Growth, Habit Formation, and Catching-up with the Joneses

Transcription

1 Growh, Habi Formaion, and Caching-up wih he Joneses Jaime Alonso-Carrera Deparameno de Economia Aplicada Universidade de Vigo Jordi Caballé Unia de Fonamens de l Anàlisi Economica and CODE Universia Auònoma de Barcelona Revised version: June 6, 2003 Xavier Raurich Deparamen d Economia Universia de Girona Absrac When habis are inroduced muliplicaively in a capial accumulaion model, he consumers objecive funcion migh fail o be concave. In his paper we provide condiions aimed a guaraneeing he exisence of inerior soluions o he consumers problem. We also characerize he equilibrium pah of wo growh models wih muliplicaive habis: he inernal habi formaion model, where individual habis coincide wih own pas consumpion, and he exernal habi formaion (or cachingup wih he Joneses) model, where habis arise from he average pas consumpion in he economy. We show ha he inroducion of exernal habis makes he equilibrium pah inefficien during he ransiion owards he balanced growh pah. We characerize in his conex he opimal ax policy. Key words: endogenous growh, habi formaion, consumpion exernaliies, opimal axaion. JEL ClassiÞcaion Number: E2, O40. We hank he anonymous referees of his review and he seminar paricipans a CEMFI for heir valuable commens. Of course, hey should no bear any responsibiliy for he remaining errors. Financial suppor from he Spanish Minisry of Science and Technology hrough grans SEC , BEC and HI , he Generalia of Caalonia hrough he Barcelona Economics program (CREA) and gran SGR is graefully acknowledged. Correspondence address: Jordi Caballé. Universia Auònoma de Barcelona. Deparamen d Economia id Hisòria Econòmica. EdiÞci B Bellaerra (Barcelona). Spain. Phone: (34) Fax: (34) Jordi.Caballe@uab.es

2 Inroducion Several recen papers have inroduced habi formaion in he agens uiliy funcion in order o explain some empirical facs ha canno be reconciled wih he radiional models displaying ime-separable preferences. Examples of his srand of he lieraure are he papers of Abel (990, 999), who provides a possible explanaion of he equiy premium puzzle; Leau and Uhlig (2000), who ry o Þ some sylized facs of business cycles; Ljungqvis and Uhlig (2000), who examine he effecs of Þscal policy under habi formaion; Fuhrer (2000), who sudies he implicaions of habi formaion for moneary policy; and Carroll e al. (997, 2000) and Shieh e al. (2000), who sudy how he paerns of growh are modiþed when habis are presen. The aim of he presen paper is wofold. Firs, o characerize he equilibrium pah of a class of endogenous growh models under habi formaion and, second, o characerize he opimal ax raes ha solve he inefficiencies brough abou by he habis associaed wih he average pas consumpion of he economy. In order o allow for susained growh, we will assume ha he producion funcion is asympoically linear in capial in spie of poenially exhibiing diminishing reurns wih respec o capial (as in Jones and Manuelli, 990). This ype of Sobelow producion funcion consiues one of he main differences of our analysis wih he relaed lieraure. On he one hand, Carroll e al. (997, 2000) characerize he equilibrium pah when he producion funcion is linear in capial and, hence, ransiional dynamics is driven only by habis. In conras, in our paper, ransiional dynamics is driven by boh habi formaion and diminishing reurns o scale. We show ha his difference modiþes he paerns of growh along he ransiion. On he oher hand, Fisher and Hof (2000) analyze he opimal ax policy in a model wihou habi formaion where agens are exposed o conemporaneous consumpion spillovers. Therefore, unlike in our model, ransiional dynamics is driven only by he neoclassical producion funcion hey consider, which in urn prevens he economy from exhibiing susained growh. In our model we will assume ha consumers uiliy depends boh on own curren consumpion and a reference level. This reference is a sandard of living deermined by he own pas consumpion and by he pas average consumpion of he economy. While he isoelasic funcional form for he individuals insananeous uiliy has been exensively used in he lieraure, wo alernaive forms have been used o inroduce habis. One form is he addiive one, according o which habis play in fac he role of a minimum level of consumpion. The oher funcional form is he muliplicaive one, where consumers uiliy depends on heir curren level of consumpion relaive o a reference level deermined by habis. Boh funcional forms exhibi some echnical problems. On he one hand, Carroll (2000) poins ou ha he addiive model may give riseoanowelldeþned uiliy in sochasic economies under plausible calibraions. On he oher hand, we argue in his paper ha he insananeous uiliy funcion is no concave under our muliplicaive formulaion for inernal habis. In his case, he convexiy of he consumers maximizaion problem is no ensured and he opimal pah chosen by consumers migh fail o be inerior. However, we provide a se of assumpions under which he sandard Þrs order condiions characerize inerior equilibrium pahs. We also show ha hose assumpions are me by wo famous models, which are in

