6) Assuming ha firms are price akers in he facor markes and he inernaional energy price (p o ) is exogenous o he economy, profi maximizaion by each firm

4 4 YAN CHEN AND YAN ZHANG leads o he following firs-order condiions: y r = a k, k (7) y w = a n, n (8) p o y (1 + τ ) = a o, o (9) where τ denoes he ariff rae levied on he impored energy. Wihou governmen spending, he governmen ransfers he ariff revenue o he households, which means ha T = p o τ o = a oτ 1+τ y holds in each period. 3 By combining equaions (2), (7), (8), and (9) and he governmen budge consrain, we obain he aggregae resource consrain k +1 = (1 δ)k + y (1 a o 1+τ ) c. As in ACW (2008), we can subsiue ou z and he opimal demand for o in he producion funcion o obain he reduced-form producion funcion in equilibrium: y = A k α k αn n, (10) a o p o (1 + τ ) ] αo where α k a k (1+η), α n a n (1+η), and α o a o (1+η). A = [ acs as he echnology coefficien in a neoclassical growh model, which is inversely relaed o he energy price and he ariff rae. In he reduced-form producion funcion, he effecive reurns o scale are measured by α k + α n 1 α o. Provided ha η>0, he social echnology exhibis increasing reurns o scale (α k + α n + α o > 1) DYNAMICS WITH CONSTANT TARIFF/SUBSIDY RATES We firs consider he benchmark ariff policy ha is an alernaive o he opimal fiscal policy and closes he gap beween he social and privae marginal producs of he impored energy. PROPOSITION 1. The wedge beween he social and privae marginal producs of he impored energy is eliminaed when τ = 1 1 for all and 1 + η T = a o ηy for all. Proof. The social marginal producs from equaions (5) and (6) is y / o = α o y /o. The afer-ariff privae marginal produc is (1 + τ ) 1 y a o o. Wih τ = η 1 < 0, we have (1 + τ ) 1 y a o o = α o y /o. The lump-sum ariff revenue follows direcly from he equaion T = τ p o o = a oτ 1 + τ y (= a o ηy ). The benchmark policy involves consan subsidy raes ha are governed only by he exernaliy parameer η. A similar resul is also obained in Guo and Lansing (2002). In he following numerical exercise, we assume ha he ariff rae is consan, which implies ha he governmen revenue is endogenous.

5 TABLE 1. Baseline parameers TARIFF POLICY AND ENDOGENOUS FLUCTUATIONS 5 Parameer Value Descripion γ 0 Indivisible labor; see Hansen (1985) β 0.99 Discoun facor; see ACW (2005, 2008) a n 0.7 Labor s share a o 0.16 Oil s share; see ACW (2008) a k 1 a n a o Capial s share δ Depreciaion rae; see ACW (2008) B Implies fracion of ime spen working = η Implies local indeerminacy in he laissez-faire version of he model Following he exising real business cycle (RBC) lieraure, we use he srucural parameers for a quarerly model, as in ACW (2005, 2007, 2008), Hansen (1985), and Guo and Lansing (2002). Table 1 summarizes he baseline parameer values. The baseline parameer values are commonly used in real business cycle models excep he exernaliy parameer η. 6 The degree of reurns o scale in he model is 1+η. Given our baseline parameer values, i requires reurns o scale o be a leas o exhibi local indeerminacy. We se η = for our quaniaive exercise, which implies reurns o scale of around (he laissez-faire economy is locally indeerminae) and he benchmark fiscal policy parameer τ b = In he appendix, we show ha he equilibrium condiions in our model can be described by he following log-linear sysem: [ ] λ ln(k+1 /k) 1 λ [ ] 2 ln(k /k) = λ 1 λ 3 1 λ 2 λ 3, k 0 given, (11) ln(c +1 /c) ln(c /c) λ 4 λ }{{ 4 } J where k and c are seady-sae values of capial and consumpion and J denoes a 2 2 Jacobian marix of parial derivaives evaluaed a he seady sae. The wo eigenvalues of J deermine he sabiliy of he log-linear sysem. The oil price p o and he labor disuiliy parameer B do no appear in J and hus do no affec he model s local sabiliy properies. We observe ha he ariff rae no only affecs he rade-off beween consumpion and leisure a a given dae [see equaion (3)] bu also affecs he rade-off beween consumpion goods a differen daes [see equaion (4)]. And he ineremporal rade-off is crucial for generaing muliple equilibria in his open economy because agens expecaions of fuure reurns mus be self-fulfilling. Table 2 summarizes he model s local sabiliy properies as we vary he ariff rae over he whole real line.