3 fac exreme cases of our original model: he inernal habi formaion model, where he reference variable coincides wih own pas consumpion, and he exernal habi formaion (or caching-up wih he Joneses ) model, where he reference is an exernaliy accruing from he average pas consumpion in he economy. We also characerize he equilibrium pah of he previous wo exreme models and show ha in boh cases he dynamic equilibrium converges asympoically o a balanced growh pah, along which oupu, consumpion and capial grow a a common consan rae. We show ha he inroducion of habis increases his long run rae of growh because i raises he consumers willingness o shif consumpion from he presen o he fuure. We also prove ha he growh raes of boh models exhibi a monoonic behavior along he ransiion when he producion funcion is linear in capial, whereas hey could exhibi a non-monoonic behavior when he producion funcion has diminishing reurns o scale wih respec o capial. In he laer case, we show ha he ransiional dynamics depends on he values of boh capial and he reference level of consumpion, whereas he ransiion only depends on he raio of capial o he reference level of consumpion when he producion funcion has consan reurns o scale. This means ha he speciþcaion of he producion funcion has ineresing implicaions on cross-counry convergence. Our analysis shows ha he inroducion of a consumpion exernaliy makes he equilibrium of he caching-up wih he Joneses model inefficien during he ransiion owards he balanced growh pah. In order o prove his resul, we use he fac ha he equilibrium soluion of he inernal habi formaion model corresponds o he efficien soluion of he caching-up wih he Joneses model. Then, we show ha he equilibrium soluions obained in he wo models coincide in he long run, whereas hese wo soluions are differen during he ransiion. Inefficiency arises because he consumers willingness o subsiue consumpion across periods in he caching-up wih he Joneses model is no opimal. This source of inefficiency can be correced by means of an appropriae ax policy and, in paricular, we show ha eiher a consumpion ax or an income ax may resore efficiency. If he consumers willingness o shif presen consumpion o he fuure is subopimally low, hen he opimal Þscal policy will consis of eiher a decreasing sequence of consumpion axes or a subsidy on income (or oupu). A decreasing sequence of consumpion axes implies ha fuure consumpion purchases will be cheaper. Therefore, his Þscal policy increases consumers willingness o pospone consumpion. A subsidy on oupu also correcs he inefficiency because i encourages consumers o shif consumpion from he presen o he fuure. Conversely, eiher an increasing sequence of consumpion axes or a ax on income will be he opimal Þscal policies when consumers willingness o shif presen consumpion o he fuure is subopimally large. Finally, we show ha if he marginal produciviy of capial is consan during he ransiion, which occurs when he producion funcion is of he Ak ype, hen he opimal ax raes will depend only on he iniial value of he raio of capial o habis. However, if he marginal produciviy of capial is changing during he ransiion, hen he value of he opimal ax raes will depend on he paricular iniial values of capial and habis and no only on heir raio. In his case, he opimal pah of ax raes could exhibi a non-monoonic behavior along is ransiion. The characerizaion of he opimal income ax rae highlighs he dynamic naure of he inefficiency, which affecs he willingness o subsiue consumpion across periods This is in conras o he keeping-up wih he Joneses model, where he exernaliy accrues from average curren consumpion (see Gal õ, 994) 2

4 and, hus, modiþes he paern of capial accumulaion. In his respec, le us menion ha Ljungqvis and Uhlig (2000) have also analyzed he inefficiency accruing from exernal habis. However, hey consider a model wihou capial accumulaion where he exernaliy disors he inraemporal choice beween consumpion and leisure and, hus, heir efficiency analysis canno be exended o a growh model like ours. The res of he paper is organized as follows. Secion 2 describes he model and provides condiions ha guaranee he exisence of inerior equilibria characerized by sandard Þrs order condiions. Secion 3 and 4 characerize he equilibrium pah of he inernal habi formaion model and of he caching-up wih he Joneses model, respecively. Secion 5 shows ha he equilibrium of he laer model is inefficien and derives he opimal ax policy aimed a resoring efficiency. Secion 6 concludes he paper. Some lenghy proofs appear in he Appendix. 2 The Model Consider an economy in discree ime populaed by idenical dynasies facing an inþnie horizon. The members of each dynasy are also idenical. We assume ha populaion grows a a consan exogenous rae n>. We also assume ha consumers uiliy in period depends boh on consumpion c andonavariablev represening a sandard of living ha i is used as a reference wih respec which presen consumpion is compared o. This sandard of living is deermined by he pas consumpion experience. Following Abel (990), we assume ha v = c θ c θ, () where θ [0, ], c is he own consumpion in period, and c is he average consumpion of he economy in period. When θ = his formulaion coincides wih ha of he inernal habi formaion (IH) model, where he reference is jus he own pas consumpion. On he conrary, he case where θ = 0 corresponds o he caching-up wih he Joneses (CJ) model, where consumers uiliy depends boh on presen consumpion and on an exernaliy accruing from he ohers pas consumpion. Following Abel (990) and Carroll e al. (997), we inroduce muliplicaively he reference variable. This means ha consumers uiliy depends on own curren consumpion relaive o he sandard of living summarized by he variable v. Accordingly, he insananeous uiliy funcion akes he following funcional form: u = µ σ c σ v γ, wih σ > 0andγ (0, ), (2) where γ is a parameer measuring he imporance of he consumpion reference and σ coincides wih he inverse of he elasiciy of ineremporal subsiuion of consumpion when γ = 0. The assumpion γ > 0 agrees wih our noion of a reference for consumpion, whereas we mus impose ha γ < since, oherwise, he uiliy funcion would no be sricly increasing in consumpion along a balanced growh pah. Noe ha if γ = 0 he uiliy funcion u is ime-independen and concave. However, when γ > 0 he uiliy funcion is ime-dependen and i is no joinly concave wih respec o he wo variables c and v. In fac, he necessary condiions for join concaviy are + γ( σ) 0and γ + σ( γ) 0. Obviously, he laer inequaliy canno hold under our parameric assumpions. 3

5 Each consumer is endowed wih k unis of capial ha are used o produce a cerain amoun of oupu according o he following Sobelow gross producion funcion per capia: f(k )=Ak + Bk β, (3) where A>0, B 0andβ (0, ). This producion funcion allows for susained growh provided he asympoic marginal produciviy A of capial is sufficienly high (Jones and Manuelli, 990). The oupu may be used eiher for consumpion or for invesmen in new capial. Thus, he resource consrain per capia is given by f(k ) c +(+n) k + ( δ) k, for all =0,,... (4) where δ [0, ] is he depreciaion rae of capial. 2 The objecive of each dynasy is o maximize he discouned sum of uiliies of each of is idenical members, X µ u, (5) +ρ =0 where ρ > 0 is he subjecive discoun rae. A ime = 0, each dynasy chooses {c,k + } =0 o maximize (5) subjec o (4), aking as given he pah of average consumpion { c } = and he wo iniial condiions on capial k 0 > 0 and pas consumpion c > 0. Boh c and k are resriced o be non-negaive in all periods. Noe ha in his dynamic opimizaion problem k and v are he sae variables. While he former variable is only affeced by he individual decisions of consumers, he laer is deermined by boh individual decisions and he exogenous pah of average consumpion. The Lagrangian associaed wih he dynasy problem is L(c, k, λ) = X =0 µ u + +ρ X λ [f(k ) c ( + n) k + +( δ) k ], (6) =0 where c = {c } =0,k= {k } =0 and λ = {λ } =0 are non-negaive pahs, and he average consumpion pah { c, } = is aken as given. Compuing he derivaive of he previous Lagrangian wih respec o c, we obain he following necessary Þrs order condiions for opimaliy: µ L µ u + u + v + = + λ 0, (7) c +ρ c +ρ v + c wih c 0, and L c =0. (8) c The corresponding ransversaliy condiion is lim ( µ ) u λ 0, (9) +ρ c wih lim (" µ # ) u λ c =0. (0) +ρ c 2 From now on, he expression for all =0,,... will be skipped as long as he meaning is clear. 4

6 Differeniaing (6) wih respec o k +, we also ge he following necessary Þrs order condiions: L = ( + n)λ + +f 0 (k + ) δ λ + 0, k + () wih k + 0, and L k + =0. k + (2) The corresponding ransversaliy condiion is lim { ( + n)λ } 0, (3) wih lim { ( + n)λ k + } =0. (4) Finally, by aking he derivaive of (6) wih respec o he Lagrange muliplier λ, he soluion o he opimizaion problem involves also o saisfy he resource consrain (4), and λ [f(k ) c ( + n) k + +( δ) k ]=0, (5) wih λ 0. The following lemma provides a necessary condiion o be saisþed by an inerior soluion o his dynamic opimizaion problem: Lemma If he pah {c,k + } =0 chosen by a dynasy is sricly posiive, hen he following condiion mus be saisþed: +ρ u + c + + u c + +ρ +ρ u+2 v+2 v +2 c + u+ = v + v + c +n +f 0 (k + ) δ. (6) Proof. Since, by assumpion, c > 0andk > 0, for all, (8) implies ha (7) holds wih equaliy, and similarly (2) implies ha () also holds wih equaliy. From combining he Þrs order condiions (7) and (), i is sraighforward o obain he equaion (6). The opimaliy condiion (6), dubbed he Keynes-Ramsey equaion, equaes he marginal rae of subsiuion of consumpion beween periods and + (MRS, +, henceforh) wih he corresponding marginal rae of ransformaion (MRT, +, henceforh). Noe ha he MRS, + depends on he own consumpion and on he exernaliies arising from average pas consumpion. 3 More precisely, he MRS, + is a funcion of c,c,c +,c +2, and of he average consumpions c, c and c +. Since he pah {c,k + } =0 chosen by a dynasy is a funcion of he average consumpion pah { c } =, he nex deþniion makes clear he Þxed poin naure of a compeiive equilibrium: DeÞniion An equilibrium pah {c,k + } =0 is a soluion o he dynasic opimizaion problem when c = c, for all. 3 I should be poined ou ha, if σ =, hen he MRS,+ does no depend on he exernaliies, so ha average pas consumpion does no modify he pah chosen by he dynasy. 5

7 From he previous deþniion, i follows ha along he equilibrium pah he MRS, + depends only on c,c,c + and c +2. Noe ha his equilibrium MRS, + differs from he MRS, + appearing in sandard models of capial accumulaion because here consumers ake ino accoun he effec ha presen consumpion has in seing he reference for nex period consumpion. Because of he dependence of he MRS, + on consumpion in differen ime periods, he analysis of he equilibrium is simpliþed by inroducing he following ransformed variables: x = c c,h = u + u,z = k c, and m = f(k ) k. Noe ha he average produciviy m of capial and he raio z of capial o he reference level of consumpion are he sae variables, whereas he gross rae x of consumpion growh and he gross rae h of growh of he uiliy are he conrol variables. Noe also ha, for given values c and k 0 of iniial pas consumpion and iniial capial, respecively, here is a one-o-one correspondence beween he equilibrium values of he original variables c and k and he values of he ransformed variables m, x,z and h. Thus, given he iniial condiions m 0 = f (k 0 )/ k 0 and z 0 = k 0 / c, we can rewrie he equilibrium pah in erms of he ransformed variables. We deþne a saionary pah in erms of he previous ransformed variables as follows: DeÞniion 2 A saionary pah {x,h,z,m } =0 isapahalongwhichx,h,z and m are all consan. From he previous deþniion and ha of z, i follows ha along a saionary pah consumpion and capial grow a he same consan growh rae. Noe ha no equilibrium condiion is imposed in he deþniion of a saionary pah. DeÞniion 3 A balanced growh pah (BGP) {x,h,z,m } =0 ha is saionary. is an equilibrium pah Noe ha from he deþniion of he variable m, a BGP involves a consan marginal produciviy of capial. I is hen obvious from he funcional form of he producion funcion (3) ha a BGP is never reached in Þnie ime when B>0andβ > 0. However, we will say ha a pah {x,h,z,m } =0 converges o a BGP when lim x = x, lim h = h, lim z = z, and lim m = m, where x, h, z and m are he BGP values of x,h,z and m, respecively. Le us deþne he following parameers: ε γ +ρ, γ + σ ( γ), and ϕ +A δ ( + n)(+ρ). Our nex proposiion presens necessary condiions o be saisþed by an inerior equilibrium pah converging owards a BGP wih posiive growh. Proposiion Le ϕ >. Assume ha, for given iniial values z 0 > 0 and m 0 > 0, here is only one sricly posiive equilibrium pah {x,h,z,m } =0 and ha his pah converges o a sricly posiive BGP. Then, 6

8 (a) he following condiions are saisþed along he equilibrium pah: µ µ z m + δ z + = +n +n, (7) µ m + = A +(m A) +n and µ µ h µ θεh+ +ρ x + θεh x x + =(h ) σ (x ) γ, (8) β µ β, (9) = m + δ x z +n +A ( β)+βm + δ ; (20) (b) he sricly posiive BGP which he equilibrium pah converges o saisþes x = ϕ >, (2) and m = A, (22) x z = ( + A δ) ( + n) x, (23) h = ϕ. (24) Proof. See he Appendix. Equaion (2) ells us ha he equilibrium pah exhibis susained growh in he long run. Noe ha he value of he parameer ϕ is crucial for he exisence of posiive growh, ha is, for x>. Equaion (7) follows from he budge consrain and saes ha i is binding along he equilibrium pah. In fac, his equaion is he budge consrain deþned in erms of he ransformed variables. Equaion (8) follows from he deþniion of he ransformed variable h and equaion (9) follows from he deþniion of he ransformed variable m. Finally, (20) is he Keynes-Ramsey equaion in equilibrium deþned in erms of he ransformed variables. The lef hand side of his equaion corresponds o he MRS,+ and he righ hand side is he MRT,+. Proposiion esablishes he necessiy of equaions (7)-(20) in order o obain a sricly posiive equilibrium pah converging o a BGP exhibiing susained growh. If he consumers maximizaion problem were convex, hese four equaions and he corresponding iniial and ransversaliy condiions would no be only necessary bu also sufficien for obaining ha equilibrium pah when ϕ >. Given he assumpion of non-increasing reurns o scale, he resource consrain (4) deþnes a convex se of feasible soluions. Thus, he consumers problem would be convex if he objecive funcion (5) were concave. Sokey e al. (989, ch. 4) have shown ha concaviy of he insananeous uiliy funcion is a sufficien, alhough no necessary, condiion ha guaranees he concaviy of he objecive funcion. However, as follows from our previous discussion, he insananeous uiliy funcion is no concave in his model when θ 6= 0. Therefore, since he convexiy of he consumers maximizaion problem is no guaraneed, he equilibrium pah could be non-inerior (i.e., no sricly posiive) and, in his case, he sysem of difference equaions (7)-(20) would no characerize ha equilibrium pah. The following proposiions provide condiions aimed a ensuring he 7