6 6 YAN CHEN AND YAN ZHANG TABLE 2. Sabiliy properies near he seady sae Tariff rae Eigenvalues of Jacobian marix Seady sae τ< Complex µ 1 = µ 2 > 1 Source τ = (Hopf bifurcaion) Complex µ 1 = µ 2 =1 Source changes o sink <τ<τ R = Complex µ 1 = µ 2 < 1 Sink τ R <τ< Real µ 1 < 1, µ 2 < 1 Sink τ = (flip bifurcaion) Real µ 1 = 1, µ 2 < 1 Sink changes o saddle τ> Real µ 1 < 1, µ 2 < 1 Saddle We observe in Table 2 ha local indeerminacy occurs only for ariff raes in he range <τ< When τ > , he model exhibis a locally unique equilibrium. However, wihin a small open neighborhood of τ flip = (in our case τ flip + ε), he seady sae goes from being a sink o being a saddle surrounded by an aracing period-2 cycle as τ increases. In he following global analysis, we show ha he flip bifurcaion poin is supercriical using numerical simulaions, which means ha local deerminacy can coexis wih global indeerminacy when τ> Anoher ineresing finding is ha he dynamic sysem undergoes a Hopf bifurcaion as τ is decreased pas he value τ Hopf = We can also use numerical simulaions o esablish ha he Hopf bifurcaion is supercriical. In he supercriical Hopf bifurcaion, an aracing closed orbi emerges on he side of τ Hopf where he seady sae is unsable (in our case a source), ha is, in he small neighborhood o he lef of τ Hopf. To simulae he global dynamics, we describe he model s perfec foresigh dynamics as follows (see he appendix): αn β c +1 a k Ak α k +αo 1 +1 a na B α k k+1 c +1 (1+γ )() αn + 1 δ = 1, (12) c k +1 = ( 1 a ) o Ak 1 + τ α k a na B k α k c αn (1+γ )() αn + (1 δ) k c. (13) Following Guo and Lansing (2002), we ierae he above map for a range of values of τ. We disurb he seady sae by an arbirary amoun and se our iniial values (k 0, c 0 ), and hen we solve equaion (13) for k 1. Subsiuing he value of k 1 ino equaion (12) yields a nonlinear equaion ha can be used o solve c 1. We can repea he procedure o compue (k 2, c 2 ) and so on. Figure 1 plos he bifurcaion diagram and he larges Lyapunov exponen over he range τ The bifurcaion diagram summarizes he

7 TARIFF POLICY AND ENDOGENOUS FLUCTUATIONS 7 FIGURE 1A. Bifurcaion diagram. Larges Lynapunov Exponen LE> Tariff Rae τ FIGURE 1B. Larges Lyapunov exponen. long-run behavior of he model by ploing he las 250 poins of a simulaion comprising a specified number (5,250) of ieraions. Figures 2 and 3 show ha he Hopf bifurcaion is supercriical because he invarian closed orbi is aracing. As τ = τ Hopf 2.1e 4, raional expecaions equilibrium pahs evenually converge o he invarian closed orbi for arbirary saring poins eiher inside or ouside he circle. Figure 4 shows ha he invarian closed orbi sars o break up ino a regular 15-cycle when he ariff rae is decreased o some poin in he lef-hand side of τ Hopf. In he small neighborhood of he flip bifurcaion poin, he model exhibis sable 2-, 4-, and 8-cycles for ariff raes in he range of τ flip <τ< Figure 5 depics hese hree kinds of cycles and heir corresponding ime-series simulaed daa. When we increase he ariff rae o τ = , a ype of chaoic