9 exisence of an inerior equilibrium pah characerized by he previous dynamic sysem. We sar by imposing a resricion on he values of σ compaible wih he exisence of an inerior (i.e., sricly posiive) equilibrium pah. 4 Proposiion 2 Le θ > 0 and assume ha here exiss a sricly posiive equilibrium pah {x,h,z,m } =0 for given z 0 > 0 and m 0 > 0. Then,σ. Proof. We proceed by conradicion and assume ha σ <. In his case, i is easy o check ha u + = when c =0 and c + > 0, since hen v + =0. Moreover, u =0whenc =0andc = c > 0. This implies ha pahs for which consumpion and he gross rae x of consumpion are equal o zero in some, bu no all, periods deliver higher discouned uiliy han any sricly posiive pah. Noe ha o achieve zero consumpion in some periods is always feasible (see (4)). In view of Proposiion 2, if σ < he soluion o he dynamic opimizaion problem canno be inerior when θ > 0. 5 The following proposiion provides sufficien condiions for an inerior equilibrium pah: Proposiion 3 Le σ and ϕ >. Assume ha, for all iniial values z 0 > 0 and m 0 > 0, here is only one pah {x,h,z,m } =0 solving he sysem of difference equaions (7)-(20), and ha his pah is sricly posiive and converges o a sricly posiive saionary pah. Then, he pah {x,h,z,m } =0 is an equilibrium pah. Moreover, he saionary pah given by expressions (2)-(24)isheuniquesriclyposiiveBGPof he economy. Proof. See he Appendix. The previous proposiion ells us ha, when σ andϕ >, an equilibrium pah ha converges o an inerior BGP is fully characerized by he dynamic sysem composed by he difference equaions (7)-(20), ogeher wih he iniial condiions, for any value of he parameer θ in he closed inerval [0, ]. In paricular, we will use he previous sysem of equaions o characerize he equilibrium dynamics corresponding o he following wo exreme models, which are commonly found in he lieraure: he IH model and he CJ model. We will see ha boh models exhibi saddle pah sabiliy owards a unique BGP and, hus, he assumpions in Proposiion 3 are clearly me. According o he resuls of his secion, we will mainain he assumpions σ andϕ > hroughou he res of he paper. Concerning he properies of he BGP, noe ha he saionary rae x of economic growh given in (2) increases wih he value of he parameer A measuring oal facor produciviy (TFP, henceforh) in he long run. The inuiion behind his resul can be obained from he Keynes-Ramsey equaion (6). From ha equaion we observe ha an increase in TFP reduces he cos of shifing resources o fuure periods and, hus, drives he price of fuure consumpion in erms of presen consumpion down. This encourages consumers o shif presen consumpion o he fuure and, hus, he rae of economic growh mus increase. 4 Noe ha, if we had assumed ha γ < 0, he insananeous uiliy funcion could be concave and he consumers maximizaion problem would be convex. In his case, he condiion saed in Proposiion 2 is no required o guaranee an inerior equilibrium pah. 5 Noe ha he argumens o rule ou he case σ < do no apply when he reference variable does no depend on own pas consumpion (θ =0). Thus, in he caching-up wih he Joneses model, he equilibrium could be characerized by (7)-(20) even if σ <. 8

10 Noe also ha, if habis become more imporan (which amouns o an increase in he value of he parameer γ), hen he growh rae goes up when σ >. Moreover, he effec on he growh rae of an increase in TFP becomes larger for higher values of γ. 6 This occurs because he inroducion of habis makes he ineremporal elasiciy of subsiuion larger and his acceleraes economic growh as he raio of presen o pas consumpion is forced o increase. 7 This resul is in sark conras o ha obained by Shieh e al. (2000), where he inroducion of inernal habis could deer growh in some cases. This difference arises because Shieh e al. (2000) did no inroduce condiions ha guaranee he exisence of inerior soluions. 8 3 Equilibrium under Inernal Habis (IH) In his secion we assume ha consumers view only heir own pas consumpion as he sandard of living o be used as a reference. Therefore, we impose θ = in expression () and, hus, he reference variable becomes simply v = c. In his case he Keynes- Ramsey equaion (20) simpliþes o µ µ µ h εh+ +n = +ρ x + εh +A ( β)+βm + δ, (25) where he lef hand side of he equaion is he marginal rae of subsiuion in he inernal habi formaion model (MRS,+ IH, henceforh). Given he iniial condiions m 0 = f (k 0 )/ k 0 and z 0 = k 0 / c, we can hus deþne an inerior equilibrium pah of he IH model as a sricly posiive pah { m,x,z,h } =0 saisfying he difference equaions (7), (8), (9) and (25), and he corresponding ransversaliy condiions. The BGP of he IH model is given by he expressions (2)-(24), since hese expressions do no depend on he value aken by he parameer θ. The nex wo proposiions characerize he ransiional dynamics of he economic sysem in he neighborhood of he BGP. This ransiional dynamics was already esablished by Carroll e al. (997) when he echnology is represened by an Ak producion funcion. We exend he analysis o he Sobelow producion funcion, where he marginal produciviy of capial is ime-varying. Proposiion 4 The BGP of he IH model is saddle pah sable. Proof. See he Appendix. Proposiions 3 and 4 allow us o conclude ha, for a given pair of iniial condiions z 0 and m 0 sufficienly close o he saionary values z and m, respecively, here is a unique equilibrium pah. Moreover, his equilibrium pah is he saddle pah converging o he BGP. Proposiion 5 Given he iniial condiions z 0 > 0 and m 0 > 0, he following holds for he IH model: 6 Observe ha x > 0and x γ A γ > 0whenσ >. 7 Noe ha, if we deþne he saionary ineremporal elasiciy of subsiuion as he elasiciy of he saionary rae of growh wih respec o he asympoic reurn o capial, his saionary elasiciy is given by he value of /. Clearly, his elasiciy is sricly increasing in γ when σ >. 8 In heir paper, he inroducion of habis may reduce he long run growh rae because hey do no assume ha σ. In our paper, his assumpion is required o rule ou corner soluions yielding unbounded uiliy. 9

11 (a) If B =0hen he variables x and h will boh exhibi a monoonic behavior along he ransiion owards he BGP. In paricular, if z 0 <z(z 0 >z), hen he variables x and h will increase (decrease) oward heir respecive saionary values. (b) If B>0 hen he variables x and h could exhibi a non-monoonic behavior along he ransiion owards he BGP. Proof. See he Appendix. When B = 0 he echnology is characerized by an Ak producion funcion and, hence, m = A. In his case, here is only one sae variable, z, and he behavior of x and h only depends on he iniial value z 0 of he sae variable. However, when B>0 here are wo sae variables, z and m, and he ransiion of x and h depends on he paricular iniial values of hese wo variables. This ransiion could hen be nonmonoonic. Hence, our model could give rise o a ransiory non-monoonic behavior of he growh rae. 9 Therefore, while Carroll e al. (997) have shown ha he consumpion growh rae in a model wih an Ak producion funcion displays a monoonic convergence owards he BGP when preferences are no ime-separable, we show ha his convergence could be non-monoonic when he producion funcion exhibis diminishing reurns o scale. In his case, a reducion in he sock of capial may cause eiher an increase or a decrease in he consumpion growh rae depending on he iniial sock of habis. In conras, Carroll e al. (997) have shown ha a reducion in he sock of capial causes an unambiguous reducion in he consumpion growh rae when he producion funcion exhibis consan reurns o scale. The inuiion behind our resul lies in he fac ha, when he capial sock becomes smaller, he reurn on invesmen increases under diminishing reurns o scale and his has a posiive effec on he growh rae. However, he reducion in he capial sock makes he amouns of boh capial and oupu small relaive o he consumpion reference, so ha agens would be forced o choose a consumpion level so large ha i would no be susainable in he long run. Therefore, such a consumpion level will have o decrease in he fuure. This means ha habis make he growh rae decrease as a response o a reducion in he capial sock, while diminishing reurns accoun for he opposie effec. Obviously, hese wo opposie forces explain boh he ambiguiy of he response of he growh rae o changes in he sock of capial and he non-monoonic behavior during he ransiion. The previous resul has also implicaions for he cross-counry convergence. Consider he original model in erms of he variables c and k. When B>0he ransiional dynamics of he growh rae depends on he paricular iniial values of boh he capial level and he reference level of consumpion. This means ha, under diminishing reurns o capial, wo economies wih differen iniial capial socks will follow differen equilibrium pahs for x even if hey share a common iniial value of he raio z of capial o consumpion reference. On he conrary, when he producion funcion is Ak, he ransiional dynamics of he growh rae x only depends on he iniial value of he raio z. In his case, wo economies wih he same iniial value of z will follow equilibrium pahs wih idenical growh raes regardless of heir iniial levels of capial. I follows ha cross-counry differences in he growh rae can only be explained by differences in he raio z when he producion funcion is Ak, whereas hey can be explained by differences on he values of boh he sock of capial and he reference level of consumpion when he producion funcion exhibis diminishing reurns o scale. 9 In he proof of Proposiion 5 we provide an example of an economy exhibiing such a non-monoonic behavior. 0

12 4 Equilibrium under Caching-up wih he Joneses (CJ) In his secion we make θ = 0 in expression (). This means ha he average aggregae consumpion of he previous period is now he reference level of consumpion, ha is, v = c. Therefore, he model displays he ypical caching-up wih he Joneses feaure, since average pas consumpion eners ino he consumers uiliy as a negaive exernaliy. We nex derive he equaions characerizing he dynamic equilibrium of his paricular model. Since θ = 0, he Keynes-Ramsey equaion (20) is simply µ µ h = +ρ x + +n +A ( β)+βm + δ, (26) where he lef hand side of equaion (26) is he marginal rae of subsiuion in he caching-up wih he Joneses model (MRS,+ CJ, henceforh). Using he deþniion of h inroduced in Secion 2, (26) becomes γ(σ ) σ x + = x µ +A ( β)+βm+ δ ( + n)(+ρ) σ. (27) In conras o he IH model, he equilibrium is now fully described by only hree variables: z,m, and x. The Þrs wo variables are he sae variables, whereas he hird one is he conrol variable. Hence, given he iniial condiions m 0 = f (k 0 )/ k 0 and z 0 = k 0 / c, we deþne an equilibrium pah of he CJ model as a sricly posiive pah { m,x,z } =0, saisfying he difference equaions (7), (9) and (27), and he corresponding ransversaliy condiions. A BGP will be hus an equilibrium pah along which he variables m,x and z are consan. Obviously he gross rae of growh h of he insananeous uiliy u is also consan along a BGP. I is hus clear from he expressions appearing in par (b) of Proposiion ha he BGP of he CJ model is he same as ha of he IH model, since he BGP of he general model of Secion 2 is independen of he parameer θ. We nex discuss he inuiion for obaining idenical saionary soluions for boh models. In he CJ model consumers do no inernalize he spillovers accruing from he average pas consumpion. On he conrary, consumers in he IH model ake ino accoun he fuure effecs of heir curren decisions on consumpion. This difference ranslaes ino differences beween he marginal rae of subsiuion of boh models during he ransiion, as one can easily see by comparing equaions (25) and (26). However, since he discouned sum of uiliies is bounded, he growh raes of boh insananeous uiliy and consumpion mus converge o a consan value. Hence, i is immediae o see from (25) and (26) ha he marginal raes of subsiuion of boh models coincide along a saionary pah (i.e., when x and h are consan for all ). The nex wo proposiions characerize he ransiional dynamics of he CJ model in he neighborhood of he BGP: Proposiion 6 The BGP of he CJ model is saddle pah sable. Proof. See he Appendix. The previous resul esablishes ha he equilibrium pah of he CJ model is unique for a given pair of iniial condiions z 0 and m 0 sufficienly close o heir respecive saionary values z and m. Moreover, he equilibrium pah converges o he unique BGP.