8 8 YAN CHEN AND YAN ZHANG FIGURE 2. Phase diagram (aracing circle sar inside): τ = τ Hopf 2.1e 4. aracors emerges, which shows ha pushing he ariff rae beyond τ flip evenually leads o chaos, as indicaed by a significanly posiive Lyapunov exponen [see Figure (6)]. And his ransiion o chaos akes place via a period-doubling roue in he high-ariff region (τ >τ flip ) Hours Worked h Capial Sock k FIGURE 3. Phase diagram (aracing circle sar ouside): τ = τ Hopf 2.1e 4.

9 TARIFF POLICY AND ENDOGENOUS FLUCTUATIONS 9 Hours Worked h Capial Sock k FIGURE 4. Long-run phase diagram (aracing 15-cycle): τ = τ Hopf 3.7e 4. Hours Worked h Hours Worked h Hours Worked h cycle (τ=0.3008) Capial Sock k 4-cycle (τ=0.3858) Capial Sock k 8-cycle (τ=0.3953) Capial Sock k a % Deviaion b % Deviaion c % Deviaion Time-series plo (2-cycle) Quarer Time-series plo (4-cycle) Quarer Time-series plo (8-cycle) Quarer FIGURE 5. 2-, 4-, and 8-Cycles in he viciniy of flip bifurcaion poin.

10 10 YAN CHEN AND YAN ZHANG Hours Worked h Capial Sock k FIGURE 6. Phase diagram (chaoic aracor): τ = From hese figures, we observe ha changes in τ affec he ampliude of hese cycles or oscillaions. In wha follows, we inerpre he occurrence of endogenous flucuaions by explaining why sable cycles can emerge due o he presence of increasing reurns and disorionary ariffs. In he high-subsidy region, he supercriical Hopf bifurcaion can be viewed as a sylized business cycle and characerized by inermien spikes in hours worked and oupu. This reflecs a bunching effec in producion and/or consumpion when agens opimal decisions inernalize he increasing reurns. 7 To make his poin clear, le us consider an increase in he capial sock a he equilibrium. The impac of his is o lead o a decrease in consumpion oday, and he capial sock ends o fall back o is former level. A he same ime, he iniial increase in he capial sock leads o an increase in oupu and/or consumpion in he nex period (because he capial sock in he nex period increases). In order o make Euler equaion (4) hold, he capial sock should be reduced. Tha is because he marginal produc of capial is decreasing in he capial sock (k +1 ) and c +1 /c increases. The laer effec reinforces he decrease of he capial sock. Depending on he srengh of hese effecs, he capial sock may overshoo is long-run equilibrium level. When he capial sock keeps decreasing beyond is equilibrium level, he effecs on he oher variable are reversed. Tha is, consumpion falls. Considering he effec of he capial sock on he invesmen, he fall of he capial sock will be reversed. This process describes possibly convergen or divergen oscillaions around he unique seady sae. And oscillaions ha change sabiliy from being a sink o being a source as he ariff rae changes can be sable, forming closed orbis [hanks o Benhabib and Miyao (1981, p. 591)]. In he high-ariff region, he sable 2n-cycle can emerge as he subsiuion effec generaed by expeced movemens in he afer-ariff ineres rae