13 Proposiion 7 Given he iniial condiions z 0 > 0 and m 0 > 0, he following holds for he CJ model: (a) If B =0hen he variable x will exhibi a monoonic behavior along he ransiion owards he BGP. In paricular, if z 0 <z(z 0 >z), hen he variable x will increase (decrease) oward is saionary values. (b) If B>0 hen he variable x could exhibi a non-monoonic behavior along he ransiion owards he BGP. Proof. See he Appendix. Proposiion 7 has he same qualiaive implicaions for he ransiional dynamics and he cross-counry convergence of he CJ model as hose esablished by Proposiion 5 for he IH model. In oher words, he policy funcions racing ou he relaionship beween he sae variables and he opimal value of he conrol variables in he CJ model are qualiaively similar o hose of he IH model. However, he efficiency analysis of he nex secion will show ha he relaionship beween sae and conrol variables differs quaniaively from one model o he oher. 0 5 Efficiency and Opimal Policy The equilibrium of he CJ model could be inefficien because consumers do no inernalize he spillover effecs from average pas consumpion. This source of inefficiency has been sudied by Ljungqvis and Uhlig (2000) in a model wihou capial accumulaion. In his secion, we exend he efficiency analysis ino a growh model wih capial accumulaion. To his end, noe ha he equilibrium of he IH model described in Secion 3 coincides wih he soluion of he CJ model ha a benevolen social planner would implemen, since ha planner would ake ino accoun all he exernal effecs accruing from average pas consumpion. This means ha, in order o deal wih efficiency issues, we jus have o compare he equilibrium soluion of he CJ model wih ha of he IH model. The only difference beween he equaions ha characerize he equilibrium pahs of he wo models lies in he MRS,+ appearing in he lef hand side of he Keynes-Ramsey equaions (25) and (26). Efficiency of he compeiive soluion of he CJ model requires ha he MRS,+ obained in he wo models be idenical, i.e., MRS,+ CJ = MRSIH,+, where MRS,+ IH is he efficien MRS,+. Given our assumpions on preferences, he previous efficiency condiion is obviously saisþed when σ =. However, if σ >, hen he efficiency condiion becomes simply h + = h. This equaliy holds along he BGP, which implies ha he equilibrium of he CJ model is asympoically efficien. This is consisen wih he fac ha he CJ and IH models share he same BGP, as shown in he previous secions. Finally, he dynamic equilibrium of he CJ model is obviously inefficien during he ransiion when σ >. In wha follows we will show ha efficiency can be resored in he CJ model by means of an appropriae ax policy. We presen wo alernaive ax insrumens ha 0 InhecasewhereheproducionfuncionisAk, Carroll e al. (997) show ha he slope of he equilibrium saddle pah (or he policy funcion) in he CJ model differs from ha of he IH model. Fisher and Hof (2000) have also analyzed equilibrium efficiency in a neoclassical growh model when he source of inefficiency is an exernaliy arising from average curren consumpion. In conras, in our paper he exernaliy is associaed o he average pas consumpion level and he resuls on efficiency are in sark conras wih hose obained by Fisher and Hof (2000). Acually, Ljungqvis and Uhlig (2000) have sressed he fac ha inefficiency depends on he iming of he consumpion exernaliy. 2

14 make idenical he wo Keynes-Ramsey equaions (25) and (26): a ax on ne income (or ne oupu) and a ax on consumpion. By using a procedure similar o ha of Fisher and Hof (2000) and Alonso-Carrera e al. (200), we derive he corresponding opimal ax raes. We assume ha he ax revenues are reurned o consumers hrough a lumpsum subsidy. This assumpion implies ha he resource consrain of a represenaive dynasy becomes now ( + τ c ) c +(+n) k + k =( τ y )[f (k ) δk ]+T, (28) where τ c and τ y are he ax raes on consumpion and on ne income, respecively, and T is he lump-sum subsidy ha saisþes he following governmen budge consrain: 2 T = τ y [f (k ) δk ]+τ c c. (29) Combining (28) and (29) and using he ransformed variables, we obain he resource consrain. Therefore, because all he ax revenue is reurned o he consumers as a lump-sum subsidy, he inroducion of axes only modiþes he Keynes-Ramsey equaion (26). Thus, his equaion becomes µ Ã h CJ! +ρ x CJ = + µ +τ c + +τ c Ã! +n + τ+ y ( β) A + βm CJ + δ, (30) where he superscrip CJ is used o denoe he variables of he CJ model. Noe ha he LHS of he previous equaion is he marginal rae of subsiuion MRS,+ CJ h CJ,x CJ + of he CJ model. Evaluaing (30) a he efficien equilibrium pah, and dividing he resuling equaion by he Keynes-Ramsey equaion (25) of he IH model, we obain he following opimal axaion condiion: MRS,+ CJ h IH MRS IH,+ h IH +,h IH,x IH +,x IH + +ˆτ c µ = + +ˆτ c +( β)a+βm IH + δ +( ˆτ y +)[( β)a+βm IH + δ], (3) where he superscrip IH is used o denoe he equilibrium value of he variables in he IH model; ˆτ c and ˆτ y are he opimal values of he ax raes on consumpion and income, respecively; and MRS,+ IH h IH +,hih,x IH + and MRS,+ CJ h IH,x IH + are he MRS,+ corresponding o he IH model and he CJ model, respecively, when hey are evaluaed along he efficien equilibrium pah. We see from (3) ha opimal axes display ime-varying raes off he BGP, while he opimal consumpion ax rae is consan and he opimal income ax rae is zero a he BGP. This resul abou he opimal income ax rae in he long run resembles hose obained by Judd (985) and Chamley (986) in models wih sandard preferences. 3 Thus, we see ha he inroducion of habi formaion affecs he opimal ax raes only during he ransiion. If he MRS,+ of he CJ model evaluaed along he efficien consumpion pah urns ou o be smaller han he efficien MRS,+ along he same pah, hen he 2 Noe ha a ax on income has he same effecs on capial accumulaion as a ax on capial income provided he ax revenue is enirely reurned o consumers hrough a lump-sum ransfer. Even if he amoun of axes colleced for a given ax rae is no he same, he marginal produciviy of capial is modiþed idenically under he wo ax schemes. 3 I should be poined ou ha in he papers of Judd and Chamley he governmen inends o Þnance opimally a given sream of spending. However, in our model he governmen jus uses opimal axes aimed a correcing for he ineffciencies brough abou by consumpions spillovers. 3