11 TARIFF POLICY AND ENDOGENOUS FLUCTUATIONS 11 exceeds he corresponding income effec by an amoun ha is sufficien o induce deerminisic cycles in our dynamic model [see Euler equaion (4)]. 8 Similar o Guo and Lansing (2002), we describe some policy mechanisms ha are used o eliminae sunspo flucuaions near he seady sae. For example, explici adjusmen coss for capial invesmen can be used o selec a locally unique equilibrium. We can consider an economy in which he household budge consrain is described by he following new equaion: c + i 1 + ψ ( ) 2 k k }{{ = r k + w n ; (14) } τ ( ) he adjusmen cos parameer ψ can be used as he bifurcaion parameer. Given oher parameer values in our baseline model, he dynamical sysem undergoes a supercriical flip bifurcaion as ψ is increased pas he value This implies ha global indeerminacy coexiss wih local deerminacy afer he seady sae changes sabiliy a he flip bifurcaion poin. The equilibrium selecion mechanisms designed using he log-linear approximaion mehod are unsuccessful when we observe he rue nonlinear dynamics. 4. CONCLUSION In his paper, we use he ACW model wih (exogenous) ariff raes o show ha capial income axes and ariffs are equivalen in generaing global indeerminacy. Global analysis reveals ha a rich se of endogenous flucuaions including bifurcaions and/or chaos can arise in his model. Our resuls indicae ha cauion should be exercised when linearized versions of his class of RBC models are used because global indeerminacy may coexis wih local deerminacy in some region of he parameer space. NOTES 1. See, for example, Bizer and Suar (1987), Roemberg and Woodford (1994), and de Miguel and Manzano (2006). 2. ACW (2007, 2008) inroduced capaciy uilizaion rae ino he RBC model, as in Wen (1998), in order o reduce he magniude of increasing reurns required for indeerminacy. In his paper, o remain comparable wih Guo and Lansing s analysis, we do no ake ino accoun his facor, which makes he model exhibi local indeerminacy only for quie large exernaliies. We claim ha adding capaciy uilizaion will no qualiaively change he main resul of his paper. However, i will reduce he required exernaliies o an empirically reasonable range, and he model predicions could be compared wih real daa when we calibrae his model. 3. Negaive ariff raes represen energy subsidies, and a negaive value of T represens a lump-sum ax received from he governmen. 4. We consider only he case α k < 1, which says ha he exernaliy is no srong enough o generae susained endogenous growh.

12 12 YAN CHEN AND YAN ZHANG 5. Because our exercise is a numerical exercise, we simply se he implied fracion of ime spen on working o be 0.3, which was used in Guo and Lansing (2002). We claim ha our resuls will be robus for reasonable parameer selecion. Usually, people work 8 hours per day. 6. Our resuls are robus o he parameer p o. In our numerical case, we se p o o be We should menion ha in a subcriical Hopf bifurcaion, a repelling orbi emerges in he small neighborhood o he righ of τ Hopf. The repelling orbi associaed wih an aracive seady sae resembles he corridor of sabiliy concep, as noed by Leijonhufvud (1973) and Benhabib and Miyao (1981). 8. In a subcriical flip bifurcaion, a repelling wo-period cycle emerges in he small neighborhood o he lef of τ flip. REFERENCES Aguiar-Conraria, L. and Y. Wen (2005) Foreign Trade and Equilibrium Indeerminacy. Working paper a, Federal Reserve Bank of S. Louis. Aguiar-Conraria, L. and Y. Wen (2007) Undersanding he large negaive impac of oil shocks. Journal of Money, Credi, and Banking 39, Aguiar-Conraria, L. and Y. Wen (2008) A noe on oil dependence and economic insabiliy. Macroeconomic Dynamics 12, Benhabib, J. and T. Miyao (1981) Some new resuls on he dynamics of he generalized Tobin model. Inernaional Economic Review 22, Bizer, D. and C. Suar (1987) The public finance of a proecive ariff: The case of an oil impor fee. American Economic Review 77(5), Cazzavillan, G. (1996) Public spending, endogenous growh, and endogenous flucuaions. Journal of Economic Theory 71, Chrisiano, L.J. and S.G. Harrison (1999) Chaos, sunspos, and auomaic sabilizers in a business cycle model. Journal of Moneary Economics 44, Coury, T. and Y. Wen (2009) Global indeerminacy in locally deerminae RBC models. Inernaional Journal of Economic Theory 5, de Miguel, C. and B. Manzano (2006) Opimal oil axaion in a small open economy. Review of Economic Dynamics 9, Guo, J.T. and K.J. Lansing (2002) Fiscal policy, increasing reurns and endogenous flucuaions. Macroeconomic Dynamics 6, Hansen, G.D. (1985) Indivisible labor and he business cycle, Journal of Moneary Economics 16, Leijonhufvud, A. (1973) Effecive demand failures. Swedish Journal of Economics 75, Pinus, P.A., D.J. Sands, and R. de Vilder (2000) On he ransiion from local regular o global irregular flucuaions. Journal of Economic Dynamics and Conrol 24, Roemberg, J. and M. Woodford (1994) Energy axes and aggregae economic aciviy. In J. Poerba (ed.), Tax Policy and he Economy, vol. 8, pp Cambridge, MA: MIT Press. Schmi-Grohe, S. and M. Uribe (1997) Balanced-budge rules, disorionary axes and aggregae insabiliy. Journal of Poliical Economy 105, Wen, Y. (1998) Capaciy uilizaion under increasing reurns o scale. Journal of Economic Theory 81, Zhang, Y. (2009) Tariff and Equilibrium Indeerminacy. Mimeo, New York Universiy. hp://mpra. ub.uni-muenchen.de/13099/. APPENDIX This secion summarizes he equaions used o sudy he model s equilibrium dynamics under he assumpion of consan ariff/subsidy raes where τ = τ holds for all. The