15 consumers willingness o shif presen consumpion o he fuure will be oo small. In his case, condiion (3) ells us ha he efficien pah can be reached by means of eiher subsidizing oupu or inroducing a ax on consumpion wih ˆτ c > ˆτ + c. These Þscal policies correc he inefficiency because hey make presen consumpion purchases more expensive han he fuure ones and, hence, hey encourage consumers o pospone consumpion. A decreasing sequence of ax raes on consumpion direcly drives he afer-ax price of fuure consumpion in erms of presen consumpion down. Moreover, a subsidy on oupu also reduces he relaive price of fuure consumpion because his policy reduces he cos of shifing resources o fuure periods. Therefore, if he MRS,+ of he CJ model along an efficien pah is larger (smaller) han he efficien MRS,+, hen a welfare-maximizing governmen mus impose eiher a ax (subsidy) on income or a ax on consumpion wih a rae ha rises (falls) over ime. Finally, we can also characerize he dynamic behavior of he opimal ax raes by expressing hem as funcions of he sae variables of he model. As a Þrs sep owards his goal, we show ha boh raes depend only on he efficien value h IH of he uiliy growh rae. On he one hand, making ˆτ y + = 0 and from he deþniion of he variable, condiion (3) can be rewrien as follows: h IH ˆτ c + ˆτ c +ˆτ c = ε à h IH + h IH! εh IH. (32) + We hus see ha he opimal ax on consumpion increases (decreases) when h IH increases (decreases). Also noe ha ˆτ + c =ˆτ c along he BGP. Therefore, any sequence of consan ax raes (no necessarily equal o zero) on consumpion is opimal along a BGP. On he oher hand, imposing ˆτ c =0forall in condiion (3), we obain ha he opimal rae of he income ax is à ˆτ y h IH + = ε + h IH εh IH!à +( β) A + βm IH + δ ( β) A + βm IH + δ!. (33) Noe ha his opimal rae equals zero along he BGP as h IH + = hih. However, his ax is posiive when he growh rae of uiliy increases wih ime, h IH + >hih, and i is negaive oherwise. We have hen shown ha he evoluion of he opimal raes of boh axes is qualiaively deermined only by he ransiion of he variable h IH along he efficien equilibrium. This occurs because, when habis are modeled in a muliplicaive way, here is a direc relaion beween he variable h IH being increasing (decreasing) along he ransiion and he MRS,+ being subopimally large (small). Proposiion 5 describes he behavior of h IH during he ransiion, so ha we can derive he evoluion of he opimal ax raes direcly from his Proposiion. Corollary (a) The sequence of opimal consumpion ax raes {ˆτ c } = around he BGP could be eiher monoonic or non-monoonic for a given arbirary value of ˆτ 0 c. This sequence converges o a consan. (b) The sequence of opimal income ax raes {ˆτ y } = around he BGP could eiher exhibi he same sign or change is sign. This sequence converges o zero. Proof. Obvious from Proposiion 5 and expressions (32) and (33). 4

16 Par (a) of Proposiion 5 ells us ha if z 0 <z,henh IH will increase. In his case, he MRS,+ will be subopimally large and he opimal ax policy will consis on eiher an increasing sequence of ax raes on consumpion or a posiive ax rae on income. Theopposiewilloccurifz 0 >z.however, Par (b) of Proposiion 5 ells us ha, if he marginal produciviy of capial is no consan, namely, when B > 0, hen he variable h IH could exhibi a non-monoonic behavior along he ransiion. This implies ha he opimal consumpion ax could grow during a number of periods and decrease aferwards. Similarly, he opimal ax on income could be posiive during some periods and become negaive laer on, or vice versa. When B > 0 here are wo sae variables, z and m, and he ransiion of he depends on he iniial value of hese wo variables. Therefore, he opimal Þscal policy depends also on he iniial values of hese wo sae variables or, equivalenly, on he iniial values of boh capial and he reference level of consumpion. This means ha, under sricly decreasing reurns o capial, wo economies wih differen iniial capial socks will have differen opimal ax raes even if hey share a common iniial value of he raio z of capial o consumpion reference. On he conrary, when he producion funcion is Ak, he opimal Þscal policy only depends on he iniial value of he raio z. In his case, wo economies wih he same iniial value of z will exhibi he same opimal ax raes regardless of heir iniial levels of capial. We can provide a numerical example o compare he opimal income axes for wo economies ha are idenical excep on he iniial value of he sae variable m.leus assume ha he parameers characerizing boh economies ake he following values: 4 variable h IH A =0.83, σ =5, δ =0.09, n=0, ρ =0.03, γ =0.5, and β =0.2. Boh economies share he same iniial value of he raio of capial o consumpion reference, namely, z 0 =0.99z. We also assume ha in one economy m 0 = m and, since m = A, he echnology is characerized by an Ak producion funcion from =0on. In he oher economy we se m 0 =.0m and, hence, capial exhibis sricly decreasing reurns o capial. The opimal rae of he income ax when he echnology is Ak urns ou o be always posiive along he ransiion and converges o zero. However, he opimal ax rae on income when he producion funcion exhibis diminishing reurns o scale akes negaive values for 6, while for >6 i akes posiive values. Similarly, under diminishing reurns o scale he ax rae on consumpion is decreasing for 6 and increasing for >6for any arbirarily given iniial ax rae. Noe ha, when =6.708, i holds ha h IH + = hih, which is consisen wih he expressions (32) and (33) characerizing opimal ax raes. This numerical example has hus illusraed clearly he poenial non-monooniciy of opimal ax raes when he echnology exhibis sricly decreasing reurns o capial. The previous resuls on he opimal income ax rae are in a sark conras wih he resuls obained by Ljungqvis and Uhlig (2000) in a caching-up wih he Joneses model wihou capial accumulaion. These auhors show ha he opimal income ax rae is posiive when here is a high realizaion of a produciviy shock raising he growh rae, and i is negaive oherwise. In his paper we show ha his resul does no hold when capial accumulaion is inroduced. On he one hand, if he producion funcion exhibis 4 We se he values of δ, n,ρ, σ, γ as in Caroll e al. (2000). In paricular, he values of σ and γ are such ha he inverse of he saionary ineremporal elasiciy of subsiuion akes he reasonable value = 3 (see foonoe 7), and he value of A is such ha yields a long-run growh rae equal o 2%. The value of β allows us o obain a speed of convergence of.6%. This conþguraion of parameer values is also used in he proof of Proposiion 5 (see (46) in he Appendix). 5