13 TARIFF POLICY AND ENDOGENOUS FLUCTUATIONS 13 equilibrium condiions can be shown by he following four equaions: y = Ak α k [ αn n, where A = a o p o (1 + τ) ] αo, (E-1) Bn 1+γ = a n y c, 1 = β c c +1 (E-2) ( ) y +1 a k + 1 δ, (E-3) k +1 c + k +1 = (1 δ) k + y ( 1 a o 1 + τ ). (E-4) For he parameer values in Table 1, here is a unique inerior seady sae in his economy. Equaions (E-1) and (E-2) imply ha ) ( an A (1+γ )() αn α k n = k (1+γ )() αn c (1+γ )() αn, B which can be used o subsiue ou n in equaions (E-3) and (E-4). Then we can obain he dynamical sysem equaions (12) and (13). In he small neighborhood of he seady sae, he dynamic sysem can be approximaed by he log-linearizaion mehod: [ ] λ ln(k+1 /k) 1 λ [ ] 2 ln(k /k) = λ 1 λ 3 1 λ 2 λ 3, ln(c +1 /c) ln(c /c) λ 4 λ }{{ 4 } J c where four elemens in J are λ 1 = δm δ + M 4, λ k 2 = δm 1 + (1 + M 1 ) c, k α λ 3 = M 3 β(ρ + δ),andλ 4 = 1 + M 1 β(ρ + δ). M 1 = n (1 + γ)() (1 + γ)(1 α o) α n, M 2 = (1 + γ)(1 α o) α n, M 3 = (1 + γ )(α k + α o 1)+α n α (1 + γ)(1 α o) α n,andm 4 = k (1 + γ) (1 + γ)(1 α o) α n are he elemens ha make up λ i (i = 1, 2, 3, and 4). The seady-sae capial labor raio is c/k = ρ+δ a k (1 ao ) δ. 1+τ ρ 1/β 1 is he household s rae of ime preference. The deerminan and race of J are given by de(j ) = λ 1 λ 4, (E-5) r(j ) = λ λ 2λ 3. (E-6) λ 4 As in Guo and Lansing (2002), in he Hopf bifurcaion, de(j ) = 1 implies ha τ Hopf = a om 4 (ρ + δ) M 4 (ρ + δ) a k [δ + M 1 1. In he flip bifurcaion, de(j ) + r(j ) = 1 implies ha β(ρ + δ)] τ flip a = oh 1 (ρ + δ) 1, where H H 1 (ρ + δ) + a k (H 2 δh 1 ) 1 = 2M 4 + β(ρ + δ)[m 1 (M 4 M 3 ) M 3 ]and H 2 = 4 2δ + 2δM 4 + β(ρ + δ)[δm 1 (M 4 M 3 ) + (2 δ)m 1 ].