17 consan reurns o scale, he opimal income ax rae will ake negaive values when he consumpion growh rae is above is BGP value and akes posiive values oherwise. On he oher hand, if he producion funcion exhibis diminishing reurns o scale, he opimal income ax rae will be eiher procyclical or counercyclical, depending on he reference level of consumpion. Thus, we conclude ha he resuls on opimal Þscal policy obained in a model wihou capial do no hold in a model exhibiing capial accumulaion. 6 Conclusion We have analyzed he dynamic equilibrium of an endogenous growh model where preferences are ime-dependen. In paricular, we have assumed he exisence of inernal and exernal habi formaion in consumpion. Thus, uiliy depends on own consumpion relaive o a reference level, which grows wih boh pas own consumpion and pas average consumpion. The presence of inernal habis makes he insananeous uiliy funcion non-concave and, hence, concaviy of he objecive funcion is no guaraneed. We have provided condiions under which he equilibrium pah is he soluion o a dynamic sysem formed by sandard Þrs order condiions. We have hen sudied he equilibrium of wo growh models, namely, he IH model, where he reference is he own pas consumpion, and he CJ model, where he reference akes he form of an exernaliy accruing from average pas consumpion. The inroducion of a consumpion exernaliy makes he equilibrium of he CJ model inefficien during he ransiion. We have hen characerized he opimal Þscal policy. In paricular, we have derived he opimal ax raes on income and on consumpion. The opimal ax rae on income is zero along he BGP, whereas i is differen from zero along he ransiion. The opimal rae of consumpion ax is consan a he BGP, while i could eiher increase or decrease wih ime along he ransiion. We have shown how he value of opimal ax raes during he ransiion depends on he iniial values of boh he reference variable and capial. More precisely, if he marginal produciviy of capial is consan, hen he opimal ax raes will only depend on he iniial value of he raio of capial o consumpion reference. However, if ha marginal produciviy is no consan during he ransiion, hen he opimal ax raes will depend on he iniial values of boh capial and consumpion and hey could exhibi a non-monoonic dynamics. 6

18 References [] Abel, A., (990). Asse Prices under Habi Formaion and Caching up wih he Joneses, American Economic Review 80, [2] Abel, A., (999). Risk Premia and Term Premia in General Equilibrium, Journal of Moneary Economics 43: [3] Alonso-Carrera, J, J. Caballé, and X. Raurich, (200). Consumpion Exernaliies, Habi Formaion, and Equilibrium Efficiency, UAB-IAE Working Paper [4] Carroll, C., (2000). Solving Consumpion Models wih Muliplicaive Habis, Economics Leers 68: [5] Carroll, C., J. Overland, and D. Weil, (997). Comparison Uiliy in a Growh Model, Journal of Economic Growh 2: [6] Carroll, C., J. Overland, and D. Weil, (2000). Saving and Growh wih Habi Formaion, American Economic Review 90: -5. [7] Chamley, C., (986). Opimal Taxaion of Capial Income in General Equilibrium wih InÞnie Lives, Economerica 54: [8] Fisher, W., and F. Hof, (2000). Relaive Consumpion, Economic Growh, and Taxaion, Journal of Economics 72: [9] Fuhrer, J.C., (2000). Habi Formaion in Consumpion and is Implicaions for Moneary Policy, American Economic Review 90: [0] Gal õ, J., (994). Keeping up wih he Joneses: Consumpion Exernaliies, Porfolio Choice, and Asse Prices, Journal of Money Credi and Banking 26: -8. [] Jones, L., and R. Manuelli, (990). A Convex Model of Economic Growh: Theory and Policy Implicaions, Journal of Poliical Economy 98: [2] Judd, K., (985). Redisribuive Taxaion in a Simple Perfec Foresigh Model, Journal of Publics Economics 28: [3] Leau, M., and H. Uhlig, (2000). Can Habi Formaion Be Reconciled wih Business Cycles Facs?, Review of Economic Dynamics 3: [4] Ljungqvis, L., and H. Uhlig, (2000). Tax Policy and Aggregae Demand Managemen Under Caching Up wih he Joneses, American Economic Review 90: [5] Shieh, J., C. Lai, and W. Chang, (2000). Addicive Behaviour and Endogenous Growh, Journal of Economics 72: [6] Sokey, N., R. Lucas, and E. Presco, (989). Recursive Mehods in Economic Dynamics, Harvard Universiy Press. 7

19 A Appendix ProofofProposiion. (a)weuseheransformedvariablesx and h, and he equilibrium condiion c = c, in order o rewrie equaion (6) as follows µ µ µ h θεh+ +n = +ρ x + θεh +f 0 (k + ) δ. (34) Noe ha, by using he ransformed variable m, he marginal produc of capial is hen f 0 (k + )=A ( β)+βm +. Hence, afer rearranging erms, equaion (34) becomes equaion (20) in he saemen of he proposiion. Evaluaing equaion (20) when ends o inþniy, and using he deþniions of h and x, and he funcional form of he uiliy funcion, we ge Since γ + σ( γ) > 0and x [γ+σ( γ)] =lim ( + n)(+ρ) +A ( β)+βm + δ. (35) lim ( + n)(+ρ) +A ( β)+βm + δ = ( + n)(+ρ) +A δ = ϕ <, i follows from (35) ha (2) holds. Since x> implies ha lim c = and c = c, we conclude from (0) ha (9) holds wih equaliy. Therefore, lim λ =lim " µ +ρ # u c x γ( σ) lim =( σ) lim as γ + σ( γ) > 0. Rewriing (), we ge +f 0 (k + ) δ λ λ +. +n " µ +ρ # u = c " µ # (c ) [γ+σ( γ)] =0, (36) +ρ Since + f 0 (k + ) δ +A δ > ( + n)(+ρ) > +n, we have ha λ > λ +, which ogeher wih (36), implies ha λ > 0 for all Þnie. Therefore, (5) implies ha he resource consrain (4) is saisþed wih equaliy and, by using he ransformed variables, we ge equaion (7). From he deþniion of h and he funcional form of he uiliy funcion u, i follows ha equaion (8) mus also hold. Combining he deþniion of m wih he funcional form of f (k + ) and he resource consrain (7), we obain equaion (9). (b) We have already proved ha (2) holds. From he deþniion of he variable z, we have ha lim (k + /k )=x> and, hus, lim m = A (see (3)). The saionary values z and h are obained from a direc compuaion aimed a obaining heir limiing values according o he dynamic sysem formed by equaions (7)-(20). 8